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<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC">
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    <dcterms:identifier>ECHO:1X8T70WB.xml</dcterms:identifier>
    <dcterms:creator identifier="GND:11864548X">Appolonius Pergaeus</dcterms:creator>
    <dcterms:title xml:lang="la">Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit et notas vberiores in vniuersum opus adiecit</dcterms:title>
    <dcterms:alternative xml:lang="la">Apollonii Pergaei Conicorum Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additus in calce Archimedis assumptorum liber, ex codicibus arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alphonsus Borellus curam in geometricis versione contulit et notas uberiores in universum opus adiecit</dcterms:alternative>
    <dcterms:date xsi:type="dcterms:W3CDTF">1661</dcterms:date>
    <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language>
    <dcterms:rights>CC-BY-SA</dcterms:rights>
    <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license>
    <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder>
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<div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/>
<pb file="0002" n="2"/>
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<handwritten/>
<handwritten/>
<pb file="0003" n="3"/>
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<div xml:id="echoid-div2" type="section" level="1" n="2">
<head xml:id="echoid-head1" xml:space="preserve">APOLLONII <lb/>PERGÆI <lb/>CONICORVM</head>
<head xml:id="echoid-head2" xml:space="preserve">LIB. V. VI. VII. <lb/>&amp; <lb/>ARCHIMEDIS <lb/>ASVMPTOR VM LIBER.</head>
<pb file="0004" n="4"/>
<pb file="0005" n="5"/>
</div>
<div xml:id="echoid-div3" type="section" level="1" n="3">
<head xml:id="echoid-head3" xml:space="preserve"><emph style="red">APOLLONII PERGÆI</emph></head>
<head xml:id="echoid-head4" xml:space="preserve">CONICORVM LIB. V. VI. VII.</head>
<head xml:id="echoid-head5" xml:space="preserve">PARAPHRASTE</head>
<head xml:id="echoid-head6" xml:space="preserve"><emph style="red">ABALPHATO ASPHAHANENSI</emph></head>
<p>
  <s xml:id="echoid-s1" xml:space="preserve">Nunc primùm editi.</s>
  <s xml:id="echoid-s2" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div4" type="section" level="1" n="4">
<head xml:id="echoid-head7" xml:space="preserve">ADDITVS IN CALCE</head>
<head xml:id="echoid-head8" xml:space="preserve"><emph style="red">ARCHIMEDIS ASSVMPTORVM LIBER,</emph></head>
<head xml:id="echoid-head9" xml:space="preserve">EX CODICIBVS ARABICIS M.SS.</head>
<head xml:id="echoid-head10" xml:space="preserve">SERENISSIMI</head>
<head xml:id="echoid-head11" xml:space="preserve">MAGNI DVCIS ETRVRIÆ</head>
<head xml:id="echoid-head12" xml:space="preserve"><emph style="red">ABRAHAMVS ECCHELLENSIS MARONITA</emph></head>
<p>
  <s xml:id="echoid-s3" xml:space="preserve">In Alma Vrbe Linguar. </s>
  <s xml:id="echoid-s4" xml:space="preserve">Orient. </s>
  <s xml:id="echoid-s5" xml:space="preserve">Profeſſor Latinos reddidit.</s>
  <s xml:id="echoid-s6" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div5" type="section" level="1" n="5">
<head xml:id="echoid-head13" xml:space="preserve"><emph style="red">IO: ALFONSVS BORELLVS</emph></head>
<p>
  <s xml:id="echoid-s7" xml:space="preserve">In Piſana Academia Matheſeos Profeſſor curam in Geometricis verſioni <lb/>contulit, &amp; </s>
  <s xml:id="echoid-s8" xml:space="preserve">notas vberiores in vniuerſum opus adiecit.</s>
  <s xml:id="echoid-s9" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div6" type="section" level="1" n="6">
<head xml:id="echoid-head14" xml:space="preserve">AD SERENISSIMVM</head>
<head xml:id="echoid-head15" xml:space="preserve"><emph style="red">COSMVM III.</emph></head>
<head xml:id="echoid-head16" xml:space="preserve">ETRVRIÆ PRINCIPEM</head>
<head xml:id="echoid-head17" xml:space="preserve"><emph style="red">FLORENTIÆ,</emph></head>
<head xml:id="echoid-head18" xml:space="preserve">Ex Typographia Ioſephi Cocchini ad inſigne Stellæ MDCLXI.</head>
<head xml:id="echoid-head19" xml:space="preserve">SVPERIORVM PERMISSV.</head>
<pb file="0006" n="6"/>
<pb file="0007" n="7" rhead="AD SERENISSIMVM"/>
</div>
<div xml:id="echoid-div7" type="section" level="1" n="7">
<head xml:id="echoid-head20" xml:space="preserve">COSMVM TERTIVM <lb/>ETRVRIÆ PRINCIPEM.</head>
<head xml:id="echoid-head21" style="it" xml:space="preserve">10: AL FONSVS BORELLIVS F.</head>
<p>
  <s xml:id="echoid-s10" xml:space="preserve">H A V D puto, Sereniſsime Princeps, <lb/>timorem cœleſtis irę, ſed Amorem <lb/>potius, &amp; </s>
  <s xml:id="echoid-s11" xml:space="preserve">beneficentiam primùm <lb/>in orbe Deos feciſse; </s>
  <s xml:id="echoid-s12" xml:space="preserve">nec alios ab <lb/>initio habitos cum Prodico cenſeo, <lb/>quàm res humano generi ſummo-<lb/>pere vtiles, &amp; </s>
  <s xml:id="echoid-s13" xml:space="preserve">ſalutares. </s>
  <s xml:id="echoid-s14" xml:space="preserve">Et ſa-<lb/>nè conſentaneum eſt in primorum hominum men-<lb/>tibus, quibus reuelationis lumen non affulſit, excita-<lb/>tam fuiſſe notitiam cuiuſdam naturæ, quæ eſſet mun-<lb/>di veluti Princeps, &amp; </s>
  <s xml:id="echoid-s15" xml:space="preserve">Parens, quotieſcumque non <lb/>perfunctoriè attenderet animum præcipuè ad boni-<lb/>tatis affluentiam, mirabiliumque, &amp; </s>
  <s xml:id="echoid-s16" xml:space="preserve">inſignium vti-<lb/>litatum comprehenſionem, qua Solaris ſplendidiſ-<lb/>ſima machina lumine ſuo ordinatiſsimè circumacto <lb/>cuncta viuificat, fouet, ac nutrit; </s>
  <s xml:id="echoid-s17" xml:space="preserve">mirarenturque li-<lb/>beralitatem Telluris, cùm tot opes, ac copias plan-<lb/>tarum, fructuum, animalium è ſinu ſuo veluti mater <lb/>benigna mortalibus præbet. </s>
  <s xml:id="echoid-s18" xml:space="preserve">Hæc &amp; </s>
  <s xml:id="echoid-s19" xml:space="preserve">fimilia dum <lb/>priſci homines contemplarentur, fieri non potuit, <lb/>quin tantorum munerum largitores grato affectu <lb/>proſequerentur. </s>
  <s xml:id="echoid-s20" xml:space="preserve">Neque alia ratione cúm viri heroi-<lb/>ca virtute præditi artes, &amp; </s>
  <s xml:id="echoid-s21" xml:space="preserve">inuenta præclara valdè <lb/>vtilia ingeniosè iuxta, ac liberaliter mortalibus con-
<pb file="0008" n="8"/>
tuliſſent, ſumma veneratione talem, ac tantam bo-<lb/>nitatem ſuſceperunt, &amp; </s>
  <s xml:id="echoid-s22" xml:space="preserve">Diuinitatis honores eis deſi-<lb/>gnarunt, vt Cereri, Baccho, Herculi, Mercurio, <lb/>&amp; </s>
  <s xml:id="echoid-s23" xml:space="preserve">alijs. </s>
  <s xml:id="echoid-s24" xml:space="preserve">Horum autem illi præſtantiora bona attu-<lb/>liſſe humano generi cenſendi ſunt, non qui fragi-<lb/>lem, &amp; </s>
  <s xml:id="echoid-s25" xml:space="preserve">limo affixam noſtram partem, ſed qui ani-<lb/>mum Diuinæ auræ participem eruditione, ac ſapi-<lb/>entia perfecerunt, &amp; </s>
  <s xml:id="echoid-s26" xml:space="preserve">ornarunt. </s>
  <s xml:id="echoid-s27" xml:space="preserve">Hìnc artem, &amp; </s>
  <s xml:id="echoid-s28" xml:space="preserve"><lb/>facultatem illam tradentes, qua vaſti maris plani-<lb/>tiem intrepidè perambulare non dubitamus coactis <lb/>ventis imperata facere, ibidemque verſantes acu <lb/>magnetica itinera ad vnguem menſuramus, &amp; </s>
  <s xml:id="echoid-s29" xml:space="preserve">terræ <lb/>plagas, &amp; </s>
  <s xml:id="echoid-s30" xml:space="preserve">cœli, ſtellarumque loca, &amp; </s>
  <s xml:id="echoid-s31" xml:space="preserve">ſitus medijs <lb/>in tenebris conſtituti clarè conſpicimus. </s>
  <s xml:id="echoid-s32" xml:space="preserve">Vel hìnc <lb/>qua pondera immenſa puſillis noſtris viribus tanta <lb/>facilitate mouemus, vt terram vniuerſam è ſuo loco <lb/>transferre ſe poſſe non diffiteretur magnus ille Ar-<lb/>chimedes, ſi haberet, vbi pedem extra illam fige-<lb/>ret. </s>
  <s xml:id="echoid-s33" xml:space="preserve">Aut qua naturæ miracula in elementis, plantis, <lb/>animantibus perſcrutamur. </s>
  <s xml:id="echoid-s34" xml:space="preserve">Quaue ex fragili vitro lin-<lb/>ceos oculos veluti efformantes adeò cœlo proximi <lb/>efficimur, vt ferè ſummas mundi partes, &amp; </s>
  <s xml:id="echoid-s35" xml:space="preserve">ſtellas <lb/>innumeras hactenus inconſpicuas contrectare videa-<lb/>mur. </s>
  <s xml:id="echoid-s36" xml:space="preserve">Aut eam tandem doctrinam Aſtronomicam, <lb/>qua in Cęlum transuolamus, duabus nimirum alis <lb/>Geometriæ, &amp; </s>
  <s xml:id="echoid-s37" xml:space="preserve">Arithmeticæ, quibus Diuinæ Sa-<lb/>pientiæ theſauros contemplando, ſumma dulcedine <lb/>in hac mortali vita, gloriæ, felicitatiſque illius ineffa-<lb/>bilis participes efficimur.</s>
  <s xml:id="echoid-s38" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s39" xml:space="preserve">Sed quia felices admirabilium rerum inuentores <lb/>vel fortunæ, vel temporum iniuria plerumque neque-<lb/>unt ſua ſtudia, licet illuſtria, &amp; </s>
  <s xml:id="echoid-s40" xml:space="preserve">ſalutaria poſteritati
<pb file="0009" n="9"/>
tranſmittere, ideo viris principibus ſingulari virtute <lb/>præditis, ſine quorum auctoritate, &amp; </s>
  <s xml:id="echoid-s41" xml:space="preserve">munificentia <lb/>bonæ illæ artes omnino depreſſæ, contemptæ, &amp; </s>
  <s xml:id="echoid-s42" xml:space="preserve"><lb/>ſqualidæ deperirent, dum eas diuino inſtinctu pro-<lb/>mouent, augent, atque in vitam reuocant, ne dum <lb/>pares, ſed maiores gratias ijs habendas priſci homines <lb/>cenſuerunt, quàm inuentoribus ipſis, cùm ipſi bonis <lb/>illis alioqui non duraturis genus hominum beauerint.</s>
  <s xml:id="echoid-s43" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s44" xml:space="preserve">Atqui inter iſtos Heroas digniſsimum ſibi meritò <lb/>locum vindicarunt Maiores tui, Princeps Sereniſsime, <lb/>quibus gratitudinis perpetuam deberi memoriam eru-<lb/>diti omnes fatentur. </s>
  <s xml:id="echoid-s45" xml:space="preserve">Quippe poſtquam Barbarorum <lb/>incurſionibus Europa vniuerſa, &amp; </s>
  <s xml:id="echoid-s46" xml:space="preserve">Italia Princeps eius <lb/>prouincia priſco nitore amiſſo, omni ornatu litterarũ, <lb/>artium, bibliothecarum, lycęorum, imo humani-<lb/>tatis, &amp; </s>
  <s xml:id="echoid-s47" xml:space="preserve">politiæ ſpoliata diù iacuiſſet, Diuino fauore <lb/>primus omnium ſurrexit Magnus ille Coſmus Medi-<lb/>ceus, qui viros doctrina eximios cum vniuerſa ſupel-<lb/>lectile Gręcæ ſapientiæ Conſtantinopolitani Imperij <lb/>calamitatem fugientes eo affectu complexus eſt, vt <lb/>omnium Muſarum parens appellari deberet, qui ob <lb/>liberalitatem pluſquam regiam, &amp; </s>
  <s xml:id="echoid-s48" xml:space="preserve">beneficentiam vbi-<lb/>que terrarum effuſam, atque ob alia heroica geſta <lb/>Pater Patrię prius ſalutatus fuerat. </s>
  <s xml:id="echoid-s49" xml:space="preserve">In eius locum ſuc-<lb/>ceſsit Laurentius nepos, qui non ferro, &amp; </s>
  <s xml:id="echoid-s50" xml:space="preserve">cęde, ſed <lb/>ciuili prudentia, &amp; </s>
  <s xml:id="echoid-s51" xml:space="preserve">alto conſilio Patriam, &amp; </s>
  <s xml:id="echoid-s52" xml:space="preserve">pene <lb/>Europam moderatus eſt: </s>
  <s xml:id="echoid-s53" xml:space="preserve">nec modo Poéticis lepori-<lb/>bus ornatus, ſed profundiſsimæ Philoſophiæ Plato-<lb/>nicæ innutritus, eamdem doctrinam opera, &amp; </s>
  <s xml:id="echoid-s54" xml:space="preserve">ſtudio <lb/>potiſsimum Marſilij Ficini è Gręco translatam illu-<lb/>ſtratamque poſteris tranſtulit. </s>
  <s xml:id="echoid-s55" xml:space="preserve">Bibliothecam inſuper <lb/>Laurentianam à maioribus inchoatam comparatis
<pb file="0010" n="10"/>
vndique manuſcriptis codicibus ſummo impendio, <lb/>ſummaque cura locupletauit. </s>
  <s xml:id="echoid-s56" xml:space="preserve">Iſq; </s>
  <s xml:id="echoid-s57" xml:space="preserve">filium reliquit Leo-<lb/>nem X. </s>
  <s xml:id="echoid-s58" xml:space="preserve">Pont. </s>
  <s xml:id="echoid-s59" xml:space="preserve">Max.</s>
  <s xml:id="echoid-s60" xml:space="preserve">, qui vniuerſi orbis viros eruditos <lb/>dilexit, fouit, amplificauit: </s>
  <s xml:id="echoid-s61" xml:space="preserve">Bibliothecam Vaticanã <lb/>mirificè inſtruxit: </s>
  <s xml:id="echoid-s62" xml:space="preserve">Vrbis Lyceum à fundamentis ere-<lb/>xit, codicibus, &amp; </s>
  <s xml:id="echoid-s63" xml:space="preserve">viris doctrina magnis ornauit, atq; <lb/></s>
  <s xml:id="echoid-s64" xml:space="preserve">priſca barbarie omnino deleta aureum litterarum ſæ-<lb/>culum reſtituit. </s>
  <s xml:id="echoid-s65" xml:space="preserve">Sed Coſmus ille primus Magnus Dux <lb/>Etruriæ mihi nunc non reticendus, qui præter præcla-<lb/>ra bellica, &amp; </s>
  <s xml:id="echoid-s66" xml:space="preserve">politica facinora, quibus Etruſcum Im-<lb/>perium auxit, atque firmauit, promouendis diſcipli-<lb/>nis ſedulò intentus Athenæum Piſanum, vt cum ma-<lb/>ximè reparauit, vt profeſſoribus diſciplinarum fama <lb/>præſtantibus nobilitauit: </s>
  <s xml:id="echoid-s67" xml:space="preserve">Florentinam Academiam <lb/>inſtituit, Pandectarum libros ad fidem egregij, &amp; </s>
  <s xml:id="echoid-s68" xml:space="preserve">ve-<lb/>tuſtiſsimi codicis manuſcripti ampliſsimè excudi iuſ-<lb/>ſit: </s>
  <s xml:id="echoid-s69" xml:space="preserve">tot inſignes Græci, Latini, Etruſci idiomatis ſcri-<lb/>ptores vigilijs, &amp; </s>
  <s xml:id="echoid-s70" xml:space="preserve">labore eruditiſsimorum virorum illu-<lb/>ftratos typis edendos curauit: </s>
  <s xml:id="echoid-s71" xml:space="preserve">Paulum Iouium cum <lb/>primis, &amp; </s>
  <s xml:id="echoid-s72" xml:space="preserve">Io: </s>
  <s xml:id="echoid-s73" xml:space="preserve">Baptiſtam Adrianum ad ſui temporis <lb/>hiſtorias conſcribendas ampliſsimis oblatis præmijs <lb/>perſuaſit. </s>
  <s xml:id="echoid-s74" xml:space="preserve">Virtutes, atque opera tam Magni Paren-<lb/>tis imitatus eſt Franciſcus, qui in Imperio ſucceſsit, &amp; </s>
  <s xml:id="echoid-s75" xml:space="preserve"><lb/>antiquitatis ſtudio maximè delectatus, præclaras, atq; </s>
  <s xml:id="echoid-s76" xml:space="preserve"><lb/>innumeas venerandæ vetuſtatis reliquias, lapides, <lb/>gemmas, numiſmata collegit. </s>
  <s xml:id="echoid-s77" xml:space="preserve">Hunc excepit Ferdi-<lb/>nandus primus verè litteratorũ Mecoenas, qui Biblio-<lb/>thecam codicibus Hæbreis, Chaldæis, Syriacis, Egy-<lb/>ptijs, Perſis, &amp; </s>
  <s xml:id="echoid-s78" xml:space="preserve">Arabicis (inter quos hi libri Apollo-<lb/>nij, &amp; </s>
  <s xml:id="echoid-s79" xml:space="preserve">Archimedis extant) feliciſsimè ditatam reli-<lb/>quit, atque eruditiſsimos viros Hieronymum Mercu-<lb/>rialem, Petrum An<unsure/>gelum, Iacobum Mazzonum, Io:</s>
  <s xml:id="echoid-s80" xml:space="preserve">
<pb file="0011" n="11"/>
Baptiſtam Raimundum, totq; </s>
  <s xml:id="echoid-s81" xml:space="preserve">alios largiſsimis ſtipen-<lb/>pendijs euocauit, atq; </s>
  <s xml:id="echoid-s82" xml:space="preserve">aluit; </s>
  <s xml:id="echoid-s83" xml:space="preserve">Sacroſanctaq; </s>
  <s xml:id="echoid-s84" xml:space="preserve">Euangelia <lb/>fidei propagandæ ſtudio imprimi, Euclidem quoque, <lb/>Auicennam, Geographiam Nubienſem typis nitidiſ-<lb/>ſimis Arabicè omnia edi curauit. </s>
  <s xml:id="echoid-s85" xml:space="preserve">Non abſimilis litte-<lb/>rarum amore Coſmus Secundus, cuius nomen, ac glo-<lb/>riam magnus ille Galilæus erga Principem de ſe opti-<lb/>mè meritum gratiſsimus in cœlum vexit, ac inſculpſit; <lb/></s>
  <s xml:id="echoid-s86" xml:space="preserve">Vir nempe (vtGaſſendus ait) ſuper æthera notus; </s>
  <s xml:id="echoid-s87" xml:space="preserve">quo <lb/>alium non extulit ætas hæc noſtra glorioſiorem; </s>
  <s xml:id="echoid-s88" xml:space="preserve">quip-<lb/>pe tametſi orbis terrarum laudatis virorum illuſtrium <lb/>dictis, factiſq; </s>
  <s xml:id="echoid-s89" xml:space="preserve">circumſtrepit, horum tamen omnium <lb/>memoriam ſilentium altum breui inuoluet: </s>
  <s xml:id="echoid-s90" xml:space="preserve">nomen, <lb/>quod ille cœlo inſcripſit, donec cœleſtia curæ erunt, <lb/>apud homines perennabit. </s>
  <s xml:id="echoid-s91" xml:space="preserve">Tandem Ferdinandus <lb/>Secundus ingenij perſpicacia mirabilis, maieſtate im-<lb/>perij præclarus, virtutibus, &amp; </s>
  <s xml:id="echoid-s92" xml:space="preserve">Philoſophia illuſtrior fe-<lb/>liciter regnat: </s>
  <s xml:id="echoid-s93" xml:space="preserve">is eſt, cuius munificentia, ac fauore Eu-<lb/>ropa vniuerſa in Etruſca hac regia (ne aulicum decus, <lb/>aut cultum, nobilium obſequia, &amp; </s>
  <s xml:id="echoid-s94" xml:space="preserve">famulitium, Muſæ-<lb/>um ampliſsimum, ac ditiſsimum referam) eruditorum <lb/>frequentiam philoſophantium, diſceptationes, ac per-<lb/>petua exercitia literaria æſtimari, ac florere merito <lb/>ſuſpicit, &amp; </s>
  <s xml:id="echoid-s95" xml:space="preserve">veneratur; </s>
  <s xml:id="echoid-s96" xml:space="preserve">cum Muſæ reliquis in aulis <lb/>tantũ non neglectæ huc ſe ſe recepiſſe veluti in ſedem <lb/>ſuam videantur; </s>
  <s xml:id="echoid-s97" xml:space="preserve">hìc enim in delicijs habentur ſectio-<lb/>nes anatomicæ, cœleſtes obſeruationes, chimica eſpe-<lb/>rimenta, vniuerſęque naturalis philoſophiæ accurata <lb/>inquiſitio. </s>
  <s xml:id="echoid-s98" xml:space="preserve">Vno verbo hinc credula philoſophia exu-<lb/>lat; </s>
  <s xml:id="echoid-s99" xml:space="preserve">non hominum libri in pretio habentur, ſed Dei <lb/>volumen, ſcilicet rerum natura veris, accuratiſq; </s>
  <s xml:id="echoid-s100" xml:space="preserve">expe-<lb/>rimentis ſummo ſtudio indagatur, &amp; </s>
  <s xml:id="echoid-s101" xml:space="preserve">colitur. </s>
  <s xml:id="echoid-s102" xml:space="preserve">Præcla-
<pb file="0012" n="12"/>
ris hiſce ſtudijs lactatus, &amp; </s>
  <s xml:id="echoid-s103" xml:space="preserve">innutritus es, Princeps Se-<lb/>reniſsimè, tot tantorumq; </s>
  <s xml:id="echoid-s104" xml:space="preserve">heroum progenies, quorum <lb/>virtutes incomparabiles, &amp; </s>
  <s xml:id="echoid-s105" xml:space="preserve">egregia geſta conſentaneũ <lb/>eſt in te vno veluti foco ſpeculi parabolici ſimul colle-<lb/>cta, &amp; </s>
  <s xml:id="echoid-s106" xml:space="preserve">vnita ſplendeſcere, vt totas vires ſuas ſumma <lb/>virtus experiatur, atq; </s>
  <s xml:id="echoid-s107" xml:space="preserve">ineffabilem bonitatem, benefi-<lb/>centiæq; </s>
  <s xml:id="echoid-s108" xml:space="preserve">ſtudium, virtutum, artium, ſcientiarum cul-<lb/>tum à maioribus acceptum ſtudiosè, &amp; </s>
  <s xml:id="echoid-s109" xml:space="preserve">religiosè con-<lb/>ſerues, atq; </s>
  <s xml:id="echoid-s110" xml:space="preserve">ad poſteros auctum tranſmittas.</s>
  <s xml:id="echoid-s111" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s112" xml:space="preserve">Si igitur hominum genus natura dictante primum <lb/>Deo Op. </s>
  <s xml:id="echoid-s113" xml:space="preserve">Max.</s>
  <s xml:id="echoid-s114" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s115" xml:space="preserve">beneficentiſsimo gratias iuſtis ho-<lb/>noribus, &amp; </s>
  <s xml:id="echoid-s116" xml:space="preserve">memori mente perſoluendas eſſe decreuit; <lb/></s>
  <s xml:id="echoid-s117" xml:space="preserve">atq; </s>
  <s xml:id="echoid-s118" xml:space="preserve">ne memoria beneficiorum deleretur templa, fa-<lb/>na, feſtos dies, &amp; </s>
  <s xml:id="echoid-s119" xml:space="preserve">ludos inſtituit. </s>
  <s xml:id="echoid-s120" xml:space="preserve">Secundo loco eoſ-<lb/>dem ferè honores Heroibus, ac Principibus ſtatuit, nõ <lb/>his qui armis, &amp; </s>
  <s xml:id="echoid-s121" xml:space="preserve">cęde potentiam violenter ſibi vindi-<lb/>carunt, ſed qui præſtantibus virtutibus ornati magna <lb/>beneficia in homines contulerunt, ſique eos non hu-<lb/>manis, ſed diuinis laudibus celebrari iuſsit, potiori iure <lb/>tibi, Princeps Glorioſiſsimè, pręclariſsimorũ heroum, <lb/>ac virtutum hęredi plauſus debitus, honores, laudes, <lb/>&amp; </s>
  <s xml:id="echoid-s122" xml:space="preserve">grati animi monumenta ab eruditis Europæ viris <lb/>offeruntur. </s>
  <s xml:id="echoid-s123" xml:space="preserve">Quandoquidem magna, &amp; </s>
  <s xml:id="echoid-s124" xml:space="preserve">certa illos ſpes <lb/>tenet ampliſsimum patrimonium heroicarum virtutũ, <lb/>quod Coſmus Pater Patriæ, Laurentius magnificen-<lb/>tiæ exemplar, Leo ſui ſęculi felicitas, inſequenteſque <lb/>generoſiſsimi Principes, atq; </s>
  <s xml:id="echoid-s125" xml:space="preserve">Heroes de genere huma-<lb/>no, &amp; </s>
  <s xml:id="echoid-s126" xml:space="preserve">bonis litteris optimè meriti tibi reliquerunt non <lb/>ad ſaſtum, ſed ad imitationem, &amp; </s>
  <s xml:id="echoid-s127" xml:space="preserve">ſtimulum gloriæ, <lb/>nec externè, ſed in animo, &amp; </s>
  <s xml:id="echoid-s128" xml:space="preserve">cordis ſacrario piè a te, <lb/>ac reuerenter curandum, ſeruandum, amplificandum <lb/>ea pręcipuè qua polles pręclara indole, ingenijq; </s>
  <s xml:id="echoid-s129" xml:space="preserve">acu-
<pb file="0013" n="13"/>
mine, ac felicitate, amoreq; </s>
  <s xml:id="echoid-s130" xml:space="preserve">ſcientiarum, ac bonarum <lb/>artium, quibus te Deus, &amp; </s>
  <s xml:id="echoid-s131" xml:space="preserve">Natura indulgentiſsimè <lb/>cumulauit. </s>
  <s xml:id="echoid-s132" xml:space="preserve">Hoc quidem ſummopere precatur, &amp; </s>
  <s xml:id="echoid-s133" xml:space="preserve"><lb/>vouet eruditorum Reſpublica, ò Princeps longe in-<lb/>comparabilis, idque vaticinatur ex hoc tuo pręclaro <lb/>decore, &amp; </s>
  <s xml:id="echoid-s134" xml:space="preserve">ſummæ bonitatis ſpecimine: </s>
  <s xml:id="echoid-s135" xml:space="preserve">Quippe, ò <lb/>Principum decus, &amp; </s>
  <s xml:id="echoid-s136" xml:space="preserve">ſtudioſorum delicium, perbellè <lb/>docuiſti virtutis heroicę magis proprium eſſe benefa-<lb/>cere, &amp; </s>
  <s xml:id="echoid-s137" xml:space="preserve">alijs prodeſſe, quàm laudes meritas captare, &amp; </s>
  <s xml:id="echoid-s138" xml:space="preserve"><lb/>exigere; </s>
  <s xml:id="echoid-s139" xml:space="preserve">dum veluti epulo lautiſsimo in hac ſolemni <lb/>pompa tuarum nuptiarum, ſcientiarum cultores dona-<lb/>tos voluiſti; </s>
  <s xml:id="echoid-s140" xml:space="preserve">quid enim pretioſius, &amp; </s>
  <s xml:id="echoid-s141" xml:space="preserve">magis expetitum <lb/>veritatis ſtudioſis præbere poſſes, quàm Quintum, <lb/>Sextum, &amp; </s>
  <s xml:id="echoid-s142" xml:space="preserve">Septimum libros Conicorum Apollonij <lb/>Pergæi hactenus deploratos, atq; </s>
  <s xml:id="echoid-s143" xml:space="preserve">lemmata Archime-<lb/>dis, quæ Sereniſsimus Ferdinandus Secundus inclytus, <lb/>atque optimus parens tuus ex Arabico verti, &amp; </s>
  <s xml:id="echoid-s144" xml:space="preserve">ty pis <lb/>excudi ad communem reipublicæ litterariæ bonum <lb/>iuſsit? </s>
  <s xml:id="echoid-s145" xml:space="preserve">Tanto ergo pro beneficio</s>
</p>
<p>
  <s xml:id="echoid-s146" xml:space="preserve">-- grates perſoluere dignas</s>
</p>
<p>
  <s xml:id="echoid-s147" xml:space="preserve">-- Non opis eſt noſtræ, <lb/>Numina tibi</s>
</p>
<p>
  <s xml:id="echoid-s148" xml:space="preserve">-- pręmia digna ferant, quę te tam lęta tulerunt</s>
</p>
<p>
  <s xml:id="echoid-s149" xml:space="preserve">-- ſęcula. </s>
  <s xml:id="echoid-s150" xml:space="preserve">qui tanti talem genuere parentes.</s>
  <s xml:id="echoid-s151" xml:space="preserve"/>
</p>
<pb file="0014" n="14"/>
</div>
<div xml:id="echoid-div8" type="section" level="1" n="8">
<head xml:id="echoid-head22" xml:space="preserve">CAVE CHRISTIANE LECTOR.</head>
<p>
  <s xml:id="echoid-s152" xml:space="preserve">ABalphatus Aſphahanenſis Apollonij Paraphra-<lb/>ſtes religione Maumedanus fuit; </s>
  <s xml:id="echoid-s153" xml:space="preserve">quapropter <lb/>aliquot locis more ſuę Gentis non modo Regi ſuo <lb/>Abicaligiar Carſciaſeph nimium adulatur, verùm <lb/>etiam impiè loquitur. </s>
  <s xml:id="echoid-s154" xml:space="preserve">Nihil tamen omiſsum eſt, vt <lb/>antiquus Codex integrè, fideliterq; </s>
  <s xml:id="echoid-s155" xml:space="preserve">exhiberetur. </s>
  <s xml:id="echoid-s156" xml:space="preserve">Hęc <lb/>eadem de Archimedis interprete dicta ſunto. </s>
  <s xml:id="echoid-s157" xml:space="preserve">De <lb/>his te præmonitum volui, ne inter legendum piæ <lb/>aures tuæ vel minimùm offenderentur.</s>
  <s xml:id="echoid-s158" xml:space="preserve"/>
</p>
<pb file="0015" n="15"/>
</div>
<div xml:id="echoid-div9" type="section" level="1" n="9">
<head xml:id="echoid-head23" xml:space="preserve">IN NOMINE DEI <lb/>MISERICORDIS <lb/>MISERATORIS.</head>
<head xml:id="echoid-head24" style="it" xml:space="preserve">PROOE MIVM</head>
<head xml:id="echoid-head25" xml:space="preserve">ABALPHATHI FILII MAHMVDI, FILII ALCASEMI, <lb/>FILII ALPHADHALI ASPHAHANENSIS.</head>
<head xml:id="echoid-head26" style="it" xml:space="preserve">LAVS DEO VTRIVSQVE SECVLI DOMINO.</head>
<p>
  <s xml:id="echoid-s159" xml:space="preserve">MATHEMATICA quamuis pra-<lb/>ctica ſit ſcientia, ac diſciplina, cu-<lb/>ius legibus, &amp; </s>
  <s xml:id="echoid-s160" xml:space="preserve">præceptionibus diſ-<lb/>ponitur, atq; </s>
  <s xml:id="echoid-s161" xml:space="preserve">dirigitur intellectiua <lb/>potentia ad abſolutam, perfectam-<lb/>que imaginum cognitionem, præ-<lb/>ſcindendo à materijs, qui eſt pri-<lb/>mus gradus aſcenſionis à ſenſibilibus ad intelligi-<lb/>qilia; </s>
  <s xml:id="echoid-s162" xml:space="preserve">nihilominus ſuarum claritate demonſtratio-<lb/>num, non ſolùm ab alijs differt ſcientijs verùm
<pb file="0016" n="16" rhead="ABALPHATI"/>
etiam longiſsimè ijs præſtat, atq; </s>
  <s xml:id="echoid-s163" xml:space="preserve">præcellit, eò quòd <lb/>fæcium, ſordiumque dubitationum, &amp; </s>
  <s xml:id="echoid-s164" xml:space="preserve">aliorum hu-<lb/>iuſmodi generis accidentium expers omninò ſit, atq; <lb/></s>
  <s xml:id="echoid-s165" xml:space="preserve">libera. </s>
  <s xml:id="echoid-s166" xml:space="preserve">Ea autem propter ſe habet ad ſcientificam <lb/>potentiam, quemadmodùm habent ſe limpidiſsima <lb/>quæque orbi ſolis oppoſita ad viſiuam potentiam. </s>
  <s xml:id="echoid-s167" xml:space="preserve"><lb/>Ex quo ad illam comparandam, conſequendamq; </s>
  <s xml:id="echoid-s168" xml:space="preserve"><lb/>non excitatur intellectiua duntaxat vis, verùm etiam <lb/>multùm exacuitur, atq; </s>
  <s xml:id="echoid-s169" xml:space="preserve">delectatur, ponderatis præ-<lb/>ſertim, expenſisq; </s>
  <s xml:id="echoid-s170" xml:space="preserve">illius demonſtrationibus, &amp; </s>
  <s xml:id="echoid-s171" xml:space="preserve">cer-<lb/>tiſsima earum comprehenſa, &amp; </s>
  <s xml:id="echoid-s172" xml:space="preserve">cognita veritate. </s>
  <s xml:id="echoid-s173" xml:space="preserve"><lb/>Tunc quippè huius veritatis percepta animus odo-<lb/>rationis ſuauitate, auidè, &amp; </s>
  <s xml:id="echoid-s174" xml:space="preserve">ardentiùs appetit con-<lb/>ſequi ea omnia, quæ illius ſuggerunt demonſtra-<lb/>tiones, earumque potiri. </s>
  <s xml:id="echoid-s175" xml:space="preserve">Subindè verò procedere <lb/>conatur vltrò ad vltimum finem, nempè ad pro-<lb/>prietatum, &amp; </s>
  <s xml:id="echoid-s176" xml:space="preserve">obiecti illius cognitionem, excelſita-<lb/>tem, atque præſtantiam comparandam, tandemque <lb/>ad ea omnia, quæ ad ipſam ſpectant. </s>
  <s xml:id="echoid-s177" xml:space="preserve">Quod qui-<lb/>dem luminis cùm ipſius affulſerit ſtudioſis, &amp; </s>
  <s xml:id="echoid-s178" xml:space="preserve">quàm <lb/>præcellens ſit, animaduerterint, omnes ſuos con-<lb/>tulerunt conatus ad libros componendos, conſcri-<lb/>bendoſq; </s>
  <s xml:id="echoid-s179" xml:space="preserve">de ipſius elementis, principijs, ac omni-<lb/>bus ijs, quę indè deriuantur, &amp; </s>
  <s xml:id="echoid-s180" xml:space="preserve">eò ſpectant. </s>
  <s xml:id="echoid-s181" xml:space="preserve">Soli-<lb/>diora porrò profeſsionis huius fundamenta omnium <lb/>primus iecit Euclides in eo libro, quem de elemen-<lb/>tis inſcripſit, in quo fundamentales continentur ra-<lb/>tiones linearum tam rectarum, quàm curuarum, <lb/>nec non ſuperficierum prouenientium vel ex earum <lb/>ſingulis vel ex omnibus ſimul ſumptis. </s>
  <s xml:id="echoid-s182" xml:space="preserve">Rationes <lb/>prætereà habentur ſolidorum prouenientium, vel
<pb file="0017" n="17" rhead="PROEMIVM."/>
ex ſuperficiebus rectilineis, qualia ſunt habentia <lb/>baſes; </s>
  <s xml:id="echoid-s183" xml:space="preserve">vel ex curuis, qualia ſunt ſphœrica; </s>
  <s xml:id="echoid-s184" xml:space="preserve">vel ex <lb/>hiſce compoſitis, quales ſunt ſuperficies Cylindro-<lb/>rum, &amp; </s>
  <s xml:id="echoid-s185" xml:space="preserve">Conorum. </s>
  <s xml:id="echoid-s186" xml:space="preserve">Verùm enim verò figuris ex <lb/>ſegmentis ſuperficierum planarum prouenientibus, <lb/>&amp; </s>
  <s xml:id="echoid-s187" xml:space="preserve">cuiuſlibet etiam Solidorum Sphœricorum, Cy-<lb/>lindricorum, atque Conicorum nullum hactenùs ia-<lb/>ctum erat fundamentum, aut præmiſſa elementa, <lb/>vel fundamenta aliqua. </s>
  <s xml:id="echoid-s188" xml:space="preserve">Ex quo illi priſci librorum <lb/>Scriptores aliquid de ijs innuebant duntaxat, &amp; </s>
  <s xml:id="echoid-s189" xml:space="preserve"><lb/>quidem leuiter. </s>
  <s xml:id="echoid-s190" xml:space="preserve">De Sphœricis autem aliquid ex eo-<lb/>rum legebant proprietatibus, &amp; </s>
  <s xml:id="echoid-s191" xml:space="preserve">paſsionibus; </s>
  <s xml:id="echoid-s192" xml:space="preserve">ſiue ex <lb/>proprietatibus ſegmentorum indè prouenientium; <lb/></s>
  <s xml:id="echoid-s193" xml:space="preserve">vel figurarum in ea incidentium; </s>
  <s xml:id="echoid-s194" xml:space="preserve">vel ex accidenti-<lb/>bus quibuſdam ipſius Sphœræ, quæ ex eius proce-<lb/>dunt motibus; </s>
  <s xml:id="echoid-s195" xml:space="preserve">vel quia ſe inuicem includunt, &amp; </s>
  <s xml:id="echoid-s196" xml:space="preserve"><lb/>componunt. </s>
  <s xml:id="echoid-s197" xml:space="preserve">Nam Sphœra aliqua opus illi erat ad <lb/>Sphœræ vniuerſalis cognitionem conſequendam vna <lb/>cum eius orbibus, ac motibus, &amp; </s>
  <s xml:id="echoid-s198" xml:space="preserve">ad inuicem at-<lb/>que ſua centra applicatione. </s>
  <s xml:id="echoid-s199" xml:space="preserve">Et id tandem, donec <lb/>librum Almageſti compoſuit Ptolomæus, in quo <lb/>ea omnia recondidit copiosè, quæ illi anguſtè, &amp; </s>
  <s xml:id="echoid-s200" xml:space="preserve"><lb/>leuiter hoc de argumento ſuis innuebant ſcriptis, <lb/>tradens non ſolùm methodum, ac rationem eorum <lb/>aſſequendi cognitionem, ſed, &amp; </s>
  <s xml:id="echoid-s201" xml:space="preserve">inſtrumentorum <lb/>etiam vſum. </s>
  <s xml:id="echoid-s202" xml:space="preserve">Quod profectò iactum fuit tamquàm <lb/>vniuerſale quoddam fundamentum, ac principium <lb/>ea omnia comprehendens, quæ ad Sphœrica perti-<lb/>nent; </s>
  <s xml:id="echoid-s203" xml:space="preserve">vndè hac in re ſatis abundè ſtudioſorum ſiti, <lb/>&amp; </s>
  <s xml:id="echoid-s204" xml:space="preserve">deſiderio conſultum fuit. </s>
  <s xml:id="echoid-s205" xml:space="preserve">Porrò Appollonius <lb/>profeſsionẽ, &amp; </s>
  <s xml:id="echoid-s206" xml:space="preserve">diſciplinam hanc ad ſupremum per-
<pb file="0018" n="18" rhead="ABALPHATI"/>
fectionis perduxit gradum, Conicorum componen-<lb/>do librum, qui Conicarum ſectionum complecti-<lb/>tur proprietates, quæ ſublimiorem, eminentiorem-<lb/>que diſciplinæ huius ſibi vindicant locum. </s>
  <s xml:id="echoid-s207" xml:space="preserve">Et ſane <lb/>tot propoſitionibus, totque figuris illum ditauit, vt <lb/>admirabiles illæ nuncupari meruerint, eò quòd <lb/>contineant lineas curuas, ſeu medias inter rectas, <lb/>ac circulares ſeſe inuicem ſecantes; </s>
  <s xml:id="echoid-s208" xml:space="preserve">adeoque miros <lb/>quidem fundunt ſenſus, &amp; </s>
  <s xml:id="echoid-s209" xml:space="preserve">proprietates. </s>
  <s xml:id="echoid-s210" xml:space="preserve">Quos qui-<lb/>dem omneslibros, qui diſciplinæ huius fundamenta <lb/>ſunt, ad Arabicam tranſtulere linguam illius ſtu-<lb/>dioſi. </s>
  <s xml:id="echoid-s211" xml:space="preserve">Quamuis autem liber iſte Conicorum præ-<lb/>ſtantiſsimus ſit, tam ratione ſui, quàm præclariſ-<lb/>ſimi Auctoris, nihilominùs nimiam ob illius obſcu-<lb/>ritatem, difficultateſque obuiam occurrentes, ac <lb/>profundiſsimos, quos continet ſenſus; </s>
  <s xml:id="echoid-s212" xml:space="preserve">tum etiam <lb/>ob innumeras, &amp; </s>
  <s xml:id="echoid-s213" xml:space="preserve">admirabiles figuras, &amp; </s>
  <s xml:id="echoid-s214" xml:space="preserve">propoſi-<lb/>tiones; </s>
  <s xml:id="echoid-s215" xml:space="preserve">tandemque ob temporis diuturnitatem, in-<lb/>genteſque perferendos labores ab interprete, qui <lb/>eùm ex Græca transferat lingua, dudum neglectus <lb/>fuit, ac penè etærnæ datus obliuioni, vt nemò ha-<lb/>ctenùs illum, vel Commentarijs illuſtrauerit, vel <lb/>congeſſerit in ordinem, quamquàm ſummè ſit ne-<lb/>ceſſarius, ac vtiliſsimas complectatur propoſitiones, <lb/>&amp; </s>
  <s xml:id="echoid-s216" xml:space="preserve">figuras. </s>
  <s xml:id="echoid-s217" xml:space="preserve">Quapropter diù ſepultus, &amp; </s>
  <s xml:id="echoid-s218" xml:space="preserve">ignotus <lb/>iacuit, &amp; </s>
  <s xml:id="echoid-s219" xml:space="preserve">penè ad defectum vſque, ac interitum, <lb/>cùm apud Diſciplinæ ſtudioſos, tum etiam ipſos <lb/>profeſſores, &amp; </s>
  <s xml:id="echoid-s220" xml:space="preserve">fragmenta ex illo circumferebantur <lb/>aliqua, &amp; </s>
  <s xml:id="echoid-s221" xml:space="preserve">ea ſanè faciliora, quia obſcuriora euita-<lb/>bant omnes, atque declinabant; </s>
  <s xml:id="echoid-s222" xml:space="preserve">vno verbo inte-<lb/>grum hactenùs viderat nemo. </s>
  <s xml:id="echoid-s223" xml:space="preserve">Hinc mihi famulo
<pb file="0019" n="19" rhead="PROEMIVM."/>
viſum eſt, me Reipublicæ Literariæ gratam rem <lb/>facturum, ſi eum in integrum reſtituam, ac in <lb/>vnum congeram volumen, vt ita redactus facilis ſit <lb/>portatu, ſub omnium verſetur oculis, omnium te-<lb/>ratur manibus, &amp; </s>
  <s xml:id="echoid-s224" xml:space="preserve">ad reliqua facilior reddatur adi-<lb/>tus. </s>
  <s xml:id="echoid-s225" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/>Quem etiam librum comparare ſtudui Biblio-
<anchor type="note" xlink:label="note-0019-01a" xlink:href="note-0019-01"/>
thecæ domini noſtri Regis præſtantiſsimi, munifi-<lb/>centiſsimi, doctiſsimi, iuſtiſsimi, victoris, trium-<lb/>phatoris, Fidei defenſoris, celſitudinis Monarcha-<lb/>rum, gloriationis ſui generis, gloriæ religionis, ſolis <lb/>Regum, Abicaligiar Carſciaſeph Filij Alì, Filij <lb/>Phrami, Filij Haſami, Principis Fidelium, quem <lb/>incolumem, ac ſoſpitem ſeruet Deus, eiuſque de-<lb/>primat hoſtes, &amp; </s>
  <s xml:id="echoid-s226" xml:space="preserve">proterat oſores. </s>
  <s xml:id="echoid-s227" xml:space="preserve">Nunc autem ali-<lb/>quid de ordine, &amp; </s>
  <s xml:id="echoid-s228" xml:space="preserve">rerum diſpoſitione, ac conciſa <lb/>breuitate dicendum nobis ſupereſt. </s>
  <s xml:id="echoid-s229" xml:space="preserve">Nam rerum <lb/>ordo, &amp; </s>
  <s xml:id="echoid-s230" xml:space="preserve">accommodata diſpoſitio id intelligentiæ <lb/>afferunt auxilij, quod in ſcientijs comparandis lu-<lb/>culentiſsimæ demonſtrationes; </s>
  <s xml:id="echoid-s231" xml:space="preserve">conciſa verò breui-<lb/>tas, ac ſuis terminis neceſſarijs expedita, &amp; </s>
  <s xml:id="echoid-s232" xml:space="preserve">ritè di-<lb/>ſpoſita, eandem penè proportionem habet ad in-<lb/>telligentiam, ac cauſa ad cauſatum. </s>
  <s xml:id="echoid-s233" xml:space="preserve">Ea autem <lb/>propter ordinis conſeruatrix virtus venatio dici ſo-<lb/>lita eſt, &amp; </s>
  <s xml:id="echoid-s234" xml:space="preserve">ſatis quidem appoſitè, &amp; </s>
  <s xml:id="echoid-s235" xml:space="preserve">eleganter. <lb/></s>
  <s xml:id="echoid-s236" xml:space="preserve">Nam concepti ſenſus, &amp; </s>
  <s xml:id="echoid-s237" xml:space="preserve">in mente comparati, ſi <lb/>intra ordinis cancellos includantur, ſingulos ſuis di-<lb/>ſpenſare momentis procliuè poterit conſeruatrix re-<lb/>rum illa virtus. </s>
  <s xml:id="echoid-s238" xml:space="preserve">Simillimi, alioquin erunt feris per <lb/>vaſtas vagantibus ſolitudines, ac nullo coércitis val-<lb/>lo, quorum imagines, &amp; </s>
  <s xml:id="echoid-s239" xml:space="preserve">motus ita ſeſe offerunt <lb/>conſpicienti, &amp; </s>
  <s xml:id="echoid-s240" xml:space="preserve">contemplanti, vt nullo negotio eas
<pb file="0020" n="20" rhead="ABALPHATI"/>
capere, &amp; </s>
  <s xml:id="echoid-s241" xml:space="preserve">aucupari ſe poſſe arbitretur, at cum id <lb/>præſtare tentat, ſtatim dilabuntur, atque euane-<lb/>ſcunt. </s>
  <s xml:id="echoid-s242" xml:space="preserve">Ea planè ratione termini rerum ſingulos in <lb/>mente conceptos ſenſus deſignantes, niſi ſuo coér-<lb/>ceantur ordine dilabuntur, &amp; </s>
  <s xml:id="echoid-s243" xml:space="preserve">euaneſcunt; </s>
  <s xml:id="echoid-s244" xml:space="preserve">præci-<lb/>puè cùm modò hanc, modò illam fundant ſigniſi-<lb/>cationem, cùm iuxta labentis temporis varietatem, <lb/>tùm diuerſitatem regionum, &amp; </s>
  <s xml:id="echoid-s245" xml:space="preserve">prouinciarum, vt <lb/>non eadem vbique, &amp; </s>
  <s xml:id="echoid-s246" xml:space="preserve">ſemper ſit par ratio, licet <lb/>ijdem in anima maneant habitus. </s>
  <s xml:id="echoid-s247" xml:space="preserve">Ex quo palam, <lb/>&amp; </s>
  <s xml:id="echoid-s248" xml:space="preserve">planè relinquitur, quòd acquiſiti illi termini non <lb/>inhæreant, quemadmodùm ſubſiſtenti eſſentiales <lb/>inhærent differentiæ; </s>
  <s xml:id="echoid-s249" xml:space="preserve">neque etiam quemadmobùm <lb/>proprietates neceſſariò conſequentes ſuo inhærent <lb/>ſubiecto; </s>
  <s xml:id="echoid-s250" xml:space="preserve">ſed ea inhærent ratione, quá accidentia <lb/>difficilè, ac tardè amouibilia. </s>
  <s xml:id="echoid-s251" xml:space="preserve">Quandoquidem ter-<lb/>mini eiuſmodi vocabula ſunt quædam rebus impo-<lb/>ſita, &amp; </s>
  <s xml:id="echoid-s252" xml:space="preserve">applicata ad ſenſus commodè eliciendos, <lb/>atque eruendos. </s>
  <s xml:id="echoid-s253" xml:space="preserve">Quod autem vel diuino factitatum <lb/>eſt inſtinctu, vel Prophetica inſpiratione edoctum, <lb/>ſicut indicat nobis Altiſsimus Deus dicens:</s>
  <s xml:id="echoid-s254" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> ( in Al-
<anchor type="note" xlink:label="note-0020-01a" xlink:href="note-0020-01"/>
corano ) &amp; </s>
  <s xml:id="echoid-s255" xml:space="preserve">docuit Adamum cuncta nomina; </s>
  <s xml:id="echoid-s256" xml:space="preserve">vel <lb/>iudicio, &amp; </s>
  <s xml:id="echoid-s257" xml:space="preserve">calculo ſapientum virorum, quemad-<lb/>modùm præſtitiſſe legimus primos illos artium in-<lb/>uentores. </s>
  <s xml:id="echoid-s258" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s259" xml:space="preserve">ſcientiarum; </s>
  <s xml:id="echoid-s260" xml:space="preserve">vel magna aliqua neceſ-<lb/>ſitas hominum coégit vulgus ad eiuſmodi excogi-<lb/>tandos terminos, rebuſque imponendos, ac tranſ-<lb/>latione quadam vocabula mutuanda, &amp; </s>
  <s xml:id="echoid-s261" xml:space="preserve">ad alias, <lb/>atque alias res transferenda, ex quo ſynonymo-<lb/>rum ea enata eſt ccpia. </s>
  <s xml:id="echoid-s262" xml:space="preserve">Nec vllus profectò ſapien-<lb/>tum, qui has profeſsi ſunt Diſciplinas, aut qui ip-
<pb file="0021" n="21" rhead="PROEMIVM."/>
ſorum ſecuti ſunt veſtigia, hanc imponendorum <lb/>terminorum rationem aſpernatus ſubindè eſt, aut <lb/>ab illa abhorruit; </s>
  <s xml:id="echoid-s263" xml:space="preserve">quinimò acceptiſsima ſemper om-<lb/>nibus fuit, vt quæ maximum rerum intelligentiæ <lb/>ſplendorem affert, &amp; </s>
  <s xml:id="echoid-s264" xml:space="preserve">claritatem. </s>
  <s xml:id="echoid-s265" xml:space="preserve">Eandem igitur <lb/>hanc ob cauſam in colligendis, digerendiſque hiſce <lb/>famulus libris, antiquorum ſapientum, &amp; </s>
  <s xml:id="echoid-s266" xml:space="preserve">artium <lb/>profeſſorum, inuentorumque inſiſtens veſtigijs, ter-<lb/>minos, &amp; </s>
  <s xml:id="echoid-s267" xml:space="preserve">vocabula ſingulis rebus imponere, &amp; </s>
  <s xml:id="echoid-s268" xml:space="preserve"><lb/>earum vim breui declarare definitione cenſuit, vt <lb/>ita ſuis coércita omnia limitibus nequeant in varias <lb/>partes, &amp; </s>
  <s xml:id="echoid-s269" xml:space="preserve">ſenſus diffluere, ad conciliandam lecto-<lb/>ri inter legendum hos Apollonij libros eam, quæ <lb/>fieri poteſt, facilitatem. </s>
  <s xml:id="echoid-s270" xml:space="preserve">Innui prætereà eandem <lb/>etiam ob cauſam obſcurioribus in locis expoſitionem <lb/>aliquam, ne vlla ſubindè relinqueretur difficultas <lb/>ad mentem Auctoris cumulatè aſſequendam. <lb/></s>
  <s xml:id="echoid-s271" xml:space="preserve">Tandem lectorem meum enixè rogo, vt <lb/>excuſatum me habeat, ſi mendum <lb/>aliquod, aut erratum meam <lb/>ſubterfugerit diligentiam. </s>
  <s xml:id="echoid-s272" xml:space="preserve"><lb/>Interea Deum ſup-<lb/>pliciter depre-<lb/>cor <lb/>Altiſsimum, vt nos ad ea, quæ vtiliora <lb/>nobis ſunt, demúm <lb/>perducat.</s>
  <s xml:id="echoid-s273" xml:space="preserve"/>
</p>
<div xml:id="echoid-div9" type="float" level="2" n="1">
<note symbol="*" position="right" xlink:label="note-0019-01" xlink:href="note-0019-01a" xml:space="preserve">Impiè <lb/>adulatur <lb/>Regi ſuo <lb/>Paraphra <lb/>ſtes Ara-<lb/>bicus.</note>
<note symbol="*" position="left" xlink:label="note-0020-01" xlink:href="note-0020-01a" xml:space="preserve">Inſulsè <lb/>ex Alco-<lb/>rano pro-<lb/>fert, quæ <lb/>ſunt in Sa <lb/>cra Gene-ſi.</note>
</div>
<pb file="0022" n="22"/>
<p style="it">
  <s xml:id="echoid-s274" xml:space="preserve">Ne vacaret pagina ipſiusmet Apollonĳ Pergæi ex Epiſtola ad Eude-<lb/>mum Argumenta in quatuor Conicorum libros poſteriores, qui Græcà <lb/>linguà iniuria temporum perierunt, hìc apponuntur, quorum tres ex <lb/>Arabicis M.</s>
  <s xml:id="echoid-s275" xml:space="preserve">SS. </s>
  <s xml:id="echoid-s276" xml:space="preserve">nunc exhibentur.</s>
  <s xml:id="echoid-s277" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s278" xml:space="preserve">Reliqui autem quatuor libri ad abundatiorem <lb/>ſcientiam pertinent. </s>
  <s xml:id="echoid-s279" xml:space="preserve">Quintus de Minimis, &amp; </s>
  <s xml:id="echoid-s280" xml:space="preserve">Ma-<lb/>ximis magna ex parte agit. </s>
  <s xml:id="echoid-s281" xml:space="preserve">Sextus de Æqualibus, <lb/>&amp; </s>
  <s xml:id="echoid-s282" xml:space="preserve">Similibus coni ſectionibus. </s>
  <s xml:id="echoid-s283" xml:space="preserve">Septimus continet <lb/>Theoremata quæ determinandi vim habent. <lb/></s>
  <s xml:id="echoid-s284" xml:space="preserve">Octauus Problemata conica determinata.</s>
  <s xml:id="echoid-s285" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s286" xml:space="preserve">Hæc eadem Pappus Alexandrinus lib. </s>
  <s xml:id="echoid-s287" xml:space="preserve">7. </s>
  <s xml:id="echoid-s288" xml:space="preserve">Mathemat. </s>
  <s xml:id="echoid-s289" xml:space="preserve">Collect.</s>
  <s xml:id="echoid-s290" xml:space="preserve">, atq; <lb/></s>
  <s xml:id="echoid-s291" xml:space="preserve">Eutocius in Commentar. </s>
  <s xml:id="echoid-s292" xml:space="preserve">ad Apollonium.</s>
  <s xml:id="echoid-s293" xml:space="preserve"/>
</p>
<pb file="0023" n="23" rhead="PRÆFATIO."/>
</div>
<div xml:id="echoid-div11" type="section" level="1" n="10">
<head xml:id="echoid-head27" xml:space="preserve">ABRAHAMI ECCHELLENSIS</head>
<head xml:id="echoid-head28" xml:space="preserve">IN LATINAM EX ARABICIS</head>
<head xml:id="echoid-head29" xml:space="preserve">Librorum Apollonij Pergæi verſionem</head>
<head xml:id="echoid-head30" xml:space="preserve">PRÆFATIO.</head>
<p>
  <s xml:id="echoid-s294" xml:space="preserve">APOLLONIVS Pergæus vetuſtiſſimus, <lb/>ac magni nominis Græcus auctor otto de <lb/>Sectionibus Conicis conſcripſit libros. <lb/></s>
  <s xml:id="echoid-s295" xml:space="preserve">Horum priores quatuor hactenus omnium <lb/>teruntur manibus; </s>
  <s xml:id="echoid-s296" xml:space="preserve">poſteriores verò, ne-<lb/>ſcio quo fato, &amp; </s>
  <s xml:id="echoid-s297" xml:space="preserve">rerum viciſſitudine ſunt <lb/>amiſſi, ac non ſine magno literatorum <lb/>animi mcerore iamdudum deplorati, &amp; </s>
  <s xml:id="echoid-s298" xml:space="preserve"><lb/>nuſquam perdiligenter non quæſiti ab ijs præſertim, qui Geo-<lb/>metriæ, &amp; </s>
  <s xml:id="echoid-s299" xml:space="preserve">Matheſeos operam nauant ſtudijs, ſed fuſtra diu. </s>
  <s xml:id="echoid-s300" xml:space="preserve"><lb/>Tandem deprehenſum eſt, hos, quemadmodum, &amp; </s>
  <s xml:id="echoid-s301" xml:space="preserve">reliquam <lb/>penè Grecæ ſapientiæ ſupellectilem ad Arabum migraſſe ſcho-<lb/>las, ibique Arabicè conuerſos, &amp; </s>
  <s xml:id="echoid-s302" xml:space="preserve">Arabicis indutos ornamen-<lb/>tis, in illius gentis tamquam extorres, &amp; </s>
  <s xml:id="echoid-s303" xml:space="preserve">inquilinos latitaſſe <lb/>Bibliothecis. </s>
  <s xml:id="echoid-s304" xml:space="preserve">Quamobrem eorum miſerti vicem Sereniſſimi Ma-
<pb file="0024" n="24" rhead="ABRAHAMI ECCHELLENSIS"/>
gni Etruriæ Duces, inde magno ſoluto pretio redemerunt, ip-<lb/>ſorumque tam præclara opera quaſi iure poſtliminij vindica-<lb/>runt, ac demum patrio ſolo reddiderunt. </s>
  <s xml:id="echoid-s305" xml:space="preserve">Attamen ſat non-<lb/>fuit, aut viſum eſt ſummis iſtis Principibus Apollonium in liber-<lb/>tatem afferuiſſe, &amp; </s>
  <s xml:id="echoid-s306" xml:space="preserve">ex Barbarorum eripuiſſe manibus, ac in ce-<lb/>leberrrima tctius Europæ Auorum repoſuiſſe Biblioteca; </s>
  <s xml:id="echoid-s307" xml:space="preserve">ſed <lb/>omnem nauarunt operam, &amp; </s>
  <s xml:id="echoid-s308" xml:space="preserve">ſtudium, vt Latina etiam donati <lb/>linguà in literatorum gratiam publici iuris fierent. </s>
  <s xml:id="echoid-s309" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/>Ea propter
<anchor type="note" xlink:label="note-0024-01a" xlink:href="note-0024-01"/>
verè Magnus ille in omnibus Ferdinandus primus celeberrimam <lb/>eam erexit Typographiam è nomine gentilitio Sereniſſimæ fa-<lb/>miliæ Mediceam nuncupatam, cui nullam ſimilem, aut parem <lb/>vidit Chriſtianus Orbis, aut viſurus vnquàm eſt; </s>
  <s xml:id="echoid-s310" xml:space="preserve">ſiue characte-<lb/>rum, præſertim Arabicorum, ſpectes copiam, ſiue varietatem, <lb/>ſiue nitorem, ſiue elegantiam. </s>
  <s xml:id="echoid-s311" xml:space="preserve">Dictis hiſce profectò noſtris <lb/>ſpectatiſſimam, ac manifectiſſimam fidem faciunt Sacroſanti <lb/>Euangeliorum libri, Auicennæ, Euclidis, aliaque Arabica ope-<lb/>ra ijs edita typis, quæ omnibus Orientis gentibus admirationi <lb/>ſunt, atque adeo auidiſſimè expetuntur, ac magno comparan-<lb/>tur pretio. </s>
  <s xml:id="echoid-s312" xml:space="preserve">Sed hæc non typis duntaxat excudi iuſſit munifi-<lb/>centiſſimu s Princeps, verùm etiam viros linguarum peritiſſi-<lb/>mos ingenti conduxit ſtipendio, qui Arabicorũ Codicum va-<lb/>carent verſionibus. </s>
  <s xml:id="echoid-s313" xml:space="preserve">Hos autem inter principem obtinebat locum <lb/>Ioannes Baptiſta Raimundus vir, &amp; </s>
  <s xml:id="echoid-s314" xml:space="preserve">ſcientiarum cognitione, &amp; </s>
  <s xml:id="echoid-s315" xml:space="preserve"><lb/>linguarum peritia omnium ore celebratiſſimus. </s>
  <s xml:id="echoid-s316" xml:space="preserve">Is autem, &amp; </s>
  <s xml:id="echoid-s317" xml:space="preserve"><lb/>ſcriptis literis, &amp; </s>
  <s xml:id="echoid-s318" xml:space="preserve">familiaribus cum amicis colloquijs Apollonij <lb/>librorum verſionem ſæpenumerò pollicitus eſt. </s>
  <s xml:id="echoid-s319" xml:space="preserve">Imò erant, qui <lb/>libris editis verſionem iam à Raimundo confectam, &amp; </s>
  <s xml:id="echoid-s320" xml:space="preserve">perfe-<lb/>ctam eſſe, in vulgus iactarent. </s>
  <s xml:id="echoid-s321" xml:space="preserve">Verùm cum nunquam viſa fue-<lb/>rit eiuſmodi verſio, neque inter ipſius ſcripta reperta, neque <lb/>in Aduerſarijs notata, aut catalogo ipſius librorum adſcripta, <lb/>quæ omnia religiosè hactenùs conſeruantur, hoc vnum creden-<lb/>dum ſupereſt, eam votis ſolùm ſuſceptam, &amp; </s>
  <s xml:id="echoid-s322" xml:space="preserve">cogitatione deli-<lb/>neatam fuiſſe; </s>
  <s xml:id="echoid-s323" xml:space="preserve">rem autem, aut quòd per otium ipſi non licuit, aut <lb/>ob Codicis lectionem, &amp; </s>
  <s xml:id="echoid-s324" xml:space="preserve">ſcripturæ difficultatem, quæ maxima <lb/>eſt, vel ob orationis abſtruſæ, &amp; </s>
  <s xml:id="echoid-s325" xml:space="preserve">ſermonis ancipitem, ac mul-<lb/>tiplicem verborum poteſtatem, vel tandem aliquam aliam ob <lb/>cauſam, quàm, conijcere difficile eſt, perficere non potuiſſe.</s>
  <s xml:id="echoid-s326" xml:space="preserve">
<pb file="0025" n="25" rhead="PRÆFATIO."/>
Nihilo tamen minùs Magni Principis, Magni Filij, Magni Ne-<lb/>potes aut ab incœptis deſtiterunt, aut generoſiſſimi animi du-<lb/>dum conceptum ſtudium remiſerunt. </s>
  <s xml:id="echoid-s327" xml:space="preserve">Quamobrem ante bien-<lb/>nium ſcriptis à Sereniſſimo Principe Leopoldo literis officij ple-<lb/>nis, &amp; </s>
  <s xml:id="echoid-s328" xml:space="preserve">humanitate, tam proprio, quàm Magni Ferdinandi <lb/>II. </s>
  <s xml:id="echoid-s329" xml:space="preserve">fratris nomine, impoſita mihi fuit hæc prouincia optatæ diu, <lb/>&amp; </s>
  <s xml:id="echoid-s330" xml:space="preserve">penè deſperatæ verſionis. </s>
  <s xml:id="echoid-s331" xml:space="preserve">Quo ſanè, vt ingenuè fatear, ni-<lb/>hil iucundius, nihil carius, nihil antiquius accidere mihi po-<lb/>terat; </s>
  <s xml:id="echoid-s332" xml:space="preserve">quòd hac data occaſione, aliquam grati animi ſignifica-<lb/>tionem exhibere me poſſe putabam Sereniſſimo illi Principi, <lb/>cuius ampliſſima in me beneficia ſum expertus. </s>
  <s xml:id="echoid-s333" xml:space="preserve">Memini profe-<lb/>ctò, nec ex animo meo excidet, imo clauo fixum trabali ma-<lb/>net, quanta in me contulit Magnus Ferdinandus Secundus or-<lb/>namenta, quanta in me vſus eſt liberalitate, &amp; </s>
  <s xml:id="echoid-s334" xml:space="preserve">beneficentia, <lb/>non tantùm dum fortuna mihi arridebat, non ſolùm dum res <lb/>ſuccedebant proſperè, non modò dum ad illum ab Amiro Fa-<lb/>chraddino miſſus ſingulari felicitate fiuebar, ſed etiam in nau-<lb/>fragio, &amp; </s>
  <s xml:id="echoid-s335" xml:space="preserve">iactura illa barbarica, in Carrellina coniuratione, <lb/>&amp; </s>
  <s xml:id="echoid-s336" xml:space="preserve">proditione, in aduerſiſſima fortuna. </s>
  <s xml:id="echoid-s337" xml:space="preserve">Sed hæc omnia magis <lb/>à me exprimi poſſunt profundiſſimo ſilentio, quàm verborum, <lb/>copia, aut oratione altius exaggerata. </s>
  <s xml:id="echoid-s338" xml:space="preserve">Verùm enim verò dum <lb/>arbitrabar, mirificam nactum me eſſe opportunitatem gratifi-<lb/>candi Principi de me optimè merito, &amp; </s>
  <s xml:id="echoid-s339" xml:space="preserve">exhibendi aliquod gra-<lb/>ti animi ſignum, penè concepta excidi ſpe. </s>
  <s xml:id="echoid-s340" xml:space="preserve">Nam aperto Apol-<lb/>lonij Codice, &amp; </s>
  <s xml:id="echoid-s341" xml:space="preserve">coniectis in eum oculis duæ primo ferè intuitu <lb/>ſeſe mihi obtulerunt difficultates, quas à me neque ſuperari, <lb/>neque vinci poſſe prorsùs exiſtimaui. </s>
  <s xml:id="echoid-s342" xml:space="preserve">Hinc ſummus, &amp; </s>
  <s xml:id="echoid-s343" xml:space="preserve">abſtru-<lb/>ſus pudor, hinc plurimus ſudor ingenuè omnia fateor. </s>
  <s xml:id="echoid-s344" xml:space="preserve">Et eò <lb/>magis intimis animi ſenſibus angebar, quod ea verſio non in <lb/>ſeceſſu aliquo fiebat, &amp; </s>
  <s xml:id="echoid-s345" xml:space="preserve">remotis arbitris, vbi aciem mentis ab-<lb/>ducere, difficultates commodè expendere, animoque intento, <lb/>&amp; </s>
  <s xml:id="echoid-s346" xml:space="preserve">libero luſtrare quæ in rem eſſent, ac per otium poſſem, ſed <lb/>præſentibus grauiſſimis viris, &amp; </s>
  <s xml:id="echoid-s347" xml:space="preserve">quidem, ex tempore, &amp; </s>
  <s xml:id="echoid-s348" xml:space="preserve">nulla <lb/>data præmeditandi facultate, interpretationem facere compel-<lb/>lebar. </s>
  <s xml:id="echoid-s349" xml:space="preserve">Ea fortè illorũ præclariſſimorũ virorũ de me erat opinio, <lb/>&amp; </s>
  <s xml:id="echoid-s350" xml:space="preserve">exiſtimatio, quàm tamen parum abfuit, quin penitus perdi-<lb/>diſſem, cùm vix, &amp; </s>
  <s xml:id="echoid-s351" xml:space="preserve">ne vix quidem ſcripturam illam legere poſ-
<pb file="0026" n="26" rhead="ABRAHAMI ECCHELLENSIS"/>
ſem, quæ prima erat difficultas. </s>
  <s xml:id="echoid-s352" xml:space="preserve">Nam puncta aberant diacriti-<lb/>ca imprimis (de punctis vocalibus hic non loquimur, nec eorũ <lb/>inter legendum à peritis linguæ habetur ratio, aut negotium <lb/>aliquod faceſſunt), nempè ea, quæ formam dant literis, lite-<lb/>raſque conſtituunt, &amp; </s>
  <s xml:id="echoid-s353" xml:space="preserve">ſine quibus literæ ſunt pura, ac nuda <lb/>materia omni ſpoliatæ forma. </s>
  <s xml:id="echoid-s354" xml:space="preserve">Quid autem ſit materia omni <lb/>ſpoliata forma, neque ipſi ſciunt Philoſophi, quorum id ſcire <lb/>intereſt. </s>
  <s xml:id="echoid-s355" xml:space="preserve">Eodem prorſus ſe habent modo Arabum literæ, ſeu <lb/>potiùs literarum ductus, &amp; </s>
  <s xml:id="echoid-s356" xml:space="preserve">lineæ diacriticis hiſce carentes pun-<lb/>ctis. </s>
  <s xml:id="echoid-s357" xml:space="preserve">Eadem enim figura, ſeu linea, exempli gratia, ſi vnum <lb/>ei ſuperponitur punctum erit N. </s>
  <s xml:id="echoid-s358" xml:space="preserve">ſi verò ſupponatur, B. </s>
  <s xml:id="echoid-s359" xml:space="preserve">ſi duo <lb/>ſuperponuntur, T. </s>
  <s xml:id="echoid-s360" xml:space="preserve">ſi tria Th. </s>
  <s xml:id="echoid-s361" xml:space="preserve">ſi duo ſupponantur, I. </s>
  <s xml:id="echoid-s362" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s363" xml:space="preserve">ſic de <lb/>cæteris ferè omuibus arguendum eſt. </s>
  <s xml:id="echoid-s364" xml:space="preserve">Si quis autem percontabi-<lb/>tur, quid erit illa figura, &amp; </s>
  <s xml:id="echoid-s365" xml:space="preserve">linea, ſi nullum adſit punctum? <lb/></s>
  <s xml:id="echoid-s366" xml:space="preserve">reſpondetur materia ſine forma, &amp; </s>
  <s xml:id="echoid-s367" xml:space="preserve">quid ſit prorsùs ignoratur. </s>
  <s xml:id="echoid-s368" xml:space="preserve"><lb/>Augebant etiam lectionis difficultatem ipſæ literarum figuræ, <lb/>quæ ita raptim, &amp; </s>
  <s xml:id="echoid-s369" xml:space="preserve">curſim, licet elegantiſſimè, ductæ erant, vt <lb/>vix ab inuicem quandoque, &amp; </s>
  <s xml:id="echoid-s370" xml:space="preserve">identim diſtinguerentur. </s>
  <s xml:id="echoid-s371" xml:space="preserve">Hæc <lb/>autem difficultas terruit quidem primo aſpectu ſed breui, &amp; </s>
  <s xml:id="echoid-s372" xml:space="preserve">ci-<lb/>tius quàm credebam, ſuperata fuit, tum ſtudio, &amp; </s>
  <s xml:id="echoid-s373" xml:space="preserve">diligentia, <lb/>tum experientia, quàm ab ipſa ineunte ætate ex lectione eiuſmo-<lb/>di ſcriptorum generis comparauimus.</s>
  <s xml:id="echoid-s374" xml:space="preserve"/>
</p>
<div xml:id="echoid-div11" type="float" level="2" n="1">
<note symbol="*" position="left" xlink:label="note-0024-01" xlink:href="note-0024-01a" xml:space="preserve">Fallitur <lb/>C.V. Ger. <lb/>10: V oſſius <lb/>hoc tribu-<lb/>ens Sixto <lb/>V. P. M. <lb/>C. 17. 29. <lb/>de ſcient. <lb/>Matbe-<lb/>mat.</note>
</div>
<p>
  <s xml:id="echoid-s375" xml:space="preserve">Altera difficultas, quæ ſe nobis obtulerat, maioris quidem <lb/>erat ponderis, &amp; </s>
  <s xml:id="echoid-s376" xml:space="preserve">momenti; </s>
  <s xml:id="echoid-s377" xml:space="preserve">verſabatur quippè circa diſciplinæ <lb/>vocabulorum intelligentiam, &amp; </s>
  <s xml:id="echoid-s378" xml:space="preserve">notionem, quorum ignari era-<lb/>mus, &amp; </s>
  <s xml:id="echoid-s379" xml:space="preserve">penitùs ieiuni. </s>
  <s xml:id="echoid-s380" xml:space="preserve">At hanc quoque difficultatem facili ne-<lb/>gotio ſuperauimus ope, &amp; </s>
  <s xml:id="echoid-s381" xml:space="preserve">opera Clariſſimi, atque Doctiſſimi <lb/>Viri D. </s>
  <s xml:id="echoid-s382" xml:space="preserve">Ioannis Alphonſi Borelli Matheſeos in Piſana Acade-<lb/>mia profeſſoris celeberrimi, qui, &amp; </s>
  <s xml:id="echoid-s383" xml:space="preserve">verſionem ipſam promo-<lb/>uerat apud Sereniſſimos Principes, &amp; </s>
  <s xml:id="echoid-s384" xml:space="preserve">Codicem comportauerat <lb/>idem Romam, ac perpetuus mihi aderat Dux, &amp; </s>
  <s xml:id="echoid-s385" xml:space="preserve">Magiſter. <lb/></s>
  <s xml:id="echoid-s386" xml:space="preserve">Et ita ſanè ea omnia, quæ ad Diſciplinæ, eiuſque vocabulo-<lb/>rum notionem pertinebant, clarè, dilucidè, &amp; </s>
  <s xml:id="echoid-s387" xml:space="preserve">explicatè ordi-<lb/>ne inſinuauit, vt breui meis auditoribus Matheſeos profeſſor vi-<lb/>derer. </s>
  <s xml:id="echoid-s388" xml:space="preserve">Porrò quod hac in re magis mirandum eſt, nec ſilentio <lb/>prætereundum, ea erat Viro illi Doctiſſimo ſingularis ingenij <lb/>perſpicacitas, vt ſæpe in abſtruſis quibusdam locis, non ex in-
<pb file="0027" n="27" rhead="PRÆFATIO."/>
tegris, inquam, præmiſſis, ſed ex vnica dictione totam illatio-<lb/>nem inde colligeret, non ſenſu, ſed totidem penè verbis, ac <lb/>ſi Arabica legeret verba, &amp; </s>
  <s xml:id="echoid-s389" xml:space="preserve">linguæ veteranus eſſet profeſſor. </s>
  <s xml:id="echoid-s390" xml:space="preserve">Pro-<lb/>indè verius ipſi, quàm mihi adſcribenda eſt hæc verſio, longè <lb/>tamen abſit omnis adulatio, &amp; </s>
  <s xml:id="echoid-s391" xml:space="preserve">animi propenſio in virum ami-<lb/>ciſſimum. </s>
  <s xml:id="echoid-s392" xml:space="preserve">Hac mutua contentione, &amp; </s>
  <s xml:id="echoid-s393" xml:space="preserve">interpretandi, &amp; </s>
  <s xml:id="echoid-s394" xml:space="preserve">verten-<lb/>di trium Menſium ſpatio verſio noſtra confecta, &amp; </s>
  <s xml:id="echoid-s395" xml:space="preserve">abſoluta <lb/>eſt, in qua horis tantummodò matutinis propter nimios calo-<lb/>res æſtiuos conſumpſimus. </s>
  <s xml:id="echoid-s396" xml:space="preserve">Et hæc de ratione verſionis poſterio-<lb/>rum librorum Apollonij, &amp; </s>
  <s xml:id="echoid-s397" xml:space="preserve">methodo ſatis dicta ſint. </s>
  <s xml:id="echoid-s398" xml:space="preserve">Nunc de <lb/>ipſo Apollonio, eiuſqne librorum Arabica verſione, &amp; </s>
  <s xml:id="echoid-s399" xml:space="preserve">illius au-<lb/>ctoribus nonnihil dicere, par, &amp; </s>
  <s xml:id="echoid-s400" xml:space="preserve">conſentaneum eſt.</s>
  <s xml:id="echoid-s401" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s402" xml:space="preserve">Apollonium ſub Achaz Filio Ioatham regis Iuda poſt Tha-<lb/>letem Mileſium Floruiſſe, Arabes perhibent Scriptores. </s>
  <s xml:id="echoid-s403" xml:space="preserve">Sic <lb/>enim lib. </s>
  <s xml:id="echoid-s404" xml:space="preserve">3. </s>
  <s xml:id="echoid-s405" xml:space="preserve">Chronicorum in Achaz ſcriptum reliquit Gregorius <lb/>Barhebræus: </s>
  <s xml:id="echoid-s406" xml:space="preserve">Poſt Thaletem celebris fuit in Geometricis præcipuè diſci-<lb/>plinis Apollonius Naggiar. </s>
  <s xml:id="echoid-s407" xml:space="preserve">(ideſt faber lignarius) Is cornpoſuit Tra-<lb/>ctatum de ſcientia Conicor. </s>
  <s xml:id="echoid-s408" xml:space="preserve">nempè de lineis, quæ neque rectæ ſunt, ne-<lb/>que arcuatæ, ſeu curuæ, ſed inclinatæ. </s>
  <s xml:id="echoid-s409" xml:space="preserve">Notandum hìc eſt vocem <lb/>Naggiar, quæ Apollonio tribuitur, vt cognomen, &amp; </s>
  <s xml:id="echoid-s410" xml:space="preserve">nos fa-<lb/>brum lignarium vertimus, poni (vt opinor) pro Geometra, &amp; </s>
  <s xml:id="echoid-s411" xml:space="preserve">id <lb/>fortè exindè, quòd inſtrumenta, quibus vtebantur Geometræ <lb/>ex lignis olim conficiebantur. </s>
  <s xml:id="echoid-s412" xml:space="preserve">Quod, &amp; </s>
  <s xml:id="echoid-s413" xml:space="preserve">indè conijcio, quia <lb/>hoc idem vocabulum Euclidi quoque tribuitur apud eundem <lb/>Gregorium ſic de illo ſcribentem. </s>
  <s xml:id="echoid-s414" xml:space="preserve">At Euclides Naggiar ex Vrbe <lb/>Tyro erat.</s>
  <s xml:id="echoid-s415" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s416" xml:space="preserve">De verſione autem librorum Apollonij in Arabicam linguam <lb/>ita ſtatim ſubdit mox laudatus Gregorius: </s>
  <s xml:id="echoid-s417" xml:space="preserve">Ex his autem verſi <lb/>ſunt in Arabicam linguam tempore Almamuni ſeptem libri, eius tamen <lb/>præfatio indicat, octo fuiſſe libros; </s>
  <s xml:id="echoid-s418" xml:space="preserve">qui quidem Tractatus cum alio Tra-<lb/>ctatu eiuſdẽ Apollonĳ cauſam dedere Euclidi ſuorum componendorum li-<lb/>brorũ longum poſt tempus. </s>
  <s xml:id="echoid-s419" xml:space="preserve">In his longè videtur diſcrepare Grego-<lb/>rius à communi Chronologorum ſententia, &amp; </s>
  <s xml:id="echoid-s420" xml:space="preserve">opinione, qui <lb/>Apollonium Floruiſſe ſcribunt anno periodi Iulianę 4474. </s>
  <s xml:id="echoid-s421" xml:space="preserve">ideſt <lb/>annis ante Chriſtum Dominum 240. </s>
  <s xml:id="echoid-s422" xml:space="preserve">adeoque multò iunior eſt, <lb/>quàm facit illum Gregorius. </s>
  <s xml:id="echoid-s423" xml:space="preserve">Diſcrepat prætereà ab ijſdem Chro-<lb/>nologis in ætate Euclidis, quem Apollonio iuniorem agnoſcit,
<pb file="0028" n="28" rhead="ABRAHAMI ECCHELLENSIS"/>
vbi illi eum collocant in anno periodi Iulianæ 4430. </s>
  <s xml:id="echoid-s424" xml:space="preserve">ideſt ante <lb/>Chriſtum Dominum annis 284. </s>
  <s xml:id="echoid-s425" xml:space="preserve">iuxta quàm opinionem Apollo-<lb/>nius iunior erit Euclide annis 44.</s>
  <s xml:id="echoid-s426" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s427" xml:space="preserve">Almamun autem ſub quo facta eſt librorum Apollonij verſio <lb/>in Arabicam linguam ex laudato Gregorio Chalipha ſecundò <lb/>ſalutatus eſt An. </s>
  <s xml:id="echoid-s428" xml:space="preserve">Heg. </s>
  <s xml:id="echoid-s429" xml:space="preserve">203. </s>
  <s xml:id="echoid-s430" xml:space="preserve">ex omnium ſcriptorum ſententia, <lb/>qui annus ex Tabula Aerarum Iſmaelis Sciahinſciah, quàm re-<lb/>fert in hiſtoria Gentium, reſpondet Anno Chriſti Domini ſola-<lb/>ri 826. </s>
  <s xml:id="echoid-s431" xml:space="preserve">plùs minuſue. </s>
  <s xml:id="echoid-s432" xml:space="preserve">Nam Hegiram accidiſſe anno Chriſti <lb/>631. </s>
  <s xml:id="echoid-s433" xml:space="preserve">habet Iſmaèlis Tabula contra omnium Chronologorum <lb/>Orientalium opinionem, qui eam reponunt in ann. </s>
  <s xml:id="echoid-s434" xml:space="preserve">Chriſti 622. <lb/></s>
  <s xml:id="echoid-s435" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s436" xml:space="preserve">vndecim Heraclij, vno excepto Eutychio Alexandrino, qui <lb/>eam reponit in ſua hiſt. </s>
  <s xml:id="echoid-s437" xml:space="preserve">Eccles. </s>
  <s xml:id="echoid-s438" xml:space="preserve">in an. </s>
  <s xml:id="echoid-s439" xml:space="preserve">Chriſti 614. </s>
  <s xml:id="echoid-s440" xml:space="preserve">ſcribit enim <lb/>ibi: </s>
  <s xml:id="echoid-s441" xml:space="preserve">A Chriſto Domino noſtro vſque ad Hegiram ſunt anni 614. </s>
  <s xml:id="echoid-s442" xml:space="preserve">In <lb/>quo octennio integro diſcrepat ab alijs Chronologicis. </s>
  <s xml:id="echoid-s443" xml:space="preserve">Sed hæc <lb/>leuiter tetigiſſe, ſatis eſt; </s>
  <s xml:id="echoid-s444" xml:space="preserve">non eſt enim animus hic temporum <lb/>apices data opera excutere, nec id ſanè vacat, nec huius lo-<lb/>ci eſt.</s>
  <s xml:id="echoid-s445" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s446" xml:space="preserve">Principem autem Almamunum, eam procuraſſe verſionem <lb/>librorum Apollonij, non ſolùm facilè, ſed procerto credendum <lb/>eſt. </s>
  <s xml:id="echoid-s447" xml:space="preserve">Nam is omnium ſcientiarum ſtudijs vehementiſſimè arde-<lb/>bat, proindeque congerendorum vndique librorum nunquàm <lb/>finem faciebat, eratque in eorum interpretes prolixiſſimus. <lb/></s>
  <s xml:id="echoid-s448" xml:space="preserve">Mira ſanè, quæ de illius, ac proaui Abugiahphar Almanſur <lb/>animi propenſione in literas, &amp; </s>
  <s xml:id="echoid-s449" xml:space="preserve">literatos viros refert Sahadus <lb/>Filius Ahmedi Andaluſij in Hiſt. </s>
  <s xml:id="echoid-s450" xml:space="preserve">Arabum. </s>
  <s xml:id="echoid-s451" xml:space="preserve">Is, inquit ibi, erat <lb/>ſtatus Arabum in gentilitate. </s>
  <s xml:id="echoid-s452" xml:space="preserve">Poſtquàm verò fauoribus proſequutus eſt <lb/>Deus Altiſsimus Hacſemitas, deuoluitque ad eos imperium, conuerſæ <lb/>mentes ſunt, &amp; </s>
  <s xml:id="echoid-s453" xml:space="preserve">intellectus à ſtupore, in quo iacebant, &amp; </s>
  <s xml:id="echoid-s454" xml:space="preserve">exſuſcitata <lb/>ingeniorum acumina poſtquàm extincta erant. </s>
  <s xml:id="echoid-s455" xml:space="preserve">Horum autem primus, <lb/>qui promouendis ſcientĳs operam nauauit, erat Abugiahphar Almanſur <lb/>ſecundus Chalipha. </s>
  <s xml:id="echoid-s456" xml:space="preserve">Qui tametſi luriſprudentiæ deditiſsimus eſſet, &amp; </s>
  <s xml:id="echoid-s457" xml:space="preserve"><lb/>peritiſsimus; </s>
  <s xml:id="echoid-s458" xml:space="preserve">nihilominus, &amp; </s>
  <s xml:id="echoid-s459" xml:space="preserve">Philoſophtæ vacabat ſtudio, ſed arden-<lb/>tius Aſtronomiæ. </s>
  <s xml:id="echoid-s460" xml:space="preserve">Cùm verò Imperĳ ſuſcepiſſet ſceptra Chalipha ſepti-<lb/>mus Abdalla Almanſun filius Aaronis Raſcidi, abſoluit ea, quæ ince-<lb/>perat Auus ipſius Almanſur, operamque dedit ſcientĳs vbique inquiren-<lb/>dis. </s>
  <s xml:id="echoid-s461" xml:space="preserve">Hinc Græcorum ſcripſit Imperatoribus rogans ſibi mitti quotquot
<pb file="0029" n="29" rhead="PRÆFATIO."/>
haberi poſſunt Philoſophorum libri, qui quotquot comparare potuerunt mi-<lb/>ſerunt ipſi. </s>
  <s xml:id="echoid-s462" xml:space="preserve">Quibus ille vertendis peritiſsimos quoſque ſelegit interpretes, <lb/>&amp; </s>
  <s xml:id="echoid-s463" xml:space="preserve">curam iniunxit interpretandi, &amp; </s>
  <s xml:id="echoid-s464" xml:space="preserve">verſi ſunt eo ſtudio maiori, quod <lb/>fieri poteſt. </s>
  <s xml:id="echoid-s465" xml:space="preserve">Quo autem facto homines non ſolùm incitabat, ſed &amp; </s>
  <s xml:id="echoid-s466" xml:space="preserve">co-<lb/>gebat quodammodò, vt ĳs legendis, &amp; </s>
  <s xml:id="echoid-s467" xml:space="preserve">ediſcendis operam darent. <lb/></s>
  <s xml:id="echoid-s468" xml:space="preserve">Ipſe verò ſapientes viros familiariſsimè conueniebat, eorumque perami-<lb/>ce vtebatur conſuetudine, atque plurimum illorum delectabatur collo-<lb/>quĳs. </s>
  <s xml:id="echoid-s469" xml:space="preserve">Nouerat, &amp; </s>
  <s xml:id="echoid-s470" xml:space="preserve">quippe optime, ſapientes viros Deo Altiſsimo mor-<lb/>talium eſſe cariſsimos, ac ipſi coniunctiſsimos, eo quod ſeſe dederunt <lb/>animæ rationalis virtutibus comparandis, poſthabitis, &amp; </s>
  <s xml:id="echoid-s471" xml:space="preserve">contemptis <lb/>ĳs, quibus Sinenſes, ac Turcæ, eorumque ſimiles incumbunt. </s>
  <s xml:id="echoid-s472" xml:space="preserve">Hi <lb/>enim oſtentare amant artium Mechanicarum ſubtilitatem, animæ ira-<lb/>ſcibilis gloriantur potentĳs, &amp; </s>
  <s xml:id="echoid-s473" xml:space="preserve">concupiſcibilis iactant ſe ſe facultatibus. </s>
  <s xml:id="echoid-s474" xml:space="preserve"><lb/>Cum tamen hæc omnia communia cum ĳs ipſa habere bruta, ſcire de-<lb/>beant; </s>
  <s xml:id="echoid-s475" xml:space="preserve">imò longè ab illis ſuperantur. </s>
  <s xml:id="echoid-s476" xml:space="preserve">Peritia, &amp; </s>
  <s xml:id="echoid-s477" xml:space="preserve">ſubtilitate artis ab <lb/>Apibus, quæ ſua examina, ſeu penarium ſexangula mirà conſtruunt <lb/>arte. </s>
  <s xml:id="echoid-s478" xml:space="preserve">Audacia, &amp; </s>
  <s xml:id="echoid-s479" xml:space="preserve">fortitudine à Leonibus, alĳſque feris, quibus in <lb/>hiſce haud comparandus eſt homo. </s>
  <s xml:id="echoid-s480" xml:space="preserve">Libidine, &amp; </s>
  <s xml:id="echoid-s481" xml:space="preserve">Luxuria à ſuibus, at-<lb/>que alĳs, quæ hic memorari neceſſe non eſt. </s>
  <s xml:id="echoid-s482" xml:space="preserve">Hacque de cauſa ſapien-<lb/>tes viri ſunt lampades in tenebris, &amp; </s>
  <s xml:id="echoid-s483" xml:space="preserve">mortalium omnium Domini. </s>
  <s xml:id="echoid-s484" xml:space="preserve"><lb/>Et heu quàm turpe, atque deforme eſt terrarum hoc orbis theatrum, <lb/>quoties ſuis caret ſapientibus. </s>
  <s xml:id="echoid-s485" xml:space="preserve">Hæc Sahedus in Hiſtoria Arabum.</s>
  <s xml:id="echoid-s486" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s487" xml:space="preserve">Noſtram autem verſionem hanc Arabicam quod attinet, alia <lb/>prorsùs eſt ab ea, quæ ſub Almamuno confecta eſt. </s>
  <s xml:id="echoid-s488" xml:space="preserve">Hoc planè <lb/>patet ex ipfius Auctoris Abalphathi in præfatione verbis. </s>
  <s xml:id="echoid-s489" xml:space="preserve">Dicit <lb/>quippè ibi, ſe eam adornaſſe verſionem pro regis Abicaligiar <lb/>Bibliotheca. </s>
  <s xml:id="echoid-s490" xml:space="preserve">Abicaligiar autem rex ſalutatus eſt, teſte Sciahin-<lb/>ſciah, Gregorio, &amp; </s>
  <s xml:id="echoid-s491" xml:space="preserve">alijs, Hegiræ anno 372. </s>
  <s xml:id="echoid-s492" xml:space="preserve">nempè annis <lb/>169. </s>
  <s xml:id="echoid-s493" xml:space="preserve">poſt Almamuni inaugurationem ijsdem, quos mox lauda-<lb/>uimus, auctoribus.</s>
  <s xml:id="echoid-s494" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s495" xml:space="preserve">Verſionem tamen illam, quæ ſub Almamuno facta eſt, ne-<lb/>quaquàm vidit noſtræ huius verſionis auctor Abalphath, quemad-<lb/>modùm ex verbis eius, quæ ad ſeptimi libri adiecit calcem, <lb/>patet luculenter. </s>
  <s xml:id="echoid-s496" xml:space="preserve">Ibi enim, puto inquit, me in hoc, nempè in <lb/>hac verſione concinnanda, quoſcunque alios anteuertiſſe. </s>
  <s xml:id="echoid-s497" xml:space="preserve">Itaque <lb/>exiſtimat is noſter Auctor, ſe omnium primum Apollonij ver-<lb/>ſionem Reipublicæ literariæ dediſſe. </s>
  <s xml:id="echoid-s498" xml:space="preserve">Quod, &amp; </s>
  <s xml:id="echoid-s499" xml:space="preserve">in ipſa quoque
<pb file="0030" n="30" rhead="ABRAHAMI ECCHELLENSIS"/>
innuit præfatione, aſſerens vſque ad ſua tempora nullam inte-<lb/>gram librorum Apollonij extitiſſe inter Arabes verſionem; </s>
  <s xml:id="echoid-s500" xml:space="preserve">ſed <lb/>fragmenta quædam. </s>
  <s xml:id="echoid-s501" xml:space="preserve">Ex quo arguere eſt, aut eum minimè an-<lb/>tiquiorem Almamuni vidiſſe verſionem, aut iſtam non fuiſſe in-<lb/>tegram, ſed Epitomem aliquam ex ſeptem Apollonij decer-<lb/>ptam libris, de qua ille in præfatione. </s>
  <s xml:id="echoid-s502" xml:space="preserve">Vt vt ſit noſtra hæc <lb/>alia prorsùs eſt ab ea, &amp; </s>
  <s xml:id="echoid-s503" xml:space="preserve">ad ipſius auctoris calculos redacta, <lb/>atque adeò integra, &amp; </s>
  <s xml:id="echoid-s504" xml:space="preserve">omnium perfectiſſima, atque abſolutiſ-<lb/>ſima.</s>
  <s xml:id="echoid-s505" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s506" xml:space="preserve">Cæterũ admonitum volumus benignum lectorem, nos in hac <lb/>verſione adornanda ſatis preſsè Arabicam ſecutos eſſe phraſim, <lb/>nec omninò elegantiam, &amp; </s>
  <s xml:id="echoid-s507" xml:space="preserve">venuſtatem linguæ expreſſiſſe, ar-<lb/>bitrantes id maximè pertinere ad fidelis interpretis partes, &amp; </s>
  <s xml:id="echoid-s508" xml:space="preserve"><lb/>officium.</s>
  <s xml:id="echoid-s509" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s510" xml:space="preserve">Ea autem quæ occurrunt circa ipſam phraſim, &amp; </s>
  <s xml:id="echoid-s511" xml:space="preserve">vocabula <lb/>nonnulla obſeruanda, Arabicæ Editioni reſeruauimus, rati ea <lb/>commodius, &amp; </s>
  <s xml:id="echoid-s512" xml:space="preserve">magis ad rem ibi exponenda eſſe, &amp; </s>
  <s xml:id="echoid-s513" xml:space="preserve">ſuis ex-<lb/>primenda characteribus. </s>
  <s xml:id="echoid-s514" xml:space="preserve">Interim benè vale, &amp; </s>
  <s xml:id="echoid-s515" xml:space="preserve">hoc qualicunque <lb/>fruere ſtudio, &amp; </s>
  <s xml:id="echoid-s516" xml:space="preserve">labore.</s>
  <s xml:id="echoid-s517" xml:space="preserve"/>
</p>
<pb file="0031" n="31" rhead="IO: ALFONSI BORELLI"/>
</div>
<div xml:id="echoid-div13" type="section" level="1" n="11">
<head xml:id="echoid-head31" xml:space="preserve">PRÆFATIO AD LECTOREM.</head>
<p style="it">
  <s xml:id="echoid-s518" xml:space="preserve">ACCIPE tandem, ſtudioſe Lector, in ſolemni hac pompa <lb/>nuptiarum Sereniſsimi Principis Etruriæ Regio ſplendore <lb/>à Sereniſsimo Magno Duce parata tamdiu deploratos, <lb/>&amp; </s>
  <s xml:id="echoid-s519" xml:space="preserve">expetitos libros poſtremos Conicorum Apollonĳ Per-<lb/>gæi, vtque ſine mora mens tua epulis hiſce lautiſsimis <lb/>ſaturari poſsit, non te demorari diutine patiar in limine, recenſendo ſci-<lb/>licet nomen Apollonĳ, patriam, ætatem, &amp; </s>
  <s xml:id="echoid-s520" xml:space="preserve">opera ab eo conſcripta, ne-<lb/>que inſuper doctrinæ conicæ ortum, &amp; </s>
  <s xml:id="echoid-s521" xml:space="preserve">progreſſum à primis incunabulis <lb/>ad virilem vſque, &amp; </s>
  <s xml:id="echoid-s522" xml:space="preserve">vegetam ætatem, ad quàm Apollonius eam <lb/>euexit, propter quod facimus magnus Geometra cognominatus eſt; </s>
  <s xml:id="echoid-s523" xml:space="preserve">hæc <lb/>enim trita iam ſunt, &amp; </s>
  <s xml:id="echoid-s524" xml:space="preserve">vulgaria: </s>
  <s xml:id="echoid-s525" xml:space="preserve">breuiter tantummodo percurram, <lb/>quæ ad notitiam horum librorum facere videntur.</s>
  <s xml:id="echoid-s526" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s527" xml:space="preserve">Illius pretioſiſsimæ bibliothecæ orientalis, quàm Sereniſsimo Ferdinan-<lb/>do Primo gratitudinis ergo reliquerat lgnatius Neama Patriarcha Anti-<lb/>ochenſis libellum nitidiſsimè Arabicè ſcriptum mihi oſtenderat Sereniſ-<lb/>ſimus Princeps Leopoldus Muſarum decus, &amp; </s>
  <s xml:id="echoid-s528" xml:space="preserve">gloria, noſtrique ſæculi <lb/>lumen eruditum. </s>
  <s xml:id="echoid-s529" xml:space="preserve">Codici inſcripſerat Raimundus, ſiue quis alius: </s>
  <s xml:id="echoid-s530" xml:space="preserve">Otto libri <lb/>de Conici d'Apollonio del Patriarca. </s>
  <s xml:id="echoid-s531" xml:space="preserve">Summa lætitia libellum exoſcu-<lb/>latus, licet Arabici idiomatis ſim prorſus ignarus, non potui me conti-<lb/>nere, quin ſaltem contrectarem, atque reuoluerem paginas illas; </s>
  <s xml:id="echoid-s532" xml:space="preserve">cumque <lb/>præter figuras mihi ſatis notas quatuor priorum Apollonĳ librorum vidiſ-<lb/>ſem alias conicas figuras, in quibus ab vno puncto in eis collocato edu-<lb/>ctæ erant plurimæ rectæ lineæ ad coniſectionem, illico in mentem venere <lb/>illa Eutocĳ verba in expoſitione epiſtolæ Apollonĳ ad Eudemum: </s>
  <s xml:id="echoid-s533" xml:space="preserve">Quin-<lb/>tus, inquit, liber de Minimis, &amp; </s>
  <s xml:id="echoid-s534" xml:space="preserve">Maximis magna ex parte agit; <lb/></s>
  <s xml:id="echoid-s535" xml:space="preserve">quemadmodum enim in elementis didicimus, ſi ab aliquo pun-<lb/>cto in circulum lineæ ducantur, earum quidem, quæ ad conca-<lb/>uam ipſius circumferentiam pertinent, maximam eſſe, quæ per <lb/>centrum tranſit, earum vero, quæ ad conuexam, minimam eſſe, <lb/>quæ inter dictum punctum, &amp; </s>
  <s xml:id="echoid-s536" xml:space="preserve">diametrum interijcitur, ita &amp; </s>
  <s xml:id="echoid-s537" xml:space="preserve">de
<pb file="0032" n="32" rhead="Io Alfonſi Borelli"/>
coniſectionibus in quinto libro inquirit. </s>
  <s xml:id="echoid-s538" xml:space="preserve">Sexti, ſeptimi, &amp; </s>
  <s xml:id="echoid-s539" xml:space="preserve">octa-<lb/>ui libri propoſitum manifeſtè ab ipſo Apollonio explicatur. </s>
  <s xml:id="echoid-s540" xml:space="preserve">Cùmq; <lb/></s>
  <s xml:id="echoid-s541" xml:space="preserve">poſteà à quodam Maronita Arabicè callente accepiſſem tractatum, ſeu li-<lb/>brum quintum Apollonĳ eſſe illum, in quo figuræ prædictæ delineatæ erant, <lb/>pariterque in ſubſequenti libro ſexto conſpexiſſem figuras alias exprimentes <lb/>æqualitatem, &amp; </s>
  <s xml:id="echoid-s542" xml:space="preserve">ſimilitudinem ſectionum conicarum, mihi certum fuit, <lb/>verè Apollonĳ eſſe libros illos. </s>
  <s xml:id="echoid-s543" xml:space="preserve">Haud tamen negabo ſcrupulum, ac du-<lb/>bitationem iniectam, ex eo quod textus ille Arabicus non præferebat <lb/>in fronte Apollonĳ, vel vllius alterius nomen, &amp; </s>
  <s xml:id="echoid-s544" xml:space="preserve">definitiones primi libri <lb/>centuriam ſuperabant, cum Apollonius non niſi nouendecim ſuo primo li-<lb/>bro appoſuißet. </s>
  <s xml:id="echoid-s545" xml:space="preserve">Inſuper in prioribus quatuor libris non totidem figuras con-<lb/>ſpiciebam, nec omnino ſimiles, eaſdemque, nec eodem ordine diſpoſitas, <lb/>ac in textu Græco Eutocĳ videre eſt; </s>
  <s xml:id="echoid-s546" xml:space="preserve">quare cenſui librum prædictum <lb/>epitomen eſſe Conicorum Apollonĳ ab aliquo alio conſcriptam. </s>
  <s xml:id="echoid-s547" xml:space="preserve">Hanc quoq; </s>
  <s xml:id="echoid-s548" xml:space="preserve"><lb/>præclariſsimi Torricellĳ fuiſſe ſententiam poſtea didici ex eius Epiſtola ad <lb/>eruditiſsimum Michaelem Angelum Riccium miſſam. </s>
  <s xml:id="echoid-s549" xml:space="preserve">Perſtiti tamen de-<lb/>bere latinè verti lucubrationem tam eximiam, eruditiſq; </s>
  <s xml:id="echoid-s550" xml:space="preserve">optatiſsimam, <lb/>nam niſi ipſiſsimum opus eſſet Apollonĳ, ſaltem ex ĳſdemmet libris epi-<lb/>tome illa deſumpta, &amp; </s>
  <s xml:id="echoid-s551" xml:space="preserve">tranſcripta exiſtimari debuerat.</s>
  <s xml:id="echoid-s552" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s553" xml:space="preserve">Igitur Sereniſsimus Ferdinandus Secundus Magnus Dux munificen-<lb/>tia verè regia, qua bonas artes promouere ſtudet, annuente, &amp; </s>
  <s xml:id="echoid-s554" xml:space="preserve">ſummo-<lb/>pere coadiuuante Sereniſsimo Principe Leopoldo fratre Matheſeos, atque <lb/>omnigenæ Sapientiæ perito cultore, atq; </s>
  <s xml:id="echoid-s555" xml:space="preserve">egregio vindice, præcepit, vt vo-<lb/>lumen Arabicum Romæ latinè redderetur ab Abrahamo Ecchellenſe lingua-<lb/>rum Orientalium doctiſsimo, &amp; </s>
  <s xml:id="echoid-s556" xml:space="preserve">peritiſsimo profeſſore. </s>
  <s xml:id="echoid-s557" xml:space="preserve">Is quidem ſumma <lb/>alacritate negotio ſuſcepto primùm bono me eſſe animo iuſsit; </s>
  <s xml:id="echoid-s558" xml:space="preserve">monuit enim <lb/>nouum non eße apud Arabes libros nomine auctoris in fronte carere, oſten-<lb/>ditque in proemio eiuſdem codicis apertiſsimè declarari eſſe libros Conico-<lb/>rum Apollonĳ paraphraſticè expoſitos: </s>
  <s xml:id="echoid-s559" xml:space="preserve">deinde ex translatione priorum qua-<lb/>tuor librorum patuit demonſtrationes propoſitionum penè non differre quoad <lb/>doctrinam à textu Græco Eutocĳ, licet verbum verbo non reſponderet: <lb/></s>
  <s xml:id="echoid-s560" xml:space="preserve">nec mirari paucitatem figurarum, quandoquidem vna, eademq; </s>
  <s xml:id="echoid-s561" xml:space="preserve">figura <lb/>quatuor, aut quinque propoſitionibus inſeruiret. </s>
  <s xml:id="echoid-s562" xml:space="preserve">Incomparabili igitur gau-<lb/>dio perfuſus Apollonium penè è manibus ſublatum iterum amplexibus ſtrin-<lb/>xi, &amp; </s>
  <s xml:id="echoid-s563" xml:space="preserve">exoſculatus ſum. </s>
  <s xml:id="echoid-s564" xml:space="preserve">Sed moleſtum ſummopere fuit octauum librum <lb/>deeſſe: </s>
  <s xml:id="echoid-s565" xml:space="preserve">collegi tamen lo: </s>
  <s xml:id="echoid-s566" xml:space="preserve">Baptiſtam Raimundum opuſculum arithmeticum <lb/>(quod in hoc codice Arabico ſubſequitur libro Septimo Apollonĳ) pro octauo
<pb file="0033" n="33" rhead="Præfatio."/>
eiuſdem libro accepiſſe, pariterq; </s>
  <s xml:id="echoid-s567" xml:space="preserve">Hieronymum Lunadorum in libro de Ro-<lb/>mana Curia nobis impoſuiſse, cum octo Apollonĳ libros ex Arabico tran-<lb/>ſtuliſſe latinè Raimundum typis publicauit; </s>
  <s xml:id="echoid-s568" xml:space="preserve">quì enim fieri potuit, vt octo <lb/>libros dediſſet is, qui an ſeptem, aut octo libri eſſent non animaduerterat?</s>
  <s xml:id="echoid-s569" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s570" xml:space="preserve">Modo operæ pretium erit ante oculos ponere formam, &amp; </s>
  <s xml:id="echoid-s571" xml:space="preserve">diſpoſitionem <lb/>huius paraphraſis ab interprete Abalphatho editæ. </s>
  <s xml:id="echoid-s572" xml:space="preserve">Et primo ſciendum eſt <lb/>eum collegiſſe ſimul ſeptem integros libros Conicorum Apollonĳ ex fragmen-<lb/>tis, quæ hactenus apud Arabes ſparſim circumferebantur, diſpoſuiſſeque <lb/>propoſitiones eorumdem librorum alio ordine, ac diuerſo ab Apolloniano, <lb/>relictis tamen numeris antiquis, nam in primo libro poſt primam, &amp; </s>
  <s xml:id="echoid-s573" xml:space="preserve"><lb/>ſecundam propoſitiones ſubſequuntur vndecima, tertia, quarta, ſeptima, <lb/>&amp; </s>
  <s xml:id="echoid-s574" xml:space="preserve">ſic vlterius ſemper ordine perturbato procedendo. </s>
  <s xml:id="echoid-s575" xml:space="preserve">Hac nempe ratione <lb/>ſimul collectis in eadem figura pluribus propoſitionibus, quas in locis diſsi-<lb/>tis collocauerat Apollonius, putauit Abalphathus breuiùs ſe eas demonſtra-<lb/>turum retenta ſemper Apollonĳ ſententia, ſcilicet ĳſdem medĳs, &amp; </s>
  <s xml:id="echoid-s576" xml:space="preserve">eodem <lb/>progreſſu, quo vſus eſt Apollonius, demonſtrat Paraphraſtes eaſdem pro-<lb/>poſitiones. </s>
  <s xml:id="echoid-s577" xml:space="preserve">An vero variare noluerit reuerentia retentus, vel potius ne-<lb/>quiuerit virium defectu, ( quippe ingenio non admodum felici, et inue-<lb/>niendi ſagaci à natura donatus ) non auſim affirmare. </s>
  <s xml:id="echoid-s578" xml:space="preserve">Superaddit quoq; <lb/></s>
  <s xml:id="echoid-s579" xml:space="preserve">numeroſam farraginem aliarum definitionum, quibus compendioſiùs, &amp; </s>
  <s xml:id="echoid-s580" xml:space="preserve"><lb/>clariùs demonſirationes abſolui poſſe profitetur, quod quidem non rarò ipſe <lb/>aßequitur; </s>
  <s xml:id="echoid-s581" xml:space="preserve">aliquaudo verò ob affectatam nimiam breuitatem obſcurior effi-<lb/>citur : </s>
  <s xml:id="echoid-s582" xml:space="preserve">accidit quoque, vt aliquæ definitiones inutiles, &amp; </s>
  <s xml:id="echoid-s583" xml:space="preserve">otioſæ ſint, vel <lb/>repetitio declarationis earumdem prolixitatem creet maiorem.</s>
  <s xml:id="echoid-s584" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s585" xml:space="preserve">Animaduerſione dignum eſt, quod Manuſcriptum licet non diſtin-<lb/>guatur capitibus, aut paragraphis, ſed continuo, perpetuoquè ſermone proce-<lb/>dat more Arabum, in eo tamen numerorum tria genera paſsim occurrunt, <lb/>qui omnes ferè interlineares, pauci quidem in margine poſiti, aliqui ru-<lb/>bris characteribus depicti, alĳ vero poſiti ſuper alios numeros in eadem <lb/>linea, veluti fractiones numerorum deſcribi ſolent, hac ratione {9/49.</s>
  <s xml:id="echoid-s586" xml:space="preserve">} 50. <lb/></s>
  <s xml:id="echoid-s587" xml:space="preserve">vel {16/68}. </s>
  <s xml:id="echoid-s588" xml:space="preserve">69. </s>
  <s xml:id="echoid-s589" xml:space="preserve">70. </s>
  <s xml:id="echoid-s590" xml:space="preserve">71.</s>
  <s xml:id="echoid-s591" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s592" xml:space="preserve">licet rarò ſynceri, &amp; </s>
  <s xml:id="echoid-s593" xml:space="preserve">veridici ſint, conie-<lb/>ci tamen ſupremos numeros indicare partes, ſeu ſectiones, in quas Abal-<lb/>phathus librum diſtribuit, atq; </s>
  <s xml:id="echoid-s594" xml:space="preserve">partitur: </s>
  <s xml:id="echoid-s595" xml:space="preserve">infimi verò numeri docent quot-<lb/>nam propoſitiones in vnaquaque ſectione contineantur: </s>
  <s xml:id="echoid-s596" xml:space="preserve">itaque hi nume-<lb/>ri {16/68}. </s>
  <s xml:id="echoid-s597" xml:space="preserve">69. </s>
  <s xml:id="echoid-s598" xml:space="preserve">70. </s>
  <s xml:id="echoid-s599" xml:space="preserve">71. </s>
  <s xml:id="echoid-s600" xml:space="preserve">ſignificant in lib. </s>
  <s xml:id="echoid-s601" xml:space="preserve">5. </s>
  <s xml:id="echoid-s602" xml:space="preserve">Sect. </s>
  <s xml:id="echoid-s603" xml:space="preserve">16. </s>
  <s xml:id="echoid-s604" xml:space="preserve">contineri Apollonĳ <lb/>Propoſitiones 68. </s>
  <s xml:id="echoid-s605" xml:space="preserve">69. </s>
  <s xml:id="echoid-s606" xml:space="preserve">70. </s>
  <s xml:id="echoid-s607" xml:space="preserve">71. </s>
  <s xml:id="echoid-s608" xml:space="preserve">reliqui numeri interlineares ſic diſpoſiti 24. </s>
  <s xml:id="echoid-s609" xml:space="preserve"><lb/>ex 5.</s>
  <s xml:id="echoid-s610" xml:space="preserve">, vel 37. </s>
  <s xml:id="echoid-s611" xml:space="preserve">ex 6. </s>
  <s xml:id="echoid-s612" xml:space="preserve">citationes ſunt, indicantque Prop. </s>
  <s xml:id="echoid-s613" xml:space="preserve">24. </s>
  <s xml:id="echoid-s614" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s615" xml:space="preserve">5. </s>
  <s xml:id="echoid-s616" xml:space="preserve">Conic.</s>
  <s xml:id="echoid-s617" xml:space="preserve">
<pb file="0034" n="34" rhead="Io: Alfonſi Borelli"/>
Apoll.</s>
  <s xml:id="echoid-s618" xml:space="preserve">, vel Prop. </s>
  <s xml:id="echoid-s619" xml:space="preserve">37. </s>
  <s xml:id="echoid-s620" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s621" xml:space="preserve">6. </s>
  <s xml:id="echoid-s622" xml:space="preserve">Sed mirum quàm mendoſi ſint omnes fere <lb/>numeri huius codicis ! in ſolo enim quinto libro frequenter duæ, vel tres <lb/>propoſitiones diuerſæ vno, &amp; </s>
  <s xml:id="echoid-s623" xml:space="preserve">eodem numero deſignantur, &amp; </s>
  <s xml:id="echoid-s624" xml:space="preserve">è contra <lb/>plures, &amp; </s>
  <s xml:id="echoid-s625" xml:space="preserve">ſeparati numeri nulli propoſitioni tribuuntur; </s>
  <s xml:id="echoid-s626" xml:space="preserve">nuſpiam enim <lb/>reperies propoſitiones 16. </s>
  <s xml:id="echoid-s627" xml:space="preserve">17. </s>
  <s xml:id="echoid-s628" xml:space="preserve">18. </s>
  <s xml:id="echoid-s629" xml:space="preserve">24. </s>
  <s xml:id="echoid-s630" xml:space="preserve">40.</s>
  <s xml:id="echoid-s631" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s632" xml:space="preserve">quamplurimas alias. <lb/></s>
  <s xml:id="echoid-s633" xml:space="preserve">Citationes poſtea inter propoſitiones interpoſitæ mendoſisſimæ, obſcuriores tene-<lb/>bras obducunt, quare non parum laboris, &amp; </s>
  <s xml:id="echoid-s634" xml:space="preserve">moleſtiæ habui, vt propoſitioni-<lb/>bus horum ſubſequentium librorum numeros debitos, &amp; </s>
  <s xml:id="echoid-s635" xml:space="preserve">legitimos aſsigna-<lb/>rem; </s>
  <s xml:id="echoid-s636" xml:space="preserve">nam prioribus quatuor in libris propoſitionum numeri licet perturba-<lb/>to ordine diſpoſitarum facilè reſtitui, &amp; </s>
  <s xml:id="echoid-s637" xml:space="preserve">corrigi potuerunt ex Græco exem-<lb/>plari, at in libris 5. </s>
  <s xml:id="echoid-s638" xml:space="preserve">6. </s>
  <s xml:id="echoid-s639" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s640" xml:space="preserve">7. </s>
  <s xml:id="echoid-s641" xml:space="preserve">numeros erroneos ſerie propoſitionum alte-<lb/>rata niſi ariolando aßequi quis poterit ? </s>
  <s xml:id="echoid-s642" xml:space="preserve">Cum ex Arabico codice mendas <lb/>haſce numericas corrigi poſſe Excellentiſsimus Abrahamus Ecchellenſis <lb/>deſperaſſet, repetitis litteris, vt coniecturis negotium perficerem, iuſsit; </s>
  <s xml:id="echoid-s643" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s644" xml:space="preserve"><lb/>ſiquidem propoſitiones Apollonĳ vno, vel altero tantum ordine diſponi po-<lb/>tuiſſent, forſan mentem auctoris conĳcere arduum non fuiſſet, ſed inter <lb/>multas, &amp; </s>
  <s xml:id="echoid-s645" xml:space="preserve">varias ſeries, quibus conica doctrina exponi poſſet, ſi eam, <lb/>quàm Abalphathus elegit, aſſecutus fuero, fortunæ tribuendum erit.</s>
  <s xml:id="echoid-s646" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s647" xml:space="preserve">Sed quid ego minutias numerorum conſector, cum in textu ipſo inſu-<lb/>perabiles ferè, &amp; </s>
  <s xml:id="echoid-s648" xml:space="preserve">maioris momenti difficultates ſuperſint? </s>
  <s xml:id="echoid-s649" xml:space="preserve">nulla propoſi-<lb/>tio fuit, in qua ſententiæ, verba, aut numeri, aut litteræ non fuerint <lb/>multifariam permutatæ, mutilatæ, aliæ pro alĳs repoſitæ, atque in propo-<lb/>ſitionibus pleriſque tituli ipſi, &amp; </s>
  <s xml:id="echoid-s650" xml:space="preserve">expoſitiones ſummopere deprauatæ, vt <lb/>prorſus ignoraretur quid nam demonſtrandum propoſuerit Apollonius. </s>
  <s xml:id="echoid-s651" xml:space="preserve">Ita-<lb/>que verba, litteræ, numeri, citationes, imò ſententiæ deficientes, aut per-<lb/>mutatæ vna cum affectata Paraphraſtis Arabici breuitate, &amp; </s>
  <s xml:id="echoid-s652" xml:space="preserve">multipli-<lb/>ci, &amp; </s>
  <s xml:id="echoid-s653" xml:space="preserve">noua nomenclatura cimmerias tenebras effundebant. </s>
  <s xml:id="echoid-s654" xml:space="preserve">Haſce in an-<lb/>guſtias redactus, quod potui, feci, vt germanum ſenſum Apollonĳ, &amp; </s>
  <s xml:id="echoid-s655" xml:space="preserve"><lb/>correctiſsimum exhiberem textum.</s>
  <s xml:id="echoid-s656" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s657" xml:space="preserve">Hanc tamen cautionem adhibui, vt in notis ſemper bona fide appo-<lb/>nerem ipſiſsima verba textus, quæ tranſtulerat ex codice Arabico me præ-<lb/>ſente Excellentiſsimus Ecchellenſis, ibidemque rationes appoſui mutatio-<lb/>nis, &amp; </s>
  <s xml:id="echoid-s658" xml:space="preserve">correctionis factæ. </s>
  <s xml:id="echoid-s659" xml:space="preserve">Itaque perſæpe vbi ſententia videbatur obſcu-<lb/>ra, neque diſtinctè explanata, tunc quidem meis verbisdeclaraui. </s>
  <s xml:id="echoid-s660" xml:space="preserve">Et quia <lb/>multoties ob nimiam paraphraſtis breuitatem, vellibrariorum vitio propoſi-<lb/>tiones nõ ſolide demonſtrantur, vel nequeunt ex præcedentibus deduci, addidi <lb/>ex meo penu lemmata nonnulla, quibus euidenter confirmantur, quæ in
<pb file="0035" n="35" rhead="Præfatio."/>
textu ambiguitatem aliquam præſeferebant. </s>
  <s xml:id="echoid-s661" xml:space="preserve">Appoſui quoque prolixè pro-<lb/>poſitionum caſus omnes neglectos in textu, eorumque demonſtrationes. <lb/></s>
  <s xml:id="echoid-s662" xml:space="preserve">Sed hiſce omnibus in rebus religioſus adeò fui, vt omnia diuerſo chara-<lb/>ctere in notis memorauerim, exceptis tamenĳs, quæ minoris momenti <lb/>ſunt, vt litteræ tranſpoſitæ, &amp; </s>
  <s xml:id="echoid-s663" xml:space="preserve">deficientes, &amp; </s>
  <s xml:id="echoid-s664" xml:space="preserve">verba aliqua impro-<lb/>pria, &amp; </s>
  <s xml:id="echoid-s665" xml:space="preserve">non ſignificantia, quæ commemorare non cenſui, ne volumen <lb/>in immenſum excreſceret.</s>
  <s xml:id="echoid-s666" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s667" xml:space="preserve">Tandem potuiſſem quidem abundantioris doctrinæ gratia non pauca <lb/>meo marte hiſce libris ſuperaddere non omnino forſan contemnenda, ſed <lb/>parcus adeo fui, vt tantummodo quæ ad illuſtrationem, &amp; </s>
  <s xml:id="echoid-s668" xml:space="preserve">ornatum <lb/>operis facere videbantur, adiecerim ſuntq; </s>
  <s xml:id="echoid-s669" xml:space="preserve">nonnullæ propoſitiones additæ, <lb/>quæ nouæ, &amp; </s>
  <s xml:id="echoid-s670" xml:space="preserve">forſan inelegantes non erunt.</s>
  <s xml:id="echoid-s671" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s672" xml:space="preserve">Conſiderandæ modo ſunt difficultates à præſtantiſsimo, et doctisſimo Clau-<lb/>dio Midorgio propoſitæ contra Manuſcriptum Arabicum Apollonĳ, quod Cla-<lb/>riſsimus, &amp; </s>
  <s xml:id="echoid-s673" xml:space="preserve">de bonis litteris optimè meritus Golius ex oriente detulit, <lb/>eædemq; </s>
  <s xml:id="echoid-s674" xml:space="preserve">difficultates eodem iure noſtrum Manuſcriptum, quod Golianum, <lb/>petunt. </s>
  <s xml:id="echoid-s675" xml:space="preserve">Verba Merſenni in præfatione Conicorum Apollonĳ ſuæ ſynopſis Ma-<lb/>thematicæ hæc ſunt. </s>
  <s xml:id="echoid-s676" xml:space="preserve">Suſpicatur autem Claudius Midorgius hos tres <lb/>libros, (ſcilicet 5. </s>
  <s xml:id="echoid-s677" xml:space="preserve">6. </s>
  <s xml:id="echoid-s678" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s679" xml:space="preserve">7. </s>
  <s xml:id="echoid-s680" xml:space="preserve">Conicorum Apollonĳ) eſſe cuiuſdam Ara-<lb/>bis ſub Apollonio latentis, quòd in quinto ſuo libro primam <lb/>propoſitionem ſexti Apollonij ſuperius allatam non ſolum in <lb/>cono recto, ſed in quouis etiam ſcaleno, &amp; </s>
  <s xml:id="echoid-s681" xml:space="preserve">illorum portioni-<lb/>bus quibuſcumque datis poſſibilia quæque demonſtrat. </s>
  <s xml:id="echoid-s682" xml:space="preserve">Hæc qui-<lb/>dem ratio quanti ponderis ſit æqui rerum æſtimatores iudicent, &amp; </s>
  <s xml:id="echoid-s683" xml:space="preserve">ſi qui-<lb/>dem omnes, qui in Geometricis mediocriter verſati ſunt optimè norunt <lb/>ſucceſsiuè aliquid vlteriùs inueniri præter ea, quæ diuini Præceptores <lb/>Euclides, Archimedes, Apollonius, &amp; </s>
  <s xml:id="echoid-s684" xml:space="preserve">Ptolemæus ediderunt, facile enim <lb/>eſſe inuentis addere quis ignorat? </s>
  <s xml:id="echoid-s685" xml:space="preserve">Nulli vnquam venit in mentem librum <lb/>Spiralium non ab Archimede, ſed ab aliquo alio ſcriptum fuiſſe, propterea <lb/>quod vniuerſaliùs quarumcumque ſpiralium paſsiones Neoterici demon-<lb/>ſtrarunt; </s>
  <s xml:id="echoid-s686" xml:space="preserve">Nec quia admirabilis Maurolicus in ſuo quinto Conicorum libro, <lb/>&amp; </s>
  <s xml:id="echoid-s687" xml:space="preserve">alĳ recentiores, ſicuti præclarus Phyloſophus, &amp; </s>
  <s xml:id="echoid-s688" xml:space="preserve">Mathematicus Vin-<lb/>centius Viuianus Patritius Florentinus in ſuo erudito libro de Maximis, <lb/>&amp; </s>
  <s xml:id="echoid-s689" xml:space="preserve">Minimis alia longè diuerſa ab Apollonĳ ſpeculationibus excogitarunt, <lb/>hos libros adulterinos eße auſi ſunt affirmare. </s>
  <s xml:id="echoid-s690" xml:space="preserve">Et ſicuti ipſemet Midor-<lb/>gius non repudiauit librum primum Conicorum ab Eutocio editum, licet <lb/>ipſe in ſuo libro tertio melius ſe demonſtraſſe propoſitiones 52. </s>
  <s xml:id="echoid-s691" xml:space="preserve">53. </s>
  <s xml:id="echoid-s692" xml:space="preserve">54.</s>
  <s xml:id="echoid-s693" xml:space="preserve">
<pb file="0036" n="36" rhead="Io: Alfonſi Borelli"/>
libri primi ſummopere glorietur, pari iure hi libri adulterini cenſendi non <lb/>erunt non alia de cauſa, niſi quia propoſitiones horum librorum non cor-<lb/>reſpondent, nec aſsimilantur admirandis cogitationibus in eius ſublimi <lb/>mente repoſitis. </s>
  <s xml:id="echoid-s694" xml:space="preserve">Et ſane non dubito, quòd ſi Midorgius ipſe hos libros <lb/>vidiſſet, &amp; </s>
  <s xml:id="echoid-s695" xml:space="preserve">contrectaßet, omnino illius magni Apollonĳ eſſe abſq; </s>
  <s xml:id="echoid-s696" xml:space="preserve">vlla <lb/>hæſitatione affirmaſſet. </s>
  <s xml:id="echoid-s697" xml:space="preserve">Nam primi quatuor libri continent eaſdem pro-<lb/>poſitiones, &amp; </s>
  <s xml:id="echoid-s698" xml:space="preserve">ſæpe numero eadem verba, quæ in textu Græco Eutocĳ <lb/>leguntur: </s>
  <s xml:id="echoid-s699" xml:space="preserve">reliqui libri ſubſequentes docent ea, quæ in epiſtola ad Eudemum <lb/>propoſuerat ſe demonſtraturum Apollonius, &amp; </s>
  <s xml:id="echoid-s700" xml:space="preserve">quæ Pappus, &amp; </s>
  <s xml:id="echoid-s701" xml:space="preserve">Eutocius <lb/>diſtinctè, &amp; </s>
  <s xml:id="echoid-s702" xml:space="preserve">expreſsè ibidem tractari affirmant. </s>
  <s xml:id="echoid-s703" xml:space="preserve">Rurſus profunda men-<lb/>tis perſpicacia, methodus ſcribendi, &amp; </s>
  <s xml:id="echoid-s704" xml:space="preserve">genius Apollonĳ adhuc ibidem <lb/>conſpicitur, nec fieri potuit, vt à translatoribus, à Paraphraſte, à tem <lb/>poris diuturnitate prorſus deleretur, atque mirandum ingenium Apollonĳ <lb/>à tanta barbarie omnino occultaretur. </s>
  <s xml:id="echoid-s705" xml:space="preserve">Rurſus in confeſſo eſt opera Euclidis, <lb/>Archimedis, Apollonĳ, Ptolomæi, &amp; </s>
  <s xml:id="echoid-s706" xml:space="preserve">aliorum magnorum virorum Ara-<lb/>bicè translata fuisſe, &amp; </s>
  <s xml:id="echoid-s707" xml:space="preserve">expreſsè grauisſimi ſcriptores Arabi, præcipuè <lb/>Gregorius Bar-Hebræus lib. </s>
  <s xml:id="echoid-s708" xml:space="preserve">9. </s>
  <s xml:id="echoid-s709" xml:space="preserve">Chronicorum ait, opera Apollonĳ Arabicè <lb/>translata primò fuisſe anno 200. </s>
  <s xml:id="echoid-s710" xml:space="preserve">AEgyræ Maumettanæ ſub Almen Kalypha <lb/>à loanne Patricida, &amp; </s>
  <s xml:id="echoid-s711" xml:space="preserve">poſtea ab alĳs recentioribus. </s>
  <s xml:id="echoid-s712" xml:space="preserve">Quare dubitandum <lb/>non eſt hos eſſe veros, atque legitimos tres poſtremos Conicorum libros <lb/>Apollonĳ Pergæi Paraphraſticè ab Abalphatho deſcriptos.</s>
  <s xml:id="echoid-s713" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s714" xml:space="preserve">Fruere modo, mi lector, præclaro, &amp; </s>
  <s xml:id="echoid-s715" xml:space="preserve">admirando beneficio Serenisſi-<lb/>mi Principis Etruriæ, qui regali magnificentia, et liberalitate pretioſisſimum <lb/>hunc theſaurum humanisſimè largitur. </s>
  <s xml:id="echoid-s716" xml:space="preserve">Vale.</s>
  <s xml:id="echoid-s717" xml:space="preserve"/>
</p>
<pb file="0037" n="37"/>
</div>
<div xml:id="echoid-div14" type="section" level="1" n="12">
<head xml:id="echoid-head32" xml:space="preserve">INDEX</head>
<p>
  <s xml:id="echoid-s718" xml:space="preserve">Propoſitionum Lib. </s>
  <s xml:id="echoid-s719" xml:space="preserve">V. </s>
  <s xml:id="echoid-s720" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s721" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s722" xml:space="preserve">Conic. </s>
  <s xml:id="echoid-s723" xml:space="preserve">iuxta ſeriem numerorum <lb/>ab Apoll, ſeruatam, cum Lemmatibus, &amp; </s>
  <s xml:id="echoid-s724" xml:space="preserve">Propoſition, additis,</s>
</p>
<p style="it">
  <s xml:id="echoid-s725" xml:space="preserve">Vbi indicantur ſectiones, &amp; </s>
  <s xml:id="echoid-s726" xml:space="preserve">paginę, in quibus propoſitiones reperiri debent.</s>
  <s xml:id="echoid-s727" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve"> <lb/>### Lib. V. <lb/>Propoſ. # Sect. # Pag. <lb/>i # 1 # 5 <lb/>ii # 1 # 5 <lb/>iii # 1 # 6 <lb/>iv # 2 # 8 <lb/>v # 2 # 8 <lb/>vi # 2 # 8 <lb/>vii # 4 # 24 <lb/>viii # 3 # 16 <lb/>ix # 3 # 18 <lb/>x # 3 # 18 <lb/>xi # 5 # 26 <lb/>xii # 4 # 24 <lb/>xiii # 6 # 27 <lb/>xiv # 6 # 27 <lb/>xv # 6 # 27 <lb/>xvi # 16 # 112 <lb/>xvii # 16 # 112 <lb/>xviii # 16 # 112 <lb/>xix # 17 # 116 <lb/>xx # 17 # 117 <lb/>xxi # 17 # 117 <lb/>xxii # 17 # 117 <lb/>xxiii # 17 # 118 <lb/>xxiv # 17 # 118 <lb/>xxv # 17 # 119 <lb/>xxvi # 7 # 29 <lb/>xxvii # 7 # 29 <lb/>xxviii # 7 # 29 <lb/>xxix # 12 # 72 <lb/>xxx # 12 # 72 <lb/>xxxi # 12 # 72 <lb/>xxxii # 18 # 124 <lb/>xxxiii # 18 # 125 <lb/>xxxiv # 18 # 125 <lb/>xxxv # 18 # 125 <lb/>xxxvi # 18 # 126 <lb/>xxxvii # 18 # 126 <lb/>xxxviii # 18 # 127 <lb/>xxxix # 18 # 128 <lb/>xxxx # 18 # 128 <lb/>xxxxi # 15 # 109 <lb/>xxxxii # 15 # 109 <lb/>xxxxiii # 15 # 110 <lb/>xxxxiv # 10 # 67 <lb/>xxxxv # 10 # 68 <lb/>Prop. # Sect. # Pag. <lb/>xxxxvi # 18 # 126 <lb/>xxxxvii # 18 # 128 <lb/>xxxxviii # 18 # 129 <lb/>xxxxix # 8 # 32 33 <lb/>l # 8 # 33 <lb/>lj # 8 # 34 <lb/>lii # 8 # 35 <lb/>liii # 8 # 35 <lb/>liv # 8 # 39 <lb/>lv # 8 # 39 <lb/>lvi # 8 # 39 <lb/>lvii # 8 # 40 <lb/>lviii # 9 # 60 <lb/>lix # 9 # 60 <lb/>lx # 9 # 62 <lb/>lxi # 9 # 62 <lb/>lxii # 9 # 60 <lb/>lxiii # 9 # 60 <lb/>lxiv # 13 # 74 <lb/>lxv # 13 # 74 <lb/>lxvi # 13 # 75 <lb/>lxvii # 13 # 76 <lb/>lxviii # 11 # 70 <lb/>lxix # 11 # 70 <lb/>lxx # 11 # 71 <lb/>lxxi # 11 # 71 <lb/>lxxii # 13 # 77 <lb/>lxxiii # 14 # 88 89 <lb/>lxxiv # 14 # 90 <lb/>lxxv # 14 # 90 <lb/>lxxvi # 14 # 91 <lb/>lxxvii # 14 # 92 <lb/>## Lib. V. <lb/>Lemm. addita # Paginæ. <lb/>i # 13 <lb/>ii # 14 <lb/>iii # 15 <lb/>iv # 15 <lb/>v # 30 <lb/>vi # 31 <lb/>vii # 31 <lb/>viii # 57 <lb/>ix # 78 <lb/>x # 78 <lb/>xi # 79 <lb/>xii # 92 <lb/>## Lib. V. <lb/>Prop. additæ # Paginæ <lb/>i # 11 <lb/>ii # 11 <lb/>iii # 22 <lb/>iv # 23 <lb/>v # 54 <lb/>vi # 86 <lb/>vii # 101 <lb/>viii # 103 <lb/>ix # 103 <lb/>x # 104 <lb/>xi # 105 <lb/>xii # 106 <lb/>xiii # 107 <lb/>xiv # 107 <lb/>### Lib. VI. <lb/>Propoſ. # Sect. # Pag. <lb/>i # 1 # 138 <lb/>ii # 1 # 139 <lb/>iii # 2 # 146 <lb/>iv # 1 # 141 <lb/>v # 3 # 152 <lb/>vi # 2 # 147 <lb/>vii # 2 # 147 <lb/>viii # 3 # 153 <lb/>ix # 2 # 148 <lb/>x # 1 # 141 <lb/>xi # 4 # 154 <lb/>xii # 4 # 155 <lb/>xiii # 4 # 156 <lb/>xiv # 4 # 157 <lb/>xv # 6 # 175 <lb/>xvi # 6 # 177 <lb/>xvii # 6 # 178 <lb/>xviii # 7 # 191 <lb/>xix # 7 # 191 <lb/>xx # 8 # 193 <lb/>xxi # 8 # 195 <lb/>xxii # 8 # 197 <lb/>xxiii # 8 # 198 <lb/>xxiv # 8 # 198 <lb/>xxv # 9 # 207 <lb/>xxvi # 10 # 237 <lb/>xxvii # 10 # 238 <lb/>xxviii # 10 # 240 <lb/></note>
<pb file="0038" n="38"/>
<note position="right" xml:space="preserve"> <lb/>Prop. # Sect. # Pag. <lb/>xxix # 11 # 247 <lb/>xxx # 11 # 248 <lb/>xxxi # 11 # 251 <lb/>Antiquæ # Propoſ. # Præmiſſæ <lb/>i # 5 # 168 <lb/>ii # 5 # 168 <lb/>iii # 5 # 168 <lb/>iv # 5 # 168 <lb/>v # 5 # 168 <lb/>vi # 5 # 171 <lb/>## Lib. VI. <lb/>Lemm. addita. # Pag. <lb/>i # 150 <lb/>ii # 158 <lb/>iii # 159 <lb/>iv # 160 <lb/>v # 161 <lb/>vi # 183 <lb/>vii # 184 <lb/>viii # 186 <lb/>ix # 229 <lb/>x # 246 <lb/>## Lib. VI. <lb/>Prop. additæ. # Pag. <lb/>i # 151 <lb/>ii # 210 <lb/>iii # 211 <lb/>iv # 214 <lb/>v # 216 <lb/>vi # 219 <lb/>vii # 220 <lb/>viii # 222 <lb/>ix # 226 <lb/>x # 227 <lb/>xi # 230 <lb/>xii # 231 <lb/>xiii # 233 <lb/>xiv # 236 <lb/>xv # 261 <lb/>xvi # 262 <lb/>xvii # 265 <lb/>xviii # 267 <lb/>xix # 267 <lb/>Prop. # Pag. <lb/>xx # 268 <lb/>xxi # 269 <lb/>xxii # 270 <lb/>### Lib. VII. <lb/>Propoſ. # Sect. # Pag. <lb/>i # 1 # 273 <lb/>ii # 2 # 276 <lb/>iii # 2 # 276 <lb/>iv # 2 # 277 <lb/>v # 1 # 274 <lb/>vi # 2 # 278 <lb/>vii # 2 # 278 <lb/>viii # 3 # 282 <lb/>ix # 3 # 283 <lb/>x # 3 # 283 <lb/>xi # 3 # 283 <lb/>xii # 4 # 291 <lb/>xiii # 4 # 291 <lb/>xiv # 4 # 291 <lb/>xv # 3 # 283 <lb/>xvi # 3 # 283 <lb/>xvii # 3 # 283 <lb/>xviii # 3 # 283 <lb/>xix # 3 # 283 <lb/>xx # 3 # 283 <lb/>xxi # 5 # 299 <lb/>xxii # 4 # 291 <lb/>xxiii # 1 # 274 <lb/>xxiv # 5 # 298 303 <lb/>xxv # 4 # 291 <lb/>xxvi # 5 # 298 300 <lb/>xxvii # 4 # 291 <lb/>xxviii # 5 # 299 300 <lb/>xxix # 4 # 291 <lb/>xxx # 4 # 291 <lb/>xxxi # 11 # 370 <lb/>xxxii # 11 # 370 <lb/>xxxiii # 6 # 314 <lb/>xxxiv # 6 # 315 <lb/>xxxv # 6 # 316 <lb/>xxxvi # 6 # 316 <lb/>xxxvii # 5 # 304 <lb/>xxxviii # 7 # 323 <lb/>xxxix # 7 # 324 <lb/>xxxx # 7 # 325 <lb/>xxxxi # 9 # 341 343 <lb/>Prop. # Sect. # Pag. <lb/>xxxxii # 5 # 301 <lb/>xxxxiii # 5 # 298 302 <lb/>xxxxiv # 8 # 333 <lb/>xxxxv # 8 # 333 <lb/>xxxxvi # 8 # 335 <lb/>xxxxvii # 9 # 342 344 <lb/>xxxxviii # 9 # 342 347 <lb/>xxxxix # 10 # 358 <lb/>L # 10 # 358 <lb/>Lj # 10 # 358 <lb/>## Lib. VII. <lb/>Lemm. addita. # Pag. <lb/>i # 306 <lb/>ii # 318 <lb/>iii # 318 <lb/>iv # 318 <lb/>v # 319 <lb/>vi # 327 <lb/>vii # 327 <lb/>viii # 328 <lb/>ix # 328 <lb/>x # 336 <lb/>xi # 336 <lb/>xii # 337 <lb/>xiii # 349 <lb/>xiv # 350 <lb/>xv # 350 <lb/>xvi # 361 <lb/>xvii # 361 <lb/>xviii # 364 <lb/>## Lib. VII. <lb/>Prop. additæ. # Pag. <lb/>i # 322 <lb/>ii # 323 <lb/>iii # 331 <lb/>iv # 332 <lb/>v # 341 <lb/>vi # 341 <lb/>vii # 357 <lb/>viii # 357 <lb/>ix # 368 <lb/>x # 368 <lb/></note>
<pb o="1" file="0039" n="39"/>
</div>
<div xml:id="echoid-div15" type="section" level="1" n="13">
<head xml:id="echoid-head33" xml:space="preserve">APOLLONII PERGAEI <lb/>CONICORVM LIB. V.</head>
<head xml:id="echoid-head34" xml:space="preserve">DEFINITIONES.</head>
<head xml:id="echoid-head35" xml:space="preserve">I.</head>
<p>
  <s xml:id="echoid-s728" xml:space="preserve">SI à puncto aliquo in axe ſectionis conicæ ſumpto <lb/>egrediantur aliquę rectæ lineæ ad ſectionem, <lb/>vocabo punctum illud, ORIGINEM.</s>
  <s xml:id="echoid-s729" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div16" type="section" level="1" n="14">
<head xml:id="echoid-head36" xml:space="preserve">II.</head>
<p>
  <s xml:id="echoid-s730" xml:space="preserve">Et lineas, RAMOS.</s>
  <s xml:id="echoid-s731" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div17" type="section" level="1" n="15">
<head xml:id="echoid-head37" xml:space="preserve">III.</head>
<p>
  <s xml:id="echoid-s732" xml:space="preserve">Segmentum autem axis intèr illud, &amp; </s>
  <s xml:id="echoid-s733" xml:space="preserve">verticem ſectionis ei pro-<lb/>ximiorem, MENSVRAM.</s>
  <s xml:id="echoid-s734" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div18" type="section" level="1" n="16">
<head xml:id="echoid-head38" xml:space="preserve">IV.</head>
<p>
  <s xml:id="echoid-s735" xml:space="preserve">Sed ſi fuerit menſura æqualis ſemiſſi erecti, vocabo illam, <lb/>COMPARATAM.</s>
  <s xml:id="echoid-s736" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div19" type="section" level="1" n="17">
<head xml:id="echoid-head39" xml:space="preserve">V.</head>
<p>
  <s xml:id="echoid-s737" xml:space="preserve">Et perpendiculares cadentes ab extremitatibus ramorum ſuper <lb/>axim vocabo, POTENTES illorum ramorum.</s>
  <s xml:id="echoid-s738" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div20" type="section" level="1" n="18">
<head xml:id="echoid-head40" xml:space="preserve">VI.</head>
<p>
  <s xml:id="echoid-s739" xml:space="preserve">Abſciſſa verò illarum potentium, ABSCISSA ramorum.</s>
  <s xml:id="echoid-s740" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div21" type="section" level="1" n="19">
<head xml:id="echoid-head41" xml:space="preserve">VII.</head>
<p>
  <s xml:id="echoid-s741" xml:space="preserve">Et inuerſa illarum potentium, INVERSA ramorum.</s>
  <s xml:id="echoid-s742" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div22" type="section" level="1" n="20">
<head xml:id="echoid-head42" xml:space="preserve">VIII.</head>
<p>
  <s xml:id="echoid-s743" xml:space="preserve">Atque rectangulum contentum ſub inclinato, &amp; </s>
  <s xml:id="echoid-s744" xml:space="preserve">aggregato in-<lb/>clinati, &amp; </s>
  <s xml:id="echoid-s745" xml:space="preserve">erecti, vel differentia tranſuerſi, &amp; </s>
  <s xml:id="echoid-s746" xml:space="preserve">erecti vocabo, FI-<lb/>GVRAM COMPARATAM.</s>
  <s xml:id="echoid-s747" xml:space="preserve"/>
</p>
<pb o="2" file="0040" n="40" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div23" type="section" level="1" n="21">
<head xml:id="echoid-head43" xml:space="preserve">IX.</head>
<p>
  <s xml:id="echoid-s748" xml:space="preserve">In quolibet rectangulo applicato ad ſegmentum axis, ſi illud <lb/>ſegmentum ad latitudinem illius rectanguli eandem proportio-<lb/>nem habuerit, quam axis ad latitudinem figurę comparatæ vocabo <lb/>illud, EXEMPLAR.</s>
  <s xml:id="echoid-s749" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div24" type="section" level="1" n="22">
<head xml:id="echoid-head44" xml:space="preserve">X.</head>
<p>
  <s xml:id="echoid-s750" xml:space="preserve">Si ex puncto ſuper axim educatur perpendicularis ad vtraſque <lb/>partes ſectionis, &amp; </s>
  <s xml:id="echoid-s751" xml:space="preserve">ex puncto aliquo illius perpendicularis educan-<lb/>tur lineæ terminatæ ad ſectionem ex vtraque parte, vocabo pun-<lb/>ctum illud in perpendiculari ſumptum, CONCVRSVM.</s>
  <s xml:id="echoid-s752" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div25" type="section" level="1" n="23">
<head xml:id="echoid-head45" xml:space="preserve">XI.</head>
<p>
  <s xml:id="echoid-s753" xml:space="preserve">Et lineas etiam, RAMOS.</s>
  <s xml:id="echoid-s754" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div26" type="section" level="1" n="24">
<head xml:id="echoid-head46" xml:space="preserve">XII.</head>
<p>
  <s xml:id="echoid-s755" xml:space="preserve">Et qui ſecant menſuram, &amp; </s>
  <s xml:id="echoid-s756" xml:space="preserve">terminantur ad ſectionem ex altera <lb/>parte concurſus, RAMOS SECANTES.</s>
  <s xml:id="echoid-s757" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div27" type="section" level="1" n="25">
<head xml:id="echoid-head47" xml:space="preserve">XIII.</head>
<p>
  <s xml:id="echoid-s758" xml:space="preserve">At qui non ſecat illam, &amp; </s>
  <s xml:id="echoid-s759" xml:space="preserve">tranſit per concurſum, &amp; </s>
  <s xml:id="echoid-s760" xml:space="preserve">terminatur <lb/>ad axim, &amp; </s>
  <s xml:id="echoid-s761" xml:space="preserve">ſectionem ſimul, RAMVM TERMINATVM.</s>
  <s xml:id="echoid-s762" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div28" type="section" level="1" n="26">
<head xml:id="echoid-head48" xml:space="preserve">XIV.</head>
<p>
  <s xml:id="echoid-s763" xml:space="preserve">Sed cuiuſcumque rami ſecantis, cuius portio interſectionem, &amp; </s>
  <s xml:id="echoid-s764" xml:space="preserve"><lb/>axim intercepta eſt linea breuiſſima, vocabo illum, BREVISE-<lb/>CANTEM.</s>
  <s xml:id="echoid-s765" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div29" type="section" level="1" n="27">
<head xml:id="echoid-head49" xml:space="preserve">XV.</head>
<p>
  <s xml:id="echoid-s766" xml:space="preserve">Et vocabo ſegmentum axis inter perpendicularem, &amp; </s>
  <s xml:id="echoid-s767" xml:space="preserve">verticem <lb/>ſectionis proximior em interceptum, MENSVRAM, quoque.</s>
  <s xml:id="echoid-s768" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div30" type="section" level="1" n="28">
<head xml:id="echoid-head50" xml:space="preserve">XIV.</head>
<p>
  <s xml:id="echoid-s769" xml:space="preserve">Et portionem ſectionis conicæ diſſectam ab ordinatione axis <lb/>tranſeuntis per originem, ſiuè per coneurſum propè verticem pro-<lb/>ximiorem ſectionis, vocabo, SEGMENTVM illius puncti.</s>
  <s xml:id="echoid-s770" xml:space="preserve"/>
</p>
<pb o="3" file="0041" n="41" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div31" type="section" level="1" n="29">
<head xml:id="echoid-head51" xml:space="preserve">NOTÆ.</head>
<p style="it">
  <s xml:id="echoid-s771" xml:space="preserve">HAE definitiones non ſunt Apollonij, ſed Interpretis Arabici, qui in proe-<lb/>mio huius operis apertè ait, addidiſſe plurimas definitiones in libris Apol-<lb/>lonij, quibus theoremata breuiſsimè propo-<lb/>
<anchor type="figure" xlink:label="fig-0041-01a" xlink:href="fig-0041-01"/>
ni poſſe profitetur, vt in prioribus quatuor <lb/>libris videre eſt. </s>
  <s xml:id="echoid-s772" xml:space="preserve">Eas autem exemplis illu-<lb/>ſtrare conabor.</s>
  <s xml:id="echoid-s773" xml:space="preserve"/>
</p>
<div xml:id="echoid-div31" type="float" level="2" n="1">
  <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a">
    <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0041-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s774" xml:space="preserve">I. </s>
  <s xml:id="echoid-s775" xml:space="preserve">Sit quælibet coni ſectio A B C, cuius <lb/>axis B D, &amp; </s>
  <s xml:id="echoid-s776" xml:space="preserve">in eo ſumatur quodlibet pun-<lb/>ctum D intrà ſectionem, à quo educantur <lb/>rectæ lineæ D A, D E, D F, D C vſque ad <lb/>ſectionem. </s>
  <s xml:id="echoid-s777" xml:space="preserve">Tùnc vocatnr punctum D, Origo.</s>
  <s xml:id="echoid-s778" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s779" xml:space="preserve">II. </s>
  <s xml:id="echoid-s780" xml:space="preserve">Et lineæ D A, D E, &amp; </s>
  <s xml:id="echoid-s781" xml:space="preserve">cæteræ vo-<lb/>cantur, Rami.</s>
  <s xml:id="echoid-s782" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s783" xml:space="preserve">III. </s>
  <s xml:id="echoid-s784" xml:space="preserve">Portio verò axis B D intèr origi-<lb/>nem D, &amp; </s>
  <s xml:id="echoid-s785" xml:space="preserve">verticem B interpoſita vocatur <lb/>Menſura. </s>
  <s xml:id="echoid-s786" xml:space="preserve">Sed in ellipſi A B C G, ſi axis <lb/>portiones D B, &amp; </s>
  <s xml:id="echoid-s787" xml:space="preserve">D G inæquales fuerint, <lb/>tantummodò minor portio B D vocatur Mẽ-<lb/>ſura, non autem maior D G.</s>
  <s xml:id="echoid-s788" xml:space="preserve"/>
</p>
  <figure>
    <image file="0041-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0041-02"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s789" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s790" xml:space="preserve">Sit poſteà recta B I ſemiſsis lateris <lb/>recti B H iam ſi menſura D B æqualis fue-<lb/>rit ſemierecto B I, vocatur D B, Menfura <lb/>comparata.</s>
  <s xml:id="echoid-s791" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s792" xml:space="preserve">V. </s>
  <s xml:id="echoid-s793" xml:space="preserve">At ſi à terminis ramorum A, E, F <lb/>C educantur ad axim perpendiculares A K, <lb/>E L, F M, C N, ipſum ſecantes in K, L, <lb/>M, N vocantur illærectæ lineæ Potentes illo-<lb/>rum ramorum.</s>
  <s xml:id="echoid-s794" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s795" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s796" xml:space="preserve">Recta verò K B vocatur Abſciſſa <lb/>rami D A, &amp; </s>
  <s xml:id="echoid-s797" xml:space="preserve">L B Abſciſſa rami D E, &amp; </s>
  <s xml:id="echoid-s798" xml:space="preserve"><lb/>ſic reliquæ omnes.</s>
  <s xml:id="echoid-s799" xml:space="preserve"/>
</p>
  <figure>
    <image file="0041-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0041-03"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s800" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s801" xml:space="preserve">Sit poſteà O centrum ſectionis, iam <lb/>axis portio ex centro O vſquè ad potentia-<lb/>lem A K educta, ſcilicet O K vocatur In-<lb/>uerſa rami D A, pariterque O M eſt Inuer-<lb/>ſa rami D F.</s>
  <s xml:id="echoid-s802" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s803" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s804" xml:space="preserve">Si ponatur recta linea B P ad <lb/>axim perpendicularis, quæ in hyperbola <lb/>fiat æqualis aggregato, in ellipſi verò fiat <lb/>æqualis differentiæ laterum recti B H, &amp; </s>
  <s xml:id="echoid-s805" xml:space="preserve"><lb/>tranſuerſi G B, tunc rectangulum contentum <lb/>ſub G B, &amp; </s>
  <s xml:id="echoid-s806" xml:space="preserve">B P vocatur, Figura comparata.</s>
  <s xml:id="echoid-s807" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s808" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s809" xml:space="preserve">Poſteà ſi, vt G B ad B P ità ſiat ſeg-
<pb o="4" file="0042" n="42" rhead="Apollonij Pergæi"/>
mentum axis D B ad D R, &amp; </s>
  <s xml:id="echoid-s810" xml:space="preserve">compleatur <lb/>parallelogrãmum rectãgulum B R, tunc ſpa-<lb/>tium B R vocatur Exemplar. </s>
  <s xml:id="echoid-s811" xml:space="preserve">Pari ratione <lb/>ſi, vt G B ad D P ità fiat ſegmentum axis <lb/>D K ad latitudinem K S, compleaturque <lb/>parallelogrammum rectangulum D S, voca-<lb/>bitur paritèr D S Exemplar.</s>
  <s xml:id="echoid-s812" xml:space="preserve"/>
</p>
  <figure>
    <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0042-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s813" xml:space="preserve">X. </s>
  <s xml:id="echoid-s814" xml:space="preserve">Et ſi C D perpendicularis fuerit ad <lb/>axim B D, &amp; </s>
  <s xml:id="echoid-s815" xml:space="preserve">producatur vltrà axim in <lb/>E, atquè à puncto E extendantur vſquè ad <lb/>ſectionem rectæ lineæ E B, E F, E G, vo-<lb/>cabitur E punctum Concurſus.</s>
  <s xml:id="echoid-s816" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s817" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s818" xml:space="preserve">Et lineæ rectæ E B, E F, E G vo-<lb/>cantur etiam Rami.</s>
  <s xml:id="echoid-s819" xml:space="preserve"/>
</p>
  <figure>
    <image file="0042-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0042-02"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s820" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s821" xml:space="preserve">Atquè linea recta E F ſecans axim <lb/>in H vocatur Ramus ſecans.</s>
  <s xml:id="echoid-s822" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s823" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s824" xml:space="preserve">Et recta linea E B conueniens <lb/>cum axi in vertice ſectionis vocatur Ra-<lb/>mus terminatus.</s>
  <s xml:id="echoid-s825" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s826" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s827" xml:space="preserve">Si verò rami ſecantis E F por-<lb/>tio cius H F inter ſectionem, &amp; </s>
  <s xml:id="echoid-s828" xml:space="preserve">axim in-<lb/>tercepta fuerit breuiſsima omnium linea-<lb/>rum, quæ ex puncto H ad ſectionem duci <lb/>poſſunt, tunc ramus E F vocabitur Breui-<lb/>ſecans. </s>
  <s xml:id="echoid-s829" xml:space="preserve">In textu Arabico ſecans ramus vo-<lb/>cabatur, mendosè, vt arbitror, non enim <lb/>hæc definitio diſtingueretur à duodecima, <lb/>definitione.</s>
  <s xml:id="echoid-s830" xml:space="preserve"/>
</p>
  <figure>
    <image file="0042-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0042-03"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s831" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s832" xml:space="preserve">Similitèr ſegmentum axis D B ſe-<lb/>ctum à perpendiculari ad axim ex origine <lb/>E ducta, vocatur quoquè Menſura.</s>
  <s xml:id="echoid-s833" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s834" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s835" xml:space="preserve">Tandem ſi per punctum originis <lb/>D, vel concurſus E ducatur ordinata A C, <lb/>tunc figura contenta ab ordinata A C, &amp; </s>
  <s xml:id="echoid-s836" xml:space="preserve"><lb/>ſectione conica A B C, vocatur Segmentum <lb/>illius puncti.</s>
  <s xml:id="echoid-s837" xml:space="preserve"/>
</p>
<pb o="5" file="0043" n="43" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div33" type="section" level="1" n="30">
<head xml:id="echoid-head52" xml:space="preserve">SECTIO PRIMA</head>
<head xml:id="echoid-head53" xml:space="preserve">Continens propoſitiones I. II. &amp; III. Apollonij.</head>
<head xml:id="echoid-head54" xml:space="preserve">PROPOSITIO I.</head>
<p>
  <s xml:id="echoid-s838" xml:space="preserve">Si ex centro D ſectionis A B (habentis centrum) egrediatur <lb/>linea recta D F H bifariam diuidens A E erectum illius axis, <lb/>quod ſit perpendiculare ſuper axim C A G, ſecans axis ordina-<lb/>tionem B G I; </s>
  <s xml:id="echoid-s839" xml:space="preserve">vtiquè dimidium illius ordinationis, videlicet B <lb/>G, poterit duplum plani, quod producit illa linea cum axi in-<lb/>ter erectum, &amp; </s>
  <s xml:id="echoid-s840" xml:space="preserve">illam ordinationem, nempè duplum A G H F.</s>
  <s xml:id="echoid-s841" xml:space="preserve"/>
</p>
  <figure>
    <image file="0043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0043-01"/>
  </figure>
<p>
  <s xml:id="echoid-s842" xml:space="preserve">QVia B G poteſt comparatum applicatum ad abſciſſam A G, &amp; </s>
  <s xml:id="echoid-s843" xml:space="preserve">pla-<lb/>
<anchor type="note" xlink:label="note-0043-01a" xlink:href="note-0043-01"/>
num G F dimidium eſt illius comparati; </s>
  <s xml:id="echoid-s844" xml:space="preserve">ergò B G poterit duplum <lb/>
<anchor type="note" xlink:label="note-0043-02a" xlink:href="note-0043-02"/>
plani G F; </s>
  <s xml:id="echoid-s845" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s846" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s847" xml:space="preserve"/>
</p>
<div xml:id="echoid-div33" type="float" level="2" n="1">
<note position="left" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0043-02" xlink:href="note-0043-02a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div35" type="section" level="1" n="31">
<head xml:id="echoid-head55" xml:space="preserve">PROPOS. II.</head>
  <figure>
    <image file="0043-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0043-02"/>
  </figure>
<p>
  <s xml:id="echoid-s848" xml:space="preserve">PAritèr quoquè oſtendetur, ſi potens <lb/>tranſierit per centrum ellipſis, quod <lb/>B G poterit duplum trianguli A F G.</s>
  <s xml:id="echoid-s849" xml:space="preserve"/>
</p>
<pb o="6" file="0044" n="44" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div36" type="section" level="1" n="32">
<head xml:id="echoid-head56" xml:space="preserve">PROPOS. III.</head>
  <figure>
    <image file="0044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0044-01"/>
  </figure>
<p>
  <s xml:id="echoid-s850" xml:space="preserve">SI verò in ellipſi cadat B G infrà cen-<lb/>trum, poterit duplum differentię duo-<lb/>rum triangulorum D A F, &amp; </s>
  <s xml:id="echoid-s851" xml:space="preserve">D G H, nem-<lb/>pè duplum plani G L. </s>
  <s xml:id="echoid-s852" xml:space="preserve">Et hoc erat pro-<lb/>poſitum.</s>
  <s xml:id="echoid-s853" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div37" type="section" level="1" n="33">
<head xml:id="echoid-head57" xml:space="preserve">Notæ in Propoſitionem primam.</head>
<p style="it">
  <s xml:id="echoid-s854" xml:space="preserve">VOcat in primo libro interpres ſectiones habentes centrum hyperbolem, &amp; </s>
  <s xml:id="echoid-s855" xml:space="preserve"><lb/>ellipſim, &amp; </s>
  <s xml:id="echoid-s856" xml:space="preserve">vocat erectum latus rectum ſectionis, vocat etiam ordina-<lb/>tionem axis eam, quam nos ordinatim ad axim applicatam appellamus.</s>
  <s xml:id="echoid-s857" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s858" xml:space="preserve">Quia BG poteſt comparatum applicatum ad abſciſſam AG, &amp;</s>
  <s xml:id="echoid-s859" xml:space="preserve">c. </s>
  <s xml:id="echoid-s860" xml:space="preserve">Vocat <lb/>
<anchor type="note" xlink:label="note-0044-01a" xlink:href="note-0044-01"/>
inſuper parallelogrammum comparatum applicatum ad axis abſciſſam A G re-<lb/>ctangulum ipſum A G I, quod quidem adiacet lateri recto A E latitudinem ha-<lb/>
<anchor type="note" xlink:label="note-0044-02a" xlink:href="note-0044-02"/>
bens abſciſſam A G excedens in hyperbola, &amp; </s>
  <s xml:id="echoid-s861" xml:space="preserve">deficiens in ellipſi rectangulo ſi-<lb/>mile ei, quod latere recto, &amp; </s>
  <s xml:id="echoid-s862" xml:space="preserve">tranſuerſo continetur; </s>
  <s xml:id="echoid-s863" xml:space="preserve">ſcilicèt rectangulo C A E.</s>
  <s xml:id="echoid-s864" xml:space="preserve"/>
</p>
<div xml:id="echoid-div37" type="float" level="2" n="1">
<note position="right" xlink:label="note-0044-01" xlink:href="note-0044-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0044-02" xlink:href="note-0044-02a" xml:space="preserve">12. 13. lib. <lb/>primi.</note>
</div>
  <figure>
    <image file="0044-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0044-02"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s865" xml:space="preserve">Et planum G F dimidium eſt illius comparati, &amp;</s>
  <s xml:id="echoid-s866" xml:space="preserve">c. </s>
  <s xml:id="echoid-s867" xml:space="preserve">Non erit inutile <lb/>
<anchor type="note" xlink:label="note-0044-03a" xlink:href="note-0044-03"/>
paulo fuſius oſtendere id quod ob nimiam facilitatem Apollonius tantummodò in-<lb/>nuit. </s>
  <s xml:id="echoid-s868" xml:space="preserve">Ducatur recta linea F K parallela axi D A ſecans ordinatam B G produ-<lb/>ctam in K: </s>
  <s xml:id="echoid-s869" xml:space="preserve">quia figuræ latera C A, &amp; </s>
  <s xml:id="echoid-s870" xml:space="preserve">A E ſunt ipſarum D A, A F duplicia <lb/>ergo C E, &amp; </s>
  <s xml:id="echoid-s871" xml:space="preserve">D F H parallelæ ſunt, eſtque K H parallela A E, cum ambo poſitæ <lb/>ſint perpendiculares ad axim, &amp; </s>
  <s xml:id="echoid-s872" xml:space="preserve">C A, F K ſunt quoquè æquidiſtantes, ergò <lb/>triangulum F K H ſimile eſt triangulo C A E, &amp; </s>
  <s xml:id="echoid-s873" xml:space="preserve">proptereà parallelogramma <lb/>rectangula F K H, &amp; </s>
  <s xml:id="echoid-s874" xml:space="preserve">C A E ſimilia erunt. </s>
  <s xml:id="echoid-s875" xml:space="preserve">Et quoniam quadratum ordinatæ <lb/>
<anchor type="note" xlink:label="note-0044-04a" xlink:href="note-0044-04"/>
B G æquale eſt rectangulo contento ſub latere recto E A, &amp; </s>
  <s xml:id="echoid-s876" xml:space="preserve">abſciſſa A G exce-
<pb o="7" file="0045" n="45" rhead="Conicor. Lib. V."/>
dente in hyperbola, &amp; </s>
  <s xml:id="echoid-s877" xml:space="preserve">deficiente in ellipſi rectangulo F K H ſimile ei, quod la-<lb/>teribus recto, &amp; </s>
  <s xml:id="echoid-s878" xml:space="preserve">tranſuerſo continetur, ſcilicet G A E, &amp; </s>
  <s xml:id="echoid-s879" xml:space="preserve">eſt A F ſemiſsis la-<lb/>teris recti, igitur quadratum B G æquale eſt ſummæ in hyperbole, &amp; </s>
  <s xml:id="echoid-s880" xml:space="preserve">differen-<lb/>tiæ in ellipſi rectanguli G A F bis ſumpti, &amp; </s>
  <s xml:id="echoid-s881" xml:space="preserve">rectanguli F K H, quod eſt æqua-<lb/>le duplo trianguli F K H: </s>
  <s xml:id="echoid-s882" xml:space="preserve">ſed quadrilaterum A G H F æquale eſt aggregato in <lb/>hyperbola, &amp; </s>
  <s xml:id="echoid-s883" xml:space="preserve">differentiæ in ellipſi rectanguli G A F, &amp; </s>
  <s xml:id="echoid-s884" xml:space="preserve">trianguli F K H, ergò <lb/>quadratum B G æquale eſt duplo quadrilateri A G H F, ſeù diſſerentiæ triangu-<lb/>lorum D A F, &amp; </s>
  <s xml:id="echoid-s885" xml:space="preserve">D G H.</s>
  <s xml:id="echoid-s886" xml:space="preserve"/>
</p>
<div xml:id="echoid-div38" type="float" level="2" n="2">
<note position="right" xlink:label="note-0044-03" xlink:href="note-0044-03a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0044-04" xlink:href="note-0044-04a" xml:space="preserve">Ibidem.</note>
</div>
  <figure>
    <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0045-01"/>
  </figure>
</div>
<div xml:id="echoid-div40" type="section" level="1" n="34">
<head xml:id="echoid-head58" xml:space="preserve">Notæ in Propoſitionem <lb/>ſecundam.</head>
<p>
  <s xml:id="echoid-s887" xml:space="preserve">SEcunda propoſitio facilè ex prima deducitur; <lb/></s>
  <s xml:id="echoid-s888" xml:space="preserve">nam, quando ordinata B G H I tranſit per cen-<lb/>trum D ellipſis; </s>
  <s xml:id="echoid-s889" xml:space="preserve">tunc tria puncta G, D, H conue-<lb/>niunt, &amp; </s>
  <s xml:id="echoid-s890" xml:space="preserve">triangulum D G H euaneſcit, &amp; </s>
  <s xml:id="echoid-s891" xml:space="preserve">ideò <lb/>differentia trianguli D A F, &amp; </s>
  <s xml:id="echoid-s892" xml:space="preserve">trianguli D G H <lb/>nullum ſpatium habentis, erit triangulum ipſum <lb/>D A F.</s>
  <s xml:id="echoid-s893" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div41" type="section" level="1" n="35">
<head xml:id="echoid-head59" xml:space="preserve">Notæ in Propoſitionem <lb/>tertiam.</head>
  <figure>
    <image file="0045-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0045-02"/>
  </figure>
<p>
  <s xml:id="echoid-s894" xml:space="preserve">IN tertia propoſitione ſimilitèr, quandò ordinata <lb/>B H G I cadit infrà centrum D ellipſis, tunc <lb/>ducta C L parallela ipſi A E, erunt duo triangula <lb/>D A F, &amp; </s>
  <s xml:id="echoid-s895" xml:space="preserve">D C L æqualia inter ſe, cum ſint ſimi-<lb/>lia, &amp; </s>
  <s xml:id="echoid-s896" xml:space="preserve">latera homologa D A, D C ſint æqualia, <lb/>quia ſunt ſemiaxes; </s>
  <s xml:id="echoid-s897" xml:space="preserve">proptereà differentia triangu-<lb/>lorum D G H, &amp; </s>
  <s xml:id="echoid-s898" xml:space="preserve">D A F, ſeù D C L erit trapezium <lb/>C G H L, quod ſubduplum eſt quadrati ordinatæ <lb/>B G.</s>
  <s xml:id="echoid-s899" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div42" type="section" level="1" n="36">
<head xml:id="echoid-head60" xml:space="preserve">SECTIO SECVNDA</head>
<head xml:id="echoid-head61" xml:space="preserve">Continens propoſitiones IV. V. VI. Apollonij.</head>
<p>
  <s xml:id="echoid-s900" xml:space="preserve">COmparata eſt minima ramorum egredientium ex ſua origine <lb/>(4) in parabola (5) &amp; </s>
  <s xml:id="echoid-s901" xml:space="preserve">hyperbola (6) pariterque in ellipſi (ſi <lb/>comparata fuerit portio maioris duorum axium, &amp; </s>
  <s xml:id="echoid-s902" xml:space="preserve">tunc maxi-<lb/>mus eſt reſiduum tranſuerſi axis.) </s>
  <s xml:id="echoid-s903" xml:space="preserve">Reliquorum verò propinquior
<pb o="8" file="0046" n="46" rhead="Apollonij Pergæi"/>
minimo remotiore minor eſt. </s>
  <s xml:id="echoid-s904" xml:space="preserve">Quadratum autem menſuræ mi-<lb/>nus eſt quadrato cuiuslibet rami aſſignati (4) in parabola qui-<lb/>dem quadrato ſuæ abſciſſæ (5) &amp; </s>
  <s xml:id="echoid-s905" xml:space="preserve">in hyperbola (6) &amp; </s>
  <s xml:id="echoid-s906" xml:space="preserve">ellipſi <lb/>exemplari applicato ad abſciſſam illius rami.</s>
  <s xml:id="echoid-s907" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div43" type="section" level="1" n="37">
<head xml:id="echoid-head62" xml:space="preserve">PROPOSITIO IV.</head>
<p>
  <s xml:id="echoid-s908" xml:space="preserve">SIt ſectio A B C, &amp; </s>
  <s xml:id="echoid-s909" xml:space="preserve">axis eius C E, &amp; </s>
  <s xml:id="echoid-s910" xml:space="preserve">inclinatus, ſiue tranſuerſa D C <lb/>centrum G, atque erectum C F, &amp; </s>
  <s xml:id="echoid-s911" xml:space="preserve">ex C E ſecetur C I æqualis C H <lb/>
<anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/>
(quæ ſit ſemiſſis erecti) &amp; </s>
  <s xml:id="echoid-s912" xml:space="preserve">ex puncto <lb/>originis I educantur rami I B perpen-<lb/>dicularis, &amp; </s>
  <s xml:id="echoid-s913" xml:space="preserve">I K, I L, I A, &amp; </s>
  <s xml:id="echoid-s914" xml:space="preserve">per H, I <lb/>in hyperbola, &amp; </s>
  <s xml:id="echoid-s915" xml:space="preserve">ellipſi ducatur H I P, <lb/>&amp; </s>
  <s xml:id="echoid-s916" xml:space="preserve">per H, G recta H G T, ad quam ex <lb/>A, B, K, L extendantur A P E T, B I S, <lb/>K N R, L M O Q perpendiculares ſuper <lb/>C E. </s>
  <s xml:id="echoid-s917" xml:space="preserve">Dico, quod C I, comparata mi-<lb/>nor eſt, quam I L, &amp; </s>
  <s xml:id="echoid-s918" xml:space="preserve"><lb/>I L, quam I K, &amp; </s>
  <s xml:id="echoid-s919" xml:space="preserve">I K, <lb/>quam I B, &amp; </s>
  <s xml:id="echoid-s920" xml:space="preserve">maximus <lb/>ramorum in ellipſi eſt <lb/>I D, &amp; </s>
  <s xml:id="echoid-s921" xml:space="preserve">quod quadra-<lb/>tum menſuræ I C mi-<lb/>nus eſt quadrato I L, <lb/>in parabola quidem <lb/>quadrato C M, &amp; </s>
  <s xml:id="echoid-s922" xml:space="preserve">in <lb/>hyperbola, &amp; </s>
  <s xml:id="echoid-s923" xml:space="preserve">ellipſi <lb/>exemplari applicato <lb/>ad C M. </s>
  <s xml:id="echoid-s924" xml:space="preserve">Quoniam in <lb/>parabola L M poteſt <lb/>
<anchor type="note" xlink:label="note-0046-01a" xlink:href="note-0046-01"/>
duplum M C in C H, nempè C I (12. </s>
  <s xml:id="echoid-s925" xml:space="preserve">ex primo) &amp; </s>
  <s xml:id="echoid-s926" xml:space="preserve">quadratum I L ęqua-<lb/>le eſt aggregato duorum quadratorum L M, &amp; </s>
  <s xml:id="echoid-s927" xml:space="preserve">M I, quadratum itaque L <lb/>I æquale eſt quadrato M I, &amp; </s>
  <s xml:id="echoid-s928" xml:space="preserve">M C in C I bis, quæ ſunt æqualia duobus <lb/>quadratis C I, M C. </s>
  <s xml:id="echoid-s929" xml:space="preserve">Quadratum igitur C I minus eſt quadrato L I qua-<lb/>drato ipſius M C, quæ eſt eius abſciſſa, &amp; </s>
  <s xml:id="echoid-s930" xml:space="preserve">pariter oſtendetur, quod qua-<lb/>dratum C I minus eſt quadrato I K quadrato N C, &amp; </s>
  <s xml:id="echoid-s931" xml:space="preserve">minus quadrato I <lb/>B quadrato C I, &amp; </s>
  <s xml:id="echoid-s932" xml:space="preserve">minus quadrato A I quadrato E C.</s>
  <s xml:id="echoid-s933" xml:space="preserve"/>
</p>
<div xml:id="echoid-div43" type="float" level="2" n="1">
  <figure xlink:label="fig-0046-01" xlink:href="fig-0046-01a">
    <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0046-01"/>
  </figure>
<note position="right" xlink:label="note-0046-01" xlink:href="note-0046-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div45" type="section" level="1" n="38">
<head xml:id="echoid-head63" xml:space="preserve">PROPOSITIO V. &amp; VI.</head>
<p>
  <s xml:id="echoid-s934" xml:space="preserve">AT verò in hyperbola, &amp; </s>
  <s xml:id="echoid-s935" xml:space="preserve">ellipſi producantur ex Q, O, H lineæ pa-<lb/>rallelæ ipſi M C, &amp; </s>
  <s xml:id="echoid-s936" xml:space="preserve">quia I C ex hypotheſi æqualis eſt H C, erit I <lb/>
<anchor type="note" xlink:label="note-0046-02a" xlink:href="note-0046-02"/>
M æqualis M O, quadratum itaque I M duplum eſt trianguli I M O, &amp; </s>
  <s xml:id="echoid-s937" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0046-03a" xlink:href="note-0046-03"/>
quadratum L M duplum eſt trapezij C M Q H (prima ex 5.) </s>
  <s xml:id="echoid-s938" xml:space="preserve">ergo quadra-
<pb o="9" file="0047" n="47" rhead="Conicor. Lib. V."/>
tum I L duplum eſt trianguli I C H vnà cum duplo trianguli Q H O, nem-<lb/>pe cum plano rectanguli QZ; </s>
  <s xml:id="echoid-s939" xml:space="preserve">ſed quadratum I C eſt duplum trianguli I <lb/>H C (eò quod C H æqualis eſt C I) ergo quadratum C I minus eſt qua-<lb/>drato L I plano rectanguli Q Z.</s>
  <s xml:id="echoid-s940" xml:space="preserve"/>
</p>
<div xml:id="echoid-div45" type="float" level="2" n="1">
<note position="right" xlink:label="note-0046-02" xlink:href="note-0046-02a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0046-03" xlink:href="note-0046-03a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s941" xml:space="preserve">Deindè ponamus in ellipſi Y F æqualem differentiæ, &amp; </s>
  <s xml:id="echoid-s942" xml:space="preserve">in hyperbola <lb/>
<anchor type="note" xlink:label="note-0047-01a" xlink:href="note-0047-01"/>
æqualem aggregato D C, C F; </s>
  <s xml:id="echoid-s943" xml:space="preserve">ergo propter ſimilitudinem duorum trian-<lb/>
<anchor type="note" xlink:label="note-0047-02a" xlink:href="note-0047-02"/>
gulorum G M Q, H V Q, &amp; </s>
  <s xml:id="echoid-s944" xml:space="preserve">H V O, M I O, erit H V æqualis V O, &amp; </s>
  <s xml:id="echoid-s945" xml:space="preserve">H <lb/>V, vel ei æqualis O V ad V Q eſt, vt M G ad M Q, nempe vt G C ad <lb/>
<anchor type="note" xlink:label="note-0047-03a" xlink:href="note-0047-03"/>
<anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/>
H C, ſeù vt D C ad C F, igi-<lb/>tur V O ad V Q eſt vt D C <lb/>
<anchor type="note" xlink:label="note-0047-04a" xlink:href="note-0047-04"/>
ad CF, &amp; </s>
  <s xml:id="echoid-s946" xml:space="preserve">comparando ſum-<lb/>mas terminorum ad antece-<lb/>dentes in hyperbola, &amp; </s>
  <s xml:id="echoid-s947" xml:space="preserve">dif-<lb/>ferentias eorundem ad ante-<lb/>cedentes in ellipſi fiet O Q <lb/>ad V O (quæ æqualis eſt O <lb/>Z, nempè M C) vt Y F ad <lb/>
<anchor type="note" xlink:label="note-0047-05a" xlink:href="note-0047-05"/>
Y C, &amp; </s>
  <s xml:id="echoid-s948" xml:space="preserve">eſt Y C, æqualis D <lb/>C, &amp; </s>
  <s xml:id="echoid-s949" xml:space="preserve">Y F æqualis ſummæ <lb/>in hyperbola, &amp; </s>
  <s xml:id="echoid-s950" xml:space="preserve">differentiæ <lb/>in ellipſi ipſarum D C, &amp; </s>
  <s xml:id="echoid-s951" xml:space="preserve">C <lb/>F; </s>
  <s xml:id="echoid-s952" xml:space="preserve">quadratum igitur I C mi-<lb/>
<anchor type="note" xlink:label="note-0047-06a" xlink:href="note-0047-06"/>
<anchor type="note" xlink:label="note-0047-07a" xlink:href="note-0047-07"/>
nus eſt quadrato I L rectangulo Q Z, quod eſt exemplar ſimile <lb/>plano rectanguli C D in Y F, quæ eſt figura comparata. </s>
  <s xml:id="echoid-s953" xml:space="preserve">Atque ſic de-<lb/>monſtrabitur, quod quadratum I C minus ſit quadrato I K exemplari ap-<lb/>plicato ad N C, &amp; </s>
  <s xml:id="echoid-s954" xml:space="preserve">minus eſt quadrato B I exemplari applicato ad I C, <lb/>&amp; </s>
  <s xml:id="echoid-s955" xml:space="preserve">minus quadrato A I exemplari applicato ad E C: </s>
  <s xml:id="echoid-s956" xml:space="preserve">Eſtque M C minor, <lb/>quàm N C, &amp; </s>
  <s xml:id="echoid-s957" xml:space="preserve">N C, quam C I, &amp; </s>
  <s xml:id="echoid-s958" xml:space="preserve">C I, quàm C E; </s>
  <s xml:id="echoid-s959" xml:space="preserve">igitur L I maior eſt, <lb/>quàm I C, &amp; </s>
  <s xml:id="echoid-s960" xml:space="preserve">I K maior, quàm L I, &amp; </s>
  <s xml:id="echoid-s961" xml:space="preserve">I B maior, quàm I K, &amp; </s>
  <s xml:id="echoid-s962" xml:space="preserve">I A, quàm <lb/>I B. </s>
  <s xml:id="echoid-s963" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s964" xml:space="preserve"/>
</p>
<div xml:id="echoid-div46" type="float" level="2" n="2">
<note position="left" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0047-02" xlink:href="note-0047-02a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0047-03" xlink:href="note-0047-03a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0047-01" xlink:href="fig-0047-01a">
    <image file="0047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0047-01"/>
  </figure>
<note position="left" xlink:label="note-0047-04" xlink:href="note-0047-04a" xml:space="preserve">f</note>
<note position="left" xlink:label="note-0047-05" xlink:href="note-0047-05a" xml:space="preserve">g</note>
<note position="left" xlink:label="note-0047-06" xlink:href="note-0047-06a" xml:space="preserve">h</note>
<note position="right" xlink:label="note-0047-07" xlink:href="note-0047-07a" xml:space="preserve">Def. 8. 9. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div48" type="section" level="1" n="39">
<head xml:id="echoid-head64" xml:space="preserve">Notæ in pro poſitionem quartam.</head>
<p style="it">
  <s xml:id="echoid-s965" xml:space="preserve">QVoniam in parabola L M poteſt <lb/>
<anchor type="note" xlink:label="note-0047-08a" xlink:href="note-0047-08"/>
<anchor type="figure" xlink:label="fig-0047-02a" xlink:href="fig-0047-02"/>
duplum M C, &amp;</s>
  <s xml:id="echoid-s966" xml:space="preserve">c. </s>
  <s xml:id="echoid-s967" xml:space="preserve">Quadratum <lb/>enim L M æquale eſt rectangu-<lb/>lo ſub abſciſſa M C, &amp; </s>
  <s xml:id="echoid-s968" xml:space="preserve">latere recto C F, <lb/>eſtque C H ſemiſsis erecti C F; </s>
  <s xml:id="echoid-s969" xml:space="preserve">ergo L M <lb/>poteſt duplum rectanguli M C H.</s>
  <s xml:id="echoid-s970" xml:space="preserve"/>
</p>
<div xml:id="echoid-div48" type="float" level="2" n="1">
<note position="left" xlink:label="note-0047-08" xlink:href="note-0047-08a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0047-02" xlink:href="fig-0047-02a">
    <image file="0047-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0047-02"/>
  </figure>
</div>
<pb o="10" file="0048" n="48" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div50" type="section" level="1" n="40">
<head xml:id="echoid-head65" xml:space="preserve">Notæ in propoſitionem quintam.</head>
<p style="it">
  <s xml:id="echoid-s971" xml:space="preserve">ERit I M æqualis M O, &amp;</s>
  <s xml:id="echoid-s972" xml:space="preserve">c. </s>
  <s xml:id="echoid-s973" xml:space="preserve">Propter parallelas M O, C H, &amp; </s>
  <s xml:id="echoid-s974" xml:space="preserve">ſimilitudi-<lb/>
<anchor type="note" xlink:label="note-0048-01a" xlink:href="note-0048-01"/>
nem triangulorum I M O, &amp; </s>
  <s xml:id="echoid-s975" xml:space="preserve">I C H.</s>
  <s xml:id="echoid-s976" xml:space="preserve"/>
</p>
<div xml:id="echoid-div50" type="float" level="2" n="1">
<note position="right" xlink:label="note-0048-01" xlink:href="note-0048-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s977" xml:space="preserve">Ergo quadratum <lb/>
<anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/>
<anchor type="note" xlink:label="note-0048-02a" xlink:href="note-0048-02"/>
I L duplum eſt triã-<lb/>guli I C H, &amp;</s>
  <s xml:id="echoid-s978" xml:space="preserve">c. </s>
  <s xml:id="echoid-s979" xml:space="preserve">Eo <lb/>quod quadratum I L <lb/>æquale eſt duobus qua-<lb/>dratis I M, M L in <lb/>rectangulo triangulo I <lb/>M L; </s>
  <s xml:id="echoid-s980" xml:space="preserve">Quadratis au-<lb/>tẽ I M, &amp; </s>
  <s xml:id="echoid-s981" xml:space="preserve">L M æqua-<lb/>lia ſunt triangulum <lb/>I M O bis ſumptum <lb/>cum trapezio C M Q <lb/>H bis ſumpto; </s>
  <s xml:id="echoid-s982" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s983" xml:space="preserve">quia <lb/>
<anchor type="note" xlink:label="note-0048-03a" xlink:href="note-0048-03"/>
trapezium C M Q H <lb/>æquale eſt trapezio C <lb/>M O H, cum triangu-<lb/>lo H O Q; </s>
  <s xml:id="echoid-s984" xml:space="preserve">at triangulo I M O, <lb/>&amp; </s>
  <s xml:id="echoid-s985" xml:space="preserve">trapezio C M Q H ſimul ſum-<lb/>ptis æqualia ſunt triangulum <lb/>I C H, cum triangulo H O Q. <lb/></s>
  <s xml:id="echoid-s986" xml:space="preserve">Ergo quadratum L I æquale erit <lb/>duplo trianguli I C H cum duplo <lb/>trianguli H O Q.</s>
  <s xml:id="echoid-s987" xml:space="preserve"/>
</p>
<div xml:id="echoid-div51" type="float" level="2" n="2">
  <figure xlink:label="fig-0048-01" xlink:href="fig-0048-01a">
    <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0048-01"/>
  </figure>
<note position="right" xlink:label="note-0048-02" xlink:href="note-0048-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0048-03" xlink:href="note-0048-03a" xml:space="preserve">1. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s988" xml:space="preserve">Deindè ponamus in ellipſi <lb/>
<anchor type="note" xlink:label="note-0048-04a" xlink:href="note-0048-04"/>
Y F æqualem D C, &amp; </s>
  <s xml:id="echoid-s989" xml:space="preserve">in hy-<lb/>perbola, &amp;</s>
  <s xml:id="echoid-s990" xml:space="preserve">c. </s>
  <s xml:id="echoid-s991" xml:space="preserve">Textus videtur <lb/>corruptus, quem ſic corrigendum <lb/>puto. </s>
  <s xml:id="echoid-s992" xml:space="preserve">Ponamus γ F in ellipſi æ-<lb/>qualem differentiæ, &amp; </s>
  <s xml:id="echoid-s993" xml:space="preserve">in hyper-<lb/>bola æqualem aggregato D C, &amp; </s>
  <s xml:id="echoid-s994" xml:space="preserve">C F.</s>
  <s xml:id="echoid-s995" xml:space="preserve"/>
</p>
<div xml:id="echoid-div52" type="float" level="2" n="3">
<note position="right" xlink:label="note-0048-04" xlink:href="note-0048-04a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s996" xml:space="preserve">Propter ſimilitudinem triangulorum, &amp;</s>
  <s xml:id="echoid-s997" xml:space="preserve">c. </s>
  <s xml:id="echoid-s998" xml:space="preserve">Sunt enim duæ rectæ lineæ C G, <lb/>
<anchor type="note" xlink:label="note-0048-05a" xlink:href="note-0048-05"/>
&amp; </s>
  <s xml:id="echoid-s999" xml:space="preserve">V H æquidiſtantes, quæ ſecant rectas lineas conuenientes in Q, &amp; </s>
  <s xml:id="echoid-s1000" xml:space="preserve">O.</s>
  <s xml:id="echoid-s1001" xml:space="preserve"/>
</p>
<div xml:id="echoid-div53" type="float" level="2" n="4">
<note position="right" xlink:label="note-0048-05" xlink:href="note-0048-05a" xml:space="preserve">d</note>
</div>
<p style="it">
  <s xml:id="echoid-s1002" xml:space="preserve">Erit H V æqualis V O, &amp;</s>
  <s xml:id="echoid-s1003" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1004" xml:space="preserve">Eo quòd M I oſtenſa eſt æqualis M O, eſtque <lb/>
<anchor type="note" xlink:label="note-0048-06a" xlink:href="note-0048-06"/>
H V ad V O in eadem proportione æqualitatis propter iam dictam ſimilitudinem <lb/>triangulorum.</s>
  <s xml:id="echoid-s1005" xml:space="preserve"/>
</p>
<div xml:id="echoid-div54" type="float" level="2" n="5">
<note position="right" xlink:label="note-0048-06" xlink:href="note-0048-06a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s1006" xml:space="preserve">Igitur V O ad V Q eſt, vt D C ad C F, &amp; </s>
  <s xml:id="echoid-s1007" xml:space="preserve">conuerſa proportione dein-<lb/>
<anchor type="note" xlink:label="note-0048-07a" xlink:href="note-0048-07"/>
dè componendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s1008" xml:space="preserve">inuertendo in ellipſi fiet in hyperbola <lb/>Q O ad O V, &amp;</s>
  <s xml:id="echoid-s1009" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1010" xml:space="preserve">Textum corruptum, atque confuſum clariùs exponi poſſe <lb/>cenſeo per Lemma inferius appoſitum hac ratione. </s>
  <s xml:id="echoid-s1011" xml:space="preserve">Et comparando ſummas in <lb/>hyperbola, &amp; </s>
  <s xml:id="echoid-s1012" xml:space="preserve">differentias terminorum in ellipſi ad antecedentes.</s>
  <s xml:id="echoid-s1013" xml:space="preserve"/>
</p>
<div xml:id="echoid-div55" type="float" level="2" n="6">
<note position="right" xlink:label="note-0048-07" xlink:href="note-0048-07a" xml:space="preserve">f</note>
</div>
<p>
  <s xml:id="echoid-s1014" xml:space="preserve">Vt Y F ad Y C, &amp; </s>
  <s xml:id="echoid-s1015" xml:space="preserve">in ellipſi, vt F C ad C F, &amp; </s>
  <s xml:id="echoid-s1016" xml:space="preserve">Y F in ellipſi æqualis <lb/>
<anchor type="note" xlink:label="note-0048-08a" xlink:href="note-0048-08"/>
<pb o="11" file="0049" n="49" rhead="Conicor. Lib. V."/>
D C, quad<unsure/>ratum igitur, &amp;</s>
  <s xml:id="echoid-s1017" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1018" xml:space="preserve">Textum corruptum ſic corrigendum puto; </s>
  <s xml:id="echoid-s1019" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1020" xml:space="preserve">eſt <lb/>r C æqualis D C, atque γ F æqualis ſummæ in hyperbola, &amp; </s>
  <s xml:id="echoid-s1021" xml:space="preserve">differentiæ in elli-<lb/>pſi laterum D C, &amp; </s>
  <s xml:id="echoid-s1022" xml:space="preserve">C F.</s>
  <s xml:id="echoid-s1023" xml:space="preserve"/>
</p>
<div xml:id="echoid-div56" type="float" level="2" n="7">
<note position="right" xlink:label="note-0048-08" xlink:href="note-0048-08a" xml:space="preserve">g</note>
</div>
<p>
  <s xml:id="echoid-s1024" xml:space="preserve">Exemplar ſimile plano rectanguli C D in Y F in hyperbola, &amp; </s>
  <s xml:id="echoid-s1025" xml:space="preserve">Y C in <lb/>ellipſi, &amp;</s>
  <s xml:id="echoid-s1026" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1027" xml:space="preserve">Hæc poſtrema verba expungenda duxi, tanquam ſuperuacanea.</s>
  <s xml:id="echoid-s1028" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1029" xml:space="preserve">Poteſt etiam ad imitationem Euclidis reperiri multitudo ramorum inter ſe-<lb/>æqualium, qui ex origine duci poſſunt in eadem coniſectione. </s>
  <s xml:id="echoid-s1030" xml:space="preserve">Itaque quoties <lb/>
<anchor type="note" xlink:label="note-0049-01a" xlink:href="note-0049-01"/>
menſura fuerit comparata, ſcilicet aqualis ſemiſsi lateris recti, tunc duo tan-<lb/>tum rami inter ſe æquales a puncto originis ad vtraſque partes axis duci poſ-<lb/>ſunt in qualibet coniſectione, eruntque illi, qui ad terminos L l cuiuslibet or-<lb/>dinatim applicatæ L l ducuntur ab origine <lb/>
<anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/>
I, nam efſiciuntur duo triangula I M L, &amp; </s>
  <s xml:id="echoid-s1031" xml:space="preserve"><lb/>I M l, quæ circa angulos æquales ad M, nẽ-<lb/>pe rectos, habent latera æqualia, ſcilicet L <lb/>M, &amp; </s>
  <s xml:id="echoid-s1032" xml:space="preserve">l M medietates ordinatim applicatæ, <lb/>&amp; </s>
  <s xml:id="echoid-s1033" xml:space="preserve">ſegmentum axis I M inter ordinatam, &amp; </s>
  <s xml:id="echoid-s1034" xml:space="preserve"><lb/>originem eſt latus commune; </s>
  <s xml:id="echoid-s1035" xml:space="preserve">ergobaſes, ſeu <lb/>rami I L, &amp; </s>
  <s xml:id="echoid-s1036" xml:space="preserve">I l ſunt æquales. </s>
  <s xml:id="echoid-s1037" xml:space="preserve">Reliquiverò <lb/>rami ſupra, vel infra terminum eiuſdem ordinatim applicatæ minores, aut ma-<lb/>iores ſunt ramo ad eius terminum ducto; </s>
  <s xml:id="echoid-s1038" xml:space="preserve">quare duo tantum rami ad vtraſque <lb/>partes axis inter ſe æquales duci poſſunt.</s>
  <s xml:id="echoid-s1039" xml:space="preserve"/>
</p>
<div xml:id="echoid-div57" type="float" level="2" n="8">
<note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">PROP. I. <lb/>Additar.</note>
  <figure xlink:label="fig-0049-01" xlink:href="fig-0049-01a">
    <image file="0049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0049-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1040" xml:space="preserve">Rurſus quadratum rami I A remotioris a comparata ſuperat quadratum ra-<lb/>
<anchor type="note" xlink:label="note-0049-02a" xlink:href="note-0049-02"/>
mì I L propinquioris (in parabola quidem) rectangulo ſub differentia, &amp; </s>
  <s xml:id="echoid-s1041" xml:space="preserve">ſub <lb/>aggregato abſciſſarum eorundem ramorum; </s>
  <s xml:id="echoid-s1042" xml:space="preserve">in reliquis verò ſectionibus rectan-<lb/>gulo ſub differentia abſciſſarum, &amp; </s>
  <s xml:id="echoid-s1043" xml:space="preserve">ſub recta linea, ad quam ſumma abſcißa-<lb/>rum eandem proportionem habet, quam latus tranſuerſum ad ſummam in hy-<lb/>perbola, &amp; </s>
  <s xml:id="echoid-s1044" xml:space="preserve">ad differentiam in ellipſi laterum tranſuerſi, &amp; </s>
  <s xml:id="echoid-s1045" xml:space="preserve">recti.</s>
  <s xml:id="echoid-s1046" xml:space="preserve"/>
</p>
<div xml:id="echoid-div58" type="float" level="2" n="9">
<note position="right" xlink:label="note-0049-02" xlink:href="note-0049-02a" xml:space="preserve">PROP. <lb/>II.Add.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1047" xml:space="preserve">Et primò in parabola, quia quadratum I A æquale eſt quadrato I C cum qua-<lb/>
<anchor type="note" xlink:label="note-0049-03a" xlink:href="note-0049-03"/>
drato abſciſſæ C E; </s>
  <s xml:id="echoid-s1048" xml:space="preserve">pariterque quadratum I L æquale eſt quadrato eiuſdem I C <lb/>cum quadrato abſciſſæ C M; </s>
  <s xml:id="echoid-s1049" xml:space="preserve">ergo exceſſus quadrati I A ſupra quadratum I L <lb/>
<anchor type="note" xlink:label="note-0049-04a" xlink:href="note-0049-04"/>
æqualis eſt differentiæ quadratorum E C, &amp; </s>
  <s xml:id="echoid-s1050" xml:space="preserve">C M; </s>
  <s xml:id="echoid-s1051" xml:space="preserve">ſed exceſſus quadrati E C <lb/>ſupra quadratum M C æqualis eſt rectangulo, cuius baſis æqualis eſt ſummæ la-<lb/>terum E C, &amp; </s>
  <s xml:id="echoid-s1052" xml:space="preserve">C M; </s>
  <s xml:id="echoid-s1053" xml:space="preserve">altitudo verò æqualis eſt E M differentiæ laterum eorun-<lb/>dem quadratorum (vt de-<lb/>ducitur ex elementis) igitur <lb/>
<anchor type="figure" xlink:label="fig-0049-02a" xlink:href="fig-0049-02"/>
exceſſus quadrati I A ſupra <lb/>quadratum I L æqualis eſt <lb/>rectangulo, cuius baſis eſt <lb/>ſumma abſciſſarum E C, C <lb/>M, altitudo verò E M dif-<lb/>ferentia earundem abſciſſa-<lb/>rum.</s>
  <s xml:id="echoid-s1054" xml:space="preserve"/>
</p>
<div xml:id="echoid-div59" type="float" level="2" n="10">
<note position="right" xlink:label="note-0049-03" xlink:href="note-0049-03a" xml:space="preserve">4. huius.</note>
<note position="right" xlink:label="note-0049-04" xlink:href="note-0049-04a" xml:space="preserve">ibidem.</note>
  <figure xlink:label="fig-0049-02" xlink:href="fig-0049-02a">
    <image file="0049-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0049-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1055" xml:space="preserve">Secundò in hyperbola, &amp; </s>
  <s xml:id="echoid-s1056" xml:space="preserve"><lb/>ellipſi fiat exemplar N T ap-<lb/>plicatum ab abſciſſam C E. <lb/></s>
  <s xml:id="echoid-s1057" xml:space="preserve">Et quia quadratum I A æ-<lb/>quale eſt quadrato eiuſdem
<pb o="12" file="0050" n="50" rhead="Apollonij Pergæi"/>
I C cum exemplari N T, &amp; </s>
  <s xml:id="echoid-s1058" xml:space="preserve">quadratum I L æquale eſt quadrato eiuſdem I C cum <lb/>exemplari Q Z. </s>
  <s xml:id="echoid-s1059" xml:space="preserve">Ergò exceſſus quadrati I A ſupra quadratum I L æqualis eſt <lb/>differentiæ exemplarium N T, &amp; </s>
  <s xml:id="echoid-s1060" xml:space="preserve">Q Z. </s>
  <s xml:id="echoid-s1061" xml:space="preserve">Poſteà ducatur recta Q N: </s>
  <s xml:id="echoid-s1062" xml:space="preserve">quia trian-<lb/>gula Q N S, O N Q. </s>
  <s xml:id="echoid-s1063" xml:space="preserve">æqualia ſunt triangulo, cuius baſis æqualis eſt ſummæ re-<lb/>ctarum N S, &amp; </s>
  <s xml:id="echoid-s1064" xml:space="preserve">O Q. <lb/></s>
  <s xml:id="echoid-s1065" xml:space="preserve">altitudo verò V R, vel <lb/>
<anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/>
M E, ſuntque illa duo <lb/>triãgula æqualia tra-<lb/>pezio N O Q ſiue-<lb/>exceſſui trianguli N <lb/>H S, ſupra triangu-<lb/>lum H O Q: </s>
  <s xml:id="echoid-s1066" xml:space="preserve">ergo triã-<lb/>gulum cuius baſis æ-<lb/>quatur ſumme ipſa-<lb/>rum N S, O Q alti-<lb/>tudo verò E M, æqua-<lb/>le eſt differentiæ triã-<lb/>gulorum N H S, O H <lb/>Q. </s>
  <s xml:id="echoid-s1067" xml:space="preserve">Et ſimiliter eorum dupla, ſcilicet rectangulum, cuius baſis æqualis eſt ſum-<lb/>mæ N S, O Q altitudo verò æqualis M E, erit differentia exemplarium rectã-<lb/>gulorum N T, &amp; </s>
  <s xml:id="echoid-s1068" xml:space="preserve">Q Z; </s>
  <s xml:id="echoid-s1069" xml:space="preserve">ſed ſumma altitudinum V H, H R, ſeu ſumma abſciſ-<lb/>ſarum C M, C E ad ſum mam baſium N S, O Q eandem proportionem habet, <lb/>quam vna H V ad vnam O Q, ſeu quam latus tranſuerſum D C ad ſummam-<lb/>in hyperbola, &amp; </s>
  <s xml:id="echoid-s1070" xml:space="preserve">ad differentiam in ellipſi laterum tranſuerſi D C, &amp; </s>
  <s xml:id="echoid-s1071" xml:space="preserve">recti C F: <lb/></s>
  <s xml:id="echoid-s1072" xml:space="preserve">Igitur differentia exemplar ium N T, Q Z, ſeu exceſſus quadrati I A ſupra-<lb/>quadratum I L æqualis eſt rectangulo contento ſub E M differentia abſciſſarum, <lb/>&amp; </s>
  <s xml:id="echoid-s1073" xml:space="preserve">ſub ſumma ipſarum N S, &amp; </s>
  <s xml:id="echoid-s1074" xml:space="preserve">O Q, ad quam ſumma abſcißarum eandem pro-<lb/>portionem habet, quam latus tranſuerſum ad ſummam in hyperbola, &amp; </s>
  <s xml:id="echoid-s1075" xml:space="preserve">ad dif-<lb/>ferentiam in ellipſi laterum tranſuerſi, &amp; </s>
  <s xml:id="echoid-s1076" xml:space="preserve">recti, quod fuerat propoſitum.</s>
  <s xml:id="echoid-s1077" xml:space="preserve"/>
</p>
<div xml:id="echoid-div60" type="float" level="2" n="11">
  <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a">
    <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0050-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div62" type="section" level="1" n="41">
<head xml:id="echoid-head66" xml:space="preserve">MONITVM.</head>
<p style="it">
  <s xml:id="echoid-s1078" xml:space="preserve">E X varia diſpoſitione terminorum proportionalitatis ſcilicet duo-<lb/>rum antecedentium, &amp; </s>
  <s xml:id="echoid-s1079" xml:space="preserve">duorum conſequentium conſurgunt <lb/>plures modi argumentandi, quorum aliqui in elementis ex-<lb/>poſiti non ſunt, aliqui verò ſignificantiſsimis vocibus, &amp; </s>
  <s xml:id="echoid-s1080" xml:space="preserve"><lb/>breuiùs indicantur in textu Arabico, igitur, ne ſepius repetatur prolixa-<lb/>expoſitio modorum argumentandi in proportionalibus, &amp; </s>
  <s xml:id="echoid-s1081" xml:space="preserve">non proportiona-<lb/>libus, qui cumulatè inſeruntur in demonſirationibus Apollonĳ opere pre-<lb/>tium erit eos ſemel hìc exponere.</s>
  <s xml:id="echoid-s1082" xml:space="preserve"/>
</p>
<pb o="13" file="0051" n="51" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div63" type="section" level="1" n="42">
<head xml:id="echoid-head67" xml:space="preserve">LEMMA I.</head>
<p style="it">
  <s xml:id="echoid-s1083" xml:space="preserve">Si quatuor quantitates eandem proportionem habuerint, antecedentes, <lb/>vel cońſequentes ad terminorum ſummas, vel differentias in eadem ra-<lb/>tione erunt; </s>
  <s xml:id="echoid-s1084" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1085" xml:space="preserve">è contra.</s>
  <s xml:id="echoid-s1086" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1087" xml:space="preserve">HAbeat A B ad B C eandem proportionem, quàm D E ad E H: </s>
  <s xml:id="echoid-s1088" xml:space="preserve">ſequitur pri-<lb/>mò, quod A C ad C B ſit, vt D H ad H E; </s>
  <s xml:id="echoid-s1089" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1090" xml:space="preserve">huiuſmodi argumentatio <lb/>vocatur in elementis compoſitio terminorum proportionis: </s>
  <s xml:id="echoid-s1091" xml:space="preserve">itaque ſummæ antece-<lb/>dentium, &amp; </s>
  <s xml:id="echoid-s1092" xml:space="preserve">conſequentium ad eaſdem conſequentes ſunt etiam proportionales: <lb/></s>
  <s xml:id="echoid-s1093" xml:space="preserve">ſi vero ex eadem hypotbeſi concludai<unsure/>ur, quod A C ad A B, ſit vt D H ad D E, <lb/>vt nimirum ſummæ terminorum proportionis ad antecedentes ſint proportiona-<lb/>les: </s>
  <s xml:id="echoid-s1094" xml:space="preserve">quod quidem manifeſtum eſt, nam poſita fuit A B ad B C, vt D E ad E H; </s>
  <s xml:id="echoid-s1095" xml:space="preserve"><lb/>erit inuertendo C B ad B A, vt H E ad E D, &amp; </s>
  <s xml:id="echoid-s1096" xml:space="preserve">componendo C A ad A B erit <lb/>vt H D ad D E: </s>
  <s xml:id="echoid-s1097" xml:space="preserve">modo huiuſmodi argumentandi forma innominata eſt; </s>
  <s xml:id="echoid-s1098" xml:space="preserve">poteſt <lb/>autem breuitatis gratia appellari, Per comparationem ſummæ terminorum ad <lb/>antecedentes.</s>
  <s xml:id="echoid-s1099" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1100" xml:space="preserve">Secundò concludi poteſt, quod A B ad A <lb/>C ſit vt D E ad D H; </s>
  <s xml:id="echoid-s1101" xml:space="preserve">quia, vt in prima <lb/>
<anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/>
parte dictum eſt, A C ad A B erat vt D H <lb/>ad D E, ergo inuertendo A B ad A C erit <lb/>vt D E ad D H: </s>
  <s xml:id="echoid-s1102" xml:space="preserve">hæc argumentandi forma <lb/>vocari poteſt, Per comparationem antece-<lb/>dentium ad terminorum ſummas.</s>
  <s xml:id="echoid-s1103" xml:space="preserve"/>
</p>
<div xml:id="echoid-div63" type="float" level="2" n="1">
  <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a">
    <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0051-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1104" xml:space="preserve">Tertiò concludi poteſt: </s>
  <s xml:id="echoid-s1105" xml:space="preserve">quod B C ad C A, ſit vt E H ad H D; </s>
  <s xml:id="echoid-s1106" xml:space="preserve">nam componen-<lb/>do A C ad C B, erat vt D H ad H E, quare inuertendo B C ad C A erit vt E <lb/>H ad H D, &amp; </s>
  <s xml:id="echoid-s1107" xml:space="preserve">hæc argumentatio fieri dicetur comparando conſequentes ad ter-<lb/>minorum ſummas.</s>
  <s xml:id="echoid-s1108" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1109" xml:space="preserve">Deindè ſint eædem quatuor proportiona-<lb/>les in ſecunda figura, nimirum totum A B <lb/>
<anchor type="figure" xlink:label="fig-0051-02a" xlink:href="fig-0051-02"/>
ad ſegmentum eius B C ſit vt totum D E <lb/>ad portionem eius E H; </s>
  <s xml:id="echoid-s1110" xml:space="preserve">tunc reſiduum A C <lb/>ad C B erit, vt reſiduum D H ad H E; </s>
  <s xml:id="echoid-s1111" xml:space="preserve">hæc <lb/>argumentatio ſieri dicitur in elementis, di-<lb/>uidendo terminos proportionis, eſtque comparatio differentiarum terminorum ad <lb/>conſequentes.</s>
  <s xml:id="echoid-s1112" xml:space="preserve"/>
</p>
<div xml:id="echoid-div64" type="float" level="2" n="2">
  <figure xlink:label="fig-0051-02" xlink:href="fig-0051-02a">
    <image file="0051-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0051-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1113" xml:space="preserve">At ſi concludatur ex eadem hypotbeſi quod A B ad A C ſit vt D E ad D H; <lb/></s>
  <s xml:id="echoid-s1114" xml:space="preserve">hæc argumentatio in elementis fieri dicitur per conuerſionem rationis eſtque <lb/>comparatio antecedentium ad differentias terminorum.</s>
  <s xml:id="echoid-s1115" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1116" xml:space="preserve">Poſtea ex eadem hypotbeſi ſequitur quod A C ad A B ſit vt D H ad D E: </s>
  <s xml:id="echoid-s1117" xml:space="preserve">quia <lb/>per conuerſionem rationis, ſeu referendo antecedentes ad differentias terminorum <lb/>eſt A B ad A C, vt D E ad D H; </s>
  <s xml:id="echoid-s1118" xml:space="preserve">ergo inuertendo A C ad A B erit vt D H ad <lb/>D E, &amp; </s>
  <s xml:id="echoid-s1119" xml:space="preserve">hæc argumentatio innominata fiet comparando differentias terminorum <lb/>ad antecedentes.</s>
  <s xml:id="echoid-s1120" xml:space="preserve"/>
</p>
<pb o="14" file="0052" n="52" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s1121" xml:space="preserve">Tandem ex eadem hypotheſi ſequitur, quod C B ad C A ſit vt E H ad H D: <lb/></s>
  <s xml:id="echoid-s1122" xml:space="preserve">nam diuidenda eſt vt A C ad C B, ita D H ad H E; </s>
  <s xml:id="echoid-s1123" xml:space="preserve">ergo inuertendo B C ad C A <lb/>erit vt E H ad H D: </s>
  <s xml:id="echoid-s1124" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1125" xml:space="preserve">hæc argumentatio innominata fieri dicetur comparan-<lb/>do conſequentes ad derenifftias terminorum.</s>
  <s xml:id="echoid-s1126" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div66" type="section" level="1" n="43">
<head xml:id="echoid-head68" xml:space="preserve">LEMMA II.</head>
<p style="it">
  <s xml:id="echoid-s1127" xml:space="preserve">Si prima A B ad ſecundam B C maiorem proportionem habuerit quàm <lb/>tertia D E ad quartam E H: </s>
  <s xml:id="echoid-s1128" xml:space="preserve">comparando antecedentes ad terminorum. <lb/></s>
  <s xml:id="echoid-s1129" xml:space="preserve">ſummas habebit AB ad AC maiorem proportionem quàm D E ad D H.</s>
  <s xml:id="echoid-s1130" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1131" xml:space="preserve">FIat A B ad B F, vt D E ad E H; </s>
  <s xml:id="echoid-s1132" xml:space="preserve">erit B F maior quàm B C, atque A F ma-<lb/>
<anchor type="note" xlink:label="note-0052-01a" xlink:href="note-0052-01"/>
ior quàm A C; </s>
  <s xml:id="echoid-s1133" xml:space="preserve">ergo A B ad A F eandem proportionem habebit quàm D E <lb/>ad D H; </s>
  <s xml:id="echoid-s1134" xml:space="preserve">ſed eadem A B ad minorem A C maiorem proportionem habet quàm <lb/>ad A F maiorem, ergo A B ad A C maiorem proportionem habet quàm D E <lb/>ad D H.</s>
  <s xml:id="echoid-s1135" xml:space="preserve"/>
</p>
<div xml:id="echoid-div66" type="float" level="2" n="1">
<note position="left" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">Lem. I.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1136" xml:space="preserve">Secundò ĳſdem poſitis, dico com-<lb/>
<anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/>
parando terminorum ſummas ad an-<lb/>tecedẽtes A C ad A B habere minorem <lb/>proportionem quàm D H ad D E. <lb/></s>
  <s xml:id="echoid-s1137" xml:space="preserve">Quoniam ex præcedenti caſu A B ad <lb/>A C maiorem proportionem habebat <lb/>quàm D E ad D H; </s>
  <s xml:id="echoid-s1138" xml:space="preserve">igitur inuertendo <lb/>C A ad A B minorem proportionem <lb/>habebit quàm D H ad D E.</s>
  <s xml:id="echoid-s1139" xml:space="preserve"/>
</p>
<div xml:id="echoid-div67" type="float" level="2" n="2">
  <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a">
    <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0052-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1140" xml:space="preserve">Tertiò, dico quod comparando con-<lb/>ſequentes adterminorum ſummas B C <lb/>ad C A minorem proportionem habe-<lb/>bit quàm E H ad H D; </s>
  <s xml:id="echoid-s1141" xml:space="preserve">quia (ex hy-<lb/>pothcſi) A B ad B C maiorem proportionem habet quàm D E ad E H componen-<lb/>do A C ad C B maiorem proportionem habebit quàm D H ad H E, &amp; </s>
  <s xml:id="echoid-s1142" xml:space="preserve">inuerten-<lb/>do B C ad C A minorem proportionem habebit, quàm E H ad H D.</s>
  <s xml:id="echoid-s1143" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1144" xml:space="preserve">Quartò, ĳ ſdem poſitis in quarta figura, dico quod comparando differentias <lb/>terminorum ad conſequentes A C ad C B maiorem proportionem habebit quàm <lb/>D H ad H E: </s>
  <s xml:id="echoid-s1145" xml:space="preserve">quia ex conſtructione A B ad B F eſt, vt D E ad E H, diuiden-<lb/>do A F ad F B erit vt D H ad H E; </s>
  <s xml:id="echoid-s1146" xml:space="preserve">ſed A C maior eſt quàm A F, &amp; </s>
  <s xml:id="echoid-s1147" xml:space="preserve">C B mi-<lb/>nor, quàm F B; </s>
  <s xml:id="echoid-s1148" xml:space="preserve">igitur A C ad C B maiorem proportionem habebit quàm A F ad <lb/>F B; </s>
  <s xml:id="echoid-s1149" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1150" xml:space="preserve">propterea A C ad C B maiorem proportionem habebit, quàm D H ad H E.</s>
  <s xml:id="echoid-s1151" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1152" xml:space="preserve">Quintò, dico quod è contra, comparando conſequentes ad differentias termi-<lb/>norum C B ad C A minorem proportionem habebit quàm E H ad H D. </s>
  <s xml:id="echoid-s1153" xml:space="preserve">Quia <lb/>(ex præcedenti caſu) A C ad C B maiorem proportionem habebat quàm D H ad <lb/>H E; </s>
  <s xml:id="echoid-s1154" xml:space="preserve">ergo inuertendo C B ad C A minorem proportionem habebit quàm E H <lb/>ad H D.</s>
  <s xml:id="echoid-s1155" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1156" xml:space="preserve">Sextò, dico quod comparando antecedentes ad differentias terminorum B A ad <lb/>A C minorem proportionem habebit quàm E D ad D H. </s>
  <s xml:id="echoid-s1157" xml:space="preserve">Quia ex conſtructione <lb/>
<anchor type="note" xlink:label="note-0052-02a" xlink:href="note-0052-02"/>
<pb o="15" file="0053" n="53" rhead="Conicor. Lib. V."/>
A B ad B F eſt, vt D E ad E H; </s>
  <s xml:id="echoid-s1158" xml:space="preserve">ergo A B ad A F eſt, vt E D ad D H; </s>
  <s xml:id="echoid-s1159" xml:space="preserve">ſed B A <lb/>ad maiorem C A habet minorem proportionem quàm ad F A; </s>
  <s xml:id="echoid-s1160" xml:space="preserve">igitur B A ad A C <lb/>minorem proportionem habet quàm E D ad D H.</s>
  <s xml:id="echoid-s1161" xml:space="preserve"/>
</p>
<div xml:id="echoid-div68" type="float" level="2" n="3">
<note position="left" xlink:label="note-0052-02" xlink:href="note-0052-02a" xml:space="preserve">Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1162" xml:space="preserve">Septimò, dico è contra, quod comparando differentias terminorum ad ante-<lb/>cedentes C A ad A B maiorem proportionem habebit quàm H D ad D E. </s>
  <s xml:id="echoid-s1163" xml:space="preserve">Quo-<lb/>niam, ex præcedenti caſu, B A ad A C minorem proportionem habebat quàm E <lb/>D ad D H; </s>
  <s xml:id="echoid-s1164" xml:space="preserve">igitur inuertendo C A ad A B maiorem proportionem habebit quàm <lb/>H D ad D E.</s>
  <s xml:id="echoid-s1165" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div70" type="section" level="1" n="44">
<head xml:id="echoid-head69" xml:space="preserve">LEMMA III.</head>
<p style="it">
  <s xml:id="echoid-s1166" xml:space="preserve">Si quatuor quantitates eandem rationem habuerint homologorum ſum-<lb/>mæ, vel differentiæ in eadem ratione erunt.</s>
  <s xml:id="echoid-s1167" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1168" xml:space="preserve">OStenſum enim fuit in elemen-<lb/>
<anchor type="figure" xlink:label="fig-0053-01a" xlink:href="fig-0053-01"/>
tis, quod proportionalium om-<lb/>nes antecedentes ad omnes conſequen-<lb/>tes eandem proportionem habent, <lb/>quàm vna antecedentium ad vnam <lb/>conſequentium. </s>
  <s xml:id="echoid-s1169" xml:space="preserve">Similiter oſtenſum <lb/>fuit, quod ſi totum ad totum eandem <lb/>rationem habuerit, quàm ablatum <lb/>ad ablatum, &amp; </s>
  <s xml:id="echoid-s1170" xml:space="preserve">reliquum ad reliquũ, <lb/>vt totum ad totum ſe habebit; </s>
  <s xml:id="echoid-s1171" xml:space="preserve">ſed <lb/>vno verbo homologorum ſummæ, vel <lb/>differentiæ in eadem ratione erunt <lb/>iuxtà Arabici expoſitoris compendium.</s>
  <s xml:id="echoid-s1172" xml:space="preserve"/>
</p>
<div xml:id="echoid-div70" type="float" level="2" n="1">
  <figure xlink:label="fig-0053-01" xlink:href="fig-0053-01a">
    <image file="0053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0053-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div72" type="section" level="1" n="45">
<head xml:id="echoid-head70" xml:space="preserve">LEMMA IV.</head>
<p style="it">
  <s xml:id="echoid-s1173" xml:space="preserve">Si prima A B ad ſecundam D E maiorem proportionem habuerit, <lb/>quàm tertia B C ad qnartam E H: </s>
  <s xml:id="echoid-s1174" xml:space="preserve">dico, quod comparando homologorum <lb/>ſummas A B ad D E maiorem proportionem habebit, quàm prima cum <lb/>tertia, ideſt A C ad ſecundam cum quarta, ideſt D H.</s>
  <s xml:id="echoid-s1175" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1176" xml:space="preserve">FIat B F ad E H, vt A B ad D E: </s>
  <s xml:id="echoid-s1177" xml:space="preserve">ergo A B ad D E eſt, vt A F ad D H; </s>
  <s xml:id="echoid-s1178" xml:space="preserve">ſed <lb/>
<anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/>
A F maior eſt quàm A C, igitur A F ad eandem D H maiorem proportio-<lb/>nem habet, quàm A C: </s>
  <s xml:id="echoid-s1179" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1180" xml:space="preserve">ideo A B ad D E maiorem proportionem habet, quàm <lb/>A C ad D H.</s>
  <s xml:id="echoid-s1181" xml:space="preserve"/>
</p>
<div xml:id="echoid-div72" type="float" level="2" n="1">
<note position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">Lem. 3.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1182" xml:space="preserve">Secundò ĳſdem poſitis, dico, quod tertia B C ad quartam E H minorem pro-<lb/>portionem habet quàm A C ad D H.</s>
  <s xml:id="echoid-s1183" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1184" xml:space="preserve">Fiat vt B C ad E H, ita I B ad D E, ergo C B ad E H eſt, vt C I ad H D; <lb/></s>
  <s xml:id="echoid-s1185" xml:space="preserve">
<anchor type="note" xlink:label="note-0053-02a" xlink:href="note-0053-02"/>
ſed A B maior eſt quàm I B, &amp; </s>
  <s xml:id="echoid-s1186" xml:space="preserve">ideo C A maior quàm C I; </s>
  <s xml:id="echoid-s1187" xml:space="preserve">igitur I C ad eandem
<pb o="16" file="0054" n="54" rhead="Apollonij Pergæi"/>
D H minorem proportionem habet quàm A C, &amp; </s>
  <s xml:id="echoid-s1188" xml:space="preserve">propterea B C ad E H minorem <lb/>proportionem habebit quàm A C ad D H.</s>
  <s xml:id="echoid-s1189" xml:space="preserve"/>
</p>
<div xml:id="echoid-div73" type="float" level="2" n="2">
<note position="right" xlink:label="note-0053-02" xlink:href="note-0053-02a" xml:space="preserve">Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1190" xml:space="preserve">Tertiò ĳſdem poſitis in ſexta fi-<lb/>gura, dico quod comparando homolo-<lb/>
<anchor type="figure" xlink:label="fig-0054-01a" xlink:href="fig-0054-01"/>
gorum differentias prima A B ad ſe-<lb/>cundam D E minorem proportionem <lb/>habet quàm differentia A C ad diffe-<lb/>rentiam D H.</s>
  <s xml:id="echoid-s1191" xml:space="preserve"/>
</p>
<div xml:id="echoid-div74" type="float" level="2" n="3">
  <figure xlink:label="fig-0054-01" xlink:href="fig-0054-01a">
    <image file="0054-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0054-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1192" xml:space="preserve">Fiat B F ad E H, vt A B ad D <lb/>E, ergo A F ad D H eſt vt A B ad <lb/>
<anchor type="note" xlink:label="note-0054-01a" xlink:href="note-0054-01"/>
D E, ſed A F minor eſt quam A C, <lb/>ergo A F ad eandem D H minorem <lb/>proportionem habet quàm A C: </s>
  <s xml:id="echoid-s1193" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1194" xml:space="preserve"><lb/>propterea A B ad D E minorem pro-<lb/>portionem habet quàm A C ad D H.</s>
  <s xml:id="echoid-s1195" xml:space="preserve"/>
</p>
<div xml:id="echoid-div75" type="float" level="2" n="4">
<note position="left" xlink:label="note-0054-01" xlink:href="note-0054-01a" xml:space="preserve">Lem.3.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1196" xml:space="preserve">Quartò, dico, quod tertia C B ad quartam H E minorem proportionem habet <lb/>
<anchor type="note" xlink:label="note-0054-02a" xlink:href="note-0054-02"/>
quàm differentia A C ad differentiam D H. </s>
  <s xml:id="echoid-s1197" xml:space="preserve">Quoniam ex conſtructione A B ad <lb/>D E eſt vt F B ad H E, erit F B ad H E, vt A F ad D H; </s>
  <s xml:id="echoid-s1198" xml:space="preserve">ſed C B minor <lb/>eſt quàm F B, atque A C maior quàm A F, &amp; </s>
  <s xml:id="echoid-s1199" xml:space="preserve">A F ad eandem D H minorem <lb/>proportionem habet quàm A C; </s>
  <s xml:id="echoid-s1200" xml:space="preserve">igitur C B ad H E eo magis habebit minorem <lb/>proportionem quàm A C ad D H quæ erant oſtendenda.</s>
  <s xml:id="echoid-s1201" xml:space="preserve"/>
</p>
<div xml:id="echoid-div76" type="float" level="2" n="5">
<note position="left" xlink:label="note-0054-02" xlink:href="note-0054-02a" xml:space="preserve">Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div78" type="section" level="1" n="46">
<head xml:id="echoid-head71" xml:space="preserve">SECTIO TERTIA</head>
<head xml:id="echoid-head72" xml:space="preserve">Continens VIII. IX. X. Propoſ. Apollonij.</head>
<p>
  <s xml:id="echoid-s1202" xml:space="preserve">SI menſura fuerit maior comparata, dummodo in ellipſi minor <lb/>ſit medietate axis tranſuerſi, tunc minimus ramorum in ſe-<lb/>ctionibus eſt, cuius potentialis abſcindit à menſura verſus origi-<lb/>nem in parabola (8) lineam æqualem comparatæ, in hyperbo-<lb/>la verò (9) &amp; </s>
  <s xml:id="echoid-s1203" xml:space="preserve">in ellipſi (10.) </s>
  <s xml:id="echoid-s1204" xml:space="preserve">lineam, cuius inuerſæ proportio <lb/>ad illam eſt, vt proportio figuræ &amp; </s>
  <s xml:id="echoid-s1205" xml:space="preserve">reliqui rami, quo accedunt <lb/>ad minimum ſunt minores remotioribus; </s>
  <s xml:id="echoid-s1206" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1207" xml:space="preserve">quadratum minimæ <lb/>minus eſt quadrato cuiuslibet rami aſſignati in parabola quidem <lb/>(8) quadrato exceſſus ſuarum abſciſſarum, &amp; </s>
  <s xml:id="echoid-s1208" xml:space="preserve">in hyperbola (9) <lb/>&amp; </s>
  <s xml:id="echoid-s1209" xml:space="preserve">ellipſi (10.) </s>
  <s xml:id="echoid-s1210" xml:space="preserve">exemplari applicato ad exceſſum ſuarum inuer-<lb/>ſarum.</s>
  <s xml:id="echoid-s1211" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1212" xml:space="preserve">SIt itaque ſectio A B C, &amp; </s>
  <s xml:id="echoid-s1213" xml:space="preserve">menſura I C, inclinatus, ſiue tranſuerſa E C, <lb/>
<anchor type="note" xlink:label="note-0054-03a" xlink:href="note-0054-03"/>
dimidium erecti C G, centrum F, origo I, &amp; </s>
  <s xml:id="echoid-s1214" xml:space="preserve">I H in parabola ſit equa-<lb/>lis C G, &amp; </s>
  <s xml:id="echoid-s1215" xml:space="preserve">in hyperbola, &amp; </s>
  <s xml:id="echoid-s1216" xml:space="preserve">ellipſi F H ad H I ſit, vt F C dimidium incli-<lb/>nati, ſeu tranſuerſæ ad C G, dimidium erecti, &amp; </s>
  <s xml:id="echoid-s1217" xml:space="preserve">educta ex H perpendi-<lb/>culari H N, &amp; </s>
  <s xml:id="echoid-s1218" xml:space="preserve">coniuncta recta N I; </s>
  <s xml:id="echoid-s1219" xml:space="preserve">Dico N I minimum eſſe ramorum
<pb o="17" file="0055" n="55" rhead="Conicor. Lib. V."/>
egredientium ex I, &amp; </s>
  <s xml:id="echoid-s1220" xml:space="preserve">inſuper, propinquiores illi minores eſſe remotiori-<lb/>bus ramis ex vtraque parte, &amp; </s>
  <s xml:id="echoid-s1221" xml:space="preserve">quod quadratum IN minus eſt quadrato <lb/>MI (exempli gratia) in parabola quadrato QH, in hyperbola, &amp; </s>
  <s xml:id="echoid-s1222" xml:space="preserve">ellipſi <lb/>exemplari applicato ad QH. </s>
  <s xml:id="echoid-s1223" xml:space="preserve">Quoniam quadratum HN in parabola ęqua-<lb/>
<anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/>
le eſt HI, nempe C G in HC bis (11. </s>
  <s xml:id="echoid-s1224" xml:space="preserve">ex primo) erit quadratum IN ęqua-<lb/>le IH in HC bis cum quadrato HI; </s>
  <s xml:id="echoid-s1225" xml:space="preserve">at ꝗuadratum M Q æquale eſt HI <lb/>
<anchor type="figure" xlink:label="fig-0055-01a" xlink:href="fig-0055-01"/>
in QC bis (11. </s>
  <s xml:id="echoid-s1226" xml:space="preserve">ex primo) <lb/>igitur quadratum MI ęqua-<lb/>le eſt IH in QC bis cum <lb/>quadrato IQ; </s>
  <s xml:id="echoid-s1227" xml:space="preserve">hoc autem <lb/>
<anchor type="note" xlink:label="note-0055-02a" xlink:href="note-0055-02"/>
eſt ęquale duobus quadra-<lb/>tis IH, HQ, &amp; </s>
  <s xml:id="echoid-s1228" xml:space="preserve">IH in H <lb/>Q bis; </s>
  <s xml:id="echoid-s1229" xml:space="preserve">igitur quadratum I <lb/>M æquale eſt IH in HC <lb/>bis cum quadrato IH, quę <lb/>ſunt æqualia quadrato NI <lb/>vnà cum quadrato HQ. <lb/></s>
  <s xml:id="echoid-s1230" xml:space="preserve">Quadratum igitur MI ex-<lb/>cedit quadratum NI qua-<lb/>drato HQ. </s>
  <s xml:id="echoid-s1231" xml:space="preserve">Et conſtat quo-<lb/>que, quadratum I L exce-<lb/>dere quadratum I N quadrato P H; </s>
  <s xml:id="echoid-s1232" xml:space="preserve">atque P H maior eſt, quàm Q H, <lb/>ergo I L maior eſt, quàm I M, &amp; </s>
  <s xml:id="echoid-s1233" xml:space="preserve">I M, quàm N I. </s>
  <s xml:id="echoid-s1234" xml:space="preserve">Ponamus iam B I <lb/>perpendicularem ſuper C I, ergo quadratum B I ęquale eſt I C <lb/>in I H bis (11. </s>
  <s xml:id="echoid-s1235" xml:space="preserve">ex primo); </s>
  <s xml:id="echoid-s1236" xml:space="preserve">quadratum igitur I N minus eſt <lb/>
<anchor type="note" xlink:label="note-0055-03a" xlink:href="note-0055-03"/>
quàm quadratum B I quadrato I H. </s>
  <s xml:id="echoid-s1237" xml:space="preserve">Et quia quadra-<lb/>
<anchor type="note" xlink:label="note-0055-04a" xlink:href="note-0055-04"/>
tum O R ęquale eſt C R in I H bis excedet qua-<lb/>dratum I N (quod eſt ęquale quadrato I H, <lb/>&amp; </s>
  <s xml:id="echoid-s1238" xml:space="preserve">I H in H C bis) duobus quadratis <lb/>HI, IR, &amp; </s>
  <s xml:id="echoid-s1239" xml:space="preserve">IH in IR bis, nem-<lb/>pè quadrato R H; </s>
  <s xml:id="echoid-s1240" xml:space="preserve">atquè ſic <lb/>conſtat, quadratum.</s>
  <s xml:id="echoid-s1241" xml:space="preserve"/>
</p>
<div xml:id="echoid-div78" type="float" level="2" n="1">
<note position="right" xlink:label="note-0054-03" xlink:href="note-0054-03a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0055-01" xlink:href="fig-0055-01a">
    <image file="0055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0055-01"/>
  </figure>
<note position="left" xlink:label="note-0055-02" xlink:href="note-0055-02a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0055-03" xlink:href="note-0055-03a" xml:space="preserve">e</note>
<note position="left" xlink:label="note-0055-04" xlink:href="note-0055-04a" xml:space="preserve">f</note>
</div>
<p>
  <s xml:id="echoid-s1242" xml:space="preserve">A I excedere <lb/>quadratum I N quadrato D H; </s>
  <s xml:id="echoid-s1243" xml:space="preserve">eſtque <lb/>D H maior, quàm R H, igitur <lb/>I A maior eſt, quàm I O, <lb/>&amp; </s>
  <s xml:id="echoid-s1244" xml:space="preserve">I O quàm I N. </s>
  <s xml:id="echoid-s1245" xml:space="preserve">Et <lb/>hoc propofitum <lb/>fuerat.</s>
  <s xml:id="echoid-s1246" xml:space="preserve"/>
</p>
<pb o="18" file="0056" n="56" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div80" type="section" level="1" n="47">
<head xml:id="echoid-head73" xml:space="preserve">PROPOSITIO IX. &amp; X.</head>
<p>
  <s xml:id="echoid-s1247" xml:space="preserve">AT in hyper-<lb/>
<anchor type="note" xlink:label="note-0056-01a" xlink:href="note-0056-01"/>
<anchor type="figure" xlink:label="fig-0056-01a" xlink:href="fig-0056-01"/>
bola (10.) <lb/></s>
  <s xml:id="echoid-s1248" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1249" xml:space="preserve">ellipſi educa-<lb/>mus rectas lineas, <lb/>G F quidem ſecã-<lb/>tem A D in a, &amp; </s>
  <s xml:id="echoid-s1250" xml:space="preserve"><lb/>N H occurrẽtem <lb/>F G in S, &amp; </s>
  <s xml:id="echoid-s1251" xml:space="preserve">I S <lb/>ſecantem C G in <lb/>T, pariterque M <lb/>Q ſecantem F G <lb/>in m, &amp; </s>
  <s xml:id="echoid-s1252" xml:space="preserve">I T in X, <lb/>&amp; </s>
  <s xml:id="echoid-s1253" xml:space="preserve">ex punctis m, S, <lb/>x educamus inter <lb/>N S, M X rectas <lb/>m y, X n, S Z pa-<lb/>rallelas ipſi C I. </s>
  <s xml:id="echoid-s1254" xml:space="preserve"><lb/>Et quia C F ad C <lb/>G, nempe F H ad <lb/>H S poſita eſt, vt <lb/>F H ad H I erit H I æqualis H S; </s>
  <s xml:id="echoid-s1255" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0056-02a" xlink:href="note-0056-02"/>
<anchor type="figure" xlink:label="fig-0056-02a" xlink:href="fig-0056-02"/>
quadratum igitur I H eſt æquale <lb/>duplo trianguli I H S, &amp; </s>
  <s xml:id="echoid-s1256" xml:space="preserve">quadra-<lb/>tum N H ęquale eſt duplo trape-<lb/>zij H G; </s>
  <s xml:id="echoid-s1257" xml:space="preserve">quare quadratum N I <lb/>
<anchor type="note" xlink:label="note-0056-03a" xlink:href="note-0056-03"/>
æquale eſt duplo trapezij I G; <lb/></s>
  <s xml:id="echoid-s1258" xml:space="preserve">ſimiliter quadratum I Q ęquale eſt <lb/>
<anchor type="note" xlink:label="note-0056-04a" xlink:href="note-0056-04"/>
duplo trianguli I Q X, &amp; </s>
  <s xml:id="echoid-s1259" xml:space="preserve">quadra-<lb/>tum M Q eſt æquale duplo trape-<lb/>zij Q G; </s>
  <s xml:id="echoid-s1260" xml:space="preserve">itaque quadratum ex I M <lb/>æquale eſt duplo trapezij I G cum <lb/>duplo trianguli m S X, quod eſt æ-<lb/>quale plano m n: </s>
  <s xml:id="echoid-s1261" xml:space="preserve">Et C F ad C G, <lb/>nempe proportio figuræ eſt, vt S Z, <lb/>nempe Z X ad Z m (&amp; </s>
  <s xml:id="echoid-s1262" xml:space="preserve">hoc quidem <lb/>propter ſimilitudinem triangulorũ) <lb/>quare comparãdo priores ad ſum-<lb/>
<anchor type="note" xlink:label="note-0056-05a" xlink:href="note-0056-05"/>
mas terminorum in hyperbola, &amp; </s>
  <s xml:id="echoid-s1263" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0056-06a" xlink:href="note-0056-06"/>
ad eorundem differentias in ellipſi <lb/>fiet X Z (quæ eſt æqualis ipſi X n) <lb/>ad X m, vt proportio inclinati, ſiue <lb/>
<anchor type="note" xlink:label="note-0056-07a" xlink:href="note-0056-07"/>
tranſuerſæ ad latitudinem figuræ <lb/>comparatæ; </s>
  <s xml:id="echoid-s1264" xml:space="preserve">igitur planum m n eſt exemplar, eſtque applicatum ad X n, <lb/>
<anchor type="note" xlink:label="note-0056-08a" xlink:href="note-0056-08"/>
<pb o="19" file="0057" n="57" rhead="Conicor. Lib. V."/>
nempe ad QH. </s>
  <s xml:id="echoid-s1265" xml:space="preserve">Eodem modo conſtat, quod quadratum IL excedit qua-<lb/>dratum I N quantitate exemplaris applicati ad H P, &amp; </s>
  <s xml:id="echoid-s1266" xml:space="preserve">quod quadratum <lb/>B I excedit quadratum I N exemplari applicato ad I H, &amp; </s>
  <s xml:id="echoid-s1267" xml:space="preserve">quod quadra-<lb/>tum I O excedit quadratum I N exemplari applicato ad R H (eo quod <lb/>
<anchor type="note" xlink:label="note-0057-01a" xlink:href="note-0057-01"/>
quadratum R I æquale eſt duplo trianguli R V I, &amp; </s>
  <s xml:id="echoid-s1268" xml:space="preserve">quadratum O R ęqua-<lb/>
<anchor type="note" xlink:label="note-0057-02a" xlink:href="note-0057-02"/>
le eſt duplo trapezij R G, at in ellipſi quando O R cadit infra centrum F <lb/>æquale eſt duplo trapezij R K; </s>
  <s xml:id="echoid-s1269" xml:space="preserve">quadratum igitur O I in ellipſi æquale eſt <lb/>
<anchor type="note" xlink:label="note-0057-03a" xlink:href="note-0057-03"/>
duplo trianguli K E F, quod eſt æquale F C G cum duplo trapezij V F, <lb/>
<anchor type="note" xlink:label="note-0057-04a" xlink:href="note-0057-04"/>
igitur quadratum O I in hyperbola, &amp; </s>
  <s xml:id="echoid-s1270" xml:space="preserve">ellipſi excedit duplum trapezij I G <lb/>(quod eſt æquale quadrato N I) duplo trianguli V S<emph style="sub">0</emph>, quod eſt æquale <lb/>
<anchor type="note" xlink:label="note-0057-05a" xlink:href="note-0057-05"/>
exemplari applicato ad R H: </s>
  <s xml:id="echoid-s1271" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1272" xml:space="preserve">ſimiliter patet, quod quadratum A I ex-<lb/>cedit quadratum N I exemplari applicato ad D H, eſtque D H maior <lb/>quàm R H, &amp; </s>
  <s xml:id="echoid-s1273" xml:space="preserve">R H maior quàm I H; </s>
  <s xml:id="echoid-s1274" xml:space="preserve">quare A I maior eſt, quàm O I, &amp; </s>
  <s xml:id="echoid-s1275" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0057-06a" xlink:href="note-0057-06"/>
O I maior, quàm B I, &amp; </s>
  <s xml:id="echoid-s1276" xml:space="preserve">B I, quàm N I, &amp; </s>
  <s xml:id="echoid-s1277" xml:space="preserve">quodlibet horum duorum ex-<lb/>cedit N I poteſtate plano iam dicto, &amp; </s>
  <s xml:id="echoid-s1278" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s1279" xml:space="preserve"/>
</p>
<div xml:id="echoid-div80" type="float" level="2" n="1">
<note position="right" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve">g</note>
  <figure xlink:label="fig-0056-01" xlink:href="fig-0056-01a">
    <image file="0056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0056-01"/>
  </figure>
<note position="right" xlink:label="note-0056-02" xlink:href="note-0056-02a" xml:space="preserve">h</note>
  <figure xlink:label="fig-0056-02" xlink:href="fig-0056-02a">
    <image file="0056-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0056-02"/>
  </figure>
<note position="left" xlink:label="note-0056-03" xlink:href="note-0056-03a" xml:space="preserve">Prop. I. h.</note>
<note position="right" xlink:label="note-0056-04" xlink:href="note-0056-04a" xml:space="preserve">i</note>
<note position="left" xlink:label="note-0056-05" xlink:href="note-0056-05a" xml:space="preserve">Lem. 1. h.</note>
<note position="right" xlink:label="note-0056-06" xlink:href="note-0056-06a" xml:space="preserve">k</note>
<note position="right" xlink:label="note-0056-07" xlink:href="note-0056-07a" xml:space="preserve">l</note>
<note position="left" xlink:label="note-0056-08" xlink:href="note-0056-08a" xml:space="preserve">Def 9.</note>
<note position="left" xlink:label="note-0057-01" xlink:href="note-0057-01a" xml:space="preserve">m</note>
<note position="right" xlink:label="note-0057-02" xlink:href="note-0057-02a" xml:space="preserve">Prop. 1. h.</note>
<note position="right" xlink:label="note-0057-03" xlink:href="note-0057-03a" xml:space="preserve">Prop. 3. h.</note>
<note position="left" xlink:label="note-0057-04" xlink:href="note-0057-04a" xml:space="preserve">n</note>
<note position="left" xlink:label="note-0057-05" xlink:href="note-0057-05a" xml:space="preserve">o</note>
<note position="left" xlink:label="note-0057-06" xlink:href="note-0057-06a" xml:space="preserve">p</note>
</div>
</div>
<div xml:id="echoid-div82" type="section" level="1" n="48">
<head xml:id="echoid-head74" xml:space="preserve">Notæ in Propoſitionem VIII.</head>
<p>
  <s xml:id="echoid-s1280" xml:space="preserve">S I menſura fuerit maior comparata, dummodò in ellipſi ſit portio tran-<lb/>
<anchor type="note" xlink:label="note-0057-07a" xlink:href="note-0057-07"/>
ſuerſæ, non maior medietate ipſius, tunc minimus, &amp;</s>
  <s xml:id="echoid-s1281" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1282" xml:space="preserve">Sic puto le-<lb/>gendum: </s>
  <s xml:id="echoid-s1283" xml:space="preserve">Si menſura fuerit maior comparata, dummodo in ellipſi minor ſit me-<lb/>dietate axis tranſuerſi, tunc minimus, &amp;</s>
  <s xml:id="echoid-s1284" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1285" xml:space="preserve">Nam ſi menſura ſumi poſſet æqua-<lb/>lis ſemitranſuerſo, tunc qui-<lb/>
<anchor type="figure" xlink:label="fig-0057-01a" xlink:href="fig-0057-01"/>
dem origo eßet in centro elli-<lb/>pſis, quare undecima propo-<lb/>ſitio huius eſſet ſuperflua, in <lb/>qua ſupponitur origo in ipſo-<lb/>met centro ellipſis. </s>
  <s xml:id="echoid-s1286" xml:space="preserve">Animad-<lb/>uertendum eſt quod in hac <lb/>propoſitione menſura neceſſa-<lb/>riò ſumi debet in axe maiori <lb/>ellipſis; </s>
  <s xml:id="echoid-s1287" xml:space="preserve">quandoquidem menſu-<lb/>ra I C ponitur maior, quàm <lb/>C G, &amp; </s>
  <s xml:id="echoid-s1288" xml:space="preserve">C F maior quàm C I, <lb/>ergo C F maior eſt quàm C G, <lb/>&amp; </s>
  <s xml:id="echoid-s1289" xml:space="preserve">illius duplum ſcilicet axis <lb/>E C maior erit duplo huius, ſed ut E C ad duplum C G, ita eſt quadratum E C <lb/>ad quadratum Recti axis eiuſdem ellipſis: </s>
  <s xml:id="echoid-s1290" xml:space="preserve">ergo E C eſt maior duorum axium <lb/>ellipſis A B C.</s>
  <s xml:id="echoid-s1291" xml:space="preserve"/>
</p>
<div xml:id="echoid-div82" type="float" level="2" n="1">
<note position="left" xlink:label="note-0057-07" xlink:href="note-0057-07a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0057-01" xlink:href="fig-0057-01a">
    <image file="0057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0057-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s1292" xml:space="preserve">Et educta ex H perpendiculari H N, &amp;</s>
  <s xml:id="echoid-s1293" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1294" xml:space="preserve">Ideſt ex H educta H N per-<lb/>
<anchor type="note" xlink:label="note-0057-08a" xlink:href="note-0057-08"/>
pendiculari ad axim C I, quæ ſecet ſectionem in N, &amp; </s>
  <s xml:id="echoid-s1295" xml:space="preserve">iuncta recta N I, pari-<lb/>terque ductis reliquis ramis I M, I L, I B, I A, atque ab eorum terminis ad <lb/>axim extenſis perpendicularibus, vt in propoſitionibus quarta, quinta, ſexta <lb/>factum eſt.</s>
  <s xml:id="echoid-s1296" xml:space="preserve"/>
</p>
<div xml:id="echoid-div83" type="float" level="2" n="2">
<note position="left" xlink:label="note-0057-08" xlink:href="note-0057-08a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s1297" xml:space="preserve">Quadratum H N in parabola æquale eſt H I nempè C G in H C bis <lb/>
<anchor type="note" xlink:label="note-0057-09a" xlink:href="note-0057-09"/>
(prima ex quinto) &amp;</s>
  <s xml:id="echoid-s1298" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1299" xml:space="preserve">Hoc deduci non poteſt ex prima propoſitione huius libri,
<pb o="20" file="0058" n="58" rhead="Apollonij Pergæi"/>
ſed potius ex vndecima libri primi; <lb/></s>
  <s xml:id="echoid-s1300" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/>
eſt enim quadratum H N æquale re-<lb/>ctangulo contento ſub abſciſſa H C, <lb/>&amp; </s>
  <s xml:id="echoid-s1301" xml:space="preserve">ſub latere recto, eſtque rectangu-<lb/>lum ſub H C, &amp; </s>
  <s xml:id="echoid-s1302" xml:space="preserve">ſub ſemierecto C G <lb/>ſemiſsis illius; </s>
  <s xml:id="echoid-s1303" xml:space="preserve">igitur quadratum H <lb/>N æquale eſt duplo rectanguli H C G.</s>
  <s xml:id="echoid-s1304" xml:space="preserve"/>
</p>
<div xml:id="echoid-div84" type="float" level="2" n="3">
<note position="left" xlink:label="note-0057-09" xlink:href="note-0057-09a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a">
    <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0058-01"/>
  </figure>
</div>
<note position="right" xml:space="preserve">d</note>
<p style="it">
  <s xml:id="echoid-s1305" xml:space="preserve">Hoc autem eſt æquale duobus <lb/>quadratis I H, H Q, &amp; </s>
  <s xml:id="echoid-s1306" xml:space="preserve">I H in H <lb/>Q bis, &amp;</s>
  <s xml:id="echoid-s1307" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1308" xml:space="preserve">Poſt hæc verba ſubiun-<lb/>go claritatis gratia, atque C H in H <lb/>I bis æquale eſt duplo C Q in H I <lb/>vna cum duplo Q H in H I.</s>
  <s xml:id="echoid-s1309" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1310" xml:space="preserve">Ergo quadratum B I æquale eſt <lb/>
<anchor type="note" xlink:label="note-0058-02a" xlink:href="note-0058-02"/>
I C in I H bis, &amp;</s>
  <s xml:id="echoid-s1311" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1312" xml:space="preserve">Hìc pariter, vt <lb/>clarior reddatur demõſtratio, ſubiun-<lb/>go, ſcilicet duplo rectãguli C H I vna <lb/>cum duplo quadrati H I; </s>
  <s xml:id="echoid-s1313" xml:space="preserve">erat autem <lb/>quadratum N I æquale duplo rectan-<lb/>guli C H I, &amp; </s>
  <s xml:id="echoid-s1314" xml:space="preserve">vnico quadrato H I, <lb/>ergo, &amp;</s>
  <s xml:id="echoid-s1315" xml:space="preserve">c.</s>
  <s xml:id="echoid-s1316" xml:space="preserve"/>
</p>
<div xml:id="echoid-div85" type="float" level="2" n="4">
<note position="right" xlink:label="note-0058-02" xlink:href="note-0058-02a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s1317" xml:space="preserve">Et quia quadratum OR æqua-<lb/>
<anchor type="note" xlink:label="note-0058-03a" xlink:href="note-0058-03"/>
le eſt C R in I H bis, &amp;</s>
  <s xml:id="echoid-s1318" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s1319" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0058-02a" xlink:href="fig-0058-02"/>
Subiungo hanc declarationem. <lb/></s>
  <s xml:id="echoid-s1320" xml:space="preserve">Scilicet duplo rectanguli C H <lb/>I, &amp; </s>
  <s xml:id="echoid-s1321" xml:space="preserve">duplo quadrati H I cum <lb/>duplo rectanguli R I H. </s>
  <s xml:id="echoid-s1322" xml:space="preserve">Qua-<lb/>re quadratum I O æquale eſt <lb/>quadrato R I, duplo quadrati <lb/>H I, duplo rectanguli R I H, <lb/>&amp; </s>
  <s xml:id="echoid-s1323" xml:space="preserve">duplo rectanguli C H I: </s>
  <s xml:id="echoid-s1324" xml:space="preserve">ſed <lb/>quadratũ H R æquale eſt qua-<lb/>drato R I, quadrato I H cum <lb/>duplo rectanguli R I H. </s>
  <s xml:id="echoid-s1325" xml:space="preserve">Ergo <lb/>quadratum I O æquale eſt qua-<lb/>drato H R, quadrato H I cum duplo rectanguli C H I; </s>
  <s xml:id="echoid-s1326" xml:space="preserve">erat autem prius qua-<lb/>dratum I N æquale quadrato I H cum duplo rectanguli C H I. </s>
  <s xml:id="echoid-s1327" xml:space="preserve">Igitur exceßus <lb/>quadrati I O ſupra quadratum I N eſt quadratum H R.</s>
  <s xml:id="echoid-s1328" xml:space="preserve"/>
</p>
<div xml:id="echoid-div86" type="float" level="2" n="5">
<note position="right" xlink:label="note-0058-03" xlink:href="note-0058-03a" xml:space="preserve">f</note>
  <figure xlink:label="fig-0058-02" xlink:href="fig-0058-02a">
    <image file="0058-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0058-02"/>
  </figure>
</div>
<pb o="21" file="0059" n="59" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div88" type="section" level="1" n="49">
<head xml:id="echoid-head75" xml:space="preserve">Notæ in Propoſitionem IX. &amp; X.</head>
<p>
  <s xml:id="echoid-s1329" xml:space="preserve">AT in hyper-<lb/>
<anchor type="note" xlink:label="note-0059-01a" xlink:href="note-0059-01"/>
<anchor type="figure" xlink:label="fig-0059-01a" xlink:href="fig-0059-01"/>
bola, &amp; </s>
  <s xml:id="echoid-s1330" xml:space="preserve">el-<lb/>lipſi educamus G <lb/>F ad a ex A D, &amp; </s>
  <s xml:id="echoid-s1331" xml:space="preserve"><lb/>H N ad s ex F G, <lb/>&amp; </s>
  <s xml:id="echoid-s1332" xml:space="preserve">I S ad T ex C <lb/>G, ſi educta oc-<lb/>currat ſectioni ad <lb/>A, &amp; </s>
  <s xml:id="echoid-s1333" xml:space="preserve">M Q poſita <lb/>ad m ex a, F G, <lb/>&amp; </s>
  <s xml:id="echoid-s1334" xml:space="preserve">X in I T, &amp; </s>
  <s xml:id="echoid-s1335" xml:space="preserve">ex <lb/>m, S X, m y, x n, <lb/>S Z inter N S, M <lb/>X, &amp;</s>
  <s xml:id="echoid-s1336" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1337" xml:space="preserve">Eadẽ phraſi <lb/>inconcinna exponi-<lb/>tur vniuerſa con-<lb/>ſtructio buius pro-<lb/>poſitionis, ideo cu-<lb/>raui eam reddere <lb/>clariorem, dicendo; <lb/></s>
  <s xml:id="echoid-s1338" xml:space="preserve">Educamus rectas lineas G F quidem ſec antem A D in a, &amp;</s>
  <s xml:id="echoid-s1339" xml:space="preserve">c.</s>
  <s xml:id="echoid-s1340" xml:space="preserve"/>
</p>
<div xml:id="echoid-div88" type="float" level="2" n="1">
<note position="left" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">g</note>
  <figure xlink:label="fig-0059-01" xlink:href="fig-0059-01a">
    <image file="0059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0059-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s1341" xml:space="preserve">Quadratum igitur I H eſt æquale triangulo I H S, &amp;</s>
  <s xml:id="echoid-s1342" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1343" xml:space="preserve">Qaia nimirum. <lb/></s>
  <s xml:id="echoid-s1344" xml:space="preserve">
<anchor type="note" xlink:label="note-0059-02a" xlink:href="note-0059-02"/>
Quadratum I H eſt æquale duplo iſoſcelei, &amp; </s>
  <s xml:id="echoid-s1345" xml:space="preserve">rectanguli trianguli I H S.</s>
  <s xml:id="echoid-s1346" xml:space="preserve"/>
</p>
<div xml:id="echoid-div89" type="float" level="2" n="2">
<note position="left" xlink:label="note-0059-02" xlink:href="note-0059-02a" xml:space="preserve">h</note>
</div>
<p>
  <s xml:id="echoid-s1347" xml:space="preserve">Et ſimiliter quadratum I Q æquale eſt duplo trianguli I Q X, &amp;</s>
  <s xml:id="echoid-s1348" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1349" xml:space="preserve">Sci-<lb/>
<anchor type="note" xlink:label="note-0059-03a" xlink:href="note-0059-03"/>
licet duplo trapezĳ I S m Q cum duplo trianguli S m X.</s>
  <s xml:id="echoid-s1350" xml:space="preserve"/>
</p>
<div xml:id="echoid-div90" type="float" level="2" n="3">
<note position="left" xlink:label="note-0059-03" xlink:href="note-0059-03a" xml:space="preserve">i</note>
</div>
<p>
  <s xml:id="echoid-s1351" xml:space="preserve">Et hoc quidem propter ſimilitudinem triangulorum, at componendo <lb/>
<anchor type="note" xlink:label="note-0059-04a" xlink:href="note-0059-04"/>
proportionem in hyperbola, tum inuertendo, &amp; </s>
  <s xml:id="echoid-s1352" xml:space="preserve">reflectendo in ellipſi <lb/>fit, &amp;</s>
  <s xml:id="echoid-s1353" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1354" xml:space="preserve">Huiuſmodi verba inepta ad concluſionem inferendam commutaui di-<lb/>cendo; </s>
  <s xml:id="echoid-s1355" xml:space="preserve">Quare comparando priores ad ſummas terminorum in hyperbola, &amp; </s>
  <s xml:id="echoid-s1356" xml:space="preserve">ad <lb/>eorum differentias in ellipſi fit, &amp;</s>
  <s xml:id="echoid-s1357" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1358" xml:space="preserve">Quæ quidem expeditè (vt in primo præce-<lb/>cedentium Lemmatum oſtenſum eſt) progreſſum declarant.</s>
  <s xml:id="echoid-s1359" xml:space="preserve"/>
</p>
<div xml:id="echoid-div91" type="float" level="2" n="4">
<note position="left" xlink:label="note-0059-04" xlink:href="note-0059-04a" xml:space="preserve">k</note>
</div>
<note position="left" xml:space="preserve">l</note>
<p>
  <s xml:id="echoid-s1360" xml:space="preserve">Vt proportio inclinati, ſiue tranſuerſæ ad latitudinem figuræ compara-<lb/>tæ; </s>
  <s xml:id="echoid-s1361" xml:space="preserve">igitur planum m n eſt exemplar, &amp;</s>
  <s xml:id="echoid-s1362" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1363" xml:space="preserve">Subiungo: </s>
  <s xml:id="echoid-s1364" xml:space="preserve">nam, vt dictum eſt in <lb/>quinta, &amp; </s>
  <s xml:id="echoid-s1365" xml:space="preserve">ſexta huius, poteſt hìc demonſtrari, quod figura m n ſimilis eſt ei, <lb/>quæ continetur latere tranſuerſo E C, &amp; </s>
  <s xml:id="echoid-s1366" xml:space="preserve">ſumma in hyperbola, &amp; </s>
  <s xml:id="echoid-s1367" xml:space="preserve">differentia in <lb/>ellipſi laterum tranſuerſi, &amp; </s>
  <s xml:id="echoid-s1368" xml:space="preserve">recti iuxta definitiones octauam, &amp; </s>
  <s xml:id="echoid-s1369" xml:space="preserve">nonam.</s>
  <s xml:id="echoid-s1370" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1371" xml:space="preserve">Quadratum R I æquale eſt duplo trianguli R V I, &amp; </s>
  <s xml:id="echoid-s1372" xml:space="preserve">quadratum O R in <lb/>
<anchor type="note" xlink:label="note-0059-06a" xlink:href="note-0059-06"/>
hyperbola æquale eſt duplo trapezij R G, &amp; </s>
  <s xml:id="echoid-s1373" xml:space="preserve">in ellipſi æquale eſt duplo <lb/>trapezij R K, &amp;</s>
  <s xml:id="echoid-s1374" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1375" xml:space="preserve">Legendum puto quadratum R I æquale eſt duplo trianguli <lb/>
<anchor type="note" xlink:label="note-0059-07a" xlink:href="note-0059-07"/>
R V I, &amp; </s>
  <s xml:id="echoid-s1376" xml:space="preserve">quadratum O R æquale eſt duplo trapezĳ R G, at in ellipſi quando <lb/>O R cadit infra centrum F æquale eſt duplo trapezĳ R K, &amp;</s>
  <s xml:id="echoid-s1377" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1378" xml:space="preserve">Deindè <lb/>quum triangulum R V I ſimile ſit triangulo I H S propter parallelas V R, S <lb/>H; </s>
  <s xml:id="echoid-s1379" xml:space="preserve">ideò triangulum R V I erit quoque iſoſceleum, &amp; </s>
  <s xml:id="echoid-s1380" xml:space="preserve">rectangulum. </s>
  <s xml:id="echoid-s1381" xml:space="preserve">Poſtea qua-
<pb o="22" file="0060" n="60" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0060-01a" xlink:href="fig-0060-01"/>
dratum O R æquale eſt duplo trapezĳ R C G O; <lb/></s>
  <s xml:id="echoid-s1382" xml:space="preserve">
<anchor type="note" xlink:label="note-0060-01a" xlink:href="note-0060-01"/>
Sed in ellipſi quando ordinata O R cadit infra <lb/>centrum F, tunc quidem ducta E K parallela <lb/>C G, quæ ſecet G F in K, erit quadratum O R <lb/>æquale duplo differentiæ triangulorum F R <emph style="sub">o</emph>, &amp; </s>
  <s xml:id="echoid-s1383" xml:space="preserve"><lb/>F C G, ſeu F E K, quæ differentia æqualis eſt <lb/>trapezio R E K <emph style="sub">o</emph>, ideoque duo quadrata ex I R, <lb/>&amp; </s>
  <s xml:id="echoid-s1384" xml:space="preserve">ex R O, ideſt quadratum ex I O æquale erit <lb/>triangulis F C G, &amp; </s>
  <s xml:id="echoid-s1385" xml:space="preserve">I R V bis ſumptis dempto <lb/>duplo trianguli F R <emph style="sub">o</emph>.</s>
  <s xml:id="echoid-s1386" xml:space="preserve"/>
</p>
<div xml:id="echoid-div92" type="float" level="2" n="5">
<note position="left" xlink:label="note-0059-06" xlink:href="note-0059-06a" xml:space="preserve">m</note>
<note position="right" xlink:label="note-0059-07" xlink:href="note-0059-07a" xml:space="preserve">1. huius.</note>
  <figure xlink:label="fig-0060-01" xlink:href="fig-0060-01a">
    <image file="0060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0060-01"/>
  </figure>
<note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve">Prop. 1. h.</note>
</div>
<p>
  <s xml:id="echoid-s1387" xml:space="preserve">Quod eſt ęquale triangulo F C G cum <lb/>
<anchor type="note" xlink:label="note-0060-02a" xlink:href="note-0060-02"/>
duplo trapezij V F, &amp;</s>
  <s xml:id="echoid-s1388" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1389" xml:space="preserve">Addo, quævidentur <lb/>in textu deficere, ſeu cum duplo differentiæ triã-<lb/>gulorum I V R, &amp; </s>
  <s xml:id="echoid-s1390" xml:space="preserve">F R <emph style="sub">o</emph>. </s>
  <s xml:id="echoid-s1391" xml:space="preserve">In hyperbola verò <lb/>quadratum O I æquale eſt ſpatio rectilineo V I C G <emph style="sub">o</emph> bis ſumpto, quare in hyperbo-<lb/>la, &amp; </s>
  <s xml:id="echoid-s1392" xml:space="preserve">ellipſi quadratũ O I æquale eſt duplo trapezĳ I C G S cum duplo triãguli V <emph style="sub">o</emph> S.</s>
  <s xml:id="echoid-s1393" xml:space="preserve"/>
</p>
<div xml:id="echoid-div93" type="float" level="2" n="6">
<note position="right" xlink:label="note-0060-02" xlink:href="note-0060-02a" xml:space="preserve">n</note>
</div>
<p>
  <s xml:id="echoid-s1394" xml:space="preserve">Quod eſt æquale exemplari applicato ad R H, &amp;</s>
  <s xml:id="echoid-s1395" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1396" xml:space="preserve">Hoc enim conſtat ex <lb/>
<anchor type="note" xlink:label="note-0060-03a" xlink:href="note-0060-03"/>
ĳs, quæ ſupra dicta ſunt.</s>
  <s xml:id="echoid-s1397" xml:space="preserve"/>
</p>
<div xml:id="echoid-div94" type="float" level="2" n="7">
<note position="right" xlink:label="note-0060-03" xlink:href="note-0060-03a" xml:space="preserve">o</note>
</div>
<p>
  <s xml:id="echoid-s1398" xml:space="preserve">Eſtque D H maior in hyperbola, quàm R H, itaque A I maior, quàm <lb/>
<anchor type="note" xlink:label="note-0060-04a" xlink:href="note-0060-04"/>
OI, &amp; </s>
  <s xml:id="echoid-s1399" xml:space="preserve">O I in omnibus maior, quàm B I, &amp;</s>
  <s xml:id="echoid-s1400" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1401" xml:space="preserve">Textum hunc corruptum ſic <lb/>reſtituo: </s>
  <s xml:id="echoid-s1402" xml:space="preserve">Eſtque D H maior, quàm R H, &amp; </s>
  <s xml:id="echoid-s1403" xml:space="preserve">R H maior quàm I H; </s>
  <s xml:id="echoid-s1404" xml:space="preserve">itaque A I <lb/>maior eſt, quàm O I, &amp; </s>
  <s xml:id="echoid-s1405" xml:space="preserve">O I maior quàm B I.</s>
  <s xml:id="echoid-s1406" xml:space="preserve"/>
</p>
<div xml:id="echoid-div95" type="float" level="2" n="8">
<note position="right" xlink:label="note-0060-04" xlink:href="note-0060-04a" xml:space="preserve">p</note>
</div>
<p style="it">
  <s xml:id="echoid-s1407" xml:space="preserve">Similiter, vt in præcedenti ſectione factum eſt, reperietur multitudo ramo-<lb/>rum inter ſe æqualium, qui ex origine ad ſectionem duci poſſunt. </s>
  <s xml:id="echoid-s1408" xml:space="preserve">Exiſtente <lb/>menſura I C maiore, quàm comparata, ſi differentia abſcißarum rami maioris, <lb/>
<anchor type="note" xlink:label="note-0060-05a" xlink:href="note-0060-05"/>
&amp; </s>
  <s xml:id="echoid-s1409" xml:space="preserve">breuiſsimi æqualis fuerit abſciſſæ rami breuiſsimi, erunt tantummodo tres <lb/>rami inter ſe æquales; </s>
  <s xml:id="echoid-s1410" xml:space="preserve">ſi verò maior fuerit, duo rami ſolummodo æquales erunt; <lb/></s>
  <s xml:id="echoid-s1411" xml:space="preserve">at ſi fuerit minor eadem abſciſſa, erunt quatuor rami tantùm æquales inter ſe.</s>
  <s xml:id="echoid-s1412" xml:space="preserve"/>
</p>
<div xml:id="echoid-div96" type="float" level="2" n="9">
<note position="left" xlink:label="note-0060-05" xlink:href="note-0060-05a" xml:space="preserve">PROP. <lb/>III. Add.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1413" xml:space="preserve">Et primò ramorum I O, &amp; </s>
  <s xml:id="echoid-s1414" xml:space="preserve"><lb/>
<anchor type="figure" xlink:label="fig-0060-02a" xlink:href="fig-0060-02"/>
breuiſsimi I N abſciſſæ ſint R <lb/>C, H C, &amp; </s>
  <s xml:id="echoid-s1415" xml:space="preserve">eorum differen-<lb/>tia R H, ſitque R H æqualis <lb/>H C, &amp; </s>
  <s xml:id="echoid-s1416" xml:space="preserve">producatur O R per-<lb/>pendicularis ad axim quouſ-<lb/>que ſecet ſectionem ex altera <lb/>parte in puncto o, coniunga-<lb/>turque ramus 10. </s>
  <s xml:id="echoid-s1417" xml:space="preserve">Dico quod <lb/>tres rami I O, 10, I C tan-<lb/>tũmodo inter ſe æquales ſunt; <lb/></s>
  <s xml:id="echoid-s1418" xml:space="preserve">quoniam quadrata in para-<lb/>bola rectarum R H, &amp; </s>
  <s xml:id="echoid-s1419" xml:space="preserve">H C, <lb/>
<anchor type="note" xlink:label="note-0060-06a" xlink:href="note-0060-06"/>
ſeu in hyperbola, &amp; </s>
  <s xml:id="echoid-s1420" xml:space="preserve">ellipſi, <lb/>
<anchor type="note" xlink:label="note-0060-07a" xlink:href="note-0060-07"/>
rectangula exemplaria inter ſe ſimilia applicata ad R H, &amp; </s>
  <s xml:id="echoid-s1421" xml:space="preserve">H C æqualia ſunt <lb/>inter ſe, cum eorum latera homologa R H, H C æqualia ſuppoſita ſint; </s>
  <s xml:id="echoid-s1422" xml:space="preserve">eſtque <lb/>exceſſus quadrati rami I O, vel 10, ſeu I C ſupra quadratum rami bre-<lb/>uiſsimi I N æqualis quadrato R H, vel C H in parabola, &amp; </s>
  <s xml:id="echoid-s1423" xml:space="preserve">in reliquis <lb/>ſectionibus, exemplaribus ſimilibus applicatis ad eaſdem rectas æquales R H,
<pb o="23" file="0061" n="61" rhead="Conicor. Lib. V."/>
H C; </s>
  <s xml:id="echoid-s1424" xml:space="preserve">igitur prædisti exceſſus tam in parabola, quàm in reliquis ſectioni-<lb/>bus æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s1425" xml:space="preserve">ideò quadrata ramorum I O, 10, I C, &amp; </s>
  <s xml:id="echoid-s1426" xml:space="preserve">rami ipſi <lb/>æquales erunt: </s>
  <s xml:id="echoid-s1427" xml:space="preserve">cumque quilibet alius ramus ſupra, vel infra ramum I O maior, <lb/>vel minor ſit illo, non crunt plures, quam tres rami inter ſe æquales.</s>
  <s xml:id="echoid-s1428" xml:space="preserve"/>
</p>
<div xml:id="echoid-div97" type="float" level="2" n="10">
  <figure xlink:label="fig-0060-02" xlink:href="fig-0060-02a">
    <image file="0060-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0060-02"/>
  </figure>
<note position="left" xlink:label="note-0060-06" xlink:href="note-0060-06a" xml:space="preserve">8. huius.</note>
<note position="left" xlink:label="note-0060-07" xlink:href="note-0060-07a" xml:space="preserve">9. 10. h.</note>
</div>
<p style="it">
  <s xml:id="echoid-s1429" xml:space="preserve">Secundò H D differentia abſciſſarum rami I A, &amp; </s>
  <s xml:id="echoid-s1430" xml:space="preserve">breniſsimi I N ſupponatur <lb/>maior, quàm H C quæ eſt abſciſſa breuiſsimi rami I N; </s>
  <s xml:id="echoid-s1431" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1432" xml:space="preserve">producta ſimiliter <lb/>ordinata D A vltra axim ad ſectionem in a, &amp; </s>
  <s xml:id="echoid-s1433" xml:space="preserve">coniuncta I a; </s>
  <s xml:id="echoid-s1434" xml:space="preserve">Dico, quod duo <lb/>rami tantummodo I A, &amp; </s>
  <s xml:id="echoid-s1435" xml:space="preserve">I a inter ſe æquales ſunt: </s>
  <s xml:id="echoid-s1436" xml:space="preserve">Quia H D maior eſt, quàm <lb/>H C, erit quadratum ex H D maius quadrato H C; </s>
  <s xml:id="echoid-s1437" xml:space="preserve">pariterque exemplar appli-<lb/>catum ad H D maius erit exemplari ei ſimili applicato ad H C, &amp; </s>
  <s xml:id="echoid-s1438" xml:space="preserve">ideo tam. <lb/></s>
  <s xml:id="echoid-s1439" xml:space="preserve">quadratum I A, quàm I a maius erit quadrato I C, cum quodlibet illorum ma-<lb/>iori exceſſu ſuperet quadratum breuiſsimi rami I N quam quadratqm I C, qua-<lb/>re tam ramus I A, quàm I a (qui æquales ſunt) maiores erunt, quàm I C, &amp; </s>
  <s xml:id="echoid-s1440" xml:space="preserve"><lb/>ideo maiores quàm intercepti inter I C, &amp; </s>
  <s xml:id="echoid-s1441" xml:space="preserve">I N, pariterque maiores, quàm in-<lb/>terpoſiti inter I N, &amp; </s>
  <s xml:id="echoid-s1442" xml:space="preserve">I A, &amp; </s>
  <s xml:id="echoid-s1443" xml:space="preserve">minores omnibus alĳs, qui infra ipſos cadunt. </s>
  <s xml:id="echoid-s1444" xml:space="preserve"><lb/>Quapropter duo tantùm rami I A, I a ab origine ad ſectionem duci poſſunt in-<lb/>ter ſe æquales.</s>
  <s xml:id="echoid-s1445" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1446" xml:space="preserve">Tertiò ſint duæ abſciſſarum differentiæ H P, &amp; </s>
  <s xml:id="echoid-s1447" xml:space="preserve">H I æquales inter ſe, &amp; </s>
  <s xml:id="echoid-s1448" xml:space="preserve">quæ-<lb/>libet earum minor H C abſciſſa rami breuiſsimi, &amp; </s>
  <s xml:id="echoid-s1449" xml:space="preserve">producantur perpendicula-<lb/>res ad axim L P, B I, donec conueniant ex altera parte cum ſectione in l, &amp; </s>
  <s xml:id="echoid-s1450" xml:space="preserve">b, <lb/>coniunganturque rami ad l, b. </s>
  <s xml:id="echoid-s1451" xml:space="preserve">Dico, quatuor ramos I B, I L, I l, I b æquales <lb/>inter ſe tantummodo duci poſſe; </s>
  <s xml:id="echoid-s1452" xml:space="preserve">quia, vt dictum eſt, quilibet eorum ſuperat ra-<lb/>mum breuiſsimum I N potentia eodem exceſſu, erunt radĳ ipſi I B, I L, I l, I b <lb/>æquales inter ſe, reliqui verò ſupra, &amp; </s>
  <s xml:id="echoid-s1453" xml:space="preserve">infra ipſos maiores, aut minores erunt, <lb/>&amp; </s>
  <s xml:id="echoid-s1454" xml:space="preserve">ideo non poſſunt duci plures, quàm quatuor rami iam dicti æquales. </s>
  <s xml:id="echoid-s1455" xml:space="preserve">Quod <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s1456" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1457" xml:space="preserve">Et inſuper quadratum rami <lb/>
<anchor type="note" xlink:label="note-0061-01a" xlink:href="note-0061-01"/>
à breuiſsimo remotioris ſuper at <lb/>quadratum rami propinquioris, <lb/>
<anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/>
in parabola quidem rectangulo <lb/>ſub exceſſu, &amp; </s>
  <s xml:id="echoid-s1458" xml:space="preserve">ſub aggregato <lb/>differẽtiali ſuarum abſciſſarum <lb/>ab abſciſſa rami breuiſsimi, in <lb/>reliquis verò ſectionibus rectã-<lb/>gulo ſub codem exceſſu differen-<lb/>tiali, &amp; </s>
  <s xml:id="echoid-s1459" xml:space="preserve">ſub recta linea, ad quam <lb/>ſumma differentialis eandem <lb/>proportionem habet, quam latus <lb/>tranſuer ſum ad ſummam in hy-<lb/>perbola, &amp; </s>
  <s xml:id="echoid-s1460" xml:space="preserve">ad differentiam in ellipſi laterum recti, &amp; </s>
  <s xml:id="echoid-s1461" xml:space="preserve">tranſuerſi.</s>
  <s xml:id="echoid-s1462" xml:space="preserve"/>
</p>
<div xml:id="echoid-div98" type="float" level="2" n="11">
<note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve">PROP. <lb/>IV. Add.</note>
  <figure xlink:label="fig-0061-01" xlink:href="fig-0061-01a">
    <image file="0061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0061-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1463" xml:space="preserve">Quoniam in parabola quadratum I L ſuperat quadratum I M eodem exceſſu, <lb/>quo quadratum H P ſuperat quadratum H Q (cum quadratum H P, atque qua-<lb/>
<anchor type="note" xlink:label="note-0061-02a" xlink:href="note-0061-02"/>
dratum I N ſimul ſumpta æqualia ſint quadrato L I, &amp; </s>
  <s xml:id="echoid-s1464" xml:space="preserve">quadrata ex H Q, &amp; </s>
  <s xml:id="echoid-s1465" xml:space="preserve"><lb/>ex I N æqualia ſint quadrato I M) ſed exceſſus quadrati H P ſupra quadratum <lb/>H Q æqualis eſt rectangulo ſub P Q differentia, &amp; </s>
  <s xml:id="echoid-s1466" xml:space="preserve">P H, H Q, ſumma laterum <lb/>eorundem quadratorum contento; </s>
  <s xml:id="echoid-s1467" xml:space="preserve">igitur quadratum I L ſuperat quadratum ra-<lb/>mi I M propinquioris breuiſsimo I N rectangulo ſub P Q exceſſu, &amp; </s>
  <s xml:id="echoid-s1468" xml:space="preserve">P H Q
<pb o="24" file="0062" n="62" rhead="Apollonij Pergæi"/>
aggregato differentiali ab-<lb/>
<anchor type="figure" xlink:label="fig-0062-01a" xlink:href="fig-0062-01"/>
ſciſſarum ramorum I L, I <lb/>M ab abſciſſa rami breuiſ-<lb/>ſimi.</s>
  <s xml:id="echoid-s1469" xml:space="preserve"/>
</p>
<div xml:id="echoid-div99" type="float" level="2" n="12">
<note position="right" xlink:label="note-0061-02" xlink:href="note-0061-02a" xml:space="preserve">Ex 8. hu.</note>
  <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a">
    <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0062-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1470" xml:space="preserve">Pari modo in hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s1471" xml:space="preserve">ellipſi quadratum I L ſu-<lb/>perat quadratum I M eodẽ <lb/>exceſſu, quo exemplar ap-<lb/>
<anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/>
plicatum ad H P ſuperat <lb/>exemplar applicatum ad H <lb/>L; </s>
  <s xml:id="echoid-s1472" xml:space="preserve">ſed differentia exem-<lb/>plarium applicatorum ad H <lb/>P, &amp; </s>
  <s xml:id="echoid-s1473" xml:space="preserve">H Q æqualis eſt re-<lb/>ctangulo ſub P Q exceſſu <lb/>differentiali, &amp; </s>
  <s xml:id="echoid-s1474" xml:space="preserve">recta linea <lb/>compoſita ex X m, &amp; </s>
  <s xml:id="echoid-s1475" xml:space="preserve">u l, ad quam ſumma <lb/>differentialis P H Q eandem proportionem <lb/>
<anchor type="figure" xlink:label="fig-0062-02a" xlink:href="fig-0062-02"/>
habet, quam latus trãſuerſum ad ſummam <lb/>in hyperbola, &amp; </s>
  <s xml:id="echoid-s1476" xml:space="preserve">ad differentiam in ellipſi <lb/>laterum tranſuerſi, &amp; </s>
  <s xml:id="echoid-s1477" xml:space="preserve">recti, vt in nota <lb/>propoſitionis 5. </s>
  <s xml:id="echoid-s1478" xml:space="preserve">oſtenſum eſt; </s>
  <s xml:id="echoid-s1479" xml:space="preserve">igitur quadra-<lb/>tum I L ſuperat quadratum I M iam dicto <lb/>rectangulo ſub P Q, &amp; </s>
  <s xml:id="echoid-s1480" xml:space="preserve">ſub X m, &amp; </s>
  <s xml:id="echoid-s1481" xml:space="preserve">u l, <lb/>quod erat oſtendendum.</s>
  <s xml:id="echoid-s1482" xml:space="preserve"/>
</p>
<div xml:id="echoid-div100" type="float" level="2" n="13">
<note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve">Ex 9. 10. h.</note>
  <figure xlink:label="fig-0062-02" xlink:href="fig-0062-02a">
    <image file="0062-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0062-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div102" type="section" level="1" n="50">
<head xml:id="echoid-head76" xml:space="preserve">SECTIO IV.</head>
<head xml:id="echoid-head77" xml:space="preserve">Continens Propoſit. VII. <lb/>&amp; XII. Apollonij.</head>
<p>
  <s xml:id="echoid-s1483" xml:space="preserve">SIfuerit menſura A <lb/>
<anchor type="figure" xlink:label="fig-0062-03a" xlink:href="fig-0062-03"/>
D minor com-<lb/>
<anchor type="note" xlink:label="note-0062-02a" xlink:href="note-0062-02"/>
parata A E, (12.) </s>
  <s xml:id="echoid-s1484" xml:space="preserve">aut <lb/>ſit pars lineæ breuiſſi-<lb/>mæ, &amp; </s>
  <s xml:id="echoid-s1485" xml:space="preserve">axis in ellipſi <lb/>ſit maior, erit A D <lb/>breuiſſimus ramorum <lb/>egredientium ex ori-<lb/>gine eius in omnibus <lb/>ſectionibus, vt ſunt F <lb/>D, G D, B D, C D, <lb/>&amp; </s>
  <s xml:id="echoid-s1486" xml:space="preserve">proximior illi minor eſt remotiore, nempe F D quam G D, &amp; </s>
  <s xml:id="echoid-s1487" xml:space="preserve">G <lb/>D, quàm B D.</s>
  <s xml:id="echoid-s1488" xml:space="preserve"/>
</p>
<div xml:id="echoid-div102" type="float" level="2" n="1">
  <figure xlink:label="fig-0062-03" xlink:href="fig-0062-03a">
    <image file="0062-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0062-03"/>
  </figure>
<note position="right" xlink:label="note-0062-02" xlink:href="note-0062-02a" xml:space="preserve">a</note>
</div>
<pb o="25" file="0063" n="63" rhead="Conicor. Lib. V."/>
<p>
  <s xml:id="echoid-s1489" xml:space="preserve">QVia A E eſt line a breuiſſima, igi-<lb/>
<anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/>
<anchor type="figure" xlink:label="fig-0063-01a" xlink:href="fig-0063-01"/>
tur F E maior eſt illa; </s>
  <s xml:id="echoid-s1490" xml:space="preserve">itaque an-<lb/>gulus F A E maior eſt, quàm <lb/>
<anchor type="note" xlink:label="note-0063-02a" xlink:href="note-0063-02"/>
A F E; </s>
  <s xml:id="echoid-s1491" xml:space="preserve">Ergo ille eſt multò maior quàm <lb/>A F D, quare F D maior eſt; </s>
  <s xml:id="echoid-s1492" xml:space="preserve">atque ſic <lb/>patet quod G E maior ſit quàm E F, &amp; </s>
  <s xml:id="echoid-s1493" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0063-03a" xlink:href="note-0063-03"/>
ideo angulus G F E maior eſt, quàm E <lb/>G F; </s>
  <s xml:id="echoid-s1494" xml:space="preserve">igitur angulus G F D multò maior <lb/>eſt, quàm F G D, &amp; </s>
  <s xml:id="echoid-s1495" xml:space="preserve">propterea G D ma-<lb/>ior eſt, quàm D F, &amp; </s>
  <s xml:id="echoid-s1496" xml:space="preserve">ſimiliter B D, <lb/>quàm G D, &amp; </s>
  <s xml:id="echoid-s1497" xml:space="preserve">D C, quàm A D, &amp; </s>
  <s xml:id="echoid-s1498" xml:space="preserve">hoc <lb/>erat propoſitum.</s>
  <s xml:id="echoid-s1499" xml:space="preserve"/>
</p>
<div xml:id="echoid-div103" type="float" level="2" n="2">
<note position="left" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0063-01" xlink:href="fig-0063-01a">
    <image file="0063-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0063-01"/>
  </figure>
<note position="left" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0063-03" xlink:href="note-0063-03a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div105" type="section" level="1" n="51">
<head xml:id="echoid-head78" xml:space="preserve">NOTÆ.</head>
<p style="it">
  <s xml:id="echoid-s1500" xml:space="preserve">SI fuerit menſura A D minor comparata A E, &amp;</s>
  <s xml:id="echoid-s1501" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1502" xml:space="preserve">Senſus propoſitionis <lb/>
<anchor type="note" xlink:label="note-0063-04a" xlink:href="note-0063-04"/>
clarior ſic reddetur; </s>
  <s xml:id="echoid-s1503" xml:space="preserve">Si fuerit menſura A D minor comparata A E, quæ in <lb/>ellipſi ſumi debet in axi maiori eius (12.) </s>
  <s xml:id="echoid-s1504" xml:space="preserve">aut ſit pars lineæ breuiſsimæ; </s>
  <s xml:id="echoid-s1505" xml:space="preserve">erit <lb/>A D minimus ramorum F D, G D, B D, C D, egredientium ex origine eius in <lb/>omnibus ſectionibus, &amp; </s>
  <s xml:id="echoid-s1506" xml:space="preserve">proximior illi, &amp;</s>
  <s xml:id="echoid-s1507" xml:space="preserve">c.</s>
  <s xml:id="echoid-s1508" xml:space="preserve"/>
</p>
<div xml:id="echoid-div105" type="float" level="2" n="1">
<note position="left" xlink:label="note-0063-04" xlink:href="note-0063-04a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s1509" xml:space="preserve">Quia A E eſt linea breuiſſima, igitur, &amp;</s>
  <s xml:id="echoid-s1510" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1511" xml:space="preserve">Vt conſtructio compleatur ſu-<lb/>
<anchor type="note" xlink:label="note-0063-05a" xlink:href="note-0063-05"/>
biungo: </s>
  <s xml:id="echoid-s1512" xml:space="preserve">Igitur ſi coniungantur rectæ lineæ E F, E G, E C, E B, &amp; </s>
  <s xml:id="echoid-s1513" xml:space="preserve">rectæ lineæ <lb/>A F, F G, G B, A C erit F E maior, quàm A E.</s>
  <s xml:id="echoid-s1514" xml:space="preserve"/>
</p>
<div xml:id="echoid-div106" type="float" level="2" n="2">
<note position="left" xlink:label="note-0063-05" xlink:href="note-0063-05a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s1515" xml:space="preserve">Ergo hic eſt multò maior, quàm A F E, &amp;</s>
  <s xml:id="echoid-s1516" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1517" xml:space="preserve">Senſus clarior reddetur hac <lb/>
<anchor type="note" xlink:label="note-0063-06a" xlink:href="note-0063-06"/>
ratione: </s>
  <s xml:id="echoid-s1518" xml:space="preserve">Ergo angulus F A E multò maior erit, quàm A F D, qui eſt portio mi-<lb/>noris anguli, quarè F D ſubtendens angulum maiorem eſt maior, quàm A D.</s>
  <s xml:id="echoid-s1519" xml:space="preserve"/>
</p>
<div xml:id="echoid-div107" type="float" level="2" n="3">
<note position="left" xlink:label="note-0063-06" xlink:href="note-0063-06a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s1520" xml:space="preserve">Igitur ipſe multò maior eſt, &amp;</s>
  <s xml:id="echoid-s1521" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1522" xml:space="preserve">Superaddo rationem illationis dicendo; <lb/></s>
  <s xml:id="echoid-s1523" xml:space="preserve">
<anchor type="note" xlink:label="note-0063-07a" xlink:href="note-0063-07"/>
Et propterea angulus G F D maiorem excedens erit multò maior, quàm F G D, <lb/>qui portio minoris eſt.</s>
  <s xml:id="echoid-s1524" xml:space="preserve"/>
</p>
<div xml:id="echoid-div108" type="float" level="2" n="4">
<note position="left" xlink:label="note-0063-07" xlink:href="note-0063-07a" xml:space="preserve">d</note>
</div>
<p style="it">
  <s xml:id="echoid-s1525" xml:space="preserve">Manifeſtum eſt in prima figura propoſitionis 7. </s>
  <s xml:id="echoid-s1526" xml:space="preserve">quando A D eſt portio axis <lb/>minor comparata, quod tunc ex origine D duo tantummodo rami inter ſe æqua-<lb/>les ad vtraſque partes axis duci poſſunt ad ſectionem, &amp; </s>
  <s xml:id="echoid-s1527" xml:space="preserve">erunt illi, qui ad ter-<lb/>minos eiuſdem ordinatim ad axim applicatæ iunguntur ab origine D, vt conſtat <lb/>ex ſuperiùs dictis.</s>
  <s xml:id="echoid-s1528" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1529" xml:space="preserve">At in ſecunda figura propoſitionis 12. </s>
  <s xml:id="echoid-s1530" xml:space="preserve">poſſunt quidem ab origine D ad ſectio-<lb/>nem duci hinc indè à breuiſsima D A, aliquando duo tantùm rami inter ſe <lb/>æquales, aliquando tres, atque etiam quatuor inter ſe æquales, quæcognitio pen-<lb/>det ex propoſitione 72. </s>
  <s xml:id="echoid-s1531" xml:space="preserve">huius libri.</s>
  <s xml:id="echoid-s1532" xml:space="preserve"/>
</p>
<pb o="26" file="0064" n="64" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div110" type="section" level="1" n="52">
<head xml:id="echoid-head79" xml:space="preserve">SECTIO QVINTA</head>
<head xml:id="echoid-head80" xml:space="preserve">Continens XI. Propoſit. Apollonij.</head>
<p>
  <s xml:id="echoid-s1533" xml:space="preserve">LInearum egredientium ex D centro ellipſis A B C, breuiſſi-<lb/>ma eſt ſemiaxis minor rectus <lb/>illius, qui ſit B D, maxima verò eſt <lb/>
<anchor type="figure" xlink:label="fig-0064-01a" xlink:href="fig-0064-01"/>
ſemiaxis tranſuerſus, qui ſit A D, &amp; </s>
  <s xml:id="echoid-s1534" xml:space="preserve"><lb/>propinquiores maiori ſunt maiores <lb/>remotioribus, vt H D, quam G D, <lb/>&amp; </s>
  <s xml:id="echoid-s1535" xml:space="preserve">quadratum cuiuslibet rami, vt G <lb/>D (exempli gratia) excedit quadra-<lb/>
<anchor type="note" xlink:label="note-0064-01a" xlink:href="note-0064-01"/>
tum breuiſſimę B D exemplari appli-<lb/>cato ad inuerſam illius I D.</s>
  <s xml:id="echoid-s1536" xml:space="preserve"/>
</p>
<div xml:id="echoid-div110" type="float" level="2" n="1">
  <figure xlink:label="fig-0064-01" xlink:href="fig-0064-01a">
    <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0064-01"/>
  </figure>
<note position="right" xlink:label="note-0064-01" xlink:href="note-0064-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s1537" xml:space="preserve">EDucamus itaque E A æqualem A D, &amp; </s>
  <s xml:id="echoid-s1538" xml:space="preserve">abſcindamus ex illa A F ęqua-<lb/>
<anchor type="note" xlink:label="note-0064-02a" xlink:href="note-0064-02"/>
lem dimidio erecti, &amp; </s>
  <s xml:id="echoid-s1539" xml:space="preserve">iungamus D F, D E, &amp; </s>
  <s xml:id="echoid-s1540" xml:space="preserve">perducamus ex G, H <lb/>perpendiculares ad D A, &amp; </s>
  <s xml:id="echoid-s1541" xml:space="preserve">ſint G I M, H L N. </s>
  <s xml:id="echoid-s1542" xml:space="preserve">Quia quadratum G I æ-<lb/>
<anchor type="note" xlink:label="note-0064-03a" xlink:href="note-0064-03"/>
quale eſt duplo trapezij I F (prima ex quinto) &amp; </s>
  <s xml:id="echoid-s1543" xml:space="preserve">quadratum I D eſt æqua-<lb/>le duplo trianguli I D M, eo quod I D eſt æqualis I M, erit quadratum <lb/>
<anchor type="note" xlink:label="note-0064-04a" xlink:href="note-0064-04"/>
D G æquale duplo trianguli A D F (quod eſt æquale quadrato B D (2. </s>
  <s xml:id="echoid-s1544" xml:space="preserve">ex <lb/>quinto) vnà cum duplo trianguli Q M D, quod eſt æquale rectangulo Q <lb/>P; </s>
  <s xml:id="echoid-s1545" xml:space="preserve">igitur quadrati G D exceſſus ſupra quadratum B D eſt æqualis plano <lb/>Q P, &amp; </s>
  <s xml:id="echoid-s1546" xml:space="preserve">quia D A, nempe E A ad A F eſt, vt D I, nempe M I ad I Q, <lb/>
<anchor type="note" xlink:label="note-0064-05a" xlink:href="note-0064-05"/>
&amp; </s>
  <s xml:id="echoid-s1547" xml:space="preserve">per conuerſionem rationis A E ad E F, ſcilicet dimidium tranſuerſæ <lb/>ad illius exceſſum ſuper A F dimidium erecti, eſt, vt M I, nempe M P <lb/>ad M Q; </s>
  <s xml:id="echoid-s1548" xml:space="preserve">igitur planum Q P ſimile eſt figuræ comparatæ, &amp; </s>
  <s xml:id="echoid-s1549" xml:space="preserve">M P æqua-<lb/>lis eſt D I. </s>
  <s xml:id="echoid-s1550" xml:space="preserve">Similiter patet, quod quadratum D H excedit quadratum B <lb/>
<anchor type="note" xlink:label="note-0064-06a" xlink:href="note-0064-06"/>
D exemplari applicato ad D L, &amp; </s>
  <s xml:id="echoid-s1551" xml:space="preserve">quadratum D A ſuperat quadratum <lb/>B D exemplari applicato ad D A: </s>
  <s xml:id="echoid-s1552" xml:space="preserve">Eſt verò D I minor, quàm D L, &amp; </s>
  <s xml:id="echoid-s1553" xml:space="preserve"><lb/>D L, quàm D A; </s>
  <s xml:id="echoid-s1554" xml:space="preserve">igitur B D (quæ eſt dimidium recti) minor eſt, quàm <lb/>
<anchor type="note" xlink:label="note-0064-07a" xlink:href="note-0064-07"/>
G D, &amp; </s>
  <s xml:id="echoid-s1555" xml:space="preserve">G D, quàm D H, &amp; </s>
  <s xml:id="echoid-s1556" xml:space="preserve">D H quàm D A, quod erat oſtendendum.</s>
  <s xml:id="echoid-s1557" xml:space="preserve"/>
</p>
<div xml:id="echoid-div111" type="float" level="2" n="2">
<note position="right" xlink:label="note-0064-02" xlink:href="note-0064-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0064-03" xlink:href="note-0064-03a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0064-04" xlink:href="note-0064-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0064-05" xlink:href="note-0064-05a" xml:space="preserve">e</note>
<note position="left" xlink:label="note-0064-06" xlink:href="note-0064-06a" xml:space="preserve">Def. 8. 9. <lb/>huius.</note>
<note position="right" xlink:label="note-0064-07" xlink:href="note-0064-07a" xml:space="preserve">f</note>
</div>
</div>
<div xml:id="echoid-div113" type="section" level="1" n="53">
<head xml:id="echoid-head81" xml:space="preserve">NOTÆ.</head>
<p style="it">
  <s xml:id="echoid-s1558" xml:space="preserve">ET debet eſſe linea breuiſſima perpendicularis ad menſuram, nempe B <lb/>
<anchor type="note" xlink:label="note-0064-08a" xlink:href="note-0064-08"/>
D perpendicularis D A, &amp;</s>
  <s xml:id="echoid-s1559" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1560" xml:space="preserve">Hæc omnino expungi debent, tanquam <lb/>ſuperuacanea, axes enim eſſe nequeunt, niſi ad inuicem perpendiculares ſint; <lb/></s>
  <s xml:id="echoid-s1561" xml:space="preserve">quare cenſeo ab aliquo verba illa addita textui Apollonĳ fuiſſe.</s>
  <s xml:id="echoid-s1562" xml:space="preserve"/>
</p>
<div xml:id="echoid-div113" type="float" level="2" n="1">
<note position="right" xlink:label="note-0064-08" xlink:href="note-0064-08a" xml:space="preserve">a</note>
</div>
<pb o="27" file="0065" n="65" rhead="Conicor. Lib. V."/>
<p>
  <s xml:id="echoid-s1563" xml:space="preserve">Educamus itaque E A, &amp;</s>
  <s xml:id="echoid-s1564" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1565" xml:space="preserve">Lego: </s>
  <s xml:id="echoid-s1566" xml:space="preserve">Educamus itaq; </s>
  <s xml:id="echoid-s1567" xml:space="preserve">E A perpendicularem, &amp; </s>
  <s xml:id="echoid-s1568" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/>
æqualem A D.</s>
  <s xml:id="echoid-s1569" xml:space="preserve"/>
</p>
<div xml:id="echoid-div114" type="float" level="2" n="2">
<note position="left" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s1570" xml:space="preserve">Et perducamus ex G, H perpendiculares, &amp;</s>
  <s xml:id="echoid-s1571" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1572" xml:space="preserve">Et perducamus ex G, H <lb/>
<anchor type="note" xlink:label="note-0065-02a" xlink:href="note-0065-02"/>
perpendiculares ad D A, &amp; </s>
  <s xml:id="echoid-s1573" xml:space="preserve">ſint H L N, &amp; </s>
  <s xml:id="echoid-s1574" xml:space="preserve">G I M, quæ ſecent F D in Q, &amp; </s>
  <s xml:id="echoid-s1575" xml:space="preserve">D <lb/>E in M, &amp; </s>
  <s xml:id="echoid-s1576" xml:space="preserve">N, atque à punctis Q, M educantur M P, Q O, parallelæ D A, <lb/>quæ ſecent rectum axem B D in O, P. </s>
  <s xml:id="echoid-s1577" xml:space="preserve">Addidi hæc poſtrema verba, vt conſtru-<lb/>ctio completa ſit.</s>
  <s xml:id="echoid-s1578" xml:space="preserve"/>
</p>
<div xml:id="echoid-div115" type="float" level="2" n="3">
<note position="left" xlink:label="note-0065-02" xlink:href="note-0065-02a" xml:space="preserve">c</note>
</div>
<p>
  <s xml:id="echoid-s1579" xml:space="preserve">Eo quod I D eſt æqualis I M, &amp;</s>
  <s xml:id="echoid-s1580" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1581" xml:space="preserve">Quoniam ſicuti in triangulo D A E <lb/>
<anchor type="note" xlink:label="note-0065-03a" xlink:href="note-0065-03"/>
ſimili triangulo D I M (propter angulum D communem, &amp; </s>
  <s xml:id="echoid-s1582" xml:space="preserve">rectos angulos ad I, <lb/>&amp; </s>
  <s xml:id="echoid-s1583" xml:space="preserve">A) latus D A æquale erat E A, ita latus D I æquale eſt I M.</s>
  <s xml:id="echoid-s1584" xml:space="preserve"/>
</p>
<div xml:id="echoid-div116" type="float" level="2" n="4">
<note position="left" xlink:label="note-0065-03" xlink:href="note-0065-03a" xml:space="preserve">d</note>
</div>
<p>
  <s xml:id="echoid-s1585" xml:space="preserve">Nempe M I ad I Q, &amp; </s>
  <s xml:id="echoid-s1586" xml:space="preserve">è contra, &amp;</s>
  <s xml:id="echoid-s1587" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1588" xml:space="preserve">Lego: </s>
  <s xml:id="echoid-s1589" xml:space="preserve">Nempe M I ad I Q, &amp; </s>
  <s xml:id="echoid-s1590" xml:space="preserve">per <lb/>
<anchor type="note" xlink:label="note-0065-04a" xlink:href="note-0065-04"/>
conuerſionem rationis.</s>
  <s xml:id="echoid-s1591" xml:space="preserve"/>
</p>
<div xml:id="echoid-div117" type="float" level="2" n="5">
<note position="left" xlink:label="note-0065-04" xlink:href="note-0065-04a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s1592" xml:space="preserve">Cumque B D ſit dimidium axis recti erit perpendicularis ad A D men-<lb/>
<anchor type="note" xlink:label="note-0065-05a" xlink:href="note-0065-05"/>
ſuram, &amp;</s>
  <s xml:id="echoid-s1593" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1594" xml:space="preserve">Hæc verba poſtrema pariter expungi debent, niſi fortè corollarium <lb/>propoſitionis exponunt, &amp; </s>
  <s xml:id="echoid-s1595" xml:space="preserve">tunc textus ſic reſtitui deberet. </s>
  <s xml:id="echoid-s1596" xml:space="preserve">Ex dictis conſtat, li-<lb/>neam breuiſsimam è centro ellipſis ad ſectionem ductam, perpendicularem eße <lb/>ad axim eius maiorem.</s>
  <s xml:id="echoid-s1597" xml:space="preserve"/>
</p>
<div xml:id="echoid-div118" type="float" level="2" n="6">
<note position="left" xlink:label="note-0065-05" xlink:href="note-0065-05a" xml:space="preserve">f</note>
</div>
<p style="it">
  <s xml:id="echoid-s1598" xml:space="preserve">Manifeſtum eſt ex centro ellipſis ad ſectionem duci non poſſe plures, quàm <lb/>quatuor ramos inter ſe æquales, neque pauciores duobus; </s>
  <s xml:id="echoid-s1599" xml:space="preserve">tres autem nequaquam; <lb/></s>
  <s xml:id="echoid-s1600" xml:space="preserve">nam duæ medietates cuiuslibet axis æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s1601" xml:space="preserve">quatuor rami ad <lb/>extremitates duarum applicatarum ad axim æqualiter è centro diſtantium ducti <lb/>æquales ſunt inter ſe.</s>
  <s xml:id="echoid-s1602" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div120" type="section" level="1" n="54">
<head xml:id="echoid-head82" xml:space="preserve">SECTIO SEXTA</head>
<head xml:id="echoid-head83" xml:space="preserve">Continens Propoſit. XIII. XIV. XV. Apollonij.</head>
<p>
  <s xml:id="echoid-s1603" xml:space="preserve">OStendamus modò cõ-<lb/>uerſum harum pro-<lb/>
<anchor type="figure" xlink:label="fig-0065-01a" xlink:href="fig-0065-01"/>
poſitionum; </s>
  <s xml:id="echoid-s1604" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1605" xml:space="preserve">eſt, quod li-<lb/>nea breuiſſima B F continet <lb/>cum ſua menſura A F angu-<lb/>lum acutum, vt B F A in <lb/>omnibus ſectionibus, &amp; </s>
  <s xml:id="echoid-s1606" xml:space="preserve">el-<lb/>lipſi (ſi tamen non egre-<lb/>diatur ex eius centro) eiuſ-<lb/>que potentialis abſcindet <lb/>menſuram (13) in parabola æqualem comparatæ (14) &amp; </s>
  <s xml:id="echoid-s1607" xml:space="preserve">in <lb/>
<anchor type="note" xlink:label="note-0065-06a" xlink:href="note-0065-06"/>
hyperbola (15) &amp; </s>
  <s xml:id="echoid-s1608" xml:space="preserve">ellipſi lineam, ad quam inuerſa eſt, vt pro-<lb/>portio figuræ.</s>
  <s xml:id="echoid-s1609" xml:space="preserve"/>
</p>
<div xml:id="echoid-div120" type="float" level="2" n="1">
  <figure xlink:label="fig-0065-01" xlink:href="fig-0065-01a">
    <image file="0065-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0065-01"/>
  </figure>
<note position="left" xlink:label="note-0065-06" xlink:href="note-0065-06a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s1610" xml:space="preserve">SIt centrum D, &amp; </s>
  <s xml:id="echoid-s1611" xml:space="preserve">dimidium erecti A C. </s>
  <s xml:id="echoid-s1612" xml:space="preserve">Quia B F eſt linea breuiſſima, <lb/>erit A F maior quàm A C, eo quòd ſi eſſet æqualis (4. </s>
  <s xml:id="echoid-s1613" xml:space="preserve">6. </s>
  <s xml:id="echoid-s1614" xml:space="preserve">ex quinto)
<pb o="28" file="0066" n="66" rhead="Apollonij Pergæi"/>
aut minor illa (7. </s>
  <s xml:id="echoid-s1615" xml:space="preserve">ex quinto) eſ-<lb/>ſet linea breuiſſima A F, aut pars <lb/>
<anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/>
illius, quod eſt falſum, igitur <lb/>maior eſt, quàm A C; </s>
  <s xml:id="echoid-s1616" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1617" xml:space="preserve">pro-<lb/>pterea A D ad A C maiorem <lb/>proportionem habet, quàm ad <lb/>A F; </s>
  <s xml:id="echoid-s1618" xml:space="preserve">ponamus ergo, vt A D ad <lb/>A C, ita D G ad G F in hyper-<lb/>bola, &amp; </s>
  <s xml:id="echoid-s1619" xml:space="preserve">ellipſi; </s>
  <s xml:id="echoid-s1620" xml:space="preserve">at in parabola <lb/>
<anchor type="note" xlink:label="note-0066-01a" xlink:href="note-0066-01"/>
ponamus G F æqualem A C, &amp; </s>
  <s xml:id="echoid-s1621" xml:space="preserve"><lb/>ducatur ex G perpendicularis ad <lb/>ſectionem. </s>
  <s xml:id="echoid-s1622" xml:space="preserve">Dico, quod ei oc-<lb/>curret ad B. </s>
  <s xml:id="echoid-s1623" xml:space="preserve">Nam ſi occurrat <lb/>ſectioni ad aliud punctum, vt H co-<lb/>
<anchor type="figure" xlink:label="fig-0066-02a" xlink:href="fig-0066-02"/>
niuncta H F erit H F breuiſſima (8. </s>
  <s xml:id="echoid-s1624" xml:space="preserve">9. <lb/></s>
  <s xml:id="echoid-s1625" xml:space="preserve">10. </s>
  <s xml:id="echoid-s1626" xml:space="preserve">ex quinto) ſed ſuppoſuimus B F eſſe <lb/>breuiſſimam, quod eſt abſurdum, ergo <lb/>perpendicularis occurrit ſectioni in B. </s>
  <s xml:id="echoid-s1627" xml:space="preserve"><lb/>Et quia angulus B G F eſt rectus, erit <lb/>angulus B F G acutus, quod erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s1628" xml:space="preserve"/>
</p>
<div xml:id="echoid-div121" type="float" level="2" n="2">
  <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a">
    <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0066-01"/>
  </figure>
<note position="right" xlink:label="note-0066-01" xlink:href="note-0066-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0066-02" xlink:href="fig-0066-02a">
    <image file="0066-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0066-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div123" type="section" level="1" n="55">
<head xml:id="echoid-head84" xml:space="preserve">NOTÆ.</head>
<p style="it">
  <s xml:id="echoid-s1629" xml:space="preserve">ET eius potentialis ſecet menſuram <lb/>in parabola, &amp;</s>
  <s xml:id="echoid-s1630" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1631" xml:space="preserve">Ideſt, &amp; </s>
  <s xml:id="echoid-s1632" xml:space="preserve">eius po-<lb/>
<anchor type="note" xlink:label="note-0066-02a" xlink:href="note-0066-02"/>
<anchor type="figure" xlink:label="fig-0066-03a" xlink:href="fig-0066-03"/>
tentialis abſcindet ex menſura vſque ad originem, <lb/>in parabola quidem ſegmentum æquale compara-<lb/>tæ, &amp; </s>
  <s xml:id="echoid-s1633" xml:space="preserve">in hyperbola, &amp; </s>
  <s xml:id="echoid-s1634" xml:space="preserve">ellipſi lineam, ad quam <lb/>inuerſa eandem proporportionem habet, quam la-<lb/>tus tranſuerſum ad rectum.</s>
  <s xml:id="echoid-s1635" xml:space="preserve"/>
</p>
<div xml:id="echoid-div123" type="float" level="2" n="1">
<note position="right" xlink:label="note-0066-02" xlink:href="note-0066-02a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0066-03" xlink:href="fig-0066-03a">
    <image file="0066-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0066-03"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s1636" xml:space="preserve">Et ducatur ex G perpendicularis ad ſectio-<lb/>
<anchor type="note" xlink:label="note-0066-03a" xlink:href="note-0066-03"/>
nem, &amp;</s>
  <s xml:id="echoid-s1637" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1638" xml:space="preserve">Et ducatur ex G recta linea perpendi-<lb/>cularis ad axim, &amp; </s>
  <s xml:id="echoid-s1639" xml:space="preserve">producatur vſque ad ſectio-<lb/>nem.</s>
  <s xml:id="echoid-s1640" xml:space="preserve"/>
</p>
<div xml:id="echoid-div124" type="float" level="2" n="2">
<note position="right" xlink:label="note-0066-03" xlink:href="note-0066-03a" xml:space="preserve">b</note>
</div>
<pb o="29" file="0067" n="67" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div126" type="section" level="1" n="56">
<head xml:id="echoid-head85" xml:space="preserve">SECTIO SEPTIMA</head>
<head xml:id="echoid-head86" xml:space="preserve">Continens XXVI. XXVII. XXVIII. Propoſ. <lb/>Apollonij.</head>
<head xml:id="echoid-head87" xml:space="preserve">PROPOSITIO XXVI. &amp; XXVII.</head>
<p>
  <s xml:id="echoid-s1641" xml:space="preserve">ANgulorum ab axi ſectionis A H, &amp; </s>
  <s xml:id="echoid-s1642" xml:space="preserve">à lineis breuiſſimis F <lb/>B, H G contentorum proximiores vertici ſectionis mi-<lb/>nores ſunt remotioribus, nempe angulus AFB minor eſt AHG.</s>
  <s xml:id="echoid-s1643" xml:space="preserve"/>
</p>
  <figure>
    <image file="0067-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0067-01"/>
  </figure>
<p>
  <s xml:id="echoid-s1644" xml:space="preserve">SIt itaque centrum D, &amp; </s>
  <s xml:id="echoid-s1645" xml:space="preserve">ſemi inclinatus axis A D, ſiue ſemitranſuer-<lb/>ſus, &amp; </s>
  <s xml:id="echoid-s1646" xml:space="preserve">dimidium erecti A C: </s>
  <s xml:id="echoid-s1647" xml:space="preserve">educamus itaque duas perpendiculares <lb/>
<anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/>
GL, BI, &amp; </s>
  <s xml:id="echoid-s1648" xml:space="preserve">ſi ſectio fuerit parabole, erit FI æqualis LH, quia quælibet <lb/>earum æqualis eſt A C (13. </s>
  <s xml:id="echoid-s1649" xml:space="preserve">ex quinto) &amp; </s>
  <s xml:id="echoid-s1650" xml:space="preserve">L G maior eſt, quàm BI; </s>
  <s xml:id="echoid-s1651" xml:space="preserve">an-<lb/>
<anchor type="note" xlink:label="note-0067-02a" xlink:href="note-0067-02"/>
gulus igitur F minor quàm H; </s>
  <s xml:id="echoid-s1652" xml:space="preserve">ſi verò ſectio fuerit hyperbole, aut ellipſis, <lb/>erit FI ad ID, vt HL ad LD, quia quælibet earum eſt, vt AC ad AD <lb/>
<anchor type="note" xlink:label="note-0067-03a" xlink:href="note-0067-03"/>
(14. </s>
  <s xml:id="echoid-s1653" xml:space="preserve">15. </s>
  <s xml:id="echoid-s1654" xml:space="preserve">ex quinto) &amp; </s>
  <s xml:id="echoid-s1655" xml:space="preserve">permutando, erit I D ad L D nempe B I ad M L, <lb/>
<anchor type="note" xlink:label="note-0067-04a" xlink:href="note-0067-04"/>
vt I F ad L H, &amp; </s>
  <s xml:id="echoid-s1656" xml:space="preserve">anguli I, &amp; </s>
  <s xml:id="echoid-s1657" xml:space="preserve">L ſunt recti; </s>
  <s xml:id="echoid-s1658" xml:space="preserve">igitur duo triangula BIF, M <lb/>L H ſunt ſimilia, ideoque angulus A H G maior eſt, quàm angulus A F <lb/>B, &amp; </s>
  <s xml:id="echoid-s1659" xml:space="preserve">hoc erat propoſitum.</s>
  <s xml:id="echoid-s1660" xml:space="preserve"/>
</p>
<div xml:id="echoid-div126" type="float" level="2" n="1">
<note position="left" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0067-02" xlink:href="note-0067-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0067-03" xlink:href="note-0067-03a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0067-04" xlink:href="note-0067-04a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div128" type="section" level="1" n="57">
<head xml:id="echoid-head88" xml:space="preserve">PROPOSITIO XXVIII.</head>
<p>
  <s xml:id="echoid-s1661" xml:space="preserve">Hinc patet, lineas breuiſſimas ſibi occurrere ad partes axis <lb/>ſectionis.</s>
  <s xml:id="echoid-s1662" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1663" xml:space="preserve">QVia angulus AFB minor eſt, quàm angulus AHG; </s>
  <s xml:id="echoid-s1664" xml:space="preserve">quare ſibi oc-<lb/>
<anchor type="note" xlink:label="note-0067-05a" xlink:href="note-0067-05"/>
<anchor type="note" xlink:label="note-0067-06a" xlink:href="note-0067-06"/>
currunt ad partes F, H, &amp; </s>
  <s xml:id="echoid-s1665" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s1666" xml:space="preserve"/>
</p>
<div xml:id="echoid-div128" type="float" level="2" n="1">
<note position="right" xlink:label="note-0067-05" xlink:href="note-0067-05a" xml:space="preserve">26. 27. h.</note>
<note position="left" xlink:label="note-0067-06" xlink:href="note-0067-06a" xml:space="preserve">e</note>
</div>
<pb o="30" file="0068" n="68" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div130" type="section" level="1" n="58">
<head xml:id="echoid-head89" xml:space="preserve">NOTÆ.</head>
<p style="it">
  <s xml:id="echoid-s1667" xml:space="preserve">EDucamus itaque duas perpendiculares, &amp;</s>
  <s xml:id="echoid-s1668" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1669" xml:space="preserve">Educamus itaque ex pun-<lb/>
<anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/>
ctis B, G duas G L, B I perpendiculares ad axim ei occurrentes in L, I. <lb/></s>
  <s xml:id="echoid-s1670" xml:space="preserve">Et LG maior eſt, quàm B I, &amp;</s>
  <s xml:id="echoid-s1671" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1672" xml:space="preserve">Subiungo: </s>
  <s xml:id="echoid-s1673" xml:space="preserve">Eo quod potentialis G L ma-<lb/>
<anchor type="note" xlink:label="note-0068-02a" xlink:href="note-0068-02"/>
gis recedit à vertice, quàm B I; </s>
  <s xml:id="echoid-s1674" xml:space="preserve">ſi iam ducatur B M parallela axi in parabola, <lb/>&amp; </s>
  <s xml:id="echoid-s1675" xml:space="preserve">ex centro educta in reliquis ſectionibus, ſecans G L in M, coniungaturque H <lb/>M, erit in parabola M L minor quàm G L, &amp; </s>
  <s xml:id="echoid-s1676" xml:space="preserve">æqualis B I, &amp; </s>
  <s xml:id="echoid-s1677" xml:space="preserve">ideo angulus M <lb/>H L minor erit angulo G H L, &amp; </s>
  <s xml:id="echoid-s1678" xml:space="preserve">æqualis angulo F, &amp; </s>
  <s xml:id="echoid-s1679" xml:space="preserve">propterea angulus F mi-<lb/>nor eſt, quàm G H L.</s>
  <s xml:id="echoid-s1680" xml:space="preserve"/>
</p>
<div xml:id="echoid-div130" type="float" level="2" n="1">
<note position="right" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">b</note>
</div>
  <figure>
    <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0068-01"/>
  </figure>
<p>
  <s xml:id="echoid-s1681" xml:space="preserve">Si verò ſectio fuerit hyperbole, aut ellipſis, &amp;</s>
  <s xml:id="echoid-s1682" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1683" xml:space="preserve">Addo: </s>
  <s xml:id="echoid-s1684" xml:space="preserve">Manifeſtum eſt <lb/>
<anchor type="note" xlink:label="note-0068-03a" xlink:href="note-0068-03"/>
<anchor type="note" xlink:label="note-0068-04a" xlink:href="note-0068-04"/>
rectam B D ex centro ductam ſectionem ſecare in B, &amp; </s>
  <s xml:id="echoid-s1685" xml:space="preserve">propterea occurrere po-<lb/>tentiali G L à vertice remotiori, quàm B I inter puncta G, &amp; </s>
  <s xml:id="echoid-s1686" xml:space="preserve">L, &amp; </s>
  <s xml:id="echoid-s1687" xml:space="preserve">erit F I, <lb/>&amp; </s>
  <s xml:id="echoid-s1688" xml:space="preserve">cætera.</s>
  <s xml:id="echoid-s1689" xml:space="preserve"/>
</p>
<div xml:id="echoid-div131" type="float" level="2" n="2">
<note position="left" xlink:label="note-0068-03" xlink:href="note-0068-03a" xml:space="preserve">31. lib. I.</note>
<note position="right" xlink:label="note-0068-04" xlink:href="note-0068-04a" xml:space="preserve">C</note>
</div>
<p>
  <s xml:id="echoid-s1690" xml:space="preserve">Erit ID ad LD, nempe B I ad M L, &amp;</s>
  <s xml:id="echoid-s1691" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1692" xml:space="preserve">Addo (propter parallelas B I, <lb/>
<anchor type="note" xlink:label="note-0068-05a" xlink:href="note-0068-05"/>
M L, &amp; </s>
  <s xml:id="echoid-s1693" xml:space="preserve">ſimilitudincm triangulorum D B I, &amp; </s>
  <s xml:id="echoid-s1694" xml:space="preserve">D M L.)</s>
  <s xml:id="echoid-s1695" xml:space="preserve"/>
</p>
<div xml:id="echoid-div132" type="float" level="2" n="3">
<note position="right" xlink:label="note-0068-05" xlink:href="note-0068-05a" xml:space="preserve">d</note>
</div>
<p>
  <s xml:id="echoid-s1696" xml:space="preserve">Quia angulus A F B minor eſt, quàm angulus A H G, &amp;</s>
  <s xml:id="echoid-s1697" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1698" xml:space="preserve">Addo: </s>
  <s xml:id="echoid-s1699" xml:space="preserve">Et <lb/>
<anchor type="note" xlink:label="note-0068-06a" xlink:href="note-0068-06"/>
ſumpto communi angulo F H N erunt A F B, ſeu H F N, &amp; </s>
  <s xml:id="echoid-s1700" xml:space="preserve">F H N ſimul ſumpti <lb/>minores duobus angulis G H A, F H N, qui duobus rectis æquales ſunt; </s>
  <s xml:id="echoid-s1701" xml:space="preserve">quare <lb/>B F, G H, concurrunt ad partes F, &amp; </s>
  <s xml:id="echoid-s1702" xml:space="preserve">H, vt in N.</s>
  <s xml:id="echoid-s1703" xml:space="preserve"/>
</p>
<div xml:id="echoid-div133" type="float" level="2" n="4">
<note position="right" xlink:label="note-0068-06" xlink:href="note-0068-06a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s1704" xml:space="preserve">Pro intelligentia ſequentium propoſitionum hæc præmitti debent.</s>
  <s xml:id="echoid-s1705" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div135" type="section" level="1" n="59">
<head xml:id="echoid-head90" xml:space="preserve">LEMMA V.</head>
<p style="it">
  <s xml:id="echoid-s1706" xml:space="preserve">Habeat A ad B maiorem proportionem, quàm C ad D. </s>
  <s xml:id="echoid-s1707" xml:space="preserve">Dico, re-<lb/>ctangulum ſub extremis A, D contentum maius eſſe eo, quod ſub me-<lb/>dijs B, C continetur, &amp; </s>
  <s xml:id="echoid-s1708" xml:space="preserve">è conuerſo.</s>
  <s xml:id="echoid-s1709" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1710" xml:space="preserve">Flat vt C ad D, ita E ad B; </s>
  <s xml:id="echoid-s1711" xml:space="preserve">patet ex elementis, A excedere ipſam E; </s>
  <s xml:id="echoid-s1712" xml:space="preserve">qua-<lb/>re rectangulum A D maius erit rectangulo E D: </s>
  <s xml:id="echoid-s1713" xml:space="preserve">eſt verò rectangulum B,
<pb o="31" file="0069" n="69" rhead="Conicor. Lib. V."/>
C ſub intermedĳs contentum æquale ei, quod <lb/>
<anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/>
ſub extremis E, D quatuor proportionaliũ con-<lb/>tinetur ; </s>
  <s xml:id="echoid-s1714" xml:space="preserve">ergo rectangulum A D maius eſt re-<lb/>ctangulo B C. </s>
  <s xml:id="echoid-s1715" xml:space="preserve">Poſtea ſit rectangulũ A D ma-<lb/>ius rectangulo B C; </s>
  <s xml:id="echoid-s1716" xml:space="preserve">Dico A ad B maiorem pro-<lb/>portionem habere, quàm C ad D; </s>
  <s xml:id="echoid-s1717" xml:space="preserve">Si enim hoc <lb/>verum non eſt, habebit A ad B eandem, aut <lb/>minorem proportionem quàm C ad D, quare rectangulum A D æquale, aut mi-<lb/>nus erit rectangulo B C, quæ ſunt contra hypotheſim ; </s>
  <s xml:id="echoid-s1718" xml:space="preserve">igitur A ad B maiorem <lb/>proportionem babet, quàm C ad D.</s>
  <s xml:id="echoid-s1719" xml:space="preserve"/>
</p>
<div xml:id="echoid-div135" type="float" level="2" n="1">
  <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a">
    <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0069-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div137" type="section" level="1" n="60">
<head xml:id="echoid-head91" xml:space="preserve">LEMMA. VI.</head>
<p style="it">
  <s xml:id="echoid-s1720" xml:space="preserve">SIrectæ linea A B ſecetur bifariam in C, &amp; </s>
  <s xml:id="echoid-s1721" xml:space="preserve">non bifariam in D: </s>
  <s xml:id="echoid-s1722" xml:space="preserve">Dico, <lb/>quod ſemiſsis C B ad alterum ſegmentorum inæqualium D B habet <lb/>maiorẽ proportionẽ, quàm reliquum inæqualiũ AD ad alter ã medietatẽ AC.</s>
  <s xml:id="echoid-s1723" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1724" xml:space="preserve">Quoniam quadratum ſemiſſis C B, ſeu re-<lb/>
<anchor type="figure" xlink:label="fig-0069-02a" xlink:href="fig-0069-02"/>
ctangulum B C A maius eſt rectangulo A D B <lb/>ſub inæqualibus ſegmentis contento;</s>
  <s xml:id="echoid-s1725" xml:space="preserve">ergo ex præ-<lb/>cedenti lemmate C B ad D B maiorem propor-<lb/>tionem habet, quàm A D ad A C; </s>
  <s xml:id="echoid-s1726" xml:space="preserve">Aſſumitur <lb/>in ſequenti prop. </s>
  <s xml:id="echoid-s1727" xml:space="preserve">52. </s>
  <s xml:id="echoid-s1728" xml:space="preserve">problema antiquum in-<lb/>uentionis duarum mediarum continuè proportionalium inter duas rectas lineas <lb/>
<anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/>
datas, cuius conſtructio, &amp; </s>
  <s xml:id="echoid-s1729" xml:space="preserve">demonſtratio ab Apollonio inuenta adhuc legitur apud <lb/>Eutocium, ſed organica quidem illa eſt, &amp; </s>
  <s xml:id="echoid-s1730" xml:space="preserve">ad manuum operationes maximè ac-<lb/>comodata, non omnino diuerſa ab ea, quàm Hero, &amp; </s>
  <s xml:id="echoid-s1731" xml:space="preserve">philo ediderunt. </s>
  <s xml:id="echoid-s1732" xml:space="preserve">At Par-<lb/>menion aliam eiuſdem problematis demonſtrationem Apollonio tribuit paulò di-<lb/>uerſam ab ea , quàm Eutocius recenſuit : </s>
  <s xml:id="echoid-s1733" xml:space="preserve">eam ſane nec percepit, nec rite expo-<lb/>
<anchor type="note" xlink:label="note-0069-02a" xlink:href="note-0069-02"/>
ſuit, Philoponus, quàm enim petitionem non demonſtratam ipſe vocat conſequẽ-<lb/>tia eſt neceſſaria ex deſcriptione hyperboles, quæ omnino ſubintelligi, &amp; </s>
  <s xml:id="echoid-s1734" xml:space="preserve">adiun-<lb/>gi debet, vt colligitur ex Pappi verbis : </s>
  <s xml:id="echoid-s1735" xml:space="preserve">hi enim (ſcilicet Hero, &amp; </s>
  <s xml:id="echoid-s1736" xml:space="preserve">Philo) <lb/>
<anchor type="note" xlink:label="note-0069-03a" xlink:href="note-0069-03"/>
aßerentes problema ſolidum eße, ipſius conſtructionem inſtrumentis tantum per-<lb/>fecerunt congruenter Apollonio Pergæo, qui reſolutionem eius fecit per coniſe-<lb/>ctiones. </s>
  <s xml:id="echoid-s1737" xml:space="preserve">Erit igitur Apollonĳ propoſitio huiuſmodi.</s>
  <s xml:id="echoid-s1738" xml:space="preserve"/>
</p>
<div xml:id="echoid-div137" type="float" level="2" n="1">
  <figure xlink:label="fig-0069-02" xlink:href="fig-0069-02a">
    <image file="0069-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0069-02"/>
  </figure>
<note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">Cõm. lib. <lb/>2. Arch. de <lb/>Sphę a, &amp; <lb/>Cylin. <lb/>Prop. 2.</note>
<note position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">In lib. 5. <lb/>Poſt Ana-<lb/>lit. comm. <lb/>36.</note>
<note position="right" xlink:label="note-0069-03" xlink:href="note-0069-03a" xml:space="preserve">Coll. lib. 3. <lb/>Prop. 4.</note>
</div>
</div>
<div xml:id="echoid-div139" type="section" level="1" n="61">
<head xml:id="echoid-head92" xml:space="preserve">LEMMA VII.</head>
<p style="it">
  <s xml:id="echoid-s1739" xml:space="preserve">INter rectam lineam A C maiorem , &amp; </s>
  <s xml:id="echoid-s1740" xml:space="preserve">B C minorem duas medias <lb/>proportionales reperire.</s>
  <s xml:id="echoid-s1741" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1742" xml:space="preserve">Conueniant illæ ad angulos rectos in A , &amp; </s>
  <s xml:id="echoid-s1743" xml:space="preserve">compleatur Parallelogrammum <lb/>
<anchor type="note" xlink:label="note-0069-04a" xlink:href="note-0069-04"/>
A B D C, cui circumſcribatur circulus diametro D A, &amp; </s>
  <s xml:id="echoid-s1744" xml:space="preserve">per punctum D circa <lb/>aſymptotos C A B deſcribatur hyperbole D F, &amp; </s>
  <s xml:id="echoid-s1745" xml:space="preserve">ducatur recta D M circulum <lb/>
<anchor type="note" xlink:label="note-0069-05a" xlink:href="note-0069-05"/>
tangens in D, &amp; </s>
  <s xml:id="echoid-s1746" xml:space="preserve">recta I D K ſectionem ibidem contingens , occurrens aſym-<lb/>ptotis in I , &amp; </s>
  <s xml:id="echoid-s1747" xml:space="preserve">K, erunt quidem I D, &amp; </s>
  <s xml:id="echoid-s1748" xml:space="preserve">I K æquales inter ſe, &amp; </s>
  <s xml:id="echoid-s1749" xml:space="preserve">D C paral-<lb/>
<anchor type="note" xlink:label="note-0069-06a" xlink:href="note-0069-06"/>
lela eſt A K , ergo I C æqualis eſt C A : </s>
  <s xml:id="echoid-s1750" xml:space="preserve">pari ratione K B æqualis erit B A, <lb/>ſed poſita fuit C A maior quàm A B, ergo in triangulis I A D, &amp; </s>
  <s xml:id="echoid-s1751" xml:space="preserve">K D A baſis <lb/>I A maior erit, quàm A K, &amp; </s>
  <s xml:id="echoid-s1752" xml:space="preserve">latera I D, D K æqualia ſunt, &amp; </s>
  <s xml:id="echoid-s1753" xml:space="preserve">D A eſt commune, <lb/>igitur angulus A D I maior erit angulo A D K, &amp; </s>
  <s xml:id="echoid-s1754" xml:space="preserve">propterearecta line a I K ſectionẽ
<pb o="32" file="0070" n="70" rhead="Apollonij Pergæi"/>
contingens in D intra circulũ cadet ad <lb/>
<anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/>
partes acuti anguli ADK, ſed quælibet <lb/>recta linea ex D inter tangentes K D, <lb/>&amp; </s>
  <s xml:id="echoid-s1755" xml:space="preserve">D M incedens ſecat circulum, &amp; </s>
  <s xml:id="echoid-s1756" xml:space="preserve"><lb/>hyperbolam D F, ergo circuli periphe-<lb/>
<anchor type="note" xlink:label="note-0070-01a" xlink:href="note-0070-01"/>
ria, &amp; </s>
  <s xml:id="echoid-s1757" xml:space="preserve">hyperbole non ad eaſdem par-<lb/>tes cauæ ſe mutuo ſecant in duobus pun-<lb/>
<anchor type="note" xlink:label="note-0070-02a" xlink:href="note-0070-02"/>
ctis : </s>
  <s xml:id="echoid-s1758" xml:space="preserve">concurrant in D, &amp; </s>
  <s xml:id="echoid-s1759" xml:space="preserve">F, &amp; </s>
  <s xml:id="echoid-s1760" xml:space="preserve">co-<lb/>niungatur recta linea D F, quæ pro-<lb/>ducta ſecet aſymptotos in punctis G , <lb/>
<anchor type="note" xlink:label="note-0070-03a" xlink:href="note-0070-03"/>
&amp; </s>
  <s xml:id="echoid-s1761" xml:space="preserve">H : </s>
  <s xml:id="echoid-s1762" xml:space="preserve">oſtendendũ eſt rectas B H, &amp; </s>
  <s xml:id="echoid-s1763" xml:space="preserve">G C <lb/>eſſe duas medias proportionales quæſitas. <lb/></s>
  <s xml:id="echoid-s1764" xml:space="preserve">Quoniã eiuſdem rectæ lincæ portiones G <lb/>
<anchor type="note" xlink:label="note-0070-04a" xlink:href="note-0070-04"/>
D, &amp; </s>
  <s xml:id="echoid-s1765" xml:space="preserve">F H inter hyperbolen, &amp; </s>
  <s xml:id="echoid-s1766" xml:space="preserve">aſym-<lb/>ptotos interceptæ æquales ſunt inter ſe, addita communi D F, erunt F G, &amp; </s>
  <s xml:id="echoid-s1767" xml:space="preserve">G H <lb/>inter ſe quoq; </s>
  <s xml:id="echoid-s1768" xml:space="preserve">æquales quare rectangulum D H F æquale erit rectangulo F G D, ſed <lb/>rectangulũ A H B æquale eſt rectangulo D H F , (eo quod ab eodem puncto H extra <lb/>circulum poſito ducuntur duæ rectæ lineæ circulum ſecantes): </s>
  <s xml:id="echoid-s1769" xml:space="preserve">ſimili modo rectangulũ <lb/>A G C æquale eſt rectangulo F G D, igitur duo rectangula A G C, &amp; </s>
  <s xml:id="echoid-s1770" xml:space="preserve">A H B æqualia <lb/>inter ſe erunt, &amp; </s>
  <s xml:id="echoid-s1771" xml:space="preserve">ideo vt G A ad A H, ita erit reciprocè B H ad G C, ſed vt G A ad <lb/>A H; </s>
  <s xml:id="echoid-s1772" xml:space="preserve">ita eſt D B ad B H, nec non G C ad C D, (propter æquidiſtantiã ipſarum D B, <lb/>G A, &amp; </s>
  <s xml:id="echoid-s1773" xml:space="preserve">ipſarum C D, &amp; </s>
  <s xml:id="echoid-s1774" xml:space="preserve">A H, &amp; </s>
  <s xml:id="echoid-s1775" xml:space="preserve">ſimilitudin<unsure/>em triangulorum), quare D B, ſeu <lb/>C A ad B H eandem proportionem habebit, quam B H ad G C, &amp; </s>
  <s xml:id="echoid-s1776" xml:space="preserve">eandem , <lb/>quàm habet G C ad C D, ſeu ad A B, &amp; </s>
  <s xml:id="echoid-s1777" xml:space="preserve">propterea quatuor rectæ lineæ C A, <lb/>B H , C G , &amp; </s>
  <s xml:id="echoid-s1778" xml:space="preserve">B A erunt in continua proportionalitate , quod erat propoſitum.</s>
  <s xml:id="echoid-s1779" xml:space="preserve"/>
</p>
<div xml:id="echoid-div139" type="float" level="2" n="1">
<note position="right" xlink:label="note-0069-04" xlink:href="note-0069-04a" xml:space="preserve">Prop. 4. <lb/>lib. 2.</note>
<note position="right" xlink:label="note-0069-05" xlink:href="note-0069-05a" xml:space="preserve">Prop. 34. <lb/>lib. 1.</note>
<note position="right" xlink:label="note-0069-06" xlink:href="note-0069-06a" xml:space="preserve">3. lib. 1.</note>
  <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a">
    <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0070-01"/>
  </figure>
<note position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">36. lib. 1.</note>
<note position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">33. lib. 4.</note>
<note position="left" xlink:label="note-0070-03" xlink:href="note-0070-03a" xml:space="preserve">8. lib. 2.</note>
<note position="left" xlink:label="note-0070-04" xlink:href="note-0070-04a" xml:space="preserve">Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div141" type="section" level="1" n="62">
<head xml:id="echoid-head93" xml:space="preserve">SECTIO OCTAVA</head>
<head xml:id="echoid-head94" xml:space="preserve">Continens Prop. IL. L. LI. LII. LIII. Apoll.</head>
<p>
  <s xml:id="echoid-s1780" xml:space="preserve">SI menſura non excedit comparatam, nullus ramorum ſecantiũ <lb/>ex concurſu egredientium erit Breuiſecans: </s>
  <s xml:id="echoid-s1781" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1782" xml:space="preserve">lineæ breuiſſimæ <lb/>ab extremitatibus ramorum ductæ in ſectione abſcindunt ex axi li-<lb/>neam maiorem, quàm abſcindunt rami (51. </s>
  <s xml:id="echoid-s1783" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1784" xml:space="preserve">52.) </s>
  <s xml:id="echoid-s1785" xml:space="preserve">Si verò menſura <lb/>
<anchor type="note" xlink:label="note-0070-05a" xlink:href="note-0070-05"/>
excedit comparatã exponi debet linea certis quibuſdam legibus in-<lb/>uenienda, quæ vocabitur TRVTINA. </s>
  <s xml:id="echoid-s1786" xml:space="preserve">Et ſiquidẽ perpendicularis <lb/>maior fuerit illa, tunc rami habebunt proprietates memoratas; </s>
  <s xml:id="echoid-s1787" xml:space="preserve">ſi ve-<lb/>rò æqualis fuerit, tunc inter ramos vnicus breuiſecans aſſignari po-<lb/>teſt, &amp; </s>
  <s xml:id="echoid-s1788" xml:space="preserve">propietates reliquorũ ramorũ erunt illæ eædem ſuperius ex-<lb/>poſitæ ſi verò minor eſt illa, ramorũ omniũ duo tantum breuiſecan-<lb/>tes erunt, reliquorum verò, qui non intercipiuntur inter duosbre-<lb/>uiſecantes, eædem propietates erunt; </s>
  <s xml:id="echoid-s1789" xml:space="preserve">eorũ verò, qui intercipiuntur, <lb/>lineæ breuiſſimæ egredientes ab earum extremitatibus abſcindunt <lb/>ex axi lineas minores , quàm ſecant rami ipſi. </s>
  <s xml:id="echoid-s1790" xml:space="preserve">Oportet autem
<pb o="33" file="0071" n="71" rhead="Conicor. Lib. V."/>
in ellipſi, vt menſura ſumatur in maiori duorum axium, &amp; </s>
  <s xml:id="echoid-s1791" xml:space="preserve">rami <lb/>egrediantur ad eius ſectionem.</s>
  <s xml:id="echoid-s1792" xml:space="preserve"/>
</p>
<div xml:id="echoid-div141" type="float" level="2" n="1">
<note position="right" xlink:label="note-0070-05" xlink:href="note-0070-05a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div143" type="section" level="1" n="63">
<head xml:id="echoid-head95" xml:space="preserve">PROPOSITIO IL. &amp; L.</head>
<p>
  <s xml:id="echoid-s1793" xml:space="preserve">EXE concurſu ſuper perpendicularem ED educamus E B ſe-<lb/>cantem menſuram A D in F, &amp; </s>
  <s xml:id="echoid-s1794" xml:space="preserve">ſectionem A B in B, &amp; </s>
  <s xml:id="echoid-s1795" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/>
ſit A H dimidium erecti; </s>
  <s xml:id="echoid-s1796" xml:space="preserve">ſitque menſura A D non maior, quàm <lb/>H A. </s>
  <s xml:id="echoid-s1797" xml:space="preserve">Dico quod BF non erit breuiſſima, &amp; </s>
  <s xml:id="echoid-s1798" xml:space="preserve">minima egrediens <lb/>
<anchor type="note" xlink:label="note-0071-02a" xlink:href="note-0071-02"/>
ex B abſcindit ex ſagitta maiorem lineam, quàm F A: </s>
  <s xml:id="echoid-s1799" xml:space="preserve">at ſi fue-<lb/>rit A D maior , quàm A H, tunc B F poteſt eſſe linea breuiſ-<lb/>ſima.</s>
  <s xml:id="echoid-s1800" xml:space="preserve"/>
</p>
<div xml:id="echoid-div143" type="float" level="2" n="1">
<note position="left" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0071-02" xlink:href="note-0071-02a" xml:space="preserve">c</note>
</div>
  <figure>
    <image file="0071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0071-01"/>
  </figure>
<p>
  <s xml:id="echoid-s1801" xml:space="preserve">EDucamus iam B I perpendicularem ad axim, &amp; </s>
  <s xml:id="echoid-s1802" xml:space="preserve">ſupponamus prius A <lb/>D non maiorem quàm A H , &amp; </s>
  <s xml:id="echoid-s1803" xml:space="preserve">ſit ſectio parabole ; </s>
  <s xml:id="echoid-s1804" xml:space="preserve">igitur D I mi-<lb/>
<anchor type="note" xlink:label="note-0071-03a" xlink:href="note-0071-03"/>
nor eſt , quàm A H, &amp; </s>
  <s xml:id="echoid-s1805" xml:space="preserve">ponatur G I æqualis A H, erit B G minima (8. <lb/></s>
  <s xml:id="echoid-s1806" xml:space="preserve">ex quinto) &amp; </s>
  <s xml:id="echoid-s1807" xml:space="preserve">abſcindit G A ex ſagitta maiorem , quàm A F; </s>
  <s xml:id="echoid-s1808" xml:space="preserve">ſi verò ſe-<lb/>ctio fuerit hyperbole, aut ellipſis, ſit centrum C; </s>
  <s xml:id="echoid-s1809" xml:space="preserve">ergo A C ad A H non <lb/>
<anchor type="note" xlink:label="note-0071-04a" xlink:href="note-0071-04"/>
habet maiorem proportionem, quàm ad A D, quare C I ad I F maiorem <lb/>proportionem habet, quàm C A ad A H; </s>
  <s xml:id="echoid-s1810" xml:space="preserve">ponatur ergo I C ad I G , vt <lb/>A C ad A H; </s>
  <s xml:id="echoid-s1811" xml:space="preserve">ergo B G eſt minima , &amp; </s>
  <s xml:id="echoid-s1812" xml:space="preserve">abſcindit (9. </s>
  <s xml:id="echoid-s1813" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1814" xml:space="preserve">10. </s>
  <s xml:id="echoid-s1815" xml:space="preserve">ex quinto) <lb/>G A maiorem , quam F A, quod erat oſtendendum.</s>
  <s xml:id="echoid-s1816" xml:space="preserve"/>
</p>
<div xml:id="echoid-div144" type="float" level="2" n="2">
<note position="left" xlink:label="note-0071-03" xlink:href="note-0071-03a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0071-04" xlink:href="note-0071-04a" xml:space="preserve">e</note>
</div>
<pb o="34" file="0072" n="72" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div146" type="section" level="1" n="64">
<head xml:id="echoid-head96" xml:space="preserve">PROPOSITIO LI.</head>
<p>
  <s xml:id="echoid-s1817" xml:space="preserve">DEindè ſit D A maior quàm A C , ſitque prius ſectio pa-<lb/>rabole , &amp; </s>
  <s xml:id="echoid-s1818" xml:space="preserve">ſecetur ex D A ipſa D F æqualis A C, &amp; </s>
  <s xml:id="echoid-s1819" xml:space="preserve">A <lb/>G fiat pars tertia ipſius A F, educaturque B G perpendicularis <lb/>ad axim, &amp; </s>
  <s xml:id="echoid-s1820" xml:space="preserve">vt D F ad F G, ita fiat B G ad lineam H (&amp; </s>
  <s xml:id="echoid-s1821" xml:space="preserve">hæc <lb/>eſt Trutina) coniungaturque B E ; </s>
  <s xml:id="echoid-s1822" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1823" xml:space="preserve">ſiquidem D E fuerit ma-<lb/>
<anchor type="note" xlink:label="note-0072-01a" xlink:href="note-0072-01"/>
ior quàm H. </s>
  <s xml:id="echoid-s1824" xml:space="preserve">Dico, quod nullus ramus breuiſecans duci poteſt.</s>
  <s xml:id="echoid-s1825" xml:space="preserve"/>
</p>
<div xml:id="echoid-div146" type="float" level="2" n="1">
<note position="right" xlink:label="note-0072-01" xlink:href="note-0072-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s1826" xml:space="preserve">Quoniam D E maior eſt, quàm H habebit D E ad B G, nempe D I <lb/>
<anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/>
ad I G maiorem rationem , quàm G F ad F D, &amp; </s>
  <s xml:id="echoid-s1827" xml:space="preserve">componatur propor-<lb/>tio , vt demonſtretur , quod I G minor ſit , quàm D F, quæ æqualis <lb/>eſt ipſi A C; </s>
  <s xml:id="echoid-s1828" xml:space="preserve">breuiſſima itaque egrediens ex B abſcindit ex ſagitta A <lb/>D maiorem lineam , quàm A I (13. </s>
  <s xml:id="echoid-s1829" xml:space="preserve">ex quinto) ; </s>
  <s xml:id="echoid-s1830" xml:space="preserve">poſtea ducamus ex E <lb/>ad ſectionem ramos E K, E L ad vtramque partem B E, &amp; </s>
  <s xml:id="echoid-s1831" xml:space="preserve">duas per-<lb/>pendiculares <lb/>
<anchor type="figure" xlink:label="fig-0072-01a" xlink:href="fig-0072-01"/>
KM, LN, pro-<lb/>ducamus vſq; <lb/></s>
  <s xml:id="echoid-s1832" xml:space="preserve">ad QO tan-<lb/>gẽtem ſectio-<lb/>nem in B; </s>
  <s xml:id="echoid-s1833" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1834" xml:space="preserve"><lb/>quoniã ſectio <lb/>eſt, parabole, <lb/>&amp; </s>
  <s xml:id="echoid-s1835" xml:space="preserve">OQ tãgens <lb/>eſt, igitur OG <lb/>
<anchor type="note" xlink:label="note-0072-03a" xlink:href="note-0072-03"/>
eſt dupla ip-<lb/>ſius A G, quę <lb/>eſt ſemiſſis ip-<lb/>ſi us F G; </s>
  <s xml:id="echoid-s1836" xml:space="preserve">ergo <lb/>
<anchor type="note" xlink:label="note-0072-04a" xlink:href="note-0072-04"/>
G F æqualis <lb/>eſt G O, erit <lb/>igitur G O ad <lb/>O M, nempe <lb/>B G ad P M <lb/>in maiori pro-<lb/>portione, quã <lb/>M F ad F G; <lb/></s>
  <s xml:id="echoid-s1837" xml:space="preserve">
<anchor type="note" xlink:label="note-0072-05a" xlink:href="note-0072-05"/>
itaque M K in F M minus eſt , quàm B G in G F, quod eſt minus quàm <lb/>E D in D F propterea quod E D maior eſt quàm H; </s>
  <s xml:id="echoid-s1838" xml:space="preserve">igitur E D in D F <lb/>multò maius eſt, quàm K M in MF, quare ED ad M K, nempe D R ad <lb/>R M maiorem rationem habet, quàm M F ad F D, &amp; </s>
  <s xml:id="echoid-s1839" xml:space="preserve">componendo patet, <lb/>
<anchor type="note" xlink:label="note-0072-06a" xlink:href="note-0072-06"/>
quod D F maior ſit, quàm R M. </s>
  <s xml:id="echoid-s1840" xml:space="preserve">Igitur breuiſſima egrediens ex K (13. <lb/></s>
  <s xml:id="echoid-s1841" xml:space="preserve">
<anchor type="note" xlink:label="note-0072-07a" xlink:href="note-0072-07"/>
ex quinto) cadit extra R K; </s>
  <s xml:id="echoid-s1842" xml:space="preserve">Et ſimili modo conſtat, quod breuiſſima
<pb o="35" file="0073" n="73" rhead="Conicor. Lib. V."/>
egrediens ex puncto L cadit extra L S, quapropter duci non poteſt ex E <lb/>ad ſectionem L B A linea, aliqua cuius portio intercepta inter axim, &amp; </s>
  <s xml:id="echoid-s1843" xml:space="preserve"><lb/>ſectionem, ſit linea breuiſſima.</s>
  <s xml:id="echoid-s1844" xml:space="preserve"/>
</p>
<div xml:id="echoid-div147" type="float" level="2" n="2">
<note position="right" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0072-01" xlink:href="fig-0072-01a">
    <image file="0072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0072-01"/>
  </figure>
<note position="left" xlink:label="note-0072-03" xlink:href="note-0072-03a" xml:space="preserve">35. lib. 1.</note>
<note position="right" xlink:label="note-0072-04" xlink:href="note-0072-04a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0072-05" xlink:href="note-0072-05a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0072-06" xlink:href="note-0072-06a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0072-07" xlink:href="note-0072-07a" xml:space="preserve">f</note>
</div>
<p>
  <s xml:id="echoid-s1845" xml:space="preserve">Pariter demonſtrabitur, quemadmodum iam oſtenſum eſt, quod ſi E D <lb/>fuerit æqualis H, tunc GI æqualis erit D F, quæ eſt æqualis ipſi A C; </s>
  <s xml:id="echoid-s1846" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1847" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0073-01a" xlink:href="note-0073-01"/>
ideo B I (8. </s>
  <s xml:id="echoid-s1848" xml:space="preserve">ex quinto) vna eſt ex breuiſſimis, non autem R K, quia de-<lb/>monſtrabitur, quod E D ad M K, nempe D R ad R M maiorem rationem <lb/>habet, quàm M F ad F D, &amp; </s>
  <s xml:id="echoid-s1849" xml:space="preserve">propterea D F maior erit, quàm R M; </s>
  <s xml:id="echoid-s1850" xml:space="preserve">bre-<lb/>uiſſima ergo cadit extra R K. </s>
  <s xml:id="echoid-s1851" xml:space="preserve">(13. </s>
  <s xml:id="echoid-s1852" xml:space="preserve">ex quinto) Et S L quoque non eſt ex <lb/>breuiſſimis, quod ita demonſtrabimus; </s>
  <s xml:id="echoid-s1853" xml:space="preserve">Si N S minor eſt, quàm D F; </s>
  <s xml:id="echoid-s1854" xml:space="preserve">ergo <lb/>breuiſſima egrediens ex L cadit extra S L; </s>
  <s xml:id="echoid-s1855" xml:space="preserve">Non igitur ex E duci poteſt <lb/>ad ſectionem linea breuiſecans præter E B, &amp; </s>
  <s xml:id="echoid-s1856" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s1857" xml:space="preserve"/>
</p>
<div xml:id="echoid-div148" type="float" level="2" n="3">
<note position="left" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">g</note>
</div>
<p>
  <s xml:id="echoid-s1858" xml:space="preserve">Tertio loco ſit E D minor quàm H, &amp; </s>
  <s xml:id="echoid-s1859" xml:space="preserve">oſtendetur quod E D in D F <lb/>minus eſt, quàm B G in G F; </s>
  <s xml:id="echoid-s1860" xml:space="preserve">poſtea ponamus T G in G F æquale illi, &amp; </s>
  <s xml:id="echoid-s1861" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0073-02a" xlink:href="note-0073-02"/>
erigamus ſuper F perpendicularem F V, &amp; </s>
  <s xml:id="echoid-s1862" xml:space="preserve">ducamus per T ſectionem <lb/>
<anchor type="note" xlink:label="note-0073-03a" xlink:href="note-0073-03"/>
hyperbolicam circa duas continentes A F, &amp; </s>
  <s xml:id="echoid-s1863" xml:space="preserve">F V; </s>
  <s xml:id="echoid-s1864" xml:space="preserve">duæ ſectiones ſe mu-<lb/>tuò ſecabunt in duobus punctis, &amp; </s>
  <s xml:id="echoid-s1865" xml:space="preserve">ſint K, L, &amp; </s>
  <s xml:id="echoid-s1866" xml:space="preserve">educamus ex illis duas <lb/>L N, P K M perpendiculares ad A D. </s>
  <s xml:id="echoid-s1867" xml:space="preserve">Et quoniam perpendiculares K M, <lb/>T G, L N parallelæ ſunt continenti V F, erit K M in M F æquale L N in <lb/>N F (12. </s>
  <s xml:id="echoid-s1868" xml:space="preserve">ex ſecundo) &amp; </s>
  <s xml:id="echoid-s1869" xml:space="preserve">quodlibet eorum æquale eſt T G in G F, quod fa-<lb/>ctum eſt æquale E D in D F; </s>
  <s xml:id="echoid-s1870" xml:space="preserve">igitur E D ad K M, nempe D R ad R M eſt <lb/>vt M F ad F D, &amp; </s>
  <s xml:id="echoid-s1871" xml:space="preserve">componendo patet, quod D F eſt æqualis R M, &amp; </s>
  <s xml:id="echoid-s1872" xml:space="preserve">pro-<lb/>
<anchor type="note" xlink:label="note-0073-04a" xlink:href="note-0073-04"/>
pterea K R eſt linea breuiſſima (8. </s>
  <s xml:id="echoid-s1873" xml:space="preserve">ex quinto.)</s>
  <s xml:id="echoid-s1874" xml:space="preserve"/>
</p>
<div xml:id="echoid-div149" type="float" level="2" n="4">
<note position="left" xlink:label="note-0073-02" xlink:href="note-0073-02a" xml:space="preserve">h</note>
<note position="right" xlink:label="note-0073-03" xlink:href="note-0073-03a" xml:space="preserve">4. lib. 2.</note>
<note position="left" xlink:label="note-0073-04" xlink:href="note-0073-04a" xml:space="preserve">i</note>
</div>
<p>
  <s xml:id="echoid-s1875" xml:space="preserve">Et ſimiliter patebit, quod L S ſit breuiſſima.</s>
  <s xml:id="echoid-s1876" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">k</note>
<p>
  <s xml:id="echoid-s1877" xml:space="preserve">Et cum B I intercipiatur inter illas patet etiam, quod B G in G F ma-<lb/>
<anchor type="note" xlink:label="note-0073-06a" xlink:href="note-0073-06"/>
ius ſit, quàm E D in D F, oſtendetur vt dictum eſt, quod I G maior ſit, <lb/>quàm D F; </s>
  <s xml:id="echoid-s1878" xml:space="preserve">breuiſſima ergo ducta ex B cadit inter I, &amp; </s>
  <s xml:id="echoid-s1879" xml:space="preserve">A.</s>
  <s xml:id="echoid-s1880" xml:space="preserve"/>
</p>
<div xml:id="echoid-div150" type="float" level="2" n="5">
<note position="left" xlink:label="note-0073-06" xlink:href="note-0073-06a" xml:space="preserve">l</note>
</div>
<p>
  <s xml:id="echoid-s1881" xml:space="preserve">Deindè ex concurſu E ad ſectionem parobolicam A B Z educamus E X, <lb/>
<anchor type="note" xlink:label="note-0073-07a" xlink:href="note-0073-07"/>
E Z; </s>
  <s xml:id="echoid-s1882" xml:space="preserve">quas interſecant l Z, X Y perpendiculares ad A D, quæ parallelæ <lb/>ſunt continenti F V ſecantes K T L hyperbolen, ergo a Y in Y F æquale <lb/>eſt G T in G F, quod factum eſt æquale E D in D F, itaque E D in D F <lb/>maius eſt, quàm X Y in Y F; </s>
  <s xml:id="echoid-s1883" xml:space="preserve">igitur E D ad X Y, quæ eſt vt D b ad b Y <lb/>maiorem rationem habet, quàm Y F ad F D, &amp; </s>
  <s xml:id="echoid-s1884" xml:space="preserve">componendo patet, quod <lb/>F D maior eſt quàm b Y; </s>
  <s xml:id="echoid-s1885" xml:space="preserve">itaque breuiſſima egrediens ex X abſcindit ex <lb/>A D lineam maiorem, quàm b A; </s>
  <s xml:id="echoid-s1886" xml:space="preserve">Simili modo demonſtrabitur, quod Z c <lb/>non ſit breuiſſima, &amp; </s>
  <s xml:id="echoid-s1887" xml:space="preserve">quod breuiſſima egrediens ex Z abſcindit ex A D <lb/>
<anchor type="note" xlink:label="note-0073-08a" xlink:href="note-0073-08"/>
lineam maiorem, quàm A c, &amp; </s>
  <s xml:id="echoid-s1888" xml:space="preserve">hoc erat propoſitum.</s>
  <s xml:id="echoid-s1889" xml:space="preserve"/>
</p>
<div xml:id="echoid-div151" type="float" level="2" n="6">
<note position="left" xlink:label="note-0073-07" xlink:href="note-0073-07a" xml:space="preserve">m</note>
<note position="left" xlink:label="note-0073-08" xlink:href="note-0073-08a" xml:space="preserve">n</note>
</div>
</div>
<div xml:id="echoid-div153" type="section" level="1" n="65">
<head xml:id="echoid-head97" xml:space="preserve">PROPOSITIO LII. LIII.</head>
<p>
  <s xml:id="echoid-s1890" xml:space="preserve">Deindè ſit ſectio hyperbole, aut ellipſis A B, &amp; </s>
  <s xml:id="echoid-s1891" xml:space="preserve">axis illius C <lb/>A D, centrum C, &amp; </s>
  <s xml:id="echoid-s1892" xml:space="preserve">D A menſura, quæ ſit maior dimidio ere-<lb/>cti, &amp; </s>
  <s xml:id="echoid-s1893" xml:space="preserve">perpendicularis E D. </s>
  <s xml:id="echoid-s1894" xml:space="preserve">Dico, quod rami egredientes ex E <lb/>habent ſuperiùs expoſitas proprietates.</s>
  <s xml:id="echoid-s1895" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">a</note>
<pb o="36" file="0074" n="74" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s1896" xml:space="preserve">ITaque per C producamus C I parallelam perpendiculari E D, &amp; </s>
  <s xml:id="echoid-s1897" xml:space="preserve">pona-<lb/>
<anchor type="note" xlink:label="note-0074-01a" xlink:href="note-0074-01"/>
mus quamlibet duarum proportionum C F ad F D, &amp; </s>
  <s xml:id="echoid-s1898" xml:space="preserve">E K ad K D, vt <lb/>proportio figuræ, &amp; </s>
  <s xml:id="echoid-s1899" xml:space="preserve">educamus ex E, K rectas E I, K S parallelas ipſi C <lb/>AD, &amp; </s>
  <s xml:id="echoid-s1900" xml:space="preserve">interponamus inter F C, C A duas medias proportionales C N, <lb/>
<anchor type="note" xlink:label="note-0074-02a" xlink:href="note-0074-02"/>
<anchor type="note" xlink:label="note-0074-03a" xlink:href="note-0074-03"/>
C O, &amp; </s>
  <s xml:id="echoid-s1901" xml:space="preserve">erigamus per O perpendicularem B O, quæ occurrat ſectioni in <lb/>B; </s>
  <s xml:id="echoid-s1902" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1903" xml:space="preserve">ponamus proportionem alicuius lineæ, vt Q ad B O compoſitam <lb/>
<anchor type="note" xlink:label="note-0074-04a" xlink:href="note-0074-04"/>
ex C D ad D F, &amp; </s>
  <s xml:id="echoid-s1904" xml:space="preserve">F O ad O C, &amp; </s>
  <s xml:id="echoid-s1905" xml:space="preserve">ſit E D maior, quàm Q Trutina: </s>
  <s xml:id="echoid-s1906" xml:space="preserve">Di-<lb/>co, quod nulla breuiſecans egreditur ex E ad ſectionem, &amp; </s>
  <s xml:id="echoid-s1907" xml:space="preserve">linea breuiſ-<lb/>ſima, egrediens ab extremitate cuiuslibet rami aſſignati, abſcindit cum <lb/>A ab axi maiorem lineam, quàm ſecant illi rami. </s>
  <s xml:id="echoid-s1908" xml:space="preserve">Producatur priùs E B <lb/>
<anchor type="note" xlink:label="note-0074-05a" xlink:href="note-0074-05"/>
ſecans axim in H, &amp; </s>
  <s xml:id="echoid-s1909" xml:space="preserve">quia E D maior eſt, quàm Q, ergo proportio E D <lb/>
<anchor type="note" xlink:label="note-0074-06a" xlink:href="note-0074-06"/>
ad B O (quæ componitur ex E D ad D K, nempe I C ad C S, &amp; </s>
  <s xml:id="echoid-s1910" xml:space="preserve">ex D <lb/>K, nempe G O ad O B) maior eſt proportione, quàm habet Q ad B O, <lb/>quæ ex hypotheſi componebatur ex C D ad D F, &amp; </s>
  <s xml:id="echoid-s1911" xml:space="preserve">ex F O ad O C; </s>
  <s xml:id="echoid-s1912" xml:space="preserve">ſed <lb/>
<anchor type="note" xlink:label="note-0074-07a" xlink:href="note-0074-07"/>
E D ad D K eſt, vt C D ad D F (quia quælibet earum eſt, vt proportio <lb/>figuræ compoſitæ, vel diuiſæ) remanet proportio O G ad O B maior ea, <lb/>quàm habet F O ad O C; </s>
  <s xml:id="echoid-s1913" xml:space="preserve">igitur O G in O C, nempe rectangulum C G <lb/>
<anchor type="note" xlink:label="note-0074-08a" xlink:href="note-0074-08"/>
maius eſt, quàm B O in O F: </s>
  <s xml:id="echoid-s1914" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1915" xml:space="preserve">ponamus rectangulum F G commune, <lb/>
<anchor type="note" xlink:label="note-0074-09a" xlink:href="note-0074-09"/>
erit rectangulum F S maius, quàm B G in G M; </s>
  <s xml:id="echoid-s1916" xml:space="preserve">eſt verò rectangulum <lb/>F S æquale rectangulo E M (eo quod E K ad K D, nempe ad F M eſt, vt <lb/>S M ad M K, quia quælibet earum eſt, vt proportio figuræ; </s>
  <s xml:id="echoid-s1917" xml:space="preserve">itaque re-<lb/>
<anchor type="note" xlink:label="note-0074-10a" xlink:href="note-0074-10"/>
ctangulum E M maius eſt, quàm M G in G B, &amp; </s>
  <s xml:id="echoid-s1918" xml:space="preserve">propterea E K ad B G, <lb/>
<anchor type="note" xlink:label="note-0074-11a" xlink:href="note-0074-11"/>
nempe K R ad R G maiorem rationem habet, quàm G M ad M K, ergo <lb/>componendo, patet, quod K M, nempe D F maior eſt, quàm G R, &amp; </s>
  <s xml:id="echoid-s1919" xml:space="preserve"><lb/>ideo E I ad K M, nempe C D ad D F, ſeu I C ad C S minorem propor-<lb/>tionem habet, quàm E I ad G R, quæ eſt, vt I T ad B G, propter ſimi-<lb/>litudinem duorum triangulorum E I T, B G R, ergo I T ad B G maiorem <lb/>
<anchor type="note" xlink:label="note-0074-12a" xlink:href="note-0074-12"/>
rationem habet, quàm I C ad C S, ſeu ad O G; </s>
  <s xml:id="echoid-s1920" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1921" xml:space="preserve">comparando homo-<lb/>
<anchor type="note" xlink:label="note-0074-13a" xlink:href="note-0074-13"/>
logorum differentias in hyperbola, &amp; </s>
  <s xml:id="echoid-s1922" xml:space="preserve">eorum ſummas in ellipſi, habebit <lb/>C T ad B O, nempe C H ad H O maiorem rationem, quàm I C ad C S, <lb/>nempe C D ad D F, &amp; </s>
  <s xml:id="echoid-s1923" xml:space="preserve">diuidendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s1924" xml:space="preserve">componendo in elli-<lb/>pſi C O ad O H, habebit maiorem proportionem quàm C F ad F D, quæ <lb/>eſt, vt proportio figuræ; </s>
  <s xml:id="echoid-s1925" xml:space="preserve">igitur breuiſſima egrediens ex B (9. </s>
  <s xml:id="echoid-s1926" xml:space="preserve">10. </s>
  <s xml:id="echoid-s1927" xml:space="preserve">ex quinto) <lb/>abſcindit cum A maiorem lineam, quàm A H.</s>
  <s xml:id="echoid-s1928" xml:space="preserve"/>
</p>
<div xml:id="echoid-div153" type="float" level="2" n="1">
<note position="right" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0074-02" xlink:href="note-0074-02a" xml:space="preserve">Lem. 7.</note>
<note position="right" xlink:label="note-0074-03" xlink:href="note-0074-03a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0074-04" xlink:href="note-0074-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0074-05" xlink:href="note-0074-05a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0074-06" xlink:href="note-0074-06a" xml:space="preserve">f</note>
<note position="right" xlink:label="note-0074-07" xlink:href="note-0074-07a" xml:space="preserve">g</note>
<note position="left" xlink:label="note-0074-08" xlink:href="note-0074-08a" xml:space="preserve">Lem. 5. <lb/>præmiſſ.</note>
<note position="right" xlink:label="note-0074-09" xlink:href="note-0074-09a" xml:space="preserve">h</note>
<note position="right" xlink:label="note-0074-10" xlink:href="note-0074-10a" xml:space="preserve">i</note>
<note position="left" xlink:label="note-0074-11" xlink:href="note-0074-11a" xml:space="preserve">ibidem.</note>
<note position="right" xlink:label="note-0074-12" xlink:href="note-0074-12a" xml:space="preserve">K</note>
<note position="left" xlink:label="note-0074-13" xlink:href="note-0074-13a" xml:space="preserve">Lem. 4. <lb/>præm</note>
</div>
<p>
  <s xml:id="echoid-s1929" xml:space="preserve">Poſteà educamus ex E lineam occurrentem ſectioni in V, &amp; </s>
  <s xml:id="echoid-s1930" xml:space="preserve">produca-<lb/>mus eam, quouſque occurrat C I ad X, &amp; </s>
  <s xml:id="echoid-s1931" xml:space="preserve">ducamus per B lineam tan-<lb/>
<anchor type="note" xlink:label="note-0074-14a" xlink:href="note-0074-14"/>
gentem ſectionem, quæ occurrat inclinato, ſiue tranſuerſæ in a, &amp; </s>
  <s xml:id="echoid-s1932" xml:space="preserve">per V <lb/>ducamus perpendicularem ſuper axim, cui occurrat ad c, &amp; </s>
  <s xml:id="echoid-s1933" xml:space="preserve">occurrat tan-<lb/>genti B a in d; </s>
  <s xml:id="echoid-s1934" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1935" xml:space="preserve">quoniam O G ad O B, quemadmodum demonſtraui-<lb/>mus, maiorem proportionem habet, quàm F O ad O C, ponamus fO ad <lb/>O B, vt F O ad O C, &amp; </s>
  <s xml:id="echoid-s1936" xml:space="preserve">per f producamus f g h parallelam axi A D: </s>
  <s xml:id="echoid-s1937" xml:space="preserve">Et <lb/>
<anchor type="note" xlink:label="note-0074-15a" xlink:href="note-0074-15"/>
quia f O ad O B eſt, vt F O ad O C, erit rectangulum f O C æquale B O <lb/>in O F, &amp; </s>
  <s xml:id="echoid-s1938" xml:space="preserve">ponamus rectangulum f F communiter fiet B f in f g æquale g <lb/>
<anchor type="note" xlink:label="note-0074-16a" xlink:href="note-0074-16"/>
F in F C, &amp; </s>
  <s xml:id="echoid-s1939" xml:space="preserve">quia C O inuerſa in trutinatam C a æquale eſt quadrato C <lb/>A dimidij inclinati, ſiue tranſuerſæ (39. </s>
  <s xml:id="echoid-s1940" xml:space="preserve">ex primo) erit O C ad C A, vt <lb/>C A ad C a; </s>
  <s xml:id="echoid-s1941" xml:space="preserve">igitur C a eſt linea quinta proportionalis aliarum quatuor <lb/>
<anchor type="note" xlink:label="note-0074-17a" xlink:href="note-0074-17"/>
<anchor type="note" xlink:label="note-0074-18a" xlink:href="note-0074-18"/>
linearum proportionalium aſſignatarum; </s>
  <s xml:id="echoid-s1942" xml:space="preserve">ergo F C ad C O eſt, vt C O ad
<pb o="37" file="0075" n="75" rhead="Conicor. Lib. V."/>
<anchor type="figure" xlink:label="fig-0075-01a" xlink:href="fig-0075-01"/>
C a, &amp; </s>
  <s xml:id="echoid-s1943" xml:space="preserve">comparando homologorum differentias erit F O ad O a, vt F C <lb/>
<anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/>
ad C O, quæ eſt, vt f B ad B O, nempe f h ad O a; </s>
  <s xml:id="echoid-s1944" xml:space="preserve">igitur proportiones <lb/>ipſarum F O, f h ad eandem O a eædem ſunt; </s>
  <s xml:id="echoid-s1945" xml:space="preserve">ergo ſunt æquales; </s>
  <s xml:id="echoid-s1946" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1947" xml:space="preserve">pro-<lb/>pterea f i ad i h maiorem proportionem habet, quàm ad f g, &amp; </s>
  <s xml:id="echoid-s1948" xml:space="preserve">compo-<lb/>
<anchor type="note" xlink:label="note-0075-02a" xlink:href="note-0075-02"/>
nendo f h ad i h, nempe B f ad V i maiorem proportionem habet, quàm <lb/>i g ad g f; </s>
  <s xml:id="echoid-s1949" xml:space="preserve">ergo B f in f g, nempe rectangulum g C maius eſt quàm i V <lb/>in i g, &amp; </s>
  <s xml:id="echoid-s1950" xml:space="preserve">ponamus rectangulum g e commune, erit aggregatum rectan-<lb/>
<anchor type="note" xlink:label="note-0075-03a" xlink:href="note-0075-03"/>
<pb o="38" file="0076" n="76" rhead="Apollonij Pergæi"/>
gulorum C g, g e, in hyperbola, vel eorum exceſſus in ellip ſi maior, <lb/>quàm M e in e V, ergo rectangulum C M, nempe rectangulum E M mul-<lb/>tò maius eſt, quàm V e in e M, &amp; </s>
  <s xml:id="echoid-s1951" xml:space="preserve">propterea E K ad e V, nempe K Y ad <lb/>Y e maiorem proportionem habet, quàm e M ad M K, &amp; </s>
  <s xml:id="echoid-s1952" xml:space="preserve">componendo <lb/>
<anchor type="note" xlink:label="note-0076-01a" xlink:href="note-0076-01"/>
<anchor type="note" xlink:label="note-0076-02a" xlink:href="note-0076-02"/>
patet, quod e Y minor ſit, quàm K M, &amp; </s>
  <s xml:id="echoid-s1953" xml:space="preserve">conſtat (quemadmodum antea <lb/>demonſtrauimus) quod breuiſſima egrediens ex V abſcindit ab axi maio-<lb/>
<anchor type="note" xlink:label="note-0076-03a" xlink:href="note-0076-03"/>
rem lineam quàm c Z.</s>
  <s xml:id="echoid-s1954" xml:space="preserve"/>
</p>
<div xml:id="echoid-div154" type="float" level="2" n="2">
<note position="right" xlink:label="note-0074-14" xlink:href="note-0074-14a" xml:space="preserve">l</note>
<note position="right" xlink:label="note-0074-15" xlink:href="note-0074-15a" xml:space="preserve">m</note>
<note position="right" xlink:label="note-0074-16" xlink:href="note-0074-16a" xml:space="preserve">n</note>
<note position="left" xlink:label="note-0074-17" xlink:href="note-0074-17a" xml:space="preserve">37. primi.</note>
<note position="right" xlink:label="note-0074-18" xlink:href="note-0074-18a" xml:space="preserve">o</note>
  <figure xlink:label="fig-0075-01" xlink:href="fig-0075-01a">
    <image file="0075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0075-01"/>
  </figure>
<note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">Lem. 4.</note>
<note position="left" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">p</note>
<note position="left" xlink:label="note-0075-03" xlink:href="note-0075-03a" xml:space="preserve">q</note>
<note position="left" xlink:label="note-0076-01" xlink:href="note-0076-01a" xml:space="preserve">Lem. 5.</note>
<note position="right" xlink:label="note-0076-02" xlink:href="note-0076-02a" xml:space="preserve">r</note>
<note position="right" xlink:label="note-0076-03" xlink:href="note-0076-03a" xml:space="preserve">s</note>
</div>
<p>
  <s xml:id="echoid-s1955" xml:space="preserve">Simili modo conſtat, quod breuiſſima egrediens ex l eiuſdem ſit rationis.</s>
  <s xml:id="echoid-s1956" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">t</note>
<p>
  <s xml:id="echoid-s1957" xml:space="preserve">DEindè ſit E D æqualis Q, inde demonſtrabitur, (quemadmodum ſu-<lb/>pra factum eſt) quod B H tantùm ſit linea breuiſſima, &amp; </s>
  <s xml:id="echoid-s1958" xml:space="preserve">quod mi-<lb/>
<anchor type="note" xlink:label="note-0076-05a" xlink:href="note-0076-05"/>
nima egrediens ex V abſcindit ab axi cum A maiorem lineam, quàm A <lb/>Z, &amp; </s>
  <s xml:id="echoid-s1959" xml:space="preserve">quod minima egrediens ex l ſecet maiorem lineam, quàm A m.</s>
  <s xml:id="echoid-s1960" xml:space="preserve"/>
</p>
<div xml:id="echoid-div155" type="float" level="2" n="3">
<note position="right" xlink:label="note-0076-05" xlink:href="note-0076-05a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s1961" xml:space="preserve">Tandem pona-<lb/>
<anchor type="figure" xlink:label="fig-0076-01a" xlink:href="fig-0076-01"/>
mus E D minorẽ, <lb/>quàm Q, ergo E <lb/>D ad B O minorẽ <lb/>proportionem ha-<lb/>bet, quàm Q ad <lb/>eandem; </s>
  <s xml:id="echoid-s1962" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1963" xml:space="preserve">demõ-<lb/>ſtrabitur (quemad-<lb/>
<anchor type="note" xlink:label="note-0076-06a" xlink:href="note-0076-06"/>
modum dictũ eſt) <lb/>quod G O ad O B <lb/>minorem propor-<lb/>tionem habeat, <lb/>quàm F O ad O C; <lb/></s>
  <s xml:id="echoid-s1964" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1965" xml:space="preserve">ponamus O G <lb/>ad O o, vt F O ad <lb/>O C; </s>
  <s xml:id="echoid-s1966" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1967" xml:space="preserve">produca-<lb/>mus per o ſectionẽ <lb/>hyperbolicam cir-<lb/>ca duas continen-<lb/>tes S M, M F, quę <lb/>ſecet ſectionem A <lb/>B in V, l, &amp; </s>
  <s xml:id="echoid-s1968" xml:space="preserve">iun-<lb/>gamus E V, E l, <lb/>
<anchor type="note" xlink:label="note-0076-07a" xlink:href="note-0076-07"/>
&amp; </s>
  <s xml:id="echoid-s1969" xml:space="preserve">producamus ex <lb/>V, l duas perpendiculares V c, l P, quæ parallelæ ſint continenti M F, <lb/>ergo o G in G M eſt æquale V e in e M (12. </s>
  <s xml:id="echoid-s1970" xml:space="preserve">ex ſecundo) &amp; </s>
  <s xml:id="echoid-s1971" xml:space="preserve">quia G O ad <lb/>O o eſt, vt F O ad O C erit o O in O F æquale rectangulo G C, &amp; </s>
  <s xml:id="echoid-s1972" xml:space="preserve">pona-<lb/>mus rectangulum F G commune fiet rectangulum C M (quod erat ęquale <lb/>rectangulo M E) æquale ipſi o G in G M, quod eſt æquale ipſi V e in e <lb/>
<anchor type="note" xlink:label="note-0076-08a" xlink:href="note-0076-08"/>
M; </s>
  <s xml:id="echoid-s1973" xml:space="preserve">ergo rectangulum E M æquale eſt ipſi V e in e M. </s>
  <s xml:id="echoid-s1974" xml:space="preserve">Tandem proſe-<lb/>quamur ſuperiorem demonſtrationem, vt oſtendatur veritas reliquarum <lb/>
<anchor type="note" xlink:label="note-0076-09a" xlink:href="note-0076-09"/>
propoſitionum, &amp; </s>
  <s xml:id="echoid-s1975" xml:space="preserve">hoc erat propoſitum.</s>
  <s xml:id="echoid-s1976" xml:space="preserve"/>
</p>
<div xml:id="echoid-div156" type="float" level="2" n="4">
  <figure xlink:label="fig-0076-01" xlink:href="fig-0076-01a">
    <image file="0076-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0076-01"/>
  </figure>
<note position="right" xlink:label="note-0076-06" xlink:href="note-0076-06a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0076-07" xlink:href="note-0076-07a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0076-08" xlink:href="note-0076-08a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0076-09" xlink:href="note-0076-09a" xml:space="preserve">e</note>
</div>
<pb o="39" file="0077" n="77" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div158" type="section" level="1" n="66">
<head xml:id="echoid-head98" xml:space="preserve">PROPOSITIO LIV. LV.</head>
<p>
  <s xml:id="echoid-s1977" xml:space="preserve">ITaque oſtenſum eſt, vti memorauimus, quod ex concurſu <lb/>duarum breuiſſimarum ad coniſectionem non egrediatur alia <lb/>
<anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/>
breuiſecans præter illas duas, &amp; </s>
  <s xml:id="echoid-s1978" xml:space="preserve">quod reliqui rami ex eorum <lb/>concurſu educti ad ſectionem habent proprietates ſuperiùs ex-<lb/>poſitas.</s>
  <s xml:id="echoid-s1979" xml:space="preserve"/>
</p>
<div xml:id="echoid-div158" type="float" level="2" n="1">
<note position="left" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div160" type="section" level="1" n="67">
<head xml:id="echoid-head99" xml:space="preserve">PROPOSITIO LVI.</head>
<p>
  <s xml:id="echoid-s1980" xml:space="preserve">In ellipſi ramorum, ſecantium vtrumque axim, à concurſu vl-<lb/>tra centrum poſito egredientium, vnius tantum portio, inter <lb/>axim maiorem, &amp; </s>
  <s xml:id="echoid-s1981" xml:space="preserve">ſectionem intercepta, erit linea breuiſsima, <lb/>
<anchor type="note" xlink:label="note-0077-02a" xlink:href="note-0077-02"/>
ſiue menſura ipſam comparatam, nec non perpendicularis ipſam <lb/>trutinam ſuperet, æquet, vel ab ea deficiat.</s>
  <s xml:id="echoid-s1982" xml:space="preserve"/>
</p>
<div xml:id="echoid-div160" type="float" level="2" n="1">
<note position="left" xlink:label="note-0077-02" xlink:href="note-0077-02a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s1983" xml:space="preserve">SIt ſectio ellipſis A C B, &amp; </s>
  <s xml:id="echoid-s1984" xml:space="preserve">axis maior tranſuerſus A B perpendicularis <lb/>E F, centrum D, &amp; </s>
  <s xml:id="echoid-s1985" xml:space="preserve">ponamus D G ad G F, vt proportio figuræ, &amp; </s>
  <s xml:id="echoid-s1986" xml:space="preserve">ſi-<lb/>
<anchor type="note" xlink:label="note-0077-03a" xlink:href="note-0077-03"/>
militer E H ad H F, &amp; </s>
  <s xml:id="echoid-s1987" xml:space="preserve">producamus per H rectam I H K parallelam ipſi A B, <lb/>&amp; </s>
  <s xml:id="echoid-s1988" xml:space="preserve">per G rectã I G L ipſi <lb/>
<anchor type="figure" xlink:label="fig-0077-01a" xlink:href="fig-0077-01"/>
E F, quæ ſibi occurrant <lb/>in I, &amp; </s>
  <s xml:id="echoid-s1989" xml:space="preserve">ducamus per <lb/>
<anchor type="note" xlink:label="note-0077-04a" xlink:href="note-0077-04"/>
punctum E ſectionem <lb/>
<anchor type="note" xlink:label="note-0077-05a" xlink:href="note-0077-05"/>
hyperbolen E M C cir-<lb/>ca duas eius continen-<lb/>tes L I, I K, quæ oc-<lb/>curret ſectioni A C B <lb/>ellipticæ, quia I L, I K <lb/>ſunt duæ cõtinentes ſe-<lb/>ctionem E M C, &amp; </s>
  <s xml:id="echoid-s1990" xml:space="preserve">pro-<lb/>portio E H ad H F po-<lb/>ſita eſt, vt D G ad G F; <lb/></s>
  <s xml:id="echoid-s1991" xml:space="preserve">
<anchor type="note" xlink:label="note-0077-06a" xlink:href="note-0077-06"/>
ergo E H prima proportionalium in H I, nempe G F quartam, æquale <lb/>eſt D G ſecundæ in I G, nempe F H tertiam; </s>
  <s xml:id="echoid-s1992" xml:space="preserve">ergo punctum M eſt in il-<lb/>lius diametro, &amp; </s>
  <s xml:id="echoid-s1993" xml:space="preserve">propterea ſectio hyperbole E M C tranſit per centrum <lb/>ſectionis ellipſis A C B; </s>
  <s xml:id="echoid-s1994" xml:space="preserve">quare duæ ſectiones ſe inuicem ſecant, ſitque <lb/>concurſus in C, &amp; </s>
  <s xml:id="echoid-s1995" xml:space="preserve">producamus per E, C lineam occurrentem duabus con-<lb/>
<anchor type="note" xlink:label="note-0077-07a" xlink:href="note-0077-07"/>
tinentibus ſectionem in L, K, &amp; </s>
  <s xml:id="echoid-s1996" xml:space="preserve">producamus duas perpendiculares C N, <lb/>K O ſuper A B. </s>
  <s xml:id="echoid-s1997" xml:space="preserve">Et quia K C, L E ſunt æquales (16. </s>
  <s xml:id="echoid-s1998" xml:space="preserve">exſecundo) erit G F <lb/>
<anchor type="note" xlink:label="note-0077-08a" xlink:href="note-0077-08"/>
æqualis O N; </s>
  <s xml:id="echoid-s1999" xml:space="preserve">quare F O æqualis eſt ipſi G N; </s>
  <s xml:id="echoid-s2000" xml:space="preserve">atque E H ad H F, nempe <lb/>
<anchor type="note" xlink:label="note-0077-09a" xlink:href="note-0077-09"/>
E K ad K P, ſeu F O (quæ eſt æqualis ipſi G N) ad O P eandem propor-<lb/>tionem habet, quàm D G ad G F, quę eſt ęqualis ipſi O N, &amp; </s>
  <s xml:id="echoid-s2001" xml:space="preserve">ideo G N <lb/>ad O P eſt, vt D G ad O N, &amp; </s>
  <s xml:id="echoid-s2002" xml:space="preserve">comparando homologum differentias D N <lb/>
<anchor type="note" xlink:label="note-0077-10a" xlink:href="note-0077-10"/>
<pb o="40" file="0078" n="78" rhead="Apollonij Pergæi"/>
ad N P erit, vt D G ad G F, quæ eſt proportio figuræ; </s>
  <s xml:id="echoid-s2003" xml:space="preserve">ergo C P eſt li-<lb/>nea breuiſſima. </s>
  <s xml:id="echoid-s2004" xml:space="preserve">(10. </s>
  <s xml:id="echoid-s2005" xml:space="preserve">ex quinto) Et hoc ſuit ptopoſitum.</s>
  <s xml:id="echoid-s2006" xml:space="preserve"/>
</p>
<div xml:id="echoid-div161" type="float" level="2" n="2">
<note position="left" xlink:label="note-0077-03" xlink:href="note-0077-03a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0077-01" xlink:href="fig-0077-01a">
    <image file="0077-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0077-01"/>
  </figure>
<note position="left" xlink:label="note-0077-04" xlink:href="note-0077-04a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0077-05" xlink:href="note-0077-05a" xml:space="preserve">4. lib. 2</note>
<note position="left" xlink:label="note-0077-06" xlink:href="note-0077-06a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0077-07" xlink:href="note-0077-07a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0077-08" xlink:href="note-0077-08a" xml:space="preserve">8. lib. 2.</note>
<note position="left" xlink:label="note-0077-09" xlink:href="note-0077-09a" xml:space="preserve">f</note>
<note position="right" xlink:label="note-0077-10" xlink:href="note-0077-10a" xml:space="preserve">Lem. 3.</note>
</div>
</div>
<div xml:id="echoid-div163" type="section" level="1" n="68">
<head xml:id="echoid-head100" xml:space="preserve">PROPOSITIO LVII.</head>
<p>
  <s xml:id="echoid-s2007" xml:space="preserve">Et dico, quod non reperiatur vllus alius ramus, à quo ab-<lb/>
<anchor type="note" xlink:label="note-0078-01a" xlink:href="note-0078-01"/>
ſcindi poſſit inter ſectionem, &amp; </s>
  <s xml:id="echoid-s2008" xml:space="preserve">D B linea breuiſſima.</s>
  <s xml:id="echoid-s2009" xml:space="preserve"/>
</p>
<div xml:id="echoid-div163" type="float" level="2" n="1">
<note position="right" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">g</note>
</div>
<p>
  <s xml:id="echoid-s2010" xml:space="preserve">NAm ſi producantur E H, E G ad vtraſque partes ipſius E C ſecan-<lb/>
<anchor type="note" xlink:label="note-0078-02a" xlink:href="note-0078-02"/>
tes D B in K, I, &amp; </s>
  <s xml:id="echoid-s2011" xml:space="preserve">producamus per D perpendicularem ad A B, <lb/>quæ occurrat ſectioni ad L, <lb/>
<anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/>
&amp; </s>
  <s xml:id="echoid-s2012" xml:space="preserve">ipſi E C ad M, quia iam <lb/>
<anchor type="note" xlink:label="note-0078-03a" xlink:href="note-0078-03"/>
productæ ſunt ex concurſu <lb/>M duæ breuiſecantes M C, <lb/>M L (51. </s>
  <s xml:id="echoid-s2013" xml:space="preserve">ex quinto) igitur <lb/>linea educta ex M ad H ab-<lb/>ſcindit ex D B cum B ma-<lb/>iorem lineam, quàm ſecat <lb/>
<anchor type="note" xlink:label="note-0078-04a" xlink:href="note-0078-04"/>
breuiſſima egrediens ex H <lb/>(11. </s>
  <s xml:id="echoid-s2014" xml:space="preserve">ex quinto) &amp; </s>
  <s xml:id="echoid-s2015" xml:space="preserve">linea edu-<lb/>cta ex M ad G abſcindit ex <lb/>D B lineam minorem ea, <lb/>quàm ſecat linea breuiſſima egrediens ex G (51. </s>
  <s xml:id="echoid-s2016" xml:space="preserve">ex quinto) ſed E H, &amp; </s>
  <s xml:id="echoid-s2017" xml:space="preserve"><lb/>E G efficiunt abſciſſas oppoſito modo; </s>
  <s xml:id="echoid-s2018" xml:space="preserve">ergo non ſunt duæ breuiſecantes, <lb/>&amp; </s>
  <s xml:id="echoid-s2019" xml:space="preserve">propterea non reperitur alius ramus, cui competat proprietas ipſius E <lb/>C, &amp; </s>
  <s xml:id="echoid-s2020" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s2021" xml:space="preserve"/>
</p>
<div xml:id="echoid-div164" type="float" level="2" n="2">
<note position="right" xlink:label="note-0078-02" xlink:href="note-0078-02a" xml:space="preserve">h</note>
  <figure xlink:label="fig-0078-01" xlink:href="fig-0078-01a">
    <image file="0078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0078-01"/>
  </figure>
<note position="right" xlink:label="note-0078-03" xlink:href="note-0078-03a" xml:space="preserve">i</note>
<note position="right" xlink:label="note-0078-04" xlink:href="note-0078-04a" xml:space="preserve">k</note>
</div>
</div>
<div xml:id="echoid-div166" type="section" level="1" n="69">
<head xml:id="echoid-head101" xml:space="preserve">Notæ in Propoſit. IL. L.</head>
<p>
  <s xml:id="echoid-s2022" xml:space="preserve">SI verò menſura excedit comparatam educatur linea, ad quam com-<lb/>
<anchor type="note" xlink:label="note-0078-05a" xlink:href="note-0078-05"/>
paratur perpendicularis, &amp; </s>
  <s xml:id="echoid-s2023" xml:space="preserve">vocabo lineam illam Trutinam, &amp;</s>
  <s xml:id="echoid-s2024" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2025" xml:space="preserve">Sic <lb/>legendum puto: </s>
  <s xml:id="echoid-s2026" xml:space="preserve">Si verò menſura excedit comparatam exponi debet linea certis <lb/>quibuſdam legibus inuenienda, quæ vocabitur Trutina.</s>
  <s xml:id="echoid-s2027" xml:space="preserve"/>
</p>
<div xml:id="echoid-div166" type="float" level="2" n="1">
<note position="right" xlink:label="note-0078-05" xlink:href="note-0078-05a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s2028" xml:space="preserve">Ex E concurſu ſuper perpendicularem, &amp;</s>
  <s xml:id="echoid-s2029" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2030" xml:space="preserve">Ideſt. </s>
  <s xml:id="echoid-s2031" xml:space="preserve">Ex E concurſu per-<lb/>
<anchor type="note" xlink:label="note-0078-06a" xlink:href="note-0078-06"/>
pendicularis E D ad axim A G, &amp; </s>
  <s xml:id="echoid-s2032" xml:space="preserve">ramoram ſecantium educamus E B ſecantem <lb/>menſuram, &amp;</s>
  <s xml:id="echoid-s2033" xml:space="preserve">c.</s>
  <s xml:id="echoid-s2034" xml:space="preserve"/>
</p>
<div xml:id="echoid-div167" type="float" level="2" n="2">
<note position="right" xlink:label="note-0078-06" xlink:href="note-0078-06a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s2035" xml:space="preserve">Tunc B F non eſt ex minimis, &amp;</s>
  <s xml:id="echoid-s2036" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2037" xml:space="preserve">Dico quod B F non erit recta linea <lb/>
<anchor type="note" xlink:label="note-0078-07a" xlink:href="note-0078-07"/>
minima earum, quæ inter punctum ſectionis B, &amp; </s>
  <s xml:id="echoid-s2038" xml:space="preserve">axim intercipitur.</s>
  <s xml:id="echoid-s2039" xml:space="preserve"/>
</p>
<div xml:id="echoid-div168" type="float" level="2" n="3">
<note position="right" xlink:label="note-0078-07" xlink:href="note-0078-07a" xml:space="preserve">c</note>
</div>
<p>
  <s xml:id="echoid-s2040" xml:space="preserve">Et ponatur G I æqualis A H, &amp;</s>
  <s xml:id="echoid-s2041" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2042" xml:space="preserve">Et ponatur G I æqualis A H, iungatur-<lb/>
<anchor type="note" xlink:label="note-0078-08a" xlink:href="note-0078-08"/>
que B G, cumque A D poſita ſit non maior, quàm H A, erit illius portio F I <lb/>
<anchor type="note" xlink:label="note-0078-09a" xlink:href="note-0078-09"/>
minor, quàm A H, ſeu quàm G I, ergo B G eſt breuiſsima, &amp;</s>
  <s xml:id="echoid-s2043" xml:space="preserve">c.</s>
  <s xml:id="echoid-s2044" xml:space="preserve"/>
</p>
<div xml:id="echoid-div169" type="float" level="2" n="4">
<note position="right" xlink:label="note-0078-08" xlink:href="note-0078-08a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0078-09" xlink:href="note-0078-09a" xml:space="preserve">8. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2045" xml:space="preserve">Ergo C A ad A H non habet maiorem proportionem, quàm ad A D; <lb/></s>
  <s xml:id="echoid-s2046" xml:space="preserve">
<anchor type="note" xlink:label="note-0078-10a" xlink:href="note-0078-10"/>
quare D I ad I F, &amp;</s>
  <s xml:id="echoid-s2047" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2048" xml:space="preserve">Ergo G A ad A H non habet maiorem proportionem, <lb/>quàm ad A D, &amp; </s>
  <s xml:id="echoid-s2049" xml:space="preserve">addatur indirectum recta A L æqualis A H in hyperbola, &amp;</s>
  <s xml:id="echoid-s2050" xml:space="preserve">
<pb o="41" file="0079" n="79" rhead="Conicor. Lib. V."/>
<anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/>
auferatur in ellipſi; </s>
  <s xml:id="echoid-s2051" xml:space="preserve">quare C A ad A L non habet maiorem proportionem, quàm <lb/>ad A D, &amp; </s>
  <s xml:id="echoid-s2052" xml:space="preserve">componendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s2053" xml:space="preserve">diuidendo in ellipſi C L ad A L, non <lb/>habet maiorem proportionem, quàm C D ad D A, ſed C D ad A D minorem <lb/>proportionem habet, quam ad eius ſegmentum I D, ergo diuidendo in hyperbo-<lb/>la, &amp; </s>
  <s xml:id="echoid-s2054" xml:space="preserve">componendo in ellipſi habebit A C ad A D, &amp; </s>
  <s xml:id="echoid-s2055" xml:space="preserve">adhuc ad A L, ſeu A H <lb/>minorem proportionem, quàm C I ad I D, habet verò C I ad I D minorem ra-<lb/>tionem, quàm ad eius ſegmentum I F; </s>
  <s xml:id="echoid-s2056" xml:space="preserve">igitur C I ad I F maiorem proportionem <lb/>habet, quàm C A ad A H.</s>
  <s xml:id="echoid-s2057" xml:space="preserve"/>
</p>
<div xml:id="echoid-div170" type="float" level="2" n="5">
<note position="right" xlink:label="note-0078-10" xlink:href="note-0078-10a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0079-01" xlink:href="fig-0079-01a">
    <image file="0079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0079-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div172" type="section" level="1" n="70">
<head xml:id="echoid-head102" xml:space="preserve">Notæ in Propoſit. LI.</head>
<p style="it">
  <s xml:id="echoid-s2058" xml:space="preserve">DIco quod nul-<lb/>
<anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/>
<anchor type="figure" xlink:label="fig-0079-02a" xlink:href="fig-0079-02"/>
lus ramus bre <lb/>uiſecans duci po-<lb/>teſt, &amp;</s>
  <s xml:id="echoid-s2059" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2060" xml:space="preserve">Dico, quod <lb/>ex concurſu E ad ſe-<lb/>ctionem nullus ra-<lb/>musbreuiſecans duci <lb/>poteſt.</s>
  <s xml:id="echoid-s2061" xml:space="preserve"/>
</p>
<div xml:id="echoid-div172" type="float" level="2" n="1">
<note position="left" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0079-02" xlink:href="fig-0079-02a">
    <image file="0079-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0079-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s2062" xml:space="preserve">Quoniam D E <lb/>
<anchor type="note" xlink:label="note-0079-02a" xlink:href="note-0079-02"/>
maior eſt, quàm H, <lb/>&amp;</s>
  <s xml:id="echoid-s2063" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2064" xml:space="preserve">Quoniam D E <lb/>maior eſt, quàm H <lb/>habebit E D ad B G <lb/>maiorem rationem, <lb/>quàm H ad eandem <lb/>B G; </s>
  <s xml:id="echoid-s2065" xml:space="preserve">poſita autem fuit <lb/>inuersè G F ad F D, <lb/>vt H ad B G; </s>
  <s xml:id="echoid-s2066" xml:space="preserve">ergo E D <lb/>ad B G maiorem ra-<lb/>tionem habet, quàm <lb/>G F ad F D; </s>
  <s xml:id="echoid-s2067" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2068" xml:space="preserve">pro-
<pb o="42" file="0080" n="80" rhead="Apollonij Pergæi"/>
pter parallelas D E, <lb/>
<anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/>
B G, &amp; </s>
  <s xml:id="echoid-s2069" xml:space="preserve">ſimilitudinẽ <lb/>triangulorum E D I, <lb/>&amp; </s>
  <s xml:id="echoid-s2070" xml:space="preserve">B G I, eſt D I ad I <lb/>G, vt E D ad B G; <lb/></s>
  <s xml:id="echoid-s2071" xml:space="preserve">igitur D I ad I G ma-<lb/>iorem proportionem <lb/>habet, quàm G F ad <lb/>F D, &amp; </s>
  <s xml:id="echoid-s2072" xml:space="preserve">componendo <lb/>D G ad G I maio rem <lb/>rationem habebit, <lb/>quàm eadem G D ad <lb/>D F; </s>
  <s xml:id="echoid-s2073" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2074" xml:space="preserve">Ideo I G mi-<lb/>nor eſt, quàm D F.</s>
  <s xml:id="echoid-s2075" xml:space="preserve"/>
</p>
<div xml:id="echoid-div173" type="float" level="2" n="2">
<note position="left" xlink:label="note-0079-02" xlink:href="note-0079-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a">
    <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0080-01"/>
  </figure>
</div>
<note position="right" xml:space="preserve">c</note>
<p style="it">
  <s xml:id="echoid-s2076" xml:space="preserve">Igitur G F æqua-<lb/>lis eſt GO, ergo G <lb/>O ad O M, &amp;</s>
  <s xml:id="echoid-s2077" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2078" xml:space="preserve">Igi-<lb/>tur G F æqualis eſt G <lb/>O, &amp; </s>
  <s xml:id="echoid-s2079" xml:space="preserve">quia F O ſecatur <lb/>bifariam in G, &amp; </s>
  <s xml:id="echoid-s2080" xml:space="preserve">non <lb/>bifariam in M (ex <lb/>lemmate ſexto huius <lb/>libri) habebit ſemisſis G O ad vnum ſegmentorum inæqualium M O maiorem pro-<lb/>portionem, quàm reliquum ſegmentum M F ad alteram medietatem F G, ſed pro-<lb/>pter parallelas P M, B G, &amp; </s>
  <s xml:id="echoid-s2081" xml:space="preserve">ſimilitudinem triangulorum B G O, P M O eſt G O ad <lb/>O M, vt B G ad P M, ergo B G ad P M maiorem proportionem habet, quàm M F ad <lb/>F G: </s>
  <s xml:id="echoid-s2082" xml:space="preserve">habet verò B G ad minorem M K maiorem proportionem, quàm ad M P (cum <lb/>punctum P tangentis cadat extra ſectionem); </s>
  <s xml:id="echoid-s2083" xml:space="preserve">ergo B G ad K M adhuc maiorem pro-<lb/>portionem habet, quàm M F ad F G.</s>
  <s xml:id="echoid-s2084" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2085" xml:space="preserve">Itaque K M in M F minus eſt, quàm B G in G F, &amp;</s>
  <s xml:id="echoid-s2086" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2087" xml:space="preserve">Quoniam prima B G <lb/>
<anchor type="note" xlink:label="note-0080-02a" xlink:href="note-0080-02"/>
ad ſecundam K M maiorem proportionem habet, quàm tertia M F ad quartam F G; <lb/></s>
  <s xml:id="echoid-s2088" xml:space="preserve">ergo ex lemmate quinto huius librirectangulum ſub intermedĳs contentum K M F <lb/>minus erit rectangulo B G F ſub extremis cõtento; </s>
  <s xml:id="echoid-s2089" xml:space="preserve">poſtea, quia H ad B G ex hypotheſi <lb/>erat, vt G F ad F D, poſita autem fuit E D maior, quàm H, quæ eſt prima propor-<lb/>tionalium; </s>
  <s xml:id="echoid-s2090" xml:space="preserve">ergo E D ad B G maiorem proportionem habet, quàm G F ad F D, &amp; </s>
  <s xml:id="echoid-s2091" xml:space="preserve">pro-<lb/>
<anchor type="note" xlink:label="note-0080-03a" xlink:href="note-0080-03"/>
pterea rectangulum ſub extremis E D F maius erit rectangulo ſub intermedĳs con-<lb/>tento B G F; </s>
  <s xml:id="echoid-s2092" xml:space="preserve">fuit autem rectangulum B G F maius rectangulo K M F; </s>
  <s xml:id="echoid-s2093" xml:space="preserve">igitur rectan-<lb/>gulum E D F multò maius eſt, quàm rectangulum K M F, &amp; </s>
  <s xml:id="echoid-s2094" xml:space="preserve">ideo, ex eodem lemma-<lb/>te quinto, E D ad M K, nempe D R ad R M (propter ſimilitudinem triangulorum <lb/>E D R, &amp; </s>
  <s xml:id="echoid-s2095" xml:space="preserve">K M R) maiorem rationem habet, quàm M F ad F D.</s>
  <s xml:id="echoid-s2096" xml:space="preserve"/>
</p>
<div xml:id="echoid-div174" type="float" level="2" n="3">
<note position="right" xlink:label="note-0080-02" xlink:href="note-0080-02a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0080-03" xlink:href="note-0080-03a" xml:space="preserve">Lem. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2097" xml:space="preserve">Et componendo patet, quod D F, &amp;</s>
  <s xml:id="echoid-s2098" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2099" xml:space="preserve">Quoniam D R ad R M maiorem ratio-<lb/>
<anchor type="note" xlink:label="note-0080-04a" xlink:href="note-0080-04"/>
nem habet, quàm M F ad F D, componendo D M ad M R habebit maiorem propor-<lb/>tionem, quàm eadem M D ad D F, &amp; </s>
  <s xml:id="echoid-s2100" xml:space="preserve">propterea D F mator eſt, quàm R M, eſt verò <lb/>ſemisſis erecti A C æqualis D F ex conſtructione, igitur M R minor eſt A C medieta-<lb/>te lateris recti, &amp; </s>
  <s xml:id="echoid-s2101" xml:space="preserve">propterea breuiſsima educta ex K ſecat ex axi ſegmentum maius, <lb/>
<anchor type="note" xlink:label="note-0080-05a" xlink:href="note-0080-05"/>
quàm M R; </s>
  <s xml:id="echoid-s2102" xml:space="preserve">ideoque cadit extra, ſcilicet infra ramum K R E.</s>
  <s xml:id="echoid-s2103" xml:space="preserve"/>
</p>
<div xml:id="echoid-div175" type="float" level="2" n="4">
<note position="right" xlink:label="note-0080-04" xlink:href="note-0080-04a" xml:space="preserve">e</note>
<note position="left" xlink:label="note-0080-05" xlink:href="note-0080-05a" xml:space="preserve">8. huius.</note>
</div>
<pb o="43" file="0081" n="81" rhead="Conicor. Lib. V."/>
  <figure>
    <image file="0081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0081-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s2104" xml:space="preserve">Et ſimili modo conſtat, quod breuiſſima egrediens ex puncto L cadit <lb/>
<anchor type="note" xlink:label="note-0081-01a" xlink:href="note-0081-01"/>
extra S L, &amp;</s>
  <s xml:id="echoid-s2105" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2106" xml:space="preserve">Ad vitandam confuſionem figuræ, &amp; </s>
  <s xml:id="echoid-s2107" xml:space="preserve">prolixitatem demonſtrationis <lb/>appoſui duas figuras, in quibus duo caſus ĳſdem caracteribus notantur, itaque abſq; <lb/></s>
  <s xml:id="echoid-s2108" xml:space="preserve">nouo labore, ſi inſpiciatur ſecunda figura, ĳſdem verbis prioris caſus, oſtendetur ca-<lb/>ſus ſecundus.</s>
  <s xml:id="echoid-s2109" xml:space="preserve"/>
</p>
<div xml:id="echoid-div176" type="float" level="2" n="5">
<note position="left" xlink:label="note-0081-01" xlink:href="note-0081-01a" xml:space="preserve">f</note>
</div>
<p style="it">
  <s xml:id="echoid-s2110" xml:space="preserve">Pariter demonſtrabitur, quemadmodum iam oſtenſumeſt, &amp;</s>
  <s xml:id="echoid-s2111" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2112" xml:space="preserve">Pars ſecun-<lb/>
<anchor type="note" xlink:label="note-0081-02a" xlink:href="note-0081-02"/>
da huius propoſitionis innuitur tantummodo pauciſsimis verbis; </s>
  <s xml:id="echoid-s2113" xml:space="preserve">quare maioris cla-<lb/>ritatis gratia integram demonſtrationem hìc afferre libuit.</s>
  <s xml:id="echoid-s2114" xml:space="preserve"/>
</p>
<div xml:id="echoid-div177" type="float" level="2" n="6">
<note position="left" xlink:label="note-0081-02" xlink:href="note-0081-02a" xml:space="preserve">g</note>
</div>
</div>
<div xml:id="echoid-div179" type="section" level="1" n="71">
<head xml:id="echoid-head103" xml:space="preserve">Demonſtratio ſecundæ partis.</head>
<head xml:id="echoid-head104" xml:space="preserve">PROPOSITIONIS LI.</head>
<p style="it">
  <s xml:id="echoid-s2115" xml:space="preserve">Eſto E D æqualis trutinæ H: </s>
  <s xml:id="echoid-s2116" xml:space="preserve">Dico ex concurſu E vnicum tantùm breui-<lb/>ſecantem ramum duci poſſe.</s>
  <s xml:id="echoid-s2117" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2118" xml:space="preserve">In eadem ſigura, quia ex conſtructione H ad B G eſt, vt G F ad F D, ponitur verò <lb/>E D æqualis H; </s>
  <s xml:id="echoid-s2119" xml:space="preserve">ergo E D ad B G, ſeu D I ad I G (propter ſimilitudinem triangulo-<lb/>rum E D I, B G I) eſt, vt G F ad F D, &amp; </s>
  <s xml:id="echoid-s2120" xml:space="preserve">componendo D G ad G I eſt, vt eadem G D <lb/>ad D F; </s>
  <s xml:id="echoid-s2121" xml:space="preserve">ideoque I G æqualis eſt D F, ſeu A C ſemierecto; </s>
  <s xml:id="echoid-s2122" xml:space="preserve">igitur B I eſt breuiſsima.</s>
  <s xml:id="echoid-s2123" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">8. huius.</note>
<p style="it">
  <s xml:id="echoid-s2124" xml:space="preserve">Poſte a ducto quolibet ramo E K ſupra breuiſecantem E B (in prima figura, &amp; </s>
  <s xml:id="echoid-s2125" xml:space="preserve">in-<lb/>fra in ſecunda) occurrente axi in R, &amp; </s>
  <s xml:id="echoid-s2126" xml:space="preserve">ducta K M perpendiculari ad axim, quæ eum <lb/>ſecet in M, &amp; </s>
  <s xml:id="echoid-s2127" xml:space="preserve">tangentem O B in P. </s>
  <s xml:id="echoid-s2128" xml:space="preserve">Quoniam (vt dictum eſt) O F ſecatur bifariam
<pb o="44" file="0082" n="82" rhead="Apollonij Pergæi"/>
in G, &amp; </s>
  <s xml:id="echoid-s2129" xml:space="preserve">non bifariam in M, ergo (ex lemmate ſexto huius libri) G O ad O M, ſeu <lb/>G B ad P M (propter ſimilitudinem triangulorum B G O, &amp; </s>
  <s xml:id="echoid-s2130" xml:space="preserve">P M O) &amp; </s>
  <s xml:id="echoid-s2131" xml:space="preserve">multo magis <lb/>G B ad illius portionem K M habebit maiorem proportionem, quàm M F, ad F G; <lb/></s>
  <s xml:id="echoid-s2132" xml:space="preserve">ideoque rectangulum K M F ſub intermedĳs contentum minus erit rectangulo B G F <lb/>
<anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/>
contento ſub extremis nõ proportionalium; </s>
  <s xml:id="echoid-s2133" xml:space="preserve">ſed rectangulum B G F æquale eſt rectan-<lb/>gulo E D F (propterea quod D F, ad F G erat, vt B G ad H, ſeu ad ei æqualæm E D) <lb/>
<anchor type="note" xlink:label="note-0082-02a" xlink:href="note-0082-02"/>
igitur rectangulum K M F minus erit rectangulo E D F, &amp; </s>
  <s xml:id="echoid-s2134" xml:space="preserve">propterea E D ad K M, <lb/>ſeu D R ad R M (propter ſimilitudinem triangulorum E D R, K M R) maiorem ra-<lb/>tionem habebit, quàm M F ad F D, &amp; </s>
  <s xml:id="echoid-s2135" xml:space="preserve">componendo, eadem D M maiorem rationem <lb/>habebit ad R M, quàm ad F D, &amp; </s>
  <s xml:id="echoid-s2136" xml:space="preserve">propterea R M minor erit, quàm F D, ſeu quàm <lb/>A C; </s>
  <s xml:id="echoid-s2137" xml:space="preserve">igitur minimus ramorum ex K ad axim cadentium fertur infra K R; </s>
  <s xml:id="echoid-s2138" xml:space="preserve">Quapro-<lb/>
<anchor type="note" xlink:label="note-0082-03a" xlink:href="note-0082-03"/>
pter ramus E K ſupra, vel infra breuiſecantem E B ad ſectionem ductus non eſt bre-<lb/>uiſecans, &amp; </s>
  <s xml:id="echoid-s2139" xml:space="preserve">abſcindit ex axi ſegmentum A R minus, quàm abſcindat breuiſsima ex <lb/>K ad axim ducta, quod erat oſtendendum.</s>
  <s xml:id="echoid-s2140" xml:space="preserve"/>
</p>
<div xml:id="echoid-div179" type="float" level="2" n="1">
<note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">Lem. 5. <lb/>pręmiſ.</note>
<note position="left" xlink:label="note-0082-02" xlink:href="note-0082-02a" xml:space="preserve">Lem. 5. <lb/>pręmiſ.</note>
<note position="left" xlink:label="note-0082-03" xlink:href="note-0082-03a" xml:space="preserve">ex 8. 13. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2141" xml:space="preserve">Tertio loco ſit E D minor, quàm H, &amp; </s>
  <s xml:id="echoid-s2142" xml:space="preserve">oſtendetur, &amp;</s>
  <s xml:id="echoid-s2143" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2144" xml:space="preserve">Quia H ad B G eſt, <lb/>
<anchor type="note" xlink:label="note-0082-04a" xlink:href="note-0082-04"/>
vt G F ad F D, eſtque E D minor, quàm H; </s>
  <s xml:id="echoid-s2145" xml:space="preserve">ergo E D ad B G minorem proportionem <lb/>habet, quàm G F ad F D; </s>
  <s xml:id="echoid-s2146" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2147" xml:space="preserve">ideo rectangulum E D F ſub extremis contentum minus <lb/>
<anchor type="note" xlink:label="note-0082-05a" xlink:href="note-0082-05"/>
eſt rectangulo B G F, quod ſub intermedĳs continetur; </s>
  <s xml:id="echoid-s2148" xml:space="preserve">ponatur iam rectangulum T <lb/>G F æquale rectangulo E D F, &amp; </s>
  <s xml:id="echoid-s2149" xml:space="preserve">per F ducatur F V perpendicularis ſuper axim <lb/>A D.</s>
  <s xml:id="echoid-s2150" xml:space="preserve"/>
</p>
<div xml:id="echoid-div180" type="float" level="2" n="2">
<note position="right" xlink:label="note-0082-04" xlink:href="note-0082-04a" xml:space="preserve">h</note>
<note position="left" xlink:label="note-0082-05" xlink:href="note-0082-05a" xml:space="preserve">Lem. 5. <lb/>pręmiſ.</note>
</div>
  <figure>
    <image file="0082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0082-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s2151" xml:space="preserve">Et componendo, patet, quod D F eſt æqualis R M, &amp;</s>
  <s xml:id="echoid-s2152" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2153" xml:space="preserve">Nam D Rad R M <lb/>
<anchor type="note" xlink:label="note-0082-06a" xlink:href="note-0082-06"/>
eſt, vt M F ad F D, &amp; </s>
  <s xml:id="echoid-s2154" xml:space="preserve">componendo, eadem D M ad R M, atque ad D F, ſeuad ſemi-<lb/>erectum A C eandem proportionem habebit, &amp; </s>
  <s xml:id="echoid-s2155" xml:space="preserve">ideo D F eſt æqualis R M.</s>
  <s xml:id="echoid-s2156" xml:space="preserve"/>
</p>
<div xml:id="echoid-div181" type="float" level="2" n="3">
<note position="right" xlink:label="note-0082-06" xlink:href="note-0082-06a" xml:space="preserve">i</note>
</div>
<pb o="45" file="0083" n="83" rhead="Conicor. Lib. V."/>
<p style="it">
  <s xml:id="echoid-s2157" xml:space="preserve">Et ſimiliter patebit, quod L S ſit breuiſſima, &amp;</s>
  <s xml:id="echoid-s2158" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2159" xml:space="preserve">Secundus caſus abſque vllo <lb/>
<anchor type="note" xlink:label="note-0083-01a" xlink:href="note-0083-01"/>
labore oſtenſus erit ĳſdem verbis, &amp; </s>
  <s xml:id="echoid-s2160" xml:space="preserve">caracteribus, quibus caſus primus expoſitus <lb/>fuit, ſi inſpiciatur ſecunda figura.</s>
  <s xml:id="echoid-s2161" xml:space="preserve"/>
</p>
<div xml:id="echoid-div182" type="float" level="2" n="4">
<note position="left" xlink:label="note-0083-01" xlink:href="note-0083-01a" xml:space="preserve">k</note>
</div>
<p style="it">
  <s xml:id="echoid-s2162" xml:space="preserve">Et cum B I intercipiatur inter illas patebit etiam, &amp;</s>
  <s xml:id="echoid-s2163" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2164" xml:space="preserve">Et cum B I intercipia-<lb/>
<anchor type="note" xlink:label="note-0083-02a" xlink:href="note-0083-02"/>
tur inter duos ramos breuiſecantes E K, qui ducuntur ex punctis K, in quibus hy-<lb/>perbole K T L ſecat parabolen A B L, cadet punctum T hyperboles intra parabolen; <lb/></s>
  <s xml:id="echoid-s2165" xml:space="preserve">quare rectangulum B G F maius erit rectangulo T G F, ſeu K M F, quod æquale eſt <lb/>rectangulo E D F, vt dictum eſt, quare E D ad B G, ſeu D I ad I G (propter ſimili-<lb/>
<anchor type="note" xlink:label="note-0083-03a" xlink:href="note-0083-03"/>
tudinem triangulorum E D I, B G I) habebit minorem proportionem, quàm G F ad <lb/>F D, &amp; </s>
  <s xml:id="echoid-s2166" xml:space="preserve">componendo, eadem D G ad G I minorem proportionem habebit, quàm ad <lb/>F D, ſiue ad A C, &amp; </s>
  <s xml:id="echoid-s2167" xml:space="preserve">ideo I G maior erit, quàm A C.</s>
  <s xml:id="echoid-s2168" xml:space="preserve"/>
</p>
<div xml:id="echoid-div183" type="float" level="2" n="5">
<note position="left" xlink:label="note-0083-02" xlink:href="note-0083-02a" xml:space="preserve">l</note>
<note position="right" xlink:label="note-0083-03" xlink:href="note-0083-03a" xml:space="preserve">Lem. 5. <lb/>præmiſ.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2169" xml:space="preserve">Deinde ex con-<lb/>
<anchor type="note" xlink:label="note-0083-04a" xlink:href="note-0083-04"/>
<anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/>
curſu E ad ſectio-<lb/>nem, &amp;</s>
  <s xml:id="echoid-s2170" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2171" xml:space="preserve">Deinde <lb/>ex concurſu E ad ſe-<lb/>ctionem A B parabo-<lb/>len educantur duo ra-<lb/>mi E X ſupra breui-<lb/>ſecantem E K in pri-<lb/>ma figura, &amp; </s>
  <s xml:id="echoid-s2172" xml:space="preserve">infra <lb/>eamdem in figura ſe-<lb/>cunda, &amp; </s>
  <s xml:id="echoid-s2173" xml:space="preserve">ex punct is <lb/>X ducantur due X Y <lb/>perpendiculares ad <lb/>axim, ſecantes axim <lb/>in Y, &amp; </s>
  <s xml:id="echoid-s2174" xml:space="preserve">hyperbolen K <lb/>T in a exiſtẽte extra <lb/>parabolen; </s>
  <s xml:id="echoid-s2175" xml:space="preserve">cumque <lb/>duæ rectæ a Y, necnõ <lb/>T G parallelæ ſint cõ-<lb/>tinenti F V, &amp; </s>
  <s xml:id="echoid-s2176" xml:space="preserve">inter-<lb/>ponātur inter hyper-<lb/>bolẽ K T, &amp; </s>
  <s xml:id="echoid-s2177" xml:space="preserve">reliquã <lb/>continentem F A eritrectangulum a Y F æquale rectangulo T G F, quod factum <lb/>
<anchor type="note" xlink:label="note-0083-05a" xlink:href="note-0083-05"/>
eſt æquale rectangulo E D F, eſtque X Y portio ipſius a Y; </s>
  <s xml:id="echoid-s2178" xml:space="preserve">igitur rectangulum E D F <lb/>maius erit rectangulo X Y F, &amp; </s>
  <s xml:id="echoid-s2179" xml:space="preserve">ideo E D ad X Y, ſeu D b, ad b Y (propter ſimilitu-<lb/>
<anchor type="note" xlink:label="note-0083-06a" xlink:href="note-0083-06"/>
dinem triangulorum E D b, X Y b) maiorem rationem habet, quàm Y F ad F D, &amp; </s>
  <s xml:id="echoid-s2180" xml:space="preserve"><lb/>componendo eadem D Y ad Y b maiorem proportionem habebit, quàm ad D F, ſeu <lb/>C A.</s>
  <s xml:id="echoid-s2181" xml:space="preserve"/>
</p>
<div xml:id="echoid-div184" type="float" level="2" n="6">
<note position="left" xlink:label="note-0083-04" xlink:href="note-0083-04a" xml:space="preserve">m</note>
  <figure xlink:label="fig-0083-01" xlink:href="fig-0083-01a">
    <image file="0083-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0083-01"/>
  </figure>
<note position="right" xlink:label="note-0083-05" xlink:href="note-0083-05a" xml:space="preserve">12. lib. 2.</note>
<note position="right" xlink:label="note-0083-06" xlink:href="note-0083-06a" xml:space="preserve">Lem. 5. <lb/>præmiſ.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2182" xml:space="preserve">Simili modo demonſtrabitur, &amp;</s>
  <s xml:id="echoid-s2183" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2184" xml:space="preserve">Abſquenoua demonſtratione propoſitum <lb/>
<anchor type="note" xlink:label="note-0083-07a" xlink:href="note-0083-07"/>
oſtendetur inſpiciendo ſecundam ſiguram.</s>
  <s xml:id="echoid-s2185" xml:space="preserve"/>
</p>
<div xml:id="echoid-div185" type="float" level="2" n="7">
<note position="left" xlink:label="note-0083-07" xlink:href="note-0083-07a" xml:space="preserve">n</note>
</div>
</div>
<div xml:id="echoid-div187" type="section" level="1" n="72">
<head xml:id="echoid-head105" xml:space="preserve">Notæ in Propoſ. LII. LIII.</head>
<p style="it">
  <s xml:id="echoid-s2186" xml:space="preserve">DIco, quod rami egredientes ex E habent ſuperiùs expoſitas proprieta-<lb/>
<anchor type="note" xlink:label="note-0083-08a" xlink:href="note-0083-08"/>
tes, &amp;</s>
  <s xml:id="echoid-s2187" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2188" xml:space="preserve">Ideſt eaſdem, quas habent rami in parabola educti iuxta compara-<lb/>tionem perpendicularis E D ad T rutinam.</s>
  <s xml:id="echoid-s2189" xml:space="preserve"/>
</p>
<div xml:id="echoid-div187" type="float" level="2" n="1">
<note position="left" xlink:label="note-0083-08" xlink:href="note-0083-08a" xml:space="preserve">a</note>
</div>
<pb o="46" file="0084" n="84" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s2190" xml:space="preserve">Et ponamus quamlibet duarum proportionum C F ad F D, &amp; </s>
  <s xml:id="echoid-s2191" xml:space="preserve">I S ad S C, <lb/>
<anchor type="note" xlink:label="note-0084-01a" xlink:href="note-0084-01"/>
vt proportio figuræ, &amp; </s>
  <s xml:id="echoid-s2192" xml:space="preserve">educamus ex E, S, &amp;</s>
  <s xml:id="echoid-s2193" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2194" xml:space="preserve">Ideſt fiat diſtantia ex centro <lb/>vſque ad perpendicularem E D ad eius portionem D F in hyperbola, vt ſumma late-<lb/>ris tranſuerſi, &amp; </s>
  <s xml:id="echoid-s2195" xml:space="preserve">recti ad latus rectum, &amp; </s>
  <s xml:id="echoid-s2196" xml:space="preserve">vt eorum differentia in ellipſi ad latus <lb/>rectum ita fiat C D ad eius productionem D F; </s>
  <s xml:id="echoid-s2197" xml:space="preserve">tunc enim C F ad F D diuidendo in <lb/>hyperbola, &amp; </s>
  <s xml:id="echoid-s2198" xml:space="preserve">compo-<lb/>
<anchor type="figure" xlink:label="fig-0084-01a" xlink:href="fig-0084-01"/>
nendo in ellipſi habe-<lb/>bit eandem propor-<lb/>tionem, quàm latus <lb/>tranſuerſum ad re-<lb/>ctum; </s>
  <s xml:id="echoid-s2199" xml:space="preserve">pariterq; </s>
  <s xml:id="echoid-s2200" xml:space="preserve">fiat <lb/>E K ad K D in eadẽ <lb/>proportione figuræ, <lb/>&amp; </s>
  <s xml:id="echoid-s2201" xml:space="preserve">ex E, K educamus <lb/>rectas E I, K S pa-<lb/>rallelas axi A C D, <lb/>ſecantes I C, &amp; </s>
  <s xml:id="echoid-s2202" xml:space="preserve">L F <lb/>parallelas ipſi E D <lb/>in I, S, L, &amp; </s>
  <s xml:id="echoid-s2203" xml:space="preserve">M. <lb/></s>
  <s xml:id="echoid-s2204" xml:space="preserve">Immutaui poſtremã <lb/>partem conſtructio-<lb/>nis, vt manifeſte er-<lb/>roneã in textu Ara-<lb/>bico; </s>
  <s xml:id="echoid-s2205" xml:space="preserve">Si enim I C ad <lb/>libitum ſumpta ſeca-<lb/>tur in S in ratione <lb/>C F ad F D non ca-<lb/>det neceſſariò E L <lb/>parallela C D ſuper <lb/>punctum I.</s>
  <s xml:id="echoid-s2206" xml:space="preserve"/>
</p>
<div xml:id="echoid-div188" type="float" level="2" n="2">
<note position="right" xlink:label="note-0084-01" xlink:href="note-0084-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0084-01" xlink:href="fig-0084-01a">
    <image file="0084-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0084-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s2207" xml:space="preserve">Et interponamus <lb/>
<anchor type="note" xlink:label="note-0084-02a" xlink:href="note-0084-02"/>
inter F C, C A du-<lb/>as C N, C O pro-<lb/>portionales illis duabus, &amp;</s>
  <s xml:id="echoid-s2208" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2209" xml:space="preserve">Textum corruptum ſic reſtituo: </s>
  <s xml:id="echoid-s2210" xml:space="preserve">Interponamus in-<lb/>ter F C, &amp; </s>
  <s xml:id="echoid-s2211" xml:space="preserve">A C duas medias proportionales, itaut F C, N C, C O, C A ſint continuè <lb/>proportionales, quod fieri poſſe conſtat ex lemmate 7. </s>
  <s xml:id="echoid-s2212" xml:space="preserve">huius librt.</s>
  <s xml:id="echoid-s2213" xml:space="preserve"/>
</p>
<div xml:id="echoid-div189" type="float" level="2" n="3">
<note position="right" xlink:label="note-0084-02" xlink:href="note-0084-02a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s2214" xml:space="preserve">Et ponamus proportionem lineæ alicuius, vt eſt Q compoſitam, &amp;</s>
  <s xml:id="echoid-s2215" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2216" xml:space="preserve">Vo-<lb/>
<anchor type="note" xlink:label="note-0084-03a" xlink:href="note-0084-03"/>
catur Trutina in hyperbola, &amp; </s>
  <s xml:id="echoid-s2217" xml:space="preserve">ellipſi linea recta Q, quæ ad B O compoſitam propor-<lb/>tionem habet ex C D ad D F, &amp; </s>
  <s xml:id="echoid-s2218" xml:space="preserve">ex ratione F O ad O C.</s>
  <s xml:id="echoid-s2219" xml:space="preserve"/>
</p>
<div xml:id="echoid-div190" type="float" level="2" n="4">
<note position="right" xlink:label="note-0084-03" xlink:href="note-0084-03a" xml:space="preserve">d</note>
</div>
<p style="it">
  <s xml:id="echoid-s2220" xml:space="preserve">Producatur priùs E B ſecans axim in H, &amp;</s>
  <s xml:id="echoid-s2221" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2222" xml:space="preserve">Producatur priùs E B ſecans <lb/>
<anchor type="note" xlink:label="note-0084-04a" xlink:href="note-0084-04"/>
axim in H, &amp; </s>
  <s xml:id="echoid-s2223" xml:space="preserve">rectam S K in R, nec non rectam I C in puncto T.</s>
  <s xml:id="echoid-s2224" xml:space="preserve"/>
</p>
<div xml:id="echoid-div191" type="float" level="2" n="5">
<note position="right" xlink:label="note-0084-04" xlink:href="note-0084-04a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s2225" xml:space="preserve">Ergo E D ad B O, quæ componitur ex E D ad D K, &amp;</s>
  <s xml:id="echoid-s2226" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2227" xml:space="preserve">Nam poſita inter-<lb/>
<anchor type="note" xlink:label="note-0084-05a" xlink:href="note-0084-05"/>
media D K, proportio E D ad B O compoſita erit ex ratione E D ad D K, &amp; </s>
  <s xml:id="echoid-s2228" xml:space="preserve">ex ra-<lb/>tione D K ad B O; </s>
  <s xml:id="echoid-s2229" xml:space="preserve">eſt verò I C ad C S, vt E D ad D K (propter parallelas I E, S K, <lb/>C D) atque D K eſt æqualis G O in parallelogrammo G D; </s>
  <s xml:id="echoid-s2230" xml:space="preserve">ergo proportio E D ad B O <lb/>componitur ex ratione I C ad C S, &amp; </s>
  <s xml:id="echoid-s2231" xml:space="preserve">ex ratione G O ad O B.</s>
  <s xml:id="echoid-s2232" xml:space="preserve"/>
</p>
<div xml:id="echoid-div192" type="float" level="2" n="6">
<note position="right" xlink:label="note-0084-05" xlink:href="note-0084-05a" xml:space="preserve">f</note>
</div>
<p style="it">
  <s xml:id="echoid-s2233" xml:space="preserve">Sed E D ad D K eſt, vt CD ad DF, quia quælibet earum vt proportio <lb/>
<anchor type="note" xlink:label="note-0084-06a" xlink:href="note-0084-06"/>
<pb o="47" file="0085" n="85" rhead="Conicor. Lib. V."/>
figuræ compoſitæ, vel diuiſæ, &amp;</s>
  <s xml:id="echoid-s2234" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2235" xml:space="preserve">Quia E K ad K D, atque C F ad F D eandem <lb/>proportionem habebant, quàm latus tranſuerſum ad rectum; </s>
  <s xml:id="echoid-s2236" xml:space="preserve">ergo componendo in <lb/>hyperbola, &amp; </s>
  <s xml:id="echoid-s2237" xml:space="preserve">diuidendo in ellipſi erit E D ad D K, vt C D ad D F.</s>
  <s xml:id="echoid-s2238" xml:space="preserve"/>
</p>
<div xml:id="echoid-div193" type="float" level="2" n="7">
<note position="right" xlink:label="note-0084-06" xlink:href="note-0084-06a" xml:space="preserve">g</note>
</div>
<p>
  <s xml:id="echoid-s2239" xml:space="preserve">Et ponamus re-<lb/>
<anchor type="note" xlink:label="note-0085-01a" xlink:href="note-0085-01"/>
<anchor type="figure" xlink:label="fig-0085-01a" xlink:href="fig-0085-01"/>
ctangulum F G cõ-<lb/>mune, &amp;</s>
  <s xml:id="echoid-s2240" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2241" xml:space="preserve">Scilicet <lb/>rectangulũ F G ad-<lb/>datur in hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s2242" xml:space="preserve">auferatur cõmu-<lb/>niter in ellipſi.</s>
  <s xml:id="echoid-s2243" xml:space="preserve"/>
</p>
<div xml:id="echoid-div194" type="float" level="2" n="8">
<note position="left" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">h</note>
  <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a">
    <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0085-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s2244" xml:space="preserve">Et propterea E <lb/>
<anchor type="note" xlink:label="note-0085-02a" xlink:href="note-0085-02"/>
K ad B G, nempe <lb/>K R ad R G, &amp;</s>
  <s xml:id="echoid-s2245" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s2246" xml:space="preserve">Quia propter ſimili-<lb/>tudinem triangulo-<lb/>rum E K R, &amp; </s>
  <s xml:id="echoid-s2247" xml:space="preserve">B G <lb/>R erit E K ad B G, <lb/>vt K R ad R G; </s>
  <s xml:id="echoid-s2248" xml:space="preserve">qua-<lb/>re K R ad R G maio-<lb/>rem proportionẽ ha-<lb/>bet, quàm G M ad <lb/>M K; </s>
  <s xml:id="echoid-s2249" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2250" xml:space="preserve">componen-<lb/>do K G ad G R ma-<lb/>iorem rationem ha-<lb/>bet, quam eadem G <lb/>K ad K M, quare <lb/>K M, nẽpe e i æqua-<lb/>lis D F maior eſt, <lb/>quàm G R.</s>
  <s xml:id="echoid-s2251" xml:space="preserve"/>
</p>
<div xml:id="echoid-div195" type="float" level="2" n="9">
<note position="left" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">i</note>
</div>
<p>
  <s xml:id="echoid-s2252" xml:space="preserve">Et auferẽdo ho-<lb/>
<anchor type="note" xlink:label="note-0085-03a" xlink:href="note-0085-03"/>
mologũ ab homo-<lb/>logo in hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s2253" xml:space="preserve">coniungendo e <lb/>a in ellipſi, habebit, &amp;</s>
  <s xml:id="echoid-s2254" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2255" xml:space="preserve">Scilicet comparando homologorum differentias in hy-<lb/>
<anchor type="note" xlink:label="note-0085-04a" xlink:href="note-0085-04"/>
perbola, eorundem ſummas in ellipſi, ideſt C T ad B O, nempe C H ad H O (pro-<lb/>pter ſimilitudinem triangulorum C H T, &amp; </s>
  <s xml:id="echoid-s2256" xml:space="preserve">O H B) habebit maiorem proportionem, <lb/>quàm I C ad C S, nempe C D ad D F.</s>
  <s xml:id="echoid-s2257" xml:space="preserve"/>
</p>
<div xml:id="echoid-div196" type="float" level="2" n="10">
<note position="left" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">k</note>
<note position="right" xlink:label="note-0085-04" xlink:href="note-0085-04a" xml:space="preserve">Lem. 4. <lb/>præmiſ.</note>
</div>
<p>
  <s xml:id="echoid-s2258" xml:space="preserve">Poſtea educamus ex E lineam occurrentem ſectioni in V, &amp;</s>
  <s xml:id="echoid-s2259" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2260" xml:space="preserve">Educamus <lb/>
<anchor type="note" xlink:label="note-0085-05a" xlink:href="note-0085-05"/>
ex E lineam occurrentem ſectioni in V, quæ ſecet axim in Z, &amp; </s>
  <s xml:id="echoid-s2261" xml:space="preserve">S M in Y.</s>
  <s xml:id="echoid-s2262" xml:space="preserve"/>
</p>
<div xml:id="echoid-div197" type="float" level="2" n="11">
<note position="left" xlink:label="note-0085-05" xlink:href="note-0085-05a" xml:space="preserve">l</note>
</div>
<p>
  <s xml:id="echoid-s2263" xml:space="preserve">Et per f producamus f g h parallelam axi A D, &amp;</s>
  <s xml:id="echoid-s2264" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2265" xml:space="preserve">Et per f ducamus f g pa-<lb/>
<anchor type="note" xlink:label="note-0085-06a" xlink:href="note-0085-06"/>
rallelam axi A D, quæ ſecet tangentem B a in h, &amp; </s>
  <s xml:id="echoid-s2266" xml:space="preserve">L F in g, atque V c ſecet illam in <lb/>i, &amp; </s>
  <s xml:id="echoid-s2267" xml:space="preserve">S M in e.</s>
  <s xml:id="echoid-s2268" xml:space="preserve"/>
</p>
<div xml:id="echoid-div198" type="float" level="2" n="12">
<note position="left" xlink:label="note-0085-06" xlink:href="note-0085-06a" xml:space="preserve">m</note>
</div>
<p style="it">
  <s xml:id="echoid-s2269" xml:space="preserve">Et ponamus rectangulum F f communiter, &amp;</s>
  <s xml:id="echoid-s2270" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2271" xml:space="preserve">Et communiter addamus in <lb/>
<anchor type="note" xlink:label="note-0085-07a" xlink:href="note-0085-07"/>
hyperbola, &amp; </s>
  <s xml:id="echoid-s2272" xml:space="preserve">auferamus in ellipſi rectangulum F f, fiet rectangulum B fg æquale <lb/>rectangulo g F C. </s>
  <s xml:id="echoid-s2273" xml:space="preserve">Nomina Inuerſi, &amp; </s>
  <s xml:id="echoid-s2274" xml:space="preserve">Trutinatæ definita fuerunt in primo libro ab <lb/>interprete Arabico.</s>
  <s xml:id="echoid-s2275" xml:space="preserve"/>
</p>
<div xml:id="echoid-div199" type="float" level="2" n="13">
<note position="left" xlink:label="note-0085-07" xlink:href="note-0085-07a" xml:space="preserve">n</note>
</div>
<pb o="48" file="0086" n="86" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s2276" xml:space="preserve">Igitur C a eſt li-<lb/>
<anchor type="note" xlink:label="note-0086-01a" xlink:href="note-0086-01"/>
<anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/>
nea quinta propor-<lb/>tionalis aliarum. <lb/></s>
  <s xml:id="echoid-s2277" xml:space="preserve">quatuor, &amp;</s>
  <s xml:id="echoid-s2278" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2279" xml:space="preserve">Quia <lb/>poſitæ fuerunt qua-<lb/>tuor rectæ lineæ F C, <lb/>N C, O C, C A con-<lb/>tinuè proportionales, <lb/>eſt que C A ad C a, vt <lb/>O C ad C A; </s>
  <s xml:id="echoid-s2280" xml:space="preserve">ergò pri-<lb/>
<anchor type="note" xlink:label="note-0086-02a" xlink:href="note-0086-02"/>
ma F C ad tertiam, <lb/>O C eamdem propor-<lb/>tionem habet, quàm <lb/>O C ad quintam C a <lb/>continuè proportio-<lb/>nalium, quare com-<lb/>parando homologorũ <lb/>
<anchor type="note" xlink:label="note-0086-03a" xlink:href="note-0086-03"/>
differentias F O ad <lb/>O a eſt, vt F C ad C <lb/>O; </s>
  <s xml:id="echoid-s2281" xml:space="preserve">ſedfacta fuit vt <lb/>F O, ad O C, ita f O <lb/>ad O B; </s>
  <s xml:id="echoid-s2282" xml:space="preserve">ergo compo-<lb/>nendo in hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s2283" xml:space="preserve">comparando dif-<lb/>ferentias terminorũ <lb/>
<anchor type="note" xlink:label="note-0086-04a" xlink:href="note-0086-04"/>
ad conſequentes in, <lb/>ellipſi, eſt F C ad C O, ſeu F O ad O a, vt f B ad B O; </s>
  <s xml:id="echoid-s2284" xml:space="preserve">nempe vt f h ad eandem O a, <lb/>propter ſimilitudinẽ triangulorum B fh, &amp; </s>
  <s xml:id="echoid-s2285" xml:space="preserve">B O a; </s>
  <s xml:id="echoid-s2286" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2287" xml:space="preserve">ideo F O, &amp; </s>
  <s xml:id="echoid-s2288" xml:space="preserve">fh æquales ſunt.</s>
  <s xml:id="echoid-s2289" xml:space="preserve"/>
</p>
<div xml:id="echoid-div200" type="float" level="2" n="14">
<note position="right" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve">o</note>
  <figure xlink:label="fig-0086-01" xlink:href="fig-0086-01a">
    <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0086-01"/>
  </figure>
<note position="left" xlink:label="note-0086-02" xlink:href="note-0086-02a" xml:space="preserve">37. lib. 1.</note>
<note position="left" xlink:label="note-0086-03" xlink:href="note-0086-03a" xml:space="preserve">Lem. 4. <lb/>præmiff.</note>
<note position="left" xlink:label="note-0086-04" xlink:href="note-0086-04a" xml:space="preserve">Lem. 2. <lb/>præm.</note>
</div>
<note position="right" xml:space="preserve">p</note>
<p>
  <s xml:id="echoid-s2290" xml:space="preserve">Et propterea fi ad i h maiorem proportionem habet, quàm ad f g, &amp;</s>
  <s xml:id="echoid-s2291" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s2292" xml:space="preserve">Quia F O, ſeu g f oſtenſa fuit æqualis fh erit g h ſecta bifariam in f, &amp; </s>
  <s xml:id="echoid-s2293" xml:space="preserve">non bifa-<lb/>riam in i propterea (ex lemmate ſexto huius lib.) </s>
  <s xml:id="echoid-s2294" xml:space="preserve">habebit fh ad ih, ſcilicet B f ad <lb/>di (propter ſimilitudinem triangulorum B fh, dih) maiorem proportionem, quàm <lb/>ig ad gf, ſed B f ad V i portionem ipſius d i habet maiorem proportionem, quàm ad <lb/>
<anchor type="note" xlink:label="note-0086-06a" xlink:href="note-0086-06"/>
di; </s>
  <s xml:id="echoid-s2295" xml:space="preserve">ergo B f ad V i habet maiorem proportionem, quàm i g ad g f, ergo rectangulum <lb/>B f g, nempe rectangulum g C (quod eſt oſtenſum ei æquale) maius eſt rectangulo <lb/>V i g.</s>
  <s xml:id="echoid-s2296" xml:space="preserve"/>
</p>
<div xml:id="echoid-div201" type="float" level="2" n="15">
<note position="left" xlink:label="note-0086-06" xlink:href="note-0086-06a" xml:space="preserve">Lem. 5. <lb/>In nota <lb/>litere n <lb/>præm.</note>
</div>
<p>
  <s xml:id="echoid-s2297" xml:space="preserve">Et ponamus rectangulum g e commune, &amp;</s>
  <s xml:id="echoid-s2298" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2299" xml:space="preserve">Et addamus in hyperbola, &amp; </s>
  <s xml:id="echoid-s2300" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0086-07a" xlink:href="note-0086-07"/>
auferamus in ellipſi rectangulum g e communiter.</s>
  <s xml:id="echoid-s2301" xml:space="preserve"/>
</p>
<div xml:id="echoid-div202" type="float" level="2" n="16">
<note position="right" xlink:label="note-0086-07" xlink:href="note-0086-07a" xml:space="preserve">q</note>
</div>
<p>
  <s xml:id="echoid-s2302" xml:space="preserve">Et propterea E K ad e V, nempe K ad Y e, &amp;</s>
  <s xml:id="echoid-s2303" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2304" xml:space="preserve">Sunt enim triangula E K Y, <lb/>
<anchor type="note" xlink:label="note-0086-08a" xlink:href="note-0086-08"/>
&amp; </s>
  <s xml:id="echoid-s2305" xml:space="preserve">V e Y ſimilia, ergo E K ad e V eſt, vt K Y ad Y e, quarè K Y ad Y e maiorem pro-<lb/>portionem habet, quàm e M ad M K, &amp; </s>
  <s xml:id="echoid-s2306" xml:space="preserve">componendo, eadem K e maiorem propor-<lb/>tionem habet ad e Y, quàm ad M K, ſeu ad F D; </s>
  <s xml:id="echoid-s2307" xml:space="preserve">vnde patet, quod e Y minor ſit, <lb/>quam F D.</s>
  <s xml:id="echoid-s2308" xml:space="preserve"/>
</p>
<div xml:id="echoid-div203" type="float" level="2" n="17">
<note position="right" xlink:label="note-0086-08" xlink:href="note-0086-08a" xml:space="preserve">r</note>
</div>
<p>
  <s xml:id="echoid-s2309" xml:space="preserve">Et conſtat quemadmodum antea demonſtrauimus, &amp;</s>
  <s xml:id="echoid-s2310" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2311" xml:space="preserve">Quoniam e Y mi-<lb/>
<anchor type="note" xlink:label="note-0086-09a" xlink:href="note-0086-09"/>
nor oſtenſa eſt, quam K M ergo eadem E I ad r e, ſeu I X ad V e (propter ſimilitu-<lb/>dinem triangulorum E I X, r e V) maiorem proportionem habebit, quàm E I ad <lb/>M K, ſeu I C ad C S, veladei æqualem c e; </s>
  <s xml:id="echoid-s2312" xml:space="preserve">igitur comparando homologorum ſum-
<pb o="49" file="0087" n="87" rhead="Conicor. Lib. V."/>
mas in ellipſi, &amp; </s>
  <s xml:id="echoid-s2313" xml:space="preserve">eo-<lb/>
<anchor type="figure" xlink:label="fig-0087-01a" xlink:href="fig-0087-01"/>
rundem differentias <lb/>in hyperbola C X ad <lb/>
<anchor type="note" xlink:label="note-0087-01a" xlink:href="note-0087-01"/>
c V, vel (propter <lb/>ſimilitudinem triã-<lb/>gulorum X C Z, V c <lb/>Z) C Z ad Z c ma-<lb/>iorem proportionem <lb/>habet, quàm I C ad <lb/>C S, vel C D ad D <lb/>F; </s>
  <s xml:id="echoid-s2314" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2315" xml:space="preserve">componendo <lb/>in ellipſi, &amp; </s>
  <s xml:id="echoid-s2316" xml:space="preserve">diui-<lb/>dendo in hyperbola <lb/>C c ad c Z maiorẽ <lb/>proportionem habe-<lb/>bit, quàm C F ad <lb/>
<anchor type="note" xlink:label="note-0087-02a" xlink:href="note-0087-02"/>
F D, &amp; </s>
  <s xml:id="echoid-s2317" xml:space="preserve">ideo breuiſ-<lb/>ſima egrediens ex V <lb/>abſcindit lineã ma-<lb/>iorem, quàm A Z.</s>
  <s xml:id="echoid-s2318" xml:space="preserve"/>
</p>
<div xml:id="echoid-div204" type="float" level="2" n="18">
<note position="right" xlink:label="note-0086-09" xlink:href="note-0086-09a" xml:space="preserve">ſ</note>
  <figure xlink:label="fig-0087-01" xlink:href="fig-0087-01a">
    <image file="0087-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0087-01"/>
  </figure>
<note position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve">Lem. 4.</note>
<note position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">9. 10. <lb/>huius.</note>
</div>
<p>
  <s xml:id="echoid-s2319" xml:space="preserve">Simili modo cõ-<lb/>ſtat, quod breuiſ-<lb/>
<anchor type="note" xlink:label="note-0087-03a" xlink:href="note-0087-03"/>
ſima egrediens ex <lb/>l eiuſdem ſit ratio-<lb/>nis, &amp;</s>
  <s xml:id="echoid-s2320" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2321" xml:space="preserve">Abſque no-<lb/>ua demonſtratione <lb/>in ſecunda, &amp; </s>
  <s xml:id="echoid-s2322" xml:space="preserve">quar <lb/>ta figura propoſitum oſtenſum erit.</s>
  <s xml:id="echoid-s2323" xml:space="preserve"/>
</p>
<div xml:id="echoid-div205" type="float" level="2" n="19">
<note position="left" xlink:label="note-0087-03" xlink:href="note-0087-03a" xml:space="preserve">t</note>
</div>
<p>
  <s xml:id="echoid-s2324" xml:space="preserve">Deinde ſit E D æqualis Q, inde demonſtrabitur (quemadmodum ſu-<lb/>
<anchor type="note" xlink:label="note-0087-04a" xlink:href="note-0087-04"/>
pra factum eſt) quod B H tantum ſit linea breuiſſima, &amp;</s>
  <s xml:id="echoid-s2325" xml:space="preserve">c.</s>
  <s xml:id="echoid-s2326" xml:space="preserve"/>
</p>
<div xml:id="echoid-div206" type="float" level="2" n="20">
<note position="left" xlink:label="note-0087-04" xlink:href="note-0087-04a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div208" type="section" level="1" n="73">
<head xml:id="echoid-head106" style="it" xml:space="preserve">Secunda pars buius propoſitionis, quam Apollonius non expoſuit hac <lb/>ratione ſuppleri poteſt.</head>
<p style="it">
  <s xml:id="echoid-s2327" xml:space="preserve">Sit E D æqualis Trutinæ Q habebunt E D, atque Q eandem proportionem <lb/>ad B O, componitur verò proportio E D ad B O ex rationibus E D ad D K, &amp; </s>
  <s xml:id="echoid-s2328" xml:space="preserve"><lb/>D K ad B O, ſeu O G ad B O; </s>
  <s xml:id="echoid-s2329" xml:space="preserve">componebatur autem proportio Trutinæ Q ad B O <lb/>ex rationibus C D ad D F, &amp; </s>
  <s xml:id="echoid-s2330" xml:space="preserve">F O ad O C; </s>
  <s xml:id="echoid-s2331" xml:space="preserve">ergo ablata communiter proportione <lb/>E D ad D K, vel C D ad D F, relinquetur proportio G O ad O B eadem propor-<lb/>tioni F O ad O C; </s>
  <s xml:id="echoid-s2332" xml:space="preserve">ergo rectangulum G O C ſub extremis contentum æquale erit <lb/>rectangulo B O F ſub intermedĳs compræbenſo, addatur in hyperbola, &amp; </s>
  <s xml:id="echoid-s2333" xml:space="preserve">aufe-<lb/>ratur in ellipſi communiter rectangulum F G, erit rectangulum F S æquale re-<lb/>ctangulo B G M; </s>
  <s xml:id="echoid-s2334" xml:space="preserve">Et quia I S ad S C, vel E K ad K D, velad F M erat, vt C <lb/>F ad F D, vel vt S M ad M K; </s>
  <s xml:id="echoid-s2335" xml:space="preserve">ergo rectangulum E M æquale eſt rectangulo <lb/>F S; </s>
  <s xml:id="echoid-s2336" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2337" xml:space="preserve">propterea rectangulum E M æquale erit rectangulo B G M; </s>
  <s xml:id="echoid-s2338" xml:space="preserve">quapropter <lb/>vt E K ad B G, ſeu K R ad R G, ita erit G M ad M K, &amp; </s>
  <s xml:id="echoid-s2339" xml:space="preserve">componendo, eadem
<pb o="50" file="0088" n="88" rhead="Apollonij Pergæi"/>
K G eandem propor-<lb/>tionem habebit ad R <lb/>
<anchor type="figure" xlink:label="fig-0088-01a" xlink:href="fig-0088-01"/>
G, atque ad M K, <lb/>vnde R G æqualis e-<lb/>rit M K, vel F D, <lb/>quare eadem E I ad <lb/>K M, vel C D ad <lb/>D F, ſiue I C ad C <lb/>S eandem proportio-<lb/>nem habebit, quam <lb/>eadem E I ad R G, <lb/>vel I T ad B G (pro-<lb/>pter ſimilitudinem <lb/>triangulorum I E T, <lb/>&amp; </s>
  <s xml:id="echoid-s2340" xml:space="preserve">G R B) ergo com-<lb/>parando homologo-<lb/>rum ſummas in elli-<lb/>pſi, vel differentias <lb/>
<anchor type="note" xlink:label="note-0088-01a" xlink:href="note-0088-01"/>
in hyperbola C T ad <lb/>B O, vel C H ad H <lb/>O (propter ſimilitu-<lb/>dinem triangulorum <lb/>C H T, &amp; </s>
  <s xml:id="echoid-s2341" xml:space="preserve">O H B) <lb/>eandem proportionẽ <lb/>habebit, quàm I C <lb/>ad C S, vel C D ad <lb/>D F, &amp; </s>
  <s xml:id="echoid-s2342" xml:space="preserve">diuidendo <lb/>in hyperbola, &amp; </s>
  <s xml:id="echoid-s2343" xml:space="preserve">cõ-<lb/>ponendo in ellipſi C O ad O H eandem proportionem habebit, quàm C F ad F D, <lb/>ſiue quàm habet latus tranſuerſum ad rectum; </s>
  <s xml:id="echoid-s2344" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2345" xml:space="preserve">propterea B H eſt breuiſsima <lb/>
<anchor type="note" xlink:label="note-0088-02a" xlink:href="note-0088-02"/>
linearum ex B ad axim cadentium.</s>
  <s xml:id="echoid-s2346" xml:space="preserve"/>
</p>
<div xml:id="echoid-div208" type="float" level="2" n="1">
  <figure xlink:label="fig-0088-01" xlink:href="fig-0088-01a">
    <image file="0088-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0088-01"/>
  </figure>
<note position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve">Lem. 4.</note>
<note position="left" xlink:label="note-0088-02" xlink:href="note-0088-02a" xml:space="preserve">9. 10. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2347" xml:space="preserve">Deinde educatur quilibet ramus E V ſupra, velinfr a breuiſecantem E B, qui <lb/>productus ſecet rectam I C in X, &amp; </s>
  <s xml:id="echoid-s2348" xml:space="preserve">C A in Z, atque S M in γ, &amp; </s>
  <s xml:id="echoid-s2349" xml:space="preserve">educatur ex <lb/>V recta V e perpendicularis ad axim, ſecans D F in c, &amp; </s>
  <s xml:id="echoid-s2350" xml:space="preserve">S M in e, atque <lb/>contingentem ſectionem in puncto B, ſcilicet ipſam B a ſecet in d. </s>
  <s xml:id="echoid-s2351" xml:space="preserve">Et quia (vt <lb/>modo oſtenſum eſt) rectangulum F S æquale eſt rectangulo B G M, ſuntque pa-<lb/>riter oſtenſæ O C, A C, C a proportionales; </s>
  <s xml:id="echoid-s2352" xml:space="preserve">ergo C a eſt quinta proportionalis poſt <lb/>quatuor præcedentes F C, N C, O C, A C continuè proportionales; </s>
  <s xml:id="echoid-s2353" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2354" xml:space="preserve">ideo F C ad <lb/>C O eſt, vt C O ad C a; </s>
  <s xml:id="echoid-s2355" xml:space="preserve">ergo comparando homologorum differentias tam in hyper-<lb/>
<anchor type="note" xlink:label="note-0088-03a" xlink:href="note-0088-03"/>
bola, quàm in ellipſi erit, F O ad O a, vt F C ad C O: </s>
  <s xml:id="echoid-s2356" xml:space="preserve">eſt autem G B ad B O, <lb/>vt F C ad C O, vt antea oſtenſum eſt; </s>
  <s xml:id="echoid-s2357" xml:space="preserve">ergo G B ad B O erit, vt F O ad O a; </s>
  <s xml:id="echoid-s2358" xml:space="preserve">ſed <lb/>propter ſimilitudinem triangulorum B G b, B O a eſt G B ad B O, vt G b ad O a; <lb/></s>
  <s xml:id="echoid-s2359" xml:space="preserve">ergo F O, ſeu M G ad O a eandem proportionem habet, quàm G b ad eandem O a; </s>
  <s xml:id="echoid-s2360" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s2361" xml:space="preserve">propterea M G æqualis eſt G b; </s>
  <s xml:id="echoid-s2362" xml:space="preserve">cumque M b ſecetur æqualiter in G, &amp; </s>
  <s xml:id="echoid-s2363" xml:space="preserve">inæqua-<lb/>liter in e (ex lemmate 6. </s>
  <s xml:id="echoid-s2364" xml:space="preserve">huius) G b ad e b, ſeu B G, ad d e, propter ſimilitu-<lb/>dinem triangulorum B G b, &amp; </s>
  <s xml:id="echoid-s2365" xml:space="preserve">B O a, &amp; </s>
  <s xml:id="echoid-s2366" xml:space="preserve">multo magis B G ad V e portionem <lb/>ipſius d e habebit maiorem proportionem, quàm, e M ad G M; </s>
  <s xml:id="echoid-s2367" xml:space="preserve">ergo rectangulum
<pb o="51" file="0089" n="89" rhead="Conicor. Lib. V."/>
BG M ſub extremis <lb/>
<anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/>
<anchor type="figure" xlink:label="fig-0089-01a" xlink:href="fig-0089-01"/>
cõtentum maius erit <lb/>rectãgulo V e M ſub <lb/>medĳ s compræhenſo; <lb/></s>
  <s xml:id="echoid-s2368" xml:space="preserve">erat autem prius re-<lb/>ctangulum B G M <lb/>æquale rectangulo E <lb/>M; </s>
  <s xml:id="echoid-s2369" xml:space="preserve">ergo rectangulũ <lb/>E M maius eſt re-<lb/>ctangulo V e M, &amp; </s>
  <s xml:id="echoid-s2370" xml:space="preserve"><lb/>propterea E K ad V <lb/>
<anchor type="note" xlink:label="note-0089-02a" xlink:href="note-0089-02"/>
e, ſeu K γ ad γ e <lb/>(propter ſimilitudi-<lb/>nem triangulorum <lb/>E Y K, &amp; </s>
  <s xml:id="echoid-s2371" xml:space="preserve">V e Y) ma-<lb/>iorem proportionem <lb/>habebit, quàm e M <lb/>ad M K, &amp; </s>
  <s xml:id="echoid-s2372" xml:space="preserve">compo-<lb/>nendo, eadem K e <lb/>ad Y e maiorem pro-<lb/>portionem habebit, <lb/>quàm ad M K; </s>
  <s xml:id="echoid-s2373" xml:space="preserve">ergo <lb/>Y e minor eſt, quàm <lb/>M K, quare E I ad <lb/>Y e, ſeu I X ad e V <lb/>(propter ſimilitudi-<lb/>nem triangulorum I <lb/>E X, e Y V) habebit <lb/>maiorem proportio-<lb/>nem, quàm eadem. <lb/></s>
  <s xml:id="echoid-s2374" xml:space="preserve">E I ad M K, ſeu I C ad C S, velad c e; </s>
  <s xml:id="echoid-s2375" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2376" xml:space="preserve">propterea comparando homologorum <lb/>
<anchor type="note" xlink:label="note-0089-03a" xlink:href="note-0089-03"/>
ſummas in ellipſi, &amp; </s>
  <s xml:id="echoid-s2377" xml:space="preserve">earundem differentias in hyperbola C X ad c V, vel C Z <lb/>ad Z c (propter ſimilitudinem triangulorum C Z X, V c Z) maiorem proportio-<lb/>nem habebit, quàm S K, ad K M, ſeu C D ad D F, &amp; </s>
  <s xml:id="echoid-s2378" xml:space="preserve">diuidendo in hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s2379" xml:space="preserve">componendo in ellipſi C c ad c Z habebit maiorem proportionem, quàm C F <lb/>ad F D, ſeu quàm latus tranſuerſum ad rectum, &amp; </s>
  <s xml:id="echoid-s2380" xml:space="preserve">propterea breuiſsima linea-<lb/>
<anchor type="note" xlink:label="note-0089-04a" xlink:href="note-0089-04"/>
rum cadentium ex puncto V ad axim abſcindet ſegmentum maius, quàm A Z, <lb/>&amp; </s>
  <s xml:id="echoid-s2381" xml:space="preserve">ramus E V non erit breuiſecans, quod ſuerat oſtendendum.</s>
  <s xml:id="echoid-s2382" xml:space="preserve"/>
</p>
<div xml:id="echoid-div209" type="float" level="2" n="2">
<note position="left" xlink:label="note-0088-03" xlink:href="note-0088-03a" xml:space="preserve">Lem. 3.</note>
<note position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">Lem. 5.</note>
  <figure xlink:label="fig-0089-01" xlink:href="fig-0089-01a">
    <image file="0089-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0089-01"/>
  </figure>
<note position="right" xlink:label="note-0089-02" xlink:href="note-0089-02a" xml:space="preserve">Lem. 5.</note>
<note position="right" xlink:label="note-0089-03" xlink:href="note-0089-03a" xml:space="preserve">Lem. 4.</note>
<note position="right" xlink:label="note-0089-04" xlink:href="note-0089-04a" xml:space="preserve">ex 9. 10. <lb/>huius.</note>
</div>
<p>
  <s xml:id="echoid-s2383" xml:space="preserve">Et demonſtrabitur, quemadmodum dictum eſt, quod G O ad B O mi-<lb/>
<anchor type="note" xlink:label="note-0089-05a" xlink:href="note-0089-05"/>
norem proportionem habet, quàm F O ad O C, &amp;</s>
  <s xml:id="echoid-s2384" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2385" xml:space="preserve">Nam proportio E D ad <lb/>B O componitur ex rationibus E D ad D K, &amp; </s>
  <s xml:id="echoid-s2386" xml:space="preserve">D K, ſeu G O ad B O. </s>
  <s xml:id="echoid-s2387" xml:space="preserve">Pariterque <lb/>proportio Trutinæ Q quæ erat maior quàm E D ad B O componitur ex ratio-<lb/>nibus C D ad D F, &amp; </s>
  <s xml:id="echoid-s2388" xml:space="preserve">F O ad O C, auferatur communis proportio E D ad D K, <lb/>vel C D ad D F, remanet proportio G O ad O B minor proportione F O ad O C.</s>
  <s xml:id="echoid-s2389" xml:space="preserve"/>
</p>
<div xml:id="echoid-div210" type="float" level="2" n="3">
<note position="left" xlink:label="note-0089-05" xlink:href="note-0089-05a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s2390" xml:space="preserve">Et producamus ex V, l duas perpendiculares V e, l P, quæ, &amp;</s>
  <s xml:id="echoid-s2391" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2392" xml:space="preserve">Et <lb/>
<anchor type="note" xlink:label="note-0089-06a" xlink:href="note-0089-06"/>
producamus ex V, &amp; </s>
  <s xml:id="echoid-s2393" xml:space="preserve">V duas perpendiculares V e, quæ parallelæ ſint continenti <lb/>F M, &amp; </s>
  <s xml:id="echoid-s2394" xml:space="preserve">ſecent reliquas lineas in ſignis antea expoſitis; </s>
  <s xml:id="echoid-s2395" xml:space="preserve">Rectangulum ergo V e
<pb o="52" file="0090" n="90" rhead="Apollonij Pergæi"/>
in e M æquale eſt <lb/>
<anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/>
rectangulo V e M, <lb/>alterius figuræ, &amp;</s>
  <s xml:id="echoid-s2396" xml:space="preserve">c.</s>
  <s xml:id="echoid-s2397" xml:space="preserve"/>
</p>
<div xml:id="echoid-div211" type="float" level="2" n="4">
<note position="left" xlink:label="note-0089-06" xlink:href="note-0089-06a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a">
    <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0090-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s2398" xml:space="preserve">Et ponamus re-<lb/>
<anchor type="note" xlink:label="note-0090-01a" xlink:href="note-0090-01"/>
ctangulum F G cõ-<lb/>mune, &amp;</s>
  <s xml:id="echoid-s2399" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2400" xml:space="preserve">Scili-<lb/>cet, addatur in hy-<lb/>perbola, &amp; </s>
  <s xml:id="echoid-s2401" xml:space="preserve">aufera-<lb/>ratur in ellipſi com-<lb/>muniter rectangulis <lb/>F G.</s>
  <s xml:id="echoid-s2402" xml:space="preserve"/>
</p>
<div xml:id="echoid-div212" type="float" level="2" n="5">
<note position="right" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve">d</note>
</div>
<p>
  <s xml:id="echoid-s2403" xml:space="preserve">Tandem proſe-<lb/>
<anchor type="note" xlink:label="note-0090-02a" xlink:href="note-0090-02"/>
quamur ſuperiorẽ <lb/>demonſtrationem, <lb/>vt oſtendatur veri-<lb/>tas reliquarũ pro-<lb/>poſitionum, &amp;</s>
  <s xml:id="echoid-s2404" xml:space="preserve">c.</s>
  <s xml:id="echoid-s2405" xml:space="preserve"/>
</p>
<div xml:id="echoid-div213" type="float" level="2" n="6">
<note position="right" xlink:label="note-0090-02" xlink:href="note-0090-02a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s2406" xml:space="preserve">Demonſtratio ab <lb/>Apollonio breuitatis <lb/>gratia neglecta ſic <lb/>perficietur.</s>
  <s xml:id="echoid-s2407" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2408" xml:space="preserve">Quoniam rectã-<lb/>gulum E M æquale <lb/>eſt rectangulo V e <lb/>M, igitur vt E K ad <lb/>V e, ſeu K γ ad γ e <lb/>(propter ſimilitudinem triangulorum E K γ, &amp; </s>
  <s xml:id="echoid-s2409" xml:space="preserve">V e γ) ita erit e M ad M K, <lb/>&amp; </s>
  <s xml:id="echoid-s2410" xml:space="preserve">componendo, eadem e K habebit ad e γ, atque ad M K eandem proportionem, <lb/>ideoque e γ æqualis eſt M K; </s>
  <s xml:id="echoid-s2411" xml:space="preserve">quare E I ad K M, ſeu I C ad C S eandem pro-<lb/>portionem habebit, quàm E I ad e γ, ſeu quàm I X ad e V (propter ſimilitudi-<lb/>nem triangulorum I E X, &amp; </s>
  <s xml:id="echoid-s2412" xml:space="preserve">e γ V) quare comparando homologorum differentias <lb/>in hyperbola, &amp; </s>
  <s xml:id="echoid-s2413" xml:space="preserve">eorundem ſummas in ellipſi C X ad c V, vel C Z ad Z c (propter <lb/>
<anchor type="note" xlink:label="note-0090-03a" xlink:href="note-0090-03"/>
ſimilitudinem triangulorum C Z X, c Z V) habebit eandem proportionem, quàm I <lb/>C ad C S, vel C D ad D F, &amp; </s>
  <s xml:id="echoid-s2414" xml:space="preserve">diuidendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s2415" xml:space="preserve">componendo in ellipſi C c <lb/>ad c Z eandem proportionem habebit, quàm C F ad F D, ſeu quàm habet latus <lb/>
<anchor type="note" xlink:label="note-0090-04a" xlink:href="note-0090-04"/>
tranſuerſum ad rectum, &amp; </s>
  <s xml:id="echoid-s2416" xml:space="preserve">propterea recta linea V Z eſt breuiſsima omnium, <lb/>quæ ex V ad axim A D duci poſſunt.</s>
  <s xml:id="echoid-s2417" xml:space="preserve"/>
</p>
<div xml:id="echoid-div214" type="float" level="2" n="7">
<note position="left" xlink:label="note-0090-03" xlink:href="note-0090-03a" xml:space="preserve">Lem. 3.</note>
<note position="left" xlink:label="note-0090-04" xlink:href="note-0090-04a" xml:space="preserve">9. 10. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2418" xml:space="preserve">Iiſdem prorſus verbis oſtenſum erit, quod recta linea l m ſit breuiſsima om-<lb/>nium cadentium ex puncto l ad axim, ſi nimirum apponãtur caracteres prioris <lb/>caſus, vt patet in ſecunda, &amp; </s>
  <s xml:id="echoid-s2419" xml:space="preserve">quarta figura.</s>
  <s xml:id="echoid-s2420" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2421" xml:space="preserve">Iiſdem poſitis oſtendendum eſt, ramum B E, interceptum inter duos breuiſe-<lb/>cantes E V, non eſſe breuiſecantem, atque lineam breuiſsimam ex B ad axim <lb/>A D extenſam cadere ſupra ramum B E verſus verticem A.</s>
  <s xml:id="echoid-s2422" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2423" xml:space="preserve">Quoniam rectangulum B G M maius eſt rectangulo O G M, atque oſtenſum ſuit <lb/>rectangulum E M æquale rectangulo O G M; </s>
  <s xml:id="echoid-s2424" xml:space="preserve">ergo rectangulum B G M maius eſt <lb/>rectangulo E M, &amp; </s>
  <s xml:id="echoid-s2425" xml:space="preserve">propterea E K ad B G, ſeu K R ad R G (propter ſimilitudi-<lb/>
<anchor type="note" xlink:label="note-0090-05a" xlink:href="note-0090-05"/>
nem triangulorum) minorem proportionem habet, quàm G M ad M K, &amp; </s>
  <s xml:id="echoid-s2426" xml:space="preserve">com-
<pb o="53" file="0091" n="91" rhead="Conicor. Lib. V."/>
<anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/>
ponendo eadem K G <lb/>ad G R minorẽ pro-<lb/>portionem habebit, <lb/>quãad K M, &amp; </s>
  <s xml:id="echoid-s2427" xml:space="preserve">pro-<lb/>pterea G R maior e-<lb/>rit, quàm K M, vnde <lb/>E I ad G R, ſeu I T <lb/>ad G B (propter ſi-<lb/>militudinem trian-<lb/>gulorum E I T, R G <lb/>B) minorem propor-<lb/>tionem habet, quàm <lb/>E I ad K M, ſeu I C <lb/>ad C S; </s>
  <s xml:id="echoid-s2428" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2429" xml:space="preserve">ideo com-<lb/>parando homologarũ <lb/>ſummas in ellipſi, &amp; </s>
  <s xml:id="echoid-s2430" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/>
eorundem differen-<lb/>tias in hyperbola C <lb/>T ad O B, ſiue C H <lb/>ad H O (propter ſi-<lb/>militudinem trian-<lb/>gulorũ) habebit mi-<lb/>norẽ proportionem, <lb/>quàm I C ad C S, <lb/>vel C D ad D F, &amp; </s>
  <s xml:id="echoid-s2431" xml:space="preserve"><lb/>diuidendo in hyper-<lb/>bola, &amp; </s>
  <s xml:id="echoid-s2432" xml:space="preserve">componendo in ellipſi C O ad O H habebit minorem proportionem, quàm <lb/>
<anchor type="note" xlink:label="note-0091-02a" xlink:href="note-0091-02"/>
C F ad F D, ſiue quàm latus tranſuerſum habet ad rectum; </s>
  <s xml:id="echoid-s2433" xml:space="preserve">ergo breuiſsima ex <lb/>B ad axim ducta eum ſecat ſupra punctum H, &amp; </s>
  <s xml:id="echoid-s2434" xml:space="preserve">abſcindit lineam minorem, <lb/>quàm A H.</s>
  <s xml:id="echoid-s2435" xml:space="preserve"/>
</p>
<div xml:id="echoid-div215" type="float" level="2" n="8">
<note position="left" xlink:label="note-0090-05" xlink:href="note-0090-05a" xml:space="preserve">Lem. 5.</note>
  <figure xlink:label="fig-0091-01" xlink:href="fig-0091-01a">
    <image file="0091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0091-01"/>
  </figure>
<note position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve">Lem. 4.</note>
<note position="right" xlink:label="note-0091-02" xlink:href="note-0091-02a" xml:space="preserve">Ex 9. 10. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2436" xml:space="preserve">Rurſus ĳſdem poſitis, oſtendendum eſt, ramum E p cadentem ſupra ramum <lb/>E V verſus verticem, velinfra infimum breuiſecantem E V non eße breuiſecan-<lb/>tem, &amp; </s>
  <s xml:id="echoid-s2437" xml:space="preserve">abſcindere ex axi minorem lineam, quàm abſcindit breuiſsima ex pun-<lb/>cto p ad axim ducta. </s>
  <s xml:id="echoid-s2438" xml:space="preserve">Ducatur ex p recta linea p x perpendicularis ad axim, <lb/>eum ſecans in x, &amp; </s>
  <s xml:id="echoid-s2439" xml:space="preserve">ſecans S M in r, &amp; </s>
  <s xml:id="echoid-s2440" xml:space="preserve">hyperbolen V o in t, pariterque ramus <lb/>E p ſecet S M in z, &amp; </s>
  <s xml:id="echoid-s2441" xml:space="preserve">A F in q, atque I C in f. </s>
  <s xml:id="echoid-s2442" xml:space="preserve">Quoniam hyperbole V o ſe-<lb/>cat coniſectionem A B in V, &amp; </s>
  <s xml:id="echoid-s2443" xml:space="preserve">p ponitur ſupra V ad partes A; </s>
  <s xml:id="echoid-s2444" xml:space="preserve">ergo t cadit <lb/>extra ſectionem A B, &amp; </s>
  <s xml:id="echoid-s2445" xml:space="preserve">propterea t r maior erit, quàm p r; </s>
  <s xml:id="echoid-s2446" xml:space="preserve">vnde rectangulum <lb/>p r M minus erit rectangulo t r M; </s>
  <s xml:id="echoid-s2447" xml:space="preserve">ſed propter aſymptotos S M, M F eſt rectan-<lb/>
<anchor type="note" xlink:label="note-0091-03a" xlink:href="note-0091-03"/>
gulum t r M æquale rectangulo o G M, ſeu rectangulo E M, vt dictum eſt; </s>
  <s xml:id="echoid-s2448" xml:space="preserve">ergo <lb/>rectangulum p r M minus eſt rectangulo E K M, &amp; </s>
  <s xml:id="echoid-s2449" xml:space="preserve">propterea E K ad p r, ſeu <lb/>
<anchor type="note" xlink:label="note-0091-04a" xlink:href="note-0091-04"/>
K z ad z r (propter ſimilitudinem triangulorum) maiorem proportionem habet, <lb/>quàm r M ad M K, &amp; </s>
  <s xml:id="echoid-s2450" xml:space="preserve">componendo, eadé K r ad r z maioré proportioné habet, <lb/>quàm ad M K; </s>
  <s xml:id="echoid-s2451" xml:space="preserve">ergo r z minor eſt, quàm M K; </s>
  <s xml:id="echoid-s2452" xml:space="preserve">ideoque E I ad r z, ſeu I f ad <lb/>r p (propter ſimilitudinem triangulorum E I ſ, &amp; </s>
  <s xml:id="echoid-s2453" xml:space="preserve">r p z) maiorem proportionem <lb/>habet, quàm E I ad M K, ſeu I C ad C S, vel ad r x; </s>
  <s xml:id="echoid-s2454" xml:space="preserve">ergo comparando homo-<lb/>
<anchor type="note" xlink:label="note-0091-05a" xlink:href="note-0091-05"/>
logorum ſummas in ellipſi, &amp; </s>
  <s xml:id="echoid-s2455" xml:space="preserve">eorundem differentias in hyperbola C ſ ad x p,
<pb o="54" file="0092" n="92" rhead="Apollonij Pergæi"/>
ſiue C q ad q x (propter ſimilitudinem triangulorum) maiorem proportionem <lb/>habebit, quàm I C ad C S, vel C D ad D F, &amp; </s>
  <s xml:id="echoid-s2456" xml:space="preserve">diuidendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s2457" xml:space="preserve">com-<lb/>ponendo in ellipſi, C x ad x q maiorem proportionem habebit, quàm C F ad F <lb/>D, ſiue quàm latus tranſuerſum ad rectum, quapropter breuiſsima ex p ad axim <lb/>
<anchor type="note" xlink:label="note-0092-01a" xlink:href="note-0092-01"/>
ducta ſecat maiorem lineam, quàm A q. </s>
  <s xml:id="echoid-s2458" xml:space="preserve">Quæ omnia oſtendenda fuerant.</s>
  <s xml:id="echoid-s2459" xml:space="preserve"/>
</p>
<div xml:id="echoid-div216" type="float" level="2" n="9">
<note position="right" xlink:label="note-0091-03" xlink:href="note-0091-03a" xml:space="preserve">12. lib.2.</note>
<note position="right" xlink:label="note-0091-04" xlink:href="note-0091-04a" xml:space="preserve">Lem. 5.</note>
<note position="right" xlink:label="note-0091-05" xlink:href="note-0091-05a" xml:space="preserve">Lem. 4.</note>
<note position="left" xlink:label="note-0092-01" xlink:href="note-0092-01a" xml:space="preserve">Ex 9. 10. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div218" type="section" level="1" n="74">
<head xml:id="echoid-head107" xml:space="preserve">Notæ in Propoſ. LIV. LV.</head>
<p>
  <s xml:id="echoid-s2460" xml:space="preserve">ITaque oſtenſum eſt, vti memorauimus, quod ex concurſu duarum, <lb/>
<anchor type="note" xlink:label="note-0092-02a" xlink:href="note-0092-02"/>
breuiſſimarum ad illam ſectionem non egrediatur alia breuiſecans prę-<lb/>ter illas duas, &amp; </s>
  <s xml:id="echoid-s2461" xml:space="preserve">quod reliqui rami ex eorum concurſu educti ad ſectio-<lb/>nem habent proprietates ſuperius expoſitas.</s>
  <s xml:id="echoid-s2462" xml:space="preserve"/>
</p>
<div xml:id="echoid-div218" type="float" level="2" n="1">
<note position="right" xlink:label="note-0092-02" xlink:href="note-0092-02a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s2463" xml:space="preserve">Senſum germanum huius conſectarĳ, in quo duæ propoſitiones Apollonĳ con-<lb/>tinentur, non eſt facile diuinare in tanta Apolloĳ breuitate, &amp; </s>
  <s xml:id="echoid-s2464" xml:space="preserve">textus Arabici <lb/>inſigni corruptione; </s>
  <s xml:id="echoid-s2465" xml:space="preserve">videtur enim recenſere, &amp; </s>
  <s xml:id="echoid-s2466" xml:space="preserve">recolligere concluſionem quam-<lb/>dam præcedentium propoſitionum: </s>
  <s xml:id="echoid-s2467" xml:space="preserve">at hoc fieri nullo modo debebat in duabus pro-<lb/>poſitionibus 44. </s>
  <s xml:id="echoid-s2468" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2469" xml:space="preserve">45. </s>
  <s xml:id="echoid-s2470" xml:space="preserve">Rurſus ſi theoremata ſunt, demonſtrari non poterant ante <lb/>propoſitiones 51. </s>
  <s xml:id="echoid-s2471" xml:space="preserve">52. </s>
  <s xml:id="echoid-s2472" xml:space="preserve">53; </s>
  <s xml:id="echoid-s2473" xml:space="preserve">ſed for ſan numeri Arabici non 44. </s>
  <s xml:id="echoid-s2474" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2475" xml:space="preserve">45; </s>
  <s xml:id="echoid-s2476" xml:space="preserve">ſed 54. </s>
  <s xml:id="echoid-s2477" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2478" xml:space="preserve"><lb/>55. </s>
  <s xml:id="echoid-s2479" xml:space="preserve">eſſe debent, quod mirum non eſt, cum numeri paſsim in hoc codice Arabico <lb/>deformati reperiantur. </s>
  <s xml:id="echoid-s2480" xml:space="preserve">Itaque in hac ambiguitate ſuſpicor, textum ſic reſtitui <lb/>poſſe.</s>
  <s xml:id="echoid-s2481" xml:space="preserve"/>
</p>
  <figure>
    <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0092-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s2482" xml:space="preserve">Si in coniſectione duæ breuiſecantes ductæ fuerint ab eorum concurſu, <lb/>
<anchor type="note" xlink:label="note-0092-03a" xlink:href="note-0092-03"/>
nullus alius ramus ductus erit breuiſecans: </s>
  <s xml:id="echoid-s2483" xml:space="preserve">Et ramorum ab eodem con-<lb/>curſu extenſorum, qui inter breuiſecantes intercipiuntur, abſcindunt axis <lb/>ſegmenta maiora, &amp; </s>
  <s xml:id="echoid-s2484" xml:space="preserve">qui non intercipiuntur, minora, quàm abſcindant <lb/>lineæ breuiſsimæ ab eorum terminis ad axim ductæ: </s>
  <s xml:id="echoid-s2485" xml:space="preserve">oportet autem in, <lb/>ellipſi, vt duo rami, &amp; </s>
  <s xml:id="echoid-s2486" xml:space="preserve">perpendicularis cadant inter axis maioris ver-<lb/>ticem, &amp; </s>
  <s xml:id="echoid-s2487" xml:space="preserve">centrum ſectionis.</s>
  <s xml:id="echoid-s2488" xml:space="preserve"/>
</p>
<div xml:id="echoid-div219" type="float" level="2" n="2">
<note position="left" xlink:label="note-0092-03" xlink:href="note-0092-03a" xml:space="preserve">PROP. 5. <lb/>Addit.</note>
</div>
<pb o="55" file="0093" n="93" rhead="Conicor. Lib. V."/>
<p style="it">
  <s xml:id="echoid-s2489" xml:space="preserve">Sit coniſectio A B C, cuius axis A D, &amp; </s>
  <s xml:id="echoid-s2490" xml:space="preserve">in hyperbola, &amp; </s>
  <s xml:id="echoid-s2491" xml:space="preserve">ellipſi centrum <lb/>E; </s>
  <s xml:id="echoid-s2492" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2493" xml:space="preserve">ſumantur quælibet duo puncta B, &amp; </s>
  <s xml:id="echoid-s2494" xml:space="preserve">C, quæ in ellipſi ſint in eodem eius <lb/>quadrante, &amp; </s>
  <s xml:id="echoid-s2495" xml:space="preserve">ducantur B F, C H perpendiculares ad axim, &amp; </s>
  <s xml:id="echoid-s2496" xml:space="preserve">in parabola, <lb/>fiant F G, &amp; </s>
  <s xml:id="echoid-s2497" xml:space="preserve">H I æquales ſemiſsi lateris recti; </s>
  <s xml:id="echoid-s2498" xml:space="preserve">at in hyperbola, &amp; </s>
  <s xml:id="echoid-s2499" xml:space="preserve">ellipſi fiat <lb/>E F ad F G, nec non E H ad H I, vt latus tranſuerſum ad rectum, coniun-<lb/>ganturq; </s>
  <s xml:id="echoid-s2500" xml:space="preserve">rectæ B G, &amp; </s>
  <s xml:id="echoid-s2501" xml:space="preserve">C I. </s>
  <s xml:id="echoid-s2502" xml:space="preserve">Manifeſtum eſt B G, &amp; </s>
  <s xml:id="echoid-s2503" xml:space="preserve">C I eſſe lineas breuiſsimas, <lb/>quæ ſi producantur vltra axim (ex 28. </s>
  <s xml:id="echoid-s2504" xml:space="preserve">propoſitione huius libri) conuenient <lb/>
<anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/>
alicubi, vt in K. </s>
  <s xml:id="echoid-s2505" xml:space="preserve">Dico, quod ex concurſu K nullus alius ramus breuiſecans <lb/>duci poteſt ad ſectionem A B C. </s>
  <s xml:id="echoid-s2506" xml:space="preserve">Extendatur ex K ſuper axim A D perpendi-<lb/>cularis K D, &amp; </s>
  <s xml:id="echoid-s2507" xml:space="preserve">reperiatur ſectionis Trutina L competens menſuræ A D ipſius <lb/>concurſus K, vt in propoſitionibus 51. </s>
  <s xml:id="echoid-s2508" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2509" xml:space="preserve">52. </s>
  <s xml:id="echoid-s2510" xml:space="preserve">præcipitur. </s>
  <s xml:id="echoid-s2511" xml:space="preserve">Et certè perpendicu-<lb/>laris K D non erit maior, quàm L, aliàs duci non poſſet ramus vllus breui-<lb/>
<anchor type="note" xlink:label="note-0093-02a" xlink:href="note-0093-02"/>
ſecans ex concurſu K ad ſectionem A B C, quod eſt falſum; </s>
  <s xml:id="echoid-s2512" xml:space="preserve">factæ enim fuerunt <lb/>K B, &amp; </s>
  <s xml:id="echoid-s2513" xml:space="preserve">K C breuiſecantes; </s>
  <s xml:id="echoid-s2514" xml:space="preserve">Similiter K D non exit æqualis Trutinæ L, quan-<lb/>doquidem tunc vnica tantummodo breuiſecans ex K ad ſectionem A B C duci <lb/>poßet, quod rurſus falſum eſt, poſitæ enim fuerunt duæ breuiſecantes; </s>
  <s xml:id="echoid-s2515" xml:space="preserve">igitur per-<lb/>pendicularis K D neceſſario minor erit Trutina L, &amp; </s>
  <s xml:id="echoid-s2516" xml:space="preserve">ideo ex concurſu K duæ <lb/>
<anchor type="note" xlink:label="note-0093-03a" xlink:href="note-0093-03"/>
tantummodo breuiſecantes ad ſectionem A B C duci poſſunt, quæ ſunt B K, C K; <lb/></s>
  <s xml:id="echoid-s2517" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2518" xml:space="preserve">propterea nullus alius ramus breuiſecans ex concurſu. </s>
  <s xml:id="echoid-s2519" xml:space="preserve">K ad ſectionem A B C <lb/>duci poteſt præter duos K B, &amp; </s>
  <s xml:id="echoid-s2520" xml:space="preserve">K C; </s>
  <s xml:id="echoid-s2521" xml:space="preserve">quod erat primo loco oſtendendum.</s>
  <s xml:id="echoid-s2522" xml:space="preserve"/>
</p>
<div xml:id="echoid-div220" type="float" level="2" n="3">
<note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">8. 9. 10. <lb/>huius.</note>
<note position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">51. 52. <lb/>huius.</note>
<note position="right" xlink:label="note-0093-03" xlink:href="note-0093-03a" xml:space="preserve">51. 52. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2523" xml:space="preserve">Secundo ĳſdem poſitis, dico, quod rami ducti inter K B, &amp; </s>
  <s xml:id="echoid-s2524" xml:space="preserve">K C cadunt infra <lb/>lineas breuiſsimas ab eorom terminis ad axim ductas, &amp; </s>
  <s xml:id="echoid-s2525" xml:space="preserve">quod rami producti ex <lb/>K ſupra breuiſccantem K B verſus A verticem ſectionis, vel infra ramum bre-<lb/>uiſecantem K C abſcindunt axis ſegmenta ex vertice minora, quàm abſcindant <lb/>lineæ breuiſsimæ ab eorum terminis ad axim ductæ. </s>
  <s xml:id="echoid-s2526" xml:space="preserve">Reperiatur denuo Trutina <lb/>L, oſtendetur, vt prius perpendicularis K D minor, quàm L, &amp; </s>
  <s xml:id="echoid-s2527" xml:space="preserve">duæ tantummo-<lb/>do breuiſecantes K B, &amp; </s>
  <s xml:id="echoid-s2528" xml:space="preserve">K C; </s>
  <s xml:id="echoid-s2529" xml:space="preserve">quare quilibet ramus ex K ad ſectionis punctum, <lb/>
<anchor type="note" xlink:label="note-0093-04a" xlink:href="note-0093-04"/>
inter B, C poſitum extenſus, ſecat ſegmentum axis ex vertice A maius quàm ab-<lb/>ſcindat linea breniſsima ab eius termino ad axim ducta: </s>
  <s xml:id="echoid-s2530" xml:space="preserve">pariterque quilibet ra-<lb/>mus ex K ad punctum ſectionis ſupra B, poſitum, vel infra ramum K C exten-<lb/>ſus, abſcindet ſegmentum axis ex A minus, quàm ſecet linea breuiſsima ab <lb/>eius termino ad axim ducta; </s>
  <s xml:id="echoid-s2531" xml:space="preserve">quod erat oſtendendum.</s>
  <s xml:id="echoid-s2532" xml:space="preserve"/>
</p>
<div xml:id="echoid-div221" type="float" level="2" n="4">
<note position="right" xlink:label="note-0093-04" xlink:href="note-0093-04a" xml:space="preserve">51. 52. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div223" type="section" level="1" n="75">
<head xml:id="echoid-head108" xml:space="preserve">Notæ in Propoſit. LVI.</head>
<p>
  <s xml:id="echoid-s2533" xml:space="preserve">R Eperitur quidem in ramis aggregati ſecantis bifariam inclinatum, <lb/>
<anchor type="note" xlink:label="note-0093-05a" xlink:href="note-0093-05"/>
ſuper quod non cadit perpendicularis, breuiſecans vna tantum, quo-<lb/>modocumque ſe habeant perpendicularis, &amp; </s>
  <s xml:id="echoid-s2534" xml:space="preserve">menſura, &amp;</s>
  <s xml:id="echoid-s2535" xml:space="preserve">c.</s>
  <s xml:id="echoid-s2536" xml:space="preserve"/>
</p>
<div xml:id="echoid-div223" type="float" level="2" n="1">
<note position="left" xlink:label="note-0093-05" xlink:href="note-0093-05a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s2537" xml:space="preserve">Senſum huius propoſitionis nec Apollonius quidem ſi reuiuiſceret inſigni bar-<lb/>barie corruptum perciperet, cenſeo tamen, ſic reſtitui debere.</s>
  <s xml:id="echoid-s2538" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2539" xml:space="preserve">In ellipſi ramorum ſecantium vtrumque axim à concur ſu vltra centrum po-<lb/>ſito egredientium, vnius tantùm portio inter axim maiorem, &amp; </s>
  <s xml:id="echoid-s2540" xml:space="preserve">ſectionem inter-<lb/>cepta erit linea breuiſsima; </s>
  <s xml:id="echoid-s2541" xml:space="preserve">ſiue menſura ipſam comparatam, nec non perpendi-<lb/>cularis ipſam Trutinam ſuperet, æquet, vel ab ea deficiat.</s>
  <s xml:id="echoid-s2542" xml:space="preserve"/>
</p>
<pb o="56" file="0094" n="94" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s2543" xml:space="preserve">Sit ſectio ellipſis <lb/>
<anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/>
A C B tranſuerſa A <lb/>
<anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/>
B, &amp;</s>
  <s xml:id="echoid-s2544" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2545" xml:space="preserve">Lego; </s>
  <s xml:id="echoid-s2546" xml:space="preserve">Sit ſe-<lb/>ctio ellipſis A C B, &amp; </s>
  <s xml:id="echoid-s2547" xml:space="preserve"><lb/>axis maior A B, cen-<lb/>trum D, &amp; </s>
  <s xml:id="echoid-s2548" xml:space="preserve">perpendi-<lb/>cularis E F ſecans a-<lb/>xim in F inter cen-<lb/>trũ ellipſis D, &amp; </s>
  <s xml:id="echoid-s2549" xml:space="preserve">ver-<lb/>ticem A.</s>
  <s xml:id="echoid-s2550" xml:space="preserve"/>
</p>
<div xml:id="echoid-div224" type="float" level="2" n="2">
<note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a">
    <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0094-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s2551" xml:space="preserve">Et ducamus per <lb/>
<anchor type="note" xlink:label="note-0094-02a" xlink:href="note-0094-02"/>
punctum E ſectionẽ <lb/>hyperbolicam E M <lb/>C circa duas eius continentes, &amp;</s>
  <s xml:id="echoid-s2552" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2553" xml:space="preserve">Ideſt circa duas asymptotos I L, I H per <lb/>E deſcribatur hyperbole E M C, quæ ſecet axim A B æquidiſtantem alteri asym-<lb/>
<anchor type="note" xlink:label="note-0094-03a" xlink:href="note-0094-03"/>
ptoton in aliquo puncto vt in M; </s>
  <s xml:id="echoid-s2554" xml:space="preserve">oſtendetur punctum M ſuper ellipſis centrum <lb/>D cadere.</s>
  <s xml:id="echoid-s2555" xml:space="preserve"/>
</p>
<div xml:id="echoid-div225" type="float" level="2" n="3">
<note position="left" xlink:label="note-0094-02" xlink:href="note-0094-02a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">12. &amp; 13. <lb/>lib. 2.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2556" xml:space="preserve">Ergo E H prima in proportione in IH ſubſequentem, nempe G F ſub-<lb/>
<anchor type="note" xlink:label="note-0094-04a" xlink:href="note-0094-04"/>
ſequens ipſam M G quartam, æquale eſt ſubſequenti D G ſecundæ in, <lb/>I G nempe F H tertiam. </s>
  <s xml:id="echoid-s2557" xml:space="preserve">Ergo punctum N, &amp;</s>
  <s xml:id="echoid-s2558" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2559" xml:space="preserve">Textus corruptus ſic reſti-<lb/>tui poſſe cenſeo; </s>
  <s xml:id="echoid-s2560" xml:space="preserve">Ergo E H prima proportionalium in H I, nempe G F quartam <lb/>æquale eſt D G ſecundæ in I G, nempe F H tertiam, &amp;</s>
  <s xml:id="echoid-s2561" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2562" xml:space="preserve">Propterea quod E H ad <lb/>F H, atque D G ad G F poſitæ fuerunt, vt latus tranſuerſum ad rectum; </s>
  <s xml:id="echoid-s2563" xml:space="preserve">ergo re-<lb/>ctangulum ſub D G, &amp; </s>
  <s xml:id="echoid-s2564" xml:space="preserve">H F, ſeu I G, extremis quatuor proportionalium, æqua-<lb/>le eſt rectangulo ſub intermedĳs E H, &amp; </s>
  <s xml:id="echoid-s2565" xml:space="preserve">F G, ſeu H I, eſt que punctum E in, <lb/>hyperbola E M C cuius aſymptoti K I, L I; </s>
  <s xml:id="echoid-s2566" xml:space="preserve">ergo punctum D in eadem hyperbola <lb/>exiſtit; </s>
  <s xml:id="echoid-s2567" xml:space="preserve">ſed erat prius in ellipſis diametro A B, ſcilicet in centro; </s>
  <s xml:id="echoid-s2568" xml:space="preserve">quare in eorum <lb/>communi ſectione exiſtet: </s>
  <s xml:id="echoid-s2569" xml:space="preserve">erat autem punctum M communis ſectio hyperboles <lb/>E C, &amp; </s>
  <s xml:id="echoid-s2570" xml:space="preserve">axis ellipſis A B; </s>
  <s xml:id="echoid-s2571" xml:space="preserve">igitur puncta M, &amp; </s>
  <s xml:id="echoid-s2572" xml:space="preserve">D coincidunt, &amp; </s>
  <s xml:id="echoid-s2573" xml:space="preserve">hyperbole E D C <lb/>tranſit per centrũ ſectionis ellipticæ A C B, &amp; </s>
  <s xml:id="echoid-s2574" xml:space="preserve">ideo hyperbole E D C, quæ in infinitũ <lb/>
<anchor type="note" xlink:label="note-0094-05a" xlink:href="note-0094-05"/>
extendi, &amp; </s>
  <s xml:id="echoid-s2575" xml:space="preserve">dilatari poteſt neceſſario ſecabit finitam ellipſim alicubi, vt in C.</s>
  <s xml:id="echoid-s2576" xml:space="preserve"/>
</p>
<div xml:id="echoid-div226" type="float" level="2" n="4">
<note position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">8. lib. I.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2577" xml:space="preserve">Et producamus per E C lineam, &amp;</s>
  <s xml:id="echoid-s2578" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2579" xml:space="preserve">Et producamus per E C rectam li-<lb/>
<anchor type="note" xlink:label="note-0094-06a" xlink:href="note-0094-06"/>
neam, quæ occurrat continentibus in L, K, &amp; </s>
  <s xml:id="echoid-s2580" xml:space="preserve">ſecet axim ellipſis in P.</s>
  <s xml:id="echoid-s2581" xml:space="preserve"/>
</p>
<div xml:id="echoid-div227" type="float" level="2" n="5">
<note position="right" xlink:label="note-0094-06" xlink:href="note-0094-06a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s2582" xml:space="preserve">Erit G F æqualis O N, quare F O, &amp;</s>
  <s xml:id="echoid-s2583" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2584" xml:space="preserve">Quia duæ rectæ lineæ A O, L K <lb/>ſecantur à parallelis I L, F E, C N, K O proportionaliter, &amp; </s>
  <s xml:id="echoid-s2585" xml:space="preserve">ſunt K C, L E <lb/>æquales, ergo O N, F G inter ſe æquales erunt, &amp; </s>
  <s xml:id="echoid-s2586" xml:space="preserve">addita communiter N F erit <lb/>
<anchor type="note" xlink:label="note-0094-07a" xlink:href="note-0094-07"/>
F O æqualis N G; </s>
  <s xml:id="echoid-s2587" xml:space="preserve">Et quoniam E H ad H F eſt vt E K ad K P (propter pa-<lb/>rallelas K I, O A) nempe vt F O, ſeu ei æqualis G N ad O P (propter paral-<lb/>lelas E F, O K) ſed eandem proportionẽ habet D G ad G F, quàm E H ad H F; <lb/></s>
  <s xml:id="echoid-s2588" xml:space="preserve">ergo G N ad O P eandem proportionem habet quàm D G ad G F, &amp; </s>
  <s xml:id="echoid-s2589" xml:space="preserve">compa-<lb/>rando homologorum differentias D N ad N P erit vt D G ad G F, ſeu vt latus <lb/>
<anchor type="note" xlink:label="note-0094-08a" xlink:href="note-0094-08"/>
tranſuerſum ad rectum; </s>
  <s xml:id="echoid-s2590" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2591" xml:space="preserve">ideo C P eſt breuiſsima.</s>
  <s xml:id="echoid-s2592" xml:space="preserve"/>
</p>
<div xml:id="echoid-div228" type="float" level="2" n="6">
<note position="left" xlink:label="note-0094-07" xlink:href="note-0094-07a" xml:space="preserve">8. lib. 2.</note>
<note position="left" xlink:label="note-0094-08" xlink:href="note-0094-08a" xml:space="preserve">Lem. 3. <lb/>10. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2593" xml:space="preserve">Quia in ſequenti propoſitione 57; </s>
  <s xml:id="echoid-s2594" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2595" xml:space="preserve">in alĳs adhibetur propoſitio non adhuc <lb/>demonſtrata; </s>
  <s xml:id="echoid-s2596" xml:space="preserve">nimirum poſita C P linea breuiſsima, pariter que I D ſemiſsi axis <lb/>recti minoris etiam breuiſsima (ex II. </s>
  <s xml:id="echoid-s2597" xml:space="preserve">huius) quæ occurrant vltra axim in, <lb/>M deducuntur ea omnia, quæ in propoſitionibus 51. </s>
  <s xml:id="echoid-s2598" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2599" xml:space="preserve">52. </s>
  <s xml:id="echoid-s2600" xml:space="preserve">ex hypotheſi omni-
<pb o="57" file="0095" n="95" rhead="Conicor. Lib. V."/>
no diuerſa eliciebantur; </s>
  <s xml:id="echoid-s2601" xml:space="preserve">nam in dictis propoſitionibus perpendicularis ex concur-<lb/>ſu ad axim ducta efficiebat in ellipſi menſuram (iuxta deſinitionem 15. </s>
  <s xml:id="echoid-s2602" xml:space="preserve">huius <lb/>libri) minorem medietate axis tranſuerſi, ideſt perpendicularis ex concurſu ca-<lb/>debat inter centrum ſectionis, &amp; </s>
  <s xml:id="echoid-s2603" xml:space="preserve">proximiorem verticem: </s>
  <s xml:id="echoid-s2604" xml:space="preserve">hic vero perpendicu-<lb/>laris ex concurſu M per centrum D ellipſis tranſit.</s>
  <s xml:id="echoid-s2605" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2606" xml:space="preserve">Animaduertendum eſt hoc theorema demonſtratum fuiſſe ab Apollonio Propoſ. <lb/></s>
  <s xml:id="echoid-s2607" xml:space="preserve">35. </s>
  <s xml:id="echoid-s2608" xml:space="preserve">huius libri, quod tamen paraphraſtes neſcio an iure in fine huius voluminis <lb/>tranſpoſuit; </s>
  <s xml:id="echoid-s2609" xml:space="preserve">Sed quia predicta propoſitio 35. </s>
  <s xml:id="echoid-s2610" xml:space="preserve">omnino hic eſt neceſſaria, &amp; </s>
  <s xml:id="echoid-s2611" xml:space="preserve">pendet <lb/>ex alĳs præcedentibus, libuit potius aliam independentem demonſtrationem af-<lb/>ferre quam ordinem propoſitionum ſatis alter atum denuo perturbare.</s>
  <s xml:id="echoid-s2612" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div230" type="section" level="1" n="76">
<head xml:id="echoid-head109" xml:space="preserve">LEMMA VIII.</head>
<p style="it">
  <s xml:id="echoid-s2613" xml:space="preserve">IN ellipſi ABC linea breuiſsima F G, &amp; </s>
  <s xml:id="echoid-s2614" xml:space="preserve">ſemiaxis minor rectus B <lb/>D conueniant in E, erunt E F, &amp; </s>
  <s xml:id="echoid-s2615" xml:space="preserve">E B duæ breuiſecantes, duca-<lb/>tur quilibet ramus E H inter eos: </s>
  <s xml:id="echoid-s2616" xml:space="preserve">Dico E H non eſſe breuiſecantem, &amp; </s>
  <s xml:id="echoid-s2617" xml:space="preserve"><lb/>cadere infra lineam breuiſsimam ductam ex puncto H ad axim.</s>
  <s xml:id="echoid-s2618" xml:space="preserve"/>
</p>
  <figure>
    <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0095-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s2619" xml:space="preserve">Ducantur ex F, &amp; </s>
  <s xml:id="echoid-s2620" xml:space="preserve">H rectæ F K, H L perpendiculares aa axim rectum B <lb/>D eum ſecantes in K, &amp; </s>
  <s xml:id="echoid-s2621" xml:space="preserve">L, pariterque ducantur F M, H N perpendiculares ad <lb/>axim tranſuerſum A D eum ſecantes in M, N. </s>
  <s xml:id="echoid-s2622" xml:space="preserve">Et quia F G eſt breuiſsima, ergo <lb/>D M ad M G eandem proportionem habet, quàm latus tranſuerſum C A ad eius <lb/>
<anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/>
latus rectum; </s>
  <s xml:id="echoid-s2623" xml:space="preserve">ſed propter parallelas D E, M F, eſt D M ad M G, vt E F ad F <lb/>G, ſeu E K ad K D (propter parallelas G D, F K) quare E K ad K D eandem <lb/>proportionem habet, quàm latus tranſuerſum ad rectum, &amp; </s>
  <s xml:id="echoid-s2624" xml:space="preserve">diuidendo E D ad <lb/>D K eandem proportionem habebit, quàm differentia lateris tranuerſi, &amp; </s>
  <s xml:id="echoid-s2625" xml:space="preserve">recti <lb/>ad latus rectum, eſt vero D L maior, quàm D K (cum H L parallela ipſi F K <lb/>cadat inter punctum K, &amp; </s>
  <s xml:id="echoid-s2626" xml:space="preserve">B) igitur E D ad maiorem D L minorem proportio-<lb/>nem habet, quàm ad D K, &amp; </s>
  <s xml:id="echoid-s2627" xml:space="preserve">propterea componendo E L ad L D minorem pro-<lb/>portionem habebit, quàm latus tranſuerſum ad rectum: </s>
  <s xml:id="echoid-s2628" xml:space="preserve">eſt vero E H ad H I,
<pb o="58" file="0096" n="96" rhead="Apollonij Pergæi"/>
vt E L ad L D (propter parallelas I D, H L) pariterque D N ad N I eſt, vt E H <lb/>ad H I (porpter parallelas E D, N H) quare D N ad N I erit vt E L ad L D, <lb/>&amp; </s>
  <s xml:id="echoid-s2629" xml:space="preserve">propterea D N ad N I minorem proportionem habebit, quàm latus tranſuer-<lb/>ſum C A ad eius latus rectum, &amp; </s>
  <s xml:id="echoid-s2630" xml:space="preserve">ideo linea breuiſsima ex puncto H ad axim <lb/>
<anchor type="note" xlink:label="note-0096-01a" xlink:href="note-0096-01"/>
A D ducta cadet ſupra ramum H I E verſus verticem A, atq; </s>
  <s xml:id="echoid-s2631" xml:space="preserve">E H non erit bre-<lb/>uiſecans, quod erat primo loco oſtendendum.</s>
  <s xml:id="echoid-s2632" xml:space="preserve"/>
</p>
<div xml:id="echoid-div230" type="float" level="2" n="1">
<note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">15. huius.</note>
<note position="left" xlink:label="note-0096-01" xlink:href="note-0096-01a" xml:space="preserve">10. huius.</note>
</div>
  <figure>
    <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0096-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s2633" xml:space="preserve">Secundo ducatur ramus E O ſecans maiorem axim in P inter verticem A, &amp; </s>
  <s xml:id="echoid-s2634" xml:space="preserve"><lb/>breuiſecantem E F; </s>
  <s xml:id="echoid-s2635" xml:space="preserve">Dico E O non eſſe breuiſecantem, &amp; </s>
  <s xml:id="echoid-s2636" xml:space="preserve">breuiſsimam ex puncto <lb/>O ad axim A D ductam cadere infra ramum O P E; </s>
  <s xml:id="echoid-s2637" xml:space="preserve">Ducantur O Q, O R per-<lb/>pendiculares ad axes, ſecantes eos in Q, R. </s>
  <s xml:id="echoid-s2638" xml:space="preserve">Manifeſtum eſt Q D minorem eſſe, <lb/>quàm K D, &amp; </s>
  <s xml:id="echoid-s2639" xml:space="preserve">propterea E D ad D Q maiorem proportionem habebit, quàm ad <lb/>D K, &amp; </s>
  <s xml:id="echoid-s2640" xml:space="preserve">componendo E Q ad Q D maiorem proportionem habebit, quàm E K <lb/>ad K D: </s>
  <s xml:id="echoid-s2641" xml:space="preserve">oſtenſa autem fuit E K ad K D, vt latus tranſuerſum C A ad eius la-<lb/>tus rectum; </s>
  <s xml:id="echoid-s2642" xml:space="preserve">igitur E Q ad Q D maiorem proportionem habebit, quàm latus <lb/>tranſuerſum ad rectum; </s>
  <s xml:id="echoid-s2643" xml:space="preserve">ſed (propter parallelas P D, O Q) vt E Q ad Q D <lb/>ita eſt E O ad O P, &amp; </s>
  <s xml:id="echoid-s2644" xml:space="preserve">propter parallelas E D, R O, vt E O ad O P, ita eſt D R <lb/>ad R P; </s>
  <s xml:id="echoid-s2645" xml:space="preserve">ergo D R ad R P eſt, vt E Q ad Q D, &amp; </s>
  <s xml:id="echoid-s2646" xml:space="preserve">propterea D R ad R P ma-<lb/>iorem proportionem habebit, quàm latus tranuer ſum C A ad eius latus rectum; <lb/></s>
  <s xml:id="echoid-s2647" xml:space="preserve">igitur E O non erit breuiſecans, &amp; </s>
  <s xml:id="echoid-s2648" xml:space="preserve">breuiſsima ex puncto O ad axim ducta cadit <lb/>infra ramum E O verſus D, quod erat oſtendendum.</s>
  <s xml:id="echoid-s2649" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">ex 10. <lb/>huius.</note>
</div>
<div xml:id="echoid-div232" type="section" level="1" n="77">
<head xml:id="echoid-head110" xml:space="preserve">Notæ in Propoſ. LVII.</head>
<p style="it">
  <s xml:id="echoid-s2650" xml:space="preserve">ET dico, quod non repe-<lb/>
<anchor type="figure" xlink:label="fig-0096-02a" xlink:href="fig-0096-02"/>
riatur vllus alius ramus, <lb/>&amp;</s>
  <s xml:id="echoid-s2651" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2652" xml:space="preserve">Ideſt ſit rurſus linea bre-<lb/>uiſsima C M, quæ producta, <lb/>concurrat cum perpendiculari E <lb/>F in E, quæ ſecet axim in F <lb/>vltra centrum D ad partes ver-<lb/>ticis A. </s>
  <s xml:id="echoid-s2653" xml:space="preserve">Dico, quod præter ra-
<pb o="59" file="0097" n="97" rhead="Conicor. Lib. V."/>
mum E C nullus alius ramus breuiſecans ex concurſu E ad ſectionem duci poteſt, <lb/>qui cadat in eodem quadrante B L, quem breuiſecans interſecat.</s>
  <s xml:id="echoid-s2654" xml:space="preserve"/>
</p>
<div xml:id="echoid-div232" type="float" level="2" n="1">
  <figure xlink:label="fig-0096-02" xlink:href="fig-0096-02a">
    <image file="0096-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0096-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s2655" xml:space="preserve">Nam ſi producantur E H, E G, &amp;</s>
  <s xml:id="echoid-s2656" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2657" xml:space="preserve">Ducantur quilibet rami E H, E G ad <lb/>
<anchor type="note" xlink:label="note-0097-01a" xlink:href="note-0097-01"/>
vtraſque partes breuiſecantis E C intra quadrantem B L, qui ſecent D B in K, <lb/>&amp; </s>
  <s xml:id="echoid-s2658" xml:space="preserve">I, &amp; </s>
  <s xml:id="echoid-s2659" xml:space="preserve">producatur per centrum D recta M D L perpendicularis ad axim B A, <lb/>quæ ſecet ſectionem in L, &amp; </s>
  <s xml:id="echoid-s2660" xml:space="preserve">ramum E C in M.</s>
  <s xml:id="echoid-s2661" xml:space="preserve"/>
</p>
<div xml:id="echoid-div233" type="float" level="2" n="2">
<note position="left" xlink:label="note-0097-01" xlink:href="note-0097-01a" xml:space="preserve">h</note>
</div>
<p style="it">
  <s xml:id="echoid-s2662" xml:space="preserve">Et quia iam productæ ſunt ex concurſu M duæ breuiſecantes, &amp;</s>
  <s xml:id="echoid-s2663" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s2664" xml:space="preserve">
<anchor type="note" xlink:label="note-0097-02a" xlink:href="note-0097-02"/>
Quia C M breuiſsima ex hypotheſi occurrit ſemiaxi minori recto L D breuiſsi-<lb/>mæ pariter (ex 11. </s>
  <s xml:id="echoid-s2665" xml:space="preserve">huius) in M, ſequitur (non quidem ex 51. </s>
  <s xml:id="echoid-s2666" xml:space="preserve">52. </s>
  <s xml:id="echoid-s2667" xml:space="preserve">huius, ſed <lb/>ex lemmate 8. </s>
  <s xml:id="echoid-s2668" xml:space="preserve">præmiſſo) quod linea recta ex M ad H coniuncta cadat infra <lb/>breuiſsimam ex puncto H ad axim B A ductam, &amp; </s>
  <s xml:id="echoid-s2669" xml:space="preserve">coniuncta recta M G cadit <lb/>ſupra breuiſsimam ex puncto G ad axim ductam.</s>
  <s xml:id="echoid-s2670" xml:space="preserve"/>
</p>
<div xml:id="echoid-div234" type="float" level="2" n="3">
<note position="left" xlink:label="note-0097-02" xlink:href="note-0097-02a" xml:space="preserve">i</note>
</div>
<p style="it">
  <s xml:id="echoid-s2671" xml:space="preserve">Sed E H, &amp; </s>
  <s xml:id="echoid-s2672" xml:space="preserve">E G efficiunt abſciſsas oppoſito modo, &amp;</s>
  <s xml:id="echoid-s2673" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2674" xml:space="preserve">Quia ab eodem <lb/>
<anchor type="note" xlink:label="note-0097-03a" xlink:href="note-0097-03"/>
puncto H ſectionis ducuntur tres rectæ lineæ. </s>
  <s xml:id="echoid-s2675" xml:space="preserve">H E, H M, &amp; </s>
  <s xml:id="echoid-s2676" xml:space="preserve">breuiſsima ex H ad <lb/>axim B A ducta, quarum intermedia eſt H M, eo quod breuiſsima ex H ad <lb/>axim A B cadit ſupra H M ad partes B, vt dictum eſt, &amp; </s>
  <s xml:id="echoid-s2677" xml:space="preserve">H E cadit <lb/>
<anchor type="note" xlink:label="note-0097-04a" xlink:href="note-0097-04"/>
infra H M ad partes A; </s>
  <s xml:id="echoid-s2678" xml:space="preserve">ergo H E cadit infra breuiſsimam ex <lb/>H ad A B ductam, &amp; </s>
  <s xml:id="echoid-s2679" xml:space="preserve">propterea E H nan erit breuiſecans: <lb/></s>
  <s xml:id="echoid-s2680" xml:space="preserve">Similiter breuiſsimaex G ad A B extenſa cadit infra <lb/>G M ad partes A, vt dictum eſt; </s>
  <s xml:id="echoid-s2681" xml:space="preserve">at E G cadit <lb/>
<anchor type="note" xlink:label="note-0097-05a" xlink:href="note-0097-05"/>
ſupra G M ad partes B; </s>
  <s xml:id="echoid-s2682" xml:space="preserve">ergo E G cadit <lb/>ſupra breuiſsimam ex G ad axim <lb/>A B ductam, quare E G non <lb/>eſt breuiſecans.</s>
  <s xml:id="echoid-s2683" xml:space="preserve"/>
</p>
<div xml:id="echoid-div235" type="float" level="2" n="4">
<note position="left" xlink:label="note-0097-03" xlink:href="note-0097-03a" xml:space="preserve">k</note>
<note position="right" xlink:label="note-0097-04" xlink:href="note-0097-04a" xml:space="preserve">Lem 8.</note>
<note position="right" xlink:label="note-0097-05" xlink:href="note-0097-05a" xml:space="preserve">1 bidem.</note>
</div>
  <figure>
    <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0097-01"/>
  </figure>
<pb o="60" file="0098" n="98" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div237" type="section" level="1" n="78">
<head xml:id="echoid-head111" xml:space="preserve">SECTIO NONA <lb/>Continens Propoſ. LVIII. LIX. LX. LXI. <lb/>LXII. &amp; LXIII.</head>
<p>
  <s xml:id="echoid-s2684" xml:space="preserve">I Am ex puncto dato C extra, vel intra ſectionem A B (quod <lb/>
<anchor type="note" xlink:label="note-0098-01a" xlink:href="note-0098-01"/>
in axi I A non ſit) poſſumus rectam lineam ducere, cuius <lb/>portio intercepta inter ſectionem, &amp; </s>
  <s xml:id="echoid-s2685" xml:space="preserve">axim ſit linea breuiſſima.</s>
  <s xml:id="echoid-s2686" xml:space="preserve"/>
</p>
<div xml:id="echoid-div237" type="float" level="2" n="1">
<note position="right" xlink:label="note-0098-01" xlink:href="note-0098-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div239" type="section" level="1" n="79">
<head xml:id="echoid-head112" xml:space="preserve">PROPOSITIO LVIII.</head>
<p>
  <s xml:id="echoid-s2687" xml:space="preserve">Sit ſectio parabole, &amp; </s>
  <s xml:id="echoid-s2688" xml:space="preserve">producamus perpendicularem C E ſu-<lb/>per I E A, &amp; </s>
  <s xml:id="echoid-s2689" xml:space="preserve">ponamus E F æqualem dimidio erecti, &amp; </s>
  <s xml:id="echoid-s2690" xml:space="preserve">du-<lb/>camus G F parallelam ipſi C E, &amp; </s>
  <s xml:id="echoid-s2691" xml:space="preserve">per C ducamus hyperbolen <lb/>
<anchor type="note" xlink:label="note-0098-02a" xlink:href="note-0098-02"/>
<anchor type="note" xlink:label="note-0098-03a" xlink:href="note-0098-03"/>
H C B circa duas continentes illam G F, I F, quæ occurat ſe-<lb/>ctioni A B in B, &amp; </s>
  <s xml:id="echoid-s2692" xml:space="preserve">per B, C producatur linea occurrens con-<lb/>tinenti I A in I, &amp; </s>
  <s xml:id="echoid-s2693" xml:space="preserve">continenti G F in G: </s>
  <s xml:id="echoid-s2694" xml:space="preserve">Dico, quod B I eſt <lb/>linea breuiſsima.</s>
  <s xml:id="echoid-s2695" xml:space="preserve"/>
</p>
<div xml:id="echoid-div239" type="float" level="2" n="1">
<note position="left" xlink:label="note-0098-02" xlink:href="note-0098-02a" xml:space="preserve">4. lib. 2.</note>
<note position="right" xlink:label="note-0098-03" xlink:href="note-0098-03a" xml:space="preserve">b</note>
</div>
  <figure>
    <image file="0098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0098-01"/>
  </figure>
<p>
  <s xml:id="echoid-s2696" xml:space="preserve">Producatur perpendicularis B K. </s>
  <s xml:id="echoid-s2697" xml:space="preserve">Quoniam C I æqualis eſt B G (sexta <lb/>
<anchor type="note" xlink:label="note-0098-04a" xlink:href="note-0098-04"/>
ex ſecundo) erit E I æqualis K F, &amp; </s>
  <s xml:id="echoid-s2698" xml:space="preserve">E F, K I erunt æquales, atque ſup-<lb/>
<anchor type="note" xlink:label="note-0098-05a" xlink:href="note-0098-05"/>
poſita, eſt E F æqualis dimidio erecti; </s>
  <s xml:id="echoid-s2699" xml:space="preserve">ergo K I ita eſt pariter; </s>
  <s xml:id="echoid-s2700" xml:space="preserve">Quare <lb/>B I eſt breuiſsima, (octaua ex quinto) &amp; </s>
  <s xml:id="echoid-s2701" xml:space="preserve">hoc erat probandum.</s>
  <s xml:id="echoid-s2702" xml:space="preserve"/>
</p>
<div xml:id="echoid-div240" type="float" level="2" n="2">
<note position="right" xlink:label="note-0098-04" xlink:href="note-0098-04a" xml:space="preserve">C</note>
<note position="left" xlink:label="note-0098-05" xlink:href="note-0098-05a" xml:space="preserve">8. lib. 2.</note>
</div>
</div>
<div xml:id="echoid-div242" type="section" level="1" n="80">
<head xml:id="echoid-head113" xml:space="preserve">PROPOSITIO LIX. LXII. &amp; LXIII.</head>
<p>
  <s xml:id="echoid-s2703" xml:space="preserve">D Einde fit ſectio hyperbole, aut ellipſis, cuius centrum D, &amp; </s>
  <s xml:id="echoid-s2704" xml:space="preserve">lineis, <lb/>
<anchor type="note" xlink:label="note-0098-06a" xlink:href="note-0098-06"/>
atque ſignis in eodem ſtatu manentibus, ponamus D F ad F E, &amp;</s>
  <s xml:id="echoid-s2705" xml:space="preserve">
<pb o="61" file="0099" n="99" rhead="Conicor. Lib. V."/>
<anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/>
ſimiliter C L ad L E, vt proportio figuræ, &amp; </s>
  <s xml:id="echoid-s2706" xml:space="preserve">producamus per L ip-<lb/>ſam O M parallelam A I F, &amp; </s>
  <s xml:id="echoid-s2707" xml:space="preserve">per F ipſam G M parallelam C E, &amp; </s>
  <s xml:id="echoid-s2708" xml:space="preserve">fa-<lb/>ciamus ſectionem H C B hyperbolen tranſeuntem per punctum C circa <lb/>
<anchor type="note" xlink:label="note-0099-01a" xlink:href="note-0099-01"/>
continentes G M, O M, quæ occurret ſectioni A B (in ellipſi quidem vt <lb/>
<anchor type="note" xlink:label="note-0099-02a" xlink:href="note-0099-02"/>
demonſtrauimus) in hyberbola vero eo quod O M parallela axi D A in-<lb/>
<anchor type="note" xlink:label="note-0099-03a" xlink:href="note-0099-03"/>
clinato ſubtendit, ſi producatur, angulum ſubſequentem continentiæ an-<lb/>gulum ſecabit A B, &amp; </s>
  <s xml:id="echoid-s2709" xml:space="preserve">corda, ſi producatur, occurret ſectioni; </s>
  <s xml:id="echoid-s2710" xml:space="preserve">Ergo O <lb/>M ingreditur ſectionem A B, &amp; </s>
  <s xml:id="echoid-s2711" xml:space="preserve">ampliatur ſectio A B per extenſionem, <lb/>longè à duabus lineis O M, M G, &amp; </s>
  <s xml:id="echoid-s2712" xml:space="preserve">ſectio B C prope illas ducitur (deci-<lb/>
<anchor type="note" xlink:label="note-0099-04a" xlink:href="note-0099-04"/>
moſexta, ex ſecundo) igitur duæ ſectiones A B, C B ſibi occurrunt, vt <lb/>in B, &amp; </s>
  <s xml:id="echoid-s2713" xml:space="preserve">ducamus per B, C lineam occurrentem D F A in I, &amp; </s>
  <s xml:id="echoid-s2714" xml:space="preserve">G F in G; <lb/></s>
  <s xml:id="echoid-s2715" xml:space="preserve">Et quia B O æqualis eſt ipſi C G (octaua ex ſecundo) erit O N æqualis <lb/>
<anchor type="note" xlink:label="note-0099-05a" xlink:href="note-0099-05"/>
ipſi M L, &amp; </s>
  <s xml:id="echoid-s2716" xml:space="preserve">O L ipſi N M; </s>
  <s xml:id="echoid-s2717" xml:space="preserve">ergo O L, nempe N M, ſeu K F ad E I eſt, <lb/>vt C L ad C E, nempe D F ad D E, ergo K F ad E I eſt, vt D F <lb/>ad E D comparando homologorum ſummas in hyperbola, &amp; </s>
  <s xml:id="echoid-s2718" xml:space="preserve">eorundem <lb/>
<anchor type="note" xlink:label="note-0099-06a" xlink:href="note-0099-06"/>
<anchor type="note" xlink:label="note-0099-07a" xlink:href="note-0099-07"/>
differentias in ellipſi, &amp; </s>
  <s xml:id="echoid-s2719" xml:space="preserve">iterum comparando antecedentes ad differen-<lb/>
<anchor type="note" xlink:label="note-0099-08a" xlink:href="note-0099-08"/>
tias terminorum <lb/>
<anchor type="figure" xlink:label="fig-0099-02a" xlink:href="fig-0099-02"/>
fiet D K ad K <lb/>I, vt D F ad F <lb/>E, quæ eſt vt <lb/>proportio figu-<lb/>ræ; </s>
  <s xml:id="echoid-s2720" xml:space="preserve">igitur B I eſt <lb/>linea breuiſſima <lb/>(9. </s>
  <s xml:id="echoid-s2721" xml:space="preserve">10. </s>
  <s xml:id="echoid-s2722" xml:space="preserve">ex quin-<lb/>to) &amp; </s>
  <s xml:id="echoid-s2723" xml:space="preserve">hoc erat <lb/>probandum.</s>
  <s xml:id="echoid-s2724" xml:space="preserve"/>
</p>
<div xml:id="echoid-div242" type="float" level="2" n="1">
<note position="right" xlink:label="note-0098-06" xlink:href="note-0098-06a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0099-01" xlink:href="fig-0099-01a">
    <image file="0099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0099-01"/>
  </figure>
<note position="right" xlink:label="note-0099-01" xlink:href="note-0099-01a" xml:space="preserve">4. lib. 2.</note>
<note position="right" xlink:label="note-0099-02" xlink:href="note-0099-02a" xml:space="preserve">56. <lb/>huius.</note>
<note position="left" xlink:label="note-0099-03" xlink:href="note-0099-03a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0099-04" xlink:href="note-0099-04a" xml:space="preserve">14. lib. 2.</note>
<note position="left" xlink:label="note-0099-05" xlink:href="note-0099-05a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0099-06" xlink:href="note-0099-06a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0099-07" xlink:href="note-0099-07a" xml:space="preserve">Lem. 3.</note>
<note position="right" xlink:label="note-0099-08" xlink:href="note-0099-08a" xml:space="preserve">Lem. 1.</note>
  <figure xlink:label="fig-0099-02" xlink:href="fig-0099-02a">
    <image file="0099-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0099-02"/>
  </figure>
</div>
<pb o="62" file="0100" n="100" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div244" type="section" level="1" n="81">
<head xml:id="echoid-head114" xml:space="preserve">PROPOSITIO LX.</head>
<p>
  <s xml:id="echoid-s2725" xml:space="preserve">D Einde perpendicularis egrediens ex <lb/>
<anchor type="figure" xlink:label="fig-0100-01a" xlink:href="fig-0100-01"/>
<anchor type="note" xlink:label="note-0100-01a" xlink:href="note-0100-01"/>
C cadat ad centrum D ſectionis A B <lb/>hyperboles, &amp; </s>
  <s xml:id="echoid-s2726" xml:space="preserve">ponamus C E ad E D, vt <lb/>proportio figuræ, &amp; </s>
  <s xml:id="echoid-s2727" xml:space="preserve">producamus ex E ad <lb/>ſectionem rectã lineam E B, quæ parallela <lb/>ſit D E, producaturque C B, quæ occur-<lb/>rat axi in G. </s>
  <s xml:id="echoid-s2728" xml:space="preserve">Et quia C E ad E D, nempe <lb/>
<anchor type="note" xlink:label="note-0100-02a" xlink:href="note-0100-02"/>
C B ad B G, nempe D H ad H G eſt, vt <lb/>proportio figuræ; </s>
  <s xml:id="echoid-s2729" xml:space="preserve">erit G B linea breuiſſima <lb/>(nona ex quinto) quod erat oſtenden-<lb/>dum.</s>
  <s xml:id="echoid-s2730" xml:space="preserve"/>
</p>
<div xml:id="echoid-div244" type="float" level="2" n="1">
  <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a">
    <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0100-01"/>
  </figure>
<note position="right" xlink:label="note-0100-01" xlink:href="note-0100-01a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0100-02" xlink:href="note-0100-02a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div246" type="section" level="1" n="82">
<head xml:id="echoid-head115" xml:space="preserve">PROPOSITIO LXI.</head>
<p>
  <s xml:id="echoid-s2731" xml:space="preserve">S It poſtea punctum C, &amp; </s>
  <s xml:id="echoid-s2732" xml:space="preserve"><lb/>
<anchor type="figure" xlink:label="fig-0100-02a" xlink:href="fig-0100-02"/>
perpendicularis C F, &amp; </s>
  <s xml:id="echoid-s2733" xml:space="preserve"><lb/>F remotius à vertice ſectio-<lb/>nis, quàm ſit centrum, &amp; </s>
  <s xml:id="echoid-s2734" xml:space="preserve">po-<lb/>namus C E ad E F, vt eſt <lb/>proportio figuræ, &amp; </s>
  <s xml:id="echoid-s2735" xml:space="preserve">ſimiliter <lb/>D G ad G F, &amp; </s>
  <s xml:id="echoid-s2736" xml:space="preserve">ex E pro-<lb/>ducamus E H, quæ ſit paral-<lb/>lela ipſi F A, &amp; </s>
  <s xml:id="echoid-s2737" xml:space="preserve">ex G, D. <lb/></s>
  <s xml:id="echoid-s2738" xml:space="preserve">ad illam G I, D K, quæ ſint <lb/>parallelæ ipſi C F; </s>
  <s xml:id="echoid-s2739" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2740" xml:space="preserve">duca-<lb/>mus ſectionem hyperbolen <lb/>
<anchor type="note" xlink:label="note-0100-03a" xlink:href="note-0100-03"/>
tranſeuntem per D, quam <lb/>contineant I H, I G, quæ occurret ſectioni A B ſimiliter in B; </s>
  <s xml:id="echoid-s2741" xml:space="preserve">Itaque <lb/>
<anchor type="note" xlink:label="note-0100-04a" xlink:href="note-0100-04"/>
per B, C producamus lineam, quæ occurrat axi F A in L, &amp; </s>
  <s xml:id="echoid-s2742" xml:space="preserve">ipſi E H <lb/>in M. </s>
  <s xml:id="echoid-s2743" xml:space="preserve">Dico, quod B L eſt linea breuiſſima. </s>
  <s xml:id="echoid-s2744" xml:space="preserve">quia ducta perpendiculari <lb/>
<anchor type="note" xlink:label="note-0100-05a" xlink:href="note-0100-05"/>
H N, C E ad E F, ſeu ad K D, eſt vt D G ad G F, nempe vt K I ad <lb/>I E, &amp; </s>
  <s xml:id="echoid-s2745" xml:space="preserve">propterea E C in E I erit æquale rectangulo D I ſubſequenti <lb/>(octaua ex ſecundo) nempe rectangulo B I conſequenti; </s>
  <s xml:id="echoid-s2746" xml:space="preserve">Ergo C E in <lb/>
<anchor type="note" xlink:label="note-0100-06a" xlink:href="note-0100-06"/>
E I eſt æquale B H in H I, &amp; </s>
  <s xml:id="echoid-s2747" xml:space="preserve">propterea B H ad C E, nempe H M ad <lb/>M E eſt, vt E I ad I H; </s>
  <s xml:id="echoid-s2748" xml:space="preserve">ergo H I, nempe N G æqualis eſt E M, &amp; </s>
  <s xml:id="echoid-s2749" xml:space="preserve">ideo <lb/>L F ad E M, nempe ad N G eſt, vt C F ad E C, nempe D F ad D G, <lb/>quia quælibet earum aſſignata eſt, vt proportio figuræ; </s>
  <s xml:id="echoid-s2750" xml:space="preserve">ergo L F ad N <lb/>G eſt, vt D F ad D G; </s>
  <s xml:id="echoid-s2751" xml:space="preserve">itaq; </s>
  <s xml:id="echoid-s2752" xml:space="preserve">comparando homologorum differentias L <lb/>D ad D N, vt D F ad D G; </s>
  <s xml:id="echoid-s2753" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2754" xml:space="preserve">per conuerſionem rationis, &amp; </s>
  <s xml:id="echoid-s2755" xml:space="preserve">poſtea <lb/>diuidendo D N ad N L erit, vt D G, ad G F, quæ eſt vt propor-<lb/>tio figuræ; </s>
  <s xml:id="echoid-s2756" xml:space="preserve">Ergo B L eſt linea breuiſſima ( nona ex quinto ) &amp; </s>
  <s xml:id="echoid-s2757" xml:space="preserve">hoc erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s2758" xml:space="preserve"/>
</p>
<div xml:id="echoid-div246" type="float" level="2" n="1">
  <figure xlink:label="fig-0100-02" xlink:href="fig-0100-02a">
    <image file="0100-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0100-02"/>
  </figure>
<note position="left" xlink:label="note-0100-03" xlink:href="note-0100-03a" xml:space="preserve">4 lib. 2.</note>
<note position="right" xlink:label="note-0100-04" xlink:href="note-0100-04a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0100-05" xlink:href="note-0100-05a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0100-06" xlink:href="note-0100-06a" xml:space="preserve">12. lib. 2.</note>
</div>
<pb o="63" file="0101" n="101" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div248" type="section" level="1" n="83">
<head xml:id="echoid-head116" xml:space="preserve">Notæ in Propoſit. LVIII.</head>
<p>
  <s xml:id="echoid-s2759" xml:space="preserve">I Am poſſumus producere ex puncto aſſignato C extra datam ſectionem <lb/>
<anchor type="note" xlink:label="note-0101-01a" xlink:href="note-0101-01"/>
A B, aut intra (ſi punctum non fuerit ad axim I A) lineam diuiden-<lb/>tem ex illo inter ſectionem, &amp; </s>
  <s xml:id="echoid-s2760" xml:space="preserve">axim lineam breuiſſimam, &amp;</s>
  <s xml:id="echoid-s2761" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2762" xml:space="preserve">Sic legen-<lb/>dum puto. </s>
  <s xml:id="echoid-s2763" xml:space="preserve">Ex punto dato C extra, vel intra ſectionem A B, quod in axi non <lb/>ſit, lineam rectam ducere, cuius portio incercepta inter ſectionem, &amp; </s>
  <s xml:id="echoid-s2764" xml:space="preserve">axim ſit <lb/>linea breuiſsima.</s>
  <s xml:id="echoid-s2765" xml:space="preserve"/>
</p>
<div xml:id="echoid-div248" type="float" level="2" n="1">
<note position="left" xlink:label="note-0101-01" xlink:href="note-0101-01a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0101-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s2766" xml:space="preserve">Et per C ducamus ſectionem H C B circa duas continentes illam G F, <lb/>
<anchor type="note" xlink:label="note-0101-02a" xlink:href="note-0101-02"/>
I F, quæ occurrat ſectioni A B (16. </s>
  <s xml:id="echoid-s2767" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s2768" xml:space="preserve">in B, &amp;</s>
  <s xml:id="echoid-s2769" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2770" xml:space="preserve">Scilicet ducamus per <lb/>C hyperbolen H C B circa aſymptots G F, F I, &amp; </s>
  <s xml:id="echoid-s2771" xml:space="preserve">quia aſymptoti, &amp; </s>
  <s xml:id="echoid-s2772" xml:space="preserve">hyperbo-<lb/>
<anchor type="note" xlink:label="note-0101-03a" xlink:href="note-0101-03"/>
le H C B productæ ad ſe ipſas ſemper proprius accedunt, atque parabole A B <lb/>
<anchor type="note" xlink:label="note-0101-04a" xlink:href="note-0101-04"/>
producta ſemper magis ab axi A I remouetur; </s>
  <s xml:id="echoid-s2773" xml:space="preserve">igitur hyperbole H C B, &amp; </s>
  <s xml:id="echoid-s2774" xml:space="preserve">para-<lb/>bola A B ſe mutuo ſecabunt; </s>
  <s xml:id="echoid-s2775" xml:space="preserve">ſecent ſe ſe in puncto B. </s>
  <s xml:id="echoid-s2776" xml:space="preserve">Animaduertendum eſt, <lb/>quod in textu Arabico aſſumitur hæc concluſio, vt demonſtrata in propoſitione <lb/>16. </s>
  <s xml:id="echoid-s2777" xml:space="preserve">huius quinti libri; </s>
  <s xml:id="echoid-s2778" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2779" xml:space="preserve">ſiquidem numeri huius citationis mendoſi non ſunt, <lb/>hæc propoſitio ſexta decima deſideratur in hoc libro.</s>
  <s xml:id="echoid-s2780" xml:space="preserve"/>
</p>
<div xml:id="echoid-div249" type="float" level="2" n="2">
<note position="left" xlink:label="note-0101-02" xlink:href="note-0101-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0101-03" xlink:href="note-0101-03a" xml:space="preserve">4. lib. 2.</note>
<note position="right" xlink:label="note-0101-04" xlink:href="note-0101-04a" xml:space="preserve">14. 2. <lb/>Ex 8. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2781" xml:space="preserve">Producatur perpendicularis B K. </s>
  <s xml:id="echoid-s2782" xml:space="preserve">Quoniam C I, &amp;</s>
  <s xml:id="echoid-s2783" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2784" xml:space="preserve">Ex puncto B ad <lb/>
<anchor type="note" xlink:label="note-0101-05a" xlink:href="note-0101-05"/>
axim ducatur perpendicularis B K, ſecans eum in K; </s>
  <s xml:id="echoid-s2785" xml:space="preserve">quoniam quando punctum <lb/>C ponitur intra parabolen, tunc B G æqualis eſt I C; </s>
  <s xml:id="echoid-s2786" xml:space="preserve">quando vero cadit extra, <lb/>
<anchor type="note" xlink:label="note-0101-06a" xlink:href="note-0101-06"/>
tunc C G eſt æqualis B I, &amp; </s>
  <s xml:id="echoid-s2787" xml:space="preserve">addita communi B C erit I C æqualis B G, cumq; <lb/></s>
  <s xml:id="echoid-s2788" xml:space="preserve">duæ rectæ lineæ I G, I F conuenientes in I ſecentur à rectis lineis K B, E C, <lb/>F G inter ſe parallelis, eo quod ſunt perpendiculares ad eundem axim; </s>
  <s xml:id="echoid-s2789" xml:space="preserve">ergo I G, <lb/>&amp; </s>
  <s xml:id="echoid-s2790" xml:space="preserve">I F ſecantur in ĳſdem rationibus, &amp; </s>
  <s xml:id="echoid-s2791" xml:space="preserve">propterea E I æqualis erit K F; </s>
  <s xml:id="echoid-s2792" xml:space="preserve">ſicuti <lb/>I C æqualis erat B g, pariterque I K æqualis erit E F, ſicuti I B æqualis erat <lb/>C G; </s>
  <s xml:id="echoid-s2793" xml:space="preserve">poſita autem fuit E F æqualis ſemierecto; </s>
  <s xml:id="echoid-s2794" xml:space="preserve">igitur K I ſemiſsi lateris recti <lb/>pariter æqualis erit.</s>
  <s xml:id="echoid-s2795" xml:space="preserve"/>
</p>
<div xml:id="echoid-div250" type="float" level="2" n="3">
<note position="left" xlink:label="note-0101-05" xlink:href="note-0101-05a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0101-06" xlink:href="note-0101-06a" xml:space="preserve">8. lib. 2.</note>
</div>
<pb o="64" file="0102" n="102" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div252" type="section" level="1" n="84">
<head xml:id="echoid-head117" xml:space="preserve">Notæ in Propoſit. LIX. LXII. &amp; LXIII.</head>
<p style="it">
  <s xml:id="echoid-s2796" xml:space="preserve">E T lineis, atque ſignis eodem ſtatu manentibus, &amp;</s>
  <s xml:id="echoid-s2797" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2798" xml:space="preserve">Ideſt punctum <lb/>
<anchor type="note" xlink:label="note-0102-01a" xlink:href="note-0102-01"/>
C extra, aut intra ſectionem ponatur, dummodo non ſit in axi, ducaturq; <lb/></s>
  <s xml:id="echoid-s2799" xml:space="preserve">C E perpendicularis ad axim, ſecans eum in E, &amp; </s>
  <s xml:id="echoid-s2800" xml:space="preserve">vt latus tranſuerſum ad re-<lb/>ctum, ita ſiat D F ad F E, atque C L ad L E, &amp; </s>
  <s xml:id="echoid-s2801" xml:space="preserve">per L producatur O L M pa-<lb/>rallela A I, &amp; </s>
  <s xml:id="echoid-s2802" xml:space="preserve">per F ducatur F M G parallela C E, quæ ſecet O M in M, &amp; </s>
  <s xml:id="echoid-s2803" xml:space="preserve">per <lb/>C deſcribatur hyperbole H C B circa aſymptotos G M O, quæ in ellipſi per eius <lb/>
<anchor type="note" xlink:label="note-0102-02a" xlink:href="note-0102-02"/>
centrum D tranſibit, &amp; </s>
  <s xml:id="echoid-s2804" xml:space="preserve">ideo eam ſecabit ſicuti oſtenſum eſt in 56. </s>
  <s xml:id="echoid-s2805" xml:space="preserve">huius.</s>
  <s xml:id="echoid-s2806" xml:space="preserve"/>
</p>
<div xml:id="echoid-div252" type="float" level="2" n="1">
<note position="right" xlink:label="note-0102-01" xlink:href="note-0102-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0102-02" xlink:href="note-0102-02a" xml:space="preserve">4. lib. 2.</note>
</div>
  <figure>
    <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0102-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s2807" xml:space="preserve">Eo quod O M parallela axi D A inclinato ſubtendit, &amp;</s>
  <s xml:id="echoid-s2808" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2809" xml:space="preserve">Quoniam <lb/>
<anchor type="note" xlink:label="note-0102-03a" xlink:href="note-0102-03"/>
in hyperbola O M parallela axi ſecat vtrãque linearum continentium angulum, <lb/>qui deinceps eſt ei, qui hyperbolen continet ſectioni occurret, &amp; </s>
  <s xml:id="echoid-s2810" xml:space="preserve">producta ſectio-<lb/>
<anchor type="note" xlink:label="note-0102-04a" xlink:href="note-0102-04"/>
nem A B ſecabit, &amp; </s>
  <s xml:id="echoid-s2811" xml:space="preserve">ideo O M cadit intra ſectionem A B, atque hyperbole A B <lb/>producta ſemper magis, ac magis recedit tum ab M O parallela axi, cum ab M <lb/>G parallela tangenti verticali, &amp; </s>
  <s xml:id="echoid-s2812" xml:space="preserve">ſectio H C B, &amp; </s>
  <s xml:id="echoid-s2813" xml:space="preserve">asymptoti O M G ad ſe ip-<lb/>
<anchor type="note" xlink:label="note-0102-05a" xlink:href="note-0102-05"/>
ſas jemper propius accedunt, igitur ſectiones A B, B C conueniunt; </s>
  <s xml:id="echoid-s2814" xml:space="preserve">ſecent ſe <lb/>ſe in B, &amp; </s>
  <s xml:id="echoid-s2815" xml:space="preserve">ducamus per B, C lineam occurrentem axi in I, ipſi M O in O, &amp; </s>
  <s xml:id="echoid-s2816" xml:space="preserve"><lb/>M G in G.</s>
  <s xml:id="echoid-s2817" xml:space="preserve"/>
</p>
<div xml:id="echoid-div253" type="float" level="2" n="2">
<note position="right" xlink:label="note-0102-03" xlink:href="note-0102-03a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0102-04" xlink:href="note-0102-04a" xml:space="preserve">11. lib. 2.</note>
<note position="left" xlink:label="note-0102-05" xlink:href="note-0102-05a" xml:space="preserve">14. lib. 2.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2818" xml:space="preserve">Et quia B O æqualis eſt ipſi C G, &amp;</s>
  <s xml:id="echoid-s2819" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2820" xml:space="preserve">Cum lineæ rectæ O M, O G ſe ſe-<lb/>
<anchor type="note" xlink:label="note-0102-06a" xlink:href="note-0102-06"/>
cantes in O, ſecentur à parallelis E C, K B, F G proportionaliter, erit O N <lb/>æqualis M L, ſicuti O B æqualis erat C G, &amp; </s>
  <s xml:id="echoid-s2821" xml:space="preserve">O L, æqualis erit N M, ſicuti <lb/>O C æqualis erat B G, cumque triangula O C L, &amp; </s>
  <s xml:id="echoid-s2822" xml:space="preserve">I C E ſint ſimilia propter <lb/>
<anchor type="note" xlink:label="note-0102-07a" xlink:href="note-0102-07"/>
parallelas O L, I E, erit O L ad E I, vt L C ad C E; </s>
  <s xml:id="echoid-s2823" xml:space="preserve">eſt vero M N, ſeu F <lb/>K æqualis ipſi L O, igitur F K ad E I eſt, vt L C ad E C, ſed ex conſtru-<lb/>ctione erat D F ad F E, vt C L ad L E, ſciluet vt latus tranſuerſum ad <lb/>rectum; </s>
  <s xml:id="echoid-s2824" xml:space="preserve">ergo antecedentes ad ſummas terminorum in hyperbola, &amp; </s>
  <s xml:id="echoid-s2825" xml:space="preserve">ad <lb/>
<anchor type="note" xlink:label="note-0102-08a" xlink:href="note-0102-08"/>
<pb o="65" file="0103" n="103" rhead="Conicor. Lib. V."/>
eorundẽ differen-<lb/>
<anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/>
tias in ellipſi ſci-<lb/>licet C L ad C E <lb/>erit vt D F ad D <lb/>E, &amp; </s>
  <s xml:id="echoid-s2826" xml:space="preserve">propterea <lb/>K F ad E I erit, <lb/>vt D F ad D E, <lb/>
<anchor type="note" xlink:label="note-0103-01a" xlink:href="note-0103-01"/>
&amp; </s>
  <s xml:id="echoid-s2827" xml:space="preserve">cõparando ho-<lb/>mologorum ſum-<lb/>mas in hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s2828" xml:space="preserve">eorundem dif-<lb/>ferentias in elli-<lb/>pſi, K D ad D I <lb/>erit, vt D F ad D E, &amp; </s>
  <s xml:id="echoid-s2829" xml:space="preserve">iterum comparando antecedentes ad differentias ter <lb/>
<anchor type="note" xlink:label="note-0103-02a" xlink:href="note-0103-02"/>
minorum fiet D K ad K I, vt D F ad F E, ſeu vt latus tranſuer ſum ad rectum; <lb/></s>
  <s xml:id="echoid-s2830" xml:space="preserve">igitur B I eſt linea breuiſsima.</s>
  <s xml:id="echoid-s2831" xml:space="preserve"/>
</p>
<div xml:id="echoid-div254" type="float" level="2" n="3">
<note position="right" xlink:label="note-0102-06" xlink:href="note-0102-06a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0102-07" xlink:href="note-0102-07a" xml:space="preserve">8. lib. 2.</note>
<note position="left" xlink:label="note-0102-08" xlink:href="note-0102-08a" xml:space="preserve">Lem. 1.</note>
  <figure xlink:label="fig-0103-01" xlink:href="fig-0103-01a">
    <image file="0103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0103-01"/>
  </figure>
<note position="right" xlink:label="note-0103-01" xlink:href="note-0103-01a" xml:space="preserve">Lem. 3.</note>
<note position="right" xlink:label="note-0103-02" xlink:href="note-0103-02a" xml:space="preserve">Lem. 1.</note>
</div>
<note position="right" xml:space="preserve">Ex 9. 10. <lb/>huius.</note>
<p>
  <s xml:id="echoid-s2832" xml:space="preserve">Si autem componamus proportionem in hyperbola deinde abſcinda-<lb/>
<anchor type="note" xlink:label="note-0103-04a" xlink:href="note-0103-04"/>
mus, &amp; </s>
  <s xml:id="echoid-s2833" xml:space="preserve">reijciamus oppoſitum ab oppoſito in ellipſi, deinde inuertamus <lb/>fiet K D ad K I, vt D F ad F E, &amp;</s>
  <s xml:id="echoid-s2834" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2835" xml:space="preserve">Sed textum mendoſum corrigi debere, <lb/>vt ſupra factum eſt conſtat ex præcedenti nota.</s>
  <s xml:id="echoid-s2836" xml:space="preserve"/>
</p>
<div xml:id="echoid-div255" type="float" level="2" n="4">
<note position="left" xlink:label="note-0103-04" xlink:href="note-0103-04a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div257" type="section" level="1" n="85">
<head xml:id="echoid-head118" xml:space="preserve">Notæ in Propoſit. LX.</head>
<p style="it">
  <s xml:id="echoid-s2837" xml:space="preserve">DEinde ſit perpendicularis ex C, &amp;</s>
  <s xml:id="echoid-s2838" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2839" xml:space="preserve">Siex puncto C extra hyperbolen po-<lb/>
<anchor type="note" xlink:label="note-0103-05a" xlink:href="note-0103-05"/>
ſito perpendicularis ad axim ducta ad centrum eius D pertingat, duci de-<lb/>bet pariter ex puncto C recta linea ad ſectionem, cuius portio inter axim D F, <lb/>&amp; </s>
  <s xml:id="echoid-s2840" xml:space="preserve">ſectionem A B ſit linea breuiſsima; </s>
  <s xml:id="echoid-s2841" xml:space="preserve">fiat C E ad E D, vt latus tranſuer ſum ad <lb/>rectum, &amp; </s>
  <s xml:id="echoid-s2842" xml:space="preserve">ex E ducatur E B par allela axi, ſecans hyperbolen in B, &amp; </s>
  <s xml:id="echoid-s2843" xml:space="preserve">ex B du-<lb/>catur B H perpendicularis ad axim, ſecans eum in H.</s>
  <s xml:id="echoid-s2844" xml:space="preserve"/>
</p>
<div xml:id="echoid-div257" type="float" level="2" n="1">
<note position="left" xlink:label="note-0103-05" xlink:href="note-0103-05a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s2845" xml:space="preserve">Et quia C E ad E D, nempe C B ad B G, <lb/>
<anchor type="figure" xlink:label="fig-0103-02a" xlink:href="fig-0103-02"/>
<anchor type="note" xlink:label="note-0103-06a" xlink:href="note-0103-06"/>
&amp;</s>
  <s xml:id="echoid-s2846" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2847" xml:space="preserve">Quia propter parallelas B E, F D eſt C E ad <lb/>E D, vt C B ad B G, &amp; </s>
  <s xml:id="echoid-s2848" xml:space="preserve">propter parallelas D C, <lb/>H B, eſt D H ad H G, vt C B ad B G, quare D H <lb/>ad H G erit, vt C E ad E D: </s>
  <s xml:id="echoid-s2849" xml:space="preserve">poſita autem fuit C <lb/>E ad E D, vt latus tranſuer ſum ad rectum; </s>
  <s xml:id="echoid-s2850" xml:space="preserve">igi-<lb/>tur D H ex centro hyperboles ad H G eandem <lb/>proportionem habet, quàm latus tranſuerſum ad <lb/>rectum, &amp; </s>
  <s xml:id="echoid-s2851" xml:space="preserve">propterea G B erit linea breuiſsima.</s>
  <s xml:id="echoid-s2852" xml:space="preserve"/>
</p>
<div xml:id="echoid-div258" type="float" level="2" n="2">
  <figure xlink:label="fig-0103-02" xlink:href="fig-0103-02a">
    <image file="0103-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0103-02"/>
  </figure>
<note position="left" xlink:label="note-0103-06" xlink:href="note-0103-06a" xml:space="preserve">b</note>
</div>
<note position="right" xml:space="preserve">9. huius.</note>
</div>
<div xml:id="echoid-div260" type="section" level="1" n="86">
<head xml:id="echoid-head119" xml:space="preserve">Notæ in Propoſit. LXI.</head>
<p style="it">
  <s xml:id="echoid-s2853" xml:space="preserve">SIt poſtea punctum C, &amp; </s>
  <s xml:id="echoid-s2854" xml:space="preserve">perpendicularis C F, &amp;</s>
  <s xml:id="echoid-s2855" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2856" xml:space="preserve">Si à puncto C extra <lb/>
<anchor type="note" xlink:label="note-0103-08a" xlink:href="note-0103-08"/>
hyperbolen A B poſito, C F perpendicularis ad axim efficiat F A ſegmentũ <lb/>tranſuerſi axis maius ſemiſse eius D A, &amp; </s>
  <s xml:id="echoid-s2857" xml:space="preserve">ponantur C E ad E F, atque D G
<pb o="66" file="0104" n="104" rhead="Apollonij Pergæi"/>
ad G F, vt latus tranuer ſum ad <lb/>
<anchor type="figure" xlink:label="fig-0104-01a" xlink:href="fig-0104-01"/>
rectum, &amp; </s>
  <s xml:id="echoid-s2858" xml:space="preserve">ducatur ex E recta <lb/>E H parallela F A, quæ ſecetur <lb/>à rectis D K, G I ad axim per-<lb/>pendicularibus in K, &amp; </s>
  <s xml:id="echoid-s2859" xml:space="preserve">I, &amp; </s>
  <s xml:id="echoid-s2860" xml:space="preserve"><lb/>per D ducatur hyperbole D B <lb/>
<anchor type="note" xlink:label="note-0104-01a" xlink:href="note-0104-01"/>
circa aſymptotos H I G, occur-<lb/>ret hyperbole A B (vt in Prop. <lb/></s>
  <s xml:id="echoid-s2861" xml:space="preserve">59. </s>
  <s xml:id="echoid-s2862" xml:space="preserve">62. </s>
  <s xml:id="echoid-s2863" xml:space="preserve">63. </s>
  <s xml:id="echoid-s2864" xml:space="preserve">oſtenſum eſt) ali-<lb/>cubi, vt in B, coniungatur rect a <lb/>linea B C, quæ occurrat axi in <lb/>L, &amp; </s>
  <s xml:id="echoid-s2865" xml:space="preserve">ipſi E H in M, duca-<lb/>turque ex B perpendicularis ad <lb/>axim eum ſecans in N, &amp; </s>
  <s xml:id="echoid-s2866" xml:space="preserve">re-<lb/>ctam E M in H. </s>
  <s xml:id="echoid-s2867" xml:space="preserve">Dico, quod B L eſt linea breuiſsima.</s>
  <s xml:id="echoid-s2868" xml:space="preserve"/>
</p>
<div xml:id="echoid-div260" type="float" level="2" n="1">
<note position="left" xlink:label="note-0103-08" xlink:href="note-0103-08a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0104-01" xlink:href="fig-0104-01a">
    <image file="0104-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0104-01"/>
  </figure>
<note position="left" xlink:label="note-0104-01" xlink:href="note-0104-01a" xml:space="preserve">4. lib. 2.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2869" xml:space="preserve">C E ad E F, nempe K D eſt, vt D G ad G F, &amp;</s>
  <s xml:id="echoid-s2870" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2871" xml:space="preserve">Quoniam ex conſtru-<lb/>
<anchor type="note" xlink:label="note-0104-02a" xlink:href="note-0104-02"/>
ctione C E ad E F, ſeu ad ei æqualem K D, in parallelogrammo D E, eſt vt <lb/>D G ad G F, ſcilicet vt latus @ anſuerſum ad rectum, eſtque K I ad I E, vt D <lb/>G ad G F propter parallelas D K, G I, F E; </s>
  <s xml:id="echoid-s2872" xml:space="preserve">ergo vt prima C E ad ſecundam <lb/>D K, ita eſt tertia K I ad quartam I E, &amp; </s>
  <s xml:id="echoid-s2873" xml:space="preserve">propterea rectangulum C E I ſub <lb/>extremis contentum æquale eſt rectangulo D K I ſub intermedĳs compræhenſo; <lb/></s>
  <s xml:id="echoid-s2874" xml:space="preserve">eſt vero rectangulum B I æquale rectangulo D I cum compræhendantur ab hyper-<lb/>bole D B, &amp; </s>
  <s xml:id="echoid-s2875" xml:space="preserve">aſymptotis H I G; </s>
  <s xml:id="echoid-s2876" xml:space="preserve">ergo rectangulum C E I æquale eſt rectangulo <lb/>
<anchor type="note" xlink:label="note-0104-03a" xlink:href="note-0104-03"/>
B H I; </s>
  <s xml:id="echoid-s2877" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2878" xml:space="preserve">propterea B H ad C E, nempe H M ad M E (propter ſimilitudinem <lb/>triangulorum B H M, C E M) eandem proportionem habebit, quàm E I ad I <lb/>H, &amp; </s>
  <s xml:id="echoid-s2879" xml:space="preserve">componendo eadem H E ad H I, atque ad E M eandem proportioner<unsure/> <lb/>habebit; </s>
  <s xml:id="echoid-s2880" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2881" xml:space="preserve">ideo H I ſeu ei æqualis N G æqualis erit E M, quare eadem <lb/>L F ad N G, atque ad E M eandem proportionem habebit: </s>
  <s xml:id="echoid-s2882" xml:space="preserve">ſed propter ſimi-<lb/>litudinem triangulorum L C F, M C E eſt F C ad E C, vt F L ad M E, <lb/>ſeu ad N G, &amp; </s>
  <s xml:id="echoid-s2883" xml:space="preserve">erat C E ad E F, necnon D G ad G F in eadem propor-<lb/>tione lateris tranſuerſi ad rectum, &amp; </s>
  <s xml:id="echoid-s2884" xml:space="preserve">ſummæ terminorum ad antece-<lb/>
<anchor type="note" xlink:label="note-0104-04a" xlink:href="note-0104-04"/>
dentes terminos, ſcilicet F C ad E C, necnon F D ad D G ean-<lb/>dem proportionem habent; </s>
  <s xml:id="echoid-s2885" xml:space="preserve">quare L F ad N G eandem <lb/>proportionem habet, quàm F D ad D G, &amp; </s>
  <s xml:id="echoid-s2886" xml:space="preserve">compa-<lb/>rando homologorum differentias L D ad D N <lb/>
<anchor type="note" xlink:label="note-0104-05a" xlink:href="note-0104-05"/>
eandem proportionem habebit, quàm F D <lb/>ad D G, &amp; </s>
  <s xml:id="echoid-s2887" xml:space="preserve">comparando conſe-<lb/>quentes ad differentias termi-<lb/>
<anchor type="note" xlink:label="note-0104-06a" xlink:href="note-0104-06"/>
norum D N ad L N erit, <lb/>vt D G ad F G, <lb/>ſcilicet <lb/>vt latus tranſuer ſum ad rectum; <lb/></s>
  <s xml:id="echoid-s2888" xml:space="preserve">quapropter B L eſt linea <lb/>
<anchor type="note" xlink:label="note-0104-07a" xlink:href="note-0104-07"/>
breuiſsima.</s>
  <s xml:id="echoid-s2889" xml:space="preserve"/>
</p>
<div xml:id="echoid-div261" type="float" level="2" n="2">
<note position="right" xlink:label="note-0104-02" xlink:href="note-0104-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0104-03" xlink:href="note-0104-03a" xml:space="preserve">12. lib. 2.</note>
<note position="left" xlink:label="note-0104-04" xlink:href="note-0104-04a" xml:space="preserve">Lem. 1.</note>
<note position="left" xlink:label="note-0104-05" xlink:href="note-0104-05a" xml:space="preserve">Lem. 3.</note>
<note position="left" xlink:label="note-0104-06" xlink:href="note-0104-06a" xml:space="preserve">Lem. 1.</note>
<note position="left" xlink:label="note-0104-07" xlink:href="note-0104-07a" xml:space="preserve">9. huius.</note>
</div>
<pb o="67" file="0105" n="105" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div263" type="section" level="1" n="87">
<head xml:id="echoid-head120" xml:space="preserve">SECTIO DECIMA</head>
<head xml:id="echoid-head121" xml:space="preserve">Continens Propof. XXXXIV. XXXXV. <lb/>Apollonij.</head>
<p>
  <s xml:id="echoid-s2890" xml:space="preserve">SI ex axe recto ellipſis ſumatur menſura ab origine, quæ ad <lb/>
<anchor type="note" xlink:label="note-0105-01a" xlink:href="note-0105-01"/>
ſemiaxim rectum non habeat minorem proportionem, quàm <lb/>habet figura ſuæ tranſuerſæ, tunc quicumque ramus ſecans, ab <lb/>illa origine ad fectionem ductus, abſcindit ex axe tranſuerſo ad <lb/>verticem ſectionis lineam minorem ea, quàm abſcindit linea <lb/>breuiſsima egrediens ab eius termino in ſectione poſito ad tran-<lb/>ſuerſum axim; </s>
  <s xml:id="echoid-s2891" xml:space="preserve">ſi vero fuerit proportio ad ſemirectum minor, <lb/>tunc ramorum ſecantium vnus eſt breuiſecans; </s>
  <s xml:id="echoid-s2892" xml:space="preserve">reliqui vero, qui <lb/>ſequuntur extremum tranſuerſæ habent proprietates ſuperius ex-<lb/>poſitas, &amp; </s>
  <s xml:id="echoid-s2893" xml:space="preserve">qui ſequuntur extremitatem recti, ſecant ex tranſuer-<lb/>ſa lineam maiorem ea, quàm abſcindit breuiſsima egrediens ab <lb/>eius termino.</s>
  <s xml:id="echoid-s2894" xml:space="preserve"/>
</p>
<div xml:id="echoid-div263" type="float" level="2" n="1">
<note position="left" xlink:label="note-0105-01" xlink:href="note-0105-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div265" type="section" level="1" n="88">
<head xml:id="echoid-head122" xml:space="preserve">PROPOSITIO XXXXIV.</head>
<p>
  <s xml:id="echoid-s2895" xml:space="preserve">Sit A D dimidium axis recti, &amp; </s>
  <s xml:id="echoid-s2896" xml:space="preserve">minoris ſectionis ellipticæ <lb/>
<anchor type="note" xlink:label="note-0105-02a" xlink:href="note-0105-02"/>
A B C, &amp; </s>
  <s xml:id="echoid-s2897" xml:space="preserve">meuſura A E, quæ ſit maior, quàm A D, &amp; </s>
  <s xml:id="echoid-s2898" xml:space="preserve">pro-<lb/>portio illius ad iſtam non ſit minor proportione figuræ ſectionis; <lb/></s>
  <s xml:id="echoid-s2899" xml:space="preserve">Dico, quod linea breuiſsima egrediens ab extremitate cuiuſcum-<lb/>que rami ſecantis educti ex E ad ſectionem A B C, ſecat ex <lb/>tranuerſa B C cum vertice B, vel C lineam maiorem ea, quàm <lb/>abſcindit ille ramus.</s>
  <s xml:id="echoid-s2900" xml:space="preserve"/>
</p>
<div xml:id="echoid-div265" type="float" level="2" n="1">
<note position="left" xlink:label="note-0105-02" xlink:href="note-0105-02a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s2901" xml:space="preserve">Ponatur ramus E F, &amp; </s>
  <s xml:id="echoid-s2902" xml:space="preserve">ducamus ex F ad vtrum-<lb/>
<anchor type="figure" xlink:label="fig-0105-01a" xlink:href="fig-0105-01"/>
<anchor type="note" xlink:label="note-0105-03a" xlink:href="note-0105-03"/>
que axim duas perpendiculares F H, F I. </s>
  <s xml:id="echoid-s2903" xml:space="preserve">Et quia <lb/>proportio E A ad A D non eſt minor proportio-<lb/>ne ſiguræ, ſed minor eſt, quàm E H ad H D, nem-<lb/>pe E F ad F G, ſeu D I ad I G, erit proportio ſigu-<lb/>ræ minor, quàm D I ad I G, &amp; </s>
  <s xml:id="echoid-s2904" xml:space="preserve">ponamus D I ad <lb/>I K, vt eſt proportio figuræ, &amp; </s>
  <s xml:id="echoid-s2905" xml:space="preserve">iungamus F K; <lb/></s>
  <s xml:id="echoid-s2906" xml:space="preserve">erit ergo F K linea breuiſſima (10. </s>
  <s xml:id="echoid-s2907" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s2908" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2909" xml:space="preserve">iam <lb/>
<anchor type="note" xlink:label="note-0105-04a" xlink:href="note-0105-04"/>
ſecat K B maiorem, quàm B G, &amp; </s>
  <s xml:id="echoid-s2910" xml:space="preserve">G F non erit <lb/>breuiſſima; </s>
  <s xml:id="echoid-s2911" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2912" xml:space="preserve">hoc erat propoſitum.</s>
  <s xml:id="echoid-s2913" xml:space="preserve"/>
</p>
<div xml:id="echoid-div266" type="float" level="2" n="2">
  <figure xlink:label="fig-0105-01" xlink:href="fig-0105-01a">
    <image file="0105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0105-01"/>
  </figure>
<note position="left" xlink:label="note-0105-03" xlink:href="note-0105-03a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0105-04" xlink:href="note-0105-04a" xml:space="preserve">10. <lb/>huius.</note>
</div>
<pb o="68" file="0106" n="106" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div268" type="section" level="1" n="89">
<head xml:id="echoid-head123" xml:space="preserve">PROPOSITIO XXXXV.</head>
<p>
  <s xml:id="echoid-s2914" xml:space="preserve">SI autem fuerit ratio E A ad A D minor, <lb/>
<anchor type="figure" xlink:label="fig-0106-01a" xlink:href="fig-0106-01"/>
<anchor type="note" xlink:label="note-0106-01a" xlink:href="note-0106-01"/>
quàm proportio figuræ, ponamus E H ad H <lb/>D in proportione figuræ, &amp; </s>
  <s xml:id="echoid-s2915" xml:space="preserve">producamus per-<lb/>pendicularem H F, &amp; </s>
  <s xml:id="echoid-s2916" xml:space="preserve">iungamus F E, &amp; </s>
  <s xml:id="echoid-s2917" xml:space="preserve">duca-<lb/>mus perpendicularem F I. </s>
  <s xml:id="echoid-s2918" xml:space="preserve">Et quoniam E H ad <lb/>H D, nempe D I ad I G eſt, vt proportio figu-<lb/>ræ, erit F G linea breuiſſima (10. </s>
  <s xml:id="echoid-s2919" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s2920" xml:space="preserve">Et quo-<lb/>niam iam educti ſunt ex E duo breuiſecantes <lb/>F E, &amp; </s>
  <s xml:id="echoid-s2921" xml:space="preserve">E A (11. </s>
  <s xml:id="echoid-s2922" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s2923" xml:space="preserve">tunc à terminis ramo-<lb/>rum egredientium ex E, qui terminantur ad ſe-<lb/>ctionem B F, linea breuiſſima egrediens erit re-<lb/>motior ab ipſo B, &amp; </s>
  <s xml:id="echoid-s2924" xml:space="preserve">qui terminatur ad ſectio-<lb/>nem A F, breuiſſima egrediens ab extremitate illius erit proximior, ipſi <lb/>B (51. </s>
  <s xml:id="echoid-s2925" xml:space="preserve">52. </s>
  <s xml:id="echoid-s2926" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s2927" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2928" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s2929" xml:space="preserve"/>
</p>
<div xml:id="echoid-div268" type="float" level="2" n="1">
  <figure xlink:label="fig-0106-01" xlink:href="fig-0106-01a">
    <image file="0106-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0106-01"/>
  </figure>
<note position="right" xlink:label="note-0106-01" xlink:href="note-0106-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div270" type="section" level="1" n="90">
<head xml:id="echoid-head124" xml:space="preserve">Notæ in Propoſ. XXXXIV.</head>
<p style="it">
  <s xml:id="echoid-s2930" xml:space="preserve">PVto, numeros 53. </s>
  <s xml:id="echoid-s2931" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2932" xml:space="preserve">54. </s>
  <s xml:id="echoid-s2933" xml:space="preserve">Propoſitionum huius ſe-<lb/>
<anchor type="figure" xlink:label="fig-0106-02a" xlink:href="fig-0106-02"/>
ctionis mendoſos eſſe, nam Propoſitio 53. </s>
  <s xml:id="echoid-s2934" xml:space="preserve">poſita <lb/>fuit in præmiſſa ſectione, &amp; </s>
  <s xml:id="echoid-s2935" xml:space="preserve">Propoſitio 54. </s>
  <s xml:id="echoid-s2936" xml:space="preserve">inferius <lb/>appoſita reperitur; </s>
  <s xml:id="echoid-s2937" xml:space="preserve">Cenſeo igitur, eſſe Propoſitiones <lb/>XXXXIV. </s>
  <s xml:id="echoid-s2938" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2939" xml:space="preserve">XXXXV.</s>
  <s xml:id="echoid-s2940" xml:space="preserve"/>
</p>
<div xml:id="echoid-div270" type="float" level="2" n="1">
  <figure xlink:label="fig-0106-02" xlink:href="fig-0106-02a">
    <image file="0106-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0106-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s2941" xml:space="preserve">Si ex axe recto ellipſis ſumatur menſura, &amp;</s>
  <s xml:id="echoid-s2942" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s2943" xml:space="preserve">
<anchor type="note" xlink:label="note-0106-02a" xlink:href="note-0106-02"/>
Hoc eſt ſi ex axe minori, recto ellipſis ſumatur menſu-<lb/>ra, quæ habeat non minorem proportionem ad ſemi-<lb/>axim rectum, quàm habet axis tranſuerſus ad ſuum <lb/>latus rectum, quilibet ramus ſecans, ab origine ad ſe-<lb/>ctionem ductus, abſcindit ex axe tranſuerſo ad ver-<lb/>ticem ſectionis minorem lineam, quàm ſecat linea breuiſsima ab eius termi-<lb/>no ad axim tranſuer ſum ducta. </s>
  <s xml:id="echoid-s2944" xml:space="preserve">Si vero menſura ad minorem ſemiaxim re-<lb/>ctum proportionem minorem habuerit, quàm latus tranſuer ſum ad rectum, tunc <lb/>vnicus ramus erit breuiſecans; </s>
  <s xml:id="echoid-s2945" xml:space="preserve">reliqui vero ſequentes terminum tranſuerſi, ha-<lb/>bent ſuperius expoſitas proprietates, &amp; </s>
  <s xml:id="echoid-s2946" xml:space="preserve">ſequentes extr emitates axis recti, ſecant <lb/>ex tranſuer ſa maiorem lineam, quàm ſecet breuiſsima ab eius termino ad axim <lb/>tranſuer ſum ducta. </s>
  <s xml:id="echoid-s2947" xml:space="preserve">Quod autem menſura neceßario ſumi debeat in axe minori <lb/>ellipſis patet, nàm ex hypotheſi rami ſunt ſecantes non quidem ex concurſu, ſed <lb/>ex origine ducti igitur origo cadit infra centrum, &amp; </s>
  <s xml:id="echoid-s2948" xml:space="preserve">menſura maior erit medie-<lb/>tate axis vt in textu habetur; </s>
  <s xml:id="echoid-s2949" xml:space="preserve">debet autem habere menſura ad ſemiaxim rectum <lb/>maiorem aut eandem proportionem, quàm axis tranſuerſus habet ad eius latus <lb/>rectum, ergo proportio axis tranſuerſi ad ſuum latus rectum erit maioris inæqua-<lb/>litatis, &amp; </s>
  <s xml:id="echoid-s2950" xml:space="preserve">propterea tranſuerſus axis erit maior quàm axis rectus.</s>
  <s xml:id="echoid-s2951" xml:space="preserve"/>
</p>
<div xml:id="echoid-div271" type="float" level="2" n="2">
<note position="right" xlink:label="note-0106-02" xlink:href="note-0106-02a" xml:space="preserve">a</note>
</div>
<pb o="69" file="0107" n="107" rhead="Conicor. Lib. V."/>
<p style="it">
  <s xml:id="echoid-s2952" xml:space="preserve">Sit A D dimidium axis recti ſectionis ellipticæ A B C, &amp;</s>
  <s xml:id="echoid-s2953" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2954" xml:space="preserve">Sit A D di-<lb/>
<anchor type="note" xlink:label="note-0107-01a" xlink:href="note-0107-01"/>
midium axis minoris, &amp; </s>
  <s xml:id="echoid-s2955" xml:space="preserve">recti ellipſis A B C, ſitque menſura A E maior, quàm <lb/>A D, &amp; </s>
  <s xml:id="echoid-s2956" xml:space="preserve">E A ad A D habeat maiorem, aut eandem proportionem, quàm habet <lb/>latus tranſuerſum B C ad eius rectum latus.</s>
  <s xml:id="echoid-s2957" xml:space="preserve"/>
</p>
<div xml:id="echoid-div272" type="float" level="2" n="3">
<note position="left" xlink:label="note-0107-01" xlink:href="note-0107-01a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s2958" xml:space="preserve">Ponatur ramus E F, &amp; </s>
  <s xml:id="echoid-s2959" xml:space="preserve">producamus ex F, &amp;</s>
  <s xml:id="echoid-s2960" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2961" xml:space="preserve">Ducatur quilibet ramus <lb/>
<anchor type="note" xlink:label="note-0107-02a" xlink:href="note-0107-02"/>
ſecans E F, &amp; </s>
  <s xml:id="echoid-s2962" xml:space="preserve">ex F ad vtrumque axim perpendiculares F H, F I, quæ ſecent <lb/>eos in H, &amp; </s>
  <s xml:id="echoid-s2963" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2964" xml:space="preserve">Et quia D H minor eſt, quàm D A, habebit eadem E D ad <lb/>D H maiorem proportionem, quàm ad D A, &amp; </s>
  <s xml:id="echoid-s2965" xml:space="preserve">componendo E H ad H D, ma-<lb/>iorem proportionem habebit, quàm E A ad A D; </s>
  <s xml:id="echoid-s2966" xml:space="preserve">eſt vero E F ad F G, vt E <lb/>H ad H D (propter parallelas D G, H F) nec non D I ad I G eſt, vt E F ad <lb/>F G (propter parallelas E D, I F) ergo D I ad I G maiorem proportionem ha-<lb/>bet, quàm E A ad A D: </s>
  <s xml:id="echoid-s2967" xml:space="preserve">habebat autem E A ad A D maiorem, aut eandem <lb/>proportionem, quàm latus tranſuer ſum B C ad eius rectum latus; </s>
  <s xml:id="echoid-s2968" xml:space="preserve">igitur D I ad <lb/>I G maiorem proportionem habebit, quàm latus tranſuer ſum B C ad eius rectum <lb/>latus: </s>
  <s xml:id="echoid-s2969" xml:space="preserve">fiat iam D I ad I K, vt latus tranſuer ſum B C ad eius latus rectum, <lb/>iungaturque F K, erit I K maior, quàm I G, &amp; </s>
  <s xml:id="echoid-s2970" xml:space="preserve">F K linea breuiſsima, quæ ſe-<lb/>
<anchor type="note" xlink:label="note-0107-03a" xlink:href="note-0107-03"/>
cat ſegmentum axis K B maius, quàm B G, vnde E F non erit breuiſcans.</s>
  <s xml:id="echoid-s2971" xml:space="preserve"/>
</p>
<div xml:id="echoid-div273" type="float" level="2" n="4">
<note position="left" xlink:label="note-0107-02" xlink:href="note-0107-02a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0107-03" xlink:href="note-0107-03a" xml:space="preserve">10. huius.</note>
</div>
</div>
<div xml:id="echoid-div275" type="section" level="1" n="91">
<head xml:id="echoid-head125" xml:space="preserve">Notæ in Propoſ. XLV.</head>
<p style="it">
  <s xml:id="echoid-s2972" xml:space="preserve">SI autem fuerit ratio E A ad A D minor, quàm proportio figuræ, &amp;</s>
  <s xml:id="echoid-s2973" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s2974" xml:space="preserve">
<anchor type="note" xlink:label="note-0107-04a" xlink:href="note-0107-04"/>
Habeat E A ad A D minorẽ proportionem, quàm latus tranſuer ſum B C ad <lb/>eius rectum latus, &amp; </s>
  <s xml:id="echoid-s2975" xml:space="preserve">fiat E H ad H D, vt latus tranſuer ſum ad rectum; </s>
  <s xml:id="echoid-s2976" xml:space="preserve">ha-<lb/>bebit E H ad H D maiorem proportionem, quàm E A ad A D, &amp; </s>
  <s xml:id="echoid-s2977" xml:space="preserve">diuidendo <lb/>eadem E D ad D H habebit maiorem proportionem, quàm ad D A; </s>
  <s xml:id="echoid-s2978" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2979" xml:space="preserve">pro-<lb/>pterea D H minor erit, quàm D A; </s>
  <s xml:id="echoid-s2980" xml:space="preserve">vnde ex puncto H ſi eleuetur H F perpen-<lb/>dicularis ad D A intra ſectionem cadet, &amp; </s>
  <s xml:id="echoid-s2981" xml:space="preserve">ſecabit eam alicubi, vt in F: </s>
  <s xml:id="echoid-s2982" xml:space="preserve">duca-<lb/>tur poſtea ex F recta F E, quæ ſecet axim in G, &amp; </s>
  <s xml:id="echoid-s2983" xml:space="preserve">F I perpendicularis ad axim <lb/>B C eum ſecans in I. </s>
  <s xml:id="echoid-s2984" xml:space="preserve">Et quoniam, propter parallelas G D, F H, eſt E F ad F <lb/>G, vt E H ad H D, pariterque, propter parallelas E D, I F, eſt D I ad I G, vt <lb/>E F ad F G, quare D I ad I G eandem proportionem habet, quàm E H ad H <lb/>D, ſeu quàm latus tranſuer ſum B C ad eius latus rectum; </s>
  <s xml:id="echoid-s2985" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2986" xml:space="preserve">propterea F G eſt <lb/>
<anchor type="note" xlink:label="note-0107-05a" xlink:href="note-0107-05"/>
breuiſsima.</s>
  <s xml:id="echoid-s2987" xml:space="preserve"/>
</p>
<div xml:id="echoid-div275" type="float" level="2" n="1">
<note position="left" xlink:label="note-0107-04" xlink:href="note-0107-04a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0107-05" xlink:href="note-0107-05a" xml:space="preserve">10. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2988" xml:space="preserve">Et quoniam iam eductæ ſunt ex E duæ breuiſecantes, &amp;</s>
  <s xml:id="echoid-s2989" xml:space="preserve">c. </s>
  <s xml:id="echoid-s2990" xml:space="preserve">Textus Ara-<lb/>
<anchor type="note" xlink:label="note-0107-06a" xlink:href="note-0107-06"/>
bicus vſque ad finem propoſitionis eſt omnino corruptus, cum ſupponat propoſi-<lb/>tionem non demonſtratam, vt in propoſitione 56. </s>
  <s xml:id="echoid-s2991" xml:space="preserve">notaui; </s>
  <s xml:id="echoid-s2992" xml:space="preserve">Itaque, ſic eum reſti-<lb/>tui poſſe cenſeo. </s>
  <s xml:id="echoid-s2993" xml:space="preserve">Quoniam ex conſurſu E breuiſsimæ F G, &amp; </s>
  <s xml:id="echoid-s2994" xml:space="preserve">ſemiaxis recti <lb/>minoris D A rami educti ad ſectionem F A ſecant axis ſegmenta vſque ad <lb/>verticem B maiora, quàm abſcindant breuiſsimæ ab eorum terminis ad axim <lb/>ductæ, ſcilicet breuiſsimæ cadunt ſupra ramos (ex Lemmate 8. </s>
  <s xml:id="echoid-s2995" xml:space="preserve">præmiſſo) ſimi-<lb/>liter rami ex concur ſu E ad ſectionem B F ducti cadunt ſupra breuiſsimas ab <lb/>eorum terminis ad axim extenſas (ex eodem Lemmate 8.) </s>
  <s xml:id="echoid-s2996" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2997" xml:space="preserve">hoc erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s2998" xml:space="preserve"/>
</p>
<div xml:id="echoid-div276" type="float" level="2" n="2">
<note position="left" xlink:label="note-0107-06" xlink:href="note-0107-06a" xml:space="preserve">b</note>
</div>
<pb o="70" file="0108" n="108" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div278" type="section" level="1" n="92">
<head xml:id="echoid-head126" xml:space="preserve">SECTIO VNDECIMA</head>
<head xml:id="echoid-head127" xml:space="preserve">Continens Propoſ. LXVIII. LXIX. LXX. <lb/>&amp; LXXI. Apollonij.</head>
<head xml:id="echoid-head128" xml:space="preserve">PROPOSITIO LXVIII. LXIX.</head>
<p>
  <s xml:id="echoid-s2999" xml:space="preserve">SI occurrant duæ tangentes alicui ſectioni A B C, vt ſunt A <lb/>
<anchor type="note" xlink:label="note-0108-01a" xlink:href="note-0108-01"/>
F, E F, vtique quod abſcinditur ex tangente proximiori <lb/>vertici ſectionis, qui eſt B minus eſt ſegmento abſciſſo ex alia, <lb/>nempe E F minor eſt, quàm A F.</s>
  <s xml:id="echoid-s3000" xml:space="preserve"/>
</p>
<div xml:id="echoid-div278" type="float" level="2" n="1">
<note position="right" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s3001" xml:space="preserve">Iuncta enim A E, <lb/>
<anchor type="note" xlink:label="note-0108-02a" xlink:href="note-0108-02"/>
<anchor type="figure" xlink:label="fig-0108-01a" xlink:href="fig-0108-01"/>
&amp; </s>
  <s xml:id="echoid-s3002" xml:space="preserve">in parabola ex F <lb/>producta linea F I <lb/>parallela axi B D e-<lb/>rit illa diameter, bi-<lb/>fariam ſecans E A in <lb/>G (34. </s>
  <s xml:id="echoid-s3003" xml:space="preserve">ex 2.) </s>
  <s xml:id="echoid-s3004" xml:space="preserve">Simi-<lb/>
<anchor type="note" xlink:label="note-0108-03a" xlink:href="note-0108-03"/>
liter ex centro H pro-<lb/>ducamus H F G, quæ <lb/>eſt quoque diameter <lb/>(34. </s>
  <s xml:id="echoid-s3005" xml:space="preserve">ex 2.) </s>
  <s xml:id="echoid-s3006" xml:space="preserve">bifariam <lb/>
<anchor type="note" xlink:label="note-0108-04a" xlink:href="note-0108-04"/>
ſecans E A in G, &amp; </s>
  <s xml:id="echoid-s3007" xml:space="preserve"><lb/>ducamus A D in pa-<lb/>rabola, &amp; </s>
  <s xml:id="echoid-s3008" xml:space="preserve">hyperbola perpendicularem ſuper axim D B. </s>
  <s xml:id="echoid-s3009" xml:space="preserve">Ergo angulus <lb/>A I G in parabola eſt rectus, &amp; </s>
  <s xml:id="echoid-s3010" xml:space="preserve">in hyperbola obtuſus; </s>
  <s xml:id="echoid-s3011" xml:space="preserve">ergo F G A erit <lb/>obtuſus in illis omnibus; </s>
  <s xml:id="echoid-s3012" xml:space="preserve">quare maior eſt, quàm angulus F G E, &amp; </s>
  <s xml:id="echoid-s3013" xml:space="preserve">A <lb/>G æqualis eſt ipſi G E, &amp; </s>
  <s xml:id="echoid-s3014" xml:space="preserve">F G communis; </s>
  <s xml:id="echoid-s3015" xml:space="preserve">igitur E F minor eſt, quàm <lb/>F A.</s>
  <s xml:id="echoid-s3016" xml:space="preserve"/>
</p>
<div xml:id="echoid-div279" type="float" level="2" n="2">
<note position="right" xlink:label="note-0108-02" xlink:href="note-0108-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0108-01" xlink:href="fig-0108-01a">
    <image file="0108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0108-01"/>
  </figure>
<note position="left" xlink:label="note-0108-03" xlink:href="note-0108-03a" xml:space="preserve">30. lib. 2.</note>
<note position="left" xlink:label="note-0108-04" xlink:href="note-0108-04a" xml:space="preserve">Ibidem.</note>
</div>
  <figure>
    <image file="0108-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0108-02"/>
  </figure>
<pb o="71" file="0109" n="109" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div281" type="section" level="1" n="93">
<head xml:id="echoid-head129" xml:space="preserve">PROPOSITIO LXX.</head>
<p>
  <s xml:id="echoid-s3017" xml:space="preserve">P Oſtea in ellipſi iungamus E H, A H, &amp; </s>
  <s xml:id="echoid-s3018" xml:space="preserve">C <lb/>
<anchor type="figure" xlink:label="fig-0109-01a" xlink:href="fig-0109-01"/>
ſit extremitas axis recti; </s>
  <s xml:id="echoid-s3019" xml:space="preserve">erit A H minor <lb/>quàm E H (11. </s>
  <s xml:id="echoid-s3020" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3021" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3022" xml:space="preserve">angulus EGH, nempe <lb/>
<anchor type="note" xlink:label="note-0109-01a" xlink:href="note-0109-01"/>
A G F maior erit, quàm A G H, ſeu E G F, <lb/>ergo E F minor eſt, quàm F A, &amp; </s>
  <s xml:id="echoid-s3023" xml:space="preserve">hoc erat <lb/>propoſitum.</s>
  <s xml:id="echoid-s3024" xml:space="preserve"/>
</p>
<div xml:id="echoid-div281" type="float" level="2" n="1">
  <figure xlink:label="fig-0109-01" xlink:href="fig-0109-01a">
    <image file="0109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0109-01"/>
  </figure>
<note position="left" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">c</note>
</div>
</div>
<div xml:id="echoid-div283" type="section" level="1" n="94">
<head xml:id="echoid-head130" xml:space="preserve">PROPOSITIO LXXI.</head>
<p>
  <s xml:id="echoid-s3025" xml:space="preserve">P Atet ex hoc, quod ſi producantur ex duo-<lb/>
<anchor type="note" xlink:label="note-0109-02a" xlink:href="note-0109-02"/>
bus punctis contactus in ellipſi perpendi-<lb/>culares E M, A L, &amp; </s>
  <s xml:id="echoid-s3026" xml:space="preserve">fuerit E M minor, <lb/>exempli gratia, tunc tangens educta ab eius <lb/>extremitate minor quoque eſt, quemadmodum demonſtrauimus, &amp; </s>
  <s xml:id="echoid-s3027" xml:space="preserve">hoc <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s3028" xml:space="preserve"/>
</p>
<div xml:id="echoid-div283" type="float" level="2" n="1">
<note position="left" xlink:label="note-0109-02" xlink:href="note-0109-02a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div285" type="section" level="1" n="95">
<head xml:id="echoid-head131" xml:space="preserve">Notæ in Propoſit. LXVIII. LXIX. LXX. <lb/>&amp; LXXI.</head>
<p style="it">
  <s xml:id="echoid-s3029" xml:space="preserve">S I occurrant duæ tangentes alicui fectioni A B C, aut circulo, vt ſunt, <lb/>
<anchor type="note" xlink:label="note-0109-03a" xlink:href="note-0109-03"/>
&amp;</s>
  <s xml:id="echoid-s3030" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3031" xml:space="preserve">Ideſt ſi coniſectionem A B C contingant duæ rectæ A F, E F in pun-<lb/>ctis A, &amp; </s>
  <s xml:id="echoid-s3032" xml:space="preserve">E concurrentes in F, erit portio tangentis inter occurſum, &amp; </s>
  <s xml:id="echoid-s3033" xml:space="preserve">conta-<lb/>ctum vertici B proximiorem intercepta, minor ea, quæ inter occur ſum, &amp; </s>
  <s xml:id="echoid-s3034" xml:space="preserve">re-<lb/>motiorem à vertice contactum continetur: </s>
  <s xml:id="echoid-s3035" xml:space="preserve">oportet autem in ellipſi B verticem, <lb/>eſſe axis maioris. </s>
  <s xml:id="echoid-s3036" xml:space="preserve">Expungo verba, aut circulo, tanquam erronea, &amp; </s>
  <s xml:id="echoid-s3037" xml:space="preserve">incaute <lb/>ab aliquo textui ſuperaddita. </s>
  <s xml:id="echoid-s3038" xml:space="preserve">Circulum enim tangentes ab eodem puncto ductæ <lb/>inæquales eſſe nequeunt.</s>
  <s xml:id="echoid-s3039" xml:space="preserve"/>
</p>
<div xml:id="echoid-div285" type="float" level="2" n="1">
<note position="left" xlink:label="note-0109-03" xlink:href="note-0109-03a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s3040" xml:space="preserve">Et ducamus A D in parabola, &amp; </s>
  <s xml:id="echoid-s3041" xml:space="preserve">hyperbola, &amp;</s>
  <s xml:id="echoid-s3042" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3043" xml:space="preserve">Et ducamus A D in <lb/>
<anchor type="note" xlink:label="note-0109-04a" xlink:href="note-0109-04"/>
parabola, &amp; </s>
  <s xml:id="echoid-s3044" xml:space="preserve">hyperbola perpendicularem ſuper axim B D, ſecantem eum in D, <lb/>atque G F H in I; </s>
  <s xml:id="echoid-s3045" xml:space="preserve">cumque in parabola diameter F G I ſit parallela axi B D, <lb/>erit angulus A I G rectus æqualis interno, &amp; </s>
  <s xml:id="echoid-s3046" xml:space="preserve">oppoſito ad eaſdem partes, angu-<lb/>lo D; </s>
  <s xml:id="echoid-s3047" xml:space="preserve">in hyperbola vero cum triangulum H D I ſit rectangulum in D, erit ex-<lb/>ternus A I G obtuſus, eſtque in triangulo G I A angulus externus A G F maior <lb/>interno, &amp; </s>
  <s xml:id="echoid-s3048" xml:space="preserve">oppoſito A I G, recto in parabola, &amp; </s>
  <s xml:id="echoid-s3049" xml:space="preserve">obtuſo in hyperbola; </s>
  <s xml:id="echoid-s3050" xml:space="preserve">erit quo-<lb/>que angulus F G A obtuſus in parabola, &amp; </s>
  <s xml:id="echoid-s3051" xml:space="preserve">hyperbola.</s>
  <s xml:id="echoid-s3052" xml:space="preserve"/>
</p>
<div xml:id="echoid-div286" type="float" level="2" n="2">
<note position="left" xlink:label="note-0109-04" xlink:href="note-0109-04a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s3053" xml:space="preserve">Et angulus E G H, &amp;</s>
  <s xml:id="echoid-s3054" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3055" xml:space="preserve">Zuia F H eſt diameter ſecans bifariam E A in <lb/>
<anchor type="note" xlink:label="note-0109-05a" xlink:href="note-0109-05"/>
<anchor type="note" xlink:label="note-0109-06a" xlink:href="note-0109-06"/>
G; </s>
  <s xml:id="echoid-s3056" xml:space="preserve">ergo triangula E G H, &amp; </s>
  <s xml:id="echoid-s3057" xml:space="preserve">A G H habent àuo latera ægualia E G, A G, &amp; </s>
  <s xml:id="echoid-s3058" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0109-07a" xlink:href="note-0109-07"/>
G H, commune; </s>
  <s xml:id="echoid-s3059" xml:space="preserve">eſtque H E, vertici B axis maioris ellipſis propinquior, maior <lb/>remotiore H A; </s>
  <s xml:id="echoid-s3060" xml:space="preserve">ergo angulus E G H maior erit angulo A G H; </s>
  <s xml:id="echoid-s3061" xml:space="preserve">eſtque angulus <lb/>A G F æqualis E G H maiori, &amp; </s>
  <s xml:id="echoid-s3062" xml:space="preserve">E G F æqualis minori A G H; </s>
  <s xml:id="echoid-s3063" xml:space="preserve">igitur angulus <lb/>A G F maior eſt angulo E G F, &amp; </s>
  <s xml:id="echoid-s3064" xml:space="preserve">latera circa inæquales angulos ſunt æqualia <lb/>ſingula ſingulis, ergo tangens A F maior eſt, quàm E F.</s>
  <s xml:id="echoid-s3065" xml:space="preserve"/>
</p>
<div xml:id="echoid-div287" type="float" level="2" n="3">
<note position="left" xlink:label="note-0109-05" xlink:href="note-0109-05a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0109-06" xlink:href="note-0109-06a" xml:space="preserve">30. ex 2. <lb/>Com.</note>
<note position="right" xlink:label="note-0109-07" xlink:href="note-0109-07a" xml:space="preserve">11. huius.</note>
</div>
<pb o="72" file="0110" n="110" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s3066" xml:space="preserve">Patet ex hoc, quod ſi producantur ex duo-<lb/>
<anchor type="note" xlink:label="note-0110-01a" xlink:href="note-0110-01"/>
<anchor type="figure" xlink:label="fig-0110-01a" xlink:href="fig-0110-01"/>
bus punctis contactus in ellipſi perpendiculares <lb/>E M, A L; </s>
  <s xml:id="echoid-s3067" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3068" xml:space="preserve">fuerit E M minor, exempli gra-<lb/>tia, tunc tangens educta ab eius extremitate, <lb/>quæ eſt in ſectione, minor quoque eſt, &amp;</s>
  <s xml:id="echoid-s3069" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3070" xml:space="preserve">Si <lb/>enim ex punctis E, A contactuum in ellipſi ducan-<lb/>tur ad axim minorem K C perpendiculares E M, <lb/>&amp; </s>
  <s xml:id="echoid-s3071" xml:space="preserve">A L ſecantes eum in M, &amp; </s>
  <s xml:id="echoid-s3072" xml:space="preserve">L, fueritque E M <lb/>minor, quàm A L, tunc quidem punctum E magis <lb/>recedit à vertice B axis maioris, quàm punctum <lb/>A; </s>
  <s xml:id="echoid-s3073" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3074" xml:space="preserve">propterea, ex præmiſſa 70. </s>
  <s xml:id="echoid-s3075" xml:space="preserve">huius libri, erit <lb/>tangens E F minor, quàm A F. </s>
  <s xml:id="echoid-s3076" xml:space="preserve">Expungo deter-<lb/>minationem ab aliquo incaute additam (quæ eſt in <lb/>ſectione) manifeſtum enim eſt ducinon poſſe contin-<lb/>gentem ellipſim à perpendicularis termino M in axi minori poſito, ſed à termi-<lb/>no E in ſectionis peripheria conſtituto.</s>
  <s xml:id="echoid-s3077" xml:space="preserve"/>
</p>
<div xml:id="echoid-div288" type="float" level="2" n="4">
<note position="right" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">d</note>
  <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a">
    <image file="0110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0110-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div290" type="section" level="1" n="96">
<head xml:id="echoid-head132" xml:space="preserve">SECTIO DVODECIMA</head>
<head xml:id="echoid-head133" xml:space="preserve">Continens XXIX. XXX. XXXI.</head>
<head xml:id="echoid-head134" xml:space="preserve">Propoſ. Appollonij.</head>
<p>
  <s xml:id="echoid-s3078" xml:space="preserve">Q Vælibet linea recta A E D tangens fectionem aliquam A <lb/>F B in A extremitate lineæ breuiſſimæ A C eſt perpeudi-<lb/>cularis ſuper illam, nẽpe D A C eſt angulus rectus. <lb/></s>
  <s xml:id="echoid-s3079" xml:space="preserve">Et ſi fuerit perpendicularis ſuper illam vtique tanget ſectio-<lb/>nem.</s>
  <s xml:id="echoid-s3080" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3081" xml:space="preserve">Alioquin producatur perpendicu-<lb/>
<anchor type="note" xlink:label="note-0110-02a" xlink:href="note-0110-02"/>
<anchor type="figure" xlink:label="fig-0110-02a" xlink:href="fig-0110-02"/>
laris C E ſuper A D, erit A C maior, <lb/>quàm E C, ergo maior eſt, quàm F <lb/>C; </s>
  <s xml:id="echoid-s3082" xml:space="preserve">ſed eſt minor, cũ ſit minor, quàm <lb/>C F, quod eſt abſurdum. </s>
  <s xml:id="echoid-s3083" xml:space="preserve">Igitur an-<lb/>gulus D A C, eſt rectus, quod erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s3084" xml:space="preserve"/>
</p>
<div xml:id="echoid-div290" type="float" level="2" n="1">
<note position="right" xlink:label="note-0110-02" xlink:href="note-0110-02a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0110-02" xlink:href="fig-0110-02a">
    <image file="0110-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0110-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s3085" xml:space="preserve">Si verò fuerit D A C rectus, erit <lb/>
<anchor type="note" xlink:label="note-0110-03a" xlink:href="note-0110-03"/>
A D tangens, alioquin ſit tangens A <lb/>G; </s>
  <s xml:id="echoid-s3086" xml:space="preserve">ergo C A G erit rectus, ſed erat <lb/>C A D rectus, quod eſt abſurdum; <lb/></s>
  <s xml:id="echoid-s3087" xml:space="preserve">ergo A D eſt tangens, &amp; </s>
  <s xml:id="echoid-s3088" xml:space="preserve">hoc erat <lb/>probandum.</s>
  <s xml:id="echoid-s3089" xml:space="preserve"/>
</p>
<div xml:id="echoid-div291" type="float" level="2" n="2">
<note position="right" xlink:label="note-0110-03" xlink:href="note-0110-03a" xml:space="preserve">b</note>
</div>
<pb o="73" file="0111" n="111" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div293" type="section" level="1" n="97">
<head xml:id="echoid-head135" xml:space="preserve">Notæ in Propoſit. XXIX. XXX. <lb/>&amp; XXXI.</head>
<p style="it">
  <s xml:id="echoid-s3090" xml:space="preserve">A Lioquin producatur perpendicularis C E, &amp;</s>
  <s xml:id="echoid-s3091" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3092" xml:space="preserve">Exiſtente C A lineæ <lb/>
<anchor type="note" xlink:label="note-0111-01a" xlink:href="note-0111-01"/>
breuiſsima, &amp; </s>
  <s xml:id="echoid-s3093" xml:space="preserve">A D tangente, ſi C A non eſt perpendicularis ad tangen-<lb/>tem ducatur ex origine C recta C E perpendicularis ad tangentem A D, ſecans <lb/>eam in E, &amp; </s>
  <s xml:id="echoid-s3094" xml:space="preserve">ſectionem in F, erit in triangulo A C E angulus C A E acutus, <lb/>&amp; </s>
  <s xml:id="echoid-s3095" xml:space="preserve">minor angulo recto E, &amp; </s>
  <s xml:id="echoid-s3096" xml:space="preserve">propterea C A ſubtendens maiorem angulum re-<lb/>ctum, maior erit quàm C E, quæ acutum ſubtendit: </s>
  <s xml:id="echoid-s3097" xml:space="preserve">cumque punctum E tan-<lb/>gentis cadat extra ſectionem, erit C F minor, quàm C E; </s>
  <s xml:id="echoid-s3098" xml:space="preserve">ideoque C A multo <lb/>maior eſt, quàm C F, quapropter C A non erit breuiſsima, quod eſt contra, <lb/>hypotheſin.</s>
  <s xml:id="echoid-s3099" xml:space="preserve"/>
</p>
<div xml:id="echoid-div293" type="float" level="2" n="1">
<note position="left" xlink:label="note-0111-01" xlink:href="note-0111-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s3100" xml:space="preserve">Si vero fuerit D A C rectus, &amp;</s>
  <s xml:id="echoid-s3101" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3102" xml:space="preserve">Quia C A ſupponitur breuiſsima, <lb/>
<anchor type="note" xlink:label="note-0111-02a" xlink:href="note-0111-02"/>
<anchor type="note" xlink:label="note-0111-03a" xlink:href="note-0111-03"/>
&amp; </s>
  <s xml:id="echoid-s3103" xml:space="preserve">angulus D A C rectus, erit A D tangens; </s>
  <s xml:id="echoid-s3104" xml:space="preserve">nam ſi hoc verum non eſt, <lb/>ducatur ex puncto A recta linea A G, contingens ſectionem in <lb/>A; </s>
  <s xml:id="echoid-s3105" xml:space="preserve">ſecabit vtique tangens A G ipſam D A, &amp; </s>
  <s xml:id="echoid-s3106" xml:space="preserve">erit an-<lb/>gulus C A G rectus nimirum contentus à breuiſsima <lb/>C A, &amp; </s>
  <s xml:id="echoid-s3107" xml:space="preserve">tangente A G, ex proxime demon-<lb/>ſtrata propoſitione; </s>
  <s xml:id="echoid-s3108" xml:space="preserve">ergo duo anguli recti <lb/>C A D, &amp; </s>
  <s xml:id="echoid-s3109" xml:space="preserve">C A G æquales ſunt <lb/>inter ſe, pars, &amp; </s>
  <s xml:id="echoid-s3110" xml:space="preserve">totum, quod <lb/>eſt abſurdum.</s>
  <s xml:id="echoid-s3111" xml:space="preserve"/>
</p>
<div xml:id="echoid-div294" type="float" level="2" n="2">
<note position="left" xlink:label="note-0111-02" xlink:href="note-0111-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0111-03" xlink:href="note-0111-03a" xml:space="preserve">33. 34. <lb/>lib. 2.</note>
</div>
  <figure>
    <image file="0111-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0111-01"/>
  </figure>
<pb o="74" file="0112" n="112" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div296" type="section" level="1" n="98">
<head xml:id="echoid-head136" xml:space="preserve">SECTIO DECIMATERTIA</head>
<head xml:id="echoid-head137" xml:space="preserve">Continens Propoſ. LXIV. LXV. LXVI. <lb/>LXVII. &amp; LXXII. Apollonij.</head>
<head xml:id="echoid-head138" xml:space="preserve">PROPOSITIO LXIV. LXV.</head>
<p>
  <s xml:id="echoid-s3112" xml:space="preserve">S I ramorum ſecantium D C, D B, D A eductorum ex con-<lb/>curſu D ad fectionem C A non fuerint duo breuiſecantes, <lb/>vtique minimus eorum eſt, ramus terminatus D A, qui ambit <lb/>cum axe A E angulum acutum; </s>
  <s xml:id="echoid-s3113" xml:space="preserve">nempe D A E, &amp; </s>
  <s xml:id="echoid-s3114" xml:space="preserve">reliquorum <lb/>propinquior illi minor eſt remotiore, ſcilicet D B maior, eſt <lb/>quàm D A, &amp; </s>
  <s xml:id="echoid-s3115" xml:space="preserve">D C quàm D B.</s>
  <s xml:id="echoid-s3116" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3117" xml:space="preserve">Si vero inter illos fuerint duo breuiſecantes tunc vicinior <lb/>vertici ſectionis eſt maximus ramorum, &amp; </s>
  <s xml:id="echoid-s3118" xml:space="preserve">maiori proximior, <lb/>eſt maior, &amp; </s>
  <s xml:id="echoid-s3119" xml:space="preserve">minori propinquior eſt minor.</s>
  <s xml:id="echoid-s3120" xml:space="preserve"/>
</p>
  <figure>
    <image file="0112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0112-01"/>
  </figure>
<p>
  <s xml:id="echoid-s3121" xml:space="preserve">Producamus perpendicularem D E ſuper axim E A, &amp; </s>
  <s xml:id="echoid-s3122" xml:space="preserve">reperiatur Tru-<lb/>
<anchor type="note" xlink:label="note-0112-01a" xlink:href="note-0112-01"/>
tina F. </s>
  <s xml:id="echoid-s3123" xml:space="preserve">Et primo loco nullus ramus ſit breuiſecans, iam ſi D B, non eſt <lb/>maior, quàm D A, ſit æqualis illi, &amp; </s>
  <s xml:id="echoid-s3124" xml:space="preserve">ducamus duas perpendiculares
<pb o="75" file="0113" n="113" rhead="Conicor. Lib. V."/>
A G, A H ſuper E A, &amp; </s>
  <s xml:id="echoid-s3125" xml:space="preserve">D A. </s>
  <s xml:id="echoid-s3126" xml:space="preserve">Et quia A G tangit ſectionem, cadet <lb/>A H intra ſectionem, &amp; </s>
  <s xml:id="echoid-s3127" xml:space="preserve">ducamus rectam B I tangentem ſectionem in <lb/>
<anchor type="note" xlink:label="note-0113-01a" xlink:href="note-0113-01"/>
<anchor type="note" xlink:label="note-0113-02a" xlink:href="note-0113-02"/>
B. </s>
  <s xml:id="echoid-s3128" xml:space="preserve">Quoniam ex D non educitur ad ſectionem A C vllus breuiſecans, <lb/>erit E A non maior dimidio erecti (49. </s>
  <s xml:id="echoid-s3129" xml:space="preserve">50. </s>
  <s xml:id="echoid-s3130" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3131" xml:space="preserve">aut erit D E maior <lb/>quàm F (52. </s>
  <s xml:id="echoid-s3132" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3133" xml:space="preserve">Iis poſitis vtique linea breuiſſima ex B educta abſcin-<lb/>dit cum A ex axi lineam maiorem, quàm A K (49. </s>
  <s xml:id="echoid-s3134" xml:space="preserve">50. </s>
  <s xml:id="echoid-s3135" xml:space="preserve">51. </s>
  <s xml:id="echoid-s3136" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3137" xml:space="preserve">ex 5.) <lb/></s>
  <s xml:id="echoid-s3138" xml:space="preserve">verùm linea breuiſſima continet cum tangente B I angulum rectum (29. </s>
  <s xml:id="echoid-s3139" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0113-03a" xlink:href="note-0113-03"/>
30. </s>
  <s xml:id="echoid-s3140" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3141" xml:space="preserve">igitur D B I eſt acutus, quare ſi centro D, interuallo D B cir-<lb/>culus deſcribatur, tunc B I cadit intra circulum, &amp; </s>
  <s xml:id="echoid-s3142" xml:space="preserve">A H cadit extra id <lb/>
<anchor type="note" xlink:label="note-0113-04a" xlink:href="note-0113-04"/>
ipſum, quia eſt perpendicularis ad D A; </s>
  <s xml:id="echoid-s3143" xml:space="preserve">igitur circulus ſecat coniſectio-<lb/>nem; </s>
  <s xml:id="echoid-s3144" xml:space="preserve">ſecet eam in L, &amp; </s>
  <s xml:id="echoid-s3145" xml:space="preserve">iungamus L D, ducamuſque L G ſectionem, <lb/>
<anchor type="note" xlink:label="note-0113-05a" xlink:href="note-0113-05"/>
tangentem. </s>
  <s xml:id="echoid-s3146" xml:space="preserve">Pater (vt dictũ eſt) quod D L G ſit acutus; </s>
  <s xml:id="echoid-s3147" xml:space="preserve">ergo L G cadit <lb/>
<anchor type="note" xlink:label="note-0113-06a" xlink:href="note-0113-06"/>
intra circulum B L A, ſed cadit extra, quod eſt abſurdum; </s>
  <s xml:id="echoid-s3148" xml:space="preserve">ergo B D <lb/>non eſt æqualis ipſi A D. </s>
  <s xml:id="echoid-s3149" xml:space="preserve">Neque minor illo eſſe poteſt; </s>
  <s xml:id="echoid-s3150" xml:space="preserve">quia ſi ſecetur <lb/>D M maior, quàm D B, &amp; </s>
  <s xml:id="echoid-s3151" xml:space="preserve">minor, quàm D A, &amp; </s>
  <s xml:id="echoid-s3152" xml:space="preserve">centro D, interuallo <lb/>D M, circulus M L N deſcribatur, tunc D N, nempe D M maior eſt, <lb/>quàm D B, &amp; </s>
  <s xml:id="echoid-s3153" xml:space="preserve">propterea circulus N L M ſecat coniſectionem. </s>
  <s xml:id="echoid-s3154" xml:space="preserve">Subinde, <lb/>
<anchor type="note" xlink:label="note-0113-07a" xlink:href="note-0113-07"/>
patebit (quemadmodũ demoſtrauimus) quod D B non ſit minor, quàm <lb/>D A; </s>
  <s xml:id="echoid-s3155" xml:space="preserve">igitur D B maior eſt, quàm D A.</s>
  <s xml:id="echoid-s3156" xml:space="preserve"/>
</p>
<div xml:id="echoid-div296" type="float" level="2" n="1">
<note position="right" xlink:label="note-0112-01" xlink:href="note-0112-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0113-01" xlink:href="note-0113-01a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0113-02" xlink:href="note-0113-02a" xml:space="preserve">33. 34. <lb/>lib. 1.</note>
<note position="left" xlink:label="note-0113-03" xlink:href="note-0113-03a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0113-04" xlink:href="note-0113-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0113-05" xlink:href="note-0113-05a" xml:space="preserve">33. 34. <lb/>lib, 1.</note>
<note position="left" xlink:label="note-0113-06" xlink:href="note-0113-06a" xml:space="preserve">e</note>
<note position="left" xlink:label="note-0113-07" xlink:href="note-0113-07a" xml:space="preserve">f</note>
</div>
<p>
  <s xml:id="echoid-s3157" xml:space="preserve">Poſtea dico, quod D C maior eſt, quàm D B; </s>
  <s xml:id="echoid-s3158" xml:space="preserve">quia demonſtrauimus, <lb/>
<anchor type="note" xlink:label="note-0113-08a" xlink:href="note-0113-08"/>
angulũ D B O eſſe obtuſum, &amp; </s>
  <s xml:id="echoid-s3159" xml:space="preserve">patet, quod D C P eſt acutus, &amp; </s>
  <s xml:id="echoid-s3160" xml:space="preserve">proce-<lb/>dendo trito iam itinere demonſtrabimus, quod Q O neceſſe eſt, vt cadat <lb/>intra circulum C Q B. </s>
  <s xml:id="echoid-s3161" xml:space="preserve">Et quod ſi fuerit D C minor, quàm D B, aut æ-<lb/>qualis, neceſſe eſt, vt Q O cadat intra circulum C Q B; </s>
  <s xml:id="echoid-s3162" xml:space="preserve">ſed cecidit ex-<lb/>tra, quod eſt abſurdum; </s>
  <s xml:id="echoid-s3163" xml:space="preserve">igitur D C maior eſt, quàm D B, &amp; </s>
  <s xml:id="echoid-s3164" xml:space="preserve">D B ma-<lb/>ior, quàm D A, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3165" xml:space="preserve"/>
</p>
<div xml:id="echoid-div297" type="float" level="2" n="2">
<note position="left" xlink:label="note-0113-08" xlink:href="note-0113-08a" xml:space="preserve">g</note>
</div>
</div>
<div xml:id="echoid-div299" type="section" level="1" n="99">
<head xml:id="echoid-head139" xml:space="preserve">PROPOSITIO LXVI.</head>
<p>
  <s xml:id="echoid-s3166" xml:space="preserve">IN ſectione elliptica A B C, <lb/>
<anchor type="figure" xlink:label="fig-0113-01a" xlink:href="fig-0113-01"/>
cuius axis maior A C eius <lb/>centrum D, &amp; </s>
  <s xml:id="echoid-s3167" xml:space="preserve">D B dimidium <lb/>recti, duci nequeat ex E ad <lb/>quadrantem A B breuiſecans, <lb/>&amp; </s>
  <s xml:id="echoid-s3168" xml:space="preserve">producatur perpendicularis <lb/>E F; </s>
  <s xml:id="echoid-s3169" xml:space="preserve">Dico punctum F cadere <lb/>inter D A.</s>
  <s xml:id="echoid-s3170" xml:space="preserve"/>
</p>
<div xml:id="echoid-div299" type="float" level="2" n="1">
  <figure xlink:label="fig-0113-01" xlink:href="fig-0113-01a">
    <image file="0113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0113-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s3171" xml:space="preserve">Quia ſi caderet inter C, D du-<lb/>
<anchor type="note" xlink:label="note-0113-09a" xlink:href="note-0113-09"/>
ci poſſet ex E ad ſectionem A B <lb/>
<anchor type="note" xlink:label="note-0113-10a" xlink:href="note-0113-10"/>
aliqua breuiſecans (56. </s>
  <s xml:id="echoid-s3172" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3173" xml:space="preserve">quod eſt contra ſuppoſitionem. </s>
  <s xml:id="echoid-s3174" xml:space="preserve">Deinde <lb/>patet, quemadmodum demonſtrauimus in parabola, &amp; </s>
  <s xml:id="echoid-s3175" xml:space="preserve">hyperbola, quod <lb/>
<anchor type="note" xlink:label="note-0113-11a" xlink:href="note-0113-11"/>
E A minima ſit linearum, &amp; </s>
  <s xml:id="echoid-s3176" xml:space="preserve">ramorum ad ſectionem B A cadentium, &amp; </s>
  <s xml:id="echoid-s3177" xml:space="preserve"><lb/>propinquior illi, minor ſit remotiore, &amp; </s>
  <s xml:id="echoid-s3178" xml:space="preserve">hoc erat propoſitum.</s>
  <s xml:id="echoid-s3179" xml:space="preserve"/>
</p>
<div xml:id="echoid-div300" type="float" level="2" n="2">
<note position="left" xlink:label="note-0113-09" xlink:href="note-0113-09a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0113-10" xlink:href="note-0113-10a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0113-11" xlink:href="note-0113-11a" xml:space="preserve">pr. 64. 65. <lb/>huius.</note>
</div>
<pb o="76" file="0114" n="114" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div302" type="section" level="1" n="100">
<head xml:id="echoid-head140" xml:space="preserve">PROPOSITIO LXVII.</head>
<p>
  <s xml:id="echoid-s3180" xml:space="preserve">P Oſtea repetamus figuras, paraboles, &amp; </s>
  <s xml:id="echoid-s3181" xml:space="preserve">hyperboles, &amp; </s>
  <s xml:id="echoid-s3182" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0114-01a" xlink:href="note-0114-01"/>
quoquot ſunt illius ſigna, &amp; </s>
  <s xml:id="echoid-s3183" xml:space="preserve">ſupponamus quod ipſius D B <lb/>portio B K, ſit tantummodo linea breuiſſima; </s>
  <s xml:id="echoid-s3184" xml:space="preserve">Dico, quod D A <lb/>quoque minima eſt linearum egredientium ex D ad ſectionem <lb/>
<anchor type="note" xlink:label="note-0114-02a" xlink:href="note-0114-02"/>
A C, &amp; </s>
  <s xml:id="echoid-s3185" xml:space="preserve">illi propinquiores ſunt minores remotioribus.</s>
  <s xml:id="echoid-s3186" xml:space="preserve"/>
</p>
<div xml:id="echoid-div302" type="float" level="2" n="1">
<note position="right" xlink:label="note-0114-01" xlink:href="note-0114-01a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0114-02" xlink:href="note-0114-02a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s3187" xml:space="preserve">Quia educitur ex D vnus tantum <lb/>
<anchor type="note" xlink:label="note-0114-03a" xlink:href="note-0114-03"/>
<anchor type="figure" xlink:label="fig-0114-01a" xlink:href="fig-0114-01"/>
breuiſecans erit menſura E A maior <lb/>dimidio erecti, &amp; </s>
  <s xml:id="echoid-s3188" xml:space="preserve">D E æqualis F <lb/>Trutinæ (51. </s>
  <s xml:id="echoid-s3189" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3190" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3191" xml:space="preserve">vnde ſequi-<lb/>tur, quod lineæ breuiſſimæ eductæ <lb/>ab extremitatibus reliquorum ramo-<lb/>rum abſcindunt cum A ab axi line-<lb/>as maiores, quàm ſecant illi rami. <lb/></s>
  <s xml:id="echoid-s3192" xml:space="preserve">Ducamus prius ad ſectionem B A <lb/>ramum D G, inde conſtat D G ma-<lb/>
<anchor type="note" xlink:label="note-0114-04a" xlink:href="note-0114-04"/>
iorem eſſe, quàm D A (64. </s>
  <s xml:id="echoid-s3193" xml:space="preserve">65. </s>
  <s xml:id="echoid-s3194" xml:space="preserve">ex <lb/>5.) </s>
  <s xml:id="echoid-s3195" xml:space="preserve">Dico iam, quod D B maior eſt <lb/>illa, alioquin eſſet æqualis, vel mi-<lb/>nor illa, &amp; </s>
  <s xml:id="echoid-s3196" xml:space="preserve">producamus D H ad ſectionem B G; </s>
  <s xml:id="echoid-s3197" xml:space="preserve">ergo D H maior eſt, <lb/>quàm D G, quia remotior eſt ab D A (64. </s>
  <s xml:id="echoid-s3198" xml:space="preserve">65. </s>
  <s xml:id="echoid-s3199" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3200" xml:space="preserve">quare maior eſt, <lb/>quàm D B, &amp; </s>
  <s xml:id="echoid-s3201" xml:space="preserve">ex illo ſecetur D I maior, quàm D B, &amp; </s>
  <s xml:id="echoid-s3202" xml:space="preserve">minor, quàm, <lb/>D H, &amp; </s>
  <s xml:id="echoid-s3203" xml:space="preserve">centro D interuallo D I deſcriptus circulus ſecabit ſectionem, <lb/>B G, ſecet eam in M, &amp; </s>
  <s xml:id="echoid-s3204" xml:space="preserve">iungamus D M; </s>
  <s xml:id="echoid-s3205" xml:space="preserve">ergo D M, nempe D I, quæ <lb/>
<anchor type="note" xlink:label="note-0114-05a" xlink:href="note-0114-05"/>
conceſſa fuit maior, quàm D B eſt etiam maior, quàm D H, propterea <lb/>quod eſt remotior ab D A, quàm D H (64. </s>
  <s xml:id="echoid-s3206" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3207" xml:space="preserve">igitur D I maior eſt <lb/>quàm D H, quod eſt abſurdum; </s>
  <s xml:id="echoid-s3208" xml:space="preserve">quare D B maior eſt, quàm D H.</s>
  <s xml:id="echoid-s3209" xml:space="preserve"/>
</p>
<div xml:id="echoid-div303" type="float" level="2" n="2">
<note position="right" xlink:label="note-0114-03" xlink:href="note-0114-03a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0114-01" xlink:href="fig-0114-01a">
    <image file="0114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0114-01"/>
  </figure>
<note position="right" xlink:label="note-0114-04" xlink:href="note-0114-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0114-05" xlink:href="note-0114-05a" xml:space="preserve">e</note>
</div>
<p>
  <s xml:id="echoid-s3210" xml:space="preserve">Patet etiam, quod D B minor ſit, quàm D C, alioquin eſſet vel illi <lb/>
<anchor type="note" xlink:label="note-0114-06a" xlink:href="note-0114-06"/>
æqualis, aut maior, &amp; </s>
  <s xml:id="echoid-s3211" xml:space="preserve">ducamus D N ad ſectionem C B; </s>
  <s xml:id="echoid-s3212" xml:space="preserve">ergo D N mi-<lb/>nor eſt, quàm D C, eò quod proximior eſt D A (64. </s>
  <s xml:id="echoid-s3213" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3214" xml:space="preserve">quare mi-<lb/>nor eſt, quàm D B, &amp; </s>
  <s xml:id="echoid-s3215" xml:space="preserve">fecetur D O ex D B maior, quàm D N, &amp; </s>
  <s xml:id="echoid-s3216" xml:space="preserve">mi-<lb/>nor quàm D B, &amp; </s>
  <s xml:id="echoid-s3217" xml:space="preserve">centro D, interuallo D O circulus deſcriptus ſecabit <lb/>
<anchor type="note" xlink:label="note-0114-07a" xlink:href="note-0114-07"/>
ſectionem exempli gratia, in Q, &amp; </s>
  <s xml:id="echoid-s3218" xml:space="preserve">iungamus D Q, igitur D Q minor eſt <lb/>quàm D N, ſed eſt æqualis D O, quæ ſuppoſita fuit maior, quàm D N, <lb/>ergo D Q maior eſt, quàm D N; </s>
  <s xml:id="echoid-s3219" xml:space="preserve">verum eſt minor illo, quod eſt abſur-<lb/>dum; </s>
  <s xml:id="echoid-s3220" xml:space="preserve">igitur D C non eſt minor D B, neque æqualis; </s>
  <s xml:id="echoid-s3221" xml:space="preserve">quare maior illa. <lb/></s>
  <s xml:id="echoid-s3222" xml:space="preserve">eſt. </s>
  <s xml:id="echoid-s3223" xml:space="preserve">Atque ſic patet, quod D B minor ſit omnibus lineis egredientibus <lb/>ex D ad ſectionem B C, &amp; </s>
  <s xml:id="echoid-s3224" xml:space="preserve">illi proximiores ex illa parte, minores ſunt <lb/>remotioribus. </s>
  <s xml:id="echoid-s3225" xml:space="preserve">Quapropter manifeſtum eſt, quod D A ſit minimus omni-<lb/>um ramorum egredientium ex D ad ſectionem A B C, &amp; </s>
  <s xml:id="echoid-s3226" xml:space="preserve">reliqui proxi-<lb/>miores illi, minores ſunt remotioribus, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3227" xml:space="preserve"/>
</p>
<div xml:id="echoid-div304" type="float" level="2" n="3">
<note position="right" xlink:label="note-0114-06" xlink:href="note-0114-06a" xml:space="preserve">f</note>
<note position="right" xlink:label="note-0114-07" xlink:href="note-0114-07a" xml:space="preserve">g</note>
</div>
<pb o="77" file="0115" n="115" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div306" type="section" level="1" n="101">
<head xml:id="echoid-head141" xml:space="preserve">PROPOSITIO LXXII.</head>
<p>
  <s xml:id="echoid-s3228" xml:space="preserve">SI eductæ fuerint ex D duæ <lb/>
<anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/>
breuiſecantes D C, D B, <lb/>quorum ſegmenta G C, B K <lb/>ſint breuiſſima, &amp; </s>
  <s xml:id="echoid-s3229" xml:space="preserve">D B propin-<lb/>quior ſit vertici ſectionis; </s>
  <s xml:id="echoid-s3230" xml:space="preserve">Di-<lb/>co, quod D B maximus eſt ra-<lb/>morum egredientium ad ſectio-<lb/>
<anchor type="note" xlink:label="note-0115-01a" xlink:href="note-0115-01"/>
nem A B C, &amp; </s>
  <s xml:id="echoid-s3231" xml:space="preserve">minimus eorũ <lb/>D C, &amp; </s>
  <s xml:id="echoid-s3232" xml:space="preserve">ramorum egredientiũ <lb/>ad ſectionem A C, qui D B <lb/>propinquiores maiores ſunt <lb/>remotioribus, &amp; </s>
  <s xml:id="echoid-s3233" xml:space="preserve">propinquiores <lb/>D C (ex ramis egredientibus ad ſectionem in ea parte) mino-<lb/>res ſunt remotioribus.</s>
  <s xml:id="echoid-s3234" xml:space="preserve"/>
</p>
<div xml:id="echoid-div306" type="float" level="2" n="1">
  <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a">
    <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0115-01"/>
  </figure>
<note position="left" xlink:label="note-0115-01" xlink:href="note-0115-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s3235" xml:space="preserve">Sit F Trutina, &amp; </s>
  <s xml:id="echoid-s3236" xml:space="preserve">quia iam ducti ſunt ex D duo breuiſecantes, ideo <lb/>E A excedit dimidium erecti, &amp; </s>
  <s xml:id="echoid-s3237" xml:space="preserve">D E minor eſt, quàm F (51. </s>
  <s xml:id="echoid-s3238" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3239" xml:space="preserve">ex 5.) <lb/></s>
  <s xml:id="echoid-s3240" xml:space="preserve">his poſitis, vtique lineæ breuiſſimæ egredientes ab extremitatibus ramo-<lb/>rum qui ſunt in ſectione B C abſcindunt ab axi EA minores lineas, quàm <lb/>abſcindunt rami (51. </s>
  <s xml:id="echoid-s3241" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3242" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3243" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3244" xml:space="preserve">qui ducuntur ab extremitatibus egre-<lb/>dientium ad reliquas ſectiones abſcindunt lineas maiores. </s>
  <s xml:id="echoid-s3245" xml:space="preserve">Educamus ita-<lb/>que ramos D H, D I ad ſectionem B C, &amp; </s>
  <s xml:id="echoid-s3246" xml:space="preserve">ducamus B L, L H M, &amp; </s>
  <s xml:id="echoid-s3247" xml:space="preserve">I <lb/>M tangentes ſectionem in punctis B, H, I; </s>
  <s xml:id="echoid-s3248" xml:space="preserve">quia B K eſt breuiſsima erit <lb/>
<anchor type="note" xlink:label="note-0115-02a" xlink:href="note-0115-02"/>
I. </s>
  <s xml:id="echoid-s3249" xml:space="preserve">B D angulus rectus, &amp; </s>
  <s xml:id="echoid-s3250" xml:space="preserve">quia breuiſſima egrediens ex H abſcindit cum <lb/>A ab axi E A lineam minorem, quàm ſecat D H erit L H D obtuſus, &amp; </s>
  <s xml:id="echoid-s3251" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0115-03a" xlink:href="note-0115-03"/>
iungamus D L; </s>
  <s xml:id="echoid-s3252" xml:space="preserve">igitur duo quadrata D H, H L minora ſunt, quàm qua-<lb/>dratum D L, quod eſt æquale duobus quadratis L B, D B; </s>
  <s xml:id="echoid-s3253" xml:space="preserve">verum L B <lb/>minor eſt, quàm H L (68. </s>
  <s xml:id="echoid-s3254" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3255" xml:space="preserve">ergo D B maior eſt, quàm D H. </s>
  <s xml:id="echoid-s3256" xml:space="preserve">atq; <lb/></s>
  <s xml:id="echoid-s3257" xml:space="preserve">
<anchor type="note" xlink:label="note-0115-04a" xlink:href="note-0115-04"/>
ſic patet, quod D H maior ſit, quàm D I, quia D H M eſt acutus, &amp; </s>
  <s xml:id="echoid-s3258" xml:space="preserve">D <lb/>
<anchor type="note" xlink:label="note-0115-05a" xlink:href="note-0115-05"/>
I M obtuſus: </s>
  <s xml:id="echoid-s3259" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3260" xml:space="preserve">D I maior ſit, quàm D C. </s>
  <s xml:id="echoid-s3261" xml:space="preserve">Quare B D maximus eſt ra-<lb/>morum egredientium ad B C, &amp; </s>
  <s xml:id="echoid-s3262" xml:space="preserve">iam demonſtratum eſt, quod ſit maxi-<lb/>
<anchor type="note" xlink:label="note-0115-06a" xlink:href="note-0115-06"/>
mus ramorum egredientium ad B A (64. </s>
  <s xml:id="echoid-s3263" xml:space="preserve">65. </s>
  <s xml:id="echoid-s3264" xml:space="preserve">ex 5.)</s>
  <s xml:id="echoid-s3265" xml:space="preserve"/>
</p>
<div xml:id="echoid-div307" type="float" level="2" n="2">
<note position="right" xlink:label="note-0115-02" xlink:href="note-0115-02a" xml:space="preserve">29. 30. <lb/>huius.</note>
<note position="right" xlink:label="note-0115-03" xlink:href="note-0115-03a" xml:space="preserve">Ex 29. 30. <lb/>huius.</note>
<note position="right" xlink:label="note-0115-04" xlink:href="note-0115-04a" xml:space="preserve">Ibidem.</note>
<note position="left" xlink:label="note-0115-05" xlink:href="note-0115-05a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0115-06" xlink:href="note-0115-06a" xml:space="preserve">c</note>
</div>
<p>
  <s xml:id="echoid-s3266" xml:space="preserve">Ponamus poſtea N extra ſectionem B C, &amp; </s>
  <s xml:id="echoid-s3267" xml:space="preserve">iungamus D N, itaque, <lb/>linea breuiſſima egrediens ex N abſcindit ab axi E A maiorem lineam, <lb/>
<anchor type="note" xlink:label="note-0115-07a" xlink:href="note-0115-07"/>
<anchor type="note" xlink:label="note-0115-08a" xlink:href="note-0115-08"/>
quàm ſecat D N; </s>
  <s xml:id="echoid-s3268" xml:space="preserve">ergo tangens in N continet cum D N angulum acu-<lb/>
<anchor type="note" xlink:label="note-0115-09a" xlink:href="note-0115-09"/>
tum: </s>
  <s xml:id="echoid-s3269" xml:space="preserve">poſtea oſtendetur, quemadmodum hic dictum eſt, quod D C mi-<lb/>nimus ſit reliquorum ramorum egredientium ad reliquas ſectiones, &amp; </s>
  <s xml:id="echoid-s3270" xml:space="preserve">ſit <lb/>minimus ramorum egredientium ad A C, quare manifeſtum eſt, quod <lb/>D B ſit maximus ramorum, &amp; </s>
  <s xml:id="echoid-s3271" xml:space="preserve">D C minimus, &amp; </s>
  <s xml:id="echoid-s3272" xml:space="preserve">quod maioribus pro-<lb/>pinquiores ſunt maiores remotioribus, &amp; </s>
  <s xml:id="echoid-s3273" xml:space="preserve">minoribus propinquiores, mi-<lb/>nores ſunt remotioribus, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3274" xml:space="preserve"/>
</p>
<div xml:id="echoid-div308" type="float" level="2" n="3">
<note position="right" xlink:label="note-0115-07" xlink:href="note-0115-07a" xml:space="preserve">51. 52. <lb/>huius.</note>
<note position="left" xlink:label="note-0115-08" xlink:href="note-0115-08a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0115-09" xlink:href="note-0115-09a" xml:space="preserve">Ex 29. 30. <lb/>huius.</note>
</div>
<pb o="78" file="0116" n="116" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div310" type="section" level="1" n="102">
<head xml:id="echoid-head142" xml:space="preserve">MONITVM.</head>
<p style="it">
  <s xml:id="echoid-s3275" xml:space="preserve">ANtequam huius Decimætertiæ Sectionis explicationes, atque <lb/>emendationes aggrediamur, vt Notæ breuiores, clarioreſque <lb/>reddentatur, &amp; </s>
  <s xml:id="echoid-s3276" xml:space="preserve">teſtus Arabici menda facilius corrigi poſſent, operæ <lb/>pretium duximus (amice lector) Lemmata ſequentia præmittere.</s>
  <s xml:id="echoid-s3277" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div311" type="section" level="1" n="103">
<head xml:id="echoid-head143" xml:space="preserve">LEMMA IX.</head>
<p style="it">
  <s xml:id="echoid-s3278" xml:space="preserve">Si ad coniſectionem, atque ad vnum quadrantem ellipſis A B C à <lb/>concurſu D nullus ramus duci poſsit, qui ſit breuiſecans; </s>
  <s xml:id="echoid-s3279" xml:space="preserve">Dico, quod <lb/>quilibet ſecans ramus D B cum tangente H B G per eius terminum B <lb/>ducta efficit angulum D B H ad partes verticis A acutum, &amp; </s>
  <s xml:id="echoid-s3280" xml:space="preserve">D B <lb/>G, qui deinceps eſt, obtuſum.</s>
  <s xml:id="echoid-s3281" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s3282" xml:space="preserve">Quoniam nullus ramus ex concurſu <lb/>
<anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/>
D ad ſectionem A C ductus eſt breui-<lb/>ſecans, erit (ex conuerſa propoſitionis <lb/>49. </s>
  <s xml:id="echoid-s3283" xml:space="preserve">50. </s>
  <s xml:id="echoid-s3284" xml:space="preserve">51. </s>
  <s xml:id="echoid-s3285" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3286" xml:space="preserve">huius) menſura A E <lb/>aut non maior ſemiſſe lateris recti, aut <lb/>perpendicularis D E maior Trutina, <lb/>quæ ſit F, &amp; </s>
  <s xml:id="echoid-s3287" xml:space="preserve">ideo quilibet ramus ſe-<lb/>cans D B cadit ſupra breuiſsimam ex <lb/>puncto B ad axim ductam, eſt verò <lb/>breuiſsima ex puncto B ad axim ducta <lb/>perpendicularis ad G B H tangentem <lb/>
<anchor type="note" xlink:label="note-0116-01a" xlink:href="note-0116-01"/>
ſectionem in B; </s>
  <s xml:id="echoid-s3288" xml:space="preserve">ergo angulus D B H, <lb/>verticem A reſpiciens eſt acutus, &amp; </s>
  <s xml:id="echoid-s3289" xml:space="preserve">qui deinceps eſt D B G erit obtuſus.</s>
  <s xml:id="echoid-s3290" xml:space="preserve"/>
</p>
<div xml:id="echoid-div311" type="float" level="2" n="1">
  <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a">
    <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0116-01"/>
  </figure>
<note position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">29. 30. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div313" type="section" level="1" n="104">
<head xml:id="echoid-head144" xml:space="preserve">LEMMA X.</head>
<p style="it">
  <s xml:id="echoid-s3291" xml:space="preserve">Iiſdem poſitis, ſi à concurſu D vnicus tantum ramus D B breuiſe-<lb/>cans ad ſectionem A B duci poteſt; </s>
  <s xml:id="echoid-s3292" xml:space="preserve">Dico, quod quilibet alius ramus <lb/>ſecans D 1 ſupra, vel infra breuiſecantem D B poſitus efficit cum recta <lb/>L I H tangente ſectionem in I angulum D I L, verticem reſpicien-<lb/>tem, acutum, &amp; </s>
  <s xml:id="echoid-s3293" xml:space="preserve">D I H, qui deinceps eſt, obtuſum.</s>
  <s xml:id="echoid-s3294" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s3295" xml:space="preserve">Nam ex conuerſa propoſitione 51. </s>
  <s xml:id="echoid-s3296" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3297" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3298" xml:space="preserve">huius perpendicularis D E æqualis <lb/>c<unsure/>rit Trutinæ F, &amp; </s>
  <s xml:id="echoid-s3299" xml:space="preserve">ideo quilibet ramus D I poſitus ſupra, velinſra breuiſecantẽ
<pb o="79" file="0117" n="117" rhead="Conicor. Lib. V."/>
(qui eſt D B) cadit ſupra breuiſsimam ex puncto I ad axim ductam, quæ per-<lb/>
<anchor type="note" xlink:label="note-0117-01a" xlink:href="note-0117-01"/>
pendicularis eſt ad tangentem L I H, &amp; </s>
  <s xml:id="echoid-s3300" xml:space="preserve">propterea angulus D I L, verticem <lb/>
<anchor type="note" xlink:label="note-0117-02a" xlink:href="note-0117-02"/>
A reſpiciens erit acutus, &amp; </s>
  <s xml:id="echoid-s3301" xml:space="preserve">conſequens angulus D I H obtuſus.</s>
  <s xml:id="echoid-s3302" xml:space="preserve"/>
</p>
<div xml:id="echoid-div313" type="float" level="2" n="1">
<note position="right" xlink:label="note-0117-01" xlink:href="note-0117-01a" xml:space="preserve">51. 52. <lb/>huius.</note>
<note position="right" xlink:label="note-0117-02" xlink:href="note-0117-02a" xml:space="preserve">29. 30. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div315" type="section" level="1" n="105">
<head xml:id="echoid-head145" xml:space="preserve">LEMMA XI.</head>
<p style="it">
  <s xml:id="echoid-s3303" xml:space="preserve">Iiſdem poſitis, ſi à concurſu D duo breuiſecantes D C, D B ad ſe-<lb/>ctionem A B duci poſſunt; </s>
  <s xml:id="echoid-s3304" xml:space="preserve">Dico, quod quilibet ramus ſecans D I poſi-<lb/>tus ſupra breuiſecantem D B vertici proximiorem, vel infra infimum <lb/>breuiſecantem D C, efficit cum recta L I H tangente ſectionem in I an-<lb/>gulum D I L, reſpicientem verticem A, acutum, &amp; </s>
  <s xml:id="echoid-s3305" xml:space="preserve">conſequentem D <lb/>I H obtuſum, &amp; </s>
  <s xml:id="echoid-s3306" xml:space="preserve">quilibet ramus D O inter breuiſecantes poſitus efficit <lb/>cum recta G O N ſectionem tangente in O angulum D O G verticem <lb/>reſpicientem obtuſum, conſequentem vero D O N acutum.</s>
  <s xml:id="echoid-s3307" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s3308" xml:space="preserve">Quia (ex conuerſa propoſitione 51. </s>
  <s xml:id="echoid-s3309" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3310" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3311" xml:space="preserve">huius) perpendicularis D E mi-<lb/>
<anchor type="note" xlink:label="note-0117-03a" xlink:href="note-0117-03"/>
nor eſſe debet Trutina F, &amp; </s>
  <s xml:id="echoid-s3312" xml:space="preserve">propterea quilibet ramus D I ſupra breuiſecantem <lb/>D B, vel infra breuiſecãtem D C cadit ſupra breuiſsimam ex puncto I ad axim <lb/>
<anchor type="note" xlink:label="note-0117-04a" xlink:href="note-0117-04"/>
ductam, cum qua contingens L I angulum rectum conſtituit; </s>
  <s xml:id="echoid-s3313" xml:space="preserve">ergo angulus D I <lb/>L verticem reſpiciens, eſt acutus, &amp; </s>
  <s xml:id="echoid-s3314" xml:space="preserve">conſequens D I H obtuſus; </s>
  <s xml:id="echoid-s3315" xml:space="preserve">Similiter qui-<lb/>libet ramus D O inter breuiſecantes poſitus cadit infra breuiſsimam ex puncto <lb/>O ad axim ductam, &amp; </s>
  <s xml:id="echoid-s3316" xml:space="preserve">cum illa ſectionem contingens G O efſicit angulos rectos, <lb/>
<anchor type="note" xlink:label="note-0117-05a" xlink:href="note-0117-05"/>
igitur angulus D O G verticem reſpiciens, eſt obtuſus, &amp; </s>
  <s xml:id="echoid-s3317" xml:space="preserve">conſequens D O N <lb/>acutus.</s>
  <s xml:id="echoid-s3318" xml:space="preserve"/>
</p>
<div xml:id="echoid-div315" type="float" level="2" n="1">
<note position="right" xlink:label="note-0117-03" xlink:href="note-0117-03a" xml:space="preserve">51. 52. <lb/>huius.</note>
<note position="right" xlink:label="note-0117-04" xlink:href="note-0117-04a" xml:space="preserve">29. 30. <lb/>huius.</note>
<note position="right" xlink:label="note-0117-05" xlink:href="note-0117-05a" xml:space="preserve">Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div317" type="section" level="1" n="106">
<head xml:id="echoid-head146" xml:space="preserve">Notæ in Propoſ. LXIV. <lb/>&amp; LXV.</head>
<p style="it">
  <s xml:id="echoid-s3319" xml:space="preserve">ANtea Apollonius docuit qui nam rami ab origine ad coniſectionem ducti <lb/>eſſent minimi, &amp; </s>
  <s xml:id="echoid-s3320" xml:space="preserve">quo ordine reliqui rami ſe ſe excederent, modo agit <lb/>de ramis axim ſecantibus à concurſu ductis, &amp; </s>
  <s xml:id="echoid-s3321" xml:space="preserve">quærit qui minimus, &amp; </s>
  <s xml:id="echoid-s3322" xml:space="preserve">qui <lb/>maximus ſit, &amp; </s>
  <s xml:id="echoid-s3323" xml:space="preserve">quo ordine diſponantur.</s>
  <s xml:id="echoid-s3324" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s3325" xml:space="preserve">Producamus perpendicularem D E ſuper axim, &amp;</s>
  <s xml:id="echoid-s3326" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3327" xml:space="preserve">Si nullus ramus <lb/>
<anchor type="note" xlink:label="note-0117-06a" xlink:href="note-0117-06"/>
breuiſecans à concurſu D ad ſectionem A C duci poteſt; </s>
  <s xml:id="echoid-s3328" xml:space="preserve">Dico, quod ramus ter-<lb/>minatus D A eſt minimus omnium ramorum ſecantium D B, D C, &amp; </s>
  <s xml:id="echoid-s3329" xml:space="preserve">propin-<lb/>quiores vertici A minores ſunt remotioribus; </s>
  <s xml:id="echoid-s3330" xml:space="preserve">ducatur D E perpendicularis ad <lb/>axim eum ſecans in E, &amp; </s>
  <s xml:id="echoid-s3331" xml:space="preserve">reperiatur Trutina F. </s>
  <s xml:id="echoid-s3332" xml:space="preserve">Et ſiquidem D A non eſt <lb/>minor quolibet alio ramo ſecante D B infra ipſum poſito erit æqualis, aut maior <lb/>illo; </s>
  <s xml:id="echoid-s3333" xml:space="preserve">ſitque prius D A æqualis D B, ſi fieri poteſt, &amp; </s>
  <s xml:id="echoid-s3334" xml:space="preserve">ex puncto A verticis du-<lb/>catur A G perpendicularis ad axim A E, quæ continget ſectionem in A, pari-<lb/>
<anchor type="note" xlink:label="note-0117-07a" xlink:href="note-0117-07"/>
terque ducatur recta A H perpendicularis ad ramum A D inclinatum ad axim;</s>
  <s xml:id="echoid-s3335" xml:space="preserve">
<pb o="80" file="0118" n="118" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0118-01a" xlink:href="fig-0118-01"/>
&amp; </s>
  <s xml:id="echoid-s3336" xml:space="preserve">quia A H cadit infra A G ad partes axis cum D A, ad quam illa perpen-<lb/>dicularis eſt, extendatur vltra axim A E, nec poſsit inter tangentem A G, &amp; </s>
  <s xml:id="echoid-s3337" xml:space="preserve"><lb/>ſectionem conicam A B, aliqua recta linea intercipi; </s>
  <s xml:id="echoid-s3338" xml:space="preserve">igitur A H cadit intra <lb/>coniſectionem, &amp; </s>
  <s xml:id="echoid-s3339" xml:space="preserve">angulus E A H eſt acutus.</s>
  <s xml:id="echoid-s3340" xml:space="preserve"/>
</p>
<div xml:id="echoid-div317" type="float" level="2" n="1">
<note position="right" xlink:label="note-0117-06" xlink:href="note-0117-06a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0117-07" xlink:href="note-0117-07a" xml:space="preserve">17. lib. 1. <lb/>32. pr.</note>
  <figure xlink:label="fig-0118-01" xlink:href="fig-0118-01a">
    <image file="0118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0118-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s3341" xml:space="preserve">Quoniam ex D non educitur ad ſectionem A C vllus breuiſecans, &amp; </s>
  <s xml:id="echoid-s3342" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s3343" xml:space="preserve">
<anchor type="note" xlink:label="note-0118-01a" xlink:href="note-0118-01"/>
Sequitur quidem ex hac hypotheſi, quod menſura E A non ſit maior ſemierecto <lb/>
<anchor type="note" xlink:label="note-0118-02a" xlink:href="note-0118-02"/>
aut ſi maior eſt, ſit quoque perpendicularis D E maior Trutina F, ex conuerſa <lb/>propoſitione 51. </s>
  <s xml:id="echoid-s3344" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3345" xml:space="preserve">huius per deductionem ad inconueniens.</s>
  <s xml:id="echoid-s3346" xml:space="preserve"/>
</p>
<div xml:id="echoid-div318" type="float" level="2" n="2">
<note position="right" xlink:label="note-0118-01" xlink:href="note-0118-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0118-02" xlink:href="note-0118-02a" xml:space="preserve">Ex 49. 50. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3347" xml:space="preserve">Quare ſi centro D interuallo D B, &amp;</s>
  <s xml:id="echoid-s3348" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3349" xml:space="preserve">Circulus enim B I L H A ra-<lb/>
<anchor type="note" xlink:label="note-0118-03a" xlink:href="note-0118-03"/>
dio D B deſcriptus tranſibit per verticem A cum radius D B poſitus ſit æqualis <lb/>D A, cumque angulus D B I ſit acutus, ex Lemmate nono, cadet neceſſario B <lb/>I intra circulum B I L.</s>
  <s xml:id="echoid-s3350" xml:space="preserve"/>
</p>
<div xml:id="echoid-div319" type="float" level="2" n="3">
<note position="right" xlink:label="note-0118-03" xlink:href="note-0118-03a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s3351" xml:space="preserve">Ig tur circulus ſecat coniſectionem, &amp;</s>
  <s xml:id="echoid-s3352" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3353" xml:space="preserve">Quia B I cadit extra coniſe-<lb/>
<anchor type="note" xlink:label="note-0118-04a" xlink:href="note-0118-04"/>
ctionem, quàm tangit, &amp; </s>
  <s xml:id="echoid-s3354" xml:space="preserve">intra circulum B L A, vt dictum eſt, è contra re-<lb/>cta A H cadit intra eandem coniſectionem, &amp; </s>
  <s xml:id="echoid-s3355" xml:space="preserve">extra ipſum circulum, quem, <lb/>tangit, cum H A perpendicularis ſit ad circuli radium D A; </s>
  <s xml:id="echoid-s3356" xml:space="preserve">igitur circulus B <lb/>I L A fertur extra coniſectionem ad partes B I, &amp; </s>
  <s xml:id="echoid-s3357" xml:space="preserve">intra eandem ad partes A <lb/>H; </s>
  <s xml:id="echoid-s3358" xml:space="preserve">quare neceſſario coniſectionem ſecat.</s>
  <s xml:id="echoid-s3359" xml:space="preserve"/>
</p>
<div xml:id="echoid-div320" type="float" level="2" n="4">
<note position="right" xlink:label="note-0118-04" xlink:href="note-0118-04a" xml:space="preserve">d</note>
</div>
<p style="it">
  <s xml:id="echoid-s3360" xml:space="preserve">Patet, vt dictum eſt, quod D L G ſit acutus, &amp;</s>
  <s xml:id="echoid-s3361" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3362" xml:space="preserve">Hoc enim ſequitur ex <lb/>
<anchor type="note" xlink:label="note-0118-05a" xlink:href="note-0118-05"/>
nono Lemmate præmiſſo, reſpicit enim angulus D L G verticem A; </s>
  <s xml:id="echoid-s3363" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3364" xml:space="preserve">ideo eſt <lb/>acutus, &amp; </s>
  <s xml:id="echoid-s3365" xml:space="preserve">cadit neceſſario recta L G intra circulum B L A radio D L deſcri-<lb/>ptum ad partes L A; </s>
  <s xml:id="echoid-s3366" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3367" xml:space="preserve">portio circuli L H A cadit intra coniſectionem L A; <lb/></s>
  <s xml:id="echoid-s3368" xml:space="preserve">igitur recta L G cadit intra coniſectionem L A, ſed cadit extra eandem ſectio-<lb/>
<anchor type="note" xlink:label="note-0118-06a" xlink:href="note-0118-06"/>
nem, cum contingat eam in L, quod eſt abſurdum.</s>
  <s xml:id="echoid-s3369" xml:space="preserve"/>
</p>
<div xml:id="echoid-div321" type="float" level="2" n="5">
<note position="right" xlink:label="note-0118-05" xlink:href="note-0118-05a" xml:space="preserve">e</note>
<note position="left" xlink:label="note-0118-06" xlink:href="note-0118-06a" xml:space="preserve">35. 36. <lb/>lib. 1.</note>
</div>
<pb o="81" file="0119" n="119" rhead="Conicor. Lib. V."/>
<p style="it">
  <s xml:id="echoid-s3370" xml:space="preserve">Deinde patebit, quemadmodum demonſtrauimus, &amp;</s>
  <s xml:id="echoid-s3371" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3372" xml:space="preserve">Quia D M fa-<lb/>
<anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/>
cta eſt maior, quàm D B, &amp; </s>
  <s xml:id="echoid-s3373" xml:space="preserve">minor quàm D A, eſtque circuli radius D N <lb/>æqualis D M; </s>
  <s xml:id="echoid-s3374" xml:space="preserve">ergo punctum M cadit intra coniſectionem, N vero extra ip-<lb/>ſam; </s>
  <s xml:id="echoid-s3375" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3376" xml:space="preserve">propterea circulus M L N ſectionem conicam ſecabit alicubi, vt in L, <lb/>&amp; </s>
  <s xml:id="echoid-s3377" xml:space="preserve">portio circuli M L intra coniſectionem A L incidet: </s>
  <s xml:id="echoid-s3378" xml:space="preserve">rurſus ducatur radius <lb/>D L, &amp; </s>
  <s xml:id="echoid-s3379" xml:space="preserve">L G coniſectionem tangens in L erit, vt priùs angulus D L G acu-<lb/>
<anchor type="note" xlink:label="note-0119-02a" xlink:href="note-0119-02"/>
tus; </s>
  <s xml:id="echoid-s3380" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3381" xml:space="preserve">ideo L G cadit intra circulum L M, &amp; </s>
  <s xml:id="echoid-s3382" xml:space="preserve">propterea intra coniſectionem <lb/>A L, ſed eadem L G cadit extra ipſam, quia eam contingit in L, quod eſt ab-<lb/>ſurdum; </s>
  <s xml:id="echoid-s3383" xml:space="preserve">quare ramus D A non eſt maior, quàm D B; </s>
  <s xml:id="echoid-s3384" xml:space="preserve">ſed priùs neque illi <lb/>æqualis erat; </s>
  <s xml:id="echoid-s3385" xml:space="preserve">igitur ramus terminatus D A minor eſt quolibet ramo ſecante <lb/>D B infra ipſum poſito, &amp; </s>
  <s xml:id="echoid-s3386" xml:space="preserve">propterea minimus erit omnium ſecantium.</s>
  <s xml:id="echoid-s3387" xml:space="preserve"/>
</p>
<div xml:id="echoid-div322" type="float" level="2" n="6">
<note position="left" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">f</note>
<note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">33. 34. <lb/>lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3388" xml:space="preserve">Poſtea dico, quod D C maior eſt, quàm D B, &amp;</s>
  <s xml:id="echoid-s3389" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3390" xml:space="preserve">Demonſtratio ſe-<lb/>
<anchor type="note" xlink:label="note-0119-03a" xlink:href="note-0119-03"/>
cundæ partis huius propoſitionis, quàm Apollonius innuit (quia conſtructione, <lb/>ac progreſſu ſimili ſuperiori perſici poteſt) hac ratione reſtituitur. </s>
  <s xml:id="echoid-s3391" xml:space="preserve">Demonſtran-<lb/>dum eſt quemlibet ramum D B vertici A proximiorem eße minorem quolibet <lb/>ramo D C remotiore. </s>
  <s xml:id="echoid-s3392" xml:space="preserve">Ducantur recta C P contingens ſectionem in C, &amp; </s>
  <s xml:id="echoid-s3393" xml:space="preserve">O B <lb/>tangens ſectionem in B, &amp; </s>
  <s xml:id="echoid-s3394" xml:space="preserve">recta B R perpendicularis ad ramum D B; </s>
  <s xml:id="echoid-s3395" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3396" xml:space="preserve">ſi <lb/>quidem ramus D C non concedatur maior, quàm D B, ſit primo ei æqualis, ſi <lb/>fieri poteſt, &amp; </s>
  <s xml:id="echoid-s3397" xml:space="preserve">centro D interuallo D C deſcribatur circulus C P R, qui tran-<lb/>ſibit per punctum B, ob æqualitatem radiorum D C, D B; </s>
  <s xml:id="echoid-s3398" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3399" xml:space="preserve">quia (ex Lem-<lb/>mate nono) angulus D C P verticem reſpiciens, eſt acutus, recta C P cadet <lb/>intra circulum C P R; </s>
  <s xml:id="echoid-s3400" xml:space="preserve">ſed cadit extra coniſectionem, cum ſit contingens; </s>
  <s xml:id="echoid-s3401" xml:space="preserve">igi-<lb/>tur portio circularis peripheriæ C P ducitur extra coniſectionem C Q B: </s>
  <s xml:id="echoid-s3402" xml:space="preserve">rur-<lb/>ſus, quia angulus D B O eſt obtuſus (ex nono Lemmate, cum verticem A non reſpi-<lb/>ciat) ergo R B perpendicularis ad D B cadit intra coniſectionẽ, cum B O poſita ſit eã <lb/>contingens: </s>
  <s xml:id="echoid-s3403" xml:space="preserve">cadit verò eadem B R extra circulum B R Q, cum ſit perpendicu-<lb/>laris ad circuli radium D B; </s>
  <s xml:id="echoid-s3404" xml:space="preserve">igitur circuli portio B R intra coniſectionem ca-<lb/>det: </s>
  <s xml:id="echoid-s3405" xml:space="preserve">ſed priùs eiuſdem circuli portio C P extra eandem ſectionem ducebatur; <lb/></s>
  <s xml:id="echoid-s3406" xml:space="preserve">igitur idem circulus ſecat coniſectionem alicubi, vt in Q, ducaturque denuo <lb/>ramus D Q, &amp; </s>
  <s xml:id="echoid-s3407" xml:space="preserve">Q O contingens ſectionem in Q; </s>
  <s xml:id="echoid-s3408" xml:space="preserve">Vnde (ex nono Lemmate) <lb/>
<anchor type="note" xlink:label="note-0119-04a" xlink:href="note-0119-04"/>
angulus D Q O erit acutus; </s>
  <s xml:id="echoid-s3409" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3410" xml:space="preserve">propterea recta Q O intra circuli portionem; <lb/></s>
  <s xml:id="echoid-s3411" xml:space="preserve">Q R conſtituta intra coniſectionem cadet, quod eſt abſurdum; </s>
  <s xml:id="echoid-s3412" xml:space="preserve">recta enim Q <lb/>O extra coniſectionem Q A cadit, quàm contingit in Q; </s>
  <s xml:id="echoid-s3413" xml:space="preserve">non ergo ramus D <lb/>C æqualis eſt ipſi D B. </s>
  <s xml:id="echoid-s3414" xml:space="preserve">Sit ſecundò D C minor, quàm D B (ſi fieri poteſt) ſe-<lb/>ceturque D T minor quàm D B, ſed maior quàm D C; </s>
  <s xml:id="echoid-s3415" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3416" xml:space="preserve">centro D interuallo <lb/>D T deſcribatur circulus T Q S; </s>
  <s xml:id="echoid-s3417" xml:space="preserve">is quidem ad partes B cadet intra, ad par-<lb/>tes vero C extra coniſectionem; </s>
  <s xml:id="echoid-s3418" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3419" xml:space="preserve">propterea eam alicubi ſecabit, vt in Q; </s>
  <s xml:id="echoid-s3420" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s3421" xml:space="preserve">ducto ramo D Q, &amp; </s>
  <s xml:id="echoid-s3422" xml:space="preserve">Q O contingente ſectionem in Q, erit angulus D Q <lb/>
<anchor type="note" xlink:label="note-0119-05a" xlink:href="note-0119-05"/>
O acutus, &amp; </s>
  <s xml:id="echoid-s3423" xml:space="preserve">ideo recta Q O cadet intra circulum T Q, &amp; </s>
  <s xml:id="echoid-s3424" xml:space="preserve">propterea intra <lb/>coniſectionem, quod eſt abſurdum; </s>
  <s xml:id="echoid-s3425" xml:space="preserve">Q O enim cadit extra ſectionem Q A, <lb/>quàm contingit in Q; </s>
  <s xml:id="echoid-s3426" xml:space="preserve">non ergo ramus D C minor eſt, quàm D B, ſed neque <lb/>æqualis priùs oſtenſus fuit; </s>
  <s xml:id="echoid-s3427" xml:space="preserve">igitur quilibet ramus D B vertici A propinquior <lb/>minor eſt quolibet ramo remotiore D C, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3428" xml:space="preserve"/>
</p>
<div xml:id="echoid-div323" type="float" level="2" n="7">
<note position="left" xlink:label="note-0119-03" xlink:href="note-0119-03a" xml:space="preserve">g</note>
<note position="right" xlink:label="note-0119-04" xlink:href="note-0119-04a" xml:space="preserve">33. 34. <lb/>lib. 1.</note>
<note position="right" xlink:label="note-0119-05" xlink:href="note-0119-05a" xml:space="preserve">Lem. 9.</note>
</div>
<pb o="82" file="0120" n="120" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div325" type="section" level="1" n="107">
<head xml:id="echoid-head147" xml:space="preserve">Notæ in Propoſ. LXVI.</head>
<p style="it">
  <s xml:id="echoid-s3429" xml:space="preserve">QVia ſi caderet inter C, D ducipoſ-<lb/>
<anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/>
<anchor type="note" xlink:label="note-0120-01a" xlink:href="note-0120-01"/>
ſet, &amp;</s>
  <s xml:id="echoid-s3430" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3431" xml:space="preserve">Quotieſcumq; </s>
  <s xml:id="echoid-s3432" xml:space="preserve">enim perpen-<lb/>dicularis E F cadit ſuper centrũ <lb/>D, vel ſecat ſemiaxim D C inter D, &amp; </s>
  <s xml:id="echoid-s3433" xml:space="preserve">C, tũc <lb/>ex concurſu E vnicus ramus breuiſecans du-<lb/>ci poteſt ad ſectionem B A, qui nimirum ca-<lb/>
<anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/>
dit inter verticem remotiorem A, &amp; </s>
  <s xml:id="echoid-s3434" xml:space="preserve">axim <lb/>minorem D B: </s>
  <s xml:id="echoid-s3435" xml:space="preserve">ſed ex hypotheſi nullus ra-<lb/>mus ex concurſu E ad quadrantem ellipſis A <lb/>B duci poteſt, qui ſit breuiſecans; </s>
  <s xml:id="echoid-s3436" xml:space="preserve">igitur per-<lb/>pendicularis E F ſecat ſemiaxim A D in <lb/>puncto F poſito inter A, &amp; </s>
  <s xml:id="echoid-s3437" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3438" xml:space="preserve"/>
</p>
<div xml:id="echoid-div325" type="float" level="2" n="1">
  <figure xlink:label="fig-0120-01" xlink:href="fig-0120-01a">
    <image file="0120-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0120-01"/>
  </figure>
<note position="right" xlink:label="note-0120-01" xlink:href="note-0120-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">45. 56. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3439" xml:space="preserve">Deinde patet, quemadmodum demon-<lb/>ſtrauimus in vtraque hyperbola, &amp;</s>
  <s xml:id="echoid-s3440" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3441" xml:space="preserve">Permuto particulam [vtraque] vt <lb/>manifeſtè erroneam, legi enim debet in parabola, &amp; </s>
  <s xml:id="echoid-s3442" xml:space="preserve">hyperbola. </s>
  <s xml:id="echoid-s3443" xml:space="preserve">Quod vero ra-<lb/>mus terminatus E A minimus ſit omnium ramorum ſecantium manifeſtum eſt <lb/>ex demonſtratione propoſitionis 64. </s>
  <s xml:id="echoid-s3444" xml:space="preserve">65.</s>
  <s xml:id="echoid-s3445" xml:space="preserve">, quæ compræhendit etiam ellipſim, <lb/>quando menſura F A minor eſt ſemiaxi A D, vt ex propoſitione 52. </s>
  <s xml:id="echoid-s3446" xml:space="preserve">patet. </s>
  <s xml:id="echoid-s3447" xml:space="preserve">Et ſi-<lb/>militer ramorũ ſecantium ex concurſu E ad ſectionem A B ductorum propinquio-<lb/>res vertici A minores ſunt remotioribus ex eadem demonſtratione 64. </s>
  <s xml:id="echoid-s3448" xml:space="preserve">65. </s>
  <s xml:id="echoid-s3449" xml:space="preserve">huius.</s>
  <s xml:id="echoid-s3450" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div327" type="section" level="1" n="108">
<head xml:id="echoid-head148" style="it" xml:space="preserve">Ex demonſtratione præmiſſa propoſitionum 64. &amp; 65. <lb/>deduci poteſt conſectarium, à quo notæ ſubſe-<lb/>quentes breuiores reddantur.</head>
<head xml:id="echoid-head149" xml:space="preserve">COROLLARIVM PROPOSIT. <lb/>LXIV. &amp; LXV.</head>
<p style="it">
  <s xml:id="echoid-s3451" xml:space="preserve">SI in aliqua peripheria cuiuslibet coniſectio-<lb/>
<anchor type="figure" xlink:label="fig-0120-02a" xlink:href="fig-0120-02"/>
nis omnes rami ſecantes, qui à concurſu <lb/>duci poſſunt, cum tangentibus ab eorum ter-<lb/>minis ductis conſtituunt angulos, qui verti-<lb/>cem reſpiciunt, acutos; </s>
  <s xml:id="echoid-s3452" xml:space="preserve">rami proximiores ver-<lb/>tici ſectionis minores erunt remotioribus.</s>
  <s xml:id="echoid-s3453" xml:space="preserve"/>
</p>
<div xml:id="echoid-div327" type="float" level="2" n="1">
  <figure xlink:label="fig-0120-02" xlink:href="fig-0120-02a">
    <image file="0120-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0120-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s3454" xml:space="preserve">Ex eo enim, quod ïn propoſitionibus 64. </s>
  <s xml:id="echoid-s3455" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3456" xml:space="preserve"><lb/>65.</s>
  <s xml:id="echoid-s3457" xml:space="preserve">, omnes rami D A, D L, D B, D Q, D <lb/>C, &amp; </s>
  <s xml:id="echoid-s3458" xml:space="preserve">reliqui omnes, qui duci poſſunt ex con-<lb/>curſu D ad ſectionem A B C efficiunt cum <lb/>tangentibus ſectionẽ à terminis A, L, B, Q, C <lb/>angulos, verticem A reſpicientes, acutos, vt
<pb o="83" file="0121" n="121" rhead="Conicor. Lib. V."/>
ſunt D A V, D L G, D B I, D Q O, D C P, oſtenſus eſt ramus D A minor <lb/>quàm D B, &amp; </s>
  <s xml:id="echoid-s3459" xml:space="preserve">D B propinquior vertici A, minor ramo D C remotiore.</s>
  <s xml:id="echoid-s3460" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div329" type="section" level="1" n="109">
<head xml:id="echoid-head150" xml:space="preserve">Notæ in Propoſ. LXVII.</head>
<p style="it">
  <s xml:id="echoid-s3461" xml:space="preserve">POſtea repetamus figuram vtrã-<lb/>
<anchor type="figure" xlink:label="fig-0121-01a" xlink:href="fig-0121-01"/>
<anchor type="note" xlink:label="note-0121-01a" xlink:href="note-0121-01"/>
que hyperboles, &amp;</s>
  <s xml:id="echoid-s3462" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3463" xml:space="preserve">Lego; <lb/></s>
  <s xml:id="echoid-s3464" xml:space="preserve">Repetamus figuras paraboles, &amp; </s>
  <s xml:id="echoid-s3465" xml:space="preserve">hy-<lb/>perboles, &amp; </s>
  <s xml:id="echoid-s3466" xml:space="preserve">ſupponantur denuo eædem <lb/>lineæ æductæ ex concurſu D ad ſectio-<lb/>nem; </s>
  <s xml:id="echoid-s3467" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3468" xml:space="preserve">perpendicularis D E, atque <lb/>Trutina F, &amp; </s>
  <s xml:id="echoid-s3469" xml:space="preserve">omnium ramorum ſe-<lb/>cantium vnicus tantummodo D B ſit <lb/>breuiſecans.</s>
  <s xml:id="echoid-s3470" xml:space="preserve"/>
</p>
<div xml:id="echoid-div329" type="float" level="2" n="1">
  <figure xlink:label="fig-0121-01" xlink:href="fig-0121-01a">
    <image file="0121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0121-01"/>
  </figure>
<note position="left" xlink:label="note-0121-01" xlink:href="note-0121-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s3471" xml:space="preserve">Et illi propinquiores ſint maio-<lb/>
<anchor type="note" xlink:label="note-0121-02a" xlink:href="note-0121-02"/>
res remotioribus, &amp;</s>
  <s xml:id="echoid-s3472" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3473" xml:space="preserve">Sed mendo-<lb/>sè; </s>
  <s xml:id="echoid-s3474" xml:space="preserve">legi debet: </s>
  <s xml:id="echoid-s3475" xml:space="preserve">Et illi propinquiores <lb/>ſint minores remotioribus.</s>
  <s xml:id="echoid-s3476" xml:space="preserve"/>
</p>
<div xml:id="echoid-div330" type="float" level="2" n="2">
<note position="left" xlink:label="note-0121-02" xlink:href="note-0121-02a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s3477" xml:space="preserve">Quia educitur ex D vnus tantum breuiſecans, &amp;</s>
  <s xml:id="echoid-s3478" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3479" xml:space="preserve">Legi debet. </s>
  <s xml:id="echoid-s3480" xml:space="preserve">Quia <lb/>
<anchor type="note" xlink:label="note-0121-03a" xlink:href="note-0121-03"/>
<anchor type="note" xlink:label="note-0121-04a" xlink:href="note-0121-04"/>
educitur ex concurſu D vnus tantum breuiſecans, erit menſura E A maior di-<lb/>midio erecti, &amp; </s>
  <s xml:id="echoid-s3481" xml:space="preserve">D E perpendicularis ad axim æqualis erit Trutinæ F.</s>
  <s xml:id="echoid-s3482" xml:space="preserve"/>
</p>
<div xml:id="echoid-div331" type="float" level="2" n="3">
<note position="right" xlink:label="note-0121-03" xlink:href="note-0121-03a" xml:space="preserve">Conuerſ. <lb/>51. 52. <lb/>huius.</note>
<note position="left" xlink:label="note-0121-04" xlink:href="note-0121-04a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s3483" xml:space="preserve">Inde conſtat D G maiorem eſſe, quàm D A, &amp;</s>
  <s xml:id="echoid-s3484" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3485" xml:space="preserve">Quia ex concurſu D <lb/>
<anchor type="note" xlink:label="note-0121-05a" xlink:href="note-0121-05"/>
ad ſectionem A C vnicus ramus D B breuiſecans ſupponitur igitur omnes rami <lb/>cadentes inter A, &amp; </s>
  <s xml:id="echoid-s3486" xml:space="preserve">B præter infimum D B conſtituunt cum tangentibus ſectio-<lb/>nem, ab eorum terminis ductis, angulos reſpicientes verticem A acutos; </s>
  <s xml:id="echoid-s3487" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3488" xml:space="preserve">pro-<lb/>
<anchor type="note" xlink:label="note-0121-06a" xlink:href="note-0121-06"/>
pterea ramus terminatus D A minor eſt quolibet ramo D G infra ipſum, &amp; </s>
  <s xml:id="echoid-s3489" xml:space="preserve">ſu-<lb/>pra ramum D B poſito; </s>
  <s xml:id="echoid-s3490" xml:space="preserve">atque ramus D G minor eſt quolibet alio à vertice re-<lb/>
<anchor type="note" xlink:label="note-0121-07a" xlink:href="note-0121-07"/>
motiore ducto ex D ad peripheriam A B. </s>
  <s xml:id="echoid-s3491" xml:space="preserve">Dico iam, quod ramus D B maior <lb/>eſt quolibet ramo D G, poſito infra verticem A, &amp; </s>
  <s xml:id="echoid-s3492" xml:space="preserve">ſupra breuiſecantem D B; <lb/></s>
  <s xml:id="echoid-s3493" xml:space="preserve">Si enim hoc verum non eſt, erit D B æqualis, aut minor, quàm D G, &amp; </s>
  <s xml:id="echoid-s3494" xml:space="preserve">tunc <lb/>ducto quolibet ramo D H ad ſectionem G B infra ramum D G, erit D H re-<lb/>
<anchor type="note" xlink:label="note-0121-08a" xlink:href="note-0121-08"/>
motior à vertice A maior propinquiore D G, &amp; </s>
  <s xml:id="echoid-s3495" xml:space="preserve">propterea ramus D B adhuc <lb/>minor erit ramo D H.</s>
  <s xml:id="echoid-s3496" xml:space="preserve"/>
</p>
<div xml:id="echoid-div332" type="float" level="2" n="4">
<note position="left" xlink:label="note-0121-05" xlink:href="note-0121-05a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0121-06" xlink:href="note-0121-06a" xml:space="preserve">Lem. 10.</note>
<note position="right" xlink:label="note-0121-07" xlink:href="note-0121-07a" xml:space="preserve">Coro 11.<unsure/> <lb/>64. 65. <lb/>huius.</note>
<note position="right" xlink:label="note-0121-08" xlink:href="note-0121-08a" xml:space="preserve">Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3497" xml:space="preserve">Ergo D M nempe D I, &amp;</s>
  <s xml:id="echoid-s3498" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3499" xml:space="preserve">Quia D M, vt remotior à vertice A, eſt ma-<lb/>
<anchor type="note" xlink:label="note-0121-09a" xlink:href="note-0121-09"/>
ior, quàm propinquior D H eſt vero D L, atque D I æqualis D M cum ſint <lb/>
<anchor type="note" xlink:label="note-0121-10a" xlink:href="note-0121-10"/>
radĳ eiuſdem circuli; </s>
  <s xml:id="echoid-s3500" xml:space="preserve">ergo D I portio maior eſt, quàm totum D H, quod eſt <lb/>abſurdum; </s>
  <s xml:id="echoid-s3501" xml:space="preserve">quare D B maior eſt quolibet ramo D G infra verticem A, &amp; </s>
  <s xml:id="echoid-s3502" xml:space="preserve">ſu-<lb/>pra ramum D B poſito; </s>
  <s xml:id="echoid-s3503" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3504" xml:space="preserve">propterea D B multo maior erit, quàm D A.</s>
  <s xml:id="echoid-s3505" xml:space="preserve"/>
</p>
<div xml:id="echoid-div333" type="float" level="2" n="5">
<note position="left" xlink:label="note-0121-09" xlink:href="note-0121-09a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0121-10" xlink:href="note-0121-10a" xml:space="preserve">Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3506" xml:space="preserve">Ergo D N minor eſt, quàm D C, &amp;</s>
  <s xml:id="echoid-s3507" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3508" xml:space="preserve">Dubitare quis poſſet, an ramus <lb/>
<anchor type="note" xlink:label="note-0121-11a" xlink:href="note-0121-11"/>
D N, quia propinquior eſt vertici A ſit minor remotiore ramo D C, vt in pro-<lb/>poſitione 64. </s>
  <s xml:id="echoid-s3509" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3510" xml:space="preserve">65. </s>
  <s xml:id="echoid-s3511" xml:space="preserve">verificabatur; </s>
  <s xml:id="echoid-s3512" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3513" xml:space="preserve">ratio eſt, quia hypotheſes ſunt diuerſæ, <lb/>nam ibi nullus ramus breuiſecans à concurſu D ad ſectionem A C duci poſſe <lb/>ſupponebatur, in hac vero propoſitione 67. </s>
  <s xml:id="echoid-s3514" xml:space="preserve">ponitur vnicus breuiſecans D B, at <lb/>ſcrupulus omnis tolletur, ſi dicatur, non quidem ex propoſitionibus 64. </s>
  <s xml:id="echoid-s3515" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3516" xml:space="preserve">65. <lb/></s>
  <s xml:id="echoid-s3517" xml:space="preserve">ſed ex demonſtratione ibi allata, ſeu ex Corollario in fine notarum appoſito,
<pb o="84" file="0122" n="122" rhead="Apollonij Pergæi"/>
propoſitum deduci, nam duo rami D <lb/>
<anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/>
C, &amp; </s>
  <s xml:id="echoid-s3518" xml:space="preserve">D N poſiti infra ſingularem <lb/>breuiſecantem D B efficiunt cum re-<lb/>
<anchor type="note" xlink:label="note-0122-01a" xlink:href="note-0122-01"/>
ctis tangentibus ſectionẽ angulos ver-<lb/>ticem reſpicientes acutos; </s>
  <s xml:id="echoid-s3519" xml:space="preserve">igitur vt <lb/>in ſecunda parte propoſitionum 64. <lb/></s>
  <s xml:id="echoid-s3520" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3521" xml:space="preserve">65. </s>
  <s xml:id="echoid-s3522" xml:space="preserve">demonſtratum eſt, eritramus <lb/>D N vertici propinquior minor re-<lb/>motiore ramo D C.</s>
  <s xml:id="echoid-s3523" xml:space="preserve"/>
</p>
<div xml:id="echoid-div334" type="float" level="2" n="6">
<note position="left" xlink:label="note-0121-11" xlink:href="note-0121-11a" xml:space="preserve">f</note>
  <figure xlink:label="fig-0122-01" xlink:href="fig-0122-01a">
    <image file="0122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0122-01"/>
  </figure>
<note position="left" xlink:label="note-0122-01" xlink:href="note-0122-01a" xml:space="preserve">I em. 10.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3524" xml:space="preserve">Et centro D, interuallo D O <lb/>circulus deſcriptus ſecabit ſectio-<lb/>nem exempli gratia in Q (56. </s>
  <s xml:id="echoid-s3525" xml:space="preserve">ex <lb/>
<anchor type="note" xlink:label="note-0122-02a" xlink:href="note-0122-02"/>
5.) </s>
  <s xml:id="echoid-s3526" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3527" xml:space="preserve">iungamus, &amp;</s>
  <s xml:id="echoid-s3528" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3529" xml:space="preserve">Videtur om-<lb/>nino expungenda citatio in textu appoſita; </s>
  <s xml:id="echoid-s3530" xml:space="preserve">(56. </s>
  <s xml:id="echoid-s3531" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3532" xml:space="preserve">nam circulum O Q ma-<lb/>nifeſtum eſt, ſecare coniſectionem alicubi, vt in Q, cum radius D O poſitus <lb/>ſit minor D B, &amp; </s>
  <s xml:id="echoid-s3533" xml:space="preserve">maior D N; </s>
  <s xml:id="echoid-s3534" xml:space="preserve">poſtea, quia D Q propinquior eſt vertici A, <lb/>quàm D N, &amp; </s>
  <s xml:id="echoid-s3535" xml:space="preserve">omnes rami à D ad peripheriam ſectionis N Q ducti, effici-<lb/>
<anchor type="note" xlink:label="note-0122-03a" xlink:href="note-0122-03"/>
unt cum ſuis tangentibus angulos, verticem reſpicientes, acutos; </s>
  <s xml:id="echoid-s3536" xml:space="preserve">igitur D Q mi-<lb/>nor eſt, quàm D N, quod eſt abſurdum; </s>
  <s xml:id="echoid-s3537" xml:space="preserve">poſita enim fuit D O, ſeu ei æqualis <lb/>D Q, &amp; </s>
  <s xml:id="echoid-s3538" xml:space="preserve">D P maior, quàm D N.</s>
  <s xml:id="echoid-s3539" xml:space="preserve"/>
</p>
<div xml:id="echoid-div335" type="float" level="2" n="7">
<note position="right" xlink:label="note-0122-02" xlink:href="note-0122-02a" xml:space="preserve">g</note>
<note position="left" xlink:label="note-0122-03" xlink:href="note-0122-03a" xml:space="preserve">Lem. 10. <lb/>Coroll. <lb/>64. 65. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div337" type="section" level="1" n="110">
<head xml:id="echoid-head151" xml:space="preserve">COROLLARIVM PROPOSIT. <lb/>LXVII.</head>
<p style="it">
  <s xml:id="echoid-s3540" xml:space="preserve">A Ngulorum à ramis ſecantibus, qui à cõ-<lb/>
<anchor type="figure" xlink:label="fig-0122-02a" xlink:href="fig-0122-02"/>
curſu ad coniſectionem duci poſſunt, <lb/>cum tangentibus ab eorum terminis ductis cõ-<lb/>præhenſorum, ſi vnus tantnm rectus fuerit, <lb/>reliqui omnes verticem reſpicientes acuti; </s>
  <s xml:id="echoid-s3541" xml:space="preserve">ra-<lb/>mi proximiores vertici ſectionis, minores erũt <lb/>remotioribus.</s>
  <s xml:id="echoid-s3542" xml:space="preserve"/>
</p>
<div xml:id="echoid-div337" type="float" level="2" n="1">
  <figure xlink:label="fig-0122-02" xlink:href="fig-0122-02a">
    <image file="0122-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0122-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s3543" xml:space="preserve">Ex eo enim, quod in propoſitione 67. </s>
  <s xml:id="echoid-s3544" xml:space="preserve">om-<lb/>nes rami D A, D L, D C, &amp; </s>
  <s xml:id="echoid-s3545" xml:space="preserve">reliqui om-<lb/>nes, qui duci poſſunt ex concurſu D ad ſectio-<lb/>nem A B C, cum tangentibus ſectionem à ter-<lb/>minis A, L, C compræhenderunt angulos ver-<lb/>ticem reſpicientes D A V, D L G, D C P <lb/>acutos, &amp; </s>
  <s xml:id="echoid-s3546" xml:space="preserve">tantummodo vnus D B I rectus fuit <lb/>oſtenſus eſt ramus D A minor, quàm D L, &amp; </s>
  <s xml:id="echoid-s3547" xml:space="preserve">D L vertici A propinquior, mi-<lb/>nor, quàm D B, atq; </s>
  <s xml:id="echoid-s3548" xml:space="preserve">D B minor quolibet remotiore D C.</s>
  <s xml:id="echoid-s3549" xml:space="preserve"/>
</p>
<pb o="85" file="0123" n="123" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div339" type="section" level="1" n="111">
<head xml:id="echoid-head152" xml:space="preserve">Notæ in Propoſit. LXXII.</head>
<p style="it">
  <s xml:id="echoid-s3550" xml:space="preserve">ET minimus eorum D C, &amp;</s>
  <s xml:id="echoid-s3551" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s3552" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/>
Textus videtur mendoſus; </s>
  <s xml:id="echoid-s3553" xml:space="preserve">nam <lb/>
<anchor type="note" xlink:label="note-0123-01a" xlink:href="note-0123-01"/>
vt inferius oſtendetur, ramus breuiſe-<lb/>cans D C à vertice remotior, non ſem-<lb/>per minimus eſt omnium ramorum ca-<lb/>dentium ex concurſu D ad ſectionem <lb/>A B C; </s>
  <s xml:id="echoid-s3554" xml:space="preserve">itaque legendum puto; </s>
  <s xml:id="echoid-s3555" xml:space="preserve">D C <lb/>eſt minimus ramorum cadentium ad <lb/>peripheriam ſectionis B C N; </s>
  <s xml:id="echoid-s3556" xml:space="preserve">quod <lb/>manifeſtè indicatur ex determinatione <lb/>in fine propoſitionis appoſita; </s>
  <s xml:id="echoid-s3557" xml:space="preserve">inquit <lb/>enim: </s>
  <s xml:id="echoid-s3558" xml:space="preserve">propinquiores D C (ex ramis <lb/>egredientibus ad ſectionem in ea par-<lb/>te) minores ſunt remotioribus, vbi <lb/>conĳcitur, Apollonium noluiſſe pronũ-<lb/>ciare, ramum D C minimum eße omnium, qui in ſectione A C N duci poſſunt, <lb/>neque propinquiores D C minores eſſe quolibet remotiori ad partes verticis A <lb/>conſtituto, ſed tantummodo eorum, qui in ſectione C B, &amp; </s>
  <s xml:id="echoid-s3559" xml:space="preserve">in inferiori C N <lb/>ducuntur minimum eſſe D C, &amp; </s>
  <s xml:id="echoid-s3560" xml:space="preserve">ei propinquiores minores eſſe remotioribus.</s>
  <s xml:id="echoid-s3561" xml:space="preserve"/>
</p>
<div xml:id="echoid-div339" type="float" level="2" n="1">
  <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a">
    <image file="0123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0123-01"/>
  </figure>
<note position="left" xlink:label="note-0123-01" xlink:href="note-0123-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s3562" xml:space="preserve">Atque ſic patet, quod D H maior ſit, quàm D I, &amp;</s>
  <s xml:id="echoid-s3563" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3564" xml:space="preserve">Ex vndecimo <lb/>
<anchor type="note" xlink:label="note-0123-02a" xlink:href="note-0123-02"/>
enim Lemmate angulus D H M eſt acutus, &amp; </s>
  <s xml:id="echoid-s3565" xml:space="preserve">D I M obtuſus, &amp; </s>
  <s xml:id="echoid-s3566" xml:space="preserve">coniuncta <lb/>D M erunt duo quadrata D H, H M maiora quadrato D M, quæ ſubtendit <lb/>angulum acutum; </s>
  <s xml:id="echoid-s3567" xml:space="preserve">quadratum verò D M maius eſt duobus quadratis D I, I M, <lb/>ergo multo magis duo quadrata D H, H M ſimul ſumpta maiora ſunt duobus <lb/>quadratis D I, I M ſimul ſumptis, &amp; </s>
  <s xml:id="echoid-s3568" xml:space="preserve">auferatur ex aggregato maiori quadra-<lb/>tum minus H M, &amp; </s>
  <s xml:id="echoid-s3569" xml:space="preserve">ex minori tollatur quadratum maius I M (cum contin-<lb/>gens H M propinquior vertici A minor ſit remotiore M I) remanet quadratũ <lb/>
<anchor type="note" xlink:label="note-0123-03a" xlink:href="note-0123-03"/>
D H maius quadrato D I, &amp; </s>
  <s xml:id="echoid-s3570" xml:space="preserve">propterea ramus D H maior erit ramo D I, &amp; </s>
  <s xml:id="echoid-s3571" xml:space="preserve"><lb/>ſimili modo ramus D I maior oſtendetur ramo D C.</s>
  <s xml:id="echoid-s3572" xml:space="preserve"/>
</p>
<div xml:id="echoid-div340" type="float" level="2" n="2">
<note position="left" xlink:label="note-0123-02" xlink:href="note-0123-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0123-03" xlink:href="note-0123-03a" xml:space="preserve">68. 69. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3573" xml:space="preserve">Et iam demonſtratũ eſt, &amp;</s>
  <s xml:id="echoid-s3574" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3575" xml:space="preserve">Scilicet: </s>
  <s xml:id="echoid-s3576" xml:space="preserve">quia omnesrami ex D ad peripheriã <lb/>
<anchor type="note" xlink:label="note-0123-04a" xlink:href="note-0123-04"/>
<anchor type="note" xlink:label="note-0123-05a" xlink:href="note-0123-05"/>
A B ducti efficiunt cum ſuis tangentibus angulos verticem reſpicientes acutos; <lb/></s>
  <s xml:id="echoid-s3577" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3578" xml:space="preserve">propterea ramus D B maior erit quolibet alio ramo inter B, &amp; </s>
  <s xml:id="echoid-s3579" xml:space="preserve">A ducto; </s>
  <s xml:id="echoid-s3580" xml:space="preserve"><lb/>ideoque D B erit maximus cadentium in peripheria A B.</s>
  <s xml:id="echoid-s3581" xml:space="preserve"/>
</p>
<div xml:id="echoid-div341" type="float" level="2" n="3">
<note position="left" xlink:label="note-0123-04" xlink:href="note-0123-04a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0123-05" xlink:href="note-0123-05a" xml:space="preserve">Lem. 11. <lb/>Coroll. <lb/>64. 65. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3582" xml:space="preserve">Poſtea oſtendetur, quemadmodum hìc dictum eſt, &amp;</s>
  <s xml:id="echoid-s3583" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3584" xml:space="preserve">Textus eſt val-<lb/>
<anchor type="note" xlink:label="note-0123-06a" xlink:href="note-0123-06"/>
de corruptus, ſic reſtituendum puto; </s>
  <s xml:id="echoid-s3585" xml:space="preserve">Oſtendetur, quemadmodum ſupra dictum <lb/>eſt, (scilicet in ſecunda parte propoſ. </s>
  <s xml:id="echoid-s3586" xml:space="preserve">67.) </s>
  <s xml:id="echoid-s3587" xml:space="preserve">quod D C minimus ſit omnium ra-<lb/>morum ad ſectionem infimam C N cadentium, &amp; </s>
  <s xml:id="echoid-s3588" xml:space="preserve">vt hic oſtenſum eſt, ſit mi-<lb/>nimus ramorum egredientium ad ſectionem B C; </s>
  <s xml:id="echoid-s3589" xml:space="preserve">quare patet, quod D B ſit <lb/>maximus ramorum cadentium ad ſectionem A C, &amp; </s>
  <s xml:id="echoid-s3590" xml:space="preserve">D C ſit minimus caden-<lb/>tium ad ſectionem B C N, &amp; </s>
  <s xml:id="echoid-s3591" xml:space="preserve">quod propinquiores maioribus, ſunt maiores re-<lb/>motioribus in peripheria ſectionis A C, &amp; </s>
  <s xml:id="echoid-s3592" xml:space="preserve">propinquiores minoribus, ſunt mi-<lb/>nores remotioribus in peripheria ſectionis B C N, &amp; </s>
  <s xml:id="echoid-s3593" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s3594" xml:space="preserve"/>
</p>
<div xml:id="echoid-div342" type="float" level="2" n="4">
<note position="left" xlink:label="note-0123-06" xlink:href="note-0123-06a" xml:space="preserve">d</note>
</div>
<pb o="86" file="0124" n="124" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s3595" xml:space="preserve">Quod autem infimus ramus breuiſecans D C non ſit neceſſario minimus om-<lb/>nium ramorum cadentium ad peripheriam ſectionis A B, modò oſtendetur.</s>
  <s xml:id="echoid-s3596" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s3597" xml:space="preserve">In coniſectione duos ramos hreuiſecantes, ducere, quorum infimus <lb/>
<anchor type="note" xlink:label="note-0124-01a" xlink:href="note-0124-01"/>
maior ſit ramo ſecante poſito in peripheria à vertice, &amp; </s>
  <s xml:id="echoid-s3598" xml:space="preserve">ſuprema bre-<lb/>uiſecante compræhenſa: </s>
  <s xml:id="echoid-s3599" xml:space="preserve">oportet autem in ellipſi, vt rami ſecantes ad <lb/>vnum eius quadrantem ducantur à concurſu, inter axim minorem, &amp; </s>
  <s xml:id="echoid-s3600" xml:space="preserve"><lb/>verticem collocato.</s>
  <s xml:id="echoid-s3601" xml:space="preserve"/>
</p>
<div xml:id="echoid-div343" type="float" level="2" n="5">
<note position="left" xlink:label="note-0124-01" xlink:href="note-0124-01a" xml:space="preserve">PROB.6. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3602" xml:space="preserve">In coniſectione A B C, cuius ver-<lb/>
<anchor type="figure" xlink:label="fig-0124-01a" xlink:href="fig-0124-01"/>
tex A axis A D, &amp; </s>
  <s xml:id="echoid-s3603" xml:space="preserve">in hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s3604" xml:space="preserve">ellipſi centrum E ducatur quæli-<lb/>
<anchor type="note" xlink:label="note-0124-02a" xlink:href="note-0124-02"/>
bet breuiſsima F B: </s>
  <s xml:id="echoid-s3605" xml:space="preserve">poſtea ſecetur <lb/>F G ex axi, ita vt punctum G non <lb/>cadat ſupra verticem A, ſeceturque <lb/>F H non maior, quam F G, ducan-<lb/>turque rectæ H C, G G parallelæ ipſi <lb/>F B occurrentes ſectioni in C, &amp; </s>
  <s xml:id="echoid-s3606" xml:space="preserve"><lb/>G, coniungaturque recta C G ſecans <lb/>F B in I: </s>
  <s xml:id="echoid-s3607" xml:space="preserve">patet, C I maiorem non <lb/>eſſe, quàm I G; </s>
  <s xml:id="echoid-s3608" xml:space="preserve">propterea quod G C, <lb/>G H à parallelis ſecantur proportio-<lb/>naliter; </s>
  <s xml:id="echoid-s3609" xml:space="preserve">Deinde ex C ducatur alia <lb/>
<anchor type="note" xlink:label="note-0124-03a" xlink:href="note-0124-03"/>
breuiſsima C K, occurrens B F vl-<lb/>tra axim in L, iungaturque ramus <lb/>G L: </s>
  <s xml:id="echoid-s3610" xml:space="preserve">oſtendendum eſt L C maiorem <lb/>eſſe, quàm L G. </s>
  <s xml:id="echoid-s3611" xml:space="preserve">Secetur C G bifa-<lb/>riam in M, atque per M ducatur ſe-<lb/>ctionis diameter M N parallela axi <lb/>in parabola, &amp; </s>
  <s xml:id="echoid-s3612" xml:space="preserve">per centrum exſten-<lb/>ſa in reliquis ſectionibus, occurrens <lb/>ſectioni in N, ducaturque O N ſe-<lb/>ctionem contingens in N, iungantur-<lb/>
<anchor type="note" xlink:label="note-0124-04a" xlink:href="note-0124-04"/>
que L M, &amp; </s>
  <s xml:id="echoid-s3613" xml:space="preserve">L N, quæ ſecet G C in <lb/>P. </s>
  <s xml:id="echoid-s3614" xml:space="preserve">Quoniam G I æqualis, aut ma-<lb/>ior eſt, quàm I C, cadet punctum <lb/>M bipartitæ diuiſionis totius C G, <lb/>vel in I, vel inter I, G, &amp; </s>
  <s xml:id="echoid-s3615" xml:space="preserve">in vtro-<lb/>que caſu punctum N cadet inter G, <lb/>&amp; </s>
  <s xml:id="echoid-s3616" xml:space="preserve">B (eoquod diameter M N paral-<lb/>lela axi in parabola, aut ex centro <lb/>E educta in reliquis ſectionibus effi-<lb/>cit angulum N M L ad partes ver-<lb/>ticis A) &amp; </s>
  <s xml:id="echoid-s3617" xml:space="preserve">ideo ramus L N cadens <lb/>ſupra duos breuiſecantes L C, L B <lb/>ad partes verticis efficit cum tangen-
<pb o="87" file="0125" n="125" rhead="Conicor. Lib. V."/>
te O N angulum acutum L N O ver-<lb/>
<anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/>
<anchor type="note" xlink:label="note-0125-01a" xlink:href="note-0125-01"/>
ticem A reſpicientem; </s>
  <s xml:id="echoid-s3618" xml:space="preserve">eſtque G C or-<lb/>dinatim applicata ad diametrum N <lb/>
<anchor type="note" xlink:label="note-0125-02a" xlink:href="note-0125-02"/>
M parallela tangenti verticali O N; <lb/></s>
  <s xml:id="echoid-s3619" xml:space="preserve">ergo angulus L P G externus æqua-<lb/>lis erit angulo L N O interno, &amp; </s>
  <s xml:id="echoid-s3620" xml:space="preserve">op-<lb/>poſito, &amp; </s>
  <s xml:id="echoid-s3621" xml:space="preserve">ad eaſdem partes conſtitu-<lb/>to; </s>
  <s xml:id="echoid-s3622" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3623" xml:space="preserve">ideo angulus G P L acutus <lb/>quoque erit, at in triangulo P M <lb/>L angulus internus L M P, &amp; </s>
  <s xml:id="echoid-s3624" xml:space="preserve">oppo-<lb/>ſitus minor eſt externo L P G acuto; </s>
  <s xml:id="echoid-s3625" xml:space="preserve"><lb/>igitur angulus L M P acutus pariter <lb/>erit, &amp; </s>
  <s xml:id="echoid-s3626" xml:space="preserve">L M C obtuſus; </s>
  <s xml:id="echoid-s3627" xml:space="preserve">ſuntq; </s>
  <s xml:id="echoid-s3628" xml:space="preserve">intrian-<lb/>gulis L M G, &amp; </s>
  <s xml:id="echoid-s3629" xml:space="preserve">L M C circa an-<lb/>gulos inæquales, latera G M, M C <lb/>æqualia, &amp; </s>
  <s xml:id="echoid-s3630" xml:space="preserve">L M commune; </s>
  <s xml:id="echoid-s3631" xml:space="preserve">ergo L <lb/>C maior eſt, quàm L G, quod erat <lb/>faciendum.</s>
  <s xml:id="echoid-s3632" xml:space="preserve"/>
</p>
<div xml:id="echoid-div344" type="float" level="2" n="6">
  <figure xlink:label="fig-0124-01" xlink:href="fig-0124-01a">
    <image file="0124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0124-01"/>
  </figure>
<note position="left" xlink:label="note-0124-02" xlink:href="note-0124-02a" xml:space="preserve">8. 9. 10. <lb/>huius.</note>
<note position="left" xlink:label="note-0124-03" xlink:href="note-0124-03a" xml:space="preserve">8. 9. 10. <lb/>26. 27. 28. <lb/>huius.</note>
<note position="left" xlink:label="note-0124-04" xlink:href="note-0124-04a" xml:space="preserve">33. 34. <lb/>lib. 1.</note>
  <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a">
    <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0125-01"/>
  </figure>
<note position="right" xlink:label="note-0125-01" xlink:href="note-0125-01a" xml:space="preserve">Lem. 11.</note>
<note position="right" xlink:label="note-0125-02" xlink:href="note-0125-02a" xml:space="preserve">5. lib. 2.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3633" xml:space="preserve">E contra fieri poteſt, vt infimus <lb/>breuiſecans ramus L C æqualis, aut <lb/>minor ſit ramo aliquo ſupra breuiſe-<lb/>cantem reliquum B L poſito. </s>
  <s xml:id="echoid-s3634" xml:space="preserve">Nam L C minor eſt, quàm B L, &amp; </s>
  <s xml:id="echoid-s3635" xml:space="preserve">maior effici <lb/>poteſt ramo non vltra ſectionis verticem A collocato ex prima parte huius pro-<lb/>poſitionis, ſed rami à concurſu L educti cadentes inter puncta A, &amp; </s>
  <s xml:id="echoid-s3636" xml:space="preserve">B ſucceſ-<lb/>ſiuè augentur quo magis à vertice A recedunt; </s>
  <s xml:id="echoid-s3637" xml:space="preserve">Ergo ramus L C æqualis, <lb/>aut minor erit aliquo ramo ab eodem concurſu L educto inter puncta <lb/>A, &amp; </s>
  <s xml:id="echoid-s3638" xml:space="preserve">B cadente; </s>
  <s xml:id="echoid-s3639" xml:space="preserve">igitur manifeſtum eſt ramum breuiſecantem <lb/>C L infimum duorum breuiſecantium, non eſſe ſemper <lb/>minimum omnium ramorum cadentium ex concurſis<unsure/> <lb/>L ad peripheriam ſectionis A B C, ſed tan-<lb/>tummodo minorem eſſe eorum, qui inter <lb/>duo breuiſecantes B L, C L cadunt, <lb/>&amp; </s>
  <s xml:id="echoid-s3640" xml:space="preserve">reliquorum infra ramum <lb/>C L cadentium, atque <lb/>aliquorum in pe-<lb/>pheria <lb/>A N exiſtentium propè maximum L B; <lb/></s>
  <s xml:id="echoid-s3641" xml:space="preserve">quapropter exiſtimandum eſt, in-<lb/>curia alicuius verba illa non <lb/>ſine Apollonĳ iniuria <lb/>textui irrepſiſſe.</s>
  <s xml:id="echoid-s3642" xml:space="preserve"/>
</p>
  <figure>
    <image file="0125-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0125-02"/>
  </figure>
<pb o="88" file="0126" n="126" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div346" type="section" level="1" n="112">
<head xml:id="echoid-head153" xml:space="preserve">SECTIO DECIMAQVARTA</head>
<head xml:id="echoid-head154" xml:space="preserve">Continens Propoſ. LXXIII. LXXIV. LXXV. <lb/>LXXVI. &amp; LXXVII.</head>
<head xml:id="echoid-head155" xml:space="preserve">PROPOSITIO LXXIII.</head>
<p>
  <s xml:id="echoid-s3643" xml:space="preserve">SI ex concurſu E non exiſtente ſuper rectum minorem elli-<lb/>
<anchor type="note" xlink:label="note-0126-01a" xlink:href="note-0126-01"/>
pſis A B C ducatur ad ſectionem A B vnicus ramus vtrum-<lb/>que axim ſecans, cuius portio G I inter ſectionem, &amp; </s>
  <s xml:id="echoid-s3644" xml:space="preserve">axim <lb/>maiorem A C ſit breuiſſima, vel duo breuiſecantes; </s>
  <s xml:id="echoid-s3645" xml:space="preserve">vtique ra-<lb/>morum ſecantium ex illo concurſu egredientium maximus erit <lb/>breuiſecans, qui ſectionis rectum ſecat, nempe E G, &amp; </s>
  <s xml:id="echoid-s3646" xml:space="preserve">illi <lb/>proximior maior eſt remotiore; </s>
  <s xml:id="echoid-s3647" xml:space="preserve">minimus verò eorum eſt, qui <lb/>terminatur à vertice ſectionis proximiori concurſui, nempe E <lb/>C, &amp; </s>
  <s xml:id="echoid-s3648" xml:space="preserve">illi propinquiores minores ſunt remotioribus, nempe in-<lb/>ter C G. </s>
  <s xml:id="echoid-s3649" xml:space="preserve">Si autem egrediantur ex illo tres breuiſecantes, &amp; </s>
  <s xml:id="echoid-s3650" xml:space="preserve"><lb/>duo illorum ſecuerint menſuram, &amp; </s>
  <s xml:id="echoid-s3651" xml:space="preserve">vnus ſecuerit rectum, vtique <lb/>qui rectum ſecat eſt maximus ramorum ſecantium: </s>
  <s xml:id="echoid-s3652" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3653" xml:space="preserve">ramorum <lb/>inter mediam breuiſecantem, &amp; </s>
  <s xml:id="echoid-s3654" xml:space="preserve">remotiorem verticem ſectionis <lb/>à concurſu cadentium, proximior illi, eſt maior remotiore, &amp; </s>
  <s xml:id="echoid-s3655" xml:space="preserve"><lb/>maximus duorum reliquorum breuiſecantium eſt ille, qui vertici <lb/>proximus eſt, &amp; </s>
  <s xml:id="echoid-s3656" xml:space="preserve">ramorum, inter proximiorem verticem ſectio-<lb/>nis, &amp; </s>
  <s xml:id="echoid-s3657" xml:space="preserve">intermedium breuiſecantem cadentium, vicinior illi, ma-<lb/>ior eſt remotiore.</s>
  <s xml:id="echoid-s3658" xml:space="preserve"/>
</p>
<div xml:id="echoid-div346" type="float" level="2" n="1">
<note position="right" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0126-01"/>
  </figure>
<pb o="89" file="0127" n="127" rhead="Conicor. Lib. V."/>
<p>
  <s xml:id="echoid-s3659" xml:space="preserve">Erigamus itaque ſuper D perpendicularem D B occurrentem E G in, <lb/>
<anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/>
L; </s>
  <s xml:id="echoid-s3660" xml:space="preserve">ergo eſt dimidium recti, &amp; </s>
  <s xml:id="echoid-s3661" xml:space="preserve">E non eſt indirectum, quia non egredi-<lb/>tur ex E, niſi vnicus breuiſecans; </s>
  <s xml:id="echoid-s3662" xml:space="preserve">inſuper lineæ breuiſſimæ egredien-<lb/>
<anchor type="note" xlink:label="note-0127-02a" xlink:href="note-0127-02"/>
tes ab extremitatibus reliquorum ramorum abſcindunt ab axi A C cum <lb/>C, lineam maiorem, quàm ſecant rami illi. </s>
  <s xml:id="echoid-s3663" xml:space="preserve">(51. </s>
  <s xml:id="echoid-s3664" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3665" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3666" xml:space="preserve">His po-<lb/>ſitis manifeſtum eſt, quod E C F eſt acutus; </s>
  <s xml:id="echoid-s3667" xml:space="preserve">atque E C minima eſt linea-<lb/>rum egredientium ex E ad quadrantem E B, &amp; </s>
  <s xml:id="echoid-s3668" xml:space="preserve">illi propinquior, minor <lb/>eſt remotiore; </s>
  <s xml:id="echoid-s3669" xml:space="preserve">modo demonſtrandum eſt, quod E K maior quoque eſt, <lb/>
<anchor type="note" xlink:label="note-0127-03a" xlink:href="note-0127-03"/>
quàm E B, producamus itaque B M, M K tangentes, ergo M B E eſt <lb/>obtuſus, &amp; </s>
  <s xml:id="echoid-s3670" xml:space="preserve">M K E acutus (29. </s>
  <s xml:id="echoid-s3671" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3672" xml:space="preserve">quia breuiſſima egrediens ex K <lb/>abſcindit cum A minorem lineam, quàm ſecat K E (57. </s>
  <s xml:id="echoid-s3673" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3674" xml:space="preserve">eo quod <lb/>K cadit inter duas lineas L B, L G; </s>
  <s xml:id="echoid-s3675" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3676" xml:space="preserve">iungamus M E; </s>
  <s xml:id="echoid-s3677" xml:space="preserve">ergo duo qua-<lb/>drata M B, B E minora ſunt, quàm quadratum M E, quare minora, <lb/>erunt duobus quadratis M K, K E, &amp; </s>
  <s xml:id="echoid-s3678" xml:space="preserve">M B maior eſt, quàm M K, ergo <lb/>
<anchor type="note" xlink:label="note-0127-04a" xlink:href="note-0127-04"/>
B E minor eſt, quàm K E; </s>
  <s xml:id="echoid-s3679" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3680" xml:space="preserve">ſic demonſtratur, quod G E maior ſit, <lb/>quàm K E; </s>
  <s xml:id="echoid-s3681" xml:space="preserve">Nam ſi producamus G N tangentem, tunc N G E eſt re-<lb/>ctus, quia G I eſt breuiſſima, &amp; </s>
  <s xml:id="echoid-s3682" xml:space="preserve">N K E obtuſus; </s>
  <s xml:id="echoid-s3683" xml:space="preserve">ergo G E maior eſt, <lb/>
<anchor type="note" xlink:label="note-0127-05a" xlink:href="note-0127-05"/>
quàm E K; </s>
  <s xml:id="echoid-s3684" xml:space="preserve">itaque G E maximus eſt ramorum egredientium ex E ad ſe-<lb/>ctionem G C, &amp; </s>
  <s xml:id="echoid-s3685" xml:space="preserve">minimus eorum E C, atque propinquior E C minor <lb/>eſt remotiore.</s>
  <s xml:id="echoid-s3686" xml:space="preserve"/>
</p>
<div xml:id="echoid-div347" type="float" level="2" n="2">
<note position="left" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0127-03" xlink:href="note-0127-03a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0127-04" xlink:href="note-0127-04a" xml:space="preserve">70. huius.</note>
<note position="right" xlink:label="note-0127-05" xlink:href="note-0127-05a" xml:space="preserve">30. huius.</note>
</div>
<p>
  <s xml:id="echoid-s3687" xml:space="preserve">Educamus ex E ad ſectionem A G, E A, E O, oſtendetur quod <lb/>
<anchor type="note" xlink:label="note-0127-06a" xlink:href="note-0127-06"/>
E G maior ſit, quàm E O, &amp; </s>
  <s xml:id="echoid-s3688" xml:space="preserve">E O, quàm E A. </s>
  <s xml:id="echoid-s3689" xml:space="preserve">Erigamus <lb/>itaque ad A C perpendicularem A P; </s>
  <s xml:id="echoid-s3690" xml:space="preserve">ergo E A P eſt <lb/>obtuſus, &amp; </s>
  <s xml:id="echoid-s3691" xml:space="preserve">producamus P O Q tangentem; </s>
  <s xml:id="echoid-s3692" xml:space="preserve">ergo <lb/>P O E eſt acutus, quia linea breuiſſima egre-<lb/>
<anchor type="note" xlink:label="note-0127-07a" xlink:href="note-0127-07"/>
diens ex O ſecat cum A lineam maiorem; <lb/></s>
  <s xml:id="echoid-s3693" xml:space="preserve">ergo E O maior eſt, quàm E A: </s>
  <s xml:id="echoid-s3694" xml:space="preserve">atq; </s>
  <s xml:id="echoid-s3695" xml:space="preserve"><lb/>ſic patet, quod E G maior ſit, <lb/>quàm E O (29. </s>
  <s xml:id="echoid-s3696" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3697" xml:space="preserve">quia <lb/>Q G E eſt rectus, &amp; </s>
  <s xml:id="echoid-s3698" xml:space="preserve"><lb/>Q O E obtuſus, <lb/>&amp; </s>
  <s xml:id="echoid-s3699" xml:space="preserve">G Q <lb/>maior, quàm O Q, ergo E G maximus eſt ramorum <lb/>egredientium ex E ad ſectionem A B C, &amp; </s>
  <s xml:id="echoid-s3700" xml:space="preserve"><lb/>minimus eorum E C, &amp; </s>
  <s xml:id="echoid-s3701" xml:space="preserve">propinquiores <lb/>minimo, remotioribus minores ſunt, <lb/>&amp; </s>
  <s xml:id="echoid-s3702" xml:space="preserve">propinquiores maximo, ma-<lb/>iores ſunt remotioribus; </s>
  <s xml:id="echoid-s3703" xml:space="preserve"><lb/>quod erat oſtenden-<lb/>dum.</s>
  <s xml:id="echoid-s3704" xml:space="preserve"/>
</p>
<div xml:id="echoid-div348" type="float" level="2" n="3">
<note position="left" xlink:label="note-0127-06" xlink:href="note-0127-06a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0127-07" xlink:href="note-0127-07a" xml:space="preserve">57. huius.</note>
</div>
  <figure>
    <image file="0127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0127-01"/>
  </figure>
<pb o="90" file="0128" n="128" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div350" type="section" level="1" n="113">
<head xml:id="echoid-head156" xml:space="preserve">PROPOSITO LXXIV.</head>
<p>
  <s xml:id="echoid-s3705" xml:space="preserve">DEinde ſint E H, E G duo breuiſecantes, &amp; </s>
  <s xml:id="echoid-s3706" xml:space="preserve">E G ſecet <lb/>rectum B D. </s>
  <s xml:id="echoid-s3707" xml:space="preserve">Dico, quod E G eſt maximus ramorum, <lb/>egredientium ex E ad ſectioncm A B C, &amp; </s>
  <s xml:id="echoid-s3708" xml:space="preserve">E C eſt minimus.</s>
  <s xml:id="echoid-s3709" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3710" xml:space="preserve">Producatur perpendicularis E F, quæ non cadet ſuper centrum; </s>
  <s xml:id="echoid-s3711" xml:space="preserve">ſi e-<lb/>nim per centrum duceretur, duci poſſet ex E, aut vnicus breuiſecans <lb/>
<anchor type="note" xlink:label="note-0128-01a" xlink:href="note-0128-01"/>
tantum (44. </s>
  <s xml:id="echoid-s3712" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3713" xml:space="preserve">aut tres (45. </s>
  <s xml:id="echoid-s3714" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3715" xml:space="preserve">quod eſt contra hypotheſin; </s>
  <s xml:id="echoid-s3716" xml:space="preserve">er-<lb/>
<anchor type="note" xlink:label="note-0128-02a" xlink:href="note-0128-02"/>
go E F per centrum non tranſit, cadat ſuper C D; </s>
  <s xml:id="echoid-s3717" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3718" xml:space="preserve">quia ducuntur ex <lb/>E duo breuiſecantes, erit C F maior dimidio erecti, &amp; </s>
  <s xml:id="echoid-s3719" xml:space="preserve">E F æqualis Tru-<lb/>tinæ (52. </s>
  <s xml:id="echoid-s3720" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3721" xml:space="preserve">patet itaquè, vti antea demonſtrauimus, quod E G <lb/>maximus ſit ramorũ, &amp; </s>
  <s xml:id="echoid-s3722" xml:space="preserve">E C minimus; </s>
  <s xml:id="echoid-s3723" xml:space="preserve">atquè propinquior maximo, maior <lb/>eſt, &amp; </s>
  <s xml:id="echoid-s3724" xml:space="preserve">propinquior minimo, eſt minor.</s>
  <s xml:id="echoid-s3725" xml:space="preserve"/>
</p>
<div xml:id="echoid-div350" type="float" level="2" n="1">
<note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve">Ex 45. <lb/>huius.</note>
<note position="right" xlink:label="note-0128-02" xlink:href="note-0128-02a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0128-01"/>
  </figure>
</div>
<div xml:id="echoid-div352" type="section" level="1" n="114">
<head xml:id="echoid-head157" xml:space="preserve">PROPOSITO LXXV.</head>
<p>
  <s xml:id="echoid-s3726" xml:space="preserve">POſtea educamus ex E tres breuiſecantes E G, E H, E I, <lb/>
<anchor type="note" xlink:label="note-0128-03a" xlink:href="note-0128-03"/>
&amp; </s>
  <s xml:id="echoid-s3727" xml:space="preserve">ſecent E I, E H menſuram, &amp; </s>
  <s xml:id="echoid-s3728" xml:space="preserve">E G ſecet rectum in L. <lb/></s>
  <s xml:id="echoid-s3729" xml:space="preserve">Dico, quod E G eſt maximus ramorum egredientium ex E<unsure/> ad <lb/>ſectionem A B C, &amp; </s>
  <s xml:id="echoid-s3730" xml:space="preserve">ramorum inter A H cadentium propin-<lb/>quiores illi, maiores ſunt remotioribus, &amp; </s>
  <s xml:id="echoid-s3731" xml:space="preserve">E I eſt maximus ra-<lb/>morum egredientium ad ſectionem H C, &amp; </s>
  <s xml:id="echoid-s3732" xml:space="preserve">illi propinquiores <lb/>maiores ſunt remotioribus.</s>
  <s xml:id="echoid-s3733" xml:space="preserve"/>
</p>
<div xml:id="echoid-div352" type="float" level="2" n="1">
<note position="right" xlink:label="note-0128-03" xlink:href="note-0128-03a" xml:space="preserve">a</note>
</div>
<pb o="91" file="0129" n="129" rhead="Conicor. Lib. V."/>
  <figure>
    <image file="0129-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0129-01"/>
  </figure>
<p>
  <s xml:id="echoid-s3734" xml:space="preserve">Quoniam I K, H M ſunt duæ breuiſſimæ conſtat, quod E I maximus <lb/>
<anchor type="note" xlink:label="note-0129-01a" xlink:href="note-0129-01"/>
ſit ramorum cadentium ad illam ſectionem (72. </s>
  <s xml:id="echoid-s3735" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3736" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3737" xml:space="preserve">propinquior <lb/>illi maior eſt remotiore: </s>
  <s xml:id="echoid-s3738" xml:space="preserve">nec non; </s>
  <s xml:id="echoid-s3739" xml:space="preserve">quia H M, G N ſunt duæ breuiſſimæ <lb/>
<anchor type="note" xlink:label="note-0129-02a" xlink:href="note-0129-02"/>
<anchor type="note" xlink:label="note-0129-03a" xlink:href="note-0129-03"/>
conſtat, vt dictum eſt, quod G E ſit maximus ramorum cadentium vtrin-<lb/>que ad ſectionẽ A H. </s>
  <s xml:id="echoid-s3740" xml:space="preserve">Dico etiam, E G maiorem eſſe, quàm E I; </s>
  <s xml:id="echoid-s3741" xml:space="preserve">nam <lb/>
<anchor type="note" xlink:label="note-0129-04a" xlink:href="note-0129-04"/>
ſi producatur I O parallela ipſi A C, &amp; </s>
  <s xml:id="echoid-s3742" xml:space="preserve">iungatur E O, ducanturque per-<lb/>
<anchor type="note" xlink:label="note-0129-05a" xlink:href="note-0129-05"/>
pendiculares I P, O Q, G R, E F S, quia G N, I K ſunt breuiſſimæ er it <lb/>D P ad P K, quæ eſt, vt proportio figuræ, vt D R ad R N; </s>
  <s xml:id="echoid-s3743" xml:space="preserve">ergo F P <lb/>ad P K minorem proportionem habet, quàm F R ad R N, &amp; </s>
  <s xml:id="echoid-s3744" xml:space="preserve">diuidendo <lb/>F K ad K P, nempe F E ad I P, minorem proportionem habet, quàm, <lb/>F N ad N R, nempe F E ad G R: </s>
  <s xml:id="echoid-s3745" xml:space="preserve">ergo F E ad I P minorem proportio-<lb/>nem habet, quàm ad G R, &amp; </s>
  <s xml:id="echoid-s3746" xml:space="preserve">propterea G R minor eſt, quàm I P, quæ <lb/>eſt æqualis O Q, cuius punctum O remotior eſt à vertice, quàm G, <lb/>&amp; </s>
  <s xml:id="echoid-s3747" xml:space="preserve">ideo E G maior eſt, quàm E O. </s>
  <s xml:id="echoid-s3748" xml:space="preserve">(74. </s>
  <s xml:id="echoid-s3749" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3750" xml:space="preserve">Et quia O T æqualis <lb/>eſt T I erit O S maior quàm S I, &amp; </s>
  <s xml:id="echoid-s3751" xml:space="preserve">S E perpendicularis ad O I eſt com-<lb/>munis; </s>
  <s xml:id="echoid-s3752" xml:space="preserve">igitur O E maior eſt, quàm E I; </s>
  <s xml:id="echoid-s3753" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3754" xml:space="preserve">oſtenſa eſt E G maior, quàm <lb/>O E; </s>
  <s xml:id="echoid-s3755" xml:space="preserve">Ergo E G maior eſt, quàm E I, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3756" xml:space="preserve"/>
</p>
<div xml:id="echoid-div353" type="float" level="2" n="2">
<note position="left" xlink:label="note-0129-01" xlink:href="note-0129-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0129-02" xlink:href="note-0129-02a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0129-03" xlink:href="note-0129-03a" xml:space="preserve">74. huius.</note>
<note position="left" xlink:label="note-0129-04" xlink:href="note-0129-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0129-05" xlink:href="note-0129-05a" xml:space="preserve">15, huius.</note>
</div>
</div>
<div xml:id="echoid-div355" type="section" level="1" n="115">
<head xml:id="echoid-head158" xml:space="preserve">PROPOSITIO LXXVI.</head>
<p>
  <s xml:id="echoid-s3757" xml:space="preserve">SI ex concurſu E in recto E B <lb/>
<anchor type="figure" xlink:label="fig-0129-02a" xlink:href="fig-0129-02"/>
<anchor type="note" xlink:label="note-0129-06a" xlink:href="note-0129-06"/>
poſito ellipſis A B C nõ edu-<lb/>catur breuiſecans præter E B, qui <lb/>tranſeat per centrum; </s>
  <s xml:id="echoid-s3758" xml:space="preserve">erit E B ma-<lb/>ximus ramorum ſecantium ex con-<lb/>curſu ad ſectionem egredientium.</s>
  <s xml:id="echoid-s3759" xml:space="preserve"/>
</p>
<div xml:id="echoid-div355" type="float" level="2" n="1">
  <figure xlink:label="fig-0129-02" xlink:href="fig-0129-02a">
    <image file="0129-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0129-02"/>
  </figure>
<note position="left" xlink:label="note-0129-06" xlink:href="note-0129-06a" xml:space="preserve">a</note>
</div>
<pb o="92" file="0130" n="130" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s3760" xml:space="preserve">Si vero ex illo educatur alius bre-<lb/>
<anchor type="figure" xlink:label="fig-0130-01a" xlink:href="fig-0130-01"/>
uiſecans erit æqualis vni breuiſecan-<lb/>ti ex altera parte recti poſito, &amp; </s>
  <s xml:id="echoid-s3761" xml:space="preserve"><lb/>omnium reliquorum erit maximus.</s>
  <s xml:id="echoid-s3762" xml:space="preserve"/>
</p>
<div xml:id="echoid-div356" type="float" level="2" n="2">
  <figure xlink:label="fig-0130-01" xlink:href="fig-0130-01a">
    <image file="0130-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0130-01"/>
  </figure>
</div>
<note position="right" xml:space="preserve">b</note>
<p>
  <s xml:id="echoid-s3763" xml:space="preserve">Quia breuiſſimæ egredientes ab ex-<lb/>tremitatibus reliquorum ramorum ab-<lb/>ſcindunt cum C, vel A lineas maiores, <lb/>quàm ſecent rami (illi 44. </s>
  <s xml:id="echoid-s3764" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3765" xml:space="preserve">de-<lb/>monſtrabitur ductis tangentibus, per <lb/>extremitates illorum (quemadmodum, <lb/>antea oſtenſum eſt) quod E B ſit maximus ramorum egredientium ad <lb/>duos quadrantes C B, B A, &amp; </s>
  <s xml:id="echoid-s3766" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s3767" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div358" type="section" level="1" n="116">
<head xml:id="echoid-head159" xml:space="preserve">PROPOSITIO LXXVII.</head>
<p>
  <s xml:id="echoid-s3768" xml:space="preserve">POſtea educatur alius breuiſe-<lb/>
<anchor type="figure" xlink:label="fig-0130-02a" xlink:href="fig-0130-02"/>
<anchor type="note" xlink:label="note-0130-02a" xlink:href="note-0130-02"/>
cans E F; </s>
  <s xml:id="echoid-s3769" xml:space="preserve">Dico, quod eſt æ-<lb/>qualis vni breuiſecanti E G æque <lb/>remoto à recto D B, &amp; </s>
  <s xml:id="echoid-s3770" xml:space="preserve">eſt maxi-<lb/>mus reliquorum omnium.</s>
  <s xml:id="echoid-s3771" xml:space="preserve"/>
</p>
<div xml:id="echoid-div358" type="float" level="2" n="1">
  <figure xlink:label="fig-0130-02" xlink:href="fig-0130-02a">
    <image file="0130-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0130-02"/>
  </figure>
<note position="right" xlink:label="note-0130-02" xlink:href="note-0130-02a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s3772" xml:space="preserve">Quia B D, F H ſunt duæ breuiſſimæ, <lb/>
<anchor type="note" xlink:label="note-0130-03a" xlink:href="note-0130-03"/>
ergo rami egredientes ad ſectionem B <lb/>F abſcindunt cum A maiores lineas, <lb/>quàm ſecent breuiſſimæ, egredientes ab <lb/>eorum extremitatibus: </s>
  <s xml:id="echoid-s3773" xml:space="preserve">idem dicendum eſt de ramis educti ad ſectionis <lb/>peripheriam B G, &amp; </s>
  <s xml:id="echoid-s3774" xml:space="preserve">rami educti ad peripherias C G, A F abſcindunt <lb/>cum C, vel A lineas minores (45. </s>
  <s xml:id="echoid-s3775" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3776" xml:space="preserve">conſtat itaque adhibitis li-<lb/>
<anchor type="note" xlink:label="note-0130-04a" xlink:href="note-0130-04"/>
neis tangentibus, vt dictum eſt, quod E F ſit maximus ramorum ſecan-<lb/>tium ex E ad C B A egredientium, excepto vno E G, cui eſt æqualis, <lb/>quod erat oſtendendum.</s>
  <s xml:id="echoid-s3777" xml:space="preserve"/>
</p>
<div xml:id="echoid-div359" type="float" level="2" n="2">
<note position="right" xlink:label="note-0130-03" xlink:href="note-0130-03a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0130-04" xlink:href="note-0130-04a" xml:space="preserve">c</note>
</div>
</div>
<div xml:id="echoid-div361" type="section" level="1" n="117">
<head xml:id="echoid-head160" xml:space="preserve">Notæ in Propoſit. LXXIII.</head>
<p>
  <s xml:id="echoid-s3778" xml:space="preserve">PR O clariori intelligentia propoſitionum huius ſectionis hæc præmitto.</s>
  <s xml:id="echoid-s3779" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div362" type="section" level="1" n="118">
<head xml:id="echoid-head161" xml:space="preserve">LEMMA XII.</head>
<p style="it">
  <s xml:id="echoid-s3780" xml:space="preserve">Si in ellipſi A B C à concurſu E ductus fuerit ramus E G ſecans <lb/>vtrumque axim in H, &amp; </s>
  <s xml:id="echoid-s3781" xml:space="preserve">1, cuius portio G 1, inter axim maiorem <lb/>A C, &amp; </s>
  <s xml:id="echoid-s3782" xml:space="preserve">ſectionem intercepta, ſit linea breuiſsima; </s>
  <s xml:id="echoid-s3783" xml:space="preserve">dico, quod quili-<lb/>bet alius ramus E K inter breuiſecantem G E, &amp; </s>
  <s xml:id="echoid-s3784" xml:space="preserve">axim minorem in-<lb/>terceptus, efficit cum ſectionem tangente K P angulum E K P acutum,
<pb o="93" file="0131" n="131" rhead="Conicor. Lib. V."/>
reſpicientem verticem C concurſui propinquiorem: </s>
  <s xml:id="echoid-s3785" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3786" xml:space="preserve">quilibet. </s>
  <s xml:id="echoid-s3787" xml:space="preserve">ramus E <lb/>L inter breuiſecantem G E, &amp; </s>
  <s xml:id="echoid-s3788" xml:space="preserve">axim maiorem poſitus efficit cum tan-<lb/>gente L M angulum E L M reſpicientem eundem verticem A acu-<lb/>tum.</s>
  <s xml:id="echoid-s3789" xml:space="preserve"/>
</p>
  <figure>
    <image file="0131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0131-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s3790" xml:space="preserve">Ducatur E F perpendicularis ad axim maiorem, eum ſecans inter verticem <lb/>c, &amp; </s>
  <s xml:id="echoid-s3791" xml:space="preserve">centrum D in F, &amp; </s>
  <s xml:id="echoid-s3792" xml:space="preserve">ex concurſu axis minoris B H, &amp; </s>
  <s xml:id="echoid-s3793" xml:space="preserve">breuiſsimæ G E, <lb/>scilicet ex H ducantur rectæ H K, &amp; </s>
  <s xml:id="echoid-s3794" xml:space="preserve">H L; </s>
  <s xml:id="echoid-s3795" xml:space="preserve">pariterque ex punctis, K, &amp; </s>
  <s xml:id="echoid-s3796" xml:space="preserve">L <lb/>ducantur ad axim maiorem A C lineæ breuiſsimæ K N, L O, ei occurrentes in <lb/>N, &amp; </s>
  <s xml:id="echoid-s3797" xml:space="preserve">O. </s>
  <s xml:id="echoid-s3798" xml:space="preserve">Luoniam (ex præmiſſo Lemmate 8.) </s>
  <s xml:id="echoid-s3799" xml:space="preserve">à concurſu H ducitur ramus <lb/>H K inter breuiſecantes H B, H G interceptus; </s>
  <s xml:id="echoid-s3800" xml:space="preserve">ergo H K cadit infra breuiſ-<lb/>ſimam K N ad partes verticis C; </s>
  <s xml:id="echoid-s3801" xml:space="preserve">eſt vero angulus N K P rectus à tangente, <lb/>
<anchor type="note" xlink:label="note-0131-01a" xlink:href="note-0131-01"/>
&amp; </s>
  <s xml:id="echoid-s3802" xml:space="preserve">breuiſsima contentus; </s>
  <s xml:id="echoid-s3803" xml:space="preserve">ergo angulus H K P erit acutus, cum H K cadat in-<lb/>ter N K, &amp; </s>
  <s xml:id="echoid-s3804" xml:space="preserve">tangentem K P; </s>
  <s xml:id="echoid-s3805" xml:space="preserve">cadit vero E K infra ramum H K verſus C; </s>
  <s xml:id="echoid-s3806" xml:space="preserve">igi-<lb/>tur angulus E K P reſpiciens verticem C proximiorem concurſui E erit acutus.</s>
  <s xml:id="echoid-s3807" xml:space="preserve"/>
</p>
<div xml:id="echoid-div362" type="float" level="2" n="1">
<note position="right" xlink:label="note-0131-01" xlink:href="note-0131-01a" xml:space="preserve">29. 30. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3808" xml:space="preserve">Similiter (ex eodem Lemmate 8.) </s>
  <s xml:id="echoid-s3809" xml:space="preserve">quia ramus H L ducitur inter breuiſecan-<lb/>tem H G, &amp; </s>
  <s xml:id="echoid-s3810" xml:space="preserve">verticem A à concurſu E remotiorem, cadet ipſe ſupra breuiſsimã <lb/>
<anchor type="note" xlink:label="note-0131-02a" xlink:href="note-0131-02"/>
L O, eſtque angulus O L M ad partes verticis A rectus; </s>
  <s xml:id="echoid-s3811" xml:space="preserve">ergo H L M acutus erit, <lb/>cumque E L cadat ſupra H L verſus A; </s>
  <s xml:id="echoid-s3812" xml:space="preserve">igitur angulus E L M, verticem A re-<lb/>motiorem reſpiciens, erit acutus, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3813" xml:space="preserve"/>
</p>
<div xml:id="echoid-div363" type="float" level="2" n="2">
<note position="right" xlink:label="note-0131-02" xlink:href="note-0131-02a" xml:space="preserve">Ibidem.</note>
</div>
<p>
  <s xml:id="echoid-s3814" xml:space="preserve">Si à concurſu E non exiſtente ſuper recto ellipſis A C, producatur vni-<lb/>
<anchor type="note" xlink:label="note-0131-03a" xlink:href="note-0131-03"/>
cus ramus ſecans ipſam A C, vt E G, cuius ſegmentum G I, &amp; </s>
  <s xml:id="echoid-s3815" xml:space="preserve">A C ſit <lb/>breuiſsimum, vel duo breuiſecantes; </s>
  <s xml:id="echoid-s3816" xml:space="preserve">vtique maximus ſecantium ramorum <lb/>egredientium ex illo concurſu, eſt breuiſecans, qui rectum ſectionis ab-<lb/>ſcindit, nempe E G, &amp;</s>
  <s xml:id="echoid-s3817" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3818" xml:space="preserve">Textum mendoſum ſic reſtituendum cenſeo. </s>
  <s xml:id="echoid-s3819" xml:space="preserve">Si ex
<pb o="94" file="0132" n="132" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/>
concur ſu E non exiſtente ſuper axim rectum minorem ellipſis A B C ducatur ad <lb/>ſectionem A B vnicus ramus vtrumque axim ſecans, cuius portio G I inter ſe-<lb/>ctionem, &amp; </s>
  <s xml:id="echoid-s3820" xml:space="preserve">axim maiorem A C intercepta ſit linea breuiſsima; </s>
  <s xml:id="echoid-s3821" xml:space="preserve">vel ducatur præ-<lb/>ter E G alius ramus breuiſecans, menſuram tantummodo abſcindens; </s>
  <s xml:id="echoid-s3822" xml:space="preserve">vtique, <lb/>ramorum ſecantium, ex illo concurſu egredientium, maximus erit ille, qui axim <lb/>rectum ſectionis diuidit, &amp;</s>
  <s xml:id="echoid-s3823" xml:space="preserve">c.</s>
  <s xml:id="echoid-s3824" xml:space="preserve"/>
</p>
<div xml:id="echoid-div364" type="float" level="2" n="3">
<note position="left" xlink:label="note-0131-03" xlink:href="note-0131-03a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0132-01" xlink:href="fig-0132-01a">
    <image file="0132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0132-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s3825" xml:space="preserve">Erigamus itaque ſuper D perpendicularem, &amp;</s>
  <s xml:id="echoid-s3826" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3827" xml:space="preserve">Scilicet ex centro ſectio-<lb/>
<anchor type="note" xlink:label="note-0132-01a" xlink:href="note-0132-01"/>
nis D eleuetur D B perpendicularis ad axim maiorem A C, occurrens ſectioni <lb/>in B, &amp; </s>
  <s xml:id="echoid-s3828" xml:space="preserve">ipſi E G in L, &amp; </s>
  <s xml:id="echoid-s3829" xml:space="preserve">propterea D B erit ſemiſsis recti axis, &amp; </s>
  <s xml:id="echoid-s3830" xml:space="preserve">punctum <lb/>E in axi B D non exiſtit ex hypotheſi, &amp;</s>
  <s xml:id="echoid-s3831" xml:space="preserve">c.</s>
  <s xml:id="echoid-s3832" xml:space="preserve"/>
</p>
<div xml:id="echoid-div365" type="float" level="2" n="4">
<note position="right" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s3833" xml:space="preserve">Quoniam non egreditur ex E niſi vnus breuiſecans, ergo lineæ breuiſsi-<lb/>
<anchor type="note" xlink:label="note-0132-02a" xlink:href="note-0132-02"/>
mæ egredientes ab extremitatibus reliquorum ramorum, abſcindunt ab axi <lb/>cum A C, L A lineam maiorem, quàm ſecent illorum rami (51. </s>
  <s xml:id="echoid-s3834" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3835" xml:space="preserve">ex <lb/>5.) </s>
  <s xml:id="echoid-s3836" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3837" xml:space="preserve">iam patet, quod ſi ita ſe res habet L E C eſt acutus; </s>
  <s xml:id="echoid-s3838" xml:space="preserve">quia E C <lb/>breuiſsima eſt linearum egredientium ex E ad quadrantem A B, &amp; </s>
  <s xml:id="echoid-s3839" xml:space="preserve">pro-<lb/>pinquior illi, minor eſt remotiore, &amp;</s>
  <s xml:id="echoid-s3840" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3841" xml:space="preserve">Sic legendum puto; </s>
  <s xml:id="echoid-s3842" xml:space="preserve">Luia præter E <lb/>G, vtrumque axim ſecantem nullus alius breuiſecans duci poſſe à concurſu E ad <lb/>ſectionem ſupponitur, ergo lineæ breniſsimæ egredientes ab axtremitatibus reli-<lb/>quorum ramorum in quadrante C B abſcindunt ab axi A C cum vertice C li-<lb/>neas maiores, quàm ſecent rami (51 52. </s>
  <s xml:id="echoid-s3843" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3844" xml:space="preserve">pariterque conſtat, quod an-<lb/>gulus E C F ſit acutus, atque ramus E C eſt minimus egredientium ex E ad qua-<lb/>
<anchor type="note" xlink:label="note-0132-03a" xlink:href="note-0132-03"/>
drantem C B, &amp; </s>
  <s xml:id="echoid-s3845" xml:space="preserve">propinquior minimæ, minor eſt remotiore. </s>
  <s xml:id="echoid-s3846" xml:space="preserve">Demonſtrandum, <lb/>modo eſt, quod K E maior quoque eſt, quàm E B, &amp;</s>
  <s xml:id="echoid-s3847" xml:space="preserve">c.</s>
  <s xml:id="echoid-s3848" xml:space="preserve"/>
</p>
<div xml:id="echoid-div366" type="float" level="2" n="5">
<note position="right" xlink:label="note-0132-02" xlink:href="note-0132-02a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0132-03" xlink:href="note-0132-03a" xml:space="preserve">64. 65, <lb/>huius.</note>
</div>
<p>
  <s xml:id="echoid-s3849" xml:space="preserve">Producamus itaque M B, M K tangentes; </s>
  <s xml:id="echoid-s3850" xml:space="preserve">ergo M B E eſt obtuſus, &amp; </s>
  <s xml:id="echoid-s3851" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0132-04a" xlink:href="note-0132-04"/>
M K E eſt acutus (29. </s>
  <s xml:id="echoid-s3852" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3853" xml:space="preserve">quia breuiſsima egrediens ex K abſcindit A <lb/>lineam minorem, quàm A E (57. </s>
  <s xml:id="echoid-s3854" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3855" xml:space="preserve">eo quod K eſt inter duo ſegmen-<lb/>ta L B, L G: </s>
  <s xml:id="echoid-s3856" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3857" xml:space="preserve">iungamus M E; </s>
  <s xml:id="echoid-s3858" xml:space="preserve">ergo duo quadrata M B, B E minora, <lb/>ſunt, quàm quadratum M E, quæ minora ſunt duobus quadratis M K, <lb/>K E, &amp;</s>
  <s xml:id="echoid-s3859" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3860" xml:space="preserve">Ideſt: </s>
  <s xml:id="echoid-s3861" xml:space="preserve">ex punctis B, K ducantur duæ tangentes ſectionem M B, K M
<pb o="95" file="0133" n="133" rhead="Conicor. Lib. V."/>
occurrentes in M, &amp; </s>
  <s xml:id="echoid-s3862" xml:space="preserve">quia angulus D B M rectus eſt contentus ab axe, &amp; </s>
  <s xml:id="echoid-s3863" xml:space="preserve">tangen-<lb/>
<anchor type="note" xlink:label="note-0133-01a" xlink:href="note-0133-01"/>
te, &amp; </s>
  <s xml:id="echoid-s3864" xml:space="preserve">cadit B E inter C, &amp; </s>
  <s xml:id="echoid-s3865" xml:space="preserve">D ergo angulus E B M eſt obtuſus; </s>
  <s xml:id="echoid-s3866" xml:space="preserve">poſtea, quia E <lb/>K cadit infra breuiſsimam E G, &amp; </s>
  <s xml:id="echoid-s3867" xml:space="preserve">ſupra minorem axim B D, ergo angulus <lb/>
<anchor type="note" xlink:label="note-0133-02a" xlink:href="note-0133-02"/>
E K M reſpiciens verticem C propinquiorem concurſui, erit acutus, &amp; </s>
  <s xml:id="echoid-s3868" xml:space="preserve">iuncta <lb/>M E erunt duo quadrata E B, B M minora quadrato E M, eſtque quadratum <lb/>E M minus duobus quadratis E K, K M circa acutum angulum (cum prior a <lb/>angulum obtuſum compræhendant,) Igitur duo quadrata E B, B M ſimul ſum-<lb/>pta minora ſunt duobus quadratis E K, K M: </s>
  <s xml:id="echoid-s3869" xml:space="preserve">eſtque quadratum M B maius <lb/>quadrato M K, cum contingens M K, proximior vertici A axis maioris minor <lb/>
<anchor type="note" xlink:label="note-0133-03a" xlink:href="note-0133-03"/>
ſit remotiore B M; </s>
  <s xml:id="echoid-s3870" xml:space="preserve">igitur quadratum E B, ſcilicet reſiduum minoris ſummæ mi-<lb/>nus erit quadrato E K, &amp; </s>
  <s xml:id="echoid-s3871" xml:space="preserve">propterea ramus E B minor erit, quàm E K.</s>
  <s xml:id="echoid-s3872" xml:space="preserve"/>
</p>
<div xml:id="echoid-div367" type="float" level="2" n="6">
<note position="right" xlink:label="note-0132-04" xlink:href="note-0132-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">Conue ſ. <lb/>32. lib. 1.</note>
<note position="right" xlink:label="note-0133-02" xlink:href="note-0133-02a" xml:space="preserve">Lem. 12.</note>
<note position="right" xlink:label="note-0133-03" xlink:href="note-0133-03a" xml:space="preserve">70. huius.</note>
</div>
<p>
  <s xml:id="echoid-s3873" xml:space="preserve">Et educamus ex E ad ſectionem A G, E A, E O, &amp; </s>
  <s xml:id="echoid-s3874" xml:space="preserve">patebit, quod E <lb/>
<anchor type="note" xlink:label="note-0133-04a" xlink:href="note-0133-04"/>
G maior fit, quàm E O, &amp; </s>
  <s xml:id="echoid-s3875" xml:space="preserve">E O, quàm E A: </s>
  <s xml:id="echoid-s3876" xml:space="preserve">erigamus itaque ad A C <lb/>perpendicularem A P; </s>
  <s xml:id="echoid-s3877" xml:space="preserve">ergo E A P eſt obtuſus: </s>
  <s xml:id="echoid-s3878" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3879" xml:space="preserve">ducamus P O Q tan-<lb/>gentem; </s>
  <s xml:id="echoid-s3880" xml:space="preserve">ergo P O E eſt acutus, quia linea breuiſsima egrediens ex O ab-<lb/>fcindit cum A lineam maiorem, &amp; </s>
  <s xml:id="echoid-s3881" xml:space="preserve">P O eſt maior, quàm P A; </s>
  <s xml:id="echoid-s3882" xml:space="preserve">ergo E O <lb/>maior eſt quàm E A, atque ſic patet, quod E G maior ſit, quàm E O, &amp;</s>
  <s xml:id="echoid-s3883" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s3884" xml:space="preserve">Demonſtratio poſtremæ partis huius propoſitionis neglecta ab Apollonio ob ſui fa-<lb/>cilitatem occaſionem errandi alicui præbere poſſet, propter verba illa poſtrema <lb/>textui ſuperaddita; </s>
  <s xml:id="echoid-s3885" xml:space="preserve">non enim ex maiori ſumma duorum laterum P O, O E ſi au-<lb/>feratur maior O P, &amp; </s>
  <s xml:id="echoid-s3886" xml:space="preserve">ex minori ſumma P A, A E auferatur minor P A, neceſſa-<lb/>rio reſiduum maioris, ideſt E O maior erit quam E A reſiduum minoris; </s>
  <s xml:id="echoid-s3887" xml:space="preserve">itaque <lb/>ſenſus huius contextus talis erit.</s>
  <s xml:id="echoid-s3888" xml:space="preserve"/>
</p>
<div xml:id="echoid-div368" type="float" level="2" n="7">
<note position="left" xlink:label="note-0133-04" xlink:href="note-0133-04a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s3889" xml:space="preserve">Ex concurſu E ad ſectionem A G ducantur rami E A, &amp; </s>
  <s xml:id="echoid-s3890" xml:space="preserve">quilibet alius E O; <lb/></s>
  <s xml:id="echoid-s3891" xml:space="preserve">oſtendendum eſt, E G maiorem eſſe, quàm E O, &amp; </s>
  <s xml:id="echoid-s3892" xml:space="preserve">E O maiorem, quàm E A: </s>
  <s xml:id="echoid-s3893" xml:space="preserve">du-<lb/>cantur A P, Q O tangentes ſectionem in A, &amp; </s>
  <s xml:id="echoid-s3894" xml:space="preserve">O conuenientes in P, &amp; </s>
  <s xml:id="echoid-s3895" xml:space="preserve">tangenti <lb/>
<anchor type="note" xlink:label="note-0133-05a" xlink:href="note-0133-05"/>
G Q in Q. </s>
  <s xml:id="echoid-s3896" xml:space="preserve">manifectum eſt angulum E A P obtuſum eſſe, cum angulus C A P ſit <lb/>rectus pariterque quilibet ramus E O inter breuiſecantem E G, &amp; </s>
  <s xml:id="echoid-s3897" xml:space="preserve">verticem A <lb/>
<anchor type="note" xlink:label="note-0133-06a" xlink:href="note-0133-06"/>
remotiorem interceptus efficit angulum E O P, verticem A reſpicientem acutum, <lb/>&amp; </s>
  <s xml:id="echoid-s3898" xml:space="preserve">ſic reliqui omnes rami inter puncta G, &amp; </s>
  <s xml:id="echoid-s3899" xml:space="preserve">A cadentes; </s>
  <s xml:id="echoid-s3900" xml:space="preserve">quare (ex Corollario <lb/>propoſitionum 64. </s>
  <s xml:id="echoid-s3901" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3902" xml:space="preserve">65.) </s>
  <s xml:id="echoid-s3903" xml:space="preserve">ramus E A minor erit quolibet ramo E O inter verti-<lb/>cem A, &amp; </s>
  <s xml:id="echoid-s3904" xml:space="preserve">G cadente: </s>
  <s xml:id="echoid-s3905" xml:space="preserve">rurſus, quoniam breuiſecans E G conſtituit cum tangente <lb/>
<anchor type="note" xlink:label="note-0133-07a" xlink:href="note-0133-07"/>
angulũ E G Q rectum; </s>
  <s xml:id="echoid-s3906" xml:space="preserve">quare ex concurſu E ad ſectionis peripheriam G A omnes <lb/>
<anchor type="note" xlink:label="note-0133-08a" xlink:href="note-0133-08"/>
rami cadentes efficiunt cum tangentibus angulos, verticem A reſpicientes, acutos, <lb/>&amp; </s>
  <s xml:id="echoid-s3907" xml:space="preserve">vnus tantummodo E G Q eſt rectus; </s>
  <s xml:id="echoid-s3908" xml:space="preserve">igitur (ex Coroll. </s>
  <s xml:id="echoid-s3909" xml:space="preserve">propoſ. </s>
  <s xml:id="echoid-s3910" xml:space="preserve">67. </s>
  <s xml:id="echoid-s3911" xml:space="preserve">huius) ramus <lb/>E O vertici A propinquior minor eſt remotiore E G; </s>
  <s xml:id="echoid-s3912" xml:space="preserve">Quapropter ramus breuiſecãs <lb/>E G maximus eſt omnium ramorum ſecantium ad peripheriam A B C cadentium.</s>
  <s xml:id="echoid-s3913" xml:space="preserve"/>
</p>
<div xml:id="echoid-div369" type="float" level="2" n="8">
<note position="right" xlink:label="note-0133-05" xlink:href="note-0133-05a" xml:space="preserve">Conuerſ. <lb/>32. lib. 1.</note>
<note position="right" xlink:label="note-0133-06" xlink:href="note-0133-06a" xml:space="preserve">Lem. 12.</note>
<note position="right" xlink:label="note-0133-07" xlink:href="note-0133-07a" xml:space="preserve">29. 30. <lb/>huius.</note>
<note position="right" xlink:label="note-0133-08" xlink:href="note-0133-08a" xml:space="preserve">Lem. 12.</note>
</div>
<p style="it">
  <s xml:id="echoid-s3914" xml:space="preserve">At adhuc non conſtat, ramum E C minimum eſſe prædictorum ramorum om-<lb/>nium, niſi priùs oſtendatur, E C minorem eſſe quolibet ramo ad peripheriam <lb/>A G educto: </s>
  <s xml:id="echoid-s3915" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3916" xml:space="preserve">hoc etiam ob ſui facilitatem neglectum fuit ab Apollonio. </s>
  <s xml:id="echoid-s3917" xml:space="preserve">Abſol-<lb/>uetur tamen hac ratione.</s>
  <s xml:id="echoid-s3918" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s3919" xml:space="preserve">Quoniam perpendicularis E F cadit inter C, &amp; </s>
  <s xml:id="echoid-s3920" xml:space="preserve">D, igitur A F maior eſt, quàm <lb/>C F, &amp; </s>
  <s xml:id="echoid-s3921" xml:space="preserve">F E eſt communis circa angulos rectos in triangulis C F E, A F E, igi-<lb/>tur C E minor eſt, quàm E A: </s>
  <s xml:id="echoid-s3922" xml:space="preserve">eſtque E A minor quolibet alio E O inter A, &amp; </s>
  <s xml:id="echoid-s3923" xml:space="preserve">G <lb/>cadente, igitur E C minor eſt omnium ramorum cadentium ad peripheriam A G, <lb/>ſed priùs minor oſtenſus fuit reliquis omnibus cadentibus ad peripheriam C B G; <lb/></s>
  <s xml:id="echoid-s3924" xml:space="preserve">igitur ramus E C minimus eſt omnium ſecantium, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3925" xml:space="preserve"/>
</p>
<pb o="96" file="0134" n="134" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div371" type="section" level="1" n="119">
<head xml:id="echoid-head162" xml:space="preserve">Notæ in Propoſ. LXXIV.</head>
<p>
  <s xml:id="echoid-s3926" xml:space="preserve">ERgo E F per centrum non tranſit, cadat ſuper C D, &amp; </s>
  <s xml:id="echoid-s3927" xml:space="preserve">quia produ-<lb/>
<anchor type="note" xlink:label="note-0134-01a" xlink:href="note-0134-01"/>
cti ſunt ex E duo breuiſecantes; </s>
  <s xml:id="echoid-s3928" xml:space="preserve">ergo C F excedit dimidium erecti, <lb/>&amp; </s>
  <s xml:id="echoid-s3929" xml:space="preserve">E F æqualis eſt Trutinæ (52. </s>
  <s xml:id="echoid-s3930" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3931" xml:space="preserve">patet itaque, vt antea demonſtra-<lb/>uimus, quod E G ſit maximus ramorum, &amp; </s>
  <s xml:id="echoid-s3932" xml:space="preserve">E C minimus, &amp;</s>
  <s xml:id="echoid-s3933" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s3934" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/>
Quoniam in 11. </s>
  <s xml:id="echoid-s3935" xml:space="preserve">huius oſtenſum eſt, quod ſemiaxis minor ellipſis eſt ramus bre-<lb/>uiſsimus, ergo ſi incidentia perpendicularis E F ſuper axim A C, ideſt punctum <lb/>F eſt centrum ellipſis educerentur ex concurſu E tres breuiſecantes, nimirum <lb/>E H, E G, &amp; </s>
  <s xml:id="echoid-s3936" xml:space="preserve">E F producta, quæ eſſet axis minor ellipſis: </s>
  <s xml:id="echoid-s3937" xml:space="preserve">hoc autem eſt con-<lb/>tra hypotheſim, cum ducti ſint ex E duo breuiſecantes: </s>
  <s xml:id="echoid-s3938" xml:space="preserve">ergo eorum vnus E H <lb/>menſuram C F ſecat, quæ minor eſſe debet ſemiſſe axis maioris C D; </s>
  <s xml:id="echoid-s3939" xml:space="preserve">igitur <lb/>ex conuerſa propoſitione 50. </s>
  <s xml:id="echoid-s3940" xml:space="preserve">huius, menſura C F maior erit ſemiſſe lateris re-<lb/>cti, &amp; </s>
  <s xml:id="echoid-s3941" xml:space="preserve">(ex conuerſa propoſ. </s>
  <s xml:id="echoid-s3942" xml:space="preserve">52. </s>
  <s xml:id="echoid-s3943" xml:space="preserve">huius) erit perpendicularis E F æqualis Tru-<lb/>tinæ. </s>
  <s xml:id="echoid-s3944" xml:space="preserve">Demonſtratio huius propoſitionis neglecta ab Apollonio, propterea quod <lb/>eodem ferè modo, ac præcedens oſtendi poteſt, breuiſsimè perficietur in hunc <lb/>modum.</s>
  <s xml:id="echoid-s3945" xml:space="preserve"/>
</p>
<div xml:id="echoid-div371" type="float" level="2" n="1">
<note position="left" xlink:label="note-0134-01" xlink:href="note-0134-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0134-01" xlink:href="fig-0134-01a">
    <image file="0134-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0134-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s3946" xml:space="preserve">Quoniam à concurſu E vnicus tantum breuiſecans E H ad quadrantem C B <lb/>
<anchor type="note" xlink:label="note-0134-02a" xlink:href="note-0134-02"/>
ducitur; </s>
  <s xml:id="echoid-s3947" xml:space="preserve">igitur C E minimus eſt omnium ramorum cadentium ad ſectionis pe-<lb/>ripheriam C B, &amp; </s>
  <s xml:id="echoid-s3948" xml:space="preserve">E C vertici B propinquior minor eſt remotiore E H, &amp; </s>
  <s xml:id="echoid-s3949" xml:space="preserve">E <lb/>H minor, quàm E B: </s>
  <s xml:id="echoid-s3950" xml:space="preserve">rurſus, quia ramorum cadentium ex E ad peripheriam <lb/>
<anchor type="note" xlink:label="note-0134-03a" xlink:href="note-0134-03"/>
B G vnus tantummodo breuiſecans E G conſtituit cum tangente N G angulum
<pb o="97" file="0135" n="135" rhead="Conicor. Lib. V."/>
rectum, &amp; </s>
  <s xml:id="echoid-s3951" xml:space="preserve">reliqui omnes rami cadentes ſuper totum arcũ G B, conſtituunt cum <lb/>
<anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/>
ſuis tangentibus angulos acutos, reſpicientes verticem C; </s>
  <s xml:id="echoid-s3952" xml:space="preserve">igitur quilibet ramus <lb/>
<anchor type="note" xlink:label="note-0135-02a" xlink:href="note-0135-02"/>
E B propinquior vertici C minor eſt quolibet remotiore ramo E K, &amp; </s>
  <s xml:id="echoid-s3953" xml:space="preserve">E K mi-<lb/>nor eſt remotiore E G: </s>
  <s xml:id="echoid-s3954" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3955" xml:space="preserve">propterea ramus E G maximus eſt omnium cadentium <lb/>ad peripheriam C B G. </s>
  <s xml:id="echoid-s3956" xml:space="preserve">Poſtremò, quia ramorum cadentium inter breuiſecan-<lb/>tem E G, &amp; </s>
  <s xml:id="echoid-s3957" xml:space="preserve">remotiorem verticem A axis maioris, vnicus tantũ E G efficit cum <lb/>
<anchor type="note" xlink:label="note-0135-03a" xlink:href="note-0135-03"/>
ſua tangente angulum E G N rectum; </s>
  <s xml:id="echoid-s3958" xml:space="preserve">reliqui vero omnes cadentes inter G, &amp; </s>
  <s xml:id="echoid-s3959" xml:space="preserve"><lb/>A efficiunt cum ſuis tangentibus angulos, reſpicientes verticem A remotiorem, <lb/>
<anchor type="note" xlink:label="note-0135-04a" xlink:href="note-0135-04"/>
acutos; </s>
  <s xml:id="echoid-s3960" xml:space="preserve">igitur (ex Corollario propoſ. </s>
  <s xml:id="echoid-s3961" xml:space="preserve">67. </s>
  <s xml:id="echoid-s3962" xml:space="preserve">huius) ramus E G maior eſt quolibet <lb/>ramo E O vertici A propinquiore, &amp; </s>
  <s xml:id="echoid-s3963" xml:space="preserve">E O maior eſt, quàm E A: </s>
  <s xml:id="echoid-s3964" xml:space="preserve">quapropter <lb/>breuiſecans E G vtrumque axim abſcindens maximus eſt omnium ex E caden-<lb/>tium ad ſemiperipheriam ellipſis C B A, &amp; </s>
  <s xml:id="echoid-s3965" xml:space="preserve">ramus E C, vt in præcedenti dictũ <lb/>eſt, minimus erit omnium, atque propinquiores maximo ex eadem parte maio-<lb/>res erunt remotioribus, &amp; </s>
  <s xml:id="echoid-s3966" xml:space="preserve">cadentium ad peripheriam C B G minimo C E pro-<lb/>pinquiores, minores erunt remotioribus, quod erat oſtendendum.</s>
  <s xml:id="echoid-s3967" xml:space="preserve"/>
</p>
<div xml:id="echoid-div372" type="float" level="2" n="2">
<note position="left" xlink:label="note-0134-02" xlink:href="note-0134-02a" xml:space="preserve">Propoſ. <lb/>67. huius.</note>
<note position="left" xlink:label="note-0134-03" xlink:href="note-0134-03a" xml:space="preserve">Ex 29. 30. <lb/>huius.</note>
<note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">Lem. 12.</note>
<note position="right" xlink:label="note-0135-02" xlink:href="note-0135-02a" xml:space="preserve">Coroll. <lb/>prop. 67. <lb/>huius.</note>
<note position="right" xlink:label="note-0135-03" xlink:href="note-0135-03a" xml:space="preserve">29. 30. <lb/>huius.</note>
<note position="right" xlink:label="note-0135-04" xlink:href="note-0135-04a" xml:space="preserve">I em. 12. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div374" type="section" level="1" n="120">
<head xml:id="echoid-head163" xml:space="preserve">Notæ in Propoſit. LXXV.</head>
<note position="left" xml:space="preserve">a</note>
<p>
  <s xml:id="echoid-s3968" xml:space="preserve">POſtea ducamus ex E tres breuiſecantes E G, E I, E H, &amp; </s>
  <s xml:id="echoid-s3969" xml:space="preserve">ſecent E <lb/>I menſuram, &amp; </s>
  <s xml:id="echoid-s3970" xml:space="preserve">E G ſecet rectum in L, &amp;</s>
  <s xml:id="echoid-s3971" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3972" xml:space="preserve">Ideſt: </s>
  <s xml:id="echoid-s3973" xml:space="preserve">Poſtea ſi ex concur-<lb/>ſu E ducti fuerint tres breuiſecantes E G, E I, E H; </s>
  <s xml:id="echoid-s3974" xml:space="preserve">quorum duo E I, E H ſe-<lb/>cent menſuram in K, &amp; </s>
  <s xml:id="echoid-s3975" xml:space="preserve">M: </s>
  <s xml:id="echoid-s3976" xml:space="preserve">E G vero ſecet axim rectum in L, &amp; </s>
  <s xml:id="echoid-s3977" xml:space="preserve">axim ma-<lb/>iorem A C in N. </s>
  <s xml:id="echoid-s3978" xml:space="preserve">Dico, &amp;</s>
  <s xml:id="echoid-s3979" xml:space="preserve">c.</s>
  <s xml:id="echoid-s3980" xml:space="preserve"/>
</p>
  <figure>
    <image file="0135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0135-01"/>
  </figure>
<p>
  <s xml:id="echoid-s3981" xml:space="preserve">Quoniam I K, N M ſunt duæ breuiſſimæ conſtat, quod E I maximus ſit <lb/>
<anchor type="note" xlink:label="note-0135-06a" xlink:href="note-0135-06"/>
ramorum egredientium ad illius ſectionem (52. </s>
  <s xml:id="echoid-s3982" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s3983" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3984" xml:space="preserve">reliquorum ra-<lb/>morum propinquior illi, maior eſt remotiore, &amp;</s>
  <s xml:id="echoid-s3985" xml:space="preserve">c. </s>
  <s xml:id="echoid-s3986" xml:space="preserve">Ideſt: </s>
  <s xml:id="echoid-s3987" xml:space="preserve">Quia in quadran-
<pb o="98" file="0136" n="136" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/>
te ellipſis C B ducuntur à concurſu E duo breuiſecantes E I, E H; </s>
  <s xml:id="echoid-s3988" xml:space="preserve">igitur (ex <lb/>propoſitione 72. </s>
  <s xml:id="echoid-s3989" xml:space="preserve">huius) erit breuiſecans E I vertici A propinquior maximus om-<lb/>nium ramorum cadentium ex concurſu E ad ellipſis peripheriam C H; </s>
  <s xml:id="echoid-s3990" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3991" xml:space="preserve">pro-<lb/>pinquior maximo E I maior erit remotiore, ſed non omnium ramorũ cadentium <lb/>ad quadrantem C B, ſed eorum ſolummodo, qui inter verticem C, &amp; </s>
  <s xml:id="echoid-s3992" xml:space="preserve">infimum <lb/>breuiſecantem E H, &amp; </s>
  <s xml:id="echoid-s3993" xml:space="preserve">aliquorum propè ipſum; </s>
  <s xml:id="echoid-s3994" xml:space="preserve">nam rami ſecantes cadentes pro-<lb/>pè punctum H hinc inde ſucceſsiuè augentur, vt dictum eſt in notis propoſ. </s>
  <s xml:id="echoid-s3995" xml:space="preserve">67. <lb/></s>
  <s xml:id="echoid-s3996" xml:space="preserve">in eiuſque Corollario.</s>
  <s xml:id="echoid-s3997" xml:space="preserve"/>
</p>
<div xml:id="echoid-div374" type="float" level="2" n="1">
<note position="left" xlink:label="note-0135-06" xlink:href="note-0135-06a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0136-01" xlink:href="fig-0136-01a">
    <image file="0136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0136-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s3998" xml:space="preserve">Nec non, quia H M, G N ſunt duæ breuiſſimæ, conſtat, vt dictũ eſt, quod <lb/>
<anchor type="note" xlink:label="note-0136-01a" xlink:href="note-0136-01"/>
G E ſit maximus ramorũ egredientiũ ex vtroque latere eius ad A H, &amp;</s>
  <s xml:id="echoid-s3999" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s4000" xml:space="preserve">Quorũ verborũ ſenſus hic eſt. </s>
  <s xml:id="echoid-s4001" xml:space="preserve">Quiaex concurſu E ducuntur duæ breuiſecantes E G <lb/>&amp; </s>
  <s xml:id="echoid-s4002" xml:space="preserve">E H ad ſemiellipſim A B C, quarum E G ſecat vtrumq; </s>
  <s xml:id="echoid-s4003" xml:space="preserve">axim, at E H ſecat <lb/>tantummodo menſuram; </s>
  <s xml:id="echoid-s4004" xml:space="preserve">ergo, ſicuti in præcedenti propoſ. </s>
  <s xml:id="echoid-s4005" xml:space="preserve">74. </s>
  <s xml:id="echoid-s4006" xml:space="preserve">oſtenſum eſt, erit <lb/>ramus E G maximus omniũ cadentiũ ad peripheriam H A, &amp;</s>
  <s xml:id="echoid-s4007" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4008" xml:space="preserve">At quia dubitari <lb/>poſſet de certitudine huius conſequentiæ, quandoquidem hypotheſes non ſunt om-<lb/>nino eædem; </s>
  <s xml:id="echoid-s4009" xml:space="preserve">in propoſitione enim 74. </s>
  <s xml:id="echoid-s4010" xml:space="preserve">non tres, ſed duo tantummodo breuiſecan-<lb/>tes ex concurſu E ad ſectionem C B A ducebãtur, hic vero etiam tertia breui-<lb/>ſecans ducitur: </s>
  <s xml:id="echoid-s4011" xml:space="preserve">ſed ſi conſideretur progreſſus Apollonĳ, eandem concluſionem ex <lb/>vtraque hypotheſi deduci poſſe percipitur; </s>
  <s xml:id="echoid-s4012" xml:space="preserve">nam (ex propoſitione 72. </s>
  <s xml:id="echoid-s4013" xml:space="preserve">huius) bre-<lb/>uiſecans E H, infra breuiſecantem, E I poſitus, minimus eſt omnium ramorum <lb/>cadentium ex E ad peripheriam H B ellipſis, &amp; </s>
  <s xml:id="echoid-s4014" xml:space="preserve">propinquior minimo E H mi-<lb/>nor eſt remotiore, reliquorum vero ramorum cadentium ad quadrantem B A ma-<lb/>ximus eſt breuiſecans E G, vt oſtenſum eſt in præcedenti propoſit. </s>
  <s xml:id="echoid-s4015" xml:space="preserve">74. </s>
  <s xml:id="echoid-s4016" xml:space="preserve">ex Lemma-<lb/>te 12. </s>
  <s xml:id="echoid-s4017" xml:space="preserve">huius, &amp; </s>
  <s xml:id="echoid-s4018" xml:space="preserve">ex Corollario propoſit. </s>
  <s xml:id="echoid-s4019" xml:space="preserve">67, atque propinquior ramus maximo <lb/>E G eorum, qui ad quadrantem B A cadunt maior eſt remotiore; </s>
  <s xml:id="echoid-s4020" xml:space="preserve">quapropter ra-<lb/>mus E G maximus eſt omnium ramorum ex E ad ellipſis peripheriam H A ca-<lb/>dentium.</s>
  <s xml:id="echoid-s4021" xml:space="preserve"/>
</p>
<div xml:id="echoid-div375" type="float" level="2" n="2">
<note position="right" xlink:label="note-0136-01" xlink:href="note-0136-01a" xml:space="preserve">c</note>
</div>
<pb o="99" file="0137" n="137" rhead="Conicor. Lib. V."/>
<note position="left" xml:space="preserve">d</note>
<p style="it">
  <s xml:id="echoid-s4022" xml:space="preserve">Dico etiam, quod E G maior ſit, quàm E I, &amp;</s>
  <s xml:id="echoid-s4023" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4024" xml:space="preserve">Ideſt: </s>
  <s xml:id="echoid-s4025" xml:space="preserve">Oſtendetur etiam, <lb/>quod ramus E G maximus etiam ſit omnium ramorũ cadentium ad peripheriam <lb/>C H, propterea quod E G oſtendetur maior E I maximo eorum, qui ad periphe-<lb/>riam C H duci poſſunt. </s>
  <s xml:id="echoid-s4026" xml:space="preserve">Ducatur ex puncto I recta I O parallela axi matori A <lb/>C, quæ ſecabit axim minorem, &amp; </s>
  <s xml:id="echoid-s4027" xml:space="preserve">ſectionem, cum punctum I cadat inter ver-<lb/>tices C, &amp; </s>
  <s xml:id="echoid-s4028" xml:space="preserve">B duorum axium; </s>
  <s xml:id="echoid-s4029" xml:space="preserve">ſecet igitur ſectionem in O, coniungaturque E O, <lb/>atque ex punctis I, O, G, E ducantur perpendiculares ad axim I P, O Q, G <lb/>R, E F S, quæ ſecent axim in P, Q, R, F, &amp; </s>
  <s xml:id="echoid-s4030" xml:space="preserve">I O in S, &amp; </s>
  <s xml:id="echoid-s4031" xml:space="preserve">quia G N, &amp; </s>
  <s xml:id="echoid-s4032" xml:space="preserve"><lb/>I K ſunt breuiſsimæ; </s>
  <s xml:id="echoid-s4033" xml:space="preserve">ergo D R ad R N, atque D P ad P K eandem proportio-<lb/>
<anchor type="note" xlink:label="note-0137-02a" xlink:href="note-0137-02"/>
nem habent, nimirum eam, quàm habet latus tranſuerſum ad rectum; </s>
  <s xml:id="echoid-s4034" xml:space="preserve">eſt verò <lb/>K F minor, quàm D K, atque R F maior, quàm D R; </s>
  <s xml:id="echoid-s4035" xml:space="preserve">igitur F P ad P K mi-<lb/>norem proportionem habet, quàm D P ad P K, ſeu quàm D R ad R N, &amp; </s>
  <s xml:id="echoid-s4036" xml:space="preserve">mul-<lb/>to minorem, quàm F R ad R N; </s>
  <s xml:id="echoid-s4037" xml:space="preserve">quare diuidendo F K ad K P minorem pro-<lb/>portionem habebit, quàm F N ad N R, &amp; </s>
  <s xml:id="echoid-s4038" xml:space="preserve">propter parallelas F E, I P, &amp; </s>
  <s xml:id="echoid-s4039" xml:space="preserve">ſi-<lb/>militudinem triangulorum E K F, I K P eſt E F ad I P, vt F K ad K P; </s>
  <s xml:id="echoid-s4040" xml:space="preserve">igi-<lb/>tur E F ad I P minorem proportionem habet, quàm F N ad N R; </s>
  <s xml:id="echoid-s4041" xml:space="preserve">ſed propter <lb/>ſimilitudinem triangulorum E F N, G R N eſt E F ad G R, vt F N ad R N; <lb/></s>
  <s xml:id="echoid-s4042" xml:space="preserve">igitur eadem E F ad I P minorem proportionem habet, quàm ad G R; </s>
  <s xml:id="echoid-s4043" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4044" xml:space="preserve">pro-<lb/>pterea I P, ſeu ei æqualis O Q (in parallegrammo rectangulo P O) maior erit, <lb/>quàm G R, &amp; </s>
  <s xml:id="echoid-s4045" xml:space="preserve">propterea punctum O recedit à puncto G verſus B, ideoq; </s>
  <s xml:id="echoid-s4046" xml:space="preserve">ramus <lb/>
<anchor type="note" xlink:label="note-0137-03a" xlink:href="note-0137-03"/>
E G maximus, maior erit ramo E O, &amp;</s>
  <s xml:id="echoid-s4047" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4048" xml:space="preserve"/>
</p>
<div xml:id="echoid-div376" type="float" level="2" n="3">
<note position="right" xlink:label="note-0137-02" xlink:href="note-0137-02a" xml:space="preserve">15. huius.</note>
<note position="right" xlink:label="note-0137-03" xlink:href="note-0137-03a" xml:space="preserve">74. huius.</note>
</div>
</div>
<div xml:id="echoid-div378" type="section" level="1" n="121">
<head xml:id="echoid-head164" xml:space="preserve">Notæ in Propoſ. LXXVI.</head>
<p>
  <s xml:id="echoid-s4049" xml:space="preserve">SI autem non educatur ex concurſu E ad rectum E B ellipſis A B C <lb/>
<anchor type="note" xlink:label="note-0137-04a" xlink:href="note-0137-04"/>
breuiſecans præter tranſeuntem per centrum, vt E B, vtique erit ma-<lb/>ximus ramorum ſecantium egredientium ex concurſu ad ſectionem.</s>
  <s xml:id="echoid-s4050" xml:space="preserve"/>
</p>
<div xml:id="echoid-div378" type="float" level="2" n="1">
<note position="left" xlink:label="note-0137-04" xlink:href="note-0137-04a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s4051" xml:space="preserve">Si vero eductus fuerit ex illo alius <lb/>
<anchor type="figure" xlink:label="fig-0137-01a" xlink:href="fig-0137-01"/>
breuiſecans, ipſe erit ramus maximus, <lb/>&amp;</s>
  <s xml:id="echoid-s4052" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4053" xml:space="preserve">Imperceptibilis eſt ſenſus huius textus, <lb/>quia, præter phraſis Arabicæ difficultatem, <lb/>nonnulla verba in textu deſiderantnr; </s>
  <s xml:id="echoid-s4054" xml:space="preserve">itaq; <lb/></s>
  <s xml:id="echoid-s4055" xml:space="preserve">ſic legendum puto. </s>
  <s xml:id="echoid-s4056" xml:space="preserve">Si ex concurſu E in re-<lb/>cto E B poſito ellipſis A B C non educatur <lb/>breuiſecans præter E B tranſeuntem per cen-<lb/>trum, erit E B maximus ramorum ſecan-<lb/>tium ex concurſu ad ſectionem egredientiũ.</s>
  <s xml:id="echoid-s4057" xml:space="preserve"/>
</p>
<div xml:id="echoid-div379" type="float" level="2" n="2">
  <figure xlink:label="fig-0137-01" xlink:href="fig-0137-01a">
    <image file="0137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0137-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s4058" xml:space="preserve">Si vero ex illo educatur alius breuiſe-<lb/>cans, erit æqualis vni breuiſecanti ex altera parte recti poſito, &amp; </s>
  <s xml:id="echoid-s4059" xml:space="preserve">omnium re-<lb/>liquorum erit maximus: </s>
  <s xml:id="echoid-s4060" xml:space="preserve">Si enim hæc extrema verba non opponerentur, propo-<lb/>ſitio non eſſet vera, vt oſtendetur.</s>
  <s xml:id="echoid-s4061" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s4062" xml:space="preserve">Quia breuiſſimæ egredientes ab extremitatibus reliquorum ramorum <lb/>
<anchor type="note" xlink:label="note-0137-05a" xlink:href="note-0137-05"/>
abſcindunt cum A, vel B lineam maiorem, quàm ſecet ramus illius (49. <lb/></s>
  <s xml:id="echoid-s4063" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4064" xml:space="preserve">demonſtratum ergo eſt in lineis tangentibus ad extremitatem il-<lb/>lius, quemadmodum antea, &amp;</s>
  <s xml:id="echoid-s4065" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4066" xml:space="preserve">Mendoſe citatur quadrageſima nona huius, <lb/>debet potius legi 43. </s>
  <s xml:id="echoid-s4067" xml:space="preserve">in qua oſtenſum eſt, quod quotieſcunque ramus E B ad ſe-
<pb o="100" file="0138" n="138" rhead="Apollonij Pergæi"/>
miaxim minorem B D habet eandem, aut <lb/>
<anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/>
maiorem proportionem, quàm latus tran-<lb/>ſuerſum A C ad eius latus rectum; </s>
  <s xml:id="echoid-s4068" xml:space="preserve">tunc <lb/>nullus alius ramus ad ſectionem A B C <lb/>breuiſecans duci poteſt, &amp; </s>
  <s xml:id="echoid-s4069" xml:space="preserve">quælibet linea, <lb/>breuiſsima vt F H ducta ex puncto F ad <lb/>axim A C cadit infra ramum E F adpar-<lb/>tes centri, &amp; </s>
  <s xml:id="echoid-s4070" xml:space="preserve">propterea ſi per F ducatur <lb/>F I contingens ellipſin quilibet ramus E <lb/>
<anchor type="note" xlink:label="note-0138-01a" xlink:href="note-0138-01"/>
F efficiet cum tangente angulum E F I reſ-<lb/>picientem verticem A acutum: </s>
  <s xml:id="echoid-s4071" xml:space="preserve">Similiter ſi <lb/>ducatur A K contingens ſectionem in A co-<lb/>
<anchor type="note" xlink:label="note-0138-02a" xlink:href="note-0138-02"/>
niungaturque E A, erit quoque angulus E A K acutus, &amp; </s>
  <s xml:id="echoid-s4072" xml:space="preserve">ducta B L contingente <lb/>ſectionem in B erit angulus E B L rectus; </s>
  <s xml:id="echoid-s4073" xml:space="preserve">quapropter omnes rami ex concurſu <lb/>E ad quadrantem A B ducti efficiunt cum ſuis tangentibus angulos reſpicientes <lb/>verticem A acutos, &amp; </s>
  <s xml:id="echoid-s4074" xml:space="preserve">vnus tantummodo E B L eſt rectus; </s>
  <s xml:id="echoid-s4075" xml:space="preserve">igitur ramorum ca-<lb/>
<anchor type="note" xlink:label="note-0138-03a" xlink:href="note-0138-03"/>
dentium ex E ad quadrantem B A minimus eſt E A, &amp; </s>
  <s xml:id="echoid-s4076" xml:space="preserve">quilibet ramus E F <lb/>propinquior vertici A minor eſt quolibet remotiore; </s>
  <s xml:id="echoid-s4077" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4078" xml:space="preserve">propterea E B erit ma-<lb/>ximus: </s>
  <s xml:id="echoid-s4079" xml:space="preserve">ſimili modo E B maior erit quolibet ramo E G in quadrante B C exiſten-<lb/>te; </s>
  <s xml:id="echoid-s4080" xml:space="preserve">Et hic eſt ſenſus, ni fallor illorum verborum; </s>
  <s xml:id="echoid-s4081" xml:space="preserve">demonſtrabitur in lineis <lb/>tangentibus, quemadmodum antea oſtenſum eſt, &amp;</s>
  <s xml:id="echoid-s4082" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4083" xml:space="preserve"/>
</p>
<div xml:id="echoid-div380" type="float" level="2" n="3">
<note position="left" xlink:label="note-0137-05" xlink:href="note-0137-05a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a">
    <image file="0138-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0138-01"/>
  </figure>
<note position="left" xlink:label="note-0138-01" xlink:href="note-0138-01a" xml:space="preserve">ex 29. 30. <lb/>huius.</note>
<note position="left" xlink:label="note-0138-02" xlink:href="note-0138-02a" xml:space="preserve">ex 32. <lb/>lib. 1.</note>
<note position="left" xlink:label="note-0138-03" xlink:href="note-0138-03a" xml:space="preserve">Coroll. <lb/>67. huius.</note>
</div>
</div>
<div xml:id="echoid-div382" type="section" level="1" n="122">
<head xml:id="echoid-head165" xml:space="preserve">Notæ in Propoſit. LXXVII.</head>
<p style="it">
  <s xml:id="echoid-s4084" xml:space="preserve">POſtea educatur E F, qui eſt maxi-<lb/>
<anchor type="figure" xlink:label="fig-0138-02a" xlink:href="fig-0138-02"/>
musramorum, &amp;</s>
  <s xml:id="echoid-s4085" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4086" xml:space="preserve">Repono hic ſimi-<lb/>liter verba, quæ in textu deſiderantur; </s>
  <s xml:id="echoid-s4087" xml:space="preserve">Po-<lb/>ſtea educatur alius breuiſecans E F; </s>
  <s xml:id="echoid-s4088" xml:space="preserve">Dico, <lb/>quod eſt æqualis vni breuiſecanti E G æquè <lb/>remoto à recto D B, &amp; </s>
  <s xml:id="echoid-s4089" xml:space="preserve">eſt maximus reli-<lb/>quorum omnium.</s>
  <s xml:id="echoid-s4090" xml:space="preserve"/>
</p>
<div xml:id="echoid-div382" type="float" level="2" n="1">
  <figure xlink:label="fig-0138-02" xlink:href="fig-0138-02a">
    <image file="0138-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0138-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s4091" xml:space="preserve">Quia B D, F H ſunt duæ breuiſſimæ; <lb/></s>
  <s xml:id="echoid-s4092" xml:space="preserve">ergo rami egredientes ad ſectionem B F <lb/>abſcindunt cum A lineas maiores, quàm <lb/>ſecent breuiſſimæ egredientes ab eorum extremitatibus, &amp; </s>
  <s xml:id="echoid-s4093" xml:space="preserve">rami egredien-<lb/>tes ad duas peripherias C B, F A abſcindunt cum A, vel C lineas mino-<lb/>res (52. </s>
  <s xml:id="echoid-s4094" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4095" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s4096" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4097" xml:space="preserve">Quia in ellipſi ſemiaxis minor B D, &amp; </s>
  <s xml:id="echoid-s4098" xml:space="preserve">breuiſsima F H <lb/>concurrunt in E; </s>
  <s xml:id="echoid-s4099" xml:space="preserve">ergo quilibet ramus ex E ad peripheriam F B ductus cadit <lb/>
<anchor type="note" xlink:label="note-0138-04a" xlink:href="note-0138-04"/>
infra breuiſsimam ab eius termino ad axim A C ductam: </s>
  <s xml:id="echoid-s4100" xml:space="preserve">ſimiliter, quia ramus <lb/>E G æquè recedit ab axi D B, ac ramus E F; </s>
  <s xml:id="echoid-s4101" xml:space="preserve">propterea, ne dum ramus F E <lb/>æqualis erit ramo E G, ſed ſimiliter quilibet alius ramus incidens inter E B, <lb/>&amp; </s>
  <s xml:id="echoid-s4102" xml:space="preserve">E G eadet infra breuiſsimam ab eius termino ad axim A C ductam verſus <lb/>
<anchor type="note" xlink:label="note-0138-05a" xlink:href="note-0138-05"/>
D, &amp; </s>
  <s xml:id="echoid-s4103" xml:space="preserve">rami cadentes ad peripherias A F, &amp; </s>
  <s xml:id="echoid-s4104" xml:space="preserve">C G cadunt ſupra breuiſsimas ab <lb/>
<anchor type="note" xlink:label="note-0138-06a" xlink:href="note-0138-06"/>
eorum terminis ad axim C A ductas ad partes A, &amp; </s>
  <s xml:id="echoid-s4105" xml:space="preserve">C.</s>
  <s xml:id="echoid-s4106" xml:space="preserve"/>
</p>
<div xml:id="echoid-div383" type="float" level="2" n="2">
<note position="left" xlink:label="note-0138-04" xlink:href="note-0138-04a" xml:space="preserve">Lem. 8. <lb/>huius.</note>
<note position="left" xlink:label="note-0138-05" xlink:href="note-0138-05a" xml:space="preserve">Ibidem.</note>
<note position="left" xlink:label="note-0138-06" xlink:href="note-0138-06a" xml:space="preserve">Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4107" xml:space="preserve">Conſtat itaque, vt dictum eſt de lineis tangentibus, quod E F ſit ma-<lb/>ximus ramorum ſecantium egredientium ex E ad A B C, quod erat oſten-
<pb o="101" file="0139" n="139" rhead="Conicor. Lib. V."/>
dendum, &amp;</s>
  <s xml:id="echoid-s4108" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4109" xml:space="preserve">Quæ poſtrema verba ſic intelligi, ac corrigi debent. </s>
  <s xml:id="echoid-s4110" xml:space="preserve">Quia qui-<lb/>
<anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/>
libet ramus ex E ad A F ductus cadit ſupra breuiſsimam ad partes A ab eius <lb/>termino ad axim C A ductam; </s>
  <s xml:id="echoid-s4111" xml:space="preserve">igitur, vt multoties dictum eſt, conſtituit cum <lb/>ſua tangente angulum reſpicientem verticem A acutum, ſicuti angulus E A K <lb/>acutus quoque eſt, &amp; </s>
  <s xml:id="echoid-s4112" xml:space="preserve">omnium ramorum ad peripheriam A F cadentiũ tantum-<lb/>modo angulus E F 1 eſt rectus; </s>
  <s xml:id="echoid-s4113" xml:space="preserve">igitur omnium ramorum ex E ad peripheriam <lb/>
<anchor type="note" xlink:label="note-0139-02a" xlink:href="note-0139-02"/>
A F cadentium maximus eſt F E remotiſsimus à vertice A, eſtque ramus E G <lb/>æqualis E F, &amp; </s>
  <s xml:id="echoid-s4114" xml:space="preserve">E G maximus eſt ramorum cadentium ex E ad peripheriam <lb/>G C; </s>
  <s xml:id="echoid-s4115" xml:space="preserve">igitur ramus E F maximus etiam eſt ramorum cadentium ad peripheriam <lb/>G C: </s>
  <s xml:id="echoid-s4116" xml:space="preserve">poſtea ducto quolibet ramo E M inter F, B, &amp; </s>
  <s xml:id="echoid-s4117" xml:space="preserve">M N tangente ſectionem <lb/>in M, quæ conueniat cum tangente I F in N, quia E M, vt dictum eſt, cadit <lb/>infra breuiſsimam ex M ad axim B A ductam, cum qua contingens N M an-<lb/>gulum rectũ conſtituit, (ex 30. </s>
  <s xml:id="echoid-s4118" xml:space="preserve">huius) ergo angulus E M N reſpiciens verticem <lb/>A eſt obtuſus, &amp; </s>
  <s xml:id="echoid-s4119" xml:space="preserve">angulus E F N eſt rectus, cum F O ſit breuiſsima, igitur duo <lb/>quadrata E F, F N maiora ſunt duobus quadratis E M, M N ſimul ſumptis, <lb/>&amp; </s>
  <s xml:id="echoid-s4120" xml:space="preserve">ablatum quadratum M N ex minori ſumma maius eſt ablato quadrato N F, <lb/>cum contingens N F vertici A maioris axis propinquior ſit; </s>
  <s xml:id="echoid-s4121" xml:space="preserve">ergo quadratum <lb/>
<anchor type="note" xlink:label="note-0139-03a" xlink:href="note-0139-03"/>
E F maius ex quadrato E M, ideoque ramus E F maior erit quolibet ramo E <lb/>M inter F, &amp; </s>
  <s xml:id="echoid-s4122" xml:space="preserve">B poſito. </s>
  <s xml:id="echoid-s4123" xml:space="preserve">Non ſecus oſtendetur E M maior quàm E B; </s>
  <s xml:id="echoid-s4124" xml:space="preserve">quare <lb/>ramus E F maximus erit omnium cadentium ad peripheriam F B. </s>
  <s xml:id="echoid-s4125" xml:space="preserve">Eodem mo-<lb/>do ramus breuiſecans E G maximus erit omnium cadentium ad peripheriam G <lb/>B; </s>
  <s xml:id="echoid-s4126" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4127" xml:space="preserve">propterea ramus E F maximus erit omnium ad peripheriam F B G ca-<lb/>dentium; </s>
  <s xml:id="echoid-s4128" xml:space="preserve">Quapropter ramus breuiſecans E F æqualis erit vni tantummodo E <lb/>G æquè ab axi remoto, &amp; </s>
  <s xml:id="echoid-s4129" xml:space="preserve">maximus omnium ramorum ex concurſu E ad ſemi-<lb/>ellipſim A B C cadentium, quod erat oſtendendum.</s>
  <s xml:id="echoid-s4130" xml:space="preserve"/>
</p>
<div xml:id="echoid-div384" type="float" level="2" n="3">
<note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">Lem. 8. <lb/>huius.</note>
<note position="right" xlink:label="note-0139-02" xlink:href="note-0139-02a" xml:space="preserve">Coroll. <lb/>Prop. 67. <lb/>huius.</note>
<note position="right" xlink:label="note-0139-03" xlink:href="note-0139-03a" xml:space="preserve">70. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4131" xml:space="preserve">Sicuti in prioribus propoſitionibus factum eſt, reperientur, quotnam rami in-<lb/>ter ſe æquales à puncto concurſus ad coniſectionem duci poſſunt, qua occaſione <lb/>afferam propoſitiones aliquas non iniucundas, quarum prima erit.</s>
  <s xml:id="echoid-s4132" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s4133" xml:space="preserve">Si ad coniſectionem B A à concurſu D vnicus tantum breuiſecans D <lb/>
<anchor type="note" xlink:label="note-0139-04a" xlink:href="note-0139-04"/>
A duci poſsit, &amp; </s>
  <s xml:id="echoid-s4134" xml:space="preserve">ducatur quælibet F C parallela perpendiculari D E <lb/>
<anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/>
inter productionem breuiſsimæ, &amp; </s>
  <s xml:id="echoid-s4135" xml:space="preserve">axim intercepta quem ſecet in F, re-
<pb o="102" file="0140" n="140" rhead="Apollonij Pergæi"/>
periaturque Trutina K minoris, vel maioris menſuræ F B: </s>
  <s xml:id="echoid-s4136" xml:space="preserve">dico perpen-<lb/>dicularem C F minorem eſſe Trutina K.</s>
  <s xml:id="echoid-s4137" xml:space="preserve"/>
</p>
<div xml:id="echoid-div385" type="float" level="2" n="4">
<note position="right" xlink:label="note-0139-04" xlink:href="note-0139-04a" xml:space="preserve">PROP.7. <lb/>Addit.</note>
  <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a">
    <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0139-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s4138" xml:space="preserve">Secentur primo in parabola abciſsæ B H, &amp; </s>
  <s xml:id="echoid-s4139" xml:space="preserve">B N æquales trienti exceſſus inæ-<lb/>qualium menſurarum ſupra ſemierectum (vt præcipitur in propoſitione 51. </s>
  <s xml:id="echoid-s4140" xml:space="preserve">hu-<lb/>ius) manifeſtum eſt, abſciſſam B N minorem eſſe ipſa B H, quando B F minor <lb/>eſt, quàm B E, &amp; </s>
  <s xml:id="echoid-s4141" xml:space="preserve">maior, quando B F ſuperat ipſam B E; </s>
  <s xml:id="echoid-s4142" xml:space="preserve">eo quod eorum tri-<lb/>plæ, vna cum ſemierecto, ideſt menſura B F minor fuerat in primo caſu, &amp; </s>
  <s xml:id="echoid-s4143" xml:space="preserve"><lb/>maior in ſecundo, quàm menſura B E.</s>
  <s xml:id="echoid-s4144" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s4145" xml:space="preserve">In hyperbola vero, &amp; </s>
  <s xml:id="echoid-s4146" xml:space="preserve">ellipſi fiat proportio rectæ H L ad ſemiaxim tranſuer-<lb/>
<anchor type="note" xlink:label="note-0140-01a" xlink:href="note-0140-01"/>
ſum L B ſubtriplicata eius, quàm inuerſæ L E ſegmentum L G homologum la-<lb/>teri tranſuerſo habet ad ſemiaxim tranſuerſum (ex præſcripto propoſit. </s>
  <s xml:id="echoid-s4147" xml:space="preserve">52. </s>
  <s xml:id="echoid-s4148" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4149" xml:space="preserve"><lb/>53. </s>
  <s xml:id="echoid-s4150" xml:space="preserve">huius) pariterque fiat proportio N L ad L B ſubtriplicata eius quàm inuer-<lb/>ſæ minoris L F in primo caſu, &amp; </s>
  <s xml:id="echoid-s4151" xml:space="preserve">maioris in ſecundo, ſegmentum homologum <lb/>lateri tranſuerſo habet ad L B.</s>
  <s xml:id="echoid-s4152" xml:space="preserve"/>
</p>
<div xml:id="echoid-div386" type="float" level="2" n="5">
<note position="left" xlink:label="note-0140-01" xlink:href="note-0140-01a" xml:space="preserve">Lem. 7. <lb/>huius.</note>
</div>
  <figure>
    <image file="0140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0140-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s4153" xml:space="preserve">Quoniam in primo caſu maius ſegmentum G L ad eandem L B habet maio-<lb/>rem proportionem, quàm minus ſegmentum ex L F diſſectum; </s>
  <s xml:id="echoid-s4154" xml:space="preserve">igitur earum; <lb/></s>
  <s xml:id="echoid-s4155" xml:space="preserve">ſubtriplicatæ proportiones inæquales erunt, videlicet H L ad L B maiorem pro-<lb/>portionem habebit, quàm N L ad ipſam L B, &amp; </s>
  <s xml:id="echoid-s4156" xml:space="preserve">propterea H L maior erit, <lb/>quàm N L, &amp; </s>
  <s xml:id="echoid-s4157" xml:space="preserve">ablata communi L B, erit H B abſciſſa maioris menſuræ ma-<lb/>ior, quàm N B abſcißa menſuræ minoris. </s>
  <s xml:id="echoid-s4158" xml:space="preserve">Similiter oſtendetur in ſecundo ca-<lb/>ſu, quod abſciſſa N B maioris menſuræ maior eſt, quàm B H. </s>
  <s xml:id="echoid-s4159" xml:space="preserve">Oſtendedum <lb/>modo eſt, perpendicularem C F in vtroque caſu minorem eſſe trutina K; </s>
  <s xml:id="echoid-s4160" xml:space="preserve">Si <lb/>
<anchor type="note" xlink:label="note-0140-02a" xlink:href="note-0140-02"/>
enim hoc verum non eſt, ſi fieri poteſt, ſit C F maior trutina K; </s>
  <s xml:id="echoid-s4161" xml:space="preserve">igitur ex con-<lb/>curſu C ad ſectionem B A nullus ramus breuiſecans duci poteſt, quod eſt contra <lb/>hypotheſim; </s>
  <s xml:id="echoid-s4162" xml:space="preserve">erat enim A I breuiſsima; </s>
  <s xml:id="echoid-s4163" xml:space="preserve">quare C F non erit maior trutina K. <lb/></s>
  <s xml:id="echoid-s4164" xml:space="preserve">Sit ſecundo C F æqualis K, ſi fieri poteſt, ergo ramus principalis C O ductus <lb/>legibus propoſit. </s>
  <s xml:id="echoid-s4165" xml:space="preserve">51. </s>
  <s xml:id="echoid-s4166" xml:space="preserve">52. </s>
  <s xml:id="echoid-s4167" xml:space="preserve">huius cui competit trutina K erit breuiſecans ſin-<lb/>gularis eorum, qui ad ſectionem duci poſſunt, nec vllus alius, præter C O, bre-<lb/>uiſecans erit: </s>
  <s xml:id="echoid-s4168" xml:space="preserve">cadit vero ramus C A infra, vel ſupra ramum C O, propterea <lb/>quod abſciſſæ B H, &amp; </s>
  <s xml:id="echoid-s4169" xml:space="preserve">B N inæquales oſtenſæ ſunt; </s>
  <s xml:id="echoid-s4170" xml:space="preserve">igitur ramus C A diuerſus <lb/>à breuiſecante ſingulari C O non erit breuiſecans, quod eſt contra hypotheſin;</s>
  <s xml:id="echoid-s4171" xml:space="preserve">
<pb o="103" file="0141" n="141" rhead="Conicor. Lib. V."/>
non ergo perpendicularis C F æqualis erit Trutinæ K, ſed priùs, neque maior <lb/>illa erat; </s>
  <s xml:id="echoid-s4172" xml:space="preserve">igitur perpendicularis C F neceſſario minor erit Trutina K; </s>
  <s xml:id="echoid-s4173" xml:space="preserve">quod <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s4174" xml:space="preserve"/>
</p>
<div xml:id="echoid-div387" type="float" level="2" n="6">
<note position="left" xlink:label="note-0140-02" xlink:href="note-0140-02a" xml:space="preserve">51. 52. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4175" xml:space="preserve">Iiſdem poſitis, ſi in productione breuiſsimæ A I ſumatur quodlibet <lb/>
<anchor type="note" xlink:label="note-0141-01a" xlink:href="note-0141-01"/>
punctum C citra terminum D perpendicularis D E, à puncto C duci <lb/>poterit alter ramus breuiſecans ſupra C A incedens; </s>
  <s xml:id="echoid-s4176" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4177" xml:space="preserve">ſi punctum C <lb/>ſumatur vltra punctum D poterit ex C duci alter ramus breuiſecans <lb/>infra ipſum C A.</s>
  <s xml:id="echoid-s4178" xml:space="preserve"/>
</p>
<div xml:id="echoid-div388" type="float" level="2" n="7">
<note position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve">PROP. 8. <lb/>Addit.</note>
</div>
  <figure>
    <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0141-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s4179" xml:space="preserve">Quoniam quælibet recta C F parallela perpendiculari D E interpoſita inter <lb/>productionem breuiſsimæ A I, &amp; </s>
  <s xml:id="echoid-s4180" xml:space="preserve">axim minor eſt Trutina K nouæ menſuræ B <lb/>F (ex præcedenti propoſ.) </s>
  <s xml:id="echoid-s4181" xml:space="preserve">propterea ramus principalis C O cadit ſupra ipſum <lb/>C A, quando B F minor eſt, quàm B E, &amp; </s>
  <s xml:id="echoid-s4182" xml:space="preserve">tunc quidem duci poteſt hyperbola <lb/>ex puncto A circa aſymptotos (vt in propoſitione 51. </s>
  <s xml:id="echoid-s4183" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4184" xml:space="preserve">52. </s>
  <s xml:id="echoid-s4185" xml:space="preserve">factum eſt) quæ pro-<lb/>ducta occurret ſectioni B A inter B, &amp; </s>
  <s xml:id="echoid-s4186" xml:space="preserve">O, vt in P, &amp; </s>
  <s xml:id="echoid-s4187" xml:space="preserve">coniuncto radio C P, <lb/>
<anchor type="note" xlink:label="note-0141-02a" xlink:href="note-0141-02"/>
erunt duo rami C A, &amp; </s>
  <s xml:id="echoid-s4188" xml:space="preserve">C P breuiſecantes, quorum infimus eſt C A. </s>
  <s xml:id="echoid-s4189" xml:space="preserve">Si vero <lb/>punctum C ſumatur vltra punctum D, tunc quidem menſura B F maior erit, <lb/>quàm B E, &amp; </s>
  <s xml:id="echoid-s4190" xml:space="preserve">propterea abſciſſa N B maior, quàm H B, &amp; </s>
  <s xml:id="echoid-s4191" xml:space="preserve">ideo principalis <lb/>ramus C O cadet infra ramum C A; </s>
  <s xml:id="echoid-s4192" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4193" xml:space="preserve">denuo facta eadem conſtructione propo-<lb/>ſit. </s>
  <s xml:id="echoid-s4194" xml:space="preserve">51. </s>
  <s xml:id="echoid-s4195" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4196" xml:space="preserve">52. </s>
  <s xml:id="echoid-s4197" xml:space="preserve">huius, erunt duo rami C P, &amp; </s>
  <s xml:id="echoid-s4198" xml:space="preserve">C A breuiſecantes, quorũ ſupre-<lb/>mus verſus B erit C A, quod erat probandum.</s>
  <s xml:id="echoid-s4199" xml:space="preserve"/>
</p>
<div xml:id="echoid-div389" type="float" level="2" n="8">
<note position="right" xlink:label="note-0141-02" xlink:href="note-0141-02a" xml:space="preserve">51. 52. 53. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4200" xml:space="preserve">Sit coniſectio, vel ellipſis portio quadrantis B A G, cuius axis B <lb/>
<anchor type="note" xlink:label="note-0141-03a" xlink:href="note-0141-03"/>
E, perpendicularis E D, euiuſque Trutina L ſit minor perpendiculari <lb/>D E, &amp; </s>
  <s xml:id="echoid-s4201" xml:space="preserve">centro D, interuallo cuiuslibet rami ſecantis D A circulus Z <lb/>A γ deſcribatur, &amp; </s>
  <s xml:id="echoid-s4202" xml:space="preserve">ex puncto A ducatur recta A x contingens ſectio-
<pb o="104" file="0142" n="142" rhead="104 Apollonij Pergæi"/>
nem: </s>
  <s xml:id="echoid-s4203" xml:space="preserve">Dico, quod circumpherentia Z γ ſecat tangentem rectam lineam <lb/>x A, &amp; </s>
  <s xml:id="echoid-s4204" xml:space="preserve">coniſectionem B G in puncto A.</s>
  <s xml:id="echoid-s4205" xml:space="preserve"/>
</p>
<div xml:id="echoid-div390" type="float" level="2" n="9">
<note position="right" xlink:label="note-0141-03" xlink:href="note-0141-03a" xml:space="preserve">PROP. 9. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4206" xml:space="preserve">Quoniam perpendicularis D E ponitur ma-<lb/>
<anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/>
ior trutina L; </s>
  <s xml:id="echoid-s4207" xml:space="preserve">ergo quilibet ramus D A cadit <lb/>
<anchor type="note" xlink:label="note-0142-01a" xlink:href="note-0142-01"/>
ſupra breuiſsimam ex puncto A ad axim B E <lb/>ductam: </s>
  <s xml:id="echoid-s4208" xml:space="preserve">efficit vero breuiſsima cum tangente <lb/>A x angulum rectum; </s>
  <s xml:id="echoid-s4209" xml:space="preserve">ergo angulus D A x eſt <lb/>
<anchor type="note" xlink:label="note-0142-02a" xlink:href="note-0142-02"/>
acutus; </s>
  <s xml:id="echoid-s4210" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4211" xml:space="preserve">propterea recta A x cadit intracir-<lb/>culum A Z; </s>
  <s xml:id="echoid-s4212" xml:space="preserve">ſed A x cadit extra coniſectio-<lb/>
<anchor type="note" xlink:label="note-0142-03a" xlink:href="note-0142-03"/>
nem B A, quàm contingit; </s>
  <s xml:id="echoid-s4213" xml:space="preserve">ergo circumferen-<lb/>tia Z A cadit extra ſectionem B A, &amp; </s>
  <s xml:id="echoid-s4214" xml:space="preserve">extra <lb/>tangentem A x: </s>
  <s xml:id="echoid-s4215" xml:space="preserve">poſtea ducatur quilibet ramus <lb/>D G infra ramum D A ſecans circumferentiã <lb/>circuli in r: </s>
  <s xml:id="echoid-s4216" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4217" xml:space="preserve">quia ramus D A propinquior <lb/>eſt vertici B, quàm D G, erit D A minor, <lb/>
<anchor type="note" xlink:label="note-0142-04a" xlink:href="note-0142-04"/>
quàm D G; </s>
  <s xml:id="echoid-s4218" xml:space="preserve">eſtque D γ æqualis D A (cum ſint ambo radĳ eiuſdem circuli) ergo <lb/>D γ minor erit, quàm D G: </s>
  <s xml:id="echoid-s4219" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4220" xml:space="preserve">propterea quodlibet punctum γ peripheriæ cir-<lb/>cularis infra punctum A poſitum cadet intra coniſectionem B G; </s>
  <s xml:id="echoid-s4221" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4222" xml:space="preserve">ideo cir-<lb/>cumferentia Z A γ ſecat tangentẽ, &amp; </s>
  <s xml:id="echoid-s4223" xml:space="preserve">coniſectionẽ in A, quod erat propoſitum.</s>
  <s xml:id="echoid-s4224" xml:space="preserve"/>
</p>
<div xml:id="echoid-div391" type="float" level="2" n="10">
  <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a">
    <image file="0142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0142-01"/>
  </figure>
<note position="left" xlink:label="note-0142-01" xlink:href="note-0142-01a" xml:space="preserve">51. 52. <lb/>huius.</note>
<note position="left" xlink:label="note-0142-02" xlink:href="note-0142-02a" xml:space="preserve">29. 30. <lb/>huius.</note>
<note position="left" xlink:label="note-0142-03" xlink:href="note-0142-03a" xml:space="preserve">35. 36. <lb/>Lib. 1.</note>
<note position="left" xlink:label="note-0142-04" xlink:href="note-0142-04a" xml:space="preserve">64. 65. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4225" xml:space="preserve">Iſdem poſitis, ſit perpendicularis D E æqualis Trutinæ L, &amp; </s>
  <s xml:id="echoid-s4226" xml:space="preserve">ſit D <lb/>
<anchor type="note" xlink:label="note-0142-05a" xlink:href="note-0142-05"/>
A ſingularis ille ramus breuiſecans, qui ex concurſu D ad ſectionem <lb/>B G duci poteſt; </s>
  <s xml:id="echoid-s4227" xml:space="preserve">perficiaturque conſtructio, vt antea factum eſt; </s>
  <s xml:id="echoid-s4228" xml:space="preserve">Dico, <lb/>
<anchor type="note" xlink:label="note-0142-06a" xlink:href="note-0142-06"/>
circulum Z A γ ſecare coniſectionem in A, &amp; </s>
  <s xml:id="echoid-s4229" xml:space="preserve">contingere rectam Ax.</s>
  <s xml:id="echoid-s4230" xml:space="preserve"/>
</p>
<div xml:id="echoid-div392" type="float" level="2" n="11">
<note position="left" xlink:label="note-0142-05" xlink:href="note-0142-05a" xml:space="preserve">PR. 10. <lb/>Addit.</note>
<note position="left" xlink:label="note-0142-06" xlink:href="note-0142-06a" xml:space="preserve">51. 52. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4231" xml:space="preserve">Ducatur quilibet ramus D F ſupra breuiſe-<lb/>
<anchor type="figure" xlink:label="fig-0142-02a" xlink:href="fig-0142-02"/>
cantem D A, ſecans circuli peripheriam in Z, <lb/>&amp; </s>
  <s xml:id="echoid-s4232" xml:space="preserve">quilibet alius ramus D G infra D A ſecans <lb/>eandem peripheriam in γ. </s>
  <s xml:id="echoid-s4233" xml:space="preserve">Et quia ex con-<lb/>curſu D ad ſectionem B G vnicus tantum bre-<lb/>
<anchor type="note" xlink:label="note-0142-07a" xlink:href="note-0142-07"/>
uiſecans D A duci poteſt; </s>
  <s xml:id="echoid-s4234" xml:space="preserve">igitur ramus D F <lb/>propinquio<unsure/>r vertici B minor eſt remotiore D <lb/>
<anchor type="note" xlink:label="note-0142-08a" xlink:href="note-0142-08"/>
A, &amp; </s>
  <s xml:id="echoid-s4235" xml:space="preserve">D A propinquior vertici B minor eſt <lb/>remotiore D G: </s>
  <s xml:id="echoid-s4236" xml:space="preserve">ſuntque rectæ D Z, D γ æ-<lb/>quales eidem D A (cum ſint radĳ eiuſdem, <lb/>circuli) ergo D Z maior eſt, quàm D F, &amp; </s>
  <s xml:id="echoid-s4237" xml:space="preserve"><lb/>D γ minor, quàm D G; </s>
  <s xml:id="echoid-s4238" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4239" xml:space="preserve">propterea quodli-<lb/>bet punctum Z circuli ſupra A ſumptum ca-<lb/>dit extra coniſectionem B F A, &amp; </s>
  <s xml:id="echoid-s4240" xml:space="preserve">quodlibet <lb/>infimum punctum γ eiuſdem circuli cadit intra eandem coniſectionem A G; <lb/></s>
  <s xml:id="echoid-s4241" xml:space="preserve">quapropter circumferentia circuli Z A γ ſecat coniſectionem B A G in A. </s>
  <s xml:id="echoid-s4242" xml:space="preserve">Po-<lb/>ſtea quia recta A x contingens ſectionem in A perpendicularis eſt ad breuiſe-<lb/>cantem D A, cum I A ſit breuiſsima; </s>
  <s xml:id="echoid-s4243" xml:space="preserve">igitur recta linea x A, quæ perpendicu-<lb/>
<anchor type="note" xlink:label="note-0142-09a" xlink:href="note-0142-09"/>
laris eſt ad radium D A, continget circulum Z Y γ. </s>
  <s xml:id="echoid-s4244" xml:space="preserve">Quapropter circulus Z <lb/>A γ ſecant coniſectionem B A G in A, &amp; </s>
  <s xml:id="echoid-s4245" xml:space="preserve">tangit eandem rectam lineam A x, <lb/>quàm contingit ſectio conica B A G, &amp; </s>
  <s xml:id="echoid-s4246" xml:space="preserve">in eodem puncto A, quod erat oſtendendũ.</s>
  <s xml:id="echoid-s4247" xml:space="preserve"/>
</p>
<div xml:id="echoid-div393" type="float" level="2" n="12">
  <figure xlink:label="fig-0142-02" xlink:href="fig-0142-02a">
    <image file="0142-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0142-02"/>
  </figure>
<note position="left" xlink:label="note-0142-07" xlink:href="note-0142-07a" xml:space="preserve">Ibidem.</note>
<note position="left" xlink:label="note-0142-08" xlink:href="note-0142-08a" xml:space="preserve">67. huius.</note>
<note position="left" xlink:label="note-0142-09" xlink:href="note-0142-09a" xml:space="preserve">29. 30. <lb/>huius.</note>
</div>
<pb o="105" file="0143" n="143" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div395" type="section" level="1" n="123">
<head xml:id="echoid-head166" xml:space="preserve">COROLLARIVM.</head>
<p style="it">
  <s xml:id="echoid-s4248" xml:space="preserve">HInc conſtat, ſupremam circuli peripheriam A Z cadere in locum à tan-<lb/>gente X A, &amp; </s>
  <s xml:id="echoid-s4249" xml:space="preserve">coniſectionem B A contentum, infimam vero circuferen-<lb/>tiam A γ cadere ne dum infra tangentem, ſed etiam infra coniſectionem A G; <lb/></s>
  <s xml:id="echoid-s4250" xml:space="preserve">eoquod recta A X cadit extra circuli peripheriam A Z, quàm contingit in A, <lb/>&amp; </s>
  <s xml:id="echoid-s4251" xml:space="preserve">eadem circumferentia A Z cadit extra ſectionem A B, quàm ſecat in A, vt <lb/>dictum eſt.</s>
  <s xml:id="echoid-s4252" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s4253" xml:space="preserve">Mirabile quidem hoc videri poterit aliquibus, qui contingentiæ angulos, quos <lb/>vocant, verè angulos eſſe cenſent; </s>
  <s xml:id="echoid-s4254" xml:space="preserve">nam hic duæ circumſerentiæ curuæ, conica <lb/>nimirum B A G, &amp; </s>
  <s xml:id="echoid-s4255" xml:space="preserve">circularis Z A γ ſe mutuo ſecant in A, &amp; </s>
  <s xml:id="echoid-s4256" xml:space="preserve">tamen ambo <lb/>tanguntur ab eadẽ recta linea A X in eodem puncto A, in quo illæ ſe mutuò ſecant. <lb/></s>
  <s xml:id="echoid-s4257" xml:space="preserve">Vnde colligent etiam, quod anguli contingentiæ facti à coniſectione B A G, &amp; </s>
  <s xml:id="echoid-s4258" xml:space="preserve"><lb/>recta linea X A non ſunt æquales inter ſe, quando punctum A in vertice axis <lb/>non exiſtit; </s>
  <s xml:id="echoid-s4259" xml:space="preserve">nam duo anguli contingentiæ circumſerentiæ circularis, &amp; </s>
  <s xml:id="echoid-s4260" xml:space="preserve">rectæ <lb/>tangentis X A æquales ſunt inter ſe: </s>
  <s xml:id="echoid-s4261" xml:space="preserve">at angulus contingentiæ ſectionis conicæ ſu-<lb/>premus reſpiciens verticem B maior eſt angulo contingentiæ circularis, vt dictũ <lb/>eſt: </s>
  <s xml:id="echoid-s4262" xml:space="preserve">infimus vero angulus contingentiæ à ſectione conica, &amp; </s>
  <s xml:id="echoid-s4263" xml:space="preserve">eadem tangente <lb/>contentus minor eſt eodem angulo contingentiæ circularis, &amp; </s>
  <s xml:id="echoid-s4264" xml:space="preserve">propterea ſupremus <lb/>angnlus contingentiæ ſectionis conicæ maior erit inferiori.</s>
  <s xml:id="echoid-s4265" xml:space="preserve"/>
</p>
  <figure>
    <image file="0143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0143-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s4266" xml:space="preserve">Sit perpendicularis D E <lb/>
<anchor type="note" xlink:label="note-0143-01a" xlink:href="note-0143-01"/>
minor trutina L, ſintque D <lb/>A, &amp; </s>
  <s xml:id="echoid-s4267" xml:space="preserve">D C duo illi rami, <lb/>qui tantummodo breuiſecantes <lb/>eſſe poſſunt omnium ramorum <lb/>ex concurſu D ad ſectionem <lb/>B C cadentium; </s>
  <s xml:id="echoid-s4268" xml:space="preserve">atque cen-<lb/>tro D, interuallo D A deſcri-<lb/>batur circulus Z A γ; </s>
  <s xml:id="echoid-s4269" xml:space="preserve">pari-<lb/>terque centro D, interuallo D <lb/>C deſcribatur circulus O C Q; <lb/></s>
  <s xml:id="echoid-s4270" xml:space="preserve">ducanturque rectæ X P, M <lb/>P contingentes coniſectionem <lb/>in A, &amp; </s>
  <s xml:id="echoid-s4271" xml:space="preserve">C. </s>
  <s xml:id="echoid-s4272" xml:space="preserve">Dico, circulũ <lb/>Z A γ contingere coniſectio-<lb/>nem in A, &amp; </s>
  <s xml:id="echoid-s4273" xml:space="preserve">extra ipſam <lb/>cadere, at circulum O C Q contingere eandem coniſectionem in C, &amp; </s>
  <s xml:id="echoid-s4274" xml:space="preserve"><lb/>intra ipſam cadere.</s>
  <s xml:id="echoid-s4275" xml:space="preserve"/>
</p>
<div xml:id="echoid-div395" type="float" level="2" n="1">
<note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">PROP. <lb/>II. <lb/>Addit. <lb/>Ex 51. 52. <lb/>53. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4276" xml:space="preserve">Ducantur quilibet rami D F, D G ſupra, &amp; </s>
  <s xml:id="echoid-s4277" xml:space="preserve">infra breuiſecantem D A, ſe-<lb/>cantes circulum Z A γ in Z, &amp; </s>
  <s xml:id="echoid-s4278" xml:space="preserve">γ; </s>
  <s xml:id="echoid-s4279" xml:space="preserve">pariterque ducantur quilibet rami D G
<pb o="106" file="0144" n="144" rhead="Apollonij Pergæi"/>
D N ſupra, &amp; </s>
  <s xml:id="echoid-s4280" xml:space="preserve">infra breuiſe-<lb/>
<anchor type="figure" xlink:label="fig-0144-01a" xlink:href="fig-0144-01"/>
cantem D C, ſecantes circulum <lb/>O C Q, in O, &amp; </s>
  <s xml:id="echoid-s4281" xml:space="preserve">Q, dummo-<lb/>do D G non ducatur infra D C <lb/>in primo caſu, nec ſupra D A <lb/>in ſecundo. </s>
  <s xml:id="echoid-s4282" xml:space="preserve">Quoniam ramus D <lb/>A ſupremus duorum breuiſecan-<lb/>tium maximus eſt omnium ra-<lb/>morum cadentium ad periphe-<lb/>riam B A C; </s>
  <s xml:id="echoid-s4283" xml:space="preserve">igitur D A maior <lb/>
<anchor type="note" xlink:label="note-0144-01a" xlink:href="note-0144-01"/>
erit, quàm D F, &amp; </s>
  <s xml:id="echoid-s4284" xml:space="preserve">quàm D G; <lb/></s>
  <s xml:id="echoid-s4285" xml:space="preserve">ſunt verò D Z, &amp; </s>
  <s xml:id="echoid-s4286" xml:space="preserve">D γ æqua-<lb/>les eidem D A (cum ſint radĳ <lb/>eiuſdem circuli) ergo D Z ma-<lb/>ior eſt, quàm D F; </s>
  <s xml:id="echoid-s4287" xml:space="preserve">pariterque <lb/>D γ maior eſt quàm D G: </s>
  <s xml:id="echoid-s4288" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4289" xml:space="preserve"><lb/>propterea duo quælibet puncta <lb/>Z, γ eiuſdem circuli Z A γ ca-<lb/>dunt extra coniſectionem B A <lb/>G; </s>
  <s xml:id="echoid-s4290" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4291" xml:space="preserve">ideo circulus Z A γ tan-<lb/>tummodo in puncto A coniſectio-<lb/>nem extrinſecus tangit.</s>
  <s xml:id="echoid-s4292" xml:space="preserve"/>
</p>
<div xml:id="echoid-div396" type="float" level="2" n="2">
  <figure xlink:label="fig-0144-01" xlink:href="fig-0144-01a">
    <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0144-01"/>
  </figure>
<note position="left" xlink:label="note-0144-01" xlink:href="note-0144-01a" xml:space="preserve">72. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4293" xml:space="preserve">Poſtea quia ramus D C infimus breuiſecantium eſt minimus omnium ramo-<lb/>rum cadentium ex D ad peripheriam A C N, ergo ramus D C minor eſt, quàm <lb/>
<anchor type="note" xlink:label="note-0144-02a" xlink:href="note-0144-02"/>
D G, &amp; </s>
  <s xml:id="echoid-s4294" xml:space="preserve">quàm D N: </s>
  <s xml:id="echoid-s4295" xml:space="preserve">ſunt vero D O, D Q æquales eidem D C (cum ſint radĳ <lb/>eiuſdem circuli) igitur D O minor eſt, quàm D G: </s>
  <s xml:id="echoid-s4296" xml:space="preserve">pariterque D Q minor eſt, <lb/>quàm D N: </s>
  <s xml:id="echoid-s4297" xml:space="preserve">quare quælibet duo puncta O, Q circuli O C Q hinc inde à puncto <lb/>C cadunt intra coniſectionem B C N, &amp; </s>
  <s xml:id="echoid-s4298" xml:space="preserve">ideo circulus O C Q intrinſecus con-<lb/>tingit coniſectionem in C, quod erat oſtendendum.</s>
  <s xml:id="echoid-s4299" xml:space="preserve"/>
</p>
<div xml:id="echoid-div397" type="float" level="2" n="3">
<note position="left" xlink:label="note-0144-02" xlink:href="note-0144-02a" xml:space="preserve">72. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4300" xml:space="preserve">Si ad coniſectionem, <lb/>
<anchor type="note" xlink:label="note-0144-03a" xlink:href="note-0144-03"/>
<anchor type="figure" xlink:label="fig-0144-02a" xlink:href="fig-0144-02"/>
vel ad portionem qua-<lb/>drantis ellipſis B A C, <lb/>ex concurſu D duci non <lb/>poſsit, niſi vnicus tan-<lb/>tum breuiſecans D A, <lb/>atque centro D, interual-<lb/>lo D A circulus Z A γ <lb/>deſcribatur; </s>
  <s xml:id="echoid-s4301" xml:space="preserve">Dico, om-<lb/>nium circulorum tangen-<lb/>tium eandem rectam li-<lb/>neam X A P (quàm <lb/>cõtingit quoque coniſectio <lb/>in A) vnicum eſſe cir-
<pb o="107" file="0145" n="145" rhead="Conicor. Lib. V."/>
culum Z A γ, qui coniſectionem in puncto A ſecat.</s>
  <s xml:id="echoid-s4302" xml:space="preserve"/>
</p>
<div xml:id="echoid-div398" type="float" level="2" n="4">
<note position="left" xlink:label="note-0144-03" xlink:href="note-0144-03a" xml:space="preserve">PROP. <lb/>12. <lb/>Addit.</note>
  <figure xlink:label="fig-0144-02" xlink:href="fig-0144-02a">
    <image file="0144-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0144-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s4303" xml:space="preserve">Sumatur enim quodlibet punctum G in productione breuiſsimæ A I ſupra, <lb/>vel infra punctum D: </s>
  <s xml:id="echoid-s4304" xml:space="preserve">manifeſtum eſt (ex 8. </s>
  <s xml:id="echoid-s4305" xml:space="preserve">præcedentium propoſit.) </s>
  <s xml:id="echoid-s4306" xml:space="preserve">à puncto <lb/>G duci poſſe duos breuiſecantes ramos, quorum A G erit infimus, ſi punctum G <lb/>cadit ſupra punctum D, &amp; </s>
  <s xml:id="echoid-s4307" xml:space="preserve">tunc circulus radio G A deſcriptus continget coniſe-<lb/>
<anchor type="note" xlink:label="note-0145-01a" xlink:href="note-0145-01"/>
ctionem intrinſecus in A: </s>
  <s xml:id="echoid-s4308" xml:space="preserve">ſi vero punctum g cadat infra punctum D, tunc pa-<lb/>
<anchor type="note" xlink:label="note-0145-02a" xlink:href="note-0145-02"/>
riter ex g duo breuiſecantes duci poſſunt ad ſectionem, quorum ſupremus erit <lb/>g A; </s>
  <s xml:id="echoid-s4309" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4310" xml:space="preserve">propterea circulus radio g A deſcriptus continget coniſectionem B AC <lb/>
<anchor type="note" xlink:label="note-0145-03a" xlink:href="note-0145-03"/>
extrinſecus in A; </s>
  <s xml:id="echoid-s4311" xml:space="preserve">quaproptcr circulus radio D A deſcriptus (quem contingit <lb/>eadem recta linea X A quæ tangebat ſectionem in A) vnicus erit, qui ſectionem <lb/>B C ſecet in A, quod erat oſtendendum.</s>
  <s xml:id="echoid-s4312" xml:space="preserve"/>
</p>
<div xml:id="echoid-div399" type="float" level="2" n="5">
<note position="right" xlink:label="note-0145-01" xlink:href="note-0145-01a" xml:space="preserve">11. <lb/>Additarũ.</note>
<note position="right" xlink:label="note-0145-02" xlink:href="note-0145-02a" xml:space="preserve">8. <lb/>Additarũ.</note>
<note position="right" xlink:label="note-0145-03" xlink:href="note-0145-03a" xml:space="preserve">11. <lb/>Additarũ.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4313" xml:space="preserve">Circulorum omnium intrinſecus tangentium coniſectionem non in axis <lb/>
<anchor type="note" xlink:label="note-0145-04a" xlink:href="note-0145-04"/>
vertice, aſsignari non poteſt maximus: </s>
  <s xml:id="echoid-s4314" xml:space="preserve">tangentium vero intrinſecus ſe-<lb/>ctionem in termino axis maximus erit, cuius radius æqualis eſt ſemie-<lb/>recto.</s>
  <s xml:id="echoid-s4315" xml:space="preserve"/>
</p>
<div xml:id="echoid-div400" type="float" level="2" n="6">
<note position="right" xlink:label="note-0145-04" xlink:href="note-0145-04a" xml:space="preserve">PROP. <lb/>13. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4316" xml:space="preserve">Repetatur figura, &amp; </s>
  <s xml:id="echoid-s4317" xml:space="preserve"><lb/>
<anchor type="figure" xlink:label="fig-0145-01a" xlink:href="fig-0145-01"/>
hypotheſis præcedẽtis pro <lb/>poſitionis. </s>
  <s xml:id="echoid-s4318" xml:space="preserve">Quoniã qui-<lb/>libet circulus radio G A <lb/>minori, quàm D A de-<lb/>ſcriptus ſemper intrin-<lb/>ſecus tangit coniſectio-<lb/>nem in A (vt in præce-<lb/>dẽti propoſitione dictum <lb/>eſt) vbicumque ponatur <lb/>centrum G ſupra punctũ <lb/>D; </s>
  <s xml:id="echoid-s4319" xml:space="preserve">neque augendo ra-<lb/>dium G A efſicitur alius <lb/>contactus circuli, &amp; </s>
  <s xml:id="echoid-s4320" xml:space="preserve">ſe-<lb/>ctionis, quàm intrinſe-<lb/>cus, &amp; </s>
  <s xml:id="echoid-s4321" xml:space="preserve">tunc primo cir-<lb/>culus deſinit intrinſecus <lb/>tangere ſectionem in A, <lb/>quando D A efſicitur <lb/>radius, ſcilicet quando <lb/>non amplius intrinſecus ſectionem tangit, ſed eam ſecat in A; </s>
  <s xml:id="echoid-s4322" xml:space="preserve">quapropter aſsi-<lb/>gnari non poteſt maximus circulorum tangentium intrinſecus ſectionem in A. <lb/></s>
  <s xml:id="echoid-s4323" xml:space="preserve">Quod verò circulorum intrinſecus tangentium eandem ſectionem in vertice axis <lb/>B, ille, cuius radius B K æqualis eſt ſemierecto B H ſit maximus, oſtenſum eſt <lb/>à Maurolico propoſ: </s>
  <s xml:id="echoid-s4324" xml:space="preserve">5. </s>
  <s xml:id="echoid-s4325" xml:space="preserve">8. </s>
  <s xml:id="echoid-s4326" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4327" xml:space="preserve">11. </s>
  <s xml:id="echoid-s4328" xml:space="preserve">libri 5. </s>
  <s xml:id="echoid-s4329" xml:space="preserve">Conicorum. </s>
  <s xml:id="echoid-s4330" xml:space="preserve">Patet ergo propoſitum.</s>
  <s xml:id="echoid-s4331" xml:space="preserve"/>
</p>
<div xml:id="echoid-div401" type="float" level="2" n="7">
  <figure xlink:label="fig-0145-01" xlink:href="fig-0145-01a">
    <image file="0145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0145-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s4332" xml:space="preserve">Iiſdem poſitis: </s>
  <s xml:id="echoid-s4333" xml:space="preserve">dico circulorum omnium extrinſecus tangentium coni-<lb/>
<anchor type="note" xlink:label="note-0145-05a" xlink:href="note-0145-05"/>
ſectionem minimum aſsignari non poſſe.</s>
  <s xml:id="echoid-s4334" xml:space="preserve"/>
</p>
<div xml:id="echoid-div402" type="float" level="2" n="8">
<note position="right" xlink:label="note-0145-05" xlink:href="note-0145-05a" xml:space="preserve">PROP. <lb/>14. <lb/>Addit.</note>
</div>
<pb o="108" file="0146" n="146" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s4335" xml:space="preserve">Sumpto<unsure/> in eadem ſi-<lb/>
<anchor type="figure" xlink:label="fig-0146-01a" xlink:href="fig-0146-01"/>
gura quolibet puncto g <lb/>
<anchor type="note" xlink:label="note-0146-01a" xlink:href="note-0146-01"/>
infra punctum D, quo-<lb/>niam circulus radio g A <lb/>deſcriptus contingit ex-<lb/>trinſecus coniſectionem <lb/>in A, nec vnquam ceſ-<lb/>ſabit prædictus cõtactus <lb/>extrinſecus, licet magis, <lb/>ac magis in infinitum, <lb/>punctum g ipſi D pro-<lb/>pinquior fiat, &amp; </s>
  <s xml:id="echoid-s4336" xml:space="preserve">tunc de-<lb/>mu<unsure/>m ceſſat huiuſmodi <lb/>extrinſecus contactus, <lb/>quando deſcribitur cir-<lb/>culus radio D A, qui <lb/>quidem ſectionem ſecat <lb/>in A, vt dictũ eſt; </s>
  <s xml:id="echoid-s4337" xml:space="preserve">qua-<lb/>propter minimus omniũ <lb/>extrinſecus ſectionem, <lb/>tangentium in A aſsignari nequit. </s>
  <s xml:id="echoid-s4338" xml:space="preserve">Quodvero extrinſecus tangentium eandem <lb/>ſectionem in vertice axis B non poſsit aſsignari minimus, patet; </s>
  <s xml:id="echoid-s4339" xml:space="preserve">nam omnes <lb/>circuli, quorum radĳ maiores ſunt ſemierecto ſectionis, eam extrinſecus tan-<lb/>
<anchor type="note" xlink:label="note-0146-02a" xlink:href="note-0146-02"/>
gunt; </s>
  <s xml:id="echoid-s4340" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4341" xml:space="preserve">tunc demum eiuſmodi contactus extrinſecus ceſſat, quando radius cir-<lb/>culi æqualis efſicitur ſemierecto: </s>
  <s xml:id="echoid-s4342" xml:space="preserve">at tunc intrinſecus ſectionem tangit; </s>
  <s xml:id="echoid-s4343" xml:space="preserve">quapro-<lb/>pter reperiri non poteſt minimus circulorum coniſectionem extrinſecus tangenti-<lb/>um: </s>
  <s xml:id="echoid-s4344" xml:space="preserve">quod erat oſtendendum.</s>
  <s xml:id="echoid-s4345" xml:space="preserve"/>
</p>
<div xml:id="echoid-div403" type="float" level="2" n="9">
  <figure xlink:label="fig-0146-01" xlink:href="fig-0146-01a">
    <image file="0146-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0146-01"/>
  </figure>
<note position="left" xlink:label="note-0146-01" xlink:href="note-0146-01a" xml:space="preserve">11. Addit.</note>
<note position="left" xlink:label="note-0146-02" xlink:href="note-0146-02a" xml:space="preserve">Maurol. 4. <lb/>7. &amp; 10. <lb/>lib. 5. <lb/>Conic.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4346" xml:space="preserve">Ex dictis colligitur, quod ex concurſu ad quamlibet coniſectionem poßunt du-<lb/>ci tres, vel quatuor ramiſecantes inter ſe æquales: </s>
  <s xml:id="echoid-s4347" xml:space="preserve">in ellipſi vero, &amp; </s>
  <s xml:id="echoid-s4348" xml:space="preserve">in reliquis <lb/>ſectionibus ſi rami ſecantes non fuerint, duci poteſt vnus, vel duo rami inter <lb/>ſe æquales.</s>
  <s xml:id="echoid-s4349" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s4350" xml:space="preserve">Nam circulus radio alicuius breuiſecantis deſcriptus tangit, vel ſecat coni-<lb/>ſectionem, &amp; </s>
  <s xml:id="echoid-s4351" xml:space="preserve">ſiquidem eam extrinſecus tangit, neceſſario eandem bis ſecat, ſi <lb/>fuerit parabole, aut hyperbole, quæ infinitè augẽtur, &amp; </s>
  <s xml:id="echoid-s4352" xml:space="preserve">dilatãtur; </s>
  <s xml:id="echoid-s4353" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4354" xml:space="preserve">propterea <lb/>radĳ circuli ad occurſus, &amp; </s>
  <s xml:id="echoid-s4355" xml:space="preserve">contactum ducti æquales ſunt interſe; </s>
  <s xml:id="echoid-s4356" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4357" xml:space="preserve">ideo tres <lb/>rami tantum erunt æquales: </s>
  <s xml:id="echoid-s4358" xml:space="preserve">ſi vero deſcribatur circulus, cuius centrum eſt con-<lb/>curſus, radius vero minor eſt maximo, &amp; </s>
  <s xml:id="echoid-s4359" xml:space="preserve">maior minimo duorum breuiſecan-<lb/>tium: </s>
  <s xml:id="echoid-s4360" xml:space="preserve">tunc quidem neceſſario circulus quatuor in punctis ſectioni conicæ occur-<lb/>ret: </s>
  <s xml:id="echoid-s4361" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4362" xml:space="preserve">propterea quatuor radĳ ad occurſus ducti erunt inter ſe æquales.</s>
  <s xml:id="echoid-s4363" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s4364" xml:space="preserve">At in ellipſi ſi concurſus ſiat circuli centrum, radius vero breuiſecans maxi-<lb/>mus trium, qui in ea duci puſſunt, circulus prædicto radio deſcriptus continget <lb/>quidem exterius ellipſim, neque deinceps vnquam ei occurret: </s>
  <s xml:id="echoid-s4365" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4366" xml:space="preserve">propterea ra-<lb/>mus ille maximus erit vnicus, cum nullus alius ei æqualis duci poſsit in eadem <lb/>ellipſi: </s>
  <s xml:id="echoid-s4367" xml:space="preserve">ſi verò à concurſu in productione axis ellipſis poſito deſcribatur circulus, <lb/>cuius radius minor ſit maximo ramo, ſed maior vtroque terminato; </s>
  <s xml:id="echoid-s4368" xml:space="preserve">tunc qui-<lb/>dem circulus duobus in locis ellipſi occurret; </s>
  <s xml:id="echoid-s4369" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4370" xml:space="preserve">propterea duo tantum rami inter <lb/>ſe æquales erunt; </s>
  <s xml:id="echoid-s4371" xml:space="preserve">pari modo, quando à concurſu tres breuiſecantes ad ellipſin,
<pb o="109" file="0147" n="147" rhead="Conicor. Lib. V."/>
educuntur, tunc quidem circulus, cuius centrum eſt concurſus, radius vero mi-<lb/>nor maximo breuiſecantium, &amp; </s>
  <s xml:id="echoid-s4372" xml:space="preserve">maior duobus reliquis neceßariò ellipſin duobus <lb/>in locis ſecabit; </s>
  <s xml:id="echoid-s4373" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4374" xml:space="preserve">ideo duo tantummodo rami inter ſe æquales erunt.</s>
  <s xml:id="echoid-s4375" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div405" type="section" level="1" n="124">
<head xml:id="echoid-head167" xml:space="preserve">SECTIO DECIMAQVINTA</head>
<head xml:id="echoid-head168" xml:space="preserve">Continens Propoſ. XXXXI. XXXXII. <lb/>XXXXIII. Apollonij. <lb/>PROPOSITIO XXXXI.</head>
<p>
  <s xml:id="echoid-s4376" xml:space="preserve">IN hyperhola angulus contentus à linea breuiſſima, &amp; </s>
  <s xml:id="echoid-s4377" xml:space="preserve">à men-<lb/>
<anchor type="note" xlink:label="note-0147-01a" xlink:href="note-0147-01"/>
ſura minor eſt angulo compræhenſo à linea diſtante cum cõ-<lb/>tinente.</s>
  <s xml:id="echoid-s4378" xml:space="preserve"/>
</p>
<div xml:id="echoid-div405" type="float" level="2" n="1">
<note position="left" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s4379" xml:space="preserve">Sit hyberbole A B, eius axis D <lb/>
<anchor type="figure" xlink:label="fig-0147-01a" xlink:href="fig-0147-01"/>
<anchor type="note" xlink:label="note-0147-02a" xlink:href="note-0147-02"/>
C, linea breuiſſima B C, duo con-<lb/>tinentes D E, D F, &amp; </s>
  <s xml:id="echoid-s4380" xml:space="preserve">diſtantia ſit <lb/>A E, &amp; </s>
  <s xml:id="echoid-s4381" xml:space="preserve">dimidium erecti A G: </s>
  <s xml:id="echoid-s4382" xml:space="preserve">Di-<lb/>co, angulum B C D minorem eſſe <lb/>angulo D E A. </s>
  <s xml:id="echoid-s4383" xml:space="preserve">Educamus itaque <lb/>perdendicularem B H, &amp; </s>
  <s xml:id="echoid-s4384" xml:space="preserve">iungamus <lb/>B D, quæ ſecet A E in I. </s>
  <s xml:id="echoid-s4385" xml:space="preserve">Quia. <lb/></s>
  <s xml:id="echoid-s4386" xml:space="preserve">D A ad A G eſt, vt D H ad H C <lb/>(14. </s>
  <s xml:id="echoid-s4387" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4388" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4389" xml:space="preserve">I A ad A D eſt, vt <lb/>
<anchor type="note" xlink:label="note-0147-03a" xlink:href="note-0147-03"/>
B H ad H D; </s>
  <s xml:id="echoid-s4390" xml:space="preserve">ergo ex æqualitate, <lb/>I A ad A G, eandem proportionẽ <lb/>habebit, quàm B H ad H C, &amp; </s>
  <s xml:id="echoid-s4391" xml:space="preserve"><lb/>propterea E A ad A G, nempe D <lb/>A ad diſtantiam A E maiorẽ pro-<lb/>portionem habebit, quàm B H ad H C igitur angulus B C H minor eſt, <lb/>quàm D E A, quod erat oſtendendum.</s>
  <s xml:id="echoid-s4392" xml:space="preserve"/>
</p>
<div xml:id="echoid-div406" type="float" level="2" n="2">
  <figure xlink:label="fig-0147-01" xlink:href="fig-0147-01a">
    <image file="0147-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0147-01"/>
  </figure>
<note position="left" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0147-03" xlink:href="note-0147-03a" xml:space="preserve">9. huius.</note>
</div>
</div>
<div xml:id="echoid-div408" type="section" level="1" n="125">
<head xml:id="echoid-head169" xml:space="preserve">PROPOSITO XXXXII.</head>
<p>
  <s xml:id="echoid-s4393" xml:space="preserve">IN parabola lineæ breuiſſimæ productæ occurrunt ſectioni ex <lb/>
<anchor type="note" xlink:label="note-0147-04a" xlink:href="note-0147-04"/>
vtraque parte.</s>
  <s xml:id="echoid-s4394" xml:space="preserve"/>
</p>
<div xml:id="echoid-div408" type="float" level="2" n="1">
<note position="left" xlink:label="note-0147-04" xlink:href="note-0147-04a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s4395" xml:space="preserve">Quoniam breuiſſima eſt linea recta ſecans diametrum paraboles intra <lb/>ſectionem; </s>
  <s xml:id="echoid-s4396" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4397" xml:space="preserve">propterea ſectioni occurret ex vtraque parte (28. </s>
  <s xml:id="echoid-s4398" xml:space="preserve">ex pr.) <lb/></s>
  <s xml:id="echoid-s4399" xml:space="preserve">
<anchor type="note" xlink:label="note-0147-05a" xlink:href="note-0147-05"/>
&amp; </s>
  <s xml:id="echoid-s4400" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s4401" xml:space="preserve"/>
</p>
<div xml:id="echoid-div409" type="float" level="2" n="2">
<note position="right" xlink:label="note-0147-05" xlink:href="note-0147-05a" xml:space="preserve">27 lib. 1.</note>
</div>
<pb o="110" file="0148" n="148" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div411" type="section" level="1" n="126">
<head xml:id="echoid-head170" xml:space="preserve">PROPOSITIO XXXXIII.</head>
<p>
  <s xml:id="echoid-s4402" xml:space="preserve">SI inclinatus axis hyperboles erectum non excedit, nulla li-<lb/>
<anchor type="note" xlink:label="note-0148-01a" xlink:href="note-0148-01"/>
nearum breuiſſimarum ſectioni ex altera parte occurret: </s>
  <s xml:id="echoid-s4403" xml:space="preserve">ſi <lb/>verò maior illo fuerit, tunc breuiſſimarum linearum aliquæ oc-<lb/>currunt ſectioni, aliquæ verò non occurrunt.</s>
  <s xml:id="echoid-s4404" xml:space="preserve"/>
</p>
<div xml:id="echoid-div411" type="float" level="2" n="1">
<note position="right" xlink:label="note-0148-01" xlink:href="note-0148-01a" xml:space="preserve">a</note>
</div>
<note position="right" xml:space="preserve">b</note>
<p>
  <s xml:id="echoid-s4405" xml:space="preserve">Sit priùs D A non maior, quàm A G: </s>
  <s xml:id="echoid-s4406" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4407" xml:space="preserve">quia D A ad A G eandẽ pro-<lb/>portionem habet, quàm quadratum D A ad quadratum A E, erit D A <lb/>non maior quàm A E; </s>
  <s xml:id="echoid-s4408" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4409" xml:space="preserve">propterea angulus D E A non crit maior an-<lb/>gulo E D A: </s>
  <s xml:id="echoid-s4410" xml:space="preserve">ſed maior fuerat angulo B C H (41. </s>
  <s xml:id="echoid-s4411" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4412" xml:space="preserve">ergo angulus <lb/>E D A, nempe A D F maior eſt, quàm B C D, &amp; </s>
  <s xml:id="echoid-s4413" xml:space="preserve">propterea B C, D F <lb/>non conueniunt ad partes C, F; </s>
  <s xml:id="echoid-s4414" xml:space="preserve">igitur B C non occurrit ſectioni ad par-<lb/>tes K, A; </s>
  <s xml:id="echoid-s4415" xml:space="preserve">eo quod ſi illam ſecaret, etiam ipſi D F occurreret (8. </s>
  <s xml:id="echoid-s4416" xml:space="preserve">ex 2.) <lb/></s>
  <s xml:id="echoid-s4417" xml:space="preserve">quare non occurrit ſectioni in duobus punctis.</s>
  <s xml:id="echoid-s4418" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4419" xml:space="preserve">Deinde ſit D A maior, quàm A <lb/>
<anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/>
<anchor type="note" xlink:label="note-0148-03a" xlink:href="note-0148-03"/>
G habebit E A ad A G maiorem <lb/>proportionẽ, quàm ad A D; </s>
  <s xml:id="echoid-s4420" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4421" xml:space="preserve">po-<lb/>natur I A ad A G, vt E A ad A <lb/>D; </s>
  <s xml:id="echoid-s4422" xml:space="preserve">ergo I A minor eſt, quàm E A, <lb/>
<anchor type="note" xlink:label="note-0148-04a" xlink:href="note-0148-04"/>
quare recta D B, illam diuidens, <lb/>occurret ſectioni, &amp; </s>
  <s xml:id="echoid-s4423" xml:space="preserve">cadat in B, du-<lb/>caturque linea breuiſſima B C, &amp; </s>
  <s xml:id="echoid-s4424" xml:space="preserve"><lb/>B H perpendicularis ad D C; </s>
  <s xml:id="echoid-s4425" xml:space="preserve">erit <lb/>I A ad A D, vt B H ad H D; </s>
  <s xml:id="echoid-s4426" xml:space="preserve">eſt-<lb/>que D A ad A G, vt D H ad H C; <lb/></s>
  <s xml:id="echoid-s4427" xml:space="preserve">
<anchor type="note" xlink:label="note-0148-05a" xlink:href="note-0148-05"/>
ergo I A ad A G, nempe E A ad <lb/>A D eſt, vt B H ad H C, &amp; </s>
  <s xml:id="echoid-s4428" xml:space="preserve">pro-<lb/>pterea duo triangula E A D, B H <lb/>C ſunt ſimilia; </s>
  <s xml:id="echoid-s4429" xml:space="preserve">igitur angulus B C <lb/>H æqualis eſt E D A, nempe F D A; </s>
  <s xml:id="echoid-s4430" xml:space="preserve">quare B C, D F ſunt parallelæ, <lb/>nec poſſunt ſe ſe mutuo ſecare; </s>
  <s xml:id="echoid-s4431" xml:space="preserve">ergo B C non occurret ſectioni K A. <lb/></s>
  <s xml:id="echoid-s4432" xml:space="preserve">
<anchor type="note" xlink:label="note-0148-06a" xlink:href="note-0148-06"/>
Lineæ vero breuiſſimæ, quæ in peripheria A B cadunt, continent cum. <lb/></s>
  <s xml:id="echoid-s4433" xml:space="preserve">C A angulos minores angulo B C D (26. </s>
  <s xml:id="echoid-s4434" xml:space="preserve">27. </s>
  <s xml:id="echoid-s4435" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4436" xml:space="preserve">vnde non occurrent <lb/>
<anchor type="note" xlink:label="note-0148-07a" xlink:href="note-0148-07"/>
ipſi D F, &amp; </s>
  <s xml:id="echoid-s4437" xml:space="preserve">propterea neque ſectioni occurrent. </s>
  <s xml:id="echoid-s4438" xml:space="preserve">At ille, qui cadit extra <lb/>hanc ſectionis peripheriam; </s>
  <s xml:id="echoid-s4439" xml:space="preserve">ſi producatur continet cum C D angulum. <lb/></s>
  <s xml:id="echoid-s4440" xml:space="preserve">maiorem angulo B C D (26. </s>
  <s xml:id="echoid-s4441" xml:space="preserve">27. </s>
  <s xml:id="echoid-s4442" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4443" xml:space="preserve">igitur productus occurrit D F, <lb/>&amp; </s>
  <s xml:id="echoid-s4444" xml:space="preserve">occurrit ſectioni A K: </s>
  <s xml:id="echoid-s4445" xml:space="preserve">quod erat oſtendendum.</s>
  <s xml:id="echoid-s4446" xml:space="preserve"/>
</p>
<div xml:id="echoid-div412" type="float" level="2" n="2">
  <figure xlink:label="fig-0148-01" xlink:href="fig-0148-01a">
    <image file="0148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0148-01"/>
  </figure>
<note position="right" xlink:label="note-0148-03" xlink:href="note-0148-03a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0148-04" xlink:href="note-0148-04a" xml:space="preserve">Secunda <lb/>lib. 2.</note>
<note position="left" xlink:label="note-0148-05" xlink:href="note-0148-05a" xml:space="preserve">14. huius.</note>
<note position="left" xlink:label="note-0148-06" xlink:href="note-0148-06a" xml:space="preserve">13. lib. 2.</note>
<note position="left" xlink:label="note-0148-07" xlink:href="note-0148-07a" xml:space="preserve">Conuerſ. <lb/>8. lib. 2.</note>
</div>
</div>
<div xml:id="echoid-div414" type="section" level="1" n="127">
<head xml:id="echoid-head171" xml:space="preserve">Notæ in Propoſ. XXXXI.</head>
<p>
  <s xml:id="echoid-s4447" xml:space="preserve">ANgulus contentus à breuiſſima linea, &amp; </s>
  <s xml:id="echoid-s4448" xml:space="preserve">menſura minor eſt angulo <lb/>contento à diſtante cum continente in ſectione, &amp;</s>
  <s xml:id="echoid-s4449" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4450" xml:space="preserve">Adĳcio par-
<pb o="111" file="0149" n="149" rhead="Conicor. Lib. V."/>
ticulam in hyperbole, quæ in textu deſideratur. </s>
  <s xml:id="echoid-s4451" xml:space="preserve">Vocat interpres Arabicus li-<lb/>neam diſtantem ipſam A E, quæ contingit hyperbolem in vertice axis A, &amp; </s>
  <s xml:id="echoid-s4452" xml:space="preserve"><lb/>interponitur inter verticem A, &amp; </s>
  <s xml:id="echoid-s4453" xml:space="preserve">continentem, ſeu asymptoton D E.</s>
  <s xml:id="echoid-s4454" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s4455" xml:space="preserve">Sit ſectio, D C diameter illius, &amp;</s>
  <s xml:id="echoid-s4456" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4457" xml:space="preserve">Legendum puto; </s>
  <s xml:id="echoid-s4458" xml:space="preserve">Sit hyperbole A B <lb/>
<anchor type="note" xlink:label="note-0149-01a" xlink:href="note-0149-01"/>
eius axis D C. </s>
  <s xml:id="echoid-s4459" xml:space="preserve">Poſtea quia D A, ad A G, ſeu latus tranſuer ſum ad rectum eſt, <lb/>
<anchor type="note" xlink:label="note-0149-02a" xlink:href="note-0149-02"/>
vt D H ad H C, atque I A ad A D eſt, vt B H ad H D (propter ſimilitudi-<lb/>nem triangulorum I A D, &amp; </s>
  <s xml:id="echoid-s4460" xml:space="preserve">B H D) ergo ex æqualitate ordinata I A ad A <lb/>G eſt vt B H ad H C: </s>
  <s xml:id="echoid-s4461" xml:space="preserve">deinde quia linea A E media proportionalis eſt inter ſe-<lb/>miaxim tranſuerſum D A, &amp; </s>
  <s xml:id="echoid-s4462" xml:space="preserve">ſemierectum A G, cum quadratum ipſius A E <lb/>quadrans ſit figuræ quæ ad diametrum per A ductum conſtituitur; </s>
  <s xml:id="echoid-s4463" xml:space="preserve">igitur E A <lb/>
<anchor type="note" xlink:label="note-0149-03a" xlink:href="note-0149-03"/>
ad A G erit, vt D A ad A E, eſt vero E A maior, quàm I A; </s>
  <s xml:id="echoid-s4464" xml:space="preserve">igitur I A ad A <lb/>G minorem proportionem habet, quàm E A ad A G, ſeu quàm D A ad A E: <lb/></s>
  <s xml:id="echoid-s4465" xml:space="preserve">erat autem B H ad H C, vt I A ad A G: </s>
  <s xml:id="echoid-s4466" xml:space="preserve">igitur B H ad H C minorem propor-<lb/>tionem habet, quàm D A ad A E: </s>
  <s xml:id="echoid-s4467" xml:space="preserve">fiat poſtea L A ad A E, vt B H ad H C <lb/>circa angulos rectos A, H, coniungaturq; </s>
  <s xml:id="echoid-s4468" xml:space="preserve">L E, manifeſtum eſt, L A minorem <lb/>eſſe-D A, &amp; </s>
  <s xml:id="echoid-s4469" xml:space="preserve">angulum A E L minorem eſſe angulo A E D: </s>
  <s xml:id="echoid-s4470" xml:space="preserve">ſed propter ſimili-<lb/>tudinem triangulorum B H C, L A E eſt angulus C æqualis angulo A E L; </s>
  <s xml:id="echoid-s4471" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4472" xml:space="preserve"><lb/>proptrea angulus A E D maior eſt angulo B C H.</s>
  <s xml:id="echoid-s4473" xml:space="preserve"/>
</p>
<div xml:id="echoid-div414" type="float" level="2" n="1">
<note position="left" xlink:label="note-0149-01" xlink:href="note-0149-01a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0149-02" xlink:href="note-0149-02a" xml:space="preserve">Ex 14. <lb/>huius.</note>
<note position="right" xlink:label="note-0149-03" xlink:href="note-0149-03a" xml:space="preserve">3. lib. 2.</note>
</div>
</div>
<div xml:id="echoid-div416" type="section" level="1" n="128">
<head xml:id="echoid-head172" xml:space="preserve">Notæ in Propoſ. XXXXII.</head>
<p>
  <s xml:id="echoid-s4474" xml:space="preserve">QVia eſt linea recta ſecans diametrum paraboles; </s>
  <s xml:id="echoid-s4475" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s4476" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4477" xml:space="preserve">Addo illam par-<lb/>
<anchor type="note" xlink:label="note-0149-04a" xlink:href="note-0149-04"/>
ticulam breuiſſimam, quæ in textu deſiderari videtur.</s>
  <s xml:id="echoid-s4478" xml:space="preserve"/>
</p>
<div xml:id="echoid-div416" type="float" level="2" n="1">
<note position="left" xlink:label="note-0149-04" xlink:href="note-0149-04a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div418" type="section" level="1" n="129">
<head xml:id="echoid-head173" xml:space="preserve">Notæ in Propoſit. XXXXIII.</head>
<p style="it">
  <s xml:id="echoid-s4479" xml:space="preserve">INclinatum ſi non excedit erectum, nulla linearum, &amp;</s>
  <s xml:id="echoid-s4480" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4481" xml:space="preserve">Addo, quæeui-<lb/>
<anchor type="note" xlink:label="note-0149-05a" xlink:href="note-0149-05"/>
denter deſiciunt in textu, legi enim debet: </s>
  <s xml:id="echoid-s4482" xml:space="preserve">Axis inclinatus ideſt tranſuer-<lb/>ſus ſi non excedit erectum, &amp;</s>
  <s xml:id="echoid-s4483" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4484" xml:space="preserve"/>
</p>
<div xml:id="echoid-div418" type="float" level="2" n="1">
<note position="left" xlink:label="note-0149-05" xlink:href="note-0149-05a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s4485" xml:space="preserve">Et quia D A ad A G eſt vt quadratum D A ad quadratum A E, &amp;</s>
  <s xml:id="echoid-s4486" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s4487" xml:space="preserve">
<anchor type="note" xlink:label="note-0149-06a" xlink:href="note-0149-06"/>
Eo quod quadratum A E æquale eſt quartæ parti figuræ, quæ ad duplam ſemia-<lb/>
<anchor type="note" xlink:label="note-0149-07a" xlink:href="note-0149-07"/>
xis D A applicatur, ſcilicet æquale eſt rectangulo D A G; </s>
  <s xml:id="echoid-s4488" xml:space="preserve">igitur D A, A E, <lb/>A G ſunt continuæ proportionales: </s>
  <s xml:id="echoid-s4489" xml:space="preserve">ponitur vero D A æqualis, aut minor, quàm <lb/>A G; </s>
  <s xml:id="echoid-s4490" xml:space="preserve">igitur D A æqualis, aut minor quoque erit, quàm A E; </s>
  <s xml:id="echoid-s4491" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4492" xml:space="preserve">propterea in <lb/>triangulo D E A erit angulus D E A æqualis, aut maior angulo A D E, ſeu <lb/>A D F (cum angulus continentiæ ſecetur bifariam ab axi) &amp; </s>
  <s xml:id="echoid-s4493" xml:space="preserve">prius erat an-<lb/>
<anchor type="note" xlink:label="note-0149-08a" xlink:href="note-0149-08"/>
gulus C minor angulo A E D; </s>
  <s xml:id="echoid-s4494" xml:space="preserve">igitur angulus B C D minor erit alterno angulo <lb/>F D C; </s>
  <s xml:id="echoid-s4495" xml:space="preserve">vnde conſtat rectas lineas F D, C B concurrere poſſe, ſi vlterius pro-<lb/>ducantur ad partes D, B; </s>
  <s xml:id="echoid-s4496" xml:space="preserve">non autem ad partes C, &amp; </s>
  <s xml:id="echoid-s4497" xml:space="preserve">F.</s>
  <s xml:id="echoid-s4498" xml:space="preserve"/>
</p>
<div xml:id="echoid-div419" type="float" level="2" n="2">
<note position="left" xlink:label="note-0149-06" xlink:href="note-0149-06a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0149-07" xlink:href="note-0149-07a" xml:space="preserve">3. lib. 2.</note>
<note position="right" xlink:label="note-0149-08" xlink:href="note-0149-08a" xml:space="preserve">41. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s4499" xml:space="preserve">Quia ſi occurreret illi occurreret D F (7. </s>
  <s xml:id="echoid-s4500" xml:space="preserve">ex 2.) </s>
  <s xml:id="echoid-s4501" xml:space="preserve">ſecaretque ſectionem <lb/>
<anchor type="note" xlink:label="note-0149-09a" xlink:href="note-0149-09"/>
in duobus punctis, &amp;</s>
  <s xml:id="echoid-s4502" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4503" xml:space="preserve">Senſus huius textus talis eſt. </s>
  <s xml:id="echoid-s4504" xml:space="preserve">Quoniam, vt oſtensũ <lb/>eſt, recta B C inſinite producta non occurrit asymptoto D F ad partes F C; </s>
  <s xml:id="echoid-s4505" xml:space="preserve">igi-<lb/>
<anchor type="note" xlink:label="note-0149-10a" xlink:href="note-0149-10"/>
tur recta C B producta non ſecabit peripheriam hyperboles ad partes K; </s>
  <s xml:id="echoid-s4506" xml:space="preserve">nam <lb/>ſi ipſam ſecaret, ſecaret quoque asymptoton D F ad partes F, quod non poni-<lb/>
<anchor type="note" xlink:label="note-0149-11a" xlink:href="note-0149-11"/>
tur. </s>
  <s xml:id="echoid-s4507" xml:space="preserve">Ex his inferri debet concluſio principalis, nimirum, quod B C non occurrit <lb/>ſectioni duobus in punctis: </s>
  <s xml:id="echoid-s4508" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4509" xml:space="preserve">hac ratione textum alioqui corruptum emendaui.</s>
  <s xml:id="echoid-s4510" xml:space="preserve"/>
</p>
<div xml:id="echoid-div420" type="float" level="2" n="3">
<note position="left" xlink:label="note-0149-09" xlink:href="note-0149-09a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0149-10" xlink:href="note-0149-10a" xml:space="preserve">8. lib. 2.</note>
<note position="right" xlink:label="note-0149-11" xlink:href="note-0149-11a" xml:space="preserve">Ibidem.</note>
</div>
<pb o="112" file="0150" n="150" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s4511" xml:space="preserve">Lineæ vero breuiſſimæ, quæ ca-<lb/>
<anchor type="figure" xlink:label="fig-0150-01a" xlink:href="fig-0150-01"/>
<anchor type="note" xlink:label="note-0150-01a" xlink:href="note-0150-01"/>
dunt ad peripheriam ſectionis B <lb/>A, continent angulos minores, <lb/>quàm B C D, vtique non occur-<lb/>runt D F, &amp;</s>
  <s xml:id="echoid-s4512" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4513" xml:space="preserve">Ideſt: </s>
  <s xml:id="echoid-s4514" xml:space="preserve">quia quælibet <lb/>breuiſsima ex puncto peripheriæ A B <lb/>ad axim ducta efſicit angulum propin-<lb/>
<anchor type="note" xlink:label="note-0150-02a" xlink:href="note-0150-02"/>
quiorem vertici, minorem ipſo angulo <lb/>C; </s>
  <s xml:id="echoid-s4515" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4516" xml:space="preserve">propterea quælibet breuiſsima, <lb/>ad peripheriam A B extenſa ſecabit ne-<lb/>
<anchor type="note" xlink:label="note-0150-03a" xlink:href="note-0150-03"/>
ceſſario ipſam B C vlterius productam <lb/>ad partes C: </s>
  <s xml:id="echoid-s4517" xml:space="preserve">ſed prius oſtenſa fuit B <lb/>C parallela asymptoto D F; </s>
  <s xml:id="echoid-s4518" xml:space="preserve">igitur quæ-<lb/>libet breuiſsima ad peripheriam A B <lb/>educta ideſt inter parallelas poſita non <lb/>occurret alteri æquidiſtantium D F ad partes F, ſed ad partes oppoſitas verſus <lb/>D; </s>
  <s xml:id="echoid-s4519" xml:space="preserve">eo quod quælibet recta linea intra hyperbolam ducta non ſecat peripheriam ſe-<lb/>
<anchor type="note" xlink:label="note-0150-04a" xlink:href="note-0150-04"/>
ctionis in ea parte, in qua continentem D F nõ ſecat; </s>
  <s xml:id="echoid-s4520" xml:space="preserve">At quælibet alia breuiſsima <lb/>infra C B ducta, neceſſario efſiciet ad axim angulum maiorem, quàm C; </s>
  <s xml:id="echoid-s4521" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4522" xml:space="preserve">pro-<lb/>pterea vlterius producta ſecabit ipſam B C ad partes C; </s>
  <s xml:id="echoid-s4523" xml:space="preserve">ſed quælibet breuiſsima <lb/>extra parallelas poſita quæ ſecat vnam æquidiſtantium B C, ſecabit quoq; </s>
  <s xml:id="echoid-s4524" xml:space="preserve">reli-<lb/>quam ad eaſdem partes F C; </s>
  <s xml:id="echoid-s4525" xml:space="preserve">quare prius ſectioni occurret, vt dictum eſt.</s>
  <s xml:id="echoid-s4526" xml:space="preserve"/>
</p>
<div xml:id="echoid-div421" type="float" level="2" n="4">
  <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a">
    <image file="0150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0150-01"/>
  </figure>
<note position="right" xlink:label="note-0150-01" xlink:href="note-0150-01a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0150-02" xlink:href="note-0150-02a" xml:space="preserve">26. 27. <lb/>huius.</note>
<note position="left" xlink:label="note-0150-03" xlink:href="note-0150-03a" xml:space="preserve">28. huius.</note>
<note position="left" xlink:label="note-0150-04" xlink:href="note-0150-04a" xml:space="preserve">Conuerſ. <lb/>8. lib. 2. <lb/>26. 27. <lb/>huius. <lb/>28. huius.</note>
</div>
</div>
<div xml:id="echoid-div423" type="section" level="1" n="130">
<head xml:id="echoid-head174" xml:space="preserve">SECTIO DECIMASEXTA</head>
<head xml:id="echoid-head175" xml:space="preserve">Continens XVI. XVII. XVIII. Propoſ. <lb/>Apollonij.</head>
<p>
  <s xml:id="echoid-s4527" xml:space="preserve">SI menſura comparata ſumpta fuerit in axe recto minore elli-<lb/>
<anchor type="note" xlink:label="note-0150-05a" xlink:href="note-0150-05"/>
pſis, erit maximus ramorum ab eius origine egredientium, <lb/>&amp; </s>
  <s xml:id="echoid-s4528" xml:space="preserve">illi propinquior maior eſt remotiore: </s>
  <s xml:id="echoid-s4529" xml:space="preserve">minimus vero ramorũ <lb/>eſt differentia recti, &amp; </s>
  <s xml:id="echoid-s4530" xml:space="preserve">comparatæ, &amp; </s>
  <s xml:id="echoid-s4531" xml:space="preserve">illi propinquior, minor <lb/>eſt remotiore, atque exceſſus quadrati comparatæ ſupra qua-<lb/>dratum cuiuſcunque rami aſſignati æqualis eſt exemplari appli-<lb/>cato ad abſciſſam illius rami, ſiue comparata ſit minor, aut <lb/>æqualis, aut maior recto.</s>
  <s xml:id="echoid-s4532" xml:space="preserve"/>
</p>
<div xml:id="echoid-div423" type="float" level="2" n="1">
<note position="right" xlink:label="note-0150-05" xlink:href="note-0150-05a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s4533" xml:space="preserve">Sit D C rectus a-<lb/>
<anchor type="figure" xlink:label="fig-0150-02a" xlink:href="fig-0150-02"/>
xis minor ſectionis <lb/>ellipticæ A B C ſit-<lb/>que C I comparata, <lb/>&amp; </s>
  <s xml:id="echoid-s4534" xml:space="preserve">rami I H, I K, I <lb/>B, I L, I A, I D, &amp; </s>
  <s xml:id="echoid-s4535" xml:space="preserve"><lb/>ſemiſſis erecti ſit C <lb/>F, &amp; </s>
  <s xml:id="echoid-s4536" xml:space="preserve">centrum E, &amp;</s>
  <s xml:id="echoid-s4537" xml:space="preserve">
<pb o="113" file="0151" n="151" rhead="Conicor. Lib. V."/>
educamus F E quouſque ſecet D M perpendicularem ad axim in M, &amp; </s>
  <s xml:id="echoid-s4538" xml:space="preserve"><lb/>F I occurrat D M in N, &amp; </s>
  <s xml:id="echoid-s4539" xml:space="preserve">ducantur ad axim perpendiculares H O T S, <lb/>K P V, B E, L Q, A R: </s>
  <s xml:id="echoid-s4540" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4541" xml:space="preserve">ſit in prima figura C I minor recto, in ſecun-<lb/>da æqualis, in tertia vero maior. </s>
  <s xml:id="echoid-s4542" xml:space="preserve">Conſtat, quemadmodum demonſtra-<lb/>
<anchor type="note" xlink:label="note-0151-01a" xlink:href="note-0151-01"/>
uimus in propoſitione ſexta huius, quod quadratum I C æquale ſit du-<lb/>plo trianguli I C F; </s>
  <s xml:id="echoid-s4543" xml:space="preserve">at quadratum O H duplum eſt trapezij O T F C <lb/>(1. </s>
  <s xml:id="echoid-s4544" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4545" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4546" xml:space="preserve">quadratum I O duplum eſt trianguli O I S; </s>
  <s xml:id="echoid-s4547" xml:space="preserve">ergo quadra-<lb/>tum I C, nempe duplum trianguli I F C excedit quadratum I H duplo <lb/>trianguli F T S, quod eſt æquale rectangulo T a: </s>
  <s xml:id="echoid-s4548" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4549" xml:space="preserve">conſtat, vti dictum <lb/>
<anchor type="note" xlink:label="note-0151-02a" xlink:href="note-0151-02"/>
<anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/>
eſt, quod ſit exemplar applicatum ad O C; </s>
  <s xml:id="echoid-s4550" xml:space="preserve">ergo quadratum I C excedit <lb/>quadratum I H exemplari applicato ad O C abſciſſam ipſius I H. </s>
  <s xml:id="echoid-s4551" xml:space="preserve">Patet <lb/>etiam, quod quadratum I C excedit quadratum I K exemplari applica-<lb/>to ad P C; </s>
  <s xml:id="echoid-s4552" xml:space="preserve">idemque conſtat in I B; </s>
  <s xml:id="echoid-s4553" xml:space="preserve">igitur I C maior eſt, quàm I H, &amp; </s>
  <s xml:id="echoid-s4554" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0151-03a" xlink:href="note-0151-03"/>
I H, quàm I K, &amp; </s>
  <s xml:id="echoid-s4555" xml:space="preserve">I K, quàm I B: </s>
  <s xml:id="echoid-s4556" xml:space="preserve">poſtea, in figura prima, &amp; </s>
  <s xml:id="echoid-s4557" xml:space="preserve">tertia.</s>
  <s xml:id="echoid-s4558" xml:space="preserve">, <lb/>
<anchor type="figure" xlink:label="fig-0151-02a" xlink:href="fig-0151-02"/>
quia triangulum F C E æquale eſt triangulo D E M; </s>
  <s xml:id="echoid-s4559" xml:space="preserve">ergo quadratum. <lb/></s>
  <s xml:id="echoid-s4560" xml:space="preserve">
<anchor type="note" xlink:label="note-0151-04a" xlink:href="note-0151-04"/>
I C æquale eſt duplo trianguli N F M cum duplo trianguli D I N, qua-<lb/>dratum vero I D æquale eſt duplo trianguli D I N; </s>
  <s xml:id="echoid-s4561" xml:space="preserve">igitur quadratum.</s>
  <s xml:id="echoid-s4562" xml:space="preserve">
<pb o="114" file="0152" n="152" rhead="Apollonij Pergæi"/>
I D minus eſt, quàm quadratũ I C duplo trianguli N F M, quod æqua-<lb/>le eſt exemplari applicato ad D C, &amp; </s>
  <s xml:id="echoid-s4563" xml:space="preserve">quadratum I R æquale eſt duplo <lb/>trianguli I X R, &amp; </s>
  <s xml:id="echoid-s4564" xml:space="preserve">quadratum A R æquale eſt duplo trapezij R M (3. </s>
  <s xml:id="echoid-s4565" xml:space="preserve">ex <lb/>5.) </s>
  <s xml:id="echoid-s4566" xml:space="preserve">ergo quadratũ I A minus eſt, quàm quadratum I C duplo trianguli <lb/>F Z X, quod æquale ex exemplari applicato ad C R (6. </s>
  <s xml:id="echoid-s4567" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4568" xml:space="preserve">ſimiliter <lb/>quadratum I L minus eſt, quàm quadratum I C exemplari applicato ad <lb/>C Q; </s>
  <s xml:id="echoid-s4569" xml:space="preserve">eſtque C D maior, quàm C R, &amp; </s>
  <s xml:id="echoid-s4570" xml:space="preserve">C R quàm C Q; </s>
  <s xml:id="echoid-s4571" xml:space="preserve">ergo I A ma-<lb/>ior eſt, quàm I D, &amp; </s>
  <s xml:id="echoid-s4572" xml:space="preserve">I L, quàm I A; </s>
  <s xml:id="echoid-s4573" xml:space="preserve">quod erat propoſitum.</s>
  <s xml:id="echoid-s4574" xml:space="preserve"/>
</p>
<div xml:id="echoid-div424" type="float" level="2" n="2">
  <figure xlink:label="fig-0150-02" xlink:href="fig-0150-02a">
    <image file="0150-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0150-02"/>
  </figure>
<note position="left" xlink:label="note-0151-01" xlink:href="note-0151-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0151-02" xlink:href="note-0151-02a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a">
    <image file="0151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0151-01"/>
  </figure>
<note position="left" xlink:label="note-0151-03" xlink:href="note-0151-03a" xml:space="preserve">d</note>
  <figure xlink:label="fig-0151-02" xlink:href="fig-0151-02a">
    <image file="0151-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0151-02"/>
  </figure>
<note position="left" xlink:label="note-0151-04" xlink:href="note-0151-04a" xml:space="preserve">e</note>
</div>
</div>
<div xml:id="echoid-div426" type="section" level="1" n="131">
<head xml:id="echoid-head176" xml:space="preserve">Notæ in Propoſit. XVI. XVII. XVIII.</head>
<p style="it">
  <s xml:id="echoid-s4575" xml:space="preserve">COmparata ſi fuerit ex recto duorum axium ellipſis crit maximus ra-<lb/>
<anchor type="note" xlink:label="note-0152-01a" xlink:href="note-0152-01"/>
morum, &amp;</s>
  <s xml:id="echoid-s4576" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4577" xml:space="preserve">Addidi particulam illam axis minoris, quæ in textu defi-<lb/>ciebat, nunquam enim C F ſemiſsis lateris recti, eſſe poteſt maior C E ſemiſſe <lb/>lateris tranſuerſi, niſi C D fuerit axis minor ellipſis.</s>
  <s xml:id="echoid-s4578" xml:space="preserve"/>
</p>
<div xml:id="echoid-div426" type="float" level="2" n="1">
<note position="right" xlink:label="note-0152-01" xlink:href="note-0152-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s4579" xml:space="preserve">Conſtat, quemadmodum demonſtrauimus in propoſitione 6. </s>
  <s xml:id="echoid-s4580" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s4581" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4582" xml:space="preserve">Quo-<lb/>
<anchor type="note" xlink:label="note-0152-02a" xlink:href="note-0152-02"/>
niã menſura I C ſupponitur cõparata, ideſt æqualis ipſi C F ſemiſsi lateris recti; <lb/></s>
  <s xml:id="echoid-s4583" xml:space="preserve">propterea triangulum I C F iſoſceleum erit, &amp; </s>
  <s xml:id="echoid-s4584" xml:space="preserve">rectangulum in C; </s>
  <s xml:id="echoid-s4585" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4586" xml:space="preserve">ideo qua-<lb/>dratum I C æquale erit duplo trianguli I C F: </s>
  <s xml:id="echoid-s4587" xml:space="preserve">eadem ratione propter parallelas <lb/>S O, &amp; </s>
  <s xml:id="echoid-s4588" xml:space="preserve">C F, erit triangulum I O S ſimile triangulo I C F, &amp; </s>
  <s xml:id="echoid-s4589" xml:space="preserve">propterea illud <lb/>quoque iſoſceleum erit, &amp; </s>
  <s xml:id="echoid-s4590" xml:space="preserve">rectangulum in O, &amp; </s>
  <s xml:id="echoid-s4591" xml:space="preserve">ideo quadratum I O æquale, <lb/>erit duplo trianguli I O S: </s>
  <s xml:id="echoid-s4592" xml:space="preserve">eſt verò quadratum O H æquale duplo trapezĳ F T <lb/>
<anchor type="note" xlink:label="note-0152-03a" xlink:href="note-0152-03"/>
O C; </s>
  <s xml:id="echoid-s4593" xml:space="preserve">igitur quadratum I H ( quod eſt æquale duobus quadratis I O, O H circa <lb/>angulum rectum O) æquale erit duplo trianguli I O S cum duplo trapezĳ F T <lb/>O C, ſed hæc duo ſpatia minora ſunt duplo integri trianguli I C F, eſtque de-<lb/>fectus duplum trianguli F T S, ſiue rectangulum S T b a; </s>
  <s xml:id="echoid-s4594" xml:space="preserve">igitur duplum trian-<lb/>guli I C F, ſiue quadratum I C maius eſt quadrato I H, &amp; </s>
  <s xml:id="echoid-s4595" xml:space="preserve">exceſſus eſt rectan-<lb/>gulum T a: </s>
  <s xml:id="echoid-s4596" xml:space="preserve">quod vero rectangulum T a ſit exemplar demonſtrabitur modo, vt <lb/>in ſexta propoſitione huius.</s>
  <s xml:id="echoid-s4597" xml:space="preserve"/>
</p>
<div xml:id="echoid-div427" type="float" level="2" n="2">
<note position="right" xlink:label="note-0152-02" xlink:href="note-0152-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0152-03" xlink:href="note-0152-03a" xml:space="preserve">1. huius.</note>
</div>
  <figure>
    <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0152-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s4598" xml:space="preserve">Et conſtat, vt dictum eſt, quod ſit exemplar applicatum ad O C, &amp;</s>
  <s xml:id="echoid-s4599" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s4600" xml:space="preserve">
<anchor type="note" xlink:label="note-0152-04a" xlink:href="note-0152-04"/>
Quoniam rectæ S a, T b, I C ſunt parallelæ, erunt triangula I C F, &amp; </s>
  <s xml:id="echoid-s4601" xml:space="preserve">S a F,
<pb o="115" file="0153" n="153" rhead="Conicor. Lib. V."/>
ſimilia; </s>
  <s xml:id="echoid-s4602" xml:space="preserve">pariterque duo triangula E F C, T b F ſimilia erunt; </s>
  <s xml:id="echoid-s4603" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4604" xml:space="preserve">propterea S a <lb/>ad a F eandem æqualitatis proportionem habebit, quàm I C habebat ad C F, ſi-<lb/>militer T b ad b F eandem proportionem habebit, quàm E C ad C F, ſeu quàm <lb/>latus tranſuerſum D C ad eius latus rectum: </s>
  <s xml:id="echoid-s4605" xml:space="preserve">eſtvero T b æqualis S a, ſeu a F; <lb/></s>
  <s xml:id="echoid-s4606" xml:space="preserve">ergo F a ad F b eandem proportionem habet, quàm latus tranſuerſum D C ad <lb/>eius latus rectum; </s>
  <s xml:id="echoid-s4607" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4608" xml:space="preserve">comparando antecedentes ad differentias terminorum.</s>
  <s xml:id="echoid-s4609" xml:space="preserve">, <lb/>
<anchor type="note" xlink:label="note-0153-01a" xlink:href="note-0153-01"/>
erit F a, ſeu b T ad b a, vt latus tranſuerſum D C ad differentiam eiuſdem <lb/>tranſuerſi, &amp; </s>
  <s xml:id="echoid-s4610" xml:space="preserve">recti lateris; </s>
  <s xml:id="echoid-s4611" xml:space="preserve">quare parallelogrammũ rectangulum S b, erit exem-<lb/>
<anchor type="note" xlink:label="note-0153-02a" xlink:href="note-0153-02"/>
plar applicatum ad abſciſſam O C.</s>
  <s xml:id="echoid-s4612" xml:space="preserve"/>
</p>
<div xml:id="echoid-div428" type="float" level="2" n="3">
<note position="right" xlink:label="note-0152-04" xlink:href="note-0152-04a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0153-01" xlink:href="note-0153-01a" xml:space="preserve">Lem. 1. <lb/>huius.</note>
<note position="right" xlink:label="note-0153-02" xlink:href="note-0153-02a" xml:space="preserve">Defin. 9. <lb/>huius.</note>
</div>
  <figure>
    <image file="0153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0153-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s4613" xml:space="preserve">Igitur I C maior eſt, quàm I H, &amp; </s>
  <s xml:id="echoid-s4614" xml:space="preserve">I H, quàm I K, &amp;</s>
  <s xml:id="echoid-s4615" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4616" xml:space="preserve">Eo quod abſciſ-<lb/>
<anchor type="note" xlink:label="note-0153-03a" xlink:href="note-0153-03"/>
ſa O C minor eſt, quàm C P, &amp; </s>
  <s xml:id="echoid-s4617" xml:space="preserve">C P minor, quàm C E: </s>
  <s xml:id="echoid-s4618" xml:space="preserve">ſuntque prædictæ ab-<lb/>ſciſſæ latera homologa exemplarium, quæ ad eaſdem abſciſſas applicantur; </s>
  <s xml:id="echoid-s4619" xml:space="preserve">at-<lb/>
<anchor type="note" xlink:label="note-0153-04a" xlink:href="note-0153-04"/>
que prædicta exemplaria ſimilia ſunt inter ſe, cum circa angulos rectos latera <lb/>habeant eandem proportionem, quàm latus tranſuerſum D C ad differentiam. <lb/></s>
  <s xml:id="echoid-s4620" xml:space="preserve">eiuſdem tranſuerſi, &amp; </s>
  <s xml:id="echoid-s4621" xml:space="preserve">recti lateris; </s>
  <s xml:id="echoid-s4622" xml:space="preserve">quare exceſſus quadrati I C ſupra quadra-<lb/>tum I H minus eſt exceſſu eiuſdem quadrati I C ſupra quadratum I K; </s>
  <s xml:id="echoid-s4623" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4624" xml:space="preserve">ad <lb/>huc minus exceſſu quadrati I C ſupra quadratum I B, &amp; </s>
  <s xml:id="echoid-s4625" xml:space="preserve">propterea recta I C <lb/>minori exceſſu ipſam I H ſuperabit, quàm ipſam I K; </s>
  <s xml:id="echoid-s4626" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4627" xml:space="preserve">adhuc minori exceſ-<lb/>ſu ſuperabit I K, quàm excedat I B; </s>
  <s xml:id="echoid-s4628" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4629" xml:space="preserve">ideo I C maior erit, quàm I H, &amp; </s>
  <s xml:id="echoid-s4630" xml:space="preserve">I <lb/>H maior, quàm I K, &amp; </s>
  <s xml:id="echoid-s4631" xml:space="preserve">I K maior, quàm I B.</s>
  <s xml:id="echoid-s4632" xml:space="preserve"/>
</p>
<div xml:id="echoid-div429" type="float" level="2" n="4">
<note position="left" xlink:label="note-0153-03" xlink:href="note-0153-03a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0153-04" xlink:href="note-0153-04a" xml:space="preserve">Defin. 9. <lb/>huius.</note>
</div>
  <figure>
    <image file="0153-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0153-02"/>
  </figure>
<pb o="116" file="0154" n="154" rhead="Apollonij Pergæi"/>
  <figure>
    <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0154-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s4633" xml:space="preserve">Ergo quadratum I C æquale eſt duplo trianguli N F M cum duplo <lb/>
<anchor type="note" xlink:label="note-0154-01a" xlink:href="note-0154-01"/>
trianguli D I N, &amp;</s>
  <s xml:id="echoid-s4634" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4635" xml:space="preserve">Quoniam quadratum I C æquale eſt duplo trianguli I <lb/>C F, ſeu duplo trianguli I F E vna cum duplo trianguli E F C; </s>
  <s xml:id="echoid-s4636" xml:space="preserve">eſtque duplum <lb/>trianguli E D M æquale duplo trianguli E C F; </s>
  <s xml:id="echoid-s4637" xml:space="preserve">igitur quadratum I C æquale <lb/>eſt duplo trianguli I F E vna cum duplo trianguli E M D: </s>
  <s xml:id="echoid-s4638" xml:space="preserve">ĳs vero triangulis <lb/>æquatur duplum trianguli N F M vna cum duplo trianguli D I N; </s>
  <s xml:id="echoid-s4639" xml:space="preserve">igitur qua-<lb/>dratum I C æquale eſt duplo trianguli N F M vna cum duplo trianguli D I N: <lb/></s>
  <s xml:id="echoid-s4640" xml:space="preserve">eſt vero quadratum I D æquale duplo trianguli D I N; </s>
  <s xml:id="echoid-s4641" xml:space="preserve">igitur exceſſus quadrati <lb/>I C ſupra quadratum I D eſt triangulum N F M bis ſumptum; </s>
  <s xml:id="echoid-s4642" xml:space="preserve">ſcilicet exem-<lb/>plar applicatum ad latus tranſuerſum D C.</s>
  <s xml:id="echoid-s4643" xml:space="preserve"/>
</p>
<div xml:id="echoid-div430" type="float" level="2" n="5">
<note position="right" xlink:label="note-0154-01" xlink:href="note-0154-01a" xml:space="preserve">e</note>
</div>
</div>
<div xml:id="echoid-div432" type="section" level="1" n="132">
<head xml:id="echoid-head177" xml:space="preserve">SECTIO DECIMASEPTIMA <lb/>Continens XIX. XX. XXI. XXII. XXIII. <lb/>XXIV. &amp; XXV. Propoſ. Apollonij.</head>
<head xml:id="echoid-head178" xml:space="preserve">PROPOSITIO XIX.</head>
<p>
  <s xml:id="echoid-s4644" xml:space="preserve">SI menſura E C ſumatur in axe minori ellipſis A B C, ſitque <lb/>
<anchor type="note" xlink:label="note-0154-02a" xlink:href="note-0154-02"/>
maior comparata; </s>
  <s xml:id="echoid-s4645" xml:space="preserve">erit maximus omniũ ramorũ egredientiũ <lb/>ex ſua origine, vt E F, E B, E G; </s>
  <s xml:id="echoid-s4646" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4647" xml:space="preserve">maximo propinquior, <lb/>maior erit remotiore, nempe E F, quàm E B, &amp; </s>
  <s xml:id="echoid-s4648" xml:space="preserve">E B, quàm E G.</s>
  <s xml:id="echoid-s4649" xml:space="preserve"/>
</p>
<div xml:id="echoid-div432" type="float" level="2" n="1">
<note position="right" xlink:label="note-0154-02" xlink:href="note-0154-02a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s4650" xml:space="preserve">Coniungamus rectas A G, G B, B F, <lb/>
<anchor type="figure" xlink:label="fig-0154-02a" xlink:href="fig-0154-02"/>
<anchor type="note" xlink:label="note-0154-03a" xlink:href="note-0154-03"/>
F C; </s>
  <s xml:id="echoid-s4651" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4652" xml:space="preserve">ſecetur C H æqualis compara-<lb/>tæ: </s>
  <s xml:id="echoid-s4653" xml:space="preserve">iungãturque F H, H B, H G.</s>
  <s xml:id="echoid-s4654" xml:space="preserve"/>
</p>
<div xml:id="echoid-div433" type="float" level="2" n="2">
  <figure xlink:label="fig-0154-02" xlink:href="fig-0154-02a">
    <image file="0154-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0154-02"/>
  </figure>
<note position="right" xlink:label="note-0154-03" xlink:href="note-0154-03a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s4655" xml:space="preserve">Et quoniam H C maior eſt, quàm H <lb/>F, (16. </s>
  <s xml:id="echoid-s4656" xml:space="preserve">17. </s>
  <s xml:id="echoid-s4657" xml:space="preserve">18. </s>
  <s xml:id="echoid-s4658" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4659" xml:space="preserve">erit angulus H C <lb/>F minor, quàm H F C; </s>
  <s xml:id="echoid-s4660" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4661" xml:space="preserve">ideo multo <lb/>minor erit, quàm E F C, quare E C <lb/>maior eſt, quàm E F: </s>
  <s xml:id="echoid-s4662" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4663" xml:space="preserve">ſic conſtat, quod <lb/>E F maior ſit, quàm E B, &amp; </s>
  <s xml:id="echoid-s4664" xml:space="preserve">E B, quàm <lb/>E G, &amp; </s>
  <s xml:id="echoid-s4665" xml:space="preserve">E G, quàm A E; </s>
  <s xml:id="echoid-s4666" xml:space="preserve">quod erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s4667" xml:space="preserve"/>
</p>
<pb o="117" file="0155" n="155" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div435" type="section" level="1" n="133">
<head xml:id="echoid-head179" xml:space="preserve">PROPOSITIO XX. XXI. <lb/>&amp; XXII.</head>
<p>
  <s xml:id="echoid-s4668" xml:space="preserve">SI in ellipſi A B C menſura I C in axe minori C D ſumpta <lb/>
<anchor type="note" xlink:label="note-0155-01a" xlink:href="note-0155-01"/>
minor fuerit comparata, C F, &amp; </s>
  <s xml:id="echoid-s4669" xml:space="preserve">maior dimidio axis E C, <lb/>( perficiaturque figura, vt antea ) dico, quod omnium ramorum <lb/>I A, I B, I K, I H, I C egredientium ex origine I maximus <lb/>
<anchor type="figure" xlink:label="fig-0155-01a" xlink:href="fig-0155-01"/>
eſt I B, cuius potentialis B G abſcindit à menſura verſus origi-<lb/>nem rectam G I, ad quàm inuerſa E G eandem proportionem <lb/>habet, quàm D C ad eius erectum; </s>
  <s xml:id="echoid-s4670" xml:space="preserve">Et quadratum maximi I B ſu-<lb/>perat quadratum cuiuslibet alterius rami I K exemplari applica-<lb/>to ad G P differentiam eorum abſciſſarum.</s>
  <s xml:id="echoid-s4671" xml:space="preserve"/>
</p>
<div xml:id="echoid-div435" type="float" level="2" n="1">
<note position="left" xlink:label="note-0155-01" xlink:href="note-0155-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0155-01" xlink:href="fig-0155-01a">
    <image file="0155-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0155-01"/>
  </figure>
</div>
  <figure>
    <image file="0155-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0155-02"/>
  </figure>
<pb o="118" file="0156" n="156" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s4672" xml:space="preserve">Quoniam proportio E G ad G I facta eſt, vt E C ad C F, nempè E <lb/>
<anchor type="note" xlink:label="note-0156-01a" xlink:href="note-0156-01"/>
G ad G V, erit G V æqualis G I; </s>
  <s xml:id="echoid-s4673" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4674" xml:space="preserve">propterea quadratum G I æquale. <lb/></s>
  <s xml:id="echoid-s4675" xml:space="preserve">eſt duplo trianguli G I V, &amp; </s>
  <s xml:id="echoid-s4676" xml:space="preserve">quadratum G B æquale eſt duplo trapezij <lb/>G F (1. </s>
  <s xml:id="echoid-s4677" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4678" xml:space="preserve">ergo quadratum I B æquale eſt duplo trianguli I C S cum <lb/>
<anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/>
duplo trianguli F S V; </s>
  <s xml:id="echoid-s4679" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4680" xml:space="preserve">ſic conſtat, quod quadratum I K æquale eſt du-<lb/>plo trianguli I C S cum duplo trapezij S L; </s>
  <s xml:id="echoid-s4681" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4682" xml:space="preserve">propterea quadrati I B ex-<lb/>ceſſus ſupra quadratũ I K æqualis erit duplo trianguli L T V, quæ æqua-<lb/>lia ſunt exemplari applicato ad G P (6. </s>
  <s xml:id="echoid-s4683" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4684" xml:space="preserve">atque ſic oſtendetur, quod <lb/>I B potentia ſuperat I H; </s>
  <s xml:id="echoid-s4685" xml:space="preserve">eſtque exceſſus exemplar applicatum ad G O, <lb/>&amp; </s>
  <s xml:id="echoid-s4686" xml:space="preserve">ſuperat quoque I A poteſtate, eſtque exceſſus æqualis exemplari ap-<lb/>plicato ad G Q; </s>
  <s xml:id="echoid-s4687" xml:space="preserve">eſt vero G O maior, quàm G P; </s>
  <s xml:id="echoid-s4688" xml:space="preserve">ergo I B maior eſt quã <lb/>I K, &amp; </s>
  <s xml:id="echoid-s4689" xml:space="preserve">quàm I H; </s>
  <s xml:id="echoid-s4690" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4691" xml:space="preserve">ſic oſtendetur, quod I B maior ſit, quàm I A; </s>
  <s xml:id="echoid-s4692" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4693" xml:space="preserve"><lb/>hoc erat oſtendendum.</s>
  <s xml:id="echoid-s4694" xml:space="preserve"/>
</p>
<div xml:id="echoid-div436" type="float" level="2" n="2">
<note position="right" xlink:label="note-0156-01" xlink:href="note-0156-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a">
    <image file="0156-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0156-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div438" type="section" level="1" n="134">
<head xml:id="echoid-head180" xml:space="preserve">PROPOSITIO XXIII. &amp; XXIV.</head>
<p>
  <s xml:id="echoid-s4695" xml:space="preserve">EContra, ſi maximi rami origo <lb/>
<anchor type="figure" xlink:label="fig-0156-02a" xlink:href="fig-0156-02"/>
<anchor type="note" xlink:label="note-0156-02a" xlink:href="note-0156-02"/>
ponatur in axi minore, at non in <lb/>cẽtro ellipſis, nec ſit menſura continet <lb/>cum ipſa menſura angulum acutum, <lb/>&amp; </s>
  <s xml:id="echoid-s4696" xml:space="preserve">eius inuerſa ad abſciſſam à poten-<lb/>tiali cum origine habet eandem pro-<lb/>portionem figuræ axis recti minoris: <lb/></s>
  <s xml:id="echoid-s4697" xml:space="preserve">ſi vero educatur ex centro, erit per-<lb/>pendicularis ſuper rectum.</s>
  <s xml:id="echoid-s4698" xml:space="preserve"/>
</p>
<div xml:id="echoid-div438" type="float" level="2" n="1">
  <figure xlink:label="fig-0156-02" xlink:href="fig-0156-02a">
    <image file="0156-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0156-02"/>
  </figure>
<note position="right" xlink:label="note-0156-02" xlink:href="note-0156-02a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s4699" xml:space="preserve">Sit ſectio elliptica A B C centrum D, &amp; </s>
  <s xml:id="echoid-s4700" xml:space="preserve">E origo, quæ ſit in axi mino-<lb/>
<anchor type="note" xlink:label="note-0156-03a" xlink:href="note-0156-03"/>
ri C A, &amp; </s>
  <s xml:id="echoid-s4701" xml:space="preserve">E F ramus omnium maximus; </s>
  <s xml:id="echoid-s4702" xml:space="preserve">erit vtique E C, vel maior
<pb o="119" file="0157" n="157" rhead="Conicor. Lib. V."/>
ſemierecto, aut æqualis, aut minor illo; </s>
  <s xml:id="echoid-s4703" xml:space="preserve">ſed ſi eſſet æqualis, aut maior eſ-<lb/>ſet quoque E C maximus ramorum (16. </s>
  <s xml:id="echoid-s4704" xml:space="preserve">17. </s>
  <s xml:id="echoid-s4705" xml:space="preserve">18. </s>
  <s xml:id="echoid-s4706" xml:space="preserve">19. </s>
  <s xml:id="echoid-s4707" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4708" xml:space="preserve">ergo C E mi-<lb/>nor eſt dimidio erecti, &amp; </s>
  <s xml:id="echoid-s4709" xml:space="preserve">ideo aliqua minor, quàm D C ad reſiduam vſq; <lb/></s>
  <s xml:id="echoid-s4710" xml:space="preserve">
<anchor type="note" xlink:label="note-0157-01a" xlink:href="note-0157-01"/>
ad E eandem proportionẽ habebit, quàm D C ad ſemiſſim erecti; </s>
  <s xml:id="echoid-s4711" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4712" xml:space="preserve">ſit D G <lb/>ad G E, &amp; </s>
  <s xml:id="echoid-s4713" xml:space="preserve">ex G ad axim ducamus perpendicularem: </s>
  <s xml:id="echoid-s4714" xml:space="preserve">hanc, dico, occur-<lb/>rere ſectioni in F; </s>
  <s xml:id="echoid-s4715" xml:space="preserve">alioquin occurrat ei in H, &amp; </s>
  <s xml:id="echoid-s4716" xml:space="preserve">iungamus E H; </s>
  <s xml:id="echoid-s4717" xml:space="preserve">igitur E <lb/>H eſt maximus ramus (20. </s>
  <s xml:id="echoid-s4718" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4719" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4720" xml:space="preserve">propterea maior, quàm E F, qui <lb/>maximus ſuppoſitus fuit, &amp; </s>
  <s xml:id="echoid-s4721" xml:space="preserve">hoc eſt abſurdum; </s>
  <s xml:id="echoid-s4722" xml:space="preserve">igitur occurrit ſectioni in <lb/>F; </s>
  <s xml:id="echoid-s4723" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4724" xml:space="preserve">quia G eſt rectus angulus, erit F E G acutus. </s>
  <s xml:id="echoid-s4725" xml:space="preserve">Siverò ramus maxi-<lb/>
<anchor type="note" xlink:label="note-0157-02a" xlink:href="note-0157-02"/>
mus educatur ex cẽtro, vt D B erit perpendicularis ſuper A C; </s>
  <s xml:id="echoid-s4726" xml:space="preserve">alioquin <lb/>educatur D I perpẽdicularis ad axim; </s>
  <s xml:id="echoid-s4727" xml:space="preserve">igitur D I eſt ſemiſsis axis tranſuer-<lb/>ſi (11. </s>
  <s xml:id="echoid-s4728" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4729" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4730" xml:space="preserve">propterea eſt ramus omnium maximus, ſed D B ſuppo-<lb/>ſitus fuit maximus, quod eſt abſurdum, vti dictum eſt; </s>
  <s xml:id="echoid-s4731" xml:space="preserve">quare patet pro-<lb/>poſitum.</s>
  <s xml:id="echoid-s4732" xml:space="preserve"/>
</p>
<div xml:id="echoid-div439" type="float" level="2" n="2">
<note position="right" xlink:label="note-0156-03" xlink:href="note-0156-03a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0157-01" xlink:href="note-0157-01a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0157-02" xlink:href="note-0157-02a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div441" type="section" level="1" n="135">
<head xml:id="echoid-head181" xml:space="preserve">PROPOSITIO XXV.</head>
<p>
  <s xml:id="echoid-s4733" xml:space="preserve">SI in ellipſi ramus <lb/>
<anchor type="note" xlink:label="note-0157-03a" xlink:href="note-0157-03"/>
<anchor type="figure" xlink:label="fig-0157-01a" xlink:href="fig-0157-01"/>
maximus E B mẽ-<lb/>ſuram ſecans vltra ori-<lb/>ginem E, in axe eius <lb/>minori exiſtentem, pro-<lb/>ducatur ad F, fiet F B <lb/>maximus omniũ ramo-<lb/>rum F G, F H, FI, ab <lb/>eodem puncto, ad ſe-<lb/>ctionem A B C caden-<lb/>tium, &amp; </s>
  <s xml:id="echoid-s4734" xml:space="preserve">propinquior <lb/>maximo maior eſt remotiore.</s>
  <s xml:id="echoid-s4735" xml:space="preserve"/>
</p>
<div xml:id="echoid-div441" type="float" level="2" n="1">
<note position="left" xlink:label="note-0157-03" xlink:href="note-0157-03a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a">
    <image file="0157-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0157-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s4736" xml:space="preserve">Educamus B G, B H, H I, I A, E G, E H, E I; </s>
  <s xml:id="echoid-s4737" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4738" xml:space="preserve">quia E B maior <lb/>
<anchor type="note" xlink:label="note-0157-04a" xlink:href="note-0157-04"/>
eſt, quàm E H, erit angulus B H E maior, quàm E B H; </s>
  <s xml:id="echoid-s4739" xml:space="preserve">igitur angulus <lb/>B H F multo maior erit, quàm H B F, &amp; </s>
  <s xml:id="echoid-s4740" xml:space="preserve">propterea B F maior, eſt quàm <lb/>F H; </s>
  <s xml:id="echoid-s4741" xml:space="preserve">atque ſic demonſtrabitur, quod H F maior ſit, quàm F I, &amp; </s>
  <s xml:id="echoid-s4742" xml:space="preserve">F I, <lb/>quàm F A; </s>
  <s xml:id="echoid-s4743" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4744" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s4745" xml:space="preserve"/>
</p>
<div xml:id="echoid-div442" type="float" level="2" n="2">
<note position="left" xlink:label="note-0157-04" xlink:href="note-0157-04a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div444" type="section" level="1" n="136">
<head xml:id="echoid-head182" xml:space="preserve">Notæ in Propoſit. XIX.</head>
<p style="it">
  <s xml:id="echoid-s4746" xml:space="preserve">SI vero fuerit menſura E C ex recto duorum axium ellipſis A B C, <lb/>
<anchor type="note" xlink:label="note-0157-05a" xlink:href="note-0157-05"/>
fed ſit maior comparata, &amp;</s>
  <s xml:id="echoid-s4747" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4748" xml:space="preserve">Similiter bic declarari debet, quod axis <lb/>rectus ſit minor; </s>
  <s xml:id="echoid-s4749" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4750" xml:space="preserve">propterea lego: </s>
  <s xml:id="echoid-s4751" xml:space="preserve">Si menſura E C ſumatur in axe minori <lb/>ellipſis, &amp;</s>
  <s xml:id="echoid-s4752" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4753" xml:space="preserve"/>
</p>
<div xml:id="echoid-div444" type="float" level="2" n="1">
<note position="left" xlink:label="note-0157-05" xlink:href="note-0157-05a" xml:space="preserve">a</note>
</div>
<pb o="120" file="0158" n="158" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s4754" xml:space="preserve">Nam ſi coniungamus A G, B G, B F, <lb/>
<anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/>
<anchor type="note" xlink:label="note-0158-01a" xlink:href="note-0158-01"/>
F C, &amp;</s>
  <s xml:id="echoid-s4755" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4756" xml:space="preserve">Ideſt; </s>
  <s xml:id="echoid-s4757" xml:space="preserve">ſecetur C H æqualis com-<lb/>paratæ, ſeu ſemiſsi lateris recti axis A C; <lb/></s>
  <s xml:id="echoid-s4758" xml:space="preserve">quia menſura E C ſuppoſita eſt maior compa-<lb/>rata, erit quoque E C maior, quàm C H, &amp; </s>
  <s xml:id="echoid-s4759" xml:space="preserve"><lb/>propterea recta linea E F cadet infra H F; </s>
  <s xml:id="echoid-s4760" xml:space="preserve"><lb/>ideoque angulus C F E maior erit angulo C <lb/>F H: </s>
  <s xml:id="echoid-s4761" xml:space="preserve">eadem ratione angulus F B E maior <lb/>erit angulo F B H, atque angulus B F E mi-<lb/>nor erit angulo B F H, &amp; </s>
  <s xml:id="echoid-s4762" xml:space="preserve">ſic de reliquis, <lb/>cumque C H ſit æqualis comparatæ, &amp; </s>
  <s xml:id="echoid-s4763" xml:space="preserve">ſit <lb/>maior C D ſemiße axis recti minoris, omnium ramorum ex origine H ad elli-<lb/>
<anchor type="note" xlink:label="note-0158-02a" xlink:href="note-0158-02"/>
pſim C F B G, cadentium maximus erit H C; </s>
  <s xml:id="echoid-s4764" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4765" xml:space="preserve">propterea H C maior erit, <lb/>quàm H F, &amp; </s>
  <s xml:id="echoid-s4766" xml:space="preserve">in triangulo H F C angulus H F C oppoſitus maiori lateri ma-<lb/>ior erit angulo C; </s>
  <s xml:id="echoid-s4767" xml:space="preserve">eſtque oſtenſus angulus E F C maior angulo H F C; </s>
  <s xml:id="echoid-s4768" xml:space="preserve">igitur <lb/>in triangulo C E F erit angulus C F E maior angulo F C E; </s>
  <s xml:id="echoid-s4769" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4770" xml:space="preserve">propterea ra-<lb/>mus E C maior erit, quàm E F: </s>
  <s xml:id="echoid-s4771" xml:space="preserve">ſimili modo, quia ramus H F propinquior ma-<lb/>
<anchor type="note" xlink:label="note-0158-03a" xlink:href="note-0158-03"/>
ximo maior eſt remotiore H B, erit angulus H F B minor angulo H B F: </s>
  <s xml:id="echoid-s4772" xml:space="preserve">ideo-<lb/>que angulus E F B, pars minoris, adbuc minor erit angulo E B F, maiorem <lb/>excedente; </s>
  <s xml:id="echoid-s4773" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4774" xml:space="preserve">propterea in triangulo E F B erit ramus E F propinquior maxi-<lb/>mo E C, maior remotiore E B, &amp;</s>
  <s xml:id="echoid-s4775" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4776" xml:space="preserve"/>
</p>
<div xml:id="echoid-div445" type="float" level="2" n="2">
  <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a">
    <image file="0158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0158-01"/>
  </figure>
<note position="right" xlink:label="note-0158-01" xlink:href="note-0158-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0158-02" xlink:href="note-0158-02a" xml:space="preserve">16. 17. 18. <lb/>huius.</note>
<note position="left" xlink:label="note-0158-03" xlink:href="note-0158-03a" xml:space="preserve">Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div447" type="section" level="1" n="137">
<head xml:id="echoid-head183" xml:space="preserve">Notæ in Propoſit. XX. XXI. XXII.</head>
<p>
  <s xml:id="echoid-s4777" xml:space="preserve">SI vero fuerit menſura I C minor comparata, quæ ſit C F, nempe ſe-<lb/>
<anchor type="note" xlink:label="note-0158-04a" xlink:href="note-0158-04"/>
miſſe erecti, &amp; </s>
  <s xml:id="echoid-s4778" xml:space="preserve">maior dimidio recti E C, &amp; </s>
  <s xml:id="echoid-s4779" xml:space="preserve">origo ſit in recto, aut in <lb/>eius productione, vt in I; </s>
  <s xml:id="echoid-s4780" xml:space="preserve">tunc maximus ramorum egredientium ex origi-<lb/>ne, vt I A, I B, I K, I H eſt cuius inuerſi proportio E G (poſt abſolu-<lb/>tionem figuræ cum perpendicularibus, &amp; </s>
  <s xml:id="echoid-s4781" xml:space="preserve">lineis præcedentibus) ad ab-<lb/>
<anchor type="figure" xlink:label="fig-0158-02a" xlink:href="fig-0158-02"/>
ſciſſam eius potentialis ex menſura cum origine, vt I G eſt, vt propor-<lb/>tio figuræ recti, vt D C ad erectum illius, &amp; </s>
  <s xml:id="echoid-s4782" xml:space="preserve">quadratum eius, nẽpe qua-
<pb o="121" file="0159" n="159" rhead="Conicor. Lib. V."/>
dratum maximi, qui eſt I B, ſuperat quadratum cuiuslibet illorum exem-<lb/>plari applicato abſciſſionibus eorum potentialium, &amp;</s>
  <s xml:id="echoid-s4783" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4784" xml:space="preserve">Senſus buius tex-<lb/>tus penè vix diuinari poteſt inter tot menda, &amp; </s>
  <s xml:id="echoid-s4785" xml:space="preserve">phraſis Arabicæ obſcuritatem; <lb/></s>
  <s xml:id="echoid-s4786" xml:space="preserve">puto tamen, eum eſſe, quem in textu appoſui, vbi paucula verba immutaui, <lb/>quæ deſiderari videbantur, aliqua verò tranſpoſui, vt ſenſus continuari poſſet.</s>
  <s xml:id="echoid-s4787" xml:space="preserve"/>
</p>
<div xml:id="echoid-div447" type="float" level="2" n="1">
<note position="right" xlink:label="note-0158-04" xlink:href="note-0158-04a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0158-02" xlink:href="fig-0158-02a">
    <image file="0158-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0158-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s4788" xml:space="preserve">Cæterum animaduertendum eſt in biſce propoſitionibus, ſicuti in 8. </s>
  <s xml:id="echoid-s4789" xml:space="preserve">9. </s>
  <s xml:id="echoid-s4790" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4791" xml:space="preserve">10. <lb/></s>
  <s xml:id="echoid-s4792" xml:space="preserve">buius libri ſupponi vt res manifeſta intra ſectionem duci poſſe à puncto originis <lb/>ramum maximum, vel breuiſsimum, ideſt neceſſario reperiri debere ramum, <lb/>cuius potentialis abſcindit à menſura verſus originem rectam lineam, ad quàm <lb/>inuerſa eandem proportionem babeant quàm axis tranſuerſus ad ſuum erectum: </s>
  <s xml:id="echoid-s4793" xml:space="preserve"><lb/>boc autem ſine demonſtratione admittere nefas eſt. </s>
  <s xml:id="echoid-s4794" xml:space="preserve">Ergo quod in textu deſidera-<lb/>tur ſuppleri poteſt bac ratione. </s>
  <s xml:id="echoid-s4795" xml:space="preserve">Quia C I maior eſt, quàm C E, ſed minor, <lb/>quàm C F; </s>
  <s xml:id="echoid-s4796" xml:space="preserve">ergo eadem E C ad minorem C I maiorem proportionẽ babet, quàm <lb/>ad C F; </s>
  <s xml:id="echoid-s4797" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4798" xml:space="preserve">comparando antecedentes ad differentias terminorum C E ad E I <lb/>maiorem proportionem babebit, quàm E C ad differentiam ipſius C F à C E; </s>
  <s xml:id="echoid-s4799" xml:space="preserve"><lb/>quare aliqua magnitudo minor quàm prima ſcilicet G E ad E I eandem propor-<lb/>tionem habebit, quàm C E ad differentiam ipſarum C F, &amp; </s>
  <s xml:id="echoid-s4800" xml:space="preserve">C E: </s>
  <s xml:id="echoid-s4801" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4802" xml:space="preserve">iterum <lb/>comparando antecedentes ad ſummas terminorum E G ad G I eandem proportio-<lb/>nem babebit, quàm E C ad C F; </s>
  <s xml:id="echoid-s4803" xml:space="preserve">quare punctum G cadet intra ſectionem, pa-<lb/>riterq; </s>
  <s xml:id="echoid-s4804" xml:space="preserve">G B ad axim perpendicularis occurrens ſectioni in B cadet intra eandem <lb/>ſectionem: </s>
  <s xml:id="echoid-s4805" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4806" xml:space="preserve">ideo duci poterit ramus I B, qui oſtendetur maximus reliquorum <lb/>omnium.</s>
  <s xml:id="echoid-s4807" xml:space="preserve"/>
</p>
  <figure>
    <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0159-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s4808" xml:space="preserve">Quoniam proportio G E ad E I facta eſt, vt E C ad C F, &amp;</s>
  <s xml:id="echoid-s4809" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4810" xml:space="preserve">Nam <lb/>
<anchor type="note" xlink:label="note-0159-01a" xlink:href="note-0159-01"/>
vt axis D C ad eius erectum, ſeu vt ſemiaxis E C ad ſemierectum C F, ita <lb/>facta eſt E G ad G I: </s>
  <s xml:id="echoid-s4811" xml:space="preserve">ſed propter parallelas G V, &amp; </s>
  <s xml:id="echoid-s4812" xml:space="preserve">F C: </s>
  <s xml:id="echoid-s4813" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4814" xml:space="preserve">ſimilitudinem <lb/>triangulorum E G V, E C F eſt E G ad G V, vt E C ad C F; </s>
  <s xml:id="echoid-s4815" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4816" xml:space="preserve">propterea <lb/>eadem E G ad duas G V, &amp; </s>
  <s xml:id="echoid-s4817" xml:space="preserve">G I babebit eandem proportionem, &amp; </s>
  <s xml:id="echoid-s4818" xml:space="preserve">ideo I G æ-<lb/>qualis erit G V, &amp; </s>
  <s xml:id="echoid-s4819" xml:space="preserve">triangulum I G V iſoſceleum, &amp; </s>
  <s xml:id="echoid-s4820" xml:space="preserve">rectangulum erit in G; <lb/></s>
  <s xml:id="echoid-s4821" xml:space="preserve">quare quadratum I G duplum erit trianguli I G V: </s>
  <s xml:id="echoid-s4822" xml:space="preserve">eſt verò quadratum B G <lb/>æquale duplo trapezĳ G C F V; </s>
  <s xml:id="echoid-s4823" xml:space="preserve">ideſt duplo trapezĳ G C S V, cum duplo trian-<lb/>
<anchor type="note" xlink:label="note-0159-02a" xlink:href="note-0159-02"/>
guli F S V; </s>
  <s xml:id="echoid-s4824" xml:space="preserve">igitur quadratum I B (quod eſt æquale duobus quadratis I G, G <lb/>B circa angulum rectum G) æquale eſt duplo trianguli I G V duplo trapezĳ G
<pb o="122" file="0160" n="160" rhead="Apollonij Pergæi"/>
C S V cum duplo trianguli F S V; </s>
  <s xml:id="echoid-s4825" xml:space="preserve">ideſt quadratum I B æquale eſt duplo trian-<lb/>guli I S C cum duplo trianguli F S V; </s>
  <s xml:id="echoid-s4826" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4827" xml:space="preserve">quoniam propter parallelas C S, &amp; </s>
  <s xml:id="echoid-s4828" xml:space="preserve"><lb/>G V, triangulum I C S ſimile eſt iſoſcelio, &amp; </s>
  <s xml:id="echoid-s4829" xml:space="preserve">rectangulo triangulo I G V, erit, <lb/>quadratum I C æquale duplo trianguli I C S iſoſcelei, &amp; </s>
  <s xml:id="echoid-s4830" xml:space="preserve">rectanguli in C; </s>
  <s xml:id="echoid-s4831" xml:space="preserve">ergo <lb/>exceſſus quadrati I B ſupra quadratum I C æquale eſt duplo trianguli F S V; <lb/></s>
  <s xml:id="echoid-s4832" xml:space="preserve">eſt verò rectangulum, cuius baſis F S, altitudo verò C G æquale duplo trianguli <lb/>F S V; </s>
  <s xml:id="echoid-s4833" xml:space="preserve">atque buiuſmodi rectangulum eſt exemplar applicatum ad abſciſſam G <lb/>C, vt in notis prop. </s>
  <s xml:id="echoid-s4834" xml:space="preserve">16. </s>
  <s xml:id="echoid-s4835" xml:space="preserve">17. </s>
  <s xml:id="echoid-s4836" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4837" xml:space="preserve">18. </s>
  <s xml:id="echoid-s4838" xml:space="preserve">litera c. </s>
  <s xml:id="echoid-s4839" xml:space="preserve">oſtenſum eſt igitur quadrati I B <lb/>exceßus ſupra quadratum I C eſt exemplar applicatum ad abſciſſam G C: </s>
  <s xml:id="echoid-s4840" xml:space="preserve">Simili <lb/>
<anchor type="figure" xlink:label="fig-0160-01a" xlink:href="fig-0160-01"/>
modo quadratum I K oſtendetur æquale duplo trianguli I C S vna cum duplo <lb/>trapezĳ L T S F; </s>
  <s xml:id="echoid-s4841" xml:space="preserve">atque dupli trianguli I C S cum duplo trianguli F S V ex-<lb/>ceſſus ſupra duplum trianguli I C S cum duplo trapezĳ L T S F eſt duplum <lb/>trianguli L T V; </s>
  <s xml:id="echoid-s4842" xml:space="preserve">ergo quadrati I B exceſſus ſupra quadratum I K eſt duplum <lb/>trianguli L T V, ſeu exemplar applicatum ad G P differentiam abſciſſarum. <lb/></s>
  <s xml:id="echoid-s4843" xml:space="preserve">Poſtea quia triangula ſimilia E C F, E D M ſunt æqualia, cum eorum bomologa <lb/>latera E C, E D æqualia ſint; </s>
  <s xml:id="echoid-s4844" xml:space="preserve">ergo addito communi triangulo I E V, erit trian-<lb/>gulum E C F cum triangulo E I V, ſeu triangulũ I C S cum triangulo F S V <lb/>æquale duobus triaugulis E D M, &amp; </s>
  <s xml:id="echoid-s4845" xml:space="preserve">I E V, ſeu duobus triangulis M V N, &amp; </s>
  <s xml:id="echoid-s4846" xml:space="preserve"><lb/>N I D: </s>
  <s xml:id="echoid-s4847" xml:space="preserve">erat autem quadratum I B æquale duplo trianguli I C S cum duplo tri-<lb/>anguli F S V; </s>
  <s xml:id="echoid-s4848" xml:space="preserve">igitur quadratum I B æquale erit duplo trianguli M N V cum <lb/>duplo trianguli N I D; </s>
  <s xml:id="echoid-s4849" xml:space="preserve">eſtque quadratum I D æquale duplo trianguli iſoſcelei, <lb/>rectanguli I D N; </s>
  <s xml:id="echoid-s4850" xml:space="preserve">igitur quadratum I B ſuperat quadratum I D, eſtque exceſ-<lb/>ſus duplum trianguli M N V ſeu exemplar applicatum ad G D. </s>
  <s xml:id="echoid-s4851" xml:space="preserve">Tandem quia <lb/>quadratum I Q æquale eſt duplo trianguli iſoſcelei rectanguli I Q X, atque <lb/>quadratum Q A æquale eſt duplo trapezĳ Q M; </s>
  <s xml:id="echoid-s4852" xml:space="preserve">igitur quadratũ bypotbenuſæ I <lb/>A æquale eſt duplo trianguli I D N cum duplo trapezĳ X N M Z; </s>
  <s xml:id="echoid-s4853" xml:space="preserve">ergo exceſ-<lb/>ſus quadrati I A ſupra quadratnm I D æqualis eſt duplo trapezĳ X N M Z; </s>
  <s xml:id="echoid-s4854" xml:space="preserve">exceſ-<lb/>ſus autem trianguli N M V ſupra trapezium N Z eſt triangulum X Z V; </s>
  <s xml:id="echoid-s4855" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4856" xml:space="preserve"><lb/>erat quadrati I B exceſſus ſupra quadratum I D, triangulum ipſum M V N bis <lb/>ſumptum. </s>
  <s xml:id="echoid-s4857" xml:space="preserve">Igitur quadrati I B exceſſus ſupra quadratum I A eſt duplum trian-<lb/>guli X Z V, ſeu exemplar applicatum ad G Q. </s>
  <s xml:id="echoid-s4858" xml:space="preserve">Quod autem exemplaria æqualia <lb/>ſint prædictis triangulis bis ſumptis, oſtenſum eſt in prop. </s>
  <s xml:id="echoid-s4859" xml:space="preserve">6. </s>
  <s xml:id="echoid-s4860" xml:space="preserve">buius.</s>
  <s xml:id="echoid-s4861" xml:space="preserve"/>
</p>
<div xml:id="echoid-div448" type="float" level="2" n="2">
<note position="left" xlink:label="note-0159-01" xlink:href="note-0159-01a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0159-02" xlink:href="note-0159-02a" xml:space="preserve">1. huius.</note>
  <figure xlink:label="fig-0160-01" xlink:href="fig-0160-01a">
    <image file="0160-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0160-01"/>
  </figure>
</div>
<pb o="123" file="0161" n="161" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div450" type="section" level="1" n="138">
<head xml:id="echoid-head184" xml:space="preserve">Notæ in Propoſ. XXIII. XXIV.</head>
<p>
  <s xml:id="echoid-s4862" xml:space="preserve">EContra linea maxima, ſi non egredia-<lb/>
<anchor type="note" xlink:label="note-0161-01a" xlink:href="note-0161-01"/>
<anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/>
tur ex centro, continet cum mẽſura <lb/>angulum acutum, &amp; </s>
  <s xml:id="echoid-s4863" xml:space="preserve">proportio illius in-<lb/>uerſæ ad abſciſſam eius potentialis ex mẽ-<lb/>ſura cum origine, eſt vt proportio figuræ <lb/>recti. </s>
  <s xml:id="echoid-s4864" xml:space="preserve">Si verò fuerit extra centrum, erit <lb/>perpendicularis ſuper rectum, &amp;</s>
  <s xml:id="echoid-s4865" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4866" xml:space="preserve">Mani-<lb/>feſtè nõ nulla in textu Arabicc<unsure/> deficiunt; </s>
  <s xml:id="echoid-s4867" xml:space="preserve">ali-<lb/>qua verò immutari debent; </s>
  <s xml:id="echoid-s4868" xml:space="preserve">alioquin propo-<lb/>ſitio vera non eſſet, itaque legendum puto: </s>
  <s xml:id="echoid-s4869" xml:space="preserve">E <lb/>contra ſi maximi rami origo ponatur in axi <lb/>minore, &amp;</s>
  <s xml:id="echoid-s4870" xml:space="preserve">c: </s>
  <s xml:id="echoid-s4871" xml:space="preserve">Vt in textu babetur.</s>
  <s xml:id="echoid-s4872" xml:space="preserve"/>
</p>
<div xml:id="echoid-div450" type="float" level="2" n="1">
<note position="left" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a">
    <image file="0161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0161-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s4873" xml:space="preserve">Sit ſectio A B C elliptica, &amp; </s>
  <s xml:id="echoid-s4874" xml:space="preserve">E origo, &amp; </s>
  <s xml:id="echoid-s4875" xml:space="preserve">E F linea maxima, &amp;</s>
  <s xml:id="echoid-s4876" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4877" xml:space="preserve">Ad-<lb/>
<anchor type="note" xlink:label="note-0161-02a" xlink:href="note-0161-02"/>
didi pariter in bac expoſitione verba, quæ deſiciunt; </s>
  <s xml:id="echoid-s4878" xml:space="preserve">nimirum: </s>
  <s xml:id="echoid-s4879" xml:space="preserve">Sit centrum <lb/>D, &amp; </s>
  <s xml:id="echoid-s4880" xml:space="preserve">origo E, quæ ſit in axi minori A C.</s>
  <s xml:id="echoid-s4881" xml:space="preserve"/>
</p>
<div xml:id="echoid-div451" type="float" level="2" n="2">
<note position="left" xlink:label="note-0161-02" xlink:href="note-0161-02a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s4882" xml:space="preserve">Et ideo D C ad dimidium erecti eſt linea minor, quàm D C, &amp; </s>
  <s xml:id="echoid-s4883" xml:space="preserve">ſit D <lb/>
<anchor type="note" xlink:label="note-0161-03a" xlink:href="note-0161-03"/>
G ad G E, &amp;</s>
  <s xml:id="echoid-s4884" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4885" xml:space="preserve">Nonnulla adiungi debent buic textui corruptiſsimo, ne ſint <lb/>verba nil prorſus ſignificantia, itaque ſic legendum puto. </s>
  <s xml:id="echoid-s4886" xml:space="preserve">Et ideo aliqua minor, <lb/>quàm D C ad reſiduam vſque ad E eandem proportionem babebit, quàm D C ad <lb/>ſemiſſem erecti; </s>
  <s xml:id="echoid-s4887" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4888" xml:space="preserve">ſit D G ad G E, &amp;</s>
  <s xml:id="echoid-s4889" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4890" xml:space="preserve">Quæ verba breuiſsimè more Apollonĳ <lb/>expoſita ſic confirmantur. </s>
  <s xml:id="echoid-s4891" xml:space="preserve">Quia E C oſtenſa eſt minor dimidio erecti axis mi-<lb/>noris C A, fiat C K æqualis dimidio erecti; </s>
  <s xml:id="echoid-s4892" xml:space="preserve">erit E C minor quàm C K, <lb/>&amp; </s>
  <s xml:id="echoid-s4893" xml:space="preserve">ablata communi D C erit D E minor, quàm K D; </s>
  <s xml:id="echoid-s4894" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4895" xml:space="preserve">propterea D E ad ean-<lb/>dem D C minorem proportionem babebit, quàm K D: </s>
  <s xml:id="echoid-s4896" xml:space="preserve">fiat E D a d D G, vt K <lb/>D ad D C, erit D G minor, quàm D C: </s>
  <s xml:id="echoid-s4897" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4898" xml:space="preserve">componendo, E G ad G D eandem <lb/>proportionem babebit, quàm K C ad C D, &amp; </s>
  <s xml:id="echoid-s4899" xml:space="preserve">inuertendo, D G ad G E eandem <lb/>proportionem babebit, quàm D C ſemiſsis axis recti ad C K ſemiſsim erecti <lb/>eiuſdem axis; </s>
  <s xml:id="echoid-s4900" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4901" xml:space="preserve">ex G ducatur G F perpendicularis ad axim, quàm, dico, oc-<lb/>currere ſectioni in F termino maximi rami E F.</s>
  <s xml:id="echoid-s4902" xml:space="preserve"/>
</p>
<div xml:id="echoid-div452" type="float" level="2" n="3">
<note position="left" xlink:label="note-0161-03" xlink:href="note-0161-03a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s4903" xml:space="preserve">Et ſi maxima fuerit extra centrum, vt D B erit perpendicularis, &amp;</s>
  <s xml:id="echoid-s4904" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s4905" xml:space="preserve">
<anchor type="note" xlink:label="note-0161-04a" xlink:href="note-0161-04"/>
Textus euidenter corruptus ſic corrigi debet. </s>
  <s xml:id="echoid-s4906" xml:space="preserve">Si verò ramus maximus educatur <lb/>ex centro, vt D B, &amp;</s>
  <s xml:id="echoid-s4907" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4908" xml:space="preserve"/>
</p>
<div xml:id="echoid-div453" type="float" level="2" n="4">
<note position="left" xlink:label="note-0161-04" xlink:href="note-0161-04a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div455" type="section" level="1" n="139">
<head xml:id="echoid-head185" xml:space="preserve">Notæ in Propoſ. XXXV.</head>
<p>
  <s xml:id="echoid-s4909" xml:space="preserve">SI producatur vna linearum maximarum, vt E B ad latus illius originis <lb/>
<anchor type="note" xlink:label="note-0161-05a" xlink:href="note-0161-05"/>
E ad punctum F, fiet maxima linearum egredientium ab illo puncto <lb/>F G, F H, F I, F A ad ſectionem B I A in directum, &amp; </s>
  <s xml:id="echoid-s4910" xml:space="preserve">propinquior illi <lb/>maior eſt remotiore, &amp;</s>
  <s xml:id="echoid-s4911" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4912" xml:space="preserve">Immutaui nonnulla, quæ ad propoſitionis integrita-<lb/>tem facerc videbantur: </s>
  <s xml:id="echoid-s4913" xml:space="preserve">vt in textu babetur.</s>
  <s xml:id="echoid-s4914" xml:space="preserve"/>
</p>
<div xml:id="echoid-div455" type="float" level="2" n="1">
<note position="left" xlink:label="note-0161-05" xlink:href="note-0161-05a" xml:space="preserve">a</note>
</div>
<pb o="124" file="0162" n="162" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s4915" xml:space="preserve">Erit angulus BHE ma-<lb/>
<anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/>
<anchor type="note" xlink:label="note-0162-01a" xlink:href="note-0162-01"/>
ior, i<unsure/>quàm E B H, &amp;</s>
  <s xml:id="echoid-s4916" xml:space="preserve">c. </s>
  <s xml:id="echoid-s4917" xml:space="preserve">Eo <lb/>quod ramorum omnium ab <lb/>origine E ad ellipſim C B H <lb/>cadentium maximus ſuppo-<lb/>nitur E B; </s>
  <s xml:id="echoid-s4918" xml:space="preserve">ergo maior erit, <lb/>quàm E H, &amp; </s>
  <s xml:id="echoid-s4919" xml:space="preserve">propterea <lb/>angulus E B H minori late-<lb/>ri oppoſitus minor erit angu-<lb/>lo E H B: </s>
  <s xml:id="echoid-s4920" xml:space="preserve">cadit vero recta <lb/>H F infra H E; </s>
  <s xml:id="echoid-s4921" xml:space="preserve">propterea <lb/>quod punctum F infra pun-<lb/>ctum E exiſtit; </s>
  <s xml:id="echoid-s4922" xml:space="preserve">igitur angu-<lb/>lus F H B maior eſt angulo <lb/>E H B; </s>
  <s xml:id="echoid-s4923" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4924" xml:space="preserve">ideo angulus F H B multo maior erit angulo F B H; </s>
  <s xml:id="echoid-s4925" xml:space="preserve">igitur ramus <lb/>F B, maiorem angulum ſubtendens, maior erit, quàm F H, &amp;</s>
  <s xml:id="echoid-s4926" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4927" xml:space="preserve"/>
</p>
<div xml:id="echoid-div456" type="float" level="2" n="2">
  <figure xlink:label="fig-0162-01" xlink:href="fig-0162-01a">
    <image file="0162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0162-01"/>
  </figure>
<note position="right" xlink:label="note-0162-01" xlink:href="note-0162-01a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div458" type="section" level="1" n="140">
<head xml:id="echoid-head186" xml:space="preserve">SECTIO DECIMAOCTAVA</head>
<head xml:id="echoid-head187" xml:space="preserve">Continens XXXII. XXXIII. XXXIV. XXXV. <lb/>XXXVI. XXXVII. XXXVIII. XXXIX. <lb/>XXXX. XXXXVII. XXXXVIII.</head>
<head xml:id="echoid-head188" xml:space="preserve">Propoſit. Apollonij.</head>
<head xml:id="echoid-head189" xml:space="preserve">PROPOSITIO XXXII.</head>
<p>
  <s xml:id="echoid-s4928" xml:space="preserve">IN ellipſi A B C rami cuiuslibet <lb/>
<anchor type="figure" xlink:label="fig-0162-02a" xlink:href="fig-0162-02"/>
<anchor type="note" xlink:label="note-0162-02a" xlink:href="note-0162-02"/>
maximi G H vtrumque axim ſe-<lb/>cantis portio N H inter axim maio-<lb/>rem, &amp; </s>
  <s xml:id="echoid-s4929" xml:space="preserve">ſectionem intercepta, eſt li-<lb/>nea breuiſsima.</s>
  <s xml:id="echoid-s4930" xml:space="preserve"/>
</p>
<div xml:id="echoid-div458" type="float" level="2" n="1">
  <figure xlink:label="fig-0162-02" xlink:href="fig-0162-02a">
    <image file="0162-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0162-02"/>
  </figure>
<note position="right" xlink:label="note-0162-02" xlink:href="note-0162-02a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s4931" xml:space="preserve">Producatur rectus axis minor A D vl-<lb/>tra centrum D ad I, G, &amp; </s>
  <s xml:id="echoid-s4932" xml:space="preserve">ex I, G ad ſe-<lb/>ctionem ducantur duo rami maximi G H, <lb/>I K, qui ſecent tranſuerſum B D in N, <lb/>M, &amp; </s>
  <s xml:id="echoid-s4933" xml:space="preserve">ſit B E dimidium erecti axis B D, <lb/>&amp; </s>
  <s xml:id="echoid-s4934" xml:space="preserve">A F dimidium erecti axis A G; </s>
  <s xml:id="echoid-s4935" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4936" xml:space="preserve">edu-<lb/>cantur perpendiculares ad axes H O, H <lb/>P, K Q, K R. </s>
  <s xml:id="echoid-s4937" xml:space="preserve">Dico, N H breuiſsimum <lb/>eſſe ramorum egredientium ex H. </s>
  <s xml:id="echoid-s4938" xml:space="preserve">Quia <lb/>G H eſt linea maxima, erit D A ad A F, <lb/>nempe B E ad B D, vt D O ad O G (22. <lb/></s>
  <s xml:id="echoid-s4939" xml:space="preserve">
<anchor type="note" xlink:label="note-0162-03a" xlink:href="note-0162-03"/>
ex 5.) </s>
  <s xml:id="echoid-s4940" xml:space="preserve">nempe N H ad H G, ſeu N P ad
<pb o="125" file="0163" n="163" rhead="Conicor. Lib. V."/>
P D; </s>
  <s xml:id="echoid-s4941" xml:space="preserve">ergo B E ſemiſsis erecti ad B D ſemiſsim tranſuerſi eſt, vt N P ad <lb/>P D, &amp; </s>
  <s xml:id="echoid-s4942" xml:space="preserve">ideo N H eſt breuiſsima linearum egredientium ex N (10. <lb/></s>
  <s xml:id="echoid-s4943" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4944" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4945" xml:space="preserve">fic oſtendetur, quod ſi K I fuerit maximus, erit K M breuiſſima.</s>
  <s xml:id="echoid-s4946" xml:space="preserve"/>
</p>
<div xml:id="echoid-div459" type="float" level="2" n="2">
<note position="right" xlink:label="note-0162-03" xlink:href="note-0162-03a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div461" type="section" level="1" n="141">
<head xml:id="echoid-head190" xml:space="preserve">PROPOSITIO XXXIII. XXXIV.</head>
<p>
  <s xml:id="echoid-s4947" xml:space="preserve">EContra oſtendetur, quod duæ breuiſſimæ, ſi producantur <lb/>
<anchor type="note" xlink:label="note-0163-01a" xlink:href="note-0163-01"/>
ad partes ſuarum originum vſque ad axim minorem rectũ <lb/>ellipſis, fient duo maximi; </s>
  <s xml:id="echoid-s4948" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4949" xml:space="preserve">lineæ maximæ mutuò ſe ſecant in-<lb/>ter tranſuerſum, &amp; </s>
  <s xml:id="echoid-s4950" xml:space="preserve">rectum in eadem parte, &amp; </s>
  <s xml:id="echoid-s4951" xml:space="preserve">quod continent <lb/>cum menſura angulos, quorum proximior vertici ſectiouis ma-<lb/>ior eſt.</s>
  <s xml:id="echoid-s4952" xml:space="preserve"/>
</p>
<div xml:id="echoid-div461" type="float" level="2" n="1">
<note position="left" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s4953" xml:space="preserve">Quia D Q ad Q I eſt, vt D O ad O G, quia quælibet earum eſt, vt <lb/>
<anchor type="note" xlink:label="note-0163-02a" xlink:href="note-0163-02"/>
D A ad A F (22. </s>
  <s xml:id="echoid-s4954" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4955" xml:space="preserve">diuidendo, &amp; </s>
  <s xml:id="echoid-s4956" xml:space="preserve">permutando, fiet D Q minor ad <lb/>D O maiorem, vt D I ad D G; </s>
  <s xml:id="echoid-s4957" xml:space="preserve">ergo D I minor eſt, quàm D G, &amp; </s>
  <s xml:id="echoid-s4958" xml:space="preserve">K Q <lb/>maior, quàm H O; </s>
  <s xml:id="echoid-s4959" xml:space="preserve">quare angulus I maior eſt, quàm G; </s>
  <s xml:id="echoid-s4960" xml:space="preserve">igitur H G, K I, <lb/>ſe mutuo ſecantes, conueniunt in L.</s>
  <s xml:id="echoid-s4961" xml:space="preserve"/>
</p>
<div xml:id="echoid-div462" type="float" level="2" n="2">
<note position="left" xlink:label="note-0163-02" xlink:href="note-0163-02a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s4962" xml:space="preserve">Et conſtat, quod occurſus duarum breuiſsimarum (ſi producantur ver-<lb/>ſus ſuam originem) erit intra angulum contentum à duabus medietati-<lb/>bus axium ellipſis B D, D C ſupra vnum eorum, nempe punctum L ca-<lb/>dit intra angulum B D C. </s>
  <s xml:id="echoid-s4963" xml:space="preserve">Quoniam breuiſsimæ N H, M K ſe mutuò ſe-<lb/>cant, ſi producantur ad partes ſuæ originis (28. </s>
  <s xml:id="echoid-s4964" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4965" xml:space="preserve">occurrent vtique <lb/>extra B D, &amp; </s>
  <s xml:id="echoid-s4966" xml:space="preserve">intra A G (33. </s>
  <s xml:id="echoid-s4967" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4968" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4969" xml:space="preserve">hoc erat oſtendendum.</s>
  <s xml:id="echoid-s4970" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div464" type="section" level="1" n="142">
<head xml:id="echoid-head191" xml:space="preserve">PROPOSITIO XXXV.</head>
<p>
  <s xml:id="echoid-s4971" xml:space="preserve">SI per centrũ ellipfis tranſierit vna <lb/>
<anchor type="note" xlink:label="note-0163-03a" xlink:href="note-0163-03"/>
<anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/>
duarum breuiſſimarum, vtique <lb/>rami egrediẽtes ab eorum occurſu ad <lb/>ſectionis quadrantem alterius breuiſſi-<lb/>mæ habebunt proprietates expoſitas <lb/>in propoſitionibus 54. </s>
  <s xml:id="echoid-s4972" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4973" xml:space="preserve">55.</s>
  <s xml:id="echoid-s4974" xml:space="preserve"/>
</p>
<div xml:id="echoid-div464" type="float" level="2" n="1">
<note position="left" xlink:label="note-0163-03" xlink:href="note-0163-03a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a">
    <image file="0163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0163-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s4975" xml:space="preserve">In ellipſi A B C ſit punctum E occur-<lb/>ſus duarum breuiſſimarum B D, C I, &amp; </s>
  <s xml:id="echoid-s4976" xml:space="preserve"><lb/>centrum ſectionis D: </s>
  <s xml:id="echoid-s4977" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4978" xml:space="preserve">ex E educamus E F, quæ ſecet tranſuerſum a-<lb/>xim in H. </s>
  <s xml:id="echoid-s4979" xml:space="preserve">Dico, quod H F nõ eſt breuiſſima, &amp; </s>
  <s xml:id="echoid-s4980" xml:space="preserve">quod breuiſſima egre-<lb/>diens ex F abſcindit ex ſagitta A C cum A lineam maiorem, quàm A <lb/>H. </s>
  <s xml:id="echoid-s4981" xml:space="preserve">Quoniam G I eſt breuiſſima; </s>
  <s xml:id="echoid-s4982" xml:space="preserve">igitur F H, ſi eſſet quoque breuiſſima, <lb/>
<anchor type="note" xlink:label="note-0163-04a" xlink:href="note-0163-04"/>
occurreret ipſi G I intra angulum A D E: </s>
  <s xml:id="echoid-s4983" xml:space="preserve">ſed non occurrit ei, niſi in E, <lb/>ergo F H non eſt breuiſſima; </s>
  <s xml:id="echoid-s4984" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4985" xml:space="preserve">quia F E non cadit inter duas breuiſe-<lb/>cantes E B, E G; </s>
  <s xml:id="echoid-s4986" xml:space="preserve">ergo breuiſſima, egrediens ex F, abſcindit ex ſagitta <lb/>lineam maiorem, quàm A H (54. </s>
  <s xml:id="echoid-s4987" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s4988" xml:space="preserve">quod erat oſtenden@um.</s>
  <s xml:id="echoid-s4989" xml:space="preserve"/>
</p>
<div xml:id="echoid-div465" type="float" level="2" n="2">
<note position="right" xlink:label="note-0163-04" xlink:href="note-0163-04a" xml:space="preserve">34. Huius.</note>
</div>
<pb o="126" file="0164" n="164" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div467" type="section" level="1" n="143">
<head xml:id="echoid-head192" xml:space="preserve">PROPOSITIO XXXVI.</head>
<p>
  <s xml:id="echoid-s4990" xml:space="preserve">IN ſectione elliptica quatuor lineæ <lb/>
<anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/>
breuiſſimæ, vt B D, F I, G K, <lb/>H L, non conueniunt omnes in vno <lb/>puncto.</s>
  <s xml:id="echoid-s4991" xml:space="preserve"/>
</p>
<div xml:id="echoid-div467" type="float" level="2" n="1">
  <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a">
    <image file="0164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0164-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s4992" xml:space="preserve">Alioquin ſit occurſus in E, &amp; </s>
  <s xml:id="echoid-s4993" xml:space="preserve">prius ſit <lb/>B D perpendicularis ſuper A C, tranſi-<lb/>ens per D centrum ſectionis; </s>
  <s xml:id="echoid-s4994" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4995" xml:space="preserve">quia E <lb/>eſt occurſus duarum breuiſſimarum B D, <lb/>
<anchor type="note" xlink:label="note-0164-01a" xlink:href="note-0164-01"/>
F I, &amp; </s>
  <s xml:id="echoid-s4996" xml:space="preserve">B E tranſit per centrum; </s>
  <s xml:id="echoid-s4997" xml:space="preserve">igitur <lb/>
<anchor type="figure" xlink:label="fig-0164-02a" xlink:href="fig-0164-02"/>
G K non eſt linea breuiſſima, quod eſt <lb/>contra hypotheſim. </s>
  <s xml:id="echoid-s4998" xml:space="preserve">Si vero nullus eorũ <lb/>tranſit per centrum, educamus per cen-<lb/>trum D O perpendicularem ad A C; </s>
  <s xml:id="echoid-s4999" xml:space="preserve">qua-<lb/>re duæ breuiſſimæ F I, G K conueniunt <lb/>intra angulum A D O (34. </s>
  <s xml:id="echoid-s5000" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s5001" xml:space="preserve">ſimi-<lb/>liter H L, M N breuiſſimæ occurrunt in-<lb/>tra angulum C D O (34. </s>
  <s xml:id="echoid-s5002" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s5003" xml:space="preserve">ſed cõ-<lb/>ueniunt in E, quod eſt abſurdum; </s>
  <s xml:id="echoid-s5004" xml:space="preserve">igitur <lb/>quatuor lineæ breuiſſimæ non cõueniunt in vno puncto; </s>
  <s xml:id="echoid-s5005" xml:space="preserve">quod erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s5006" xml:space="preserve"/>
</p>
<div xml:id="echoid-div468" type="float" level="2" n="2">
<note position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" xml:space="preserve">35. huius.</note>
  <figure xlink:label="fig-0164-02" xlink:href="fig-0164-02a">
    <image file="0164-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0164-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div470" type="section" level="1" n="144">
<head xml:id="echoid-head193" xml:space="preserve">PROPOSITIO XXXVII. XLVI.</head>
<p>
  <s xml:id="echoid-s5007" xml:space="preserve">IN coniſectione A B, cuius centrum D duci non poſſunt-duæ <lb/>lineæ maximæ in ellipſi, neque duæbreuiſſimæ in omnibus <lb/>ſectionibus, vt A E, A F ad vnum punctum A circumferentiæ <lb/>ſectionis terminatæ.</s>
  <s xml:id="echoid-s5008" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5009" xml:space="preserve">Educamus A G perpendicularem ad axim B E. </s>
  <s xml:id="echoid-s5010" xml:space="preserve">Si itaque ſectio fue-<lb/>rit parabole, fiet E G æqualis F G, quia quælibet earum eſt æqualis di-<lb/>midio erecti (13. </s>
  <s xml:id="echoid-s5011" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s5012" xml:space="preserve">ſi vero fuerit hyperbole, aut ellipſis, fiet D G <lb/>ad G E, vt D G ad G F; </s>
  <s xml:id="echoid-s5013" xml:space="preserve">quia quælibet earum eſt, vt proportio figuræ <lb/>(14. </s>
  <s xml:id="echoid-s5014" xml:space="preserve">15. </s>
  <s xml:id="echoid-s5015" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s5016" xml:space="preserve">igitur G F æqualis eſt G E, quod eſt abſurdum. </s>
  <s xml:id="echoid-s5017" xml:space="preserve">Simi-<lb/>liter ſi B G fuerit minor duarum axium ellipſis, &amp; </s>
  <s xml:id="echoid-s5018" xml:space="preserve">fuerint A E, A F <lb/>rami maximi oſtendetur, quod G F æqualis ſit G E (23. </s>
  <s xml:id="echoid-s5019" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s5020" xml:space="preserve">Patet <lb/>igitur, vt dictum eſt, quod ex vno puncto ſectionis educi non poſſunt <lb/>ad axim illius duæ lineæ maximæ, neque breuiſſimæ, &amp; </s>
  <s xml:id="echoid-s5021" xml:space="preserve">hoc erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s5022" xml:space="preserve"/>
</p>
<pb o="127" file="0165" n="165" rhead="Conicor. Lib. V."/>
  <figure>
    <image file="0165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0165-01"/>
  </figure>
</div>
<div xml:id="echoid-div471" type="section" level="1" n="145">
<head xml:id="echoid-head194" xml:space="preserve">PROPOSITIO XXXVIII.</head>
<p>
  <s xml:id="echoid-s5023" xml:space="preserve">SI linea maxima, aut breuiſſima, <lb/>
<anchor type="figure" xlink:label="fig-0165-02a" xlink:href="fig-0165-02"/>
vt C B, producatur extra ſe-<lb/>ctionem A B ad D, erit eius portio <lb/>B D extra ſectionem abſciſſa mini-<lb/>ma omnium linearum D E, D F, <lb/>D A egredientium ab illo pnncto ad <lb/>circumferentiam ſectionis: </s>
  <s xml:id="echoid-s5024" xml:space="preserve">reliqua-<lb/>rũ vero propinquior, illi minor eſt <lb/>remotiore.</s>
  <s xml:id="echoid-s5025" xml:space="preserve"/>
</p>
<div xml:id="echoid-div471" type="float" level="2" n="1">
  <figure xlink:label="fig-0165-02" xlink:href="fig-0165-02a">
    <image file="0165-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0165-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s5026" xml:space="preserve">Educatur B G, tangens ſectionem in <lb/>
<anchor type="note" xlink:label="note-0165-01a" xlink:href="note-0165-01"/>
B; </s>
  <s xml:id="echoid-s5027" xml:space="preserve">erit D B minor, quàm D H; </s>
  <s xml:id="echoid-s5028" xml:space="preserve">ergo mul-<lb/>to minor eſt, quàm D E: </s>
  <s xml:id="echoid-s5029" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5030" xml:space="preserve">iungamus <lb/>F E, F A, erit angulus F E D obtuſus, <lb/>&amp; </s>
  <s xml:id="echoid-s5031" xml:space="preserve">propterea D E minor eſt, quàm D F, <lb/>&amp; </s>
  <s xml:id="echoid-s5032" xml:space="preserve">ſimiliter D F minor, quàm D A; </s>
  <s xml:id="echoid-s5033" xml:space="preserve">quod <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s5034" xml:space="preserve"/>
</p>
<div xml:id="echoid-div472" type="float" level="2" n="2">
<note position="left" xlink:label="note-0165-01" xlink:href="note-0165-01a" xml:space="preserve">a</note>
</div>
<pb o="128" file="0166" n="166" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div474" type="section" level="1" n="146">
<head xml:id="echoid-head195" xml:space="preserve">PR OPOSITIO XXXIX.</head>
<p>
  <s xml:id="echoid-s5035" xml:space="preserve">IN ſectione A B elliptica quælibet <lb/>
<anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/>
perpendicularis F D ad lineam <lb/>maximam C D, ab eius termino D <lb/>in ſectione poſito educta, continget <lb/>coniſectionem.</s>
  <s xml:id="echoid-s5036" xml:space="preserve"/>
</p>
<div xml:id="echoid-div474" type="float" level="2" n="1">
  <figure xlink:label="fig-0166-01" xlink:href="fig-0166-01a">
    <image file="0166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0166-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s5037" xml:space="preserve">Alioquin ſecet illam, &amp; </s>
  <s xml:id="echoid-s5038" xml:space="preserve">in eius produ-<lb/>ctione D G ſumatur punctum G intra ſe-<lb/>
<anchor type="note" xlink:label="note-0166-01a" xlink:href="note-0166-01"/>
ctionem: </s>
  <s xml:id="echoid-s5039" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5040" xml:space="preserve">educamus B G C, igitur G <lb/>C maior eſt, quàm C D, quia ſubtendit <lb/>rectum angulum C D G, &amp; </s>
  <s xml:id="echoid-s5041" xml:space="preserve">propterea B C multo maior eſt, quàm C D, <lb/>quod eſt abſurdum; </s>
  <s xml:id="echoid-s5042" xml:space="preserve">igitur educta illa linea eſt tangens; </s>
  <s xml:id="echoid-s5043" xml:space="preserve">quod erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s5044" xml:space="preserve"/>
</p>
<div xml:id="echoid-div475" type="float" level="2" n="2">
<note position="right" xlink:label="note-0166-01" xlink:href="note-0166-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div477" type="section" level="1" n="147">
<head xml:id="echoid-head196" xml:space="preserve">PROPOSITIO XXXX.</head>
<p>
  <s xml:id="echoid-s5045" xml:space="preserve">E Contra ſi fuerit F D tangens, erit perpendicularis ſuper <lb/>maximam D C.</s>
  <s xml:id="echoid-s5046" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5047" xml:space="preserve">Alioquin educamus aliam E D perpendicularem ſuper illam; </s>
  <s xml:id="echoid-s5048" xml:space="preserve">ergo E <lb/>D tangit ſectionem in puncto D (39. </s>
  <s xml:id="echoid-s5049" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s5050" xml:space="preserve">ſed F D ſuppoſita fuit tan-<lb/>gens; </s>
  <s xml:id="echoid-s5051" xml:space="preserve">igitur duæ D F, &amp; </s>
  <s xml:id="echoid-s5052" xml:space="preserve">D E tangunt ſectionem in vno puncto, quod <lb/>eſt abſurdum (36. </s>
  <s xml:id="echoid-s5053" xml:space="preserve">ex I.)</s>
  <s xml:id="echoid-s5054" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div478" type="section" level="1" n="148">
<head xml:id="echoid-head197" xml:space="preserve">PROPOSITIO XXXXVII.</head>
<p>
  <s xml:id="echoid-s5055" xml:space="preserve">Q Vælibet linea D E ex puncto <lb/>
<anchor type="figure" xlink:label="fig-0166-02a" xlink:href="fig-0166-02"/>
contactus D ad axim alicuius <lb/>ſectionis A B educta per-<lb/>pendicularis ad tangentem D C, <lb/>erit linea breuiſſima, aut maxima.</s>
  <s xml:id="echoid-s5056" xml:space="preserve"/>
</p>
<div xml:id="echoid-div478" type="float" level="2" n="1">
  <figure xlink:label="fig-0166-02" xlink:href="fig-0166-02a">
    <image file="0166-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0166-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s5057" xml:space="preserve">Alioquin educamus D F breuiſſimam, <lb/>
<anchor type="note" xlink:label="note-0166-02a" xlink:href="note-0166-02"/>
vel maximam; </s>
  <s xml:id="echoid-s5058" xml:space="preserve">ergo D C perpendicularis <lb/>eſt ſuper D F; </s>
  <s xml:id="echoid-s5059" xml:space="preserve">ſed C D ſuppoſita fuit per-<lb/>
<anchor type="note" xlink:label="note-0166-03a" xlink:href="note-0166-03"/>
pendicularis ſuper D E; </s>
  <s xml:id="echoid-s5060" xml:space="preserve">quod eſt abſur-<lb/>dum: </s>
  <s xml:id="echoid-s5061" xml:space="preserve">quapropter demonſtratũ eſt, quod <lb/>fuerat propoſitum.</s>
  <s xml:id="echoid-s5062" xml:space="preserve"/>
</p>
<div xml:id="echoid-div479" type="float" level="2" n="2">
<note position="left" xlink:label="note-0166-02" xlink:href="note-0166-02a" xml:space="preserve">Ex 10. &amp; 20. huius.</note>
<note position="left" xlink:label="note-0166-03" xlink:href="note-0166-03a" xml:space="preserve">40. huius.</note>
</div>
<pb o="129" file="0167" n="167" rhead="Conicor. Lib. V."/>
</div>
<div xml:id="echoid-div481" type="section" level="1" n="149">
<head xml:id="echoid-head198" xml:space="preserve">PROPOSITIO XXXXVIII.</head>
<p>
  <s xml:id="echoid-s5063" xml:space="preserve">T Res lineæ maximæ E F, G H, <lb/>
<anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/>
<anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/>
I K ad vnum ellipſis quadrã-<lb/>tem A F B cadentens non cõueniunt <lb/>in vno puncto.</s>
  <s xml:id="echoid-s5064" xml:space="preserve"/>
</p>
<div xml:id="echoid-div481" type="float" level="2" n="1">
<note position="left" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a">
    <image file="0167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0167-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s5065" xml:space="preserve">Alioquin cõueniant in O, &amp; </s>
  <s xml:id="echoid-s5066" xml:space="preserve">quia ſunt <lb/>lineæ maximæ erunt M K, H N, L F, li-<lb/>neæ breuiſſimæ (32. </s>
  <s xml:id="echoid-s5067" xml:space="preserve">ex 5.) </s>
  <s xml:id="echoid-s5068" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5069" xml:space="preserve">conueniunt <lb/>in puncto O; </s>
  <s xml:id="echoid-s5070" xml:space="preserve">quod eſt abſurdũ (54. </s>
  <s xml:id="echoid-s5071" xml:space="preserve">ex 5.) <lb/></s>
  <s xml:id="echoid-s5072" xml:space="preserve">oſtenſum ergo eſt, quod fuerat propoſitũ.</s>
  <s xml:id="echoid-s5073" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div483" type="section" level="1" n="150">
<head xml:id="echoid-head199" xml:space="preserve">Notæ in Propoſit. XXXII.</head>
<p>
  <s xml:id="echoid-s5074" xml:space="preserve">L Inea maxima ſecat tranſuerſam in pũ-<lb/>
<anchor type="note" xlink:label="note-0167-02a" xlink:href="note-0167-02"/>
<anchor type="figure" xlink:label="fig-0167-02a" xlink:href="fig-0167-02"/>
cto, cuius intercepta inter punctum <lb/>illud, &amp; </s>
  <s xml:id="echoid-s5075" xml:space="preserve">ſectionem, eſt linea breuiſſima, <lb/>&amp;</s>
  <s xml:id="echoid-s5076" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5077" xml:space="preserve">Verba, quæ in textu Arabico deſideran-<lb/>tur ſupplenda cenſui, vt æquiuocationes tolle-<lb/>rentur.</s>
  <s xml:id="echoid-s5078" xml:space="preserve"/>
</p>
<div xml:id="echoid-div483" type="float" level="2" n="1">
<note position="left" xlink:label="note-0167-02" xlink:href="note-0167-02a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0167-02" xlink:href="fig-0167-02a">
    <image file="0167-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0167-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5079" xml:space="preserve">Quia G H eſt linea maxima, erit D A <lb/>
<anchor type="note" xlink:label="note-0167-03a" xlink:href="note-0167-03"/>
ad A F, nempe B E ad B D, &amp;</s>
  <s xml:id="echoid-s5080" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5081" xml:space="preserve">Quia <lb/>in 22. </s>
  <s xml:id="echoid-s5082" xml:space="preserve">huius oſtenſum eſt, lineæ maximæ G <lb/>H potentialem H O ſecare ſemiaxim minorẽ <lb/>A D in O, vt ſit D O ad O G in eadẽ propor-<lb/>tione figuræ axis minoris A C; </s>
  <s xml:id="echoid-s5083" xml:space="preserve">ſcilicet erit, <lb/>vt D A ſemiaxis minor ad A F eius ſemie-<lb/>rectum; </s>
  <s xml:id="echoid-s5084" xml:space="preserve">ſed vt A D ad A F, ita eſt B E ſe-<lb/>miſsis lateris recti axis tranſuerſi ad B D <lb/>ſemiſſem eiuſdem tranſuerſi; </s>
  <s xml:id="echoid-s5085" xml:space="preserve">igitur D O ad <lb/>O G eandem proportionem habebit, quàm E <lb/>B ad B D; </s>
  <s xml:id="echoid-s5086" xml:space="preserve">ſed propter parallelas N D, H O, <lb/>eſt N H ad H G, vt D O ad O G; </s>
  <s xml:id="echoid-s5087" xml:space="preserve">pariter-<lb/>que propter parallelas D G, H P, erit N P ad P D, vt N H ad H G; </s>
  <s xml:id="echoid-s5088" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5089" xml:space="preserve">pro-<lb/>pterea N P ad P D eandem proportionem habebit, quàm D O ad O G, ſeu <lb/>quàm E B ad B D; </s>
  <s xml:id="echoid-s5090" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5091" xml:space="preserve">permutando D P ad P N erit, vt D B ad B E, ſeu vt <lb/>
<anchor type="note" xlink:label="note-0167-04a" xlink:href="note-0167-04"/>
axis tranſuerſus ad eius, erectum; </s>
  <s xml:id="echoid-s5092" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5093" xml:space="preserve">propterea linea N H erit breuiſsima.</s>
  <s xml:id="echoid-s5094" xml:space="preserve"/>
</p>
<div xml:id="echoid-div484" type="float" level="2" n="2">
<note position="left" xlink:label="note-0167-03" xlink:href="note-0167-03a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0167-04" xlink:href="note-0167-04a" xml:space="preserve">15. huius.</note>
</div>
</div>
<div xml:id="echoid-div486" type="section" level="1" n="151">
<head xml:id="echoid-head200" xml:space="preserve">Notæ in Propoſit. XXXIII. XXXIV.</head>
<p>
  <s xml:id="echoid-s5095" xml:space="preserve">E Contra oſtendetur, quod duæ breuiſſimæ, ſi educantur ex parte ſuæ <lb/>
<anchor type="note" xlink:label="note-0167-05a" xlink:href="note-0167-05"/>
originis ad rectum, fient duo maximi cum relatione ad rectum: </s>
  <s xml:id="echoid-s5096" xml:space="preserve">Et
<pb o="130" file="0168" n="168" rhead="Apollonij Pergæi"/>
oſtendetur ex dictis, quod lineæ maximæ mutuò ſe ſecant inter diame <lb/>trum, &amp; </s>
  <s xml:id="echoid-s5097" xml:space="preserve">rectum, &amp;</s>
  <s xml:id="echoid-s5098" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5099" xml:space="preserve">Textũ corrigi debere maniſeſtum eſt ex dictis ſuperius</s>
</p>
<div xml:id="echoid-div486" type="float" level="2" n="1">
<note position="left" xlink:label="note-0167-05" xlink:href="note-0167-05a" xml:space="preserve">a</note>
</div>
<note position="right" xml:space="preserve">b</note>
<p style="it">
  <s xml:id="echoid-s5100" xml:space="preserve">Quia D Q ad Q I eſt, vt D O ad O <lb/>
<anchor type="figure" xlink:label="fig-0168-01a" xlink:href="fig-0168-01"/>
G, &amp;</s>
  <s xml:id="echoid-s5101" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5102" xml:space="preserve">In eadem figura propoſitionis 32. <lb/></s>
  <s xml:id="echoid-s5103" xml:space="preserve">præcedentis perſiciatur conſtructio, vt priùs <lb/>quia duæ K M, H N ſunt breuiſsimæ li-<lb/>
<anchor type="note" xlink:label="note-0168-02a" xlink:href="note-0168-02"/>
neæ; </s>
  <s xml:id="echoid-s5104" xml:space="preserve">ergo M R ad R D, nec non N P ad <lb/>P D eandem proportionem habent, ſcilicet <lb/>eam quàm habent latus rectum ad tranſuer-<lb/>ſum, ſeu eandem quàm habet ſemierectus <lb/>
<anchor type="note" xlink:label="note-0168-03a" xlink:href="note-0168-03"/>
E B ad ſemiaxim B D; </s>
  <s xml:id="echoid-s5105" xml:space="preserve">eſt verò C A ad eius <lb/>latus rectum, ſeu D A ad A F, vt E B ad <lb/>B D; </s>
  <s xml:id="echoid-s5106" xml:space="preserve">igitur tam M R ad R D, quàm N P <lb/>ad P D eandem proporiionem habent, quàm <lb/>D A ad A F; </s>
  <s xml:id="echoid-s5107" xml:space="preserve">ſed propter parallelas C D, R <lb/>K, P H, c<unsure/>ſt M K ad K I, vt M R ad R D; <lb/></s>
  <s xml:id="echoid-s5108" xml:space="preserve">pariterque N H ad H G eandem proportionẽ <lb/>habet, quàm N P ad P D; </s>
  <s xml:id="echoid-s5109" xml:space="preserve">atque propter pa-<lb/>rallelas D B, Q K, O H eſt D Q, ad Q I <lb/>vt M K ad K I, &amp; </s>
  <s xml:id="echoid-s5110" xml:space="preserve">D O ad O G eſt vt N H <lb/>ad H G; </s>
  <s xml:id="echoid-s5111" xml:space="preserve">ergo tam D Q ad Q I, quàm D <lb/>O ad O G eandem proportionem habent, quàm D A ad A F, ſeu quàm axis mi-<lb/>
<anchor type="note" xlink:label="note-0168-04a" xlink:href="note-0168-04"/>
nor A C ad ſuum erectum, &amp; </s>
  <s xml:id="echoid-s5112" xml:space="preserve">propterea tam K I, quàm H G eſt ramus maxi-<lb/>mus; </s>
  <s xml:id="echoid-s5113" xml:space="preserve">igitur ſi duæ lineæ breuiſſimæ H G, &amp; </s>
  <s xml:id="echoid-s5114" xml:space="preserve">K I producantur quouſque axim <lb/>minorem ſecent in punctis G, &amp; </s>
  <s xml:id="echoid-s5115" xml:space="preserve">I efficientur rami omnium maximi. </s>
  <s xml:id="echoid-s5116" xml:space="preserve">Poſtea quia <lb/>D Q ad Q I, eſt vt D O ad O G; </s>
  <s xml:id="echoid-s5117" xml:space="preserve">permutando D Q ad D O eandem propor-<lb/>tionem habebit, quàm Q I ad O G; </s>
  <s xml:id="echoid-s5118" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5119" xml:space="preserve">permutando, &amp; </s>
  <s xml:id="echoid-s5120" xml:space="preserve">comparando antecedentes ad <lb/>differentias terminorum erit D Q ad D I, vt D O ad D G: </s>
  <s xml:id="echoid-s5121" xml:space="preserve">eſtque D Q minor <lb/>quàm D O; </s>
  <s xml:id="echoid-s5122" xml:space="preserve">igitur Q I minor eſt, quàm O G; </s>
  <s xml:id="echoid-s5123" xml:space="preserve">pariterque D I minor eſt, quàm <lb/>D G; </s>
  <s xml:id="echoid-s5124" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5125" xml:space="preserve">propterea punctum I cadit inter exim B D, &amp; </s>
  <s xml:id="echoid-s5126" xml:space="preserve">ramum H G; </s>
  <s xml:id="echoid-s5127" xml:space="preserve">eſtque <lb/>etiam potentialis K Q propinquior &amp; </s>
  <s xml:id="echoid-s5128" xml:space="preserve">parallela axi maiori, &amp; </s>
  <s xml:id="echoid-s5129" xml:space="preserve">ideo maior re-<lb/>motiore H O; </s>
  <s xml:id="echoid-s5130" xml:space="preserve">igitur punctum K cadit inter axim B D, &amp; </s>
  <s xml:id="echoid-s5131" xml:space="preserve">ramum H G; </s>
  <s xml:id="echoid-s5132" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5133" xml:space="preserve"><lb/>propterea ramus K I ſecat ramum H G in puncto L inter puncta H, &amp; </s>
  <s xml:id="echoid-s5134" xml:space="preserve">G: <lb/></s>
  <s xml:id="echoid-s5135" xml:space="preserve">
<anchor type="note" xlink:label="note-0168-05a" xlink:href="note-0168-05"/>
ſed duæ breuiſsimæ K M, H N ſe ſecant vltra axim B D: </s>
  <s xml:id="echoid-s5136" xml:space="preserve">igitur occurſus L <lb/>cadit intra angulum B D C ab axibus compræhenſum. </s>
  <s xml:id="echoid-s5137" xml:space="preserve">Tandem quia K I ſecat <lb/>H G inter puncta G, &amp; </s>
  <s xml:id="echoid-s5138" xml:space="preserve">H; </s>
  <s xml:id="echoid-s5139" xml:space="preserve">ergo efficit angulum externum K I A maio-<lb/>rem interno, &amp; </s>
  <s xml:id="echoid-s5140" xml:space="preserve">oppoſito G: </s>
  <s xml:id="echoid-s5141" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5142" xml:space="preserve">propterea ramus K I propinquior vertici B, <lb/>quàm H G efficiet cum axe minore C A angulum A I K maiorem.</s>
  <s xml:id="echoid-s5143" xml:space="preserve"/>
</p>
<div xml:id="echoid-div487" type="float" level="2" n="2">
  <figure xlink:label="fig-0168-01" xlink:href="fig-0168-01a">
    <image file="0168-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0168-01"/>
  </figure>
<note position="left" xlink:label="note-0168-02" xlink:href="note-0168-02a" xml:space="preserve">Pr. 15-<lb/>huius.</note>
<note position="left" xlink:label="note-0168-03" xlink:href="note-0168-03a" xml:space="preserve">15. lib. I.</note>
<note position="left" xlink:label="note-0168-04" xlink:href="note-0168-04a" xml:space="preserve">20. 21. 22. <lb/>huius.</note>
<note position="left" xlink:label="note-0168-05" xlink:href="note-0168-05a" xml:space="preserve">36. huius.</note>
</div>
</div>
<div xml:id="echoid-div489" type="section" level="1" n="152">
<head xml:id="echoid-head201" xml:space="preserve">Notæ in Propoſit. XXXV.</head>
<p style="it">
  <s xml:id="echoid-s5144" xml:space="preserve">SI tranſeat per centrum ellipſis vna duarum breuiſſimarum; </s>
  <s xml:id="echoid-s5145" xml:space="preserve">vtique ra-<lb/>
<anchor type="note" xlink:label="note-0168-06a" xlink:href="note-0168-06"/>
mi, &amp;</s>
  <s xml:id="echoid-s5146" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5147" xml:space="preserve">Hæc propoſitio parum differt à 54. </s>
  <s xml:id="echoid-s5148" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5149" xml:space="preserve">55. </s>
  <s xml:id="echoid-s5150" xml:space="preserve">buius, vbi oſtenſum <lb/>eſt, quod ſi duo rami E B, E G breuiſecantes ex eodem concurſu E ad ellipſim <lb/>A B ducuntur, quilibet alius ramus E F, extra breuiſecantes poſitus, cadet ſu-<lb/>pra breuiſsimam ex puncto F ad axim A C ductam: </s>
  <s xml:id="echoid-s5151" xml:space="preserve">hic vero ſupponuntur duæ
<pb o="131" file="0169" n="169" rhead="Conicor. Lib. V."/>
breuiſsimæ B D, GI, quarum B D per centrũ <lb/>
<anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/>
tranſit, quæ productæ concurrunt in puncto E <lb/>axis minoris, &amp; </s>
  <s xml:id="echoid-s5152" xml:space="preserve">concluditur, quodrami E F, <lb/>portio F H, nedũ breuiſsima non eſt, ſed ſupra <lb/>ipſam breuiſsimã ex puncto F eductam cadit.</s>
  <s xml:id="echoid-s5153" xml:space="preserve"/>
</p>
<div xml:id="echoid-div489" type="float" level="2" n="1">
<note position="right" xlink:label="note-0168-06" xlink:href="note-0168-06a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0169-01" xlink:href="fig-0169-01a">
    <image file="0169-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0169-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5154" xml:space="preserve">Sed duo hic notanda ſunt. </s>
  <s xml:id="echoid-s5155" xml:space="preserve">Primo, quod hæc <lb/>prop. </s>
  <s xml:id="echoid-s5156" xml:space="preserve">35. </s>
  <s xml:id="echoid-s5157" xml:space="preserve">non poterat poſtponi, nã vſum habet <lb/>in 57. </s>
  <s xml:id="echoid-s5158" xml:space="preserve">huius vbi malè citatur prop. </s>
  <s xml:id="echoid-s5159" xml:space="preserve">52. </s>
  <s xml:id="echoid-s5160" xml:space="preserve">loco hu-<lb/>ius 35.</s>
  <s xml:id="echoid-s5161" xml:space="preserve">, vt ibidem inſinuatum eſt. </s>
  <s xml:id="echoid-s5162" xml:space="preserve">Secundo, <lb/>quod hæc demonſtratio non videtur omnino <lb/>perſecta nam pendet ex prop. </s>
  <s xml:id="echoid-s5163" xml:space="preserve">34.</s>
  <s xml:id="echoid-s5164" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s5165" xml:space="preserve">ex eius <lb/>conuerſa, quæ demonſtrata non reperitur qua-<lb/>re ſuperuacanea non fuit noua demonſtratio in <lb/>Lemmat. </s>
  <s xml:id="echoid-s5166" xml:space="preserve">8. </s>
  <s xml:id="echoid-s5167" xml:space="preserve">appoſita.</s>
  <s xml:id="echoid-s5168" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div491" type="section" level="1" n="153">
<head xml:id="echoid-head202" xml:space="preserve">Notæ in Prop. XXXVI.</head>
<p style="it">
  <s xml:id="echoid-s5169" xml:space="preserve">SI verò nulla earum tranſit per centrũ, <lb/>
<anchor type="note" xlink:label="note-0169-01a" xlink:href="note-0169-01"/>
educamus D O, &amp;</s>
  <s xml:id="echoid-s5170" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5171" xml:space="preserve">Si enim fuerint <lb/>quatuor lineæ breuiſſimæ G K, F I, H L, M <lb/>N, quarum nulla per centrum D tranſit, ſi-<lb/>militer oſtendetur, quod non conueniunt in <lb/>vno puncto E; </s>
  <s xml:id="echoid-s5172" xml:space="preserve">nam ducto ſemiaxe minori <lb/>D O neceſſe eſt, vt punctum E concurſus duorũ <lb/>breuiſecantiũ E G, E F cadat intra angulũ A <lb/>D O; </s>
  <s xml:id="echoid-s5173" xml:space="preserve">pariterque idem punctum E concurſus <lb/>
<anchor type="note" xlink:label="note-0169-02a" xlink:href="note-0169-02"/>
duorum breuiſec antium E H, E M, cadet ne-<lb/>ceſſario intra angulum C D O, ſed idem pun-<lb/>ctum E nequit duobus in locis reperiri, ni-<lb/>mirũ intra angulum A D O, &amp; </s>
  <s xml:id="echoid-s5174" xml:space="preserve">intra angu-<lb/>lum C D O, igitur non poſſunt ab eodẽ puncto <lb/>educi ad ellipſim quatuor rami breuiſecantes.</s>
  <s xml:id="echoid-s5175" xml:space="preserve"/>
</p>
<div xml:id="echoid-div491" type="float" level="2" n="1">
<note position="left" xlink:label="note-0169-01" xlink:href="note-0169-01a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0169-02" xlink:href="note-0169-02a" xml:space="preserve">34. huius. <lb/>Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div493" type="section" level="1" n="154">
<head xml:id="echoid-head203" xml:space="preserve">Notæ in Prop. XXXVIII.</head>
<p style="it">
  <s xml:id="echoid-s5176" xml:space="preserve">NAm ſi educamus B G tangentem erit <lb/>
<anchor type="note" xlink:label="note-0169-03a" xlink:href="note-0169-03"/>
<anchor type="note" xlink:label="note-0169-04a" xlink:href="note-0169-04"/>
B D minor quàm D H, &amp;</s>
  <s xml:id="echoid-s5177" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5178" xml:space="preserve">Quo-<lb/>niam C B eſt linea breuiſſima, aut ſi maxima <lb/>
<anchor type="note" xlink:label="note-0169-05a" xlink:href="note-0169-05"/>
eſt, eius portio erit breuiſſima, &amp; </s>
  <s xml:id="echoid-s5179" xml:space="preserve">G B cõtin-<lb/>gens ſectionem in eius termino B perpendicu-<lb/>laris ad B C; </s>
  <s xml:id="echoid-s5180" xml:space="preserve">propterea in triangulo B D H <lb/>latus H D, ſubtendens angulum rectum B, <lb/>maius erit latere D B; </s>
  <s xml:id="echoid-s5181" xml:space="preserve">eſt verò D E maior, <lb/>quàm D H, eo quod punctum H contingentis <lb/>B G cadit extra ſectionem; </s>
  <s xml:id="echoid-s5182" xml:space="preserve">igitur linea B D <lb/>minor eſt, quàm D E, &amp; </s>
  <s xml:id="echoid-s5183" xml:space="preserve">propterea angulus <lb/>D E B acutus erit, quare eſt minor obtuſo
<pb o="132" file="0170" n="170" rhead="Apollonij Pergæi"/>
angulo D B E; </s>
  <s xml:id="echoid-s5184" xml:space="preserve">cadit verò F E infra rectam B E, quam ſecat in E, propter <lb/>curuitatem ſectionis F E B; </s>
  <s xml:id="echoid-s5185" xml:space="preserve">igitur angulus D E F obtuſus quoque erit, &amp; </s>
  <s xml:id="echoid-s5186" xml:space="preserve">an-<lb/>gulus D F E acutus; </s>
  <s xml:id="echoid-s5187" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5188" xml:space="preserve">propterea recta linea D E minor erit, quàm D F; </s>
  <s xml:id="echoid-s5189" xml:space="preserve">ea-<lb/>dem ratione oſtendetur D F minor, quàm D A.</s>
  <s xml:id="echoid-s5190" xml:space="preserve"/>
</p>
<div xml:id="echoid-div493" type="float" level="2" n="1">
<note position="left" xlink:label="note-0169-03" xlink:href="note-0169-03a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0169-04" xlink:href="note-0169-04a" xml:space="preserve">32. huius.</note>
<note position="right" xlink:label="note-0169-05" xlink:href="note-0169-05a" xml:space="preserve">29. 30. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div495" type="section" level="1" n="155">
<head xml:id="echoid-head204" xml:space="preserve">Notæ in Propoſit. XXXIX.</head>
<p style="it">
  <s xml:id="echoid-s5191" xml:space="preserve">ALioquin ſecet illam, &amp; </s>
  <s xml:id="echoid-s5192" xml:space="preserve">ſecemus ex, <lb/>
<anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/>
&amp; </s>
  <s xml:id="echoid-s5193" xml:space="preserve">D G intra ſectionem, &amp;</s>
  <s xml:id="echoid-s5194" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5195" xml:space="preserve">Si e-<lb/>nim recta F D non contingit ellipſim A B, <lb/>ſecet eam ſi fieri poteſt in D: </s>
  <s xml:id="echoid-s5196" xml:space="preserve">quare F D pro-<lb/>ducta in directum cadet intra ſectionem, &amp; </s>
  <s xml:id="echoid-s5197" xml:space="preserve"><lb/>in producta recta linea F D G ſumatur quod-<lb/>libet punctum G dummodo intra ſectionem <lb/>exiſtat, &amp; </s>
  <s xml:id="echoid-s5198" xml:space="preserve">per G ad concurſum C coniunga-<lb/>tur recta linea G C, quæ producta occurrat <lb/>ſectioni in B: </s>
  <s xml:id="echoid-s5199" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5200" xml:space="preserve">quia ex hypotheſi recta F <lb/>D G perpendicularis erat ad maximum ramum D C, ergo in triangulo D G C <lb/>rectangulo erit hypothenuſa G C maior quàm D C, &amp; </s>
  <s xml:id="echoid-s5201" xml:space="preserve">ideo B C multo maior <lb/>erit quàm D C; </s>
  <s xml:id="echoid-s5202" xml:space="preserve">quod eſt abſurdum, ſuppoſita enim fuit D C omnium maxima <lb/>earum, quæ ex C ad ſectionem A B duci poſſunt.</s>
  <s xml:id="echoid-s5203" xml:space="preserve"/>
</p>
<div xml:id="echoid-div495" type="float" level="2" n="1">
  <figure xlink:label="fig-0170-01" xlink:href="fig-0170-01a">
    <image file="0170-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0170-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div497" type="section" level="1" n="156">
<head xml:id="echoid-head205" xml:space="preserve">Notæ in Propoſit. XXXXVIII.</head>
<p style="it">
  <s xml:id="echoid-s5204" xml:space="preserve">ALioquin occurtant in O, quia iſtæ <lb/>
<anchor type="figure" xlink:label="fig-0170-02a" xlink:href="fig-0170-02"/>
lineæ ſunt maximæ, &amp;</s>
  <s xml:id="echoid-s5205" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5206" xml:space="preserve">Secant e-<lb/>nim lineæ maximæ ſemiaxim maiorem D A <lb/>in punctis M, N, &amp; </s>
  <s xml:id="echoid-s5207" xml:space="preserve">L: </s>
  <s xml:id="echoid-s5208" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5209" xml:space="preserve">ſiquidem tres li-<lb/>neæ maximæ conueniunt in vnico puncto O, <lb/>erunt ſegmenta inter axim maiorem, &amp; </s>
  <s xml:id="echoid-s5210" xml:space="preserve">ſe-<lb/>
<anchor type="note" xlink:label="note-0170-01a" xlink:href="note-0170-01"/>
ctionem intercepta, nimirum M K, N H, L <lb/>F lineæ breuiſſimæ; </s>
  <s xml:id="echoid-s5211" xml:space="preserve">quarum duæ quæquè L <lb/>F, N H educuntur ab eodem puncto concur-<lb/>ſus O: </s>
  <s xml:id="echoid-s5212" xml:space="preserve">igitur (ex 54. </s>
  <s xml:id="echoid-s5213" xml:space="preserve">55. </s>
  <s xml:id="echoid-s5214" xml:space="preserve">huius) tertius ra-<lb/>mus O K ab eodem concurſu O eductus non erit breuiſecans; </s>
  <s xml:id="echoid-s5215" xml:space="preserve">quod eſt contra <lb/>hypotheſim.</s>
  <s xml:id="echoid-s5216" xml:space="preserve"/>
</p>
<div xml:id="echoid-div497" type="float" level="2" n="1">
  <figure xlink:label="fig-0170-02" xlink:href="fig-0170-02a">
    <image file="0170-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0170-02"/>
  </figure>
<note position="left" xlink:label="note-0170-01" xlink:href="note-0170-01a" xml:space="preserve">32. buius.</note>
</div>
</div>
<div xml:id="echoid-div499" type="section" level="1" n="157">
<head xml:id="echoid-head206" xml:space="preserve">LIBRI QVINTI FINIS.</head>
<pb o="133" file="0171" n="171"/>
</div>
<div xml:id="echoid-div500" type="section" level="1" n="158">
<head xml:id="echoid-head207" xml:space="preserve">APOLLONII PERGAEI</head>
<head xml:id="echoid-head208" xml:space="preserve">CONICORVM LIB VI.</head>
<head xml:id="echoid-head209" xml:space="preserve">DEFINITIONES.</head>
<head xml:id="echoid-head210" xml:space="preserve">I.</head>
<p>
  <s xml:id="echoid-s5217" xml:space="preserve">SEctiones ÆQVALES ſunt, quæ ad inuicem ſu-<lb/>perpoſitæ ſibi mutuò congruunt.</s>
  <s xml:id="echoid-s5218" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div501" type="section" level="1" n="159">
<head xml:id="echoid-head211" xml:space="preserve">II.</head>
<p>
  <s xml:id="echoid-s5219" xml:space="preserve">SIMILES verò ſunt, in quibus omnes po-<lb/>tentiales ad axium abſciſſas vtrobique ſunt in <lb/>ijſdem rationibus, tum abſciſſæ ad abſciſſas.</s>
  <s xml:id="echoid-s5220" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div502" type="section" level="1" n="160">
<head xml:id="echoid-head212" xml:space="preserve">III.</head>
<p>
  <s xml:id="echoid-s5221" xml:space="preserve">Et linea, quæ ſubtendit ſegmentum circumferentiæ circuli, <lb/>aut ſectionis coni vocatur BASIS illius ſegmenti.</s>
  <s xml:id="echoid-s5222" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div503" type="section" level="1" n="161">
<head xml:id="echoid-head213" xml:space="preserve">IV.</head>
<p>
  <s xml:id="echoid-s5223" xml:space="preserve">Et linea, quæ bifariam diuidit ordinationes æquidiſtantes baſi <lb/>illius, vocatur DIAMETER illius ſegmenti.</s>
  <s xml:id="echoid-s5224" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div504" type="section" level="1" n="162">
<head xml:id="echoid-head214" xml:space="preserve">V.</head>
<p>
  <s xml:id="echoid-s5225" xml:space="preserve">Et eius terminus, qui eſt ad ſectionem, VERTEX ſegmenti.</s>
  <s xml:id="echoid-s5226" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div505" type="section" level="1" n="163">
<head xml:id="echoid-head215" xml:space="preserve">VI.</head>
<p>
  <s xml:id="echoid-s5227" xml:space="preserve">Et SEGMENTA ÆQVALIA ſunt, quæ ſuperpoſita ſibi mu-<lb/>tuò congruunt.</s>
  <s xml:id="echoid-s5228" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div506" type="section" level="1" n="164">
<head xml:id="echoid-head216" xml:space="preserve">VII.</head>
<p>
  <s xml:id="echoid-s5229" xml:space="preserve">Et SIMILIA ſunt, quorum baſes cum diametris æquales an-<lb/>gulos continent, &amp; </s>
  <s xml:id="echoid-s5230" xml:space="preserve">in eorum ſingulis ductæ lineæ baſi parallelæ <lb/>numero æquales ad abſciſſas diametrorum ſunt in ijſdem ratio-<lb/>nibus tum abſciſsæ ad abſciſsas.</s>
  <s xml:id="echoid-s5231" xml:space="preserve"/>
</p>
<pb o="134" file="0172" n="172" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div507" type="section" level="1" n="165">
<head xml:id="echoid-head217" xml:space="preserve">VIII.</head>
<p>
  <s xml:id="echoid-s5232" xml:space="preserve">CONI SIMILES ſunt, quorum axes æquè ad baſes inclinati, <lb/>ad diametros baſium proportionales ſunt.</s>
  <s xml:id="echoid-s5233" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div508" type="section" level="1" n="166">
<head xml:id="echoid-head218" xml:space="preserve">IX.</head>
<p>
  <s xml:id="echoid-s5234" xml:space="preserve">Et dicitur conus continere ſectionem, &amp; </s>
  <s xml:id="echoid-s5235" xml:space="preserve">ſectio in cono po-<lb/>ſita eſse, ſi ſectio tota fuerit in ſuperſicie coni, aut cadat in illa, <lb/>ſi producatur ex parte baſis.</s>
  <s xml:id="echoid-s5236" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div509" type="section" level="1" n="167">
<head xml:id="echoid-head219" xml:space="preserve">NOTÆ.</head>
<p style="it">
  <s xml:id="echoid-s5237" xml:space="preserve">DEſinitiones huius ſeſti libri ferè omnes ſunt Appollonĳ, in paucis quidem <lb/>alteratæ ab interprete Arabico: </s>
  <s xml:id="echoid-s5238" xml:space="preserve">quod quidem conſtat teſtimonio Eutocĳ <lb/>Aſcalonitæ, qui in tertiam propoſitionem ſecundi æquiponder antium Archime-<lb/>dis affert definitionem ſimilium portionum conicarum ſectionum, traditam ab <lb/>Apollonio in eius ſeſto libro: </s>
  <s xml:id="echoid-s5239" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5240" xml:space="preserve">ſanè ordo doctrinæ exigebat, vt prius ſectio-<lb/>nes æquales, &amp; </s>
  <s xml:id="echoid-s5241" xml:space="preserve">ſimiles definirentur, vt poſtea earum symptomata demonſtrari <lb/>poſſent: </s>
  <s xml:id="echoid-s5242" xml:space="preserve">ſed animaduertendum eſt, hactenus nomen ſectionis conicæ ſignificaſſe <lb/>quamlibet indeterminatam portionem curuæ lineæ in coni ſuper ſicie ortam ex ſe-<lb/>ctione alicuius plani non per verticem coni ducti, non conſiderando termiuos eius <lb/>neque menſuram. </s>
  <s xml:id="echoid-s5243" xml:space="preserve">Segmentum verò ſignificat portionem aliquam ſectionis conicæ <lb/>determinatæ menſuræ, &amp; </s>
  <s xml:id="echoid-s5244" xml:space="preserve">certis finibus terminatam; </s>
  <s xml:id="echoid-s5245" xml:space="preserve">at multoties ſignificat ſu-<lb/>perficiem à coniſectione, &amp; </s>
  <s xml:id="echoid-s5246" xml:space="preserve">recta linea eam ſubtendente contenta. </s>
  <s xml:id="echoid-s5247" xml:space="preserve">Igitur ad <lb/>confuſionem vitandam vocabo huiuſmodi ſuperficiem planam, Mixtam ſuperficiẽ <lb/>ſectionis conicæ. </s>
  <s xml:id="echoid-s5248" xml:space="preserve">Modò in relatis definitionibus prius quænam coniſectiones vo-<lb/>cari debeant inter ſe æquales exponit Apollonius.</s>
  <s xml:id="echoid-s5249" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5250" xml:space="preserve">I. </s>
  <s xml:id="echoid-s5251" xml:space="preserve">Et primo; </s>
  <s xml:id="echoid-s5252" xml:space="preserve">Si fuerint duæ quælibet coni-<lb/>
<anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/>
ſectiones B A C, E D F, quarum axes A G, <lb/>D H; </s>
  <s xml:id="echoid-s5253" xml:space="preserve">vertices verò A, &amp; </s>
  <s xml:id="echoid-s5254" xml:space="preserve">D, &amp; </s>
  <s xml:id="echoid-s5255" xml:space="preserve">ſiquidem <lb/>intelligatur ſectio B A C ſuperpoſita ſectioni <lb/>E D F, vt nimirum vertex A ſuper verti-<lb/>cem D cadat, atque axis A G ſuper axim <lb/>D H, atque pariter peripheriæ B A C, &amp; </s>
  <s xml:id="echoid-s5256" xml:space="preserve">E <lb/>D F ſibi mutuò congruant: </s>
  <s xml:id="echoid-s5257" xml:space="preserve">tunc quidem vo-<lb/>cantur duæ dictæ ſectiones conicæ æquales in-<lb/>ter ſe. </s>
  <s xml:id="echoid-s5258" xml:space="preserve">V bi notandum eſt, non oportere lon-<lb/>gitudinem curuæ B A C æqualem eſſe longi-<lb/>tudini curuæ E D F; </s>
  <s xml:id="echoid-s5259" xml:space="preserve">ſicuti, vt duo anguli <lb/>rectilinei dicantur æquales, &amp; </s>
  <s xml:id="echoid-s5260" xml:space="preserve">ſibi mu-<lb/>tuò congruentes, neceſſe non eſt, vt rectæ li-<lb/>neæ, angulos continentes, ſint æquales longi-<lb/>tudine, dummodo certum ſit, quod lineæ ipſæ <lb/>vlterius productæ ſemper ſibi mutuò congruant; </s>
  <s xml:id="echoid-s5261" xml:space="preserve">ſic pariter peripheriæ conicarũ <lb/>ſectionum A B, &amp; </s>
  <s xml:id="echoid-s5262" xml:space="preserve">D E, ſi vlterius producantur, ſemper ſibi mutuò congruent.</s>
  <s xml:id="echoid-s5263" xml:space="preserve"/>
</p>
<div xml:id="echoid-div509" type="float" level="2" n="1">
  <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a">
    <image file="0172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0172-01"/>
  </figure>
</div>
<pb o="135" file="0173" n="173" rhead="Conicor. Lib. VI."/>
<p style="it">
  <s xml:id="echoid-s5264" xml:space="preserve">II. </s>
  <s xml:id="echoid-s5265" xml:space="preserve">Codex Arabicus habet. </s>
  <s xml:id="echoid-s5266" xml:space="preserve">Similes verò ſunt, quarum proportio po-<lb/>tentium in vna earum ad ſua abſciſſa eſt eadem proportioni aliarum po-<lb/>tentium ad ſua abſciſſa, &amp; </s>
  <s xml:id="echoid-s5267" xml:space="preserve">proportio abſciſſarum in vna earum ad ſua op-<lb/>poſita abſciſſa eadem eſt. </s>
  <s xml:id="echoid-s5268" xml:space="preserve">Putabit forte quiſpiam, me nimis licentiosè tran-<lb/>sformaſſe potius, quàm emendaſſe textum in <lb/>
<anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/>
hac ſecunda definitione; </s>
  <s xml:id="echoid-s5269" xml:space="preserve">ſed is ſciat velim, <lb/>non meo arbitratu id feciſſe ſed ex præſcripto <lb/>eiuſdem Apollonĳ pluribus in locis; </s>
  <s xml:id="echoid-s5270" xml:space="preserve">non qui-<lb/>dem in hiſce compendioſiſſimis definitionibus, <lb/>in quibus vna particula omiſſa, vel addita <lb/>(vt paſſim cõtingit in codicibus vetuſtiſſimis) <lb/>ſenſum omninò permutat; </s>
  <s xml:id="echoid-s5271" xml:space="preserve">ſed ĳs in locis in <lb/>quibus oratione continua exponit, &amp; </s>
  <s xml:id="echoid-s5272" xml:space="preserve">exem-<lb/>plis declarat germanum ſenſum huius ſecun-<lb/>dæ definitionis, &amp; </s>
  <s xml:id="echoid-s5273" xml:space="preserve">ſeptimæ ſubſequentis, vt <lb/>ſuis in locis monebitur. </s>
  <s xml:id="echoid-s5274" xml:space="preserve">Primo igitur ſupple-<lb/>ri debent particulæ ad conterminas axium <lb/>abſciſſas, quæ in textu omnino ſubintelligi <lb/>debent vt expreſsè declaratur in propoſ. </s>
  <s xml:id="echoid-s5275" xml:space="preserve">11. <lb/></s>
  <s xml:id="echoid-s5276" xml:space="preserve">12. </s>
  <s xml:id="echoid-s5277" xml:space="preserve">15. </s>
  <s xml:id="echoid-s5278" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5279" xml:space="preserve">16. </s>
  <s xml:id="echoid-s5280" xml:space="preserve">huius libri, quibus in locis <lb/>ſemper in ſectionibus ſimilibus præcipitur vt abſciſſæ tantummodo in axibus ſu-<lb/>mantur, aut æquè ſint inclinatæ ad conterminas potentiales. </s>
  <s xml:id="echoid-s5281" xml:space="preserve">Secundò poſtrema <lb/>verba ſunt in ijſdem rationibus tum abſciſſæ ad abſciſſas poſſent retineri cũ <lb/>ſenſum definitionis non omnino intollerabilẽ reddant: </s>
  <s xml:id="echoid-s5282" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5283" xml:space="preserve">inſuper in textu gre-<lb/>co Eutocy repetantur, &amp; </s>
  <s xml:id="echoid-s5284" xml:space="preserve">eius ſenſus talis eſt. </s>
  <s xml:id="echoid-s5285" xml:space="preserve">In coniſectionibus B A C, E D <lb/>F, quarum axes A G, D H ſi ductæ fuerint quotcunq; </s>
  <s xml:id="echoid-s5286" xml:space="preserve">potentiales, ſeu ad axim <lb/>applicatæ B C, E F, I L, M O occurrentes axibus in G, H, K, N hac lege, vt <lb/>potentialis B C ad abſciſſam G A eandem proportionem habeat quàm potentialis <lb/>E F ad abſcißam H D, &amp; </s>
  <s xml:id="echoid-s5287" xml:space="preserve">potentialis I L ad abſciſſam K A ſit, vt M O ad N <lb/>D, &amp; </s>
  <s xml:id="echoid-s5288" xml:space="preserve">tandem abſciſſa G A ad K A ſit, vt abſciſſa H D ad N D: </s>
  <s xml:id="echoid-s5289" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5290" xml:space="preserve">hoc v eri-<lb/>ficetur in omnibus alĳs potentialibus eadem lege ductis; </s>
  <s xml:id="echoid-s5291" xml:space="preserve">tunc quidem duæ illæ <lb/>ſectiones ſimiles appellantur iuxta Eutocĳ, &amp; </s>
  <s xml:id="echoid-s5292" xml:space="preserve">Mydorgĳ ſententiam.</s>
  <s xml:id="echoid-s5293" xml:space="preserve"/>
</p>
<div xml:id="echoid-div510" type="float" level="2" n="2">
  <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a">
    <image file="0173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0173-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5294" xml:space="preserve">Ego contra puto, hanc expoſitionem neq. </s>
  <s xml:id="echoid-s5295" xml:space="preserve">Apollonio, neq. </s>
  <s xml:id="echoid-s5296" xml:space="preserve">veritati conciliari <lb/>poße, vt ad propoſ. </s>
  <s xml:id="echoid-s5297" xml:space="preserve">12. </s>
  <s xml:id="echoid-s5298" xml:space="preserve">oſtendetur attamen exiſtimo, definitionem hac ratione <lb/>formari poße.</s>
  <s xml:id="echoid-s5299" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5300" xml:space="preserve">Similes coniſectiones ſunt, in quibus quælibet axium abſcißæ erectis pro-<lb/>portionales etiam ad conterminas potentiales eandẽ rationem habent: </s>
  <s xml:id="echoid-s5301" xml:space="preserve">quæ omni-<lb/>no conformis eſt præcedenti definitioni, præterquam in poſtrema particula, vbi <lb/>enim ait. </s>
  <s xml:id="echoid-s5302" xml:space="preserve">Sunt in ijſdem rationibus tum abſciſſæ ad abſciſſas. </s>
  <s xml:id="echoid-s5303" xml:space="preserve">Legendum <lb/>eſſet: </s>
  <s xml:id="echoid-s5304" xml:space="preserve">ſunt in ijſdem rationibus tum abſciſſæ ad erecta. </s>
  <s xml:id="echoid-s5305" xml:space="preserve">Sed an hæc parti-<lb/>cula corrigi debeat, vel non, alĳ videant.</s>
  <s xml:id="echoid-s5306" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5307" xml:space="preserve">III. </s>
  <s xml:id="echoid-s5308" xml:space="preserve">Si verò fuerit portio ſectionis conicæ B A C, vel circunferentiæ circuli, <lb/>atq. </s>
  <s xml:id="echoid-s5309" xml:space="preserve">recta linea B C eam ſubtendat, &amp; </s>
  <s xml:id="echoid-s5310" xml:space="preserve">ſecet in duobus punctis B, &amp; </s>
  <s xml:id="echoid-s5311" xml:space="preserve">C, voca-<lb/>tur B C, Baſis prædicti ſegmenti B A C.</s>
  <s xml:id="echoid-s5312" xml:space="preserve"/>
</p>
<pb o="136" file="0174" n="174" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s5313" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s5314" xml:space="preserve">Et ſi in eodem ſegmento ducantur or-<lb/>
<anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/>
dinatæ parallelæ baſi B C, atque recta linea <lb/>A M ſecet omnes æquidiſtantes ipſi B C bifa-<lb/>riam in punctis M, N, &amp; </s>
  <s xml:id="echoid-s5315" xml:space="preserve">O vocabitur A M: <lb/></s>
  <s xml:id="echoid-s5316" xml:space="preserve">Diameter eiuſdem ſegmenti.</s>
  <s xml:id="echoid-s5317" xml:space="preserve"/>
</p>
<div xml:id="echoid-div511" type="float" level="2" n="3">
  <figure xlink:label="fig-0174-01" xlink:href="fig-0174-01a">
    <image file="0174-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0174-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5318" xml:space="preserve">V. </s>
  <s xml:id="echoid-s5319" xml:space="preserve">Et terminus eiuſdem diametri A ad <lb/>ſectionem poſitus, vocatur Vertex ſegmenti.</s>
  <s xml:id="echoid-s5320" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5321" xml:space="preserve">Tres prædictæ definitiones ſuperadditæ ab <lb/>interprete Arabico fuerunt, vt ego puto, quandoquidem omnino neceſſariæ non <lb/>ſunt.</s>
  <s xml:id="echoid-s5322" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5323" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s5324" xml:space="preserve">Sicuti in prima definitione ſectiones ſibi mutuò congruentes æquales vo-<lb/>cabantur, ſic pariter, ſi ſegmentum B A C ſuperpoſitum ſegmento E D F ſibi <lb/>mutuò congruant, ſunt duæ illæ lineæ curuæ æquales inter ſe.</s>
  <s xml:id="echoid-s5325" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5326" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s5327" xml:space="preserve">Declarat Apollonius in hac definitio-<lb/>
<anchor type="figure" xlink:label="fig-0174-02a" xlink:href="fig-0174-02"/>
ne ſeptima, quænam ſegmenta conica ſimilia <lb/>inter ſe cenſeri debeant. </s>
  <s xml:id="echoid-s5328" xml:space="preserve">Vt ſi fuerint dua-<lb/>rum conicarum ſectionum ſegmenta B A C, <lb/>&amp; </s>
  <s xml:id="echoid-s5329" xml:space="preserve">E D F, quarum diametri A M, &amp; </s>
  <s xml:id="echoid-s5330" xml:space="preserve">D L <lb/>eſſiciant cum ordinatim applicatis, ſeu cum <lb/>baſibus B C, &amp; </s>
  <s xml:id="echoid-s5331" xml:space="preserve">E F angulos æquales in M, <lb/>&amp; </s>
  <s xml:id="echoid-s5332" xml:space="preserve">L, &amp; </s>
  <s xml:id="echoid-s5333" xml:space="preserve">in vnaquaque earum ductæ fuerint <lb/>pares multitudines applicatarum, quæ ſint ba-<lb/>ſibus æquidiſtantes, vt G H, &amp; </s>
  <s xml:id="echoid-s5334" xml:space="preserve">I K, &amp; </s>
  <s xml:id="echoid-s5335" xml:space="preserve">in <lb/>eis veriſicentur hæ conditiones, vt habeat B <lb/>C ad abſciſſam M A eandem proportionem, <lb/>quàm E F ad abſcißam L D, &amp; </s>
  <s xml:id="echoid-s5336" xml:space="preserve">G H ad ab-<lb/>ciſſam N A eandem proportionem habeat, <lb/>quàm I K ad abciſsam O D, &amp; </s>
  <s xml:id="echoid-s5337" xml:space="preserve">tandem ab-<lb/>ciſsa M A ad abſciſſam A N eandem propor-<lb/>tionem habeat, quàm abſcißa L D ad abſciſ-<lb/>ſam D O; </s>
  <s xml:id="echoid-s5338" xml:space="preserve">tunc quidem vocat Apollonius duo <lb/>ſegmenta B A C, &amp; </s>
  <s xml:id="echoid-s5339" xml:space="preserve">E D F ſimilia inter ſe. </s>
  <s xml:id="echoid-s5340" xml:space="preserve">Et hic primo animaduertendum <lb/>eſt, dìfinitionem ſegmentorum ſimilium relatam ab Eutocio Aſcalonita in 3. </s>
  <s xml:id="echoid-s5341" xml:space="preserve">prop. <lb/></s>
  <s xml:id="echoid-s5342" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s5343" xml:space="preserve">2. </s>
  <s xml:id="echoid-s5344" xml:space="preserve">æquipond. </s>
  <s xml:id="echoid-s5345" xml:space="preserve">Archimedis, non eße integram: </s>
  <s xml:id="echoid-s5346" xml:space="preserve">in ea enim deſiderantur illa <lb/>verba, quarum baſes cumdiametris continent angulos æquales, ſine quibus <lb/>definitio eſſet erro-<lb/>
<anchor type="figure" xlink:label="fig-0174-03a" xlink:href="fig-0174-03"/>
nea, vt optime notat <lb/>Mydorgius. </s>
  <s xml:id="echoid-s5347" xml:space="preserve">Hoc au-<lb/>tem ita eße verba <lb/>textus Arabici aper-<lb/>te declarant, habent <lb/>enim. </s>
  <s xml:id="echoid-s5348" xml:space="preserve">Et ſimilia <lb/>ſunt quorum baſes <lb/>continent cum dia <lb/>metris angulos re-<lb/>ctos legẽdum æqua-
<pb o="137" file="0175" n="175" rhead="Conicor. Lib. VI."/>
les, &amp; </s>
  <s xml:id="echoid-s5349" xml:space="preserve">educantur in quolibet eorum ordinationes ad ſuas baſes numero <lb/>æquales, quarum proportio cum diametris eſt, vti diximus in ſectioni-<lb/>bus ſimilibus. </s>
  <s xml:id="echoid-s5350" xml:space="preserve">Idem repetit in propoſ. </s>
  <s xml:id="echoid-s5351" xml:space="preserve">15. </s>
  <s xml:id="echoid-s5352" xml:space="preserve">huius lib. </s>
  <s xml:id="echoid-s5353" xml:space="preserve">rurſus in propoſ. </s>
  <s xml:id="echoid-s5354" xml:space="preserve">16. </s>
  <s xml:id="echoid-s5355" xml:space="preserve">li-<lb/>tera a inquit: </s>
  <s xml:id="echoid-s5356" xml:space="preserve">Et quod <lb/>
<anchor type="figure" xlink:label="fig-0175-01a" xlink:href="fig-0175-01"/>
anguli à potentialibus, <lb/>&amp; </s>
  <s xml:id="echoid-s5357" xml:space="preserve">abſciſſis contenti ſint <lb/>æquales in duobus ſeg-<lb/>mentis, erit ſegmentum <lb/>H A G ſimile ſegmento <lb/>ICK: </s>
  <s xml:id="echoid-s5358" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s5359" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5360" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5361" xml:space="preserve">propoſ. </s>
  <s xml:id="echoid-s5362" xml:space="preserve">17. <lb/></s>
  <s xml:id="echoid-s5363" xml:space="preserve">litera c ait: </s>
  <s xml:id="echoid-s5364" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5365" xml:space="preserve">anguli <lb/>comprehenſi à potenti-<lb/>bus, &amp; </s>
  <s xml:id="echoid-s5366" xml:space="preserve">abſciſſis ſunt æ-<lb/>quales; </s>
  <s xml:id="echoid-s5367" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s5368" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5369" xml:space="preserve">propterea <lb/>duo ſegmenta ſunt ſimi-<lb/>lia; </s>
  <s xml:id="echoid-s5370" xml:space="preserve">Et in eadem propoſ. </s>
  <s xml:id="echoid-s5371" xml:space="preserve"><lb/>litera d dicit. </s>
  <s xml:id="echoid-s5372" xml:space="preserve">Quia propter ſimilitudinem duorum ſegmentorum conti-<lb/>nebunt potentes cum ſuis abſciſſis angulos æquales. </s>
  <s xml:id="echoid-s5373" xml:space="preserve">Et codem modo ſem-<lb/>per loquitur Apollonius; </s>
  <s xml:id="echoid-s5374" xml:space="preserve">quare dubitandum non eſt, in Eutocĳ definitione hæc <lb/>eadem verba deſiderari.</s>
  <s xml:id="echoid-s5375" xml:space="preserve"/>
</p>
<div xml:id="echoid-div512" type="float" level="2" n="4">
  <figure xlink:label="fig-0174-02" xlink:href="fig-0174-02a">
    <image file="0174-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0174-02"/>
  </figure>
  <figure xlink:label="fig-0174-03" xlink:href="fig-0174-03a">
    <image file="0174-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0174-03"/>
  </figure>
  <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a">
    <image file="0175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0175-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5376" xml:space="preserve">Immutaui poſtea verba ſubſequentia; </s>
  <s xml:id="echoid-s5377" xml:space="preserve">nam ordinationes, ſeu ordinatim ap-<lb/>plicatæ ducuntur ad diametros, non ad baſes, &amp; </s>
  <s xml:id="echoid-s5378" xml:space="preserve">debent eſſe baſibus æquidiſtan-<lb/>tes. </s>
  <s xml:id="echoid-s5379" xml:space="preserve">Deinde breuitas affectata poſtremæ partis huius definitionis non Apollonio, <lb/>ſed Arabico Interpreti tribui debet, nam eadem expreſſe, &amp; </s>
  <s xml:id="echoid-s5380" xml:space="preserve">extenſe declaratur <lb/>in textu Eutocĳ his verbis. </s>
  <s xml:id="echoid-s5381" xml:space="preserve">In quarum ſingulis ductis lineis baſi parallelis <lb/>numero æqualibus, ſint ipſæ parallelæ, &amp; </s>
  <s xml:id="echoid-s5382" xml:space="preserve">baſes ad abſciſſas diametrorum <lb/>partes ſumptas à verticibus in ijſdem rationibus, tum abſciſſe ipſæ ad ab-<lb/>ſciſſas. </s>
  <s xml:id="echoid-s5383" xml:space="preserve">In textu verò Arabico hæc non habentur expreſsè, ſicut in ſecunda de-<lb/>finitione, quàm citat hiſce verbis. </s>
  <s xml:id="echoid-s5384" xml:space="preserve">Et educantur ex quolibet eorum ordina-<lb/>tiones baſibus parallelæ numero æquales, quarum proportio cum diame-<lb/>tris eſt, vti diximus in ſectionibus ſimilibus.</s>
  <s xml:id="echoid-s5385" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div514" type="section" level="1" n="168">
<head xml:id="echoid-head220" xml:space="preserve">MONITVM.</head>
<p style="it">
  <s xml:id="echoid-s5386" xml:space="preserve">AMOR veritatis, &amp; </s>
  <s xml:id="echoid-s5387" xml:space="preserve">muneris ſuſcepti ratio exigere vide-<lb/>tur, vt definitiones ſectionum conicarum ſimilium, quæ cir-<lb/>cunferuntur, accuratius examinentur, ne (vt Mydorgĳ <lb/>verbis vtar) à magnis nominibus (Eutocium dico, Com-<lb/>mandinum, &amp; </s>
  <s xml:id="echoid-s5388" xml:space="preserve">Mydorgium) præiudicium diutius fiat veritati, hoc au-<lb/>tem ad propoſ. </s>
  <s xml:id="echoid-s5389" xml:space="preserve">11. </s>
  <s xml:id="echoid-s5390" xml:space="preserve">12. </s>
  <s xml:id="echoid-s5391" xml:space="preserve">huius lib. </s>
  <s xml:id="echoid-s5392" xml:space="preserve">præſtabo. </s>
  <s xml:id="echoid-s5393" xml:space="preserve">Interim monendus es Le-<lb/>ctor, in definitione ab Eutocio relata aliqua verba deficere (nimirum <lb/>quod abſciſſæ in axibus, aut diametris æquè ad ordinatas inclinatis <lb/>ſumantur) in definitiombus Commandini aliquod deſiderari, &amp; </s>
  <s xml:id="echoid-s5394" xml:space="preserve">eas me-
<pb o="138" file="0176" n="176" rhead="Apollonij Pergæi"/>
rito reiectas à Mydorgio ſuiſſe, nam licet latera tranſuerſa proportiona-<lb/>lia ſint lateribus rectis, non tamen duæ eiuſdem nominis ſectiones ſimi-<lb/>les erunt, niſi diametri æquè inclinatæ ſint ad ordinatim ad eas applica-<lb/>tas: </s>
  <s xml:id="echoid-s5395" xml:space="preserve">tandem deſinitionem Mydorgĳ ſimilium ſectionum pariter imperfe-<lb/>ctam eſſe ſuſpicor; </s>
  <s xml:id="echoid-s5396" xml:space="preserve">nam licet duæ ſectiones, quibus competit tradita de-<lb/>finitio, ſeu paſsio eiuſdem definitionis, ſint reuera ſimiles, non tamen è <lb/>conuerſo ſimilibus ſectionibus conuenit ſolummodo definitio, ſeu eius paſ-<lb/>ſio, curn aliquando appoſita paſsio in eiſdem reperiatur: </s>
  <s xml:id="echoid-s5397" xml:space="preserve">quod perinde eſt, <lb/>ac ſi quis putaret triangulum æquilaterum aliquando latera inæqualia ha-<lb/>bere poſſe.</s>
  <s xml:id="echoid-s5398" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5399" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s5400" xml:space="preserve">In hac deſinitione manifeſtè aliquid deſideratur: </s>
  <s xml:id="echoid-s5401" xml:space="preserve">inquit enim (Coni <lb/>fimiles ſunt quorum axium proportio ad diametros ſuarum baſium eadem <lb/>eſt.) </s>
  <s xml:id="echoid-s5402" xml:space="preserve">Quod quidem verificatur tantummodo in conis rectis: </s>
  <s xml:id="echoid-s5403" xml:space="preserve">at in ſcalenis de-<lb/>bent neceſſario axes conorum efficere æquales inclinationes ſuper baſes: </s>
  <s xml:id="echoid-s5404" xml:space="preserve">Quod <lb/>quidem in ſequentibus propoſitionibus manifeſtè ab Apollonio declaratur. </s>
  <s xml:id="echoid-s5405" xml:space="preserve">Ita-<lb/>que textum hac ratione reſtitui debere puto. </s>
  <s xml:id="echoid-s5406" xml:space="preserve">Coni ſimiles ſunt, quorum axes æ-<lb/>que ad baſes inclinati ad diametros baſium proportionales ſunt.</s>
  <s xml:id="echoid-s5407" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s5408" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s5409" xml:space="preserve">Sectio genita in ſuperſicie coni à plano eum ſecante, non per verticem <lb/>eius ducto dicitur in dicto cono poſita, &amp; </s>
  <s xml:id="echoid-s5410" xml:space="preserve">contenta; </s>
  <s xml:id="echoid-s5411" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5412" xml:space="preserve">conus ille continere di-<lb/>citur eandem ſectionem: </s>
  <s xml:id="echoid-s5413" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5414" xml:space="preserve">licet coniſectio exhibeatur extra conum; </s>
  <s xml:id="echoid-s5415" xml:space="preserve">dicetur ni-<lb/>hilominus contineri ab illo cono, in quo ſectio illa accomodari poteſt, ſeu in quo <lb/>ab aliquo plano ſecante effici poteſt in coni ſuperficie eadem illa coniſectio.</s>
  <s xml:id="echoid-s5416" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div515" type="section" level="1" n="169">
<head xml:id="echoid-head221" xml:space="preserve">SECTIO PRIMA</head>
<head xml:id="echoid-head222" xml:space="preserve">Continens Propoſit. I. II. IV. &amp; X.</head>
<head xml:id="echoid-head223" xml:space="preserve">PROPOSITIO I.</head>
<p>
  <s xml:id="echoid-s5417" xml:space="preserve">QVælibet duæ ſectiones parabolicæ A B, C D, ſi habue-<lb/>
<anchor type="note" xlink:label="note-0176-01a" xlink:href="note-0176-01"/>
rint axium erectos A I, C N æquales: </s>
  <s xml:id="echoid-s5418" xml:space="preserve">erunt inter ſe æ-<lb/>quales. </s>
  <s xml:id="echoid-s5419" xml:space="preserve">Si verò duæ illæ ſectiones fuerint æquales, <lb/>erunt axium erecta æqualia inter ſe.</s>
  <s xml:id="echoid-s5420" xml:space="preserve"/>
</p>
<div xml:id="echoid-div515" type="float" level="2" n="1">
<note position="right" xlink:label="note-0176-01" xlink:href="note-0176-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s5421" xml:space="preserve">Quoniam ſuperpoſita axi C H ſuper axim A G, cadet ſectio C D ſu-<lb/>
<anchor type="note" xlink:label="note-0176-02a" xlink:href="note-0176-02"/>
per ſectionem A B: </s>
  <s xml:id="echoid-s5422" xml:space="preserve">ſi enim cadere non concedatur ſuper illam, ſigne-<lb/>tur (ſi fieri poteſt) punctum eius D, extra ſectionem A B cadens: </s>
  <s xml:id="echoid-s5423" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5424" xml:space="preserve"><lb/>educatur D F perpendicularis ad axim; </s>
  <s xml:id="echoid-s5425" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5426" xml:space="preserve">perficiatur planum rectangu-<lb/>lum F N, &amp; </s>
  <s xml:id="echoid-s5427" xml:space="preserve">ab axi A G ſecetur A E æqualis C F; </s>
  <s xml:id="echoid-s5428" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5429" xml:space="preserve">educatur ex E
<pb o="139" file="0177" n="177" rhead="Conicor. Lib. VI."/>
pespendicularis B E, &amp; </s>
  <s xml:id="echoid-s5430" xml:space="preserve">perficiatur <lb/>planũ E I. </s>
  <s xml:id="echoid-s5431" xml:space="preserve">Et quia A I, A E æquã-<lb/>
<anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/>
tur C N, C F, vnaquæque ſuo ho-<lb/>mologo: </s>
  <s xml:id="echoid-s5432" xml:space="preserve">igitur planum I E, nempe <lb/>
<anchor type="note" xlink:label="note-0177-01a" xlink:href="note-0177-01"/>
(12. </s>
  <s xml:id="echoid-s5433" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5434" xml:space="preserve">quadratum B E æquale <lb/>eſt rectangulo F N, nempe quadrato <lb/>D F (12. </s>
  <s xml:id="echoid-s5435" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5436" xml:space="preserve">ergo B E æqualis <lb/>eſt D F; </s>
  <s xml:id="echoid-s5437" xml:space="preserve">ſi autem ſuperponatur axis <lb/>axi cadet D ſuper B, quæ tamẽhaud <lb/>cadere conceſſum fuerat: </s>
  <s xml:id="echoid-s5438" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5439" xml:space="preserve">hoc eſt <lb/>abſurdum; </s>
  <s xml:id="echoid-s5440" xml:space="preserve">ergo fieri non poteſt, vt <lb/>duæ ſectiones æquales non ſint.</s>
  <s xml:id="echoid-s5441" xml:space="preserve"/>
</p>
<div xml:id="echoid-div516" type="float" level="2" n="2">
<note position="right" xlink:label="note-0176-02" xlink:href="note-0176-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0177-01" xlink:href="fig-0177-01a">
    <image file="0177-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0177-01"/>
  </figure>
<note position="right" xlink:label="note-0177-01" xlink:href="note-0177-01a" xml:space="preserve">11. lib. 1. <lb/>Ibidcm.</note>
</div>
<p>
  <s xml:id="echoid-s5442" xml:space="preserve">Præterea ſupponamus duas illas ſe-<lb/>
<anchor type="note" xlink:label="note-0177-02a" xlink:href="note-0177-02"/>
ctiones æquales eſſe inter ſe, &amp; </s>
  <s xml:id="echoid-s5443" xml:space="preserve">fiat <lb/>F C æqualis E A, &amp; </s>
  <s xml:id="echoid-s5444" xml:space="preserve">educamus ad <lb/>axes perpendiculares B E, D F, &amp; </s>
  <s xml:id="echoid-s5445" xml:space="preserve">per-<lb/>ficiamus plana rectangula F N, E I. <lb/></s>
  <s xml:id="echoid-s5446" xml:space="preserve">Quia ſectio A B cadit ſuper ſectionem C D, &amp; </s>
  <s xml:id="echoid-s5447" xml:space="preserve">A E ſuper C F cadet; </s>
  <s xml:id="echoid-s5448" xml:space="preserve"><lb/>alioquin eſſent in eadem parabola duo axes: </s>
  <s xml:id="echoid-s5449" xml:space="preserve">ergo F cadit ſuper E, &amp; </s>
  <s xml:id="echoid-s5450" xml:space="preserve">D <lb/>ſuper B, &amp; </s>
  <s xml:id="echoid-s5451" xml:space="preserve">propterea B E potens planum E I (12. </s>
  <s xml:id="echoid-s5452" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5453" xml:space="preserve">æqualis erit <lb/>
<anchor type="note" xlink:label="note-0177-03a" xlink:href="note-0177-03"/>
D F potenti planum F N (12. </s>
  <s xml:id="echoid-s5454" xml:space="preserve">ex 1.)</s>
  <s xml:id="echoid-s5455" xml:space="preserve">; </s>
  <s xml:id="echoid-s5456" xml:space="preserve">ergo duo plana ſunt æqualia; </s>
  <s xml:id="echoid-s5457" xml:space="preserve">ſed <lb/>
<anchor type="note" xlink:label="note-0177-04a" xlink:href="note-0177-04"/>
ſunt applicata ad æquales F C, A E; </s>
  <s xml:id="echoid-s5458" xml:space="preserve">igitur C N, A I erectæ æquales <lb/>
<anchor type="note" xlink:label="note-0177-05a" xlink:href="note-0177-05"/>
ſunt. </s>
  <s xml:id="echoid-s5459" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s5460" xml:space="preserve"/>
</p>
<div xml:id="echoid-div517" type="float" level="2" n="3">
<note position="left" xlink:label="note-0177-02" xlink:href="note-0177-02a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0177-03" xlink:href="note-0177-03a" xml:space="preserve">11 lib. 1.</note>
<note position="right" xlink:label="note-0177-04" xlink:href="note-0177-04a" xml:space="preserve">Ibidem.</note>
<note position="left" xlink:label="note-0177-05" xlink:href="note-0177-05a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div519" type="section" level="1" n="170">
<head xml:id="echoid-head224" xml:space="preserve">PROPOSITIO II.</head>
<p>
  <s xml:id="echoid-s5461" xml:space="preserve">SI duæ ſectiones hyperbolicæ, aut duæ ellipſes A B C, D E <lb/>F habuerint axium figuras G I, H K ſimiles, &amp; </s>
  <s xml:id="echoid-s5462" xml:space="preserve">æquales; <lb/></s>
  <s xml:id="echoid-s5463" xml:space="preserve">duæ illæ ſectiones æquales erunt. </s>
  <s xml:id="echoid-s5464" xml:space="preserve">Si verò duæ ſectiones æquales <lb/>
<anchor type="note" xlink:label="note-0177-06a" xlink:href="note-0177-06"/>
fuerint, earũ figuræ axiũ erunt æquales, ſimiles, &amp; </s>
  <s xml:id="echoid-s5465" xml:space="preserve">ſimiliter poſitæ.</s>
  <s xml:id="echoid-s5466" xml:space="preserve"/>
</p>
<div xml:id="echoid-div519" type="float" level="2" n="1">
<note position="left" xlink:label="note-0177-06" xlink:href="note-0177-06a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0177-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0177-02"/>
  </figure>
  <figure>
    <image file="0177-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0177-03"/>
  </figure>
<pb o="140" file="0178" n="178" rhead="Apollonij Pergæi"/>
  <figure>
    <image file="0178-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0178-01"/>
  </figure>
<p>
  <s xml:id="echoid-s5467" xml:space="preserve">Quoniam facta conuenienti ſuperpoſitione axis A M ſuper axim D <lb/>O, cadet quoque ſectio A B ſuper ſectionem D E: </s>
  <s xml:id="echoid-s5468" xml:space="preserve">ſi enim non cadit ſu-<lb/>per illam, ſumatur (ſi fieri poteſt) eius punctum B, extra ſectionem. <lb/></s>
  <s xml:id="echoid-s5469" xml:space="preserve">D E cadens; </s>
  <s xml:id="echoid-s5470" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5471" xml:space="preserve">producatur ad axim perpendicularis B L vſque ad P: </s>
  <s xml:id="echoid-s5472" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5473" xml:space="preserve"><lb/>perficiatur planum A P applicatum comparatum; </s>
  <s xml:id="echoid-s5474" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5475" xml:space="preserve">ſecetur D N æqua-<lb/>lis A L, &amp; </s>
  <s xml:id="echoid-s5476" xml:space="preserve">erigatur per N ad axim perpendicularis N E, &amp; </s>
  <s xml:id="echoid-s5477" xml:space="preserve">producatur <lb/>vſque ad R, perficiendo planum D R applicatum comparatum; </s>
  <s xml:id="echoid-s5478" xml:space="preserve">Et quia <lb/>A I æqualis eſt D K, &amp; </s>
  <s xml:id="echoid-s5479" xml:space="preserve">A L æqualis D N: </s>
  <s xml:id="echoid-s5480" xml:space="preserve">erit planum I L, æquale pla-<lb/>no K N; </s>
  <s xml:id="echoid-s5481" xml:space="preserve">cumque G I, H K ſint duæ figuræ ſimiles, &amp; </s>
  <s xml:id="echoid-s5482" xml:space="preserve">æquales, pariter-<lb/>
<anchor type="note" xlink:label="note-0178-01a" xlink:href="note-0178-01"/>
que I P, K R; </s>
  <s xml:id="echoid-s5483" xml:space="preserve">ergo duo plana A P, D R ſunt æqualia: </s>
  <s xml:id="echoid-s5484" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5485" xml:space="preserve">propterea E <lb/>N, B L, quæ illa ſpatia poſſunt (13. </s>
  <s xml:id="echoid-s5486" xml:space="preserve">14. </s>
  <s xml:id="echoid-s5487" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5488" xml:space="preserve">ſunt æquales. </s>
  <s xml:id="echoid-s5489" xml:space="preserve">Si autem <lb/>
<anchor type="note" xlink:label="note-0178-02a" xlink:href="note-0178-02"/>
ſuperponatur axis axi cadet B L ſuper E N, eoquod duo anguli N, &amp; </s>
  <s xml:id="echoid-s5490" xml:space="preserve">L <lb/>ſunt æquales; </s>
  <s xml:id="echoid-s5491" xml:space="preserve">igitur B cadit ſuper E, quod prius cadere non concedeba-<lb/>tur: </s>
  <s xml:id="echoid-s5492" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5493" xml:space="preserve">hoc eſt abſurdum. </s>
  <s xml:id="echoid-s5494" xml:space="preserve">Quapropter ſectio ſectioni æqualis eſt.</s>
  <s xml:id="echoid-s5495" xml:space="preserve"/>
</p>
<div xml:id="echoid-div520" type="float" level="2" n="2">
<note position="right" xlink:label="note-0178-01" xlink:href="note-0178-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0178-02" xlink:href="note-0178-02a" xml:space="preserve">12. 13. <lb/>lib. I.</note>
</div>
<p>
  <s xml:id="echoid-s5496" xml:space="preserve">Deinde ponamus duas ſe-<lb/>
<anchor type="figure" xlink:label="fig-0178-02a" xlink:href="fig-0178-02"/>
ctiones æquales, vtique con-<lb/>gruet ſectio A B ſectioni D E, <lb/>&amp; </s>
  <s xml:id="echoid-s5497" xml:space="preserve">axis A L axi D N, quia ſi <lb/>non cadit ſuper illum, eſſent <lb/>
<anchor type="note" xlink:label="note-0178-03a" xlink:href="note-0178-03"/>
in hyperbola duo axes, &amp; </s>
  <s xml:id="echoid-s5498" xml:space="preserve">in <lb/>ellipſi tres axes, quod eſt ab-<lb/>ſurdum (52. </s>
  <s xml:id="echoid-s5499" xml:space="preserve">53. </s>
  <s xml:id="echoid-s5500" xml:space="preserve">ex 2.) </s>
  <s xml:id="echoid-s5501" xml:space="preserve">Et fi-<lb/>
<anchor type="note" xlink:label="note-0178-04a" xlink:href="note-0178-04"/>
at A L æqualis D N, &amp; </s>
  <s xml:id="echoid-s5502" xml:space="preserve">reli-<lb/>qua perficiantur, vt prius ca-<lb/>dent duo puncta L, B ſuper <lb/>N, E; </s>
  <s xml:id="echoid-s5503" xml:space="preserve">ideoque B L æqualis <lb/>
<anchor type="note" xlink:label="note-0178-05a" xlink:href="note-0178-05"/>
erit E N; </s>
  <s xml:id="echoid-s5504" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5505" xml:space="preserve">poterunt æqua-<lb/>lia rectangula A P, D R applicata ad æquales A L, D N (13. </s>
  <s xml:id="echoid-s5506" xml:space="preserve">14. </s>
  <s xml:id="echoid-s5507" xml:space="preserve">ex 1.) <lb/></s>
  <s xml:id="echoid-s5508" xml:space="preserve">
<anchor type="note" xlink:label="note-0178-06a" xlink:href="note-0178-06"/>
ergo L P æqualis eſt N R. </s>
  <s xml:id="echoid-s5509" xml:space="preserve">Similiter ponatur A M æqualis D O, &amp; </s>
  <s xml:id="echoid-s5510" xml:space="preserve">edu-<lb/>cantur C M Q, F O S duæ ordinationes, oſtendetur, quod M Q æqua-<lb/>lis eſt O S, &amp; </s>
  <s xml:id="echoid-s5511" xml:space="preserve">L M æqualis N O; </s>
  <s xml:id="echoid-s5512" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5513" xml:space="preserve">propterea duo plana P Q, R S ſunt <lb/>æqualia, &amp; </s>
  <s xml:id="echoid-s5514" xml:space="preserve">ſimilia; </s>
  <s xml:id="echoid-s5515" xml:space="preserve">igitur duo plana G P, H R ſunt æqualia, &amp; </s>
  <s xml:id="echoid-s5516" xml:space="preserve">ſimilia, <lb/>&amp; </s>
  <s xml:id="echoid-s5517" xml:space="preserve">L P oſtenſa eſt æqualis N R: </s>
  <s xml:id="echoid-s5518" xml:space="preserve">ergo G L æqualis eſt H N, &amp; </s>
  <s xml:id="echoid-s5519" xml:space="preserve">A L æ-<lb/>qualis D N; </s>
  <s xml:id="echoid-s5520" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5521" xml:space="preserve">propterea G A æqualis eſt D H, &amp; </s>
  <s xml:id="echoid-s5522" xml:space="preserve">A I æqualis D K.</s>
  <s xml:id="echoid-s5523" xml:space="preserve">
<pb o="141" file="0179" n="179" rhead="Conicor. Lib. VI."/>
Quapropter duæ figuræ G I, H K ſunt æquales, &amp; </s>
  <s xml:id="echoid-s5524" xml:space="preserve">ſimiles. </s>
  <s xml:id="echoid-s5525" xml:space="preserve">Quod erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s5526" xml:space="preserve"/>
</p>
<div xml:id="echoid-div521" type="float" level="2" n="3">
  <figure xlink:label="fig-0178-02" xlink:href="fig-0178-02a">
    <image file="0178-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0178-02"/>
  </figure>
<note position="right" xlink:label="note-0178-03" xlink:href="note-0178-03a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0178-04" xlink:href="note-0178-04a" xml:space="preserve">48. lib. 2.</note>
<note position="right" xlink:label="note-0178-05" xlink:href="note-0178-05a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0178-06" xlink:href="note-0178-06a" xml:space="preserve">12. 13. <lb/>lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div523" type="section" level="1" n="171">
<head xml:id="echoid-head225" xml:space="preserve">PROPOSITIO IV.</head>
<p>
  <s xml:id="echoid-s5527" xml:space="preserve">SImili modo demõſtrabitur, quod <lb/>
<anchor type="note" xlink:label="note-0179-01a" xlink:href="note-0179-01"/>
duæ ſectiones oppoſitæ ſintſimi-<lb/>
<anchor type="figure" xlink:label="fig-0179-01a" xlink:href="fig-0179-01"/>
les, &amp; </s>
  <s xml:id="echoid-s5528" xml:space="preserve">æquales.</s>
  <s xml:id="echoid-s5529" xml:space="preserve"/>
</p>
<div xml:id="echoid-div523" type="float" level="2" n="1">
<note position="right" xlink:label="note-0179-01" xlink:href="note-0179-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0179-01" xlink:href="fig-0179-01a">
    <image file="0179-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0179-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s5530" xml:space="preserve">Eo quod axis inclinatus eſt communis', <lb/>&amp; </s>
  <s xml:id="echoid-s5531" xml:space="preserve">erecti ſunt æquales (16. </s>
  <s xml:id="echoid-s5532" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5533" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5534" xml:space="preserve">prot <lb/>
<anchor type="note" xlink:label="note-0179-02a" xlink:href="note-0179-02"/>
pterea earum figuræ æquales quoque ſun-<lb/>inter ſe. </s>
  <s xml:id="echoid-s5535" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s5536" xml:space="preserve"/>
</p>
<div xml:id="echoid-div524" type="float" level="2" n="2">
<note position="right" xlink:label="note-0179-02" xlink:href="note-0179-02a" xml:space="preserve">14. lib. 4.</note>
</div>
</div>
<div xml:id="echoid-div526" type="section" level="1" n="172">
<head xml:id="echoid-head226" xml:space="preserve">PROPOSITIO X.</head>
<p>
  <s xml:id="echoid-s5537" xml:space="preserve">PAriter conſtat, quod ſi poten-<lb/>
<anchor type="note" xlink:label="note-0179-03a" xlink:href="note-0179-03"/>
tiales cum ſuis abſciſſis cõpræ-<lb/>hendant angulos æquales obliquos, <lb/>eadem conſequentur, quæ prius dicta ſunt.</s>
  <s xml:id="echoid-s5538" xml:space="preserve"/>
</p>
<div xml:id="echoid-div526" type="float" level="2" n="1">
<note position="right" xlink:label="note-0179-03" xlink:href="note-0179-03a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s5539" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s5540" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div528" type="section" level="1" n="173">
<head xml:id="echoid-head227" xml:space="preserve">Notæ in Propoſit. I.</head>
<p>
  <s xml:id="echoid-s5541" xml:space="preserve">QVælibet duæ ſectiones parabolicæ, <lb/>
<anchor type="figure" xlink:label="fig-0179-02a" xlink:href="fig-0179-02"/>
<anchor type="note" xlink:label="note-0179-04a" xlink:href="note-0179-04"/>
vt A B, C D, quarum relationes <lb/>ſunt duo plana A L, C M, &amp; </s>
  <s xml:id="echoid-s5542" xml:space="preserve"><lb/>erecti earum A I, C N æquales. </s>
  <s xml:id="echoid-s5543" xml:space="preserve">ipſæ quo-<lb/>que ſunt æquales. </s>
  <s xml:id="echoid-s5544" xml:space="preserve">Si verò duæ illæ ſectio-<lb/>nes fuerint æquales, vtique earum appli-<lb/>cata, &amp; </s>
  <s xml:id="echoid-s5545" xml:space="preserve">erecti erunt æquales, &amp;</s>
  <s xml:id="echoid-s5546" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5547" xml:space="preserve">Verba <lb/>illa propoſitionis (applicata ſunt duo plana <lb/>A L, C M, &amp;</s>
  <s xml:id="echoid-s5548" xml:space="preserve">c.) </s>
  <s xml:id="echoid-s5549" xml:space="preserve">caſu in textum irrepſiſſe <lb/>puto, eo quod rectangula illa A L, C M, ne-<lb/>dum æqualia non ſupponuntur, ſed è contra. <lb/></s>
  <s xml:id="echoid-s5550" xml:space="preserve">conſtruuntur, atque demonſtrantur æqualia eſ-<lb/>ſe inter ſe.</s>
  <s xml:id="echoid-s5551" xml:space="preserve"/>
</p>
<div xml:id="echoid-div528" type="float" level="2" n="1">
  <figure xlink:label="fig-0179-02" xlink:href="fig-0179-02a">
    <image file="0179-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0179-02"/>
  </figure>
<note position="right" xlink:label="note-0179-04" xlink:href="note-0179-04a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s5552" xml:space="preserve">Quia ſi ponamus ſagittam C H ſuper ſa-<lb/>
<anchor type="note" xlink:label="note-0179-05a" xlink:href="note-0179-05"/>
gittã A G, cadet ſectio C D ſuper ſectio-<lb/>nem A B: </s>
  <s xml:id="echoid-s5553" xml:space="preserve">11 verò non cadit ſuper illam, <lb/>ſignemus ſuper literam, in quam non ca-<lb/>dit punctum D: </s>
  <s xml:id="echoid-s5554" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s5555" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5556" xml:space="preserve">Sic legendũ puto. </s>
  <s xml:id="echoid-s5557" xml:space="preserve">Quo-
<pb o="142" file="0180" n="180" rhead="Apollonij Pergæi"/>
niam, ſuperpoſita axi C H ſuper axim A G, <lb/>&amp;</s>
  <s xml:id="echoid-s5558" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5559" xml:space="preserve">vt in textu habetur. </s>
  <s xml:id="echoid-s5560" xml:space="preserve">Si enim axis C H <lb/>
<anchor type="figure" xlink:label="fig-0180-01a" xlink:href="fig-0180-01"/>
ſuper axim A G applicatur, ita vt vertices A, <lb/>C coincidant, neceſſariò ſectio C D cadet ſu-<lb/>per ſectionem A B alias aſſignari poſſet pun-<lb/>ctum eius D, extra ſectionem A B cadens.</s>
  <s xml:id="echoid-s5561" xml:space="preserve"/>
</p>
<div xml:id="echoid-div529" type="float" level="2" n="2">
<note position="right" xlink:label="note-0179-05" xlink:href="note-0179-05a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0180-01" xlink:href="fig-0180-01a">
    <image file="0180-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0180-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5562" xml:space="preserve">Præterea ponamus duas ſectiones æqua-<lb/>
<anchor type="note" xlink:label="note-0180-01a" xlink:href="note-0180-01"/>
les, &amp; </s>
  <s xml:id="echoid-s5563" xml:space="preserve">C F æqualis A E, &amp;</s>
  <s xml:id="echoid-s5564" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5565" xml:space="preserve">Textum cor-<lb/>ruptum ſic reſtituendum cenſeo. </s>
  <s xml:id="echoid-s5566" xml:space="preserve">Præterea ſup-<lb/>ponamus, duas illas ſectiones æquales eſſe in-<lb/>ter ſe, &amp; </s>
  <s xml:id="echoid-s5567" xml:space="preserve">fiat C F æqualis A E, educamus ad <lb/>axes perpendiculares B E, D F, &amp;</s>
  <s xml:id="echoid-s5568" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5569" xml:space="preserve">Sic enim <lb/>diſtinguitur hypotheſis propoſitionis à conſtru-<lb/>ctione eius.</s>
  <s xml:id="echoid-s5570" xml:space="preserve"/>
</p>
<div xml:id="echoid-div530" type="float" level="2" n="3">
<note position="right" xlink:label="note-0180-01" xlink:href="note-0180-01a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s5571" xml:space="preserve">Ergo ſectio A B cadit ſuper ſectionem. <lb/></s>
  <s xml:id="echoid-s5572" xml:space="preserve">
<anchor type="note" xlink:label="note-0180-02a" xlink:href="note-0180-02"/>
C D, &amp; </s>
  <s xml:id="echoid-s5573" xml:space="preserve">A E ſuper C F: </s>
  <s xml:id="echoid-s5574" xml:space="preserve">alioqui eſſent ſe-<lb/>ctioni parabolicæ duo axes; </s>
  <s xml:id="echoid-s5575" xml:space="preserve">ergo F cadit <lb/>ſuper E, &amp;</s>
  <s xml:id="echoid-s5576" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5577" xml:space="preserve">Quoniam (ex hypotheſi) ſectio-<lb/>nes A B, &amp; </s>
  <s xml:id="echoid-s5578" xml:space="preserve">C D æquales ſunt, facta intellectuali conuenienti ſuperpoſitione, ſi-<lb/>bi mutuò congruent, &amp; </s>
  <s xml:id="echoid-s5579" xml:space="preserve">vertex A cadet ſuper verticcm C. </s>
  <s xml:id="echoid-s5580" xml:space="preserve">Dico iam, axim A <lb/>E cadere ſuper axim C F: </s>
  <s xml:id="echoid-s5581" xml:space="preserve">alioquin in eadem parabola, ſcilicet in duabus pa-<lb/>rabolis ſibi congruentibus à communi vertice C, vel A, duo axes A E, &amp; </s>
  <s xml:id="echoid-s5582" xml:space="preserve">C F <lb/>ducerentur: </s>
  <s xml:id="echoid-s5583" xml:space="preserve">quod eſt impoſſibile. </s>
  <s xml:id="echoid-s5584" xml:space="preserve">Quare axis A E cadit ſuper axim C F.</s>
  <s xml:id="echoid-s5585" xml:space="preserve"/>
</p>
<div xml:id="echoid-div531" type="float" level="2" n="4">
<note position="right" xlink:label="note-0180-02" xlink:href="note-0180-02a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div533" type="section" level="1" n="174">
<head xml:id="echoid-head228" xml:space="preserve">Notæ in Propoſit. II.</head>
<p>
  <s xml:id="echoid-s5586" xml:space="preserve">SI fuerint figuræ duarum ſectionem hyperbolicarum, aut duarum elli-<lb/>
<anchor type="note" xlink:label="note-0180-03a" xlink:href="note-0180-03"/>
pſium, vt duo plana G I, H K in A B, D E ſimiles, &amp; </s>
  <s xml:id="echoid-s5587" xml:space="preserve">æquales; <lb/></s>
  <s xml:id="echoid-s5588" xml:space="preserve">vtique duæ ſectiones æquales erunt: </s>
  <s xml:id="echoid-s5589" xml:space="preserve">ſi vero duæ ſectiones ſint æquales <lb/>earum figuræ erunt æquales, ſimiles, &amp;</s>
  <s xml:id="echoid-s5590" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5591" xml:space="preserve">In duabus ſectionibus A B, &amp; </s>
  <s xml:id="echoid-s5592" xml:space="preserve"><lb/>D E ſumi debent figuræ G I, &amp; </s>
  <s xml:id="echoid-s5593" xml:space="preserve">H K, non qualeſcunque, ſed illæ, quæ ad axes <lb/>fiunt, nimirum debent eſſe G A, &amp; </s>
  <s xml:id="echoid-s5594" xml:space="preserve">H D axes inclinati, ſeu tranſuerſi, &amp; </s>
  <s xml:id="echoid-s5595" xml:space="preserve">A <lb/>I, atque D K eorum latera recta; </s>
  <s xml:id="echoid-s5596" xml:space="preserve">tunc quidem, ſi figuræ axium G I, H K fue-<lb/>rint ſimiles, &amp; </s>
  <s xml:id="echoid-s5597" xml:space="preserve">æquales, conicæ ſectiones B A, D E æquales quoque oſtenduntur <lb/>in propoſitione. </s>
  <s xml:id="echoid-s5598" xml:space="preserve">Quod verò particula illa (axium) deſideretur in textu propo-<lb/>ſitionis, conſtat ex primis verbis immediatè ſequentis conſtructionis. </s>
  <s xml:id="echoid-s5599" xml:space="preserve">Inquit <lb/>enim. </s>
  <s xml:id="echoid-s5600" xml:space="preserve">Quoniam ſi ponamus axim A M ſuper axim D O, &amp;</s>
  <s xml:id="echoid-s5601" xml:space="preserve">c.</s>
  <s xml:id="echoid-s5602" xml:space="preserve"/>
</p>
<div xml:id="echoid-div533" type="float" level="2" n="1">
<note position="right" xlink:label="note-0180-03" xlink:href="note-0180-03a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s5603" xml:space="preserve">Cumque G I, H K ſint duæ figuræ ſimiles, &amp; </s>
  <s xml:id="echoid-s5604" xml:space="preserve">æquales, pariterque <lb/>
<anchor type="note" xlink:label="note-0180-04a" xlink:href="note-0180-04"/>
I P, K R; </s>
  <s xml:id="echoid-s5605" xml:space="preserve">ergo duo plana A P, D R ſunt æqualia, &amp;</s>
  <s xml:id="echoid-s5606" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5607" xml:space="preserve">Quia rectangula <lb/>I P, G I circa communcm diametrum G I P conſiſtunt, erunt inter ſe ſimilia: <lb/></s>
  <s xml:id="echoid-s5608" xml:space="preserve">pariterque K R ſimile erit rectangulo K H: </s>
  <s xml:id="echoid-s5609" xml:space="preserve">quare duo rectangula I P, &amp; </s>
  <s xml:id="echoid-s5610" xml:space="preserve">K R <lb/>ſimilia ſunt duobus rectangulis G I, H K inter ſe ſimilibus; </s>
  <s xml:id="echoid-s5611" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5612" xml:space="preserve">ideo illa inter <lb/>ſe quoque ſimilia erunt, &amp; </s>
  <s xml:id="echoid-s5613" xml:space="preserve">habent latera homologa æqualia, illa nimirum, quæ <lb/>opponuntur æqualibus abciſsis A L, &amp; </s>
  <s xml:id="echoid-s5614" xml:space="preserve">D N, igitur rectangula P I, &amp; </s>
  <s xml:id="echoid-s5615" xml:space="preserve">R K
<pb o="143" file="0181" n="181" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/>
æqualia ſunt inter ſe: </s>
  <s xml:id="echoid-s5616" xml:space="preserve">ſunt verò rectangula N K, &amp; </s>
  <s xml:id="echoid-s5617" xml:space="preserve">L I æqualia quoque (cum <lb/>latera circa angulos rectos æqualia habeant, ſingula ſingulis) ergo duo rectangu-<lb/>la A P, &amp; </s>
  <s xml:id="echoid-s5618" xml:space="preserve">D R æqualia ſunt inter ſe.</s>
  <s xml:id="echoid-s5619" xml:space="preserve"/>
</p>
<div xml:id="echoid-div534" type="float" level="2" n="2">
<note position="right" xlink:label="note-0180-04" xlink:href="note-0180-04a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a">
    <image file="0181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0181-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5620" xml:space="preserve">Quia, ſi non cadit ſuper illum, eſſent ſectioni hyperbolicæ duo axes, <lb/>
<anchor type="note" xlink:label="note-0181-01a" xlink:href="note-0181-01"/>
&amp; </s>
  <s xml:id="echoid-s5621" xml:space="preserve">in ellipſi tres axes, &amp;</s>
  <s xml:id="echoid-s5622" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5623" xml:space="preserve">Q<unsure/>uoniam æquales ſectiones B A, E D ſibi mutuò <lb/>congruunt, &amp; </s>
  <s xml:id="echoid-s5624" xml:space="preserve">vertices A, &amp; </s>
  <s xml:id="echoid-s5625" xml:space="preserve">D coincidunt, ſiquidem axis A L non cadit ſuper <lb/>axim D N (cum ambo tamen axes ſint) haberet vnica ſectio, ſcilicet duæ ſe-<lb/>ctiones congruentes, duos axes A L, &amp; </s>
  <s xml:id="echoid-s5626" xml:space="preserve">D N conuenientes in eodem puncto ver-<lb/>
<anchor type="figure" xlink:label="fig-0181-02a" xlink:href="fig-0181-02"/>
ticis, quod in hyperbola eſt im-<lb/>
<anchor type="note" xlink:label="note-0181-02a" xlink:href="note-0181-02"/>
poſſibile; </s>
  <s xml:id="echoid-s5627" xml:space="preserve">in ellipſi verò, in qua <lb/>ſemper duo axes reperiuntur ſe <lb/>ſe ſecantes in centro ad angulos <lb/>rectos, reperietur tertius axis, <lb/>ille nimirum, qui ab eodem ver-<lb/>tice A ducitur in eadem ſectione <lb/>A B, &amp; </s>
  <s xml:id="echoid-s5628" xml:space="preserve">non coincidit cum axi <lb/>A L.</s>
  <s xml:id="echoid-s5629" xml:space="preserve"/>
</p>
<div xml:id="echoid-div535" type="float" level="2" n="3">
<note position="left" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve">C</note>
  <figure xlink:label="fig-0181-02" xlink:href="fig-0181-02a">
    <image file="0181-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0181-02"/>
  </figure>
<note position="right" xlink:label="note-0181-02" xlink:href="note-0181-02a" xml:space="preserve">48. lib. 2.</note>
</div>
<p style="it">
  <s xml:id="echoid-s5630" xml:space="preserve">Ideoque B L æqualis eſt N <lb/>
<anchor type="note" xlink:label="note-0181-03a" xlink:href="note-0181-03"/>
E, &amp; </s>
  <s xml:id="echoid-s5631" xml:space="preserve">poterunt A P, D R, ap-<lb/>plicata ad A L, D N æqualia <lb/>&amp;</s>
  <s xml:id="echoid-s5632" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5633" xml:space="preserve">Q<unsure/>uia quadrata æqualium. <lb/></s>
  <s xml:id="echoid-s5634" xml:space="preserve">B L, E N æqualia ſunt rectangulis A P, D R; </s>
  <s xml:id="echoid-s5635" xml:space="preserve">erunt illa æqualia, &amp; </s>
  <s xml:id="echoid-s5636" xml:space="preserve">corum <lb/>latera A L, D N facta ſunt æqualia; </s>
  <s xml:id="echoid-s5637" xml:space="preserve">igitur reliqua duo latera L P, N R æ-<lb/>qualia quoque ſunt. </s>
  <s xml:id="echoid-s5638" xml:space="preserve">Simili modo oſtendetur, quod M Q æqualis eſt O S, ſeù L <lb/>T æqualis eſt N V, &amp; </s>
  <s xml:id="echoid-s5639" xml:space="preserve">L M, ſeu T Q æqualis eſt N O, ſeu V S; </s>
  <s xml:id="echoid-s5640" xml:space="preserve">erant autem. </s>
  <s xml:id="echoid-s5641" xml:space="preserve"><lb/>prius L P, N R æquales; </s>
  <s xml:id="echoid-s5642" xml:space="preserve">igitur reſiduæ P T, &amp; </s>
  <s xml:id="echoid-s5643" xml:space="preserve">R V æquales erunt, ſed quia <lb/>T Q, &amp; </s>
  <s xml:id="echoid-s5644" xml:space="preserve">G L ſunt parallelæ pariterque V S, &amp; </s>
  <s xml:id="echoid-s5645" xml:space="preserve">H N; </s>
  <s xml:id="echoid-s5646" xml:space="preserve">ergo vt T P ad P L ita <lb/>eſt Q T ad L G, ſimili modo vt V R ad R N ita eſt S V ad N H; </s>
  <s xml:id="echoid-s5647" xml:space="preserve">habent ve-<lb/>rò duæ æquales T P, &amp; </s>
  <s xml:id="echoid-s5648" xml:space="preserve">V R ad duas æquales P L, &amp; </s>
  <s xml:id="echoid-s5649" xml:space="preserve">R N eandem proportio-<lb/>nem, igitur duæ æquales Q T, &amp; </s>
  <s xml:id="echoid-s5650" xml:space="preserve">S V eandem proportionem habent ad L G, &amp; </s>
  <s xml:id="echoid-s5651" xml:space="preserve"><lb/>N H, &amp; </s>
  <s xml:id="echoid-s5652" xml:space="preserve">propterea hæ erunt æquales, &amp; </s>
  <s xml:id="echoid-s5653" xml:space="preserve">ablatis æqualibus A L, D N, erunt reliquæ <lb/>A G, &amp; </s>
  <s xml:id="echoid-s5654" xml:space="preserve">D H inter ſe æquales, &amp; </s>
  <s xml:id="echoid-s5655" xml:space="preserve">habet G A ad A I eandem proportionẽ, quàm <lb/>Q T ad T P, ſeu quàm S V ad V R; </s>
  <s xml:id="echoid-s5656" xml:space="preserve">pariterq; </s>
  <s xml:id="echoid-s5657" xml:space="preserve">H D ad D K eſt vt S V ad V R <lb/>(propter parallelas &amp; </s>
  <s xml:id="echoid-s5658" xml:space="preserve">ſimilitudinẽ triangulorũ) igitur vt G A ad A I itaerit H D
<pb o="144" file="0182" n="182" rhead="Apollonij Pergæi"/>
ad D K, &amp; </s>
  <s xml:id="echoid-s5659" xml:space="preserve">propterea etiam conſequentes A I, &amp; </s>
  <s xml:id="echoid-s5660" xml:space="preserve">D K æquales ſunt inter ſe, <lb/>&amp; </s>
  <s xml:id="echoid-s5661" xml:space="preserve">compræhendunt angulos rectos A, &amp; </s>
  <s xml:id="echoid-s5662" xml:space="preserve">D; </s>
  <s xml:id="echoid-s5663" xml:space="preserve">ergo ſiguræ G A I, &amp; </s>
  <s xml:id="echoid-s5664" xml:space="preserve">H D K ſimi-<lb/>les ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s5665" xml:space="preserve">æquales.</s>
  <s xml:id="echoid-s5666" xml:space="preserve"/>
</p>
<div xml:id="echoid-div536" type="float" level="2" n="4">
<note position="left" xlink:label="note-0181-03" xlink:href="note-0181-03a" xml:space="preserve">d</note>
</div>
</div>
<div xml:id="echoid-div538" type="section" level="1" n="175">
<head xml:id="echoid-head229" xml:space="preserve">Notæ in Propoſit. IV.</head>
<p style="it">
  <s xml:id="echoid-s5667" xml:space="preserve">I Am ergo demonſtratum eſt, quod duo <lb/>
<anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/>
vertices tympani ſunt ſimiles, &amp; </s>
  <s xml:id="echoid-s5668" xml:space="preserve">æqua-<lb/>les, &amp; </s>
  <s xml:id="echoid-s5669" xml:space="preserve">inclinatus communis inter vtrum-<lb/>que verticem (16. </s>
  <s xml:id="echoid-s5670" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5671" xml:space="preserve">ergo figura eſt <lb/>communis, &amp;</s>
  <s xml:id="echoid-s5672" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5673" xml:space="preserve">Hæc propoſitio eſt veluti Co-<lb/>rollarium primæ partis ſecundæ propoſitionis in <lb/>qua oſtenſum eſt, quod ſi duæ hyperbolæ habue-<lb/>rint axium ſiguras æquales, &amp; </s>
  <s xml:id="echoid-s5674" xml:space="preserve">ſimiles, erunt <lb/>quoque ſectiones ipſæ æquales, &amp; </s>
  <s xml:id="echoid-s5675" xml:space="preserve">congruentes; <lb/></s>
  <s xml:id="echoid-s5676" xml:space="preserve">habent verò ſectiones oppoſitæ A B, &amp; </s>
  <s xml:id="echoid-s5677" xml:space="preserve">D E <lb/>(quæ vocantur Vertices Tympani ab Arabico <lb/>interprete) figuras D A H, &amp; </s>
  <s xml:id="echoid-s5678" xml:space="preserve">A D I axis D <lb/>A æquales, &amp; </s>
  <s xml:id="echoid-s5679" xml:space="preserve">ſimiles (vt in 14. </s>
  <s xml:id="echoid-s5680" xml:space="preserve">primi libri <lb/>demonſtrauit Apollonius); </s>
  <s xml:id="echoid-s5681" xml:space="preserve">ergo ſectiones oppo-<lb/>ſitæ æquales erunt inter ſe, &amp; </s>
  <s xml:id="echoid-s5682" xml:space="preserve">congruentes.</s>
  <s xml:id="echoid-s5683" xml:space="preserve"/>
</p>
<div xml:id="echoid-div538" type="float" level="2" n="1">
  <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a">
    <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0182-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div540" type="section" level="1" n="176">
<head xml:id="echoid-head230" xml:space="preserve">Notæ in Propoſit. X.</head>
<p style="it">
  <s xml:id="echoid-s5684" xml:space="preserve">SImiliter conſtat, quod ſi potentes contineant cum ſuis abſciſſis angu-<lb/>los equales obliquos, iudicium eſt, quod memorauimus in ſectioni-<lb/>
<anchor type="note" xlink:label="note-0182-01a" xlink:href="note-0182-01"/>
bus, &amp;</s>
  <s xml:id="echoid-s5685" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5686" xml:space="preserve">Senſus huius propoſitionis talis eſt. </s>
  <s xml:id="echoid-s5687" xml:space="preserve">In duabus ſectionibus conicis, ſi <lb/>cum earum diametris ordinatim applicatæ contineant angulos æquales, non re-<lb/>ctos, &amp; </s>
  <s xml:id="echoid-s5688" xml:space="preserve">earum latera recta ſint æqualia in parabolis, in reliquis verò ſectioni-<lb/>
<anchor type="figure" xlink:label="fig-0182-02a" xlink:href="fig-0182-02"/>
bus latera recta, &amp; </s>
  <s xml:id="echoid-s5689" xml:space="preserve">tran-<lb/>ſuerſa æqualia, itaut figuræ <lb/>ipſæ æquales ſint; </s>
  <s xml:id="echoid-s5690" xml:space="preserve">erunt ſe-<lb/>ctiones ipſæ inter ſe æqua-<lb/>les: </s>
  <s xml:id="echoid-s5691" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5692" xml:space="preserve">è conuer ſo, ſi ſectio-<lb/>nes æquales fuerint, habe-<lb/>bunt latera æqualia earum <lb/>diametrorum, cum quibus <lb/>ordinatim applicatæ angulos <lb/>æquales, non rectos continent.</s>
  <s xml:id="echoid-s5693" xml:space="preserve"/>
</p>
<div xml:id="echoid-div540" type="float" level="2" n="1">
<note position="right" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0182-02" xlink:href="fig-0182-02a">
    <image file="0182-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0182-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5694" xml:space="preserve">Demonſtrationes non apponuntur ab Apollonio, quia ĳſdem verbis omnino in <lb/>eiſdem figuris ab ſolui poßunt. </s>
  <s xml:id="echoid-s5695" xml:space="preserve">Sint enim primo duæ parabolæ A B, &amp; </s>
  <s xml:id="echoid-s5696" xml:space="preserve">C D, at-<lb/>que earum diametri A G, &amp; </s>
  <s xml:id="echoid-s5697" xml:space="preserve">C H efficiant æquales angulos F, &amp; </s>
  <s xml:id="echoid-s5698" xml:space="preserve">E, cum ordi-<lb/>natim ductis D F, &amp; </s>
  <s xml:id="echoid-s5699" xml:space="preserve">B E, ſintque latera recta A I, C N æqualia. </s>
  <s xml:id="echoid-s5700" xml:space="preserve">Dico,
<pb o="145" file="0183" n="183" rhead="Conicor. Lib. VI."/>
ſectiones æquales eſſe. </s>
  <s xml:id="echoid-s5701" xml:space="preserve">Sumatur quodlibet punctum B in ſectione B A ducaturque <lb/>ordinatim applicata B E, ſeceturque C F æqualis A E, &amp; </s>
  <s xml:id="echoid-s5702" xml:space="preserve">ducatur ordinatim <lb/>D F. </s>
  <s xml:id="echoid-s5703" xml:space="preserve">Maniſeſtum eſt, rectangula E A I, &amp; </s>
  <s xml:id="echoid-s5704" xml:space="preserve">F C N æqualia eße (cum latera <lb/>ſint æqualia, ſingula ſingulis); </s>
  <s xml:id="echoid-s5705" xml:space="preserve">his verò rectangulis æqualia ſunt quadrata or-<lb/>
<anchor type="note" xlink:label="note-0183-01a" xlink:href="note-0183-01"/>
dinatim applicatarum B E, D F; </s>
  <s xml:id="echoid-s5706" xml:space="preserve">ergo &amp; </s>
  <s xml:id="echoid-s5707" xml:space="preserve">quadrata ſunt æqualia, atque eorum <lb/>latera B E, D F æqualia quoque. </s>
  <s xml:id="echoid-s5708" xml:space="preserve">Si igitur parabolæ ſuperponantur ita, vt <lb/>punctum E ſuper F, &amp; </s>
  <s xml:id="echoid-s5709" xml:space="preserve">diameter A E ſuper C F cadat, neceſſariò punctum A <lb/>ſuper C cadet (propter æqualitatem abſcißarum) atque punctum B ſuper punctũ <lb/>D incidet (propterea quod anguli E, &amp; </s>
  <s xml:id="echoid-s5710" xml:space="preserve">F æquales ſunt, pariterque rectæ B E, <lb/>&amp; </s>
  <s xml:id="echoid-s5711" xml:space="preserve">D F ſunt æquales), &amp; </s>
  <s xml:id="echoid-s5712" xml:space="preserve">quia quodlibet punctum B parabolæ A B cadit ſemper <lb/>ſuper ſectionem C D; </s>
  <s xml:id="echoid-s5713" xml:space="preserve">ergo duæ ſectiones B A, &amp; </s>
  <s xml:id="echoid-s5714" xml:space="preserve">D C ſibi mutuò congruunt, &amp; </s>
  <s xml:id="echoid-s5715" xml:space="preserve"><lb/>ideo æquales ſunt. </s>
  <s xml:id="echoid-s5716" xml:space="preserve">Non ſecus conuerſum huius propoſitionis demonſtrari poteſt.</s>
  <s xml:id="echoid-s5717" xml:space="preserve"/>
</p>
<div xml:id="echoid-div541" type="float" level="2" n="2">
<note position="right" xlink:label="note-0183-01" xlink:href="note-0183-01a" xml:space="preserve">11. lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s5718" xml:space="preserve">Altera verò pars propoſitionis breuius de-<lb/>
<anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/>
monſtrabitur hac ratione. </s>
  <s xml:id="echoid-s5719" xml:space="preserve">In duabus hyperbo-<lb/>lis, aut ellipſibus efficiant ordinatim applicatæ <lb/>B E, D F cum diametris A E, &amp; </s>
  <s xml:id="echoid-s5720" xml:space="preserve">C F angu-<lb/>los æquales, &amp; </s>
  <s xml:id="echoid-s5721" xml:space="preserve">non rectos; </s>
  <s xml:id="echoid-s5722" xml:space="preserve">ſintque tranſuerſa <lb/>latera G A, &amp; </s>
  <s xml:id="echoid-s5723" xml:space="preserve">H C æqualia, pariterque late-<lb/>ra recta A I, &amp; </s>
  <s xml:id="echoid-s5724" xml:space="preserve">C N æqualia. </s>
  <s xml:id="echoid-s5725" xml:space="preserve">Dico, ſectiones <lb/>B A, C D æquales eſſe. </s>
  <s xml:id="echoid-s5726" xml:space="preserve">Sumatur quodlibet <lb/>punctum B ſectionis B A, ducaturque ad A E <lb/>diametrum ordinatim applicata B E, ſecetur-<lb/>que C F æqualis abſciſſæ A E, ducaturque F D <lb/>ad H C F diametrũ ordinatim applicata. </s>
  <s xml:id="echoid-s5727" xml:space="preserve">Erit <lb/>rectangulum G E A ad quadr atum B E, vt la-<lb/>tus tranſuerſum G A ad rectum A I; </s>
  <s xml:id="echoid-s5728" xml:space="preserve">pariter-<lb/>que rectangulum H F C ad quadratum F D <lb/>erit, vt H C ad C N: </s>
  <s xml:id="echoid-s5729" xml:space="preserve">habent vero duæ æqua-<lb/>les G A, &amp; </s>
  <s xml:id="echoid-s5730" xml:space="preserve">H C eandem proportionem ad duas <lb/>æquales A I, &amp; </s>
  <s xml:id="echoid-s5731" xml:space="preserve">C N; </s>
  <s xml:id="echoid-s5732" xml:space="preserve">igitur rectangulum G E <lb/>A ad quadratum B E eandem proportionem ha-<lb/>
<anchor type="figure" xlink:label="fig-0183-02a" xlink:href="fig-0183-02"/>
bebit, quàm rectangulum. <lb/></s>
  <s xml:id="echoid-s5733" xml:space="preserve">H F C ad quadratum D F, <lb/>ſunt verò rectangula G E <lb/>A, H F C æqualia inter, ſe <lb/>(quandoquidem eorum la-<lb/>tera A E, C F facta ſunt <lb/>æqualia) quæ addita ipſis <lb/>A G, &amp; </s>
  <s xml:id="echoid-s5734" xml:space="preserve">C H æqualibus eſ-<lb/>eſſiciunt latera E G, &amp; </s>
  <s xml:id="echoid-s5735" xml:space="preserve">F <lb/>H æqualia; </s>
  <s xml:id="echoid-s5736" xml:space="preserve">ergo quadrat a <lb/>d a um B E, &amp; </s>
  <s xml:id="echoid-s5737" xml:space="preserve">D F æqua-<lb/>lia ſunt inter ſe; </s>
  <s xml:id="echoid-s5738" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5739" xml:space="preserve">ideo ordinatim applicatæ B E, &amp; </s>
  <s xml:id="echoid-s5740" xml:space="preserve">D F æquales erunt. </s>
  <s xml:id="echoid-s5741" xml:space="preserve"><lb/>Quare facta, vt prius, intellectuali ſuperpoſitione; </s>
  <s xml:id="echoid-s5742" xml:space="preserve">nedum veriex A ſuper C, <lb/>ſed etiam quodlibet punctum B ſectionis A B ſuper ſectionem C D cadet; </s>
  <s xml:id="echoid-s5743" xml:space="preserve">ideo-<lb/>que ſectiones ſibi mutuò congruent, &amp; </s>
  <s xml:id="echoid-s5744" xml:space="preserve">æquales erunt.</s>
  <s xml:id="echoid-s5745" xml:space="preserve"/>
</p>
<div xml:id="echoid-div542" type="float" level="2" n="3">
  <figure xlink:label="fig-0183-01" xlink:href="fig-0183-01a">
    <image file="0183-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0183-01"/>
  </figure>
  <figure xlink:label="fig-0183-02" xlink:href="fig-0183-02a">
    <image file="0183-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0183-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5746" xml:space="preserve">E conuerſo, ſi ſectiones B A, &amp; </s>
  <s xml:id="echoid-s5747" xml:space="preserve">C D æquales ſupponantur, ſibi mutuò con-
<pb o="146" file="0184" n="184" rhead="Apollonij Pergæi"/>
gruent, &amp; </s>
  <s xml:id="echoid-s5748" xml:space="preserve">ideo à communi vertice A, <lb/>
<anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/>
ducta qualibet diametro A E, vel C <lb/>F, ad quàm ordinatim applicetur quæ-<lb/>libet B E, ſeu D F in angulo non re-<lb/>cto; </s>
  <s xml:id="echoid-s5749" xml:space="preserve">ſintque latera tranſuerſa, &amp; </s>
  <s xml:id="echoid-s5750" xml:space="preserve">recta <lb/>G A, A I, atque H C, C N. </s>
  <s xml:id="echoid-s5751" xml:space="preserve">Dico, <lb/>huinſmodi latera, &amp; </s>
  <s xml:id="echoid-s5752" xml:space="preserve">ſiguræ ſeu rectã-<lb/>gula G A I, H C N æqualia, &amp; </s>
  <s xml:id="echoid-s5753" xml:space="preserve">ſimi-<lb/>lia eſſe inter ſe, &amp; </s>
  <s xml:id="echoid-s5754" xml:space="preserve">ſibi mutuò congru-<lb/>entia. </s>
  <s xml:id="echoid-s5755" xml:space="preserve">Si enim hoc verum non eſt, eo-<lb/>rum diametri G I, &amp; </s>
  <s xml:id="echoid-s5756" xml:space="preserve">H N ſimiliter <lb/>poſitæ, &amp; </s>
  <s xml:id="echoid-s5757" xml:space="preserve">ſubtendentes communem an-<lb/>gulum A non coincident; </s>
  <s xml:id="echoid-s5758" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5759" xml:space="preserve">ideo æquidiſtantes erunt aut ſe mutuò ſecabunt in <lb/>vno puncto: </s>
  <s xml:id="echoid-s5760" xml:space="preserve">ducatur ergo à termino E alicuius ordinatim applicatæ B E recta <lb/>linea E M parallela lateribus rectis A I, C N, ita vt ſecet diametros ſigurarum <lb/>ſupra aut inſra occurſum in duobus punctis M, &amp; </s>
  <s xml:id="echoid-s5761" xml:space="preserve">O. </s>
  <s xml:id="echoid-s5762" xml:space="preserve">Igitur in ſectione A B <lb/>idem quadratum ordinatim applicatæ B E, ſeu D F æquale erit rectangulo A E <lb/>M, &amp; </s>
  <s xml:id="echoid-s5763" xml:space="preserve">in ſectione D C æquale erit rectangulo C F O, ſuntque abſciſſæ A E, &amp; </s>
  <s xml:id="echoid-s5764" xml:space="preserve"><lb/>C F æquales; </s>
  <s xml:id="echoid-s5765" xml:space="preserve">ergo M E, &amp; </s>
  <s xml:id="echoid-s5766" xml:space="preserve">O F æquales inter ſe ſunt: </s>
  <s xml:id="echoid-s5767" xml:space="preserve">pars, &amp; </s>
  <s xml:id="echoid-s5768" xml:space="preserve">totum quod <lb/>eſt abſurdum: </s>
  <s xml:id="echoid-s5769" xml:space="preserve">Non ergo latera ſigurarum inequalia ſunt. </s>
  <s xml:id="echoid-s5770" xml:space="preserve">Quod erat oſtenden-<lb/>dum.</s>
  <s xml:id="echoid-s5771" xml:space="preserve"/>
</p>
<div xml:id="echoid-div543" type="float" level="2" n="4">
  <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a">
    <image file="0184-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0184-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div545" type="section" level="1" n="177">
<head xml:id="echoid-head231" xml:space="preserve">SECTIO SECVNDA <lb/>Continens Propoſit. III. VI. VII. &amp; IX. <lb/>PROPOSITIO III.</head>
<p>
  <s xml:id="echoid-s5772" xml:space="preserve">COniſectio non eſt æqualis ſectioni quæ eiuſdem generis cũ <lb/>illa non ſit.</s>
  <s xml:id="echoid-s5773" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5774" xml:space="preserve">Etenim elli-<lb/>
<anchor type="figure" xlink:label="fig-0184-02a" xlink:href="fig-0184-02"/>
pſis non erit æ-<lb/>qualis alicui pa-<lb/>rabolæ, aut hy-<lb/>perbolæ quia <lb/>illa eſt termina-<lb/>ta, hæ verò ſunt <lb/>indeterminatæ. <lb/></s>
  <s xml:id="echoid-s5775" xml:space="preserve">At parabola D <lb/>E F, cuius axis <lb/>D I non erit æ-<lb/>qualis hyperbolæ A B C, cuius axis A G, &amp; </s>
  <s xml:id="echoid-s5776" xml:space="preserve">inclinatus A H. </s>
  <s xml:id="echoid-s5777" xml:space="preserve">Quia ſi <lb/>abſcindantur A K, K G æquales D L, L I, &amp; </s>
  <s xml:id="echoid-s5778" xml:space="preserve">educamus ad axes perpen-<lb/>diculares B K, C G, E L, F I: </s>
  <s xml:id="echoid-s5779" xml:space="preserve">Dico, quod ſectio D F non eſt æqualis
<pb o="147" file="0185" n="185" rhead="Conicor. Lib. VI."/>
ſectioni A C; </s>
  <s xml:id="echoid-s5780" xml:space="preserve">quia ſi eſſet æqualis illi, facta ſuperpoſitione, ſibi mutuò <lb/>congruerent, &amp; </s>
  <s xml:id="echoid-s5781" xml:space="preserve">caderent puncta E, F, L, I, ſuper B, C, G, K, &amp; </s>
  <s xml:id="echoid-s5782" xml:space="preserve">eſſet <lb/>F I æqualis C G, atque E L æqualis B K; </s>
  <s xml:id="echoid-s5783" xml:space="preserve">ideoque quadratũ F I ad qua-<lb/>dratum E L eſſet, vt D I ad D L (19. </s>
  <s xml:id="echoid-s5784" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5785" xml:space="preserve">eſſetque quadratum C G <lb/>
<anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/>
ad quadratum K B, vt A G ad K A, quod eſt abſurdum; </s>
  <s xml:id="echoid-s5786" xml:space="preserve">quia illius pro-<lb/>portio ad iſtam eſt, vt H G in G A ad H K in K A (20. </s>
  <s xml:id="echoid-s5787" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s5788" xml:space="preserve">Igitur <lb/>
<anchor type="note" xlink:label="note-0185-02a" xlink:href="note-0185-02"/>
ſectio parabolica non eſt æqualis ſectioni hyperbolæ, nec ſectio aliqua. <lb/></s>
  <s xml:id="echoid-s5789" xml:space="preserve">æqualis eſt ſectioni, quæ non ſit eiuſdem generis; </s>
  <s xml:id="echoid-s5790" xml:space="preserve">Et hoc erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s5791" xml:space="preserve"/>
</p>
<div xml:id="echoid-div545" type="float" level="2" n="1">
  <figure xlink:label="fig-0184-02" xlink:href="fig-0184-02a">
    <image file="0184-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0184-02"/>
  </figure>
<note position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve">20. lib. 1.</note>
<note position="right" xlink:label="note-0185-02" xlink:href="note-0185-02a" xml:space="preserve">21. lib. 1</note>
</div>
</div>
<div xml:id="echoid-div547" type="section" level="1" n="178">
<head xml:id="echoid-head232" xml:space="preserve">PROPOSITIO VI.</head>
<p>
  <s xml:id="echoid-s5792" xml:space="preserve">QVælibet duæ ſectiones A B C, &amp; </s>
  <s xml:id="echoid-s5793" xml:space="preserve">D H F, quarum portio <lb/>
<anchor type="note" xlink:label="note-0185-03a" xlink:href="note-0185-03"/>
vnius ſuperpoſita portioni alterius congruit, ſunt æquales <lb/>inter ſe.</s>
  <s xml:id="echoid-s5794" xml:space="preserve"/>
</p>
<div xml:id="echoid-div547" type="float" level="2" n="1">
<note position="left" xlink:label="note-0185-03" xlink:href="note-0185-03a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0185-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0185-01"/>
  </figure>
<p>
  <s xml:id="echoid-s5795" xml:space="preserve">Alioquin congruat portio B C portio-<lb/>
<anchor type="figure" xlink:label="fig-0185-02a" xlink:href="fig-0185-02"/>
ni E F, at non cadat portio A B ſuper <lb/>D E, ſed cadat in ſitu E G, &amp; </s>
  <s xml:id="echoid-s5796" xml:space="preserve">educamus <lb/>lineam tangentem duas ſectiones in H, &amp; </s>
  <s xml:id="echoid-s5797" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0185-04a" xlink:href="note-0185-04"/>
educamus E I, D G F parallelas tangen-<lb/>ti; </s>
  <s xml:id="echoid-s5798" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5799" xml:space="preserve">ex H ad ſemipartitionem ipſius E I <lb/>ducatur H K, quæ occurrat D F in L. <lb/></s>
  <s xml:id="echoid-s5800" xml:space="preserve">Et quia H L ſecat bifariam lineam paral-<lb/>lelam tangenti ab eius termino ductæ; </s>
  <s xml:id="echoid-s5801" xml:space="preserve"><lb/>ergo eſt diameter vniuerſæ ſectionis (5. </s>
  <s xml:id="echoid-s5802" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0185-05a" xlink:href="note-0185-05"/>
ex 2.) </s>
  <s xml:id="echoid-s5803" xml:space="preserve">quare bifariam ſecat vnamquan-<lb/>que ex D F, G F, &amp; </s>
  <s xml:id="echoid-s5804" xml:space="preserve">fiet D L æqualis G <lb/>L, quod eſt abſurdum: </s>
  <s xml:id="echoid-s5805" xml:space="preserve">igitur ſectio A B <lb/>C tota congruit ſectioni D H F. </s>
  <s xml:id="echoid-s5806" xml:space="preserve">Quod <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s5807" xml:space="preserve"/>
</p>
<div xml:id="echoid-div548" type="float" level="2" n="2">
  <figure xlink:label="fig-0185-02" xlink:href="fig-0185-02a">
    <image file="0185-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0185-02"/>
  </figure>
<note position="right" xlink:label="note-0185-04" xlink:href="note-0185-04a" xml:space="preserve">34. lib. 1.</note>
<note position="right" xlink:label="note-0185-05" xlink:href="note-0185-05a" xml:space="preserve">7. lib. 2.</note>
</div>
</div>
<div xml:id="echoid-div550" type="section" level="1" n="179">
<head xml:id="echoid-head233" xml:space="preserve">PROPOSITIO VII.</head>
<p>
  <s xml:id="echoid-s5808" xml:space="preserve">DVæ ordinationes axis in qualibet coniſectione abſcindunt <lb/>
<anchor type="note" xlink:label="note-0185-06a" xlink:href="note-0185-06"/>
à ſectione ex vtraque parte axis duas portiones, quarum <lb/>ſi vna alteri ſuperponatur ſibi mutuò congruent, nec congruunt <lb/>alicui aliæ portioni ſectionis.</s>
  <s xml:id="echoid-s5809" xml:space="preserve"/>
</p>
<div xml:id="echoid-div550" type="float" level="2" n="1">
<note position="left" xlink:label="note-0185-06" xlink:href="note-0185-06a" xml:space="preserve">a</note>
</div>
<pb o="148" file="0186" n="186" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s5810" xml:space="preserve">Sit coniſectio A B C, &amp; </s>
  <s xml:id="echoid-s5811" xml:space="preserve">eius axis B D, &amp; </s>
  <s xml:id="echoid-s5812" xml:space="preserve">ſu-<lb/>
<anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/>
mantur in ſectione puncta G, C, ab eis educã-<lb/>tur duæ ordinationes G H, C A occurrentes axi <lb/>in I, D. </s>
  <s xml:id="echoid-s5813" xml:space="preserve">Dico, quod B G congruit B H, &amp; </s>
  <s xml:id="echoid-s5814" xml:space="preserve">G <lb/>C ipſi H A, &amp; </s>
  <s xml:id="echoid-s5815" xml:space="preserve">ſuperſicies B D C ſuperficiei B <lb/>D A, &amp; </s>
  <s xml:id="echoid-s5816" xml:space="preserve">ſegmentum B G C ſegmento B H A. <lb/></s>
  <s xml:id="echoid-s5817" xml:space="preserve">Quoniam axis B D bifariam diuidit G H, A C <lb/>in I, D, vtique G I ipſi I H congruet, &amp; </s>
  <s xml:id="echoid-s5818" xml:space="preserve">D C <lb/>
<anchor type="note" xlink:label="note-0186-01a" xlink:href="note-0186-01"/>
ipſi D A, &amp; </s>
  <s xml:id="echoid-s5819" xml:space="preserve">duo puncta G, C ſuper duobus <lb/>punctis H, A cadent, &amp; </s>
  <s xml:id="echoid-s5820" xml:space="preserve">portio ſectionis conicæ <lb/>G C ſuper portionem H A, &amp; </s>
  <s xml:id="echoid-s5821" xml:space="preserve">G B ſuper H B: <lb/></s>
  <s xml:id="echoid-s5822" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0186-02a" xlink:href="fig-0186-02"/>
Et dico, quod portio H A non congruit <lb/>alicui alteri portioni, quàm G C: </s>
  <s xml:id="echoid-s5823" xml:space="preserve">ſi enim <lb/>poſſibile eſt cõgruat portioni C K, &amp; </s>
  <s xml:id="echoid-s5824" xml:space="preserve">por-<lb/>tio H B congruet portioni, quæ continua-<lb/>tur ipſi K C; </s>
  <s xml:id="echoid-s5825" xml:space="preserve">ergo cadet B ex H B non ſu-<lb/>per B ex C G B; </s>
  <s xml:id="echoid-s5826" xml:space="preserve">quia portio H B non eſt <lb/>æqualis portioni C B; </s>
  <s xml:id="echoid-s5827" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5828" xml:space="preserve">propterea incidet <lb/>axis B D in alium locum, eſſentque eidem <lb/>ſectioni plures axes: </s>
  <s xml:id="echoid-s5829" xml:space="preserve">quod eſt abſurdum; <lb/></s>
  <s xml:id="echoid-s5830" xml:space="preserve">(51. </s>
  <s xml:id="echoid-s5831" xml:space="preserve">52. </s>
  <s xml:id="echoid-s5832" xml:space="preserve">ex 2.) </s>
  <s xml:id="echoid-s5833" xml:space="preserve">igitur non cadit H A niſi <lb/>
<anchor type="note" xlink:label="note-0186-02a" xlink:href="note-0186-02"/>
ſuper C G. </s>
  <s xml:id="echoid-s5834" xml:space="preserve">Vt fuerat propoſitum.</s>
  <s xml:id="echoid-s5835" xml:space="preserve"/>
</p>
<div xml:id="echoid-div551" type="float" level="2" n="2">
  <figure xlink:label="fig-0186-01" xlink:href="fig-0186-01a">
    <image file="0186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0186-01"/>
  </figure>
<note position="right" xlink:label="note-0186-01" xlink:href="note-0186-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0186-02" xlink:href="fig-0186-02a">
    <image file="0186-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0186-02"/>
  </figure>
<note position="left" xlink:label="note-0186-02" xlink:href="note-0186-02a" xml:space="preserve">48. lib. 2.</note>
</div>
</div>
<div xml:id="echoid-div553" type="section" level="1" n="180">
<head xml:id="echoid-head234" xml:space="preserve">PROPOSITIO IX.</head>
<p>
  <s xml:id="echoid-s5836" xml:space="preserve">M Anifeſtum eſt ex demoſtratis, quod portiones ſectionum <lb/>
<anchor type="note" xlink:label="note-0186-03a" xlink:href="note-0186-03"/>
æqualium non congruunt ſibi inuicem, niſi earum di-<lb/>ſtantiæ à verticibus ſint æquales.</s>
  <s xml:id="echoid-s5837" xml:space="preserve"/>
</p>
<div xml:id="echoid-div553" type="float" level="2" n="1">
<note position="right" xlink:label="note-0186-03" xlink:href="note-0186-03a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s5838" xml:space="preserve">Oſtenſum enim eſt ſibi non congruere, quarum diſtantiæ à verticibus <lb/>non ſunt æquales, quia portio H A, ſi caderet ſuper portionem C K, &amp; </s>
  <s xml:id="echoid-s5839" xml:space="preserve"><lb/>earum diſtantiæ à B non eſſent æquales, conſequitur, quod in hyperbola <lb/>ſint duo axes, &amp; </s>
  <s xml:id="echoid-s5840" xml:space="preserve">in ellipſi tres axes: </s>
  <s xml:id="echoid-s5841" xml:space="preserve">quod eſt abſurdum (51. </s>
  <s xml:id="echoid-s5842" xml:space="preserve">52. </s>
  <s xml:id="echoid-s5843" xml:space="preserve">53. <lb/></s>
  <s xml:id="echoid-s5844" xml:space="preserve">
<anchor type="note" xlink:label="note-0186-04a" xlink:href="note-0186-04"/>
ex 2.)</s>
  <s xml:id="echoid-s5845" xml:space="preserve"/>
</p>
<div xml:id="echoid-div554" type="float" level="2" n="2">
<note position="left" xlink:label="note-0186-04" xlink:href="note-0186-04a" xml:space="preserve">48. lib. 2</note>
</div>
  <figure>
    <image file="0186-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0186-03"/>
  </figure>
<p>
  <s xml:id="echoid-s5846" xml:space="preserve">Si autem in ellipſi cadit axis A E tranſuer-<lb/>
<anchor type="note" xlink:label="note-0186-05a" xlink:href="note-0186-05"/>
<anchor type="figure" xlink:label="fig-0186-04a" xlink:href="fig-0186-04"/>
ſus ſuper axim rectum illius, vtique differunt <lb/>inter ſe, &amp; </s>
  <s xml:id="echoid-s5847" xml:space="preserve">non ſibi inuicem congruunt ſectio-<lb/>nes.</s>
  <s xml:id="echoid-s5848" xml:space="preserve"/>
</p>
<div xml:id="echoid-div555" type="float" level="2" n="3">
<note position="right" xlink:label="note-0186-05" xlink:href="note-0186-05a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0186-04" xlink:href="fig-0186-04a">
    <image file="0186-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0186-04"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s5849" xml:space="preserve">Conſtat etiam, quod in ſectionibus inæ-<lb/>qualibus, vt A B C, D E F portio vnius ea-<lb/>rum non congruit portioni alterius.</s>
  <s xml:id="echoid-s5850" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5851" xml:space="preserve">Alioqui congruet B A ipſi D E, &amp; </s>
  <s xml:id="echoid-s5852" xml:space="preserve">congrue-<lb/>ret etiam E F ipſi B C (6. </s>
  <s xml:id="echoid-s5853" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s5854" xml:space="preserve">eſſetque ſe-<lb/>ctio C B A æqualis ſectioni F E D: </s>
  <s xml:id="echoid-s5855" xml:space="preserve">at ſuppo-<lb/>ſuimus, non eſſe æquales, quod eſt abſurdum:</s>
  <s xml:id="echoid-s5856" xml:space="preserve">
<pb o="149" file="0187" n="187" rhead="Conicor. Lib. V."/>
ergo non congruit portio alicuius <lb/>
<anchor type="figure" xlink:label="fig-0187-01a" xlink:href="fig-0187-01"/>
ſectionis portioni alterius ſectionis, <lb/>cui æqualis non eſt. </s>
  <s xml:id="echoid-s5857" xml:space="preserve">Et hoc erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s5858" xml:space="preserve"/>
</p>
<div xml:id="echoid-div556" type="float" level="2" n="4">
  <figure xlink:label="fig-0187-01" xlink:href="fig-0187-01a">
    <image file="0187-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0187-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div558" type="section" level="1" n="181">
<head xml:id="echoid-head235" xml:space="preserve">Notæ in Propoſit. III.</head>
<p>
  <s xml:id="echoid-s5859" xml:space="preserve">ETenim ellipſis non eſt æqualis alicui hyperbolæ, &amp;</s>
  <s xml:id="echoid-s5860" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5861" xml:space="preserve">Suppleri debet in <lb/>textu verbum (parabolæ) dicendo. </s>
  <s xml:id="echoid-s5862" xml:space="preserve">Etenim ellipſis non eſt æqualis alicui <lb/>
<anchor type="note" xlink:label="note-0187-01a" xlink:href="note-0187-01"/>
parabolæ, aut hyperbolæ, quia illa eſt determinata; </s>
  <s xml:id="echoid-s5863" xml:space="preserve">hæ verò ſunt indeterminatæ, <lb/>ſcilicet ellipſis eſt finita parabole verò, &amp; </s>
  <s xml:id="echoid-s5864" xml:space="preserve">hyperbole in infinitum extendi poſ-<lb/>ſunt, &amp; </s>
  <s xml:id="echoid-s5865" xml:space="preserve">propterea nulla ratione æquales oſtendentur.</s>
  <s xml:id="echoid-s5866" xml:space="preserve"/>
</p>
<div xml:id="echoid-div558" type="float" level="2" n="1">
<note position="left" xlink:label="note-0187-01" xlink:href="note-0187-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div560" type="section" level="1" n="182">
<head xml:id="echoid-head236" xml:space="preserve">Notæ in Propoſit. VI.</head>
<p>
  <s xml:id="echoid-s5867" xml:space="preserve">QVælibet duæ ſectiones A B C, D E <lb/>
<anchor type="note" xlink:label="note-0187-02a" xlink:href="note-0187-02"/>
F, quarum vnaquæque literarum <lb/>ſuperpoſita literis alterius con-<lb/>
<anchor type="figure" xlink:label="fig-0187-02a" xlink:href="fig-0187-02"/>
gruit; </s>
  <s xml:id="echoid-s5868" xml:space="preserve">vtique ſunt æquales, &amp;</s>
  <s xml:id="echoid-s5869" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5870" xml:space="preserve">Legendum <lb/>puto. </s>
  <s xml:id="echoid-s5871" xml:space="preserve">Quælibet duæ ſectiones A B C, &amp; </s>
  <s xml:id="echoid-s5872" xml:space="preserve">D <lb/>E F, quarum portio vnius, alterius portioni <lb/>ſuperpoſita congruit ſunt æquales inter ſe.</s>
  <s xml:id="echoid-s5873" xml:space="preserve"/>
</p>
<div xml:id="echoid-div560" type="float" level="2" n="1">
<note position="left" xlink:label="note-0187-02" xlink:href="note-0187-02a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0187-02" xlink:href="fig-0187-02a">
    <image file="0187-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0187-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div562" type="section" level="1" n="183">
<head xml:id="echoid-head237" xml:space="preserve">Notæ in Propoſit. VII.</head>
<p>
  <s xml:id="echoid-s5874" xml:space="preserve">ORdinationes axis in qualibet hyper-<lb/>
<anchor type="note" xlink:label="note-0187-03a" xlink:href="note-0187-03"/>
<anchor type="figure" xlink:label="fig-0187-03a" xlink:href="fig-0187-03"/>
bolarum abſcindunt à ſectione ex <lb/>vtraque parte axis duo ſegmenta, quæ, <lb/>ſi cadit vnum ſuper alterum, ſibi mutuò <lb/>congruunt, nec excedunt, nec deficiunt, <lb/>nec congruunt alicui portioni ſectionis, <lb/>&amp;</s>
  <s xml:id="echoid-s5875" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5876" xml:space="preserve">Expungi debent verba aliqua huius te-<lb/>xtus ſuperuacanea, &amp; </s>
  <s xml:id="echoid-s5877" xml:space="preserve">aliqua adiungi, vt ſenſus continuus talis ſit. </s>
  <s xml:id="echoid-s5878" xml:space="preserve">Duæ <lb/>ordinationes axis in qualibet coniſectione abſcindunt à ſectione ex vtraque <lb/>parte, axis duas portiones, quarum vna alteri ſuperpoſita ſibi mutuò congruent, <lb/>nec cõgruunt alicui aliæ portioni ſectionis.</s>
  <s xml:id="echoid-s5879" xml:space="preserve"/>
</p>
<div xml:id="echoid-div562" type="float" level="2" n="1">
<note position="left" xlink:label="note-0187-03" xlink:href="note-0187-03a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0187-03" xlink:href="fig-0187-03a">
    <image file="0187-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0187-03"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s5880" xml:space="preserve">Quoniam axis B D bifariam diuidit G H, A C, &amp;</s>
  <s xml:id="echoid-s5881" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5882" xml:space="preserve">Ex eo <lb/>
<anchor type="note" xlink:label="note-0187-04a" xlink:href="note-0187-04"/>
enim quod omnes applicatæ ad axim B D ſecantur bifariam ab
<pb o="150" file="0188" n="188" rhead="Apollonij Pergæi"/>
illo, &amp; </s>
  <s xml:id="echoid-s5883" xml:space="preserve">ad angulos rectos, ſi intelligatur ſuperficies B I G, ſuperpoſita ſuperfi-<lb/>ciei B I H, itaut axis ſuper axim cadat, atque vertex B ſit communis neceſ-<lb/>ſario punctum I commune erit, atque recta I G cadet ſuper I H, cum anguli G <lb/>I B, &amp; </s>
  <s xml:id="echoid-s5884" xml:space="preserve">H I B recti ſint, atque punctum G cadet in H, propter æqualitatem <lb/>duarum ordinatim applicatarum I G, I H: </s>
  <s xml:id="echoid-s5885" xml:space="preserve">eadem ratione quælibet alia puncta <lb/>ſectionis G B inter G, &amp; </s>
  <s xml:id="echoid-s5886" xml:space="preserve">B ſumpta cadent ſuper B H; </s>
  <s xml:id="echoid-s5887" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5888" xml:space="preserve">ideo portio ſectionis <lb/>conicæ G B congruet portioni B H, &amp; </s>
  <s xml:id="echoid-s5889" xml:space="preserve">eidem æqualis erit. </s>
  <s xml:id="echoid-s5890" xml:space="preserve">Simili modo conſtat, <lb/>portionem G C æqualem eße portioni H A, &amp; </s>
  <s xml:id="echoid-s5891" xml:space="preserve">ſic <lb/>
<anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/>
ſuperficies ipſæ. </s>
  <s xml:id="echoid-s5892" xml:space="preserve">Quod verò portio H A non con-<lb/>gruat alicui alteri ſegmento C K præter G C, con-<lb/>ſtat ex eo, quod ſi portiones K C, &amp; </s>
  <s xml:id="echoid-s5893" xml:space="preserve">A H ſibi mu-<lb/>tuò congruunt, vt nimirum punctum C ſuper H, &amp; </s>
  <s xml:id="echoid-s5894" xml:space="preserve"><lb/>punctum K ſuper A cadat: </s>
  <s xml:id="echoid-s5895" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5896" xml:space="preserve">concipiatur punctũ <lb/>C idem ac N, &amp; </s>
  <s xml:id="echoid-s5897" xml:space="preserve">K idem ac O, &amp; </s>
  <s xml:id="echoid-s5898" xml:space="preserve">portio O N L <lb/>æqualis immo eadem ſectio K C B, &amp; </s>
  <s xml:id="echoid-s5899" xml:space="preserve">illius axis <lb/>L M omnino idem ac axis B D: </s>
  <s xml:id="echoid-s5900" xml:space="preserve">tunc quidem (ex <lb/>precedenti prop. </s>
  <s xml:id="echoid-s5901" xml:space="preserve">6.) </s>
  <s xml:id="echoid-s5902" xml:space="preserve">ſectiones ipſæ A B, &amp; </s>
  <s xml:id="echoid-s5903" xml:space="preserve">K B, ſeu O L æquales erunt, &amp; </s>
  <s xml:id="echoid-s5904" xml:space="preserve">ſi-<lb/>bi mutuò congruentes: </s>
  <s xml:id="echoid-s5905" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5906" xml:space="preserve">propterea H B cadet ſuper portionem maiorem C B <lb/>ſeu ei æqualem N B L (cum H B æqualis oſtenſa ſit ipſi G B) &amp; </s>
  <s xml:id="echoid-s5907" xml:space="preserve">ideo vertices <lb/>B, &amp; </s>
  <s xml:id="echoid-s5908" xml:space="preserve">L duarum axium B D, &amp; </s>
  <s xml:id="echoid-s5909" xml:space="preserve">L M in duabus ſectionibus A B, &amp; </s>
  <s xml:id="echoid-s5910" xml:space="preserve">K B ſeu <lb/>O N L inæqualibus non conuenient: </s>
  <s xml:id="echoid-s5911" xml:space="preserve">quapropter in duabus congruentibus, ſeu in <lb/>eadem ſectione duo axes B D, &amp; </s>
  <s xml:id="echoid-s5912" xml:space="preserve">L M exiſtent, quod eſt abſurdum, quia eſt <lb/>contra propoſ: </s>
  <s xml:id="echoid-s5913" xml:space="preserve">48. </s>
  <s xml:id="echoid-s5914" xml:space="preserve">libri 2.</s>
  <s xml:id="echoid-s5915" xml:space="preserve"/>
</p>
<div xml:id="echoid-div563" type="float" level="2" n="2">
<note position="left" xlink:label="note-0187-04" xlink:href="note-0187-04a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a">
    <image file="0188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0188-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div565" type="section" level="1" n="184">
<head xml:id="echoid-head238" xml:space="preserve">Notæ in Propoſit. IX.</head>
<p>
  <s xml:id="echoid-s5916" xml:space="preserve">MAnifeſtum eſt ex demonſtratis, quod portiones ſectionum æqua-<lb/>
<anchor type="note" xlink:label="note-0188-01a" xlink:href="note-0188-01"/>
lium non congruunt, &amp;</s>
  <s xml:id="echoid-s5917" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5918" xml:space="preserve">Sicuti in propoſ. </s>
  <s xml:id="echoid-s5919" xml:space="preserve">7. </s>
  <s xml:id="echoid-s5920" xml:space="preserve">dictum eſt, quod duæ <lb/>portiones non æqualiter à vertice axis diſtantes ſibi mutuò congruere nõ poſſunt, <lb/>ita hic in duabus quibuslibet æqualibus coniſectionibus idem verificari oſtendi-<lb/>tur, quod nimirum duæ portiones cuiuslibet ſectionis conicæ, vel duarum æqua-<lb/>lium ſectionum inæqualiter à vertice axis diſtantes non ſint congruentes. </s>
  <s xml:id="echoid-s5921" xml:space="preserve">Hoc <lb/>autem alia ratione demonſtrare ſuperuacaneum non erit, cum demonſtratio, quæ <lb/>in textu Arabico corrupto affertur non omnino ſufficiens videatur, ſed prius <lb/>oſtendendum eſt.</s>
  <s xml:id="echoid-s5922" xml:space="preserve"/>
</p>
<div xml:id="echoid-div565" type="float" level="2" n="1">
<note position="right" xlink:label="note-0188-01" xlink:href="note-0188-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div567" type="section" level="1" n="185">
<head xml:id="echoid-head239" xml:space="preserve">LEMMAI.</head>
<p style="it">
  <s xml:id="echoid-s5923" xml:space="preserve">IN duabus æqualibus coniſectionibus A B C, &amp; </s>
  <s xml:id="echoid-s5924" xml:space="preserve">D E F, quarum <lb/>axes A G, D H deſcribere duos circulos æquales contingentes coni-<lb/>cas ſectiones, quorum is, qui propinquior eſt vertici extrinſecùs, reli-<lb/>quus verò intrinſecùs ſectionem tangat.</s>
  <s xml:id="echoid-s5925" xml:space="preserve"/>
</p>
<pb o="151" file="0189" n="189" rhead="Conicor. Lib. VI."/>
<p style="it">
  <s xml:id="echoid-s5926" xml:space="preserve">In ſestione A B C ducatur ramus breuiſe-<lb/>
<anchor type="figure" xlink:label="fig-0189-01a" xlink:href="fig-0189-01"/>
cans ſingularis I L ſecans axem in G, ſitque <lb/>
<anchor type="note" xlink:label="note-0189-01a" xlink:href="note-0189-01"/>
I punctum concur ſus perpendicularis I K, &amp; </s>
  <s xml:id="echoid-s5927" xml:space="preserve"><lb/>breuiſecantis; </s>
  <s xml:id="echoid-s5928" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5929" xml:space="preserve">à quolibet puncto B inter <lb/>L, &amp; </s>
  <s xml:id="echoid-s5930" xml:space="preserve">verticem A ducatur alius ramus bre-<lb/>uiſecans B M, qui occurret L I vltra axim <lb/>in M, &amp; </s>
  <s xml:id="echoid-s5931" xml:space="preserve">inter puncta G, &amp; </s>
  <s xml:id="echoid-s5932" xml:space="preserve">I; </s>
  <s xml:id="echoid-s5933" xml:space="preserve">coniungatur-<lb/>
<anchor type="note" xlink:label="note-0189-02a" xlink:href="note-0189-02"/>
que recta linea B 1. </s>
  <s xml:id="echoid-s5934" xml:space="preserve">Quoniam angulus L G A acutus eſt, erit angnlus G M N <lb/>internus, &amp; </s>
  <s xml:id="echoid-s5935" xml:space="preserve">oppoſitus in triangulo G M N minor illò, &amp; </s>
  <s xml:id="echoid-s5936" xml:space="preserve">ideo acutus, &amp; </s>
  <s xml:id="echoid-s5937" xml:space="preserve">pro-<lb/>
<anchor type="note" xlink:label="note-0189-03a" xlink:href="note-0189-03"/>
pterea qui deinceps eſt angulus B M I erit obtuſus, &amp; </s>
  <s xml:id="echoid-s5938" xml:space="preserve">ideo in triangulo I B M <lb/>latus I B ſubtendens maximum angulum obtuſum maius erit latera B M; </s>
  <s xml:id="echoid-s5939" xml:space="preserve">ſedra-<lb/>mus I L maior eſt, quàm I B, propterea quod remotior eſt à vertice A, igitur <lb/>
<anchor type="note" xlink:label="note-0189-04a" xlink:href="note-0189-04"/>
ramus I L maior erit, quàm B M: </s>
  <s xml:id="echoid-s5940" xml:space="preserve">Secari ergo poterunt æquales rectæ lineæ L R, <lb/>B S, quæ ſint minores quidẽ, quàm I L, ſed maiores, quàm M B; </s>
  <s xml:id="echoid-s5941" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5942" xml:space="preserve">deſcribantur <lb/>duo circuli, quorum radij ſint S B, &amp; </s>
  <s xml:id="echoid-s5943" xml:space="preserve">R L æquales, atque centra ſint S, &amp; </s>
  <s xml:id="echoid-s5944" xml:space="preserve">R; <lb/></s>
  <s xml:id="echoid-s5945" xml:space="preserve">
<anchor type="note" xlink:label="note-0189-05a" xlink:href="note-0189-05"/>
Manifeſtum eſt circulum, cuius radius B S contingere coniſectionem A C in <lb/>puncto B, &amp; </s>
  <s xml:id="echoid-s5946" xml:space="preserve">extrinſecùs incedere, propterea quod radius B S maior eſt maximo <lb/>breuiſecantium M B à concurſu M educto; </s>
  <s xml:id="echoid-s5947" xml:space="preserve">è contra circulus radio R L deſcri-<lb/>
<anchor type="note" xlink:label="note-0189-06a" xlink:href="note-0189-06"/>
ptus intrinſecùs continget eandem coniſectionem in L cum ramus M L minor ſit <lb/>ſingulari breuiſecante L I. </s>
  <s xml:id="echoid-s5948" xml:space="preserve">Tandẽ in ſectione D E F ſecetur axis abſcißa D H <lb/>æqualis A N, &amp; </s>
  <s xml:id="echoid-s5949" xml:space="preserve">in angulo D H P æquali angulo A N B ducatur radius γ H P, <lb/>qui fiat æqualis S B, &amp; </s>
  <s xml:id="echoid-s5950" xml:space="preserve">cẽtro γ radio verò γ P circulus deſcribatur. </s>
  <s xml:id="echoid-s5951" xml:space="preserve">Et quia in <lb/>ſectionibus æqualibus abſciſſæ, breuiſecantes, anguli ab eis contenti, &amp; </s>
  <s xml:id="echoid-s5952" xml:space="preserve">circu-<lb/>li deſcripti ſunt æquales, &amp; </s>
  <s xml:id="echoid-s5953" xml:space="preserve">congruentes; </s>
  <s xml:id="echoid-s5954" xml:space="preserve">igitur circulus radio γ P deſcriptus, <lb/>contingit coniſectionem D E F extrinſecùs; </s>
  <s xml:id="echoid-s5955" xml:space="preserve">ſicuti circulus radij S B tangebat <lb/>ſectionem A B C in B extrinſecùs. </s>
  <s xml:id="echoid-s5956" xml:space="preserve">Vterat propoſitum.</s>
  <s xml:id="echoid-s5957" xml:space="preserve"/>
</p>
<div xml:id="echoid-div567" type="float" level="2" n="1">
  <figure xlink:label="fig-0189-01" xlink:href="fig-0189-01a">
    <image file="0189-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0189-01"/>
  </figure>
<note position="right" xlink:label="note-0189-01" xlink:href="note-0189-01a" xml:space="preserve">51. 52. 53. <lb/>lib. 5,</note>
<note position="right" xlink:label="note-0189-02" xlink:href="note-0189-02a" xml:space="preserve">28. lib. 5. <lb/>8. Addir. <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0189-03" xlink:href="note-0189-03a" xml:space="preserve">13. 14. 15. <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0189-04" xlink:href="note-0189-04a" xml:space="preserve">67. lib. 5.</note>
<note position="right" xlink:label="note-0189-05" xlink:href="note-0189-05a" xml:space="preserve">Ex 12. <lb/>Addit. <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0189-06" xlink:href="note-0189-06a" xml:space="preserve">8. Addit. <lb/>lib. 5. <lb/>Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s5958" xml:space="preserve">Hoc demonſtrat o oſtendetur, quod in duabus coniſectionibus A B C, <lb/>
<anchor type="note" xlink:label="note-0189-07a" xlink:href="note-0189-07"/>
D E F æqualibus, quarum axes A G, D H duæ portiones B C, &amp; </s>
  <s xml:id="echoid-s5959" xml:space="preserve"><lb/>E F non æquè ab axium verticibus remotæ non erunt ſibi congruentes.</s>
  <s xml:id="echoid-s5960" xml:space="preserve"/>
</p>
<div xml:id="echoid-div568" type="float" level="2" n="2">
<note position="right" xlink:label="note-0189-07" xlink:href="note-0189-07a" xml:space="preserve">PROP. 1. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s5961" xml:space="preserve">Si enim poſſibile eſt B C, &amp; </s>
  <s xml:id="echoid-s5962" xml:space="preserve">E F ſibi mutuò congruant, &amp; </s>
  <s xml:id="echoid-s5963" xml:space="preserve">ſumatur interme-<lb/>dium punctum commune, vel duo puncta coincidentia L, &amp; </s>
  <s xml:id="echoid-s5964" xml:space="preserve">P, &amp; </s>
  <s xml:id="echoid-s5965" xml:space="preserve">quia portio-<lb/>nes B C, E F inæqualiter diſtant à verticibus, ergo puncta coincidentia L, P <lb/>non erunt æquè à verticibus remota; </s>
  <s xml:id="echoid-s5966" xml:space="preserve">ſit ergo P propinquius vertici D, quàm eſt <lb/>L vertici A, &amp; </s>
  <s xml:id="echoid-s5967" xml:space="preserve">per L, &amp; </s>
  <s xml:id="echoid-s5968" xml:space="preserve">P ducantur rectæ <lb/>
<anchor type="figure" xlink:label="fig-0189-02a" xlink:href="fig-0189-02"/>
lineæ L O, P Q tangentes ſectiones, &amp; </s>
  <s xml:id="echoid-s5969" xml:space="preserve">ex lẽ-<lb/>
<anchor type="note" xlink:label="note-0189-08a" xlink:href="note-0189-08"/>
matæ præcedenti deſcribantur duo circuli æ-<lb/>quales Z P T, &amp; </s>
  <s xml:id="echoid-s5970" xml:space="preserve">V L X radijs I L &amp; </s>
  <s xml:id="echoid-s5971" xml:space="preserve">S <lb/>P, quorum Z T extrinſecus tangat ſectionẽ <lb/>in P, &amp; </s>
  <s xml:id="echoid-s5972" xml:space="preserve">V X intrinſecus in L, cumque eo-<lb/>rum radij I L, S P ſint breuiſecantes, erunt <lb/>perpendiculares ad L O, P Q contingentes <lb/>
<anchor type="note" xlink:label="note-0189-09a" xlink:href="note-0189-09"/>
ſectionem in L, &amp; </s>
  <s xml:id="echoid-s5973" xml:space="preserve">P; </s>
  <s xml:id="echoid-s5974" xml:space="preserve">atque portiones B C, E F ſibi mutuò congruunt, ideſt <lb/>
<anchor type="note" xlink:label="note-0189-10a" xlink:href="note-0189-10"/>
conſtituunt vnicam communem peripheriam, ergo rectæ lineæ L O, P Q <lb/>contingentes eandem ſectionem ſibi mutuò congruent, pariterque breuiſe-<lb/>cantes æquales L I, P M ad illas perpendiculariter inſiſtentes crunt congruentes <lb/>quoque; </s>
  <s xml:id="echoid-s5975" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5976" xml:space="preserve">propterea circuli V X, Z T ab ĳs radĳs geniti erunt quoque congru-
<pb o="152" file="0190" n="190" rhead="Apollonij Pergæi"/>
entes; </s>
  <s xml:id="echoid-s5977" xml:space="preserve">ideoque ſi vnus eorum, nempe Z T extrinſecùs tangit communem portio-<lb/>nem conicam B C, reliquus V X extrinſecùs quoque eam langet, ſed ex conſtru-<lb/>ctione intrinſecùs ſectionem tangebat, quod eſt abſurdum: </s>
  <s xml:id="echoid-s5978" xml:space="preserve">Non ergo duæ por-<lb/>tiones B C, &amp; </s>
  <s xml:id="echoid-s5979" xml:space="preserve">E F non æquè à verticibus axium remotæ ſibi mutuo congruent-<lb/>Quod erat oſtendendum.</s>
  <s xml:id="echoid-s5980" xml:space="preserve"/>
</p>
<div xml:id="echoid-div569" type="float" level="2" n="3">
  <figure xlink:label="fig-0189-02" xlink:href="fig-0189-02a">
    <image file="0189-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0189-02"/>
  </figure>
<note position="right" xlink:label="note-0189-08" xlink:href="note-0189-08a" xml:space="preserve">33. 34. <lb/>lib. 1.</note>
<note position="right" xlink:label="note-0189-09" xlink:href="note-0189-09a" xml:space="preserve">29. 30. <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0189-10" xlink:href="note-0189-10a" xml:space="preserve">35. 36. <lb/>lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s5981" xml:space="preserve">Si autem cadit in ellipſi axis A C tranſuerſus ſuper axim rectum illius; <lb/></s>
  <s xml:id="echoid-s5982" xml:space="preserve">
<anchor type="note" xlink:label="note-0190-01a" xlink:href="note-0190-01"/>
vtique excedit illam, &amp; </s>
  <s xml:id="echoid-s5983" xml:space="preserve">non ſibi mutuò congruunt ſectiones, &amp; </s>
  <s xml:id="echoid-s5984" xml:space="preserve">quædam <lb/>congruunt, &amp;</s>
  <s xml:id="echoid-s5985" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5986" xml:space="preserve">Senſus eſt. </s>
  <s xml:id="echoid-s5987" xml:space="preserve">Si intelligantur duæ ellipſes, habentes axes tran-<lb/>ſuerſos A B, &amp; </s>
  <s xml:id="echoid-s5988" xml:space="preserve">G H æquales inier ſe, pariterque <lb/>
<anchor type="figure" xlink:label="fig-0190-01a" xlink:href="fig-0190-01"/>
axes rectos C D, I K æquales: </s>
  <s xml:id="echoid-s5989" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5990" xml:space="preserve">axis A B tran-<lb/>ſuerſus vnius ponatur ſuper I K axim rectum al-<lb/>terius, ita vt centra ſibi mutuò congruant in E: <lb/></s>
  <s xml:id="echoid-s5991" xml:space="preserve">tunc quidem, quia axes in ellipſi inæquales ſunt <lb/>(alias eſſet circulus) igitur extremitates axis tran-<lb/>ſuerſi A B non cadunt ſuper extremitaites axis re-<lb/>cti K I, neque G, H cadunt ſuper C, D; </s>
  <s xml:id="echoid-s5992" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5993" xml:space="preserve">ideo <lb/>circumferentiæ ellipſium ſe ſe mutuò ſecant qua-<lb/>tuor in locis, vt in libro 4. </s>
  <s xml:id="echoid-s5994" xml:space="preserve">oſtenſnm eſt.</s>
  <s xml:id="echoid-s5995" xml:space="preserve"/>
</p>
<div xml:id="echoid-div570" type="float" level="2" n="4">
<note position="right" xlink:label="note-0190-01" xlink:href="note-0190-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0190-01" xlink:href="fig-0190-01a">
    <image file="0190-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0190-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div572" type="section" level="1" n="186">
<head xml:id="echoid-head240" xml:space="preserve">SECTIO TERTIA <lb/>Continens Propoſit. V. &amp; VIII. <lb/>PROPOSITIO V.</head>
<p>
  <s xml:id="echoid-s5996" xml:space="preserve">SI per centrum E ellipſis A B, C D tranſeat linea recta A <lb/>C vſque ad ſectionem; </s>
  <s xml:id="echoid-s5997" xml:space="preserve">vtique bifariam diuidit ſuperſiciem <lb/>ſectionis, &amp; </s>
  <s xml:id="echoid-s5998" xml:space="preserve">circumferentiam illius, ſcilicet erit ſuperſicies A B <lb/>C æqualis ſuperficiei A D C.</s>
  <s xml:id="echoid-s5999" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6000" xml:space="preserve">Nam ſi A C fuerit axis ſectio-<lb/>
<anchor type="figure" xlink:label="fig-0190-02a" xlink:href="fig-0190-02"/>
nis, vtique circumferentia A B C <lb/>congruet A D C, nam ſi non cõ-<lb/>gruit ſignemus locum B, quod al-<lb/>teri ſectioni nõ coincidat, &amp; </s>
  <s xml:id="echoid-s6001" xml:space="preserve">pro-<lb/>ducamus ex illo perpendicularem <lb/>B F ſuper A C vſque ad D. </s>
  <s xml:id="echoid-s6002" xml:space="preserve">Er-<lb/>go B D ordinata eſt ad C A, &amp; </s>
  <s xml:id="echoid-s6003" xml:space="preserve"><lb/>propterea B F ſuperpoſita cõgru-<lb/>et ipſi D F, &amp; </s>
  <s xml:id="echoid-s6004" xml:space="preserve">cadet B ſuper D, <lb/>quia B F æqualis eſt D F (8. </s>
  <s xml:id="echoid-s6005" xml:space="preserve">ex <lb/>1.)</s>
  <s xml:id="echoid-s6006" xml:space="preserve">; </s>
  <s xml:id="echoid-s6007" xml:space="preserve">ſed non cadebat ſuper illum; </s>
  <s xml:id="echoid-s6008" xml:space="preserve">quod eſt abſurdum. </s>
  <s xml:id="echoid-s6009" xml:space="preserve">Igitur circumfe-
<pb o="153" file="0191" n="191" rhead="Conicor. Lib. VI."/>
rentia A B C æqualis eſt circumferentiæ A D C, &amp; </s>
  <s xml:id="echoid-s6010" xml:space="preserve">ſuperficies illius æ-<lb/>qualis ſuperficiei.</s>
  <s xml:id="echoid-s6011" xml:space="preserve"/>
</p>
<div xml:id="echoid-div572" type="float" level="2" n="1">
  <figure xlink:label="fig-0190-02" xlink:href="fig-0190-02a">
    <image file="0190-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0190-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s6012" xml:space="preserve">Iam linea G H tranſiens per centrum ellipſis non ſit axis. </s>
  <s xml:id="echoid-s6013" xml:space="preserve">Ducamus <lb/>ex G, H ſuper axim C A duas perpendiculares G I, H K, quæ pertin-<lb/>gant ad L, M. </s>
  <s xml:id="echoid-s6014" xml:space="preserve">Et quia ſi ponatur A D C ſuper A B C, congruit G I <lb/>ſuper L I (7. </s>
  <s xml:id="echoid-s6015" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s6016" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6017" xml:space="preserve">cadet G ſuper L, quia G I æqualis eſt I L, &amp; </s>
  <s xml:id="echoid-s6018" xml:space="preserve"><lb/>cadit circumferentia C G ſuper circumferentiam C L; </s>
  <s xml:id="echoid-s6019" xml:space="preserve">ergo ſuperſicies C <lb/>I G æqualis eſt ſuperficiei C I L: </s>
  <s xml:id="echoid-s6020" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6021" xml:space="preserve">quia B C D congruit B A D, &amp; </s>
  <s xml:id="echoid-s6022" xml:space="preserve">ſu-<lb/>perficies ſuperficiei, cadet C I ſuper A K, &amp; </s>
  <s xml:id="echoid-s6023" xml:space="preserve">L I ſuper K H, &amp; </s>
  <s xml:id="echoid-s6024" xml:space="preserve">circum-<lb/>ferentia C L ſuper circumferentiam A H (quia E I æqualis eſt E K) &amp; </s>
  <s xml:id="echoid-s6025" xml:space="preserve"><lb/>ſuperficies C I L congruit ſuperficiei A K H; </s>
  <s xml:id="echoid-s6026" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6027" xml:space="preserve">propterea ſuperficies A <lb/>K H æqualis eſt G I C, &amp; </s>
  <s xml:id="echoid-s6028" xml:space="preserve">triangulum E G I æquale eſt triangulo E K H; <lb/></s>
  <s xml:id="echoid-s6029" xml:space="preserve">igitur ſuperficies A E H æqualis eſt ſuperficiei G E C, &amp; </s>
  <s xml:id="echoid-s6030" xml:space="preserve">circumferentia <lb/>A H æqualis eſt circumferentiæ G C, eritque circumferentia C D H, &amp; </s>
  <s xml:id="echoid-s6031" xml:space="preserve"><lb/>ſuperficies eius æqualis A B G, &amp; </s>
  <s xml:id="echoid-s6032" xml:space="preserve">ſuperficiei illius. </s>
  <s xml:id="echoid-s6033" xml:space="preserve">Quare G H tranſiens <lb/>per centrum ſectionis A B C D bifariam eam diuidit. </s>
  <s xml:id="echoid-s6034" xml:space="preserve">Et hoc erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s6035" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div574" type="section" level="1" n="187">
<head xml:id="echoid-head241" xml:space="preserve">PROPOSITIO VIII.</head>
<p>
  <s xml:id="echoid-s6036" xml:space="preserve">SImiliter conſtat, quod ſi ex quolibet quadrante ellipſis ſe-<lb/>centur circumferentiæ, per quarum extremitates rectæ li-<lb/>neæ coniunctæ ſint ad eundem axim ordinatim applicatæ, &amp; </s>
  <s xml:id="echoid-s6037" xml:space="preserve"><lb/>æquè à centro remotæ; </s>
  <s xml:id="echoid-s6038" xml:space="preserve">vtique ſunt congruentes, &amp; </s>
  <s xml:id="echoid-s6039" xml:space="preserve">æquales, <lb/>nec alicui portioni eiuſdem ſectionis vna illarum æqualis eſt.</s>
  <s xml:id="echoid-s6040" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6041" xml:space="preserve">Nam demonſtrauimus, quod duæ ſuperficies <lb/>
<anchor type="note" xlink:label="note-0191-01a" xlink:href="note-0191-01"/>
<anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/>
G I C, L I C ſibi congruunt, nec non congru-<lb/>unt, duabus ſuperficiebus H A K, M A K (5. <lb/></s>
  <s xml:id="echoid-s6042" xml:space="preserve">ex 6.)</s>
  <s xml:id="echoid-s6043" xml:space="preserve">; </s>
  <s xml:id="echoid-s6044" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6045" xml:space="preserve">ſi eduxerimus duas ordinationes N <lb/>O, P Q, quarum diſtantiæ à centro ſint æqua-<lb/>les, ſimili modo oſtendetur, quod ſuperficies <lb/>N R C, O R C, A S Q, A S P ſint congruen-<lb/>tes (5. </s>
  <s xml:id="echoid-s6046" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s6047" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6048" xml:space="preserve">quod circumferentiæ N C, C <lb/>O, A Q, A P ſint congruentes, remanebunt <lb/>quatuor ſegmenta G N, L O, H Q, M P con-<lb/>gruentia, &amp; </s>
  <s xml:id="echoid-s6049" xml:space="preserve">ſuperficies quoque eorum congru-<lb/>entes. </s>
  <s xml:id="echoid-s6050" xml:space="preserve">Et inſuper dico, quod quodlibet horum <lb/>
<anchor type="note" xlink:label="note-0191-02a" xlink:href="note-0191-02"/>
ſegmentorum non congruit alicui alio ſegmen-<lb/>to; </s>
  <s xml:id="echoid-s6051" xml:space="preserve">nam ſequeretur, quod in eadem ellipſi ſint <lb/>
<anchor type="note" xlink:label="note-0191-03a" xlink:href="note-0191-03"/>
tres axes, vti dictum eſt, Quare patet propoſitum.</s>
  <s xml:id="echoid-s6052" xml:space="preserve"/>
</p>
<div xml:id="echoid-div574" type="float" level="2" n="1">
<note position="left" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0191-01" xlink:href="fig-0191-01a">
    <image file="0191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0191-01"/>
  </figure>
<note position="left" xlink:label="note-0191-02" xlink:href="note-0191-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0191-03" xlink:href="note-0191-03a" xml:space="preserve">48. lib. 2.</note>
</div>
<pb o="154" file="0192" n="192" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div576" type="section" level="1" n="188">
<head xml:id="echoid-head242" xml:space="preserve">Notæ in Propoſit. V.</head>
<p>
  <s xml:id="echoid-s6053" xml:space="preserve">ATque B C D congruit B A D, &amp; </s>
  <s xml:id="echoid-s6054" xml:space="preserve">ſuperficies ſuperficiei, &amp;</s>
  <s xml:id="echoid-s6055" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6056" xml:space="preserve">Quo-<lb/>
<anchor type="note" xlink:label="note-0192-01a" xlink:href="note-0192-01"/>
niam in ſecunda figura B D eſt axis ellipſis per centrum E ductus; </s>
  <s xml:id="echoid-s6057" xml:space="preserve">ergò <lb/>vt in prima parte huius propoſitionis dictum eſt, ſibi mutuò congruent ſemielli-<lb/>pſes B C D, &amp; </s>
  <s xml:id="echoid-s6058" xml:space="preserve">B A D.</s>
  <s xml:id="echoid-s6059" xml:space="preserve"/>
</p>
<div xml:id="echoid-div576" type="float" level="2" n="1">
<note position="right" xlink:label="note-0192-01" xlink:href="note-0192-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div578" type="section" level="1" n="189">
<head xml:id="echoid-head243" xml:space="preserve">Notæ in Propoſit. VIII.</head>
<p>
  <s xml:id="echoid-s6060" xml:space="preserve">NAm demonſtrauimus, &amp;</s>
  <s xml:id="echoid-s6061" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6062" xml:space="preserve">Expoſitio huius <lb/>
<anchor type="figure" xlink:label="fig-0192-01a" xlink:href="fig-0192-01"/>
<anchor type="note" xlink:label="note-0192-02a" xlink:href="note-0192-02"/>
propoſitionis hæc erit. </s>
  <s xml:id="echoid-s6063" xml:space="preserve">Sit ellipſis A B C D, <lb/>cuius axes C A, &amp; </s>
  <s xml:id="echoid-s6064" xml:space="preserve">B D, &amp; </s>
  <s xml:id="echoid-s6065" xml:space="preserve">in quolibet eius qua-<lb/>drante ſignentur tales circumferentiæ N G, O L, H <lb/>Q, M P, vt coniunctæ rectæ lineæ O N, G L, H <lb/>M, Q P ſint ad axim A C ordinatim applicatæ ſe-<lb/>cantes eum in R, I, K, S; </s>
  <s xml:id="echoid-s6066" xml:space="preserve">ſintque binarum extre-<lb/>marum N O, P Q à centro E diſtantiæ æquales E R, <lb/>E S, &amp; </s>
  <s xml:id="echoid-s6067" xml:space="preserve">binarum intermediarum L G, H M æquales à <lb/>centro diſtantiæ E I, E K oſtendendum eſt ſegmenta <lb/>G N, L O, H Q, M P æqualia eße.</s>
  <s xml:id="echoid-s6068" xml:space="preserve"/>
</p>
<div xml:id="echoid-div578" type="float" level="2" n="1">
  <figure xlink:label="fig-0192-01" xlink:href="fig-0192-01a">
    <image file="0192-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0192-01"/>
  </figure>
<note position="right" xlink:label="note-0192-02" xlink:href="note-0192-02a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s6069" xml:space="preserve">Et inſuper dico, quod quodlibet horum ſeg-<lb/>
<anchor type="note" xlink:label="note-0192-03a" xlink:href="note-0192-03"/>
mentorum non congruet alicui alio ſegmento, <lb/>&amp;</s>
  <s xml:id="echoid-s6070" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6071" xml:space="preserve">Si enim in eodem, vel in duabus ellipſis qua-<lb/>drantibus ſumantur ſegmenta G N, &amp; </s>
  <s xml:id="echoid-s6072" xml:space="preserve">M P non æque ab axis vertice B vel à <lb/>verticibus A, C eiuſdem axis remota, non erunt congruentia, vt deducitur ex <lb/>propoſ. </s>
  <s xml:id="echoid-s6073" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6074" xml:space="preserve">additarum huius.</s>
  <s xml:id="echoid-s6075" xml:space="preserve"/>
</p>
<div xml:id="echoid-div579" type="float" level="2" n="2">
<note position="right" xlink:label="note-0192-03" xlink:href="note-0192-03a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div581" type="section" level="1" n="190">
<head xml:id="echoid-head244" xml:space="preserve">SECTIO QVARTA <lb/>Continens Propoſit. XI. XII. XIII. &amp; XIV. <lb/>PROPOSITIO XI.</head>
<p>
  <s xml:id="echoid-s6076" xml:space="preserve">QVælibet ſectio parabolica, vt A B, cuius axis B C, &amp; </s>
  <s xml:id="echoid-s6077" xml:space="preserve">ere-<lb/>ctum B D ſimilis eſt cuilibet ſectioni parabolicæ, vt E F, <lb/>cuius axis F H, &amp; </s>
  <s xml:id="echoid-s6078" xml:space="preserve">erectum F I.</s>
  <s xml:id="echoid-s6079" xml:space="preserve"/>
</p>
<pb o="155" file="0193" n="193" rhead="Conicor. Lib. VI."/>
  <figure>
    <image file="0193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0193-01"/>
  </figure>
<p>
  <s xml:id="echoid-s6080" xml:space="preserve">Ponamus itaque C B ad B D, vt H F ad F I, &amp; </s>
  <s xml:id="echoid-s6081" xml:space="preserve">diuidantur tam B C, <lb/>quàm F H in punctis K, L, M, N in eiſdem rationibus, &amp; </s>
  <s xml:id="echoid-s6082" xml:space="preserve">educamus ſu-<lb/>per eas ordinationes O P, Q R, A S, T V, X Y , E Z. </s>
  <s xml:id="echoid-s6083" xml:space="preserve">Quia B C ad <lb/>B D eſt vt H F ad F I, &amp; </s>
  <s xml:id="echoid-s6084" xml:space="preserve">A C eſt media proportionalis inter C B, B D <lb/>
<anchor type="note" xlink:label="note-0193-01a" xlink:href="note-0193-01"/>
(12. </s>
  <s xml:id="echoid-s6085" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6086" xml:space="preserve">pariterque E H inter H F, F I (12. </s>
  <s xml:id="echoid-s6087" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6088" xml:space="preserve">igitur A C ad C <lb/>B eſt, vt E H ad H F , &amp; </s>
  <s xml:id="echoid-s6089" xml:space="preserve">A S dupla ipſius A C ad C B eſt, vt E Z ad <lb/>H F; </s>
  <s xml:id="echoid-s6090" xml:space="preserve">cumque B C ad B L poſita ſit, vt H F ad F N, erit B D ad B L, vt <lb/>
<anchor type="note" xlink:label="note-0193-02a" xlink:href="note-0193-02"/>
I F ad F N; </s>
  <s xml:id="echoid-s6091" xml:space="preserve">igitur Q R ad L B eſt vt X Y ad N F; </s>
  <s xml:id="echoid-s6092" xml:space="preserve">atque ſic oſtendetur, <lb/>quod O P ad K B eſt, vt T V ad M F, quare proportio ordinationum <lb/>axis vnius ſectionum ad ſua abſciſſa eſt, vt proportio ordinationum alte-<lb/>rius ad ſua abſciſſa, &amp; </s>
  <s xml:id="echoid-s6093" xml:space="preserve">proportiones abſciſſarum vnius ſectionis ad abſciſ-<lb/>ſa alterius ſectionis eædem ſunt. </s>
  <s xml:id="echoid-s6094" xml:space="preserve">Quare ſectio A B ſimilis eſt ſectioni E <lb/>
<anchor type="note" xlink:label="note-0193-03a" xlink:href="note-0193-03"/>
F. </s>
  <s xml:id="echoid-s6095" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s6096" xml:space="preserve"/>
</p>
<div xml:id="echoid-div581" type="float" level="2" n="1">
<note position="right" xlink:label="note-0193-01" xlink:href="note-0193-01a" xml:space="preserve">Ex 11. <lb/>Lib. 1.</note>
<note position="left" xlink:label="note-0193-02" xlink:href="note-0193-02a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0193-03" xlink:href="note-0193-03a" xml:space="preserve">Defin. 2. huius.</note>
</div>
</div>
<div xml:id="echoid-div583" type="section" level="1" n="191">
<head xml:id="echoid-head245" xml:space="preserve">PROPOSITIO XII.</head>
<p>
  <s xml:id="echoid-s6097" xml:space="preserve">SI duarum hyperbolarum, aut ellipſium duæ axium figuræ <lb/>fuerint ſimiles, vtique ſectiones ſimiles erunt: </s>
  <s xml:id="echoid-s6098" xml:space="preserve">Si verò fue-<lb/>rint ſectiones ſimiles, figuræ etiam ſimiles erunt.</s>
  <s xml:id="echoid-s6099" xml:space="preserve"/>
</p>
  <figure>
    <image file="0193-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0193-02"/>
  </figure>
<p>
  <s xml:id="echoid-s6100" xml:space="preserve">Sint ſectiones A B, E F, earum axes inclinati, vel tranſuerſi B a, F b, <lb/>&amp; </s>
  <s xml:id="echoid-s6101" xml:space="preserve">erecti earum B D, F I, &amp; </s>
  <s xml:id="echoid-s6102" xml:space="preserve">maneant ſigna, ordinationes, &amp; </s>
  <s xml:id="echoid-s6103" xml:space="preserve">proportio-<lb/>
<anchor type="note" xlink:label="note-0193-04a" xlink:href="note-0193-04"/>
nes eædem, quæ in præcedenti propoſitione. </s>
  <s xml:id="echoid-s6104" xml:space="preserve">Quoniam figura ſectionis <lb/>
<anchor type="note" xlink:label="note-0193-05a" xlink:href="note-0193-05"/>
<pb o="156" file="0194" n="194" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0194-01a" xlink:href="fig-0194-01"/>
A B ſimilis eſt figuræ ſectionis E F, erit quadratum H E ad H b in H F, <lb/>vt quadratum A C ad C a in C B; </s>
  <s xml:id="echoid-s6105" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6106" xml:space="preserve">b H in H F ad quadratum H F, <lb/>vt a C in C B ad quadratum C B (nam poſuimus H F ad F b, vt C B ad <lb/>B a) ergo ex æqualitate, quadratũ E H ad quadratũ H F eſt, vt quadra-<lb/>tum A C ad quadratum C B: </s>
  <s xml:id="echoid-s6107" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6108" xml:space="preserve">propterea E Z ad H F eſt vt A S ad C <lb/>B; </s>
  <s xml:id="echoid-s6109" xml:space="preserve">Atque ſic oſtendetur, quod X Y ad N F ſit vt Q R ad L B, &amp; </s>
  <s xml:id="echoid-s6110" xml:space="preserve">T V <lb/>ad M F ſit vt O P ad K B; </s>
  <s xml:id="echoid-s6111" xml:space="preserve">ergo proportiones ordinationum axis vnius <lb/>earum ad ſua abſciſſa ſunt eædem rationibus aliarum ordinationum axis <lb/>ad ſua abſciſſa, &amp; </s>
  <s xml:id="echoid-s6112" xml:space="preserve">alternatiuè. </s>
  <s xml:id="echoid-s6113" xml:space="preserve">Quare duæ ſectiones ſunt ſimiles.</s>
  <s xml:id="echoid-s6114" xml:space="preserve"/>
</p>
<div xml:id="echoid-div583" type="float" level="2" n="1">
<note position="left" xlink:label="note-0193-04" xlink:href="note-0193-04a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0193-05" xlink:href="note-0193-05a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0194-01" xlink:href="fig-0194-01a">
    <image file="0194-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0194-01"/>
  </figure>
</div>
<note position="left" xml:space="preserve">Defin. 2. <lb/>huius.</note>
<p>
  <s xml:id="echoid-s6115" xml:space="preserve">E contra oſtendetur, quod <lb/>ſi duæ ſectiones fuerint ſimi-<lb/>
<anchor type="figure" xlink:label="fig-0194-02a" xlink:href="fig-0194-02"/>
les, earũ figuræ ſimiles quo-<lb/>que erunt. </s>
  <s xml:id="echoid-s6116" xml:space="preserve">Quia eſt A C ad <lb/>
<anchor type="note" xlink:label="note-0194-02a" xlink:href="note-0194-02"/>
C B, vt E H ad H F, &amp; </s>
  <s xml:id="echoid-s6117" xml:space="preserve">ean-<lb/>dem proportionem habent <lb/>earum quadrata, atque <lb/>quadratum H F ad H F in <lb/>H b eſt, vt quadratum C B <lb/>ad C B in C a (eo quod <lb/>H F ad F b poſita fuit, vt <lb/>C B ad B a); </s>
  <s xml:id="echoid-s6118" xml:space="preserve">ergo ex æ-<lb/>qualitate quadratum E H ad <lb/>b H in H F, nempe I F <lb/>ad F b (20. </s>
  <s xml:id="echoid-s6119" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6120" xml:space="preserve">eſt, vt quadratum A C ad a C in C B, nempe vt <lb/>
<anchor type="note" xlink:label="note-0194-03a" xlink:href="note-0194-03"/>
D B ad B a (20. </s>
  <s xml:id="echoid-s6121" xml:space="preserve">ex 1.)</s>
  <s xml:id="echoid-s6122" xml:space="preserve">; </s>
  <s xml:id="echoid-s6123" xml:space="preserve">quare figuræ duarum ſectionum ſunt ſimiles. <lb/></s>
  <s xml:id="echoid-s6124" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s6125" xml:space="preserve"/>
</p>
<div xml:id="echoid-div584" type="float" level="2" n="2">
  <figure xlink:label="fig-0194-02" xlink:href="fig-0194-02a">
    <image file="0194-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0194-02"/>
  </figure>
<note position="left" xlink:label="note-0194-02" xlink:href="note-0194-02a" xml:space="preserve">Ex def. 2. <lb/>buius.</note>
<note position="left" xlink:label="note-0194-03" xlink:href="note-0194-03a" xml:space="preserve">21. lib. 1. <lb/>Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div586" type="section" level="1" n="192">
<head xml:id="echoid-head246" xml:space="preserve">PROPOSITIO XIII.</head>
<p>
  <s xml:id="echoid-s6126" xml:space="preserve">PArabola non eſt ſimilis hyperbolæ, neque ellipſi.</s>
  <s xml:id="echoid-s6127" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6128" xml:space="preserve">Hyperbolæ, ſeu ellipſis A B ſit axis B C, &amp; </s>
  <s xml:id="echoid-s6129" xml:space="preserve">inclinatus, ſeu tranſuerſus <lb/>B a, &amp; </s>
  <s xml:id="echoid-s6130" xml:space="preserve">E F ſit ſectio parabolæ, cuius axis F H. </s>
  <s xml:id="echoid-s6131" xml:space="preserve">Dico, quod ſectio E F <lb/>non eſt ſimilis ſectioni A B hyperbolicæ, aut ellipticæ, alioquin ſit ſimi-
<pb o="157" file="0195" n="195" rhead="Conicor. Lib. VI."/>
lis alicui earum (ſi poſ-<lb/>
<anchor type="note" xlink:label="note-0195-01a" xlink:href="note-0195-01"/>
ſibile eſt) ergo poſſu-<lb/>
<anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/>
mus educere in ſingulis <lb/>ſectionibus potentes, <lb/>quæ habeant ad ſua ab-<lb/>ſciſſa axium eaſdẽ pro-<lb/>portiones, &amp; </s>
  <s xml:id="echoid-s6132" xml:space="preserve">abſciſſæ in <lb/>ter ſe ſint proportiona-<lb/>les; </s>
  <s xml:id="echoid-s6133" xml:space="preserve">ſintque illæ V M, <lb/>Y N, P K, R L. </s>
  <s xml:id="echoid-s6134" xml:space="preserve">Quia <lb/>Y N ad N F poſita fuit, <lb/>vt R L ad L B, &amp; </s>
  <s xml:id="echoid-s6135" xml:space="preserve">N F <lb/>ad F M, vt L B ad BK, <lb/>&amp; </s>
  <s xml:id="echoid-s6136" xml:space="preserve">F M ad M V, vt B <lb/>K ad K P; </s>
  <s xml:id="echoid-s6137" xml:space="preserve">ergo Y N ad <lb/>M V in potentia, nem-<lb/>pe N F ad M F (cum <lb/>ſectio ſit parabola 19. <lb/></s>
  <s xml:id="echoid-s6138" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6139" xml:space="preserve">nempe L B ad B K ex contructione erit, vt R L ad K P potentia, <lb/>
<anchor type="note" xlink:label="note-0195-02a" xlink:href="note-0195-02"/>
quæ eandem proportionem habent, quàm a L in L B ad a K in K B; <lb/></s>
  <s xml:id="echoid-s6140" xml:space="preserve">
<anchor type="note" xlink:label="note-0195-03a" xlink:href="note-0195-03"/>
quia ſectio eſt hyperbolæ, aut ellipſis (20. </s>
  <s xml:id="echoid-s6141" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6142" xml:space="preserve">quare a L in L B ad a <lb/>K in K B eſt, vt L B ad B K; </s>
  <s xml:id="echoid-s6143" xml:space="preserve">quare a L eſt æqualis a K: </s>
  <s xml:id="echoid-s6144" xml:space="preserve">quod eſt abſur-<lb/>dum. </s>
  <s xml:id="echoid-s6145" xml:space="preserve">Igitur parabole non eſt ſimilis vlli reliquarum ſectionum. </s>
  <s xml:id="echoid-s6146" xml:space="preserve">Et hoc <lb/>erat probandum.</s>
  <s xml:id="echoid-s6147" xml:space="preserve"/>
</p>
<div xml:id="echoid-div586" type="float" level="2" n="1">
<note position="left" xlink:label="note-0195-01" xlink:href="note-0195-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0195-01" xlink:href="fig-0195-01a">
    <image file="0195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0195-01"/>
  </figure>
<note position="right" xlink:label="note-0195-02" xlink:href="note-0195-02a" xml:space="preserve">20. lib. 1.</note>
<note position="right" xlink:label="note-0195-03" xlink:href="note-0195-03a" xml:space="preserve">21. lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div588" type="section" level="1" n="193">
<head xml:id="echoid-head247" xml:space="preserve">PROPOSITIO XIV.</head>
<p>
  <s xml:id="echoid-s6148" xml:space="preserve">ET ſic oſtendetur, quod hyperbolæ non eſt ſimilis ellipſi.</s>
  <s xml:id="echoid-s6149" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6150" xml:space="preserve">Alioquin ſequitur, quod quadratum <lb/>
<anchor type="figure" xlink:label="fig-0195-02a" xlink:href="fig-0195-02"/>
<anchor type="note" xlink:label="note-0195-04a" xlink:href="note-0195-04"/>
R L ad quadratum K P, nempe a L in <lb/>L B ad a K in K B in hyperbola eſt, vt <lb/>quadratum Y N ad quadratum M V, <lb/>ſeu vt b N in N F ad b M in M F in el-<lb/>lipſi. </s>
  <s xml:id="echoid-s6151" xml:space="preserve">His poſitis: </s>
  <s xml:id="echoid-s6152" xml:space="preserve">quia L B ad B K po-<lb/>
<anchor type="note" xlink:label="note-0195-05a" xlink:href="note-0195-05"/>
ſita fuit, vt N F ad M F; </s>
  <s xml:id="echoid-s6153" xml:space="preserve">ergo a L ad <lb/>a K eandem proportionem habet, quàm <lb/>b N ad b M: </s>
  <s xml:id="echoid-s6154" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6155" xml:space="preserve">hoc eſt abſurdum. </s>
  <s xml:id="echoid-s6156" xml:space="preserve">Qua-<lb/>re ſectio A B non eſt ſimilis E F; </s>
  <s xml:id="echoid-s6157" xml:space="preserve">vt fue-<lb/>rat propoſitum.</s>
  <s xml:id="echoid-s6158" xml:space="preserve"/>
</p>
<div xml:id="echoid-div588" type="float" level="2" n="1">
  <figure xlink:label="fig-0195-02" xlink:href="fig-0195-02a">
    <image file="0195-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0195-02"/>
  </figure>
<note position="left" xlink:label="note-0195-04" xlink:href="note-0195-04a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0195-05" xlink:href="note-0195-05a" xml:space="preserve">21. lib. 1.</note>
</div>
<pb o="158" file="0196" n="196" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div590" type="section" level="1" n="194">
<head xml:id="echoid-head248" xml:space="preserve">MONITVM.</head>
<p style="it">
  <s xml:id="echoid-s6159" xml:space="preserve">IN principio huius libri monuimus, definitionem ſimilium conicarum <lb/>ſectionum, quæ circunfertur, vitioſam eſſe; </s>
  <s xml:id="echoid-s6160" xml:space="preserve">quod hic oſtendendum <lb/>ſuſcepimus: </s>
  <s xml:id="echoid-s6161" xml:space="preserve">ſed prius hæc demonſtranda ſunt.</s>
  <s xml:id="echoid-s6162" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div591" type="section" level="1" n="195">
<head xml:id="echoid-head249" xml:space="preserve">LEMMA II.</head>
<p style="it">
  <s xml:id="echoid-s6163" xml:space="preserve">IN duabus coniſectionibus A B, E F eiuſdem nominis ſint axium <lb/>figuræ G B D, K F I ſimiles inter ſe, ideſt tranſuerſa latera G B, <lb/>K F proportionalia ſint lateribus rectis B D, F I : </s>
  <s xml:id="echoid-s6164" xml:space="preserve">duci debent in ſingu-<lb/>lis ſectionibus ſeries applicatarum ad axes, ita vt axium abſciſſæ (quæ <lb/>proportionales ſunt inter ſe) ad conterminas potentiales non ſint in ĳſdem <lb/>rationibus.</s>
  <s xml:id="echoid-s6165" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6166" xml:space="preserve">Sumantur duæ abſciſſæ B C, F H, quarum C B ad B D habeat maiorem pro-<lb/>portionem, quàm habet H F ad F I, &amp; </s>
  <s xml:id="echoid-s6167" xml:space="preserve">C B, H F ſecentur proportionaliter in <lb/>R, V.</s>
  <s xml:id="echoid-s6168" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s6169" xml:space="preserve">per ea puncta ducantur ad axes ordinatim applicatæ A C, E H, Q <lb/>R, T V. </s>
  <s xml:id="echoid-s6170" xml:space="preserve">Quoniam quadratum A C ad rectangulum G C B eandem proportio-<lb/>
<anchor type="figure" xlink:label="fig-0196-01a" xlink:href="fig-0196-01"/>
nem babet, quàm latus rectum D B ad tranſuerſum G B, pariterq; </s>
  <s xml:id="echoid-s6171" xml:space="preserve">quadratum <lb/>
<anchor type="note" xlink:label="note-0196-01a" xlink:href="note-0196-01"/>
E H ad rectangulum K H F eſt vt I F ad F K; </s>
  <s xml:id="echoid-s6172" xml:space="preserve">atq; </s>
  <s xml:id="echoid-s6173" xml:space="preserve">D B ad B G ex hypotheſi, <lb/>eſt vt I F ad F K; </s>
  <s xml:id="echoid-s6174" xml:space="preserve">ergo quadratum A C ad rectangulum G C B eandem pro-<lb/>portionem habet quàm quadratum E H ad rectangulum K H F : </s>
  <s xml:id="echoid-s6175" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6176" xml:space="preserve">quia G B <lb/>ad B D eſt vt K F ad F I, &amp; </s>
  <s xml:id="echoid-s6177" xml:space="preserve">D B ad B C minorem proportionẽ habet quàm <lb/>I F ad F H, ergo ex æquali G B ad B C, minorem proportionem habet quàm <lb/>K F ad F H, &amp; </s>
  <s xml:id="echoid-s6178" xml:space="preserve">componendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s6179" xml:space="preserve">diuidendo in ellipſi G C ad C B <lb/>ſeu rectangulum G C B ad quadratum B C minorem proportionẽ habebit quàm <lb/>K H ad H F, ſeu quàm rectangulum K H F ad quadratum F H : </s>
  <s xml:id="echoid-s6180" xml:space="preserve">erat autem <lb/>quadratum A C ad rectangulum G C B vt quadratum E H ad rectangulum K <lb/>H F ; </s>
  <s xml:id="echoid-s6181" xml:space="preserve">igitur ex æquali, quadratum A C, ad quadratum C B minorem propor-<lb/>tionem habet quàm quaàratum E H ad quadratum H F, &amp; </s>
  <s xml:id="echoid-s6182" xml:space="preserve">ideo A C ad C B
<pb o="159" file="0197" n="197" rhead="Conicor. Lib. VI."/>
minorem proportionem habebit, quàm E H ad H F. </s>
  <s xml:id="echoid-s6183" xml:space="preserve">Poſtea quia C B ad B R <lb/>erat vt H F ad F V, &amp; </s>
  <s xml:id="echoid-s6184" xml:space="preserve">prius G B ad B C minorẽ proportionem habebat, quàm <lb/>K F ad F H, ergo ex æquali G B ad B R minorem proportionem habet, quàm <lb/>K F ad F V, &amp; </s>
  <s xml:id="echoid-s6185" xml:space="preserve">componendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s6186" xml:space="preserve">diuidenào in ellipſi G R ad R B, <lb/>ſeu rectangulum G R B ad quadratum B R minorem proportionem habet, quàm <lb/>K V ad V F, ſeu rectangulum K V F ad quadratum V F ; </s>
  <s xml:id="echoid-s6187" xml:space="preserve">ſed propter ſimili-<lb/>tudinem figurarum, vt prius quadratum Q R ad rectangulum G R B eſt vt qua-<lb/>dratũ T V ad rectangulum K V F; </s>
  <s xml:id="echoid-s6188" xml:space="preserve">ergo ex æquali quadratum Q R ad quadra-<lb/>tum R B minorem proportionem habet, quàm quadratum T V ad quadratum <lb/>V F, &amp; </s>
  <s xml:id="echoid-s6189" xml:space="preserve">Q R ad R B minorem proportionem habebit, quàm T V ad V F. </s>
  <s xml:id="echoid-s6190" xml:space="preserve">Et <lb/>ſic reliquæ omnes abſciſſæ : </s>
  <s xml:id="echoid-s6191" xml:space="preserve">quapropter patet propoſitum.</s>
  <s xml:id="echoid-s6192" xml:space="preserve"/>
</p>
<div xml:id="echoid-div591" type="float" level="2" n="1">
  <figure xlink:label="fig-0196-01" xlink:href="fig-0196-01a">
    <image file="0196-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0196-01"/>
  </figure>
<note position="left" xlink:label="note-0196-01" xlink:href="note-0196-01a" xml:space="preserve">21. lib. 5.</note>
</div>
</div>
<div xml:id="echoid-div593" type="section" level="1" n="196">
<head xml:id="echoid-head250" xml:space="preserve">COROLLARIVM.</head>
<p style="it">
  <s xml:id="echoid-s6193" xml:space="preserve">HInc conſtat in duabus ſimilibus coniſectionibus duci poſſe duas ſeries appli-<lb/>catarum ad axes, itaut abſciſſæ axium, quæ inter ſe proportionales ſunt, <lb/>ad ſuas potentiales nonſint in ĳſdem rationibus. </s>
  <s xml:id="echoid-s6194" xml:space="preserve">Quandoquidẽ ex prima parte <lb/>propoſitionis 12. </s>
  <s xml:id="echoid-s6195" xml:space="preserve">quotieſcunque axium figuræ ſimiles ſunt etiam ſectiones ipſæ <lb/>ſunt ſimiles.</s>
  <s xml:id="echoid-s6196" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div594" type="section" level="1" n="197">
<head xml:id="echoid-head251" xml:space="preserve">LEMMA III.</head>
<p style="it">
  <s xml:id="echoid-s6197" xml:space="preserve">IN ĳſdem figuris habeat G B ad B D maiorem proportionem, quàm <lb/>K F ad F I duci debent duæ ordinatim ad axes applicatæ, quæ ad <lb/>conterminas abſciſſas eandem proportionem habeant.</s>
  <s xml:id="echoid-s6198" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6199" xml:space="preserve">Ducatur quælibet ordinata E H, producanturq; </s>
  <s xml:id="echoid-s6200" xml:space="preserve">vt ſecet coniunctam K I in <lb/>L, &amp; </s>
  <s xml:id="echoid-s6201" xml:space="preserve">vt D B ad B G ita fiat L H ad H N, atq; </s>
  <s xml:id="echoid-s6202" xml:space="preserve">fiat G C ad B C, vt N H ad <lb/>H F, ducaturque ordinata A C; </s>
  <s xml:id="echoid-s6203" xml:space="preserve">quæ producta ſecet coniunctam G D in P. </s>
  <s xml:id="echoid-s6204" xml:space="preserve">Di-<lb/>co A C, &amp; </s>
  <s xml:id="echoid-s6205" xml:space="preserve">E H eße quæſitas. </s>
  <s xml:id="echoid-s6206" xml:space="preserve">Quoniam quadratum A C ad rectangulum G C <lb/>
<anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/>
B eandem proportionem habet, quàm D B ad B G, ſeu L H ad H N, &amp; </s>
  <s xml:id="echoid-s6207" xml:space="preserve">rectã. <lb/></s>
  <s xml:id="echoid-s6208" xml:space="preserve">gulum G C B ad quadratum B C eſt <lb/>
<anchor type="figure" xlink:label="fig-0197-01a" xlink:href="fig-0197-01"/>
vt G C ad C B, ſeu vt N H ad H <lb/>F, ergo ex æqualitate quadratum <lb/>A C ad quadratum C B eſt vt L H <lb/>ad H F, ſeu vt rectangnlum L H <lb/>F ad quadratum H F; </s>
  <s xml:id="echoid-s6209" xml:space="preserve">vel potius <lb/>vt quadratum E H ad quadratum <lb/>
<anchor type="note" xlink:label="note-0197-02a" xlink:href="note-0197-02"/>
H F; </s>
  <s xml:id="echoid-s6210" xml:space="preserve">ideoque A C ad C B erit vt <lb/>E H ad H F. </s>
  <s xml:id="echoid-s6211" xml:space="preserve">Quod erat propoſi-<lb/>tum.</s>
  <s xml:id="echoid-s6212" xml:space="preserve"/>
</p>
<div xml:id="echoid-div594" type="float" level="2" n="1">
<note position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">21. lib. 1.</note>
  <figure xlink:label="fig-0197-01" xlink:href="fig-0197-01a">
    <image file="0197-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0197-01"/>
  </figure>
<note position="right" xlink:label="note-0197-02" xlink:href="note-0197-02a" xml:space="preserve">12. 13. <lb/>lib. 1.</note>
</div>
<pb o="160" file="0198" n="198" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div596" type="section" level="1" n="198">
<head xml:id="echoid-head252" xml:space="preserve">LEMMA IV.</head>
<p style="it">
  <s xml:id="echoid-s6213" xml:space="preserve">SI G B ad B D maiorem proportionem habuerit, quàm K F ad F <lb/>I: </s>
  <s xml:id="echoid-s6214" xml:space="preserve">Dico in ſingulis ſectionibus reperiri non poſſe binas axium ab-<lb/>ſciſſas inter ſe proportionales, quæ ad conterminas potentiales ſint in eiſ-<lb/>dem rationibus.</s>
  <s xml:id="echoid-s6215" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6216" xml:space="preserve">Si enim fieri poteſt, ſit A C ad <lb/>
<anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/>
C B, vt E H ad H F, &amp; </s>
  <s xml:id="echoid-s6217" xml:space="preserve">Q R ad <lb/>R B ſit, vt T V ad V F, atque C <lb/>B ad B R ſit vt H F ad F V; </s>
  <s xml:id="echoid-s6218" xml:space="preserve">con-<lb/>iungantur rectæ G D, K I quæ ſecẽt <lb/>ordinatas in S, P, X, L; </s>
  <s xml:id="echoid-s6219" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6220" xml:space="preserve">ſecen-<lb/>tur C a æqualis R S, &amp; </s>
  <s xml:id="echoid-s6221" xml:space="preserve">H b æqualis <lb/>V X, ſuntq; </s>
  <s xml:id="echoid-s6222" xml:space="preserve">æquidiſtantes; </s>
  <s xml:id="echoid-s6223" xml:space="preserve">ergo co-<lb/>niungentes S a, R C æquales ſunt, <lb/>&amp; </s>
  <s xml:id="echoid-s6224" xml:space="preserve">parallelæ, &amp; </s>
  <s xml:id="echoid-s6225" xml:space="preserve">ſic etiam coniun-<lb/>gentes X b, &amp; </s>
  <s xml:id="echoid-s6226" xml:space="preserve">V H, quare quadratum A C, ſeu rectangulum P C B ad qua-<lb/>dratum C B eandem proportionem habet, quàm quadratum E H, ſeu rectangu-<lb/>
<anchor type="note" xlink:label="note-0198-01a" xlink:href="note-0198-01"/>
lum L H F ad quadratum H F; </s>
  <s xml:id="echoid-s6227" xml:space="preserve">ideoque P C ad C B eandem proportionem ha-<lb/>bet, quàm L H ad H F; </s>
  <s xml:id="echoid-s6228" xml:space="preserve">eſt verò C B ad B R, vt H F ad F V, &amp; </s>
  <s xml:id="echoid-s6229" xml:space="preserve">per conuerſio-<lb/>nem rationis C B ad C R eſt vt H F ad H V, ergo ex æquali C P ad C R eſt <lb/>vt L H ad H V: </s>
  <s xml:id="echoid-s6230" xml:space="preserve">Eodem modo oſtendetur, quod S R, ſeu a C ad R C eſt, vt <lb/>X V, ſeu b H ad V H; </s>
  <s xml:id="echoid-s6231" xml:space="preserve">erat autem P C ad C R vt L H ad H V; </s>
  <s xml:id="echoid-s6232" xml:space="preserve">ergo a P dif-<lb/>ferentia ipſarum S R, P C ad G R, ſeu ad S a eſt vt b L differentia ipſarum <lb/>X V, L H ad H V, ſeu ad X b; </s>
  <s xml:id="echoid-s6233" xml:space="preserve">eſtque D B ad B G vt P a ad S a (propter pa-<lb/>rallelas a S, C G, &amp; </s>
  <s xml:id="echoid-s6234" xml:space="preserve">parallelas a P, &amp; </s>
  <s xml:id="echoid-s6235" xml:space="preserve">B D) pariterque I F ad F K eſt vt L <lb/>b ad b X, ergo D B ad B G eandem proportionem habet, quàm I F ad F K; <lb/></s>
  <s xml:id="echoid-s6236" xml:space="preserve">quod eſt contra hypotheſim, non ergo binæ axium abſciſſæ inter ſe proportionales <lb/>reperiri poſſunt in ſectionibus A B, &amp; </s>
  <s xml:id="echoid-s6237" xml:space="preserve">E F, quæ ad conterminas potentiales ſint <lb/>in eiſdem rationibus; </s>
  <s xml:id="echoid-s6238" xml:space="preserve">quod erat oſtendendum.</s>
  <s xml:id="echoid-s6239" xml:space="preserve"/>
</p>
<div xml:id="echoid-div596" type="float" level="2" n="1">
  <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a">
    <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0198-01"/>
  </figure>
<note position="left" xlink:label="note-0198-01" xlink:href="note-0198-01a" xml:space="preserve">12. 13. <lb/>lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div598" type="section" level="1" n="199">
<head xml:id="echoid-head253" xml:space="preserve">COROLLARIVM.</head>
<p style="it">
  <s xml:id="echoid-s6240" xml:space="preserve">HInc conſtat in duabus ſectionibus eiuſdem nominis ſi axium figuræ G B D, <lb/>&amp; </s>
  <s xml:id="echoid-s6241" xml:space="preserve">K F I non ſuerint ſimiles, neque ſectiones A B, &amp; </s>
  <s xml:id="echoid-s6242" xml:space="preserve">E F, ſimiles eſſe. <lb/></s>
  <s xml:id="echoid-s6243" xml:space="preserve">Nam eſt impoſſibile, vt omnes, ideſt infinitæ axium abſciſſæ inter ſe proportio-<lb/>nales ad conterminas potentiales ſint in eiſdem rationibus, cum neque bine in <lb/>ſingulis reperiri poſſint ex hac propoſitione.</s>
  <s xml:id="echoid-s6244" xml:space="preserve"/>
</p>
<pb o="161" file="0199" n="199" rhead="Conicor. Lib. VI."/>
</div>
<div xml:id="echoid-div599" type="section" level="1" n="200">
<head xml:id="echoid-head254" xml:space="preserve">LEMMAV.</head>
<p style="it">
  <s xml:id="echoid-s6245" xml:space="preserve">IN eiſdem figuris rurſus G B ad B D maiorem proportionem habeat, <lb/>qnàm K F ad F 1 : </s>
  <s xml:id="echoid-s6246" xml:space="preserve">Dico quod minimè reperiri poſſunt axium ab-<lb/>ſcißæ erectis proportionales, quæ habeant eandem rationem ad contermi-<lb/>nas potentiales.</s>
  <s xml:id="echoid-s6247" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6248" xml:space="preserve">Secentur quælibet abſciſſæ, B C, F H ita vt C B ad B D ſit vt H F ad F I, <lb/>&amp; </s>
  <s xml:id="echoid-s6249" xml:space="preserve">ducantur ordinatim ad axes applicatæ A C, E H, quæ productæ ſecent, con-<lb/>iunctas G D, K I in P, L, atque fiat γ B ad B D vt K F ad F I, iungatur-<lb/>que γ D ſecans A P in M. </s>
  <s xml:id="echoid-s6250" xml:space="preserve">Manifeſtum eſt rectam C M inæqualem eſſe C P, <lb/>(propterea quod γ B minor eſt, quàm G B, cum ad eandem B D minorem pro-<lb/>portionem habeat, quàm G B, ideoque punctum Y<unsure/>, &amp; </s>
  <s xml:id="echoid-s6251" xml:space="preserve">recta γ D cadent intra, <lb/>triangulum G B D, &amp; </s>
  <s xml:id="echoid-s6252" xml:space="preserve">punctum M intra ipſum cadet, aut extra G D pro-<lb/>ductam). </s>
  <s xml:id="echoid-s6253" xml:space="preserve">Quoniam D B ad B γ eſt vt I F ad F K, &amp; </s>
  <s xml:id="echoid-s6254" xml:space="preserve">erat C B ad B D vt <lb/>H F ad F I ; </s>
  <s xml:id="echoid-s6255" xml:space="preserve">ergo ex æquali C B ad B γ erit vt H F ad F K, &amp; </s>
  <s xml:id="echoid-s6256" xml:space="preserve">comparando <lb/>terminorum ſummas in hyperbola, &amp; </s>
  <s xml:id="echoid-s6257" xml:space="preserve">differentias in ellipſi ad antecedentes, γ C <lb/>ad C B erit vt K H ad H F; </s>
  <s xml:id="echoid-s6258" xml:space="preserve">eſt verò M C ad C R<unsure/> vt L H ad H K (eoquod <lb/>triãgula M C R<unsure/>, &amp; </s>
  <s xml:id="echoid-s6259" xml:space="preserve">L H K ſimilia ſunt triangulis ſimilibus B D Y<unsure/>, I F K,) ergo <lb/>ex æquali M C ad C B erit vt L H ad H F, &amp; </s>
  <s xml:id="echoid-s6260" xml:space="preserve">rectangulum M C B ad quadra-<lb/>tum C B eandem proportionem habebit, quàrn rectangulum L H F ad quadra-<lb/>tũ H F; </s>
  <s xml:id="echoid-s6261" xml:space="preserve">ſed rectangulũ M C B æquale nõ eſt rectangulo P C B (cum M C oſtenſa <lb/>ſit inæqualis P C); </s>
  <s xml:id="echoid-s6262" xml:space="preserve">ergo rectangulum P C B, ſeu quadratum A C ad quadratum <lb/>
<anchor type="note" xlink:label="note-0199-01a" xlink:href="note-0199-01"/>
C B non eandem proportionem habet, quàm rectangulum L H F, ſeu quadratum <lb/>E H ad quadratum H F; </s>
  <s xml:id="echoid-s6263" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6264" xml:space="preserve">propterea A C ad C B non eandem proportionem <lb/>habebit quàm E H ad H F. </s>
  <s xml:id="echoid-s6265" xml:space="preserve">Idem oſtendetur in reliquis omnibus abſciſſis ſimi-<lb/>liter poſitis. </s>
  <s xml:id="echoid-s6266" xml:space="preserve">Quare patet propoſitum.</s>
  <s xml:id="echoid-s6267" xml:space="preserve"/>
</p>
<div xml:id="echoid-div599" type="float" level="2" n="1">
<note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve">12. 13. <lb/>lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div601" type="section" level="1" n="201">
<head xml:id="echoid-head255" xml:space="preserve">COROLLARIVM I.</head>
<p style="it">
  <s xml:id="echoid-s6268" xml:space="preserve">MAnifeſtum eſt in coniſectionibus non ſimilibus duci poſſe duas ſeries appli-<lb/>catarum ad axes, itaut abſciſſæ ſimiles, ſeu proportionales inter ſe adcõ-<lb/>terminas potentiales non ſint in ĳſdem rationibus.</s>
  <s xml:id="echoid-s6269" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div602" type="section" level="1" n="202">
<head xml:id="echoid-head256" xml:space="preserve">COROLLARIVM II.</head>
<p style="it">
  <s xml:id="echoid-s6270" xml:space="preserve">Colligitur pariter conuertendo, quod in duabus ſectionibus eiuſdem nominis <lb/>ſi duæ ſeries abſciſſarum ſimilium in axibus poſitæ fuerint, &amp; </s>
  <s xml:id="echoid-s6271" xml:space="preserve">in vna ſe-<lb/>rie abſciſſæ ad conterminas potentiales maiorem proportionem habeant, quàm in <lb/>altera ſerie, fieri poteſt vt ſiguræ axium non ſint inter ſe ſimiles: </s>
  <s xml:id="echoid-s6272" xml:space="preserve">Quod verifi-<lb/>catur ſaltem in caſu præcedentis propoſitionis.</s>
  <s xml:id="echoid-s6273" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6274" xml:space="preserve">His præmiſſis, quoniam paſſo in definitione poſita eſſentialiter conuenit defini-<lb/>to eſt impoſſibile, vt eidem ſubiecto definito competant duæ paſſiones diuerſæ, &amp; </s>
  <s xml:id="echoid-s6275" xml:space="preserve"><lb/>inter ſe oppoſitæ, exempli gratia, fieri non poteſt, vt in triangulis ſimilibus ali-
<pb o="162" file="0200" n="200" rhead="Apollonij Pergæi"/>
quando anguli vnius inæquales ſint angulis alterius, aut aliquaudo latera circa <lb/>angulos æquales non ſint proportionalia; </s>
  <s xml:id="echoid-s6276" xml:space="preserve">ita in definitione Mydorgiana, quia co-<lb/>niſectiones dicuntur ſimiles in quibus omnes axium abſcißæ, quæ proportionales <lb/>ſunt inter ſe in ĳsdem ſunt rationibus ad conterminas potentiales, igitur eidem <lb/>ſubiecto deſinito, ideſt in duabus ſectionibus conicis ſimilibus, eſt impoſſibile, vt <lb/>reperiatur ſeries aliqua infinitarum ſimilium abſciſſarum in axibus, quæ ad con-<lb/>terminas potentiales non ſint in ĳſdem rationibus, &amp; </s>
  <s xml:id="echoid-s6277" xml:space="preserve">ſiquidem duæ paſſiones op-<lb/>poſitæ eidem ſubiecto definito conueniant nulla earum erit eius paſſio eſſentialis, <lb/>&amp; </s>
  <s xml:id="echoid-s6278" xml:space="preserve">ideo definitio bona non erit: </s>
  <s xml:id="echoid-s6279" xml:space="preserve">vt exempli gratia quia in duobus ſimilibus cir-<lb/>culorum ſegmentis duo triangula inſcripta poſſunt eſſe æquiangula, &amp; </s>
  <s xml:id="echoid-s6280" xml:space="preserve">etiam non <lb/>æquiangula; </s>
  <s xml:id="echoid-s6281" xml:space="preserve">ergo ſimilitudo inſcriptorum triangulorum non eſt paſſio eſſentialis <lb/>ſegmentorum circularium ſimilium inter ſe, &amp; </s>
  <s xml:id="echoid-s6282" xml:space="preserve">ideo non erit bæc bona definitio: <lb/></s>
  <s xml:id="echoid-s6283" xml:space="preserve">Similia circulorũ ſegmenta ſunt in quibus deſcribi poſſunt duo triangula ſi-<lb/>milia, &amp; </s>
  <s xml:id="echoid-s6284" xml:space="preserve">ratio eſt, quia per definitionem nedum natura rei declaratur, &amp; </s>
  <s xml:id="echoid-s6285" xml:space="preserve">indi-<lb/>catur, ſed etiam diftinguitur, &amp; </s>
  <s xml:id="echoid-s6286" xml:space="preserve">diuerſificatur à qualibet alia; </s>
  <s xml:id="echoid-s6287" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6288" xml:space="preserve">quoniam in <lb/>
<anchor type="note" xlink:label="note-0200-01a" xlink:href="note-0200-01"/>
ſectionibus ſimilibus reperiuntur duæ ſeries ſimilium abſciſſarum, quæ ad con-<lb/>terminas potentiales non ſunt in ĳſdem rationibus; </s>
  <s xml:id="echoid-s6289" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6290" xml:space="preserve">è contra ex definitione, <lb/>Mydorgĳ duæ ſeries ſimilium abſciſſarum, quæ ad conterminas potentiales ſunt <lb/>in ĳſdem rationibus, eſſentialiter conueniunt definito; </s>
  <s xml:id="echoid-s6291" xml:space="preserve">igitur hæ duæ oppoſitæ <lb/>paſſiones conueniunt eidem ſubiecto definito, ſcilicet ſectionibus ſimilibus iu-<lb/>xta Mydorgĳ ſententiam : </s>
  <s xml:id="echoid-s6292" xml:space="preserve">quapropter tradita definitio ſectionum ſimilium vi-<lb/>tioſa erit, &amp; </s>
  <s xml:id="echoid-s6293" xml:space="preserve">manca.</s>
  <s xml:id="echoid-s6294" xml:space="preserve"/>
</p>
<div xml:id="echoid-div602" type="float" level="2" n="1">
<note position="left" xlink:label="note-0200-01" xlink:href="note-0200-01a" xml:space="preserve">Coroll. <lb/>Lem. 2. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s6295" xml:space="preserve">Vt autem hoc clarius pateat ex-<lb/>
<anchor type="figure" xlink:label="fig-0200-01a" xlink:href="fig-0200-01"/>
ponantur duæ ſectiones A B, E F <lb/>eiuſdem nominis, quarum axes B <lb/>C, F H, &amp; </s>
  <s xml:id="echoid-s6296" xml:space="preserve">propoſitum primò ſit de-<lb/>monſtrare ſectiones illas eſſe ſimiles <lb/>inter ſe; </s>
  <s xml:id="echoid-s6297" xml:space="preserve">ergo oſtendendum eſt paſ-<lb/>ſionem definitionis traditæ conueni-<lb/>re ſectionibus A B, E F; </s>
  <s xml:id="echoid-s6298" xml:space="preserve">quod ni-<lb/>mirum ſimiles axium abſcißæ in, <lb/>ĳſdem rationibus debent eſſe adcõ-<lb/>terminas potentiales, &amp; </s>
  <s xml:id="echoid-s6299" xml:space="preserve">quia in, <lb/>definitione nulla cautio, vel determinatio adhibetur, igitur ſumi poſſunt quæ-<lb/>libet axium abſciſſæ B C, F H, &amp; </s>
  <s xml:id="echoid-s6300" xml:space="preserve">hæc ſecari proportionaliter in R, V, &amp; </s>
  <s xml:id="echoid-s6301" xml:space="preserve">à <lb/>punctis diuiſionum duci poßunt ad axes ordinatim applicatæ A C, E H, Q R, <lb/>T V; </s>
  <s xml:id="echoid-s6302" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6303" xml:space="preserve">ſupponamus demonſtratum eſſe, quod B C ad C A ſit vt F H ad H E, <lb/>pariterque vt B R ad R Q ſit vt F V ad V T, tunc quidem ex vi definitionis <lb/>deducitur, quod ſimiles ſint ſectiones A B, &amp; </s>
  <s xml:id="echoid-s6304" xml:space="preserve">E F. </s>
  <s xml:id="echoid-s6305" xml:space="preserve">At quia demonſtrari poteſt <lb/>
<anchor type="note" xlink:label="note-0200-02a" xlink:href="note-0200-02"/>
in ĳſdem ſectionibus (ſumendo abſciſſas B C, F H ad libitum, &amp; </s>
  <s xml:id="echoid-s6306" xml:space="preserve">proportiona-<lb/>liter diuidendo eas in R, &amp; </s>
  <s xml:id="echoid-s6307" xml:space="preserve">V) quod B C ad C A habet maiorem proportionem, <lb/>
<anchor type="note" xlink:label="note-0200-03a" xlink:href="note-0200-03"/>
quàm F H ad H E; </s>
  <s xml:id="echoid-s6308" xml:space="preserve">pariterque B R ad R Q maiorem proportionẽ habeat, quàm <lb/>
<anchor type="note" xlink:label="note-0200-04a" xlink:href="note-0200-04"/>
F V ad V T, &amp; </s>
  <s xml:id="echoid-s6309" xml:space="preserve">ſic ſemper; </s>
  <s xml:id="echoid-s6310" xml:space="preserve">ergo non poterit deduci ſimilitudo potius quàm non <lb/>ſimilitudo; </s>
  <s xml:id="echoid-s6311" xml:space="preserve">ideoque definitio ſimilium ſectionum erit vitioſa, quandoquidem ex <lb/>ea duæ contradictoriæ deducuntur.</s>
  <s xml:id="echoid-s6312" xml:space="preserve"/>
</p>
<div xml:id="echoid-div603" type="float" level="2" n="2">
  <figure xlink:label="fig-0200-01" xlink:href="fig-0200-01a">
    <image file="0200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0200-01"/>
  </figure>
<note position="left" xlink:label="note-0200-02" xlink:href="note-0200-02a" xml:space="preserve">ex Lem. 2. <lb/>huius.</note>
<note position="left" xlink:label="note-0200-03" xlink:href="note-0200-03a" xml:space="preserve">Coroll. 2. <lb/>Lem. 5. <lb/>huius.</note>
<note position="left" xlink:label="note-0200-04" xlink:href="note-0200-04a" xml:space="preserve">Coroll. 2. <lb/>Lem. 5. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s6313" xml:space="preserve">Secundo loco ſupponantur duæ ſectiones A B, &amp; </s>
  <s xml:id="echoid-s6314" xml:space="preserve">E F ſimiles inter ſe, &amp; </s>
  <s xml:id="echoid-s6315" xml:space="preserve">pro-<lb/>poſitum, ſit demonſtrare quod axium figuræ, ſeu rectangula G B D, &amp; </s>
  <s xml:id="echoid-s6316" xml:space="preserve">K F I
<pb o="163" file="0201" n="201" rhead="Conicor. Lib. VI."/>
ſint ſimilia, quæ quidem, eſt propoſitio 3. </s>
  <s xml:id="echoid-s6317" xml:space="preserve">libri 4. </s>
  <s xml:id="echoid-s6318" xml:space="preserve">Mydorgĳ, eiuſque præparatio, <lb/>ſeu conſtructio talis eſt (, &amp; </s>
  <s xml:id="echoid-s6319" xml:space="preserve">appono eius verba immutatis tantummodo literis fi-<lb/>gurarũ) ſint à ſectione A B ordinatim ad axim B C applicatæ binæ quæ-<lb/>quæ A C, Q R, &amp; </s>
  <s xml:id="echoid-s6320" xml:space="preserve">vt C B ad B R ita ſit, H F ad F V, ordinatimque à ſe-<lb/>ctione E F applicentur E H, T V ( ſubſequitur poſtea demonſtratio ſic.) <lb/></s>
  <s xml:id="echoid-s6321" xml:space="preserve">Quoniam igitur ſimiles ponuntur ſectiones A B, E F, &amp; </s>
  <s xml:id="echoid-s6322" xml:space="preserve">ſunt H F, F V <lb/>portiones portionibus C B, B R fimiles, (ideſt proportionales) vt B C <lb/>ad C A, ita erit F H ad H E, &amp; </s>
  <s xml:id="echoid-s6323" xml:space="preserve">vt B R ad R Q, ita erit F V ad V T, <lb/>&amp; </s>
  <s xml:id="echoid-s6324" xml:space="preserve">c.</s>
  <s xml:id="echoid-s6325" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6326" xml:space="preserve">Huiuſmodi verba ſubtiliori trutina expendenda ſunt. </s>
  <s xml:id="echoid-s6327" xml:space="preserve">In præparatione, ſeu <lb/>conſtructione aſſumit abſcißas B C, &amp; </s>
  <s xml:id="echoid-s6328" xml:space="preserve">F H abſque vlla lege, aut determinatione; <lb/></s>
  <s xml:id="echoid-s6329" xml:space="preserve">ergo ſumi poſſunt cuiuſcunq; </s>
  <s xml:id="echoid-s6330" xml:space="preserve">longitudinis: </s>
  <s xml:id="echoid-s6331" xml:space="preserve">quare fieri poteſt vt C B ad latus re-<lb/>ctum B D non habeat eandem proportionem quàm habet F H ad F I, &amp; </s>
  <s xml:id="echoid-s6332" xml:space="preserve">tunc <lb/>
<anchor type="note" xlink:label="note-0201-01a" xlink:href="note-0201-01"/>
licet C B , H F diuidantur proportionaliter, &amp; </s>
  <s xml:id="echoid-s6333" xml:space="preserve">ducantur potentiales, &amp; </s>
  <s xml:id="echoid-s6334" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6335" xml:space="preserve">A C <lb/>ad C B habebit maiorem, aut minorem proportionem quàm E H ad H F, &amp; </s>
  <s xml:id="echoid-s6336" xml:space="preserve">pa-<lb/>riter Q R ad R B non habebit eandem rationem, quàm T V ad V F, &amp; </s>
  <s xml:id="echoid-s6337" xml:space="preserve">ſit vl-<lb/>terius in tota ſerie; </s>
  <s xml:id="echoid-s6338" xml:space="preserve">ſed ex hoc ſequitur, quod poſſint eſſe figuræ axium inter ſe <lb/>
<anchor type="note" xlink:label="note-0201-02a" xlink:href="note-0201-02"/>
non ſimiles; </s>
  <s xml:id="echoid-s6339" xml:space="preserve">Mydorgius autem ſimiles eſſe concludit; </s>
  <s xml:id="echoid-s6340" xml:space="preserve">igitur ex eadem hypotheſi, <lb/>&amp; </s>
  <s xml:id="echoid-s6341" xml:space="preserve">ex eadem definitione deducitur, quod ſectiones ſimiles habent figuras axium, <lb/>ſimiles inter ſe, &amp; </s>
  <s xml:id="echoid-s6342" xml:space="preserve">non ſimiles, quod eſt impoſſibile; </s>
  <s xml:id="echoid-s6343" xml:space="preserve">non igitur definitio à My-<lb/>dorgio tradita legitima, &amp; </s>
  <s xml:id="echoid-s6344" xml:space="preserve">perfecta eſt: </s>
  <s xml:id="echoid-s6345" xml:space="preserve">quod fuerat oſtendendum.</s>
  <s xml:id="echoid-s6346" xml:space="preserve"/>
</p>
<div xml:id="echoid-div604" type="float" level="2" n="3">
<note position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve">Lem. 2. <lb/>huius.</note>
<note position="right" xlink:label="note-0201-02" xlink:href="note-0201-02a" xml:space="preserve">Coroll. 2. <lb/>Lem. 5. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s6347" xml:space="preserve">Quod vero deſinitio à me reformata tribui poſſit Apollonio conĳcitur præcipuè <lb/>ex demonſtratione ſecundæ partis propor. </s>
  <s xml:id="echoid-s6348" xml:space="preserve">12. </s>
  <s xml:id="echoid-s6349" xml:space="preserve">ibi enim ex hac ſuppoſitione, quod <lb/>
<anchor type="figure" xlink:label="fig-0201-01a" xlink:href="fig-0201-01"/>
ſcilicet duæ ſectiones A B, &amp; </s>
  <s xml:id="echoid-s6350" xml:space="preserve">E F ſint ſimiles deducit earum figuras ſimiles eſſe. <lb/></s>
  <s xml:id="echoid-s6351" xml:space="preserve">Ait enim: </s>
  <s xml:id="echoid-s6352" xml:space="preserve">quia eſt A C ad C B vt E H ad H F, &amp; </s>
  <s xml:id="echoid-s6353" xml:space="preserve">eandem proportioné <lb/>habent earum quadrata, atque quadratum H F ad rectangulum: </s>
  <s xml:id="echoid-s6354" xml:space="preserve">F H b <lb/>eandem proportionem habet quàm quadratum C B ad rectangulũ B C a <lb/>( c<unsure/>o quod H F ad F b poſita fuit vt C B ad B a) ergo, &amp;</s>
  <s xml:id="echoid-s6355" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6356" xml:space="preserve">Modo ſi ac-<lb/>curatè hæc verba perpendantur non poterit hic vſurpari vulgata definitio Euto-<lb/>cĳ, vel Mydorgĳ nam cum ſectiones A B, E F ſupponantur ſimiles, ea tan-<lb/>tummodo quæ in definitione ſimilium ſectionum perhibentur concedi poßunt, &amp; </s>
  <s xml:id="echoid-s6357" xml:space="preserve"><lb/>nihil amplius; </s>
  <s xml:id="echoid-s6358" xml:space="preserve">igitur ſi in definitione non includitur particula illa [ abſciſſæ H <lb/>F, C B’ ad erecta, vel tranſuerſa latera F b, B a ſint proportionalia ] deliran-
<pb o="164" file="0202" n="202" rhead="Apollonij Pergæi"/>
tis potius, quàm demonſtrantis <lb/>
<anchor type="figure" xlink:label="fig-0202-01a" xlink:href="fig-0202-01"/>
eſſet dicere. </s>
  <s xml:id="echoid-s6359" xml:space="preserve">Eo quod H F, ad <lb/>F b poſita fuit vt C B ad B a; <lb/></s>
  <s xml:id="echoid-s6360" xml:space="preserve">vbi nam, aut quando hoc ſuppo-<lb/>ſitum eſt, ſi in definitione non <lb/>continetur? </s>
  <s xml:id="echoid-s6361" xml:space="preserve">Nec ſuspicari po-<lb/>teſt caſu hæc verba in textu ir-<lb/>repſiß, cum in alĳs locis repe-<lb/>tantur, &amp; </s>
  <s xml:id="echoid-s6362" xml:space="preserve">ab eis pendeat tota <lb/>demonſtratio; </s>
  <s xml:id="echoid-s6363" xml:space="preserve">igitur in defini-<lb/>tione vulgata addenda eſt illa <lb/>particula, abſciſſæ fint in ea-<lb/>dem ratione ad erecta;</s>
  <s xml:id="echoid-s6364" xml:space="preserve"/>
</p>
<div xml:id="echoid-div605" type="float" level="2" n="4">
  <figure xlink:label="fig-0201-01" xlink:href="fig-0201-01a">
    <image file="0201-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0201-01"/>
  </figure>
  <figure xlink:label="fig-0202-01" xlink:href="fig-0202-01a">
    <image file="0202-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0202-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s6365" xml:space="preserve">Rurſus in propoſ. </s>
  <s xml:id="echoid-s6366" xml:space="preserve">II. </s>
  <s xml:id="echoid-s6367" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6368" xml:space="preserve">I. <lb/></s>
  <s xml:id="echoid-s6369" xml:space="preserve">parte 12. </s>
  <s xml:id="echoid-s6370" xml:space="preserve">quando concluſio demonſtrationis eſt quod ſectiones A B, E F ſimi-<lb/>les ſint: </s>
  <s xml:id="echoid-s6371" xml:space="preserve">tunc quidem quia tenetur oſtendere Apollonius definitionem traditam, <lb/>conuenire ſectionibus A B, E F, non aßumit incautè abſciſſas homologas C B, <lb/>H F, ſed ait in II. </s>
  <s xml:id="echoid-s6372" xml:space="preserve">propoſitionc ponamus C B ad B D vt H F ad F I, &amp; </s>
  <s xml:id="echoid-s6373" xml:space="preserve"><lb/>in 12. </s>
  <s xml:id="echoid-s6374" xml:space="preserve">inquit, nam pofuimus H F ad F b vt C B ad B a &amp;</s>
  <s xml:id="echoid-s6375" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6376" xml:space="preserve">Poſtea in pro-<lb/>poſitione 16. </s>
  <s xml:id="echoid-s6377" xml:space="preserve">litera a: </s>
  <s xml:id="echoid-s6378" xml:space="preserve">ergo M A ad A P, ideſt abſciſſa ad erectum eſt vt O <lb/>C ad C Q, ſeu vt homologa abſcißa ad latus rectum, &amp; </s>
  <s xml:id="echoid-s6379" xml:space="preserve">angulus O æqualis <lb/>eſt M: </s>
  <s xml:id="echoid-s6380" xml:space="preserve">patet igitur, vt diximus in II. </s>
  <s xml:id="echoid-s6381" xml:space="preserve">ex 6. </s>
  <s xml:id="echoid-s6382" xml:space="preserve">quod ſi, &amp;</s>
  <s xml:id="echoid-s6383" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6384" xml:space="preserve">Ex quibus locis <lb/>ſatis apertè colligitur ( ni fallor ) id quod ſupra rationibus non leuibus inſi-<lb/>nuaui, quod abſciſſæ proportionales eſſe debent erectis in ſectionibus ſimilibus.</s>
  <s xml:id="echoid-s6385" xml:space="preserve"/>
</p>
  <figure>
    <image file="0202-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0202-02"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s6386" xml:space="preserve">Sed hic animaduertendum eſt, eandem definitionem non poſſe æquè aptari ſe-<lb/>ctionibus conicis, atque ſegmentis conicis ſimilibus, vt perperam cenſuit Mydor-<lb/>gius: </s>
  <s xml:id="echoid-s6387" xml:space="preserve">nam in ſegmentis conicis ſimilibus A B C, &amp; </s>
  <s xml:id="echoid-s6388" xml:space="preserve">D E F diametrorum æquè <lb/>ad baſes inclinatarum abſciſſæ homologæ ex ſui natura determinatæ ſunt, quan-<lb/>doquidem non poßunt eße maiores, neque minores quàm G B, &amp; </s>
  <s xml:id="echoid-s6389" xml:space="preserve">H E, quæ inter <lb/>baſes A C, &amp; </s>
  <s xml:id="echoid-s6390" xml:space="preserve">D F ſegmentorum conicorum, &amp; </s>
  <s xml:id="echoid-s6391" xml:space="preserve">vertices B, E intercipiuntur; <lb/></s>
  <s xml:id="echoid-s6392" xml:space="preserve">at ſi in conicis ſectionibus A B S, &amp; </s>
  <s xml:id="echoid-s6393" xml:space="preserve">K F G ſint axes tranſuerſis a B, &amp; </s>
  <s xml:id="echoid-s6394" xml:space="preserve">b F <lb/>
<anchor type="note" xlink:label="note-0202-01a" xlink:href="note-0202-01"/>
ad ſua latera recta B D, &amp; </s>
  <s xml:id="echoid-s6395" xml:space="preserve">F I in eadem proportione, tunc quidem ſimiles e-<lb/>runt curuæ lineæ A B S, &amp; </s>
  <s xml:id="echoid-s6396" xml:space="preserve">K F G, quæ poßunt habere indeterminatas, &amp; </s>
  <s xml:id="echoid-s6397" xml:space="preserve">mul-<lb/>tiplices longitudines, immo poßunt in inſinitum prolongari, ſi fuerint parabolæ
<pb o="165" file="0203" n="203" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0203-01a" xlink:href="fig-0203-01"/>
vel hyperbolæ, nec habent baſes, à quibus circumſcribantur, igitur in ſectionibus <lb/>ſimilibus A B, &amp; </s>
  <s xml:id="echoid-s6398" xml:space="preserve">G F homolegæ axium abſciſſæ B C, F H non ſupponuntur iam <lb/>dißectæ, &amp; </s>
  <s xml:id="echoid-s6399" xml:space="preserve">determinatæ; </s>
  <s xml:id="echoid-s6400" xml:space="preserve">quare poßunt eße cuiuſcunque menſuræ, &amp; </s>
  <s xml:id="echoid-s6401" xml:space="preserve">habere poſ-<lb/>ſunt eandem, &amp; </s>
  <s xml:id="echoid-s6402" xml:space="preserve">non eandem proportionem ad conterminas potentiales; </s>
  <s xml:id="echoid-s6403" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6404" xml:space="preserve">ideo <lb/>ad vitandam incertitudinem adiungi debet determinatio, quod prædictæ homo-<lb/>logæ abſcißæ B C, F H proportionales ſint lateribus rectis B D, F I, at in ſeg-<lb/>mentis, ſeu portionibus ſectionum conicarum ſimilium inutilis omnino eßet illa <lb/>determinatio. </s>
  <s xml:id="echoid-s6405" xml:space="preserve">An verò hæc mea ſententia omninò reĳci debeat alĳs iudicandũ <lb/>relinquo.</s>
  <s xml:id="echoid-s6406" xml:space="preserve"/>
</p>
<div xml:id="echoid-div606" type="float" level="2" n="5">
<note position="left" xlink:label="note-0202-01" xlink:href="note-0202-01a" xml:space="preserve">Propof. <lb/>12. huius <lb/>lib. I.</note>
  <figure xlink:label="fig-0203-01" xlink:href="fig-0203-01a">
    <image file="0203-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0203-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div608" type="section" level="1" n="203">
<head xml:id="echoid-head257" xml:space="preserve">Notæ in Propoſit. XI.</head>
<p style="it">
  <s xml:id="echoid-s6407" xml:space="preserve">CVmque B C ad B L poſita ſit vt H F ad F N, &amp;</s>
  <s xml:id="echoid-s6408" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6409" xml:space="preserve">Quia inuertendo <lb/>
<anchor type="note" xlink:label="note-0203-01a" xlink:href="note-0203-01"/>
D B ad B C eandem proportionem habet quàm I F ad F H, &amp; </s>
  <s xml:id="echoid-s6410" xml:space="preserve">C B ad B <lb/>L eſt vt H F ad F N; </s>
  <s xml:id="echoid-s6411" xml:space="preserve">ergo ex æquali ordinata D B ad B L eandem proportio-<lb/>nem habebit, quàm I F ad F N; </s>
  <s xml:id="echoid-s6412" xml:space="preserve">eſtque ordinatim applicata Q L media pro. <lb/></s>
  <s xml:id="echoid-s6413" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0203-02a" xlink:href="fig-0203-02"/>
portionatis inter abſciſſam B L, &amp; </s>
  <s xml:id="echoid-s6414" xml:space="preserve">latus rectum B D ( cum in parabola quadra-<lb/>tum Q L æquale ſit rectangulo L B D ) pariterque X N media proportionalis eſt <lb/>
<anchor type="note" xlink:label="note-0203-02a" xlink:href="note-0203-02"/>
inter F N, &amp; </s>
  <s xml:id="echoid-s6415" xml:space="preserve">I F; </s>
  <s xml:id="echoid-s6416" xml:space="preserve">ergo Q L ad L B eſt vt X N ad N F, &amp; </s>
  <s xml:id="echoid-s6417" xml:space="preserve">antecedentium, <lb/>duplæ, ſcilicet Q R ad L B, atque X r<unsure/> ad N F in eadem ratione erunt. </s>
  <s xml:id="echoid-s6418" xml:space="preserve">Non <lb/>ſecus oſtendetur O P ad K B vt T V ad M F.</s>
  <s xml:id="echoid-s6419" xml:space="preserve"/>
</p>
<div xml:id="echoid-div608" type="float" level="2" n="1">
<note position="left" xlink:label="note-0203-01" xlink:href="note-0203-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0203-02" xlink:href="fig-0203-02a">
    <image file="0203-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0203-02"/>
  </figure>
<note position="right" xlink:label="note-0203-02" xlink:href="note-0203-02a" xml:space="preserve">11. lib. I.</note>
</div>
<pb o="166" file="0204" n="204" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div610" type="section" level="1" n="204">
<head xml:id="echoid-head258" xml:space="preserve">Notæ in Propoſit. XII.</head>
<p style="it">
  <s xml:id="echoid-s6420" xml:space="preserve">SVpponamus itaque ſectiones A B, E F, earum inclinati, vel tran-<lb/>
<anchor type="note" xlink:label="note-0204-01a" xlink:href="note-0204-01"/>
ſuerſi B a, F b, &amp; </s>
  <s xml:id="echoid-s6421" xml:space="preserve">erecti eorum B D, F I ordinationes, &amp; </s>
  <s xml:id="echoid-s6422" xml:space="preserve">propoſitio-<lb/>nes, vti diximus, &amp;</s>
  <s xml:id="echoid-s6423" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6424" xml:space="preserve">Ideſt. </s>
  <s xml:id="echoid-s6425" xml:space="preserve">Sint axes inclinati, ſiue tranſuerſi B a, F b, &amp; </s>
  <s xml:id="echoid-s6426" xml:space="preserve"><lb/>maneant ſigna, ordinationes, &amp; </s>
  <s xml:id="echoid-s6427" xml:space="preserve">proportiones eædem, quæ in præcedenti propoſi-<lb/>tione; </s>
  <s xml:id="echoid-s6428" xml:space="preserve">ſcilicet fiat C B ad B D, vt H F ad F I, &amp; </s>
  <s xml:id="echoid-s6429" xml:space="preserve">quia D B ad B a eſt vt I <lb/>F ad F b ( propter ſimilitudinem figurarum D B a, I F b ) ergo ex æquali C <lb/>B ad B a erit vt H F ad F b; </s>
  <s xml:id="echoid-s6430" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6431" xml:space="preserve">comparando antecedentes ad ſummas termino-<lb/>rum in hyperbola, &amp; </s>
  <s xml:id="echoid-s6432" xml:space="preserve">ad differentias in ellipſi erit B C ad C a vt F H ad H b: <lb/></s>
  <s xml:id="echoid-s6433" xml:space="preserve">poſtea diuidantur tam B C, quàm F H in ĳſdem rationibus in punctis K, L, <lb/>M, N, &amp; </s>
  <s xml:id="echoid-s6434" xml:space="preserve">educantur ordinatim applicatæ, ſeu æquidiſtantes baſibus O P, Q R, <lb/>A S, T V, X r<unsure/>, E Z.</s>
  <s xml:id="echoid-s6435" xml:space="preserve"/>
</p>
<div xml:id="echoid-div610" type="float" level="2" n="1">
<note position="right" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s6436" xml:space="preserve">Quoniam figura ſectionis A B ſimilis eſt figuræ ſectionis E F erit qua-<lb/>
<anchor type="note" xlink:label="note-0204-02a" xlink:href="note-0204-02"/>
dratum H E ad H b in H F, vt quadratum A C ad C a in C B, &amp; </s>
  <s xml:id="echoid-s6437" xml:space="preserve">b H <lb/>in H F ad quadratum H F, vt C a in C B ad quadratnm C B ( nam po-<lb/>ſuimus H F ad F b, vt C B ad B a, &amp;</s>
  <s xml:id="echoid-s6438" xml:space="preserve">c.) </s>
  <s xml:id="echoid-s6439" xml:space="preserve">Quouiam in figuris, ſeu rectan-<lb/>gulis ſimilibus D B a, &amp; </s>
  <s xml:id="echoid-s6440" xml:space="preserve">I F b habet D B ad B a eandem proportionem, quàm <lb/>
<anchor type="note" xlink:label="note-0204-03a" xlink:href="note-0204-03"/>
I F ad F b, &amp; </s>
  <s xml:id="echoid-s6441" xml:space="preserve">vt D B ad B a, ita eſt quadratum A C ad rectangulum B C a, <lb/>pariterque vt I F ad F b ita eſt quadratum E H ad rectangulũ F H b ſed ( ſi-<lb/>cut in præcedenti nota dictum eſt) C a ad C B, ſeu rectangulum B C a ad qua-<lb/>dratum C B eandem proportionem habet, quàm H b ad H F, ſeu quàm rectan-<lb/>gulum F H b ad quadratum F H; </s>
  <s xml:id="echoid-s6442" xml:space="preserve">igitur ex æqualitate quadratum A C ad qua-<lb/>dratum C B eandem proportionem habet, quàm quadratum E H ad quadratum <lb/>H F.</s>
  <s xml:id="echoid-s6443" xml:space="preserve"/>
</p>
<div xml:id="echoid-div611" type="float" level="2" n="2">
<note position="right" xlink:label="note-0204-02" xlink:href="note-0204-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0204-03" xlink:href="note-0204-03a" xml:space="preserve">21. lib. I.</note>
</div>
<p style="it">
  <s xml:id="echoid-s6444" xml:space="preserve">Atque quadratum H F ad H F in H b eſt vt quadratum C B ad B C in <lb/>
<anchor type="note" xlink:label="note-0204-04a" xlink:href="note-0204-04"/>
C a (eo quod H F ad F b poſita fuit C B ad B a), ergo ex æqualitate, &amp;</s>
  <s xml:id="echoid-s6445" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s6446" xml:space="preserve">Ideſt ſumã tur axium abſcißæ C B, H F, quæ ſint proportionales lateribus rectis <lb/>B D, &amp; </s>
  <s xml:id="echoid-s6447" xml:space="preserve">F I, ſeu proportionales ſint lateribus tranſuerſis B a, &amp; </s>
  <s xml:id="echoid-s6448" xml:space="preserve">F b, &amp; </s>
  <s xml:id="echoid-s6449" xml:space="preserve">ſecẽtur <lb/>abſciſſæ B C, &amp; </s>
  <s xml:id="echoid-s6450" xml:space="preserve">F H proportionaliter in punctis K, L, M, N, &amp; </s>
  <s xml:id="echoid-s6451" xml:space="preserve">per puncta <lb/>diuiſionum ducantur ordinatim applicatæ A C, Q L, E H, X N, &amp; </s>
  <s xml:id="echoid-s6452" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6453" xml:space="preserve">Quia ſe-<lb/>ctiones A B, E F ſupponuntur ſimiles; </s>
  <s xml:id="echoid-s6454" xml:space="preserve">ergo ex definitione 2. </s>
  <s xml:id="echoid-s6455" xml:space="preserve">huius A C ad C B <lb/>eandem proportionem habebit, quàm E H ad H F, nec non Q L ad L B erit vt <lb/>X N ad N F; </s>
  <s xml:id="echoid-s6456" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6457" xml:space="preserve">ideo quadratum A C ad quadratum C B eandem proportionẽ <lb/>habet, quàm quadratum E H ad quadratum H F; </s>
  <s xml:id="echoid-s6458" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6459" xml:space="preserve">quia ex conſtructione, <lb/>iuxta leges definitionis 2. </s>
  <s xml:id="echoid-s6460" xml:space="preserve">vt C B ad B a ita erat H F ad F b, &amp; </s>
  <s xml:id="echoid-s6461" xml:space="preserve">comparando <lb/>antecedentes ad terminorũ ſummas in hyperbolis, &amp; </s>
  <s xml:id="echoid-s6462" xml:space="preserve">ad differentias in ellipſibus, <lb/>habebit B C ad C a, ſeu quadratum B C ad rectangulum B C a eandẽ propor-<lb/>tionem quàm F H habet ad H b, ſeu quàm quadratum F H habet ad rectangu-<lb/>lum F H b; </s>
  <s xml:id="echoid-s6463" xml:space="preserve">ergo ex æqualitate quadratum A C ad rectangulum B C a eãdem <lb/>proportionem habet, quàm quadratum E H ad rectangulum F H b; </s>
  <s xml:id="echoid-s6464" xml:space="preserve">eſt verò la-<lb/>tus rectum D B ad latus tranſuerſum B a, vt quadratum A C ad rectangulum
<pb o="167" file="0205" n="205" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0205-01a" xlink:href="fig-0205-01"/>
B C a, pariterque latus re <lb/>
<anchor type="figure" xlink:label="fig-0205-02a" xlink:href="fig-0205-02"/>
ctum I F ad tranſuer ſum F <lb/>
<anchor type="note" xlink:label="note-0205-01a" xlink:href="note-0205-01"/>
b eſt vt quadratum E H ad <lb/>rectangulum F H b, igitur <lb/>D B ad B a eandem propor-<lb/>tionem habebit quàm I F ad <lb/>F b, &amp; </s>
  <s xml:id="echoid-s6465" xml:space="preserve">ideo figuræ axium <lb/>ſimiles erunt.</s>
  <s xml:id="echoid-s6466" xml:space="preserve"/>
</p>
<div xml:id="echoid-div612" type="float" level="2" n="3">
<note position="right" xlink:label="note-0204-04" xlink:href="note-0204-04a" xml:space="preserve">C</note>
  <figure xlink:label="fig-0205-01" xlink:href="fig-0205-01a">
    <image file="0205-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0205-01"/>
  </figure>
  <figure xlink:label="fig-0205-02" xlink:href="fig-0205-02a">
    <image file="0205-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0205-02"/>
  </figure>
<note position="right" xlink:label="note-0205-01" xlink:href="note-0205-01a" xml:space="preserve">21. lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div614" type="section" level="1" n="205">
<head xml:id="echoid-head259" xml:space="preserve">Notæ in Propoſit. XIII.</head>
<p>
  <s xml:id="echoid-s6467" xml:space="preserve">SInt axes earum B C, &amp; </s>
  <s xml:id="echoid-s6468" xml:space="preserve">inclinatus, ſeu tranſuerſus B a, &amp;</s>
  <s xml:id="echoid-s6469" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6470" xml:space="preserve">Addidi <lb/>
<anchor type="note" xlink:label="note-0205-02a" xlink:href="note-0205-02"/>
verba, quæ in expoſitione propoſitionis deficiunt. </s>
  <s xml:id="echoid-s6471" xml:space="preserve">Hyperbole, ſeu ellipſis A <lb/>B ſit axis B C, &amp; </s>
  <s xml:id="echoid-s6472" xml:space="preserve">inclinatus, ſeu tranſuerſus B a, &amp; </s>
  <s xml:id="echoid-s6473" xml:space="preserve">E F ſit parabole, cuius <lb/>axis F H, &amp;</s>
  <s xml:id="echoid-s6474" xml:space="preserve">c.</s>
  <s xml:id="echoid-s6475" xml:space="preserve"/>
</p>
<div xml:id="echoid-div614" type="float" level="2" n="1">
<note position="left" xlink:label="note-0205-02" xlink:href="note-0205-02a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s6476" xml:space="preserve">Alioquin ſit (ſi poſſi-<lb/>
<anchor type="note" xlink:label="note-0205-03a" xlink:href="note-0205-03"/>
<anchor type="figure" xlink:label="fig-0205-03a" xlink:href="fig-0205-03"/>
bile eſt) ſimilis vni ea-<lb/>rum, &amp; </s>
  <s xml:id="echoid-s6477" xml:space="preserve">minima ſimilis <lb/>earum figuræ, quæ non <lb/>ſunt ſimiles ſuis figuris: <lb/></s>
  <s xml:id="echoid-s6478" xml:space="preserve">deinde poſſumus produ-<lb/>cere in ſingulis ſectioni-<lb/>bus potentes, &amp;</s>
  <s xml:id="echoid-s6479" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6480" xml:space="preserve">Non <lb/>nulla verba ex hoc textu <lb/>expunxi vt ſuperuacanea <lb/>eiuſq; </s>
  <s xml:id="echoid-s6481" xml:space="preserve">ſenſus hic eſt. </s>
  <s xml:id="echoid-s6482" xml:space="preserve">Sie-<lb/>nim par abolæ E F ſimilis <lb/>eſt hyperbolæ, aut ellipſi A <lb/>B (ex definitione ſimilium <lb/>
<anchor type="note" xlink:label="note-0205-04a" xlink:href="note-0205-04"/>
figurarum) duci poßunt <lb/>in vnaquaque duarum ſi-<lb/>milium ſectionum ordina-
<pb o="168" file="0206" n="206" rhead="Apollonij Pergæi"/>
natim ad axium applicatæ, numero pares, quæ ad abſciſſas ſint proportionales, <lb/>tum abſcißæ inter ſe: </s>
  <s xml:id="echoid-s6483" xml:space="preserve">V nde ſequitur poſtrema concluſio, quæ in textu habetur, <lb/>quod nimirum rectangulum a L B ad rectangulum a K B eandem proportionem <lb/>habeat, quàm abſciſſa, L B ad abſciſſam K B: </s>
  <s xml:id="echoid-s6484" xml:space="preserve">ſed quotieſcunque duo rectangu-<lb/>la eandem proportionem habent, quàm baſes, illa ſunt æque alta: </s>
  <s xml:id="echoid-s6485" xml:space="preserve">igitur altitu-<lb/>dines a L, &amp; </s>
  <s xml:id="echoid-s6486" xml:space="preserve">a K æquales ſunt inter ſe, pars, &amp; </s>
  <s xml:id="echoid-s6487" xml:space="preserve">totum: </s>
  <s xml:id="echoid-s6488" xml:space="preserve">quod eſt absurdum.</s>
  <s xml:id="echoid-s6489" xml:space="preserve"/>
</p>
<div xml:id="echoid-div615" type="float" level="2" n="2">
<note position="left" xlink:label="note-0205-03" xlink:href="note-0205-03a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0205-03" xlink:href="fig-0205-03a">
    <image file="0205-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0205-03"/>
  </figure>
<note position="right" xlink:label="note-0205-04" xlink:href="note-0205-04a" xml:space="preserve">Defin. 2.</note>
</div>
</div>
<div xml:id="echoid-div617" type="section" level="1" n="206">
<head xml:id="echoid-head260" xml:space="preserve">Notæ in Propoſit. XIV.</head>
<p style="it">
  <s xml:id="echoid-s6490" xml:space="preserve">ALioquin ſequitur, quod quadratum R L ad quadratum K P, &amp;</s>
  <s xml:id="echoid-s6491" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6492" xml:space="preserve">In <lb/>
<anchor type="note" xlink:label="note-0206-01a" xlink:href="note-0206-01"/>
propoſitione deficit expoſitio, quæ talis eſt. </s>
  <s xml:id="echoid-s6493" xml:space="preserve">Sit A B quælibet hyperbolc, <lb/>&amp; </s>
  <s xml:id="echoid-s6494" xml:space="preserve">E F quælibet ellipſis. </s>
  <s xml:id="echoid-s6495" xml:space="preserve">Dico A B ipſi E <lb/>
<anchor type="figure" xlink:label="fig-0206-01a" xlink:href="fig-0206-01"/>
F ſimilem non eße. </s>
  <s xml:id="echoid-s6496" xml:space="preserve">Sint eorum axes late-<lb/>ra tranſuerſa, &amp; </s>
  <s xml:id="echoid-s6497" xml:space="preserve">recta eadem, quæ in præ-<lb/>cedenti propoſitione poſita ſunt. </s>
  <s xml:id="echoid-s6498" xml:space="preserve">Et ſiqui-<lb/>dem ſectiones A B, &amp; </s>
  <s xml:id="echoid-s6499" xml:space="preserve">E F ſimiles credan-<lb/>tur, neceßario ex definitione ſecunda, duci <lb/>poterunt ad axes ordinatim applicatæ nu-<lb/>mero pares proportionales abſciſſis, tum <lb/>abſciſſæ inter ſe proportionales: </s>
  <s xml:id="echoid-s6500" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6501" xml:space="preserve">vt in <lb/>præcedenti propoſitione oſtenſum eſt, qua-<lb/>dratum R L ad quadratum P K, ſcilicet <lb/>rectangulum a L B ad rectangulum a K B in hyperbola eandem proportionem <lb/>
<anchor type="note" xlink:label="note-0206-02a" xlink:href="note-0206-02"/>
habebit, quàm quadratum γ N ad quadratum V M, ſeu quàm rectangulum b <lb/>
<anchor type="note" xlink:label="note-0206-03a" xlink:href="note-0206-03"/>
N F ad rectangulum b M F in ellipſi, ergo rectangulum a L B ad rectangulum <lb/>a K B eandem proportionem habet, quàm rectangulum b N F ad rectangulum <lb/>b M F: </s>
  <s xml:id="echoid-s6502" xml:space="preserve">ſed eorundem rectangulorum baſes proportionales ſunt, eo quod L B ad <lb/>B K erat vt N F ad F M; </s>
  <s xml:id="echoid-s6503" xml:space="preserve">igitur eorundem altitudines proportionales erunt, <lb/>ſcilicet a L ad a K eandem proportionem habebit, quàm b N ad b M, ſed in <lb/>hyperqola a L maior eſt, quàm a K; </s>
  <s xml:id="echoid-s6504" xml:space="preserve">in ellipſi verò contra b N minor eſt, quã <lb/>b M; </s>
  <s xml:id="echoid-s6505" xml:space="preserve">igitur maior a L ad minorem a K eandem proportionem habebit, quàm <lb/>minor b N ad maiorem b M. </s>
  <s xml:id="echoid-s6506" xml:space="preserve">Luod erat abſurdum.</s>
  <s xml:id="echoid-s6507" xml:space="preserve"/>
</p>
<div xml:id="echoid-div617" type="float" level="2" n="1">
<note position="right" xlink:label="note-0206-01" xlink:href="note-0206-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0206-01" xlink:href="fig-0206-01a">
    <image file="0206-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0206-01"/>
  </figure>
<note position="left" xlink:label="note-0206-02" xlink:href="note-0206-02a" xml:space="preserve">21. lib. 1.</note>
<note position="left" xlink:label="note-0206-03" xlink:href="note-0206-03a" xml:space="preserve">Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div619" type="section" level="1" n="207">
<head xml:id="echoid-head261" xml:space="preserve">SECTIO QVINTA</head>
<head xml:id="echoid-head262" xml:space="preserve">Continens ſex Propoſitiones Præmiſſas, <lb/>PROPOSITIO I. II. III. IV. &amp; V.</head>
<p>
  <s xml:id="echoid-s6508" xml:space="preserve">SI in triangulis A B C, D E F in duobus circulorum ſeg-<lb/>
<anchor type="note" xlink:label="note-0206-04a" xlink:href="note-0206-04"/>
mentis A T C, D G F deſcriptis, à duobus angulis B, <lb/>E, educantur duæ rectæ lineæ B T H, E G I efficientes cum <lb/>baſibus A C, D F duos angulos H, I æquales (incidentes in
<pb o="169" file="0207" n="207" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0207-01a" xlink:href="fig-0207-01"/>
prima figura extra duo ſegmenta, &amp; </s>
  <s xml:id="echoid-s6509" xml:space="preserve">in ſecunda intra, at in ter-<lb/>tia intra duos ſemicirculos), &amp; </s>
  <s xml:id="echoid-s6510" xml:space="preserve">fuerit proportio plani rectan-<lb/>guli ex portionibus lineæ baſis inter angulum prouenientem, &amp; </s>
  <s xml:id="echoid-s6511" xml:space="preserve"><lb/>duos angulos reliquos trianguli, nempe A H in H C ad qua-<lb/>
<anchor type="note" xlink:label="note-0207-01a" xlink:href="note-0207-01"/>
dratum interceptæ inter prouenientem angulum, &amp; </s>
  <s xml:id="echoid-s6512" xml:space="preserve">circuli peri-<lb/>pheriam, nempe ad quadratum H B in quolibet caſu eadem <lb/>ſit, quàm D I in I F ad quadratum I E, vel H A in H C ad <lb/>quadratum H T ſit, vt D I in I F ad quadratum I G; </s>
  <s xml:id="echoid-s6513" xml:space="preserve">ſintque <lb/>duo priores anguli, inter ſe æquales, &amp; </s>
  <s xml:id="echoid-s6514" xml:space="preserve">prouenientes extra duo <lb/>triangula poſiti: </s>
  <s xml:id="echoid-s6515" xml:space="preserve">vel duo priores recti, &amp; </s>
  <s xml:id="echoid-s6516" xml:space="preserve">prouenientes intra <lb/>
<anchor type="note" xlink:label="note-0207-02a" xlink:href="note-0207-02"/>
duos angulos non ſint recti; </s>
  <s xml:id="echoid-s6517" xml:space="preserve">aut duo priores non recti, &amp; </s>
  <s xml:id="echoid-s6518" xml:space="preserve">pro-<lb/>
<anchor type="note" xlink:label="note-0207-03a" xlink:href="note-0207-03"/>
uenientes recti intra duo triangula: </s>
  <s xml:id="echoid-s6519" xml:space="preserve">vel duo priores diuerſæ, <lb/>
<anchor type="note" xlink:label="note-0207-04a" xlink:href="note-0207-04"/>
aut eiuſdem ſpeciei, ſed duæ lineæ efficiant duos angulos æqua-<lb/>les cum lateribus duorum triangulorum ſubtendentibus angulos <lb/>prouenientes: </s>
  <s xml:id="echoid-s6520" xml:space="preserve">vtique duo priora triangula ſunt ſimilia.</s>
  <s xml:id="echoid-s6521" xml:space="preserve"/>
</p>
<div xml:id="echoid-div619" type="float" level="2" n="1">
<note position="right" xlink:label="note-0206-04" xlink:href="note-0206-04a" xml:space="preserve">I</note>
  <figure xlink:label="fig-0207-01" xlink:href="fig-0207-01a">
    <image file="0207-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0207-01"/>
  </figure>
<note position="left" xlink:label="note-0207-01" xlink:href="note-0207-01a" xml:space="preserve">2</note>
<note position="left" xlink:label="note-0207-02" xlink:href="note-0207-02a" xml:space="preserve">3</note>
<note position="left" xlink:label="note-0207-03" xlink:href="note-0207-03a" xml:space="preserve">4</note>
<note position="left" xlink:label="note-0207-04" xlink:href="note-0207-04a" xml:space="preserve">5</note>
</div>
<p>
  <s xml:id="echoid-s6522" xml:space="preserve">Quia C H in H A; </s>
  <s xml:id="echoid-s6523" xml:space="preserve">nempe T H in H B ad quadratum H B, quod eſt, <lb/>vt H T ad H B eandem proportionem habet, quàm D I in I F, nempe <lb/>
<anchor type="figure" xlink:label="fig-0207-02a" xlink:href="fig-0207-02"/>
G I in I E ad quadratum I E, quod eſt vt I G ad I E, erit B H ad H T, <lb/>vt E I ad I G; </s>
  <s xml:id="echoid-s6524" xml:space="preserve">ſimiliter, &amp; </s>
  <s xml:id="echoid-s6525" xml:space="preserve">eorum quadrata; </s>
  <s xml:id="echoid-s6526" xml:space="preserve">oſtendetur igitur ex æqua-
<pb o="170" file="0208" n="208" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0208-01a" xlink:href="fig-0208-01"/>
litate, quod ſi fuerit A H in H C ad quadratum H B, vt D I in I F ad <lb/>quadratum I E, quod A H in H C ad quadratum H T ſit etiam, vt I D <lb/>in I F ad quadratum I G. </s>
  <s xml:id="echoid-s6527" xml:space="preserve">Dico iam, quod triangulum A B C ſimile eſt <lb/>triangulo D E F. </s>
  <s xml:id="echoid-s6528" xml:space="preserve">Si enim hoc verum non eſt, non erit angulus A æqua-<lb/>lis vni duorum angulorum D, vel F: </s>
  <s xml:id="echoid-s6529" xml:space="preserve">ſitque angulus D maior, quàm A, <lb/>&amp; </s>
  <s xml:id="echoid-s6530" xml:space="preserve">fiat angulus K D F æqualis A, iungaturque F K; </s>
  <s xml:id="echoid-s6531" xml:space="preserve">quia angulus K, ve-<lb/>luti E, eſt æqualis angulo B; </s>
  <s xml:id="echoid-s6532" xml:space="preserve">ſimilia erunt triangula A B C, D K F, &amp; </s>
  <s xml:id="echoid-s6533" xml:space="preserve">e-<lb/>ducamus K L parallelam E I: </s>
  <s xml:id="echoid-s6534" xml:space="preserve">quare K L F ſimile quoque erit B H C <lb/>
<anchor type="note" xlink:label="note-0208-01a" xlink:href="note-0208-01"/>
ideoque H A ad H B eſt vt D L ad L K, &amp; </s>
  <s xml:id="echoid-s6535" xml:space="preserve">H C ad H B, vt F L ad L <lb/>K; </s>
  <s xml:id="echoid-s6536" xml:space="preserve">igitur H A in H C, nempe B H in H T ad quadratum H B, quod eſt, <lb/>vt H T ad H B, quæ oſtenſa eſt; </s>
  <s xml:id="echoid-s6537" xml:space="preserve">vt I G ad I E, erit vt D L in L F, nẽ-<lb/>pe K L in L M ad qua-<lb/>
<anchor type="figure" xlink:label="fig-0208-02a" xlink:href="fig-0208-02"/>
dratum K L: </s>
  <s xml:id="echoid-s6538" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6539" xml:space="preserve">propte-<lb/>rea M L ad L K erit vt G <lb/>I ad I E in omnibus fi-<lb/>guris; </s>
  <s xml:id="echoid-s6540" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6541" xml:space="preserve">hoc eſt abſurdũ <lb/>
<anchor type="note" xlink:label="note-0208-02a" xlink:href="note-0208-02"/>
in prima figura: </s>
  <s xml:id="echoid-s6542" xml:space="preserve">in ſecun-<lb/>
<anchor type="note" xlink:label="note-0208-03a" xlink:href="note-0208-03"/>
da verò ſecentur bifariam <lb/>E G, K M in N, O, &amp; </s>
  <s xml:id="echoid-s6543" xml:space="preserve"><lb/>iungatur N O, quæ pa-<lb/>rallela erit L I, quia ſunt <lb/>duæ perpendiculares ſu-<lb/>per K M, E G, quæ ſunt <lb/>parallelæ; </s>
  <s xml:id="echoid-s6544" xml:space="preserve">ergo I N eſt <lb/>æqualis L O, &amp; </s>
  <s xml:id="echoid-s6545" xml:space="preserve">quia E G ad E I iam oſtenſa eſt vt K M ad K L; </s>
  <s xml:id="echoid-s6546" xml:space="preserve">ergo <lb/>E N ad E I eſt, vt O K ad K L: </s>
  <s xml:id="echoid-s6547" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6548" xml:space="preserve">diuidendo erit N I ad I E, vt O L, <lb/>quæ eſt æqualis N I ad L K. </s>
  <s xml:id="echoid-s6549" xml:space="preserve">Et hoc quoque eſt abſurdum.</s>
  <s xml:id="echoid-s6550" xml:space="preserve"/>
</p>
<div xml:id="echoid-div620" type="float" level="2" n="2">
  <figure xlink:label="fig-0207-02" xlink:href="fig-0207-02a">
    <image file="0207-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0207-02"/>
  </figure>
  <figure xlink:label="fig-0208-01" xlink:href="fig-0208-01a">
    <image file="0208-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0208-01"/>
  </figure>
<note position="right" xlink:label="note-0208-01" xlink:href="note-0208-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0208-02" xlink:href="fig-0208-02a">
    <image file="0208-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0208-02"/>
  </figure>
<note position="right" xlink:label="note-0208-02" xlink:href="note-0208-02a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0208-03" xlink:href="note-0208-03a" xml:space="preserve">d</note>
</div>
<p>
  <s xml:id="echoid-s6551" xml:space="preserve">In figura autem tertia educamus duas perpendiculares E P Q, K R S <lb/>
<anchor type="note" xlink:label="note-0208-04a" xlink:href="note-0208-04"/>
ſuper diametrum D F, cui occurrant in P, R: </s>
  <s xml:id="echoid-s6552" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6553" xml:space="preserve">iungamus G Q, M S, <lb/>quia erat G E ad E I, vt M K ad L K, &amp; </s>
  <s xml:id="echoid-s6554" xml:space="preserve">propter ſimilitudinem trian-<lb/>gulorum I E P, K L R, E I ad E P eſt, vt L K ad K R, atque E P ad E <lb/>Q eſt, vt R K ad K S, &amp; </s>
  <s xml:id="echoid-s6555" xml:space="preserve">angulus G E Q æqualis eſt M K S; </s>
  <s xml:id="echoid-s6556" xml:space="preserve">ergo E G
<pb o="171" file="0209" n="209" rhead="Conicor. Lib. VI."/>
Q ſimile eſt M K S, <lb/>
<anchor type="figure" xlink:label="fig-0209-01a" xlink:href="fig-0209-01"/>
quare angulus G æ-<lb/>qualis eſt angulo M, <lb/>&amp; </s>
  <s xml:id="echoid-s6557" xml:space="preserve">propterea periphe-<lb/>riæ E F Q, &amp; </s>
  <s xml:id="echoid-s6558" xml:space="preserve">K F S, <lb/>quibus inſiſtunt, æ-<lb/>quales erunt, quod <lb/>eſt abſurdũ: </s>
  <s xml:id="echoid-s6559" xml:space="preserve">eſt enim <lb/>E F Q maior, quàm <lb/>K F S; </s>
  <s xml:id="echoid-s6560" xml:space="preserve">ergo duo triã-<lb/>gula A B C, D E F <lb/>in omnibus figuris <lb/>ſunt ſimilia. </s>
  <s xml:id="echoid-s6561" xml:space="preserve">Quod e-<lb/>rat oſtendendum.</s>
  <s xml:id="echoid-s6562" xml:space="preserve"/>
</p>
<div xml:id="echoid-div621" type="float" level="2" n="3">
<note position="right" xlink:label="note-0208-04" xlink:href="note-0208-04a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0209-01" xlink:href="fig-0209-01a">
    <image file="0209-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0209-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div623" type="section" level="1" n="208">
<head xml:id="echoid-head263" xml:space="preserve">PROPOSITIO <lb/>Præmiſſa VI.</head>
<p>
  <s xml:id="echoid-s6563" xml:space="preserve">DEinde ſint duo anguli B, E qualeſcunque; </s>
  <s xml:id="echoid-s6564" xml:space="preserve">ſed angulus <lb/>
<anchor type="note" xlink:label="note-0209-01a" xlink:href="note-0209-01"/>
A B H, vel C B H æqualis angulo D E I, aut F E I: <lb/></s>
  <s xml:id="echoid-s6565" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6566" xml:space="preserve">ſupponantur reliqua omnia iam dicta.</s>
  <s xml:id="echoid-s6567" xml:space="preserve"/>
</p>
<div xml:id="echoid-div623" type="float" level="2" n="1">
<note position="left" xlink:label="note-0209-01" xlink:href="note-0209-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s6568" xml:space="preserve">Quia proportio C H in H A ad quadratum H B ſuppoſita eſt, vt F I <lb/>in I D ad quadratum I E, &amp; </s>
  <s xml:id="echoid-s6569" xml:space="preserve">H C, vel H A ad H B eſt, vt F I, vel D I <lb/>ad I E; </s>
  <s xml:id="echoid-s6570" xml:space="preserve">erit etiam H A ad H B, vt I D ad I E, &amp; </s>
  <s xml:id="echoid-s6571" xml:space="preserve">duo anguli H, I ſunt <lb/>æquales; </s>
  <s xml:id="echoid-s6572" xml:space="preserve">igitur triangulum H B A, aut H B C ſimile eſt triangulo E D <lb/>I, aut E F I, quare duo triangula A B C, D E F ſimilia ſunt; </s>
  <s xml:id="echoid-s6573" xml:space="preserve">Et hoc <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s6574" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div625" type="section" level="1" n="209">
<head xml:id="echoid-head264" xml:space="preserve">Notæ in Propoſit. Præmiſſas <lb/>I. II. III. IV. &amp; V.</head>
<p style="it">
  <s xml:id="echoid-s6575" xml:space="preserve">AFferuntur in hac ſectione aliquæ propoſitiones ſimul coaceruatæ, quæ lem-<lb/>maticæ ſunt, &amp; </s>
  <s xml:id="echoid-s6576" xml:space="preserve">vſum habent in ſequentibus propoſitionibus; </s>
  <s xml:id="echoid-s6577" xml:space="preserve">ſanè conij-<lb/>citur ex hoc titulo PRAEMISS AE rubeis characteribus inſcripto, huiuſmodi lẽ-<lb/>mata T extui Apollonij ab Arabico Interprete, vel ab aliquo alio ſuperaddita fuiſſe; <lb/></s>
  <s xml:id="echoid-s6578" xml:space="preserve">licet Pappus Alexandrinus libro 7. </s>
  <s xml:id="echoid-s6579" xml:space="preserve">afferat eadem ferè lemmata, tanquã propria, <lb/>&amp; </s>
  <s xml:id="echoid-s6580" xml:space="preserve">conferentia ad Apollonij ſexti libri intelligentiam.</s>
  <s xml:id="echoid-s6581" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6582" xml:space="preserve">Poteſt tamen propoſitio vniuerſalis breuius exponi hac ratione. </s>
  <s xml:id="echoid-s6583" xml:space="preserve">Si à vertici-<lb/>bus duorum triangulorum à duobus circulis compræhenſorum rectæ lineæ ductæ <lb/>efficiant cum baſibus angulos æquales; </s>
  <s xml:id="echoid-s6584" xml:space="preserve">atque eorundem ſegmentorum inter baſim, <lb/>&amp; </s>
  <s xml:id="echoid-s6585" xml:space="preserve">peripheriam interceptorum quadrata ad rectangula ſub factis ſegmentis ba-
<pb o="172" file="0210" n="210" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0210-01a" xlink:href="fig-0210-01"/>
ſium eandem proportionem habeant, fuerintque anguli verticales inter ſe æquales, <lb/>vel qui à lateribus, &amp; </s>
  <s xml:id="echoid-s6586" xml:space="preserve">à vertice ductis continentur, ſint æquales: </s>
  <s xml:id="echoid-s6587" xml:space="preserve">ſemper trian-<lb/>gula erunt ſimilia.</s>
  <s xml:id="echoid-s6588" xml:space="preserve"/>
</p>
<div xml:id="echoid-div625" type="float" level="2" n="1">
  <figure xlink:label="fig-0210-01" xlink:href="fig-0210-01a">
    <image file="0210-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0210-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s6589" xml:space="preserve">Dico iam, quod triangulum A B C ſimile eſt triangulo D E F, ſi enim <lb/>
<anchor type="note" xlink:label="note-0210-01a" xlink:href="note-0210-01"/>
hoc verum non eſt, ſit angulus D maior, quàm angulus A, &amp;</s>
  <s xml:id="echoid-s6590" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6591" xml:space="preserve">Textus <lb/>alterari debuit, nam duo triangula B A C, &amp; </s>
  <s xml:id="echoid-s6592" xml:space="preserve">E D F ponuntur non ſimilia, &amp; </s>
  <s xml:id="echoid-s6593" xml:space="preserve"><lb/>propterea æquiangula non erunt, ſcilicet non habebunt duos angulos æquales duo-<lb/>bus angulis alterius trianguli; </s>
  <s xml:id="echoid-s6594" xml:space="preserve">ſed ex hypotheſi anguli verticales A B C, &amp; </s>
  <s xml:id="echoid-s6595" xml:space="preserve">D E <lb/>F æquales erant; </s>
  <s xml:id="echoid-s6596" xml:space="preserve">ergo angulus B A C non erit æqualis angulo E D F, neque <lb/>angulo E F D; </s>
  <s xml:id="echoid-s6597" xml:space="preserve">alias dicta triangula eßent æquiangula, &amp; </s>
  <s xml:id="echoid-s6598" xml:space="preserve">ſimilia, quod non <lb/>ponitur; </s>
  <s xml:id="echoid-s6599" xml:space="preserve">igitur neceſſe eſt, vt angulus A non ſit æqualis vni duorum angulorum <lb/>D, vel F, poſtea rectangulorum A H C, &amp; </s>
  <s xml:id="echoid-s6600" xml:space="preserve">D I F tam latus A H ipſius H C <lb/>non ſit maius, quàm D I ipſius I F, &amp; </s>
  <s xml:id="echoid-s6601" xml:space="preserve">ad punctũ D fiat angulus F D K æqua-<lb/>lis angulo A.</s>
  <s xml:id="echoid-s6602" xml:space="preserve"/>
</p>
<div xml:id="echoid-div626" type="float" level="2" n="2">
<note position="right" xlink:label="note-0210-01" xlink:href="note-0210-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s6603" xml:space="preserve">Quare K L F ſimile quoq; </s>
  <s xml:id="echoid-s6604" xml:space="preserve">erit B H C, &amp;</s>
  <s xml:id="echoid-s6605" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6606" xml:space="preserve">Luoniã angulus F D K æqualis <lb/>
<anchor type="note" xlink:label="note-0210-02a" xlink:href="note-0210-02"/>
eſt factus angulo C A B, &amp; </s>
  <s xml:id="echoid-s6607" xml:space="preserve">angulus F K D ſeu ei æqualis F E. </s>
  <s xml:id="echoid-s6608" xml:space="preserve">D eſt ipſi angu-<lb/>lo A B C æqualis (cum in ſimilibus circulorum ſegmentis exiſtant), igitur in <lb/>triangulis F K D, &amp; </s>
  <s xml:id="echoid-s6609" xml:space="preserve">C B A tertius angulus K F D æqualis erit tertio angulo <lb/>C; </s>
  <s xml:id="echoid-s6610" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6611" xml:space="preserve">propter parallelas K L, E I eſt angulus D L K æqualis angulo D I E; </s>
  <s xml:id="echoid-s6612" xml:space="preserve">eſt <lb/>verò angulus A H B ex hypotheſi æqualis eidem angulo D I E; </s>
  <s xml:id="echoid-s6613" xml:space="preserve">ergò angulus D <lb/>L K æqualis eſt angulo A H B, &amp; </s>
  <s xml:id="echoid-s6614" xml:space="preserve">F L K æqualis angulo C H B: </s>
  <s xml:id="echoid-s6615" xml:space="preserve">at oſtenſus fuit <lb/>angulus K F L æqualis angulo B C H; </s>
  <s xml:id="echoid-s6616" xml:space="preserve">ergo angulo C B H æqualis eſt angulus <lb/>F K L; </s>
  <s xml:id="echoid-s6617" xml:space="preserve">ideoque triangula C B H, &amp; </s>
  <s xml:id="echoid-s6618" xml:space="preserve">F K L ſimilia erunt. </s>
  <s xml:id="echoid-s6619" xml:space="preserve">Pariterq; </s>
  <s xml:id="echoid-s6620" xml:space="preserve">duo trian-<lb/>gula B A H, &amp; </s>
  <s xml:id="echoid-s6621" xml:space="preserve">K D L ſimilia erunt, cum angulus L æqualis ſit angulo H, &amp; </s>
  <s xml:id="echoid-s6622" xml:space="preserve"><lb/>angulus K D L æqualis ſit interno B A H.</s>
  <s xml:id="echoid-s6623" xml:space="preserve"/>
</p>
<div xml:id="echoid-div627" type="float" level="2" n="3">
<note position="right" xlink:label="note-0210-02" xlink:href="note-0210-02a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s6624" xml:space="preserve">Et hoc eſt abſurdum in prima figura, &amp;</s>
  <s xml:id="echoid-s6625" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6626" xml:space="preserve">Luoniam ſunt rectæ lineæ in <lb/>
<anchor type="note" xlink:label="note-0210-03a" xlink:href="note-0210-03"/>
circulo applicatæ K M, E G parallelæ inter ſe; </s>
  <s xml:id="echoid-s6627" xml:space="preserve">ergo coniunctæ rectæ lineæ E K, <lb/>G M parallelæ erunt inter ſe, aut conuenient extra circulum cum diametro bifa-<lb/>riam, &amp; </s>
  <s xml:id="echoid-s6628" xml:space="preserve">ad angulos rectos diuidente applicatas E G, K M; </s>
  <s xml:id="echoid-s6629" xml:space="preserve">ſed eadem rectæ lineæ <lb/>G M ſecat trianguli baſim F A I intra circulũ, aut extra ipſum inter puncta I, A, &amp; </s>
  <s xml:id="echoid-s6630" xml:space="preserve"><lb/>F (propterea quod angulus E I F conſtituitur à duabus in circulo applicatis extra <lb/>ipſum concurrentibus); </s>
  <s xml:id="echoid-s6631" xml:space="preserve">ergo tres coniunctæ rectæ lineæ K E, M G, &amp; </s>
  <s xml:id="echoid-s6632" xml:space="preserve">I L, nec ſunt <lb/>omnes inter ſe parallelæ, nec in vno puncto cõueniunt, &amp; </s>
  <s xml:id="echoid-s6633" xml:space="preserve">propterea E I, &amp; </s>
  <s xml:id="echoid-s6634" xml:space="preserve">K L,
<pb o="173" file="0211" n="211" rhead="Conicor. Lib. VI."/>
ſectæ non erunt proportionaliter in punctis G, [<unsure/>&amp; </s>
  <s xml:id="echoid-s6635" xml:space="preserve">M, ſed prius oſtenſa fuit E <lb/>I ad I G vt K L ad L M; </s>
  <s xml:id="echoid-s6636" xml:space="preserve">quod eſt abſurdum.</s>
  <s xml:id="echoid-s6637" xml:space="preserve"/>
</p>
<div xml:id="echoid-div628" type="float" level="2" n="4">
<note position="right" xlink:label="note-0210-03" xlink:href="note-0210-03a" xml:space="preserve">c</note>
</div>
  <figure>
    <image file="0211-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0211-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s6638" xml:space="preserve">In ſecunda verò ſecentur bifariam E G, K M in N O, &amp;</s>
  <s xml:id="echoid-s6639" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6640" xml:space="preserve">Sunt enim <lb/>
<anchor type="note" xlink:label="note-0211-01a" xlink:href="note-0211-01"/>
in tertio caſu K M, &amp; </s>
  <s xml:id="echoid-s6641" xml:space="preserve">E G perpendiculares ad baſim D F; </s>
  <s xml:id="echoid-s6642" xml:space="preserve">igitur ſi ſecentur <lb/>bifariam in O, &amp; </s>
  <s xml:id="echoid-s6643" xml:space="preserve">N coniuncta recta linea N O diameter circuli erit, quando-<lb/>quidem diuidit bifariam duas equidiſtantes in circulo applicatas; </s>
  <s xml:id="echoid-s6644" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6645" xml:space="preserve">ideo eas <lb/>ſecat ad angulos rectos, ſicuti D F eaſdem perpendiculariter ſecabat; </s>
  <s xml:id="echoid-s6646" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6647" xml:space="preserve">propte-<lb/>rea I N O L parallelogram-<lb/>mum erit, cuius latera op-<lb/>
<anchor type="figure" xlink:label="fig-0211-02a" xlink:href="fig-0211-02"/>
poſita N I, &amp; </s>
  <s xml:id="echoid-s6648" xml:space="preserve">O L æqualia <lb/>crunt. </s>
  <s xml:id="echoid-s6649" xml:space="preserve">Poſtea quia oſtenſa <lb/>fuit I G ad I E, vt L M <lb/>ad L K; </s>
  <s xml:id="echoid-s6650" xml:space="preserve">ergo ſummæ termi-<lb/>
<anchor type="note" xlink:label="note-0211-02a" xlink:href="note-0211-02"/>
norum ad conſequentespro <lb/>portionales erunt; </s>
  <s xml:id="echoid-s6651" xml:space="preserve">ſcilice <lb/>G E ad E I erit vt M K ad <lb/>K L, &amp; </s>
  <s xml:id="echoid-s6652" xml:space="preserve">antecedentiũ ſemiſ-<lb/>ſes N E ad E I, vt O K ad <lb/>K L: </s>
  <s xml:id="echoid-s6653" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6654" xml:space="preserve">diuidendo, duæ æ-<lb/>quates N I, O L eandem <lb/>proportionem habebunt ad I E, &amp; </s>
  <s xml:id="echoid-s6655" xml:space="preserve">L K; </s>
  <s xml:id="echoid-s6656" xml:space="preserve">ideoq; </s>
  <s xml:id="echoid-s6657" xml:space="preserve">I E æqualis eſt L K. </s>
  <s xml:id="echoid-s6658" xml:space="preserve">Et quoniã <lb/>triangulum A B H ſimile eſt triangulo D K L; </s>
  <s xml:id="echoid-s6659" xml:space="preserve">ergo A H ad H B eandem pro-<lb/>portionem habet, quàm D L ad L K; </s>
  <s xml:id="echoid-s6660" xml:space="preserve">eſtque triangulum B H C ſimile triangu-<lb/>lo K L F; </s>
  <s xml:id="echoid-s6661" xml:space="preserve">ergo B H ad H C eſt vt K L ad L F, &amp; </s>
  <s xml:id="echoid-s6662" xml:space="preserve">ex æqualitate vt A H ad H C <lb/>ita eſt D L ad L F; </s>
  <s xml:id="echoid-s6663" xml:space="preserve">erat autem ſegmentum A H non maius ſegmento H C; </s>
  <s xml:id="echoid-s6664" xml:space="preserve">ergo <lb/>D L maius non erit ſegmento L F; </s>
  <s xml:id="echoid-s6665" xml:space="preserve">ſed erat ſegmentum D I non maius ſegmen-<lb/>to I F, igitur duo ſegmenta D I, &amp; </s>
  <s xml:id="echoid-s6666" xml:space="preserve">D L non ſunt maiora, ideſt non ſunt ma-<lb/>iora medietate totius D F, ſed diameter parallela ipſis K M, &amp; </s>
  <s xml:id="echoid-s6667" xml:space="preserve">E G ſecat D F <lb/>biſariam; </s>
  <s xml:id="echoid-s6668" xml:space="preserve">ergo K M, E G ad eaſdem partes diametri cadunt verſus D, &amp; </s>
  <s xml:id="echoid-s6669" xml:space="preserve">ſunt <lb/>inter ſe parallelæ; </s>
  <s xml:id="echoid-s6670" xml:space="preserve">ergo inæqualiter à centro diſtant; </s>
  <s xml:id="echoid-s6671" xml:space="preserve">ideoque inæquales erunt in-<lb/>ter ſe, &amp; </s>
  <s xml:id="echoid-s6672" xml:space="preserve">earum meditates N E, O K inæquales erunt; </s>
  <s xml:id="echoid-s6673" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6674" xml:space="preserve">ablatis æqualibus N
<pb o="174" file="0212" n="212" rhead="Apollonij Pergæi"/>
I, O L remanebunt I E, L K inæquales. </s>
  <s xml:id="echoid-s6675" xml:space="preserve">Quod eſt abſurdum: </s>
  <s xml:id="echoid-s6676" xml:space="preserve">oſtenſæ enim fue-<lb/>runt prius æquales inter ſe.</s>
  <s xml:id="echoid-s6677" xml:space="preserve"/>
</p>
<div xml:id="echoid-div629" type="float" level="2" n="5">
<note position="left" xlink:label="note-0211-01" xlink:href="note-0211-01a" xml:space="preserve">d</note>
  <figure xlink:label="fig-0211-02" xlink:href="fig-0211-02a">
    <image file="0211-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0211-02"/>
  </figure>
<note position="right" xlink:label="note-0211-02" xlink:href="note-0211-02a" xml:space="preserve">Lem. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s6678" xml:space="preserve">In figura autem tertia ducamus duas perpendiculares, &amp;</s>
  <s xml:id="echoid-s6679" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6680" xml:space="preserve">In quarto <lb/>
<anchor type="note" xlink:label="note-0212-01a" xlink:href="note-0212-01"/>
caſu ſupponuntur baſes A C, &amp; </s>
  <s xml:id="echoid-s6681" xml:space="preserve">D F per centra circulorum tranſire, eo quod <lb/>anguli A B C, &amp; </s>
  <s xml:id="echoid-s6682" xml:space="preserve">D E F recti ſupponuntur, atque rectæ lineæ B H, E I non <lb/>ſunt perpendiculares ſuper eaſdem baſes, licet intra circulos efficiant angulos B <lb/>H C, &amp; </s>
  <s xml:id="echoid-s6683" xml:space="preserve">E I F inter ſe æqua-<lb/>les: </s>
  <s xml:id="echoid-s6684" xml:space="preserve">perſecta igitur conſiru-<lb/>
<anchor type="figure" xlink:label="fig-0212-01a" xlink:href="fig-0212-01"/>
ctione, vt prius ad diame-<lb/>trũ D F, ducãtur ex punctis <lb/>E, &amp; </s>
  <s xml:id="echoid-s6685" xml:space="preserve">K perpendiculares E <lb/>Q, K S, quæ diuidẽtur bi-<lb/>fariã, &amp; </s>
  <s xml:id="echoid-s6686" xml:space="preserve">ad angulos rectos <lb/>in P, &amp; </s>
  <s xml:id="echoid-s6687" xml:space="preserve">R. </s>
  <s xml:id="echoid-s6688" xml:space="preserve">Et quoniam <lb/>(vt in præcedenti caſu oſtẽ-<lb/>ſum eſt) G E ad E I ean-<lb/>dem proportionem habet, <lb/>quàm M K ad K L, cum-<lb/>que latera I E, L K ſint <lb/>parallela, pariterque P E, &amp; </s>
  <s xml:id="echoid-s6689" xml:space="preserve">K R æquidiſtent, atque baſes I P, L R in dire-<lb/>ctum poſitæ ſint, erunt triangula I E P, &amp; </s>
  <s xml:id="echoid-s6690" xml:space="preserve">L K R æquiangula, &amp; </s>
  <s xml:id="echoid-s6691" xml:space="preserve">ſimilia: </s>
  <s xml:id="echoid-s6692" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6693" xml:space="preserve"><lb/>propterea I E ad E P erit, vt. </s>
  <s xml:id="echoid-s6694" xml:space="preserve">L K ad K R: </s>
  <s xml:id="echoid-s6695" xml:space="preserve">eſt verò P E ad eius duplam E Q, <lb/>vt R K ad eius duplam K S (cum diameter ſecet eas bifariam, quas perpendi-<lb/>culariter prius ſecabat) ergo, ex æquali ordinata, erit G E ad E Q, vt M K ad <lb/>K S; </s>
  <s xml:id="echoid-s6696" xml:space="preserve">ſuntq; </s>
  <s xml:id="echoid-s6697" xml:space="preserve">anguli verticales G E Q, &amp; </s>
  <s xml:id="echoid-s6698" xml:space="preserve">M K S æquales, propterea quod conti-<lb/>nẽtur à rectis lineis quæ binæ binis ſunt æquidiſtantes; </s>
  <s xml:id="echoid-s6699" xml:space="preserve">ergo triangula G E Q, &amp; </s>
  <s xml:id="echoid-s6700" xml:space="preserve"><lb/>M K S ſimilia ſunt inter ſe: </s>
  <s xml:id="echoid-s6701" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6702" xml:space="preserve">propterea angulus E G Q æqualis erit angulo K M S.</s>
  <s xml:id="echoid-s6703" xml:space="preserve"/>
</p>
<div xml:id="echoid-div630" type="float" level="2" n="6">
<note position="right" xlink:label="note-0212-01" xlink:href="note-0212-01a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0212-01" xlink:href="fig-0212-01a">
    <image file="0212-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0212-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s6704" xml:space="preserve">Et propterea ſegmentum E F Q maius ſimile erit ſegmento K F S mi-<lb/>
<anchor type="note" xlink:label="note-0212-02a" xlink:href="note-0212-02"/>
nori: </s>
  <s xml:id="echoid-s6705" xml:space="preserve">quod eſt abſurdum, &amp;</s>
  <s xml:id="echoid-s6706" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6707" xml:space="preserve">Legendum puto. </s>
  <s xml:id="echoid-s6708" xml:space="preserve">Et propterea periheriæ E F <lb/>Q, &amp; </s>
  <s xml:id="echoid-s6709" xml:space="preserve">K F S, quibus inſiſtunt æquales erunt: </s>
  <s xml:id="echoid-s6710" xml:space="preserve">quod eſt abſurdum. </s>
  <s xml:id="echoid-s6711" xml:space="preserve">Eſt enim E <lb/>F Q ma<unsure/>ior, quàm K F S.</s>
  <s xml:id="echoid-s6712" xml:space="preserve"/>
</p>
<div xml:id="echoid-div631" type="float" level="2" n="7">
<note position="right" xlink:label="note-0212-02" xlink:href="note-0212-02a" xml:space="preserve">f</note>
</div>
</div>
<div xml:id="echoid-div633" type="section" level="1" n="210">
<head xml:id="echoid-head265" xml:space="preserve">Notæ in Propoſit. Præmiſſ. VI.</head>
<p style="it">
  <s xml:id="echoid-s6713" xml:space="preserve">DEinde ſint duo anguli B, E qualeſcumque; </s>
  <s xml:id="echoid-s6714" xml:space="preserve">ſed angulus A B H, vel <lb/>
<anchor type="note" xlink:label="note-0212-03a" xlink:href="note-0212-03"/>
C B H æqualis angulo D E I vel F E I, &amp; </s>
  <s xml:id="echoid-s6715" xml:space="preserve">condictiones, vti dixi-<lb/>
<anchor type="figure" xlink:label="fig-0212-02a" xlink:href="fig-0212-02"/>
<pb o="175" file="0213" n="213" rhead="Conicor. Lib. VI."/>
mus, &amp;</s>
  <s xml:id="echoid-s6716" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6717" xml:space="preserve">Expoſitio, atque demonſtratio huius propoſitionis obſcura eſt propter <lb/>nimiam eius breuitatem: </s>
  <s xml:id="echoid-s6718" xml:space="preserve">itaque duo eius caſus diſtingui debent hac ratione. </s>
  <s xml:id="echoid-s6719" xml:space="preserve">In <lb/>duobus triangulis A B C, D E F ſupponantur anguli H, &amp; </s>
  <s xml:id="echoid-s6720" xml:space="preserve">I æquales, pariter-<lb/>que anguli H B A, I E D æquales inter ſe; </s>
  <s xml:id="echoid-s6721" xml:space="preserve">ideoque duo triangula A B H, &amp; </s>
  <s xml:id="echoid-s6722" xml:space="preserve"><lb/>D E I ſimilia erunt, &amp; </s>
  <s xml:id="echoid-s6723" xml:space="preserve">propterea A H ad H B eandem proportionem habebit, <lb/>quàm D I ad I E; </s>
  <s xml:id="echoid-s6724" xml:space="preserve">ſed ex vniuerſali hypotheſi rectangulum C A H ad quadra-<lb/>tum H B eandem proportionem habet, quãm rectangulum F I D ad quadratum <lb/>I E, &amp; </s>
  <s xml:id="echoid-s6725" xml:space="preserve">componuntur proportiones rectangulorum ad quadrata iam dicta ex ra-<lb/>tionibus laterum circa angulos æquales H, &amp; </s>
  <s xml:id="echoid-s6726" xml:space="preserve">I, ſuntque oſtenſæ proportiones A <lb/>H ad H B, atque D I ad I E eædem inter ſe; </s>
  <s xml:id="echoid-s6727" xml:space="preserve">igitur reliquæ componentes pro-<lb/>portiones, ſcilicet C H ad H B, atque F I ad I E eædem quoque erunt inter ſe, <lb/>&amp; </s>
  <s xml:id="echoid-s6728" xml:space="preserve">compræhendunt angulos æquales H, &amp; </s>
  <s xml:id="echoid-s6729" xml:space="preserve">I; </s>
  <s xml:id="echoid-s6730" xml:space="preserve">igitur triangula C H B, &amp; </s>
  <s xml:id="echoid-s6731" xml:space="preserve">F I E <lb/>ſimilia ſunt inter ſe: </s>
  <s xml:id="echoid-s6732" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6733" xml:space="preserve">propterea angulus B C A æqualis erit angulo E F D, <lb/>ſed anguli B A C, &amp; </s>
  <s xml:id="echoid-s6734" xml:space="preserve">E D F æquales ſunt inter ſe, quia eorum conſequentes <lb/>æquales erant in triangulis æquiangulis B A H, &amp; </s>
  <s xml:id="echoid-s6735" xml:space="preserve">E D I, igitur duo triangu-<lb/>la B A C, &amp; </s>
  <s xml:id="echoid-s6736" xml:space="preserve">E D F æquiangula, &amp; </s>
  <s xml:id="echoid-s6737" xml:space="preserve">ſimilia inter ſe erunt.</s>
  <s xml:id="echoid-s6738" xml:space="preserve"/>
</p>
<div xml:id="echoid-div633" type="float" level="2" n="1">
<note position="right" xlink:label="note-0212-03" xlink:href="note-0212-03a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0212-02" xlink:href="fig-0212-02a">
    <image file="0212-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0212-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s6739" xml:space="preserve">Simili modo ſi ſupponantur anguli C B H, &amp; </s>
  <s xml:id="echoid-s6740" xml:space="preserve">F E I æquales, cum anguli H, <lb/>&amp; </s>
  <s xml:id="echoid-s6741" xml:space="preserve">I æquales ſint, erunt triangula B C H, &amp; </s>
  <s xml:id="echoid-s6742" xml:space="preserve">E F I ſimilia inter ſe, &amp; </s>
  <s xml:id="echoid-s6743" xml:space="preserve">vt prius, <lb/>oſtendentur quoque triangula ablata B A H, E D I æquiangula, &amp; </s>
  <s xml:id="echoid-s6744" xml:space="preserve">ſimilia in-<lb/>ter ſe (propterea quod circa angulos æquales H, &amp; </s>
  <s xml:id="echoid-s6745" xml:space="preserve">I babent latera proportiona-<lb/>lia); </s>
  <s xml:id="echoid-s6746" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6747" xml:space="preserve">ideo reſidua triangula C A B, &amp; </s>
  <s xml:id="echoid-s6748" xml:space="preserve">F D E erunt quoque ſimilia, vt <lb/>propoſitum fuerat.</s>
  <s xml:id="echoid-s6749" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div635" type="section" level="1" n="211">
<head xml:id="echoid-head266" xml:space="preserve">SECTIO SEXTA</head>
<head xml:id="echoid-head267" xml:space="preserve">Continens Propoſit. XV. XVI. &amp; XVII.</head>
<head xml:id="echoid-head268" xml:space="preserve">PROPOSITIO XV.</head>
<p>
  <s xml:id="echoid-s6750" xml:space="preserve">DVarum hyperbolarum, aut ellipſium, ſi figuræ diametro-<lb/>rum, quæ axes non ſint, fuerint ſimiles, atque potentes <lb/>contineant cum diametris angulos æquales: </s>
  <s xml:id="echoid-s6751" xml:space="preserve">vtique ſectiones <lb/>ſunt ſimiles.</s>
  <s xml:id="echoid-s6752" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6753" xml:space="preserve">Sint ſectiones A B, C D hyperbolicæ, vel ellipticæ earum diametri, <lb/>quæ non ſint axes I A K, L C M, &amp; </s>
  <s xml:id="echoid-s6754" xml:space="preserve">earum centra G, H, &amp; </s>
  <s xml:id="echoid-s6755" xml:space="preserve">duo axes <lb/>ſint E B, F D: </s>
  <s xml:id="echoid-s6756" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6757" xml:space="preserve">educamus duas tangentes A R, C S ad duos axes, <lb/>quæ continebunt cum duabus diametris A K, C M duos angulos æqua-<lb/>les, eo quod parallelæ ſunt potentialibus ad diametros eductis; </s>
  <s xml:id="echoid-s6758" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6759" xml:space="preserve">edu-<lb/>camus à B, D ad duabus diametros A K, C M tangentes B N, D O, &amp; </s>
  <s xml:id="echoid-s6760" xml:space="preserve"><lb/>circumducamus ſuper triangula B N G, H D O duos circulos, &amp; </s>
  <s xml:id="echoid-s6761" xml:space="preserve">ex A, <lb/>C educamus ad axes duas potentiales A P, C Q, &amp; </s>
  <s xml:id="echoid-s6762" xml:space="preserve">per B, D ducamus <lb/>I B T, L D V parallelas ipſis A R, C S, quæ ſecent duos circulos in B, <lb/>T, D, V: </s>
  <s xml:id="echoid-s6763" xml:space="preserve">eritque G I in I N, ſcilicet ei æquale T I in I B ad quadra-<lb/>
<anchor type="note" xlink:label="note-0213-01a" xlink:href="note-0213-01"/>
<pb o="176" file="0214" n="214" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0214-01a" xlink:href="fig-0214-01"/>
tum potentialis I B, vt H L in L O, ſeu L V in L D ad quadratum L <lb/>D, eò quod quælibet ex dictis proportionibus eadem eſt proportioni fi-<lb/>guræ K A, &amp; </s>
  <s xml:id="echoid-s6764" xml:space="preserve">M C (39. </s>
  <s xml:id="echoid-s6765" xml:space="preserve">ex 1.)</s>
  <s xml:id="echoid-s6766" xml:space="preserve">, ergo T I ad I B eſt, vt V L ad L D, &amp; </s>
  <s xml:id="echoid-s6767" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0214-01a" xlink:href="note-0214-01"/>
angulus I, qui æqualis eſt ipſi R A G æqualis eſt angulo L, qui æqualis <lb/>eſt S C H; </s>
  <s xml:id="echoid-s6768" xml:space="preserve">igitur angulus G æqualis etiam eſt angulo H: </s>
  <s xml:id="echoid-s6769" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6770" xml:space="preserve">propterea <lb/>
<anchor type="note" xlink:label="note-0214-02a" xlink:href="note-0214-02"/>
G A R ſimile eſt H C S, &amp; </s>
  <s xml:id="echoid-s6771" xml:space="preserve">pariter G A P, H C Q ſunt ſimilia, quia P, Q <lb/>ſunt recti, vnde A P R, C Q S ſunt etiã ſimilia, &amp; </s>
  <s xml:id="echoid-s6772" xml:space="preserve">proportio vniuſcuiuſq; <lb/></s>
  <s xml:id="echoid-s6773" xml:space="preserve">eorum, nempe G P, P R ad P A, eſt, vt proportio H Q, S Q ad C Q; </s>
  <s xml:id="echoid-s6774" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0214-03a" xlink:href="note-0214-03"/>
igitur G P in P R ad quadratum P A, nempe B E ad erectum illius (39. <lb/></s>
  <s xml:id="echoid-s6775" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6776" xml:space="preserve">eſt vt H Q in Q S ad quadratum C Q, nempe D F ad erectum <lb/>illius (39. </s>
  <s xml:id="echoid-s6777" xml:space="preserve">ex 1.)</s>
  <s xml:id="echoid-s6778" xml:space="preserve">; </s>
  <s xml:id="echoid-s6779" xml:space="preserve">igitur <lb/>
<anchor type="note" xlink:label="note-0214-04a" xlink:href="note-0214-04"/>
<anchor type="figure" xlink:label="fig-0214-02a" xlink:href="fig-0214-02"/>
figuræ duorum axiũ ſunt <lb/>ſimiles, &amp; </s>
  <s xml:id="echoid-s6780" xml:space="preserve">duæ ſectiones <lb/>ſimiles ſunt (12. </s>
  <s xml:id="echoid-s6781" xml:space="preserve">ex 6.</s>
  <s xml:id="echoid-s6782" xml:space="preserve">( <lb/>ſed oportet in ellipſi, vt <lb/>duæ diametri, ideoque <lb/>duo axes ſint ſimul aut <lb/>tranſuerſi, aut ſimul re-<lb/>cti. </s>
  <s xml:id="echoid-s6783" xml:space="preserve">Et hoc erat propoſi-<lb/>tum.</s>
  <s xml:id="echoid-s6784" xml:space="preserve"/>
</p>
<div xml:id="echoid-div635" type="float" level="2" n="1">
<note position="left" xlink:label="note-0213-01" xlink:href="note-0213-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0214-01" xlink:href="fig-0214-01a">
    <image file="0214-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0214-01"/>
  </figure>
<note position="left" xlink:label="note-0214-01" xlink:href="note-0214-01a" xml:space="preserve">37. lib. 1.</note>
<note position="left" xlink:label="note-0214-02" xlink:href="note-0214-02a" xml:space="preserve">Propoſ. 2. <lb/>præmiſſ.</note>
<note position="right" xlink:label="note-0214-03" xlink:href="note-0214-03a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0214-04" xlink:href="note-0214-04a" xml:space="preserve">37. lib. 1.</note>
  <figure xlink:label="fig-0214-02" xlink:href="fig-0214-02a">
    <image file="0214-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0214-02"/>
  </figure>
</div>
<pb o="177" file="0215" n="215" rhead="Conicor. Lib. VI."/>
</div>
<div xml:id="echoid-div637" type="section" level="1" n="212">
<head xml:id="echoid-head269" xml:space="preserve">PROPOSITIO XVI.</head>
<p>
  <s xml:id="echoid-s6785" xml:space="preserve">SI ſectiones A B, C D ſimiles inter ſe, quæ ſint prius para-<lb/>bolæ, tangant lineæ A E, C F terminatæ ad earum axes <lb/>E B, F D, &amp; </s>
  <s xml:id="echoid-s6786" xml:space="preserve">contineant cum illis angulos æquales E, F, &amp; </s>
  <s xml:id="echoid-s6787" xml:space="preserve"><lb/>in qualibet earum educantur ordinationes G H, I K ad diame-<lb/>tros L A M, N C O tranſeuntes per puncta contactus axibus <lb/>
<anchor type="figure" xlink:label="fig-0215-01a" xlink:href="fig-0215-01"/>
æquidiſtantes, &amp; </s>
  <s xml:id="echoid-s6788" xml:space="preserve">fuerit proportio ſuarum abſciſſarum A M, C <lb/>O ad lineas tangentes A E, C F eadem; </s>
  <s xml:id="echoid-s6789" xml:space="preserve">vtique ordinationes <lb/>abſcindent ex ſectionibus ſimilia ſegmenta, &amp; </s>
  <s xml:id="echoid-s6790" xml:space="preserve">ſimiliter poſita, vt <lb/>G A H, I C K. </s>
  <s xml:id="echoid-s6791" xml:space="preserve">Si verò ordinationes ſecuerint ſimilia ſegmen-<lb/>ta; </s>
  <s xml:id="echoid-s6792" xml:space="preserve">vtique ſectiones ſimiles erunt, &amp; </s>
  <s xml:id="echoid-s6793" xml:space="preserve">abſciſſarum ad lineas tan-<lb/>gentes proportio erit eadem, atque lineæ tangentes continebunt <lb/>cum axibus angulos æquales.</s>
  <s xml:id="echoid-s6794" xml:space="preserve"/>
</p>
<div xml:id="echoid-div637" type="float" level="2" n="1">
  <figure xlink:label="fig-0215-01" xlink:href="fig-0215-01a">
    <image file="0215-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0215-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s6795" xml:space="preserve">Educamus enim duas B L, D N ſuper duos axes B E, F D perpendi-<lb/>culares, quæ tangent ſectiones in B, D: </s>
  <s xml:id="echoid-s6796" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6797" xml:space="preserve">ponamus A P ad duplam A <lb/>
<anchor type="note" xlink:label="note-0215-01a" xlink:href="note-0215-01"/>
E, vt R A aſſumpta ad A L ei ſimilem, nec non C Q ad duplam C F, <lb/>vt aſſumpta S C ad C N; </s>
  <s xml:id="echoid-s6798" xml:space="preserve">igitur P A, Q C ſunt erecti duarum diametro-<lb/>rum L M, N O (52. </s>
  <s xml:id="echoid-s6799" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6800" xml:space="preserve">ergo G M poteſt P A in A M, (12. </s>
  <s xml:id="echoid-s6801" xml:space="preserve">ex 1.) <lb/></s>
  <s xml:id="echoid-s6802" xml:space="preserve">
<anchor type="note" xlink:label="note-0215-02a" xlink:href="note-0215-02"/>
&amp; </s>
  <s xml:id="echoid-s6803" xml:space="preserve">ſimiliter I O poteſt O C in C Q, (12. </s>
  <s xml:id="echoid-s6804" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6805" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6806" xml:space="preserve">propter æquidiſtan-<lb/>
<anchor type="note" xlink:label="note-0215-03a" xlink:href="note-0215-03"/>
tiam E B, L A, atque F D, C N ſunt ſimilia E R B, R L A, atque D <lb/>S F, S N C; </s>
  <s xml:id="echoid-s6807" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6808" xml:space="preserve">duo anguli E, F ſuppoſiti ſunt æquales; </s>
  <s xml:id="echoid-s6809" xml:space="preserve">igitur angulus R <lb/>A L æqualis eſt S C N, &amp; </s>
  <s xml:id="echoid-s6810" xml:space="preserve">N, L ſunt recti; </s>
  <s xml:id="echoid-s6811" xml:space="preserve">quare R A ad A L, nempe <lb/>P A ad duplam A E eſt, vt S C ad N C, nempe vt Q C ad duplam <lb/>C F, &amp; </s>
  <s xml:id="echoid-s6812" xml:space="preserve">M A ad A E ſuppoſita eſt, vt O C ad C F: </s>
  <s xml:id="echoid-s6813" xml:space="preserve">ergo M A ad A P <lb/>eſt, vt O C ad C Q, &amp; </s>
  <s xml:id="echoid-s6814" xml:space="preserve">angulus O æqualis eſt M. </s>
  <s xml:id="echoid-s6815" xml:space="preserve">Oſtendetur igitur (vt <lb/>
<anchor type="note" xlink:label="note-0215-04a" xlink:href="note-0215-04"/>
<pb o="178" file="0216" n="216" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0216-01a" xlink:href="fig-0216-01"/>
diximus in 11. </s>
  <s xml:id="echoid-s6816" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s6817" xml:space="preserve">quod ſi ad abſciſſas A M, C O egrediantur quælibet <lb/>potentes, ad ſua abſciſſa eandẽ proportionẽ habebunt ſi abſciſſæ ad abſciſ-<lb/>ſas ſint in cadem proportione, &amp; </s>
  <s xml:id="echoid-s6818" xml:space="preserve">quod anguli à potentialibus, &amp; </s>
  <s xml:id="echoid-s6819" xml:space="preserve">ab-<lb/>
<anchor type="note" xlink:label="note-0216-01a" xlink:href="note-0216-01"/>
ſciſſis contenti, erunt æquales in duabus ſectionibus: </s>
  <s xml:id="echoid-s6820" xml:space="preserve">quare erit ſegmen-<lb/>tum H A G ſimile ſegmento I C K atque ſimiliter poſitum.</s>
  <s xml:id="echoid-s6821" xml:space="preserve"/>
</p>
<div xml:id="echoid-div638" type="float" level="2" n="2">
<note position="right" xlink:label="note-0215-01" xlink:href="note-0215-01a" xml:space="preserve">32. lib. 1.</note>
<note position="right" xlink:label="note-0215-02" xlink:href="note-0215-02a" xml:space="preserve">49 lib. 1.</note>
<note position="right" xlink:label="note-0215-03" xlink:href="note-0215-03a" xml:space="preserve">11. lib. 1. <lb/>lbidem.</note>
<note position="left" xlink:label="note-0215-04" xlink:href="note-0215-04a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0216-01" xlink:href="fig-0216-01a">
    <image file="0216-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0216-01"/>
  </figure>
<note position="left" xlink:label="note-0216-01" xlink:href="note-0216-01a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
</div>
<p>
  <s xml:id="echoid-s6822" xml:space="preserve">Deinde ijſdem ſignis in eiſdem figuris manẽtibus, vt prius de-<lb/>ſignatis ſupponatur, ſegmentum H A G ſimile ipſi K C I. </s>
  <s xml:id="echoid-s6823" xml:space="preserve">Dico, <lb/>quod angulus E æqualis erit F, &amp; </s>
  <s xml:id="echoid-s6824" xml:space="preserve">M A ad A E erit, vt O C ad <lb/>C F.</s>
  <s xml:id="echoid-s6825" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6826" xml:space="preserve">Quoniam duo ſegmenta ſunt ſimilia erit angulus O æqualis M, &amp; </s>
  <s xml:id="echoid-s6827" xml:space="preserve">duo <lb/>
<anchor type="note" xlink:label="note-0216-02a" xlink:href="note-0216-02"/>
anguli E A L, F C N illis æquales, ſunt quoque inter ſe æquales; </s>
  <s xml:id="echoid-s6828" xml:space="preserve">ergo <lb/>duo anguli F, E, qui illis æquales ſunt, erunt inter ſe æquales, eoquod <lb/>A E, C F parallelæ ſunt G H, I K, &amp; </s>
  <s xml:id="echoid-s6829" xml:space="preserve">anguli N, L ſunt recti; </s>
  <s xml:id="echoid-s6830" xml:space="preserve">ergo duo <lb/>triangula proportionis ſunt ſimilia, ideoque R A ad A L, nempe P A ad <lb/>
<anchor type="note" xlink:label="note-0216-03a" xlink:href="note-0216-03"/>
duplam A E eſt, vt C S ad C N, nempe Q C ad duplam C F: </s>
  <s xml:id="echoid-s6831" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6832" xml:space="preserve">quia <lb/>G M poteſt P A in A M (12. </s>
  <s xml:id="echoid-s6833" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s6834" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6835" xml:space="preserve">ſimiliter I O poteſt Q C in C O; <lb/></s>
  <s xml:id="echoid-s6836" xml:space="preserve">
<anchor type="note" xlink:label="note-0216-04a" xlink:href="note-0216-04"/>
ergo P A ad G M eſt, vt Q C ad O I, &amp; </s>
  <s xml:id="echoid-s6837" xml:space="preserve">G M ad M A eſt, vt I O ad <lb/>O C; </s>
  <s xml:id="echoid-s6838" xml:space="preserve">quia duo ſegmenta ſunt ſimilia, &amp; </s>
  <s xml:id="echoid-s6839" xml:space="preserve">E A ad A M eſt, vt C F ad C <lb/>O: </s>
  <s xml:id="echoid-s6840" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6841" xml:space="preserve">iam oſtenſum eſt, quod duo anguli E, F ſunt æquales. </s>
  <s xml:id="echoid-s6842" xml:space="preserve">Et hoc erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s6843" xml:space="preserve"/>
</p>
<div xml:id="echoid-div639" type="float" level="2" n="3">
<note position="left" xlink:label="note-0216-02" xlink:href="note-0216-02a" xml:space="preserve">Defin. 7.</note>
<note position="left" xlink:label="note-0216-03" xlink:href="note-0216-03a" xml:space="preserve">49. lib. 1. <lb/>11. lib. 1.</note>
<note position="right" xlink:label="note-0216-04" xlink:href="note-0216-04a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div641" type="section" level="1" n="213">
<head xml:id="echoid-head270" xml:space="preserve">PROPOSITIO XVII.</head>
<p>
  <s xml:id="echoid-s6844" xml:space="preserve">DEinde ſectiones ſint hyperbolicæ, aut ellipticæ, &amp; </s>
  <s xml:id="echoid-s6845" xml:space="preserve">reliqua <lb/>
<anchor type="note" xlink:label="note-0216-05a" xlink:href="note-0216-05"/>
ſupponantur, vt prius.</s>
  <s xml:id="echoid-s6846" xml:space="preserve"/>
</p>
<div xml:id="echoid-div641" type="float" level="2" n="1">
<note position="right" xlink:label="note-0216-05" xlink:href="note-0216-05a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s6847" xml:space="preserve">Educamus C c perpendicularẽ ſuper axim D F, &amp; </s>
  <s xml:id="echoid-s6848" xml:space="preserve">A a perpendicula-<lb/>rem ſuper axim B E; </s>
  <s xml:id="echoid-s6849" xml:space="preserve">atque V, Y ſint duo centra. </s>
  <s xml:id="echoid-s6850" xml:space="preserve">Ergo (propter ſimi-<lb/>litudinem duarum ſectionum) erit V a in a E ad quadratum A a potentis,
<pb o="179" file="0217" n="217" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0217-01a" xlink:href="fig-0217-01"/>
vt Y c in c F ad quadratum C c ( 39. </s>
  <s xml:id="echoid-s6851" xml:space="preserve">ex 1. </s>
  <s xml:id="echoid-s6852" xml:space="preserve">) quæ habent eandem pro-<lb/>
<anchor type="note" xlink:label="note-0217-01a" xlink:href="note-0217-01"/>
portionem, quàm figuræ axis habent, &amp; </s>
  <s xml:id="echoid-s6853" xml:space="preserve">angulus F ſuppoſitus eſt æqualis <lb/>E: </s>
  <s xml:id="echoid-s6854" xml:space="preserve">ergò Y c C ſimile eſt V a A: </s>
  <s xml:id="echoid-s6855" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6856" xml:space="preserve">propterea angulus Y æqualis eſt V, <lb/>
<anchor type="note" xlink:label="note-0217-02a" xlink:href="note-0217-02"/>
<anchor type="note" xlink:label="note-0217-03a" xlink:href="note-0217-03"/>
&amp; </s>
  <s xml:id="echoid-s6857" xml:space="preserve">angulus F C Y æqualis E A V: </s>
  <s xml:id="echoid-s6858" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6859" xml:space="preserve">propter ſimilitudinem N D Y, L B <lb/>V æquales ſunt duo anguli C N S, A L R; </s>
  <s xml:id="echoid-s6860" xml:space="preserve">ergo ſimilia ſunt C N S, A L <lb/>R. </s>
  <s xml:id="echoid-s6861" xml:space="preserve">Quare C S aſſumpta ad ei coniugatam C N eſt vt R A ad A L: </s>
  <s xml:id="echoid-s6862" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6863" xml:space="preserve">po-<lb/>namus C Q ad duplam C F, vt C S ad C N, nec non A P ad duplam <lb/>A E, vt A R ad A L; </s>
  <s xml:id="echoid-s6864" xml:space="preserve">igitur Q C, A P ſunt erecti duarum diametrorum <lb/>C Y X, A V T ( 53. </s>
  <s xml:id="echoid-s6865" xml:space="preserve">54. </s>
  <s xml:id="echoid-s6866" xml:space="preserve">ex I. </s>
  <s xml:id="echoid-s6867" xml:space="preserve">) ſed C F ad C X duplam ipſius C Y eſt <lb/>
<anchor type="note" xlink:label="note-0217-04a" xlink:href="note-0217-04"/>
vt A E ad A T duplam ipſius A V, propter ſimilitudinem C F Y, A E V: <lb/></s>
  <s xml:id="echoid-s6868" xml:space="preserve">ergo ex æqualitate Q C ad C X diametrum inclinatam, ſeu tranſuerſam <lb/>eſt vt A P ad A T; </s>
  <s xml:id="echoid-s6869" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6870" xml:space="preserve">propterea figuræ earundem diametrorumſunt ſimi-<lb/>
<anchor type="note" xlink:label="note-0217-05a" xlink:href="note-0217-05"/>
les, &amp; </s>
  <s xml:id="echoid-s6871" xml:space="preserve">quia CO <lb/>
<anchor type="figure" xlink:label="fig-0217-02a" xlink:href="fig-0217-02"/>
ad C F ſuppoſi-<lb/>ta eſt, vt A M <lb/>ad A E: </s>
  <s xml:id="echoid-s6872" xml:space="preserve">ergo ex <lb/>æqualitate Q C <lb/>ad C O eſt, vt <lb/>P A ad A M: <lb/></s>
  <s xml:id="echoid-s6873" xml:space="preserve">Quare potentes <lb/>ad duo eius ab-<lb/>ſciſſa C O, A M, <lb/>à quibus diuidũ-<lb/>tur bifariam, eã-<lb/>dem proportio-<lb/>nem habent: </s>
  <s xml:id="echoid-s6874" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6875" xml:space="preserve"><lb/>proportio abſciſ
<pb o="180" file="0218" n="218" rhead="Apollonij Pergæi"/>
ſarum in vna ſectionum ad homologa abſciſſa alterius eſt eadem ( 12. </s>
  <s xml:id="echoid-s6876" xml:space="preserve">ex <lb/>6. </s>
  <s xml:id="echoid-s6877" xml:space="preserve">), &amp; </s>
  <s xml:id="echoid-s6878" xml:space="preserve">anguli compræhenſi à potentibus, &amp; </s>
  <s xml:id="echoid-s6879" xml:space="preserve">abſciſſis ſunt æquales; </s>
  <s xml:id="echoid-s6880" xml:space="preserve">quia <lb/>æquales ſunt duobus angulis R A L, S C N æqualibus, &amp; </s>
  <s xml:id="echoid-s6881" xml:space="preserve">propterea duo <lb/>
<anchor type="note" xlink:label="note-0218-01a" xlink:href="note-0218-01"/>
ſegmenta ſunt ſimilia.</s>
  <s xml:id="echoid-s6882" xml:space="preserve"/>
</p>
<div xml:id="echoid-div642" type="float" level="2" n="2">
  <figure xlink:label="fig-0217-01" xlink:href="fig-0217-01a">
    <image file="0217-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0217-01"/>
  </figure>
<note position="right" xlink:label="note-0217-01" xlink:href="note-0217-01a" xml:space="preserve">37. lib. I. <lb/>12. huius.</note>
<note position="left" xlink:label="note-0217-02" xlink:href="note-0217-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0217-03" xlink:href="note-0217-03a" xml:space="preserve">6. præmiſ. <lb/>huius.</note>
<note position="right" xlink:label="note-0217-04" xlink:href="note-0217-04a" xml:space="preserve">50. lib. I.</note>
<note position="left" xlink:label="note-0217-05" xlink:href="note-0217-05a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0217-02" xlink:href="fig-0217-02a">
    <image file="0217-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0217-02"/>
  </figure>
<note position="left" xlink:label="note-0218-01" xlink:href="note-0218-01a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
</div>
<p>
  <s xml:id="echoid-s6883" xml:space="preserve">Poſtea oſtendetur, quod ſi duo ſegmenta fuerint ſimilia, erit <lb/>angulus F æqualis E, &amp; </s>
  <s xml:id="echoid-s6884" xml:space="preserve">A M ad A E, vt O C ad C F.</s>
  <s xml:id="echoid-s6885" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6886" xml:space="preserve">Quia propter ſimilitudinem duorum ſegmentorum continebunt poten-<lb/>
<anchor type="note" xlink:label="note-0218-02a" xlink:href="note-0218-02"/>
tes cum ſuis abſciſſis angulos æquales, &amp; </s>
  <s xml:id="echoid-s6887" xml:space="preserve">erit proportio potentium ad ab-<lb/>
<anchor type="note" xlink:label="note-0218-03a" xlink:href="note-0218-03"/>
ſciſſas eadem, &amp; </s>
  <s xml:id="echoid-s6888" xml:space="preserve">proportio abſciſſarum, in vna earum ad ſua homologa in <lb/>altera, erit eadem. </s>
  <s xml:id="echoid-s6889" xml:space="preserve">Et quia V a in a E ad quadratũ a A eandem propor-<lb/>
<anchor type="figure" xlink:label="fig-0218-01a" xlink:href="fig-0218-01"/>
tionem habet, quàm Y c in c F ad quadratum c C, &amp; </s>
  <s xml:id="echoid-s6890" xml:space="preserve">duo anguli a, &amp; </s>
  <s xml:id="echoid-s6891" xml:space="preserve">c <lb/>ſunt recti; </s>
  <s xml:id="echoid-s6892" xml:space="preserve">atque angulus C, nempe O æqualis eſt A, nempe M, propter <lb/>ſimilitudinem ſegmentorum: </s>
  <s xml:id="echoid-s6893" xml:space="preserve">ergo triangulum A E V ſimile eſt C F Y, <lb/>&amp; </s>
  <s xml:id="echoid-s6894" xml:space="preserve">angulus V æqualis eſt angulo Y; </s>
  <s xml:id="echoid-s6895" xml:space="preserve">pariterque angulus E æqualis eſt F, <lb/>&amp; </s>
  <s xml:id="echoid-s6896" xml:space="preserve">A V ad A E eandem proportionem habet, quàm Y C ad C F. </s>
  <s xml:id="echoid-s6897" xml:space="preserve">Po-<lb/>namus iam P A ad duplam A E, vt Q C ad duplam C F; </s>
  <s xml:id="echoid-s6898" xml:space="preserve">ergo ex æqua-<lb/>litate A T diameter ad A P erectum eius eſt, vt C X diameter ad C Q <lb/>erectum eius ( 53. </s>
  <s xml:id="echoid-s6899" xml:space="preserve">54. </s>
  <s xml:id="echoid-s6900" xml:space="preserve">ex I. </s>
  <s xml:id="echoid-s6901" xml:space="preserve">) &amp; </s>
  <s xml:id="echoid-s6902" xml:space="preserve">T M in M A ad quadratum M G eandẽ <lb/>
<anchor type="note" xlink:label="note-0218-04a" xlink:href="note-0218-04"/>
proportionem habet, quàm X O in O C ad quadratum O I: </s>
  <s xml:id="echoid-s6903" xml:space="preserve">at ſuppoſi-<lb/>tum eſt quadratum A M ad quadratum M G, vt quadratum C O ad qua-<lb/>dratum O I; </s>
  <s xml:id="echoid-s6904" xml:space="preserve">ergo ex æqualitate T M in M A ad quadratum A M, nem-<lb/>pe T M ad M A, eandem proportionem habet, quàm X O in O C ad
<pb o="181" file="0219" n="219" rhead="Conicor. Lib. VI."/>
quadratũ O C, <lb/>
<anchor type="figure" xlink:label="fig-0219-01a" xlink:href="fig-0219-01"/>
nempe X O ad <lb/>O C; </s>
  <s xml:id="echoid-s6905" xml:space="preserve">quare di-<lb/>uidendo, vel cõ-<lb/>ponendo, &amp; </s>
  <s xml:id="echoid-s6906" xml:space="preserve">ex <lb/>æqualitate A M <lb/>ad A E eſt vt C <lb/>O ad C F: </s>
  <s xml:id="echoid-s6907" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6908" xml:space="preserve">iã <lb/>oſtenſũ eſt, quod <lb/>duo anguli F, <lb/>&amp; </s>
  <s xml:id="echoid-s6909" xml:space="preserve">E ſunt æqua-<lb/>les. </s>
  <s xml:id="echoid-s6910" xml:space="preserve">Quare pa-<lb/>tet propoſitum.</s>
  <s xml:id="echoid-s6911" xml:space="preserve"/>
</p>
<div xml:id="echoid-div643" type="float" level="2" n="3">
<note position="right" xlink:label="note-0218-02" xlink:href="note-0218-02a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0218-03" xlink:href="note-0218-03a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
  <figure xlink:label="fig-0218-01" xlink:href="fig-0218-01a">
    <image file="0218-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0218-01"/>
  </figure>
<note position="left" xlink:label="note-0218-04" xlink:href="note-0218-04a" xml:space="preserve">21. lib. I.</note>
  <figure xlink:label="fig-0219-01" xlink:href="fig-0219-01a">
    <image file="0219-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0219-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div645" type="section" level="1" n="214">
<head xml:id="echoid-head271" xml:space="preserve">Notæ in Propoſit. XV.</head>
<p>
  <s xml:id="echoid-s6912" xml:space="preserve">SI figuræ diametrorum hyperbolarum, aut ellipſium fuerint ſimiles diſ-<lb/>
<anchor type="note" xlink:label="note-0219-01a" xlink:href="note-0219-01"/>
ſimilium axium, &amp; </s>
  <s xml:id="echoid-s6913" xml:space="preserve">potentes illarum diametrorum contineant ſimul <lb/>angulos rectos, vtique ſectiones ſimiles ſunt, &amp;</s>
  <s xml:id="echoid-s6914" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6915" xml:space="preserve">Textus mendoſus huius <lb/>propoſitionis ex ſubſequenti expoſitione, &amp; </s>
  <s xml:id="echoid-s6916" xml:space="preserve">demonſtratione corrigi debuit.</s>
  <s xml:id="echoid-s6917" xml:space="preserve"/>
</p>
<div xml:id="echoid-div645" type="float" level="2" n="1">
<note position="left" xlink:label="note-0219-01" xlink:href="note-0219-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s6918" xml:space="preserve">Et G I in I N æquale ipſi T I in I B ad quadratum I B potentis eſt, vt <lb/>
<anchor type="note" xlink:label="note-0219-02a" xlink:href="note-0219-02"/>
H L in L O æquale ipſi V L in L D ad quadratum L D; </s>
  <s xml:id="echoid-s6919" xml:space="preserve">quia, &amp;</s>
  <s xml:id="echoid-s6920" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6921" xml:space="preserve">Quo-<lb/>niam à puncto B ſectionis A B ad diametrum K A I ducuntur ordinatim appli-<lb/>cata B I, &amp; </s>
  <s xml:id="echoid-s6922" xml:space="preserve">B N contingens ſectionem in B ſecantes diametrum in I, &amp; </s>
  <s xml:id="echoid-s6923" xml:space="preserve">N; <lb/></s>
  <s xml:id="echoid-s6924" xml:space="preserve">igitur rectangulum G I N ad quadratum ordinatim applicatæ I B eandem pro-<lb/>
<anchor type="note" xlink:label="note-0219-03a" xlink:href="note-0219-03"/>
<anchor type="figure" xlink:label="fig-0219-02a" xlink:href="fig-0219-02"/>
<pb o="182" file="0220" n="220" rhead="Apollonij Pergæi"/>
portionem habebit, quàm latus tranſuerſum K A ad eius latus rectum: </s>
  <s xml:id="echoid-s6925" xml:space="preserve">eadem <lb/>ractione in ſectione C D erit rectangulum H L O ad quadratum ordinatim ap-<lb/>plicatæ D L, vt latus tranſuerſum M C ad eius latus rectum; </s>
  <s xml:id="echoid-s6926" xml:space="preserve">propt<unsure/>erea quod <lb/>à puncto D ducitur D O ſectionem contingens, &amp; </s>
  <s xml:id="echoid-s6927" xml:space="preserve">D L ordinatim applicata ad <lb/>diametrum M C, ei occurrentes in L, &amp; </s>
  <s xml:id="echoid-s6928" xml:space="preserve">O. </s>
  <s xml:id="echoid-s6929" xml:space="preserve">Et quoniam ex hypotheſi latus <lb/>tranſuerſum K A ad eius latus rectum eandem proportionem habet, quàm latus <lb/>tranſuerſum M C ad eius latus rectum, cum figuræ harum diametrorum ſup-<lb/>poſitæ ſint ſimiles; </s>
  <s xml:id="echoid-s6930" xml:space="preserve">ergo rectangulum G I N ad quadratum I B eandem propor-<lb/>tionem habet, quàm rectangulum H L O ad quadratum L D: </s>
  <s xml:id="echoid-s6931" xml:space="preserve">deinde quia in <lb/>duobus triangulis G B N, &amp; </s>
  <s xml:id="echoid-s6932" xml:space="preserve">H O D ſunt duo anguli G B N, &amp; </s>
  <s xml:id="echoid-s6933" xml:space="preserve">H D O equales, <lb/>nẽpe recti ( cum B N, &amp; </s>
  <s xml:id="echoid-s6934" xml:space="preserve">D O ſectiones contingentes in terminis axium E B, &amp; </s>
  <s xml:id="echoid-s6935" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0220-01a" xlink:href="note-0220-01"/>
F D efficiant cum ipſis angulos rectos ) atq; </s>
  <s xml:id="echoid-s6936" xml:space="preserve">à verticalibus angulis B, &amp; </s>
  <s xml:id="echoid-s6937" xml:space="preserve">D du-<lb/>cuntur ad baſes rectæ lineæ B I, D L efficientes angulos I, &amp; </s>
  <s xml:id="echoid-s6938" xml:space="preserve">L æquales, eo <lb/>quod æquales ſunt angulis æqualibus R A G, &amp; </s>
  <s xml:id="echoid-s6939" xml:space="preserve">S C H propter æquidiſtantiam <lb/>linearum B I, A R, atque <lb/>
<anchor type="figure" xlink:label="fig-0220-01a" xlink:href="fig-0220-01"/>
linearum D L, S C, &amp; </s>
  <s xml:id="echoid-s6940" xml:space="preserve">in <lb/>ſuper rectangulum G I N ad <lb/>quadratum I B eandem pro-<lb/>portionem habet, quàm re-<lb/>ctangulum H L O ad qua-<lb/>dratum L D; </s>
  <s xml:id="echoid-s6941" xml:space="preserve">igitur trian-<lb/>
<anchor type="note" xlink:label="note-0220-02a" xlink:href="note-0220-02"/>
gula G B N, &amp; </s>
  <s xml:id="echoid-s6942" xml:space="preserve">H D O ſi-<lb/>milia ſunt inter ſe; </s>
  <s xml:id="echoid-s6943" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6944" xml:space="preserve">pro-<lb/>pterea angulus G æqualis e-<lb/>rit angulo H.</s>
  <s xml:id="echoid-s6945" xml:space="preserve"/>
</p>
<div xml:id="echoid-div646" type="float" level="2" n="2">
<note position="left" xlink:label="note-0219-02" xlink:href="note-0219-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0219-03" xlink:href="note-0219-03a" xml:space="preserve">37. lib. I.</note>
  <figure xlink:label="fig-0219-02" xlink:href="fig-0219-02a">
    <image file="0219-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0219-02"/>
  </figure>
<note position="left" xlink:label="note-0220-01" xlink:href="note-0220-01a" xml:space="preserve">Coruerſ. <lb/>32. lib. I.</note>
  <figure xlink:label="fig-0220-01" xlink:href="fig-0220-01a">
    <image file="0220-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0220-01"/>
  </figure>
<note position="left" xlink:label="note-0220-02" xlink:href="note-0220-02a" xml:space="preserve">Propoſ. 2. <lb/>pręmiſſ.</note>
</div>
<p style="it">
  <s xml:id="echoid-s6946" xml:space="preserve">Et proportio vniuſcu-<lb/>inſque eorum, nempe G <lb/>P, P R ad P A eſt, vt <lb/>proportio H Q, Q S ad <lb/>C O; </s>
  <s xml:id="echoid-s6947" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s6948" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6949" xml:space="preserve">In triangulis enim ſimilibus G P A, &amp; </s>
  <s xml:id="echoid-s6950" xml:space="preserve">H Q C circa angulos rectos <lb/>P, &amp; </s>
  <s xml:id="echoid-s6951" xml:space="preserve">Qerit G P ad P A, vt H Q ad Q C: </s>
  <s xml:id="echoid-s6952" xml:space="preserve">pariter in duobus triangulis ſi-<lb/>milibus R P A, &amp; </s>
  <s xml:id="echoid-s6953" xml:space="preserve">S Q C habebit R P ad P A eandem porportionem quàm, S <lb/>Q ad Q C; </s>
  <s xml:id="echoid-s6954" xml:space="preserve">proportio verò rectanguli G P R ad quadratum P A componitur ex <lb/>ĳſdem rationibus laterum circa angulum rectum P: </s>
  <s xml:id="echoid-s6955" xml:space="preserve">pariterque proportio rectan-<lb/>guli H Q S ad quadratum Q C ex rationibus laterum circa angulum rectum <lb/>Q componitur, ſuntque oſtenſæ prædictæ componentes proportiones eædem inter <lb/>ſe; </s>
  <s xml:id="echoid-s6956" xml:space="preserve">igitur rectangulum G P R ad quadratum P A eandem proportionem habe-<lb/>bit, quàm rectangulum H Q S ad quadratum Q C; </s>
  <s xml:id="echoid-s6957" xml:space="preserve">ſed habet rectangulum G <lb/>P R ad quadratum P A eandem proportionem, quàm axis tranſuerſus E B ad <lb/>
<anchor type="note" xlink:label="note-0220-03a" xlink:href="note-0220-03"/>
eius latus rectum ( propterea quod ab eodem puncto A ſectionis ducitur contin-<lb/>gens A R, &amp; </s>
  <s xml:id="echoid-s6958" xml:space="preserve">ordinatim applicata ad axim A P) atque eodem modo rectangu-<lb/>
<anchor type="note" xlink:label="note-0220-04a" xlink:href="note-0220-04"/>
lum H Q S ad quadratum Q C eandem proportionem habet, quàm axis tran-<lb/>ſuerſus F D ad eius latus rectum; </s>
  <s xml:id="echoid-s6959" xml:space="preserve">igitur axis tranſuerſus E B ad eius latus <lb/>rectum eandem proportionem habet, quàm latus tranſuerſum F D ad eius latus <lb/>rectum; </s>
  <s xml:id="echoid-s6960" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6961" xml:space="preserve">propterea figuræ axium duarum ſectionum A B, &amp; </s>
  <s xml:id="echoid-s6962" xml:space="preserve">C D ſimiles in-<lb/>ter ſe erunt; </s>
  <s xml:id="echoid-s6963" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6964" xml:space="preserve">ideo conicæ ſectiones ſimiles erunt.</s>
  <s xml:id="echoid-s6965" xml:space="preserve"/>
</p>
<div xml:id="echoid-div647" type="float" level="2" n="3">
<note position="left" xlink:label="note-0220-03" xlink:href="note-0220-03a" xml:space="preserve">37. lib. I.</note>
<note position="left" xlink:label="note-0220-04" xlink:href="note-0220-04a" xml:space="preserve">Ibidem.</note>
</div>
<note position="left" xml:space="preserve">12. huius.</note>
<pb o="183" file="0221" n="221" rhead="Conicor. Lib. VI."/>
  <figure>
    <image file="0221-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0221-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s6966" xml:space="preserve">Sed oportet in ellipſi, vt duo axes ſint ſimul, aut tranſuerſi, aut recti-<lb/>ſimul, &amp;</s>
  <s xml:id="echoid-s6967" xml:space="preserve">c. </s>
  <s xml:id="echoid-s6968" xml:space="preserve">Addidi verba, quæ videntur in textu deficere. </s>
  <s xml:id="echoid-s6969" xml:space="preserve">Sed oportet in elli-<lb/>pſi, vt duæ diametri, ideòque duo axes ſint ſimul, aut tranſuerſi, aut ſimul re-<lb/>cti. </s>
  <s xml:id="echoid-s6970" xml:space="preserve">Licet enim multoties diametri coniugatæ ellipſium æquales eße poſſint, ni-<lb/>hilominus eæ ſumi debent, quæ ad eaſdem partes reſpiciunt axes tranſuerſos, <lb/>alias conſtructio, atque demonſtratio non ſequeretur, vt manifeſtum eſt.</s>
  <s xml:id="echoid-s6971" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div649" type="section" level="1" n="215">
<head xml:id="echoid-head272" xml:space="preserve">MONITVM.</head>
<p style="it">
  <s xml:id="echoid-s6972" xml:space="preserve">PRo intelligentia propoſ. </s>
  <s xml:id="echoid-s6973" xml:space="preserve">16. </s>
  <s xml:id="echoid-s6974" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6975" xml:space="preserve">17. </s>
  <s xml:id="echoid-s6976" xml:space="preserve">præmitti debent tria hæc lem-<lb/>mata.</s>
  <s xml:id="echoid-s6977" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div650" type="section" level="1" n="216">
<head xml:id="echoid-head273" xml:space="preserve">LEMMA VI.</head>
<p style="it">
  <s xml:id="echoid-s6978" xml:space="preserve">SI in duobus parabolicis ſegmentis A B C, &amp; </s>
  <s xml:id="echoid-s6979" xml:space="preserve">D E F baſes A C, <lb/>&amp; </s>
  <s xml:id="echoid-s6980" xml:space="preserve">D F cum diametris G B, &amp; </s>
  <s xml:id="echoid-s6981" xml:space="preserve">H E æquales angulos G, &amp; </s>
  <s xml:id="echoid-s6982" xml:space="preserve"><lb/>H non rectos contineant, atque efficiant abſciſſas G B, &amp; </s>
  <s xml:id="echoid-s6983" xml:space="preserve">H E dia-<lb/>metrorum ad latera recta B I, &amp; </s>
  <s xml:id="echoid-s6984" xml:space="preserve">E K proportionalia; </s>
  <s xml:id="echoid-s6985" xml:space="preserve">erunt ſegmenta <lb/>ſimilia inter ſe.</s>
  <s xml:id="echoid-s6986" xml:space="preserve"/>
</p>
<pb o="184" file="0222" n="222" rhead="Apollonij Pergæi"/>
  <figure>
    <image file="0222-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0222-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s6987" xml:space="preserve">Secentur diametrorum abſciſſæ G B, &amp; </s>
  <s xml:id="echoid-s6988" xml:space="preserve">H E in ĳſdem rationibus in L, M, <lb/>N, O, &amp; </s>
  <s xml:id="echoid-s6989" xml:space="preserve">ab ĳſdem punctis educantur baſibus æquiſtantes, ſeu ad diametros or-<lb/>dinatim applicatæ P Q, R S, T V, X Y<unsure/>. </s>
  <s xml:id="echoid-s6990" xml:space="preserve">Quoniam ex hypotheſi G B ad B I <lb/>eſt, vt H E ad E K; </s>
  <s xml:id="echoid-s6991" xml:space="preserve">eſtque A G media proportionalis inter G B, &amp; </s>
  <s xml:id="echoid-s6992" xml:space="preserve">B I; </s>
  <s xml:id="echoid-s6993" xml:space="preserve">pari-<lb/>
<anchor type="note" xlink:label="note-0222-01a" xlink:href="note-0222-01"/>
terque D H media proportionalis eſt inter H E, &amp; </s>
  <s xml:id="echoid-s6994" xml:space="preserve">E K; </s>
  <s xml:id="echoid-s6995" xml:space="preserve">igitur A G ad G B <lb/>eſt, vt D H ad H E; </s>
  <s xml:id="echoid-s6996" xml:space="preserve">Et quoniam inuertendo L B ad B G eſt, vt N E ad E H, <lb/>atque B G ad B I poſita fuit, vt H E ad E K; </s>
  <s xml:id="echoid-s6997" xml:space="preserve">ergo ex æquali ordinata L B ad <lb/>B I erit, vt N E ad E K, quare vt L B ad P L, mediã proportionalẽ inter L B, <lb/>&amp; </s>
  <s xml:id="echoid-s6998" xml:space="preserve">I B, ita erit N E ad N T mediam proportionalem inter N E, &amp; </s>
  <s xml:id="echoid-s6999" xml:space="preserve">E K. </s>
  <s xml:id="echoid-s7000" xml:space="preserve">Eo-<lb/>dem modo oſtendetur, quod R M ad M B eandem proportionem habet, quàm X <lb/>O ad O E: </s>
  <s xml:id="echoid-s7001" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7002" xml:space="preserve">hoc ſemper continget in quibuslibet alĳs diuiſionibus proportiona-<lb/>libus abſciſſarum, ſuntque anguli G, &amp; </s>
  <s xml:id="echoid-s7003" xml:space="preserve">H æquales; </s>
  <s xml:id="echoid-s7004" xml:space="preserve">igitur ſegmenta A B C, &amp; </s>
  <s xml:id="echoid-s7005" xml:space="preserve"><lb/>D E F ſimilia ſunt inter ſe. </s>
  <s xml:id="echoid-s7006" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s7007" xml:space="preserve"/>
</p>
<div xml:id="echoid-div650" type="float" level="2" n="1">
<note position="left" xlink:label="note-0222-01" xlink:href="note-0222-01a" xml:space="preserve">II. lib. I.</note>
</div>
<note position="left" xml:space="preserve">Defin. 7. <lb/>huius.</note>
</div>
<div xml:id="echoid-div652" type="section" level="1" n="217">
<head xml:id="echoid-head274" xml:space="preserve">LEMMA VII.</head>
<p style="it">
  <s xml:id="echoid-s7008" xml:space="preserve">S I in duobus ſegmentis A B C, &amp; </s>
  <s xml:id="echoid-s7009" xml:space="preserve">D E F hyperbolicis, aut ellipti-<lb/>cis, baſes A C, &amp; </s>
  <s xml:id="echoid-s7010" xml:space="preserve">D F cum diametris G B, &amp; </s>
  <s xml:id="echoid-s7011" xml:space="preserve">H E, æquales <lb/>angulos G, &amp; </s>
  <s xml:id="echoid-s7012" xml:space="preserve">H obliquos continentes, efficiant abſciſſas G B, &amp; </s>
  <s xml:id="echoid-s7013" xml:space="preserve">H E <lb/>proportionales lateribus rectis B I, &amp; </s>
  <s xml:id="echoid-s7014" xml:space="preserve">E K, atque tranſuerſis B Z, &amp; </s>
  <s xml:id="echoid-s7015" xml:space="preserve"><lb/>E a, erunt ſegmenta ſimilia inter ſe.</s>
  <s xml:id="echoid-s7016" xml:space="preserve"/>
</p>
  <figure>
    <image file="0222-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0222-02"/>
  </figure>
<pb o="185" file="0223" n="223" rhead="Conicor. Lib. VI."/>
<p style="it">
  <s xml:id="echoid-s7017" xml:space="preserve">Secentur abſcißæ G B, &amp; </s>
  <s xml:id="echoid-s7018" xml:space="preserve">H E in ĳſdem rationibus, ducanturque ordinatim <lb/>applicatæ vt in precedenti factum eſt. </s>
  <s xml:id="echoid-s7019" xml:space="preserve">Quoniam G B ad B I eſt, vt H E ad E <lb/>K, &amp; </s>
  <s xml:id="echoid-s7020" xml:space="preserve">inuertendo Z B ad B G eſt, vt a E ad E H, ergo ex æquali ordinata Z <lb/>B latus tranſuerſum ad B I latus rectum erit, vt a E latus tranſuerſum alte-<lb/>rius ſectionis ad E K eius latus rectum: </s>
  <s xml:id="echoid-s7021" xml:space="preserve">eſt verò rectangulum Z G B ad qua-<lb/>dratum ordinatim applicatæ G A, vt latus tranſuerſum Z B ad rectum B I; <lb/></s>
  <s xml:id="echoid-s7022" xml:space="preserve">pariterque rectangulum a H E ad quadratum ordinatim applicatæ D H, vt <lb/>tranſuerſum a E ad latus rectum E K, ſuntque prædicta latera figurarum oſtẽ-<lb/>ſa proportionalia; </s>
  <s xml:id="echoid-s7023" xml:space="preserve">igitur rectangulum Z G B ad quadratum A G eandem pro-<lb/>portionem habet, quàm rectangulum a H E ad quadratum D H; </s>
  <s xml:id="echoid-s7024" xml:space="preserve">ſed quadratum <lb/>B G ad rectangulum Z G B eandem proportionem habet, quàm G B ad G Z <lb/>(propterea quod G B eſt illorum altitudo communis) pariterque quadratum E <lb/>H ad rectangulum a H E eſt, vt H E ad H a, ſeu vt G B ad G Z; </s>
  <s xml:id="echoid-s7025" xml:space="preserve">igitur qua-<lb/>dratum G B ad rectangulum Z G B eandem proportionem habebit, quàm qua-<lb/>dratum E H ad rectangulum a H E; </s>
  <s xml:id="echoid-s7026" xml:space="preserve">quare ex æquali quadratum G B ad qua-<lb/>dratum G A eandem proportionem habebit, quàm quadratum E H ad quadratũ <lb/>H D; </s>
  <s xml:id="echoid-s7027" xml:space="preserve">ideoque inuertendo A G ad G B erit vt D H ad H E. </s>
  <s xml:id="echoid-s7028" xml:space="preserve">Rurſus, quia in-<lb/>uertendo L B ad B G eſt vt N E ad E H; </s>
  <s xml:id="echoid-s7029" xml:space="preserve">ſed G B, atque H E ad latera trã-<lb/>ſuerſa proportionalia ſunt; </s>
  <s xml:id="echoid-s7030" xml:space="preserve">igitur L B ad B Z erit vt N E ad E a; </s>
  <s xml:id="echoid-s7031" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7032" xml:space="preserve">propte-<lb/>rea, vt prius quadratum L B ad rectangulum Z L B erit, vt quadratum E N <lb/>ad rectangulum a N E; </s>
  <s xml:id="echoid-s7033" xml:space="preserve">eſtque rectangulum Z L B ad quadratum ordinatim <lb/>
<anchor type="figure" xlink:label="fig-0223-01a" xlink:href="fig-0223-01"/>
applicatæ P L, vt rectangulum a N E ad quadratum T N, (ſcilicet vt latera <lb/>tranſuerſa ad recta, quæ proportionalia oſtenſa ſunt); </s>
  <s xml:id="echoid-s7034" xml:space="preserve">igitur ex æquali ordinata <lb/>quadratũ B L ad quadratum P L eandem proportionẽ habebit, quàm quadratũ <lb/>E N ad quadratum T N; </s>
  <s xml:id="echoid-s7035" xml:space="preserve">quare vt prius dictum eſt, P L ad L B eandem pro-<lb/>portionem habebit, quàm T N ad N E; </s>
  <s xml:id="echoid-s7036" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7037" xml:space="preserve">hoc ſemper contingit in reliquis om-<lb/>nibus diuiſionibus abſciſſarum in eiſdem rationibus ſectis; </s>
  <s xml:id="echoid-s7038" xml:space="preserve">ſuntque anguli G, &amp; </s>
  <s xml:id="echoid-s7039" xml:space="preserve"><lb/>H æquales inter ſe, licet non recti, igitur (ex definitione 7.) </s>
  <s xml:id="echoid-s7040" xml:space="preserve">ſegmenta A B C, <lb/>&amp; </s>
  <s xml:id="echoid-s7041" xml:space="preserve">D E F ſimilia ſunt inter ſe. </s>
  <s xml:id="echoid-s7042" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s7043" xml:space="preserve"/>
</p>
<div xml:id="echoid-div652" type="float" level="2" n="1">
  <figure xlink:label="fig-0223-01" xlink:href="fig-0223-01a">
    <image file="0223-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0223-01"/>
  </figure>
</div>
<pb o="186" file="0224" n="224" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div654" type="section" level="1" n="218">
<head xml:id="echoid-head275" xml:space="preserve">LEMMA VIII.</head>
<p style="it">
  <s xml:id="echoid-s7044" xml:space="preserve">SI duo hyperbolica, aut elliptica ſegmenta A B C, D E F fuerint <lb/>ſimilia, quorum baſes A C, D F efficiant cum diametrorum ab-<lb/>ſciſsis B M, E O angulos æquales M, &amp; </s>
  <s xml:id="echoid-s7045" xml:space="preserve">O; </s>
  <s xml:id="echoid-s7046" xml:space="preserve">ſintque eorum tranſ-<lb/>uerſa latera T B, Z E, recta vero B L, E Q. </s>
  <s xml:id="echoid-s7047" xml:space="preserve">Dico figuras eorum; <lb/></s>
  <s xml:id="echoid-s7048" xml:space="preserve">ſiue rectangula T B L, &amp; </s>
  <s xml:id="echoid-s7049" xml:space="preserve">Z E Q ſimilia eße.</s>
  <s xml:id="echoid-s7050" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s7051" xml:space="preserve">Secentur ſegmentorum abſciſſæ M B, O E proportionaliter in N, P, &amp; </s>
  <s xml:id="echoid-s7052" xml:space="preserve">per <lb/>ea puncta ducantur ordinatim ad diametros applicatæ G N, I P æquidiſtantes <lb/>baſibus, efficientes abſciſſas B N, E P, coniunganturq; </s>
  <s xml:id="echoid-s7053" xml:space="preserve">duæ rectæ lineæ T L, Z <lb/>Q ſecantes rectas lineas N H, M V, P K, O S æquidiſtantes lateribus rectis B <lb/>L, E Q in punctis H, V, <lb/>
<anchor type="figure" xlink:label="fig-0224-01a" xlink:href="fig-0224-01"/>
K, S, atque à punctis H, &amp; </s>
  <s xml:id="echoid-s7054" xml:space="preserve"><lb/>K ducantur rectæ lineæ H X, <lb/>K R parallelæ diametris occur-<lb/>rentes ipſis M V, O S in X, <lb/>
<anchor type="note" xlink:label="note-0224-01a" xlink:href="note-0224-01"/>
&amp; </s>
  <s xml:id="echoid-s7055" xml:space="preserve">R. </s>
  <s xml:id="echoid-s7056" xml:space="preserve">Quoniam ſegmenta ſup-<lb/>ponuntur ſimilia erit A M ad <lb/>M B, vt D O ad O E, &amp; </s>
  <s xml:id="echoid-s7057" xml:space="preserve">G <lb/>N ad N B erit vt I P ad P <lb/>E, atque quadratum A M, ſeu <lb/>ei æquale rectangulum B M V, <lb/>
<anchor type="note" xlink:label="note-0224-02a" xlink:href="note-0224-02"/>
ad quadratum M B eandem <lb/>proportionem habebit, quàm, <lb/>
<anchor type="note" xlink:label="note-0224-03a" xlink:href="note-0224-03"/>
quadratum D O, ſeu ei æquale <lb/>rectangulum E O S ad quadratum O E; </s>
  <s xml:id="echoid-s7058" xml:space="preserve">ſed vt rectangulum B M V ad quadra-<lb/>tum M B ita eſt M V ad M B (cum M B ſit eorum altitudo communis) pari-<lb/>terque vt rectangulum E O S ad quadratum O E, ita eſt O S ad O E; </s>
  <s xml:id="echoid-s7059" xml:space="preserve">quare <lb/>M V ad M B eandem proportionem habebit, quàm O S ad O E; </s>
  <s xml:id="echoid-s7060" xml:space="preserve">non aliter oſten-<lb/>detur N H ad N B eandem proportionem <lb/>
<anchor type="figure" xlink:label="fig-0224-02a" xlink:href="fig-0224-02"/>
habere, quàm P K ad P E: </s>
  <s xml:id="echoid-s7061" xml:space="preserve">erat autem <lb/>
<anchor type="note" xlink:label="note-0224-04a" xlink:href="note-0224-04"/>
M B ad B N vt O E ad E P; </s>
  <s xml:id="echoid-s7062" xml:space="preserve">ergo compa-<lb/>rando antecedentes, &amp; </s>
  <s xml:id="echoid-s7063" xml:space="preserve">poſtea conſequentes <lb/>ad differentias terminorum erit B M ad M <lb/>N vt E O ad O P; </s>
  <s xml:id="echoid-s7064" xml:space="preserve">atque B N ad N M eã-<lb/>dem proportionem habebit, quàm E P ad P <lb/>O. </s>
  <s xml:id="echoid-s7065" xml:space="preserve">Quare ex æquali V M ad M N erit vt <lb/>S O ad O P, atque H N ad N M erit vt K <lb/>P ad P O; </s>
  <s xml:id="echoid-s7066" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7067" xml:space="preserve">differentia ipſarum V M &amp; </s>
  <s xml:id="echoid-s7068" xml:space="preserve"><lb/>H N ideſt X V ad M N, ſeu ad X H ean-<lb/>dem proportionem habebit, quàm differentia ipſarum S O, &amp; </s>
  <s xml:id="echoid-s7069" xml:space="preserve">K P, ideſt S R <lb/>ad O P, ſeu ad R K; </s>
  <s xml:id="echoid-s7070" xml:space="preserve">quapropter V X ad X H erit vt S R ad R K; </s>
  <s xml:id="echoid-s7071" xml:space="preserve">ſed quia <lb/>X V, L B inter ſe, nec non X H, &amp; </s>
  <s xml:id="echoid-s7072" xml:space="preserve">B T ſunt parallelæ, atq; </s>
  <s xml:id="echoid-s7073" xml:space="preserve">etiam S R, Q E <lb/>inter ſe, nec nõ R K, &amp; </s>
  <s xml:id="echoid-s7074" xml:space="preserve">E Z ſunt æquidiſtantes; </s>
  <s xml:id="echoid-s7075" xml:space="preserve">erunt triangula V X H, &amp; </s>
  <s xml:id="echoid-s7076" xml:space="preserve">L B
<pb o="187" file="0225" n="225" rhead="Conicor. Lib. VI."/>
T ſimilia, pariterque triangula S R K, &amp; </s>
  <s xml:id="echoid-s7077" xml:space="preserve">Q E Z inter ſe ſimilia; </s>
  <s xml:id="echoid-s7078" xml:space="preserve">ideoque erit <lb/>L B ad B T vt V X ad X H, pariterque Q E ad E Z erit vt S R ad R K; <lb/></s>
  <s xml:id="echoid-s7079" xml:space="preserve">erat autem prius V X ad X H, vt S R ad R K; </s>
  <s xml:id="echoid-s7080" xml:space="preserve">igitur L B ad B T eandem <lb/>proportionem habebit, quàm Q E ad E Z; </s>
  <s xml:id="echoid-s7081" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7082" xml:space="preserve">propterea circa roctos angulos B, <lb/>E, figuræ ſectionum ſimiles erunt inter ſe. </s>
  <s xml:id="echoid-s7083" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s7084" xml:space="preserve"/>
</p>
<div xml:id="echoid-div654" type="float" level="2" n="1">
  <figure xlink:label="fig-0224-01" xlink:href="fig-0224-01a">
    <image file="0224-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0224-01"/>
  </figure>
<note position="left" xlink:label="note-0224-01" xlink:href="note-0224-01a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
<note position="left" xlink:label="note-0224-02" xlink:href="note-0224-02a" xml:space="preserve">12. 13. <lb/>lib. 1.</note>
<note position="left" xlink:label="note-0224-03" xlink:href="note-0224-03a" xml:space="preserve">Ibidem.</note>
  <figure xlink:label="fig-0224-02" xlink:href="fig-0224-02a">
    <image file="0224-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0224-02"/>
  </figure>
<note position="left" xlink:label="note-0224-04" xlink:href="note-0224-04a" xml:space="preserve">Lem. 1. <lb/>lib. 5.</note>
</div>
</div>
<div xml:id="echoid-div656" type="section" level="1" n="219">
<head xml:id="echoid-head276" xml:space="preserve">Notæ in Propoſit. XVI.</head>
<p style="it">
  <s xml:id="echoid-s7085" xml:space="preserve">ERgo M A ad A P eſt vt O C ad C Q, &amp; </s>
  <s xml:id="echoid-s7086" xml:space="preserve">angulus O æqualis eſt M, <lb/>
<anchor type="note" xlink:label="note-0225-01a" xlink:href="note-0225-01"/>
oſtendetur (vt diximus in 11. </s>
  <s xml:id="echoid-s7087" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s7088" xml:space="preserve">quod, &amp;</s>
  <s xml:id="echoid-s7089" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7090" xml:space="preserve">Sequitur enim ex <lb/>æqualitate ordinata, quod M A ad A P eandem proportionem habet, quàm O C <lb/>ad C Q, cumque ſint duo ſegmenta parabolica H A G, &amp; </s>
  <s xml:id="echoid-s7091" xml:space="preserve">K C I, quorũ diame-<lb/>tri A M, &amp; </s>
  <s xml:id="echoid-s7092" xml:space="preserve">C O efficiunt cum baſibus G H, &amp; </s>
  <s xml:id="echoid-s7093" xml:space="preserve">K I angulos M, &amp; </s>
  <s xml:id="echoid-s7094" xml:space="preserve">O æquales <lb/>inter ſe (cum ſint æquales angulis R A L, &amp; </s>
  <s xml:id="echoid-s7095" xml:space="preserve">S C N æqualibus à contingentibus <lb/>
<anchor type="figure" xlink:label="fig-0225-01a" xlink:href="fig-0225-01"/>
verticalibus parallelis baſibus, &amp; </s>
  <s xml:id="echoid-s7096" xml:space="preserve">à diametris contentis) atque abſcißa M A ad <lb/>latus rectum A P eandem proportionem habet, quàm altera abſcißa O C ad C Q <lb/>latus rectum alterius ſectionis; </s>
  <s xml:id="echoid-s7097" xml:space="preserve">igitur duo ſegmenta H A G, &amp; </s>
  <s xml:id="echoid-s7098" xml:space="preserve">K C I ſimilia <lb/>
<anchor type="note" xlink:label="note-0225-02a" xlink:href="note-0225-02"/>
ſunt inter ſe.</s>
  <s xml:id="echoid-s7099" xml:space="preserve"/>
</p>
<div xml:id="echoid-div656" type="float" level="2" n="1">
<note position="left" xlink:label="note-0225-01" xlink:href="note-0225-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0225-01" xlink:href="fig-0225-01a">
    <image file="0225-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0225-01"/>
  </figure>
<note position="right" xlink:label="note-0225-02" xlink:href="note-0225-02a" xml:space="preserve">Lem. 6. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7100" xml:space="preserve">Et quia G M poteſt A P in A M, &amp; </s>
  <s xml:id="echoid-s7101" xml:space="preserve">ſimiliter I O poteſt C Q in C <lb/>
<anchor type="note" xlink:label="note-0225-03a" xlink:href="note-0225-03"/>
O; </s>
  <s xml:id="echoid-s7102" xml:space="preserve">ergo P A ad G M eſt, vt C Q ad I O, &amp; </s>
  <s xml:id="echoid-s7103" xml:space="preserve">G M ad M A eſt, vt I O <lb/>ad O C; </s>
  <s xml:id="echoid-s7104" xml:space="preserve">quia duo ſegmenta ſunt ſimilia, &amp; </s>
  <s xml:id="echoid-s7105" xml:space="preserve">E A ad A M, eſt vt F C ad <lb/>C O; </s>
  <s xml:id="echoid-s7106" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s7107" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7108" xml:space="preserve">Senſus huius textus confuſi, talis eſt. </s>
  <s xml:id="echoid-s7109" xml:space="preserve">Quia ſegmenta H A G, &amp; </s>
  <s xml:id="echoid-s7110" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0225-04a" xlink:href="note-0225-04"/>
K C I ſimilia ſupponuntur erit A M ad M G, vt C O ad O I, &amp; </s>
  <s xml:id="echoid-s7111" xml:space="preserve">quadratum <lb/>A M ad quadratum M G erit vt quadratum C O ad quadratum O I; </s>
  <s xml:id="echoid-s7112" xml:space="preserve">eſt verò <lb/>
<anchor type="note" xlink:label="note-0225-05a" xlink:href="note-0225-05"/>
rectangulum P A M æquale quadrato G M; </s>
  <s xml:id="echoid-s7113" xml:space="preserve">pariterque rectangulum Q C O eſt <lb/>æquale quadrato I O; </s>
  <s xml:id="echoid-s7114" xml:space="preserve">igitur quadratum A M ad rectangulum P A M eandem <lb/>proportionem habet, quàm quadratum C O ad rectangulum Q C O; </s>
  <s xml:id="echoid-s7115" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7116" xml:space="preserve">propte-<lb/>rea M A ad A P eandem proportionem habebit, quàm C O ad C Q; </s>
  <s xml:id="echoid-s7117" xml:space="preserve">ſed prius <lb/>oſt enſa fuit P A ad A E, vt Q C ad C F; </s>
  <s xml:id="echoid-s7118" xml:space="preserve">igitur ex æquali ordinata erit M A
<pb o="188" file="0226" n="226" rhead="Apollonij Pergæi"/>
ad A E, vt O C ad C F, ſuntque anguli E, &amp; </s>
  <s xml:id="echoid-s7119" xml:space="preserve">F æquales, vt dictum eſt. </s>
  <s xml:id="echoid-s7120" xml:space="preserve">Et <lb/>hoc erat propoſitum.</s>
  <s xml:id="echoid-s7121" xml:space="preserve"/>
</p>
<div xml:id="echoid-div657" type="float" level="2" n="2">
<note position="left" xlink:label="note-0225-03" xlink:href="note-0225-03a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0225-04" xlink:href="note-0225-04a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
<note position="right" xlink:label="note-0225-05" xlink:href="note-0225-05a" xml:space="preserve">11. lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div659" type="section" level="1" n="220">
<head xml:id="echoid-head277" xml:space="preserve">Notæ in Propoſit. XVII.</head>
<p style="it">
  <s xml:id="echoid-s7122" xml:space="preserve">DEinde ſint ſectiones hyperbolicæ, aut ellipticæ, &amp; </s>
  <s xml:id="echoid-s7123" xml:space="preserve">reliqua in ſuo <lb/>
<anchor type="note" xlink:label="note-0226-01a" xlink:href="note-0226-01"/>
ſtatu, &amp;</s>
  <s xml:id="echoid-s7124" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7125" xml:space="preserve">Ideſt. </s>
  <s xml:id="echoid-s7126" xml:space="preserve">Supponantur ſectiones hyperbolicæ, vel ellipticæ A B, <lb/>&amp; </s>
  <s xml:id="echoid-s7127" xml:space="preserve">C D ſimiles inter ſe, ſcilicet figuræ axium V B, &amp; </s>
  <s xml:id="echoid-s7128" xml:space="preserve">γ D ſint ſimiles inter ſe, <lb/>atque à verticibus A, &amp; </s>
  <s xml:id="echoid-s7129" xml:space="preserve">C duarum diametrorum A M, &amp; </s>
  <s xml:id="echoid-s7130" xml:space="preserve">C O ductæ ſint re-<lb/>
<anchor type="figure" xlink:label="fig-0226-01a" xlink:href="fig-0226-01"/>
ctæ lineæ contingentes A E, &amp; </s>
  <s xml:id="echoid-s7131" xml:space="preserve">C F, efficientes cum axibus angulos A E B, &amp; </s>
  <s xml:id="echoid-s7132" xml:space="preserve"><lb/>C F D æquales, ſintque H G, &amp; </s>
  <s xml:id="echoid-s7133" xml:space="preserve">K I ordinatim ad diametros applicatæ, ſcili-<lb/>cet æquidiſtantes contingentibus verticalibus; </s>
  <s xml:id="echoid-s7134" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7135" xml:space="preserve">habeat abſciſſa M A ad portio-<lb/>nem contingentis A E eandem proportionem, quàm abſcißa O C habet ad por-<lb/>tionem contingentis C F; </s>
  <s xml:id="echoid-s7136" xml:space="preserve">Dico ſegmenta H A G, &amp; </s>
  <s xml:id="echoid-s7137" xml:space="preserve">K C I ſimlia eſſe inter ſe.</s>
  <s xml:id="echoid-s7138" xml:space="preserve"/>
</p>
<div xml:id="echoid-div659" type="float" level="2" n="1">
<note position="right" xlink:label="note-0226-01" xlink:href="note-0226-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0226-01" xlink:href="fig-0226-01a">
    <image file="0226-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0226-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7139" xml:space="preserve">Ergo Y c C ſimile eſt V a A, &amp;</s>
  <s xml:id="echoid-s7140" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7141" xml:space="preserve">Quoniam duæ ordinatim ad axes ap-<lb/>
<anchor type="note" xlink:label="note-0226-02a" xlink:href="note-0226-02"/>
plicatæ A a, &amp; </s>
  <s xml:id="echoid-s7142" xml:space="preserve">C c perpendiculares ſunt ad axes, erunt in triangulis A a E, <lb/>&amp; </s>
  <s xml:id="echoid-s7143" xml:space="preserve">C c F duo anguli a, &amp; </s>
  <s xml:id="echoid-s7144" xml:space="preserve">c recti: </s>
  <s xml:id="echoid-s7145" xml:space="preserve">atque ex hypotheſi duo reliqui anguli E, &amp; </s>
  <s xml:id="echoid-s7146" xml:space="preserve"><lb/>F æquales quoque ſunt; </s>
  <s xml:id="echoid-s7147" xml:space="preserve">igitur tertius angulus a A E æqualis eſt tertio angulo c <lb/>C F, cumque in duobus triangulis V A E, atque γ C F ab eorum verticibus A, <lb/>&amp; </s>
  <s xml:id="echoid-s7148" xml:space="preserve">C ducuntur ad baſes V E, &amp; </s>
  <s xml:id="echoid-s7149" xml:space="preserve">γ F duæ rectæ lineæ A a, &amp; </s>
  <s xml:id="echoid-s7150" xml:space="preserve">C c continentes <lb/>cum baſibus angulos æquales, nempe rectos, &amp; </s>
  <s xml:id="echoid-s7151" xml:space="preserve">rectangulum V a E ad quadra-<lb/>tum a A eandem proportionem habet, quàm rectangulum γ c F ad quadratum <lb/>c C, vt in textu oſtenſum eſt: </s>
  <s xml:id="echoid-s7152" xml:space="preserve">atq; </s>
  <s xml:id="echoid-s7153" xml:space="preserve">duo anguli a A E, &amp; </s>
  <s xml:id="echoid-s7154" xml:space="preserve">c C F æquales oſten-<lb/>
<anchor type="note" xlink:label="note-0226-03a" xlink:href="note-0226-03"/>
ſi ſunt inter ſe; </s>
  <s xml:id="echoid-s7155" xml:space="preserve">igitur erunt triangula V A E, &amp; </s>
  <s xml:id="echoid-s7156" xml:space="preserve">γ C F ſimilia inter ſe; </s>
  <s xml:id="echoid-s7157" xml:space="preserve">ergo <lb/>
<anchor type="note" xlink:label="note-0226-04a" xlink:href="note-0226-04"/>
angulus V æqualis eſt angulo γ, atque angulus E A V æqualis erit angulo F C
<pb o="189" file="0227" n="227" rhead="Conicor. Lib. VI."/>
γ: </s>
  <s xml:id="echoid-s7158" xml:space="preserve">poſtea, quia B <lb/>
<anchor type="figure" xlink:label="fig-0227-01a" xlink:href="fig-0227-01"/>
L, &amp; </s>
  <s xml:id="echoid-s7159" xml:space="preserve">D N con-<lb/>tingunt ſectiones <lb/>in verticibus a-<lb/>
<anchor type="note" xlink:label="note-0227-01a" xlink:href="note-0227-01"/>
xium efficient an-<lb/>gulos V B L, &amp; </s>
  <s xml:id="echoid-s7160" xml:space="preserve"><lb/>γ D N rectos, cũ-<lb/>que duo anguli V, <lb/>&amp; </s>
  <s xml:id="echoid-s7161" xml:space="preserve">γ oſtenſi ſint æ-<lb/>quales, in trian-<lb/>gulis V B L, γ <lb/>D N, anguli V <lb/>L B, &amp; </s>
  <s xml:id="echoid-s7162" xml:space="preserve">γ N D <lb/>æquales erunt in-<lb/>ter ſe, &amp; </s>
  <s xml:id="echoid-s7163" xml:space="preserve">qui de-<lb/>inceps A L R, &amp; </s>
  <s xml:id="echoid-s7164" xml:space="preserve">C N S ſunt æquales inter ſe; </s>
  <s xml:id="echoid-s7165" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7166" xml:space="preserve">ideo triangula A R L, &amp; </s>
  <s xml:id="echoid-s7167" xml:space="preserve">C <lb/>S N ſimilia ſunt inter ſe.</s>
  <s xml:id="echoid-s7168" xml:space="preserve"/>
</p>
<div xml:id="echoid-div660" type="float" level="2" n="2">
<note position="right" xlink:label="note-0226-02" xlink:href="note-0226-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0226-03" xlink:href="note-0226-03a" xml:space="preserve">ex 37. <lb/>lib. 1.</note>
<note position="left" xlink:label="note-0226-04" xlink:href="note-0226-04a" xml:space="preserve">Propoſ. 6 <lb/>præmiſſ.</note>
  <figure xlink:label="fig-0227-01" xlink:href="fig-0227-01a">
    <image file="0227-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0227-01"/>
  </figure>
<note position="right" xlink:label="note-0227-01" xlink:href="note-0227-01a" xml:space="preserve">Conuerſ. <lb/>32. lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7169" xml:space="preserve">Et propterea figuræ earundem diametrorum ſunt ſimiles, &amp;</s>
  <s xml:id="echoid-s7170" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7171" xml:space="preserve">Quia <lb/>
<anchor type="note" xlink:label="note-0227-02a" xlink:href="note-0227-02"/>
ex hypotheſi M A ad A E erat, vt O C ad C F; </s>
  <s xml:id="echoid-s7172" xml:space="preserve">atque (propter ſimilitudinem <lb/>triangulorum A E V, &amp; </s>
  <s xml:id="echoid-s7173" xml:space="preserve">C F γ) vt E A ad duplam ipſius A V, ſeu ad latus <lb/>tranſuerſum A T, ita eſt F C ad duplam ipſius C γ, ſeu ad latus tranſuerſum <lb/>C X alterius ſectionis; </s>
  <s xml:id="echoid-s7174" xml:space="preserve">ergo ex æquali ordinata erit M A ad A T, vt O C ad <lb/>C X; </s>
  <s xml:id="echoid-s7175" xml:space="preserve">oſtenſum autem fuit latus tranſuerſum T A ad A P latus rectum eius ha-<lb/>bere eandem proportionem, quàm alterius ſectionis latus tranſuerſum X C ad <lb/>eius latus rectum C Q; </s>
  <s xml:id="echoid-s7176" xml:space="preserve">ergo ex æquali ordinata M A ad A P eandem propor-<lb/>tionem habet, quàm O C ad C Q; </s>
  <s xml:id="echoid-s7177" xml:space="preserve">quare duæ abſciſſæ A M, &amp; </s>
  <s xml:id="echoid-s7178" xml:space="preserve">O C eandem <lb/>proportionem habent ad latera recta, atque ad tranſuerſa earundem diametro-<lb/>rum, atque efficiunt baſes H G, &amp; </s>
  <s xml:id="echoid-s7179" xml:space="preserve">K I cum diametris angulos M, &amp; </s>
  <s xml:id="echoid-s7180" xml:space="preserve">O æqua-<lb/>
<anchor type="note" xlink:label="note-0227-03a" xlink:href="note-0227-03"/>
les inter ſe: </s>
  <s xml:id="echoid-s7181" xml:space="preserve">propterea quod æquales ſunt angulis E A V, &amp; </s>
  <s xml:id="echoid-s7182" xml:space="preserve">F C γ æqualibus <lb/>(propter æquidiſtantiam rectarum H G, &amp; </s>
  <s xml:id="echoid-s7183" xml:space="preserve">A E; </s>
  <s xml:id="echoid-s7184" xml:space="preserve">nec non K I, &amp; </s>
  <s xml:id="echoid-s7185" xml:space="preserve">C F) igitur <lb/>erunt duo ſegmenta H A G, &amp; </s>
  <s xml:id="echoid-s7186" xml:space="preserve">K C I ſimilia inter ſe.</s>
  <s xml:id="echoid-s7187" xml:space="preserve"/>
</p>
<div xml:id="echoid-div661" type="float" level="2" n="3">
<note position="left" xlink:label="note-0227-02" xlink:href="note-0227-02a" xml:space="preserve">C</note>
<note position="right" xlink:label="note-0227-03" xlink:href="note-0227-03a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7188" xml:space="preserve">Quia propter ſimilitudinem duorum ſegmentorum continebunt poten-<lb/>
<anchor type="note" xlink:label="note-0227-04a" xlink:href="note-0227-04"/>
tes cum ſuis abſciſſis angulos æquales: </s>
  <s xml:id="echoid-s7189" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7190" xml:space="preserve">erit proportio potẽtium ad ab-<lb/>ſciſſa eadem, &amp; </s>
  <s xml:id="echoid-s7191" xml:space="preserve">proportio abſciſſarum in vna earum ad alia ſimilia eadẽ, <lb/>quia V a in a E ad quadratum A a, eſt vt Y c in c F ad quadratum C c, <lb/>&amp; </s>
  <s xml:id="echoid-s7192" xml:space="preserve">duo anguli a, &amp; </s>
  <s xml:id="echoid-s7193" xml:space="preserve">c ſunt æquales; </s>
  <s xml:id="echoid-s7194" xml:space="preserve">ergo angulus Y æqualis eſt angulo <lb/>V, &amp; </s>
  <s xml:id="echoid-s7195" xml:space="preserve">angulus C, nempe O æqualis A, nempe M propter ſimilitudinem <lb/>duorum ſegmentorum; </s>
  <s xml:id="echoid-s7196" xml:space="preserve">igitur A E V ſimile eſt Y F C, &amp; </s>
  <s xml:id="echoid-s7197" xml:space="preserve">angulus E; </s>
  <s xml:id="echoid-s7198" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s7199" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s7200" xml:space="preserve">In hoc textu nonnulla verba deficiunt, aliqua verò tranſpoſita ſunt, vt nullus <lb/>ſenſus colligi poſſit: </s>
  <s xml:id="echoid-s7201" xml:space="preserve">tamen eum reſtitui poße cenſeo vt ibidem videre eſt. </s>
  <s xml:id="echoid-s7202" xml:space="preserve">Quo-<lb/>niam duo ſegmenta H A G, &amp; </s>
  <s xml:id="echoid-s7203" xml:space="preserve">K C I ſupponuntur ſimilia efficient diametri A <lb/>M, &amp; </s>
  <s xml:id="echoid-s7204" xml:space="preserve">C O cum baſibus G H, &amp; </s>
  <s xml:id="echoid-s7205" xml:space="preserve">K I angulos M, &amp; </s>
  <s xml:id="echoid-s7206" xml:space="preserve">O æquales, licet non rectos; </s>
  <s xml:id="echoid-s7207" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0227-05a" xlink:href="note-0227-05"/>
eruntque figuræ earumdem diametrorum ſimiles inter ſe: </s>
  <s xml:id="echoid-s7208" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7209" xml:space="preserve">propterea habebit <lb/>T A ad eius erectum eandem proportionem, quàm X C ad eius latus rectum; <lb/></s>
  <s xml:id="echoid-s7210" xml:space="preserve">
<anchor type="note" xlink:label="note-0227-06a" xlink:href="note-0227-06"/>
igitur ſectiones A B, &amp; </s>
  <s xml:id="echoid-s7211" xml:space="preserve">C D ſimiles ſunt, ideſt ductis axibus V B, &amp; </s>
  <s xml:id="echoid-s7212" xml:space="preserve">γ D <lb/>
<anchor type="note" xlink:label="note-0227-07a" xlink:href="note-0227-07"/>
erunt figuræ axium ſimiles inter ſe: </s>
  <s xml:id="echoid-s7213" xml:space="preserve">ducuntur verò à punctis A, &amp; </s>
  <s xml:id="echoid-s7214" xml:space="preserve">C ad axes <lb/>
<anchor type="note" xlink:label="note-0227-08a" xlink:href="note-0227-08"/>
ordinatim applicati A a, &amp; </s>
  <s xml:id="echoid-s7215" xml:space="preserve">C c, atque contingentes A E, &amp; </s>
  <s xml:id="echoid-s7216" xml:space="preserve">C F; </s>
  <s xml:id="echoid-s7217" xml:space="preserve">igitur re-<lb/>
<anchor type="note" xlink:label="note-0227-09a" xlink:href="note-0227-09"/>
<pb o="190" file="0228" n="228" rhead="Apollonij Pergæi"/>
ctangulum V a E ad quadratum a A eandem proportionem habebit, quàm axis <lb/>tranſuerſus ad eius erectum, ſeu quàm axis tranſuerſus alterius ſectionis C D <lb/>ad eius erectum: </s>
  <s xml:id="echoid-s7218" xml:space="preserve">ſed in eadem proportione eſt rectangulum γ c F ad quadratũ <lb/>
<anchor type="note" xlink:label="note-0228-01a" xlink:href="note-0228-01"/>
c C; </s>
  <s xml:id="echoid-s7219" xml:space="preserve">igitur in duobus triangulis A V E, &amp; </s>
  <s xml:id="echoid-s7220" xml:space="preserve">C γ F rectæ A a, &amp; </s>
  <s xml:id="echoid-s7221" xml:space="preserve">C c cũ baſibus <lb/>angulos æquales a, &amp; </s>
  <s xml:id="echoid-s7222" xml:space="preserve">c, nempe rectos efficiunt, cum ordinatim applicatæ ſint ad <lb/>axes; </s>
  <s xml:id="echoid-s7223" xml:space="preserve">atque duo anguli verticales V A E, &amp; </s>
  <s xml:id="echoid-s7224" xml:space="preserve">γ C F æquales ſint inter ſe, cum <lb/>propter parallelas æquales ſint angulis O, &amp; </s>
  <s xml:id="echoid-s7225" xml:space="preserve">M æqualibus in ſegmentis ſimilibus; <lb/></s>
  <s xml:id="echoid-s7226" xml:space="preserve">
<anchor type="note" xlink:label="note-0228-02a" xlink:href="note-0228-02"/>
igitur duo triangula A E V, &amp; </s>
  <s xml:id="echoid-s7227" xml:space="preserve">C F γ æquiangula, &amp; </s>
  <s xml:id="echoid-s7228" xml:space="preserve">ſimilia ſunt inter ſe: </s>
  <s xml:id="echoid-s7229" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7230" xml:space="preserve"><lb/>proptered V A ad A E erit, vt γ C ad C F, &amp;</s>
  <s xml:id="echoid-s7231" xml:space="preserve">c.</s>
  <s xml:id="echoid-s7232" xml:space="preserve"/>
</p>
<div xml:id="echoid-div662" type="float" level="2" n="4">
<note position="left" xlink:label="note-0227-04" xlink:href="note-0227-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0227-05" xlink:href="note-0227-05a" xml:space="preserve">Lem. 8. <lb/>huius.</note>
<note position="right" xlink:label="note-0227-06" xlink:href="note-0227-06a" xml:space="preserve">15. huius.</note>
<note position="right" xlink:label="note-0227-07" xlink:href="note-0227-07a" xml:space="preserve">47. lib. 2.</note>
<note position="right" xlink:label="note-0227-08" xlink:href="note-0227-08a" xml:space="preserve">12. huius.</note>
<note position="right" xlink:label="note-0227-09" xlink:href="note-0227-09a" xml:space="preserve">37. lib. 1.</note>
<note position="left" xlink:label="note-0228-01" xlink:href="note-0228-01a" xml:space="preserve">37. lib. 1.</note>
<note position="left" xlink:label="note-0228-02" xlink:href="note-0228-02a" xml:space="preserve">Propoſ. 7. <lb/>præmiſſ.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7233" xml:space="preserve">Ponamus iam P A ad duplam A E, vt Q C ad duplam C F: </s>
  <s xml:id="echoid-s7234" xml:space="preserve">ergo ex <lb/>
<anchor type="note" xlink:label="note-0228-03a" xlink:href="note-0228-03"/>
æqualitate A T diameter ad A P erectum eius, &amp;</s>
  <s xml:id="echoid-s7235" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7236" xml:space="preserve">In hoc textu nonnulla <lb/>videntur deficere, eiuſq; </s>
  <s xml:id="echoid-s7237" xml:space="preserve">ſenſus talis erit. </s>
  <s xml:id="echoid-s7238" xml:space="preserve">Quia veluti ſupra dictum eſt, triã-<lb/>gula R A L, &amp; </s>
  <s xml:id="echoid-s7239" xml:space="preserve">S C N ſimilia ſunt inter ſe, habebit R A ad A L eandem pro-<lb/>portionem, quàm S C ad C N: </s>
  <s xml:id="echoid-s7240" xml:space="preserve">Ponamus iam P A ad duplam A E, vt R A ad <lb/>A L, &amp; </s>
  <s xml:id="echoid-s7241" xml:space="preserve">Q C ad duplam C F, vt S C ad C N, erunt A P, &amp; </s>
  <s xml:id="echoid-s7242" xml:space="preserve">C Q latera re-<lb/>cta diametrorum A M, &amp; </s>
  <s xml:id="echoid-s7243" xml:space="preserve">O C; </s>
  <s xml:id="echoid-s7244" xml:space="preserve">ſed earundem diametrorum figuræ oſtenſæ ſunt <lb/>
<anchor type="note" xlink:label="note-0228-04a" xlink:href="note-0228-04"/>
ſimiles; </s>
  <s xml:id="echoid-s7245" xml:space="preserve">igitur latus tranſuerſum A T ad A P erectum eius eſt, vt latus tran-<lb/>uerſum X C ad C Q erectum eius. </s>
  <s xml:id="echoid-s7246" xml:space="preserve">Et quia vt latus tranſuerſum ad rectum <lb/>
<anchor type="figure" xlink:label="fig-0228-01a" xlink:href="fig-0228-01"/>
ita eſt rectangulum T M A ad quadratum M G, &amp; </s>
  <s xml:id="echoid-s7247" xml:space="preserve">ſimiliter rectangulum X O <lb/>
<anchor type="note" xlink:label="note-0228-05a" xlink:href="note-0228-05"/>
C ad quadratum O I eandem proportionem habebit, quàm latus tranſuerſum ad <lb/>rectum, ſcilicet eandem, quàm habent latera figurarũ earundẽ diametrorũ; </s>
  <s xml:id="echoid-s7248" xml:space="preserve">igi-<lb/>tur rectangulum T M A ad quadratum M G eandem proportionẽ habebit, quàm <lb/>rectangulum X O C ad quadratum O I; </s>
  <s xml:id="echoid-s7249" xml:space="preserve">habet verò M G ad M A eandem pro-<lb/>portionem, quàm I O ad O C propter ſimilitudinem ſegmentorum; </s>
  <s xml:id="echoid-s7250" xml:space="preserve">ergo quadra-<lb/>tum G M ad quadratum M A erit vt quadratum I O ad quadratum O C: </s>
  <s xml:id="echoid-s7251" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7252" xml:space="preserve"><lb/>propterea ex æquali ordinata rectangulum T M A ad quadratum M A, ſeu T M
<pb o="191" file="0229" n="229" rhead="Conicor. Lib. VI."/>
ad A M eandem <lb/>
<anchor type="figure" xlink:label="fig-0229-01a" xlink:href="fig-0229-01"/>
proportionem ha-<lb/>bebit, quàm X O <lb/>C ad quadratum <lb/>O C, ſeu eandẽ, <lb/>quàm habet X O <lb/>ad C O, &amp; </s>
  <s xml:id="echoid-s7253" xml:space="preserve">com-<lb/>parando conſequẽ <lb/>tes ad differẽtias <lb/>terminorum M A <lb/>ad A T eandem <lb/>proportionem ha-<lb/>bebit, quàm O C <lb/>ad C X: </s>
  <s xml:id="echoid-s7254" xml:space="preserve">erat autẽ <lb/>prius T A ad A <lb/>E, vt X C ad C F; </s>
  <s xml:id="echoid-s7255" xml:space="preserve">igitur ex æquali M A ad A E erit, vt O C ad C F, &amp; </s>
  <s xml:id="echoid-s7256" xml:space="preserve">fue-<lb/>runt oſtenſi anguli E, &amp; </s>
  <s xml:id="echoid-s7257" xml:space="preserve">F æquales. </s>
  <s xml:id="echoid-s7258" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s7259" xml:space="preserve"/>
</p>
<div xml:id="echoid-div663" type="float" level="2" n="5">
<note position="right" xlink:label="note-0228-03" xlink:href="note-0228-03a" xml:space="preserve">e</note>
<note position="left" xlink:label="note-0228-04" xlink:href="note-0228-04a" xml:space="preserve">50 lib. 1. <lb/>Lem. 8.</note>
  <figure xlink:label="fig-0228-01" xlink:href="fig-0228-01a">
    <image file="0228-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0228-01"/>
  </figure>
<note position="left" xlink:label="note-0228-05" xlink:href="note-0228-05a" xml:space="preserve">21. lib. 1.</note>
  <figure xlink:label="fig-0229-01" xlink:href="fig-0229-01a">
    <image file="0229-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0229-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div665" type="section" level="1" n="221">
<head xml:id="echoid-head278" xml:space="preserve">SECTIO SEPTIMA</head>
<head xml:id="echoid-head279" xml:space="preserve">Continens Propoſit. XVIII. &amp; XIX.</head>
<p>
  <s xml:id="echoid-s7260" xml:space="preserve">CViuslibet ſectionis A B C duo ſegmenta C F, A E ca-<lb/>dentia inter duas ordinationes A C, E F ad vtraſque par-<lb/>tes axis B V ſunt inter ſe ſimilia, &amp; </s>
  <s xml:id="echoid-s7261" xml:space="preserve">ſimiliter poſita, nec ſunt <lb/>ſimilia alteri ſegmento (niſi <lb/>
<anchor type="figure" xlink:label="fig-0229-02a" xlink:href="fig-0229-02"/>
in ellipſi, in qua quatuor ſeg <lb/>menta memorata in propo-<lb/>ſitione 8. </s>
  <s xml:id="echoid-s7262" xml:space="preserve">ſunt æqualia, ſimi-<lb/>lia, &amp; </s>
  <s xml:id="echoid-s7263" xml:space="preserve">ſimiliter poſita, quæ al-<lb/>teri ſegmẽto ſimilia nõ ſunt.</s>
  <s xml:id="echoid-s7264" xml:space="preserve"/>
</p>
<div xml:id="echoid-div665" type="float" level="2" n="1">
  <figure xlink:label="fig-0229-02" xlink:href="fig-0229-02a">
    <image file="0229-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0229-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s7265" xml:space="preserve">Quoniam vnumquodque eo-<lb/>
<anchor type="note" xlink:label="note-0229-01a" xlink:href="note-0229-01"/>
rum alteri congruit, nec non cõ-<lb/>gruunt duo ſegmenta GI, K H <lb/>in ellipſi (7. </s>
  <s xml:id="echoid-s7266" xml:space="preserve">8. </s>
  <s xml:id="echoid-s7267" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s7268" xml:space="preserve">at non ſunt <lb/>ſimilia alteri ſegmento: </s>
  <s xml:id="echoid-s7269" xml:space="preserve">ſi enim <lb/>hoc fieri poteſt, ſit ſegmentum <lb/>L M ſimile ſegmento F C. </s>
  <s xml:id="echoid-s7270" xml:space="preserve">Et <lb/>quia F C congruit A E. </s>
  <s xml:id="echoid-s7271" xml:space="preserve">Ergo <lb/>duo ſegmenta L M, A E ſunt <lb/>ſimilia, producamus A E, L M <lb/>quouſque occurrant axi in N, <lb/>
<anchor type="note" xlink:label="note-0229-02a" xlink:href="note-0229-02"/>
O, erit angulus N æqualis O (vti <lb/>demonſtrauimus in 16. </s>
  <s xml:id="echoid-s7272" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7273" xml:space="preserve">17.</s>
  <s xml:id="echoid-s7274" xml:space="preserve">
<pb o="192" file="0230" n="230" rhead="Apollonij Pergæi"/>
huius) atque A N pa-<lb/>
<anchor type="figure" xlink:label="fig-0230-01a" xlink:href="fig-0230-01"/>
rallela erit L O. </s>
  <s xml:id="echoid-s7275" xml:space="preserve">Edu-<lb/>catur iam R Q bifariã <lb/>diuidens A E, L M in <lb/>P, Q: </s>
  <s xml:id="echoid-s7276" xml:space="preserve">quare erit diame <lb/>
<anchor type="note" xlink:label="note-0230-01a" xlink:href="note-0230-01"/>
ter ſectionis (32. </s>
  <s xml:id="echoid-s7277" xml:space="preserve">ex 2.) <lb/></s>
  <s xml:id="echoid-s7278" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7279" xml:space="preserve">educatur R V paral-<lb/>lela A N, quæ ſectionẽ <lb/>
<anchor type="note" xlink:label="note-0230-02a" xlink:href="note-0230-02"/>
continget (18. </s>
  <s xml:id="echoid-s7280" xml:space="preserve">ex 1.)</s>
  <s xml:id="echoid-s7281" xml:space="preserve">. <lb/></s>
  <s xml:id="echoid-s7282" xml:space="preserve">Et quia duo ſegmen-<lb/>ta L M, A E ſunt ſi-<lb/>milia habebit maior <lb/>
<anchor type="note" xlink:label="note-0230-03a" xlink:href="note-0230-03"/>
Q R ad eandem R V <lb/>eandem proportionẽ, <lb/>quàm habet minor R <lb/>P; </s>
  <s xml:id="echoid-s7283" xml:space="preserve">quod eſt abſurdum. <lb/></s>
  <s xml:id="echoid-s7284" xml:space="preserve">Quare non ſunt ſimilia <lb/>duo ſegmenta A E, C <lb/>F alteri ſegmento. </s>
  <s xml:id="echoid-s7285" xml:space="preserve"><lb/>Quod erat oſtenden-<lb/>dum.</s>
  <s xml:id="echoid-s7286" xml:space="preserve"/>
</p>
<div xml:id="echoid-div666" type="float" level="2" n="2">
<note position="left" xlink:label="note-0229-01" xlink:href="note-0229-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0229-02" xlink:href="note-0229-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0230-01" xlink:href="fig-0230-01a">
    <image file="0230-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0230-01"/>
  </figure>
<note position="left" xlink:label="note-0230-01" xlink:href="note-0230-01a" xml:space="preserve">28. lib. 2.</note>
<note position="left" xlink:label="note-0230-02" xlink:href="note-0230-02a" xml:space="preserve">17. lib. 1.</note>
<note position="left" xlink:label="note-0230-03" xlink:href="note-0230-03a" xml:space="preserve">16. 17. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div668" type="section" level="1" n="222">
<head xml:id="echoid-head280" xml:space="preserve">Notæ in Propoſit. XVIII. &amp; XIX.</head>
<p style="it">
  <s xml:id="echoid-s7287" xml:space="preserve">QVuoniam vnumquodque corum alteri congruit, nec non congruunt <lb/>duo ſegmenta G I, K H in ellipſi (7. </s>
  <s xml:id="echoid-s7288" xml:space="preserve">8. </s>
  <s xml:id="echoid-s7289" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s7290" xml:space="preserve">at non ſunt ſimilia <lb/>
<anchor type="note" xlink:label="note-0230-04a" xlink:href="note-0230-04"/>
alteri ſegmento, &amp;</s>
  <s xml:id="echoid-s7291" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7292" xml:space="preserve">Ideſt. </s>
  <s xml:id="echoid-s7293" xml:space="preserve">Sit prius ſectio A B C parabole, vel <lb/>hyperbole. </s>
  <s xml:id="echoid-s7294" xml:space="preserve">Quoniam duæ A C, &amp; </s>
  <s xml:id="echoid-s7295" xml:space="preserve">E F ordinatim ad axim B D applicatæ ab-<lb/>
<anchor type="note" xlink:label="note-0230-05a" xlink:href="note-0230-05"/>
ſcindunt ex vtraque parte axis duo ſegmen-<lb/>
<anchor type="figure" xlink:label="fig-0230-02a" xlink:href="fig-0230-02"/>
ta A E, &amp; </s>
  <s xml:id="echoid-s7296" xml:space="preserve">C F congruentia, propterea ſi-<lb/>milia erunt, atque ſimiliter poſita. </s>
  <s xml:id="echoid-s7297" xml:space="preserve">Secundo, <lb/>in ellipſi ductæ ſint ad axim quatuor ordina-<lb/>tim applicatæ, quarum binæ extremæ E F, <lb/>&amp; </s>
  <s xml:id="echoid-s7298" xml:space="preserve">I K æqualiter à centro D diſtent; </s>
  <s xml:id="echoid-s7299" xml:space="preserve">pari-<lb/>terque binæ intermediæ A C, &amp; </s>
  <s xml:id="echoid-s7300" xml:space="preserve">G H æqua-<lb/>
<anchor type="note" xlink:label="note-0230-06a" xlink:href="note-0230-06"/>
liter diſtent ab eodem centro: </s>
  <s xml:id="echoid-s7301" xml:space="preserve">quare quatuor <lb/>ſegmenta G I, H K, C F, &amp; </s>
  <s xml:id="echoid-s7302" xml:space="preserve">A E æqualia <lb/>erunt, &amp; </s>
  <s xml:id="echoid-s7303" xml:space="preserve">ſibi mutuo congruent, &amp; </s>
  <s xml:id="echoid-s7304" xml:space="preserve">propterea <lb/>ſimilid quoque inter ſe erunt.</s>
  <s xml:id="echoid-s7305" xml:space="preserve"/>
</p>
<div xml:id="echoid-div668" type="float" level="2" n="1">
<note position="right" xlink:label="note-0230-04" xlink:href="note-0230-04a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0230-05" xlink:href="note-0230-05a" xml:space="preserve">7. huius.</note>
  <figure xlink:label="fig-0230-02" xlink:href="fig-0230-02a">
    <image file="0230-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0230-02"/>
  </figure>
<note position="left" xlink:label="note-0230-06" xlink:href="note-0230-06a" xml:space="preserve">8. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7306" xml:space="preserve">Erit angulus N æqualis O, vti demõ-<lb/>
<anchor type="note" xlink:label="note-0230-07a" xlink:href="note-0230-07"/>
ſtrauimus, &amp;</s>
  <s xml:id="echoid-s7307" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7308" xml:space="preserve">Quoniam duo ſegmenta L <lb/>M, &amp; </s>
  <s xml:id="echoid-s7309" xml:space="preserve">A E, ponuntur ſimilia, atque eorum <lb/>baſes L M, &amp; </s>
  <s xml:id="echoid-s7310" xml:space="preserve">A E productæ occurrunt axi <lb/>in O, &amp; </s>
  <s xml:id="echoid-s7311" xml:space="preserve">N: </s>
  <s xml:id="echoid-s7312" xml:space="preserve">igitur vt demonſtratum eſt, <lb/>
<anchor type="note" xlink:label="note-0230-08a" xlink:href="note-0230-08"/>
anguli à contingentibus verticalibus ſegmen-<lb/>torum ſimilium L M, &amp; </s>
  <s xml:id="echoid-s7313" xml:space="preserve">A E cum axi com-<lb/>muni B D eiuſdem ſectionis continebunt an-
<pb o="193" file="0231" n="231" rhead="Conicor. Lib. VI."/>
gulos æquales; </s>
  <s xml:id="echoid-s7314" xml:space="preserve">ĳ verò anguli æquales ſunt angulis O, &amp; </s>
  <s xml:id="echoid-s7315" xml:space="preserve">N, cum baſes L M, &amp; </s>
  <s xml:id="echoid-s7316" xml:space="preserve"><lb/>A E parallelæ ſint contingentibus verticalibus eorundem ſegmentorum; </s>
  <s xml:id="echoid-s7317" xml:space="preserve">igitur an-<lb/>guli L O B, &amp; </s>
  <s xml:id="echoid-s7318" xml:space="preserve">A N B æquales ſunt inter ſe; </s>
  <s xml:id="echoid-s7319" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7320" xml:space="preserve">propterea duorum ſegmentorũ <lb/>baſes L M, &amp; </s>
  <s xml:id="echoid-s7321" xml:space="preserve">A E parallelæ ſunt inter ſe.</s>
  <s xml:id="echoid-s7322" xml:space="preserve"/>
</p>
<div xml:id="echoid-div669" type="float" level="2" n="2">
<note position="right" xlink:label="note-0230-07" xlink:href="note-0230-07a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0230-08" xlink:href="note-0230-08a" xml:space="preserve">Prop 16. <lb/>17. huius.</note>
</div>
</div>
<div xml:id="echoid-div671" type="section" level="1" n="223">
<head xml:id="echoid-head281" xml:space="preserve">SECTIO OCTAVA</head>
<head xml:id="echoid-head282" xml:space="preserve">Continens Propoſit. XX. &amp; XXI. <lb/>Apollonij.</head>
<head xml:id="echoid-head283" xml:space="preserve">PROPOSITIO XX.</head>
<p>
  <s xml:id="echoid-s7323" xml:space="preserve">SI in quibuslibet ſimilibus coniſectionibus A B C, &amp; </s>
  <s xml:id="echoid-s7324" xml:space="preserve">D E F <lb/>
<anchor type="note" xlink:label="note-0231-01a" xlink:href="note-0231-01"/>
ductæ fuerint ad axes B O, E Q ordinatim applicatæ A C, <lb/>D F, N L, P M, quarum illæ, quæ ad eaſdem partes verticum <lb/>B, &amp; </s>
  <s xml:id="echoid-s7325" xml:space="preserve">E ducuntur efficiant abſciſſas erectis proportionales, ſci-<lb/>licet I B ad B G ſit, vt K E ad E H, nec non L B ad B G, vt <lb/>M E ad E H: </s>
  <s xml:id="echoid-s7326" xml:space="preserve">Dico ſegmenta facta ab ordinatis ſimiliter poſi-<lb/>tis eſſe inter ſe ſimilia, ac ſimiliter poſita, ſcilicet N A ipſi P D, <lb/>atque A B ipſi D E, nec non N B ipſi P E.</s>
  <s xml:id="echoid-s7327" xml:space="preserve"/>
</p>
<div xml:id="echoid-div671" type="float" level="2" n="1">
<note position="left" xlink:label="note-0231-01" xlink:href="note-0231-01a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0231-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0231-01"/>
  </figure>
<p>
  <s xml:id="echoid-s7328" xml:space="preserve">Sintque primò ſectiones parabolæ; </s>
  <s xml:id="echoid-s7329" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7330" xml:space="preserve">educamus N A ad B L in O, &amp; </s>
  <s xml:id="echoid-s7331" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0231-02a" xlink:href="note-0231-02"/>
P D ad M E in Q. </s>
  <s xml:id="echoid-s7332" xml:space="preserve">Et quia G B ad B I eſt, vt H E ad E K, &amp; </s>
  <s xml:id="echoid-s7333" xml:space="preserve">B L ad <lb/>B G eſt vt M E ad E H; </s>
  <s xml:id="echoid-s7334" xml:space="preserve">ergo L B ad B I, nempe L N ad I A potentia <lb/>(19. </s>
  <s xml:id="echoid-s7335" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s7336" xml:space="preserve">nempe L N ad O I eandem proportionem habet, quàm M E <lb/>
<anchor type="note" xlink:label="note-0231-03a" xlink:href="note-0231-03"/>
<pb o="194" file="0232" n="232" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0232-01a" xlink:href="fig-0232-01"/>
ad E K, nempe P M ad D K potentia, nempe M Q ad Q K, &amp; </s>
  <s xml:id="echoid-s7337" xml:space="preserve">per con-<lb/>uerſionem rationis O L ad L I erit, vt Q M ad M K: </s>
  <s xml:id="echoid-s7338" xml:space="preserve">eſtque I L ad L B, <lb/>vt K M ad M E; </s>
  <s xml:id="echoid-s7339" xml:space="preserve">ergo O L ad L B eſt, vt Q M ad M E, &amp; </s>
  <s xml:id="echoid-s7340" xml:space="preserve">L B ad L N <lb/>eſt, vt E M ad M P (propter ſimilitudinem duarum ſectionum) ergo ex <lb/>
<anchor type="note" xlink:label="note-0232-01a" xlink:href="note-0232-01"/>
<anchor type="note" xlink:label="note-0232-02a" xlink:href="note-0232-02"/>
æqualitate O L ad L N erit, vt Q M ad M P; </s>
  <s xml:id="echoid-s7341" xml:space="preserve">ſuntque M, &amp; </s>
  <s xml:id="echoid-s7342" xml:space="preserve">L duo an-<lb/>guli recti; </s>
  <s xml:id="echoid-s7343" xml:space="preserve">ergo N L O ſimile eſt P M Q; </s>
  <s xml:id="echoid-s7344" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7345" xml:space="preserve">per R, S ſemipartitiones ip-<lb/>ſarum N A, D P ducamus ipſas T V, X Y parallelas duobus axibus, &amp; </s>
  <s xml:id="echoid-s7346" xml:space="preserve"><lb/>ex duobus punctis V, Y, educamus perpendiculares V Z, Y a ſuper duos <lb/>axes. </s>
  <s xml:id="echoid-s7347" xml:space="preserve">Et quia N O ad O A eſt, vt P Q ad Q D comparando antecedẽ-<lb/>tes ad ſemiſſes differentiarum terminorum vel ad ſemiſummas eorũ fiet N <lb/>
<anchor type="note" xlink:label="note-0232-03a" xlink:href="note-0232-03"/>
O ad R O, nempe N L ad L T, quæ eſt æqualis ipſi V Z, nempe L B <lb/>ad B Z longitudine (19. </s>
  <s xml:id="echoid-s7348" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s7349" xml:space="preserve">vt P Q ad Q S, nempe P M ad X M æ-<lb/>qualem ipſi Y a, nempe longitudine, vt M E ad E a (19. </s>
  <s xml:id="echoid-s7350" xml:space="preserve">ex 1) igitur <lb/>
<anchor type="note" xlink:label="note-0232-04a" xlink:href="note-0232-04"/>
comparando differentias terminorum ad antecedentes, erit Z L ad L B, <lb/>vt a M ad M E, &amp; </s>
  <s xml:id="echoid-s7351" xml:space="preserve">L B ad L O eſt, vt M E ad M Q; </s>
  <s xml:id="echoid-s7352" xml:space="preserve">ergo ex æqualitate <lb/>L Z ad L O, nempe N b ad N O eſt, vt M a ad M Q, nempe P c ad P Q <lb/>
<anchor type="note" xlink:label="note-0232-05a" xlink:href="note-0232-05"/>
crat autem prius N R ad N O, vt S P ad P Q, &amp; </s>
  <s xml:id="echoid-s7353" xml:space="preserve">comparando ſemisũ-<lb/>
<anchor type="note" xlink:label="note-0232-06a" xlink:href="note-0232-06"/>
mas, vel ſemidifferentias terminorum ad eorundem differentias O R ad <lb/>R b erit, vt Q S ad S c, &amp; </s>
  <s xml:id="echoid-s7354" xml:space="preserve">R b ad R V eſt, vt S c ad S Y; </s>
  <s xml:id="echoid-s7355" xml:space="preserve">quia <lb/>duo triangula V R b, Y S c ſunt ſimilia; </s>
  <s xml:id="echoid-s7356" xml:space="preserve">ergo R O ad R V eandem pro-<lb/>portionem habet, quàm Q S ad S Y; </s>
  <s xml:id="echoid-s7357" xml:space="preserve">ſed tangens in V perueniens ad L O <lb/>
<anchor type="note" xlink:label="note-0232-07a" xlink:href="note-0232-07"/>
æqualis eſt O R, cui parallela eſt; </s>
  <s xml:id="echoid-s7358" xml:space="preserve">quia cadit inter duas lineas parallelas; <lb/></s>
  <s xml:id="echoid-s7359" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7360" xml:space="preserve">ſimiliter tangens in Y parallela eſt S Q, &amp; </s>
  <s xml:id="echoid-s7361" xml:space="preserve">ei æqualis; </s>
  <s xml:id="echoid-s7362" xml:space="preserve">ergo V R ab-<lb/>ſciſſa ad tangentem eſt, vt abſciſſa S Y ad eius tangentem, &amp; </s>
  <s xml:id="echoid-s7363" xml:space="preserve">angulus Q <lb/>æqualis eſt angulo O; </s>
  <s xml:id="echoid-s7364" xml:space="preserve">igitur duo ſegmenta N V A, P Y D ſunt ſimilia <lb/>(16. </s>
  <s xml:id="echoid-s7365" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s7366" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7367" xml:space="preserve">pariter duo ſegmenta A B C, D E F, atque duo ſegmen-<lb/>
<anchor type="note" xlink:label="note-0232-08a" xlink:href="note-0232-08"/>
ta N B, P E ſunt ſimilia inter ſe, &amp; </s>
  <s xml:id="echoid-s7368" xml:space="preserve">ſimiliter poſita.</s>
  <s xml:id="echoid-s7369" xml:space="preserve"/>
</p>
<div xml:id="echoid-div672" type="float" level="2" n="2">
<note position="left" xlink:label="note-0231-02" xlink:href="note-0231-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0231-03" xlink:href="note-0231-03a" xml:space="preserve">20. lib. 1.</note>
  <figure xlink:label="fig-0232-01" xlink:href="fig-0232-01a">
    <image file="0232-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0232-01"/>
  </figure>
<note position="right" xlink:label="note-0232-01" xlink:href="note-0232-01a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0232-02" xlink:href="note-0232-02a" xml:space="preserve">Defin. 2.</note>
<note position="right" xlink:label="note-0232-03" xlink:href="note-0232-03a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0232-04" xlink:href="note-0232-04a" xml:space="preserve">20. lib. 1.</note>
<note position="left" xlink:label="note-0232-05" xlink:href="note-0232-05a" xml:space="preserve">Ibidem.</note>
<note position="right" xlink:label="note-0232-06" xlink:href="note-0232-06a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0232-07" xlink:href="note-0232-07a" xml:space="preserve">f</note>
<note position="right" xlink:label="note-0232-08" xlink:href="note-0232-08a" xml:space="preserve">g</note>
</div>
<p>
  <s xml:id="echoid-s7370" xml:space="preserve">Deinde ponamus aliud ſegmentum P d. </s>
  <s xml:id="echoid-s7371" xml:space="preserve">Dico non eſſe ſimile alicui <lb/>
<anchor type="note" xlink:label="note-0232-09a" xlink:href="note-0232-09"/>
prædictorum ſegmentorum, quia non abſcinduntur à duabus ordinationi-<lb/>bus vnius axis (18. </s>
  <s xml:id="echoid-s7372" xml:space="preserve">ex 6.)</s>
  <s xml:id="echoid-s7373" xml:space="preserve">. </s>
  <s xml:id="echoid-s7374" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s7375" xml:space="preserve"/>
</p>
<div xml:id="echoid-div673" type="float" level="2" n="3">
<note position="right" xlink:label="note-0232-09" xlink:href="note-0232-09a" xml:space="preserve">h</note>
</div>
<pb o="195" file="0233" n="233" rhead="Conicor. Lib. VI."/>
  <figure>
    <image file="0233-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0233-01"/>
  </figure>
</div>
<div xml:id="echoid-div675" type="section" level="1" n="224">
<head xml:id="echoid-head284" xml:space="preserve">PROPOSITIO XXI.</head>
<p>
  <s xml:id="echoid-s7376" xml:space="preserve">SInt poſtea duæ illæ ſectiones hyperbolicæ, &amp; </s>
  <s xml:id="echoid-s7377" xml:space="preserve">ellipticæ ſi-<lb/>miles, &amp; </s>
  <s xml:id="echoid-s7378" xml:space="preserve">earum centra T, X (remanentibus lineis, &amp; </s>
  <s xml:id="echoid-s7379" xml:space="preserve">ſi-<lb/>gnis, vt prius) &amp; </s>
  <s xml:id="echoid-s7380" xml:space="preserve">ducantur duæ contingentes V e, &amp; </s>
  <s xml:id="echoid-s7381" xml:space="preserve">Y f.</s>
  <s xml:id="echoid-s7382" xml:space="preserve"/>
</p>
  <figure>
    <image file="0233-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0233-02"/>
  </figure>
<p>
  <s xml:id="echoid-s7383" xml:space="preserve">Quoniam B G ad B I ſuppoſita eſt, vt H E ad E K, &amp; </s>
  <s xml:id="echoid-s7384" xml:space="preserve">pariter G B ad <lb/>
<anchor type="note" xlink:label="note-0233-01a" xlink:href="note-0233-01"/>
B L, vt H E ad E M; </s>
  <s xml:id="echoid-s7385" xml:space="preserve">ergo ex æqualitate, &amp; </s>
  <s xml:id="echoid-s7386" xml:space="preserve">per conuerſionem rationis <lb/>B L ad L I eſt vt E M ad M K; </s>
  <s xml:id="echoid-s7387" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7388" xml:space="preserve">propter ſimilitudinem duarum ſectio-<lb/>
<anchor type="note" xlink:label="note-0233-02a" xlink:href="note-0233-02"/>
num N L ad A I nempe L O ad O I eſt, vt M P ad D K, nempe M Q <lb/>ad Q K, &amp; </s>
  <s xml:id="echoid-s7389" xml:space="preserve">antecedentes ad ſummas vel differentias terminorum, ſcilicet <lb/>
<anchor type="note" xlink:label="note-0233-03a" xlink:href="note-0233-03"/>
O L ad L I eandem proportionem habebit, quàm Q M ad M K, &amp; </s>
  <s xml:id="echoid-s7390" xml:space="preserve">ex <lb/>
<anchor type="note" xlink:label="note-0233-04a" xlink:href="note-0233-04"/>
æqualitate O L ad L B erit, vt Q M ad M E, ſed B L ad L N eſt, vt E <lb/>M ad M P, cum ex ſuppoſitione ſectiones ſint ſimiles; </s>
  <s xml:id="echoid-s7391" xml:space="preserve">ergo O L ad L N <lb/>eſt, vt Q M ad M P; </s>
  <s xml:id="echoid-s7392" xml:space="preserve">ſuntque L, M duo anguli recti: </s>
  <s xml:id="echoid-s7393" xml:space="preserve">ergo anguli O, Q,
<pb o="196" file="0234" n="234" rhead="Apollonij Pergæi"/>
nempe e, f ſunt æquales: </s>
  <s xml:id="echoid-s7394" xml:space="preserve">deinde ducantur V Z, Y a ad axes ordinatæ; <lb/></s>
  <s xml:id="echoid-s7395" xml:space="preserve">ergo (propter ſimilitudinem duarum ſectionum) T Z in Z e ad quadra-<lb/>
<anchor type="note" xlink:label="note-0234-01a" xlink:href="note-0234-01"/>
tum Z V eandem proportionem habebit, quam X a in a f ad quadratum <lb/>a Y, &amp; </s>
  <s xml:id="echoid-s7396" xml:space="preserve">angulus e æqualis eſt angulo f; </s>
  <s xml:id="echoid-s7397" xml:space="preserve">igitur V e T ſimile eſt Y f X, &amp; </s>
  <s xml:id="echoid-s7398" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0234-02a" xlink:href="note-0234-02"/>
pariter O T R, Q X S; </s>
  <s xml:id="echoid-s7399" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7400" xml:space="preserve">propterea O e ad R V eandem proportionem <lb/>habebit, quàm Q f ad Y S, &amp; </s>
  <s xml:id="echoid-s7401" xml:space="preserve">propter ſimilitudinem duarum ſectionum <lb/>B I ad I A eſt, vt E K ad K D, &amp; </s>
  <s xml:id="echoid-s7402" xml:space="preserve">A I ad I O, vt D K ad K Q propter <lb/>ſimilitudinem duorum triangulorum; </s>
  <s xml:id="echoid-s7403" xml:space="preserve">ergo (ex æqualitate, &amp; </s>
  <s xml:id="echoid-s7404" xml:space="preserve">comparan-<lb/>
<anchor type="note" xlink:label="note-0234-03a" xlink:href="note-0234-03"/>
do antecedentes ad ſummas vel differentias terminorum) erit B I ad B O, <lb/>
<anchor type="note" xlink:label="note-0234-04a" xlink:href="note-0234-04"/>
vt E K ad E Q, ſed B T ad B I erat, vt X E ad E K (propter ſimilitu-<lb/>dinem duarum ſectionum) <lb/>
<anchor type="figure" xlink:label="fig-0234-01a" xlink:href="fig-0234-01"/>
ergo ex æqualitate, &amp; </s>
  <s xml:id="echoid-s7405" xml:space="preserve">rurſus <lb/>comparando antecedẽtes ad <lb/>ſummas vel differentias ter-<lb/>
<anchor type="note" xlink:label="note-0234-05a" xlink:href="note-0234-05"/>
minorum B T ad T O erit, <lb/>vt X E ad X Q, cumque T <lb/>Z in Z e ad quadratum V Z <lb/>
<anchor type="note" xlink:label="note-0234-06a" xlink:href="note-0234-06"/>
ſit vt X a in a f ad quadra-<lb/>tum a Y (39. </s>
  <s xml:id="echoid-s7406" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s7407" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7408" xml:space="preserve">qua-<lb/>dratum V Z ad quadratum <lb/>Z e eſt, vt quadratum a Y ad <lb/>quadratũ a f erit T Z in Z e, <lb/>ad quadratũ Z e, nempe T Z <lb/>ad Z e vt X a in a f ad quadra <lb/>tum a f nempe G a ad a f, &amp; </s>
  <s xml:id="echoid-s7409" xml:space="preserve"><lb/>comparãdo antecedentes ad <lb/>differnntias terminorũ in hy-<lb/>perbola, &amp; </s>
  <s xml:id="echoid-s7410" xml:space="preserve">ad eorum ſummas <lb/>in ellipſi, fiet Z T ad T e, nẽ-<lb/>pe quadratum B T (quod eſt <lb/>æquale ipſi Z T in T e (39 ex 1.) </s>
  <s xml:id="echoid-s7411" xml:space="preserve">ad quadratnm T e eſt, vt X a ad X f, <lb/>
<anchor type="note" xlink:label="note-0234-07a" xlink:href="note-0234-07"/>
nempe a X in X f, quod eſt æquale quadrato E X (39. </s>
  <s xml:id="echoid-s7412" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s7413" xml:space="preserve">ad qua-<lb/>dratum X f; </s>
  <s xml:id="echoid-s7414" xml:space="preserve">ergo B T ad T e potentia eſt, vt E X ad X f; </s>
  <s xml:id="echoid-s7415" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7416" xml:space="preserve">propterea <lb/>
<anchor type="note" xlink:label="note-0234-08a" xlink:href="note-0234-08"/>
<anchor type="figure" xlink:label="fig-0234-02a" xlink:href="fig-0234-02"/>
<pb o="197" file="0235" n="235" rhead="Conicor. Lib. VI."/>
T B ad T e erit, vt E X ad X f; </s>
  <s xml:id="echoid-s7417" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7418" xml:space="preserve">iam oſtendimus, quod B T ad T O <lb/>eſt, vt E X ad X Q; </s>
  <s xml:id="echoid-s7419" xml:space="preserve">igitur ex æqualitate, &amp; </s>
  <s xml:id="echoid-s7420" xml:space="preserve">comparando terminorum <lb/>differentias ad conſequentes erit O e ad e T, vt Q f ad f X; </s>
  <s xml:id="echoid-s7421" xml:space="preserve">ſed T e ad e <lb/>
<anchor type="note" xlink:label="note-0235-01a" xlink:href="note-0235-01"/>
V eandem proportionem habet quam X f ad f Y, eo quod oſtenſa ſunt <lb/>ſimilia triangula V T e, Y X f; </s>
  <s xml:id="echoid-s7422" xml:space="preserve">quare O e ad e V eſt vt Q f ad f Y; </s>
  <s xml:id="echoid-s7423" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7424" xml:space="preserve"><lb/>iam oſtendimus, quod O e ad R V eandem proportionem habet, quàm <lb/>Q f ad S Y; </s>
  <s xml:id="echoid-s7425" xml:space="preserve">ergo R V ad V e eſt, vt S Y ad Y f, &amp; </s>
  <s xml:id="echoid-s7426" xml:space="preserve">angulus e æqualis <lb/>eſt angulo f; </s>
  <s xml:id="echoid-s7427" xml:space="preserve">igitur duo ſegmenta N V A, P Y D ſimilia ſunt inter ſe <lb/>(17. </s>
  <s xml:id="echoid-s7428" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s7429" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7430" xml:space="preserve">ſimiliter poſita. </s>
  <s xml:id="echoid-s7431" xml:space="preserve">Inſuper dico, non eſſe ſimilia alicui alte-<lb/>ri ſegmento; </s>
  <s xml:id="echoid-s7432" xml:space="preserve">quia non abſcinduntur ab vna ordinatione, aut duabus, &amp; </s>
  <s xml:id="echoid-s7433" xml:space="preserve"><lb/>earum diſtantia in ellipſi à centro non eſt æqualis (18. </s>
  <s xml:id="echoid-s7434" xml:space="preserve">ex 6.)</s>
  <s xml:id="echoid-s7435" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s7436" xml:space="preserve">hoc erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s7437" xml:space="preserve"/>
</p>
<div xml:id="echoid-div675" type="float" level="2" n="1">
<note position="left" xlink:label="note-0233-01" xlink:href="note-0233-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0233-02" xlink:href="note-0233-02a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0233-03" xlink:href="note-0233-03a" xml:space="preserve">Lem. 1. <lb/>lib. 5.</note>
<note position="left" xlink:label="note-0233-04" xlink:href="note-0233-04a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0234-01" xlink:href="note-0234-01a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0234-02" xlink:href="note-0234-02a" xml:space="preserve">Propoſ. 6. <lb/>pręmiſſ.</note>
<note position="left" xlink:label="note-0234-03" xlink:href="note-0234-03a" xml:space="preserve">Lem. 1. <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0234-04" xlink:href="note-0234-04a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0234-01" xlink:href="fig-0234-01a">
    <image file="0234-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0234-01"/>
  </figure>
<note position="left" xlink:label="note-0234-05" xlink:href="note-0234-05a" xml:space="preserve">Ibldem.</note>
<note position="left" xlink:label="note-0234-06" xlink:href="note-0234-06a" xml:space="preserve">37. lib. 1.</note>
<note position="left" xlink:label="note-0234-07" xlink:href="note-0234-07a" xml:space="preserve">37. lib. 1.</note>
<note position="left" xlink:label="note-0234-08" xlink:href="note-0234-08a" xml:space="preserve">Ibidem.</note>
  <figure xlink:label="fig-0234-02" xlink:href="fig-0234-02a">
    <image file="0234-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0234-02"/>
  </figure>
<note position="right" xlink:label="note-0235-01" xlink:href="note-0235-01a" xml:space="preserve">Lem. 1. <lb/>lib. 5.</note>
</div>
</div>
<div xml:id="echoid-div677" type="section" level="1" n="225">
<head xml:id="echoid-head285" xml:space="preserve">PROPOSITIO XXII.</head>
<p>
  <s xml:id="echoid-s7438" xml:space="preserve">SEctionum non ſimilium A B C, D E F vnum ſegmentum <lb/>vnius non eſt ſimile alicui ſegmento alterius.</s>
  <s xml:id="echoid-s7439" xml:space="preserve"/>
</p>
  <figure>
    <image file="0235-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0235-01"/>
  </figure>
<p>
  <s xml:id="echoid-s7440" xml:space="preserve">Si enim hoc verum non eſt, ſit ſegmentum G C ſectionis A B C (ſi <lb/>fieri poteſt) ſimile ipſi H F alterius ſectionis D E F, &amp; </s>
  <s xml:id="echoid-s7441" xml:space="preserve">iungamus G C, <lb/>H F, eaſdẽq; </s>
  <s xml:id="echoid-s7442" xml:space="preserve">bifariam ſecemus in I, K; </s>
  <s xml:id="echoid-s7443" xml:space="preserve">iungamuſque L I, M K; </s>
  <s xml:id="echoid-s7444" xml:space="preserve">quæ ſint <lb/>
<anchor type="note" xlink:label="note-0235-02a" xlink:href="note-0235-02"/>
duæ diametri, &amp; </s>
  <s xml:id="echoid-s7445" xml:space="preserve">ſecent ſegmenta in B, E: </s>
  <s xml:id="echoid-s7446" xml:space="preserve">ſi itaque fuerint duo axes, cũ <lb/>duo ſegmenta ſint ſimilia, vtique egrederentur in eorum ſingulis ordina-<lb/>
<anchor type="note" xlink:label="note-0235-03a" xlink:href="note-0235-03"/>
tiones ad duos axes, numero æquales, continentes cum axibus angulos <lb/>rectos, &amp; </s>
  <s xml:id="echoid-s7447" xml:space="preserve">proportiones ordinationum ad ſua abſciſſa in qualibet earum <lb/>eſſent æedem, ac abſciſſæ ad abſciſſas proportionales quoque eſſent. </s>
  <s xml:id="echoid-s7448" xml:space="preserve">Et <lb/>
<anchor type="note" xlink:label="note-0235-04a" xlink:href="note-0235-04"/>
<anchor type="note" xlink:label="note-0235-05a" xlink:href="note-0235-05"/>
propterea duæ ſectiones A B C, D E F ſimiles erunt, ſed iam ſuppoſitæ <lb/>fuerunt non ſimiles; </s>
  <s xml:id="echoid-s7449" xml:space="preserve">quod eſt abſurdum. </s>
  <s xml:id="echoid-s7450" xml:space="preserve">Si verò I L, M K non fuerint <lb/>axes, educamus ex B, E ad duos axes L P, M Q duas perpendiculares <lb/>B P, E Q, &amp; </s>
  <s xml:id="echoid-s7451" xml:space="preserve">duas tangentes B N, &amp; </s>
  <s xml:id="echoid-s7452" xml:space="preserve">E O: </s>
  <s xml:id="echoid-s7453" xml:space="preserve">itaque (propter ſimilitudinẽ <lb/>
<anchor type="note" xlink:label="note-0235-06a" xlink:href="note-0235-06"/>
duorum ſegmentorum) ſimilia erunt B N L, E O M; </s>
  <s xml:id="echoid-s7454" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7455" xml:space="preserve">pariter L B P, <lb/>M E Q; </s>
  <s xml:id="echoid-s7456" xml:space="preserve">atque quadratum B P ad L B in P N, nempe in eadem propor-
<pb o="198" file="0236" n="236" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0236-01a" xlink:href="fig-0236-01"/>
tione figuræ diametri A L (40. </s>
  <s xml:id="echoid-s7457" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s7458" xml:space="preserve">erit vt quadratum E Q ad M Q <lb/>
<anchor type="note" xlink:label="note-0236-01a" xlink:href="note-0236-01"/>
in O Q, nempe in eadem proportione figuræ diametri D M (40. </s>
  <s xml:id="echoid-s7459" xml:space="preserve">ex 1.) <lb/></s>
  <s xml:id="echoid-s7460" xml:space="preserve">
<anchor type="note" xlink:label="note-0236-02a" xlink:href="note-0236-02"/>
quapropter duæ proportiones figurarum earundem ſectionum ſunt eædem <lb/>inter ſe; </s>
  <s xml:id="echoid-s7461" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7462" xml:space="preserve">propterea duæ ſectiones ſunt ſimiles (12. </s>
  <s xml:id="echoid-s7463" xml:space="preserve">ex 6.) </s>
  <s xml:id="echoid-s7464" xml:space="preserve">at ſuppoſitæ <lb/>fuerunt non ſimiles. </s>
  <s xml:id="echoid-s7465" xml:space="preserve">Quod eſt abſurdum.</s>
  <s xml:id="echoid-s7466" xml:space="preserve"/>
</p>
<div xml:id="echoid-div677" type="float" level="2" n="1">
<note position="right" xlink:label="note-0235-02" xlink:href="note-0235-02a" xml:space="preserve">44. lib. 2.</note>
<note position="right" xlink:label="note-0235-03" xlink:href="note-0235-03a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
<note position="right" xlink:label="note-0235-04" xlink:href="note-0235-04a" xml:space="preserve">Defin. 2. <lb/>huius.</note>
<note position="left" xlink:label="note-0235-05" xlink:href="note-0235-05a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0235-06" xlink:href="note-0235-06a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0236-01" xlink:href="fig-0236-01a">
    <image file="0236-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0236-01"/>
  </figure>
<note position="left" xlink:label="note-0236-01" xlink:href="note-0236-01a" xml:space="preserve">37. lib. 1.</note>
<note position="left" xlink:label="note-0236-02" xlink:href="note-0236-02a" xml:space="preserve">Ibidem.</note>
</div>
</div>
<div xml:id="echoid-div679" type="section" level="1" n="226">
<head xml:id="echoid-head286" xml:space="preserve">PROPOSITIO XXIII.</head>
<p>
  <s xml:id="echoid-s7467" xml:space="preserve">SI autem ſectio A B C fuerit parabola, &amp; </s>
  <s xml:id="echoid-s7468" xml:space="preserve">ſectio D E F hy-<lb/>perbola, aut ellipſis: </s>
  <s xml:id="echoid-s7469" xml:space="preserve">manifeſtum eſt, ſectiones non eſſe in-<lb/>
<anchor type="note" xlink:label="note-0236-03a" xlink:href="note-0236-03"/>
ter ſe ſimiles. </s>
  <s xml:id="echoid-s7470" xml:space="preserve">Et dico quod duo ſegmenta G C, H F non ſunt <lb/>ſimilia.</s>
  <s xml:id="echoid-s7471" xml:space="preserve"/>
</p>
<div xml:id="echoid-div679" type="float" level="2" n="1">
<note position="left" xlink:label="note-0236-03" xlink:href="note-0236-03a" xml:space="preserve">13. huius.</note>
</div>
<p>
  <s xml:id="echoid-s7472" xml:space="preserve">Si enim ſimilia eſſent haberent conditiones ſimilitudinis, quod eſt im-<lb/>
<anchor type="note" xlink:label="note-0236-04a" xlink:href="note-0236-04"/>
poſſibile, quemadmodum oſtenſum eſt in omnibus ſectionibus ad propo-<lb/>ſitionem 13. </s>
  <s xml:id="echoid-s7473" xml:space="preserve">ſi vero vna earum fuerit hyperbole, altera verò ellipſis, <lb/>idipſum oſtenſum eſt ad propoſitionem 14. </s>
  <s xml:id="echoid-s7474" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s7475" xml:space="preserve"/>
</p>
<div xml:id="echoid-div680" type="float" level="2" n="2">
<note position="right" xlink:label="note-0236-04" xlink:href="note-0236-04a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div682" type="section" level="1" n="227">
<head xml:id="echoid-head287" xml:space="preserve">PROPOSITIO XXIV.</head>
<p>
  <s xml:id="echoid-s7476" xml:space="preserve">CViuslibet coniſectionis A C D portio B A C D non erit <lb/>arcus circuli.</s>
  <s xml:id="echoid-s7477" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7478" xml:space="preserve">Si enim hoc verum non eſt educamus in illa chordas A B, C D, A C, <lb/>quarum nulla alteri ſit parallela: </s>
  <s xml:id="echoid-s7479" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7480" xml:space="preserve">educamus E F parallelam A B, &amp; </s>
  <s xml:id="echoid-s7481" xml:space="preserve">E <lb/>G parallelam A C, atque G H parallelam C D, &amp; </s>
  <s xml:id="echoid-s7482" xml:space="preserve">per ſingularum dua-<lb/>rum æquidiſtantium ſemipartitiones iungamus K I, L M, N O, quæ qui-
<pb o="199" file="0237" n="237" rhead="Conicor. Lib. VI."/>
dem lineæ perpendiculares ſunt ad præ-<lb/>
<anchor type="figure" xlink:label="fig-0237-01a" xlink:href="fig-0237-01"/>
dictas chordas, ſuntque etiam diametri <lb/>ſectionis, ergo I K, L M, N O ſunt axes, <lb/>nec ſibi in directum coincidunt; </s>
  <s xml:id="echoid-s7483" xml:space="preserve">quia <lb/>chordæ primo eductæ inter ſe parallelæ <lb/>non erant: </s>
  <s xml:id="echoid-s7484" xml:space="preserve">hoc autem eſt abſurdum, <lb/>quia in qualibet ſectione reperiri non <lb/>poſſunt plures, quàm duo axes (52. </s>
  <s xml:id="echoid-s7485" xml:space="preserve">ex <lb/>
<anchor type="note" xlink:label="note-0237-01a" xlink:href="note-0237-01"/>
2.)</s>
  <s xml:id="echoid-s7486" xml:space="preserve">; </s>
  <s xml:id="echoid-s7487" xml:space="preserve">ergo fieri non poteſt, vt ſectionis <lb/>conicæ portio ſit arcus circuli. </s>
  <s xml:id="echoid-s7488" xml:space="preserve">Quod <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s7489" xml:space="preserve"/>
</p>
<div xml:id="echoid-div682" type="float" level="2" n="1">
  <figure xlink:label="fig-0237-01" xlink:href="fig-0237-01a">
    <image file="0237-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0237-01"/>
  </figure>
<note position="right" xlink:label="note-0237-01" xlink:href="note-0237-01a" xml:space="preserve">48. lib. 2.</note>
</div>
</div>
<div xml:id="echoid-div684" type="section" level="1" n="228">
<head xml:id="echoid-head288" xml:space="preserve">Notæ in Propoſit. XX.</head>
<p>
  <s xml:id="echoid-s7490" xml:space="preserve">QVodlibet duorum ſegmentorum, vt A B C, D E F in duobus ſeg-<lb/>
<anchor type="note" xlink:label="note-0237-02a" xlink:href="note-0237-02"/>
mentis ſimilibus, vt N A C, P D F abſciſſa ſint ab ordinatis duo-<lb/>rum axium ſectionum, vt A C, D F, N L, P M, A M, A S, <lb/>K M ad latus ſuarum verticum vt B, E; </s>
  <s xml:id="echoid-s7491" xml:space="preserve">ſitque proportio earum abſciſ-<lb/>ſarum ad erecta duorum ſegmentorum eadem, nempe I B ad B G, vt K <lb/>E ad E H, &amp; </s>
  <s xml:id="echoid-s7492" xml:space="preserve">L B ad B G, vt M E ad E H: </s>
  <s xml:id="echoid-s7493" xml:space="preserve">vtique duo ſegmenta A B <lb/>C, D E F, N B, P E ſimilia ſunt, &amp; </s>
  <s xml:id="echoid-s7494" xml:space="preserve">ſimilia poſitione: </s>
  <s xml:id="echoid-s7495" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s7496" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7497" xml:space="preserve">Textus hic <lb/>
<anchor type="figure" xlink:label="fig-0237-02a" xlink:href="fig-0237-02"/>
adeo corruptus eſt, vt ne Apollonius quidem, ſi reuiuiſceret, ſenſum ex verbis <lb/>tam inconcinnis, &amp; </s>
  <s xml:id="echoid-s7498" xml:space="preserve">non coherentibus elicere poſſet. </s>
  <s xml:id="echoid-s7499" xml:space="preserve">Itaque diuinando eam eſſe <lb/>veram lectionem cenſeo; </s>
  <s xml:id="echoid-s7500" xml:space="preserve">quàm in textu appoſui.</s>
  <s xml:id="echoid-s7501" xml:space="preserve"/>
</p>
<div xml:id="echoid-div684" type="float" level="2" n="1">
<note position="left" xlink:label="note-0237-02" xlink:href="note-0237-02a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0237-02" xlink:href="fig-0237-02a">
    <image file="0237-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0237-02"/>
  </figure>
</div>
<pb o="200" file="0238" n="238" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s7502" xml:space="preserve">Educamus itaque N A ad O ex B L, &amp; </s>
  <s xml:id="echoid-s7503" xml:space="preserve">P D ad Q ex M E, quia B G <lb/>
<anchor type="note" xlink:label="note-0238-01a" xlink:href="note-0238-01"/>
ad B I eſt, vt H E ad E K, &amp; </s>
  <s xml:id="echoid-s7504" xml:space="preserve">B G ad B L eſt vt H E ad E M; </s>
  <s xml:id="echoid-s7505" xml:space="preserve">ergo L B <lb/>ad B I, nempe L N ad A I (19. </s>
  <s xml:id="echoid-s7506" xml:space="preserve">ex 1. </s>
  <s xml:id="echoid-s7507" xml:space="preserve">(nempe L O ad O I eſt vt M E <lb/>ad E K, nempe P M ad D K, nempe M Q ad Q K; </s>
  <s xml:id="echoid-s7508" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7509" xml:space="preserve">contra O L ad L <lb/>I, vt V M ad M K, &amp;</s>
  <s xml:id="echoid-s7510" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7511" xml:space="preserve">Addenda non nulla verba, quæ deficiunt, &amp; </s>
  <s xml:id="echoid-s7512" xml:space="preserve">reliqua <lb/>reſtituenda cenſui, vt in textu leguntur. </s>
  <s xml:id="echoid-s7513" xml:space="preserve">Zuoniam B G ad B I eſt vt H E ad <lb/>E K, &amp; </s>
  <s xml:id="echoid-s7514" xml:space="preserve">B L ad B G eſt vt M E ad E H; </s>
  <s xml:id="echoid-s7515" xml:space="preserve">ergo, ex æqualitate, L B ad B I <lb/>eandem proportionem habet, quàm M E ad E K, ſed quadratum N L ad qua-<lb/>dratum A I eſt in parabola, vt abſcißa L B ad B I; </s>
  <s xml:id="echoid-s7516" xml:space="preserve">pariterque quadratum P <lb/>
<anchor type="note" xlink:label="note-0238-02a" xlink:href="note-0238-02"/>
M ad quadratum D K eſt, vt M E ad E K: </s>
  <s xml:id="echoid-s7517" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7518" xml:space="preserve">propterea quadratum N L ad <lb/>quadratum A I eandem proportionem habebit quàm quadratum P M ad quadra-<lb/>tum D K; </s>
  <s xml:id="echoid-s7519" xml:space="preserve">igitur N L ad A I eandem proportionem habebit, quàm P M ad D <lb/>
<anchor type="figure" xlink:label="fig-0238-01a" xlink:href="fig-0238-01"/>
K; </s>
  <s xml:id="echoid-s7520" xml:space="preserve">ſed vt N L ad A I ita eſt L O ad O I (propter parallelas A I, N L, &amp; </s>
  <s xml:id="echoid-s7521" xml:space="preserve">ſimi-<lb/>litudinem triangulorũ A I O, &amp; </s>
  <s xml:id="echoid-s7522" xml:space="preserve">O N L) pariterg; </s>
  <s xml:id="echoid-s7523" xml:space="preserve">vt P M ad D K ita eſt M <lb/>Z ad Z K (propter ſimilitudinem triangulorum Q M P, &amp; </s>
  <s xml:id="echoid-s7524" xml:space="preserve">Q K D) igitur <lb/>L O ad O I eandem proportionem habebit, quàm M Q ad Q K; </s>
  <s xml:id="echoid-s7525" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7526" xml:space="preserve">compa-<lb/>rando antecedentes ad differentias, vel ſummas terminorum O L ad L I eandem <lb/>proportionem habebit, quàm Q M ad M K.</s>
  <s xml:id="echoid-s7527" xml:space="preserve"/>
</p>
<div xml:id="echoid-div685" type="float" level="2" n="2">
<note position="left" xlink:label="note-0238-01" xlink:href="note-0238-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0238-02" xlink:href="note-0238-02a" xml:space="preserve">20. lib. 1.</note>
  <figure xlink:label="fig-0238-01" xlink:href="fig-0238-01a">
    <image file="0238-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0238-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7528" xml:space="preserve">Et B L ad L N eſt vt E M ad M P (propter ſimilitudinem duorum <lb/>ſegmentorum) ergo ex æqualitate O L ad L N, &amp;</s>
  <s xml:id="echoid-s7529" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7530" xml:space="preserve">Sequitur quidem hoc <lb/>
<anchor type="note" xlink:label="note-0238-03a" xlink:href="note-0238-03"/>
non propter ſimilitudinem ſegmentorum, quandoquidem ſegmenta ſimilia non <lb/>ſupponuntur ſed quia ſemper parabolæ ſunt ſimiles, &amp; </s>
  <s xml:id="echoid-s7531" xml:space="preserve">in eis poſitæ ſunt axium <lb/>abſciſſæ L B, &amp; </s>
  <s xml:id="echoid-s7532" xml:space="preserve">M E proportionales lateribus rectis B G, &amp; </s>
  <s xml:id="echoid-s7533" xml:space="preserve">E H, propterea <lb/>
<anchor type="note" xlink:label="note-0238-04a" xlink:href="note-0238-04"/>
(vt in prop. </s>
  <s xml:id="echoid-s7534" xml:space="preserve">11. </s>
  <s xml:id="echoid-s7535" xml:space="preserve">huius oſtenſum eſt ) B L ad L N eandem proportionem habebit <lb/>quàm E M ad M P; </s>
  <s xml:id="echoid-s7536" xml:space="preserve">ſed prius L B ad B I erat vt M E ad E K, ergo comparã-<lb/>do differentias terminorum ad antecedentes erit I L ad L B vt K M ad M E, <lb/>eſtq; </s>
  <s xml:id="echoid-s7537" xml:space="preserve">oſtenſa O L ad L I vt Q M ad M K, ergo ex æquali ordinata O L ad L B <lb/>erit vt Q M ad M E.</s>
  <s xml:id="echoid-s7538" xml:space="preserve"/>
</p>
<div xml:id="echoid-div686" type="float" level="2" n="3">
<note position="right" xlink:label="note-0238-03" xlink:href="note-0238-03a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0238-04" xlink:href="note-0238-04a" xml:space="preserve">11. huius.</note>
</div>
<pb o="201" file="0239" n="239" rhead="Conicor. Lib. VI."/>
<p style="it">
  <s xml:id="echoid-s7539" xml:space="preserve">Et quia N O ad O A eſt vt P Q ad Q D inuertamus proportionem, <lb/>
<anchor type="note" xlink:label="note-0239-01a" xlink:href="note-0239-01"/>
deinde bifariam ſecemus duas tertias partes, &amp; </s>
  <s xml:id="echoid-s7540" xml:space="preserve">inuertamus eas quoque <lb/>fiet N O ad O R, nempe N L ad L T in eadem ratione ipſi V Z, nempe <lb/>L B ad B Z, vt D Q ad Q T, nempe P M ad P X æqualem ipſi Y a, <lb/>nempe M E ad E a, &amp;</s>
  <s xml:id="echoid-s7541" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7542" xml:space="preserve">Quoniam L O ad O I oſtenſa fuit vt M Q ad Q <lb/>K, &amp; </s>
  <s xml:id="echoid-s7543" xml:space="preserve">propter parallelas I A, L N, nec non D K, M P eſt N O ad O A, vt L O <lb/>ad O I; </s>
  <s xml:id="echoid-s7544" xml:space="preserve">pariterq; </s>
  <s xml:id="echoid-s7545" xml:space="preserve">P Q ad Q D eſt vt M Q ad Q K; </s>
  <s xml:id="echoid-s7546" xml:space="preserve">igitur N O ad O A eandẽ <lb/>proportionẽ habet, quàm P Q ad Q D, &amp; </s>
  <s xml:id="echoid-s7547" xml:space="preserve">comparando antecedentes ad ſemidif-<lb/>ferentias, vel ſemisũmas terminorũ erit N O ad R A, vt P Q ad S D: </s>
  <s xml:id="echoid-s7548" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7549" xml:space="preserve">pro-<lb/>
<anchor type="figure" xlink:label="fig-0239-01a" xlink:href="fig-0239-01"/>
pterea N O ad O R ſummã, vel differentiã conſequentium eandem proportionem <lb/>habebit, quàm P Q ad Q S; </s>
  <s xml:id="echoid-s7550" xml:space="preserve">ſed propter parallelas R T, &amp; </s>
  <s xml:id="echoid-s7551" xml:space="preserve">O L eſt L N ad T L, <lb/>vt N O ad O R: </s>
  <s xml:id="echoid-s7552" xml:space="preserve">pariterque (propter parallelas S X, &amp; </s>
  <s xml:id="echoid-s7553" xml:space="preserve">Q M) eſt P M ad X <lb/>M, vt P Q ad Q S; </s>
  <s xml:id="echoid-s7554" xml:space="preserve">igitur N L ad L T eandem proportionem habet, quàm <lb/>P M ad M X: </s>
  <s xml:id="echoid-s7555" xml:space="preserve">ſuntque in parallelogrammis V L, &amp; </s>
  <s xml:id="echoid-s7556" xml:space="preserve">γ M latera oppoſita æqua-<lb/>lia V Z ipſi T L, atque a γ ipſi X M; </s>
  <s xml:id="echoid-s7557" xml:space="preserve">igitur N L ad V Z eandem proportio-<lb/>nem habet, quàm P M ad γ a, &amp; </s>
  <s xml:id="echoid-s7558" xml:space="preserve">ita erunt earum quadrata; </s>
  <s xml:id="echoid-s7559" xml:space="preserve">ſed vt quadratũ <lb/>
<anchor type="note" xlink:label="note-0239-02a" xlink:href="note-0239-02"/>
N L ad quadratum V Z ita eſt abſciſſa L B ad abſcißam B Z, pariterque vt <lb/>quadratum P M ad quadratum γ a, ita eſt abſciſſa M E ad abſcißam E a; </s>
  <s xml:id="echoid-s7560" xml:space="preserve">er-<lb/>go L B ad B Z eandem proportiònem habet, quàm M E ad E a.</s>
  <s xml:id="echoid-s7561" xml:space="preserve"/>
</p>
<div xml:id="echoid-div687" type="float" level="2" n="4">
<note position="left" xlink:label="note-0239-01" xlink:href="note-0239-01a" xml:space="preserve">d</note>
  <figure xlink:label="fig-0239-01" xlink:href="fig-0239-01a">
    <image file="0239-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0239-01"/>
  </figure>
<note position="right" xlink:label="note-0239-02" xlink:href="note-0239-02a" xml:space="preserve">20 lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7562" xml:space="preserve">Et occurrere faciamus par pari remanet O R ad R b, vt Q S ad S c, &amp;</s>
  <s xml:id="echoid-s7563" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s7564" xml:space="preserve">
<anchor type="note" xlink:label="note-0239-03a" xlink:href="note-0239-03"/>
Quoniam oſtenſa fuit O N ad O R, vt Q P ad Q S, per conuerſionem rationis <lb/>O N ad N R erit vt Q P ad P S, pariterque oſtenſa fuit b N ad N O, vt <lb/>c P ad P Q; </s>
  <s xml:id="echoid-s7565" xml:space="preserve">ergo ex æquali b N ad N R eſt vt c P ad S P, &amp; </s>
  <s xml:id="echoid-s7566" xml:space="preserve">diuidendo b R <lb/>ad R N erit vt c S ad S P; </s>
  <s xml:id="echoid-s7567" xml:space="preserve">ſed erat inuertendo R N ad N O, vt S P ad P Q; <lb/></s>
  <s xml:id="echoid-s7568" xml:space="preserve">quare comparando antecedentes ad differentias terminorum erit N R ad R O vt <lb/>P S ad S Q; </s>
  <s xml:id="echoid-s7569" xml:space="preserve">ideoq; </s>
  <s xml:id="echoid-s7570" xml:space="preserve">rurſus ex æqualitate b R ad R O erit vt c S ad S Q; </s>
  <s xml:id="echoid-s7571" xml:space="preserve">eſtq; </s>
  <s xml:id="echoid-s7572" xml:space="preserve"><lb/>V R ad R b vt γ S ad S c (eo quod triangula V R b, &amp; </s>
  <s xml:id="echoid-s7573" xml:space="preserve">γ S c ſunt ſimilia <lb/>triangulis ſimilibus O N L, &amp; </s>
  <s xml:id="echoid-s7574" xml:space="preserve">Q M P propter æquidiſtantes) ergo ex æquali <lb/>ordinata V R ad R O eandem proportionem habet, quàm γ S ad S Q.</s>
  <s xml:id="echoid-s7575" xml:space="preserve"/>
</p>
<div xml:id="echoid-div688" type="float" level="2" n="5">
<note position="left" xlink:label="note-0239-03" xlink:href="note-0239-03a" xml:space="preserve">e</note>
</div>
<pb o="202" file="0240" n="240" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s7576" xml:space="preserve">Sed tangens in V perueniens ad L O, &amp;</s>
  <s xml:id="echoid-s7577" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7578" xml:space="preserve">Si enim ex punctis γ, V du-<lb/>
<anchor type="note" xlink:label="note-0240-01a" xlink:href="note-0240-01"/>
cantur V e, &amp; </s>
  <s xml:id="echoid-s7579" xml:space="preserve">γ f tangentes parabolas, &amp; </s>
  <s xml:id="echoid-s7580" xml:space="preserve">producantur quouſque ſecent axes <lb/>in e, &amp; </s>
  <s xml:id="echoid-s7581" xml:space="preserve">f eſſicientur duo parallelogramma V e O R, &amp; </s>
  <s xml:id="echoid-s7582" xml:space="preserve">γ f Q S, in quibus tã-<lb/>gentes V e, &amp; </s>
  <s xml:id="echoid-s7583" xml:space="preserve">γ f efficientur æquales ipſis O R, &amp; </s>
  <s xml:id="echoid-s7584" xml:space="preserve">Q S: </s>
  <s xml:id="echoid-s7585" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7586" xml:space="preserve">propterea inuer-<lb/>tendo R V abſciſſa ad contingentem V e æqualem ipſi R O eandem proportionem <lb/>habebit, quàm abſcißa S γ ad contingentem γ f æqualem ipſi S Q, atque effi-<lb/>ciunt prædictæ contingentes cum axibus angulos e, f æquales ipſis O, &amp; </s>
  <s xml:id="echoid-s7587" xml:space="preserve">Q æ-<lb/>qualibus propter parallelas; </s>
  <s xml:id="echoid-s7588" xml:space="preserve">igitur ſegmenta N V A, &amp; </s>
  <s xml:id="echoid-s7589" xml:space="preserve">P γ D ſimilia ſunt in-<lb/>
<anchor type="note" xlink:label="note-0240-02a" xlink:href="note-0240-02"/>
ter ſe.</s>
  <s xml:id="echoid-s7590" xml:space="preserve"/>
</p>
<div xml:id="echoid-div689" type="float" level="2" n="6">
<note position="right" xlink:label="note-0240-01" xlink:href="note-0240-01a" xml:space="preserve">f</note>
<note position="left" xlink:label="note-0240-02" xlink:href="note-0240-02a" xml:space="preserve">Prop. 16. <lb/>huius.</note>
</div>
  <figure>
    <image file="0240-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0240-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s7591" xml:space="preserve">Et pariter duo ſegmenta A B C, D E F, atque duo ſegmenta N B, P <lb/>
<anchor type="note" xlink:label="note-0240-03a" xlink:href="note-0240-03"/>
E ſunt ſimilia inter ſe, &amp; </s>
  <s xml:id="echoid-s7592" xml:space="preserve">ſimiliter poſita, &amp;</s>
  <s xml:id="echoid-s7593" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7594" xml:space="preserve">Hoc manifeſtum eſt, ſi enim <lb/>coniungantur rectæ lineæ N C, &amp; </s>
  <s xml:id="echoid-s7595" xml:space="preserve">P F, &amp; </s>
  <s xml:id="echoid-s7596" xml:space="preserve">bifariam diuidantur, atque ducan-<lb/>tur diametri, &amp;</s>
  <s xml:id="echoid-s7597" xml:space="preserve">c, vti fecimus in ſectione N A, oſtendetur ſimiliter (ex ea-<lb/>dem 16. </s>
  <s xml:id="echoid-s7598" xml:space="preserve">propoſitione) ſegmenta N C, P F ſimilia eſſe inter ſe. </s>
  <s xml:id="echoid-s7599" xml:space="preserve">Non ſecus ſi <lb/>coniungantur rectæ lineæ N B, &amp; </s>
  <s xml:id="echoid-s7600" xml:space="preserve">P E, &amp; </s>
  <s xml:id="echoid-s7601" xml:space="preserve">bifariam diuidantur, atque ducan-<lb/>tur diametri, &amp; </s>
  <s xml:id="echoid-s7602" xml:space="preserve">reliqua perficiantur, vt prius, oſtendentur codem modo, ſegmẽ-<lb/>ta N B, &amp; </s>
  <s xml:id="echoid-s7603" xml:space="preserve">P E ſimilia inter ſe.</s>
  <s xml:id="echoid-s7604" xml:space="preserve"/>
</p>
<div xml:id="echoid-div690" type="float" level="2" n="7">
<note position="right" xlink:label="note-0240-03" xlink:href="note-0240-03a" xml:space="preserve">g</note>
</div>
<p style="it">
  <s xml:id="echoid-s7605" xml:space="preserve">Deinde ponamus ſegmentũ P d; </s>
  <s xml:id="echoid-s7606" xml:space="preserve">quia non abſcindunt illa duæ ordina-<lb/>
<anchor type="note" xlink:label="note-0240-04a" xlink:href="note-0240-04"/>
tiones vnius axis (18. </s>
  <s xml:id="echoid-s7607" xml:space="preserve">ex 6.)</s>
  <s xml:id="echoid-s7608" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s7609" xml:space="preserve">hoc erat, &amp;</s>
  <s xml:id="echoid-s7610" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7611" xml:space="preserve">Sed legendum puto vt in <lb/>textu apparet. </s>
  <s xml:id="echoid-s7612" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7613" xml:space="preserve">horum verborũ ſenſus erit; </s>
  <s xml:id="echoid-s7614" xml:space="preserve">fieri non poteſt, vt ſegmentũ p d ſit <lb/>ſimile ipſi N A, vel N B, propterea quod in ſectione P F ſegmenta P d vni tan-<lb/>tummodo portioni ſimile eſt (præter quàm in ellipſi), &amp; </s>
  <s xml:id="echoid-s7615" xml:space="preserve">ambo intercipi debent à <lb/>duabus ordinatim applicatis ad axim E Q: </s>
  <s xml:id="echoid-s7616" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7617" xml:space="preserve">propterea ſegmenta P D, vel P E <lb/>non erunt ſimilia ipſi P d, &amp; </s>
  <s xml:id="echoid-s7618" xml:space="preserve">quia N A oſtenſum eſt ſimile P D, pariterque N <lb/>B oſtenſum eſt ſimile P E; </s>
  <s xml:id="echoid-s7619" xml:space="preserve">igitur ſegmentum P d ſimile non eſt, neque N A, <lb/>neque ſegmento N B; </s>
  <s xml:id="echoid-s7620" xml:space="preserve">quod erat oſtendundum.</s>
  <s xml:id="echoid-s7621" xml:space="preserve"/>
</p>
<div xml:id="echoid-div691" type="float" level="2" n="8">
<note position="left" xlink:label="note-0240-04" xlink:href="note-0240-04a" xml:space="preserve">h</note>
</div>
<pb o="203" file="0241" n="241" rhead="Conicor. Lib. VI."/>
  <figure>
    <image file="0241-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0241-01"/>
  </figure>
</div>
<div xml:id="echoid-div693" type="section" level="1" n="229">
<head xml:id="echoid-head289" xml:space="preserve">Notæ in Propoſit. XXI.</head>
<p style="it">
  <s xml:id="echoid-s7622" xml:space="preserve">QVoniam G B ad B I, ſuppoſita eſt vt H E ad E K, &amp;</s>
  <s xml:id="echoid-s7623" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7624" xml:space="preserve">Quia L B <lb/>
<anchor type="note" xlink:label="note-0241-01a" xlink:href="note-0241-01"/>
ad B G ex bypotheſi erat, vt M E ad E H, &amp; </s>
  <s xml:id="echoid-s7625" xml:space="preserve">inuertendo G B ad B I <lb/>erat vt H E ad E K; </s>
  <s xml:id="echoid-s7626" xml:space="preserve">ergo ex æqualitate L B ad B I erit vt M E <lb/>ad E K; </s>
  <s xml:id="echoid-s7627" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7628" xml:space="preserve">per conuerſionem rationis B L ad L I erit vt E M ad M K.</s>
  <s xml:id="echoid-s7629" xml:space="preserve"/>
</p>
<div xml:id="echoid-div693" type="float" level="2" n="1">
<note position="left" xlink:label="note-0241-01" xlink:href="note-0241-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s7630" xml:space="preserve">Et propter ſimilitudinem duarum ſectionum N L ad A I, nempe L O <lb/>
<anchor type="note" xlink:label="note-0241-02a" xlink:href="note-0241-02"/>
ad O I eſt, vt P M ad F K, nempe M Q ad Q K, &amp;</s>
  <s xml:id="echoid-s7631" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7632" xml:space="preserve">Quoniam duæ ſe-<lb/>ctiones N B, &amp; </s>
  <s xml:id="echoid-s7633" xml:space="preserve">P E ſimiles ſuppoſitæ ſunt, &amp; </s>
  <s xml:id="echoid-s7634" xml:space="preserve">axiũ abſciſſæ L B, M E, nec non <lb/>I B, K E ad latera recta B G, <lb/>
<anchor type="figure" xlink:label="fig-0241-02a" xlink:href="fig-0241-02"/>
&amp; </s>
  <s xml:id="echoid-s7635" xml:space="preserve">H E proportionales ſunt;</s>
  <s xml:id="echoid-s7636" xml:space="preserve">
igitur N L ad A I eandem pro-
<anchor type="note" xlink:label="note-0241-03a" xlink:href="note-0241-03"/>
portionem habebit, quàm P M
ad D K: </s>
  <s xml:id="echoid-s7637" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7638" xml:space="preserve">quia triangula N
L O, &amp; </s>
  <s xml:id="echoid-s7639" xml:space="preserve">A I O ſimilia ſunt pro-
pter parallelas N L, &amp; </s>
  <s xml:id="echoid-s7640" xml:space="preserve">I A,
pariterque triangula P M Q,
&amp; </s>
  <s xml:id="echoid-s7641" xml:space="preserve">D K Q ſimilia ſunt; </s>
  <s xml:id="echoid-s7642" xml:space="preserve">igitur
L O ad O I erit vt N L ad I
A; </s>
  <s xml:id="echoid-s7643" xml:space="preserve">pariterque M Q ad Q K
erit vt P M ad D I, ſeu vt
N L ad A I: </s>
  <s xml:id="echoid-s7644" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7645" xml:space="preserve">propterea L
O ad O I erit vt M Q ad Q
K.</s>
  <s xml:id="echoid-s7646" xml:space="preserve"/>
</p>
<div xml:id="echoid-div694" type="float" level="2" n="2">
<note position="left" xlink:label="note-0241-02" xlink:href="note-0241-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0241-02" xlink:href="fig-0241-02a">
    <image file="0241-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0241-02"/>
<caption xml:id="echoid-caption1" xml:space="preserve">Cc 2</caption>
  </figure>
<note position="right" xlink:label="note-0241-03" xlink:href="note-0241-03a" xml:space="preserve">ex 12. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7647" xml:space="preserve">Et ex æqualitate L O ad <lb/>
<anchor type="note" xlink:label="note-0241-04a" xlink:href="note-0241-04"/>
L B erit vt Q M ad M E, ſed <lb/>L B ad L N eſt vt M E ad <lb/>M P, cum ex ſuppoſitione <lb/>ſectiones ſint ſimiles, &amp;</s>
  <s xml:id="echoid-s7648" xml:space="preserve">c,
<pb o="204" file="0242" n="242" rhead="Apollonij Pergæi"/>
Quoniam O L ad L I oſtenſa fuit, vt Q M ad M K; </s>
  <s xml:id="echoid-s7649" xml:space="preserve">atque prius oſtenſa ſuit <lb/>B L ad L I, vt E M ad M K; </s>
  <s xml:id="echoid-s7650" xml:space="preserve">ergo inuertendo I L ad L B erit, vt K M ad <lb/>M E; </s>
  <s xml:id="echoid-s7651" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7652" xml:space="preserve">propterea ex æqualitate O L ad L B erit vt Q M ad M E; </s>
  <s xml:id="echoid-s7653" xml:space="preserve">ſed B L <lb/>
<anchor type="note" xlink:label="note-0242-01a" xlink:href="note-0242-01"/>
ad L N eſt, vt E M ad M P; </s>
  <s xml:id="echoid-s7654" xml:space="preserve">igitur ex æqualitate O L ad L N erit vt Q M <lb/>ad M P; </s>
  <s xml:id="echoid-s7655" xml:space="preserve">ſuntque duo anguli L, &amp; </s>
  <s xml:id="echoid-s7656" xml:space="preserve">M recti; </s>
  <s xml:id="echoid-s7657" xml:space="preserve">ergo triangula O L N, &amp; </s>
  <s xml:id="echoid-s7658" xml:space="preserve">Q M P <lb/>æquiangula erunt; </s>
  <s xml:id="echoid-s7659" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7660" xml:space="preserve">propterea anguli O, &amp; </s>
  <s xml:id="echoid-s7661" xml:space="preserve">Qæquales inter ſe erunt; </s>
  <s xml:id="echoid-s7662" xml:space="preserve">ſed quia <lb/>contingentes verticales V e, &amp; </s>
  <s xml:id="echoid-s7663" xml:space="preserve">γ f parallelæ ſunt or dinatim applicatis N A, P <lb/>D ad diametros V R, &amp; </s>
  <s xml:id="echoid-s7664" xml:space="preserve">γ S; </s>
  <s xml:id="echoid-s7665" xml:space="preserve">igitur angulus V e B æqualis erit angulo N O L; <lb/></s>
  <s xml:id="echoid-s7666" xml:space="preserve">pariterque angulus γ f E æqualis erit angulo P Q M; </s>
  <s xml:id="echoid-s7667" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7668" xml:space="preserve">propterea anguli e, &amp; </s>
  <s xml:id="echoid-s7669" xml:space="preserve"><lb/>f æquales erunt inter ſe.</s>
  <s xml:id="echoid-s7670" xml:space="preserve"/>
</p>
<div xml:id="echoid-div695" type="float" level="2" n="3">
<note position="left" xlink:label="note-0241-04" xlink:href="note-0241-04a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0242-01" xlink:href="note-0242-01a" xml:space="preserve">ex 12. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7671" xml:space="preserve">Ergo propter ſimilitudinem duarum ſectionum T Z in Z e ad quadra-<lb/>
<anchor type="note" xlink:label="note-0242-02a" xlink:href="note-0242-02"/>
tum Z V eandem proportionem habebit quàm X a in a f ad quadratum <lb/>a Y, &amp; </s>
  <s xml:id="echoid-s7672" xml:space="preserve">angulus e æqualis eſt angulo f; </s>
  <s xml:id="echoid-s7673" xml:space="preserve">igitur V e T ſimile eſt Y f X, <lb/>&amp; </s>
  <s xml:id="echoid-s7674" xml:space="preserve">pariter, &amp;</s>
  <s xml:id="echoid-s7675" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7676" xml:space="preserve">Quoniam in ſectionibus ſimilibus V B, &amp; </s>
  <s xml:id="echoid-s7677" xml:space="preserve">γ E axes tranſuerſi <lb/>
<anchor type="note" xlink:label="note-0242-03a" xlink:href="note-0242-03"/>
lateribus rectis proportionales ſunt, &amp; </s>
  <s xml:id="echoid-s7678" xml:space="preserve">ductæ ſunt ad axes ordinatim applicatæ <lb/>V Z, γ a, &amp; </s>
  <s xml:id="echoid-s7679" xml:space="preserve">contingentes V e, γ f, eſtque rectangulum T Z e ad quadratum <lb/>
<anchor type="note" xlink:label="note-0242-04a" xlink:href="note-0242-04"/>
Z V, vt latus tranſuerſum ad rectum, pariterque rectangulum X a f ad qua-<lb/>dratum a γ, vt axis tranſuerſus ad erectum; </s>
  <s xml:id="echoid-s7680" xml:space="preserve">igitur rectangulũ T Z e adqua-<lb/>dratum Z V eandem proportionem habet, quàm rectangulum X a f ad quadra-<lb/>tum a γ, &amp; </s>
  <s xml:id="echoid-s7681" xml:space="preserve">à verticibus V, γ duorum triangulorum V e T, &amp; </s>
  <s xml:id="echoid-s7682" xml:space="preserve">γ f X ductæ <lb/>ſunt ad baſes rectæ linæ V Z, γ a efficientes angulos rectos, cum ordinatim <lb/>
<anchor type="figure" xlink:label="fig-0242-01a" xlink:href="fig-0242-01"/>
applicatæ ſint ad axes; </s>
  <s xml:id="echoid-s7683" xml:space="preserve">atque angulus V e Z oſtenſus eſt æqualis angulo γ f a, <lb/>igitur tertius angulus Z V e æqualis erit tertio angulo a γ f; </s>
  <s xml:id="echoid-s7684" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7685" xml:space="preserve">ideo duo triã-<lb/>
<anchor type="note" xlink:label="note-0242-05a" xlink:href="note-0242-05"/>
gula V T e, &amp; </s>
  <s xml:id="echoid-s7686" xml:space="preserve">γ X f ſimilia erunt inter ſe: </s>
  <s xml:id="echoid-s7687" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7688" xml:space="preserve">propterea circa angulos æquales <lb/>T, &amp; </s>
  <s xml:id="echoid-s7689" xml:space="preserve">X latus e T ad T V eandem proportionem habebit, quàm f X ad X γ: <lb/></s>
  <s xml:id="echoid-s7690" xml:space="preserve">cumque duæ contingentes verticales V e, γ f parallelæ ſint ordinatim applicatis <lb/>N A, &amp; </s>
  <s xml:id="echoid-s7691" xml:space="preserve">P D ad diametros V R, γ S, erit O e ad R V, vt e T ad T V; </s>
  <s xml:id="echoid-s7692" xml:space="preserve">pa-<lb/>riterque Q f ad S γ erit, vt f X ad X r: </s>
  <s xml:id="echoid-s7693" xml:space="preserve">erat autem e T ad T V, vt f X ad <lb/>X γ; </s>
  <s xml:id="echoid-s7694" xml:space="preserve">igitur pariter O e ad R V eandem proportionem habebit, quàm Q f ad <lb/>
<anchor type="note" xlink:label="note-0242-06a" xlink:href="note-0242-06"/>
S γ; </s>
  <s xml:id="echoid-s7695" xml:space="preserve">ſed B I ad I A eſt, vt E K ad K D.</s>
  <s xml:id="echoid-s7696" xml:space="preserve"/>
</p>
<div xml:id="echoid-div696" type="float" level="2" n="4">
<note position="right" xlink:label="note-0242-02" xlink:href="note-0242-02a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0242-03" xlink:href="note-0242-03a" xml:space="preserve">12. huius.</note>
<note position="left" xlink:label="note-0242-04" xlink:href="note-0242-04a" xml:space="preserve">37. lib. 1.</note>
  <figure xlink:label="fig-0242-01" xlink:href="fig-0242-01a">
    <image file="0242-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0242-01"/>
  </figure>
<note position="left" xlink:label="note-0242-05" xlink:href="note-0242-05a" xml:space="preserve">Propof. 6 <lb/>præmiſſ.</note>
<note position="left" xlink:label="note-0242-06" xlink:href="note-0242-06a" xml:space="preserve">12. huius.</note>
</div>
<pb o="205" file="0243" n="243" rhead="Conicor. Lib. VI."/>
<p style="it">
  <s xml:id="echoid-s7697" xml:space="preserve">Sed B T ad B I erat vt X E ad E K propter ſimilitudinem duarum ſe-<lb/>
<anchor type="note" xlink:label="note-0243-01a" xlink:href="note-0243-01"/>
ctionum, &amp;</s>
  <s xml:id="echoid-s7698" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7699" xml:space="preserve">Quoniam ex hypotheſi abſcißa axis I B ad latus rectum B G <lb/>erat vt abſciſſa K E ad latus rectum E H; </s>
  <s xml:id="echoid-s7700" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7701" xml:space="preserve">propter ſimilitudinem ſectionum <lb/>
<anchor type="note" xlink:label="note-0243-02a" xlink:href="note-0243-02"/>
latera erecta G B, &amp; </s>
  <s xml:id="echoid-s7702" xml:space="preserve">H E ad axes tranſuerſos, &amp; </s>
  <s xml:id="echoid-s7703" xml:space="preserve">ideo ad eorum ſemißes T B <lb/>&amp; </s>
  <s xml:id="echoid-s7704" xml:space="preserve">E X eandem proportionem habebunt; </s>
  <s xml:id="echoid-s7705" xml:space="preserve">ergo ex æquali I B ad B T erit vt K <lb/>E ad E X, &amp; </s>
  <s xml:id="echoid-s7706" xml:space="preserve">inuertendo T B <lb/>
<anchor type="figure" xlink:label="fig-0243-01a" xlink:href="fig-0243-01"/>
ad B I erit vt X E ad E K. <lb/></s>
  <s xml:id="echoid-s7707" xml:space="preserve">Sed libet aliam expoſitionem <lb/>afferre Apollonĳ principĳs cõue-<lb/>nientiorẽ. </s>
  <s xml:id="echoid-s7708" xml:space="preserve">Quia ex definitione <lb/>2. </s>
  <s xml:id="echoid-s7709" xml:space="preserve">huius libri legitime inter pre-<lb/>tata, &amp; </s>
  <s xml:id="echoid-s7710" xml:space="preserve">ſicuticõſtat ex 12. </s>
  <s xml:id="echoid-s7711" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s7712" xml:space="preserve"><lb/>huius. </s>
  <s xml:id="echoid-s7713" xml:space="preserve">In ſectionibus ſimilibus <lb/>non quælibet axium abſcißæ ad <lb/>conterminas potentiales habent <lb/>eandem rationem; </s>
  <s xml:id="echoid-s7714" xml:space="preserve">ſed illæ tan-<lb/>tummodo, quæ figuræ lateribus <lb/>proportionales ſunt: </s>
  <s xml:id="echoid-s7715" xml:space="preserve">itaq; </s>
  <s xml:id="echoid-s7716" xml:space="preserve">in ſe-<lb/>ctionibus ſimilibus A B, D E <lb/>vt quælibet axium, abſcißæ B <lb/>I, E K ad conterminas poten-<lb/>tiales I A, K D ſint proportio-<lb/>nales, neceße eſt, vt eædem I B, <lb/>&amp; </s>
  <s xml:id="echoid-s7717" xml:space="preserve">E K lateribus figurarum B <lb/>T, E X proportionales ſint.</s>
  <s xml:id="echoid-s7718" xml:space="preserve"/>
</p>
<div xml:id="echoid-div697" type="float" level="2" n="5">
<note position="left" xlink:label="note-0243-01" xlink:href="note-0243-01a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0243-02" xlink:href="note-0243-02a" xml:space="preserve">12. huius.</note>
  <figure xlink:label="fig-0243-01" xlink:href="fig-0243-01a">
    <image file="0243-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0243-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7719" xml:space="preserve">Et quadratum V Z ad quadratum Z e eſt, vt quadratum a Y ad qua-<lb/>
<anchor type="note" xlink:label="note-0243-03a" xlink:href="note-0243-03"/>
dratum a f, &amp;</s>
  <s xml:id="echoid-s7720" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7721" xml:space="preserve">oſtenſa enim fuerunt duo trìangula V Z e, &amp; </s>
  <s xml:id="echoid-s7722" xml:space="preserve">γ a f ſimilia <lb/>inter ſe; </s>
  <s xml:id="echoid-s7723" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7724" xml:space="preserve">ideo latera circa angulos rectos Z, &amp; </s>
  <s xml:id="echoid-s7725" xml:space="preserve">a proportionalia erunt; </s>
  <s xml:id="echoid-s7726" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7727" xml:space="preserve"><lb/>pariter eorum quadrata.</s>
  <s xml:id="echoid-s7728" xml:space="preserve"/>
</p>
<div xml:id="echoid-div698" type="float" level="2" n="6">
<note position="left" xlink:label="note-0243-03" xlink:href="note-0243-03a" xml:space="preserve">f</note>
</div>
<p style="it">
  <s xml:id="echoid-s7729" xml:space="preserve">Inſuper dico non eſſe ſimilia alicui alteri ſegmento, &amp;</s>
  <s xml:id="echoid-s7730" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7731" xml:space="preserve">Sicutì in præ-<lb/>
<anchor type="note" xlink:label="note-0243-04a" xlink:href="note-0243-04"/>
cedenti propoſitione factum eſt oſtendetur, quod ſegmentum N C non eſt ſimile <lb/>alicui alio ſegmento in altera ſectione P E, quando non compræhenduntur ab <lb/>ordinatim ad axes applicatis, &amp; </s>
  <s xml:id="echoid-s7732" xml:space="preserve">in ellipſibus æqualiter à centris diſtant.</s>
  <s xml:id="echoid-s7733" xml:space="preserve"/>
</p>
<div xml:id="echoid-div699" type="float" level="2" n="7">
<note position="left" xlink:label="note-0243-04" xlink:href="note-0243-04a" xml:space="preserve">g</note>
</div>
</div>
<div xml:id="echoid-div701" type="section" level="1" n="230">
<head xml:id="echoid-head290" xml:space="preserve">Notæ in Propoſit. XXII.</head>
<p style="it">
  <s xml:id="echoid-s7734" xml:space="preserve">ET propterea duo ſectiones A B C, D E F ſimiles erunt, &amp;</s>
  <s xml:id="echoid-s7735" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7736" xml:space="preserve">Quo-<lb/>
<anchor type="note" xlink:label="note-0243-05a" xlink:href="note-0243-05"/>
niam ſegmenta G B C, &amp; </s>
  <s xml:id="echoid-s7737" xml:space="preserve">H E F poſita ſunt ſimilia, erunt diamctrorum <lb/>
<anchor type="figure" xlink:label="fig-0243-02a" xlink:href="fig-0243-02"/>
<pb o="206" file="0244" n="244" rhead="Apollonij Pergæi"/>
ſeu axium (in hoc caſu) L B, &amp; </s>
  <s xml:id="echoid-s7738" xml:space="preserve">M E ſiguræ ſimiles inter ſe; </s>
  <s xml:id="echoid-s7739" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7740" xml:space="preserve">ideò ſectiones <lb/>
<anchor type="note" xlink:label="note-0244-01a" xlink:href="note-0244-01"/>
A B C, &amp; </s>
  <s xml:id="echoid-s7741" xml:space="preserve">D E F ſimiles erunt.</s>
  <s xml:id="echoid-s7742" xml:space="preserve"/>
</p>
<div xml:id="echoid-div701" type="float" level="2" n="1">
<note position="right" xlink:label="note-0243-05" xlink:href="note-0243-05a" xml:space="preserve">Lem. 8. <lb/>huius.</note>
  <figure xlink:label="fig-0243-02" xlink:href="fig-0243-02a">
    <image file="0243-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0243-02"/>
  </figure>
<note position="left" xlink:label="note-0244-01" xlink:href="note-0244-01a" xml:space="preserve">ex 11. 12. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7743" xml:space="preserve">Itaque propter ſimilitudinem duorum ſegmẽtorum ſimlia erunt B N L, <lb/>
<anchor type="note" xlink:label="note-0244-02a" xlink:href="note-0244-02"/>
E O M, &amp; </s>
  <s xml:id="echoid-s7744" xml:space="preserve">pariter L B P, &amp; </s>
  <s xml:id="echoid-s7745" xml:space="preserve">M E Q atque quadratum B P ad L P in P <lb/>N nempe, &amp;</s>
  <s xml:id="echoid-s7746" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7747" xml:space="preserve">Huius ſecundæ partis demonſtrationem, quàm non ſinceram Pa-<lb/>raphraſtes Arabicus nobis tranſmiſit omittere opere pretium erit, eandemq; </s>
  <s xml:id="echoid-s7748" xml:space="preserve">bre-<lb/>
<anchor type="figure" xlink:label="fig-0244-01a" xlink:href="fig-0244-01"/>
uius demonſtrare hac ratione. </s>
  <s xml:id="echoid-s7749" xml:space="preserve">Quia ſegmenta C B G, &amp; </s>
  <s xml:id="echoid-s7750" xml:space="preserve">F E H ſimilia ponun-<lb/>tur; </s>
  <s xml:id="echoid-s7751" xml:space="preserve">ergo erunt figuræ diametrorum B I, E K ſimiles inter ſe in angulis I, K <lb/>
<anchor type="note" xlink:label="note-0244-03a" xlink:href="note-0244-03"/>
æqualibus, &amp; </s>
  <s xml:id="echoid-s7752" xml:space="preserve">ſectiones ipſæ C B G, &amp; </s>
  <s xml:id="echoid-s7753" xml:space="preserve">F E H ſimiles inter ſe erunt; </s>
  <s xml:id="echoid-s7754" xml:space="preserve">quod eſt <lb/>
<anchor type="note" xlink:label="note-0244-04a" xlink:href="note-0244-04"/>
contra hypotheſin.</s>
  <s xml:id="echoid-s7755" xml:space="preserve"/>
</p>
<div xml:id="echoid-div702" type="float" level="2" n="2">
<note position="right" xlink:label="note-0244-02" xlink:href="note-0244-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0244-01" xlink:href="fig-0244-01a">
    <image file="0244-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0244-01"/>
  </figure>
<note position="left" xlink:label="note-0244-03" xlink:href="note-0244-03a" xml:space="preserve">Lem. 8. <lb/>huius.</note>
<note position="left" xlink:label="note-0244-04" xlink:href="note-0244-04a" xml:space="preserve">Prop. 15. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div704" type="section" level="1" n="231">
<head xml:id="echoid-head291" xml:space="preserve">Notæ in Propoſit. XXIII.</head>
<p style="it">
  <s xml:id="echoid-s7756" xml:space="preserve">SI enim ſimilia eſſent haberent conditiones ſimilitudinis, quod eſt im-<lb/>
<anchor type="note" xlink:label="note-0244-05a" xlink:href="note-0244-05"/>
poſſibile, &amp;</s>
  <s xml:id="echoid-s7757" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7758" xml:space="preserve">Si enim concedantur ſegmenta G B C in parabola, &amp; </s>
  <s xml:id="echoid-s7759" xml:space="preserve">H E <lb/>F in hyperbole, vel ellipſi, ſimilia inter ſe; </s>
  <s xml:id="echoid-s7760" xml:space="preserve">igitur in vnaquaque earũ duci poſ-<lb/>
<anchor type="note" xlink:label="note-0244-06a" xlink:href="note-0244-06"/>
ſent ad diametros ordinatim applicatæ numero æquales, efficientes angulos æqua-<lb/>
<anchor type="figure" xlink:label="fig-0244-02a" xlink:href="fig-0244-02"/>
<pb o="207" file="0245" n="245" rhead="Conicor. Lib. VI."/>
les cum diametris, quæ abſciſſis ſint proportionales, &amp; </s>
  <s xml:id="echoid-s7761" xml:space="preserve">abſciſſæ quoque inter ſe. <lb/></s>
  <s xml:id="echoid-s7762" xml:space="preserve">Vnde ſequitur, quod portiones eiuſdem diametri E K à centro M ad omnes or-<lb/>dinatim ad diametros applicatas ſint æquales inter ſe, vt oſtenſum eſt in propo-<lb/>ſitione 13. </s>
  <s xml:id="echoid-s7763" xml:space="preserve">huius: </s>
  <s xml:id="echoid-s7764" xml:space="preserve">quod eſt impoſſibile.</s>
  <s xml:id="echoid-s7765" xml:space="preserve"/>
</p>
<div xml:id="echoid-div704" type="float" level="2" n="1">
<note position="right" xlink:label="note-0244-05" xlink:href="note-0244-05a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0244-06" xlink:href="note-0244-06a" xml:space="preserve">Defin. 7. <lb/>huius.</note>
  <figure xlink:label="fig-0244-02" xlink:href="fig-0244-02a">
    <image file="0244-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0244-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7766" xml:space="preserve">Quando verò ſectio A C eſt byperbole, ac ſectio D F eſt ellipſis, ſimiliter, <lb/>vt in 14. </s>
  <s xml:id="echoid-s7767" xml:space="preserve">propoſitione huius, oſtendetur; </s>
  <s xml:id="echoid-s7768" xml:space="preserve">quo abſciſſæ in hyperbola, &amp; </s>
  <s xml:id="echoid-s7769" xml:space="preserve">ellipſi ſint <lb/>proportionales; </s>
  <s xml:id="echoid-s7770" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7771" xml:space="preserve">propterea omnes habebunt rationes maioris inæqualitatis, aut <lb/>omnes habebunt, proportiones inæqualitatis minoris, quod tamen in prædicta 14. <lb/></s>
  <s xml:id="echoid-s7772" xml:space="preserve">propoſitione impoſſibile eſſe oſtenditur.</s>
  <s xml:id="echoid-s7773" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div706" type="section" level="1" n="232">
<head xml:id="echoid-head292" xml:space="preserve">Notæ in Propoſit. XXIV.</head>
<p style="it">
  <s xml:id="echoid-s7774" xml:space="preserve">SI enim hoc verum non eſt, &amp;</s>
  <s xml:id="echoid-s7775" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7776" xml:space="preserve">Quod quælibet portio B A D ſectionis <lb/>
<anchor type="note" xlink:label="note-0245-01a" xlink:href="note-0245-01"/>
conicæ A B G nullo pacto circumferentia circuli eſſe poſſit ſic oſtendetur. <lb/></s>
  <s xml:id="echoid-s7777" xml:space="preserve">Quia in circulo rectæ lineæ diuidentes bifariam duas parallelas inter ſe ſunt <lb/>neceſſariò diametri circuli, qui perpendicu-<lb/>
<anchor type="figure" xlink:label="fig-0245-01a" xlink:href="fig-0245-01"/>
lariter ſecant prædictas parallelas applica-<lb/>tas; </s>
  <s xml:id="echoid-s7778" xml:space="preserve">igitur ſi curua linea B G D fuerit cir-<lb/>culi peripheria rectæ lineæ K I, L M, &amp; </s>
  <s xml:id="echoid-s7779" xml:space="preserve"><lb/>N O diametri circuli, erunt perpendicula-<lb/>res ad ordinatim applicatas æquidiſtantes <lb/>inter ſe; </s>
  <s xml:id="echoid-s7780" xml:space="preserve">ſed quia etiam A B G ſupponitur <lb/>ſectio conica, erunt K I, L M, N O axes <lb/>prædictæ ſectionis conicæ eo quod bifariam, <lb/>&amp; </s>
  <s xml:id="echoid-s7781" xml:space="preserve">ad angulos rectos diuidunt ordinatim ap-<lb/>plicatas. </s>
  <s xml:id="echoid-s7782" xml:space="preserve">Rurſus quia prædictæ ordinatim <lb/>applicatæ non ſunt omnes inter ſe parallelæ, <lb/>eo quodex conſtructione applicatæ A B, A C, <lb/>C D non fuerunt ductæ æquidiſtantes; </s>
  <s xml:id="echoid-s7783" xml:space="preserve">igi-<lb/>tur tres axes I K, L M, N O indirectum <lb/>
<anchor type="note" xlink:label="note-0245-02a" xlink:href="note-0245-02"/>
non coincidunt; </s>
  <s xml:id="echoid-s7784" xml:space="preserve">quare in ſectione conica B A G reperiri poſſent tres axes; </s>
  <s xml:id="echoid-s7785" xml:space="preserve">quod <lb/>eſt impoſſibile.</s>
  <s xml:id="echoid-s7786" xml:space="preserve"/>
</p>
<div xml:id="echoid-div706" type="float" level="2" n="1">
<note position="left" xlink:label="note-0245-01" xlink:href="note-0245-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0245-01" xlink:href="fig-0245-01a">
    <image file="0245-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0245-01"/>
  </figure>
<note position="right" xlink:label="note-0245-02" xlink:href="note-0245-02a" xml:space="preserve">48. lib. 2.</note>
</div>
</div>
<div xml:id="echoid-div708" type="section" level="1" n="233">
<head xml:id="echoid-head293" xml:space="preserve">SECTIO NONA</head>
<head xml:id="echoid-head294" xml:space="preserve">Continens Propoſit. XXV.</head>
<p>
  <s xml:id="echoid-s7787" xml:space="preserve">SI duo plana æquidiſtantia conum aliquem ſecuerint, atque <lb/>
<anchor type="note" xlink:label="note-0245-03a" xlink:href="note-0245-03"/>
in eo efficiant duas hyperbolas, aut ellipſes; </s>
  <s xml:id="echoid-s7788" xml:space="preserve">vtique ſectio-<lb/>nes ſimiles inter ſe erunt, ſed non erunt neceſſariò æquales.</s>
  <s xml:id="echoid-s7789" xml:space="preserve"/>
</p>
<div xml:id="echoid-div708" type="float" level="2" n="1">
<note position="left" xlink:label="note-0245-03" xlink:href="note-0245-03a" xml:space="preserve">b</note>
</div>
<pb o="208" file="0246" n="246" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s7790" xml:space="preserve">Efficiant duo plana parallela D <lb/>
<anchor type="note" xlink:label="note-0246-01a" xlink:href="note-0246-01"/>
<anchor type="figure" xlink:label="fig-0246-01a" xlink:href="fig-0246-01"/>
E N F, G H P I in baſim coni A C <lb/>duas rectas lineas D F, G I, &amp; </s>
  <s xml:id="echoid-s7791" xml:space="preserve">pla-<lb/>num per axim coniductum efficiat <lb/>triangulum A B C perpendiculare <lb/>ad duo illa plana parallela; </s>
  <s xml:id="echoid-s7792" xml:space="preserve">quæ ab <lb/>illo ſecentur in E K, H L. </s>
  <s xml:id="echoid-s7793" xml:space="preserve">Erunt <lb/>D F, I G perpendiculares ad A C, <lb/>&amp; </s>
  <s xml:id="echoid-s7794" xml:space="preserve">educamus B M parallelam ipſis <lb/>E K, H L; </s>
  <s xml:id="echoid-s7795" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7796" xml:space="preserve">vt quadratum B M ad <lb/>A M in M C; </s>
  <s xml:id="echoid-s7797" xml:space="preserve">ita ponatur N E ad <lb/>E O, &amp; </s>
  <s xml:id="echoid-s7798" xml:space="preserve">ita P H ſiat ad H Q, erunt <lb/>
<anchor type="note" xlink:label="note-0246-02a" xlink:href="note-0246-02"/>
N E, P H inclinata duarũ ſectionũ <lb/>F E D, I H G, aut eorum tranſuer-<lb/>ſæ; </s>
  <s xml:id="echoid-s7799" xml:space="preserve">igitur O E, H Q erunt eorum <lb/>erecta, &amp; </s>
  <s xml:id="echoid-s7800" xml:space="preserve">propterea figuræ duarum ſectionũ ſunt ſimiles; </s>
  <s xml:id="echoid-s7801" xml:space="preserve">igitur duæ ſectio-<lb/>
<anchor type="note" xlink:label="note-0246-03a" xlink:href="note-0246-03"/>
<anchor type="figure" xlink:label="fig-0246-02a" xlink:href="fig-0246-02"/>
nes ſimiles ſunt. </s>
  <s xml:id="echoid-s7802" xml:space="preserve">Et ſi quidem fuerint N E, P H æquales; </s>
  <s xml:id="echoid-s7803" xml:space="preserve">ipſæ quoque <lb/>
<anchor type="note" xlink:label="note-0246-04a" xlink:href="note-0246-04"/>
æquales erunt, alias non; </s>
  <s xml:id="echoid-s7804" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s7805" xml:space="preserve"/>
</p>
<div xml:id="echoid-div709" type="float" level="2" n="2">
<note position="right" xlink:label="note-0246-01" xlink:href="note-0246-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0246-01" xlink:href="fig-0246-01a">
    <image file="0246-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0246-01"/>
  </figure>
<note position="left" xlink:label="note-0246-02" xlink:href="note-0246-02a" xml:space="preserve">12. 13. <lb/>lib. 1.</note>
<note position="left" xlink:label="note-0246-03" xlink:href="note-0246-03a" xml:space="preserve">12. huius.</note>
  <figure xlink:label="fig-0246-02" xlink:href="fig-0246-02a">
    <image file="0246-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0246-02"/>
  </figure>
<note position="left" xlink:label="note-0246-04" xlink:href="note-0246-04a" xml:space="preserve">2. &amp; 10. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div711" type="section" level="1" n="234">
<head xml:id="echoid-head295" xml:space="preserve">Notæ in Propoſit. XXV.</head>
<p>
  <s xml:id="echoid-s7806" xml:space="preserve">SI abſcindant conum aliquem duo plana parallela prouenient duæ ſe-<lb/>
<anchor type="note" xlink:label="note-0246-05a" xlink:href="note-0246-05"/>
ctiones hyperbolicæ, vel quia duæ ſectiones ſunt ſimiles, &amp;</s>
  <s xml:id="echoid-s7807" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7808" xml:space="preserve">Quæ, <lb/>immutanda cenſui vt in textu videre eſt.</s>
  <s xml:id="echoid-s7809" xml:space="preserve"/>
</p>
<div xml:id="echoid-div711" type="float" level="2" n="1">
<note position="right" xlink:label="note-0246-05" xlink:href="note-0246-05a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s7810" xml:space="preserve">Sint abſciſſiones duorum planorum æquidiſtantium cum baſi I G, F D, <lb/>
<anchor type="note" xlink:label="note-0246-06a" xlink:href="note-0246-06"/>
&amp; </s>
  <s xml:id="echoid-s7811" xml:space="preserve">ſecet conum planum tranſiens per eius axim, &amp;</s>
  <s xml:id="echoid-s7812" xml:space="preserve">c. </s>
  <s xml:id="echoid-s7813" xml:space="preserve">Addidi verba, quæ <lb/>in textu d ſiderantur, quæ expoſitionem perſiciunt. </s>
  <s xml:id="echoid-s7814" xml:space="preserve">Animaduertendum eſt, hanc <lb/>propoſitionem conuertibilem non eſſe; </s>
  <s xml:id="echoid-s7815" xml:space="preserve">licet enim plana parallela in eodem cono <lb/>eſſiciant ſectiones ſimiles, verum non eſt, quod quotieſcunque in eodem cono duæ
<pb o="209" file="0247" n="247" rhead="Conicor. Lib. VI."/>
ſe{ct}iones ſunt æquales, vel ſimiles inter ſe, tunc quidem earum plana ſunt æqui-<lb/>diſtantia: </s>
  <s xml:id="echoid-s7816" xml:space="preserve">Sicuti enim in eodem cono ſcaleno deſignari poßunt circuli æquales <lb/>ſubcontrariè poſiti, ſic etiam reliquæ coniſe{ct}iones ſubcontrariè conſtitutæ effici <lb/>poſſunt æquales, &amp; </s>
  <s xml:id="echoid-s7817" xml:space="preserve">ſimiles inter ſe: </s>
  <s xml:id="echoid-s7818" xml:space="preserve">hæc autem, ſicuti etiam quamplurima vi-<lb/>deri poſſunt in libris neotericorum.</s>
  <s xml:id="echoid-s7819" xml:space="preserve"/>
</p>
<div xml:id="echoid-div712" type="float" level="2" n="2">
<note position="right" xlink:label="note-0246-06" xlink:href="note-0246-06a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s7820" xml:space="preserve">Sed non alienum erit à noſtro inſtituto hic paucis conſiderare paſſiones, &amp; </s>
  <s xml:id="echoid-s7821" xml:space="preserve">de-<lb/>ſcriptiones ſe{ct}ionum conicarum ſimilium, vel æqualium, quæ æquidiſtantes, <lb/>ſeu asymptoticæ vocantur. </s>
  <s xml:id="echoid-s7822" xml:space="preserve">Et licet hæ ab alĳs inuentæ, &amp; </s>
  <s xml:id="echoid-s7823" xml:space="preserve">traditæ ſint, non nul-<lb/>la tamen noua in medium afferam: </s>
  <s xml:id="echoid-s7824" xml:space="preserve">non enim rerum nouitas ex ſubie{ct}i nouita-<lb/>te tantummodò arguitur, imo de ſubie{ct}o antiquo poſſunt nouæ ſpeculationes af-<lb/>ferri, atque corrigi, &amp; </s>
  <s xml:id="echoid-s7825" xml:space="preserve">cõpleri ea, quæ apicem perfe{ct}ionis non attingunt, &amp; </s>
  <s xml:id="echoid-s7826" xml:space="preserve"><lb/>hæc quidem omnia noua dici poterunt, &amp; </s>
  <s xml:id="echoid-s7827" xml:space="preserve">poſſunt, &amp; </s>
  <s xml:id="echoid-s7828" xml:space="preserve">debent zelo veritatis e-<lb/>uulgari, nec propterea prædeceßorum nominibus, ant inuentionibus iniuria in-<lb/>fertur.</s>
  <s xml:id="echoid-s7829" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s7830" xml:space="preserve">Primus itaque omnium ( quod ſciam ) Pappus Alexandrinus libro ſeptimo col-<lb/>le{ct}ionum Mathematic arum propoſitione 208. </s>
  <s xml:id="echoid-s7831" xml:space="preserve">lemmate ſexto in quintum librum <lb/>Apollonĳ, conſiderauit concentricas hyperbolas inter ſe ſimiles, eundẽ axim habentes, <lb/>ad eaſdem partes cauas inter ſe ſe non concurrere, ſed ſemper ad ſe ipſas vi-<lb/>cinius accedere. </s>
  <s xml:id="echoid-s7832" xml:space="preserve">Poſtea Gregorius à Santo Vincentio oſtendit, quod duæ parabo-<lb/>
<anchor type="note" xlink:label="note-0247-01a" xlink:href="note-0247-01"/>
læ inter ſe æquales, ſimiliter poſitæ circa communem axim, vel diametrum, pa-<lb/>riter nunquàm conueniunt, &amp; </s>
  <s xml:id="echoid-s7833" xml:space="preserve">parallelæ ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s7834" xml:space="preserve">in infinitum produ{ct}æ <lb/>ſemper magis ad inuicem accedunt; </s>
  <s xml:id="echoid-s7835" xml:space="preserve">atque propoſit. </s>
  <s xml:id="echoid-s7836" xml:space="preserve">139. </s>
  <s xml:id="echoid-s7837" xml:space="preserve">de Hyperbola conſidera-<lb/>uit duas hyperbolas æquales, &amp; </s>
  <s xml:id="echoid-s7838" xml:space="preserve">ſimiles, quæ pariter in infinitd extensæ nunquàm <lb/>conueniunt, &amp; </s>
  <s xml:id="echoid-s7839" xml:space="preserve">ſimul cum Pappo putat, rite co@cludi poſſe, quod prædi{ct}æ ſe{ct}io-<lb/>nes, in infinitum extenſæ, ſint asymptoti, &amp; </s>
  <s xml:id="echoid-s7840" xml:space="preserve">ſemper magis, ac magis ad inui-<lb/>cem appropinquentur ex eo, quod re{ct}æ lineæ inter ſe æquidiſtantes inter duas <lb/>ſe{ct}iones interceptæ, ſucceſſiuè ſemper diminuantur. </s>
  <s xml:id="echoid-s7841" xml:space="preserve">Propoſitiones quidem recon-<lb/>ditæ, &amp; </s>
  <s xml:id="echoid-s7842" xml:space="preserve">ſcitu iucundæ, ſed an æquè certæ, &amp; </s>
  <s xml:id="echoid-s7843" xml:space="preserve">indubitatæ cenſeri debeant, in-<lb/>quiremus, aliquibus tamen præmiſſis.</s>
  <s xml:id="echoid-s7844" xml:space="preserve"/>
</p>
<div xml:id="echoid-div713" type="float" level="2" n="3">
<note position="right" xlink:label="note-0247-01" xlink:href="note-0247-01a" xml:space="preserve">Parab. <lb/>pr 344.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7845" xml:space="preserve">In qualibet hyperbola I E, cuius asymptoti C A B, duarum re{ct}arum linea-<lb/>
<anchor type="note" xlink:label="note-0247-02a" xlink:href="note-0247-02"/>
<anchor type="figure" xlink:label="fig-0247-01a" xlink:href="fig-0247-01"/>
rum F I, G K inter ſe æquidiſtantium, ab vna asymptoto A C ad hyperbolen, <lb/>edu{ct}arum, ſit F I propinquior centro, quàm G K, quando ambo cadunt infra <lb/>centrum A ad partes C; </s>
  <s xml:id="echoid-s7846" xml:space="preserve">vel F I magis à centro recedat, quando ambo cadunt
<pb o="210" file="0248" n="248" rhead="Apollonij Pergæi"/>
vltra centrum in eadem asymptoti produ{ct}ione A Z; </s>
  <s xml:id="echoid-s7847" xml:space="preserve">aut F I ſupra, &amp; </s>
  <s xml:id="echoid-s7848" xml:space="preserve">G K in-<lb/>fra centrum A exiſtat: </s>
  <s xml:id="echoid-s7849" xml:space="preserve">In quo libet caſu dicetur, F I vlterius tendere ad partes <lb/>centri, vel asymptoti A B, quàm G K.</s>
  <s xml:id="echoid-s7850" xml:space="preserve"/>
</p>
<div xml:id="echoid-div714" type="float" level="2" n="4">
<note position="right" xlink:label="note-0247-02" xlink:href="note-0247-02a" xml:space="preserve">DEFINI <lb/>TIO <lb/>Addita.</note>
  <figure xlink:label="fig-0247-01" xlink:href="fig-0247-01a">
    <image file="0247-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0247-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7851" xml:space="preserve">Non ſecus ſi ab eadem asymptoto A C educantur quatuor rectæ lineæ inter ſe <lb/>æquidiſtantes F I, G K, H L, C E, quarum duæ priores F I, G K, centro pro-<lb/>pinquiores ſint, quando omnes infra centrum A collocantur; </s>
  <s xml:id="echoid-s7852" xml:space="preserve">vel magis à centro <lb/>recedant, quando omnes in productione A Z exiſtunt; </s>
  <s xml:id="echoid-s7853" xml:space="preserve">aut certe duæ F I, G K <lb/>ſupra centrum, &amp; </s>
  <s xml:id="echoid-s7854" xml:space="preserve">H L, C E infra centrum exiſtant: </s>
  <s xml:id="echoid-s7855" xml:space="preserve">Tunc ſimiliter in quoli-<lb/>bet caſu dicentur rectæ lineæ F I, G K vlterius tendere ad partes centri, &amp; </s>
  <s xml:id="echoid-s7856" xml:space="preserve"><lb/>asympoti A B, quàm duæ aliæ H L, C E.</s>
  <s xml:id="echoid-s7857" xml:space="preserve"/>
</p>
  <figure>
    <image file="0248-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0248-01"/>
  </figure>
<note position="left" xml:space="preserve">PROP.2. <lb/>Addit.</note>
<p style="it">
  <s xml:id="echoid-s7858" xml:space="preserve">Si in vna aſymptoto A C, hyperboles D E ſumantur duo ſegmenta <lb/>æqualia F G, H C, &amp; </s>
  <s xml:id="echoid-s7859" xml:space="preserve">à punctis diuiſionum ducantur quatuor rectæ <lb/>lineæ F I, G K, H L, C E parallelæ inter ſe, vſque ad hyperbolen: <lb/></s>
  <s xml:id="echoid-s7860" xml:space="preserve">Dico quod differentia duarum æquidiſtantium F I, G K ad partes cen-<lb/>tri, &amp; </s>
  <s xml:id="echoid-s7861" xml:space="preserve">alterius aſymptoti A B vlterius tendentium, maior erit differen-<lb/>tia reliquarum H L, C E.</s>
  <s xml:id="echoid-s7862" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s7863" xml:space="preserve">Ducantnr à punctis E, K rectæ <lb/>
<anchor type="figure" xlink:label="fig-0248-02a" xlink:href="fig-0248-02"/>
lineæ E S, K R parallelæ asympto-<lb/>to A C, quæ efficiant parallelogrã-<lb/>ma C S, G R. </s>
  <s xml:id="echoid-s7864" xml:space="preserve">Patet I R eſſe dif-<lb/>ferentiam æquidiſtantium F I, &amp; </s>
  <s xml:id="echoid-s7865" xml:space="preserve"><lb/>G K; </s>
  <s xml:id="echoid-s7866" xml:space="preserve">pariterque L S eſſe differen-<lb/>tiam æquidiſtantium H L, C E; <lb/></s>
  <s xml:id="echoid-s7867" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7868" xml:space="preserve">coniungantur rectæ lineæ E I, <lb/>&amp; </s>
  <s xml:id="echoid-s7869" xml:space="preserve">K I, ducaturque E O parallela <lb/>I K, ſecans H L in O. </s>
  <s xml:id="echoid-s7870" xml:space="preserve">Et quia <lb/>recta linea E I cadit intra curuam <lb/>ſectionem conicam E K I, &amp; </s>
  <s xml:id="echoid-s7871" xml:space="preserve">pun-<lb/>ctum K eiuſdem conicæ ſectionis
<pb o="211" file="0249" n="249" rhead="Conicor. Lib. VI."/>
inter E, &amp; </s>
  <s xml:id="echoid-s7872" xml:space="preserve">I exiſtit; </s>
  <s xml:id="echoid-s7873" xml:space="preserve">ergo recta linea I K poſita intra conicũ ſegmentum E K I <lb/>ſupra eius baſim E I cadit; </s>
  <s xml:id="echoid-s7874" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7875" xml:space="preserve">ideo ei parallela E O cadit infra eandem ſeg-<lb/>menti conici baſim E I, &amp; </s>
  <s xml:id="echoid-s7876" xml:space="preserve">propterea occurret ipſi H L intra coniſectionem, &amp; </s>
  <s xml:id="echoid-s7877" xml:space="preserve"><lb/>infra punctum L in ſectione poſitum, vt in O; </s>
  <s xml:id="echoid-s7878" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7879" xml:space="preserve">ideo O S maior erit, quàm, <lb/>S L. </s>
  <s xml:id="echoid-s7880" xml:space="preserve">Et quoniam S E, &amp; </s>
  <s xml:id="echoid-s7881" xml:space="preserve">R K ſunt inter ſe parallelæ ( quia eidem A C æqui-<lb/>diſtant) pariterque E O, &amp; </s>
  <s xml:id="echoid-s7882" xml:space="preserve">K I factæ ſunt parallelæ, atque S O, &amp; </s>
  <s xml:id="echoid-s7883" xml:space="preserve">R I (ex <lb/>hypotheſi) æquidiſtantes erant; </s>
  <s xml:id="echoid-s7884" xml:space="preserve">igitur duo triangula E S O, &amp; </s>
  <s xml:id="echoid-s7885" xml:space="preserve">K R I ſimilia <lb/>ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s7886" xml:space="preserve">eorũ latera homologa E S, &amp; </s>
  <s xml:id="echoid-s7887" xml:space="preserve">K R æqualia ſunt inter ſe (quiæ <lb/>in parallelogrãmis C S, &amp; </s>
  <s xml:id="echoid-s7888" xml:space="preserve">G R latera E S, R K æqualia ſunt oppoſitis C H, G <lb/>F inter ſe æqualibus, ex hypotheſi) igitur reliqua latera homologa S O, &amp; </s>
  <s xml:id="echoid-s7889" xml:space="preserve">R I <lb/>æqualia ſunt inter ſe; </s>
  <s xml:id="echoid-s7890" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7891" xml:space="preserve">propterea R I differentia æquidiſtantium F I, G K ad <lb/>partes centri A, &amp; </s>
  <s xml:id="echoid-s7892" xml:space="preserve">asymptoti A B vlterius tendentium, maior erit, quàm S L, <lb/>quæ portio eſt ipſius S O, &amp; </s>
  <s xml:id="echoid-s7893" xml:space="preserve">eſt differentia æquidiſtantium H L, &amp; </s>
  <s xml:id="echoid-s7894" xml:space="preserve">C E alte-<lb/>rius ſegmenti H C. </s>
  <s xml:id="echoid-s7895" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s7896" xml:space="preserve"/>
</p>
<div xml:id="echoid-div715" type="float" level="2" n="5">
  <figure xlink:label="fig-0248-02" xlink:href="fig-0248-02a">
    <image file="0248-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0248-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7897" xml:space="preserve">Ex conſtructione, &amp; </s>
  <s xml:id="echoid-s7898" xml:space="preserve">demonſtratione huius propoſitionis colligitur, quod ſi à <lb/>
<anchor type="note" xlink:label="note-0249-01a" xlink:href="note-0249-01"/>
duobus punctis eiuſdem asymptoti A C ad hyperbolen ducantur duæ rectæ lineæ <lb/>inter ſe parallelæ; </s>
  <s xml:id="echoid-s7899" xml:space="preserve">illa, quæ ad partes centri A, &amp; </s>
  <s xml:id="echoid-s7900" xml:space="preserve">asymptoti A B vlterius ten-<lb/>dit, maior eſt reliqua. </s>
  <s xml:id="echoid-s7901" xml:space="preserve">Nam recta linea K R, asymptoto A C parallela cadit ex-<lb/>tra ſectionem, &amp; </s>
  <s xml:id="echoid-s7902" xml:space="preserve">ideo ſecat interceptam parallelam F I, quæ erit maior, quàm <lb/>F R, ſeu G K; </s>
  <s xml:id="echoid-s7903" xml:space="preserve">igitur F I ad partes centri A vlterius tendens maior eſt quali-<lb/>bet alia parallela G K ad partes oppoſitas tendente. </s>
  <s xml:id="echoid-s7904" xml:space="preserve">Eadem ratione F I maior <lb/>erit quàm H L, &amp; </s>
  <s xml:id="echoid-s7905" xml:space="preserve">H L maior, quàm C E. </s>
  <s xml:id="echoid-s7906" xml:space="preserve">Vnde patet propoſitum.</s>
  <s xml:id="echoid-s7907" xml:space="preserve"/>
</p>
<div xml:id="echoid-div716" type="float" level="2" n="6">
<note position="right" xlink:label="note-0249-01" xlink:href="note-0249-01a" xml:space="preserve">COROL <lb/>LAR.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7908" xml:space="preserve">Si fuerint duæ hyperbolæ A B, &amp; </s>
  <s xml:id="echoid-s7909" xml:space="preserve">D E æquales, &amp; </s>
  <s xml:id="echoid-s7910" xml:space="preserve">ſimiles ad eaſ-<lb/>
<anchor type="note" xlink:label="note-0249-02a" xlink:href="note-0249-02"/>
dem partes cauæ, quarum centra H, &amp; </s>
  <s xml:id="echoid-s7911" xml:space="preserve">L, &amp; </s>
  <s xml:id="echoid-s7912" xml:space="preserve">aſymptoti G H I, &amp; </s>
  <s xml:id="echoid-s7913" xml:space="preserve"><lb/>K L M, nec non axes A H, &amp; </s>
  <s xml:id="echoid-s7914" xml:space="preserve">D L ſint parallelæ inter ſe, &amp; </s>
  <s xml:id="echoid-s7915" xml:space="preserve">rectæ <lb/>lineæ B E, &amp; </s>
  <s xml:id="echoid-s7916" xml:space="preserve">C F ab hyperbolis interceptæ parallelæ fuerint rectæ H <lb/>L centra coniungenti; </s>
  <s xml:id="echoid-s7917" xml:space="preserve">erunt B E, &amp; </s>
  <s xml:id="echoid-s7918" xml:space="preserve">C F æquales ipſi H L, &amp; </s>
  <s xml:id="echoid-s7919" xml:space="preserve">in-<lb/>ter ſe.</s>
  <s xml:id="echoid-s7920" xml:space="preserve"/>
</p>
<div xml:id="echoid-div717" type="float" level="2" n="7">
<note position="right" xlink:label="note-0249-02" xlink:href="note-0249-02a" xml:space="preserve">PROP.3. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s7921" xml:space="preserve">Si autem parallelæ ſint alicui rectæ lineæ L f diuidenti angulum K L <lb/>
<anchor type="figure" xlink:label="fig-0249-01a" xlink:href="fig-0249-01"/>
H contentum à recta linea L H cen-
tra coniungente, &amp; </s>
  <s xml:id="echoid-s7922" xml:space="preserve">interiore aſympto-
to L K, in qua B E, &amp; </s>
  <s xml:id="echoid-s7923" xml:space="preserve">C F poſitæ
ſunt: </s>
  <s xml:id="echoid-s7924" xml:space="preserve">Dico B E vlterius tendentem.</s>
  <s xml:id="echoid-s7925" xml:space="preserve">
ad partes reliquæ aſymptoti L M ma-
iorem eſſe, quàm C F.</s>
  <s xml:id="echoid-s7926" xml:space="preserve"/>
</p>
<div xml:id="echoid-div718" type="float" level="2" n="8">
  <figure xlink:label="fig-0249-01" xlink:href="fig-0249-01a">
    <image file="0249-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0249-01"/>
<caption xml:id="echoid-caption2" xml:space="preserve">Dd 2</caption>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7927" xml:space="preserve">Si vero B E, &amp; </s>
  <s xml:id="echoid-s7928" xml:space="preserve">C F parallelæ <lb/>ſint alicui rectæ lineæ H g diuidenti <lb/>angulum L H G à recta linea L H <lb/>centra coniungente, &amp; </s>
  <s xml:id="echoid-s7929" xml:space="preserve">eadem aſym-<lb/>ptoto H G contentum: </s>
  <s xml:id="echoid-s7930" xml:space="preserve">Dico B E vl-<lb/>terius tendentẽ ad partes reliquæ aſym-<lb/>ptoti H I minorem eſſe, quàm C F.</s>
  <s xml:id="echoid-s7931" xml:space="preserve">
<pb o="212" file="0250" n="250" rhead="Apollonij Pergæi"/>
Rectæ lineæ parallelæ B E, C F ſe-<lb/>
<anchor type="figure" xlink:label="fig-0250-01a" xlink:href="fig-0250-01"/>
cent æquidiſtantes aſymptotos H G, <lb/>L K in punctis N<unsure/>, O, P, Q. </s>
  <s xml:id="echoid-s7932" xml:space="preserve">De-<lb/>bent autem coniſectiones in eodem pla-<lb/>no collocari ſicuti aliæ omnes, quæ in. <lb/></s>
  <s xml:id="echoid-s7933" xml:space="preserve">ſequentibus propoſitionibus 4. </s>
  <s xml:id="echoid-s7934" xml:space="preserve">5. </s>
  <s xml:id="echoid-s7935" xml:space="preserve">6. </s>
  <s xml:id="echoid-s7936" xml:space="preserve">7. </s>
  <s xml:id="echoid-s7937" xml:space="preserve"><lb/>8. </s>
  <s xml:id="echoid-s7938" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7939" xml:space="preserve">9. </s>
  <s xml:id="echoid-s7940" xml:space="preserve">vſurpantur ſemper in vno <lb/>plano poſitæ intelligi debent.</s>
  <s xml:id="echoid-s7941" xml:space="preserve"/>
</p>
<div xml:id="echoid-div719" type="float" level="2" n="9">
  <figure xlink:label="fig-0250-01" xlink:href="fig-0250-01a">
    <image file="0250-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0250-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s7942" xml:space="preserve">Et primo duæ rectæ B E, C F paralle-<lb/>læ ſint rectæ lineæ H L centra coniungen-<lb/>ti. </s>
  <s xml:id="echoid-s7943" xml:space="preserve">Quoniam hyperbolæ A B, D E æqua-<lb/>les ſunt, &amp; </s>
  <s xml:id="echoid-s7944" xml:space="preserve">congruentes; </s>
  <s xml:id="echoid-s7945" xml:space="preserve">atque æquidiſtan-<lb/>tes asymptoti H N, L P æque inclinan-<lb/>tur ad æquales ſemiaxes tranſuerſos H <lb/>A, &amp; </s>
  <s xml:id="echoid-s7946" xml:space="preserve">L D; </s>
  <s xml:id="echoid-s7947" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7948" xml:space="preserve">ſegmenta asymptotorum H N, L P æqualia ſunt in paralle-<lb/>logrammo H P, nec non duo anguli H N B, &amp; </s>
  <s xml:id="echoid-s7949" xml:space="preserve">L P E æquales ſunt inter ſe, pro-<lb/>pter parallelas asymptotos: </s>
  <s xml:id="echoid-s7950" xml:space="preserve">igitur duæ figuræ A H N B A, &amp; </s>
  <s xml:id="echoid-s7951" xml:space="preserve">D L P E D æquales <lb/>erunt, &amp; </s>
  <s xml:id="echoid-s7952" xml:space="preserve">congruentes: </s>
  <s xml:id="echoid-s7953" xml:space="preserve">quapropter interpoſitæ rectæ lineæ N B &amp; </s>
  <s xml:id="echoid-s7954" xml:space="preserve">P E congruẽ-<lb/>tes, &amp; </s>
  <s xml:id="echoid-s7955" xml:space="preserve">æquales erunt; </s>
  <s xml:id="echoid-s7956" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7957" xml:space="preserve">addita vel ablata communi B P, erit N P æqualis <lb/>B E: </s>
  <s xml:id="echoid-s7958" xml:space="preserve">eſt verò N P æqualis H L, eo quod H P parallelogrammum eſt; </s>
  <s xml:id="echoid-s7959" xml:space="preserve">igitur <lb/>intercepta B E æqualis eſt rectæ lineæ H L centra coniungenti. </s>
  <s xml:id="echoid-s7960" xml:space="preserve">Eadem ratione <lb/>quælibet alia intercepta C F parallela ipſi H L eidem æqualis oſtendetur: </s>
  <s xml:id="echoid-s7961" xml:space="preserve">qua-<lb/>propter duæ interceptæ æquidiſtantes B E, &amp; </s>
  <s xml:id="echoid-s7962" xml:space="preserve">C F inter ſe æquales erunt.</s>
  <s xml:id="echoid-s7963" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s7964" xml:space="preserve">Secundo B E, C F parallelæ ſint alicui rectæ lineæ L f diuidenti angulum K <lb/>L H; </s>
  <s xml:id="echoid-s7965" xml:space="preserve">ideoque P L f N, &amp; </s>
  <s xml:id="echoid-s7966" xml:space="preserve">Q L f O parallelogramma erunt: </s>
  <s xml:id="echoid-s7967" xml:space="preserve">ſecetur L T æqua-<lb/>
<anchor type="figure" xlink:label="fig-0250-02a" xlink:href="fig-0250-02"/>
lis H N, atque L V æqualis H O; </s>
  <s xml:id="echoid-s7968" xml:space="preserve">ducan-<lb/>turque T X, V Z parallelæ ipſis N B, O <lb/>C ſecantes reliquam hyperbolen in X, Z; <lb/></s>
  <s xml:id="echoid-s7969" xml:space="preserve">eritque ( vt in prima parte oſtenſum eſt) <lb/>T X æqualis N B, atque V Z æqualis O C. </s>
  <s xml:id="echoid-s7970" xml:space="preserve"><lb/>Et ſiquidem B E, C F cadunt infra cen-<lb/>tra H, L ad partes G, K, cadent quoque <lb/>infra L f eis parallelam per L ductam in-<lb/>fra centrum H incidentem, &amp; </s>
  <s xml:id="echoid-s7971" xml:space="preserve">ideo N f, <lb/>ſeu ei æqualis P L in parallelogrãmo P f <lb/>minor erit, quàm H N; </s>
  <s xml:id="echoid-s7972" xml:space="preserve">eſtque L T æqua-<lb/>lis H N; </s>
  <s xml:id="echoid-s7973" xml:space="preserve">igitur L P minor erit, quàm L T ; </s>
  <s xml:id="echoid-s7974" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7975" xml:space="preserve">propterea punctum P propin-<lb/>quius erit centro L, quàm T: </s>
  <s xml:id="echoid-s7976" xml:space="preserve">Eadem ratione oſtendetur, quod punctum Q pro-<lb/>pinquius ſit centro L, quàm V, &amp; </s>
  <s xml:id="echoid-s7977" xml:space="preserve">P propinquius centro quàm Q; </s>
  <s xml:id="echoid-s7978" xml:space="preserve">ergo quatuor <lb/>
<anchor type="note" xlink:label="note-0250-01a" xlink:href="note-0250-01"/>
æquidiſtantium P E, Q F, T X, V Z cadentium infra centrum ad partes K, <lb/>duæ P E, T X vlterius ad partes centri, vel asymptoti L M tendunt, quàm, <lb/>duæ Q F, V Z. </s>
  <s xml:id="echoid-s7979" xml:space="preserve">At ſi B E, C F ſecent rectã lineam centra coniungentem inter <lb/>duo centra H, &amp; </s>
  <s xml:id="echoid-s7980" xml:space="preserve">L, manifeſtum eſt puncta P, &amp; </s>
  <s xml:id="echoid-s7981" xml:space="preserve">Q cadere ſupra centrum L, <lb/>atque duo puncta N, &amp; </s>
  <s xml:id="echoid-s7982" xml:space="preserve">O cadere infra centrnm H alterius hyperboles, cumque <lb/>L T ſecta ſit æqualis ipſi H N ad eaſdem partes; </s>
  <s xml:id="echoid-s7983" xml:space="preserve">pariterque L V æqualis ipſi
<pb o="213" file="0251" n="251" rhead="Conicor. Lib. VI."/>
H O cadent puncta T, &amp; </s>
  <s xml:id="echoid-s7984" xml:space="preserve">V infra centrum L; </s>
  <s xml:id="echoid-s7985" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7986" xml:space="preserve">P vlterius tendit quàm Q ad <lb/>partes, eiuſdem centri L. </s>
  <s xml:id="echoid-s7987" xml:space="preserve">igitur in tali caſit quatuor æquidiſtantium duæ P E, <lb/>
<anchor type="note" xlink:label="note-0251-01a" xlink:href="note-0251-01"/>
T X vlterius tendent ad partes centri, &amp; </s>
  <s xml:id="echoid-s7988" xml:space="preserve">asymptoti L M, quàm duæ aliæ æqui-<lb/>diſtantes Q F, V Z. </s>
  <s xml:id="echoid-s7989" xml:space="preserve">Quando verò B E, &amp; </s>
  <s xml:id="echoid-s7990" xml:space="preserve">C F cadunt vltra centra H, &amp; </s>
  <s xml:id="echoid-s7991" xml:space="preserve"><lb/>L in productionibus æquidiſtantium asymptotorum G H, K L: </s>
  <s xml:id="echoid-s7992" xml:space="preserve">quia N P cadit <lb/>
<anchor type="figure" xlink:label="fig-0251-01a" xlink:href="fig-0251-01"/>
ſupra, &amp; </s>
  <s xml:id="echoid-s7993" xml:space="preserve">L f infra centrũ H, ergo in parallelogrammo P f recta N f, ſeu ei æ-<lb/>qualis L P maior erit quàm N H: </s>
  <s xml:id="echoid-s7994" xml:space="preserve">facta autem fuit L T æqualis H N; </s>
  <s xml:id="echoid-s7995" xml:space="preserve">igitur <lb/>L T minor eſt, quàm L P; </s>
  <s xml:id="echoid-s7996" xml:space="preserve">Eadem ratione L V minor erit, quàm L Q, at-<lb/>que P vlterius tendit quàm Q ad partes centri L, &amp; </s>
  <s xml:id="echoid-s7997" xml:space="preserve">ab ĳſdem punctis caden-<lb/>tibus ſupra centrum L in productione asymptoti K L ducuntur quatuor rectæ <lb/>lineæ inter ſe æquidiſtantes vſque ad hyperbolen D Z; </s>
  <s xml:id="echoid-s7998" xml:space="preserve">igitur duæ P E, T X vl-<lb/>
<anchor type="note" xlink:label="note-0251-02a" xlink:href="note-0251-02"/>
terius tendunt ad partes centri, vel asymptoti L M, quàm duæ Q F, V Z. <lb/></s>
  <s xml:id="echoid-s7999" xml:space="preserve">Secetur poſtea P a æqualis N B, atque Q b æqualis O C. </s>
  <s xml:id="echoid-s8000" xml:space="preserve">Et quia T X æqua-<lb/>lis oſtenſa fuit N B erit P a æqualis ipſi T X; </s>
  <s xml:id="echoid-s8001" xml:space="preserve">eſtque P E maior quàm T X; </s>
  <s xml:id="echoid-s8002" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0251-03a" xlink:href="note-0251-03"/>
propterea quod illa vlterius tendit ad partes cẽtri L, quàm T X; </s>
  <s xml:id="echoid-s8003" xml:space="preserve">igitur P E ma-<lb/>ior erit, quàm P a, &amp; </s>
  <s xml:id="echoid-s8004" xml:space="preserve">earum differentia erit E a. </s>
  <s xml:id="echoid-s8005" xml:space="preserve">Simili modo oſtendetur Q <lb/>b æqualis V Z, &amp; </s>
  <s xml:id="echoid-s8006" xml:space="preserve">minor quàm Q F, quarum differentia F b: </s>
  <s xml:id="echoid-s8007" xml:space="preserve">cumque Q P <lb/>æqualis ſit ipſi N O, propterea quod ſunt latera oppoſita eiuſdem parallelogram-<lb/>mi; </s>
  <s xml:id="echoid-s8008" xml:space="preserve">igitur T V, quæ oſtenſa fuit æqualis O N erit quoque æqualis Q P, &amp; </s>
  <s xml:id="echoid-s8009" xml:space="preserve">sũ-<lb/>pta communiter Q T erit Q V æqualis T P, atque à terminis æqualium ſeg-<lb/>mentorum eiuſdem asymptoti L K ducuntur vſque ad hyperbolen E Z quatuor <lb/>rectæ lineæ inter ſe æquidiſtantes, &amp; </s>
  <s xml:id="echoid-s8010" xml:space="preserve">earum binæ P E, T X vlterius tendunt <lb/>ad partes centri, &amp; </s>
  <s xml:id="echoid-s8011" xml:space="preserve">asymptoti L M, quàm binæ Q F, V Z; </s>
  <s xml:id="echoid-s8012" xml:space="preserve">igitur differentia <lb/>
<anchor type="note" xlink:label="note-0251-04a" xlink:href="note-0251-04"/>
priorum, ſcilicet E a maior erit poſteriorum differentia F b; </s>
  <s xml:id="echoid-s8013" xml:space="preserve">eſtque B a æqua-<lb/>lis N P, propterea quod æqualibus N B, &amp; </s>
  <s xml:id="echoid-s8014" xml:space="preserve">P a ponitur communiter B P; </s>
  <s xml:id="echoid-s8015" xml:space="preserve">pa-<lb/>riterque O Q æqualis eſt C b; </s>
  <s xml:id="echoid-s8016" xml:space="preserve">ſuntque N P, &amp; </s>
  <s xml:id="echoid-s8017" xml:space="preserve">O Q æquales inter ſe, nempe <lb/>latera oppoſita eiuſdem parallelogrammi; </s>
  <s xml:id="echoid-s8018" xml:space="preserve">igitur B a, &amp; </s>
  <s xml:id="echoid-s8019" xml:space="preserve">C b æquales ſunt inter <lb/>ſe: </s>
  <s xml:id="echoid-s8020" xml:space="preserve">ĳs verò adduntur exceßus inæquales E a, F b efficietur E B vlterius ten-<lb/>dens ad partes asymptoti H I maior, quàm F C. </s>
  <s xml:id="echoid-s8021" xml:space="preserve">Quod erat primum.</s>
  <s xml:id="echoid-s8022" xml:space="preserve"/>
</p>
<div xml:id="echoid-div720" type="float" level="2" n="10">
  <figure xlink:label="fig-0250-02" xlink:href="fig-0250-02a">
    <image file="0250-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0250-02"/>
  </figure>
<note position="left" xlink:label="note-0250-01" xlink:href="note-0250-01a" xml:space="preserve">Def. add.</note>
<note position="right" xlink:label="note-0251-01" xlink:href="note-0251-01a" xml:space="preserve">Def. add.</note>
  <figure xlink:label="fig-0251-01" xlink:href="fig-0251-01a">
    <image file="0251-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0251-01"/>
  </figure>
<note position="right" xlink:label="note-0251-02" xlink:href="note-0251-02a" xml:space="preserve">Ibidem.</note>
<note position="right" xlink:label="note-0251-03" xlink:href="note-0251-03a" xml:space="preserve">Coroll. <lb/>Propoſ. 2. <lb/>addit.</note>
<note position="right" xlink:label="note-0251-04" xlink:href="note-0251-04a" xml:space="preserve">Propoſ. 2. <lb/>addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8023" xml:space="preserve">Tertio ĳſdem poſitis N E, O F ſint parallelæ alicui rectæ lineæ H g diuidẽti <lb/>angulum L H G, &amp; </s>
  <s xml:id="echoid-s8024" xml:space="preserve">propterea extensæ productionem asymptoti M L ſecabunt,
<pb o="214" file="0252" n="252" rhead="Apollonij Pergæi"/>
&amp; </s>
  <s xml:id="echoid-s8025" xml:space="preserve">parallelæ erunt alicui recta lineæ ex L <lb/>
<anchor type="figure" xlink:label="fig-0252-01a" xlink:href="fig-0252-01"/>
diuidenti angulum H L M, eo quod paral-<lb/>lelæ erãt rectæ H g diuidenti angulum L H <lb/>G, &amp; </s>
  <s xml:id="echoid-s8026" xml:space="preserve">prius B E vlterius, quàm C F ten-<lb/>debat ad partes asymptoti H I; </s>
  <s xml:id="echoid-s8027" xml:space="preserve">ergo è con-<lb/>tra C F vlterius tendet ad partes asymptoti <lb/>H G, &amp; </s>
  <s xml:id="echoid-s8028" xml:space="preserve">educũtur ab asymptoto L M producta, <lb/>&amp; </s>
  <s xml:id="echoid-s8029" xml:space="preserve">parallelæ ſunt rectæ lineæ ex L diuidenti <lb/>angulũ H L M, contentum à recta linea cen-<lb/>tra coniungente, &amp; </s>
  <s xml:id="echoid-s8030" xml:space="preserve">a symptoto M L, in qua <lb/>illæ cadunt; </s>
  <s xml:id="echoid-s8031" xml:space="preserve">igitur ( ex prima parte huius propoſitionis) C F maior erit, quàm <lb/>B E; </s>
  <s xml:id="echoid-s8032" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8033" xml:space="preserve">è contra B E vlterius tendens ad partes asymptoti H I minor erit, quã <lb/>C F; </s>
  <s xml:id="echoid-s8034" xml:space="preserve">vt propoſitum fuerat.</s>
  <s xml:id="echoid-s8035" xml:space="preserve"/>
</p>
<div xml:id="echoid-div721" type="float" level="2" n="11">
  <figure xlink:label="fig-0252-01" xlink:href="fig-0252-01a">
    <image file="0252-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0252-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s8036" xml:space="preserve">Sint duæ æquales parabolæ A B, D E ad eaſdem partes cauæ, qua-<lb/>
<anchor type="note" xlink:label="note-0252-01a" xlink:href="note-0252-01"/>
rum diametri G I, H K ſint congruentes aut parallelæ inter ſe, nec nõ <lb/>ad eas ordinatim applicatæ B Z K, L X N<unsure/> ſint parallelæ alicui rectæ <lb/>diuidenti angulum G H K à recta linea G H vertices coniungenti, &amp; </s>
  <s xml:id="echoid-s8037" xml:space="preserve"><lb/>diametro H K interioris ſectionis D H contentum, ſi diametri congruentes <lb/>non fuerint. </s>
  <s xml:id="echoid-s8038" xml:space="preserve">Dico quod, B E, L M portiones applicatarum à ſectioni-<lb/>bus ad eaſdem partes interceptæ, ſemper magis diminuentur, quo magis <lb/>à verticibus recedunt; </s>
  <s xml:id="echoid-s8039" xml:space="preserve">efficienturque minores quacumque recta linea pro-<lb/>poſita, ſi diametri ſunt congruentes: </s>
  <s xml:id="echoid-s8040" xml:space="preserve">ſi verò ſunt parallelæ nunquam mi-<lb/>nores erunt portione ordinatæ inter diametros intercepta. </s>
  <s xml:id="echoid-s8041" xml:space="preserve">At ſi paral-<lb/>lelæ fuerint alicui rectæ lineæ diuidenti angulum H G I à recta G H, <lb/>&amp; </s>
  <s xml:id="echoid-s8042" xml:space="preserve">diametro I G exterioris ſectionis A G contentum, ſemper magis au-<lb/>gentur, ſed erunt ſemper minores ea quæ à diametris intercipitur. </s>
  <s xml:id="echoid-s8043" xml:space="preserve">Vel ſi <lb/>fuerint parallelæ diametris non congruentibus, ſemper magis augentur, <lb/>quo magis à concurſu recedunt.</s>
  <s xml:id="echoid-s8044" xml:space="preserve"/>
</p>
<div xml:id="echoid-div722" type="float" level="2" n="12">
<note position="left" xlink:label="note-0252-01" xlink:href="note-0252-01a" xml:space="preserve">PROP. 4. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8045" xml:space="preserve">Sit F G latus rectum diametri G I in, <lb/>
<anchor type="figure" xlink:label="fig-0252-02a" xlink:href="fig-0252-02"/>
parabola G B, ordinatim applicatæ B E K, <lb/>&amp; </s>
  <s xml:id="echoid-s8046" xml:space="preserve">L M N ſecent diametrum G I in X, Z, <lb/>&amp; </s>
  <s xml:id="echoid-s8047" xml:space="preserve">diametrum H K in N, K, &amp; </s>
  <s xml:id="echoid-s8048" xml:space="preserve">ſecetur <lb/>abſciſſa G I æqualis H K, &amp; </s>
  <s xml:id="echoid-s8049" xml:space="preserve">G R æqualis <lb/>H N; </s>
  <s xml:id="echoid-s8050" xml:space="preserve">ideoque R I æqualis erit N K, ſeu <lb/>X Z (propterea quod in parallelogrammo <lb/>N Z oppoſita latera æqualia ſunt) ducan-<lb/>turque ordinatæ O I, Q R, quæ erunt æqua-<lb/>
<anchor type="note" xlink:label="note-0252-02a" xlink:href="note-0252-02"/>
les, &amp; </s>
  <s xml:id="echoid-s8051" xml:space="preserve">congruentes ipſis E K, M N pro-<lb/>pter æqualitatem ſectionum, &amp; </s>
  <s xml:id="echoid-s8052" xml:space="preserve">abſciſſarũ <lb/>ſimilium diametrorum; </s>
  <s xml:id="echoid-s8053" xml:space="preserve">ducanturque à pun-<lb/>ctis E, L, Q rectæ lineæ E S, L T, Q V <lb/>parallelæ diametris occurrentes ipſis B E, <lb/>&amp; </s>
  <s xml:id="echoid-s8054" xml:space="preserve">O I in S, T, V: </s>
  <s xml:id="echoid-s8055" xml:space="preserve">manifeſtum eſt S M
<pb o="215" file="0253" n="253" rhead="Conicor. Lib. VI."/>
æqualem eße O V, eo quod in perallelogrammis Q I, &amp; </s>
  <s xml:id="echoid-s8056" xml:space="preserve">S K latera oppoſita ſunt <lb/>æqualia, &amp; </s>
  <s xml:id="echoid-s8057" xml:space="preserve">ipſæ ordinatæ E K O I; </s>
  <s xml:id="echoid-s8058" xml:space="preserve">nec non M N, Q R æquales oſtenſæ ſunt: <lb/></s>
  <s xml:id="echoid-s8059" xml:space="preserve">Deinde producantur, B E, O I ad ſectionem in C, P; </s>
  <s xml:id="echoid-s8060" xml:space="preserve">Et quia differentia qua-<lb/>dratorum B Z, L X, ſeu T Z, ideſt rectangulum B T C æquale eſt differentiæ <lb/>
<anchor type="note" xlink:label="note-0253-01a" xlink:href="note-0253-01"/>
rectangulorum Z G F, &amp; </s>
  <s xml:id="echoid-s8061" xml:space="preserve">X G F ſeu rectangulo ſub abſciſſarum differentia X Z, <lb/>&amp; </s>
  <s xml:id="echoid-s8062" xml:space="preserve">latere recto G F. </s>
  <s xml:id="echoid-s8063" xml:space="preserve">Simili modo rectangulum O V P æquale erit rectangulo ſub <lb/>abſciſſarum differentia R I, &amp; </s>
  <s xml:id="echoid-s8064" xml:space="preserve">latere recto G F: </s>
  <s xml:id="echoid-s8065" xml:space="preserve">ſuntque rectangula contenta <lb/>ſub X Z, G F, &amp; </s>
  <s xml:id="echoid-s8066" xml:space="preserve">ſub R I, G F æqualia, propterea quod later a X Z, R I æqua-<lb/>lia oſtenſa ſunt, &amp; </s>
  <s xml:id="echoid-s8067" xml:space="preserve">latus rectum G F eſt commune; </s>
  <s xml:id="echoid-s8068" xml:space="preserve">igitur rectangula B T C, &amp; </s>
  <s xml:id="echoid-s8069" xml:space="preserve"><lb/>O V P æqualia ſunt; </s>
  <s xml:id="echoid-s8070" xml:space="preserve">ideoque vt T C ad V P, ita reciprocè erit O V ad B T. <lb/></s>
  <s xml:id="echoid-s8071" xml:space="preserve">Et primò quia diametri G Z, H K coincidunt, &amp; </s>
  <s xml:id="echoid-s8072" xml:space="preserve">parabolæ H D compræhendi-<lb/>tur ab A G: </s>
  <s xml:id="echoid-s8073" xml:space="preserve">erit G Z maior quàm H K, ſeu quàm G I, &amp; </s>
  <s xml:id="echoid-s8074" xml:space="preserve">B Z maior quàm <lb/>E K, &amp; </s>
  <s xml:id="echoid-s8075" xml:space="preserve">L X quàm M N. </s>
  <s xml:id="echoid-s8076" xml:space="preserve">Si verò B E, L M parallelæ ſunt alicui rectæ lineæ <lb/>H Y diuidenti angulum G H K; </s>
  <s xml:id="echoid-s8077" xml:space="preserve">ergo Y Z, ſeu ei æqualis H K, vel G I minor <lb/>erit, quàm G Z. </s>
  <s xml:id="echoid-s8078" xml:space="preserve">Eadem ratione G X maior erit, quàm G R; </s>
  <s xml:id="echoid-s8079" xml:space="preserve">quare ordinatim <lb/>applicata B Z maior erit, quàm O I, &amp; </s>
  <s xml:id="echoid-s8080" xml:space="preserve">Z C maior, quàm I P; </s>
  <s xml:id="echoid-s8081" xml:space="preserve">pariterque L <lb/>X, ſeu T Z maior erit, quàm Q R, ſeu V I; </s>
  <s xml:id="echoid-s8082" xml:space="preserve">ideoque T C maior erit, quàm <lb/>V P: </s>
  <s xml:id="echoid-s8083" xml:space="preserve">erat autem O V ad B T reciprocè, vt T C ad V P; </s>
  <s xml:id="echoid-s8084" xml:space="preserve">ergo O V, ſeu ei æqua-<lb/>lis S M maior erit, quàm B T: </s>
  <s xml:id="echoid-s8085" xml:space="preserve">ĳ verò addantur æquales L S, T E, quæ in <lb/>parallelogrammo S T ſunt latera oppoſita, igitur L M, maior erit quàm B E.</s>
  <s xml:id="echoid-s8086" xml:space="preserve"/>
</p>
<div xml:id="echoid-div723" type="float" level="2" n="13">
  <figure xlink:label="fig-0252-02" xlink:href="fig-0252-02a">
    <image file="0252-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0252-02"/>
  </figure>
<note position="left" xlink:label="note-0252-02" xlink:href="note-0252-02a" xml:space="preserve">ex 10. <lb/>ex 21. <lb/>huius.</note>
<note position="right" xlink:label="note-0253-01" xlink:href="note-0253-01a" xml:space="preserve">ex II. <lb/>lib. I.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8087" xml:space="preserve">Deinde quando diametri G I, H K ſibi mutuo congruunt ſit b minor qualibet <lb/>data recta linea, &amp; </s>
  <s xml:id="echoid-s8088" xml:space="preserve">à vertice H ducatur H d cuius quadratũ æquale ſit rectangulo <lb/>H G F, &amp; </s>
  <s xml:id="echoid-s8089" xml:space="preserve">fiat vt b ad H d, ita H d ad aliam rectam lineam æqualem C E; </s>
  <s xml:id="echoid-s8090" xml:space="preserve">atq; <lb/></s>
  <s xml:id="echoid-s8091" xml:space="preserve">vt H d ad ſemiſſem sũmæ C E, &amp; </s>
  <s xml:id="echoid-s8092" xml:space="preserve">b potentia, ita fiat longitudine H G ad G K, <lb/>ducaturque B K C ordinatim applicata ad diametrum G I. </s>
  <s xml:id="echoid-s8093" xml:space="preserve">Quoniam quadra-<lb/>
<anchor type="note" xlink:label="note-0253-02a" xlink:href="note-0253-02"/>
tum E K æquale eſt parallelogrammo H K, G F (propterea quod parabolæ ſunt <lb/>æquales, &amp; </s>
  <s xml:id="echoid-s8094" xml:space="preserve">diametri ſimiles) &amp; </s>
  <s xml:id="echoid-s8095" xml:space="preserve">ĳs adduntur inter ſe æqualia quadratum d H, <lb/>&amp; </s>
  <s xml:id="echoid-s8096" xml:space="preserve">rectangulum H G F, erunt duo quadrata E K, &amp; </s>
  <s xml:id="echoid-s8097" xml:space="preserve">d H ſimul ſumpta æqualia <lb/>rectãgulo K G F, ſeu quadrato B Z; </s>
  <s xml:id="echoid-s8098" xml:space="preserve">quare differentia quadratorũ B K, &amp; </s>
  <s xml:id="echoid-s8099" xml:space="preserve">E K, <lb/>ideſt rectanguli B E C æqualis erit quadrato d H; </s>
  <s xml:id="echoid-s8100" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8101" xml:space="preserve">propterea d H media pro-<lb/>portionalis eſt inter C E, B E, ſed facta fuit media proportionalis inter C E, <lb/>&amp; </s>
  <s xml:id="echoid-s8102" xml:space="preserve">b; </s>
  <s xml:id="echoid-s8103" xml:space="preserve">Ergo B E æqualis eſt b; </s>
  <s xml:id="echoid-s8104" xml:space="preserve">ideoque R E minor @@ qu@libet recta linea data. <lb/></s>
  <s xml:id="echoid-s8105" xml:space="preserve">Quando verò diametri G Z, H K ſunt æquidiſtantes, ĳsdem poſitis ducatur O <lb/>n parallela diametris ſecans B E in n. </s>
  <s xml:id="echoid-s8106" xml:space="preserve">Quia n Z eſt æqualis O I. </s>
  <s xml:id="echoid-s8107" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8108" xml:space="preserve">erat E K <lb/>æqualis O I, ergo n Z, &amp; </s>
  <s xml:id="echoid-s8109" xml:space="preserve">E K æquales ſunt, &amp; </s>
  <s xml:id="echoid-s8110" xml:space="preserve">addita, vel ablata comm@ni Z <lb/>E erit n E æqualis Z K; </s>
  <s xml:id="echoid-s8111" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8112" xml:space="preserve">propterea quælibet intercepta B E @@ior erit in <lb/>ſecundo caſu, &amp; </s>
  <s xml:id="echoid-s8113" xml:space="preserve">minor in tertio, quàm n E, ſeu Z K à diametris compræben-<lb/>ſa.</s>
  <s xml:id="echoid-s8114" xml:space="preserve"/>
</p>
<div xml:id="echoid-div724" type="float" level="2" n="14">
<note position="right" xlink:label="note-0253-02" xlink:href="note-0253-02a" xml:space="preserve">II. lib. I.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8115" xml:space="preserve">Tertio quando B E, L M parallelæ ſunt alicui rectæ G a diuidenti angulum <lb/>H G I, erit K a, ſeu ei æqualis G Z minor, quàm H K, ſeu quàm G I, atq; </s>
  <s xml:id="echoid-s8116" xml:space="preserve">vt <lb/>prius rectangula B T C, &amp; </s>
  <s xml:id="echoid-s8117" xml:space="preserve">O V P æqualia erunt, &amp; </s>
  <s xml:id="echoid-s8118" xml:space="preserve">eorum latera reciprocè <lb/>proportionalia, eſtque S M æqualis minori O V, ergo S M minor erit quàm B <lb/>T; </s>
  <s xml:id="echoid-s8119" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8120" xml:space="preserve">additis æqualibus L S, &amp; </s>
  <s xml:id="echoid-s8121" xml:space="preserve">T E, erit L M minor quàm B E.</s>
  <s xml:id="echoid-s8122" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s8123" xml:space="preserve">Tandem ſint interceptæ B E, L M parallelæ G V, H C portionibus interce-<lb/>ptarum diametrorum non congruentium, &amp; </s>
  <s xml:id="echoid-s8124" xml:space="preserve">à terminis B, E, L, M, ducan-<lb/>tur ad diametros ordinatim applicatæ, eas ſecantes in Z, K, I, N, O, S, &amp; </s>
  <s xml:id="echoid-s8125" xml:space="preserve"><lb/>ſectiones in P, &amp; </s>
  <s xml:id="echoid-s8126" xml:space="preserve">R; </s>
  <s xml:id="echoid-s8127" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8128" xml:space="preserve">cadat B E inter duas diametros. </s>
  <s xml:id="echoid-s8129" xml:space="preserve">Quoniam punctum
<pb o="216" file="0254" n="254" rhead="Apollonij Pergæi"/>
B cadit inter verticem G, &amp; </s>
  <s xml:id="echoid-s8130" xml:space="preserve">punctum <lb/>
<anchor type="figure" xlink:label="fig-0254-01a" xlink:href="fig-0254-01"/>
C eiuſdem parabolæ G C; </s>
  <s xml:id="echoid-s8131" xml:space="preserve">igitur Z B <lb/>K ordinatim applicata ad diametrum <lb/>G I neceßario ſecabit diametrum G I <lb/>intra ſectionem in Z, &amp; </s>
  <s xml:id="echoid-s8132" xml:space="preserve">producta <lb/>occurret K N extra eandem in K. <lb/></s>
  <s xml:id="echoid-s8133" xml:space="preserve">Non ſecus oſtendetur, quod E N I or-<lb/>dinatim applicatæ ad diametrum H <lb/>N, punctum N cadit intra, &amp; </s>
  <s xml:id="echoid-s8134" xml:space="preserve">I ex-<lb/>tra eandem ſectionem H E, &amp; </s>
  <s xml:id="echoid-s8135" xml:space="preserve">pro-<lb/>pterea recta C H minor erit, quàm K <lb/>N, ſeu B E ei æqualis in parallelo-<lb/>grammo E K; </s>
  <s xml:id="echoid-s8136" xml:space="preserve">pariterque Z I, ſeu ei <lb/>æqualis B E minor erit, quàm G V. </s>
  <s xml:id="echoid-s8137" xml:space="preserve"><lb/>Cadat poſtea L M extra duas diame-<lb/>tros ad eaſdem partes. </s>
  <s xml:id="echoid-s8138" xml:space="preserve">Quoniam in parallelogrammo L S latera L O, M S æqua-<lb/>lia ſunt; </s>
  <s xml:id="echoid-s8139" xml:space="preserve">eſtque S R maior quàm M S, ſeu quàm O L; </s>
  <s xml:id="echoid-s8140" xml:space="preserve">ergo (vt in prima parte <lb/>huius propoſitionis oſtenſum eſt) rectangulum M S R, ſeu rectangulum ſub S V, <lb/>&amp; </s>
  <s xml:id="echoid-s8141" xml:space="preserve">latere recto G F maius erit quadrato L O, ſeu rectãgulo O G F, &amp; </s>
  <s xml:id="echoid-s8142" xml:space="preserve">propterea <lb/>
<anchor type="note" xlink:label="note-0254-01a" xlink:href="note-0254-01"/>
S V maior erit, quàm O G, &amp; </s>
  <s xml:id="echoid-s8143" xml:space="preserve">addita communi O V; </s>
  <s xml:id="echoid-s8144" xml:space="preserve">erit O S, ſeu ei æqualis <lb/>L M, in parallellogrammo L S, maior quàm G V. </s>
  <s xml:id="echoid-s8145" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s8146" xml:space="preserve"/>
</p>
<div xml:id="echoid-div725" type="float" level="2" n="15">
  <figure xlink:label="fig-0254-01" xlink:href="fig-0254-01a">
    <image file="0254-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0254-01"/>
  </figure>
<note position="left" xlink:label="note-0254-01" xlink:href="note-0254-01a" xml:space="preserve">II. lib. I.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8147" xml:space="preserve">Idem omnino verificari in ellipſibus demonſtrari facile poſſet, quod breuitati <lb/>
<anchor type="note" xlink:label="note-0254-02a" xlink:href="note-0254-02"/>
ſtudens libens omitto.</s>
  <s xml:id="echoid-s8148" xml:space="preserve"/>
</p>
<div xml:id="echoid-div726" type="float" level="2" n="16">
<note position="left" xlink:label="note-0254-02" xlink:href="note-0254-02a" xml:space="preserve">SCHO-<lb/>LIVM.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8149" xml:space="preserve">Si fuerint duæ quælibet coniſectiones A B C, D E F æquales, &amp; </s>
  <s xml:id="echoid-s8150" xml:space="preserve">ſi-<lb/>
<anchor type="note" xlink:label="note-0254-03a" xlink:href="note-0254-03"/>
miles ad eaſdemque partes cauæ, quarum diametri B H, E I (æquè in-<lb/>clinatæ ad ordinatim ad eas applicatas) æquidiſtantes ſint inter ſe, vel <lb/>congruentes; </s>
  <s xml:id="echoid-s8151" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8152" xml:space="preserve">ducantur quælibet rectæ lineæ A D, K L à ſectionibus <lb/>interceptæ, parallelæ rectæ lineæ B E vertices coniungenti: </s>
  <s xml:id="echoid-s8153" xml:space="preserve">erunt illæ <lb/>æquales inter ſe.</s>
  <s xml:id="echoid-s8154" xml:space="preserve"/>
</p>
<div xml:id="echoid-div727" type="float" level="2" n="17">
<note position="left" xlink:label="note-0254-03" xlink:href="note-0254-03a" xml:space="preserve">PROP. 5. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8155" xml:space="preserve">Si enim hoc verum non eſt, <lb/>
<anchor type="figure" xlink:label="fig-0254-02a" xlink:href="fig-0254-02"/>
ſit A D ſi fieri pote@t maior, <lb/>aut minor, quàm B E, &amp; </s>
  <s xml:id="echoid-s8156" xml:space="preserve">ſe-<lb/>t@tur A R æqualis B E: </s>
  <s xml:id="echoid-s8157" xml:space="preserve">pa-<lb/>tet punctum R cadere intra, <lb/>aut extra ſectionem D E (ſed <lb/>in eius plano cum ſectiones in <lb/>eodem plano exiſtant) iungan-<lb/>turque rectæ lineæ A B, E <lb/>R, quæ æquales erunt, &amp; </s>
  <s xml:id="echoid-s8158" xml:space="preserve">pa-<lb/>rallelæ inter ſe, cum ſint con-<lb/>iungentes æqualium, &amp; </s>
  <s xml:id="echoid-s8159" xml:space="preserve">æqui-<lb/>diſtantium B E, &amp; </s>
  <s xml:id="echoid-s8160" xml:space="preserve">A R. </s>
  <s xml:id="echoid-s8161" xml:space="preserve">Po-<lb/>ſtea ducatur A H ordinatim <lb/>applicata ad diametrum B H efficiens abſcißam H B; </s>
  <s xml:id="echoid-s8162" xml:space="preserve">ſeceturque abſciſſa E I in <lb/>altera ſectione æqualis B H; </s>
  <s xml:id="echoid-s8163" xml:space="preserve">iunganturque H I, I D, &amp; </s>
  <s xml:id="echoid-s8164" xml:space="preserve">I R. </s>
  <s xml:id="echoid-s8165" xml:space="preserve">Et quoniam B H,
<pb o="217" file="0255" n="255" rhead="Conicor. Lib. VI."/>
E I ſunt æquales, &amp; </s>
  <s xml:id="echoid-s8166" xml:space="preserve">parallelæ; </s>
  <s xml:id="echoid-s8167" xml:space="preserve">ergo H I æqualis erit, &amp; </s>
  <s xml:id="echoid-s8168" xml:space="preserve">parallela ipſi B E (vel <lb/>quia additur communis H E, vel propter parallelogrammum B I) ſed prius A <lb/>R æqualis erat, &amp; </s>
  <s xml:id="echoid-s8169" xml:space="preserve">parallela eidem B E; </s>
  <s xml:id="echoid-s8170" xml:space="preserve">igitur A R, &amp; </s>
  <s xml:id="echoid-s8171" xml:space="preserve">H I æquales ſunt inter <lb/>ſe, &amp; </s>
  <s xml:id="echoid-s8172" xml:space="preserve">æquidiſtantes; </s>
  <s xml:id="echoid-s8173" xml:space="preserve">ideoque coniungentes A H, R I erunt æquales, &amp; </s>
  <s xml:id="echoid-s8174" xml:space="preserve">paral-<lb/>lelæ; </s>
  <s xml:id="echoid-s8175" xml:space="preserve">ſuntque anguli A H B, &amp; </s>
  <s xml:id="echoid-s8176" xml:space="preserve">R I E æquales inter ſe, cum ab æqualibus la-<lb/>teribus in triangulis A B H, &amp; </s>
  <s xml:id="echoid-s8177" xml:space="preserve">R E I æquilateris inter ſe contineantur; </s>
  <s xml:id="echoid-s8178" xml:space="preserve">ergo <lb/>R I ordinatim quoque applicata eſt ad àiametrum E I; </s>
  <s xml:id="echoid-s8179" xml:space="preserve">atque in ſectionibus æ-<lb/>qualibus abſciſsæ B H, E I <lb/>
<anchor type="figure" xlink:label="fig-0255-01a" xlink:href="fig-0255-01"/>
diametrorum ſimilium, ſci-<lb/>licet æque inclinatarum ad <lb/>ſuas ordinatas æquales ſunt <lb/>inter ſe; </s>
  <s xml:id="echoid-s8180" xml:space="preserve">nec non ordinatæ A <lb/>H, I R æquales ſunt oſten-<lb/>sæ; </s>
  <s xml:id="echoid-s8181" xml:space="preserve">igitur ſicut punctum A in <lb/>
<anchor type="note" xlink:label="note-0255-01a" xlink:href="note-0255-01"/>
ſectione A B cadit, ita pun-<lb/>ctum R in ſectione E D exi-<lb/>ſtit; </s>
  <s xml:id="echoid-s8182" xml:space="preserve">ſed poſitus fuit intra, <lb/>aut extra ipſam, quod eſt ab-<lb/>ſurdũ: </s>
  <s xml:id="echoid-s8183" xml:space="preserve">Non igitur recta linea <lb/>A D maior, aut minor eſſe <lb/>poteſt, quàm B E; </s>
  <s xml:id="echoid-s8184" xml:space="preserve">ideoque ei <lb/>quælibet alia intercepta K L æqualis omnino erit. </s>
  <s xml:id="echoid-s8185" xml:space="preserve">Simili ratiocinio oſtendetur <lb/>æquidiſtans ipſi B E eidem <lb/>
<anchor type="figure" xlink:label="fig-0255-02a" xlink:href="fig-0255-02"/>
æqualis; </s>
  <s xml:id="echoid-s8186" xml:space="preserve">quapropter interce-<lb/>ptæ A D, K L, &amp; </s>
  <s xml:id="echoid-s8187" xml:space="preserve">B E æqua-<lb/>les erunt inter ſe: </s>
  <s xml:id="echoid-s8188" xml:space="preserve">Quod erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s8189" xml:space="preserve"/>
</p>
<div xml:id="echoid-div728" type="float" level="2" n="18">
  <figure xlink:label="fig-0254-02" xlink:href="fig-0254-02a">
    <image file="0254-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0254-02"/>
  </figure>
  <figure xlink:label="fig-0255-01" xlink:href="fig-0255-01a">
    <image file="0255-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0255-01"/>
  </figure>
<note position="left" xlink:label="note-0255-01" xlink:href="note-0255-01a" xml:space="preserve">ex 10. <lb/>huius.</note>
  <figure xlink:label="fig-0255-02" xlink:href="fig-0255-02a">
    <image file="0255-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0255-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s8190" xml:space="preserve">Si duæ parabolæ B A C, <lb/>F D E æquales ad eaſdem <lb/>
<anchor type="note" xlink:label="note-0255-02a" xlink:href="note-0255-02"/>
partes cauæ, conſtitutæ ſue-<lb/>rint circa axes A K, D G <lb/>æquidiſtantes, &amp; </s>
  <s xml:id="echoid-s8191" xml:space="preserve">non con-<lb/>gruentes ſe mutuo ſecabunt.</s>
  <s xml:id="echoid-s8192" xml:space="preserve"/>
</p>
<div xml:id="echoid-div729" type="float" level="2" n="19">
<note position="right" xlink:label="note-0255-02" xlink:href="note-0255-02a" xml:space="preserve">SCHO-<lb/>LIVM.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8193" xml:space="preserve">Ex vertice D axis G D ducatur D H perpendicularis ad axim A K, eum ſe-<lb/>cans in H, &amp; </s>
  <s xml:id="echoid-s8194" xml:space="preserve">deſcribatur alia parabolæ I H L æqualis prioribus B A, vel E <lb/>D, cuius axis ſit K H, &amp; </s>
  <s xml:id="echoid-s8195" xml:space="preserve">ver-<lb/>
<anchor type="figure" xlink:label="fig-0255-03a" xlink:href="fig-0255-03"/>
tex H, &amp; </s>
  <s xml:id="echoid-s8196" xml:space="preserve">ſicuti in propoſi-<lb/>tione 4. </s>
  <s xml:id="echoid-s8197" xml:space="preserve">additarum factum <lb/>eſt, reperiatur B F C ordina-<lb/>tim ad axes applicata ſecans <lb/>parabolas in E, B, I, &amp; </s>
  <s xml:id="echoid-s8198" xml:space="preserve">axes <lb/>in G, K, ita vt intercepta <lb/>B I æqualis ſit D H, ſen G <lb/>K, quæ in parallelogrammo <lb/>D K ei æqualis eſt. </s>
  <s xml:id="echoid-s8199" xml:space="preserve">Quoniā <lb/>parabolæ E D, &amp; </s>
  <s xml:id="echoid-s8200" xml:space="preserve">I H æqua-
<pb o="218" file="0256" n="256" rhead="Apollonij Pergæi"/>
les ſunt, &amp; </s>
  <s xml:id="echoid-s8201" xml:space="preserve">axium abſciſſæ D G, H K æquales cum ſint latera oppoſita paralle-<lb/>
<anchor type="note" xlink:label="note-0256-01a" xlink:href="note-0256-01"/>
logrammi D K; </s>
  <s xml:id="echoid-s8202" xml:space="preserve">ergo ordinatim ad axes applicatæ E G, &amp; </s>
  <s xml:id="echoid-s8203" xml:space="preserve">I K æquales ſunt, &amp; </s>
  <s xml:id="echoid-s8204" xml:space="preserve"><lb/>ablata communi I G, erit E I æqualis G K, ſeu D H; </s>
  <s xml:id="echoid-s8205" xml:space="preserve">erat autem intercepta <lb/>B I æqualis eidem D H; </s>
  <s xml:id="echoid-s8206" xml:space="preserve">igitur B I erit æqualis E I; </s>
  <s xml:id="echoid-s8207" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8208" xml:space="preserve">propterea punctum E <lb/>parabolæ E D F cadet ſuper punctum B parabolæ B A C; </s>
  <s xml:id="echoid-s8209" xml:space="preserve">ergo duæ parabolæ B <lb/>
<anchor type="note" xlink:label="note-0256-02a" xlink:href="note-0256-02"/>
A C, &amp; </s>
  <s xml:id="echoid-s8210" xml:space="preserve">E D F conueniunt in vno puncto, &amp; </s>
  <s xml:id="echoid-s8211" xml:space="preserve">in eo ſe mutuo tangere non poſ-<lb/>ſunt; </s>
  <s xml:id="echoid-s8212" xml:space="preserve">igitur ſe mutuo ſecant. </s>
  <s xml:id="echoid-s8213" xml:space="preserve">Quare patet propoſitum.</s>
  <s xml:id="echoid-s8214" xml:space="preserve"/>
</p>
<div xml:id="echoid-div730" type="float" level="2" n="20">
  <figure xlink:label="fig-0255-03" xlink:href="fig-0255-03a">
    <image file="0255-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0255-03"/>
  </figure>
<note position="left" xlink:label="note-0256-01" xlink:href="note-0256-01a" xml:space="preserve">ex prop@@. <lb/>huius.</note>
<note position="left" xlink:label="note-0256-02" xlink:href="note-0256-02a" xml:space="preserve">Maurol. <lb/>27. lib <lb/>Conic.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8215" xml:space="preserve">His demonſiratis manifeſtè percipitur, quod ex ſucceſſiua diminutione rectarũ <lb/>æquidiſtantium, inter coniſectiones interceptarum, deduci non poteſt, coniſe-<lb/>ctiones magis ad ſe ipſas propius accedere; </s>
  <s xml:id="echoid-s8216" xml:space="preserve">propterea quod in ĳ ſdem ſectionibus <lb/>aſymptoticis duci poßunt interceptæ rectæ lineæ inter ſe æquidiſtantes, quæ ſint <lb/>omnes æquales inter ſe, nimirum illæ, quæ parallelæ ſunt alicui communi dia-<lb/>metro, vel rectæ lineæ vertices earum coniungenti, vt in propoſitione 5. </s>
  <s xml:id="echoid-s8217" xml:space="preserve">additarũ <lb/>oſtenſum eſt. </s>
  <s xml:id="echoid-s8218" xml:space="preserve">Similiter aliæ interceptæ rectæ lineæ, inter ſe æquidiſtantes ſucceſſiuè <lb/>augentur aliæ verò ſucceſſiuè diminuuntur verſus caſdem partes, vt in propoſitione <lb/>3. </s>
  <s xml:id="echoid-s8219" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8220" xml:space="preserve">4. </s>
  <s xml:id="echoid-s8221" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s8222" xml:space="preserve">oſtenſum eſt. </s>
  <s xml:id="echoid-s8223" xml:space="preserve">Et hoc nedũ verificatur in ſectionibus non congruen-<lb/>tibus, &amp; </s>
  <s xml:id="echoid-s8224" xml:space="preserve">asymptoticis, ſed etiã in duabus æqualibus, &amp; </s>
  <s xml:id="echoid-s8225" xml:space="preserve">inter ſe ſimilibus ſectioni-<lb/>bus ſe mutuo ſecantibus, dummodo earum axes paralleli ſint, in ĳs enim inter-<lb/>ceptæ rectæ lineæ inter ſe æquidiſtantes, tendentes ad eaſdem partes, etiam illæ, <lb/>quæ proprius ad punctum occurſus ſcctionum conicarum accedunt, poßunt dimi-<lb/>nui, pariterque inter ſe æquales eße, &amp; </s>
  <s xml:id="echoid-s8226" xml:space="preserve">quod mirum eſt poßunt ſemper magis <lb/>augeri. </s>
  <s xml:id="echoid-s8227" xml:space="preserve">Si igitur æquidiſtantes interceptæ ſunt menſuræ diſtantiarũ duarum ſe-<lb/>ctionum, eædem coniſectiones cenſeri debent modo parallelæ, &amp; </s>
  <s xml:id="echoid-s8228" xml:space="preserve">æqualibus inter-<lb/>uallis inter ſe diſtantes, modo ad eaſdem partes ſtringi, &amp; </s>
  <s xml:id="echoid-s8229" xml:space="preserve">coanguſtari, &amp; </s>
  <s xml:id="echoid-s8230" xml:space="preserve">ſi-<lb/>mul dilatari magis, ac magis, quod omnino videtur abſurdum. </s>
  <s xml:id="echoid-s8231" xml:space="preserve">Non igitur ex <lb/>eo qnod omnes interceptæ rectæ lineæ inter ſe æquidiſtantes ſunt æquales inter ſe; <lb/></s>
  <s xml:id="echoid-s8232" xml:space="preserve">propterea ſectiones ipſæ crunt parallelæ, &amp; </s>
  <s xml:id="echoid-s8233" xml:space="preserve">asymptoticæ, &amp; </s>
  <s xml:id="echoid-s8234" xml:space="preserve">ſemper æquali in-<lb/>teruallo ad inuicem ſeparatæ; </s>
  <s xml:id="echoid-s8235" xml:space="preserve">neque ex eo quod prædictæ parallelæ magis augẽ-<lb/>tur, vel diminuuntur interualla augeri, vel ſtringi cenſendum eſt.</s>
  <s xml:id="echoid-s8236" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s8237" xml:space="preserve">Et præcipuè præſtantiſſimus Gregorius à Sancto Vincentio neſcio an iure de-<lb/>monſtrationem propoſitionis 14. </s>
  <s xml:id="echoid-s8238" xml:space="preserve">libri 2. </s>
  <s xml:id="echoid-s8239" xml:space="preserve">ipſiuſmet Apollonĳ inſufficientem repu-<lb/>tauerit, propterea quod Apollonius deduxit rectas lineas hyperbolen compræbendẽ-<lb/>tes, quæ aſymptoti vocantur ſemper magis, ac magis ſectioni viciniores fieri ex eo <lb/>quod rectæ lineæ inter ſe æquidiſtãtes, interceptæ inter rectas asymptotos vocatas, <lb/>&amp; </s>
  <s xml:id="echoid-s8240" xml:space="preserve">hyperbolen contentam ſucceſſiuè ſemper magis, ac magis diminuantur; </s>
  <s xml:id="echoid-s8241" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8242" xml:space="preserve">è <lb/>contra aßeruit cum Cardano, &amp; </s>
  <s xml:id="echoid-s8243" xml:space="preserve">quodam Rabino Moſe diſtantiam hyperbolæ à re-<lb/>ctis asymptotis ſumi debere, non à quibu ſcunque rectis lineis interceptis inter <lb/>ſe parallelis, ſed tantummodo à rectis lineis perpendicularibus ad aſymptotos, <lb/>quæ ſolummodo, inquiunt ipſi, diſtantias determinant; </s>
  <s xml:id="echoid-s8244" xml:space="preserve">at reuera hæc animad-<lb/>nerſio non videtur neceßaria: </s>
  <s xml:id="echoid-s8245" xml:space="preserve">perinde enim eſt conſiderare rectas lineas ab hy-<lb/>perbole ad vnam rectam lineam continentium ductas, quæ efficiat cum illa an-<lb/>gulos æquales, ac ſi perpendiculares eßent ad eandem: </s>
  <s xml:id="echoid-s8246" xml:space="preserve">at quando rectæ lineæ in-<lb/>terceptæ ſunt inter ſe æquidiſtantes, tunc omnes efficiunt ſuper rectam lineam <lb/>continentem hyperbolen angulos æquales ad eaſdem partes; </s>
  <s xml:id="echoid-s8247" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8248" xml:space="preserve">propterca (ex inæ-<lb/>qualitate prædictarum æquidiſt antium) optimè concluditur cum Apollonio inæ-<lb/>qualitas perpendicularium, ſeu diſtantiarum. </s>
  <s xml:id="echoid-s8249" xml:space="preserve">Quando verò conſiderantur duæ <lb/>lineæ curuæ veluti ſunt duæ parabolæ, vel duæ hyperbolæ, vel ellipſes, tunc qui-
<pb o="219" file="0257" n="257" rhead="Conicor. Lib. VI."/>
dem nulla ratione rectæ lineæ inter ſe æquidiſtantes, inter curuas interceptæ de-<lb/>terminare poſſunt prædictarum curuarum diſtantias; </s>
  <s xml:id="echoid-s8250" xml:space="preserve">quandoquidem inæquali-<lb/>ter ſemper inclinantur ad quamlibet prædictarum curuarum, &amp; </s>
  <s xml:id="echoid-s8251" xml:space="preserve">rectæ lineæ in-<lb/>terceptæ, quæ ſunt perpendiculares ad vnam ipſarum, non erunt inter ſe æqui-<lb/>diſtantes. </s>
  <s xml:id="echoid-s8252" xml:space="preserve">Et quia, vt dictum eſt, prædictæ perpendiculares ſunt diſtantiarum <lb/>legitimæ menſuræ, nunquàm concludi poteſt certo, quod prædictæ ſectiones ſint <lb/>æquidiſtantes. </s>
  <s xml:id="echoid-s8253" xml:space="preserve">vel ſibi ipſis ſucceſſiuè viciniores ſiant, niſi conſiderentur rectæ <lb/>lineæ interceptæ ad vnam ſectionum perpendiculares: </s>
  <s xml:id="echoid-s8254" xml:space="preserve">quod quidem hucuſque <lb/>quod ſciam factum non eſt, neque forſan huiuſmodi ſpeculatio inuentu facilis <lb/>erit, aut iniucunda.</s>
  <s xml:id="echoid-s8255" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s8256" xml:space="preserve">In parabola, vel hyperbola A B C ad eius axim E A I ducere ra-<lb/>
<anchor type="note" xlink:label="note-0257-01a" xlink:href="note-0257-01"/>
mum breuiſſimum æquidiſtantem alicui rectæ lineæ E F, quæ oportet, <lb/>vt efficiat cum axi ad partes ſectionis angulum A E F acutum, ſed in <lb/>hyperbola ſit minor ſemiße vnius recti, &amp; </s>
  <s xml:id="echoid-s8257" xml:space="preserve">angulus F E X ab vna <lb/>asymptoto, &amp; </s>
  <s xml:id="echoid-s8258" xml:space="preserve">recta linea E F contentus ſit acutus.</s>
  <s xml:id="echoid-s8259" xml:space="preserve"/>
</p>
<div xml:id="echoid-div731" type="float" level="2" n="21">
<note position="right" xlink:label="note-0257-01" xlink:href="note-0257-01a" xml:space="preserve">PROP. 6. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8260" xml:space="preserve">Fiat angulus A E D æqualis an-<lb/>
<anchor type="figure" xlink:label="fig-0257-01a" xlink:href="fig-0257-01"/>
gulo A E F, &amp; </s>
  <s xml:id="echoid-s8261" xml:space="preserve">ex vertice A du-<lb/>catur recta linea A B efficiens an-<lb/>gulum I A B, qui ſimul cum an-<lb/>gulo A E F vnum rectum angulũ <lb/>compleat; </s>
  <s xml:id="echoid-s8262" xml:space="preserve">ſed in hyperbola, quia <lb/>vterq; </s>
  <s xml:id="echoid-s8263" xml:space="preserve">angulus X E A, &amp; </s>
  <s xml:id="echoid-s8264" xml:space="preserve">A E F <lb/>deficit à ſemirecto erũt ambo mino-<lb/>res ſumma præcedentium, ſcilicet <lb/>vno angulo recto; </s>
  <s xml:id="echoid-s8265" xml:space="preserve">ergo ablato cõmuni <lb/>angulo A E F, erit angulus I A B <lb/>maior angulo A E X. </s>
  <s xml:id="echoid-s8266" xml:space="preserve">Poſtea, quia <lb/>tam A E F, quàm A E D minor eſt ſemiſſe vnius anguli recti, &amp; </s>
  <s xml:id="echoid-s8267" xml:space="preserve">A E F cum <lb/>angulo I A B vnum rectum angulum complent; </s>
  <s xml:id="echoid-s8268" xml:space="preserve">ergo angulus I A B maior erit <lb/>angulo D E A: </s>
  <s xml:id="echoid-s8269" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8270" xml:space="preserve">propterea recta linea A B producta neceſſario ſecabit vtram-<lb/>que rectam lineam E D, &amp; </s>
  <s xml:id="echoid-s8271" xml:space="preserve">E X asymptotum extra ſectionem cadentem ad par-<lb/>tes D, X; </s>
  <s xml:id="echoid-s8272" xml:space="preserve">ideoque A B hyperbolen ſecabit in aliquo alio puncto B. </s>
  <s xml:id="echoid-s8273" xml:space="preserve">In parabola <lb/>verò, quia recta linea A B axim <lb/>
<anchor type="figure" xlink:label="fig-0257-02a" xlink:href="fig-0257-02"/>
ſecat in vertice A non ad angulos <lb/>rectos (cum anguli I A B, &amp; </s>
  <s xml:id="echoid-s8274" xml:space="preserve">A <lb/>
<anchor type="note" xlink:label="note-0257-02a" xlink:href="note-0257-02"/>
E F rectum compleant) ergo A B <lb/>ſectioni occurrit in duobus pun-<lb/>ctis. </s>
  <s xml:id="echoid-s8275" xml:space="preserve">Secetur iam A B bifariam <lb/>in L, &amp; </s>
  <s xml:id="echoid-s8276" xml:space="preserve">per L ducatur diameter <lb/>ſectionis L G ſectioni occurrens in <lb/>
<anchor type="note" xlink:label="note-0257-03a" xlink:href="note-0257-03"/>
G, &amp; </s>
  <s xml:id="echoid-s8277" xml:space="preserve">per G ducatur contingens G <lb/>H, ſeu parallela A B ſecans axim <lb/>in H, &amp; </s>
  <s xml:id="echoid-s8278" xml:space="preserve">per G ducatur I G O per-<lb/>pendicularis ad G H. </s>
  <s xml:id="echoid-s8279" xml:space="preserve">Dico I G <lb/>problema efficere. </s>
  <s xml:id="echoid-s8280" xml:space="preserve">Quoniam pro-
<pb o="220" file="0258" n="258" rhead="Apollonij Pergæi"/>
pter parallelas G H, B A, eſt an-<lb/>
<anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/>
gulus G H A, ſeu E H N æqualis <lb/>angulo B A I; </s>
  <s xml:id="echoid-s8281" xml:space="preserve">ſed anguli B A I, <lb/>&amp; </s>
  <s xml:id="echoid-s8282" xml:space="preserve">A E F vnicum rectum com-<lb/>plent; </s>
  <s xml:id="echoid-s8283" xml:space="preserve">ergo duo anguli N H E, &amp; </s>
  <s xml:id="echoid-s8284" xml:space="preserve"><lb/>N E H ſimul ſumpti vni recto æ-<lb/>quales ſunt, &amp; </s>
  <s xml:id="echoid-s8285" xml:space="preserve">propterea in trian-<lb/>gulo E N H reliquus angulus N <lb/>rectus erit: </s>
  <s xml:id="echoid-s8286" xml:space="preserve">erat quoque angulus <lb/>I G H rectus; </s>
  <s xml:id="echoid-s8287" xml:space="preserve">igitur I G (qui eſt <lb/>
<anchor type="note" xlink:label="note-0258-01a" xlink:href="note-0258-01"/>
ramus breuiſſimus cum ſit perpen-<lb/>dicularis ad tangentem G H) eſt <lb/>æquidiſtans rectæ lineæ E F; </s>
  <s xml:id="echoid-s8288" xml:space="preserve">quod erat propoſitum.</s>
  <s xml:id="echoid-s8289" xml:space="preserve"/>
</p>
<div xml:id="echoid-div732" type="float" level="2" n="22">
  <figure xlink:label="fig-0257-01" xlink:href="fig-0257-01a">
    <image file="0257-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0257-01"/>
  </figure>
  <figure xlink:label="fig-0257-02" xlink:href="fig-0257-02a">
    <image file="0257-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0257-02"/>
  </figure>
<note position="right" xlink:label="note-0257-02" xlink:href="note-0257-02a" xml:space="preserve">17. 27. <lb/>lib. I.</note>
<note position="right" xlink:label="note-0257-03" xlink:href="note-0257-03a" xml:space="preserve">35. 36. <lb/>lib. I. <lb/>5. lib. 2.</note>
  <figure xlink:label="fig-0258-01" xlink:href="fig-0258-01a">
    <image file="0258-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0258-01"/>
  </figure>
<note position="left" xlink:label="note-0258-01" xlink:href="note-0258-01a" xml:space="preserve">31. lib. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8290" xml:space="preserve">Facile deducitur, quod ſi angulus A E F fuerit rectus in parabola, &amp; </s>
  <s xml:id="echoid-s8291" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0258-02a" xlink:href="note-0258-02"/>
non fuerit ſemirecto minor in hyperbole facta eadem conſtructione quilibet <lb/>ramus breuiſſimus I G æquidiſtans erit rectæ lineæ diuidenti angulũ A E F.</s>
  <s xml:id="echoid-s8292" xml:space="preserve"/>
</p>
<div xml:id="echoid-div733" type="float" level="2" n="23">
<note position="left" xlink:label="note-0258-02" xlink:href="note-0258-02a" xml:space="preserve">SCHO-<lb/>LIVM.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8293" xml:space="preserve">Nam angulus A I G ab axi, &amp; </s>
  <s xml:id="echoid-s8294" xml:space="preserve">ramo breuiſſimo contentus eſt acutus, ſed an-<lb/>gulus F E A in parabola eſt re-<lb/>
<anchor type="figure" xlink:label="fig-0258-02a" xlink:href="fig-0258-02"/>
<anchor type="note" xlink:label="note-0258-03a" xlink:href="note-0258-03"/>
ctus; </s>
  <s xml:id="echoid-s8295" xml:space="preserve">ergo recta linea I G paralle-<lb/>la eſt alicui rectæ lineæ diuidenti <lb/>angulum A E F, in hyperbela ve-<lb/>rò factus eſt angulus A E D æqua-<lb/>lis angulo A E F, qui ſemirecto <lb/>minor non eſt; </s>
  <s xml:id="echoid-s8296" xml:space="preserve">propterea erit totus <lb/>angulus D E F rectus, aut obtu-<lb/>ſus; </s>
  <s xml:id="echoid-s8297" xml:space="preserve">ergo in triangulo E M N ex-<lb/>ternus angulus F N M maior in-<lb/>terno, &amp; </s>
  <s xml:id="echoid-s8298" xml:space="preserve">oppoſito angulo E recto, <lb/>vel obtuſo, erit quoque obtuſus, &amp; </s>
  <s xml:id="echoid-s8299" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0258-04a" xlink:href="note-0258-04"/>
angulus I G N rectus eſt; </s>
  <s xml:id="echoid-s8300" xml:space="preserve">igitur I <lb/>G, F N ſe viciſſim ſecabunt vltra punctum E, &amp; </s>
  <s xml:id="echoid-s8301" xml:space="preserve">ideo I G parallela erit rectæ <lb/>lineæ diuidenti angulum A E F. </s>
  <s xml:id="echoid-s8302" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s8303" xml:space="preserve"/>
</p>
<div xml:id="echoid-div734" type="float" level="2" n="24">
  <figure xlink:label="fig-0258-02" xlink:href="fig-0258-02a">
    <image file="0258-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0258-02"/>
  </figure>
<note position="left" xlink:label="note-0258-03" xlink:href="note-0258-03a" xml:space="preserve">13. 14. 15. <lb/>lib. 5.</note>
<note position="left" xlink:label="note-0258-04" xlink:href="note-0258-04a" xml:space="preserve">31. lib. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8304" xml:space="preserve">Sint duæ parabolæ, vel duæ hyperbo-<lb/>
<anchor type="figure" xlink:label="fig-0258-03a" xlink:href="fig-0258-03"/>
<anchor type="note" xlink:label="note-0258-05a" xlink:href="note-0258-05"/>
læ æquales, &amp; </s>
  <s xml:id="echoid-s8305" xml:space="preserve">ſimiliter poſitæ H B D, <lb/>&amp; </s>
  <s xml:id="echoid-s8306" xml:space="preserve">I F G circa communem axim A H I: <lb/></s>
  <s xml:id="echoid-s8307" xml:space="preserve">intercepta axis portio erit diſtantia ſectio-<lb/>num omnium maxima, &amp; </s>
  <s xml:id="echoid-s8308" xml:space="preserve">ei propinquior <lb/>remotiore maior erit.</s>
  <s xml:id="echoid-s8309" xml:space="preserve"/>
</p>
<div xml:id="echoid-div735" type="float" level="2" n="25">
  <figure xlink:label="fig-0258-03" xlink:href="fig-0258-03a">
    <image file="0258-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0258-03"/>
  </figure>
<note position="left" xlink:label="note-0258-05" xlink:href="note-0258-05a" xml:space="preserve">PROP. 7. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8310" xml:space="preserve">Sint centra E, &amp; </s>
  <s xml:id="echoid-s8311" xml:space="preserve">K, asymptoti P E O, <lb/>Q K R, &amp; </s>
  <s xml:id="echoid-s8312" xml:space="preserve">à vertice H, &amp; </s>
  <s xml:id="echoid-s8313" xml:space="preserve">à quibuslibet <lb/>punctis interiores ſectionis B D eleuentur <lb/>
<anchor type="note" xlink:label="note-0258-06a" xlink:href="note-0258-06"/>
lineæ breuiſſimæ, ſeu perpendiculares ad rectas <lb/>curuam B D contingentes in eiſdem punctis, <lb/>quæ ſint H A, B A, &amp; </s>
  <s xml:id="echoid-s8314" xml:space="preserve">D C, quæ ſecent re-<lb/>liquam ſectionem in punctis I, F, &amp; </s>
  <s xml:id="echoid-s8315" xml:space="preserve">G.</s>
  <s xml:id="echoid-s8316" xml:space="preserve">
<pb o="221" file="0259" n="259" rhead="Conicor. Lib. VI."/>
Manifeſtum eſt interceptas I H, F B, G D eſſe minimas linearum rectarum, <lb/>quæ à punctis I, F, G ad ſectionem B D duci poßunt; </s>
  <s xml:id="echoid-s8317" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8318" xml:space="preserve">ideo eædem interce-<lb/>
<anchor type="note" xlink:label="note-0259-01a" xlink:href="note-0259-01"/>
ptæ erunt diſtantiæ quorunlibet punctorum ſectionis I F G à ſectione B D: </s>
  <s xml:id="echoid-s8319" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8320" xml:space="preserve"><lb/>propterea erunt diſtantiæ prædictarum curuarum. </s>
  <s xml:id="echoid-s8321" xml:space="preserve">Oſtendendum modo eſt H I <lb/>maiorem eſſe, quàm B F, &amp; </s>
  <s xml:id="echoid-s8322" xml:space="preserve">B F maiorem, quàm D G, &amp; </s>
  <s xml:id="echoid-s8323" xml:space="preserve">ſic ſemper. </s>
  <s xml:id="echoid-s8324" xml:space="preserve">Duca-<lb/>tur à puncto F intercepta recta linea F M parallela axi I H, atque à puncto G <lb/>ducatur recta linea G N parallela ipſi F B, quæ occurrant ſectioni B D in M, <lb/>N. </s>
  <s xml:id="echoid-s8325" xml:space="preserve">Et quoniam F M æquidiſtat vertices coniungenti I H, erit intercepta F M <lb/>
<anchor type="note" xlink:label="note-0259-02a" xlink:href="note-0259-02"/>
æqualis I H, ſed cum ramus B A ſit breuiſſimus, &amp; </s>
  <s xml:id="echoid-s8326" xml:space="preserve">eius portio F B erit quoque <lb/>breuiſſima omnium, quæ ex puncto F ad eandem ſectionem B H duci poſſunt; <lb/></s>
  <s xml:id="echoid-s8327" xml:space="preserve">quare B F minor erit quàm F M, &amp; </s>
  <s xml:id="echoid-s8328" xml:space="preserve">F M oſtenſa fuit æqualis I H; </s>
  <s xml:id="echoid-s8329" xml:space="preserve">igitur di-<lb/>ſtantia intercepta F B minor erit quàm I H.</s>
  <s xml:id="echoid-s8330" xml:space="preserve"/>
</p>
<div xml:id="echoid-div736" type="float" level="2" n="26">
<note position="left" xlink:label="note-0258-06" xlink:href="note-0258-06a" xml:space="preserve">8. 9. 10. 30. <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0259-01" xlink:href="note-0259-01a" xml:space="preserve">38. lib. 5.</note>
<note position="right" xlink:label="note-0259-02" xlink:href="note-0259-02a" xml:space="preserve">5. aiddit. <lb/>huus. <lb/>38. lib. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8331" xml:space="preserve">Secundò quia duæ interceptæ B F, N G parallelæ inter ſe productæ occurrunt <lb/>axi intra ſectiones ad partes A C, &amp; </s>
  <s xml:id="echoid-s8332" xml:space="preserve">in parabola, quàm ſecabunt in binis pun-<lb/>
<anchor type="note" xlink:label="note-0259-03a" xlink:href="note-0259-03"/>
ctis, erunt ſaltem ordinatim applicatæ ad aliquàm diametrum: </s>
  <s xml:id="echoid-s8333" xml:space="preserve">in byperbolis verò <lb/>
<anchor type="figure" xlink:label="fig-0259-01a" xlink:href="fig-0259-01"/>
parallelæ erunt rectæ lineæ diuidenti angulum P E K à recta linea E K centra <lb/>coniungente, &amp; </s>
  <s xml:id="echoid-s8334" xml:space="preserve">E P interiore asymptoto contentum; </s>
  <s xml:id="echoid-s8335" xml:space="preserve">propterea tam in parabo-<lb/>
<anchor type="note" xlink:label="note-0259-04a" xlink:href="note-0259-04"/>
lis, quàm in hyperbolis intercepta B F, quæ vlterius tendit ad partes reliquæ <lb/>asymptoti E O maior erit intercepta N G; </s>
  <s xml:id="echoid-s8336" xml:space="preserve">ſed quia G D eſt linea breuiſſima om-<lb/>
<anchor type="note" xlink:label="note-0259-05a" xlink:href="note-0259-05"/>
nium, quæ ad ſectienem H D duci poſſunt, cum ſit portio breuiſſimæ D C, quæ <lb/>perpendicularis eſt ad rectam contingentem in D, igitur G D minor erit, <lb/>quàm G N; </s>
  <s xml:id="echoid-s8337" xml:space="preserve">eſtque G N oſtenſa minor, quàm F B; </s>
  <s xml:id="echoid-s8338" xml:space="preserve">ergo G D minor erit, quàm <lb/>F B.</s>
  <s xml:id="echoid-s8339" xml:space="preserve"/>
</p>
<div xml:id="echoid-div737" type="float" level="2" n="27">
<note position="right" xlink:label="note-0259-03" xlink:href="note-0259-03a" xml:space="preserve">27. lib. 1.</note>
  <figure xlink:label="fig-0259-01" xlink:href="fig-0259-01a">
    <image file="0259-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0259-01"/>
  </figure>
<note position="right" xlink:label="note-0259-04" xlink:href="note-0259-04a" xml:space="preserve">3. &amp; 4. <lb/>addit.</note>
<note position="right" xlink:label="note-0259-05" xlink:href="note-0259-05a" xml:space="preserve">38. lib. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8340" xml:space="preserve">In parabolis autem, quia duci poteſt aliqua recta linea, vt N G parallela <lb/>cuilibet interceptæ B F; </s>
  <s xml:id="echoid-s8341" xml:space="preserve">itaut ſit N G minor quacunque recta linea data (quan-<lb/>
<anchor type="note" xlink:label="note-0259-06a" xlink:href="note-0259-06"/>
do nimirum ad aliquam diametrum ordinatim ſunt applicatæ, ſcilicet, quando <lb/>vna ipſarum, puta B F occurrat axi intra ſectiones; </s>
  <s xml:id="echoid-s8342" xml:space="preserve">quod quidem neceſſario <lb/>
<anchor type="note" xlink:label="note-0259-07a" xlink:href="note-0259-07"/>
eueniet, quando B A eſt ramus breuiſſimus) eſtque ramus breuiſſimus D G mi-
<pb o="222" file="0260" n="260" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0260-01a" xlink:href="fig-0260-01"/>
nor eadem G N; </s>
  <s xml:id="echoid-s8343" xml:space="preserve">igitur diſtantia ſectionum G D minor erit quacunque recta <lb/>linea propoſita. </s>
  <s xml:id="echoid-s8344" xml:space="preserve">Quia verò (vt conſtat ex demonſtratione caſus 2. </s>
  <s xml:id="echoid-s8345" xml:space="preserve">propoſ. </s>
  <s xml:id="echoid-s8346" xml:space="preserve">3. <lb/></s>
  <s xml:id="echoid-s8347" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s8348" xml:space="preserve">huius) quælibet recta linea G D intercepta inter hyperbolas conueniens <lb/>cum axi intra ſectiones maior eſt portione eiuſdem rectæ lineæ C D G inter æ-<lb/>quidiſtantes asymptotos E P, &amp; </s>
  <s xml:id="echoid-s8349" xml:space="preserve">K Q intercepta; </s>
  <s xml:id="echoid-s8350" xml:space="preserve">igitur interuallum inter duas <lb/>hyperbolas, licet ſucceſſiuè ſemper magis, ac magis diminuatur, nunquàm ta-<lb/>men minor effici poterit interuallo duarum æquidiſtantium hyperbolas continen-<lb/>tium E P, &amp; </s>
  <s xml:id="echoid-s8351" xml:space="preserve">K Q; </s>
  <s xml:id="echoid-s8352" xml:space="preserve">Quod quidem eſt perpendiculare ad vtramque rectam con-<lb/>tinentem E P, &amp; </s>
  <s xml:id="echoid-s8353" xml:space="preserve">K Q; </s>
  <s xml:id="echoid-s8354" xml:space="preserve">eſtque prædicta perpendicularis minima omnium in-<lb/>terceptarum inter eas.</s>
  <s xml:id="echoid-s8355" xml:space="preserve"/>
</p>
<div xml:id="echoid-div738" type="float" level="2" n="28">
<note position="right" xlink:label="note-0259-06" xlink:href="note-0259-06a" xml:space="preserve">Prop. 4. <lb/>addit.</note>
<note position="right" xlink:label="note-0259-07" xlink:href="note-0259-07a" xml:space="preserve">27. lib. 1.</note>
  <figure xlink:label="fig-0260-01" xlink:href="fig-0260-01a">
    <image file="0260-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0260-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s8356" xml:space="preserve">Duarum parabolarum, vel hyperbolarum A B, D E æqualium, &amp; </s>
  <s xml:id="echoid-s8357" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0260-01a" xlink:href="note-0260-01"/>
ſimilium, quarum axes A O, D Y, nec non asymptoti H I K, L M N <lb/>ſint parallelæ inter ſe, &amp; </s>
  <s xml:id="echoid-s8358" xml:space="preserve">ſimiliter poſitæ: </s>
  <s xml:id="echoid-s8359" xml:space="preserve">Sectionum diſtantia maxima <lb/>parallela erit vertices coniungenti, &amp; </s>
  <s xml:id="echoid-s8360" xml:space="preserve">ei propinquiores ex vtraq; </s>
  <s xml:id="echoid-s8361" xml:space="preserve">parte <lb/>maiores ſunt remotioribus vſq; </s>
  <s xml:id="echoid-s8362" xml:space="preserve">ad concurſum: </s>
  <s xml:id="echoid-s8363" xml:space="preserve">ſi veró diſtantiam ma-<lb/>ximam non habent ſemper augentur quo magis à concurſu recedunt.</s>
  <s xml:id="echoid-s8364" xml:space="preserve"/>
</p>
<div xml:id="echoid-div739" type="float" level="2" n="29">
<note position="left" xlink:label="note-0260-01" xlink:href="note-0260-01a" xml:space="preserve">PROP. 8. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8365" xml:space="preserve">Cadat concurſus ſectionum Z <lb/>
<anchor type="figure" xlink:label="fig-0260-02a" xlink:href="fig-0260-02"/>
inter axes A G, &amp; </s>
  <s xml:id="echoid-s8366" xml:space="preserve">D Y, &amp; </s>
  <s xml:id="echoid-s8367" xml:space="preserve">aſym-<lb/>ptoti I K, M N coincidant, aut <lb/>ſibi ſint viciniores, quàm I H; </s>
  <s xml:id="echoid-s8368" xml:space="preserve">M <lb/>L. </s>
  <s xml:id="echoid-s8369" xml:space="preserve">Et primò angulus Y D A ab <lb/>axe Y D, &amp; </s>
  <s xml:id="echoid-s8370" xml:space="preserve">D A vertices con-<lb/>iungente contentus ſemirecto minor <lb/>non ſit in hyperbola, ſitque rectus <lb/>in parabola, &amp; </s>
  <s xml:id="echoid-s8371" xml:space="preserve">vltra concurſum. <lb/></s>
  <s xml:id="echoid-s8372" xml:space="preserve">Z, ad partes axis D Y, &amp; </s>
  <s xml:id="echoid-s8373" xml:space="preserve">asym-<lb/>ptotorum magis diſſitorum H I, <lb/>L M: </s>
  <s xml:id="echoid-s8374" xml:space="preserve">ſumantur in compræhenſa ſectione A B quælibet puncta, B, P, à quibus
<pb o="223" file="0261" n="261" rhead="Conicor. Lib. VI."/>
ad axim ducantur rami breuiſſimi O B, Q P præter axim A O, &amp; </s>
  <s xml:id="echoid-s8375" xml:space="preserve">ſecent ex-<lb/>
<anchor type="note" xlink:label="note-0261-01a" xlink:href="note-0261-01"/>
ternam curuam in G, E, R, &amp; </s>
  <s xml:id="echoid-s8376" xml:space="preserve">occurſui Z, vel communi asymptoto I M N, <lb/>
<anchor type="figure" xlink:label="fig-0261-01a" xlink:href="fig-0261-01"/>
aut vicinioribus asymptotis I K, M N ſit A G propinquior, quàm E B, &amp; </s>
  <s xml:id="echoid-s8377" xml:space="preserve">E B <lb/>propinquior, quàm R P: </s>
  <s xml:id="echoid-s8378" xml:space="preserve">Oſtendendum eſt curuarum diſtantiam A G minorem <lb/>eße, quàm B E, &amp; </s>
  <s xml:id="echoid-s8379" xml:space="preserve">B E, quàm P R. </s>
  <s xml:id="echoid-s8380" xml:space="preserve">Ducantur interceptæ G S parallela E B, <lb/>&amp; </s>
  <s xml:id="echoid-s8381" xml:space="preserve">E X parallela R P. </s>
  <s xml:id="echoid-s8382" xml:space="preserve">Et quia in parabola angulus Y D A rectus ſupponitur, <lb/>
<anchor type="note" xlink:label="note-0261-02a" xlink:href="note-0261-02"/>
&amp; </s>
  <s xml:id="echoid-s8383" xml:space="preserve">in hyperbola non eſt minor ſemirecto, ergo quilibet ramus breuiſſimus E B, <lb/>vel R P æquidiſtans erit rectæ lineæ diuidenti angulum A D Y in parabsla, &amp; </s>
  <s xml:id="echoid-s8384" xml:space="preserve"><lb/>angulum M I H in hyperbola; </s>
  <s xml:id="echoid-s8385" xml:space="preserve">ſed duarum parallelarum E B, G S, vel R P, <lb/>E X eſt G S vertici propinquior, vel vlterius tendit ad partes asymptoti I K, <lb/>quàm E B; </s>
  <s xml:id="echoid-s8386" xml:space="preserve">ergo G S minor eſt, quàm E B; </s>
  <s xml:id="echoid-s8387" xml:space="preserve">eſtque G A minor, quàm G S, quia <lb/>
<anchor type="note" xlink:label="note-0261-03a" xlink:href="note-0261-03"/>
illa eſt portio, vel productio lineæ breuiſſimæ O A; </s>
  <s xml:id="echoid-s8388" xml:space="preserve">igitur G A adhuc minor erit <lb/>
<anchor type="figure" xlink:label="fig-0261-02a" xlink:href="fig-0261-02"/>
quàm E B. </s>
  <s xml:id="echoid-s8389" xml:space="preserve">Eadem ratione E B minor oſtendetur, quàm R P. </s>
  <s xml:id="echoid-s8390" xml:space="preserve">Poſtea ſi occur-<lb/>ſus Z cadit extra duos axes, inter axim A G, &amp; </s>
  <s xml:id="echoid-s8391" xml:space="preserve">occurſum aut ad partes asym-
<pb o="224" file="0262" n="262" rhead="Apollonij Pergæi"/>
Protorum coincidentium, vel propinquiorum, ad oppoſitas partes citra axim G A, <lb/>ſumantur duo puncta C, T, &amp; </s>
  <s xml:id="echoid-s8392" xml:space="preserve">ab cis ducantur ad axim rami breuiſſimi O C, <lb/>Q T ſecantcs externam ſectionem in F, V, &amp; </s>
  <s xml:id="echoid-s8393" xml:space="preserve">ab occurſu, vel communi asym-<lb/>ptoto, vel ab asymptotis vicinioribus I K, M N magis recedat A G, quàm C F, <lb/>&amp; </s>
  <s xml:id="echoid-s8394" xml:space="preserve">C F, quàm T V; </s>
  <s xml:id="echoid-s8395" xml:space="preserve">Dico G A maiorem eſſe, quàm C F, &amp; </s>
  <s xml:id="echoid-s8396" xml:space="preserve">C F maiorem, <lb/>quàm T V. </s>
  <s xml:id="echoid-s8397" xml:space="preserve">Ducantur interceptæ F a parallela G A, &amp; </s>
  <s xml:id="echoid-s8398" xml:space="preserve">V b parallela C F. <lb/></s>
  <s xml:id="echoid-s8399" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0262-01a" xlink:href="fig-0262-01"/>
Et quia in parabola F a propinquior eſt occur ſui ſectionum, &amp; </s>
  <s xml:id="echoid-s8400" xml:space="preserve">parallela eſt dia-<lb/>
<anchor type="note" xlink:label="note-0262-01a" xlink:href="note-0262-01"/>
metro G A; </s>
  <s xml:id="echoid-s8401" xml:space="preserve">at in hyperbola F a parallela eſt axi G A, vel D Y diuidenti an-<lb/>gulum M I H, &amp; </s>
  <s xml:id="echoid-s8402" xml:space="preserve">F a vlterius tendit ad partes asymptoti I K, quàm G A; </s>
  <s xml:id="echoid-s8403" xml:space="preserve">ergo <lb/>
<anchor type="note" xlink:label="note-0262-02a" xlink:href="note-0262-02"/>
F a minor eſt, quàm G A: </s>
  <s xml:id="echoid-s8404" xml:space="preserve">eſtque C F productio rami breuiſſiimi minor quàm <lb/>
<anchor type="note" xlink:label="note-0262-03a" xlink:href="note-0262-03"/>
F a; </s>
  <s xml:id="echoid-s8405" xml:space="preserve">ergo A G maior erit, quàm C F. </s>
  <s xml:id="echoid-s8406" xml:space="preserve">Eodem ratiocinio oſtendetur C F maior, <lb/>quàm T V.</s>
  <s xml:id="echoid-s8407" xml:space="preserve"/>
</p>
<div xml:id="echoid-div740" type="float" level="2" n="30">
  <figure xlink:label="fig-0260-02" xlink:href="fig-0260-02a">
    <image file="0260-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0260-02"/>
  </figure>
<note position="right" xlink:label="note-0261-01" xlink:href="note-0261-01a" xml:space="preserve">8. 9. &amp; 10. <lb/>lib. 5.</note>
  <figure xlink:label="fig-0261-01" xlink:href="fig-0261-01a">
    <image file="0261-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0261-01"/>
  </figure>
<note position="right" xlink:label="note-0261-02" xlink:href="note-0261-02a" xml:space="preserve">SCHO-<lb/>LIVM. <lb/>Prop. 6. <lb/>addit.</note>
<note position="right" xlink:label="note-0261-03" xlink:href="note-0261-03a" xml:space="preserve">Prop. 3. 4. <lb/>addit. <lb/>7. &amp; 38. <lb/>lib. 5.</note>
  <figure xlink:label="fig-0261-02" xlink:href="fig-0261-02a">
    <image file="0261-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0261-02"/>
  </figure>
  <figure xlink:label="fig-0262-01" xlink:href="fig-0262-01a">
    <image file="0262-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0262-01"/>
  </figure>
<note position="left" xlink:label="note-0262-01" xlink:href="note-0262-01a" xml:space="preserve">Poſtr. pars <lb/>pr. 4. add. <lb/>huius.</note>
<note position="left" xlink:label="note-0262-02" xlink:href="note-0262-02a" xml:space="preserve">Pars 3.</note>
<note position="left" xlink:label="note-0262-03" xlink:href="note-0262-03a" xml:space="preserve">prop. 3. <lb/>addit. <lb/>huius.</note>
</div>
<note position="left" xml:space="preserve">38. lib. 5.</note>
<p style="it">
  <s xml:id="echoid-s8408" xml:space="preserve">Secundò angulus Y D A ſit acutus in parabolis, at in hyperbolis minor ſe-<lb/>mirecto, &amp; </s>
  <s xml:id="echoid-s8409" xml:space="preserve">M I H ab asymptoto I H, &amp; </s>
  <s xml:id="echoid-s8410" xml:space="preserve">recta linea centra coniungente con-<lb/>tentus ſit acutus: </s>
  <s xml:id="echoid-s8411" xml:space="preserve">Manifeſtum eſt duci poſſe ramum breuiſſimum, vt O B ad ſe-<lb/>
<anchor type="note" xlink:label="note-0262-05a" xlink:href="note-0262-05"/>
ctionem interiorem A B, qui parallelus ſit rectæ lineæ D A vertices coniungenti, <lb/>vel I M centra coniungenti; </s>
  <s xml:id="echoid-s8412" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8413" xml:space="preserve">ex vtraque parte ipſius rami O B præter axim <lb/>A G ducantur quilibet breuiſſimi rami Q P, d e, i l, O C, qui ſecent exter-<lb/>
<anchor type="note" xlink:label="note-0262-06a" xlink:href="note-0262-06"/>
nam peripheriam in R, f, m, F. </s>
  <s xml:id="echoid-s8414" xml:space="preserve">Oſtendendum modò eſt in eiſdem coniſectio-<lb/>nibus E B eſſe diſtantiam omnium maximam, &amp; </s>
  <s xml:id="echoid-s8415" xml:space="preserve">R P propinquiorem maximæ <lb/>maiorem eſſe remotiore f e; </s>
  <s xml:id="echoid-s8416" xml:space="preserve">pariterque m l maiorem eſſe quàm G A. </s>
  <s xml:id="echoid-s8417" xml:space="preserve">Ducantur <lb/>interceptæ R g, m n parallelæ E B, &amp; </s>
  <s xml:id="echoid-s8418" xml:space="preserve">f h parallela R P, nec non G S paral-<lb/>lela m l, &amp; </s>
  <s xml:id="echoid-s8419" xml:space="preserve">F a parallela G a. </s>
  <s xml:id="echoid-s8420" xml:space="preserve">Quoniam interceptæ R g, m n parallelæ ſunt <lb/>eidem E B, &amp; </s>
  <s xml:id="echoid-s8421" xml:space="preserve">recta linea D A vertices coniungens, vel I M centra coniun-<lb/>
<anchor type="note" xlink:label="note-0262-07a" xlink:href="note-0262-07"/>
gens parallela facta fuit eidem E B; </s>
  <s xml:id="echoid-s8422" xml:space="preserve">ergo E B, R g, m n erunt omnes inter <lb/>ſe æquales; </s>
  <s xml:id="echoid-s8423" xml:space="preserve">eſtque R P minor, quàm R g; </s>
  <s xml:id="echoid-s8424" xml:space="preserve">pariterque m l minor, quàm m n, <lb/>
<anchor type="note" xlink:label="note-0262-08a" xlink:href="note-0262-08"/>
quia iliæ ſunt productiones breuiſſimorum ramorum Q P, &amp; </s>
  <s xml:id="echoid-s8425" xml:space="preserve">i l; </s>
  <s xml:id="echoid-s8426" xml:space="preserve">igitur quæ-<lb/>libet diſtantia R P, vel l m ex vtraque parte ipſius E B ſumpta minor eſt, <lb/>quàm E B; </s>
  <s xml:id="echoid-s8427" xml:space="preserve">ideoque E B erit omnium maxima. </s>
  <s xml:id="echoid-s8428" xml:space="preserve">Deinde quia O B parallela eſt <lb/>A D, vel M I, &amp; </s>
  <s xml:id="echoid-s8429" xml:space="preserve">rami breuiſſimi O B, Q P ſe ſecant vltra axim A O; </s>
  <s xml:id="echoid-s8430" xml:space="preserve">ergo <lb/>recta linea R P Q producta ſecabit quoque reliquam parallelarum D A, vel <lb/>
<anchor type="note" xlink:label="note-0262-09a" xlink:href="note-0262-09"/>
<pb o="225" file="0263" n="263" rhead="Conicor. Lib. VI."/>
I M ad partes O A M; </s>
  <s xml:id="echoid-s8431" xml:space="preserve">ideoque interceptæ R P, f h parallelæ erunt alicui re-<lb/>ctæ lineæ diuidenti angulum D A O ab axe interioris parabola, &amp; </s>
  <s xml:id="echoid-s8432" xml:space="preserve">vertices <lb/>coniungente contentum, vel angulum I M L ab asymptoto interioris hyperbolæ, <lb/>&amp; </s>
  <s xml:id="echoid-s8433" xml:space="preserve">centra coniungente contentum; </s>
  <s xml:id="echoid-s8434" xml:space="preserve">igitur R P propinquior verticibus, vel vlte-<lb/>
<anchor type="note" xlink:label="note-0263-01a" xlink:href="note-0263-01"/>
rius tendens ad partes reliquæ asymptoti M N maior erit quàm f h; </s>
  <s xml:id="echoid-s8435" xml:space="preserve">eſtque f h <lb/>
<anchor type="figure" xlink:label="fig-0263-01a" xlink:href="fig-0263-01"/>
maior f e quæ eſt productio rami breuiſſimi; </s>
  <s xml:id="echoid-s8436" xml:space="preserve">ergo diſtãtia R P propinquior maximæ <lb/>
<anchor type="note" xlink:label="note-0263-02a" xlink:href="note-0263-02"/>
E B maior erit, quàm f e. </s>
  <s xml:id="echoid-s8437" xml:space="preserve">E contra quia breuiſſimus ramus i l m cadit inter <lb/>duas parallelas E B, &amp; </s>
  <s xml:id="echoid-s8438" xml:space="preserve">D A, &amp; </s>
  <s xml:id="echoid-s8439" xml:space="preserve">ſecat ramũ breuiſſimum E B ad partes O i; <lb/></s>
  <s xml:id="echoid-s8440" xml:space="preserve">
<anchor type="note" xlink:label="note-0263-03a" xlink:href="note-0263-03"/>
ergo l m occurrit A D, vel M I ad partes D, vel I; </s>
  <s xml:id="echoid-s8441" xml:space="preserve">ideoque intercepta m l, <lb/>&amp; </s>
  <s xml:id="echoid-s8442" xml:space="preserve">ei parallela G S erunt æquidiſtantes alicui rectæ lineæ diuidenti angulum Y <lb/>D A, in parabolis, vel H I M in hyperbolis: </s>
  <s xml:id="echoid-s8443" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8444" xml:space="preserve">propterea G S propinquior ver-<lb/>
<anchor type="note" xlink:label="note-0263-04a" xlink:href="note-0263-04"/>
tici parabolæ, vel vlterius tendens ad partes reliquæ asymptoti M N minor <lb/>erit, quàm m l; </s>
  <s xml:id="echoid-s8445" xml:space="preserve">eſtque G A productio rami breuiſſimi minor quàm G S; </s>
  <s xml:id="echoid-s8446" xml:space="preserve">ergo <lb/>
<anchor type="note" xlink:label="note-0263-05a" xlink:href="note-0263-05"/>
m l maior erit, quàm G A; </s>
  <s xml:id="echoid-s8447" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8448" xml:space="preserve">ſic vlterius G A maior erit C F, quando oc-<lb/>curſus Z ſectionum cadit vltra interceptam F C ad partes T V; </s>
  <s xml:id="echoid-s8449" xml:space="preserve">vt in prima <lb/>parte oſtenſum eſt.</s>
  <s xml:id="echoid-s8450" xml:space="preserve"/>
</p>
<div xml:id="echoid-div741" type="float" level="2" n="31">
<note position="left" xlink:label="note-0262-05" xlink:href="note-0262-05a" xml:space="preserve">Propoſ. 6. <lb/>addit. <lb/>huius.</note>
<note position="left" xlink:label="note-0262-06" xlink:href="note-0262-06a" xml:space="preserve">8. 9. &amp; 10. <lb/>lib. 5.</note>
<note position="left" xlink:label="note-0262-07" xlink:href="note-0262-07a" xml:space="preserve">Propoſ. 5. <lb/>addit. <lb/>huius.</note>
<note position="left" xlink:label="note-0262-08" xlink:href="note-0262-08a" xml:space="preserve">38. lib. 5.</note>
<note position="left" xlink:label="note-0262-09" xlink:href="note-0262-09a" xml:space="preserve">38. lib. 5.</note>
<note position="right" xlink:label="note-0263-01" xlink:href="note-0263-01a" xml:space="preserve">3. 4. addit.</note>
  <figure xlink:label="fig-0263-01" xlink:href="fig-0263-01a">
    <image file="0263-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0263-01"/>
  </figure>
<note position="right" xlink:label="note-0263-02" xlink:href="note-0263-02a" xml:space="preserve">38. lib. 5.</note>
<note position="right" xlink:label="note-0263-03" xlink:href="note-0263-03a" xml:space="preserve">28. lib. 5.</note>
<note position="right" xlink:label="note-0263-04" xlink:href="note-0263-04a" xml:space="preserve">3. 4. addit.</note>
<note position="right" xlink:label="note-0263-05" xlink:href="note-0263-05a" xml:space="preserve">38. lib. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8451" xml:space="preserve">Iiſdem manentibus: </s>
  <s xml:id="echoid-s8452" xml:space="preserve">dico poſtea, quod vltra diſtantiam maximam E B ad <lb/>partes R P, diſtantiæ, licet ſemper diminuantur non efficiuntur minores inter-<lb/>uallo diametrorum æquidiſtantium D Y, A O in parabolis, vel interuallo asym-<lb/>ptotorum collateralium I H, M L in hyperbolis, vt facile deducitur ex 3. </s>
  <s xml:id="echoid-s8453" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8454" xml:space="preserve">4. <lb/></s>
  <s xml:id="echoid-s8455" xml:space="preserve">additarum. </s>
  <s xml:id="echoid-s8456" xml:space="preserve">At ad partes asymptotorum congruentium hyperbolæ ad ſe ſe ipſas <lb/>propius accedunt, interuallo minori quolibet dato: </s>
  <s xml:id="echoid-s8457" xml:space="preserve">Nam in locum ab hyperbole <lb/>B A C, &amp; </s>
  <s xml:id="echoid-s8458" xml:space="preserve">asymptoto M N contentum extenditur altera hyperbole E D F; </s>
  <s xml:id="echoid-s8459" xml:space="preserve">ſed <lb/>diſtantia hyperbolæ B A C ab asymptoto M N efficitur minor qualibet data: </s>
  <s xml:id="echoid-s8460" xml:space="preserve">igi-<lb/>tur diſtantia hyperbolæ D G F compræhenſæ ab hyperbole intercipiente minor erit <lb/>qualibet data diſtantia.</s>
  <s xml:id="echoid-s8461" xml:space="preserve"/>
</p>
<pb o="226" file="0264" n="264" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s8462" xml:space="preserve">Tandem ĳſdem poſitis ducantur ex altera parte concurſus Z rami breuiſſimi <lb/>O C, Q T, qui eſſiciant diſtantias F C, T V. </s>
  <s xml:id="echoid-s8463" xml:space="preserve">Dico F C propinquiorem con-<lb/>curſui Z minorem eße, quàm T V. </s>
  <s xml:id="echoid-s8464" xml:space="preserve">Quoniam angulus Y D A, vel Y I M ſup-<lb/>
<anchor type="figure" xlink:label="fig-0264-01a" xlink:href="fig-0264-01"/>
ponitur acutus; </s>
  <s xml:id="echoid-s8465" xml:space="preserve">ſuntque I D Y, M A O inter ſe, parallelæ; </s>
  <s xml:id="echoid-s8466" xml:space="preserve">ergo angulus D A O, <lb/>vel I M O, &amp; </s>
  <s xml:id="echoid-s8467" xml:space="preserve">multo magis I M N erit obtuſus; </s>
  <s xml:id="echoid-s8468" xml:space="preserve">ſed quilibet ramus breuiſſimus <lb/>
<anchor type="note" xlink:label="note-0264-01a" xlink:href="note-0264-01"/>
Q V T parallelus F a eſſicit cum axi A O angulũ acutũ; </s>
  <s xml:id="echoid-s8469" xml:space="preserve">igitur ramus breuiſſimus <lb/>Q T, &amp; </s>
  <s xml:id="echoid-s8470" xml:space="preserve">ei parallelus F a ſunt æquidiſtantes alicui rectæ lineæ diuidenti angu-<lb/>lum D A O, vel I M N; </s>
  <s xml:id="echoid-s8471" xml:space="preserve">ideoque F a propinquior concurſui, vel vlterius ten-<lb/>
<anchor type="note" xlink:label="note-0264-02a" xlink:href="note-0264-02"/>
dens ad partes reliquæ asymptoti I H minor eſt, quàm T V; </s>
  <s xml:id="echoid-s8472" xml:space="preserve">eſtque F C minor <lb/>quàm F a (quia illa eſt portio rami breuiſſimi) ergo F C minor eſt, quàm T V. <lb/></s>
  <s xml:id="echoid-s8473" xml:space="preserve">Quod erat propoſitum.</s>
  <s xml:id="echoid-s8474" xml:space="preserve"/>
</p>
<div xml:id="echoid-div742" type="float" level="2" n="32">
  <figure xlink:label="fig-0264-01" xlink:href="fig-0264-01a">
    <image file="0264-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0264-01"/>
  </figure>
<note position="left" xlink:label="note-0264-01" xlink:href="note-0264-01a" xml:space="preserve">13. 14. <lb/>lib. 5.</note>
<note position="left" xlink:label="note-0264-02" xlink:href="note-0264-02a" xml:space="preserve">Propoſ. 3. <lb/>&amp; 4. add.</note>
</div>
<note position="left" xml:space="preserve">12. lib. 5.</note>
<p style="it">
  <s xml:id="echoid-s8475" xml:space="preserve">In duabus hyperbolis C A D, <lb/>
<anchor type="note" xlink:label="note-0264-04a" xlink:href="note-0264-04"/>
<anchor type="figure" xlink:label="fig-0264-02a" xlink:href="fig-0264-02"/>
H G I ſimilibus, concentricis, <lb/>&amp; </s>
  <s xml:id="echoid-s8476" xml:space="preserve">ſimiliter poſitis circa com-<lb/>munem axim B A G, ideſt <lb/>conſiſtant circa cõmunes asym-<lb/>ptotos E B F: </s>
  <s xml:id="echoid-s8477" xml:space="preserve">Dico ſectionum <lb/>C A D, H G I interualla sẽ-<lb/>per minui, quo magis ab axis <lb/>vertice recedunt; </s>
  <s xml:id="echoid-s8478" xml:space="preserve">atque effici <lb/>poſſe minora interuallo quolibet <lb/>dato.</s>
  <s xml:id="echoid-s8479" xml:space="preserve"/>
</p>
<div xml:id="echoid-div743" type="float" level="2" n="33">
<note position="left" xlink:label="note-0264-04" xlink:href="note-0264-04a" xml:space="preserve">PROP. 9. <lb/>Addit.</note>
  <figure xlink:label="fig-0264-02" xlink:href="fig-0264-02a">
    <image file="0264-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0264-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s8480" xml:space="preserve">Deſcribatur hyperbole M G N <lb/>
<anchor type="note" xlink:label="note-0264-05a" xlink:href="note-0264-05"/>
æqualis, ſimilis, &amp; </s>
  <s xml:id="echoid-s8481" xml:space="preserve">ſimiliter poſita
<pb o="227" file="0265" n="265" rhead="Conicor. Lib. VI."/>
ipſi C A D circa communem axim A G. </s>
  <s xml:id="echoid-s8482" xml:space="preserve">Et quoniam hyperbolæ H G 1 ſemiaxis <lb/>tranſuer ſus B G maior eſt tranſuer ſo ſemiaxe B A, hyperboles C A D, pariter-<lb/>què latus rectum illius maius erit buius latere recto (cum later a figurarum ſint <lb/>
<anchor type="note" xlink:label="note-0265-01a" xlink:href="note-0265-01"/>
proportionalia in hyperbolis ſimilibus:) </s>
  <s xml:id="echoid-s8483" xml:space="preserve">igitur hyperbole H G I maior eſt hyper-<lb/>bola M G N (quod ab alĳs oſtenſum eſt), &amp; </s>
  <s xml:id="echoid-s8484" xml:space="preserve">conſiſtunt circa communē axim A G, <lb/>&amp; </s>
  <s xml:id="echoid-s8485" xml:space="preserve">vertex G eſt communis; </s>
  <s xml:id="echoid-s8486" xml:space="preserve">igitur hyperbole H G I compræbendit hyperbolen M <lb/>G N; </s>
  <s xml:id="echoid-s8487" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8488" xml:space="preserve">ideo hyperbole H G I cadit inter duas hyperbolas G M, &amp; </s>
  <s xml:id="echoid-s8489" xml:space="preserve">A C : </s>
  <s xml:id="echoid-s8490" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8491" xml:space="preserve"><lb/>propterea hyperbole G H multo magis ſucceſſiuè vicinior efficitur hyperbolæ A C, <lb/>quàm hyperbole G M; </s>
  <s xml:id="echoid-s8492" xml:space="preserve">ſed duæ hyperbole æquales, &amp; </s>
  <s xml:id="echoid-s8493" xml:space="preserve">ſimiliter poſitæ A C, &amp; </s>
  <s xml:id="echoid-s8494" xml:space="preserve">G <lb/>
<anchor type="note" xlink:label="note-0265-02a" xlink:href="note-0265-02"/>
M ſemper magis, ac magis ad inuicem approximantur, igitur multo magis hy-<lb/>perbolæ concentricæ A C, &amp; </s>
  <s xml:id="echoid-s8495" xml:space="preserve">G H ſemper magis, ac magis ad ſe ſe ipſas appro-<lb/>
<anchor type="note" xlink:label="note-0265-03a" xlink:href="note-0265-03"/>
pinquantur, &amp; </s>
  <s xml:id="echoid-s8496" xml:space="preserve">inter ſe non conuenient vt Pappus demonſtrauit. </s>
  <s xml:id="echoid-s8497" xml:space="preserve">Tandem, quoniã <lb/>lineæ breuiſſimæ, quæ perpendicularis eſt ad tangentem hyperbolem G H portio <lb/>ab asymptoto E B, &amp; </s>
  <s xml:id="echoid-s8498" xml:space="preserve">ſectione H G compræ henſa effici poteſt minor quacunque <lb/>recta linea propoſita; </s>
  <s xml:id="echoid-s8499" xml:space="preserve">cadit verò hyperbole A C inter ſectionem G H, &amp; </s>
  <s xml:id="echoid-s8500" xml:space="preserve">continen-<lb/>
<anchor type="note" xlink:label="note-0265-04a" xlink:href="note-0265-04"/>
tem B E; </s>
  <s xml:id="echoid-s8501" xml:space="preserve">igitur multo magis diſtantia inter hyperbolas G H, &amp; </s>
  <s xml:id="echoid-s8502" xml:space="preserve">A C minor <lb/>erit quacunque recta linea propofita. </s>
  <s xml:id="echoid-s8503" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s8504" xml:space="preserve"/>
</p>
<div xml:id="echoid-div744" type="float" level="2" n="34">
<note position="left" xlink:label="note-0264-05" xlink:href="note-0264-05a" xml:space="preserve">12. huius. <lb/>&amp; ex 53. <lb/>lib. 1.</note>
<note position="right" xlink:label="note-0265-01" xlink:href="note-0265-01a" xml:space="preserve">12. huius.</note>
<note position="right" xlink:label="note-0265-02" xlink:href="note-0265-02a" xml:space="preserve">Propoſ. 7. <lb/>addit.</note>
<note position="right" xlink:label="note-0265-03" xlink:href="note-0265-03a" xml:space="preserve">lib. 7. <lb/>prop. 208. <lb/>29. 30. <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0265-04" xlink:href="note-0265-04a" xml:space="preserve">Propof. 4. <lb/>lib. 2.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8505" xml:space="preserve">Si in duobus conis ducta fuerint duo triangula per axes A B C, D E <lb/>
<anchor type="note" xlink:label="note-0265-05a" xlink:href="note-0265-05"/>
F ſimilia, &amp; </s>
  <s xml:id="echoid-s8506" xml:space="preserve">ſimiliter poſita, atq; </s>
  <s xml:id="echoid-s8507" xml:space="preserve">ſectionum I G H, &amp; </s>
  <s xml:id="echoid-s8508" xml:space="preserve">N L M dia-<lb/>metri G O, L K æque ad baſes inclinatæ intercipiant cũ triangulorum la-<lb/>teribus A B, D E eiſdem G O, L K parallelis, portiones O B, K E æquales; <lb/></s>
  <s xml:id="echoid-s8509" xml:space="preserve">vel cum axibus conorum Aγ, D Z diametris æquidiſtantibus intercipiant <lb/>portiones O Y, K Z æquales, &amp; </s>
  <s xml:id="echoid-s8510" xml:space="preserve">efficiant angulos A Y C, D Z F <lb/>aquales : </s>
  <s xml:id="echoid-s8511" xml:space="preserve">erunt conicæ ſectiones inter ſe æquales, &amp; </s>
  <s xml:id="echoid-s8512" xml:space="preserve">in qualibet earum, <lb/>duplum interceptæ poterit figuram ſectionis.</s>
  <s xml:id="echoid-s8513" xml:space="preserve"/>
</p>
<div xml:id="echoid-div745" type="float" level="2" n="35">
<note position="right" xlink:label="note-0265-05" xlink:href="note-0265-05a" xml:space="preserve">PROP. <lb/>10. Add.</note>
</div>
  <figure>
    <image file="0265-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0265-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s8514" xml:space="preserve">Primò in parabolis, quia triangula A B C, D E F ſunt ſimilia, erit B C <lb/>ad C A vt E F ad F D, &amp; </s>
  <s xml:id="echoid-s8515" xml:space="preserve">G O, L K ſunt parallelæ homologis A B, D E; <lb/></s>
  <s xml:id="echoid-s8516" xml:space="preserve">ergo O C ad C G, &amp; </s>
  <s xml:id="echoid-s8517" xml:space="preserve">B O ad G A eandem proportionem habebunt, quàm B C <lb/>ad C A, ſeu eandem, quàm habet E F ad F D; </s>
  <s xml:id="echoid-s8518" xml:space="preserve">eſtque E K ad L D vt E F <lb/>ad F D; </s>
  <s xml:id="echoid-s8519" xml:space="preserve">ergo B O ad G A eſt vt E K ad L D; </s>
  <s xml:id="echoid-s8520" xml:space="preserve">ſuntque B O, E K æquales;</s>
  <s xml:id="echoid-s8521" xml:space="preserve">
<pb o="228" file="0266" n="266" rhead="Apollonij Pergæi"/>
igitur G A æqualis eſt L D: </s>
  <s xml:id="echoid-s8522" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8523" xml:space="preserve">quia in triangulis ſimilibus rectangulum B A <lb/>C ad quadratum B C, ſeu A G ad latus rectum G R eandem proportionem ha-<lb/>
<anchor type="note" xlink:label="note-0266-01a" xlink:href="note-0266-01"/>
bet; </s>
  <s xml:id="echoid-s8524" xml:space="preserve">quàm rectangulum E D F ad quadratum E F, ſeu quàm D L habet ad la-<lb/>tus rectum L S; </s>
  <s xml:id="echoid-s8525" xml:space="preserve">igitur A G ad G R erit vt D L ad L S; </s>
  <s xml:id="echoid-s8526" xml:space="preserve">ſuntq; </s>
  <s xml:id="echoid-s8527" xml:space="preserve">A G, D L <lb/>oſtenſæ æquales ergo G R, &amp; </s>
  <s xml:id="echoid-s8528" xml:space="preserve">L S latera recta æqualia ſunt, &amp; </s>
  <s xml:id="echoid-s8529" xml:space="preserve">diametri ſectio-<lb/>num eſſiciunt angulos G O H, L K M æquales; </s>
  <s xml:id="echoid-s8530" xml:space="preserve">ergo parabolæ H G I, &amp; </s>
  <s xml:id="echoid-s8531" xml:space="preserve">M L N <lb/>
<anchor type="note" xlink:label="note-0266-02a" xlink:href="note-0266-02"/>
æquales ſunt inter ſe.</s>
  <s xml:id="echoid-s8532" xml:space="preserve"/>
</p>
<div xml:id="echoid-div746" type="float" level="2" n="36">
<note position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">11. lib. 1.</note>
<note position="left" xlink:label="note-0266-02" xlink:href="note-0266-02a" xml:space="preserve">Prop 10. <lb/>huius.</note>
</div>
  <figure>
    <image file="0266-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0266-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s8533" xml:space="preserve">In hyperbolis verò, quoniam P G parallela eſt axi A Y, &amp; </s>
  <s xml:id="echoid-s8534" xml:space="preserve">A V parallela, <lb/>eſt baſi B C, &amp; </s>
  <s xml:id="echoid-s8535" xml:space="preserve">latera P B, &amp; </s>
  <s xml:id="echoid-s8536" xml:space="preserve">A C ſunt communia; </s>
  <s xml:id="echoid-s8537" xml:space="preserve">igitur P V ad V A eſt vt <lb/>A Y ad Y B, &amp; </s>
  <s xml:id="echoid-s8538" xml:space="preserve">G V ad V A eſt vt Y A ad Y C: </s>
  <s xml:id="echoid-s8539" xml:space="preserve">habet verò eadem A Y ad <lb/>æquales Y B, Y C eandem rationem ergò P V, &amp; </s>
  <s xml:id="echoid-s8540" xml:space="preserve">G V ad eandem V A habent <lb/>eandem proportionem, &amp; </s>
  <s xml:id="echoid-s8541" xml:space="preserve">ideo P V æqualis eſt V G, atq; </s>
  <s xml:id="echoid-s8542" xml:space="preserve">punctum V erit cen-<lb/>trum ſectionis, &amp; </s>
  <s xml:id="echoid-s8543" xml:space="preserve">quadratum A Y æquale erit quadrato V O (propter paral-<lb/>lelogrammum V Y), &amp; </s>
  <s xml:id="echoid-s8544" xml:space="preserve">quadratum V O æquale eſt rectangulo P O G cum qua-<lb/>drato V G; </s>
  <s xml:id="echoid-s8545" xml:space="preserve">pariterque quadratum C Y æquale eſt rectangulo C O B cum qua <lb/>drato O Y, &amp; </s>
  <s xml:id="echoid-s8546" xml:space="preserve">habet quadratum A Y ad quadratum C Y eandem proportionem, <lb/>quàm latus tranſuer ſum P G ad latus rectum G R, ſeu eandem, quàm habet <lb/>
<anchor type="note" xlink:label="note-0266-03a" xlink:href="note-0266-03"/>
rectangulum P O G ad rectangulum C O B, ergo diuidendo quadratum V G ad <lb/>quadratũ O Y eandem proportionem habebit, quàm quadratum A Y ad quadratũ <lb/>Y C, ſeu vt P G ad G R, ſeu vt quadratum P G ad rectangulum P G R, <lb/>&amp; </s>
  <s xml:id="echoid-s8547" xml:space="preserve">ideo quadratum duplæ V G, ſeu P G eandem proportionem habebit ad re-<lb/>ctangulum P G R, atq; </s>
  <s xml:id="echoid-s8548" xml:space="preserve">ad quadratum duplæ ipſius Y O; </s>
  <s xml:id="echoid-s8549" xml:space="preserve">quare quadratum duplæ <lb/>ipſius O Y æquale erit figuræ ſectionis ſeu rectangulo P G R. </s>
  <s xml:id="echoid-s8550" xml:space="preserve">Eodem modo <lb/>oſtendetur X centrum hyperbolæ M L N, &amp; </s>
  <s xml:id="echoid-s8551" xml:space="preserve">quadratum L Z ad quadratum du-<lb/>ple K Z eſſe vt quadratum D Z ad quadratum Z F, ſeu vt Z L ad L S, &amp; </s>
  <s xml:id="echoid-s8552" xml:space="preserve"><lb/>ideo quadratum duplæ ipſius K Z æquale erit figuræ ſectionis, ſeu rectangulo Z <lb/>L S. </s>
  <s xml:id="echoid-s8553" xml:space="preserve">Tandem, quia propter ſimilitudinem triangulorum per axes, ſunt anguli <lb/>C, F æquales, &amp; </s>
  <s xml:id="echoid-s8554" xml:space="preserve">anguli Y, Z pariter æquales ( cum ex hypotheſi diametri G O, <lb/>L K parallelæ axibus AY, D Z efficiant angulos G O C, L K F æquales); </s>
  <s xml:id="echoid-s8555" xml:space="preserve">ergo <lb/>A Y ad Y C erit vt D Z ad Z F, &amp; </s>
  <s xml:id="echoid-s8556" xml:space="preserve">earum quadrata etiam proportionalia <lb/>erunt; </s>
  <s xml:id="echoid-s8557" xml:space="preserve">ſed P G ad G R eſt vt quadratum A Y ad quadratum Y C, atque Z L
<pb o="229" file="0267" n="267" rhead="Conicor. Lib. VI."/>
ad L S eſt vt quadratum D Z ad quadratum Z F ; </s>
  <s xml:id="echoid-s8558" xml:space="preserve">igitur P G ad G R ean-<lb/>dem proportionem habet, quàm Z L ad L S, &amp; </s>
  <s xml:id="echoid-s8559" xml:space="preserve">propterea figuræ ſectionem <lb/>
<anchor type="note" xlink:label="note-0267-01a" xlink:href="note-0267-01"/>
erunt ſimiles; </s>
  <s xml:id="echoid-s8560" xml:space="preserve">ĳs autẽ figuris æqualia oſtenſa ſunt quadrata dupliciũ O Y, &amp; </s>
  <s xml:id="echoid-s8561" xml:space="preserve">K <lb/>Z, quæ ſuppoſitæ fuerunt æquales; </s>
  <s xml:id="echoid-s8562" xml:space="preserve">igitur figuræ P G R, &amp; </s>
  <s xml:id="echoid-s8563" xml:space="preserve">Z L S ſimiles, &amp; </s>
  <s xml:id="echoid-s8564" xml:space="preserve"><lb/>æquales ſunt inter ſe, atque diametri æquæ inclinatæ ſunt ad ordinatim ad eas <lb/>applicatas H I, M N; </s>
  <s xml:id="echoid-s8565" xml:space="preserve">igitur ſectiones H G I, M L N æquales ſunt inter ſe, <lb/>
<anchor type="note" xlink:label="note-0267-02a" xlink:href="note-0267-02"/>
ſimiles, &amp; </s>
  <s xml:id="echoid-s8566" xml:space="preserve">congruentes, quarum figuræ æquales ſunt quadratis duplicium inter-<lb/>ceptarum O Y, &amp; </s>
  <s xml:id="echoid-s8567" xml:space="preserve">K Z, quod erat propoſitum.</s>
  <s xml:id="echoid-s8568" xml:space="preserve"/>
</p>
<div xml:id="echoid-div747" type="float" level="2" n="37">
<note position="left" xlink:label="note-0266-03" xlink:href="note-0266-03a" xml:space="preserve">21. lib.1.</note>
<note position="right" xlink:label="note-0267-01" xlink:href="note-0267-01a" xml:space="preserve">ex 12. <lb/>huius.</note>
<note position="right" xlink:label="note-0267-02" xlink:href="note-0267-02a" xml:space="preserve">Prop. 10. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div749" type="section" level="1" n="235">
<head xml:id="echoid-head296" xml:space="preserve">LEMMA IX.</head>
<p style="it">
  <s xml:id="echoid-s8569" xml:space="preserve">S I in duobus conis A B C, D E F, baſes ſint in eodem plano, &amp; </s>
  <s xml:id="echoid-s8570" xml:space="preserve"><lb/>duo triangula per axes A B C, D E F fuerint ſimilia, &amp; </s>
  <s xml:id="echoid-s8571" xml:space="preserve">ſimi-<lb/>liter poſita, &amp; </s>
  <s xml:id="echoid-s8572" xml:space="preserve">in eodem plano exiſtentia, erunt coni ſimiles inter ſe.</s>
  <s xml:id="echoid-s8573" xml:space="preserve"/>
</p>
  <figure>
    <image file="0267-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0267-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s8574" xml:space="preserve">Ducantur à verticibus A, &amp; </s>
  <s xml:id="echoid-s8575" xml:space="preserve">D duæ rectæ A G, &amp; </s>
  <s xml:id="echoid-s8576" xml:space="preserve">D H perpendiculares ad <lb/>baſes conorũ, &amp; </s>
  <s xml:id="echoid-s8577" xml:space="preserve">à terminis axium A Y, &amp; </s>
  <s xml:id="echoid-s8578" xml:space="preserve">D Z coniungantur rectæ lineæ Y G, <lb/>&amp; </s>
  <s xml:id="echoid-s8579" xml:space="preserve">Z H. </s>
  <s xml:id="echoid-s8580" xml:space="preserve">Quoniã planum, in quo exiſtunt duo triangula A B C, D E F ſecat <lb/>planum, in quo baſes conorum iacent in vna recta linea, quæ baſis eſt vtriuſque <lb/>trianguli per axes conorum ducti; </s>
  <s xml:id="echoid-s8581" xml:space="preserve">ideoque B C, &amp; </s>
  <s xml:id="echoid-s8582" xml:space="preserve">E F in directum conſtitutæ <lb/>erunt, &amp; </s>
  <s xml:id="echoid-s8583" xml:space="preserve">circa angulos æquales B, &amp; </s>
  <s xml:id="echoid-s8584" xml:space="preserve">E latera A B ad B C, atque D E ad E <lb/>F ſunt proportionalia ( propter triangulorum A B C, &amp; </s>
  <s xml:id="echoid-s8585" xml:space="preserve">D E F ſimilitudinem) <lb/>erunt quoque ad conſequẽtium ſemiſſes proportionales, ſcilicet A B ad B Y erit, <lb/>vt D E ad E Z circa angulos æquales, &amp; </s>
  <s xml:id="echoid-s8586" xml:space="preserve">propterea triangula A B Y, &amp; </s>
  <s xml:id="echoid-s8587" xml:space="preserve">D E <lb/>Z ſimilia erunt: </s>
  <s xml:id="echoid-s8588" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8589" xml:space="preserve">ideò duo anguli B Y A, &amp; </s>
  <s xml:id="echoid-s8590" xml:space="preserve">E Z D, externus interno, æqua-<lb/>les erunt inter ſe; </s>
  <s xml:id="echoid-s8591" xml:space="preserve">igitur Y A, &amp; </s>
  <s xml:id="echoid-s8592" xml:space="preserve">Z D in eodem plano exiſtentes, parallelæ <lb/>erunt inter ſe; </s>
  <s xml:id="echoid-s8593" xml:space="preserve">ſunt quoque A G, D H inter ſe parallelæ ( cum ſint perpendicu-<lb/>lares ad idem planum baſium ) ergo duo anguli Y A G, &amp; </s>
  <s xml:id="echoid-s8594" xml:space="preserve">Z D H æquales ſunt <lb/>inter ſe; </s>
  <s xml:id="echoid-s8595" xml:space="preserve">atquè anguli G, &amp; </s>
  <s xml:id="echoid-s8596" xml:space="preserve">H æquales ſunt, nempe recti; </s>
  <s xml:id="echoid-s8597" xml:space="preserve">igitur in triangu-<lb/>lis A Y G, &amp; </s>
  <s xml:id="echoid-s8598" xml:space="preserve">D Z H, duo poſtremi anguli A Y G, &amp; </s>
  <s xml:id="echoid-s8599" xml:space="preserve">D Z H æquales ſunt
<pb o="230" file="0268" n="268" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0268-01a" xlink:href="fig-0268-01"/>
inter ſe: </s>
  <s xml:id="echoid-s8600" xml:space="preserve">hi autem anguli inclinationes ſunt axium conorum ad ſuas baſes; </s>
  <s xml:id="echoid-s8601" xml:space="preserve">igi-<lb/>tur axes A Y, &amp; </s>
  <s xml:id="echoid-s8602" xml:space="preserve">D Z æque ſunt inclinati ad ſuas baſes: </s>
  <s xml:id="echoid-s8603" xml:space="preserve">ſuntque proportiona-<lb/>les ad baſium ſemidiametros Y B, &amp; </s>
  <s xml:id="echoid-s8604" xml:space="preserve">Z E ( cum triangula A B Y, D E Z ſi-<lb/>
<anchor type="note" xlink:label="note-0268-01a" xlink:href="note-0268-01"/>
milia oſtenſa ſint ); </s>
  <s xml:id="echoid-s8605" xml:space="preserve">igitur coni A B C, &amp; </s>
  <s xml:id="echoid-s8606" xml:space="preserve">D E F ſimiles ſunt inter ſe. </s>
  <s xml:id="echoid-s8607" xml:space="preserve">Quod <lb/>erat oſtendendum.</s>
  <s xml:id="echoid-s8608" xml:space="preserve"/>
</p>
<div xml:id="echoid-div749" type="float" level="2" n="1">
  <figure xlink:label="fig-0268-01" xlink:href="fig-0268-01a">
    <image file="0268-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0268-01"/>
  </figure>
<note position="left" xlink:label="note-0268-01" xlink:href="note-0268-01a" xml:space="preserve">Defin. 8. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8609" xml:space="preserve">Data parabola Z duos conos ſimiles exhibere, vt idem planum ef-<lb/>
<anchor type="note" xlink:label="note-0268-02a" xlink:href="note-0268-02"/>
ficiat in eis duas parabolas æquales eidem datæ parabolæ, quæ asympto-<lb/>ticæ ſint, &amp; </s>
  <s xml:id="echoid-s8610" xml:space="preserve">ſibi ipſis viciniores fiant diſtantia minore quacunque <lb/>data.</s>
  <s xml:id="echoid-s8611" xml:space="preserve"/>
</p>
<div xml:id="echoid-div750" type="float" level="2" n="2">
<note position="left" xlink:label="note-0268-02" xlink:href="note-0268-02a" xml:space="preserve">PROP. <lb/>11. <lb/>Addit.</note>
</div>
  <figure>
    <image file="0268-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0268-02"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s8612" xml:space="preserve">In quolibet plano fiat angulus I H C æqualis angulo inclinationis diametri, <lb/>&amp; </s>
  <s xml:id="echoid-s8613" xml:space="preserve">baſis parabolæ Z , &amp; </s>
  <s xml:id="echoid-s8614" xml:space="preserve">per H C extenſo alio quolibet plano ducatur in eo B H <lb/>G perpendicularis ad X H C; </s>
  <s xml:id="echoid-s8615" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8616" xml:space="preserve">fiat quodlibet triangulum H G K, &amp; </s>
  <s xml:id="echoid-s8617" xml:space="preserve">vt qua-<lb/>dratum H G ad rectangulum H K G, ita fiat latus rectum parabolæ Z ad pro-
<pb o="231" file="0269" n="269" rhead="Conicor. Lib. VI."/>
ductionem K E, &amp; </s>
  <s xml:id="echoid-s8618" xml:space="preserve">ab E ducatur A E B parallela 1 H, quæ ſecet G H in B: <lb/></s>
  <s xml:id="echoid-s8619" xml:space="preserve">poſtea producatur H K, vt cumq; </s>
  <s xml:id="echoid-s8620" xml:space="preserve">in I, &amp; </s>
  <s xml:id="echoid-s8621" xml:space="preserve">per I ducatur A 1 D parallela E G, <lb/>quæ ſecet B G in D; </s>
  <s xml:id="echoid-s8622" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8623" xml:space="preserve">in plano B X D C, diametris B G, B D, fiant duo <lb/>circuli, qui ſint baſes duorum conorum, quorum vertices A, &amp; </s>
  <s xml:id="echoid-s8624" xml:space="preserve">E, &amp; </s>
  <s xml:id="echoid-s8625" xml:space="preserve">in eo-<lb/>rum ſuperficiebus planum per X I C ductum, efficiat ſectiones C I X, &amp; </s>
  <s xml:id="echoid-s8626" xml:space="preserve">F K <lb/>T. </s>
  <s xml:id="echoid-s8627" xml:space="preserve">Dico eas eße parabolas quæſitas. </s>
  <s xml:id="echoid-s8628" xml:space="preserve">Quoniam recta E G facta eſt parallela. </s>
  <s xml:id="echoid-s8629" xml:space="preserve"><lb/>ipſi A D; </s>
  <s xml:id="echoid-s8630" xml:space="preserve">igitur duo triangula A B D, &amp; </s>
  <s xml:id="echoid-s8631" xml:space="preserve">E B G per axes conorum ducta ſi-<lb/>milia, &amp; </s>
  <s xml:id="echoid-s8632" xml:space="preserve">ſimiliter poſita in eodem ſunt plano; </s>
  <s xml:id="echoid-s8633" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8634" xml:space="preserve">duo circuli baſium in eodem <lb/>ſunt plano; </s>
  <s xml:id="echoid-s8635" xml:space="preserve">ergo coni A B D, &amp; </s>
  <s xml:id="echoid-s8636" xml:space="preserve">E B G ſimiles erunt: </s>
  <s xml:id="echoid-s8637" xml:space="preserve">poſtea quia triangula. </s>
  <s xml:id="echoid-s8638" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0269-01a" xlink:href="note-0269-01"/>
A B D, &amp; </s>
  <s xml:id="echoid-s8639" xml:space="preserve">E B G ſimilia ſunt, &amp; </s>
  <s xml:id="echoid-s8640" xml:space="preserve">I K H communis diameter ſectionum ad <lb/>coincidentes baſes C X, F T æque inclinata, &amp; </s>
  <s xml:id="echoid-s8641" xml:space="preserve">recta linea A E B à verticibus <lb/>conorum ducta parallelæ ſunt inter ſe, atque intercipiunt in angulis æqualibus <lb/>A B H, &amp; </s>
  <s xml:id="echoid-s8642" xml:space="preserve">E B H communem portionem B H baſium triangulorum ſimilium. <lb/></s>
  <s xml:id="echoid-s8643" xml:space="preserve">per axes; </s>
  <s xml:id="echoid-s8644" xml:space="preserve">ergo parabolæ C I X, &amp; </s>
  <s xml:id="echoid-s8645" xml:space="preserve">F K T æquales ſunt inter ſe. </s>
  <s xml:id="echoid-s8646" xml:space="preserve">Secundò, quia <lb/>
<anchor type="note" xlink:label="note-0269-02a" xlink:href="note-0269-02"/>
propter parallelas E B, K H ſunt triangula E B G, H K G ſimilia; </s>
  <s xml:id="echoid-s8647" xml:space="preserve">ergo qua-<lb/>dratum B G ad rectangulum B E G ſcilicet latus rectum parabolæ F K T ad K <lb/>
<anchor type="note" xlink:label="note-0269-03a" xlink:href="note-0269-03"/>
E eſt, vt quadratum H G ad rectangulum H K G, ſed latus rectum parabolæ <lb/>Z ad K E fuit vt qtadratum H G ad rectangulum H K G; </s>
  <s xml:id="echoid-s8648" xml:space="preserve">igitur duo latera <lb/>recta, parabole Z, atq; </s>
  <s xml:id="echoid-s8649" xml:space="preserve">parabole F K T ad eandem K E habent eandem pro-<lb/>portionem, &amp; </s>
  <s xml:id="echoid-s8650" xml:space="preserve">propterea æqualia ſunt, &amp; </s>
  <s xml:id="echoid-s8651" xml:space="preserve">diametri, ad baſes æque inclinatæ <lb/>ſunt ex conſtructione; </s>
  <s xml:id="echoid-s8652" xml:space="preserve">igitur parabole F K T, &amp; </s>
  <s xml:id="echoid-s8653" xml:space="preserve">ei æqualis C I X erit æqua-<lb/>
<anchor type="note" xlink:label="note-0269-04a" xlink:href="note-0269-04"/>
lis eidem parabolæ Z. </s>
  <s xml:id="echoid-s8654" xml:space="preserve">Tertiò quia ſectionum plano, &amp; </s>
  <s xml:id="echoid-s8655" xml:space="preserve">communi diametro I <lb/>K H æquidiſtat cummune lateris A E B, in quo duo coni ſe ſe contingunt; </s>
  <s xml:id="echoid-s8656" xml:space="preserve">ergo <lb/>latus A E B nunquàm occurret plano C I X: </s>
  <s xml:id="echoid-s8657" xml:space="preserve">ſed duæ ſuperficies conicæ tantum-<lb/>modò ſe ſe tangunt in latere A E B, &amp; </s>
  <s xml:id="echoid-s8658" xml:space="preserve">reliquis omnibus in locis ſeparatæ ſunt; <lb/></s>
  <s xml:id="echoid-s8659" xml:space="preserve">igitur duæ parabolæ C I X, F K T in illo plano poſitæ per contactum A E B <lb/>non tranſeunte, &amp; </s>
  <s xml:id="echoid-s8660" xml:space="preserve">extenſæ in duabus conicis ſuperficiebus nunquàm conuenien-<lb/>tibus, erunt asymptoticæ. </s>
  <s xml:id="echoid-s8661" xml:space="preserve">Quartò quia duæ parabole C I X, F K T æquales <lb/>ſunt, &amp; </s>
  <s xml:id="echoid-s8662" xml:space="preserve">ſimiliter poſitæ circa communem diametrum I K H; </s>
  <s xml:id="echoid-s8663" xml:space="preserve">ergo earum di-<lb/>
<anchor type="note" xlink:label="note-0269-05a" xlink:href="note-0269-05"/>
ſtantiæ ſemper magis, ac magis diminuuntur quouſque ſint minores qualibet <lb/>recta linea data. </s>
  <s xml:id="echoid-s8664" xml:space="preserve">Quod erat faciendum.</s>
  <s xml:id="echoid-s8665" xml:space="preserve"/>
</p>
<div xml:id="echoid-div751" type="float" level="2" n="3">
<note position="right" xlink:label="note-0269-01" xlink:href="note-0269-01a" xml:space="preserve">Lem. 9. <lb/>huius.</note>
<note position="right" xlink:label="note-0269-02" xlink:href="note-0269-02a" xml:space="preserve">Prop. 10. <lb/>addit.</note>
<note position="right" xlink:label="note-0269-03" xlink:href="note-0269-03a" xml:space="preserve">11. lib. 1.</note>
<note position="right" xlink:label="note-0269-04" xlink:href="note-0269-04a" xml:space="preserve">Prop. 10. <lb/>huius.</note>
<note position="right" xlink:label="note-0269-05" xlink:href="note-0269-05a" xml:space="preserve">Propof. 7. <lb/>addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8666" xml:space="preserve">Data hyperbola Z duos conos ſimiles exhibere, vt idem planum in, <lb/>
<anchor type="note" xlink:label="note-0269-06a" xlink:href="note-0269-06"/>
eis efſiciat duas hyperbolas æquales, &amp; </s>
  <s xml:id="echoid-s8667" xml:space="preserve">ſimiles datæ, quæ aſymptoticæ <lb/>ſint, &amp; </s>
  <s xml:id="echoid-s8668" xml:space="preserve">ſibi ipſis ſemper viciniores fiant, non tamen interuallo minore <lb/>recta linea data.</s>
  <s xml:id="echoid-s8669" xml:space="preserve"/>
</p>
<div xml:id="echoid-div752" type="float" level="2" n="4">
<note position="right" xlink:label="note-0269-06" xlink:href="note-0269-06a" xml:space="preserve">PRO 1. <lb/>12. <lb/>Addit</note>
</div>
<p style="it">
  <s xml:id="echoid-s8670" xml:space="preserve">In quolibet plano fiat angulus H I M æqualis angulo inclinationis diametri, <lb/>&amp; </s>
  <s xml:id="echoid-s8671" xml:space="preserve">baſis datæ hyperboles Z, &amp; </s>
  <s xml:id="echoid-s8672" xml:space="preserve">per M I extenſo quolibet alio plano ducatur in <lb/>eo B I C perpendicularis ad M I K; </s>
  <s xml:id="echoid-s8673" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8674" xml:space="preserve">ſumpto quolibet puncto O in recta linea <lb/>I H producta, ducatur à puncto O in plano per O I B extenſo recta linea O A <lb/>parallela ipſi B I, &amp; </s>
  <s xml:id="echoid-s8675" xml:space="preserve">ſecetur O A æqualis ſemiſſi potentis figuram ſectionis Z, <lb/>cuius rectum latus ad tranſuerſum eandem proportionem habeat quàm quadra-<lb/>tum A O ad quadratum O H; </s>
  <s xml:id="echoid-s8676" xml:space="preserve">atque à puncto A àucatur recta linea A D G <lb/>parallela ipſi H I, &amp; </s>
  <s xml:id="echoid-s8677" xml:space="preserve">coniungatur A H, quæ ſecent rectam lineam G I in pun-<lb/>ctis G, &amp; </s>
  <s xml:id="echoid-s8678" xml:space="preserve">C, &amp; </s>
  <s xml:id="echoid-s8679" xml:space="preserve">ſectur recta linea G B æqualis G C iungaturq; </s>
  <s xml:id="echoid-s8680" xml:space="preserve">A B, &amp; </s>
  <s xml:id="echoid-s8681" xml:space="preserve">à <lb/>quolibet puncto D in recta A G ſumpto ducãtur in eodem plano A B C duæ re-<lb/>ctæ lineæ D E, &amp; </s>
  <s xml:id="echoid-s8682" xml:space="preserve">D F @ parallelæ lateribus A B, &amp; </s>
  <s xml:id="echoid-s8683" xml:space="preserve">A C; </s>
  <s xml:id="echoid-s8684" xml:space="preserve">eruntque triangula
<pb o="232" file="0270" n="270" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0270-01a" xlink:href="fig-0270-01"/>
A B C, &amp; </s>
  <s xml:id="echoid-s8685" xml:space="preserve">D E F ſimilia, &amp; </s>
  <s xml:id="echoid-s8686" xml:space="preserve">ſimiliter poſita: </s>
  <s xml:id="echoid-s8687" xml:space="preserve">poſtea in plano per B C, M K <lb/>ducto, diametris B C, &amp; </s>
  <s xml:id="echoid-s8688" xml:space="preserve">E F, fiant duo circuli B K C, E L F, qui ſint ba-<lb/>ſes duorum conorum, quorum vertices ſint A, &amp; </s>
  <s xml:id="echoid-s8689" xml:space="preserve">D, &amp; </s>
  <s xml:id="echoid-s8690" xml:space="preserve">in eorum ſuper ficie-<lb/>bus planum per H I, M K ductum efficiat ſectiones K H M, &amp; </s>
  <s xml:id="echoid-s8691" xml:space="preserve">L X S: </s>
  <s xml:id="echoid-s8692" xml:space="preserve">Dico <lb/>eas eſſe quæſitas. </s>
  <s xml:id="echoid-s8693" xml:space="preserve">Quoniam duo triangula A B C, D E F ſimilia, &amp; </s>
  <s xml:id="echoid-s8694" xml:space="preserve">ſimiliter <lb/>poſita in eodem ſunt plano, pariterque duo circuli baſium in vno plano exiſtunt; <lb/></s>
  <s xml:id="echoid-s8695" xml:space="preserve">
<anchor type="note" xlink:label="note-0270-01a" xlink:href="note-0270-01"/>
ergo duo coni A B C, &amp; </s>
  <s xml:id="echoid-s8696" xml:space="preserve">D E F ſimiles erunt; </s>
  <s xml:id="echoid-s8697" xml:space="preserve">poſtea quia triangula A B C, <lb/>&amp; </s>
  <s xml:id="echoid-s8698" xml:space="preserve">D E F ſimilia ſunt, &amp; </s>
  <s xml:id="echoid-s8699" xml:space="preserve">communis ſectionum diameter H X I æque inclina-<lb/>tur ad coincidentes baſes M K, S L, &amp; </s>
  <s xml:id="echoid-s8700" xml:space="preserve">axi communi A D G æquidiſtat, &amp; </s>
  <s xml:id="echoid-s8701" xml:space="preserve"><lb/>in angulis æqualibus intercipiunt G I communem portionem baſium triangulorum <lb/>
<anchor type="note" xlink:label="note-0270-02a" xlink:href="note-0270-02"/>
ſimilium per axes; </s>
  <s xml:id="echoid-s8702" xml:space="preserve">igitur hyperbolæ K H M, &amp; </s>
  <s xml:id="echoid-s8703" xml:space="preserve">L X S æquales ſunt, &amp; </s>
  <s xml:id="echoid-s8704" xml:space="preserve">ſimi-<lb/>les inter ſe, &amp; </s>
  <s xml:id="echoid-s8705" xml:space="preserve">earum figuris æqualia ſunt quadrata ex dupla interceptæ G I <lb/>deſcripta. </s>
  <s xml:id="echoid-s8706" xml:space="preserve">Secundò quia ( propter parallelas A O, &amp; </s>
  <s xml:id="echoid-s8707" xml:space="preserve">B C ) triangula H O A, <lb/>&amp; </s>
  <s xml:id="echoid-s8708" xml:space="preserve">A G C ſimilia ſunt; </s>
  <s xml:id="echoid-s8709" xml:space="preserve">igitur quadratum A G ad quadratum G C, ſeu ad re-<lb/>ctangulum B G C eandem proportionem habebit, quàm quadratum H O ad qua-<lb/>dratum O A, ſeu quàm latus tranſuerſum ad rectum figuræ Z; </s>
  <s xml:id="echoid-s8710" xml:space="preserve">ſed vt quadra-<lb/>
<anchor type="note" xlink:label="note-0270-03a" xlink:href="note-0270-03"/>
tum A G ad rectangulum B G C, ita eſt latus tranſuerſum ad rectum hyperbo-<lb/>les K H M; </s>
  <s xml:id="echoid-s8711" xml:space="preserve">igitur duæ hyperbolæ Z, &amp; </s>
  <s xml:id="echoid-s8712" xml:space="preserve">K H M, habent figurarum latera, <lb/>porportionalia; </s>
  <s xml:id="echoid-s8713" xml:space="preserve">ſuntq; </s>
  <s xml:id="echoid-s8714" xml:space="preserve">prædictæ figuræ æquales cum ſint æquales quadratis ex du-<lb/>plis ipsarum A O, &amp; </s>
  <s xml:id="echoid-s8715" xml:space="preserve">interceptæ G I: </s>
  <s xml:id="echoid-s8716" xml:space="preserve">quæ ſunt æquales in parallelogrammo G <lb/>O, &amp; </s>
  <s xml:id="echoid-s8717" xml:space="preserve">habent angulos à diametris, &amp; </s>
  <s xml:id="echoid-s8718" xml:space="preserve">baſibus contenti, æquales inter ſe: </s>
  <s xml:id="echoid-s8719" xml:space="preserve">erunt <lb/>
<anchor type="note" xlink:label="note-0270-04a" xlink:href="note-0270-04"/>
hyperbolæ K H M, &amp; </s>
  <s xml:id="echoid-s8720" xml:space="preserve">Z æquales, &amp; </s>
  <s xml:id="echoid-s8721" xml:space="preserve">ſimiles inter ſe: </s>
  <s xml:id="echoid-s8722" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8723" xml:space="preserve">propterea ſectio L X S, <lb/>quæ ſimilis, &amp; </s>
  <s xml:id="echoid-s8724" xml:space="preserve">æqualis oſtenſa eſt ipſi K H M, erit quoque æqualis, &amp; </s>
  <s xml:id="echoid-s8725" xml:space="preserve">ſimilis <lb/>eidem ſectioni Z. </s>
  <s xml:id="echoid-s8726" xml:space="preserve">Tertiò, quia in duobus conis ſimilibus, &amp; </s>
  <s xml:id="echoid-s8727" xml:space="preserve">ſimiliter poſitis <lb/>circa communem axim A D G, ſuperficies nunquàm conueniunt, propterea, <lb/>quod latera A B, &amp; </s>
  <s xml:id="echoid-s8728" xml:space="preserve">D E, à quibus generantur in tota reuolutione inter fc,
<pb o="233" file="0271" n="271" rhead="Conicor. Lib. VI."/>
parallela conſeruantur; </s>
  <s xml:id="echoid-s8729" xml:space="preserve">igitur duæ ſectiones K H M, &amp; </s>
  <s xml:id="echoid-s8730" xml:space="preserve">L X S, exiſtentes in <lb/>eodem plano ſecante duas ſuperficies, quæ licet in infinitum producantur vbique <lb/>ſeparatæ ſunt, erunt aſymptoticæ. </s>
  <s xml:id="echoid-s8731" xml:space="preserve">Quartò, quia duæ hyperbolæ H K M, &amp; </s>
  <s xml:id="echoid-s8732" xml:space="preserve">L <lb/>X S ſunt æquales, ſimiles, &amp; </s>
  <s xml:id="echoid-s8733" xml:space="preserve">ſimiliter poſitæ circa communem diametrum H X <lb/>
<anchor type="note" xlink:label="note-0271-01a" xlink:href="note-0271-01"/>
I, earum diſtantiæ ſemper magis, ac magis diminuuntur; </s>
  <s xml:id="echoid-s8734" xml:space="preserve">nunquam tamen mi-<lb/>nores efſici poſſunt interuallo duarum æquidiſtantium, hyperbolas continentium. <lb/></s>
  <s xml:id="echoid-s8735" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s8736" xml:space="preserve"/>
</p>
<div xml:id="echoid-div753" type="float" level="2" n="5">
  <figure xlink:label="fig-0270-01" xlink:href="fig-0270-01a">
    <image file="0270-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0270-01"/>
  </figure>
<note position="left" xlink:label="note-0270-01" xlink:href="note-0270-01a" xml:space="preserve">Lem. 9. <lb/>huius.</note>
<note position="left" xlink:label="note-0270-02" xlink:href="note-0270-02a" xml:space="preserve">Prop. 10. <lb/>add.</note>
<note position="left" xlink:label="note-0270-03" xlink:href="note-0270-03a" xml:space="preserve">12. lib. 1.</note>
<note position="left" xlink:label="note-0270-04" xlink:href="note-0270-04a" xml:space="preserve">10. 12. <lb/>huus.</note>
<note position="right" xlink:label="note-0271-01" xlink:href="note-0271-01a" xml:space="preserve">Prop. 7. <lb/>addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8737" xml:space="preserve">Data hyperbola X duos conos ſimiles exhibere vt idem planum in eis <lb/>
<anchor type="note" xlink:label="note-0271-02a" xlink:href="note-0271-02"/>
efficiat duas hyperbolas ſimiles, &amp; </s>
  <s xml:id="echoid-s8738" xml:space="preserve">æquales datæ, quæ aſymptoticæ ſint, <lb/>&amp; </s>
  <s xml:id="echoid-s8739" xml:space="preserve">ex vna parte ſibi ipſis viciniores fiant interuallo minori quolibet da-<lb/>to: </s>
  <s xml:id="echoid-s8740" xml:space="preserve">ex altera verò parte ad ſe ipſas propius accedant interuallo tamen <lb/>maiore dato: </s>
  <s xml:id="echoid-s8741" xml:space="preserve">oportet autem vt angulus ab aſymptotis ſectionis X con-<lb/>tentus ſit acutus.</s>
  <s xml:id="echoid-s8742" xml:space="preserve"/>
</p>
<div xml:id="echoid-div754" type="float" level="2" n="6">
<note position="right" xlink:label="note-0271-02" xlink:href="note-0271-02a" xml:space="preserve">PROP. <lb/>13. <lb/>Addit.</note>
</div>
  <figure>
    <image file="0271-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0271-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s8743" xml:space="preserve">In quolibet @l@no fiat angulus A d O æqualis angulo inclinationis diametri, <lb/>&amp; </s>
  <s xml:id="echoid-s8744" xml:space="preserve">baſis hyperb l@ X; </s>
  <s xml:id="echoid-s8745" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8746" xml:space="preserve">per o d extenſo quolibet alio planol, ducatur in eo re-<lb/>cta linea B d C perpendicularis ad O d G, &amp; </s>
  <s xml:id="echoid-s8747" xml:space="preserve">ſumpto quolibet alio puncto b in <lb/>recta linea G O in plano per B G C O ducto, centris d, &amp; </s>
  <s xml:id="echoid-s8748" xml:space="preserve">b deſcribantur duo
<pb o="234" file="0272" n="272" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0272-01a" xlink:href="fig-0272-01"/>
cireuli G C O B, &amp; </s>
  <s xml:id="echoid-s8749" xml:space="preserve">G Q P L ſe ſe contingentes in communi puncto G rectæ li-<lb/>neæ G O ducaturque diameter L b Q æquidiſtans ipſi B C: </s>
  <s xml:id="echoid-s8750" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8751" xml:space="preserve">vt latus rectum <lb/>ad tranſuer ſum ſectionis X, ita fiat quadratum G d ad quadratum d A; </s>
  <s xml:id="echoid-s8752" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8753" xml:space="preserve"><lb/>coniungantur rectæ lineæ A G, &amp; </s>
  <s xml:id="echoid-s8754" xml:space="preserve">A O, ducaturque ex puncto P recta linea P <lb/>N parallela ipſi O A occurrens G A in N, atque A, &amp; </s>
  <s xml:id="echoid-s8755" xml:space="preserve">N fiant vertices duorum <lb/>conorum A B C, &amp; </s>
  <s xml:id="echoid-s8756" xml:space="preserve">N L Q, &amp; </s>
  <s xml:id="echoid-s8757" xml:space="preserve">ſecetur D d æqualis ſemiſſi potentis figuram <lb/>ſectionis X; </s>
  <s xml:id="echoid-s8758" xml:space="preserve">ducaturque per punctum D planum E M F æquidiſtans plano com-<lb/>muni A G O per axes ducto, efficiens in conicis ſuperficiebus ſectiones H I K, &amp; </s>
  <s xml:id="echoid-s8759" xml:space="preserve"><lb/>T V c; </s>
  <s xml:id="echoid-s8760" xml:space="preserve">Dico eas eſſe hyperbolas quæſitas. </s>
  <s xml:id="echoid-s8761" xml:space="preserve">Quoniam propter parallelas A O, N <lb/>P eſt A G ad G O, vt N G ad G P, &amp; </s>
  <s xml:id="echoid-s8762" xml:space="preserve">ad ſemißes conſequentium, ſcilicet A G <lb/>ad G d, atque N G ad G b proportionales erunt, ideoque A d, N b erunt pa-<lb/>rallelæ, &amp; </s>
  <s xml:id="echoid-s8763" xml:space="preserve">A d ad d G, ſeu ad d C eſt vt N b ad b G, ſeu ad b Q; </s>
  <s xml:id="echoid-s8764" xml:space="preserve">eſtque <lb/>d C etiam parallela b Q; </s>
  <s xml:id="echoid-s8765" xml:space="preserve">ergo plana A B C, &amp; </s>
  <s xml:id="echoid-s8766" xml:space="preserve">N L Q parallela ſunt, &amp; </s>
  <s xml:id="echoid-s8767" xml:space="preserve"><lb/>anguli A d C, &amp; </s>
  <s xml:id="echoid-s8768" xml:space="preserve">N b Q æquales ſunt, atque triangula A d C, &amp; </s>
  <s xml:id="echoid-s8769" xml:space="preserve">N b Q <lb/>ſimilia crunt inter ſe; </s>
  <s xml:id="echoid-s8770" xml:space="preserve">ideoque circa angulos æquales C, &amp; </s>
  <s xml:id="echoid-s8771" xml:space="preserve">Q erit A C ad C d, <lb/>vt N Q ad Q b, &amp; </s>
  <s xml:id="echoid-s8772" xml:space="preserve">ad conſequentium duplas, ſcilicet A C ad C B, atq; </s>
  <s xml:id="echoid-s8773" xml:space="preserve">N Q <lb/>ad Q L proportionales erunt; </s>
  <s xml:id="echoid-s8774" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8775" xml:space="preserve">propterea triangula A B C, &amp; </s>
  <s xml:id="echoid-s8776" xml:space="preserve">N L Q ſimilia <lb/>exunt, &amp; </s>
  <s xml:id="echoid-s8777" xml:space="preserve">ſimiliter poſita, &amp; </s>
  <s xml:id="echoid-s8778" xml:space="preserve">inter ſe parallela; </s>
  <s xml:id="echoid-s8779" xml:space="preserve">ergo efficient in duobus planis A O <lb/>G, &amp; </s>
  <s xml:id="echoid-s8780" xml:space="preserve">M E F inter ſe æquidiſtantibus ſectionũ diametros I D, &amp; </s>
  <s xml:id="echoid-s8781" xml:space="preserve">V a parallelas <lb/>conorũ axibus A d, &amp; </s>
  <s xml:id="echoid-s8782" xml:space="preserve">N b, &amp; </s>
  <s xml:id="echoid-s8783" xml:space="preserve">inter ſe; </s>
  <s xml:id="echoid-s8784" xml:space="preserve">quare conſtituent cum ſectionũ baſibus
<pb o="235" file="0273" n="273" rhead="Conicor. Lib. VI."/>
soincidentibus angulos æquales I D H, &amp; </s>
  <s xml:id="echoid-s8785" xml:space="preserve">V a T &amp; </s>
  <s xml:id="echoid-s8786" xml:space="preserve">cum ipſis D d, &amp; </s>
  <s xml:id="echoid-s8787" xml:space="preserve">a b etiã <lb/>parallelis inter ſe continebunt angulos æquales I D d, &amp; </s>
  <s xml:id="echoid-s8788" xml:space="preserve">V a b, eruntque in-<lb/>terceptæ D d, a b æquales ( cum ſint latera oppoſita parallelogrammi D b); <lb/></s>
  <s xml:id="echoid-s8789" xml:space="preserve">
<anchor type="note" xlink:label="note-0273-01a" xlink:href="note-0273-01"/>
igitur hyperbole H I K, &amp; </s>
  <s xml:id="echoid-s8790" xml:space="preserve">T V e æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s8791" xml:space="preserve">ſimiles atq; </s>
  <s xml:id="echoid-s8792" xml:space="preserve">earum <lb/>figuris æqualia ſunt quadrata ex duplis interceptarum D d, &amp; </s>
  <s xml:id="echoid-s8793" xml:space="preserve">a b. </s>
  <s xml:id="echoid-s8794" xml:space="preserve">Et quia <lb/>triangula A G O, N G P ſunt ſimilia in eodem plano, ſuntque pariter duo cir-<lb/>culi baſium in vno plano extenſi; </s>
  <s xml:id="echoid-s8795" xml:space="preserve">igitur coni A B C, &amp; </s>
  <s xml:id="echoid-s8796" xml:space="preserve">N L Q ſimiles ſunt <lb/>
<anchor type="note" xlink:label="note-0273-02a" xlink:href="note-0273-02"/>
inter ſe. </s>
  <s xml:id="echoid-s8797" xml:space="preserve">Secundo quia vt quadratum A d ad rectangulum G d O, ſeu ad re-<lb/>ctangulum B d C ita eſt latus tranſuerſum ad rectum ſectionis H I K, &amp; </s>
  <s xml:id="echoid-s8798" xml:space="preserve">(ex <lb/>conſtructione) in eadem proportione erat latus tranſuer ſum ad rectum hyperbo-<lb/>les X, atque anguli I D K, &amp; </s>
  <s xml:id="echoid-s8799" xml:space="preserve">A d O æquales ſunt inter ſe (propterea quod <lb/>D I, d A parallelæ ſunt, pariterque D K, d O parallelæ ſunt inter ſe, cum <lb/>communes ſectiones ſint plani baſis, &amp; </s>
  <s xml:id="echoid-s8800" xml:space="preserve">duorum planorum æquidiſtantium K I <lb/>H, &amp; </s>
  <s xml:id="echoid-s8801" xml:space="preserve">O A G): </s>
  <s xml:id="echoid-s8802" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8803" xml:space="preserve">erat angulus inclinationis diametri, &amp; </s>
  <s xml:id="echoid-s8804" xml:space="preserve">baſis hyperbolæ X æ-<lb/>qualis angulo A d O; </s>
  <s xml:id="echoid-s8805" xml:space="preserve">igitur diametri ſectionum X, &amp; </s>
  <s xml:id="echoid-s8806" xml:space="preserve">H I K ad ſuas baſes <lb/>æque inclinantur, &amp; </s>
  <s xml:id="echoid-s8807" xml:space="preserve">habebant latera earundem figurarum proportionalia; </s>
  <s xml:id="echoid-s8808" xml:space="preserve">ſuntq; <lb/></s>
  <s xml:id="echoid-s8809" xml:space="preserve">prædictæ figuræ æquales, cum ſint æquales quadrato ex dupla interceptæ D d vt <lb/>dictum eſt: </s>
  <s xml:id="echoid-s8810" xml:space="preserve">igitur ſectiones H I K, &amp; </s>
  <s xml:id="echoid-s8811" xml:space="preserve">X ſimiles ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s8812" xml:space="preserve">æquales; </s>
  <s xml:id="echoid-s8813" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0273-03a" xlink:href="note-0273-03"/>
ideoque reliqua ſectio T V d, quæ æqualis, &amp; </s>
  <s xml:id="echoid-s8814" xml:space="preserve">congruens oſtenſa eſt ipſi H I K, <lb/>erit quoque ſimilis, &amp; </s>
  <s xml:id="echoid-s8815" xml:space="preserve">æqualis eidem hyperbolæ X. </s>
  <s xml:id="echoid-s8816" xml:space="preserve">Tertiò quoniam plana H I <lb/>K, &amp; </s>
  <s xml:id="echoid-s8817" xml:space="preserve">G A O æquidiſtantia ſunt, nunquam conuenient; </s>
  <s xml:id="echoid-s8818" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8819" xml:space="preserve">ideo plannum H I K <lb/>nunquam lateri A N G alterius plani occurret; </s>
  <s xml:id="echoid-s8820" xml:space="preserve">ſed ſuperficies conicæ ſe ſe tan-<lb/>tummodo tangunt in communi latere A N G, &amp; </s>
  <s xml:id="echoid-s8821" xml:space="preserve">alibi perpetuo ſeparatæ incedunt; <lb/></s>
  <s xml:id="echoid-s8822" xml:space="preserve">igitur duæ ſectiones H I K, &amp; </s>
  <s xml:id="echoid-s8823" xml:space="preserve">T V e in plano E I K exiſtentes, quæ infinitè <lb/>producuntur in ſuperficiebus conicis, nunquam ſe ſe mutuo ſecant; </s>
  <s xml:id="echoid-s8824" xml:space="preserve">igitur ſectio-<lb/>nes ipſæ aſymptoticæ ſunt. </s>
  <s xml:id="echoid-s8825" xml:space="preserve">Quartò ducantur rectæ lineæ G E, O F, P R tan-<lb/>gentes circulos in extremitatibus communis diametri G P O, quæ parallelæ erunt <lb/>inter ſe (cum perpendiculares ſint ad communem diametrum G P O): </s>
  <s xml:id="echoid-s8826" xml:space="preserve">poſtea <lb/>producantur plana E G A, F O A, R P N tangentia conos in lateribus G A, <lb/>O A, &amp; </s>
  <s xml:id="echoid-s8827" xml:space="preserve">P N, &amp; </s>
  <s xml:id="echoid-s8828" xml:space="preserve">extendantur quouſque ſecent planum conicæ ſectionis H I Kin <lb/>rectis lineis E S M, F M, R S. </s>
  <s xml:id="echoid-s8829" xml:space="preserve">Et quoniam duo plana æquidiſtantia G A O, <lb/>et E M F efficiunt in eodem plano E G A, vtrumque conum contingente, duas <lb/>rectas lineas G A, E M æquidiſtantes inter ſe: </s>
  <s xml:id="echoid-s8830" xml:space="preserve">pari ratione in plano tangente <lb/>F O A erunt rectæ lineæ F M, et O A parallelæ inter ſe: </s>
  <s xml:id="echoid-s8831" xml:space="preserve">ſimili modo in plano <lb/>R P N erunt P N, et R S inter ſe æquidiſtantes, cumque A O, et N P paral-<lb/>lelæ ſint, erunt quoque F M, et R S inter ſe æquidiſtantes; </s>
  <s xml:id="echoid-s8832" xml:space="preserve">ſuntque E M, et <lb/>M F aſymptoti continentes hyperbolen E I K pariterq; </s>
  <s xml:id="echoid-s8833" xml:space="preserve">rectæ lineæ E S, S R ſunt <lb/>
<anchor type="note" xlink:label="note-0273-04a" xlink:href="note-0273-04"/>
aſymptoti hyperboles T V e: </s>
  <s xml:id="echoid-s8834" xml:space="preserve">quare duæ hyperbolæ H I K, et T V e, ſimiles ei-<lb/>dem X, et æquales, &amp; </s>
  <s xml:id="echoid-s8835" xml:space="preserve">ſimiliter poſitæ, quarum duæ asymptoti F M, R S æqui-<lb/>diſtantes ſunt; </s>
  <s xml:id="echoid-s8836" xml:space="preserve">reliquæ verò E M, &amp; </s>
  <s xml:id="echoid-s8837" xml:space="preserve">E S coincidunt (cum exiſtant in eodem <lb/>plano tangente E A), &amp; </s>
  <s xml:id="echoid-s8838" xml:space="preserve">angulus ab eis contenctus E M F, vel E S R eſt acu-<lb/>tus (cum æqualis ſit acuto angulo ab asymptotis ſectionis X contento, propter ſi-<lb/>
<anchor type="note" xlink:label="note-0273-05a" xlink:href="note-0273-05"/>
militudinẽ ſectionũ, vt ab alĳs oſtenſum eſt): </s>
  <s xml:id="echoid-s8839" xml:space="preserve">poterit ergo duciramus breuiſſimus <lb/>in ſectione T V e adpartes V e qui æquidiſtãs ſit rectæ lineæ V I vertices ſectionũ <lb/>coniungenti: </s>
  <s xml:id="echoid-s8840" xml:space="preserve">eritque illius breuiſſimæ portio inter ſectiones compræhenſa diſtantia <lb/>
<anchor type="note" xlink:label="note-0273-06a" xlink:href="note-0273-06"/>
omniũ maxima; </s>
  <s xml:id="echoid-s8841" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8842" xml:space="preserve">propterea interualla ſectionũ ad vtraſq; </s>
  <s xml:id="echoid-s8843" xml:space="preserve">partes maximæ diſtã-<lb/>tiæ ſucceſſiuè diminuuntur, &amp; </s>
  <s xml:id="echoid-s8844" xml:space="preserve">ad partes æquidiſtantiũ asymptotorũ F M, R S dimi-
<pb o="236" file="0274" n="274" rhead="Apollonij Pergæi"/>
nuuntur quidem; </s>
  <s xml:id="echoid-s8845" xml:space="preserve">ſed non efficiuntur minora interuallo quo parallelæ asymptoti <lb/>diſtant inter ſe; </s>
  <s xml:id="echoid-s8846" xml:space="preserve">ex altera verò parte perueniri poteſt ad interuallum minus <lb/>quolibet dato. </s>
  <s xml:id="echoid-s8847" xml:space="preserve">Et hoc erat faciendum.</s>
  <s xml:id="echoid-s8848" xml:space="preserve"/>
</p>
<div xml:id="echoid-div755" type="float" level="2" n="7">
  <figure xlink:label="fig-0272-01" xlink:href="fig-0272-01a">
    <image file="0272-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0272-01"/>
  </figure>
<note position="right" xlink:label="note-0273-01" xlink:href="note-0273-01a" xml:space="preserve">Prop. 10. <lb/>addit. <lb/>huius.</note>
<note position="right" xlink:label="note-0273-02" xlink:href="note-0273-02a" xml:space="preserve">Lem. 9. <lb/>huius.</note>
<note position="right" xlink:label="note-0273-03" xlink:href="note-0273-03a" xml:space="preserve">10. 12. <lb/>huius.</note>
<note position="right" xlink:label="note-0273-04" xlink:href="note-0273-04a" xml:space="preserve">Maurol. <lb/>lib. 3. de <lb/>lin. horar. <lb/>ca. 6. 7.</note>
<note position="right" xlink:label="note-0273-05" xlink:href="note-0273-05a" xml:space="preserve">Propoſ. 6. <lb/>addit. <lb/>huius.</note>
<note position="right" xlink:label="note-0273-06" xlink:href="note-0273-06a" xml:space="preserve">Propoſ. 8. <lb/>addit. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s8849" xml:space="preserve">Data hyperbola eadem X præcedentis propoſitionis deſcribere duos ſi-<lb/>
<anchor type="note" xlink:label="note-0274-01a" xlink:href="note-0274-01"/>
miles conos, vt idem planum in eis efficiat duas hyperbolas ſimiles da-<lb/>tæ ſectioni, quæ asymptoticæ ſint, &amp; </s>
  <s xml:id="echoid-s8850" xml:space="preserve">ex vtraque parte ſibi ipſis vici-<lb/>niores fiant interuallo minori quolibet dato.</s>
  <s xml:id="echoid-s8851" xml:space="preserve"/>
</p>
<div xml:id="echoid-div756" type="float" level="2" n="8">
<note position="left" xlink:label="note-0274-01" xlink:href="note-0274-01a" xml:space="preserve">PROP. <lb/>14. Add.</note>
</div>
  <figure>
    <image file="0274-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0274-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s8852" xml:space="preserve">In quolibet plano fiat angulus A d G æqualis angulo inclinationis diametri, <lb/>&amp; </s>
  <s xml:id="echoid-s8853" xml:space="preserve">baſis hyperbolæ datæ X, &amp; </s>
  <s xml:id="echoid-s8854" xml:space="preserve">per G d extenſo quolibet alio plano, ducatur in <lb/>eo recta linea B d C perpendicularis ad G d O, &amp; </s>
  <s xml:id="echoid-s8855" xml:space="preserve">ſumpto quolibet alio puncto <lb/>b in recta linea B C in plano per B G O extenſo, centris d, &amp; </s>
  <s xml:id="echoid-s8856" xml:space="preserve">b, deſcribãtur <lb/>duo circuli inter ſe æquales G C O B, &amp; </s>
  <s xml:id="echoid-s8857" xml:space="preserve">S Q P L ſe ſe ſecantes in duobus punctis <lb/>R, a: </s>
  <s xml:id="echoid-s8858" xml:space="preserve">atq; </s>
  <s xml:id="echoid-s8859" xml:space="preserve">vt latus rectum ad tranſuerſum ſectionis datæ X, ita fiat quadratũ <lb/>G d ad quadratũ d A, &amp; </s>
  <s xml:id="echoid-s8860" xml:space="preserve">ducatur recta linea A N M parallela ipſi B C, quæ ſecet <lb/>b N æquidiſtantẽ d A in N, &amp; </s>
  <s xml:id="echoid-s8861" xml:space="preserve">coniungantur rectæ lineæ A B, A C, N L, N Q, <lb/>&amp; </s>
  <s xml:id="echoid-s8862" xml:space="preserve">fiant A, &amp; </s>
  <s xml:id="echoid-s8863" xml:space="preserve">N vertices duorũ conorũ A B C, N L Q, &amp; </s>
  <s xml:id="echoid-s8864" xml:space="preserve">in eorũ ſuper ficiebus <lb/>planum M c T æquidiſtans planis A G O, &amp; </s>
  <s xml:id="echoid-s8865" xml:space="preserve">N S P efficiat ſectiones H I K, <lb/>&amp; </s>
  <s xml:id="echoid-s8866" xml:space="preserve">T V c, quarum diametri D V I genitæ à triangulis A B C, &amp; </s>
  <s xml:id="echoid-s8867" xml:space="preserve">N L Q per <lb/>axes in eodem plano exiſbentibus ſunt æquidiſtantes axibus conorum A d, N b, <lb/>propter planorum æquidiſtantiam: </s>
  <s xml:id="echoid-s8868" xml:space="preserve">Dico, eas eſſe hyperbolas quæſitas. </s>
  <s xml:id="echoid-s8869" xml:space="preserve">Qnoniam <lb/>(propter æquidiſtantiam oppoſitarum linearum) eſt ſpatium A b parallelogram-<lb/>mum; </s>
  <s xml:id="echoid-s8870" xml:space="preserve">igitur conorum axes A d, N b æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s8871" xml:space="preserve">æquè inclinan-<lb/>tur ad communem rectam lineam B C Q (propter æquidiſtantiam earundem <lb/>A d, N b); </s>
  <s xml:id="echoid-s8872" xml:space="preserve">ſuntque æqualium circulorum radĳ d B, d C, b L, b Q æqua-<lb/>les inter ſe; </s>
  <s xml:id="echoid-s8873" xml:space="preserve">igitur triangula A B C, N L Q ſimilia ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s8874" xml:space="preserve">ſimili-
<pb o="237" file="0275" n="275" rhead="Conicor. Lib. VI."/>
ter poſita in eodem plano; </s>
  <s xml:id="echoid-s8875" xml:space="preserve">ſuntquè etiam duo circuli baſium in vno plano extenſi; <lb/></s>
  <s xml:id="echoid-s8876" xml:space="preserve">igitur coni A B C, &amp; </s>
  <s xml:id="echoid-s8877" xml:space="preserve">N L Q ſimiles ſunt inter ſe; </s>
  <s xml:id="echoid-s8878" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8879" xml:space="preserve">quoniam, vt latus <lb/>
<anchor type="note" xlink:label="note-0275-01a" xlink:href="note-0275-01"/>
tranſuerſum ad rectum ſectionis datæ X, ita eſt quadratum A d ad quadratum <lb/>radĳ G d, &amp; </s>
  <s xml:id="echoid-s8880" xml:space="preserve">ita eſt latus tranſuerſum ad rectum ſectionis H I K; </s>
  <s xml:id="echoid-s8881" xml:space="preserve">pariterque <lb/>vt quadratum N b ad quadratum radĳ L b ita eſt latus tranſuerſum ad rectũ <lb/>hyperbolæ T V c; </s>
  <s xml:id="echoid-s8882" xml:space="preserve">Et quadrata axium ad quadrata radiorum baſeos eandem <lb/>proportionem habet ideo latus tranſuerſum ad rectum ſectionis H I K eandem <lb/>proportionem habebit, quàm latus tranſuerſum ad rectum alterius ſectionis T <lb/>V c, ſeu eandem, quàm babet latus tranſuerſum ad rectum datæ ſectionis X; <lb/></s>
  <s xml:id="echoid-s8883" xml:space="preserve">atque diametri I V D, &amp; </s>
  <s xml:id="echoid-s8884" xml:space="preserve">diameter ſectionis X æquè inclinantur ad baſes, vt <lb/>dictum eſt; </s>
  <s xml:id="echoid-s8885" xml:space="preserve">igitur duæ ſectiones H I K, &amp; </s>
  <s xml:id="echoid-s8886" xml:space="preserve">T V c, nedum datæ hyperbolæ X; </s>
  <s xml:id="echoid-s8887" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0275-02a" xlink:href="note-0275-02"/>
ſed etiam inter ſe ſimiles ſunt. </s>
  <s xml:id="echoid-s8888" xml:space="preserve">Secundò quoniam duæ peripheriæ circulorum <lb/>baſium circa communem diametrum B C Q ſe ſe mutuo ſecant in duobus pun-<lb/>ctis R, &amp; </s>
  <s xml:id="echoid-s8889" xml:space="preserve">a, quæ neceſſario cadunt inter duas circulorum diametros G O, S P <lb/>perpendiculares ad communem diametrum B C Q; </s>
  <s xml:id="echoid-s8890" xml:space="preserve">igitur ſuperficies conorum <lb/>viciſſim ſe ſecant ſemper inter duo triangula, per conorum axes A G O, &amp; </s>
  <s xml:id="echoid-s8891" xml:space="preserve">N <lb/>S P, in reliquis autem locis ſeparatæ ſunt; </s>
  <s xml:id="echoid-s8892" xml:space="preserve">planum verò efficiens ſectiones H I <lb/>K, T V c cadit nõ inter axes A d, &amp; </s>
  <s xml:id="echoid-s8893" xml:space="preserve">N b; </s>
  <s xml:id="echoid-s8894" xml:space="preserve">igitur duæ ſectiones H I K, &amp; </s>
  <s xml:id="echoid-s8895" xml:space="preserve">T <lb/>V c exiſtentes in duabus conicis ſuperficiebus, non ſe ſecantibus, nunquàm con-<lb/>uenient, &amp; </s>
  <s xml:id="echoid-s8896" xml:space="preserve">asymptoticæ erunt. </s>
  <s xml:id="echoid-s8897" xml:space="preserve">Tertiò quoniam recta linea N A M per verti-<lb/>ces conorum ducta parallela eſt communi baſi B Q triangulorum per axes, &amp; </s>
  <s xml:id="echoid-s8898" xml:space="preserve"><lb/>ſecat diametrum communem D V I in M: </s>
  <s xml:id="echoid-s8899" xml:space="preserve">ergo (ſicuti oſtenſum eſt in prop. </s>
  <s xml:id="echoid-s8900" xml:space="preserve">10. <lb/></s>
  <s xml:id="echoid-s8901" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s8902" xml:space="preserve">huius) erit punctum M centrum ſectionis H I K, atq; </s>
  <s xml:id="echoid-s8903" xml:space="preserve">centrum alterius <lb/>ſectionis T V c; </s>
  <s xml:id="echoid-s8904" xml:space="preserve">ergo duæ ſectiones H I K, &amp; </s>
  <s xml:id="echoid-s8905" xml:space="preserve">T V c ſimiles ſunt inter ſe, <lb/>concentricæ, &amp; </s>
  <s xml:id="echoid-s8906" xml:space="preserve">ſimiliter poſitæ circa communem diametrum D V I; </s>
  <s xml:id="echoid-s8907" xml:space="preserve">igitur ſe-<lb/>
<anchor type="note" xlink:label="note-0275-03a" xlink:href="note-0275-03"/>
ctionum interualla ſemper magis, ac magis in infinitum minuuntur, &amp; </s>
  <s xml:id="echoid-s8908" xml:space="preserve">repe-<lb/>riri poßunt minora quolibet interuallo dato. </s>
  <s xml:id="echoid-s8909" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s8910" xml:space="preserve"/>
</p>
<div xml:id="echoid-div757" type="float" level="2" n="9">
<note position="right" xlink:label="note-0275-01" xlink:href="note-0275-01a" xml:space="preserve">Lem. 9. <lb/>huius.</note>
<note position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">Prop. 12. <lb/>huius.</note>
<note position="right" xlink:label="note-0275-03" xlink:href="note-0275-03a" xml:space="preserve">Propoſ. 9. <lb/>addit. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div759" type="section" level="1" n="236">
<head xml:id="echoid-head297" xml:space="preserve">SECTIO DECIMA <lb/>Continens Propoſit. XXVI. XXVII. <lb/>&amp; XXVIII. <lb/>PROPOSITIO XXVI.</head>
<p>
  <s xml:id="echoid-s8911" xml:space="preserve">IN cono recto, cuius triangulum per axim ſit A B C reperi-<lb/>re ſectionem datæ parabolæ D E æqualem, cuius axis E F, <lb/>&amp; </s>
  <s xml:id="echoid-s8912" xml:space="preserve">erectum E G.</s>
  <s xml:id="echoid-s8913" xml:space="preserve"/>
</p>
<pb o="238" file="0276" n="276" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s8914" xml:space="preserve">Vt quadratum A C ad C B in BA, <lb/>
<anchor type="figure" xlink:label="fig-0276-01a" xlink:href="fig-0276-01"/>
ita ponatur E G ad B H: </s>
  <s xml:id="echoid-s8915" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8916" xml:space="preserve">educa-<lb/>mus H I parallelam B C, &amp; </s>
  <s xml:id="echoid-s8917" xml:space="preserve">exten-<lb/>datur per H I planum eleuatum ſuper <lb/>triangulum A B C ad angulos rectos <lb/>efficiens in cono ſectionem K H L. <lb/></s>
  <s xml:id="echoid-s8918" xml:space="preserve">Dico eam æqualem eſſe ſectioni D E. </s>
  <s xml:id="echoid-s8919" xml:space="preserve"><lb/>Quia quadratum A C ad C B in B <lb/>A eſt, vt E G ad B H; </s>
  <s xml:id="echoid-s8920" xml:space="preserve">ergo poten-<lb/>tes eductæ ad axim H I in ſectione <lb/>
<anchor type="note" xlink:label="note-0276-01a" xlink:href="note-0276-01"/>
K H L poſſunt applicata contenta ab <lb/>abſciſſis illarum potentium, &amp; </s>
  <s xml:id="echoid-s8921" xml:space="preserve">ab E <lb/>G; </s>
  <s xml:id="echoid-s8922" xml:space="preserve">quare E G erit erectum ſectionis <lb/>K H, &amp; </s>
  <s xml:id="echoid-s8923" xml:space="preserve">idem etiam eſt erectum ſectionis D E; </s>
  <s xml:id="echoid-s8924" xml:space="preserve">ergo duo erecta duarum <lb/>ſectionum ſunt æqualia, &amp; </s>
  <s xml:id="echoid-s8925" xml:space="preserve">propterea ſectiones æquales ſunt (1. </s>
  <s xml:id="echoid-s8926" xml:space="preserve">ex 6.)</s>
  <s xml:id="echoid-s8927" xml:space="preserve"/>
</p>
<div xml:id="echoid-div759" type="float" level="2" n="1">
  <figure xlink:label="fig-0276-01" xlink:href="fig-0276-01a">
    <image file="0276-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0276-01"/>
  </figure>
<note position="right" xlink:label="note-0276-01" xlink:href="note-0276-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s8928" xml:space="preserve">Et dico, quod in cono A B C reperiri non poteſt ſectio alia parabo-<lb/>
<anchor type="note" xlink:label="note-0276-02a" xlink:href="note-0276-02"/>
lica, cuius vertex ſit ſuper A B, quæ eidem D E ſit æqualis. </s>
  <s xml:id="echoid-s8929" xml:space="preserve">Si enim <lb/>hoc eſt poſſibile, ſit axis illius ſectionis M N, qui quidem cadet in trian-<lb/>gulo A B C; </s>
  <s xml:id="echoid-s8930" xml:space="preserve">quia conus eſt rectus, &amp; </s>
  <s xml:id="echoid-s8931" xml:space="preserve">erectum illius ſit M O; </s>
  <s xml:id="echoid-s8932" xml:space="preserve">atq; </s>
  <s xml:id="echoid-s8933" xml:space="preserve">M O <lb/>ad M B erit, vt G E ad B H; </s>
  <s xml:id="echoid-s8934" xml:space="preserve">eſtque B H maior, quàm B M; </s>
  <s xml:id="echoid-s8935" xml:space="preserve">ergo M O <lb/>
<anchor type="note" xlink:label="note-0276-03a" xlink:href="note-0276-03"/>
minor eſt, quàm G E; </s>
  <s xml:id="echoid-s8936" xml:space="preserve">quare ſectio, cuius axis eſt M N non eſt æqualis <lb/>ſectioni D E; </s>
  <s xml:id="echoid-s8937" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8938" xml:space="preserve">tamen ſuppoſita fuit æqualis illi, quod eſt abſurdum. <lb/></s>
  <s xml:id="echoid-s8939" xml:space="preserve">Quare patet propoſitum.</s>
  <s xml:id="echoid-s8940" xml:space="preserve"/>
</p>
<div xml:id="echoid-div760" type="float" level="2" n="2">
<note position="right" xlink:label="note-0276-02" xlink:href="note-0276-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0276-03" xlink:href="note-0276-03a" xml:space="preserve">ex conu. <lb/>Prop. 1. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div762" type="section" level="1" n="237">
<head xml:id="echoid-head298" xml:space="preserve">PROPOSITIO XXVII.</head>
<p>
  <s xml:id="echoid-s8941" xml:space="preserve">SIt deinde hyperbole A B, cuius axis C D, inclinatus B <lb/>
<anchor type="note" xlink:label="note-0276-04a" xlink:href="note-0276-04"/>
D, &amp; </s>
  <s xml:id="echoid-s8942" xml:space="preserve">erectus B E; </s>
  <s xml:id="echoid-s8943" xml:space="preserve">atque quadratum axis F G dati coni <lb/>recti F H I ad quadratum G H ſemidiametri baſis eius, non <lb/>habeat maiorem proportionem, quàm habet figura, ſcilicet <lb/>quàm habet D B ad B E.</s>
  <s xml:id="echoid-s8944" xml:space="preserve"/>
</p>
<div xml:id="echoid-div762" type="float" level="2" n="1">
<note position="left" xlink:label="note-0276-04" xlink:href="note-0276-04a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s8945" xml:space="preserve">Sit prius proportio eadem, &amp; </s>
  <s xml:id="echoid-s8946" xml:space="preserve">producamus I F ad K; </s>
  <s xml:id="echoid-s8947" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8948" xml:space="preserve">ducamus K <lb/>L ſubtendentem angulum H F K, quæ parallela ſit ipſi F G, &amp; </s>
  <s xml:id="echoid-s8949" xml:space="preserve">æqualis <lb/>exiſtat ipſi D B; </s>
  <s xml:id="echoid-s8950" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8951" xml:space="preserve">per K L planum extendatur eleuatum ad angulos re-<lb/>ctos ſuper planum trianguli H F I, quod efficiet in ſuperſicie conica ſe-<lb/>ctionem hyperbolicam, cuius axis erit L M, &amp; </s>
  <s xml:id="echoid-s8952" xml:space="preserve">inclinatus K L. </s>
  <s xml:id="echoid-s8953" xml:space="preserve">Et quia <lb/>F G parallela eſt K L, erit quadratum F G ad G I in G H, vt K L in-<lb/>
<anchor type="note" xlink:label="note-0276-05a" xlink:href="note-0276-05"/>
clinatus ad illius erectum, ſiue vt D B ad B E; </s>
  <s xml:id="echoid-s8954" xml:space="preserve">facta autem fuit K L æ-<lb/>qualis D B; </s>
  <s xml:id="echoid-s8955" xml:space="preserve">ergo erectus inclinati K L æqualis eſt B E; </s>
  <s xml:id="echoid-s8956" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8957" xml:space="preserve">propterea ſe-<lb/>
<anchor type="note" xlink:label="note-0276-06a" xlink:href="note-0276-06"/>
<anchor type="note" xlink:label="note-0276-07a" xlink:href="note-0276-07"/>
ctio, cuius axis eſt L M æqualis eſt ſectioni A B. </s>
  <s xml:id="echoid-s8958" xml:space="preserve">Nec reperiri poterit <lb/>in cono H F I alia ſectio hyperbolica, cuius vertex ſit ſuper H F, quæ <lb/>æqualis ſit A B; </s>
  <s xml:id="echoid-s8959" xml:space="preserve">quia, ſi reperiri poſſet eſſet illius axis in plano trianguli <lb/>H F I, &amp; </s>
  <s xml:id="echoid-s8960" xml:space="preserve">eius inclinatus, ſubtendens angulum H F K æqualis eſſet D B, <lb/>nec tamen eſſet K L, nequè ipſi æquidiſtans (eo quod, ſi æquidiſtaret
<pb o="239" file="0277" n="277" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0277-01a" xlink:href="fig-0277-01"/>
ipſi K L, non eſſet eidem æqualis.) </s>
  <s xml:id="echoid-s8961" xml:space="preserve">His poſitis ſi educatur ex F linea ipſi <lb/>patallela cadet inter F G, F H, aut inter F I, F G; </s>
  <s xml:id="echoid-s8962" xml:space="preserve">ſitque F N; </s>
  <s xml:id="echoid-s8963" xml:space="preserve">igitur <lb/>
<anchor type="note" xlink:label="note-0277-01a" xlink:href="note-0277-01"/>
quadratum F N ad I N in N H eſt, vt D B ad B E: </s>
  <s xml:id="echoid-s8964" xml:space="preserve">quod eſt abſurdum; <lb/></s>
  <s xml:id="echoid-s8965" xml:space="preserve">quia quadratum F N maius eſt, quàm quadratum F G, &amp; </s>
  <s xml:id="echoid-s8966" xml:space="preserve">N H in N I <lb/>minus eſt, quàm quadratum G H.</s>
  <s xml:id="echoid-s8967" xml:space="preserve"/>
</p>
<div xml:id="echoid-div763" type="float" level="2" n="2">
<note position="left" xlink:label="note-0276-05" xlink:href="note-0276-05a" xml:space="preserve">12. lib. 1.</note>
<note position="left" xlink:label="note-0276-06" xlink:href="note-0276-06a" xml:space="preserve">2. huius.</note>
<note position="right" xlink:label="note-0276-07" xlink:href="note-0276-07a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0277-01" xlink:href="fig-0277-01a">
    <image file="0277-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0277-01"/>
  </figure>
<note position="right" xlink:label="note-0277-01" xlink:href="note-0277-01a" xml:space="preserve">12. lib. 1.</note>
</div>
<p>
  <s xml:id="echoid-s8968" xml:space="preserve">Poſtea habeat quadratum F G ad quadratum G H minorem propor-<lb/>tionem quàm babet D B ad B E; </s>
  <s xml:id="echoid-s8969" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8970" xml:space="preserve">circumſcribamus circa triangulum. <lb/></s>
  <s xml:id="echoid-s8971" xml:space="preserve">H F I circulum ; </s>
  <s xml:id="echoid-s8972" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8973" xml:space="preserve">producamus F G quouſque occurrat circuli circum-<lb/>ferentię in O; </s>
  <s xml:id="echoid-s8974" xml:space="preserve">ergo quadratum F G ad quadratum G H, nempe ad F G <lb/>in G O habet minorem proportionem, quàm D B ad B E: </s>
  <s xml:id="echoid-s8975" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8976" xml:space="preserve">ponamus <lb/>F G ad G P, vt D B ad B E ; </s>
  <s xml:id="echoid-s8977" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8978" xml:space="preserve">per P ducamus P Q parallellam H I ; </s>
  <s xml:id="echoid-s8979" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s8980" xml:space="preserve">coniungamus F R, F Q; </s>
  <s xml:id="echoid-s8981" xml:space="preserve">quæ occurrant H I in S, N: </s>
  <s xml:id="echoid-s8982" xml:space="preserve">quare D B ad <lb/>B E eſt, vt F G ad G P, quæ eſt, vt F N ad N Q; </s>
  <s xml:id="echoid-s8983" xml:space="preserve">nempe vt quadra-<lb/>tum F N ad F N in N Q æquale ipſi I N in N H, atque vt quadra-<lb/>tum F S ad F S in S R, nempe vt quadratum F S ad I S in S H; </s>
  <s xml:id="echoid-s8984" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8985" xml:space="preserve">edu-<lb/>camus T V, K L, quæ ſubtendant duos angulos H F K, I F T, &amp; </s>
  <s xml:id="echoid-s8986" xml:space="preserve">ſint <lb/>
<anchor type="note" xlink:label="note-0277-02a" xlink:href="note-0277-02"/>
parallelæ ipſis F N, &amp; </s>
  <s xml:id="echoid-s8987" xml:space="preserve">F S, &amp; </s>
  <s xml:id="echoid-s8988" xml:space="preserve">æquales ipſi D B; </s>
  <s xml:id="echoid-s8989" xml:space="preserve">igitur duo plana per K <lb/>
<anchor type="note" xlink:label="note-0277-03a" xlink:href="note-0277-03"/>
L, T V extenſa ſuper triangulum H F I ad angulos rectos eleuata, pro-<lb/>ducunt in cono H F I ſectiones hyperbolicas, quarum axes L M, V X, <lb/>&amp; </s>
  <s xml:id="echoid-s8990" xml:space="preserve">inclinati ipſarum L K, T V, &amp; </s>
  <s xml:id="echoid-s8991" xml:space="preserve">ſinguli earum ad ſuos erectos eandem <lb/>proportionem habent, quàm D B ad B E, &amp; </s>
  <s xml:id="echoid-s8992" xml:space="preserve">propterea figuræ ſectionum <lb/>
<anchor type="note" xlink:label="note-0277-04a" xlink:href="note-0277-04"/>
ſimiles ſunt, &amp; </s>
  <s xml:id="echoid-s8993" xml:space="preserve">æquales, ideoque ſectiones, quarum axes ſunt L M, V <lb/>X ſunt æquales ſectioni A B.</s>
  <s xml:id="echoid-s8994" xml:space="preserve"/>
</p>
<div xml:id="echoid-div764" type="float" level="2" n="3">
<note position="left" xlink:label="note-0277-02" xlink:href="note-0277-02a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0277-03" xlink:href="note-0277-03a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0277-04" xlink:href="note-0277-04a" xml:space="preserve">2. huius.</note>
</div>
<p>
  <s xml:id="echoid-s8995" xml:space="preserve">Nec reperitur ſectio præter iam dictas, cuius vertex ſit ſuper aliquam <lb/>
<anchor type="note" xlink:label="note-0277-05a" xlink:href="note-0277-05"/>
duarum linearum H F, F I, &amp; </s>
  <s xml:id="echoid-s8996" xml:space="preserve">ſit æqualis ſectioni A B. </s>
  <s xml:id="echoid-s8997" xml:space="preserve">Quia ſi reperiri <lb/>poſſet, caderet eius axis in planum trianguli H F I, illiuſque axi educa-<lb/>tur parallela F Z a, quæ non cadet ſuper F R, neque ſuper F Q, eritq; <lb/></s>
  <s xml:id="echoid-s8998" xml:space="preserve">quadratum F Z ad I Z in Z H, quod eſt æquale ipſi F Z in Z a, nempe <lb/>F Z ad Z a eandem proportionem haberet, quàm D B ad B E; </s>
  <s xml:id="echoid-s8999" xml:space="preserve">ſed D <lb/>B ad B E eſt, vt F G ad G P, nempe F Z ad Z b; </s>
  <s xml:id="echoid-s9000" xml:space="preserve">ergo proportio F Z
<pb o="240" file="0278" n="278" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0278-01a" xlink:href="fig-0278-01"/>
ad Z b, &amp; </s>
  <s xml:id="echoid-s9001" xml:space="preserve">ad Z a eſt eadem; </s>
  <s xml:id="echoid-s9002" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9003" xml:space="preserve">propterea Z b æqualis eſt Z a, quod eſt <lb/>abſurdum.</s>
  <s xml:id="echoid-s9004" xml:space="preserve"/>
</p>
<div xml:id="echoid-div765" type="float" level="2" n="4">
<note position="left" xlink:label="note-0277-05" xlink:href="note-0277-05a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0278-01" xlink:href="fig-0278-01a">
    <image file="0278-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0278-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s9005" xml:space="preserve">Ponamus iam quadratum F G ad G H in G I maiorem proportionem <lb/>habere, quàm D B ad B E. </s>
  <s xml:id="echoid-s9006" xml:space="preserve">Dico in cono H F I exhiberi non poſſe ſe-<lb/>ctionem æqualem hyperbolæ A B. </s>
  <s xml:id="echoid-s9007" xml:space="preserve">Si enim exhiberi poſſet illius axi ali-<lb/>qua parallela reperiretur vt F N: </s>
  <s xml:id="echoid-s9008" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9009" xml:space="preserve">quadratum F N ad I N in N H ma-<lb/>iorem proportionem habens, quàm quadratum F G ad quadratum G H, <lb/>erit vt D B ad B E; </s>
  <s xml:id="echoid-s9010" xml:space="preserve">quæ minor eſt proportione quadrati F G ad qua-<lb/>dratum G H: </s>
  <s xml:id="echoid-s9011" xml:space="preserve">quod eſt abſurdum. </s>
  <s xml:id="echoid-s9012" xml:space="preserve">Non ergo reperitur in cono H F I ſe-<lb/>ctio æqualis hyperbolæ A B. </s>
  <s xml:id="echoid-s9013" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s9014" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div767" type="section" level="1" n="238">
<head xml:id="echoid-head299" xml:space="preserve">PROPOSITIO XXVIII.</head>
<p>
  <s xml:id="echoid-s9015" xml:space="preserve">SIt iam ſectio elliptica A B, cuius axis tranſuerſus B D, &amp; </s>
  <s xml:id="echoid-s9016" xml:space="preserve"><lb/>erectus illius B E, &amp; </s>
  <s xml:id="echoid-s9017" xml:space="preserve">circa coni triangulum H F I deſcri-<lb/>
<anchor type="note" xlink:label="note-0278-01a" xlink:href="note-0278-01"/>
<anchor type="figure" xlink:label="fig-0278-02a" xlink:href="fig-0278-02"/>
<pb o="241" file="0279" n="279" rhead="Conicor. Lib. VI."/>
bamus circulum, &amp; </s>
  <s xml:id="echoid-s9018" xml:space="preserve">ex F ducamus lineam ad H I, occurrentem <lb/>ipſi extra circulum in K, &amp; </s>
  <s xml:id="echoid-s9019" xml:space="preserve">occurrat circulo in L, itaut ſit F K <lb/>ad K L, vt D B ad B E (&amp; </s>
  <s xml:id="echoid-s9020" xml:space="preserve">hoc eſt facile, vti demonſtraui-<lb/>mus in 59. </s>
  <s xml:id="echoid-s9021" xml:space="preserve">ex 1.)</s>
  <s xml:id="echoid-s9022" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s9023" xml:space="preserve">educamus in triangulo chordam M N <lb/>
<anchor type="note" xlink:label="note-0279-01a" xlink:href="note-0279-01"/>
parallelam F K, &amp; </s>
  <s xml:id="echoid-s9024" xml:space="preserve">æqualem D B; </s>
  <s xml:id="echoid-s9025" xml:space="preserve">Aio quod planum tranſiens <lb/>
<anchor type="note" xlink:label="note-0279-02a" xlink:href="note-0279-02"/>
per M N erectum ſuper triangulum coni producit in cono H F I <lb/>ſectionem ellipticam, æqualem ſectioni A B.</s>
  <s xml:id="echoid-s9026" xml:space="preserve"/>
</p>
<div xml:id="echoid-div767" type="float" level="2" n="1">
<note position="right" xlink:label="note-0278-01" xlink:href="note-0278-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0278-02" xlink:href="fig-0278-02a">
    <image file="0278-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0278-02"/>
  </figure>
<note position="left" xlink:label="note-0279-01" xlink:href="note-0279-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0279-02" xlink:href="note-0279-02a" xml:space="preserve">c</note>
</div>
<p>
  <s xml:id="echoid-s9027" xml:space="preserve">Quia D B tranſuerſus ad eius erectum B E eandem proportionem habe-<lb/>bat, quàm F K ad K L, nempe quàm quadratum F K habet ad F K in-<lb/>K L, quod eſt æquale ipſi I K in K H; </s>
  <s xml:id="echoid-s9028" xml:space="preserve">eſtque vt M N parallela ipſi F K <lb/>
<anchor type="note" xlink:label="note-0279-03a" xlink:href="note-0279-03"/>
ad illius erectum; </s>
  <s xml:id="echoid-s9029" xml:space="preserve">quare D B ad B E eandem proportionem habet, quàm <lb/>M N ad illius erectum; </s>
  <s xml:id="echoid-s9030" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9031" xml:space="preserve">M N æqualis eſt D B; </s>
  <s xml:id="echoid-s9032" xml:space="preserve">igitur figuræ dua-<lb/>
<anchor type="note" xlink:label="note-0279-04a" xlink:href="note-0279-04"/>
rum ſectionum A B D, M O N P ſunt æquales, &amp; </s>
  <s xml:id="echoid-s9033" xml:space="preserve">ſimiles, &amp; </s>
  <s xml:id="echoid-s9034" xml:space="preserve">ideo <lb/>
<anchor type="note" xlink:label="note-0279-05a" xlink:href="note-0279-05"/>
duæ illæ ſectiones ſunt æquales. </s>
  <s xml:id="echoid-s9035" xml:space="preserve">Dico inſuper, quod non reperitur in. <lb/></s>
  <s xml:id="echoid-s9036" xml:space="preserve">
<anchor type="note" xlink:label="note-0279-06a" xlink:href="note-0279-06"/>
cono H F I vlla alia ſectio elliptica, habens verticem ſuper F I, cuius <lb/>axis non æquidiſter alicui duarum F L K, quæ æqualis ſit eidem B A D. <lb/></s>
  <s xml:id="echoid-s9037" xml:space="preserve">Quia ſi poſſibile eſſet, oſtenderetur axis eius cadere in planum trianguli <lb/>H F I, quia ſectio eſt elliptica, &amp; </s>
  <s xml:id="echoid-s9038" xml:space="preserve">æqualis ſectioni A B, vtiq; </s>
  <s xml:id="echoid-s9039" xml:space="preserve">eius axis <lb/>occurret F I, F H, &amp; </s>
  <s xml:id="echoid-s9040" xml:space="preserve">æqualis eſt D B; </s>
  <s xml:id="echoid-s9041" xml:space="preserve">cumque vertex illius ſit ſuper F <lb/>I, non cadet axis eius ſuper M N, nec ipſi erit parallelus; </s>
  <s xml:id="echoid-s9042" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9043" xml:space="preserve">ideo edu-<lb/>cta F Q parallela axi eius non cadet F Q ſuper F K, &amp; </s>
  <s xml:id="echoid-s9044" xml:space="preserve">ſecabit arcum <lb/>F H in R; </s>
  <s xml:id="echoid-s9045" xml:space="preserve">eritque proportio axis illius ſectionis ad eius erectum, nempe <lb/>
<anchor type="note" xlink:label="note-0279-07a" xlink:href="note-0279-07"/>
quadratum F Q ad I Q in Q H, quod eſt æquale ipſi Q F in Q R, nẽ-<lb/>pe vt F Q ad Q R, ita erit D B ad B E, quæ eandem proportionem ha-<lb/>bet quàm F K ad K L, &amp; </s>
  <s xml:id="echoid-s9046" xml:space="preserve">diuidendo permutandoq; </s>
  <s xml:id="echoid-s9047" xml:space="preserve">F R maior ſubtenſa <lb/>
<anchor type="note" xlink:label="note-0279-08a" xlink:href="note-0279-08"/>
ad minorem F L eandem proportionem habebit, quàm R Q minor in-<lb/>tercepta ad maiorem K L; </s>
  <s xml:id="echoid-s9048" xml:space="preserve">quod eſt abſurdum: </s>
  <s xml:id="echoid-s9049" xml:space="preserve">non ergo reperitur in co-<lb/>no H F I ſectio elliptica, verticem habens in F I, quæ ſit æqualis ſe-<lb/>ctioni A B, præter ſuperius expoſitam. </s>
  <s xml:id="echoid-s9050" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s9051" xml:space="preserve"/>
</p>
<div xml:id="echoid-div768" type="float" level="2" n="2">
<note position="right" xlink:label="note-0279-03" xlink:href="note-0279-03a" xml:space="preserve">13. lib. 1.</note>
<note position="left" xlink:label="note-0279-04" xlink:href="note-0279-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0279-05" xlink:href="note-0279-05a" xml:space="preserve">2. huius.</note>
<note position="left" xlink:label="note-0279-06" xlink:href="note-0279-06a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0279-07" xlink:href="note-0279-07a" xml:space="preserve">13. lib. 1.</note>
<note position="left" xlink:label="note-0279-08" xlink:href="note-0279-08a" xml:space="preserve">f</note>
</div>
</div>
<div xml:id="echoid-div770" type="section" level="1" n="239">
<head xml:id="echoid-head300" xml:space="preserve">Notæ in Propoſit. XXVI.</head>
<p style="it">
  <s xml:id="echoid-s9052" xml:space="preserve">ERgo potentes egredientes ex ſe-<lb/>
<anchor type="note" xlink:label="note-0279-09a" xlink:href="note-0279-09"/>
<anchor type="figure" xlink:label="fig-0279-01a" xlink:href="fig-0279-01"/>
ctione L H K ad axim H I pote-<lb/>runt applicatum, quod continet ab-<lb/>ſciſſum illius potentis cum G E; </s>
  <s xml:id="echoid-s9053" xml:space="preserve">ergo <lb/>G E eſt erectus ſectionis L H; </s>
  <s xml:id="echoid-s9054" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9055" xml:space="preserve">eſt <lb/>etiã erectus ſectionis D E; </s>
  <s xml:id="echoid-s9056" xml:space="preserve">igitur duo <lb/>applicata duarum ſectionũ ſunt æqua-<lb/>lia, &amp; </s>
  <s xml:id="echoid-s9057" xml:space="preserve">ideo ſectio D E congruit ſe-<lb/>ctioni K H L, &amp; </s>
  <s xml:id="echoid-s9058" xml:space="preserve">propterea æquales <lb/>ſunt, &amp;</s>
  <s xml:id="echoid-s9059" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9060" xml:space="preserve">Ex eo quod quadratum A C <lb/>baſis trianguli per axim coni recti ad <lb/>rectangulum C B A, ſub eius lateribus
<pb o="242" file="0280" n="280" rhead="Apollonij Pergæi"/>
contentum, habet eandẽ rationem, quam <lb/>
<anchor type="figure" xlink:label="fig-0280-01a" xlink:href="fig-0280-01"/>
G E ad H B, ſufficienter deducitur, quod <lb/>G E ſit latus rectum tàm parabolæ L H <lb/>
<anchor type="note" xlink:label="note-0280-01a" xlink:href="note-0280-01"/>
K, quàm D E; </s>
  <s xml:id="echoid-s9061" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9062" xml:space="preserve">ideo erit parabole L <lb/>
<anchor type="note" xlink:label="note-0280-02a" xlink:href="note-0280-02"/>
H æqualis D E. </s>
  <s xml:id="echoid-s9063" xml:space="preserve">Non igitur neceſſe eſt, <lb/>vt rectangula ſub abſciſſis, &amp; </s>
  <s xml:id="echoid-s9064" xml:space="preserve">lateribus <lb/>rectis æqualibus oſtendãtur æqualia inter <lb/>ſe, &amp; </s>
  <s xml:id="echoid-s9065" xml:space="preserve">inde eliciatur æqualitas, &amp; </s>
  <s xml:id="echoid-s9066" xml:space="preserve">con-<lb/>gruentia ſectionum. </s>
  <s xml:id="echoid-s9067" xml:space="preserve">Quapropter caſu il-<lb/>la verba in Codice Arabico irrepſiße. <lb/></s>
  <s xml:id="echoid-s9068" xml:space="preserve">puto.</s>
  <s xml:id="echoid-s9069" xml:space="preserve"/>
</p>
<div xml:id="echoid-div770" type="float" level="2" n="1">
<note position="left" xlink:label="note-0279-09" xlink:href="note-0279-09a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0279-01" xlink:href="fig-0279-01a">
    <image file="0279-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0279-01"/>
  </figure>
  <figure xlink:label="fig-0280-01" xlink:href="fig-0280-01a">
    <image file="0280-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0280-01"/>
  </figure>
<note position="left" xlink:label="note-0280-01" xlink:href="note-0280-01a" xml:space="preserve">11. lib. 1.</note>
<note position="left" xlink:label="note-0280-02" xlink:href="note-0280-02a" xml:space="preserve">Propoſ. 1. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9070" xml:space="preserve">Et dico, quod non reperiatur in. <lb/></s>
  <s xml:id="echoid-s9071" xml:space="preserve">ſectione A B C alia ſectio parabolica; </s>
  <s xml:id="echoid-s9072" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0280-03a" xlink:href="note-0280-03"/>
quia ſi reperiretur, &amp;</s>
  <s xml:id="echoid-s9073" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9074" xml:space="preserve">Verba, quæ in hoc textu addidi ex ſerie demonſtra-<lb/>tionis facile colliguntur: </s>
  <s xml:id="echoid-s9075" xml:space="preserve">Sed animaduertendum eſt, quod ne dum in cono recto, <lb/>ſed in quolibet cono ſcaleno quomodolibet per axim ſecetur triangulo A B C, de-<lb/>ſignari poteſt in eius ſuper ficie parabole æqualis datæ D E.</s>
  <s xml:id="echoid-s9076" xml:space="preserve"/>
</p>
<div xml:id="echoid-div771" type="float" level="2" n="2">
<note position="right" xlink:label="note-0280-03" xlink:href="note-0280-03a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s9077" xml:space="preserve">Ducatur C P contingens circulum baſis in C, &amp; </s>
  <s xml:id="echoid-s9078" xml:space="preserve">in parabola D E ducatur <lb/>diameter E F, &amp; </s>
  <s xml:id="echoid-s9079" xml:space="preserve">contingens verticalis, quæ contineat angulum F E G æqua-<lb/>
<anchor type="note" xlink:label="note-0280-04a" xlink:href="note-0280-04"/>
lem angulo B C P; </s>
  <s xml:id="echoid-s9080" xml:space="preserve">ſitque G E latus rectum diametri F E; </s>
  <s xml:id="echoid-s9081" xml:space="preserve">atque vt quadratum <lb/>C A ad rectangulum C B A, ita fiat G E ad H B, &amp; </s>
  <s xml:id="echoid-s9082" xml:space="preserve">per H extendatur pla-<lb/>num L H K æquidiſtans plano per B C P ducto. </s>
  <s xml:id="echoid-s9083" xml:space="preserve">Dico ſectionem L H K eße pa-<lb/>rabolen quæſitam. </s>
  <s xml:id="echoid-s9084" xml:space="preserve">Quia plana æquidiſtantia L H K, &amp; </s>
  <s xml:id="echoid-s9085" xml:space="preserve">B C P efficiunt in cir-<lb/>culo baſis rectas P C, L K inter ſe parallelas, &amp; </s>
  <s xml:id="echoid-s9086" xml:space="preserve">in plano A B C efficiunt re-<lb/>ctas H I, B C inter ſe parallelas; </s>
  <s xml:id="echoid-s9087" xml:space="preserve">ergo anguli B C P, &amp; </s>
  <s xml:id="echoid-s9088" xml:space="preserve">H I L æquales ſunt, <lb/>ſed in parabola D E diameter E F eſſicit cum ordinatis ad eam applicatis angulos <lb/>æquales F E G, ſcilicet ei, qui cum tangente verticali conſtituit, ſeu angulo B C <lb/>
<anchor type="note" xlink:label="note-0280-05a" xlink:href="note-0280-05"/>
P; </s>
  <s xml:id="echoid-s9089" xml:space="preserve">ergo duarum ſectionum L H K, &amp; </s>
  <s xml:id="echoid-s9090" xml:space="preserve">D E, diametri H I, &amp; </s>
  <s xml:id="echoid-s9091" xml:space="preserve">E F æque ſunt <lb/>inclinatæ ad ſuas baſes, cumquè latus rectum parabolæ L H K ad H B ſit, vt <lb/>quadratum C A ad rectangulum C B A, ſeu vt G E ad H B; </s>
  <s xml:id="echoid-s9092" xml:space="preserve">igitur duo late-<lb/>ra recta ſimilium diametrorum I H, &amp; </s>
  <s xml:id="echoid-s9093" xml:space="preserve">F E ad H B eandem proportionem ha-<lb/>bent; </s>
  <s xml:id="echoid-s9094" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9095" xml:space="preserve">ideo æqualia ſunt inter ſe; </s>
  <s xml:id="echoid-s9096" xml:space="preserve">quare ſectiones ipſæ æquales, &amp; </s>
  <s xml:id="echoid-s9097" xml:space="preserve">congruen-<lb/>tes erunt. </s>
  <s xml:id="echoid-s9098" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s9099" xml:space="preserve"/>
</p>
<div xml:id="echoid-div772" type="float" level="2" n="3">
<note position="left" xlink:label="note-0280-04" xlink:href="note-0280-04a" xml:space="preserve">51. lib. 2.</note>
<note position="left" xlink:label="note-0280-05" xlink:href="note-0280-05a" xml:space="preserve">Conu. 46. <lb/>lib. 1.</note>
</div>
<note position="left" xml:space="preserve">10. huius.</note>
<p style="it">
  <s xml:id="echoid-s9100" xml:space="preserve">Multoties in eodem cono duæ parabolæ æquales ſnbcontrariæ duci poßunt, <lb/>vt Mydorgius demonſtrauit.</s>
  <s xml:id="echoid-s9101" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div774" type="section" level="1" n="240">
<head xml:id="echoid-head301" xml:space="preserve">Notæ in Propoſit. XXVII.</head>
<p style="it">
  <s xml:id="echoid-s9102" xml:space="preserve">DEinde ſit hyperbole, vt A B, &amp; </s>
  <s xml:id="echoid-s9103" xml:space="preserve">axis illius C D, &amp; </s>
  <s xml:id="echoid-s9104" xml:space="preserve">inclinatus B <lb/>
<anchor type="note" xlink:label="note-0280-07a" xlink:href="note-0280-07"/>
D, &amp; </s>
  <s xml:id="echoid-s9105" xml:space="preserve">erectus B E, ita vt non ſit proportio quadrati axis coni ad <lb/>quadratum dimidij diametri illius baſis, vt quadratum F G ad quadratum <lb/>G H, maior, quàm proportio figuræ ſectionis: </s>
  <s xml:id="echoid-s9106" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s9107" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9108" xml:space="preserve">Senſus huius propoſi-<lb/>tionis hic erit. </s>
  <s xml:id="echoid-s9109" xml:space="preserve">In cono recto F H I, cuius triangulum per axim H F I repe-<lb/>rire ſectionem æqualem hyperbole datæ A B, cuius tranſuerſus axis D B, &amp; </s>
  <s xml:id="echoid-s9110" xml:space="preserve"><lb/>latus rectum B E. </s>
  <s xml:id="echoid-s9111" xml:space="preserve">Oportet autem, vt quadratum F G axis dati coni ad qua-<lb/>dratum radĳ G H circuli baſis non habeant maiorem proportionem, quàm ha-
<pb o="243" file="0281" n="281" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0281-01a" xlink:href="fig-0281-01"/>
bent figuræ latera, ſcilicet, quàm habet D B ad B E. </s>
  <s xml:id="echoid-s9112" xml:space="preserve">At quomoao duci de-<lb/>beat ſubtenſa K L quæ æqualis ſit ipſi D B, &amp; </s>
  <s xml:id="echoid-s9113" xml:space="preserve">parallela alteri F G, oſtendetur <lb/>inferius.</s>
  <s xml:id="echoid-s9114" xml:space="preserve"/>
</p>
<div xml:id="echoid-div774" type="float" level="2" n="1">
<note position="right" xlink:label="note-0280-07" xlink:href="note-0280-07a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0281-01" xlink:href="fig-0281-01a">
    <image file="0281-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0281-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s9115" xml:space="preserve">Et non reperitur in cono H F I alia ſectio hyperbolica ſuper F H, &amp; </s>
  <s xml:id="echoid-s9116" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0281-01a" xlink:href="note-0281-01"/>
æqualis A B, &amp;</s>
  <s xml:id="echoid-s9117" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9118" xml:space="preserve">Addidi verba quæ ad huius textus integritatem facere vi-<lb/>debantur.</s>
  <s xml:id="echoid-s9119" xml:space="preserve"/>
</p>
<div xml:id="echoid-div775" type="float" level="2" n="2">
<note position="left" xlink:label="note-0281-01" xlink:href="note-0281-01a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s9120" xml:space="preserve">Et educamus T V, K L, quæ ſubtendant duos angulos L F K, I F <lb/>
<anchor type="note" xlink:label="note-0281-02a" xlink:href="note-0281-02"/>
T, &amp; </s>
  <s xml:id="echoid-s9121" xml:space="preserve">ſint parallelæ ipſis F N, F S, &amp; </s>
  <s xml:id="echoid-s9122" xml:space="preserve">æquales D B, &amp;</s>
  <s xml:id="echoid-s9123" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9124" xml:space="preserve">Quomodo au-<lb/>tem hoc fieri poſſit modo oſtendemus. </s>
  <s xml:id="echoid-s9125" xml:space="preserve">Sumatur in recta linea H F quodlibet <lb/>punctum c inter F, &amp; </s>
  <s xml:id="echoid-s9126" xml:space="preserve">H; </s>
  <s xml:id="echoid-s9127" xml:space="preserve">atque à puncto c ducatur recta linea c d parallela <lb/>ipſi F N, vel F S, quæ ſecet productionem alterius lateris I F in d, &amp; </s>
  <s xml:id="echoid-s9128" xml:space="preserve">quàm <lb/>proportionem habet c d ad D B, eandem habeat C F ad F L, &amp; </s>
  <s xml:id="echoid-s9129" xml:space="preserve">per punctum <lb/>L ducatur recta L K parallela ipſi c d. </s>
  <s xml:id="echoid-s9130" xml:space="preserve">Manifeſtum eſt c d ad L K eandem pro-<lb/>portionem habere, quàm c F ad F L, ſeu quàm c d ad B D; </s>
  <s xml:id="echoid-s9131" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9132" xml:space="preserve">ideo K L æ-<lb/>qualis erit B D, &amp; </s>
  <s xml:id="echoid-s9133" xml:space="preserve">ſubtendit angulum L F K, eſtque parallela ipſi c d, ſeu <lb/>ipſi F N, vel F S. </s>
  <s xml:id="echoid-s9134" xml:space="preserve">Et hoc erat faciendum.</s>
  <s xml:id="echoid-s9135" xml:space="preserve"/>
</p>
<div xml:id="echoid-div776" type="float" level="2" n="3">
<note position="left" xlink:label="note-0281-02" xlink:href="note-0281-02a" xml:space="preserve">c</note>
</div>
  <figure>
    <image file="0281-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0281-02"/>
  </figure>
<pb o="244" file="0282" n="282" rhead="Apollonij Pergæi"/>
  <figure>
    <image file="0282-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0282-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s9136" xml:space="preserve">Igitur duo plana tranſeuntia per K L, T V eleuata ſuper triangulum. <lb/></s>
  <s xml:id="echoid-s9137" xml:space="preserve">
<anchor type="note" xlink:label="note-0282-01a" xlink:href="note-0282-01"/>
H F I ad angulos rectos producunt in cono H F I duas ſectiones hypor-<lb/>bolicas, quarum axes L M, V X, &amp; </s>
  <s xml:id="echoid-s9138" xml:space="preserve">inclinati ipſarum L K, V T, &amp; </s>
  <s xml:id="echoid-s9139" xml:space="preserve"><lb/>ſingulì eorum ad ſuos erectos ſunt, vt D B ad B E; </s>
  <s xml:id="echoid-s9140" xml:space="preserve">ergo figuræ trium. <lb/></s>
  <s xml:id="echoid-s9141" xml:space="preserve">ſectionum ſunt ſimiles, &amp; </s>
  <s xml:id="echoid-s9142" xml:space="preserve">æquales; </s>
  <s xml:id="echoid-s9143" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9144" xml:space="preserve">propterea duæ ſectiones, qua-<lb/>rum axes ſunt L M, V X ſunt æquales ſectioni A B, &amp;</s>
  <s xml:id="echoid-s9145" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9146" xml:space="preserve">Ex textu men-<lb/>doſo expungi debent ſuperuacanea aliqua verba, ſicut in contextu habetur. </s>
  <s xml:id="echoid-s9147" xml:space="preserve"><lb/>Non enim verum eſt, quod duæ tantummodo hyperbole æquales eidem A B duci <lb/>poſſunt in cono recto H F I, vertices habentes in lateribus H F, &amp; </s>
  <s xml:id="echoid-s9148" xml:space="preserve">F I, ſed <lb/>quatuor inter ſe æquales eße poßunt; </s>
  <s xml:id="echoid-s9149" xml:space="preserve">nam ſuper latus F H duci poſſunt duæ <lb/>hyperbole, quarum axes tranſuerſi K L æquales ſint ipſi B D, &amp; </s>
  <s xml:id="echoid-s9150" xml:space="preserve">æquidiſtan-<lb/>tes ſint rectis lineis F N, &amp; </s>
  <s xml:id="echoid-s9151" xml:space="preserve">F S. </s>
  <s xml:id="echoid-s9152" xml:space="preserve">Quod ſic oſtendetur. </s>
  <s xml:id="echoid-s9153" xml:space="preserve">Quoniam recta linea <lb/>Q R ducta eſt parallela ipſi H I erunt duo arcus circuli intercepti H Q, I R <lb/>æquales inter ſe; </s>
  <s xml:id="echoid-s9154" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9155" xml:space="preserve">ideo duo anguli ad peripheriam H F Q, &amp; </s>
  <s xml:id="echoid-s9156" xml:space="preserve">I F R æquales <lb/>erunt inter ſe; </s>
  <s xml:id="echoid-s9157" xml:space="preserve">poſita autem fuit K L æqualis, &amp; </s>
  <s xml:id="echoid-s9158" xml:space="preserve">parallela ipſi F N; </s>
  <s xml:id="echoid-s9159" xml:space="preserve">igitur <lb/>duo anguli alterni K L F, &amp; </s>
  <s xml:id="echoid-s9160" xml:space="preserve">H F N æquales ſunt inter ſe: </s>
  <s xml:id="echoid-s9161" xml:space="preserve">pari ratione; </s>
  <s xml:id="echoid-s9162" xml:space="preserve">quia <lb/>reliqua K L ducta eſt parallela ipſi F S, erit angulus externus S F I æqualis <lb/>interno, &amp; </s>
  <s xml:id="echoid-s9163" xml:space="preserve">oppoſito, &amp; </s>
  <s xml:id="echoid-s9164" xml:space="preserve">ad eaſdem partes L K F; </s>
  <s xml:id="echoid-s9165" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9166" xml:space="preserve">ideo duo triangula L F K <lb/>habent angulum F, communem, &amp; </s>
  <s xml:id="echoid-s9167" xml:space="preserve">duos angolos in ſingulis triangulis K, &amp; </s>
  <s xml:id="echoid-s9168" xml:space="preserve"><lb/>L æquales; </s>
  <s xml:id="echoid-s9169" xml:space="preserve">igitur ſunt æquiangula, &amp; </s>
  <s xml:id="echoid-s9170" xml:space="preserve">ſimilia, &amp;</s>
  <s xml:id="echoid-s9171" xml:space="preserve">, vt antea dictum eſt, fieri <lb/>poſſunt duæ rectæ lineæ K L æquales eidem D B, &amp; </s>
  <s xml:id="echoid-s9172" xml:space="preserve">inter ſe: </s>
  <s xml:id="echoid-s9173" xml:space="preserve">ſi igitur per duas <lb/>rectas lineas K L ducantur plana perpendicularia ad planum trianguli per axim <lb/>H F I, eſſicientur in cono recto duæ hyperbole, quarum bini axes tranſuerſi K L <lb/>ſunt æquales: </s>
  <s xml:id="echoid-s9174" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9175" xml:space="preserve">quia, propter parallelas H I, Q R, eſt F N ad N Q ſeu qua-<lb/>dratum F N ad rectangulum F N Q vt F S æd S R ſeu vt quadratum F S ad <lb/>rectangum F S R; </s>
  <s xml:id="echoid-s9176" xml:space="preserve">ſed rectangulum H N I æquale eſt rectangulo F N Q, &amp; </s>
  <s xml:id="echoid-s9177" xml:space="preserve"><lb/>rectangulum H S I æquale eſt rectangulo F S R: </s>
  <s xml:id="echoid-s9178" xml:space="preserve">ergo quadratum F N ad re-<lb/>ctangulum H N I eandem proportionem habet, quàm quaàratum F S ad rectã-<lb/>gulum H S I; </s>
  <s xml:id="echoid-s9179" xml:space="preserve">eſtque latus tranſuerſum K L ad ſuum latus rectum, vt quadra-<lb/>
<anchor type="note" xlink:label="note-0282-02a" xlink:href="note-0282-02"/>
tum F N ad rectangulum H N I, pariterque latus tranſuerſum K L alterius <lb/>ſectionis ad ſuum latus rectum eſt vt quadratum F S ad rectangulum H S I: <lb/></s>
  <s xml:id="echoid-s9180" xml:space="preserve">
<anchor type="note" xlink:label="note-0282-03a" xlink:href="note-0282-03"/>
<pb o="245" file="0283" n="283" rhead="Conicor. Lib. VI."/>
igitur duo æqualia latera tranſuerſa K L ad ſua latera recta eandem proportio-<lb/>nem habent, &amp; </s>
  <s xml:id="echoid-s9181" xml:space="preserve">ideo huiuſmodi latera recta æqualia ſunt inter ſe; </s>
  <s xml:id="echoid-s9182" xml:space="preserve">ideoque duæ <lb/>hyperbole genitæ, habentes vertices in eodem latere F H, æquales ſunt inter ſe, <lb/>quas vocat Mydorgius ſubcontrarias. </s>
  <s xml:id="echoid-s9183" xml:space="preserve">Simili modo duæ aliæ hyperbole inter ſe, <lb/>
<anchor type="note" xlink:label="note-0283-01a" xlink:href="note-0283-01"/>
&amp; </s>
  <s xml:id="echoid-s9184" xml:space="preserve">prioribus æquales in eodem cono duci poßunt, vertices habentes in latere <lb/>F I.</s>
  <s xml:id="echoid-s9185" xml:space="preserve"/>
</p>
<div xml:id="echoid-div777" type="float" level="2" n="4">
<note position="right" xlink:label="note-0282-01" xlink:href="note-0282-01a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0282-02" xlink:href="note-0282-02a" xml:space="preserve">12. lib. 1.</note>
<note position="left" xlink:label="note-0282-03" xlink:href="note-0282-03a" xml:space="preserve">Ibidem.</note>
<note position="right" xlink:label="note-0283-01" xlink:href="note-0283-01a" xml:space="preserve">10. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9186" xml:space="preserve">Nec reperitur tertia, cuius vertex ſit ſuper aliqua duarum linearum <lb/>
<anchor type="note" xlink:label="note-0283-02a" xlink:href="note-0283-02"/>
H F.</s>
  <s xml:id="echoid-s9187" xml:space="preserve">, F I, &amp; </s>
  <s xml:id="echoid-s9188" xml:space="preserve">ſit æqualis ſectioni A B, quia, &amp; </s>
  <s xml:id="echoid-s9189" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9190" xml:space="preserve">Immutaui particulam, <lb/>quæ propoſitionem reddebat falſam, id quod colligitur ex conſtructione, &amp; </s>
  <s xml:id="echoid-s9191" xml:space="preserve">progreßu <lb/>demonſtrationis: </s>
  <s xml:id="echoid-s9192" xml:space="preserve">Quælibet enim alia ſectio, præter quatuor aſſignatas, habebit <lb/>axem æquidiſtantem alicui rectæ vt F Z, quæ cadit inter F N, &amp; </s>
  <s xml:id="echoid-s9193" xml:space="preserve">F S; </s>
  <s xml:id="echoid-s9194" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9195" xml:space="preserve"><lb/>hæc oſtendetur inæqualis prædictis ſectionibus, &amp; </s>
  <s xml:id="echoid-s9196" xml:space="preserve">ipſi A B.</s>
  <s xml:id="echoid-s9197" xml:space="preserve"/>
</p>
<div xml:id="echoid-div778" type="float" level="2" n="5">
<note position="left" xlink:label="note-0283-02" xlink:href="note-0283-02a" xml:space="preserve">e</note>
</div>
<p>
  <s xml:id="echoid-s9198" xml:space="preserve">Deinde ponamus quadratum F G ad GH maius, quàm D B ad B E. <lb/></s>
  <s xml:id="echoid-s9199" xml:space="preserve">
<anchor type="note" xlink:label="note-0283-03a" xlink:href="note-0283-03"/>
Dico, non reperiri in cono H F I ſectionem æqualem ſectioni A B: </s>
  <s xml:id="echoid-s9200" xml:space="preserve">nam, <lb/>ſi reperiretur, eſſet vel æqualis parallela ſuo axi, &amp; </s>
  <s xml:id="echoid-s9201" xml:space="preserve">erit quadratum N <lb/>F ad I N in N H, &amp;</s>
  <s xml:id="echoid-s9202" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9203" xml:space="preserve">Legendum eße vt in textu dixi conſtat ex progreſſis<unsure/> <lb/>totius propoſitionis. </s>
  <s xml:id="echoid-s9204" xml:space="preserve">I am facili negotio demonſtratio perfici poteſt, nam axis F <lb/>G minor eſt quàm F N, quæ ſubtendit angulum rectum G, quadratum vero <lb/>G H ſemiſſius totius H I maius eſt rectangulo I N H, ſub inæqualibus ſegmen-<lb/>tis contentum; </s>
  <s xml:id="echoid-s9205" xml:space="preserve">propterea quadratum F N ad rectangulum I N H maiorem pro-<lb/>portionem habebit, quàm quadratum G F ad quadratum G H: </s>
  <s xml:id="echoid-s9206" xml:space="preserve">eſtque D B ad <lb/>B E, vt quadratum F N ad rectangulum I N H; </s>
  <s xml:id="echoid-s9207" xml:space="preserve">propterea quod F N paral-<lb/>
<anchor type="note" xlink:label="note-0283-04a" xlink:href="note-0283-04"/>
lela eſt axi illius ſectionis, quæ poſita fuit æqualis A B; </s>
  <s xml:id="echoid-s9208" xml:space="preserve">igitur D B ad B E <lb/>maiorem proportionem habet, quàm quadratum F G ad quadratum G H; </s>
  <s xml:id="echoid-s9209" xml:space="preserve">quod <lb/>eſt contra hypotheſin: </s>
  <s xml:id="echoid-s9210" xml:space="preserve">habebat enim quadratum F G ad quadratum G H maio-<lb/>rem proportionem, quàm D B ad B E. </s>
  <s xml:id="echoid-s9211" xml:space="preserve">Non ergo reperitur in cono; </s>
  <s xml:id="echoid-s9212" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s9213" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s9214" xml:space="preserve"/>
</p>
<div xml:id="echoid-div779" type="float" level="2" n="6">
<note position="left" xlink:label="note-0283-03" xlink:href="note-0283-03a" xml:space="preserve">f</note>
<note position="right" xlink:label="note-0283-04" xlink:href="note-0283-04a" xml:space="preserve">12. lib. I.</note>
</div>
  <figure>
    <image file="0283-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0283-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s9215" xml:space="preserve">Sicutì in præcedenti propoſitione factum eſt, nedum in cono recto, ſed etiam <lb/>in quolibet cono ſcaleno, quomodolibet per axim ſectio à triangulo H F I deter-<lb/>minari poßet, quando, &amp; </s>
  <s xml:id="echoid-s9216" xml:space="preserve">quomodo in eo deſignari poſſet ſectio æqualis datæ hy-<lb/>perbole A B. </s>
  <s xml:id="echoid-s9217" xml:space="preserve">Quod ab alĳs factum eſt.</s>
  <s xml:id="echoid-s9218" xml:space="preserve"/>
</p>
<pb o="246" file="0284" n="284" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div781" type="section" level="1" n="241">
<head xml:id="echoid-head302" xml:space="preserve">Notæ in Propoſit. XXVIII.</head>
<p>
  <s xml:id="echoid-s9219" xml:space="preserve">DEinde ſit ſectio elliptica, vt A B, &amp; </s>
  <s xml:id="echoid-s9220" xml:space="preserve">axis eius tranſuerſus B D, &amp; </s>
  <s xml:id="echoid-s9221" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0284-01a" xlink:href="note-0284-01"/>
erectus illius B E; </s>
  <s xml:id="echoid-s9222" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9223" xml:space="preserve">ſit triãgulum coni H F I, &amp; </s>
  <s xml:id="echoid-s9224" xml:space="preserve">circumducamus <lb/>circa illum circulum, &amp; </s>
  <s xml:id="echoid-s9225" xml:space="preserve">educamus ex F lineam F L K occurrentem ipſi <lb/>extra circulum in K; </s>
  <s xml:id="echoid-s9226" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9227" xml:space="preserve">occurrat circulo in L ita vt ſit F K ad K L, vt <lb/>D B ad B E; </s>
  <s xml:id="echoid-s9228" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9229" xml:space="preserve">eſt facile ( vti demonſtrauimus in 59. </s>
  <s xml:id="echoid-s9230" xml:space="preserve">ex I.)</s>
  <s xml:id="echoid-s9231" xml:space="preserve">, &amp;</s>
  <s xml:id="echoid-s9232" xml:space="preserve">c.</s>
  <s xml:id="echoid-s9233" xml:space="preserve"/>
</p>
<div xml:id="echoid-div781" type="float" level="2" n="1">
<note position="right" xlink:label="note-0284-01" xlink:href="note-0284-01a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0284-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0284-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s9234" xml:space="preserve">Senſus propoſitionis hic erit. </s>
  <s xml:id="echoid-s9235" xml:space="preserve">In cono recto, cuius triangulum per axim H F I <lb/>reperire ſectionem æqualem datæ ellipſi A B, cuius axis tranſuerſus D B, &amp; </s>
  <s xml:id="echoid-s9236" xml:space="preserve"><lb/>latus rectum B E. </s>
  <s xml:id="echoid-s9237" xml:space="preserve">In conſtructione poſtea duci debet recta linea F L K extra <lb/>circulum, &amp; </s>
  <s xml:id="echoid-s9238" xml:space="preserve">triangulum ad vtraſque partes, alias conſtructio non eſſet perfecta.</s>
  <s xml:id="echoid-s9239" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s9240" xml:space="preserve">Lemma verò, quod repoſuiſſe, dicit Arabicus interpres in I. </s>
  <s xml:id="echoid-s9241" xml:space="preserve">libro, ab hoc <lb/>ſequenti for ſam diuerſum non erit.</s>
  <s xml:id="echoid-s9242" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div783" type="section" level="1" n="242">
<head xml:id="echoid-head303" xml:space="preserve">LEMMAX.</head>
<p style="it">
  <s xml:id="echoid-s9243" xml:space="preserve">SEcetur latus F I in S, vt ſit F I <lb/>
<anchor type="figure" xlink:label="fig-0284-02a" xlink:href="fig-0284-02"/>
ad I S in eadem ratione, quàm <lb/>habet axis tranſuerſus D B ad latus re-<lb/>ctum B E: </s>
  <s xml:id="echoid-s9244" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9245" xml:space="preserve">ducatur S L æquidiſtans <lb/>trianguli baſi H I, quæ ſecet circulum ex <lb/>vtraque parte in L, &amp; </s>
  <s xml:id="echoid-s9246" xml:space="preserve">coniungantur re-<lb/>ctæ lineæ F L, producanturque quoſquè <lb/>ſecent baſim H I in punctis K.</s>
  <s xml:id="echoid-s9247" xml:space="preserve"/>
</p>
<div xml:id="echoid-div783" type="float" level="2" n="1">
  <figure xlink:label="fig-0284-02" xlink:href="fig-0284-02a">
    <image file="0284-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0284-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9248" xml:space="preserve">Quoniam in triangulo F I K ducitur recta <lb/>linea S L æquidiſtans baſi I K, erit F I ad
<pb o="247" file="0285" n="285" rhead="Conicor. Lib. VI."/>
I S, vt F K ad K L: </s>
  <s xml:id="echoid-s9249" xml:space="preserve">ſed erat D B ad B E, vt F I ad I S; </s>
  <s xml:id="echoid-s9250" xml:space="preserve">igitur F K ad K L <lb/>eandem proportionem habebit: </s>
  <s xml:id="echoid-s9251" xml:space="preserve">quàm D B ad D E.</s>
  <s xml:id="echoid-s9252" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9253" xml:space="preserve">Et educamus in triangulo chordam M N parallelam K F, &amp; </s>
  <s xml:id="echoid-s9254" xml:space="preserve">æqualem <lb/>
<anchor type="note" xlink:label="note-0285-01a" xlink:href="note-0285-01"/>
D B, &amp;</s>
  <s xml:id="echoid-s9255" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9256" xml:space="preserve">Non vna, ſed duplex recta linea M N duci poteſt parallela cuilibet <lb/>duarum F K, quæ interius ſubtendat angulum verticis F trianguli H F I per <lb/>axim ducti. </s>
  <s xml:id="echoid-s9257" xml:space="preserve">Et poteſt etiam effici M N æqualis ipſi D B, vt in expoſitione præ-<lb/>cedentis propoſitionis oſtenſum eſt.</s>
  <s xml:id="echoid-s9258" xml:space="preserve"/>
</p>
<div xml:id="echoid-div784" type="float" level="2" n="2">
<note position="left" xlink:label="note-0285-01" xlink:href="note-0285-01a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s9259" xml:space="preserve">Itaque planum, tranſiens per M N, producit in cono H F I ſectionem <lb/>
<anchor type="note" xlink:label="note-0285-02a" xlink:href="note-0285-02"/>
ellipticam æqualem ſectioni A B; </s>
  <s xml:id="echoid-s9260" xml:space="preserve">quia, &amp;</s>
  <s xml:id="echoid-s9261" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9262" xml:space="preserve">Addidi verba, quæ in textu <lb/>deſiderantur, vt ſenſus perfectus ſit.</s>
  <s xml:id="echoid-s9263" xml:space="preserve"/>
</p>
<div xml:id="echoid-div785" type="float" level="2" n="3">
<note position="left" xlink:label="note-0285-02" xlink:href="note-0285-02a" xml:space="preserve">c</note>
</div>
<p>
  <s xml:id="echoid-s9264" xml:space="preserve">Ergo duæ illæ ſectiones ſunt æquales, &amp;</s>
  <s xml:id="echoid-s9265" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9266" xml:space="preserve">Concipi debet ſectio N O M <lb/>
<anchor type="note" xlink:label="note-0285-03a" xlink:href="note-0285-03"/>
P, duplex, quia nimirum duæ ſectiones ſub contrariæ, æquales ſunt, vt faci-<lb/>le cum Mydorgìo oſtendi poteſt.</s>
  <s xml:id="echoid-s9267" xml:space="preserve"/>
</p>
<div xml:id="echoid-div786" type="float" level="2" n="4">
<note position="left" xlink:label="note-0285-03" xlink:href="note-0285-03a" xml:space="preserve">d</note>
</div>
<p>
  <s xml:id="echoid-s9268" xml:space="preserve">Et dico, quod non reperiatur in cono H F I ſectio elliptica, habens <lb/>
<anchor type="note" xlink:label="note-0285-04a" xlink:href="note-0285-04"/>
verticem ſuper F I; </s>
  <s xml:id="echoid-s9269" xml:space="preserve">quia ſi poſſibile eſſet, &amp;</s>
  <s xml:id="echoid-s9270" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9271" xml:space="preserve">Textus valde corruptus ex-<lb/>poſito modo reſtitui debere conſtat ex progreſſu demonſtrationis.</s>
  <s xml:id="echoid-s9272" xml:space="preserve"/>
</p>
<div xml:id="echoid-div787" type="float" level="2" n="5">
<note position="left" xlink:label="note-0285-04" xlink:href="note-0285-04a" xml:space="preserve">e</note>
</div>
<p>
  <s xml:id="echoid-s9273" xml:space="preserve">Et diuidendo F R maior ad minorem R Q eſt vt F L minor ad maio-<lb/>
<anchor type="note" xlink:label="note-0285-05a" xlink:href="note-0285-05"/>
rem K L, &amp;</s>
  <s xml:id="echoid-s9274" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9275" xml:space="preserve">Supplendæ fuerunt particulæ aliquæ ad tollendam equiuocatio-<lb/>nem.</s>
  <s xml:id="echoid-s9276" xml:space="preserve"/>
</p>
<div xml:id="echoid-div788" type="float" level="2" n="6">
<note position="left" xlink:label="note-0285-05" xlink:href="note-0285-05a" xml:space="preserve">f</note>
</div>
</div>
<div xml:id="echoid-div790" type="section" level="1" n="243">
<head xml:id="echoid-head304" xml:space="preserve">SECTIO VNDECIMA</head>
<head xml:id="echoid-head305" xml:space="preserve">Continens Propoſit. XXIX. XXX. <lb/>&amp; XXXI.</head>
<head xml:id="echoid-head306" xml:space="preserve">PROPOSTIO XXIX.</head>
<p>
  <s xml:id="echoid-s9277" xml:space="preserve">DAto cono recto A B C, conum exhibere ei ſimilem, qui <lb/>datam ſectionem D E F contineat, cuius axis E G, &amp; </s>
  <s xml:id="echoid-s9278" xml:space="preserve"><lb/>erectus E H; </s>
  <s xml:id="echoid-s9279" xml:space="preserve">ſitque prius ſectio parabole.</s>
  <s xml:id="echoid-s9280" xml:space="preserve"/>
</p>
  <figure>
    <image file="0285-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0285-01"/>
  </figure>
<p>
  <s xml:id="echoid-s9281" xml:space="preserve">Super E G educatur planum ad ſectionem D <lb/>E F ad angulos rectos eleuatum, in quo duca-<lb/>tur E I K, quæ contineat cum E G angulum <lb/>æqualem ipſi angulo C: </s>
  <s xml:id="echoid-s9282" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9283" xml:space="preserve">ponamus E H ad E <lb/>
<anchor type="note" xlink:label="note-0285-06a" xlink:href="note-0285-06"/>
K, vt A C ad C B, &amp; </s>
  <s xml:id="echoid-s9284" xml:space="preserve">faciamus ſuper E K tri-<lb/>angulum E L K ſimile triangulo A B C, vt an-<lb/>gulus verticalis L æqualis ſit angulo B. </s>
  <s xml:id="echoid-s9285" xml:space="preserve">Facia-<lb/>mus etiam conum, cuius vertex ſit L, eiuſque <lb/>baſis circulus, cuius diameter ſit E K, qui ſit <lb/>eleuatus ſuper triangulum E L K ad angulos re-<lb/>ctos: </s>
  <s xml:id="echoid-s9286" xml:space="preserve">erit igitur angulus E K L æqualis ipſi C,
<pb o="248" file="0286" n="286" rhead="Apollonij Pergæi"/>
ſed angulus K E G factus fuit etiam eidẽ æqua-<lb/>
<anchor type="figure" xlink:label="fig-0286-01a" xlink:href="fig-0286-01"/>
lis; </s>
  <s xml:id="echoid-s9287" xml:space="preserve">igitur L K, quod eſt latus trianguli per a-<lb/>
<anchor type="note" xlink:label="note-0286-01a" xlink:href="note-0286-01"/>
xim coni tranſeuntis, parallelum erit ipſi E G: <lb/></s>
  <s xml:id="echoid-s9288" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9289" xml:space="preserve">propterea planum, in quo eſt ſectio D E F <lb/>
<anchor type="note" xlink:label="note-0286-02a" xlink:href="note-0286-02"/>
producit in cono ſectionem parabolicam; </s>
  <s xml:id="echoid-s9290" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9291" xml:space="preserve"><lb/>quia A C ad C B eſt, vt H E ad E K, &amp; </s>
  <s xml:id="echoid-s9292" xml:space="preserve">vt E <lb/>K ad K L; </s>
  <s xml:id="echoid-s9293" xml:space="preserve">igitur H E ad E L (quæ eſt æqualis <lb/>
<anchor type="note" xlink:label="note-0286-03a" xlink:href="note-0286-03"/>
ipſi K L) eandem proportionem habet, quàm <lb/>quadratum E K ad quadratum K L, nempe ad <lb/>K L in L E: </s>
  <s xml:id="echoid-s9294" xml:space="preserve">quaproptor H E eſt erectus ſectio-<lb/>
<anchor type="note" xlink:label="note-0286-04a" xlink:href="note-0286-04"/>
nis prouenientis in cono, ſed eſt etiam erectus <lb/>ſectionis D E F; </s>
  <s xml:id="echoid-s9295" xml:space="preserve">igitur D E F exiſtit in ſuperfi-<lb/>cie coni, cuius vertex eſt L, qui ſimilis eſt co-<lb/>
<anchor type="note" xlink:label="note-0286-05a" xlink:href="note-0286-05"/>
no A B C: </s>
  <s xml:id="echoid-s9296" xml:space="preserve">eo quod triangulum A B C ſimi-<lb/>le eſt triangulo E L K. </s>
  <s xml:id="echoid-s9297" xml:space="preserve">Dico etiam, quod ſectio D E F contineri non <lb/>poteſt ab aliquo alio cono, ſimili cono A B C, cuius vertex ſit ex eadẽ <lb/>parte ſectionis præter conum iam exhibitum. </s>
  <s xml:id="echoid-s9298" xml:space="preserve">Nam (ſi poſſibile eſt) ſit <lb/>conus habens verticem M, &amp; </s>
  <s xml:id="echoid-s9299" xml:space="preserve">triangulum eius erectum ſit ſuper planum <lb/>ſectionis D E F, &amp; </s>
  <s xml:id="echoid-s9300" xml:space="preserve">communis ſectio illius, &amp; </s>
  <s xml:id="echoid-s9301" xml:space="preserve">coni ſectionis erit axis eius; <lb/></s>
  <s xml:id="echoid-s9302" xml:space="preserve">eſtque E G illius axis; </s>
  <s xml:id="echoid-s9303" xml:space="preserve">ergo hæc eſt abſciſſio communis eorundem pla-<lb/>norum; </s>
  <s xml:id="echoid-s9304" xml:space="preserve">ſed eſt E G abſciſſio communis plani ſectionis, &amp; </s>
  <s xml:id="echoid-s9305" xml:space="preserve">plani trianguli <lb/>K E L, ſuper quod eſt etiam erectum; </s>
  <s xml:id="echoid-s9306" xml:space="preserve">igitur duo triangula E L K, E M <lb/>I ſunt in eodem plano, &amp; </s>
  <s xml:id="echoid-s9307" xml:space="preserve">angulus L æqualis eſt M (propter ſimilitudinẽ <lb/>
<anchor type="note" xlink:label="note-0286-06a" xlink:href="note-0286-06"/>
duorum conorum); </s>
  <s xml:id="echoid-s9308" xml:space="preserve">ergo E M eſt indirectum ipſi E L, &amp; </s>
  <s xml:id="echoid-s9309" xml:space="preserve">educta E K ad <lb/>
<anchor type="note" xlink:label="note-0286-07a" xlink:href="note-0286-07"/>
I ſectio D E F continebitur in cono, cuius vertex eſt M: </s>
  <s xml:id="echoid-s9310" xml:space="preserve">ſi autem pona-<lb/>
<anchor type="note" xlink:label="note-0286-08a" xlink:href="note-0286-08"/>
mus proportionem lineæ alicuius ad E M, eandem quàm habet quadra-<lb/>tum E I ad I M in M E, linea illa eſſet erectus ſectionis D E F; </s>
  <s xml:id="echoid-s9311" xml:space="preserve">ſed H <lb/>
<anchor type="note" xlink:label="note-0286-09a" xlink:href="note-0286-09"/>
E erat erectus ſectionis D E F; </s>
  <s xml:id="echoid-s9312" xml:space="preserve">igitur H E eſt illa linea, hæc autem ad <lb/>E L eandem proportionem habebat, quàm quadratum E K ad K L in <lb/>L E; </s>
  <s xml:id="echoid-s9313" xml:space="preserve">ergo quadratum E K ad K L in L E eandem proportionem habet, <lb/>quàm quadratũ E I ad I M in M E; </s>
  <s xml:id="echoid-s9314" xml:space="preserve">igitur H E ad E M, &amp; </s>
  <s xml:id="echoid-s9315" xml:space="preserve">ad E L ean-<lb/>dem proportionem habet: </s>
  <s xml:id="echoid-s9316" xml:space="preserve">quod eſt abſurdum. </s>
  <s xml:id="echoid-s9317" xml:space="preserve">Non ergo in aliquo alio <lb/>cono ſectio contineri poteſt, vt diximus. </s>
  <s xml:id="echoid-s9318" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s9319" xml:space="preserve"/>
</p>
<div xml:id="echoid-div790" type="float" level="2" n="1">
<note position="left" xlink:label="note-0285-06" xlink:href="note-0285-06a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0286-01" xlink:href="fig-0286-01a">
    <image file="0286-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0286-01"/>
  </figure>
<note position="right" xlink:label="note-0286-01" xlink:href="note-0286-01a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0286-02" xlink:href="note-0286-02a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0286-03" xlink:href="note-0286-03a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0286-04" xlink:href="note-0286-04a" xml:space="preserve">11. lib. 1.</note>
<note position="left" xlink:label="note-0286-05" xlink:href="note-0286-05a" xml:space="preserve">Def. 8. <lb/>huius.</note>
<note position="left" xlink:label="note-0286-06" xlink:href="note-0286-06a" xml:space="preserve">Def. 8.</note>
<note position="right" xlink:label="note-0286-07" xlink:href="note-0286-07a" xml:space="preserve">f</note>
<note position="left" xlink:label="note-0286-08" xlink:href="note-0286-08a" xml:space="preserve">Def. 9.</note>
<note position="left" xlink:label="note-0286-09" xlink:href="note-0286-09a" xml:space="preserve">11. lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div792" type="section" level="1" n="244">
<head xml:id="echoid-head307" xml:space="preserve">PROPOSITIO XXX.</head>
<p>
  <s xml:id="echoid-s9320" xml:space="preserve">SI ſectio hyperbolica D E F, cuius axis E G inclinatus E H, &amp; </s>
  <s xml:id="echoid-s9321" xml:space="preserve">erectus <lb/>
<anchor type="note" xlink:label="note-0286-10a" xlink:href="note-0286-10"/>
E I (oportet autem, vt quadratum axis B Q coni recti ad quadratũ ſe-<lb/>midiametri baſis illius A Q non maiorẽ proportionẽ habeat, quàm habent fi-<lb/>guræ latera). </s>
  <s xml:id="echoid-s9322" xml:space="preserve">Et habeat prius eandem proportionẽ, quàm H E ad E I, &amp; </s>
  <s xml:id="echoid-s9323" xml:space="preserve"><lb/>producamus A B ad M, &amp; </s>
  <s xml:id="echoid-s9324" xml:space="preserve">ſuper H E in plano erecto ad ſectionẽ D E F <lb/>deſcribamus ſegmentũ circuli E L H, quod capiat angulum æqualem an-<lb/>gulo M B C, &amp; </s>
  <s xml:id="echoid-s9325" xml:space="preserve">bifariam ſecemus arcum E O H in O, &amp; </s>
  <s xml:id="echoid-s9326" xml:space="preserve">educamus per-<lb/>pendicularem O N ſuper H E; </s>
  <s xml:id="echoid-s9327" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9328" xml:space="preserve">producamus illam, quouſque occur-
<pb o="249" file="0287" n="287" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0287-01a" xlink:href="fig-0287-01"/>
rat circumferentiæ in L, &amp; </s>
  <s xml:id="echoid-s9329" xml:space="preserve">iungamus E L, &amp; </s>
  <s xml:id="echoid-s9330" xml:space="preserve">L H, quæ occurrat in K <lb/>perpendiculari ex puncto E ſuper lineam E H. </s>
  <s xml:id="echoid-s9331" xml:space="preserve">Et quia E K parallela eſt <lb/>L O erit angulus K æqualis H L O, qui eſt ſemiſſis anguli H L E, &amp; </s>
  <s xml:id="echoid-s9332" xml:space="preserve">hic <lb/>eſt æqualis duobus angulis K, K E L; </s>
  <s xml:id="echoid-s9333" xml:space="preserve">igitur ſunt æquales; </s>
  <s xml:id="echoid-s9334" xml:space="preserve">quare K L E <lb/>eſt æquicrus, &amp; </s>
  <s xml:id="echoid-s9335" xml:space="preserve">angulus K L E æqualis eſt A B C; </s>
  <s xml:id="echoid-s9336" xml:space="preserve">quia angulus H L E <lb/>æqualis eſt M B C; </s>
  <s xml:id="echoid-s9337" xml:space="preserve">quapropter K L E ſimile eſt A B C, quia æqualia <lb/>
<anchor type="note" xlink:label="note-0287-01a" xlink:href="note-0287-01"/>
crura etiam habet! Si autem ponamus K L E triangulum coni, cuius <lb/>vertex L, &amp; </s>
  <s xml:id="echoid-s9338" xml:space="preserve">planum illius trianguli erectum ad planum D E F; </s>
  <s xml:id="echoid-s9339" xml:space="preserve">vtique <lb/>planum ſectionis producit in cono hyperbolen, cuius axis E G, inclina-<lb/>tus E H; </s>
  <s xml:id="echoid-s9340" xml:space="preserve">eo quod ſi educamus L P, B Q perpendiculares in duobus <lb/>triangulis, habebit quadratum B Q ad C Q in Q A (quod eſt vt H E <lb/>ad E I) eandem proportionem, quàm quadratum L P ad P K in P E: <lb/></s>
  <s xml:id="echoid-s9341" xml:space="preserve">quare potentes æductæ in illa ſectione ad axim E G, poterunt compa-<lb/>rata, applicata ad E I erectum; </s>
  <s xml:id="echoid-s9342" xml:space="preserve">ſed potentes, eductæ in ſectione D E F, <lb/>
<anchor type="note" xlink:label="note-0287-02a" xlink:href="note-0287-02"/>
poſſunt quoque illa applicata; </s>
  <s xml:id="echoid-s9343" xml:space="preserve">ergo ſectio D E F æqualis eſt ſectioni, <lb/>prouenienti in cono, cuius vertex eſt L, &amp; </s>
  <s xml:id="echoid-s9344" xml:space="preserve">exiſtit in eodem plano, ha-<lb/>betque eundem axim: </s>
  <s xml:id="echoid-s9345" xml:space="preserve">quare conus, cuius vertex L continet ſectionem <lb/>
<anchor type="note" xlink:label="note-0287-03a" xlink:href="note-0287-03"/>
D E F, &amp; </s>
  <s xml:id="echoid-s9346" xml:space="preserve">eſt ſimilis cono A B C.</s>
  <s xml:id="echoid-s9347" xml:space="preserve"/>
</p>
<div xml:id="echoid-div792" type="float" level="2" n="1">
<note position="right" xlink:label="note-0286-10" xlink:href="note-0286-10a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0287-01" xlink:href="fig-0287-01a">
    <image file="0287-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0287-01"/>
  </figure>
<note position="left" xlink:label="note-0287-01" xlink:href="note-0287-01a" xml:space="preserve">c</note>
<note position="right" xlink:label="note-0287-02" xlink:href="note-0287-02a" xml:space="preserve">12. lib. 1.</note>
<note position="right" xlink:label="note-0287-03" xlink:href="note-0287-03a" xml:space="preserve">Defin. 9.</note>
</div>
<p>
  <s xml:id="echoid-s9348" xml:space="preserve">Dico rurſus, quod nullus alius conus ſimilis cono A B C, cuius ver-<lb/>tex ſit in ea parte, in qua eſt L, præter iam dictum, continebit hanc <lb/>eandem ſectionem. </s>
  <s xml:id="echoid-s9349" xml:space="preserve">Si enim hoc verum non eſt, contineat illam alius <lb/>
<anchor type="note" xlink:label="note-0287-04a" xlink:href="note-0287-04"/>
conus ſimilis cono A B C, cuius vertex R in plano L E G; </s>
  <s xml:id="echoid-s9350" xml:space="preserve">atque latera <lb/>illius ſint E R, R T. </s>
  <s xml:id="echoid-s9351" xml:space="preserve">Quia angulus E R T æqualis eſt E L K, &amp; </s>
  <s xml:id="echoid-s9352" xml:space="preserve">eorum <lb/>conſequentes æquales inter ſe in eodem circuli ſegmento E L H exiſtent, <lb/>eo quod T R produſta occurrit axi tranſuerſo E H in H, &amp; </s>
  <s xml:id="echoid-s9353" xml:space="preserve">iungamus R <lb/>O, &amp; </s>
  <s xml:id="echoid-s9354" xml:space="preserve">ex E educamus E T, quæ ſit parallela coniunctæ rectæ lineæ O R; <lb/></s>
  <s xml:id="echoid-s9355" xml:space="preserve">vnde angulus O R H æqualis eſt O R E) propter æqualitatem arcuum <lb/>ſuorum, &amp; </s>
  <s xml:id="echoid-s9356" xml:space="preserve">ſunt æquales duobus angulis R T E, R E T, ergo E R T eſt <lb/>æquicrus, &amp; </s>
  <s xml:id="echoid-s9357" xml:space="preserve">angulus T R E æqualis eſt A B C: </s>
  <s xml:id="echoid-s9358" xml:space="preserve">educatur iam R S pa-<lb/>rallela H E, tunc quadratum R S ad T S in S E eandem proportionem <lb/>habebit, quàm E H inclinatus ſectionis D E F ad E I erectum illius; </s>
  <s xml:id="echoid-s9359" xml:space="preserve">eo <lb/>quod ſectionem D E F continet conus, cuius vertex eſt R; </s>
  <s xml:id="echoid-s9360" xml:space="preserve">ſed H E ad
<pb o="250" file="0288" n="288" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0288-01a" xlink:href="fig-0288-01"/>
E I eandem proportionẽ habet, quàm quadratum B Q ad C Q in Q A <lb/>eſtq; </s>
  <s xml:id="echoid-s9361" xml:space="preserve">C Q æqualis Q A, atq; </s>
  <s xml:id="echoid-s9362" xml:space="preserve">T S æqualis S E, &amp; </s>
  <s xml:id="echoid-s9363" xml:space="preserve">T S ad S E eandẽ pro-<lb/>
<anchor type="note" xlink:label="note-0288-01a" xlink:href="note-0288-01"/>
portionẽ habet, quã T R ad R H, ſeu quàm E V ad V H; </s>
  <s xml:id="echoid-s9364" xml:space="preserve">igitur E V æqua-<lb/>lis eſt V H; </s>
  <s xml:id="echoid-s9365" xml:space="preserve">quod eſt abſurdum; </s>
  <s xml:id="echoid-s9366" xml:space="preserve">propterea quo L O diameter, quæ ad illã <lb/>perpendicularis eſt, bifariam ſecat eam in N. </s>
  <s xml:id="echoid-s9367" xml:space="preserve">Oſtenſum igitur eſt, non repe-<lb/>riri conum alium continentem ſectionem D E F, præter ſuperius expoſi-<lb/>tum. </s>
  <s xml:id="echoid-s9368" xml:space="preserve">Tandem ſupponamus, quadratum B Q ad quadratum Q A habere <lb/>minorem proportionem, quàm E H ad E I. </s>
  <s xml:id="echoid-s9369" xml:space="preserve">Patet quadratum L P, nẽ-<lb/>
<anchor type="note" xlink:label="note-0288-02a" xlink:href="note-0288-02"/>
pe N E, ſeu O N in N L ad quadratum E P, nempe ad quadratum N <lb/>L, ſcilicet O N ad N L habere minorem proportionem, quàm H E ad <lb/>E I: </s>
  <s xml:id="echoid-s9370" xml:space="preserve">ponamus iam O N ad N X, vt H E ad E I, &amp; </s>
  <s xml:id="echoid-s9371" xml:space="preserve">per X ducamus R <lb/>X Y parallelam H E, &amp; </s>
  <s xml:id="echoid-s9372" xml:space="preserve">iungamus E R, O R, &amp; </s>
  <s xml:id="echoid-s9373" xml:space="preserve">H R producatur ad T <lb/>quouſque ſecet E T parallelam ipſi O R. </s>
  <s xml:id="echoid-s9374" xml:space="preserve">Oſtendetur (quemadmodum <lb/>
<anchor type="note" xlink:label="note-0288-03a" xlink:href="note-0288-03"/>
ſupra dictum eſt) quod E T R, B A C ſunt iſoſcelia, &amp; </s>
  <s xml:id="echoid-s9375" xml:space="preserve">ſimilia. </s>
  <s xml:id="echoid-s9376" xml:space="preserve">Et quia <lb/>E H ad E I eſt vt O N ad N X; </s>
  <s xml:id="echoid-s9377" xml:space="preserve">nempe vt O V ad V R, nempe vt O V <lb/>in V R, quod eſt æquale ipſi E V in V H ad quadratum V R; </s>
  <s xml:id="echoid-s9378" xml:space="preserve">hæc au-<lb/>tem proportio componitur ex E V, nempe S R ad V R, nempe ad E S, <lb/>&amp; </s>
  <s xml:id="echoid-s9379" xml:space="preserve">ex proportione V H ad V R, nempe S R ad S T, ex quibus compo-<lb/>nitur proportio quadrati R S ad S T in S E; </s>
  <s xml:id="echoid-s9380" xml:space="preserve">igitur quadratum R S ad E <lb/>S in S T eandẽ proportionem habet, quàm H E ad E I; </s>
  <s xml:id="echoid-s9381" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9382" xml:space="preserve">propterea <lb/>planum ſectionis D E F in cono, cuius vertex eſt R, &amp; </s>
  <s xml:id="echoid-s9383" xml:space="preserve">illius trianguli <lb/>latera R E, R T, producit ſectionem hyperbolicam, cuius inclinatus eſt <lb/>E H, &amp; </s>
  <s xml:id="echoid-s9384" xml:space="preserve">erectus E I; </s>
  <s xml:id="echoid-s9385" xml:space="preserve">quare conus cuius vertex eſt R, continet ſectionẽ D E <lb/>F, nec non continet illam alius conus, huic cono ſimilis, cuius vertex <lb/>eſt Y; </s>
  <s xml:id="echoid-s9386" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9387" xml:space="preserve">hi duo coni ſunt ſimiles cono A B C, nec continet illam ter-<lb/>tius alius conus, qui ſimilis ſit cono A B C, nam (ſi hoc ſieri poſſibile <lb/>eſt) contineat illam alius conus, cuius vertex Z, &amp; </s>
  <s xml:id="echoid-s9388" xml:space="preserve">punctum verticis <lb/>illius incidet in arcum E L H, &amp; </s>
  <s xml:id="echoid-s9389" xml:space="preserve">iungamus O Z, quæ ſecet H E in e: <lb/></s>
  <s xml:id="echoid-s9390" xml:space="preserve">
<anchor type="note" xlink:label="note-0288-04a" xlink:href="note-0288-04"/>
<pb o="251" file="0289" n="289" rhead="Conicor. Lib. VI."/>
Inde demonſtrabitur, quod H E ad E I habebit neceſſario eandem pro-<lb/>portionem, quàm O e ad e Z; </s>
  <s xml:id="echoid-s9391" xml:space="preserve">quod eſt abſurdum, quia haberet eandem <lb/>proportionem, quàm O N ad N X. </s>
  <s xml:id="echoid-s9392" xml:space="preserve">Quapropter non continet illam ter-<lb/>tius alius conus ſimilis cono A B C.</s>
  <s xml:id="echoid-s9393" xml:space="preserve"/>
</p>
<div xml:id="echoid-div793" type="float" level="2" n="2">
<note position="right" xlink:label="note-0287-04" xlink:href="note-0287-04a" xml:space="preserve">d</note>
  <figure xlink:label="fig-0288-01" xlink:href="fig-0288-01a">
    <image file="0288-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0288-01"/>
  </figure>
<note position="right" xlink:label="note-0288-01" xlink:href="note-0288-01a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0288-02" xlink:href="note-0288-02a" xml:space="preserve">f</note>
<note position="right" xlink:label="note-0288-03" xlink:href="note-0288-03a" xml:space="preserve">g</note>
<note position="right" xlink:label="note-0288-04" xlink:href="note-0288-04a" xml:space="preserve">h</note>
</div>
<p>
  <s xml:id="echoid-s9394" xml:space="preserve">Supponamus iam, quadratum B Q ad quadratum Q A maiorem pro-<lb/>portionem habere, quàm H E ad E I. </s>
  <s xml:id="echoid-s9395" xml:space="preserve">Dico, exhiberi non poſſe conum <lb/>
<anchor type="note" xlink:label="note-0289-01a" xlink:href="note-0289-01"/>
ſimilem cono A B C, qui contineat ſectionem D E F. </s>
  <s xml:id="echoid-s9396" xml:space="preserve">Alioquin conti-<lb/>neat illam conus, cuius vertex eſt R, &amp; </s>
  <s xml:id="echoid-s9397" xml:space="preserve">demonſtrabitur, quod O V ad <lb/>V R ſit, vt H E ad E I, quæ habet minorem proportionem, quàm qua-<lb/>dratum B Q ad quadratum Q A, quæ oſtenſa eſt eadem, quàm O N ad <lb/>N L; </s>
  <s xml:id="echoid-s9398" xml:space="preserve">ergo O V ad V R; </s>
  <s xml:id="echoid-s9399" xml:space="preserve">nempe O N ad N X minorem, proportionem <lb/>habet, quàm eadẽ O N ad N L, quod eſt abſurdum. </s>
  <s xml:id="echoid-s9400" xml:space="preserve">Non igitur conti-<lb/>nebit ſectionem D E F conus ſimilis cono A B C. </s>
  <s xml:id="echoid-s9401" xml:space="preserve">Vt propoſitũ fuerat.</s>
  <s xml:id="echoid-s9402" xml:space="preserve"/>
</p>
<div xml:id="echoid-div794" type="float" level="2" n="3">
<note position="left" xlink:label="note-0289-01" xlink:href="note-0289-01a" xml:space="preserve">i</note>
</div>
</div>
<div xml:id="echoid-div796" type="section" level="1" n="245">
<head xml:id="echoid-head308" xml:space="preserve">PROPOSITIO XXXI.</head>
<p>
  <s xml:id="echoid-s9403" xml:space="preserve">SIt tandem ſectio elliptica A B C, eiuſque tranſuerſus axis A C, &amp; </s>
  <s xml:id="echoid-s9404" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0289-02a" xlink:href="note-0289-02"/>
erectus A D, &amp; </s>
  <s xml:id="echoid-s9405" xml:space="preserve">in plano perpendiculariter erecto ad ſectionis pla-<lb/>num A B C, fiat ſuper A C ſegmentum circuli, quod capiat angulum. <lb/></s>
  <s xml:id="echoid-s9406" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0289-01a" xlink:href="fig-0289-01"/>
æqualem angulo F, eumque bifariam diuidamus in H, &amp; </s>
  <s xml:id="echoid-s9407" xml:space="preserve">iungamus A H, <lb/>C H, &amp; </s>
  <s xml:id="echoid-s9408" xml:space="preserve">ex H educamus H I, quæ ſecet circulum in K, &amp; </s>
  <s xml:id="echoid-s9409" xml:space="preserve">occurrat ſub-<lb/>
<anchor type="note" xlink:label="note-0289-03a" xlink:href="note-0289-03"/>
tenſæ extra circulum in I; </s>
  <s xml:id="echoid-s9410" xml:space="preserve">ſitque H I ad I K, vt A C ad A D: </s>
  <s xml:id="echoid-s9411" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9412" xml:space="preserve">e-<lb/>ducamus H L M eaſdem conditiones habens; </s>
  <s xml:id="echoid-s9413" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9414" xml:space="preserve">iungamus C K, A K, <lb/>ducaturque K N parallela A C, &amp; </s>
  <s xml:id="echoid-s9415" xml:space="preserve">A N parallela H I, quæ ſecet K C <lb/>
<anchor type="note" xlink:label="note-0289-04a" xlink:href="note-0289-04"/>
in O. </s>
  <s xml:id="echoid-s9416" xml:space="preserve">Quia H I in I K (quod eſt æquale ipſi C I in A I ad quadratum <lb/>I K) eſt vt A C ed A D; </s>
  <s xml:id="echoid-s9417" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9418" xml:space="preserve">proportio C I in A I ad quadratum I K <lb/>componitur ex ratione C I ad I K, nempe K N ad N O (propter ſimili-
<pb o="252" file="0290" n="290" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0290-01a" xlink:href="fig-0290-01"/>
tudinem duorum triangulorum), &amp; </s>
  <s xml:id="echoid-s9419" xml:space="preserve">ex ratione A I, nempe K N ad I K, <lb/>nempe ad A N ( propter parallelas ), &amp; </s>
  <s xml:id="echoid-s9420" xml:space="preserve">ex his duabus proportionibus <lb/>componitur proportio quadrati K N ad A N in N O; </s>
  <s xml:id="echoid-s9421" xml:space="preserve">ergo quadratum. <lb/></s>
  <s xml:id="echoid-s9422" xml:space="preserve">K N ad A N in N O eandem proportionem habet, quàm A C tranſuer-<lb/>ſus ad A D erectum; </s>
  <s xml:id="echoid-s9423" xml:space="preserve">igitur planum, in quo eſt ſectio A B C, in cono <lb/>cuius vertex eſt K, &amp; </s>
  <s xml:id="echoid-s9424" xml:space="preserve">baſis circulus, cuius diameter A O producit ſe-<lb/>
<anchor type="note" xlink:label="note-0290-01a" xlink:href="note-0290-01"/>
ctionem ellipticam, cuius tranſuerſus eſt A C, &amp; </s>
  <s xml:id="echoid-s9425" xml:space="preserve">erectus A D: </s>
  <s xml:id="echoid-s9426" xml:space="preserve">quare <lb/>ſectionem B A C continet; </s>
  <s xml:id="echoid-s9427" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9428" xml:space="preserve">quia angulus H K C, nempe A O K æ-<lb/>
<anchor type="note" xlink:label="note-0290-02a" xlink:href="note-0290-02"/>
qualis eſt H A C, &amp; </s>
  <s xml:id="echoid-s9429" xml:space="preserve">angulus C H A æqualis eſt C K A, remanet angu-<lb/>lus H C A æqualis O A K; </s>
  <s xml:id="echoid-s9430" xml:space="preserve">eritque H C A, quod ſimile eſt F E G, ſi-<lb/>mile quoque O K A; </s>
  <s xml:id="echoid-s9431" xml:space="preserve">quapropter O K A iſoſceleum, &amp; </s>
  <s xml:id="echoid-s9432" xml:space="preserve">ſimile eſt ipſi <lb/>F E G; </s>
  <s xml:id="echoid-s9433" xml:space="preserve">igitur conus, cuius vertex eſt K, ſimilis eſt dato cono F E G, <lb/>
<anchor type="note" xlink:label="note-0290-03a" xlink:href="note-0290-03"/>
&amp; </s>
  <s xml:id="echoid-s9434" xml:space="preserve">quidem continet ſectionem A B C, vti diximus. </s>
  <s xml:id="echoid-s9435" xml:space="preserve">Similiter quoque <lb/>oſtendemus, quod eandem ſectionem continebit alius conus, cuius ver-<lb/>tex eſt L, ſi educantur A L, L C. </s>
  <s xml:id="echoid-s9436" xml:space="preserve">Et alius conus, præter hos duos, <lb/>iuxta hanc hypotheſin non continebit illam: </s>
  <s xml:id="echoid-s9437" xml:space="preserve">Alioquin contineat illam, <lb/>
<anchor type="note" xlink:label="note-0290-04a" xlink:href="note-0290-04"/>
alius conus, cuius vertex ſit Q, &amp; </s>
  <s xml:id="echoid-s9438" xml:space="preserve">triangulum A Q P: </s>
  <s xml:id="echoid-s9439" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9440" xml:space="preserve">oſtendetur, <lb/>quemadmodum ſupra dictum eſt, quod communis ſectio plani, per axim <lb/>illius coni ducti, erecti ad planum ſectionis A B C, &amp; </s>
  <s xml:id="echoid-s9441" xml:space="preserve">plani ſectionis <lb/>eſt A C, &amp; </s>
  <s xml:id="echoid-s9442" xml:space="preserve">quod punctum verticis illius coni ſit in circumferentia ſeg-<lb/>menti A H C, &amp; </s>
  <s xml:id="echoid-s9443" xml:space="preserve">ſit Q, ducamus per H Q rectam H R, &amp; </s>
  <s xml:id="echoid-s9444" xml:space="preserve">iungamus <lb/>C Q, A Q, &amp; </s>
  <s xml:id="echoid-s9445" xml:space="preserve">educamus A S parallelam H Q R, &amp; </s>
  <s xml:id="echoid-s9446" xml:space="preserve">Q S parallelam A <lb/>C, erit Q A P triangulum illius coni, &amp; </s>
  <s xml:id="echoid-s9447" xml:space="preserve">eſt iſoſceleum, erit quadratum <lb/>Q S ad A S in S P, vt C R in R A; </s>
  <s xml:id="echoid-s9448" xml:space="preserve">quod eſt æquale ipſi H R in R Q <lb/>ad quadratum R Q, nempe H R ad R Q; </s>
  <s xml:id="echoid-s9449" xml:space="preserve">ergo H R ad R Q eſt, vt A C <lb/>
<anchor type="note" xlink:label="note-0290-05a" xlink:href="note-0290-05"/>
ad A D, quæ eſt, vt H I ad I K; </s>
  <s xml:id="echoid-s9450" xml:space="preserve">ergo diuidendo permutandoq; </s>
  <s xml:id="echoid-s9451" xml:space="preserve">H K <lb/>maior ad H Q minorem, eandem proportionem habebit, quàm K I mi-<lb/>nor ad R Q maiorem: </s>
  <s xml:id="echoid-s9452" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9453" xml:space="preserve">hoc eſt abſurdum. </s>
  <s xml:id="echoid-s9454" xml:space="preserve">Non ergo reperiri poteſt <lb/>tertius conus, continens ſectionem B A C. </s>
  <s xml:id="echoid-s9455" xml:space="preserve">Et hoc erat oſtendendum,</s>
  </p>
<pb o="253" file="0291" n="291" rhead="Conicor. Lib. VI."/>
</div>
<div xml:id="echoid-div797" type="section" level="1" n="246">
<head xml:id="echoid-head309" xml:space="preserve">Notæ in Propoſit. XXIX.</head>
<p>
<s xml:id="echoid-s9456" xml:space="preserve">ET faciamus ſuper E K triangulum ſimile triangulo A B C, &amp;</s>
  <s xml:id="echoid-s9457" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9458" xml:space="preserve">Ni-
<anchor type="note" xlink:label="note-0291-01a" xlink:href="note-0291-01"/>
mirum, fiat angulus K E L æqualis angulo A, &amp; </s>
  <s xml:id="echoid-s9459" xml:space="preserve">angulus L fiat æqualis
angulo B.</s>
  <s xml:id="echoid-s9460" xml:space="preserve"/>
</p>
<div xml:id="echoid-div797" type="float" level="2" n="1">
<note position="left" xlink:label="note-0289-02" xlink:href="note-0289-02a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0289-01" xlink:href="fig-0289-01a">
    <image file="0289-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0289-01"/>
  </figure>
<note position="right" xlink:label="note-0289-03" xlink:href="note-0289-03a" xml:space="preserve">Lem. 10. <lb/>huius.</note>
<note position="left" xlink:label="note-0289-04" xlink:href="note-0289-04a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0290-01" xlink:href="fig-0290-01a">
    <image file="0290-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0290-01"/>
  </figure>
<note position="left" xlink:label="note-0290-01" xlink:href="note-0290-01a" xml:space="preserve">13. &amp; 54. <lb/>lib. 1. <lb/>Defin. 9. <lb/>huius.</note>
<note position="right" xlink:label="note-0290-02" xlink:href="note-0290-02a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0290-03" xlink:href="note-0290-03a" xml:space="preserve">Defin. 8. <lb/>huus.</note>
<note position="right" xlink:label="note-0290-04" xlink:href="note-0290-04a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0290-05" xlink:href="note-0290-05a" xml:space="preserve">e</note>
<note position="left" xlink:label="note-0291-01" xlink:href="note-0291-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s9461" xml:space="preserve">Ergo L K, quæ eſt latus trianguli tranſeuntis per axim E G para llelũ <lb/>
<anchor type="note" xlink:label="note-0291-02a" xlink:href="note-0291-02"/>
eſt E G, &amp;</s>
  <s xml:id="echoid-s9462" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9463" xml:space="preserve">Legi debet, vt in textu videre eſt. </s>
  <s xml:id="echoid-s9464" xml:space="preserve">Hoc conſtat ex conſtructio-<lb/>ne; </s>
  <s xml:id="echoid-s9465" xml:space="preserve">nam duo anguli alterni G E K,, &amp; </s>
  <s xml:id="echoid-s9466" xml:space="preserve">L K E æquales ſunt eidem angulo C.</s>
  <s xml:id="echoid-s9467" xml:space="preserve"/>
</p>
<div xml:id="echoid-div798" type="float" level="2" n="2">
<note position="left" xlink:label="note-0291-02" xlink:href="note-0291-02a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s9468" xml:space="preserve">Et propterea planum, in quo eſt ſectio D E <lb/>
<anchor type="note" xlink:label="note-0291-03a" xlink:href="note-0291-03"/>
<anchor type="figure" xlink:label="fig-0291-01a" xlink:href="fig-0291-01"/>
F producit in cono ſectionem parabolicam, &amp;</s>
  <s xml:id="echoid-s9469" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s9470" xml:space="preserve">Quoniam planum circuli, cuius diameter E K <lb/>perpendiculare eſt ad planum trianguli L E K: </s>
  <s xml:id="echoid-s9471" xml:space="preserve">igi-<lb/>tur ſi ducatur planum N F O æquidiſtans circulo E <lb/>K ſecans planum D E F in recta linea D G F, erit <lb/>quoque circulus, &amp; </s>
  <s xml:id="echoid-s9472" xml:space="preserve">perpendicularis ad planum triã-<lb/>guli per axim L E K: </s>
  <s xml:id="echoid-s9473" xml:space="preserve">ſed ex conſtructione planum <lb/>D E F perpendiculare quoque erat ad idem trian-<lb/>gulum per axim E L K; </s>
  <s xml:id="echoid-s9474" xml:space="preserve">igitur D F communis ſectio <lb/>eorundem planorum perpendicularis quoque erit ad <lb/>idem planum L N O, &amp; </s>
  <s xml:id="echoid-s9475" xml:space="preserve">efficiet angulos rectos cum <lb/>diametro circuli N O, &amp; </s>
  <s xml:id="echoid-s9476" xml:space="preserve">cum E G, quæ in eodẽ pla-<lb/>no exiſtunt, &amp; </s>
  <s xml:id="echoid-s9477" xml:space="preserve">cũ illo conueniunt in puncto G; </s>
  <s xml:id="echoid-s9478" xml:space="preserve">ſuntq; </s>
  <s xml:id="echoid-s9479" xml:space="preserve">E G, &amp; </s>
  <s xml:id="echoid-s9480" xml:space="preserve">L O parallelæ: </s>
  <s xml:id="echoid-s9481" xml:space="preserve">igitur <lb/>
<anchor type="note" xlink:label="note-0291-04a" xlink:href="note-0291-04"/>
planum ſectionis D E F producit neceſſariò in cono L N O producto parabolam.</s>
  <s xml:id="echoid-s9482" xml:space="preserve"/>
</p>
<div xml:id="echoid-div799" type="float" level="2" n="3">
<note position="left" xlink:label="note-0291-03" xlink:href="note-0291-03a" xml:space="preserve">C</note>
  <figure xlink:label="fig-0291-01" xlink:href="fig-0291-01a">
    <image file="0291-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0291-01"/>
  </figure>
<note position="right" xlink:label="note-0291-04" xlink:href="note-0291-04a" xml:space="preserve">11. lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9483" xml:space="preserve">Igitur H E ad E L, quæ eſt æqualis ipſi L K eamdem proportionem, <lb/>
<anchor type="note" xlink:label="note-0291-05a" xlink:href="note-0291-05"/>
habet, quàm quadratum E K ad quadratum K L, &amp;</s>
  <s xml:id="echoid-s9484" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9485" xml:space="preserve">Quoniam conus <lb/>L E K ſimilis eſt cono recto A B C erit quoque rectus: </s>
  <s xml:id="echoid-s9486" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9487" xml:space="preserve">propterea duo latera <lb/>trianguli per axim E L, &amp; </s>
  <s xml:id="echoid-s9488" xml:space="preserve">L K æqualia erunt inter ſe, &amp; </s>
  <s xml:id="echoid-s9489" xml:space="preserve">ideo E K ad K L, <lb/>atque ad E L eandem proportionem habebit, &amp;</s>
  <s xml:id="echoid-s9490" xml:space="preserve">c.</s>
  <s xml:id="echoid-s9491" xml:space="preserve"/>
</p>
<div xml:id="echoid-div800" type="float" level="2" n="4">
<note position="left" xlink:label="note-0291-05" xlink:href="note-0291-05a" xml:space="preserve">d</note>
</div>
<p style="it">
  <s xml:id="echoid-s9492" xml:space="preserve">Et dico, quod ſectio D E F non reperitur in alio cono ſimili cono A <lb/>
<anchor type="note" xlink:label="note-0291-06a" xlink:href="note-0291-06"/>
B C, cuius vertex ſit ex parte plani ſectionis præter hunc conum, &amp;</s>
  <s xml:id="echoid-s9493" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s9494" xml:space="preserve">Ideſt. </s>
  <s xml:id="echoid-s9495" xml:space="preserve">Nullus alius conus rectus continebit eandem parabolam D E F, qui ſit <lb/>ſinilis cono A B C, &amp; </s>
  <s xml:id="echoid-s9496" xml:space="preserve">vertex E parabole magis, aut minus recedat à vertice <lb/>coni, quàm E L.</s>
  <s xml:id="echoid-s9497" xml:space="preserve"/>
</p>
<div xml:id="echoid-div801" type="float" level="2" n="5">
<note position="left" xlink:label="note-0291-06" xlink:href="note-0291-06a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s9498" xml:space="preserve">Ergo E M eſt indirectum ipſi E L, &amp;</s>
  <s xml:id="echoid-s9499" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9500" xml:space="preserve">Quia D G baſis ſectionis conicæ <lb/>
<anchor type="note" xlink:label="note-0291-07a" xlink:href="note-0291-07"/>
perpendicularis eße debet ad G O, &amp; </s>
  <s xml:id="echoid-s9501" xml:space="preserve">ad G E, &amp; </s>
  <s xml:id="echoid-s9502" xml:space="preserve">ideo ad triangulum per axim <lb/>vtriuſque coni recti L E K, &amp; </s>
  <s xml:id="echoid-s9503" xml:space="preserve">M E I; </s>
  <s xml:id="echoid-s9504" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9505" xml:space="preserve">conueniunt plana eorundem trian-<lb/>gulorum in E G axi conicæ ſectionis geniti ab eis; </s>
  <s xml:id="echoid-s9506" xml:space="preserve">ergo dicta triangula in eo-<lb/>dem plano exiſtunt per rectas E G, &amp; </s>
  <s xml:id="echoid-s9507" xml:space="preserve">G O ducto; </s>
  <s xml:id="echoid-s9508" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9509" xml:space="preserve">in vtroquè cono triangu-<lb/>lorum per axes latera L K, &amp; </s>
  <s xml:id="echoid-s9510" xml:space="preserve">M I parallela ſunt eidem axi E G paraboles: <lb/></s>
  <s xml:id="echoid-s9511" xml:space="preserve">ergo L K, M I parallelæ ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s9512" xml:space="preserve">anguli L, &amp; </s>
  <s xml:id="echoid-s9513" xml:space="preserve">M æquales ſunt pro-<lb/>pter ſimilitudinem triangulorum per axes in conis ſimilibus: </s>
  <s xml:id="echoid-s9514" xml:space="preserve">igitur L E, &amp; </s>
  <s xml:id="echoid-s9515" xml:space="preserve">M <lb/>E ſunt quoq; </s>
  <s xml:id="echoid-s9516" xml:space="preserve">parallelæ, &amp; </s>
  <s xml:id="echoid-s9517" xml:space="preserve">conueniunt in E vertice paraboles; </s>
  <s xml:id="echoid-s9518" xml:space="preserve">ergo in directum <lb/>ſunt conſtitutæ.</s>
  <s xml:id="echoid-s9519" xml:space="preserve"/>
</p>
<div xml:id="echoid-div802" type="float" level="2" n="6">
<note position="left" xlink:label="note-0291-07" xlink:href="note-0291-07a" xml:space="preserve">f</note>
</div>
<pb o="254" file="0292" n="292" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div804" type="section" level="1" n="247">
<head xml:id="echoid-head310" xml:space="preserve">Notæ in Propoſit. XXX.</head>
<p style="it">
  <s xml:id="echoid-s9520" xml:space="preserve">ITa vt non ſit proportio quadrati axis coni, B Q ad quadratum ſemi-<lb/>
<anchor type="note" xlink:label="note-0292-01a" xlink:href="note-0292-01"/>
diametri baſis illius vt C Q minor proportione figuræ ſectionis, &amp;</s>
  <s xml:id="echoid-s9521" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s9522" xml:space="preserve">Rurſus datus ſit conus rectus A B C, cuius axis B Q ſemidiameter circuli ba-<lb/>
<anchor type="figure" xlink:label="fig-0292-01a" xlink:href="fig-0292-01"/>
ſis ſit C Q, exhiberi aebet alius conus ſimilis dato, qui datam byperbolen D E <lb/>F contineat; </s>
  <s xml:id="echoid-s9523" xml:space="preserve">oportet autem, vt quadratum axis coni B Q ad quadratum ſemi-<lb/>diametri illius Q A non babeat maiorem proportionem, quàm habet axis tran-<lb/>ſuerſus H E ad latus rectum E I.</s>
  <s xml:id="echoid-s9524" xml:space="preserve"/>
</p>
<div xml:id="echoid-div804" type="float" level="2" n="1">
<note position="right" xlink:label="note-0292-01" xlink:href="note-0292-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0292-01" xlink:href="fig-0292-01a">
    <image file="0292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0292-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9525" xml:space="preserve">Et producamus L H ad E I occurret in K perpendiculari rectæ ad pun-<lb/>
<anchor type="note" xlink:label="note-0292-02a" xlink:href="note-0292-02"/>
ctum E linea H, &amp;</s>
  <s xml:id="echoid-s9526" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9527" xml:space="preserve">Ideſt ſi ducatur recta linea E K in plano circuli H L E <lb/>perpendicularis ad H E, ſeu parallela ipſi L N coniuncta recta linea H L ſeca-<lb/>bit reliquam æquidiſtantium E K in K.</s>
  <s xml:id="echoid-s9528" xml:space="preserve"/>
</p>
<div xml:id="echoid-div805" type="float" level="2" n="2">
<note position="right" xlink:label="note-0292-02" xlink:href="note-0292-02a" xml:space="preserve">b</note>
</div>
<p style="it">
  <s xml:id="echoid-s9529" xml:space="preserve">Quapropter K L E ſimile eſt A B C, quia æquicrus etiam eſt: </s>
  <s xml:id="echoid-s9530" xml:space="preserve">ſi au-<lb/>
<anchor type="note" xlink:label="note-0292-03a" xlink:href="note-0292-03"/>
tem ponamus K L E triangulum coni, cuius vertex L, &amp; </s>
  <s xml:id="echoid-s9531" xml:space="preserve">planum trian-<lb/>guli illius erectum ad planum D E F; </s>
  <s xml:id="echoid-s9532" xml:space="preserve">vtique planum, quod eſt in ſectione <lb/>producit in cono ſectionẽ hyperbolicã, cuius axis E G, &amp; </s>
  <s xml:id="echoid-s9533" xml:space="preserve">inclinatus E H, <lb/>&amp;</s>
  <s xml:id="echoid-s9534" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9535" xml:space="preserve">Quoniam in duobus triangulis A B C, &amp; </s>
  <s xml:id="echoid-s9536" xml:space="preserve">E L K ſunt anguli verticales B, &amp; </s>
  <s xml:id="echoid-s9537" xml:space="preserve"><lb/>L æquales inter ſe, cũ externi M B C, &amp; </s>
  <s xml:id="echoid-s9538" xml:space="preserve">H L E æquales facti ſint; </s>
  <s xml:id="echoid-s9539" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9540" xml:space="preserve">angulus H <lb/>L N æqualis ſit interno, &amp; </s>
  <s xml:id="echoid-s9541" xml:space="preserve">oppoſito K, &amp; </s>
  <s xml:id="echoid-s9542" xml:space="preserve">angulus N L E æqualis eſt alterno angulo <lb/>L E K propter parallelas N L, E K, &amp; </s>
  <s xml:id="echoid-s9543" xml:space="preserve">quilibet eorũ eſt medietas externi anguli <lb/>H L E; </s>
  <s xml:id="echoid-s9544" xml:space="preserve">ergo angulus K æqualis erit angulo L E K, &amp; </s>
  <s xml:id="echoid-s9545" xml:space="preserve">trianguliũ L E K erit iſoſceliũ, <lb/>ſed triangulum A B C per axim coni recti ductum eſt quoque iſoſcelium; </s>
  <s xml:id="echoid-s9546" xml:space="preserve">igitur <lb/>duo anguli ſupra baſim A, &amp; </s>
  <s xml:id="echoid-s9547" xml:space="preserve">C æquales ſunt inter ſe; </s>
  <s xml:id="echoid-s9548" xml:space="preserve">erant autem prius ver-<lb/>ticales anguli B, &amp; </s>
  <s xml:id="echoid-s9549" xml:space="preserve">L æquales; </s>
  <s xml:id="echoid-s9550" xml:space="preserve">igitur triangula A B C, &amp; </s>
  <s xml:id="echoid-s9551" xml:space="preserve">E L K æquiangula, <lb/>&amp; </s>
  <s xml:id="echoid-s9552" xml:space="preserve">ſimilia ſunt. </s>
  <s xml:id="echoid-s9553" xml:space="preserve">Ducatur poſtea recta linea L P perpendicularis ad baſim E K, <lb/>quæ eam ſecabit bifariam in P, &amp; </s>
  <s xml:id="echoid-s9554" xml:space="preserve">ducatur planum per E K perpendiculare ad <lb/>planum E L K, &amp; </s>
  <s xml:id="echoid-s9555" xml:space="preserve">in eo diametro E K fiat circulus, qui ſit baſis coni, cuius <lb/>vertex L, &amp; </s>
  <s xml:id="echoid-s9556" xml:space="preserve">ducatur planum F D a æquidiſtans plano circuli E K; </s>
  <s xml:id="echoid-s9557" xml:space="preserve">efficietur
<pb o="255" file="0293" n="293" rhead="Conicor. Lib. VI."/>
alius circulus F D a perpendicularis ad planum trianguli per axim L E K; </s>
  <s xml:id="echoid-s9558" xml:space="preserve">erat <lb/>autem ex conſtructione planum byperboles D E F perpendiculare ad idem planum <lb/>per axim E L K; </s>
  <s xml:id="echoid-s9559" xml:space="preserve">igitur duorum planorum communis ſectio, quæ ſit F G D per-<lb/>pendicularis quoque erit ad planum trianguli L E K: </s>
  <s xml:id="echoid-s9560" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9561" xml:space="preserve">ideo efficiet angulos F <lb/>G E, &amp; </s>
  <s xml:id="echoid-s9562" xml:space="preserve">F G a rectos, &amp; </s>
  <s xml:id="echoid-s9563" xml:space="preserve">G E H producta ſubtendit angulum externum trian-<lb/>guli conici E L K; </s>
  <s xml:id="echoid-s9564" xml:space="preserve">quapropter planum D E F efficiet in cono E L K byperbolen, <lb/>cuius axis tranſnerſus erit H E.</s>
  <s xml:id="echoid-s9565" xml:space="preserve"/>
</p>
<div xml:id="echoid-div806" type="float" level="2" n="3">
<note position="right" xlink:label="note-0292-03" xlink:href="note-0292-03a" xml:space="preserve">c</note>
</div>
<p style="it">
  <s xml:id="echoid-s9566" xml:space="preserve">Alias eontineat illam alius conus ſimilis cono A B C, ſitque vertex <lb/>
<anchor type="note" xlink:label="note-0293-01a" xlink:href="note-0293-01"/>
eius R in plano L E G, &amp; </s>
  <s xml:id="echoid-s9567" xml:space="preserve">duo latera trianguli illius ſint E R, T R; </s>
  <s xml:id="echoid-s9568" xml:space="preserve">ergo <lb/>angulus E R T æqualis eſt E L K, &amp; </s>
  <s xml:id="echoid-s9569" xml:space="preserve">eſt in cir cumferentia arcus E L H; <lb/></s>
  <s xml:id="echoid-s9570" xml:space="preserve">ergo T R ſi producatur, occurret H: </s>
  <s xml:id="echoid-s9571" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s9572" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9573" xml:space="preserve">Senſus buius textus corrupti ta-<lb/>lis eſt: </s>
  <s xml:id="echoid-s9574" xml:space="preserve">Si enim fieri poteſt, vt aliquis alius conus, vt E R T, qui ſimilis ſit <lb/>cono A B C, vel E L K, contineat eandem byperbolam D E F, &amp; </s>
  <s xml:id="echoid-s9575" xml:space="preserve">conorum, <lb/>vertices R, &amp; </s>
  <s xml:id="echoid-s9576" xml:space="preserve">L ad eaſdem partes tendant, erunt duo plana iriangulorum per <lb/>axes conorum ducta perpendicularia ad planum ſectionis D E F; </s>
  <s xml:id="echoid-s9577" xml:space="preserve">alias E G non <lb/>eßet axis hyperbole D E F; </s>
  <s xml:id="echoid-s9578" xml:space="preserve">Et quia coni ſitpponuntur ſimiles erunt quoque <lb/>
<anchor type="note" xlink:label="note-0293-02a" xlink:href="note-0293-02"/>
triangula per axes E L K, &amp; </s>
  <s xml:id="echoid-s9579" xml:space="preserve">E R T ſimilia int er ſe; </s>
  <s xml:id="echoid-s9580" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9581" xml:space="preserve">ideo anguli verticales. <lb/></s>
  <s xml:id="echoid-s9582" xml:space="preserve">L K, &amp; </s>
  <s xml:id="echoid-s9583" xml:space="preserve">E R T æquales inter ſe erunt, atque ſu bſequentes anguli E L H, &amp; </s>
  <s xml:id="echoid-s9584" xml:space="preserve">E R <lb/>H æquales quoque inter ſe erunt, &amp; </s>
  <s xml:id="echoid-s9585" xml:space="preserve">ſubtendunt commune latus tranſuerſum H <lb/>E; </s>
  <s xml:id="echoid-s9586" xml:space="preserve">igitur duo anguli E L H, &amp; </s>
  <s xml:id="echoid-s9587" xml:space="preserve">E R H in eodem circuli ſegmento conſiſtunt. </s>
  <s xml:id="echoid-s9588" xml:space="preserve"><lb/>Textus igitur corrigi debebat vt dictum eſt.</s>
  <s xml:id="echoid-s9589" xml:space="preserve"/>
</p>
<div xml:id="echoid-div807" type="float" level="2" n="4">
<note position="left" xlink:label="note-0293-01" xlink:href="note-0293-01a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0293-02" xlink:href="note-0293-02a" xml:space="preserve">E ex Def. 8.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9590" xml:space="preserve">Atque T S æqualis eſt ipſi E, &amp; </s>
  <s xml:id="echoid-s9591" xml:space="preserve">T S ad S E eſt, vt T R ad R H, quæ <lb/>
<anchor type="note" xlink:label="note-0293-03a" xlink:href="note-0293-03"/>
eſt vt E V ad V N; </s>
  <s xml:id="echoid-s9592" xml:space="preserve">ergo E V æqualis eſt V H, &amp;</s>
  <s xml:id="echoid-s9593" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9594" xml:space="preserve">In duobus triangulis <lb/>iſoſcelĳs inter ſe ſimilibus A B C, &amp; </s>
  <s xml:id="echoid-s9595" xml:space="preserve">E R T ab æqualibus angulis verticalibus <lb/>A B C, &amp; </s>
  <s xml:id="echoid-s9596" xml:space="preserve">E R T ducuntur rectæ lineæ B Q, R S ſecantes baſes in Q, &amp; </s>
  <s xml:id="echoid-s9597" xml:space="preserve">S: <lb/></s>
  <s xml:id="echoid-s9598" xml:space="preserve">eſtque quadratum R S ad rectangulum E S T, vt quadratum B Q ad rectangu-<lb/>lum A Q C, &amp; </s>
  <s xml:id="echoid-s9599" xml:space="preserve">ſecatur A C bifariam in Q; </s>
  <s xml:id="echoid-s9600" xml:space="preserve">oſtendendum eſt E T in duas par-<lb/>tes æquales in S quoque ſecari. </s>
  <s xml:id="echoid-s9601" xml:space="preserve">Si enim boc verum non eſt E T in alio puncto <lb/>bifariam diuidetur vt in b iungaturquè R <lb/>
<anchor type="figure" xlink:label="fig-0293-01a" xlink:href="fig-0293-01"/>
b. </s>
  <s xml:id="echoid-s9602" xml:space="preserve">Quoniam à verticibus triangulorum, <lb/>A B C, &amp; </s>
  <s xml:id="echoid-s9603" xml:space="preserve">R E T iſoſcelium ducuntur re-<lb/>ctæ lineæ B Q, R b diuidentes baſes bifa-<lb/>riam in Q, b, ergo anguli ad Q, &amp; </s>
  <s xml:id="echoid-s9604" xml:space="preserve">b <lb/>ſunt recti, &amp; </s>
  <s xml:id="echoid-s9605" xml:space="preserve">erant anguli A, &amp; </s>
  <s xml:id="echoid-s9606" xml:space="preserve">E æquales <lb/>(propter ſimilitudinem eorundem triangu-<lb/>lorum) igitur triangula A B Q, &amp; </s>
  <s xml:id="echoid-s9607" xml:space="preserve">E R b <lb/>ſimilia ſunt, ideoq; </s>
  <s xml:id="echoid-s9608" xml:space="preserve">B Q ad Q A erit vt R b <lb/>ad b E, &amp; </s>
  <s xml:id="echoid-s9609" xml:space="preserve">quadratũ B Q ad quadratum Q A erit vt quadratũ R b ad quadratũ <lb/>b E; </s>
  <s xml:id="echoid-s9610" xml:space="preserve">erat autem quadratum R S ad rectangulum E S T vt quadratum B Q ad <lb/>quadratum Q A; </s>
  <s xml:id="echoid-s9611" xml:space="preserve">ergo quadratum R b ad quadratum b E eandem proportionem <lb/>habet, quàm quadratum R S ad rectangulum E S T; </s>
  <s xml:id="echoid-s9612" xml:space="preserve">eſtque quadratum R b <lb/>minus quadrato R S (cum perpendicularis R b minor ſit quàm R S) quarè qua-<lb/>dratum ex b E ſemiſſe totius E T minus erit rectangulo E S T ſub ſegmentis <lb/>inæqualibus eiusdem E T contento; </s>
  <s xml:id="echoid-s9613" xml:space="preserve">quod eſt abſurdum: </s>
  <s xml:id="echoid-s9614" xml:space="preserve">quarè neceſſario E T <lb/>bifariam ſecatur in S. </s>
  <s xml:id="echoid-s9615" xml:space="preserve">Poſtea propter parallela R S, &amp; </s>
  <s xml:id="echoid-s9616" xml:space="preserve">H E, vt T S ad S E <lb/>ita erit T R ad R H; </s>
  <s xml:id="echoid-s9617" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9618" xml:space="preserve">propter parallelas R V, &amp; </s>
  <s xml:id="echoid-s9619" xml:space="preserve">E T erit E V ad V H, vt <lb/>T R ad R H, ſeu T S ad S E: </s>
  <s xml:id="echoid-s9620" xml:space="preserve">oſtenſa autem fuit T S æqualis S E; </s>
  <s xml:id="echoid-s9621" xml:space="preserve">igitur E
<pb o="256" file="0294" n="294" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0294-01a" xlink:href="fig-0294-01"/>
V æqualis eſt V H, quod eſt abſurdum.</s>
  <s xml:id="echoid-s9622" xml:space="preserve"/>
</p>
<div xml:id="echoid-div808" type="float" level="2" n="5">
<note position="left" xlink:label="note-0293-03" xlink:href="note-0293-03a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0293-01" xlink:href="fig-0293-01a">
    <image file="0293-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0293-01"/>
  </figure>
  <figure xlink:label="fig-0294-01" xlink:href="fig-0294-01a">
    <image file="0294-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0294-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9623" xml:space="preserve">Patet quadratum L P nempe N E, ſeu O N in N L ad quadratum E P, <lb/>
<anchor type="note" xlink:label="note-0294-01a" xlink:href="note-0294-01"/>
nempe ad quadratum N L, ſcilicet O N ad N L habere minorem pro-<lb/>portionem, quàm H E ad E I: </s>
  <s xml:id="echoid-s9624" xml:space="preserve">ponamus iam O N ad Z X, vt H E ad E <lb/>I; </s>
  <s xml:id="echoid-s9625" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9626" xml:space="preserve">per X ducamus X R, &amp; </s>
  <s xml:id="echoid-s9627" xml:space="preserve">iungamus E R, &amp;</s>
  <s xml:id="echoid-s9628" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9629" xml:space="preserve">Suppoſita conſtructione <lb/>prioris caſus, quandò conus rectus E L K factus eſt ſimilis cono A B C quadra-<lb/>tum L P ad quadratum E P habebat eandem proportionem, quàm O N ad N L, <lb/>ſeu quàm quadratum B Q ad quadratum Q A: </s>
  <s xml:id="echoid-s9630" xml:space="preserve">modò in hac altera ſuppoſitione <lb/>conceditur quadratum B Q ad quadratum Q A habere minorem proportionem, <lb/>quàm E H ad E I; </s>
  <s xml:id="echoid-s9631" xml:space="preserve">igitur O N ad N L minorem proportionem habebit, quàm, <lb/>H E ad E I; </s>
  <s xml:id="echoid-s9632" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9633" xml:space="preserve">fiat O N ad N X vt H E ad E I, erit N X minor quàm N L, <lb/>&amp; </s>
  <s xml:id="echoid-s9634" xml:space="preserve">ideo punctum X intra circulum cadet, &amp; </s>
  <s xml:id="echoid-s9635" xml:space="preserve">per X ducta R X Y parallelæ H E; <lb/></s>
  <s xml:id="echoid-s9636" xml:space="preserve">vtique ſecabit circulum in duobus punctis, vt in R, &amp; </s>
  <s xml:id="echoid-s9637" xml:space="preserve">Y. </s>
  <s xml:id="echoid-s9638" xml:space="preserve">Quod verò recta, <lb/>R X Y duci debeat parallela ipſi H E, non quomodocunque, patet ex contextu <lb/>ſequenti, nam debent O X, O R ſecari in N, &amp; </s>
  <s xml:id="echoid-s9639" xml:space="preserve">V proportionaliter, quarè tex-<lb/>tus debuit omnino corrigi.</s>
  <s xml:id="echoid-s9640" xml:space="preserve"/>
</p>
<div xml:id="echoid-div809" type="float" level="2" n="6">
<note position="left" xlink:label="note-0294-01" xlink:href="note-0294-01a" xml:space="preserve">f</note>
</div>
<p style="it">
  <s xml:id="echoid-s9641" xml:space="preserve">Oſtendetur, quemadmodum dictum eſt, quod E T R, &amp; </s>
  <s xml:id="echoid-s9642" xml:space="preserve">A B C ſunt <lb/>
<anchor type="note" xlink:label="note-0294-02a" xlink:href="note-0294-02"/>
iſoſcelia, &amp; </s>
  <s xml:id="echoid-s9643" xml:space="preserve">ſimilia, &amp;</s>
  <s xml:id="echoid-s9644" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9645" xml:space="preserve">Quoniam arcus circuli E O, &amp; </s>
  <s xml:id="echoid-s9646" xml:space="preserve">O H æquales ſunt <lb/>inter ſe ex conſtructione, erunt anguli E R O, &amp; </s>
  <s xml:id="echoid-s9647" xml:space="preserve">O R H æquales inter ſe, &amp; </s>
  <s xml:id="echoid-s9648" xml:space="preserve"><lb/>propter parallelas O R, &amp; </s>
  <s xml:id="echoid-s9649" xml:space="preserve">E T eſt angulus O R E æqualis alterno T E R; </s>
  <s xml:id="echoid-s9650" xml:space="preserve">at-<lb/>què externus H R O æqualis eſt interno, &amp; </s>
  <s xml:id="echoid-s9651" xml:space="preserve">oppoſito R T E; </s>
  <s xml:id="echoid-s9652" xml:space="preserve">igitur duo anguli <lb/>R E T, &amp; </s>
  <s xml:id="echoid-s9653" xml:space="preserve">R T E æquales ſunt inter ſe; </s>
  <s xml:id="echoid-s9654" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9655" xml:space="preserve">propterea triangulum E R T erit <lb/>iſoſcelium. </s>
  <s xml:id="echoid-s9656" xml:space="preserve">Rurſus quia duo anguli E L H, E R H in eodem circuli ſegmento <lb/>couſtituti æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s9657" xml:space="preserve">erat ex conſtructione angulus M B C æqualis <lb/>angulo H L E; </s>
  <s xml:id="echoid-s9658" xml:space="preserve">igitur anguli H R E, &amp; </s>
  <s xml:id="echoid-s9659" xml:space="preserve">M B C æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s9660" xml:space="preserve">ideo <lb/>conſequentes anguli verticales E R T, &amp; </s>
  <s xml:id="echoid-s9661" xml:space="preserve">A B C æquales erunt inter ſe, eſt quo-<lb/>que triangulum A B C per axim coni recti iſoſcelium igitur duo triangula, <lb/>E R T, &amp; </s>
  <s xml:id="echoid-s9662" xml:space="preserve">A B C ſimilia ſunt inter ſe. </s>
  <s xml:id="echoid-s9663" xml:space="preserve">Et quia vt dictum eſt O N ad N X <lb/>eandem proportionem habet, quàm H E ad E I, atque propter parallelas V N, <lb/>&amp; </s>
  <s xml:id="echoid-s9664" xml:space="preserve">R X eſt O V ad V R vt O N ad N X, &amp; </s>
  <s xml:id="echoid-s9665" xml:space="preserve">ſumpta cõmuni altitudine V R erit
<pb o="257" file="0295" n="295" rhead="Conicor. Lib. VI."/>
rectangulum O V R ad quadratum V R, vt H E ad E I: </s>
  <s xml:id="echoid-s9666" xml:space="preserve">eſt verò rectangulum <lb/>H V E æquale rectangulo O V R (propterea quod duæ rect æ line æ O R, H E ſe ſe ſe-<lb/>cant intra circulum in V) igitur rectangulum H V E ad quadratum V R eandẽ <lb/>proportionẽ habet quàm H E ad E I; </s>
  <s xml:id="echoid-s9667" xml:space="preserve">cumq; </s>
  <s xml:id="echoid-s9668" xml:space="preserve">proportio rectanguli H V E ad qua. <lb/></s>
  <s xml:id="echoid-s9669" xml:space="preserve">dratum V R compoſita ſit ex duabus rationibus, ipſius E V ad V R, ſeu R S ad <lb/>S E, (propter parallelogrammum V E S R), &amp; </s>
  <s xml:id="echoid-s9670" xml:space="preserve">ex proportione H V ad V R, <lb/>quæ eadem eſt proportioni ipſius R S ad S T (propterea quod triangula H V R, <lb/>&amp; </s>
  <s xml:id="echoid-s9671" xml:space="preserve">R S T ſimilia conſtituuntur ab æquidiſtantibus H V, R S, &amp; </s>
  <s xml:id="echoid-s9672" xml:space="preserve">V R, S T) <lb/>quapropter duæ proportiones R S ad S E, &amp; </s>
  <s xml:id="echoid-s9673" xml:space="preserve">R S ad S T componentes proportio-<lb/>nem quadrati R S ad rectangulum E S T eædem ſunt rationibus, ex quibus <lb/>componitur proportio rectanguli H V E ad quadratum V R; </s>
  <s xml:id="echoid-s9674" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9675" xml:space="preserve">ideo quadratum <lb/>R S ad rectangulum E S T eandem proportionem habebit, quàm rectangulum <lb/>H V E ad quadratum V R, ſeu eandem quàm habet H E ad E I; </s>
  <s xml:id="echoid-s9676" xml:space="preserve">igitur ſi fiat <lb/>conus, cuius vertex R, &amp; </s>
  <s xml:id="echoid-s9677" xml:space="preserve">baſis circulus diametro E T, cuius planum perpen-<lb/>diculare ſit ad planum trianguli E R T, erit triangulum E R T iſoſcelium per <lb/>axim prædicti coni extenſum, atq; </s>
  <s xml:id="echoid-s9678" xml:space="preserve">ad ipſum ſectionis D E F planum eſt quo-<lb/>que perpendiculare, &amp; </s>
  <s xml:id="echoid-s9679" xml:space="preserve">eius axis G E ſubtendit angulum E R H, qui deinceps <lb/>eſt angulo verticis; </s>
  <s xml:id="echoid-s9680" xml:space="preserve">igitur planum D E F in cono E R T generat hyperbolen, <lb/>cuius axis inclinatus eſt E H, &amp; </s>
  <s xml:id="echoid-s9681" xml:space="preserve">erectus E I: </s>
  <s xml:id="echoid-s9682" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9683" xml:space="preserve">propterea conus E R T com-<lb/>prehendit hyperbolen D E F. </s>
  <s xml:id="echoid-s9684" xml:space="preserve">Rurſus ſi recta R X producatur quouſque ſecet <lb/>peripheriam circuli L E ex altera parte in puncto Y; </s>
  <s xml:id="echoid-s9685" xml:space="preserve">atque denuò coniungantur <lb/>rectæ lineæ E Y, &amp; </s>
  <s xml:id="echoid-s9686" xml:space="preserve">H Y, quæ extendatur quouſquè conueniat cum recta linea <lb/>ex puncto E parallela ipſi O Y in puncto aliquo, quod concipiatur eſſe d; </s>
  <s xml:id="echoid-s9687" xml:space="preserve">fieri <lb/>poterit alius conus (cuius vertex Y, baſis circulus diametro E d erectus ad <lb/>planum trianguli) ſimilis cono E R T, ſiue A B C: </s>
  <s xml:id="echoid-s9688" xml:space="preserve">Oſtendetur ſicuti modo di-<lb/>ctum eſt, quod idem planum H D F eſſiciet in cono γ d E eandem hyperbolen <lb/>D E F.</s>
  <s xml:id="echoid-s9689" xml:space="preserve"/>
</p>
<div xml:id="echoid-div810" type="float" level="2" n="7">
<note position="right" xlink:label="note-0294-02" xlink:href="note-0294-02a" xml:space="preserve">g</note>
</div>
<p style="it">
  <s xml:id="echoid-s9690" xml:space="preserve">Inde demonſtrabitur quod E H ad E I neceſſe eſt, vt habeat eandem <lb/>
<anchor type="note" xlink:label="note-0295-01a" xlink:href="note-0295-01"/>
proportionem, quàm O e ad e Z: </s>
  <s xml:id="echoid-s9691" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9692" xml:space="preserve">hoc eſt abſurdum, &amp;</s>
  <s xml:id="echoid-s9693" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9694" xml:space="preserve">quia conus <lb/>Z E f continet hyperbolen D E F neceſſariò eius axis tranſuerſus E H ſubten-<lb/>det angulum H Z E, qui deinceps eſt anguli verticis trianguli per axim; </s>
  <s xml:id="echoid-s9695" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9696" xml:space="preserve"><lb/>propter ſimilitudinẽ conorũ rectorum, ſunt triangula per axes A B C, E R T, &amp; </s>
  <s xml:id="echoid-s9697" xml:space="preserve"><lb/>E Z f ſimilia inter ſe, &amp; </s>
  <s xml:id="echoid-s9698" xml:space="preserve">anguli verticales B, Z, &amp; </s>
  <s xml:id="echoid-s9699" xml:space="preserve">R æquales erunt inter ſe; <lb/></s>
  <s xml:id="echoid-s9700" xml:space="preserve">ideo conſequentes anguli M B C, &amp; </s>
  <s xml:id="echoid-s9701" xml:space="preserve">H R E, nec non H Z E æquales erunt in-<lb/>ter ſe, &amp; </s>
  <s xml:id="echoid-s9702" xml:space="preserve">ſubtenduntur ab eadem recta linea H E; </s>
  <s xml:id="echoid-s9703" xml:space="preserve">ergo in eodem circuli ſeg-<lb/>mento conſiſtunt: </s>
  <s xml:id="echoid-s9704" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9705" xml:space="preserve">propterea punctum Z in circuli peripheria H Z E cadit. </s>
  <s xml:id="echoid-s9706" xml:space="preserve"><lb/>Poſtea (vt in propoſitione 53. </s>
  <s xml:id="echoid-s9707" xml:space="preserve">primi libri, &amp; </s>
  <s xml:id="echoid-s9708" xml:space="preserve">in hac eadem propoſitione demon-<lb/>ſtrauit Apollonius) conſtat quod H E ad E I habet eandem proportionem, quàm <lb/>O e ad e Z; </s>
  <s xml:id="echoid-s9709" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9710" xml:space="preserve">prius O V ad V R erat vt H E ad E I; </s>
  <s xml:id="echoid-s9711" xml:space="preserve">ergo O V ad V R eã-<lb/>dem proportionem habet quàm O e ad e Z; </s>
  <s xml:id="echoid-s9712" xml:space="preserve">ſed quia punctum Z non cadit in <lb/>R, neque in γ alias conus E Z f non eſſet alius à præcedentibus E R T, &amp; </s>
  <s xml:id="echoid-s9713" xml:space="preserve">E <lb/>γ d; </s>
  <s xml:id="echoid-s9714" xml:space="preserve">ergo O e ad e Z non habet eandem proportionem, quàm O V ad V R, quod <lb/>eſt abſurdum.</s>
  <s xml:id="echoid-s9715" xml:space="preserve"/>
</p>
<div xml:id="echoid-div811" type="float" level="2" n="8">
<note position="left" xlink:label="note-0295-01" xlink:href="note-0295-01a" xml:space="preserve">h</note>
</div>
<p style="it">
  <s xml:id="echoid-s9716" xml:space="preserve">Et demonſtrabitur quod O V ad V R ſit vt H E ad E I, &amp;</s>
  <s xml:id="echoid-s9717" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9718" xml:space="preserve">Repeta-<lb/>
<anchor type="note" xlink:label="note-0295-02a" xlink:href="note-0295-02"/>
tur denuo conſtructio primi caſus huius propoſitionis, vt fiat conus rectus L E <lb/>K ſim lis cono A B C, tunc quidem quadratum L P ad quadratum E P habe-<lb/>bit eandem proportionem, quàm O N ad N L, ſeu quàm quadratum B Q ad
<pb o="258" file="0296" n="296" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0296-01a" xlink:href="fig-0296-01"/>
quadratum Q A; </s>
  <s xml:id="echoid-s9719" xml:space="preserve">ſed in hac poſtrema ſuppoſitione conceditur quadratum B Q <lb/>ad quadratum Q A habere maiorem proportionem, quàm H E ad E I; </s>
  <s xml:id="echoid-s9720" xml:space="preserve">igitur <lb/>O N ad N L maiorem proportionem habebit, quàm H E ad E I; </s>
  <s xml:id="echoid-s9721" xml:space="preserve">ſed quia co-<lb/>nus E R T ponitur continere ſectionem D E F: </s>
  <s xml:id="echoid-s9722" xml:space="preserve">habebit O V ad V R eandem <lb/>proportionem, quàm H E ad E I (vt ex 53. </s>
  <s xml:id="echoid-s9723" xml:space="preserve">primi deducitur, &amp; </s>
  <s xml:id="echoid-s9724" xml:space="preserve">in hac pro-<lb/>poſitione denuò factum eſt): </s>
  <s xml:id="echoid-s9725" xml:space="preserve">igitur O N ad N L maiorem proportionem habebit <lb/>quàm O V ad V R; </s>
  <s xml:id="echoid-s9726" xml:space="preserve">oſtenſa autem fuit O N ad N X, vt O V ad V R; </s>
  <s xml:id="echoid-s9727" xml:space="preserve">ergo O <lb/>N ad N L maiorem proportionem habebit, quàm O N ad N X: </s>
  <s xml:id="echoid-s9728" xml:space="preserve">quod eſt abſur-<lb/>dum, nam N X minor eſt, quàm N L.</s>
  <s xml:id="echoid-s9729" xml:space="preserve"/>
</p>
<div xml:id="echoid-div812" type="float" level="2" n="9">
<note position="left" xlink:label="note-0295-02" xlink:href="note-0295-02a" xml:space="preserve">i</note>
  <figure xlink:label="fig-0296-01" xlink:href="fig-0296-01a">
    <image file="0296-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0296-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div814" type="section" level="1" n="248">
<head xml:id="echoid-head311" xml:space="preserve">Notæ in Propoſit. XXXI.</head>
<p>
  <s xml:id="echoid-s9730" xml:space="preserve">DEinde ſit ſectio elliptica A B C, &amp; </s>
  <s xml:id="echoid-s9731" xml:space="preserve">tranſuerſa illius A C, &amp; </s>
  <s xml:id="echoid-s9732" xml:space="preserve">erectus <lb/>
<anchor type="note" xlink:label="note-0296-01a" xlink:href="note-0296-01"/>
A D, &amp; </s>
  <s xml:id="echoid-s9733" xml:space="preserve">circunducamus ſuper A C in plano erecto ad ſectionis <lb/>planum A B C ſegmentum circuli, quod capiat angulum æqualem an-<lb/>gulo F: </s>
  <s xml:id="echoid-s9734" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s9735" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9736" xml:space="preserve">Rurſus conus exhiberi debet ſimilis cono dato E F G, qui datam <lb/>ellipſim A B C contineat, ſitque axis tranſuerſus ellipſis C A, eiuſque latus <lb/>rectum A D.</s>
  <s xml:id="echoid-s9737" xml:space="preserve"/>
</p>
<div xml:id="echoid-div814" type="float" level="2" n="1">
<note position="right" xlink:label="note-0296-01" xlink:href="note-0296-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s9738" xml:space="preserve">Quia H I in I K, quod eſt æquale ipſi C I in I A, ad quadratum I A <lb/>
<anchor type="note" xlink:label="note-0296-02a" xlink:href="note-0296-02"/>
eſt, vt A C ad A D, &amp; </s>
  <s xml:id="echoid-s9739" xml:space="preserve">C I in A I ad quadratum I K nempe K N ad <lb/>N O propter ſimilitudinem duorum triangulorum, &amp; </s>
  <s xml:id="echoid-s9740" xml:space="preserve">ex A I, nempe N <lb/>K ad I K nempe A N vt parallelas conſtituamus lineas, &amp; </s>
  <s xml:id="echoid-s9741" xml:space="preserve">ex his dua-<lb/>bus proportionibus componitur proportio quadrati N K ad A N in N O, <lb/>&amp;</s>
  <s xml:id="echoid-s9742" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9743" xml:space="preserve">Senſus huius textus valdè corrupti hic eſt. </s>
  <s xml:id="echoid-s9744" xml:space="preserve">Quia ex conſtructione H I ad <lb/>I K erat vt C A ad A D, &amp; </s>
  <s xml:id="echoid-s9745" xml:space="preserve">ſumpta communi altitudine I K, erit rectangu-
<pb o="259" file="0297" n="297" rhead="Conicor. Lib. VI."/>
<anchor type="figure" xlink:label="fig-0297-01a" xlink:href="fig-0297-01"/>
lum H I K ad quadratum I K, vt H I ad I K ſeu vt C A ad A D; </s>
  <s xml:id="echoid-s9746" xml:space="preserve">eſtque <lb/>rectangulum C I A æquale rectangulo H I K; </s>
  <s xml:id="echoid-s9747" xml:space="preserve">igitur rectangulum C I A ad qua-<lb/>dratum I K eandem proportionem habet, quàm C A ad A D; </s>
  <s xml:id="echoid-s9748" xml:space="preserve">componitur verò <lb/>proportio rectanguli C I A ad quadratum I K ex duabus proportionibus laterum <lb/>C I ad I K, &amp; </s>
  <s xml:id="echoid-s9749" xml:space="preserve">A I ad I K: </s>
  <s xml:id="echoid-s9750" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9751" xml:space="preserve">propter parallelas N O, I K, atque K N, &amp; </s>
  <s xml:id="echoid-s9752" xml:space="preserve"><lb/>C I, &amp; </s>
  <s xml:id="echoid-s9753" xml:space="preserve">latus commune C O K duo triangula C I K, &amp; </s>
  <s xml:id="echoid-s9754" xml:space="preserve">K O N ſimilia ſunt; <lb/></s>
  <s xml:id="echoid-s9755" xml:space="preserve">igitur K N ad N O eſt, vt C I ad I K; </s>
  <s xml:id="echoid-s9756" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9757" xml:space="preserve">quia in parallelogrammo I N la-<lb/>tera oppoſita ſunt æqualia K N ad N A eandem proportionem habebit quàm A I <lb/>ad I K; </s>
  <s xml:id="echoid-s9758" xml:space="preserve">quapropter duæ rationes K N ad N O, &amp; </s>
  <s xml:id="echoid-s9759" xml:space="preserve">K N ad N A componunt <lb/>proportionem quadrati K N ad rectangulum A N O, quæ eadem eſt proportioni <lb/>rectanguli C I A ad quadratum I K; </s>
  <s xml:id="echoid-s9760" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9761" xml:space="preserve">propterea quadratum K N ad rectan-<lb/>gulum A N O eandem proportionem habebit, quàm A G ad A D. </s>
  <s xml:id="echoid-s9762" xml:space="preserve">Si igitur <lb/>fiat conus, cuius vertex K baſis circulus diametro A O deſcriptus, cuius pla-<lb/>num perpendiculare ſit ad planum A K C; </s>
  <s xml:id="echoid-s9763" xml:space="preserve">atque per rectam A C æquidiſtan-<lb/>tem ipſi K N planum ducatur perpendiculare ad idem planum A K C genera-<lb/>bitur ellipſis, cuius axis tranſuerſus erit A C, &amp; </s>
  <s xml:id="echoid-s9764" xml:space="preserve">latus rectum A D. </s>
  <s xml:id="echoid-s9765" xml:space="preserve">Textus <lb/>igitur corrigi debere ex dictis manifeſtum eſt.</s>
  <s xml:id="echoid-s9766" xml:space="preserve"/>
</p>
<div xml:id="echoid-div815" type="float" level="2" n="2">
<note position="right" xlink:label="note-0296-02" xlink:href="note-0296-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0297-01" xlink:href="fig-0297-01a">
    <image file="0297-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0297-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9767" xml:space="preserve">Et quia angulus H K C nempe A O K æqualis eſt H A C, &amp; </s>
  <s xml:id="echoid-s9768" xml:space="preserve">angulus <lb/>
<anchor type="note" xlink:label="note-0297-01a" xlink:href="note-0297-01"/>
C H A æqualis eſt C K A remanet angulus H C A æqualis O A K erit <lb/>H C A ſimile F E G ſimile quoque O K A; </s>
  <s xml:id="echoid-s9769" xml:space="preserve">ergo, &amp;</s>
  <s xml:id="echoid-s9770" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9771" xml:space="preserve">Quoniam ex con-<lb/>ſtructione ſegmentum A H C capax eſt anguli æqualis angulo F erit angulus A <lb/>H C æqualis angulo F; </s>
  <s xml:id="echoid-s9772" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9773" xml:space="preserve">quia peripheria A H C ſecta eſt bifariam in H; </s>
  <s xml:id="echoid-s9774" xml:space="preserve">ergo <lb/>ſubtenſa latera A H, &amp; </s>
  <s xml:id="echoid-s9775" xml:space="preserve">H C æqualia ſunt: </s>
  <s xml:id="echoid-s9776" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9777" xml:space="preserve">propterea triangulum A H C <lb/>iſoſcelium, &amp; </s>
  <s xml:id="echoid-s9778" xml:space="preserve">ſimile erit triangulo E F G; </s>
  <s xml:id="echoid-s9779" xml:space="preserve">propterea quod anguli verticales æ-<lb/>quales ſunt inter ſe; </s>
  <s xml:id="echoid-s9780" xml:space="preserve">ſunt verò duo anguli A H C, &amp; </s>
  <s xml:id="echoid-s9781" xml:space="preserve">A K C in eodem circuli <lb/>ſegmento; </s>
  <s xml:id="echoid-s9782" xml:space="preserve">ergo æquales ſunt inter ſe; </s>
  <s xml:id="echoid-s9783" xml:space="preserve">pariterque duo anguli C A H, &amp; </s>
  <s xml:id="echoid-s9784" xml:space="preserve">C K H <lb/>in eodem circuli ſegmento conſtituti, æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s9785" xml:space="preserve">propter paralle-
<pb o="260" file="0298" n="298" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0298-01a" xlink:href="fig-0298-01"/>
las A O, K H ſunt anguli alterni A O K, &amp; </s>
  <s xml:id="echoid-s9786" xml:space="preserve">H K O æquales inter ſe; </s>
  <s xml:id="echoid-s9787" xml:space="preserve">igitur <lb/>angulus A O K æqualis erit angulo C A H; </s>
  <s xml:id="echoid-s9788" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9789" xml:space="preserve">propterea in duobus triangulis <lb/>K A O, &amp; </s>
  <s xml:id="echoid-s9790" xml:space="preserve">H C A tertius angulus A C H æqualis erit tertio angulo K A O, <lb/>&amp; </s>
  <s xml:id="echoid-s9791" xml:space="preserve">propterea triangulum K A O iſoſcelium, &amp; </s>
  <s xml:id="echoid-s9792" xml:space="preserve">ſimile erit triangulo H A C, <lb/>ſiuè F G E; </s>
  <s xml:id="echoid-s9793" xml:space="preserve">igitur conus, cuius vertex K baſis circulus A O perpendicularis <lb/>ad planum trianguli A K O erit conus rectus, &amp; </s>
  <s xml:id="echoid-s9794" xml:space="preserve">ſimilis cono E F G dato.</s>
  <s xml:id="echoid-s9795" xml:space="preserve"/>
</p>
<div xml:id="echoid-div816" type="float" level="2" n="3">
<note position="left" xlink:label="note-0297-01" xlink:href="note-0297-01a" xml:space="preserve">C</note>
  <figure xlink:label="fig-0298-01" xlink:href="fig-0298-01a">
    <image file="0298-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0298-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9796" xml:space="preserve">Alioquin contineat illum conus alius, cuius vertex ſit Q, &amp; </s>
  <s xml:id="echoid-s9797" xml:space="preserve">triangu-<lb/>
<anchor type="note" xlink:label="note-0298-01a" xlink:href="note-0298-01"/>
lum Q A P, &amp; </s>
  <s xml:id="echoid-s9798" xml:space="preserve">oſtendetur quemadmodum dictum eſt, quod planum <lb/>tranſiens per axim illius coni erectum ad planum ſectionis A B C ſectio <lb/>communis cum plano ſectionis eſt A C, &amp; </s>
  <s xml:id="echoid-s9799" xml:space="preserve">quod punctum verticis illius <lb/>coni ſit in circumferentia ſegmenti A H C, &amp;</s>
  <s xml:id="echoid-s9800" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9801" xml:space="preserve">Quia ſupponitur, quod <lb/>conus Q A P ſimilis cono E F G contineat ellipſim A B C, cuius axis tranſuer-<lb/>ſus C A, &amp; </s>
  <s xml:id="echoid-s9802" xml:space="preserve">latus rectum A D; </s>
  <s xml:id="echoid-s9803" xml:space="preserve">igitur triangulum per axim coni ductum Q <lb/>A P, nedum ſimile erit triangulo E F G, ſed etiam perpendiculare erit ad pla-<lb/>num ellipſis A B C, &amp; </s>
  <s xml:id="echoid-s9804" xml:space="preserve">propterea conſiſtet in plano circularis ſegmenti A H C <lb/>pariter erecti ad planum A B C, per idem axim A C extenſum, &amp; </s>
  <s xml:id="echoid-s9805" xml:space="preserve">eſt angu-<lb/>lus A Q C æqualis angulo verticali F propter ſimilitudinem duorum triangu-<lb/>lorum, &amp; </s>
  <s xml:id="echoid-s9806" xml:space="preserve">ex conſtructione primæ partis huius propoſitionis, eſt ſegmentum A <lb/>H C capax anguli æqualis angulo F; </s>
  <s xml:id="echoid-s9807" xml:space="preserve">ſecaturque bifariam in H; </s>
  <s xml:id="echoid-s9808" xml:space="preserve">igitur angulus <lb/>A Q C æqualis ipſi F in peripheria ſegmenti A H C exiſtit. </s>
  <s xml:id="echoid-s9809" xml:space="preserve">Ducatur poſtea <lb/>Q S parallela lateri tranſuer ſo ellipſis A C, quæ ſecet baſim trianguli per axim <lb/>Q A P productam in S, &amp; </s>
  <s xml:id="echoid-s9810" xml:space="preserve">à puncto H bipartitæ diuiſionis ſegmenti A H C <lb/>coniungatur recta linea H Q producaturq; </s>
  <s xml:id="echoid-s9811" xml:space="preserve">quouſq; </s>
  <s xml:id="echoid-s9812" xml:space="preserve">occurratrectæ lineæ C A in R. <lb/></s>
  <s xml:id="echoid-s9813" xml:space="preserve">Quoniã duo anguli A H C, &amp; </s>
  <s xml:id="echoid-s9814" xml:space="preserve">A Q C in eodẽ circuli ſegmento conſtituti æqua-<lb/>les ſunt inter ſe; </s>
  <s xml:id="echoid-s9815" xml:space="preserve">pariterq; </s>
  <s xml:id="echoid-s9816" xml:space="preserve">duo anguli C A H, &amp; </s>
  <s xml:id="echoid-s9817" xml:space="preserve">C Q H in eodẽ circuli ſegmento <lb/>exiſtentes ſunt æquales, &amp; </s>
  <s xml:id="echoid-s9818" xml:space="preserve">eſt angulus A P Q æqualis angulo P A Q in triangu-<lb/>lo iſoſcelio Q A P; </s>
  <s xml:id="echoid-s9819" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9820" xml:space="preserve">angulus P A Q æqualis angulo C A H in triangulis ſimi-<lb/>libus; </s>
  <s xml:id="echoid-s9821" xml:space="preserve">igitur angulus A P Q æqualis eſt alterno angulo P Q H; </s>
  <s xml:id="echoid-s9822" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9823" xml:space="preserve">propterea
<pb o="261" file="0299" n="299" rhead="Conicor. Lib. VI."/>
recta linea H R parallela eſt ipſi A S; </s>
  <s xml:id="echoid-s9824" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9825" xml:space="preserve">erat prius Q S parallela ipſi C R, <lb/>&amp; </s>
  <s xml:id="echoid-s9826" xml:space="preserve">recta linea C P Q eſt communis; </s>
  <s xml:id="echoid-s9827" xml:space="preserve">igitur triangula C R Q, &amp; </s>
  <s xml:id="echoid-s9828" xml:space="preserve">Q S P ſimi-<lb/>lia ſunt, &amp; </s>
  <s xml:id="echoid-s9829" xml:space="preserve">ſpatium R S parallelogrammum eſt; </s>
  <s xml:id="echoid-s9830" xml:space="preserve">eritque vt prius dictum eſt <lb/>proportio quadrati Q S ad rectangulum A S P eadem proportioni rectangnli C <lb/>R A ad quadratum R Q; </s>
  <s xml:id="echoid-s9831" xml:space="preserve">eſt vero quadratum Q S ad rectangulum A S P, vt <lb/>ellipſis axis tranſuerſus C A ad eius latus rectùm A D, propterea quod conus <lb/>A Q P ſupponitur continere ellipſim A B C; </s>
  <s xml:id="echoid-s9832" xml:space="preserve">igitur rectangulum C R A ad qua-<lb/>dratum R Q eandem proportionem habet, quàm C A ad A D; </s>
  <s xml:id="echoid-s9833" xml:space="preserve">eſt verò rectan-<lb/>gulum H R Q æquale rectangulo C R A; </s>
  <s xml:id="echoid-s9834" xml:space="preserve">igitur rectangulum H R Q ad qua-<lb/>dratum R Q ſeu H R ad R Q eandem proportionem habebit, quàm C A ad A <lb/>D; </s>
  <s xml:id="echoid-s9835" xml:space="preserve">ſed in priori caſu facta eſt H I ad I K in eadem proportione, quàm C A <lb/>ad A D; </s>
  <s xml:id="echoid-s9836" xml:space="preserve">igitur H R ad R Q eandem proportionem habebit quàm H I ad I K.</s>
  <s xml:id="echoid-s9837" xml:space="preserve"/>
</p>
<div xml:id="echoid-div817" type="float" level="2" n="4">
<note position="right" xlink:label="note-0298-01" xlink:href="note-0298-01a" xml:space="preserve">d</note>
</div>
<p style="it">
  <s xml:id="echoid-s9838" xml:space="preserve">Ergo diuidendo H K maior ad minorem K I erit vt minor H Q ad ma-<lb/>
<anchor type="note" xlink:label="note-0299-01a" xlink:href="note-0299-01"/>
iorem Q R, &amp;</s>
  <s xml:id="echoid-s9839" xml:space="preserve">c. </s>
  <s xml:id="echoid-s9840" xml:space="preserve">Ideſt quia H R ad R Q eſt vt H I ad I K, &amp; </s>
  <s xml:id="echoid-s9841" xml:space="preserve">diuiden-<lb/>do H Q ad Q R eandem proportionem habebit quàm H K ad K I, &amp; </s>
  <s xml:id="echoid-s9842" xml:space="preserve">permu-<lb/>tando H Q ad H K erit vt Q R ad K I: </s>
  <s xml:id="echoid-s9843" xml:space="preserve">quod eſt abſurdum; </s>
  <s xml:id="echoid-s9844" xml:space="preserve">quandoquidem <lb/>in circulo ſubtenſa H Q à centro remotior minor eſt, quàm H K, at exterius <lb/>comprehenſa Q R maior eſt, quàm K I. </s>
  <s xml:id="echoid-s9845" xml:space="preserve">Quapropter fieri non poteſt, vt ali-<lb/>quis alius conus A Q P præter iam dictos contineat ellipſim A B C, &amp; </s>
  <s xml:id="echoid-s9846" xml:space="preserve">ſit ſi-<lb/>milis dato cono E F G. </s>
  <s xml:id="echoid-s9847" xml:space="preserve">Textus ergo confuſus corrigi debebat.</s>
  <s xml:id="echoid-s9848" xml:space="preserve"/>
</p>
<div xml:id="echoid-div818" type="float" level="2" n="5">
<note position="left" xlink:label="note-0299-01" xlink:href="note-0299-01a" xml:space="preserve">e</note>
</div>
<p style="it">
  <s xml:id="echoid-s9849" xml:space="preserve">Ad propoſitionem 77. </s>
  <s xml:id="echoid-s9850" xml:space="preserve">libri quinti egi de <lb/>
<anchor type="figure" xlink:label="fig-0299-01a" xlink:href="fig-0299-01"/>
contactibus circulorum, &amp; </s>
  <s xml:id="echoid-s9851" xml:space="preserve">ſectionum coni-<lb/>carum, eorumque admirabilia ſymptomata à <lb/>nemine adhuc quod ſciam excogitata patefeci, <lb/>non tamen prædicta diſceptatio omnino perfe-<lb/>cta, &amp; </s>
  <s xml:id="echoid-s9852" xml:space="preserve">abſoluta fuit: </s>
  <s xml:id="echoid-s9853" xml:space="preserve">itaque iuxta loci exigen-<lb/>tiam hic afferam coronidis loco eiuſdem doctri-<lb/>næ complementum.</s>
  <s xml:id="echoid-s9854" xml:space="preserve"/>
</p>
<div xml:id="echoid-div819" type="float" level="2" n="6">
  <figure xlink:label="fig-0299-01" xlink:href="fig-0299-01a">
    <image file="0299-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0299-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9855" xml:space="preserve">Per rectam lineam coniungentem ver-<lb/>
<anchor type="note" xlink:label="note-0299-02a" xlink:href="note-0299-02"/>
tices duorum conorum eandem baſim ha-<lb/>bentium ducere duo plana vtrumque co-<lb/>num tangentia: </s>
  <s xml:id="echoid-s9856" xml:space="preserve">oportet autem rectam li-<lb/>neam vertices coniungentem extra peri-<lb/>pheriam circuli communis baſis cadere.</s>
  <s xml:id="echoid-s9857" xml:space="preserve"/>
</p>
<div xml:id="echoid-div820" type="float" level="2" n="7">
<note position="right" xlink:label="note-0299-02" xlink:href="note-0299-02a" xml:space="preserve">PROP. <lb/>15. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9858" xml:space="preserve">Circulus A M C ſit communis baſis duorum <lb/>conorum, quorum vertices B, &amp; </s>
  <s xml:id="echoid-s9859" xml:space="preserve">E, &amp; </s>
  <s xml:id="echoid-s9860" xml:space="preserve">co-<lb/>niuncta recta linea B E extra peripheriam <lb/>circuli A M C cadat: </s>
  <s xml:id="echoid-s9861" xml:space="preserve">duci debent duo plana <lb/>tangentia vtroſque conos per eandem rectam <lb/>lineam B E extenſa. </s>
  <s xml:id="echoid-s9862" xml:space="preserve">Et primo recta linea <lb/>E B plano circuli A M C æquidiſtet, &amp; </s>
  <s xml:id="echoid-s9863" xml:space="preserve">ducto <lb/>quolibet plano per E B circulum ſecante in <lb/>recta linea N O erit ipſa N O pirallela E B; <lb/></s>
  <s xml:id="echoid-s9864" xml:space="preserve">tunc ducatur diameter A M perpendicularis <lb/>ad N O, &amp; </s>
  <s xml:id="echoid-s9865" xml:space="preserve">per A, &amp; </s>
  <s xml:id="echoid-s9866" xml:space="preserve">M ducantur A D, M <lb/>V tangentes circulum, ſiue perpendiculares ad
<pb o="262" file="0300" n="300" rhead="Apollonij Pergæi"/>
idem diametrum M A; </s>
  <s xml:id="echoid-s9867" xml:space="preserve">erunt igitur tangentes <lb/>
<anchor type="figure" xlink:label="fig-0300-01a" xlink:href="fig-0300-01"/>
A D, &amp; </s>
  <s xml:id="echoid-s9868" xml:space="preserve">M V parallelæ eidem N O, erat au-<lb/>tem E B parallela ipſi N O; </s>
  <s xml:id="echoid-s9869" xml:space="preserve">igitur duæ cir-<lb/>culum tangentes A B, &amp; </s>
  <s xml:id="echoid-s9870" xml:space="preserve">M V parallelæ ſunt <lb/>idem E B; </s>
  <s xml:id="echoid-s9871" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9872" xml:space="preserve">propterea A D, &amp; </s>
  <s xml:id="echoid-s9873" xml:space="preserve">E B in eo-<lb/>dem ſunt plano, vtrumque conum tangente <lb/>cum per vertices E, &amp; </s>
  <s xml:id="echoid-s9874" xml:space="preserve">B ducatur, &amp; </s>
  <s xml:id="echoid-s9875" xml:space="preserve">per A <lb/>D baſis circulum tangentem. </s>
  <s xml:id="echoid-s9876" xml:space="preserve">Eadem ratione <lb/>M V, &amp; </s>
  <s xml:id="echoid-s9877" xml:space="preserve">E B ineodem plano vtrumque conum <lb/>tangente exiſtent. </s>
  <s xml:id="echoid-s9878" xml:space="preserve">Si verò recta E B plano cir-<lb/>culi non æquidiſtat producta alicubi planum <lb/>eiuſdem circuli ſecabit extra circulum ipſum, <lb/>vt in γ, &amp; </s>
  <s xml:id="echoid-s9879" xml:space="preserve">tunc quidem à puncto γ extra, <lb/>circulum poſito ducantur duæ contingentes γ A, <lb/>&amp; </s>
  <s xml:id="echoid-s9880" xml:space="preserve">γ M. </s>
  <s xml:id="echoid-s9881" xml:space="preserve">Manifeſtum eſt, rectas lineas A γ, <lb/>B E in eodem plano iacere: </s>
  <s xml:id="echoid-s9882" xml:space="preserve">tranſit verò præ-<lb/>dictum planum per vertices B, &amp; </s>
  <s xml:id="echoid-s9883" xml:space="preserve">E duorum <lb/>conorum, atque per γ A tangentem circulum <lb/>baſis communis; </s>
  <s xml:id="echoid-s9884" xml:space="preserve">igitur planum A E B vtrum-<lb/>que conum contingit. </s>
  <s xml:id="echoid-s9885" xml:space="preserve">Eodem modo planum E <lb/>B M ex altera parte vtrumq; </s>
  <s xml:id="echoid-s9886" xml:space="preserve">conum tanget. <lb/></s>
  <s xml:id="echoid-s9887" xml:space="preserve">Et hoc erat faciendum.</s>
  <s xml:id="echoid-s9888" xml:space="preserve"/>
</p>
<div xml:id="echoid-div821" type="float" level="2" n="8">
  <figure xlink:label="fig-0300-01" xlink:href="fig-0300-01a">
    <image file="0300-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0300-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9889" xml:space="preserve">In qualibet coniſectione H A I <lb/>
<anchor type="note" xlink:label="note-0300-01a" xlink:href="note-0300-01"/>
cuius diameter A L non ſit axis, <lb/>per eius verticem A aliam coniſe-<lb/>ctionem in eodem plano deſcribere, <lb/>quæ priorem abſcindat, atque eadem <lb/>recta linea vtramq; </s>
  <s xml:id="echoid-s9890" xml:space="preserve">ſectionem tangat <lb/>in puncto mutuæ earum abſcisſionis.</s>
  <s xml:id="echoid-s9891" xml:space="preserve"/>
</p>
<div xml:id="echoid-div822" type="float" level="2" n="9">
<note position="left" xlink:label="note-0300-01" xlink:href="note-0300-01a" xml:space="preserve">PROP <lb/>16. <lb/>Addit</note>
</div>
<p style="it">
  <s xml:id="echoid-s9892" xml:space="preserve">Sicut in conſtructione prop. </s>
  <s xml:id="echoid-s9893" xml:space="preserve">11. </s>
  <s xml:id="echoid-s9894" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9895" xml:space="preserve">12. <lb/></s>
  <s xml:id="echoid-s9896" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s9897" xml:space="preserve">factum eſt, deſcribatur conus B A <lb/>C comprehendens ſectionem H A I, cu <lb/>ius vertex B baſis circulus A M C per <lb/>ſectionis verticem A ductus, &amp; </s>
  <s xml:id="echoid-s9898" xml:space="preserve">trian-<lb/>gulum per axim B A C efficiat diame-<lb/>trum A L: </s>
  <s xml:id="echoid-s9899" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9900" xml:space="preserve">in duobus circulis æqui-<lb/>diſtantibus A C M, &amp; </s>
  <s xml:id="echoid-s9901" xml:space="preserve">in eo, qui per <lb/>ſectionis baſim H I ducitur idẽ planum <lb/>ſectionis conicæ deſignet duas parallelas <lb/>A D, H I, &amp; </s>
  <s xml:id="echoid-s9902" xml:space="preserve">planum trianguli per axim <lb/>efficiat circulorũ diamctros C A, &amp; </s>
  <s xml:id="echoid-s9903" xml:space="preserve">eum, <lb/>qui per L ducitur æquidiſtantes inter ſe: </s>
  <s xml:id="echoid-s9904" xml:space="preserve"><lb/>ergo ſicuti baſis H I perpendicularis eſt <lb/>ad circuli diametrum per L ductam, ſeu <lb/>ad baſim trianguli per axim, ita D A
<pb o="263" file="0301" n="301" rhead="Conicor. Lib. VI."/>
perpendicularis eſt ad circuli diametrum C A, &amp; </s>
  <s xml:id="echoid-s9905" xml:space="preserve">propterea A D, planorum <lb/>H A I, &amp; </s>
  <s xml:id="echoid-s9906" xml:space="preserve">A C M communis ſectio, tanget circulum A C, &amp; </s>
  <s xml:id="echoid-s9907" xml:space="preserve">ideo ſuperficiem <lb/>ipſam conicam, &amp; </s>
  <s xml:id="echoid-s9908" xml:space="preserve">ſectionem in ea exiſtentem continget; </s>
  <s xml:id="echoid-s9909" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9910" xml:space="preserve">diameter A L non <lb/>erit perpendicularis ad tangentem, ſeu ordinatim applicatam A D per verticem <lb/>A, alias A L eſſet axis, quod non ponitur. </s>
  <s xml:id="echoid-s9911" xml:space="preserve">Deinde in plano D A B ex A du-<lb/>catur recta linea A E perpendicularis ad A D ſupra, vel infra circulum, &amp; </s>
  <s xml:id="echoid-s9912" xml:space="preserve"><lb/>vertice quolibet puncto E ſumpto in recta linea A E, &amp; </s>
  <s xml:id="echoid-s9913" xml:space="preserve">baſi circulo A C M fiat <lb/>alter conus E A C, in cuius ſuperficie planũ D A H I deſignet ſectionẽ F A G, &amp; </s>
  <s xml:id="echoid-s9914" xml:space="preserve"><lb/>in ea triangulum per axim E A C efficiat diametrum A K: </s>
  <s xml:id="echoid-s9915" xml:space="preserve">Et quia eadem re-<lb/>cta linea D A perpendicularis eſt ad A C, atque ad A E ſe ſecantes in A; </s>
  <s xml:id="echoid-s9916" xml:space="preserve">ergo <lb/>D A perpendicularis eſt ad planum C E A, atque planum D A C extenſum <lb/>per perpendicularem D A, erit quoque perpendiculare ad planum trianguli per <lb/>axim C E A, quare triangulum per axim efficiet diametrum A K, quæ erit <lb/>
<anchor type="figure" xlink:label="fig-0301-01a" xlink:href="fig-0301-01"/>
axis ſectionis F A G, atque D A perpendicularis erit ad axim A K exiſtentem <lb/>in plano C E A, ad quod D A eſt perpendicularis, &amp; </s>
  <s xml:id="echoid-s9917" xml:space="preserve">cum ea conuenit: </s>
  <s xml:id="echoid-s9918" xml:space="preserve">quare <lb/>D A ordinatim ad axim applicata perverticem A tanget ſectionem F A G, quæ <lb/>
<anchor type="note" xlink:label="note-0301-01a" xlink:href="note-0301-01"/>
prius in eodem puncto A tangebat ſectionem H A I in eodem plano exiſtentem; <lb/></s>
  <s xml:id="echoid-s9919" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9920" xml:space="preserve">propterea eadem recta A D vtramque ſectionem tangit in puncto A. </s>
  <s xml:id="echoid-s9921" xml:space="preserve">Poſtea <lb/>coniungatur recta linea B E, &amp; </s>
  <s xml:id="echoid-s9922" xml:space="preserve">quia rectæ lineæ B A, A D, A E ſunt in eo-<lb/>dem plano tangente vtrumque conum (cum per vertices B, &amp; </s>
  <s xml:id="echoid-s9923" xml:space="preserve">E, atque per D <lb/>A contingentem circulum baſis communis ducatur) &amp; </s>
  <s xml:id="echoid-s9924" xml:space="preserve">E A, &amp; </s>
  <s xml:id="echoid-s9925" xml:space="preserve">B A angulum <lb/>conſtituunt, cum E A poſita ſit perpendicularis ad D A, at B A ad eandem ſit <lb/>inclinata, &amp; </s>
  <s xml:id="echoid-s9926" xml:space="preserve">exiſtunt in eodem plano; </s>
  <s xml:id="echoid-s9927" xml:space="preserve">ergo recta B E parallela eſt, aut ſecat <lb/>contingentem D A extra circulum vt in D. </s>
  <s xml:id="echoid-s9928" xml:space="preserve">Poterit igitur ex propoſ. </s>
  <s xml:id="echoid-s9929" xml:space="preserve">15. </s>
  <s xml:id="echoid-s9930" xml:space="preserve">addi-<lb/>tarum duci per rectam B E planum aliud B E M V vtrumq; </s>
  <s xml:id="echoid-s9931" xml:space="preserve">conum contingens,
<pb o="264" file="0302" n="302" rhead="Apollonij Pergæi"/>
&amp; </s>
  <s xml:id="echoid-s9932" xml:space="preserve">per rectam B E extendatur aliud planum E N O B inter duo plana contin-<lb/>gentia prope verticem A vbicumq; </s>
  <s xml:id="echoid-s9933" xml:space="preserve">cadens, quod ſecet vtrumque conum, &amp; </s>
  <s xml:id="echoid-s9934" xml:space="preserve">cir-<lb/>culum baſis in recta linea N O, &amp; </s>
  <s xml:id="echoid-s9935" xml:space="preserve">ſuperficies duorum conorum in lateribus B <lb/>N Q, E N, B O, E O R, quarum B N occurret ſemiſectioni A H in quolibet <lb/>eius puncto Q prope verticem A, eo quod portio A H, &amp; </s>
  <s xml:id="echoid-s9936" xml:space="preserve">peripheria A N C ex <lb/>cepto puncto eius A totæ inter duo plana conos tangentia intercipiuntur; </s>
  <s xml:id="echoid-s9937" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9938" xml:space="preserve">eadem <lb/>ratione E O occurret ſemiſectioni A G in quolibet eius puncto R vltra verticem <lb/>A ad partes G. </s>
  <s xml:id="echoid-s9939" xml:space="preserve">Et quoniam in eo-<lb/>
<anchor type="figure" xlink:label="fig-0302-01a" xlink:href="fig-0302-01"/>
dem plano trianguli E N B (ſcili-<lb/>cet plani B N O E ſecantis vtrum-<lb/>que conum) à puncto E ducitur re-<lb/>cta linea E O intra angulum N E B; <lb/></s>
  <s xml:id="echoid-s9940" xml:space="preserve">ergo vlterius producta ſecabit latus <lb/>B N ſubtendentem angulum N E B <lb/>inter puncta N, &amp; </s>
  <s xml:id="echoid-s9941" xml:space="preserve">B, vt in X, &amp; </s>
  <s xml:id="echoid-s9942" xml:space="preserve"><lb/>propterearecta linea N X intra triã-<lb/>gulum E N O, &amp; </s>
  <s xml:id="echoid-s9943" xml:space="preserve">ideo intra conum <lb/>E A C intercepta erit; </s>
  <s xml:id="echoid-s9944" xml:space="preserve">ſimiliter re-<lb/>cta linea O X intra triangulum B N <lb/>O, &amp; </s>
  <s xml:id="echoid-s9945" xml:space="preserve">intra conum B A C interclu-<lb/>ſa erit: </s>
  <s xml:id="echoid-s9946" xml:space="preserve">quare quodlibet aliud punctũ <lb/>Qlateris conici B N citra, vel vltra <lb/>interclusã portionẽ N X cadet neceſ-<lb/>ario extra ſuperficiem coni E A C, <lb/>&amp; </s>
  <s xml:id="echoid-s9947" xml:space="preserve">ideo quodlibet punctum Q in pro-<lb/>ductione lateris coni B N ſumptum <lb/>&amp; </s>
  <s xml:id="echoid-s9948" xml:space="preserve">in ſemiſſe ſectionis conicæ H A <lb/>prope verticem A cadet extra ſemiſ-<lb/>ſem ſectionis F A, quæ in ſuperfi-<lb/>cie coni E A C exiſtit, &amp; </s>
  <s xml:id="echoid-s9949" xml:space="preserve">ad eaſ-<lb/>dem partes vergit. </s>
  <s xml:id="echoid-s9950" xml:space="preserve">Pari modo quod-<lb/>libet aliud punctum R lateris conici <lb/>E O citra, vel vltra intercluſam <lb/>portionẽ X O cadet extra ſuperſiciem <lb/>coni B A C, &amp; </s>
  <s xml:id="echoid-s9951" xml:space="preserve">ideo quodlibet punctũ <lb/>R ſumptum in medietate ſectionis <lb/>conicæ A G prope verticem A cadet <lb/>extra medietatem ſectionis A I, quæ <lb/>in ſuperficie coni B A C exiſtit, &amp; </s>
  <s xml:id="echoid-s9952" xml:space="preserve"><lb/>ad eaſdem partes vergit. </s>
  <s xml:id="echoid-s9953" xml:space="preserve">Igitur ſe-<lb/>ctio H A I abſcindit coniſectionem <lb/>F A G in vertice communi A, vbi <lb/>ambo tanguntur ab eadem recta li-<lb/>nea A D. </s>
  <s xml:id="echoid-s9954" xml:space="preserve">Quod erat faciendum.</s>
  <s xml:id="echoid-s9955" xml:space="preserve"/>
</p>
<div xml:id="echoid-div823" type="float" level="2" n="10">
  <figure xlink:label="fig-0301-01" xlink:href="fig-0301-01a">
    <image file="0301-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0301-01"/>
  </figure>
<note position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">32. lib. I.</note>
  <figure xlink:label="fig-0302-01" xlink:href="fig-0302-01a">
    <image file="0302-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0302-01"/>
  </figure>
</div>
<pb o="265" file="0303" n="303" rhead="Conicor. Lib. VI."/>
<p style="it">
  <s xml:id="echoid-s9956" xml:space="preserve">Si fuerint quotcunque coni <lb/>
<anchor type="note" xlink:label="note-0303-01a" xlink:href="note-0303-01"/>
<anchor type="figure" xlink:label="fig-0303-01a" xlink:href="fig-0303-01"/>
ſuper circulum communem ba-<lb/>ſis deſcripti, habentes latus com-<lb/>mune indefinitè extenſum in-<lb/>triangulis per axes ad baſes <lb/>perpendicularibus, atque per ter-<lb/>minum lateris communis duca-<lb/>tur planum efficiens coni ſectio-<lb/>nes tangentes baſim: </s>
  <s xml:id="echoid-s9957" xml:space="preserve">habebunt <lb/>illæ latera recta æqualia inter <lb/>ſe, eritquè ſectio ſingularis, ſi <lb/>fuerit par abole, vel circulus: <lb/></s>
  <s xml:id="echoid-s9958" xml:space="preserve">ſi verò fuerit ellipſis, aut hy-<lb/>perbole erunt infinitæ.</s>
  <s xml:id="echoid-s9959" xml:space="preserve"/>
</p>
<div xml:id="echoid-div824" type="float" level="2" n="11">
<note position="right" xlink:label="note-0303-01" xlink:href="note-0303-01a" xml:space="preserve">PROP. <lb/>17. <lb/>Addit.</note>
  <figure xlink:label="fig-0303-01" xlink:href="fig-0303-01a">
    <image file="0303-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0303-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s9960" xml:space="preserve">Sit conus A D C ſingularis, &amp; </s>
  <s xml:id="echoid-s9961" xml:space="preserve"><lb/>A B C ſit multiplex, habentes cir-<lb/>culum A C baſeos communem, &amp; </s>
  <s xml:id="echoid-s9962" xml:space="preserve"><lb/>latus A B D productum commu-<lb/>ne ſumptum ſit in triangulis per <lb/>axes conorum perpendicularibus ad <lb/>circulum baſis B C, atque à ter-<lb/>mino A ducatur planũ ſecans cir-<lb/>culi A C planum in recta linea, <lb/>quæ perpendicularis ſit ad diame-<lb/>trum C A, quod efficiat in cono <lb/>quidem A B C ſectionem A N, <lb/>cuius latus rectum ſit X, &amp; </s>
  <s xml:id="echoid-s9963" xml:space="preserve">latus <lb/>tranſuerſum A F: </s>
  <s xml:id="echoid-s9964" xml:space="preserve">in cono verò <lb/>A D C efficiat ſectionem A M, cu-<lb/>ius latus rectum Z, &amp; </s>
  <s xml:id="echoid-s9965" xml:space="preserve">diameter <lb/>communis A E; </s>
  <s xml:id="echoid-s9966" xml:space="preserve">ſitque ſectio A N <lb/>hyperbole, circulus, aut ellipſis <lb/>circa axim maiorem, aut mino-<lb/>rem; </s>
  <s xml:id="echoid-s9967" xml:space="preserve">Sectio verò ſingularis A M in cono D A C ſit parabole, &amp; </s>
  <s xml:id="echoid-s9968" xml:space="preserve">ducatur B H <lb/>parallela diametro ſectionis A E ſecans circuli diametrum A C in H: </s>
  <s xml:id="echoid-s9969" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9970" xml:space="preserve">du-<lb/>catur C O parallela D A ſecans A E in O. </s>
  <s xml:id="echoid-s9971" xml:space="preserve">Dico latus rectum Z paraboles A M <lb/>æquale eſſe lateri recto X cuiuſlibet alterius ſectionis A N; </s>
  <s xml:id="echoid-s9972" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9973" xml:space="preserve">ſupponantur tres <lb/>parabolæ A M inter ſe æquales earumq; </s>
  <s xml:id="echoid-s9974" xml:space="preserve">latera recta Z æqualia, quæ in tribus fi-<lb/>guris apponẽtur, vt confuſio euitetur. </s>
  <s xml:id="echoid-s9975" xml:space="preserve">Quoniam vt latus rectum X ad tran-<lb/>ſuerſum A F ſectionis A N, ita eſt rectangulum A H C ad quadratum B H: <lb/></s>
  <s xml:id="echoid-s9976" xml:space="preserve">
<anchor type="note" xlink:label="note-0303-02a" xlink:href="note-0303-02"/>
hæc verò proportio componitur ex ratione C H ad H B, &amp; </s>
  <s xml:id="echoid-s9977" xml:space="preserve">ex ratione A H ad <lb/>H B: </s>
  <s xml:id="echoid-s9978" xml:space="preserve">eſtque C A ad A F, vt C H ad H B (propter parallelas F A, H B, &amp; </s>
  <s xml:id="echoid-s9979" xml:space="preserve"><lb/>ſimilitudinem triangulorum) &amp; </s>
  <s xml:id="echoid-s9980" xml:space="preserve">vt A H ad H B, ita eſt A C ad C D, ſeu ad
<pb o="266" file="0304" n="304" rhead="Apollonij Pergæi"/>
A O (cum C D, &amp; </s>
  <s xml:id="echoid-s9981" xml:space="preserve">H B ſint parallelæ, atque D O ſit parallelogrammum) com-<lb/>ponunt verò hæ duæ proportiones rationem quadrati C A ad rectangulum F <lb/>A O: </s>
  <s xml:id="echoid-s9982" xml:space="preserve">ergo vt rectangulum A H C ad quadratum H B; </s>
  <s xml:id="echoid-s9983" xml:space="preserve">ita eſt quadratum C A <lb/>ad rectangulum F A O, &amp; </s>
  <s xml:id="echoid-s9984" xml:space="preserve">pro-<lb/>
<anchor type="figure" xlink:label="fig-0304-01a" xlink:href="fig-0304-01"/>
pterea vt X ad A F, ita erit qua-<lb/>dratum A C ad rectangulum F A <lb/>O, ſed vt F A ad A D (ſum-<lb/>ptis æqualibus altitudinibus A O, <lb/>C D) ita eſt rectangulum F A O <lb/>ad rectangulum A D C; </s>
  <s xml:id="echoid-s9985" xml:space="preserve">quare ex <lb/>æquali X ad A D erit vt quadra-<lb/>tum A C ad rectangulum A D C; <lb/></s>
  <s xml:id="echoid-s9986" xml:space="preserve">tandem vt Z latus rectum para-<lb/>boles A M ad D A ita eſt quadra-<lb/>
<anchor type="note" xlink:label="note-0304-01a" xlink:href="note-0304-01"/>
tum A C ad rectangulum A D C; <lb/></s>
  <s xml:id="echoid-s9987" xml:space="preserve">igitur X, &amp; </s>
  <s xml:id="echoid-s9988" xml:space="preserve">Z ad eandem D A <lb/>habent eandem proportionem quàm <lb/>quadr atum A C ad rectangulum <lb/>A D C, &amp; </s>
  <s xml:id="echoid-s9989" xml:space="preserve">propterea latera recta <lb/>X, &amp; </s>
  <s xml:id="echoid-s9990" xml:space="preserve">Z æqualia ſunt inter ſe. </s>
  <s xml:id="echoid-s9991" xml:space="preserve"><lb/>Et quoniam in quolibet caſu ſectio-<lb/>nis conicæ A N latus rectum X <lb/>ſemper æquale eſt Z lateri recto <lb/>vnius eiuſdemq; </s>
  <s xml:id="echoid-s9992" xml:space="preserve">paraboles A M; </s>
  <s xml:id="echoid-s9993" xml:space="preserve"><lb/>ergo latera recta X reliquarum <lb/>omnium ſectionum æqualia ſunt <lb/>inter ſe, licet ſectiones illæ ſint <lb/>inæquales, &amp; </s>
  <s xml:id="echoid-s9994" xml:space="preserve">habeant latera trã-<lb/>ſuerſa inæqualia, imò neque eiuſ-<lb/>dem ſpeciei ſint. </s>
  <s xml:id="echoid-s9995" xml:space="preserve">Quod erat pro-<lb/>poſitum.</s>
  <s xml:id="echoid-s9996" xml:space="preserve"/>
</p>
<div xml:id="echoid-div825" type="float" level="2" n="12">
<note position="right" xlink:label="note-0303-02" xlink:href="note-0303-02a" xml:space="preserve">12. &amp; 13 <lb/>lib. I.</note>
  <figure xlink:label="fig-0304-01" xlink:href="fig-0304-01a">
    <image file="0304-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0304-01"/>
  </figure>
<note position="left" xlink:label="note-0304-01" xlink:href="note-0304-01a" xml:space="preserve">II. lib. I.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9997" xml:space="preserve">Admiratione dignum præcipuè <lb/>eſt in hac propoſitione, quod ſi ſe-<lb/>ctio A N fuerit circulus, vnicus <lb/>tantummodò erit; </s>
  <s xml:id="echoid-s9998" xml:space="preserve">nam circuli la-<lb/>tus rectum X æquale erit eius dia-<lb/>metro, ſeu axi tranſuerſo A F; </s>
  <s xml:id="echoid-s9999" xml:space="preserve">eſt-<lb/>que ſemper latus rectum eiuſdem <lb/>menſuræ, vt aſtenſum eſt; </s>
  <s xml:id="echoid-s10000" xml:space="preserve">igitur <lb/>circuli diameter F A idem ſemper erit; </s>
  <s xml:id="echoid-s10001" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10002" xml:space="preserve">propterea circulus, qui à tali plano <lb/>generari poteſt ſingularis erit, nimirum ille, qui in vnico cono A B C efficit <lb/>triangula per axim ſimilia, &amp; </s>
  <s xml:id="echoid-s10003" xml:space="preserve">ſubcontraria B A C, &amp; </s>
  <s xml:id="echoid-s10004" xml:space="preserve">B F A. </s>
  <s xml:id="echoid-s10005" xml:space="preserve">Manifeſtum <lb/>quoq; </s>
  <s xml:id="echoid-s10006" xml:space="preserve">eſt parabolem A M ſingularem eße, nam ſupponitur idem circulus baſis A <lb/>C, &amp; </s>
  <s xml:id="echoid-s10007" xml:space="preserve">in plano per axim coni cõmune latus A D B ſemper eoſdẽ angulos D A E, <lb/>&amp; </s>
  <s xml:id="echoid-s10008" xml:space="preserve">D A C efficere conceditur; </s>
  <s xml:id="echoid-s10009" xml:space="preserve">igitur vt ſectio A M ſit parabole neceßariò recta à <lb/>puncto C duci debet parallela diametro par aboles A E; </s>
  <s xml:id="echoid-s10010" xml:space="preserve">cum ergo in triangulo per <lb/>axim D A C detur baſis A C inuariabilis quia circulus vnicus ſupponitur eiuſ-
<pb o="267" file="0305" n="305" rhead="Conicor. Lib. VI."/>
què anguli D, &amp; </s>
  <s xml:id="echoid-s10011" xml:space="preserve">D A C; </s>
  <s xml:id="echoid-s10012" xml:space="preserve">dabitur quoq; </s>
  <s xml:id="echoid-s10013" xml:space="preserve">eius ſpecies ſemper eadem, immo triã-<lb/>gulum per axim inuariabile erit, qui ſemper eodem modo inclinatur ad circu-<lb/>lum baſis C A: </s>
  <s xml:id="echoid-s10014" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10015" xml:space="preserve">propterea conus D A C ſemper idem erit, &amp; </s>
  <s xml:id="echoid-s10016" xml:space="preserve">eodem modo <lb/>ſectus, vnde ſectio par aboles A M eadem ſemper omnino erit, habens idem latus <lb/>rectum Z. </s>
  <s xml:id="echoid-s10017" xml:space="preserve">In hyperbole verò, aut ellipſi latera C B poſſunt ſupra, vel infra <lb/>C D parallelam ipſi A E à puncto C ductam, extendi, &amp; </s>
  <s xml:id="echoid-s10018" xml:space="preserve">ſic efficientur tranſuer-<lb/>ſa latera A F inæqualia inter ſe, cumque coni ſectiones A N habeant latera <lb/>
<anchor type="note" xlink:label="note-0305-01a" xlink:href="note-0305-01"/>
recta X æqualia inter ſe, latera verò tranſuerſa A F inæqualia, &amp; </s>
  <s xml:id="echoid-s10019" xml:space="preserve">hyperbola-<lb/>rum commune latus rectum habentium illa maior eſt, cuius axis tranſuerſus eſt <lb/>minor: </s>
  <s xml:id="echoid-s10020" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10021" xml:space="preserve">duarum ellipſium commune latus rectum habentium, illa maior eſt <lb/>cuius axis tranſuerſus eſt maior; </s>
  <s xml:id="echoid-s10022" xml:space="preserve">igitur ellipſes, aut byperbole, quæ in conis <lb/>prædicta lege conſtructis deſcribuntur non ſingulares ſed infinitæ eſſe poßunt. <lb/></s>
  <s xml:id="echoid-s10023" xml:space="preserve">Vbi notandum eſt, quod ellipſes poßunt eſſe eæ quæ ad maiores, aut ad minores <lb/>axes adiacent. </s>
  <s xml:id="echoid-s10024" xml:space="preserve">Pari modo conſtat quod ſi in conis ſuperius expoſitis fiant ſe-<lb/>ctiones conicæ conſtituentur ad eundem axim quinque ſectiones commune latus <lb/>rectum habentes ſe ſe in eodem vertice tangentes, &amp; </s>
  <s xml:id="echoid-s10025" xml:space="preserve">earum intima erit elli-<lb/>
<anchor type="note" xlink:label="note-0305-02a" xlink:href="note-0305-02"/>
pſis, quæ ad axim minorem adiacet, &amp; </s>
  <s xml:id="echoid-s10026" xml:space="preserve">non erit vnica, ſed multiplex, &amp; </s>
  <s xml:id="echoid-s10027" xml:space="preserve">om-<lb/>nes cadent intra circulum, circulus verò intra ellipſim ad axim maiorem acco-<lb/>modatam cadet, hæc verò intra parabolen conſtituetur, &amp; </s>
  <s xml:id="echoid-s10028" xml:space="preserve">inter circulum, &amp; </s>
  <s xml:id="echoid-s10029" xml:space="preserve"><lb/>parabolen infinitæ ellipſes ſe in eodem puncto verticis tangentes collocari poſ-<lb/>ſunt. </s>
  <s xml:id="echoid-s10030" xml:space="preserve">T andem parabole compræhendetur ab infinitis alĳs hyperbolis ſe ſe in eo-<lb/>dem puncto tangentibus.</s>
  <s xml:id="echoid-s10031" xml:space="preserve"/>
</p>
<div xml:id="echoid-div826" type="float" level="2" n="13">
<note position="right" xlink:label="note-0305-01" xlink:href="note-0305-01a" xml:space="preserve">Maurol. <lb/>2. lib. 5. <lb/>Conic.</note>
<note position="right" xlink:label="note-0305-02" xlink:href="note-0305-02a" xml:space="preserve">Maurol. <lb/>prop. 28. <lb/>lib. 5. <lb/>Conic.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10032" xml:space="preserve">Si in qualibet coniſectione B A C <lb/>
<anchor type="note" xlink:label="note-0305-03a" xlink:href="note-0305-03"/>
<anchor type="figure" xlink:label="fig-0305-01a" xlink:href="fig-0305-01"/>
ducatur breuiſecans ſingularis D A, <lb/>tunc quælibet alia coniſectio M A <lb/>N, cuius axis ſit eadem breuiſe-<lb/>cans, &amp; </s>
  <s xml:id="echoid-s10033" xml:space="preserve">A L ſemiſſis erecti eius <lb/>minor ſit eadem ſingulari breuiſecan-<lb/>te A D. </s>
  <s xml:id="echoid-s10034" xml:space="preserve">Dico ſectionem M A N <lb/>interius contingere priorem ſectionem <lb/>B A C in A.</s>
  <s xml:id="echoid-s10035" xml:space="preserve"/>
</p>
<div xml:id="echoid-div827" type="float" level="2" n="14">
<note position="right" xlink:label="note-0305-03" xlink:href="note-0305-03a" xml:space="preserve">PROP. <lb/>18. <lb/>Addit. <lb/>ex 51. 52. <lb/>lib. 5.</note>
  <figure xlink:label="fig-0305-01" xlink:href="fig-0305-01a">
    <image file="0305-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0305-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s10036" xml:space="preserve">Quia A L minor eſt, quàm A D <lb/>ſumi poterit recta A O maior quidem quàm A L, &amp; </s>
  <s xml:id="echoid-s10037" xml:space="preserve">minor quàm A D, &amp; </s>
  <s xml:id="echoid-s10038" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0305-04a" xlink:href="note-0305-04"/>
centro O interuallo O A deſcribatur circulus P A Q. </s>
  <s xml:id="echoid-s10039" xml:space="preserve">Manifeſtum eſt, quod <lb/>circulus P A Q ſectionem M A N exterius continget in A, at circulus P A <lb/>Q interius priorem ſectionem B A C tanget, vt oſtenſum eſt, igitur coni ſe-<lb/>
<anchor type="note" xlink:label="note-0305-05a" xlink:href="note-0305-05"/>
ctio M A N continget ſectionem B A C interius in A. </s>
  <s xml:id="echoid-s10040" xml:space="preserve">Quod erat oſtenden-<lb/>dum.</s>
  <s xml:id="echoid-s10041" xml:space="preserve"/>
</p>
<div xml:id="echoid-div828" type="float" level="2" n="15">
<note position="right" xlink:label="note-0305-04" xlink:href="note-0305-04a" xml:space="preserve">Maurol. <lb/>pr.4.7.10. <lb/>14. lib. 5.</note>
<note position="right" xlink:label="note-0305-05" xlink:href="note-0305-05a" xml:space="preserve">Conic. <lb/>Prop. 12. <lb/>Addit. <lb/>lib. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10042" xml:space="preserve">Iiſdem poſitis ſi ſectionis T A V, cuius axis A D ſemiſſis eius e-<lb/>
<anchor type="note" xlink:label="note-0305-06a" xlink:href="note-0305-06"/>
recti fuerit A R maior quàm D A, quæ eſt ſingularis breuiſecans ſe-<lb/>ctionis B A C. </s>
  <s xml:id="echoid-s10043" xml:space="preserve">Dico, quod T A V exterius contingit ſectionem B A C <lb/>in A.</s>
  <s xml:id="echoid-s10044" xml:space="preserve"/>
</p>
<div xml:id="echoid-div829" type="float" level="2" n="16">
<note position="right" xlink:label="note-0305-06" xlink:href="note-0305-06a" xml:space="preserve">PROP. <lb/>19. Add.</note>
</div>
<pb o="268" file="0306" n="306" rhead="Apollonij Pergæi"/>
  <figure>
    <image file="0306-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0306-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s10045" xml:space="preserve">Quoniam A R maior ponitur quã <lb/>A D ſumi poterit recta A X minor <lb/>quidem, quàm A R, ſed maior quã <lb/>A D, &amp; </s>
  <s xml:id="echoid-s10046" xml:space="preserve">centro X interuallo X A <lb/>deſcribatur circulus I A S. </s>
  <s xml:id="echoid-s10047" xml:space="preserve">Patet <lb/>
<anchor type="note" xlink:label="note-0306-01a" xlink:href="note-0306-01"/>
(ex demonſtratis ſuperius) circulum <lb/>I S extrinſecus tangere coniſectionem <lb/>B A C; </s>
  <s xml:id="echoid-s10048" xml:space="preserve">at ſectio T V extrinſecus <lb/>
<anchor type="note" xlink:label="note-0306-02a" xlink:href="note-0306-02"/>
circulum I S tangit in eodem puncto <lb/>verticis A, ergo ſectio T V extrin-<lb/>ſecus tangit coniſectionem B A C in <lb/>eodem puncto A. </s>
  <s xml:id="echoid-s10049" xml:space="preserve">Quod erat oſten-<lb/>dendum.</s>
  <s xml:id="echoid-s10050" xml:space="preserve"/>
</p>
<div xml:id="echoid-div830" type="float" level="2" n="17">
<note position="left" xlink:label="note-0306-01" xlink:href="note-0306-01a" xml:space="preserve">ex pr. 14. <lb/>addit. <lb/>lib. 5.</note>
<note position="left" xlink:label="note-0306-02" xlink:href="note-0306-02a" xml:space="preserve">Maurol. <lb/>pr. 3. 6. 9. <lb/>13. lib. 5. <lb/>Conic.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10051" xml:space="preserve">Si in eodem plano circulus F A G ſecuerit coniſectionem H A I in <lb/>
<anchor type="note" xlink:label="note-0306-03a" xlink:href="note-0306-03"/>
puncto A quod non ſit vertex axis eius, atque eadem recta linea D A <lb/>contingat circulum, &amp; </s>
  <s xml:id="echoid-s10052" xml:space="preserve">ſectionem in eodem puncto A; </s>
  <s xml:id="echoid-s10053" xml:space="preserve">Dico quod quæ-<lb/>libet alia coniſectio S A Z in eodem plano cum illis poſita cuius axis ſit <lb/>idem circuli diameter A K habens Y ſemiſſem lateris recti axis æqualẽ radio <lb/>circuli F A G: </s>
  <s xml:id="echoid-s10054" xml:space="preserve">ſecabit quoque eandem coniſectionem H A I in eodem <lb/>puncto A, atque continget eandem rectam lineam A D in A.</s>
  <s xml:id="echoid-s10055" xml:space="preserve"/>
</p>
<div xml:id="echoid-div831" type="float" level="2" n="18">
<note position="left" xlink:label="note-0306-03" xlink:href="note-0306-03a" xml:space="preserve">PROP. <lb/>20. <lb/>Addit. <lb/>ex 16. <lb/>addit. <lb/>huius.</note>
</div>
  <figure>
    <image file="0306-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0306-02"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s10056" xml:space="preserve">Deſcribantur (vt in 16. </s>
  <s xml:id="echoid-s10057" xml:space="preserve">additarum huius libri factum eſt) duo coni A B C, <lb/>Scalenus comprehendens ſectionem H A I, &amp; </s>
  <s xml:id="echoid-s10058" xml:space="preserve">conus rectus E A C comprehen-<lb/>dens circularem ſubcontrariam ſectionem F A G, quorum baſis communis ſit
<pb o="269" file="0307" n="307" rhead="Conicor. Lib. VI."/>
circulus A M C, ita vt idem planum per vertices conorum B, &amp; </s>
  <s xml:id="echoid-s10059" xml:space="preserve">E, &amp; </s>
  <s xml:id="echoid-s10060" xml:space="preserve">per <lb/>A D contingentem eundem circulum baſis extenſum tangat vtrumque conum <lb/>in lateribus A B, &amp; </s>
  <s xml:id="echoid-s10061" xml:space="preserve">A E. </s>
  <s xml:id="echoid-s10062" xml:space="preserve">Poſiea ſi S A Z optatur parabole ducatur in plano <lb/>A E C ex C recta C N parallela A K axi ſectionis F A G; </s>
  <s xml:id="echoid-s10063" xml:space="preserve">ſi verò S A Z <lb/>dſideratur hyperbole, aut ellipſis producatur axis A K in directum extra aut intra <lb/>ſectionem, &amp; </s>
  <s xml:id="echoid-s10064" xml:space="preserve">in recta linea K A O ſecetur portio A O æqualis lateri tranſuer-<lb/>ſo ſectionis S A Z, coniungaturque recta linea C O, ſecans E A in N (eo <lb/>quod axis K A in plano A E C erecto ad circulũ A M C, exiſtit) &amp; </s>
  <s xml:id="echoid-s10065" xml:space="preserve">vertice N <lb/>fiat alter conus N C A. </s>
  <s xml:id="echoid-s10066" xml:space="preserve">Manifeſtum eſt in cono recto E A C deſignari ab eo-<lb/>dem plano D A K circulum F A G, at in cono recto N A C efficietur alia ſe-<lb/>ctio conica circa communem axim A K, quæ ſe ſe mutuo, &amp; </s>
  <s xml:id="echoid-s10067" xml:space="preserve">eandem rectam <lb/>lineam D A tangent, in communi vertice A, atque circuli F A G, &amp; </s>
  <s xml:id="echoid-s10068" xml:space="preserve">ſectio-<lb/>
<anchor type="note" xlink:label="note-0307-01a" xlink:href="note-0307-01"/>
nis genitæ in cono N A C duo latera recta erunt æqualia, &amp; </s>
  <s xml:id="echoid-s10069" xml:space="preserve">propterea ſectio-<lb/>nis genitæ in cono N A C ſemilatus rectum æquale erit radio circuli γ ſeu di-<lb/>midio erecti ſectionis H A I, &amp; </s>
  <s xml:id="echoid-s10070" xml:space="preserve">ſi habuerit latus tranſuerſum erit æquale A <lb/>O; </s>
  <s xml:id="echoid-s10071" xml:space="preserve">ergo ſectio genita in cono N A C, &amp; </s>
  <s xml:id="echoid-s10072" xml:space="preserve">ſectio S A Z circa communem axim <lb/>A K habent latus rectum cummune duplum ipſius γ, &amp; </s>
  <s xml:id="echoid-s10073" xml:space="preserve">etiam commune latus <lb/>tranſuerſum A O: </s>
  <s xml:id="echoid-s10074" xml:space="preserve">Quare ſectio genita in cono N A C, &amp; </s>
  <s xml:id="echoid-s10075" xml:space="preserve">S A Z æquales ſunt <lb/>
<anchor type="note" xlink:label="note-0307-02a" xlink:href="note-0307-02"/>
inter ſe, &amp; </s>
  <s xml:id="echoid-s10076" xml:space="preserve">congruentes; </s>
  <s xml:id="echoid-s10077" xml:space="preserve">quapropter idem planum D A K, quod efficit in cono <lb/>Scaleno B A C ſectionem H A I, deſignat quoque in cono recto N A C ſectio-<lb/>nem S A Z: </s>
  <s xml:id="echoid-s10078" xml:space="preserve">habent verò hi duo coni circulum baſis communem, &amp; </s>
  <s xml:id="echoid-s10079" xml:space="preserve">idem pla-<lb/>num per contingentem A D, &amp; </s>
  <s xml:id="echoid-s10080" xml:space="preserve">per vertices B, &amp; </s>
  <s xml:id="echoid-s10081" xml:space="preserve">N ductum vtrumque co-<lb/>num tangit; </s>
  <s xml:id="echoid-s10082" xml:space="preserve">igitur (vt demonſtratum eſt in 16. </s>
  <s xml:id="echoid-s10083" xml:space="preserve">Addit. </s>
  <s xml:id="echoid-s10084" xml:space="preserve">huius) ſectio conica <lb/>S A Z abſcindet aliam ſectionem H A I, &amp; </s>
  <s xml:id="echoid-s10085" xml:space="preserve">ambæ tangentur ab eadem recta <lb/>linea D A in eodem puncto mutuæ abſciſſionis A. </s>
  <s xml:id="echoid-s10086" xml:space="preserve">Quod erat propoſitum.</s>
  <s xml:id="echoid-s10087" xml:space="preserve"/>
</p>
<div xml:id="echoid-div832" type="float" level="2" n="19">
<note position="right" xlink:label="note-0307-01" xlink:href="note-0307-01a" xml:space="preserve">Prop. 17. <lb/>addit. <lb/>huius.</note>
<note position="right" xlink:label="note-0307-02" xlink:href="note-0307-02a" xml:space="preserve">10. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10088" xml:space="preserve">Si in qualibet coniſectione B A C <lb/>
<anchor type="figure" xlink:label="fig-0307-01a" xlink:href="fig-0307-01"/>
<anchor type="note" xlink:label="note-0307-03a" xlink:href="note-0307-03"/>
ducatur breuiſecans ſingularis D A, <lb/>&amp; </s>
  <s xml:id="echoid-s10089" xml:space="preserve">quælibet alia coniſectio I A K, <lb/>cuius axis ſit D A, atque ſemiſſis <lb/>lateris recti axis ſectionis I A K ſit <lb/>æqualis breuiſecanti D A. </s>
  <s xml:id="echoid-s10090" xml:space="preserve">Dico, <lb/>ſectionem I A K contingere eandem <lb/>rectam lineam G A, quàm tangit <lb/>ſectio B A C, &amp; </s>
  <s xml:id="echoid-s10091" xml:space="preserve">abſcindere reli-<lb/>quam coniſectionem in eodem pun-<lb/>cto A.</s>
  <s xml:id="echoid-s10092" xml:space="preserve"/>
</p>
<div xml:id="echoid-div833" type="float" level="2" n="20">
  <figure xlink:label="fig-0307-01" xlink:href="fig-0307-01a">
    <image file="0307-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0307-01"/>
  </figure>
<note position="right" xlink:label="note-0307-03" xlink:href="note-0307-03a" xml:space="preserve">PROP. <lb/>21. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10093" xml:space="preserve">Deſcribatur centro D interuallo D <lb/>A circulus T A S conſtat (ex prop. </s>
  <s xml:id="echoid-s10094" xml:space="preserve">10. </s>
  <s xml:id="echoid-s10095" xml:space="preserve">additarum libri quinti) circulum T <lb/>A S ſecare coniſectionem B A C in A, cumque circa eundem axim D A po-<lb/>nantur circulus T A S, atque coniſectio I A K, cuius lateris recti ſemiſſis æ-<lb/>qualis eſt D A radio circuli T A S, ergo coniſectio I A K abſcindit coniſectio-<lb/>
<anchor type="note" xlink:label="note-0307-04a" xlink:href="note-0307-04"/>
nem B A C in eodem puncto A, in quo ſecatur à circulo T A S, &amp; </s>
  <s xml:id="echoid-s10096" xml:space="preserve">tanguntur <lb/>ab eadem contingente G A in puncto A. </s>
  <s xml:id="echoid-s10097" xml:space="preserve">Quod erat, &amp;</s>
  <s xml:id="echoid-s10098" xml:space="preserve">c.</s>
  <s xml:id="echoid-s10099" xml:space="preserve"/>
</p>
<div xml:id="echoid-div834" type="float" level="2" n="21">
<note position="right" xlink:label="note-0307-04" xlink:href="note-0307-04a" xml:space="preserve">20. addit. <lb/>huius.</note>
</div>
<pb o="270" file="0308" n="308" rhead="Apollonij Pergæi Conicor. Lib. VI."/>
<p style="it">
  <s xml:id="echoid-s10100" xml:space="preserve">Sectionum conicarum circa axim communem poſitarum datam coniſe-<lb/>
<anchor type="note" xlink:label="note-0308-01a" xlink:href="note-0308-01"/>
ctionem abſcindentium non in eius vertice, quas omnes eadem recta li-<lb/>nea contingat, erunt ſingulares tantummodo parabolæ, &amp; </s>
  <s xml:id="echoid-s10101" xml:space="preserve">circulus, elli-<lb/>pſes verò, &amp; </s>
  <s xml:id="echoid-s10102" xml:space="preserve">hyperbole erunt infinitæ.</s>
  <s xml:id="echoid-s10103" xml:space="preserve"/>
</p>
<div xml:id="echoid-div835" type="float" level="2" n="22">
<note position="left" xlink:label="note-0308-01" xlink:href="note-0308-01a" xml:space="preserve">PROP. <lb/>22. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10104" xml:space="preserve">Quoniam circa communem axim D <lb/>
<anchor type="figure" xlink:label="fig-0308-01a" xlink:href="fig-0308-01"/>
A conſtitui poßunt parabolæ, circulus, <lb/>infinitæ hyperbolæ, &amp; </s>
  <s xml:id="echoid-s10105" xml:space="preserve">infinitæ ellipſes <lb/>
<anchor type="note" xlink:label="note-0308-02a" xlink:href="note-0308-02"/>
habentes ſemilatus rectum axis æqualẽ <lb/>ſingulari breuiſecanti D A in ſectione <lb/>conica B A C educto, &amp; </s>
  <s xml:id="echoid-s10106" xml:space="preserve">hæ omnes ab-<lb/>
<anchor type="note" xlink:label="note-0308-03a" xlink:href="note-0308-03"/>
ſcindunt coniſectionem B A C in A. <lb/></s>
  <s xml:id="echoid-s10107" xml:space="preserve">Ergo patet propoſitum.</s>
  <s xml:id="echoid-s10108" xml:space="preserve"/>
</p>
<div xml:id="echoid-div836" type="float" level="2" n="23">
  <figure xlink:label="fig-0308-01" xlink:href="fig-0308-01a">
    <image file="0308-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0308-01"/>
  </figure>
<note position="left" xlink:label="note-0308-02" xlink:href="note-0308-02a" xml:space="preserve">Prop. 17. <lb/>addit. <lb/>huius.</note>
<note position="left" xlink:label="note-0308-03" xlink:href="note-0308-03a" xml:space="preserve">Prop. 21. <lb/>addit. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10109" xml:space="preserve">Hinc colligitur dari non poſſe coniſe-<lb/>ctionem minimam extrinſecus tangen-<lb/>tium, neque maximam intrinſecus tã-<lb/>gentium eandem coniſectionem in pun-<lb/>cto A extra verticem axis poſito.</s>
  <s xml:id="echoid-s10110" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10111" xml:space="preserve">Nam quælibet coniſectio, cuius ſemie-<lb/>rectum axis minus eſt breuiſecante ſingulari D A intrinſecus tangit ſectionem <lb/>
<anchor type="note" xlink:label="note-0308-04a" xlink:href="note-0308-04"/>
B A C in A, &amp; </s>
  <s xml:id="echoid-s10112" xml:space="preserve">ſi ſemierectum maius fuerit eadem D A extrinſecus eandem <lb/>ſectionem B A C continget, neque vnquam ceſſant prædicti contactus extrin-<lb/>
<anchor type="note" xlink:label="note-0308-05a" xlink:href="note-0308-05"/>
ſeci, vel intrinſeci quouſque ſemierectum axis efficitur æquale breuiſecanti D <lb/>A: </s>
  <s xml:id="echoid-s10113" xml:space="preserve">at tunc non amplius contingit, ſed ſecat eam in A. </s>
  <s xml:id="echoid-s10114" xml:space="preserve">Quare patet propoſi-<lb/>
<anchor type="note" xlink:label="note-0308-06a" xlink:href="note-0308-06"/>
tum.</s>
  <s xml:id="echoid-s10115" xml:space="preserve"/>
</p>
<div xml:id="echoid-div837" type="float" level="2" n="24">
<note position="left" xlink:label="note-0308-04" xlink:href="note-0308-04a" xml:space="preserve">Prop. 18. <lb/>addit. <lb/>huius.</note>
<note position="left" xlink:label="note-0308-05" xlink:href="note-0308-05a" xml:space="preserve">Prop. 19. <lb/>addit. <lb/>huius.</note>
<note position="left" xlink:label="note-0308-06" xlink:href="note-0308-06a" xml:space="preserve">Prop. 21. <lb/>addit. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10116" xml:space="preserve">Conſtat etiam quod parabolarum vnica tantummodò, &amp; </s>
  <s xml:id="echoid-s10117" xml:space="preserve">circulorum vnicus <lb/>etiam abſcindit coniſectionem B A C in A, &amp; </s>
  <s xml:id="echoid-s10118" xml:space="preserve">contingit eandem contingentem <lb/>A G in A.</s>
  <s xml:id="echoid-s10119" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10120" xml:space="preserve">At hyperbolarum, atque ellipſium abſcindentium eandem ſectionem B A C in <lb/>A, quas omnes eadem recta linea A G tangit in A non poteſt affignari maxi-<lb/>ma, neque minima.</s>
  <s xml:id="echoid-s10121" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10122" xml:space="preserve">Nam vt dictum eſt ad 17. </s>
  <s xml:id="echoid-s10123" xml:space="preserve">Additarum huius libri infinitæ hyperbolæ ſe ſe <lb/>contingentes in vertice axis deſinunt in parabolam vnicam, &amp; </s>
  <s xml:id="echoid-s10124" xml:space="preserve">poſt parabolam <lb/>interius ſe ſe ſucceſſiuè contingunt infinitæ ellipſes ad axim maiorem adiacen-<lb/>tes, quæ deſinunt in circulum vnicum, ac poſt circulum interius eum contin-<lb/>gunt inſinitæ ellipſes ad axim minorem adiacentes, quarum omnium ſemiere-<lb/>cta latera axium æqualia ſunt breuiſecanti ſingulari D A datæ ſectionis B A C. <lb/></s>
  <s xml:id="echoid-s10125" xml:space="preserve">Quare patet propoſitum.</s>
  <s xml:id="echoid-s10126" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div839" type="section" level="1" n="249">
<head xml:id="echoid-head312" xml:space="preserve">LIBRI SEXTI FINIS.</head>
<pb o="271" file="0309" n="309" rhead="APOLLONII PERGAEI CONICORVM LIB. VII."/>
</div>
<div xml:id="echoid-div840" type="section" level="1" n="250">
<head xml:id="echoid-head313" xml:space="preserve">DEFINITIONES.</head>
<head xml:id="echoid-head314" xml:space="preserve">I.</head>
<p>
  <s xml:id="echoid-s10127" xml:space="preserve">SI diuidatur inclinatum ſecundum proportionem <lb/>figuræ, aut addatur vni axium ellipſis linea, <lb/>earumque differentia, aut aggregatum ad ean-<lb/>dem lineam habeat eandem proportionem fi-<lb/>guræ: </s>
  <s xml:id="echoid-s10128" xml:space="preserve">vocabo homologam inclinati PRÆSE-<lb/>CTAM.</s>
  <s xml:id="echoid-s10129" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div841" type="section" level="1" n="251">
<head xml:id="echoid-head315" xml:space="preserve">II.</head>
<p>
  <s xml:id="echoid-s10130" xml:space="preserve">Et homologam erecti INTERCEPTAM.</s>
  <s xml:id="echoid-s10131" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div842" type="section" level="1" n="252">
<head xml:id="echoid-head316" xml:space="preserve">III.</head>
<p>
  <s xml:id="echoid-s10132" xml:space="preserve">Atque punctum, quod eſt extremum ipſius interceptæ, &amp; </s>
  <s xml:id="echoid-s10133" xml:space="preserve">dia-<lb/>metri: </s>
  <s xml:id="echoid-s10134" xml:space="preserve">vocabo TERMINVM COMMVNEM.</s>
  <s xml:id="echoid-s10135" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div843" type="section" level="1" n="253">
<head xml:id="echoid-head317" xml:space="preserve">IV.</head>
<p>
  <s xml:id="echoid-s10136" xml:space="preserve">Reliquum verò TERMINVM DIVIDENTEM.</s>
  <s xml:id="echoid-s10137" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div844" type="section" level="1" n="254">
<head xml:id="echoid-head318" xml:space="preserve">V.</head>
<p>
  <s xml:id="echoid-s10138" xml:space="preserve">Et differentiam, vel ſummam lateris, &amp; </s>
  <s xml:id="echoid-s10139" xml:space="preserve">interceptæ: </s>
  <s xml:id="echoid-s10140" xml:space="preserve">vocabo IN-<lb/>TERCEPTAM COMPARATAM.</s>
  <s xml:id="echoid-s10141" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div845" type="section" level="1" n="255">
<head xml:id="echoid-head319" xml:space="preserve">VI.</head>
<p>
  <s xml:id="echoid-s10142" xml:space="preserve">Differentiam verò, aut ſummam lateris, &amp; </s>
  <s xml:id="echoid-s10143" xml:space="preserve">præſectę: </s>
  <s xml:id="echoid-s10144" xml:space="preserve">vocabo <lb/>PRÆSECTAM COMPARATAM: </s>
  <s xml:id="echoid-s10145" xml:space="preserve">hoc autem latus refer-<lb/>tur ad diametrum, quæ bifariam diuidit lineam coniungen-<lb/>tem verticem ſectionis, &amp; </s>
  <s xml:id="echoid-s10146" xml:space="preserve">terminum potentis huius lateris:</s>
  <s xml:id="echoid-s10147" xml:space="preserve">
<pb o="272" file="0310" n="310" rhead="Apollonij Pergæi"/>
reliquæ verò lineæ referuntur ad hoc latus.</s>
  <s xml:id="echoid-s10148" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div846" type="section" level="1" n="256">
<head xml:id="echoid-head320" xml:space="preserve">VII.</head>
<p>
  <s xml:id="echoid-s10149" xml:space="preserve">Inſuper vocabo duas diametros coniugatas, &amp; </s>
  <s xml:id="echoid-s10150" xml:space="preserve">æquales in elli-<lb/>pſi, ÆQVALES.</s>
  <s xml:id="echoid-s10151" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10152" xml:space="preserve">Et ſi quidem ad vtraſque partes axis ſectionis duæ diame-<lb/>tri educantur, quæ ad ſua erecta eandem proportionem ha-<lb/>beant, vtique vocabo c<unsure/>as ÆQVALES.</s>
  <s xml:id="echoid-s10153" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div847" type="section" level="1" n="257">
<head xml:id="echoid-head321" xml:space="preserve">VIII.</head>
<p>
  <s xml:id="echoid-s10154" xml:space="preserve">Diametros verò æquales ad vtraſque partes duarum axium elli-<lb/>pſis cadentes, voco Homologas illius axis: </s>
  <s xml:id="echoid-s10155" xml:space="preserve">ſuntque homo-<lb/>logæ diametri in ellipſi tranſuerſa ad tranſuerſam, &amp; </s>
  <s xml:id="echoid-s10156" xml:space="preserve">recta <lb/>ad rectam.</s>
  <s xml:id="echoid-s10157" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div848" type="section" level="1" n="258">
<head xml:id="echoid-head322" xml:space="preserve">NOTÆ.</head>
<p style="it">
  <s xml:id="echoid-s10158" xml:space="preserve">I. </s>
  <s xml:id="echoid-s10159" xml:space="preserve">P Rima definitio breuiſſimè exponi poteſt hac ratione. </s>
  <s xml:id="echoid-s10160" xml:space="preserve">Si axis tranſuerſus <lb/>interius in hyperbola diuidatur, aut exterius in ellipſi, ſecundum pro-<lb/>portionem figuræ, ſegmentum homologum axis tranſuerſi vocabo Præſectum, vt <lb/>ſi fuerit hyperbole, vel ellipſis A B, cuius axis tranſuerſus A C, centrum D, <lb/>latus rectũ A F, &amp; </s>
  <s xml:id="echoid-s10161" xml:space="preserve">in hyperbola ſecetur C A inter vertices A, &amp; </s>
  <s xml:id="echoid-s10162" xml:space="preserve">C; </s>
  <s xml:id="echoid-s10163" xml:space="preserve">in ellipſi <lb/>verò ſecetur exterius in puncto G, ita vt ſumma, vel differentia ipſarum G A, <lb/>&amp; </s>
  <s xml:id="echoid-s10164" xml:space="preserve">axis C A, ideſt C G ad G A habeat proportionem figuræ ſcilicet eandem, <lb/>quàm habet latus tranſuerſum C A ad latus rectum A F; </s>
  <s xml:id="echoid-s10165" xml:space="preserve">tunc quidem vocatur <lb/>recta linea C G Præſecta.</s>
  <s xml:id="echoid-s10166" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10167" xml:space="preserve">II. </s>
  <s xml:id="echoid-s10168" xml:space="preserve">Atque G A vocatur Intercepta.</s>
  <s xml:id="echoid-s10169" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10170" xml:space="preserve">III. </s>
  <s xml:id="echoid-s10171" xml:space="preserve">Punctum verò A extremum <lb/>
<anchor type="figure" xlink:label="fig-0310-01a" xlink:href="fig-0310-01"/>
interceptæ G A, &amp; </s>
  <s xml:id="echoid-s10172" xml:space="preserve">diametri C A <lb/>vocabitur terminus communis dua-<lb/>rum linearum, ſcilicet axis C A, &amp; </s>
  <s xml:id="echoid-s10173" xml:space="preserve"><lb/>additæ, vel ablatæ A G.</s>
  <s xml:id="echoid-s10174" xml:space="preserve"/>
</p>
<div xml:id="echoid-div848" type="float" level="2" n="1">
  <figure xlink:label="fig-0310-01" xlink:href="fig-0310-01a">
    <image file="0310-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0310-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s10175" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s10176" xml:space="preserve">Punctum verò G, in quo axis <lb/>A C interius, vel exterius diuiditur <lb/>ſecundum proportionem figuræ voca-<lb/>tur terminus diuidens; </s>
  <s xml:id="echoid-s10177" xml:space="preserve">Si verò ſece-<lb/>tur C H æqualis A G vocabitur etiã <lb/>C H intercepta, &amp; </s>
  <s xml:id="echoid-s10178" xml:space="preserve">A H præſecta, <lb/>atque C terminus communis, &amp; </s>
  <s xml:id="echoid-s10179" xml:space="preserve">H <lb/>terminus diuidens.</s>
  <s xml:id="echoid-s10180" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10181" xml:space="preserve">V. </s>
  <s xml:id="echoid-s10182" xml:space="preserve">Si diameter I L ſecuerit biſa-<lb/>riam ſubtenſam A B à ſectionis ver <lb/>tice A eductam, atque à termino B
<pb o="273" file="0311" n="311" rhead="Conicor. Lib. VII."/>
ducatur B E perpendicularis ad axim eum ſecans in E, tunc quidem axis ſeg-<lb/>mentum C E ab oppoſito vertice C ductum, vocat interpres Latus. </s>
  <s xml:id="echoid-s10183" xml:space="preserve">Poſtea ſum-<lb/>mam in prima ellipſi, &amp; </s>
  <s xml:id="echoid-s10184" xml:space="preserve">differentiam in reliquis figuris lateris C E, &amp; </s>
  <s xml:id="echoid-s10185" xml:space="preserve">inter-<lb/>ceptæ H C, nimirum ipſam lineam H E, vocat Interceptam comparatam.</s>
  <s xml:id="echoid-s10186" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10187" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s10188" xml:space="preserve">Et lateris C E, &amp; </s>
  <s xml:id="echoid-s10189" xml:space="preserve">præſectæ G C differentia in tribus prioribus figuris, <lb/>&amp; </s>
  <s xml:id="echoid-s10190" xml:space="preserve">ſumma in figura quarta, ideſt G E, vocatur Præſecta comparata.</s>
  <s xml:id="echoid-s10191" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s10192" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s10193" xml:space="preserve">Ducantur in ellipſi A B C duæ diametri coniugatæ I L, &amp; </s>
  <s xml:id="echoid-s10194" xml:space="preserve">N O, quæ <lb/>inter ſe ſint æquales. </s>
  <s xml:id="echoid-s10195" xml:space="preserve">Vel tranſuerſa I L ad eius latus rectum eandem propor-<lb/>tionem habeat, quàm eius coniugata N O ad ſuum latus rectum; </s>
  <s xml:id="echoid-s10196" xml:space="preserve">tunc quidem <lb/>vocat pariter diametros coniugatas I L, N O AEquales.</s>
  <s xml:id="echoid-s10197" xml:space="preserve"/>
</p>
  <figure>
    <image file="0311-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0311-01"/>
  </figure>
</div>
<div xml:id="echoid-div850" type="section" level="1" n="259">
<head xml:id="echoid-head323" xml:space="preserve">SECTIO PRIMA</head>
<head xml:id="echoid-head324" xml:space="preserve">Continens Propoſit. I. V. &amp; XXIII. <lb/>Apollonij.</head>
<head xml:id="echoid-head325" xml:space="preserve">PROPOSITIO I.</head>
<p>
  <s xml:id="echoid-s10198" xml:space="preserve">SI in parabola A B à termino <lb/>
<anchor type="figure" xlink:label="fig-0311-02a" xlink:href="fig-0311-02"/>
axis A D educatur recta linea <lb/>A B ſubtendens ſegmentum @ectionis <lb/>A B, &amp; </s>
  <s xml:id="echoid-s10199" xml:space="preserve">ab eius termino ducatur B <lb/>D ad axim perpendicularis; </s>
  <s xml:id="echoid-s10200" xml:space="preserve">vtiquè <lb/>illa chorda poterit eius abſciſſam D <lb/>A in aggregatum abſciſſæ, &amp; </s>
  <s xml:id="echoid-s10201" xml:space="preserve">erecti.</s>
  <s xml:id="echoid-s10202" xml:space="preserve"/>
</p>
<div xml:id="echoid-div850" type="float" level="2" n="1">
  <figure xlink:label="fig-0311-02" xlink:href="fig-0311-02a">
    <image file="0311-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0311-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s10203" xml:space="preserve">Fiat A F æqualis erecto A E. </s>
  <s xml:id="echoid-s10204" xml:space="preserve">Quia <lb/>
<anchor type="note" xlink:label="note-0311-01a" xlink:href="note-0311-01"/>
quadratum A B eſt æquale quadrato D A
<pb o="274" file="0312" n="312" rhead="Apollonij Pergæi"/>
cum quadrato D B, quod eſt æquale ipſi A D in A F; </s>
  <s xml:id="echoid-s10205" xml:space="preserve">igitur eſt æqua-<lb/>le ipſi F D in D A. </s>
  <s xml:id="echoid-s10206" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s10207" xml:space="preserve"/>
</p>
<div xml:id="echoid-div851" type="float" level="2" n="2">
<note position="left" xlink:label="note-0311-01" xlink:href="note-0311-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div853" type="section" level="1" n="260">
<head xml:id="echoid-head326" xml:space="preserve">PROPOSITIO V. &amp; XXIII.</head>
<p>
  <s xml:id="echoid-s10208" xml:space="preserve">IN parabola A B C cuiuſcumque diametri B F erectus B H ex-<lb/>cedit axis A D erectum A E quadruplo abciſſæ A D potentis <lb/>à termino illius diametri ad axim ductæ 23. </s>
  <s xml:id="echoid-s10209" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10210" xml:space="preserve">diametri C G, re-<lb/>
<anchor type="note" xlink:label="note-0312-01a" xlink:href="note-0312-01"/>
motioris ab axe, erectus C I maior eſt erecto B H diametri propin-<lb/>quioris B F quadruplo differentiæ axis abſciſſarum potentium à <lb/>terminis diametrorum ad axim ductorum.</s>
  <s xml:id="echoid-s10211" xml:space="preserve"/>
</p>
<div xml:id="echoid-div853" type="float" level="2" n="1">
<note position="right" xlink:label="note-0312-01" xlink:href="note-0312-01a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0312-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0312-01"/>
  </figure>
<p>
  <s xml:id="echoid-s10212" xml:space="preserve">Educamus A L, B K tangentes in A, B, &amp; </s>
  <s xml:id="echoid-s10213" xml:space="preserve">B N perpendicularem ad <lb/>B K, erit K D in D N æquale quadrato D B, quod eſt æquale ipſi A E <lb/>
<anchor type="note" xlink:label="note-0312-02a" xlink:href="note-0312-02"/>
in A D; </s>
  <s xml:id="echoid-s10214" xml:space="preserve">ergo K D ad D A eandem proportionem habet, quàm A E ad <lb/>D N: </s>
  <s xml:id="echoid-s10215" xml:space="preserve">eſtque D K dupla ipſius A D (37. </s>
  <s xml:id="echoid-s10216" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s10217" xml:space="preserve">igitur A E eſt dupla <lb/>
<anchor type="note" xlink:label="note-0312-03a" xlink:href="note-0312-03"/>
ipſius D N; </s>
  <s xml:id="echoid-s10218" xml:space="preserve">quarè A E cum duplo D K, nempe cum quadruplo A D eſt <lb/>
<anchor type="note" xlink:label="note-0312-04a" xlink:href="note-0312-04"/>
æqualis duplo K N, nempe B H (eo quod N K ad B K tangentem ean-<lb/>dem proportionem habet, quàm aſſumpta M B ad B L coniugatam (57. <lb/></s>
  <s xml:id="echoid-s10219" xml:space="preserve">
<anchor type="note" xlink:label="note-0312-05a" xlink:href="note-0312-05"/>
ex 1.) </s>
  <s xml:id="echoid-s10220" xml:space="preserve">(propter ſimilitudinem duorum triangulorum); </s>
  <s xml:id="echoid-s10221" xml:space="preserve">ergo B H æqualis <lb/>eſt quadruplo A D cum A E; </s>
  <s xml:id="echoid-s10222" xml:space="preserve">quarè erectus diametri B F excedit A E <lb/>quadruplo A D. </s>
  <s xml:id="echoid-s10223" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10224" xml:space="preserve">A O maior eſt, quàm A D; </s>
  <s xml:id="echoid-s10225" xml:space="preserve">ergo erectus diametri <lb/>
<anchor type="note" xlink:label="note-0312-06a" xlink:href="note-0312-06"/>
C G remotioris maior eſt, quàm erectus B F proximioris quadruplo D <lb/>O differentiæ abſciſſarum. </s>
  <s xml:id="echoid-s10226" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s10227" xml:space="preserve"/>
</p>
<div xml:id="echoid-div854" type="float" level="2" n="2">
<note position="left" xlink:label="note-0312-02" xlink:href="note-0312-02a" xml:space="preserve">11. lib. 1.</note>
<note position="left" xlink:label="note-0312-03" xlink:href="note-0312-03a" xml:space="preserve">35. lib. 1.</note>
<note position="right" xlink:label="note-0312-04" xlink:href="note-0312-04a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0312-05" xlink:href="note-0312-05a" xml:space="preserve">44. lib. 1.</note>
<note position="right" xlink:label="note-0312-06" xlink:href="note-0312-06a" xml:space="preserve">c</note>
</div>
</div>
<div xml:id="echoid-div856" type="section" level="1" n="261">
<head xml:id="echoid-head327" xml:space="preserve">Notæ in Propoſit. I.</head>
<p style="it">
  <s xml:id="echoid-s10228" xml:space="preserve">QVia quadratum A B eſt æquale quadrato D A, &amp;</s>
  <s xml:id="echoid-s10229" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10230" xml:space="preserve">Quoniam re-<lb/>
<anchor type="note" xlink:label="note-0312-07a" xlink:href="note-0312-07"/>
ctangulum F D A æquale eſt rectangulo F A D ſubſegmentis vna cum <lb/>quadrato reliqui ſegmenti D A; </s>
  <s xml:id="echoid-s10231" xml:space="preserve">eſtque latus rectum A E æquale
<pb o="275" file="0313" n="313" rhead="Conicor. Lib. VII."/>
A F; </s>
  <s xml:id="echoid-s10232" xml:space="preserve">igitur rectangulum F D A æquale eſt <lb/>
<anchor type="figure" xlink:label="fig-0313-01a" xlink:href="fig-0313-01"/>
rectangulo D A E vna cum quadrato D A; <lb/></s>
  <s xml:id="echoid-s10233" xml:space="preserve">ſed quadratum ordinatim ad axim applicatæ <lb/>
<anchor type="note" xlink:label="note-0313-01a" xlink:href="note-0313-01"/>
B D æquale eſt rectangulo D A E ſub abſciſ-<lb/>ſa &amp; </s>
  <s xml:id="echoid-s10234" xml:space="preserve">latere recto contento; </s>
  <s xml:id="echoid-s10235" xml:space="preserve">igitur rectangu-<lb/>lum F D A æquale eſt duobus quadratis B D, <lb/>&amp; </s>
  <s xml:id="echoid-s10236" xml:space="preserve">D A: </s>
  <s xml:id="echoid-s10237" xml:space="preserve">eſtquè quadratum A B ſubtenden-<lb/>tis rectum angulum D æquale duobus quadra-<lb/>tis B D, &amp; </s>
  <s xml:id="echoid-s10238" xml:space="preserve">D A; </s>
  <s xml:id="echoid-s10239" xml:space="preserve">igitur quadratum ſubten-<lb/>ſæ A B æquale eſt rectangulo A D E ſub ab-<lb/>ſciſſa D A, &amp; </s>
  <s xml:id="echoid-s10240" xml:space="preserve">ſub D F, quæ æqualis eſt ei-<lb/>dem abſciſſæ cum latere recto.</s>
  <s xml:id="echoid-s10241" xml:space="preserve"/>
</p>
<div xml:id="echoid-div856" type="float" level="2" n="1">
<note position="right" xlink:label="note-0312-07" xlink:href="note-0312-07a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0313-01" xlink:href="fig-0313-01a">
    <image file="0313-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0313-01"/>
  </figure>
<note position="right" xlink:label="note-0313-01" xlink:href="note-0313-01a" xml:space="preserve">2 1. lib. 1.</note>
</div>
</div>
<div xml:id="echoid-div858" type="section" level="1" n="262">
<head xml:id="echoid-head328" xml:space="preserve">Notæ in Propoſit. V. &amp; XXIII.</head>
<p style="it">
  <s xml:id="echoid-s10242" xml:space="preserve">ET diametri G C remotioris ab axe erectus C I maior eſt erecto B H <lb/>
<anchor type="note" xlink:label="note-0313-02a" xlink:href="note-0313-02"/>
diametri propinquioris B F, &amp;</s>
  <s xml:id="echoid-s10243" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10244" xml:space="preserve">Videtur hæc 23. </s>
  <s xml:id="echoid-s10245" xml:space="preserve">propoſitio deficiens; <lb/></s>
  <s xml:id="echoid-s10246" xml:space="preserve">cum omnino inueriſimile ſit Apollonium non animaduertiſſe rem adeo facilem; </s>
  <s xml:id="echoid-s10247" xml:space="preserve"><lb/>quod nimirum diametri G C remotioris ab axe erectus C I maior ſit erecto B <lb/>H diametri B F proximioris quadruplo differentiæ axis abſciſſarum potentium <lb/>à terminis diametrorum ad axim ductorum.</s>
  <s xml:id="echoid-s10248" xml:space="preserve"/>
</p>
<div xml:id="echoid-div858" type="float" level="2" n="1">
<note position="left" xlink:label="note-0313-02" xlink:href="note-0313-02a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s10249" xml:space="preserve">Quare A E cum duplo K D, nempe cum quadruplo A D eſt æqualis <lb/>
<anchor type="note" xlink:label="note-0313-03a" xlink:href="note-0313-03"/>
duplo K N, nempe dimidio B H, &amp;</s>
  <s xml:id="echoid-s10250" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10251" xml:space="preserve">Zuoniam B H latus rectum diame-<lb/>
<anchor type="note" xlink:label="note-0313-04a" xlink:href="note-0313-04"/>
tri B F ad duplum contingentis B K eſt vt M B ad B L, ſed (propter æqui-<lb/>diſtantes, &amp; </s>
  <s xml:id="echoid-s10252" xml:space="preserve">ſimilitudinem triangulorum L B M, &amp; </s>
  <s xml:id="echoid-s10253" xml:space="preserve">K N B) vt M B ad B <lb/>L, ita eſt duplum N K ad duplum R B; </s>
  <s xml:id="echoid-s10254" xml:space="preserve">ergo latus rectum B H æquale eſt du-<lb/>plo K N; </s>
  <s xml:id="echoid-s10255" xml:space="preserve">ſed prius oſtenſum eſt quod D A æqualis eſt medietati ipſius D K, &amp; </s>
  <s xml:id="echoid-s10256" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0313-05a" xlink:href="note-0313-05"/>
D N æqualis medietati ipſius A E; </s>
  <s xml:id="echoid-s10257" xml:space="preserve">igitur duplum K N æquale eſt duplo K D, <lb/>ſeu quadruplo A D cum duplo D N, ſeu cum A E.</s>
  <s xml:id="echoid-s10258" xml:space="preserve"/>
</p>
<div xml:id="echoid-div859" type="float" level="2" n="2">
<note position="left" xlink:label="note-0313-03" xlink:href="note-0313-03a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0313-04" xlink:href="note-0313-04a" xml:space="preserve">49. lib. 1.</note>
<note position="right" xlink:label="note-0313-05" xlink:href="note-0313-05a" xml:space="preserve">35. .lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10259" xml:space="preserve">Et A O maior eſt, quàm A D; </s>
  <s xml:id="echoid-s10260" xml:space="preserve">ergo erectus diametri C G remotioris <lb/>
<anchor type="note" xlink:label="note-0313-06a" xlink:href="note-0313-06"/>
maior eſt quàm erectus B F proximioris, &amp;</s>
  <s xml:id="echoid-s10261" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10262" xml:space="preserve">Addidi in bac concluſione <lb/>verba bæc (quadruplo D O differētiæ abſciſſarum) quæ videntur deficere. </s>
  <s xml:id="echoid-s10263" xml:space="preserve">Ma-<lb/>nifeſtum enim eſt, quod C I latus rectum diametri C G ab axe remotioris ſu-<lb/>perat latus rectum B H diametri F B axi propinguioris quadruplo D O diffe-<lb/>rentiæ abſeiſſarum axis ab ordinatis à verticibus earũdem diametrorum ductis; <lb/></s>
  <s xml:id="echoid-s10264" xml:space="preserve">nam B H æqualis oſtenſa eſt E A vna cum quadruplo A D, eademque ratione <lb/>C I æqualis eſt eidem axis lateri recto E A cum quadruplo A O; </s>
  <s xml:id="echoid-s10265" xml:space="preserve">ergo exceſſus <lb/>C I ſupra B H erit æqualis quadruplo differentiæ D O.</s>
  <s xml:id="echoid-s10266" xml:space="preserve"/>
</p>
<div xml:id="echoid-div860" type="float" level="2" n="3">
<note position="left" xlink:label="note-0313-06" xlink:href="note-0313-06a" xml:space="preserve">c</note>
</div>
<pb o="276" file="0314" n="314" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div862" type="section" level="1" n="263">
<head xml:id="echoid-head329" xml:space="preserve">SECTIO SECVNDA</head>
<head xml:id="echoid-head330" xml:space="preserve">Continens Propoſit. II. III. IV. VI. <lb/>&amp; VII. Apollonij.</head>
<head xml:id="echoid-head331" xml:space="preserve">PROPOSITIO II. &amp; III.</head>
<p>
  <s xml:id="echoid-s10267" xml:space="preserve">SI in ſectione A B à termino cõmuni A vtriuslibet interceptæ <lb/>
<anchor type="note" xlink:label="note-0314-01a" xlink:href="note-0314-01"/>
educatur linea recta A B vſq; </s>
  <s xml:id="echoid-s10268" xml:space="preserve">ad ſectionem, atquè ab eius <lb/>termino B ad axim A E ducatur perpendicularis B E; </s>
  <s xml:id="echoid-s10269" xml:space="preserve">erit qua-<lb/>dratum A B ad rectangulum contentum à rectis lineis inter per-<lb/>pendicularis incidentiam, &amp; </s>
  <s xml:id="echoid-s10270" xml:space="preserve">terminos interceptæ, nempe A E <lb/>in G E habebit eandem proportionem, quàm habet inclinatus, <lb/>ſiuè tranſuerſus A C ad præſectam C G.</s>
  <s xml:id="echoid-s10271" xml:space="preserve"/>
</p>
<div xml:id="echoid-div862" type="float" level="2" n="1">
<note position="right" xlink:label="note-0314-01" xlink:href="note-0314-01a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0314-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0314-01"/>
  </figure>
<p>
  <s xml:id="echoid-s10272" xml:space="preserve">Sit itaque A F erectus A C, &amp; </s>
  <s xml:id="echoid-s10273" xml:space="preserve">ponamus A E in E H æquale quadra-<lb/>to B E; </s>
  <s xml:id="echoid-s10274" xml:space="preserve">igitur A E in E H ad A E in E C, nempe H E ad E C eſt vt <lb/>
<anchor type="figure" xlink:label="fig-0314-02a" xlink:href="fig-0314-02"/>
<pb o="277" file="0315" n="315" rhead="Conicor. Lib. VII."/>
A F ad A C, &amp; </s>
  <s xml:id="echoid-s10275" xml:space="preserve">vt A G ad G C; </s>
  <s xml:id="echoid-s10276" xml:space="preserve">ergo H E ad E C eſt vt A G ad G <lb/>C; </s>
  <s xml:id="echoid-s10277" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10278" xml:space="preserve">componendo in hyperbolis, &amp; </s>
  <s xml:id="echoid-s10279" xml:space="preserve">diuidendo in ellipſibus, deinde <lb/>
<anchor type="note" xlink:label="note-0315-01a" xlink:href="note-0315-01"/>
comparando homologorum differentias in duabus figuris prioribus, &amp; </s>
  <s xml:id="echoid-s10280" xml:space="preserve"><lb/>ſummas homologorum in reliquis, fiet A H ad G E, vt C A ad C G; <lb/></s>
  <s xml:id="echoid-s10281" xml:space="preserve">ergo A H in A E; </s>
  <s xml:id="echoid-s10282" xml:space="preserve">nempe quadratum A B ad G E in A E eſt vt C A <lb/>inclinatus, ſiue tranſuerſus ad C G præſectam. </s>
  <s xml:id="echoid-s10283" xml:space="preserve">Quod fuerat propoſi-<lb/>tum.</s>
  <s xml:id="echoid-s10284" xml:space="preserve"/>
</p>
<div xml:id="echoid-div863" type="float" level="2" n="2">
  <figure xlink:label="fig-0314-02" xlink:href="fig-0314-02a">
    <image file="0314-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0314-02"/>
  </figure>
<note position="left" xlink:label="note-0315-01" xlink:href="note-0315-01a" xml:space="preserve">b</note>
</div>
</div>
<div xml:id="echoid-div865" type="section" level="1" n="264">
<head xml:id="echoid-head332" xml:space="preserve">PROPOSITIO IV.</head>
<p>
  <s xml:id="echoid-s10285" xml:space="preserve">SI hyperbolen, aut ellipſin A B tangat recta linea I M in I, <lb/>
<anchor type="note" xlink:label="note-0315-02a" xlink:href="note-0315-02"/>
&amp; </s>
  <s xml:id="echoid-s10286" xml:space="preserve">occurrat axi A C in M; </s>
  <s xml:id="echoid-s10287" xml:space="preserve">vtique ipſius I M quadratum <lb/>ad quadratum ſemidiametri ND coniugatæ ipſi I L habebit eã-<lb/>dem proportionem, quàm axis contenta M S ad eius inuerſam <lb/>S D.</s>
  <s xml:id="echoid-s10288" xml:space="preserve"/>
</p>
<div xml:id="echoid-div865" type="float" level="2" n="1">
<note position="left" xlink:label="note-0315-02" xlink:href="note-0315-02a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0315-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0315-01"/>
  </figure>
<p>
  <s xml:id="echoid-s10289" xml:space="preserve">Educantur A Q, M R perpendiculares ad axim vſque ad I L, ponatur-<lb/>que linea P, quæ ad I M eandem proportionem habeat, quàm K I ad <lb/>Q I, ſeu eandem, quàm habet M I ad I R; </s>
  <s xml:id="echoid-s10290" xml:space="preserve">Ergo P eſt ſemiſſis erecti <lb/>
<anchor type="note" xlink:label="note-0315-03a" xlink:href="note-0315-03"/>
diametri I L (52. </s>
  <s xml:id="echoid-s10291" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s10292" xml:space="preserve">atque D N dimidium coniugatæ diametri N O <lb/>poterit P in I D, atque I M poterit P in I R; </s>
  <s xml:id="echoid-s10293" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10294" xml:space="preserve">ideo I R ad I D, <lb/>nempe M S contenta ad S D inuerſam eandem proportionem habet, quã <lb/>quadratum tangentis I M ad quadratum N D ſemiſſis coniugatæ ipſius I <lb/>L. </s>
  <s xml:id="echoid-s10295" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s10296" xml:space="preserve"/>
</p>
<div xml:id="echoid-div866" type="float" level="2" n="2">
<note position="right" xlink:label="note-0315-03" xlink:href="note-0315-03a" xml:space="preserve">50. lib. 1.</note>
</div>
<pb o="278" file="0316" n="316" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div868" type="section" level="1" n="265">
<head xml:id="echoid-head333" xml:space="preserve">PROPOSITIO VI. &amp; VII.</head>
<p>
  <s xml:id="echoid-s10297" xml:space="preserve">SI in hyperbole, aut ellipſi addantur axi tranſuerſo, vel au-<lb/>
<anchor type="note" xlink:label="note-0316-01a" xlink:href="note-0316-01"/>
ferantur ab inclinato duæ interceptæ A G, C H ab eius <lb/>terminis A, C, atque à vertice ſectionis A educatur recta linea <lb/>A B ad terminum alicuius potentialis B E, &amp; </s>
  <s xml:id="echoid-s10298" xml:space="preserve">per centrum D <lb/>
<anchor type="figure" xlink:label="fig-0316-01a" xlink:href="fig-0316-01"/>
ducãtur diametri coniugatæ I L, N O, ita vt rectus N O æqui-<lb/>diſtet ipſi lineæ A B: </s>
  <s xml:id="echoid-s10299" xml:space="preserve">vtiquè proportio figuræ inclinatæ, vel <lb/>tranſuerſæ coniugatarum, quæ eſt eadem proportioni quadrati <lb/>I L ad quadratum N O, erit quoquè eadem, quàm habent li-<lb/>neæ inter incidentiam illius ordinatim applicatæ ad axim, &amp; </s>
  <s xml:id="echoid-s10300" xml:space="preserve">ter-<lb/>minos diuidentes duarum interceptarũ, ſcilicet vt H E ad E G.</s>
  <s xml:id="echoid-s10301" xml:space="preserve"/>
</p>
<div xml:id="echoid-div868" type="float" level="2" n="1">
<note position="right" xlink:label="note-0316-01" xlink:href="note-0316-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0316-01" xlink:href="fig-0316-01a">
    <image file="0316-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0316-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s10302" xml:space="preserve">Educamus I M tangentem, &amp; </s>
  <s xml:id="echoid-s10303" xml:space="preserve">I S perpendicularem. </s>
  <s xml:id="echoid-s10304" xml:space="preserve">Et quia A D eſt <lb/>
<anchor type="note" xlink:label="note-0316-02a" xlink:href="note-0316-02"/>
æqualis D C, &amp; </s>
  <s xml:id="echoid-s10305" xml:space="preserve">A K æqualis K B (eo quod I L cum ſit coniugata N O <lb/>bifariam diuidit A B) erit C B parallela ipſi I D, &amp; </s>
  <s xml:id="echoid-s10306" xml:space="preserve">propterea M S ad <lb/>S D, nempè A E ad E C (propter ſimilitudinem triangulorum) eſt vt <lb/>quadratum I M ad quadratum N D (4. </s>
  <s xml:id="echoid-s10307" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s10308" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10309" xml:space="preserve">quadratum I D ad qua-<lb/>dratum I M eſt vt quadratum C B ad quadratum B A (propter ſimilitu-<lb/>dinem triangulorum); </s>
  <s xml:id="echoid-s10310" xml:space="preserve">ergo proportio quadrati I D ad quadratum N D <lb/>eſt compoſita ex ratione A E ad E C, &amp; </s>
  <s xml:id="echoid-s10311" xml:space="preserve">ex ratione quadrati C B ad qua-<lb/>dratum B A; </s>
  <s xml:id="echoid-s10312" xml:space="preserve">ſed proportio quadrati C B ad quadratum B A eſt compo-<lb/>ſita ex ratione quadrati C B ad C E in E H, &amp; </s>
  <s xml:id="echoid-s10313" xml:space="preserve">ex ratione C E in E H <lb/>ad A E in E G, &amp; </s>
  <s xml:id="echoid-s10314" xml:space="preserve">ex ratione A E in E G ad quadratum A B; </s>
  <s xml:id="echoid-s10315" xml:space="preserve">eſt vero <lb/>quadratum C B ad C E in E H, vt C A ad A H (3. </s>
  <s xml:id="echoid-s10316" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s10317" xml:space="preserve">atquè A E <lb/>in E G ad quadratum A B eſt vt G C ad C A (2. </s>
  <s xml:id="echoid-s10318" xml:space="preserve">ex 7.)</s>
  <s xml:id="echoid-s10319" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s10320" xml:space="preserve">proportio <lb/>C E in E H ad A E in E G, componitur ex ratione C E ad A E, &amp; </s>
  <s xml:id="echoid-s10321" xml:space="preserve">ex
<pb o="279" file="0317" n="317" rhead="Conicor. Lib. VII."/>
<anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/>
H E ad E G; </s>
  <s xml:id="echoid-s10322" xml:space="preserve">igitur proportio quadrati I D ad quadratum N D compo-<lb/>ſita eſt ex proportione C A ad A H, &amp; </s>
  <s xml:id="echoid-s10323" xml:space="preserve">ex G C ad C A, atque ex C E <lb/>ad E A, &amp; </s>
  <s xml:id="echoid-s10324" xml:space="preserve">A E ad E C, &amp; </s>
  <s xml:id="echoid-s10325" xml:space="preserve">tandem ex H E ad E G; </s>
  <s xml:id="echoid-s10326" xml:space="preserve">ſed C A ad A H, <lb/>&amp; </s>
  <s xml:id="echoid-s10327" xml:space="preserve">G C ad C A componunt proportionem C A ad ei æqualem A C: </s>
  <s xml:id="echoid-s10328" xml:space="preserve">ſi-<lb/>militer C E ad E A, &amp; </s>
  <s xml:id="echoid-s10329" xml:space="preserve">A E ad E C eſt vt E C ad ſe ipſam: </s>
  <s xml:id="echoid-s10330" xml:space="preserve">quare ſi hæ <lb/>proportiones auſerantur, remanebit E H ad E G, vt quadratum I D ad <lb/>quadratum N D: </s>
  <s xml:id="echoid-s10331" xml:space="preserve">nempe erit eadem ac proportio figuræ diametri I L. <lb/></s>
  <s xml:id="echoid-s10332" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s10333" xml:space="preserve"/>
</p>
<div xml:id="echoid-div869" type="float" level="2" n="2">
<note position="right" xlink:label="note-0316-02" xlink:href="note-0316-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0317-01" xlink:href="fig-0317-01a">
    <image file="0317-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0317-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div871" type="section" level="1" n="266">
<head xml:id="echoid-head334" xml:space="preserve">Notæ in Propoſit. II. III.</head>
<p style="it">
  <s xml:id="echoid-s10334" xml:space="preserve">SI in ſectione A B à termino communi A interceptæ, &amp;</s>
  <s xml:id="echoid-s10335" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10336" xml:space="preserve">Addidi par-<lb/>ticulam vtriuslibet interceptæ vt propoſitio efficiatur vniuerſalis compræhen-<lb/>
<anchor type="note" xlink:label="note-0317-01a" xlink:href="note-0317-01"/>
<anchor type="figure" xlink:label="fig-0317-02a" xlink:href="fig-0317-02"/>
<pb o="280" file="0318" n="318" rhead="Apollonij Pergæi"/>
dens quartum caſum in poſtrema figura, quàm ſuperaddidi, vti neceſſariam, <lb/>pro intelligentia octauæ propoſitionis.</s>
  <s xml:id="echoid-s10337" xml:space="preserve"/>
</p>
<div xml:id="echoid-div871" type="float" level="2" n="1">
<note position="left" xlink:label="note-0317-01" xlink:href="note-0317-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0317-02" xlink:href="fig-0317-02a">
    <image file="0317-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0317-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s10338" xml:space="preserve">Et componendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s10339" xml:space="preserve">diuidendo in ellipſi prima deindè <lb/>
<anchor type="note" xlink:label="note-0318-01a" xlink:href="note-0318-01"/>
coniungendo in duabus figuris prioribus, &amp; </s>
  <s xml:id="echoid-s10340" xml:space="preserve">occurrere faciamus reſpe-<lb/>ctiuum cum reſpectiuo in reliquis figuris poſt inuerſionem, vt fiat, &amp;</s>
  <s xml:id="echoid-s10341" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s10342" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0318-01a" xlink:href="fig-0318-01"/>
Ideſt componendo in byperbolis, &amp; </s>
  <s xml:id="echoid-s10343" xml:space="preserve">in ellipſibus comparando differentias termi <lb/>norum ad conſequentes, deinde comparando homologorum differentias in duabus <lb/>figuris prioribus, &amp; </s>
  <s xml:id="echoid-s10344" xml:space="preserve">ſumas in reliquis, innc enim A H ad G E eſt, vt A C <lb/>ad C G, &amp; </s>
  <s xml:id="echoid-s10345" xml:space="preserve">ſumpta communi altitudine E A, erit tectangulum H A E ad re-<lb/>ctangulum G E A, vt A C ad C G. </s>
  <s xml:id="echoid-s10346" xml:space="preserve">Seà rectangulum H A E æquale eſt qua-<lb/>drato A E vna cum rectangulo H E A, cui æquale eſt quadratum B E, ergo <lb/>quadratum A B æquale eſt rectangulo H A E (propterea quod A B ſubtendit <lb/>angulum rectum E in triangulo B A E) quare quadratũ A B ad rectangulum <lb/>A G E eandem proportionẽ habet quàm C A ad C G.</s>
  <s xml:id="echoid-s10347" xml:space="preserve"/>
</p>
<div xml:id="echoid-div872" type="float" level="2" n="2">
<note position="right" xlink:label="note-0318-01" xlink:href="note-0318-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0318-01" xlink:href="fig-0318-01a">
    <image file="0318-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0318-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div874" type="section" level="1" n="267">
<head xml:id="echoid-head335" xml:space="preserve">Notæ in Propoſit. IV.</head>
<p style="it">
  <s xml:id="echoid-s10348" xml:space="preserve">SI hyperbolen, aut ellipſim A B tangat recta linea I M, &amp; </s>
  <s xml:id="echoid-s10349" xml:space="preserve">occurrat <lb/>
<anchor type="note" xlink:label="note-0318-02a" xlink:href="note-0318-02"/>
axi A C in M, vtique ipſius I M quadratum, &amp;</s>
  <s xml:id="echoid-s10350" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10351" xml:space="preserve">Suppleri debet <lb/>
<anchor type="figure" xlink:label="fig-0318-02a" xlink:href="fig-0318-02"/>
<pb o="281" file="0319" n="319" rhead="Conicor. Lib. VII."/>
conſtructio, quæ deficit in hac propoſitione, vt nimirum ſenſus continuatus ſit <lb/>à punctis M, A, I educatur ad axim perpẽdiculares M R, A Q, &amp; </s>
  <s xml:id="echoid-s10352" xml:space="preserve">I S ſecãtes <lb/>diametros in R, Q, &amp; </s>
  <s xml:id="echoid-s10353" xml:space="preserve">S, &amp; </s>
  <s xml:id="echoid-s10354" xml:space="preserve">A Q, I M ſe mutuò ſecent in K, erit I S <lb/>ordinatim ad axim applicata, &amp; </s>
  <s xml:id="echoid-s10355" xml:space="preserve">A Q, ſicuti etiam I M contingit ſectionem. <lb/></s>
  <s xml:id="echoid-s10356" xml:space="preserve">vocat autem Interpres rectam lineam M S, quæ inter tangentem, &amp; </s>
  <s xml:id="echoid-s10357" xml:space="preserve">ordinatam <lb/>interijcitur Contentam, atque D S vocat Inuerſam.</s>
  <s xml:id="echoid-s10358" xml:space="preserve"/>
</p>
<div xml:id="echoid-div874" type="float" level="2" n="1">
<note position="left" xlink:label="note-0318-02" xlink:href="note-0318-02a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0318-02" xlink:href="fig-0318-02a">
    <image file="0318-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0318-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div876" type="section" level="1" n="268">
<head xml:id="echoid-head336" xml:space="preserve">Notæ in Propoſit. VI. &amp; VII.</head>
<p>
  <s xml:id="echoid-s10359" xml:space="preserve">SI addatur duabus extremitatibus tranſuerſæ, aut inſiſtant ad duas ex-<lb/>
<anchor type="note" xlink:label="note-0319-01a" xlink:href="note-0319-01"/>
tremitates recti, aut diminuatur à duabus extremitatibus inclinati A, <lb/>
<anchor type="figure" xlink:label="fig-0319-01a" xlink:href="fig-0319-01"/>
&amp; </s>
  <s xml:id="echoid-s10360" xml:space="preserve">C duo intercepta, &amp;</s>
  <s xml:id="echoid-s10361" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10362" xml:space="preserve">Expungo verba appoſititia. </s>
  <s xml:id="echoid-s10363" xml:space="preserve">Aut inſiſtat ad duas <lb/>extremitates recti; </s>
  <s xml:id="echoid-s10364" xml:space="preserve">quæ ſenſum perturbant.</s>
  <s xml:id="echoid-s10365" xml:space="preserve"/>
</p>
<div xml:id="echoid-div876" type="float" level="2" n="1">
<note position="left" xlink:label="note-0319-01" xlink:href="note-0319-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0319-01" xlink:href="fig-0319-01a">
    <image file="0319-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0319-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s10366" xml:space="preserve">Educamus I M tangentem, &amp; </s>
  <s xml:id="echoid-s10367" xml:space="preserve">I S perpendicularem. </s>
  <s xml:id="echoid-s10368" xml:space="preserve">Et quia A D eſt <lb/>
<anchor type="note" xlink:label="note-0319-02a" xlink:href="note-0319-02"/>
æqualis D C, &amp;</s>
  <s xml:id="echoid-s10369" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10370" xml:space="preserve">Ideſt Educamus I M contingentem ſectionem in I, quæ <lb/>
<anchor type="figure" xlink:label="fig-0319-02a" xlink:href="fig-0319-02"/>
<pb o="282" file="0320" n="320" rhead="Apollonij Pergæi"/>
ſecet axim in M, &amp; </s>
  <s xml:id="echoid-s10371" xml:space="preserve">I S ad axim perpendicularem, ſeu ordinatim applica-<lb/>tam, eum ſecans in S. </s>
  <s xml:id="echoid-s10372" xml:space="preserve">Et quia trianguli A C B duo latera A C, A B ſecan-<lb/>tur proportionaliter, ſcilicet bifariam in D, &amp; </s>
  <s xml:id="echoid-s10373" xml:space="preserve">K; </s>
  <s xml:id="echoid-s10374" xml:space="preserve">ergo I D parallela eſt baſi <lb/>C B: </s>
  <s xml:id="echoid-s10375" xml:space="preserve">eſtquè tangens I M parallela ipſi B A, cum ambo ad diametrum I L ſint <lb/>
<anchor type="note" xlink:label="note-0320-01a" xlink:href="note-0320-01"/>
ordinatim applicatæ; </s>
  <s xml:id="echoid-s10376" xml:space="preserve">pariterquè I S parallela eſt B E ( cum ſint ad axim per-<lb/>pendiculares ) igitur triangula M I S, A B E ſimilia erunt; </s>
  <s xml:id="echoid-s10377" xml:space="preserve">pariterquè trian-<lb/>gula D I S, C B E erunt ſimilia: </s>
  <s xml:id="echoid-s10378" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10379" xml:space="preserve">ideo M S ad S I erit vt A E ad E B, &amp; </s>
  <s xml:id="echoid-s10380" xml:space="preserve"><lb/>S I ad S D erit, vt B E ad E C: </s>
  <s xml:id="echoid-s10381" xml:space="preserve">quarè ex æquali ordinata M S ad S D ean-<lb/>dem proportionem habebit, quàm A E ad E C: </s>
  <s xml:id="echoid-s10382" xml:space="preserve">eſtquè quadratum I M ad qua-<lb/>dratum N D, vt M S ad S D; </s>
  <s xml:id="echoid-s10383" xml:space="preserve">ergo quadratum I M ad quadratum N D eſt, <lb/>
<anchor type="note" xlink:label="note-0320-02a" xlink:href="note-0320-02"/>
vt A E ad E C, &amp;</s>
  <s xml:id="echoid-s10384" xml:space="preserve">c.</s>
  <s xml:id="echoid-s10385" xml:space="preserve"/>
</p>
<div xml:id="echoid-div877" type="float" level="2" n="2">
<note position="left" xlink:label="note-0319-02" xlink:href="note-0319-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0319-02" xlink:href="fig-0319-02a">
    <image file="0319-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0319-02"/>
  </figure>
<note position="left" xlink:label="note-0320-01" xlink:href="note-0320-01a" xml:space="preserve">Prop. 5. <lb/>lib. 2.</note>
<note position="left" xlink:label="note-0320-02" xlink:href="note-0320-02a" xml:space="preserve">Prop. 4. <lb/>huius.</note>
</div>
  <figure>
    <image file="0320-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0320-01"/>
  </figure>
</div>
<div xml:id="echoid-div879" type="section" level="1" n="269">
<head xml:id="echoid-head337" xml:space="preserve">SECTIO TERTIA</head>
<head xml:id="echoid-head338" xml:space="preserve">Continens Propoſit. Apollonij VIII. IX. X. <lb/>XI. XV. XIX. XVI. XVIII. <lb/>XVII. &amp; XX.</head>
<p>
  <s xml:id="echoid-s10386" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s10387" xml:space="preserve">IN hyperbola, vel ellipſi quadratum axis inclinati, ſiue <lb/>tranſuerſi ad quadratum ſummæ duarum diametrorum <lb/>coniugatarum eiuſdem ſectionis habebit eandem proportionem, <lb/>quàm productum præſectæ axis in ſuam interceptam compara-<lb/>tam ad quadratum ſummæ ſuæ interceptæ, &amp; </s>
  <s xml:id="echoid-s10388" xml:space="preserve">potentis compa-<lb/>ratarum.</s>
  <s xml:id="echoid-s10389" xml:space="preserve"/>
</p>
<pb o="283" file="0321" n="321" rhead="Conicor. Lib. VII."/>
  <figure>
    <image file="0321-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0321-01"/>
  </figure>
<p>
  <s xml:id="echoid-s10390" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s10391" xml:space="preserve">Vel ad quadratum <lb/>differẽtiæ coniugatarum eã-<lb/>dem proportionem habet, <lb/>quàm productum præſectæ in <lb/>ſuam interceptam compara-<lb/>tam ad quadratum differen-<lb/>tiæ interceptæ, &amp; </s>
  <s xml:id="echoid-s10392" xml:space="preserve">potentis <lb/>comparatarum.</s>
  <s xml:id="echoid-s10393" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10394" xml:space="preserve">X. </s>
  <s xml:id="echoid-s10395" xml:space="preserve">Vel ad rectangulum <lb/>ſub duabus coniugatis con-<lb/>tentum eandem proportionem habet, quàm præſecta axis ad <lb/>ſuam potentem comparatam.</s>
  <s xml:id="echoid-s10396" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10397" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s10398" xml:space="preserve">Ad ſummam verò duorum quadratorum ex coniugatis <lb/>eandem proportionem habet, quàm præſecta ad ſummam præ-<lb/>ſectæ, &amp; </s>
  <s xml:id="echoid-s10399" xml:space="preserve">interceptæ comparatarum.</s>
  <s xml:id="echoid-s10400" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10401" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s10402" xml:space="preserve">Sed ad quadratum erecti vnius coniugatæ eandem pro-<lb/>portionem habet, quàm præſecta axis in ſuam interceptam com-<lb/>paratam ad quadratum ſuæ præſectæ comparatæ.</s>
  <s xml:id="echoid-s10403" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10404" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s10405" xml:space="preserve">Sed ad quadratum differentiæ vnius coniugatarum, &amp; </s>
  <s xml:id="echoid-s10406" xml:space="preserve"><lb/>eius erecti eandem proportionem habet, quàm productum præ-<lb/>ſectę axis illi diametro homologę in ſuam interceptam compa-<lb/>ratam ad quadratum exceſſus præſectæ, &amp; </s>
  <s xml:id="echoid-s10407" xml:space="preserve">interceptæ compara-<lb/>tarum.</s>
  <s xml:id="echoid-s10408" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10409" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s10410" xml:space="preserve">Ad quadratum verò ſummæ inclinatæ diametri, &amp; </s>
  <s xml:id="echoid-s10411" xml:space="preserve">eius <lb/>erecti eandem proportionem habet, qnàm præſecta axis in ſuam <lb/>interceptam comparatam ad quadratum ſummæ interceptæ, &amp; </s>
  <s xml:id="echoid-s10412" xml:space="preserve"><lb/>præſectæ comparatarum.</s>
  <s xml:id="echoid-s10413" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10414" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s10415" xml:space="preserve">Sed ad figuram inclinatæ vnius coniugatarum ean-<lb/>dem proportionem habet, quàm axis præſecta ad præſectam <lb/>comparatam.</s>
  <s xml:id="echoid-s10416" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10417" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s10418" xml:space="preserve">Et ad ſummam duorum quadratorum inclinatæ, &amp; </s>
  <s xml:id="echoid-s10419" xml:space="preserve"><lb/>erecti vnius coniugatarum eandem proportionem habet, quàm <lb/>præſecta in interceptam comparatam ad duo quadrata præſectæ, <lb/>&amp; </s>
  <s xml:id="echoid-s10420" xml:space="preserve">interceptæ comparatarum.</s>
  <s xml:id="echoid-s10421" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10422" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s10423" xml:space="preserve">Et tandem ad exceſſum duorum quadratorum laterum <lb/>figuræ inclinatæ duarum coniugatarum eandem proportionem <lb/>habet, quàm productum præſectæ in interceptam comparatã ad <lb/>exceſſum quadratorum præſectæ, &amp; </s>
  <s xml:id="echoid-s10424" xml:space="preserve">interceptæ comparatarum.</s>
  <s xml:id="echoid-s10425" xml:space="preserve"/>
</p>
<pb o="284" file="0322" n="322" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s10426" xml:space="preserve">Iiſdem figuris manentibus ſit H V potens comparata, &amp; </s>
  <s xml:id="echoid-s10427" xml:space="preserve">I P ſit erectũ <lb/>
<anchor type="note" xlink:label="note-0322-01a" xlink:href="note-0322-01"/>
ipſius I L. </s>
  <s xml:id="echoid-s10428" xml:space="preserve">Dico quod quadratum A C ad quadratum ſummæ I L, &amp; </s>
  <s xml:id="echoid-s10429" xml:space="preserve">N <lb/>O eſt vt C G in E H ad quadratum E H V. </s>
  <s xml:id="echoid-s10430" xml:space="preserve">Quia quadratũ A D æquale <lb/>
<anchor type="figure" xlink:label="fig-0322-01a" xlink:href="fig-0322-01"/>
eſt S D in D M (39. </s>
  <s xml:id="echoid-s10431" xml:space="preserve">ex I.) </s>
  <s xml:id="echoid-s10432" xml:space="preserve">ergo S D in D M ad quadratum I D, nem-<lb/>
<anchor type="note" xlink:label="note-0322-02a" xlink:href="note-0322-02"/>
<anchor type="note" xlink:label="note-0322-03a" xlink:href="note-0322-03"/>
pe E C in C A ad quadratum C B (propter ſimilitudinem triangulorũ) <lb/>eſt vt quadratum A D ad quadratum I D, nempe vt quadratum A C ad <lb/>quadratum I L: </s>
  <s xml:id="echoid-s10433" xml:space="preserve">eſtque quadratum C B ad C E in E H, vt C A ad A <lb/>H, ſeu ad C G (2. </s>
  <s xml:id="echoid-s10434" xml:space="preserve">3. </s>
  <s xml:id="echoid-s10435" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s10436" xml:space="preserve">ideſt vt A C in C E ad C G in C E, &amp; </s>
  <s xml:id="echoid-s10437" xml:space="preserve"><lb/>permutando; </s>
  <s xml:id="echoid-s10438" xml:space="preserve">igitur A C in C E ad quadratum C B, quod habebat <lb/>(vt oſtenſum eſt) eandem proportionem, quàm quadratum A C ad <lb/>quadratum I L, erit vt G C in C E ad C E in E H, nempe vt C <lb/>G ad E H, ſeu C G in E H ad quadratum E H; </s>
  <s xml:id="echoid-s10439" xml:space="preserve">igitur quadratum. <lb/></s>
  <s xml:id="echoid-s10440" xml:space="preserve">A C ad quadratum I L eandem proportionem habet, quàm C G in. </s>
  <s xml:id="echoid-s10441" xml:space="preserve"><lb/>E H ad quadratum E H. </s>
  <s xml:id="echoid-s10442" xml:space="preserve">Et quadratum I L ad quadratum N O, ſeu L I <lb/>ad I P eſt vt H E ad E G (6. </s>
  <s xml:id="echoid-s10443" xml:space="preserve">7. </s>
  <s xml:id="echoid-s10444" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s10445" xml:space="preserve">ſcilicet vt quadratum E H ad <lb/>H E in E G, quod æquale ſuppoſitum fuit quadrato H V; </s>
  <s xml:id="echoid-s10446" xml:space="preserve">Ideoque <lb/>I I. </s>
  <s xml:id="echoid-s10447" xml:space="preserve">ad N O eandem proportionem habebit, quàm E H ad H V; </s>
  <s xml:id="echoid-s10448" xml:space="preserve">qua-<lb/>propter quadratum I L, ſiue ad quadratum ſummæ ipſarum I L, N O eſt <lb/>vt quadratum H E ad quadratum E H V; </s>
  <s xml:id="echoid-s10449" xml:space="preserve">ſiue ad quadratum differentiæ <lb/>I L, &amp; </s>
  <s xml:id="echoid-s10450" xml:space="preserve">N O erit vt quadratum E H ad quadratum differentiæ E H, &amp; </s>
  <s xml:id="echoid-s10451" xml:space="preserve"><lb/>H V, ſiue ad I L in N O habebit eandem proportionem, quàm E H ad <lb/>H V; </s>
  <s xml:id="echoid-s10452" xml:space="preserve">ſiue ad duo quadrata I L, N O eandem proportionem habebit, <lb/>quàm E H ad ſummam E H, E G; </s>
  <s xml:id="echoid-s10453" xml:space="preserve">eo quod quadratum I L ad quadra-<lb/>tum N O eſt vt E H ad E G; </s>
  <s xml:id="echoid-s10454" xml:space="preserve">ſiue inſuper ad quadratum I P eandem <lb/>proportionem habebit, quàm quadratum E H ad quadratum E G; </s>
  <s xml:id="echoid-s10455" xml:space="preserve">vel <lb/>potius ad quadratum differentiæ I L, &amp; </s>
  <s xml:id="echoid-s10456" xml:space="preserve">I P erit vt quadratum E H ad <lb/>quadratum differentiæ E H, &amp; </s>
  <s xml:id="echoid-s10457" xml:space="preserve">E G, vel rurſus ad quadratum rectæ li-<lb/>neæ ex L I, &amp; </s>
  <s xml:id="echoid-s10458" xml:space="preserve">I P compoſitæ, erit vt quadratum H E ad quadratum <lb/>ſummæ duarum H E, E G, atque ad L I in I P eandem proportionem <lb/>habebit, quàm H E ad E G; </s>
  <s xml:id="echoid-s10459" xml:space="preserve">vel ad quadratum ipſius L I cum quadrato <lb/>I P habebit eandem proportionem, quàm quadratum H E ad duo qua-
<pb o="285" file="0323" n="323" rhead="Conicor. Lib. VII."/>
drata H E, &amp; </s>
  <s xml:id="echoid-s10460" xml:space="preserve">ipſius E G, ſiue ad differentiam duorum quadratorum L <lb/>I, &amp; </s>
  <s xml:id="echoid-s10461" xml:space="preserve">ipſius I P eandem proportionem habebit, quàm quadratum H E <lb/>ad differentiam duorum quadratorum H E, &amp; </s>
  <s xml:id="echoid-s10462" xml:space="preserve">E G. </s>
  <s xml:id="echoid-s10463" xml:space="preserve">Et iam oſtenſum eſt <lb/>quod quadratum A C ad quadratum I L eandem proportionem habet, <lb/>quàm C G in H E ad quadratum H E; </s>
  <s xml:id="echoid-s10464" xml:space="preserve">8. </s>
  <s xml:id="echoid-s10465" xml:space="preserve">ergo ex æqualitate quadratum <lb/>A C, fiue ad quadratum ſummæ I L, N O eſt, vt C G in H E ad qua-<lb/>dratum E H V; </s>
  <s xml:id="echoid-s10466" xml:space="preserve">9. </s>
  <s xml:id="echoid-s10467" xml:space="preserve">ſiue ad quadratum differentiæ eius, quæ eſt inter I <lb/>
<anchor type="note" xlink:label="note-0323-01a" xlink:href="note-0323-01"/>
L, N O eſt vt C G in H E ad quadratum exceſſus E H ſupra H V: </s>
  <s xml:id="echoid-s10468" xml:space="preserve">10. <lb/></s>
  <s xml:id="echoid-s10469" xml:space="preserve">
<anchor type="note" xlink:label="note-0323-02a" xlink:href="note-0323-02"/>
ſiue ad I L in N O erit, vt C G ad H V: </s>
  <s xml:id="echoid-s10470" xml:space="preserve">11. </s>
  <s xml:id="echoid-s10471" xml:space="preserve">ſiue ad duorum quadrato-<lb/>
<anchor type="note" xlink:label="note-0323-03a" xlink:href="note-0323-03"/>
<anchor type="figure" xlink:label="fig-0323-01a" xlink:href="fig-0323-01"/>
rum I L, N O ſummam, erit vt <lb/>C G ad ſummam G E, E H; </s>
  <s xml:id="echoid-s10472" xml:space="preserve">12. <lb/></s>
  <s xml:id="echoid-s10473" xml:space="preserve">
<anchor type="note" xlink:label="note-0323-04a" xlink:href="note-0323-04"/>
ſiue ad quadratum I P erit, vt <lb/>C G in H E ad quadratum E G: <lb/></s>
  <s xml:id="echoid-s10474" xml:space="preserve">13. </s>
  <s xml:id="echoid-s10475" xml:space="preserve">ſiue ad quadratum differen-<lb/>
<anchor type="note" xlink:label="note-0323-05a" xlink:href="note-0323-05"/>
tiæ L I, I P erit, vt C G in E <lb/>H ad quadratum differentiæ H <lb/>E, E G: </s>
  <s xml:id="echoid-s10476" xml:space="preserve">14. </s>
  <s xml:id="echoid-s10477" xml:space="preserve">ſiue ad quadratum <lb/>
<anchor type="note" xlink:label="note-0323-06a" xlink:href="note-0323-06"/>
ex recta linea æquali sũmæ dua-<lb/>rum L I, I P, erit vt C G in <lb/>E H ad quadratum ex recta li-<lb/>nea compoſita ex H E, E G: <lb/></s>
  <s xml:id="echoid-s10478" xml:space="preserve">
<anchor type="note" xlink:label="note-0323-07a" xlink:href="note-0323-07"/>
15. </s>
  <s xml:id="echoid-s10479" xml:space="preserve">ſiue ad L I in I P erit vt C G ad G E: </s>
  <s xml:id="echoid-s10480" xml:space="preserve">16. </s>
  <s xml:id="echoid-s10481" xml:space="preserve">ſiue ad duo quadrata ex <lb/>L I, &amp; </s>
  <s xml:id="echoid-s10482" xml:space="preserve">ex I P erit vt C G in E H ad duo quadrata E G, &amp; </s>
  <s xml:id="echoid-s10483" xml:space="preserve">E H: </s>
  <s xml:id="echoid-s10484" xml:space="preserve">17. <lb/></s>
  <s xml:id="echoid-s10485" xml:space="preserve">
<anchor type="note" xlink:label="note-0323-08a" xlink:href="note-0323-08"/>
ſiue ad differentiam duorum quadratorum ex L I, &amp; </s>
  <s xml:id="echoid-s10486" xml:space="preserve">ex I P erit vt C G <lb/>
<anchor type="note" xlink:label="note-0323-09a" xlink:href="note-0323-09"/>
in E H ad differentiam duorum quadratorum ex H E, &amp; </s>
  <s xml:id="echoid-s10487" xml:space="preserve">ex E G. </s>
  <s xml:id="echoid-s10488" xml:space="preserve">Et <lb/>hoc erat propoſitum.</s>
  <s xml:id="echoid-s10489" xml:space="preserve"/>
</p>
<div xml:id="echoid-div879" type="float" level="2" n="1">
<note position="right" xlink:label="note-0322-01" xlink:href="note-0322-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0322-01" xlink:href="fig-0322-01a">
    <image file="0322-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0322-01"/>
  </figure>
<note position="left" xlink:label="note-0322-02" xlink:href="note-0322-02a" xml:space="preserve">37. lib. I.</note>
<note position="right" xlink:label="note-0322-03" xlink:href="note-0322-03a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0323-01" xlink:href="note-0323-01a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0323-02" xlink:href="note-0323-02a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0323-03" xlink:href="note-0323-03a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0323-01" xlink:href="fig-0323-01a">
    <image file="0323-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0323-01"/>
  </figure>
<note position="left" xlink:label="note-0323-04" xlink:href="note-0323-04a" xml:space="preserve">f</note>
<note position="left" xlink:label="note-0323-05" xlink:href="note-0323-05a" xml:space="preserve">g</note>
<note position="left" xlink:label="note-0323-06" xlink:href="note-0323-06a" xml:space="preserve">h</note>
<note position="left" xlink:label="note-0323-07" xlink:href="note-0323-07a" xml:space="preserve">i</note>
<note position="left" xlink:label="note-0323-08" xlink:href="note-0323-08a" xml:space="preserve">k</note>
<note position="left" xlink:label="note-0323-09" xlink:href="note-0323-09a" xml:space="preserve">l</note>
</div>
</div>
<div xml:id="echoid-div881" type="section" level="1" n="270">
<head xml:id="echoid-head339" xml:space="preserve">Notæ in Propoſit. VIII.</head>
<p>
  <s xml:id="echoid-s10490" xml:space="preserve">IIſdem figuris manentibus ſit H V potens comparata, &amp;</s>
  <s xml:id="echoid-s10491" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10492" xml:space="preserve">Præter defi-<lb/>
<anchor type="note" xlink:label="note-0323-10a" xlink:href="note-0323-10"/>
nitiones ſuperius expoſitas hic duæ aliæ declarari debent, ignotum enim eſt <lb/>quid nam nomine Figuræ comparatæ, &amp; </s>
  <s xml:id="echoid-s10493" xml:space="preserve">Potentis comparatæ intelligi debeat. <lb/></s>
  <s xml:id="echoid-s10494" xml:space="preserve">Itaq; </s>
  <s xml:id="echoid-s10495" xml:space="preserve">rectangulum ſub præſecta comparata, &amp; </s>
  <s xml:id="echoid-s10496" xml:space="preserve">intercepta comparata contentum, <lb/>ideſt rectangulum H E G vocatur Figura comparata: </s>
  <s xml:id="echoid-s10497" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10498" xml:space="preserve">ſi quadratum rectæ li-<lb/>neæ H V æquale fuerit rectangulo H E G vocatur H V Potens comparata.</s>
  <s xml:id="echoid-s10499" xml:space="preserve"/>
</p>
<div xml:id="echoid-div881" type="float" level="2" n="1">
<note position="left" xlink:label="note-0323-10" xlink:href="note-0323-10a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s10500" xml:space="preserve">Ergo S D in D M ad quadratum D I, nempe E C in C A ad qua-<lb/>
<anchor type="note" xlink:label="note-0323-11a" xlink:href="note-0323-11"/>
dratũ C E, &amp;</s>
  <s xml:id="echoid-s10501" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10502" xml:space="preserve">AEqualia enim ſpatia, ſcilicet rectangulũ S D M, &amp; </s>
  <s xml:id="echoid-s10503" xml:space="preserve">quadratũ <lb/>
<anchor type="note" xlink:label="note-0323-12a" xlink:href="note-0323-12"/>
D A ad idem quadratum I D habent eandem proportionem; </s>
  <s xml:id="echoid-s10504" xml:space="preserve">ſed quia triangula <lb/>M I D, &amp; </s>
  <s xml:id="echoid-s10505" xml:space="preserve">A B C ſimilia ſunt, propterea quod latera homologa ſunt parallela <lb/>inter ſe; </s>
  <s xml:id="echoid-s10506" xml:space="preserve">pariterquè triangula D S I, &amp; </s>
  <s xml:id="echoid-s10507" xml:space="preserve">C E B ſunt ſimilia, vt oſtenſum eſt <lb/>in 6. </s>
  <s xml:id="echoid-s10508" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10509" xml:space="preserve">7. </s>
  <s xml:id="echoid-s10510" xml:space="preserve">huius; </s>
  <s xml:id="echoid-s10511" xml:space="preserve">ergo S D ad D I erit vt E C ad C B, atquè M D ad D I <lb/>eſt vt A C ad C B erunt compoſitæ proportiones eædem inter ſe, ſcilicet rectan-<lb/>gulum S D M ad quadratum D I eandem proportionem habebit, quàm rectan-<lb/>gulum E C A ad quadratum C B; </s>
  <s xml:id="echoid-s10512" xml:space="preserve">quare vt quadratum A D ad quadratum <lb/>D I, ſeu vt quadruplum ad quadruplum, ſcilicet vt quadratum A C ad qua-<lb/>dratum I L, co quod A D, &amp; </s>
  <s xml:id="echoid-s10513" xml:space="preserve">I D ſemiſſes ſunt diametrorum A C, I L.</s>
  <s xml:id="echoid-s10514" xml:space="preserve"/>
</p>
<div xml:id="echoid-div882" type="float" level="2" n="2">
<note position="left" xlink:label="note-0323-11" xlink:href="note-0323-11a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0323-12" xlink:href="note-0323-12a" xml:space="preserve">37. lib. I.</note>
</div>
<pb o="286" file="0324" n="324" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div884" type="section" level="1" n="271">
<head xml:id="echoid-head340" xml:space="preserve">Notæ in Propoſit. IX.</head>
<p>
  <s xml:id="echoid-s10515" xml:space="preserve">SIue ad quadratum differentiæ eius, quæ eſt inter I L, N O eſt vt C <lb/>
<anchor type="note" xlink:label="note-0324-01a" xlink:href="note-0324-01"/>
G in H E ad quadratum E H, H V, &amp;</s>
  <s xml:id="echoid-s10516" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10517" xml:space="preserve">Licet nouem ſubſequentes <lb/>propoſitiones facile ex octaua deducantur, nequeunt tamen omnes ſimul conglo-<lb/>batæ vnico bauſtu deuorari; </s>
  <s xml:id="echoid-s10518" xml:space="preserve">itaque opere prætium erit aliquantisper breuita-<lb/>tem nimiam Arabici Interpretis relinquere. </s>
  <s xml:id="echoid-s10519" xml:space="preserve">Tria demonſtrata ſunt in propoſi-<lb/>tione octaua, quæ in ſequentibus nouem propoſitionibus vſum babent. </s>
  <s xml:id="echoid-s10520" xml:space="preserve">Primò <lb/>quod quadratum A C ad quadratum I L eandem proportionem habeat, quàm <lb/>rectangulum C G in H E ad quudratum H E. </s>
  <s xml:id="echoid-s10521" xml:space="preserve">Secundò quod I L ad N O ean-<lb/>dem proportionem habeat, quàm H E intercepta comparata ad H V potentem <lb/>comparatam. </s>
  <s xml:id="echoid-s10522" xml:space="preserve">Tertio quod quadratum I L ad quadratum N O, ſeu L I ad eius <lb/>
<anchor type="note" xlink:label="note-0324-02a" xlink:href="note-0324-02"/>
<anchor type="figure" xlink:label="fig-0324-01a" xlink:href="fig-0324-01"/>
latus rectum I P, ſit vt H E ad E G, vel vt quadratum H E ad rectangulum <lb/>H E G, vel ad quadratũ H V. </s>
  <s xml:id="echoid-s10523" xml:space="preserve">Modo propoſitio nona ſic demonſtrabitur. </s>
  <s xml:id="echoid-s10524" xml:space="preserve">Quia <lb/>I L ad N O eandem rationem habet quàm H E ad H V, erunt antecedentes ad <lb/>differentias terminorum proportionales, ideſt I L ad differentiam ipſarum I L, <lb/>&amp; </s>
  <s xml:id="echoid-s10525" xml:space="preserve">N O eandem proportionem habebit, quàm H E ad differentiam ipſarum E <lb/>H, &amp; </s>
  <s xml:id="echoid-s10526" xml:space="preserve">H V: </s>
  <s xml:id="echoid-s10527" xml:space="preserve">atquè quadratum I L ad quadratum ex differentia ipſarum I L, <lb/>&amp; </s>
  <s xml:id="echoid-s10528" xml:space="preserve">N O deſcriptum eandem proportionem habebit, quàm quadratum H E ad <lb/>quadratum ex differentia ipſarum E H, &amp; </s>
  <s xml:id="echoid-s10529" xml:space="preserve">H V deſcriptum: </s>
  <s xml:id="echoid-s10530" xml:space="preserve">erat autem qua-<lb/>
<anchor type="note" xlink:label="note-0324-03a" xlink:href="note-0324-03"/>
dratum A C ad quadratum I L, vt rectangulum C G in H E ad quadratum <lb/>E H; </s>
  <s xml:id="echoid-s10531" xml:space="preserve">ergo ex æquali ordinata quadratum A C ad quadratum ex differentia ip-<lb/>ſarum I L, &amp; </s>
  <s xml:id="echoid-s10532" xml:space="preserve">N O deſcriptum eandem proportionem habebit, quàm rectangu-<lb/>lum C G in H E ad quadratum ex differentia ipſarum E H, &amp; </s>
  <s xml:id="echoid-s10533" xml:space="preserve">H V.</s>
  <s xml:id="echoid-s10534" xml:space="preserve"/>
</p>
<div xml:id="echoid-div884" type="float" level="2" n="1">
<note position="right" xlink:label="note-0324-01" xlink:href="note-0324-01a" xml:space="preserve">c</note>
<note position="left" xlink:label="note-0324-02" xlink:href="note-0324-02a" xml:space="preserve">15. &amp; 16. <lb/>lib. I.</note>
  <figure xlink:label="fig-0324-01" xlink:href="fig-0324-01a">
    <image file="0324-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0324-01"/>
  </figure>
<note position="left" xlink:label="note-0324-03" xlink:href="note-0324-03a" xml:space="preserve">8. huius.</note>
</div>
<pb o="287" file="0325" n="325" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div886" type="section" level="1" n="272">
<head xml:id="echoid-head341" xml:space="preserve">Notæ in Propoſit. X.</head>
<p style="it">
  <s xml:id="echoid-s10535" xml:space="preserve">SIue ad I L in N O erit vt C G ad H V, &amp;</s>
  <s xml:id="echoid-s10536" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10537" xml:space="preserve">Quia I L ad N O habe-<lb/>
<anchor type="note" xlink:label="note-0325-01a" xlink:href="note-0325-01"/>
bat eandem proportionem, quàm E H ad H V poſitis communibus altitudi-<lb/>nibus I L, &amp; </s>
  <s xml:id="echoid-s10538" xml:space="preserve">E H habebit quadratum I L ad rectangulum I L in N O eandẽ <lb/>proportionem, quàm quadratum E H ad rectangulum E H in H V; </s>
  <s xml:id="echoid-s10539" xml:space="preserve">ſed qua-<lb/>
<anchor type="note" xlink:label="note-0325-02a" xlink:href="note-0325-02"/>
dratum A C ad quadratum I L habebat eandem proportionem, quàm rectangu-<lb/>lum C G in E H ad quadratum E H; </s>
  <s xml:id="echoid-s10540" xml:space="preserve">ergo ex æqualitate quadratum A C ad <lb/>rectangulum ſub I L in N O eandem proportionem habet, quàm rectangulum <lb/>C G in H E ad rectangulum E H in H V, ſiue quàm habet C G, ad H V.</s>
  <s xml:id="echoid-s10541" xml:space="preserve"/>
</p>
<div xml:id="echoid-div886" type="float" level="2" n="1">
<note position="left" xlink:label="note-0325-01" xlink:href="note-0325-01a" xml:space="preserve">d</note>
<note position="right" xlink:label="note-0325-02" xlink:href="note-0325-02a" xml:space="preserve">ex prop. 8. <lb/>huius.</note>
</div>
  <figure>
    <image file="0325-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0325-01"/>
  </figure>
</div>
<div xml:id="echoid-div888" type="section" level="1" n="273">
<head xml:id="echoid-head342" xml:space="preserve">Notæ in Propoſit. XI.</head>
<p style="it">
  <s xml:id="echoid-s10542" xml:space="preserve">SIue ad duorum quadratorum I L, N O ſummam erit vt C G ad ſum-<lb/>
<anchor type="note" xlink:label="note-0325-03a" xlink:href="note-0325-03"/>
mam G E, &amp; </s>
  <s xml:id="echoid-s10543" xml:space="preserve">E H, &amp;</s>
  <s xml:id="echoid-s10544" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10545" xml:space="preserve">Quia quadratum I L ad quadratum N O erat, <lb/>vt H E ad E G, antecedentes ad ſummas terminorum erunt proportionales, <lb/>ſcilicet quadratum I L ad quadratum I L ſimul cum quadrato N O eandem pro-<lb/>
<anchor type="note" xlink:label="note-0325-04a" xlink:href="note-0325-04"/>
portionem babebit, quàm H E ad ſummam ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10546" xml:space="preserve">E G; </s>
  <s xml:id="echoid-s10547" xml:space="preserve">erat au-<lb/>tem quadratum C A ad quadratum I L, vt C G ad E H; </s>
  <s xml:id="echoid-s10548" xml:space="preserve">ergo ex æqualitate <lb/>quadratum A C ad quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s10549" xml:space="preserve">ex N O ſimul ſumpta eandem pro-<lb/>portionem babebit, quàm C G, vel H A ad ſummam ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10550" xml:space="preserve">G E.</s>
  <s xml:id="echoid-s10551" xml:space="preserve"/>
</p>
<div xml:id="echoid-div888" type="float" level="2" n="1">
<note position="left" xlink:label="note-0325-03" xlink:href="note-0325-03a" xml:space="preserve">e</note>
<note position="right" xlink:label="note-0325-04" xlink:href="note-0325-04a" xml:space="preserve">Prop. 8. <lb/>huius.</note>
</div>
<pb o="288" file="0326" n="326" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div890" type="section" level="1" n="274">
<head xml:id="echoid-head343" xml:space="preserve">Notæ in Propoſit. XV.</head>
<p style="it">
  <s xml:id="echoid-s10552" xml:space="preserve">SIue ad quadratum I P erit vt C G in E H ad quadratum E G, &amp;</s>
  <s xml:id="echoid-s10553" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s10554" xml:space="preserve">
<anchor type="note" xlink:label="note-0326-01a" xlink:href="note-0326-01"/>
Quoniam I L ad I P erat vt H E ad E G; </s>
  <s xml:id="echoid-s10555" xml:space="preserve">ergo quadratum I L ad qua-<lb/>dratum I P erit vt quadratum H E ad quadratum E G; </s>
  <s xml:id="echoid-s10556" xml:space="preserve">erat autem quadra-<lb/>tum A C ad quadratum I L, vt rectangulum C G, ſeu A H in H E ad qua-<lb/>dratum E H; </s>
  <s xml:id="echoid-s10557" xml:space="preserve">igitur ex æqualitate quadratum A C ad quadratum I P ean-<lb/>dem proportionem habebit, quàm rectangulum A H E ad quadratum G E.</s>
  <s xml:id="echoid-s10558" xml:space="preserve"/>
</p>
<div xml:id="echoid-div890" type="float" level="2" n="1">
<note position="left" xlink:label="note-0326-01" xlink:href="note-0326-01a" xml:space="preserve">f</note>
</div>
</div>
<div xml:id="echoid-div892" type="section" level="1" n="275">
<head xml:id="echoid-head344" xml:space="preserve">Notæ in Propoſit. XIX.</head>
<p style="it">
  <s xml:id="echoid-s10559" xml:space="preserve">SIue ad quadratum differentiæ L I, &amp; </s>
  <s xml:id="echoid-s10560" xml:space="preserve">I P erit vt C G in E H ad qua-<lb/>
<anchor type="note" xlink:label="note-0326-02a" xlink:href="note-0326-02"/>
dratum differentiæ H E, E G, &amp;</s>
  <s xml:id="echoid-s10561" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10562" xml:space="preserve">Quia I L ad I P erat vt H E ad <lb/>E G, comparando antecedentes ad terminorum differentias, ſcilicet I L ad dif-<lb/>ferentiam ipſarum I L, &amp; </s>
  <s xml:id="echoid-s10563" xml:space="preserve">I P eandem proportionem habebit, quàm E H ad <lb/>
<anchor type="figure" xlink:label="fig-0326-01a" xlink:href="fig-0326-01"/>
differentiam ipſarum E H, &amp; </s>
  <s xml:id="echoid-s10564" xml:space="preserve">E G, &amp; </s>
  <s xml:id="echoid-s10565" xml:space="preserve">quadratum I L ad quadratum ex dif-<lb/>ferentia ipſarum I L, &amp; </s>
  <s xml:id="echoid-s10566" xml:space="preserve">I P deſcriptum eandem proportionem habebit, quàm <lb/>quadrætum H E ad quadratum ex differentia ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10567" xml:space="preserve">G E deſcriptũ: <lb/></s>
  <s xml:id="echoid-s10568" xml:space="preserve">erat autem quadratum C A ad quadratum I L, vt rectangulum A H E ad <lb/>quadratum H E; </s>
  <s xml:id="echoid-s10569" xml:space="preserve">ergo ex æqualitate quadratum A C ad quadratum ex diffe-<lb/>rentia ipſarum I L, &amp; </s>
  <s xml:id="echoid-s10570" xml:space="preserve">I P eandem proportionem habebit, quàm rectangulum <lb/>A H E ad quadratum ex differentia ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10571" xml:space="preserve">E G.</s>
  <s xml:id="echoid-s10572" xml:space="preserve"/>
</p>
<div xml:id="echoid-div892" type="float" level="2" n="1">
<note position="left" xlink:label="note-0326-02" xlink:href="note-0326-02a" xml:space="preserve">g</note>
  <figure xlink:label="fig-0326-01" xlink:href="fig-0326-01a">
    <image file="0326-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0326-01"/>
  </figure>
</div>
<pb o="289" file="0327" n="327" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div894" type="section" level="1" n="276">
<head xml:id="echoid-head345" xml:space="preserve">Notæ in Propoſit. XVI.</head>
<p style="it">
  <s xml:id="echoid-s10573" xml:space="preserve">SIue ad quadratum ex recta linea æquali ſummæ duarum I L, &amp; </s>
  <s xml:id="echoid-s10574" xml:space="preserve">I P <lb/>
<anchor type="note" xlink:label="note-0327-01a" xlink:href="note-0327-01"/>
erit, vt C G in H E ad quadratum ex recta linea compoſita ex H E, <lb/>E G, &amp;</s>
  <s xml:id="echoid-s10575" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10576" xml:space="preserve">Quia I L ad I P erat vt H E ad E G comparando, antecedentes ad <lb/>ſummas terminorum, erit I L ad I L, &amp; </s>
  <s xml:id="echoid-s10577" xml:space="preserve">I P ſimul ſumptas, vt H E ad H E, <lb/>&amp; </s>
  <s xml:id="echoid-s10578" xml:space="preserve">E G ſimul ſumptas, &amp; </s>
  <s xml:id="echoid-s10579" xml:space="preserve">quadratum I L ad quadratum ex ſumma ipſarum <lb/>I L, &amp; </s>
  <s xml:id="echoid-s10580" xml:space="preserve">I P deſcriptum, erit vt quadratum H E ad quadratum ex ſumma <lb/>duarum H E, &amp; </s>
  <s xml:id="echoid-s10581" xml:space="preserve">E G deſcriptum; </s>
  <s xml:id="echoid-s10582" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10583" xml:space="preserve">erat prius quadratum A C ad quadra-<lb/>tum I L, vt rectangulum A H E ad quadratum H E; </s>
  <s xml:id="echoid-s10584" xml:space="preserve">igitur ex æqualitate <lb/>quadratum A C ad quadratum ex ſumma ipſarum I L, &amp; </s>
  <s xml:id="echoid-s10585" xml:space="preserve">I P deſcriptum eã-<lb/>dem proportionem habebit, quàm rectangulum A H E ad quadratum ex ſumma <lb/>ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10586" xml:space="preserve">E G deſcriptum.</s>
  <s xml:id="echoid-s10587" xml:space="preserve"/>
</p>
<div xml:id="echoid-div894" type="float" level="2" n="1">
<note position="left" xlink:label="note-0327-01" xlink:href="note-0327-01a" xml:space="preserve">h</note>
</div>
  <figure>
    <image file="0327-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0327-01"/>
  </figure>
</div>
<div xml:id="echoid-div896" type="section" level="1" n="277">
<head xml:id="echoid-head346" xml:space="preserve">Notæ in Propoſit. XVIII.</head>
<p style="it">
  <s xml:id="echoid-s10588" xml:space="preserve">SIue ad I L in I P erit, vt C G in G E, &amp;</s>
  <s xml:id="echoid-s10589" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10590" xml:space="preserve">Quia I L ad I P eſt vt H <lb/>
<anchor type="note" xlink:label="note-0327-02a" xlink:href="note-0327-02"/>
E ad G E poſitis communibus altitudinibus I L, H E habebit quadratum <lb/>I L ad rectangulum ſub I L, &amp; </s>
  <s xml:id="echoid-s10591" xml:space="preserve">I P eandem proportionem, quàm quadratum <lb/>H E ad rectangulum H E G: </s>
  <s xml:id="echoid-s10592" xml:space="preserve">ſed quadratum A C ad quadratum I L eandem <lb/>proportionem habebat, quàm rectangulum A H E ad quadratum H E; </s>
  <s xml:id="echoid-s10593" xml:space="preserve">ergo ex <lb/>æqualitate quadratum A C ad rectangulum L I P eandem proportionem habebit <lb/>quàm rectangulum A H E ad rectangulum H E G, ſeu vt A H, vel C G ad <lb/>G E.</s>
  <s xml:id="echoid-s10594" xml:space="preserve"/>
</p>
<div xml:id="echoid-div896" type="float" level="2" n="1">
<note position="left" xlink:label="note-0327-02" xlink:href="note-0327-02a" xml:space="preserve">i</note>
</div>
<pb o="290" file="0328" n="328" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div898" type="section" level="1" n="278">
<head xml:id="echoid-head347" xml:space="preserve">Notæ in Propoſit. XVII.</head>
<p style="it">
  <s xml:id="echoid-s10595" xml:space="preserve">SIue ad duo quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s10596" xml:space="preserve">I P erit, vt C G in E H ad duo qua-<lb/>
<anchor type="note" xlink:label="note-0328-01a" xlink:href="note-0328-01"/>
drata E G, &amp; </s>
  <s xml:id="echoid-s10597" xml:space="preserve">E H, &amp;</s>
  <s xml:id="echoid-s10598" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10599" xml:space="preserve">Quoniam I L ad I P erat vt H E ad E G, <lb/>&amp; </s>
  <s xml:id="echoid-s10600" xml:space="preserve">quadratum I L ad quadratum I P erit vt quadratum H E ad quadratum <lb/>E G: </s>
  <s xml:id="echoid-s10601" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10602" xml:space="preserve">comparando antecedentes ad terminorũ ſummas quadratum I L ad qua-<lb/>dratum I L vna cum quadrato I P habebit eandem proportionem, quàm qua-<lb/>dratum H E ad ſummam quadrati H E cum quadrato E G: </s>
  <s xml:id="echoid-s10603" xml:space="preserve">ſed prius quadra-<lb/>tum A C ad quadratum I L erat vt rectangulum A H E ad quadratum H E; <lb/></s>
  <s xml:id="echoid-s10604" xml:space="preserve">igitur quadratum A C ad ſummam quadrati I L cum quadrato I P eãdem pro-<lb/>portionem babebit quàm rectangulum A H E ad quadratum E G vna cum qua-<lb/>drato E H.</s>
  <s xml:id="echoid-s10605" xml:space="preserve"/>
</p>
<div xml:id="echoid-div898" type="float" level="2" n="1">
<note position="left" xlink:label="note-0328-01" xlink:href="note-0328-01a" xml:space="preserve">k</note>
</div>
  <figure>
    <image file="0328-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0328-01"/>
  </figure>
</div>
<div xml:id="echoid-div900" type="section" level="1" n="279">
<head xml:id="echoid-head348" xml:space="preserve">Notæ in Propoſit. XX.</head>
<p style="it">
  <s xml:id="echoid-s10606" xml:space="preserve">SIue ad differentiam duorum quadratorum I L, I P erit, vt C G in H <lb/>
<anchor type="note" xlink:label="note-0328-02a" xlink:href="note-0328-02"/>
E ad differentiam duorum quadratorum ex H E, &amp; </s>
  <s xml:id="echoid-s10607" xml:space="preserve">ex E G, &amp;</s>
  <s xml:id="echoid-s10608" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s10609" xml:space="preserve">Quoniam vt dictum eſt quadratum I L ad quadratum I P eandem proportionẽ <lb/>habet, quàm quadratum H E ad quadratum G E, &amp; </s>
  <s xml:id="echoid-s10610" xml:space="preserve">comparando anteceden-<lb/>tes ad terminorum differentias quadratum I P ad differentiam quadrati I L à <lb/>quadrato I P eandem proportionem habebit, quàm quadratum H E ad diffe-<lb/>rentiam inter quadratum H E, &amp; </s>
  <s xml:id="echoid-s10611" xml:space="preserve">quadratum E G: </s>
  <s xml:id="echoid-s10612" xml:space="preserve">eſtque quadratum C A <lb/>ad quadratum I L, vt rectangulum A H E ad quadratũ H E; </s>
  <s xml:id="echoid-s10613" xml:space="preserve">ergo ex æquali <lb/>quadratum A C ad quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10614" xml:space="preserve">ex I P differentiam eandem pro-<lb/>portionem habebit, quàm rectangulum A H E ad quadratorum ex E G, &amp; </s>
  <s xml:id="echoid-s10615" xml:space="preserve">ex <lb/>E H differentiam.</s>
  <s xml:id="echoid-s10616" xml:space="preserve"/>
</p>
<div xml:id="echoid-div900" type="float" level="2" n="1">
<note position="left" xlink:label="note-0328-02" xlink:href="note-0328-02a" xml:space="preserve">l</note>
</div>
<pb o="291" file="0329" n="329" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div902" type="section" level="1" n="280">
<head xml:id="echoid-head349" xml:space="preserve">SECTIO QVARTA</head>
<head xml:id="echoid-head350" xml:space="preserve">Continens Propoſit. Apollonij XII. XIII. <lb/>XXIX. XVII. XXII. XXX. <lb/>XIV. &amp; XXV.</head>
<p>
  <s xml:id="echoid-s10617" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s10618" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s10619" xml:space="preserve">DIfferentia quadratorum duorum axium hy-<lb/>XXV. </s>
  <s xml:id="echoid-s10620" xml:space="preserve">perboles æqualis eſt differentiæ quadra-<lb/>torum quarumlibet duarum diametrorum coniugatarum.</s>
  <s xml:id="echoid-s10621" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10622" xml:space="preserve">XXVIIII. </s>
  <s xml:id="echoid-s10623" xml:space="preserve">Nempe differẽtiæ inter quadrata à figuris earumdẽ <lb/>diametrorum æquales ſunt.</s>
  <s xml:id="echoid-s10624" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10625" xml:space="preserve">XXVII. </s>
  <s xml:id="echoid-s10626" xml:space="preserve">Et differentia duorum axium maior eſt differentia <lb/>quarumlibet duarum diametrorum coniugatarum.</s>
  <s xml:id="echoid-s10627" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10628" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s10629" xml:space="preserve">Et ſumma quadratorũ duorum axium ellipſis æqualis <lb/>eſt ſummæ quadratorum quarumlibet duarum diametrorum con-<lb/>iugatarum.</s>
  <s xml:id="echoid-s10630" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10631" xml:space="preserve">XXX. </s>
  <s xml:id="echoid-s10632" xml:space="preserve">Nempe ſummæ quadratorum, &amp; </s>
  <s xml:id="echoid-s10633" xml:space="preserve">figurarum earundem <lb/>diametrorum homologarum ſunt æquales.</s>
  <s xml:id="echoid-s10634" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10635" xml:space="preserve">XIIII. </s>
  <s xml:id="echoid-s10636" xml:space="preserve">Axis verò tranſuerſi quadratũ ad differentiam quadra-<lb/>torum duarum diametrorum coniugatarum eandem proportio-<lb/>nem habet, quàm præſecta ad duplam inuerſæ.</s>
  <s xml:id="echoid-s10637" xml:space="preserve"/>
</p>
  <figure>
    <image file="0329-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0329-01"/>
  </figure>
<pb file="0329a" n="330"/>
  <figure>
    <image file="0329a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0329a-01"/>
  </figure>
<pb o="292" file="0330" n="331" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s10638" xml:space="preserve">In eiſdem figuris, quia quadratum A C ad quadratum ſui coniugati <lb/>
<anchor type="note" xlink:label="note-0330-01a" xlink:href="note-0330-01"/>
(in propoſitione 12. </s>
  <s xml:id="echoid-s10639" xml:space="preserve">13. </s>
  <s xml:id="echoid-s10640" xml:space="preserve">25.) </s>
  <s xml:id="echoid-s10641" xml:space="preserve">nempe C A ad A F erectum ipſius eſt, <lb/>
<anchor type="note" xlink:label="note-0330-02a" xlink:href="note-0330-02"/>
vt Præſecta C G ad Interceptam G A, ſiue ad C H; </s>
  <s xml:id="echoid-s10642" xml:space="preserve">ergo quadratum <lb/>A C in hyperbola ad differentiam quadratorum axium ipſius, &amp; </s>
  <s xml:id="echoid-s10643" xml:space="preserve">in elli-<lb/>pſi ad eorundem ſummam eandem proportionem habet, quàm C G ad <lb/>H G. </s>
  <s xml:id="echoid-s10644" xml:space="preserve">Demonſtratum autem prius fuit, quadratum C A ad quadratum <lb/>
<anchor type="note" xlink:label="note-0330-03a" xlink:href="note-0330-03"/>
I L eandem proportionem habere, quàm C G ad H E, &amp; </s>
  <s xml:id="echoid-s10645" xml:space="preserve">quadratum <lb/>
<anchor type="figure" xlink:label="fig-0330-01a" xlink:href="fig-0330-01"/>
I L ad quadratum N O eandem proportionem habet, quàm H E ad E <lb/>
<anchor type="note" xlink:label="note-0330-04a" xlink:href="note-0330-04"/>
G; </s>
  <s xml:id="echoid-s10646" xml:space="preserve">Inſuper quudratum I L ad ſummam quadratorum I L, N O in elli-<lb/>pſi, aut ad eorundem differentiam in hyperbola eandem proportionem <lb/>habebit, quàm H E ad H G; </s>
  <s xml:id="echoid-s10647" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10648" xml:space="preserve">in propoſitione 14. </s>
  <s xml:id="echoid-s10649" xml:space="preserve">vt H E ad exceffum <lb/>H E, E G, quod eſt duplum D G; </s>
  <s xml:id="echoid-s10650" xml:space="preserve">igitur ex æqualitate quadratum A <lb/>C, ſiue ad ſummam duorum quadratorum I L, N O, quemadmodum <lb/>habetur in propoſitione 22. </s>
  <s xml:id="echoid-s10651" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10652" xml:space="preserve">30. </s>
  <s xml:id="echoid-s10653" xml:space="preserve">ſiue ad eorundem differentiam, veluti <lb/>habetur in propoſitionibus 12. </s>
  <s xml:id="echoid-s10654" xml:space="preserve">13. </s>
  <s xml:id="echoid-s10655" xml:space="preserve">14. </s>
  <s xml:id="echoid-s10656" xml:space="preserve">eandem proportionem habebit, <lb/>quàm C G ad H G, ſiue ad duplum D G, vt in propofitione 14. </s>
  <s xml:id="echoid-s10657" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10658" xml:space="preserve">de-<lb/>monſtratum fuit in eadem proportione eſſe quadratum A C ad ſummam <lb/>quadratorum A C, &amp; </s>
  <s xml:id="echoid-s10659" xml:space="preserve">eius coniugati, &amp; </s>
  <s xml:id="echoid-s10660" xml:space="preserve">eſt propoſitio 25. </s>
  <s xml:id="echoid-s10661" xml:space="preserve">aut ad eorun-<lb/>dem differentiam, &amp; </s>
  <s xml:id="echoid-s10662" xml:space="preserve">eſt propoſitio 12. </s>
  <s xml:id="echoid-s10663" xml:space="preserve">quapropter ſumma quadratorum <lb/>I L, N O coniugatarum in ellipſi, nempe quadratum I L vna cum eius <lb/>figura eſt æquale aggregato quadrati A C vna cum quadrato eius coniu-<lb/>gati 30. </s>
  <s xml:id="echoid-s10664" xml:space="preserve">nempe quadrato A C, &amp; </s>
  <s xml:id="echoid-s10665" xml:space="preserve">illius figuræ, &amp; </s>
  <s xml:id="echoid-s10666" xml:space="preserve">in hyperbola diffe-<lb/>rentia quadratorum I L, N O nempe exceſſus quadrati I L ſuper illius <lb/>figuram æqualis eſt differentiæ duorum quadratorum A C, &amp; </s>
  <s xml:id="echoid-s10667" xml:space="preserve">recti illius <lb/>nempe quadrato A C, &amp; </s>
  <s xml:id="echoid-s10668" xml:space="preserve">illius figuræ 27. </s>
  <s xml:id="echoid-s10669" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10670" xml:space="preserve">oſtenſum iam eſt, quod I <lb/>
<anchor type="note" xlink:label="note-0330-05a" xlink:href="note-0330-05"/>
L in hyperbola maior eſt, quàm A C; </s>
  <s xml:id="echoid-s10671" xml:space="preserve">ergo differentia A C &amp; </s>
  <s xml:id="echoid-s10672" xml:space="preserve">illius con-<lb/>iugati maior quàm differentia I L, &amp; </s>
  <s xml:id="echoid-s10673" xml:space="preserve">N O: </s>
  <s xml:id="echoid-s10674" xml:space="preserve">atquè fic oſtendetur, quod
<pb o="293" file="0331" n="332" rhead="Conicor. Lib. VII."/>
differentia I L, &amp; </s>
  <s xml:id="echoid-s10675" xml:space="preserve">N O maior ſit, quàm differentia quarumlibet duarum <lb/>coniugatarum ab axi remotiorum. </s>
  <s xml:id="echoid-s10676" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s10677" xml:space="preserve"/>
</p>
<div xml:id="echoid-div902" type="float" level="2" n="1">
<note position="left" xlink:label="note-0330-01" xlink:href="note-0330-01a" xml:space="preserve">a</note>
<note position="left" xlink:label="note-0330-02" xlink:href="note-0330-02a" xml:space="preserve">ex Def. 1. <lb/>&amp; 2.</note>
<note position="right" xlink:label="note-0330-03" xlink:href="note-0330-03a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0330-01" xlink:href="fig-0330-01a">
    <image file="0330-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0330-01"/>
  </figure>
<note position="left" xlink:label="note-0330-04" xlink:href="note-0330-04a" xml:space="preserve">6. &amp; 7. <lb/>huius.</note>
<note position="left" xlink:label="note-0330-05" xlink:href="note-0330-05a" xml:space="preserve">c</note>
</div>
  <figure>
    <image file="0331-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0331-01"/>
  </figure>
</div>
<div xml:id="echoid-div904" type="section" level="1" n="281">
<head xml:id="echoid-head351" xml:space="preserve">Notæ in Propoſit. XII.</head>
<p style="it">
  <s xml:id="echoid-s10678" xml:space="preserve">IN eiſdem figuris, quia quadratum A C ad quadratum ſui coniugati in <lb/>
<anchor type="note" xlink:label="note-0331-01a" xlink:href="note-0331-01"/>
propoſitione 12. </s>
  <s xml:id="echoid-s10679" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10680" xml:space="preserve">25. </s>
  <s xml:id="echoid-s10681" xml:space="preserve">nempe A C ad A F erectum ipſius eſt vt præ-<lb/>ſecta C G ad Interceptam G A, ſeu C H: </s>
  <s xml:id="echoid-s10682" xml:space="preserve">ergo quadratum A C in hy-<lb/>perbola ad differentiam quadratorum axium ipſius, &amp; </s>
  <s xml:id="echoid-s10683" xml:space="preserve">in ellipſi ad illo-<lb/>rum ſnmmam eſt, vt C G ad H G, &amp;</s>
  <s xml:id="echoid-s10684" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10685" xml:space="preserve">Ideſt. </s>
  <s xml:id="echoid-s10686" xml:space="preserve">Quia quadratum A C ad <lb/>quadratum axis ei coniugati Q R, ſiue C A ad eius erectum A F eandem pro-<lb/>
<anchor type="note" xlink:label="note-0331-02a" xlink:href="note-0331-02"/>
portionem habet, quàm præſecta C G ad Interceptam G A, vel ad C H, &amp; </s>
  <s xml:id="echoid-s10687" xml:space="preserve"><lb/>comparando antecedentes ad terminorum differentias in hyperbola, &amp; </s>
  <s xml:id="echoid-s10688" xml:space="preserve">ad ter-<lb/>minorum ſummas in ellipſi, quadratum C A ad differentiam quadratorum ex axi <lb/>A C, &amp; </s>
  <s xml:id="echoid-s10689" xml:space="preserve">ex axi Q R habebit in hyperbola eandem proportionem, quàm C G <lb/>ad differentiam inter C G, &amp; </s>
  <s xml:id="echoid-s10690" xml:space="preserve">C H: </s>
  <s xml:id="echoid-s10691" xml:space="preserve">in ellipſi verò quadratum A C ad ſum-<lb/>mam quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s10692" xml:space="preserve">ex Q R eandem proportionem habebit, quàm <lb/>C G ad ſummam ipſius C G cum C H.</s>
  <s xml:id="echoid-s10693" xml:space="preserve"/>
</p>
<div xml:id="echoid-div904" type="float" level="2" n="1">
<note position="left" xlink:label="note-0331-01" xlink:href="note-0331-01a" xml:space="preserve">a</note>
<note position="right" xlink:label="note-0331-02" xlink:href="note-0331-02a" xml:space="preserve">Defin. 1. <lb/>&amp; 2. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s10694" xml:space="preserve">Et quia iam demonſtratum eſt, quod quadratum C A ad quadratum <lb/>
<anchor type="note" xlink:label="note-0331-03a" xlink:href="note-0331-03"/>
I L ſit, vt C G ad E H, &amp;</s>
  <s xml:id="echoid-s10695" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10696" xml:space="preserve">Relicta abſtruſa complicatione propoſitionum <lb/>Arabici Interpretis diſtinctiori methodo, ſicuti in præcedenti ſectione factum eſt <lb/>
<anchor type="note" xlink:label="note-0331-04a" xlink:href="note-0331-04"/>
propoſitiones declarabimus. </s>
  <s xml:id="echoid-s10697" xml:space="preserve">Quoniam in hyperbola quadratum I L ad quadra-<lb/>tum N O eandem proportionem habet, quàm H E ad E G comparando antece-<lb/>dentes ad terminorum differentias, quadratum I L ad differentiam quadrati <lb/>I L à quadrato N O eandem proportionem habebit, quàm H E ad ipſarum H <lb/>E, &amp; </s>
  <s xml:id="echoid-s10698" xml:space="preserve">E G differentiam; </s>
  <s xml:id="echoid-s10699" xml:space="preserve">ſed quadratum A C ad quadratum I L eſt vt C G <lb/>ad H E (veluti in propoſitione 8. </s>
  <s xml:id="echoid-s10700" xml:space="preserve">oſtenſum eſt) ergo ex æqualitate quadratum <lb/>A C ad quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10701" xml:space="preserve">ex N O differentiam eandem proportionem
<pb o="294" file="0332" n="333" rhead="Apollonij Pergæi"/>
habebit, quàm C G ad ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10702" xml:space="preserve">E G differentiam, ſeu ad H G: </s>
  <s xml:id="echoid-s10703" xml:space="preserve">ſed <lb/>in eadem hyperbola quadratum A C ad quadratorum A C, &amp; </s>
  <s xml:id="echoid-s10704" xml:space="preserve">Q R differen-<lb/>tiam eandem proportionem habet, quàm C G ad ipſarum C G, &amp; </s>
  <s xml:id="echoid-s10705" xml:space="preserve">C H diffe-<lb/>rentiam, ſeu ad H G (veluti in principio huius propoſitionis dictum eſt) ergo <lb/>quadratum A C ad quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s10706" xml:space="preserve">ex Q R differentiam, eandem <lb/>proportionem habebit, quàm ad quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10707" xml:space="preserve">ex N O differentiam; <lb/></s>
  <s xml:id="echoid-s10708" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10709" xml:space="preserve">ideo in hyperbola differentiæ quadratorum axium A C, &amp; </s>
  <s xml:id="echoid-s10710" xml:space="preserve">Q R æqualis <lb/>eſt diffcrentiæ quadratorum I L, &amp; </s>
  <s xml:id="echoid-s10711" xml:space="preserve">N O coniugatarum.</s>
  <s xml:id="echoid-s10712" xml:space="preserve"/>
</p>
<div xml:id="echoid-div905" type="float" level="2" n="2">
<note position="left" xlink:label="note-0331-03" xlink:href="note-0331-03a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0331-04" xlink:href="note-0331-04a" xml:space="preserve">6. huius.</note>
</div>
</div>
<div xml:id="echoid-div907" type="section" level="1" n="282">
<head xml:id="echoid-head352" xml:space="preserve">Notæ in Propoſit. XIII.</head>
<p style="it">
  <s xml:id="echoid-s10713" xml:space="preserve">QVoniam in ellipſi quadratum I L ad quadratum N O eandem proportio-<lb/>
<anchor type="note" xlink:label="note-0332-01a" xlink:href="note-0332-01"/>
nem habet, quàm H E ad G E; </s>
  <s xml:id="echoid-s10714" xml:space="preserve">comparando antecedentes ad terminorũ <lb/>ſummas quadratum I L ad quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10715" xml:space="preserve">ex N O ſum-<lb/>mam eandem proportionem habebit, quàm H E ad ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10716" xml:space="preserve">E G ſum-<lb/>mam: </s>
  <s xml:id="echoid-s10717" xml:space="preserve">ſed quadratum A C ad quadratum I L eſt, vt C G ad H E (vt in octa-<lb/>ua propoſitione dictum eſt) ergo ex æquali quadratum A C ad quadratorum ex <lb/>
<anchor type="figure" xlink:label="fig-0332-01a" xlink:href="fig-0332-01"/>
I L, &amp; </s>
  <s xml:id="echoid-s10718" xml:space="preserve">ex N O ſummam eandem proportionem habebit, quàm C G ad ſum-<lb/>mam ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10719" xml:space="preserve">E G, ſeu ad G H: </s>
  <s xml:id="echoid-s10720" xml:space="preserve">ſed in principio præcedentis notæ <lb/>oſtenſum eſt, quod in ellipſi quadratum A C ad quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s10721" xml:space="preserve">ex Q <lb/>R ſummam eandem proportionem habet, quàm C G ad ſummam ipſarum C G, <lb/>&amp; </s>
  <s xml:id="echoid-s10722" xml:space="preserve">C H, ſeu ad G H: </s>
  <s xml:id="echoid-s10723" xml:space="preserve">quarè quadratum A C eãdem proportionem habet ad ſum-<lb/>mam quadratorum ex C A, &amp; </s>
  <s xml:id="echoid-s10724" xml:space="preserve">ex Q R, quàm ad ſummam quadratorum ex I <lb/>L, &amp; </s>
  <s xml:id="echoid-s10725" xml:space="preserve">ex N O; </s>
  <s xml:id="echoid-s10726" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10727" xml:space="preserve">propterea in ellipſi quadrata duorum axium A C, &amp; </s>
  <s xml:id="echoid-s10728" xml:space="preserve">Q R <lb/>ſimul ſumpta æqualia ſunt quadratis duarum coniugatarum diametrorum I L, <lb/>&amp; </s>
  <s xml:id="echoid-s10729" xml:space="preserve">N O ſimul ſumptis.</s>
  <s xml:id="echoid-s10730" xml:space="preserve"/>
</p>
<div xml:id="echoid-div907" type="float" level="2" n="1">
<note position="left" xlink:label="note-0332-01" xlink:href="note-0332-01a" xml:space="preserve">7. huius.</note>
  <figure xlink:label="fig-0332-01" xlink:href="fig-0332-01a">
    <image file="0332-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0332-01"/>
  </figure>
</div>
<pb o="295" file="0333" n="334" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div909" type="section" level="1" n="283">
<head xml:id="echoid-head353" xml:space="preserve">Notæ in Propoſit. XXIX.</head>
<p style="it">
  <s xml:id="echoid-s10731" xml:space="preserve">QVoniam in hyperbola differentia quadratorum ex axi A C, &amp; </s>
  <s xml:id="echoid-s10732" xml:space="preserve">ex axi Q <lb/>
<anchor type="note" xlink:label="note-0333-01a" xlink:href="note-0333-01"/>
R æqualis eſt differentiæ inter quadratum I L à quadrato eius coniugatæ <lb/>N O; </s>
  <s xml:id="echoid-s10733" xml:space="preserve">eſtque Q R media proportionalis inter ſiguræ latera A C, &amp; </s>
  <s xml:id="echoid-s10734" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0333-02a" xlink:href="note-0333-02"/>
A F; </s>
  <s xml:id="echoid-s10735" xml:space="preserve">ergo rectangulum C A F ſub extremis contentum æquale eſt quadrato in-<lb/>termediæ Q R: </s>
  <s xml:id="echoid-s10736" xml:space="preserve">Et propterea differentia inter quadratum A C, &amp; </s>
  <s xml:id="echoid-s10737" xml:space="preserve">rectangu-<lb/>lum C A F æqualis erit differentiæ inter quadratum A C à quadrato Q R. <lb/></s>
  <s xml:id="echoid-s10738" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0333-01a" xlink:href="fig-0333-01"/>
pari ratione erit differentia quadrati I L à rectangulo L I P æqualis differen-<lb/>tiæ quadrati I L à quadrato N O; </s>
  <s xml:id="echoid-s10739" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10740" xml:space="preserve">propterea in hyperbole differentia qua-<lb/>drati axis A C à rectangulo ſub figuræ lateribus contentum C A F æqualis <lb/>eſt differentiæ quadrati diametri I L à rectangulo L I P ſub lateribus figuræ <lb/>eius.</s>
  <s xml:id="echoid-s10741" xml:space="preserve"/>
</p>
<div xml:id="echoid-div909" type="float" level="2" n="1">
<note position="right" xlink:label="note-0333-01" xlink:href="note-0333-01a" xml:space="preserve">12. huius.</note>
<note position="right" xlink:label="note-0333-02" xlink:href="note-0333-02a" xml:space="preserve">16. lib. 1.</note>
  <figure xlink:label="fig-0333-01" xlink:href="fig-0333-01a">
    <image file="0333-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0333-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div911" type="section" level="1" n="284">
<head xml:id="echoid-head354" xml:space="preserve">Notæ in Propoſit. XXX.</head>
<p style="it">
  <s xml:id="echoid-s10742" xml:space="preserve">QVoniam in ellipſi quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s10743" xml:space="preserve">ex Q R ſumma æqualis eſt <lb/>
<anchor type="note" xlink:label="note-0333-03a" xlink:href="note-0333-03"/>
ſummæ quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10744" xml:space="preserve">ex N O: </s>
  <s xml:id="echoid-s10745" xml:space="preserve">eſtque rectangulum C A F <lb/>æquale quadrato Q R, &amp; </s>
  <s xml:id="echoid-s10746" xml:space="preserve">rectangulum L I P æquale quadrato N O <lb/>(vt in præcedenti nota dictum eſt) igitur in ellipſi quadratum axis A C, &amp; </s>
  <s xml:id="echoid-s10747" xml:space="preserve"><lb/>rectangulum C A F ſub eius lateribus cõtentum ſimul ſumpta æqualia ſunt qua-<lb/>drato ex I L cum rectangulo figuræ eius L I P.</s>
  <s xml:id="echoid-s10748" xml:space="preserve"/>
</p>
<div xml:id="echoid-div911" type="float" level="2" n="1">
<note position="right" xlink:label="note-0333-03" xlink:href="note-0333-03a" xml:space="preserve">Prop. 13. <lb/>huius. <lb/>ex 15. <lb/>lib. 1.</note>
</div>
<pb o="296" file="0334" n="335" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div913" type="section" level="1" n="285">
<head xml:id="echoid-head355" xml:space="preserve">Notæ in Propoſit. XIV. &amp; XXV.</head>
<p style="it">
  <s xml:id="echoid-s10749" xml:space="preserve">QVoniam nedum in hyperbola, ſed etiam in ellipſi quadratum A C ad ſum-<lb/>mam quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10750" xml:space="preserve">ex N O eandem proportionem habet, quã <lb/>A H ad ſummam ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10751" xml:space="preserve">E G, atque quadratorum ex I <lb/>L, &amp; </s>
  <s xml:id="echoid-s10752" xml:space="preserve">ex N O ſumma ad eorundem quadratorum differentiam eandem propor-<lb/>tionem habet, quàm ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10753" xml:space="preserve">E G ſumma ad earundem differentiam; <lb/></s>
  <s xml:id="echoid-s10754" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0334-01a" xlink:href="fig-0334-01"/>
evgo ex æquali quadratum A C ad quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10755" xml:space="preserve">ex N O differen-<lb/>tiam eandem proportionem habet, quàm C G, ſiue H A ad ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10756" xml:space="preserve"><lb/>E G differentiam; </s>
  <s xml:id="echoid-s10757" xml:space="preserve">ſed in ellipſi ipſarum H E, &amp; </s>
  <s xml:id="echoid-s10758" xml:space="preserve">E G differentia æqualis eſt <lb/>duplo E D; </s>
  <s xml:id="echoid-s10759" xml:space="preserve">igitur in ellipſi quadratum A C ad quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10760" xml:space="preserve">ex <lb/>N O differentiam eandem proportionem habebit, quàm præſecta C G ad duplum <lb/>inuerſæ E D.</s>
  <s xml:id="echoid-s10761" xml:space="preserve"/>
</p>
<div xml:id="echoid-div913" type="float" level="2" n="1">
  <figure xlink:label="fig-0334-01" xlink:href="fig-0334-01a">
    <image file="0334-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0334-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div915" type="section" level="1" n="286">
<head xml:id="echoid-head356" xml:space="preserve">Notæ in Propoſit. XXVII.</head>
<p style="it">
  <s xml:id="echoid-s10762" xml:space="preserve">ET oſtenſum iam eſt, quod I L in hyperbola maior eſt, quàm A C; <lb/></s>
  <s xml:id="echoid-s10763" xml:space="preserve">
<anchor type="note" xlink:label="note-0334-01a" xlink:href="note-0334-01"/>
ergo differentia A C, &amp; </s>
  <s xml:id="echoid-s10764" xml:space="preserve">illius coniugati maior eſt, quàm differen-<lb/>tia homologorum ſuorum à ſuis coniugatis, &amp; </s>
  <s xml:id="echoid-s10765" xml:space="preserve">differentia proximioris ho-<lb/>mologi ad ſuam coniugatam maior eſt differentia remotioris à ſua coniu-<lb/>gata, &amp;</s>
  <s xml:id="echoid-s10766" xml:space="preserve">c. </s>
  <s xml:id="echoid-s10767" xml:space="preserve">Hoc autem ſic demonſtrabitur. </s>
  <s xml:id="echoid-s10768" xml:space="preserve">In diametris A C, &amp; </s>
  <s xml:id="echoid-s10769" xml:space="preserve">I L produca-<lb/>tur A M æqualis Q R, &amp; </s>
  <s xml:id="echoid-s10770" xml:space="preserve">I K æqualis N O, &amp; </s>
  <s xml:id="echoid-s10771" xml:space="preserve">ab ĳsdem ſecentur A S æqua-<lb/>lis Q R, &amp; </s>
  <s xml:id="echoid-s10772" xml:space="preserve">I T æqualis N O. </s>
  <s xml:id="echoid-s10773" xml:space="preserve">Quoniam M S bifariam ſecatur in A, &amp; </s>
  <s xml:id="echoid-s10774" xml:space="preserve">e<unsure/>i
<pb o="297" file="0335" n="336" rhead="Conicor. Lib. VII."/>
indirectum additur S C, <lb/>
<anchor type="figure" xlink:label="fig-0335-01a" xlink:href="fig-0335-01"/>
erit rectangulum M C S <lb/>cum quadrato ex A S, ſeu <lb/>ex Q R æquale quadrato <lb/>ipſius A C; </s>
  <s xml:id="echoid-s10775" xml:space="preserve">ergo rectangu-<lb/>lum M C S æquale eſt dif-<lb/>ferentiæ quadrati A C à <lb/>quadrato Q R: </s>
  <s xml:id="echoid-s10776" xml:space="preserve">pariratione <lb/>rectangulum K L T vna <lb/>cum quadrato N O æquale <lb/>erit quadrato I L: </s>
  <s xml:id="echoid-s10777" xml:space="preserve">ergo ſi-<lb/>militer rectangulum K L T æquale eſt differentiæ quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10778" xml:space="preserve">ex <lb/>N O; </s>
  <s xml:id="echoid-s10779" xml:space="preserve">eſtquè quadratum I L maius quadrato A C, cum diameter I L in hyper-<lb/>bola maior ſit, quàm axis C A; </s>
  <s xml:id="echoid-s10780" xml:space="preserve">igitur rectangulum K L T vna cum quadrato <lb/>N O maius erit rectangulo M C S vna cum quadrato Q R: </s>
  <s xml:id="echoid-s10781" xml:space="preserve">eſt verò rectangu-<lb/>lum M C S æquale rectangulo K L T (cum ſint differentiæ quadratorum ex con-<lb/>
<anchor type="note" xlink:label="note-0335-01a" xlink:href="note-0335-01"/>
iugatis diametris, quæ in hyperbola oſtenſæ ſunt æquales); </s>
  <s xml:id="echoid-s10782" xml:space="preserve">ergo quadratum N <lb/>
<anchor type="figure" xlink:label="fig-0335-02a" xlink:href="fig-0335-02"/>
O, ſcilicet reſiduum maioris ſummæ, maius erit quadrato Q R, quod eſt reſi-<lb/>duum ſummæ minoris: </s>
  <s xml:id="echoid-s10783" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10784" xml:space="preserve">propterea N O maior erit, quàm Q R: </s>
  <s xml:id="echoid-s10785" xml:space="preserve">erat autem <lb/>I L maior quàm C A; </s>
  <s xml:id="echoid-s10786" xml:space="preserve">igitur I L cum N O, ſeu K L maior erit, quàm A C, <lb/>&amp; </s>
  <s xml:id="echoid-s10787" xml:space="preserve">Q R ſimul, ſiue quàm M C: </s>
  <s xml:id="echoid-s10788" xml:space="preserve">ſed in rectangulis M C S, &amp; </s>
  <s xml:id="echoid-s10789" xml:space="preserve">K L T æquali-<lb/>bus, vt K L ad M C, ita reciprocè C S ad L T; </s>
  <s xml:id="echoid-s10790" xml:space="preserve">igitur C S, ſeu differentia <lb/>ipſarum A C, &amp; </s>
  <s xml:id="echoid-s10791" xml:space="preserve">Q R maior eſt, quàm L T, ſeu differentia ipſarum I L, &amp; </s>
  <s xml:id="echoid-s10792" xml:space="preserve"><lb/>N O in hyperbola.</s>
  <s xml:id="echoid-s10793" xml:space="preserve"/>
</p>
<div xml:id="echoid-div915" type="float" level="2" n="1">
<note position="right" xlink:label="note-0334-01" xlink:href="note-0334-01a" xml:space="preserve">C</note>
  <figure xlink:label="fig-0335-01" xlink:href="fig-0335-01a">
    <image file="0335-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0335-01"/>
  </figure>
<note position="right" xlink:label="note-0335-01" xlink:href="note-0335-01a" xml:space="preserve">Prop. 12. <lb/>huius.</note>
  <figure xlink:label="fig-0335-02" xlink:href="fig-0335-02a">
    <image file="0335-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0335-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s10794" xml:space="preserve">Si poſtea præter I L ponatur alia diameter ab axe remotior cum ſua coniu-<lb/>gata erit ſimiliter differentia quadratorum ex diametris coniugatis remotiori-<lb/>bus ab axi æqualis differentiæ quadratorum axium A C, &amp; </s>
  <s xml:id="echoid-s10795" xml:space="preserve">Q R, &amp; </s>
  <s xml:id="echoid-s10796" xml:space="preserve">ideo
<pb o="298" file="0336" n="337" rhead="Apollonij Pergæi"/>
æqualis erit differentiæ quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s10797" xml:space="preserve">ex N O; </s>
  <s xml:id="echoid-s10798" xml:space="preserve">eſtque pariter diame-<lb/>ter illa remotior ab axe maior quàm I L; </s>
  <s xml:id="echoid-s10799" xml:space="preserve">ergo ſimili ratiocinio oſtendetur, quod <lb/>differentia coniugatarum diametrorum ab axe remotiorum minor eſt, quàm dif-<lb/>ferentia propinquiorum I L, &amp; </s>
  <s xml:id="echoid-s10800" xml:space="preserve">N O.</s>
  <s xml:id="echoid-s10801" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div917" type="section" level="1" n="287">
<head xml:id="echoid-head357" xml:space="preserve">SECTIO QVINTA</head>
<head xml:id="echoid-head358" xml:space="preserve">Continens Propoſit. XXI. XXVIII. XXXXII. <lb/>XXXXIII. XXIV. &amp; XXXVII.</head>
<p>
  <s xml:id="echoid-s10802" xml:space="preserve">AXes hyperboles ſi fuerint æquales, tunc quælibet diame-<lb/>tri coniugatæ in illa ſectione æquales ſunt 21. </s>
  <s xml:id="echoid-s10803" xml:space="preserve">ſi verò fue-<lb/>rit 28. </s>
  <s xml:id="echoid-s10804" xml:space="preserve">vnus duorum axium in hyperbola, aut ellipſi maior, <lb/>
<anchor type="note" xlink:label="note-0336-01a" xlink:href="note-0336-01"/>
tunc eius diameter homologa maior erit ſua coniugata, quouſ-<lb/>què ad duas æquales diametros coniugatas in ellipſi peruenia-<lb/>tur, &amp; </s>
  <s xml:id="echoid-s10805" xml:space="preserve">axis maior ad ſuum coniugatum, ſiuè ad erectum eius <lb/>maiorem proportionem habet, quàm quælibet alia diameter <lb/>eiuſdem ſectionis ad ſibi coniugatam, ſiue ad eius erectum; <lb/></s>
  <s xml:id="echoid-s10806" xml:space="preserve">eritque proportio maioris diametri axi proximioris ad ſibi con-<lb/>iugatam, ſiue ad eius erectum maior proportione maioris con-<lb/>iugatarum ab illo remotioris ad minorem, ſiue ad eius erectũ. </s>
  <s xml:id="echoid-s10807" xml:space="preserve"><lb/>Et minima figurarum diametrorum erit figura axis inclinati, ſiue <lb/>tranſuerſi, &amp; </s>
  <s xml:id="echoid-s10808" xml:space="preserve">maxima erit figura recti in ellipſi: </s>
  <s xml:id="echoid-s10809" xml:space="preserve">atque figuræ <lb/>reliquarum diametrorum (ſiue diametri ſint inclinatæ, vel tran-<lb/>ſuerſæ) maiores ſunt, quã figuræ diametrorũ ab axi remotiorũ 24. </s>
  <s xml:id="echoid-s10810" xml:space="preserve"><lb/>Et in ellipſi erectus axis tranſuerſi minor eſt, quã erectus cuiuslibet <lb/>alterius diametri, &amp; </s>
  <s xml:id="echoid-s10811" xml:space="preserve">erectus proximioris diametri minor eſt erecto <lb/>cuiuslibet remotioris 37. </s>
  <s xml:id="echoid-s10812" xml:space="preserve">Et <lb/>
<anchor type="figure" xlink:label="fig-0336-01a" xlink:href="fig-0336-01"/>
exceſſus axis tranſuerſi ſuper e-<lb/>ius coniugatum maior eſt, quã <lb/>exceſſus homologarum diame-<lb/>trorum, ſuper ſuas coniugatas, <lb/>&amp; </s>
  <s xml:id="echoid-s10813" xml:space="preserve">exceſſus proximioris homo-<lb/>logæ ſuper ſuam coniugatam <lb/>maior eſt, quàm exceſſus re-<lb/>motioris ſuper eius coniugatã. <lb/></s>
  <s xml:id="echoid-s10814" xml:space="preserve">Et differentia duorum laterum <lb/>figuræ axis maior eſt, quàm
<pb o="299" file="0337" n="338" rhead="Conicor. Lib. VII."/>
differentia duorum laterum figuræ ſui homologi; </s>
  <s xml:id="echoid-s10815" xml:space="preserve">pariterque pro-<lb/>ximioris axi homologi differentia duorum laterum figuræ eius <lb/>maior eſt, quàm differentia duorum laterum figuræ remotioris.</s>
  <s xml:id="echoid-s10816" xml:space="preserve"/>
</p>
<div xml:id="echoid-div917" type="float" level="2" n="1">
<note position="right" xlink:label="note-0336-01" xlink:href="note-0336-01a" xml:space="preserve">a</note>
  <figure xlink:label="fig-0336-01" xlink:href="fig-0336-01a">
    <image file="0336-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0336-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div919" type="section" level="1" n="288">
<head xml:id="echoid-head359" xml:space="preserve">PROPOSITIO XXI. &amp; XXVIII.</head>
<p>
  <s xml:id="echoid-s10817" xml:space="preserve">SIt itaque ſectio A B P, &amp; </s>
  <s xml:id="echoid-s10818" xml:space="preserve">duo axes coniugati eius A C, Q <lb/>R, centrum D; </s>
  <s xml:id="echoid-s10819" xml:space="preserve">ſintque I L, N O duæ aliæ diametri con-<lb/>iugatæ; </s>
  <s xml:id="echoid-s10820" xml:space="preserve">pariterque S T, V X, &amp; </s>
  <s xml:id="echoid-s10821" xml:space="preserve">educamus ad axim C A M <lb/>perpendiculares B E, P M. </s>
  <s xml:id="echoid-s10822" xml:space="preserve">Dico quod ſi fuerit A C æqualis <lb/>Q R; </s>
  <s xml:id="echoid-s10823" xml:space="preserve">erit quoque I L æqualis ipſi N O, &amp; </s>
  <s xml:id="echoid-s10824" xml:space="preserve">S T ipſi V X. </s>
  <s xml:id="echoid-s10825" xml:space="preserve">Si <lb/>verò fuerit eorum aliquis reliquo major, vtique eius homologa <lb/>diameter maior quoque erit ſua coniugata, &amp; </s>
  <s xml:id="echoid-s10826" xml:space="preserve">ſimiliter in reli-<lb/>quis propoſitionibus.</s>
  <s xml:id="echoid-s10827" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10828" xml:space="preserve">Sit prius alter axis A C maior in prima figura, ſed Q R in ſecunda; <lb/></s>
  <s xml:id="echoid-s10829" xml:space="preserve">ſintque A G, C H duæ interceptæ diametri A C. </s>
  <s xml:id="echoid-s10830" xml:space="preserve">Et quia quadratum <lb/>A C ad quadratum Q R, nempe A C ad eius erectum eſt vt A H ad H <lb/>C, ſeu ad A G; </s>
  <s xml:id="echoid-s10831" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10832" xml:space="preserve">habet H A ad A G maiorem proportionem in prima <lb/>
<anchor type="note" xlink:label="note-0337-01a" xlink:href="note-0337-01"/>
figura, &amp; </s>
  <s xml:id="echoid-s10833" xml:space="preserve">minorem in ſecunda, quàm H E ad E G, quæ oſtenſa eſt <lb/>( 6. </s>
  <s xml:id="echoid-s10834" xml:space="preserve">7. </s>
  <s xml:id="echoid-s10835" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s10836" xml:space="preserve">) vt quadratum I L ad <lb/>
<anchor type="figure" xlink:label="fig-0337-01a" xlink:href="fig-0337-01"/>
quadratum N O, nempe I L ad eius <lb/>erectum. </s>
  <s xml:id="echoid-s10837" xml:space="preserve">Et ſimiliter proportio illa <lb/>maior, aut minor eſt, quam H M ad <lb/>M G, quæ eſt vt quadratum S T ad <lb/>quadratum V X; </s>
  <s xml:id="echoid-s10838" xml:space="preserve">igitur A C ad Q R, <lb/>ſiue ad erectum ipſius A C in prima <lb/>maiorem proportionem habet, &amp; </s>
  <s xml:id="echoid-s10839" xml:space="preserve">in <lb/>ſecunda minorem, quàm I L ad N O, <lb/>ſiue ad erectum ipſius I L, ſiue quàm <lb/>S T ad V X, vel ad erectum ipſius <lb/>S T; </s>
  <s xml:id="echoid-s10840" xml:space="preserve">ſed quia H E ad E G in prima <lb/>figura maiorem proportionem, &amp; </s>
  <s xml:id="echoid-s10841" xml:space="preserve">in <lb/>ſecunda minorem, quàm H M ad M <lb/>G habebit I L ad N O maiorem pro-<lb/>portionem in prima, &amp; </s>
  <s xml:id="echoid-s10842" xml:space="preserve">minorem in <lb/>ſecunda, quàm S T ad V X, cum-<lb/>que H E in prima figura ſit maior, &amp; </s>
  <s xml:id="echoid-s10843" xml:space="preserve"><lb/>in ſecunda minor, quàm E G, pari-<lb/>terque H M, quàm M G, erit I L in <lb/>prima maior, &amp; </s>
  <s xml:id="echoid-s10844" xml:space="preserve">in ſecunda minor, <lb/>quàm N O, ſimiliterque S T, quàm <lb/>V X.</s>
  <s xml:id="echoid-s10845" xml:space="preserve"/>
</p>
<div xml:id="echoid-div919" type="float" level="2" n="1">
<note position="right" xlink:label="note-0337-01" xlink:href="note-0337-01a" xml:space="preserve">ex Def. I. <lb/>huius.</note>
  <figure xlink:label="fig-0337-01" xlink:href="fig-0337-01a">
    <image file="0337-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0337-01"/>
  </figure>
</div>
<pb o="300" file="0338" n="339" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s10846" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s10847" xml:space="preserve">Deinde ſit A C æqualis QR in hyperbola fiet A C æqualis ere-<lb/>cto, &amp; </s>
  <s xml:id="echoid-s10848" xml:space="preserve">conuenient duo puncta H, &amp; </s>
  <s xml:id="echoid-s10849" xml:space="preserve">G in puncto D, eritque A C ad <lb/>
<anchor type="note" xlink:label="note-0338-01a" xlink:href="note-0338-01"/>
<anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/>
Q R vt A D ad ſe ipſam, ſiue vt A C ad ſe ipſam, quæ eſt vt D E ad <lb/>ſe ipſam, &amp; </s>
  <s xml:id="echoid-s10850" xml:space="preserve">hæc oſtenſa eſt, vt quadratum I L ad quadratum N O; </s>
  <s xml:id="echoid-s10851" xml:space="preserve">igi-<lb/>
<anchor type="note" xlink:label="note-0338-02a" xlink:href="note-0338-02"/>
tur I L, &amp; </s>
  <s xml:id="echoid-s10852" xml:space="preserve">N O ſunt æquales, &amp; </s>
  <s xml:id="echoid-s10853" xml:space="preserve">ſic demonſtrabitur, quod S T, V X ſunt <lb/>æquales, &amp; </s>
  <s xml:id="echoid-s10854" xml:space="preserve">hoc erat propoſitum.</s>
  <s xml:id="echoid-s10855" xml:space="preserve"/>
</p>
<div xml:id="echoid-div920" type="float" level="2" n="2">
<note position="right" xlink:label="note-0338-01" xlink:href="note-0338-01a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0338-01" xlink:href="fig-0338-01a">
    <image file="0338-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0338-01"/>
  </figure>
<note position="left" xlink:label="note-0338-02" xlink:href="note-0338-02a" xml:space="preserve">Prop. 6. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div922" type="section" level="1" n="289">
<head xml:id="echoid-head360" xml:space="preserve">PROPOSITIO XXVI</head>
<p>
  <s xml:id="echoid-s10856" xml:space="preserve"><emph style="sc">At</emph> in ellipſi fieri po-<lb/>
<anchor type="figure" xlink:label="fig-0338-02a" xlink:href="fig-0338-02"/>
teſt, vt H E ſit æ-<lb/>qualis E G, ſi nimirum <lb/>punctum B cadat in Q, &amp; </s>
  <s xml:id="echoid-s10857" xml:space="preserve"><lb/>tunc B E cadetſuper Q D, <lb/>&amp; </s>
  <s xml:id="echoid-s10858" xml:space="preserve">erit diameter I L æqua-<lb/>lis ſuæ coniugatæ; </s>
  <s xml:id="echoid-s10859" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10860" xml:space="preserve">vo-<lb/>cabo eas æquales.</s>
  <s xml:id="echoid-s10861" xml:space="preserve"/>
</p>
<div xml:id="echoid-div922" type="float" level="2" n="1">
  <figure xlink:label="fig-0338-02" xlink:href="fig-0338-02a">
    <image file="0338-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0338-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s10862" xml:space="preserve">Quia C G ad C H, nempe <lb/>quadratum A C ad ſuam fi-<lb/>guram maiorem proportionem <lb/>habet in primis figuris, &amp; </s>
  <s xml:id="echoid-s10863" xml:space="preserve">mi-<lb/>norem in ſecunda ellipſi, quàm <lb/>C G ad G E, nempe quàm <lb/>quadratum A C ad figuram <lb/>ipſius I L ( 18. </s>
  <s xml:id="echoid-s10864" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s10865" xml:space="preserve">) &amp; </s>
  <s xml:id="echoid-s10866" xml:space="preserve">C <lb/>G ad G E in primis figurisma-<lb/>iorem proportionem habet, &amp;</s>
  <s xml:id="echoid-s10867" xml:space="preserve">
<pb o="301" file="0339" n="340" rhead="Conicor. Lib. VII."/>
in ſecunda ellipſi minorem, quàm C G ad G M, nempe quàm quadra-<lb/>tum A C ad figuram ipſius S T ( 18. </s>
  <s xml:id="echoid-s10868" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s10869" xml:space="preserve">) ergo figura ipſius A C eſt <lb/>minor; </s>
  <s xml:id="echoid-s10870" xml:space="preserve">in ſecunda verò maior quàm figura ipſius I L; </s>
  <s xml:id="echoid-s10871" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10872" xml:space="preserve">ſimiliter figura <lb/>ipſius I L maior, aut minor eſt figura S T. </s>
  <s xml:id="echoid-s10873" xml:space="preserve">Et hoc eſt propoſitum.</s>
  <s xml:id="echoid-s10874" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div924" type="section" level="1" n="290">
<head xml:id="echoid-head361" xml:space="preserve">PROPOSITIO XXXXII.</head>
<p>
  <s xml:id="echoid-s10875" xml:space="preserve"><emph style="sc">In</emph> hyperbole, &amp; </s>
  <s xml:id="echoid-s10876" xml:space="preserve">ellipſi sũ-<lb/>
<anchor type="figure" xlink:label="fig-0339-01a" xlink:href="fig-0339-01"/>
ma duorum axium minor eſt <lb/>ſumma quarumlibet duarum cõ-<lb/>iugatarum diametrorum eiuſdẽ <lb/>ſectionis.</s>
  <s xml:id="echoid-s10877" xml:space="preserve"/>
</p>
<div xml:id="echoid-div924" type="float" level="2" n="1">
  <figure xlink:label="fig-0339-01" xlink:href="fig-0339-01a">
    <image file="0339-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0339-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s10878" xml:space="preserve">XXXXIII. </s>
  <s xml:id="echoid-s10879" xml:space="preserve">Et planum ab eis <lb/>contentũ minus eſt plano à dua-<lb/>bus coniugatis contento, &amp; </s>
  <s xml:id="echoid-s10880" xml:space="preserve"><lb/>planum à proximioribus axi <lb/>coniugatis contentum minus <lb/>eſt plano à remotioribus con-<lb/>tento.</s>
  <s xml:id="echoid-s10881" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10882" xml:space="preserve">Iiſdem figuris manentibus, quia in hyperbole A C minor eſt quàm I <lb/>L, &amp; </s>
  <s xml:id="echoid-s10883" xml:space="preserve">I L, quàm S T; </s>
  <s xml:id="echoid-s10884" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10885" xml:space="preserve">ſiquidem <lb/>
<anchor type="figure" xlink:label="fig-0339-02a" xlink:href="fig-0339-02"/>
A C æqualis fuerit Q R, erit quo-<lb/>que I L æqualis N O, &amp; </s>
  <s xml:id="echoid-s10886" xml:space="preserve">S T æqua-<lb/>lis V X ( 21. </s>
  <s xml:id="echoid-s10887" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s10888" xml:space="preserve">) ergo ſumma <lb/>ipſorum A C, Q R minor eſt, quã <lb/>ſumma I L, N O, &amp; </s>
  <s xml:id="echoid-s10889" xml:space="preserve">quàm S T, <lb/>V X: </s>
  <s xml:id="echoid-s10890" xml:space="preserve">ſi verò A C non fuerit æqua-<lb/>lis ipſi Q R, vtique differentia duo-<lb/>
<anchor type="note" xlink:label="note-0339-01a" xlink:href="note-0339-01"/>
rum quadratorum A C, Q R æqua-<lb/>lis erit differentiæ quadratorum I L, <lb/>N O: </s>
  <s xml:id="echoid-s10891" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10892" xml:space="preserve">propterea ſumma ipſorum <lb/>
<anchor type="note" xlink:label="note-0339-02a" xlink:href="note-0339-02"/>
A C, Q R minor erit, quàm ſum-<lb/>ma I L, N O: </s>
  <s xml:id="echoid-s10893" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10894" xml:space="preserve">hæc ſumma ex <lb/>hac eadem demonſtratione minor <lb/>etiam erit, quàm ſumma duarum <lb/>S T, V X. </s>
  <s xml:id="echoid-s10895" xml:space="preserve">At in ellipſi; </s>
  <s xml:id="echoid-s10896" xml:space="preserve">quia A <lb/>C ad Q R maiorem proportionem <lb/>
<anchor type="note" xlink:label="note-0339-03a" xlink:href="note-0339-03"/>
habet, quàm I L ad N O ( 28. </s>
  <s xml:id="echoid-s10897" xml:space="preserve">ex <lb/>7. </s>
  <s xml:id="echoid-s10898" xml:space="preserve">) habebit quadratum ex ſumma <lb/>A C, Q R ad earundem duarum <lb/>ſummam quadratorum maiorem <lb/>proportionem, quàm quadratum <lb/>ſummæ I L, N O ad quadratorum
<pb o="302" file="0340" n="341" rhead="Apollonij Pergæi"/>
ſummam earundem: </s>
  <s xml:id="echoid-s10899" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10900" xml:space="preserve">ſumma duorum quadratorum ipſarum æqualis eſt <lb/>ſummæ duorum quadratorum A C, Q R ( 22. </s>
  <s xml:id="echoid-s10901" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s10902" xml:space="preserve">) ergo ſumma A C, <lb/>Q R minor eſt, quàm ſumma I L, N O, atque ſic oſtendetur, quod sũ-<lb/>ma I L, N O minor eſt, quàm ſumma S T, V X. </s>
  <s xml:id="echoid-s10903" xml:space="preserve">Quod erat propoſitũ.</s>
  <s xml:id="echoid-s10904" xml:space="preserve"/>
</p>
<div xml:id="echoid-div925" type="float" level="2" n="2">
  <figure xlink:label="fig-0339-02" xlink:href="fig-0339-02a">
    <image file="0339-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0339-02"/>
  </figure>
<note position="right" xlink:label="note-0339-01" xlink:href="note-0339-01a" xml:space="preserve">12. 13. <lb/>huius.</note>
<note position="left" xlink:label="note-0339-02" xlink:href="note-0339-02a" xml:space="preserve">d</note>
<note position="left" xlink:label="note-0339-03" xlink:href="note-0339-03a" xml:space="preserve">e</note>
</div>
</div>
<div xml:id="echoid-div927" type="section" level="1" n="291">
<head xml:id="echoid-head362" xml:space="preserve">PROPOSITIO XXXXIII.</head>
<p>
  <s xml:id="echoid-s10905" xml:space="preserve">D Einde in ellipſi quadratum ſummæ A C, Q R minus eſt quadrato <lb/>ſummæ I L, N O; </s>
  <s xml:id="echoid-s10906" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10907" xml:space="preserve">ſumma duorum quadratorum A C, Q R <lb/>
<anchor type="figure" xlink:label="fig-0340-01a" xlink:href="fig-0340-01"/>
æqualis eſt ſummæ duorum quadratorum I L, N O (22. </s>
  <s xml:id="echoid-s10908" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s10909" xml:space="preserve">) igitur <lb/>remanet A C in Q R minus quàm I L in N O, &amp; </s>
  <s xml:id="echoid-s10910" xml:space="preserve">ſimiliter I L in N O <lb/>
<anchor type="note" xlink:label="note-0340-01a" xlink:href="note-0340-01"/>
minus erit, quàm S T in V X.</s>
  <s xml:id="echoid-s10911" xml:space="preserve"/>
</p>
<div xml:id="echoid-div927" type="float" level="2" n="1">
  <figure xlink:label="fig-0340-01" xlink:href="fig-0340-01a">
    <image file="0340-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0340-01"/>
  </figure>
<note position="left" xlink:label="note-0340-01" xlink:href="note-0340-01a" xml:space="preserve">f</note>
</div>
<p>
  <s xml:id="echoid-s10912" xml:space="preserve">Sed in hyperbola, quia quilibet axium minor eſt homologa diame-<lb/>tro coniugatarum; </s>
  <s xml:id="echoid-s10913" xml:space="preserve">igitur planum rectangulum ab axibus contentum mi-<lb/>nus eſt eo quod à duabus coniugatis continetur hoc igitur in hyperbo-<lb/>le manifeſtum eſt.</s>
  <s xml:id="echoid-s10914" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10915" xml:space="preserve">In ellipſi autem, quia A C ad Q R maiorem proportionem habet; <lb/></s>
  <s xml:id="echoid-s10916" xml:space="preserve">
<anchor type="note" xlink:label="note-0340-02a" xlink:href="note-0340-02"/>
quàm I L ad N O per conuerſionem rationis, &amp; </s>
  <s xml:id="echoid-s10917" xml:space="preserve">permutando maior A C <lb/>ad minorem I L minorem proportionem habebit, quàm differentia ipſa-<lb/>rum A C, Q R ad differentiam ipſarum I L &amp; </s>
  <s xml:id="echoid-s10918" xml:space="preserve">N O; </s>
  <s xml:id="echoid-s10919" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10920" xml:space="preserve">propterea diffe-<lb/>rentia ipſarum A C, &amp; </s>
  <s xml:id="echoid-s10921" xml:space="preserve">Q R maior erit differentia reliquarum I L, &amp; </s>
  <s xml:id="echoid-s10922" xml:space="preserve">N <lb/>O. </s>
  <s xml:id="echoid-s10923" xml:space="preserve">Et ſimiliter oſtendetur, quod exceſſus I L ſuper N O maior ſit, quàm <lb/>exceſſus S T ſuper V X.</s>
  <s xml:id="echoid-s10924" xml:space="preserve"/>
</p>
<div xml:id="echoid-div928" type="float" level="2" n="2">
<note position="left" xlink:label="note-0340-02" xlink:href="note-0340-02a" xml:space="preserve">g</note>
</div>
<pb o="303" file="0341" n="342" rhead="Conicor. Lib. VII."/>
  <figure>
    <image file="0341-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0341-01"/>
  </figure>
</div>
<div xml:id="echoid-div930" type="section" level="1" n="292">
<head xml:id="echoid-head363" xml:space="preserve">PROPOSITIO XXIV.</head>
<p>
  <s xml:id="echoid-s10925" xml:space="preserve"><emph style="sc">Et</emph> quia in ellipſi qua-<lb/>
<anchor type="figure" xlink:label="fig-0341-02a" xlink:href="fig-0341-02"/>
dratum Q R, nempe <lb/>figura axis A C minor eſt <lb/>in prima, &amp; </s>
  <s xml:id="echoid-s10926" xml:space="preserve">maior in ſe-<lb/>cunda ellipſi, qdàm qua-<lb/>dratum N O, nempe quã <lb/>figura I L ( 28. </s>
  <s xml:id="echoid-s10927" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s10928" xml:space="preserve">) <lb/>eſtque A C maior in pri-<lb/>ma, &amp; </s>
  <s xml:id="echoid-s10929" xml:space="preserve">minor in ſecunda <lb/>figura quàm I L ; </s>
  <s xml:id="echoid-s10930" xml:space="preserve">igitur <lb/>
<anchor type="note" xlink:label="note-0341-01a" xlink:href="note-0341-01"/>
erectum ipſius A C minus <lb/>eſt in prima figura, &amp; </s>
  <s xml:id="echoid-s10931" xml:space="preserve">ma-<lb/>ius in ſecunda, quàm ere-<lb/>ctum I L. </s>
  <s xml:id="echoid-s10932" xml:space="preserve">Et ſic oſtende-<lb/>tur, quod ereæum ipſius <lb/>I L maius ſit, ſiue minus, <lb/>quàm erectum S T.</s>
  <s xml:id="echoid-s10933" xml:space="preserve"/>
</p>
<div xml:id="echoid-div930" type="float" level="2" n="1">
  <figure xlink:label="fig-0341-02" xlink:href="fig-0341-02a">
    <image file="0341-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0341-02"/>
  </figure>
<note position="left" xlink:label="note-0341-01" xlink:href="note-0341-01a" xml:space="preserve">h</note>
</div>
<p>
  <s xml:id="echoid-s10934" xml:space="preserve">Et quia erectum ipſius <lb/>A C minus eſt in prima <lb/>ellipſi, &amp; </s>
  <s xml:id="echoid-s10935" xml:space="preserve">maius in ſecun-<lb/>da, quàm erectum ipſius <lb/>I L, &amp; </s>
  <s xml:id="echoid-s10936" xml:space="preserve">A C maior eſt in <lb/>prima, &amp; </s>
  <s xml:id="echoid-s10937" xml:space="preserve">minor in ſecun-<lb/>da figura quàm I L, igi-<lb/>tur differentia A C, eiuſq; <lb/></s>
  <s xml:id="echoid-s10938" xml:space="preserve">erecti, quæ ſunt duo la-<lb/>tera figuræ A C, in quo-
<pb o="304" file="0342" n="343" rhead="Apollonij Pergæi."/>
libet caſu maior erit differentia I L, eiuſque erecti. </s>
  <s xml:id="echoid-s10939" xml:space="preserve">Pari modo oſtende-<lb/>tur quod differentia ipſius I L, &amp; </s>
  <s xml:id="echoid-s10940" xml:space="preserve">eius erecti maior ſit differentia ipſius S <lb/>T, eiuſque erecti. </s>
  <s xml:id="echoid-s10941" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s10942" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div932" type="section" level="1" n="293">
<head xml:id="echoid-head364" xml:space="preserve">PROPOSITIO XXXVII.</head>
<p>
  <s xml:id="echoid-s10943" xml:space="preserve"><emph style="sc">In</emph> hyperbole differentia la-<lb/>terum figuræ axis inclinati <lb/>maior eſt differentia laterũ figu-<lb/>rę ſui homologi eiuſdẽ ſectionis: <lb/></s>
  <s xml:id="echoid-s10944" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10945" xml:space="preserve">differẽtia laterum figuræ in-<lb/>
<anchor type="figure" xlink:label="fig-0342-01a" xlink:href="fig-0342-01"/>
clinati proximioris axi maior <lb/>eſt differentia laterum figuræ <lb/>inclinati ab illo remotioris.</s>
  <s xml:id="echoid-s10946" xml:space="preserve"/>
</p>
<div xml:id="echoid-div932" type="float" level="2" n="1">
  <figure xlink:label="fig-0342-01" xlink:href="fig-0342-01a">
    <image file="0342-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0342-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s10947" xml:space="preserve">In hyperbole A B P ſit axis C <lb/>A, &amp; </s>
  <s xml:id="echoid-s10948" xml:space="preserve">I L, S T ſit duæ aliæ dia-<lb/>metri, &amp; </s>
  <s xml:id="echoid-s10949" xml:space="preserve">centrum D; </s>
  <s xml:id="echoid-s10950" xml:space="preserve">atque ere-<lb/>ctus ipſius A C ſit A F, &amp; </s>
  <s xml:id="echoid-s10951" xml:space="preserve">ipſius <lb/>I L ſit I K, atque ipſius S T ſit S <lb/>Z: </s>
  <s xml:id="echoid-s10952" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10953" xml:space="preserve">educamus C B, C P, pa-<lb/>rallelas duabus homologis diame-<lb/>tris I L, S T, &amp; </s>
  <s xml:id="echoid-s10954" xml:space="preserve">duas ad axim <lb/>perpendiculares B E, P M, ſece-<lb/>muſque duas interceptas C H, A <lb/>G, &amp; </s>
  <s xml:id="echoid-s10955" xml:space="preserve">ſit inclinatus A C in prima <lb/>figura maior, quàm A F, in ſecũ-<lb/>da verò minor. </s>
  <s xml:id="echoid-s10956" xml:space="preserve">Et quoniam A C <lb/>ad A F ſupponitur vt H A ad A G <lb/>
<anchor type="figure" xlink:label="fig-0342-02a" xlink:href="fig-0342-02"/>
<pb o="305" file="0343" n="344" rhead="Conicor. Lib. VII."/>
erit quadratum A C ad quadratum differentię ipſarum A C, A F, vt <lb/>quadratum H A ad quadratum H G, at ad quadratum differentię ipſa-<lb/>rum I L, I K eſt, vt E H in H A ad quadratum H G (19. </s>
  <s xml:id="echoid-s10957" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s10958" xml:space="preserve">ad <lb/>quadratum verò differentię S T, S Z eſt, vt H M in H A ad quadratum <lb/>H G (19. </s>
  <s xml:id="echoid-s10959" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s10960" xml:space="preserve">eſt verò M H in H A maius quàm E H in H A, atque <lb/>E H in H A maius quàm quadratum H A; </s>
  <s xml:id="echoid-s10961" xml:space="preserve">igitur A C ad differentiam <lb/>ipſarum A C, A F minorem proportionem habet, quàm ad differentiam <lb/>ipſarum I L, I K, &amp; </s>
  <s xml:id="echoid-s10962" xml:space="preserve">ad differentiam earundem I L, I K minorem pro-<lb/>portionem habet, quam ad differentiam ipſarum S T, S Z; </s>
  <s xml:id="echoid-s10963" xml:space="preserve">igitur diffe-<lb/>rentia ipſarum A C, A F maior eſt, quàm differentia ipſarum I L, I K, <lb/>atquè differentia earundem I L, I K maior eſt quàm differentia S T, S <lb/>Z. </s>
  <s xml:id="echoid-s10964" xml:space="preserve">Quod erat propoſitum.</s>
  <s xml:id="echoid-s10965" xml:space="preserve"/>
</p>
<div xml:id="echoid-div933" type="float" level="2" n="2">
  <figure xlink:label="fig-0342-02" xlink:href="fig-0342-02a">
    <image file="0342-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0342-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div935" type="section" level="1" n="294">
<head xml:id="echoid-head365" xml:space="preserve">Notę in Propoſit. XXVIII.</head>
<p style="it">
  <s xml:id="echoid-s10966" xml:space="preserve">S It in primis figuris axis A C maior, quàm axis Q R. </s>
  <s xml:id="echoid-s10967" xml:space="preserve">Quia quadratum <lb/>
<anchor type="note" xlink:label="note-0343-01a" xlink:href="note-0343-01"/>
A C ad quadratum Q R eandem proportionem habet, quàm H A ad A G: <lb/></s>
  <s xml:id="echoid-s10968" xml:space="preserve">eſtque G A minor quàm G E; </s>
  <s xml:id="echoid-s10969" xml:space="preserve">ergo H G ad G A maiorem proportionem habet <lb/>quàm ad G E: </s>
  <s xml:id="echoid-s10970" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10971" xml:space="preserve">componendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s10972" xml:space="preserve">diuidendo in ellipſi H A ad A <lb/>G maiorem proportionem habet, quàm H E ad E G; </s>
  <s xml:id="echoid-s10973" xml:space="preserve">ſed H E ad E G eandem <lb/>
<anchor type="note" xlink:label="note-0343-02a" xlink:href="note-0343-02"/>
<anchor type="figure" xlink:label="fig-0343-01a" xlink:href="fig-0343-01"/>
proportionem habet, quàm quadratum <lb/>I L ad quadratum N O; </s>
  <s xml:id="echoid-s10974" xml:space="preserve">ergo quadra-<lb/>tum A C ad quadratum Q R maiorem <lb/>proportionem habet, quàm quadratum <lb/>I L ad quadratum N O : </s>
  <s xml:id="echoid-s10975" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10976" xml:space="preserve">propterea <lb/>A C ad Q R maiorem proportionem <lb/>habet, quàm I L ad N O : </s>
  <s xml:id="echoid-s10977" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10978" xml:space="preserve">ſunt <lb/>quoquè earundem proportionum dupli-<lb/>catę pariter inęquales, nimirum axis <lb/>
<anchor type="note" xlink:label="note-0343-03a" xlink:href="note-0343-03"/>
A C ad eius latus rectum A F maio-<lb/>rem proportionem habebit, quàm dia-<lb/>meter I L ad eius latus rectum I K. <lb/></s>
  <s xml:id="echoid-s10979" xml:space="preserve">Secundò quia G E minor eſt, quàm <lb/>G M ; </s>
  <s xml:id="echoid-s10980" xml:space="preserve">ergo H G ad G E maiorem pro-<lb/>portionem habet, quàm ad G M ; </s>
  <s xml:id="echoid-s10981" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10982" xml:space="preserve">componendo in hyperbola, &amp; </s>
  <s xml:id="echoid-s10983" xml:space="preserve">diuidendo in <lb/>ellipſi H E ad E G maiorem proportionem habebit, quàm H M ad M G, &amp; </s>
  <s xml:id="echoid-s10984" xml:space="preserve"><lb/>quadratum I L ad quadratum N O habet eandem proportionem, quàm H E ad <lb/>E G ; </s>
  <s xml:id="echoid-s10985" xml:space="preserve">nec non quadratum S T ad quadratum V X eandem proportionem habet, <lb/>
<anchor type="note" xlink:label="note-0343-04a" xlink:href="note-0343-04"/>
quàm H M ad M G ; </s>
  <s xml:id="echoid-s10986" xml:space="preserve">ergo quadratum I L ad quadratum N O maiorem pro-<lb/>portionem habet, quàm quadratum S T ad quadratum V X, &amp; </s>
  <s xml:id="echoid-s10987" xml:space="preserve">I L ad N O <lb/>maiorem proportionem habebit, quàm S T ad V X, &amp; </s>
  <s xml:id="echoid-s10988" xml:space="preserve">earundem proportio-<lb/>num duplicatę inęquales quoque erunt, ſcilicet I L ad eius latus rectum maio-<lb/>rem proportionem habebit, quàm S T ad eius latus rectum. </s>
  <s xml:id="echoid-s10989" xml:space="preserve">Deindè in ſecun-<lb/>dis figuris ſit axis A C minor quàm Q R. </s>
  <s xml:id="echoid-s10990" xml:space="preserve">Quia H A minor eſt, quàm H E;</s>
  <s xml:id="echoid-s10991" xml:space="preserve">
<pb o="306" file="0344" n="345" rhead="Apollonij Pergęi"/>
nec non H E minor quàm H M ergo H <lb/>
<anchor type="figure" xlink:label="fig-0344-01a" xlink:href="fig-0344-01"/>
A ad eandem H G minorem proportio-<lb/>nem habebit, quàm H E, &amp; </s>
  <s xml:id="echoid-s10992" xml:space="preserve">compa-<lb/>rando antecedentes, ad terminorum <lb/>ſummas vel ad differentias H A ad A <lb/>
<anchor type="note" xlink:label="note-0344-01a" xlink:href="note-0344-01"/>
G minorem proportionem habet, quàm <lb/>H E ad E G, &amp; </s>
  <s xml:id="echoid-s10993" xml:space="preserve">ſimiliter H E ad E G <lb/>minorem proportionem habet, quàm H <lb/>M ad M G : </s>
  <s xml:id="echoid-s10994" xml:space="preserve">eſt verò quadratum A C <lb/>
<anchor type="note" xlink:label="note-0344-02a" xlink:href="note-0344-02"/>
ad quadratum Q R, vt H A ad A G, <lb/>&amp; </s>
  <s xml:id="echoid-s10995" xml:space="preserve">quadratum I L ad quadratum N O, <lb/>vt H E ad E G ; </s>
  <s xml:id="echoid-s10996" xml:space="preserve">pariterquè quadratum <lb/>S T ad quadratum V X eſt, vt H M <lb/>ad M G ; </s>
  <s xml:id="echoid-s10997" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10998" xml:space="preserve">ideo A C ad Q R mino-<lb/>rem proportionem habebit, quàm I L ad <lb/>N O, &amp; </s>
  <s xml:id="echoid-s10999" xml:space="preserve">I L ad N O minorem propor-<lb/>tionem habebit, quàm S T ad V X; </s>
  <s xml:id="echoid-s11000" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11001" xml:space="preserve"><lb/>ſimiliter earundem proportionum dupli-<lb/>
<anchor type="note" xlink:label="note-0344-03a" xlink:href="note-0344-03"/>
catę eodem ordine inęquales erunt, ſci-<lb/>licet A C ad eius latus rectum minorem <lb/>proportionem habebit quàm I L ad etus <lb/>rectum latus, &amp;</s>
  <s xml:id="echoid-s11002" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11003" xml:space="preserve">Ad perfectionem <lb/>partis ſecundę propoſitionis 28. </s>
  <s xml:id="echoid-s11004" xml:space="preserve">requiri-<lb/>tur hoc.</s>
  <s xml:id="echoid-s11005" xml:space="preserve"/>
</p>
<div xml:id="echoid-div935" type="float" level="2" n="1">
<note position="right" xlink:label="note-0343-01" xlink:href="note-0343-01a" xml:space="preserve">ex 15. 16. <lb/>lib. 1. <lb/>Defin. 1. <lb/>huius.</note>
<note position="right" xlink:label="note-0343-02" xlink:href="note-0343-02a" xml:space="preserve">6. &amp; 7. <lb/>huius.</note>
  <figure xlink:label="fig-0343-01" xlink:href="fig-0343-01a">
    <image file="0343-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0343-01"/>
  </figure>
<note position="right" xlink:label="note-0343-03" xlink:href="note-0343-03a" xml:space="preserve">ex 15. 16. <lb/>huius.</note>
<note position="right" xlink:label="note-0343-04" xlink:href="note-0343-04a" xml:space="preserve">6. &amp; 7. <lb/>huius.</note>
  <figure xlink:label="fig-0344-01" xlink:href="fig-0344-01a">
    <image file="0344-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0344-01"/>
  </figure>
<note position="left" xlink:label="note-0344-01" xlink:href="note-0344-01a" xml:space="preserve">Lem. 2. <lb/>lib. 5.</note>
<note position="left" xlink:label="note-0344-02" xlink:href="note-0344-02a" xml:space="preserve">ex 15. 16. <lb/>lib. 1. <lb/>Defin. 1. <lb/>huius. <lb/>Prop. 7. <lb/>huius.</note>
<note position="left" xlink:label="note-0344-03" xlink:href="note-0344-03a" xml:space="preserve">ex 15. 16. <lb/>lib. 1.</note>
</div>
  <figure>
    <image file="0344-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0344-02"/>
  </figure>
</div>
<div xml:id="echoid-div937" type="section" level="1" n="295">
<head xml:id="echoid-head366" xml:space="preserve">LEMMA. I.</head>
<p style="it">
  <s xml:id="echoid-s11006" xml:space="preserve">I N ellipſi cuius axes inęquales ſunt, duas diametros coniugatas inter <lb/>ſe ęquales reperire.</s>
  <s xml:id="echoid-s11007" xml:space="preserve"/>
</p>
<pb o="307" file="0345" n="346" rhead="Conicor. Lib. VII."/>
<p style="it">
  <s xml:id="echoid-s11008" xml:space="preserve">In eadem figura coniungatur recta linèa A Q terminos axium coniungens, <lb/>&amp; </s>
  <s xml:id="echoid-s11009" xml:space="preserve">per centrum huic parallela ſit e d, perq; </s>
  <s xml:id="echoid-s11010" xml:space="preserve">idem centrum, &amp; </s>
  <s xml:id="echoid-s11011" xml:space="preserve">ſemipartitionem <lb/>
<anchor type="figure" xlink:label="fig-0345-01a" xlink:href="fig-0345-01"/>
<anchor type="figure" xlink:label="fig-0345-02a" xlink:href="fig-0345-02"/>
applicatę A Q ducatur diameter a b: </s>
  <s xml:id="echoid-s11012" xml:space="preserve">Dico diametros coniugatas a b, &amp; </s>
  <s xml:id="echoid-s11013" xml:space="preserve">e d <lb/>ęquales eſſe inter ſe. </s>
  <s xml:id="echoid-s11014" xml:space="preserve">Quoniam à termino Q ordinatim applicatę A Q ad dia-<lb/>metrum a b ducitur ad axim perpendicularis Q D cadens in centrum D; </s>
  <s xml:id="echoid-s11015" xml:space="preserve">ergo <lb/>
<anchor type="note" xlink:label="note-0345-01a" xlink:href="note-0345-01"/>
H D ad D G eandem proportionem habet, quàm quadratum diametri a b ad <lb/>quadratum eius coniugatę c d; </s>
  <s xml:id="echoid-s11016" xml:space="preserve">ſuntquè H D, &amp; </s>
  <s xml:id="echoid-s11017" xml:space="preserve">G D ęquales inter ſe, cum <lb/>ſemiaxes, atquè interceptę ſint ęquales inter ſe; </s>
  <s xml:id="echoid-s11018" xml:space="preserve">ergo diametri coniugatę a b, <lb/>&amp; </s>
  <s xml:id="echoid-s11019" xml:space="preserve">c d ęquales erunt inter ſe hoc pręmiſſo.</s>
  <s xml:id="echoid-s11020" xml:space="preserve"/>
</p>
<div xml:id="echoid-div937" type="float" level="2" n="1">
  <figure xlink:label="fig-0345-01" xlink:href="fig-0345-01a">
    <image file="0345-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0345-01"/>
  </figure>
  <figure xlink:label="fig-0345-02" xlink:href="fig-0345-02a">
    <image file="0345-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0345-02"/>
  </figure>
<note position="right" xlink:label="note-0345-01" xlink:href="note-0345-01a" xml:space="preserve">Prop. 7. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11021" xml:space="preserve">Reperiantur in ellipſi duę diametri coniugatę inter ſe ęquales a b, e d, &amp; </s>
  <s xml:id="echoid-s11022" xml:space="preserve"><lb/>inter a, &amp; </s>
  <s xml:id="echoid-s11023" xml:space="preserve">A ponantur diametri I L, S T, quarum coniugatę N O, &amp; </s>
  <s xml:id="echoid-s11024" xml:space="preserve">V X, <lb/>
<anchor type="figure" xlink:label="fig-0345-03a" xlink:href="fig-0345-03"/>
&amp; </s>
  <s xml:id="echoid-s11025" xml:space="preserve">ducãtur reliquę rectę lineę, <lb/>vt prius factum eſt, &amp; </s>
  <s xml:id="echoid-s11026" xml:space="preserve">pona-<lb/>tur primo loco axis A C maior <lb/>quàm Q R: </s>
  <s xml:id="echoid-s11027" xml:space="preserve">Dico I L maiorem <lb/>eſſe ipſa N O, &amp; </s>
  <s xml:id="echoid-s11028" xml:space="preserve">S T maiorem <lb/>V X. </s>
  <s xml:id="echoid-s11029" xml:space="preserve">Quia quadratum A C ad <lb/>quadratum Q R eandem propor-<lb/>
<anchor type="note" xlink:label="note-0345-02a" xlink:href="note-0345-02"/>
tionem habet, quàm H A ad A <lb/>G, &amp; </s>
  <s xml:id="echoid-s11030" xml:space="preserve">quadratum I L ad qua-<lb/>dratum N O eandem proportio-<lb/>nem habet, quàm H E ad E G; <lb/></s>
  <s xml:id="echoid-s11031" xml:space="preserve">pariterquè quadratum S T ad <lb/>quadratum V X eandem propor-<lb/>
<anchor type="note" xlink:label="note-0345-03a" xlink:href="note-0345-03"/>
tionem habet, quàm H M ad <lb/>M G ; </s>
  <s xml:id="echoid-s11032" xml:space="preserve">ſed in prima hyperbola, <lb/>&amp; </s>
  <s xml:id="echoid-s11033" xml:space="preserve">prima ellipſi H A maior eſt, <lb/>quàm A G, &amp; </s>
  <s xml:id="echoid-s11034" xml:space="preserve">H E maior, quã <lb/>E G, atquè H M maior, quàm <lb/>M G; </s>
  <s xml:id="echoid-s11035" xml:space="preserve">igitnr quadratum I L ma-
<pb o="308" file="0346" n="347" rhead="Apollonij Pergęi"/>
ius eſt quadrato N O, &amp; </s>
  <s xml:id="echoid-s11036" xml:space="preserve">qua-<lb/>
<anchor type="figure" xlink:label="fig-0346-01a" xlink:href="fig-0346-01"/>
dratum S T maius quadrato V <lb/>X ; </s>
  <s xml:id="echoid-s11037" xml:space="preserve">ideoquè quando axis A C <lb/>maior eſt, quàm Q R, crit dia-<lb/>meter I L maior eius coniugata <lb/>N O, &amp; </s>
  <s xml:id="echoid-s11038" xml:space="preserve">S T maior quàm V X. <lb/></s>
  <s xml:id="echoid-s11039" xml:space="preserve">Pari ratione, quandò axis A C <lb/>minor eſt, quàm Q R erit H A <lb/>minor, quàm A G, &amp; </s>
  <s xml:id="echoid-s11040" xml:space="preserve">H E mi-<lb/>nor, quàm E G, atque H M mi-<lb/>nor, quàm M G : </s>
  <s xml:id="echoid-s11041" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11042" xml:space="preserve">propterea <lb/>in ſecunda hyperbola, &amp; </s>
  <s xml:id="echoid-s11043" xml:space="preserve">ſecun-<lb/>da ellipſi etiam diameter I L <lb/>minor erit, quàm N O, &amp; </s>
  <s xml:id="echoid-s11044" xml:space="preserve">S T <lb/>minor erit quàm V X. </s>
  <s xml:id="echoid-s11045" xml:space="preserve">Idem, <lb/>contingit in reliquis diametris, <lb/>dummodò in ellipſi cadant inter <lb/>A, &amp; </s>
  <s xml:id="echoid-s11046" xml:space="preserve">a, nam a b eſt ęqualis <lb/>ſuę coniugatę e d: </s>
  <s xml:id="echoid-s11047" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11048" xml:space="preserve">vltra pũ-<lb/>ctum a ad partes Q diametri <lb/>cadentes minores ſunt ſuis coniugatis in prima ellipſi, &amp; </s>
  <s xml:id="echoid-s11049" xml:space="preserve">maiores in ſecunda, <lb/>cum propinquiores ſint axi Q R.</s>
  <s xml:id="echoid-s11050" xml:space="preserve"/>
</p>
<div xml:id="echoid-div938" type="float" level="2" n="2">
  <figure xlink:label="fig-0345-03" xlink:href="fig-0345-03a">
    <image file="0345-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0345-03"/>
  </figure>
<note position="right" xlink:label="note-0345-02" xlink:href="note-0345-02a" xml:space="preserve">Defin. 1. <lb/>huius.</note>
<note position="right" xlink:label="note-0345-03" xlink:href="note-0345-03a" xml:space="preserve">Prop. 7. <lb/>huius.</note>
  <figure xlink:label="fig-0346-01" xlink:href="fig-0346-01a">
    <image file="0346-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0346-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s11051" xml:space="preserve">Si verò fuerit vnus duorum axium in hyperbola aut ellipſi maior, tunc <lb/>
<anchor type="note" xlink:label="note-0346-01a" xlink:href="note-0346-01"/>
eius homologa diameter coniugata maior eſt, &amp;</s>
  <s xml:id="echoid-s11052" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11053" xml:space="preserve">Non nulla in hoc texta <lb/>deficiunt; </s>
  <s xml:id="echoid-s11054" xml:space="preserve">non enim omnes diametri in ellipſi ſunt inęquales vt in Lemmate I. <lb/></s>
  <s xml:id="echoid-s11055" xml:space="preserve">oſtenſum eſt, &amp; </s>
  <s xml:id="echoid-s11056" xml:space="preserve">ideo textus corrigi debuit.</s>
  <s xml:id="echoid-s11057" xml:space="preserve"/>
</p>
<div xml:id="echoid-div939" type="float" level="2" n="3">
<note position="right" xlink:label="note-0346-01" xlink:href="note-0346-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div941" type="section" level="1" n="296">
<head xml:id="echoid-head367" xml:space="preserve">Notę in Propoſit. XXI.</head>
<p style="it">
  <s xml:id="echoid-s11058" xml:space="preserve">ET conuenient duo puncta H, &amp; </s>
  <s xml:id="echoid-s11059" xml:space="preserve">G in puncto D ; </s>
  <s xml:id="echoid-s11060" xml:space="preserve">eritque A C ad Q <lb/>
<anchor type="note" xlink:label="note-0346-02a" xlink:href="note-0346-02"/>
R, vt A D ad ſe ipſam, ſiue vt A C ad ſe ipſam, &amp;</s>
  <s xml:id="echoid-s11061" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11062" xml:space="preserve">Quia qua-<lb/>
<anchor type="figure" xlink:label="fig-0346-02a" xlink:href="fig-0346-02"/>
dratum A C ad quadratum Q R eſt <lb/>vt C G ad G A, &amp; </s>
  <s xml:id="echoid-s11063" xml:space="preserve">vt quadratum, <lb/>
<anchor type="note" xlink:label="note-0346-03a" xlink:href="note-0346-03"/>
I L ad quadratum N O, ita eſt H E <lb/>ad E G, nec non quadratum S T ad <lb/>quadratum V X eſt vt H M ad M G; <lb/></s>
  <s xml:id="echoid-s11064" xml:space="preserve">ſed quandò axium quadrata ſunt inter <lb/>ſe ęqualia, tunc quidem pręſecta C G, <lb/>ſeu H A ęqualis eſt interceptę G A, &amp; </s>
  <s xml:id="echoid-s11065" xml:space="preserve"><lb/>terminus G, ſeu H cadit in cẽtro D; </s>
  <s xml:id="echoid-s11066" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11067" xml:space="preserve"><lb/>ideo H E vel D E ęqualis eſt E G vel <lb/>E D : </s>
  <s xml:id="echoid-s11068" xml:space="preserve">pariterq; </s>
  <s xml:id="echoid-s11069" xml:space="preserve">H M ęqualis eſt M G: </s>
  <s xml:id="echoid-s11070" xml:space="preserve"><lb/>quarè coniugatarũ diametrorũ quadra-<lb/>ta ęqualia ſunt inter ſe; </s>
  <s xml:id="echoid-s11071" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11072" xml:space="preserve">etiã tranſ-<lb/>uer ſa latera ſuis erectis ęqualia erunt.</s>
  <s xml:id="echoid-s11073" xml:space="preserve"/>
</p>
<div xml:id="echoid-div941" type="float" level="2" n="1">
<note position="right" xlink:label="note-0346-02" xlink:href="note-0346-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0346-02" xlink:href="fig-0346-02a">
    <image file="0346-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0346-02"/>
  </figure>
<note position="left" xlink:label="note-0346-03" xlink:href="note-0346-03a" xml:space="preserve">Defin. 1. <lb/>Prop. 7. <lb/>huius.</note>
</div>
<pb o="309" file="0347" n="348" rhead="Conicor. Lib. VII."/>
<p style="it">
  <s xml:id="echoid-s11074" xml:space="preserve">Quia C G ad A G, nempe quadratum A C ad ſuam figuram in ma-<lb/>
<anchor type="note" xlink:label="note-0347-01a" xlink:href="note-0347-01"/>
iori, &amp; </s>
  <s xml:id="echoid-s11075" xml:space="preserve">in figura ſecunda ellipſi in minori proportione, &amp;</s>
  <s xml:id="echoid-s11076" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11077" xml:space="preserve">Ideſt. </s>
  <s xml:id="echoid-s11078" xml:space="preserve">In, <lb/>
<anchor type="figure" xlink:label="fig-0347-01a" xlink:href="fig-0347-01"/>
prima, &amp; </s>
  <s xml:id="echoid-s11079" xml:space="preserve">ſecunda figura hyperboles, <lb/>&amp; </s>
  <s xml:id="echoid-s11080" xml:space="preserve">in prima figura ellipſis habet C G ad <lb/>G A maiorem proportionem, quàm ad <lb/>G E, eo quod G E maior eſt, quàm G <lb/>A: </s>
  <s xml:id="echoid-s11081" xml:space="preserve">at in ſecunda figura ellipſis propor-<lb/>tio minor eſt; </s>
  <s xml:id="echoid-s11082" xml:space="preserve">quia G E minor eſt, quã <lb/>A G. </s>
  <s xml:id="echoid-s11083" xml:space="preserve">Propoſitum verò aliter oſtendetur <lb/>hac ratione.</s>
  <s xml:id="echoid-s11084" xml:space="preserve"/>
</p>
<div xml:id="echoid-div942" type="float" level="2" n="2">
<note position="left" xlink:label="note-0347-01" xlink:href="note-0347-01a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0347-01" xlink:href="fig-0347-01a">
    <image file="0347-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0347-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11085" xml:space="preserve">Quoniam ex demonſtratis in nota <lb/>propoſit. </s>
  <s xml:id="echoid-s11086" xml:space="preserve">27. </s>
  <s xml:id="echoid-s11087" xml:space="preserve">in hyperbola, atquè ex <lb/>propoſitione 11. </s>
  <s xml:id="echoid-s11088" xml:space="preserve">libri quinti in ellipſi <lb/>erit axis minor, &amp; </s>
  <s xml:id="echoid-s11089" xml:space="preserve">rectus Q R minor <lb/>diametro recta N O, &amp; </s>
  <s xml:id="echoid-s11090" xml:space="preserve">N O minor <lb/>remotiore V X, ideoquè quadratum Q <lb/>R minus erit quadrato N O, &amp; </s>
  <s xml:id="echoid-s11091" xml:space="preserve">qua-<lb/>dratum N O minus quàm quadratum <lb/>V X : </s>
  <s xml:id="echoid-s11092" xml:space="preserve">eſt verò figura, ſeu rectangulum <lb/>C A F ſub extremis contentum ęquale <lb/>quadrato Q R ex media proportionali <lb/>
<anchor type="note" xlink:label="note-0347-02a" xlink:href="note-0347-02"/>
inter illas deſcriptum: </s>
  <s xml:id="echoid-s11093" xml:space="preserve">pariterquè re-<lb/>ctangulum L I K ęquale eſt quadrato <lb/>diametri ei coniugatę N O, nec non, <lb/>rectangulum T S Z ęquale erit qua-<lb/>drato V X, ergo rectangulum C A F <lb/>minus eſt rectangulo L I K, atque rectangulum L I K minus eſt rectangulo T <lb/>
<anchor type="figure" xlink:label="fig-0347-02a" xlink:href="fig-0347-02"/>
S Z. </s>
  <s xml:id="echoid-s11094" xml:space="preserve">E contra in ellipſi ſecunda. </s>
  <s xml:id="echoid-s11095" xml:space="preserve">Quia. </s>
  <s xml:id="echoid-s11096" xml:space="preserve">Q R maior eſt, quàm N O, &amp; </s>
  <s xml:id="echoid-s11097" xml:space="preserve">hęc <lb/>maior, quàm V X ; </s>
  <s xml:id="echoid-s11098" xml:space="preserve">ergo rectangulum C A F maius eſt rectangulo L I K, &amp; </s>
  <s xml:id="echoid-s11099" xml:space="preserve"><lb/>hoc maius erit rectangulo T S Z.</s>
  <s xml:id="echoid-s11100" xml:space="preserve"/>
</p>
<div xml:id="echoid-div943" type="float" level="2" n="3">
<note position="right" xlink:label="note-0347-02" xlink:href="note-0347-02a" xml:space="preserve">15. lib. 1.</note>
  <figure xlink:label="fig-0347-02" xlink:href="fig-0347-02a">
    <image file="0347-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0347-02"/>
  </figure>
</div>
<pb o="310" file="0348" n="349" rhead="Apollonij Pergęi"/>
</div>
<div xml:id="echoid-div945" type="section" level="1" n="297">
<head xml:id="echoid-head368" xml:space="preserve">Notę in Propoſit. XXXXII.</head>
<p>
  <s xml:id="echoid-s11101" xml:space="preserve">E Rit igitur aggregatum A C, Q R minus quàm aggregatum I L, N <lb/>
<anchor type="note" xlink:label="note-0348-01a" xlink:href="note-0348-01"/>
O, &amp;</s>
  <s xml:id="echoid-s11102" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11103" xml:space="preserve">Hoc oſtenſum eſt in nota propoſit. </s>
  <s xml:id="echoid-s11104" xml:space="preserve">27. </s>
  <s xml:id="echoid-s11105" xml:space="preserve">huius.</s>
  <s xml:id="echoid-s11106" xml:space="preserve"/>
</p>
<div xml:id="echoid-div945" type="float" level="2" n="1">
<note position="right" xlink:label="note-0348-01" xlink:href="note-0348-01a" xml:space="preserve">d</note>
</div>
<p style="it">
  <s xml:id="echoid-s11107" xml:space="preserve">At in ellipſi, quia A C ad Q R maiorem proportionem habet, quàm <lb/>I L ad N O, erit quadratum aggregati A C, Q R ad ſummam duorum <lb/>
<anchor type="note" xlink:label="note-0348-02a" xlink:href="note-0348-02"/>
<anchor type="figure" xlink:label="fig-0348-01a" xlink:href="fig-0348-01"/>
quadratorum ipſarum in maiori proportione, quàm quadratum aggregati <lb/>I L, N O ad ſummam duorum quadratorum earundem, &amp; </s>
  <s xml:id="echoid-s11108" xml:space="preserve">ſumma duo-<lb/>rum quadratorum ipſarum, &amp;</s>
  <s xml:id="echoid-s11109" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11110" xml:space="preserve">Fiat A R ęqualis duabus A C &amp; </s>
  <s xml:id="echoid-s11111" xml:space="preserve">Q R, <lb/>I O fiat ęqualis duabus I L, &amp; </s>
  <s xml:id="echoid-s11112" xml:space="preserve">N O ; </s>
  <s xml:id="echoid-s11113" xml:space="preserve">atquè ſecetur A R in m, vt ſit A m <lb/>
<anchor type="note" xlink:label="note-0348-03a" xlink:href="note-0348-03"/>
ad m R, vt I L ad L O. </s>
  <s xml:id="echoid-s11114" xml:space="preserve">Quia in prima ellipſi A C ad Q R, vel ad C R <lb/>(in hac figura) maiorem proportionem habet, quàm I L ad N O, ſeu ad L O (in <lb/>
<anchor type="figure" xlink:label="fig-0348-02a" xlink:href="fig-0348-02"/>
pręſenti figura); </s>
  <s xml:id="echoid-s11115" xml:space="preserve">Ergo A C ad C R <lb/>maiorem proportionem habet, quàm <lb/>A m ad m R; </s>
  <s xml:id="echoid-s11116" xml:space="preserve">ideoq; </s>
  <s xml:id="echoid-s11117" xml:space="preserve">A C ad ean-<lb/>
<anchor type="note" xlink:label="note-0348-04a" xlink:href="note-0348-04"/>
dem A R maiorem proportionem ha-<lb/>bebit quàm A m; </s>
  <s xml:id="echoid-s11118" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11119" xml:space="preserve">propterea A m <lb/>minor erit, quàm A C : </s>
  <s xml:id="echoid-s11120" xml:space="preserve">ſed A m <lb/>
<anchor type="figure" xlink:label="fig-0348-03a" xlink:href="fig-0348-03"/>
maior eſt quàm M R, eo quod I L <lb/>priori homologa maior eſt, quàm L <lb/>O : </s>
  <s xml:id="echoid-s11121" xml:space="preserve">at in ſecunda ellipſi A C ad C R <lb/>minorem proportionem habet, quàm
<pb o="311" file="0349" n="350" rhead="Conicor. Lib. VII."/>
I L ad L O, ſeu quàm A m ad m R; </s>
  <s xml:id="echoid-s11122" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11123" xml:space="preserve">A C ad eandem A R minorem pro-<lb/>portionem habet quàm A m; </s>
  <s xml:id="echoid-s11124" xml:space="preserve">ideoque A C minor erit, quàm A m, &amp; </s>
  <s xml:id="echoid-s11125" xml:space="preserve">A m <lb/>
<anchor type="note" xlink:label="note-0349-01a" xlink:href="note-0349-01"/>
minor quàm m R, ſicuti I L minor eſt, quàm L O ; </s>
  <s xml:id="echoid-s11126" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11127" xml:space="preserve">propterea ſecta A R <lb/>bifariam in n in vtroq; </s>
  <s xml:id="echoid-s11128" xml:space="preserve">caſu C n ſemidifferentia maximè, &amp; </s>
  <s xml:id="echoid-s11129" xml:space="preserve">minimè ſcilicet <lb/>A C, &amp; </s>
  <s xml:id="echoid-s11130" xml:space="preserve">C R maior erit, quàm m n ſemidifferentia inæqualium intermedia-<lb/>rum A m, &amp; </s>
  <s xml:id="echoid-s11131" xml:space="preserve">R m: </s>
  <s xml:id="echoid-s11132" xml:space="preserve">ſuntque duo quaarata ex A C, &amp; </s>
  <s xml:id="echoid-s11133" xml:space="preserve">ex C R æqualia qua-<lb/>dratis ex R n, &amp; </s>
  <s xml:id="echoid-s11134" xml:space="preserve">ex C n bis ſumptis, atquè quadrata ex A m, &amp; </s>
  <s xml:id="echoid-s11135" xml:space="preserve">ex R m <lb/>æqualia ſunt quadratis ex R n, &amp; </s>
  <s xml:id="echoid-s11136" xml:space="preserve">ex m n bis ſumptis, ſed duplum quadrati <lb/>n C cum duplo quadrati n R maiora ſunt duplo quadrati n m cum duplo qua-<lb/>drati n R (cum n R ſit communis, &amp; </s>
  <s xml:id="echoid-s11137" xml:space="preserve">n C maior ſit n m); </s>
  <s xml:id="echoid-s11138" xml:space="preserve">igitur in vtroque <lb/>caſu duo quadrata ex maxima, &amp; </s>
  <s xml:id="echoid-s11139" xml:space="preserve">ex minima, ſcilicet quadratum A C vna <lb/>cum quadrato C R maiora ſunt quadrato A m, &amp; </s>
  <s xml:id="echoid-s11140" xml:space="preserve">quadrato m R ſimul ſum-<lb/>ptis: </s>
  <s xml:id="echoid-s11141" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11142" xml:space="preserve">quadratum A R minorem proportionem habet ad ſummam quadrato-<lb/>rum ex A C, &amp; </s>
  <s xml:id="echoid-s11143" xml:space="preserve">ex C R, quàm ad ſummam quadrati A m, &amp; </s>
  <s xml:id="echoid-s11144" xml:space="preserve">quadrati m <lb/>R; </s>
  <s xml:id="echoid-s11145" xml:space="preserve">ſed quadratum I O ad quadratum I L vna cum quadraio L O eandem pro-<lb/>portionem habet, quàm quadratum A R ad ſummam duorum quadratorum ex <lb/>A m, &amp; </s>
  <s xml:id="echoid-s11146" xml:space="preserve">ex m R (propterea quod A R, &amp; </s>
  <s xml:id="echoid-s11147" xml:space="preserve">I O diuiduntur proportionaliter in <lb/>m, &amp; </s>
  <s xml:id="echoid-s11148" xml:space="preserve">L): </s>
  <s xml:id="echoid-s11149" xml:space="preserve">igitur quadratum A R ad ſummam quadrati A C vna cum qua-<lb/>drato C R minorem proportionem habet, quàm quadratum IO ad ſummam qua-<lb/>drati I L cum quadrato L O.</s>
  <s xml:id="echoid-s11150" xml:space="preserve"/>
</p>
<div xml:id="echoid-div946" type="float" level="2" n="2">
<note position="right" xlink:label="note-0348-02" xlink:href="note-0348-02a" xml:space="preserve">e</note>
  <figure xlink:label="fig-0348-01" xlink:href="fig-0348-01a">
    <image file="0348-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0348-01"/>
  </figure>
<note position="left" xlink:label="note-0348-03" xlink:href="note-0348-03a" xml:space="preserve">Prop. 21. <lb/>hu us.</note>
  <figure xlink:label="fig-0348-02" xlink:href="fig-0348-02a">
    <image file="0348-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0348-02"/>
  </figure>
<note position="left" xlink:label="note-0348-04" xlink:href="note-0348-04a" xml:space="preserve">Lem. 2. <lb/>lib. 5.</note>
  <figure xlink:label="fig-0348-03" xlink:href="fig-0348-03a">
    <image file="0348-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0348-03"/>
  </figure>
<note position="right" xlink:label="note-0349-01" xlink:href="note-0349-01a" xml:space="preserve">Lem. 2. <lb/>Lib. 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11151" xml:space="preserve">Non ſecus oſtendetur, quod quadratum ſumme I L, &amp; </s>
  <s xml:id="echoid-s11152" xml:space="preserve">N O ad quadrati ex <lb/>I L, &amp; </s>
  <s xml:id="echoid-s11153" xml:space="preserve">quadrati ex N O ſummam habet minorem proportionem, quàm qua-<lb/>dratum ſumme S T, &amp; </s>
  <s xml:id="echoid-s11154" xml:space="preserve">V X ad quadratorum ex S T, atquè ex V X ſum-<lb/>
<anchor type="note" xlink:label="note-0349-02a" xlink:href="note-0349-02"/>
mam: </s>
  <s xml:id="echoid-s11155" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11156" xml:space="preserve">ideo I L cum N O minores erunt, quàm S T cum V X.</s>
  <s xml:id="echoid-s11157" xml:space="preserve"/>
</p>
<div xml:id="echoid-div947" type="float" level="2" n="3">
<note position="right" xlink:label="note-0349-02" xlink:href="note-0349-02a" xml:space="preserve">ex 22. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div949" type="section" level="1" n="298">
<head xml:id="echoid-head369" xml:space="preserve">Notæ in Propoſit. XXXXIII.</head>
<note position="left" xml:space="preserve">f</note>
<p style="it">
  <s xml:id="echoid-s11158" xml:space="preserve">R Emanet A C in Q R minus quàm I L in N O, &amp; </s>
  <s xml:id="echoid-s11159" xml:space="preserve">pariter I L in N <lb/>
<anchor type="note" xlink:label="note-0349-04a" xlink:href="note-0349-04"/>
O minus quàm S T in V X, &amp;</s>
  <s xml:id="echoid-s11160" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11161" xml:space="preserve">Quia ſi ex quadrato ſummæ A C, <lb/>
<anchor type="figure" xlink:label="fig-0349-01a" xlink:href="fig-0349-01"/>
&amp; </s>
  <s xml:id="echoid-s11162" xml:space="preserve">Q R quferantur duo quadrata ex <lb/>C A, &amp; </s>
  <s xml:id="echoid-s11163" xml:space="preserve">ex Q R ſimul ſumpta, re-<lb/>manent duo rectangula ſub C A, &amp; </s>
  <s xml:id="echoid-s11164" xml:space="preserve"><lb/>Q R contenta: </s>
  <s xml:id="echoid-s11165" xml:space="preserve">pariterque duplum re-<lb/>ctanguli ex I L in N O eſt rcſiduum <lb/>quadrati ex ſumma ipſarum I L, &amp; </s>
  <s xml:id="echoid-s11166" xml:space="preserve"><lb/>N O deſcripti, poſtquàm ablata ſunt <lb/>quadratum ex I L, &amp; </s>
  <s xml:id="echoid-s11167" xml:space="preserve">quadratum ex <lb/>
<anchor type="note" xlink:label="note-0349-05a" xlink:href="note-0349-05"/>
N O ſimul; </s>
  <s xml:id="echoid-s11168" xml:space="preserve">ſed bina quadrata vtrinq; <lb/></s>
  <s xml:id="echoid-s11169" xml:space="preserve">ablata ſunt æqualia inter ſe in ellipſi; </s>
  <s xml:id="echoid-s11170" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s11171" xml:space="preserve">ſumma A C, Q R minor eſt quàm <lb/>
<anchor type="note" xlink:label="note-0349-06a" xlink:href="note-0349-06"/>
ſumma I L, N O; </s>
  <s xml:id="echoid-s11172" xml:space="preserve">Ergo duplum re-<lb/>ctanguli ſub C A &amp; </s>
  <s xml:id="echoid-s11173" xml:space="preserve">ſub Q R mi-<lb/>nus eſt duplo rectanguli I L in N O, <lb/>&amp; </s>
  <s xml:id="echoid-s11174" xml:space="preserve">rectangulum ſub A C, &amp; </s>
  <s xml:id="echoid-s11175" xml:space="preserve">Q R minus eſt rectangulo ſub I L, &amp; </s>
  <s xml:id="echoid-s11176" xml:space="preserve">N O.</s>
  <s xml:id="echoid-s11177" xml:space="preserve"/>
</p>
<div xml:id="echoid-div949" type="float" level="2" n="1">
<note position="left" xlink:label="note-0349-04" xlink:href="note-0349-04a" xml:space="preserve">f</note>
  <figure xlink:label="fig-0349-01" xlink:href="fig-0349-01a">
    <image file="0349-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0349-01"/>
  </figure>
<note position="right" xlink:label="note-0349-05" xlink:href="note-0349-05a" xml:space="preserve">Prop. 22. <lb/>huius.</note>
<note position="right" xlink:label="note-0349-06" xlink:href="note-0349-06a" xml:space="preserve">Prop 42. <lb/>huius.</note>
</div>
<pb o="312" file="0350" n="351" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s11178" xml:space="preserve">Q Via A C ad Q R maiorem pro-<lb/>
<anchor type="note" xlink:label="note-0350-01a" xlink:href="note-0350-01"/>
<anchor type="figure" xlink:label="fig-0350-01a" xlink:href="fig-0350-01"/>
portionem habet, quàm I L <lb/>ad N O poſt cõuerſionem <lb/>rationis, &amp; </s>
  <s xml:id="echoid-s11179" xml:space="preserve">permutationem A C ma-<lb/>ior ad I L, minorem, habebit pro-<lb/>portionem minorem, quàm exceſſus <lb/>A C ſuper Q R ad exceſſum I L ſu-<lb/>per N O, &amp;</s>
  <s xml:id="echoid-s11180" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11181" xml:space="preserve">Hoc quidem verum <lb/>eſt in ellipſi, (veluti dictum eſt ad <lb/>propoſ. </s>
  <s xml:id="echoid-s11182" xml:space="preserve">28. </s>
  <s xml:id="echoid-s11183" xml:space="preserve">huius) quandò maior axis <lb/>eſt A C, ſed quandò A C eſt minor, <lb/>atque A C ad Q R minorem proportio-<lb/>nem habet, quàm I L ad N O, opere <lb/>prætium erit, demonſtrare, quod tunc <lb/>etiam differentia axium A C, &amp; </s>
  <s xml:id="echoid-s11184" xml:space="preserve">Q R <lb/>maior ſit differentia diametrorum I L, <lb/>&amp; </s>
  <s xml:id="echoid-s11185" xml:space="preserve">N O. </s>
  <s xml:id="echoid-s11186" xml:space="preserve">Quoniam exiſtente C A mi-<lb/>nore, quàm Q R (ex 28. </s>
  <s xml:id="echoid-s11187" xml:space="preserve">huius) A C <lb/>ad Q R minorem proportionem habet, <lb/>quàm I L ad N O; </s>
  <s xml:id="echoid-s11188" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11189" xml:space="preserve">inuertendo Q R <lb/>ad A C maiorem proportionem habebit, <lb/>qu àm N O ad I L, &amp; </s>
  <s xml:id="echoid-s11190" xml:space="preserve">per conuerſioné <lb/>rationis Q R ad differentiam ipſarum <lb/>Q R, &amp; </s>
  <s xml:id="echoid-s11191" xml:space="preserve">A C minorem proportionem <lb/>habebit, quàm N O ad differentiam ipſarum N O, &amp; </s>
  <s xml:id="echoid-s11192" xml:space="preserve">I L; </s>
  <s xml:id="echoid-s11193" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11194" xml:space="preserve">permutando Q <lb/>R maior ad minorem N O habebit proportionem minorem, quàm differentia <lb/>ipſarum Q R, &amp; </s>
  <s xml:id="echoid-s11195" xml:space="preserve">A C ad differentiam ipſarum N O, &amp; </s>
  <s xml:id="echoid-s11196" xml:space="preserve">I L: </s>
  <s xml:id="echoid-s11197" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11198" xml:space="preserve">propterea <lb/>differentia ipſarum Q R, &amp; </s>
  <s xml:id="echoid-s11199" xml:space="preserve">A C maior erit, quàm differentia ipſarum N O, <lb/>&amp; </s>
  <s xml:id="echoid-s11200" xml:space="preserve">I L.</s>
  <s xml:id="echoid-s11201" xml:space="preserve"/>
</p>
<div xml:id="echoid-div950" type="float" level="2" n="2">
<note position="right" xlink:label="note-0350-01" xlink:href="note-0350-01a" xml:space="preserve">g</note>
  <figure xlink:label="fig-0350-01" xlink:href="fig-0350-01a">
    <image file="0350-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0350-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11202" xml:space="preserve">Poſtea quandò C A eſt maior axis, tunc I L ad N O maiorem proportionem <lb/>
<anchor type="note" xlink:label="note-0350-02a" xlink:href="note-0350-02"/>
habet, quàm S T ad V X; </s>
  <s xml:id="echoid-s11203" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11204" xml:space="preserve">ſimiliter per conuerſionem rationis, &amp; </s>
  <s xml:id="echoid-s11205" xml:space="preserve">permu-<lb/>tando maior I L ad minorem S D habebit minorem proportionem, quàm diffe-<lb/>rentia coniugatarum diametrorum I L, &amp; </s>
  <s xml:id="echoid-s11206" xml:space="preserve">N O ad differentiam coniugatarum <lb/>S T, &amp; </s>
  <s xml:id="echoid-s11207" xml:space="preserve">V X, quapropter axi propinquiorum diametrorum I L, &amp; </s>
  <s xml:id="echoid-s11208" xml:space="preserve">N O diffe-<lb/>rentia maior erit, quàm remotiorum coniugatarum S T, &amp; </s>
  <s xml:id="echoid-s11209" xml:space="preserve">V X differentia.</s>
  <s xml:id="echoid-s11210" xml:space="preserve"/>
</p>
<div xml:id="echoid-div951" type="float" level="2" n="3">
<note position="left" xlink:label="note-0350-02" xlink:href="note-0350-02a" xml:space="preserve">28. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11211" xml:space="preserve">E contra quandò C A eſt axis minor idem concludetur, vti paulo ante fa-<lb/>ctum eſt.</s>
  <s xml:id="echoid-s11212" xml:space="preserve"/>
</p>
<pb o="313" file="0351" n="352" rhead="Conicor. Lib. VII."/>
  <figure>
    <image file="0351-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0351-01"/>
  </figure>
</div>
<div xml:id="echoid-div953" type="section" level="1" n="299">
<head xml:id="echoid-head370" xml:space="preserve">Notæ in Propoſit. XXIV.</head>
<p style="it">
  <s xml:id="echoid-s11213" xml:space="preserve">I Gitur erectum ipſius A C mi-<lb/>
<anchor type="figure" xlink:label="fig-0351-02a" xlink:href="fig-0351-02"/>
<anchor type="note" xlink:label="note-0351-01a" xlink:href="note-0351-01"/>
nus eſt in prima, &amp; </s>
  <s xml:id="echoid-s11214" xml:space="preserve">maius in-<lb/>ſecunda, quàm I L, &amp; </s>
  <s xml:id="echoid-s11215" xml:space="preserve">ſic oſten-<lb/>detur, quod erectum ipſius I L ma-<lb/>ius ſit, ſiue minus quàm erectum. <lb/></s>
  <s xml:id="echoid-s11216" xml:space="preserve">S T, &amp;</s>
  <s xml:id="echoid-s11217" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11218" xml:space="preserve">Quoniam in prima ellipſi <lb/>rectangulum C A F minus eſt rectan-<lb/>
<anchor type="note" xlink:label="note-0351-02a" xlink:href="note-0351-02"/>
gulo L I K; </s>
  <s xml:id="echoid-s11219" xml:space="preserve">ergo A C ad I L mino-<lb/>rem proportionem habet reciproce, quã <lb/>I @ ad A F; </s>
  <s xml:id="echoid-s11220" xml:space="preserve">quare I K ad aliquam <lb/>aliam quantitatem maiorem, quàm. <lb/></s>
  <s xml:id="echoid-s11221" xml:space="preserve">A F eandem proportionem habebit, <lb/>quàm A C ad I L; </s>
  <s xml:id="echoid-s11222" xml:space="preserve">eſtquè A C maior <lb/>quàm I L in prima ellipſi; </s>
  <s xml:id="echoid-s11223" xml:space="preserve">ergo multò <lb/>magis I K maior erit quàm A F. </s>
  <s xml:id="echoid-s11224" xml:space="preserve">Pari ratione in eadem prima ellipſi rectan-<lb/>gulum L I K minus eſt rectangulo T S Z, &amp; </s>
  <s xml:id="echoid-s11225" xml:space="preserve">I L axi maiori propinquior ma-<lb/>ior eſt, quàm S T; </s>
  <s xml:id="echoid-s11226" xml:space="preserve">ergo S Z maior erit, quàm I K.</s>
  <s xml:id="echoid-s11227" xml:space="preserve"/>
</p>
<div xml:id="echoid-div953" type="float" level="2" n="1">
  <figure xlink:label="fig-0351-02" xlink:href="fig-0351-02a">
    <image file="0351-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0351-02"/>
  </figure>
<note position="left" xlink:label="note-0351-01" xlink:href="note-0351-01a" xml:space="preserve">h</note>
<note position="right" xlink:label="note-0351-02" xlink:href="note-0351-02a" xml:space="preserve">Pro p. 28. <lb/>h uius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11228" xml:space="preserve">E contra in ſecunda ellipſi rectangulum L I K minus erit rectangulo C A F; <lb/></s>
  <s xml:id="echoid-s11229" xml:space="preserve">
<anchor type="note" xlink:label="note-0351-03a" xlink:href="note-0351-03"/>
&amp; </s>
  <s xml:id="echoid-s11230" xml:space="preserve">rectangulum T S Z minus erit rectangulo L I K; </s>
  <s xml:id="echoid-s11231" xml:space="preserve">eſtquè T S maior quàm <lb/>I L, &amp; </s>
  <s xml:id="echoid-s11232" xml:space="preserve">I L maior, quàm A C; </s>
  <s xml:id="echoid-s11233" xml:space="preserve">igitur reciprocè A F maior erit, quàm I K, <lb/>&amp; </s>
  <s xml:id="echoid-s11234" xml:space="preserve">I K maior, quàm S Z.</s>
  <s xml:id="echoid-s11235" xml:space="preserve"/>
</p>
<div xml:id="echoid-div954" type="float" level="2" n="2">
<note position="right" xlink:label="note-0351-03" xlink:href="note-0351-03a" xml:space="preserve">Ibidem.</note>
</div>
<pb o="314" file="0352" n="353" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div956" type="section" level="1" n="300">
<head xml:id="echoid-head371" xml:space="preserve">SECTIO SEXTA</head>
<head xml:id="echoid-head372" xml:space="preserve">Continens Propoſit. XXXIII. XXXIV. <lb/>XXXV. &amp; XXXVI.</head>
<head xml:id="echoid-head373" xml:space="preserve">PROPOSITIO XXXIII.</head>
<p>
  <s xml:id="echoid-s11236" xml:space="preserve">A Xis inclinatus ſi non fuerit minor dimidio ſui erecti, vti-<lb/>que eius erectus minor eſt erecto cæterarum diametrorum <lb/>inclinatarum eiuſdem ſectionis, &amp; </s>
  <s xml:id="echoid-s11237" xml:space="preserve">axi proximioris inclinati ere-<lb/>ctus minor eſt, quàm erectus remotioris.</s>
  <s xml:id="echoid-s11238" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11239" xml:space="preserve">XXXV. </s>
  <s xml:id="echoid-s11240" xml:space="preserve">Et ſi ſuerit axis inclinatus minor dimidio erecti, vti-<lb/>que ad vtraſque eius partes cadent duæ inclinatæ, quarum quæ-<lb/>libet æqualis eſt ſemiſſi erecti ipſius, atque eius erectus minor <lb/>eſt erecto cuiuslibet inclinati ad vtraſque partes eius poſitæ, &amp; </s>
  <s xml:id="echoid-s11241" xml:space="preserve"><lb/>erectus proximioris minor eſt erecto remotioris.</s>
  <s xml:id="echoid-s11242" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11243" xml:space="preserve">In hyperbole A B N ſint A C, <lb/>
<anchor type="figure" xlink:label="fig-0352-01a" xlink:href="fig-0352-01"/>
I L, P Q, S T diametri inclinatæ, <lb/>&amp; </s>
  <s xml:id="echoid-s11244" xml:space="preserve">A F ſit erectus ipſius A C, I <lb/>K ipſius I L, P R ipſius P Q, &amp; </s>
  <s xml:id="echoid-s11245" xml:space="preserve"><lb/>S Z ipſius S T: </s>
  <s xml:id="echoid-s11246" xml:space="preserve">ſitquè axis A C <lb/>non minor medietate ipſius A F. <lb/></s>
  <s xml:id="echoid-s11247" xml:space="preserve">Dico, quod A F minor eſt, quàm <lb/>I K, &amp; </s>
  <s xml:id="echoid-s11248" xml:space="preserve">I K minor quàm P R, &amp; </s>
  <s xml:id="echoid-s11249" xml:space="preserve"><lb/>P R minor quàm S Z. </s>
  <s xml:id="echoid-s11250" xml:space="preserve">Educantur <lb/>C B parallela I L, &amp; </s>
  <s xml:id="echoid-s11251" xml:space="preserve">C N ipſi P <lb/>Q &amp; </s>
  <s xml:id="echoid-s11252" xml:space="preserve">C X ipſi S T: </s>
  <s xml:id="echoid-s11253" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11254" xml:space="preserve">ducantur <lb/>B E, N M, X V perpendiculares <lb/>ad axim C A E. </s>
  <s xml:id="echoid-s11255" xml:space="preserve">Quoniam ſi A C <lb/>æqualis eſt ipſi A F, etiam I L æ-<lb/>qualis eſt ipſi I K (21. </s>
  <s xml:id="echoid-s11256" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11257" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11258" xml:space="preserve"><lb/>P Q ipſi P R; </s>
  <s xml:id="echoid-s11259" xml:space="preserve">eſtque A C minor <lb/>
<anchor type="note" xlink:label="note-0352-01a" xlink:href="note-0352-01"/>
quam I L, &amp; </s>
  <s xml:id="echoid-s11260" xml:space="preserve">I L, quàm P Q;</s>
  <s xml:id="echoid-s11261" xml:space="preserve">
<pb o="315" file="0353" n="354" rhead="Conicor. Lib. VII."/>
ergo A F minor eſt, quàm I K, &amp; </s>
  <s xml:id="echoid-s11262" xml:space="preserve">I K minor quàm P R. </s>
  <s xml:id="echoid-s11263" xml:space="preserve">Si verò A C <lb/>
<anchor type="note" xlink:label="note-0353-01a" xlink:href="note-0353-01"/>
maior eſt, quàm A F eſſet I L maior, quàm I K: </s>
  <s xml:id="echoid-s11264" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11265" xml:space="preserve">I L ad I K mino-<lb/>rem proportionem habebit, quàm A C ad A F (28. </s>
  <s xml:id="echoid-s11266" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11267" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11268" xml:space="preserve">I L ma-<lb/>ior eſt quàm A C; </s>
  <s xml:id="echoid-s11269" xml:space="preserve">igitur A F minor eſt, quàm I K: </s>
  <s xml:id="echoid-s11270" xml:space="preserve">atquè ſimiliter pa-<lb/>tebit I K minorem eſſe quàm P R, &amp; </s>
  <s xml:id="echoid-s11271" xml:space="preserve">P R, quàm S Z.</s>
  <s xml:id="echoid-s11272" xml:space="preserve"/>
</p>
<div xml:id="echoid-div956" type="float" level="2" n="1">
  <figure xlink:label="fig-0352-01" xlink:href="fig-0352-01a">
    <image file="0352-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0352-01"/>
  </figure>
<note position="left" xlink:label="note-0352-01" xlink:href="note-0352-01a" xml:space="preserve">ex 38 <lb/>lib. 5.</note>
<note position="right" xlink:label="note-0353-01" xlink:href="note-0353-01a" xml:space="preserve">21. huins.</note>
</div>
</div>
<div xml:id="echoid-div958" type="section" level="1" n="301">
<head xml:id="echoid-head374" xml:space="preserve">PROPOSITIO XXXIV.</head>
<p>
  <s xml:id="echoid-s11273" xml:space="preserve">D Einde ſit A C minor, quàm A F, dummodò minor non ſit dimi-<lb/>dio eius: </s>
  <s xml:id="echoid-s11274" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11275" xml:space="preserve">ſecentur duæ præſectæ A H, C G, quæ erunt æqua-<lb/>les; </s>
  <s xml:id="echoid-s11276" xml:space="preserve">pariterque A G, C H interceptæ æquales; </s>
  <s xml:id="echoid-s11277" xml:space="preserve">ponaturque linea γ æqua-<lb/>lis ſummæ G E, G A. </s>
  <s xml:id="echoid-s11278" xml:space="preserve">Et quia A G non eſt maior duplo A H, &amp; </s>
  <s xml:id="echoid-s11279" xml:space="preserve">γ maior <lb/>
<anchor type="figure" xlink:label="fig-0353-01a" xlink:href="fig-0353-01"/>
eſt duplo A G, erit γ in A H maius, quàm quadratũ A G; </s>
  <s xml:id="echoid-s11280" xml:space="preserve">igitur γ in A <lb/>E ad γ in A H, nempe E A ad A H minorem proportionẽ habebit, quã <lb/>γ in A E ad quadratum A G; </s>
  <s xml:id="echoid-s11281" xml:space="preserve">ideoquè E H ad H A, nẽpe E H in H A ad <lb/>quadratum A H minorẽ proportionẽ habebit, quàm γ, ſeu eidem æqules <lb/>E G, G A in A E, cum quadrato A G (quæ ſunt æqualia quadrato G E) <lb/>ad quadratum A G; </s>
  <s xml:id="echoid-s11282" xml:space="preserve">ergo E H in H A ad quadratum E G, ſeu (vt <lb/>oſtenſum eſt in 15. </s>
  <s xml:id="echoid-s11283" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11284" xml:space="preserve">quadratum A C ad quadratum I K minorem <lb/>proportionem habebit, quàm quadratum A H ad quadratũ A G, ſeu quã <lb/>quadratum A C ad quadratum A F. </s>
  <s xml:id="echoid-s11285" xml:space="preserve">Igitur A C ad I K minorem pro-<lb/>portionem habet, quàm ad A F; </s>
  <s xml:id="echoid-s11286" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11287" xml:space="preserve">propterea A F minor eſt quàm I K.</s>
  <s xml:id="echoid-s11288" xml:space="preserve">
<pb o="316" file="0354" n="355" rhead="Apollonij Pergæi"/>
Simili modo oſtendetur quod I K minor ſit, quam P R: </s>
  <s xml:id="echoid-s11289" xml:space="preserve">etenim ſi pona-<lb/>tur linea f æqualis ſummæ M G, G E: </s>
  <s xml:id="echoid-s11290" xml:space="preserve">cum G E non ſit maior duplo E <lb/>H, &amp; </s>
  <s xml:id="echoid-s11291" xml:space="preserve">f maior ſit duplo G E; </s>
  <s xml:id="echoid-s11292" xml:space="preserve">igitur f in E H maius eſt quadrato G E. <lb/></s>
  <s xml:id="echoid-s11293" xml:space="preserve">Poſtea oſtendetur (quemadmodum antea dictum eſt) quod M H ad H <lb/>E, nempe M H in H A ad E H in H A minorem proportionem habet <lb/>
<anchor type="figure" xlink:label="fig-0354-01a" xlink:href="fig-0354-01"/>
quàm quadratum M G ad quadratum G E; </s>
  <s xml:id="echoid-s11294" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11295" xml:space="preserve">permutando M H in H A <lb/>ad quadratum M G, ſeu quadratum A C ad quadratum P R(15. </s>
  <s xml:id="echoid-s11296" xml:space="preserve">ex 7.) <lb/></s>
  <s xml:id="echoid-s11297" xml:space="preserve">minorem proportionem habebit, quàm E H in H A ad quadratum G E, <lb/>nempe quàm quadratum A C ad quadratum I K: </s>
  <s xml:id="echoid-s11298" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11299" xml:space="preserve">propterea A C ad <lb/>P R minorem proportionem habebit, quàm ad I K; </s>
  <s xml:id="echoid-s11300" xml:space="preserve">ideoquè I K minor <lb/>eſt, quàm P R: </s>
  <s xml:id="echoid-s11301" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11302" xml:space="preserve">pariter P R minor, quàm S Z.</s>
  <s xml:id="echoid-s11303" xml:space="preserve"/>
</p>
<div xml:id="echoid-div958" type="float" level="2" n="1">
  <figure xlink:label="fig-0353-01" xlink:href="fig-0353-01a">
    <image file="0353-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0353-01"/>
  </figure>
  <figure xlink:label="fig-0354-01" xlink:href="fig-0354-01a">
    <image file="0354-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0354-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div960" type="section" level="1" n="302">
<head xml:id="echoid-head375" xml:space="preserve">PROPOSITIO XXXV. &amp; <lb/>XXXVI.</head>
<p>
  <s xml:id="echoid-s11304" xml:space="preserve">S It poſtea A C minor dimidio A F; </s>
  <s xml:id="echoid-s11305" xml:space="preserve">erit A G maior duplo A H, &amp; </s>
  <s xml:id="echoid-s11306" xml:space="preserve"><lb/>ideo H G maior eſt, quàm H A: </s>
  <s xml:id="echoid-s11307" xml:space="preserve">ponatur iam H M æqualis H G, <lb/>ducaturque ad axim perpendicularis N M ; </s>
  <s xml:id="echoid-s11308" xml:space="preserve">iungaturque N C, &amp; </s>
  <s xml:id="echoid-s11309" xml:space="preserve">educa-<lb/>tur diameter P Q parallela N C. </s>
  <s xml:id="echoid-s11310" xml:space="preserve">Et quia M H medietas eſt ipſius M G, <lb/>erit P Q dimidium ipſius P R (6. </s>
  <s xml:id="echoid-s11311" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11312" xml:space="preserve">Inter duas diametros P Q, A C <lb/>ducatur diameter I I.</s>
  <s xml:id="echoid-s11313" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s11314" xml:space="preserve">C B ei parallela, &amp; </s>
  <s xml:id="echoid-s11315" xml:space="preserve">ad axim perpendicularis <lb/>B E. </s>
  <s xml:id="echoid-s11316" xml:space="preserve">Quoniam M H in H E minus eſt quadrato H G; </s>
  <s xml:id="echoid-s11317" xml:space="preserve">addito communi
<pb o="317" file="0355" n="356" rhead="Conicor. Lib. VII."/>
producto ex G E, &amp; </s>
  <s xml:id="echoid-s11318" xml:space="preserve">G H in E H, erit M H in H E cum E G, atquè <lb/>G H in H E, nempe ſumma M G, G E, quæ eſt æqualis ipſi f in E H <lb/>minus erit, quàm quadratum H G cum aggregato E G, G H in E H, <lb/>quæ ſunt æqualia quadrato G E; </s>
  <s xml:id="echoid-s11319" xml:space="preserve">igitur f in E H minus eſt quadrato E <lb/>G. </s>
  <s xml:id="echoid-s11320" xml:space="preserve">Poſtea vti prius dictum eſt oſtendetur, quod quadratum A C ad <lb/>quadratum P R maiorem proportionem habet, quàm ad quadratum I K: <lb/></s>
  <s xml:id="echoid-s11321" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11322" xml:space="preserve">propterea P R minor eſt, quàm I K. </s>
  <s xml:id="echoid-s11323" xml:space="preserve">Non aliter oſtendetur quod I K <lb/>minor ſit, quàm A F. </s>
  <s xml:id="echoid-s11324" xml:space="preserve">Ponatur poſtea diameter S T extra locum inter <lb/>P Q, A C compræhenſum, ducaturque C X ei parallela, &amp; </s>
  <s xml:id="echoid-s11325" xml:space="preserve">ad axim <lb/>perpendicularis X V. </s>
  <s xml:id="echoid-s11326" xml:space="preserve">Igitur V H M maius erit quàm quadratum H G, <lb/>
<anchor type="figure" xlink:label="fig-0355-01a" xlink:href="fig-0355-01"/>
&amp; </s>
  <s xml:id="echoid-s11327" xml:space="preserve">eodem modo procedendo, tandem oſtendetur quod quadratum A C ad <lb/>quadratum S Z minorem proportionem habet, quàm ad quadratum P <lb/>R, &amp; </s>
  <s xml:id="echoid-s11328" xml:space="preserve">ideo P R minor erit quàm S Z. </s>
  <s xml:id="echoid-s11329" xml:space="preserve">Non ſecus oſtendetur quod S Z <lb/>minor eſt erecto cuiuslibet inclinati cadentis ad partem S T extra illam. <lb/></s>
  <s xml:id="echoid-s11330" xml:space="preserve">Itaque demonſtratum eſt, quod P R minor ſit erecto cuiuslibet diametri <lb/>ſectionis cadentis ad vtraſque partes ipſius P Q verſus A, &amp; </s>
  <s xml:id="echoid-s11331" xml:space="preserve">X, &amp; </s>
  <s xml:id="echoid-s11332" xml:space="preserve">ere-<lb/>cti proximiores diametro P Q minores ſunt remotioribus. </s>
  <s xml:id="echoid-s11333" xml:space="preserve">Et hoc erat <lb/>propoſitum.</s>
  <s xml:id="echoid-s11334" xml:space="preserve"/>
</p>
<div xml:id="echoid-div960" type="float" level="2" n="1">
  <figure xlink:label="fig-0355-01" xlink:href="fig-0355-01a">
    <image file="0355-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0355-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div962" type="section" level="1" n="303">
<head xml:id="echoid-head376" xml:space="preserve">In Sectionem VI.</head>
<p style="it">
  <s xml:id="echoid-s11335" xml:space="preserve">IN Expoſitione ſequentium Propoſitionum difficultas, quæ à nimia prolixitate <lb/>oritur, ineuitabilis eſt, niſi Methodus in textu ſeruata aliquantisper relin-<lb/>quatur: </s>
  <s xml:id="echoid-s11336" xml:space="preserve">propterea non nulla lemmata præmittam, ex quibus ſemel demonſtra-<lb/>tis caſus omnes ſequentium propoſitionum facillime, &amp; </s>
  <s xml:id="echoid-s11337" xml:space="preserve">breuiſſime deducnntur.</s>
  <s xml:id="echoid-s11338" xml:space="preserve"/>
</p>
<pb o="318" file="0356" n="357" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div963" type="section" level="1" n="304">
<head xml:id="echoid-head377" xml:space="preserve">LEMMA II.</head>
<p style="it">
  <s xml:id="echoid-s11339" xml:space="preserve">SI recta linea H G producatur in A &amp; </s>
  <s xml:id="echoid-s11340" xml:space="preserve">E, ita vt A H, pariter-<lb/>que E H, non maior ſit H G: </s>
  <s xml:id="echoid-s11341" xml:space="preserve">Dico rectangulum ex A G E <lb/>ſumma inæqualium ſegmentorum in E H intermediam ſectionem, mi-<lb/>nus eſſe quadrato ex ſegmento intermedio minore E G.</s>
  <s xml:id="echoid-s11342" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11343" xml:space="preserve">Fiat H M æqualis H G, &amp; </s>
  <s xml:id="echoid-s11344" xml:space="preserve">quia A <lb/>E æqualis, aut minor eſt, quàm M E; <lb/></s>
  <s xml:id="echoid-s11345" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0356-01a" xlink:href="fig-0356-01"/>
&amp; </s>
  <s xml:id="echoid-s11346" xml:space="preserve">E G maior, quàm E H, ergo A E <lb/>ad M E minorem proportionem babet, <lb/>quàm E G ad E H, &amp; </s>
  <s xml:id="echoid-s11347" xml:space="preserve">permutando <lb/>A E ad E G minorem proportionem <lb/>habebit, quàm M E ad E H, &amp; </s>
  <s xml:id="echoid-s11348" xml:space="preserve">cõ-<lb/>ponendo A G ad G E minorem proportionem habebit, quàm M H, ſeu ei æqua-<lb/>lis G H aà H E, &amp; </s>
  <s xml:id="echoid-s11349" xml:space="preserve">iterum componendo A G E ad G E minorem proportionem <lb/>habebit, quàm G E ad E H: </s>
  <s xml:id="echoid-s11350" xml:space="preserve">quare rectangulum ex ſumma A G E in H E <lb/>minus erit quadrato ex intermedia G E, vt propoſitum fuerat.</s>
  <s xml:id="echoid-s11351" xml:space="preserve"/>
</p>
<div xml:id="echoid-div963" type="float" level="2" n="1">
  <figure xlink:label="fig-0356-01" xlink:href="fig-0356-01a">
    <image file="0356-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0356-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div965" type="section" level="1" n="305">
<head xml:id="echoid-head378" xml:space="preserve">LEMMA III.</head>
<p style="it">
  <s xml:id="echoid-s11352" xml:space="preserve">IIſdem poſitis ſint A H, <lb/>&amp; </s>
  <s xml:id="echoid-s11353" xml:space="preserve">E H non minores <lb/>
<anchor type="figure" xlink:label="fig-0356-02a" xlink:href="fig-0356-02"/>
quàm G H, vel H M: <lb/></s>
  <s xml:id="echoid-s11354" xml:space="preserve">Dico rectangulum ex A G <lb/>E in E H maius eſſe quadrato ex E G.</s>
  <s xml:id="echoid-s11355" xml:space="preserve"/>
</p>
<div xml:id="echoid-div965" type="float" level="2" n="1">
  <figure xlink:label="fig-0356-02" xlink:href="fig-0356-02a">
    <image file="0356-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0356-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11356" xml:space="preserve">Quia A G maior eſt quàm E G, &amp; </s>
  <s xml:id="echoid-s11357" xml:space="preserve">G H non maior ipſa H E, ergo A G ad <lb/>G E maiorem proportionem habet, quàm G H ad H E, &amp; </s>
  <s xml:id="echoid-s11358" xml:space="preserve">componendo A G E <lb/>ad E G maiorem proportionem habebit, quàm G E ad E H, &amp; </s>
  <s xml:id="echoid-s11359" xml:space="preserve">ideo rectangu-<lb/>lum ex A G E in E H maius erit quadrato ex G E.</s>
  <s xml:id="echoid-s11360" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div967" type="section" level="1" n="306">
<head xml:id="echoid-head379" xml:space="preserve">LEMMA IV.</head>
<p style="it">
  <s xml:id="echoid-s11361" xml:space="preserve">IIſdem poſitis ſit A H ma-<lb/>ior, ſed E H minor ea-<lb/>dem M H ſemiſſe totius M <lb/>
<anchor type="figure" xlink:label="fig-0356-03a" xlink:href="fig-0356-03"/>
G: </s>
  <s xml:id="echoid-s11362" xml:space="preserve">Dico quod ſi proportio ip-<lb/>ſius A G ad G E fuerit eadem <lb/>rationi G H ad H E, erit
<pb o="319" file="0357" n="358" rhead="Conicor. Lib. VII."/>
rectangulum ſub A G E in E H æquale quadrato ex G E, &amp; </s>
  <s xml:id="echoid-s11363" xml:space="preserve">ſi pro-<lb/>portio illa maior fuerit, erit quoque rectangulum maius quadrato; </s>
  <s xml:id="echoid-s11364" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11365" xml:space="preserve"><lb/>ſi illa proportio minor fuerit, Rectangulum quadrato miuus erit.</s>
  <s xml:id="echoid-s11366" xml:space="preserve"/>
</p>
<div xml:id="echoid-div967" type="float" level="2" n="1">
  <figure xlink:label="fig-0356-03" xlink:href="fig-0356-03a">
    <image file="0356-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0356-03"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11367" xml:space="preserve">Et primo, quia A G ad G E ponitur vt G H ad H E; </s>
  <s xml:id="echoid-s11368" xml:space="preserve">componendo A G E <lb/>ad G E, erit vt G E ad E H, &amp; </s>
  <s xml:id="echoid-s11369" xml:space="preserve">rectangulum ſub extremis contentum, ni-<lb/>mirum ſub A G E in E H, æquale erit quadrato ex intermedia G E.</s>
  <s xml:id="echoid-s11370" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11371" xml:space="preserve">Secundo, ſi A G ad G E maiorem proportionem habuerit, quàm G H ad H <lb/>E, componendo A G E ad G E maiorem proportionem habebit, quàm G E ad <lb/>E H, &amp; </s>
  <s xml:id="echoid-s11372" xml:space="preserve">ideo Rectangulum ſub A G E in E H maius erit quadrato ex G E. <lb/></s>
  <s xml:id="echoid-s11373" xml:space="preserve">pari ratione ſi A G ad G E minorem proportionem habuerit, quàm G H ad <lb/>H E, oſtendetur Rectangulum ex A G E in E H minus quadrato G E.</s>
  <s xml:id="echoid-s11374" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div969" type="section" level="1" n="307">
<head xml:id="echoid-head380" xml:space="preserve">LEMMA V.</head>
<p style="it">
  <s xml:id="echoid-s11375" xml:space="preserve">IN hyperbola, cuius axis C A, &amp; </s>
  <s xml:id="echoid-s11376" xml:space="preserve">erectus A F, præſecta H A, in-<lb/>tercepta G A, diameter L I, cuius erectus I K, latus C E, &amp; </s>
  <s xml:id="echoid-s11377" xml:space="preserve"><lb/>
<anchor type="figure" xlink:label="fig-0357-01a" xlink:href="fig-0357-01"/>
diameter Q P, cuius erectus P R, latus C O: </s>
  <s xml:id="echoid-s11378" xml:space="preserve">Dico quod erectus P <lb/>R ab ipſo erecto I K, vel ab A F atque rectangulum ſub O G E in <lb/>G H ab ipſo quadrato G E, vel rectangulum ex O G A in A H ab <lb/>ipſo quadrato G A, vna deficiunt, vel vna æqualia ſunt, aut vna <lb/>excedunt.</s>
  <s xml:id="echoid-s11379" xml:space="preserve"/>
</p>
<div xml:id="echoid-div969" type="float" level="2" n="1">
  <figure xlink:label="fig-0357-01" xlink:href="fig-0357-01a">
    <image file="0357-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0357-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11380" xml:space="preserve">Et primo ponatur rectangulum ſub O G E in E H æquale quadrato E G, er-<lb/>go idem rectangulum ſub O G E in E O ad rectangulum ſub E G O in E H, <lb/>ſeu E O ad E H eandem proportionem habet, quàm ad quadratum G E, &amp; </s>
  <s xml:id="echoid-s11381" xml:space="preserve"><lb/>propterea E O ad E H erit vt rectangulum ſub E G O in E O ad quadratum
<pb o="320" file="0358" n="359" rhead="Apollonij Pergæi"/>
G E, &amp; </s>
  <s xml:id="echoid-s11382" xml:space="preserve">componendo O H ad E H, ſeu rectangulum O H A ad rectangulum <lb/>E H A, erit vt rectangulum ſub G E, &amp; </s>
  <s xml:id="echoid-s11383" xml:space="preserve">G O in O E vna cum quadrato E <lb/>G, ſeu vt quadratum ex O G ad quadratum ex G E, &amp; </s>
  <s xml:id="echoid-s11384" xml:space="preserve">permutando rectangu-<lb/>lum A H O ad quadratum O G, erit vt rectangulum E H A ad quadratum G <lb/>E, ſed vt rectangulum O H A ad quadratum O G, ita eſt quadratum A C ad <lb/>
<anchor type="note" xlink:label="note-0358-01a" xlink:href="note-0358-01"/>
quadratum P K, &amp; </s>
  <s xml:id="echoid-s11385" xml:space="preserve">vt rectangulum E H A ad quadratnm ex G E, ſeu vt <lb/>quadratum A C ad quadratum A F, vel ex I K; </s>
  <s xml:id="echoid-s11386" xml:space="preserve">quapropter idem quadratum <lb/>A C ad quadratum ex P K, atque ad quadratum ex A F vel I K eandem pro-<lb/>portionem habet, &amp; </s>
  <s xml:id="echoid-s11387" xml:space="preserve">ideo quadrata ipſa æqualia ſunt, &amp; </s>
  <s xml:id="echoid-s11388" xml:space="preserve">eorum latera P K; </s>
  <s xml:id="echoid-s11389" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11390" xml:space="preserve"><lb/>A F, vel I K pariter æqualia erunt.</s>
  <s xml:id="echoid-s11391" xml:space="preserve"/>
</p>
<div xml:id="echoid-div970" type="float" level="2" n="2">
<note position="left" xlink:label="note-0358-01" xlink:href="note-0358-01a" xml:space="preserve">15. huius. <lb/>ex Def. &amp; <lb/>15. huius.</note>
</div>
  <figure>
    <image file="0358-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0358-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s11392" xml:space="preserve">Eodem modo quando rectangulum ſub O G E in E H maius eſt quadrato G <lb/>E, tunc quidem idem rectangulum, cuius altitudo O G E, baſis vero O E, ad <lb/>rectangulum, cuius altitudo O G E, baſis verò E H, ſeu O E ad E H, mino-<lb/>rem proportionem habebit, quàm ad quadratum E G, &amp; </s>
  <s xml:id="echoid-s11393" xml:space="preserve">componendo, atque <lb/>permutando, vt prius factum eſt, habebit rectangulum O H A ad quadratum <lb/>O G, ſiue quadratum A C ad quadratum P K minorem proportionem, quàm <lb/>rectangulum E H A ad quadratum G E, ſeu quàm quadratum A C ad qua-<lb/>
<anchor type="note" xlink:label="note-0358-02a" xlink:href="note-0358-02"/>
dratum A F, vel I K, &amp; </s>
  <s xml:id="echoid-s11394" xml:space="preserve">propterea P K maior erit, quàm A F, vel I K.</s>
  <s xml:id="echoid-s11395" xml:space="preserve"/>
</p>
<div xml:id="echoid-div971" type="float" level="2" n="3">
<note position="left" xlink:label="note-0358-02" xlink:href="note-0358-02a" xml:space="preserve">15. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11396" xml:space="preserve">Quando verò rectangulum ſub E G O in E H minus eſt quadrato E G, tunc <lb/>quidem oſtendetur eodem progreſſu quadratum P K minus eſſe quadrato A F, <lb/>vel I K, quod erat propoſitum.</s>
  <s xml:id="echoid-s11397" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div973" type="section" level="1" n="308">
<head xml:id="echoid-head381" xml:space="preserve">Notæ in Propof. XXXIII. &amp; XXXIV.</head>
<p style="it">
  <s xml:id="echoid-s11398" xml:space="preserve">QVoniam ex hypoteſi C A minor non eſt medietate ipſius A F, eſtque A H <lb/>ad A G, vt C A, ad A F, ergo A H maior, aut æqualis eſt medietati <lb/>
<anchor type="note" xlink:label="note-0358-03a" xlink:href="note-0358-03"/>
ipſius A G, &amp; </s>
  <s xml:id="echoid-s11399" xml:space="preserve">ideo A H maior, aut æqualis eſt reſiduo H G, quare
<pb o="321" file="0359" n="360" rhead="Conicor. Lib. VII."/>
E H, atque eius portio A H non <lb/>
<anchor type="figure" xlink:label="fig-0359-01a" xlink:href="fig-0359-01"/>
<anchor type="note" xlink:label="note-0359-01a" xlink:href="note-0359-01"/>
minores ſunt eadem G H; </s>
  <s xml:id="echoid-s11400" xml:space="preserve">ergo re-<lb/>ctangulum ſub E G A in A H ma-<lb/>ius erit quadrato A G, atque I K <lb/>maior erit quàm A F.</s>
  <s xml:id="echoid-s11401" xml:space="preserve"/>
</p>
<div xml:id="echoid-div973" type="float" level="2" n="1">
<note position="left" xlink:label="note-0358-03" xlink:href="note-0358-03a" xml:space="preserve">Def. 2. <lb/>huius.</note>
  <figure xlink:label="fig-0359-01" xlink:href="fig-0359-01a">
    <image file="0359-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0359-01"/>
  </figure>
<note position="right" xlink:label="note-0359-01" xlink:href="note-0359-01a" xml:space="preserve">Lem. 3. <lb/>huius.</note>
</div>
<note position="right" xml:space="preserve">Lem. 5.</note>
<p style="it">
  <s xml:id="echoid-s11402" xml:space="preserve">Simili modo, quia tam M H, <lb/>quam E H excedunt ipſam G H, <lb/>
<anchor type="note" xlink:label="note-0359-03a" xlink:href="note-0359-03"/>
erit rectangulum ſub M G E in E <lb/>H maius quadrato A G, atque P <lb/>
<anchor type="note" xlink:label="note-0359-04a" xlink:href="note-0359-04"/>
R maior, quam I K.</s>
  <s xml:id="echoid-s11403" xml:space="preserve"/>
</p>
<div xml:id="echoid-div974" type="float" level="2" n="2">
<note position="right" xlink:label="note-0359-03" xlink:href="note-0359-03a" xml:space="preserve">Lem. 3. <lb/>huius.</note>
<note position="right" xlink:label="note-0359-04" xlink:href="note-0359-04a" xml:space="preserve">Lem. 5. <lb/>huius.</note>
</div>
  <figure>
    <image file="0359-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0359-02"/>
  </figure>
</div>
<div xml:id="echoid-div976" type="section" level="1" n="309">
<head xml:id="echoid-head382" xml:space="preserve">Notæ in Propoſit. XXXV.</head>
<p style="it">
  <s xml:id="echoid-s11404" xml:space="preserve">QVia ex hypoteſi axis A C minor eſt ſemi A F, erit A H minor medieta-<lb/>te ipſius A G, &amp; </s>
  <s xml:id="echoid-s11405" xml:space="preserve">ideo A H minor erit H G: </s>
  <s xml:id="echoid-s11406" xml:space="preserve">fiat igitur M H æqualis <lb/>H G, &amp; </s>
  <s xml:id="echoid-s11407" xml:space="preserve">per M (quæ intra ſuctionẽ cadet) ad axim ordinatim applicata
<pb o="322" file="0360" n="361" rhead="Apollonij Pergæi"/>
ducatur N n occurrens ſectioni in N, &amp; </s>
  <s xml:id="echoid-s11408" xml:space="preserve">n, à quibus iungantur N C, n C, &amp; </s>
  <s xml:id="echoid-s11409" xml:space="preserve"><lb/>eis æquidiſtantes diametri P Q, &amp; </s>
  <s xml:id="echoid-s11410" xml:space="preserve">p q extendantur, quarũ erecta P R, &amp; </s>
  <s xml:id="echoid-s11411" xml:space="preserve">p r. <lb/></s>
  <s xml:id="echoid-s11412" xml:space="preserve">Oſtendendum eſt P Q ſubduplam eſſe ipſius P R, atq; </s>
  <s xml:id="echoid-s11413" xml:space="preserve">P R, &amp; </s>
  <s xml:id="echoid-s11414" xml:space="preserve">p r æquales eße <lb/>inter ſe, &amp; </s>
  <s xml:id="echoid-s11415" xml:space="preserve">minima eſſe erectorum quarumlibet Diametrorum eiuſdem ſectio-<lb/>nis. </s>
  <s xml:id="echoid-s11416" xml:space="preserve">Quoniam vt H M ad M G ita eſt P Q ad P R, &amp; </s>
  <s xml:id="echoid-s11417" xml:space="preserve">p q ad p r, erat au-<lb/>
<anchor type="note" xlink:label="note-0360-01a" xlink:href="note-0360-01"/>
tem H M ſubdupla ipſius M G, ergo Diameter P Q ſubdupla eſt erecti eius P R, <lb/>pariterque p q ſubdupla eſt ipſius p r: </s>
  <s xml:id="echoid-s11418" xml:space="preserve">atque Diametri P Q, &amp; </s>
  <s xml:id="echoid-s11419" xml:space="preserve">p q æquales <lb/>ſunt inter ſe, cum æque recedant ab axi A C, atque earum commune latus ſit <lb/>C M. </s>
  <s xml:id="echoid-s11420" xml:space="preserve">Poſtea quia tam E H, quàm M H maiores non ſunt eadem H M, vel <lb/>G H, ergo rectangulum ſub M G E in E H minus eſt quadrato E G, &amp; </s>
  <s xml:id="echoid-s11421" xml:space="preserve">ex <lb/>
<anchor type="note" xlink:label="note-0360-02a" xlink:href="note-0360-02"/>
lem. </s>
  <s xml:id="echoid-s11422" xml:space="preserve">5. </s>
  <s xml:id="echoid-s11423" xml:space="preserve">P R minor eſt I K.</s>
  <s xml:id="echoid-s11424" xml:space="preserve"/>
</p>
<div xml:id="echoid-div976" type="float" level="2" n="1">
<note position="left" xlink:label="note-0360-01" xlink:href="note-0360-01a" xml:space="preserve">Prop. 6. <lb/>huius.</note>
<note position="left" xlink:label="note-0360-02" xlink:href="note-0360-02a" xml:space="preserve">Lem. 2. <lb/>huius.</note>
</div>
  <figure>
    <image file="0360-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0360-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s11425" xml:space="preserve">Similiter quia tam E H, quàm A H minor eſt eadem H M, ergo rectan-<lb/>
<anchor type="note" xlink:label="note-0360-03a" xlink:href="note-0360-03"/>
gulum ſub E G A in A H minus eſt quadrato A G, &amp; </s>
  <s xml:id="echoid-s11426" xml:space="preserve">I K minor erit, quàm <lb/>A F. </s>
  <s xml:id="echoid-s11427" xml:space="preserve">tandem, quia tam V H, quàm M H non eſt minor eadem G H, ergo re-<lb/>ctangulum V G M in M H maius erit quadrato G M, &amp; </s>
  <s xml:id="echoid-s11428" xml:space="preserve">ideo S Z maior erit, <lb/>
<anchor type="note" xlink:label="note-0360-04a" xlink:href="note-0360-04"/>
quàm P R, &amp; </s>
  <s xml:id="echoid-s11429" xml:space="preserve">ſic vlterius: </s>
  <s xml:id="echoid-s11430" xml:space="preserve">quare P R minimum eſt laterum rectorum quarum-<lb/>
<anchor type="note" xlink:label="note-0360-05a" xlink:href="note-0360-05"/>
libet Diametrorum eiuſdem hyperboles.</s>
  <s xml:id="echoid-s11431" xml:space="preserve"/>
</p>
<div xml:id="echoid-div977" type="float" level="2" n="2">
<note position="left" xlink:label="note-0360-03" xlink:href="note-0360-03a" xml:space="preserve">Lem. 2. <lb/>&amp; 5. hui.</note>
<note position="left" xlink:label="note-0360-04" xlink:href="note-0360-04a" xml:space="preserve">Lem. 3.</note>
<note position="left" xlink:label="note-0360-05" xlink:href="note-0360-05a" xml:space="preserve">Lem 5.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11432" xml:space="preserve">In hyperbole latus rectum alicuius Diametri reperire, quod æquale <lb/>
<anchor type="note" xlink:label="note-0360-06a" xlink:href="note-0360-06"/>
ſit lateri recto axis; </s>
  <s xml:id="echoid-s11433" xml:space="preserve">ſed oportet, vt axis tranſuerſus A C minor ſit me-<lb/>dietate eius erecti A F.</s>
  <s xml:id="echoid-s11434" xml:space="preserve"/>
</p>
<div xml:id="echoid-div978" type="float" level="2" n="3">
<note position="left" xlink:label="note-0360-06" xlink:href="note-0360-06a" xml:space="preserve">PROP. 1. <lb/>Addit</note>
</div>
<p style="it">
  <s xml:id="echoid-s11435" xml:space="preserve">Reperiatur Diameter P Q, quæ ſubdupla ſit eius erecti P R, ſitque C M la-<lb/>
<anchor type="note" xlink:label="note-0360-07a" xlink:href="note-0360-07"/>
tus, &amp; </s>
  <s xml:id="echoid-s11436" xml:space="preserve">fiat e G ad G A, vt M H ad H A, &amp; </s>
  <s xml:id="echoid-s11437" xml:space="preserve">ducatur ordinatim applicata <lb/>ad axim e d, coniungaturque recta d C, &amp; </s>
  <s xml:id="echoid-s11438" xml:space="preserve">extendatur diameter a b paralle-<lb/>la ipſi d C, cuius latus rectum ſit a c. </s>
  <s xml:id="echoid-s11439" xml:space="preserve">Dico a c æquale eſſe A F: </s>
  <s xml:id="echoid-s11440" xml:space="preserve">quia e G <lb/>ad G A facta fuit vt M H, ſiue G H ad H A, ergo rectangulum ſub e G A in <lb/>
<anchor type="note" xlink:label="note-0360-08a" xlink:href="note-0360-08"/>
A H æquale eſt quadrato G A, ideoque erectum a c æquale erit erecto A F, <lb/>
<anchor type="note" xlink:label="note-0360-09a" xlink:href="note-0360-09"/>
quod erat propoſitum.</s>
  <s xml:id="echoid-s11441" xml:space="preserve"/>
</p>
<div xml:id="echoid-div979" type="float" level="2" n="4">
<note position="left" xlink:label="note-0360-07" xlink:href="note-0360-07a" xml:space="preserve">ex 35. hu.</note>
<note position="left" xlink:label="note-0360-08" xlink:href="note-0360-08a" xml:space="preserve">Lem. 4. <lb/>huius.</note>
<note position="left" xlink:label="note-0360-09" xlink:href="note-0360-09a" xml:space="preserve">Lem. 5. <lb/>huius.</note>
</div>
<pb o="323" file="0361" n="362" rhead="Conicor. Lib. VII."/>
<p style="it">
  <s xml:id="echoid-s11442" xml:space="preserve">Dato latere recto I K diametri hyperboles I L reperire latus rectum <lb/>
<anchor type="note" xlink:label="note-0361-01a" xlink:href="note-0361-01"/>
alterius Diametri, quod æquale ſit lateri recto I K: </s>
  <s xml:id="echoid-s11443" xml:space="preserve">oportet autem, <lb/>vt Diameter I L cadat inter axim, @ aliam Diametrum, quæ ſub-<lb/>dupla ſit ſui erecti.</s>
  <s xml:id="echoid-s11444" xml:space="preserve"/>
</p>
<div xml:id="echoid-div980" type="float" level="2" n="5">
<note position="right" xlink:label="note-0361-01" xlink:href="note-0361-01a" xml:space="preserve">PROP. 2. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11445" xml:space="preserve">Reperiatur Diameter Q P, quæ ſubdupla ſit ſui erecti P R, eiuſque latus <lb/>
<anchor type="note" xlink:label="note-0361-02a" xlink:href="note-0361-02"/>
ſit M C; </s>
  <s xml:id="echoid-s11446" xml:space="preserve">ergo ex hypotheſi I L cadet inter axim A C, &amp; </s>
  <s xml:id="echoid-s11447" xml:space="preserve">Diametrum P Q, <lb/>&amp; </s>
  <s xml:id="echoid-s11448" xml:space="preserve">propterea terminus E lateris C E cadet inter A, &amp; </s>
  <s xml:id="echoid-s11449" xml:space="preserve">M, igitur reperiri po-<lb/>terit V G, quæ ad G E eandem proportionem habeat, quàm maior M H ad <lb/>minorem H E, &amp; </s>
  <s xml:id="echoid-s11450" xml:space="preserve">vt prius, lateris C V ducatur diameter S T, cuius latus <lb/>rectum S Z: </s>
  <s xml:id="echoid-s11451" xml:space="preserve">dico S Z æquale eße I K: </s>
  <s xml:id="echoid-s11452" xml:space="preserve">quia V G ad G E eſt, vt M H, ſeu <lb/>
<anchor type="note" xlink:label="note-0361-03a" xlink:href="note-0361-03"/>
G H ad H E, ergo rectangulum ſub V G E in E H æquale eſt quadrato G E, <lb/>ideoque S Z æquale I K; </s>
  <s xml:id="echoid-s11453" xml:space="preserve">quod erat propoſitum.</s>
  <s xml:id="echoid-s11454" xml:space="preserve"/>
</p>
<div xml:id="echoid-div981" type="float" level="2" n="6">
<note position="right" xlink:label="note-0361-02" xlink:href="note-0361-02a" xml:space="preserve">ex 35. hu.</note>
<note position="right" xlink:label="note-0361-03" xlink:href="note-0361-03a" xml:space="preserve">Lem. 4. <lb/>huius. <lb/>Lem. 5. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11455" xml:space="preserve">Deducitur ex prima propoſitione additarum quod in aliqua hyperbola reperi-<lb/>ri poßunt tria diametrorum latera recta æqualia inter ſe; </s>
  <s xml:id="echoid-s11456" xml:space="preserve">ſi nimirum in hyper-<lb/>bola, cuius axis C A minor ſit medietate eius lateris recti, reperiantur vtrin-<lb/>que duæ diametri b a, quarum latera recta a c æqualia ſint ipſi A F; </s>
  <s xml:id="echoid-s11457" xml:space="preserve">tunc <lb/>quidem tria illa latera recta æqualia erunt inter ſe: </s>
  <s xml:id="echoid-s11458" xml:space="preserve">reliqua verò latera recta <lb/>diametrorum cadentium inter A, &amp; </s>
  <s xml:id="echoid-s11459" xml:space="preserve">a maiora erunt latere recto A F; </s>
  <s xml:id="echoid-s11460" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11461" xml:space="preserve">la-<lb/>tera recta diametrorum cadentium vltra punctum a ad partes B maiora ſunt <lb/>
<anchor type="note" xlink:label="note-0361-04a" xlink:href="note-0361-04"/>
latere recto a c, propterea quod magis recedunt ab omnium minimo latere re-<lb/>cto P R.</s>
  <s xml:id="echoid-s11462" xml:space="preserve"/>
</p>
<div xml:id="echoid-div982" type="float" level="2" n="7">
<note position="right" xlink:label="note-0361-04" xlink:href="note-0361-04a" xml:space="preserve">ex 35. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11463" xml:space="preserve">Simili modo in eadem hyperbola reperiri poßunt quatuor diametrorum latera <lb/>recta æqualia inter ſe, ſi nimirum ex ſecunda propoſitione additarum dato la-<lb/>tere recto I K diametri I L reperiatur æquale latus rectum S Z alterius diame-<lb/>tri S T, &amp; </s>
  <s xml:id="echoid-s11464" xml:space="preserve">ex altera parte axis ducantur duæ aliæ diametri æquè ab axi re-<lb/>motæ ac illæ, erunt quatuor recta latera earum æqualia inter ſe, &amp; </s>
  <s xml:id="echoid-s11465" xml:space="preserve">maiora <lb/>quolibet latere recto diametri cadentis inter I, &amp; </s>
  <s xml:id="echoid-s11466" xml:space="preserve">S ad vtraſque partes axis: <lb/></s>
  <s xml:id="echoid-s11467" xml:space="preserve">minora verò erunt quolibet latere recto diametri cadentis vltra punctum I ad <lb/>partes verticis A, vel infra puncta S ad partes a, vt deducitur ex 35. </s>
  <s xml:id="echoid-s11468" xml:space="preserve">huius.</s>
  <s xml:id="echoid-s11469" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div984" type="section" level="1" n="310">
<head xml:id="echoid-head383" xml:space="preserve">SECTIO SEPTIMA</head>
<head xml:id="echoid-head384" xml:space="preserve">Continens Propoſit. XXXVIII. XXXIX. <lb/>&amp; XXXX.</head>
<head xml:id="echoid-head385" xml:space="preserve">PROPOSITIO XXXVIII.</head>
<p>
  <s xml:id="echoid-s11470" xml:space="preserve">IN hyperbole axis inclinatus ſi non fuerit minortriente erecti <lb/>ipſius, erunt duo latera figuræ axis minora, quàm duo late-<lb/>ra figuræ cuiuslibet inclinatæ coniugatarum, quæ in eadem ſe-<lb/>ctione conſiſtunt, &amp; </s>
  <s xml:id="echoid-s11471" xml:space="preserve">duo latera figuræ inclinati proximioris axi <lb/>minora ſunt, quàm duo latera figuræ remotioris inclinati.</s>
  <s xml:id="echoid-s11472" xml:space="preserve"/>
</p>
<pb o="324" file="0362" n="363" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s11473" xml:space="preserve">Si verò fuerit axis minor parte tertia ſui erecti aſſignari po-<lb/>terunt ad vtraſque eius partes duo æquales diametri, quarum <lb/>quælibet pars tertia ſit ſui erecti, atque duo latera figuræ eiuſ-<lb/>dem minora ſunt duobus lateribus figuræ cuiuslibet alterius dia-<lb/>metri ad vtraſque eius partes in eadem ſectione cadentis: </s>
  <s xml:id="echoid-s11474" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11475" xml:space="preserve">duo <lb/>latera figuræ diametri ei propinquiores minora ſunt duobus la-<lb/>teribus figuræ remotioris.</s>
  <s xml:id="echoid-s11476" xml:space="preserve"/>
</p>
  <figure>
    <image file="0362-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0362-01"/>
  </figure>
<p>
  <s xml:id="echoid-s11477" xml:space="preserve">In eadem figura ſupponatur prius hyperboles axis A C non minor ſuo <lb/>erecto, erit P Q maior quàm A C, &amp; </s>
  <s xml:id="echoid-s11478" xml:space="preserve">S T maior quàm P Q: </s>
  <s xml:id="echoid-s11479" xml:space="preserve">ideoquè <lb/>erectus ipſius A C minor erit erecto ipſius P Q (33. </s>
  <s xml:id="echoid-s11480" xml:space="preserve">ex 7.)</s>
  <s xml:id="echoid-s11481" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s11482" xml:space="preserve">erectus <lb/>ipſius P Q minor eſt erecto ipſius S T; </s>
  <s xml:id="echoid-s11483" xml:space="preserve">igitur duo latera figuræ A C mi-<lb/>nora ſunt, quàm duo latera figuræ P Q, &amp; </s>
  <s xml:id="echoid-s11484" xml:space="preserve">duo latera figuræ P Q mino-<lb/>ra, quàm duo latera figuræ S T.</s>
  <s xml:id="echoid-s11485" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div985" type="section" level="1" n="311">
<head xml:id="echoid-head386" xml:space="preserve">PROPOSITIO XXXIX.</head>
<p>
  <s xml:id="echoid-s11486" xml:space="preserve">D Einde ſit A C minor quàm A F, ſed non ſit minor tertia parte <lb/>
<anchor type="note" xlink:label="note-0362-01a" xlink:href="note-0362-01"/>
eius; </s>
  <s xml:id="echoid-s11487" xml:space="preserve">igitur A H non erit minor tertia parte ipſius H C: </s>
  <s xml:id="echoid-s11488" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11489" xml:space="preserve">pro-<lb/>pterea non eſt minor quadrante ipſius A C; </s>
  <s xml:id="echoid-s11490" xml:space="preserve">ideoque C A in A H non. <lb/></s>
  <s xml:id="echoid-s11491" xml:space="preserve">eſt minus quarta parte quadrati A C; </s>
  <s xml:id="echoid-s11492" xml:space="preserve">quare C A in A M quater ſum-<lb/>ptum ad C A in A H quater, nempe M A ad A H non habet maiorem <lb/>proportionem, quàm quadruplum ipſius A C in A M ad quadratum A <lb/>C. </s>
  <s xml:id="echoid-s11493" xml:space="preserve">Et ponamus M m æqualem M A, componendo M H, ad H A, <lb/>nempe M H in H A ad quadratum H A non habebit maiorem propor-
<pb o="325" file="0363" n="364" rhead="Conicor. Lib. VII."/>
tionem, quàm C M in M A quater ſumptum vna cum quadrato C A, <lb/>nempe quàm quadratum C m ad quadratum A C; </s>
  <s xml:id="echoid-s11494" xml:space="preserve">ideoque M H in H <lb/>A ad quadrarum H A minorem proportionem habet quàm quadratum. <lb/></s>
  <s xml:id="echoid-s11495" xml:space="preserve">C m ad quadratum A C. </s>
  <s xml:id="echoid-s11496" xml:space="preserve">Et permutando M H in H A ad quadratum. </s>
  <s xml:id="echoid-s11497" xml:space="preserve"><lb/>C m, ſeu ad quadratum ex ſumma ipſarum G M; </s>
  <s xml:id="echoid-s11498" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11499" xml:space="preserve">M H, ad quod <lb/>habet eandem proportionem quàm quadratum C A ad quadratum ſum-<lb/>mæ P Q, &amp; </s>
  <s xml:id="echoid-s11500" xml:space="preserve">P R (17. </s>
  <s xml:id="echoid-s11501" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11502" xml:space="preserve">habebit minorem proportionem, quàm <lb/>quadratum A H ad quadratum A C, ſeu quàm quadratum A C ad qua-<lb/>dratum ſummæ ipſarum A C, &amp; </s>
  <s xml:id="echoid-s11503" xml:space="preserve">A F; </s>
  <s xml:id="echoid-s11504" xml:space="preserve">igitur ſumma ipſarum A C, &amp; </s>
  <s xml:id="echoid-s11505" xml:space="preserve"><lb/>A F minor eſt quàm ſumma ipſarum P Q, &amp; </s>
  <s xml:id="echoid-s11506" xml:space="preserve">P R. </s>
  <s xml:id="echoid-s11507" xml:space="preserve">Et quia M H maior <lb/>eſt quarta parte ſummæ ipſarum M G, &amp; </s>
  <s xml:id="echoid-s11508" xml:space="preserve">M H; </s>
  <s xml:id="echoid-s11509" xml:space="preserve">ergo quadruplum C m <lb/>in M H maius eſt quadrato C m, &amp; </s>
  <s xml:id="echoid-s11510" xml:space="preserve">ponatur V u æqualis A V; </s>
  <s xml:id="echoid-s11511" xml:space="preserve">igitur <lb/>quadruplũ V M in C m ad quadruplum M H in C m, ſcilicet V M ad <lb/>M H minorem proportionem habebit, quàm quadruplum V M in C m <lb/>ad quadratum C m: </s>
  <s xml:id="echoid-s11512" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11513" xml:space="preserve">componendo V H ad H M, nempe V H in H <lb/>A ad M H in H A minorem proportionem habebit, quàm V M in C m <lb/>quater ſumptum, vel u m in m C bis ſumptum cum quadrato C m (eo <lb/>quod u m dupla eſt ipſius V M quæ omnia ſimul ad idem quadratum C <lb/>m minorem proportionem habet, quàm quadratum C u. </s>
  <s xml:id="echoid-s11514" xml:space="preserve">Ergo V H in <lb/>H A ad quadratum C u, ſcilicet quadratum A C ad quadratum ſummæ <lb/>ipſarum S T, &amp; </s>
  <s xml:id="echoid-s11515" xml:space="preserve">S Z (17. </s>
  <s xml:id="echoid-s11516" xml:space="preserve">ex 7. </s>
  <s xml:id="echoid-s11517" xml:space="preserve">) minorem proportionem habet quàm <lb/>M H in H A ad quadratum C m, ſeu qnàm quadratum A C ad quadra-<lb/>tum ſummæ ipſarum P Q, P R (17. </s>
  <s xml:id="echoid-s11518" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11519" xml:space="preserve">quapropter P Q, &amp; </s>
  <s xml:id="echoid-s11520" xml:space="preserve">P R ſi-<lb/>mul ſumptæ minores ſunt, quàm S T, &amp; </s>
  <s xml:id="echoid-s11521" xml:space="preserve">S Z ſimul ſumptæ.</s>
  <s xml:id="echoid-s11522" xml:space="preserve"/>
</p>
<div xml:id="echoid-div985" type="float" level="2" n="1">
<note position="right" xlink:label="note-0362-01" xlink:href="note-0362-01a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div987" type="section" level="1" n="312">
<head xml:id="echoid-head387" xml:space="preserve">PROPOSITIO XXXX.</head>
<p>
  <s xml:id="echoid-s11523" xml:space="preserve">S It A C minor triente ipſius A F, erit A H minor dimidio <lb/>ipſius H G, &amp; </s>
  <s xml:id="echoid-s11524" xml:space="preserve">ponatur M H æqualis dimidio H G, &amp; </s>
  <s xml:id="echoid-s11525" xml:space="preserve">du-<lb/>
<anchor type="figure" xlink:label="fig-0363-01a" xlink:href="fig-0363-01"/>
<pb o="326" file="0364" n="365" rhead="Apollonij Pergæi"/>
camus perpendicularem, &amp; </s>
  <s xml:id="echoid-s11526" xml:space="preserve">diametrum. </s>
  <s xml:id="echoid-s11527" xml:space="preserve">Dico, quod P Q æ-<lb/>qualis eſt trienti ipſius P R.</s>
  <s xml:id="echoid-s11528" xml:space="preserve"/>
</p>
<div xml:id="echoid-div987" type="float" level="2" n="1">
  <figure xlink:label="fig-0363-01" xlink:href="fig-0363-01a">
    <image file="0363-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0363-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s11529" xml:space="preserve">Educamus inter P Q, A C diametrum I L, &amp; </s>
  <s xml:id="echoid-s11530" xml:space="preserve">educamus C B ei æ-<lb/>quidiſtantem, &amp; </s>
  <s xml:id="echoid-s11531" xml:space="preserve">perpendicularem B E, &amp; </s>
  <s xml:id="echoid-s11532" xml:space="preserve">ſecemus E l æqualem E A <lb/>erit ſumma ipſarum G E, &amp; </s>
  <s xml:id="echoid-s11533" xml:space="preserve">E H æqualis C l; </s>
  <s xml:id="echoid-s11534" xml:space="preserve">eſtque H E minor quam <lb/>M H, quæ quarta pars eſt ipſius C m; </s>
  <s xml:id="echoid-s11535" xml:space="preserve">ergo ſumma ipſarum M G, H E <lb/>in M H quater ſumptum minus eſt quadrato C m: </s>
  <s xml:id="echoid-s11536" xml:space="preserve">auferatur communi-<lb/>ter M G, H E in M E quater ſumptum remanebit quadruplum ſummæ <lb/>M G, H E in H E minus quàm quadratum C l (quia M G, H E ſimul <lb/>ſumptæ, nempe M C vna cum A E in M E quater ſumptum æquale eſt <lb/>quadrato l m; </s>
  <s xml:id="echoid-s11537" xml:space="preserve">quod eſt duplum M E, &amp; </s>
  <s xml:id="echoid-s11538" xml:space="preserve">aggregatum C E, A E, nem-<lb/>pe C l in l m bis ſumptum ) igitur aggregatum M G, &amp; </s>
  <s xml:id="echoid-s11539" xml:space="preserve">H E in M E <lb/>quater ſumptum ad aggregatum M G, H E in H E quater ſumptum, nẽ-<lb/>pe G E ad H E maiorem proportionem habebit, quàm ad quadratum l <lb/>C. </s>
  <s xml:id="echoid-s11540" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11541" xml:space="preserve">componendo M H ad H E, ſeu M H in H A ad E H in H A <lb/>habebit maiorem proportionem, quàm M G, H E in M E quater ſum-<lb/>ptum cum quadrato l C (quæ æqualia ſunt quadrato C m) ad quadra-<lb/>tum l C: </s>
  <s xml:id="echoid-s11542" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11543" xml:space="preserve">permutando erit M H in H A ad quadratum C m, nempe <lb/>ad quadratum ſummæ ipſarum M G, &amp; </s>
  <s xml:id="echoid-s11544" xml:space="preserve">M H, ſeu quadratum A C ad <lb/>quadratum ſummæ ipſarum P Q, P R (17. </s>
  <s xml:id="echoid-s11545" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11546" xml:space="preserve">maiorem proportio-<lb/>nem habebit, quàm E H in H A ad quadratum l C (quod eſt æquale <lb/>quadrato ſummæ ipſarum G E, E H ) quod erit vt quadratum A C ad <lb/>quadratum aggregati ipſarum I L, I K: </s>
  <s xml:id="echoid-s11547" xml:space="preserve">quapropter A C ad duo latera <lb/>figuræ P Q maiorem proportionem habet, quàm ad duo latera figuræ I <lb/>L. </s>
  <s xml:id="echoid-s11548" xml:space="preserve">Et propterea duo latera figuræ P Q minora ſunt, quàm duo latera <lb/>
<anchor type="figure" xlink:label="fig-0364-01a" xlink:href="fig-0364-01"/>
<pb o="327" file="0365" n="366" rhead="Conicor. Lib. VII."/>
figuræ I L. </s>
  <s xml:id="echoid-s11549" xml:space="preserve">Simili modo eſtendetur, quod duo latera figuræ I L minora <lb/>ſunt, quàm duo latera figuræ A C.</s>
  <s xml:id="echoid-s11550" xml:space="preserve"/>
</p>
<div xml:id="echoid-div988" type="float" level="2" n="2">
  <figure xlink:label="fig-0364-01" xlink:href="fig-0364-01a">
    <image file="0364-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0364-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s11551" xml:space="preserve">Educamus poſtea C X extra ſegmentum A N; </s>
  <s xml:id="echoid-s11552" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11553" xml:space="preserve">educamus diametrũ <lb/>S T ei parallelam, &amp; </s>
  <s xml:id="echoid-s11554" xml:space="preserve">ad axim perpendicularem X V, erit aggregatum. <lb/></s>
  <s xml:id="echoid-s11555" xml:space="preserve">G V, M H in M H quater ſumptum maius quàm quadratum C m; </s>
  <s xml:id="echoid-s11556" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11557" xml:space="preserve">ad-<lb/>damus communiter aggregatuin M H, G V in M H quater ſumptum; </s>
  <s xml:id="echoid-s11558" xml:space="preserve"><lb/>oſtendetur vt antea, quod duo latera ſiguræ S T maiora ſunt, quàm duo <lb/>latera figuræ P Q.</s>
  <s xml:id="echoid-s11559" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11560" xml:space="preserve">Oſtendetur quoque in reliquis diametris cadentibus ad vtraſque par-<lb/>tes ipſius P Q in eadem ſectione, quod duo latera ſiguræ diametri ipſi <lb/>P Q proximioris minora ſunt, quàm duo latera figuræ remotioris.</s>
  <s xml:id="echoid-s11561" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div990" type="section" level="1" n="313">
<head xml:id="echoid-head388" xml:space="preserve">In Sectionem VII. Propoſit: XXXVIII. <lb/>XXXIX. &amp; XXXX. <lb/>LEMMA VI.</head>
<p style="it">
  <s xml:id="echoid-s11562" xml:space="preserve">S I recta linea H G bifariam ſecta in D producatur vtcumque ad A, <lb/>@ E, ita vt D H non maior ſit quàm H E, vel H A, @ <lb/>E D maior ſit, quàm D A: </s>
  <s xml:id="echoid-s11563" xml:space="preserve">dico rectangulum ſub E D A in H A <lb/>maius eſſe quadrato D A.</s>
  <s xml:id="echoid-s11564" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11565" xml:space="preserve">Quia E D maior ad minorem <lb/>
<anchor type="figure" xlink:label="fig-0365-01a" xlink:href="fig-0365-01"/>
D A habet maiorem proportionem, <lb/>quàm D H non maior ipſa H A, <lb/>ad H A, ergo componendo E D A <lb/>ad D A maiorem proportionem ha-<lb/>bet, quàm D A ad A H, &amp; </s>
  <s xml:id="echoid-s11566" xml:space="preserve">pro-<lb/>pterea rectangulum ſub extremis contentum, ſcilicet ſub E D A in A H, ma-<lb/>ins eſt quadrato D A.</s>
  <s xml:id="echoid-s11567" xml:space="preserve"/>
</p>
<div xml:id="echoid-div990" type="float" level="2" n="1">
  <figure xlink:label="fig-0365-01" xlink:href="fig-0365-01a">
    <image file="0365-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0365-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div992" type="section" level="1" n="314">
<head xml:id="echoid-head389" xml:space="preserve">LEMMA VII.</head>
<p style="it">
  <s xml:id="echoid-s11568" xml:space="preserve">I Iſdem poſitis, ſi D H <lb/>
<anchor type="figure" xlink:label="fig-0365-02a" xlink:href="fig-0365-02"/>
non minor fuerit quàm <lb/>H A, vel H E, ſitque <lb/>H E maior, quàm H A: <lb/></s>
  <s xml:id="echoid-s11569" xml:space="preserve">dico rectangulam ſub E D <lb/>A in A H minus eſſe quadrato D A.</s>
  <s xml:id="echoid-s11570" xml:space="preserve"/>
</p>
<div xml:id="echoid-div992" type="float" level="2" n="1">
  <figure xlink:label="fig-0365-02" xlink:href="fig-0365-02a">
    <image file="0365-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0365-02"/>
  </figure>
</div>
<pb o="328" file="0366" n="367" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s11571" xml:space="preserve">Fiat H M æqualis maiori H D, erit E A differentia minimæ H A, &amp; </s>
  <s xml:id="echoid-s11572" xml:space="preserve">in-<lb/>termediæ H E minor, quàm M A, quæ eſt differentia maximæ M H, &amp; </s>
  <s xml:id="echoid-s11573" xml:space="preserve">mi-<lb/>nimæ H A, &amp; </s>
  <s xml:id="echoid-s11574" xml:space="preserve">A D maior eſt quàm A H, ergo E A ad M A minorem pro-<lb/>portionem habet, quàm D A ad A H, &amp; </s>
  <s xml:id="echoid-s11575" xml:space="preserve">permutando E A ad A D habebit mi-<lb/>norem proportionem, quàm M A ad A H, &amp; </s>
  <s xml:id="echoid-s11576" xml:space="preserve">componendo E D ad D A mino-<lb/>proportionem habebit, quàm M H, ſiue D H ad A H, &amp; </s>
  <s xml:id="echoid-s11577" xml:space="preserve">iterum componendo <lb/>E D A ad D A minorem proportionem habebit, quàm eadem D A ad A H, &amp; </s>
  <s xml:id="echoid-s11578" xml:space="preserve"><lb/>propterea rectangulum ſub E D A in A H minus erit quadrato D A.</s>
  <s xml:id="echoid-s11579" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div994" type="section" level="1" n="315">
<head xml:id="echoid-head390" xml:space="preserve">LEMMA VIII.</head>
<p style="it">
  <s xml:id="echoid-s11580" xml:space="preserve">I Iſdem poſitis ſi D H maior fuerit, quàm A H ſed minor quàm E <lb/>H, fueritque proportio E A ad A D eadem proportioni M A ad A <lb/>H, dico rectangulum ſub E D A in A H æquale eſſe quadrato D A: <lb/></s>
  <s xml:id="echoid-s11581" xml:space="preserve">ſi verò proportio illa maior fuerit, vel minor rectangulum ſimiliter qua-<lb/>drato maius, vel minus erit.</s>
  <s xml:id="echoid-s11582" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11583" xml:space="preserve">Quia E A ad A D po-<lb/>
<anchor type="figure" xlink:label="fig-0366-01a" xlink:href="fig-0366-01"/>
nitur vt M A ad A H, <lb/>componendo E D ad D A, <lb/>erit vt M H, ſeu D H ad <lb/>H A, &amp; </s>
  <s xml:id="echoid-s11584" xml:space="preserve">iterum componen-<lb/>do E D A ad D A, erit vt <lb/>D A ad A H, &amp; </s>
  <s xml:id="echoid-s11585" xml:space="preserve">propterea rectangulum ſub E D A in A H æquale erit qua-<lb/>drato D A.</s>
  <s xml:id="echoid-s11586" xml:space="preserve"/>
</p>
<div xml:id="echoid-div994" type="float" level="2" n="1">
  <figure xlink:label="fig-0366-01" xlink:href="fig-0366-01a">
    <image file="0366-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0366-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11587" xml:space="preserve">Quando verò E A ad A D maiorem proportionem habet, quàm M A ad A <lb/>H, t@nc bis componendo E D A ad D A maiorem proportionem habebit, quàm <lb/>D A ad A H, &amp; </s>
  <s xml:id="echoid-s11588" xml:space="preserve">propterea rectangulum ſub extremis; </s>
  <s xml:id="echoid-s11589" xml:space="preserve">ſcilicet ſub E D A in <lb/>A H maius erit quadrato intermediæ D A: </s>
  <s xml:id="echoid-s11590" xml:space="preserve">non ſecus quando E A ad A D <lb/>minorem peoportionem habet, quàm M A ad A H, oſtendetur rectangulum ſub <lb/>E D A in A H minus quadrato ex D A.</s>
  <s xml:id="echoid-s11591" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div996" type="section" level="1" n="316">
<head xml:id="echoid-head391" xml:space="preserve">LEMMA IX.</head>
<p style="it">
  <s xml:id="echoid-s11592" xml:space="preserve">I N hyperbola, cuius axis A C, erectus A F, præſecta H A, in-<lb/>tercepta G A, centrum D, diameter I L, eiuſque erectus I K, <lb/>@ C E ſit latus eiuſdem, ſitque diameter Q P, cuius erectus P R, <lb/>@ latus L O: </s>
  <s xml:id="echoid-s11593" xml:space="preserve">dico quod rectangulum ſub O D E in E H ab ipſo qua-<lb/>drato D E, atque Q P R ſumma laterum figuræ Diametri P Q ab L <lb/>I K ſumma laterum figuræ I L, vel ab ipſa C A F ſumma laterum <lb/>figuræ axis, vna deficiunt, vel vna æqualia ſunt, aut vna excedunt.</s>
  <s xml:id="echoid-s11594" xml:space="preserve"/>
</p>
<pb o="329" file="0367" n="368" rhead="Conicor. Lib. VII."/>
  <figure>
    <image file="0367-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0367-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s11595" xml:space="preserve">Et primo rectangulum ſub O D E in E H æquale ſit quadrato D E, ergo <lb/>ad hæc duo ſpatia æqualia eandem proportionem habebit idem rectangulum ſub <lb/>E D O in O E, ſed vt rectangulum ſub E D O in O E ad rectangulum ſub E <lb/>D O in E H, ita eſt O E ad E H, (propterea quod æquales altitudines ha-<lb/>bent), igitur vt O E ad E H, ita eſt rectangulum ſub E D O in O E ad <lb/>quadratum D E, &amp; </s>
  <s xml:id="echoid-s11596" xml:space="preserve">componendo O H ad E H, ſiue rectangulum O H A ad <lb/>rectangulum E H A eandem proportionẽ habebit, quàm rectangulum ſub E D <lb/>O in O E vna cum quadrato D E, ſeu quàm quadratum D O ad quadratum <lb/>D E, vel potius vt quadratum ex dupla D O ad quadratum ex dupla D E, <lb/>nempe vt quadratum ex G O H ad quadratum ex G E H, quare permutando <lb/>rectangulum O H A ad quadratum ex G O H eandem proportionem habebit, <lb/>quàm rectangulum ex E H A ad quadratum ex G E H, ſeu vt quadratum ex <lb/>
<anchor type="note" xlink:label="note-0367-01a" xlink:href="note-0367-01"/>
A C ad quadratum ex C A F, vel ex L I K; </s>
  <s xml:id="echoid-s11597" xml:space="preserve">ſed vt rectangulum A H O ad <lb/>quadratum ex G O H, ita eſt quadratum ex A C ad quadratum ex Q P R: <lb/></s>
  <s xml:id="echoid-s11598" xml:space="preserve">quare idem quadratum A C eandem proportionem habet ad quadratum ex Q P <lb/>R, quàm ad quadratum ex C A F, vel ex I R L, &amp; </s>
  <s xml:id="echoid-s11599" xml:space="preserve">propterea quadrata ipſa <lb/>æqualia ſunt, &amp; </s>
  <s xml:id="echoid-s11600" xml:space="preserve">ſumma laterum Q P R æqualis eſt ſummæ laterum C A F, <lb/>vel I L K.</s>
  <s xml:id="echoid-s11601" xml:space="preserve"/>
</p>
<div xml:id="echoid-div996" type="float" level="2" n="1">
<note position="right" xlink:label="note-0367-01" xlink:href="note-0367-01a" xml:space="preserve">Prop. 16. <lb/>huius. <lb/>Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11602" xml:space="preserve">Secundo ſit rectangulũ ſub E D O in E H maius quadrato D E, tunc quidem <lb/>idem rectangulum ſub E D O in O E ad rectangulum ſub O D E in E H mi-<lb/>norem proportionẽ habebit, quàm ad quadratum ex D E, ſeu O E ad E H mi-<lb/>norem proportionem habebit, quàm ad quadratum ex D E; </s>
  <s xml:id="echoid-s11603" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11604" xml:space="preserve">componendo <lb/>ſumpta eadem altitudine H A, quadruplicando poſtrema quadrata, &amp; </s>
  <s xml:id="echoid-s11605" xml:space="preserve">permu-<lb/>tando, &amp; </s>
  <s xml:id="echoid-s11606" xml:space="preserve">ex 16. </s>
  <s xml:id="echoid-s11607" xml:space="preserve">huius, idem quadratum A C ad quadratum ex Q P R mi-<lb/>norem proportionem habebit, quàm ad quadratum ex C A F, vel ex L I K, <lb/>&amp; </s>
  <s xml:id="echoid-s11608" xml:space="preserve">propterea ſumma Q P R maior erit, quàm C A F, ſeu quàm L I K.</s>
  <s xml:id="echoid-s11609" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11610" xml:space="preserve">Tertio ſit rectangulum ſub E D O in E H minus quadrato D E, patet quod <lb/>idem rectangulum ſub E D O in O E ad rectangulum ſub E D O in E H, ſeu <lb/>O E ad E H maiorem proportionem habet, quàm ad quadratum D E, &amp; </s>
  <s xml:id="echoid-s11611" xml:space="preserve">com-<lb/>ponendo ductis prioribus terminis in A H, quadruplicando poſtrema quadrata,
<pb o="330" file="0368" n="369" rhead="Apollonij Pergæi"/>
permutando vt prius, idem quadratum A C ad quadratum ex Q P R, maiorem <lb/>proportionem habebit, quàm ad quadratum ex C A F, ſeu ex L I K, &amp; </s>
  <s xml:id="echoid-s11612" xml:space="preserve">pro-<lb/>pterea ſumma Q P R minor erit, quàm C A F, vel L I K, quæ erat oſten-<lb/>denda.</s>
  <s xml:id="echoid-s11613" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div998" type="section" level="1" n="317">
<head xml:id="echoid-head392" xml:space="preserve">Notæ in Propoſit. XXXVIII. XXXIX.</head>
<p style="it">
  <s xml:id="echoid-s11614" xml:space="preserve">QVia axis C A minor non eſt triente eius erecti A F, eſtq; </s>
  <s xml:id="echoid-s11615" xml:space="preserve">H A ad A G vt C A <lb/>ad A F, ergo H A æqualis, aut maior eſt parte tertia ipſius A G; </s>
  <s xml:id="echoid-s11616" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11617" xml:space="preserve">A H <lb/>æqualis, aut maior erit, quàm ſemiſſis ipſius H G differentiæ illa-<lb/>rum, eſtque G H ſecta bifariam in D, ergo H A æqualis, aut maior erit, <lb/>
<anchor type="figure" xlink:label="fig-0368-01a" xlink:href="fig-0368-01"/>
quàm D H, eſtque H E maior quàm H A, ergo pariter H E maior eſt, quàm <lb/>
<anchor type="note" xlink:label="note-0368-01a" xlink:href="note-0368-01"/>
D H, quare rectangulum ſub E D A in A H maius erit quadrato D A, atque <lb/>ſumma laterum figuræ L I K maior, quàm ſumma laterum figuræ axis C A F.</s>
  <s xml:id="echoid-s11618" xml:space="preserve"/>
</p>
<div xml:id="echoid-div998" type="float" level="2" n="1">
  <figure xlink:label="fig-0368-01" xlink:href="fig-0368-01a">
    <image file="0368-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0368-01"/>
  </figure>
<note position="left" xlink:label="note-0368-01" xlink:href="note-0368-01a" xml:space="preserve">Lem. 6.</note>
</div>
<note position="left" xml:space="preserve">Lem. 9.</note>
<p style="it">
  <s xml:id="echoid-s11619" xml:space="preserve">Similiter quia H M maior eſt, quàm H E, erit quoque H M maior, quàm <lb/>D H, &amp; </s>
  <s xml:id="echoid-s11620" xml:space="preserve">propterea ex lemma 6. </s>
  <s xml:id="echoid-s11621" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11622" xml:space="preserve">9. </s>
  <s xml:id="echoid-s11623" xml:space="preserve">ſumma Q P R maior erit, quàm ſum-<lb/>ma L I K.</s>
  <s xml:id="echoid-s11624" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1000" type="section" level="1" n="318">
<head xml:id="echoid-head393" xml:space="preserve">Notæ in Propoſit. XXXX.</head>
<p style="it">
  <s xml:id="echoid-s11625" xml:space="preserve">QVia C A minor eſt triente ipſius A F, eſtque H A ad A G vt C A ad A <lb/>F, ergo H A minor eſt tertia parte ipſius A G, &amp; </s>
  <s xml:id="echoid-s11626" xml:space="preserve">minor ſemiſſe diffe-
<pb o="331" file="0369" n="370" rhead="Conicor. Lib. VII."/>
rentiæ H G, &amp; </s>
  <s xml:id="echoid-s11627" xml:space="preserve">ideo H A minor erit, quàm H D: </s>
  <s xml:id="echoid-s11628" xml:space="preserve">ſecari ergo poterit H M <lb/>æqualis D H, quæmaior erit, quàm A H, ducaturq; </s>
  <s xml:id="echoid-s11629" xml:space="preserve">per M ad axim ordinatim <lb/>applicata N M n occurrens ſectioni in punctis N n, à quibus iungãtur C N, &amp; </s>
  <s xml:id="echoid-s11630" xml:space="preserve">C <lb/>
<anchor type="figure" xlink:label="fig-0369-01a" xlink:href="fig-0369-01"/>
n, ĳſdemque æquidiſtantes ducantur duæ diametri P Q, &amp; </s>
  <s xml:id="echoid-s11631" xml:space="preserve">p q, quarum la-<lb/>tera recta P R, &amp; </s>
  <s xml:id="echoid-s11632" xml:space="preserve">p r. </s>
  <s xml:id="echoid-s11633" xml:space="preserve">Oſtenàendum eſt P Q ſut erecti P R, atque p q ſui <lb/>erecti p r ſubtriplam eße, ſed duo figuræ latera P Q, P R æqualia eſſe alterius <lb/>figuræ lateribus p q, p r, &amp; </s>
  <s xml:id="echoid-s11634" xml:space="preserve">inſuper P Q, P R minima eſſe laterum figuræ <lb/>cuiuſlibet alterius diametri eiuſdem ſectionis, &amp; </s>
  <s xml:id="echoid-s11635" xml:space="preserve">latera figurarum minimis pro-<lb/>ximiora, eſſe minora lateribus figurarum remotiorum.</s>
  <s xml:id="echoid-s11636" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1000" type="float" level="2" n="1">
  <figure xlink:label="fig-0369-01" xlink:href="fig-0369-01a">
    <image file="0369-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0369-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11637" xml:space="preserve">Quia H M ad M G eandem proportionem habet quàm P Q ad P R, vel p <lb/>
<anchor type="note" xlink:label="note-0369-01a" xlink:href="note-0369-01"/>
q ad p r, eſtque H M ſubtripla ipſius M G (cum M H facta ſit æqualis H D) <lb/>ergo P Q ipſius P R, pariterque p q ipſius p r ſubtripla eſt: </s>
  <s xml:id="echoid-s11638" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11639" xml:space="preserve">ſunt latera <lb/>figuræ Q P R æqualia lateribus q p r alterius figuræ, cum diametri Q P, &amp; </s>
  <s xml:id="echoid-s11640" xml:space="preserve"><lb/>q p æquè recedant ab axi, &amp; </s>
  <s xml:id="echoid-s11641" xml:space="preserve">habeant latus commune C M.</s>
  <s xml:id="echoid-s11642" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1001" type="float" level="2" n="2">
<note position="right" xlink:label="note-0369-01" xlink:href="note-0369-01a" xml:space="preserve">Prop. 6. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11643" xml:space="preserve">Quod verò ſumma laterum figuræ Q P R minima ſit reliquarum ſummarũ <lb/>laterum figuræ cuiuſlibet diametri ſic oſtendetur.</s>
  <s xml:id="echoid-s11644" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11645" xml:space="preserve">Quia A H, &amp; </s>
  <s xml:id="echoid-s11646" xml:space="preserve">E H minora ſunt, quàm H M, ſiue D H, ergo rectangulum <lb/>
<anchor type="note" xlink:label="note-0369-02a" xlink:href="note-0369-02"/>
ſub E D A in A H minus eſt quadrato D A, &amp; </s>
  <s xml:id="echoid-s11647" xml:space="preserve">ſumma L I K minor eſt ſum-<lb/>
<anchor type="note" xlink:label="note-0369-03a" xlink:href="note-0369-03"/>
ma C A F.</s>
  <s xml:id="echoid-s11648" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1002" type="float" level="2" n="3">
<note position="right" xlink:label="note-0369-02" xlink:href="note-0369-02a" xml:space="preserve">Lem. 7.</note>
<note position="right" xlink:label="note-0369-03" xlink:href="note-0369-03a" xml:space="preserve">Lem. 9.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11649" xml:space="preserve">Pariter quia M H æqualia eſt H D, &amp; </s>
  <s xml:id="echoid-s11650" xml:space="preserve">H E minor eadem, ergo ambo non <lb/>
<anchor type="note" xlink:label="note-0369-04a" xlink:href="note-0369-04"/>
<anchor type="note" xlink:label="note-0369-05a" xlink:href="note-0369-05"/>
erunt maiores eadem D H, ergo rectangulum ſub M D E in E H minus erit <lb/>quadrato D E, atque ſumma Q P R minor erit, quàm L I K.</s>
  <s xml:id="echoid-s11651" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1003" type="float" level="2" n="4">
<note position="right" xlink:label="note-0369-04" xlink:href="note-0369-04a" xml:space="preserve">Lem. 7.</note>
<note position="right" xlink:label="note-0369-05" xlink:href="note-0369-05a" xml:space="preserve">Lem. 9.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11652" xml:space="preserve">Rurſus quia V H maior, eſt quàm M H, ſeu quàm D H, erunt illæ non, <lb/>
<anchor type="note" xlink:label="note-0369-06a" xlink:href="note-0369-06"/>
<anchor type="note" xlink:label="note-0369-07a" xlink:href="note-0369-07"/>
minores eadem D H, ergo rectangulum ſub V D M in H M maius erit qua-<lb/>drato D M, atque ſumma T S Z maior erit, quàm ſumma Q P R.</s>
  <s xml:id="echoid-s11653" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1004" type="float" level="2" n="5">
<note position="right" xlink:label="note-0369-06" xlink:href="note-0369-06a" xml:space="preserve">Lem. 6.</note>
<note position="right" xlink:label="note-0369-07" xlink:href="note-0369-07a" xml:space="preserve">Lem. 9.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11654" xml:space="preserve">In hyperbola reperire diametrum, cuius figuræ latera æqualia ſint lateribus <lb/>
<anchor type="note" xlink:label="note-0369-08a" xlink:href="note-0369-08"/>
figuræ axis: </s>
  <s xml:id="echoid-s11655" xml:space="preserve">oportet autem vt axis A C minor ſit triente erecti eius. </s>
  <s xml:id="echoid-s11656" xml:space="preserve">Reperia-<lb/>tur diameter P Q ſubtripla erecti eius P R, eiuſque latus ſit C M, &amp; </s>
  <s xml:id="echoid-s11657" xml:space="preserve">fiat e <lb/>A ad A D, vt M A ad A H, &amp; </s>
  <s xml:id="echoid-s11658" xml:space="preserve">lateris C e ducatur diameter a b, cuius ere-<lb/>ctus a c. </s>
  <s xml:id="echoid-s11659" xml:space="preserve">Dico hanc eße diametrum quæſitam: </s>
  <s xml:id="echoid-s11660" xml:space="preserve">quia e A ad A D eandem pro-<lb/>portionem habet, quàm M A ad A H, erit rectangulum ſub e D A in A H
<pb o="332" file="0370" n="371" rhead="Apollonij Pergæi"/>
æquale quadrato D A, &amp; </s>
  <s xml:id="echoid-s11661" xml:space="preserve">ſumma laterum b a c æqualis erit laterum figuræ <lb/>
<anchor type="note" xlink:label="note-0370-01a" xlink:href="note-0370-01"/>
<anchor type="note" xlink:label="note-0370-02a" xlink:href="note-0370-02"/>
axis ſummæ C A F.</s>
  <s xml:id="echoid-s11662" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1005" type="float" level="2" n="6">
<note position="right" xlink:label="note-0369-08" xlink:href="note-0369-08a" xml:space="preserve">PROP. 3. <lb/>Addit. <lb/>ex 40. <lb/>huius.</note>
<note position="left" xlink:label="note-0370-01" xlink:href="note-0370-01a" xml:space="preserve">Lem. 8.</note>
<note position="left" xlink:label="note-0370-02" xlink:href="note-0370-02a" xml:space="preserve">Lem. 9.</note>
</div>
  <figure>
    <image file="0370-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0370-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s11663" xml:space="preserve">In eadem hyperbola data diametro I L reperire aliam diametrum, ita v@ <lb/>
<anchor type="note" xlink:label="note-0370-03a" xlink:href="note-0370-03"/>
eius figuræ latera æqualia ſint lateribus figuræ datæ diametri I L: </s>
  <s xml:id="echoid-s11664" xml:space="preserve">oportet au-<lb/>tem vt I L cadat inter axim, &amp; </s>
  <s xml:id="echoid-s11665" xml:space="preserve">diametrum P Q ſubtriplam eius erecti. </s>
  <s xml:id="echoid-s11666" xml:space="preserve">Sit <lb/>
<anchor type="note" xlink:label="note-0370-04a" xlink:href="note-0370-04"/>
C E latus diametri I L, &amp; </s>
  <s xml:id="echoid-s11667" xml:space="preserve">C M, ſit latus diametri P Q, &amp; </s>
  <s xml:id="echoid-s11668" xml:space="preserve">quia punctum <lb/>E cadit inter M, &amp; </s>
  <s xml:id="echoid-s11669" xml:space="preserve">A, erit H E minor, quàm H M, vel D H: </s>
  <s xml:id="echoid-s11670" xml:space="preserve">fiat V E <lb/>
<anchor type="note" xlink:label="note-0370-05a" xlink:href="note-0370-05"/>
ad E D, vt M E ad E H, ergo rectangulum ſub V D E in E H æquale erit <lb/>quadrato E D, &amp; </s>
  <s xml:id="echoid-s11671" xml:space="preserve">ex lemma 9. </s>
  <s xml:id="echoid-s11672" xml:space="preserve">ſumma laterum T S Z æqualis erit ſummæ la-<lb/>terum L I K; </s>
  <s xml:id="echoid-s11673" xml:space="preserve">quod erat propoſitum.</s>
  <s xml:id="echoid-s11674" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1006" type="float" level="2" n="7">
<note position="left" xlink:label="note-0370-03" xlink:href="note-0370-03a" xml:space="preserve">PROP. 4. <lb/>Addit.</note>
<note position="left" xlink:label="note-0370-04" xlink:href="note-0370-04a" xml:space="preserve">ex 40. <lb/>huius.</note>
<note position="left" xlink:label="note-0370-05" xlink:href="note-0370-05a" xml:space="preserve">Lem. 8.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11675" xml:space="preserve">Facile colligitur ex 3. </s>
  <s xml:id="echoid-s11676" xml:space="preserve">additarum, quod in hyperbola cuius axis ſubtripla ſit <lb/>erecti eius aſſignari poſſunt tres ſummæ laterum figurarum trium Diametrorum <lb/>quæ æquales ſint inter ſe. </s>
  <s xml:id="echoid-s11677" xml:space="preserve">Ex 4. </s>
  <s xml:id="echoid-s11678" xml:space="preserve">verò additarum in eadem Hyperbola aſſignari, <lb/>poßunt quatuor ſummæ laterum figurarum quatuor diametrorum, quæ æquales <lb/>ſint inter ſe.</s>
  <s xml:id="echoid-s11679" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11680" xml:space="preserve">Deinde ſit A C minor, quàm A F, ſed non ſit minor eius triplo, er-<lb/>go A H non erit minor triplo H C, &amp;</s>
  <s xml:id="echoid-s11681" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11682" xml:space="preserve">Textus mendoſus omnino corrigi <lb/>
<anchor type="note" xlink:label="note-0370-06a" xlink:href="note-0370-06"/>
debuit, nam ex contextu ſequenti deducitur A C non tripla minor, ſed minor <lb/>parte tertia ſupponi debere ipſius A F.</s>
  <s xml:id="echoid-s11683" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1007" type="float" level="2" n="8">
<note position="right" xlink:label="note-0370-06" xlink:href="note-0370-06a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0370-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0370-02"/>
  </figure>
<pb o="333" file="0371" n="372" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div1009" type="section" level="1" n="319">
<head xml:id="echoid-head394" xml:space="preserve">SECTIO OCTAVA</head>
<head xml:id="echoid-head395" xml:space="preserve">Continens Propoſit. XXXXIIII. XXXXV. <lb/>&amp; XXXXVI.</head>
<p>
  <s xml:id="echoid-s11684" xml:space="preserve">IN hyperbole ſi quadratum axis inclinati minus non fuerit di-<lb/>midio quadrati ex differentia ipſius, &amp; </s>
  <s xml:id="echoid-s11685" xml:space="preserve">ſui erecti, vtique <lb/>quadratum diametri figuræ eius minus eſt, quàm quadratum <lb/>diametri figuræ cuiuſcumque alterius inclinati eiuſdem ſectionis.</s>
  <s xml:id="echoid-s11686" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11687" xml:space="preserve">XXXXVI. </s>
  <s xml:id="echoid-s11688" xml:space="preserve">Si verò minus fuerit cadent ad vtraſque partes <lb/>eius duæ inter ſe æquales diametri, quarum vniuscuiuſlibet qua-<lb/>dratum æquale eſt quadrato exceſſus ſui erecti, &amp; </s>
  <s xml:id="echoid-s11689" xml:space="preserve">quadratum <lb/>diametri figuræ ipſius minus eſt quàm quadratum diametri figu-<lb/>ræ cuiuſlibet alterius inclinati ad vtraſque eius partes cadentis: <lb/></s>
  <s xml:id="echoid-s11690" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11691" xml:space="preserve">diameter figuræ inclinati proximioris illi minor eſt quàm dia-<lb/>meter figuræ inclinati remotioris.</s>
  <s xml:id="echoid-s11692" xml:space="preserve"/>
</p>
  <figure>
    <image file="0371-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0371-01"/>
  </figure>
<p>
  <s xml:id="echoid-s11693" xml:space="preserve">Iiſdem figuris manentibus ſupponatur prius A C non minor quàm A Demonſt <lb/>F; </s>
  <s xml:id="echoid-s11694" xml:space="preserve">ergo P Q non erit minor quàm P R (28. </s>
  <s xml:id="echoid-s11695" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11696" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11697" xml:space="preserve">duo quadrata A prop. </s>
  <s xml:id="echoid-s11698" xml:space="preserve">44. <lb/></s>
  <s xml:id="echoid-s11699" xml:space="preserve">C, A F nempe diameter figuræ A C minor eſt quàm diameter figuræ P
<pb o="334" file="0372" n="373" rhead="Apollonij Pergæi"/>
Q; </s>
  <s xml:id="echoid-s11700" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11701" xml:space="preserve">pariter diameter figuræ P Q minor eſt, quàm diameter figuræ S <lb/>T. </s>
  <s xml:id="echoid-s11702" xml:space="preserve">Sit iam A C minor quàm A F, &amp; </s>
  <s xml:id="echoid-s11703" xml:space="preserve">eius quadratum non minus dimi-<lb/>
<anchor type="note" xlink:label="note-0372-01a" xlink:href="note-0372-01"/>
dio quadrati exceſſus ipſius A F ſuper A C. </s>
  <s xml:id="echoid-s11704" xml:space="preserve">Et quia A C ad A F ean-<lb/>dem proportionem habet, quàm A H ad A G; </s>
  <s xml:id="echoid-s11705" xml:space="preserve">ergo duplum quadrati <lb/>A H non eſt minus quadrato H G; </s>
  <s xml:id="echoid-s11706" xml:space="preserve">ergo M H in H A bis ſumptum ma-<lb/>ius eſt quadrato H G, &amp; </s>
  <s xml:id="echoid-s11707" xml:space="preserve">addatur communiter duplum G A in A H fiet <lb/>duplum ſummæ G A, M H, vel C M in A H maius quàm duplum G A <lb/>in A H cum quadrato H G, ſeu quàm quadratum G A cum quadrato A <lb/>
<anchor type="figure" xlink:label="fig-0372-01a" xlink:href="fig-0372-01"/>
H: </s>
  <s xml:id="echoid-s11708" xml:space="preserve">quare duplum C M in M A ad duplum C M in A H, ſeu M A ad <lb/>A H minorem proportionem habet, quàm duplum C M in M A ad qua-<lb/>dratum G A vna cum quadrato A H: </s>
  <s xml:id="echoid-s11709" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11710" xml:space="preserve">componendo habebit M H ad <lb/>H A, ſeu M H in H A ad quadratum A H minorem proportionem quàm <lb/>duplum C M in M A cum duobus quadratis ipſarum G A, &amp; </s>
  <s xml:id="echoid-s11711" xml:space="preserve">A H (quæ <lb/>omnia ſimul æqualia ſunt quadrato M G cum quadrato M H) ad qua-<lb/>dratum A G cum quadrato A H: </s>
  <s xml:id="echoid-s11712" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11713" xml:space="preserve">permutando M H in H A ad qua-<lb/>dratum G M cum quadrato M H (nempe quadratum A C ad duo qua-<lb/>drata laterum figuræ P Q) ſiue ad quadratum diametri figuræ P Q (17. <lb/></s>
  <s xml:id="echoid-s11714" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s11715" xml:space="preserve">minorem proportionem habebit, quàm quadratum H A ad qua-<lb/>dratum A G cum quadrato A H, ſeu quàm quadratum A C ad quadra-<lb/>tum diametri figuræ eius; </s>
  <s xml:id="echoid-s11716" xml:space="preserve">igitur quadratum A C ad diametrum figuræ <lb/>P Q minorem proportionem habet, quàm ad diametrum figuræ A C: </s>
  <s xml:id="echoid-s11717" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11718" xml:space="preserve"><lb/>ideo diameter figuræ P Q maior erit diametro figuræ A C. </s>
  <s xml:id="echoid-s11719" xml:space="preserve">Præterea, <lb/>quia duplum quadrati M H maius eſt quadrato H G; </s>
  <s xml:id="echoid-s11720" xml:space="preserve">ergo V H in M H <lb/>bis maius erit, quàm quadratum H G: </s>
  <s xml:id="echoid-s11721" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11722" xml:space="preserve">oſtendetur (quemadmodum <lb/>diximus) quod diameter figuræ S T maior ſit quàm diameter figuræ P Q.</s>
  <s xml:id="echoid-s11723" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1009" type="float" level="2" n="1">
<note position="left" xlink:label="note-0372-01" xlink:href="note-0372-01a" xml:space="preserve">Demonſt. <lb/>prop. 45.</note>
  <figure xlink:label="fig-0372-01" xlink:href="fig-0372-01a">
    <image file="0372-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0372-01"/>
  </figure>
</div>
<pb o="335" file="0373" n="374" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div1011" type="section" level="1" n="320">
<head xml:id="echoid-head396" xml:space="preserve">PROPOSITIO XXXXVI.</head>
<p>
  <s xml:id="echoid-s11724" xml:space="preserve">SIt poſtea quadratum A C minus dimidio quadrati ex differentia ipſa-<lb/>rum C A, &amp; </s>
  <s xml:id="echoid-s11725" xml:space="preserve">A F; </s>
  <s xml:id="echoid-s11726" xml:space="preserve">erit duplum quadrati A H minus quadrato H G <lb/>&amp; </s>
  <s xml:id="echoid-s11727" xml:space="preserve">ponamus duplum quadrati M H æquale quadrato H G: </s>
  <s xml:id="echoid-s11728" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11729" xml:space="preserve">educamus <lb/>ad axim perpendicularem N M, &amp; </s>
  <s xml:id="echoid-s11730" xml:space="preserve">iungamus N C; </s>
  <s xml:id="echoid-s11731" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11732" xml:space="preserve">ducamus diame-<lb/>trum P Q parallelã ipſi N C, erit H M ad M G, vt P Q ad P R, &amp; </s>
  <s xml:id="echoid-s11733" xml:space="preserve">pro-<lb/>
<anchor type="note" xlink:label="note-0373-01a" xlink:href="note-0373-01"/>
pterea quadratum P Q dimidium erit quadrati exceſſus ipſius P R; </s>
  <s xml:id="echoid-s11734" xml:space="preserve">ergo <lb/>
<anchor type="note" xlink:label="note-0373-02a" xlink:href="note-0373-02"/>
P Q eſt vna æqualium: </s>
  <s xml:id="echoid-s11735" xml:space="preserve">ponatur inſuper inter A, &amp; </s>
  <s xml:id="echoid-s11736" xml:space="preserve">P diameter I L, &amp; </s>
  <s xml:id="echoid-s11737" xml:space="preserve"><lb/>conſtructio perficiatur, vt prius. </s>
  <s xml:id="echoid-s11738" xml:space="preserve">Et quia duplum quadrati M H æquale <lb/>eſt quadrato H G, erit duplum M H in H E minus quadrato H G, &amp; </s>
  <s xml:id="echoid-s11739" xml:space="preserve"><lb/>ponatur communiter duplum G E in E H; </s>
  <s xml:id="echoid-s11740" xml:space="preserve">igitur duplum aggregati M G <lb/>in E H minus eſt quadrato G E cum quadrato E H; </s>
  <s xml:id="echoid-s11741" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11742" xml:space="preserve">oſtendetur que-<lb/>madmodum diximus antea, quod quadratum diametri figuræ P Q mi-<lb/>nus ſit quadrato diametri figuræ I L; </s>
  <s xml:id="echoid-s11743" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11744" xml:space="preserve">quadratum diametri figuræ I L <lb/>minus ſit quadrato diametri figure A C.</s>
  <s xml:id="echoid-s11745" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1011" type="float" level="2" n="1">
<note position="right" xlink:label="note-0373-01" xlink:href="note-0373-01a" xml:space="preserve">6. huius.</note>
<note position="left" xlink:label="note-0373-02" xlink:href="note-0373-02a" xml:space="preserve">a</note>
</div>
  <figure>
    <image file="0373-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0373-01"/>
  </figure>
<p>
  <s xml:id="echoid-s11746" xml:space="preserve">Deindè ducatur diameter inclinata S T extra ſegmentum A P, &amp; </s>
  <s xml:id="echoid-s11747" xml:space="preserve">C X ei <lb/>parallela, &amp; </s>
  <s xml:id="echoid-s11748" xml:space="preserve">ad axim perpendicularis X V: </s>
  <s xml:id="echoid-s11749" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11750" xml:space="preserve">quia duplum quadrati M H <lb/>æquale eſt quadrato H G erit duplum V H in H M maius quadrato H <lb/>G: </s>
  <s xml:id="echoid-s11751" xml:space="preserve">ponatur communiter duplum G M in M H, fiet duplum aggregati <lb/>V G, M H, in M H maius quadrato M G cum quadrato M H: </s>
  <s xml:id="echoid-s11752" xml:space="preserve">quare <lb/>duplum aggregati V G, &amp; </s>
  <s xml:id="echoid-s11753" xml:space="preserve">M H in M V ad duplum aggregati V G, &amp; </s>
  <s xml:id="echoid-s11754" xml:space="preserve"><lb/>M H in M H, nempe M V ad M H minorem proportionem habebit, <lb/>quàm duplum aggregati V G, &amp; </s>
  <s xml:id="echoid-s11755" xml:space="preserve">M H in M V ad quadratum G M cum <lb/>quadrato M H: </s>
  <s xml:id="echoid-s11756" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11757" xml:space="preserve">componendo oſtendetur (quemadmodum antea di-<lb/>ctum eſt) quod quadratum A C ad diametrum figuræ P Q maiorem pro-<lb/>portionem habeat, quàm ad diametrum figuræ S T. </s>
  <s xml:id="echoid-s11758" xml:space="preserve">Eadem prorſus cõ-<lb/>tingent in reliquis omnibus diametris. </s>
  <s xml:id="echoid-s11759" xml:space="preserve">Quapropter diameter figuræ P Q <lb/>minor eſt diametro figuræ cuiuslibet diametri ad vtraſque eius partes in <lb/>eadem ſectione exiſtente. </s>
  <s xml:id="echoid-s11760" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s11761" xml:space="preserve"/>
</p>
<pb o="336" file="0374" n="375" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div1013" type="section" level="1" n="321">
<head xml:id="echoid-head397" xml:space="preserve">In Sectionem VIII. Propoſit. XXXXIIII. <lb/>XXXXV. &amp; XXXXVI.</head>
<head xml:id="echoid-head398" xml:space="preserve">LEMM A.X.</head>
<p style="it">
  <s xml:id="echoid-s11762" xml:space="preserve">SI rectæ lineæ G H bifariam ſectæ in D addantur ſegmenta H A, <lb/>&amp; </s>
  <s xml:id="echoid-s11763" xml:space="preserve">H E atque proportio dupli E H ad H G eadem fuerit propor-<lb/>tioni G H ad H A: </s>
  <s xml:id="echoid-s11764" xml:space="preserve">dico duplum rectanguli ex G A, &amp; </s>
  <s xml:id="echoid-s11765" xml:space="preserve">H E in H <lb/>A æquale eſſe quadratis ex G A, &amp; </s>
  <s xml:id="echoid-s11766" xml:space="preserve">ex A H: </s>
  <s xml:id="echoid-s11767" xml:space="preserve">ſi verò proportio illa <lb/>maior fuerit, erit quoque rectangulum maius quadratis: </s>
  <s xml:id="echoid-s11768" xml:space="preserve">ſi verò propor-<lb/>tio fuerit minor, rectangulum minus erit quadratis.</s>
  <s xml:id="echoid-s11769" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11770" xml:space="preserve">Primo quia ſi duplum E H ad H G, eſt vt G H ad H A, ergo duplum re-<lb/>ctanguli E H A æquale erit quadrato G H, &amp; </s>
  <s xml:id="echoid-s11771" xml:space="preserve">addatur communiter duplum <lb/>
<anchor type="figure" xlink:label="fig-0374-01a" xlink:href="fig-0374-01"/>
rectanguli G A H, erit duplum <lb/>rectanguli ex ſumma G A, &amp; </s>
  <s xml:id="echoid-s11772" xml:space="preserve">E H <lb/>in A H æquale duplo rectanguli G <lb/>A H cum quadrato G H; </s>
  <s xml:id="echoid-s11773" xml:space="preserve">his verò <lb/>ſpatĳs æquantur quadrata ex G A, <lb/>&amp; </s>
  <s xml:id="echoid-s11774" xml:space="preserve">ex A H, ergo duplum rectan-<lb/>guli ex ſumma G A, E H in A H æquale erit duobus quadratis ex G A, &amp; </s>
  <s xml:id="echoid-s11775" xml:space="preserve"><lb/>ex A H.</s>
  <s xml:id="echoid-s11776" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1013" type="float" level="2" n="1">
  <figure xlink:label="fig-0374-01" xlink:href="fig-0374-01a">
    <image file="0374-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0374-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11777" xml:space="preserve">Secundo, quia duplum E H ad H G, maiorem proportionem habet, quàm <lb/>G H ad A H, ergo duplum rectanguli E H A maius eſt quadrato G H, &amp; </s>
  <s xml:id="echoid-s11778" xml:space="preserve">ad-<lb/>dito communiter duplo rectanguli G A H, erit duplum rectanguli ex G A, E <lb/>H in A H maius duobus quadratis ex G A, &amp; </s>
  <s xml:id="echoid-s11779" xml:space="preserve">ex A H.</s>
  <s xml:id="echoid-s11780" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11781" xml:space="preserve">Tertio, quia duplum E H ad H G minorem proportionem habet, quàm G H <lb/>ad A H, ergo duplum rectanguli E H A minus eſt quadrato G H, &amp; </s>
  <s xml:id="echoid-s11782" xml:space="preserve">addito <lb/>duplo rectanguli G A H, erit duplum rectanguli ex G A, E H in A H minus <lb/>quadratis ex G A, &amp; </s>
  <s xml:id="echoid-s11783" xml:space="preserve">ex A H.</s>
  <s xml:id="echoid-s11784" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1015" type="section" level="1" n="322">
<head xml:id="echoid-head399" xml:space="preserve">LEMM A XI.</head>
<p style="it">
  <s xml:id="echoid-s11785" xml:space="preserve">SI recta linea G H ſecetur exterius in A, E, &amp; </s>
  <s xml:id="echoid-s11786" xml:space="preserve">ſit eadem G H <lb/>differentia nedum ſegmentorum G E, &amp; </s>
  <s xml:id="echoid-s11787" xml:space="preserve">E H, ſed etiam duo-<lb/>rum ſegmentorum G A, &amp; </s>
  <s xml:id="echoid-s11788" xml:space="preserve">A H: </s>
  <s xml:id="echoid-s11789" xml:space="preserve">dico quod quadrata ex maximo, <lb/>&amp; </s>
  <s xml:id="echoid-s11790" xml:space="preserve">ex vno intermediorum ſegmentorum, ſcilicet ex G E, &amp; </s>
  <s xml:id="echoid-s11791" xml:space="preserve">ex E H <lb/>
<anchor type="figure" xlink:label="fig-0374-02a" xlink:href="fig-0374-02"/>
æqualia ſunt quadratis ex <lb/>reliquo intermediorum, &amp; </s>
  <s xml:id="echoid-s11792" xml:space="preserve"><lb/>ex minimo ſegmento, ſci-<lb/>licet ex G A, &amp; </s>
  <s xml:id="echoid-s11793" xml:space="preserve">ex A <lb/>H vna cum duplo rectã-
<pb o="337" file="0375" n="376" rhead="Conicor. Lib. VII."/>
guli ex ſumma extremorum, vel intermediorum in differentiam mini-<lb/>morum ſegmentorum, ſcilicet ex G A cum H E in E A.</s>
  <s xml:id="echoid-s11794" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1015" type="float" level="2" n="1">
  <figure xlink:label="fig-0374-02" xlink:href="fig-0374-02a">
    <image file="0374-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0374-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s11795" xml:space="preserve">Quia duplum rectanguli G A H cum duplo rectanguli G A E æquatur duplo <lb/>rectanguli ſub G A in H E, addito cõmuniter duplo rectanguli H E A erit du-<lb/>plum rectanguli G E H æquale duplo rectanguli G A H cum duplo rectanguli ex <lb/>ſumma G A, H E in E A; </s>
  <s xml:id="echoid-s11796" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11797" xml:space="preserve">addito communi quadrato G H, erit duplum <lb/>rectanguli G E H cum quadrato G H, ſcilicet duo quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s11798" xml:space="preserve">ex E <lb/>
<anchor type="figure" xlink:label="fig-0375-01a" xlink:href="fig-0375-01"/>
H, erunt æqualia illis om-<lb/>nibus ſpatĳs, ſcilicet duplo <lb/>rectanguli ex ſumma G A, <lb/>H E in E A cum duplo re-<lb/>ctanguli G A H ſimul cum <lb/>quadrato ex G H: </s>
  <s xml:id="echoid-s11799" xml:space="preserve">ſed duplo <lb/>rectanguli G A H cum quadrato G H æqualia ſunt duo quadrata ex G A, <lb/>&amp; </s>
  <s xml:id="echoid-s11800" xml:space="preserve">ex A H, ergo duo quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s11801" xml:space="preserve">ex E H æqualia erunt quadratis ex <lb/>G A, &amp; </s>
  <s xml:id="echoid-s11802" xml:space="preserve">ex A H cum duplo rectanguli ex G A; </s>
  <s xml:id="echoid-s11803" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11804" xml:space="preserve">H E in E A, quod erat <lb/>oſtendendum.</s>
  <s xml:id="echoid-s11805" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1016" type="float" level="2" n="2">
  <figure xlink:label="fig-0375-01" xlink:href="fig-0375-01a">
    <image file="0375-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0375-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1018" type="section" level="1" n="323">
<head xml:id="echoid-head400" xml:space="preserve">LEMM A XII.</head>
<p style="it">
  <s xml:id="echoid-s11806" xml:space="preserve">IN hyperbola, cuius axis A C, erectus A F, præſectæ C G, H A, <lb/>centrum D, atque diameter I L, eiuſque erectus I K, &amp; </s>
  <s xml:id="echoid-s11807" xml:space="preserve">latus <lb/>C E, pariterque altera diameter Q P, cuius erectus P R, &amp; </s>
  <s xml:id="echoid-s11808" xml:space="preserve">latus <lb/>C O: </s>
  <s xml:id="echoid-s11809" xml:space="preserve">dico quod duplum rectanguli ex G E cũ O H in H E à duobus <lb/>quadratis ex G E, &amp; </s>
  <s xml:id="echoid-s11810" xml:space="preserve">ex E H; </s>
  <s xml:id="echoid-s11811" xml:space="preserve">nec non quadrata Q P, &amp; </s>
  <s xml:id="echoid-s11812" xml:space="preserve">P R late-<lb/>rum figuræ diametri Q P à quadratis ex L I, &amp; </s>
  <s xml:id="echoid-s11813" xml:space="preserve">ex I K, vel ex C A, <lb/>&amp; </s>
  <s xml:id="echoid-s11814" xml:space="preserve">ex A F, vna deficiunt, aut vna æqualia ſunt, vel vna exce-<lb/>dunt.</s>
  <s xml:id="echoid-s11815" xml:space="preserve"/>
</p>
  <figure>
    <image file="0375-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0375-02"/>
  </figure>
<pb o="338" file="0376" n="377" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s11816" xml:space="preserve">Quia duplum rectanguli ex G E, O H in H E æquale eſt quadratis ex G E <lb/>&amp; </s>
  <s xml:id="echoid-s11817" xml:space="preserve">ex E H, ergo idem rectangulum, cuius altitudo G E, &amp; </s>
  <s xml:id="echoid-s11818" xml:space="preserve">O H, baſis verò <lb/>O E bis ſumptum ad duplum rectanguli, cuius altitudo G E, O H, baſis verò <lb/>H E, ſeu O E ad H E eandem proportionem habet, quàm duplum rectanguli <lb/>ex G E, &amp; </s>
  <s xml:id="echoid-s11819" xml:space="preserve">O H in O E ad quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s11820" xml:space="preserve">ex E H: </s>
  <s xml:id="echoid-s11821" xml:space="preserve">quare componen-<lb/>do O H ad E H, ſeu O H A ad E H A eandem proportionem habebit, quàm <lb/>
<anchor type="note" xlink:label="note-0376-01a" xlink:href="note-0376-01"/>
duo quadrata ex G O, &amp; </s>
  <s xml:id="echoid-s11822" xml:space="preserve">ex O H ad duo quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s11823" xml:space="preserve">ex E H, &amp; </s>
  <s xml:id="echoid-s11824" xml:space="preserve"><lb/>permutando O H A ad quadrata ex G O, &amp; </s>
  <s xml:id="echoid-s11825" xml:space="preserve">ex O H, ſeu quadratum ex A C <lb/>
<anchor type="note" xlink:label="note-0376-02a" xlink:href="note-0376-02"/>
ad quadrata ex Q P, &amp; </s>
  <s xml:id="echoid-s11826" xml:space="preserve">ex P R eandem proportionem habebit, quàm rectan-<lb/>gulũ E H A ad quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s11827" xml:space="preserve">ex E H, ſeu erit vt quadratum A C ad <lb/>
<anchor type="note" xlink:label="note-0376-03a" xlink:href="note-0376-03"/>
quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s11828" xml:space="preserve">ex I K, vel ad quadrata ex C A &amp; </s>
  <s xml:id="echoid-s11829" xml:space="preserve">ex A F: </s>
  <s xml:id="echoid-s11830" xml:space="preserve">quare <lb/>duo quadrata ex Q P, &amp; </s>
  <s xml:id="echoid-s11831" xml:space="preserve">ex R P æqualia ſunt duobus quadratis ex I L, &amp; </s>
  <s xml:id="echoid-s11832" xml:space="preserve"><lb/>ex I K, vel ex C A, &amp; </s>
  <s xml:id="echoid-s11833" xml:space="preserve">A F.</s>
  <s xml:id="echoid-s11834" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1018" type="float" level="2" n="1">
<note position="left" xlink:label="note-0376-01" xlink:href="note-0376-01a" xml:space="preserve">Lem. 11. <lb/>huius.</note>
<note position="left" xlink:label="note-0376-02" xlink:href="note-0376-02a" xml:space="preserve">17. huius.</note>
<note position="left" xlink:label="note-0376-03" xlink:href="note-0376-03a" xml:space="preserve">Ibidem.</note>
</div>
  <figure>
    <image file="0376-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0376-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s11835" xml:space="preserve">Secundo quia duplum rectanguli ex G E, O H in H E minus ponitur quadratis <lb/>ex G E, &amp; </s>
  <s xml:id="echoid-s11836" xml:space="preserve">ex E H, igitur idem ſpatium ſcilicet duplum rectanguli ex G E, &amp; </s>
  <s xml:id="echoid-s11837" xml:space="preserve"><lb/>O H in O E ad duplum rectanguli ex G E, &amp; </s>
  <s xml:id="echoid-s11838" xml:space="preserve">O H in H E, ſiue O E ad H E <lb/>maiorem proportionem habet, quàm duplum rectanguli ex G E, O H in O E ad <lb/>quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s11839" xml:space="preserve">O H, &amp; </s>
  <s xml:id="echoid-s11840" xml:space="preserve">vt prius componendo, ex lemmate 11. </s>
  <s xml:id="echoid-s11841" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11842" xml:space="preserve">permu-<lb/>tando, ex 17. </s>
  <s xml:id="echoid-s11843" xml:space="preserve">huius; </s>
  <s xml:id="echoid-s11844" xml:space="preserve">idem quadratum A C ad quadrata ex Q P, &amp; </s>
  <s xml:id="echoid-s11845" xml:space="preserve">ex P R <lb/>maiorem proportionem habebit quàm ad quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s11846" xml:space="preserve">ex I K, vel ad <lb/>quadrata, ex C A, &amp; </s>
  <s xml:id="echoid-s11847" xml:space="preserve">ex A F: </s>
  <s xml:id="echoid-s11848" xml:space="preserve">quapropter quadrata ex Q P, &amp; </s>
  <s xml:id="echoid-s11849" xml:space="preserve">ex P R mi-<lb/>nora erunt quadratis ex I L, &amp; </s>
  <s xml:id="echoid-s11850" xml:space="preserve">ex I K, vel quadratis ex C A, &amp; </s>
  <s xml:id="echoid-s11851" xml:space="preserve">ex A F.</s>
  <s xml:id="echoid-s11852" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11853" xml:space="preserve">Tertio quia duplum rectanguli ex G E, O H in H E maius eſt ſumma qua-<lb/>dratorum ex G E, &amp; </s>
  <s xml:id="echoid-s11854" xml:space="preserve">ex E H, igitur, eodem progreſſu, habebit quadratum A C <lb/>ad ſummam quadratorum ex Q P, &amp; </s>
  <s xml:id="echoid-s11855" xml:space="preserve">ex P R minorem proportionem, quàm <lb/>ad ſummam quadraterum ex I L, &amp; </s>
  <s xml:id="echoid-s11856" xml:space="preserve">ex I K, vel ex C A, &amp; </s>
  <s xml:id="echoid-s11857" xml:space="preserve">ex A F: </s>
  <s xml:id="echoid-s11858" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11859" xml:space="preserve"><lb/>propterea ſumma priorum quadratorum maior erit ſumma poſteriorum, vt fue-<lb/>rat propoſitum.</s>
  <s xml:id="echoid-s11860" xml:space="preserve"/>
</p>
<pb o="339" file="0377" n="378" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div1020" type="section" level="1" n="324">
<head xml:id="echoid-head401" xml:space="preserve">Notæ in Propoſit. XXXXIV. &amp; XXXXV.</head>
<p style="it">
  <s xml:id="echoid-s11861" xml:space="preserve">QVia C A maior eſt, quàm A F, vel ſi minor eſt quadratum ex C A, <lb/>minor non eſt dimidio quadrati ex differentia C A, &amp; </s>
  <s xml:id="echoid-s11862" xml:space="preserve">A F, eſtque H <lb/>A ad A G vt A C ad A F, &amp; </s>
  <s xml:id="echoid-s11863" xml:space="preserve">H A ad G H, vt A C ad differen-<lb/>tiam ipſarum A C, A F, ergo quadratum H A ad dimidium quadrati G H <lb/>erit vt quadratum A C ad dimidium quadrati ex differentia ipſarum A C, &amp; </s>
  <s xml:id="echoid-s11864" xml:space="preserve"><lb/>A F, quare quadratum ex H A minor non erit ſemiſſe quadrati H G, ideoq; <lb/></s>
  <s xml:id="echoid-s11865" xml:space="preserve">
<anchor type="figure" xlink:label="fig-0377-01a" xlink:href="fig-0377-01"/>
duplum quadrati A H minor non erit quadrato H G, eſtque duplum rectanguli <lb/>E H A, vel M H E maius duplo quadrati A H, ſeu maius quadrato H G; <lb/></s>
  <s xml:id="echoid-s11866" xml:space="preserve">propterea duplum E H ad H G maiorem proportionem habebit, quàm G H <lb/>
<anchor type="note" xlink:label="note-0377-01a" xlink:href="note-0377-01"/>
ad H A, ideoque duplum rectanguli ex G A, H A in A H maius erit quadra-<lb/>
<anchor type="note" xlink:label="note-0377-02a" xlink:href="note-0377-02"/>
tis ex G A, &amp; </s>
  <s xml:id="echoid-s11867" xml:space="preserve">ex A H, &amp; </s>
  <s xml:id="echoid-s11868" xml:space="preserve">inſuper ſumma quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s11869" xml:space="preserve">ex I K <lb/>maior erit, quàm ſumma quadratorum ex C A, &amp; </s>
  <s xml:id="echoid-s11870" xml:space="preserve">ex A F.</s>
  <s xml:id="echoid-s11871" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1020" type="float" level="2" n="1">
  <figure xlink:label="fig-0377-01" xlink:href="fig-0377-01a">
    <image file="0377-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0377-01"/>
  </figure>
<note position="right" xlink:label="note-0377-01" xlink:href="note-0377-01a" xml:space="preserve">Lem. 10.</note>
<note position="right" xlink:label="note-0377-02" xlink:href="note-0377-02a" xml:space="preserve">Lem 12.</note>
</div>
</div>
<div xml:id="echoid-div1022" type="section" level="1" n="325">
<head xml:id="echoid-head402" xml:space="preserve">Notæ in Propoſit. XXXXVI.</head>
<p style="it">
  <s xml:id="echoid-s11872" xml:space="preserve">QVia quadratum axis C A minus eſt ſemiße quadrati ex differentia ipſa-<lb/>rum A C, &amp; </s>
  <s xml:id="echoid-s11873" xml:space="preserve">A @, eſtque H A ad A G, vt C A ad A F, atque G H <lb/>eſt differentia ipſarum A H, &amp; </s>
  <s xml:id="echoid-s11874" xml:space="preserve">A G, igitur quadratum ex A H
<pb o="340" file="0378" n="379" rhead="Apollonij Pergæi"/>
minus eſt ſemiſſe quadrati G H: </s>
  <s xml:id="echoid-s11875" xml:space="preserve">fiat iam quadratum ex M H æquale ſemiqua-<lb/>drato ex G H, &amp; </s>
  <s xml:id="echoid-s11876" xml:space="preserve">lateris C M fiant duo diametri Q P, &amp; </s>
  <s xml:id="echoid-s11877" xml:space="preserve">q p, eorumque <lb/>erecta ſint P R, &amp; </s>
  <s xml:id="echoid-s11878" xml:space="preserve">p r: </s>
  <s xml:id="echoid-s11879" xml:space="preserve">dico ductas diametros æquales eſſe, &amp; </s>
  <s xml:id="echoid-s11880" xml:space="preserve">quadratum <lb/>ex P Q æquale eſſe quadrato ex differentia ipſarum P Q, &amp; </s>
  <s xml:id="echoid-s11881" xml:space="preserve">P R.</s>
  <s xml:id="echoid-s11882" xml:space="preserve"/>
</p>
  <figure>
    <image file="0378-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0378-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s11883" xml:space="preserve">Quia vt M H ad G M, ita eſt diameter Q P ad eius erectum P R, ergo <lb/>comparando antecedentes ad terminorum differentias, erit M H ad H G, vt <lb/>
<anchor type="note" xlink:label="note-0378-01a" xlink:href="note-0378-01"/>
P Q ad differentiam ipſarum P Q, &amp; </s>
  <s xml:id="echoid-s11884" xml:space="preserve">P R, &amp; </s>
  <s xml:id="echoid-s11885" xml:space="preserve">pariter eorundem quadrata <lb/>proportionalia erunt, eſtque quadratum ex H M æquale ſemiquadrato ex <lb/>G H, ergo quadratum ex P Q æquale erit ſemiquadrato ex differentia P Q, <lb/>&amp; </s>
  <s xml:id="echoid-s11886" xml:space="preserve">P R, &amp; </s>
  <s xml:id="echoid-s11887" xml:space="preserve">ſic quadratum ex p q æquale erit ſemiquadrato ex differentia ip-<lb/>ſarum p q &amp; </s>
  <s xml:id="echoid-s11888" xml:space="preserve">p r; </s>
  <s xml:id="echoid-s11889" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11890" xml:space="preserve">ſunt diametri P Q, &amp; </s>
  <s xml:id="echoid-s11891" xml:space="preserve">p q æquales, cum æquè rece-<lb/>dant ab axi, &amp; </s>
  <s xml:id="echoid-s11892" xml:space="preserve">habeant latus commune C M.</s>
  <s xml:id="echoid-s11893" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1022" type="float" level="2" n="1">
<note position="left" xlink:label="note-0378-01" xlink:href="note-0378-01a" xml:space="preserve">ex 6. hu.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11894" xml:space="preserve">Secundo dico quod ſumma quadratorum ex Q P, &amp; </s>
  <s xml:id="echoid-s11895" xml:space="preserve">ex P R minor eſt qua-<lb/>libet alia ſumma quadratorum laterum figuræ alterius diametri.</s>
  <s xml:id="echoid-s11896" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11897" xml:space="preserve">Quia duplum rectanguli M H E minus eſt duplo quadrati M H, ſeu ſingu-<lb/>lari quadrato ex G H, ergo duplum M H ad H G minorem proportionem ha-<lb/>bet, quàm G H ad H E, ergo duplum rectanguli ex G E, &amp; </s>
  <s xml:id="echoid-s11898" xml:space="preserve">M H in E H <lb/>
<anchor type="note" xlink:label="note-0378-02a" xlink:href="note-0378-02"/>
minus erit ſumma quadratorum ex G E, &amp; </s>
  <s xml:id="echoid-s11899" xml:space="preserve">ex E H &amp; </s>
  <s xml:id="echoid-s11900" xml:space="preserve">propterea ſumma qua-<lb/>dratorum ex Q P, &amp; </s>
  <s xml:id="echoid-s11901" xml:space="preserve">ex P R minor erit ſumma quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s11902" xml:space="preserve">ex <lb/>
<anchor type="note" xlink:label="note-0378-03a" xlink:href="note-0378-03"/>
I K.</s>
  <s xml:id="echoid-s11903" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1023" type="float" level="2" n="2">
<note position="left" xlink:label="note-0378-02" xlink:href="note-0378-02a" xml:space="preserve">Lem. 10. <lb/>huius.</note>
<note position="left" xlink:label="note-0378-03" xlink:href="note-0378-03a" xml:space="preserve">Lem. 12. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11904" xml:space="preserve">Tertio, quia duplum rectanguli ex E H A minus eſt duplo quadrati M H, <lb/>ſeu ſingulari quadrato ex G H, ergo duplum E H ad H G minorem proportio-<lb/>
<anchor type="note" xlink:label="note-0378-04a" xlink:href="note-0378-04"/>
nem habet, quàm G H ad H A, ergo duplum rectanguli ex G A, E H in A H <lb/>minus erit ſumma quadratorum ex G A, &amp; </s>
  <s xml:id="echoid-s11905" xml:space="preserve">ex A H: </s>
  <s xml:id="echoid-s11906" xml:space="preserve">quare ſumma quadra-<lb/>
<anchor type="note" xlink:label="note-0378-05a" xlink:href="note-0378-05"/>
torum ex I L, &amp; </s>
  <s xml:id="echoid-s11907" xml:space="preserve">ex I K minor erit, quàm quadratorum ſumma ex A C, &amp; </s>
  <s xml:id="echoid-s11908" xml:space="preserve"><lb/>ex A F.</s>
  <s xml:id="echoid-s11909" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1024" type="float" level="2" n="3">
<note position="left" xlink:label="note-0378-04" xlink:href="note-0378-04a" xml:space="preserve">Lem. 10. <lb/>huius.</note>
<note position="left" xlink:label="note-0378-05" xlink:href="note-0378-05a" xml:space="preserve">Lem. 12. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11910" xml:space="preserve">Quarto quia duplum rectanguli V H M maius eſt duplo quadrati ex M H, <lb/>ſeu ſingulari quadrato ex G H, ergo duplum V H ad H G maiorem proportio-<lb/>nem habet, quàm H G ad H M, &amp; </s>
  <s xml:id="echoid-s11911" xml:space="preserve">propterea duplum rectanguli ex G M, &amp; </s>
  <s xml:id="echoid-s11912" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0378-06a" xlink:href="note-0378-06"/>
V H in M H maius erit ſumma quadratorum ex G M, &amp; </s>
  <s xml:id="echoid-s11913" xml:space="preserve">ex M H, &amp; </s>
  <s xml:id="echoid-s11914" xml:space="preserve">ideo <lb/>ſumma quadratorum ex T S, &amp; </s>
  <s xml:id="echoid-s11915" xml:space="preserve">S Z maior erit quadratorum ſumma ex Q <lb/>
<anchor type="note" xlink:label="note-0378-07a" xlink:href="note-0378-07"/>
P, &amp; </s>
  <s xml:id="echoid-s11916" xml:space="preserve">ex P R, &amp; </s>
  <s xml:id="echoid-s11917" xml:space="preserve">ſic de reliquis: </s>
  <s xml:id="echoid-s11918" xml:space="preserve">quare ſumma quadratorum ex Q P, &amp; </s>
  <s xml:id="echoid-s11919" xml:space="preserve">ex <lb/>P R minima eſt omnium, vt fuit propoſitum.</s>
  <s xml:id="echoid-s11920" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1025" type="float" level="2" n="4">
<note position="left" xlink:label="note-0378-06" xlink:href="note-0378-06a" xml:space="preserve">Lem 10 <lb/>huius.</note>
<note position="left" xlink:label="note-0378-07" xlink:href="note-0378-07a" xml:space="preserve">Lem. 12. <lb/>huius.</note>
</div>
<pb o="341" file="0379" n="380" rhead="Conicor. Lib. VII."/>
<p style="it">
  <s xml:id="echoid-s11921" xml:space="preserve">In hyperbola reperire diametrum, cuius figuræ duo quadrata laterum <lb/>
<anchor type="note" xlink:label="note-0379-01a" xlink:href="note-0379-01"/>
æqualia ſint quadratis laterum figuræ axis: </s>
  <s xml:id="echoid-s11922" xml:space="preserve">oportet autem vt quadra-<lb/>tum axis C A minus ſit ſemiquadrato ex differentia laterum ſiguræ eius <lb/>C A, &amp; </s>
  <s xml:id="echoid-s11923" xml:space="preserve">A F.</s>
  <s xml:id="echoid-s11924" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1026" type="float" level="2" n="5">
<note position="right" xlink:label="note-0379-01" xlink:href="note-0379-01a" xml:space="preserve">PROP. <lb/>5. Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11925" xml:space="preserve">Quia ex hypotheſi quadratum axis A C minus eſt ſemiquadrato ex differen-<lb/>tia laterum figuræ A C, A F, vt in nota propoſit. </s>
  <s xml:id="echoid-s11926" xml:space="preserve">46. </s>
  <s xml:id="echoid-s11927" xml:space="preserve">dictum eſt, quadratum <lb/>ex A H minus eſt ſemiquadrato ex G H: </s>
  <s xml:id="echoid-s11928" xml:space="preserve">fiat duplum e H ad H G, vt G H <lb/>
<anchor type="note" xlink:label="note-0379-02a" xlink:href="note-0379-02"/>
ad H A, &amp; </s>
  <s xml:id="echoid-s11929" xml:space="preserve">lateris C e ducatur diameter b a, cuius erectus c a, ergo duplum <lb/>rectanguli ex ſumma G A, e H in A H æquale eſt ſummæ quadratorum ex G A, <lb/>
<anchor type="note" xlink:label="note-0379-03a" xlink:href="note-0379-03"/>
&amp; </s>
  <s xml:id="echoid-s11930" xml:space="preserve">ex A H, &amp; </s>
  <s xml:id="echoid-s11931" xml:space="preserve">ſumma quadratorum ex a b, &amp; </s>
  <s xml:id="echoid-s11932" xml:space="preserve">ex a c æqualis erit quadrato-<lb/>rum ſummæ ex A C, &amp; </s>
  <s xml:id="echoid-s11933" xml:space="preserve">ex A F, quod erat oſtendendum.</s>
  <s xml:id="echoid-s11934" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1027" type="float" level="2" n="6">
<note position="right" xlink:label="note-0379-02" xlink:href="note-0379-02a" xml:space="preserve">Lem. 10. <lb/>huius.</note>
<note position="right" xlink:label="note-0379-03" xlink:href="note-0379-03a" xml:space="preserve">I em. 12. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11935" xml:space="preserve">In eadem hyperbola diametrum reperire, cuius figuræ duo quadrata, <lb/>
<anchor type="note" xlink:label="note-0379-04a" xlink:href="note-0379-04"/>
laterum æqualia ſint quadratis laterum figuræ datæ diametri I L: </s>
  <s xml:id="echoid-s11936" xml:space="preserve">opor-<lb/>tet autem vt I L cadat inter axim, &amp; </s>
  <s xml:id="echoid-s11937" xml:space="preserve">diametrum P Q, cuius qua-<lb/>dratum ſubduplum ſit quadrati ex differentia P Q, &amp; </s>
  <s xml:id="echoid-s11938" xml:space="preserve">ex P R.</s>
  <s xml:id="echoid-s11939" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1028" type="float" level="2" n="7">
<note position="right" xlink:label="note-0379-04" xlink:href="note-0379-04a" xml:space="preserve">PROP. 6. <lb/>Addit</note>
</div>
<p style="it">
  <s xml:id="echoid-s11940" xml:space="preserve">Sit C E latus diametri I L, &amp; </s>
  <s xml:id="echoid-s11941" xml:space="preserve">fiat duplum V H ad H G, vt G H ad H E, <lb/>&amp; </s>
  <s xml:id="echoid-s11942" xml:space="preserve">ponatur S T diameter lateris C V, cuius erectus ſit S Z: </s>
  <s xml:id="echoid-s11943" xml:space="preserve">erit igitur duplũ <lb/>
<anchor type="note" xlink:label="note-0379-05a" xlink:href="note-0379-05"/>
rectanguli ex G E, &amp; </s>
  <s xml:id="echoid-s11944" xml:space="preserve">V H in E H æquale quadratis ex G E, &amp; </s>
  <s xml:id="echoid-s11945" xml:space="preserve">ex E H, &amp; </s>
  <s xml:id="echoid-s11946" xml:space="preserve"><lb/>propterea ſumma quadratorum ex T S, &amp; </s>
  <s xml:id="echoid-s11947" xml:space="preserve">ex S Z æqualis erit quadratorum, <lb/>
<anchor type="note" xlink:label="note-0379-06a" xlink:href="note-0379-06"/>
ſummæ ex L I, &amp; </s>
  <s xml:id="echoid-s11948" xml:space="preserve">ex I K, quod erat propoſitum.</s>
  <s xml:id="echoid-s11949" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1029" type="float" level="2" n="8">
<note position="right" xlink:label="note-0379-05" xlink:href="note-0379-05a" xml:space="preserve">Lem. 10.</note>
<note position="right" xlink:label="note-0379-06" xlink:href="note-0379-06a" xml:space="preserve">Lem. 12. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s11950" xml:space="preserve">Deducitur pariter ex 5. </s>
  <s xml:id="echoid-s11951" xml:space="preserve">propoſitione additarum in eadem hyperbola tres dia-<lb/>metros reperiri poſſe, quarum laterum ſummæ quadratorum æquales ſint in-<lb/>ter ſe.</s>
  <s xml:id="echoid-s11952" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11953" xml:space="preserve">Et ex 6. </s>
  <s xml:id="echoid-s11954" xml:space="preserve">propoſitione additarum deducitur, quod quatuor diametrorum eiuſ-<lb/>dem hyperbolæ laterum ſummæ quadratorum æquales eße posſunt inter ſe.</s>
  <s xml:id="echoid-s11955" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s11956" xml:space="preserve">Et educamus inter A P inclinatam I L: </s>
  <s xml:id="echoid-s11957" xml:space="preserve">quia quadruplum quadrati M <lb/>
<anchor type="note" xlink:label="note-0379-07a" xlink:href="note-0379-07"/>
H æquale eſt quadrato H G, &amp;</s>
  <s xml:id="echoid-s11958" xml:space="preserve">c. </s>
  <s xml:id="echoid-s11959" xml:space="preserve">Suppleri debent ea, quæ deficiunt, alioqui <lb/>conſtructio imperfecta eßet: </s>
  <s xml:id="echoid-s11960" xml:space="preserve">duci igitur debet C B parallela diametro I L, <lb/>quæ occurrat ſectioni ad punctum B, à quo ad axim perpendicularis ducatur <lb/>B E ſecans axim in E.</s>
  <s xml:id="echoid-s11961" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1030" type="float" level="2" n="9">
<note position="left" xlink:label="note-0379-07" xlink:href="note-0379-07a" xml:space="preserve">a</note>
</div>
</div>
<div xml:id="echoid-div1032" type="section" level="1" n="326">
<head xml:id="echoid-head403" xml:space="preserve">SECTIO NONA</head>
<head xml:id="echoid-head404" xml:space="preserve">Continens Propoſit. XXXXI. XXXXVII. <lb/>&amp; XXXXVIII.</head>
<p>
  <s xml:id="echoid-s11962" xml:space="preserve">IN ellipſi duo latera figuræ maioris axis tranſuerſi minora ſunt <lb/>
<anchor type="note" xlink:label="note-0379-08a" xlink:href="note-0379-08"/>
duobus lateribus figuræ cuiuslibet alterius diametri, &amp; </s>
  <s xml:id="echoid-s11963" xml:space="preserve">duo <lb/>latera figuræ diametri axi maiori proximioris minora ſunt duo-<lb/>bus lateribus figuræ diametri remotioris.</s>
  <s xml:id="echoid-s11964" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1032" type="float" level="2" n="1">
<note position="left" xlink:label="note-0379-08" xlink:href="note-0379-08a" xml:space="preserve">a</note>
</div>
<pb o="342" file="0380" n="381" rhead="Apollonij Pergæi"/>
<p>
  <s xml:id="echoid-s11965" xml:space="preserve">XXXXVII. </s>
  <s xml:id="echoid-s11966" xml:space="preserve">Si verò duplum quadrati A C maius non fuerit <lb/>quadrato ex ſumma duorum laterum ſuæ figuræ; </s>
  <s xml:id="echoid-s11967" xml:space="preserve">vtique quadra-<lb/>tum diametri ſuæ figuræ minus erit quadrato diametri figuræ cu-<lb/>iuſlibet alterius diametri eiuſdem ſectionis, &amp; </s>
  <s xml:id="echoid-s11968" xml:space="preserve">quadratum dia-<lb/>metri figuræ proximioris axi minus erit quadrato diametri figu-<lb/>ræ remotioris.</s>
  <s xml:id="echoid-s11969" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11970" xml:space="preserve">XXXXVIII. </s>
  <s xml:id="echoid-s11971" xml:space="preserve">Si autem duplum quadrati axis tranſuerſi maius <lb/>fuerit quadrato ex ſumma duorum laterum ſuæ figuræ, æquidem <lb/>reperientur ad vtraſque eius partes duæ diametri æquales, &amp; </s>
  <s xml:id="echoid-s11972" xml:space="preserve">cu-<lb/>
<anchor type="figure" xlink:label="fig-0380-01a" xlink:href="fig-0380-01"/>
iuslibet earum quadratum bis ſumptum æquale erit quadrato ex <lb/>ſumma duorum laterum ſuæ figuræ; </s>
  <s xml:id="echoid-s11973" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11974" xml:space="preserve">quadratum diametri ſuæ <lb/>figuræ minus eſt quadrato diametri figuræ alterius cuiuſcunque <lb/>diametri exiſtentis in eodem quadrante eiuſdem ſectionis; </s>
  <s xml:id="echoid-s11975" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11976" xml:space="preserve"><lb/>diameter figuræ proximioris minor eſt diametro figuræ remo-<lb/>tioris.</s>
  <s xml:id="echoid-s11977" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1033" type="float" level="2" n="2">
  <figure xlink:label="fig-0380-01" xlink:href="fig-0380-01a">
    <image file="0380-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0380-01"/>
  </figure>
</div>
<pb o="343" file="0381" n="382" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div1035" type="section" level="1" n="327">
<head xml:id="echoid-head405" xml:space="preserve">PROPOSITIO XXXXI.</head>
<p>
  <s xml:id="echoid-s11978" xml:space="preserve">IN ellipſi A B C ſit A C axis maior, &amp; </s>
  <s xml:id="echoid-s11979" xml:space="preserve">y O minor, &amp; </s>
  <s xml:id="echoid-s11980" xml:space="preserve">ſint P <lb/>Q, &amp; </s>
  <s xml:id="echoid-s11981" xml:space="preserve">S T duæ aliæ diametri, ſitque A F erectus ipſius A <lb/>C, &amp; </s>
  <s xml:id="echoid-s11982" xml:space="preserve">P R erectus ipſius P Q, &amp; </s>
  <s xml:id="echoid-s11983" xml:space="preserve">O f ipſius y O. </s>
  <s xml:id="echoid-s11984" xml:space="preserve">Dico quod <lb/>C F minor eſt, quàm Q R, &amp; </s>
  <s xml:id="echoid-s11985" xml:space="preserve">Q R, quàm T Z, &amp; </s>
  <s xml:id="echoid-s11986" xml:space="preserve">T Z, <lb/>quàm y f.</s>
  <s xml:id="echoid-s11987" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11988" xml:space="preserve">Ducantur A N, A X ordinatim applicatæ ad diametros P Q, S T, <lb/>&amp; </s>
  <s xml:id="echoid-s11989" xml:space="preserve">duæ ad axim perpendiculares N M, X V, &amp; </s>
  <s xml:id="echoid-s11990" xml:space="preserve">interceptæ A G, C H. <lb/></s>
  <s xml:id="echoid-s11991" xml:space="preserve">Quia quadratum A C ad quadratum y O, nempe A C ad A F eandem, <lb/>
<anchor type="note" xlink:label="note-0381-01a" xlink:href="note-0381-01"/>
proportionem habet, quàm C G ad G A, ſeu ad C H habebit quadra-<lb/>
<anchor type="note" xlink:label="note-0381-02a" xlink:href="note-0381-02"/>
tum C A ad quadratum C F ſummæ ipſius C A, eiuſque erecti eandem <lb/>proportionem, quàm quadratum C G, nempe C G in A H ad quadra-<lb/>tum G H: </s>
  <s xml:id="echoid-s11992" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11993" xml:space="preserve">quadratum A C ad quadratum y O eandem proportionem, <lb/>habet, quàm G C in C H ad quadratum C H: </s>
  <s xml:id="echoid-s11994" xml:space="preserve">eſtquè quadratum y O ad <lb/>quadratum ſummæ y f, vt quadra-<lb/>
<anchor type="figure" xlink:label="fig-0381-01a" xlink:href="fig-0381-01"/>
tum C H ad quadratum H G; </s>
  <s xml:id="echoid-s11995" xml:space="preserve">er-<lb/>go quadratum A C ad quadratum <lb/>y f eſt, vt C G in C H minorem <lb/>ad quadratum H G; </s>
  <s xml:id="echoid-s11996" xml:space="preserve">ſed quadra-<lb/>tum A C ad quadratum C F ean-<lb/>dem proportionem habet, quàm. <lb/></s>
  <s xml:id="echoid-s11997" xml:space="preserve">G C in maiorem A H ad quadra-<lb/>tum G H; </s>
  <s xml:id="echoid-s11998" xml:space="preserve">igitur A C ad C F ma-<lb/>iorem proportionem habet, quàm <lb/>ad y f: </s>
  <s xml:id="echoid-s11999" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12000" xml:space="preserve">propterea C F ſumma, <lb/>A C, &amp; </s>
  <s xml:id="echoid-s12001" xml:space="preserve">erecti illius minor eſt, <lb/>quàm y f, quæ eſt ſumma y O, &amp; </s>
  <s xml:id="echoid-s12002" xml:space="preserve"><lb/>erecti illius. </s>
  <s xml:id="echoid-s12003" xml:space="preserve">Et quoniam C G in, <lb/>M H, quod minus eſt, quàm C G <lb/>in A H ad quadratum H G eandem <lb/>proportionem habet, quàm qua-<lb/>dratum A C ad quadratum Q R <lb/>ſummæ diametri, &amp; </s>
  <s xml:id="echoid-s12004" xml:space="preserve">erecti ipſius <lb/>P Q (16. </s>
  <s xml:id="echoid-s12005" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s12006" xml:space="preserve">quare quadratum <lb/>A C ad quadratnm C F maiorem <lb/>proportionem babebit, quàm ad <lb/>quadratum Q R, &amp; </s>
  <s xml:id="echoid-s12007" xml:space="preserve">propterea C F <lb/>minor erit, quam Q R. </s>
  <s xml:id="echoid-s12008" xml:space="preserve">Et quoniam <lb/>C G in V H ad quadratum H G eſt vt quadratum A C ad quadratum <lb/>T Z ad quàm ordinatim applicatur A X (16. </s>
  <s xml:id="echoid-s12009" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s12010" xml:space="preserve">erit C F minor quàm <lb/>T Z: </s>
  <s xml:id="echoid-s12011" xml:space="preserve">cumque C G in H M ad quadratum H G maiorem proportionem, <lb/>habeat, quàm G C in V H ad quadratum idipſum H G habebit quadra-
<pb o="344" file="0382" n="383" rhead="Apollonij Pergæi"/>
tum A C ad quadratum Q R maiorem proportionem quàm ad quadratũ <lb/>T Z. </s>
  <s xml:id="echoid-s12012" xml:space="preserve">Et pariter oſtendetur, quod quadratum A C ad quadratum T Z <lb/>maiorem proportionem habet, quàm ad quadratum y f; </s>
  <s xml:id="echoid-s12013" xml:space="preserve">quapropter C F <lb/>minor eſt quàm Q R, &amp; </s>
  <s xml:id="echoid-s12014" xml:space="preserve">Q R minor, quàm T Z, &amp; </s>
  <s xml:id="echoid-s12015" xml:space="preserve">T Z minor, quàm <lb/>y f. </s>
  <s xml:id="echoid-s12016" xml:space="preserve">Quod erat oſtendendum.</s>
  <s xml:id="echoid-s12017" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1035" type="float" level="2" n="1">
<note position="left" xlink:label="note-0381-01" xlink:href="note-0381-01a" xml:space="preserve">b</note>
<note position="right" xlink:label="note-0381-02" xlink:href="note-0381-02a" xml:space="preserve">Defin. 1. <lb/>huius.</note>
  <figure xlink:label="fig-0381-01" xlink:href="fig-0381-01a">
    <image file="0381-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0381-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1037" type="section" level="1" n="328">
<head xml:id="echoid-head406" xml:space="preserve">PROPOSITIO XXXXVII.</head>
<p>
  <s xml:id="echoid-s12018" xml:space="preserve">IN eadem figura ſi duplum quadrati A C maius non fuerit <lb/>quadrato ſummæ C F. </s>
  <s xml:id="echoid-s12019" xml:space="preserve">Dico, quod diameter figuræ eius <lb/>minor eſt diametro figuræ Q P R, &amp; </s>
  <s xml:id="echoid-s12020" xml:space="preserve">diameter figuræ Q P R <lb/>minor eſt diametro figuræ T S Z.</s>
  <s xml:id="echoid-s12021" xml:space="preserve"/>
</p>
  <figure>
    <image file="0382-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0382-01"/>
  </figure>
<p>
  <s xml:id="echoid-s12022" xml:space="preserve">Quoniam duplum quadrati A C non excedit quadratum ſummæ C A <lb/>F; </s>
  <s xml:id="echoid-s12023" xml:space="preserve">ergo duplum quadrati C G, nempe G C in A H bis ſumptum non, <lb/>excedit quadratum H G, &amp; </s>
  <s xml:id="echoid-s12024" xml:space="preserve">propterea C G in H M bis ſumptum minus <lb/>eſt quadrato H G: </s>
  <s xml:id="echoid-s12025" xml:space="preserve">tollatur communiter duplum M G in H M remanebit
<pb o="345" file="0383" n="384" rhead="Conicor. Lib. VII."/>
duplum H M in C M minus duobus quadratis ex M H, &amp; </s>
  <s xml:id="echoid-s12026" xml:space="preserve">ex G M: </s>
  <s xml:id="echoid-s12027" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12028" xml:space="preserve"><lb/>propterea A M in M C bis ſumptum ad H M in M C bis ſumptum, nẽ-<lb/>pe A M ad M H habebit maiorem proportionem, quam duplum A M <lb/>in M C ad duo quadrata ex H M, &amp; </s>
  <s xml:id="echoid-s12029" xml:space="preserve">ex G M: </s>
  <s xml:id="echoid-s12030" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12031" xml:space="preserve">componendo A H ad <lb/>H M, ſeu quadratum A H ad A H in H M maiorem proportionem ha-<lb/>bebit quàm duplum A M in M C cum duobus quadratis ex H M, &amp; </s>
  <s xml:id="echoid-s12032" xml:space="preserve">ex <lb/>M G (quæ omnia ſimul æqualia ſunt duobus quadratis C G, &amp; </s>
  <s xml:id="echoid-s12033" xml:space="preserve">H C) <lb/>ad duo quadrata M H, &amp; </s>
  <s xml:id="echoid-s12034" xml:space="preserve">M G; </s>
  <s xml:id="echoid-s12035" xml:space="preserve">igitur quadratum A H ad A H in H M <lb/>maiorem proportionem habet, quàm duo quadrata C G, &amp; </s>
  <s xml:id="echoid-s12036" xml:space="preserve">C H ad duo <lb/>quadrata H M, &amp; </s>
  <s xml:id="echoid-s12037" xml:space="preserve">G M, &amp; </s>
  <s xml:id="echoid-s12038" xml:space="preserve">permutando quadratum A H ad duo qua-<lb/>drata C G, &amp; </s>
  <s xml:id="echoid-s12039" xml:space="preserve">H C, ſcilicet quadratum A C ad quadratum diametri <lb/>figuræ eius maiorem proportionem habet, quàm A H in H M ad duo <lb/>quadrata M G, &amp; </s>
  <s xml:id="echoid-s12040" xml:space="preserve">M H, ſeu quàm quadratum A C ad quadratum dia-<lb/>metri figuræ P Q (19. </s>
  <s xml:id="echoid-s12041" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s12042" xml:space="preserve">quapropter diameter figuræ P Q maior <lb/>eſt diametro figuræ A C. </s>
  <s xml:id="echoid-s12043" xml:space="preserve">Ducatur poſtea diameter S T, &amp; </s>
  <s xml:id="echoid-s12044" xml:space="preserve">ad eam or-<lb/>dinatim applicata A X, &amp; </s>
  <s xml:id="echoid-s12045" xml:space="preserve">ad axim <lb/>perpendicularem X V. </s>
  <s xml:id="echoid-s12046" xml:space="preserve">Et ſiqui-<lb/>dem G M minor eſt, quàm V H <lb/>
<anchor type="figure" xlink:label="fig-0383-01a" xlink:href="fig-0383-01"/>
cum A G, &amp; </s>
  <s xml:id="echoid-s12047" xml:space="preserve">C H ſint æquales, <lb/>erunt duo quadrata H M, &amp; </s>
  <s xml:id="echoid-s12048" xml:space="preserve">M G <lb/>maiora duobus quadratis H V, V <lb/>G: </s>
  <s xml:id="echoid-s12049" xml:space="preserve">hæc autem maiora ſunt quàm <lb/>duplum V H in V d: </s>
  <s xml:id="echoid-s12050" xml:space="preserve">ergo duplũ <lb/>M V in V d ad duplum H V in V <lb/>d, nempe V M ad V H maiorem <lb/>proportionem habet, quàm duplũ <lb/>M V in V d ad duo quadrata ex <lb/>V H, &amp; </s>
  <s xml:id="echoid-s12051" xml:space="preserve">ex V G: </s>
  <s xml:id="echoid-s12052" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12053" xml:space="preserve">componendo <lb/>M H ad H V, ſeu M H in H A <lb/>ad V H in H A maiorem propor-<lb/>tionem habebit, quàm duplum M <lb/>V in V d cum duobus quadratis ex <lb/>V H, &amp; </s>
  <s xml:id="echoid-s12054" xml:space="preserve">ex V G, quæ omnia ſi-<lb/>mul ſunt vt duo quadrata M H, &amp; </s>
  <s xml:id="echoid-s12055" xml:space="preserve"><lb/>M G ad duo quadrata V H, &amp; </s>
  <s xml:id="echoid-s12056" xml:space="preserve">V <lb/>G: </s>
  <s xml:id="echoid-s12057" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12058" xml:space="preserve">permutando M H in H A <lb/>ad duo quadrata H M, &amp; </s>
  <s xml:id="echoid-s12059" xml:space="preserve">G M, <lb/>ſeu vt quadratum A C ad quadra-<lb/>tum diametri figuræ P Q (19. </s>
  <s xml:id="echoid-s12060" xml:space="preserve">ex <lb/>7.) </s>
  <s xml:id="echoid-s12061" xml:space="preserve">maiorem proportionem habebit, quàm V H in H A ad duo quadrata <lb/>V H, &amp; </s>
  <s xml:id="echoid-s12062" xml:space="preserve">V G, ſeu quàm quadratum A C ad quadratum diametri figuræ <lb/>S T (19. </s>
  <s xml:id="echoid-s12063" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s12064" xml:space="preserve">quare diameter figuræ S T maior eſt diametro figuræ <lb/>P Q. </s>
  <s xml:id="echoid-s12065" xml:space="preserve">Poſtea quia y O eſt media proportionalis inter A C, &amp; </s>
  <s xml:id="echoid-s12066" xml:space="preserve">A F erit <lb/>quadratum A C ad quadratum y O, vt A C ad A F, nempe vt C G ad <lb/>C H, ſeu vt C G in C H ad quadratum C H, &amp; </s>
  <s xml:id="echoid-s12067" xml:space="preserve">quadratum y O ad ſum-<lb/>mam quadratorum y O, &amp; </s>
  <s xml:id="echoid-s12068" xml:space="preserve">O f, nempe ad quadratum diametri ſuæ figuræ <lb/>eſt vt quadratum H C ad quadratum C G cum quadrato H C: </s>
  <s xml:id="echoid-s12069" xml:space="preserve">quare ex
<pb o="346" file="0384" n="385" rhead="Apollonij Pergæi"/>
æqualitate quadratum A C ad quadratum @diametri figuræ y O eandem, <lb/>proportionem habet, quàm C G, ſeu A H in H C ad duo quadrata ip-<lb/>ſius C G, atque ipſius C H: </s>
  <s xml:id="echoid-s12070" xml:space="preserve">igitur A H in H V maiorem ad duo qua-<lb/>
<anchor type="figure" xlink:label="fig-0384-01a" xlink:href="fig-0384-01"/>
drata ex V G minori, &amp; </s>
  <s xml:id="echoid-s12071" xml:space="preserve">ex V H, ſeu vt quadratum A C ad quadratum <lb/>diametri figuræ S T (19. </s>
  <s xml:id="echoid-s12072" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s12073" xml:space="preserve">maiorem proportionem habebit, quàm <lb/>A H in H C minorem ad duo quadrata ex G C, &amp; </s>
  <s xml:id="echoid-s12074" xml:space="preserve">C H maiora, ſcili-<lb/>cet vt quadratum A C ad quadratum diametri figuræ y O (19. </s>
  <s xml:id="echoid-s12075" xml:space="preserve">ex 7.)</s>
  <s xml:id="echoid-s12076" xml:space="preserve">; <lb/></s>
  <s xml:id="echoid-s12077" xml:space="preserve">igitur quadratum diametri figuræ y O maior eſt quàm quadratum diametri <lb/>figuræ S T. </s>
  <s xml:id="echoid-s12078" xml:space="preserve">Si verò G M non fuerit minor quàm V H; </s>
  <s xml:id="echoid-s12079" xml:space="preserve">vtique duo qua-<lb/>drata ex G M, &amp; </s>
  <s xml:id="echoid-s12080" xml:space="preserve">M H non erunt maiora duobus quadratis ex V G, &amp; </s>
  <s xml:id="echoid-s12081" xml:space="preserve"><lb/>ex V H: </s>
  <s xml:id="echoid-s12082" xml:space="preserve">at A H in M H ad duo quadrata ex G M, &amp; </s>
  <s xml:id="echoid-s12083" xml:space="preserve">ex M H, nempe <lb/>quadratum A C ad quadratum diametri figuræ P Q habebit maiorem, <lb/>proportionem, quàm A H ad H V ad duo quadrata ex V H, &amp; </s>
  <s xml:id="echoid-s12084" xml:space="preserve">ex V <lb/>G, ſcilicet vt quadratum A C ad quadratum diametri figuræ S T; </s>
  <s xml:id="echoid-s12085" xml:space="preserve">igi-<lb/>tur diameter figuræ S T maior eſt diametro figuræ P Q. </s>
  <s xml:id="echoid-s12086" xml:space="preserve">Eadem prorſus <lb/>oſtendentur, quando punctum V cadit vltra punctum D ad partes A in-<lb/>ter puncta D, &amp; </s>
  <s xml:id="echoid-s12087" xml:space="preserve">M. </s>
  <s xml:id="echoid-s12088" xml:space="preserve">Et hoc erat propoſitum.</s>
  <s xml:id="echoid-s12089" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1037" type="float" level="2" n="1">
  <figure xlink:label="fig-0383-01" xlink:href="fig-0383-01a">
    <image file="0383-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0383-01"/>
  </figure>
  <figure xlink:label="fig-0384-01" xlink:href="fig-0384-01a">
    <image file="0384-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0384-01"/>
  </figure>
</div>
<pb o="347" file="0385" n="386" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div1039" type="section" level="1" n="329">
<head xml:id="echoid-head407" xml:space="preserve">PROPOSITIO XXXXVIII.</head>
<p>
  <s xml:id="echoid-s12090" xml:space="preserve">S It iam duplum quadrati A C maius quadrato C A F, erit duplum <lb/>quadrati A H maius quadrato G H: </s>
  <s xml:id="echoid-s12091" xml:space="preserve">ponatur duplum quadrati H M <lb/>æquale quadrato G H: </s>
  <s xml:id="echoid-s12092" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12093" xml:space="preserve">ducatur ad axim perpendicularis M N; </s>
  <s xml:id="echoid-s12094" xml:space="preserve">iun-<lb/>
<anchor type="figure" xlink:label="fig-0385-01a" xlink:href="fig-0385-01"/>
gaturque A N, eiuſque diameter P Q extendatur, erit H M ad M G, <lb/>vt P Q ad P R (7. </s>
  <s xml:id="echoid-s12095" xml:space="preserve">ex 7.)</s>
  <s xml:id="echoid-s12096" xml:space="preserve">; </s>
  <s xml:id="echoid-s12097" xml:space="preserve">ergo, &amp; </s>
  <s xml:id="echoid-s12098" xml:space="preserve">quadratum H M ad quadratum H <lb/>G erit, vt quadratum P Q ad quadratum P R, &amp; </s>
  <s xml:id="echoid-s12099" xml:space="preserve">quadratum H M ad <lb/>duo quadrata ex H M, &amp; </s>
  <s xml:id="echoid-s12100" xml:space="preserve">ex M G eandem proportionẽ habebit, quàm <lb/>quadratum P Q ad quadratum diametri ſuæ figuræ: </s>
  <s xml:id="echoid-s12101" xml:space="preserve">educatur poſtea dia-<lb/>meter I L inter A, &amp; </s>
  <s xml:id="echoid-s12102" xml:space="preserve">B, &amp; </s>
  <s xml:id="echoid-s12103" xml:space="preserve">erectum illius ſit I K ad quàm ordinatim <lb/>ducta ſit A B, &amp; </s>
  <s xml:id="echoid-s12104" xml:space="preserve">ad axim perpendicularis ſit B E erit quadratum M H, <lb/>nec non G H in H D æquale dimidio quadrati H G; </s>
  <s xml:id="echoid-s12105" xml:space="preserve">igitur G H ad M
<pb o="348" file="0386" n="387" rhead="Apollonij Pergæi"/>
H erit vt M H ad H D: </s>
  <s xml:id="echoid-s12106" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12107" xml:space="preserve">comparando homologorum differentias erit <lb/>M G ad M D, vt G H ad H M: </s>
  <s xml:id="echoid-s12108" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12109" xml:space="preserve">propterea duplum G H in M D, ſeu <lb/>quadruplum H D in D M eſt æquale duplo G M in M H: </s>
  <s xml:id="echoid-s12110" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12111" xml:space="preserve">propterea <lb/>duplum G M in M H maius erit quàm duplum G E in M H; </s>
  <s xml:id="echoid-s12112" xml:space="preserve">ponatur <lb/>communiter duplum E M in H M cum quadruplo quadrati M D, &amp; </s>
  <s xml:id="echoid-s12113" xml:space="preserve">fiat <lb/>D d æqualis D M, fiet duplum E d in M H maius quadrato H M cum <lb/>
<anchor type="figure" xlink:label="fig-0386-01a" xlink:href="fig-0386-01"/>
quadrato M G; </s>
  <s xml:id="echoid-s12114" xml:space="preserve">igitur d E in E M bis ſumptum ad duplum E d in M H. <lb/></s>
  <s xml:id="echoid-s12115" xml:space="preserve">nempe E M ad M H minorem proportionem habebit, quàm duplum d <lb/>E in E M ad duo quadrata ex M G, &amp; </s>
  <s xml:id="echoid-s12116" xml:space="preserve">ex M H: </s>
  <s xml:id="echoid-s12117" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12118" xml:space="preserve">componendo E H <lb/>ad M H, ſeu E H in H A ad M H in H A minorem proportionem habe-<lb/>bit, quàm duplum d E in E M vna cum quadratis ex M H, &amp; </s>
  <s xml:id="echoid-s12119" xml:space="preserve">ex <lb/>M G, quæ æqualia ſunt duobus quadratis H E, &amp; </s>
  <s xml:id="echoid-s12120" xml:space="preserve">G E ad duo quadra-<lb/>ta ex M G, &amp; </s>
  <s xml:id="echoid-s12121" xml:space="preserve">ex H M. </s>
  <s xml:id="echoid-s12122" xml:space="preserve">Et ſic pariter oſtendetur, quod quadratum H <lb/>A ad H E in H A minorem proportionem habebit, quàm duo quadrata <lb/>ex H A, &amp; </s>
  <s xml:id="echoid-s12123" xml:space="preserve">ex A G ad duo quadrata ex H E, &amp; </s>
  <s xml:id="echoid-s12124" xml:space="preserve">ex E G. </s>
  <s xml:id="echoid-s12125" xml:space="preserve">Atque de-<lb/>monſtrabitur quemadmodum antea dictum eſt, quod quadratum diame-
<pb o="349" file="0387" n="388" rhead="Conicor. Lib. VII."/>
tri figuræ P Q minus eſt quadrato diametri figuræ I L, &amp; </s>
  <s xml:id="echoid-s12126" xml:space="preserve">quadratum <lb/>diametri figuræ I L minus eſt quadrato diametri figuræ A C. </s>
  <s xml:id="echoid-s12127" xml:space="preserve">Ponãtur <lb/>poſtea diametri S T, &amp; </s>
  <s xml:id="echoid-s12128" xml:space="preserve">γ O vltra diametrum P Q, ſitque A X ordinatim <lb/>applicata ad diametrum S T, &amp; </s>
  <s xml:id="echoid-s12129" xml:space="preserve">V X ad axim perpendicularis ſit, oſten-<lb/>detur (quemadmodum in præcedentibus dictum eſt) quod diameter fi-<lb/>guræ P Q minor ſit diametro figuræ S T, &amp; </s>
  <s xml:id="echoid-s12130" xml:space="preserve">diameter figuræ S T minor <lb/>ſit diametro figuræ γ O, vbicunque ſecet ad axim perpendicularis X V <lb/>ipſam A C. </s>
  <s xml:id="echoid-s12131" xml:space="preserve">Et hoc erat oſtendendum.</s>
  <s xml:id="echoid-s12132" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1039" type="float" level="2" n="1">
  <figure xlink:label="fig-0385-01" xlink:href="fig-0385-01a">
    <image file="0385-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0385-01"/>
  </figure>
  <figure xlink:label="fig-0386-01" xlink:href="fig-0386-01a">
    <image file="0386-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0386-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1041" type="section" level="1" n="330">
<head xml:id="echoid-head408" xml:space="preserve">In Sectionem IX. Propoſit. XXXXI. <lb/>XXXXVII. &amp; XXXXVIII.</head>
<head xml:id="echoid-head409" xml:space="preserve">LEMMA. XIII.</head>
<p style="it">
  <s xml:id="echoid-s12133" xml:space="preserve">Sl recta linea G H ſecetur bifariam in D, &amp; </s>
  <s xml:id="echoid-s12134" xml:space="preserve">non bifariam in O, <lb/>E, atque fiat G a æqualis H E; </s>
  <s xml:id="echoid-s12135" xml:space="preserve">ſi quidem proportio dupli O H <lb/>ad H G eadem fuerit proportioni G H ad H E, erit duplum rectan-<lb/>guli ex differentia ipſarum E H, G O in H O æquale quadratis ex G <lb/>O, &amp; </s>
  <s xml:id="echoid-s12136" xml:space="preserve">ex O H: </s>
  <s xml:id="echoid-s12137" xml:space="preserve">ſi verò proportio illa maior fueri erit rectangulum ma-<lb/>ius quadratis; </s>
  <s xml:id="echoid-s12138" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12139" xml:space="preserve">ſi eadem proportio fuerit minor, idipſum rectangulum <lb/>quadratis minus erit.</s>
  <s xml:id="echoid-s12140" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12141" xml:space="preserve">Et primo quia duplum O H <lb/>
<anchor type="figure" xlink:label="fig-0387-01a" xlink:href="fig-0387-01"/>
ad H G eſt vt G H ad H E, <lb/>ergo duplum rectanguli O H <lb/>E æquale erit quadrato ex G <lb/>H; </s>
  <s xml:id="echoid-s12142" xml:space="preserve">auferatur cõmuniter du-<lb/>plum rectanguli H O G, quia <lb/>H O eſt communis rectangulo-<lb/>rum altitudo, remanet duplũ <lb/>rectanguli ex differentia ipſa-<lb/>rum E H, G O, ſeu ex diffe-<lb/>rentia ipſarum G a, &amp; </s>
  <s xml:id="echoid-s12143" xml:space="preserve">G O <lb/>in H O, ſeu remanet duplum rectanguli a O H æquale quaàrato H G minus <lb/>duplo rectanguli G O H: </s>
  <s xml:id="echoid-s12144" xml:space="preserve">huic verò differentiæ æqualia ſunt duo quadrata ex <lb/>G O, &amp; </s>
  <s xml:id="echoid-s12145" xml:space="preserve">ex H O, ergo duplum rectanguli a O H æquale eſt ſummæ quadrato-<lb/>rum ex G O, &amp; </s>
  <s xml:id="echoid-s12146" xml:space="preserve">ex O H.</s>
  <s xml:id="echoid-s12147" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1041" type="float" level="2" n="1">
  <figure xlink:label="fig-0387-01" xlink:href="fig-0387-01a">
    <image file="0387-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0387-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s12148" xml:space="preserve">Secundo, quia duplum O H ad H G maiorem proportionem habet, quàm <lb/>G H ad H E, ergo duplum rectanguli O H E maius erit quadrato G H, &amp; </s>
  <s xml:id="echoid-s12149" xml:space="preserve"><lb/>ablato communiter duplo rectanguli G O H erit duplum rectanguli ex differen-<lb/>tia ipſarum E H, &amp; </s>
  <s xml:id="echoid-s12150" xml:space="preserve">G O in H O maius, quàm ſumma quadratorum ex G O, <lb/>&amp; </s>
  <s xml:id="echoid-s12151" xml:space="preserve">ex H O.</s>
  <s xml:id="echoid-s12152" xml:space="preserve"/>
</p>
<pb o="350" file="0388" n="389" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s12153" xml:space="preserve">Tertio ſi duplum O H ad H G minorem proportionem habuerit, quàm G H <lb/>ad H E, eodem progreſſu oſtendetur, quod duplum rectanguli ex differentia <lb/>ipſarum E H, &amp; </s>
  <s xml:id="echoid-s12154" xml:space="preserve">G O in H O minus eſt quadratis ex G O, &amp; </s>
  <s xml:id="echoid-s12155" xml:space="preserve">ex H O, quod erat <lb/>propoſitum.</s>
  <s xml:id="echoid-s12156" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1043" type="section" level="1" n="331">
<head xml:id="echoid-head410" xml:space="preserve">LEMMA XIV.</head>
<p style="it">
  <s xml:id="echoid-s12157" xml:space="preserve">Ilſdem poſitis ſit G E minimum ſegmentorum, dico quod duo qua-<lb/>drata ex E H, &amp; </s>
  <s xml:id="echoid-s12158" xml:space="preserve">ex G E, ſcilicet ex maximo, &amp; </s>
  <s xml:id="echoid-s12159" xml:space="preserve">minimo ſeg-<lb/>mentorum æqualia ſunt duobus quadratis ex O H, &amp; </s>
  <s xml:id="echoid-s12160" xml:space="preserve">ex G O inter-<lb/>medĳs ſegmentis vna cum duplo rectanguli ſub differentĳs minimæ G <lb/>E à duabus intermedĳs G O, &amp; </s>
  <s xml:id="echoid-s12161" xml:space="preserve">H O.</s>
  <s xml:id="echoid-s12162" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12163" xml:space="preserve">Fiat H a æqualis G E, <lb/>
<anchor type="figure" xlink:label="fig-0388-01a" xlink:href="fig-0388-01"/>
ergo O a erit differentia ipſa-<lb/>rum E H, &amp; </s>
  <s xml:id="echoid-s12164" xml:space="preserve">G E, ſicuti O <lb/>E eſt differentia ipſarum G O, <lb/>&amp; </s>
  <s xml:id="echoid-s12165" xml:space="preserve">G E. </s>
  <s xml:id="echoid-s12166" xml:space="preserve">Et quia duo quadra-<lb/>ta ex maximo, &amp; </s>
  <s xml:id="echoid-s12167" xml:space="preserve">ex mini-<lb/>mo ſegmentorum, ſcilicet ex <lb/>H E, &amp; </s>
  <s xml:id="echoid-s12168" xml:space="preserve">ex E G æqualia ſunt <lb/>duplo quadrati ex G D ſe-<lb/>miße totius, cũ duplo quadrati <lb/>ex E D intermedia ſectione; <lb/></s>
  <s xml:id="echoid-s12169" xml:space="preserve">eſtque duplum quadrati ex E D ſemiſſe ipſius E a æquale duplo rectanguli E O <lb/>a ex inæqualibus ſegmentis vna cum duplo quadrati ex intermedia ſectione O <lb/>D, ergo duo quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s12170" xml:space="preserve">ex E H æqualia ſunt his omnibus ſpatĳs, <lb/>ſcilicet duplo quadrati ex G D, &amp; </s>
  <s xml:id="echoid-s12171" xml:space="preserve">duplo quadrati ex D O cum duplo rectan-<lb/>guli E O a, ſed duo quadrata ex inæqualibus ſegmentis G O, &amp; </s>
  <s xml:id="echoid-s12172" xml:space="preserve">ex O H æqua-<lb/>lia ſunt duplo quadrati ex ſemiſſe totius G D cum duplo quadrati ex interme-<lb/>dia ſectione O D, igitur exceßus ſummæ quadratorum ex G E, &amp; </s>
  <s xml:id="echoid-s12173" xml:space="preserve">ex E H, <lb/>ſupra ſummam quadratorum ex G O, &amp; </s>
  <s xml:id="echoid-s12174" xml:space="preserve">O H æqualis eſt duplo rectanguli ex E <lb/>O a, quod erat oſtendendum.</s>
  <s xml:id="echoid-s12175" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1043" type="float" level="2" n="1">
  <figure xlink:label="fig-0388-01" xlink:href="fig-0388-01a">
    <image file="0388-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0388-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1045" type="section" level="1" n="332">
<head xml:id="echoid-head411" xml:space="preserve">LEMMA XV.</head>
<p style="it">
  <s xml:id="echoid-s12176" xml:space="preserve">IN ellypſi, cuius axis A C, erectus A F, diameter I L, eiuſq; </s>
  <s xml:id="echoid-s12177" xml:space="preserve">erectus <lb/>I K, &amp; </s>
  <s xml:id="echoid-s12178" xml:space="preserve">latus C E, &amp; </s>
  <s xml:id="echoid-s12179" xml:space="preserve">ſimiliter altera diameter Q P, cuius ere-<lb/>ctus P R, &amp; </s>
  <s xml:id="echoid-s12180" xml:space="preserve">latus C O: </s>
  <s xml:id="echoid-s12181" xml:space="preserve">dico quod duplum rectanguli ex differentia <lb/>ipſarum E H, G O, in H O à duobus quadratis ex G O, &amp; </s>
  <s xml:id="echoid-s12182" xml:space="preserve">ex O
<pb o="351" file="0389" n="390" rhead="Conicor. Lib. VII."/>
H, atque aggregatum quadratorum larerum I L, &amp; </s>
  <s xml:id="echoid-s12183" xml:space="preserve">I K figuræ dia-<lb/>metri I L ab aggregato quadratorum laterum P Q, &amp; </s>
  <s xml:id="echoid-s12184" xml:space="preserve">P R fignræ al-<lb/>terius diametri, vna deficiunt, aut vna æqualia ſunt, vel vna exce-<lb/>dunt.</s>
  <s xml:id="echoid-s12185" xml:space="preserve"/>
</p>
  <figure>
    <image file="0389-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0389-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s12186" xml:space="preserve">Fiat O d differentia ipſarum E H, &amp; </s>
  <s xml:id="echoid-s12187" xml:space="preserve">G O, &amp; </s>
  <s xml:id="echoid-s12188" xml:space="preserve">primo quia duplum rectan-<lb/>guli ex d O H æquale eſt quadratis ex G O, &amp; </s>
  <s xml:id="echoid-s12189" xml:space="preserve">ex H O, ergo duplum rectan-<lb/>guli d O E ad duplum rectanguli d O H, ſeu O E ad H O eandem proportio-<lb/>nem habet, quàm duplum rectanguli d O E ad duo quadrata ex G O, &amp; </s>
  <s xml:id="echoid-s12190" xml:space="preserve">ex H <lb/>O, &amp; </s>
  <s xml:id="echoid-s12191" xml:space="preserve">componendo, erit E H ad H O, ſeu rectangulum E H A ad rectangu-<lb/>
<anchor type="note" xlink:label="note-0389-01a" xlink:href="note-0389-01"/>
lum O H A vt àuo quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s12192" xml:space="preserve">ex E H ad duo quadrata ex G O, &amp; </s>
  <s xml:id="echoid-s12193" xml:space="preserve"><lb/>ex H O, &amp; </s>
  <s xml:id="echoid-s12194" xml:space="preserve">permutando rectangulum E H A ad quadrata ex G E, &amp; </s>
  <s xml:id="echoid-s12195" xml:space="preserve">ex E <lb/>
<anchor type="note" xlink:label="note-0389-02a" xlink:href="note-0389-02"/>
H, ſeu quadratum ex A C ad quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12196" xml:space="preserve">ex I K, vel ad quadrata <lb/>ex A C, &amp; </s>
  <s xml:id="echoid-s12197" xml:space="preserve">ex A F eandem proportionem habebit, quàm rectangulum O H A <lb/>
<anchor type="note" xlink:label="note-0389-03a" xlink:href="note-0389-03"/>
ad quadrata ex G O, &amp; </s>
  <s xml:id="echoid-s12198" xml:space="preserve">ex H O, vel quadratum A C ad duo quadrata ex P <lb/>Q, &amp; </s>
  <s xml:id="echoid-s12199" xml:space="preserve">ex P R, quapropter duo quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12200" xml:space="preserve">ex I K, ſeu ex A C, &amp; </s>
  <s xml:id="echoid-s12201" xml:space="preserve"><lb/>A F æqualia erunt duobus quadratis ex P Q, &amp; </s>
  <s xml:id="echoid-s12202" xml:space="preserve">ex P R.</s>
  <s xml:id="echoid-s12203" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1045" type="float" level="2" n="1">
<note position="right" xlink:label="note-0389-01" xlink:href="note-0389-01a" xml:space="preserve">Lem. 14. <lb/>huius.</note>
<note position="right" xlink:label="note-0389-02" xlink:href="note-0389-02a" xml:space="preserve">17. huius.</note>
<note position="right" xlink:label="note-0389-03" xlink:href="note-0389-03a" xml:space="preserve">Ibidem.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12204" xml:space="preserve">Secundo ſit duplum rectanguli d O H minus quadratis ex G O, &amp; </s>
  <s xml:id="echoid-s12205" xml:space="preserve">ex H O. <lb/></s>
  <s xml:id="echoid-s12206" xml:space="preserve">duplum rectanguli d O E ad duplum rectanguli d O H, ſeu O E ad H O ha-<lb/>bebit maiorem proportionem, quàm duplum rectanguli d O E ad duo quadrata <lb/>ex G O, &amp; </s>
  <s xml:id="echoid-s12207" xml:space="preserve">ex H O, &amp; </s>
  <s xml:id="echoid-s12208" xml:space="preserve">rurſus componendo ex lem. </s>
  <s xml:id="echoid-s12209" xml:space="preserve">2. </s>
  <s xml:id="echoid-s12210" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s12211" xml:space="preserve">5. </s>
  <s xml:id="echoid-s12212" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12213" xml:space="preserve">ex lem. </s>
  <s xml:id="echoid-s12214" xml:space="preserve">14. </s>
  <s xml:id="echoid-s12215" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12216" xml:space="preserve"><lb/>permutando, atque ex 17. </s>
  <s xml:id="echoid-s12217" xml:space="preserve">propoſit. </s>
  <s xml:id="echoid-s12218" xml:space="preserve">huius habebit idem quadratum A C ad duo <lb/>quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12219" xml:space="preserve">ex I K maiorem proportionem, quàm ad duo quadrata ex <lb/>P Q, &amp; </s>
  <s xml:id="echoid-s12220" xml:space="preserve">ex P R: </s>
  <s xml:id="echoid-s12221" xml:space="preserve">quapropter duo quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12222" xml:space="preserve">ex I K minora erunt <lb/>duobus quadratis ex P Q, &amp; </s>
  <s xml:id="echoid-s12223" xml:space="preserve">ex P R.</s>
  <s xml:id="echoid-s12224" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12225" xml:space="preserve">Tertio ſit rectangulum d O H maius duobus quadratis ex G O, &amp; </s>
  <s xml:id="echoid-s12226" xml:space="preserve">ex H O. <lb/></s>
  <s xml:id="echoid-s12227" xml:space="preserve">duplum rectanguli ex d O E ad duplum rectanguli d O H, ſeu O E ad H O ha-
<pb o="352" file="0390" n="391" rhead="Apollonij Pergæi"/>
bebit minorem proportionem, quàm duplum rectanguli d O E ad duo quadrata <lb/>ex G O, &amp; </s>
  <s xml:id="echoid-s12228" xml:space="preserve">ex O H, &amp; </s>
  <s xml:id="echoid-s12229" xml:space="preserve">componendo ex lem. </s>
  <s xml:id="echoid-s12230" xml:space="preserve">14. </s>
  <s xml:id="echoid-s12231" xml:space="preserve">permutando, &amp; </s>
  <s xml:id="echoid-s12232" xml:space="preserve">ex 17. </s>
  <s xml:id="echoid-s12233" xml:space="preserve">hu-<lb/>
<anchor type="figure" xlink:label="fig-0390-01a" xlink:href="fig-0390-01"/>
ius, tandem erunt duo quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12234" xml:space="preserve">ex I K maiora duobus quadratis <lb/>ex P Q, &amp; </s>
  <s xml:id="echoid-s12235" xml:space="preserve">ex P R.</s>
  <s xml:id="echoid-s12236" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1046" type="float" level="2" n="2">
  <figure xlink:label="fig-0390-01" xlink:href="fig-0390-01a">
    <image file="0390-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0390-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s12237" xml:space="preserve">Sl in ellypſi termini E, O laterum <lb/>
<anchor type="figure" xlink:label="fig-0390-02a" xlink:href="fig-0390-02"/>
C E, C O, diametrorum I L, &amp; </s>
  <s xml:id="echoid-s12238" xml:space="preserve"><lb/>P Q cadant hinc inde à centro D, ſitque <lb/>D O maior quàm D E, dico quod qua-<lb/>drata ex P Q, &amp; </s>
  <s xml:id="echoid-s12239" xml:space="preserve">ex P R maiora ſunt <lb/>quadratis ex I L, &amp; </s>
  <s xml:id="echoid-s12240" xml:space="preserve">ex I K.</s>
  <s xml:id="echoid-s12241" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1047" type="float" level="2" n="3">
  <figure xlink:label="fig-0390-02" xlink:href="fig-0390-02a">
    <image file="0390-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0390-02"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s12242" xml:space="preserve">Quia O H minor eſt, quàm E H, ſed duo <lb/>quadrata ex G O maximo, &amp; </s>
  <s xml:id="echoid-s12243" xml:space="preserve">O H minimo <lb/>ſegmentorum eiuſdem rectæ lineæ G H maio-<lb/>ra ſunt duobus quadratis ex G E, &amp; </s>
  <s xml:id="echoid-s12244" xml:space="preserve">ex E <lb/>H intermedĳs ſegmentis; </s>
  <s xml:id="echoid-s12245" xml:space="preserve">ergo O H ad E H, <lb/>minor ad maiorem ſeu rectangulum O H A <lb/>ad rectangulum E H A minorem proportionem <lb/>habet, quàm maior ſumma quadratorum ex <lb/>G O, &amp; </s>
  <s xml:id="echoid-s12246" xml:space="preserve">ex O H ad minorem ſummam qua-<lb/>dratorum ex G E, &amp; </s>
  <s xml:id="echoid-s12247" xml:space="preserve">ex E H, &amp; </s>
  <s xml:id="echoid-s12248" xml:space="preserve">per-<lb/>mutando rectangulum O H A ad duo qua-<lb/>drata ex G O, &amp; </s>
  <s xml:id="echoid-s12249" xml:space="preserve">ex O H, ſeu quadratum <lb/>
<anchor type="note" xlink:label="note-0390-01a" xlink:href="note-0390-01"/>
A C ad duo quadrata ex P Q, &amp; </s>
  <s xml:id="echoid-s12250" xml:space="preserve">ex P R
<pb o="353" file="0391" n="392" rhead="Conicor. Lib. VII."/>
minorem proportionem habebit, quàm rectangulum E H A ad duo quadrata <lb/>ex G E, &amp; </s>
  <s xml:id="echoid-s12251" xml:space="preserve">ex E H, ſen quàm quadratum A C ad duo quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12252" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0391-01a" xlink:href="note-0391-01"/>
ex I K: </s>
  <s xml:id="echoid-s12253" xml:space="preserve">igitur duo quadrata ex P Q, &amp; </s>
  <s xml:id="echoid-s12254" xml:space="preserve">ex\P R maiora ſunt duobus quadra-<lb/>tis ex I L, &amp; </s>
  <s xml:id="echoid-s12255" xml:space="preserve">ex I K, quod erat oſtendendum.</s>
  <s xml:id="echoid-s12256" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1048" type="float" level="2" n="4">
<note position="left" xlink:label="note-0390-01" xlink:href="note-0390-01a" xml:space="preserve">17. huius.</note>
<note position="right" xlink:label="note-0391-01" xlink:href="note-0391-01a" xml:space="preserve">17. huíus.</note>
</div>
</div>
<div xml:id="echoid-div1050" type="section" level="1" n="333">
<head xml:id="echoid-head412" xml:space="preserve">Notæ in Propoſit. XXXXI.</head>
<p style="it">
  <s xml:id="echoid-s12257" xml:space="preserve">IN ellypſi, cuius axis maior A C, quia rectangulum A H E ad quadratum <lb/>H G eſt, vt quadratum A C ad quadratum ex L I K, vel ad quadratum <lb/>
<anchor type="note" xlink:label="note-0391-02a" xlink:href="note-0391-02"/>
ex C A F, atq; </s>
  <s xml:id="echoid-s12258" xml:space="preserve">quadratum ex G H ad rectangulum A H M eandem proportio-<lb/>
<anchor type="figure" xlink:label="fig-0391-01a" xlink:href="fig-0391-01"/>
nem habet, quàm quadratum ex Q P R ad quadratum A C, igitur ex æquali <lb/>perturbata rectangulum A H E maius ad minus rectangulum A H M eandem <lb/>proportionem habet, quàm quadratum ex Q P R ad quadratum ex L I K, vel <lb/>ad quadratum ex C A F: </s>
  <s xml:id="echoid-s12259" xml:space="preserve">eſtque rectangulum A H E maius rectangulo A H <lb/>M, ergo quadratũ ex ſumma Q P R maius eſt quadrato ex ſumma L I K, &amp; </s>
  <s xml:id="echoid-s12260" xml:space="preserve"><lb/>propterea linearũ sũma Q P R maior erit, quàm sũma L I K, vel quàm ſum-
<pb o="354" file="0392" n="393" rhead="Apollonij Pergæi"/>
ma C A F. </s>
  <s xml:id="echoid-s12261" xml:space="preserve">Tandem quia rectangulum A H M ad quadratum ex ſumma H <lb/>
<anchor type="note" xlink:label="note-0392-01a" xlink:href="note-0392-01"/>
M G eandem proportionem habet, quàm quadratum A C ad quadratum ex Q <lb/>P R, ſed quadratum ex H C G ad rectangulum ex A H C eandem proportionẽ <lb/>habet, quàm quadratnm ex sũma Y O f ad quadratum A C, (@o quod H C eſt <lb/>intercepta comparata diametri Y O, cum Y O ſecet bifariam ad eam ordinatim <lb/>applicatam A C, atque ab eodem pun-<lb/>
<anchor type="figure" xlink:label="fig-0392-01a" xlink:href="fig-0392-01"/>
cto C perpendicularis ad axim ducta <lb/>cadat ſuper idem punctum C), igitur <lb/>ex æquali perturbata rectangulum A H <lb/>M maius ad minus rectangulum ex A <lb/>H C eandem proportionem habet, quàm <lb/>quadratum ex ſumma Y O f ad qua-<lb/>dratum ex ſumma Q P R, &amp; </s>
  <s xml:id="echoid-s12262" xml:space="preserve">propte-<lb/>rea ſumma laterum Y O f maior erit, <lb/>quàm ſumma Q P R.</s>
  <s xml:id="echoid-s12263" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1050" type="float" level="2" n="1">
<note position="right" xlink:label="note-0391-02" xlink:href="note-0391-02a" xml:space="preserve">Prop. 16. <lb/>huius.</note>
  <figure xlink:label="fig-0391-01" xlink:href="fig-0391-01a">
    <image file="0391-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0391-01"/>
  </figure>
<note position="left" xlink:label="note-0392-01" xlink:href="note-0392-01a" xml:space="preserve">ex 16. <lb/>huius. <lb/>lbidem.</note>
  <figure xlink:label="fig-0392-01" xlink:href="fig-0392-01a">
    <image file="0392-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0392-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1052" type="section" level="1" n="334">
<head xml:id="echoid-head413" xml:space="preserve">Notæ in Propoſit. XXXXVII.</head>
<p style="it">
  <s xml:id="echoid-s12264" xml:space="preserve">QVia duplum quadrati A C non eſt maius quadrato ex C A F, ergo du-<lb/>plum quadrati ex A H æquale, aut minus erit quadrato ex ſumma G <lb/>H, eſtque duplum rectanguli ex E H A, vel ex E H M minus <lb/>duplo quadrati A H, igitur minus quoque erit quadrato ex G H, igitur du-<lb/>plum M H ad G H minorem proportionem habet, quàm G H ad E H, ergo <lb/>
<anchor type="note" xlink:label="note-0392-02a" xlink:href="note-0392-02"/>
duplum rectanguli ex differentia ipſarum E H G M in M H minus eſt duobus <lb/>quadratis ex G M, &amp; </s>
  <s xml:id="echoid-s12265" xml:space="preserve">ex H M: </s>
  <s xml:id="echoid-s12266" xml:space="preserve">quare duo quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12267" xml:space="preserve">ex I K minora <lb/>
<anchor type="note" xlink:label="note-0392-03a" xlink:href="note-0392-03"/>
erunt duobus quadratis ex Q P, &amp; </s>
  <s xml:id="echoid-s12268" xml:space="preserve">ex P R, &amp; </s>
  <s xml:id="echoid-s12269" xml:space="preserve">ſic duo quadrata ex Q P, &amp; </s>
  <s xml:id="echoid-s12270" xml:space="preserve"><lb/>ex P R minora ſunt duobus quadratis ex T S, &amp; </s>
  <s xml:id="echoid-s12271" xml:space="preserve">ex S Z.</s>
  <s xml:id="echoid-s12272" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1052" type="float" level="2" n="1">
<note position="left" xlink:label="note-0392-02" xlink:href="note-0392-02a" xml:space="preserve">Lem. 13. <lb/>huius.</note>
<note position="left" xlink:label="note-0392-03" xlink:href="note-0392-03a" xml:space="preserve">Lem. 15. <lb/>huius.</note>
</div>
<pb o="355" file="0393" n="394" rhead="Conicor. Lib. VII."/>
  <figure>
    <image file="0393-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0393-01"/>
  </figure>
</div>
<div xml:id="echoid-div1054" type="section" level="1" n="335">
<head xml:id="echoid-head414" xml:space="preserve">Notæ in Propoſit. XXXXVIII.</head>
<p style="it">
  <s xml:id="echoid-s12273" xml:space="preserve">QVia ex hypotheſi duplum quadrati A C maius eſt quadrato ex C A F, <lb/>ergo duplum quadrati ex A H maius erit quadrato ex H G. </s>
  <s xml:id="echoid-s12274" xml:space="preserve">Fiat igitur <lb/>quadratum ex M H æquale ſemiquadrato G H, &amp; </s>
  <s xml:id="echoid-s12275" xml:space="preserve">lateris C M fiant <lb/>duæ diametri Q P, &amp; </s>
  <s xml:id="echoid-s12276" xml:space="preserve">q p, quarum erecta ſint P R, &amp; </s>
  <s xml:id="echoid-s12277" xml:space="preserve">p r: </s>
  <s xml:id="echoid-s12278" xml:space="preserve">Dico duplum <lb/>quadrati Q P æquale eße quadrato ex ſumma laterum Q P R: </s>
  <s xml:id="echoid-s12279" xml:space="preserve">Quia Q P ad <lb/>P R eſt vt H M ad M G, &amp; </s>
  <s xml:id="echoid-s12280" xml:space="preserve">antecedentes ad terminorum ſummas, &amp; </s>
  <s xml:id="echoid-s12281" xml:space="preserve">eorum <lb/>
<anchor type="note" xlink:label="note-0393-01a" xlink:href="note-0393-01"/>
quadrata proportionalia erunt, ſcilicet quadratum Q P ad quadratum ex Q P <lb/>R eandem proportionem habebit, quàm quadratum ex M H ad quadratum ex <lb/>H G: </s>
  <s xml:id="echoid-s12282" xml:space="preserve">erat autem quadratum M H ſubduplum quadrati ex H G, igitur qua-<lb/>dratum ex P Q ſubduplum eſt quadrati ex Q P R. </s>
  <s xml:id="echoid-s12283" xml:space="preserve">Eadem ratione quadra-<lb/>tum ex q p ſubduplum erit quadrati ex q p r, &amp; </s>
  <s xml:id="echoid-s12284" xml:space="preserve">diametri Q P, &amp; </s>
  <s xml:id="echoid-s12285" xml:space="preserve">q p æqua-<lb/>les erunt, cum æque recedant ab axi, &amp; </s>
  <s xml:id="echoid-s12286" xml:space="preserve">habeant commune latus C M.</s>
  <s xml:id="echoid-s12287" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1054" type="float" level="2" n="1">
<note position="right" xlink:label="note-0393-01" xlink:href="note-0393-01a" xml:space="preserve">Prop. 7. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12288" xml:space="preserve">Poſtea quia punctum E cadit inter M, &amp; </s>
  <s xml:id="echoid-s12289" xml:space="preserve">A, erit duplum rectanguli M H <lb/>E maius duplo quadrati ex M H, ſeu maius quadrato G H, &amp; </s>
  <s xml:id="echoid-s12290" xml:space="preserve">propterea du-<lb/>plum M H ad H G maiorem proportionem habebit, quàm G H ad H E, ergo
<pb o="356" file="0394" n="395" rhead="Apollonij Pergæi"/>
duplum rectanguli ex differentia ipſarum E H, &amp; </s>
  <s xml:id="echoid-s12291" xml:space="preserve">G M in M H maius erit <lb/>
<anchor type="note" xlink:label="note-0394-01a" xlink:href="note-0394-01"/>
duobus quadratis ex G M, &amp; </s>
  <s xml:id="echoid-s12292" xml:space="preserve">ex M H, &amp; </s>
  <s xml:id="echoid-s12293" xml:space="preserve">propterea duo quadrata ex I L, &amp; </s>
  <s xml:id="echoid-s12294" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0394-02a" xlink:href="note-0394-02"/>
ex I K ſimul ſumpta maiora erunt duobus quadratis ex Q P, &amp; </s>
  <s xml:id="echoid-s12295" xml:space="preserve">ex P R.</s>
  <s xml:id="echoid-s12296" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1055" type="float" level="2" n="2">
<note position="left" xlink:label="note-0394-01" xlink:href="note-0394-01a" xml:space="preserve">Lem. 13.</note>
<note position="left" xlink:label="note-0394-02" xlink:href="note-0394-02a" xml:space="preserve">Lem. 15. <lb/>huius.</note>
</div>
  <figure>
    <image file="0394-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0394-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s12297" xml:space="preserve">Similiter duplum rectanguli E H A maius erit quadrato ex G H, &amp; </s>
  <s xml:id="echoid-s12298" xml:space="preserve">pro-<lb/>pterea duplum E H ad H G maiorem proportionem habebit, quàm G H ad H <lb/>A, &amp; </s>
  <s xml:id="echoid-s12299" xml:space="preserve">ideo duplum rectanguli ex differentia ipſarum A H, &amp; </s>
  <s xml:id="echoid-s12300" xml:space="preserve">G E in E H <lb/>
<anchor type="note" xlink:label="note-0394-03a" xlink:href="note-0394-03"/>
maius erit duobus quadratis ex G E, &amp; </s>
  <s xml:id="echoid-s12301" xml:space="preserve">ex E H: </s>
  <s xml:id="echoid-s12302" xml:space="preserve">igitur duo quadrata ex C A, <lb/>
<anchor type="note" xlink:label="note-0394-04a" xlink:href="note-0394-04"/>
&amp; </s>
  <s xml:id="echoid-s12303" xml:space="preserve">A F maiora erunt duobus quadratis ex I L, &amp; </s>
  <s xml:id="echoid-s12304" xml:space="preserve">ex I K.</s>
  <s xml:id="echoid-s12305" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1056" type="float" level="2" n="3">
<note position="left" xlink:label="note-0394-03" xlink:href="note-0394-03a" xml:space="preserve">Lem. 13. <lb/>huius.</note>
<note position="left" xlink:label="note-0394-04" xlink:href="note-0394-04a" xml:space="preserve">Lem. 15. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12306" xml:space="preserve">Rurſus quia V H minor eſt, quàm M H erit duplum rectanguli V H M mi-<lb/>nus duplo quadrati M H, ſeu minus quadrato G H, igitur duplum V H ad <lb/>H G minorem proportionem habet, quàm G H ad H M, &amp; </s>
  <s xml:id="echoid-s12307" xml:space="preserve">propterea duplum <lb/>
<anchor type="note" xlink:label="note-0394-05a" xlink:href="note-0394-05"/>
rectanguli ex differentia ipſarum M H, &amp; </s>
  <s xml:id="echoid-s12308" xml:space="preserve">G V in V H minus erit duobus <lb/>quadratis ex G V, &amp; </s>
  <s xml:id="echoid-s12309" xml:space="preserve">ex V H, &amp; </s>
  <s xml:id="echoid-s12310" xml:space="preserve">propterea duo quadrata ex Q P, &amp; </s>
  <s xml:id="echoid-s12311" xml:space="preserve">ex P <lb/>
<anchor type="note" xlink:label="note-0394-06a" xlink:href="note-0394-06"/>
R minora erunt duobus quadratis ex T S, &amp; </s>
  <s xml:id="echoid-s12312" xml:space="preserve">ex S Z: </s>
  <s xml:id="echoid-s12313" xml:space="preserve">ſi verò D V maior fue-<lb/>rit quàm D M, erunt duo quadrata ex Q P, &amp; </s>
  <s xml:id="echoid-s12314" xml:space="preserve">ex P R minora duobus qua-<lb/>
<anchor type="note" xlink:label="note-0394-07a" xlink:href="note-0394-07"/>
<pb o="357" file="0395" n="396" rhead="Conicor. Lib. VII."/>
dratis' ex T S, &amp; </s>
  <s xml:id="echoid-s12315" xml:space="preserve">S Z: </s>
  <s xml:id="echoid-s12316" xml:space="preserve">igitur ſumma duorum quadratorum ex Q P, &amp; </s>
  <s xml:id="echoid-s12317" xml:space="preserve">ex <lb/>P R minor eſt ſumma quadratorum duorum laterum figuræ cuiuſlibet alterius <lb/>diametri eiuſdem ellipſis.</s>
  <s xml:id="echoid-s12318" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1057" type="float" level="2" n="4">
<note position="left" xlink:label="note-0394-05" xlink:href="note-0394-05a" xml:space="preserve">Lem. 13. <lb/>huius.</note>
<note position="left" xlink:label="note-0394-06" xlink:href="note-0394-06a" xml:space="preserve">Lem. 15. <lb/>huius.</note>
<note position="left" xlink:label="note-0394-07" xlink:href="note-0394-07a" xml:space="preserve">Lem 16. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12319" xml:space="preserve">In ellipſi reperire diametrum, cuius duo quadrata laterum figuræ eius <lb/>
<anchor type="note" xlink:label="note-0395-01a" xlink:href="note-0395-01"/>
æqualia ſint quadratis laterum figuræ axis maioris: </s>
  <s xml:id="echoid-s12320" xml:space="preserve">oportet autem Vt <lb/>quadratum axis maioris A C maius ſit ſemiquadrato ex ſumma laterum <lb/>C A F figuræ eius.</s>
  <s xml:id="echoid-s12321" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1058" type="float" level="2" n="5">
<note position="right" xlink:label="note-0395-01" xlink:href="note-0395-01a" xml:space="preserve">PROP. 7. <lb/>Addit</note>
</div>
<p style="it">
  <s xml:id="echoid-s12322" xml:space="preserve">Quia ex hypotheſi quadratum axis maioris A C maius eſt ſemiquadrato ex <lb/>ſumma C A F, ergo, vt in nota prop. </s>
  <s xml:id="echoid-s12323" xml:space="preserve">48. </s>
  <s xml:id="echoid-s12324" xml:space="preserve">dictum eſt, duplum quadrati ex A <lb/>H maius eſt quadrato ex H G; </s>
  <s xml:id="echoid-s12325" xml:space="preserve">fiat duplum rectanguli e H A æquale quadra-<lb/>to ex G H, &amp; </s>
  <s xml:id="echoid-s12326" xml:space="preserve">lateris C e fiat diameter a b cuius erectus a c. </s>
  <s xml:id="echoid-s12327" xml:space="preserve">Dico hanc eſſe <lb/>diametrum quæſitam.</s>
  <s xml:id="echoid-s12328" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12329" xml:space="preserve">Quoniam duplum rectanguli e H A æquale eſt quadrato ex G H, ergo dup-<lb/>lum e H ad H G eſt vt G H ad H A, eritq; </s>
  <s xml:id="echoid-s12330" xml:space="preserve">duplum rectanguli ex differentia <lb/>
<anchor type="note" xlink:label="note-0395-02a" xlink:href="note-0395-02"/>
ipſarum A H, &amp; </s>
  <s xml:id="echoid-s12331" xml:space="preserve">G e in e H æquale quadratis ex G e, &amp; </s>
  <s xml:id="echoid-s12332" xml:space="preserve">ex e H, &amp; </s>
  <s xml:id="echoid-s12333" xml:space="preserve">ſum-<lb/>
<anchor type="note" xlink:label="note-0395-03a" xlink:href="note-0395-03"/>
ma quadratorum ex b a, &amp; </s>
  <s xml:id="echoid-s12334" xml:space="preserve">ex a c æqualis erit quadratorum ſummæ ex A C, <lb/>&amp; </s>
  <s xml:id="echoid-s12335" xml:space="preserve">ex A F, quod erat oſtendendum.</s>
  <s xml:id="echoid-s12336" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1059" type="float" level="2" n="6">
<note position="right" xlink:label="note-0395-02" xlink:href="note-0395-02a" xml:space="preserve">Lem 13.</note>
<note position="right" xlink:label="note-0395-03" xlink:href="note-0395-03a" xml:space="preserve">Lem. 15.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12337" xml:space="preserve">In eadem ellypſi diametrum reperire, cuius duo quadrata laterum <lb/>
<anchor type="note" xlink:label="note-0395-04a" xlink:href="note-0395-04"/>
figuræ eius æqualia ſint quadratis laterum figuræ datæ diametri I L: <lb/></s>
  <s xml:id="echoid-s12338" xml:space="preserve">oportet autem vt I L cadat inter axim, &amp; </s>
  <s xml:id="echoid-s12339" xml:space="preserve">diametrum P Q, cuius <lb/>quadratum ſubduplum ſit quadrati ex ſumma laterum Q P R.</s>
  <s xml:id="echoid-s12340" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1060" type="float" level="2" n="7">
<note position="right" xlink:label="note-0395-04" xlink:href="note-0395-04a" xml:space="preserve">PROP. <lb/>8. Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12341" xml:space="preserve">Sit C E latus diametri I L, &amp; </s>
  <s xml:id="echoid-s12342" xml:space="preserve">fiat duplum V H ad H G, vt G H ad H <lb/>E, &amp; </s>
  <s xml:id="echoid-s12343" xml:space="preserve">ponatur S T diameter lateris C V, cuius erectus ſit S Z: </s>
  <s xml:id="echoid-s12344" xml:space="preserve">erit igitur <lb/>
<anchor type="note" xlink:label="note-0395-05a" xlink:href="note-0395-05"/>
duplum rectanguli ex differentia ipſarum E H, &amp; </s>
  <s xml:id="echoid-s12345" xml:space="preserve">G V in V H æquale qua-<lb/>
<anchor type="note" xlink:label="note-0395-06a" xlink:href="note-0395-06"/>
dratis ex G V, &amp; </s>
  <s xml:id="echoid-s12346" xml:space="preserve">ex V H, ideoque ſumma quadratorum ex L I, &amp; </s>
  <s xml:id="echoid-s12347" xml:space="preserve"><lb/>ex I K æqualis erit quadratorum ſummæ ex T S, &amp; </s>
  <s xml:id="echoid-s12348" xml:space="preserve">S Z, quod propoſitum <lb/>ſuerat.</s>
  <s xml:id="echoid-s12349" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1061" type="float" level="2" n="8">
<note position="right" xlink:label="note-0395-05" xlink:href="note-0395-05a" xml:space="preserve">Lem. 13. <lb/>huius.</note>
<note position="right" xlink:label="note-0395-06" xlink:href="note-0395-06a" xml:space="preserve">Lem. 15. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12350" xml:space="preserve">Colligitur ſimiliter ex 7. </s>
  <s xml:id="echoid-s12351" xml:space="preserve">propoſit. </s>
  <s xml:id="echoid-s12352" xml:space="preserve">additarum, quod in vna ellypſi tres dia-<lb/>metri reperiri poßunt, quarum ſummæ quadratorum laterum æquales ſint inter <lb/>ſe: </s>
  <s xml:id="echoid-s12353" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12354" xml:space="preserve">ex 8. </s>
  <s xml:id="echoid-s12355" xml:space="preserve">propoſit. </s>
  <s xml:id="echoid-s12356" xml:space="preserve">additarum deducitur, quod quatuor diametrorum eiuſ-<lb/>dem ellypſis laterum ſummæ quadratorum æquales poſſunt eſſe inter ſe, ſed <lb/>oportet vt quadratum axis maioris datæ ellypſis maius ſit, quàm dimidium qua-<lb/>drati ex ſumma laterum figuræ axis C A F.</s>
  <s xml:id="echoid-s12357" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12358" xml:space="preserve">Duo latera figuræ axis tranſuerſi minora ſunt duobus lateribus ſiguræ <lb/>
<anchor type="note" xlink:label="note-0395-07a" xlink:href="note-0395-07"/>
cæterarum diametrorum, &amp; </s>
  <s xml:id="echoid-s12359" xml:space="preserve">duo latera figuræ diametri axi proximioris <lb/>minora ſunt duobus lateribus figuræ remotioris, &amp;</s>
  <s xml:id="echoid-s12360" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12361" xml:space="preserve">Addidi ea, quæ defi-<lb/>cere videbantur in hoc textu.</s>
  <s xml:id="echoid-s12362" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1062" type="float" level="2" n="9">
<note position="left" xlink:label="note-0395-07" xlink:href="note-0395-07a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s12363" xml:space="preserve">Iiſdem figuris manentibus cum ſuis ſignis oſtendatur quod duplum <lb/>
<anchor type="note" xlink:label="note-0395-08a" xlink:href="note-0395-08"/>
quadrati A C, ſi non exceſſerit F, quod diameter eſt illius figuræ minor, <lb/>quàm diameter ſiguræ I L, &amp; </s>
  <s xml:id="echoid-s12364" xml:space="preserve">diameter figuræ I L, quàm diameter figuræ <lb/>P Q, &amp;</s>
  <s xml:id="echoid-s12365" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12366" xml:space="preserve">Legendum puto vt in textu apparet.</s>
  <s xml:id="echoid-s12367" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1063" type="float" level="2" n="10">
<note position="left" xlink:label="note-0395-08" xlink:href="note-0395-08a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s12368" xml:space="preserve">Et ſic oſtendetur quod ſi punctum V inciderit ſuper D A, &amp; </s>
  <s xml:id="echoid-s12369" xml:space="preserve">oſtende-<lb/>
<anchor type="note" xlink:label="note-0395-09a" xlink:href="note-0395-09"/>
tur D, &amp; </s>
  <s xml:id="echoid-s12370" xml:space="preserve">M, &amp;</s>
  <s xml:id="echoid-s12371" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12372" xml:space="preserve">Legendum puto, vt in textu videre eſt.</s>
  <s xml:id="echoid-s12373" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1064" type="float" level="2" n="11">
<note position="left" xlink:label="note-0395-09" xlink:href="note-0395-09a" xml:space="preserve">c</note>
</div>
<pb o="358" file="0396" n="397" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div1066" type="section" level="1" n="336">
<head xml:id="echoid-head415" xml:space="preserve">SECTIO DECIMA</head>
<head xml:id="echoid-head416" xml:space="preserve">Continens Propoſit. XXXXIX. XXXXX. <lb/>&amp; XXXXXI.</head>
<p>
  <s xml:id="echoid-s12374" xml:space="preserve">XXXXXI. </s>
  <s xml:id="echoid-s12375" xml:space="preserve">IN hyperbola, &amp; </s>
  <s xml:id="echoid-s12376" xml:space="preserve">ellipſi, ſi axis tranſuerſus minor <lb/>fuerit ſuo erecto, differentia quadratorum duorum <lb/>
<anchor type="note" xlink:label="note-0396-01a" xlink:href="note-0396-01"/>
laterum figuræ axis eius maior eſt, quàm differentia quadrato-<lb/>rum laterum figuræ cuiuslibet alterius diametri ei homologæ. </s>
  <s xml:id="echoid-s12377" xml:space="preserve">Et <lb/>differentia quadratorum laterum figure homologæ proximioris <lb/>axi ſemper maior eſt in hyperbola, quàm differentia quadratorum <lb/>laterum figuræ remotioris: </s>
  <s xml:id="echoid-s12378" xml:space="preserve">at in ellypſi quouſque diameter tran-<lb/>ſuerſa æqualis non fiat ſuo erecto.</s>
  <s xml:id="echoid-s12379" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1066" type="float" level="2" n="1">
<note position="right" xlink:label="note-0396-01" xlink:href="note-0396-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s12380" xml:space="preserve">XXXXX. </s>
  <s xml:id="echoid-s12381" xml:space="preserve">Et in hyperbola differentia quadrati axis inclinati <lb/>ab eius figura minor erit ſemidifferentia quadratorum duorum <lb/>laterum figuræ ſui homologi.</s>
  <s xml:id="echoid-s12382" xml:space="preserve"/>
</p>
  <figure>
    <image file="0396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0396-01"/>
  </figure>
<p>
  <s xml:id="echoid-s12383" xml:space="preserve">XXXXIX. </s>
  <s xml:id="echoid-s12384" xml:space="preserve">Si verò in hyperbole axis inclinatus maior fuerit <lb/>ſuo erecto, vtique differentia quadratorum duorum laterum fi-<lb/>guræ axis minor erit differentia quadratorum laterum figuræ al-
<pb o="359" file="0397" n="398" rhead="Conicor. Lib. VII."/>
terius homologæ diametri, atque differentia quadrati axis ab <lb/>eius figura maior erit ſemidifferentia quadratorum duorum late-<lb/>rum figuræ ſuæ homologæ, &amp; </s>
  <s xml:id="echoid-s12385" xml:space="preserve">minor erit integra differentia eo-<lb/>rundem quadratorum.</s>
  <s xml:id="echoid-s12386" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12387" xml:space="preserve">In fectione A B N ſit axis A C maior in figura prima, &amp; </s>
  <s xml:id="echoid-s12388" xml:space="preserve">in ſecunda <lb/>minor, ſintquè I L, P Q duæ aliæ diametri, quæ in ellipſi cadant inter <lb/>axim, &amp; </s>
  <s xml:id="echoid-s12389" xml:space="preserve">vnã æqualium; </s>
  <s xml:id="echoid-s12390" xml:space="preserve">ducanturque duæ ordinationes A B, A N ad <lb/>
<anchor type="note" xlink:label="note-0397-01a" xlink:href="note-0397-01"/>
diametros I L, P Q, &amp; </s>
  <s xml:id="echoid-s12391" xml:space="preserve">duas ad axim perpendiculares B E, N M; </s>
  <s xml:id="echoid-s12392" xml:space="preserve">ſit-<lb/>que A F erectus ipſius A C, &amp; </s>
  <s xml:id="echoid-s12393" xml:space="preserve">A G, C H duæ interceptæ: </s>
  <s xml:id="echoid-s12394" xml:space="preserve">ponaturque <lb/>in ellipſi X D æqualis E D, habebit E H ad H A minorem proportio-<lb/>
<anchor type="note" xlink:label="note-0397-02a" xlink:href="note-0397-02"/>
nem in prima hyperbola, &amp; </s>
  <s xml:id="echoid-s12395" xml:space="preserve">maiorem in reliquis, quàm E D ad D A, <lb/>ſeu quàm E X, quæ eſt ſumma in hyperbola, &amp; </s>
  <s xml:id="echoid-s12396" xml:space="preserve">differentia in ellipſi <lb/>ipſarum E G, &amp; </s>
  <s xml:id="echoid-s12397" xml:space="preserve">E H ad A C differentiam ipſarum H A, A G; </s>
  <s xml:id="echoid-s12398" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12399" xml:space="preserve">qua-<lb/>
<anchor type="figure" xlink:label="fig-0397-01a" xlink:href="fig-0397-01"/>
dratum A C in omnibus figuris ad differentiam quadratorum A C, &amp; </s>
  <s xml:id="echoid-s12400" xml:space="preserve"><lb/>A F eandem proportionem habet, quàm quadratum A H ad differentiam <lb/>duorum quadratorum A H, &amp; </s>
  <s xml:id="echoid-s12401" xml:space="preserve">G A: </s>
  <s xml:id="echoid-s12402" xml:space="preserve">atque E H ad H A minorem pro-<lb/>portionem habet in duabus primis figuris, &amp; </s>
  <s xml:id="echoid-s12403" xml:space="preserve">maiorem proportionem in <lb/>duabus ſecundis, quàm E G ad G A, comparando homologorum ſum-<lb/>mas, erit E H ad H A, vt E H cum E G ad H A cum G A, nempe ag-<lb/>gregatum E H, E G in earundem differentiam ad aggregatum H A, A <lb/>G in earundem differentiam, quod eſt æquale differentiæ duorum qua-<lb/>dratorum E H, E G; </s>
  <s xml:id="echoid-s12404" xml:space="preserve">nempe quadratum A C ad differentiam quadrato-<lb/>rum duorum laterum figuræ I L minorem proportionem habet (in prima <lb/>ellipſi), &amp; </s>
  <s xml:id="echoid-s12405" xml:space="preserve">maiorem (in ſecunda) quàm quadratum A H ad aggrega-<lb/>tum H A, A G in earundem differentiam, quod eſt æquale differentiæ <lb/>quadratorum H A, A G, nempe quadratum A C ad differentiam qua-
<pb o="360" file="0398" n="399" rhead="Apollonij Pergæi"/>
dratorum duorum laterum figuræ eius; </s>
  <s xml:id="echoid-s12406" xml:space="preserve">igitur quadratum A C ad diffe-<lb/>rentiam quadratorum duorum laterum figuræ I L minorem proportionem <lb/>habet, in prima ellipſi, &amp; </s>
  <s xml:id="echoid-s12407" xml:space="preserve">maiorem in reliquis, quam ad differentiam <lb/>quadratorum duorum laterum figuræ A C; </s>
  <s xml:id="echoid-s12408" xml:space="preserve">ergo differentia quadratorum <lb/>duorum laterum figuræ A C minor eſt in prima ellipſi, &amp; </s>
  <s xml:id="echoid-s12409" xml:space="preserve">maior in cæ-<lb/>teris, quàm differentia quadratorum duorum laterum figuræ I L. </s>
  <s xml:id="echoid-s12410" xml:space="preserve">Præte-<lb/>rea M H ad H E minorem proportionem, aut maiorem habet, quàm M <lb/>G ad G E: </s>
  <s xml:id="echoid-s12411" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12412" xml:space="preserve">ponamus in ellipſi Y D æqualem D M, oſtendeturquè <lb/>
<anchor type="figure" xlink:label="fig-0398-01a" xlink:href="fig-0398-01"/>
quod M H in H A minus ſit in prima ellipſi, &amp; </s>
  <s xml:id="echoid-s12413" xml:space="preserve">maior in cæteris, quàm <lb/>duarum M G, M H ſumma in earum differentiam M Y: </s>
  <s xml:id="echoid-s12414" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12415" xml:space="preserve">oſtendetur <lb/>quemadmodum dictum eſt, quod differentia quadratorum duorum late-<lb/>rum figuræ I L maior eſt, quàm differentia quadratorum duorum late-<lb/>rum figuræ P Q.</s>
  <s xml:id="echoid-s12416" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1067" type="float" level="2" n="2">
<note position="left" xlink:label="note-0397-01" xlink:href="note-0397-01a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0397-02" xlink:href="note-0397-02a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0397-01" xlink:href="fig-0397-01a">
    <image file="0397-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0397-01"/>
  </figure>
  <figure xlink:label="fig-0398-01" xlink:href="fig-0398-01a">
    <image file="0398-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0398-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s12417" xml:space="preserve">Deinde in hyperbola ponamus I K erectum ipſius I L, erit differentia <lb/>quadratorum duarum I L, I K (quæ eſt æqualis K L in ſummam L I, I <lb/>K) maior illa, quàm I L in L K, quod eſt æquale differentiæ quadrari <lb/>I L, &amp; </s>
  <s xml:id="echoid-s12418" xml:space="preserve">eius figuræ, nempe differentiæ quadrati A C, &amp; </s>
  <s xml:id="echoid-s12419" xml:space="preserve">eius figuræ <lb/>(29. </s>
  <s xml:id="echoid-s12420" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s12421" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12422" xml:space="preserve">non eſt maior in prima, quàm duplum, &amp; </s>
  <s xml:id="echoid-s12423" xml:space="preserve">in ſecunda ma-<lb/>ior duplo, &amp; </s>
  <s xml:id="echoid-s12424" xml:space="preserve">hoc eſt propoſitum.</s>
  <s xml:id="echoid-s12425" xml:space="preserve"/>
</p>
<pb o="361" file="0399" n="400" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div1069" type="section" level="1" n="337">
<head xml:id="echoid-head417" xml:space="preserve">In Sectionem X. Propoſit. XXXXIX. <lb/>XXXXX. &amp; XXXXXI.</head>
<head xml:id="echoid-head418" xml:space="preserve">LEMMA XVI.</head>
<p style="it">
  <s xml:id="echoid-s12426" xml:space="preserve">S I rectæ lineæ A B bifariam ſectæ in C vtrinque addantur æquales <lb/>portiones A D, &amp; </s>
  <s xml:id="echoid-s12427" xml:space="preserve">B E, dico rectangulum ſub tota D E, &amp; </s>
  <s xml:id="echoid-s12428" xml:space="preserve"><lb/>ſub intermedia A B æquale eſſe differentiæ quadratorum ex A E, &amp; </s>
  <s xml:id="echoid-s12429" xml:space="preserve"><lb/>ex A D.</s>
  <s xml:id="echoid-s12430" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12431" xml:space="preserve">Apponatur F D æqualis D <lb/>
<anchor type="figure" xlink:label="fig-0399-01a" xlink:href="fig-0399-01"/>
A, vel B E: </s>
  <s xml:id="echoid-s12432" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12433" xml:space="preserve">quia F D æ-<lb/>qualis eſt B E addita communi <lb/>B D, erit F B æqualis D E, <lb/>&amp; </s>
  <s xml:id="echoid-s12434" xml:space="preserve">ideo rectangulum F B A æ-<lb/>quale erit rectangulo ſub D E, <lb/>&amp; </s>
  <s xml:id="echoid-s12435" xml:space="preserve">ſub A B, ſed quadratum <lb/>B D æquale eſt quadrato D A cum rectangulo F B A, (eo quod F A ſecta eſt <lb/>bifariam in D, &amp; </s>
  <s xml:id="echoid-s12436" xml:space="preserve">ei in directum additur A B), ergo quadratum D B æquale <lb/>eſt quadrato D A vna cum rectangulo ſub D E, &amp; </s>
  <s xml:id="echoid-s12437" xml:space="preserve">ſub A B, &amp; </s>
  <s xml:id="echoid-s12438" xml:space="preserve">propterea re-<lb/>ctangulum ſub D E, &amp; </s>
  <s xml:id="echoid-s12439" xml:space="preserve">ſub A B contentum æquale eſt differentiæ quadrati B D, <lb/>ſeu A E à quadrato D A, quod erat oſtendendum.</s>
  <s xml:id="echoid-s12440" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1069" type="float" level="2" n="1">
  <figure xlink:label="fig-0399-01" xlink:href="fig-0399-01a">
    <image file="0399-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0399-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1071" type="section" level="1" n="338">
<head xml:id="echoid-head419" xml:space="preserve">LEMMA XVII.</head>
<p style="it">
  <s xml:id="echoid-s12441" xml:space="preserve">IN hyperbola, &amp; </s>
  <s xml:id="echoid-s12442" xml:space="preserve">ellypſi, cuius centrum D, axis A C, erectus A <lb/>F, præſectæ A H, G C, &amp; </s>
  <s xml:id="echoid-s12443" xml:space="preserve">in ea diameter I L, cuius erectus <lb/>
<anchor type="figure" xlink:label="fig-0399-02a" xlink:href="fig-0399-02"/>
<pb o="362" file="0400" n="401" rhead="Apollonij Pergæi"/>
I K, &amp; </s>
  <s xml:id="echoid-s12444" xml:space="preserve">latus C E, pariterque diameter Q P, cuius erectus P R, <lb/>eiuſque latus C M, ſi fuerit proportio ipſius H M ad M D eadem <lb/>proportioni H E ad D E, vel eadem proportioni H A ad D A, erit <lb/>differentia quadratorum ex lateribus Q P, &amp; </s>
  <s xml:id="echoid-s12445" xml:space="preserve">ex P R figuræ diametri Q <lb/>P æqualis differentiæ quadratorum ex lateribus figuræ diametri I L, vel <lb/>A C: </s>
  <s xml:id="echoid-s12446" xml:space="preserve">ſi verò proportio illa minor fuerit erit prior differentia quadrato-<lb/>rum maior reliqua, &amp; </s>
  <s xml:id="echoid-s12447" xml:space="preserve">ſi illa proportio maior fuerit, erit prima quadra-<lb/>torum differentia minor reliqua.</s>
  <s xml:id="echoid-s12448" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1071" type="float" level="2" n="1">
  <figure xlink:label="fig-0399-02" xlink:href="fig-0399-02a">
    <image file="0399-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0399-02"/>
  </figure>
</div>
  <figure>
    <image file="0400-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0400-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s12449" xml:space="preserve">Fiat D X æqualis D E, &amp; </s>
  <s xml:id="echoid-s12450" xml:space="preserve">D γ æqualis D M, &amp; </s>
  <s xml:id="echoid-s12451" xml:space="preserve">primo quia H M ad M D <lb/>eſt vt H E ad D E, permutando M H ad H E erit vt D M ad D E, ſeu vt duplũ <lb/>M γ ad duplum E X, &amp; </s>
  <s xml:id="echoid-s12452" xml:space="preserve">ſumptis altitudinibus H A, &amp; </s>
  <s xml:id="echoid-s12453" xml:space="preserve">G H erit rectangulum <lb/>M H A ad rectangulum E H A vt rectangulum ſub γ M, &amp; </s>
  <s xml:id="echoid-s12454" xml:space="preserve">G H ad rectan-<lb/>
<anchor type="figure" xlink:label="fig-0400-02a" xlink:href="fig-0400-02"/>
<pb o="363" file="0401" n="402" rhead="Conicor. Lib. VII."/>
gulum ſub E X, &amp; </s>
  <s xml:id="echoid-s12455" xml:space="preserve">G H, &amp; </s>
  <s xml:id="echoid-s12456" xml:space="preserve">permutando rectangulum M H A ad rectangulum <lb/>
<anchor type="note" xlink:label="note-0401-01a" xlink:href="note-0401-01"/>
ſub γ M, &amp; </s>
  <s xml:id="echoid-s12457" xml:space="preserve">G H, ſeu ad differentiam quadratorum ex H M, &amp; </s>
  <s xml:id="echoid-s12458" xml:space="preserve">ex M G <lb/>eandem proportionem habebit, quàm rectangulum E H A ad rectangulum ſub <lb/>
<anchor type="note" xlink:label="note-0401-02a" xlink:href="note-0401-02"/>
E X, &amp; </s>
  <s xml:id="echoid-s12459" xml:space="preserve">ſub G H, ſeu ad differentiam quadratorum ex H E, &amp; </s>
  <s xml:id="echoid-s12460" xml:space="preserve">ex E G: </s>
  <s xml:id="echoid-s12461" xml:space="preserve">eſt <lb/>verò quadratum A C ad differentiam quadratorum ex P Q, &amp; </s>
  <s xml:id="echoid-s12462" xml:space="preserve">ex P R, vt <lb/>
<anchor type="note" xlink:label="note-0401-03a" xlink:href="note-0401-03"/>
rectangulum M H A ad differentiam quadratorum ex H M, &amp; </s>
  <s xml:id="echoid-s12463" xml:space="preserve">ex M G, pa-<lb/>riterque idem quadratum A C ad differentiã quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12464" xml:space="preserve">ex I K <lb/>
<anchor type="note" xlink:label="note-0401-04a" xlink:href="note-0401-04"/>
eſt, vt rectangulum E H A ad differentiam quadratorum ex H E, &amp; </s>
  <s xml:id="echoid-s12465" xml:space="preserve">ex E <lb/>G, igitur idem quadratum A C ad differentiam quadratorum ex P Q, &amp; </s>
  <s xml:id="echoid-s12466" xml:space="preserve">ex <lb/>P R eandem proportionem habet, quàm ad differentiam quadratorum ex I L, <lb/>&amp; </s>
  <s xml:id="echoid-s12467" xml:space="preserve">ex I K, &amp; </s>
  <s xml:id="echoid-s12468" xml:space="preserve">propterea differentia quadratorum ex Q P, &amp; </s>
  <s xml:id="echoid-s12469" xml:space="preserve">ex P R æqualis <lb/>eſt quadratorum differentiæ ex I L, &amp; </s>
  <s xml:id="echoid-s12470" xml:space="preserve">ex I K, ſiue æqualis eſt quadratorum <lb/>differentiæ ex A C, &amp; </s>
  <s xml:id="echoid-s12471" xml:space="preserve">ex A F.</s>
  <s xml:id="echoid-s12472" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1072" type="float" level="2" n="2">
  <figure xlink:label="fig-0400-02" xlink:href="fig-0400-02a">
    <image file="0400-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0400-02"/>
  </figure>
<note position="right" xlink:label="note-0401-01" xlink:href="note-0401-01a" xml:space="preserve">Lem. 16. <lb/>huius.</note>
<note position="right" xlink:label="note-0401-02" xlink:href="note-0401-02a" xml:space="preserve">Ibidem.</note>
<note position="right" xlink:label="note-0401-03" xlink:href="note-0401-03a" xml:space="preserve">Prop. 20. <lb/>huius.</note>
<note position="right" xlink:label="note-0401-04" xlink:href="note-0401-04a" xml:space="preserve">Ibidem.</note>
</div>
  <figure>
    <image file="0401-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0401-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s12473" xml:space="preserve">Secundo H M ad M D minorem proportionem habeat, quàm H E ad D E, <lb/>vt prius permutando habebit H M ad H E minorem proportionem, quàm D M <lb/>ad D E, ſeu quàm duplum M γ ad duplum E X, &amp; </s>
  <s xml:id="echoid-s12474" xml:space="preserve">ſumptis communibus al-<lb/>titudinibus H A ad G H, &amp; </s>
  <s xml:id="echoid-s12475" xml:space="preserve">permutando ex lem. </s>
  <s xml:id="echoid-s12476" xml:space="preserve">16. </s>
  <s xml:id="echoid-s12477" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12478" xml:space="preserve">propoſit. </s>
  <s xml:id="echoid-s12479" xml:space="preserve">20. </s>
  <s xml:id="echoid-s12480" xml:space="preserve">huius, <lb/>idem quadratum A C ad differentiam quadratorum ex P Q, &amp; </s>
  <s xml:id="echoid-s12481" xml:space="preserve">ex P R mino-<lb/>rem proportionem habebit, quàm ad differentiam quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12482" xml:space="preserve">ex <lb/>I K, quapropter differentia quadratorum ex P Q, &amp; </s>
  <s xml:id="echoid-s12483" xml:space="preserve">ex P R maior erit, quàm <lb/>differentia quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12484" xml:space="preserve">ex I K, ſeu maior, quàm differentia qua-<lb/>dratorum ex A C, &amp; </s>
  <s xml:id="echoid-s12485" xml:space="preserve">ex A F.</s>
  <s xml:id="echoid-s12486" xml:space="preserve"/>
</p>
<pb o="364" file="0402" n="403" rhead="Apollonij Pergæi"/>
<p style="it">
  <s xml:id="echoid-s12487" xml:space="preserve">Tertio habeat H M ad M D maiorem proportionem quàm H E ad D E: </s>
  <s xml:id="echoid-s12488" xml:space="preserve">vt <lb/>prius permutando, ſumptis communibus altitudinibus H A, &amp; </s>
  <s xml:id="echoid-s12489" xml:space="preserve">G H, &amp; </s>
  <s xml:id="echoid-s12490" xml:space="preserve">denuo <lb/>permutando ex lem. </s>
  <s xml:id="echoid-s12491" xml:space="preserve">16. </s>
  <s xml:id="echoid-s12492" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12493" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s12494" xml:space="preserve">20. </s>
  <s xml:id="echoid-s12495" xml:space="preserve">huius, ſequitur quod idem quadratum <lb/>
<anchor type="figure" xlink:label="fig-0402-01a" xlink:href="fig-0402-01"/>
ex A C ad differentiam quadratorum ex P Q, &amp; </s>
  <s xml:id="echoid-s12496" xml:space="preserve">ex P R maiorem proportio-<lb/>nem habet, quàm ad differentiam quadra orum ex I L, &amp; </s>
  <s xml:id="echoid-s12497" xml:space="preserve">ex I K, quare dif-<lb/>ferentia quadratorum ex P Q, &amp; </s>
  <s xml:id="echoid-s12498" xml:space="preserve">ex P R minor erit, quàm differentia qua-<lb/>dratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12499" xml:space="preserve">ex I K, ſiue minor, quàm diffi<unsure/>rentia quadratorum ex <lb/>A C, &amp; </s>
  <s xml:id="echoid-s12500" xml:space="preserve">ex A F, quæ erant oſtendenda.</s>
  <s xml:id="echoid-s12501" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1073" type="float" level="2" n="3">
  <figure xlink:label="fig-0402-01" xlink:href="fig-0402-01a">
    <image file="0402-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0402-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1075" type="section" level="1" n="339">
<head xml:id="echoid-head420" xml:space="preserve">LEMMA XVIII.</head>
<p style="it">
  <s xml:id="echoid-s12502" xml:space="preserve">IN ellipſi ſi diameter a b bifariam ſecuerit rectam lineam A O ter-<lb/>minos axium coniungentem, erit a b æqualis ſuo erecto a c. <lb/></s>
  <s xml:id="echoid-s12503" xml:space="preserve">Zuia axis A C bifariam diuiditur in centro D ab axi O D perpendiculari <lb/>ad axim A C, quæ educitur à termino O ipſius A O ordinatim applicatæ ad <lb/>diametrum a b, habebit diameter a b ad eius erectũ a c eandem proportionem <lb/>
<anchor type="note" xlink:label="note-0402-01a" xlink:href="note-0402-01"/>
æqualitatis quàm habet H D ad D G, igitur diameter a b æqualis eſt eius la-<lb/>teri recto a e, quod erat propoſitum.</s>
  <s xml:id="echoid-s12504" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1075" type="float" level="2" n="1">
<note position="left" xlink:label="note-0402-01" xlink:href="note-0402-01a" xml:space="preserve">Prop. 7. <lb/>huius.</note>
</div>
<pb o="365" file="0403" n="404" rhead="Conicor. Lib. VII."/>
  <figure>
    <image file="0403-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0403-01"/>
  </figure>
</div>
<div xml:id="echoid-div1077" type="section" level="1" n="340">
<head xml:id="echoid-head421" xml:space="preserve">Notæ in Propoſit. XXXXIX.</head>
<p style="it">
  <s xml:id="echoid-s12505" xml:space="preserve">QVia in hyperbola axis A C maior ponitur erecto eius A F, eſtque A H <lb/>ad H C vt A C ad A F, ergo præſecta A H maior portio eſt totius C A, <lb/>&amp; </s>
  <s xml:id="echoid-s12506" xml:space="preserve">ideo punctum H cadit inter C, &amp; </s>
  <s xml:id="echoid-s12507" xml:space="preserve">D, &amp; </s>
  <s xml:id="echoid-s12508" xml:space="preserve">punctum E cadit inter <lb/>M, &amp; </s>
  <s xml:id="echoid-s12509" xml:space="preserve">D, igitur eadem H D ad maiorem D M habebit minorem proportio-<lb/>
<anchor type="figure" xlink:label="fig-0403-02a" xlink:href="fig-0403-02"/>
<pb o="366" file="0404" n="405" rhead="Apollonij Pergæi"/>
<anchor type="figure" xlink:label="fig-0404-01a" xlink:href="fig-0404-01"/>
nem, quàm ad minorem D E, &amp; </s>
  <s xml:id="echoid-s12510" xml:space="preserve">componendo H M ad M D minorem propor-<lb/>tionem habebit, quàm H E ad E D, &amp; </s>
  <s xml:id="echoid-s12511" xml:space="preserve">ideo differentia quadratorum ex P Q, <lb/>&amp; </s>
  <s xml:id="echoid-s12512" xml:space="preserve">ex P R maior erit, quàm differentia quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12513" xml:space="preserve">ex I K, ſeu <lb/>
<anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/>
maior quàm differentia quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s12514" xml:space="preserve">ex A F.</s>
  <s xml:id="echoid-s12515" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1077" type="float" level="2" n="1">
  <figure xlink:label="fig-0403-02" xlink:href="fig-0403-02a">
    <image file="0403-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0403-02"/>
  </figure>
  <figure xlink:label="fig-0404-01" xlink:href="fig-0404-01a">
    <image file="0404-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0404-01"/>
  </figure>
<note position="left" xlink:label="note-0404-01" xlink:href="note-0404-01a" xml:space="preserve">Lem. 17. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12516" xml:space="preserve">Rurſus quia rectangulum C A F maius eſt quadrato A F, (propterea quod <lb/>rectangulum illud medium proportionale eſt inter maius quadratum ex A C, &amp; </s>
  <s xml:id="echoid-s12517" xml:space="preserve"><lb/>quadratum minus ex A F), ergo differentia quadrati A C à rectangulo C A <lb/>F, ſcilicet difſerentia ſpatiorum maximi, &amp; </s>
  <s xml:id="echoid-s12518" xml:space="preserve">intermedĳ, minor erit, quàm <lb/>differentia inter quadratum maximum A C, &amp; </s>
  <s xml:id="echoid-s12519" xml:space="preserve">minimum A F, ſed differen-<lb/>tia quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s12520" xml:space="preserve">ex A F minor oſtenſa eſt, quàm differentia qua-<lb/>dratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12521" xml:space="preserve">ex I K, ergo multo magis differentia quadrati A C à re-<lb/>ctangulo C A F minor erit, quàm differentia quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12522" xml:space="preserve">ex I K.</s>
  <s xml:id="echoid-s12523" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12524" xml:space="preserve">Tandem quia quadratum A C ad ſemidifferentiam quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12525" xml:space="preserve"><lb/>ex I K eandem proportionem habet, quàm rectangulum E H A ad ſemifferen-<lb/>
<anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/>
tiam quadratorum ex E H, &amp; </s>
  <s xml:id="echoid-s12526" xml:space="preserve">ex E G, vel ad ſemiſſem rectanguli ex E X in <lb/>G H, vel potius ad rectanguluw ſub E D, &amp; </s>
  <s xml:id="echoid-s12527" xml:space="preserve">ſub G H; </s>
  <s xml:id="echoid-s12528" xml:space="preserve">ſed quadrati A C à <lb/>
<anchor type="note" xlink:label="note-0404-03a" xlink:href="note-0404-03"/>
rectangulo C A F differentia ad quadratum ipſum A G, ſeu differentia A C, <lb/>&amp; </s>
  <s xml:id="echoid-s12529" xml:space="preserve">A F ad A C eandem proportionem habet, quàm H G ad H A, ſeu quàm <lb/>rectangulum E H G ad rectangulum E H A, igitur ex æquali differentia qua-<lb/>
<anchor type="note" xlink:label="note-0404-04a" xlink:href="note-0404-04"/>
drati A C à rectangulo C A F ad ſemidifferentiam quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12530" xml:space="preserve">ex <lb/>I K eandem proportionem habebit, quàm rectangulũ E H G ad rectangulum ſub <lb/>E D, &amp; </s>
  <s xml:id="echoid-s12531" xml:space="preserve">G H, eſtq; </s>
  <s xml:id="echoid-s12532" xml:space="preserve">primũ rectangulũ reliquo rectangulo æquè alto maius, cum eius <lb/>baſis E H maior ſit, quàm E D, igitur differentia quadrati A C à rectangulo <lb/>C A F maior erit, quàm ſemidifferentia quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12533" xml:space="preserve">ex I K.</s>
  <s xml:id="echoid-s12534" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1078" type="float" level="2" n="2">
<note position="left" xlink:label="note-0404-02" xlink:href="note-0404-02a" xml:space="preserve">Prop. 20 <lb/>huius.</note>
<note position="left" xlink:label="note-0404-03" xlink:href="note-0404-03a" xml:space="preserve">Lem. <lb/>16. huius.</note>
<note position="left" xlink:label="note-0404-04" xlink:href="note-0404-04a" xml:space="preserve">ex Def. 2. <lb/>huius.</note>
</div>
<pb o="367" file="0405" n="406" rhead="Conicor. Lib. VII."/>
</div>
<div xml:id="echoid-div1080" type="section" level="1" n="341">
<head xml:id="echoid-head422" xml:space="preserve">Notæ in Propoſit. XXXXX.</head>
<p style="it">
  <s xml:id="echoid-s12535" xml:space="preserve">SI hyperbole axis A C minor fuerit eius erecto A F, quia H M maior eſt, <lb/>quàm H E, &amp; </s>
  <s xml:id="echoid-s12536" xml:space="preserve">punctum H cadit inter D, &amp; </s>
  <s xml:id="echoid-s12537" xml:space="preserve">A, ergo H M ad H D ma-<lb/>
<anchor type="figure" xlink:label="fig-0405-01a" xlink:href="fig-0405-01"/>
iorem proportionem habebit, quàm H E ad eandem H D, &amp; </s>
  <s xml:id="echoid-s12538" xml:space="preserve">comparando ante-<lb/>cedentes ad terminorum ſummas H M ad M D maiorem proportionem habebit, <lb/>
<anchor type="note" xlink:label="note-0405-01a" xlink:href="note-0405-01"/>
quàm H E ad E D, quare differentia quadratorum ex P Q, &amp; </s>
  <s xml:id="echoid-s12539" xml:space="preserve">ex P R minor <lb/>erit, quàm differentta quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12540" xml:space="preserve">ex I K, ſeu minor quàm dif-<lb/>ferentia quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s12541" xml:space="preserve">ex A F.</s>
  <s xml:id="echoid-s12542" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1080" type="float" level="2" n="1">
  <figure xlink:label="fig-0405-01" xlink:href="fig-0405-01a">
    <image file="0405-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0405-01"/>
  </figure>
<note position="right" xlink:label="note-0405-01" xlink:href="note-0405-01a" xml:space="preserve">Lem. 17. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12543" xml:space="preserve">Poſtea, quia vt in precedenti nota dictũ eſt, differentia quadrati A C à rectan-<lb/>gulo C A F ad ſemidifferentiã quadratorũ ex I L, &amp; </s>
  <s xml:id="echoid-s12544" xml:space="preserve">ex I K eandem proportionẽ <lb/>habet, quàm rectangulum E H G ad rectangulum ſub E D, &amp; </s>
  <s xml:id="echoid-s12545" xml:space="preserve">ſub G H, eſt-<lb/>que illud rectangulum minus rectangulo iſto æquè alto, (cum illius baſis E H <lb/>minor ſit, quàm E D), igitur differentia quadrati A C à rectangulo C A F <lb/>minor eſt, quàm ſemidifferentia quadratorum ex I L, &amp; </s>
  <s xml:id="echoid-s12546" xml:space="preserve">ex I K.</s>
  <s xml:id="echoid-s12547" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1082" type="section" level="1" n="342">
<head xml:id="echoid-head423" xml:space="preserve">Notæ in Propoſit. XXXXXI.</head>
<p style="it">
  <s xml:id="echoid-s12548" xml:space="preserve">IN qualibet ellypſi ſit diameter a b æqualis eius erecto a c, eius latus erit C <lb/>
<anchor type="note" xlink:label="note-0405-02a" xlink:href="note-0405-02"/>
D, &amp; </s>
  <s xml:id="echoid-s12549" xml:space="preserve">diametri I L, &amp; </s>
  <s xml:id="echoid-s12550" xml:space="preserve">P Q cadant inter A C, &amp; </s>
  <s xml:id="echoid-s12551" xml:space="preserve">a b, earum laterum
<pb o="368" file="0406" n="407" rhead="Apollonij Pergæi"/>
C E, &amp; </s>
  <s xml:id="echoid-s12552" xml:space="preserve">C M, termini E, &amp; </s>
  <s xml:id="echoid-s12553" xml:space="preserve">M cadent inter D, &amp; </s>
  <s xml:id="echoid-s12554" xml:space="preserve">A, &amp; </s>
  <s xml:id="echoid-s12555" xml:space="preserve">M cadat inter <lb/>E &amp; </s>
  <s xml:id="echoid-s12556" xml:space="preserve">D, propterea M H ad M D maiorem proportionem habebit, quàm H E <lb/>
<anchor type="figure" xlink:label="fig-0406-01a" xlink:href="fig-0406-01"/>
ad E D, igitur differentia quadratorum laterum figuræ P Q minor erit diffe-<lb/>
<anchor type="note" xlink:label="note-0406-01a" xlink:href="note-0406-01"/>
rentia quadratorum laterum figuræ I L, vel figuræ A C.</s>
  <s xml:id="echoid-s12557" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1082" type="float" level="2" n="1">
<note position="right" xlink:label="note-0405-02" xlink:href="note-0405-02a" xml:space="preserve">ex Lem. <lb/>18. huius.</note>
  <figure xlink:label="fig-0406-01" xlink:href="fig-0406-01a">
    <image file="0406-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0406-01"/>
  </figure>
<note position="left" xlink:label="note-0406-01" xlink:href="note-0406-01a" xml:space="preserve">Lem. 17. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12558" xml:space="preserve">In ellypſi reperire diametrum, <lb/>
<anchor type="figure" xlink:label="fig-0406-02a" xlink:href="fig-0406-02"/>
<anchor type="note" xlink:label="note-0406-02a" xlink:href="note-0406-02"/>
cuius differentia quadratorum la-<lb/>terum figuræ eius æqualis ſit diffe-<lb/>rentiæ quadratorum laterum figuræ <lb/>axis maioris A C.</s>
  <s xml:id="echoid-s12559" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1083" type="float" level="2" n="2">
  <figure xlink:label="fig-0406-02" xlink:href="fig-0406-02a">
    <image file="0406-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0406-02"/>
  </figure>
<note position="left" xlink:label="note-0406-02" xlink:href="note-0406-02a" xml:space="preserve">PROP. 9. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12560" xml:space="preserve">Secetur H D in e, vt H e ad e D <lb/>eandem proportionem babeat, quàm <lb/>H A ad A D, &amp; </s>
  <s xml:id="echoid-s12561" xml:space="preserve">ex puncto e educa-<lb/>tur ad axim perpendicularis e h occur-<lb/>rens ſectioni in h, &amp; </s>
  <s xml:id="echoid-s12562" xml:space="preserve">coniungatur a <lb/>h, quàm bifariam ſecet diameter f d, <lb/>cuius erectus d g: </s>
  <s xml:id="echoid-s12563" xml:space="preserve">dico diametrum, <lb/>f d eße quæſitam. </s>
  <s xml:id="echoid-s12564" xml:space="preserve">Quia H e ad e D <lb/>eandem proportionem habet, quàm H <lb/>A ad A D, ergo differentia quadrato-<lb/>
<anchor type="note" xlink:label="note-0406-03a" xlink:href="note-0406-03"/>
rum ex f d, &amp; </s>
  <s xml:id="echoid-s12565" xml:space="preserve">ex d g æqualis eſt dif-<lb/>ferentiæ quadratorum ex A C, &amp; </s>
  <s xml:id="echoid-s12566" xml:space="preserve">ex <lb/>A F, quod erat propoſitum.</s>
  <s xml:id="echoid-s12567" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1084" type="float" level="2" n="3">
<note position="left" xlink:label="note-0406-03" xlink:href="note-0406-03a" xml:space="preserve">Lem. 17. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12568" xml:space="preserve">In ellypſi reperire diametrum, <lb/>
<anchor type="note" xlink:label="note-0406-04a" xlink:href="note-0406-04"/>
cuius differentia quadratorum late-<lb/>rum eius figuræ æqualis ſit diffe-<lb/>rentiæ quadratorum laterum figuræ
<pb o="369" file="0407" n="408" rhead="Conicor. Lib. VII."/>
datæ diametri I L: </s>
  <s xml:id="echoid-s12569" xml:space="preserve">oportet autem vt data diameter cadat inter axim <lb/>maiorem A C, &amp; </s>
  <s xml:id="echoid-s12570" xml:space="preserve">diametrum a b æqualem ſuo erecto a c.</s>
  <s xml:id="echoid-s12571" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1085" type="float" level="2" n="4">
<note position="left" xlink:label="note-0406-04" xlink:href="note-0406-04a" xml:space="preserve">PROP. <lb/>10. <lb/>Addit.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12572" xml:space="preserve">Sit C E latus diametri I L, &amp; </s>
  <s xml:id="echoid-s12573" xml:space="preserve">diuidatur H D in V, vt habeat H V ad V <lb/>D eandem proportionem, quàm H E habet ad E D, &amp; </s>
  <s xml:id="echoid-s12574" xml:space="preserve">ducta vt prius ad axim <lb/>perpendiculari V X occurrens ſectioni in X, &amp; </s>
  <s xml:id="echoid-s12575" xml:space="preserve">coniuncta A X, quam bifa-<lb/>riam ſecet diameter T S, cuius erectus S Z; </s>
  <s xml:id="echoid-s12576" xml:space="preserve">dico hanc eſſe quæſitam. </s>
  <s xml:id="echoid-s12577" xml:space="preserve">Quo-<lb/>
<anchor type="note" xlink:label="note-0407-01a" xlink:href="note-0407-01"/>
niam H V ad V D eandem proportionem habet, quàm H E ad E D, igitur <lb/>differentia quadratorum ex T S, &amp; </s>
  <s xml:id="echoid-s12578" xml:space="preserve">ex S Z æqualis eſt differentiæ quad<unsure/>ratorum <lb/>ex I L, &amp; </s>
  <s xml:id="echoid-s12579" xml:space="preserve">ex I K, quod propoſitum fuerat.</s>
  <s xml:id="echoid-s12580" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1086" type="float" level="2" n="5">
<note position="right" xlink:label="note-0407-01" xlink:href="note-0407-01a" xml:space="preserve">Lem. 17. <lb/>huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12581" xml:space="preserve">Deducitur ex 9. </s>
  <s xml:id="echoid-s12582" xml:space="preserve">propoſitione additarum, atque ex propoſit. </s>
  <s xml:id="echoid-s12583" xml:space="preserve">51. </s>
  <s xml:id="echoid-s12584" xml:space="preserve">huius, quod <lb/>in ellypſi exceſſus quadrati cuiuſlibet diametri tranſuerſæ ſupra quadratum ere-<lb/>cti eius ſucceſſiue decreſcit ab axi maiori A C vſque ad diametrum a b æqua-<lb/>lem ſuo erecto, atque ab hac diametro defectus quadrati cuiuſlibet tranſuerſæ <lb/>diametri à quadrato erecti eius ſucceſſiue augetur, quouſque perueniatur ad dia-<lb/>metrum f d, cuius differentia quadratorum figuræ eius æqualis ſit differentiæ <lb/>
<anchor type="note" xlink:label="note-0407-02a" xlink:href="note-0407-02"/>
quadratorum figuræ axis maioris A C, &amp; </s>
  <s xml:id="echoid-s12585" xml:space="preserve">vltra diametrum f d differentiæ præ-<lb/>dictæ ſemper magis augentur quouſque perueniatur ad axim minorem γ O cuius <lb/>differentia quadratorum figuræ eius maxima eſt omnium differentiarum inter <lb/>quadrata laterum figuræ cuiuſlibet diametri eiuſdem ellypſis.</s>
  <s xml:id="echoid-s12586" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1087" type="float" level="2" n="6">
<note position="right" xlink:label="note-0407-02" xlink:href="note-0407-02a" xml:space="preserve">ex Prop. <lb/>50. huius.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12587" xml:space="preserve">Conſtat quoque ex 9. </s>
  <s xml:id="echoid-s12588" xml:space="preserve">propoſitione additarum, quod in ellypſi tres diametri <lb/>reperiri poßunt, quarum differentia quadratorum figurarum laterum earum <lb/>æquales ſint inter ſe.</s>
  <s xml:id="echoid-s12589" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12590" xml:space="preserve">Et ex 10. </s>
  <s xml:id="echoid-s12591" xml:space="preserve">additarum reperiri poſſunt quatuor diametri, quarum differentiæ <lb/>quadrat orum laterum figurarum earum æquales ſint inter ſe: </s>
  <s xml:id="echoid-s12592" xml:space="preserve">in hyperbole verò <lb/>hoc non contingit, nam ab axi differentiæ quadratorum laterum figuræ cuiuſli-<lb/>
<anchor type="note" xlink:label="note-0407-03a" xlink:href="note-0407-03"/>
bet diametri ſucceſſiue augentur, ſi axis maior fuerit ſuo erecto, at ſi minor <lb/>
<anchor type="note" xlink:label="note-0407-04a" xlink:href="note-0407-04"/>
fuerit prædictæ differentiæ quadratorum ſucceſſiue diminuuntur.</s>
  <s xml:id="echoid-s12593" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1088" type="float" level="2" n="7">
<note position="right" xlink:label="note-0407-03" xlink:href="note-0407-03a" xml:space="preserve">ex Prop. <lb/>49. huius.</note>
<note position="right" xlink:label="note-0407-04" xlink:href="note-0407-04a" xml:space="preserve">ex Prop. <lb/>50. huius.</note>
</div>
<p>
  <s xml:id="echoid-s12594" xml:space="preserve">Differentia (8. </s>
  <s xml:id="echoid-s12595" xml:space="preserve">15.) </s>
  <s xml:id="echoid-s12596" xml:space="preserve">duorum quadratorum duorum laterum figuræ axis <lb/>
<anchor type="note" xlink:label="note-0407-05a" xlink:href="note-0407-05"/>
maior eſt in hyperbola (51.)</s>
  <s xml:id="echoid-s12597" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s12598" xml:space="preserve">ellypſi, quàm differentia quadratorum <lb/>duorum laterum figuræ homologæ diametri ſectionis, &amp; </s>
  <s xml:id="echoid-s12599" xml:space="preserve">differentia ho-<lb/>mologi proximioris axi maior eſt differentia homologi remotioris: </s>
  <s xml:id="echoid-s12600" xml:space="preserve">hoc <lb/>autem ſi axis in hyperbola minor fuerit ſuo erecto (49.)</s>
  <s xml:id="echoid-s12601" xml:space="preserve">; </s>
  <s xml:id="echoid-s12602" xml:space="preserve">ſi verò fuerit <lb/>maior oppoſitum pronunciandum eſt (50.)</s>
  <s xml:id="echoid-s12603" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s12604" xml:space="preserve">differentia quadrati axis <lb/>inclinati, &amp; </s>
  <s xml:id="echoid-s12605" xml:space="preserve">figuræ eius minor eſt ſemidifferentia quadratorum duorum <lb/>laterũ figuræ ſui homologi, ſi axis inclinatus minor eſt ſuo erecto (49.) <lb/></s>
  <s xml:id="echoid-s12606" xml:space="preserve">ſi verò fuerit maior exceſſus axis maior erit dimidio exceſſus quadrato-<lb/>rum duorum laterum figuræ homologi, &amp; </s>
  <s xml:id="echoid-s12607" xml:space="preserve">minor quàm tota, &amp;</s>
  <s xml:id="echoid-s12608" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12609" xml:space="preserve">Legen-<lb/>dum puto: </s>
  <s xml:id="echoid-s12610" xml:space="preserve">in qualibet ellypſi, &amp;</s>
  <s xml:id="echoid-s12611" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12612" xml:space="preserve">vt in textu apparet.</s>
  <s xml:id="echoid-s12613" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1089" type="float" level="2" n="8">
<note position="left" xlink:label="note-0407-05" xlink:href="note-0407-05a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s12614" xml:space="preserve">Et ſit P Q in ellypſi vna &amp;</s>
  <s xml:id="echoid-s12615" xml:space="preserve">hellip;</s>
  <s xml:id="echoid-s12616" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s12617" xml:space="preserve">educamus A B, A N, &amp;</s>
  <s xml:id="echoid-s12618" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s12619" xml:space="preserve">
<anchor type="note" xlink:label="note-0407-06a" xlink:href="note-0407-06"/>
Repleui lacunam, vt in textu videre eſt.</s>
  <s xml:id="echoid-s12620" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1090" type="float" level="2" n="9">
<note position="left" xlink:label="note-0407-06" xlink:href="note-0407-06a" xml:space="preserve">b</note>
</div>
<p>
  <s xml:id="echoid-s12621" xml:space="preserve">Ergo E H ad H A minor eſt quàm E D ad D A, nempe E X exceſſus <lb/>
<anchor type="note" xlink:label="note-0407-07a" xlink:href="note-0407-07"/>
E G, E H ad A C exceſſum H A, A G, &amp; </s>
  <s xml:id="echoid-s12622" xml:space="preserve">quadratum A C in omni-<lb/>bus figuris ad differentiam duorum quadratorum A G, A F, vt quadra-<lb/>tum A H ad differentiam duorum quadratorũ A G, &amp; </s>
  <s xml:id="echoid-s12623" xml:space="preserve">E H ad H A mi-<lb/>nor in duabus primis, &amp; </s>
  <s xml:id="echoid-s12624" xml:space="preserve">maior in duabus ſecundis, quàm E G ad G A, <lb/>&amp; </s>
  <s xml:id="echoid-s12625" xml:space="preserve">iungamus ergo E H ad H A, nempe E H ad H A, quàm aggrega-
<pb o="370" file="0408" n="409" rhead="Apollonij Pergæi"/>
tum E H, E G in ſuum exceſſum ad aggregatum H A, E G in ſuum ex-<lb/>ceſſum æqualis exceſſui duorum quadratorum E H, E G, nempe qua-<lb/>dratum A C ad exceſſum quadratorum duorum laterum figuræ I L mi-<lb/>nor in prima ellypſi, &amp; </s>
  <s xml:id="echoid-s12626" xml:space="preserve">maior in ſecunda, quàm quadratum A H ad ag-<lb/>gregatum H A, A G in eorum exceſſu æqualis, &amp;</s>
  <s xml:id="echoid-s12627" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12628" xml:space="preserve">Hæc omnia corrigi <lb/>debuiſſe nemo negabit, atque hinc manifeſtum eſt non pauca in textu arabico <lb/>deſiderari, cum propoſitio 51, vera non ſit abſque determinationibus ſuperius <lb/>expoſitis.</s>
  <s xml:id="echoid-s12629" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1091" type="float" level="2" n="10">
<note position="left" xlink:label="note-0407-07" xlink:href="note-0407-07a" xml:space="preserve">c</note>
</div>
</div>
<div xml:id="echoid-div1093" type="section" level="1" n="343">
<head xml:id="echoid-head424" xml:space="preserve">SECTIO VNDECIMA</head>
<head xml:id="echoid-head425" xml:space="preserve">Continens Propoſit. XXXII. &amp; XXXI.</head>
<head xml:id="echoid-head426" xml:space="preserve">Apollonij.</head>
<p>
  <s xml:id="echoid-s12630" xml:space="preserve">IN ellypſi, &amp; </s>
  <s xml:id="echoid-s12631" xml:space="preserve">ſectionibus coniugatis parallelogrammum ſub <lb/>
<anchor type="note" xlink:label="note-0408-01a" xlink:href="note-0408-01"/>
axibus contentum æquale eſt parallelogrammo à quibuſcun-<lb/>que duabus coniugatis diametris comprehenſo, ſi eorum anguli <lb/>æquales fuerint angulis ad centrum contentis à coniugatis dia-<lb/>metris.</s>
  <s xml:id="echoid-s12632" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1093" type="float" level="2" n="1">
<note position="right" xlink:label="note-0408-01" xlink:href="note-0408-01a" xml:space="preserve">a</note>
</div>
<p>
  <s xml:id="echoid-s12633" xml:space="preserve">Sint duo axes A B, C D in ellipſi A C <lb/>
<anchor type="figure" xlink:label="fig-0408-01a" xlink:href="fig-0408-01"/>
B D, ſiue in ſectionibus coniugatis A, B, <lb/>C, D, &amp; </s>
  <s xml:id="echoid-s12634" xml:space="preserve">ſint F G, I H aliæ duæ coniu-<lb/>gatæ diametri, &amp; </s>
  <s xml:id="echoid-s12635" xml:space="preserve">ducantur per puncta F, <lb/>I, G, H, lìneæ tangentes coniſectiones, <lb/>quæ ſibi mutuo occurrant ad puncta K, L, <lb/>M, N: </s>
  <s xml:id="echoid-s12636" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12637" xml:space="preserve">producatur A B ex vtraque <lb/>parte vſque ad tangentes, eaſque ſecet in <lb/>O, P, &amp; </s>
  <s xml:id="echoid-s12638" xml:space="preserve">ſit centrum E. </s>
  <s xml:id="echoid-s12639" xml:space="preserve">Dico quod A <lb/>B in C D æquale eſt ſpatio parallelogram-<lb/>mo M K: </s>
  <s xml:id="echoid-s12640" xml:space="preserve">ſit itaque F R perpendicularis <lb/>ad A B; </s>
  <s xml:id="echoid-s12641" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12642" xml:space="preserve">ponamus S R mediam propor-<lb/>tionalem inter O R, R E.</s>
  <s xml:id="echoid-s12643" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1094" type="float" level="2" n="2">
  <figure xlink:label="fig-0408-01" xlink:href="fig-0408-01a">
    <image file="0408-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0408-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s12644" xml:space="preserve">Et quia quadratum A E ad quadratum <lb/>E C eandem proportionem habet, quàm <lb/>
<anchor type="note" xlink:label="note-0408-02a" xlink:href="note-0408-02"/>
O R in R E, nempe quàm quadratum S <lb/>R ad quadratum F R (37. </s>
  <s xml:id="echoid-s12645" xml:space="preserve">ex 1.) </s>
  <s xml:id="echoid-s12646" xml:space="preserve">erit A E <lb/>ad E C nempe quadratum A E ad A E in <lb/>E C, vt S R ad F R, nempe S R in O E <lb/>ad F R in O E, &amp; </s>
  <s xml:id="echoid-s12647" xml:space="preserve">permutando erit qua-<lb/>dratum A E, nempe R E in O E (39. </s>
  <s xml:id="echoid-s12648" xml:space="preserve">ex 1.)</s>
  <s xml:id="echoid-s12649" xml:space="preserve">
<pb o="371" file="0409" n="410" rhead="Conicor. Lib. VII."/>
ad S R in O E, vt A E in E C ad F R in O E, &amp; </s>
  <s xml:id="echoid-s12650" xml:space="preserve">quadratum O F <lb/>
<anchor type="note" xlink:label="note-0409-01a" xlink:href="note-0409-01"/>
ad quadratum E H, nempe triangulum E O F ad triangulum E H P (24. <lb/></s>
  <s xml:id="echoid-s12651" xml:space="preserve">ex 2.) </s>
  <s xml:id="echoid-s12652" xml:space="preserve">propter ſimilitudinem duorum triangulorum eſt, vt OR ad R E <lb/>
<anchor type="figure" xlink:label="fig-0409-01a" xlink:href="fig-0409-01"/>
(4. </s>
  <s xml:id="echoid-s12653" xml:space="preserve">ex 7.)</s>
  <s xml:id="echoid-s12654" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s12655" xml:space="preserve">ſpatium parallelogrammum E K medium proportionale <lb/>eſt inter duplum trianguli E O F, &amp; </s>
  <s xml:id="echoid-s12656" xml:space="preserve">duplum trianguli E H P; </s>
  <s xml:id="echoid-s12657" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12658" xml:space="preserve">S R me-<lb/>dia proportionalis eſt inter O R, &amp; </s>
  <s xml:id="echoid-s12659" xml:space="preserve">R E, erit duplum trianguli E O F <lb/>ad parallelogrammum E K, vt S R ad R E; </s>
  <s xml:id="echoid-s12660" xml:space="preserve">nempe S R in O E ad R E, <lb/>in O E, quæ oſtendetur eſſe, vt F R in O E, quod eſt æquale duplo <lb/>trianguli O F E ad A E in E C; </s>
  <s xml:id="echoid-s12661" xml:space="preserve">ergo parallelogrammum E K æquale <lb/>eſt ipſi E A in E C, &amp; </s>
  <s xml:id="echoid-s12662" xml:space="preserve">propterea quadruplum illius ſpatij, quod eſt pa-<lb/>rallelogrammum M K æquale eſt ipſi B A in C D. </s>
  <s xml:id="echoid-s12663" xml:space="preserve">Et hoc erat propo-<lb/>ſitum.</s>
  <s xml:id="echoid-s12664" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1095" type="float" level="2" n="3">
<note position="right" xlink:label="note-0408-02" xlink:href="note-0408-02a" xml:space="preserve">b</note>
<note position="left" xlink:label="note-0409-01" xlink:href="note-0409-01a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0409-01" xlink:href="fig-0409-01a">
    <image file="0409-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0409-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s12665" xml:space="preserve">Hic eſt finis libri ſeptimi Apollonij, quemadmodum illum di-<lb/>
<anchor type="note" xlink:label="note-0409-02a" xlink:href="note-0409-02"/>
<anchor type="note" xlink:label="note-0409-03a" xlink:href="note-0409-03"/>
ſpoſui, &amp; </s>
  <s xml:id="echoid-s12666" xml:space="preserve">puto me præueniſſe in hoc quoſcunque alios, illumquè repo-<lb/>ſui in Bibliotheca Domini Noſtri Regis Glorioſiſſimi, Beneficentiſſimi, <lb/>Victorioſi; </s>
  <s xml:id="echoid-s12667" xml:space="preserve">Deus vmbram illius conſeruet ſuper omnes famulos eius, &amp; </s>
  <s xml:id="echoid-s12668" xml:space="preserve"><lb/>greges, &amp; </s>
  <s xml:id="echoid-s12669" xml:space="preserve">ad finem perducat omnia illius deſideria, &amp; </s>
  <s xml:id="echoid-s12670" xml:space="preserve">cogitationes, <lb/>&amp; </s>
  <s xml:id="echoid-s12671" xml:space="preserve">labor famuli eius ſit iuxta eius beneplacitum; </s>
  <s xml:id="echoid-s12672" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12673" xml:space="preserve">Laus Deo Domino <lb/>ſæculorum, &amp; </s>
  <s xml:id="echoid-s12674" xml:space="preserve">orationes eius ſint ſuper Maumethum, eiuſque ſequaces. <lb/></s>
  <s xml:id="echoid-s12675" xml:space="preserve">Explicit anno D XIII. </s>
  <s xml:id="echoid-s12676" xml:space="preserve">ſcribente Mahamudo filio Maſudi Medici Scira-<lb/>zeni decima die di Alkade Anno DCCCXXV.</s>
  <s xml:id="echoid-s12677" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1096" type="float" level="2" n="4">
<note position="left" xlink:label="note-0409-02" xlink:href="note-0409-02a" xml:space="preserve">*</note>
<note position="right" xlink:label="note-0409-03" xlink:href="note-0409-03a" xml:space="preserve">* In ſequẽ-<lb/>tibus Pa-<lb/>raphraſtes <lb/>Arabicus <lb/>impiè, &amp; <lb/>Maume-<lb/>danorum <lb/>more lo-<lb/>quitur.</note>
</div>
<pb o="372" file="0410" n="411" rhead="Apollonij Pergæi"/>
</div>
<div xml:id="echoid-div1098" type="section" level="1" n="344">
<head xml:id="echoid-head427" xml:space="preserve">Notæ in Propoſit. XXXI. &amp; XXXII.</head>
<p style="it">
  <s xml:id="echoid-s12678" xml:space="preserve">PLanum axium coniugatarum in ellipſi, &amp;</s>
  <s xml:id="echoid-s12679" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12680" xml:space="preserve">Ideſt in ſectionibus coniu-<lb/>
<anchor type="note" xlink:label="note-0410-01a" xlink:href="note-0410-01"/>
gatis, &amp; </s>
  <s xml:id="echoid-s12681" xml:space="preserve">in ellipſi rectangulum ſub axibus coniugatis contentum æquale <lb/>eſt parallelogrammo ſub diametris coniugatis in angulo æquali, ei qui ad cen-<lb/>trum à diametris continetur. </s>
  <s xml:id="echoid-s12682" xml:space="preserve">In textu arabico reperitur numerus 9. </s>
  <s xml:id="echoid-s12683" xml:space="preserve">in illa <lb/>propoſitione, quæ ellipſim conſiderat, ſed mendoſe, vt arbitror debet potius <lb/>cenſeri propoſit. </s>
  <s xml:id="echoid-s12684" xml:space="preserve">32.</s>
  <s xml:id="echoid-s12685" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1098" type="float" level="2" n="1">
<note position="right" xlink:label="note-0410-01" xlink:href="note-0410-01a" xml:space="preserve">a</note>
</div>
<p style="it">
  <s xml:id="echoid-s12686" xml:space="preserve">Et quia quadratum A E ad qua-<lb/>
<anchor type="note" xlink:label="note-0410-02a" xlink:href="note-0410-02"/>
<anchor type="figure" xlink:label="fig-0410-01a" xlink:href="fig-0410-01"/>
dratum E C eſt, vt O R in R E, <lb/>nempe quadratum S R ad quadra-<lb/>tum F R, &amp;</s>
  <s xml:id="echoid-s12687" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12688" xml:space="preserve">Quoniam axis rectus <lb/>D C medius proportionalis eſt inter a-<lb/>xim tranſuerſum A B, eiuſque latus <lb/>rectum, quadratum A B ad quadra-<lb/>tum D C, vel eorundem quadrantes, <lb/>ſcilicet quadratum ſemiaxis A E ad <lb/>quadratum ſemiaxis E C eandem pro-<lb/>portionem habebit, quàm axis tran-<lb/>ſuerſus A B ad eius latus rectum, ſed <lb/>rectangulum E R O ad quadratum F R <lb/>
<anchor type="note" xlink:label="note-0410-03a" xlink:href="note-0410-03"/>
eandem proportionem habet, quàm axis <lb/>tranſuerſus A B ad eius latus rectum, <lb/>atque quadratum S R æquale eſt rectan-<lb/>gulo E R O (eo quod S R facta fuit me-<lb/>dia proportionalis inter E R, &amp; </s>
  <s xml:id="echoid-s12689" xml:space="preserve">R O) <lb/>erit quadratum S R ad quadratum F <lb/>R, vt latus tranſuerſum A B ad eius <lb/>latus rectum: </s>
  <s xml:id="echoid-s12690" xml:space="preserve">quare quadratum A E <lb/>ad quadratum E C eandem proportio-<lb/>nem habebit, quàm quadratum S R ad <lb/>quadratum F R: </s>
  <s xml:id="echoid-s12691" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12692" xml:space="preserve">A E ad E C ean-<lb/>dem proportionem habebit, quàm S R ad F R: </s>
  <s xml:id="echoid-s12693" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12694" xml:space="preserve">ſumptis altitudinibus A E, <lb/>&amp; </s>
  <s xml:id="echoid-s12695" xml:space="preserve">O E erit quadratum A E, ſeu ei æquale rectangulum R E O ad rectangu-<lb/>
<anchor type="note" xlink:label="note-0410-04a" xlink:href="note-0410-04"/>
lum A E C, vt rectangulum ſub S R, &amp; </s>
  <s xml:id="echoid-s12696" xml:space="preserve">ſub O E ad rectangulum ſub F R, <lb/>&amp; </s>
  <s xml:id="echoid-s12697" xml:space="preserve">ſub O E, &amp; </s>
  <s xml:id="echoid-s12698" xml:space="preserve">permutando rectangulum R E O ad rectangulum ſub S R, &amp; </s>
  <s xml:id="echoid-s12699" xml:space="preserve"><lb/>ſub O E, ſeu vt R E ad S R eandem proportionem habebit, quàm rectangu-<lb/>lum A E C ad rectangulum ſub F R, &amp; </s>
  <s xml:id="echoid-s12700" xml:space="preserve">ſub O E: </s>
  <s xml:id="echoid-s12701" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12702" xml:space="preserve">inuertendo rectangulum <lb/>ſub F R, &amp; </s>
  <s xml:id="echoid-s12703" xml:space="preserve">ſub O E ad rectangulum A E C eandem proportionem habet quàm <lb/>S R ad R E.</s>
  <s xml:id="echoid-s12704" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1099" type="float" level="2" n="2">
<note position="right" xlink:label="note-0410-02" xlink:href="note-0410-02a" xml:space="preserve">b</note>
  <figure xlink:label="fig-0410-01" xlink:href="fig-0410-01a">
    <image file="0410-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0410-01"/>
  </figure>
<note position="left" xlink:label="note-0410-03" xlink:href="note-0410-03a" xml:space="preserve">Prop. 37. <lb/>lib. I.</note>
<note position="left" xlink:label="note-0410-04" xlink:href="note-0410-04a" xml:space="preserve">Ibidem.</note>
</div>
<pb o="373" file="0411" n="412" rhead="Conicor. Lib. VII."/>
<p style="it">
  <s xml:id="echoid-s12705" xml:space="preserve">Et quadratum F O ad quadratum E H, nempe triangulum E F O ad <lb/>
<anchor type="note" xlink:label="note-0411-01a" xlink:href="note-0411-01"/>
triangulum E H P, &amp;</s>
  <s xml:id="echoid-s12706" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12707" xml:space="preserve">Quia G F, I H ſunt diametri coniugatæ, quibus <lb/>æquidiſtant contingentes F O, &amp; </s>
  <s xml:id="echoid-s12708" xml:space="preserve">L H erunt triangula E O F, &amp; </s>
  <s xml:id="echoid-s12709" xml:space="preserve">E H P ſimi-<lb/>lia, quorum latera homologa O F, &amp; </s>
  <s xml:id="echoid-s12710" xml:space="preserve">E H; </s>
  <s xml:id="echoid-s12711" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12712" xml:space="preserve">ideo triangulum E O F ad <lb/>
<anchor type="figure" xlink:label="fig-0411-01a" xlink:href="fig-0411-01"/>
<anchor type="note" xlink:label="note-0411-02a" xlink:href="note-0411-02"/>
triangulum E H P eandem proportionem habebit, quàm quadratum O F ad <lb/>quadratum E H: </s>
  <s xml:id="echoid-s12713" xml:space="preserve">eſtque O R ad R E, vt quadratum O F ad quadratum E H, <lb/>igitur triangulum E O F ad triangulum E H P eandem proportionem habebit, <lb/>quàm O R ad R E. </s>
  <s xml:id="echoid-s12714" xml:space="preserve">Ducatur poſtea recta linea E K, erit triangulum E F K <lb/>medium proportionale inter duo ſimilia triangula E O F, &amp; </s>
  <s xml:id="echoid-s12715" xml:space="preserve">E H P (eo quod <lb/>triangulum E O F ad triangulum E F K æquè altum eandem proportionem ha-<lb/>bet quàm O F ad F K, ſeu ad latus E H ei homologum) poſita autem fuit S <lb/>R media proportionalis inter O R, &amp; </s>
  <s xml:id="echoid-s12716" xml:space="preserve">R E; </s>
  <s xml:id="echoid-s12717" xml:space="preserve">ergo triangulum E O F ad trian-<lb/>gulum E F K eſt vt S R ad R E: </s>
  <s xml:id="echoid-s12718" xml:space="preserve">eſtquè parallelogrammum E K æquale duplo <lb/>trianguli E F K; </s>
  <s xml:id="echoid-s12719" xml:space="preserve">ergo duplum trianguli E O F ad parallelogrammum E K ean-<lb/>dem proportionem habet, quàm S R ad R E; </s>
  <s xml:id="echoid-s12720" xml:space="preserve">Et quia rectangulum ſub O E, <lb/>&amp; </s>
  <s xml:id="echoid-s12721" xml:space="preserve">ſub perpendiculari R F æquale eſt duplo trianguli E O F (cum habeant baſim <lb/>O E communem, &amp; </s>
  <s xml:id="echoid-s12722" xml:space="preserve">eandem altitudinem perpendicularis R F); </s>
  <s xml:id="echoid-s12723" xml:space="preserve">igitur rectan-<lb/>gulum ſub O E, &amp; </s>
  <s xml:id="echoid-s12724" xml:space="preserve">ſub R F ad parallelogrammum E K eandem proportionem <lb/>habebit, quàm S R ad R E: </s>
  <s xml:id="echoid-s12725" xml:space="preserve">ſed prius rectangulum ſub O E, &amp; </s>
  <s xml:id="echoid-s12726" xml:space="preserve">ſub R F ad <lb/>rectangulum A E C eandem proportionem habebat, quàm S R ad R E: </s>
  <s xml:id="echoid-s12727" xml:space="preserve">ergo <lb/>idem rectangulum ſub O E, &amp; </s>
  <s xml:id="echoid-s12728" xml:space="preserve">ſub R F ad parallelogrammum E K eandem <lb/>proportionem habet, quàm ad rectangulum A E C; </s>
  <s xml:id="echoid-s12729" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12730" xml:space="preserve">propterea parallelogram-
<pb o="374" file="0412" n="413" rhead="Apollonij Pergæi"/>
mum E K æquale eſt rectangulo A E C; </s>
  <s xml:id="echoid-s12731" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12732" xml:space="preserve">eorum quadrupla erunt æqualia, <lb/>ſcilicet parallelogrammum M K æquale erit rectangulo ſub B A, &amp; </s>
  <s xml:id="echoid-s12733" xml:space="preserve">ſub D C <lb/>compræhenſo. </s>
  <s xml:id="echoid-s12734" xml:space="preserve">Quod erat propoſitum.</s>
  <s xml:id="echoid-s12735" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1100" type="float" level="2" n="3">
<note position="left" xlink:label="note-0411-01" xlink:href="note-0411-01a" xml:space="preserve">c</note>
  <figure xlink:label="fig-0411-01" xlink:href="fig-0411-01a">
    <image file="0411-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0411-01"/>
  </figure>
<note position="right" xlink:label="note-0411-02" xlink:href="note-0411-02a" xml:space="preserve">Prop. 4. <lb/>huius.</note>
</div>
</div>
<div xml:id="echoid-div1102" type="section" level="1" n="345">
<head xml:id="echoid-head428" xml:space="preserve">LIBRI SEPTIMI FINIS.</head>
  <figure>
    <image file="0412-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0412-01"/>
  </figure>
<pb file="0413" n="414"/>
<pb file="0414" n="415"/>
<pb file="0415" n="416" rhead="AR CHIMEDIS"/>
</div>
<div xml:id="echoid-div1103" type="section" level="1" n="346">
<head xml:id="echoid-head429" xml:space="preserve">LIBER ASSVMPTORVM</head>
<head xml:id="echoid-head430" xml:space="preserve">INTERPRETE <lb/>THEBIT BEN-KORA</head>
<head xml:id="echoid-head431" style="it" xml:space="preserve">EXPONENTE AL MOCHT ASSO</head>
<head xml:id="echoid-head432" xml:space="preserve">Ex Codice Arabico manuſcripto</head>
<head xml:id="echoid-head433" xml:space="preserve">SERENISS. MAGNI DV CIS ETRVRIÆ, <lb/>ABRAHAMVS ECCHELLENSIS <lb/>Latinè vertit.</head>
<head xml:id="echoid-head434" xml:space="preserve">IO: ALFONSVS BORELLVS <lb/>Notis Illuſtrauit.</head>
<pb file="0416" n="417"/>
<pb o="379" file="0417" n="418" rhead="IO: ALFONSI BORELLI"/>
</div>
<div xml:id="echoid-div1104" type="section" level="1" n="347">
<head xml:id="echoid-head435" xml:space="preserve">Præfatio ad Lectorem.</head>
<p style="it">
  <s xml:id="echoid-s12736" xml:space="preserve">SI pulchrum illud Epicharmi effatum tenes ( amice <lb/>Lector ) neruos, atque artus eſſe ſapientiæ non <lb/>temerè, ac imprudenter credere, non adeò faci-<lb/>lis eſſe debes, vt Archimedis nomen lemmata <lb/>hæc pretioſiora efficiens tibi impoſturam, aut fu-<lb/>cum facere patiaris, atque alterius contemptiſ-<lb/>ſimi auctoris opuſculum immeritò tanto viro tri-<lb/>buas; </s>
  <s xml:id="echoid-s12737" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12738" xml:space="preserve">ſiquidem maiores noſtri æquum iudi-<lb/>cium dixere, vt ſine inuidia culpa plectatur, non ita moroſus, ac dif-<lb/>ficilis eſſe debes, vt ſua ei denegare velis leui quacumque ſuſpicione, <lb/>quæ facile excuti poſsit; </s>
  <s xml:id="echoid-s12739" xml:space="preserve">verum ab omni præiudicio liberum te cupio, <lb/>&amp; </s>
  <s xml:id="echoid-s12740" xml:space="preserve">memorem illius adagĳ: </s>
  <s xml:id="echoid-s12741" xml:space="preserve">Ne quid nimis. </s>
  <s xml:id="echoid-s12742" xml:space="preserve">Tibi igitur ſic affecto no-<lb/>tionem huius controuerſiæ omnino relinquo, quod vt liberè, &amp; </s>
  <s xml:id="echoid-s12743" xml:space="preserve">ritè exe-<lb/>qui valeas, ſedato animo nullum meum iudicium interponens, afferam <lb/>primò rationes, quibus perſuaderi quis poſſet hoc opuſculum iniurià <lb/>Archimedi tributum fuiſſe, &amp; </s>
  <s xml:id="echoid-s12744" xml:space="preserve">mox coniecturas recenſebo, quæ eiuſdem <lb/>Archimedis idipſum opus eſſe fortè non inaniter probant; </s>
  <s xml:id="echoid-s12745" xml:space="preserve">ſicque penſitatis, <lb/>&amp; </s>
  <s xml:id="echoid-s12746" xml:space="preserve">compoſitis vtrinque rationum ponderibus ſententiam liberè pronuncies <lb/>tuam per me licet.</s>
  <s xml:id="echoid-s12747" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12748" xml:space="preserve">Et primò animaduerſione dignum eſt in Collect. </s>
  <s xml:id="echoid-s12749" xml:space="preserve">Mathemat. </s>
  <s xml:id="echoid-s12750" xml:space="preserve">Pappi <lb/>Alexand. </s>
  <s xml:id="echoid-s12751" xml:space="preserve">frequentiſsimè commemorari ea, quæ Archimedes conſcripſit, <lb/>præcipuè lib. </s>
  <s xml:id="echoid-s12752" xml:space="preserve">5. </s>
  <s xml:id="echoid-s12753" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12754" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s12755" xml:space="preserve">8. </s>
  <s xml:id="echoid-s12756" xml:space="preserve">De Spiralibus, de Solidis Polyedris, de Cir-<lb/>culi Menſura, de Sphæra, &amp; </s>
  <s xml:id="echoid-s12757" xml:space="preserve">Cylindro, &amp; </s>
  <s xml:id="echoid-s12758" xml:space="preserve">multoties citantur, &amp; </s>
  <s xml:id="echoid-s12759" xml:space="preserve"><lb/>tranſcribuntur Archimedeæ propoſitiones, neque vſpiam huius Opuſculi
<pb o="380" file="0418" n="419" rhead="PRÆFATIO"/>
(apud Arabes hactenus latentis) mentio vlla fit. </s>
  <s xml:id="echoid-s12760" xml:space="preserve">Neque Ptol. </s>
  <s xml:id="echoid-s12761" xml:space="preserve">in Ma-<lb/>gnæ Conſtr. </s>
  <s xml:id="echoid-s12762" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s12763" xml:space="preserve">2. </s>
  <s xml:id="echoid-s12764" xml:space="preserve">tribuit Archimedi prop. </s>
  <s xml:id="echoid-s12765" xml:space="preserve">5. </s>
  <s xml:id="echoid-s12766" xml:space="preserve">cap. </s>
  <s xml:id="echoid-s12767" xml:space="preserve">9. </s>
  <s xml:id="echoid-s12768" xml:space="preserve">ibi relatam, cum <lb/>tamen ſoleat eſſe adeò gratus, vt lib. </s>
  <s xml:id="echoid-s12769" xml:space="preserve">6. </s>
  <s xml:id="echoid-s12770" xml:space="preserve">cap. </s>
  <s xml:id="echoid-s12771" xml:space="preserve">7. </s>
  <s xml:id="echoid-s12772" xml:space="preserve">propoſitionem ab Ar-<lb/>chimede ſumpſiſſe fateatur. </s>
  <s xml:id="echoid-s12773" xml:space="preserve">Neque ipſemet Archimedes huius Opuſculi <lb/>vnquam meminit, qui alioquì valdè prolixè enumerat, &amp; </s>
  <s xml:id="echoid-s12774" xml:space="preserve">recenſet ea, <lb/>quæ in proprĳs libris continentur, &amp; </s>
  <s xml:id="echoid-s12775" xml:space="preserve">demonſtrantur. </s>
  <s xml:id="echoid-s12776" xml:space="preserve">Inexcuſabiles inſu-<lb/>per errores, atque allucinationes, quæ in huiuſmodi propoſitionibus repe-<lb/>riuntur, immò puerilia alia Opuſcula, quæ citantur vt Archimedis, ſa-<lb/>tis apertè videntur oſtendere nunquam diuinum illud ingenium buiuſmodi <lb/>minutias ſomniaſſe; </s>
  <s xml:id="echoid-s12777" xml:space="preserve">cum, vt Carpus Antiochenſis ait, referente Pappo, <lb/>quæ præcipua ſunt in Geometria, breuiter quidem, ſed diligenter conſcri-<lb/>pſerit Archimedes. </s>
  <s xml:id="echoid-s12778" xml:space="preserve">Tandem præcipuæ propoſitiones huius Opuſculi ſimiles <lb/>ſunt eis, quæ recenſentur quidem, &amp; </s>
  <s xml:id="echoid-s12779" xml:space="preserve">demonſtrantur lib. </s>
  <s xml:id="echoid-s12780" xml:space="preserve">4. </s>
  <s xml:id="echoid-s12781" xml:space="preserve">Collect. </s>
  <s xml:id="echoid-s12782" xml:space="preserve">Ma-<lb/>them. </s>
  <s xml:id="echoid-s12783" xml:space="preserve">Pappi Alex.</s>
  <s xml:id="echoid-s12784" xml:space="preserve">, eaſque Archimedis eſſe non aſſerit; </s>
  <s xml:id="echoid-s12785" xml:space="preserve">immò in quibuſ-<lb/>dam libris antiquis circumferri affirmat.</s>
  <s xml:id="echoid-s12786" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12787" xml:space="preserve">Zuod verò dictæ rationes tanti roboris, ac efficaciæ non ſint, vt pe-<lb/>nitus euincant huiuſmodi Opuſculum ab aliquo alio tributum Archimedi <lb/>fuiſſe, ex modo dicendis patebit. </s>
  <s xml:id="echoid-s12788" xml:space="preserve">Et primo optimè norunt, qui in Pappi <lb/>libris euoluendis vllam operam impenderunt lib. </s>
  <s xml:id="echoid-s12789" xml:space="preserve">7. </s>
  <s xml:id="echoid-s12790" xml:space="preserve">Collect. </s>
  <s xml:id="echoid-s12791" xml:space="preserve">recenſere <lb/>eum prolixè, &amp; </s>
  <s xml:id="echoid-s12792" xml:space="preserve">accuratè quamplurima opera Apollonĳ Pergæi, quorum <lb/>pars maxima non extat, &amp; </s>
  <s xml:id="echoid-s12793" xml:space="preserve">enumer are propoſitiones, &amp; </s>
  <s xml:id="echoid-s12794" xml:space="preserve">lemmata vſ-<lb/>que ad figuras, &amp; </s>
  <s xml:id="echoid-s12795" xml:space="preserve">tamen qui huiuſmodi minutias curat, &amp; </s>
  <s xml:id="echoid-s12796" xml:space="preserve">adnotat, <lb/>idem integra opera eiuſdem Apollonĳ non commemorat. </s>
  <s xml:id="echoid-s12797" xml:space="preserve">Sufficiant hæc in-<lb/>ſignia ſpecimina. </s>
  <s xml:id="echoid-s12798" xml:space="preserve">De admirandis aſtronomicis demonſtrationibus à Pto-<lb/>lemæo ſummoperè laudatis lib. </s>
  <s xml:id="echoid-s12799" xml:space="preserve">12. </s>
  <s xml:id="echoid-s12800" xml:space="preserve">cap. </s>
  <s xml:id="echoid-s12801" xml:space="preserve">1. </s>
  <s xml:id="echoid-s12802" xml:space="preserve">Magnæ Conſtr.</s>
  <s xml:id="echoid-s12803" xml:space="preserve">, ne verbum <lb/>quidem. </s>
  <s xml:id="echoid-s12804" xml:space="preserve">De libro Comparationis Dodecaedri, &amp; </s>
  <s xml:id="echoid-s12805" xml:space="preserve">lcoſaedri ab Y pſicle <lb/>memorato, altum ſilentium. </s>
  <s xml:id="echoid-s12806" xml:space="preserve">Si igitur idem Pappus opera Archimedis <lb/>non ex profeſſo, ſed obiter, &amp; </s>
  <s xml:id="echoid-s12807" xml:space="preserve">ſparſim commemorat, mirum non eſt ta-<lb/>cuiſſe aliqua eius opera, vt ſunt hæc lemmata.</s>
  <s xml:id="echoid-s12808" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12809" xml:space="preserve">Secundò Ptolemæus non affirmat lib. </s>
  <s xml:id="echoid-s12810" xml:space="preserve">2. </s>
  <s xml:id="echoid-s12811" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s12812" xml:space="preserve">5. </s>
  <s xml:id="echoid-s12813" xml:space="preserve">proprio marte à ſe <lb/>inuentam fuiſſe, nec eam Archimedi, aut alicui alĳ tribuit, quare fieri <lb/>potuit, vt eam ex libro antiquo deſumpſerit, à quo nomen Archimedis <lb/>caſu expunctum fuiſſet, vt poſtea oſtendetur.</s>
  <s xml:id="echoid-s12814" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12815" xml:space="preserve">Tertiò Archimedes quoque in ſuis libris exiſtentibus Græcè, &amp; </s>
  <s xml:id="echoid-s12816" xml:space="preserve">Ara-<lb/>bicè non recenſet omnia opera à ſe conſcripta, &amp; </s>
  <s xml:id="echoid-s12817" xml:space="preserve">edita, nam liber de <lb/>
<anchor type="note" xlink:label="note-0418-01a" xlink:href="note-0418-01"/>
inſidentibus humido, &amp; </s>
  <s xml:id="echoid-s12818" xml:space="preserve">de Polyedris recenſentur quidem à Pappo, non <lb/>autem ab Archimede. </s>
  <s xml:id="echoid-s12819" xml:space="preserve">Liber Mechanicus de Sphæropæia nominatur à
<pb o="381" file="0419" n="420" rhead="PRÆFATIO."/>
Carpo Antiochenſe apud Pappum. </s>
  <s xml:id="echoid-s12820" xml:space="preserve">Liber de Figuris Iſoperimetris aſſer-<lb/>
<anchor type="note" xlink:label="note-0419-01a" xlink:href="note-0419-01"/>
uatur apud Arabes tantum; </s>
  <s xml:id="echoid-s12821" xml:space="preserve">non igitur adulterina buiuſmodi lemmata <lb/>erunt, propterea quod Archimedes ea non nominat in paucis libris reſiduis, <lb/>&amp; </s>
  <s xml:id="echoid-s12822" xml:space="preserve">fortè commemorata fuerunt in aliquibus alĳs ex multis operibus eius <lb/>iniuria temporum deperditis.</s>
  <s xml:id="echoid-s12823" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1104" type="float" level="2" n="1">
<note position="left" xlink:label="note-0418-01" xlink:href="note-0418-01a" xml:space="preserve">In proh. <lb/>lib. 8. <lb/>Lib. 5. pr. <lb/>17.</note>
<note position="right" xlink:label="note-0419-01" xlink:href="note-0419-01a" xml:space="preserve">In proh. <lb/>lib. 8.</note>
</div>
<p style="it">
  <s xml:id="echoid-s12824" xml:space="preserve">Quartò ſane negari non poßunt euidentiſsimi errores in hiſce demon-<lb/>ſtrationibus, qui certè lemmatum auctori tribuendi non ſunt, vt ſuis <lb/>in locis adnotabo; </s>
  <s xml:id="echoid-s12825" xml:space="preserve">explanatorum enim imperitia ſæpenumero propoſitiones <lb/>vniuerſaliter pronunciatæ violenter in ſenſu particulari, &amp; </s>
  <s xml:id="echoid-s12826" xml:space="preserve">deformi ex-<lb/>ponuntur. </s>
  <s xml:id="echoid-s12827" xml:space="preserve">Neque mirum eſt opera antiquorum magni nominis paſsim, <lb/>&amp; </s>
  <s xml:id="echoid-s12828" xml:space="preserve">multis modis deformata fuiße tranſcriptorum incuria opponendo notas <lb/>marginales, detrahendo, &amp; </s>
  <s xml:id="echoid-s12829" xml:space="preserve">ſuperaddendo textui alienas ſententias, ac <lb/>teſtimonia, &amp; </s>
  <s xml:id="echoid-s12830" xml:space="preserve">hoc præcipuæ in codicibus Arabicis frequentiſsimè obſer-<lb/>uauit Excell. </s>
  <s xml:id="echoid-s12831" xml:space="preserve">Abrahamus Ecchellenſis. </s>
  <s xml:id="echoid-s12832" xml:space="preserve">Sed nihilominus in tanta tran-<lb/>sformatione à vetuſtate, &amp; </s>
  <s xml:id="echoid-s12833" xml:space="preserve">ignorantia amanuenſium profecta veſti-<lb/>gium aliquod ſubobſcurum admirandi, &amp; </s>
  <s xml:id="echoid-s12834" xml:space="preserve">perſpicui Archimedis ingenĳ <lb/>dignoſcitur.</s>
  <s xml:id="echoid-s12835" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12836" xml:space="preserve">Tandem non inani coniectura ex Pappi, &amp; </s>
  <s xml:id="echoid-s12837" xml:space="preserve">Eutocĳ teſtimonĳs pro-<lb/>bari poteſt idipſum, quod Arabes ratum habent, ſcilicet Archimedem <lb/>huius libelli auctorem fuiſſe. </s>
  <s xml:id="echoid-s12838" xml:space="preserve">Et primo aio præter reliqua operaiam nota <lb/>edidiſſe Archimedem librum Lemmatum, quod quidem deducitur ex <lb/>Eutocio in Comment. </s>
  <s xml:id="echoid-s12839" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s12840" xml:space="preserve">4. </s>
  <s xml:id="echoid-s12841" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s12842" xml:space="preserve">2. </s>
  <s xml:id="echoid-s12843" xml:space="preserve">de Sphæra, &amp; </s>
  <s xml:id="echoid-s12844" xml:space="preserve">Cylindro, vbi ait: <lb/></s>
  <s xml:id="echoid-s12845" xml:space="preserve">Id, quod promiſerat ſe demonſtraturum, (ſcilicrt Archimedes) in <lb/>nullis exemplaribus reperire eſt, quare etiam Dionyſodorum de-<lb/>prehendimus nunquam in ea incidiſſe, adeoque cum non potue-<lb/>rit relictum (ab Archimede) lemma attingere diuerſam viam ſu-<lb/>ſcepit vniuerſi problematis, quam deinceps deſcribemus. </s>
  <s xml:id="echoid-s12846" xml:space="preserve">Dio-<lb/>cles porrò idipſum in libro à ſe de Pyrijs inſcripto, promiſſum <lb/>fuiſſe ab Archimede nunquam præſtitum opinatus, ſupplere con-<lb/>tendit, cuius conatum mox apponemus, quod &amp; </s>
  <s xml:id="echoid-s12847" xml:space="preserve">ipſum pariter <lb/>à ſuperius propoſitis diſcedit; </s>
  <s xml:id="echoid-s12848" xml:space="preserve">itidem enim ac Dionyſodorus alia <lb/>demonſtrandi ratione problema ſtruit. </s>
  <s xml:id="echoid-s12849" xml:space="preserve">IN QVODAM AVTEM <lb/>VETERI LIBRO (neque enim diuturnæ pepercimus diligentiæ) <lb/>ſupraſcripta incidimus theoremata haud exiguam tamen haben-<lb/>tia obſcuritatem præ erratis, multiformiterque mendoſa in figu-<lb/>rationibus. </s>
  <s xml:id="echoid-s12850" xml:space="preserve">Eamdem equidem veritatem, quam inquirebamus, <lb/>atque in parte domeſticam Archimedi linguã Doricam ſeruabant,
<pb o="382" file="0420" n="421" rhead="PRÆFATIO."/>
vſitatiſque pridem rerum nominibus conſcripta erant, quæ nunc <lb/>parabola, recti coniſectione, quæ hyperbole, obtuſi anguli ſe-<lb/>ctione vocata; </s>
  <s xml:id="echoid-s12851" xml:space="preserve">vt ex his ſuſpicari liceat EADEM IPSA FOR-<lb/>TEAN ESSE, QVÆ IN FINE SCRIBENDA PROMIT-<lb/>TEBANTVR; </s>
  <s xml:id="echoid-s12852" xml:space="preserve">quare attentius incumbentes, (cum ipſam hy-<lb/>potheſim, qualiter perſcripta fuerat, præ mendarum copia (vt <lb/>diximus) ſatis incommodam, &amp; </s>
  <s xml:id="echoid-s12853" xml:space="preserve">abſtruſam reperiremus,) ſen-<lb/>ſum inde paucis elijcientes communi, &amp; </s>
  <s xml:id="echoid-s12854" xml:space="preserve">plana dictione (vt fieri <lb/>potuit) deſcribimus. </s>
  <s xml:id="echoid-s12855" xml:space="preserve">Vniuerſaliter autem primum theorema de-<lb/>ſcribetur, vt definitis manifeſtetur, deinde reſolutis in proble-<lb/>mate accomodabitur. </s>
  <s xml:id="echoid-s12856" xml:space="preserve">Inferius</s>
</p>
<p>
  <s xml:id="echoid-s12857" xml:space="preserve">Præmiſſis autem problematis, quæ hìc apponuntur, ſcilicet <lb/>duplam eſſe ipſam D B ipſius B F, &amp;</s>
  <s xml:id="echoid-s12858" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12859" xml:space="preserve">(Nota quod hic loquitur <lb/>de lemmatibus adiunctis,) &amp; </s>
  <s xml:id="echoid-s12860" xml:space="preserve">paulò poſt; </s>
  <s xml:id="echoid-s12861" xml:space="preserve">animaduertendum eſt au-<lb/>t<unsure/>em, &amp; </s>
  <s xml:id="echoid-s12862" xml:space="preserve">hæc quæ ab Archimede dicta ſunt conſonare ijs, quæ <lb/>nos reſoluimus (ſcilicet ĳdem adductis lemmatibus). </s>
  <s xml:id="echoid-s12863" xml:space="preserve">Deinde cum <lb/>dixerit, quod ſuperius dictum vniuerſaliter habet determinatio-<lb/>nem, adiectis autem problematibus ab eo inuentis, hoc eſt ip-<lb/>ſam D B duplam eſſe ipſius B F, &amp; </s>
  <s xml:id="echoid-s12864" xml:space="preserve">ipſam B F maiorem ipſa <lb/>F H, &amp;</s>
  <s xml:id="echoid-s12865" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12866" xml:space="preserve">Hìc manifeſtè Eutocius declarat propoſita lemmata in anti-<lb/>quo codice inuenta Archimedis fuiſſe.</s>
  <s xml:id="echoid-s12867" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12868" xml:space="preserve">Hæc igitur conſentanea verbis Archimedis, qua fieri potuit, <lb/>dilucidè expoſuimus.</s>
  <s xml:id="echoid-s12869" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12870" xml:space="preserve">Conſtat ergò ex Eutocĳ ſententia librum antiquum ab eo repertum, &amp; </s>
  <s xml:id="echoid-s12871" xml:space="preserve"><lb/>recognitum, eſſe opus Archimedis, licet titulo Auctoris caruerit, &amp; </s>
  <s xml:id="echoid-s12872" xml:space="preserve">men-<lb/>doſiſsimum eſſet, atque ignotum Dionyſodoro, Diocli, &amp; </s>
  <s xml:id="echoid-s12873" xml:space="preserve">pleriſque <lb/>Græcorum diù iacuiſſet; </s>
  <s xml:id="echoid-s12874" xml:space="preserve">etenim ex ſtylo, ex ſubiecto promiſſo, ex lingua <lb/>Dorica, &amp; </s>
  <s xml:id="echoid-s12875" xml:space="preserve">ex vocibns vetuſtis Archimedi familiaribus concluſit lem-<lb/>mata prædicta Archimedis fuiſſe. </s>
  <s xml:id="echoid-s12876" xml:space="preserve">Sed adhuc difficultas hæret, nam licet <lb/>concedamus ſcripſiſſe Archimedẽ, &amp; </s>
  <s xml:id="echoid-s12877" xml:space="preserve">edidiſſe librum lemmatum ab Eutocio <lb/>memoratum, diuerſus omnino erit ab eo, quem Thebitius Arabicè tranſ-<lb/>tulit, nam in iſto non reperitur lemma illud, qnod promiſerat Archime-<lb/>des ſe demonſtraturum.</s>
  <s xml:id="echoid-s12878" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12879" xml:space="preserve">Hæc difficultas duplici coniectura ſi non frangi, ac reſolui ſaltem de-<lb/>bilitari poteſt; </s>
  <s xml:id="echoid-s12880" xml:space="preserve">liber enim antiquus lemmatum Archimedis ne dum titulo <lb/>carebat ſuo, ſed erat valdè corruptus, deficiens, &amp; </s>
  <s xml:id="echoid-s12881" xml:space="preserve">mendoſus; </s>
  <s xml:id="echoid-s12882" xml:space="preserve">quarè <lb/>non ſine diuturno, ac pertinaci labore ſenſus illius lemmatis elicere potuit
<pb o="383" file="0421" n="422" rhead="PRÆFATIO."/>
Eutocius, vnde fieri potuit vt Græcus codex ad Arabes tranſmiſſus de-<lb/>terior, &amp; </s>
  <s xml:id="echoid-s12883" xml:space="preserve">magis mutilus adhuc fuerit eo exemplari, in quod incidit Eu-<lb/>tocius, vel potius incuria, aut vitio librariornm Arabum, &amp; </s>
  <s xml:id="echoid-s12884" xml:space="preserve">ama-<lb/>nuenſium eiuſdem codicis quamplurima lemmata perierunt, inter quæ <lb/>aſſumptum in prop. </s>
  <s xml:id="echoid-s12885" xml:space="preserve">4. </s>
  <s xml:id="echoid-s12886" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s12887" xml:space="preserve">2. </s>
  <s xml:id="echoid-s12888" xml:space="preserve">de Sphæra, &amp; </s>
  <s xml:id="echoid-s12889" xml:space="preserve">Cylindro excidit. </s>
  <s xml:id="echoid-s12890" xml:space="preserve">E con-<lb/>trà aliquæ propoſitiones ſimiles eis, quæ leguntur in hoc Arabico codice de <lb/>Arbelo extant apud Pappum lib. </s>
  <s xml:id="echoid-s12891" xml:space="preserve">4. </s>
  <s xml:id="echoid-s12892" xml:space="preserve">Collect. </s>
  <s xml:id="echoid-s12893" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s12894" xml:space="preserve">14. </s>
  <s xml:id="echoid-s12895" xml:space="preserve">15. </s>
  <s xml:id="echoid-s12896" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12897" xml:space="preserve">16.</s>
  <s xml:id="echoid-s12898" xml:space="preserve">, quas <lb/>ait circumferri in quibuſdam libris antiquis, ſcilicet in libro Græco incerti <lb/>Auctoris propoſitiones lemmaticas continente; </s>
  <s xml:id="echoid-s12899" xml:space="preserve">at teſtimonio Thebitĳ magni <lb/>nominis viri, &amp; </s>
  <s xml:id="echoid-s12900" xml:space="preserve">omnium Arabum, liber ex Græco translatus continens <lb/>ferè eadem lemmata, quæ recenſentur à Pappo, tribuitur Archimedi, ſi-<lb/>cuti prius Eutocius multiplici coniectura libri antiqui lemmatum à ſe re-<lb/>perti Archimedem auctorem fecit; </s>
  <s xml:id="echoid-s12901" xml:space="preserve">quare ergo nos eiſdem coniecturis per-<lb/>ſuaſi eidem Achimedi tribuere dubitabimus Opuſculum hoc ab Arabibus aſ-<lb/>ſeruatum, in quo ſi mendarum copiam ſpectes, ſimile omnino erit ei, <lb/>quod Eutocius nactus eſt? </s>
  <s xml:id="echoid-s12902" xml:space="preserve">Hæ ſunt rationes, mi lector, quas tibi exa-<lb/>minandas relinquo in hoc perplexo negotio nulla diſsimulata difficultate.</s>
  <s xml:id="echoid-s12903" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s12904" xml:space="preserve">Interim ſcito hoc manuſcriptum Arabicè elegantiſsimè exaratnm in <lb/>Bibliotheca Sereniſsimi Magni Etruriæ Ducis diù aſſeruatum fuiße; </s>
  <s xml:id="echoid-s12905" xml:space="preserve">eius <lb/>tamen editionis ſpe facta tandem anno 1658. </s>
  <s xml:id="echoid-s12906" xml:space="preserve">Sereniſsimus F erdinan-<lb/>dus Secundus Magnus Etruriæ Dux Romæ aſportandum humaniſsimè <lb/>mihi credidit, vt rei litterariæ bono latinè traduceretur, præſtitumque <lb/>fuit opera, &amp; </s>
  <s xml:id="echoid-s12907" xml:space="preserve">ſtudio celeberrimi, &amp; </s>
  <s xml:id="echoid-s12908" xml:space="preserve">peritiſsimi Orientalium linguarum <lb/>profeßoris Abrahami Ecchellenſis, ipſoque dictante religioſiſsimè, &amp; </s>
  <s xml:id="echoid-s12909" xml:space="preserve"><lb/>accuratè ipſe calamo excepi, in eoque paucula quædam in notis anima-<lb/>duertenda cenſui tum in contextu plurimis mendis corrupto, tum in <lb/>ſcholĳs Arabicis Almochtaſſo non admodum in Geometria verſati. <lb/></s>
  <s xml:id="echoid-s12910" xml:space="preserve">Addidi in fine huins libri duas alias Archimedis propoſitiones ab Euto-<lb/>cio repertas quarum altera fortaſſe illa eadem eſt quæ hìc deficit, nam <lb/>Almochtaſſo in proemio ait, propoſitiones huius Opuſculi ſexdecim eſſe, <lb/>cum tamen poſtrema ſit decimaquinta. </s>
  <s xml:id="echoid-s12911" xml:space="preserve">Et licet hæc eadem lemmata anno <lb/>præterito edita fuerint Londini, non tamen hac noſtra editione fraudan-<lb/>dus es, amice lector. </s>
  <s xml:id="echoid-s12912" xml:space="preserve">Vale.</s>
  <s xml:id="echoid-s12913" xml:space="preserve"/>
</p>
<pb file="0422" n="423"/>
<pb o="385" file="0423" n="424" rhead="IN NOMINE DEI"/>
</div>
<div xml:id="echoid-div1106" type="section" level="1" n="348">
<head xml:id="echoid-head436" xml:space="preserve">MISERICORDIS MISERATORIS</head>
<head xml:id="echoid-head437" xml:space="preserve">CVIVS OPEM IMPLORAMVS.</head>
<head xml:id="echoid-head438" xml:space="preserve">LIBER ASSVMPTORVM ARCHIMEDIS, <lb/>INTERPRETE THEBIT BEN-KORA, <lb/>Et exponente Doctore</head>
<head xml:id="echoid-head439" xml:space="preserve">ALMOCHTASSO ABILHASAN, <lb/>Halì Ben-Ahmad Noſuenſi.</head>
<head xml:id="echoid-head440" style="it" xml:space="preserve">PROPOSITIONES SEXDECIM.</head>
<p>
  <s xml:id="echoid-s12914" xml:space="preserve">ASſerit Doctor Almochtaſſo hunc librum referri ad Ar-<lb/>chimedem, in quo ſunt propoſitiones pulcherrimæ <lb/>paucæ numero, vtilitatis verò maximæ de principijs <lb/>Geometriæ, optimæ atque elegantiſſimæ, quas ad-<lb/>numerant profeſſores huius ſcientię ſummæ interme-<lb/>diorum, quæ legi oportet inter librum Euclidis, &amp; </s>
  <s xml:id="echoid-s12915" xml:space="preserve"><lb/>Almageſtum; </s>
  <s xml:id="echoid-s12916" xml:space="preserve">at verò quædam illius propoſitionum loca indi-<lb/>gent alijs propoſitionibus, quibus propoſitiones illæ clariores eua-<lb/>dant. </s>
  <s xml:id="echoid-s12917" xml:space="preserve">Et quidem ipſe Archimedes has indicauit propoſitiones, eaſ-<lb/>que retulit in alijs ſuis operibus, dum dixit quemadmodum demon-<lb/>ſtrauimus in propoſitionibus rectangulorum: </s>
  <s xml:id="echoid-s12918" xml:space="preserve">item &amp; </s>
  <s xml:id="echoid-s12919" xml:space="preserve">quemadmodũ <lb/>demonſtrauimus in noſtra expoſitione agentes de triangulis; </s>
  <s xml:id="echoid-s12920" xml:space="preserve">rurſus <lb/>quemadmodum demonſtrauimus in propoſitionibus quadrilate-<lb/>rum; </s>
  <s xml:id="echoid-s12921" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12922" xml:space="preserve">retulit in propoſitione quinta demonſtrationem hac de re <lb/>magis peculiarem. </s>
  <s xml:id="echoid-s12923" xml:space="preserve">Deinde compoſuit Abuſahal Alkuhi librum, <lb/>quem inſcripſit ordinationem libri Archimedis de aſſumptis, &amp; </s>
  <s xml:id="echoid-s12924" xml:space="preserve">tra-<lb/>ctauit demonſtractionem huius propoſitionis via vniuerſaliori, ac <lb/>meliori, nec non ea, quę dependent ex compoſitione proportionis, <lb/>quod quidẽ cum id comperi, attexui locis obſcurioribus huius libri <lb/>expoſitionem, ſeu marginales poſtillas, &amp; </s>
  <s xml:id="echoid-s12925" xml:space="preserve">confirmaui quod ille indi-<lb/>cauerat propoſitionibus, vti iudicaueram, &amp; </s>
  <s xml:id="echoid-s12926" xml:space="preserve">retuli ex propoſitioni-<lb/>bus Abiſahal duas propoſitiones, quibus opus eſt ad propoſitionem <lb/>quintã declarandam, reliquas omittens breuitatis gratia, &amp; </s>
  <s xml:id="echoid-s12927" xml:space="preserve">eo quod <lb/>non ſint neceſſarię.</s>
  <s xml:id="echoid-s12928" xml:space="preserve"/>
</p>
<pb o="386" file="0424" n="425" rhead="Archimedis"/>
</div>
<div xml:id="echoid-div1107" type="section" level="1" n="349">
<head xml:id="echoid-head441" xml:space="preserve">PROPOSITIO I.</head>
<p>
  <s xml:id="echoid-s12929" xml:space="preserve">SI mutuo ſe tangant duo circuli, vt duo circuli A E B, C E <lb/>D in E, fuerintque eorum diametri parallelæ, vt ſunt duæ <lb/>diametri A B, C D, &amp; </s>
  <s xml:id="echoid-s12930" xml:space="preserve">iungantur duo puncta B, D, &amp; </s>
  <s xml:id="echoid-s12931" xml:space="preserve">conta-<lb/>ctus E [lineis] D E, B D, erit linea B E recta.</s>
  <s xml:id="echoid-s12932" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12933" xml:space="preserve">Sint duo centra G, F, &amp; </s>
  <s xml:id="echoid-s12934" xml:space="preserve">iunga-<lb/>
<anchor type="figure" xlink:label="fig-0424-01a" xlink:href="fig-0424-01"/>
mus G F, &amp; </s>
  <s xml:id="echoid-s12935" xml:space="preserve">producamus ad E, &amp; </s>
  <s xml:id="echoid-s12936" xml:space="preserve"><lb/>educamus D H parallelam ipſi G F. <lb/></s>
  <s xml:id="echoid-s12937" xml:space="preserve">Et quia H F æqualis eſt ipſi G D, <lb/>ſuntque G D, E G æquales, ergo <lb/>ex æqualibus F B, F E remanebunt <lb/>G F, nempe D H, &amp; </s>
  <s xml:id="echoid-s12938" xml:space="preserve">H B, quæ <lb/>erunt æquales, atque duo anguli H <lb/>D B, H B D æquales. </s>
  <s xml:id="echoid-s12939" xml:space="preserve">Et quia <lb/>duo anguli E G D, E F B ſunt re-<lb/>cti, atq; </s>
  <s xml:id="echoid-s12940" xml:space="preserve">duo anguli E G D, D H <lb/>B ſunt æquales, remanebunt duo <lb/>anguli G E D, G D E, qui inter ſe, &amp; </s>
  <s xml:id="echoid-s12941" xml:space="preserve">duobus angulis H D B, H B D <lb/>æquales erunt; </s>
  <s xml:id="echoid-s12942" xml:space="preserve">ergo angulus E D G æqualis eſt angulo D B F, &amp; </s>
  <s xml:id="echoid-s12943" xml:space="preserve">com-<lb/>prehenſus angulus G D B eſt communis, ergo erunt duo anguli G D B, <lb/>F B D (qui ſunt pares duobus rectis) æquales duobus angulis G D B, <lb/>G D E: </s>
  <s xml:id="echoid-s12944" xml:space="preserve">igitur ipſi quoque ſunt æquales duobus rectis, ergo linea E D <lb/>B eſt recta, &amp; </s>
  <s xml:id="echoid-s12945" xml:space="preserve">hoc eſt, quod voluimus.</s>
  <s xml:id="echoid-s12946" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1107" type="float" level="2" n="1">
  <figure xlink:label="fig-0424-01" xlink:href="fig-0424-01a">
    <image file="0424-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0424-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1109" type="section" level="1" n="350">
<head xml:id="echoid-head442" xml:space="preserve">SCHOLIVM ALMOCHTASSO.</head>
<p>
  <s xml:id="echoid-s12947" xml:space="preserve">DIcit Doctor; </s>
  <s xml:id="echoid-s12948" xml:space="preserve">Et quidem dici poteſt cum duo anguli H D B, H B <lb/>D ſint æquales, &amp; </s>
  <s xml:id="echoid-s12949" xml:space="preserve">angulus D H B rectus, quod erit angulus B D <lb/>H ſemirectus, &amp; </s>
  <s xml:id="echoid-s12950" xml:space="preserve">ſimiliter angulus E D G, &amp; </s>
  <s xml:id="echoid-s12951" xml:space="preserve">angulus G D H rectus, <lb/>ergo tres anguli ſunt æquales duobus rectis, igitur linea E D B eſt re-<lb/>cta. </s>
  <s xml:id="echoid-s12952" xml:space="preserve">Idem ſequitur, ſi illi duo circuli ſe mutuo exterius contigerint.</s>
  <s xml:id="echoid-s12953" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1110" type="section" level="1" n="351">
<head xml:id="echoid-head443" xml:space="preserve">Notæ in Propoſit. I.</head>
<p style="it">
  <s xml:id="echoid-s12954" xml:space="preserve">HAEc eſt vna earum Propoſitionum, quas Pappus in quodam libro antiquo <lb/>reperit, qui, vt deduximus ex Eutocio, ab Archimede conſcriptus diu <lb/>apud Arabes latuit. </s>
  <s xml:id="echoid-s12955" xml:space="preserve">Hæc aſſumitur in propoſit. </s>
  <s xml:id="echoid-s12956" xml:space="preserve">14. </s>
  <s xml:id="echoid-s12957" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s12958" xml:space="preserve">4. </s>
  <s xml:id="echoid-s12959" xml:space="preserve">Collect. </s>
  <s xml:id="echoid-s12960" xml:space="preserve">Pappi, eam-<lb/>que ſupplet Commandinus, ſed extat expreſſe lib. </s>
  <s xml:id="echoid-s12961" xml:space="preserve">7. </s>
  <s xml:id="echoid-s12962" xml:space="preserve">propoſit. </s>
  <s xml:id="echoid-s12963" xml:space="preserve">110. </s>
  <s xml:id="echoid-s12964" xml:space="preserve">eiuſdem <lb/>Pappi, eſtque demonſtratio vniuer ſaliſſima comprehendens caſum neglectum <lb/>in hac demonſtratione, ſcilicet quando duo circuli ſeſe exterius contingunt, &amp;</s>
  <s xml:id="echoid-s12965" xml:space="preserve">
<pb o="387" file="0425" n="426" rhead="Aſſumpt. Liber."/>
licet non laboret vitio Arabici textus, non tamen illa omnino ſincera eſt: </s>
  <s xml:id="echoid-s12966" xml:space="preserve">con-<lb/>ueniunt tamen in vniuerſalitate propoſitionis, quàm valde peruersè ſcholiaſtes <lb/>Arabicus expoſuit; </s>
  <s xml:id="echoid-s12967" xml:space="preserve">allucinatur enim quando ait, &amp; </s>
  <s xml:id="echoid-s12968" xml:space="preserve">quia duo anguli E G <lb/>
<anchor type="figure" xlink:label="fig-0425-01a" xlink:href="fig-0425-01"/>
D, &amp; </s>
  <s xml:id="echoid-s12969" xml:space="preserve">E F B ſunt recti, &amp;</s>
  <s xml:id="echoid-s12970" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12971" xml:space="preserve">Nam inferius citatur, &amp; </s>
  <s xml:id="echoid-s12972" xml:space="preserve">vſurpatur hæc prima <lb/>propoſitio vniuerſaliſſimè, ſcilicet exiſtentibus angulis G, &amp; </s>
  <s xml:id="echoid-s12973" xml:space="preserve">F acutis, vel ob-<lb/>tuſis, &amp; </s>
  <s xml:id="echoid-s12974" xml:space="preserve">ſic reuera ſonant verba propoſitionis, vbi ait, quorum diametri A <lb/>B, C D ſunt parallelæ, &amp; </s>
  <s xml:id="echoid-s12975" xml:space="preserve">ſic pariter habetur in prædicta propoſitione Pappi; <lb/></s>
  <s xml:id="echoid-s12976" xml:space="preserve">quare textus omnino corrigi debuit, vt pronuncientur anguli E G D, &amp; </s>
  <s xml:id="echoid-s12977" xml:space="preserve">E F <lb/>B æquales, non recti. </s>
  <s xml:id="echoid-s12978" xml:space="preserve">Neſcio tamen quomodo expoſitio Almochtaſſi excuſari poſ-<lb/>ſit, qui ſupponit diametros A B, &amp; </s>
  <s xml:id="echoid-s12979" xml:space="preserve">C D perpendiculares ad rectam lineam <lb/>F G E, quod quidem in vnico caſu veriſicatur, vt dictum eſt. </s>
  <s xml:id="echoid-s12980" xml:space="preserve">Peccat poſtea <lb/>demonſtratio Pappi lib. </s>
  <s xml:id="echoid-s12981" xml:space="preserve">7. </s>
  <s xml:id="echoid-s12982" xml:space="preserve">pr. </s>
  <s xml:id="echoid-s12983" xml:space="preserve">110.</s>
  <s xml:id="echoid-s12984" xml:space="preserve">, vbi conatur oſtendere duo centra, &amp; </s>
  <s xml:id="echoid-s12985" xml:space="preserve">pun-<lb/>ctum contactus circulorum eſſe in vnica recta linea; </s>
  <s xml:id="echoid-s12986" xml:space="preserve">quod quidem in 3. </s>
  <s xml:id="echoid-s12987" xml:space="preserve">Ele-<lb/>ment. </s>
  <s xml:id="echoid-s12988" xml:space="preserve">Eucl. </s>
  <s xml:id="echoid-s12989" xml:space="preserve">oſtenſum ſupponi debuerat.</s>
  <s xml:id="echoid-s12990" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1110" type="float" level="2" n="1">
  <figure xlink:label="fig-0425-01" xlink:href="fig-0425-01a">
    <image file="0425-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0425-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1112" type="section" level="1" n="352">
<head xml:id="echoid-head444" xml:space="preserve">PROPOSITIO II.</head>
<p>
  <s xml:id="echoid-s12991" xml:space="preserve">SIt C B A ſemicirculus, quem D C, D B tangant, &amp; </s>
  <s xml:id="echoid-s12992" xml:space="preserve">B E <lb/>perpendicularis ſuper A C, &amp; </s>
  <s xml:id="echoid-s12993" xml:space="preserve">iungamus A D, erit B F <lb/>æqualis ipſi F E.</s>
  <s xml:id="echoid-s12994" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12995" xml:space="preserve">Demonſtratio. </s>
  <s xml:id="echoid-s12996" xml:space="preserve">Iungamus A B, eamque producamus in directum, &amp; </s>
  <s xml:id="echoid-s12997" xml:space="preserve"><lb/>
<anchor type="figure" xlink:label="fig-0425-02a" xlink:href="fig-0425-02"/>
educamus C D, quouſque illi occurrat <lb/>in G, &amp; </s>
  <s xml:id="echoid-s12998" xml:space="preserve">iungamus C B. </s>
  <s xml:id="echoid-s12999" xml:space="preserve">Et quia angu-<lb/>lus C B A eſt in ſemicirculo, erit re-<lb/>ctus, remanet C B G rectus, &amp; </s>
  <s xml:id="echoid-s13000" xml:space="preserve">D B E <lb/>C eſt parallelogrammum rectangulum, <lb/>ergo in triangulo G B C rectangulo edu-<lb/>citur perpendicularis B D ex B erecta <lb/>ſuper baſim, &amp; </s>
  <s xml:id="echoid-s13001" xml:space="preserve">B D, D C erunt æqua-<lb/>les, eo quod tangunt circulum, ergo C <lb/>D eſt etiam æqualis ipſi D G, quemad-<lb/>modum oſtendimus in propoſitionibus,
<pb o="388" file="0426" n="427" rhead="Archimedis"/>
quas confecimus de rectangulis. </s>
  <s xml:id="echoid-s13002" xml:space="preserve">Et quia <lb/>
<anchor type="figure" xlink:label="fig-0426-01a" xlink:href="fig-0426-01"/>
in triangulo G A C linea B E educta eſt <lb/>parallela baſi, &amp; </s>
  <s xml:id="echoid-s13003" xml:space="preserve">iam educta eſt ex D <lb/>ſemipartitione baſis linea D A ſeca ns <lb/>parallelam in F, erit B F æqualis ipſi F <lb/>E, &amp; </s>
  <s xml:id="echoid-s13004" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13005" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1112" type="float" level="2" n="1">
  <figure xlink:label="fig-0425-02" xlink:href="fig-0425-02a">
    <image file="0425-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0425-02"/>
  </figure>
  <figure xlink:label="fig-0426-01" xlink:href="fig-0426-01a">
    <image file="0426-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0426-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1114" type="section" level="1" n="353">
<head xml:id="echoid-head445" xml:space="preserve">SCHOLIVM ALMOCHTASSO.</head>
<p>
  <s xml:id="echoid-s13006" xml:space="preserve">DIcit Doctor: </s>
  <s xml:id="echoid-s13007" xml:space="preserve">Quod autem C D ſit æqualis ipſi D G, vti remittit ad <lb/>ſuum librum de propoſitionibus rectangulorum, eo quod duo angu-<lb/>li D C B, D B C æquales ſunt propter æqualitatem D B, D C, &amp; </s>
  <s xml:id="echoid-s13008" xml:space="preserve">an-<lb/>gulus D B C cum augulo D B G eſt rectus, &amp; </s>
  <s xml:id="echoid-s13009" xml:space="preserve">ſimiliter angulus D C B <lb/>cum angulo C G B: </s>
  <s xml:id="echoid-s13010" xml:space="preserve">neceſſe eſt, vt ſint duo anguli D G B, D B G æqua-<lb/>les etiam, ergo duo latera D B, D G ſunt æqualia.</s>
  <s xml:id="echoid-s13011" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13012" xml:space="preserve">Rurſus ſi dicatur quod proportio C D ad D B ſit vt proportio D B ad <lb/>D G, &amp; </s>
  <s xml:id="echoid-s13013" xml:space="preserve">D C æqualis ipſi D B, ergo D B æqualis eſt D G, eſſet para-<lb/>bola. </s>
  <s xml:id="echoid-s13014" xml:space="preserve">Dicit, quod vero B F ſit æqualis F E, hoc conſtat ex eo quod <lb/>caſus A D ſuper duas lineas B E, G C parallelas in triangulo A G C, <lb/>exigit eorum ſectio in eadem proportione, &amp; </s>
  <s xml:id="echoid-s13015" xml:space="preserve">id quidem, quia A D ad <lb/>A F eandem proportionem habet, quam G D ad B F, &amp; </s>
  <s xml:id="echoid-s13016" xml:space="preserve">quam D C ad <lb/>E F, ergo G D ad B F eſt vt D C ad E F, &amp; </s>
  <s xml:id="echoid-s13017" xml:space="preserve">permutando G D ad ei æ-<lb/>qualem D C, eſt vt B F ad E F, &amp; </s>
  <s xml:id="echoid-s13018" xml:space="preserve">propterea ipſæ etiam ſunt æquales.</s>
  <s xml:id="echoid-s13019" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1115" type="section" level="1" n="354">
<head xml:id="echoid-head446" xml:space="preserve">Notæ in Propoſ. II.</head>
<p style="it">
  <s xml:id="echoid-s13020" xml:space="preserve">HVius ſecundæ propoſitionis expoſitio, &amp; </s>
  <s xml:id="echoid-s13021" xml:space="preserve">demonſtratio inſigniter deformata <lb/>eſt; </s>
  <s xml:id="echoid-s13022" xml:space="preserve">in propoſitione enim ſupponuntur duæ rectæ D C, D B tangere cir-<lb/>culum tantummodo, non autem conſtituere angulum rectum, &amp; </s>
  <s xml:id="echoid-s13023" xml:space="preserve">ſolummodo re-<lb/>cta linea B E perpendicularis ducitur ad diametrum A C, quare male in de-<lb/>monſtratione pronunciatur quadrilaterum B D C E parallelogrammum rectan-<lb/>gulum, cum ferè ſemper ſit Trapetium: </s>
  <s xml:id="echoid-s13024" xml:space="preserve">pariterque errat, quando ait rectam <lb/>B D perpendicularem eſſe ſuper C G, quæ nunquam vera ſunt, niſi in vnico caſu, <lb/>quando ſcilicet B E cadit perpendiculariter ſuper centrum circuli.</s>
  <s xml:id="echoid-s13025" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13026" xml:space="preserve">Interim notandum eſt hanc elegantem <lb/>
<anchor type="figure" xlink:label="fig-0426-02a" xlink:href="fig-0426-02"/>
propoſitionem, inſignem vſum habere pro <lb/>inueſtigatione menſuræ circuli, &amp; </s>
  <s xml:id="echoid-s13027" xml:space="preserve">recta-<lb/>rum in eo ſubtenſarum; </s>
  <s xml:id="echoid-s13028" xml:space="preserve">deduci namque <lb/>poßunt non contemnenda problemata; </s>
  <s xml:id="echoid-s13029" xml:space="preserve">Si <lb/>enim quis cupiat circulo adſcribere duas <lb/>figuras or dinatas ſimiles, quarum circum-<lb/>ſcripta ſuperet inſcriptam exceſſu minori <lb/>quolibet dato, facile problema abſoluetur,
<pb o="389" file="0427" n="428" rhead="Aſſumpt. Liber."/>
pariterque proportio diametri ad circuli peripheriam ſatis compendioſe deduci <lb/>poteſt, quandoquidem inter figuram ordinatam eidem circulo inſcriptam, cuius <lb/>ſemilatus eſt E B, &amp; </s>
  <s xml:id="echoid-s13030" xml:space="preserve">circumſcriptam duplo laterum numero, cuius duo ſemila-<lb/>tera ſunt C D B, circulus intermediat; </s>
  <s xml:id="echoid-s13031" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13032" xml:space="preserve">Perimeter circumſcriptæ figuræ ad <lb/>Perimetrum inſcriptæ eandem proportionem habet, quam diameter C A ad A E, <lb/>quæ proportio minui ſemper magis, ac magis poteſt in infinitum; </s>
  <s xml:id="echoid-s13033" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13034" xml:space="preserve">tandem ex <lb/>3. </s>
  <s xml:id="echoid-s13035" xml:space="preserve">propoſ. </s>
  <s xml:id="echoid-s13036" xml:space="preserve">ſequenti, ex continua ſemipartitione quadrantis circuli elici poſſunt <lb/>ſubtenſæ ſucceſſiuè ſubdiuiſæ in infinitum, &amp; </s>
  <s xml:id="echoid-s13037" xml:space="preserve">propterea dabitur proportio dia-<lb/>metri A C ad ſemiſubtenſam B E, ſed datur quadratum ipſius B E, igitur da-<lb/>tur rectangulum A E C ſub ſegmentis diametri, &amp; </s>
  <s xml:id="echoid-s13038" xml:space="preserve">datur E C ex iam dicta 3. <lb/></s>
  <s xml:id="echoid-s13039" xml:space="preserve">propoſ. </s>
  <s xml:id="echoid-s13040" xml:space="preserve">igitur datur quoque E A; </s>
  <s xml:id="echoid-s13041" xml:space="preserve">eſtque B E ad C D B, vt E A ad diametrũ <lb/>A C, igitur quarta quantitas innoteſcet, ſcilicet rectæ C D B, quæ æqualia <lb/>ſunt vni lateri Poligoni circumſcripti duplo laterum numero, &amp; </s>
  <s xml:id="echoid-s13042" xml:space="preserve">ideo habebitur <lb/>menſura totius Perimetri tum Poligoni inſcripti, cum circumſcripti, quare <lb/>menſura ipſius peripheriæ circuli, quæ intermedia eſt, facili negotio inueſtiga-<lb/>bitur.</s>
  <s xml:id="echoid-s13043" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1115" type="float" level="2" n="1">
  <figure xlink:label="fig-0426-02" xlink:href="fig-0426-02a">
    <image file="0426-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0426-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1117" type="section" level="1" n="355">
<head xml:id="echoid-head447" xml:space="preserve">PROPOSITIO III.</head>
<p>
  <s xml:id="echoid-s13044" xml:space="preserve">S It C A ſegmentum circuli, &amp; </s>
  <s xml:id="echoid-s13045" xml:space="preserve">B <lb/>
<anchor type="figure" xlink:label="fig-0427-01a" xlink:href="fig-0427-01"/>
punctum ſuper illud vbicumque, <lb/>&amp; </s>
  <s xml:id="echoid-s13046" xml:space="preserve">B D perpendicularis ſuper A C, &amp; </s>
  <s xml:id="echoid-s13047" xml:space="preserve"><lb/>ſegmentum D E æquale D A, &amp; </s>
  <s xml:id="echoid-s13048" xml:space="preserve">arcus <lb/>B F æqualis arcui B A, vtique iuncta <lb/>C F erit æqualis ipſi C E.</s>
  <s xml:id="echoid-s13049" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1117" type="float" level="2" n="1">
  <figure xlink:label="fig-0427-01" xlink:href="fig-0427-01a">
    <image file="0427-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0427-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s13050" xml:space="preserve">Demonſtratio. </s>
  <s xml:id="echoid-s13051" xml:space="preserve">Iungamus lineas A B, B F, <lb/>F E, E B; </s>
  <s xml:id="echoid-s13052" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13053" xml:space="preserve">quia arcus B A æqualis eſt arcui B F, erit A B æqualis <lb/>B F, &amp; </s>
  <s xml:id="echoid-s13054" xml:space="preserve">quia A D æqualis eſt E D, &amp; </s>
  <s xml:id="echoid-s13055" xml:space="preserve">duo anguli D ſunt recti, &amp; </s>
  <s xml:id="echoid-s13056" xml:space="preserve">D B <lb/>communis, ergo A B æqualis eſt B E, &amp; </s>
  <s xml:id="echoid-s13057" xml:space="preserve">propterea B F, B E ſunt æqua-<lb/>les; </s>
  <s xml:id="echoid-s13058" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13059" xml:space="preserve">duo anguli B F E, B E F ſunt æquales. </s>
  <s xml:id="echoid-s13060" xml:space="preserve">Et quia quadrilaterum. <lb/></s>
  <s xml:id="echoid-s13061" xml:space="preserve">C F B A eſt in circulo, erit angulus C F B cum angulo C A B ipſi op-<lb/>poſito, immo cum angulo B E A, æqualis duobus rectis; </s>
  <s xml:id="echoid-s13062" xml:space="preserve">ſed angulus C <lb/>E B cum angulo B E A, æquales ſunt duobus rectis, ergo duo anguli C <lb/>F B, C E B ſunt æquales, &amp; </s>
  <s xml:id="echoid-s13063" xml:space="preserve">remanent C F E, C E F æqualas; </s>
  <s xml:id="echoid-s13064" xml:space="preserve">ergo <lb/>C E æqualis eſt C F, &amp; </s>
  <s xml:id="echoid-s13065" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13066" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1119" type="section" level="1" n="356">
<head xml:id="echoid-head448" xml:space="preserve">Notæ in Propoſit. III.</head>
<p style="it">
  <s xml:id="echoid-s13067" xml:space="preserve">HAEc eſt propoſ. </s>
  <s xml:id="echoid-s13068" xml:space="preserve">5. </s>
  <s xml:id="echoid-s13069" xml:space="preserve">cap. </s>
  <s xml:id="echoid-s13070" xml:space="preserve">9. </s>
  <s xml:id="echoid-s13071" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13072" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13073" xml:space="preserve">Almag. </s>
  <s xml:id="echoid-s13074" xml:space="preserve">Ptol.</s>
  <s xml:id="echoid-s13075" xml:space="preserve">, ſed hic vniuerſalius pro-<lb/>nunciatur; </s>
  <s xml:id="echoid-s13076" xml:space="preserve">Ptolomeus enim ſupponit ſegmentum A B C ſemicirculum <lb/>eſſe, &amp; </s>
  <s xml:id="echoid-s13077" xml:space="preserve">ex cognita circumferentia A F, &amp; </s>
  <s xml:id="echoid-s13078" xml:space="preserve">corda F C, &amp; </s>
  <s xml:id="echoid-s13079" xml:space="preserve">illius medietate A <lb/>B, quærit chordam A B; </s>
  <s xml:id="echoid-s13080" xml:space="preserve">eſt enim rectangulum ſub C A D æquale quadrato
<pb o="390" file="0428" n="429" rhead="Archimedis"/>
ipſius A B, eſtque nota A D medietas differentiæ inter diametrum A C, &amp; </s>
  <s xml:id="echoid-s13081" xml:space="preserve">chor-<lb/>dam differentiæ F C; </s>
  <s xml:id="echoid-s13082" xml:space="preserve">at propoſitio Archimedea verificatur in quolibet circuli <lb/>ſegmento ſiue maiori, ſiue minori; </s>
  <s xml:id="echoid-s13083" xml:space="preserve">ex datis enim circumferentĳs A C, A B, <lb/>
<anchor type="figure" xlink:label="fig-0428-01a" xlink:href="fig-0428-01"/>
A F, &amp; </s>
  <s xml:id="echoid-s13084" xml:space="preserve">F C vna cum cordis A C, &amp; </s>
  <s xml:id="echoid-s13085" xml:space="preserve">F C, haberi quidem poteſt chorda A B <lb/>paulo difficilius, ſi nimirum ex chorda A C tollatur chorda F C, &amp; </s>
  <s xml:id="echoid-s13086" xml:space="preserve">differen-<lb/>tia A E bifariam ſecetur in D, &amp; </s>
  <s xml:id="echoid-s13087" xml:space="preserve">ex arcu cognito B C datur angulus A, atque <lb/>angulus D rectus eſt, ergo triangulum A B D ſpecie notum erit, &amp; </s>
  <s xml:id="echoid-s13088" xml:space="preserve">propterea <lb/>proportio D A ad A B cognita erit, eſtque D A longitudine data, igitur A B <lb/>longitudine innoteſcet.</s>
  <s xml:id="echoid-s13089" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1119" type="float" level="2" n="1">
  <figure xlink:label="fig-0428-01" xlink:href="fig-0428-01a">
    <image file="0428-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0428-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s13090" xml:space="preserve">Notandum eſt quod figura appoſita in hac propoſ. </s>
  <s xml:id="echoid-s13091" xml:space="preserve">non exprimit omnes caſus <lb/>propoſitionis, quandoquidem ſemicirculus eſt A B C, &amp; </s>
  <s xml:id="echoid-s13092" xml:space="preserve">propterea ex præceden-<lb/>tibus erroribus Arabici expoſitoris ſuſpicari licet non ritè eum percepiſſe Archi-<lb/>medis mentem.</s>
  <s xml:id="echoid-s13093" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1121" type="section" level="1" n="357">
<head xml:id="echoid-head449" xml:space="preserve">PROPOSITIO IV.</head>
<p>
  <s xml:id="echoid-s13094" xml:space="preserve">A B C ſemicirculus, &amp; </s>
  <s xml:id="echoid-s13095" xml:space="preserve">fiant ſuper <lb/>
<anchor type="figure" xlink:label="fig-0428-02a" xlink:href="fig-0428-02"/>
A C diametrum duo ſemicirculi, quo-<lb/>rum vnus A D, alter vero D C, &amp; </s>
  <s xml:id="echoid-s13096" xml:space="preserve"><lb/>D B perpendicularis, vtique figura pro-<lb/>ueniens, quam vocat Archimedes AR-<lb/>BELON, eſt ſuperficies comprehenſa ab <lb/>arcu ſemicirculi maioris, &amp; </s>
  <s xml:id="echoid-s13097" xml:space="preserve">duabus cir-<lb/>cumferentijs ſemicirculorum minorum, eſt æqualis circulo, cuius <lb/>diameter eſt perpendicularis D B.</s>
  <s xml:id="echoid-s13098" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1121" type="float" level="2" n="1">
  <figure xlink:label="fig-0428-02" xlink:href="fig-0428-02a">
    <image file="0428-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0428-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s13099" xml:space="preserve">Demonſtratio. </s>
  <s xml:id="echoid-s13100" xml:space="preserve">Quia linea D B media proportionalis eſt inter duas li-<lb/>neas D A, D C, erit planum A D in D C æquale quadrato D B, &amp; </s>
  <s xml:id="echoid-s13101" xml:space="preserve"><lb/>ponamus A D in D C cum duobus quadratis A D, D C communiter, <lb/>fiet planum A D in D C bis cum duobus quadratis A D, D C, nempe <lb/>quadratum A C, æquale duplo quadrati D B cum duobus quadratis A <lb/>D, D C, &amp; </s>
  <s xml:id="echoid-s13102" xml:space="preserve">proportio circulorum eadem eſt, ac proportio quadratorum,
<pb o="391" file="0429" n="430" rhead="Aſſumpt. Liber."/>
ergo circulus, cuius diameter eſt A C, æqualis eſt duplo circuli, cuius <lb/>diameter eſt D B cum duobus circulis, quorum diametri ſunt A D, D <lb/>C, &amp; </s>
  <s xml:id="echoid-s13103" xml:space="preserve">ſemicirculus A C æqualis eſt circulo, cuius diameter eſt D B <lb/>cum duobus ſemicirculis A D, D C; </s>
  <s xml:id="echoid-s13104" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13105" xml:space="preserve">auferamus duos ſemicirculi A <lb/>D, D C communiter, remanet figura, quàm continent ſemicirculi A <lb/>C, A D, D C, &amp; </s>
  <s xml:id="echoid-s13106" xml:space="preserve">eſt figura, quàm vocauit Archimedes Arbelos æqua-<lb/>lis circulo, cuius diameter eſt D B, &amp; </s>
  <s xml:id="echoid-s13107" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13108" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1123" type="section" level="1" n="358">
<head xml:id="echoid-head450" xml:space="preserve">Notæ in Propoſit. IV.</head>
<p style="it">
  <s xml:id="echoid-s13109" xml:space="preserve">H AEc forſan eſt vna earum propoſitionum, quas Pappus legit in libro an-<lb/>tiquo de menſura ARBELI, ſeu ſpatĳ àtribus ſemicircumferentĳs circulo-<lb/>rum comprehenſi, vt ait Proclus, quæ quidem elegantiſſima eſt, eiuſque inuen-<lb/>tionis Lunulæ Hyppocratis Chĳ originem extitiße puto; </s>
  <s xml:id="echoid-s13110" xml:space="preserve">eſt enim Hyppocratis <lb/>Lunula ſuperficies plana à quadrante peripheriæ circuli maioris, &amp; </s>
  <s xml:id="echoid-s13111" xml:space="preserve">ſemiſſe pe-<lb/>ripheriæ circuli ſubdupli comprehenſa: </s>
  <s xml:id="echoid-s13112" xml:space="preserve">Arbelus vero recentiorum eſt ſpatium <lb/>à triente, &amp; </s>
  <s xml:id="echoid-s13113" xml:space="preserve">à duobus ſextantibus circumferentiarum trium circulorum æqua-<lb/>lium comprehenſum, &amp; </s>
  <s xml:id="echoid-s13114" xml:space="preserve">hiſce duobus ſpatĳs facilè quadrata æqualia reperiri <lb/>poſſunt; </s>
  <s xml:id="echoid-s13115" xml:space="preserve">at Arbeli Archimedis, &amp; </s>
  <s xml:id="echoid-s13116" xml:space="preserve">Procli hucuſque reperta non eſt quadratura; <lb/></s>
  <s xml:id="echoid-s13117" xml:space="preserve">ſed poteſt quidem aſſignari circulus prædicto ſpatio æqualis.</s>
  <s xml:id="echoid-s13118" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1124" type="section" level="1" n="359">
<head xml:id="echoid-head451" xml:space="preserve">PROPOSITIO V.</head>
<p>
  <s xml:id="echoid-s13119" xml:space="preserve">SI fuerit ſemicirculus A B, &amp; </s>
  <s xml:id="echoid-s13120" xml:space="preserve">ſignatum fuerit in eius diametro <lb/>punctum C vbicumque, &amp; </s>
  <s xml:id="echoid-s13121" xml:space="preserve">fiant ſuper diametrum duo ſe-<lb/>micirculi A C, C B, &amp; </s>
  <s xml:id="echoid-s13122" xml:space="preserve">educatur ex C perpendicularis C D ſu-<lb/>per A B, &amp; </s>
  <s xml:id="echoid-s13123" xml:space="preserve">deſcribantur ad vtraſque partes duo circuli tan-<lb/>gentes illam, &amp; </s>
  <s xml:id="echoid-s13124" xml:space="preserve">tangentes ſemicirculos, vtique illi duo circuli <lb/>ſunt æquales.</s>
  <s xml:id="echoid-s13125" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13126" xml:space="preserve">Demonſtratio. </s>
  <s xml:id="echoid-s13127" xml:space="preserve">Sit al-<lb/>
<anchor type="figure" xlink:label="fig-0429-01a" xlink:href="fig-0429-01"/>
ter circulorum tangens <lb/>D C in E, &amp; </s>
  <s xml:id="echoid-s13128" xml:space="preserve">ſemicircu-<lb/>lum A B in F, &amp; </s>
  <s xml:id="echoid-s13129" xml:space="preserve">ſemi-<lb/>circulum A C in G, &amp; </s>
  <s xml:id="echoid-s13130" xml:space="preserve"><lb/>educamus diametrũ H E, <lb/>erit parallela diametro A <lb/>B, eo quod duo anguli H <lb/>E C, A C E, ſunt recti, <lb/>&amp; </s>
  <s xml:id="echoid-s13131" xml:space="preserve">iungamus F H, H A, <lb/>ergo linea A F eſt recta, <lb/>vti dictum eſt in propo-<lb/>ſitione 1. </s>
  <s xml:id="echoid-s13132" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13133" xml:space="preserve">occurrent A F, C E in D, eo quod egrediuntur ab angulis
<pb o="392" file="0430" n="431" rhead="Archimedis"/>
A, C minoribus duobus <lb/>rectis, &amp; </s>
  <s xml:id="echoid-s13134" xml:space="preserve">iungamus etiam <lb/>F E, E B, ergo E F B <lb/>
<anchor type="figure" xlink:label="fig-0430-01a" xlink:href="fig-0430-01"/>
eſt etiam recta, vti dixi-<lb/>mus, &amp; </s>
  <s xml:id="echoid-s13135" xml:space="preserve">eſt perpendi-<lb/>cularis ſuper A D, eo <lb/>quod angulus A F B eſt <lb/>rectus, quia cadit in ſe-<lb/>micirculum A B, &amp; </s>
  <s xml:id="echoid-s13136" xml:space="preserve">iun-<lb/>gamus H G, G C, erit <lb/>H C etiam recta; </s>
  <s xml:id="echoid-s13137" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13138" xml:space="preserve">iun-<lb/>gamus E G, G A, erit <lb/>E A recta, &amp; </s>
  <s xml:id="echoid-s13139" xml:space="preserve">produca-<lb/>mus eam ad I, &amp; </s>
  <s xml:id="echoid-s13140" xml:space="preserve">iun-<lb/>gamus B I, quæ ſit etiam <lb/>perpendicularis ſuper A I, &amp; </s>
  <s xml:id="echoid-s13141" xml:space="preserve">iungamus D I; </s>
  <s xml:id="echoid-s13142" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13143" xml:space="preserve">quia A D, A B ſunt <lb/>duæ rectæ, &amp; </s>
  <s xml:id="echoid-s13144" xml:space="preserve">educta ex D ad lineam A B perpendicularis D C, &amp; </s>
  <s xml:id="echoid-s13145" xml:space="preserve">ex <lb/>B ad D A perpendicularis B F; </s>
  <s xml:id="echoid-s13146" xml:space="preserve">quæ ſe mutuo ſecant in E, &amp; </s>
  <s xml:id="echoid-s13147" xml:space="preserve">educta A <lb/>E ad I eſt perpendicularis ſuper B I, erunt B I D rectæ, quemadmo-<lb/>dum oſtendimus in Propoſitionibus, quas confecimus in expoſitione tra-<lb/>ctatus de triangulis rectangulis: </s>
  <s xml:id="echoid-s13148" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13149" xml:space="preserve">quia duo anguli A G C, A I B ſunt <lb/>recti, vtique B D, C G ſunt parallelæ, &amp; </s>
  <s xml:id="echoid-s13150" xml:space="preserve">proportio A D ad D H, <lb/>quæ eſt vt A C ad H E, eſt vt proportio A B ad B C, ergo rectangu-<lb/>lum A C in C B æquale eſt rectangulo A B in H E; </s>
  <s xml:id="echoid-s13151" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13152" xml:space="preserve">ſimiliter demon-<lb/>ſtratur in circulo L M N, quod rectangulum A C in C B æquale ſit re-<lb/>ctangulo A B in ſuam diametrum, &amp; </s>
  <s xml:id="echoid-s13153" xml:space="preserve">demonſtratur inde etiam, quod <lb/>duæ diametri circulorum E F G, L M N, ſint æquales, ergo illi duo <lb/>circuli ſunt æquales. </s>
  <s xml:id="echoid-s13154" xml:space="preserve">Et hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13155" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1124" type="float" level="2" n="1">
  <figure xlink:label="fig-0429-01" xlink:href="fig-0429-01a">
    <image file="0429-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0429-01"/>
  </figure>
  <figure xlink:label="fig-0430-01" xlink:href="fig-0430-01a">
    <image file="0430-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0430-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1126" type="section" level="1" n="360">
<head xml:id="echoid-head452" xml:space="preserve">SCHOLIVM ALMOCHTASSO.</head>
<p>
  <s xml:id="echoid-s13156" xml:space="preserve">DIcit Doctor. </s>
  <s xml:id="echoid-s13157" xml:space="preserve">Clarum quidem eſt quod citauit ex expoſi-<lb/>tione triangulorum rectangulorum in præfatione; </s>
  <s xml:id="echoid-s13158" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13159" xml:space="preserve">eſt <lb/>quidem propoſitio vtilis in principijs, ac præſertim in triangulis <lb/>acutangulis, qua opus eſt in propoſit. </s>
  <s xml:id="echoid-s13160" xml:space="preserve">6. </s>
  <s xml:id="echoid-s13161" xml:space="preserve">huius libri, &amp; </s>
  <s xml:id="echoid-s13162" xml:space="preserve">eſt hæc. <lb/></s>
  <s xml:id="echoid-s13163" xml:space="preserve">Ex triangulo A B C eduxit perpendiculares B E, C D ſe mutuo <lb/>ſecantes in F, &amp; </s>
  <s xml:id="echoid-s13164" xml:space="preserve">coniunxit A F, &amp; </s>
  <s xml:id="echoid-s13165" xml:space="preserve">produxit ad G, hæc vti-<lb/>que erit perpendicularis ſuper B C.</s>
  <s xml:id="echoid-s13166" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13167" xml:space="preserve">Iungamus itaque D E, erunt duo anguli D A F, D E F æquales, <lb/>quia circulus comprehendens triangulum A D F tranſit per punctum E, <lb/>eo quod angulus A E F eſt rectus, &amp; </s>
  <s xml:id="echoid-s13168" xml:space="preserve">cadent in illo ſuper eundem ar-<lb/>cum, &amp; </s>
  <s xml:id="echoid-s13169" xml:space="preserve">etiam angulus D E B æqualis eſt angulo D C B, quia circulus <lb/>continens triangulum B D C tranſit etiam per punctum E, ergo in duo-<lb/>bus triangulis A B G, C B D ſunt duo anguli B A G, B C D æquales;</s>
  <s xml:id="echoid-s13170" xml:space="preserve">
<pb o="393" file="0431" n="432" rhead="Aſſumpt. Liber."/>
&amp; </s>
  <s xml:id="echoid-s13171" xml:space="preserve">angulus B eſt communis, <lb/>
<anchor type="figure" xlink:label="fig-0431-01a" xlink:href="fig-0431-01"/>
ergo A G B æqualis eſt an-<lb/>gulo C D B recto, ergo A <lb/>G eſt perpendicularis ſuper <lb/>B C. </s>
  <s xml:id="echoid-s13172" xml:space="preserve">Hoc præmiſſo repe-<lb/>tamus ex propoſit. </s>
  <s xml:id="echoid-s13173" xml:space="preserve">quàm <lb/>attulit Archimedes D A, <lb/>A B, &amp; </s>
  <s xml:id="echoid-s13174" xml:space="preserve">perpendiculares D <lb/>C, A I, B F, B I, &amp; </s>
  <s xml:id="echoid-s13175" xml:space="preserve">li-<lb/>neam D I. </s>
  <s xml:id="echoid-s13176" xml:space="preserve">iam ſi B I D <lb/>non fuerit linea recta, iun-<lb/>gamus B G D rectam, erit <lb/>angulus A G B rectus ex <lb/>præmiſſa propoſitione, &amp; </s>
  <s xml:id="echoid-s13177" xml:space="preserve"><lb/>
<anchor type="figure" xlink:label="fig-0431-02a" xlink:href="fig-0431-02"/>
erat angulus A I B rectus, <lb/>ergo internus in triangulo <lb/>B I G æqualis eſt oppoſito <lb/>externo, &amp; </s>
  <s xml:id="echoid-s13178" xml:space="preserve">hoc eſt abſur-<lb/>dum, igitur linea B I D <lb/>eſt recta. </s>
  <s xml:id="echoid-s13179" xml:space="preserve">Deinde attulit <lb/>duas propoſitiones ex in-<lb/>terpretatione Alkauhi, qua-<lb/>rum prima eſt hæc.</s>
  <s xml:id="echoid-s13180" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1126" type="float" level="2" n="1">
  <figure xlink:label="fig-0431-01" xlink:href="fig-0431-01a">
    <image file="0431-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0431-01"/>
  </figure>
  <figure xlink:label="fig-0431-02" xlink:href="fig-0431-02a">
    <image file="0431-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0431-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1128" type="section" level="1" n="361">
<head xml:id="echoid-head453" xml:space="preserve">SCHOLIVM PRIMVM ALKAVHI.</head>
<p>
  <s xml:id="echoid-s13181" xml:space="preserve">S I non fuerint duo ſemicirculi tangentes, ſed mutuo ſe ſecantes, <lb/>&amp; </s>
  <s xml:id="echoid-s13182" xml:space="preserve">perpendicularis fuerit in loco mutuæ ſectionis, idem ſe-<lb/>quitur.</s>
  <s xml:id="echoid-s13183" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13184" xml:space="preserve">Sint itaque ſemicirculi A B C, A D E, F D C, &amp; </s>
  <s xml:id="echoid-s13185" xml:space="preserve">duo illi ſemicir-<lb/>culi ſe mutuo ſecantes in D, &amp; </s>
  <s xml:id="echoid-s13186" xml:space="preserve">B G perpendicularis ſuper A C inſiſtat, <lb/>
<anchor type="figure" xlink:label="fig-0431-03a" xlink:href="fig-0431-03"/>
<pb o="394" file="0432" n="433" rhead="Archimedis"/>
&amp; </s>
  <s xml:id="echoid-s13187" xml:space="preserve">circulus I H L tangat circulum A B C in H, &amp; </s>
  <s xml:id="echoid-s13188" xml:space="preserve">circulum A D E in <lb/>L, &amp; </s>
  <s xml:id="echoid-s13189" xml:space="preserve">perpendicularem in I. </s>
  <s xml:id="echoid-s13190" xml:space="preserve">Dico eſſe æqualem circulo, qui eſt in al-<lb/>tera parte. </s>
  <s xml:id="echoid-s13191" xml:space="preserve">Hoc modo, Educamus I M parallelam ipſi A C, &amp; </s>
  <s xml:id="echoid-s13192" xml:space="preserve">iungamus <lb/>A H, quæ tranſibit per M, quemadmodum demonſtrauit Archimedes, <lb/>
<anchor type="note" xlink:label="note-0432-01a" xlink:href="note-0432-01"/>
<anchor type="figure" xlink:label="fig-0432-01a" xlink:href="fig-0432-01"/>
&amp; </s>
  <s xml:id="echoid-s13193" xml:space="preserve">producamus eam quouſque occurrat perpendiculari N G in N, &amp; </s>
  <s xml:id="echoid-s13194" xml:space="preserve"><lb/>iungamus I A, quæ tranſibit per L, &amp; </s>
  <s xml:id="echoid-s13195" xml:space="preserve">producamus illam ad O, &amp; </s>
  <s xml:id="echoid-s13196" xml:space="preserve">iun-<lb/>gamus C O, O N, quæ erit linea recta, &amp; </s>
  <s xml:id="echoid-s13197" xml:space="preserve">iungamus M E, quæ tranſi-<lb/>bit per L, &amp; </s>
  <s xml:id="echoid-s13198" xml:space="preserve">iungamus C H, quæ tranſibit per I; </s>
  <s xml:id="echoid-s13199" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13200" xml:space="preserve">linea C O N pa-<lb/><gap/> a eſt lineæ E M, &amp; </s>
  <s xml:id="echoid-s13201" xml:space="preserve">proportio A N ad N M, nempe proportio A <lb/>G ad I M eſt vt C A ad C E, ergo rectangulum A G in C E æquale <lb/>eſt rectangulo C A in I M; </s>
  <s xml:id="echoid-s13202" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13203" xml:space="preserve">quia G D eſt perpendicularis in duobus <lb/>circulis C D F, E D A ſuper duas diametros C F, E A, erit rectangu-<lb/>lum C G in G F æquale quadrato G D, &amp; </s>
  <s xml:id="echoid-s13204" xml:space="preserve">rectangulum A G in G E <lb/>æquale etiam eſt illi, ergo rectangulum C G in G F æquale eſt rectan-<lb/>lo A G in G E, &amp; </s>
  <s xml:id="echoid-s13205" xml:space="preserve">proportio C G ad G A eſt vt proportio E G ad G <lb/>F, immo vt proportio C E ad F A reſiduam; </s>
  <s xml:id="echoid-s13206" xml:space="preserve">ergo rectangulum C G in <lb/>F A, eſt æquale rectangulo C A in I M cui æquale eſt rectangulum G <lb/>A in C E. </s>
  <s xml:id="echoid-s13207" xml:space="preserve">Et ſi fuerit in altera parte circulus modo præfato eadem ra-<lb/>tione oſtendemus, quod reſtangulum C A in diametrum illius circuli <lb/>æquale ſit rectangulo C G in A F, &amp; </s>
  <s xml:id="echoid-s13208" xml:space="preserve">oſtendetur quod duæ diametri duo-<lb/>rum circulorum ſint æquales.</s>
  <s xml:id="echoid-s13209" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1128" type="float" level="2" n="1">
  <figure xlink:label="fig-0431-03" xlink:href="fig-0431-03a">
    <image file="0431-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0431-03"/>
  </figure>
<note position="left" xlink:label="note-0432-01" xlink:href="note-0432-01a" xml:space="preserve">Prop. I. <lb/>huius.</note>
  <figure xlink:label="fig-0432-01" xlink:href="fig-0432-01a">
    <image file="0432-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0432-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1130" type="section" level="1" n="362">
<head xml:id="echoid-head454" xml:space="preserve">SCHOLIVM SECVNDVM ALKAVHI.</head>
<p>
  <s xml:id="echoid-s13210" xml:space="preserve">POrrò ſecunda eſt hæc. </s>
  <s xml:id="echoid-s13211" xml:space="preserve">Dicit quod ſi duo ſemicirculi non <lb/>ſint tangentes, nec ſe mutuo ſecantes, ſed ſeparati, &amp; </s>
  <s xml:id="echoid-s13212" xml:space="preserve"><lb/>perpendicularis tranſeat per concurſum duarum linearum tangen-
<pb o="395" file="0433" n="434" rhead="Aſſump. Liber."/>
tium eos, quæ ſunt æquales idem ſequetur.</s>
  <s xml:id="echoid-s13213" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13214" xml:space="preserve">Sint itaque ſemicirculi A B C, A D E, F G C, vti diſpoſuimus, &amp; </s>
  <s xml:id="echoid-s13215" xml:space="preserve"><lb/>duæ lineæ N G, N D tangentes illos duos ſemicirculos in G, D, &amp; </s>
  <s xml:id="echoid-s13216" xml:space="preserve">æ-<lb/>quales, ſibique occurrentes in N, &amp; </s>
  <s xml:id="echoid-s13217" xml:space="preserve">linea B N tranſiens per punctum <lb/>N perpendiculariter erecta ſuper A C, &amp; </s>
  <s xml:id="echoid-s13218" xml:space="preserve">tangat illam circulus M N I <lb/>in I, &amp; </s>
  <s xml:id="echoid-s13219" xml:space="preserve">idem tangat circulum A B C in H, &amp; </s>
  <s xml:id="echoid-s13220" xml:space="preserve">circulum A D E in L, <lb/>
<anchor type="figure" xlink:label="fig-0433-01a" xlink:href="fig-0433-01"/>
&amp; </s>
  <s xml:id="echoid-s13221" xml:space="preserve">educamus diametrum I M parallelam ipſi A C, &amp; </s>
  <s xml:id="echoid-s13222" xml:space="preserve">iungamus C H, <lb/>quæ tranſibit per I, &amp; </s>
  <s xml:id="echoid-s13223" xml:space="preserve">iungamus M E tranſibit per L, &amp; </s>
  <s xml:id="echoid-s13224" xml:space="preserve">iungamus A I <lb/>
<anchor type="note" xlink:label="note-0433-01a" xlink:href="note-0433-01"/>
tranſibit per L, &amp; </s>
  <s xml:id="echoid-s13225" xml:space="preserve">producamus eam ad P, &amp; </s>
  <s xml:id="echoid-s13226" xml:space="preserve">iungamus C O tranſibit <lb/>per P, eritque parallela ipſi E M, &amp; </s>
  <s xml:id="echoid-s13227" xml:space="preserve">erit proportio A O ad O M, nem-<lb/>pe proportio A N ad M I vt proportio A C ad C E, &amp; </s>
  <s xml:id="echoid-s13228" xml:space="preserve">rectangulum A <lb/>N in C E æquale rectangulo A C in I M. </s>
  <s xml:id="echoid-s13229" xml:space="preserve">Et eodem modo oſtendetur, <lb/>quod rectangulum C N in F A ſit æquale rectangulo A C in diametrum <lb/>circuli, qui eſt ex altera parte; </s>
  <s xml:id="echoid-s13230" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13231" xml:space="preserve">quia rectangulum C N in N F æqua-<lb/>le eſt quadrato G N, &amp; </s>
  <s xml:id="echoid-s13232" xml:space="preserve">eſt æquale quadrato D N, quod eſt æquale re-<lb/>ctangulo A N in N E erit rectangulum C N in N F æquale rectangulo <lb/>A N in N E, &amp; </s>
  <s xml:id="echoid-s13233" xml:space="preserve">proportio C N ad A N vt E N ad N F, &amp; </s>
  <s xml:id="echoid-s13234" xml:space="preserve">vt propor-<lb/>tio totius C E ad totum A F, ergo rectangulum A N in C E æquale eſt <lb/>rectangulo C N in F A, &amp; </s>
  <s xml:id="echoid-s13235" xml:space="preserve">iam oſtenſum eſt, quod A N in C E æqua-<lb/>le eſt rectangulo A C in I M, &amp; </s>
  <s xml:id="echoid-s13236" xml:space="preserve">quod rectangulum C N in F A ſit æqua-<lb/>le rectangulo A C in diametrum alterius circuli: </s>
  <s xml:id="echoid-s13237" xml:space="preserve">ergo duæ diametri ſunt <lb/>æquales, &amp; </s>
  <s xml:id="echoid-s13238" xml:space="preserve">duo circuli æquales, &amp; </s>
  <s xml:id="echoid-s13239" xml:space="preserve">hoc eſt quæſitum.</s>
  <s xml:id="echoid-s13240" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1130" type="float" level="2" n="1">
  <figure xlink:label="fig-0433-01" xlink:href="fig-0433-01a">
    <image file="0433-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0433-01"/>
  </figure>
<note position="right" xlink:label="note-0433-01" xlink:href="note-0433-01a" xml:space="preserve">Prop. I. <lb/>huius. <lb/>Ibidem, <lb/>Scholium <lb/>præc. <lb/>Almoc.</note>
</div>
</div>
<div xml:id="echoid-div1132" type="section" level="1" n="363">
<head xml:id="echoid-head455" xml:space="preserve">Notæ in Propoſit. V.</head>
<p style="it">
  <s xml:id="echoid-s13241" xml:space="preserve">HAEc propoſitio parum quidem differt à poſtrema parte propoſit. </s>
  <s xml:id="echoid-s13242" xml:space="preserve">14, 16. <lb/></s>
  <s xml:id="echoid-s13243" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13244" xml:space="preserve">17. </s>
  <s xml:id="echoid-s13245" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13246" xml:space="preserve">4. </s>
  <s xml:id="echoid-s13247" xml:space="preserve">Pappi Alex.</s>
  <s xml:id="echoid-s13248" xml:space="preserve">, ſi figuram, conſtructionem, &amp; </s>
  <s xml:id="echoid-s13249" xml:space="preserve">progreſſum
<pb o="396" file="0434" n="435" rhead="Archimedis"/>
demonſtrationis ſpectes; </s>
  <s xml:id="echoid-s13250" xml:space="preserve">differunt tamen in concluſione, quæ demonſtranda pro-<lb/>
<anchor type="figure" xlink:label="fig-0434-01a" xlink:href="fig-0434-01"/>
ponitur; </s>
  <s xml:id="echoid-s13251" xml:space="preserve">oſtendit enim Pap-<lb/>pus, ſicut, &amp; </s>
  <s xml:id="echoid-s13252" xml:space="preserve">Archime-<lb/>des, ſemicircularis diame-<lb/>tri ſegmentum maius A C <lb/>ad circuli intercepti dia-<lb/>metrum H E habere ean-<lb/>dem proportionem, quàm <lb/>maioris circuli diameter A <lb/>B habet ad reliquum ſeg-<lb/>mentum eius B C, pari-<lb/>terque B A ad A C ean-<lb/>dem proportionem habet, <lb/>quàm C B ad reliqui circuli intercepti L M N diametrum: </s>
  <s xml:id="echoid-s13253" xml:space="preserve">ex hiſce ſequitur <lb/>concluſio Archimedea, nam ſi A C ad H E eandem rationem habet, quàm A <lb/>B ad B C, permutando B A ad A C erit vt C B ad H E igitur eadem C B ad <lb/>duas circulorum diametros H E, &amp; </s>
  <s xml:id="echoid-s13254" xml:space="preserve">L N eandem proportionem habet, &amp; </s>
  <s xml:id="echoid-s13255" xml:space="preserve">pro-<lb/>pterea circulorum diametri H E, &amp; </s>
  <s xml:id="echoid-s13256" xml:space="preserve">L N æquales ſunt inter ſe. </s>
  <s xml:id="echoid-s13257" xml:space="preserve">Mirum ta-<lb/>men eſt hanc concluſionem, quàm præ manibus Pappus habebat, non ani-<lb/>maduertiſſe, demonſtrat tamen quamplurima ſymptomata pulcherrima circu-<lb/>lorum in Arbelo deſcriptorum, quæ tamen in hoc opuſculo Archimedi tributo <lb/>pariter recenſeri debebant, ſi hic liber eſſet idem antiquus ille à Pappo viſus, <lb/>in quo huiuſmodi lemmata circumferebantur: </s>
  <s xml:id="echoid-s13258" xml:space="preserve">ſed for ſan librariorum vitio, &amp; </s>
  <s xml:id="echoid-s13259" xml:space="preserve"><lb/>incuria codex corruptiſſimus ad Arabes tranſmißus non omnes illas admirandas <lb/>propoſitiones, ſed vnius tantum particulam continebat, ſicut è contra liber ille <lb/>antiquus, in quo Pappus prædicta lemmata reperit, carebat concluſione in hi-<lb/>ſce lemmatibus demonſtrata. </s>
  <s xml:id="echoid-s13260" xml:space="preserve">Cæterum propoſitiones in ſcholĳs additæ manifeſtæ <lb/>quidem ſunt, ſed abſque duabus prioribus poßet propoſitum facillimè demon-<lb/>ſtrari, Reliquæ duæ propoſitiones ſuperadditæ ad Arabibus faciles quidem <lb/>ſunt.</s>
  <s xml:id="echoid-s13261" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1132" type="float" level="2" n="1">
  <figure xlink:label="fig-0434-01" xlink:href="fig-0434-01a">
    <image file="0434-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0434-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1134" type="section" level="1" n="364">
<head xml:id="echoid-head456" xml:space="preserve">PROPOSITIO VI.</head>
<p>
  <s xml:id="echoid-s13262" xml:space="preserve">SI fuerit femicirculus A B C, &amp; </s>
  <s xml:id="echoid-s13263" xml:space="preserve">in eius diametro ſumatur <lb/>punctum D, &amp; </s>
  <s xml:id="echoid-s13264" xml:space="preserve">fuerit A D ipſius D C ſexqui altera, &amp; </s>
  <s xml:id="echoid-s13265" xml:space="preserve"><lb/>deſcribantur ſuper A D, D C duo ſemicirculi, &amp; </s>
  <s xml:id="echoid-s13266" xml:space="preserve">ponatur cir-<lb/>culus E F inter tres ſemicirculos tangens eos, &amp; </s>
  <s xml:id="echoid-s13267" xml:space="preserve">educatur dia-<lb/>meter E F in illo parallela diametro A C, reperiri debet pro-<lb/>portio diametri A C ad diametrum E F.</s>
  <s xml:id="echoid-s13268" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13269" xml:space="preserve">Iungamus enim duas lineas A E, E B, &amp; </s>
  <s xml:id="echoid-s13270" xml:space="preserve">duas lineas C F, F B, <lb/>erunt C B, A B rectæ, vti dictũ eſt in prima propoſit. </s>
  <s xml:id="echoid-s13271" xml:space="preserve">Deſcribamus etiam <lb/>duas lineas F G A, E H C, oſtendeturque eſſe quoque rectas; </s>
  <s xml:id="echoid-s13272" xml:space="preserve">Simili-<lb/>ter duas lineas D E, D F, &amp; </s>
  <s xml:id="echoid-s13273" xml:space="preserve">iungamus D I, D L, &amp; </s>
  <s xml:id="echoid-s13274" xml:space="preserve">E M, F N, &amp; </s>
  <s xml:id="echoid-s13275" xml:space="preserve"><lb/>producamus eas ad O, P; </s>
  <s xml:id="echoid-s13276" xml:space="preserve">Et quia in triangulo A E D, A G eſt per-
<pb o="397" file="0435" n="436" rhead="Aſſumpt. Liber."/>
pendicularis ad E D, &amp; </s>
  <s xml:id="echoid-s13277" xml:space="preserve">D I eſt quoque perpendicularis ad A E, &amp; </s>
  <s xml:id="echoid-s13278" xml:space="preserve">iam <lb/>ſe mutuo ſecuerunt in M, ergo E M O erit etiam perpendicularis, que-<lb/>madmodum oſtendimus in expoſitione, quàm confecimus de proprieta-<lb/>tibus triangulorum, &amp; </s>
  <s xml:id="echoid-s13279" xml:space="preserve">cuius demonſtratio iam quidem præceſſit in ſupe-<lb/>
<anchor type="figure" xlink:label="fig-0435-01a" xlink:href="fig-0435-01"/>
riori propoſitione; </s>
  <s xml:id="echoid-s13280" xml:space="preserve">Similiter quoque erit F P perpendicularis ſuper C A, <lb/>&amp; </s>
  <s xml:id="echoid-s13281" xml:space="preserve">quia duo anguli, qui ſunt apud L, &amp; </s>
  <s xml:id="echoid-s13282" xml:space="preserve">B ſunt recti, erit D L parallela <lb/>ipſi A B, &amp; </s>
  <s xml:id="echoid-s13283" xml:space="preserve">pariter D I ipſi C B, igitur proportio A D ad D C eſt vt <lb/>proportio A M ad F M, immo vt proportio A O ad O P, &amp; </s>
  <s xml:id="echoid-s13284" xml:space="preserve">proportio <lb/>C D ad D A vt proportio C N ad N E, immo vt proportio C P ad P <lb/>O, &amp; </s>
  <s xml:id="echoid-s13285" xml:space="preserve">erat A D ſexquialtera D C, ergo A O eſt ſexquialtera O P, &amp; </s>
  <s xml:id="echoid-s13286" xml:space="preserve"><lb/>O P ſexquialtera C P, ergo tres lineæ A O, O P, P C ſunt proportio-<lb/>nales: </s>
  <s xml:id="echoid-s13287" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13288" xml:space="preserve">in eadem menſura, in qua eſt P C quatuor, erit O P ſex, &amp; </s>
  <s xml:id="echoid-s13289" xml:space="preserve"><lb/>A O nouem, &amp; </s>
  <s xml:id="echoid-s13290" xml:space="preserve">C A nouendecim, &amp; </s>
  <s xml:id="echoid-s13291" xml:space="preserve">quia P O æ qualis eſt E F, erit <lb/>proportio A C ad E F vt nouendecim ad ſex, igitur reperimus dictam <lb/>proportionem. </s>
  <s xml:id="echoid-s13292" xml:space="preserve">Etiam ſi fuerit A D ad D C qualiſcumque vt ſexquiter-<lb/>tia, aut ſexquiquarta, aut alia, erit iudicium, &amp; </s>
  <s xml:id="echoid-s13293" xml:space="preserve">ratio, vti dictum eſt. <lb/></s>
  <s xml:id="echoid-s13294" xml:space="preserve">Et hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13295" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1134" type="float" level="2" n="1">
  <figure xlink:label="fig-0435-01" xlink:href="fig-0435-01a">
    <image file="0435-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0435-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1136" type="section" level="1" n="365">
<head xml:id="echoid-head457" xml:space="preserve">Notæ in Propoſit. VI.</head>
<p style="it">
  <s xml:id="echoid-s13296" xml:space="preserve">HAEc propoſitio nil prorſus differre videtur à 16. </s>
  <s xml:id="echoid-s13297" xml:space="preserve">propoſit. </s>
  <s xml:id="echoid-s13298" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13299" xml:space="preserve">4. </s>
  <s xml:id="echoid-s13300" xml:space="preserve">Pappi <lb/>Alex. </s>
  <s xml:id="echoid-s13301" xml:space="preserve">eſt tamen pars illius, &amp; </s>
  <s xml:id="echoid-s13302" xml:space="preserve">particulariter demonſtrata, quod quidem <lb/>peccatum alicui expoſitori tribui debet; </s>
  <s xml:id="echoid-s13303" xml:space="preserve">nunquam enim Archimedes propoſitionẽ <lb/>illam, quam vniuerſaltſſimè demonſtrare potuißet, exemplis numericis tam <lb/>pueriliter oſtendiſſet. </s>
  <s xml:id="echoid-s13304" xml:space="preserve">Pappus igitur quærit menſuram diametri illius circuli, <lb/>qui in loco inter tres circunferentias circulares interĳcitur, quod Arbelon ap-<lb/>pellatur, &amp; </s>
  <s xml:id="echoid-s13305" xml:space="preserve">oſtendit quidem diametrum ſemicirculi maioris A C ſecari in duo-<lb/>bus punctis O, &amp; </s>
  <s xml:id="echoid-s13306" xml:space="preserve">P à perpendicularibus cadentibus à terminis E, &amp; </s>
  <s xml:id="echoid-s13307" xml:space="preserve">F dia-<lb/>metri circuli in Arbelo inſcripti, ac diuidi in tria ſegmenta A O, O P, P C <lb/>continue proportionalia in eadem ratione, quàm habet A D ad D C, &amp; </s>
  <s xml:id="echoid-s13308" xml:space="preserve">in-
<pb o="398" file="0436" n="437" rhead="Archimedis"/>
ſuper oſtendit perpendicularem E O æqualem eſſe circuli diametro E F. </s>
  <s xml:id="echoid-s13309" xml:space="preserve">Itaque <lb/>in quadrato ſpatio E O P F, circuli diameter E F, ſiue O P media proportio-<lb/>nalis erit inter A O, &amp; </s>
  <s xml:id="echoid-s13310" xml:space="preserve">P C. </s>
  <s xml:id="echoid-s13311" xml:space="preserve">Zuam ergo proportionem habent tres continuè <lb/>proportionales in eadem ratione A D ad D C ſimul ſumptæ ad illarum inter-<lb/>
<anchor type="figure" xlink:label="fig-0436-01a" xlink:href="fig-0436-01"/>
mediam, eandem habebit diameter maioris ſemicirculi A C ad O P, ſiue E F. <lb/></s>
  <s xml:id="echoid-s13312" xml:space="preserve">Zuæ deinde Pappus demonſtrat perpendiculares à centris circulorum in collate-<lb/>ralibus ſpatĳs prædicti Arbeli exiſtentium eſſe multiplices diametrorum eorum <lb/>circulorum à quibus educuntur ſecundum ſeriem natur alem numerorum ab vni-<lb/>tate creſcentium, proprietas quidem eſt admirabilis, de qua in hac propoſitio-<lb/>ne Archimedis altum ſilentium, quod forte temporum iniuriæ tribuendum <lb/>eſt.</s>
  <s xml:id="echoid-s13313" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1136" type="float" level="2" n="1">
  <figure xlink:label="fig-0436-01" xlink:href="fig-0436-01a">
    <image file="0436-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0436-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s13314" xml:space="preserve">Poſſent in hiſce duabus propoſitionibus non pauca problemata ſuperaddi, quo-<lb/>modo nimirum in prædicto ſpatio à tribus ſemicirculis comprehenſo circuli in-<lb/>numerabiles deſcribi debeant, &amp; </s>
  <s xml:id="echoid-s13315" xml:space="preserve">alia quamplurima facilia, quæ lectorum ſa-<lb/>gacitati relinquuntur.</s>
  <s xml:id="echoid-s13316" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1138" type="section" level="1" n="366">
<head xml:id="echoid-head458" xml:space="preserve">PROPOSITIO VII.</head>
<p>
  <s xml:id="echoid-s13317" xml:space="preserve">SI circulus circa quadratum deſcriptus fuerit, &amp; </s>
  <s xml:id="echoid-s13318" xml:space="preserve">alius intra <lb/>illum, vtique erit circumſcriptus duplus inſcripti.</s>
  <s xml:id="echoid-s13319" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13320" xml:space="preserve">Sit itaque circulus compre-<lb/>
<anchor type="figure" xlink:label="fig-0436-02a" xlink:href="fig-0436-02"/>
hendens quadratum A B, cir-<lb/>culus A B, &amp; </s>
  <s xml:id="echoid-s13321" xml:space="preserve">inſcriptus C D, <lb/>&amp; </s>
  <s xml:id="echoid-s13322" xml:space="preserve">ſit diameter quadrati A B, &amp; </s>
  <s xml:id="echoid-s13323" xml:space="preserve"><lb/>eſt diameter circuli circumſcri-<lb/>pti, &amp; </s>
  <s xml:id="echoid-s13324" xml:space="preserve">educamus C D diame-<lb/>trum circuli inſcripti parallelam <lb/>ipſi A E, quæ eſt ei æqualis. <lb/></s>
  <s xml:id="echoid-s13325" xml:space="preserve">Et quia quadratum A B duplum <lb/>eſt quadrati A E, ſiue D C, &amp; </s>
  <s xml:id="echoid-s13326" xml:space="preserve"><lb/>proportio quadratorum ex dia-
<pb o="399" file="0437" n="438" rhead="Aſſumpt. Liber."/>
metris circulorum eſt eadem proportioni circuli ad circulum, igitur cir-<lb/>culus A B duplus eſt circuli C D, &amp; </s>
  <s xml:id="echoid-s13327" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13328" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1138" type="float" level="2" n="1">
  <figure xlink:label="fig-0436-02" xlink:href="fig-0436-02a">
    <image file="0436-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0436-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1140" type="section" level="1" n="367">
<head xml:id="echoid-head459" xml:space="preserve">SCHOLIVM ALMOCHTASSO.</head>
<p>
  <s xml:id="echoid-s13329" xml:space="preserve">DIcit Doctor Almochtaſſo. </s>
  <s xml:id="echoid-s13330" xml:space="preserve">Iam compoſui tractatum de con-<lb/>ficiendo circulo, cuius proportio ad datum circulum ſit <lb/>vt proportio data. </s>
  <s xml:id="echoid-s13331" xml:space="preserve">Qua ratione conficiendæ ſunt omnes ſiguræ <lb/>rectilineæ, &amp; </s>
  <s xml:id="echoid-s13332" xml:space="preserve">quem vſum <lb/>
<anchor type="figure" xlink:label="fig-0437-01a" xlink:href="fig-0437-01"/>
habeant in arte illæ figuræ, <lb/>&amp; </s>
  <s xml:id="echoid-s13333" xml:space="preserve">afferam hic ex illis vnam <lb/>propoſitionem, quæ cõgruit <lb/>expoſitioni huius propoſitio-<lb/>nis, &amp; </s>
  <s xml:id="echoid-s13334" xml:space="preserve">eſt tanquam epitome <lb/>illarum propoſitionum, &amp; </s>
  <s xml:id="echoid-s13335" xml:space="preserve"><lb/>illationis ex illis, &amp; </s>
  <s xml:id="echoid-s13336" xml:space="preserve">eſt hæc. <lb/></s>
  <s xml:id="echoid-s13337" xml:space="preserve">Volumus conficere circulum, <lb/>qui ſit quinta pars circuli, <lb/>exempli gratia.</s>
  <s xml:id="echoid-s13338" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1140" type="float" level="2" n="1">
  <figure xlink:label="fig-0437-01" xlink:href="fig-0437-01a">
    <image file="0437-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0437-01"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s13339" xml:space="preserve">Circulus cuius habemus diametrum eſt A B, &amp; </s>
  <s xml:id="echoid-s13340" xml:space="preserve">addamus eius partem <lb/>quintam, &amp; </s>
  <s xml:id="echoid-s13341" xml:space="preserve">eſt B C, &amp; </s>
  <s xml:id="echoid-s13342" xml:space="preserve">deſcribamus ſuper A C ſemicirculum A D C, <lb/>&amp; </s>
  <s xml:id="echoid-s13343" xml:space="preserve">educamus perpendicularem B D, &amp; </s>
  <s xml:id="echoid-s13344" xml:space="preserve">quia proportio A B ad B C eſt, <lb/>vt proportio quadrati A B ad quadratum B D, erit quilibet circulus <lb/>factus, vel, figura ſuper B D quæſita à nobis, &amp; </s>
  <s xml:id="echoid-s13345" xml:space="preserve">hoc, quia proportio <lb/>circuli, qui eſt ſuper A B, vel figuræ, quæ eſt ſuper illam, ad circu-<lb/>lum, vel figuram factam ſuper B D facit illam figuram, &amp; </s>
  <s xml:id="echoid-s13346" xml:space="preserve">ſimiliter po-<lb/>ſitam, erit vt proportio A B ad B C, &amp; </s>
  <s xml:id="echoid-s13347" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13348" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1142" type="section" level="1" n="368">
<head xml:id="echoid-head460" xml:space="preserve">PROPOSITIO VIII.</head>
<p>
  <s xml:id="echoid-s13349" xml:space="preserve">SI egrediatur in circulo linea A B vbicumque, &amp; </s>
  <s xml:id="echoid-s13350" xml:space="preserve">producatur <lb/>in directum, &amp; </s>
  <s xml:id="echoid-s13351" xml:space="preserve">ponatur B C æqualis ſemidiametro circuli <lb/>&amp; </s>
  <s xml:id="echoid-s13352" xml:space="preserve">iungatur ex C ad centrum circuli, quod eſt D, &amp; </s>
  <s xml:id="echoid-s13353" xml:space="preserve">producatur <lb/>ad E, erit arcus A E triplus arcus B F.</s>
  <s xml:id="echoid-s13354" xml:space="preserve"/>
</p>
<pb o="400" file="0438" n="439" rhead="Archimedis"/>
<p>
  <s xml:id="echoid-s13355" xml:space="preserve">Educamus igitur E G parallelam ipſi <lb/>
<anchor type="figure" xlink:label="fig-0438-01a" xlink:href="fig-0438-01"/>
A B, &amp; </s>
  <s xml:id="echoid-s13356" xml:space="preserve">iungamus D B, D G: </s>
  <s xml:id="echoid-s13357" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13358" xml:space="preserve">quia duo <lb/>anguli D E G, D G E ſunt æquales, erit <lb/>angulus G D C duplus anguli D E G, <lb/>&amp; </s>
  <s xml:id="echoid-s13359" xml:space="preserve">quia angulus B D C æqualis eſt angu-<lb/>lo B C D, &amp; </s>
  <s xml:id="echoid-s13360" xml:space="preserve">angulus C E G æqualis eſt <lb/>angulo A C E, erit angulus G D C du-<lb/>plus anguli C D B, &amp; </s>
  <s xml:id="echoid-s13361" xml:space="preserve">totus angulus B <lb/>D G triplus anguli B D C, &amp; </s>
  <s xml:id="echoid-s13362" xml:space="preserve">arcus B G <lb/>æqualis arcui A E, triplus eſt arcus B F, <lb/>&amp; </s>
  <s xml:id="echoid-s13363" xml:space="preserve">hoc eſt, quod voluimus.</s>
  <s xml:id="echoid-s13364" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1142" type="float" level="2" n="1">
  <figure xlink:label="fig-0438-01" xlink:href="fig-0438-01a">
    <image file="0438-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0438-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1144" type="section" level="1" n="369">
<head xml:id="echoid-head461" xml:space="preserve">SCHOLIVM ALMOCHTASSO.</head>
<p>
  <s xml:id="echoid-s13365" xml:space="preserve">DIcit Doctor Almoch-<lb/>
<anchor type="figure" xlink:label="fig-0438-02a" xlink:href="fig-0438-02"/>
taſſo. </s>
  <s xml:id="echoid-s13366" xml:space="preserve">Cum dicit ar-<lb/>cum B G æqualem eſſe ar-<lb/>cui A E, id ex eo eſt pro-<lb/>pter æquidiſtantiam duarum <lb/>cordarum. </s>
  <s xml:id="echoid-s13367" xml:space="preserve">Sint itaque in <lb/>circulo A B C cordæ A C, <lb/>B D parallelæ; </s>
  <s xml:id="echoid-s13368" xml:space="preserve">Dico quod <lb/>duo arcus A B, C D ſunt <lb/>æquales,</s>
</p>
<div xml:id="echoid-div1144" type="float" level="2" n="1">
  <figure xlink:label="fig-0438-02" xlink:href="fig-0438-02a">
    <image file="0438-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0438-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s13369" xml:space="preserve">Iungamus A D, ergo duo anguli C A D, A D B ſunt æquales; </s>
  <s xml:id="echoid-s13370" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13371" xml:space="preserve"><lb/>propterea duo arcus ſunt æquales, &amp; </s>
  <s xml:id="echoid-s13372" xml:space="preserve">conuerſum eodem modo demon-<lb/>ſtratur.</s>
  <s xml:id="echoid-s13373" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1146" type="section" level="1" n="370">
<head xml:id="echoid-head462" xml:space="preserve">Notæ in Propoſit. VIII.</head>
<p style="it">
  <s xml:id="echoid-s13374" xml:space="preserve">HAEc quidem propoſitio elegantiſſima eſt, quæ ſi problematicè reſolui poſ-<lb/>ſet via plana, reperta iam eßet tripartitio cuiuſlibet anguli.</s>
  <s xml:id="echoid-s13375" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13376" xml:space="preserve">Breuius tamen demonſtratio <lb/>
<anchor type="figure" xlink:label="fig-0438-03a" xlink:href="fig-0438-03"/>
perfici poteſt hac ratione. </s>
  <s xml:id="echoid-s13377" xml:space="preserve">Iuncta <lb/>recta E B, quia in triangulo Iſo-<lb/>ſcele B D C duo anguli C, &amp; </s>
  <s xml:id="echoid-s13378" xml:space="preserve">C <lb/>D B æquales ſunt, eſtque pariter <lb/>externus angulus B D C duplus an-<lb/>guli D E B in triangulo Iſoſcelio <lb/>D E B, ergo angulus C duplus eſt <lb/>anguli B E C, &amp; </s>
  <s xml:id="echoid-s13379" xml:space="preserve">propterea illi an-<lb/>guli ſimul ſumpti, ſeu externus an-<lb/>gulus A B E triplus erit anguli B <lb/>E F, &amp; </s>
  <s xml:id="echoid-s13380" xml:space="preserve">circunferentia A E tripla ipſius B F.</s>
  <s xml:id="echoid-s13381" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1146" type="float" level="2" n="1">
  <figure xlink:label="fig-0438-03" xlink:href="fig-0438-03a">
    <image file="0438-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0438-03"/>
  </figure>
</div>
<pb o="401" file="0439" n="440" rhead="Aſſumpt. Liber."/>
</div>
<div xml:id="echoid-div1148" type="section" level="1" n="371">
<head xml:id="echoid-head463" xml:space="preserve">PROPOSITIO IX.</head>
<p>
  <s xml:id="echoid-s13382" xml:space="preserve">SI mutuo ſe ſecuerint in circulo duæ lineæ A B, C D, (ſed <lb/>non in centro) ad angulos rectos, vtique duo arcus A D, <lb/>C B ſunt æquales duobus arcubus A C, D B.</s>
  <s xml:id="echoid-s13383" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13384" xml:space="preserve">Educamus diametrum E F parallelam ipſi A B, quæ ſecet C D biſa-<lb/>riam in G, erit E C æqualis ipſi E D; </s>
  <s xml:id="echoid-s13385" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13386" xml:space="preserve">quia tam arcus E D F, quam <lb/>E C F eſt ſemicirculus, &amp; </s>
  <s xml:id="echoid-s13387" xml:space="preserve">arcus <lb/>E D æqualis arcui E A cum <lb/>
<anchor type="figure" xlink:label="fig-0439-01a" xlink:href="fig-0439-01"/>
arcu A D, erit arcus C F cum <lb/>duobus arcubus E A, A D æ-<lb/>qualis ſemicirculo, &amp; </s>
  <s xml:id="echoid-s13388" xml:space="preserve">arcus E <lb/>A æqualis arcui B F, ergo ar-<lb/>cus C B cum arcu A D æqualis <lb/>eſt ſemicirculo, &amp; </s>
  <s xml:id="echoid-s13389" xml:space="preserve">remanent duo <lb/>arcus E C, E A nempe arcus A <lb/>C cum arcu D B æquales illi, <lb/>&amp; </s>
  <s xml:id="echoid-s13390" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13391" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1148" type="float" level="2" n="1">
  <figure xlink:label="fig-0439-01" xlink:href="fig-0439-01a">
    <image file="0439-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0439-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1150" type="section" level="1" n="372">
<head xml:id="echoid-head464" xml:space="preserve">PROPOSITIO X.</head>
<p>
  <s xml:id="echoid-s13392" xml:space="preserve">SI fuerit circulus A B C, &amp; </s>
  <s xml:id="echoid-s13393" xml:space="preserve">D A tangens illum, &amp; </s>
  <s xml:id="echoid-s13394" xml:space="preserve">D B ſe-<lb/>cans illum, &amp; </s>
  <s xml:id="echoid-s13395" xml:space="preserve">D C etiam tangens, &amp; </s>
  <s xml:id="echoid-s13396" xml:space="preserve">educta fuerit C E <lb/>parallela ipſi D B, &amp; </s>
  <s xml:id="echoid-s13397" xml:space="preserve">iuncta fuerit E A ſecans D B in F, &amp; </s>
  <s xml:id="echoid-s13398" xml:space="preserve"><lb/>educta fuerit ex F perpendicularis F G ſuper C E; </s>
  <s xml:id="echoid-s13399" xml:space="preserve">vtique bifa-<lb/>riam ſecabit illam in G.</s>
  <s xml:id="echoid-s13400" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13401" xml:space="preserve">Iungamus A C, &amp; </s>
  <s xml:id="echoid-s13402" xml:space="preserve">quia D A eſt tangens, &amp; </s>
  <s xml:id="echoid-s13403" xml:space="preserve">A C ſecans circulum <lb/>erit angulus D A C æqualis angulo cadenti in alterno ſegmento A C <lb/>
<anchor type="figure" xlink:label="fig-0439-02a" xlink:href="fig-0439-02"/>
<pb o="402" file="0440" n="441" rhead="Archimedis"/>
nempe angulo A E C, &amp; </s>
  <s xml:id="echoid-s13404" xml:space="preserve">eſt æqualis angulo A F D, eo quod C E, <lb/>B D ſunt parallelæ, ergo anguli D A C, A F D ſunt æquales, &amp; </s>
  <s xml:id="echoid-s13405" xml:space="preserve">in <lb/>duobus triangulis D A F, A H D ſunt duo auguli A F D, H A D æquales, <lb/>&amp; </s>
  <s xml:id="echoid-s13406" xml:space="preserve">angulus D communis, propterea erit rectãgulum F D in D H æquale <lb/>
<anchor type="figure" xlink:label="fig-0440-01a" xlink:href="fig-0440-01"/>
quadrato D A, immo quadrato D C, &amp; </s>
  <s xml:id="echoid-s13407" xml:space="preserve">quia proportio F D ad D C eſt <lb/>eadem proportioni C D ad D H, &amp; </s>
  <s xml:id="echoid-s13408" xml:space="preserve">angulus D communis, erunt triangula <lb/>D F C, D C H ſimilia, &amp; </s>
  <s xml:id="echoid-s13409" xml:space="preserve">angulus D F C æqualis D C H, qui æqualis <lb/>eſt angulo D A H, &amp; </s>
  <s xml:id="echoid-s13410" xml:space="preserve">hic eſt æqualis angulo A F D, ergo duo anguli A <lb/>F D, C F D ſunt æquales, &amp; </s>
  <s xml:id="echoid-s13411" xml:space="preserve">D F C æqualis angulo F C E, &amp; </s>
  <s xml:id="echoid-s13412" xml:space="preserve">erat D <lb/>F A æqualis angulo A E C, ergo in triangulo F E C ſunt duo anguli C, <lb/>E æquales, &amp; </s>
  <s xml:id="echoid-s13413" xml:space="preserve">duo anguli G recti, &amp; </s>
  <s xml:id="echoid-s13414" xml:space="preserve">latus G F commune, propterea <lb/>eri@ C G æqualis ipſi G E, ergo C E bifariam ſecatur in G, &amp; </s>
  <s xml:id="echoid-s13415" xml:space="preserve">hoc eſt, <lb/>quod voluimus,</s>
</p>
<div xml:id="echoid-div1150" type="float" level="2" n="1">
  <figure xlink:label="fig-0439-02" xlink:href="fig-0439-02a">
    <image file="0439-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0439-02"/>
  </figure>
  <figure xlink:label="fig-0440-01" xlink:href="fig-0440-01a">
    <image file="0440-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0440-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1152" type="section" level="1" n="373">
<head xml:id="echoid-head465" xml:space="preserve">PROPOSITIO XI.</head>
<p>
  <s xml:id="echoid-s13416" xml:space="preserve">SI mutuo ſe ſecuerint in circulo duæ lineæ A B, C D ad an-<lb/>gulos rectos in E, quod non ſit in centro, vtique omnia <lb/>quadrata A E, B E, E C, E D æqualia ſunt quadrato diametri.</s>
  <s xml:id="echoid-s13417" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13418" xml:space="preserve">Educamus diametrum A F, <lb/>
<anchor type="figure" xlink:label="fig-0440-02a" xlink:href="fig-0440-02"/>
&amp; </s>
  <s xml:id="echoid-s13419" xml:space="preserve">iungamus lineas A C, A D, <lb/>C F, D B; </s>
  <s xml:id="echoid-s13420" xml:space="preserve">Et quia angulus A <lb/>E D eſt rectus, erit æqualis an-<lb/>gulo A C F, &amp; </s>
  <s xml:id="echoid-s13421" xml:space="preserve">angulus A D C <lb/>æqualis A F C, eo quod ſunt <lb/>ſuper arcum A C, &amp; </s>
  <s xml:id="echoid-s13422" xml:space="preserve">remanent <lb/>in duobus triangulis A D E, A <lb/>F C duo anguli C A F, D A E <lb/>æquales erunt pariter duo arcus <lb/>C F, D B æquales immo, &amp; </s>
  <s xml:id="echoid-s13423" xml:space="preserve"><lb/>duæ cordæ eorum æquales, &amp; </s>
  <s xml:id="echoid-s13424" xml:space="preserve"><lb/>duo quadrata D E, E B æquantur quadrato B D, nempe C F, &amp; </s>
  <s xml:id="echoid-s13425" xml:space="preserve">duo
<pb o="403" file="0441" n="442" rhead="Aſſumpt. Liber."/>
quadrata A E, E C æquantur quadrato C A, &amp; </s>
  <s xml:id="echoid-s13426" xml:space="preserve">duo quadrata C F, C A <lb/>æquantur quadrato F A, nempe diametri, igitur quadrata A E, E B, C E, <lb/>E D omnia ſunt æqualia quadrato diametri, &amp; </s>
  <s xml:id="echoid-s13427" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13428" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1152" type="float" level="2" n="1">
  <figure xlink:label="fig-0440-02" xlink:href="fig-0440-02a">
    <image file="0440-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0440-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1154" type="section" level="1" n="374">
<head xml:id="echoid-head466" xml:space="preserve">SCHOLIVM ALMOCHTASSO.</head>
<p>
  <s xml:id="echoid-s13429" xml:space="preserve">DIcit Doctor. </s>
  <s xml:id="echoid-s13430" xml:space="preserve">Huius eſt alia facilior demonſtratio ea, quam attulit <lb/>Archimedes; </s>
  <s xml:id="echoid-s13431" xml:space="preserve">quæ eſt huiuſmodi. </s>
  <s xml:id="echoid-s13432" xml:space="preserve">Iungamus A D, C B, B D; </s>
  <s xml:id="echoid-s13433" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13434" xml:space="preserve">quia <lb/>angulus B E D eſt rectus, erunt duo <lb/>
<anchor type="figure" xlink:label="fig-0441-01a" xlink:href="fig-0441-01"/>
anguli E B D, E D B æquales vni <lb/>recto, &amp; </s>
  <s xml:id="echoid-s13435" xml:space="preserve">duo A D, B C, æqua-<lb/>les ſemicirculo, ergo duæ cordæ eo-<lb/>rum in potentia ſunt æquales diame-<lb/>tro; </s>
  <s xml:id="echoid-s13436" xml:space="preserve">ſed duo quadrata A E, D E <lb/>æqualia quadrato A D, &amp; </s>
  <s xml:id="echoid-s13437" xml:space="preserve">duo qua-<lb/>drata C E, B E ſunt æqualia qua-<lb/>drato C B, ergo quadrata A E, E <lb/>B, C E, E D æqualia ſunt quadra-<lb/>to diametri; </s>
  <s xml:id="echoid-s13438" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13439" xml:space="preserve">hoc eſt quod vo-<lb/>luimus.</s>
  <s xml:id="echoid-s13440" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1154" type="float" level="2" n="1">
  <figure xlink:label="fig-0441-01" xlink:href="fig-0441-01a">
    <image file="0441-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0441-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1156" type="section" level="1" n="375">
<head xml:id="echoid-head467" xml:space="preserve">PROPOSITIO XII.</head>
<p>
  <s xml:id="echoid-s13441" xml:space="preserve">SI fuerit ſemicirculus ſuper diametrum A B, &amp; </s>
  <s xml:id="echoid-s13442" xml:space="preserve">eductæ fue-<lb/>rint ex C duæ lineæ tangentes illum in duobus punctis D, <lb/>E, &amp; </s>
  <s xml:id="echoid-s13443" xml:space="preserve">iunctæ fuerint E A, D B ſe muto ſecantes in F, &amp; </s>
  <s xml:id="echoid-s13444" xml:space="preserve">iun cta <lb/>fuerit C F, &amp; </s>
  <s xml:id="echoid-s13445" xml:space="preserve">producatur ad G, erit C G perpendicularis ad A B.</s>
  <s xml:id="echoid-s13446" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13447" xml:space="preserve">Iungamus D A, E B. </s>
  <s xml:id="echoid-s13448" xml:space="preserve">Et quia, <lb/>angulus B D A eſt rectus, erunt duo <lb/>
<anchor type="figure" xlink:label="fig-0441-02a" xlink:href="fig-0441-02"/>
anguli D A B, D B A reliqui in, <lb/>triangulo D A B æquales vni recto, <lb/>&amp; </s>
  <s xml:id="echoid-s13449" xml:space="preserve">angulus A E B rectus, igitur ſunt <lb/>æquales ei, &amp; </s>
  <s xml:id="echoid-s13450" xml:space="preserve">ponamus angulum <lb/>F B E communem, ambo anguli D <lb/>A B, A B E ſunt æquales F B E, <lb/>F B E, immo angulo D F E exter-<lb/>no in F B E. </s>
  <s xml:id="echoid-s13451" xml:space="preserve">Et quia C D eſt tan-<lb/>gens circulum, &amp; </s>
  <s xml:id="echoid-s13452" xml:space="preserve">D B ſecans illum, <lb/>angulus C D B æquatur angulo D <lb/>A B, &amp; </s>
  <s xml:id="echoid-s13453" xml:space="preserve">pariter angulus C E F æ-<lb/>quatur angulo E B A, ergo duo an-<lb/>guli C E F, C D F ſimul æquales <lb/>ſunt angulo D F E. </s>
  <s xml:id="echoid-s13454" xml:space="preserve">Et iam quidem <lb/>planum fit ex noſtro tractatu de fi-<lb/>guris quadrilateris, quod ſi educan-
<pb o="404" file="0442" n="443" rhead="Archimedis"/>
tur inter duas lineas æquales ſibi oc-<lb/>
<anchor type="figure" xlink:label="fig-0442-01a" xlink:href="fig-0442-01"/>
currentes in aliquo puncto, vti ſunt <lb/>duæ lineæ C D, C E, duæ lineæ ſe <lb/>mutuo ſecantes, vti ſunt duæ lineæ <lb/>D F, E F, &amp; </s>
  <s xml:id="echoid-s13455" xml:space="preserve">ſuerit angulus ab illis <lb/>contentus vt eſt angulus F æqualis <lb/>duobus angulis, qui occurrunt dua-<lb/>bus [lineis] ſe inuicem ſecanti-<lb/>bus, vti ſunt duo anguli E, D ſimul, <lb/>erit linea egrediens à puncto con-<lb/>curſus ad punctum ſectionis, vti eſt <lb/>linea C F æqualis cuilibet linearum <lb/>ſibi occurrentium, vt C D, vel C <lb/>E, propterea erit C F æqualis ipſi <lb/>C D, ergo angulus C F D eſt æqua-<lb/>lis angulo C D F, nempe angulo <lb/>D A G, ſed angulus C F D cum an-<lb/>gulo D F G eſt æqualis duobus re-<lb/>ctis, ergo angulus D A G cum angulo D F G æqualis eſt duobus rectis, <lb/>&amp; </s>
  <s xml:id="echoid-s13456" xml:space="preserve">remanent in quadrilatero A D F G duo anguli A D F, A G F æqua-<lb/>les duobus rectis, ſed angulus A D B rectus eſt, ergo angulus A G C <lb/>eſt rectus, &amp; </s>
  <s xml:id="echoid-s13457" xml:space="preserve">C G perpendicularis ad A B, &amp; </s>
  <s xml:id="echoid-s13458" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13459" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1156" type="float" level="2" n="1">
  <figure xlink:label="fig-0441-02" xlink:href="fig-0441-02a">
    <image file="0441-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0441-02"/>
  </figure>
  <figure xlink:label="fig-0442-01" xlink:href="fig-0442-01a">
    <image file="0442-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0442-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1158" type="section" level="1" n="376">
<head xml:id="echoid-head468" xml:space="preserve">SCHOLIVM ALMOCHTASSO.</head>
<p>
  <s xml:id="echoid-s13460" xml:space="preserve">DIcit Doctor de demonſtratione, quàm citat ex tractatu <lb/>de figuris quadrilateris. </s>
  <s xml:id="echoid-s13461" xml:space="preserve">Sint duæ lineæ æquales ſibi oc-<lb/>currentes A B, A C, &amp; </s>
  <s xml:id="echoid-s13462" xml:space="preserve">punctum concurſus A, &amp; </s>
  <s xml:id="echoid-s13463" xml:space="preserve">ſe inuicem <lb/>ſecantes B D, D C, &amp; </s>
  <s xml:id="echoid-s13464" xml:space="preserve">punctum ſectionis D, &amp; </s>
  <s xml:id="echoid-s13465" xml:space="preserve">ſit angulus B <lb/>D C æqualis duobus angulis A B D, A C D, &amp; </s>
  <s xml:id="echoid-s13466" xml:space="preserve">iungamus A <lb/>D; </s>
  <s xml:id="echoid-s13467" xml:space="preserve">Dico quod ſit æqualis A B.</s>
  <s xml:id="echoid-s13468" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13469" xml:space="preserve">Alioquin vel eſt minor A B, vel maior <lb/>
<anchor type="figure" xlink:label="fig-0442-02a" xlink:href="fig-0442-02"/>
illa, &amp; </s>
  <s xml:id="echoid-s13470" xml:space="preserve">ſit maior, &amp; </s>
  <s xml:id="echoid-s13471" xml:space="preserve">abſcindatur A E æqua-<lb/>lis A B, &amp; </s>
  <s xml:id="echoid-s13472" xml:space="preserve">iungamus B E, ergo duo anguli <lb/>A E B, A B E ſunt æquales; </s>
  <s xml:id="echoid-s13473" xml:space="preserve">ſed angulus <lb/>A E B maior eſt angulo A D B, &amp; </s>
  <s xml:id="echoid-s13474" xml:space="preserve">pariter <lb/>angulus A E C, qui eſt æqualis A C E ma-<lb/>ior eſt angulo A D C, omnes ergo anguli <lb/>B E C, vel duo anguli ſimul A B E, B C E <lb/>maiores ſunt duobus angulis A B D, A C <lb/>D, pars ſuo toto, quod eſt abſurdum. </s>
  <s xml:id="echoid-s13475" xml:space="preserve">Dein-<lb/>de ſit A D minor quàm A B, &amp; </s>
  <s xml:id="echoid-s13476" xml:space="preserve">ponamus <lb/>A F æqualem A B, &amp; </s>
  <s xml:id="echoid-s13477" xml:space="preserve">iungamus B F, F C, <lb/>remanet, vt dictum eſt, quod angulus F,
<pb o="405" file="0443" n="444" rhead="Aſſump. Liber."/>
immo duo anguli A B F, A C F minores ſint duobus angulis A B D, <lb/>A C D, totum ſua parte, &amp; </s>
  <s xml:id="echoid-s13478" xml:space="preserve">hoc eſt abſurdum, ergo manet propoſitum.</s>
  <s xml:id="echoid-s13479" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1158" type="float" level="2" n="1">
  <figure xlink:label="fig-0442-02" xlink:href="fig-0442-02a">
    <image file="0442-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0442-02"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1160" type="section" level="1" n="377">
<head xml:id="echoid-head469" xml:space="preserve">Notæ in Propoſit. XII.</head>
<p style="it">
  <s xml:id="echoid-s13480" xml:space="preserve">LEmma aſſumptum in demonſtratione huius pulcherrimæ propoſitionis poteſt <lb/>directè oſtendi hac ratione.</s>
  <s xml:id="echoid-s13481" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13482" xml:space="preserve">Si in quadrilatero A C D B duo latera A C, &amp; </s>
  <s xml:id="echoid-s13483" xml:space="preserve">A B æqualia fuerint, atque <lb/>angulus C D B æqualis duobus angulis C, &amp; </s>
  <s xml:id="echoid-s13484" xml:space="preserve">B ſimul ſumptis. </s>
  <s xml:id="echoid-s13485" xml:space="preserve">Dico rectam A <lb/>D ipſi A C, vel A B æqualẽ eſſe. </s>
  <s xml:id="echoid-s13486" xml:space="preserve">Producatur C A, in E, vt A E fiat æqualis <lb/>A B, iungaturque B E. </s>
  <s xml:id="echoid-s13487" xml:space="preserve">Quia in triangulo Iſo-<lb/>
<anchor type="figure" xlink:label="fig-0443-01a" xlink:href="fig-0443-01"/>
ſcelio B A E angulus E æqualis eſt angulo A B <lb/>E, &amp; </s>
  <s xml:id="echoid-s13488" xml:space="preserve">angulus C D B æqualis eſt duobus angulis <lb/>C, &amp; </s>
  <s xml:id="echoid-s13489" xml:space="preserve">D B A ſimul ſumptis, ergo duo anguli C D <lb/>B, &amp; </s>
  <s xml:id="echoid-s13490" xml:space="preserve">E (oppoſiti in quadrilatero C D B E) <lb/>æquales ſunt tribus angulis C, D B A, &amp; </s>
  <s xml:id="echoid-s13491" xml:space="preserve">A B <lb/>E, ſeu duobus angulis C, &amp; </s>
  <s xml:id="echoid-s13492" xml:space="preserve">D B E, ſed qua-<lb/>tuor anguli quadrilateri E C D B æquales ſunt <lb/>quatuor rectis, ergo duo anguli oppoſiti E, C D <lb/>B duobus rectis æquales ſunt, &amp; </s>
  <s xml:id="echoid-s13493" xml:space="preserve">propterea qua-<lb/>drilaterum ipſum circulo inſcribi poteſt, cuius <lb/>circuli centrum erit A, cum tres rectæ lineæ <lb/>C A, A B, A E æquales poſitæ ſint, &amp; </s>
  <s xml:id="echoid-s13494" xml:space="preserve">propte-<lb/>rea A D radius quoque circuli erit æqualis ipſi C A.</s>
  <s xml:id="echoid-s13495" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1160" type="float" level="2" n="1">
  <figure xlink:label="fig-0443-01" xlink:href="fig-0443-01a">
    <image file="0443-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0443-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1162" type="section" level="1" n="378">
<head xml:id="echoid-head470" xml:space="preserve">PROPOSITIO XIII.</head>
<p>
  <s xml:id="echoid-s13496" xml:space="preserve">SI mutuo ſe ſecent duæ lineæ A B, C D in circulo, &amp; </s>
  <s xml:id="echoid-s13497" xml:space="preserve">fue-<lb/>rit A B diameter illius, at non C D, &amp; </s>
  <s xml:id="echoid-s13498" xml:space="preserve">educantur ex duo-<lb/>bus punctis A, B duæ per-<lb/>
<anchor type="figure" xlink:label="fig-0443-02a" xlink:href="fig-0443-02"/>
pendiculares ad C D, quæ <lb/>ſint A E, B F, vtique ab-<lb/>ſcindent ex illa C F, D E <lb/>æquales.</s>
  <s xml:id="echoid-s13499" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1162" type="float" level="2" n="1">
  <figure xlink:label="fig-0443-02" xlink:href="fig-0443-02a">
    <image file="0443-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0443-02"/>
  </figure>
</div>
<p>
  <s xml:id="echoid-s13500" xml:space="preserve">Iungamus E B, &amp; </s>
  <s xml:id="echoid-s13501" xml:space="preserve">educamus <lb/>ex I, quod eſt centrum, per-<lb/>pendicularem I G ſuper C D, <lb/>&amp; </s>
  <s xml:id="echoid-s13502" xml:space="preserve">producamus eam ad H in E <lb/>B. </s>
  <s xml:id="echoid-s13503" xml:space="preserve">Et quia I G eſt perpendicu-<lb/>laris ex centro ad C D illam bi-<lb/>fariam diuidet in G, &amp; </s>
  <s xml:id="echoid-s13504" xml:space="preserve">quia I <lb/>G, A E ſunt duæ perpendicu-<lb/>lares ſuper illam, erunt paral-
<pb o="406" file="0444" n="445" rhead="Archimedis"/>
lelæ, &amp; </s>
  <s xml:id="echoid-s13505" xml:space="preserve">quia B I æqualis eſt I A, erit B H æqualis ipſi H E, &amp; </s>
  <s xml:id="echoid-s13506" xml:space="preserve">pro-<lb/>pter earum æqualitatem, &amp; </s>
  <s xml:id="echoid-s13507" xml:space="preserve">quia B F eſt parallela ipſi H G, erit F G <lb/>æqualis ipſi G E, &amp; </s>
  <s xml:id="echoid-s13508" xml:space="preserve">ex G C, G D æqualibus remanent F C, E D æqua-<lb/>les. </s>
  <s xml:id="echoid-s13509" xml:space="preserve">Et hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13510" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1164" type="section" level="1" n="379">
<head xml:id="echoid-head471" xml:space="preserve">PROPOSITIO XIV.</head>
<p>
  <s xml:id="echoid-s13511" xml:space="preserve">SI fuerit A B ſemicirculus, &amp; </s>
  <s xml:id="echoid-s13512" xml:space="preserve">ex eius diametro A B diſſectæ <lb/>ſint A C, B D æquales, &amp; </s>
  <s xml:id="echoid-s13513" xml:space="preserve">efficiantur ſuper lineas A C, <lb/>C D, D B ſemicirculi; </s>
  <s xml:id="echoid-s13514" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13515" xml:space="preserve">ſit centrum duorum ſemicirculorum <lb/>A B, C D punctum E, &amp; </s>
  <s xml:id="echoid-s13516" xml:space="preserve">ſit E F perpendicularis ſuper A B, <lb/>&amp; </s>
  <s xml:id="echoid-s13517" xml:space="preserve">producatur ad G: </s>
  <s xml:id="echoid-s13518" xml:space="preserve">vtique circulus, cuius diameter eſt F G <lb/>æqualis eſt ſuperficiei contentæ à ſemicirculo maiori, &amp; </s>
  <s xml:id="echoid-s13519" xml:space="preserve">à duo-<lb/>bus ſemicirculis qui ſunt intra illum, &amp; </s>
  <s xml:id="echoid-s13520" xml:space="preserve">à ſemicirculo medio qui <lb/>eſt extra illum, &amp; </s>
  <s xml:id="echoid-s13521" xml:space="preserve">eſt figura, quam vocat Archimedes Salinon.</s>
  <s xml:id="echoid-s13522" xml:space="preserve"/>
</p>
  <figure>
    <image file="0444-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0444-01"/>
  </figure>
<p>
  <s xml:id="echoid-s13523" xml:space="preserve">Quia D C bifariam ſecatur in E, &amp; </s>
  <s xml:id="echoid-s13524" xml:space="preserve">addita eſt illi C A, erunt duo <lb/>quadrata D A, C A dupla duorum quadratorum D E, E A, ſed F G <lb/>æqualis eſt ipſi D A, ergo duo quadrata F G, A C dupla ſunt duorum <lb/>quadratorum D E, E A: </s>
  <s xml:id="echoid-s13525" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13526" xml:space="preserve">quia A B dupla eſt A E, &amp; </s>
  <s xml:id="echoid-s13527" xml:space="preserve">C D dupla. <lb/></s>
  <s xml:id="echoid-s13528" xml:space="preserve">quoque E D, erunt duo quadrata A B, D C quadrupla duorum qua-<lb/>dratorum D E, E A, immo dupla duorum quadratorum G F, A C ſi-<lb/>militer etiam duo circuli, quorum diametri ſunt A B, D C dupli ſunt <lb/>eorum, quorum diametri ſunt G F, A C, &amp; </s>
  <s xml:id="echoid-s13529" xml:space="preserve">dimidij eorum, quorum, <lb/>diametri ſunt A B, C D æquales duobus circulis, quorum diametri ſunt <lb/>G F, A C, ſed circulus, cuius diameter A C, eſt æqualis duobus ſe-
<pb o="407" file="0445" n="446" rhead="Aſſumpt. Liber."/>
micirculis A C, B D, ergo ſi auferamus ex illis duos ſemicirculos A C, <lb/>B D, qui ſunt communes, remanet figura contenta à quatuor ſemicircu-<lb/>lis A B, C D, D B, A C, (quæ ea eſt, quàm vocat Archimedes Sali-<lb/>non) æqualis circulo, cuius diameter eſt F G, &amp; </s>
  <s xml:id="echoid-s13530" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13531" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1165" type="section" level="1" n="380">
<head xml:id="echoid-head472" xml:space="preserve">PROPOSITIO XV.</head>
<p>
  <s xml:id="echoid-s13532" xml:space="preserve">SI fuerit A B ſemicirculus, &amp; </s>
  <s xml:id="echoid-s13533" xml:space="preserve">A C corda Pentagoni, &amp; </s>
  <s xml:id="echoid-s13534" xml:space="preserve">ſe-<lb/>miſſis arcus A C ſit A D, iungatur C D, &amp; </s>
  <s xml:id="echoid-s13535" xml:space="preserve">producatur <lb/>vt cadat ſuper E, &amp; </s>
  <s xml:id="echoid-s13536" xml:space="preserve">iungatur D B, quæ ſecet C A in F, &amp; </s>
  <s xml:id="echoid-s13537" xml:space="preserve"><lb/>ducatur ex F perpendicularis F G ſuper A B, erit linea E G <lb/>æqualis ſemidiametro circuli.</s>
  <s xml:id="echoid-s13538" xml:space="preserve"/>
</p>
  <figure>
    <image file="0445-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0445-01"/>
  </figure>
<p>
  <s xml:id="echoid-s13539" xml:space="preserve">Iungamus itaque lineam C B, &amp; </s>
  <s xml:id="echoid-s13540" xml:space="preserve">ſit centrum H, &amp; </s>
  <s xml:id="echoid-s13541" xml:space="preserve">iungamus H D, <lb/>D G, &amp; </s>
  <s xml:id="echoid-s13542" xml:space="preserve">A D. </s>
  <s xml:id="echoid-s13543" xml:space="preserve">Et quia angulus A B C, cuius baſis eſt latus Pentagoni, <lb/>eſt duæ quintæ partes recti, quilibet duorum angulorum C B D, D B <lb/>A eſt quinta pars recti, &amp; </s>
  <s xml:id="echoid-s13544" xml:space="preserve">angulus D H A duplus eſt anguli D B H, <lb/>ergo angulus D H A eſt duæ quinte partes recti. </s>
  <s xml:id="echoid-s13545" xml:space="preserve">Et quia in duobus trian-<lb/>gulis C B F, G B F duo anguli B ſunt æquales, &amp; </s>
  <s xml:id="echoid-s13546" xml:space="preserve">G, C recti, &amp; </s>
  <s xml:id="echoid-s13547" xml:space="preserve">latus <lb/>F B commune, erit B C æquale ipſi B G: </s>
  <s xml:id="echoid-s13548" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13549" xml:space="preserve">quia in duobus triangulis <lb/>C B D, G B D duo latera C B, B G ſunt æqualia, &amp; </s>
  <s xml:id="echoid-s13550" xml:space="preserve">ſimiliter duo an-<lb/>guli ad B, &amp; </s>
  <s xml:id="echoid-s13551" xml:space="preserve">latus B D commune, erunt duo anguli B C D, B G D <lb/>æquales, &amp; </s>
  <s xml:id="echoid-s13552" xml:space="preserve">quilibet eorum eſt ſex quintæ partes recti, &amp; </s>
  <s xml:id="echoid-s13553" xml:space="preserve">eſt æqualis an-<lb/>gulo D A E externo quadrilateri B A D C, quod eſt in circulo, ergo <lb/>remanet angulus D A B æqualis angulo D G A, &amp; </s>
  <s xml:id="echoid-s13554" xml:space="preserve">erit D A æqualis ip-<lb/>ſi D G. </s>
  <s xml:id="echoid-s13555" xml:space="preserve">Et quia angulus D H G eſt duæ quintæ partes recti, &amp; </s>
  <s xml:id="echoid-s13556" xml:space="preserve">angulus <lb/>D G H ſex quintæ partes recti, remanet angulus H D G duæ quintæ par-<lb/>tes recti, &amp; </s>
  <s xml:id="echoid-s13557" xml:space="preserve">erit D G æqualis G H. </s>
  <s xml:id="echoid-s13558" xml:space="preserve">Et quia A D E externus quadrila-<lb/>teri A D C B, quod eſt in circulo, eſt æqualis angulo C B A, &amp; </s>
  <s xml:id="echoid-s13559" xml:space="preserve">eſt
<pb o="408" file="0446" n="447" rhead="Archimedis"/>
duæ quintæ partes recti, &amp; </s>
  <s xml:id="echoid-s13560" xml:space="preserve">æqualis angulo G D H. </s>
  <s xml:id="echoid-s13561" xml:space="preserve">Et quia in duobus <lb/>triangulis E D A, H D G ſunt duo anguli E D A, H D G æquales, &amp; </s>
  <s xml:id="echoid-s13562" xml:space="preserve"><lb/>pariter duo anguli D G H, D A E, &amp; </s>
  <s xml:id="echoid-s13563" xml:space="preserve">duo latera D A, D G, erit E A <lb/>æquale H G, &amp; </s>
  <s xml:id="echoid-s13564" xml:space="preserve">ponamus A G commune, erit E G æquale A H, &amp; </s>
  <s xml:id="echoid-s13565" xml:space="preserve"><lb/>hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13566" xml:space="preserve"/>
</p>
  <figure>
    <image file="0446-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0446-01"/>
  </figure>
<p>
  <s xml:id="echoid-s13567" xml:space="preserve">Et hinc patet, quod linea D E æqualis ſit ſemidiametro circuli, quia <lb/>angulus A æqualis eſt angulo D G H, ideo erit linea D H æqualis li-<lb/>neæ D E. </s>
  <s xml:id="echoid-s13568" xml:space="preserve">Et dico quod E C diuiditur media, &amp; </s>
  <s xml:id="echoid-s13569" xml:space="preserve">extrema proportione <lb/>in D, &amp; </s>
  <s xml:id="echoid-s13570" xml:space="preserve">maius ſegmentum eſt D E; </s>
  <s xml:id="echoid-s13571" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13572" xml:space="preserve">hoc quia E D eſt corda hexago-<lb/>ni, &amp; </s>
  <s xml:id="echoid-s13573" xml:space="preserve">D C decagoni, &amp; </s>
  <s xml:id="echoid-s13574" xml:space="preserve">hoc iam demonſtratum eſt in libro elemento-<lb/>rum, &amp; </s>
  <s xml:id="echoid-s13575" xml:space="preserve">hoc eſt quod voluimus.</s>
  <s xml:id="echoid-s13576" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Impie vt <lb/>Mahume-<lb/>tanus Para <lb/>phr<unsure/>aſtes <lb/>loquit<unsure/>ur.</note>
<p>
  <s xml:id="echoid-s13577" xml:space="preserve">Finis libri Aſſumptorum Archimedis. </s>
  <s xml:id="echoid-s13578" xml:space="preserve">Laus Deo ſoli, &amp; </s>
  <s xml:id="echoid-s13579" xml:space="preserve">orationes eius <lb/>ſint ſuper Dominum noſtrum Mahometum, &amp; </s>
  <s xml:id="echoid-s13580" xml:space="preserve">ſuos ſocios.</s>
  <s xml:id="echoid-s13581" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1166" type="section" level="1" n="381">
<head xml:id="echoid-head473" xml:space="preserve">Notæ in Propoſit. XV.</head>
<p style="it">
  <s xml:id="echoid-s13582" xml:space="preserve">EX hac propoſitione non pauca colligi poſſunt; </s>
  <s xml:id="echoid-s13583" xml:space="preserve">Si enim coniungantur rectæ <lb/>lineæ C H, &amp; </s>
  <s xml:id="echoid-s13584" xml:space="preserve">C G, erit triangulum B C E iſoſcelium ſimile triangulo <lb/>H D E, &amp; </s>
  <s xml:id="echoid-s13585" xml:space="preserve">ſimiliter poſitum; </s>
  <s xml:id="echoid-s13586" xml:space="preserve">pariterque triangulum H C G ſimile quidem <lb/>erit ipſi G D A, &amp; </s>
  <s xml:id="echoid-s13587" xml:space="preserve">in vtriſque baſes ſimiliter ſecantur, nam angulus B C E <lb/>in tres partes æquales diuiditur à rectis lineis H C, &amp; </s>
  <s xml:id="echoid-s13588" xml:space="preserve">G C, quarum quæli-<lb/>bet duæ quintæ partes eſt vnius recti, atque angulus E C G rurſus bifariam <lb/>diuiditur à recta C A: </s>
  <s xml:id="echoid-s13589" xml:space="preserve">non ſecus tres anguli E D A, A D G, &amp; </s>
  <s xml:id="echoid-s13590" xml:space="preserve">G D H <lb/>æquales ſunt inter ſe, atque quilibet eorum duæ quintæ vnius recti. </s>
  <s xml:id="echoid-s13591" xml:space="preserve">Et effi-<lb/>ciuntur quatuor rectæ lineæ E A, A D, D G, D C, inter ſe, &amp; </s>
  <s xml:id="echoid-s13592" xml:space="preserve">lateri de-<lb/>cagoni regularis circulo inſcripti æquales. </s>
  <s xml:id="echoid-s13593" xml:space="preserve">Pari modo rectæ lineæ E D, E G, <lb/>G C, H C, H A, æquales ſunt inter ſe, &amp; </s>
  <s xml:id="echoid-s13594" xml:space="preserve">lateri hexagoni regularis circulo <lb/>inſcripti. </s>
  <s xml:id="echoid-s13595" xml:space="preserve">Tandem recta linea C B ſubtendens tres partes decimas circumfe-<lb/>rentiæ totius circuli æqualis eſt rectæ lineæ C E, ſcilicet compoſitæ ex latere <lb/>hexagoni, &amp; </s>
  <s xml:id="echoid-s13596" xml:space="preserve">latere decagoni regularium eidem circulo incſriptorum. </s>
  <s xml:id="echoid-s13597" xml:space="preserve">Præterea
<pb o="409" file="0447" n="448" rhead="Aſſumpt. Liber."/>
recta linea E G ſecatur in A extrema, ac media ratione, cuius maius ſegmen-<lb/>tum eſt E A latus decagoni, &amp; </s>
  <s xml:id="echoid-s13598" xml:space="preserve">recta A H ſimiliter diuiditur in G, cuius ma-<lb/>ius ſegmentum eſt G H decagoni latus, &amp; </s>
  <s xml:id="echoid-s13599" xml:space="preserve">tota E H ſecatur in A, &amp; </s>
  <s xml:id="echoid-s13600" xml:space="preserve">G ex-<lb/>trema, ac media ratione, pariterque recta E B ſimiliter ſecatur in H, cuius <lb/>
<anchor type="figure" xlink:label="fig-0447-01a" xlink:href="fig-0447-01"/>
minus ſegmentum H B eſt æquale lateri exagoni circulo inſcripti. </s>
  <s xml:id="echoid-s13601" xml:space="preserve">Breuius ta-<lb/>men propoſitio ſic demonſtrari poſſet.</s>
  <s xml:id="echoid-s13602" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1166" type="float" level="2" n="1">
  <figure xlink:label="fig-0447-01" xlink:href="fig-0447-01a">
    <image file="0447-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0447-01"/>
  </figure>
</div>
<p style="it">
  <s xml:id="echoid-s13603" xml:space="preserve">Quia oſtenſa eſt C D æqualis D G, &amp; </s>
  <s xml:id="echoid-s13604" xml:space="preserve">A D æqualis eſt eidem D C; </s>
  <s xml:id="echoid-s13605" xml:space="preserve">cum <lb/>ambo ſint latera decagoni, ergo D G æqualis eſt D A. </s>
  <s xml:id="echoid-s13606" xml:space="preserve">Poſtea iuncta A C, quid <lb/>angulus A H D, vel C H D quinta pars eſt duorum rectorum, ergo angulus <lb/>C D H ad baſim iſoſcelĳ, duæ quintæ partes erit duorum rectorum, &amp; </s>
  <s xml:id="echoid-s13607" xml:space="preserve">ideo an-<lb/>gulus C D H duplus erit anguli D H E, eſtque externus angulus C D H æqua-<lb/>lis duobus internis, &amp; </s>
  <s xml:id="echoid-s13608" xml:space="preserve">oppoſitis D H E, &amp; </s>
  <s xml:id="echoid-s13609" xml:space="preserve">D E H in triangulo D E H, ergo <lb/>angulus C D H duplus quoque erit reliqui anguli E, &amp; </s>
  <s xml:id="echoid-s13610" xml:space="preserve">propterea angulus D <lb/>H E æqualis erit angulo E, &amp; </s>
  <s xml:id="echoid-s13611" xml:space="preserve">ſubtenſa latera D E, D H æqualia quoque erunt, <lb/>ſed prius D A, D G æqualia erant ſubtendentia angulos æquales, &amp; </s>
  <s xml:id="echoid-s13612" xml:space="preserve">reliqui <lb/>anguli eiuſdem ſpeciei ſunt, igitur E A æqualis eſt H G. </s>
  <s xml:id="echoid-s13613" xml:space="preserve">Reliqua manifeſta <lb/>ſunt.</s>
  <s xml:id="echoid-s13614" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13615" xml:space="preserve">In præfatione huius operis memini non eße omnino improbabile hunc libellum <lb/>Archimedis non alium fuiſſe ab illo antiquo lemmatum libro ab Eutocio reper-<lb/>to, quod præcipuè ex verbis eiuſdem Eutocĳ in Comment. </s>
  <s xml:id="echoid-s13616" xml:space="preserve">propoſit. </s>
  <s xml:id="echoid-s13617" xml:space="preserve">4. </s>
  <s xml:id="echoid-s13618" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13619" xml:space="preserve">2. </s>
  <s xml:id="echoid-s13620" xml:space="preserve">de <lb/>Sphæra, &amp; </s>
  <s xml:id="echoid-s13621" xml:space="preserve">Cylindro comprobatum fuit: </s>
  <s xml:id="echoid-s13622" xml:space="preserve">illa fideliſſimè translata ex textu Græco ab <lb/>amicis doctiſſimis cum iam in præfatione excuſa eßent aliam tranſlationem ex <lb/>Arabico Manuſcripto Sereniſſimi Magni Ducis miſit Excell. </s>
  <s xml:id="echoid-s13623" xml:space="preserve">Abrahamus Ecchel-<lb/>lenſis deſumptam ex editione Abuſahli Alkuhi qui pariter librum ordinatio-<lb/>nis lemmatum Archimedis conſcripſit, vt in proemio huius operis teſtatur <lb/>Almochtaſſo. </s>
  <s xml:id="echoid-s13624" xml:space="preserve">Verba eius ſunt hæc, quæ paulo clarius propoſitum confirmare vi-<lb/>dentur: </s>
  <s xml:id="echoid-s13625" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13626" xml:space="preserve">meminit Eutocius Aſcalonita in Comment. </s>
  <s xml:id="echoid-s13627" xml:space="preserve">huius libri, quod <lb/>Archimedes promiſerit demonſtrationem huius in hoc ſuo libro, quod <lb/>in nullo exemplari reperitur, quod promiſit. </s>
  <s xml:id="echoid-s13628" xml:space="preserve">Atque ita vnuſquiſque tam <lb/>Dyoniſodorus, quàm Diocles poſt illum progreſſus eſt per aliam viam, <lb/>quàm ille (ſcilicet Archimedes) in hoc libro in diuiſione Sphæræ in <lb/>duas partes, quæ datam habeant proportionem. </s>
  <s xml:id="echoid-s13629" xml:space="preserve">Dixit, &amp; </s>
  <s xml:id="echoid-s13630" xml:space="preserve">ego reperi in
<pb o="410" file="0448" n="449" rhead="Archimedis"/>
Veteri Libro Theoremata ſatis obſcura propter multitudinem errorum, <lb/>qui in eo ſunt, nec non menda, quæ occurrunt in figuris propter igno-<lb/>rantiam amanuenſium, erantque in co Doricæ dictiones, quarum vſus <lb/>Archimedi familiaris erat, &amp; </s>
  <s xml:id="echoid-s13631" xml:space="preserve">vocabula ipſi propria; </s>
  <s xml:id="echoid-s13632" xml:space="preserve">hinc vtebatur loco <lb/>ſectionum parabolæ, &amp; </s>
  <s xml:id="echoid-s13633" xml:space="preserve">hyperbolæ, rectanguli, &amp; </s>
  <s xml:id="echoid-s13634" xml:space="preserve">obtuſanguli coni ſe-<lb/>ctionibus quamobrem operam ipſi nauaui, donec aſſecutus ſum iſtam <lb/>propoſitionem, &amp; </s>
  <s xml:id="echoid-s13635" xml:space="preserve">eſt iſta, &amp;</s>
  <s xml:id="echoid-s13636" xml:space="preserve">c.</s>
  <s xml:id="echoid-s13637" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13638" xml:space="preserve">Modo quia in prædicto libro antiquo ab Eutocio reperto recenſentur duæ pro-<lb/>poſitiones, quarum vnam promiſerat ſe demonſtraturum Archimedes, &amp; </s>
  <s xml:id="echoid-s13639" xml:space="preserve">vtra-<lb/>que in noſtro opuſculo iniuria temporum deficit: </s>
  <s xml:id="echoid-s13640" xml:space="preserve">earum altera forſan erit 16. <lb/></s>
  <s xml:id="echoid-s13641" xml:space="preserve">illa propoſitio in proemio ab Almochtaßo numerata vbi ait propoſitiones huius <lb/>opuſculi ſexdecim eſſe, cum tamen poſtrema ſit 15. </s>
  <s xml:id="echoid-s13642" xml:space="preserve">quare inutile forſan non <lb/>erit eas hic reponere, præcipuè quia Eutocius non rite eas reſtituit, nec omninò <lb/>repurgauit à mendis, quibus ſcatebat exemplar antiquum ab ipſo inuentum. </s>
  <s xml:id="echoid-s13643" xml:space="preserve">Et <lb/>primo noto, quod Eutocius eas vocat theoremata, cum potius problemata ſint, &amp; </s>
  <s xml:id="echoid-s13644" xml:space="preserve"><lb/>ſic etiam ab eodem Eutocio poſtmodum appellantur. </s>
  <s xml:id="echoid-s13645" xml:space="preserve">Forſan hoc accidit, quia <lb/>in libro illo antiquo in formam theorematum ſcripta erant, ſed Eutocius vt ad <lb/>propoſitionem Archimedis ea accomodaret, forma problematica ea expoſuit. </s>
  <s xml:id="echoid-s13646" xml:space="preserve"><lb/>Rurſus Eutocius primum theorema ſe expoſiturum pollicetur, vt deinde analyſi <lb/>problematis Archimedei accomodetur. </s>
  <s xml:id="echoid-s13647" xml:space="preserve">Vnde conĳcere licet alterum theorema <lb/>additum, vel alteratum ab Eutocio, vel ab aliquo alio fuiſſe, in quo proponit, <lb/>quod, ſi aliqua recta linea ſecta ſit in duo ſegmenta, quorum vnum duplum <lb/>ſit alterius, ſolidum parallelepipedum rectangulum contentum ſub quadrato ma-<lb/>ioris, &amp; </s>
  <s xml:id="echoid-s13648" xml:space="preserve">ſub minore ſegmento maximum erit omnium ſimilium ſolidorum, quæ <lb/>ex diuiſione eiuſdem rectæ lineæ in quolibet alio eius puncto conſurgunt. </s>
  <s xml:id="echoid-s13649" xml:space="preserve">Et <lb/>hoc quidem oſtenditur per ſectiones conicas, contra artis præcepta; </s>
  <s xml:id="echoid-s13650" xml:space="preserve">peccatum <lb/>enim eſt non paruum apud Geometras, problema planum per conicas ſectiones <lb/>reſoluere cum via plana abſolui poſſit, hoc autem preclari nonnulli viri pariter <lb/>adnotarunt, &amp; </s>
  <s xml:id="echoid-s13651" xml:space="preserve">præſtiterunt, vt nuper accepi.</s>
  <s xml:id="echoid-s13652" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1168" type="section" level="1" n="382">
<head xml:id="echoid-head474" xml:space="preserve">PROPOSITIO XVI.</head>
<p>
  <s xml:id="echoid-s13653" xml:space="preserve">SI recta linea A B ſit tripla A C, non vero tripla ipſius A <lb/>D; </s>
  <s xml:id="echoid-s13654" xml:space="preserve">Dico parallelepipedum rectangulũ contentum ſub qua-<lb/>drato C B in A C maius eſſe parallelepipedo ſub quadrato D <lb/>B in A D.</s>
  <s xml:id="echoid-s13655" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13656" xml:space="preserve">Producatur A B in E, vt ſit B E æqualis B C. </s>
  <s xml:id="echoid-s13657" xml:space="preserve">Quoniam B C dupla <lb/>erat ipſius A C, erit E C quadrupla ipſius A C, &amp; </s>
  <s xml:id="echoid-s13658" xml:space="preserve">propterea rectan-<lb/>gulum A C E æquale erit quadruplo quadrati A C, ſcilicet æquale erit <lb/>quadrato C B: </s>
  <s xml:id="echoid-s13659" xml:space="preserve">Eſt vero in primo caſu, rectangulum A D E maius re-<lb/>ctangulo A C E, in ſecundo vero minus, (eo quod punctum D in pri-<lb/>mo caſu propinquius eſt ſemipartitioni totius A E, quàm C, in ſecuudo <lb/>verò remotius); </s>
  <s xml:id="echoid-s13660" xml:space="preserve">igitur ſi fiat C D ad D O, vt quadratum C B ad rectan-
<pb o="411" file="0449" n="450" rhead="Aſſumpt. Liber."/>
gulum A D E, erit in primo caſu D O maior, quàm C D, in ſecundo <lb/>vero minor; </s>
  <s xml:id="echoid-s13661" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13662" xml:space="preserve">propterea A O minor erit, quàm A C in vtroque caſu. <lb/></s>
  <s xml:id="echoid-s13663" xml:space="preserve">Et quia quadratum C B ad rectangulum A D E eſt vt C D ad D O, igi-<lb/>tur ſolida parallelepipeda reciproca erunt æqualia, ſcilicet ſolidum qua-<lb/>
<anchor type="figure" xlink:label="fig-0449-01a" xlink:href="fig-0449-01"/>
drato C B in D O ducto æquale erit ſolido, cuius baſis rectangulum A <lb/>D E, altitudo vero C D, ſeu potius æquale erit ſolido, cuius baſis re-<lb/>ctangulum E D C, altitudo vero A D, &amp; </s>
  <s xml:id="echoid-s13664" xml:space="preserve">propterea vt quadratum B C <lb/>ad rectangulum E D C, ita erit reciproce A D ad D O, &amp; </s>
  <s xml:id="echoid-s13665" xml:space="preserve">comparando <lb/>antecedentes ad terminorum differentias in primo caſu, &amp; </s>
  <s xml:id="echoid-s13666" xml:space="preserve">ad eorundem <lb/>ſummas in ſecundo caſu, erit quadratum B C ad quadratum D B vt A <lb/>D ad A O, &amp; </s>
  <s xml:id="echoid-s13667" xml:space="preserve">denuo ſolidum parallelepipedum rectangulum contentum <lb/>ſub quadrato B C in A O æquale erit ei, cuius baſis quadratum D B, <lb/>altitudo vero A D: </s>
  <s xml:id="echoid-s13668" xml:space="preserve">Eſt vero A O oſtenſa minor, quàm A C in vtroque <lb/>caſu, igitur parallelepipedum, cuius baſis quadratum B C, altitudo A <lb/>C maius eſt eo, cuius baſis eſt idem quadratum B C, altitudo A O; <lb/></s>
  <s xml:id="echoid-s13669" xml:space="preserve">ideoque parallelepipedum, cuius baſis quadratum B C, altitudo A C <lb/>maius eſt quolibet parallelepipedo, cuius baſis quadratum B D, altitudo <lb/>A D: </s>
  <s xml:id="echoid-s13670" xml:space="preserve">quare patet propoſitum.</s>
  <s xml:id="echoid-s13671" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1168" type="float" level="2" n="1">
  <figure xlink:label="fig-0449-01" xlink:href="fig-0449-01a">
    <image file="0449-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0449-01"/>
  </figure>
</div>
</div>
<div xml:id="echoid-div1170" type="section" level="1" n="383">
<head xml:id="echoid-head475" xml:space="preserve">PROPOSITIO XVII.</head>
<p>
  <s xml:id="echoid-s13672" xml:space="preserve">SIt A B tripla ipſius A E, maior vero quàm tripla alterius C A, <lb/>ſecari debet eadem A B citra, &amp; </s>
  <s xml:id="echoid-s13673" xml:space="preserve">vltra E, in O, ita vt <lb/>parallelepipedum, cuius baſis quadratum O B, altitudo O A <lb/>æquale ſit parallelepipedo, cuius baſis quadratum E B, altitu-<lb/>do A C.</s>
  <s xml:id="echoid-s13674" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13675" xml:space="preserve">Fiat rectangulum A C B F, &amp; </s>
  <s xml:id="echoid-s13676" xml:space="preserve">producantur latera C A, F B, &amp; </s>
  <s xml:id="echoid-s13677" xml:space="preserve">fiat <lb/>rectangulum C F N æquale quadrato E B, &amp; </s>
  <s xml:id="echoid-s13678" xml:space="preserve">ducta diametro C E G com-
<pb o="412" file="0450" n="451" rhead="Archimedis"/>
pleantur parallelogramma rectangula A L, A K, L B, B K, atque axe <lb/>
<anchor type="note" xlink:label="note-0450-01a" xlink:href="note-0450-01"/>
F G, latere recto F N deſcribatur parabole F M ſecans H G in M; </s>
  <s xml:id="echoid-s13679" xml:space="preserve">erit <lb/>igitur in parabola quadratum M G æquale rectangulo G F N ſub abſciſ-<lb/>
<anchor type="note" xlink:label="note-0450-02a" xlink:href="note-0450-02"/>
ſa, &amp; </s>
  <s xml:id="echoid-s13680" xml:space="preserve">latere recto contento, ideoque idem quadratum F G ad rectangu-<lb/>lum N F G, atque ad quadratum M G eandem proportionem habebit: <lb/></s>
  <s xml:id="echoid-s13681" xml:space="preserve">eſt vero quadratum F G ad rectangulum N F G, vt F G ad F N, cum <lb/>
<anchor type="figure" xlink:label="fig-0450-01a" xlink:href="fig-0450-01"/>
F G ſit illorum altitudo communis, nec non vt C F G ad C F N ſum-<lb/>pta nimirum C F communi altitudine, ergo rectangulum C F G ad C <lb/>F N eandem proportionem habebit, quam quadratum F G ad quadra-<lb/>tum M G, &amp; </s>
  <s xml:id="echoid-s13682" xml:space="preserve">permutando rectangulum C F G ad quadratum F G erit <lb/>vt rectangulum C F N ad quadratum G M, ſed vt rectangulum C F G <lb/>ad quadratum F G, ita eſt C F ad F G, &amp; </s>
  <s xml:id="echoid-s13683" xml:space="preserve">E A ad A C, igitur E A ad <lb/>A C erit vt rectangulum C F N ad quadratum G M, ſeu vt quadratum <lb/>E B, vel K G ad quadratum G M: </s>
  <s xml:id="echoid-s13684" xml:space="preserve">eſt vero A C minor, quàm A E, <lb/>quæ triens eſt totius A B, igitur M G minor eſt, quàm G K. </s>
  <s xml:id="echoid-s13685" xml:space="preserve">Poſtea <lb/>per B circa aſymptotos A C F deſcribatur hyperbole B K, quæ tran-<lb/>
<anchor type="note" xlink:label="note-0450-03a" xlink:href="note-0450-03"/>
ſibit per punctum K, cum parallelogramma A F, &amp; </s>
  <s xml:id="echoid-s13686" xml:space="preserve">C K æqualia <lb/>ſint propter diagonalem C E G, quare punctum M paraboles cadet <lb/>intra hyperbolem B K, ſed parabole F M occurrit aſymptoto C F in ver-<lb/>tice F, &amp; </s>
  <s xml:id="echoid-s13687" xml:space="preserve">occurrit etiam aſymptoto C A in aliquo alio puncto, cum C <lb/>A ſit parallela axi F G paraboles, &amp; </s>
  <s xml:id="echoid-s13688" xml:space="preserve">hyperbole ſemper intra aſymptotos <lb/>
<anchor type="note" xlink:label="note-0450-04a" xlink:href="note-0450-04"/>
incedat, igitur parabola F M bis hyperbolæ occurrit ſupra, &amp; </s>
  <s xml:id="echoid-s13689" xml:space="preserve">inſra pun-<lb/>
<anchor type="note" xlink:label="note-0450-05a" xlink:href="note-0450-05"/>
ctum M: </s>
  <s xml:id="echoid-s13690" xml:space="preserve">ſint occurſus X, à quibus ductis parallelis ad aſymptotos com-<lb/>pleantur parallelogramma R P, &amp; </s>
  <s xml:id="echoid-s13691" xml:space="preserve">A F, quæ erunt æqualia inter aſym-<lb/>ptotos, &amp; </s>
  <s xml:id="echoid-s13692" xml:space="preserve">hyperbolen conſtituta, &amp; </s>
  <s xml:id="echoid-s13693" xml:space="preserve">propterea C O S parallelogrammo-<lb/>
<anchor type="note" xlink:label="note-0450-06a" xlink:href="note-0450-06"/>
rum diameter erit, &amp; </s>
  <s xml:id="echoid-s13694" xml:space="preserve">vna linca recta: </s>
  <s xml:id="echoid-s13695" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13696" xml:space="preserve">quia O A ad A C eſt vt C F <lb/>ad F S, ſiue vt rectangulum C F N ad rectangulum S F N: </s>
  <s xml:id="echoid-s13697" xml:space="preserve">erat autem <lb/>quadratum E B æquale rectangulo C F N ex conſtructione, &amp; </s>
  <s xml:id="echoid-s13698" xml:space="preserve">quadra-
<pb o="413" file="0451" n="452" rhead="Aſſumpt. Liber."/>
tũ O B, ſiue X S in parabola <lb/>
<anchor type="figure" xlink:label="fig-0451-01a" xlink:href="fig-0451-01"/>
<anchor type="note" xlink:label="note-0451-01a" xlink:href="note-0451-01"/>
æquale eſt rectangulo S FN, <lb/>ergo AO ad A C eſt vt qua-<lb/>dratum E B ad quadratum <lb/>O B, &amp; </s>
  <s xml:id="echoid-s13699" xml:space="preserve">propterea parallele-<lb/>pipedum, cuius baſis quadra-<lb/>tum O B, altitudo O A æ-<lb/>quale erit parallelepipedo ba-<lb/>ſe quadrato E B, altitudine <lb/>A C contento, quod erat <lb/>propoſitum.</s>
  <s xml:id="echoid-s13700" xml:space="preserve"/>
</p>
<div xml:id="echoid-div1170" type="float" level="2" n="1">
<note position="left" xlink:label="note-0450-01" xlink:href="note-0450-01a" xml:space="preserve">Prop. 52. <lb/>lib. 1.</note>
<note position="left" xlink:label="note-0450-02" xlink:href="note-0450-02a" xml:space="preserve">Prop. 11. <lb/>lib. 1.</note>
  <figure xlink:label="fig-0450-01" xlink:href="fig-0450-01a">
    <image file="0450-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0450-01"/>
  </figure>
<note position="left" xlink:label="note-0450-03" xlink:href="note-0450-03a" xml:space="preserve">Prop. 4. &amp; <lb/>12. lib. 2.</note>
<note position="left" xlink:label="note-0450-04" xlink:href="note-0450-04a" xml:space="preserve">Prop. 26. <lb/>lib. 1.</note>
<note position="left" xlink:label="note-0450-05" xlink:href="note-0450-05a" xml:space="preserve">ex 1. &amp; 2. <lb/>lib. 2.</note>
<note position="left" xlink:label="note-0450-06" xlink:href="note-0450-06a" xml:space="preserve">Prop. 12. <lb/>lib. 2.</note>
  <figure xlink:label="fig-0451-01" xlink:href="fig-0451-01a">
    <image file="0451-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0451-01"/>
  </figure>
<note position="right" xlink:label="note-0451-01" xlink:href="note-0451-01a" xml:space="preserve">Prop. 11. <lb/>lib. 1.</note>
</div>
<p style="it">
  <s xml:id="echoid-s13701" xml:space="preserve">Ex hiſce propoſitionibus de-<lb/>ducit inſuper Eutocius aliqua, <lb/>quæ non omnino firma, &amp; </s>
  <s xml:id="echoid-s13702" xml:space="preserve">cer-<lb/>ta mihi videntur, nam ex eo <lb/>quod recta linea vt I X tangit <lb/>vtramq; </s>
  <s xml:id="echoid-s13703" xml:space="preserve">coniſectionem (hyper-<lb/>bolen ſcilicet B X, &amp; </s>
  <s xml:id="echoid-s13704" xml:space="preserve">parabo-<lb/>len F X) in eodem puncto X <lb/>concludit hyperbolen interius <lb/>contingere parabolen quàm de-<lb/>inceps non ſecat ad eaſdem par-<lb/>tes axis illius. </s>
  <s xml:id="echoid-s13705" xml:space="preserve">Hoc autem omnino <lb/>neceſſarium nõ<unsure/> eſt ex demonſtra-<lb/>tis à me in prop. </s>
  <s xml:id="echoid-s13706" xml:space="preserve">20. </s>
  <s xml:id="echoid-s13707" xml:space="preserve">21. </s>
  <s xml:id="echoid-s13708" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13709" xml:space="preserve"><lb/>22. </s>
  <s xml:id="echoid-s13710" xml:space="preserve">Adàit. </s>
  <s xml:id="echoid-s13711" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13712" xml:space="preserve">6. </s>
  <s xml:id="echoid-s13713" xml:space="preserve">Apoll. </s>
  <s xml:id="echoid-s13714" xml:space="preserve">fieri <lb/>enim poteſt vt Parabole exte-<lb/>rius hyperbolen tangat in X, &amp; </s>
  <s xml:id="echoid-s13715" xml:space="preserve"><lb/>poſtea hinc inde eam ſecet. </s>
  <s xml:id="echoid-s13716" xml:space="preserve">Poteſt inſuper hyperbole ſecare eandem parabolam <lb/>in eodem puncto X, licet ambo in eodem puncto tangantur ab aliqua recta li-<lb/>nea, vt eſt I X; </s>
  <s xml:id="echoid-s13717" xml:space="preserve">quod quidem adnotaſſe fuit operepretium.</s>
  <s xml:id="echoid-s13718" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1172" type="section" level="1" n="384">
<head xml:id="echoid-head476" xml:space="preserve">FINIS.</head>
<pb o="414" file="0452" n="453"/>
<p>
  <s xml:id="echoid-s13719" xml:space="preserve">Dominus Carolus de Datis videat, &amp; </s>
  <s xml:id="echoid-s13720" xml:space="preserve">referat an in hoc opere ſit aliquid quod repugnet <lb/># fidei Catholicæ, &amp; </s>
  <s xml:id="echoid-s13721" xml:space="preserve">bonis moribus. </s>
  <s xml:id="echoid-s13722" xml:space="preserve">Die 3. </s>
  <s xml:id="echoid-s13723" xml:space="preserve">Iulij 1660.</s>
  <s xml:id="echoid-s13724" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13725" xml:space="preserve">Vinc. </s>
  <s xml:id="echoid-s13726" xml:space="preserve">de Bardis Vicar. </s>
  <s xml:id="echoid-s13727" xml:space="preserve">Gener. </s>
  <s xml:id="echoid-s13728" xml:space="preserve">Florent.</s>
  <s xml:id="echoid-s13729" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1173" type="section" level="1" n="385">
<head xml:id="echoid-head477" xml:space="preserve">Illuſtriſſime, ac Reuerendiſs, Dom.</head>
<p>
  <s xml:id="echoid-s13730" xml:space="preserve">Vidi hæc antiquorum, maximorumq; </s>
  <s xml:id="echoid-s13731" xml:space="preserve">Geometrarum Apollonij, atq; </s>
  <s xml:id="echoid-s13732" xml:space="preserve">Archimedis Ope-<lb/># ra nunquam edita, nec in ijs reperi aliquid fidei Catholicæ, &amp; </s>
  <s xml:id="echoid-s13733" xml:space="preserve">bonis moribus aduer-<lb/># ſum; </s>
  <s xml:id="echoid-s13734" xml:space="preserve">Quamobrem maximo Reip. </s>
  <s xml:id="echoid-s13735" xml:space="preserve">literariæ bono, &amp; </s>
  <s xml:id="echoid-s13736" xml:space="preserve">gloria eorum qui in ijs vertendis, <lb/># atq; </s>
  <s xml:id="echoid-s13737" xml:space="preserve">illuſtrandis ſtudium, atque operam feliciſsimè collocarunt euulganda cenſeo: <lb/></s>
  <s xml:id="echoid-s13738" xml:space="preserve"># dummodo quædam loca notentur Arabicorum interpretum, quibus Maumedanos <lb/># ſe præbent. </s>
  <s xml:id="echoid-s13739" xml:space="preserve">Florentiæ die 7. </s>
  <s xml:id="echoid-s13740" xml:space="preserve">Iulij 1660.</s>
  <s xml:id="echoid-s13741" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13742" xml:space="preserve">Carolus Dati manupropria.</s>
  <s xml:id="echoid-s13743" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13744" xml:space="preserve">Imprimatur ſeruatis ſeruandis 7. </s>
  <s xml:id="echoid-s13745" xml:space="preserve">Iulij 1660.</s>
  <s xml:id="echoid-s13746" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13747" xml:space="preserve">Vinc. </s>
  <s xml:id="echoid-s13748" xml:space="preserve">d. </s>
  <s xml:id="echoid-s13749" xml:space="preserve">Bardis Vicar. </s>
  <s xml:id="echoid-s13750" xml:space="preserve">Gener. </s>
  <s xml:id="echoid-s13751" xml:space="preserve">Flor.</s>
  <s xml:id="echoid-s13752" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13753" xml:space="preserve">Excellentiſs. </s>
  <s xml:id="echoid-s13754" xml:space="preserve">Aduocatus Dominus Auguſtinus Coltellini S. </s>
  <s xml:id="echoid-s13755" xml:space="preserve">Offic. </s>
  <s xml:id="echoid-s13756" xml:space="preserve">Florentiæ Conſultor <lb/># videat hoc opus intitulatum APOLLONII PERGÆI, &amp;</s>
  <s xml:id="echoid-s13757" xml:space="preserve">c. </s>
  <s xml:id="echoid-s13758" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13759" xml:space="preserve">referat. <lb/></s>
  <s xml:id="echoid-s13760" xml:space="preserve"># Die 7. </s>
  <s xml:id="echoid-s13761" xml:space="preserve">Iulij 1660.</s>
  <s xml:id="echoid-s13762" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13763" xml:space="preserve">Fr. </s>
  <s xml:id="echoid-s13764" xml:space="preserve">Ang. </s>
  <s xml:id="echoid-s13765" xml:space="preserve">Octau. </s>
  <s xml:id="echoid-s13766" xml:space="preserve">de Populo S. </s>
  <s xml:id="echoid-s13767" xml:space="preserve">Offic. </s>
  <s xml:id="echoid-s13768" xml:space="preserve">Flor. </s>
  <s xml:id="echoid-s13769" xml:space="preserve">Canc. </s>
  <s xml:id="echoid-s13770" xml:space="preserve">de mand.</s>
  <s xml:id="echoid-s13771" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1174" type="section" level="1" n="386">
<head xml:id="echoid-head478" xml:space="preserve">Reuerendiſs. Pater Domine.</head>
<p>
  <s xml:id="echoid-s13772" xml:space="preserve">Duorum Geometriæ luminum monumenta, quæ diu in tenebris ſepulta, adeò ſtudio-<lb/># ſorum oculos latuerunt, vt inter deperdita fruſtra deſiderarentur, &amp; </s>
  <s xml:id="echoid-s13773" xml:space="preserve">nunc Opera <lb/># Clariſs. </s>
  <s xml:id="echoid-s13774" xml:space="preserve">Virorum, verſa, &amp; </s>
  <s xml:id="echoid-s13775" xml:space="preserve">illuſtrata in lucem prodeunt remoranda non puto; </s>
  <s xml:id="echoid-s13776" xml:space="preserve">cum <lb/># etſi Ethnico fonte cadant, nihil tamen (ſalutaribus monitis Arabica interpretum ſu-<lb/># perſtitione detecta) aduerſus Chriſtianam pietatem contineant.</s>
  <s xml:id="echoid-s13777" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13778" xml:space="preserve">August. </s>
  <s xml:id="echoid-s13779" xml:space="preserve">Coltellini S. </s>
  <s xml:id="echoid-s13780" xml:space="preserve">Officij Conſultor, &amp; </s>
  <s xml:id="echoid-s13781" xml:space="preserve">librorum cenſor.</s>
  <s xml:id="echoid-s13782" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13783" xml:space="preserve">Stante ſupradicta atteſtatione Imprimatur. </s>
  <s xml:id="echoid-s13784" xml:space="preserve">Die 16. </s>
  <s xml:id="echoid-s13785" xml:space="preserve">Iulij 1660.</s>
  <s xml:id="echoid-s13786" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13787" xml:space="preserve">Fr. </s>
  <s xml:id="echoid-s13788" xml:space="preserve">Ang. </s>
  <s xml:id="echoid-s13789" xml:space="preserve">Octau. </s>
  <s xml:id="echoid-s13790" xml:space="preserve">de Populo S. </s>
  <s xml:id="echoid-s13791" xml:space="preserve">Off. </s>
  <s xml:id="echoid-s13792" xml:space="preserve">Florent. </s>
  <s xml:id="echoid-s13793" xml:space="preserve">Cancell. </s>
  <s xml:id="echoid-s13794" xml:space="preserve">demand.</s>
  <s xml:id="echoid-s13795" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13796" xml:space="preserve">Alexander Victorius Senator Sereniſs. </s>
  <s xml:id="echoid-s13797" xml:space="preserve">MagniDucis Auditor.</s>
  <s xml:id="echoid-s13798" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1175" type="section" level="1" n="387">
<head xml:id="echoid-head479" xml:space="preserve">REGISTRVM.</head>
<p>
  <s xml:id="echoid-s13799" xml:space="preserve">* ** *** **** ABCDEFGHIKLMNOPQRSTVXYZ <lb/>Aa Bb Cc Dd Ee Ff Gg Hh Ii Kk Ll Mm Nn Oo Pp Qq Rr Sſ Tt Vu Xx Yy Zz <lb/>Aaa Bbb Ccc Ddd Eee Fff <lb/>Omnes ſunt duerni, excepto * qui eſt ternus.</s>
  <s xml:id="echoid-s13800" xml:space="preserve"/>
</p>
<pb o="415" file="0453" n="454" rhead="Erraca præcipua ſic corrige."/>
<p style="it">
  <s xml:id="echoid-s13801" xml:space="preserve">PAgina 7. </s>
  <s xml:id="echoid-s13802" xml:space="preserve">linea 27. </s>
  <s xml:id="echoid-s13803" xml:space="preserve">ad margine. </s>
  <s xml:id="echoid-s13804" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s13805" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13806" xml:space="preserve">huius. </s>
  <s xml:id="echoid-s13807" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s13808" xml:space="preserve">14. </s>
  <s xml:id="echoid-s13809" xml:space="preserve">lin. </s>
  <s xml:id="echoid-s13810" xml:space="preserve">4. </s>
  <s xml:id="echoid-s13811" xml:space="preserve">ad differentiam. </s>
  <s xml:id="echoid-s13812" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13813" xml:space="preserve">24. </s>
  <s xml:id="echoid-s13814" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13815" xml:space="preserve">21. </s>
  <s xml:id="echoid-s13816" xml:space="preserve">marg. </s>
  <s xml:id="echoid-s13817" xml:space="preserve">prop. <lb/></s>
  <s xml:id="echoid-s13818" xml:space="preserve">2. </s>
  <s xml:id="echoid-s13819" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s13820" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13821" xml:space="preserve">31. </s>
  <s xml:id="echoid-s13822" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13823" xml:space="preserve">27. </s>
  <s xml:id="echoid-s13824" xml:space="preserve">marg. </s>
  <s xml:id="echoid-s13825" xml:space="preserve">in lib. </s>
  <s xml:id="echoid-s13826" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13827" xml:space="preserve">lin. </s>
  <s xml:id="echoid-s13828" xml:space="preserve">34. </s>
  <s xml:id="echoid-s13829" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13830" xml:space="preserve">B A. </s>
  <s xml:id="echoid-s13831" xml:space="preserve">lin. </s>
  <s xml:id="echoid-s13832" xml:space="preserve">40. </s>
  <s xml:id="echoid-s13833" xml:space="preserve">I D, D K. </s>
  <s xml:id="echoid-s13834" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13835" xml:space="preserve">32. </s>
  <s xml:id="echoid-s13836" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13837" xml:space="preserve">15. </s>
  <s xml:id="echoid-s13838" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13839" xml:space="preserve">D H. </s>
  <s xml:id="echoid-s13840" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13841" xml:space="preserve">36. </s>
  <s xml:id="echoid-s13842" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s13843" xml:space="preserve">21. </s>
  <s xml:id="echoid-s13844" xml:space="preserve">figuræ) p. </s>
  <s xml:id="echoid-s13845" xml:space="preserve">40. </s>
  <s xml:id="echoid-s13846" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13847" xml:space="preserve">17. </s>
  <s xml:id="echoid-s13848" xml:space="preserve">(53. </s>
  <s xml:id="echoid-s13849" xml:space="preserve">ex 5. </s>
  <s xml:id="echoid-s13850" xml:space="preserve">) 1. </s>
  <s xml:id="echoid-s13851" xml:space="preserve">33. </s>
  <s xml:id="echoid-s13852" xml:space="preserve">intercipiuntur, &amp;</s>
  <s xml:id="echoid-s13853" xml:space="preserve">. 1. </s>
  <s xml:id="echoid-s13854" xml:space="preserve">38. </s>
  <s xml:id="echoid-s13855" xml:space="preserve">ergo C A. </s>
  <s xml:id="echoid-s13856" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13857" xml:space="preserve">46. </s>
  <s xml:id="echoid-s13858" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13859" xml:space="preserve">5. </s>
  <s xml:id="echoid-s13860" xml:space="preserve">ita inquam. </s>
  <s xml:id="echoid-s13861" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13862" xml:space="preserve">49. </s>
  <s xml:id="echoid-s13863" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s13864" xml:space="preserve">35. </s>
  <s xml:id="echoid-s13865" xml:space="preserve">componebatur inſuper. </s>
  <s xml:id="echoid-s13866" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13867" xml:space="preserve">50. </s>
  <s xml:id="echoid-s13868" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13869" xml:space="preserve">46. </s>
  <s xml:id="echoid-s13870" xml:space="preserve">B G b, &amp; </s>
  <s xml:id="echoid-s13871" xml:space="preserve">d e b. </s>
  <s xml:id="echoid-s13872" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13873" xml:space="preserve">56. </s>
  <s xml:id="echoid-s13874" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13875" xml:space="preserve">15. </s>
  <s xml:id="echoid-s13876" xml:space="preserve">marg. </s>
  <s xml:id="echoid-s13877" xml:space="preserve">4. </s>
  <s xml:id="echoid-s13878" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13879" xml:space="preserve">13. </s>
  <s xml:id="echoid-s13880" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13881" xml:space="preserve">48. </s>
  <s xml:id="echoid-s13882" xml:space="preserve">pariterque L D. </s>
  <s xml:id="echoid-s13883" xml:space="preserve"><lb/>p. </s>
  <s xml:id="echoid-s13884" xml:space="preserve">62. </s>
  <s xml:id="echoid-s13885" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13886" xml:space="preserve">7. </s>
  <s xml:id="echoid-s13887" xml:space="preserve">ſit D A. </s>
  <s xml:id="echoid-s13888" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13889" xml:space="preserve">70. </s>
  <s xml:id="echoid-s13890" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13891" xml:space="preserve">14. </s>
  <s xml:id="echoid-s13892" xml:space="preserve">marg. </s>
  <s xml:id="echoid-s13893" xml:space="preserve">56. </s>
  <s xml:id="echoid-s13894" xml:space="preserve">57. </s>
  <s xml:id="echoid-s13895" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13896" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13897" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13898" xml:space="preserve">30. </s>
  <s xml:id="echoid-s13899" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13900" xml:space="preserve">2. </s>
  <s xml:id="echoid-s13901" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13902" xml:space="preserve">72. </s>
  <s xml:id="echoid-s13903" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13904" xml:space="preserve">12. </s>
  <s xml:id="echoid-s13905" xml:space="preserve">maior quam. </s>
  <s xml:id="echoid-s13906" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13907" xml:space="preserve">73. </s>
  <s xml:id="echoid-s13908" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13909" xml:space="preserve">13. </s>
  <s xml:id="echoid-s13910" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s13911" xml:space="preserve"><lb/>33. </s>
  <s xml:id="echoid-s13912" xml:space="preserve">34. </s>
  <s xml:id="echoid-s13913" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13914" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13915" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13916" xml:space="preserve">78. </s>
  <s xml:id="echoid-s13917" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13918" xml:space="preserve">4. </s>
  <s xml:id="echoid-s13919" xml:space="preserve">reddantur, &amp; </s>
  <s xml:id="echoid-s13920" xml:space="preserve">textus. </s>
  <s xml:id="echoid-s13921" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13922" xml:space="preserve">86. </s>
  <s xml:id="echoid-s13923" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13924" xml:space="preserve">17. </s>
  <s xml:id="echoid-s13925" xml:space="preserve">appliceturque recta. </s>
  <s xml:id="echoid-s13926" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13927" xml:space="preserve">96. </s>
  <s xml:id="echoid-s13928" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13929" xml:space="preserve">7. </s>
  <s xml:id="echoid-s13930" xml:space="preserve">ſuper bipartitio-<lb/>nem axis. </s>
  <s xml:id="echoid-s13931" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13932" xml:space="preserve">99. </s>
  <s xml:id="echoid-s13933" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13934" xml:space="preserve">11. </s>
  <s xml:id="echoid-s13935" xml:space="preserve">ex vero P F minor quam D P. </s>
  <s xml:id="echoid-s13936" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13937" xml:space="preserve">44. </s>
  <s xml:id="echoid-s13938" xml:space="preserve">legi 44. </s>
  <s xml:id="echoid-s13939" xml:space="preserve">45. </s>
  <s xml:id="echoid-s13940" xml:space="preserve">in qua. </s>
  <s xml:id="echoid-s13941" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13942" xml:space="preserve">109. </s>
  <s xml:id="echoid-s13943" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13944" xml:space="preserve">20. </s>
  <s xml:id="echoid-s13945" xml:space="preserve">dele poſtillam. </s>
  <s xml:id="echoid-s13946" xml:space="preserve"><lb/>p. </s>
  <s xml:id="echoid-s13947" xml:space="preserve">110. </s>
  <s xml:id="echoid-s13948" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13949" xml:space="preserve">31. </s>
  <s xml:id="echoid-s13950" xml:space="preserve">marg. </s>
  <s xml:id="echoid-s13951" xml:space="preserve">appone d. </s>
  <s xml:id="echoid-s13952" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13953" xml:space="preserve">111. </s>
  <s xml:id="echoid-s13954" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13955" xml:space="preserve">31. </s>
  <s xml:id="echoid-s13956" xml:space="preserve">aut minor angulo. </s>
  <s xml:id="echoid-s13957" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13958" xml:space="preserve">129. </s>
  <s xml:id="echoid-s13959" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13960" xml:space="preserve">35. </s>
  <s xml:id="echoid-s13961" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13962" xml:space="preserve">inuertendo. </s>
  <s xml:id="echoid-s13963" xml:space="preserve">ibidem marg. </s>
  <s xml:id="echoid-s13964" xml:space="preserve"><lb/>10. </s>
  <s xml:id="echoid-s13965" xml:space="preserve">hui. </s>
  <s xml:id="echoid-s13966" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13967" xml:space="preserve">130. </s>
  <s xml:id="echoid-s13968" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13969" xml:space="preserve">26. </s>
  <s xml:id="echoid-s13970" xml:space="preserve">dele omnia ab O G víq; </s>
  <s xml:id="echoid-s13971" xml:space="preserve">ad comparando. </s>
  <s xml:id="echoid-s13972" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13973" xml:space="preserve">138. </s>
  <s xml:id="echoid-s13974" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13975" xml:space="preserve">8. </s>
  <s xml:id="echoid-s13976" xml:space="preserve">oppoſita. </s>
  <s xml:id="echoid-s13977" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13978" xml:space="preserve">139. </s>
  <s xml:id="echoid-s13979" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13980" xml:space="preserve">18. </s>
  <s xml:id="echoid-s13981" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s13982" xml:space="preserve">d. </s>
  <s xml:id="echoid-s13983" xml:space="preserve"><lb/>p. </s>
  <s xml:id="echoid-s13984" xml:space="preserve">141. </s>
  <s xml:id="echoid-s13985" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13986" xml:space="preserve">8. </s>
  <s xml:id="echoid-s13987" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s13988" xml:space="preserve">14. </s>
  <s xml:id="echoid-s13989" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13990" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13991" xml:space="preserve">p. </s>
  <s xml:id="echoid-s13992" xml:space="preserve">146. </s>
  <s xml:id="echoid-s13993" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13994" xml:space="preserve">18. </s>
  <s xml:id="echoid-s13995" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s13996" xml:space="preserve">12. </s>
  <s xml:id="echoid-s13997" xml:space="preserve">13. </s>
  <s xml:id="echoid-s13998" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s13999" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14000" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14001" xml:space="preserve">151. </s>
  <s xml:id="echoid-s14002" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14003" xml:space="preserve">18. </s>
  <s xml:id="echoid-s14004" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14005" xml:space="preserve">8. </s>
  <s xml:id="echoid-s14006" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14007" xml:space="preserve">11. </s>
  <s xml:id="echoid-s14008" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s14009" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s14010" xml:space="preserve">5. </s>
  <s xml:id="echoid-s14011" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14012" xml:space="preserve">19. </s>
  <s xml:id="echoid-s14013" xml:space="preserve"><lb/>M L, &amp; </s>
  <s xml:id="echoid-s14014" xml:space="preserve">R L. </s>
  <s xml:id="echoid-s14015" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14016" xml:space="preserve">22. </s>
  <s xml:id="echoid-s14017" xml:space="preserve">œqualibus axium. </s>
  <s xml:id="echoid-s14018" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14019" xml:space="preserve">161. </s>
  <s xml:id="echoid-s14020" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14021" xml:space="preserve">13. </s>
  <s xml:id="echoid-s14022" xml:space="preserve">ductam in hyperbola) p. </s>
  <s xml:id="echoid-s14023" xml:space="preserve">168. </s>
  <s xml:id="echoid-s14024" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14025" xml:space="preserve">30. </s>
  <s xml:id="echoid-s14026" xml:space="preserve">quod eſt. </s>
  <s xml:id="echoid-s14027" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14028" xml:space="preserve">172. </s>
  <s xml:id="echoid-s14029" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s14030" xml:space="preserve">29. </s>
  <s xml:id="echoid-s14031" xml:space="preserve">ſed in primo caſu recta linea. </s>
  <s xml:id="echoid-s14032" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14033" xml:space="preserve">30. </s>
  <s xml:id="echoid-s14034" xml:space="preserve">baſim F I. </s>
  <s xml:id="echoid-s14035" xml:space="preserve">ibid. </s>
  <s xml:id="echoid-s14036" xml:space="preserve">puncta I. </s>
  <s xml:id="echoid-s14037" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14038" xml:space="preserve">F; </s>
  <s xml:id="echoid-s14039" xml:space="preserve">nec F I ſecat bifariam ſubtenſas G <lb/>E, M K; </s>
  <s xml:id="echoid-s14040" xml:space="preserve">propterea. </s>
  <s xml:id="echoid-s14041" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14042" xml:space="preserve">175. </s>
  <s xml:id="echoid-s14043" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14044" xml:space="preserve">26. </s>
  <s xml:id="echoid-s14045" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14046" xml:space="preserve">a. </s>
  <s xml:id="echoid-s14047" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14048" xml:space="preserve">35. </s>
  <s xml:id="echoid-s14049" xml:space="preserve">ad duas. </s>
  <s xml:id="echoid-s14050" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14051" xml:space="preserve">176. </s>
  <s xml:id="echoid-s14052" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14053" xml:space="preserve">15. </s>
  <s xml:id="echoid-s14054" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14055" xml:space="preserve">d. </s>
  <s xml:id="echoid-s14056" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14057" xml:space="preserve">183. </s>
  <s xml:id="echoid-s14058" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14059" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14060" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14061" xml:space="preserve">d. </s>
  <s xml:id="echoid-s14062" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14063" xml:space="preserve">189. </s>
  <s xml:id="echoid-s14064" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s14065" xml:space="preserve">29. </s>
  <s xml:id="echoid-s14066" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14067" xml:space="preserve">lemma 7. </s>
  <s xml:id="echoid-s14068" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14069" xml:space="preserve">47. </s>
  <s xml:id="echoid-s14070" xml:space="preserve">applicatæ. </s>
  <s xml:id="echoid-s14071" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14072" xml:space="preserve">190. </s>
  <s xml:id="echoid-s14073" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14074" xml:space="preserve">8. </s>
  <s xml:id="echoid-s14075" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14076" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s14077" xml:space="preserve">2. </s>
  <s xml:id="echoid-s14078" xml:space="preserve">præmiſ. </s>
  <s xml:id="echoid-s14079" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14080" xml:space="preserve">193. </s>
  <s xml:id="echoid-s14081" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14082" xml:space="preserve">6. </s>
  <s xml:id="echoid-s14083" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s14084" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s14085" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s14086" xml:space="preserve">XXIII. </s>
  <s xml:id="echoid-s14087" xml:space="preserve"><lb/>XXIV. </s>
  <s xml:id="echoid-s14088" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14089" xml:space="preserve">196. </s>
  <s xml:id="echoid-s14090" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14091" xml:space="preserve">25. </s>
  <s xml:id="echoid-s14092" xml:space="preserve">nempe X a. </s>
  <s xml:id="echoid-s14093" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14094" xml:space="preserve">197. </s>
  <s xml:id="echoid-s14095" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14096" xml:space="preserve">29. </s>
  <s xml:id="echoid-s14097" xml:space="preserve">ad L P. </s>
  <s xml:id="echoid-s14098" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14099" xml:space="preserve">202. </s>
  <s xml:id="echoid-s14100" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14101" xml:space="preserve">23. </s>
  <s xml:id="echoid-s14102" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14103" xml:space="preserve">18. </s>
  <s xml:id="echoid-s14104" xml:space="preserve">huius. </s>
  <s xml:id="echoid-s14105" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14106" xml:space="preserve">207. </s>
  <s xml:id="echoid-s14107" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14108" xml:space="preserve">6. </s>
  <s xml:id="echoid-s14109" xml:space="preserve">quod. </s>
  <s xml:id="echoid-s14110" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s14111" xml:space="preserve">33. </s>
  <s xml:id="echoid-s14112" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14113" xml:space="preserve">a. </s>
  <s xml:id="echoid-s14114" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14115" xml:space="preserve">213. </s>
  <s xml:id="echoid-s14116" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14117" xml:space="preserve">11. </s>
  <s xml:id="echoid-s14118" xml:space="preserve">hyperbolen E Z. </s>
  <s xml:id="echoid-s14119" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14120" xml:space="preserve">214. </s>
  <s xml:id="echoid-s14121" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14122" xml:space="preserve">38. </s>
  <s xml:id="echoid-s14123" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14124" xml:space="preserve">ex 20. </s>
  <s xml:id="echoid-s14125" xml:space="preserve">huius. </s>
  <s xml:id="echoid-s14126" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14127" xml:space="preserve">217. </s>
  <s xml:id="echoid-s14128" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14129" xml:space="preserve">21. </s>
  <s xml:id="echoid-s14130" xml:space="preserve">ideoque ei æqualis <lb/>omnino erit. </s>
  <s xml:id="echoid-s14131" xml:space="preserve">Simili ratiocinio oſtendetur quælibet alia intercepta K L æquidiſtans. </s>
  <s xml:id="echoid-s14132" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14133" xml:space="preserve">223. </s>
  <s xml:id="echoid-s14134" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14135" xml:space="preserve">6. </s>
  <s xml:id="echoid-s14136" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14137" xml:space="preserve">Schol. </s>
  <s xml:id="echoid-s14138" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s14139" xml:space="preserve"><lb/>6. </s>
  <s xml:id="echoid-s14140" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s14141" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14142" xml:space="preserve">228. </s>
  <s xml:id="echoid-s14143" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14144" xml:space="preserve">18. </s>
  <s xml:id="echoid-s14145" xml:space="preserve">ergo comparando homologorum differentias. </s>
  <s xml:id="echoid-s14146" xml:space="preserve">ibid. </s>
  <s xml:id="echoid-s14147" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14148" xml:space="preserve">lem. </s>
  <s xml:id="echoid-s14149" xml:space="preserve">3. </s>
  <s xml:id="echoid-s14150" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s14151" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14152" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14153" xml:space="preserve">233. </s>
  <s xml:id="echoid-s14154" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14155" xml:space="preserve">4. </s>
  <s xml:id="echoid-s14156" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14157" xml:space="preserve"><lb/>prop. </s>
  <s xml:id="echoid-s14158" xml:space="preserve">7. </s>
  <s xml:id="echoid-s14159" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14160" xml:space="preserve">ex 8. </s>
  <s xml:id="echoid-s14161" xml:space="preserve">addit. </s>
  <s xml:id="echoid-s14162" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14163" xml:space="preserve">235. </s>
  <s xml:id="echoid-s14164" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14165" xml:space="preserve">37. </s>
  <s xml:id="echoid-s14166" xml:space="preserve">hyperbolen H I K. </s>
  <s xml:id="echoid-s14167" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14168" xml:space="preserve">240. </s>
  <s xml:id="echoid-s14169" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14170" xml:space="preserve">3. </s>
  <s xml:id="echoid-s14171" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14172" xml:space="preserve">f. </s>
  <s xml:id="echoid-s14173" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14174" xml:space="preserve">244. </s>
  <s xml:id="echoid-s14175" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14176" xml:space="preserve">14. </s>
  <s xml:id="echoid-s14177" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14178" xml:space="preserve">I F R, ſeu H F <lb/>N, &amp; </s>
  <s xml:id="echoid-s14179" xml:space="preserve">I F S. </s>
  <s xml:id="echoid-s14180" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14181" xml:space="preserve">248. </s>
  <s xml:id="echoid-s14182" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14183" xml:space="preserve">35. </s>
  <s xml:id="echoid-s14184" xml:space="preserve">ſit ſectio. </s>
  <s xml:id="echoid-s14185" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14186" xml:space="preserve">250. </s>
  <s xml:id="echoid-s14187" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14188" xml:space="preserve">4. </s>
  <s xml:id="echoid-s14189" xml:space="preserve">quod L O. </s>
  <s xml:id="echoid-s14190" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14191" xml:space="preserve">256. </s>
  <s xml:id="echoid-s14192" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14193" xml:space="preserve">12. </s>
  <s xml:id="echoid-s14194" xml:space="preserve">parallela. </s>
  <s xml:id="echoid-s14195" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14196" xml:space="preserve">259. </s>
  <s xml:id="echoid-s14197" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14198" xml:space="preserve">12. </s>
  <s xml:id="echoid-s14199" xml:space="preserve">quàm A <lb/>C. </s>
  <s xml:id="echoid-s14200" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14201" xml:space="preserve">260. </s>
  <s xml:id="echoid-s14202" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14203" xml:space="preserve">16. </s>
  <s xml:id="echoid-s14204" xml:space="preserve">per eundem. </s>
  <s xml:id="echoid-s14205" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14206" xml:space="preserve">262. </s>
  <s xml:id="echoid-s14207" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14208" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14209" xml:space="preserve">eandem. </s>
  <s xml:id="echoid-s14210" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14211" xml:space="preserve">4. </s>
  <s xml:id="echoid-s14212" xml:space="preserve">A D, &amp; </s>
  <s xml:id="echoid-s14213" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14214" xml:space="preserve">41. </s>
  <s xml:id="echoid-s14215" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14216" xml:space="preserve">eam, quæ. </s>
  <s xml:id="echoid-s14217" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14218" xml:space="preserve">264. </s>
  <s xml:id="echoid-s14219" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14220" xml:space="preserve">13. </s>
  <s xml:id="echoid-s14221" xml:space="preserve">ſecabit <lb/>rectam. </s>
  <s xml:id="echoid-s14222" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14223" xml:space="preserve">268. </s>
  <s xml:id="echoid-s14224" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14225" xml:space="preserve">22. </s>
  <s xml:id="echoid-s14226" xml:space="preserve">conus E A C. </s>
  <s xml:id="echoid-s14227" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14228" xml:space="preserve">269. </s>
  <s xml:id="echoid-s14229" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14230" xml:space="preserve">8. </s>
  <s xml:id="echoid-s14231" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14232" xml:space="preserve">ex prop. </s>
  <s xml:id="echoid-s14233" xml:space="preserve">5. </s>
  <s xml:id="echoid-s14234" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s14235" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14236" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14237" xml:space="preserve">9. </s>
  <s xml:id="echoid-s14238" xml:space="preserve">10. </s>
  <s xml:id="echoid-s14239" xml:space="preserve">20. </s>
  <s xml:id="echoid-s14240" xml:space="preserve">expunge recto. </s>
  <s xml:id="echoid-s14241" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14242" xml:space="preserve">15. </s>
  <s xml:id="echoid-s14243" xml:space="preserve">ſectio-<lb/>nis F A G. </s>
  <s xml:id="echoid-s14244" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14245" xml:space="preserve">275. </s>
  <s xml:id="echoid-s14246" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14247" xml:space="preserve">10. </s>
  <s xml:id="echoid-s14248" xml:space="preserve">rectangulo A D F. </s>
  <s xml:id="echoid-s14249" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14250" xml:space="preserve">280. </s>
  <s xml:id="echoid-s14251" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14252" xml:space="preserve">14. </s>
  <s xml:id="echoid-s14253" xml:space="preserve">G E A eandem. </s>
  <s xml:id="echoid-s14254" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14255" xml:space="preserve">291. </s>
  <s xml:id="echoid-s14256" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14257" xml:space="preserve">3. </s>
  <s xml:id="echoid-s14258" xml:space="preserve">XXIX. </s>
  <s xml:id="echoid-s14259" xml:space="preserve">XXVII. </s>
  <s xml:id="echoid-s14260" xml:space="preserve"><lb/>p. </s>
  <s xml:id="echoid-s14261" xml:space="preserve">298. </s>
  <s xml:id="echoid-s14262" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14263" xml:space="preserve">6. </s>
  <s xml:id="echoid-s14264" xml:space="preserve">XXIIX. </s>
  <s xml:id="echoid-s14265" xml:space="preserve">XXVI. </s>
  <s xml:id="echoid-s14266" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14267" xml:space="preserve">303. </s>
  <s xml:id="echoid-s14268" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14269" xml:space="preserve">16.</s>
  <s xml:id="echoid-s14270" xml:space="preserve">. erectum. </s>
  <s xml:id="echoid-s14271" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14272" xml:space="preserve">306. </s>
  <s xml:id="echoid-s14273" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14274" xml:space="preserve">23. </s>
  <s xml:id="echoid-s14275" xml:space="preserve">ad perfectionem prop. </s>
  <s xml:id="echoid-s14276" xml:space="preserve">26. </s>
  <s xml:id="echoid-s14277" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14278" xml:space="preserve">313. </s>
  <s xml:id="echoid-s14279" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14280" xml:space="preserve">7. </s>
  <s xml:id="echoid-s14281" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14282" xml:space="preserve"><lb/>prop. </s>
  <s xml:id="echoid-s14283" xml:space="preserve">26. </s>
  <s xml:id="echoid-s14284" xml:space="preserve">huius. </s>
  <s xml:id="echoid-s14285" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14286" xml:space="preserve">318. </s>
  <s xml:id="echoid-s14287" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14288" xml:space="preserve">25. </s>
  <s xml:id="echoid-s14289" xml:space="preserve">quàm G H E ad E H, &amp; </s>
  <s xml:id="echoid-s14290" xml:space="preserve">(quando G cadit inter E, &amp; </s>
  <s xml:id="echoid-s14291" xml:space="preserve">H), multo maiorem <lb/>quàm G E. </s>
  <s xml:id="echoid-s14292" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14293" xml:space="preserve">319. </s>
  <s xml:id="echoid-s14294" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14295" xml:space="preserve">17. </s>
  <s xml:id="echoid-s14296" xml:space="preserve">E H ab ipſo quadrato G E. </s>
  <s xml:id="echoid-s14297" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14298" xml:space="preserve">321. </s>
  <s xml:id="echoid-s14299" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14300" xml:space="preserve">9. </s>
  <s xml:id="echoid-s14301" xml:space="preserve">quadrato E G. </s>
  <s xml:id="echoid-s14302" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14303" xml:space="preserve">11. </s>
  <s xml:id="echoid-s14304" xml:space="preserve">XXXV. </s>
  <s xml:id="echoid-s14305" xml:space="preserve">XXXVI, <lb/>p. </s>
  <s xml:id="echoid-s14306" xml:space="preserve">323. </s>
  <s xml:id="echoid-s14307" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14308" xml:space="preserve">2. </s>
  <s xml:id="echoid-s14309" xml:space="preserve">diametri ad eaſdem partes. </s>
  <s xml:id="echoid-s14310" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14311" xml:space="preserve">325. </s>
  <s xml:id="echoid-s14312" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14313" xml:space="preserve">7. </s>
  <s xml:id="echoid-s14314" xml:space="preserve">21. </s>
  <s xml:id="echoid-s14315" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14316" xml:space="preserve">23. </s>
  <s xml:id="echoid-s14317" xml:space="preserve">(16. </s>
  <s xml:id="echoid-s14318" xml:space="preserve">ex. </s>
  <s xml:id="echoid-s14319" xml:space="preserve">7.) </s>
  <s xml:id="echoid-s14320" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14321" xml:space="preserve">326. </s>
  <s xml:id="echoid-s14322" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14323" xml:space="preserve">11. </s>
  <s xml:id="echoid-s14324" xml:space="preserve">quæ eſt dupla. </s>
  <s xml:id="echoid-s14325" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s14326" xml:space="preserve">14. </s>
  <s xml:id="echoid-s14327" xml:space="preserve">M E ad. </s>
  <s xml:id="echoid-s14328" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14329" xml:space="preserve">20. </s>
  <s xml:id="echoid-s14330" xml:space="preserve">(16. </s>
  <s xml:id="echoid-s14331" xml:space="preserve">ex 7.) </s>
  <s xml:id="echoid-s14332" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14333" xml:space="preserve">327. </s>
  <s xml:id="echoid-s14334" xml:space="preserve">quàm D H A ad A H, &amp; </s>
  <s xml:id="echoid-s14335" xml:space="preserve">in primo caſu multo maiorem, quàm, <lb/>p. </s>
  <s xml:id="echoid-s14336" xml:space="preserve">328. </s>
  <s xml:id="echoid-s14337" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14338" xml:space="preserve">33. </s>
  <s xml:id="echoid-s14339" xml:space="preserve">latus C O. </s>
  <s xml:id="echoid-s14340" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14341" xml:space="preserve">329. </s>
  <s xml:id="echoid-s14342" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14343" xml:space="preserve">22. </s>
  <s xml:id="echoid-s14344" xml:space="preserve">quàm E D O in O E. </s>
  <s xml:id="echoid-s14345" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14346" xml:space="preserve">331. </s>
  <s xml:id="echoid-s14347" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14348" xml:space="preserve">27. </s>
  <s xml:id="echoid-s14349" xml:space="preserve">vt axis tranſuerſus A C. </s>
  <s xml:id="echoid-s14350" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14351" xml:space="preserve">335. </s>
  <s xml:id="echoid-s14352" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s14353" xml:space="preserve">7. </s>
  <s xml:id="echoid-s14354" xml:space="preserve">ipſius P R ſupra P Q. </s>
  <s xml:id="echoid-s14355" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14356" xml:space="preserve">11. </s>
  <s xml:id="echoid-s14357" xml:space="preserve">aggregati M G, H E. </s>
  <s xml:id="echoid-s14358" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14359" xml:space="preserve">338. </s>
  <s xml:id="echoid-s14360" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14361" xml:space="preserve">18. </s>
  <s xml:id="echoid-s14362" xml:space="preserve">G E, &amp; </s>
  <s xml:id="echoid-s14363" xml:space="preserve">E H. </s>
  <s xml:id="echoid-s14364" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14365" xml:space="preserve">341. </s>
  <s xml:id="echoid-s14366" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14367" xml:space="preserve">3. </s>
  <s xml:id="echoid-s14368" xml:space="preserve">axis tranſuerſi <lb/>C A. </s>
  <s xml:id="echoid-s14369" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14370" xml:space="preserve">343. </s>
  <s xml:id="echoid-s14371" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14372" xml:space="preserve">9. </s>
  <s xml:id="echoid-s14373" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14374" xml:space="preserve">dele b. </s>
  <s xml:id="echoid-s14375" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14376" xml:space="preserve">344. </s>
  <s xml:id="echoid-s14377" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14378" xml:space="preserve">7. </s>
  <s xml:id="echoid-s14379" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14380" xml:space="preserve">b. </s>
  <s xml:id="echoid-s14381" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14382" xml:space="preserve">346. </s>
  <s xml:id="echoid-s14383" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14384" xml:space="preserve">15. </s>
  <s xml:id="echoid-s14385" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14386" xml:space="preserve">c. </s>
  <s xml:id="echoid-s14387" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14388" xml:space="preserve">347. </s>
  <s xml:id="echoid-s14389" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14390" xml:space="preserve">7. </s>
  <s xml:id="echoid-s14391" xml:space="preserve">ad quadratum Q P R, <lb/>&amp;</s>
  <s xml:id="echoid-s14392" xml:space="preserve">. p. </s>
  <s xml:id="echoid-s14393" xml:space="preserve">350. </s>
  <s xml:id="echoid-s14394" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14395" xml:space="preserve">13. </s>
  <s xml:id="echoid-s14396" xml:space="preserve">O H, &amp; </s>
  <s xml:id="echoid-s14397" xml:space="preserve">G E. </s>
  <s xml:id="echoid-s14398" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14399" xml:space="preserve">356. </s>
  <s xml:id="echoid-s14400" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14401" xml:space="preserve">14. </s>
  <s xml:id="echoid-s14402" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14403" xml:space="preserve">lem. </s>
  <s xml:id="echoid-s14404" xml:space="preserve">15. </s>
  <s xml:id="echoid-s14405" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14406" xml:space="preserve">386. </s>
  <s xml:id="echoid-s14407" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14408" xml:space="preserve">31. </s>
  <s xml:id="echoid-s14409" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14410" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s14411" xml:space="preserve">4. </s>
  <s xml:id="echoid-s14412" xml:space="preserve">Coll. </s>
  <s xml:id="echoid-s14413" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s14414" xml:space="preserve">14. </s>
  <s xml:id="echoid-s14415" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14416" xml:space="preserve">391. </s>
  <s xml:id="echoid-s14417" xml:space="preserve"><lb/>1. </s>
  <s xml:id="echoid-s14418" xml:space="preserve">9. </s>
  <s xml:id="echoid-s14419" xml:space="preserve">mar. </s>
  <s xml:id="echoid-s14420" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s14421" xml:space="preserve">4. </s>
  <s xml:id="echoid-s14422" xml:space="preserve">Coll. </s>
  <s xml:id="echoid-s14423" xml:space="preserve">prop. </s>
  <s xml:id="echoid-s14424" xml:space="preserve">13. </s>
  <s xml:id="echoid-s14425" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14426" xml:space="preserve">392. </s>
  <s xml:id="echoid-s14427" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14428" xml:space="preserve">15. </s>
  <s xml:id="echoid-s14429" xml:space="preserve">quæ erit. </s>
  <s xml:id="echoid-s14430" xml:space="preserve">p. </s>
  <s xml:id="echoid-s14431" xml:space="preserve">404. </s>
  <s xml:id="echoid-s14432" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14433" xml:space="preserve">37. </s>
  <s xml:id="echoid-s14434" xml:space="preserve">A B E, A C E.</s>
  <s xml:id="echoid-s14435" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div1176" type="section" level="1" n="388">
<head xml:id="echoid-head480" xml:space="preserve">Errata in figuris.</head>
<p>
  <s xml:id="echoid-s14436" xml:space="preserve">Pag. </s>
  <s xml:id="echoid-s14437" xml:space="preserve">12. </s>
  <s xml:id="echoid-s14438" xml:space="preserve">in eius figura deeſt recta N Q, &amp; </s>
  <s xml:id="echoid-s14439" xml:space="preserve">D terminus axis. </s>
  <s xml:id="echoid-s14440" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14441" xml:space="preserve">22. </s>
  <s xml:id="echoid-s14442" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s14443" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14444" xml:space="preserve">deeſt recta I N. </s>
  <s xml:id="echoid-s14445" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14446" xml:space="preserve">30. </s>
  <s xml:id="echoid-s14447" xml:space="preserve">in <lb/>parabola decſt N in occurſu B F, G H. </s>
  <s xml:id="echoid-s14448" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14449" xml:space="preserve">37. </s>
  <s xml:id="echoid-s14450" xml:space="preserve">deeſt P in puncto incidentiæ perpendicularis à <lb/>puncto 1 ſuper S K. </s>
  <s xml:id="echoid-s14451" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14452" xml:space="preserve">46. </s>
  <s xml:id="echoid-s14453" xml:space="preserve">deeſt A in vertice axis. </s>
  <s xml:id="echoid-s14454" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14455" xml:space="preserve">93. </s>
  <s xml:id="echoid-s14456" xml:space="preserve">deeſt recta L O. </s>
  <s xml:id="echoid-s14457" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14458" xml:space="preserve">112. </s>
  <s xml:id="echoid-s14459" xml:space="preserve">in tribus <lb/>ſequentibus figuris deeſt ramus I B. </s>
  <s xml:id="echoid-s14460" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14461" xml:space="preserve">213. </s>
  <s xml:id="echoid-s14462" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s14463" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14464" xml:space="preserve">litteræ C, Q commutari debent. </s>
  <s xml:id="echoid-s14465" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14466" xml:space="preserve">240. <lb/></s>
  <s xml:id="echoid-s14467" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s14468" xml:space="preserve">2. </s>
  <s xml:id="echoid-s14469" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14470" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14471" xml:space="preserve">246. </s>
  <s xml:id="echoid-s14472" xml:space="preserve">producantur F L, H I ad K. </s>
  <s xml:id="echoid-s14473" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14474" xml:space="preserve">268. </s>
  <s xml:id="echoid-s14475" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s14476" xml:space="preserve">2. </s>
  <s xml:id="echoid-s14477" xml:space="preserve">linea curua A Z duci debet inter <lb/>A G, &amp; </s>
  <s xml:id="echoid-s14478" xml:space="preserve">A D. </s>
  <s xml:id="echoid-s14479" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s14480" xml:space="preserve">368. </s>
  <s xml:id="echoid-s14481" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s14482" xml:space="preserve">3. </s>
  <s xml:id="echoid-s14483" xml:space="preserve">in puncto I ponatur X.</s>
  <s xml:id="echoid-s14484" xml:space="preserve"/>
</p>
<pb file="0454" n="455"/>
<pb file="0455" n="456"/>
<pb file="0456" n="457"/>
  </div></text>
</echo>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Wed, 29 Nov 2017 16:55:37 +0100
parents	22d6a63640c6
children