Mercurial > hg > mpdl-xml-content
view texts/XML/echo/la/Viviani_1659_QN4GHYBF.xml @ 29:90b1eda1b0a9
Some new special instructions
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Fri, 02 Dec 2016 14:37:22 +0100 |
| parents | 22d6a63640c6 |
| children |
line wrap: on
line source
<?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"> <metadata> <dcterms:identifier>ECHO:QN4GHYBF.xml</dcterms:identifier> <dcterms:creator identifier="GND:100673287">Viviani, Vincenzio</dcterms:creator> <dcterms:title xml:lang="la">De maximis et minimis, geometrica divinatio</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1659</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> </metadata> <text xml:lang="la" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/> <pb file="0002" n="2"/> <handwritten/> <pb file="0003" n="3"/> <handwritten/> <handwritten/> <pb file="0004" n="4"/> <pb file="0005" n="5"/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">DE <lb/>MAXIMIS, <lb/>ET <lb/>MINIMIS <lb/>LIBRIDVO.</head> <pb file="0006" n="6"/> <pb file="0007" n="7"/> <handwritten/> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head2" xml:space="preserve"><emph style="red">DE MAXIMIS,</emph> <lb/>ET <lb/><emph style="red">MINIMIS</emph> <lb/>GEOMETRICA DIVINATIO <lb/><emph style="red"><emph style="sc">In</emph> <emph style="sc">Qvintvm</emph> <emph style="sc">Conicorvm</emph></emph> <lb/><emph style="red">APOLLONII PERGÆI</emph> <lb/>_ADHVC DESIDERATVM;_ <lb/>AD SERENISSIMVM <lb/><emph style="red">FERDINANDVM II.</emph> <lb/>MAGNVMDVCEM ETRVRIÆ. <lb/><emph style="red">LIBER PRIMVS.</emph> <lb/>_AVCTORE_ <lb/><emph style="red">VINCENTIO VIVIANI.</emph></head> <handwritten/> <figure> <image file="0007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0007-01"/> </figure> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head3" xml:space="preserve"><emph style="red">FLORENTIE<unsure/> MDCLIX</emph></head> <head xml:id="echoid-head4" xml:space="preserve">Apud Ioſeph Cocchini, Typis Nouis, ſub Signo STELLÆ. <lb/><emph style="red">SVPERIORVM PERMISSV.</emph></head> <handwritten/> <pb file="0008" n="8"/> <handwritten/> <handwritten/> <handwritten/> <handwritten/> <pb file="0009" n="9"/> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head5" xml:space="preserve">SERENISSIMO <lb/>FERDINANDO II. <lb/>MAGNODVCI ETRVRIÆ.</head> <p> <s xml:id="echoid-s1" xml:space="preserve">GRANDE opus aggredior, dicerem <lb/>etiam tua Celſitudine nõ indignum, <lb/>SERENISSIME MAGNEDVX, <lb/>ſi meis viribus abſolui poſſe credide-<lb/>rim. </s> <s xml:id="echoid-s2" xml:space="preserve">MAGNI GEOMETRÆ <lb/>APOLLONII Conicorum diui-<lb/>nas propè dixerim meditationes per <lb/>tot ſecula temporum iniuria nobis ablatas proprio <lb/>Marte reſtituere, Herculeus equidem labor eſt, & </s> <s xml:id="echoid-s3" xml:space="preserve">qui <lb/>diu plurima, eaque robuſtiſsima ingenia, aut ab in-<lb/>cœpto deterruit, aut in opere defatigauit. </s> <s xml:id="echoid-s4" xml:space="preserve">Audeo ta-<lb/>men auſpicijs tuis inclyte FERDINANDE; </s> <s xml:id="echoid-s5" xml:space="preserve">& </s> <s xml:id="echoid-s6" xml:space="preserve">quod <lb/>negatum imbecillitati meę timere debueram (quam per <lb/>per tot annos domeſtica incommoda, negotia publica, <lb/>& </s> <s xml:id="echoid-s7" xml:space="preserve">grauiſsimæ corporis, animique ægritudines exagita-<lb/>runt) gloriæ tuæ, maximoque in Matheſim, cognotaſ-<lb/>que ſcientias, atque artes, te fauente inſtauratas, amori <lb/>ſeruatum fuiſſe confido. </s> <s xml:id="echoid-s8" xml:space="preserve">Ibi enim Numinis fauor cla-<lb/>riùs elucet, vbi nulla hominum virtus adeſt. </s> <s xml:id="echoid-s9" xml:space="preserve">In re tam <lb/>ardua conatus mei ſi felici euentu caruerint, niſi lau-<lb/>dem, profectò excuſationem inuenient. </s> <s xml:id="echoid-s10" xml:space="preserve">At ſi fortuna <pb file="0010" n="10"/> hilari vultu votis meis arriſerit, de tuo patrocinio, ac <lb/>munificentia, pro tam illuſtri beneficio, non tantùm <lb/>hæc ætas, quæ te præſentem vnà mecum admiratur, <lb/>verùm etiam magnificè loquetur grata poſteritas. </s> <s xml:id="echoid-s11" xml:space="preserve">Hęc <lb/>igitur quidquid ſunt, benignè, vt ſoles, intuere. </s> <s xml:id="echoid-s12" xml:space="preserve">Si MI-<lb/>NIMA, quod reor, ſterilis ingenij fœtum vt foueas; </s> <s xml:id="echoid-s13" xml:space="preserve">ſi <lb/>MAXIMA, vt tuæ vegetę lucis prolem regalibus vlnis <lb/>accipias. </s> <s xml:id="echoid-s14" xml:space="preserve">Atque interim plurimos, beatoſque annos bo-<lb/>no publico viue.</s> <s xml:id="echoid-s15" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16" xml:space="preserve">Florentiæ. </s> <s xml:id="echoid-s17" xml:space="preserve">Octauo Calendas Ianuarij 1658.</s> <s xml:id="echoid-s18" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19" xml:space="preserve">SER <emph style="sc">MÆ</emph> CELS. </s> <s xml:id="echoid-s20" xml:space="preserve"><emph style="sc">NIS</emph> TVÆ</s> </p> <p style="it"> <s xml:id="echoid-s21" xml:space="preserve">Humillimus, Addictiſsimus, Obſtrictiſsimus</s> </p> <p style="it"> <s xml:id="echoid-s22" xml:space="preserve">Seruus, & </s> <s xml:id="echoid-s23" xml:space="preserve">Cliens</s> </p> <p> <s xml:id="echoid-s24" xml:space="preserve">Vincentius Viuiani.</s> <s xml:id="echoid-s25" xml:space="preserve"/> </p> <pb file="0011" n="11"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head6" xml:space="preserve">IN DIVINATIONEM GEOMETRICAM <lb/>DE MAXIMIS, ET MINIMIS <lb/>PRÆFATIO.</head> <head xml:id="echoid-head7" xml:space="preserve">AMICE LECTOR.</head> <p> <s xml:id="echoid-s26" xml:space="preserve">NEMO omnium neſcit, quieruditionis aliquid deguſta-<lb/>rint, ac geometricis potiſſimùm ſtudijs animum inten-<lb/>derint, meditationes Conicas, tum antiquiſſimas eſſe, <lb/>tum ab APOLLONIO Pergæo (qui ad annos fermè <lb/>nongentos nunc ſupra mille ſub Ptolomeo Euergete <lb/>floruit) omnia ea vſquequaque fuiſſe collecta, quæ ſparſim antea in <lb/>eo genere commentati fuerant Ariſteas Geometra, Eudoxus Cnidius, <lb/>Menæchmus, Euclides, Conon, Traſideus, Nicoteles, & </s> <s xml:id="echoid-s27" xml:space="preserve">quod ali-<lb/>qui tradunt Archimedes etiam, ac Doſitheus, multique alijvetuſtio-<lb/>res, quorum nomina cum ſcriptis periere. </s> <s xml:id="echoid-s28" xml:space="preserve">Hos inter APOLLO-<lb/>NIVS, vtiliſſimam hanc, admirabilemque doctrinam egregiè illu-<lb/>ſtrauit, ampliauitque octo libris comprehenſam, vt ipſe ad Eudemum <lb/>præfatur. </s> <s xml:id="echoid-s29" xml:space="preserve">Id autem tam ſelici ſupra cæteros excellentia perfecit, iure <lb/>vt ab omnibus ſui æui Mathematicis MAGNVS GEOMETRA au-<lb/>dierit.</s> <s xml:id="echoid-s30" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s31" xml:space="preserve">Neque illud nos fugit, ad vſque Pappi Alexandrini tempora, hos <lb/>octo libros perueniſſe, qui neceſſaria nobis Lemmata ad eorum no-<lb/>tionem conſtruxit. </s> <s xml:id="echoid-s32" xml:space="preserve">Eutocius quoq; </s> <s xml:id="echoid-s33" xml:space="preserve">Aſcalonites Pappo iunior, pręter <lb/>commentaria in quatuor APOLLONII priores, reliquas curas in toti-<lb/>dem reliquos Anthemo promittit. </s> <s xml:id="echoid-s34" xml:space="preserve">Cæterùm ex quo Eutocius floruit, <lb/>annos, vt aliqui tradunt, à condita ſaluta circiter CCCCLXXX. <lb/></s> <s xml:id="echoid-s35" xml:space="preserve">integram librorum familiam fuiſſe viſam nemo prodidit, ſed quatuor <lb/>poſterioribus inuidioſa vetuſtate diuulſis, priores tantùm meliori con-<lb/>cordia ſuperfuiſſe creditum hactenus.</s> <s xml:id="echoid-s36" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s37" xml:space="preserve">Hanc ego notitiam tacitus intra me condidi, atque iam tum, cum <pb file="0012" n="12" rhead=""/> prima Conicæ Matheſeos elementa balbutire didici, annis duodeui-<lb/>ginti iam expletis, ſtatim ea cura in animo ſedit (mihi equidem vni <lb/>placere certum, ac prurienti genio obſecundare) inueſtigandi quid <lb/>APOLLONIO propoſitum in libris deperditis, & </s> <s xml:id="echoid-s38" xml:space="preserve">qua in parte Co-<lb/>nicæ theoriæ verſarentur. </s> <s xml:id="echoid-s39" xml:space="preserve">Quinti autem libri hypotheſis præcipuè <lb/>me trahebat, vbi ex prima APOLLONII epiſtola ad Eudemum, de <lb/>MINIMIS, & </s> <s xml:id="echoid-s40" xml:space="preserve">MAXIMIS magna ſui parte agi non ignorabatur, de-<lb/>que MINIMIS, & </s> <s xml:id="echoid-s41" xml:space="preserve">MAXIMIS lineis ad ipſas Coni-ſectionum peri-<lb/>pherias, referente Eutocio. </s> <s xml:id="echoid-s42" xml:space="preserve">Qua autem ratione, aut quid ſpeciatim <lb/>colligeretur in ipſo quinti argumento, diuinationi, & </s> <s xml:id="echoid-s43" xml:space="preserve">coniecturæ tan-<lb/>tùm relinquebatur. </s> <s xml:id="echoid-s44" xml:space="preserve">In hac ergo cogitatione defixus, ſuſceptam Spar-<lb/>tam exornare vehementiori in dies ſtudio contendebam, ac biennij <lb/>ſpatio cæmenta abundè creuer@nt hiſce mox libris condendis; </s> <s xml:id="echoid-s45" xml:space="preserve">noua <lb/>ſcilicet quotidie ſuccurrebant huius diuinationis occaſione, quæ cupi-<lb/>ditatem, & </s> <s xml:id="echoid-s46" xml:space="preserve">laborem intenderent, donec paulatim in hunc, maiorem-<lb/>que numerum aucta, ad vniuerſam de MAXIMIS, & </s> <s xml:id="echoid-s47" xml:space="preserve">MINIMIS per-<lb/>tractationem ſe ſe extenderint. </s> <s xml:id="echoid-s48" xml:space="preserve">Sed vix diuinæ huius Geometriæ au-<lb/>guſtiſſimum limen ſubieram, cum inuidis caſibus, domeſticis præſer-<lb/>tim turbamentis iactatus, pedem illinc cogor referre, indolique reſ-<lb/>ponſare; </s> <s xml:id="echoid-s49" xml:space="preserve">per trina iam luſtra in alias curas proiectus, quæ inuita Ma-<lb/>theſi ſuſcipiuntur. </s> <s xml:id="echoid-s50" xml:space="preserve">Quare, & </s> <s xml:id="echoid-s51" xml:space="preserve">priuatis concoquendis negotijs diſten-<lb/>tus, & </s> <s xml:id="echoid-s52" xml:space="preserve">publicis auocatus, dum alia qualiſcunque operæ meæ obſequia <lb/>SERENISS. </s> <s xml:id="echoid-s53" xml:space="preserve">MAGNODVCI præſtare debeo, morbis ad hæc ſæpe <lb/>incurrentibus; </s> <s xml:id="echoid-s54" xml:space="preserve">atque incerta vſus valetudine, non hanc tantùm de <lb/>MAXIMIS, & </s> <s xml:id="echoid-s55" xml:space="preserve">MINIMIS ritè diſponere, ac perficere quiui, verùm <lb/>nec aliam vllam è geometricis meis commentationibus; </s> <s xml:id="echoid-s56" xml:space="preserve">quarum ta-<lb/>men, nec pauca ſchedulis commendaueram, vti per tempus ſubſe-<lb/>ciuum, & </s> <s xml:id="echoid-s57" xml:space="preserve">curis ſubtractum multiplicibus, furari induſtriam licuerat. <lb/></s> <s xml:id="echoid-s58" xml:space="preserve">Id vnum ſolatio fuit, hæc cum Amicis participaſſe honeſtiſſimis, ſicu-<lb/>ti factum memini tredecim plus minus ab hinc annis: </s> <s xml:id="echoid-s59" xml:space="preserve">cum Amicis in-<lb/>quam, & </s> <s xml:id="echoid-s60" xml:space="preserve">huiuſce ſtudij amatoribus, quibuſque haud retuſum erat pa-<lb/>latum ad noua hæc veritatum ſcitamenta. </s> <s xml:id="echoid-s61" xml:space="preserve">Vnus vtinam, vt credere <lb/>pium eſt, tardior Diuorum Comes (cur autem inuideo?) </s> <s xml:id="echoid-s62" xml:space="preserve">mihi teſtis <lb/>ſupereſſet Amicorum optimus, ac ſuauiſſimus, cui nihil iam eſt, quod <lb/>pro illius meritis in me ingentibus reddam, præter grati animi inge-<lb/>nuam profeſſionem, ac ſi quid mihi erit vnquam voc (is), aut ſoni præ-<lb/>ſtantiſſimarum eius virtutum fidam omni tempore commemorationẽ <lb/>Braccium loquor Manettum, cuius laudes piaculum eſt ignorare, ſiue <lb/>generis nobilitatem, cum morum elegantia ſumma probitate coniun- <pb file="0013" n="13" rhead=""/> ctam, ſiue eruditionis ornamenta cum Mathematicæ ſtudio, ſcientia-<lb/>que ſpectemus; </s> <s xml:id="echoid-s63" xml:space="preserve">dicam cumulatiſſimè, haud vltimum inter Audito-<lb/>res Galilei Galilei: </s> <s xml:id="echoid-s64" xml:space="preserve">quantum Heroa nomina? </s> <s xml:id="echoid-s65" xml:space="preserve">quantum Florentiæ <lb/>decus, lumen ſeculi, ingeniorum phœnicem, ſydus, Solemq; </s> <s xml:id="echoid-s66" xml:space="preserve">vniuer-<lb/>ſæ Matheſeos? </s> <s xml:id="echoid-s67" xml:space="preserve">quale dixerim numen, ac genium corrigendæ Geogra-<lb/>phię, Aſtronomię nouis phænomenis ope teleſcopij detectis illuſtran-<lb/>dæ vindicandęque Philoſophiæ, in orbis admirationem, ac poſteri-<lb/>tatis regulam natum? </s> <s xml:id="echoid-s68" xml:space="preserve">Ex huius officina prodiens Manettus, non ali-<lb/>ter coloratus apparuiſſe debuit. </s> <s xml:id="echoid-s69" xml:space="preserve">Quod vel mihi æternum incutiat ru-<lb/>borem, ac morſu pœnitentiæ aſſiduò animum lancinet, ſi tantillum <lb/>cogito profectum meum ſub eodem Præceptore Galileo, ad cuius ſa-<lb/>pientiſſimi oris dictata, laris, & </s> <s xml:id="echoid-s70" xml:space="preserve">menſæ, horarumque omnium com-<lb/>munionem per annos fere tres interiùs admitti contigerit.</s> <s xml:id="echoid-s71" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s72" xml:space="preserve">Teſtis alter accedat, quem vocare ad officium poſſit Prætor inco-<lb/>lumem, ac præſentem (ita illum fata diu ſeruent) Illuſtriſſimus, & </s> <s xml:id="echoid-s73" xml:space="preserve">Cla-<lb/>riſſimus Senator Andreas Arrighettus, cum quo dudum meos hoſce <lb/>labores communicatos volui, eiuſque examini, atque emunctiſſimo <lb/>iudicio ſubmittere; </s> <s xml:id="echoid-s74" xml:space="preserve">vt ille non tantùm eo tempore, ſed hodie quoque <lb/>Conicas diſciplinas memoriæ feliciter recolit, quas Iuuenis attentè <lb/>excolebat, cum totus Mathematicis addictus artibus eundem Gali-<lb/>leum aſſectabatur. </s> <s xml:id="echoid-s75" xml:space="preserve">Atqui ob hanc eximiam laudem, ac reliquas vir-<lb/>tutes illuſtribus hodie, primæque notæ muneribus meritò in Patria <lb/>fungitur.</s> <s xml:id="echoid-s76" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s77" xml:space="preserve">Ab eadem claſſe alium arceſſo, qui pro me aram tangat; </s> <s xml:id="echoid-s78" xml:space="preserve">Florenti-<lb/>num Patricium Carolum Datum: </s> <s xml:id="echoid-s79" xml:space="preserve">illum Matheſeos, illum liberæ, in-<lb/>deprauatæq; </s> <s xml:id="echoid-s80" xml:space="preserve">Philoſophiæ nobilem amatorem; </s> <s xml:id="echoid-s81" xml:space="preserve">cuius in ore, Græca, <lb/>Latina, Etruſca ſedet facundia; </s> <s xml:id="echoid-s82" xml:space="preserve">quem vnum inter pauciſſimos huiuſce <lb/>Vrbis demiror, qui & </s> <s xml:id="echoid-s83" xml:space="preserve">ſuæ eruditionis exemplo, & </s> <s xml:id="echoid-s84" xml:space="preserve">opera, fauore, of-<lb/>ficijs in alios, genus omne bonarum artium earundemque cultores <lb/>mirificè amplectatur, ac foueat. </s> <s xml:id="echoid-s85" xml:space="preserve">Nouit Italia, nouit Europa homi-<lb/>nem, noſcet breui vniuerſus literatorum Orbis ex amœniſſimis do-<lb/>ctiſſimiſque lucubrationibus, quas ipſe in dies eruditiſſimè molitur.</s> <s xml:id="echoid-s86" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s87" xml:space="preserve">Hic, de meis hiſce fœtibus Parente ipſo magis ſollicitus, quoties <lb/>verecundiam hanc meam, edendique moroſitatem increpuit? </s> <s xml:id="echoid-s88" xml:space="preserve">quoties <lb/>deſidiam, metumque exprobrauit? </s> <s xml:id="echoid-s89" xml:space="preserve">quoties monuit vt puſillum ali-<lb/>quod, dummodo nouum populi iudicio committerem? </s> <s xml:id="echoid-s90" xml:space="preserve">quoties à <lb/>multis annis refractario pudori calcar hortationis impegit, vt ab hoc <lb/>ſaltem Commentario de MAXIMIS, & </s> <s xml:id="echoid-s91" xml:space="preserve">MINIMIS periclitari famam <lb/>inciperem, quem magis affectum compoſitumque ſciebat? </s> <s xml:id="echoid-s92" xml:space="preserve">At ego <pb file="0014" n="14" rhead=""/> nihil edere obſtinatus, moliri aliquid lætus, ingenium, geniumque <lb/>meum ea cunctatione paſcebam; </s> <s xml:id="echoid-s93" xml:space="preserve">Amicos verò cariores detinebam <lb/>noui ſubinde aliquid è meis nugis ad eorum examen afferendo.</s> <s xml:id="echoid-s94" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s95" xml:space="preserve">Sed circa initium proximè elapſi Menſis Iunij, currentis anni 1658. <lb/></s> <s xml:id="echoid-s96" xml:space="preserve">Ioannes Alphonſus Borellus Piſis reuerſus, qua in Vrbe, & </s> <s xml:id="echoid-s97" xml:space="preserve">Academia <lb/>Clariſſimus Matheſeos Profeſſor publicè docet, Romam cogitabat. </s> <s xml:id="echoid-s98" xml:space="preserve"><lb/>Cauſa illi profectionis mihi hæc longiùs narrandi.</s> <s xml:id="echoid-s99" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s100" xml:space="preserve">Inter cætera Auguſtæ Domus inſtrumenta, quibus SEREN. </s> <s xml:id="echoid-s101" xml:space="preserve">FER-<lb/>DINANDVS II. </s> <s xml:id="echoid-s102" xml:space="preserve">MAGNVS ETRVRIÆ DVX, vel ad inuidiam <lb/>potentiſſimorum Regum prætiosè nobilitatur, loculi aſſeruantur Co-<lb/>dicum MM. </s> <s xml:id="echoid-s103" xml:space="preserve">SS. </s> <s xml:id="echoid-s104" xml:space="preserve">quos è Medicea Romæ Bibliotheca magnis pridem <lb/>ſumptibus collectos Florentiam tranſtulerunt. </s> <s xml:id="echoid-s105" xml:space="preserve">Arabicus inter hos <lb/>comparebat latina ſupernè inſcriptione. </s> <s xml:id="echoid-s106" xml:space="preserve">APOLLONII PERGÆI <lb/>CONICORVM LIBRI OCTO.</s> <s xml:id="echoid-s107" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s108" xml:space="preserve">_Exclamare libet populus, quod clamat oſiri_ <lb/> <anchor type="note" xlink:label="note-0014-01a" xlink:href="note-0014-01"/> _Inuento._</s> <s xml:id="echoid-s109" xml:space="preserve"/> </p> <div xml:id="echoid-div6" type="float" level="2" n="1"> <note position="left" xlink:label="note-0014-01" xlink:href="note-0014-01a" xml:space="preserve">Iuu. Sat. 8.</note> </div> <p> <s xml:id="echoid-s110" xml:space="preserve">Hunc Borello ſæpius tractare licuit, ſæpius diligenti oculo intueri. <lb/></s> <s xml:id="echoid-s111" xml:space="preserve">E´ numero, ac dictinctione librorum, è collatione diagrammatum, <lb/>quæ proximè congruebant tum in Arabico, tum in prioribus quatuor, <lb/>quos antea habebamus, atque è reliquorum tandem examine, quibus <lb/>conſimilis facies, ſimiliaque lineamenta Conica, haud immeritò co-<lb/>nijciebat integros eſſe APOLLONII libros diu deploratos, diu re-<lb/>quiſitos.</s> <s xml:id="echoid-s112" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s113" xml:space="preserve">Orat igitur SERENISS. </s> <s xml:id="echoid-s114" xml:space="preserve">MAGNVMDVCEM, adnitenteque SE-<lb/>RENISS. </s> <s xml:id="echoid-s115" xml:space="preserve">LEOPOLDO FRATRE, Parente muſarum inclyto, vni-<lb/>co, atq; </s> <s xml:id="echoid-s116" xml:space="preserve">aureo, ſi non aurei ſeculi Mecœnate; </s> <s xml:id="echoid-s117" xml:space="preserve">exorat ſibi, vt Romam <lb/>deferre liceat, tum APOLLONIVM, tum libellos alios quoſdam <lb/>geometricos, interpretem illic facilè nacturus inter Viros Propagan-<lb/>dæ Fidei, cui fidem veri detectam penitus exploratamque deberet.</s> <s xml:id="echoid-s118" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s119" xml:space="preserve">Commodùm Florentiæ peregrinabantur Maronitæ nonnulli, quos <lb/>huic operæ aptos ſtatim ſenſit PRINCEPS idem LEOPOLDVS. <lb/></s> <s xml:id="echoid-s120" xml:space="preserve">Accerſiti coram interpretantur, vt mihi narratũ eſt. </s> <s xml:id="echoid-s121" xml:space="preserve">Ex proęmio Ope-<lb/>ris, & </s> <s xml:id="echoid-s122" xml:space="preserve">cuiuſque libri initio, Propoſitionumque aliquot explanatione <lb/>rem ſicuti erat agnoſcunt; </s> <s xml:id="echoid-s123" xml:space="preserve">præter quatuor iam editos APOLLONII <lb/>libros, tres quoque proximos poſteriores adeſſe, compendij tamen <lb/>factos, neſcio cuius Arabis diligentia. </s> <s xml:id="echoid-s124" xml:space="preserve">Nunquam antea huc penetra-<lb/>tum, aut cognitionis tam certæ lucrifactum, quamlibet aliàs Viris, & </s> <s xml:id="echoid-s125" xml:space="preserve"><lb/>Arabicæ linguæ peritis, & </s> <s xml:id="echoid-s126" xml:space="preserve">Geometriæ conſultiſſimis ſæpe conat is <lb/>eruere: </s> <s xml:id="echoid-s127" xml:space="preserve">accurrante pręſertim SERENISS. </s> <s xml:id="echoid-s128" xml:space="preserve">eodem LEOPOLDO, <pb file="0015" n="15" rhead=""/> cuius illa inter innumeras magnanimo in pectore cura adoleſcit, noui <lb/>inſtar Triptolemi ſparſis literarum, ac beneficientiæ ſeminibus, morta-<lb/>le genus quotidre altiùs demereri. </s> <s xml:id="echoid-s129" xml:space="preserve">Indicante autem Maronita adeſſe <lb/>Romæ, vbi per Æſtatem agere Borellus decreuerat, Abrahamum Ec-<lb/>chellenſem natione Arabem, linguarum verò orientalium peritia op-<lb/>pidò celebrem, neq; </s> <s xml:id="echoid-s130" xml:space="preserve">Matheſeos ignarum; </s> <s xml:id="echoid-s131" xml:space="preserve">tunc idem Borellus (quan-<lb/>doque SERENISS. </s> <s xml:id="echoid-s132" xml:space="preserve">MAGNODVCI placuiſſet, APOLLONIVM, <lb/>ac reliqua ſcripta fidei ſuæ committere, & </s> <s xml:id="echoid-s133" xml:space="preserve">Abrahamo ocium foret in-<lb/>terpretandi) ſuam vltrò operam in rebus geometricis adhibere polli-<lb/>citus eſt. </s> <s xml:id="echoid-s134" xml:space="preserve">Satis ſuperque ſe adprobauerat Abrahami peritia, qui lin-<lb/>guarum orientalium Doctor, tunc Romæ, olim in Piſano Lyceo me-<lb/>ruerat. </s> <s xml:id="echoid-s135" xml:space="preserve">Nec minus ſpectata erat ſuæ SERENISSIMÆ CELSITV-<lb/>DINI Borelli præſtantia in geometricis, ac philoſophicis ſtudijs.</s> <s xml:id="echoid-s136" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s137" xml:space="preserve">Non cunctanter ergo SERENISS. </s> <s xml:id="echoid-s138" xml:space="preserve">MAGNVSDVX ſcripta Borel-<lb/>lo credidit, & </s> <s xml:id="echoid-s139" xml:space="preserve">qua ſolet auguſta ſapientia bonas artes tutari, ac foue-<lb/>re, operis aggreſſionem nutu firmat, ſuique SERENISSIMI NO-<lb/>MINIS auſpicio, ac maieſtate fundari permittit.</s> <s xml:id="echoid-s140" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s141" xml:space="preserve">Hæc omnia acta ſunt intra dies octo, vel minus, quibus Borellus <lb/>Florentiæ permanſit. </s> <s xml:id="echoid-s142" xml:space="preserve">Ego hinc procul, cum inſigne hoc cemelium <lb/>Reipublicæ literariæ detectum. </s> <s xml:id="echoid-s143" xml:space="preserve">Reuerſo, ſeduli Amici ſtatim nun-<lb/>ciant, ac Borellus deinceps rem totam mihi ore confirmat, paulò an-<lb/>te quàm peteret Romam. </s> <s xml:id="echoid-s144" xml:space="preserve">Exultabam animo, ac plenus gaudij geſtie-<lb/>bam, fortunatum verè me ſentiens, quod hac ætate ſpirarem, cum <lb/>magnus Geometriæ ſpiritus redderetur hoc reperto theſauro.</s> <s xml:id="echoid-s145" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s146" xml:space="preserve">Nec propterea ceſſabant Amici, quibus res meę cordi erant, horta-<lb/>tiones, ac ſtimulos ſubdere, vt hanc ſaltem de MAXIMIS, & </s> <s xml:id="echoid-s147" xml:space="preserve">MINI-<lb/>MIS lucubrationem publici iuris facerem; </s> <s xml:id="echoid-s148" xml:space="preserve">de qua actum eſſe omnino <lb/>videbam, tunc iam repertis APOLLONII libris; </s> <s xml:id="echoid-s149" xml:space="preserve">atq; </s> <s xml:id="echoid-s150" xml:space="preserve">animum ab ea <lb/>prorſus auerteram tineis iam pertundenda, aut Veneris Marito do-<lb/>nanda. </s> <s xml:id="echoid-s151" xml:space="preserve">Rarò interim, aut nunquam auditum fatebantur, mihi ſanè <lb/>(vt illis videbatur) improſperum: </s> <s xml:id="echoid-s152" xml:space="preserve">non modò librum per duodecim <lb/>iam ſecula conſepultum reuiuiſci me viuo, qui eidem aliquatenus ſup-<lb/>plendo non indiligenter vacaueram; </s> <s xml:id="echoid-s153" xml:space="preserve">ſed & </s> <s xml:id="echoid-s154" xml:space="preserve">illud damnabant qualeſ-<lb/>cunque hos labores meos delituiſſe, qui diu pridem vulgari, ante <lb/>APOLLONIVM repertum, ac ſtudioſorum manibus teri potuiſſent. <lb/></s> <s xml:id="echoid-s155" xml:space="preserve">Acriùs inquam inſtare Amici, neque incitamenta remittere; </s> <s xml:id="echoid-s156" xml:space="preserve">vno ore <lb/>adhortari, vt properatò colligerem, diſponerem, meorumque edi-<lb/>tionem anteuerterem. </s> <s xml:id="echoid-s157" xml:space="preserve">Non deerant autem illis ſpecioſa acumina ad <lb/>impellendum. </s> <s xml:id="echoid-s158" xml:space="preserve">Quod enim ad me; </s> <s xml:id="echoid-s159" xml:space="preserve">priùs fuiſſe hæc excogitata, quàm <pb file="0016" n="16" rhead=""/> illa APOLLONII reperta. </s> <s xml:id="echoid-s160" xml:space="preserve">Facilè etiam perſuadere ignarum me, <lb/>vel ipſius Arabici alphabeti, nec vnquam mihi tractatas, aut cogni-<lb/>tas noui libri figuras. </s> <s xml:id="echoid-s161" xml:space="preserve">Eſto aiebant me tantùm collineaſſe ad eundem <lb/>cum APOLLONIO ſcopum (quamuis latè ſe fundat mea de MA-<lb/>XIMIS, & </s> <s xml:id="echoid-s162" xml:space="preserve">MINIMIS ratiocinatio) non ne plures vię eandem ducunt <lb/>Corinthum? </s> <s xml:id="echoid-s163" xml:space="preserve">Quod ſi ab eo penitus abeam, dum plura conſector <lb/>geometrica, emolumenti inde tamen aliquid accedet literis, ac eo <lb/>ſaltim nomine, quia nouum commendabitur.</s> <s xml:id="echoid-s164" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s165" xml:space="preserve">Fateor autem, mihi alioquin pertinaci deditionem hæc exprimere <lb/>incipiebant: </s> <s xml:id="echoid-s166" xml:space="preserve">vehementiùs tamen a criores ſtimuli aliundè ac cedentes: <lb/></s> <s xml:id="echoid-s167" xml:space="preserve">ſed digito compeſce labellum.</s> <s xml:id="echoid-s168" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s169" xml:space="preserve">Inter hæc Borellus ſub octaua Iunij, ni fallor, Romam contendit, <lb/>cum ſuo illo diues noui APOLLONII viatico. </s> <s xml:id="echoid-s170" xml:space="preserve">At ego delinita ob-<lb/>ſtinatione meis velut ad lucem diſponendis ſenſim incumbo, quod <lb/>Viris CCLL. </s> <s xml:id="echoid-s171" xml:space="preserve">Senat. </s> <s xml:id="echoid-s172" xml:space="preserve">Arrighetto, ac Dato bona fide patefacio; </s> <s xml:id="echoid-s173" xml:space="preserve">nec prę-<lb/>ſtantiſſimo Adoleſcenti Laurentio Magalotto celatum volui, inſimul <lb/>ratus, amicitiæ candori labem inferre, ſi hæc mea qualiacunque in-<lb/>uenta feliciſſimum, atque admirabile prorſus ingenium latuiſſent, <lb/>Mathemati cis non minus, quàm Philoſophicis, atque Anatomicis <lb/>ſtudijs impensè addictum; </s> <s xml:id="echoid-s174" xml:space="preserve">Iuriſprudentiæ ſacris initiatum; </s> <s xml:id="echoid-s175" xml:space="preserve">Muſis, <lb/>quà latinis, quà Etruſcis apprimè carum; </s> <s xml:id="echoid-s176" xml:space="preserve">ad omnia egregia æque na-<lb/>tum, nulliſque demum equeſtrium exercitationum decoribus deſtitu-<lb/>tum, qui ingenuum, & </s> <s xml:id="echoid-s177" xml:space="preserve">ornatiſſimum Patricium decent, è cuius tam <lb/>clara Adoleſcentię Aurora fulgentiſſimum Virilitatis meridiem Patria <lb/>hæc meritò auguratur.</s> <s xml:id="echoid-s178" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s179" xml:space="preserve">Sic pluteos, & </s> <s xml:id="echoid-s180" xml:space="preserve">ſcrinia compilans mea confuſas pagellas in melio-<lb/>rem ordinem digero, aptiora huic tractatui ſeligo, atque in claſſes <lb/>partita tribus diſtinguo faſciculis.</s> <s xml:id="echoid-s181" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s182" xml:space="preserve">Sed interim agrum petens Solem cogor pati noxium, & </s> <s xml:id="echoid-s183" xml:space="preserve">immodi-<lb/>cum, qui febri ſtatim iniecta acutiſſima penè ad necem me afflixit; <lb/></s> <s xml:id="echoid-s184" xml:space="preserve">fuitque dies Iunij decimus octauus cum quindenis alijs inter meos <lb/>egritudinum faſtos, magnis februalibus nimiùm quantùm nefaſtos. </s> <s xml:id="echoid-s185" xml:space="preserve"><lb/>Diù inops virium omnem ſtudiorum curam abieceram: </s> <s xml:id="echoid-s186" xml:space="preserve">nec caput, <lb/>nec mens conſtabat paginis recenſendis, quæ multo punice, multaq; </s> <s xml:id="echoid-s187" xml:space="preserve"><lb/>litura indigebant, multa etiam perſcriptione; </s> <s xml:id="echoid-s188" xml:space="preserve">quippe adumbraueram <lb/>meditationes, & </s> <s xml:id="echoid-s189" xml:space="preserve">confuſanea opera, nequid interim deperiret, tan-<lb/>tummodo innueram.</s> <s xml:id="echoid-s190" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s191" xml:space="preserve">Neſcio autẽ quo pacto labores hi mei SERENISS. </s> <s xml:id="echoid-s192" xml:space="preserve">LEOPOLDO <lb/>ſuboluere, qui partitè mox de tota illorum ratione, ac proceſſu à me <pb file="0017" n="17" rhead=""/> condocefactus, non animos tantùm mihi fecit, ſed iuſſit, vt omnibus <lb/>modis publicarem. </s> <s xml:id="echoid-s193" xml:space="preserve">Verùm neceſſe eſſe prorſus admonuit, à nemine <lb/>ignorari diu mihi fuiſſe in pugillaribus meis hunc tractatum affectum <lb/>ante APOLLONII libros nuper detectos; </s> <s xml:id="echoid-s194" xml:space="preserve">ac prudenter ſuggeſſit pu-<lb/>blica teſtatione fidem confeſtim facere ſcriptis meis, quatenus ſaltim <lb/>conſtaret à me priùs detecta, atq; </s> <s xml:id="echoid-s195" xml:space="preserve">habita, quàm vllus Arabici APOL-<lb/>LONII apex in latinum verteretur. </s> <s xml:id="echoid-s196" xml:space="preserve">Adiecit ſe veritatis prædem af-<lb/>futurum vbi opus eſſet, me præter Arabicæ linguæ ignorationem nun-<lb/>quam APOLLONIVM hunc cõtrectaſſe, aut particulare quidquam <lb/>ex eo nouiſſe. </s> <s xml:id="echoid-s197" xml:space="preserve">Neque hac ſteterunt memoranda SERENISS. </s> <s xml:id="echoid-s198" xml:space="preserve">CEL-<lb/>SIT. </s> <s xml:id="echoid-s199" xml:space="preserve">beneficia, vt æquiſſimæ cauſæ patrocinaretur. </s> <s xml:id="echoid-s200" xml:space="preserve">Ne quà ſuſpicio-<lb/>nis labecula (ſi qui forte ſunt) parùm æquos mihi homines nutriat, <lb/>SERENISS. </s> <s xml:id="echoid-s201" xml:space="preserve">idem PRINCEPS videre ipſe, ac perpendere voluit <lb/>enunciata omnia, ac lineas veluti numerare, quæcunque huic tracta-<lb/>tioni inſererentur, ac ſingulis faſciculis, Mediceo ante ſigillo obſigna-<lb/>tis teſtationem inuictam his verbis propria manu exaratis inſculpere.</s> <s xml:id="echoid-s202" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s203" xml:space="preserve">In primo.</s> <s xml:id="echoid-s204" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s205" xml:space="preserve">Adì 8. </s> <s xml:id="echoid-s206" xml:space="preserve">Luglio 1658. </s> <s xml:id="echoid-s207" xml:space="preserve">furon veduti da me gli appreſſo numero quarantotto <lb/>mezi fogli di dimoſtrazioni geometriche d´ vn trattato de MASSIMI, e <lb/>MINIMI intorno alle Sezioni Coniche, di mano di Vincenzio Viuiam, fer-<lb/>mati col mio Sigillo.</s> <s xml:id="echoid-s208" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div8" type="section" level="1" n="7"> <head xml:id="echoid-head8" xml:space="preserve">Il Principe Leopoldo mano prop.</head> <p> <s xml:id="echoid-s209" xml:space="preserve">In ſecundo verò.</s> <s xml:id="echoid-s210" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s211" xml:space="preserve">Adì 8. </s> <s xml:id="echoid-s212" xml:space="preserve">Luglio 1658. </s> <s xml:id="echoid-s213" xml:space="preserve">furon veduti da me gli appreſſo numero cinquan-<lb/>totto mezi fogli di dimoſirazioni geometriche intorno à materie Coniche atte-<lb/>nenti al trattato de MASSIMI, e MINIMI, di mano di Vincenzio Vi-<lb/>uiani, fermati col mio Sigillo.</s> <s xml:id="echoid-s214" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div9" type="section" level="1" n="8"> <head xml:id="echoid-head9" xml:space="preserve">Il Principe Leopoldo mano prop.</head> <p> <s xml:id="echoid-s215" xml:space="preserve">In tertio denique.</s> <s xml:id="echoid-s216" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s217" xml:space="preserve">Adì 8. </s> <s xml:id="echoid-s218" xml:space="preserve">Luglio 1658. </s> <s xml:id="echoid-s219" xml:space="preserve">furon veduti da me gli appreſſo numero ſeſſantano-<lb/>ue mezi fogli di dimoſtrazioni geometriche d´ vn trattato de MASSIMI, <lb/>e MINIMI intorno a Problemi, e Teoremi varij, il tutto, come ne gli al-<lb/>tri faſci ſcritto in forma di bozza, di mano di Vincenzio Viuiani, fermati <lb/>col mio S gillo.</s> <s xml:id="echoid-s220" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div10" type="section" level="1" n="9"> <head xml:id="echoid-head10" xml:space="preserve">Il Principe Leopoldo mano prop.</head> <p> <s xml:id="echoid-s221" xml:space="preserve">Tam ſapientis, tam inclyti, tam generoſi Principis verendo teſti-<lb/>monio probatus, fauſtoque iuſſu excitatus, quanta animi alacritate <lb/>opus aggredior, exactiori forma, atque ordine contexendum.</s> <s xml:id="echoid-s222" xml:space="preserve"/> </p> <pb file="0018" n="18" rhead=""/> <p> <s xml:id="echoid-s223" xml:space="preserve">Roma tunc literæ à Borello vigeſima nona Iunij nunciante inter <lb/>alia, feliciter incæptam APOLLONII verſionem, cuius proxima <lb/>Hebdomade ſpecimen miſſurus foret ad SERENISS. </s> <s xml:id="echoid-s224" xml:space="preserve">LEOPOL-<lb/>DVM, vt illinc de vniuerſo opere ſpem egregiam conciperet. </s> <s xml:id="echoid-s225" xml:space="preserve">Nona <lb/>Iulij à me reſponſum, atque vnà ſignificatum quid ſtatuiſſem de meis <lb/>laboribus publico dandis, pauciſque narratum, quid, quantùmque <lb/>SERENISS. </s> <s xml:id="echoid-s226" xml:space="preserve">LEOPOLDVS egerit, & </s> <s xml:id="echoid-s227" xml:space="preserve">qua eius ſumma benignitate, <lb/>ac præſidio ad hæc animarer. </s> <s xml:id="echoid-s228" xml:space="preserve">Inſimul orabam, ne quid vel minimum, <lb/>poſthac ſuper libris APOLLONII repertis ad me ſcriberet. </s> <s xml:id="echoid-s229" xml:space="preserve">Ijſdem <lb/>præcibus SERENISS. </s> <s xml:id="echoid-s230" xml:space="preserve">LEOPOLDVM adij, vt ſacrum me, atque <lb/>inteſtabilem, & </s> <s xml:id="echoid-s231" xml:space="preserve">omni indignum colloquio cenſeret de eadem re. </s> <s xml:id="echoid-s232" xml:space="preserve">Ite-<lb/>rum Borellus ad me vigeſima eiuſdem Menſis, ſilentium paciſcens, <lb/>atque inſtitutum meum laudans (euicerant quippe Amicorum conſi-<lb/>lia, ac PRINCIPIS iuſſa) Conicas ſpeculationes typis mandandi, <lb/>diſerta ſubdens verba.</s> <s xml:id="echoid-s233" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s234" xml:space="preserve">Ed io trà gli altri teſtifico, che ella non hà hauuto minima notizia di <lb/>queſti vltimi libri d´ APOLLO´NIO.</s> <s xml:id="echoid-s235" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s236" xml:space="preserve">Magnis quotidie incrementis Romano ſemone, vt Romę par erat, <lb/>Grecus Auctor, nuper Arabs loquebatur. </s> <s xml:id="echoid-s237" xml:space="preserve">At Borellus mihi Harpo-<lb/>crates de condicto. </s> <s xml:id="echoid-s238" xml:space="preserve">Florentiam deinde reuertitur exeunte Octobri. <lb/></s> <s xml:id="echoid-s239" xml:space="preserve">Eapſe reditus die, SERENISS. </s> <s xml:id="echoid-s240" xml:space="preserve">MAGNVSDVX (qua in omnes in-<lb/>credibili humanitate ad miraculum vſque, ac diſciplinã Regnantium <lb/>vti ſolet) Borellum, me præſente, de ſilentio admonuit, donec meus <lb/>prodiret liber; </s> <s xml:id="echoid-s241" xml:space="preserve">atqui ille mecum inuiolatè ſeruauit, quod cum alijs <lb/>quoque ab eo factum non dubito.</s> <s xml:id="echoid-s242" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s243" xml:space="preserve">His itaque in anteceſſum fide vti optima maxima expoſitis, abun-<lb/>dè oſtenſum puto, ante APOLLONIVM repertum per trina iam <lb/>luſtra meas haſce, qualeſcunque cogitationes fuiſle lucubratas. </s> <s xml:id="echoid-s244" xml:space="preserve">Inde <lb/>incorruptiſſimi teſtes Arrighettus, & </s> <s xml:id="echoid-s245" xml:space="preserve">Datus adſtruunt. </s> <s xml:id="echoid-s246" xml:space="preserve">Magalottus ab <lb/>ipſa ſtatim inuentione mihi accerſitus confirmat. </s> <s xml:id="echoid-s247" xml:space="preserve">Arabicæ linguæ fa-<lb/>teor ſum ignariſſimus, quod mihi iniurato, vel incredulus credat Apel-<lb/>la. </s> <s xml:id="echoid-s248" xml:space="preserve">Neque APOLLONII poſteriores verſaſſe vnquam libros, aut <lb/>ex ijs me nouiſſe quidquam, etiam Borellus ſponſor accedit.</s> <s xml:id="echoid-s249" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s250" xml:space="preserve">Poteram ſolennem ab his formam teſtandi, ex iure Quiritum exige-<lb/>re huic præfationi ſubſcribendam, ſed omnium inſtar, ac veluti pro <lb/>muro æneo veritatis, extitit mihi SERENISS. </s> <s xml:id="echoid-s251" xml:space="preserve">LEOPOLDI lucidiſ-<lb/>ſima aſſeueratio, cui radios ſuos Apollo ſubmittit, atque illa olim, <lb/>quæ apud Sagram de veritate concedant.</s> <s xml:id="echoid-s252" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s253" xml:space="preserve">Interea non deſinam Lector quin te rogem, vt quæ hìc legeris, ijs <pb file="0019" n="19" rhead=""/> qui non legerint, vbires ferat indicare ne fugias. </s> <s xml:id="echoid-s254" xml:space="preserve">Expedit enim exi-<lb/>ſtimationis meæ cauſa, totam hanc facti ſeriem, quàm latiſſimè in <lb/>vulgus manare; </s> <s xml:id="echoid-s255" xml:space="preserve">alioquin ſilentium hìc perdet Amyclas.</s> <s xml:id="echoid-s256" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s257" xml:space="preserve">Sed inhęreamus ei, quod magis intereſt; </s> <s xml:id="echoid-s258" xml:space="preserve">ſi etenim eũdem, aut prę-<lb/>ter propter cum APOLLONIO ſcopum attigiſſe fors mihi dederit <lb/>(optare debeam, nec ne, equidem neſcio) nemo ſanus ignorat, quid <lb/>æra lupinis diſtent, nemo præſtantiſſimi Scriptoris ingenium, doctri-<lb/>nam, ſoliditatem, nemo tenuitatem meam, & </s> <s xml:id="echoid-s259" xml:space="preserve">curtam domi ſupelle-<lb/>ctilem. </s> <s xml:id="echoid-s260" xml:space="preserve">Ille omnium fermè, qui ante ſe de Conicis ſcripſerunt viden-<lb/>di commoditate vſus; </s> <s xml:id="echoid-s261" xml:space="preserve">ego illius tantùm ductu, & </s> <s xml:id="echoid-s262" xml:space="preserve">auſpicijs mea hæc <lb/>exequi conabar, & </s> <s xml:id="echoid-s263" xml:space="preserve">prioribus quatuor eiuſdem libris, hoc eſt prætio-<lb/>ſiſſimæ veſti, niſi complementum, atque integritatem, ſegmenta, & </s> <s xml:id="echoid-s264" xml:space="preserve"><lb/>lacinias ſaltim adnecterem.</s> <s xml:id="echoid-s265" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s266" xml:space="preserve">Quod ſi contra, vel in totum, vel ex parte ab eiuſdem APOLLO-<lb/>NII inſtituto aberrauero, non tamen erit pœnitendi prorſus laboris, <lb/>noui aliquid in eodem argumento Geometriæ Conicæ per me affulſiſ-<lb/>ſe. </s> <s xml:id="echoid-s267" xml:space="preserve">Neque ignoro multa, ne dicam infinita veritatum genera, admi-<lb/>randaſque MAXIMORVM, & </s> <s xml:id="echoid-s268" xml:space="preserve">MINIMORVM contemplationes à <lb/>me fuiſſe relictas: </s> <s xml:id="echoid-s269" xml:space="preserve">ſed memento benigne Lector, & </s> <s xml:id="echoid-s270" xml:space="preserve">finitum omnibus, <lb/>& </s> <s xml:id="echoid-s271" xml:space="preserve">mihi infirmiſſimum datum ingenium; </s> <s xml:id="echoid-s272" xml:space="preserve">multiſque iam annis (quod ijs <lb/>notum, qui mea norunt) partìm cum morbis, partìm cum morborum <lb/>reliquijs conflictatum, aut curis fuiſſe diſtractum alieniſſimis; </s> <s xml:id="echoid-s273" xml:space="preserve">cum ta-<lb/>men hæc ſtudia magnis olim Auctoribus creuerint, qui ſerenitatem, <lb/>atque ocium, fortunæ lautiori debebant, vel munijs opportuniori-<lb/>bus artes illas excolebant. </s> <s xml:id="echoid-s274" xml:space="preserve">Nec mirere interim ſi tot Menſes excurre-<lb/>rint, ex quo imper ata hæc editio inſtitui cæpit. </s> <s xml:id="echoid-s275" xml:space="preserve">Paucioribus abſolue-<lb/>batur, ſi valetudo, ac quies annuebant. </s> <s xml:id="echoid-s276" xml:space="preserve">Sed neque tu à me expetis <lb/>Lector, neque ego impoſſibilia capeſſo.</s> <s xml:id="echoid-s277" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s278" xml:space="preserve">Vtere interim, ac ſitanti ſunt, fruere primitijs hiſce meis, quales <lb/>iamdiu ſterili in agello prouenerunt, quas tamen non ita extenuabo, <lb/>vt ſolent cæteri, qui pręfantur; </s> <s xml:id="echoid-s279" xml:space="preserve">ſunt enim non mea, ſed Naturæ admi-<lb/>rabilia opera, ac veritates, ſicuti admirabilis illa, ac vera ſemper eſt; <lb/></s> <s xml:id="echoid-s280" xml:space="preserve">ego detexi tantùm, ac geometrico ordine concinnaui.</s> <s xml:id="echoid-s281" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s282" xml:space="preserve">At enim multis alijs erudita hæc inceſſit libido APOLLONII Co-<lb/>nica, qua deficiunt, reſtituendi ſupplendique. </s> <s xml:id="echoid-s283" xml:space="preserve">Et quidem non viles <lb/>animas, ſed mentes nobiliores, atq; </s> <s xml:id="echoid-s284" xml:space="preserve">eminentiſſimi nominis, comper-<lb/>tæq; </s> <s xml:id="echoid-s285" xml:space="preserve">auctoritatis in geometrico puluere exacuit. </s> <s xml:id="echoid-s286" xml:space="preserve">Ex his Abbas Mau-<lb/>rolicus Meſſanenſis, duobus libris, quintum, & </s> <s xml:id="echoid-s287" xml:space="preserve">ſextum APOLLONII <lb/>tunc irrepertos ſupplere, ipſorumq; </s> <s xml:id="echoid-s288" xml:space="preserve">argumenta diuinare conatus eſt, <pb file="0020" n="20" rhead=""/> (quo autem felici euentu equidem neſcio) atq; </s> <s xml:id="echoid-s289" xml:space="preserve">hi libri commentarijs <lb/>ſubijciuntur in quatuor APOLLONII priores. </s> <s xml:id="echoid-s290" xml:space="preserve">Alter fuit Claudius <lb/>Mydorgius Patricius Pariſinus, eiuſdem APOLLONII ſextum, ple-<lb/>no illo exactiſſimæ doctrinæ acumine inueſtigans, quod bini duo libri <lb/>poſtremi è quatuor hactenus à Mydorgio editis ſatis declarant. </s> <s xml:id="echoid-s291" xml:space="preserve">Vter-<lb/>que ſanè tam doctis laboribus magnam ſibi induſtriæ famam circum-<lb/>dedit. </s> <s xml:id="echoid-s292" xml:space="preserve">Non vitio tamen vllus mihi vertat, ſi ijſdem molitionibus Ado-<lb/>leſcentiæ annos ego quoq; </s> <s xml:id="echoid-s293" xml:space="preserve">impenderim. </s> <s xml:id="echoid-s294" xml:space="preserve">Etenim ipſa de MAXIMIS <lb/>& </s> <s xml:id="echoid-s295" xml:space="preserve">MINIMIS ſpeculatio, quo ad me intacta penitus ad hunc diem vo-<lb/>cari poteſt, niſi quid minimum apud quintum eiuſdem Maurolici pro-<lb/>ximis hiſce Menſibus à me notatum excipiam, vti & </s> <s xml:id="echoid-s296" xml:space="preserve">pauca nonnulla <lb/>ſparſim poſtea à me reperta in Atlantico verè opere ſummi Geometræ <lb/>Gregorij à Sancto Vincentio è doctiſſima, ſpectatiſſima, nec vnquam <lb/>ſatis laudata SOCIETATE IESV.</s> <s xml:id="echoid-s297" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s298" xml:space="preserve">Sed velim Lector antequam iſta aggrediar, illa, quæ nuper retuli, <lb/>apud laudatos Auctores adire ne recuſes; </s> <s xml:id="echoid-s299" xml:space="preserve">erit enim mox fortaſſe, vt <lb/>non tota temeritate me oneres.</s> <s xml:id="echoid-s300" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s301" xml:space="preserve">Quæ autem fata meos maneant libellos neſcio ante veſperum.</s> <s xml:id="echoid-s302" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s303" xml:space="preserve">De MAXIMIS, & </s> <s xml:id="echoid-s304" xml:space="preserve">MINIMIS ago; </s> <s xml:id="echoid-s305" xml:space="preserve">MAXIMA non anhelo, de <lb/>MINIMIS cum Prætore non curo; </s> <s xml:id="echoid-s306" xml:space="preserve">ſi vtraque componuntur, aurea <lb/>mediocritas naſcitur; </s> <s xml:id="echoid-s307" xml:space="preserve">hac ero contentus.</s> <s xml:id="echoid-s308" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s309" xml:space="preserve">Neque indictum tandem huius libri titulum volo. </s> <s xml:id="echoid-s310" xml:space="preserve">DIVINATIO-<lb/>NEM voco; </s> <s xml:id="echoid-s311" xml:space="preserve">verè enim ſolius diuinantis eſt, quid ſpeciatim APOL-<lb/>LONIO propoſitum fuerit aſſequi, quaue methodo, ſolo audito no-<lb/>mine de MAXIMIS, & </s> <s xml:id="echoid-s312" xml:space="preserve">MINIMIS. </s> <s xml:id="echoid-s313" xml:space="preserve">Non ſum ego Diſcipulus Tagis, <lb/>aut Verna Sibillę; </s> <s xml:id="echoid-s314" xml:space="preserve">diuinaculum tamen, ac Prophetam dum ago vehe-<lb/>menter cupio, vt hi labores non tantùm in hac florentiſſima Patria <lb/>mea ſint accepti, ſed exteris quoque non iniucundi, omnino autem <lb/>Reipublicæ literariæ vtiles. </s> <s xml:id="echoid-s315" xml:space="preserve">Hæc ſumma votorum. </s> <s xml:id="echoid-s316" xml:space="preserve">Vale.</s> <s xml:id="echoid-s317" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s318" xml:space="preserve">Scribebam Florentiæ Octauo Idus Decembris 1658.</s> <s xml:id="echoid-s319" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s320" xml:space="preserve">TVI</s> </p> <p style="it"> <s xml:id="echoid-s321" xml:space="preserve">Amantiſsimus</s> </p> <p> <s xml:id="echoid-s322" xml:space="preserve">Vincentius Viuiani.</s> <s xml:id="echoid-s323" xml:space="preserve"/> </p> <pb o="1" file="0021" n="21" rhead=""/> </div> <div xml:id="echoid-div11" type="section" level="1" n="10"> <head xml:id="echoid-head11" xml:space="preserve">DE MAXIMIS, ET MINIMIS</head> <head xml:id="echoid-head12" xml:space="preserve">Geometrica diuinatio in V. conic. <lb/>Apoll. Pergæi.</head> <head xml:id="echoid-head13" style="it" xml:space="preserve">LIBER PRIMVS.</head> <head xml:id="echoid-head14" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s324" xml:space="preserve">_A_NTEQV AM inſtitutum opus aggrediamur, ſiquidem in <lb/>ipſo frequenter accider vti, proferreque affectiones propoſi-<lb/>tionum 11. </s> <s xml:id="echoid-s325" xml:space="preserve">12. </s> <s xml:id="echoid-s326" xml:space="preserve">ac 13. </s> <s xml:id="echoid-s327" xml:space="preserve">primi conic. </s> <s xml:id="echoid-s328" xml:space="preserve">non erit fortaſſe omninò <lb/>incongruum meas earundem demonſtrationes hic exhibere, <lb/>quales olim, cum primùm ad elementa conica me conuerte-<lb/>rem, aliter ac breuius vnico tantùm Theoremate concludi poſſe animaduer-<lb/>ti, eaſque proponi enunciationibus, vtirebar genuinis, ac proximis ad trium <lb/>coni-ſectionum, Parabolæ, nempe, Hyperbolæ, & </s> <s xml:id="echoid-s329" xml:space="preserve">Ellipſis laterum inuen-<lb/>tionem. </s> <s xml:id="echoid-s330" xml:space="preserve">Verùm antea mihi detur, vt quibuſdam morem gerens, qui tres<unsure/> <lb/>prædictas Apollonij propoſitiones difſiciles admodum exiſtimant, ob nimium <lb/>in ea vſum 23. </s> <s xml:id="echoid-s331" xml:space="preserve">ſexti Elementorum; </s> <s xml:id="echoid-s332" xml:space="preserve">earundem demonſtr ationes ſingillatim <lb/>afferre poſsim eodem penitus modo, quo aliquibus, voce, & </s> <s xml:id="echoid-s333" xml:space="preserve">ſcriptis expli-<lb/>care ſolitus fui, hoc eſt ſine compoſita proportione, quam, neſcio quaratione <lb/>faſtidiant.</s> <s xml:id="echoid-s334" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s335" xml:space="preserve">Stantibus igitur ijſdem hypoteſibus, expoſitionibus, ac conſtructionibus<unsure/> <lb/>prædictarum Apoll. </s> <s xml:id="echoid-s336" xml:space="preserve">propoſitionum, adhibitiſque figuris, quæ ibi in Comman-<lb/>dini verſione.</s> <s xml:id="echoid-s337" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s338" xml:space="preserve">_QVo ad 11. </s> <s xml:id="echoid-s339" xml:space="preserve">primi conic. </s> <s xml:id="echoid-s340" xml:space="preserve">poſt ea verba_ Rectangulum igitur MLN æquale eſt <lb/>quadrato K L ſequatur ſic.</s> <s xml:id="echoid-s341" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s342" xml:space="preserve">Itaque, quoniam quadratum BC ad rectangulum BAC eſt vt HF <lb/>ad FA ex conſtructione, & </s> <s xml:id="echoid-s343" xml:space="preserve">rectangulum BAC ad rectangulum ACB vt AB <lb/>ad BC, vel vt ablata BF ad ablatam BG, hoc eſt vt reliqua FA ad reliquam <lb/>GC, ſiue ad LN, ergo ex æquo quadratum BC ad rectangulum ACB, vel <lb/>recta BC ad CA, vel BG ad GF, vel ML ad LF, erit vt HF ad LN, ideoque <lb/>rectangulum ſub extremis ML, LN, ſiue quadratum KL æquatur rectangu-<lb/>lo HFL. </s> <s xml:id="echoid-s344" xml:space="preserve">_Vocetur autem huiuſmodi ſectio & </s> <s xml:id="echoid-s345" xml:space="preserve">c._ </s> <s xml:id="echoid-s346" xml:space="preserve">vt ibi vſque ad finem.</s> <s xml:id="echoid-s347" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s348" xml:space="preserve">Quo ad 12. </s> <s xml:id="echoid-s349" xml:space="preserve">primi poſt ea verba, _ergo rectangulum RNS æquale eſt MN qua-_ <lb/>_drato_, ſic dicatur.</s> <s xml:id="echoid-s350" xml:space="preserve"/> </p> <pb o="2" file="0022" n="22" rhead=""/> <p> <s xml:id="echoid-s351" xml:space="preserve">Itaque, quoniam rectangulum BKC ad quadratum AK eſt vt LF ad FH <lb/>per conſtrutionem, vel vt XN ad NH, & </s> <s xml:id="echoid-s352" xml:space="preserve">quadratum AK ad rectangulum <lb/>AKC eſt vt AK ad KC, vel HG ad GC, vel HN ad NS, ergo ex æqualire-<lb/>ctangulum BKC ad rectangulum AKC, ſiue recta BK ad KA, ſiue BG ad <lb/>GF, vel RN ad NF, eſt vt XN ad NS, ac propterea rectangulum ſub extre-<lb/>mis RN, NS, hoc eſt quadratum MN æquale rectangulo ſub medijs XN, NF: <lb/></s> <s xml:id="echoid-s353" xml:space="preserve">_linea igitur MN poteſt ſpatium XF, & </s> <s xml:id="echoid-s354" xml:space="preserve">c._ </s> <s xml:id="echoid-s355" xml:space="preserve">vt ibi vſque ad finem.</s> <s xml:id="echoid-s356" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s357" xml:space="preserve">Quo tandem ad 13. </s> <s xml:id="echoid-s358" xml:space="preserve">primi poſt ea verba _ergo rectangulum PMR æquale eſt_ <lb/>_LM quadrato_ legatur ſic.</s> <s xml:id="echoid-s359" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s360" xml:space="preserve">Cumque ſit rectangulum BKC ad quadratum AK ita HE ad ED ex con-<lb/>ſtrutione, vel XM ad MD, & </s> <s xml:id="echoid-s361" xml:space="preserve">vt quadratum AK ad rectangulum AKC ita <lb/>AK ad KC, vel DG ad GC, vel vt DM ad MR, erit ex æquo rectangulum <lb/>BKC ad rectangulum AKC, vel BK ad KA, ſiue BG ad GE, vel PM ad ME <lb/>vt XM ad MR, quare rectangulum ſub extremis PM, MR, vel quadratum <lb/>ML æquatur rectangulo XME ſub medijs. </s> <s xml:id="echoid-s362" xml:space="preserve">_Liuea igitur LM poteſt ſpatinm_ <lb/>_MO &</s> <s xml:id="echoid-s363" xml:space="preserve">c._ </s> <s xml:id="echoid-s364" xml:space="preserve">vſque ad finem.</s> <s xml:id="echoid-s365" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s366" xml:space="preserve">Sed iam ad propoſitas Apollonij propoſitiones accedamus, quas ſimul ſequenti <lb/>Theoremate amplectemur, itemque ſine compoſita proportione demonſtrabimus.</s> <s xml:id="echoid-s367" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div12" type="section" level="1" n="11"> <head xml:id="echoid-head15" xml:space="preserve">THEOR. I. PROP. I.</head> <p> <s xml:id="echoid-s368" xml:space="preserve">Si conus plano per axem fecetur, fecetur autem & </s> <s xml:id="echoid-s369" xml:space="preserve">altero plano <lb/>baſi coni non æquidiſtante, quorum communis ſectio conueniat, <lb/> <anchor type="note" xlink:label="note-0022-01a" xlink:href="note-0022-01"/> vel cum vnotantum, vel cum vtroque latere trianguli per axem vl-<lb/>tra, vel infra ſui ipſius verticem, planum verò, in quo eſt baſis co-<lb/>ni, & </s> <s xml:id="echoid-s370" xml:space="preserve">ſecans planum, conueniant ſecundum rectam lineam, quæ ſit <lb/>perpendicularis, vel ad baſim trianguli per axem, vel ad eam, quæ <lb/>indirectum ipſi conſtituitur, & </s> <s xml:id="echoid-s371" xml:space="preserve">fiat, vt rectangulum ſegmentorum <lb/>diametri ſectionis inter latera, & </s> <s xml:id="echoid-s372" xml:space="preserve">baſim trianguli per axem interce-<lb/>ptorum, ad rectangulum ſegmentorum baſis, ita ſectionis diameter <lb/>ad aliam: </s> <s xml:id="echoid-s373" xml:space="preserve">recta linea, quę à ſectione coni ducitur æquidiſtans com-<lb/>muni ſectioni plani ſecantis, & </s> <s xml:id="echoid-s374" xml:space="preserve">baſis coni vſque ad ſectionis diame-<lb/>trum, poterit rectangulum adiacens lineæ quarto loco inuentæ, la-<lb/>titudinem habens lineam, quæ ex diametro abſcinditur inter ipſam, <lb/>& </s> <s xml:id="echoid-s375" xml:space="preserve">verticem ſectionis interiectam (ſi tamen ſectionis diameter ęqui-<lb/>diſtet alterutri laterum triãguli per axem) ſed ipſum excedet (ſi cum <lb/>vtroque latere vltra verticẽ conueniat) vel ab eo deficiet, (ſi ijſdem <lb/>lateribus infra verticem occurrat) rectangulo ſimili ſimiliterque po-<lb/>ſito ei, quod continetur prædicto diametri ſegmento, & </s> <s xml:id="echoid-s376" xml:space="preserve">quarta in-<lb/>uenta, iuxta quam poſſunt, quæ ad diametrum applicantur.</s> <s xml:id="echoid-s377" xml:space="preserve"/> </p> <div xml:id="echoid-div12" type="float" level="2" n="1"> <note position="left" xlink:label="note-0022-01" xlink:href="note-0022-01a" xml:space="preserve">Prop. 11. <lb/>12. 13. <lb/>primi co-<lb/>nic.</note> </div> <p> <s xml:id="echoid-s378" xml:space="preserve">SIt conus, cuius vertex A, baſis circulus BC, & </s> <s xml:id="echoid-s379" xml:space="preserve">ſecetur plano per axem, <lb/>quod ſectionem faciat triangulum B A C, ſecetur autem & </s> <s xml:id="echoid-s380" xml:space="preserve">altero <pb o="3" file="0023" n="23" rhead=""/> plano, quorum communis ſectio F G vel alterutri laterum trianguli per <lb/>axem, nempe AC æquidiſtet, vt in prima figura, vel cum vtroque latere in <lb/>F, H, extra verticem coni, vt in ſecunda; </s> <s xml:id="echoid-s381" xml:space="preserve">ſiue infra verticem, vt in tertia, & </s> <s xml:id="echoid-s382" xml:space="preserve"><lb/>quarta conueniat, & </s> <s xml:id="echoid-s383" xml:space="preserve">ſecans planum baſi non æquidiſtet, faciatque ſectio-<lb/>nem in ſuperficie coni lineam MFT, & </s> <s xml:id="echoid-s384" xml:space="preserve">communis ſectio plani ſecantis, atq; <lb/></s> <s xml:id="echoid-s385" xml:space="preserve">eius in quo eſt baſis coni ſit DGE perpendicularis ad baſim trianguli per axẽ <lb/>BC, vel ad eam, quæ indirectum ipſi conſtituitur, vt in quarta figura; </s> <s xml:id="echoid-s386" xml:space="preserve">& </s> <s xml:id="echoid-s387" xml:space="preserve">fiat <lb/>in prima figura, vt quadratum FG, in reliquis verò, vt rectangulum HGF, <lb/> <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/> ad rectangulum BGC, ita linea HF, ſegmentum diam etri ſectionis, ad aliam <lb/>FL, quæ (facilitatis, & </s> <s xml:id="echoid-s388" xml:space="preserve">commoditatis gratia tantùm ad ea, quæ à nobis in po-<lb/>ſterum ſunt pertractanda, non quod hanc, vel aliam poſitionem requirat <lb/>propoſiti demonſtratio, poteſt enim ipſa FL cum diametro FH, ad quemcun-<lb/>que angulum conſtitui) concipiatur applicari ex F, ſectionis vertice, ordi- <pb o="4" file="0024" n="24" rhead=""/> natim ductę DE ęquidiſtans. </s> <s xml:id="echoid-s389" xml:space="preserve">Patet hic ipfam FL ſectionem contingere in L, F<unsure/> <lb/>per 17. </s> <s xml:id="echoid-s390" xml:space="preserve">primi conic. </s> <s xml:id="echoid-s391" xml:space="preserve">(quæ huic aptè præponi poterat, cum ipſa, ope tan-<lb/>tum præcedentium ſeptimæ, & </s> <s xml:id="echoid-s392" xml:space="preserve">decimæ eiuſdem Librl<unsure/> demonſtretur). </s> <s xml:id="echoid-s393" xml:space="preserve">Su-<lb/>matur præterea in ſectione quodlibet punctum M, per quod agatur MN <lb/>æquidiſtans ipſi DE, vel FL, & </s> <s xml:id="echoid-s394" xml:space="preserve">producta conueniat in prima figura cum LV <lb/>parallela ad FG, in reliquis verò cum iuncta HL in X; </s> <s xml:id="echoid-s395" xml:space="preserve">& </s> <s xml:id="echoid-s396" xml:space="preserve">per L, X ipſi F N <lb/>æquidiſtantes ducantur LO, XP. </s> <s xml:id="echoid-s397" xml:space="preserve">Dico lineam MN poſſe rectangulum ſub <lb/>FN, & </s> <s xml:id="echoid-s398" xml:space="preserve">NO<unsure/>, quod quidem adiacet lineæ quarto loco inu<unsure/>entæ FL, latitudinẽ <lb/>habens FN in prima figura, in ſecunda verò prædictum<unsure/> rectangulum exce-<lb/>dens, & </s> <s xml:id="echoid-s399" xml:space="preserve">in tertia, & </s> <s xml:id="echoid-s400" xml:space="preserve">quarta ab eo deficiens rectangulo ſub LO, & </s> <s xml:id="echoid-s401" xml:space="preserve">OX ſimili <lb/>ei, quod ſub HF, & </s> <s xml:id="echoid-s402" xml:space="preserve">FL continetur.</s> <s xml:id="echoid-s403" xml:space="preserve"/> </p> <div xml:id="echoid-div13" type="float" level="2" n="2"> <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a"> <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0023-01"/> </figure> </div> <figure> <image file="0024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0024-01"/> </figure> <pb o="5" file="0025" n="25" rhead=""/> <p> <s xml:id="echoid-s404" xml:space="preserve">Ducatur enim per N linea RNS parallela ad BC, eſt autem & </s> <s xml:id="echoid-s405" xml:space="preserve">MN ipſi DE <lb/>æquidiſtans, quare angulus RNM æqualis <anchor type="note" xlink:href="" symbol="a"/> erit angulo BGD, nempe rectus, <anchor type="note" xlink:label="note-0025-01a" xlink:href="note-0025-01"/> & </s> <s xml:id="echoid-s406" xml:space="preserve">planum tranſiens per MN, RS <anchor type="note" xlink:href="" symbol="b"/> æquidiſtabit plano per BCDE, hoc eſt <anchor type="note" xlink:label="note-0025-02a" xlink:href="note-0025-02"/> baſi coni; </s> <s xml:id="echoid-s407" xml:space="preserve">ſi igitur planum per MNRS producatur ſectio circulus <anchor type="note" xlink:href="" symbol="c"/> erit, cuius <anchor type="note" xlink:label="note-0025-03a" xlink:href="note-0025-03"/> diameter RNS, atque eſt ad ipſam perpendicularis MN, ergo rectangulum <lb/>RNS æquale eſt quadrato MN, vti rectangulum BGC æquale eſt quadra-<lb/>to DG.</s> <s xml:id="echoid-s408" xml:space="preserve"/> </p> <div xml:id="echoid-div14" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0025-01" xlink:href="note-0025-01a" xml:space="preserve">10. Vn-<lb/>dec Elem.</note> <note symbol="b" position="right" xlink:label="note-0025-02" xlink:href="note-0025-02a" xml:space="preserve">15. Vn-<lb/>dec. Elem.</note> <note symbol="c" position="right" xlink:label="note-0025-03" xlink:href="note-0025-03a" xml:space="preserve">4. primi <lb/>conic.</note> </div> <p> <s xml:id="echoid-s409" xml:space="preserve">Iam cum ſit NX parallela ad GV, & </s> <s xml:id="echoid-s410" xml:space="preserve">NS ad GC, erit in prima figura GV <lb/>ad NX, vt GC ad NS, ob æqualitatem; </s> <s xml:id="echoid-s411" xml:space="preserve">in reliquis verò erit GV ad NX, vt <lb/>GH ad HN, vel GC ad NS, ob triangulorum ſimilitudinem; </s> <s xml:id="echoid-s412" xml:space="preserve">quare permu-<lb/>tando in omnibus, GV ad GC, erit vt NX ad NS.</s> <s xml:id="echoid-s413" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s414" xml:space="preserve">Amplius cum in prima figura factum ſit vt quadratum FG ad rectãgulum <lb/>BGC, ſiue ad quadratum GD, ita recta HF ad FL, vel ad GV ei æqualis, ob <lb/>parallelogrammum FV, erit FG ad GV, vt GV ad GD; </s> <s xml:id="echoid-s415" xml:space="preserve">quare rectangulum <lb/>FGV æquatur quadrato DG, ſiue rectangulo BGC. </s> <s xml:id="echoid-s416" xml:space="preserve">Item in reliquis figuris, <lb/>cum factum ſit vt rectangulum HGF, ad rectangulum BGC, ita recta HF ad <lb/>FL, vel HG ad GV, & </s> <s xml:id="echoid-s417" xml:space="preserve">idem rectangulum HGF ad rectangulum FGV ſit vt <lb/>eadem HG ad GV, erit rectangulum BGC æquale rectangulo FGV: </s> <s xml:id="echoid-s418" xml:space="preserve">cum <lb/>ergo in ſingulis figuris rectangulum BGC æquale ſit rectangulo FGV, erit <lb/>BG ad GF, ſiue RN ad NF, vt VG ad GC, ſiue vt XN ad NS: </s> <s xml:id="echoid-s419" xml:space="preserve">rectangulum <lb/>ergo RNS, ſiue quadratum MN æquatur rectangulo XNF. </s> <s xml:id="echoid-s420" xml:space="preserve">Linea igitur MN <lb/>poteſt rectangulum ſub O<unsure/>N, & </s> <s xml:id="echoid-s421" xml:space="preserve">NF, quod adiacet lineæ FL, latitudinem <lb/>habens FN, in prima figura, ſed in ſecunda ipſum rectangulum excedit, & </s> <s xml:id="echoid-s422" xml:space="preserve"><lb/>in tertia & </s> <s xml:id="echoid-s423" xml:space="preserve">quarta ab eodem deficit, rectangulo ſub LO, & </s> <s xml:id="echoid-s424" xml:space="preserve">OX, ſimili ei, <lb/>quod ſub HF, & </s> <s xml:id="echoid-s425" xml:space="preserve">FL continetur. </s> <s xml:id="echoid-s426" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s427" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div16" type="section" level="1" n="12"> <head xml:id="echoid-head16" xml:space="preserve">Definitiones Primæ.</head> <head xml:id="echoid-head17" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s428" xml:space="preserve">Sectio DFE, cuius diameter FG in prima figura æquidiſtat AC vni laterum <lb/>trianguli per axem, vocatur PARABOLE.</s> <s xml:id="echoid-s429" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div17" type="section" level="1" n="13"> <head xml:id="echoid-head18" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s430" xml:space="preserve">Et cuius diameter in ſecunda figura occrrrit vtrique lateri trianguli per axẽ, <lb/>dicitur HYPERBOLE.</s> <s xml:id="echoid-s431" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div18" type="section" level="1" n="14"> <head xml:id="echoid-head19" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s432" xml:space="preserve">Et cuius diameter, in tertia, & </s> <s xml:id="echoid-s433" xml:space="preserve">quarta conuenit cum vtroque latere infra <lb/>verticem trianguli per axem, ELLIPSIS nuncupatur.</s> <s xml:id="echoid-s434" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div19" type="section" level="1" n="15"> <head xml:id="echoid-head20" xml:space="preserve">IV.</head> <p> <s xml:id="echoid-s435" xml:space="preserve">Segmentum verò HF diametri ſectionis inter latera trianguli per axem in-<lb/>terceptum, in ſecunda, tertia, & </s> <s xml:id="echoid-s436" xml:space="preserve">quarta, dicitur LATVS TRANSVER-<lb/>SVM Hyperbolæ, vel Ellipſis, quod in ſequentibus intelligatur ſemper <lb/>extra Hyperbolen ex ipſius vertice in directum poſitum cum diametro, <lb/>licet in conſtructionibus expreſsè non dicatur.</s> <s xml:id="echoid-s437" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div20" type="section" level="1" n="16"> <head xml:id="echoid-head21" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s438" xml:space="preserve">In omnibus autem figuris linea FL, quarto loco inuenta, dicitur LATVS <lb/>RECTVM ſectionis, quod deinceps concipiatur ſemper contingenter <lb/>applicari ex ſectionis vertice, ſiue ordinatim ductis æquidiſtans.</s> <s xml:id="echoid-s439" xml:space="preserve"/> </p> <pb o="6" file="0026" n="26" rhead=""/> </div> <div xml:id="echoid-div21" type="section" level="1" n="17"> <head xml:id="echoid-head22" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s440" xml:space="preserve">Ambo ſimul latera FL, FH, FIGVRÆ LATERA nuncupantur.</s> <s xml:id="echoid-s441" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div22" type="section" level="1" n="18"> <head xml:id="echoid-head23" xml:space="preserve">VII.</head> <p> <s xml:id="echoid-s442" xml:space="preserve">Recta verò LV æquidiſtans diametro ſectionis FG, vt & </s> <s xml:id="echoid-s443" xml:space="preserve">recta HL, figuræ <lb/>latera ſub tendens dicitur FIGVRAM DETERMINANS, ſeu REGV-<lb/>LATRIX, vel REGVLA.</s> <s xml:id="echoid-s444" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div23" type="section" level="1" n="19"> <head xml:id="echoid-head24" xml:space="preserve">VIII.</head> <p> <s xml:id="echoid-s445" xml:space="preserve">Segmenta inſuper diametrorum NF, GF, licet ab ipſo Apollonio dicantur <lb/>latitudines, vocentur potius ALTITVDINES, ita vt NF dicatur altitu-<lb/>do propria ſemi-applicatæ MN &</s> <s xml:id="echoid-s446" xml:space="preserve">c.</s> <s xml:id="echoid-s447" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div24" type="section" level="1" n="20"> <head xml:id="echoid-head25" xml:space="preserve">IX.</head> <p> <s xml:id="echoid-s448" xml:space="preserve">Rectæ autem NX, GV, quæ recto lateri FL, ſiue ordinatim ductis æquidi-<lb/>ſtant, & </s> <s xml:id="echoid-s449" xml:space="preserve">inter ſectionis diametrum, & </s> <s xml:id="echoid-s450" xml:space="preserve">regulam intercipiuntur, vocentur <lb/>LATITVDINES, rectangulorum nempe FNX, FGV, quibus ſemi-ap-<lb/>plicatarum quadrata NM, GD æqualia ſunt oſtenſa, ita vt XM ſit latitu-<lb/>do propria ſemi-applicatæ MN &</s> <s xml:id="echoid-s451" xml:space="preserve">c. </s> <s xml:id="echoid-s452" xml:space="preserve">quæ ſemi-applicatæ indifferenter, <lb/>ac ſępius dicentur applicatæ, velordinatim ductæ.</s> <s xml:id="echoid-s453" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div25" type="section" level="1" n="21"> <head xml:id="echoid-head26" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s454" xml:space="preserve">HInc patet, in quacunque coni-ſectione, quamlibet ſemi-applicatam <lb/>eſſe mediam proportionalem inter propriam altitudinem, propriam-<lb/>que latitudinem: </s> <s xml:id="echoid-s455" xml:space="preserve">hoc eſt quadratum cuiuſcunque ſemi-applicatæ æquari <lb/>rectangulo ſub propria altitudine, ac propria latitudine contento: </s> <s xml:id="echoid-s456" xml:space="preserve">oſtenſum <lb/>eſt enim tàm in Parabola, quàm in Hyperbola, vel Ellipſi, vel circulo, qua-<lb/>dratum ſemi-applicatæ MN æquari rectangulo FX, quod ſub altitudine <lb/>propria FN, ac ſub propria latitudine NX continetur.</s> <s xml:id="echoid-s457" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div26" type="section" level="1" n="22"> <head xml:id="echoid-head27" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s458" xml:space="preserve">HIC animaduertendum eſt in hac propoſitione nos ſub contrariam <lb/>com-ſectionem non excluſiſſe, quam Apollonius in eius quinta <lb/>primi expendens, circulum eſſe demonſtrauit, quoniam ex eo, <lb/>quod ſuperius dictum fuit, elicitur huic etiam competere eandem <lb/>Ellipſis proprietatem, videlicet ordinatè applicatarum potentias æquarire-<lb/>ctangulis, rectæ lineæ quarto loco inuentæ applicatis, latitudinem habentibus <lb/>ea diametri ſegmenta, quæ inter ipſas applicatas, ac ſectionis verticem in-<lb/>tercipiuntur, deficientibuſque rectangulis ſimilibus contento ſub tranſuerſo re-<lb/>ctoque latere, quæ latera in hac ſub contraria ſectione inter ſe ſunt æqualia, ac <lb/>penitùs eadem cum diametro vnius circuli: </s> <s xml:id="echoid-s459" xml:space="preserve">quamobrem circulus nihil aliud <lb/>eſſe videtur quàm Ellipſis æqualium laterum, habens tamen tranſuerſum <lb/>latus, quod vicem gerit axis linearum ad ipſum ordinatè ductarum.</s> <s xml:id="echoid-s460" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s461" xml:space="preserve">Immo ſi noſtri eſſet inſtituti, hic quoque demonſtrare poſſemus non tantum <pb o="7" file="0027" n="27" rhead=""/> omnes Ellipſis affectiones circulo communes eſſe, ſed ferè omnes etiam Hy-<lb/>perbolæ, magnaque pars Parabolæ, præmittendo tamen nouas quaſdam ani-<lb/>maduerſiones, cautioneſque perutiles, nemini, quod ſciam, adhuc cognitæs, <lb/>præcipuèque vtendo methodo ab ipſo Apollonio ſatis diuerſa, certàque indu-<lb/>ſtria propoſitionum figuris characteres diſponendo, ad hoc vt eadem demon-<lb/>ſtratio cuin<unsure/>libet com-ſectioni ſimul inſeruiat, non abſimili modo ab eo, quo <lb/>in ſuperiori Theoremate vſi ſumus, ex quibus maximum doctrinæ conicæ <lb/>compendium oriretur; </s> <s xml:id="echoid-s462" xml:space="preserve">ſed quoniamid, plus laboris, ac temporis, quam in-<lb/>genij requireret, libenter opusrelinquo ijs, quibus multum ocij ſuppetit,& </s> <s xml:id="echoid-s463" xml:space="preserve"><lb/>quos magis iuuat in alienas lucubrationes commentaria ſcribere, quàm vel <lb/>ipſas latiùs promouere, vel nouas meditari, ac geometricè demonſtrare.</s> <s xml:id="echoid-s464" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s465" xml:space="preserve">Quod autem in Apollonij ſubcontraria ſectione tranſuerſum, rectumque <lb/>latus reperiatur eadem methodo, rationeque illorum rectangulorum qua vti-<lb/>mur in præcedenti, quodque hæc ipſa latera inter ſe ſint æqualia manifeſtum <lb/>fiet ex eo, quod mox demonſirabimus non tantum in prædicta ſectione ſub-<lb/>contraria, quæ recta eſt plano trianguli per axem recto plano baſis coni ſcale-<lb/>ni, ſed etiam ei quæ ſecat planum baſis com ſecundum rectam lineam perpen-<lb/>dicularem baſi cuiuſcunq; </s> <s xml:id="echoid-s466" xml:space="preserve">trianguli per axem non iſoſcelis, vel ei, quæ ipſi baſi <lb/>indirectum producitur, dummodò talis ſectio ex ipſomet triangulo, triangu-<lb/>lum auferat ſibi ſimile, ſed ſubcontr ariè poſitum.</s> <s xml:id="echoid-s467" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s468" xml:space="preserve">REpetitis igitur duabus vltimis præcedentibus figuris, intelligatur conũ <lb/>ABC ſcalenum eſſe, ſectumque plano per axem, quodcunque trian-<lb/> <anchor type="figure" xlink:label="fig-0027-01a" xlink:href="fig-0027-01"/> gulum efficiente ABC, dummodo non ſit æquicrure, (quod per doctrinam <lb/>lib. </s> <s xml:id="echoid-s469" xml:space="preserve">ſecundi Sereni, vnicum eſt) habent idcircò vnum latus altero maius, <pb o="8" file="0028" n="28" rhead=""/> ſitque ipſum AB, quod ſecetur quacunque recta linea FN intra angulum. <lb/></s> <s xml:id="echoid-s470" xml:space="preserve">BAC, efficient angulum AFN æquale angulo ACB. </s> <s xml:id="echoid-s471" xml:space="preserve">Iam dico rectam FN <lb/>productam cum reliquo latere AC conuenire, cumque baſi BC ad partem <lb/>minoris lateris AC.</s> <s xml:id="echoid-s472" xml:space="preserve"/> </p> <div xml:id="echoid-div26" type="float" level="2" n="1"> <figure xlink:label="fig-0027-01" xlink:href="fig-0027-01a"> <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0027-01"/> </figure> </div> <p> <s xml:id="echoid-s473" xml:space="preserve">Quoniam cum in triangulo BAC ſint anguli A, C, minores duobus rectis, <lb/>permutato C in F, erunt anguli FAC, AFN duobus rectis minores, ex quo <lb/>rectæ AC, FN conuenient ſimul in H, & </s> <s xml:id="echoid-s474" xml:space="preserve">reliquus angulus ABC in triangu-<lb/>lo BAC æquabitur reliquo angulo AHF in triangulo HAF, hoc eſt triangu-<lb/>la BAC, HAF erunt ſub contrariè poſita. </s> <s xml:id="echoid-s475" xml:space="preserve">Amplius cum ſit BA maior AC, <lb/>erit HA maior AF, propter ſimilitudinem triangulorum BAC, HAF, vnde <lb/>angulus AFH erit maior angulo AHF, ſiue angulo ABC, ſed anguli BFH, <lb/>AFH ſunt duobus rectis æquales, quare anguli BFH, ABC minores erunt <lb/>duobus rectis, ideoqne FH, BC ſimul conuenient, vt in G. </s> <s xml:id="echoid-s476" xml:space="preserve">Nunc verò con-<lb/> <anchor type="figure" xlink:label="fig-0028-01a" xlink:href="fig-0028-01"/> cipiatur per rectam FHG duci planum ſecans triangulum per axem ABC, <lb/>communiſque ſectio huius ſecantis plani cum plano baſis coni ſit recta DGE <lb/>perpendicularis baſi BC trianguli per axem, & </s> <s xml:id="echoid-s477" xml:space="preserve">cum conica ſuperficie ſectio-<lb/>nem efficiens MFTH, cuius diameter ſit FH. </s> <s xml:id="echoid-s478" xml:space="preserve">Itaque iam ſuperiùs oſtenſum <lb/>eſt, ſi fiat vt rectangulum FGH ad rectangulum BGC, ita diameter FH ad <lb/>aliam lineam FL, quæ ex F ordinatim in ſectione ductis æquidiſtet, iunga-<lb/>turque HL, quadratum cuiuſcunque applicatæ MN parallelæ communi ſe-<lb/>ctioni DE, æquari rectangulo NP, applicato rectę FL deficientique rectan-<lb/>gulo LX ſimili rectangulo ſub HF, FL. </s> <s xml:id="echoid-s479" xml:space="preserve">Quod verò talia latera HF, FL inter <lb/>ſe ſint æqualia ita oſtenditur.</s> <s xml:id="echoid-s480" xml:space="preserve"/> </p> <div xml:id="echoid-div27" type="float" level="2" n="2"> <figure xlink:label="fig-0028-01" xlink:href="fig-0028-01a"> <image file="0028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0028-01"/> </figure> </div> <p> <s xml:id="echoid-s481" xml:space="preserve">Cum ſit enim angulus AFH æqualis angulo ACB, erit conſequens BFG <lb/>conſequenti HCG æqualis, eſtque angulus BGF æqualis angulo HGC, cum <lb/>in tertia figura idem ſint, in quarta verò ſint ad verticem, quare in triangu-<lb/>lis BGF; </s> <s xml:id="echoid-s482" xml:space="preserve">HGC circa æquales angulos ad G erunt latera proportionalia, ſiue <lb/>vt FG ad GB ita CG ad GH, ideoque rectangulum FGH æquale erit rectan-<lb/>gulo BGC, ſed vt rectangulum FGH ad BGC, ita tranſuerſum HF ad re- <pb o="9" file="0029" n="29" rhead=""/> ctum FL, iſtaque rectangula æqualia oſtenfa funt, vnde latera quoq; </s> <s xml:id="echoid-s483" xml:space="preserve">HF, <lb/>FL æqualia crunt. </s> <s xml:id="echoid-s484" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s485" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s486" xml:space="preserve">Sed quoniam <anchor type="note" xlink:href="" symbol="a"/> eſt vt tranſuerſum HF ad rectum FL ita rectangulum.</s> <s xml:id="echoid-s487" xml:space="preserve"> HNF ad quadratum NM, atque hæc ipſa latera æqualia ſunt oſtenſa, ergo <lb/> <anchor type="note" xlink:label="note-0029-01a" xlink:href="note-0029-01"/> rectangulum HNF æquabitur quadrato NM; </s> <s xml:id="echoid-s488" xml:space="preserve">quare in qualibet ſubcontra-<lb/>ria ſectione MFTH, deducta, vt in præcedenti, ex triangulo per axem coni <lb/>ſcaleni, quod tamen non ſit æquicrure, rectangula ſub ſegmentis diametri <lb/>ſunt ſemper æqualia quadratis eorum ordinatè applicatarum, quæ quando <lb/>cum diametro FH rectos angulos conſtituent, (quod eueniet cum commu-<lb/>nis ſectio DGE perpendicularis fuerit, non ſolùm baſi BGC trianguli per <lb/>axem, ſed etiam rectæ FHG communi ſectioni plani ſecantis cum prædicto <lb/>triangulo, hoc eſt quando triangulum per axem BAC rectum fuerit baſi co-<lb/>ni BC, nam tunc DGE communis ſectio plani ſecantis FH cum plano ba-<lb/>ſis coni BC, cum poſita ſit perpendicularis rectæ BGC, quæ eſt communis <lb/>ſectio trianguli per axem cum plano baſis coni, perpendicularis etiam <anchor type="note" xlink:href="" symbol="b"/> erit plano trianguli BAC, vnde cum recta GHF rectos angulos faciet, ideoq; <lb/></s> <s xml:id="echoid-s489" xml:space="preserve"> <anchor type="note" xlink:label="note-0029-02a" xlink:href="note-0029-02"/> omnes in ſectione MFT ordinatim ductæ, ſiue ipſi DGE æquidiſtantes ei-<lb/>dem GFH erunt perpendiculares) Ellipſim efficient æqualium laterum cir-<lb/>ca axim FH, quæ eadem erit, ac circulus diametri FH. </s> <s xml:id="echoid-s490" xml:space="preserve">Si verò prædictæ <lb/>applicatæ ad obliquos angulos diametrum ſecabunt (quod accidet cum. <lb/></s> <s xml:id="echoid-s491" xml:space="preserve">DGE obliquè ſecat rectam FHG) tunc ipſa ſectio erit pariter Ellipſis æqua-<lb/>lium laterum, ſed eius tranſuerſum latus, diameter erit non autem axis.</s> <s xml:id="echoid-s492" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0029-01" xlink:href="note-0029-01a" xml:space="preserve">21. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0029-02" xlink:href="note-0029-02a" xml:space="preserve">4. def. lib. <lb/>II. Elem.</note> </div> <p style="it"> <s xml:id="echoid-s493" xml:space="preserve">Non ſemper igitur ſubcontraria ſectione coni ſcaleni efficitur circulus, ſed <lb/>ſolùm cum triangulum per axem rectum eſt baſi coni, quo in caſu, vt viſum <lb/>eſt, ei debetur eadem proprietas, ac Ellipſi, æqualium tamen laterum circa. <lb/></s> <s xml:id="echoid-s494" xml:space="preserve">axim. </s> <s xml:id="echoid-s495" xml:space="preserve">In ſectionibus autem ſubcontrarijs cuiuslibet alterius trianguli per <lb/>axem (dummodo non ſit triangulum æquicrure, quia tunc communis ſectio <lb/>plani ſecantis cum ipſo triangulo non conuenit cum baſi eiuſdem trianguli, ſed <lb/>ei æquidiſtat) oritur Ellipſis æqualium item laterum, ſed circa diametrum, <lb/>quæ oblquè ſecat applicatas. </s> <s xml:id="echoid-s496" xml:space="preserve">Hinc ergo liquidò conſtat in ſuperiori propoſitio-<lb/>ne opus non fuiſſe ſubcontrariam ſectionem reijcere, vti fit ab ipſo Apoll. </s> <s xml:id="echoid-s497" xml:space="preserve">in. </s> <s xml:id="echoid-s498" xml:space="preserve"><lb/>13. </s> <s xml:id="echoid-s499" xml:space="preserve">primi, atque ab alijs doctrinam conicam pertractantibus ſed hæc obiter <lb/>delibaſſe ſufficiat; </s> <s xml:id="echoid-s500" xml:space="preserve">quo etiam nomine liceat mihi inſequentes demonſtrationes <lb/>proferre, non tam vt deſiderio obſequar hominis mihi amiciſsimi, quam vt <lb/>alteri cuidam, quocum iam ab hinc multis annis illas, nec non plures alias <lb/>communicaui, in mentem redigam, eas, non eius, ſed quidquid ſunt ingenioli <lb/>mei eſſe inuenta; </s> <s xml:id="echoid-s501" xml:space="preserve">atque ita periculo occurram, ne ille, non dicam fidei, ſed <lb/>memoriæ forſan defectu ſibi eas aſciſcat. </s> <s xml:id="echoid-s502" xml:space="preserve">Hoc autem audentiùs faciam, <lb/>cum eæ non omnino ab inſtituto opere ſint alienæ, verſantur enim circà tan-<lb/>gentes coni-ſectionum ab Apoll. </s> <s xml:id="echoid-s503" xml:space="preserve">acutiſsimè quidem inuentas, ac negatiuè <lb/>oſtenſas in eius 33. </s> <s xml:id="echoid-s504" xml:space="preserve">ac 34. </s> <s xml:id="echoid-s505" xml:space="preserve">primi, à me autem neſcio anbreuiùs, euidentiùs <lb/>certè affirmatiuèque demonſtratas, ac Problematicè propoſitas, vt in ſe-<lb/>quentibus.</s> <s xml:id="echoid-s506" xml:space="preserve"/> </p> <pb o="10" file="0030" n="30" rhead=""/> </div> <div xml:id="echoid-div30" type="section" level="1" n="23"> <head xml:id="echoid-head28" xml:space="preserve">PROBL. I. PROP. II.</head> <p> <s xml:id="echoid-s507" xml:space="preserve">Datæ Parabolæ per punctum in ea datum lineam contingentem <lb/>ducere.</s> <s xml:id="echoid-s508" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s509" xml:space="preserve">SIt Parabole, cuius diameter AB, & </s> <s xml:id="echoid-s510" xml:space="preserve">datum in ea punctum ſit C. </s> <s xml:id="echoid-s511" xml:space="preserve">Opor-<lb/>tet ex C Parabolæ contingentem rectam lineam ducere.</s> <s xml:id="echoid-s512" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Prop. 33. <lb/>primi co-<lb/>nic.</note> <p> <s xml:id="echoid-s513" xml:space="preserve">Applicetur ordinatim recta CD, & </s> <s xml:id="echoid-s514" xml:space="preserve">diametri ſegmento DE æqualis po-<lb/>natur EA, iungaturque ACF. </s> <s xml:id="echoid-s515" xml:space="preserve">Dico ipſam eſſe tangentem quæſitam.</s> <s xml:id="echoid-s516" xml:space="preserve"/> </p> <figure> <image file="0030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0030-01"/> </figure> <p> <s xml:id="echoid-s517" xml:space="preserve">Sumpto enim in ſectione quolibet puncto G, per eum applicetur BGF <lb/>rectam AC ſecans in F, diametrum verò in B, & </s> <s xml:id="echoid-s518" xml:space="preserve">iuncta DF ex E vertice. <lb/></s> <s xml:id="echoid-s519" xml:space="preserve">ducatur EHM parallela ad AF ſecans DF in H, & </s> <s xml:id="echoid-s520" xml:space="preserve">CD in M, ſitque HL ipſi <lb/>FB æquidiſtans. </s> <s xml:id="echoid-s521" xml:space="preserve">Iam cum ſit AE æqualis ED, erit FH æqualis HD, ob pa-<lb/>rallelas AF, EH; </s> <s xml:id="echoid-s522" xml:space="preserve">itemque BL æqualis LB ob æquidiſtantes BF, LH: </s> <s xml:id="echoid-s523" xml:space="preserve">quare <lb/>fumpta EI media geometrica inter DE, & </s> <s xml:id="echoid-s524" xml:space="preserve">EB ipſa EI minor erit media. </s> <s xml:id="echoid-s525" xml:space="preserve"><lb/>arithmetica EL. </s> <s xml:id="echoid-s526" xml:space="preserve">Ampliùs quadratum GB ad CD <anchor type="note" xlink:href="" symbol="a"/> eſt vt linea EB ad ED, <anchor type="note" xlink:label="note-0030-02a" xlink:href="note-0030-02"/> vel vt quadratum mediæ geometricæ EI ad quadratum ED, ergo & </s> <s xml:id="echoid-s527" xml:space="preserve">linea. <lb/></s> <s xml:id="echoid-s528" xml:space="preserve">GB ad CD erit vt EI ad ED, cumque ſit EI minor EL, habebit EI ad ED: </s> <s xml:id="echoid-s529" xml:space="preserve"><lb/>ſiue GB ad CD, minorem rationem quam EL ad ED, vel quàm EH ad EM, <lb/>ſeu quam AF ad AC, vel quàm FB ad eandem CD, ergo GB minor eſt FB: </s> <s xml:id="echoid-s530" xml:space="preserve"><lb/>quare punctum F cadit extra Parabolen, & </s> <s xml:id="echoid-s531" xml:space="preserve">ſic de quolibet alio puncto rectæ <lb/>ACF. </s> <s xml:id="echoid-s532" xml:space="preserve">Vnde ipſa ACF Parabolen contingit in C. </s> <s xml:id="echoid-s533" xml:space="preserve">Quod faciendumerat.</s> <s xml:id="echoid-s534" xml:space="preserve"/> </p> <div xml:id="echoid-div30" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0030-02" xlink:href="note-0030-02a" xml:space="preserve">20. pri-<lb/>mi conic.</note> </div> </div> <div xml:id="echoid-div32" type="section" level="1" n="24"> <head xml:id="echoid-head29" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s535" xml:space="preserve">IIfdem poſitis, dico iterum punctum F cadere extra Parabolen. </s> <s xml:id="echoid-s536" xml:space="preserve">Nam ſe-<lb/>cta AB bifariam in H, cum eadem quoque in æqualiter ſecta ſit in E (nã <lb/>cum ſit DE æqualis EA, erit in prima figura BE maior EA, & </s> <s xml:id="echoid-s537" xml:space="preserve">in ſecunda BE <lb/>minor EA) erit rectangulum AHB maius rectangulo AEB, ac propterea. <lb/></s> <s xml:id="echoid-s538" xml:space="preserve">quadratum EA ad rectangulum AHB, ſiue ad quadratum AH minorem ha- <pb o="11" file="0031" n="31" rhead=""/> bebit rationem <lb/>quàm idem qua-<lb/> <anchor type="figure" xlink:label="fig-0031-01a" xlink:href="fig-0031-01"/> dratum E A ad <lb/>rectangulum A E <lb/>B, & </s> <s xml:id="echoid-s539" xml:space="preserve">quatuor qua-<lb/>drata E A, ſiue <lb/>vnicum quadra-<lb/>tum A D, ad qua-<lb/>tuor quadrata A <lb/>H, ſiue ad vnicum <lb/>quadratum A B <lb/>minorem habebit <lb/>rationem quàm <lb/>quadratum E A <lb/>ad rectangulum <lb/>A E B, ſed quadratum A D ad A B eſt vt quadratum C D ad F B, & </s> <s xml:id="echoid-s540" xml:space="preserve">qua-<lb/>dratum E A ad rectangulum A E B eſt vt quadratum E D ad rectangu-<lb/>lum B E D, cum ſit A E æqualis E D, vel vt recta E D ad rectam E B, <lb/>vel vt <anchor type="note" xlink:href="" symbol="a"/> quadratum C D ad quadratum G B, quare quadratum C D ad <anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/> F B minorem habebit rationem quàm idem quadratum C D ad quadra-<lb/>tum G B, ergo quadratum F B maius eſt quadrato G B, vnde punctum F <lb/>cadit extra Parabolen, & </s> <s xml:id="echoid-s541" xml:space="preserve">ſic de quolibet alio puncto rectæ A C F, præ-<lb/>ter C. </s> <s xml:id="echoid-s542" xml:space="preserve">Quare ducta eſt per datum punctum C recta A C F Parabolen <lb/>contingens. </s> <s xml:id="echoid-s543" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s544" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="1"> <figure xlink:label="fig-0031-01" xlink:href="fig-0031-01a"> <image file="0031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0031-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">20. pri-<lb/>mi conic.</note> </div> </div> <div xml:id="echoid-div34" type="section" level="1" n="25"> <head xml:id="echoid-head30" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s545" xml:space="preserve">POſitis ijſdem. </s> <s xml:id="echoid-s546" xml:space="preserve">Dico iterum, vt ſupra.</s> <s xml:id="echoid-s547" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s548" xml:space="preserve">Sumatur enim poſt D A, A B tertia proportionalis A H, erit ag-<lb/> <anchor type="figure" xlink:label="fig-0031-02a" xlink:href="fig-0031-02"/> gregatum extremarum <lb/>A D, A H maius quàm <lb/>duplum mediæ A B, <lb/>ſiue maius quàm du-<lb/>plum A E cum E B, <lb/>ſed eſt A D dupla ad <lb/>A E, ergo A H erit <lb/>maior quàm dupla E <lb/>B, ſed eſt A D dupla <lb/>D E, ergo A D ad D <lb/>E minorem habet ra-<lb/>tionem quàm A H ad <lb/>E B, & </s> <s xml:id="echoid-s549" xml:space="preserve">permutando D <lb/>A ad A H minorem <lb/>habet rationem quàm D E ad E B, ſed D A ad A H, eſt vt quadratum <lb/>D A ad quadratum A B, vel vt quadratum D C ad quadratum B F, & </s> <s xml:id="echoid-s550" xml:space="preserve"><lb/>D E ad E B, <anchor type="note" xlink:href="" symbol="b"/> eſt vt quadratum D C ad quadratum B G, ergo quadra- <anchor type="note" xlink:label="note-0031-02a" xlink:href="note-0031-02"/> tum D C ad quadratum B F minorem habet rationem quàm idem qua- <pb o="12" file="0032" n="32" rhead=""/> dratum D C ad quadratum B G, quare quadratum B F maius eſt quadra-<lb/>to B G; </s> <s xml:id="echoid-s551" xml:space="preserve">ideoque punctum F cadit extra ſectionem, vt & </s> <s xml:id="echoid-s552" xml:space="preserve">quodcunque <lb/>aliud punctum rectæ A C F, præter C. </s> <s xml:id="echoid-s553" xml:space="preserve">Erit ergo recta A C F Parabolen <lb/>contingens in in C. </s> <s xml:id="echoid-s554" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s555" xml:space="preserve"/> </p> <div xml:id="echoid-div34" type="float" level="2" n="1"> <figure xlink:label="fig-0031-02" xlink:href="fig-0031-02a"> <image file="0031-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0031-02"/> </figure> <note symbol="b" position="right" xlink:label="note-0031-02" xlink:href="note-0031-02a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div36" type="section" level="1" n="26"> <head xml:id="echoid-head31" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s556" xml:space="preserve">PRopoſitio 34. </s> <s xml:id="echoid-s557" xml:space="preserve">primi conic. </s> <s xml:id="echoid-s558" xml:space="preserve">licet ab Apollonio negatiuè ſit demon-<lb/>ſtrata, facilè tamen ad affirmatiuam reducitur, ſi ex ip-<lb/>ſa in principio demantur ea verba. </s> <s xml:id="echoid-s559" xml:space="preserve">_Si enim fieri po-_ <lb/>_teſt, ſecet vt E C F_, ad finem verò. </s> <s xml:id="echoid-s560" xml:space="preserve">_Quod fieri non_ <lb/>_poteſt_; </s> <s xml:id="echoid-s561" xml:space="preserve">nam ibi linea H G oſtenditur minor G F, vnde punctum F <lb/>cadet extra ſectionem, & </s> <s xml:id="echoid-s562" xml:space="preserve">ſic quodcunque aliud punctum rectæ E C H <lb/>præter C, quare ipſa E C H ſectionem continget in C: </s> <s xml:id="echoid-s563" xml:space="preserve">ſed vt clariùs <lb/>idem pateat, en afferemus noſtram directè concluſam demonſtrationem, <lb/>de qua in præcedenti Monito, præmiſſo tantùm (vice propoſitionis 169. <lb/></s> <s xml:id="echoid-s564" xml:space="preserve">ſeptimi Pappi, qua indiget Apolloniana propoſitio) ſequenti Lemmate, in <lb/>quo interim duæ ſimul circuli proprietates detegentur haud iniucundæ.</s> <s xml:id="echoid-s565" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div37" type="section" level="1" n="27"> <head xml:id="echoid-head32" xml:space="preserve">LEMMAI. PROP. III.</head> <p> <s xml:id="echoid-s566" xml:space="preserve">Si circuli diameter A B inæqualiter ſecetur in C, & </s> <s xml:id="echoid-s567" xml:space="preserve">ad mino-<lb/>rem partem C B producatur, ita vt ſit A D ad D B, vt A C ad <lb/>C B, & </s> <s xml:id="echoid-s568" xml:space="preserve">ex C erigatur perpendicularis C E, iungaturque D E. <lb/></s> <s xml:id="echoid-s569" xml:space="preserve">Dico quadratum ipſius D E æquari rectangulo A D B.</s> <s xml:id="echoid-s570" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s571" xml:space="preserve">Si verò in recto angulo D C E, quælibet alia ſubtenſa F G <lb/>applicetur ipſi D E æquidiſtans, productam diametri partem <lb/>ſecans in F, aut infra D, aut ſupra, & </s> <s xml:id="echoid-s572" xml:space="preserve">perpendicularem C E in <lb/>G. </s> <s xml:id="echoid-s573" xml:space="preserve">Dico ampliùs quadratum applicatæ F G ſemper excedere <lb/>rectangulum A F B.</s> <s xml:id="echoid-s574" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s575" xml:space="preserve">QVò ad primum, ſit circuli centrum H, & </s> <s xml:id="echoid-s576" xml:space="preserve">iungatur H E.</s> <s xml:id="echoid-s577" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s578" xml:space="preserve">Iam cum ſit A D ad D B, vt A C ad C B, erit componendo A D <lb/>cum D B ad D B, vt A B ad B C, & </s> <s xml:id="echoid-s579" xml:space="preserve">ſumptis antecedentium <lb/>ſubduplis, erit H D ad D B, vt H B ad B C, & </s> <s xml:id="echoid-s580" xml:space="preserve">perlconuerſionem rationis <lb/>D H ad H B, vt B H ad H C, vel vt D H ad H E (ipſi H B æqualis) ita <lb/>H E ad H C: </s> <s xml:id="echoid-s581" xml:space="preserve">quare triangula D H E, E H C, cum habeant circa com-<lb/>munem angulnm H latera proportionalia, ſimilia erunt, vnde angulus <lb/>D E H æquabitur angulo E C H, ſiue rectus erit, ideoque D E circulum <lb/>continget, hoc eſt quadratum D E æquabitur rectangulo A D B. </s> <s xml:id="echoid-s582" xml:space="preserve">Quod <lb/>primò, &</s> <s xml:id="echoid-s583" xml:space="preserve">c.</s> <s xml:id="echoid-s584" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s585" xml:space="preserve">Ampliùs iungantur E B, E A, quas, recta F G producta ſecet, in I &</s> <s xml:id="echoid-s586" xml:space="preserve"> <pb o="13" file="0033" n="33" rhead=""/> L, & </s> <s xml:id="echoid-s587" xml:space="preserve">cadat primùm applicata F G infra contingentem D E, ſitque G M <lb/>ipſi E A, & </s> <s xml:id="echoid-s588" xml:space="preserve">G N ipſi E B parallela.</s> <s xml:id="echoid-s589" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s590" xml:space="preserve">Iam cum ſit G F parallela ad E <lb/> <anchor type="figure" xlink:label="fig-0033-01a" xlink:href="fig-0033-01"/> D, G M ad E A, & </s> <s xml:id="echoid-s591" xml:space="preserve">G N ad E B, <lb/>erit triangulum A D E ſimile triã-<lb/>gulo M F G, & </s> <s xml:id="echoid-s592" xml:space="preserve">triangulum E D B <lb/>ſimile triangulo G F N, quare vt <lb/>A D ad D E, ita M F ad F G, & </s> <s xml:id="echoid-s593" xml:space="preserve"><lb/>vt E D ad D B, ita G F ad F N; <lb/></s> <s xml:id="echoid-s594" xml:space="preserve">ſuntque A D, D E, D B continuę <lb/>proportionales, vnde M F, F G, <lb/>F N, erunt quoque proportiona-<lb/>les, ſiue rectangulum M F N ęqua-<lb/>bitur quadrato F G.</s> <s xml:id="echoid-s595" xml:space="preserve"/> </p> <div xml:id="echoid-div37" type="float" level="2" n="1"> <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a"> <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0033-01"/> </figure> </div> <p> <s xml:id="echoid-s596" xml:space="preserve">Præterea cum B E circulum <lb/>contingat, & </s> <s xml:id="echoid-s597" xml:space="preserve">E B ſecet, erit an-<lb/>gulus D E B æqualis angulo B A <lb/>E, ſed (cum triangula B E C, B A <lb/>E in ſemicirculo ſint ſimilia) eſt <lb/>quoque angulus B E C, æqualis <lb/>lis angulo B A E, ergo angulus <lb/>D E B, ſiue alternus E I G æqua-<lb/>lis erit angulo B E C, ergo linea <lb/>G I ipſi G E æqualis. </s> <s xml:id="echoid-s598" xml:space="preserve">Item angu-<lb/>lus O E A æquatur angulo A B E in alterna portione, ſiue angulo A E C, <lb/>eſtque angulus O E A alterno G L E æqualis, vnde anguli A E C, G L E <lb/>æquales erunt, quare linea G L æqualis eidem G E; </s> <s xml:id="echoid-s599" xml:space="preserve">erunt ergo L G, G I <lb/>inter ſe æquales, ſed eſt G F maior I F, habebit ergo L G ad G F mino-<lb/>rem rationem quàm G I ad I F, & </s> <s xml:id="echoid-s600" xml:space="preserve">componendo L F, ad F G, ſiue A F <lb/>ad F M minorem rationem quàm G F ad F I, vel quàm N F ad F B, qua-<lb/>re rectangnlum ſub extremis A F, F B, minus <anchor type="note" xlink:href="" symbol="a"/> erit rectangulo ſub me- <anchor type="note" xlink:label="note-0033-01a" xlink:href="note-0033-01"/> dijs M F, F N, ſiue minus quadrato F G. </s> <s xml:id="echoid-s601" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s602" xml:space="preserve"/> </p> <div xml:id="echoid-div38" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0033-01" xlink:href="note-0033-01a" xml:space="preserve">16 ſept. <lb/>Pappi.</note> </div> <p> <s xml:id="echoid-s603" xml:space="preserve">Idem penitùs oſtendetur, quando applicata _F G_ productæ diametro <lb/>occurrat vltra D; </s> <s xml:id="echoid-s604" xml:space="preserve">nam adhibitis angulis ad verticem E, alterniſque pa-<lb/>rallelarum, item demonſtrabirur _I G_ ipſi _G L_ æqualem eſſe, & </s> <s xml:id="echoid-s605" xml:space="preserve">ex _G_ facta <lb/>fimili conſtructione, demonſtratio, & </s> <s xml:id="echoid-s606" xml:space="preserve">concluſio omninò erit eadem, ac <lb/>ſupra.</s> <s xml:id="echoid-s607" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div40" type="section" level="1" n="28"> <head xml:id="echoid-head33" xml:space="preserve">PROBL. II. PROP. IV.</head> <p> <s xml:id="echoid-s608" xml:space="preserve">Datæ Hyperbolæ, vel Ellipſi, per punctum in ea datum <lb/> <anchor type="note" xlink:label="note-0033-02a" xlink:href="note-0033-02"/> contingentem lineam ducere.</s> <s xml:id="echoid-s609" xml:space="preserve"/> </p> <div xml:id="echoid-div40" type="float" level="2" n="1"> <note position="right" xlink:label="note-0033-02" xlink:href="note-0033-02a" xml:space="preserve">Prop. 34. <lb/>primi co-<lb/>nic.</note> </div> <p> <s xml:id="echoid-s610" xml:space="preserve">SIt Ellipſis, vel Hyperbolæ A B K, cuius tranſuerſum latus ſit B C, & </s> <s xml:id="echoid-s611" xml:space="preserve"><lb/>datum in ſectione punctum ſit A, extra verticem B: </s> <s xml:id="echoid-s612" xml:space="preserve">oportet ex A <lb/>datæ ſectioni contingentem lineam ducere.</s> <s xml:id="echoid-s613" xml:space="preserve"/> </p> <pb o="14" file="0034" n="34" rhead=""/> <p> <s xml:id="echoid-s614" xml:space="preserve">Ex dato puncto A ordinatim applicetur A D, occurrens diametro in <lb/>D, & </s> <s xml:id="echoid-s615" xml:space="preserve">fiat vt C D ad D B, ita C E ad E B, iungaturque E A: </s> <s xml:id="echoid-s616" xml:space="preserve">dico ipſam <lb/>E A ſectionem contingere.</s> <s xml:id="echoid-s617" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s618" xml:space="preserve">Etenim ſumpto in ea quocunque puncto F, vel ſupra, vel infra A, <lb/>ordinatim agatur F H G, ſectionem ſecans in H, diametrum in G, & </s> <s xml:id="echoid-s619" xml:space="preserve">ſu-<lb/>per tranſuerſo B C deſcribatur ſemicirculus B L C, cuius diametto B C <lb/>in Ellipſi ex puncto D erigatur perpendicularis D L, iungaturque E L, <lb/>quæ, per Lemma antecedens, erit ipſi circulo contingens in L. </s> <s xml:id="echoid-s620" xml:space="preserve">At in <lb/>Hyperbola ex E puncto ducta ſit diametro C B perpendicularis E L, iun-<lb/>gaturque D L, quæ item, ob præmiſſum Lemma, ſemi-circulum B L C <lb/>continget in L, & </s> <s xml:id="echoid-s621" xml:space="preserve">ex G ipſi D L æquidiſtans ducatur G I ſemi-circulum <lb/> <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> primæ figuræ ſecans in M, in qua cum ſit E L I contingens in L, erit ap-<lb/>plicata G I maior G M, ſiue quadratum G I maius quadrato G M, vel <lb/>maius rectangulo C G B, ſed eſt quoque, per idem Lemma, quadratum <lb/>G I (in ſecunda figura) maius rectangulo C G B, quare in vtraque figu-<lb/>ra quadratum G I ad quadratum D L, vel quadratum G E ad quadratum <lb/>E D, vel quadratnm G F ad quadratum D A, maiorem habebit rationem <lb/>quàm rectangulum C G B ad idem quadratum D L, vel ad rectangulum <lb/>C D B, ſed vt rectangulum C G B ad rectangulum C D B, ita <anchor type="note" xlink:href="" symbol="a"/> quadra- <anchor type="note" xlink:label="note-0034-01a" xlink:href="note-0034-01"/> tum G H ad quadratum D A, ergo quadratum G F ad quadratum D A <lb/>maiorem habet rationem quàm quadratum G H ad idem quadratum D <lb/>A; </s> <s xml:id="echoid-s622" xml:space="preserve">quare quadratum G F maius eſt quadrato G H: </s> <s xml:id="echoid-s623" xml:space="preserve">vnde punctum F ca-<lb/>dit extra ſectionem, & </s> <s xml:id="echoid-s624" xml:space="preserve">ſic de quibuslibet alijs punctis rectæ E A F, præ-<lb/>ter A. </s> <s xml:id="echoid-s625" xml:space="preserve">Ducta eſt ergo E A ſectionem contingens in A. </s> <s xml:id="echoid-s626" xml:space="preserve">Quod erat fa-<lb/>ciendum.</s> <s xml:id="echoid-s627" xml:space="preserve"/> </p> <div xml:id="echoid-div41" type="float" level="2" n="2"> <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0034-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0034-01" xlink:href="note-0034-01a" xml:space="preserve">21. pri-<lb/>mi conic.</note> </div> <pb o="15" file="0035" n="35" rhead=""/> </div> <div xml:id="echoid-div43" type="section" level="1" n="29"> <head xml:id="echoid-head34" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s628" xml:space="preserve">VT aliquando ad rem noſtram accedamus, quoniam in hac de <lb/>MAXIMIS, & </s> <s xml:id="echoid-s629" xml:space="preserve">MINIMIS tractatione frequenter nobis eſt <lb/>opus conicas ſectiones circà datam diametrum, per datum ver-<lb/>ticem, cum datis lateribus, cumque applicatis angulum dato <lb/>æqualem cum diametro efficientibus deſcribere, quæ omnia quidem nos docet <lb/>Apoll. </s> <s xml:id="echoid-s630" xml:space="preserve">in 52. </s> <s xml:id="echoid-s631" xml:space="preserve">53. </s> <s xml:id="echoid-s632" xml:space="preserve">54. </s> <s xml:id="echoid-s633" xml:space="preserve">primi conic. </s> <s xml:id="echoid-s634" xml:space="preserve">ad quas itaque vſu exigente confugien-<lb/>dum eſſet; </s> <s xml:id="echoid-s635" xml:space="preserve">attamen cum hæo ſint forſan longiſsimæ, ac difficillimæ omnium <lb/>demonſtrationum in quatuor conicorum libris contentarum, eò quod ipſarum <lb/>quælibet in duos caſus diſtribuatur, variaque ibi Lemmata requirantur à <lb/>Pappo, Eutocio, & </s> <s xml:id="echoid-s636" xml:space="preserve">Commandino ſuppleta; </s> <s xml:id="echoid-s637" xml:space="preserve">conſentaneum viſum eſt noſtras <lb/>hic quoque horum problematum ſolutiones afferre, quæ expeditiores, admo-<lb/>dumque faciles nobis videntur, vniuerſaliter ſingulas oſtendendo, abſque <lb/>vſu prædictorum, vel aliorum Lemmatum, vt mox videre licet.</s> <s xml:id="echoid-s638" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div44" type="section" level="1" n="30"> <head xml:id="echoid-head35" xml:space="preserve">PROBL. III. PROP. V.</head> <p> <s xml:id="echoid-s639" xml:space="preserve">Data in quodam plano recta linea ad vnum punctum terminata, <lb/> <anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/> inuenire in dato plano coni-ſectionem, quæ Parabole appellatur, <lb/>cuius diameter ſit data linea, vertex eius terminus, rectum verò la-<lb/>tus ſit altera quædam linea magnitudine data, & </s> <s xml:id="echoid-s640" xml:space="preserve">diametrō<unsure/> ordina-<lb/>tim ductæ in dato angulo applicentur.</s> <s xml:id="echoid-s641" xml:space="preserve"/> </p> <div xml:id="echoid-div44" type="float" level="2" n="1"> <note position="right" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">Prop. 52. <lb/>pri. con.</note> </div> <p> <s xml:id="echoid-s642" xml:space="preserve">SIt in ſubiecto plano recta linea A B <lb/> <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/> data poſitione ad punctum A ter-<lb/>minata, altera autem recta magnitu-<lb/>dine data ſit AC, & </s> <s xml:id="echoid-s643" xml:space="preserve">datus angulus ſit <lb/>D. </s> <s xml:id="echoid-s644" xml:space="preserve">Oportet in ſubiecto plano Para-<lb/>bolen deſcribere, cuius diameter ſit <lb/>AB vertex A, rectum figuræ latus ſit <lb/>AC, & </s> <s xml:id="echoid-s645" xml:space="preserve">ordinatim ductæ ad diametrũ <lb/>in angulo D applicentur.</s> <s xml:id="echoid-s646" xml:space="preserve"/> </p> <div xml:id="echoid-div45" type="float" level="2" n="2"> <figure xlink:label="fig-0035-01" xlink:href="fig-0035-01a"> <image file="0035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0035-01"/> </figure> </div> <p> <s xml:id="echoid-s647" xml:space="preserve">Sumatur in AB quodcunque pun-<lb/>ctum B, per quod in ſubiecto plano, in <lb/>quo AB, ducatur recta EBF ad angu-<lb/>lum ABF, qui dato D ſit æqualis, ſu-<lb/>manturque hinc inde B E, & </s> <s xml:id="echoid-s648" xml:space="preserve">B F inter <lb/>ſe æquales, vtraque verò ſit media <lb/>proportionalis inter B A, & </s> <s xml:id="echoid-s649" xml:space="preserve">datam <lb/>AC, & </s> <s xml:id="echoid-s650" xml:space="preserve">per rectã EF intelligatur quod-<lb/>cunque planum GEHF, quod non ſit <lb/>idem cum plano per rectas E F, AB <pb o="16" file="0036" n="36" rhead=""/> tranſeunte, & </s> <s xml:id="echoid-s651" xml:space="preserve">horum communis ſe-<lb/> <anchor type="figure" xlink:label="fig-0036-01a" xlink:href="fig-0036-01"/> ctio ſitrecta E F, cui in plano GEH <lb/>perpendicularis ducatur recta G B H <lb/>ad vtramque partem plani A E F pro-<lb/>ducta, in qua ſumpto quocunq; </s> <s xml:id="echoid-s652" xml:space="preserve">pun-<lb/>cto G, fiat, vt G B ad B E, ita B E ad <lb/>BH; </s> <s xml:id="echoid-s653" xml:space="preserve">(& </s> <s xml:id="echoid-s654" xml:space="preserve">erit rectangulum BGH æqua-<lb/>le quadrato BE, vel BF) iungaturque <lb/>BA, & </s> <s xml:id="echoid-s655" xml:space="preserve">per H in plano per HG, & </s> <s xml:id="echoid-s656" xml:space="preserve">GA <lb/>ducto agatur recta HI ipſi BA paral-<lb/>lela.</s> <s xml:id="echoid-s657" xml:space="preserve"/> </p> <div xml:id="echoid-div46" type="float" level="2" n="3"> <figure xlink:label="fig-0036-01" xlink:href="fig-0036-01a"> <image file="0036-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0036-01"/> </figure> </div> <p> <s xml:id="echoid-s658" xml:space="preserve">Itaque cum GH, EF in vno ſint pla-<lb/>no, ac inter ſe perpendiculares, ſitque <lb/>rectangulum GBH æquale quadrato <lb/>vtriuſque EB, BF, ſi circa G H, tan-<lb/>quam diametrum deſcribatur circulus <lb/>GEHF, ipſe tranſibit per E, & </s> <s xml:id="echoid-s659" xml:space="preserve">F. </s> <s xml:id="echoid-s660" xml:space="preserve">Si <lb/>ergo intelligatur recta IAG circa pe-<lb/>ripheriam circuli GE, H F conuerti, <lb/>manente eius extremo puncto I, deſcribetur conus IGH cuius vertex I, ba-<lb/>ſis circulus GH, & </s> <s xml:id="echoid-s661" xml:space="preserve">communis ſectio conicæ ſuperficiei cum ſubiecto plano <lb/>erit linea EMANF, quam dico eſſe Parabolen quæſitam.</s> <s xml:id="echoid-s662" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s663" xml:space="preserve">Conus enim IGH, cuius vertex I, & </s> <s xml:id="echoid-s664" xml:space="preserve">baſis diameter GH ſecatur plano per <lb/>axem deſcribens triãgulum GIH; </s> <s xml:id="echoid-s665" xml:space="preserve">ſecatur autem, & </s> <s xml:id="echoid-s666" xml:space="preserve">altero plano EAF (quod <lb/>eſt datum ſubiectum planum) baſi coni non æquidiſtante, cum eam ſecet, <lb/>ſecante baſim coniſecundum rectam lineam EF, quæ ad GH baſim triangu-<lb/>li per axem eſt perpendicularis, atque eſt AB diameter ſectionis EAF vni <lb/>laterum H I trianguli per axem æquidiſtans, talis ſectio E A F per primam <lb/>huius erit Parabolæ, cuius diameter A B, vertex A, & </s> <s xml:id="echoid-s667" xml:space="preserve">ordinatim ducta EF, <lb/>quæ ipſi diametro ad angulum ABF, dato angulo D æqualem, ap-<lb/>plicata eſt, ex ipſa conſtructione. </s> <s xml:id="echoid-s668" xml:space="preserve">Et cum factum ſit vt AB, <lb/>ad BE, ita BE ad A C, erit quadratum AB ad qua-<lb/>dratum BE, vel ad rectangulum GBH, vt <lb/>AB ad AC. </s> <s xml:id="echoid-s669" xml:space="preserve">Quare AC erit rectum <lb/>latus Parabolæ EMANF, <lb/>deſcriptæ vti quære-<lb/>batur: </s> <s xml:id="echoid-s670" xml:space="preserve">Quod erat <lb/>faciendum.</s> <s xml:id="echoid-s671" xml:space="preserve"/> </p> <figure> <image file="0036-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0036-02"/> </figure> <pb o="17" file="0037" n="37" rhead=""/> </div> <div xml:id="echoid-div48" type="section" level="1" n="31"> <head xml:id="echoid-head36" xml:space="preserve">PROBL. IV. PROP. VI.</head> <p> <s xml:id="echoid-s672" xml:space="preserve">Data in quodam plano recta linea terminata, quæ ad alteram <lb/> <anchor type="note" xlink:label="note-0037-01a" xlink:href="note-0037-01"/> partem in infinitum producatur: </s> <s xml:id="echoid-s673" xml:space="preserve">inuenire in dato plano coni-ſe-<lb/>ctionem, quę dicitur Hyperbole, cuius diameter ſit producta linea, <lb/>vertex eius terminus, tranſuerſum latus ſit data linea terminata, re-<lb/>ctum verò ſit alia quæcunque data linea finita, & </s> <s xml:id="echoid-s674" xml:space="preserve">ad ipſius diametr@ <lb/>ordinatim ductæ efficiant angulos dato angulo æquales.</s> <s xml:id="echoid-s675" xml:space="preserve"/> </p> <div xml:id="echoid-div48" type="float" level="2" n="1"> <note position="right" xlink:label="note-0037-01" xlink:href="note-0037-01a" xml:space="preserve">Prop. 53. <lb/>primico-<lb/>nic.</note> </div> <p> <s xml:id="echoid-s676" xml:space="preserve">SInt datæ rectæ lineæ terminatæ AB, BC, quæ in ſubiecto plano ad angu-<lb/>lum ABC, dato angulo P æqualem conſtituantur, & </s> <s xml:id="echoid-s677" xml:space="preserve">harum altera AB <lb/>ſit vtcunque producta ad BD: </s> <s xml:id="echoid-s678" xml:space="preserve">oportet in ſubiecto plano Hyperbolen deſcri-<lb/>bere, cuius diameter ſit BD, vertex B, tranſuerſum latus AB rectum BC, & </s> <s xml:id="echoid-s679" xml:space="preserve"><lb/>ordinatim ductæ ad diametrō BD conſtituant angulos, dato, angulo P æquales.</s> <s xml:id="echoid-s680" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s681" xml:space="preserve">Iungatur AC, & </s> <s xml:id="echoid-s682" xml:space="preserve">producatur, ſu-<lb/> <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> maturq; </s> <s xml:id="echoid-s683" xml:space="preserve">in BD quodlibet punctum <lb/>D, per quod agatur in ſubiecto pla-<lb/>no recta linea DE ipſi BC parallela, <lb/>à qua, hinc inde producta, deman-<lb/>tur partes DF, DG, quæ ſint mediæ <lb/>proportionales inter BD, & </s> <s xml:id="echoid-s684" xml:space="preserve">DE; </s> <s xml:id="echoid-s685" xml:space="preserve">& </s> <s xml:id="echoid-s686" xml:space="preserve"><lb/>per rectam FG intelligatur planum <lb/>IFHG, diuerſum à plano, quod per <lb/>AD, & </s> <s xml:id="echoid-s687" xml:space="preserve">FG tranſit, quorum cõmunis <lb/>ſectio ſit recta FG, cui per D in pla-<lb/>no IFHG perpendicularis ducatur <lb/>IDH, in qua, ad partes I, ſumptum <lb/>ſit quodcunque punctum I, & </s> <s xml:id="echoid-s688" xml:space="preserve">fiat vt <lb/>ID ad DF, ita DF ad DH; </s> <s xml:id="echoid-s689" xml:space="preserve">& </s> <s xml:id="echoid-s690" xml:space="preserve">erit re-<lb/>ctangulum IDH æquale quadrato DF, uel<gap/> quadrato DG, ſed rectæ IH, FG ſe <lb/>mutuò ſecant ad rectos angulos in D, quare ſi circa IH circulus deſeribatur, <lb/>tranſibit ipſe per puncta FG. </s> <s xml:id="echoid-s691" xml:space="preserve">Tandem iungatur HA, & </s> <s xml:id="echoid-s692" xml:space="preserve">IB producatur ſe-<lb/>cans AH in L, & </s> <s xml:id="echoid-s693" xml:space="preserve">intelligatur conus cuius vertex L, baſis circulus I H, & </s> <s xml:id="echoid-s694" xml:space="preserve">cõ-<lb/>munis ſectio ſuperficiei conicæ cum ſubiecto plano ſit linea FMBNG. </s> <s xml:id="echoid-s695" xml:space="preserve">Dico <lb/>hanc eſſe quæſitam Hyperbolen.</s> <s xml:id="echoid-s696" xml:space="preserve"/> </p> <div xml:id="echoid-div49" type="float" level="2" n="2"> <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a"> <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0037-01"/> </figure> </div> <p> <s xml:id="echoid-s697" xml:space="preserve">Conus enim LIH, cuius vertex L, & </s> <s xml:id="echoid-s698" xml:space="preserve">diameter baſis, I H, plano per axem <lb/>ſecatur triangulum facient LIH, & </s> <s xml:id="echoid-s699" xml:space="preserve">ſecatur altero plano (quod eſt datum pla-<lb/>num ſubiectum) ſecante baſim coni ſecundum rectam lineam F G, quæ ad <lb/>IH baſim trianguli per axem, eſt perpendicularis, & </s> <s xml:id="echoid-s700" xml:space="preserve">communis ſectio ſubie-<lb/>cti plani, & </s> <s xml:id="echoid-s701" xml:space="preserve">trianguli per axem, hoc eſt DB, producta ad B conuenit cum al-<lb/>tero latere HL extra verticem producto in puncto A, erit, per primam hu-<lb/>ius, ſectio FBG Hyperbole, cuius vertex B, diameter BD, & </s> <s xml:id="echoid-s702" xml:space="preserve">ordinatim du-<lb/>ctæ FG cum diametro BD, ad angulum FDB, angulo CBA, ſeu dato P æ-<lb/>qualem applicantur, ex conſtructione. </s> <s xml:id="echoid-s703" xml:space="preserve">Cumque factum ſit vt BD, ad DF <lb/>ita DF ad DE, erit rectangulum EDB æquale quadrato DF, ſiue rectangulo <pb o="18" file="0038" n="38" rhead=""/> IDH: </s> <s xml:id="echoid-s704" xml:space="preserve">quare rectangulum ADB ad rectangulum EDB, erit vt idem ADB ad <lb/>IDH, ſed ADB ad EDB, eſt vt AD ad DE, vel vt AB ad BC, ergo rectan-<lb/>gulum quoque ADB ad rectangulum IDH, erit vt AB ad BC. </s> <s xml:id="echoid-s705" xml:space="preserve">Sequitur er-<lb/>go vt AB ſit tranſuerſum latus, & </s> <s xml:id="echoid-s706" xml:space="preserve">BC rectum deſcriptæ Hyperbolæ, vt in <lb/>prima huius oſtenſum eſt. </s> <s xml:id="echoid-s707" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s708" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div51" type="section" level="1" n="32"> <head xml:id="echoid-head37" xml:space="preserve">PROBL. V. PROP. VII.</head> <p> <s xml:id="echoid-s709" xml:space="preserve">Duabus datis in ſubiecto plano rectis lineis terminatis, inuenire <lb/>in eodem plano circa ipſarum alteram, tanquam circà diametrum, <lb/> <anchor type="note" xlink:label="note-0038-01a" xlink:href="note-0038-01"/> coni - ſectionem, quæ Ellipſis appellatur, cuius tranſuerſum latus <lb/>ſit prædicta diameter, rectum verò latus ſit altera data linea, & </s> <s xml:id="echoid-s710" xml:space="preserve">ad dia-<lb/>metrō<unsure/> ordinatim ductæ in dato angulo applicentur.</s> <s xml:id="echoid-s711" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="1"> <note position="left" xlink:label="note-0038-01" xlink:href="note-0038-01a" xml:space="preserve">Prop. 54. <lb/>pri. con.</note> </div> <p> <s xml:id="echoid-s712" xml:space="preserve">SInt datæ in ſubiecto plano terminate rectæ lineæ AB, BC, quæ ad datum <lb/>angulum P componantur. </s> <s xml:id="echoid-s713" xml:space="preserve">Oportet in ſubiecto plano Ellipſim deſcribe-<lb/>re, cuius diameter ſit AB, vertex B, tranſuerſum latus AB, rectum BC, & </s> <s xml:id="echoid-s714" xml:space="preserve">ad <lb/>diametrō<unsure/> AB ordinatim ductæ conſtituant angulos dato, angulo P æquales.</s> <s xml:id="echoid-s715" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s716" xml:space="preserve">Iungatur AC, ſumaturque in <lb/> <anchor type="figure" xlink:label="fig-0038-01a" xlink:href="fig-0038-01"/> AB quodcunque punctum D, à <lb/>quo ducatur, in ſubiecto plano, <lb/>recta GDFE ipſi BC parallela, è <lb/>qua ex vtraque parte abſcindan-<lb/>tur DF, DG mediæ proportiona-<lb/>les inter BD, & </s> <s xml:id="echoid-s717" xml:space="preserve">DE; </s> <s xml:id="echoid-s718" xml:space="preserve">erit vtriuſq; <lb/></s> <s xml:id="echoid-s719" xml:space="preserve">ipſarum quadratum ęquale rectã-<lb/>gulo EDB: </s> <s xml:id="echoid-s720" xml:space="preserve">per rectam autem FG <lb/>intelligatur ſecans planum IFGHG <lb/>ad vtramque partem ſubiecti pla-<lb/>ni productum, quorum commu-<lb/>nis ſectio ſit recta FG, cui per D <lb/>in plano ſecante IFHG, perpen-<lb/>dicularis ducatur IDH hic inde <lb/>producta.</s> <s xml:id="echoid-s721" xml:space="preserve"/> </p> <div xml:id="echoid-div52" type="float" level="2" n="2"> <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a"> <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0038-01"/> </figure> </div> <p> <s xml:id="echoid-s722" xml:space="preserve">Iam, veleſt CB non maior BA, vel maior. </s> <s xml:id="echoid-s723" xml:space="preserve">Si non maior, erit quoque ED <lb/>non maior ipſa DA. </s> <s xml:id="echoid-s724" xml:space="preserve">Itaque ex educta IDH infra ſubiectum planum dema-<lb/>tur DI, quæ maior ſit ipſa DB, iungatur I B, & </s> <s xml:id="echoid-s725" xml:space="preserve">ex A ducatur AO parallela <lb/>ad I B, ſecans IDH in O, & </s> <s xml:id="echoid-s726" xml:space="preserve">fiat vt ID ad DF, ita D F ad aliam DH; </s> <s xml:id="echoid-s727" xml:space="preserve">erit re-<lb/>ctangulum IDH æquale quadrato D F, ſiue rectangulo EDB, ſed rectangu-<lb/>lum EDB eſt non maius rectangulo ADB (nam eſt ED non maior recta DA) <lb/>ergo rectangulum IDH erit non maius rectangulo BDA, ſed rectangulum <lb/>IDO maius eſt rectangulo BDA (nam cum ſit vt ID ad DB, ita OD ad DA, <lb/>ſitque I D maior D B ex conſtructione, erit quoque DO maior DA) quare <lb/>IDH rectangulum minus erit rectangulo IDO, hoc eſt linea DH minor DO; <lb/></s> <s xml:id="echoid-s728" xml:space="preserve">vnde punctum H eſt inter D, & </s> <s xml:id="echoid-s729" xml:space="preserve">O, ſiue inter parallelas I B, AO; </s> <s xml:id="echoid-s730" xml:space="preserve">quare iun-<lb/>cta AH, & </s> <s xml:id="echoid-s731" xml:space="preserve">producta ſecabit productam IB ad partes B, L, vt putà in L.</s> <s xml:id="echoid-s732" xml:space="preserve"/> </p> <pb o="19" file="0039" n="39" rhead=""/> <p> <s xml:id="echoid-s733" xml:space="preserve">Siverò CB fuerit maior BA, erit quoque ED maior DA, & </s> <s xml:id="echoid-s734" xml:space="preserve">tunc ex edu-<lb/>cta IDH ſupra ſubiectum planum dematur DH, quæ minor ſit ipſa DA, & </s> <s xml:id="echoid-s735" xml:space="preserve"><lb/>iungatur AH, & </s> <s xml:id="echoid-s736" xml:space="preserve">fiat vt HD ad DF, ita DF ad DI; </s> <s xml:id="echoid-s737" xml:space="preserve">erit rectangulum HDI æ-<lb/>quale quadrato DF, ſiue rectangulo EDB, ſed rectangulum EDB maius eſt <lb/>rectangulo ADB, cum ſit ED maior DA, quare rectangulum HDI maius <lb/>erit rectangulo ADB. </s> <s xml:id="echoid-s738" xml:space="preserve">Iam ex I ducatur IR parallela ad AH, ſecans produ-<lb/>ctam AD in R; </s> <s xml:id="echoid-s739" xml:space="preserve">erit HD ad DA, vt ID ad DR; </s> <s xml:id="echoid-s740" xml:space="preserve">ſed HD facta eſt minor DA, <lb/>ergo & </s> <s xml:id="echoid-s741" xml:space="preserve">ID erit minor DR, vnde rectangulum ſub maioribus AD, DR, maius <lb/>erit rectangulo ſub minoribus HD, DI; </s> <s xml:id="echoid-s742" xml:space="preserve">ſed rectangulum HDI demonſtra-<lb/>tum eſt maius rectangulo ADB, ergo rectangulum ADR eò amplius maius <lb/>erit rectangulo ADB: </s> <s xml:id="echoid-s743" xml:space="preserve">vnde recta BR maior erit recta DB, hoc eſt punctum <lb/>B cadet inter D, & </s> <s xml:id="echoid-s744" xml:space="preserve">R, ſiue inter parallelas AH, IR; </s> <s xml:id="echoid-s745" xml:space="preserve">quare iuncta I B, & </s> <s xml:id="echoid-s746" xml:space="preserve">pro-<lb/>ducta conueniet cum producta AH ad partes B, H, veluti in L.</s> <s xml:id="echoid-s747" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s748" xml:space="preserve">His itaque conſtructis, & </s> <s xml:id="echoid-s749" xml:space="preserve">demonſtratis; </s> <s xml:id="echoid-s750" xml:space="preserve">cum factum ſit vt ID ad DF, vel <lb/>ad DG, ita DG ad DH, ſi circa diametrum IH in plano ſecante deſcribatur <lb/>circulus ipſe tranſibit per puncta F, G: </s> <s xml:id="echoid-s751" xml:space="preserve">ſi ergo intelligatur deſcriptus conus, <lb/>cuius vertex L, baſis circulus IFHG; </s> <s xml:id="echoid-s752" xml:space="preserve">& </s> <s xml:id="echoid-s753" xml:space="preserve">in infinitum productus infra baſim, <lb/>communis ſectio eius conicæ ſuperficiei cum ſubiecto plano ſit linea AMF <lb/>BGNA. </s> <s xml:id="echoid-s754" xml:space="preserve">Dico hanc eſſe Ellipſim quæſitam.</s> <s xml:id="echoid-s755" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s756" xml:space="preserve">Eſt enim conus ILH ſectus plano per axem, triangulum facient LIH, & </s> <s xml:id="echoid-s757" xml:space="preserve"><lb/>ſecatur altero plano FBGA, (nempe ſubiecto plano) quod baſi non æquidi-<lb/>ſtat (cum ſe mutuò ſecent ſecundum rectam FG) & </s> <s xml:id="echoid-s758" xml:space="preserve">communis ſectio baſis <lb/>coni I H, & </s> <s xml:id="echoid-s759" xml:space="preserve">ſecantis plani BA eſt recta linea FG, quæ ad IH baſim trianguli <lb/>per axem eſt ducta perpendicularis, erit, per primam huius, ſectio AMFBGN <lb/>Ellipſis, cuius vertex B, diameter BA, cui ordinatim ductæ, qualis eſt FG, <lb/>ad datum angulum P applicantur ex conſtructione. </s> <s xml:id="echoid-s760" xml:space="preserve">Cumque factum ſit vt <lb/>ED ad DF, ita DF ad DB, erit rectangulum EDB ęquale quadrato DF, ſiue <lb/>rectangulo IDH, vnde rectangulum ADB, ad rectangulum EDB, erit vt <lb/>idem rectangulum ADB, ad rectangulum IDH; </s> <s xml:id="echoid-s761" xml:space="preserve">ſed rectangulum ADB ad <lb/>EDB, eſt vt AD ad DE, vel vt AB ad BC, ergo rectangulum ADB, ad re-<lb/>ctangulum IDH, erit vt AB ad BC: </s> <s xml:id="echoid-s762" xml:space="preserve">vnde AB eſt latus tranſuerſum, BC ve-<lb/>rò rectum deſcriptæ Ellipſis BFAG, vt ex prima huius. </s> <s xml:id="echoid-s763" xml:space="preserve">Quod erat facien-<lb/>dum.</s> <s xml:id="echoid-s764" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div54" type="section" level="1" n="33"> <head xml:id="echoid-head38" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s765" xml:space="preserve">CVm ad MAXIMARV M, MINIMARV Mque coni-ſe-<lb/>ctionum inſcriptibilium, ac circumſcriptibilium inuentionem <lb/>nobis ſit opus admir andam illam affectionem propagare circa <lb/>lineas ſemper magis, ac magis inter ſe accedentes, nunquam <lb/>verò ſimul coeuntes, ab ipſo Apollonio præcipuè animaduerſam inter curuam <lb/>Hyperbolæ, rectamque lineam, quàm ipſe Aſymptoton appellauit, neceſsè <lb/>quidem videretur, ad hoc vt integram huius argumenti doctrinam hic ſi-<lb/>mul habeatur, addere nunc, primam, ſecundam, decimam tertiam, ac de-<lb/>cimam quartam ſecundi conicorum ad prædictam Aſymptoton ſpectantes; </s> <s xml:id="echoid-s766" xml:space="preserve">ſed <pb o="20" file="0040" n="40" rhead=""/> quoniam harum quoque habemus demonſtrationes breuiores, & </s> <s xml:id="echoid-s767" xml:space="preserve">affirmati-<lb/>uas, non indirectas, quales ab Apollonio exhibentur in prima, ſecunda, ac <lb/>decima tertia, nè noſtri libelli molem aliundè tranſcriptis demonſtratiombus <lb/>augere velle videamur, apponemus hic proprias, ita procedendo.</s> <s xml:id="echoid-s768" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div55" type="section" level="1" n="34"> <head xml:id="echoid-head39" xml:space="preserve">THEOR. II. PROP. VIII.</head> <p> <s xml:id="echoid-s769" xml:space="preserve">Si Hyperbolen recta linea ad verticem contingat, & </s> <s xml:id="echoid-s770" xml:space="preserve">ab ipſa ex <lb/>vertice ad vtramque partem diametri ſumatur æqualis ei, quæ po-<lb/> <anchor type="note" xlink:label="note-0040-01a" xlink:href="note-0040-01"/> teſt quartam figuræ partem, quæ à ſectionis centro ad ſumptos ter-<lb/>minos contingentis ducuntur cum ſectione non conuenient; </s> <s xml:id="echoid-s771" xml:space="preserve">(quæ <lb/>in poſterum cum Apollonio vocentur ASYMPTOTI) nec erit al-<lb/>tera aſymptoton, quæ diuidat angulum ab ipſis factum.</s> <s xml:id="echoid-s772" xml:space="preserve"/> </p> <div xml:id="echoid-div55" type="float" level="2" n="1"> <note position="left" xlink:label="note-0040-01" xlink:href="note-0040-01a" xml:space="preserve">Prop. 1. 2 <lb/>ſecundi <lb/>con ic.</note> </div> <p> <s xml:id="echoid-s773" xml:space="preserve">SIt Hyperbole, cuius diameter, & </s> <s xml:id="echoid-s774" xml:space="preserve">tranſuerſum latus AB, centrum C, & </s> <s xml:id="echoid-s775" xml:space="preserve"><lb/>rectum figuræ latus B F, linea verò D E ſectionem contingat in B, & </s> <s xml:id="echoid-s776" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0040-01a" xlink:href="fig-0040-01"/> quartæ parti figuræ, quæ à lateribus <lb/>AB, BF continetur æquale ſit quadra-<lb/>tum vtriuſque ipſarum BD, BE, & </s> <s xml:id="echoid-s777" xml:space="preserve">iun-<lb/>ctæ CD, CE producantur. </s> <s xml:id="echoid-s778" xml:space="preserve">Dico pri-<lb/>mum eas cum ſectione numquam con-<lb/>uenire.</s> <s xml:id="echoid-s779" xml:space="preserve"/> </p> <div xml:id="echoid-div56" type="float" level="2" n="2"> <figure xlink:label="fig-0040-01" xlink:href="fig-0040-01a"> <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0040-01"/> </figure> </div> <p> <s xml:id="echoid-s780" xml:space="preserve">Nam in altera ipſarum, vt in CD, <lb/>infra contingentem, ſumpto quolibet <lb/>puncto G, ab eo ordinatim applicetur <lb/>GIH ſectionem, ac diametrum ſecans <lb/>in I, H, quæ ipſi D B æquidiſtabit. </s> <s xml:id="echoid-s781" xml:space="preserve">Et <lb/>quoniam eſt vt latus AB ad BF, ita <lb/>quadratum AB ad rectangulum ABF, <lb/>vel ſumptis horum ſub-quadruplis, ita <lb/>quadratum CB ad quadratum BD, vel quadratum CH ad quadratum HG, <lb/>& </s> <s xml:id="echoid-s782" xml:space="preserve">vt idem latus AB ad BF ita <anchor type="note" xlink:href="" symbol="a"/> eſt rectangulum AHB ad quadratum HI, erit <anchor type="note" xlink:label="note-0040-02a" xlink:href="note-0040-02"/> quadratum CH ad HG, vt rectangulum AHB ad quadratum HI, & </s> <s xml:id="echoid-s783" xml:space="preserve">permu-<lb/>tando quadratum CH ad rectangulum AHB, vt quadratum GH, ad HI, <lb/>ſed quadratum CH maius eſt rectangulo AHB (cum eius exceſſus ſit qua-<lb/>dratum CB, nam eſt AB ſecta bifariam in C, & </s> <s xml:id="echoid-s784" xml:space="preserve">ei adiecta eſt quædam B H) <lb/>quare & </s> <s xml:id="echoid-s785" xml:space="preserve">quadratum GH quadrato IH maius erit, hoc eſt punctum G cadet <lb/>extra Hy perbolen, & </s> <s xml:id="echoid-s786" xml:space="preserve">hoc ſemper de omnibus punctis rectarum CDG, CEL <lb/>quamuis in infinitum productarum. </s> <s xml:id="echoid-s787" xml:space="preserve">Sunt igitur lineæ CD; </s> <s xml:id="echoid-s788" xml:space="preserve">CE ſectioni nun-<lb/>quam occurrentes. </s> <s xml:id="echoid-s789" xml:space="preserve">Quod erat primò demonſtrandum, taleſque lineæ vo-<lb/>centur ASYMPTOTI.</s> <s xml:id="echoid-s790" xml:space="preserve"/> </p> <div xml:id="echoid-div57" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-0040-02" xlink:href="note-0040-02a" xml:space="preserve">21. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s791" xml:space="preserve">Amplius, ijſdem manentibus, dico quamlibet aliam CM, quæ diuidat <lb/>angulum DCE, neceſſariò Hyperbolen ſecare. </s> <s xml:id="echoid-s792" xml:space="preserve">Ducta enim BM, ex vertice <lb/>B, parallcla ad CD, conueniet cum CM; </s> <s xml:id="echoid-s793" xml:space="preserve">nam & </s> <s xml:id="echoid-s794" xml:space="preserve">ipſa CM cum altera æqui-<lb/>diſtantium CD conuenit in C: </s> <s xml:id="echoid-s795" xml:space="preserve">occurrat ergo in M, per quod ordinatim ap- <pb o="21" file="0041" n="41" rhead=""/> plicetur NMO fectionem, ac diametrum ſecans in N, O.</s> <s xml:id="echoid-s796" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s797" xml:space="preserve">Quoniam igitur eodem pænitus argumento, quo ſuperius demonſtratum <lb/>eſt rectangulum AHB ad quadratum HI, eſſe vt quadratum CB ad BD, eſt <lb/>quoque rectangulum AOB ad quadratum ON, vt idem quadratum C B ad <lb/>BD, vel vt quadratum BO ad OM, erit permutando, rectangulum AOB ad <lb/>quadratum BO, vt quadratum NO ad OM, ſed rectangulum AOB ſuperat <lb/>quadratum BO, (exceſſus enim eſt rectangulum ABO) ergo & </s> <s xml:id="echoid-s798" xml:space="preserve">quadratum <lb/>NO, maius eſt quadrato MO; </s> <s xml:id="echoid-s799" xml:space="preserve">ſed punctum N eſt in ipſa ſectione, quare pun-<lb/>ctum M cadit intra: </s> <s xml:id="echoid-s800" xml:space="preserve">ideoque iuncta CM ſectionem prius ſecat. </s> <s xml:id="echoid-s801" xml:space="preserve">Non eſt ergo <lb/>altera aſymptotos, quæ diuidat angulum ab aſymptotis factum. </s> <s xml:id="echoid-s802" xml:space="preserve">Quod erat <lb/>ſecundò demonſtrandum.</s> <s xml:id="echoid-s803" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div59" type="section" level="1" n="35"> <head xml:id="echoid-head40" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s804" xml:space="preserve">HIs itaque præoſtenſis, ipſarum ope, ac tertiæ ſecundi conico-<lb/>rum demonſtremus aliter decimam quartam eiuſdem, abſq; <lb/></s> <s xml:id="echoid-s805" xml:space="preserve">auxilio præcedentium 5. </s> <s xml:id="echoid-s806" xml:space="preserve">10. </s> <s xml:id="echoid-s807" xml:space="preserve">12. </s> <s xml:id="echoid-s808" xml:space="preserve">ac 13. </s> <s xml:id="echoid-s809" xml:space="preserve">quibus ipſa 14. </s> <s xml:id="echoid-s810" xml:space="preserve">in-<lb/>diget, præmiſſo tantum ſequenti Lemmate.</s> <s xml:id="echoid-s811" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div60" type="section" level="1" n="36"> <head xml:id="echoid-head41" xml:space="preserve">LEMMA II. PROP. IX.</head> <p> <s xml:id="echoid-s812" xml:space="preserve">Sit rectangulum ABD æquale quadrato BC. </s> <s xml:id="echoid-s813" xml:space="preserve">Dico addita qua-<lb/>cunque BE, rectangulum AED maius eſſe quadrato EC.</s> <s xml:id="echoid-s814" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s815" xml:space="preserve">CVm enim rectangulum ABD æquale ſit quadrato mediæ BC, erit AB <lb/>ad BC, vt BC ad BD, & </s> <s xml:id="echoid-s816" xml:space="preserve">diuidendo, & </s> <s xml:id="echoid-s817" xml:space="preserve">permutando AC ad CD, vt <lb/> <anchor type="figure" xlink:label="fig-0041-01a" xlink:href="fig-0041-01"/> CB ad BD. </s> <s xml:id="echoid-s818" xml:space="preserve">Et cum ſit DB minor <lb/>DE, habebit CD ad DB maiorem <lb/>rationem quam ad DE, & </s> <s xml:id="echoid-s819" xml:space="preserve">compo-<lb/>nendo CB ad BD, hoc eſt AC ad CD maiorem <anchor type="note" xlink:href="" symbol="a"/> habebit rationem quam <anchor type="note" xlink:label="note-0041-01a" xlink:href="note-0041-01"/> CE ad ED, & </s> <s xml:id="echoid-s820" xml:space="preserve">permutando AC ad CE <anchor type="note" xlink:href="" symbol="b"/> maiorem rationem quam CD ad <anchor type="note" xlink:label="note-0041-02a" xlink:href="note-0041-02"/> DE, & </s> <s xml:id="echoid-s821" xml:space="preserve">componendo AE ad EC <anchor type="note" xlink:href="" symbol="c"/> maiorem quam EC ad ED. </s> <s xml:id="echoid-s822" xml:space="preserve">Si fiat ergo vt AE ad EC, ita EC ad EF, habebit quoque EC ad EF maiorem rationem <lb/> <anchor type="note" xlink:label="note-0041-03a" xlink:href="note-0041-03"/> quam EC ad ED, vnde EF erit minor ED, ſed (cum factum ſit AE ad EC, <lb/>vt EC ad EF) rectangulum AEF æquale eſt quadrato EC, quare rectangu-<lb/>lum AED maius erit quadrato EC. </s> <s xml:id="echoid-s823" xml:space="preserve">Quod erat &</s> <s xml:id="echoid-s824" xml:space="preserve">c.</s> <s xml:id="echoid-s825" xml:space="preserve"/> </p> <div xml:id="echoid-div60" 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/QN4GHYBF/figures/0041-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">28. quin-<lb/>ti elem.</note> <note symbol="b" position="right" xlink:label="note-0041-02" xlink:href="note-0041-02a" xml:space="preserve">27. quin-<lb/>ti elem.</note> <note symbol="c" position="right" xlink:label="note-0041-03" xlink:href="note-0041-03a" xml:space="preserve">28. quin-<lb/>ti elem.</note> </div> </div> <div xml:id="echoid-div62" type="section" level="1" n="37"> <head xml:id="echoid-head42" xml:space="preserve">THEOR. III. PROP. X.</head> <p> <s xml:id="echoid-s826" xml:space="preserve">Aſymptoti, & </s> <s xml:id="echoid-s827" xml:space="preserve">ſectio in infinitum productæ ad ſe propius acce-<lb/> <anchor type="note" xlink:label="note-0041-04a" xlink:href="note-0041-04"/> dunt, & </s> <s xml:id="echoid-s828" xml:space="preserve">ad interuallum perueniunt minus quolibet dato interuallo.</s> <s xml:id="echoid-s829" xml:space="preserve"/> </p> <div xml:id="echoid-div62" type="float" level="2" n="1"> <note position="right" xlink:label="note-0041-04" xlink:href="note-0041-04a" xml:space="preserve">Prop. 14. <lb/>ſec. con.</note> </div> <p> <s xml:id="echoid-s830" xml:space="preserve">SIt Hyperbole, cuius aſymptoti CD, CE, & </s> <s xml:id="echoid-s831" xml:space="preserve">datum interuallum ſit M. <lb/></s> <s xml:id="echoid-s832" xml:space="preserve">Dico aſymptotos CD, CE, & </s> <s xml:id="echoid-s833" xml:space="preserve">ſectionem productas, ad ſe ſe propius <lb/>accedere, & </s> <s xml:id="echoid-s834" xml:space="preserve">ad interuallum peruenire minus dato interuallo M.</s> <s xml:id="echoid-s835" xml:space="preserve"/> </p> <pb o="22" file="0042" n="42" rhead=""/> <p> <s xml:id="echoid-s836" xml:space="preserve">Nam ſit quæcunque recta DBE ſectionem contingens in B: </s> <s xml:id="echoid-s837" xml:space="preserve">patet per 3. <lb/></s> <s xml:id="echoid-s838" xml:space="preserve">ſec. </s> <s xml:id="echoid-s839" xml:space="preserve">conic. </s> <s xml:id="echoid-s840" xml:space="preserve">ipſam DE cum vtraque aſymptoto conuenire, & </s> <s xml:id="echoid-s841" xml:space="preserve">ad tactum B ſe-<lb/>cari bifariam, & </s> <s xml:id="echoid-s842" xml:space="preserve">quadratum vtriuſque portionis DB, BE æquale eſſe quarte <lb/>parti figuræ, quæ ad diametrum CB per tactum ducta conſtituitur; </s> <s xml:id="echoid-s843" xml:space="preserve">quare ſi <lb/>fiat CA æqualis CB, appliceturque quælibet GIH ipſi DB æquidiſtans, <lb/>aſymptoton, ſectionem, ac diametrum ſecans in G, I, H, & </s> <s xml:id="echoid-s844" xml:space="preserve">per I ducatur <lb/>IP parallela ad CD, ſecans diametrum in P infra C (nam punctum I eſt intra <lb/>angulum GCH) erit vt in præcedenti oſtenſum fuit rectangulum AHB ad <lb/>quadratum HI vt quadratum CB ad quadratum BD, vel vt quadratum PH <lb/>ad quadratum HI; </s> <s xml:id="echoid-s845" xml:space="preserve">vnde rectangulum AHB æquale erit quadrato HP, ſiue <lb/>recta HP erit media proportionalis inter AH & </s> <s xml:id="echoid-s846" xml:space="preserve">HB; </s> <s xml:id="echoid-s847" xml:space="preserve">hoc eſt punctum P ca-<lb/>det inter C & </s> <s xml:id="echoid-s848" xml:space="preserve">B; </s> <s xml:id="echoid-s849" xml:space="preserve">quare IP, quæ ipſi GC æquidiſtat contingentem BD ſeca-<lb/>bit in Q, eritque BD maior DQ, ſiue maior intercepta GI.</s> <s xml:id="echoid-s850" xml:space="preserve"/> </p> <figure> <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0042-01"/> </figure> <p> <s xml:id="echoid-s851" xml:space="preserve">Iam applicata infra G qualibet alia RN diametro occurrent in O, ex N du-<lb/>cta ſit NS parallela ad RC, quæ contingentem BD, ac diametrum ſecabit vt <lb/>ſupra in T & </s> <s xml:id="echoid-s852" xml:space="preserve">S. </s> <s xml:id="echoid-s853" xml:space="preserve">Cumque rectangulum AHB ſit æquale quadrato HP, vt mo-<lb/>dò oſtendimus, ſitque in directum ipſi AH addita quædam HO, erit, per <lb/>præcedens Lemma, rectangulum AOB maius quadrato OP, ſed rectangu-<lb/>lum AOB eadem ratione, vt ſupra, oſtenditur æquale quadrato OS; </s> <s xml:id="echoid-s854" xml:space="preserve">quare <lb/>quadratum OS maius eſt quadrato OP, hoc eſt punctum S cadit inter C, & </s> <s xml:id="echoid-s855" xml:space="preserve"><lb/>P, ſiue CP eſt maior CS, vel DQ maior DT, hoc eſt GI maior RN. </s> <s xml:id="echoid-s856" xml:space="preserve">Quare <lb/>aſymptoton<unsure/> CD, & </s> <s xml:id="echoid-s857" xml:space="preserve">ſectio BIN quæ in infinitum productæ, nunquam ſimul <lb/>conueniunt, ad ſe propiùs accedunt; </s> <s xml:id="echoid-s858" xml:space="preserve">idemque de aſymptoto CE. </s> <s xml:id="echoid-s859" xml:space="preserve">Quod erat <lb/>primò &</s> <s xml:id="echoid-s860" xml:space="preserve">c.</s> <s xml:id="echoid-s861" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s862" xml:space="preserve">Præterea dico ipſas ad interuallum peruenire minus dato interuallo M.</s> <s xml:id="echoid-s863" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s864" xml:space="preserve">Sumatur DT ex cõtingente BD, quę ſit minor interuallo M, & </s> <s xml:id="echoid-s865" xml:space="preserve">per T aga-<lb/>tur STN parallela ad CD diametro occurrens in S, ſeceturq; </s> <s xml:id="echoid-s866" xml:space="preserve">SV æqualis SB, <pb o="23" file="0043" n="43" rhead=""/> & </s> <s xml:id="echoid-s867" xml:space="preserve">fiat vt AV ad VS, ita AS ad SO, & </s> <s xml:id="echoid-s868" xml:space="preserve">per O ordinatim applicetur ONR ſe-<lb/>ctionem ſecans in N, rectam verò ST in X. </s> <s xml:id="echoid-s869" xml:space="preserve">Et cum ſit vt AS ad SO, ita AV ad <lb/>VS, erit componendo AO ad OS, vt AS ad SV, vel vt AS ad SB, & </s> <s xml:id="echoid-s870" xml:space="preserve">permu-<lb/>tando, & </s> <s xml:id="echoid-s871" xml:space="preserve">per conuerſionem rationis, vt AO ad OS, ita SO ad OB, ergo re-<lb/>ctangulum AOB æquatur quadrato OS: </s> <s xml:id="echoid-s872" xml:space="preserve">ſed rectangulum AOB ad quadra-<lb/>tum ſuæ ordinatim ductæ ON in Hyperbola ſemper eſt vt quadratum CB ad <lb/>BD (vt iam ſuperius oſtendimus) vel vt quadratum SO ad OX: </s> <s xml:id="echoid-s873" xml:space="preserve">quare permu-<lb/>tando rectangulum AOB ad quadratum SO, erit vt quadratum ON ad qua-<lb/>dratum OX, ſed eſt rectangulum AOB æquale quadrato SO, ergo & </s> <s xml:id="echoid-s874" xml:space="preserve">qua-<lb/>dratum ON quadrato OX æquale erit, quare puncta N, & </s> <s xml:id="echoid-s875" xml:space="preserve">X idem funt, ſed <lb/>eſt N in ſectione, quare recta TX conuenit cum ſectione in X, vel N, hoc eſt <lb/>RN & </s> <s xml:id="echoid-s876" xml:space="preserve">RX æquales erunt, ſed eſt RX æqualis ipſi DT, & </s> <s xml:id="echoid-s877" xml:space="preserve">DT minor M, vnde <lb/>RN, vel RX erit quoque minor M. </s> <s xml:id="echoid-s878" xml:space="preserve">Peruenit ergo aſymptoton<unsure/> CD cum ſe-<lb/>ctione ad interuallum RN minus dato interuallo M. </s> <s xml:id="echoid-s879" xml:space="preserve">Quod tandem erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s880" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div64" type="section" level="1" n="38"> <head xml:id="echoid-head43" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s881" xml:space="preserve">HInc eſt, quodlibet diametri ſegmentum inter quamcunque applicatam, <lb/>& </s> <s xml:id="echoid-s882" xml:space="preserve">rectam ex ipſius occurſu cum ſectione alteri aſymptoton<unsure/> æquidi-<lb/>ſtanter ductam, medium eſſe proportionale inter aggregatum ex tranſuer-<lb/>ſo latere cum prædicto diametri ſegmento, idemque ſegmentum. </s> <s xml:id="echoid-s883" xml:space="preserve">Demon-<lb/>ſtratum eſt enim HP eſſe mediam proportionalem inter AH, & </s> <s xml:id="echoid-s884" xml:space="preserve">HB; </s> <s xml:id="echoid-s885" xml:space="preserve">& </s> <s xml:id="echoid-s886" xml:space="preserve">OS <lb/>mediam inter AO, & </s> <s xml:id="echoid-s887" xml:space="preserve">OB.</s> <s xml:id="echoid-s888" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div65" type="section" level="1" n="39"> <head xml:id="echoid-head44" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s889" xml:space="preserve">PAtet etiam quamcunque rectam, ex puncto tranſuerſi lateris inter cen-<lb/>trum, & </s> <s xml:id="echoid-s890" xml:space="preserve">verticem ſumpto alteri aſymptoton ęquidiſtanter ductam ne-<lb/>ceſſariò ſectioni occurrere. </s> <s xml:id="echoid-s891" xml:space="preserve">Iam enim ſupra oſtendimus rectam STX, quæ <lb/>ex puncto S in tranſuerſo CB ducta eſt aſymptoton<unsure/> CD parallela, cum ſe-<lb/>ctione conuenire in N.</s> <s xml:id="echoid-s892" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div66" type="section" level="1" n="40"> <head xml:id="echoid-head45" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s893" xml:space="preserve">HInc facilè erit oſtendere 13. </s> <s xml:id="echoid-s894" xml:space="preserve">ſecundi conicorum aliter, & </s> <s xml:id="echoid-s895" xml:space="preserve">affir-<lb/>matiuè, vt videre licet in ſequenti.</s> <s xml:id="echoid-s896" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div67" type="section" level="1" n="41"> <head xml:id="echoid-head46" xml:space="preserve">THEOR. IV. PROP. XI.</head> <p> <s xml:id="echoid-s897" xml:space="preserve">Si in loco aſymptotis, & </s> <s xml:id="echoid-s898" xml:space="preserve">ſectione terminato quædam recta linea <lb/> <anchor type="note" xlink:label="note-0043-01a" xlink:href="note-0043-01"/> ducatur alteri aſymptoton æquidiſtans, in vno tantùm puncto cum <lb/>ſectione conueniet, eamque neceſſariò ſecabit.</s> <s xml:id="echoid-s899" xml:space="preserve"/> </p> <div xml:id="echoid-div67" type="float" level="2" n="1"> <note position="right" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">Prop. 13. <lb/>ſec. conic.</note> </div> <p> <s xml:id="echoid-s900" xml:space="preserve">SIt in præcedenti ſchemate in loco ab aſymptotis, & </s> <s xml:id="echoid-s901" xml:space="preserve">ſectione terminato <lb/>quodcunque punctum S, à quo ducta ſit STX aſymptoton<unsure/> CD æquidi- <pb o="24" file="0044" n="44" rhead=""/> ſtans. </s> <s xml:id="echoid-s902" xml:space="preserve">Dico ipſam cum ſectione conuenire, eamque omnino ſecare.</s> <s xml:id="echoid-s903" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s904" xml:space="preserve">Iungatur CS, quæ producta ſectioni occurret, per ſecundam partem 8. <lb/></s> <s xml:id="echoid-s905" xml:space="preserve">huius, eritque ſectionis diameter: </s> <s xml:id="echoid-s906" xml:space="preserve">quare per Coroll. </s> <s xml:id="echoid-s907" xml:space="preserve">2. </s> <s xml:id="echoid-s908" xml:space="preserve">præcedentis, ipſa STX <lb/>ſectioni occurret, vt in X.</s> <s xml:id="echoid-s909" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s910" xml:space="preserve">Præterea, cum quæcunque contingenti æquidiſtans GI ſupra RX, inter <lb/>aſymptoton, & </s> <s xml:id="echoid-s911" xml:space="preserve">ſectionem intercepta, maior ſit ipſa RX, ſiue ipſa GZ, pun-<lb/>ctum Z cadet extra ſectionem, & </s> <s xml:id="echoid-s912" xml:space="preserve">ſic de quolibet alio puncto rectæ XTS. </s> <s xml:id="echoid-s913" xml:space="preserve">E <lb/>contra cum quælibet intercepta LY infra RX, parallela ad DB, minor ſit ip-<lb/>ſa RX, ſiue LF, punctum F cadet intra ſectionem, idemque de quolibet alio <lb/>puncto rectæ XF: </s> <s xml:id="echoid-s914" xml:space="preserve">vnde recta STX ab ipſo occurſu X cum ſectione, ad partes <lb/>verticis tota cadit extra, ad oppoſitas verò partes tota cadit intra ſectionem; <lb/></s> <s xml:id="echoid-s915" xml:space="preserve">ideoque in vno tantum puncto X Hyperbolen ſecat. </s> <s xml:id="echoid-s916" xml:space="preserve">Quod erat propoſitum.</s> <s xml:id="echoid-s917" xml:space="preserve"/> </p> <figure> <image file="0044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0044-01"/> </figure> </div> <div xml:id="echoid-div69" type="section" level="1" n="42"> <head xml:id="echoid-head47" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s918" xml:space="preserve">HInc eſt, lineam alteri aſymptoton æquidiſtantem per punctum, quod <lb/>ſit, vel in ipſa ſectione, vel intra, pariter in vno tantùm puncto ſe-<lb/>ctioni occurrere, eamque ſecare.</s> <s xml:id="echoid-s919" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s920" xml:space="preserve">Nam recta XS ex puncto X, quod cſt in Hyperbola, vel recta FS ex pun-<lb/>cto F, quod eſt intra, æquidiſtanter ducta aſymptoto CD, ſi ad partes cen-<lb/>tri C producatur, alteri aſymptoto CE omnino occurrit, (quoniam EC, ſe-<lb/>cans DC vnam parallelarum ſecat quoque alteram CE) vnde aliqua pars, <lb/>ipſius rectæ XS, vel FS cadit in loco ab aſymptotis, & </s> <s xml:id="echoid-s921" xml:space="preserve">ſectione terminato, <lb/>ac ideo ex his, quæ ſuperius oſtendimus, ipſa linea in vno tantùm puncto ſe-<lb/>ctioni occurret, ac Hyperbolen ſecabit.</s> <s xml:id="echoid-s922" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s923" xml:space="preserve">Qua propter, quælibet linea alteri aſymptoton æquidiſtans, dummodo ſit <lb/>ducta ex puncto, quod ſit in angulo ab aſymptotis facto, in vno tantùm pun-<lb/>cto Hyperbolæ occurrit, atque eam ſecat.</s> <s xml:id="echoid-s924" xml:space="preserve"/> </p> <pb o="25" file="0045" n="45" rhead=""/> </div> <div xml:id="echoid-div70" type="section" level="1" n="43"> <head xml:id="echoid-head48" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s925" xml:space="preserve">EX hucuſque demonſtratis liceat animaduertere quamcumque <lb/>aſymptoton quodam-modo eſſe primam ex centro ducibilium, <lb/>ſed Hyperbolæ non occurrentium; </s> <s xml:id="echoid-s926" xml:space="preserve">itemque eſſe primam ſibi ipſi <lb/>æquidiſtantium, ſed Hyperbolen non ſecantium.</s> <s xml:id="echoid-s927" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s928" xml:space="preserve">QVæcunque enim educta ex C diuidens angulum DCE ſecat Hyperbo-<lb/>len, quæcunque verò ex C ducta extra CD, Hyperbolæ quidem non <lb/>occurrit, cum neque ipſa CD interior, cum ſectione conueniat. <lb/></s> <s xml:id="echoid-s929" xml:space="preserve">Quare angulus DCE dici poterit MINIMVS ex centro C Hyperbolen com-<lb/>prehendentium, rectis lineis nunquam ei occurrentibus.</s> <s xml:id="echoid-s930" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s931" xml:space="preserve">Item quælibet SX aſymptoto CD æquidiſtanter ducta intra angulũ DCE, <lb/>Hyperbolen ſecat, quælibet verò extra angulum ducta eidem CD parallela, <lb/>nunquam conuenit cum CD, & </s> <s xml:id="echoid-s932" xml:space="preserve">eò minus cum ſectione: </s> <s xml:id="echoid-s933" xml:space="preserve">ex quo aſymptoton<unsure/> <lb/>Hyperbolæ appellari quodammodo poſſet vltima tangentium Hyperbolen, <lb/>ad infinitum tamen interuallum. </s> <s xml:id="echoid-s934" xml:space="preserve">Nam, quæcumque contingens Hyperbo-<lb/>len ad finitam diſtantiam, ſecat ſemper diametrum CB infra C, & </s> <s xml:id="echoid-s935" xml:space="preserve">quò pun-<lb/>ctum contactus remotius fuerit à vertice eò magis occurſus contingẽtis cum <lb/>diametro, centro C fiet propior; </s> <s xml:id="echoid-s936" xml:space="preserve">donec, cum punctum contactus per infini-<lb/>tum interuallum abierit à centro, prædictus occurſus cum ipſo centro con-<lb/>ueniat.</s> <s xml:id="echoid-s937" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s938" xml:space="preserve">Sed ne ſuſcipiendam materiam interpellare nobis ſit opus, cum in ipſius <lb/>progreſſu Parabolæ quadratura indigeamus, inter alias, quas habemus, <lb/>apponemus hic̀ tantùm eam, quæ, licet expeditior non ſit, nonnulla tamen <lb/>Lemmata, ac Theoremata præmittit, quorum prima ad aliquas de MA-<lb/>XIMIS, & </s> <s xml:id="echoid-s939" xml:space="preserve">MINIMIS propoſitiones omnino ſunt neceſſaria.</s> <s xml:id="echoid-s940" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div71" type="section" level="1" n="44"> <head xml:id="echoid-head49" xml:space="preserve">LEMMA III. PROP. XII.</head> <p> <s xml:id="echoid-s941" xml:space="preserve">Si fuerit vt recta AD ad DC, ita quadratum AB ad BC. </s> <s xml:id="echoid-s942" xml:space="preserve">Dico <lb/>tres AD, DB, DC eſſe in continua eademque ratione geometrica.</s> <s xml:id="echoid-s943" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s944" xml:space="preserve">NAm ſumpta BE tertia proportionali poſt AB, BC; <lb/></s> <s xml:id="echoid-s945" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/> cum ſit in prima figura, AB ad BC, vt BC ad BE, <lb/>erit componendo AB cum BC ad BC, vt BC cum BE ad <lb/>BE, & </s> <s xml:id="echoid-s946" xml:space="preserve">permutando, AB cum BC, ſiue AC, ad BC cum <lb/>BE, ſiue ad CE, vt BC ad BE, vel vt AB ad BC, ex con-<lb/>ſtructione: </s> <s xml:id="echoid-s947" xml:space="preserve">quod memento.</s> <s xml:id="echoid-s948" xml:space="preserve"/> </p> <div xml:id="echoid-div71" type="float" level="2" n="1"> <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a"> <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0045-01"/> </figure> </div> <p> <s xml:id="echoid-s949" xml:space="preserve">Et cum ſit, ex ſuppoſitione, linea AD ad DC vt qua-<lb/>dratum AB ad BC, & </s> <s xml:id="echoid-s950" xml:space="preserve">quadratum AB ad BC, vt linea AB <lb/>ad BE, ex conſtructione, erit AD ad DC, vt AB ad BE, <lb/>& </s> <s xml:id="echoid-s951" xml:space="preserve">per conuerſionem rationis, & </s> <s xml:id="echoid-s952" xml:space="preserve">permutando, & </s> <s xml:id="echoid-s953" xml:space="preserve">iterum <lb/>per conuerſionem rationis AD ad DB, vt AC ad CE, <pb o="26" file="0046" n="46" rhead=""/> velvt AB ad BC, vt ſuperius oſtendimus: </s> <s xml:id="echoid-s954" xml:space="preserve">& </s> <s xml:id="echoid-s955" xml:space="preserve">permutan-<lb/> <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> do, & </s> <s xml:id="echoid-s956" xml:space="preserve">per conuerſionem rationis, AD ad DB vt DB ad <lb/>DC. </s> <s xml:id="echoid-s957" xml:space="preserve">Quod in prima figura oſtendendum erat.</s> <s xml:id="echoid-s958" xml:space="preserve"/> </p> <div xml:id="echoid-div72" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0046-01"/> </figure> </div> <p> <s xml:id="echoid-s959" xml:space="preserve">In ſecunda verò: </s> <s xml:id="echoid-s960" xml:space="preserve">cum ſit AB ad BC, vt BC ad BE, erit <lb/>diuidendo, & </s> <s xml:id="echoid-s961" xml:space="preserve">permutando, AC ad CE vt BC ad BE, vel <lb/>vt AB ad BC, ex conſtructione: </s> <s xml:id="echoid-s962" xml:space="preserve">quod ſerua.</s> <s xml:id="echoid-s963" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s964" xml:space="preserve">Et cum ſit AD ad DC vt quadratũ AB ad BC, & </s> <s xml:id="echoid-s965" xml:space="preserve">qua-<lb/>dratum AB ad BC vt linea AB ad BE, ex conſtructione, <lb/>erit AD ad DC vt AB ad BE, & </s> <s xml:id="echoid-s966" xml:space="preserve">per conuerſionem ratio-<lb/>nis, permutando, conuertendo, diuidendo, & </s> <s xml:id="echoid-s967" xml:space="preserve">iterum <lb/>conuertendo AD ad DB, vt AC ad CE, vel vt AB ad <lb/>BC, vt modò oſtenſum fuit, & </s> <s xml:id="echoid-s968" xml:space="preserve">permutando, conuerten-<lb/>d<unsure/>o, per conuerſionem rationis, & </s> <s xml:id="echoid-s969" xml:space="preserve">diuidendo AD ad DB vt BD ad DC. </s> <s xml:id="echoid-s970" xml:space="preserve">Quod <lb/>erat in ſecunda demonſtrandum.</s> <s xml:id="echoid-s971" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div74" type="section" level="1" n="45"> <head xml:id="echoid-head50" xml:space="preserve">ALITER idem breuiùs.</head> <p> <s xml:id="echoid-s972" xml:space="preserve">Ijſdem poſitis: </s> <s xml:id="echoid-s973" xml:space="preserve">dico iterùm vt in præcedenti.</s> <s xml:id="echoid-s974" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s975" xml:space="preserve">DEſcribatur ſuper AD ſemicirculus AED, & </s> <s xml:id="echoid-s976" xml:space="preserve">per Cerigatur CE diame-<lb/>tro AD perpendicularis, iunganturque DE, AE, & </s> <s xml:id="echoid-s977" xml:space="preserve">BE, quæ produ-<lb/>cta in ſecuuda figura, occurrat cum AF ipſi CE parallela in puncto F.</s> <s xml:id="echoid-s978" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s979" xml:space="preserve">Iam in vtraque figura, cum <lb/> <anchor type="figure" xlink:label="fig-0046-02a" xlink:href="fig-0046-02"/> ſit per hypoteſim quadratum <lb/>AB ad BC, vt recta AD ad DC, <lb/>vel vt quadratum AD ad DE, <lb/>vel vt quadratum AE ad EC, <lb/>ob triangulorum ſimilitudiné, <lb/>erit recta AB ad BC, vt recta <lb/>AE ad EC: </s> <s xml:id="echoid-s980" xml:space="preserve">quare in prima fi-<lb/>gura erit angulus AEB, æqua-<lb/>lis angulo BEC, ſed angulus <lb/>BAE æquatur angulo D E C, <lb/>quare duo ſimul AEB, BAE, <lb/>ſiue vnicus DBE, æqualis erit <lb/>duobus ſimul BEC, DEC, ſiue vnico DEB, ergo BD eſt æqualis ipſi DE. </s> <s xml:id="echoid-s981" xml:space="preserve">In <lb/>ſecunda verò, cum ſit AB ad BC, vel FA ad EC, vt AE ad EC erunt AF, <lb/>A E interſe æquales, vnde angulus AEF, æqualis angulo AFE ſiue paralle-<lb/>larum externo CEB, ſed eſt AEF æqualis duobus ſimul ABE, EAB, quare <lb/>& </s> <s xml:id="echoid-s982" xml:space="preserve">CEB ijſdem angulis ABE, EAB æqualis crit, eſtque pars CED æqualis <lb/>vnico angulo EAD, ergo reliquus angulus DEB reliquo DBE æqualis erit, <lb/>hoc eſt recta DB æqualis DE. </s> <s xml:id="echoid-s983" xml:space="preserve">Itaque in vtraque figura cum DB ſit æqualis <lb/>DE, ſitque DE media proportionalis inter AD, DC, erit quoq; </s> <s xml:id="echoid-s984" xml:space="preserve">DB media <lb/>inter eaſdem AD, DC. </s> <s xml:id="echoid-s985" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s986" xml:space="preserve"/> </p> <div xml:id="echoid-div74" type="float" level="2" n="1"> <figure xlink:label="fig-0046-02" xlink:href="fig-0046-02a"> <image file="0046-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0046-02"/> </figure> </div> <pb o="27" file="0047" n="47" rhead=""/> </div> <div xml:id="echoid-div76" type="section" level="1" n="46"> <head xml:id="echoid-head51" xml:space="preserve">ITER VM aliter breuiùs, ſed negatiuè.</head> <p> <s xml:id="echoid-s987" xml:space="preserve">Sifuerit vt recta AD ad DC, ita quadratum AB ad BC, erunt <lb/>AD, DB, DC in continua ratione geometrica.</s> <s xml:id="echoid-s988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s989" xml:space="preserve">SIenim DB non eſt media proportionalis inter AD, DC, eſto ſi fieri po-<lb/>teſt media quæcunque DE; </s> <s xml:id="echoid-s990" xml:space="preserve">erit igitur, in prima figura, tota AD ad to-<lb/>tam DE, vt pars DE ad partem DC, ergo reliqua AE ad reliquam EC, erit <lb/> <anchor type="note" xlink:label="note-0047-01a" xlink:href="note-0047-01"/> vt pars ED ad DC, vel vt tota AD ad totam DE: </s> <s xml:id="echoid-s991" xml:space="preserve">(ex cõ-<lb/>ſtructione) in ſecunda verò cum ſit AD ad DE vt DE ad <lb/> <anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/> DC, erit componendo AE ad ED, vt EC ad CD, & </s> <s xml:id="echoid-s992" xml:space="preserve">per-<lb/>mutando AE ad EC, vt ED ad DC, vel vt AD ad DE <lb/>(ex conſtructione) cum ergo in vtraque figura ſit AE ad <lb/>EC, vt AD ad DE, erit quadratum AE ad EC, vt qua-<lb/>dratum AD ad DE, vel vt recta AD ad DC, vel vt qua-<lb/>dratum AB ad BC (ex ſuppoſitione) vel recta AE ad <lb/>EC, vt recta AB ad BC, & </s> <s xml:id="echoid-s993" xml:space="preserve">in prima figura componen-<lb/>do, at in ſecunda diuidendo, AC ad CE, vt AC ad CB, <lb/>quare CE, CB inter ſe ſunt &</s> <s xml:id="echoid-s994" xml:space="preserve">quales, totum, & </s> <s xml:id="echoid-s995" xml:space="preserve">pars, <lb/>quod eſt abſurdum: </s> <s xml:id="echoid-s996" xml:space="preserve">non eſt ergo media inter AD, & </s> <s xml:id="echoid-s997" xml:space="preserve"><lb/>DC, quæ ſit maior, vel minor DB; </s> <s xml:id="echoid-s998" xml:space="preserve">vnde ipſa DB erit <lb/>media proportionalis inter AD, & </s> <s xml:id="echoid-s999" xml:space="preserve">DC. </s> <s xml:id="echoid-s1000" xml:space="preserve">Quod demon-<lb/>ſtrare oportebat.</s> <s xml:id="echoid-s1001" xml:space="preserve"/> </p> <div xml:id="echoid-div76" type="float" level="2" n="1"> <note position="right" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">Vni-<lb/>uerſalius <lb/>quàm à <lb/>Caual. in <lb/>3. prop. <lb/>exerc. 6.</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/QN4GHYBF/figures/0047-01"/> </figure> </div> </div> <div xml:id="echoid-div78" type="section" level="1" n="47"> <head xml:id="echoid-head52" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s1002" xml:space="preserve">HInc patet, quod, cum fuerint tres magnitudines continuè proportio-<lb/>nales, tùm exceſſus quibus differunt, tùm earum aggregata, erunt in <lb/>eadem ratione, in qua ſunt datæ magnitudines: </s> <s xml:id="echoid-s1003" xml:space="preserve">quando enim poſitum fuit <lb/>eſſe AD ad DE, vt DE ad DC oſtenſum quoque fuit AE ad EC eſſe vt AD <lb/>ad DE, ſed in prima figura AE, EC ſunt exceſſus datarum magnitudinum, <lb/>in ſecunda verò ſunt aggregata primæ cum ſecunda, & </s> <s xml:id="echoid-s1004" xml:space="preserve">ſecundæ cum tertia; <lb/></s> <s xml:id="echoid-s1005" xml:space="preserve">quare patet, &</s> <s xml:id="echoid-s1006" xml:space="preserve">c.</s> <s xml:id="echoid-s1007" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div79" type="section" level="1" n="48"> <head xml:id="echoid-head53" xml:space="preserve">THEOR. V. PROP. XIII.</head> <p> <s xml:id="echoid-s1008" xml:space="preserve">Si duæ Parabolæ ad eaſdem partes deſcriptæ ad idem punctum <lb/>ſimul occurrant, ſintque earum diametri inter ſe æquidiſtantes, & </s> <s xml:id="echoid-s1009" xml:space="preserve"><lb/>applicatæ ſint eædem, ac ipſarum vertices ſint in eadem recta, quæ <lb/>ducitur ex occurſu; </s> <s xml:id="echoid-s1010" xml:space="preserve">ipſæ in nullo alio puncto ſimul conuenient, & </s> <s xml:id="echoid-s1011" xml:space="preserve"><lb/>omnes, quæ ex contactu in ipſis ducuntur in eadem ratione à ſe-<lb/>ctionibus diuidentur.</s> <s xml:id="echoid-s1012" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1013" xml:space="preserve">ESto Parabole ABC, cuius diameter BF, & </s> <s xml:id="echoid-s1014" xml:space="preserve">ducta BA, ſit quælibet DE <lb/>ipſi BF æquidiſtans, & </s> <s xml:id="echoid-s1015" xml:space="preserve">AC ordinatim applicata BF, & </s> <s xml:id="echoid-s1016" xml:space="preserve">per verticem D, <pb o="28" file="0048" n="48" rhead=""/> diametro DE deſcribatur Parabole ADG, cuius AE ſit eius ſemi-applicata: <lb/></s> <s xml:id="echoid-s1017" xml:space="preserve">dico primum Parabolen ADG, etiam ſi in infinitum producatur, totam ca-<lb/>dere intra ABC, & </s> <s xml:id="echoid-s1018" xml:space="preserve">ſi ex A ducatur quæcunque ANMH, ipſam à Parabola <lb/>ADG ſecari in O, in eadem ratione, ac AC ſecatur in G, & </s> <s xml:id="echoid-s1019" xml:space="preserve">AB in D.</s> <s xml:id="echoid-s1020" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1021" xml:space="preserve">Ducta enim ex A recta AP contingente Parabolen ABC, erit FB æqualis <lb/>BP; </s> <s xml:id="echoid-s1022" xml:space="preserve">ideoque ED æqualis DR, vnde AR continget Parabolen ADG, & </s> <s xml:id="echoid-s1023" xml:space="preserve">AH <lb/>ſecabit ipſam in O.</s> <s xml:id="echoid-s1024" xml:space="preserve"/> </p> <figure> <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0048-01"/> </figure> <p> <s xml:id="echoid-s1025" xml:space="preserve">Iam ductis ex H, O, ſemi - applicatis HI, OL, erit, ob Parabolen, FB ad <lb/>BI, vt quadratum AF ad HI, vel vt quadratum FM ad quadratum MI; </s> <s xml:id="echoid-s1026" xml:space="preserve">qua-<lb/>re per Lemma præcedens, erit FB ad BM, vt BM ad BI, & </s> <s xml:id="echoid-s1027" xml:space="preserve">per Coroll. </s> <s xml:id="echoid-s1028" xml:space="preserve">eiuſ-<lb/>dem, in vtraque figura, erit FM ad MI, vt FB ad BM; </s> <s xml:id="echoid-s1029" xml:space="preserve">eadem penitus ratio-<lb/>ne oſtendetur eſſe EN ad NL, vt ED ad DN, ſed eſt FB ad BM, vt ED ad <lb/>DN, quare & </s> <s xml:id="echoid-s1030" xml:space="preserve">FM ad MI erit vt EN ad NL, ſed FM ad MI, eſt vt AM ad MH, <lb/>& </s> <s xml:id="echoid-s1031" xml:space="preserve">EN ad NL, vt AN ad NO, quare AM ad MH erit vt AN ad NO, & </s> <s xml:id="echoid-s1032" xml:space="preserve">in pri-<lb/>ma figura conuertendo, componendo, & </s> <s xml:id="echoid-s1033" xml:space="preserve">permutando, HA ad AO, vt MA <lb/>ad AN; </s> <s xml:id="echoid-s1034" xml:space="preserve">in ſecunda verò per conuerſionem rationis, conuertendo, & </s> <s xml:id="echoid-s1035" xml:space="preserve">per-<lb/>mutando HA ad AO erit vt MA ad AN. </s> <s xml:id="echoid-s1036" xml:space="preserve">Eſt igitur in vtraque figura HA ad <lb/>AO, vt MA ad AN, vel vt BA ad AD, ſed eſt BA maior AD ex conſtru-<lb/>ctione, quare & </s> <s xml:id="echoid-s1037" xml:space="preserve">HA erit maior AO, ſed HA tota eſt intra Parabolen ABC, <lb/>vnde punctum O, quod eſt in Parabola ADG erit intra Parabolen ABC, & </s> <s xml:id="echoid-s1038" xml:space="preserve"><lb/>ſic de quocunque alio puncto Parabolæ ADG, etiam ſi ducta AH cadat in-<lb/>fra AC; </s> <s xml:id="echoid-s1039" xml:space="preserve">quare ipſa cadit tota intra ABC: </s> <s xml:id="echoid-s1040" xml:space="preserve">& </s> <s xml:id="echoid-s1041" xml:space="preserve">cum ſit HA ad AO, vt BA ad <lb/>AD, vel vt FA ad AE, vel ſumptis duplis, vt CA ad AG, erit diuidendo <lb/>HO ad OA, vt CG ad GA. </s> <s xml:id="echoid-s1042" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s1043" xml:space="preserve">c.</s> <s xml:id="echoid-s1044" xml:space="preserve"/> </p> <figure> <image file="0048-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0048-02"/> </figure> <pb o="29" file="0049" n="49" rhead=""/> </div> <div xml:id="echoid-div80" type="section" level="1" n="49"> <head xml:id="echoid-head54" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s1045" xml:space="preserve">HInc patet, quod ſi recta linea in Parabola vtcunque applicata ex <lb/> <anchor type="note" xlink:label="note-0049-01a" xlink:href="note-0049-01"/> vtraque parte ſectioni occurrens cum diametro, vel intra, vel extra <lb/>ſectionem conueniat, atque ex ipſius terminis cum ſectione, ad diametrum <lb/>ducantur ordinatæ, erunt ab his abſciſſa diametri ſegmenta ex vertice ſum-<lb/>pta, extremæ, & </s> <s xml:id="echoid-s1046" xml:space="preserve">abſciſſum ab applicata, erit media trium continuè propor-<lb/>tionalium. </s> <s xml:id="echoid-s1047" xml:space="preserve">Demonſtratum eſt enim in figuris Theorematis quando AH dia-<lb/>metrum ſecat in M, & </s> <s xml:id="echoid-s1048" xml:space="preserve">ſectionem in A, H, quod ordinatim applicatis AF, <lb/>HI, eſt FB ad BM, vt BM ad BI.</s> <s xml:id="echoid-s1049" xml:space="preserve"/> </p> <div xml:id="echoid-div80" type="float" level="2" n="1"> <note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">Vni-<lb/>uerſalius <lb/>quàm in <lb/>4. prop. <lb/>exerc. 6. <lb/>Caual.</note> </div> </div> <div xml:id="echoid-div82" type="section" level="1" n="50"> <head xml:id="echoid-head55" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s1050" xml:space="preserve">EX quo etiam elicitur, quod ſi in Parabola ABC ducta AH diametrum <lb/>ſecans in M producatur vſque ad occurſum cum contingente ex verti-<lb/>ce B in S, ſemper rectangulum ſub ſegmentis AS, & </s> <s xml:id="echoid-s1051" xml:space="preserve">SH, inter ſectionem, & </s> <s xml:id="echoid-s1052" xml:space="preserve"><lb/>contingentem interceptis æqua†@ quadrato ſegmenti SM inter contingẽtem, <lb/>ac diametrum intercepti. </s> <s xml:id="echoid-s1053" xml:space="preserve">Nam cum ſit vt FB ad BM, ita BM ad BI erit quo-<lb/>que ob parallelas, AS ad SM, vt SM ad SH, quare rectangulum ASH æqua-<lb/>bitur quadrato SM.</s> <s xml:id="echoid-s1054" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div83" type="section" level="1" n="51"> <head xml:id="echoid-head56" xml:space="preserve">COROLL. III.</head> <p> <s xml:id="echoid-s1055" xml:space="preserve">HInc etiam eſt, quod, ſi ijſdem poſitis, <lb/>interior Parabole ADG habuerit ver-<lb/> <anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/> ticẽ in D puncto medio rectæ AB, ipſa quo-<lb/>que tranſibit per F medium punctum baſis <lb/>AC, & </s> <s xml:id="echoid-s1056" xml:space="preserve">quæcunque educta ex contactu A, <lb/>qualis eſt AH, bifariam ſecabitur in O ab in-<lb/>terna ſectione; </s> <s xml:id="echoid-s1057" xml:space="preserve">quare ſi ex O ducatur OLM <lb/>diametro BF æquidiſtans, ipſa erit diameter <lb/>portionis ALH, & </s> <s xml:id="echoid-s1058" xml:space="preserve">AH vna applicatarum, <lb/>AO verò ſemi-applicata. </s> <s xml:id="echoid-s1059" xml:space="preserve">Cumque ſit AP <lb/>contingens ABC in A, erit OL in trilineo <lb/>mixto ADFB, æqualis LM in trilineo mixto <lb/>ALBP, & </s> <s xml:id="echoid-s1060" xml:space="preserve">ſic de omnibus vbicunque inter-<lb/>ceptis in ijſdem trilineis.</s> <s xml:id="echoid-s1061" xml:space="preserve"/> </p> <div xml:id="echoid-div83" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0049-01"/> </figure> </div> </div> <div xml:id="echoid-div85" type="section" level="1" n="52"> <head xml:id="echoid-head57" xml:space="preserve">THEOR. VI. PROP. XIV.</head> <p> <s xml:id="echoid-s1062" xml:space="preserve">Parabolæ æqualium altitudinum inter ſe ſunt vt baſes.</s> <s xml:id="echoid-s1063" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1064" xml:space="preserve">SInt duæ Parabolæ ABC, DEF æqualium altitudinum, hoc eſt concipian-<lb/>tur diſpoſitæ inter eaſdem parallelas BE, AF: </s> <s xml:id="echoid-s1065" xml:space="preserve">dico eſſe vt baſis AC <lb/>vnius, ad baſim DF alterius, ita Parabole ABC ad Parabolen DEF.</s> <s xml:id="echoid-s1066" xml:space="preserve"/> </p> <pb o="30" file="0050" n="50" rhead=""/> <p> <s xml:id="echoid-s1067" xml:space="preserve">Nam ſi hæ Parabolæ non fuerint baſibus proportionales, erit altera Para-<lb/>bolarum minor quàm opus eſt ad hoc vt huiuſmodi magnitudines ſint pro-<lb/>portionales. </s> <s xml:id="echoid-s1068" xml:space="preserve">Eſto igitur ſi poſſibile eſt minor ABC, & </s> <s xml:id="echoid-s1069" xml:space="preserve">eius defectus ſit O; <lb/></s> <s xml:id="echoid-s1070" xml:space="preserve">ita vt baſis AC ad DF ſit vt aggregatum Parabolæ ABC cum magnitudine O <lb/>ad Parabolen DEF. </s> <s xml:id="echoid-s1071" xml:space="preserve">Iam iuxta vulgatam methodum Antiquorum circum-<lb/>ſcribatur Parabolæ ABC, ſigura ex parallelogrammis conſtans, æqualium <lb/>altitudinum, ita vt eius <lb/>exceſſus ſupra Parabo-<lb/> <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> len ſit minor O; </s> <s xml:id="echoid-s1072" xml:space="preserve">quod <lb/>fiet, nempè ſi ex circũ-<lb/>ſcripto Parabolæ paral-<lb/>lelogrammo A Y; </s> <s xml:id="echoid-s1073" xml:space="preserve">per <lb/>biſectionem diametri B <lb/>G in I, auſeratur dimi-<lb/>dium parallelogrammũ <lb/>AL, & </s> <s xml:id="echoid-s1074" xml:space="preserve">exreliquo dimi-<lb/>dium, donec ſuperſit pa-<lb/>rallelogrammum CM, <lb/>quod minus ſit ſpacio O: </s> <s xml:id="echoid-s1075" xml:space="preserve">ſic enim exceſſus circumſcriptæ figuræ ex paralle-<lb/>logrammis, ſupra inſcriptam ex æque altis parallelogrammis erit maximum <lb/>parallelogrammum CM, (vt ſatis patet) quod eſt minus ſpacio O, ac ideo ex-<lb/>ceſſus circumſcriptæ ſupra ipſam Parabolen erit adhuc minor O; </s> <s xml:id="echoid-s1076" xml:space="preserve">quapropter <lb/>addita communi Parabola ABC, erit vniuerſa figura circumſcripta minor <lb/>aggregato Parabolæ ABC cum ſpacio O: </s> <s xml:id="echoid-s1077" xml:space="preserve">itaque circumſcripta ABC ex pa-<lb/>rallelogrammis ad Parabolen DEF minorem habebit rationem, quam hu-<lb/>iuſmodi aggregatum ad eandem Parabolen DEF, ſed prædictũ aggregatum <lb/>ad DEF Parabolen ponitur eſſe vt baſis AC ad DF, vel vt circumſcripta <lb/>ABC ad circumſcriptam DEF, quæ per æquidiſtantium baſibus interſectio-<lb/>nem deſcripta, ex æquè altis, & </s> <s xml:id="echoid-s1078" xml:space="preserve">numero æqualibus, ac proportionalibus pa-<lb/>rallelogrãmis conſtabit (cum ſit quadratum A C ad QX, vt recta GB ad BN, <lb/>vel vt HE ad EP, vel vt quadratũ DF ad quadratum TZ;</s> <s xml:id="echoid-s1079" xml:space="preserve">vnde & </s> <s xml:id="echoid-s1080" xml:space="preserve">recta AC ad <lb/>QX, vel parallelogrammum CM ad QS, vt recta DF ad TZ, vel vt paralle-<lb/>logrammum DR ad TV, & </s> <s xml:id="echoid-s1081" xml:space="preserve">ſic de reliquis ſingula ſingulis, vnde vniuerſa cir-<lb/>cumſcripta ABC, ad vniuerſam DEF, eſt vt vnum CM ad vnum DR, vel vt <lb/>baſis AC ad baſim DF) quare circumſcripta ABC ad Parabolen DEF mino-<lb/>rem habebit rationem, quam eadem circumſcripta ad circumſcriptam DEF, <lb/>hoc eſt circumſcripta ex parallelogrammis erit minor ei inſcripta Parabola <lb/>DEF, totum parte; </s> <s xml:id="echoid-s1082" xml:space="preserve">quod eſt abſurdum: </s> <s xml:id="echoid-s1083" xml:space="preserve">inter has ergo Parabolas non datur <lb/>minor quàm ſit opus ad hoc vt ipſæ ſint baſibus proportionales: </s> <s xml:id="echoid-s1084" xml:space="preserve">erit ergo Pa-<lb/>rabole ABC ad DEF, vt baſis AC ad DF baſim. </s> <s xml:id="echoid-s1085" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1086" xml:space="preserve"/> </p> <div xml:id="echoid-div85" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0050-01"/> </figure> </div> </div> <div xml:id="echoid-div87" type="section" level="1" n="53"> <head xml:id="echoid-head58" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s1087" xml:space="preserve">QVod oſtenſum eſt de integris Parabolis æquè altis, idem penitus con-<lb/>ſimili conſtructione, eademque ratiocinatione demonſtrabitur de <lb/>duobus trilineis ABG, CBG ab eadem diametro BG abſciſſis; <lb/></s> <s xml:id="echoid-s1088" xml:space="preserve">item de duobus trilineis Parabolicis ABG, DEH æqualium altitudinum, à <pb o="31" file="0051" n="51" rhead=""/> curuis AB, DE; </s> <s xml:id="echoid-s1089" xml:space="preserve">diametris BG, EH; </s> <s xml:id="echoid-s1090" xml:space="preserve">& </s> <s xml:id="echoid-s1091" xml:space="preserve">ſemi-applicatis AG, DH compre-<lb/>henſis, nempe trilineum ABG ad CBG, eſſe vt baſis AG ad GC, ſib æqua-<lb/>lem; </s> <s xml:id="echoid-s1092" xml:space="preserve">ac propterea diametrum BG Parabolen ABC bifariam ſecare; </s> <s xml:id="echoid-s1093" xml:space="preserve">& </s> <s xml:id="echoid-s1094" xml:space="preserve">vnũ-<lb/>quodque trilineorum eſſe ſemi-Parabolen; </s> <s xml:id="echoid-s1095" xml:space="preserve">& </s> <s xml:id="echoid-s1096" xml:space="preserve">ſemi-Parabolen ABG ad ſe-<lb/>mi-Parabolen DEH æqualis altitudinis, eſſe vt baſis AG ad baſim DH, & </s> <s xml:id="echoid-s1097" xml:space="preserve"><lb/>integram ABC ad dimidiam DEH eſſe vt baſis AC ad ſemi-baſim DH.</s> <s xml:id="echoid-s1098" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div88" type="section" level="1" n="54"> <head xml:id="echoid-head59" xml:space="preserve">THEOR. VII. PROP. XV.</head> <p> <s xml:id="echoid-s1099" xml:space="preserve">Parabolæ æqualium baſium ſunt inter ſe vt altitudines.</s> <s xml:id="echoid-s1100" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1101" xml:space="preserve">SInt primò duæ Parabolæ ABC, ABC ſuper eandem baſim AC, & </s> <s xml:id="echoid-s1102" xml:space="preserve">circa <lb/>eandem diametrum BE. </s> <s xml:id="echoid-s1103" xml:space="preserve">Dico has eſſe inter ſe vt earum altitudines FA, <lb/>GA; </s> <s xml:id="echoid-s1104" xml:space="preserve">& </s> <s xml:id="echoid-s1105" xml:space="preserve">quod de ſemi-Parabolis EBC, EDC demonſtrabitur, idem inſe-<lb/>quetur de duplis.</s> <s xml:id="echoid-s1106" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1107" xml:space="preserve">Si enim non eſt vt FA <lb/> <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> ad AG, ita ſemi-Parabo-<lb/>le EBC ad EDC, erit al-<lb/>tera ipſarum minor quàm <lb/>ſit opus ad hoc vt ſint pro-<lb/>portionales altitudinibus <lb/>FA, AG, ſitque, ſi poſſi-<lb/>bile eſt, minor EBC de-<lb/>fectu R, & </s> <s xml:id="echoid-s1108" xml:space="preserve">bifariam ſecta <lb/>EC in H, & </s> <s xml:id="echoid-s1109" xml:space="preserve">iterum EH <lb/>bifariam in I, &</s> <s xml:id="echoid-s1110" xml:space="preserve">c. </s> <s xml:id="echoid-s1111" xml:space="preserve">circum-<lb/>ſcribatur, vt in pręceden-<lb/>ti, trilineo ſemi-Parabo-<lb/>læ ECB figura BLCE ex <lb/>parallelogrammis ęque <lb/>altis conſtans, cuius ex-<lb/>ceſſus ſupra ſemi-Parabolen ſit minor R, ita vt ipſa circumſcripta figura <lb/>BLCE ad ſemi-Parabolen EDC adhuc minorem habeat rationem quàm <lb/>altitudo FA ad AG; </s> <s xml:id="echoid-s1112" xml:space="preserve">quo facto, ſemi-Parabolæ quoque EDC per æquidi-<lb/>ſtantium diametro interſectionem altera circumſcribatur figura DMNCE <lb/>ex totidem Parallelogrammis æque altis, &</s> <s xml:id="echoid-s1113" xml:space="preserve">c. </s> <s xml:id="echoid-s1114" xml:space="preserve">Et cum ſir ob Parabolas, re-<lb/>cta BE ad OI, vt rectangulum AEC ad AIC, vel vt DE ad PI, erit permu-<lb/>tando BE ad ED, vel parallelogrammum BI ad DI, vt OI ad IP, vel vt pa-<lb/>rallelogrammum OH, ad PH, & </s> <s xml:id="echoid-s1115" xml:space="preserve">ſic de reliquis circumſcriptæ BL CE, ad re-<lb/>liqua circumſcriptæ DMCE, ſingula ſingulis, quare vniuerſa circumſcripta <lb/>ALCE ad vniuerſam DMCE, erit vt vnum parallelogrammum BI ad vnum <lb/>DI, vel vt baſis BE ad ED, vel vt FA ad AG, ſed FA ad AG habet maiorem <lb/>rationem quàm circumſcripta ALCE ad ſemi-Parabolen EDC; </s> <s xml:id="echoid-s1116" xml:space="preserve">quare cir-<lb/>cumſcripta ALCE ad circumſcriptam DMCE, habebit maiorem rationem <lb/>quàm ad ſemi-Parabolen EDC, vnde circumſcripta DMCE minor erit in-<lb/>ſcripta ſemi-Parabola EDC; </s> <s xml:id="echoid-s1117" xml:space="preserve">totum parte, quod eſt abſurdum. </s> <s xml:id="echoid-s1118" xml:space="preserve">Non datur <lb/>ergo inter has ſemi-Parabolas minor quàm ſit opus, ad hoc vt ipſæ ſint ba- <pb o="32" file="0052" n="52" rhead=""/> ſibus proportionales: </s> <s xml:id="echoid-s1119" xml:space="preserve">qua-<lb/> <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/> re ſemi-Parabole EBC ad <lb/>EDC, ſiue tota ABC ad <lb/>totam ADC, ſuper ea-<lb/>dem baſi AC, & </s> <s xml:id="echoid-s1120" xml:space="preserve">circa eã-<lb/>dem diametrum BE, eſt vt <lb/>altitudo FA ad AG. </s> <s xml:id="echoid-s1121" xml:space="preserve">At <lb/>ſi concipiatur altera Para-<lb/>bole QST, cuius baſis QT <lb/>æqualis ſit baſi AC, alti-<lb/>tudo verò SV ſit æqualis <lb/>ipſi GA (quæcunq; </s> <s xml:id="echoid-s1122" xml:space="preserve">ſit in-<lb/>clinatio baſis cum diame-<lb/>tro SZ) ipſa, per præcedẽ-<lb/>tem propoſitionem, ęqua-<lb/>lis erit Parabolę ADC, ac <lb/>ideo QST ad ABC eandem habebit rationem, quàm ADC ad ABC, vel <lb/>quàm altitudo GA, ſiue SV ad FA. </s> <s xml:id="echoid-s1123" xml:space="preserve">Vnde Parabolæ æqualium baſium <lb/>ſunt inter ſe vt altitudines. </s> <s xml:id="echoid-s1124" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s1125" xml:space="preserve">c.</s> <s xml:id="echoid-s1126" xml:space="preserve"/> </p> <div xml:id="echoid-div88" 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/QN4GHYBF/figures/0051-01"/> </figure> <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/QN4GHYBF/figures/0052-01"/> </figure> </div> </div> <div xml:id="echoid-div90" type="section" level="1" n="55"> <head xml:id="echoid-head60" xml:space="preserve">THEOR. VIII. PROP. XVI.</head> <p> <s xml:id="echoid-s1127" xml:space="preserve">Sirecta linea ſemi-Parabolen ad extremum baſis contingens <lb/>cum diametro conueniat, & </s> <s xml:id="echoid-s1128" xml:space="preserve">intra ipſam ſuper eadem baſi deſcri-<lb/>pta ſit Parabole, cuius diameter ſit dimidium diametri ſemi-Para-<lb/>bolæ, ac ei æquidiſtet; </s> <s xml:id="echoid-s1129" xml:space="preserve">erit trilineum à contingente, producta dia-<lb/>metro, & </s> <s xml:id="echoid-s1130" xml:space="preserve">conuexa ſemi-Parabolica linea contentum, æquale tri-<lb/>lineo à diametro, conuexa Parabolica, & </s> <s xml:id="echoid-s1131" xml:space="preserve">concaua ſemi-Parabo-<lb/>lica comprehenſo.</s> <s xml:id="echoid-s1132" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1133" xml:space="preserve">ESto ſemi-Parabole ABC, cuius baſis AC, & </s> <s xml:id="echoid-s1134" xml:space="preserve">contingens AE diametro <lb/>CB occurrens in E, & </s> <s xml:id="echoid-s1135" xml:space="preserve">iuncta AB, ac bifariam ſecta AC in F, agatur F <lb/>GH æquidiſtans CB, & </s> <s xml:id="echoid-s1136" xml:space="preserve">ſuper baſi AC cum diametro GF, quod eſt dimidium <lb/>CB, deſcripta ſit Parabole AGC, (quæ cadet <anchor type="note" xlink:href="" symbol="a"/> tota intra ABC:) </s> <s xml:id="echoid-s1137" xml:space="preserve">Dico tri- <anchor type="note" xlink:label="note-0052-01a" xlink:href="note-0052-01"/> lineum AEBHA æquale eſſe trilineo AHBCGA.</s> <s xml:id="echoid-s1138" xml:space="preserve"/> </p> <div xml:id="echoid-div90" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">13. h.</note> </div> <p> <s xml:id="echoid-s1139" xml:space="preserve">Sed ad hoc demonſtrandum, videndum eſt primò, quomodo cuilibet tri-<lb/>lineo ex prædictis, circumſcribi poſſint figuræ ex æquè altis, & </s> <s xml:id="echoid-s1140" xml:space="preserve">numero æ-<lb/>qualibus parallelogrammis, &</s> <s xml:id="echoid-s1141" xml:space="preserve">c.</s> <s xml:id="echoid-s1142" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1143" xml:space="preserve">Per continuam igitur biſectionem, diuidatur contingens AE, vel baſis AC <lb/>in quotcunq; </s> <s xml:id="echoid-s1144" xml:space="preserve">partes æquales CD, DL, LM, MF &</s> <s xml:id="echoid-s1145" xml:space="preserve">c.</s> <s xml:id="echoid-s1146" xml:space="preserve">: & </s> <s xml:id="echoid-s1147" xml:space="preserve">per diuiſionum pun-<lb/>cta D, L, M, F, &</s> <s xml:id="echoid-s1148" xml:space="preserve">c. </s> <s xml:id="echoid-s1149" xml:space="preserve">ducãtur ipſi CBE æquidiſtantes D1, L2, M3, F4, &</s> <s xml:id="echoid-s1150" xml:space="preserve">c. </s> <s xml:id="echoid-s1151" xml:space="preserve">quæ <lb/>ſemi-Parabolen ſecent in Q, R, K, H &</s> <s xml:id="echoid-s1152" xml:space="preserve">c. </s> <s xml:id="echoid-s1153" xml:space="preserve">Parabolen verò in N, O, P, G, &</s> <s xml:id="echoid-s1154" xml:space="preserve">c.</s> <s xml:id="echoid-s1155" xml:space="preserve">; <lb/>& </s> <s xml:id="echoid-s1156" xml:space="preserve">ex B, Q, R, K &</s> <s xml:id="echoid-s1157" xml:space="preserve">c.</s> <s xml:id="echoid-s1158" xml:space="preserve">: ducantur BY, QZ, R&</s> <s xml:id="echoid-s1159" xml:space="preserve">, KI &</s> <s xml:id="echoid-s1160" xml:space="preserve">c.</s> <s xml:id="echoid-s1161" xml:space="preserve">: ipſi AE parallelæ, quæ <lb/>intra ſemi-Parabolen ABC cadent (cum ſint contingenti æquidiſtantes) vel <lb/>extra trilineum AEBHA. </s> <s xml:id="echoid-s1162" xml:space="preserve">Hac ergo methodo circumſcribetur trilineo figu-<lb/>ra EBYZ&</s> <s xml:id="echoid-s1163" xml:space="preserve">I &</s> <s xml:id="echoid-s1164" xml:space="preserve">c. </s> <s xml:id="echoid-s1165" xml:space="preserve">ex æquè altis parallelogrammis &</s> <s xml:id="echoid-s1166" xml:space="preserve">c.</s> <s xml:id="echoid-s1167" xml:space="preserve"/> </p> <pb o="33" file="0053" n="53" rhead=""/> <p> <s xml:id="echoid-s1168" xml:space="preserve">Iam ſi iungantur AN, AO, AP, &</s> <s xml:id="echoid-s1169" xml:space="preserve">c. </s> <s xml:id="echoid-s1170" xml:space="preserve">quæ ſecent LO, MP, FG, &</s> <s xml:id="echoid-s1171" xml:space="preserve">c. </s> <s xml:id="echoid-s1172" xml:space="preserve">in 5, 6, <lb/>7, &</s> <s xml:id="echoid-s1173" xml:space="preserve">c. </s> <s xml:id="echoid-s1174" xml:space="preserve">incerceptæ N5, O6, P7, cadent totæ intra Parabolen AGC, hoc eſt <lb/>extra trilineum AGCB; </s> <s xml:id="echoid-s1175" xml:space="preserve">& </s> <s xml:id="echoid-s1176" xml:space="preserve">ſi ex punctis B, Q, R, K, &</s> <s xml:id="echoid-s1177" xml:space="preserve">c. </s> <s xml:id="echoid-s1178" xml:space="preserve">ducantur contin-<lb/>gentes BS, QT, RV, KX, &</s> <s xml:id="echoid-s1179" xml:space="preserve">c. </s> <s xml:id="echoid-s1180" xml:space="preserve">ipſæ æquidiſtabunt rectis CD, N5, O6, P7, <lb/>&</s> <s xml:id="echoid-s1181" xml:space="preserve">c. </s> <s xml:id="echoid-s1182" xml:space="preserve">(cum ductæ AC, AN, AO, AP, &</s> <s xml:id="echoid-s1183" xml:space="preserve">c. </s> <s xml:id="echoid-s1184" xml:space="preserve">ſint <anchor type="note" xlink:href="" symbol="a"/> ordinatim ductæ diametris <anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/> BC, QN, RO, KP, &</s> <s xml:id="echoid-s1185" xml:space="preserve">c) ſicque circumſcribetur trilineo AHBCGA, figura <lb/>ex æquealtis parallelogrammis BD, S5, T6, V7, &</s> <s xml:id="echoid-s1186" xml:space="preserve">c.</s> <s xml:id="echoid-s1187" xml:space="preserve"/> </p> <div xml:id="echoid-div91" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">3. Co-<lb/>roll. 13. h.</note> </div> <p> <s xml:id="echoid-s1188" xml:space="preserve">Ampliùs, ſi ipſæ ZQ, &</s> <s xml:id="echoid-s1189" xml:space="preserve">R, IK, &</s> <s xml:id="echoid-s1190" xml:space="preserve">c. </s> <s xml:id="echoid-s1191" xml:space="preserve">producantur extra ſemi-Parabolen, <lb/>cadent totæ intra trilineum AEBH, atque ita inſcribetur ei figura ex paralle-<lb/> <anchor type="figure" xlink:label="fig-0053-01a" xlink:href="fig-0053-01"/> logrammis &</s> <s xml:id="echoid-s1192" xml:space="preserve">c. </s> <s xml:id="echoid-s1193" xml:space="preserve">Et ſi ex punctis Q, R, K, &</s> <s xml:id="echoid-s1194" xml:space="preserve">c. </s> <s xml:id="echoid-s1195" xml:space="preserve">ducantur contingentibus SB <lb/>TQ, VR, &</s> <s xml:id="echoid-s1196" xml:space="preserve">c. </s> <s xml:id="echoid-s1197" xml:space="preserve">æquidiſtantes, cadent totæ intra trilineum AHBCG, cum ſint <lb/>ordinatim ductæ, & </s> <s xml:id="echoid-s1198" xml:space="preserve">ſi ex punctis N, O, P, &</s> <s xml:id="echoid-s1199" xml:space="preserve">c. </s> <s xml:id="echoid-s1200" xml:space="preserve">ducantur ad partes diametri <lb/>EBC rectæ æquidiſtantes ipſis AC, AN, AO, &</s> <s xml:id="echoid-s1201" xml:space="preserve">c. </s> <s xml:id="echoid-s1202" xml:space="preserve">(quæ erunt etiam paral-<lb/>lelæ contingentibus ex B, Q, R, &</s> <s xml:id="echoid-s1203" xml:space="preserve">c. </s> <s xml:id="echoid-s1204" xml:space="preserve">vel prædictis applicatis ex Q, R, K &</s> <s xml:id="echoid-s1205" xml:space="preserve">c.) <lb/></s> <s xml:id="echoid-s1206" xml:space="preserve">cadent hæ@ quoq; </s> <s xml:id="echoid-s1207" xml:space="preserve">intra trilineum AHBCG, nam quæ ex N ducitur ipſi AC <lb/>æquidiſtans ad partes diametri CB cadit extra Parabolen AGNC, cum ſit <lb/>AD maior DC, & </s> <s xml:id="echoid-s1208" xml:space="preserve">quæ ex O ęquidiſtat ipſi AN cadit extra AGC, cum ſit AS <lb/>maior 5N, & </s> <s xml:id="echoid-s1209" xml:space="preserve">ſic de reliquis. </s> <s xml:id="echoid-s1210" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s1211" xml:space="preserve">hac operatione inſcribetur trilineo AH-<lb/>BCG figura ex parallelogrãmis æquealtis, &</s> <s xml:id="echoid-s1212" xml:space="preserve">c. </s> <s xml:id="echoid-s1213" xml:space="preserve">Quare ex his, & </s> <s xml:id="echoid-s1214" xml:space="preserve">ex 14. </s> <s xml:id="echoid-s1215" xml:space="preserve">huius <pb o="34" file="0054" n="54" rhead=""/> patet quomodo cuilibet horum trilineorum circumſcribi poſſit figura ex <lb/>æque-altis parallelogrammis, &</s> <s xml:id="echoid-s1216" xml:space="preserve">c. </s> <s xml:id="echoid-s1217" xml:space="preserve">quæ ſuperet proprium trilineum magni-<lb/>tudine, quæ minor ſit quacunque magnitudine propoſita.</s> <s xml:id="echoid-s1218" xml:space="preserve"/> </p> <div xml:id="echoid-div92" type="float" level="2" n="3"> <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/QN4GHYBF/figures/0053-01"/> </figure> </div> <p> <s xml:id="echoid-s1219" xml:space="preserve">Iam dico huiuſmodi trilinea inter ſe eſſe æqualia. </s> <s xml:id="echoid-s1220" xml:space="preserve">Nam ſi ſint inæqualia, <lb/>alterum ipſorum, vt puta AHBCGA altero AHBE minus erit, & </s> <s xml:id="echoid-s1221" xml:space="preserve">defectus ſit <lb/>ſpacium V<unsure/>; </s> <s xml:id="echoid-s1222" xml:space="preserve">quo poſito circumſcribatur, vti nuper docuimus, trilineo AHB <lb/>CGA figura ex parallelogrammis SC, TN, VO, HP, &</s> <s xml:id="echoid-s1223" xml:space="preserve">c. </s> <s xml:id="echoid-s1224" xml:space="preserve">cuius exceſſus ſupra <lb/>trilineum ſit minor magnitudine V<unsure/>. </s> <s xml:id="echoid-s1225" xml:space="preserve">quapropter talis figura adhuc minor erit <lb/>trilinco AEBHA, cui circumſcribatur, item per eaſdem lineas ipſi CB æqui-<lb/> <anchor type="figure" xlink:label="fig-0054-01a" xlink:href="fig-0054-01"/> diſtantes, figura ex totidem parallelogrammis EY, 1Z, 2&</s> <s xml:id="echoid-s1226" xml:space="preserve">, 3I, &</s> <s xml:id="echoid-s1227" xml:space="preserve">c. </s> <s xml:id="echoid-s1228" xml:space="preserve">Patet nũc <lb/>talem circumſcriptam, alteri circumſcriptæ ABD 567, &</s> <s xml:id="echoid-s1229" xml:space="preserve">c. </s> <s xml:id="echoid-s1230" xml:space="preserve">ęqualem eſſe, cum <lb/>vtraq; </s> <s xml:id="echoid-s1231" xml:space="preserve">ipſarum ex æqualibus numero, & </s> <s xml:id="echoid-s1232" xml:space="preserve">magnitudine parallelogrammis cõ-<lb/>ſtet vtrunq; </s> <s xml:id="echoid-s1233" xml:space="preserve">vtrique: </s> <s xml:id="echoid-s1234" xml:space="preserve">(parallelogramma enim EY, BD, ſunt inter eaſdem pa-<lb/>rallelas, & </s> <s xml:id="echoid-s1235" xml:space="preserve">ſuper æqualibus baſibus EB, BE; </s> <s xml:id="echoid-s1236" xml:space="preserve">& </s> <s xml:id="echoid-s1237" xml:space="preserve">parallelogrammum IZ æqua-<lb/>tur parallelogrãmo Q5, cum inter eaſdem ſint parallelas, & </s> <s xml:id="echoid-s1238" xml:space="preserve">ſuper æqualibus <lb/>baſibus <anchor type="note" xlink:href="" symbol="a"/> IQ, QN, & </s> <s xml:id="echoid-s1239" xml:space="preserve">ſic de ſingulis) ſed circumſcripta ABD 567, &</s> <s xml:id="echoid-s1240" xml:space="preserve">c. </s> <s xml:id="echoid-s1241" xml:space="preserve">eſt mi- <anchor type="note" xlink:label="note-0054-01a" xlink:href="note-0054-01"/> nor trilineo AEBHA, vt modò oſtendimus, ergo & </s> <s xml:id="echoid-s1242" xml:space="preserve">circumſcripta AEBYZ <lb/>&</s> <s xml:id="echoid-s1243" xml:space="preserve">I, &</s> <s xml:id="echoid-s1244" xml:space="preserve">c. </s> <s xml:id="echoid-s1245" xml:space="preserve">minor erit eodem trilineo AEBHA; </s> <s xml:id="echoid-s1246" xml:space="preserve">totum parte, quod eſt abſurdũ. <lb/></s> <s xml:id="echoid-s1247" xml:space="preserve">Quare huiuſmodi trilinea inter ſe ſunt æqualia. </s> <s xml:id="echoid-s1248" xml:space="preserve">Quod oſtendere propoſi-<lb/>tum fuit.</s> <s xml:id="echoid-s1249" xml:space="preserve"/> </p> <div xml:id="echoid-div93" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0054-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0054-01" xlink:href="note-0054-01a" xml:space="preserve">3. Co-<lb/>roll. 13. h.</note> </div> <pb o="35" file="0055" n="55" rhead=""/> </div> <div xml:id="echoid-div95" type="section" level="1" n="56"> <head xml:id="echoid-head61" xml:space="preserve">THEOR. IX. PROP. XVII.</head> <p> <s xml:id="echoid-s1250" xml:space="preserve">Parabole ſeſquitertia eſt trianguli eandem ipſi baſim, & </s> <s xml:id="echoid-s1251" xml:space="preserve">ean-<lb/>dem altitudinem habentis.</s> <s xml:id="echoid-s1252" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1253" xml:space="preserve">REpetito præcedenti diagrammate, dico Parabolen AB8 ſeſquitertiam <lb/>eſſe inſcripti trianguli AB8.</s> <s xml:id="echoid-s1254" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1255" xml:space="preserve">Nam ducta G9 parallela ad AC deſcribatur ſemi Parabole 9, 8, cuius dia-<lb/>meter ſit 9C, & </s> <s xml:id="echoid-s1256" xml:space="preserve">ſemi-applicata ſit C8, æqualis baſi AC Parabolæ <lb/>AGC. </s> <s xml:id="echoid-s1257" xml:space="preserve">Et cum ſit ſemi-Parabole ABC æqualis <anchor type="note" xlink:href="" symbol="a"/> ſemi-Parabolæ CB8, &</s> <s xml:id="echoid-s1258" xml:space="preserve"> <anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/> Parabole AGC æqualis <anchor type="note" xlink:href="" symbol="b"/> ſemi-Parabolæ C98, ſitque C98 dimidium <anchor type="note" xlink:href="" symbol="c"/> CB8 <anchor type="note" xlink:label="note-0055-02a" xlink:href="note-0055-02"/> (nam eſt C9 dimidium CB &</s> <s xml:id="echoid-s1259" xml:space="preserve">c.) </s> <s xml:id="echoid-s1260" xml:space="preserve">erit Parabole AGC dimidium ſemi-Parabo-<lb/> <anchor type="note" xlink:label="note-0055-03a" xlink:href="note-0055-03"/> læ ABC, ſiue æqualis trilineo AHBCGA, ac etiam trilineo <anchor type="note" xlink:href="" symbol="d"/> AEBH; </s> <s xml:id="echoid-s1261" xml:space="preserve">quare <anchor type="note" xlink:label="note-0055-04a" xlink:href="note-0055-04"/> totum triangulum AEC ſeſqui alterum erit ſemi-Parabolæ ABC, ſiuc erit <lb/>vt 6 ad 4, ſed ad triangulum ABC eſt vt 6 ad 3, cum ſit EC dupla CB, vnde <lb/>ſemi-Parabole ABC ad triangulum ABC, hoc eſt dupla ad duplum, nempe <lb/>Parabole AB8 ad inſcriptum triangulum AB8, erit vt 4 ad 3. </s> <s xml:id="echoid-s1262" xml:space="preserve">Quod demon-<lb/>ſtrare oportebat.</s> <s xml:id="echoid-s1263" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">Coroll. <lb/>prop. 14. h.</note> <note symbol="b" position="right" xlink:label="note-0055-02" xlink:href="note-0055-02a" xml:space="preserve">Coroll. <lb/>prop. 14. h.</note> <note symbol="c" position="right" xlink:label="note-0055-03" xlink:href="note-0055-03a" xml:space="preserve">15. h.</note> <note symbol="d" position="right" xlink:label="note-0055-04" xlink:href="note-0055-04a" xml:space="preserve">16. h.</note> </div> </div> <div xml:id="echoid-div97" type="section" level="1" n="57"> <head xml:id="echoid-head62" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s1264" xml:space="preserve">VT hoc loco, ex aduerſo indirectæ Antiquorum viæ per duplicem <lb/>poſitionem, luce clarius pateat quantum facilitatis, breuitatis, <lb/>atquæ euidentiæ naſciſcatur è noua, directaque methodo (rectè <lb/>tamen cautèque vſurpata) acutiſsimi Geometræ Caualerij, <lb/>per indiuiſibilium doctrinam, nobis amiciſsimam, ex hac alteram accipe <lb/>eiuſdem theorematis demonſtr ationem, conſimili arte cōp@catam, ac in præ-<lb/>cedenti.</s> <s xml:id="echoid-s1265" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div98" type="section" level="1" n="58"> <head xml:id="echoid-head63" xml:space="preserve">THEOR. X. PROP. XVIII.</head> <p> <s xml:id="echoid-s1266" xml:space="preserve">Parabole ſeſquitertia eſt trianguli eandem ipſi baſim, & </s> <s xml:id="echoid-s1267" xml:space="preserve">ean-<lb/>dem altitudinem habentis.</s> <s xml:id="echoid-s1268" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1269" xml:space="preserve">SIt Parabole ABC, cuius diameter BD, baſis AC: </s> <s xml:id="echoid-s1270" xml:space="preserve">dico ipſam ſeſquiter-<lb/>tiam eſſe inſcripti trianguli ABC.</s> <s xml:id="echoid-s1271" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1272" xml:space="preserve">Bifariam enim ſecta AD in G, per quod ducta GF parallela ad DB, & </s> <s xml:id="echoid-s1273" xml:space="preserve">per <lb/>F, FH parallela ad AD, ac deſcriptis, vt in præcedenti figura Parabola <lb/>AED, & </s> <s xml:id="echoid-s1274" xml:space="preserve">portione Parabolæ HCD, cuius diameter ſit HD, & </s> <s xml:id="echoid-s1275" xml:space="preserve">ſemi-applica-<lb/>ta ſit DC ducatur in tota ABC quælibet applicata NI. </s> <s xml:id="echoid-s1276" xml:space="preserve">diametrum ſecans in <lb/>M, eritque NM æqualis ML, & </s> <s xml:id="echoid-s1277" xml:space="preserve">ſic de quibuslibet alijs applicatis ipſi AC æ-<lb/>quidiſtantibus, quare omnes ſimul in portione ABD, omnibus ſimul in por-<lb/>tione DBC æquales erunt, ſiue portio ABD æqualis DBC, nempè vtraque <lb/>erit ſemi-Parabole, & </s> <s xml:id="echoid-s1278" xml:space="preserve">eadem ratione oſtendetur DHC ſemi-Parabolen eſſe.</s> <s xml:id="echoid-s1279" xml:space="preserve"/> </p> <pb o="36" file="0056" n="56" rhead=""/> <p> <s xml:id="echoid-s1280" xml:space="preserve">Iam applicata quacunque OPQR, tùm in Parabola AED, tùm in ſemi-<lb/>Parabola DHC; </s> <s xml:id="echoid-s1281" xml:space="preserve">cum ſit quadratum AD ad OP vt linca GF ad FS, vel vt DH <lb/>ad HQ, vel vt quadratum DC ad QR, ſintque antecedentia AD, DC ęqua-<lb/>lia, erunt & </s> <s xml:id="echoid-s1282" xml:space="preserve">conſequentia OP, QR æqualia, nempè applicata OP æqualis ap-<lb/>plicatæ QR, & </s> <s xml:id="echoid-s1283" xml:space="preserve">ita de omnibus &</s> <s xml:id="echoid-s1284" xml:space="preserve">c. </s> <s xml:id="echoid-s1285" xml:space="preserve">quare integra Parabole AED æquatur <lb/>ſemi-Parabolæ DHC.</s> <s xml:id="echoid-s1286" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1287" xml:space="preserve">Amplius ducta quacunque TVX <lb/>parallela ad BD, erit BD ad TX, vt <lb/> <anchor type="figure" xlink:label="fig-0056-01a" xlink:href="fig-0056-01"/> rectangulum ADC ad AXC, vel vt <lb/>HD ad VX, & </s> <s xml:id="echoid-s1288" xml:space="preserve">permutando, cum ſit <lb/>BD dupla DH, & </s> <s xml:id="echoid-s1289" xml:space="preserve">TX erit dupla XV, <lb/>& </s> <s xml:id="echoid-s1290" xml:space="preserve">ſic de omnibus interceptis, & </s> <s xml:id="echoid-s1291" xml:space="preserve">æ-<lb/>quidiſtantibus in ſemi-Parabola DB <lb/>C, & </s> <s xml:id="echoid-s1292" xml:space="preserve">in ſemi-Parabola DHC, vnde <lb/>tota ſemi-Parabole DBC dupla eſt <lb/>totius ſemi-Parabolæ DHC, & </s> <s xml:id="echoid-s1293" xml:space="preserve">ſum-<lb/>ptis æqualibus; </s> <s xml:id="echoid-s1294" xml:space="preserve">ſemi-Parabole ABD <lb/>dupla Parabolæ AFD, ſiue trilineum <lb/>ANBDFA, æquale erit Parabolæ <lb/>AFD.</s> <s xml:id="echoid-s1295" xml:space="preserve"/> </p> <div xml:id="echoid-div98" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0056-01"/> </figure> </div> <p> <s xml:id="echoid-s1296" xml:space="preserve">Tandé, ſi ſit AE contingens ABC <lb/>in A, erit EB æqualis BD, & </s> <s xml:id="echoid-s1297" xml:space="preserve">ducta <lb/>in trilineo AEBDFA quacunque IKZ parallela ad ED, erit IK æqualis <anchor type="note" xlink:href="" symbol="a"/> KZ, <anchor type="note" xlink:label="note-0056-01a" xlink:href="note-0056-01"/> & </s> <s xml:id="echoid-s1298" xml:space="preserve">ſic de omnibus alijs interceptis in trilineis AEBNA, & </s> <s xml:id="echoid-s1299" xml:space="preserve">ANBDFA quare <lb/>totum trilineum AEBNA æquabitur toto trilineo ANBDFA, ſed hoc, modò <lb/>oſtenſum fuit æquale Parabolæ AFD, quapropter totum triangulum AED <lb/>erit ſeſquialterum ſemi-Parabolæ ABD, vel erit vt 6 ad 4, ſed ad triangulum <lb/>ABD eſt vt 6 ad 3; </s> <s xml:id="echoid-s1300" xml:space="preserve">quare ſemi - Parabole ABD ad inſcriptum triangulum <lb/>ABD erit vt 4 ad 3, & </s> <s xml:id="echoid-s1301" xml:space="preserve">duplum ad duplum, hoc eſt Parabole ABC ad trian-<lb/>gulum ABC, ſuper eadem baſi AC, & </s> <s xml:id="echoid-s1302" xml:space="preserve">eiuſdem altitudinis cum Parabola, <lb/>erit vt 4, ad 3, nempe ſeſquitertium. </s> <s xml:id="echoid-s1303" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1304" xml:space="preserve"/> </p> <div xml:id="echoid-div99" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve">3. Co-<lb/>roll. 13. h.</note> </div> <p style="it"> <s xml:id="echoid-s1305" xml:space="preserve">Sed iam tempus eſt vt ſuſceptum opus aggrediamur, initio facto à defini-<lb/>tionibus.</s> <s xml:id="echoid-s1306" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div101" type="section" level="1" n="59"> <head xml:id="echoid-head64" xml:space="preserve">Definitiones Secundæ. <lb/>I.</head> <p> <s xml:id="echoid-s1307" xml:space="preserve">CONI SECTIONES ÆQVALITER INCLINAT Æ <lb/> <anchor type="figure" xlink:label="fig-0056-02a" xlink:href="fig-0056-02"/> vocentur illę, quarum ordinatim ductæ æquales inuicem <lb/>angulos cum earum diametris efficiunt.</s> <s xml:id="echoid-s1308" xml:space="preserve"/> </p> <div xml:id="echoid-div101" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0056-02"/> </figure> </div> <p> <s xml:id="echoid-s1309" xml:space="preserve">Videlicet coni - ſectio ABC vocabitur æqualiter incli-<lb/>nata, vel eiuſdem inclinationis, ac ſectio conica DEF, cum <lb/>vtriuſque ordinatim ductæ AGC, DHF, earum diametros <lb/>BG, EH, ad æquales diuidunt angulos, hoc eſt cum an-<lb/>gulus AGB, angulo DHE, & </s> <s xml:id="echoid-s1310" xml:space="preserve">qui ei deinceps CGB reli-<lb/>quo FHE æqualis fuerit.</s> <s xml:id="echoid-s1311" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div103" type="section" level="1" n="60"> <head xml:id="echoid-head65" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s1312" xml:space="preserve">Coni-ſectio vel circulus, coni-ſectioni, vel circulo ſi- <pb o="37" file="0057" n="57" rhead=""/> mul adſcribi intelligatur, vel SECTIONES SIMVL ADSCRIPTÆ dican-<lb/>tur, quando cum fuerint æqualiter inclinatæ earum diametri, & </s> <s xml:id="echoid-s1313" xml:space="preserve">ordinatim <lb/>ductæ inter ſe mutuò congruant.</s> <s xml:id="echoid-s1314" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1315" xml:space="preserve">Nempe cum duę coni-<lb/> <anchor type="figure" xlink:label="fig-0057-01a" xlink:href="fig-0057-01"/> ſectiones ABC, DEF æ-<lb/>qualiter inclinatæ, ita di-<lb/>ſpoſitę fuerint vt ipſarum <lb/>diametri BG, EI ſibi mu-<lb/>tuò congruant, & </s> <s xml:id="echoid-s1316" xml:space="preserve">omnes <lb/>vnius applicatę (quarum <lb/>vna AG) congruant om-<lb/>nibus alterius applicatis, <lb/>(quarum vna ſit DI) ipſæ <lb/>vocentur ſectiones ſimul <lb/>adſcriptæ.</s> <s xml:id="echoid-s1317" xml:space="preserve"/> </p> <div xml:id="echoid-div103" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0057-01"/> </figure> </div> </div> <div xml:id="echoid-div105" type="section" level="1" n="61"> <head xml:id="echoid-head66" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s1318" xml:space="preserve">Coni-ſectio, vel circulus, coni-ſectioni, vel circulo inſcribi dicatur, vel <lb/>SECTIO SECTIONI INSCRIPTA vocetur, quando cum fuerit altera al-<lb/>teri adſcripta, ſit quoque tota intra eandem, nec alicubi ſe mutuò ſecent, <lb/>licet in infinitum producantur, quæ in infinitum extendi poſſunt.</s> <s xml:id="echoid-s1319" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1320" xml:space="preserve">Hoc eſt, ſi vt in prima, & </s> <s xml:id="echoid-s1321" xml:space="preserve">ſecunda figura vtriuſque ordinis præcedentis <lb/>ſchematis duæ coni-ſectiones ABC, DEF fuerint ſimul adſcriptæ, & </s> <s xml:id="echoid-s1322" xml:space="preserve">altera <lb/>ipſarum vt DEF tota cadat intra aliam ABC, tunc dicatur ſectio DEF inſcri-<lb/>pta ſectioni ABC.</s> <s xml:id="echoid-s1323" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div106" type="section" level="1" n="62"> <head xml:id="echoid-head67" xml:space="preserve">IV.</head> <p> <s xml:id="echoid-s1324" xml:space="preserve">Coni-ſectio, vel circulus, coni-ſectioni, vel circulo circumſcribi intelli-<lb/>gatur, vel SECTIO SECTIONI CIRCVMSCRIPTA dicatur, ſi cum fue-<lb/>rit altera alteri adſcripta, tota quoque cadat extra eandem, nec alicubi ſe <lb/>mutuò ſecent quamuis in infinitum abeant.</s> <s xml:id="echoid-s1325" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1326" xml:space="preserve">Qualis eſt in prædictis figuris ſectio ABC, quæ cum ſit adſcripta ſectioni <lb/>DEF tota cadit extra eandem DEF.</s> <s xml:id="echoid-s1327" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div107" type="section" level="1" n="63"> <head xml:id="echoid-head68" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s1328" xml:space="preserve">Coni-ſectio, vel circulus coni-ſectioni, vel circulo per verticem, vel per <lb/>punctum intra, aut extra ſectionem datum adſcribi, vel inſcribi, aut circum-<lb/>ſcribi intelligatur, ſiue SECTIO SECTIONI PER VERTICEM, vel PER <lb/>DATVM PVNCTVM INTRA, aut EXTRA SECTIONEM ADSCRI-<lb/>PTA, vel INSCRIPTA, aut CIRCVMSCRIPTA dicatur, quando cum <lb/>fuerit altera alteri adſcripta, vel inſcripta, aut circumſcripta, vnius diameter <lb/>per datum punctum educta ſit quoque diameter alterius; </s> <s xml:id="echoid-s1329" xml:space="preserve">vt videre eſt in <lb/>præcedentibus figuris.</s> <s xml:id="echoid-s1330" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div108" type="section" level="1" n="64"> <head xml:id="echoid-head69" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s1331" xml:space="preserve">HYPERBOLE, & </s> <s xml:id="echoid-s1332" xml:space="preserve">ELLIPSES, SIMILES inter ſe dicantur, quando cum <lb/>fuerint ęqualiter inclinatę ipſarum latera ſint proportionalia, hoc eſt vt tran-<lb/>ſuerſum vnius ad rectum, ita ſit tranſuerſum ad rectum alterius eiuſdem no-<lb/>minis.</s> <s xml:id="echoid-s1333" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div109" type="section" level="1" n="65"> <head xml:id="echoid-head70" xml:space="preserve">VII.</head> <p> <s xml:id="echoid-s1334" xml:space="preserve">CONGRVENTES CONI-SECTIONES dicantur illæ, quæ cum fue- <pb o="38" file="0058" n="58" rhead=""/> rint æqualiter inclinatæ, ſi ſint per vertices ſimul adſcriptæ, inter ſe mutuò <lb/>congruant.</s> <s xml:id="echoid-s1335" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div110" type="section" level="1" n="66"> <head xml:id="echoid-head71" xml:space="preserve">VIII.</head> <p> <s xml:id="echoid-s1336" xml:space="preserve">CONI-SECTIONIS VEL CIRCVLI PORTIO, SIVE SEGMENTVM <lb/>vocetur ſuperficies à quadam ſectionis ordinatim ducta, & </s> <s xml:id="echoid-s1337" xml:space="preserve">curua ſectionis, <lb/>aut circuli peripheria terminata. </s> <s xml:id="echoid-s1338" xml:space="preserve">Et ipſa ordinata dicatur <lb/>BASIS PORTIONIS, SIVE SEGMENTI.</s> <s xml:id="echoid-s1339" xml:space="preserve"/> </p> <figure> <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0058-01"/> </figure> </div> <div xml:id="echoid-div111" type="section" level="1" n="67"> <head xml:id="echoid-head72" xml:space="preserve">IX.</head> <p> <s xml:id="echoid-s1340" xml:space="preserve">MENSALIS CONI-SECTIONIS, VEL CIRCVLI <lb/>dicatur differentia duorum ſegmétorum eiuſdem coni-<lb/>ſectionis, quorum baſes ſint parallelæ.</s> <s xml:id="echoid-s1341" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1342" xml:space="preserve">Vt ſi ex coni-ſectione, vel circulo ABC abſcindan-<lb/>tur duæ portiones ABC, DBE, quarum baſes AC, DE <lb/>ſint parallelæ, ipſarum portionum differentia ADEC di-<lb/>catur menſalis, & </s> <s xml:id="echoid-s1343" xml:space="preserve">ipſæ AC, DE baſes, & </s> <s xml:id="echoid-s1344" xml:space="preserve">AD, CE late-<lb/>ra eiuſdem menſalis.</s> <s xml:id="echoid-s1345" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div112" type="section" level="1" n="68"> <head xml:id="echoid-head73" xml:space="preserve">THEOR. XI. PROP. XIX.</head> <p> <s xml:id="echoid-s1346" xml:space="preserve">Si fuerint duæ quæcunque coni-ſectiones æqualiter inclinatæ <lb/>per vertices ſimul adſcriptæ, ipſæ vel erunt in totum congruentes, <lb/>& </s> <s xml:id="echoid-s1347" xml:space="preserve">eiuſdem nominis, vel in totum diſiunctæ, præter in vertice, hoc <lb/>eſt altera alteri inſcripta, vel in duobus tantùm punctis ſe mutuò ſe-<lb/>cabunt in ipſis tamen verticibus ſe contingentes.</s> <s xml:id="echoid-s1348" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1349" xml:space="preserve">SInt in præſenti ſchematiſmo duæ quæcunque coni-ſectiones ABC, DBE <lb/> <anchor type="note" xlink:label="note-0058-01a" xlink:href="note-0058-01"/> æqualiter inclinatæ pereundem verticem B ſimul adſcriptę, quarum có-<lb/>munis diameter ſit BF: </s> <s xml:id="echoid-s1350" xml:space="preserve">dico has ſectiones, vel eſſe in totum congruentes, vel <lb/>in totum diſiunctæ, vel in duobus tantùm punctis, ſe mutuò ſecantes.</s> <s xml:id="echoid-s1351" xml:space="preserve"/> </p> <div xml:id="echoid-div112" type="float" level="2" n="1"> <note position="left" xlink:label="note-0058-01" xlink:href="note-0058-01a" xml:space="preserve">Schematif-<lb/>mus 1. & 2.</note> </div> <p> <s xml:id="echoid-s1352" xml:space="preserve">Ducatur ex vertice B cuilibet in altera ſectionum ordinatim applicatæ æ-<lb/>quidiſtans BGH, quæ vtranq; </s> <s xml:id="echoid-s1353" xml:space="preserve">ſectionem continget <anchor type="note" xlink:href="" symbol="a"/> ſuper qua ſumatur BH, <anchor type="note" xlink:label="note-0058-02a" xlink:href="note-0058-02"/> rectum latus ſectionis ABC, & </s> <s xml:id="echoid-s1354" xml:space="preserve">BG rectum ſectionis DBE, ipſarumque regu-<lb/>læ, ſectionis videlicet ABC, ſit HPL, & </s> <s xml:id="echoid-s1355" xml:space="preserve">ſectionis DBE ſit GOI.</s> <s xml:id="echoid-s1356" xml:space="preserve"/> </p> <div xml:id="echoid-div113" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0058-02" xlink:href="note-0058-02a" xml:space="preserve">32. primi <lb/>conic.</note> </div> <p> <s xml:id="echoid-s1357" xml:space="preserve">Iam, vel regulæ GOI, HPL ſibi mutuò congruunt, vel infra contingen-<lb/>tem BGH nunquam conueniunt, vel infra eandem ſe mutuò ſecant. </s> <s xml:id="echoid-s1358" xml:space="preserve">Si pri-<lb/>mùm, vt in primis 4. </s> <s xml:id="echoid-s1359" xml:space="preserve">figuris; </s> <s xml:id="echoid-s1360" xml:space="preserve">dico ſectiones in totum ſimul congruere, & </s> <s xml:id="echoid-s1361" xml:space="preserve">eiuſ-<lb/>dem nominis eſſe.</s> <s xml:id="echoid-s1362" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1363" xml:space="preserve">Sumpto enim in ſectione ABC quolibet puncto M, per ipſum ducatur ſe-<lb/>ctionum communis ordinatim applicata MNFOP, ſectionem ſecans DBE in <lb/>N, diametrum in F, regulam GI in O, NL in P. </s> <s xml:id="echoid-s1364" xml:space="preserve">Et quoniam in 4. </s> <s xml:id="echoid-s1365" xml:space="preserve">primis fi-<lb/>guris, in quibus regulæ ſunt congruentes latitudines FO, FP ſunt æquales, <lb/>& </s> <s xml:id="echoid-s1366" xml:space="preserve">altitudo eadem BF erit rectangulum BFO <anchor type="note" xlink:href="" symbol="b"/> ſiue quadratum NF in ſectione <anchor type="note" xlink:label="note-0058-03a" xlink:href="note-0058-03"/> DBE, æquale rectangulo BFP <anchor type="note" xlink:href="" symbol="c"/> ſiue quadrato MF in ſectione ABC, quare <anchor type="note" xlink:label="note-0058-04a" xlink:href="note-0058-04"/> & </s> <s xml:id="echoid-s1367" xml:space="preserve">ſemi-applicatæ NF, MF æquales erunt, hoc eſt ſectiones DBE, ABC con-<lb/>ueniunt ſimul in punctis N, & </s> <s xml:id="echoid-s1368" xml:space="preserve">M, quæ ſunt extrema communium applicata- <pb file="0059" n="59"/> <pb file="0060" n="60"/> <anchor type="figure" xlink:label="fig-0060-01a" xlink:href="fig-0060-01"/> <pb file="0061" n="61"/> <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/> <pb file="0062" n="62"/> <pb o="39" file="0063" n="63" rhead=""/> rum ex eodem diametri puncto F: </s> <s xml:id="echoid-s1369" xml:space="preserve">idemque oſtendetur de omnibus alijs ex-<lb/>tremis punctis communium applicatarum ad vtraſque diametri partes: </s> <s xml:id="echoid-s1370" xml:space="preserve">qua-<lb/>re huiuſmodi ſectiones erunt in totum congruentes: </s> <s xml:id="echoid-s1371" xml:space="preserve">eruntque eiuſdem no-<lb/>minis; </s> <s xml:id="echoid-s1372" xml:space="preserve">quoniam cum regula Parabolæ æquidiſtet diametro; </s> <s xml:id="echoid-s1373" xml:space="preserve">Hyperbolæ au-<lb/>tem conueniat cum diametro extra ſectionem; </s> <s xml:id="echoid-s1374" xml:space="preserve">Ellipſis verò eidem diametro <lb/>intra ſectionem occurrat, hoc eſt ad extremum tranſuerſi lateris, cumque <lb/>harum ſectionum diametri ſimul congruant (nam ſectiones ſunt ſimul adſcri-<lb/>ptæ) ſi diuerſi nominis eſſent ipſarum regulæ nunquam congruerent, quod <lb/>eſt contra hypoteſim. </s> <s xml:id="echoid-s1375" xml:space="preserve">Sunt ergo tales ſectiones congruentes ſimul, & </s> <s xml:id="echoid-s1376" xml:space="preserve">eiuſ-<lb/>dem nominis. </s> <s xml:id="echoid-s1377" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1378" xml:space="preserve">c.</s> <s xml:id="echoid-s1379" xml:space="preserve"/> </p> <div xml:id="echoid-div114" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-0058-03" xlink:href="note-0058-03a" xml:space="preserve">Coroll. <lb/>prop. 1. h.</note> <note symbol="c" position="left" xlink:label="note-0058-04" xlink:href="note-0058-04a" xml:space="preserve">Coroll. <lb/>brop. 1. h.</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/QN4GHYBF/figures/0060-01"/> </figure> <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/QN4GHYBF/figures/0061-01"/> </figure> </div> <p> <s xml:id="echoid-s1380" xml:space="preserve">Si verò regulæ GOI, HPL infra contingentem BGH nunquam conueniũt, <lb/>diſiunctim ſimul procedentes, vt in 26. </s> <s xml:id="echoid-s1381" xml:space="preserve">proximè ſubſequentibus figuris ap-<lb/>paret, in quarum primis quatuor, regulæ ſunt parallelæ, in reliquis autem à <lb/>contingente BGH ad partes ſectionum ſunt ſemper inter ſe recedentes, eſtq; <lb/></s> <s xml:id="echoid-s1382" xml:space="preserve">regula GOI propinquior diametro quàm HPL; </s> <s xml:id="echoid-s1383" xml:space="preserve">facta eadem conſtructione, <lb/>vt ſupra; </s> <s xml:id="echoid-s1384" xml:space="preserve">quoniam latitudo FO minor eſt latitudine FP, & </s> <s xml:id="echoid-s1385" xml:space="preserve">altitudo BF eſt ea-<lb/>dem, erit rectangulum BFO <anchor type="note" xlink:href="" symbol="a"/> ſiue quadratum applicatæ NF in ſectione <anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/> DBE, maius rectangulo BFP <anchor type="note" xlink:href="" symbol="b"/> ſiue quadrato applicatæ MF in ſectione AB <anchor type="note" xlink:label="note-0063-02a" xlink:href="note-0063-02"/> C, hoc eſt applicata NF erit minor ipſa MF: </s> <s xml:id="echoid-s1386" xml:space="preserve">quare punctum m ſectionis AB <lb/>C cadit extra ſectionem DBE: </s> <s xml:id="echoid-s1387" xml:space="preserve">idemque de omnibus alijs punctis ſectionis <lb/>ABC ad vtranque diametri partem. </s> <s xml:id="echoid-s1388" xml:space="preserve">Vnde tota ſectio ABC cadit extra ſe-<lb/>ctionem DBE; </s> <s xml:id="echoid-s1389" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s1390" xml:space="preserve">tales ſectiones ſunt in totum diſiunctæ (eò quod ſem-<lb/>per diſiunctim procedant ipſarum regulæ) & </s> <s xml:id="echoid-s1391" xml:space="preserve">in communi tantùm vertice B <lb/>ſe mutuò contingunt. </s> <s xml:id="echoid-s1392" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s1393" xml:space="preserve">c.</s> <s xml:id="echoid-s1394" xml:space="preserve"/> </p> <div xml:id="echoid-div115" type="float" level="2" n="4"> <note symbol="a" position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">Coroll. <lb/>prop. I. h.</note> <note symbol="b" position="right" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">Coroll. <lb/>prop. I. h.</note> </div> <p> <s xml:id="echoid-s1395" xml:space="preserve">Sitandem ſectionum regulę GOI, HPL infra contingentem BGH ad par-<lb/>tes ſectionum ſe mutuò ſecant in P, vt videre eſt in 9. </s> <s xml:id="echoid-s1396" xml:space="preserve">vltimis figuris; </s> <s xml:id="echoid-s1397" xml:space="preserve">duca-<lb/>tur ex P communis ſectionum applicata PFNM ſecans diametrum in F, ſe-<lb/>ctionem ABC in M, & </s> <s xml:id="echoid-s1398" xml:space="preserve">DBE in N. </s> <s xml:id="echoid-s1399" xml:space="preserve">Iam cum in ſectione ABC quadratum <lb/>applicatæ MF æquale <anchor type="note" xlink:href="" symbol="c"/> ſit rectangulo BFP, & </s> <s xml:id="echoid-s1400" xml:space="preserve">quadratum applicatæ NF in <anchor type="note" xlink:label="note-0063-03a" xlink:href="note-0063-03"/> ſectione DBE æquale ſit eidem rectangulo BFP, erunt quadrata MF, NF in-<lb/>ter ſe æqualia, hoc eſt ipſæ applicatæ æquales; </s> <s xml:id="echoid-s1401" xml:space="preserve">quare huiuſmodi ſectiones <lb/>conueniunt ſimul in puncto M. </s> <s xml:id="echoid-s1402" xml:space="preserve">Eadem omnino ratione oſtendetur has ſe-<lb/>ctiones ad alteram quoque diametri partem ſimul conuenire in extremo pũ-<lb/>cto R reliquæ ad vnam ſectionum applicatæ ex eodem diametri puncto F: <lb/></s> <s xml:id="echoid-s1403" xml:space="preserve">ergo in duobus punctis M & </s> <s xml:id="echoid-s1404" xml:space="preserve">R, præter in communi vertice B, ſimul conue-<lb/>niunt, in quibus patet has ſectiones ſe mutuò ſecare; </s> <s xml:id="echoid-s1405" xml:space="preserve">nam regulæ HL, GI <lb/>conueniunt ſimul in vnico puncto P, in quo ſe mutuò ſecantes, hinc inde di-<lb/>ſiunctim procedunt, cadens PH ſegmentum regulæ LPH remotius à diame-<lb/>tro BF, quàm PG ſegmentum regulæ GOI; </s> <s xml:id="echoid-s1406" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s1407" xml:space="preserve">ſegmentum ſectionis <lb/>ABC ſupra applicatam MR totum cadet extra ſegmentum ſectionis DBE <lb/>ſupra eandem applicatam; </s> <s xml:id="echoid-s1408" xml:space="preserve">è contra verò reliquum portionis ABC infra ap-<lb/>plicatam MR cadet totum intra reliquum portionis DBE infra eandem ap-<lb/>plicatam, cum ſegmentũ PL propriæ regulæ HPL diſiunctum ſit, & </s> <s xml:id="echoid-s1409" xml:space="preserve">propius <lb/>diametro BF quàm ſegmentum PI propriæ regulæ GOI: </s> <s xml:id="echoid-s1410" xml:space="preserve">omneque id oſten-<lb/>ditur eadem penitus ratione, ac in ſecunda parte huius Theorematis demõ-<lb/>ſtrauimus: </s> <s xml:id="echoid-s1411" xml:space="preserve">quare huiuſmodi coni-ſectiones per vertices ſimul adſcriptæ, & </s> <s xml:id="echoid-s1412" xml:space="preserve"><lb/>quarũ regulæ ſe mutuò ſecant infra contingentem ex vertice, in ipſis vertici- <pb o="40" file="0064" n="64" rhead=""/> bus ſe contingunt; </s> <s xml:id="echoid-s1413" xml:space="preserve">& </s> <s xml:id="echoid-s1414" xml:space="preserve">in duobus tantùm punctis ſe mutuò ſecant. </s> <s xml:id="echoid-s1415" xml:space="preserve">Quod tan-<lb/>dem erat demonſtrandum.</s> <s xml:id="echoid-s1416" xml:space="preserve"/> </p> <div xml:id="echoid-div116" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-0063-03" xlink:href="note-0063-03a" xml:space="preserve">Coroll. <lb/>prop. I. h.</note> </div> </div> <div xml:id="echoid-div118" type="section" level="1" n="69"> <head xml:id="echoid-head74" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s1417" xml:space="preserve">PAtet hinc, quod ſi regulæ coni-ſectionum per vertices ſimul adſcripta-<lb/>rum ſibi ipſis congruant ſectiones quoque erunt inter ſe congruentes, <lb/>vt in primis quatuor figuris præcedentis ſchematis oſtenſum eſt; </s> <s xml:id="echoid-s1418" xml:space="preserve">& </s> <s xml:id="echoid-s1419" xml:space="preserve">ſi fuerint <lb/>inter ſe congruentes, etiam ipſarum regulæ ſimul congruent: </s> <s xml:id="echoid-s1420" xml:space="preserve">ſed cum regu-<lb/>læ ſimul congruunt, congruunt quoque, & </s> <s xml:id="echoid-s1421" xml:space="preserve">latera, & </s> <s xml:id="echoid-s1422" xml:space="preserve">è conuerſo, cum ad æ-<lb/>quales angulos inter ſe diſpoſita intelligantur, quare cum latera fuerint inter <lb/>ſe congruentia ſiue æqualia, ſectiones quoque inter ſe congruentes erunt; </s> <s xml:id="echoid-s1423" xml:space="preserve">& </s> <s xml:id="echoid-s1424" xml:space="preserve"><lb/>ſi ſectiones fuerint congruentes etiam ipſarum latera æqualia erunt.</s> <s xml:id="echoid-s1425" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1426" xml:space="preserve">Si verò regulę infra recta ſectionum latera ex vertice contingenter appli-<lb/>cata diſiunctim procedentes nunquam ſimul conueniant, nec ipſæ ſectiones <lb/>vnquam conuenient, ſed in vertice ſe mutuò contingent, & </s> <s xml:id="echoid-s1427" xml:space="preserve">ea inſcripta erit, <lb/>ſiue minor, cuius regula infra prædictam contingentem diametro ſectionum <lb/>ſit propior, ſeu cadat tota inter diametrum, & </s> <s xml:id="echoid-s1428" xml:space="preserve">regulam alterius ſectionis; <lb/></s> <s xml:id="echoid-s1429" xml:space="preserve">quæ è contra circumſcripta erit, ſiue maior, vt apparet in 26. </s> <s xml:id="echoid-s1430" xml:space="preserve">figuris ſubſe-<lb/>quentibus.</s> <s xml:id="echoid-s1431" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1432" xml:space="preserve">Si tandem ipſarum regulæ infra contingentes ex vertice ſe mutuò ſecent, <lb/>ſectiones quoque, ſed in duobus tantùm punctis hinc inde à vertice (in quo <lb/>ſe tangunt) ſe mutuò ſecabunt, in illis nempe, quæ ſunt extrema eiuſdem <lb/>ordinatim applicatæ, ex regularum interſectione eductæ, ſuper qua duæ co-<lb/>ni-ſectionum portiones inerunt, quarum ea erit inſcripta, cuius regulæ ſe-<lb/>gmentum inter prædictam applicatam, & </s> <s xml:id="echoid-s1433" xml:space="preserve">contingentem interceptum, pro-<lb/>pinquius ſit diametro ſectionum, altera verò circumſcripta, ſiue maior cuius <lb/>regulæ ſegmentum à prædicta diametro magis diſtet, quod omne ſatis patet <lb/>ex reliquis eiuſdem ſchcmatis figuris.</s> <s xml:id="echoid-s1434" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div119" type="section" level="1" n="70"> <head xml:id="echoid-head75" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s1435" xml:space="preserve">PAtet quoque in Parabolis, & </s> <s xml:id="echoid-s1436" xml:space="preserve">in alijs coni-ſectionibus eiuſdem nominis <lb/>per vertices ſimul adſcriptis, cum eodem tranſuerſo latere, illam, quę <lb/>minus habet rectum latus inſcriptam, ſiue minorem eſſe ea cuius rectum la-<lb/>tus maius eſt, & </s> <s xml:id="echoid-s1437" xml:space="preserve">è contra. </s> <s xml:id="echoid-s1438" xml:space="preserve">Nam in 5. </s> <s xml:id="echoid-s1439" xml:space="preserve">13. </s> <s xml:id="echoid-s1440" xml:space="preserve">ac 14. </s> <s xml:id="echoid-s1441" xml:space="preserve">figura, in quibus ſectiones <lb/>ſunt eiuſdem nominis, vti etiam in 15. </s> <s xml:id="echoid-s1442" xml:space="preserve">& </s> <s xml:id="echoid-s1443" xml:space="preserve">16. </s> <s xml:id="echoid-s1444" xml:space="preserve">(diximus enim circulum non <lb/>incongruè haberi poſſe pro Ellipſi) demonſtratum eſt ſectionem DBE, cuius <lb/>rectum BG minus eſt recto BH ſectionis ABC, totam cadere intra ABC, vn-<lb/>de erit inſcripta, ſiue minor, & </s> <s xml:id="echoid-s1445" xml:space="preserve">è contra, ſectionem ABC cuius rectum maius <lb/>eſt totam cadere extra DBE, cuius rectum eſt minus: </s> <s xml:id="echoid-s1446" xml:space="preserve">quapropter erit ei cir-<lb/>cumſcripta, ſiue maior.</s> <s xml:id="echoid-s1447" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div120" type="section" level="1" n="71"> <head xml:id="echoid-head76" xml:space="preserve">COROLL. III.</head> <p> <s xml:id="echoid-s1448" xml:space="preserve">HInc quoque eruitur Hyperbolarum per vertices ſimul adſcriptarum <lb/>cum æqualibus rectis lateribus, illam, cuius tranſuerſum latus maius <pb o="41" file="0065" n="65" rhead=""/> eſt, inſcriptam, vel minorem eſſe ea, cuius tranſuerſum minus eſt; </s> <s xml:id="echoid-s1449" xml:space="preserve">& </s> <s xml:id="echoid-s1450" xml:space="preserve">è con-<lb/>tra eam eſſe circumſcriptam, ſiue maiorem, cuius tranſuerſum minus eſt. <lb/></s> <s xml:id="echoid-s1451" xml:space="preserve">Nam in 9. </s> <s xml:id="echoid-s1452" xml:space="preserve">figura, in qua ſectiones ſunt Hyperbolæ ſimul adſcriptæ cum eo-<lb/>dem recto latere, oſtenſum fuit Hyperbolen DBE, cuius tranſuerſum BI ma-<lb/>ius eſt, totam cadere intra Hyperbolen ABC, cuius tranſuerſum BL minus <lb/>eſt, & </s> <s xml:id="echoid-s1453" xml:space="preserve">ideo DBE erit inſcripta, ſiue minor; </s> <s xml:id="echoid-s1454" xml:space="preserve">& </s> <s xml:id="echoid-s1455" xml:space="preserve">è contra ipſa ABC, cuius tranſ-<lb/>uerſum eſt minus, erit circumſcripta, ſiue maior.</s> <s xml:id="echoid-s1456" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div121" type="section" level="1" n="72"> <head xml:id="echoid-head77" xml:space="preserve">COROLL. IV.</head> <p> <s xml:id="echoid-s1457" xml:space="preserve">PAtet etiam in Ellipſibus tantùm, vel in Ellipſibus, & </s> <s xml:id="echoid-s1458" xml:space="preserve">circulis per cundem <lb/>verticé ſimul adſcriptis cũ eodem recto latere, eam eſſe inſcriptam, ſiue <lb/>minorem, cuius tranſuerſum latus minus eſt, & </s> <s xml:id="echoid-s1459" xml:space="preserve">è contra eam circumſcriptam, <lb/>vel maiorem eſſe, cuius tranſuerſum maius eſt: </s> <s xml:id="echoid-s1460" xml:space="preserve">quoniam in 10. </s> <s xml:id="echoid-s1461" xml:space="preserve">11. </s> <s xml:id="echoid-s1462" xml:space="preserve">& </s> <s xml:id="echoid-s1463" xml:space="preserve">12. </s> <s xml:id="echoid-s1464" xml:space="preserve">figu-<lb/>ra oſtenſum fuit Ellipſim, vel circulum DBE, cuius tranſuerſum BI minus <lb/>eſt, totam cadere intra Ellipſim, vel circulum ABC, cuius latus tranſuerſum <lb/>BL maius eſt; </s> <s xml:id="echoid-s1465" xml:space="preserve">quare ipſa DBE erit inſcripta, ſiue minor: </s> <s xml:id="echoid-s1466" xml:space="preserve">& </s> <s xml:id="echoid-s1467" xml:space="preserve">è contra Ellipſis, <lb/>vel circulus ABC erit circumſcriptus, ſiue maior, &</s> <s xml:id="echoid-s1468" xml:space="preserve">c.</s> <s xml:id="echoid-s1469" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div122" type="section" level="1" n="73"> <head xml:id="echoid-head78" xml:space="preserve">COROLL. V.</head> <p> <s xml:id="echoid-s1470" xml:space="preserve">MAnifeſtum eſt etiam ſimiles coni-ſectiones per vertices ſimul adſcri-<lb/>ptas habere regulas parallelas, & </s> <s xml:id="echoid-s1471" xml:space="preserve">eam ſectionem eſſe inſcriptam, vel <lb/>minorem, cuius latera minora ſunt; </s> <s xml:id="echoid-s1472" xml:space="preserve">& </s> <s xml:id="echoid-s1473" xml:space="preserve">è contra eam eſſe circumſcriptam, vel <lb/>maiorem, cuius latera ſunt maiora. </s> <s xml:id="echoid-s1474" xml:space="preserve">Si enim in 6. </s> <s xml:id="echoid-s1475" xml:space="preserve">7. </s> <s xml:id="echoid-s1476" xml:space="preserve">& </s> <s xml:id="echoid-s1477" xml:space="preserve">8. </s> <s xml:id="echoid-s1478" xml:space="preserve">figura coni-ſectio-<lb/>nes ABC, DBE eiuſdem nominis, ac per verticem B ſimul adſcriptæ, fuerint <lb/>ſimiles, erit tranſuerſum LB ad rectum BH vt tranſuerſum IB ad rectum BG, <lb/>& </s> <s xml:id="echoid-s1479" xml:space="preserve">permutando, & </s> <s xml:id="echoid-s1480" xml:space="preserve">diuidendo, LI ad IB, vt HG ad GB, vndæ regulæ LH, <lb/>IG erunt parallelæ, ſed in hoc Theoremate demonſtratum eſt ſectioné ABE <lb/>minorum laterum, totam cadere intra ſectionem ABC maiorum laterum, <lb/>ergo ipſa DBE erit inſcripta; </s> <s xml:id="echoid-s1481" xml:space="preserve">& </s> <s xml:id="echoid-s1482" xml:space="preserve">è contra demonſtrauimus ABC maiorum <lb/>laterum totam cadere extra DBE minorum laterum, ac propterea erit ei cir-<lb/>cumſcripta.</s> <s xml:id="echoid-s1483" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div123" type="section" level="1" n="74"> <head xml:id="echoid-head79" xml:space="preserve">COROLL. VI.</head> <p> <s xml:id="echoid-s1484" xml:space="preserve">EX ipſa demum huius Theorematis demonſtratione elicitur, quod in co-<lb/>ni-ſectionibus per vertices ſimul adſcriptis, quadrata ſemi- applicata-<lb/>rum ex eodem diametri puncto inter ſe ſunt vt earundem latitudines. </s> <s xml:id="echoid-s1485" xml:space="preserve">Oſten-<lb/>dimus enim in qualibet præcedentis ſchematis figura, quadratum ſemi-ap-<lb/>plicatæ MF, in ſectione ABC, ad quadratum ſemi-applicatæ NF, in ſectio-<lb/>ne DBE, eſſe vt latitudo propria FP, ad propriam latitudinem FO.</s> <s xml:id="echoid-s1486" xml:space="preserve"/> </p> <pb o="42" file="0066" n="66" rhead=""/> </div> <div xml:id="echoid-div124" type="section" level="1" n="75"> <head xml:id="echoid-head80" xml:space="preserve">PROBL. VI. PROP. XX.</head> <p> <s xml:id="echoid-s1487" xml:space="preserve">Datæ coni ſectioni, vel circulo, per eius verticem, cum dato <lb/>tranſuerſo latere, quod in Ellipſi, vel circulo non excedat eius <lb/>tranſuerſum, MAXIMAM Ellipſim inſcribere, & </s> <s xml:id="echoid-s1488" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1489" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1490" xml:space="preserve">DAtæ Ellipſi, vel circulo, per eius verticem _MINIMAM_ coni-ſectionem <lb/>circumſcribere cum dato, pro circumſcribenda Hyperbola, quocunq; <lb/></s> <s xml:id="echoid-s1491" xml:space="preserve">tranſuerſo latere, pro Ellipſi verò, cum tranſuerſo dato, quod maius ſit tranſ-<lb/>uerſo datæ Ellipſis, vel circuli.</s> <s xml:id="echoid-s1492" xml:space="preserve"/> </p> <figure> <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0066-01"/> </figure> <p> <s xml:id="echoid-s1493" xml:space="preserve">Sit quælibet coni-ſectio, vel circulus ABC, cuius diameter BD, latus <lb/>rectum BE, regula EF; </s> <s xml:id="echoid-s1494" xml:space="preserve">oportet circa diametri ſegmentum BG per verticem <lb/>B _MAXIMAM_ Ellipſin inſcribere.</s> <s xml:id="echoid-s1495" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1496" xml:space="preserve">Adſcribatur <anchor type="note" xlink:href="" symbol="a"/> ſectioni ABC per eius verticem, & </s> <s xml:id="echoid-s1497" xml:space="preserve">circa diametrum BG <anchor type="note" xlink:label="note-0066-01a" xlink:href="note-0066-01"/> cum recto BE Ellipſis GHB. </s> <s xml:id="echoid-s1498" xml:space="preserve">Dico hanc eſſe _MAXIMAM_ quæſitam.</s> <s xml:id="echoid-s1499" xml:space="preserve"/> </p> <div xml:id="echoid-div124" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0066-01" xlink:href="note-0066-01a" xml:space="preserve">7. huius.</note> </div> <p> <s xml:id="echoid-s1500" xml:space="preserve">Nam iuncta ipſius regula GE, cum hæc diſiunctim procedat à regula EF, <lb/>ſitque propior diametro, Ellipſis quoq; </s> <s xml:id="echoid-s1501" xml:space="preserve">GHB <anchor type="note" xlink:href="" symbol="b"/> inſcripta erit ſectioni ABC, <anchor type="note" xlink:label="note-0066-02a" xlink:href="note-0066-02"/> & </s> <s xml:id="echoid-s1502" xml:space="preserve">erit _MAXIMA_ inſcriptibilium: </s> <s xml:id="echoid-s1503" xml:space="preserve">quoniam quæcunque Ellipſis cum eadem <lb/>tranſuerſa diametro BG adſcripta, & </s> <s xml:id="echoid-s1504" xml:space="preserve">cum recto BI, quod minus ſit recto BE <lb/>minor eſt <anchor type="note" xlink:href="" symbol="c"/> Ellipſi GBH, quælibet verò Ellipſis eidem diametro BG adſcri- <anchor type="note" xlink:label="note-0066-03a" xlink:href="note-0066-03"/> pta cum recto BL, quod maius ſit dato recto BE <anchor type="note" xlink:href="" symbol="d"/> maior eſt quidem Ellipſi <anchor type="note" xlink:label="note-0066-04a" xlink:href="note-0066-04"/> GHB, ſed omnino _e_ ſecat ſectionem ABC, cum eius regula GL ſecet ſectio-<lb/>nis regulam EL, infra contingentem BE. </s> <s xml:id="echoid-s1505" xml:space="preserve">Vnde Ellipſis GHB eſt _MAXIMA_. <lb/></s> <s xml:id="echoid-s1506" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1507" xml:space="preserve">c.</s> <s xml:id="echoid-s1508" xml:space="preserve"/> </p> <div xml:id="echoid-div125" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0066-02" xlink:href="note-0066-02a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="c" position="left" xlink:label="note-0066-03" xlink:href="note-0066-03a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="d" position="left" xlink:label="note-0066-04" xlink:href="note-0066-04a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <pb o="43" file="0067" n="67" rhead=""/> <p> <s xml:id="echoid-s1509" xml:space="preserve">Præterea ſit data Ellipſis, vel circulus GHB, cuius diameter BG, rectum <lb/>BE, regula EG, & </s> <s xml:id="echoid-s1510" xml:space="preserve">oporteat per verticem B, _MINIMAM_ Parabolen in pri-<lb/>ma figura, vel cum dato quocunque tranſuerſo BF, _MINIMAM_ Hyperbo-<lb/>len in ſecunda figura, ſiue cum dato tranſuerſo BF, quod in tertia, & </s> <s xml:id="echoid-s1511" xml:space="preserve">quarta <lb/>figura excedat tranſuerſum BG datæ Ellipſis, vel circuli, _MINIMAM_ Elli-<lb/>pſin circumſcribere.</s> <s xml:id="echoid-s1512" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1513" xml:space="preserve">Adſcribatur <anchor type="note" xlink:href="" symbol="a"/> Ellipſi GHB per verticem B in prima figura parabole ABC, <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> & </s> <s xml:id="echoid-s1514" xml:space="preserve">in ſecunda Hyperbole ABC, cum dato tranſuerſo BF, & </s> <s xml:id="echoid-s1515" xml:space="preserve">in tertia, & </s> <s xml:id="echoid-s1516" xml:space="preserve">quar-<lb/>ta Ellipſis ABC cum dato tranſuerſo BF; </s> <s xml:id="echoid-s1517" xml:space="preserve">& </s> <s xml:id="echoid-s1518" xml:space="preserve">harum omnium ſectionum re-<lb/>ctum latus idem ſit cum recto BE datæ Ellipſis. </s> <s xml:id="echoid-s1519" xml:space="preserve">Iam patet ipſam ſectionem <lb/>ABC datæ GHB <anchor type="note" xlink:href="" symbol="b"/> circumſcriptam eſſe. </s> <s xml:id="echoid-s1520" xml:space="preserve">Inſuper dico talem ſectionem ABC <anchor type="note" xlink:label="note-0067-02a" xlink:href="note-0067-02"/> eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s1521" xml:space="preserve"/> </p> <div xml:id="echoid-div126" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">5. 6. 7. h.</note> <note symbol="b" position="right" xlink:label="note-0067-02" xlink:href="note-0067-02a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1522" xml:space="preserve">Nam, in prima figura, quælibet parabola, vel in reliquis, quæcunque eiuſ-<lb/>dem nominis ſectio adſcripta ſectioni ABC per verticem B, cum eodem <lb/>tranſuerſo BF, ſed cum recto BL, quod excedat rectum BE ſectionis ABC <lb/>eadem ſectione <anchor type="note" xlink:href="" symbol="c"/> eſt maior, quælibet verò adſcripta ſectio cum recto BI, quod <anchor type="note" xlink:label="note-0067-03a" xlink:href="note-0067-03"/> minus ſit recto BE minor <anchor type="note" xlink:href="" symbol="d"/> eſt ſectione ABC, ſed Ellipſim GHB omninò <anchor type="note" xlink:href="" symbol="e"/> ſe- <anchor type="note" xlink:label="note-0067-04a" xlink:href="note-0067-04"/> <anchor type="note" xlink:label="note-0067-05a" xlink:href="note-0067-05"/> cat cum ipſarum regulæ IN, GE infra contingentem ex vertice ſe mutuò ſe-<lb/>cent. </s> <s xml:id="echoid-s1523" xml:space="preserve">Quare ſectio Parabolæ, vel Hyperbole, aut Ellipſis ABC eſt _MINI_-<lb/>_MA_ circumſcriptibilium datæ Ellipſi, vel circulo GHB. </s> <s xml:id="echoid-s1524" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s1525" xml:space="preserve">c.</s> <s xml:id="echoid-s1526" xml:space="preserve"/> </p> <div xml:id="echoid-div127" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-0067-03" xlink:href="note-0067-03a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="d" position="right" xlink:label="note-0067-04" xlink:href="note-0067-04a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="e" position="right" xlink:label="note-0067-05" xlink:href="note-0067-05a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> </div> <div xml:id="echoid-div129" type="section" level="1" n="76"> <head xml:id="echoid-head81" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s1527" xml:space="preserve">HInc ſolutio problematum. </s> <s xml:id="echoid-s1528" xml:space="preserve">Videlicet: </s> <s xml:id="echoid-s1529" xml:space="preserve">Datæ coni-ſectioni circa maio-<lb/>rem axem, per eius verticem _MAXIMVM_ circulum inſcribere.</s> <s xml:id="echoid-s1530" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1531" xml:space="preserve">Item datæ Ellipſi circa minorem axem, per eius verticem _MIMIMVM_ cir-<lb/>culum circumſcribere.</s> <s xml:id="echoid-s1532" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1533" xml:space="preserve">Si enim in tribus primis ſuperioribus figuris concipiatur diametrum BD <lb/>datæ Parabolæ, vel Hyperbolæ, aut Ellipſis ABC eſſe propriæ ſectionis <lb/>maiorem axem, eiuſque ſegmentum BG æquari recto lateri BE, circa quod <lb/>adſcripta <anchor type="note" xlink:href="" symbol="f"/> ſit Ellipſis GHB cũ recto BE: </s> <s xml:id="echoid-s1534" xml:space="preserve">ipſa vt ſuperius oſtensũ fuit, erit _MA_- <anchor type="note" xlink:label="note-0067-06a" xlink:href="note-0067-06"/> _XIMA_ inſcriptibilium, eritque Ellipſis æqualium laterum circa axim, quam <lb/>in Monito poſt primam huius, animaduerſum fuit circulum eſſe. </s> <s xml:id="echoid-s1535" xml:space="preserve">Vnde da-<lb/>tæ coni-ſectioni circa maiorem axim inſcriptus erit _MAXIMVS_ circulus per <lb/>verticem ſectionis. </s> <s xml:id="echoid-s1536" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1537" xml:space="preserve">c.</s> <s xml:id="echoid-s1538" xml:space="preserve"/> </p> <div xml:id="echoid-div129" type="float" level="2" n="1"> <note symbol="f" position="right" xlink:label="note-0067-06" xlink:href="note-0067-06a" xml:space="preserve">7. prop. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s1539" xml:space="preserve">Siverò, vt in quarta figura, datæ Ellipſi GHB circa minorem axim BG, & </s> <s xml:id="echoid-s1540" xml:space="preserve"><lb/>cuius rectum latus BE _MINIMVS_ circulorum ſit circumſcribendus; </s> <s xml:id="echoid-s1541" xml:space="preserve">ſumpta <lb/>BF æquali recto BE, ipſa excedet tranſuerſum latus BG datæ Ellipſis GHB <lb/>(nam ſemper in Ellipſi minor axis ad maiorem, eſt vt maior axis ad latus re-<lb/>ctum) itaque ſi circa BF Ellipſis adſcribatur ABC, cum recto BE datæ Elli-<lb/>pſis, ipſa, per ſecundam partem propoſitionis huius, erit _MINIMA_ datæ <lb/>Ellipſi circumſcriptibilium, ſed talis Ellipſis ABC per Monitũ poſt 1. </s> <s xml:id="echoid-s1542" xml:space="preserve">huius, <lb/>cum ſit æqualium laterum, & </s> <s xml:id="echoid-s1543" xml:space="preserve">circa axim, idem eſt, ac circulus. </s> <s xml:id="echoid-s1544" xml:space="preserve">Quare da-<lb/>tæ Ellipſi circa minorem axem per eius verticem _MINIMVS_ circulus circũ-<lb/>ſcriptus erit. </s> <s xml:id="echoid-s1545" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s1546" xml:space="preserve">c.</s> <s xml:id="echoid-s1547" xml:space="preserve"/> </p> <pb o="44" file="0068" n="68" rhead=""/> </div> <div xml:id="echoid-div131" type="section" level="1" n="77"> <head xml:id="echoid-head82" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s1548" xml:space="preserve">PAtet etiam quomodo datæ coni-ſectioni, vel circulo ABC per ipſius <lb/>verticem inſcribi poſſit Ellipſis, que ſit _MAXIMA_ circa idem tranſuer-<lb/>ſum, & </s> <s xml:id="echoid-s1549" xml:space="preserve">ipſius rectum latus ad verſum in Parabola, vel Hyperbola datam <lb/>quamcumque teneat rationem; </s> <s xml:id="echoid-s1550" xml:space="preserve">& </s> <s xml:id="echoid-s1551" xml:space="preserve">in Ellipſi, vel circulo data ratio non ſit <lb/>minor ratione recti BE, ad tranſuerſum BD.</s> <s xml:id="echoid-s1552" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1553" xml:space="preserve">Nam ſi exempli gratia Parabolæ, vel Hyperbolæ primæ, ac ſecundæ figu-<lb/>ræ inſcribenda ſit _MAXIMA_ Ellipſis circa idem tranſuerſum latus, & </s> <s xml:id="echoid-s1554" xml:space="preserve">cuius <lb/>rectum ad verſum datam habeat rationem, R nempe ad S: </s> <s xml:id="echoid-s1555" xml:space="preserve">fiat vt R ad S, ita <lb/>rectum EB datæ ſectionis ad BG, nam ſi cum eodem recto EB, ac tranſuerſo <lb/>BG adſcribatur per B Ellipſis GHB, ipſa erit _MAXIMA_ circa idem tranſ-<lb/>uerſum BG, per ea, quæ ſuperius demonſtrata fuerunt. </s> <s xml:id="echoid-s1556" xml:space="preserve">Siverò data ratio R <lb/>ad S non ſit minor ratione recti EB ad tranſuerſum BD; </s> <s xml:id="echoid-s1557" xml:space="preserve">in tertia, quarta, & </s> <s xml:id="echoid-s1558" xml:space="preserve"><lb/>quinta figura, fiat vt R ad S, ita EB ad BG, quod erit tranſuerſum quæſitæ <lb/>inſcriptæ Ellipſis, quæ erit _MAXIMA_, &</s> <s xml:id="echoid-s1559" xml:space="preserve">c.</s> <s xml:id="echoid-s1560" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div132" type="section" level="1" n="78"> <head xml:id="echoid-head83" xml:space="preserve">PROBL. VII. PROP. XXI.</head> <p> <s xml:id="echoid-s1561" xml:space="preserve">Datæ Hyperbolæ, per eius verticem MAXIMAM Parabolen <lb/>inſcribere, & </s> <s xml:id="echoid-s1562" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1563" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1564" xml:space="preserve">Per verticem datæ Parabolæ, cum dato tranſuerſo latere MINI-<lb/>MAM Hyperbolen circumſcribere.</s> <s xml:id="echoid-s1565" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1566" xml:space="preserve">SIt data Hyperbole ABC, cuius vertex B, diameter BD, tranſuerſum la-<lb/>tus BE, rectum BF, & </s> <s xml:id="echoid-s1567" xml:space="preserve">regula EFO oportet primùm per eius verticem B <lb/>_MAXIMAM_ Parabolen inſcribere.</s> <s xml:id="echoid-s1568" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1569" xml:space="preserve">Adſcribatur <anchor type="note" xlink:href="" symbol="a"/> Hyperbolæ ABC, <anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/> <anchor type="figure" xlink:label="fig-0068-01a" xlink:href="fig-0068-01"/> per verticem B, & </s> <s xml:id="echoid-s1570" xml:space="preserve">cum recto BF Pa-<lb/>rabole GBH. </s> <s xml:id="echoid-s1571" xml:space="preserve">Dico hanc eſſe _MAXI_-<lb/>_MAM_ quæſitam.</s> <s xml:id="echoid-s1572" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">5. prop. <lb/>huius.</note> <figure xlink:label="fig-0068-01" xlink:href="fig-0068-01a"> <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0068-01"/> </figure> </div> <p> <s xml:id="echoid-s1573" xml:space="preserve">Ducta enim ex F Parabolæ regula <lb/>FI, cum hæc infra contingentem BF, <lb/>regulę EFO, nunquam occurat, (cum <lb/>ſimul conueniãt in F) ſitque regula FI <lb/>propinquior diametro BD quam pro-<lb/>ducta regula FO, erit Parabole <anchor type="note" xlink:href="" symbol="b"/> GBH <anchor type="note" xlink:label="note-0068-02a" xlink:href="note-0068-02"/> datę Hyperbolę ABC inſcripta, eritq; <lb/></s> <s xml:id="echoid-s1574" xml:space="preserve">_MAXIMA_: </s> <s xml:id="echoid-s1575" xml:space="preserve">quoniam quælibet alia <lb/>Parabole ipſi ABC per verticem B <lb/>adſcripta cum recto BL, quod minus <lb/>ſit recto BF datę Hyperbolæ, <anchor type="note" xlink:href="" symbol="c"/> minor <anchor type="note" xlink:label="note-0068-03a" xlink:href="note-0068-03"/> eſt Parabola GBH, quælibet verò ad-<lb/>ſcripta cum recto BM, quod excedat <lb/>rectum BF datę Hyperbolæ ipſa GBH <pb o="45" file="0069" n="69" rhead=""/> eſt <anchor type="note" xlink:href="" symbol="a"/> quidem maior; </s> <s xml:id="echoid-s1576" xml:space="preserve">ſed omnino ſecat <anchor type="note" xlink:href="" symbol="b"/> Hyperbolen ABC, quoniam eius <anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> <anchor type="note" xlink:label="note-0069-02a" xlink:href="note-0069-02"/> regula MN infra contingentem BM ſecat regulam EFO. </s> <s xml:id="echoid-s1577" xml:space="preserve">Vnde Parabole <lb/>GHB erit _MAXIMA_ inſcripta, &</s> <s xml:id="echoid-s1578" xml:space="preserve">c. </s> <s xml:id="echoid-s1579" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1580" xml:space="preserve">c.</s> <s xml:id="echoid-s1581" xml:space="preserve"/> </p> <div xml:id="echoid-div133" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">3. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="c" position="left" xlink:label="note-0068-03" xlink:href="note-0068-03a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="a" position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">ibidem.</note> <note symbol="b" position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1582" xml:space="preserve">Sit verò data Parabole GBH, cuius vertex B, diameter BD, rectum BF, <lb/>& </s> <s xml:id="echoid-s1583" xml:space="preserve">regula FI, & </s> <s xml:id="echoid-s1584" xml:space="preserve">circumſcribenda ſit ei cum dato tranſuerſo BE _MINIMA_ <lb/>Hyperbole.</s> <s xml:id="echoid-s1585" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1586" xml:space="preserve">Adſcribatur <anchor type="note" xlink:href="" symbol="c"/> Parabolæ GBH, per verticem B, cum dato tranſuerſo BE <anchor type="note" xlink:label="note-0069-03a" xlink:href="note-0069-03"/> Hyperbole ABC, cuius rectum latus idem ſit, ac rectum BF datæ Parabolæ. <lb/></s> <s xml:id="echoid-s1587" xml:space="preserve">Dico huiuſmodi Hyperbolen eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s1588" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0069-03" xlink:href="note-0069-03a" xml:space="preserve">6. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1589" xml:space="preserve">Nam quælibet alia Hyperbole per verticem B datæ Hyperbolæ ABC ad-<lb/>ſcripta, cum eodem tranſuerſo BE, ſed cum recto BM, quod maius ſit recto <lb/>BF ipſa ABC <anchor type="note" xlink:href="" symbol="d"/> maior eſt; </s> <s xml:id="echoid-s1590" xml:space="preserve">quælibet verò adſcripta cum recto BL, quod defi- <anchor type="note" xlink:label="note-0069-04a" xlink:href="note-0069-04"/> ciat à recto BF, eadem ABC <anchor type="note" xlink:href="" symbol="e"/> eſt quidem minor, ſed omnino ſecat <anchor type="note" xlink:href="" symbol="f"/> Para- <anchor type="note" xlink:label="note-0069-05a" xlink:href="note-0069-05"/> <anchor type="note" xlink:label="note-0069-06a" xlink:href="note-0069-06"/> bolen GBH, cum eius regula ELP infra contingentem BF producta ſecet re-<lb/>gulam FI datæ Parabolæ GBH. </s> <s xml:id="echoid-s1591" xml:space="preserve">Quare huiuſmodi Hyperbole ABC erit _MI_-<lb/>_NIMA_ circumſcriptibilium datæ Parabolæ ABC per verticem B, & </s> <s xml:id="echoid-s1592" xml:space="preserve">cum <lb/>dato tranſuerſo BE. </s> <s xml:id="echoid-s1593" xml:space="preserve">Quod erat ſecundò, &</s> <s xml:id="echoid-s1594" xml:space="preserve">c.</s> <s xml:id="echoid-s1595" xml:space="preserve"/> </p> <div xml:id="echoid-div135" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-0069-04" xlink:href="note-0069-04a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="e" position="right" xlink:label="note-0069-05" xlink:href="note-0069-05a" xml:space="preserve">ibidem.</note> <note symbol="f" position="right" xlink:label="note-0069-06" xlink:href="note-0069-06a" xml:space="preserve">1. corol. <lb/>prop. 19. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div137" type="section" level="1" n="79"> <head xml:id="echoid-head84" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s1596" xml:space="preserve">ALiquis forſan hoc loco vereri poſſet enunciationes, vel ſaltem ar-<lb/>gumentum MAXIMARVM, MINIMARV MQVE ſe-<lb/>ctionum inſcriptibilium, ac circumſ criptibilium habuiſſe me à <lb/>celeberrimo ſui æui Mathematico Maurolico, & </s> <s xml:id="echoid-s1597" xml:space="preserve">hoc quidem, <lb/>vt fateor, haud temerè; </s> <s xml:id="echoid-s1598" xml:space="preserve">nam quod in duabus præcedentibus propoſitionibus <lb/>exponitur, profertur quoque in eius quinto conicorum libro, ab ipſo, vnà <lb/>cum ſexto iam ſupra nonaginta annos proprio Marte ſuppleto, quamuis typis <lb/>Meſſanæ tradito non antea annum 1654. </s> <s xml:id="echoid-s1599" xml:space="preserve">ſedula opera eximij Mathemati-<lb/>ci, ac Philoſophi præſtantiſsimi 10. </s> <s xml:id="echoid-s1600" xml:space="preserve">Alphonſi Borelli, qui, vt ipſemet aſſerit, <lb/>ex multis Maurolici poſthumis lucubrationibus, apud Auctoris hæredes tunc <lb/>extantibus, prædictum opus publici iuris fieri curauit, idemque mihi à duobus <lb/>circiter annis primò mdicauit, quod è Meſſana anxiè petiens, tandem ab hinc <lb/>paucis menſibus conſecutus fui. </s> <s xml:id="echoid-s1601" xml:space="preserve">At quicunque æquo, gratoque animo Mau-<lb/>rolici demonſtrationes, cum meis conferat, dum diuerſa, expeditaque metho-<lb/>do generatim oſtenditur, ex præmiſſo Theoremate Lemmatico, duobus tan-<lb/>tum Problematibus, vnicoque Corollario, totum id, quod integro libro pecu-<lb/>liariter à Maurolico 24. </s> <s xml:id="echoid-s1602" xml:space="preserve">Propoſitionibus, decẽque, aut Corollarijs, aut Scho-<lb/>lijs demonſtratur; </s> <s xml:id="echoid-s1603" xml:space="preserve">curabitque vlterius procedendo, reliquum mei operis per-<lb/>currere, in quo, tot aliæ, ex his cmanantes concluſiones reperiuntur, à nemi-<lb/>ne, quod ſciam, hactenus animaduerſæ, hic quidem, vt ſpero, à tali ſuſpi-<lb/>cione omnino remouebitur, & </s> <s xml:id="echoid-s1604" xml:space="preserve">de prædictis promeritum ius, qualecunque ſit, <lb/>mihi facilè tribuet. </s> <s xml:id="echoid-s1605" xml:space="preserve">Sed ne tempus fruſtra teramus, incæptum opus proſe-<lb/>quamur.</s> <s xml:id="echoid-s1606" xml:space="preserve"/> </p> <pb o="46" file="0070" n="70" rhead=""/> </div> <div xml:id="echoid-div138" type="section" level="1" n="80"> <head xml:id="echoid-head85" xml:space="preserve">THEOR. XII. PROP. XXII.</head> <p> <s xml:id="echoid-s1607" xml:space="preserve">MAXIMA coni-ſectionum coni-ſectioni per verticem inſcripti-<lb/>bilium cum recto datæ ſectionis, eſt quoq; </s> <s xml:id="echoid-s1608" xml:space="preserve">MAXIMA ſibi ſimilium, <lb/>eidem ſectioni per verticem inſcriptarum. </s> <s xml:id="echoid-s1609" xml:space="preserve">Et è contra.</s> <s xml:id="echoid-s1610" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1611" xml:space="preserve">MINIMA coni-ſectionum coni-ſectioni per verticem circum-<lb/>ſcriptibilium cum recto datæ ſectionis, eſt quoque MINIMA ſibi <lb/>ſimilium, eidem ſectioni circumſcriptarum.</s> <s xml:id="echoid-s1612" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1613" xml:space="preserve">SIt quælibet coni-ſectio ABC, cuius diameter BR, rectum BF, & </s> <s xml:id="echoid-s1614" xml:space="preserve">regula <lb/>GFI, ipſique inſcripta ſit per verticem B, cum recto BF coni-ſectio DB <lb/>E, quæ erit _MAXIMA_ inſcriptarum (per ea quę in præcedentibus oſtenſum <lb/> <anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/> fuit) ſitq; </s> <s xml:id="echoid-s1615" xml:space="preserve">huius regula HF. </s> <s xml:id="echoid-s1616" xml:space="preserve">Patet has regulas infra contingentem BF in totum <lb/>eſſe inter ſe diſiunctas, cum ſit altera ſectio alteri inſcripta. </s> <s xml:id="echoid-s1617" xml:space="preserve">Iam dico hanc <lb/>_MAXIMAM_ ſectionem eſſe quoq; </s> <s xml:id="echoid-s1618" xml:space="preserve">_MAXIMAM_ ſibi ſimilium, eidem datæ ſe-<lb/>ctioni per B verticem inſcriptarum.</s> <s xml:id="echoid-s1619" xml:space="preserve"/> </p> <div xml:id="echoid-div138" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0070-01"/> </figure> </div> <pb o="47" file="0071" n="71" rhead=""/> <p> <s xml:id="echoid-s1620" xml:space="preserve">Quoniam, quæcunque ſectio ſimilis ſectioni DBE adſcripta per B ſectioni <lb/>ABC, cumrecto BM, quod minus ſit recto BF, minor <anchor type="note" xlink:href="" symbol="a"/> eſt ſectione DBE, <anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/> quælibet verò adſcripta cum recto BO; </s> <s xml:id="echoid-s1621" xml:space="preserve">quod maius ſit recto BF <anchor type="note" xlink:href="" symbol="b"/> eſt quidem maior ipſa DBE, ſed datam ABC omnino ſecat; </s> <s xml:id="echoid-s1622" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> quoniam ipſius regula <anchor type="note" xlink:label="note-0071-02a" xlink:href="note-0071-02"/> ON, quæ <anchor type="note" xlink:href="" symbol="d"/> æquidiſtat regulæ FH, ſecat infra contingentem BF regulam <anchor type="note" xlink:label="note-0071-03a" xlink:href="note-0071-03"/> FIG, nam altera parallelarum FH ab eadem FIG ſecatur in F: </s> <s xml:id="echoid-s1623" xml:space="preserve">vnde ipſa <lb/>DBE eſt _MINIMA_ ſibi ſimilium, &</s> <s xml:id="echoid-s1624" xml:space="preserve">c. </s> <s xml:id="echoid-s1625" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s1626" xml:space="preserve">c.</s> <s xml:id="echoid-s1627" xml:space="preserve"/> </p> <div xml:id="echoid-div139" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">5. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="b" position="right" xlink:label="note-0071-02" xlink:href="note-0071-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="right" xlink:label="note-0071-03" xlink:href="note-0071-03a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <note symbol="d" position="right" xml:space="preserve">5. prop. <lb/>19. huius.</note> <p> <s xml:id="echoid-s1628" xml:space="preserve">Nunc verò ſit coni-ſectio DBE, cuius rectum BF, & </s> <s xml:id="echoid-s1629" xml:space="preserve">regula FH, ipſique <lb/>circumſcripta ſit cum eodem recto BF, per verticem B coni-ſectio ABC, quæ <lb/>erit _MINIMA_ circumſcripta, per iam demonſtrata, eiuſque regula ſit GFI. <lb/></s> <s xml:id="echoid-s1630" xml:space="preserve">Dico hanc _MINIMAM_ ſectionem ABC eſſe quoque _MINIMAM_ ſibi ſimi-<lb/>lium, eidem ſectioni DBE per verticem circumſcriptarum.</s> <s xml:id="echoid-s1631" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1632" xml:space="preserve">Nam quælibet coni-ſectio ſimilis ABC, adſcripta per B datæ ſectioni DB <lb/>E, cum recto BO, quod maius ſit recto BF <anchor type="note" xlink:href="" symbol="e"/> maior eſt ſectione ABC, quæ- <anchor type="note" xlink:label="note-0071-05a" xlink:href="note-0071-05"/> libet verò adſcripta cum recto BM, quod minus ſit recto BF eſt quidem <lb/>minor ipſa ABC, ſed datam ſecat DBE, quoniam ipſius regula QM, quę re-<lb/>gulæ GFI æquidiſtat, ſecat regulam FH, nam altera parallelarum GFI ſecat <lb/>infra BF ipſam FH in F. </s> <s xml:id="echoid-s1633" xml:space="preserve">Quare ipſa ABC eſt _MINIMA_ ſibi ſimilium, &</s> <s xml:id="echoid-s1634" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1635" xml:space="preserve">Quod erat ſecundò, &</s> <s xml:id="echoid-s1636" xml:space="preserve">c.</s> <s xml:id="echoid-s1637" xml:space="preserve"/> </p> <div xml:id="echoid-div140" type="float" level="2" n="3"> <note symbol="e" position="right" xlink:label="note-0071-05" xlink:href="note-0071-05a" xml:space="preserve">5. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> </div> <div xml:id="echoid-div142" type="section" level="1" n="81"> <head xml:id="echoid-head86" xml:space="preserve">PROBL. VIII. PROP. XXIII.</head> <p> <s xml:id="echoid-s1638" xml:space="preserve">Datæ Hyperbolæ, cum dato quocunque tranſuerſo latere, per <lb/>ipſius verticem MAXIMAM Hyperbolen inſcribere: </s> <s xml:id="echoid-s1639" xml:space="preserve">& </s> <s xml:id="echoid-s1640" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1641" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1642" xml:space="preserve">Datæ Hyperbolæ cum dato quolibet tranſuerſo latere per eius <lb/>verticem MINIMAM Hyperbolen circumſcribere.</s> <s xml:id="echoid-s1643" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1644" xml:space="preserve">SIt data Hyperbole ABC, cuius <lb/> <anchor type="figure" xlink:label="fig-0071-01a" xlink:href="fig-0071-01"/> vertex B, tranſuerſum latus BD, <lb/>rectum BE, & </s> <s xml:id="echoid-s1645" xml:space="preserve">regula DE: </s> <s xml:id="echoid-s1646" xml:space="preserve">oportet pri-<lb/>mò cum dato quocunque alio tranſ-<lb/>uerſo latere, per verticem B, _MAXI_-<lb/>_MAM_ Hyperbolen inſcribere.</s> <s xml:id="echoid-s1647" xml:space="preserve"/> </p> <div xml:id="echoid-div142" type="float" level="2" n="1"> <figure xlink:label="fig-0071-01" xlink:href="fig-0071-01a"> <image file="0071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0071-01"/> </figure> </div> <p> <s xml:id="echoid-s1648" xml:space="preserve">Iam, vel datum tranſuerſum latus <lb/>exceditranſuerſum BD, datę Hyper-<lb/>bolæ, vel eodem minus eſt. </s> <s xml:id="echoid-s1649" xml:space="preserve">Si pri-<lb/>mùm quale eſt BG; </s> <s xml:id="echoid-s1650" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> adſcribatur Hy- perbolę ABC per verticem B, cum <lb/> <anchor type="note" xlink:label="note-0071-06a" xlink:href="note-0071-06"/> dato tranſuerſo BG, & </s> <s xml:id="echoid-s1651" xml:space="preserve">cum eodem <lb/>recto BE Hyperbole HBI. </s> <s xml:id="echoid-s1652" xml:space="preserve">Patet ip-<lb/>ſam HBI datæ ABC <anchor type="note" xlink:href="" symbol="b"/> eſſe inſcriptam;</s> <s xml:id="echoid-s1653" xml:space="preserve"> <anchor type="note" xlink:label="note-0071-07a" xlink:href="note-0071-07"/> quàm dico eſſe _MAXIMAM_: </s> <s xml:id="echoid-s1654" xml:space="preserve">quoniam <lb/>quælibet alia ipſi HBI adſcripta cum <lb/>eodem tranſuerſo BG, ſed cumrecto, <lb/>quod ſit minus BE, ſemper minor <anchor type="note" xlink:href="" symbol="c"/> eſt <anchor type="note" xlink:label="note-0071-08a" xlink:href="note-0071-08"/> ipſa HBI, quelibet vero adſcripta cum <pb o="48" file="0072" n="72" rhead=""/> recto BI, quod excedat BL, eſt <anchor type="note" xlink:href="" symbol="a"/> qui- <anchor type="note" xlink:label="note-0072-01a" xlink:href="note-0072-01"/> <anchor type="figure" xlink:label="fig-0072-01a" xlink:href="fig-0072-01"/> dem maior ipſa HBI, ſed vel ſecat Hy-<lb/> <anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/> perbolen ABC, quod accidit <anchor type="note" xlink:href="" symbol="b"/> ſi iun- cta regula GL, ac infra contingentem <lb/>BL producta, ſecet productam regu-<lb/>lam DE; </s> <s xml:id="echoid-s1655" xml:space="preserve">vel cadit extra eandẽ ABC, <lb/>quando <anchor type="note" xlink:href="" symbol="c"/> iuncta regula GL, cum re- <anchor type="note" xlink:label="note-0072-03a" xlink:href="note-0072-03"/> gula DE infra eandem contingentem <lb/>nunquam conueniat. </s> <s xml:id="echoid-s1656" xml:space="preserve">Quare huiuſmo-<lb/>di Hyperbole HBI erit _MAXIMA_ <lb/>quæſita.</s> <s xml:id="echoid-s1657" xml:space="preserve"/> </p> <div xml:id="echoid-div143" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0071-06" xlink:href="note-0071-06a" xml:space="preserve">6. huius.</note> <note symbol="b" position="right" xlink:label="note-0071-07" xlink:href="note-0071-07a" xml:space="preserve">3. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="c" position="right" xlink:label="note-0071-08" xlink:href="note-0071-08a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="a" position="left" xlink:label="note-0072-01" xlink:href="note-0072-01a" xml:space="preserve">ibidem.</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/QN4GHYBF/figures/0072-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="c" position="left" xlink:label="note-0072-03" xlink:href="note-0072-03a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s1658" xml:space="preserve">Si deniq; </s> <s xml:id="echoid-s1659" xml:space="preserve">datũ trã ſuerſum latus BM <lb/>ſit minus tranfuerſo BD, ducatur MF <lb/>ipſi BE parallela, & </s> <s xml:id="echoid-s1660" xml:space="preserve">cũ tranſuerſo BM, <lb/>ac recto BF, per vcrticem B, Hyper-<lb/>bolæ ABC adſcribatur <anchor type="note" xlink:href="" symbol="d"/> Hyperbole <anchor type="note" xlink:label="note-0072-04a" xlink:href="note-0072-04"/> HBI, quæ ipſi ABC ſimilis erit, cum <lb/>ſit tranſuerſum DB ad rectum BE, vt <lb/>tranſuerſum MB ad rectum BF, eritq; <lb/></s> <s xml:id="echoid-s1661" xml:space="preserve">inſcripta <anchor type="note" xlink:href="" symbol="e"/> Hyperbolæ ABC, cum ſit minorum laterum. </s> <s xml:id="echoid-s1662" xml:space="preserve">Dico hanc eſſe <anchor type="note" xlink:label="note-0072-05a" xlink:href="note-0072-05"/> _MAXIMAM_ quæſitam.</s> <s xml:id="echoid-s1663" xml:space="preserve"/> </p> <div xml:id="echoid-div144" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-0072-04" xlink:href="note-0072-04a" xml:space="preserve">6. huius.</note> <note symbol="e" position="left" xlink:label="note-0072-05" xlink:href="note-0072-05a" xml:space="preserve">5. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1664" xml:space="preserve">Quoniã quælibet alia, quæ cum recto minore ipſo BF adſcribitur, ſemper <lb/> <anchor type="note" xlink:label="note-0072-06a" xlink:href="note-0072-06"/> eſt <anchor type="note" xlink:href="" symbol="f"/> minor HBI, quæ verò cum recto, quod excedat BF, eſt quidem <anchor type="note" xlink:href="" symbol="g"/> maior ipſa HBI, ſed vel ſecat Hyperbolen ABC, quod ſit cum rectum cadit inter <lb/>F, & </s> <s xml:id="echoid-s1665" xml:space="preserve">E, vt in N, <anchor type="note" xlink:href="" symbol="h"/> nam iuncta regula MN, & </s> <s xml:id="echoid-s1666" xml:space="preserve">producta, ſecat regulam DE <anchor type="note" xlink:label="note-0072-07a" xlink:href="note-0072-07"/> infra contingentem BE; </s> <s xml:id="echoid-s1667" xml:space="preserve">vel cadit tota extra ABC, quod euenit <anchor type="note" xlink:href="" symbol="i"/> cũ rectum, <anchor type="note" xlink:label="note-0072-08a" xlink:href="note-0072-08"/> velidem fuerit cum recto BE, vel maius ipſo BE, quale eſt BL; </s> <s xml:id="echoid-s1668" xml:space="preserve">tunc enim <lb/>iuncta regula ML infra contingentem BE, diſiunctim procederet à regula <lb/> <anchor type="note" xlink:label="note-0072-09a" xlink:href="note-0072-09"/> DE, cum eam ſecaret priſu4;</s> <s xml:id="echoid-s1669" xml:space="preserve">s ſupra BE. </s> <s xml:id="echoid-s1670" xml:space="preserve">Eſt igitur talis Hy perbole HBI _MA_-<lb/>_XIMA_ quæſita. </s> <s xml:id="echoid-s1671" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1672" xml:space="preserve">c.</s> <s xml:id="echoid-s1673" xml:space="preserve"/> </p> <div xml:id="echoid-div145" type="float" level="2" n="4"> <note symbol="f" position="left" xlink:label="note-0072-06" xlink:href="note-0072-06a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="g" position="left" xlink:label="note-0072-07" xlink:href="note-0072-07a" xml:space="preserve">ibidem.</note> <note symbol="h" position="left" xlink:label="note-0072-08" xlink:href="note-0072-08a" xml:space="preserve">1. Co-<lb/>rol. prop. <lb/>19. huius.</note> <note symbol="i" position="left" xlink:label="note-0072-09" xlink:href="note-0072-09a" xml:space="preserve">3. 1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1674" xml:space="preserve">Amplius, ſit data Hyperbole HBI, cuius tranſuerſum latus ſit BD, rectum <lb/>BE, & </s> <s xml:id="echoid-s1675" xml:space="preserve">regula DE, & </s> <s xml:id="echoid-s1676" xml:space="preserve">ipſi oporteat per verticem B _MINIMAM_ Hyperbolen <lb/>circumſcribere, cum dato quolibet tranſuerſo latere.</s> <s xml:id="echoid-s1677" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1678" xml:space="preserve">Si datum tranſuerſum circumſcribendæ Hyperbolæ fuerit minus ipſo BD, <lb/>quale eſt BM: </s> <s xml:id="echoid-s1679" xml:space="preserve">adſcribatur <anchor type="note" xlink:href="" symbol="l"/> datæ HBI, per verticem B Hyperbole ABC, <anchor type="note" xlink:label="note-0072-10a" xlink:href="note-0072-10"/> cuius tranſuerſum ſit BM, rectum verò ſit idem BE: </s> <s xml:id="echoid-s1680" xml:space="preserve">Nam ipſa <anchor type="note" xlink:href="" symbol="m"/> erit circum- <anchor type="note" xlink:label="note-0072-11a" xlink:href="note-0072-11"/> ſcripta, eritque _MINIMA_ quæſita; </s> <s xml:id="echoid-s1681" xml:space="preserve">quoniam quælibet alia adſcripta cum <lb/>tranſuerſo BM, ſed cum recto quod excedat BE, quale eſſet BL, eſt <anchor type="note" xlink:href="" symbol="n"/> maior ipſa ABC; </s> <s xml:id="echoid-s1682" xml:space="preserve">quælibet verò adſcripta, cum eodem tranſuerſo BM, & </s> <s xml:id="echoid-s1683" xml:space="preserve">cum re-<lb/> <anchor type="note" xlink:label="note-0072-12a" xlink:href="note-0072-12"/> cto quod minus ſit BE, eſt quidem minor <anchor type="note" xlink:href="" symbol="o"/> ipſa ABC, ſed vel ſecat <anchor type="note" xlink:href="" symbol="p"/> Hyper- bolen HBI, tum cum earum regulæ infra contingentem BE ſe mutuò ſecant, <lb/> <anchor type="note" xlink:label="note-0072-13a" xlink:href="note-0072-13"/> vel cadit intra <anchor type="note" xlink:href="" symbol="q"/> HBI, quando earundem regulæ infra prædictam contingen- <anchor type="note" xlink:label="note-0072-14a" xlink:href="note-0072-14"/> tem nunquam ſimul conueniant. </s> <s xml:id="echoid-s1684" xml:space="preserve">Quare ipſa ABC erit _MINIMA_ quæſita.</s> <s xml:id="echoid-s1685" xml:space="preserve"/> </p> <div xml:id="echoid-div146" type="float" level="2" n="5"> <note symbol="l" position="left" xlink:label="note-0072-10" xlink:href="note-0072-10a" xml:space="preserve">6. prop. <lb/>huius.</note> <note symbol="m" position="left" xlink:label="note-0072-11" xlink:href="note-0072-11a" xml:space="preserve">3. Co-<lb/>rol prop. <lb/>19. huius.</note> <note symbol="n" position="left" xlink:label="note-0072-12" xlink:href="note-0072-12a" xml:space="preserve">2. corol. <lb/>prop. 19. <lb/>huius.</note> <note symbol="o" position="left" xlink:label="note-0072-13" xlink:href="note-0072-13a" xml:space="preserve">ibidem.</note> <note symbol="p" position="left" xlink:label="note-0072-14" xlink:href="note-0072-14a" xml:space="preserve">2. Co-<lb/>roll prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1686" xml:space="preserve">Si autem datum tranſuerſum latus fuerit maius ipſo BD quale eſt BG; </s> <s xml:id="echoid-s1687" xml:space="preserve">du-<lb/> <anchor type="note" xlink:label="note-0072-15a" xlink:href="note-0072-15"/> catur GL parallela ad DE, & </s> <s xml:id="echoid-s1688" xml:space="preserve">datæ Hyperbolæ HBI cum tranſuerſo BG, re-<lb/>ctoque BL adſcribatur <anchor type="note" xlink:href="" symbol="r"/> per B Hyperbole ABC, quæ datæ HBI erit ſimilis, <anchor type="note" xlink:label="note-0072-16a" xlink:href="note-0072-16"/> cum ipſarum latera ſint proportionalia, eritque circumſcripta, cum ſit maio- <pb o="49" file="0073" n="73" rhead=""/> rum laterum. </s> <s xml:id="echoid-s1689" xml:space="preserve">Dico hanc eſſe _MINIMAM_ quæſitam: </s> <s xml:id="echoid-s1690" xml:space="preserve">Nam quælibet alia, quę <lb/>adſcribitur cum recto maiore ipſo BL, maior <anchor type="note" xlink:href="" symbol="a"/> eſt ipſa ABC, quælibet verò, <anchor type="note" xlink:label="note-0073-01a" xlink:href="note-0073-01"/> quæ adſcribitur cum recto minore ipſo BL, eſt quidem <anchor type="note" xlink:href="" symbol="b"/> minor ABC, ſed <anchor type="note" xlink:label="note-0073-02a" xlink:href="note-0073-02"/> vel ſecat Hyperbolen ABC, quod accidet, cum rectum latus terminet inter <lb/>E, & </s> <s xml:id="echoid-s1691" xml:space="preserve">L, vt in O, <anchor type="note" xlink:href="" symbol="c"/> nam iuncta regula GO, ſi producatur, infra contingentem <anchor type="note" xlink:label="note-0073-03a" xlink:href="note-0073-03"/> BO cum regula DE conueniret, ideoque, & </s> <s xml:id="echoid-s1692" xml:space="preserve">ſectiones; </s> <s xml:id="echoid-s1693" xml:space="preserve">veltota cadit intra <lb/>HBI quod fit <anchor type="note" xlink:href="" symbol="d"/> quando rectum latus, vel ſit idem cum recto BE, vel minus <anchor type="note" xlink:label="note-0073-04a" xlink:href="note-0073-04"/> ipſo BE, quale eſt BF, quoniam ſi iungatur regula GF, ipſa, atque regula <lb/>DE ſe mutuò ſecarent ſupra contingentem BE, ideoque infra diſiunctim ſi-<lb/>mul procederent. </s> <s xml:id="echoid-s1694" xml:space="preserve">Quare huiuſmodi Hyperbole ABC, quæ ſimilis eſt datæ <lb/>HBI erit _MINIMA_ circumſcripta quæſita. </s> <s xml:id="echoid-s1695" xml:space="preserve">Quod ſecundò faciendum erat.</s> <s xml:id="echoid-s1696" xml:space="preserve"/> </p> <div xml:id="echoid-div147" type="float" level="2" n="6"> <note symbol="q" position="left" xlink:label="note-0072-15" xlink:href="note-0072-15a" xml:space="preserve">ibidem.</note> <note symbol="r" position="left" xlink:label="note-0072-16" xlink:href="note-0072-16a" xml:space="preserve">6. huius.</note> <note symbol="a" position="right" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">2. Co-<lb/>rol. 19. h.</note> <note symbol="b" position="right" xlink:label="note-0073-02" xlink:href="note-0073-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="right" xlink:label="note-0073-03" xlink:href="note-0073-03a" xml:space="preserve">1. Co-<lb/>rol. 19. h.</note> <note symbol="d" position="right" xlink:label="note-0073-04" xlink:href="note-0073-04a" xml:space="preserve">3. 1. Co-<lb/>rol. 19. h.</note> </div> </div> <div xml:id="echoid-div149" type="section" level="1" n="82"> <head xml:id="echoid-head87" xml:space="preserve">PROBL. IX. PROP. XXIV.</head> <p> <s xml:id="echoid-s1697" xml:space="preserve">Datæ Hyperbolæ, cum dato recto latere, quod recto datæ ſit <lb/>minus, per eius verticem MAXIMAM Hyperbolen inſcribere, <lb/>& </s> <s xml:id="echoid-s1698" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1699" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1700" xml:space="preserve">Datæ Hyperbolæ, cum dato recto latere, quod maius ſit recto <lb/>datę, per eius verticem MAXIMAM Hyperbolen circumſcribere.</s> <s xml:id="echoid-s1701" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1702" xml:space="preserve">SIt data Hyperbole ABC, cuius tranſuerſum BD, rectum BE, & </s> <s xml:id="echoid-s1703" xml:space="preserve">regula <lb/>DE; </s> <s xml:id="echoid-s1704" xml:space="preserve">oportet per eius verticem B, cum dato recto BF, quod minus ſit re-<lb/>cto BE, _MAXIMAM_ Hyperbolen inſcribere.</s> <s xml:id="echoid-s1705" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1706" xml:space="preserve">Ducatur FG parallela ad ED, & </s> <s xml:id="echoid-s1707" xml:space="preserve">cũ <lb/> <anchor type="figure" xlink:label="fig-0073-01a" xlink:href="fig-0073-01"/> tranſuerſo BG, rectoque BF adſcriba-<lb/>tur <anchor type="note" xlink:href="" symbol="a"/> per B ipſi ABC, Hyperbole HBI, <anchor type="note" xlink:label="note-0073-05a" xlink:href="note-0073-05"/> quæ datæ ABC erit ſimilis (cum ipſa-<lb/>rum latera ſint proportionalia) eritque <lb/>inſcripta <anchor type="note" xlink:href="" symbol="b"/> (cum ſit minorum laterum) <anchor type="note" xlink:label="note-0073-06a" xlink:href="note-0073-06"/> quàm dico eſſe _MAXIMAM_ quæſitam: <lb/></s> <s xml:id="echoid-s1708" xml:space="preserve">quoniam quælibet alia adſcripta cum <lb/>recto BF, ſed cum tranſuerſo, quod ex-<lb/>cedat BG minor eſt <anchor type="note" xlink:href="" symbol="c"/> ipſa HBI, quæcũ- <anchor type="note" xlink:label="note-0073-07a" xlink:href="note-0073-07"/> que verò adſcripta cum eodem recto <lb/>BE, ſed cum tranſuerſo, quod ſit minus <lb/>tranſuerſo BG, quale eſt BL, eſt quidẽ <lb/>maior <anchor type="note" xlink:href="" symbol="d"/> ipſa HBI, ſed omnino ſecat Hy- <anchor type="note" xlink:label="note-0073-08a" xlink:href="note-0073-08"/> perbolen ABC <anchor type="note" xlink:href="" symbol="e"/> cum iũcta regula LE, <anchor type="note" xlink:label="note-0073-09a" xlink:href="note-0073-09"/> & </s> <s xml:id="echoid-s1709" xml:space="preserve">producta, regulam DE infra contin-<lb/>gentem BE omnino ſecet: </s> <s xml:id="echoid-s1710" xml:space="preserve">quare ipſa <lb/>HBI erit _MAXIMA_ inſcripta cum dato <lb/>recto BF, Quod primò, &</s> <s xml:id="echoid-s1711" xml:space="preserve">c.</s> <s xml:id="echoid-s1712" xml:space="preserve"/> </p> <div xml:id="echoid-div149" type="float" level="2" n="1"> <figure xlink:label="fig-0073-01" xlink:href="fig-0073-01a"> <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0073-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0073-05" xlink:href="note-0073-05a" xml:space="preserve">6. huius.</note> <note symbol="b" position="right" xlink:label="note-0073-06" xlink:href="note-0073-06a" xml:space="preserve">5. Co-<lb/>rol. 19. h.</note> <note symbol="c" position="right" xlink:label="note-0073-07" xlink:href="note-0073-07a" xml:space="preserve">3. Co-<lb/>rol. 19. h.</note> <note symbol="d" position="right" xlink:label="note-0073-08" xlink:href="note-0073-08a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0073-09" xlink:href="note-0073-09a" xml:space="preserve">1. Co-<lb/>rol. 19. h.</note> </div> <p> <s xml:id="echoid-s1713" xml:space="preserve">Sit iam data Hyperbole HBI, cuius <lb/>regula GF, & </s> <s xml:id="echoid-s1714" xml:space="preserve">datum rectum latus ſit BE ipſo BF maius, cum quo oporteat <lb/>_MINIMAM_ Hyperbolen circumſcribere.</s> <s xml:id="echoid-s1715" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1716" xml:space="preserve">Agatur ED ipſi FG æ quidiſtans, & </s> <s xml:id="echoid-s1717" xml:space="preserve">cum regula ED, datæ Hyperbolæ HBI <pb o="50" file="0074" n="74" rhead=""/> adſcribatur <anchor type="note" xlink:href="" symbol="a"/> per B Hyperbole ABC, quæ ipſi HBI erit ſimilis (cum earum <anchor type="note" xlink:label="note-0074-01a" xlink:href="note-0074-01"/> latera ſint proportionalia) eritque ei circumſcripta <anchor type="note" xlink:href="" symbol="b"/> (cum ſit maiorum late- rum) & </s> <s xml:id="echoid-s1718" xml:space="preserve">erit _MINIMA_ quæſita. </s> <s xml:id="echoid-s1719" xml:space="preserve">Nam quælibet alia adſcripta cum recto BE, <lb/> <anchor type="note" xlink:label="note-0074-02a" xlink:href="note-0074-02"/> ſed cum tranſuerſo, quod ipſo BD ſit minus <anchor type="note" xlink:href="" symbol="c"/> eſt maior ipfa ABC; </s> <s xml:id="echoid-s1720" xml:space="preserve">quælibet verò adſcripta cum eodem recto BE, ſed cum tranſuerſo BM, quod excedat <lb/> <anchor type="note" xlink:label="note-0074-03a" xlink:href="note-0074-03"/> BD eſt quidem minor ipſa ABC, ſed omnino ſecat Hyperbolen HBI, cum <lb/>iuncta regula ME, & </s> <s xml:id="echoid-s1721" xml:space="preserve">producta omnino ſecet regulam GF infra contingen-<lb/>tem BE. </s> <s xml:id="echoid-s1722" xml:space="preserve">Vnde ipſa ABC erit _MINIMA_ circumſcripta cum dato recto BE, <lb/>vti quærebatur. </s> <s xml:id="echoid-s1723" xml:space="preserve">Quod vltimò faciendum erat.</s> <s xml:id="echoid-s1724" xml:space="preserve"/> </p> <div xml:id="echoid-div150" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">5. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="b" position="left" xlink:label="note-0074-02" xlink:href="note-0074-02a" xml:space="preserve">3. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="c" position="left" xlink:label="note-0074-03" xlink:href="note-0074-03a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div152" type="section" level="1" n="83"> <head xml:id="echoid-head88" xml:space="preserve">PROBL. X. PROP. XXV.</head> <p> <s xml:id="echoid-s1725" xml:space="preserve">Datæ Ellipſi, cum dato latere, quodminus ſit eius recto, per ip-<lb/>ſius verticem MAXIMAM Ellipſim inſcribere: </s> <s xml:id="echoid-s1726" xml:space="preserve">& </s> <s xml:id="echoid-s1727" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1728" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1729" xml:space="preserve">Datæ Ellipſi, cum dato recto latere, quod maius ſit eius recto, <lb/>per ipſius verticem MINIMAM Ellipſim circumſcribere.</s> <s xml:id="echoid-s1730" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1731" xml:space="preserve">SIt data Ellipſis ABC, cuius tranſuerſum BD, rectum BE, regula DE: <lb/></s> <s xml:id="echoid-s1732" xml:space="preserve">oportet per verticem B, cum dato recto BF _MAXIMAM_ Ellipſim inſcri-<lb/>bere, neceſſe eſt autem, quod rectum datum BF <lb/> <anchor type="figure" xlink:label="fig-0074-01a" xlink:href="fig-0074-01"/> minus ſit recto BE (ſi enim ei æquale eſſet, vel <lb/> <anchor type="note" xlink:label="note-0074-04a" xlink:href="note-0074-04"/> maius, etiam deſcribenda Ellipſis, vel eſſet <anchor type="note" xlink:href="" symbol="a"/> ea- dem cum data ABC, vel hanc ipſam ſecaret, vt <lb/>ſatis patet, cum vel ipſarum regulæ ſimul con-<lb/>gruerent, vel ſe mutuò ſecarent.)</s> <s xml:id="echoid-s1733" xml:space="preserve"/> </p> <div xml:id="echoid-div152" type="float" level="2" n="1"> <figure xlink:label="fig-0074-01" xlink:href="fig-0074-01a"> <image file="0074-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0074-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0074-04" xlink:href="note-0074-04a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1734" xml:space="preserve">Adſcribatur <anchor type="note" xlink:href="" symbol="b"/> cum eodem trãſuerſo BD, cum- <anchor type="note" xlink:label="note-0074-05a" xlink:href="note-0074-05"/> que dato recto BF, per verticem B, Ellipſis GBL: <lb/></s> <s xml:id="echoid-s1735" xml:space="preserve">& </s> <s xml:id="echoid-s1736" xml:space="preserve">hæc erit _MAXIMA_ quæſita. </s> <s xml:id="echoid-s1737" xml:space="preserve">Nam quælibet alia <lb/>eidem ABC adſcripta cum recto BF, ſed cum <lb/>tranſuerſo, quod minus ſit BD, eſt minor <anchor type="note" xlink:href="" symbol="c"/> ipſa <anchor type="note" xlink:label="note-0074-06a" xlink:href="note-0074-06"/> GBL, quælibet verò adſcripta cum tranſuerſo <lb/>BM, quod excedat BD eſt quidem <anchor type="note" xlink:href="" symbol="d"/> maior ipſa <anchor type="note" xlink:label="note-0074-07a" xlink:href="note-0074-07"/> GBL, ſed omnino ſecat Ellipſim datam ABC <anchor type="note" xlink:href="" symbol="e"/> cum & </s> <s xml:id="echoid-s1738" xml:space="preserve">iuncta regula FM, <anchor type="note" xlink:label="note-0074-08a" xlink:href="note-0074-08"/> omnino ſecet regulam ED. </s> <s xml:id="echoid-s1739" xml:space="preserve">Quare Ellipſis GBL erit _MAXIMA_ quæſita in-<lb/>ſcripta cum recto dato BF. </s> <s xml:id="echoid-s1740" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1741" xml:space="preserve">c.</s> <s xml:id="echoid-s1742" xml:space="preserve"/> </p> <div xml:id="echoid-div153" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0074-05" xlink:href="note-0074-05a" xml:space="preserve">7. huius.</note> <note symbol="c" position="left" xlink:label="note-0074-06" xlink:href="note-0074-06a" xml:space="preserve">4. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="d" position="left" xlink:label="note-0074-07" xlink:href="note-0074-07a" xml:space="preserve">ibidem.</note> <note symbol="e" position="left" xlink:label="note-0074-08" xlink:href="note-0074-08a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1743" xml:space="preserve">Sit iam data Ellipſis GBL, cuius tranſuerſum BD, rectum BF, regula DF, <lb/>& </s> <s xml:id="echoid-s1744" xml:space="preserve">circumſcribenda ſit ci _MINIMA_ Ellipſis cum dato recto BE, quod debet <lb/>quidem eſſe maius recto BF (nam ſi æquale, vel minus eſſet, deſcribenda <lb/>quoque Ellipſis, vel eadem eſſet cum data GBL, vel huic eſſet <anchor type="note" xlink:href="" symbol="f"/> inſcripta, <anchor type="note" xlink:label="note-0074-09a" xlink:href="note-0074-09"/> cum vel harum regulæ ſimul congruerent, vel regula deſcribendæ caderet <lb/>tota intra regulam deſcriptæ GBL.)</s> <s xml:id="echoid-s1745" xml:space="preserve"/> </p> <div xml:id="echoid-div154" type="float" level="2" n="3"> <note symbol="f" position="left" xlink:label="note-0074-09" xlink:href="note-0074-09a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s1746" xml:space="preserve">Adſcribatur <anchor type="note" xlink:href="" symbol="g"/> cum tranſuerſo BD, datoque recto BE, per verticem B, El- <anchor type="note" xlink:label="note-0074-10a" xlink:href="note-0074-10"/> lipſis ABC, quæ erit _MINIMA_ circumſcripta quæſita. </s> <s xml:id="echoid-s1747" xml:space="preserve">Quoniam quæcun-<lb/>que alia adſcripta datæ GBL cum recto BE, ſed cum tranſuerſo, quod maius <lb/> <anchor type="note" xlink:label="note-0074-11a" xlink:href="note-0074-11"/> ſit BD, maior eſt <anchor type="note" xlink:href="" symbol="h"/> ipſa GBL; </s> <s xml:id="echoid-s1748" xml:space="preserve">quælibet verò adſcripta cum tranſuerſo BN, quod minus ſit tranſuerſo BD, eſt quidem minor <anchor type="note" xlink:href="" symbol="i"/> ipſa ABC, ſed omnino <anchor type="note" xlink:label="note-0074-12a" xlink:href="note-0074-12"/> <pb o="51" file="0075" n="75" rhead=""/> ſecat Elli pſim datam GBL, _a_ cum & </s> <s xml:id="echoid-s1749" xml:space="preserve">iuncta regula EN ſecet regulam FD. <lb/></s> <s xml:id="echoid-s1750" xml:space="preserve">Quare Ellipſis ABC, erit _MINIMA_ quæſita circumſcripta, cum dato recto <lb/>BE. </s> <s xml:id="echoid-s1751" xml:space="preserve">Quod vltimò, &</s> <s xml:id="echoid-s1752" xml:space="preserve">c.</s> <s xml:id="echoid-s1753" xml:space="preserve"/> </p> <div xml:id="echoid-div155" type="float" level="2" n="4"> <note symbol="g" position="left" xlink:label="note-0074-10" xlink:href="note-0074-10a" xml:space="preserve">7. huius.</note> <note symbol="h" position="left" xlink:label="note-0074-11" xlink:href="note-0074-11a" xml:space="preserve">4. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="i" position="left" xlink:label="note-0074-12" xlink:href="note-0074-12a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div157" type="section" level="1" n="84"> <head xml:id="echoid-head89" xml:space="preserve">PROBL. XI. PROP. XXVI.</head> <p> <s xml:id="echoid-s1754" xml:space="preserve">Datæ Ellipſi circa minorem axim, per eius verticem MAXI-<lb/>MVM circulum inſcribere. </s> <s xml:id="echoid-s1755" xml:space="preserve">Item.</s> <s xml:id="echoid-s1756" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1757" xml:space="preserve">Datæ Ellipſi circa maiorem axim, per eius verticem MINI-<lb/>MVM circulum circumſcribere.</s> <s xml:id="echoid-s1758" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1759" xml:space="preserve">SIt in 1. </s> <s xml:id="echoid-s1760" xml:space="preserve">fig. </s> <s xml:id="echoid-s1761" xml:space="preserve">data Ellipſis ABC, circa minorem axim BD, cuius rectũ ſit BE, <lb/>regula DE, & </s> <s xml:id="echoid-s1762" xml:space="preserve">oporteat per verticem B _MAXIMVM_ circulum inſcribere.</s> <s xml:id="echoid-s1763" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1764" xml:space="preserve">Deſcribatur circulus GBHD, cuius dimetiens ſit BD, quem dico eſſe _MA_-<lb/>_XIMVM_ quæſitum.</s> <s xml:id="echoid-s1765" xml:space="preserve"/> </p> <figure> <image file="0075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0075-01"/> </figure> <p> <s xml:id="echoid-s1766" xml:space="preserve">Sumpta enim BF æquali BD, erit ipſa rectum latus deſcripti circuli: </s> <s xml:id="echoid-s1767" xml:space="preserve">iun-<lb/>ctaque DF eius regula cum ſit axis minor BD, minor recto latere BE, erit <lb/>etiam BF minor BE, vnde regula DF cadet intra regulam DE, ideoque cir-<lb/>culus GBH inſcriptus <anchor type="note" xlink:href="" symbol="a"/> erit Ellipſi ABC, eritque _MAXIMVS_: </s> <s xml:id="echoid-s1768" xml:space="preserve">nam quilibet <anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/> alius per B adſcriptus, cuius diameter, minor ſit ipſa BD, minor eſt <anchor type="note" xlink:href="" symbol="b"/> circulo GBH, & </s> <s xml:id="echoid-s1769" xml:space="preserve">cuius diameter BI ſit maior BD <anchor type="note" xlink:href="" symbol="c"/> eſt quidem maior circulo GBH, <anchor type="note" xlink:label="note-0075-02a" xlink:href="note-0075-02"/> ſed vel ſecat, vel cadit extra Ellipſim ABC, cum punctum I quoque cadat <lb/>extra. </s> <s xml:id="echoid-s1770" xml:space="preserve">Erit ergo GBH _MAXIMVS_ circulus per verticem B minoris axis da-<lb/> <anchor type="note" xlink:label="note-0075-03a" xlink:href="note-0075-03"/> tæ Ellipſi ABC inſcriptus. </s> <s xml:id="echoid-s1771" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s1772" xml:space="preserve">c.</s> <s xml:id="echoid-s1773" xml:space="preserve"/> </p> <div xml:id="echoid-div157" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="b" position="right" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">5. Co-<lb/>rol. prop. <lb/>19. huius.</note> <note symbol="c" position="right" xlink:label="note-0075-03" xlink:href="note-0075-03a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s1774" xml:space="preserve">Sit verò in 2. </s> <s xml:id="echoid-s1775" xml:space="preserve">figura, data Ellipſis ABCD, cuius maior axis BD, rectum BE, <lb/>regula DE. </s> <s xml:id="echoid-s1776" xml:space="preserve">Oportet per verticem B _MINIMVM_ circulum circumſcribere.</s> <s xml:id="echoid-s1777" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1778" xml:space="preserve">Deſcribatur circulus GBHD, cuius diameter ſit axis maior BD. </s> <s xml:id="echoid-s1779" xml:space="preserve">Dico <lb/>hunc eſſe _MINIMVM_ quæſitum.</s> <s xml:id="echoid-s1780" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1781" xml:space="preserve">Cum ſit enim axis BD maior recto latere BE, ſumpta BF æquali BD, ipſa <lb/>erit latus rectum circuli GBH, & </s> <s xml:id="echoid-s1782" xml:space="preserve">maior BE: </s> <s xml:id="echoid-s1783" xml:space="preserve">vnde circuli regula DF cadet <lb/> <anchor type="note" xlink:label="note-0075-04a" xlink:href="note-0075-04"/> tota extra Ellipſis regulam DE, ac ideò circulus <anchor type="note" xlink:href="" symbol="d"/> erit Ellipſi circumſcriptus, eritque _MINIMVS_; </s> <s xml:id="echoid-s1784" xml:space="preserve">quoniam quilibet alius circulus GBH per B adſcriptus, <lb/> <anchor type="note" xlink:label="note-0075-05a" xlink:href="note-0075-05"/> cuius diameter ſit maior BD, eſt <anchor type="note" xlink:href="" symbol="e"/> maior ipſo GBH, & </s> <s xml:id="echoid-s1785" xml:space="preserve">quicunque alius, cuius diameter ſit minor ipſa BD, qualis eſt BI, minor eſt quidem circulo GBH, <pb o="52" file="0076" n="76" rhead=""/> ſed omnino, vel Ellipſim ſecat, velintra eam cadit, cum punctum I ſit quo-<lb/>que intra. </s> <s xml:id="echoid-s1786" xml:space="preserve">Quare circulus GBH _MINIMV S_ eſt circumſcriptibilium per ver-<lb/>ticem B maioris axis datæ Ellipſis ABC. </s> <s xml:id="echoid-s1787" xml:space="preserve">Quod ſecundò faciendum erat.</s> <s xml:id="echoid-s1788" xml:space="preserve"/> </p> <div xml:id="echoid-div158" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-0075-04" xlink:href="note-0075-04a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="e" position="right" xlink:label="note-0075-05" xlink:href="note-0075-05a" xml:space="preserve">5. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> </div> <div xml:id="echoid-div160" type="section" level="1" n="85"> <head xml:id="echoid-head90" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s1789" xml:space="preserve">HInc facilè eruitur pulcherrima de _MAXIMIS_, & </s> <s xml:id="echoid-s1790" xml:space="preserve">_MINIMIS_ circulis, <lb/>Ellipſi inſcriptis, & </s> <s xml:id="echoid-s1791" xml:space="preserve">circumſcriptis proprietas. </s> <s xml:id="echoid-s1792" xml:space="preserve">Nempe.</s> <s xml:id="echoid-s1793" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1794" xml:space="preserve">_MINIMV M_ circulum per verticem minoris axis AC Ellipſi circumſcri-<lb/>ptum, cuius diameter <anchor type="note" xlink:href="" symbol="a"/> eſt rectum latus AB.</s> <s xml:id="echoid-s1795" xml:space="preserve"> </s> </p> <note symbol="a" position="left" xml:space="preserve">1. Co-<lb/>roll. 20. h.</note> <p> <s xml:id="echoid-s1796" xml:space="preserve">_MINIMV M_ circulum per verticem maioris axis DE circumſcriptũ, cuius <lb/>diameter <anchor type="note" xlink:href="" symbol="b"/> eſt ipſe maior axis DE.</s> <s xml:id="echoid-s1797" xml:space="preserve"> </s> </p> <note symbol="b" position="left" xml:space="preserve">26. h.</note> <p> <s xml:id="echoid-s1798" xml:space="preserve">_MAXIMV M_ circulum per verticem minoris axis AC inſcriptum, cuius <lb/>diameter <anchor type="note" xlink:href="" symbol="c"/> eſt ipſe minor axis AC.</s> <s xml:id="echoid-s1799" xml:space="preserve"> </s> </p> <note symbol="c" position="left" xml:space="preserve">26. h.</note> <p> <s xml:id="echoid-s1800" xml:space="preserve">Et _MAXIMV M_ circulũ per ver-<lb/> <anchor type="figure" xlink:label="fig-0076-01a" xlink:href="fig-0076-01"/> ticem maioris axis DE inſcriptum, <lb/>cuius diameter <anchor type="note" xlink:href="" symbol="d"/> eſt rectum latus <anchor type="note" xlink:label="note-0076-04a" xlink:href="note-0076-04"/> DF, eſſe quatuor circulos in conti-<lb/>nua eademque ratione geometri-<lb/>ca; </s> <s xml:id="echoid-s1801" xml:space="preserve">nam & </s> <s xml:id="echoid-s1802" xml:space="preserve">ipſorum diametri AB, <lb/>DE, AC, DF ſunt quatuor lineæ <lb/>continuè proportionales.</s> <s xml:id="echoid-s1803" xml:space="preserve"/> </p> <div xml:id="echoid-div160" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0076-01"/> </figure> <note symbol="d" position="left" xlink:label="note-0076-04" xlink:href="note-0076-04a" xml:space="preserve">1. Co-<lb/>roll. 20. h.</note> </div> </div> <div xml:id="echoid-div162" type="section" level="1" n="86"> <head xml:id="echoid-head91" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s1804" xml:space="preserve">ELicitur quoque, Ellipſim quamcunque, mediam eſſe proportionalem <lb/>inter extremos prædictos circulos, mediamque inter medios. </s> <s xml:id="echoid-s1805" xml:space="preserve">Cum <lb/>enim quatuor lineæ AB, DE, AC, DF ſint continuè proportionales, erit <lb/>rectangulum ſub extremis AB, DF æquale rectangulo ſub medijs DE, AC, <lb/>nempè quadrato G, quæ ſit media proportionalis inter DE, AC; </s> <s xml:id="echoid-s1806" xml:space="preserve">hoc eſt vt <lb/>AB ad G, ita erit G ad DF; </s> <s xml:id="echoid-s1807" xml:space="preserve">quare circulus ex diametro AB, ad circulum ex <lb/>diametro G, erit vt circulus G, ad circulum ex DF. </s> <s xml:id="echoid-s1808" xml:space="preserve">Item cum ſit DE ad G, <lb/>ita G ad AC, erit circulus ex DE ad circulum ex G, vt circulus G ad circu-<lb/> <anchor type="note" xlink:label="note-0076-05a" xlink:href="note-0076-05"/> lum ex AC, ſed circulus ex G <anchor type="note" xlink:href="" symbol="e"/> æquatur Ellipſi; </s> <s xml:id="echoid-s1809" xml:space="preserve">vnde Ellipſis DAEC eſt media proportionalis inter extremos prædictos circulos AB, DF mediaque <lb/>inter medios DE, AC.</s> <s xml:id="echoid-s1810" xml:space="preserve"/> </p> <div xml:id="echoid-div162" type="float" level="2" n="1"> <note symbol="e" position="left" xlink:label="note-0076-05" xlink:href="note-0076-05a" xml:space="preserve">5. Arch. <lb/>de Co-<lb/>noid. & <lb/>Sphęroid.</note> </div> <pb o="53" file="0077" n="77" rhead=""/> </div> <div xml:id="echoid-div164" type="section" level="1" n="87"> <head xml:id="echoid-head92" xml:space="preserve">PROBL. XII. PROP. XXVII.</head> <p> <s xml:id="echoid-s1811" xml:space="preserve">Datæ portioni Parabolæ, cum dato quocunque tranſuerſo late-<lb/>re, vel cum dato recto, quod ſit minus recto datæ Parabolæ, per <lb/>eius verticem MAXIMAM Hyperbolæ portionem inſcribere; </s> <s xml:id="echoid-s1812" xml:space="preserve">& </s> <s xml:id="echoid-s1813" xml:space="preserve"><lb/>è contra.</s> <s xml:id="echoid-s1814" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1815" xml:space="preserve">Datæ portioni Hyperbolæ, per eius verticem MINIMAM Pa-<lb/>rabolæ portionem circumſcribere.</s> <s xml:id="echoid-s1816" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1817" xml:space="preserve">SIt data Parabolę portio ABCDE, cuius rectum CF, regula FG, baſis AE, <lb/>diameter CI, & </s> <s xml:id="echoid-s1818" xml:space="preserve">oporteat primùm cum dato quocunque tranſuerſo CH <lb/>_MAXIMAM_ Hyperbolæ portionem inſcribere.</s> <s xml:id="echoid-s1819" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1820" xml:space="preserve">Producatur applicata AI vſque ad occurſum <lb/> <anchor type="figure" xlink:label="fig-0077-01a" xlink:href="fig-0077-01"/> cum regula in G, & </s> <s xml:id="echoid-s1821" xml:space="preserve">iungatur HG, ſecans CF in <lb/>L, & </s> <s xml:id="echoid-s1822" xml:space="preserve">cum tranſuerſo CH, rectoque CL adſcriba-<lb/>tur <anchor type="note" xlink:href="" symbol="a"/> portioni ABCDE per verticem C Hyper- <anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/> bole AMCNE, quæ Parabolen ABCDE ſecabit <lb/>in <anchor type="note" xlink:href="" symbol="b"/> A, & </s> <s xml:id="echoid-s1823" xml:space="preserve">E (cum & </s> <s xml:id="echoid-s1824" xml:space="preserve">ipſarum regulæ ſe mutuò ſecẽt <anchor type="note" xlink:label="note-0077-02a" xlink:href="note-0077-02"/> in occurſu eiuſdem communis applicatæ AE) & </s> <s xml:id="echoid-s1825" xml:space="preserve"><lb/>ſupra baſim AE erit datæ Parabolę inſcripta. </s> <s xml:id="echoid-s1826" xml:space="preserve">Iam <lb/>dico hanc portionem AMCNE eſſe _MAXIMAM_ <lb/>quæſitam.</s> <s xml:id="echoid-s1827" xml:space="preserve"/> </p> <div xml:id="echoid-div164" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0077-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">6. huius.</note> <note symbol="b" position="right" xlink:label="note-0077-02" xlink:href="note-0077-02a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1828" xml:space="preserve">Quoniam quælibet alia Hyperbole adſcripta <lb/>cum eodem tranſuerſo CH, ſed cum recto, quod <lb/>minus ſit recto CL, minor eſt <anchor type="note" xlink:href="" symbol="c"/> ipſa AMCNE, <anchor type="note" xlink:label="note-0077-03a" xlink:href="note-0077-03"/> quælibet verò adſcripta cum recto, quod excedat CL, eſt quidem <anchor type="note" xlink:href="" symbol="d"/> maior <anchor type="note" xlink:label="note-0077-04a" xlink:href="note-0077-04"/> ipſa AMCNE, ſed veltota cadit extra ABCDE, cum Hyperbole, cuius re-<lb/>gula ſit quæ ducitur per H & </s> <s xml:id="echoid-s1829" xml:space="preserve">F ſit circumſcripta <anchor type="note" xlink:href="" symbol="e"/> Parabolæ ABC, & </s> <s xml:id="echoid-s1830" xml:space="preserve">eò ma- <anchor type="note" xlink:label="note-0077-05a" xlink:href="note-0077-05"/> gis, quæ cum recto CO maiore ipſo CF; </s> <s xml:id="echoid-s1831" xml:space="preserve">vel ſaltem ſecat <anchor type="note" xlink:href="" symbol="f"/> Parabolen ABC <anchor type="note" xlink:label="note-0077-06a" xlink:href="note-0077-06"/> ſupra portionis baſim AE, cum quælibet regula ducta ex H inter L, & </s> <s xml:id="echoid-s1832" xml:space="preserve">F, vt <lb/>per P, & </s> <s xml:id="echoid-s1833" xml:space="preserve">infra contingentem CF producta, ſecet regulam FG inter ipſam <lb/>contingentem, & </s> <s xml:id="echoid-s1834" xml:space="preserve">applicatam AEG. </s> <s xml:id="echoid-s1835" xml:space="preserve">Quare talis Hyperbolæ portio AMC-<lb/>NE eſt _MAXIMA_ inſcripta quæſita cũ dato tranſuerſo CH. </s> <s xml:id="echoid-s1836" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1837" xml:space="preserve">c.</s> <s xml:id="echoid-s1838" xml:space="preserve"/> </p> <div xml:id="echoid-div165" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0077-03" xlink:href="note-0077-03a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="d" position="right" xlink:label="note-0077-04" xlink:href="note-0077-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0077-05" xlink:href="note-0077-05a" xml:space="preserve">21. h.</note> <note symbol="f" position="right" xlink:label="note-0077-06" xlink:href="note-0077-06a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1839" xml:space="preserve">Si verò inſcribenda ſit _MAXIMA_ Hyperbolæ portio cum dato recto CL, <lb/>quod minus ſit recto Parabolæ CF, (nam cum æquali, vel maiori ſemper eſ-<lb/>ſet circumſcripta) iungatur GL, & </s> <s xml:id="echoid-s1840" xml:space="preserve">producatur, ipſa productam diametrum <lb/>IC ſecabit in H, cum eadem ſecet FG alteram Parallelarum ipſi diametro; <lb/></s> <s xml:id="echoid-s1841" xml:space="preserve">& </s> <s xml:id="echoid-s1842" xml:space="preserve">cum tranſuerſo HC, rectoque CL <anchor type="note" xlink:href="" symbol="g"/> adſcribatur per C Hyperbole AMC <anchor type="note" xlink:label="note-0077-07a" xlink:href="note-0077-07"/> NE, quæ ſecabit, vt ſupra, Parabolen ABCDE in <anchor type="note" xlink:href="" symbol="h"/> A, & </s> <s xml:id="echoid-s1843" xml:space="preserve">E, eique erit in- <anchor type="note" xlink:label="note-0077-08a" xlink:href="note-0077-08"/> ſcripta, eritque _MAXIMA_. </s> <s xml:id="echoid-s1844" xml:space="preserve">Nam quæ cum eodem recto CL, ſed cum tranſ-<lb/>uerſo, quod excedat CH, minor eſt <anchor type="note" xlink:href="" symbol="i"/> ipſa AMCNE; </s> <s xml:id="echoid-s1845" xml:space="preserve">quæ verò cum tranſ- <anchor type="note" xlink:label="note-0077-09a" xlink:href="note-0077-09"/> uerſo CQ minore ipſo CH, eſt quidem maior <anchor type="note" xlink:href="" symbol="l"/> Hyperbola AMCNE, ſed <anchor type="note" xlink:label="note-0077-10a" xlink:href="note-0077-10"/> omnino ſecat <anchor type="note" xlink:href="" symbol="m"/> Parabolen ABCDE ſupra baſim AE, cum & </s> <s xml:id="echoid-s1846" xml:space="preserve">eius regula QL <anchor type="note" xlink:label="note-0077-11a" xlink:href="note-0077-11"/> infra contingentem producta, ſecet regulam FG ſupra eandem applicatam. <lb/></s> <s xml:id="echoid-s1847" xml:space="preserve">Quare huiuſmodi portio Hyperbolæ AMCNE eſt _MAXIMA_ inſcripta quæ- <pb o="54" file="0078" n="78" rhead=""/> ſita cum dato recto CL. </s> <s xml:id="echoid-s1848" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s1849" xml:space="preserve">c.</s> <s xml:id="echoid-s1850" xml:space="preserve"/> </p> <div xml:id="echoid-div166" type="float" level="2" n="3"> <note symbol="g" position="right" xlink:label="note-0077-07" xlink:href="note-0077-07a" xml:space="preserve">6. huius.</note> <note symbol="h" position="right" xlink:label="note-0077-08" xlink:href="note-0077-08a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="i" position="right" xlink:label="note-0077-09" xlink:href="note-0077-09a" xml:space="preserve">3. Corol. <lb/>19. huius.</note> <note symbol="l" position="right" xlink:label="note-0077-10" xlink:href="note-0077-10a" xml:space="preserve">ibidem.</note> <note symbol="m" position="right" xlink:label="note-0077-11" xlink:href="note-0077-11a" xml:space="preserve">3. Co-<lb/>rol. 19. h.</note> </div> <p> <s xml:id="echoid-s1851" xml:space="preserve">Iam ſit data Hyperbolæ portio AMCNE, cuius tranſuerſum CH, rectum <lb/>CL, regula HLG, baſis AE, diameter CI, & </s> <s xml:id="echoid-s1852" xml:space="preserve">oporteat, per verticem C, _MI_-<lb/>_NIMAM_ Parabolæ portionem circumſcribere.</s> <s xml:id="echoid-s1853" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1854" xml:space="preserve">Producatur applicata AI, conueniens cum re-<lb/> <anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/> gula HL in G, & </s> <s xml:id="echoid-s1855" xml:space="preserve">per G ducatur GF parallela ad <lb/>IC contingentem ſecans in F, cumque recto CF <lb/>adſcribatur <anchor type="note" xlink:href="" symbol="a"/> per C Parabole ABCDE, quæ ſe- <anchor type="note" xlink:label="note-0078-01a" xlink:href="note-0078-01"/> cabit Hyperbolen in ijſdem punctis A, & </s> <s xml:id="echoid-s1856" xml:space="preserve">E, ob <lb/>rationem ſuperius allatam, & </s> <s xml:id="echoid-s1857" xml:space="preserve">datę Parabolæ AB <lb/>CD <anchor type="note" xlink:href="" symbol="b"/> erit circumſcripta; </s> <s xml:id="echoid-s1858" xml:space="preserve">eritq; </s> <s xml:id="echoid-s1859" xml:space="preserve">_MINIMA_ portio.</s> <s xml:id="echoid-s1860" xml:space="preserve"> </s> </p> <div xml:id="echoid-div167" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0078-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">5. huius.</note> </div> <note symbol="b" position="left" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <p> <s xml:id="echoid-s1861" xml:space="preserve">Quoniam, quæ cum recto maiore ipſo CF eſt <lb/>maior <anchor type="note" xlink:href="" symbol="c"/> ipſa ABCDE, quæ verò cum recto mino- <anchor type="note" xlink:label="note-0078-03a" xlink:href="note-0078-03"/> re ipſo CF eſt quidem minor <anchor type="note" xlink:href="" symbol="d"/> ABCDE, ſed vel <anchor type="note" xlink:label="note-0078-04a" xlink:href="note-0078-04"/> tota cadit intra Hyperbolen AMCN <anchor type="note" xlink:href="" symbol="e"/> ſi nempe <anchor type="note" xlink:label="note-0078-05a" xlink:href="note-0078-05"/> rectum æquale fuerit ipſo CL, & </s> <s xml:id="echoid-s1862" xml:space="preserve">eò magis ſi mi-<lb/>nus eſſet BL; </s> <s xml:id="echoid-s1863" xml:space="preserve">vel ſaltẽ ſecat Hyperbolen AMCN <lb/>ſupra applicatam AE <anchor type="note" xlink:href="" symbol="f"/> tum cum rectum ſit medium inter CF, & </s> <s xml:id="echoid-s1864" xml:space="preserve">CL, quale <anchor type="note" xlink:label="note-0078-06a" xlink:href="note-0078-06"/> eſt CP: </s> <s xml:id="echoid-s1865" xml:space="preserve">nam regula, quæ ex P, ducitur æquidiſtans CI, omninò ſecat regu-<lb/>lam LG infra contingentem CF, & </s> <s xml:id="echoid-s1866" xml:space="preserve">ſupra applicatam AG. </s> <s xml:id="echoid-s1867" xml:space="preserve">Quare ipſa Para-<lb/>bolæ portio ABCDE, eſt _MINIMA_ circumſcripta quæſita. </s> <s xml:id="echoid-s1868" xml:space="preserve">Quod tandem <lb/>faciendum erat.</s> <s xml:id="echoid-s1869" xml:space="preserve"/> </p> <div xml:id="echoid-div168" type="float" level="2" n="5"> <note symbol="c" position="left" xlink:label="note-0078-03" xlink:href="note-0078-03a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="d" position="left" xlink:label="note-0078-04" xlink:href="note-0078-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="left" xlink:label="note-0078-05" xlink:href="note-0078-05a" xml:space="preserve">21. h.</note> <note symbol="f" position="left" xlink:label="note-0078-06" xlink:href="note-0078-06a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> </div> <div xml:id="echoid-div170" type="section" level="1" n="88"> <head xml:id="echoid-head93" xml:space="preserve">PROBL. XIII. PROP. XXVIII.</head> <p> <s xml:id="echoid-s1870" xml:space="preserve">Datæ portioni Hyperbolæ, cum dato tranſuerſo vel recto, quod <lb/>minus ſit tranſuerſo, vel recto datæ Hyperbolæ, per eius verticem <lb/>MAXIMAM Hyperbolæ portionem inſcribere: </s> <s xml:id="echoid-s1871" xml:space="preserve">& </s> <s xml:id="echoid-s1872" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1873" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1874" xml:space="preserve">Datæ portioni Hyperbolæ, cum dato tranſuerſo vel recto, quod <lb/>excedat tranſuerſum, aut rectum datæ Hyperbolæ, per eius verti-<lb/>cem MINIMAM Hyperbolæ portionem circumſcribere.</s> <s xml:id="echoid-s1875" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1876" xml:space="preserve">SIt data Hyperbolæ portio ABCDE, cuius <lb/> <anchor type="figure" xlink:label="fig-0078-02a" xlink:href="fig-0078-02"/> tranſuerſum CF, rectum CG, regula FGL, <lb/>baſis AE, diameter CH. </s> <s xml:id="echoid-s1877" xml:space="preserve">Oporter primò cum <lb/>dato tranſuerſo CI, quod minus ſit ipſo CF <lb/>_MAXIMAM_ Hyporbolæ portionem inſcribere.</s> <s xml:id="echoid-s1878" xml:space="preserve"/> </p> <div xml:id="echoid-div170" type="float" level="2" n="1"> <figure xlink:label="fig-0078-02" xlink:href="fig-0078-02a"> <image file="0078-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0078-02"/> </figure> </div> <p> <s xml:id="echoid-s1879" xml:space="preserve">Producta enim applicata AH, cõueniat cum <lb/>regula FG in L, & </s> <s xml:id="echoid-s1880" xml:space="preserve">iuncta IL contingentem CG <lb/>ſecant in M, cum regula IM, per verticem C ad-<lb/>ſcribatur <anchor type="note" xlink:href="" symbol="a"/> portioni ABCDE Hyperbole ANC <anchor type="note" xlink:label="note-0078-07a" xlink:href="note-0078-07"/> OE, quę datam ABCD ſecabit <anchor type="note" xlink:href="" symbol="b"/> in A, & </s> <s xml:id="echoid-s1881" xml:space="preserve">E, at quę <anchor type="note" xlink:label="note-0078-08a" xlink:href="note-0078-08"/> ipſi erit inſcripta. </s> <s xml:id="echoid-s1882" xml:space="preserve">Dico portionẽ ANCOE eſſe <lb/>_MAXIMAM_ quæſitam.</s> <s xml:id="echoid-s1883" xml:space="preserve"/> </p> <div xml:id="echoid-div171" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0078-07" xlink:href="note-0078-07a" xml:space="preserve">6. huius.</note> <note symbol="b" position="left" xlink:label="note-0078-08" xlink:href="note-0078-08a" xml:space="preserve">1. Co-<lb/>roll prop. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1884" xml:space="preserve">Quoniam, quę adſcribitur cum eodem tranſ-<lb/>uerſo CI, ſed cum recto, quod ſit minus CM, eſt minor <anchor type="note" xlink:href="" symbol="c"/> ipſa ANCO, quæ <anchor type="note" xlink:label="note-0078-09a" xlink:href="note-0078-09"/> <pb o="55" file="0079" n="79" rhead=""/> verò cum recto, quod excedat CM, eſt quidem maior <anchor type="note" xlink:href="" symbol="a"/> ipſa ANCO, ſed <anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/> veltota cadit extra ABCDE, cum Hyperbole, cuius regula IG <anchor type="note" xlink:href="" symbol="b"/> ſit ei cir- <anchor type="note" xlink:label="note-0079-02a" xlink:href="note-0079-02"/> cumſcripta, & </s> <s xml:id="echoid-s1885" xml:space="preserve">eò ampliùs ea, quæ cum recto quod excedat CG; </s> <s xml:id="echoid-s1886" xml:space="preserve">vel ſaltem <lb/>ſecat Hyperbolen ABCD ſupra portionis baſim AE <anchor type="note" xlink:href="" symbol="c"/> ſi rectum cadat inter <anchor type="note" xlink:label="note-0079-03a" xlink:href="note-0079-03"/> M, & </s> <s xml:id="echoid-s1887" xml:space="preserve">G. </s> <s xml:id="echoid-s1888" xml:space="preserve">Vnde hæc Hyperbolæ portio ANCOE eſt _MAXIMA_ inſcripta <lb/>quæſita, cum dato tranſuerſo CI. </s> <s xml:id="echoid-s1889" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s1890" xml:space="preserve">c.</s> <s xml:id="echoid-s1891" xml:space="preserve"/> </p> <div xml:id="echoid-div172" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0078-09" xlink:href="note-0078-09a" xml:space="preserve">2. corol. <lb/>prop. 19. <lb/>huius.</note> <note symbol="a" position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="b" position="right" xlink:label="note-0079-02" xlink:href="note-0079-02a" xml:space="preserve">24. h.</note> <note symbol="c" position="right" xlink:label="note-0079-03" xlink:href="note-0079-03a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s1892" xml:space="preserve">Siautem inſcribẽda ſit _MAXIMA_ Hyperbolæ portio cum dato recto CM, <lb/>quod ſit minus recto CG (cum æ quali enim, vel maiori ſemper eſſet circum-<lb/>ſcripta) iuncta LM, & </s> <s xml:id="echoid-s1893" xml:space="preserve">producta vſque ad occurſum cum diametro in I; </s> <s xml:id="echoid-s1894" xml:space="preserve">cum <lb/>tranſuerſo latere CI, ac recto CM adſcribatur <anchor type="note" xlink:href="" symbol="d"/> per C Hyperbole ANCOE, <anchor type="note" xlink:label="note-0079-04a" xlink:href="note-0079-04"/> quæ ſecabit <anchor type="note" xlink:href="" symbol="e"/> Hyperbolæ portionem ABCD in A & </s> <s xml:id="echoid-s1895" xml:space="preserve">E, eique erit inſcripta.</s> <s xml:id="echoid-s1896" xml:space="preserve"> <anchor type="note" xlink:label="note-0079-05a" xlink:href="note-0079-05"/> Dico hanc eſſe _MAXIMAM_ quæſitam. </s> <s xml:id="echoid-s1897" xml:space="preserve">Quoniam quæ adſcribitur per C cum <lb/>eodem recto CM, ſed cum tranſuerſo, quod excedat CI, minor <anchor type="note" xlink:href="" symbol="f"/> eſt Hyper- <anchor type="note" xlink:label="note-0079-06a" xlink:href="note-0079-06"/> bola ANCO, quæ verò cum tranſuerſo, quod minus ſit ipſo CI, & </s> <s xml:id="echoid-s1898" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> quidem <anchor type="note" xlink:label="note-0079-07a" xlink:href="note-0079-07"/> maior eadem ANCO, ſed omninò ſecat <anchor type="note" xlink:href="" symbol="h"/> Hyperbolen ABCDE ſupra appli- <anchor type="note" xlink:label="note-0079-08a" xlink:href="note-0079-08"/> catam AE. </s> <s xml:id="echoid-s1899" xml:space="preserve">Eſt igitur huiuſmodi Hyperbolæ portio ANCO _MAXIMA_ in-<lb/>ſcripta quæſita cum dato recto CM. </s> <s xml:id="echoid-s1900" xml:space="preserve">Quod erat ſecundò, &</s> <s xml:id="echoid-s1901" xml:space="preserve">c.</s> <s xml:id="echoid-s1902" xml:space="preserve"/> </p> <div xml:id="echoid-div173" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-0079-04" xlink:href="note-0079-04a" xml:space="preserve">6. huius.</note> <note symbol="e" position="right" xlink:label="note-0079-05" xlink:href="note-0079-05a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="f" position="right" xlink:label="note-0079-06" xlink:href="note-0079-06a" xml:space="preserve">3. Corol. <lb/>19. huius.</note> <note symbol="g" position="right" xlink:label="note-0079-07" xlink:href="note-0079-07a" xml:space="preserve">ibidem.</note> <note symbol="h" position="right" xlink:label="note-0079-08" xlink:href="note-0079-08a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s1903" xml:space="preserve">Ampliùs, ſit data Hyperbolæ portio ANCOE, cuius verſum CI, rectum <lb/>CM, regula IML, baſis AE, ac diameter CI, cui oporteat per verticem C, <lb/>cum dato tranſuerſo CF, quod maius ſit CI _MINIMAM_ Hyperbolæ portio-<lb/>nem circumſcribere.</s> <s xml:id="echoid-s1904" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1905" xml:space="preserve">Producatur ſemi-baſis AH conueniens cum regula IM, in L, & </s> <s xml:id="echoid-s1906" xml:space="preserve">iungatur <lb/>FL contingentem CM ſecans in G, & </s> <s xml:id="echoid-s1907" xml:space="preserve">cum tranſuerſo CF, ac recto CG ad-<lb/>ſcribatur <anchor type="note" xlink:href="" symbol="i"/> per verticem C Hyperbole ABCDE, quæ occurret datæ Hyper- <anchor type="note" xlink:label="note-0079-09a" xlink:href="note-0079-09"/> bolæ ANCO in punctis A, E, eique erit circumſcripta ſupra baſim AE,& </s> <s xml:id="echoid-s1908" xml:space="preserve">erit <lb/>_MINIMA_ Hyperbolæ portio quæſita.</s> <s xml:id="echoid-s1909" xml:space="preserve"/> </p> <div xml:id="echoid-div174" type="float" level="2" n="5"> <note symbol="i" position="right" xlink:label="note-0079-09" xlink:href="note-0079-09a" xml:space="preserve">6. huius.</note> </div> <p> <s xml:id="echoid-s1910" xml:space="preserve">Quoniam, quæ adſcribitur cum eodem verſo CF, ſed cum recto maiore <lb/>ipſo CG, eſt quoque maior <anchor type="note" xlink:href="" symbol="l"/> Hyperbola ABCD, quę verò cum recto mino- <anchor type="note" xlink:label="note-0079-10a" xlink:href="note-0079-10"/> re ipſo CG, eſt quidem minor <anchor type="note" xlink:href="" symbol="m"/> eadem ABCD, ſed veltota cadit intra da- <anchor type="note" xlink:label="note-0079-11a" xlink:href="note-0079-11"/> tam ANCO <anchor type="note" xlink:href="" symbol="n"/> ſi nempe rectum æquale fuerit ipſo CM, & </s> <s xml:id="echoid-s1911" xml:space="preserve">eò magis ſi minus <anchor type="note" xlink:label="note-0079-12a" xlink:href="note-0079-12"/> eſſet CM; </s> <s xml:id="echoid-s1912" xml:space="preserve">vel ſaltem ſecat <anchor type="note" xlink:href="" symbol="o"/> Hyperbolen ANCO ſupra baſim AE, quando <anchor type="note" xlink:label="note-0079-13a" xlink:href="note-0079-13"/> rectum cadat inter CM, & </s> <s xml:id="echoid-s1913" xml:space="preserve">CG; </s> <s xml:id="echoid-s1914" xml:space="preserve">tunc enim harum regulæ ſe mutuò ſecarent, <lb/>ſupra eandem baſim AE. </s> <s xml:id="echoid-s1915" xml:space="preserve">Vnde Hyperbolæ portio ABCDE, eſt _MINIMA_ <lb/>circumſcripta quæſita cum dato tranſuerſo CF. </s> <s xml:id="echoid-s1916" xml:space="preserve">Quod tertiò, &</s> <s xml:id="echoid-s1917" xml:space="preserve">c.</s> <s xml:id="echoid-s1918" xml:space="preserve"/> </p> <div xml:id="echoid-div175" type="float" level="2" n="6"> <note symbol="l" position="right" xlink:label="note-0079-10" xlink:href="note-0079-10a" xml:space="preserve">1. Corol. <lb/>19. huius.</note> <note symbol="m" position="right" xlink:label="note-0079-11" xlink:href="note-0079-11a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="n" position="right" xlink:label="note-0079-12" xlink:href="note-0079-12a" xml:space="preserve">ibidem.</note> <note symbol="o" position="right" xlink:label="note-0079-13" xlink:href="note-0079-13a" xml:space="preserve">3. Corol. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1919" xml:space="preserve">Demùm eidem datæ Hyperbolæ ANCO, ſit circumſcribenda _MINIMA_ <lb/>Hyperbole cum dato recto CG, quod excedat datæ rectum CM. </s> <s xml:id="echoid-s1920" xml:space="preserve">Facta ea-<lb/>dem conſtructione, iungatur LG diametro occurrens in F, & </s> <s xml:id="echoid-s1921" xml:space="preserve">cum tranſuerſo <lb/>CF, ac dato recto CG <anchor type="note" xlink:href="" symbol="p"/> adſcribatur per C Hyperbole ABCDE, quæ datam <anchor type="note" xlink:label="note-0079-14a" xlink:href="note-0079-14"/> ſecabit in A, & </s> <s xml:id="echoid-s1922" xml:space="preserve">E <anchor type="note" xlink:href="" symbol="q"/> eique erit circumſcripta, & </s> <s xml:id="echoid-s1923" xml:space="preserve">erit _MINIMA_ quæſita. </s> <s xml:id="echoid-s1924" xml:space="preserve">Nam, <anchor type="note" xlink:label="note-0079-15a" xlink:href="note-0079-15"/> quæ cum eodem recto CG, ſed cum tranſuerſo, quod minus ſit CF, maior <lb/>eſt <anchor type="note" xlink:href="" symbol="r"/> ipſa ABCD, quę verò cum tranſuerſo, quod maius ſit ipſo CF, quale eſt <anchor type="note" xlink:label="note-0079-16a" xlink:href="note-0079-16"/> CP, eſt quidem <anchor type="note" xlink:href="" symbol="s"/> minor, ſed omnino ſecat, portionem ANCO ſupra baſim <anchor type="note" xlink:label="note-0079-17a" xlink:href="note-0079-17"/> AE, cum iuncta regula PG, & </s> <s xml:id="echoid-s1925" xml:space="preserve">producta, ſecet regulam IL ſupra ipſam baſim <lb/>AE. </s> <s xml:id="echoid-s1926" xml:space="preserve">Quare Hyperbolæ portio ABCD eſt _MINIMA_ circumſcripta quæſita <lb/>cum dato recto CG. </s> <s xml:id="echoid-s1927" xml:space="preserve">Quod tandem faciendum erat.</s> <s xml:id="echoid-s1928" xml:space="preserve"/> </p> <div xml:id="echoid-div176" type="float" level="2" n="7"> <note symbol="p" position="right" xlink:label="note-0079-14" xlink:href="note-0079-14a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="q" position="right" xlink:label="note-0079-15" xlink:href="note-0079-15a" xml:space="preserve">6. huius.</note> <note symbol="r" position="right" xlink:label="note-0079-16" xlink:href="note-0079-16a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="s" position="right" xlink:label="note-0079-17" xlink:href="note-0079-17a" xml:space="preserve">3. Corol. <lb/>19. huius.</note> </div> <pb o="56" file="0080" n="80" rhead=""/> </div> <div xml:id="echoid-div178" type="section" level="1" n="89"> <head xml:id="echoid-head94" xml:space="preserve">PROBL. XIV. PROP. XXIX.</head> <p> <s xml:id="echoid-s1929" xml:space="preserve">Datæ portioni circuli, vel Ellipſis, per eius verticem MAXI-<lb/>MAM Parabolæ portionem inſcribere; </s> <s xml:id="echoid-s1930" xml:space="preserve">& </s> <s xml:id="echoid-s1931" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1932" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1933" xml:space="preserve">Datæ portioni Parabolæ per eius verticem, cum dato recto, <lb/>quod excedat rectum datæ Parabolæ, vel cum dato tranſuerſo, <lb/>quod maius ſit diametro datæ portionis MINIMAM Ellipſis por-<lb/>tionem circumſcribere.</s> <s xml:id="echoid-s1934" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1935" xml:space="preserve">SIt data circuli, aut Ellipſis portio ABC, cuius diameter ſit BE, baſis AC. <lb/></s> <s xml:id="echoid-s1936" xml:space="preserve">Oporter per eius verticem B, _MAXIMAM_ Parabolæ portionem inſcri-<lb/>bere.</s> <s xml:id="echoid-s1937" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1938" xml:space="preserve">Sit BF tranſuerſum latus dati circuli, vel <lb/> <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> Ellipſis, BG rectum, & </s> <s xml:id="echoid-s1939" xml:space="preserve">FG regula, cui pro-<lb/>ducta AE occurrat in H, & </s> <s xml:id="echoid-s1940" xml:space="preserve">per H agatur LHI <lb/>ipſi BF ęquidiſtans, & </s> <s xml:id="echoid-s1941" xml:space="preserve">cum recto BI, per ver-<lb/>ticem B adſcribatur <anchor type="note" xlink:href="" symbol="a"/> portioni ADBC Para- <anchor type="note" xlink:label="note-0080-01a" xlink:href="note-0080-01"/> bole AMBC, quæ per extrema A, C <anchor type="note" xlink:href="" symbol="b"/> tran- <anchor type="note" xlink:label="note-0080-02a" xlink:href="note-0080-02"/> ſibit, ac datæ portioni ſupra baſim AC erit <lb/>inſcripta, & </s> <s xml:id="echoid-s1942" xml:space="preserve">erit _MAXIMA_: </s> <s xml:id="echoid-s1943" xml:space="preserve">quoniam, quæ <lb/>adſcribitur cum recto, quod minus ſit BI mi-<lb/>nor <anchor type="note" xlink:href="" symbol="c"/> eſt ipſa AMBC, quæ verò cum recto, <anchor type="note" xlink:label="note-0080-03a" xlink:href="note-0080-03"/> quod excedat BI, veltota cadit extra Ellipſis <lb/>portionem ADB, ſinempe <anchor type="note" xlink:href="" symbol="d"/> eius rectum ſit <anchor type="note" xlink:label="note-0080-04a" xlink:href="note-0080-04"/> idem cum recto BG, & </s> <s xml:id="echoid-s1944" xml:space="preserve">eo magis ſi ipſum ex-<lb/>cedat; </s> <s xml:id="echoid-s1945" xml:space="preserve">vel ad minus ſecat datam portionem ſupra baſim AC, ſi Parabolę re-<lb/>ctum cadat inter I, & </s> <s xml:id="echoid-s1946" xml:space="preserve">G, vt in N. </s> <s xml:id="echoid-s1947" xml:space="preserve">Nam <anchor type="note" xlink:href="" symbol="e"/> eius regula ex N ducta æquidiſtan- <anchor type="note" xlink:label="note-0080-05a" xlink:href="note-0080-05"/> ter ipſi IH omninò ſecat Ellipſis regulam HG ſupra baſim AC. </s> <s xml:id="echoid-s1948" xml:space="preserve">Quare Pa-<lb/>rabolæ portio AMBC eſt _MAXIMA_ inſcripta quæſita. </s> <s xml:id="echoid-s1949" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s1950" xml:space="preserve">c.</s> <s xml:id="echoid-s1951" xml:space="preserve"/> </p> <div xml:id="echoid-div178" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0080-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">5. huius.</note> <note symbol="b" position="left" xlink:label="note-0080-02" xlink:href="note-0080-02a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="c" position="left" xlink:label="note-0080-03" xlink:href="note-0080-03a" xml:space="preserve">2. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="d" position="left" xlink:label="note-0080-04" xlink:href="note-0080-04a" xml:space="preserve">20. h.</note> <note symbol="e" position="left" xlink:label="note-0080-05" xlink:href="note-0080-05a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s1952" xml:space="preserve">Iam ſit data Parabolæ portio AMBC, cuius rectum BI, regula IL, diame-<lb/>ter BE, baſis AC, & </s> <s xml:id="echoid-s1953" xml:space="preserve">per eius verticem B oporteat _MINIMAM_ Ellipſis por-<lb/>tionem ei circumſcribere cum dato recto BG, quod excedat rectum datæ <lb/>portionis.</s> <s xml:id="echoid-s1954" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1955" xml:space="preserve">Conueniat applicata AE cum regula IL in H, iunctaq; </s> <s xml:id="echoid-s1956" xml:space="preserve">GH, & </s> <s xml:id="echoid-s1957" xml:space="preserve">producta, <lb/>occurrat portionis diametro in F (ſecans enim vnam parallelarum IH, ſecat <lb/>alteram BE:) </s> <s xml:id="echoid-s1958" xml:space="preserve">cum tranſuerſo autem BF, ac dato recto BG <anchor type="note" xlink:href="" symbol="f"/> adſcribatur per <anchor type="note" xlink:label="note-0080-06a" xlink:href="note-0080-06"/> B Ellipſis ADBC, quæ datę Parabolæ AMB occurret in A, & </s> <s xml:id="echoid-s1959" xml:space="preserve">C, & </s> <s xml:id="echoid-s1960" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit <anchor type="note" xlink:label="note-0080-07a" xlink:href="note-0080-07"/> circumſcripta, quàm dico eſſe _MINIMAM_. </s> <s xml:id="echoid-s1961" xml:space="preserve">Nam Ellipſis quæ adſcribitur <lb/>per B, cum eodem recto BG, ſed cum tranſuerſo, quod excedat BF, maior <lb/>eſt <anchor type="note" xlink:href="" symbol="h"/> ipſa ADB; </s> <s xml:id="echoid-s1962" xml:space="preserve">quæ verò adſcribitur cum tranſuerſo, quod minus ſit ipſo <anchor type="note" xlink:label="note-0080-08a" xlink:href="note-0080-08"/> BF, eſt quidem <anchor type="note" xlink:href="" symbol="i"/> minor eadem ADB, ſed omnino ſecat Parabolen AMBC <anchor type="note" xlink:label="note-0080-09a" xlink:href="note-0080-09"/> ſupra baſim AC, <anchor type="note" xlink:href="" symbol="l"/> cum & </s> <s xml:id="echoid-s1963" xml:space="preserve">ipſarum regulę ſe mutuò ſecent ſupra eandem AC.</s> <s xml:id="echoid-s1964" xml:space="preserve"> <anchor type="note" xlink:label="note-0080-10a" xlink:href="note-0080-10"/> Quare Ellipſis portio ADBC eſt _MINIMA_ circumſcripta quæſita cum dato <lb/>recto BG. </s> <s xml:id="echoid-s1965" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s1966" xml:space="preserve">c.</s> <s xml:id="echoid-s1967" xml:space="preserve"/> </p> <div xml:id="echoid-div179" type="float" level="2" n="2"> <note symbol="f" position="left" xlink:label="note-0080-06" xlink:href="note-0080-06a" xml:space="preserve">7. huius.</note> <note symbol="g" position="left" xlink:label="note-0080-07" xlink:href="note-0080-07a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="h" position="left" xlink:label="note-0080-08" xlink:href="note-0080-08a" xml:space="preserve">4. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="i" position="left" xlink:label="note-0080-09" xlink:href="note-0080-09a" xml:space="preserve">ibidem.</note> <note symbol="l" position="left" xlink:label="note-0080-10" xlink:href="note-0080-10a" xml:space="preserve">1. Corol. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s1968" xml:space="preserve">Sit tandem circumſcribenda datæ portioni Parabolicæ AMB _MINIMA_ <pb o="57" file="0081" n="81" rhead=""/> Ellipſis portio per verticem B, cum dato tranſuerſo BF, excedent, diame-<lb/>trum BE.</s> <s xml:id="echoid-s1969" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1970" xml:space="preserve">Iungatur FH, & </s> <s xml:id="echoid-s1971" xml:space="preserve">producatur, contingentem BI ſecans in G, & </s> <s xml:id="echoid-s1972" xml:space="preserve">cum dato <lb/>tranſuerſo BF, ac recto BG <anchor type="note" xlink:href="" symbol="a"/> adſcribatur per B Ellipſis portio ADBC, quæ <anchor type="note" xlink:label="note-0081-01a" xlink:href="note-0081-01"/> item datæ portioni AMB occurret in punctis A, C <anchor type="note" xlink:href="" symbol="b"/> eritque circumſcripta:</s> <s xml:id="echoid-s1973" xml:space="preserve"> <anchor type="note" xlink:label="note-0081-02a" xlink:href="note-0081-02"/> Nam quæ cum eodem tranſuerſo BF adſcribitur, ſed cum recto maiore ipſo <lb/>BG eſt <anchor type="note" xlink:href="" symbol="c"/> quoque maior Ellipſi ADB; </s> <s xml:id="echoid-s1974" xml:space="preserve">quæ verò cum recto, quod deficiat à BG eſt <anchor type="note" xlink:href="" symbol="d"/> quidem minor ipſa ADB, ſed vel tota cadit intra AMB, quãdo re- <anchor type="note" xlink:label="note-0081-03a" xlink:href="note-0081-03"/> ctum Ellipſis <anchor type="note" xlink:href="" symbol="e"/> idem fuerit cum recto Parabolæ BI, & </s> <s xml:id="echoid-s1975" xml:space="preserve">eò magis cum fuerit minus; </s> <s xml:id="echoid-s1976" xml:space="preserve">vel ſaltem ſecat <anchor type="note" xlink:href="" symbol="f"/> Parabolen AMBC ſupra applicatam AC, cum re- <anchor type="note" xlink:label="note-0081-04a" xlink:href="note-0081-04"/> ctum cadat inter BI, & </s> <s xml:id="echoid-s1977" xml:space="preserve">BG, quale eſt BN; </s> <s xml:id="echoid-s1978" xml:space="preserve">nam iuncta regula FN ſecat om-<lb/> <anchor type="note" xlink:label="note-0081-05a" xlink:href="note-0081-05"/> nino regulam IH ſupra eandem AC. </s> <s xml:id="echoid-s1979" xml:space="preserve">Quare huiuſmodi portio Elliptica <lb/> <anchor type="note" xlink:label="note-0081-06a" xlink:href="note-0081-06"/> ADBC, eſt _MINIMA_ circumſcripta quæſita cum dato tranſuerſo BF. </s> <s xml:id="echoid-s1980" xml:space="preserve">Quod <lb/>tandem erat faciendum.</s> <s xml:id="echoid-s1981" xml:space="preserve"/> </p> <div xml:id="echoid-div180" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0081-01" xlink:href="note-0081-01a" xml:space="preserve">7. huius.</note> <note symbol="b" position="right" xlink:label="note-0081-02" xlink:href="note-0081-02a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="c" position="right" xlink:label="note-0081-03" xlink:href="note-0081-03a" xml:space="preserve">4. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="d" position="right" xlink:label="note-0081-04" xlink:href="note-0081-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0081-05" xlink:href="note-0081-05a" xml:space="preserve">20. h.</note> <note symbol="f" position="right" xlink:label="note-0081-06" xlink:href="note-0081-06a" xml:space="preserve">1. Co-<lb/>roll. prop. <lb/>19. huius.</note> </div> </div> <div xml:id="echoid-div182" type="section" level="1" n="90"> <head xml:id="echoid-head95" xml:space="preserve">PROBL. XV. PROP. XXX.</head> <p> <s xml:id="echoid-s1982" xml:space="preserve">Datæ portioni circuli, vel Ellipſis, cum dato quocunque tranſ-<lb/>uerſo latere, vel cum dato recto, quod minus ſit latitudine ſemi-ap-<lb/>plicatæ baſis portionis, per eius verticem MAXIMAM Hyperbo-<lb/>lę portionem inſcribere; </s> <s xml:id="echoid-s1983" xml:space="preserve">& </s> <s xml:id="echoid-s1984" xml:space="preserve">è contra.</s> <s xml:id="echoid-s1985" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1986" xml:space="preserve">Datæ portioni Hyperbolæ, cum dato quocunque tranſuerſo la-<lb/>tere, quod maius ſit diametro datæ portionis, vel cum dato recto, <lb/>quod excedat prædictam latitudinem, per eius verticem MINI-<lb/>MAM Ellipſis portionem circumſcribere.</s> <s xml:id="echoid-s1987" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1988" xml:space="preserve">SIt data circuli, aut Ellipſis portio AB-<lb/> <anchor type="figure" xlink:label="fig-0081-01a" xlink:href="fig-0081-01"/> CD, cui is diameter CE, baſis AD, <lb/>tranſuerſum latus CE, rectum CG, & </s> <s xml:id="echoid-s1989" xml:space="preserve">re-<lb/>gula F G. </s> <s xml:id="echoid-s1990" xml:space="preserve">Oportet per eius verticem C, <lb/>cum dato quocunque tranſuerſo CI _MA-_ <lb/>_XIMAM_ Hyperbolę portionem inſcribere.</s> <s xml:id="echoid-s1991" xml:space="preserve"/> </p> <div xml:id="echoid-div182" type="float" level="2" n="1"> <figure xlink:label="fig-0081-01" xlink:href="fig-0081-01a"> <image file="0081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0081-01"/> </figure> </div> <p> <s xml:id="echoid-s1992" xml:space="preserve">Producatur AE, vſq; </s> <s xml:id="echoid-s1993" xml:space="preserve">ad occurſum cum <lb/>regula FG in L, & </s> <s xml:id="echoid-s1994" xml:space="preserve">iungatur IL contingen-<lb/>tem CG ſecans in M, & </s> <s xml:id="echoid-s1995" xml:space="preserve">cum dato tranſ-<lb/>uerſo CI, ac recto CM adſcribatur <anchor type="note" xlink:href="" symbol="a"/> per C <anchor type="note" xlink:label="note-0081-07a" xlink:href="note-0081-07"/> Hyperbolæ portio ANCD, quæ datæ por-<lb/>tion; </s> <s xml:id="echoid-s1996" xml:space="preserve">ABCD occurret in A, & </s> <s xml:id="echoid-s1997" xml:space="preserve">D, eritq; </s> <s xml:id="echoid-s1998" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> <anchor type="note" xlink:label="note-0081-08a" xlink:href="note-0081-08"/> inſcripta; </s> <s xml:id="echoid-s1999" xml:space="preserve">quàm dico eſſe _MAXIMAM_: </s> <s xml:id="echoid-s2000" xml:space="preserve">nã <lb/>quę adſcribitur cum eodem tranſuerſo CI, <lb/>ſed cum recto minore ipſo CM, eſt quoq; <lb/></s> <s xml:id="echoid-s2001" xml:space="preserve">minor <anchor type="note" xlink:href="" symbol="c"/> Hyperbola ANCD, quę verò cum <anchor type="note" xlink:label="note-0081-09a" xlink:href="note-0081-09"/> recto maiore CM, veltota cadit extra da-<lb/>tam Ellipſim ABCD, quando <anchor type="note" xlink:href="" symbol="d"/> videlicet <anchor type="note" xlink:label="note-0081-10a" xlink:href="note-0081-10"/> eius rectum latus æquet ipſum CG, & </s> <s xml:id="echoid-s2002" xml:space="preserve">eò magis ſi rectum excedat CG; </s> <s xml:id="echoid-s2003" xml:space="preserve">vel <pb o="58" file="0082" n="82" rhead=""/> ſaltem ſecat portionem ABC ſupra baſim AD, ſi rectum cadat inter M, & </s> <s xml:id="echoid-s2004" xml:space="preserve"><lb/>G, quale eſt CO nam <anchor type="note" xlink:href="" symbol="a"/> iuncta regula IO, & </s> <s xml:id="echoid-s2005" xml:space="preserve">producta omnino ſecat regulam <anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/> GL ſupra eandem AD. </s> <s xml:id="echoid-s2006" xml:space="preserve">Quare Hyperbolæ portio ANCD eſt _MAXIMA_ in-<lb/>ſcripta quæſita cum dato tranſuerſo CI. </s> <s xml:id="echoid-s2007" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s2008" xml:space="preserve">c.</s> <s xml:id="echoid-s2009" xml:space="preserve"/> </p> <div xml:id="echoid-div183" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0081-07" xlink:href="note-0081-07a" xml:space="preserve">6. huius.</note> <note symbol="b" position="right" xlink:label="note-0081-08" xlink:href="note-0081-08a" xml:space="preserve">1. Co-<lb/>roll prop. <lb/>19. huius.</note> <note symbol="c" position="right" xlink:label="note-0081-09" xlink:href="note-0081-09a" xml:space="preserve">2. corol. <lb/>prop. 19. <lb/>huius.</note> <note symbol="d" position="right" xlink:label="note-0081-10" xlink:href="note-0081-10a" xml:space="preserve">20. h.</note> <note symbol="a" position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">1. Co-<lb/>roll. prop <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s2010" xml:space="preserve">Iam eidem Ellipticæ portioni ABCD inſcribenda ſit _MAXIMA_ Hyperbo-<lb/>læ portio cum dato recto CM, quod tamen ſit minus latitudine EL, ſemiap-<lb/>plicatæ AE (ſi enim ei æquale, vel maius eſſet, iuncta regula LM nunquam <lb/>cum diametro EC conueniret) Supra C.</s> <s xml:id="echoid-s2011" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2012" xml:space="preserve">Iungatur I.</s> <s xml:id="echoid-s2013" xml:space="preserve">M, quę ideo producta occur-<lb/> <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/> ret diametro in I, & </s> <s xml:id="echoid-s2014" xml:space="preserve">cum tranſuerſo IC, <lb/>datoq; </s> <s xml:id="echoid-s2015" xml:space="preserve">recto CM adſcribatur <anchor type="note" xlink:href="" symbol="b"/> per C Hy- <anchor type="note" xlink:label="note-0082-02a" xlink:href="note-0082-02"/> perbolæ portio ANCD, quæ datæ portio-<lb/>ni ABC occurret in A, & </s> <s xml:id="echoid-s2016" xml:space="preserve">D, & </s> <s xml:id="echoid-s2017" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit inſcri- <anchor type="note" xlink:label="note-0082-03a" xlink:href="note-0082-03"/> pta; </s> <s xml:id="echoid-s2018" xml:space="preserve">quàm dico eſſe _MAXIMAM_: </s> <s xml:id="echoid-s2019" xml:space="preserve">nam quę <lb/>adſcribitur cum eodem recto CM, ſed cum <lb/>tranſuerſo, quod excedat CI minor <anchor type="note" xlink:href="" symbol="d"/> eſt <anchor type="note" xlink:label="note-0082-04a" xlink:href="note-0082-04"/> ipſa ANC; </s> <s xml:id="echoid-s2020" xml:space="preserve">quæ verò cum tranſuerſo, quod <lb/>ſit minus CI, quale eſt CP, eſt <anchor type="note" xlink:href="" symbol="e"/> quidem <anchor type="note" xlink:label="note-0082-05a" xlink:href="note-0082-05"/> maior ipſa ANC, ſed omnino ſecat datam <lb/>portionem ABC, ſupra baſim AD <anchor type="note" xlink:href="" symbol="f"/> cum <anchor type="note" xlink:label="note-0082-06a" xlink:href="note-0082-06"/> iuncta regula PM, & </s> <s xml:id="echoid-s2021" xml:space="preserve">producta, omnino <lb/>ſecet regulam GL ſupra eandem AD. </s> <s xml:id="echoid-s2022" xml:space="preserve">Qua-<lb/>re huiuſmodi Hyperbolæ portio ANCD, <lb/>eſt _MAXIMA_ inſcripta cũ dato recto CM. <lb/></s> <s xml:id="echoid-s2023" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s2024" xml:space="preserve">c.</s> <s xml:id="echoid-s2025" xml:space="preserve"/> </p> <div xml:id="echoid-div184" type="float" level="2" n="3"> <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a"> <image file="0082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0082-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0082-02" xlink:href="note-0082-02a" xml:space="preserve">6. huius.</note> <note symbol="c" position="left" xlink:label="note-0082-03" xlink:href="note-0082-03a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="d" position="left" xlink:label="note-0082-04" xlink:href="note-0082-04a" xml:space="preserve">4. Co-<lb/>roll. 19. h.</note> <note symbol="e" position="left" xlink:label="note-0082-05" xlink:href="note-0082-05a" xml:space="preserve">ibidem.</note> <note symbol="f" position="left" xlink:label="note-0082-06" xlink:href="note-0082-06a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2026" xml:space="preserve">Ampliùs ſit data Hyperbolę portio AN <lb/>CD, cuius tranſuerſum CI, rectum CM, regula IM, diameter CE, baſis <lb/>AD: </s> <s xml:id="echoid-s2027" xml:space="preserve">oportet per verticem C _MINIMAM_ Ellipſis portionem circumſcribere <lb/>cum dato tranſuerſo CF, quod tamen excedat diametrum CE.</s> <s xml:id="echoid-s2028" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2029" xml:space="preserve">Producatur item AE occurrens regulæ IM in L, & </s> <s xml:id="echoid-s2030" xml:space="preserve">iungatur FL, quæ pro-<lb/>ducta conueniat cum contingente CM in G, & </s> <s xml:id="echoid-s2031" xml:space="preserve">cum dato tranſuerſo CF, ac <lb/>recto CG adſcribatur <anchor type="note" xlink:href="" symbol="g"/> per C Ellipſis portio ABCD, quæ datæ Hyperbolæ <anchor type="note" xlink:label="note-0082-07a" xlink:href="note-0082-07"/> occurret in A, D, eritque circumſcripta, & </s> <s xml:id="echoid-s2032" xml:space="preserve">erit _MINIMA_: </s> <s xml:id="echoid-s2033" xml:space="preserve">Nam quæ adſcri-<lb/>bitur cum eodem tranſuerſo CF, ſed cum recto, quod excedat CG eſt <anchor type="note" xlink:href="" symbol="h"/> ma- <anchor type="note" xlink:label="note-0082-08a" xlink:href="note-0082-08"/> ior ipſa ABC, quæ verò cum recto, quod minus ſit CG; </s> <s xml:id="echoid-s2034" xml:space="preserve">vel tota cadit intra <lb/>ANCD, tùm cum rectum æquet ipſum CM, & </s> <s xml:id="echoid-s2035" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> eò magis ſi ipſo ſit minus;</s> <s xml:id="echoid-s2036" xml:space="preserve"> <anchor type="note" xlink:label="note-0082-09a" xlink:href="note-0082-09"/> vel ſaltem ſecat portionem ANC ſupra baſim AD, quando <anchor type="note" xlink:href="" symbol="l"/> nempe rectum cadat inter CM, & </s> <s xml:id="echoid-s2037" xml:space="preserve">CG, quale eſt CO, nam iuncta regula FO, omnino ſecat <lb/> <anchor type="note" xlink:label="note-0082-10a" xlink:href="note-0082-10"/> Hyperbolæ regulam ML ſupra eandem AD. </s> <s xml:id="echoid-s2038" xml:space="preserve">Quapropter Ellipſis portio <lb/>ABCD, erit _MINIMA_ circumſcripta cũ dato tranſuerſo CI. </s> <s xml:id="echoid-s2039" xml:space="preserve">Quod tertiò, &</s> <s xml:id="echoid-s2040" xml:space="preserve">c.</s> <s xml:id="echoid-s2041" xml:space="preserve"/> </p> <div xml:id="echoid-div185" type="float" level="2" n="4"> <note symbol="g" position="left" xlink:label="note-0082-07" xlink:href="note-0082-07a" xml:space="preserve">7. huius.</note> <note symbol="h" position="left" xlink:label="note-0082-08" xlink:href="note-0082-08a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="i" position="left" xlink:label="note-0082-09" xlink:href="note-0082-09a" xml:space="preserve">1. Co-<lb/>rol. 19. h.</note> <note symbol="l" position="left" xlink:label="note-0082-10" xlink:href="note-0082-10a" xml:space="preserve">20. h.</note> </div> <p> <s xml:id="echoid-s2042" xml:space="preserve">Poſtremò, datis ijſdem, ſit circumſcribenda _MINIMA_ Ellipſis portio, cum <lb/>dato recto CG, quod tamen excedat latitudinem EL (ad hoc vt iuncta re-<lb/>gula GL cum diametro CE poſſit conuenire infra E) & </s> <s xml:id="echoid-s2043" xml:space="preserve">ipſa GL occurrat CE <lb/>in F, & </s> <s xml:id="echoid-s2044" xml:space="preserve">cum dato recto CG, ac tranſuerſo CF adſcribatur <anchor type="note" xlink:href="" symbol="m"/> per C Ellipſis <anchor type="note" xlink:label="note-0082-11a" xlink:href="note-0082-11"/> portio ABCD, quæ item datæ portioni occurret in A, & </s> <s xml:id="echoid-s2045" xml:space="preserve">D, eritq; </s> <s xml:id="echoid-s2046" xml:space="preserve"><anchor type="note" xlink:href="" symbol="n"/> circum- <anchor type="note" xlink:label="note-0082-12a" xlink:href="note-0082-12"/> ſcripta; </s> <s xml:id="echoid-s2047" xml:space="preserve">quàm dico eſſe _MINIMAM_: </s> <s xml:id="echoid-s2048" xml:space="preserve">quæ enim adſcribitur cum codem recto <lb/> <anchor type="note" xlink:label="note-0082-13a" xlink:href="note-0082-13"/> CG, ſed cum tranſuerſo, quod ſit maius CF, eſt etiam <anchor type="note" xlink:href="" symbol="o"/> maior ipſa ABC;</s> <s xml:id="echoid-s2049" xml:space="preserve"> <pb o="59" file="0083" n="83" rhead=""/> quæ verò cum tranſuerſo, quod deficiat à CF, eſt quidem <anchor type="note" xlink:href="" symbol="a"/> minor ipſa ABC, ſed <anchor type="note" xlink:label="note-0083-01a" xlink:href="note-0083-01"/> omnino ſecat Hyperbolæ portionem A N C ſupra baſim AD <anchor type="note" xlink:href="" symbol="b"/> cum & </s> <s xml:id="echoid-s2050" xml:space="preserve">iuncta re- <anchor type="note" xlink:label="note-0083-02a" xlink:href="note-0083-02"/> gula ſecet datæ portionis regulam M L ſupra A D. </s> <s xml:id="echoid-s2051" xml:space="preserve">Quare Ellipſis portio ABC <lb/>D eſt _MINIMA_ circumſcripta cum dato recto C E, Quod vltimò, &</s> <s xml:id="echoid-s2052" xml:space="preserve">c.</s> <s xml:id="echoid-s2053" xml:space="preserve"/> </p> <div xml:id="echoid-div186" type="float" level="2" n="5"> <note symbol="m" position="left" xlink:label="note-0082-11" xlink:href="note-0082-11a" xml:space="preserve">7. h.</note> <note symbol="n" position="left" xlink:label="note-0082-12" xlink:href="note-0082-12a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="o" position="left" xlink:label="note-0082-13" xlink:href="note-0082-13a" xml:space="preserve">4. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="a" position="right" xlink:label="note-0083-01" xlink:href="note-0083-01a" xml:space="preserve">4. Co-<lb/>roll. 19. h.</note> <note symbol="b" position="right" xlink:label="note-0083-02" xlink:href="note-0083-02a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> </div> <div xml:id="echoid-div188" type="section" level="1" n="91"> <head xml:id="echoid-head96" xml:space="preserve">PROBL. XVI. PROP. XXXI.</head> <p> <s xml:id="echoid-s2054" xml:space="preserve">Datæ portioni circuli, vel Ellipſis, cum dato tranſuerſo latere, <lb/>quod excedat verſum, vel cum dato recto, quod minus ſit recto datæ <lb/>portionis, maius verò latitudine ſemi-applicatæ baſis portionis, per <lb/>eius verticem MAXIMAM Ellipſis portionem inſcribere. </s> <s xml:id="echoid-s2055" xml:space="preserve">Item.</s> <s xml:id="echoid-s2056" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2057" xml:space="preserve">Datæ portioni circuli, vel Ellipſis, cum dato tranſuerſo, quod mi-<lb/>nus ſit tranſuerſo, ſed maius diametro datæ portionis, vel cum dato <lb/>recto, quod excedat rectum datæ portionis, per eius verticem MI-<lb/>NIMAM Ellipſis portionem circumſcribere.</s> <s xml:id="echoid-s2058" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2059" xml:space="preserve">SIt data circuli, vel Ellipſis portio A B C D, cuius diameter C E, baſis A D, <lb/>verſum CF, rectum CG, & </s> <s xml:id="echoid-s2060" xml:space="preserve">regula CF. </s> <s xml:id="echoid-s2061" xml:space="preserve">Oportet per verticẽ C _MAXIMAM_ <lb/>Ellipſis portionem inſcribere, cum dato tranſuerſo CH, quod ſit maius ipſo CF.</s> <s xml:id="echoid-s2062" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2063" xml:space="preserve">Applicata AE, & </s> <s xml:id="echoid-s2064" xml:space="preserve">producta occurrat regulæ FG <lb/> <anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/> in I; </s> <s xml:id="echoid-s2065" xml:space="preserve">iunctaque HI, conueniat producta cum con-<lb/>tingente C G in L, & </s> <s xml:id="echoid-s2066" xml:space="preserve">cum dato tranſuerſo CH, re-<lb/>ctoq; </s> <s xml:id="echoid-s2067" xml:space="preserve">CL adſcribatur <anchor type="note" xlink:href="" symbol="c"/> per C Ellipſis portio A M C <anchor type="note" xlink:label="note-0083-03a" xlink:href="note-0083-03"/> D, quæ per extrema baſis A D tranſibit <anchor type="note" xlink:href="" symbol="d"/> datæque <anchor type="note" xlink:label="note-0083-04a" xlink:href="note-0083-04"/> portioni erit inſcripta. </s> <s xml:id="echoid-s2068" xml:space="preserve">Iam dico hanc eſſe _MAXI-_ <lb/>_MAM_. </s> <s xml:id="echoid-s2069" xml:space="preserve">Nam quæ adſcribitur cum eodem verſo C <lb/>H, ſed cum recto, quod minus ſit ipſo C L, minor <lb/> <anchor type="note" xlink:label="note-0083-05a" xlink:href="note-0083-05"/> eſt <anchor type="note" xlink:href="" symbol="e"/> eſt ipſa A M C D; </s> <s xml:id="echoid-s2070" xml:space="preserve">quæ verò cum recto, quod excedat C L, eſt quidem <anchor type="note" xlink:href="" symbol="f"/> maior AMCD, ſed vel <anchor type="note" xlink:label="note-0083-06a" xlink:href="note-0083-06"/> tota cadit extra ABCD, tum <anchor type="note" xlink:href="" symbol="g"/> cum eius rectũ adæ- <anchor type="note" xlink:label="note-0083-07a" xlink:href="note-0083-07"/> quet CG, tũ cũ ipſum excedat; </s> <s xml:id="echoid-s2071" xml:space="preserve">vel ſaltim ſecat datã <lb/>portionem ABCD ſupra baſim AD quando <anchor type="note" xlink:href="" symbol="h"/> rectũ <anchor type="note" xlink:label="note-0083-08a" xlink:href="note-0083-08"/> cadat inter CL, & </s> <s xml:id="echoid-s2072" xml:space="preserve">C G, quale eſt C O, nam iuncta <lb/>regula HO, ſecat omnino regulã I G ſupra eandem <lb/>AD. </s> <s xml:id="echoid-s2073" xml:space="preserve">Vnde Ellipſis portio A M C D eſt _MAXIMA_ <lb/>inſcripta cum dato trãſuerſo CH. </s> <s xml:id="echoid-s2074" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s2075" xml:space="preserve">c.</s> <s xml:id="echoid-s2076" xml:space="preserve"/> </p> <div xml:id="echoid-div188" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0083-01"/> </figure> <note symbol="c" position="right" xlink:label="note-0083-03" xlink:href="note-0083-03a" xml:space="preserve">7. h.</note> <note symbol="d" position="right" xlink:label="note-0083-04" xlink:href="note-0083-04a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="e" position="right" xlink:label="note-0083-05" xlink:href="note-0083-05a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="f" position="right" xlink:label="note-0083-06" xlink:href="note-0083-06a" xml:space="preserve">ibidem.</note> <note symbol="g" position="right" xlink:label="note-0083-07" xlink:href="note-0083-07a" xml:space="preserve">20. h.</note> <note symbol="h" position="right" xlink:label="note-0083-08" xlink:href="note-0083-08a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2077" xml:space="preserve">Iam, ijſdem poſitis, oporteat cum dato recto C L, quod minus ſit recto CG; <lb/></s> <s xml:id="echoid-s2078" xml:space="preserve">maior verò latitudine E I, _MAXIMAM_ Ellipſis portionem inſcribere.</s> <s xml:id="echoid-s2079" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2080" xml:space="preserve">Iungatur L I, & </s> <s xml:id="echoid-s2081" xml:space="preserve">producatur, conueniens cum diametro C E in H, & </s> <s xml:id="echoid-s2082" xml:space="preserve">cum <lb/>tranſuerſo CH, datoque recto CL <anchor type="note" xlink:href="" symbol="i"/> adſcribatur per C Ellipſis portio A M C D, <anchor type="note" xlink:label="note-0083-09a" xlink:href="note-0083-09"/> quæ datæ portioni <anchor type="note" xlink:href="" symbol="l"/> occurret in A, & </s> <s xml:id="echoid-s2083" xml:space="preserve">D, eique erit inſcripta. </s> <s xml:id="echoid-s2084" xml:space="preserve">Dico hanc eſſe <anchor type="note" xlink:label="note-0083-10a" xlink:href="note-0083-10"/> _MAXIMAM_ quæſitam.</s> <s xml:id="echoid-s2085" xml:space="preserve"/> </p> <div xml:id="echoid-div189" type="float" level="2" n="2"> <note symbol="i" position="right" xlink:label="note-0083-09" xlink:href="note-0083-09a" xml:space="preserve">7. h.</note> <note symbol="l" position="right" xlink:label="note-0083-10" xlink:href="note-0083-10a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2086" xml:space="preserve">Quæ enim adſcribitur cũ eodem recto CL, ſed cum verſo, quod minus ſit ipſo <lb/> <anchor type="note" xlink:label="note-0083-11a" xlink:href="note-0083-11"/> CH eſt <anchor type="note" xlink:href="" symbol="m"/> minor portione AMCD; </s> <s xml:id="echoid-s2087" xml:space="preserve">quæ autem cum verſo, quod excedat C H, quale eſt C P, eſt quidem <anchor type="note" xlink:href="" symbol="n"/> maior ipſa AMCD, ſed omnino ſecat Ellipſim A B <anchor type="note" xlink:label="note-0083-12a" xlink:href="note-0083-12"/> C D ſupra baſim A D <anchor type="note" xlink:href="" symbol="o"/> cum iuncta regula P L, ſecet regulam I G ſupra eandem <anchor type="note" xlink:label="note-0083-13a" xlink:href="note-0083-13"/> <pb o="60" file="0084" n="84" rhead=""/> A D. </s> <s xml:id="echoid-s2088" xml:space="preserve">Quave<unsure/> Ellipſis portio A M C eſt _MAXIMA_ inſcripta cum dato recto C L. <lb/></s> <s xml:id="echoid-s2089" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s2090" xml:space="preserve">c.</s> <s xml:id="echoid-s2091" xml:space="preserve"/> </p> <div xml:id="echoid-div190" type="float" level="2" n="3"> <note symbol="m" position="right" xlink:label="note-0083-11" xlink:href="note-0083-11a" xml:space="preserve">4. Co-<lb/>roll. 19. h.</note> <note symbol="n" position="right" xlink:label="note-0083-12" xlink:href="note-0083-12a" xml:space="preserve">ibid.</note> <note symbol="o" position="right" xlink:label="note-0083-13" xlink:href="note-0083-13a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2092" xml:space="preserve">SIt verò data Ellipſis portio AMCD, cuius tranſuerſum CH, rectum C L, re-<lb/>gula LH, baſis A D, & </s> <s xml:id="echoid-s2093" xml:space="preserve">diameter C E: </s> <s xml:id="echoid-s2094" xml:space="preserve">oportet per verticem C _MINIMAM_ <lb/>Ellipſis portionem circumſcribere, cum dato tranſuerſo C F, quod minus ſit <lb/>verſo CH datæ portionis, maius verò eius diametro C E.</s> <s xml:id="echoid-s2095" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2096" xml:space="preserve">Producta ſemi - applicata A E, occurrat regulæ <lb/> <anchor type="figure" xlink:label="fig-0084-01a" xlink:href="fig-0084-01"/> LH in I, & </s> <s xml:id="echoid-s2097" xml:space="preserve">iuncta F I occurrat contingenti C L in <lb/>G, & </s> <s xml:id="echoid-s2098" xml:space="preserve">cum tranſuerſo dato C F, cumque recto C G <lb/>adſcribatur <anchor type="note" xlink:href="" symbol="a"/> per C Ellipſis portio A B C D, quæ <anchor type="note" xlink:label="note-0084-01a" xlink:href="note-0084-01"/> item per A, & </s> <s xml:id="echoid-s2099" xml:space="preserve">D tranſibit, & </s> <s xml:id="echoid-s2100" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> portioni AMC erit <anchor type="note" xlink:label="note-0084-02a" xlink:href="note-0084-02"/> circumſcripta, quàm dico eſſe _MINIMAM_. </s> <s xml:id="echoid-s2101" xml:space="preserve">Quæ-<lb/>libet enim adſcripta Ellipſis cum eodem tranſuerſo <lb/>C F, ſed cum recto, quod maius ſit ipſo C G, eſt <lb/>maior <anchor type="note" xlink:href="" symbol="c"/> eadem ABCD; </s> <s xml:id="echoid-s2102" xml:space="preserve">quæ verò cum recto, quod <anchor type="note" xlink:label="note-0084-03a" xlink:href="note-0084-03"/> minus ſit CG eſt quidem <anchor type="note" xlink:href="" symbol="d"/> minor eadem A B C, ſed <anchor type="note" xlink:label="note-0084-04a" xlink:href="note-0084-04"/> vel tota cadit intra datam AMCD, tum, cum rectũ <lb/>idem fuerit cum recto CL, aut ipſo minus; </s> <s xml:id="echoid-s2103" xml:space="preserve">vel <anchor type="note" xlink:href="" symbol="e"/> ſal- <anchor type="note" xlink:label="note-0084-05a" xlink:href="note-0084-05"/> tem ſecat portionem AMC ſupra baſim AD, quan-<lb/>do nempe illius rectum cadat inter C L, & </s> <s xml:id="echoid-s2104" xml:space="preserve">C G, <lb/>quale eſt C O, nam iuncta regula O F, omnino ſe-<lb/>cat regulam L H ſupra eandem applicatam A D. <lb/></s> <s xml:id="echoid-s2105" xml:space="preserve">Quare huiuſmodi portio Elliptica ABCD erit _MI-_ <lb/>_NIMA_ cir cumſcripta cum dato tranſuerſo CF. </s> <s xml:id="echoid-s2106" xml:space="preserve">Quod tertiò, &</s> <s xml:id="echoid-s2107" xml:space="preserve">c.</s> <s xml:id="echoid-s2108" xml:space="preserve"/> </p> <div xml:id="echoid-div191" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0084-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0084-01" xlink:href="note-0084-01a" xml:space="preserve">7. hu.</note> <note symbol="b" position="left" xlink:label="note-0084-02" xlink:href="note-0084-02a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="c" position="left" xlink:label="note-0084-03" xlink:href="note-0084-03a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="d" position="left" xlink:label="note-0084-04" xlink:href="note-0084-04a" xml:space="preserve">ibid.</note> <note symbol="e" position="left" xlink:label="note-0084-05" xlink:href="note-0084-05a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2109" xml:space="preserve">Sit tandem circumſcribenda portioni AMC _MINIMA_ Ellipſis portio cum <lb/>dato recto C G, quod tamen ſuperet rectum C L.</s> <s xml:id="echoid-s2110" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2111" xml:space="preserve">Iungatur G I, & </s> <s xml:id="echoid-s2112" xml:space="preserve">producatur, donec conueniat cum diametro in F, & </s> <s xml:id="echoid-s2113" xml:space="preserve">cum <lb/>tranſuerſo C F, datoque recto C G adſcribatur <anchor type="note" xlink:href="" symbol="f"/> per C, Elliptica portio ABC, <anchor type="note" xlink:label="note-0084-06a" xlink:href="note-0084-06"/> quæ pariter per A, & </s> <s xml:id="echoid-s2114" xml:space="preserve">D tranſibit <anchor type="note" xlink:href="" symbol="g"/> eritque datæ portioni circumſcripta: </s> <s xml:id="echoid-s2115" xml:space="preserve">inſuper <anchor type="note" xlink:label="note-0084-07a" xlink:href="note-0084-07"/> dico hanc eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s2116" xml:space="preserve"/> </p> <div xml:id="echoid-div192" type="float" level="2" n="5"> <note symbol="f" position="left" xlink:label="note-0084-06" xlink:href="note-0084-06a" xml:space="preserve">7. h.</note> <note symbol="g" position="left" xlink:label="note-0084-07" xlink:href="note-0084-07a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2117" xml:space="preserve">Ellipſis enim, quæ adſcribitur per C cum eodem recto C G, ſed cum tranſ-<lb/> <anchor type="note" xlink:label="note-0084-08a" xlink:href="note-0084-08"/> uerſo, quod excedat versũ CF eſt maior <anchor type="note" xlink:href="" symbol="h"/> ipſa ABCD, quæ verò cum trãſuerſo, quod minus ſit ipſo CF, quale eſt CR, eſt quidem <anchor type="note" xlink:href="" symbol="i"/> minor eadem A B C D, ſed <anchor type="note" xlink:label="note-0084-09a" xlink:href="note-0084-09"/> omnino ſecat portionem AMCD ſupra baſim <anchor type="note" xlink:href="" symbol="l"/> A D cum & </s> <s xml:id="echoid-s2118" xml:space="preserve">iuncta regula CR ſe- <anchor type="note" xlink:label="note-0084-10a" xlink:href="note-0084-10"/> cet datæ portionis regulam L I ſupra eandem baſim AD. </s> <s xml:id="echoid-s2119" xml:space="preserve">Quare Ellipſis portio <lb/>ABCD eſt _MINIMA_ circumſcripta cum dato recto CG. </s> <s xml:id="echoid-s2120" xml:space="preserve">Quod vltimò, &</s> <s xml:id="echoid-s2121" xml:space="preserve">c.</s> <s xml:id="echoid-s2122" xml:space="preserve"/> </p> <div xml:id="echoid-div193" type="float" level="2" n="6"> <note symbol="h" position="left" xlink:label="note-0084-08" xlink:href="note-0084-08a" xml:space="preserve">4. Co-<lb/>roll. 19. h.</note> <note symbol="i" position="left" xlink:label="note-0084-09" xlink:href="note-0084-09a" xml:space="preserve">ibidem.</note> <note symbol="l" position="left" xlink:label="note-0084-10" xlink:href="note-0084-10a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> </div> <div xml:id="echoid-div195" type="section" level="1" n="92"> <head xml:id="echoid-head97" xml:space="preserve">THEOR. XIII. PROP. XXXII.</head> <p> <s xml:id="echoid-s2123" xml:space="preserve">Parabolæ, vel Hyperbolę cum earum diametris, iuxta ordinatim ſe-<lb/>mi - applicatas ſunt ſemper ſimul recedentes, & </s> <s xml:id="echoid-s2124" xml:space="preserve">ad interuallum per-<lb/>ueniunt maius quolibet dato interuallo.</s> <s xml:id="echoid-s2125" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2126" xml:space="preserve">PRimum facilè conſtat ex 20. </s> <s xml:id="echoid-s2127" xml:space="preserve">ac 21. </s> <s xml:id="echoid-s2128" xml:space="preserve">primi Conic. </s> <s xml:id="echoid-s2129" xml:space="preserve">Secundum verò ſic.</s> <s xml:id="echoid-s2130" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2131" xml:space="preserve">Ducta enim cõtingente ex ſectionis vertice, quę quodam dato interuallo <lb/>ſit maior, atq; </s> <s xml:id="echoid-s2132" xml:space="preserve">ex eius termino ducta alia, quæ ipſi diametro ſit æquidiſtans, hæc <lb/> <anchor type="note" xlink:label="note-0084-11a" xlink:href="note-0084-11"/> omnino in vno tantùm puncto cum ſectione cõueniet, <anchor type="note" xlink:href="" symbol="m"/> à quo ſi agatur contin- <pb o="61" file="0085" n="85" rhead=""/> genti parallela, erit hæc vna ordinatim ad diametrum ſemi - applicatarum, <lb/>datumq; </s> <s xml:id="echoid-s2133" xml:space="preserve">interuallum ſuperabit: </s> <s xml:id="echoid-s2134" xml:space="preserve">vnde patet propoſitum. </s> <s xml:id="echoid-s2135" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s2136" xml:space="preserve">c.</s> <s xml:id="echoid-s2137" xml:space="preserve"/> </p> <div xml:id="echoid-div195" type="float" level="2" n="1"> <note symbol="m" position="left" xlink:label="note-0084-11" xlink:href="note-0084-11a" xml:space="preserve">26. pr. <lb/>Conic.</note> </div> </div> <div xml:id="echoid-div197" type="section" level="1" n="93"> <head xml:id="echoid-head98" xml:space="preserve">THEOR. IV. PROP. XXXIII.</head> <p> <s xml:id="echoid-s2138" xml:space="preserve">Parabolæ inæqualium laterum per eundem verticem ſimul adſcri-<lb/>ptæ, ſunt inter ſe nunquam alibi coeuntes, & </s> <s xml:id="echoid-s2139" xml:space="preserve">inſcripta eſt ea, cuius re-<lb/>ctum latus minus eſt, ſuntque, in infinitum productæ, iuxta intercepta <lb/>applicatarum ſegmenta ſemper magis recedentes, & </s> <s xml:id="echoid-s2140" xml:space="preserve">ad interuallum <lb/>perueniunt maius quolibet dato interuallo.</s> <s xml:id="echoid-s2141" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2142" xml:space="preserve">SInt duæ Parabolæ ABC, DBE, per eundem verticem B ſimul adſcriptę, qua-<lb/>rum communis diameter B H, & </s> <s xml:id="echoid-s2143" xml:space="preserve">rectum ſectionis ABC ſit linea B F, D B E <lb/>verò ſit minor B G. </s> <s xml:id="echoid-s2144" xml:space="preserve">Dico primùm has nunquam alibi ſimul conuenire, & </s> <s xml:id="echoid-s2145" xml:space="preserve">DBE <lb/>inſcriptam eſſe. </s> <s xml:id="echoid-s2146" xml:space="preserve">Hoc enim iam patet ex 2. </s> <s xml:id="echoid-s2147" xml:space="preserve">Coroll. </s> <s xml:id="echoid-s2148" xml:space="preserve">19. </s> <s xml:id="echoid-s2149" xml:space="preserve">huius.</s> <s xml:id="echoid-s2150" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2151" xml:space="preserve">Ampliùs, dico has in infinitum productas ſemper <lb/> <anchor type="figure" xlink:label="fig-0085-01a" xlink:href="fig-0085-01"/> eſſe inter ſe magis recedentes.</s> <s xml:id="echoid-s2152" xml:space="preserve"/> </p> <div xml:id="echoid-div197" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0085-01"/> </figure> </div> <p> <s xml:id="echoid-s2153" xml:space="preserve">Ductis enim regulis F O, G P, & </s> <s xml:id="echoid-s2154" xml:space="preserve">applicatis vbi-<lb/> <anchor type="figure" xlink:label="fig-0085-02a" xlink:href="fig-0085-02"/> cunque duabus ADH, IL M; </s> <s xml:id="echoid-s2155" xml:space="preserve">quæ productę ſecent <lb/>regulas in P, O, N, R; </s> <s xml:id="echoid-s2156" xml:space="preserve">manifeſtum iam eſt ex 1. </s> <s xml:id="echoid-s2157" xml:space="preserve">h. <lb/></s> <s xml:id="echoid-s2158" xml:space="preserve">has regulas inter ſe æquidiſtare: </s> <s xml:id="echoid-s2159" xml:space="preserve">& </s> <s xml:id="echoid-s2160" xml:space="preserve">cum ſit vt qua-<lb/>dratum I M ad M L, ita <anchor type="note" xlink:href="" symbol="a"/> recta R M ad M N, vel vt O <anchor type="note" xlink:label="note-0085-01a" xlink:href="note-0085-01"/> H ad H P, vel vt quadratum A H ad H D, erit etiam <lb/>recta IM ad ML, vt AH ad H D, & </s> <s xml:id="echoid-s2161" xml:space="preserve">per conuerſionẽ <lb/>rationis, & </s> <s xml:id="echoid-s2162" xml:space="preserve">permutando I M ad A H, vt I L ad A D, <lb/> <anchor type="note" xlink:label="note-0085-02a" xlink:href="note-0085-02"/> ſed eſt IM <anchor type="note" xlink:href="" symbol="b"/> maior AH, quare, & </s> <s xml:id="echoid-s2163" xml:space="preserve">IL erit maior AD.</s> <s xml:id="echoid-s2164" xml:space="preserve"> Quod ſecundò, &</s> <s xml:id="echoid-s2165" xml:space="preserve">c.</s> <s xml:id="echoid-s2166" xml:space="preserve"/> </p> <div xml:id="echoid-div198" type="float" level="2" n="2"> <figure xlink:label="fig-0085-02" xlink:href="fig-0085-02a"> <image file="0085-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0085-02"/> </figure> <note symbol="a" position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">6. Co-<lb/>rol. 19. h.</note> <note symbol="b" position="right" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">32. h.</note> </div> <p> <s xml:id="echoid-s2167" xml:space="preserve">Demũ dico, has aliquando peruenire ad interuallũ maius quolibet dato NO.</s> <s xml:id="echoid-s2168" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2169" xml:space="preserve">Fiat vt AD ad DH, vel vt IL ad LM, quod idem eſt, (modò enim oſtẽdimus <lb/>omnes huiuſmodi applicatas proportionaliter diuidi à Parabola B D L) ita datũ <lb/>interuallum N O ad aliud O P, & </s> <s xml:id="echoid-s2170" xml:space="preserve">ducta ex vertice contingente B Q R ſumatur <lb/>B R æqualis P N, & </s> <s xml:id="echoid-s2171" xml:space="preserve">B Q æqualis PO, & </s> <s xml:id="echoid-s2172" xml:space="preserve">per R agatur R I diametro B M æquidi-<lb/> <anchor type="note" xlink:label="note-0085-03a" xlink:href="note-0085-03"/> ſtans, quæ Parabolæ ABC <anchor type="note" xlink:href="" symbol="c"/> occurrat in I, & </s> <s xml:id="echoid-s2173" xml:space="preserve">per I applicetur IL M. </s> <s xml:id="echoid-s2174" xml:space="preserve">Erit ergo I M æqualis R B, ſiue æqualis NP, eſtque vt IL ad LM, ita NO ad OP, ex con-<lb/>ſtructione, quare IL ipſi N O ęqualis erit, ſed applicatę infra I L inter Parabolas <lb/>excedunt ipſam IL, vti nuper oſtendimus: </s> <s xml:id="echoid-s2175" xml:space="preserve">quare huiuſmodi Parabolæ ad inter-<lb/>uallum perueniunt maius dato iuteruallo N O. </s> <s xml:id="echoid-s2176" xml:space="preserve">Quod vltimò erat, &</s> <s xml:id="echoid-s2177" xml:space="preserve">c.</s> <s xml:id="echoid-s2178" xml:space="preserve"/> </p> <div xml:id="echoid-div199" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">26. pr. <lb/>conic.</note> </div> </div> <div xml:id="echoid-div201" type="section" level="1" n="94"> <head xml:id="echoid-head99" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s2179" xml:space="preserve">_HV_C Franciſcum Barocium ſubaudiet fortaſſe aliquis admurmurantem, <lb/>nos, qui de aſymptoticis lineis mutuam acceſsionem, vel receſsionem <lb/>perpendendam ſuſcepimus, æquidiſtantium linearum ſegmentis, inter <lb/>conuergentes, ac diuergentes aſymptotos interceptis vſos fuiſſe, veluti <lb/>in præcedenti, vbi iuxta lineas, ſiue portiones A D, I L ex ordinatim <lb/>app licatis ad communem diametrum, datarum ſectionum diſtantias commetimur; <lb/></s> <s xml:id="echoid-s2180" xml:space="preserve">dum tamen ipſæ à breuiſsimis, ſeu MINIMIS lineis ſint determinandæ, atque hæ <pb o="62" file="0086" n="86" rhead=""/> noſtræ æquidiſtantium portiones non ſint MINIMAE, quæ à punctis alterutrius <lb/>ſectionis ſuper aliam educi queant: </s> <s xml:id="echoid-s2181" xml:space="preserve">unde ob hoc Hieronymum Cardanum aſſectari <lb/>nos debuiſſe, qui iuxta perpendiculares à punctis hyperbolicæ ſectionis ſuper aſym-<lb/>pton ductas, ipſarum linearum interualla meditatus eſt; </s> <s xml:id="echoid-s2182" xml:space="preserve">quod nullos alios, admi-<lb/>randum Apollonij hoc Theorema diſcutientes, animaduerſos fuiſſe, idem Barocius <lb/>in ſuo quodam commentario Geometrico ſæpiùs admonuit, inter alia hæc proferens; <lb/></s> <s xml:id="echoid-s2183" xml:space="preserve">_quem errorem, omnes quos vidihuius rei Auctores commiſerunt, præter Car-_ <lb/>_danum_. </s> <s xml:id="echoid-s2184" xml:space="preserve">Sed bùc tã bonum Virum edoctum velimus, hoc idem iamdiu nobis innotuiſſe, <lb/>verùm deditaopera, æ libenter in hoc deſicere nobis placuiſſe cum ſummis Viris, qua-<lb/>les, eos inter quos maximè colimus, ſunt Torricellius, & </s> <s xml:id="echoid-s2185" xml:space="preserve">Gregorius à S. </s> <s xml:id="echoid-s2186" xml:space="preserve">Vincentio, <lb/>immo ipſemet Apollonius tam reconditi ſymptomatis fortaſſe primus, & </s> <s xml:id="echoid-s2187" xml:space="preserve">acutiſsimus <lb/>indagator. </s> <s xml:id="echoid-s2188" xml:space="preserve">Præterea, nos quoque ſatis agnouiſſe, vnius puncti diſtantiam à quacun-<lb/>que ſeu recta, ſeu curua linea, ſtrictim aſſumendam eſſe iuxta MINIMAM ex <lb/>eodem puncto, ſuper datam lineam eductam. </s> <s xml:id="echoid-s2189" xml:space="preserve">Inſuper ipſam MINIMAM d@m@ſſas <lb/>quel, datæ rectæ, vel contingenti ad datam curuam in puncto occurſus perpendicu-<lb/>lariter inſiſtetc quæ omnia hùc in proximo noſtro libello perſpicuè apparebunt: </s> <s xml:id="echoid-s2190" xml:space="preserve">at-<lb/>tamen hiſce, alijſque notionibus de his inſtructi, alteram methodum conſultò eli-<lb/>gere maluimus. </s> <s xml:id="echoid-s2191" xml:space="preserve">Itaquè ab humaniſsimo Barocio de hoc, cui ſanè criminis nomen <lb/>falsò tribuitur, veniam expectamus; </s> <s xml:id="echoid-s2192" xml:space="preserve">dum nos etiam, & </s> <s xml:id="echoid-s2193" xml:space="preserve">ſua, quæquæ ſint ab ipſo <lb/>ſparſim prolata reticere parati ſumus. </s> <s xml:id="echoid-s2194" xml:space="preserve">Interea concedat quæſo, vt ad inuentionem <lb/>MAXIMARVM MINIMARV Mque coni- ſectionum per diuerſos vertices inſcri-<lb/>ptibilium, aut circumſcriptibilium, vel ſaltim vt genio quodam noſtro indulgendo, <lb/>harum ſectionum diſtantias per æquidiſtantium interpoſita ſegmenta nobis perſcru-<lb/>tari liceat. </s> <s xml:id="echoid-s2195" xml:space="preserve">Perlegat inſuper, ſimulque has meditationes percipiat; </s> <s xml:id="echoid-s2196" xml:space="preserve">quod ſi ex hac <lb/>hypotheſi ipſas euidenter comprobatas inuenerit, fateatur, ſi lubet, nos abundè in <lb/>his Geometræ partem expleuiſſe; </s> <s xml:id="echoid-s2197" xml:space="preserve">ſin aliter, incuſatione dignos exiſtimet.</s> <s xml:id="echoid-s2198" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2199" xml:space="preserve">Scias itaque, ac iterum ſcias candide Lector nos, tum in decima huius, tum <lb/>in ſuperioribus, ac deinceps, vbi de non coincidentibus lineis diſſerimus, harum inter-<lb/>ualla ſemper aſſumpſiſſe iuxta interiecta parallelarum ſegmenta, licet hoc ſæpenu-<lb/>mero prætermittatur, cum ex ipſo demonſtrationum proceſſu id ſatis ſuperque ſe in <lb/>conſpectum præbeat. </s> <s xml:id="echoid-s2200" xml:space="preserve">Zuod ſi ſubijciat Barocius, non quidem intercapedines à pun-<lb/>ctis vnius ad alteram coni- ſectionem, verùm longitudines tantùm illarum æqui-<lb/>diſtantium portionum ſic meditari; </s> <s xml:id="echoid-s2201" xml:space="preserve">illum æquiter ratiocinaſſe nos ipſi profectò fa-<lb/>tebimur, quamuis & </s> <s xml:id="echoid-s2202" xml:space="preserve">eædem parallelarum portiones verè dici poſsint diſtantiæ à <lb/>punctis vnius ſectionis ad aliam; </s> <s xml:id="echoid-s2203" xml:space="preserve">prout lineæ omnes, ab vno eodemque puncto duci-<lb/>biles, dicuntur interualla ab ipſo puncto ad eandem lineam, etiamſi eductæ inter <lb/>ſe plurimùm differant longitudine.</s> <s xml:id="echoid-s2204" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2205" xml:space="preserve">Zuò verò ad methodum iuxta MINIMAS, plura ſunt, quæ à nobis huc eſſent <lb/>in medium afferenda, ſed non eſt cur in præſens futilem diſceptationem aggredia-<lb/>mur. </s> <s xml:id="echoid-s2206" xml:space="preserve">At ſi quis nos ad huius Pelagi traiectionem deuinctos cenſeret (quamuis præ-<lb/>ſtantiori fortaſſe luce orbati, nempe conica Apollonij ineditorum librorum doctri-<lb/>na) diuino tamen præſidio, noſtris quibuſdam inuentis mox huc edendis innixi, na-<lb/>ue integra ad portum appellere non dubitaremus; </s> <s xml:id="echoid-s2207" xml:space="preserve">ſed interim eum, non proprij of-<lb/>ficij cauſa, ſed pro ſua tantùm humanitate, nonnulla valde iucunda Problemata <lb/>enodanda recipere exoptaremus, quæ nobis circa nouam quandam meditationem <lb/>nuper aſſequi datum eſt. </s> <s xml:id="echoid-s2208" xml:space="preserve">Sed miſsis parergis rem iam ſuſceptam progrediamur.</s> <s xml:id="echoid-s2209" xml:space="preserve"/> </p> <pb o="63" file="0087" n="87" rhead=""/> </div> <div xml:id="echoid-div202" type="section" level="1" n="95"> <head xml:id="echoid-head100" xml:space="preserve">THEOR. XV. PROP. XXXIV.</head> <p> <s xml:id="echoid-s2210" xml:space="preserve">Si à puncto, quod eſt in Hyperbola ducatur recta linea alteri <lb/>aſymptoton æquidiſtans, ipſa, ac ſectio, quæ inter has parallelas <lb/>intercipitur, in infinitum productę ſunt infra occurſum ſemper ma-<lb/>gis recedentes, ſed tamen nunquam perueniunt ad interuallum <lb/>æquale cuidam dato interuallo; </s> <s xml:id="echoid-s2211" xml:space="preserve">dum earum diſtantia metiatur per <lb/>interceptas æquidiſtantes cuilibet rectæ, quæ ducta ſit ex occurſu <lb/>vtramque aſymptoton ſecans.</s> <s xml:id="echoid-s2212" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2213" xml:space="preserve">SIt Hyperbole ABC, cuius aſymptoti ED, EF, ſitque ex quolibet ſectio-<lb/>nis puncto B recta BGN alteri aſymptoto ED æquidiſtans, quæ intra <lb/>lectionẽ cadens, in nullo alio pũcto quam <lb/> <anchor type="figure" xlink:label="fig-0087-01a" xlink:href="fig-0087-01"/> B cum ipſa <anchor type="note" xlink:href="" symbol="a"/> conueniet. </s> <s xml:id="echoid-s2214" xml:space="preserve">Dico primùm (ſi ex <anchor type="note" xlink:label="note-0087-01a" xlink:href="note-0087-01"/> B ducatur quæcunque HBF vtranq; </s> <s xml:id="echoid-s2215" xml:space="preserve">aſym-<lb/>ptoton ſecans) ipſam, & </s> <s xml:id="echoid-s2216" xml:space="preserve">ſectionem IAM <lb/>infra BI eſſe sẽper magis inter ſerecedẽtes.</s> <s xml:id="echoid-s2217" xml:space="preserve"/> </p> <div xml:id="echoid-div202" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0087-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve">Coroll. <lb/>11. huius.</note> </div> <p> <s xml:id="echoid-s2218" xml:space="preserve">Applicentur quotcunque DAG, LMN <lb/>infra HB, ipſi æquidiſtantes: </s> <s xml:id="echoid-s2219" xml:space="preserve">patet has om-<lb/>nes LN, DG, HB inter ſe æquales eſſe, ſed <lb/>eſt DA <anchor type="note" xlink:href="" symbol="b"/> minor HI, ergo AG maior erit <anchor type="note" xlink:label="note-0087-02a" xlink:href="note-0087-02"/> IB, eſtque LM minor DA, quare & </s> <s xml:id="echoid-s2220" xml:space="preserve">MN <lb/>maior AG, & </s> <s xml:id="echoid-s2221" xml:space="preserve">hoc ſemper ſi in infinitum <lb/>producantur; </s> <s xml:id="echoid-s2222" xml:space="preserve">ergo linea BGN, & </s> <s xml:id="echoid-s2223" xml:space="preserve">ſectio <lb/>IAM ſunt ſemper ſimul recedentes. </s> <s xml:id="echoid-s2224" xml:space="preserve">Quod <lb/>primò, &</s> <s xml:id="echoid-s2225" xml:space="preserve">c.</s> <s xml:id="echoid-s2226" xml:space="preserve"/> </p> <div xml:id="echoid-div203" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">10. h.</note> </div> <p> <s xml:id="echoid-s2227" xml:space="preserve">Et quoniam earum interuallum, per eaſ-<lb/>dem interceptas metitum, ſemper minus <lb/>eſt HB interuallo parallelarum BN, HL, <lb/>cum ſit GA minos GD, NM minos NL, & </s> <s xml:id="echoid-s2228" xml:space="preserve"><lb/>omnes GD, NL, &</s> <s xml:id="echoid-s2229" xml:space="preserve">c. </s> <s xml:id="echoid-s2230" xml:space="preserve">ipſi BH equales: </s> <s xml:id="echoid-s2231" xml:space="preserve">qua-<lb/>propter, licet huiuſmodi lineæ ſint ſemper magis recedentes, non tamen <lb/>perueniunt ad interuallum æquale interuallo BH. </s> <s xml:id="echoid-s2232" xml:space="preserve">Quod erat tandem, &</s> <s xml:id="echoid-s2233" xml:space="preserve">c.</s> <s xml:id="echoid-s2234" xml:space="preserve"/> </p> <pb o="64" file="0088" n="88" rhead=""/> </div> <div xml:id="echoid-div205" type="section" level="1" n="96"> <head xml:id="echoid-head101" xml:space="preserve">THEOR. XVI. PROP. XXXV.</head> <p> <s xml:id="echoid-s2235" xml:space="preserve">Si recta linea diametro Hyperbolæ vltrà centrum occurrens, al-<lb/>teram ipſius aſymptoton ſecet, producta ſectionem quoq; </s> <s xml:id="echoid-s2236" xml:space="preserve">ſecabit.</s> <s xml:id="echoid-s2237" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2238" xml:space="preserve">ESto Hyperbole ABC, cuius cẽtrum <lb/> <anchor type="figure" xlink:label="fig-0088-01a" xlink:href="fig-0088-01"/> D, aſymptotos DE, diameter BD <lb/>F, è cuius puncto G vltrà cẽtrum aſſum-<lb/>pto ducta ſit quæpiam linea GE aſym-<lb/>ptoton ſecans in E; </s> <s xml:id="echoid-s2239" xml:space="preserve">Dico, ſi produca-<lb/>tur, ſectionem quoque ſecare.</s> <s xml:id="echoid-s2240" xml:space="preserve"/> </p> <div xml:id="echoid-div205" 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/QN4GHYBF/figures/0088-01"/> </figure> </div> <p> <s xml:id="echoid-s2241" xml:space="preserve">Ducta enim ex vertice B recta BH <lb/>parallela ad DE, ipſa ad partes A nun-<lb/>quam ſectioni <anchor type="note" xlink:href="" symbol="a"/> occurret, cum ei occur- <anchor type="note" xlink:label="note-0088-01a" xlink:href="note-0088-01"/> rat in B, ſed GE ſecat alteram Paralle-<lb/>larum DE, quare producta ſecabit, & </s> <s xml:id="echoid-s2242" xml:space="preserve"><lb/>reliquam BH, vnde neceſſariò ſectio-<lb/>nem priùs ſecabit. </s> <s xml:id="echoid-s2243" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s2244" xml:space="preserve">c.</s> <s xml:id="echoid-s2245" xml:space="preserve"/> </p> <div xml:id="echoid-div206" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve">Coroll. <lb/>11. huius.</note> </div> </div> <div xml:id="echoid-div208" type="section" level="1" n="97"> <head xml:id="echoid-head102" xml:space="preserve">THEOR. XVII. PROP. XXXVI.</head> <p> <s xml:id="echoid-s2246" xml:space="preserve">Hyperbolæ per eundem verticem ſimul adſcriptæ, æquale re-<lb/>ctum latus habentes ſunt inter ſe nunquam coeuntes, & </s> <s xml:id="echoid-s2247" xml:space="preserve">ſemper in-<lb/>ter ſe magis recedentes, & </s> <s xml:id="echoid-s2248" xml:space="preserve">in infinitum productæ ad interuallum <lb/>perueniunt maius quocunque dato interuallo.</s> <s xml:id="echoid-s2249" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2250" xml:space="preserve">SInt duæ Hyperbolæ ABC, DBE per eundem verticem B ſimul adſcriptę, <lb/>quarum rectumlatus ſit idem BF, tranſuerſum verò Hyperbolæ ABC <lb/>ſit minor recta BH, & </s> <s xml:id="echoid-s2251" xml:space="preserve">regula HF; </s> <s xml:id="echoid-s2252" xml:space="preserve">Hyperbolæ autem DBE ſit maior recta <lb/>BG eiuſque regula ſit GF: </s> <s xml:id="echoid-s2253" xml:space="preserve">dico primùm has inter ſe ſimul eſſe non coeuntes.</s> <s xml:id="echoid-s2254" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2255" xml:space="preserve">Cum enim Hyperbole DBE, maius habens trãſuerſum latus, inſcripta <anchor type="note" xlink:href="" symbol="a"/> ſit <anchor type="note" xlink:label="note-0088-02a" xlink:href="note-0088-02"/> Hyperbolæ ABC, patet ipſas, licet in infinitum producantur, nunquam in-<lb/>ter ſe conuenire, vnde erunt ſimul non coeuntes.</s> <s xml:id="echoid-s2256" xml:space="preserve"/> </p> <div xml:id="echoid-div208" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0088-02" xlink:href="note-0088-02a" xml:space="preserve">4. Corol. <lb/>19. huius.</note> </div> <p> <s xml:id="echoid-s2257" xml:space="preserve">Iam dico ipſas eſſe ſimul ſemper recedentes. </s> <s xml:id="echoid-s2258" xml:space="preserve">Applicatis enim duabus <lb/>quibuſcunque rectis CEILM, PONQR, iungatur quoque FN rectam MI <lb/>ſecans in S. </s> <s xml:id="echoid-s2259" xml:space="preserve">Cum ſit LS minor LI habebit ML ad L S maiorem rationem <lb/>quàm ML ad LI, & </s> <s xml:id="echoid-s2260" xml:space="preserve">componendo MS ad SL, ſiue RN ad NQ, hoc eſt <anchor type="note" xlink:href="" symbol="b"/> qua- <anchor type="note" xlink:label="note-0088-03a" xlink:href="note-0088-03"/> dratum PN ad NO, habebit maiorem rationem, quàm MI ad IL, hoc <anchor type="note" xlink:href="" symbol="c"/> eſt <anchor type="note" xlink:label="note-0088-04a" xlink:href="note-0088-04"/> quàm quadratum CI ad IE, ſiue applicata PN ad NO maiorem habebit ra-<lb/>tionem quàm applicata CI ad IE: </s> <s xml:id="echoid-s2261" xml:space="preserve">ſi ergo fiat vt PN ad NO, ita CI ad IT, <lb/>habebit CI ad IT maiorem rationem quàm CI ad IE, ergo IT erit minor IE, <lb/>ideoque CT maior CE: </s> <s xml:id="echoid-s2262" xml:space="preserve">cumque ſit PN ad NO vt CI ad IT, erit per conuer-<lb/>ſionem rationis, & </s> <s xml:id="echoid-s2263" xml:space="preserve">permutando PN ad CI vt PO ad CT, ſed <anchor type="note" xlink:href="" symbol="d"/> eſt PN maior <anchor type="note" xlink:label="note-0088-05a" xlink:href="note-0088-05"/> CI; </s> <s xml:id="echoid-s2264" xml:space="preserve">quare PO maior erit ipſa CT, eſtque CT maior CE, ergo PO adhuc <pb o="65" file="0089" n="89" rhead=""/> maior erit ipſa CE. </s> <s xml:id="echoid-s2265" xml:space="preserve">Quò ergo in-<lb/> <anchor type="figure" xlink:label="fig-0089-01a" xlink:href="fig-0089-01"/> terceptæ PO magis remouentur à <lb/>vertice B, eò ſunt maiores: </s> <s xml:id="echoid-s2266" xml:space="preserve">quare <lb/>huiuſmodi ſectiones inter ſe ſunt <lb/>ſemper recedentes. </s> <s xml:id="echoid-s2267" xml:space="preserve">Quod ſecun-<lb/>dò, &</s> <s xml:id="echoid-s2268" xml:space="preserve">c.</s> <s xml:id="echoid-s2269" xml:space="preserve"/> </p> <div xml:id="echoid-div209" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0088-03" xlink:href="note-0088-03a" xml:space="preserve">4. Co-<lb/>roll. prop. <lb/>19. huius.</note> <note symbol="c" position="left" xlink:label="note-0088-04" xlink:href="note-0088-04a" xml:space="preserve">ibidem.</note> <note symbol="d" position="left" xlink:label="note-0088-05" xlink:href="note-0088-05a" xml:space="preserve">32. h.</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/QN4GHYBF/figures/0089-01"/> </figure> </div> <p> <s xml:id="echoid-s2270" xml:space="preserve">Præterea ſit BY ſectionem con-<lb/>tingens in B, & </s> <s xml:id="echoid-s2271" xml:space="preserve">bifariam ſectis trãſ-<lb/>uerſis lateribus, nempe GB in V, <lb/>& </s> <s xml:id="echoid-s2272" xml:space="preserve">HB in X: </s> <s xml:id="echoid-s2273" xml:space="preserve">cum ſit tranſuerſum <lb/>GB maius BH, erit dimidium BV <lb/>maius dimidio BX: </s> <s xml:id="echoid-s2274" xml:space="preserve">iam ex centro <lb/>V ducatur VY aſymptotos inſcri-<lb/>ptæ Hyperbolæ DBE, & </s> <s xml:id="echoid-s2275" xml:space="preserve">ex cen-<lb/>tro X agatur XZ aſymptotos cir-<lb/>cumſcriptæ ABC, quæ aſymptoti <lb/>contingentem BI ſecent in Y, Z, & </s> <s xml:id="echoid-s2276" xml:space="preserve"><lb/>per X agatur X & </s> <s xml:id="echoid-s2277" xml:space="preserve">parallela ad BY contingentem ſecans in &</s> <s xml:id="echoid-s2278" xml:space="preserve">.</s> </p> <p> <s xml:id="echoid-s2279" xml:space="preserve">Itaque quadratum BY ad BZ eſt, vt rectangulum GBF ad rectangulum <lb/>HBF (vtrumque enim quadratorum <anchor type="note" xlink:href="" symbol="a"/> eſt quarta pars ſuæ figuræ) vel vt recta <anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/> GB ad BH, vel ſumptis ſubduplis, vt VB ad BX, vel ob parallelas, vt YB ad <lb/>B&</s> <s xml:id="echoid-s2280" xml:space="preserve">, quare BZ eſt media proportionalis inter BY, & </s> <s xml:id="echoid-s2281" xml:space="preserve">B&</s> <s xml:id="echoid-s2282" xml:space="preserve">: cum ergo inter pa-<lb/>rallelas VY, Z& </s> <s xml:id="echoid-s2283" xml:space="preserve">recta ZX ſecet alteram parallelarum X& </s> <s xml:id="echoid-s2284" xml:space="preserve">in X, ipſa produ-<lb/>cta ad partes Z ſecabit quoque alteram parallelam VY infra BY: </s> <s xml:id="echoid-s2285" xml:space="preserve">vnde ha-<lb/>rum ſectionum aſymptoti infra contingentem ex vertice inter ſe conueniunt.</s> <s xml:id="echoid-s2286" xml:space="preserve"/> </p> <div xml:id="echoid-div210" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">8. huius.</note> </div> <p> <s xml:id="echoid-s2287" xml:space="preserve">Amplius cum aſymptotos VY inſcriptæ occurrat diametro BG vltra cen-<lb/>trum X circumſcriptæ Hyperbolę ABC in puncto V, ipſaque aſymptotos <lb/>VY conueniat cum XZ aſymptoto circumſcriptæ ABC, vt modò oſtendi-<lb/>mus, ſi producatur, <anchor type="note" xlink:href="" symbol="b"/> ſecabit quoque Hyperbolen ABC. </s> <s xml:id="echoid-s2288" xml:space="preserve">Quare aſymptotos <anchor type="note" xlink:label="note-0089-02a" xlink:href="note-0089-02"/> inſcriptæ ſecat Hyperbolen circumſcriptam.</s> <s xml:id="echoid-s2289" xml:space="preserve"/> </p> <div xml:id="echoid-div211" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-0089-02" xlink:href="note-0089-02a" xml:space="preserve">35. h.</note> </div> <p> <s xml:id="echoid-s2290" xml:space="preserve">Tandem cum harum ſectionum aſymptoti infra contingentem BY ſe mu-<lb/>tuò ſecent, & </s> <s xml:id="echoid-s2291" xml:space="preserve">XZ aſymptotos circũſcriptæ BCP, cadat totas extra ipsã BCP, <lb/>harum aſymptoton occurſus erit extra eandem BCP, vt in 2: </s> <s xml:id="echoid-s2292" xml:space="preserve">& </s> <s xml:id="echoid-s2293" xml:space="preserve">cum VY 2 <lb/>aſymptotos inſcriptæ, ſecet Hyperbolen BCP circumſcriptam, eſto earum <lb/>communis ſectio in 3, & </s> <s xml:id="echoid-s2294" xml:space="preserve">recta 2 3, producatur ad inferiores partes 4, atque <lb/>ex 3 ducatur recta 3 5 parallela ad X 2 aſymptoton circumſcriptæ BC 3, quę <lb/>recta 3 5 nunquam conueniet <anchor type="note" xlink:href="" symbol="c"/> cum ſectione 3 7 ad inferiores partes, ſed <anchor type="note" xlink:label="note-0089-03a" xlink:href="note-0089-03"/> etiam recta 3 4 nunquã conuenit cum ſectione BE 6 ad eaſdem partes (nam <lb/>eſt eius aſymptotos) & </s> <s xml:id="echoid-s2295" xml:space="preserve">duæ rectæ 3 5, 3 4 ſunt ſemper ſimul recedentes, <lb/>& </s> <s xml:id="echoid-s2296" xml:space="preserve">ad interuallum perueniunt maius quolibet dato interuallo; </s> <s xml:id="echoid-s2297" xml:space="preserve">quare eò ma-<lb/>gis interuallum ſectionum BC7, BE6, datum quodcunque interuallum ex-<lb/>cedet. </s> <s xml:id="echoid-s2298" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s2299" xml:space="preserve"/> </p> <div xml:id="echoid-div212" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-0089-03" xlink:href="note-0089-03a" xml:space="preserve">34. h.</note> </div> </div> <div xml:id="echoid-div214" type="section" level="1" n="98"> <head xml:id="echoid-head103" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s2300" xml:space="preserve">EX hac manifeſtum fit, quod Hyperbolarum per eundem verticem ſimul <lb/>adſcriptarum, & </s> <s xml:id="echoid-s2301" xml:space="preserve">idem rectum latus habentium aſymptoti infra contin- <pb o="66" file="0090" n="90" rhead=""/> gentem ex vertice ſe mutuò ſecant, (extra tamen circumſcriptam) & </s> <s xml:id="echoid-s2302" xml:space="preserve">aſym-<lb/>ptotos inſcriptæ ſecat Hyperbolen circumſcriptam.</s> <s xml:id="echoid-s2303" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div215" type="section" level="1" n="99"> <head xml:id="echoid-head104" xml:space="preserve">THEOR. XIII. PROP. XXXVII.</head> <p> <s xml:id="echoid-s2304" xml:space="preserve">Hyperbolæ concentricæ per eundem verticem ſimul adſcriptæ, <lb/>quarum recta latera ſint inæqualia, ſunt inter ſe nunquam coeuntes, <lb/>& </s> <s xml:id="echoid-s2305" xml:space="preserve">ſemper magis recedentes, & </s> <s xml:id="echoid-s2306" xml:space="preserve">in infinitum productæ, ad interual-<lb/>lum perueniunt maius quolibet dato interuallo, & </s> <s xml:id="echoid-s2307" xml:space="preserve">aſymptotos in-<lb/>ſcriptæ ſecat Hyperbolen circumſcriptam.</s> <s xml:id="echoid-s2308" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2309" xml:space="preserve">SInt duę Hyperbolę ABC, DBE per <lb/> <anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/> eundem verticem B ſimul adſcri-<lb/>pte, quarum idem centrum ſit F, idem-<lb/>que tranſuerſum BFG, ſed tamen Hy-<lb/>perbolæ ABC rectum latus ſit BH, ma-<lb/>ius recto BI Hyperbolæ DBE. </s> <s xml:id="echoid-s2310" xml:space="preserve">Dico <lb/>primùm eas ſimul eſſe non coeuntes.</s> <s xml:id="echoid-s2311" xml:space="preserve"/> </p> <div xml:id="echoid-div215" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0090-01"/> </figure> </div> <p> <s xml:id="echoid-s2312" xml:space="preserve">Cum enim Hyperbolæ DBE, ABC <lb/>ſint per verticem ſimul adſcriptæ cum <lb/>eodem tranſuerſo BG, ipſa DBE, cuius <lb/>rectum minus eſt, inſcripta <anchor type="note" xlink:href="" symbol="a"/> erit Hy- <anchor type="note" xlink:label="note-0090-01a" xlink:href="note-0090-01"/> perbolæ ABC, cuius rectum maius eſt, <lb/>hoc eſt, ſi iſtæ ſimul in infinitum produ-<lb/>cantur, erunt ſimul non coeuntes.</s> <s xml:id="echoid-s2313" xml:space="preserve"/> </p> <div xml:id="echoid-div216" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2314" xml:space="preserve">Iam dico, has etiam eſſe ſemper in-<lb/>ter ſe recedentes. </s> <s xml:id="echoid-s2315" xml:space="preserve">Ductis enim, & </s> <s xml:id="echoid-s2316" xml:space="preserve">pro-<lb/>tractis regulis; </s> <s xml:id="echoid-s2317" xml:space="preserve">GH, GI, & </s> <s xml:id="echoid-s2318" xml:space="preserve">applicatis <lb/>duabus vbicunque rectis ADL, MON; </s> <s xml:id="echoid-s2319" xml:space="preserve">quæ regulas ſecent in Q, S, T, V, <lb/>cum ſit vt quadratum MN ad quadratũ NO, ita <anchor type="note" xlink:href="" symbol="b"/> recta VN ad NT, vel recta <anchor type="note" xlink:label="note-0090-02a" xlink:href="note-0090-02"/> SL ad SQ, vel quadratum AL ad LD, erit etiam recta MN ad NO, vt AL ad <lb/>LD, & </s> <s xml:id="echoid-s2320" xml:space="preserve">per conuerſionem rationis, & </s> <s xml:id="echoid-s2321" xml:space="preserve">permutando MN ad AL, vt MO ad <lb/>AD, ſed eſt MN <anchor type="note" xlink:href="" symbol="c"/> maior AL, quare, & </s> <s xml:id="echoid-s2322" xml:space="preserve">MO erit maior AD; </s> <s xml:id="echoid-s2323" xml:space="preserve">ſimiliter demon- <anchor type="note" xlink:label="note-0090-03a" xlink:href="note-0090-03"/> ſtrabitur quamlibet aliam interceptam applicatę portionem inter Hyperbo-<lb/>las infra MO, maiorem eſſe ipſa MO, & </s> <s xml:id="echoid-s2324" xml:space="preserve">hoc ſemper, quare huiuſmodi Hy-<lb/>perbolæ ſunt ſemper inter ſe recedentes. </s> <s xml:id="echoid-s2325" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s2326" xml:space="preserve">c.</s> <s xml:id="echoid-s2327" xml:space="preserve"/> </p> <div xml:id="echoid-div217" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-0090-02" xlink:href="note-0090-02a" xml:space="preserve">6. Co-<lb/>roll. 19. h.</note> <note symbol="c" position="left" xlink:label="note-0090-03" xlink:href="note-0090-03a" xml:space="preserve">32. h.</note> </div> <p> <s xml:id="echoid-s2328" xml:space="preserve">Ampliùs dico, has ſectiones in infinitum productas, aliquando perueni-<lb/>re, ad interuallum maius quolibet dato interuallo X. </s> <s xml:id="echoid-s2329" xml:space="preserve">Hoc autem, eadem <lb/>penitùs arte, ac in 33. </s> <s xml:id="echoid-s2330" xml:space="preserve">huius fieri poſſe demonſtrabitur. </s> <s xml:id="echoid-s2331" xml:space="preserve">Quod tertiò, &</s> <s xml:id="echoid-s2332" xml:space="preserve">c.</s> <s xml:id="echoid-s2333" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2334" xml:space="preserve">Tandem ſit FP aſymptotos inſcriptæ DBC, & </s> <s xml:id="echoid-s2335" xml:space="preserve">FR aſymptotos circumſcri-<lb/>pte, & </s> <s xml:id="echoid-s2336" xml:space="preserve">contingens HB producatur, vtranque aſymptoton ſecans in P, R: </s> <s xml:id="echoid-s2337" xml:space="preserve">erit <lb/>ergo quadratum BP ęquale quartę parti figurę GBI, <anchor type="note" xlink:href="" symbol="d"/> & </s> <s xml:id="echoid-s2338" xml:space="preserve">quadratũ BR quartę <anchor type="note" xlink:label="note-0090-04a" xlink:href="note-0090-04"/> parti figuræ GBH, ſed rectangulum GBI maius eſt rectangulo GBH, cum ſit <lb/>BI minor BH, ergo BP minor eſt BR; </s> <s xml:id="echoid-s2339" xml:space="preserve">hoc eſt FP aſymptoton inſcriptæ cadit <lb/>infra FR aſymptoton circumſcriptæ diuidens angulũ ab ipſius aſymptotis fa-<lb/>ctum, ex quo ipſa FP producta ſecabit <anchor type="note" xlink:href="" symbol="e"/> Hyperbolen circumſcriptam ABC.</s> <s xml:id="echoid-s2340" xml:space="preserve"> <anchor type="note" xlink:label="note-0090-05a" xlink:href="note-0090-05"/> Quod erat vltimò, &</s> <s xml:id="echoid-s2341" xml:space="preserve">c.</s> <s xml:id="echoid-s2342" xml:space="preserve"/> </p> <div xml:id="echoid-div218" type="float" level="2" n="4"> <note symbol="d" position="left" xlink:label="note-0090-04" xlink:href="note-0090-04a" xml:space="preserve">8. huius.</note> <note symbol="e" position="left" xlink:label="note-0090-05" xlink:href="note-0090-05a" xml:space="preserve">ibidem.</note> </div> <pb o="67" file="0091" n="91" rhead=""/> </div> <div xml:id="echoid-div220" type="section" level="1" n="100"> <head xml:id="echoid-head105" xml:space="preserve">THEOR. XIX. PROP. XXXVIII.</head> <p> <s xml:id="echoid-s2343" xml:space="preserve">In Parabolis quibuslibet, vel in ſimilibus Hyperbolis, aut ſimi-<lb/>libus Ellipſbus, ſegmenta diametrorum ſectionum lateribus pro-<lb/>portionalia, ſuſcipiunt applicatasijſdem lateribus proportionales.</s> <s xml:id="echoid-s2344" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2345" xml:space="preserve">SInt, vt in prima figura, duæquælibet Parabolæ, velvt in ſecunda, duæ <lb/>ſimiles Hyperbolæ, vel vt in tertia, duæ ſimiles Ellipſes ABC, DEF, <lb/>quarum diametrorum ſegmenta BG, EH, rectis earum lateribus BI, EL, vel <lb/>tranſuerſis BM, EN ſint proportionalia, dico & </s> <s xml:id="echoid-s2346" xml:space="preserve">applicatas GA, HD ipſis la-<lb/>teribus eſſe proportionales.</s> <s xml:id="echoid-s2347" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2348" xml:space="preserve">Nam in Parabolis pri-<lb/> <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> mùm, cum ſit rectum BI <lb/>ad rectum EL, vt ſegmẽ-<lb/>tum BG ad EH, erit per-<lb/>mutando IB ad BG, vt <lb/>LE ad EH, vnde rectan-<lb/>gulum I B G ſimile erit <lb/>rectangulo LEH, quare <lb/>rectangulum IBG ad LE <lb/>H, erit vt quadratum la-<lb/>teris I B ad quadratum <lb/>homologilateris LE, ſed <lb/>rectãgulum IBG <anchor type="note" xlink:href="" symbol="a"/> æqua- <anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/> tur quadrato GA, & </s> <s xml:id="echoid-s2349" xml:space="preserve">re-<lb/>ctãgulum LEH quadra-<lb/>to HD, vnde quadratum <lb/>GA ad HD, erit vt qua-<lb/>dratum IB ad LE, vel <lb/>applicata GA ad HD, vt <lb/>rectum IB ad rectum LE. <lb/></s> <s xml:id="echoid-s2350" xml:space="preserve">In Hyperbolis autem, & </s> <s xml:id="echoid-s2351" xml:space="preserve"><lb/>Ellipſibus cum ſit vt BI <lb/>ad EL, vel ob ſectionum ſimilitudinem, vt MB ad NE, ita BG ad EH, erit <lb/>permutando MB ad BG, vt NE ad EH, & </s> <s xml:id="echoid-s2352" xml:space="preserve">in Hyperbolis, componendo, in <lb/>Ellipſibus autem diuidendo, MG ád GB, vt NH ad HE, quare rectangulum <lb/>MGB ſimile erit rectangulo NHE, ſed rectangulum MGB ad quadratum <lb/>GA eſt, <anchor type="note" xlink:href="" symbol="b"/> vt MB ad BI, vel vt NE ad EL, vel vtrectangulum NHE ad qua- <anchor type="note" xlink:label="note-0091-02a" xlink:href="note-0091-02"/> dratum HD, quare permutando rectangulum MGB ad rectangulum NHE, <lb/>vel (ob ipſorum rectangulorum ſimilitudinem) quadratum BG ad quadra-<lb/>tum EH, vel quadratum BI ad quadratum EL, erit vt quadratum GA ad <lb/>quadratum HD, hoc eſt rectum BI ad rectum EL, vt applicata GA ad appli-<lb/>catam HD. </s> <s xml:id="echoid-s2353" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s2354" xml:space="preserve">c.</s> <s xml:id="echoid-s2355" xml:space="preserve"/> </p> <div xml:id="echoid-div220" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0091-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve">1. huius.</note> <note symbol="b" position="right" xlink:label="note-0091-02" xlink:href="note-0091-02a" xml:space="preserve">21. pri-<lb/>mi conic.</note> </div> <pb o="68" file="0092" n="92" rhead=""/> </div> <div xml:id="echoid-div222" type="section" level="1" n="101"> <head xml:id="echoid-head106" xml:space="preserve">LEMMA IV. PROP. XXXIX.</head> <p> <s xml:id="echoid-s2356" xml:space="preserve">Si duæ menſales ABCD, EFGH fuerint ſuper eadem linea AH <lb/>ad eaſdem partes deſcriptæ, ita vt ipſarum baſes AB, DC; </s> <s xml:id="echoid-s2357" xml:space="preserve">EF, <lb/>HG ſint omnes inter ſe parallelæ, ſintque proportionales lateribus <lb/>in directum poſitis; </s> <s xml:id="echoid-s2358" xml:space="preserve">nempe ſit vt AB ad EF, ita AD ad EH, & </s> <s xml:id="echoid-s2359" xml:space="preserve">DC <lb/>ad HG. </s> <s xml:id="echoid-s2360" xml:space="preserve">Dico, & </s> <s xml:id="echoid-s2361" xml:space="preserve">reliqua latera BC, FG eſſe inter ſe parallela.</s> <s xml:id="echoid-s2362" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2363" xml:space="preserve">DVctis enim diagonalibus BD, FH, productaque BC in I. </s> <s xml:id="echoid-s2364" xml:space="preserve">Cum ſit BA <lb/>ad EF, vt AD ad EH, erit permutando BA ad AD, vt FE ad EH, eſtq; <lb/></s> <s xml:id="echoid-s2365" xml:space="preserve">angulus BAD ęqualis angulo FEH, ob parallelas BA, FE, quare triangulum <lb/>BAD ſimile eſt triangulo FEH; </s> <s xml:id="echoid-s2366" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s2367" xml:space="preserve">angulus BDA æqualis angulo FHE, <lb/>& </s> <s xml:id="echoid-s2368" xml:space="preserve">totus CD A, æquatur <lb/> <anchor type="figure" xlink:label="fig-0092-01a" xlink:href="fig-0092-01"/> toto GHE, ob parallelas <lb/>CD, GH, vnde reliquus <lb/>CDB, æquatur reliquo <lb/>GHF. </s> <s xml:id="echoid-s2369" xml:space="preserve">Item cum ſit CD <lb/>ad GH, vt DA ad HE, <lb/>erit permutando CD ad <lb/>DA, vt GH ad HE, eſtq; <lb/></s> <s xml:id="echoid-s2370" xml:space="preserve">DA ad DB, vt HE ad HF, <lb/>ob triangulorum ADB, <lb/>EHF ſimilitudinem, qua-<lb/>re ex æquali CD ad DB, <lb/>erit vt GH ad HF, ſuntq; </s> <s xml:id="echoid-s2371" xml:space="preserve">anguli ad D, & </s> <s xml:id="echoid-s2372" xml:space="preserve">H æquales, ergo, & </s> <s xml:id="echoid-s2373" xml:space="preserve">angulus D CB, <lb/>ſiue HIB, æqualis angulo HGF, ideoque rectæ BC, FG inter ſe æquidiſtant. </s> <s xml:id="echoid-s2374" xml:space="preserve"><lb/>Quod erat, &</s> <s xml:id="echoid-s2375" xml:space="preserve">c.</s> <s xml:id="echoid-s2376" xml:space="preserve"/> </p> <div xml:id="echoid-div222" type="float" level="2" n="1"> <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a"> <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0092-01"/> </figure> </div> </div> <div xml:id="echoid-div224" type="section" level="1" n="102"> <head xml:id="echoid-head107" xml:space="preserve">THEOR. XX. PROP. XXXX.</head> <p> <s xml:id="echoid-s2377" xml:space="preserve">Sià terminis æqualium ſegmentorum ex diametris ſimilium Hy-<lb/>perbolarum abſciſſorum rectæ ordinatim applicentur, vſque <lb/>ad ſectionum aſymptoto<unsure/>, erit ſegmentum applicatæ in Hyperbo-<lb/>la maiorum laterum, inter ſectionem, & </s> <s xml:id="echoid-s2378" xml:space="preserve">aſymptoton interceptum, <lb/>maius ſegmento applicatæ, quod in Hyperbola minorum laterum <lb/>inter ſectionem eiuſque aſymptoton intercipitur.</s> <s xml:id="echoid-s2379" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2380" xml:space="preserve">ESto Hyperbole maiorum laterum ABC, cuius tranſuerſum DB rectum <lb/>BE, aſymptotos FG, & </s> <s xml:id="echoid-s2381" xml:space="preserve">Hyperbole minorum ſit HIL, cuius tranſuer-<lb/>ſum MG, rectum GN, aſymptotos OP, & </s> <s xml:id="echoid-s2382" xml:space="preserve">ipſarum ſectionum diametris, dem-<lb/>pta ſint ęqualia diametri ſegmenta BQ, IR, è quorũ terminis Q, R applicate <lb/>ſint (ad partes æqualium inclinationum) rectæ QAG, RHP vique ad earum <lb/>aſymptotos: </s> <s xml:id="echoid-s2383" xml:space="preserve">Dico ſegmentum GA in ſectione maiorum laterum, maius eſſe <lb/>ſegmento PH in ſectione minorum.</s> <s xml:id="echoid-s2384" xml:space="preserve"/> </p> <pb o="69" file="0093" n="93" rhead=""/> <p> <s xml:id="echoid-s2385" xml:space="preserve">Productis enim contingentibus EB, NI vſque ad aſymptotos in S, T, fiat <lb/>vt DB ad MI, ita BQ ad IV, & </s> <s xml:id="echoid-s2386" xml:space="preserve">per V applicetur VXY: </s> <s xml:id="echoid-s2387" xml:space="preserve">cum ſit DB maior <lb/>MI, erit BQ, & </s> <s xml:id="echoid-s2388" xml:space="preserve">IR maior IV, eſtque FB maior OI (cum duplum DB ſit maior <lb/>duplo MI) ergo tota FQ erit maior tota OV, & </s> <s xml:id="echoid-s2389" xml:space="preserve">QA ad VX erit <anchor type="note" xlink:href="" symbol="a"/> vt DB ad <anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/> MI, & </s> <s xml:id="echoid-s2390" xml:space="preserve">quoniam QB ad VI, eſt vt BD ad IM, vel vt dimidium BF ad dimi-<lb/>dium IO, erit per-<lb/> <anchor type="figure" xlink:label="fig-0093-01a" xlink:href="fig-0093-01"/> mutando, compo-<lb/>nendo, & </s> <s xml:id="echoid-s2391" xml:space="preserve">iterum <lb/>permutando QF ad <lb/>VO, vt BF ad IO, <lb/>vel vt DB ad MI; </s> <s xml:id="echoid-s2392" xml:space="preserve">& </s> <s xml:id="echoid-s2393" xml:space="preserve"><lb/>cum ſit quadratum <lb/>SB ad TI, vt rectan-<lb/>gulũ DBE ad MIN, <lb/>vtrunque enim eſt <lb/>quarta pars ſuæ fi-<lb/>guræ) vel vt qua-<lb/>dratum DB ad qua-<lb/>dratum MI; </s> <s xml:id="echoid-s2394" xml:space="preserve">ob rectangulorum ſimilitudinem) vel ſumptis ſubquadruplis, vt <lb/>quadratum FB ad OI, erit quoque linea SB ad TI, vt linea FB ad OI, & </s> <s xml:id="echoid-s2395" xml:space="preserve">per-<lb/>mutando SB ad BF, vt TI ad IO, ſed anguli SBF, TIO ſunt æquales per ſex-<lb/>tam ſecundarum definitionum, & </s> <s xml:id="echoid-s2396" xml:space="preserve">per conſtructionem, quare triangula SBF, <lb/>TIO erunt ſimilia, vti etiam triangula GQF, YVO, obidque homologa eo-<lb/>rum latera proportionalia erunt, hoc eſt GQ ad YV, vt FQ ad OV, ſed eſt <lb/>FQ maior OV, ergo, & </s> <s xml:id="echoid-s2397" xml:space="preserve">GQ erit maior YV, ſed FQ ad OV, eſt vt DB ad MI, <lb/>item AQ ad XV, vt DB ad MI, vt ſupra oſtendimus, quare GQ ad YV erit <lb/>vt AQ ad XV, & </s> <s xml:id="echoid-s2398" xml:space="preserve">permutando, & </s> <s xml:id="echoid-s2399" xml:space="preserve">per conuerſionem rationis, & </s> <s xml:id="echoid-s2400" xml:space="preserve">iterum per-<lb/>mutando GQ ad YV, vt GA ad YX, ſed eſt GQ maior YV, ergo, & </s> <s xml:id="echoid-s2401" xml:space="preserve">G A <lb/>maior YX, eſt autem YX maior PH, ergo eò magis GA erit maior PH. </s> <s xml:id="echoid-s2402" xml:space="preserve">Quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s2403" xml:space="preserve"/> </p> <div xml:id="echoid-div224" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">38. h.</note> <figure xlink:label="fig-0093-01" xlink:href="fig-0093-01a"> <image file="0093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0093-01"/> </figure> </div> </div> <div xml:id="echoid-div226" type="section" level="1" n="103"> <head xml:id="echoid-head108" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s2404" xml:space="preserve">EX hac patet, in ſimilibus Hyperbolis aſymptotos ad partes æqualium in-<lb/>clinationum ductas, æquales angulos cum diametris efficere, ac ideo <lb/>angulos ab aſymptotis factos eſſe inter ſe æquales. </s> <s xml:id="echoid-s2405" xml:space="preserve">Cum enim demonſtrata <lb/>ſint triangula SFB, TOI ſimilia, erunt anguli ad F, O, æquales; </s> <s xml:id="echoid-s2406" xml:space="preserve">eademque <lb/>ratione æquales etiam anguli ab alijs aſymptotis cum diametris ad alteram <lb/>partem conſtitutis; </s> <s xml:id="echoid-s2407" xml:space="preserve">vnde eorum aggregata, nempe anguli ab aſymptotis fa-<lb/>cti in ſimilibus Hyperbolis inter ſe æquales erunt.</s> <s xml:id="echoid-s2408" xml:space="preserve"/> </p> <pb o="70" file="0094" n="94" rhead=""/> </div> <div xml:id="echoid-div227" type="section" level="1" n="104"> <head xml:id="echoid-head109" xml:space="preserve">THEOR. XXI. PROP. XXXXI.</head> <p> <s xml:id="echoid-s2409" xml:space="preserve">Similes Hyperbolæ per eundem verticem ſimul adſcriptæ, ſunt <lb/>inter ſe nunquam coeuntes, & </s> <s xml:id="echoid-s2410" xml:space="preserve">ſemper magis recedentes, ſed ad in-<lb/>teruallum nunquam perueniunt æquale cuidam dato interuallo.</s> <s xml:id="echoid-s2411" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2412" xml:space="preserve">SInt duæ ſimiles Hyperbolæ ABC, DBE per eundem verticem B ſimul <lb/>adſcriptæ, & </s> <s xml:id="echoid-s2413" xml:space="preserve">Hyperbolæ ABC maiora ſint latera, tranſuerſum nempe <lb/>FB, rectũ autem BG; </s> <s xml:id="echoid-s2414" xml:space="preserve">& </s> <s xml:id="echoid-s2415" xml:space="preserve">DBE minora ſint, tranſuerſum HB, rectum verò BI. <lb/></s> <s xml:id="echoid-s2416" xml:space="preserve">Patet primùm has ſectiones eſſe inter ſe nunquam coeuntes;</s> <s xml:id="echoid-s2417" xml:space="preserve">cum enim DBE <lb/>alteri <anchor type="note" xlink:href="" symbol="a"/> ABC ſit inſcripta, ipſæ, licet in infinitum producantur, nunquam <anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/> conuenient.</s> <s xml:id="echoid-s2418" xml:space="preserve"/> </p> <div xml:id="echoid-div227" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">5. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2419" xml:space="preserve">Dico ampliùs, eaſdem in-<lb/> <anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/> ter ſe longiùs ſemper recede-<lb/>re. </s> <s xml:id="echoid-s2420" xml:space="preserve">Applicatis enim in alte-<lb/>ra ſectionum, nempe in in-<lb/>ſcripta, duabus vbicunque <lb/>rectis MN, LD, fiat vt BH ad <lb/>BF, ita BM ad BQ, & </s> <s xml:id="echoid-s2421" xml:space="preserve">BL ad <lb/>BR, & </s> <s xml:id="echoid-s2422" xml:space="preserve">per Q, R, applicentur <lb/>in circumſcripta Hyperbola <lb/>rectæ QS, RA; </s> <s xml:id="echoid-s2423" xml:space="preserve">& </s> <s xml:id="echoid-s2424" xml:space="preserve">cum dia-<lb/>metrorum ſegmẽta BM, BQ, <lb/>BL, BR ſint tranſuerſis BH, <lb/>B F proportionalia, <anchor type="note" xlink:href="" symbol="b"/> erunt <anchor type="note" xlink:label="note-0094-02a" xlink:href="note-0094-02"/> quoque applicatæ MN, QS; <lb/></s> <s xml:id="echoid-s2425" xml:space="preserve">LD, RA ijſdẽ lateribus pro-<lb/>portionales, quare M N ad <lb/>QS erit vt LD ad RA; </s> <s xml:id="echoid-s2426" xml:space="preserve">cum-<lb/>que ſit vt tota BL ad totam BR, ita pars BM ad partem BQ, erit & </s> <s xml:id="echoid-s2427" xml:space="preserve">reliqua <lb/>ML ad reliquam QR, vt tota BL ad totam BR, vel vt BH ad BF, vel vt MN <lb/>ad QS, & </s> <s xml:id="echoid-s2428" xml:space="preserve">vt LD ad RA, vt nuper oſtendimus; </s> <s xml:id="echoid-s2429" xml:space="preserve">quare iunctis rectis DN, AS <lb/>in menſalibus DM, AQ, <anchor type="note" xlink:href="" symbol="c"/> erunt ipsæ DN, AS inter ſe parallelæ. </s> <s xml:id="echoid-s2430" xml:space="preserve">Iam pro- <anchor type="note" xlink:label="note-0094-03a" xlink:href="note-0094-03"/> ductis MN, LD vſque ad circumſcriptam ſectionem ABC, in P, & </s> <s xml:id="echoid-s2431" xml:space="preserve">O, ſi iun-<lb/>gatur PO, hæc omninò ſecabit iunctam AS, vel intra ipſam ſectionem; </s> <s xml:id="echoid-s2432" xml:space="preserve">(ſi <lb/>nempe vnius iunctarum ſectioni occurſus, alterius occurſibus contineatur) <lb/>vel extra <anchor type="note" xlink:href="" symbol="d"/> (ſi nullius occurſus, alterius occurſibus amplectatur) ſed AS pro- <anchor type="note" xlink:label="note-0094-04a" xlink:href="note-0094-04"/> ducta ad partes verticis tota cadit <anchor type="note" xlink:href="" symbol="e"/> extra ſectionem in SK, & </s> <s xml:id="echoid-s2433" xml:space="preserve">punctum P eſt in ipſa ſectione, quare punctum P eſt inter parallelas lineas ASK, DN, ſed <lb/> <anchor type="note" xlink:label="note-0094-05a" xlink:href="note-0094-05"/> producta PO conuenit cum altera parallelarum AS, vt modò monitum fuit; <lb/></s> <s xml:id="echoid-s2434" xml:space="preserve">taliſq; </s> <s xml:id="echoid-s2435" xml:space="preserve">occurſus eſt omnino ad partes O infra applicatam PN, cum punctum <lb/>S cadat infra P (nam ex ipſa conſtructione applicata QS eſt infra applicatam <lb/>MP) quare eadem OP producta conueniet quoque cum altera æquidiſtan-<lb/>tium DN, ad oppoſitas tamen partes, vtputa ſupra ipſam applicatam PN, <lb/>vnde intercepta applicata OD, maior erit intercepta PN vertici B propin- <pb o="71" file="0095" n="95" rhead=""/> quiori, & </s> <s xml:id="echoid-s2436" xml:space="preserve">hoc ſemper, &</s> <s xml:id="echoid-s2437" xml:space="preserve">c. </s> <s xml:id="echoid-s2438" xml:space="preserve">Quapropter huiuſmodi Hyperbolæ ſunt ſemper <lb/>ſimul recedentes. </s> <s xml:id="echoid-s2439" xml:space="preserve">Quod ſecundò. </s> <s xml:id="echoid-s2440" xml:space="preserve">&</s> <s xml:id="echoid-s2441" xml:space="preserve">c.</s> <s xml:id="echoid-s2442" xml:space="preserve"/> </p> <div xml:id="echoid-div228" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0094-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0094-02" xlink:href="note-0094-02a" xml:space="preserve">38. h.</note> <note symbol="c" position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">39. h.</note> <note symbol="d" position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">25. ſe-<lb/>cundi co-<lb/>nic</note> <note symbol="e" position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">10. pri-<lb/>mi. conic.</note> </div> <p> <s xml:id="echoid-s2443" xml:space="preserve">Præterea ſit TX aſymptotos inſcriptæ DBE, & </s> <s xml:id="echoid-s2444" xml:space="preserve">VZ aſymptotos circum-<lb/>ſcriptæ, quæ contingentem GB productam ſecent in X, Z; </s> <s xml:id="echoid-s2445" xml:space="preserve">& </s> <s xml:id="echoid-s2446" xml:space="preserve">cum huiuſmo-<lb/>di Hyperbole ſint ſimiles, ſintque earum aſymptoti VZ, TX ad partes ęqua-<lb/>lium inclinationum ductæ, erit angulus ZVB <anchor type="note" xlink:href="" symbol="a"/> æqualis angulo XTB, quare <anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/> TX æquidiſtat VZ, ſed eſt VZ. </s> <s xml:id="echoid-s2447" xml:space="preserve">Aſymptotos circumſcriptæ, vnde TX pro-<lb/>ducta <anchor type="note" xlink:href="" symbol="b"/> ſecabit circumſcriptam Hyperbolen ABC; </s> <s xml:id="echoid-s2448" xml:space="preserve">ſecet ergo eam in 2, &</s> <s xml:id="echoid-s2449" xml:space="preserve"> <anchor type="note" xlink:label="note-0095-02a" xlink:href="note-0095-02"/> per 2 applicetur 3 2 4 5 alteram aſymptoton, inſcriptam ſectionem, ac <lb/>diametrum ſecans in 3,4,5 dico huiuſmodi Hyperbolas, licet ſemper inter <lb/>ſe magis recedant, nnnquam tamen ad interuallum peruenire æquale inter-<lb/>uallo 3 2, quod inter æquidiſtantes aſymptotos intercedit, & </s> <s xml:id="echoid-s2450" xml:space="preserve">iuxta ordina-<lb/>tim ductas metitur.</s> <s xml:id="echoid-s2451" xml:space="preserve"/> </p> <div xml:id="echoid-div229" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">Coroll. <lb/>40. h.</note> <note symbol="b" position="right" xlink:label="note-0095-02" xlink:href="note-0095-02a" xml:space="preserve">11. h.</note> </div> <p> <s xml:id="echoid-s2452" xml:space="preserve">Nam cum in ſimilibus Hyperbolis ABC, DBE, ex æqualibus, immo ex <lb/>eodem diametri ſegmento B 5, ducta ſit quædam applicata 5 4 2 3 ſimi-<lb/>limium Hyperbolarum aſymptotos ſecans in 2, 3; </s> <s xml:id="echoid-s2453" xml:space="preserve">erit <anchor type="note" xlink:href="" symbol="c"/> intercepta huius <anchor type="note" xlink:label="note-0095-03a" xlink:href="note-0095-03"/> applicatæ portio 3 2 in Hyperbola maiorum laterum, maior intercepta <lb/>portione 2 4, in Hyperbola minorum. </s> <s xml:id="echoid-s2454" xml:space="preserve">Ampliùs applicata infra 3 2 4 5, <lb/>qualibet alia 6 7 8 9; </s> <s xml:id="echoid-s2455" xml:space="preserve">erit ob eandem rationem, & </s> <s xml:id="echoid-s2456" xml:space="preserve">portio 6 7 maior por-<lb/>tione 8 9, quare addita communi 7 8; </s> <s xml:id="echoid-s2457" xml:space="preserve">erit 6 8 ſiue 3 2 maior 7 9, & </s> <s xml:id="echoid-s2458" xml:space="preserve"><lb/>hoc ſemper, vbicunque ſit intercepta 8 9 infra 2 4 licet ipſae; </s> <s xml:id="echoid-s2459" xml:space="preserve">interceptæ <lb/>continuè augeantur. </s> <s xml:id="echoid-s2460" xml:space="preserve">Vnde ſimiles Hyperbolæ per eundem verticem ſimul <lb/>adſcriptę, quamuis ſint ſemper magis recedentes ad interuallum tamen non <lb/>perueniunt æquale cuidam dato interuallo. </s> <s xml:id="echoid-s2461" xml:space="preserve">Quod erat vltimò, &</s> <s xml:id="echoid-s2462" xml:space="preserve">c.</s> <s xml:id="echoid-s2463" xml:space="preserve"/> </p> <div xml:id="echoid-div230" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-0095-03" xlink:href="note-0095-03a" xml:space="preserve">40. h.</note> </div> </div> <div xml:id="echoid-div232" type="section" level="1" n="105"> <head xml:id="echoid-head110" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s2464" xml:space="preserve">HInc eſt, quod ſimiles Hyperbolæ per eundem verticem ſimul adſcriptę <lb/>habent aſymptotos parallelas, & </s> <s xml:id="echoid-s2465" xml:space="preserve">aſymptotos inſcriptæ ſecat Hyper-<lb/>bolen circumſcriptam: </s> <s xml:id="echoid-s2466" xml:space="preserve">nam vltimò loco oſtẽdimus TX ęquidiſtare ipſi VZ, <lb/>& </s> <s xml:id="echoid-s2467" xml:space="preserve">ſecare inſcriptam in 2.</s> <s xml:id="echoid-s2468" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div233" type="section" level="1" n="106"> <head xml:id="echoid-head111" xml:space="preserve">THEOR. XXII. PROP. XXXXII.</head> <p> <s xml:id="echoid-s2469" xml:space="preserve">Parabolæ congruentes, per diuerſos vertices ſimul adſcriptæ, <lb/>ſunt inter ſe nunquam coeuntes, & </s> <s xml:id="echoid-s2470" xml:space="preserve">in infinitum productæ ad ſe pro-<lb/>pius accedunt, & </s> <s xml:id="echoid-s2471" xml:space="preserve">ad interuallum perueniunt minus quolibet dato <lb/>interuallo.</s> <s xml:id="echoid-s2472" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2473" xml:space="preserve">SInt duæ congruentes Parabolæ ABC, DEF per diuerſos vertices B, E, <lb/>ſimul adſcriptæ, quarum recta latera ſint BG, EH (quæ inter ſe æqua-<lb/>lia <anchor type="note" xlink:href="" symbol="a"/> erunt, cum ſectiones ponantur congruentes.) </s> <s xml:id="echoid-s2474" xml:space="preserve">Dico primùm has in in- <anchor type="note" xlink:label="note-0095-04a" xlink:href="note-0095-04"/> finitum productas nunquam inter ſe conuenire.</s> <s xml:id="echoid-s2475" xml:space="preserve"/> </p> <div xml:id="echoid-div233" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0095-04" xlink:href="note-0095-04a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <p> <s xml:id="echoid-s2476" xml:space="preserve">Nam producta contingente HE ſectioni ABC occurrent in A, & </s> <s xml:id="echoid-s2477" xml:space="preserve">C, hæc <lb/>erit quoque ordinatim ducta in ſectione ABC (cum ſint ſectiones ſimul ad-<lb/>ſcriptæ) & </s> <s xml:id="echoid-s2478" xml:space="preserve">Parabole DEF tota cadet infra contingentem AEC, ſumptoque <pb o="72" file="0096" n="96" rhead=""/> in ipſa DEF, quocunque puncto D, per ipſum ordinatim applicetur ODS, <lb/>alteram ſectionem ſecans in S: </s> <s xml:id="echoid-s2479" xml:space="preserve">erit quadratum SO <anchor type="note" xlink:href="" symbol="a"/> æquale rectãgulo OBG, <anchor type="note" xlink:label="note-0096-01a" xlink:href="note-0096-01"/> & </s> <s xml:id="echoid-s2480" xml:space="preserve">quadratum DO rectangulo OEH, ſed rectangulum OBG maius eſt rectã-<lb/>gulo OEH, cum latitudo BG æqualis ſit EH, altitudo verò BO maior EO, <lb/>quare SO quadratum, maius eſt quadrato DO; </s> <s xml:id="echoid-s2481" xml:space="preserve">vnde punctum D cadit intra <lb/>Parabolen BA, & </s> <s xml:id="echoid-s2482" xml:space="preserve">ſic de quibuslibet alijs pũctis Parabolæ DEF; </s> <s xml:id="echoid-s2483" xml:space="preserve">ergo huiuſ-<lb/>modi ſectiones inter ſe nunquam conueniunt. </s> <s xml:id="echoid-s2484" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s2485" xml:space="preserve">c.</s> <s xml:id="echoid-s2486" xml:space="preserve"/> </p> <div xml:id="echoid-div234" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0096-01" xlink:href="note-0096-01a" xml:space="preserve">1. huius.</note> </div> <p> <s xml:id="echoid-s2487" xml:space="preserve">Has tamen dico, licet in infinitum productas, infra contingentem EA ad <lb/>ſe propiùs accedere: </s> <s xml:id="echoid-s2488" xml:space="preserve">Ducta enim DM parallela ad ON, & </s> <s xml:id="echoid-s2489" xml:space="preserve">per M applicata <lb/>MN, fiet parallelogrammum DN, cuius oppoſita latera MN, DO æqualia <lb/>erunt. </s> <s xml:id="echoid-s2490" xml:space="preserve">Iam quadratum MN, <anchor type="note" xlink:href="" symbol="b"/> ſiue rectangulum NBG æquatur quadrato <anchor type="note" xlink:label="note-0096-02a" xlink:href="note-0096-02"/> DO, <anchor type="note" xlink:href="" symbol="c"/> ſiue rectangulo OEH, ſed horum latera BG, EH æqualia ſunt, quare <anchor type="note" xlink:label="note-0096-03a" xlink:href="note-0096-03"/> & </s> <s xml:id="echoid-s2491" xml:space="preserve">latera BN, EO æqualia erunt: </s> <s xml:id="echoid-s2492" xml:space="preserve">itaque per proſtaphereſim, erit BE æqualis <lb/>NO, ſed eſt quoque MD æqualis eidem NO, igitur BE, & </s> <s xml:id="echoid-s2493" xml:space="preserve">MD inter ſe ſunt <lb/> <anchor type="figure" xlink:label="fig-0096-01a" xlink:href="fig-0096-01"/> æquales, at ſunt quoque parallelæ, igitur coniunctæ BM, & </s> <s xml:id="echoid-s2494" xml:space="preserve">ED æquales <lb/>erunt, & </s> <s xml:id="echoid-s2495" xml:space="preserve">parallelæ, ſed BM ſecat NM, quare producta ſecabit quoque alte-<lb/>ram parallelarum OD, ſed extra ſectionem BMA (cum ſit BM intra ſectio-<lb/>nem, producta verò, tota cadat extra) ſit ergo occurſus cum ODS in P, & </s> <s xml:id="echoid-s2496" xml:space="preserve"><lb/>cum contingente EA in T; </s> <s xml:id="echoid-s2497" xml:space="preserve">& </s> <s xml:id="echoid-s2498" xml:space="preserve">in ſecunda figura, in qua contingens EA cadit <lb/>inter applicatas MN, OS, iungatur SM ſecans EA in V.</s> <s xml:id="echoid-s2499" xml:space="preserve"/> </p> <div xml:id="echoid-div235" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-0096-02" xlink:href="note-0096-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="left" xlink:label="note-0096-03" xlink:href="note-0096-03a" xml:space="preserve">ibidem.</note> <figure xlink:label="fig-0096-01" xlink:href="fig-0096-01a"> <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0096-01"/> </figure> </div> <p> <s xml:id="echoid-s2500" xml:space="preserve">Iam in prima figura, cum in parallelogrammo PE latera ET, DP, ſint æ-<lb/>qualia, ſitque EA maius ET, erit EA quoque maius DP, eſtque DP maius <lb/>intercepto ſegmento DS, quare AE, eò maius erit ipſo DS. </s> <s xml:id="echoid-s2501" xml:space="preserve">In ſecunda au-<lb/>tem figura, cum pariter ET, DP ſint æquales, ſitque ablata TV minor abla-<lb/>ta PS, erit reliqua EV maior reliqua DS, & </s> <s xml:id="echoid-s2502" xml:space="preserve">eò magis EA maior eadem DS. <lb/></s> <s xml:id="echoid-s2503" xml:space="preserve">Non abſimili modò oſtendetur quamcunque interceptam XY infra SD, mi-<lb/>norem eſſe ipſa SD; </s> <s xml:id="echoid-s2504" xml:space="preserve">ducta enim YZ æquidiſtanter EB, demonſtrabitur item <lb/>YZ æqualis eidem BE, ideoque YZ, & </s> <s xml:id="echoid-s2505" xml:space="preserve">DM inter ſe æquales erunt, & </s> <s xml:id="echoid-s2506" xml:space="preserve">pa- <pb o="73" file="0097" n="97" rhead=""/> rallelæ, ex quò ſi iungatur MZ, & </s> <s xml:id="echoid-s2507" xml:space="preserve">DY, ipſæ æquales, & </s> <s xml:id="echoid-s2508" xml:space="preserve">parallelæ erunt, & </s> <s xml:id="echoid-s2509" xml:space="preserve"><lb/>facta conſtructione vt ſupra, idem omninò demonſtrabitur, nempe interce-<lb/>ptam YX minorem adhuc eſſe ipſa DS. </s> <s xml:id="echoid-s2510" xml:space="preserve">Huiuſmodi igitur Parabolæ con-<lb/>gruentes, quò magis à tangente EA remouentur ad ſe propiùs accedunt: <lb/></s> <s xml:id="echoid-s2511" xml:space="preserve">quod ſecundò, &</s> <s xml:id="echoid-s2512" xml:space="preserve">c.</s> <s xml:id="echoid-s2513" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div237" type="section" level="1" n="107"> <head xml:id="echoid-head112" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2514" xml:space="preserve">SEd hoc idem aliter in nouo hoc ſchemate, in quo item oſtendetur inter-<lb/>ceptam contingentem EA maiorem eſſe intercepta applicata DI, & </s> <s xml:id="echoid-s2515" xml:space="preserve">DI <lb/>maiorem infra intercepta <lb/> <anchor type="figure" xlink:label="fig-0097-01a" xlink:href="fig-0097-01"/> ML, & </s> <s xml:id="echoid-s2516" xml:space="preserve">hoc ſemper, ſi ſectio-<lb/>nes in infinitum producan-<lb/>tur. </s> <s xml:id="echoid-s2517" xml:space="preserve">Ducta enim DN paral-<lb/>lela ad EB, eadem penitus <lb/>methodo, qua ſuperiùs vſi <lb/>ſumus, demonſtrabimus DN <lb/>ipſi EB æqualem eſſe, & </s> <s xml:id="echoid-s2518" xml:space="preserve">pa-<lb/>rallelam, quare, & </s> <s xml:id="echoid-s2519" xml:space="preserve">coniun-<lb/>ctæ BN, ED æquales erunt, <lb/>ac parallelæ: </s> <s xml:id="echoid-s2520" xml:space="preserve">ſi ergo BN ſe-<lb/>cetur bifariam in O, duca-<lb/>turque POT diametro BE <lb/>æquidiſtans, patet ipſam <lb/>TOP eſſe <anchor type="note" xlink:href="" symbol="a"/> vtriuſque Para- <anchor type="note" xlink:label="note-0097-01a" xlink:href="note-0097-01"/> bolæ diametrum, & </s> <s xml:id="echoid-s2521" xml:space="preserve">BN eſſe <lb/>vnam ei applicatarum in <lb/>Parabola ABC, vti etiam QDER ipſi NB æquidiſtantem: </s> <s xml:id="echoid-s2522" xml:space="preserve">quapropter QP, <lb/>& </s> <s xml:id="echoid-s2523" xml:space="preserve">PR æquales erunt, ſed eſt DP æqualis PE (ob parallelas, & </s> <s xml:id="echoid-s2524" xml:space="preserve">quia NO <lb/>æquatur OB) quare reliquæ QD, ER æquales erunt, ideoque rectangulum <lb/>QDR æquabitur rectangulo QER. </s> <s xml:id="echoid-s2525" xml:space="preserve">Ampliùs ducatur TV æquidiſtans ad <lb/>QR, vel ad NB: </s> <s xml:id="echoid-s2526" xml:space="preserve">patet TV ſectionem <anchor type="note" xlink:href="" symbol="b"/> contingere in T, & </s> <s xml:id="echoid-s2527" xml:space="preserve">contingenti GB <anchor type="note" xlink:label="note-0097-02a" xlink:href="note-0097-02"/> occurrere in V, (nam hæc, cum ſecet in B alteram parallelarum BN, ſecat <lb/>quoque reliquam TV.) </s> <s xml:id="echoid-s2528" xml:space="preserve">Cumque duæ rectæ TV, BV, ſectionem ABC con-<lb/>tingentes, in vnum conueniant, ſitque QR ipſi TV, atque IS, & </s> <s xml:id="echoid-s2529" xml:space="preserve">AC ipſi BV <lb/>æquidiſtantes, ac ſe mutuò ſecantes in D, & </s> <s xml:id="echoid-s2530" xml:space="preserve">E, erit rectangulum QDR ad <lb/>IDS, <anchor type="note" xlink:href="" symbol="c"/> vt quadratum TV ad BV quadratum, itemque rectangulum QER ad <anchor type="note" xlink:label="note-0097-03a" xlink:href="note-0097-03"/> AEC, <anchor type="note" xlink:href="" symbol="d"/> vt idem quadratum TV ad BV, quare vt rectãgulum QDR ad IDS, <anchor type="note" xlink:label="note-0097-04a" xlink:href="note-0097-04"/> ita rectangulum QER ad AEC, & </s> <s xml:id="echoid-s2531" xml:space="preserve">permutando rectangulum QDR ad QER, <lb/>vt rectangulum IDS ad AEC, ſed QDR, QER ſunt ęqualia, vt modò oſten-<lb/>dimus, ergo & </s> <s xml:id="echoid-s2532" xml:space="preserve">rectangulum IDS æquatur rectangulo AEC, ſiue quadrato <lb/>AE, quare vt SD ad EA, ita EA ad DI, ſed SD maior eſt EA, cum <anchor type="note" xlink:href="" symbol="e"/> ſit SX <anchor type="note" xlink:label="note-0097-05a" xlink:href="note-0097-05"/> maior CE ſiue EA, vnde AE quoque, maior erit DI. </s> <s xml:id="echoid-s2533" xml:space="preserve">Eadem ratione oſten-<lb/>detur rectangulum LMX æquale quadrato AE: </s> <s xml:id="echoid-s2534" xml:space="preserve">vnde rectangula IDS, LMY <lb/>inter ſe æqualia erunt, ſed eſt latus MY maius later@ DS, cum eius ſegmen-<lb/>tum ZY <anchor type="note" xlink:href="" symbol="f"/> maius ſit huius ſegmento XS, & </s> <s xml:id="echoid-s2535" xml:space="preserve">reliquum ſegmentum MZ maius <anchor type="note" xlink:label="note-0097-06a" xlink:href="note-0097-06"/> reliquo ſegmento DX, quare latus LM minus erit latere ID, & </s> <s xml:id="echoid-s2536" xml:space="preserve">ſemper, quò <pb o="74" file="0098" n="98" rhead=""/> interceptæ applicatarum portiones à contingente AE magis remouentur eò <lb/>ſunt minores, vnde tales ſectiones ad ſe propiùs accedunt. </s> <s xml:id="echoid-s2537" xml:space="preserve">Sed quod de <lb/>congruentibus, ſiue æqualibus parabolis hactenus expoſuimus, & </s> <s xml:id="echoid-s2538" xml:space="preserve">iam olim <lb/>demonſtrauimus (dum Aſymptoton doctrina promoueri poſſe animaduer-<lb/>timus) maximos poſtea Geometras, Torricellium nempe, ac Gregorium à <lb/>S. </s> <s xml:id="echoid-s2539" xml:space="preserve">Vincentio aliter quoque oſtendiſſe reperimus, quorum edita opera ad <lb/>vberiorem de hac re eruditionem conſulere ſuademus.</s> <s xml:id="echoid-s2540" xml:space="preserve"/> </p> <div xml:id="echoid-div237" type="float" level="2" n="1"> <figure xlink:label="fig-0097-01" xlink:href="fig-0097-01a"> <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0097-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0097-01" xlink:href="note-0097-01a" xml:space="preserve">46. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0097-02" xlink:href="note-0097-02a" xml:space="preserve">32. pri-<lb/>mi conic.</note> <note symbol="c" position="right" xlink:label="note-0097-03" xlink:href="note-0097-03a" xml:space="preserve">17. tertil <lb/>conic.</note> <note symbol="d" position="right" xlink:label="note-0097-04" xlink:href="note-0097-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0097-05" xlink:href="note-0097-05a" xml:space="preserve">32. h.</note> <note symbol="f" position="right" xlink:label="note-0097-06" xlink:href="note-0097-06a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s2541" xml:space="preserve">Dico tandem has con-<lb/> <anchor type="figure" xlink:label="fig-0098-01a" xlink:href="fig-0098-01"/> gruentes Parabolas ad in-<lb/>teruallum ſimul peruenire <lb/>minus quocunque dato in-<lb/>teruallo 1 2. </s> <s xml:id="echoid-s2542" xml:space="preserve">Fiat enim vt <lb/>1 2 ad AE, ita AE ad 2 3 <lb/>quæ ipſi 1 2 indirectum po-<lb/>natur, & </s> <s xml:id="echoid-s2543" xml:space="preserve">tota 1 3 bifariam <lb/>ſecetur in 4, & </s> <s xml:id="echoid-s2544" xml:space="preserve">per B appli-<lb/>cetur BK ęqualis 1 4; </s> <s xml:id="echoid-s2545" xml:space="preserve">aga-<lb/>turque KI parallela ad BX, <lb/>& </s> <s xml:id="echoid-s2546" xml:space="preserve">per I recta IDS contingẽti <lb/>BK æquidiſtans, erit ergo <lb/>IX æqualis KB, ſiue 4 1; <lb/></s> <s xml:id="echoid-s2547" xml:space="preserve">eſtque IX dimidium IS, & </s> <s xml:id="echoid-s2548" xml:space="preserve"><lb/>4 1 dimidium 1 3; </s> <s xml:id="echoid-s2549" xml:space="preserve">quare <lb/>IS, 1 3 ſunt æquales; </s> <s xml:id="echoid-s2550" xml:space="preserve">ſed <lb/>factum eſt rectangulum 1 2 3 æquale quadrato AE, & </s> <s xml:id="echoid-s2551" xml:space="preserve">rectangulum IDS <lb/>oſtenſum eſt æquale eidem quadrato AE, ergo rectangula IDS, 1 2 3 in-<lb/>ter ſe ſunt æqualia; </s> <s xml:id="echoid-s2552" xml:space="preserve">ſed rectæ IS, 1 3, ſunt æquales, quare ſegmentum ID <lb/>æquatur dato interuallo 1 2; </s> <s xml:id="echoid-s2553" xml:space="preserve">interceptæ verò infra ID ſunt minores ipſa <lb/>intercepta ID, quapropter huiuſmodi congruentes Parabolę ad interuallum <lb/>perueniunt minus dato interuallo 1 2. </s> <s xml:id="echoid-s2554" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s2555" xml:space="preserve"/> </p> <div xml:id="echoid-div238" type="float" level="2" n="2"> <figure xlink:label="fig-0098-01" xlink:href="fig-0098-01a"> <image file="0098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0098-01"/> </figure> </div> </div> <div xml:id="echoid-div240" type="section" level="1" n="108"> <head xml:id="echoid-head113" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s2556" xml:space="preserve">EXhac patet, in congruentibus Parabolis per diuerſos vertices ſimul ad-<lb/>ſcriptis, omnes, inter eas, interceptas lineas communi diametro ęqui-<lb/>diſtanter ductas, eſſe inter ſe æquales, quales ſunt EB, DN, &</s> <s xml:id="echoid-s2557" xml:space="preserve">c.</s> <s xml:id="echoid-s2558" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div241" type="section" level="1" n="109"> <head xml:id="echoid-head114" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s2559" xml:space="preserve">PAtet quoque, ex penultima parte huius, rectangula ſegmentorum ap-<lb/>plicatarum vtranque Parabolen ſecantium omnia inter ſe æqualia eſſe, <lb/>qualia ſunt rectangula LMY, IDS, &</s> <s xml:id="echoid-s2560" xml:space="preserve">c.</s> <s xml:id="echoid-s2561" xml:space="preserve"/> </p> <pb o="75" file="0099" n="99" rhead=""/> </div> <div xml:id="echoid-div242" type="section" level="1" n="110"> <head xml:id="echoid-head115" xml:space="preserve">LEMMA V. PROP. XXXXIII.</head> <p> <s xml:id="echoid-s2562" xml:space="preserve">Si duo triangula ABC, DEF, habuerint circa angulos B, E, <lb/>latera AB, BE, item@altera BC, EF inter ſe æqualia, & </s> <s xml:id="echoid-s2563" xml:space="preserve">in angulis <lb/>BAC, EDF applicatæ ſint GH, IL parallelæ ad BC, EF, ſitque <lb/>rectangulum BGH æquale rectangulo EIL. </s> <s xml:id="echoid-s2564" xml:space="preserve">Dico latera BG, EI <lb/>inter ſe æqualia eſſe.</s> <s xml:id="echoid-s2565" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2566" xml:space="preserve">SEd conſultò omiſſa, præter mei inſtituti morem, affirmatiua demonſtra-<lb/>tione, libet potiùs indirectam, ac breuiorem afferre, ſimulque egregiæ <lb/>indolis ſpecimen exhibere nobiliſsimi, ac ingenioſiſsimi Romani Adoleſcentis, <lb/>Bruti Annibali della Molara, ex ſelectiſsimis Ephebis SERENISSIMO <lb/>MAGNO DVCI miniſtrantibus, de quo non auſim aſſerere, quæ ſint ei <lb/>maioris oblectamenti, an equeſtrium exercitationum ornamenta, quibus <lb/>elegantiſsimè inſignitur, an mathematicæ contemplationes, dum, etiam <lb/>inter Aulæ ſtrepitus, pacatos ſubtilioris Geometriæ nouit inuenire receſſus, <lb/>prout varia teſtantur problemata, ac theoremata, à me identidem ei propoſi-<lb/>ta, & </s> <s xml:id="echoid-s2567" xml:space="preserve">ab ipſo quàm feliciter ſoluta, quorum, licet facillimum, poſteriori <lb/>tamen inſeruiens hic habes.</s> <s xml:id="echoid-s2568" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2569" xml:space="preserve">ESto, ſi fieri poteſt, alterum ipſorum laterum, quale eſt BG, maius alte-<lb/>ro EI: </s> <s xml:id="echoid-s2570" xml:space="preserve">habebit ergo GB ad BA, maiorem rationem quam IE ad ED ipſi <lb/>BA æqualem, & </s> <s xml:id="echoid-s2571" xml:space="preserve">componendo GA <lb/> <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/> ad AB maiorem rationem quàm ID <lb/>ad DE, ſed GA ad AB, eſt vt GH ad <lb/>BC, & </s> <s xml:id="echoid-s2572" xml:space="preserve">ID ad DE, vt IL ad EF; </s> <s xml:id="echoid-s2573" xml:space="preserve">ergo <lb/>GH ad BC habet maiorem rationem <lb/>quàm IL ad EF, hoc eſt ad ſibi æqua-<lb/>lem BC, quare GH erit maior IL, & </s> <s xml:id="echoid-s2574" xml:space="preserve"><lb/>ponitur BG maior EI, vnde rectan-<lb/>gulum BGH maius eſt rectangulo <lb/>EIL: </s> <s xml:id="echoid-s2575" xml:space="preserve">quod eſt contra hypoteſim. <lb/></s> <s xml:id="echoid-s2576" xml:space="preserve">Sunt ergo BG, EI interſe æquales. </s> <s xml:id="echoid-s2577" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s2578" xml:space="preserve">c.</s> <s xml:id="echoid-s2579" xml:space="preserve"/> </p> <div xml:id="echoid-div242" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0099-01"/> </figure> </div> <pb o="76" file="0100" n="100" rhead=""/> </div> <div xml:id="echoid-div244" type="section" level="1" n="111"> <head xml:id="echoid-head116" xml:space="preserve">THEOR. XXIII. PROP. XXXXIV.</head> <p> <s xml:id="echoid-s2580" xml:space="preserve">Hyperbolæ congruentes, per diuerſos vertices ſimul adſcriptæ, <lb/>ſunt inter ſe nunquam coeuntes, & </s> <s xml:id="echoid-s2581" xml:space="preserve">ſemper ſimul accedentes: </s> <s xml:id="echoid-s2582" xml:space="preserve">ſed <lb/>ad interuallum nunquam perueniunt æquale cuidam dato inter-<lb/>uallo.</s> <s xml:id="echoid-s2583" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2584" xml:space="preserve">SInt duæ congruentes Hyperbolæ ABC, DEF per diuerlos vertices B, E <lb/>ſimul adſcriptæ, quarum recta latera ſint BG, EH (quæ inter ſe æqua-<lb/>lia erunt) & </s> <s xml:id="echoid-s2585" xml:space="preserve">ipſarum tranſuerſa ſint BI, EL (quæ item æqualia erunt <anchor type="note" xlink:href="" symbol="a"/> cum <anchor type="note" xlink:label="note-0100-01a" xlink:href="note-0100-01"/> ſectiones ponantur congruentes.) </s> <s xml:id="echoid-s2586" xml:space="preserve">Dico primùm has ſectiones nunquam <lb/>inter ſe conuenire.</s> <s xml:id="echoid-s2587" xml:space="preserve"/> </p> <div xml:id="echoid-div244" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0100-01" xlink:href="note-0100-01a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> </div> <figure> <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0100-01"/> </figure> <p> <s xml:id="echoid-s2588" xml:space="preserve">Nam producta contingente HE donec ſectioni ABC occurrat in A, & </s> <s xml:id="echoid-s2589" xml:space="preserve">C, <lb/>ipſa quoque erit ordinata in ſectione ABC (cum ſint ſectiones ſimul adſcri-<lb/>ptæ) & </s> <s xml:id="echoid-s2590" xml:space="preserve">ſectio DEF tota cadet infra contingentem AEC; </s> <s xml:id="echoid-s2591" xml:space="preserve">ſumptoque in ipſa <lb/>ED quocunque puncto D, applicetur SDO, quæ iunctis regulis IG, LH oc-<lb/>currat in K, R; </s> <s xml:id="echoid-s2592" xml:space="preserve">& </s> <s xml:id="echoid-s2593" xml:space="preserve">cum ſit triangulum IBG ſimile triangulo LEH (habent <lb/>enim circa æquales angulos B, E, æqualia latera, vtrunque vtrique) erit <lb/>angulus BIG æqualis angulo ELH, vnde regulæ IGK, LHR æquidiſtant, <lb/>ideoque IK cadit extra LR, cum ſit punctum I ſupra L, ergo OK maior eſt <lb/>OR, ſed eſt OB maior OE, igitur rectangulum BOK <anchor type="note" xlink:href="" symbol="b"/> ſiue quadratum SO, <anchor type="note" xlink:label="note-0100-02a" xlink:href="note-0100-02"/> maius eſt rectangulo, EOR <anchor type="note" xlink:href="" symbol="c"/> ſiue quadrato DO; </s> <s xml:id="echoid-s2594" xml:space="preserve">hoc eſt punctum D cadit in- <anchor type="note" xlink:label="note-0100-03a" xlink:href="note-0100-03"/> tra ſectionem ED, & </s> <s xml:id="echoid-s2595" xml:space="preserve">ſic de quocunque alio puncto eiuſdem ſectionis infra <lb/>contingentem EA: </s> <s xml:id="echoid-s2596" xml:space="preserve">quapropter huiuſmodi Hyperbolæ inter ſe nunquam <lb/>conueniunt. </s> <s xml:id="echoid-s2597" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s2598" xml:space="preserve">c.</s> <s xml:id="echoid-s2599" xml:space="preserve"/> </p> <div xml:id="echoid-div245" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0100-02" xlink:href="note-0100-02a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="c" position="left" xlink:label="note-0100-03" xlink:href="note-0100-03a" xml:space="preserve">ibidem.</note> </div> <pb o="77" file="0101" n="101" rhead=""/> <p> <s xml:id="echoid-s2600" xml:space="preserve">Ampliùs dico ipſam DEF quò longiùs aberit à vertice E infra EA, eò <lb/>magis appropinquare ſectioni B A M. </s> <s xml:id="echoid-s2601" xml:space="preserve">quoniam ducta D M parallela ad <lb/>OEB, & </s> <s xml:id="echoid-s2602" xml:space="preserve">MN ad DO, fiet parallelogrammum DN, cuius oppoſita latera <lb/>MN, DO æqualia erunt: </s> <s xml:id="echoid-s2603" xml:space="preserve">Itaqueregulæ IG occurrat producta MN in Q, & </s> <s xml:id="echoid-s2604" xml:space="preserve"><lb/>regulæ LH producta DO in R: </s> <s xml:id="echoid-s2605" xml:space="preserve">cum ſit oſtenſa MN æqualis DO, erit qua-<lb/>dratum MN <anchor type="note" xlink:href="" symbol="a"/> ſiue rectangulum BNQ, æquale quadrato DO <anchor type="note" xlink:href="" symbol="b"/> ſiue rectangu- <anchor type="note" xlink:label="note-0101-01a" xlink:href="note-0101-01"/> <anchor type="note" xlink:label="note-0101-02a" xlink:href="note-0101-02"/> lo EOR: </s> <s xml:id="echoid-s2606" xml:space="preserve">ſed in triangulis IBG, LEH ſunt latera IB, LE, & </s> <s xml:id="echoid-s2607" xml:space="preserve">BG, EH inter ſe <lb/>æqualia, alterum alteri, quapropter æqualium rectangulorum BNQ, EOR <lb/>latera BN, & </s> <s xml:id="echoid-s2608" xml:space="preserve">EO <anchor type="note" xlink:href="" symbol="c"/> æqualia erunt: </s> <s xml:id="echoid-s2609" xml:space="preserve">quare cum diametri ſegmenta BN, EO <anchor type="note" xlink:label="note-0101-03a" xlink:href="note-0101-03"/> ſint æqualia, facta proſtaphereſi, proueniet BE æqualis NO, ſed eſt quoque <lb/>MD æqualis NO in parallelogrammo DN, igitur rectæ BE, & </s> <s xml:id="echoid-s2610" xml:space="preserve">MD inter ſe <lb/>ſunt æquales, at ſunt etiam parallelæ, ergo coniuncta BM iunctæ ED æqui-<lb/>diſtat, ſed BM ſecat NM, quare producta ſecabit quoque OD, ſed extra ſe-<lb/>ctionem BMA (cum BM ſit intra ſectionem, producta verò tota cadat extra) <lb/>ſitque occurſus in P, & </s> <s xml:id="echoid-s2611" xml:space="preserve">OD occurrat ſectioni BMA in S, PM verò contin-<lb/>gentem EA ſecet in T; </s> <s xml:id="echoid-s2612" xml:space="preserve">& </s> <s xml:id="echoid-s2613" xml:space="preserve">in ſecunda figura, in qua punctum A cadit inter <lb/>puncta S, & </s> <s xml:id="echoid-s2614" xml:space="preserve">M, iungatur SM, quæ cum tota cadat intra ſectionem, neceſſa-<lb/>riò ſecabit applicatam AE: </s> <s xml:id="echoid-s2615" xml:space="preserve">veluti in V.</s> <s xml:id="echoid-s2616" xml:space="preserve"/> </p> <div xml:id="echoid-div246" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0101-01" xlink:href="note-0101-01a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="b" position="right" xlink:label="note-0101-02" xlink:href="note-0101-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="right" xlink:label="note-0101-03" xlink:href="note-0101-03a" xml:space="preserve">43. h.</note> </div> <p> <s xml:id="echoid-s2617" xml:space="preserve">Iam, in prima figura, cum in parallelogrammo P E oppoſita latera ET, <lb/>DP ſint æqualia, ſitque EA maius ET, erit EA quoque maius ipſo DP, ſed <lb/>eſt DP maius intercepto applicatæ ſegmento DS, erit ergo AE, eò maius <lb/>ipſo DS. </s> <s xml:id="echoid-s2618" xml:space="preserve">In ſecunda autem figura cum pariter ET, DP ſint æquales, ſitque <lb/>dempta TV minor dempta PS, erit reliqua EV maior reliqua DS, & </s> <s xml:id="echoid-s2619" xml:space="preserve">eò ma-<lb/>gis EA maior eadem DS. </s> <s xml:id="echoid-s2620" xml:space="preserve">Eodè penitùs modo oſtendetur, quamlibet aliam <lb/>interceptam ZY infra SD minorem eſſe ipſa SD: </s> <s xml:id="echoid-s2621" xml:space="preserve">nam ducta YZ æquidiſtan-<lb/>ter ad EB, demonſtrabitur item YZ æqualem eſſe eidem BE, ac ideo YZ, & </s> <s xml:id="echoid-s2622" xml:space="preserve"><lb/>DM eſſe inter ſe ęquales, & </s> <s xml:id="echoid-s2623" xml:space="preserve">parallelas: </s> <s xml:id="echoid-s2624" xml:space="preserve">ex quo ſi iungantur MZ, & </s> <s xml:id="echoid-s2625" xml:space="preserve">DY, ipſę <lb/>æquales erunt, & </s> <s xml:id="echoid-s2626" xml:space="preserve">parallelæ; </s> <s xml:id="echoid-s2627" xml:space="preserve">completa igitur conſimili conſtructione, ac ſu-<lb/>pra, idem omnino inſequetur, hoc eſt interceptam YX minorem adhuc eſſe <lb/>DS: </s> <s xml:id="echoid-s2628" xml:space="preserve">tales ergo interceptæ quò magis à tangente EA remouentur continuè <lb/>decreſcunt. </s> <s xml:id="echoid-s2629" xml:space="preserve">Quare ſectiones ABC, DEF ſunt ſemper ſimul accedentes. <lb/></s> <s xml:id="echoid-s2630" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s2631" xml:space="preserve">c.</s> <s xml:id="echoid-s2632" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2633" xml:space="preserve">Præterea, ſi ad euitandam in hiſce figuris linearum implicationem, con-<lb/>cipiatur circumſcriptæ Hyperbolæ ABC centrum eſſe I, aſymptoton IG, & </s> <s xml:id="echoid-s2634" xml:space="preserve"><lb/>contingens ex vertice BG; </s> <s xml:id="echoid-s2635" xml:space="preserve">at inſcriptæ DEF centrum L, aſymptoton LH, <lb/>contingens autem ex vertice ſit EH: </s> <s xml:id="echoid-s2636" xml:space="preserve">cum harum ſectionum latera ſint data <lb/>æqualia, erunt quoque ipſorum rectangula inter ſe æqualia, ideoque, & </s> <s xml:id="echoid-s2637" xml:space="preserve">eo-<lb/>rum ſubquadrupla <anchor type="note" xlink:href="" symbol="d"/> hoc eſt quadrata contingentium BG, EH, vnde ipſæ li- <anchor type="note" xlink:label="note-0101-04a" xlink:href="note-0101-04"/> neæ BG, EH æquales erunt, ſed eſt etiam BI æqualis EL (nam vtra eſt dimi-<lb/>dium æqualium verſorum laterum) quare in triangulis IBG, LEH, cum ſint <lb/>latera IB, BG, lateribus LE, EH æqualia, & </s> <s xml:id="echoid-s2638" xml:space="preserve">anguli ad B, E æquales, etiam <lb/>anguli ad baſes I, L æquales erunt, vnde aſymptoti IG, LG inter ſe æqui-<lb/>diſtant; </s> <s xml:id="echoid-s2639" xml:space="preserve">& </s> <s xml:id="echoid-s2640" xml:space="preserve">cum ſit à puncto L, quod eſt intra angulum ab aſymptotis cir-<lb/>cumſcriptæ ſectionis factũ, ducta LH alteri aſymptoto IK æquidiſtans, pro-<lb/>ducta <anchor type="note" xlink:href="" symbol="e"/> ſecabit omnino Hyperbolen ABC: </s> <s xml:id="echoid-s2641" xml:space="preserve">quare LH aſymptotos inſcriptæ <anchor type="note" xlink:label="note-0101-05a" xlink:href="note-0101-05"/> ſecat Hyperbolen circumſcriptam; </s> <s xml:id="echoid-s2642" xml:space="preserve">ſecet ergo in 1, per quod applicetur <lb/>2 1 3: </s> <s xml:id="echoid-s2643" xml:space="preserve">Dico harum ſectionum interuallum infra applicatam 2 1 3 per in- <pb o="78" file="0102" n="102" rhead=""/> tercepta applicatarum ſegmenta metitum, licet ſemper magis, ac magis de-<lb/>creſcat, eſſe tamen non minus interuallo 1 3, quod iuxta eaſdem æquidi-<lb/>ſtantes ordinatim ſectionibus applicatas, inter vtranque aſymptoton cadit. <lb/></s> <s xml:id="echoid-s2644" xml:space="preserve">Nam per ea, quæ infra demonſtrabimus, interuallum 2 1, maius eſt inter-<lb/>uallo 1 3. </s> <s xml:id="echoid-s2645" xml:space="preserve">Pariter 4 5, eſt maius 6 7, communique addito 5 6, erit inter-<lb/>uallum 4 6, maius interuallo 5 7, ſiue 1 3, & </s> <s xml:id="echoid-s2646" xml:space="preserve">hoc ſemper vbicunque ſu-<lb/>matur harum ſectionum interuallum infra applicatam 2 1 3. </s> <s xml:id="echoid-s2647" xml:space="preserve">Quare hy-<lb/>perbolæ congruentes per diuerſos vertices ſimul adſcriptæ, licèt ſemper ma-<lb/>gis accedentes, ad interuallum nunquam perueniunt æquale cuidam dato <lb/>interuallo. </s> <s xml:id="echoid-s2648" xml:space="preserve">Quod erat vltimò, &</s> <s xml:id="echoid-s2649" xml:space="preserve">c.</s> <s xml:id="echoid-s2650" xml:space="preserve"/> </p> <div xml:id="echoid-div247" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-0101-04" xlink:href="note-0101-04a" xml:space="preserve">1. ſecú-<lb/>diconic.</note> <note symbol="e" position="right" xlink:label="note-0101-05" xlink:href="note-0101-05a" xml:space="preserve">11. h.</note> </div> </div> <div xml:id="echoid-div249" type="section" level="1" n="112"> <head xml:id="echoid-head117" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s2651" xml:space="preserve">EX his conſtat, congruentium Hyperbolarum non per eundem verticem <lb/>ſimul adſcriptarum aſymptotos eſſe inter ſe æquidiſtantes, & </s> <s xml:id="echoid-s2652" xml:space="preserve">aſympto-<lb/>ton inſcriptæ ſecare Hyperbolen circumſcriptam.</s> <s xml:id="echoid-s2653" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div250" type="section" level="1" n="113"> <head xml:id="echoid-head118" xml:space="preserve">Quod ſuperiùs promiſimus oſtendetur ſic.</head> <p> <s xml:id="echoid-s2654" xml:space="preserve">SInt duæ congruentes Hyperbolæ KBC, DEF, per diuerſos vertices B, E <lb/>ſimul adſcriptæ, & </s> <s xml:id="echoid-s2655" xml:space="preserve">circumſcriptæ KBC ſit centrum G, & </s> <s xml:id="echoid-s2656" xml:space="preserve">aſymptotos <lb/>GI, inſcriptæ verò ſit centrum H, & </s> <s xml:id="echoid-s2657" xml:space="preserve">aſymptotos HM, quæ ipſi GI æquidi-<lb/>ſtabit, per præcedens Coroll. </s> <s xml:id="echoid-s2658" xml:space="preserve">ſitque applicata quæcunque IL vtranque <lb/>aſymptoton, & </s> <s xml:id="echoid-s2659" xml:space="preserve">Hyperbolen ſecans in I, A, K, D, communemque diame-<lb/>trum in L: </s> <s xml:id="echoid-s2660" xml:space="preserve">dico interceptum applicatę ſegmentum AD inſcriptæ Hyperbolę <lb/>DEF, maius eſſe intercepto eiuſdem applicatæ ſegmento IK inter aſympto-<lb/>ton, & </s> <s xml:id="echoid-s2661" xml:space="preserve">circumſcriptam.</s> <s xml:id="echoid-s2662" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2663" xml:space="preserve">Ducta enim IM parallela ad GHO, & </s> <s xml:id="echoid-s2664" xml:space="preserve">per <lb/> <anchor type="figure" xlink:label="fig-0102-01a" xlink:href="fig-0102-01"/> Mapplicata MNO, erunt IH, IO parallelo-<lb/>gramma, ac ideò tàm GH, quàm LO ipſi IM <lb/>æquales erunt, & </s> <s xml:id="echoid-s2665" xml:space="preserve">inter ſe; </s> <s xml:id="echoid-s2666" xml:space="preserve">quare addita com-<lb/>muni HL, erit GL æqualis HO, ſed eſt GB <lb/>æqualis HE, (cum ſint ſemi-tranſuerſa late-<lb/>ra congruentium Hyperbolarum) vnde reli-<lb/>qua BL, reliquæ EO æqualis erit, & </s> <s xml:id="echoid-s2667" xml:space="preserve">ob id <lb/>ſemi-applicata LK ſemi-applicatę ON ęqua-<lb/>lis, ſed eſt tota LI æqualis totæ OM (cum ſint <lb/>oppoſitæ in parallelogrammo IO) ergo reli-<lb/>quæ KI, NM æquales erunt, ſed eſt <anchor type="note" xlink:href="" symbol="a"/> DA <anchor type="note" xlink:label="note-0102-01a" xlink:href="note-0102-01"/> maior NM, quare & </s> <s xml:id="echoid-s2668" xml:space="preserve">eadem DA erit maior <lb/>KI. </s> <s xml:id="echoid-s2669" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s2670" xml:space="preserve">c.</s> <s xml:id="echoid-s2671" xml:space="preserve"/> </p> <div xml:id="echoid-div250" type="float" level="2" n="1"> <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a"> <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0102-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0102-01" xlink:href="note-0102-01a" xml:space="preserve">10. h.</note> </div> <pb o="79" file="0103" n="103" rhead=""/> </div> <div xml:id="echoid-div252" type="section" level="1" n="114"> <head xml:id="echoid-head119" xml:space="preserve">THEOR. XXIV. PROP. XXXXV.</head> <p> <s xml:id="echoid-s2672" xml:space="preserve">Similes Hyperbolæ per diuerſos vertices ſimul adſcriptæ, & </s> <s xml:id="echoid-s2673" xml:space="preserve"><lb/>quarum eadem ſit regula, ſunt inter ſe nunquam coeuntes, & </s> <s xml:id="echoid-s2674" xml:space="preserve">in in-<lb/>finitum productæ ad ſe propiùs accedentes, ſed ad interuallum <lb/>nunquam perueniunt æquale cuidam dato interuallo.</s> <s xml:id="echoid-s2675" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2676" xml:space="preserve">SInt duæ Hyperbolæ ABC, DEF per diuerſos vertices B, E ſimul adſcri-<lb/>ptæ, quarum eadem ſit regula GH (ſic enim <anchor type="note" xlink:href="" symbol="a"/><unsure/> ſimiles erunt, quoniam <anchor type="note" xlink:label="note-0103-01a" xlink:href="note-0103-01"/> ductis contingentibus BI, EL; </s> <s xml:id="echoid-s2677" xml:space="preserve">eſt tranſuerſum GB ad rectum BI, vt tranſuer-<lb/>ſum GE ad rectum EL.) </s> <s xml:id="echoid-s2678" xml:space="preserve">Dico primùm, has in infinitum productas, nun-<lb/>quam ſimul conuenire.</s> <s xml:id="echoid-s2679" xml:space="preserve"/> </p> <div xml:id="echoid-div252" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0103-01" xlink:href="note-0103-01a" xml:space="preserve">6. ſecúd. <lb/>defin.</note> </div> <figure> <image file="0103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0103-01"/> </figure> <p> <s xml:id="echoid-s2680" xml:space="preserve">Protracta enim contingente LE, ſectionem ABC ſecane in M, N, quæ <lb/>erit ipſi ordinata, cum ſectiones ponantur ſimul adſcriptæ; </s> <s xml:id="echoid-s2681" xml:space="preserve">patet ſectionem <lb/>DEF totam cadere infra contingentem MN. </s> <s xml:id="echoid-s2682" xml:space="preserve">Iam ſumpto in ſectione DEF <lb/>quolibet puncto D, per ipſum ordinatim applicetur ADOH alteram ſectio-<lb/>nem ſecans in A, regulam verò in H: </s> <s xml:id="echoid-s2683" xml:space="preserve">erit quadratum AO, ad quadratum <lb/>DO, vt rectangulum BOH ad rectangulum EOH (ob <anchor type="note" xlink:href="" symbol="b"/> ęqualitatem) vel vt <anchor type="note" xlink:label="note-0103-02a" xlink:href="note-0103-02"/> altitudo BO ad altitudinem EO, ſed eſt BO maior EO, quare quadratum <lb/>AO maius erit quadrato DO, ex quo punctum D cadit intra Hyperbolen <lb/>ABC; </s> <s xml:id="echoid-s2684" xml:space="preserve">& </s> <s xml:id="echoid-s2685" xml:space="preserve">ſic de quibuſcunque alijs punctis Hyperbolę DEF: </s> <s xml:id="echoid-s2686" xml:space="preserve">quare huiuſmo-<lb/>di ſectiones inter ſe nunquam conueniunt. </s> <s xml:id="echoid-s2687" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s2688" xml:space="preserve">c.</s> <s xml:id="echoid-s2689" xml:space="preserve"/> </p> <div xml:id="echoid-div253" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0103-02" xlink:href="note-0103-02a" xml:space="preserve">Coroll. <lb/>1. huius.</note> </div> <p> <s xml:id="echoid-s2690" xml:space="preserve">Iam ſi datæ Hyperbolæ, per verticem E, adſcribatur Hyperbole P E Q <lb/>cuius latera ER, ES æqualia ſint lateribus BG, BI, vtrunque vtrique, ipſæ <lb/>Hyperbolæ ABC, PEQ, congruentes <anchor type="note" xlink:href="" symbol="c"/> erunt, eritque, (ob æqualitatem) <anchor type="note" xlink:label="note-0103-03a" xlink:href="note-0103-03"/> RE ad ES, vt GB ad BI, vel vt GE ad EL, quare Hyperbolæ DEF, PEQ <pb o="80" file="0104" n="104" rhead=""/> erunt ſimiles, at ſunt per verticem E ſimul adſcriptæ, vnde PEQ minorum <lb/>laterum inſcripta <anchor type="note" xlink:href="" symbol="a"/> erit Hyperbolæ DEF maiorum laterum: </s> <s xml:id="echoid-s2691" xml:space="preserve">& </s> <s xml:id="echoid-s2692" xml:space="preserve">infra ADX <anchor type="note" xlink:label="note-0104-01a" xlink:href="note-0104-01"/> applicata quacunque TVP; </s> <s xml:id="echoid-s2693" xml:space="preserve">cum Hyperbolæ ABC, PEQ ſint congruentes, <lb/>& </s> <s xml:id="echoid-s2694" xml:space="preserve">per diuerſos vertices ſimul adſcriptæ <anchor type="note" xlink:href="" symbol="b"/> erit intercepta AX maior interce- <anchor type="note" xlink:label="note-0104-02a" xlink:href="note-0104-02"/> pta TP: </s> <s xml:id="echoid-s2695" xml:space="preserve">cumque Hyperbolæ DEF, PEQ ſint ſimiles, ac per eundem verti-<lb/>cem ſimul adſcriptæ <anchor type="note" xlink:href="" symbol="c"/> erit intercepta DX minor intercepta VP, vnde reliqua <anchor type="note" xlink:label="note-0104-03a" xlink:href="note-0104-03"/> intercepta AD omnino erit maior reliqua intercepta TV; </s> <s xml:id="echoid-s2696" xml:space="preserve">& </s> <s xml:id="echoid-s2697" xml:space="preserve">hoc ſemper: <lb/></s> <s xml:id="echoid-s2698" xml:space="preserve">quare huiuſmodi Hyperbolæ ABC, DEF ſunt ad ſe propiùs accedentes. </s> <s xml:id="echoid-s2699" xml:space="preserve"><lb/>Quod erat ſecundò, &</s> <s xml:id="echoid-s2700" xml:space="preserve">c.</s> <s xml:id="echoid-s2701" xml:space="preserve"/> </p> <div xml:id="echoid-div254" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0103-03" xlink:href="note-0103-03a" xml:space="preserve">1. Co-<lb/>roll. 19. h.</note> <note symbol="a" position="left" xlink:label="note-0104-01" xlink:href="note-0104-01a" xml:space="preserve">5. Co-<lb/>roll. 19. h.</note> <note symbol="b" position="left" xlink:label="note-0104-02" xlink:href="note-0104-02a" xml:space="preserve">44. h.</note> <note symbol="c" position="left" xlink:label="note-0104-03" xlink:href="note-0104-03a" xml:space="preserve">41. h.</note> </div> <figure> <image file="0104-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0104-01"/> </figure> <p> <s xml:id="echoid-s2702" xml:space="preserve">Tandem, bifariam ſectis tranſuerſis lateribus GB, GE, RE, in Y, Z, K, <lb/>erit Y centrum Hyperbolæ ABC, Z verò centrum DEF, ac demum K cen-<lb/>trum PEQ: </s> <s xml:id="echoid-s2703" xml:space="preserve">& </s> <s xml:id="echoid-s2704" xml:space="preserve">cum ſit GB minor GE, erit dimidium GY minus dimidio GZ; <lb/></s> <s xml:id="echoid-s2705" xml:space="preserve">quare punctũ Z cadit infra Y: </s> <s xml:id="echoid-s2706" xml:space="preserve">cumq; </s> <s xml:id="echoid-s2707" xml:space="preserve">ſit EG maior ER, erit dimidiũ EZ maius <lb/>dimidio EK, vnde K punctum cadit infra Z. </s> <s xml:id="echoid-s2708" xml:space="preserve">Si ergo ex Hyperbolarum cen-<lb/>tris Y, Z, ducantur earum aſymptoti Y 2, Z 3, K 4, <anchor type="note" xlink:href="" symbol="d"/> erit Z 3, parallela ad <anchor type="note" xlink:label="note-0104-04a" xlink:href="note-0104-04"/> K 4, & </s> <s xml:id="echoid-s2709" xml:space="preserve">Y 2 <anchor type="note" xlink:href="" symbol="e"/> æquidiſtabit eidem K 4; </s> <s xml:id="echoid-s2710" xml:space="preserve">quare aſymptoti omnes Y 2, Z 3, K4, <anchor type="note" xlink:label="note-0104-05a" xlink:href="note-0104-05"/> erunt inter ſe parallelæ: </s> <s xml:id="echoid-s2711" xml:space="preserve">& </s> <s xml:id="echoid-s2712" xml:space="preserve">cum Y 2 ſit aſymptotos ABC, & </s> <s xml:id="echoid-s2713" xml:space="preserve">Z 3 ſit intra an-<lb/>gulum ab aſymptotis comprehenſum, ipſa ſectionem ABC <anchor type="note" xlink:href="" symbol="f"/> ſecabit, vt in 3, <anchor type="note" xlink:label="note-0104-06a" xlink:href="note-0104-06"/> per quod ordinatim ducta recta 2 3 4, alias aſymptotos ſecantin 2 4, infra <lb/>ipſam applicetur quælibet alia TVP, ſingulas Hyperbolas ſecans in T, V, P. <lb/></s> <s xml:id="echoid-s2714" xml:space="preserve">Erit intercepta TP <anchor type="note" xlink:href="" symbol="g"/> maior ſemper interuallo 2 4, ſed ablata intercepta VP <anchor type="note" xlink:label="note-0104-07a" xlink:href="note-0104-07"/> eſt ſemper <anchor type="note" xlink:href="" symbol="b"/> minor ablato interuallo 3 4, vnde reliqua intercepta TV inter <anchor type="note" xlink:label="note-0104-08a" xlink:href="note-0104-08"/> datas ſectiones A B C, D E F, erit omnino maior reliquo interuallo 2 3, <lb/>quod inter datarum ſectionum parallelas aſymptotos eſt interceptum, ac <lb/>iuxta ordinatim ductis æquidiſtantes metitur. </s> <s xml:id="echoid-s2715" xml:space="preserve">Quod erat vltimò demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s2716" xml:space="preserve"/> </p> <div xml:id="echoid-div255" type="float" level="2" n="4"> <note symbol="d" position="left" xlink:label="note-0104-04" xlink:href="note-0104-04a" xml:space="preserve">Coroll. <lb/>41. huius.</note> <note symbol="e" position="left" xlink:label="note-0104-05" xlink:href="note-0104-05a" xml:space="preserve">Coroll. <lb/>44. h.</note> <note symbol="f" position="left" xlink:label="note-0104-06" xlink:href="note-0104-06a" xml:space="preserve">Coroll. <lb/>11. h.</note> <note symbol="g" position="left" xlink:label="note-0104-07" xlink:href="note-0104-07a" xml:space="preserve">44. h.</note> <note symbol="b" position="left" xlink:label="note-0104-08" xlink:href="note-0104-08a" xml:space="preserve">41. h.</note> </div> <pb o="81" file="0105" n="105" rhead=""/> </div> <div xml:id="echoid-div257" type="section" level="1" n="115"> <head xml:id="echoid-head120" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s2717" xml:space="preserve">EX his patet, ſimilium Hyperbolarum per diuerſos vertices ſimul adſcri-<lb/>ptarum, & </s> <s xml:id="echoid-s2718" xml:space="preserve">quarum eadem ſit regula, aſymptotos eſſe inter ſe paralle-<lb/>las, & </s> <s xml:id="echoid-s2719" xml:space="preserve">aſymptoton inſcriptæ ſecare Hyperbolen circumſcriptam.</s> <s xml:id="echoid-s2720" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div258" type="section" level="1" n="116"> <head xml:id="echoid-head121" xml:space="preserve">LEMMA VI. PROP. XXXXVI.</head> <p> <s xml:id="echoid-s2721" xml:space="preserve">Si in quocunque triangulo ABC ducta ſit quæpiam linea DE <lb/>baſi BC parallela, rectangulum ABC ſuperabit ADE rectangu-<lb/>lo ſub DB, differentia altitudinum, & </s> <s xml:id="echoid-s2722" xml:space="preserve">ſub aggregato baſium <lb/>BC, DE.</s> <s xml:id="echoid-s2723" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2724" xml:space="preserve">PRoducta enim BC, ac ſumpta CF æ-<lb/> <anchor type="figure" xlink:label="fig-0105-01a" xlink:href="fig-0105-01"/> quali ipſi DE, & </s> <s xml:id="echoid-s2725" xml:space="preserve">completis in angulo <lb/>A B C parallelogrammis AE, AC, DF. <lb/></s> <s xml:id="echoid-s2726" xml:space="preserve">Conſtat parallelogrammum AC ſuperare <lb/>parallelogrammum AE gnomone DCG, <lb/>ſed gnomon DCG æquatur parallelogrã-<lb/>mis B E, GC, & </s> <s xml:id="echoid-s2727" xml:space="preserve">GC æquatur DC, ſiue <lb/>EF, quare AC ſuperat AE parallelogram-<lb/>mo DF, hoc eſt rectangulum ABC ſupe-<lb/>rat rectangulum ADE, rectangulo DBF; </s> <s xml:id="echoid-s2728" xml:space="preserve">ſed DB eſt differentia altitudinum, <lb/>& </s> <s xml:id="echoid-s2729" xml:space="preserve">BF aggregatum baſium BC, DE. </s> <s xml:id="echoid-s2730" xml:space="preserve">Quare patet propoſitum.</s> <s xml:id="echoid-s2731" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0105-01"/> </figure> </div> </div> <div xml:id="echoid-div260" type="section" level="1" n="117"> <head xml:id="echoid-head122" xml:space="preserve">THEOR. XXV. PROP. XXXXVII.</head> <p> <s xml:id="echoid-s2732" xml:space="preserve">Similes Hyperbolæ concentricæ per diuerſos vertices ſimul <lb/>adſcriptæ, ſunt inter ſe nunquam coeuntes, ac ſemper propiùs <lb/>accedentes, & </s> <s xml:id="echoid-s2733" xml:space="preserve">ad interuallum perueniunt minus quocunque dato <lb/>interuallo.</s> <s xml:id="echoid-s2734" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2735" xml:space="preserve">SInt duæ ſimiles Hyperbolæ ABC, DEF per diuerſos vertices B, E ſimul <lb/>adſcriptæ, quarum commune centrum ſit G, ſitque Hyperbolæ ABC <lb/>tranſuerſum latus BH, rectum BI, Hyperbolæ autem DEF ſit tranſuerſum <lb/>EL, rectum EM. </s> <s xml:id="echoid-s2736" xml:space="preserve">Dico primùm has, in infinitum productas, nunquam inter <lb/>ſe conuenire.</s> <s xml:id="echoid-s2737" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2738" xml:space="preserve">Producta enim contingente ME, donec vtrinque ſectioni ABC occurrat, <lb/>ipſa erit ordinata in ſectione ABC (cum ſint ſectiones ſimul adſcriptæ) ac <lb/>ſectio DEF cadet tota infra contingentem KEM; </s> <s xml:id="echoid-s2739" xml:space="preserve">& </s> <s xml:id="echoid-s2740" xml:space="preserve">ſumpto in DEC quoli-<lb/>bet puncto D, applicataque per D recta ADN, quæ iunctis regulis HI, LM <pb o="82" file="0106" n="106" rhead=""/> occurrat in P, O; </s> <s xml:id="echoid-s2741" xml:space="preserve">quoniam datæ ſectiones ſunt ſimiles, erit HB ad BI, vt LE <lb/>ad EM, ſuntque anguli ad B, E æquales, (cum ſectiones ſint ſimul adſcriptæ) <lb/>quare triangula HBI, LEM æquiangula erunt, ideoque regula HIP æquidi-<lb/>ſtabit regulæ LMO; </s> <s xml:id="echoid-s2742" xml:space="preserve">& </s> <s xml:id="echoid-s2743" xml:space="preserve">triangula LNO, HNP inter ſe ſimilia.</s> <s xml:id="echoid-s2744" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2745" xml:space="preserve">Iam cum ſit GE æqualis ipſi GL, & </s> <s xml:id="echoid-s2746" xml:space="preserve">ablata GB æqualis ablatæ GH, erit <lb/>reliqua BE, reliquæ HL æqualis, ſed eſt EN minor HN, quare BE ad EN <lb/>maiorem habet rationem, quàm LH ad HN, & </s> <s xml:id="echoid-s2747" xml:space="preserve">componendo BN ad NE, <lb/>maiorem item rationem quàm LN ad NH, vel quàm ON ad NP, ergo re-<lb/>ctangulum ſub extremis BN, NP, ſiue <anchor type="note" xlink:href="" symbol="a"/> quadratum applicatæ AN, maius <anchor type="note" xlink:label="note-0106-01a" xlink:href="note-0106-01"/> erit <anchor type="note" xlink:href="" symbol="b"/> rectangulo ſub medijs EN, NO, ſiue <anchor type="note" xlink:href="" symbol="c"/> quadrato applicatę DN, hoc eſt <anchor type="note" xlink:label="note-0106-02a" xlink:href="note-0106-02"/> <anchor type="note" xlink:label="note-0106-03a" xlink:href="note-0106-03"/> ipſa AN maior DN, ac propterea punctum D cadit intra Hyperbolen ABC, <lb/>idemque de quolibet alio puncto ſectionis DEF: </s> <s xml:id="echoid-s2748" xml:space="preserve">vnde ipſa DEF inſcripta <lb/>erit ipſi ABC, vel erunt nunquam ſimul coeuntes. </s> <s xml:id="echoid-s2749" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s2750" xml:space="preserve">c.</s> <s xml:id="echoid-s2751" xml:space="preserve"/> </p> <div xml:id="echoid-div260" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0106-01" xlink:href="note-0106-01a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="b" position="left" xlink:label="note-0106-02" xlink:href="note-0106-02a" xml:space="preserve">17. ſept. <lb/>Pappi.</note> <note symbol="c" position="left" xlink:label="note-0106-03" xlink:href="note-0106-03a" xml:space="preserve">Coroll. <lb/>1. huius.</note> </div> <figure> <image file="0106-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0106-01"/> </figure> <p> <s xml:id="echoid-s2752" xml:space="preserve">Ampliùs applicata infra ADT qualibet alia QRS, & </s> <s xml:id="echoid-s2753" xml:space="preserve">Hyperbolę DEF per <lb/>eundem verticem E adſcripta Hyperbola ETV, quæ ſit æqualium laterum, <lb/>ſiue congruens Hyperbolæ ABC, applicatas ſecans in T, V: </s> <s xml:id="echoid-s2754" xml:space="preserve">cum duæ Hy-<lb/>perbolæ EDR, ETV, ſint ſimiles, & </s> <s xml:id="echoid-s2755" xml:space="preserve">per eundem verticem ſimul adſcriptæ <lb/>erit ETV, cuius latera æqualia ſunt ipſis lateribus HB, BI, inſcripta <anchor type="note" xlink:href="" symbol="d"/> ſectio- <anchor type="note" xlink:label="note-0106-04a" xlink:href="note-0106-04"/> ni EDR, cuius maiora ſunt latera LE, EM: </s> <s xml:id="echoid-s2756" xml:space="preserve">ſed erunt <anchor type="note" xlink:href="" symbol="e"/> ſimul ſemper receden- <anchor type="note" xlink:label="note-0106-05a" xlink:href="note-0106-05"/> tes; </s> <s xml:id="echoid-s2757" xml:space="preserve">quare intercepta DT minor erit intercepta RV, eſt autem tota AT <anchor type="note" xlink:href="" symbol="f"/> ma- <anchor type="note" xlink:label="note-0106-06a" xlink:href="note-0106-06"/> ior tota QV; </s> <s xml:id="echoid-s2758" xml:space="preserve">quapropter reliqua AD erit omnino maior reliqua QR; </s> <s xml:id="echoid-s2759" xml:space="preserve">& </s> <s xml:id="echoid-s2760" xml:space="preserve">hoc <lb/>ſemper: </s> <s xml:id="echoid-s2761" xml:space="preserve">Vnde ſimiles concentricæ Hyperbolæ per diuerſos vertices ſimul <lb/>adſcriptæ, ſunt ad ſe propiùs accedentes. </s> <s xml:id="echoid-s2762" xml:space="preserve">Quod ſecundò erat, &</s> <s xml:id="echoid-s2763" xml:space="preserve">c.</s> <s xml:id="echoid-s2764" xml:space="preserve"/> </p> <div xml:id="echoid-div261" type="float" level="2" n="2"> <note symbol="d" position="left" xlink:label="note-0106-04" xlink:href="note-0106-04a" xml:space="preserve">5. Co-<lb/>roll. 19. h.</note> <note symbol="e" position="left" xlink:label="note-0106-05" xlink:href="note-0106-05a" xml:space="preserve">41. h.</note> <note symbol="f" position="left" xlink:label="note-0106-06" xlink:href="note-0106-06a" xml:space="preserve">44. h.</note> </div> <pb o="83" file="0107" n="107" rhead=""/> </div> <div xml:id="echoid-div263" type="section" level="1" n="118"> <head xml:id="echoid-head123" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2765" xml:space="preserve">DVcantur ex communi centro G aſymptoti GX, GZ ſectionis ABC, quę <lb/>alterius ſimilis, & </s> <s xml:id="echoid-s2766" xml:space="preserve">concentricæ ſectionis DEF <anchor type="note" xlink:href="" symbol="a"/> erunt quoque aſym- <anchor type="note" xlink:label="note-0107-01a" xlink:href="note-0107-01"/> ptoti, & </s> <s xml:id="echoid-s2767" xml:space="preserve">ipſi GX, productæ contingentes IB, ME, occurrant in X, Y, & </s> <s xml:id="echoid-s2768" xml:space="preserve">per <lb/>G ſit G 2, regulis HI, LM parallela, recta latera ſecans in 2, & </s> <s xml:id="echoid-s2769" xml:space="preserve">3; </s> <s xml:id="echoid-s2770" xml:space="preserve">cum ſit GE <lb/>æqualis GL, & </s> <s xml:id="echoid-s2771" xml:space="preserve">GB æqualis GH, erit E 3 æqualis 3 M, & </s> <s xml:id="echoid-s2772" xml:space="preserve">B 2 æqualis 2 I, <lb/>ſiue 3 4, quare E 4 eſt aggregatum E 3 cum B 2. </s> <s xml:id="echoid-s2773" xml:space="preserve">Iam cum rectangulum <lb/>GE 3 ſit quarta pars rectanguli LEM, & </s> <s xml:id="echoid-s2774" xml:space="preserve">quadratum EY eiuſdem rectangu-<lb/>li <anchor type="note" xlink:href="" symbol="b"/> ſubquadruplum, ergo quadratum EY ęquatur rectangulo GE 3: </s> <s xml:id="echoid-s2775" xml:space="preserve">eadem- <anchor type="note" xlink:label="note-0107-02a" xlink:href="note-0107-02"/> que ratione eſt quadratum BX æquale rectangulo GB 2, ſed rectangulum <lb/>GE 3 excedit rectangulum GB 2 rectangulo BE 4, ſiue <anchor type="note" xlink:href="" symbol="c"/> quadrato KE, qua- <anchor type="note" xlink:label="note-0107-03a" xlink:href="note-0107-03"/> re quadratum <anchor type="note" xlink:href="" symbol="*"/>EY ſuperat quadratum BX quadrato EK: </s> <s xml:id="echoid-s2776" xml:space="preserve">ſed productis ap- <anchor type="note" xlink:label="note-0107-04a" xlink:href="note-0107-04"/> plicatis AN, QS vſque ad communes aſymptotos, ipſas, ac ſectiones ſecan-<lb/>tibus in 5 ADFC 6, & </s> <s xml:id="echoid-s2777" xml:space="preserve">in 7 QR 8 9, eſt quadratum EY æquale <anchor type="note" xlink:href="" symbol="d"/> rectangu- <anchor type="note" xlink:label="note-0107-05a" xlink:href="note-0107-05"/> lo 5 D 6, & </s> <s xml:id="echoid-s2778" xml:space="preserve">quadratum BX ęquale rectangulo 5 A 6; </s> <s xml:id="echoid-s2779" xml:space="preserve">vnde quadratorum <lb/>exceſſus æquatur exceſſui rectãgulorum, ſed exceſſus quadratorum eſt qua-<lb/>dratum EK, & </s> <s xml:id="echoid-s2780" xml:space="preserve">exceſſus rectangulorum 5 D 6, 5 A 6 <anchor type="note" xlink:href="" symbol="e"/> eſt rectangulum <anchor type="note" xlink:label="note-0107-06a" xlink:href="note-0107-06"/> ADC; </s> <s xml:id="echoid-s2781" xml:space="preserve">vnde quadratum EK æquatur rectangulo ADC; </s> <s xml:id="echoid-s2782" xml:space="preserve">eademque ratione <lb/>oſtendetur idem quadratum EK æquale rectangulo QR 8, quare rectangu-<lb/>la ADC, QR 8 inter ſe ſunt æqualia, ideoque R 8 ad DC, vt DA ad QR, <lb/>ſed eſt R 8 maior DC (cum ſit RS maior DN, & </s> <s xml:id="echoid-s2783" xml:space="preserve">S 8 maior NC) ergo AD <lb/>erit maior QR, & </s> <s xml:id="echoid-s2784" xml:space="preserve">hoc ſemper, &</s> <s xml:id="echoid-s2785" xml:space="preserve">c. </s> <s xml:id="echoid-s2786" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> Quod iterum erat ſecundò demonſtran- <anchor type="note" xlink:label="note-0107-07a" xlink:href="note-0107-07"/> dum.</s> <s xml:id="echoid-s2787" xml:space="preserve"/> </p> <div xml:id="echoid-div263" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0107-01" xlink:href="note-0107-01a" xml:space="preserve">Coroll. <lb/>40. huius.</note> <note symbol="b" position="right" xlink:label="note-0107-02" xlink:href="note-0107-02a" xml:space="preserve">8. huius.</note> <note symbol="c" position="right" xlink:label="note-0107-03" xlink:href="note-0107-03a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="*" position="right" xlink:label="note-0107-04" xlink:href="note-0107-04a" xml:space="preserve">46. h.</note> <note symbol="d" position="right" xlink:label="note-0107-05" xlink:href="note-0107-05a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0107-06" xlink:href="note-0107-06a" xml:space="preserve">179. ſe-<lb/>pt. Pappi.</note> <note symbol="*" position="right" xlink:label="note-0107-07" xlink:href="note-0107-07a" xml:space="preserve">32. h.</note> </div> <p> <s xml:id="echoid-s2788" xml:space="preserve">Dico tandem has ſimiles concentricas Hyperbolas in infinitum produ-<lb/>ctas ad interuallum peruenire minus quolibet dato interuallo R<unsure/>. </s> <s xml:id="echoid-s2789" xml:space="preserve">Nam facta <lb/>eadem penitus conſtructione, ac in vltima parte 42. </s> <s xml:id="echoid-s2790" xml:space="preserve">huius, hoc quod expo-<lb/>nitur, non abſimili eiuſdem argumento demonſtrabitur. </s> <s xml:id="echoid-s2791" xml:space="preserve">Quod vltimò, &</s> <s xml:id="echoid-s2792" xml:space="preserve">c.</s> <s xml:id="echoid-s2793" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div265" type="section" level="1" n="119"> <head xml:id="echoid-head124" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s2794" xml:space="preserve">EX hac elicitur ſimilium, & </s> <s xml:id="echoid-s2795" xml:space="preserve">concentricarum Hyperbolarum, per diuer-<lb/>ſos vertices ſimul adſcriptarum, Aſymptotos communes eſſe.</s> <s xml:id="echoid-s2796" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div266" type="section" level="1" n="120"> <head xml:id="echoid-head125" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s2797" xml:space="preserve">COnſtat etiam ex penultima parte huius, in prædictis Hyperbolis rectan-<lb/>gula ſegmentorum applicatarum vtranque Hyperbolen ſecantium, <lb/>qualia ſunt rectangula ADC, QR8, omnia inter ſe æqualia eſſe.</s> <s xml:id="echoid-s2798" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2799" xml:space="preserve">Quod in prima parte præcedentium 44. </s> <s xml:id="echoid-s2800" xml:space="preserve">45. </s> <s xml:id="echoid-s2801" xml:space="preserve">47. </s> <s xml:id="echoid-s2802" xml:space="preserve">earumque primis Co-<lb/>rollarijs oſtendimus, vniuerſaliùs ſequenti Theoremate demonſtrabitur.</s> <s xml:id="echoid-s2803" xml:space="preserve"/> </p> <pb o="84" file="0108" n="108" rhead=""/> </div> <div xml:id="echoid-div267" type="section" level="1" n="121"> <head xml:id="echoid-head126" xml:space="preserve">THEOR. XXVI. PROP. XXXXVIII.</head> <p> <s xml:id="echoid-s2804" xml:space="preserve">Similes Hyperbolæ per diuerſos vertices ſimul adſcriptæ habent <lb/>aſymptotos parallelas, & </s> <s xml:id="echoid-s2805" xml:space="preserve">quando centrum interioris cadat vltra <lb/>centrum exterioris, tunc huius aſymptotos interiorem Hyperbolen <lb/>ſecabit, ac ipſæ Hyperbolæ neceſſariò ſe mutuò ſecabunt. </s> <s xml:id="echoid-s2806" xml:space="preserve">Cum <lb/>verò centrum interioris idem ſit cum centro exterioris, tunc vnius <lb/>aſymptotos erit aſymptotos alterius; </s> <s xml:id="echoid-s2807" xml:space="preserve">& </s> <s xml:id="echoid-s2808" xml:space="preserve">ſectiones erunt ſimul nun-<lb/>quam coeuntes. </s> <s xml:id="echoid-s2809" xml:space="preserve">Et ſi interioris centrum cadat infra centrum ex-<lb/>terioris, tunc eædem ſectiones erunt inter ſe nunquam coeuntes; </s> <s xml:id="echoid-s2810" xml:space="preserve">& </s> <s xml:id="echoid-s2811" xml:space="preserve"><lb/>aſymptotos inſcriptæ ſecabit Hyperbolen circumſcriptam.</s> <s xml:id="echoid-s2812" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2813" xml:space="preserve">SInt, vt in vtraque figura huius propoſitionis, duæ ſimiles Hyperbolæ <lb/>ABC, DEF per diuerſos vertices B, E ſimul adſcriptæ, quarum centra <lb/>ſint G, H, & </s> <s xml:id="echoid-s2814" xml:space="preserve">ſectionis ABC aſymptoti ſint GI, GO; </s> <s xml:id="echoid-s2815" xml:space="preserve">ſectionis verò DEF ſint <lb/>HM, HR; </s> <s xml:id="echoid-s2816" xml:space="preserve">Dico has aſymptotos eſſe inter ſe æquidiſtantes.</s> <s xml:id="echoid-s2817" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2818" xml:space="preserve">Nam in ſimilibus Hyperbolis ABC, DEF, anguli IGE, MHE, ab earum <lb/>aſymptotis, & </s> <s xml:id="echoid-s2819" xml:space="preserve">diametris ad homologas partes facti <anchor type="note" xlink:href="" symbol="a"/> ſunt æquales, ſuntque <anchor type="note" xlink:label="note-0108-01a" xlink:href="note-0108-01"/> alterni, quare ipſæ aſymptoti inter ſe æquidiſtabunt. </s> <s xml:id="echoid-s2820" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s2821" xml:space="preserve">c.</s> <s xml:id="echoid-s2822" xml:space="preserve"/> </p> <div xml:id="echoid-div267" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve">Coroll. <lb/>40. huius.</note> </div> <p> <s xml:id="echoid-s2823" xml:space="preserve">Iam in hac prima figura, (in <lb/> <anchor type="figure" xlink:label="fig-0108-01a" xlink:href="fig-0108-01"/> qua centrum H interioris DEF <lb/>remotius eſt à verticibus B, E, <lb/>quàm ſit centrum G exterioris <lb/>Hyperbolæ ABC) cum ſint HM, <lb/>HR aſymptoti Hyperbolæ DEF, <lb/>& </s> <s xml:id="echoid-s2824" xml:space="preserve">in loco ab eis, & </s> <s xml:id="echoid-s2825" xml:space="preserve">ſectione ter-<lb/>minato ducta ſit GI alteri aſym-<lb/>ptoton<unsure/> HM æquidiſtans, ipſa <lb/>omnino ſectionem DEF ſecabit. <lb/></s> <s xml:id="echoid-s2826" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s2827" xml:space="preserve">c.</s> <s xml:id="echoid-s2828" xml:space="preserve"/> </p> <div xml:id="echoid-div268" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0108-01"/> </figure> </div> <p> <s xml:id="echoid-s2829" xml:space="preserve">Sed ipſa GI, cum ſit aſympto-<lb/>tos ſectionis ABC, tota cadit ex-<lb/>tra ipſam BA, quare occurſus <lb/>prædictæ aſymptoton GI cum ſe-<lb/>ctione ED, erit extra ſectionem <lb/>BA, vnde ipſa interior ſectio ED <lb/>neceſſariò ſecabit priùs exterio-<lb/>rem BA. </s> <s xml:id="echoid-s2830" xml:space="preserve">Quod tertiò, &</s> <s xml:id="echoid-s2831" xml:space="preserve">c.</s> <s xml:id="echoid-s2832" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2833" xml:space="preserve">Ad pleniorem autem doctrinam, ſi quæratur, quo nam in puncto huiuſ-<lb/>modi Hyperbolæ ſe mutuò ſecent, ita id conſequetur.</s> <s xml:id="echoid-s2834" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2835" xml:space="preserve">Sumpta enim GS æquali GB, erit tota BS tranſuerſum exterioris ABC; <lb/></s> <s xml:id="echoid-s2836" xml:space="preserve">item ſumpta HT æquali HE, erit tota TE tranſuerſum interioris DEF.</s> <s xml:id="echoid-s2837" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2838" xml:space="preserve">Iam, vel H centrum interioris cadit in ipſo puncto S; </s> <s xml:id="echoid-s2839" xml:space="preserve">vel ſupra inter S, & </s> <s xml:id="echoid-s2840" xml:space="preserve"><lb/>T, vel infra inter G, &</s> <s xml:id="echoid-s2841" xml:space="preserve">S.</s> <s xml:id="echoid-s2842" xml:space="preserve"/> </p> <pb o="85" file="0109" n="109" rhead=""/> <p> <s xml:id="echoid-s2843" xml:space="preserve">Si primùm; </s> <s xml:id="echoid-s2844" xml:space="preserve">cum ſit EH æqualis HT, eſſet etiam EH æqualis ST, vnde <lb/>eius fegmentum EB mins eſſet diſtantia ST. </s> <s xml:id="echoid-s2845" xml:space="preserve">Si ſecundùm; </s> <s xml:id="echoid-s2846" xml:space="preserve">cum ſit HT æ-<lb/>qualis HE omnino ST maior eſſet eadem HE, & </s> <s xml:id="echoid-s2847" xml:space="preserve">eò maior ipſius ſegmento <lb/>BE. </s> <s xml:id="echoid-s2848" xml:space="preserve">Si tertiùm; </s> <s xml:id="echoid-s2849" xml:space="preserve">vt in hac ipſa figura, in qua centrum H interioris cadit inter <lb/>S, & </s> <s xml:id="echoid-s2850" xml:space="preserve">G; </s> <s xml:id="echoid-s2851" xml:space="preserve">cum ſit HE æqualis HT, & </s> <s xml:id="echoid-s2852" xml:space="preserve">ablata HB maior ablata HS (nam eſt to-<lb/>ta SB ſecta bifariam in G) erit reliqua BE maior reliqua ST. </s> <s xml:id="echoid-s2853" xml:space="preserve">Quapropter in <lb/>hoc caſu, in quo centrum H interioris cadit vltra centrum G exterioris, vbi-<lb/>cunq; </s> <s xml:id="echoid-s2854" xml:space="preserve">ſit eius incidentia, demonſtratum eſt ſemper diſtantiam verticum B, E, <lb/>minorem eſſe ipſa ST diſtantia inter ſuperiora extrema tranſuerſorum late-<lb/>rum ET, BS. </s> <s xml:id="echoid-s2855" xml:space="preserve">Quod memento.</s> <s xml:id="echoid-s2856" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2857" xml:space="preserve">Ampliùs ſint harum ſectionum recta latera BV, EX, & </s> <s xml:id="echoid-s2858" xml:space="preserve">regulæ TV, TX. <lb/></s> <s xml:id="echoid-s2859" xml:space="preserve">Patet ob ſectionum ſimilitudinem, vt SB ad BV, ita eſſe TE ad EX, ſed an-<lb/>guli ad B, E, ſunt æquales (cum ſectiones ſint ſimul adſcriptæ, &</s> <s xml:id="echoid-s2860" xml:space="preserve">c.) </s> <s xml:id="echoid-s2861" xml:space="preserve">quare <lb/>in triangulis SBV, TEX, anguli ad S, T, æquales erunt, ac ideò regulæ SV, <lb/>TX inter ſe æquidiſtabunt. </s> <s xml:id="echoid-s2862" xml:space="preserve">Cumque ſit ST maior BE, ſi dematur SK ipſi BE <lb/>æqualis, ducaturque SY parallela ad EX, & </s> <s xml:id="echoid-s2863" xml:space="preserve">abſcindatur EL æqualis SY, ac <lb/>iungantur KY, BL: </s> <s xml:id="echoid-s2864" xml:space="preserve">erunt in triangulis KSY, BEL, in quibus latera circum <lb/>æquales angulos S, E, ſunt æqualia, vtrunque vtrique, anguli quoq; </s> <s xml:id="echoid-s2865" xml:space="preserve">SKY, <lb/>EBL æquales; </s> <s xml:id="echoid-s2866" xml:space="preserve">ſuntque alterni, quare KY, & </s> <s xml:id="echoid-s2867" xml:space="preserve">BL inter ſe ęquidiſtant, ſed KY <lb/>ſecat TX, vnde & </s> <s xml:id="echoid-s2868" xml:space="preserve">BL producta ſecabit TX, vt in N: </s> <s xml:id="echoid-s2869" xml:space="preserve">Iam per N ordinatim <lb/>ductis æquidiſtans applicetur NQDP, regulam SV, ſecans in Z, communem <lb/>diametrum in Q, exteriorem ſectionem CBA in P, & </s> <s xml:id="echoid-s2870" xml:space="preserve">interiorem in D: </s> <s xml:id="echoid-s2871" xml:space="preserve">Cum <lb/>in triangulo BQN ſit EL ipſi QN parallela, erit BQ ad QN, vt BE ad EL, <lb/>& </s> <s xml:id="echoid-s2872" xml:space="preserve">permutando QB ad BE, vt QN ad EL, ſiue ad SY, vel ZN, & </s> <s xml:id="echoid-s2873" xml:space="preserve">per con-<lb/>uerſionem rationis BQ ad QE, vt NQ ad QZ, vnde rectangulum BQZ <anchor type="note" xlink:href="" symbol="a"/> ſiue <anchor type="note" xlink:label="note-0109-01a" xlink:href="note-0109-01"/> quadratum applicatæ PQ æquale eſt rectangulo EQN, ſiue quadrato appli-<lb/>catæ DQ ex quo puncta P, D in vnum conueniunt, hoc eſt interior Hyper-<lb/>bole FED exteriori ABC occurrit in D; </s> <s xml:id="echoid-s2874" xml:space="preserve">eademque ratione oſtendetur ipſas <lb/>ſimul occurrere in F, altero extremo eiuſdem applicatæ DQF, quare in ipſis <lb/>occurſibus ſe mutuò ſecant: </s> <s xml:id="echoid-s2875" xml:space="preserve">quoniam ſi exempli gratia, huiuſmodi ſectiones <lb/>non ſe ſecarent, ſed contigerent in D, contingerent ſe quoque in F, vt fa-<lb/>cillimum eſt demonſtrare, ſed Hyperbole ED ſecat omnino rectam GI extra <lb/>ſectionem BA, vti ſuperius oſtendimus, quare hæc inter ſectio alio in loco <lb/>cadet quàm in D, pariterque ad alteram partem ſectio EF ſecabit BC in alio <lb/>puncto, præter in F: </s> <s xml:id="echoid-s2876" xml:space="preserve">Quapropter coni-ſectio coni-ſectionem contingeret in <lb/>duobus punctis D, F, & </s> <s xml:id="echoid-s2877" xml:space="preserve">in alijs duobus punctis ſibi ipſis occurrerent, quod <lb/>eſt <anchor type="note" xlink:href="" symbol="b"/> impoſſibile: </s> <s xml:id="echoid-s2878" xml:space="preserve">vnde in ipſis occurſibus D, F ſe mutuò ſecant; </s> <s xml:id="echoid-s2879" xml:space="preserve">quod ex <anchor type="note" xlink:label="note-0109-02a" xlink:href="note-0109-02"/> abundanti oſtendere propoſuimus.</s> <s xml:id="echoid-s2880" xml:space="preserve"/> </p> <div xml:id="echoid-div269" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="b" position="right" xlink:label="note-0109-02" xlink:href="note-0109-02a" xml:space="preserve">37. 4. <lb/>conic.</note> </div> <p> <s xml:id="echoid-s2881" xml:space="preserve">Si verò centrum H interioris idem fuerit cum G centro exterioris, etiam <lb/>aſymptotos GI eadem erit cum aſymptoto HM, cum angulus IGB æqualis, <lb/>vel <anchor type="note" xlink:href="" symbol="c"/> idem ſit cum angulo MHE; </s> <s xml:id="echoid-s2882" xml:space="preserve">Ergo ſimilium concentricarum Hyperbo- <anchor type="note" xlink:label="note-0109-03a" xlink:href="note-0109-03"/> larum aſymptoti communes ſunt. </s> <s xml:id="echoid-s2883" xml:space="preserve">Quod quartò erat, &</s> <s xml:id="echoid-s2884" xml:space="preserve">c.</s> <s xml:id="echoid-s2885" xml:space="preserve"/> </p> <div xml:id="echoid-div270" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-0109-03" xlink:href="note-0109-03a" xml:space="preserve">Coroll. <lb/>40. huius.</note> </div> <p> <s xml:id="echoid-s2886" xml:space="preserve">Quod autem ſint ſimul nunquam coeuntes ſatis patet ex prima parte 47. <lb/></s> <s xml:id="echoid-s2887" xml:space="preserve">huius, vel quàm breuiſſimè ex propoſ. </s> <s xml:id="echoid-s2888" xml:space="preserve">208. </s> <s xml:id="echoid-s2889" xml:space="preserve">ſeptimi Pappi. </s> <s xml:id="echoid-s2890" xml:space="preserve">Quod quintò, &</s> <s xml:id="echoid-s2891" xml:space="preserve">c.</s> <s xml:id="echoid-s2892" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2893" xml:space="preserve">Si autem centrum H interioris DEF cadat infra G centrum exterioris <lb/>ABC, vt in ſecunda figura, per verticem E contingenter applicata CEA; <lb/></s> <s xml:id="echoid-s2894" xml:space="preserve">cum HM ſit intra angulum IGO ab aſymptotis factum, ac ipſi GI æquidiſtãs, <pb o="86" file="0110" n="110" rhead=""/> ipſa HM producta omnino ſecabit ſectionem CBA, vel ſupra contingentem <lb/>CEA, vt in N, vel in ipſo occurſu A, vel infra ad partes AL; </s> <s xml:id="echoid-s2895" xml:space="preserve">ſi in N, vel in <lb/>A, patet interiorem ſectionem totam cadere infra applicatas ex N, vel ex A, <lb/>& </s> <s xml:id="echoid-s2896" xml:space="preserve">nunquam infra N, vel A ſectioni BNA occurrere, ne priùs ſecet propriam <lb/>aſymptoton HM.</s> <s xml:id="echoid-s2897" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2898" xml:space="preserve">Si verò HM ſecet exteriorẽ <lb/> <anchor type="figure" xlink:label="fig-0110-01a" xlink:href="fig-0110-01"/> BA infra contingentem CAE, <lb/>vt in hac ipſa figura; </s> <s xml:id="echoid-s2899" xml:space="preserve">item pa-<lb/>tet ſectiones BA, ED infra LD <lb/>nunquam ſimul conuenire, ſin <lb/>aliter propriam aſymptoton <lb/>ſecaret.</s> <s xml:id="echoid-s2900" xml:space="preserve"/> </p> <div xml:id="echoid-div271" type="float" level="2" n="5"> <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/QN4GHYBF/figures/0110-01"/> </figure> </div> <p> <s xml:id="echoid-s2901" xml:space="preserve">Præterea indirectũ produ-<lb/>cta communi diametro EBG, <lb/>ſumptiſque in ea punctis V, T, <lb/>quæ ſint extrema tranſuerſo-<lb/>rum laterum datarum ſectio-<lb/>num, ductiſque regulis TX, <lb/>VZ; </s> <s xml:id="echoid-s2902" xml:space="preserve">ipſę vti ſuperiùs oſtenſum <lb/>fuit, inter ſe æquidiſtabunt. <lb/></s> <s xml:id="echoid-s2903" xml:space="preserve">Ampliùs ſumpto in portione <lb/>ED quolibet puncto K, per ip-<lb/>ſum applicetur KY vtranque Hyperbolen ſecans in P, K; </s> <s xml:id="echoid-s2904" xml:space="preserve">regulas verò in <lb/>Z, X.</s> <s xml:id="echoid-s2905" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2906" xml:space="preserve">Iam, vel recta VZ eſt regula interioris ſectionis DEF, & </s> <s xml:id="echoid-s2907" xml:space="preserve">TX exterioris; <lb/></s> <s xml:id="echoid-s2908" xml:space="preserve">vel ipſæ ſimul congruunt, ſi tamen puncta V, T, in vnum conueniant; </s> <s xml:id="echoid-s2909" xml:space="preserve">vel è <lb/>contra VZ eſt regula exterioris, TX verò interioris. </s> <s xml:id="echoid-s2910" xml:space="preserve">Si primum, cum in ſe-<lb/>ctione ABC quadratum applicatæ PY æquale <anchor type="note" xlink:href="" symbol="a"/> ſit rectangulo BYX, & </s> <s xml:id="echoid-s2911" xml:space="preserve">in <anchor type="note" xlink:label="note-0110-01a" xlink:href="note-0110-01"/> Hyperbola DEF quadratum KY ſit æquale rectangulo EYZ, ſitque rectan-<lb/>gulum BYX maius EYZ, cum ſub maioribus lateribus contineatur, erit quo-<lb/>que quadratum PY, maius quadrato KY: </s> <s xml:id="echoid-s2912" xml:space="preserve">vnde punctum K eſt intra ſectio-<lb/>nem ABC. </s> <s xml:id="echoid-s2913" xml:space="preserve">Si ſecundum nempe ſit VZ vtriuſque ſectionis communis regu-<lb/>la, erit quadratum PY æquale rectangulo BYZ, & </s> <s xml:id="echoid-s2914" xml:space="preserve">quadratum KY æquale <lb/>rectangulo EYZ, ſed rectangulum BYZ maius eſt EYZ, cum altitudo BY <lb/>maior ſit altitudine EY, quare quadratum PY maius eſt quadrato KY, ſiue <lb/>punctum K eſt intra ſectionem ABC. </s> <s xml:id="echoid-s2915" xml:space="preserve">Si denique interior recta VZ fuerit re-<lb/>gula exterioris ſectionis ABC, & </s> <s xml:id="echoid-s2916" xml:space="preserve">exterior TX, regula interioris DEF, erit <lb/>HE ipſi HT æqualis, ſed eſt ablata HB minor ablata GV (cum ponatur GB, <lb/>quæ maior eſt HB, æqualis GV) ergo reliqua BE maior erit reſiduis ſegmen-<lb/>tis GH, VT, & </s> <s xml:id="echoid-s2917" xml:space="preserve">eò maior vnico ſegmento VT, ſed eſt EY minor VY, quare <lb/>BE ad EY maiorem habebit rationẽ quàm TV ad VY, vel quàm XZ ad ZY, <lb/>& </s> <s xml:id="echoid-s2918" xml:space="preserve">componendo BY ad YE maiorem habebit rationem quàm XY ad YZ, vn-<lb/>de rectangulum BYZ <anchor type="note" xlink:href="" symbol="b"/> ſiue quadratum PY <anchor type="note" xlink:href="" symbol="c"/> maius erit rectangulo EYX ſiue <anchor type="note" xlink:label="note-0110-02a" xlink:href="note-0110-02"/> <anchor type="note" xlink:label="note-0110-03a" xlink:href="note-0110-03"/> quadrato <anchor type="note" xlink:href="" symbol="d"/> KY, hoc eſt punctum K incidet intra ſectionem ABC, & </s> <s xml:id="echoid-s2919" xml:space="preserve">ſic de <anchor type="note" xlink:label="note-0110-04a" xlink:href="note-0110-04"/> quolibet alio puncto portionis DEF; </s> <s xml:id="echoid-s2920" xml:space="preserve">Quare huiuſmodi ſimiles Hyperbolæ, <lb/>neque infra applicatam LF, neque inter lineas LF, AC ſimul conueniunt, <lb/>vnde ſunt in totum nunquam coeuntes. </s> <s xml:id="echoid-s2921" xml:space="preserve">Quod ſextò, &</s> <s xml:id="echoid-s2922" xml:space="preserve">c.</s> <s xml:id="echoid-s2923" xml:space="preserve"/> </p> <div xml:id="echoid-div272" type="float" level="2" n="6"> <note symbol="a" position="left" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="b" position="left" xlink:label="note-0110-02" xlink:href="note-0110-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="left" xlink:label="note-0110-03" xlink:href="note-0110-03a" xml:space="preserve">16. ſept. <lb/>Pappi.</note> <note symbol="d" position="left" xlink:label="note-0110-04" xlink:href="note-0110-04a" xml:space="preserve">Coroll. <lb/>1. huius.</note> </div> <pb o="87" file="0111" n="111" rhead=""/> <p> <s xml:id="echoid-s2924" xml:space="preserve">Quod tandem HI, aſymptotos inſcriptæ DEF, ſecet circumſcriptam Hy-<lb/>perbolen ABC, iam ſatis patet ex dictis. </s> <s xml:id="echoid-s2925" xml:space="preserve">Quod ſupererat demonſtrandum.</s> <s xml:id="echoid-s2926" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div274" type="section" level="1" n="122"> <head xml:id="echoid-head127" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s2927" xml:space="preserve">_E_N tibi Lector Geometra admiranda quædam Naturæ ſympto-<lb/>mata circa Aſymptoticas lineas iam olim à nobis detecta, ac ſi-<lb/>mul directa demonſtratione firmata, dum in Conicis hucuſque <lb/>animaduertimus non tantùm binas dari lineas in eodem plano <lb/>exiſtentes, quæ licet ſemper inter ſe magis accedant, nunquam tamen (quòd <lb/>ſanè mirum eſt) etiam ſi in infinitum productæ, ſimul conueniunt; </s> <s xml:id="echoid-s2928" xml:space="preserve">quales <lb/>ſunt, conuexa linea hyperbolica, & </s> <s xml:id="echoid-s2929" xml:space="preserve">celebris illa recta Aſymptotos Apoll. </s> <s xml:id="echoid-s2930" xml:space="preserve">ab <lb/>ipſo tunc negatiuè, à nobis verò in 8. </s> <s xml:id="echoid-s2931" xml:space="preserve">& </s> <s xml:id="echoid-s2932" xml:space="preserve">10. </s> <s xml:id="echoid-s2933" xml:space="preserve">huius affirmatiuè demonſtrata: <lb/></s> <s xml:id="echoid-s2934" xml:space="preserve">verùm alias quoque, eiuſdem penitus naturæ reperiri, alteram nempe con-<lb/>uexam, concauam alteram, quales ſunt binæ congruentes parabolæ, vel hy-<lb/>perbolæ; </s> <s xml:id="echoid-s2935" xml:space="preserve">item binæ ſimiles hyperbolæ, quarum centrum interioris, aut in ipſā<unsure/> <lb/>cadat, aut infra centrum exterioris, atque omnes ſint per diuerſos vertices <lb/>ſimul adſcriptæ; </s> <s xml:id="echoid-s2936" xml:space="preserve">prout vidimus in 42. </s> <s xml:id="echoid-s2937" xml:space="preserve">44. </s> <s xml:id="echoid-s2938" xml:space="preserve">45. </s> <s xml:id="echoid-s2939" xml:space="preserve">47. </s> <s xml:id="echoid-s2940" xml:space="preserve">& </s> <s xml:id="echoid-s2941" xml:space="preserve">elicitur ex ipſa 48. </s> <s xml:id="echoid-s2942" xml:space="preserve"><lb/>huius. </s> <s xml:id="echoid-s2943" xml:space="preserve">Præterea, non ſolùm rectam Aſymptoton, & </s> <s xml:id="echoid-s2944" xml:space="preserve">Hyperbolen dari, quæ <lb/>dum ad ſe propius ſemper accedunt, ad interuallum aliquando perueniunt mi-<lb/>nus quolibet dato interuallo, vti ex ipſo Apollonio, & </s> <s xml:id="echoid-s2945" xml:space="preserve">ex noſtra 10. </s> <s xml:id="echoid-s2946" xml:space="preserve">innotuit; </s> <s xml:id="echoid-s2947" xml:space="preserve"><lb/>ſed congruentes item parabolas, & </s> <s xml:id="echoid-s2948" xml:space="preserve">concentricas hyperbolas per varios ver-<lb/>tices ſimul adſcriptas hac ipſa admirabili affectione eſſe præditas, veluti in <lb/>42. </s> <s xml:id="echoid-s2949" xml:space="preserve">& </s> <s xml:id="echoid-s2950" xml:space="preserve">47. </s> <s xml:id="echoid-s2951" xml:space="preserve">à nobis fuit oſtenſum. </s> <s xml:id="echoid-s2952" xml:space="preserve">Verum enimuero haud minori ſaltem <lb/>admiratione dignum videtur, binas pariter lineas inueniri, quæ licet nun-<lb/>quam coeuntes, & </s> <s xml:id="echoid-s2953" xml:space="preserve">in infinitum productæ ad ſe propius accedentes, non ta-<lb/>men vnquam perueniunt ad interuallum cuiuſdam determinatæ magnitudi-<lb/>nis: </s> <s xml:id="echoid-s2954" xml:space="preserve">huiuſmodi enim ſunt congruentes Hyperbolæ, pariterque hyperbolæ ſimi-<lb/>les per diuerſos vertices ſimul adſcriptæ, prout didicimus in 44. </s> <s xml:id="echoid-s2955" xml:space="preserve">& </s> <s xml:id="echoid-s2956" xml:space="preserve">45. </s> <s xml:id="echoid-s2957" xml:space="preserve"><lb/>Alias amplius deteximus lineas, quarum diſtãtia perpetuò augetur, ſed nun-<lb/>quam tamen peruenit ad interuallum æquale cuidam terminato interuallo: </s> <s xml:id="echoid-s2958" xml:space="preserve">ta-<lb/>les enim ſunt recta linea alteri aſymptoton æquidiſtans, & </s> <s xml:id="echoid-s2959" xml:space="preserve">Hyperbolen ſecãs, <lb/>vna cum eadem curua hyperbolica: </s> <s xml:id="echoid-s2960" xml:space="preserve">tales item ſunt hyperbolæ ſimiles per eun-<lb/>dem verticem ſimul adſcriptæ, prout in 34. </s> <s xml:id="echoid-s2961" xml:space="preserve">& </s> <s xml:id="echoid-s2962" xml:space="preserve">41. </s> <s xml:id="echoid-s2963" xml:space="preserve">Oſtendimus denique <lb/>binas dari lineas ad eaſdem partes in infinitum productas, nunquam coeun-<lb/>tes, quæ ſimul, ac ſemel ſunt, & </s> <s xml:id="echoid-s2964" xml:space="preserve">ad ſe propiùs accedentes, & </s> <s xml:id="echoid-s2965" xml:space="preserve">inter ſe æqui-<lb/>diſtantes: </s> <s xml:id="echoid-s2966" xml:space="preserve">quales ſunt demum, parabolæ congruentes per diuerſos vertices <lb/>ſimul adſcriptæ, vti ex noſtra 42. </s> <s xml:id="echoid-s2967" xml:space="preserve">eiuſque primo Coroll. </s> <s xml:id="echoid-s2968" xml:space="preserve">iam ſatis patuit.</s> <s xml:id="echoid-s2969" xml:space="preserve"/> </p> <pb o="88" file="0112" n="112" rhead=""/> </div> <div xml:id="echoid-div275" type="section" level="1" n="123"> <head xml:id="echoid-head128" xml:space="preserve">THEOR. XXVII. PROP. XXXXIX.</head> <p> <s xml:id="echoid-s2970" xml:space="preserve">Si binæ Parabolæ, aut binæ concentricæ Hyperbolæ fuerint per <lb/>diuerſos vertices ſimul adſcriptæ, ipſæ, vel ad neutram partem ſe <lb/>vnquam ſecabunt, vel ſi ad alteram partem occurrant, occurrent <lb/>quoque ad aliam, punctaque occurſuum erunt extrema eiuſdem <lb/>communis applicatæ, ac in ijſdem occurſibus ſe mutuò ſecabunt.</s> <s xml:id="echoid-s2971" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2972" xml:space="preserve">SInt duæ Parabolæ ABC, DEF, vel duæ concentricæ Hyperbolæ per di-<lb/>uerſos vertices B, E ſimul adſcriptæ. </s> <s xml:id="echoid-s2973" xml:space="preserve">Dico primùm, ſi huiuſmodi ſectio-<lb/>nes ad alteram partium, vt ad A nunquam conueniunt, ad aliam quoque C <lb/>nunquam conuenire.</s> <s xml:id="echoid-s2974" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2975" xml:space="preserve">Nam ſumpto in ſectione DEF, ad <lb/> <anchor type="figure" xlink:label="fig-0112-01a" xlink:href="fig-0112-01"/> partes C, quocunque puncto F, per ip-<lb/>sũ ordinatim applicetur recta CFGDA, <lb/>quæ vtrinque producta, vtrique ſectio-<lb/>ni occurret (cum ipſæ ob Hypoteſim, <lb/>ſint ſectiones in infinitam diſtantiam <lb/>abeuntes ad inferiores partes) cumque <lb/>in ſectione ABC ſit ſemi-applicata AG, <lb/>æqualis GC, & </s> <s xml:id="echoid-s2976" xml:space="preserve">in ſectione DEF, ſemi-<lb/>applicata DG, æqualis GF, ſitque ante-<lb/>cedens AG maior antecedente DG (cum ad partes A nunquam cõueniant) <lb/>erit etiam conſequens GC, maior conſequenti GF, quare punctum F, ſe-<lb/>ctionis DEF cadit intra ABC, & </s> <s xml:id="echoid-s2977" xml:space="preserve">ſic de reliquis. </s> <s xml:id="echoid-s2978" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s2979" xml:space="preserve">c.</s> <s xml:id="echoid-s2980" xml:space="preserve"/> </p> <div xml:id="echoid-div275" type="float" level="2" n="1"> <figure xlink:label="fig-0112-01" xlink:href="fig-0112-01a"> <image file="0112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0112-01"/> </figure> </div> <p> <s xml:id="echoid-s2981" xml:space="preserve">Si verò ſectiones ad alteras partes, veluti ad A, & </s> <s xml:id="echoid-s2982" xml:space="preserve">D conueniant vt in H. <lb/></s> <s xml:id="echoid-s2983" xml:space="preserve">Dico ipſas ad alias quoque partes ſimul occurrere ad extrema puncta eiuſ-<lb/>de m applicatæ.</s> <s xml:id="echoid-s2984" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2985" xml:space="preserve">Nam, ducta per H communi applicata HI, ipſa producatur ſecans ſectio-<lb/>nem BC in L; </s> <s xml:id="echoid-s2986" xml:space="preserve">& </s> <s xml:id="echoid-s2987" xml:space="preserve">EF in M. </s> <s xml:id="echoid-s2988" xml:space="preserve">Erit in ſectione ABC ſemi- applicata HI æqualis <lb/>IL, & </s> <s xml:id="echoid-s2989" xml:space="preserve">in ſectione DEF eadem HI æqualis IM; </s> <s xml:id="echoid-s2990" xml:space="preserve">ergo IL, & </s> <s xml:id="echoid-s2991" xml:space="preserve">IM æquales, ideo-<lb/>que ſectionum puncta L, M in vnum conueniunt; </s> <s xml:id="echoid-s2992" xml:space="preserve">Quare cum ſectiones <lb/>ABC, DEF non per vertices ſimul adſcriptæ ad alteram partem occurrunt, <lb/>occurrent quoque ad aliam, punctaque occurſuum erunt extrema eiuſdem <lb/>communis applicatæ.</s> <s xml:id="echoid-s2993" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2994" xml:space="preserve">Quod autem in occurſibus H, & </s> <s xml:id="echoid-s2995" xml:space="preserve">L ſe mutuò ſecent, ſic demonſtratur. </s> <s xml:id="echoid-s2996" xml:space="preserve">Nam <lb/>ſi huiuſmodi ſectiones tangerent ſe mutuò in occurſu H, ita vt ſectionis, ver-<lb/>bi gratia, ED partes HO, HN cadant totæ intra ſectionem ABC: </s> <s xml:id="echoid-s2997" xml:space="preserve">ſumpto in <lb/>altera ipſarum partium, vt puta OH, quolibet puncto D, & </s> <s xml:id="echoid-s2998" xml:space="preserve">per ipſum ducta <lb/>communi applicata ADGFG vtranque ſectionem ſecane<unsure/>, erunt in ſectione <lb/>ABC rectæ AG, GC æquales, & </s> <s xml:id="echoid-s2999" xml:space="preserve">in ſectione DEF rectæ DG, GF itẽ ę;</s> <s xml:id="echoid-s3000" xml:space="preserve">qua-<lb/>les, ſed eſt AG maior GD cum ponatur peripheria OH cadere intra BH, vn-<lb/>de & </s> <s xml:id="echoid-s3001" xml:space="preserve">CG maior erit ipſa FG, hoc eſt punctum F cadet intra. </s> <s xml:id="echoid-s3002" xml:space="preserve">Idemque de-<lb/>monſtrabitur de quolibet alio extremo puncto cuiuſcunque applicatæ inter <lb/>O, & </s> <s xml:id="echoid-s3003" xml:space="preserve">N, tùm ſupra, tùm infra occurſum L: </s> <s xml:id="echoid-s3004" xml:space="preserve">quare ſectio DEF continget <pb o="89" file="0113" n="113" rhead=""/> ipſam ABC in puncto L, ſed poſitum fuit eam quoque contingere in H: </s> <s xml:id="echoid-s3005" xml:space="preserve">Er-<lb/>go in duobus punctis H, L ſe contingent; </s> <s xml:id="echoid-s3006" xml:space="preserve">quod eſt falſum; </s> <s xml:id="echoid-s3007" xml:space="preserve">nam Parabole <lb/>Parabolen, ſiue Hyperbole Hyperbolen concentricam <anchor type="note" xlink:href="" symbol="a"/> in duobus punctis <anchor type="note" xlink:label="note-0113-01a" xlink:href="note-0113-01"/> non contingit. </s> <s xml:id="echoid-s3008" xml:space="preserve">Non ergo tales ſectiones ſe tangunt in H; </s> <s xml:id="echoid-s3009" xml:space="preserve">neque in L, ob ean-<lb/>dem rationem; </s> <s xml:id="echoid-s3010" xml:space="preserve">quare ipſæ in occurſibus H, & </s> <s xml:id="echoid-s3011" xml:space="preserve">L ſe mutuò ſecant. </s> <s xml:id="echoid-s3012" xml:space="preserve">Quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s3013" xml:space="preserve"/> </p> <div xml:id="echoid-div276" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0113-01" xlink:href="note-0113-01a" xml:space="preserve">28. 31. 4. <lb/>conic.</note> </div> </div> <div xml:id="echoid-div278" type="section" level="1" n="124"> <head xml:id="echoid-head129" xml:space="preserve">THEOR. XXVIII. PROP. L.</head> <p> <s xml:id="echoid-s3014" xml:space="preserve">Impoſſibile eſt Hyperbolen Parabolæ, per eundem, vel per <lb/>diuerſos vertices inſcribere. </s> <s xml:id="echoid-s3015" xml:space="preserve">Item.</s> <s xml:id="echoid-s3016" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3017" xml:space="preserve">Impoſſibile eſt Parabolen Hyperbolæ, per eundem, vel per di-<lb/>uerſos vertices circumſcribere.</s> <s xml:id="echoid-s3018" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3019" xml:space="preserve">ESto Parabole ABC, cui per punctum D in ea ſumptum, vt in prima figu-<lb/>ra, vel intra ipſam, vt in ſecunda, adſcripta ſit quæcunque Hyperbole <lb/>EDF circa communem diametrum BDG, quæ per aliquas ſuæ peripheriæ <lb/>partes DE, DF, hinc inde à diametro ſumptas cadat intra Parabolen ABC. <lb/></s> <s xml:id="echoid-s3020" xml:space="preserve">Dico ipſam Hyperbolen, ſi producatur, ex vtraque parte Parabolen ſecare.</s> <s xml:id="echoid-s3021" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3022" xml:space="preserve">Nam ductis ex D re-<lb/>ctis DA, DC vtrique <lb/> <anchor type="figure" xlink:label="fig-0113-01a" xlink:href="fig-0113-01"/> aſymptoto Hyperbolæ <lb/>EDF ęquidiſtãtibus, hę; <lb/></s> <s xml:id="echoid-s3023" xml:space="preserve">neceſſariò Parabolen <lb/>ſecabunt, <anchor type="note" xlink:href="" symbol="a"/> vt in A, C;</s> <s xml:id="echoid-s3024" xml:space="preserve"> <anchor type="note" xlink:label="note-0113-02a" xlink:href="note-0113-02"/> ſed cum Hyperbola in <lb/>alio puncto quàm D <lb/>nunquam <anchor type="note" xlink:href="" symbol="b"/> conuenient:</s> <s xml:id="echoid-s3025" xml:space="preserve"> <anchor type="note" xlink:label="note-0113-03a" xlink:href="note-0113-03"/> quare, cum Hyperbola <lb/>EDF ex vtraque parte <lb/>in infinitum habeat, ſi <lb/>producatur, occurret <lb/>denique Parabolæ ABC inter puncta B, A, & </s> <s xml:id="echoid-s3026" xml:space="preserve">puncta B, C; </s> <s xml:id="echoid-s3027" xml:space="preserve">eamque ſeca-<lb/>bit, nam ſi tantùm eam tangeret, vel non, ſi vlteriùs producatur intra Para-<lb/>bolen, ſecaret aliquandò rectas DA, DC; </s> <s xml:id="echoid-s3028" xml:space="preserve">quod <anchor type="note" xlink:href="" symbol="c"/> eſt impoſſibile. </s> <s xml:id="echoid-s3029" xml:space="preserve">Non igi- <anchor type="note" xlink:label="note-0113-04a" xlink:href="note-0113-04"/> tur inſcribi vnquam poteſt Hyperbole datæ Parabolæ, per punctum in ea, <lb/>vel intra ipſa datum, eadem ratione demonſtrabitur non poſſe circumſcribi <lb/>Parabolen datæ Hyperbolę per punctum in ea, vel extra ipſam datum. </s> <s xml:id="echoid-s3030" xml:space="preserve">Quod <lb/>erat, &</s> <s xml:id="echoid-s3031" xml:space="preserve">c.</s> <s xml:id="echoid-s3032" xml:space="preserve"/> </p> <div xml:id="echoid-div278" 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/QN4GHYBF/figures/0113-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0113-02" xlink:href="note-0113-02a" xml:space="preserve">27. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0113-03" xlink:href="note-0113-03a" xml:space="preserve">Coroll. <lb/>11. huius.</note> <note symbol="c" position="right" xlink:label="note-0113-04" xlink:href="note-0113-04a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div280" type="section" level="1" n="125"> <head xml:id="echoid-head130" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s3033" xml:space="preserve">HInc patet non dari _MAXIMAM_ Hyperbolen datæ Parabolæ, vel per <lb/>eundem verticem, vel per diuerſos inſcriptibilem; </s> <s xml:id="echoid-s3034" xml:space="preserve">itemque non dari <lb/>_MINIMAM_ Parabolen datæ Hyperbolæ, vel per eundem, vel per diuerſos <lb/>vertices circumſcriptibilem.</s> <s xml:id="echoid-s3035" xml:space="preserve"/> </p> <pb o="90" file="0114" n="114" rhead=""/> </div> <div xml:id="echoid-div281" type="section" level="1" n="126"> <head xml:id="echoid-head131" xml:space="preserve">PROBL. XVII. PROP. LI.</head> <p> <s xml:id="echoid-s3036" xml:space="preserve">Datæ Parabolæ per punctum intra ipſam datum MAXIMAM <lb/>Parabolen inſcribere, & </s> <s xml:id="echoid-s3037" xml:space="preserve">è contra.</s> <s xml:id="echoid-s3038" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3039" xml:space="preserve">Datæ Parabolæ per punctum extra ipſam datum MINIMAM <lb/>Parabolen circumſcribere.</s> <s xml:id="echoid-s3040" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3041" xml:space="preserve">SIt data Parabole ABC, & </s> <s xml:id="echoid-s3042" xml:space="preserve">oporteat primò per punctum D intra ipſam da-<lb/>tum _MAXIMAM_ Parabolen inſcribere.</s> <s xml:id="echoid-s3043" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3044" xml:space="preserve">Ducatur diameter <lb/> <anchor type="figure" xlink:label="fig-0114-01a" xlink:href="fig-0114-01"/> BDE, cuius rectum <lb/>latus ſit BF, (quod <lb/>in poſterùm intelli-<lb/>gatur ſemper ex ver-<lb/>tice cõtingenter ap-<lb/>plicatum ſectioni, <lb/>prout in prę;</s> <s xml:id="echoid-s3045" xml:space="preserve">cedenti-<lb/>bus factum eſt, & </s> <s xml:id="echoid-s3046" xml:space="preserve">in <lb/>quinta primarũ defi-<lb/>nitionũ monuimus) <lb/>& </s> <s xml:id="echoid-s3047" xml:space="preserve">per verticem D, <lb/>circa diametrũ D E <lb/>adſcribatur <anchor type="note" xlink:href="" symbol="a"/> datæ Parabolæ ABC Parabole GDH, cuius rectum DI æqua- <anchor type="note" xlink:label="note-0114-01a" xlink:href="note-0114-01"/> le ſit recto BF; </s> <s xml:id="echoid-s3048" xml:space="preserve">nam ipſa erit congruens datæ Dico hanc eſſe _MAXIMAM_ <lb/>in ſcriptam quæſitam.</s> <s xml:id="echoid-s3049" xml:space="preserve"/> </p> <div xml:id="echoid-div281" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0114-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0114-01" xlink:href="note-0114-01a" xml:space="preserve">5. huius.</note> </div> <p> <s xml:id="echoid-s3050" xml:space="preserve">Nam cum ipſæ ſint congruentes Parabolæ per diuerſos vertices ſimul ad-<lb/>ſcriptæ <anchor type="note" xlink:href="" symbol="b"/> erunt inter ſe nunquam coeuntes: </s> <s xml:id="echoid-s3051" xml:space="preserve">quare GDH datæ ABC erit per <anchor type="note" xlink:label="note-0114-02a" xlink:href="note-0114-02"/> datum punctum D inſcripta.</s> <s xml:id="echoid-s3052" xml:space="preserve"/> </p> <div xml:id="echoid-div282" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0114-02" xlink:href="note-0114-02a" xml:space="preserve">42. h.</note> </div> <p> <s xml:id="echoid-s3053" xml:space="preserve">Ampliùs, quælibet alia Parabole per verticem D adſcripta cum recto, <lb/>quod minus ſit ipſo DI minor <anchor type="note" xlink:href="" symbol="c"/> eſt Parabola GDH, quæ verò cum recto DL, <anchor type="note" xlink:label="note-0114-03a" xlink:href="note-0114-03"/> quod excedat ipſum DI, qualis eſt MDN, eſt quidem <anchor type="note" xlink:href="" symbol="d"/> maior GDH, ſed <anchor type="note" xlink:label="note-0114-04a" xlink:href="note-0114-04"/> omnino ſecat circumſcriptam ABC. </s> <s xml:id="echoid-s3054" xml:space="preserve">Nam ſi fiat vt LI ad ID, ita BD ad DE, <lb/>& </s> <s xml:id="echoid-s3055" xml:space="preserve">per E applicetur EMA ſecans BA in A, & </s> <s xml:id="echoid-s3056" xml:space="preserve">DM in M. </s> <s xml:id="echoid-s3057" xml:space="preserve">Cum ſit BD ad DE, <lb/>vt Li ad ID, erit componendo BE ad ED, vt LD ad DI; </s> <s xml:id="echoid-s3058" xml:space="preserve">vnde rectangulum <lb/>ſub extremis BE, & </s> <s xml:id="echoid-s3059" xml:space="preserve">DI, ſiue BF, hoc eſt <anchor type="note" xlink:href="" symbol="e"/> quadratum applicatæ AE in Pa- <anchor type="note" xlink:label="note-0114-05a" xlink:href="note-0114-05"/> rabola ABC, æquale erit rectangulo ſub medijs ED, DL <anchor type="note" xlink:href="" symbol="f"/> ſiue quadrato ap- <anchor type="note" xlink:label="note-0114-06a" xlink:href="note-0114-06"/> plicatę; </s> <s xml:id="echoid-s3060" xml:space="preserve">ME in Parabola MDN, ac ideò AE, ME ſunt æquales, quapropter <lb/>Parabole DN occurrit ſibi adſcriptæ BA, per diuerſos vertices in puncto M, <lb/>& </s> <s xml:id="echoid-s3061" xml:space="preserve">ob id in eodem occurſu, & </s> <s xml:id="echoid-s3062" xml:space="preserve">ad alteram quoque partem ſe mutuò <anchor type="note" xlink:href="" symbol="g"/> ſecabũt:</s> <s xml:id="echoid-s3063" xml:space="preserve"> <anchor type="note" xlink:label="note-0114-07a" xlink:href="note-0114-07"/> Itaque congruens Parabole GDH erit _MAXIMA_ inſcripta quæſita. </s> <s xml:id="echoid-s3064" xml:space="preserve">Quod <lb/>primò, &</s> <s xml:id="echoid-s3065" xml:space="preserve">c.</s> <s xml:id="echoid-s3066" xml:space="preserve"/> </p> <div xml:id="echoid-div283" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0114-03" xlink:href="note-0114-03a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="d" position="left" xlink:label="note-0114-04" xlink:href="note-0114-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="left" xlink:label="note-0114-05" xlink:href="note-0114-05a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="f" position="left" xlink:label="note-0114-06" xlink:href="note-0114-06a" xml:space="preserve">ibidem.</note> <note symbol="g" position="left" xlink:label="note-0114-07" xlink:href="note-0114-07a" xml:space="preserve">50. h.</note> </div> <p> <s xml:id="echoid-s3067" xml:space="preserve">IAM datæ Parabolæ GDH, oporteat per punctum B extra ipſam datam <lb/>_MINIMAM_ Parabolen circumſcribere.</s> <s xml:id="echoid-s3068" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3069" xml:space="preserve">Ducatur BDE diameter datę; </s> <s xml:id="echoid-s3070" xml:space="preserve">GDH, cuius rectum ſit DI, & </s> <s xml:id="echoid-s3071" xml:space="preserve">ei adſcribatur <lb/>per B, cum recto BF, quod æquet ipſum DI, congruens Parabole ABC:</s> <s xml:id="echoid-s3072" xml:space="preserve"> <pb o="91" file="0115" n="115" rhead=""/> Dico hanc eſſe _MINIMAM_ circumſcriptam quæſitam.</s> <s xml:id="echoid-s3073" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3074" xml:space="preserve">Cum ſint enim ipſæ Parabolæ congruentes, & </s> <s xml:id="echoid-s3075" xml:space="preserve">per diuerſos vertices ad-<lb/>ſcriptæ, erunt <anchor type="note" xlink:href="" symbol="a"/> inter ſe nunquam coeuntes quare ABC datæ GDH erit cir- <anchor type="note" xlink:label="note-0115-01a" xlink:href="note-0115-01"/> cumſcripta.</s> <s xml:id="echoid-s3076" xml:space="preserve"/> </p> <div xml:id="echoid-div284" type="float" level="2" n="4"> <note symbol="a" position="right" xlink:label="note-0115-01" xlink:href="note-0115-01a" xml:space="preserve">42. h.</note> </div> <p> <s xml:id="echoid-s3077" xml:space="preserve">Præterea, quælibet alia Parabole per B adſcripta cum recto, quod exce-<lb/>dat BF, maior eſt ipſa ABC, quę verò cum recto BO, quod minus ſit ipſo BF, <lb/>qualis eſt PBQ, eſt quidem minor ipſa ABC, ſed omnino ſecat inſcriptam <lb/>GDH. </s> <s xml:id="echoid-s3078" xml:space="preserve">Quoniam ſi fiat vt FO ad OB, ita BD ad DE, ac per E applicetur <lb/>EGP ſecans DG in G, & </s> <s xml:id="echoid-s3079" xml:space="preserve">BP in P: </s> <s xml:id="echoid-s3080" xml:space="preserve">cum ſit BD ad DE, vt FO ad OB, erit com-<lb/>ponendo BE ad ED, vt FB ad BO; </s> <s xml:id="echoid-s3081" xml:space="preserve">vnde rectangulum ſub BE, & </s> <s xml:id="echoid-s3082" xml:space="preserve">BO <anchor type="note" xlink:href="" symbol="b"/> ſiue <anchor type="note" xlink:label="note-0115-02a" xlink:href="note-0115-02"/> quadratum applicatæ EP in Parabola PBQ æquale erit rectangulo ſub me-<lb/>dijs ED, & </s> <s xml:id="echoid-s3083" xml:space="preserve">BF, ſiue DI, hoc <anchor type="note" xlink:href="" symbol="c"/> eſt quadrato applicatę EG in Parabola GDH:</s> <s xml:id="echoid-s3084" xml:space="preserve"> <anchor type="note" xlink:label="note-0115-03a" xlink:href="note-0115-03"/> vnde EP, EG ſunt æquales. </s> <s xml:id="echoid-s3085" xml:space="preserve">Occurrit ergo Parabole BP, ſibi adſcriptæ DG <lb/>per diuerſos vertices, in puncto P, quare in eodem occurſu, & </s> <s xml:id="echoid-s3086" xml:space="preserve">ad alteram <lb/>partem <anchor type="note" xlink:href="" symbol="d"/> ſe mutuò ſecant. </s> <s xml:id="echoid-s3087" xml:space="preserve">Quapropter congruens Parabole ABC erit _MI-_ <anchor type="note" xlink:label="note-0115-04a" xlink:href="note-0115-04"/> _NIMA_ circumſcripta quæſita.</s> <s xml:id="echoid-s3088" xml:space="preserve"/> </p> <div xml:id="echoid-div285" type="float" level="2" n="5"> <note symbol="b" position="right" xlink:label="note-0115-02" xlink:href="note-0115-02a" xml:space="preserve">1. Co-<lb/>roll. 1. h.</note> <note symbol="c" position="right" xlink:label="note-0115-03" xlink:href="note-0115-03a" xml:space="preserve">ibidem.</note> <note symbol="d" position="right" xlink:label="note-0115-04" xlink:href="note-0115-04a" xml:space="preserve">50. h.</note> </div> </div> <div xml:id="echoid-div287" type="section" level="1" n="127"> <head xml:id="echoid-head132" xml:space="preserve">PROBL. XVIII. PROP. LII.</head> <p> <s xml:id="echoid-s3089" xml:space="preserve">Datæ Hyperbolę, per punctum intra ipſam datum MAXIMAM <lb/>Hyperbolen inſcribere, quarum eadem ſit regula.</s> <s xml:id="echoid-s3090" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3091" xml:space="preserve">ESto data Hyperbole ABC, cuius centrum D; </s> <s xml:id="echoid-s3092" xml:space="preserve">& </s> <s xml:id="echoid-s3093" xml:space="preserve">punctum intra ipſam da-<lb/>tum ſit E. </s> <s xml:id="echoid-s3094" xml:space="preserve">Oportet per E Hyperbolen inſcribere, quæ ſit _MAXIMA_, <lb/>ſed tamen eius regula ſit quoque regula datæ ſectionis.</s> <s xml:id="echoid-s3095" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3096" xml:space="preserve">Iungatur ED ſecans datã ſectionem in <lb/> <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> B, & </s> <s xml:id="echoid-s3097" xml:space="preserve">producta ſumatur DF æqualis BD, <lb/>erit <anchor type="note" xlink:href="" symbol="a"/> FB trãſuerſum ſectionis ABC, cuius <anchor type="note" xlink:label="note-0115-05a" xlink:href="note-0115-05"/> vertex B, ſitque BG eius rectum latus, & </s> <s xml:id="echoid-s3098" xml:space="preserve"><lb/>regula FG, quæ producatur, & </s> <s xml:id="echoid-s3099" xml:space="preserve">per E ſit <lb/>ducta EH parallela ad BG, & </s> <s xml:id="echoid-s3100" xml:space="preserve">per verticẽ <lb/>E, circa communem diametrum BE, da-<lb/>tę ſectioni ABC <anchor type="note" xlink:href="" symbol="b"/> adſcribatur Hyperbole <anchor type="note" xlink:label="note-0115-06a" xlink:href="note-0115-06"/> IEL, cuius latera ſint FE, EH, hoc eſt <lb/>eadem ſit regula FGH: </s> <s xml:id="echoid-s3101" xml:space="preserve">patet ipſam IEL <lb/>datæ ABC eſſe inſcriptam, cum in infini-<lb/>tum productæ ſint inter <anchor type="note" xlink:href="" symbol="c"/> ſe nunquam <anchor type="note" xlink:label="note-0115-07a" xlink:href="note-0115-07"/> coeuntes.</s> <s xml:id="echoid-s3102" xml:space="preserve"/> </p> <div xml:id="echoid-div287" 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/QN4GHYBF/figures/0115-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0115-05" xlink:href="note-0115-05a" xml:space="preserve">47. 1. <lb/>conic.</note> <note symbol="b" position="right" xlink:label="note-0115-06" xlink:href="note-0115-06a" xml:space="preserve">7. huius.</note> <note symbol="c" position="right" xlink:label="note-0115-07" xlink:href="note-0115-07a" xml:space="preserve">45. h.</note> </div> <p> <s xml:id="echoid-s3103" xml:space="preserve">Dico ampliùs ipſam IEL eſſe _MAXI-_ <lb/>_MAM_. </s> <s xml:id="echoid-s3104" xml:space="preserve">Quoniam quęlibet alia adſcripta <lb/>per verticem E, cum eodem tranſuerſo <lb/>FE, ſed cum recto, quod minus ſit recto <lb/>EH, minor <anchor type="note" xlink:href="" symbol="d"/> eſt ipſa IEL; </s> <s xml:id="echoid-s3105" xml:space="preserve">quæ verò cum recto EO, quod excedat EH, qua- <anchor type="note" xlink:label="note-0115-08a" xlink:href="note-0115-08"/> lis eſt Hyperbole PEQ, eſt quidem <anchor type="note" xlink:href="" symbol="e"/> maior ipſa IEL; </s> <s xml:id="echoid-s3106" xml:space="preserve">ſed omnino ſecat ipſam <anchor type="note" xlink:label="note-0115-09a" xlink:href="note-0115-09"/> ABC. </s> <s xml:id="echoid-s3107" xml:space="preserve">Nam ſi fiat vt OH ad HE, ita BE ad EM, & </s> <s xml:id="echoid-s3108" xml:space="preserve">per M applicetur MPA <lb/>Hyperbolen PEQ ſecans in P, BA verò in A, & </s> <s xml:id="echoid-s3109" xml:space="preserve">producta ſecet regulam <lb/>FH, in N, & </s> <s xml:id="echoid-s3110" xml:space="preserve">iunctam regulam FO deſcriptæ Hyperbolæ PEQ in R.</s> <s xml:id="echoid-s3111" xml:space="preserve"/> </p> <div xml:id="echoid-div288" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-0115-08" xlink:href="note-0115-08a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="e" position="right" xlink:label="note-0115-09" xlink:href="note-0115-09a" xml:space="preserve">ibidem.</note> </div> <pb o="92" file="0116" n="116" rhead=""/> <p> <s xml:id="echoid-s3112" xml:space="preserve">Cum ſit BE ad EM, vt OH ad HE, erit <lb/> <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> componendo BM ad ME, vt OE ad EH, <lb/>vel vt RM ad MN; </s> <s xml:id="echoid-s3113" xml:space="preserve">quapropter rectan-<lb/>gulum BMN <anchor type="note" xlink:href="" symbol="a"/> ſiue quadratum applicatæ <anchor type="note" xlink:label="note-0116-01a" xlink:href="note-0116-01"/> AM in Hyperbola ABC, æquale erit re-<lb/>ctangulo EMR, <anchor type="note" xlink:href="" symbol="b"/> ſiue quadrato applicatę <anchor type="note" xlink:label="note-0116-02a" xlink:href="note-0116-02"/> MP in Hyperbola PEQ; </s> <s xml:id="echoid-s3114" xml:space="preserve">ac ideò lineæ <lb/>MA, MP ſunt æquales, quare Hyperbo-<lb/>læ ABC, PEQ occurrunt ſimul in pun-<lb/>cto Q, in quo etiam ſe mutuò ſecabunt. <lb/></s> <s xml:id="echoid-s3115" xml:space="preserve">Nã ſumpto in ſectione QEP infra P quo-<lb/>libet puncto S, per quod applicata STV, <lb/>ſectionem ABC, diametrum, ac regulas <lb/>ſecans in T, V, X, Y: </s> <s xml:id="echoid-s3116" xml:space="preserve">cum ſit EM minor <lb/>EV, habebit BE ad EM, vel OH ad HE, <lb/>vel YX ad XV, maiorem rationem quam <lb/>BE ad EV, & </s> <s xml:id="echoid-s3117" xml:space="preserve">componendo YV ad VX <lb/>maiorem rationem, quàm BV ad VE, vnde rectangulum YVE, <anchor type="note" xlink:href="" symbol="c"/> ſiue qua- <anchor type="note" xlink:label="note-0116-03a" xlink:href="note-0116-03"/> dratum VS in Hyperbola EAS, <anchor type="note" xlink:href="" symbol="d"/> maius erit rectangulo XVB, <anchor type="note" xlink:href="" symbol="e"/> ſiue quadra- <anchor type="note" xlink:label="note-0116-04a" xlink:href="note-0116-04"/> <anchor type="note" xlink:label="note-0116-05a" xlink:href="note-0116-05"/> to VT in Hyperbola ABC: </s> <s xml:id="echoid-s3118" xml:space="preserve">vnde punctum S cadit extra Hyperbolen ABC, <lb/>ac ideò ipſæ Hyperbolæ ſe mutuò ſecant, ſicuti in altero extremo Q, eiuſ-<lb/>dem applicatæ. </s> <s xml:id="echoid-s3119" xml:space="preserve">Erit ergo Hyperbole IEL, quæ datæ ABC ſimilis eſt, & </s> <s xml:id="echoid-s3120" xml:space="preserve">ad <lb/>eandem regulam, _MAXIMA_ inſcripta quæſita. </s> <s xml:id="echoid-s3121" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s3122" xml:space="preserve">c.</s> <s xml:id="echoid-s3123" xml:space="preserve"/> </p> <div xml:id="echoid-div289" type="float" level="2" n="3"> <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/QN4GHYBF/figures/0116-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="b" position="left" xlink:label="note-0116-02" xlink:href="note-0116-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="left" xlink:label="note-0116-03" xlink:href="note-0116-03a" xml:space="preserve">Coroll. <lb/>1. huius.</note> <note symbol="d" position="left" xlink:label="note-0116-04" xlink:href="note-0116-04a" xml:space="preserve">16. ſept. <lb/>Pappi.</note> <note symbol="e" position="left" xlink:label="note-0116-05" xlink:href="note-0116-05a" xml:space="preserve">Coroll. <lb/>1. huius.</note> </div> <p style="it"> <s xml:id="echoid-s3124" xml:space="preserve">Verùm cum ad inueſtigationem MAXIMAE, & </s> <s xml:id="echoid-s3125" xml:space="preserve">MINIMAE in-<lb/>ſcriptæ, ac circumſcriptæ ſectionis, mlreferat ignorare quo nam in puncto, <lb/>maior, vel minor quæſitarum ſectionum, datæ ſectioni occurrat, ſufficit <lb/>enim oſtendere ipſas, vbicunq; </s> <s xml:id="echoid-s3126" xml:space="preserve">ſit earum occurſus, aliquando ſe mutuò ſeca-<lb/>re) ideò in proximè ſequentibus problematibus, hac omiſſa methodo per appli-<lb/>catarum potentias, tanquam prolixiori, & </s> <s xml:id="echoid-s3127" xml:space="preserve">minus concinna, hoc ipſum aliter <lb/>elegantiori induſtria demonſtr abimus, & </s> <s xml:id="echoid-s3128" xml:space="preserve">licet id pluribus, ac varijs aggreſ-<lb/>ſionibus conſequi poſsit, vt in hac, & </s> <s xml:id="echoid-s3129" xml:space="preserve">proxima propoſitione videre licet, ta-<lb/>men eas eligemus, quæ apportunæ magis nobis videbuntur.</s> <s xml:id="echoid-s3130" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div291" type="section" level="1" n="128"> <head xml:id="echoid-head133" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3131" xml:space="preserve">SEcetur igitur E F bifariam in 2, erit 2 centrum Hyperbolarum IEL, <lb/>PEQ, ex quo ductis harum aſumptotis, videlicet 2 3 inſcriptæ IEL, <lb/>& </s> <s xml:id="echoid-s3132" xml:space="preserve">2 4 circumſcriptæ PEQ, quæ cadet <anchor type="note" xlink:href="" symbol="a"/> extra aſymptoton 2 3, ex D quo- <anchor type="note" xlink:label="note-0116-06a" xlink:href="note-0116-06"/> que agatur D 5 aſymptotos Hyperbolæ ABC.</s> <s xml:id="echoid-s3133" xml:space="preserve"/> </p> <div xml:id="echoid-div291" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0116-06" xlink:href="note-0116-06a" xml:space="preserve">Exvlti-<lb/>ma parte <lb/>37. huius.</note> </div> <p> <s xml:id="echoid-s3134" xml:space="preserve">Iam cum Hyperbolæ ABC, IEL ſimiles ſint, per diuerſos vertices, & </s> <s xml:id="echoid-s3135" xml:space="preserve">ad <lb/>eandem regulam FGH ſimul adſcriptæ, erunt earum aſymptoti D 5, 2 3 <lb/>inter <anchor type="note" xlink:href="" symbol="b"/> ſe parallelæ ſed 2 4 inter ipſas cadit, & </s> <s xml:id="echoid-s3136" xml:space="preserve">alteram 2 3 ſecat in 2, qua- <anchor type="note" xlink:label="note-0116-07a" xlink:href="note-0116-07"/> re ipſa 2 4 producta ad partes 4, ſecabit & </s> <s xml:id="echoid-s3137" xml:space="preserve">reliquam D 5; </s> <s xml:id="echoid-s3138" xml:space="preserve">ſed eſt 2 4 <lb/>aſymptotos ſectionis SPEQ, & </s> <s xml:id="echoid-s3139" xml:space="preserve">quædam recta D 5 occurrit ei, ac ſectionis <pb o="93" file="0117" n="117" rhead=""/> diametro vltra centrum 2 in D, quare ſi eadem D 5 producatur, neceſſa-<lb/>riò <anchor type="note" xlink:href="" symbol="a"/> ſecabit Hyperbolen SPEQ, ſed ipſa D 5 tota cadit extra Hyperbolen <anchor type="note" xlink:label="note-0117-01a" xlink:href="note-0117-01"/> ABC, cum ſit eius aſymptotos, quapropter occurſus rectæ D 5 cum Hyper-<lb/>bola SPEQ ſiet extra ABC, ideoque ſectio EP ſecabit priùs Hyperbolen <lb/>ABC, & </s> <s xml:id="echoid-s3140" xml:space="preserve">ſic Hyperbole IEL erit _MAXIMA_ inſcripta quæſita. </s> <s xml:id="echoid-s3141" xml:space="preserve">Quod facien-<lb/>dum, ac demonſtrandum erat.</s> <s xml:id="echoid-s3142" xml:space="preserve"/> </p> <div xml:id="echoid-div292" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0116-07" xlink:href="note-0116-07a" xml:space="preserve">Coroll. <lb/>45. huius.</note> <note symbol="a" position="right" xlink:label="note-0117-01" xlink:href="note-0117-01a" xml:space="preserve">35. h.</note> </div> </div> <div xml:id="echoid-div294" type="section" level="1" n="129"> <head xml:id="echoid-head134" xml:space="preserve">ALITER breuiùs.</head> <p> <s xml:id="echoid-s3143" xml:space="preserve">PRoducatur contingens HE vſque ad circumſcriptã ſectionem ABC in K. <lb/></s> <s xml:id="echoid-s3144" xml:space="preserve">Cum Hyperbolę ABC, IEL ſimiles ſint per diuerſos vertices, & </s> <s xml:id="echoid-s3145" xml:space="preserve">ad ean-<lb/>dem regulam ſimul adſcriptæ <anchor type="note" xlink:href="" symbol="b"/> erunt infra EK ad ſe propiùs accedentes, ni- <anchor type="note" xlink:label="note-0117-02a" xlink:href="note-0117-02"/> mirum ſectio KAT recedet ab EI per interuallum minus ipſo EK; </s> <s xml:id="echoid-s3146" xml:space="preserve">Verùm <lb/>cum Hyperbolę PEQ, IEL ſint concentricæ, & </s> <s xml:id="echoid-s3147" xml:space="preserve">per eundem verticem ſimul <lb/>adſcriptæ, erunt ſemper magis recedentes, & </s> <s xml:id="echoid-s3148" xml:space="preserve">ad interuallũ peruenient maius <lb/>quocunq; </s> <s xml:id="echoid-s3149" xml:space="preserve">dato interuallo, videlicet ſectio EPS recedet ab eadem EI<anchor type="note" xlink:href="" symbol="*"/> per in- <anchor type="note" xlink:label="note-0117-03a" xlink:href="note-0117-03"/> teruallũ omnino maius eodẽ EK: </s> <s xml:id="echoid-s3150" xml:space="preserve">quapropter ſectiones KAT, EPS neceſſariò <lb/>ſe mutuò ſecabunt: </s> <s xml:id="echoid-s3151" xml:space="preserve">Vnde Hyperbole IEL erit _MAXIMA_ inſcripta quæſita.</s> <s xml:id="echoid-s3152" xml:space="preserve"/> </p> <div xml:id="echoid-div294" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-0117-02" xlink:href="note-0117-02a" xml:space="preserve">45. h.</note> <note symbol="*" position="right" xlink:label="note-0117-03" xlink:href="note-0117-03a" xml:space="preserve">37. h.</note> </div> </div> <div xml:id="echoid-div296" type="section" level="1" n="130"> <head xml:id="echoid-head135" xml:space="preserve">PROBL. XIX. PROP. LIII.</head> <p> <s xml:id="echoid-s3153" xml:space="preserve">Datæ Hyperbolæ per punctum extra ipſam datum MINIMAM <lb/>Hyperbolen circumſcribere, quarum eadem ſit regula.</s> <s xml:id="echoid-s3154" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3155" xml:space="preserve">Oportet autem datum punctum, vel eſſe in angulo aſymptotis <lb/>contento, vel in eo, quod<unsure/> eſt ad verticem, dummodo in hoc caſu, <lb/>ipſius diſtantia à centro datæ ſectionis, minor ſit eius ſemi-tranſ-<lb/>uerſo latere per datum punctum tranſeunte.</s> <s xml:id="echoid-s3156" xml:space="preserve"/> </p> <figure> <image file="0117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0117-01"/> </figure> <p> <s xml:id="echoid-s3157" xml:space="preserve">ESto data Hyperbole ABC, cuius centrum D, aſymptoti DF, DG, & </s> <s xml:id="echoid-s3158" xml:space="preserve">da-<lb/>tum extra ipſam punctum ſit E, quod tamen ſit in angulo aſymptotali <lb/>FDG, vt in prima figura; </s> <s xml:id="echoid-s3159" xml:space="preserve">vel in eò qui ipſi eſt ad verticem, vt in ſecunda, <lb/>dummodo coniuncta ED, & </s> <s xml:id="echoid-s3160" xml:space="preserve">producta vſque ad ſectionem in B, ipſa ED mi- <pb o="94" file="0118" n="118" rhead=""/> nor ſit ſemi-tranſuerſo DB: </s> <s xml:id="echoid-s3161" xml:space="preserve">(ſi enim datum punctum eſſet in angulis, qui <lb/>deinceps ſunt, recta linea per ipſum datum punctum, & </s> <s xml:id="echoid-s3162" xml:space="preserve">centrum ſectionis <lb/>ducta non eſſet eius diameter, cum nunquam ſectioni <anchor type="note" xlink:href="" symbol="a"/> occurreret, ac ideò <anchor type="note" xlink:label="note-0118-01a" xlink:href="note-0118-01"/> problema, iuxta quintam ſecundarum definitionum inſolubile eſſet: </s> <s xml:id="echoid-s3163" xml:space="preserve">& </s> <s xml:id="echoid-s3164" xml:space="preserve">cum <lb/>fuerit in angulo ad verticem, vt in ſecunda, niſi diſtantia ED minor ſit ſemi-<lb/>tranſuerſo DB, Hyperbole ad regulam datæ adſcribi minimè poſſet, vt ſatis <lb/>patet) oportet per E _MINIMAM_ Hyperbolen circumſcribere, cuius regula <lb/>eadem ſit cum regula datæ ſectionis.</s> <s xml:id="echoid-s3165" xml:space="preserve"/> </p> <div xml:id="echoid-div296" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0118-01" xlink:href="note-0118-01a" xml:space="preserve">Monit. <lb/>poſt 11. h.</note> </div> <p> <s xml:id="echoid-s3166" xml:space="preserve">Iungatur ED, & </s> <s xml:id="echoid-s3167" xml:space="preserve">ad partes ſectionis producatur donec ei occurrat in B, <lb/>ſumptaq; </s> <s xml:id="echoid-s3168" xml:space="preserve">in directum DH æquali DB, erit HB tranſuerſum <anchor type="note" xlink:href="" symbol="b"/> ſectionis ABC, <anchor type="note" xlink:label="note-0118-02a" xlink:href="note-0118-02"/> cuius vertex B: </s> <s xml:id="echoid-s3169" xml:space="preserve">ſit ergo BI eius rectum latus, & </s> <s xml:id="echoid-s3170" xml:space="preserve">regula HI; </s> <s xml:id="echoid-s3171" xml:space="preserve">ſitque EK æqui-<lb/>diſtans BI, & </s> <s xml:id="echoid-s3172" xml:space="preserve">per verticem B, cum tranſuerſo EH, & </s> <s xml:id="echoid-s3173" xml:space="preserve">recto EK, ſiue ad ean-<lb/>dem regulam HI adſcribatur Hyperbole LEM: </s> <s xml:id="echoid-s3174" xml:space="preserve">patet ipſam datæ ABC eſſe <lb/>inſcriptam, cum ſimul <anchor type="note" xlink:href="" symbol="c"/> ſint nun quam coeuntes.</s> <s xml:id="echoid-s3175" xml:space="preserve"/> </p> <div xml:id="echoid-div297" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0118-02" xlink:href="note-0118-02a" xml:space="preserve">47. pri-<lb/>mi conic.</note> </div> <note symbol="c" position="left" xml:space="preserve">45. h.</note> <figure> <image file="0118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0118-01"/> </figure> <p> <s xml:id="echoid-s3176" xml:space="preserve">Dico ampliùs ipſam LEM eſſe _MINIMAM_ quæſitam. </s> <s xml:id="echoid-s3177" xml:space="preserve">Quoniam quęlibet <lb/>alia adſcripta per verticem E, cum eodem verſo HE, ſed cum recto, quod <lb/>excedat EK, maior <anchor type="note" xlink:href="" symbol="d"/> eſt ipſa LEM; </s> <s xml:id="echoid-s3178" xml:space="preserve">quæ verò cum recto EN, quod minus ſit <anchor type="note" xlink:label="note-0118-04a" xlink:href="note-0118-04"/> EK, qualis OEQ, eſt quidem <anchor type="note" xlink:href="" symbol="e"/> minor eadem LEM, ſed omnino ſecat ipſam <anchor type="note" xlink:label="note-0118-05a" xlink:href="note-0118-05"/> ABC. </s> <s xml:id="echoid-s3179" xml:space="preserve">Nam ad productam regulam HN, ſecan@ BI in R adſcribatur per B <lb/>Hyperbole SBT; </s> <s xml:id="echoid-s3180" xml:space="preserve">hæc tota cadet <anchor type="note" xlink:href="" symbol="f"/> intra ABC, eruntque SBT, OEQ duæ ſi- <anchor type="note" xlink:label="note-0118-06a" xlink:href="note-0118-06"/> miles Hyperbolæ per diuerſos vertices adſcriptæ ad eandem regulam HR, <lb/>eſtque ABC ipſi SBT, per eundem verticem, & </s> <s xml:id="echoid-s3181" xml:space="preserve">cum maiori recto latere BI <lb/>adſcripta, quare per præce dentem <anchor type="note" xlink:href="" symbol="g"/> ſectiones ABC, OEQ ſe mutuò ſeca- <anchor type="note" xlink:label="note-0118-07a" xlink:href="note-0118-07"/> bunt: </s> <s xml:id="echoid-s3182" xml:space="preserve">Vnde Hyperbole LEM eſt _MINIMA_ circumſcripta quæſita. </s> <s xml:id="echoid-s3183" xml:space="preserve">Quod <lb/>faciendum, & </s> <s xml:id="echoid-s3184" xml:space="preserve">demonſtrandum erat.</s> <s xml:id="echoid-s3185" xml:space="preserve"/> </p> <div xml:id="echoid-div298" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-0118-04" xlink:href="note-0118-04a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="e" position="left" xlink:label="note-0118-05" xlink:href="note-0118-05a" xml:space="preserve">ibidem.</note> <note symbol="f" position="left" xlink:label="note-0118-06" xlink:href="note-0118-06a" xml:space="preserve">ibidem.</note> <note symbol="g" position="left" xlink:label="note-0118-07" xlink:href="note-0118-07a" xml:space="preserve">52. h.</note> </div> </div> <div xml:id="echoid-div300" type="section" level="1" n="131"> <head xml:id="echoid-head136" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3186" xml:space="preserve">SEcetur EH bifariam in X: </s> <s xml:id="echoid-s3187" xml:space="preserve">erit X centrum vtriuſque LEM, OEQ: </s> <s xml:id="echoid-s3188" xml:space="preserve">ſi ergo <lb/>ex centris X, D, ducantur XY, XZ, DF ſectionum LEM, OEQ, ABC <lb/>aſymptoti, hoc eſt XY circumſcriptæ LEM; </s> <s xml:id="echoid-s3189" xml:space="preserve">XZ inſcriptæ OEQ, quæ infra <lb/>XY <anchor type="note" xlink:href="" symbol="h"/> cadet; </s> <s xml:id="echoid-s3190" xml:space="preserve">& </s> <s xml:id="echoid-s3191" xml:space="preserve">DF ſectionis ABC, quæ ipſi XY æquidiſtabit; </s> <s xml:id="echoid-s3192" xml:space="preserve">cum XZ ſecet <anchor type="note" xlink:label="note-0118-08a" xlink:href="note-0118-08"/> <anchor type="handwritten" xlink:label="hd-0118-2a" xlink:href="hd-0118-2"/> <pb o="95" file="0119" n="119" rhead=""/> XY in X, producta ſecabit etiam DF aſymptoton ABC, ac ipſam quoque <lb/>ſectionem ABC, <anchor type="note" xlink:href="" symbol="a"/> ſed XZ tota cadit extra OEQ, cum ſit eius aſymptotos, <anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/> quare occurſus rectæ XZ cum ſectione ABC cadet extra OEQ, ac ideò ſe-<lb/>ctio ABC occurret priùs ſectioni OEQ. </s> <s xml:id="echoid-s3193" xml:space="preserve">Quapropter Hyperbole LEM eſt <lb/>_MINIMA_ circumſcripta quæſita. </s> <s xml:id="echoid-s3194" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s3195" xml:space="preserve">c.</s> <s xml:id="echoid-s3196" xml:space="preserve"/> </p> <div xml:id="echoid-div300" type="float" level="2" n="1"> <note symbol="h" position="left" xlink:label="note-0118-08" xlink:href="note-0118-08a" xml:space="preserve">Ex vlti-<lb/>ma partre <lb/>37. huius.</note> <handwritten xlink:label="hd-0118-2" xlink:href="hd-0118-2a"/> <note symbol="a" position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">35. h.</note> </div> </div> <div xml:id="echoid-div302" type="section" level="1" n="132"> <head xml:id="echoid-head137" xml:space="preserve">ALITER breuiùs.</head> <p> <s xml:id="echoid-s3197" xml:space="preserve">PRoducatur contingens IB vſq; </s> <s xml:id="echoid-s3198" xml:space="preserve">ad circumſcriptam ſectionem in V. </s> <s xml:id="echoid-s3199" xml:space="preserve">Cum <lb/>ſectiones BA, EL ſimiles, & </s> <s xml:id="echoid-s3200" xml:space="preserve">ad eandem regulam HI, infra BV ad ſe <lb/>propiùs <anchor type="note" xlink:href="" symbol="b"/> accedant ſectio BA recedet ab VL per interuallum aliquando mi- <anchor type="note" xlink:label="note-0119-02a" xlink:href="note-0119-02"/> nùs BV, ſed inſcripta OP recedit ab eadem VL per interuallũ maius eodem <lb/>BV, cum ſint ſemper magis recedẽtes, & </s> <s xml:id="echoid-s3201" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ad interuallum perueniant maius <anchor type="note" xlink:label="note-0119-03a" xlink:href="note-0119-03"/> quolibet dato interuallo: </s> <s xml:id="echoid-s3202" xml:space="preserve">quare BA, & </s> <s xml:id="echoid-s3203" xml:space="preserve">OP omnino ſe mutuò ſecabũt. </s> <s xml:id="echoid-s3204" xml:space="preserve">Quod <lb/>iterum erat, &</s> <s xml:id="echoid-s3205" xml:space="preserve">c.</s> <s xml:id="echoid-s3206" xml:space="preserve"/> </p> <div xml:id="echoid-div302" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">45. h.</note> <note symbol="c" position="right" xlink:label="note-0119-03" xlink:href="note-0119-03a" xml:space="preserve">37. h.</note> </div> </div> <div xml:id="echoid-div304" type="section" level="1" n="133"> <head xml:id="echoid-head138" xml:space="preserve">PROBL. XX. PROP. LIV.</head> <p> <s xml:id="echoid-s3207" xml:space="preserve">Datæ Hyperbolę, per punctum intra ipſam datum MAXIMAM <lb/>ſibi concentricam Hyperbolen inſcribere, & </s> <s xml:id="echoid-s3208" xml:space="preserve">è contra.</s> <s xml:id="echoid-s3209" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3210" xml:space="preserve">Datæ Hyperbolæ, per punctum extra ipſam datum MINIMAM <lb/>ſibi concentricam Hyperbolen circumſcribere. </s> <s xml:id="echoid-s3211" xml:space="preserve">Oportet autem <lb/>datum punctum eſſe in angulo aſymptotali.</s> <s xml:id="echoid-s3212" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3213" xml:space="preserve">ESto data Hyperbole ABC, cuius centrum D, aſymptotos DL, & </s> <s xml:id="echoid-s3214" xml:space="preserve">pun-<lb/>ctum intra ſectionem datum ſit E: </s> <s xml:id="echoid-s3215" xml:space="preserve">oportet primò per E _MAXIMAM_ ei <lb/>concentricam Hyperbolen inſcribere.</s> <s xml:id="echoid-s3216" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3217" xml:space="preserve">Iungatur ED ſecans ABC in B: </s> <s xml:id="echoid-s3218" xml:space="preserve">erit <lb/> <anchor type="figure" xlink:label="fig-0119-01a" xlink:href="fig-0119-01"/> DB <anchor type="note" xlink:href="" symbol="a"/> ſemi-tranſuerſum ſectionis ABC, <anchor type="note" xlink:label="note-0119-04a" xlink:href="note-0119-04"/> cui per E <anchor type="note" xlink:href="" symbol="b"/> cum ſemi tranſuerſo ED ad- <anchor type="note" xlink:label="note-0119-05a" xlink:href="note-0119-05"/> ſcribatur ſimilis, & </s> <s xml:id="echoid-s3219" xml:space="preserve">concentrica Hyper-<lb/>bole FEG (hoc autem ſieri poſſe mani-<lb/>feſtum eſt: </s> <s xml:id="echoid-s3220" xml:space="preserve">nam ſectionis FEG datur eius <lb/>aſymptotos DL, cum ſimiles concentri-<lb/>cæ Hyperbolæ per diuerſos vertices ad-<lb/>ſcriptæ <anchor type="note" xlink:href="" symbol="c"/> habeant communem aſympto- <anchor type="note" xlink:label="note-0119-06a" xlink:href="note-0119-06"/> ton, & </s> <s xml:id="echoid-s3221" xml:space="preserve">cum datur tranſuerſum latus, & </s> <s xml:id="echoid-s3222" xml:space="preserve"><lb/>aſymptotos datur quoque rectum) patet <lb/>hãc ſectionẽ FEG datæ ABC eſſe inſcri-<lb/>ptam, cum ſint <anchor type="note" xlink:href="" symbol="d"/> nunquã ſimul coeuntes.</s> <s xml:id="echoid-s3223" xml:space="preserve"> </s> </p> <div xml:id="echoid-div304" type="float" level="2" n="1"> <figure xlink:label="fig-0119-01" xlink:href="fig-0119-01a"> <image file="0119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0119-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0119-04" xlink:href="note-0119-04a" xml:space="preserve">47. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0119-05" xlink:href="note-0119-05a" xml:space="preserve">7. huius.</note> <note symbol="c" position="right" xlink:label="note-0119-06" xlink:href="note-0119-06a" xml:space="preserve">Coroll. <lb/>47. huius.</note> </div> <note symbol="d" position="right" xml:space="preserve">47. h.</note> <p> <s xml:id="echoid-s3224" xml:space="preserve">Dico ampliùs hanc ipſam FEG eſſe _MAXIMAM_ quæſitam: </s> <s xml:id="echoid-s3225" xml:space="preserve">quoniam quę-<lb/>libet alia per E verticem adſcripta ipſi ABC, vel FEG minor <anchor type="note" xlink:href="" symbol="e"/> eſt ipſa FEG;</s> <s xml:id="echoid-s3226" xml:space="preserve"> <anchor type="note" xlink:label="note-0119-08a" xlink:href="note-0119-08"/> quælibet, verò cum recto, quod prædictum excedat, qualis eſt HEI, eſt qui-<lb/>dem <anchor type="note" xlink:href="" symbol="f"/> maior eadem FEG, ſed omnino ſecat circumſcriptam ABC. </s> <s xml:id="echoid-s3227" xml:space="preserve">Nam <anchor type="note" xlink:label="note-0119-09a" xlink:href="note-0119-09"/> cum Hyperbolæ FEG, HEI ſint concentricæ, & </s> <s xml:id="echoid-s3228" xml:space="preserve">per eundem verticem E ſi-<lb/>mul adſcriptæ, ſitque DL aſymptotos inſcriptæ FEG, ipſa ſecabit circum- <pb o="96" file="0120" n="120" rhead=""/> ſcriptam <anchor type="note" xlink:href="" symbol="a"/> HEL, ſed eadem DL eſt aſymptotos ABC, ſiue tota cadit extra <anchor type="note" xlink:label="note-0120-01a" xlink:href="note-0120-01"/> ABC, quare DL, & </s> <s xml:id="echoid-s3229" xml:space="preserve">ſectio EH ſecabunt ſe mutuò extra ſectionem BA, qua-<lb/>propter EH ſecabit priùs ſectionem BA: </s> <s xml:id="echoid-s3230" xml:space="preserve">ex quo ſimilis, & </s> <s xml:id="echoid-s3231" xml:space="preserve">concentrica Hy-<lb/>perbole FEG erit _MAXIMA_ quæſita: </s> <s xml:id="echoid-s3232" xml:space="preserve">Quod primò faciendum, & </s> <s xml:id="echoid-s3233" xml:space="preserve">demon-<lb/>ſtrandum erat.</s> <s xml:id="echoid-s3234" xml:space="preserve"/> </p> <div xml:id="echoid-div305" type="float" level="2" n="2"> <note symbol="e" position="right" xlink:label="note-0119-08" xlink:href="note-0119-08a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="f" position="right" xlink:label="note-0119-09" xlink:href="note-0119-09a" xml:space="preserve">ibidem.</note> <note symbol="a" position="left" xlink:label="note-0120-01" xlink:href="note-0120-01a" xml:space="preserve">37. h.</note> </div> </div> <div xml:id="echoid-div307" type="section" level="1" n="134"> <head xml:id="echoid-head139" xml:space="preserve">ALITER breuiùs.</head> <p> <s xml:id="echoid-s3235" xml:space="preserve">DVcatur ex E contingens EM. </s> <s xml:id="echoid-s3236" xml:space="preserve">Sectio MA accedit <anchor type="note" xlink:href="" symbol="b"/> ſectioni EF per in- <anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/> teruallum minus quolibet dato interuallo; </s> <s xml:id="echoid-s3237" xml:space="preserve">at ſectio EH quæ cadit ex-<lb/>tra EF, <anchor type="note" xlink:href="" symbol="c"/> recedit ab eadem EF per interuallum maius eodem dato interuallo;</s> <s xml:id="echoid-s3238" xml:space="preserve"> <anchor type="note" xlink:label="note-0120-03a" xlink:href="note-0120-03"/> quare MA, & </s> <s xml:id="echoid-s3239" xml:space="preserve">EH neceſſariò ſe mutuò ſecant: </s> <s xml:id="echoid-s3240" xml:space="preserve">Vnde FEG eſt _MAXIMA_ in-<lb/>ſcripta quæſita. </s> <s xml:id="echoid-s3241" xml:space="preserve">Quod iterum, &</s> <s xml:id="echoid-s3242" xml:space="preserve">c.</s> <s xml:id="echoid-s3243" xml:space="preserve"/> </p> <div xml:id="echoid-div307" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">47. h.</note> <note symbol="c" position="left" xlink:label="note-0120-03" xlink:href="note-0120-03a" xml:space="preserve">37. h.</note> </div> <p> <s xml:id="echoid-s3244" xml:space="preserve">IAM ſit data Hyperbole FEG, cuius centrum G, aſymptotos DL, & </s> <s xml:id="echoid-s3245" xml:space="preserve">opor-<lb/>teat per datum extra ipſam punctum B (quod tamen ſit in angulo aſym-<lb/>ptotali, ob rationem in præcedenti propoſ. </s> <s xml:id="echoid-s3246" xml:space="preserve">allatam) _MINIMAM_ Hyperbo-<lb/>len circumſcribere.</s> <s xml:id="echoid-s3247" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3248" xml:space="preserve">Iungatur DB, & </s> <s xml:id="echoid-s3249" xml:space="preserve">producatur ſectioni <lb/> <anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/> FEG occurrens in E, & </s> <s xml:id="echoid-s3250" xml:space="preserve">cum ſemi-tranſ-<lb/>uerſo BD, per verticem B, adſcribatur <lb/>ſimilis, & </s> <s xml:id="echoid-s3251" xml:space="preserve">concentrica Hyperbole ABC: <lb/></s> <s xml:id="echoid-s3252" xml:space="preserve">patet hanc eſſe datæ FEG circumſcriptã, <lb/>cum nunquam <anchor type="note" xlink:href="" symbol="d"/> ſimul conueniant.</s> <s xml:id="echoid-s3253" xml:space="preserve"> </s> </p> <div xml:id="echoid-div308" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0120-01"/> </figure> </div> <note symbol="d" position="left" xml:space="preserve">47. h.</note> <p> <s xml:id="echoid-s3254" xml:space="preserve">Dico præterea ipſam eſſe _MAXIMAM_ <lb/>quæſitam. </s> <s xml:id="echoid-s3255" xml:space="preserve">Quoniam quæcunque alia <lb/>adſcripta per B ipſi FEG, vel ipſi ABC <lb/>concentrica, cum recto, quod maius ſit <lb/>recto ſectionis ABC, maior <anchor type="note" xlink:href="" symbol="e"/> eſt ipſa <anchor type="note" xlink:label="note-0120-05a" xlink:href="note-0120-05"/> ABC, quæ verò cum recto, quod præ-<lb/>dicto ſit minus, qualis eſt Hyperbole <lb/>NBO, eſt quidem <anchor type="note" xlink:href="" symbol="f"/> minor eadem ABC, ſed omnino ſecat inſcriptam FEG.</s> <s xml:id="echoid-s3256" xml:space="preserve"> <anchor type="note" xlink:label="note-0120-06a" xlink:href="note-0120-06"/> Quoniam ſectio MA accedit <anchor type="note" xlink:href="" symbol="g"/> ſectioni EF per interuallum minus quolibet <anchor type="note" xlink:label="note-0120-07a" xlink:href="note-0120-07"/> dato interuallo; </s> <s xml:id="echoid-s3257" xml:space="preserve">ſed ſectio PN eſt intra MA, & </s> <s xml:id="echoid-s3258" xml:space="preserve">ab ipſa <anchor type="note" xlink:href="" symbol="h"/> recedit per interual- <anchor type="note" xlink:label="note-0120-08a" xlink:href="note-0120-08"/> lum maius eodem dato interuallo; </s> <s xml:id="echoid-s3259" xml:space="preserve">quare PN, & </s> <s xml:id="echoid-s3260" xml:space="preserve">EF neceſſariò ſe mutuò ſe-<lb/>cant. </s> <s xml:id="echoid-s3261" xml:space="preserve">Igitur ſimilis, & </s> <s xml:id="echoid-s3262" xml:space="preserve">concentrica Hyperbole ABC eſt _MINIMA_ circum-<lb/>ſcripta quæſita. </s> <s xml:id="echoid-s3263" xml:space="preserve">Quod ſecundò faciendum erat.</s> <s xml:id="echoid-s3264" xml:space="preserve"/> </p> <div xml:id="echoid-div309" type="float" level="2" n="3"> <note symbol="e" position="left" xlink:label="note-0120-05" xlink:href="note-0120-05a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="f" position="left" xlink:label="note-0120-06" xlink:href="note-0120-06a" xml:space="preserve">ibidem.</note> <note symbol="g" position="left" xlink:label="note-0120-07" xlink:href="note-0120-07a" xml:space="preserve">47. h.</note> <note symbol="h" position="left" xlink:label="note-0120-08" xlink:href="note-0120-08a" xml:space="preserve">37. h.</note> </div> <p style="it"> <s xml:id="echoid-s3265" xml:space="preserve">Quod in hac, & </s> <s xml:id="echoid-s3266" xml:space="preserve">in duabus-præcedentibus factum eſt, idem ſimul, ac <lb/>vmuerſaliùs habebitur in ſequenti.</s> <s xml:id="echoid-s3267" xml:space="preserve"/> </p> <pb o="97" file="0121" n="121" rhead=""/> </div> <div xml:id="echoid-div311" type="section" level="1" n="135"> <head xml:id="echoid-head140" xml:space="preserve">PROBL. XXI. PROP. LV.</head> <p> <s xml:id="echoid-s3268" xml:space="preserve">Datæ Hyperbolæ, per punctum intra ipſam datum, cum dato <lb/>ſemi-tranſuerſo latere, quodtamen non excedat diſtantiam inter <lb/>datum punctum, & </s> <s xml:id="echoid-s3269" xml:space="preserve">datæ ſectionis centrum, MAXIMAM Hyper-<lb/>bolen inſcribere: </s> <s xml:id="echoid-s3270" xml:space="preserve">& </s> <s xml:id="echoid-s3271" xml:space="preserve">è contra.</s> <s xml:id="echoid-s3272" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3273" xml:space="preserve">Datæ Hyperbolæ, per punctum extra ipſam datum, cum dato <lb/>ſemi-tranſuerſo latere MINIMAM Hyperbolen circumſcribere.</s> <s xml:id="echoid-s3274" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3275" xml:space="preserve">Oportet autem datum punctum, vel eſſe in angulo aſymptotali, <lb/>vel in eo, qui eſt ad verticem; </s> <s xml:id="echoid-s3276" xml:space="preserve">& </s> <s xml:id="echoid-s3277" xml:space="preserve">ſi in primò caſu, neceſſe eſt, vt ſe-<lb/>mi-tranſuerſum excedat interuallum inter datum punctum, & </s> <s xml:id="echoid-s3278" xml:space="preserve">cen-<lb/>trum datæ ſectionis: </s> <s xml:id="echoid-s3279" xml:space="preserve">in ſecundò verò ſit cuiuslibet longitudinis.</s> <s xml:id="echoid-s3280" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3281" xml:space="preserve">ESto Hyperbole ABC, cuius centrum D, & </s> <s xml:id="echoid-s3282" xml:space="preserve">datum intra ipſam punctum <lb/>ſit E: </s> <s xml:id="echoid-s3283" xml:space="preserve">oporret primò per E, cum dato ſemi-tranſuerſo EF (quod ſit mi-<lb/>nus interuallo ED) _MAXIMAM_ Hyperbolen inſcribere.</s> <s xml:id="echoid-s3284" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3285" xml:space="preserve">Iungatur E D <lb/> <anchor type="figure" xlink:label="fig-0121-01a" xlink:href="fig-0121-01"/> ſecans ABC in B, <lb/>& </s> <s xml:id="echoid-s3286" xml:space="preserve">ex ipſa ED de-<lb/>matur EF æ qualis <lb/>dato ſemi- tranſ-<lb/>uerſo, & </s> <s xml:id="echoid-s3287" xml:space="preserve">per ver-<lb/>ticem E, cum cẽ-<lb/>tro F adſeribatur <lb/>ſectioni <anchor type="note" xlink:href="" symbol="a"/> A B C <anchor type="note" xlink:label="note-0121-01a" xlink:href="note-0121-01"/> Hyperbole EG ſi-<lb/>milis datæ ABC; <lb/></s> <s xml:id="echoid-s3288" xml:space="preserve">quæ (cum habeat <lb/>centrum F, velin <lb/>ipſo D, ſinempe datum ſemi-tran ſuerſum EF æquale fuerit iunctæ ED, vel <lb/>infra idem centrum D, ſi datum fuerit ipſa ED minus) erit <anchor type="note" xlink:href="" symbol="b"/> datæ Hyperbo- <anchor type="note" xlink:label="note-0121-02a" xlink:href="note-0121-02"/> læ ABC inſcripta. </s> <s xml:id="echoid-s3289" xml:space="preserve">Dico hanc eſſe _MAXIMAM_ quæſitam.</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-0121-01" xlink:href="fig-0121-01a"> <image file="0121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0121-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0121-01" xlink:href="note-0121-01a" xml:space="preserve">6. huius.</note> <note symbol="b" position="right" xlink:label="note-0121-02" xlink:href="note-0121-02a" xml:space="preserve">48. h.</note> </div> <p> <s xml:id="echoid-s3291" xml:space="preserve">Quoniam quælibet alia per verticem E, cum eodem tranſuerſo EF adſcri-<lb/>pta, ſed cum recto, quod ſit minus recto ſectionis EG, ipſa EG minor <anchor type="note" xlink:href="" symbol="c"/> eſt;</s> <s xml:id="echoid-s3292" xml:space="preserve"> <anchor type="note" xlink:label="note-0121-03a" xlink:href="note-0121-03"/> quæ verò cum recto, quod ipſum excedat qualis eſt EL <anchor type="note" xlink:href="" symbol="d"/> eſt quidem maior <anchor type="note" xlink:label="note-0121-04a" xlink:href="note-0121-04"/> eadem EG, ſed omnino ſecat circumſcriptam ABC. </s> <s xml:id="echoid-s3293" xml:space="preserve">Nam ducta FI aſym-<lb/>ptoto ſectionis EG, & </s> <s xml:id="echoid-s3294" xml:space="preserve">FM ſectionis EL, (quæ FM cadet extra E I, vt patet <lb/>ex vltima parte 37. </s> <s xml:id="echoid-s3295" xml:space="preserve">huius) ac DH ſectionis ABC: </s> <s xml:id="echoid-s3296" xml:space="preserve">erunt <anchor type="note" xlink:href="" symbol="e"/> DH, FI inter ſe pa- <anchor type="note" xlink:label="note-0121-05a" xlink:href="note-0121-05"/> rallelę, ſed FM aſymptotos EL producta ſecatur à DH, cum ſecetur quoque <lb/>ab altera parallelarum in F, quare ipſa DH ſecabit <anchor type="note" xlink:href="" symbol="f"/> Hyperbolen EL; </s> <s xml:id="echoid-s3297" xml:space="preserve">ſed <anchor type="note" xlink:label="note-0121-06a" xlink:href="note-0121-06"/> DH tota cadit extra ABC, cum ſit eius aſymptotos, ideò occurſus rectę DH <lb/>cum ſectione EL, erit extra ipſam ABC, vnde EL neceſſariò ſecabit priùs <lb/>circumſcriptam ABC. </s> <s xml:id="echoid-s3298" xml:space="preserve">Erit ergo EG _MAXIMA_ inſcripta quæſita, cum da-<lb/>to ſemi tranſuerſo EF. </s> <s xml:id="echoid-s3299" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s3300" xml:space="preserve">c.</s> <s xml:id="echoid-s3301" xml:space="preserve"/> </p> <div xml:id="echoid-div312" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0121-03" xlink:href="note-0121-03a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="d" position="right" xlink:label="note-0121-04" xlink:href="note-0121-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0121-05" xlink:href="note-0121-05a" xml:space="preserve">48. h.</note> <note symbol="f" position="right" xlink:label="note-0121-06" xlink:href="note-0121-06a" xml:space="preserve">35. h.</note> </div> <pb o="98" file="0122" n="122" rhead=""/> <p> <s xml:id="echoid-s3302" xml:space="preserve">IAM oporteat datæ Hyperbolæ GEN, cuius aſymptoti ſint FI, FO per da-<lb/>tum extra ipſam punctum B (quod tamen ſit, vel in agulo aſymptotali <lb/>IFO, vt in prima figura, velin eo, qui ipſi eſt ad verticem, vt in ſecunda, ob <lb/>id quod in 53. </s> <s xml:id="echoid-s3303" xml:space="preserve">huius monuimus, cum dato ſemi- tranſuerſo latere BD (quod <lb/>in primo caſu excedat diſtantiam BF, in ſecundo verò ſit cuiuslibet longitu-<lb/>dinis) _MINIMAM_ Hyperbolen circumſcribere.</s> <s xml:id="echoid-s3304" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3305" xml:space="preserve">Iungatur F B, <lb/> <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> quę protracta da-<lb/>tæ ſectioni GEN <lb/>occurrat in E, & </s> <s xml:id="echoid-s3306" xml:space="preserve"><lb/>producta E B ad <lb/>partes oppoſitæ <lb/>ſectionis, ſuma-<lb/>tur B D æqualis <lb/>dato ſemi-tranſ-<lb/>uerſo; </s> <s xml:id="echoid-s3307" xml:space="preserve">quę ex hy-<lb/>poteſi vtrobique <lb/>cadet in angulo <lb/>aſymptotali, ſiue <lb/>vltra centrum F, & </s> <s xml:id="echoid-s3308" xml:space="preserve">per verticem B, datæ Hyperbolæ GEN, adſcribatur <anchor type="note" xlink:href="" symbol="a"/> ſi- <anchor type="note" xlink:label="note-0122-01a" xlink:href="note-0122-01"/> milis Hyperbole ABC, cum ſemi-tranſuerſo dato BD, quæ ipſi GEN <anchor type="note" xlink:href="" symbol="b"/> erit <anchor type="note" xlink:label="note-0122-02a" xlink:href="note-0122-02"/> circumſcripta: </s> <s xml:id="echoid-s3309" xml:space="preserve">Dico hanc eſſe _MINIMAM_ quæſitam. </s> <s xml:id="echoid-s3310" xml:space="preserve">Quoniam quælibet <lb/>alia per B ei adſcripta cum recto, quod maius ſit eius recto latere, maior <anchor type="note" xlink:href="" symbol="c"/> eſt <anchor type="note" xlink:label="note-0122-03a" xlink:href="note-0122-03"/> ipſa GEN, quæ verò cum recto, quod prædicto ſit minus, qualis eſt PBQ, <lb/>eſt quidem <anchor type="note" xlink:href="" symbol="d"/> minor eadem GEN, ſed omnino ſecat inſcriptam GEN. </s> <s xml:id="echoid-s3311" xml:space="preserve">Ductis <anchor type="note" xlink:label="note-0122-04a" xlink:href="note-0122-04"/> enim DH, DR, FI, quæ ſint aſymptoti ſectionum ABC, PBQ, GEN: </s> <s xml:id="echoid-s3312" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit <anchor type="note" xlink:label="note-0122-05a" xlink:href="note-0122-05"/> DH ipſi FI parallela, & </s> <s xml:id="echoid-s3313" xml:space="preserve">DR cadet infra DH, ex vltima parte 37. </s> <s xml:id="echoid-s3314" xml:space="preserve">huius, ſed <lb/>ei occurrit in H, quare DR producta ſecabit alteram parallelam F I, <lb/>nempe aſymptoton ſectionis GEN, & </s> <s xml:id="echoid-s3315" xml:space="preserve">vlteriùs producta, ipſam, <lb/>& </s> <s xml:id="echoid-s3316" xml:space="preserve">ſectionem GEN ſecabit <anchor type="note" xlink:href="" symbol="f"/> ſed ipſa DR tota cadit extra <anchor type="note" xlink:label="note-0122-06a" xlink:href="note-0122-06"/> PBQ, cum ſit eius aſymptotos, quapropter occurſus <lb/>rectæ DR cum ſectione GEN cadet extra ſectio-<lb/>nem PBQ, ac ideò inſcripta ſectio GEN, ſe-<lb/>ctionem PBQ priùs ſecabit: </s> <s xml:id="echoid-s3317" xml:space="preserve">vnde ABC <lb/>erit _MINIMA_ circumſcripta quę-<lb/>ſita. </s> <s xml:id="echoid-s3318" xml:space="preserve">Quod ſecundò facien-<lb/>dum, ac demonſtran-<lb/>dum erat.</s> <s xml:id="echoid-s3319" xml:space="preserve"/> </p> <div xml:id="echoid-div313" type="float" level="2" n="3"> <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/QN4GHYBF/figures/0122-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0122-01" xlink:href="note-0122-01a" xml:space="preserve">6. huius.</note> <note symbol="b" position="left" xlink:label="note-0122-02" xlink:href="note-0122-02a" xml:space="preserve">48. h.</note> <note symbol="c" position="left" xlink:label="note-0122-03" xlink:href="note-0122-03a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="d" position="left" xlink:label="note-0122-04" xlink:href="note-0122-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="left" xlink:label="note-0122-05" xlink:href="note-0122-05a" xml:space="preserve">48. h.</note> <note symbol="f" position="left" xlink:label="note-0122-06" xlink:href="note-0122-06a" xml:space="preserve">35. h</note> </div> <pb o="99" file="0123" n="123" rhead=""/> </div> <div xml:id="echoid-div315" type="section" level="1" n="136"> <head xml:id="echoid-head141" xml:space="preserve">PROBL. XXII. PROP. LVI.</head> <p> <s xml:id="echoid-s3320" xml:space="preserve">Datæ Hyperbolę, per punctum intra ipſam datũ, cum dato recto <lb/>latere non excedent rectum Hyperbolæ, quæ ſimilis ſit, & </s> <s xml:id="echoid-s3321" xml:space="preserve">concen-<lb/>trica datæ per datum punctum adſcriptæ, MAXIMAM Hyperbo-<lb/>len inſcribere: </s> <s xml:id="echoid-s3322" xml:space="preserve">& </s> <s xml:id="echoid-s3323" xml:space="preserve">è contra.</s> <s xml:id="echoid-s3324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3325" xml:space="preserve">Datæ Hyperbolæ, per punctum extra ipſam datum, cum dato <lb/>recto latere MINIMAM Hyperbolen circumſcribere.</s> <s xml:id="echoid-s3326" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3327" xml:space="preserve">Oportet autem datum punctum, vel eſſe in angulo aſymptotali, <lb/>vel in eo, qui eſt ad verticem, dummodo in primò caſu datum re-<lb/>ctum latus non ſit minus recto eius Hyperbolæ, quæ ſimilis ſit, & </s> <s xml:id="echoid-s3328" xml:space="preserve"><lb/>concentrica datæ per datum punctum adſcriptæ, in ſecundò verò <lb/>ſit cuiuslibet magnitudinis.</s> <s xml:id="echoid-s3329" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3330" xml:space="preserve">SIt data Hyperbole ABC, cuius centrum D, & </s> <s xml:id="echoid-s3331" xml:space="preserve">datũ intra ipſam punctum <lb/>ſit E: </s> <s xml:id="echoid-s3332" xml:space="preserve">oportet primò per E, cum dato recto EF _MAXIMAM_ Hyperbo-<lb/>len inſcribere.</s> <s xml:id="echoid-s3333" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3334" xml:space="preserve">Iungatur ED ſecãs <lb/> <anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/> ABC in B, & </s> <s xml:id="echoid-s3335" xml:space="preserve">per E <lb/>concipiatur <anchor type="note" xlink:href="" symbol="a"/> adſcri- <anchor type="note" xlink:label="note-0123-01a" xlink:href="note-0123-01"/> bi Hyperbole EN ſi-<lb/>milis, & </s> <s xml:id="echoid-s3336" xml:space="preserve">concentrica <lb/>datę ABC, cuius re-<lb/>ctum ſit EG, quod ex <lb/>more, ordinatim ap-<lb/>plicetur diametro E <lb/>B, & </s> <s xml:id="echoid-s3337" xml:space="preserve">cum dato recto <lb/>EF, quod non ſit ma-<lb/>ius E G, adſcribatur <lb/>ipſi ABC ſimilis Hy, <lb/>perbole HEK, cuius centrum ſit I; </s> <s xml:id="echoid-s3338" xml:space="preserve">erunt ergo Hyperbolæ EH, EN inter ſe <lb/>ſimiles, quare vt rectum EF, ad rectum EG, ita ſemi-tranſuerſum EI ad ſe-<lb/>mi-tranſuerſum ED, & </s> <s xml:id="echoid-s3339" xml:space="preserve">ponitur EF non maius EG, quare EI non maius erit <lb/>ED, ſiue punctum I centrum ſectionis EH, vel cadet in ipſo D, vel infra D <lb/>centrum ABC, quapropter ipſa EH datæ ABC erit <anchor type="note" xlink:href="" symbol="b"/> inſcripta.</s> <s xml:id="echoid-s3340" xml:space="preserve"/> </p> <div xml:id="echoid-div315" 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/QN4GHYBF/figures/0123-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0123-01" xlink:href="note-0123-01a" xml:space="preserve">6. huius.</note> </div> <note symbol="b" position="right" xml:space="preserve">48. h.</note> <p> <s xml:id="echoid-s3341" xml:space="preserve">Ampliùs: </s> <s xml:id="echoid-s3342" xml:space="preserve">dico ipſam EH eſſe _MAXIMAM_ quæſitam. </s> <s xml:id="echoid-s3343" xml:space="preserve">Nam quælibet alia <lb/>per E adſcripta, cum eodem recto EF, ſed cum ſemi-tranſuerſo, quod ma-<lb/>ius ſit ipſo EI, eſt <anchor type="note" xlink:href="" symbol="c"/> minor ſectione EH, quæ verò cum eodem recto EF, at <anchor type="note" xlink:label="note-0123-03a" xlink:href="note-0123-03"/> cum ſemi-tranſuerſo EO, quod minus ſit EI, qualis ponatur eſſe ſectio EN, <lb/>eſt quidem <anchor type="note" xlink:href="" symbol="d"/> maior eadem EH, ſed omnino ſecat datam ABC: </s> <s xml:id="echoid-s3344" xml:space="preserve">quoniam du- <anchor type="note" xlink:label="note-0123-04a" xlink:href="note-0123-04"/> ctis DL, IM aſymptotis ſectionum ABC, EH, ipſæ erunt <anchor type="note" xlink:href="" symbol="e"/> inter ſe parallelæ:</s> <s xml:id="echoid-s3345" xml:space="preserve"> <anchor type="note" xlink:label="note-0123-05a" xlink:href="note-0123-05"/> ductaque OP aſymptoto ſectionis EN, ipſa OP ſecabit IM infra <anchor type="note" xlink:href="" symbol="f"/> contingen- <anchor type="note" xlink:label="note-0123-06a" xlink:href="note-0123-06"/> tem, ex communi ſectionum vertice E, & </s> <s xml:id="echoid-s3346" xml:space="preserve">producta alteri æquidiſtanti DL <pb o="100" file="0124" n="124" rhead=""/> occurret, ſi ergo ipſa DL producatur, omnino ſecabit <anchor type="note" xlink:href="" symbol="a"/> Hyperbolen EN, <anchor type="note" xlink:label="note-0124-01a" xlink:href="note-0124-01"/> ſed DL tota cadit extra ſectionem ABC, cum ſit eius aſymptotos, quare <lb/>occurſus rectæ DL, cum ſectione EN, cadet extra ABC, ac ideò EN ſecabit <lb/>priùs circumſcriptam ABC: </s> <s xml:id="echoid-s3347" xml:space="preserve">vnde ſectio HEK eſt _MAXIMA_ inſcripta quæſi-<lb/>ta, cum dato recto EF. </s> <s xml:id="echoid-s3348" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s3349" xml:space="preserve">c.</s> <s xml:id="echoid-s3350" xml:space="preserve"/> </p> <div xml:id="echoid-div316" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0123-03" xlink:href="note-0123-03a" xml:space="preserve">3. Co-<lb/>roll. 19. h.</note> <note symbol="d" position="right" xlink:label="note-0123-04" xlink:href="note-0123-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0123-05" xlink:href="note-0123-05a" xml:space="preserve">48. h.</note> <note symbol="f" position="right" xlink:label="note-0123-06" xlink:href="note-0123-06a" xml:space="preserve">Coroll. <lb/>36. huius.</note> <note symbol="a" position="left" xlink:label="note-0124-01" xlink:href="note-0124-01a" xml:space="preserve">35. h.</note> </div> <p> <s xml:id="echoid-s3351" xml:space="preserve">IAM oporteat datæ Hyperbolę HEK, cuius aſymptoti IM, IQ, per datum <lb/>extra ipſam punctum B, quod (per ea, quæ in 53. </s> <s xml:id="echoid-s3352" xml:space="preserve">huius) ſit vel in angulo <lb/>ad verticem aſymptotalis, vt in prima figura, vel in ipſo aſymptotali MIQ, <lb/>vt in ſecunda, cum dato recto latere _MINIM AM_ Hyperbolen circumſcri-<lb/>bere.</s> <s xml:id="echoid-s3353" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3354" xml:space="preserve">Iungatur BI, & </s> <s xml:id="echoid-s3355" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0124-01a" xlink:href="fig-0124-01"/> producatur vſque oc-<lb/>currat datæ ſectioni <lb/>HEK in E; </s> <s xml:id="echoid-s3356" xml:space="preserve">erit I E, <lb/>ipſius ſemi-tranſuer-<lb/>ſum, cuius rectum la-<lb/>tus ſit EF, & </s> <s xml:id="echoid-s3357" xml:space="preserve">ex B cõ-<lb/>cipiatur adſcribi Hy-<lb/>perbole TBV ſimilis, <lb/>& </s> <s xml:id="echoid-s3358" xml:space="preserve">concentrica datæ <lb/>HEK, cuius rectum <lb/>ſit BS; </s> <s xml:id="echoid-s3359" xml:space="preserve">& </s> <s xml:id="echoid-s3360" xml:space="preserve">datũ rectum <lb/>BR, in caſu primæ fi-<lb/>guræ (in quo datum punctum B cadit in angulo ad verticem aſymptotalis <lb/>MIQ) ſit cuiuslibet longitudinis; </s> <s xml:id="echoid-s3361" xml:space="preserve">in ſecundo verò non ſit minus BS, & </s> <s xml:id="echoid-s3362" xml:space="preserve">per B <lb/>cum recto BR adſcribatur Hyperbole ABC ſimilis datæ HEK, quæ item ſi-<lb/>milis erit TBV, & </s> <s xml:id="echoid-s3363" xml:space="preserve">ſit eius centrum D: </s> <s xml:id="echoid-s3364" xml:space="preserve">erit ergo in ſecunda figura, ob Hyper-<lb/>bolarum ABC, TBV ſimilitudinem, rectum BR ad BS vt ſemi- tranſuerſum <lb/>BD ad ſemi-tranſuerſum BI, eſtq; </s> <s xml:id="echoid-s3365" xml:space="preserve">BR non minus BS, quare BD erit non minus <lb/>BD; </s> <s xml:id="echoid-s3366" xml:space="preserve">ex quo centrum D ſectionis ABC, vel cadet in I, vel ſupra I centrum <lb/>ſimilis ſectionis HEK: </s> <s xml:id="echoid-s3367" xml:space="preserve">vnde ipſa ABC erit <anchor type="note" xlink:href="" symbol="b"/> omnino datæ HEK circumſcri- <anchor type="note" xlink:label="note-0124-02a" xlink:href="note-0124-02"/> pta.</s> <s xml:id="echoid-s3368" xml:space="preserve"/> </p> <div xml:id="echoid-div317" type="float" level="2" n="3"> <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/QN4GHYBF/figures/0124-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0124-02" xlink:href="note-0124-02a" xml:space="preserve">48. h.</note> </div> <p> <s xml:id="echoid-s3369" xml:space="preserve">Dicotandem ipſam ABC eſſe _MINIM AM_ quæſitam: </s> <s xml:id="echoid-s3370" xml:space="preserve">Quoniam alia Hy-<lb/>perbole, quæ per B adſcribitur, cum eodem recto BR, ſed cum ſemi-tranſ-<lb/>uerſo, quod minus ſit BD, eſt <anchor type="note" xlink:href="" symbol="c"/> maior ipſa ABC; </s> <s xml:id="echoid-s3371" xml:space="preserve">quæ verò cum eodem re- <anchor type="note" xlink:label="note-0124-03a" xlink:href="note-0124-03"/> cto BR, & </s> <s xml:id="echoid-s3372" xml:space="preserve">cum ſemi-tranſuerſo BX, quod excedat BD, qualis dicatur eſſe <lb/>ſectio TBV, eſt quidem <anchor type="note" xlink:href="" symbol="d"/> minor eadem ABC, ſed omnino ſecat datã KEH.</s> <s xml:id="echoid-s3373" xml:space="preserve"> <anchor type="note" xlink:label="note-0124-04a" xlink:href="note-0124-04"/> Ductis enim ſimilium Hyperbolarum ABC, HEK aſymptotis DL, IM; </s> <s xml:id="echoid-s3374" xml:space="preserve">ipſę <lb/>erunt inter ſe parallelæ; </s> <s xml:id="echoid-s3375" xml:space="preserve">ductaque XY aſymptoto ſectionis TBV; </s> <s xml:id="echoid-s3376" xml:space="preserve">cum ſint <lb/>Hyperbole ABC, TBV per eundem verticem B adſcriptæ, cum eodem re-<lb/>cto BR earum aſymptoti DL, XY infra contingentem ex vertice B ſe mutuò <lb/>ſecabunt, <anchor type="note" xlink:href="" symbol="e"/> & </s> <s xml:id="echoid-s3377" xml:space="preserve">cum XY ſecet DL, & </s> <s xml:id="echoid-s3378" xml:space="preserve">alteram huic æquidiſtantem IM ſecabit;</s> <s xml:id="echoid-s3379" xml:space="preserve"> <anchor type="note" xlink:label="note-0124-05a" xlink:href="note-0124-05"/> ſed eſt IM aſymptotos HEK, vnde XY producta <anchor type="note" xlink:href="" symbol="f"/> ſecabit quidem HEK, at <anchor type="note" xlink:label="note-0124-06a" xlink:href="note-0124-06"/> XY tota cadit extra TBV, cũ ſit eius aſymptotos; </s> <s xml:id="echoid-s3380" xml:space="preserve">quare XY conueniet cum <lb/>ſectione HEK, extra Hyperbolen TBV, vnde ipſa TBV ſecabit priùs inſcri-<lb/>ptam ſectionem HEK. </s> <s xml:id="echoid-s3381" xml:space="preserve">Quapropter ſectio ABC eſt _MINIMA_ circumſcripta <lb/>quæſita: </s> <s xml:id="echoid-s3382" xml:space="preserve">cum dato recto BR. </s> <s xml:id="echoid-s3383" xml:space="preserve">Quod ſecundò faciendum, ac demonſtrandum <lb/>erat.</s> <s xml:id="echoid-s3384" xml:space="preserve"/> </p> <div xml:id="echoid-div318" type="float" level="2" n="4"> <note symbol="c" position="left" xlink:label="note-0124-03" xlink:href="note-0124-03a" xml:space="preserve">3. Co-<lb/>19. huius.</note> <note symbol="d" position="left" xlink:label="note-0124-04" xlink:href="note-0124-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="left" xlink:label="note-0124-05" xlink:href="note-0124-05a" xml:space="preserve">Coroll. <lb/>36. huius.</note> <note symbol="f" position="left" xlink:label="note-0124-06" xlink:href="note-0124-06a" xml:space="preserve">35. h.</note> </div> <pb o="101" file="0125" n="125" rhead=""/> </div> <div xml:id="echoid-div320" type="section" level="1" n="137"> <head xml:id="echoid-head142" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s3385" xml:space="preserve">EX quinque proximè præcedentibus problematibus ſatis conſtat, _MAXI-_ <lb/>_MAM_, vel _MINIMAM_ Hyperbolen, inſcriptam, vel circumſcriptam <lb/>cuilibet datæ per punctum intra, vel extra Hyperbolen datũ in locis poſſibili-<lb/>bus, ad eandem regulam, aut concentricè adſcriptam, vel cuius centrum pro <lb/>inſcripta cadat infra centrum datæ, & </s> <s xml:id="echoid-s3386" xml:space="preserve">pro circumſcripta cadat vltra, ſemper <lb/>eidem datæ Hyperbolæ ſimilem eſſe.</s> <s xml:id="echoid-s3387" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div321" type="section" level="1" n="138"> <head xml:id="echoid-head143" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s3388" xml:space="preserve">PAtet quoque, _MAXIMAM_ Hyperbolarum datæ Hyperbolæ ſimilium, <lb/>per datum intra ipſam punctum inſcriptarum, eſſe concentricam, cum <lb/>hæc, inter ſimiles, ſit _MAXIMORVM_ laterum. </s> <s xml:id="echoid-s3389" xml:space="preserve">Item _MINIMAM_ Hyperbola-<lb/>rum datæ Hyperbolæ ſimilium per datum extra ipſam punctum in angulo <lb/>aſymptotali, circumſcriptarum, eſſe pariter concentricam, cum eadem, in-<lb/>ter ſimiles, ſit _MINIMORV M_ laterum.</s> <s xml:id="echoid-s3390" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div322" type="section" level="1" n="139"> <head xml:id="echoid-head144" xml:space="preserve">PROBL. XXIII. PROP. LVII.</head> <p> <s xml:id="echoid-s3391" xml:space="preserve">Datis magnitudine, & </s> <s xml:id="echoid-s3392" xml:space="preserve">poſitione cuiuslibet coni- ſectionis dia-<lb/>metri ſegmento, & </s> <s xml:id="echoid-s3393" xml:space="preserve">vna applicatarum, & </s> <s xml:id="echoid-s3394" xml:space="preserve">pro Hyperbola, & </s> <s xml:id="echoid-s3395" xml:space="preserve">Ellipſi <lb/>dato etiam tranſuerſo latere; </s> <s xml:id="echoid-s3396" xml:space="preserve">imperatam coni-ſectionem deſcri-<lb/>bere.</s> <s xml:id="echoid-s3397" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3398" xml:space="preserve">SIt in qualibet ſigura, pro quacunque coni-ſectione, datum magnitudine, <lb/>& </s> <s xml:id="echoid-s3399" xml:space="preserve">poſitione diametri ſegmentum AB, & </s> <s xml:id="echoid-s3400" xml:space="preserve">vna applicatarum CD, & </s> <s xml:id="echoid-s3401" xml:space="preserve">pro <lb/>Hyperbola, & </s> <s xml:id="echoid-s3402" xml:space="preserve">Ellipſi in ſecunda, & </s> <s xml:id="echoid-s3403" xml:space="preserve">tertia, datum ſit quoque rectum latus <lb/>AF, quod pro Hyperbola in ſecunda vltra BA ipſi in directum ponatur, & </s> <s xml:id="echoid-s3404" xml:space="preserve">in <lb/>tertia, ex A ad partes B: </s> <s xml:id="echoid-s3405" xml:space="preserve">oportet circa diametrum AB, ſuper applicatam <lb/>CD, quæſitam coni-ſectionem deſcribere.</s> <s xml:id="echoid-s3406" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3407" xml:space="preserve">Fiat in ſingulis figuris, vt <lb/> <anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/> AB ad BC, ita BC ad BE <lb/>ipſi BC in directum poſitã, <lb/>& </s> <s xml:id="echoid-s3408" xml:space="preserve">per E, in prima figura, <lb/>ducta EG parallela ad BA, <lb/>vel in ſecunda, & </s> <s xml:id="echoid-s3409" xml:space="preserve">tertia iun-<lb/>cta FE, occurrat AG, (quæ <lb/>ipſi CD æquidiſtet) in G, & </s> <s xml:id="echoid-s3410" xml:space="preserve"><lb/>per verticem A cum data <lb/>diametro AB, datiſque la-<lb/>teribus AF, A G deſcriba-<lb/>tur <anchor type="note" xlink:href="" symbol="a"/> quæſiti nominis ſectio <anchor type="note" xlink:label="note-0125-01a" xlink:href="note-0125-01"/> CAD, cuius ordinatim du-<lb/>ctæ ad angulum ABD ap- <pb o="102" file="0126" n="126" rhead=""/> plicentur. </s> <s xml:id="echoid-s3411" xml:space="preserve">Dico ipſam eſſe quæſitam. </s> <s xml:id="echoid-s3412" xml:space="preserve">Cum enim ſit CB media proportio <lb/> <anchor type="note" xlink:label="note-0126-01a" xlink:href="note-0126-01"/> nalis inter altitudinem BA, & </s> <s xml:id="echoid-s3413" xml:space="preserve">latitudinem BE, erit <anchor type="note" xlink:href="" symbol="a"/> BC, itemque ei æqua- lis BD, deſcriptæ ſectionis ſemi-applicata, nempe ſectio per C, & </s> <s xml:id="echoid-s3414" xml:space="preserve">D omni-<lb/>no tranſibit; </s> <s xml:id="echoid-s3415" xml:space="preserve">eſtque A vertex, AB diameter, & </s> <s xml:id="echoid-s3416" xml:space="preserve">AF tranſuerſum Hyperbole, <lb/>aut Ellipſis, ex conſtructione: </s> <s xml:id="echoid-s3417" xml:space="preserve">quare factum eſt quod erat propoſitum.</s> <s xml:id="echoid-s3418" xml:space="preserve"/> </p> <div xml:id="echoid-div322" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0125-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0125-01" xlink:href="note-0125-01a" xml:space="preserve">5. 6. 7. <lb/>huius.</note> <note symbol="a" position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">Coroll. <lb/>prop. 1. h.</note> </div> </div> <div xml:id="echoid-div324" type="section" level="1" n="140"> <head xml:id="echoid-head145" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s3419" xml:space="preserve">EX hac conſtat quomodo, magnitudine, & </s> <s xml:id="echoid-s3420" xml:space="preserve">poſitione datis tranſuerſo la-<lb/>tere, aut diametro AF, & </s> <s xml:id="echoid-s3421" xml:space="preserve">vna applicatarum CD, per terminos, A, C, <lb/>F, D, Ellipſis deſcribi poſſit.</s> <s xml:id="echoid-s3422" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div325" type="section" level="1" n="141"> <head xml:id="echoid-head146" xml:space="preserve">THEOR. XXIX. PROP. LIIX.</head> <p> <s xml:id="echoid-s3423" xml:space="preserve">Si coni-ſectionem, vel circuli circumferentiam duæ rectæ lineæ <lb/>contingant, ipſæ productæ conuenient ſimul extra ſectionem; </s> <s xml:id="echoid-s3424" xml:space="preserve">ſed <lb/>in Parabola, vel Hyperbola ſibiipſis occurrent ad partes periphe-<lb/>riæ à contactibus terminatę: </s> <s xml:id="echoid-s3425" xml:space="preserve">In Ellipſi verò ad partes ſui ipſius por-<lb/>tionis à linea tactus iungente abſciſſæ, in qua centrum nõ reperitur.</s> <s xml:id="echoid-s3426" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3427" xml:space="preserve">ESto Parabole, vel Hyperbole ABC (nam de circulo, & </s> <s xml:id="echoid-s3428" xml:space="preserve">Ellipſi id ab <lb/>Apollonio oſtenſum fuit in vigeſima ſeptima ſecũdi conicorum) quàm <lb/>in punctis A, C tangant rectæ AD, CE. <lb/></s> <s xml:id="echoid-s3429" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> Dico, ſi producantur ad partes ſectionis <lb/>ABC à contactibus A, C terminatæ ipſas <lb/>inter ſe conuenire.</s> <s xml:id="echoid-s3430" xml:space="preserve"/> </p> <div xml:id="echoid-div325" type="float" level="2" n="1"> <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a"> <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0126-01"/> </figure> </div> <p> <s xml:id="echoid-s3431" xml:space="preserve">Si enim per alterum contactuum, vt per <lb/>C, intelligatur fectro ris diameter H C G, <lb/> <anchor type="note" xlink:label="note-0126-02a" xlink:href="note-0126-02"/> certum <anchor type="note" xlink:href="" symbol="b"/> eſt contingentem A D, ſi produ- catur, cum diametro HG extra ſectionem <lb/>conuenire, hoc eſt ad partes G; </s> <s xml:id="echoid-s3432" xml:space="preserve">ſiergo AD <lb/>ſecat CG, neceſſariò ſecabit priùs tangen-<lb/>tem CE, quæ cadit inter ſectionis periphe-<lb/>riam ABC, & </s> <s xml:id="echoid-s3433" xml:space="preserve">diametrum HC: </s> <s xml:id="echoid-s3434" xml:space="preserve">quare tan-<lb/>gentes AD, CE ſibi ipſis occurrunt. </s> <s xml:id="echoid-s3435" xml:space="preserve">Quod <lb/>erat, &</s> <s xml:id="echoid-s3436" xml:space="preserve">c.</s> <s xml:id="echoid-s3437" xml:space="preserve"/> </p> <div xml:id="echoid-div326" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0126-02" xlink:href="note-0126-02a" xml:space="preserve">24. 25. <lb/>pr. conic.</note> </div> </div> <div xml:id="echoid-div328" type="section" level="1" n="142"> <head xml:id="echoid-head147" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3438" xml:space="preserve">CVm recta CE ſectioni occurrat, & </s> <s xml:id="echoid-s3439" xml:space="preserve">producta ex vtraque parte extra ſe-<lb/>ctionem cadat, ſi ex puncto A, quod eſt in ſectione, ducta ſit AF, ipſi <lb/>EC æquidiſtans, producta ex vtraque parte ſectioni <anchor type="note" xlink:href="" symbol="c"/> occurret; </s> <s xml:id="echoid-s3440" xml:space="preserve">ſed AD tota <anchor type="note" xlink:label="note-0126-03a" xlink:href="note-0126-03"/> cadit extra ſectionem, cum ſit contingens; </s> <s xml:id="echoid-s3441" xml:space="preserve">quare AD non congruit cum AF, <lb/>ſed ipſę ſe mutuò ſecant. </s> <s xml:id="echoid-s3442" xml:space="preserve">Cum ergo DA ſecet alteram æquidiſtantium AF, ſi <lb/>producatur, ſecabit, & </s> <s xml:id="echoid-s3443" xml:space="preserve">reliquam CE ad partes peripheriæ ABC à contacti-<lb/>bus A, C, terminatæ. </s> <s xml:id="echoid-s3444" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s3445" xml:space="preserve"/> </p> <div xml:id="echoid-div328" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-0126-03" xlink:href="note-0126-03a" xml:space="preserve">18. pri-<lb/>mi conic.</note> </div> <pb o="103" file="0127" n="127" rhead=""/> </div> <div xml:id="echoid-div330" type="section" level="1" n="143"> <head xml:id="echoid-head148" xml:space="preserve">THEOR. XXX. PROP. LIX.</head> <p> <s xml:id="echoid-s3446" xml:space="preserve">Si coni-ſectionem, vel circuli circumferentiam recta linea con-<lb/>tingat conueniens cum diametro, cui à tactu ſit ordinatim applica-<lb/>ta vſque ad ſectionem, recta linea iungens alterum terminum ap-<lb/>plicatæ, & </s> <s xml:id="echoid-s3447" xml:space="preserve">occurſum tangentis cum diametro, erit eidem ſectioni <lb/>ad alteram diametri partem contingens.</s> <s xml:id="echoid-s3448" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3449" xml:space="preserve">SIt coni-ſectio quæcunque, vel circuli circumferentia ABC, cuius diame-<lb/>ter ſit DE, & </s> <s xml:id="echoid-s3450" xml:space="preserve">ſit quæpiam AD ſectionem contingens in A, diametro oc-<lb/>currens in D, & </s> <s xml:id="echoid-s3451" xml:space="preserve">ex contactu A ducta ſit in ſectione diametro DE ordinatim <lb/>applicata AC, dico iunctam DC ſectionem quoque contingere.</s> <s xml:id="echoid-s3452" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3453" xml:space="preserve">Si enim poſſibile eſt, quæ ex C ducitur <lb/> <anchor type="figure" xlink:label="fig-0127-01a" xlink:href="fig-0127-01"/> contingens, non ſit CD, ſed alia CF, quæ <lb/> <anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/> cum tangente AD <anchor type="note" xlink:href="" symbol="a"/> conueniet, ſed in alio puncto quàm D, vt in F.</s> <s xml:id="echoid-s3454" xml:space="preserve"/> </p> <div xml:id="echoid-div330" type="float" level="2" n="1"> <figure xlink:label="fig-0127-01" xlink:href="fig-0127-01a"> <image file="0127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0127-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">58. h.</note> </div> <p> <s xml:id="echoid-s3455" xml:space="preserve">Iam cum FA, FC ſectionem contingant, <lb/>& </s> <s xml:id="echoid-s3456" xml:space="preserve">per contactus ducta ſit AC, quæ bifariam <lb/>ſecta eſt à diametro D E in E, ſi iungatur <lb/> <anchor type="note" xlink:label="note-0127-02a" xlink:href="note-0127-02"/> FEG ipſa <anchor type="note" xlink:href="" symbol="b"/> erit ſectionis diameter, hoc eſt bifariam ſecabit quamlibet aliã HI ipſi AC <lb/>æquidiſtanter ductam, vt in G, ſed D E L <lb/>quoque bifariam ſecat eandem HI in L, cum <lb/>DEL ſit diameter, per hypoteſim; </s> <s xml:id="echoid-s3457" xml:space="preserve">ergo ea-<lb/>dem recta HI in duobus diuerſis punctis G, <lb/>& </s> <s xml:id="echoid-s3458" xml:space="preserve">L bifariam diuiditur: </s> <s xml:id="echoid-s3459" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s3460" xml:space="preserve">Non eſt ergo ex C alia contin-<lb/>gens linea quàm CD. </s> <s xml:id="echoid-s3461" xml:space="preserve">Quod erat,</s> </p> <div xml:id="echoid-div331" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">29. ſe-<lb/>cundi co-<lb/>nic.</note> </div> <p style="it"> <s xml:id="echoid-s3462" xml:space="preserve">Cum Propoſitionum 13. </s> <s xml:id="echoid-s3463" xml:space="preserve">ac 14. </s> <s xml:id="echoid-s3464" xml:space="preserve">ſept. </s> <s xml:id="echoid-s3465" xml:space="preserve">Pappi, in hac noſtra tractatione fre-<lb/>quens ſit vſus, liceat hac eas transferre, vtranque ſimul ſequenti Theo-<lb/>remate demonſtrare.</s> <s xml:id="echoid-s3466" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div333" type="section" level="1" n="144"> <head xml:id="echoid-head149" xml:space="preserve">THEOR. XXXI. PROP. LX.</head> <p> <s xml:id="echoid-s3467" xml:space="preserve">Rectangulorum ſub partibus datæ rectę terminatæ MAXIMVM <lb/>eſt id, quod ab æqualibus ſegmentis producitur; </s> <s xml:id="echoid-s3468" xml:space="preserve">reliquorum verò <lb/>id, quod fit à partibus minus inæqualibus, maius eſt eo, quod ab <lb/>inæqualioribus continetur.</s> <s xml:id="echoid-s3469" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3470" xml:space="preserve">SIt data recta linea AB terminata bifariam ſecta in C, & </s> <s xml:id="echoid-s3471" xml:space="preserve">non bifariam <lb/>vtcunque in D, E, &</s> <s xml:id="echoid-s3472" xml:space="preserve">c. </s> <s xml:id="echoid-s3473" xml:space="preserve">Dico, &</s> <s xml:id="echoid-s3474" xml:space="preserve">c.</s> <s xml:id="echoid-s3475" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3476" xml:space="preserve">Cum enim recta AB ſecta ſit bifariam in C, & </s> <s xml:id="echoid-s3477" xml:space="preserve">non bifariam in D, erit <lb/>quadratum AC, ſiue rectangulum ACB, æquale rectangulo ADB, vna <pb o="104" file="0128" n="128" rhead=""/> cum quadrato intermediæ partis DC; </s> <s xml:id="echoid-s3478" xml:space="preserve">rectangulum ergo ACB ſuperat re-<lb/>ctangulum ADB; </s> <s xml:id="echoid-s3479" xml:space="preserve">& </s> <s xml:id="echoid-s3480" xml:space="preserve">hoc ſemper; </s> <s xml:id="echoid-s3481" xml:space="preserve">ergo rectangulum ACB, ſub æqualibus <lb/>partibus compræhenſum, eſt _MAXIMV M._ </s> <s xml:id="echoid-s3482" xml:space="preserve">Quod primo, &</s> <s xml:id="echoid-s3483" xml:space="preserve">c.</s> <s xml:id="echoid-s3484" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3485" xml:space="preserve">Item, quadrato dimidiæ A C æquatur <lb/> <anchor type="figure" xlink:label="fig-0128-01a" xlink:href="fig-0128-01"/> rectangulum A D B, vna cum quadrato <lb/>DC, & </s> <s xml:id="echoid-s3486" xml:space="preserve">eidem quadrato AC æquatur re-<lb/>ctangulum AED vna cum quadrato EC, <lb/>ergo rectangulum A D C cum quadrato <lb/>DC, æquale erit rectangulo AEB, cum <lb/>quadrato EC, eſt autem quadratum DC <lb/>minus quadrato EC, cum ſit linea DC mi-<lb/>nor EC, ex hypoteſi; </s> <s xml:id="echoid-s3487" xml:space="preserve">ergo rectangulum <lb/>ADC maius erit rectangulo AEB. </s> <s xml:id="echoid-s3488" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s3489" xml:space="preserve">c.</s> <s xml:id="echoid-s3490" xml:space="preserve"/> </p> <div xml:id="echoid-div333" type="float" level="2" n="1"> <figure xlink:label="fig-0128-01" xlink:href="fig-0128-01a"> <image file="0128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0128-01"/> </figure> </div> </div> <div xml:id="echoid-div335" type="section" level="1" n="145"> <head xml:id="echoid-head150" xml:space="preserve">THEOR. XXXII. PROP. LXI.</head> <p> <s xml:id="echoid-s3491" xml:space="preserve">Si fuerint duæ quæcunque coni-ſectiones, non excepto circulo, <lb/>eiuſdem, vel diuerſi nominis per diuerſos vertices ſimul adſcriptæ, <lb/>quæ in eiuſdem communis ordinatim ductæ extremis punctis ſimul <lb/>conueniant, è quorum altero eadem recta linea vtranque ſectionem <lb/>contingat, ea coni-ſectio cuius vertex cadit infra verticem alterius <lb/>erit alteri inſcripta, & </s> <s xml:id="echoid-s3492" xml:space="preserve">in ijſdem tantùm applicatæ extremis ſe con-<lb/>tingent.</s> <s xml:id="echoid-s3493" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3494" xml:space="preserve">SInt duæ quælibet coni-ſectiones ABC, ADC non excepto circulo, eiuſ-<lb/>dem, vel diuerſi nominis per diuerſos vertices B, D ſimul adſcriptæ, <lb/> <anchor type="figure" xlink:label="fig-0128-02a" xlink:href="fig-0128-02"/> quarum communis diameter ſit BH, communiſq; </s> <s xml:id="echoid-s3495" xml:space="preserve">applicata ſit AC, in cuius <lb/>extremis A, C, ſectiones ſimul occurrant, & </s> <s xml:id="echoid-s3496" xml:space="preserve">ex eorum altero veluti ex A <pb o="105" file="0129" n="129" rhead=""/> recta AE vtranque ſectionem contingat. </s> <s xml:id="echoid-s3497" xml:space="preserve">Dico ſectionem ADC, cuius ver-<lb/>tex D eſt infra alterius verticem B, totam cadere intra ſectionem ABC, hoc <lb/>eſt ei eſſe inſcriptam, & </s> <s xml:id="echoid-s3498" xml:space="preserve">in extremis A, C, ſe mutuò contingere.</s> <s xml:id="echoid-s3499" xml:space="preserve"/> </p> <div xml:id="echoid-div335" type="float" level="2" n="1"> <figure xlink:label="fig-0128-02" xlink:href="fig-0128-02a"> <image file="0128-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0128-02"/> </figure> </div> <p> <s xml:id="echoid-s3500" xml:space="preserve">Nam producta AE vſque ad occurſum cum diametro in E (ſi tamen ap-<lb/>plicata AC non fuerit diameter circuli, vel Ellipſis, vt ſecunda figura, quo <lb/> <anchor type="note" xlink:label="note-0129-01a" xlink:href="note-0129-01"/> in caſu contingentes AE, CE ſibi ipſis, & </s> <s xml:id="echoid-s3501" xml:space="preserve">coniugatæ diametro BT <anchor type="note" xlink:href="" symbol="a"/> æquidi- ſtabunt) iungatur ECO, quæ item vtranque ſectionem <anchor type="note" xlink:href="" symbol="b"/> continget in C: </s> <s xml:id="echoid-s3502" xml:space="preserve">&</s> <s xml:id="echoid-s3503" xml:space="preserve"> <anchor type="note" xlink:label="note-0129-02a" xlink:href="note-0129-02"/> applicetur quæcunque L O. </s> <s xml:id="echoid-s3504" xml:space="preserve">_lo_ eaſdem ſectiones ſecans in I, N, G, M. </s> <s xml:id="echoid-s3505" xml:space="preserve">_i, n,_ <lb/>_g, m_, contingentes verò in L, O. </s> <s xml:id="echoid-s3506" xml:space="preserve">_l, o_; </s> <s xml:id="echoid-s3507" xml:space="preserve">ducanturque ex verticibus tangentes <lb/>BQ, DP, quę ordinatim ductis æquidiſtabunt, & </s> <s xml:id="echoid-s3508" xml:space="preserve">iungatur AB, ſecans DP <lb/>in S.</s> <s xml:id="echoid-s3509" xml:space="preserve"/> </p> <div xml:id="echoid-div336" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0129-01" xlink:href="note-0129-01a" xml:space="preserve">27. ſec. <lb/>conic. & <lb/>6. eiuſd.</note> <note symbol="b" position="right" xlink:label="note-0129-02" xlink:href="note-0129-02a" xml:space="preserve">59. h.</note> </div> <p> <s xml:id="echoid-s3510" xml:space="preserve">Iam cum ſit AH æqualis H C, erit LF. </s> <s xml:id="echoid-s3511" xml:space="preserve">_l f_ æqualis F O. </s> <s xml:id="echoid-s3512" xml:space="preserve">_f o_, eſtque IF. </s> <s xml:id="echoid-s3513" xml:space="preserve">_if_ <lb/>æqualis FN. </s> <s xml:id="echoid-s3514" xml:space="preserve">_fn_, & </s> <s xml:id="echoid-s3515" xml:space="preserve">GF. </s> <s xml:id="echoid-s3516" xml:space="preserve">_gf_, ipſi FM. </s> <s xml:id="echoid-s3517" xml:space="preserve">_fm_ (ſunt enim ſectionum ſemi-appli-<lb/>catæ) quare reliquæ LI. </s> <s xml:id="echoid-s3518" xml:space="preserve">_li_, ON. </s> <s xml:id="echoid-s3519" xml:space="preserve">_on_, æquales erunt, itemque LG. </s> <s xml:id="echoid-s3520" xml:space="preserve">_lg_, OM. <lb/></s> <s xml:id="echoid-s3521" xml:space="preserve">_o m_ inter ſe æquales, ideoque rectangulum OIL. </s> <s xml:id="echoid-s3522" xml:space="preserve">_oil_ æquabitur rectangulo <lb/>NLI. </s> <s xml:id="echoid-s3523" xml:space="preserve">_nli_, & </s> <s xml:id="echoid-s3524" xml:space="preserve">rectangulum OGL. </s> <s xml:id="echoid-s3525" xml:space="preserve">_ogl_ rectangulo MLG. </s> <s xml:id="echoid-s3526" xml:space="preserve">_mlg_. </s> <s xml:id="echoid-s3527" xml:space="preserve">Et cum in ſe-<lb/> <anchor type="note" xlink:label="note-0129-03a" xlink:href="note-0129-03"/> ctione ABC ſit <anchor type="note" xlink:href="" symbol="c"/> quadratum BQ ad quadratum QA, hoc eſt quadratum SP ad PA, vt rectangulum NLI. </s> <s xml:id="echoid-s3528" xml:space="preserve">_nli_ ad quadratum L A. </s> <s xml:id="echoid-s3529" xml:space="preserve">_l_A, & </s> <s xml:id="echoid-s3530" xml:space="preserve">in ſectione <lb/> <anchor type="note" xlink:label="note-0129-04a" xlink:href="note-0129-04"/> ADC quadratum DP ad idem PA <anchor type="note" xlink:href="" symbol="d"/> ſit vt rectangulum MLG. </s> <s xml:id="echoid-s3531" xml:space="preserve">_mlg_ ad idem quadratum L A. </s> <s xml:id="echoid-s3532" xml:space="preserve">_l_A, habeatque quadratum SP ad PA minorem rationem <lb/>quàm DP quadratum, ad idem quadratum PA, habebit quoque rectangu-<lb/>lum NLI. </s> <s xml:id="echoid-s3533" xml:space="preserve">_nli_ ad quadratum LA. </s> <s xml:id="echoid-s3534" xml:space="preserve">_l_A minorem rationem quàm rectangulum <lb/>MLG. </s> <s xml:id="echoid-s3535" xml:space="preserve">_mlg_ ad idem quadratum LA. </s> <s xml:id="echoid-s3536" xml:space="preserve">_l_A; </s> <s xml:id="echoid-s3537" xml:space="preserve">quare rectangulum NLI. </s> <s xml:id="echoid-s3538" xml:space="preserve">_nli_, hoc <lb/>eſt OIL. </s> <s xml:id="echoid-s3539" xml:space="preserve">_oil_, minus eſt rectangulo MLG. </s> <s xml:id="echoid-s3540" xml:space="preserve">_mlg_, ſiue rectangulo OGL. </s> <s xml:id="echoid-s3541" xml:space="preserve">_ogl_; <lb/></s> <s xml:id="echoid-s3542" xml:space="preserve"> <anchor type="note" xlink:label="note-0129-05a" xlink:href="note-0129-05"/> vnde punctum I remotius <anchor type="note" xlink:href="" symbol="e"/> eſt ab ipſo F quàm pũctum G. </s> <s xml:id="echoid-s3543" xml:space="preserve">_g_, ſed I. </s> <s xml:id="echoid-s3544" xml:space="preserve">_i_ eſt in ipſa ſectione ABC; </s> <s xml:id="echoid-s3545" xml:space="preserve">quare punctum G. </s> <s xml:id="echoid-s3546" xml:space="preserve">_g_ ſectionis ADC cadet intra ABC, & </s> <s xml:id="echoid-s3547" xml:space="preserve">ſic <lb/>de quolibet alio puncto ſectionis SADCT, præter A, C: </s> <s xml:id="echoid-s3548" xml:space="preserve">vnde ipſa ADC in-<lb/>ſcripta erit ſectioni ABC, & </s> <s xml:id="echoid-s3549" xml:space="preserve">in punctis tantùm A, C extremis eiuſdem ap-<lb/>plicatæ ſe mutuò contingent. </s> <s xml:id="echoid-s3550" xml:space="preserve">Quod erat dei<unsure/>nonſtrandum.</s> <s xml:id="echoid-s3551" xml:space="preserve"/> </p> <div xml:id="echoid-div337" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0129-03" xlink:href="note-0129-03a" xml:space="preserve">16. tertij <lb/>conic.</note> <note symbol="d" position="right" xlink:label="note-0129-04" xlink:href="note-0129-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0129-05" xlink:href="note-0129-05a" xml:space="preserve">conuerſ. <lb/>60. h.</note> </div> </div> <div xml:id="echoid-div339" type="section" level="1" n="146"> <head xml:id="echoid-head151" xml:space="preserve">THEOR. XXXIII. PROP. LXII.</head> <p> <s xml:id="echoid-s3552" xml:space="preserve">Siextrema inæqualium baſium menſalis, cuiuſcunque coni- ſe-<lb/>ctionis, vel circuli, ad vtranque diametri partem rectis lineis iun-<lb/>gantur, ipſæ ſimul, & </s> <s xml:id="echoid-s3553" xml:space="preserve">in eodem diametri puncto conuenient, à <lb/>quo, ſi ad terminos ordinatim ductæ per interſectionem diagona-<lb/>lis cum diametro, ducantur aliæ rectæ lineæ, hæ omnino ſectionem <lb/>contingent.</s> <s xml:id="echoid-s3554" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3555" xml:space="preserve">SIt menſalis coni-ſectionis, vel circuli ABCD, cuius baſis, AD maior, <lb/>BC minor, diameter E F. </s> <s xml:id="echoid-s3556" xml:space="preserve">Dico ſi iungantur AB, DC, ipſas cum dia-<lb/>metro, & </s> <s xml:id="echoid-s3557" xml:space="preserve">in eodem puncto conuenire, ac ducta diagonali AC ſecant dia-<lb/>metrum in G, & </s> <s xml:id="echoid-s3558" xml:space="preserve">applicata LGM, ſi per extrema puncta L, M, ad prędictum <lb/>occurſum ducantur rectæ, ipſas ſectionem contingere.</s> <s xml:id="echoid-s3559" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3560" xml:space="preserve">Cum ſit enim AF maior BE, & </s> <s xml:id="echoid-s3561" xml:space="preserve">ipſi parallela, occurret AB cum FE ad par-<lb/> <anchor type="note" xlink:label="note-0129-06a" xlink:href="note-0129-06"/> tes B, E, vt in H; </s> <s xml:id="echoid-s3562" xml:space="preserve">itemq; </s> <s xml:id="echoid-s3563" xml:space="preserve">DC cum eadem FE, vt in I, vtraque verò <anchor type="note" xlink:href="" symbol="a"/> extra ſe- <pb o="106" file="0130" n="130" rhead=""/> ctionem, & </s> <s xml:id="echoid-s3564" xml:space="preserve">cum ſit FH ad HE, vt FA ad EB, vel vt FD ad EC, vel vt FI ad <lb/>IE, erit diuidendo FE ad EH, vt FE ad EI, quare EH, & </s> <s xml:id="echoid-s3565" xml:space="preserve">EI ſunt æquales <lb/>hoc eſt productę AB, DC in eodem pun-<lb/> <anchor type="figure" xlink:label="fig-0130-01a" xlink:href="fig-0130-01"/> cto H cum diametro conueniunt, & </s> <s xml:id="echoid-s3566" xml:space="preserve">ſi ſe-<lb/>ctio fuerit Hyperbola <anchor type="note" xlink:href="" symbol="a"/> infra angulum <anchor type="note" xlink:label="note-0130-01a" xlink:href="note-0130-01"/> ab aſymptotis factum; </s> <s xml:id="echoid-s3567" xml:space="preserve">ideoque ex H duci <lb/>poterunt Hyperbolen contingentes.</s> <s xml:id="echoid-s3568" xml:space="preserve"/> </p> <div xml:id="echoid-div339" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0129-06" xlink:href="note-0129-06a" xml:space="preserve">22. pri-<lb/>mi conic.</note> <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/QN4GHYBF/figures/0130-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0130-01" xlink:href="note-0130-01a" xml:space="preserve">25. ſec. <lb/>conic.</note> </div> <p> <s xml:id="echoid-s3569" xml:space="preserve">Iam, ſi ductæ HL, HM ſectionem non <lb/>contingunt, ducatur ex H contingens HO <lb/>ad aliud punctũ quàm L, vt ad O, & </s> <s xml:id="echoid-s3570" xml:space="preserve">per O <lb/>applicetur OPN; </s> <s xml:id="echoid-s3571" xml:space="preserve">erit <anchor type="note" xlink:href="" symbol="b"/> ergo AP ad PB, vt <anchor type="note" xlink:label="note-0130-02a" xlink:href="note-0130-02"/> AH ad HB, ſed AH ad HB, eſt vt AF ad <lb/>BE, vel ad EC, vel vt FG ad GE (ob ſimi-<lb/>litudinem triangulorum AFG, CEG) vel <lb/>vt AR ad RB, ergo AP ad PB erit vt AR <lb/>ad RB: </s> <s xml:id="echoid-s3572" xml:space="preserve">quod eſt falſum. </s> <s xml:id="echoid-s3573" xml:space="preserve">Non ergo contingens ex H ad aliud punctum per-<lb/>uenit quàm L, & </s> <s xml:id="echoid-s3574" xml:space="preserve">ſic non ad aliud quàm M. </s> <s xml:id="echoid-s3575" xml:space="preserve">Quare iunctæ HL, HM ſectio-<lb/>nem contingunt. </s> <s xml:id="echoid-s3576" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s3577" xml:space="preserve">c.</s> <s xml:id="echoid-s3578" xml:space="preserve"/> </p> <div xml:id="echoid-div340" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0130-02" xlink:href="note-0130-02a" xml:space="preserve">37. tertij <lb/>conic.</note> </div> </div> <div xml:id="echoid-div342" type="section" level="1" n="147"> <head xml:id="echoid-head152" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3579" xml:space="preserve">HInc eſt, quod ſi circa diametrum rectilineæ, vel conicæ menſalis tan-<lb/>quam circa tranſuerſum latus, & </s> <s xml:id="echoid-s3580" xml:space="preserve">per extrema applicatæ, quæ per pũ-<lb/>ctum inter ſectionis diagonalis eiuſdem menſalis cum diametro, ordinatim <lb/>ducitur, Ellipſis deſcribatur, ipſa, menſalis latera in eiuſdem applicatæ ex-<lb/>tremis omnino continget, nempe ei erit inſcripta.</s> <s xml:id="echoid-s3581" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3582" xml:space="preserve">Nam pro rectilinea menſali ABCD, & </s> <s xml:id="echoid-s3583" xml:space="preserve">pro ALBCMD coni-ſectionis, vel <lb/>circuli cuius baſis AD, maior ſit baſi BC, oſtendimus AH ad HB eſſe vt AR <lb/>ad RB, ergo & </s> <s xml:id="echoid-s3584" xml:space="preserve">FH ad HE erit vt FG ad GE, vnde Ellipſis, quæ deſcribitur <lb/>cum tranſuerſo EF, & </s> <s xml:id="echoid-s3585" xml:space="preserve">applicata RQ, vel LM à rectis HA, HD in <anchor type="note" xlink:href="" symbol="c"/> punctis <anchor type="note" xlink:label="note-0130-03a" xlink:href="note-0130-03"/> R, Q, vel à rectis HL, HM in punctis L, M contingetur; </s> <s xml:id="echoid-s3586" xml:space="preserve">ſed ipſæ HL, HM, <lb/>vti nuper oſtendimus in ijſdem punctis ſectionem quoque contingunt: </s> <s xml:id="echoid-s3587" xml:space="preserve">qua-<lb/>re huiuſmodi Ellipſis, & </s> <s xml:id="echoid-s3588" xml:space="preserve">menſalem rectilineam, & </s> <s xml:id="echoid-s3589" xml:space="preserve">conicam ALBCMD <anchor type="note" xlink:href="" symbol="d"/> in <anchor type="note" xlink:label="note-0130-04a" xlink:href="note-0130-04"/> ijſdem applicatæ extremis contiget, ac ipſi menſali, erit inſcripta, cum etiam <lb/>AD, BC ex diametri terminis F, E ordinatim ductis æquidiſtantes eandem <lb/>Ellipſim contingant.</s> <s xml:id="echoid-s3590" xml:space="preserve"/> </p> <div xml:id="echoid-div342" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-0130-03" xlink:href="note-0130-03a" xml:space="preserve">4 huius.</note> <note symbol="d" position="left" xlink:label="note-0130-04" xlink:href="note-0130-04a" xml:space="preserve">61. h.</note> </div> <p> <s xml:id="echoid-s3591" xml:space="preserve">At pro menſali coni-ſectionis ALBCMD, ſi ipſa fuerit menſalis Elliptica, <lb/>vel circularis, cuius oppoſita latera AD, BC ſint æqualia, erunt quoque eo-<lb/>rum dimidia AF, EC æqualia, ac ideo etiam FG æqualis GE, hoc eſt G cen-<lb/>trũ erit Ellipſis, quæ per ELFM deſcribitur cum tranſuerſo EF; </s> <s xml:id="echoid-s3592" xml:space="preserve">& </s> <s xml:id="echoid-s3593" xml:space="preserve">applicata <lb/>LM erit eius diameter coniugata. </s> <s xml:id="echoid-s3594" xml:space="preserve">Vnde quæ per L, & </s> <s xml:id="echoid-s3595" xml:space="preserve">M communi applicatæ <lb/>EF vtriuſque ſectionis æquidiſtantes ducentur <anchor type="note" xlink:href="" symbol="e"/> vtranque ſectionem contin- <anchor type="note" xlink:label="note-0130-05a" xlink:href="note-0130-05"/> gent, quàm contingunt quoque applicatæ AD, DC: </s> <s xml:id="echoid-s3596" xml:space="preserve">quapropter Ellipſis, <lb/>quæ per E, L, F, Q deſcribitur eidem menſali Ellipticæ, vel circulari <anchor type="note" xlink:href="" symbol="f"/> erit <anchor type="note" xlink:label="note-0130-06a" xlink:href="note-0130-06"/> inſcripta.</s> <s xml:id="echoid-s3597" xml:space="preserve"/> </p> <div xml:id="echoid-div343" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-0130-05" xlink:href="note-0130-05a" xml:space="preserve">32. pri-<lb/>mi conic.</note> <note symbol="f" position="left" xlink:label="note-0130-06" xlink:href="note-0130-06a" xml:space="preserve">61. h.</note> </div> <pb o="107" file="0131" n="131" rhead=""/> </div> <div xml:id="echoid-div345" type="section" level="1" n="148"> <head xml:id="echoid-head153" xml:space="preserve">THEOR. XXXIV. PROP. LXIII.</head> <p> <s xml:id="echoid-s3598" xml:space="preserve">In quacunque coni-ſectione, etiam in triangulo, MAXIMA <lb/>diametro æquidiſtantium inter ſectionem, & </s> <s xml:id="echoid-s3599" xml:space="preserve">quamcunque ordina-<lb/>tim applicatam interceptarum, eſt ipſa diameter; </s> <s xml:id="echoid-s3600" xml:space="preserve">aliarum verò <lb/>ea, quæ propinquior eſt diametro, maior eſt remotiori.</s> <s xml:id="echoid-s3601" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3602" xml:space="preserve">ESto triangulum, vt in prima figura, vel circuli, aut Ellipſis, vel Parabo-<lb/>læ, vel tandem Hyperbolæ portio ABC, vt in ſecunda, quarum dia-<lb/>meter ſit BD, & </s> <s xml:id="echoid-s3603" xml:space="preserve">ordinatim applicata ſit AC, ductiſque quotcunque EF, <lb/>GH, &</s> <s xml:id="echoid-s3604" xml:space="preserve">c. </s> <s xml:id="echoid-s3605" xml:space="preserve">parallelis ad BD. </s> <s xml:id="echoid-s3606" xml:space="preserve">Dico BD eſſe _MAXIMAM_, diametro reliqua-<lb/>rum verò, propinquiorem EF, maiorem eſſe remotiori.</s> <s xml:id="echoid-s3607" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3608" xml:space="preserve">Nam ſi concipiatur ex B duci <lb/> <anchor type="figure" xlink:label="fig-0131-01a" xlink:href="fig-0131-01"/> quædam linea ordinatim appli-<lb/>catæ AC æquidiſtans <anchor type="note" xlink:href="" symbol="a"/> quæ tota <anchor type="note" xlink:label="note-0131-01a" xlink:href="note-0131-01"/> cadet extra ſectionem, iungique <lb/>recta linea puncta E, B <anchor type="note" xlink:href="" symbol="b"/> quæ tota <anchor type="note" xlink:label="note-0131-02a" xlink:href="note-0131-02"/> cadet intra, patet ipſam EB ad al-<lb/>teram partem productam (cum <lb/>ſecet in B eam, quæ ducta ſit ex <lb/>B parallela ad AC) conuenire <lb/>quoque cum CA ad partes A, & </s> <s xml:id="echoid-s3609" xml:space="preserve">ſic BD maiorem eſſe recta EF, ſiue omnium <lb/>_MAXIMAM_. </s> <s xml:id="echoid-s3610" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s3611" xml:space="preserve">c.</s> <s xml:id="echoid-s3612" xml:space="preserve"/> </p> <div xml:id="echoid-div345" type="float" level="2" n="1"> <figure xlink:label="fig-0131-01" xlink:href="fig-0131-01a"> <image file="0131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0131-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0131-01" xlink:href="note-0131-01a" xml:space="preserve">17. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0131-02" xlink:href="note-0131-02a" xml:space="preserve">10. pri. <lb/>conic. & <lb/>32. eiuſd.</note> </div> <p> <s xml:id="echoid-s3613" xml:space="preserve">Item ſi puncta G, E, iungantur recta linea <anchor type="note" xlink:href="" symbol="c"/> ipſa omnino cum diametro <anchor type="note" xlink:label="note-0131-03a" xlink:href="note-0131-03"/> extra ſectionem conueniet, ac propterea ſecabit priùs eam, quæ ex B ducta <lb/>ſit ipſi A C ęquidiſtans; </s> <s xml:id="echoid-s3614" xml:space="preserve">cum ergo GE ſecet vnam parallelarum, ſecabit quo-<lb/>que, ſi producatur, alteram CA ad partes A, & </s> <s xml:id="echoid-s3615" xml:space="preserve">ſic EF erit maior ipſa GH. <lb/></s> <s xml:id="echoid-s3616" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s3617" xml:space="preserve">c.</s> <s xml:id="echoid-s3618" xml:space="preserve"/> </p> <div xml:id="echoid-div346" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0131-03" xlink:href="note-0131-03a" xml:space="preserve">22. pri. <lb/>conic. & <lb/>23. eiuſd.</note> </div> </div> <div xml:id="echoid-div348" type="section" level="1" n="149"> <head xml:id="echoid-head154" xml:space="preserve">THEOR. XXXV. PROP. LXIV.</head> <p> <s xml:id="echoid-s3619" xml:space="preserve">Ellipſium æqualium diametrorum, eidem angulo, vel Parabo-<lb/>læ, vel Hyperbolæ, aut portioni Ellipticæ, vel circulari, quæ non <lb/>ſit maior Ellipſis, vel circuli dimidio, inſcriptarum, ſe mutuò, ac <lb/>ſectionem contingentium, quæ propior eſt vertici, minor eſt re-<lb/>motiori.</s> <s xml:id="echoid-s3620" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3621" xml:space="preserve">ESto ABC, vel angulus rectilineus, vel Parabole, vel Hyperbole, aut por-<lb/>tio non maior dimidio ſemi-Ellipſis, vel ſemi-circuli, cuius vertex B, <lb/>diameter BD, & </s> <s xml:id="echoid-s3622" xml:space="preserve">circa æqualia ipſius ſegmenta DE, EF adſcriptæ ſint dato <lb/>angulo, vel ſectioni Ellipſes DVE, ETF, ope diagonalium AG, IL, & </s> <s xml:id="echoid-s3623" xml:space="preserve">ap-<lb/>plicatarum KHV, NMT, vt in præcedenti Scholio monuimus, quæ anguli <lb/>latera, vel ſectionem contingent in K, V, N, T, eique erunt inſcriptæ, & </s> <s xml:id="echoid-s3624" xml:space="preserve">ſe <lb/>mutuò contingent in E (cum applicata LEG vtranque ſectionem contingat.)</s> <s xml:id="echoid-s3625" xml:space="preserve"> <pb o="108" file="0132" n="132" rhead=""/> Dico Ellipſim ETF vertici B propiorem, minorem eſſe Ellipſi DVE ab ipſo <lb/>vertice remotiori.</s> <s xml:id="echoid-s3626" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3627" xml:space="preserve">Applicata enim ADC; </s> <s xml:id="echoid-s3628" xml:space="preserve">eſt DH <lb/> <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> <anchor type="note" xlink:label="note-0132-01a" xlink:href="note-0132-01"/> ad HE, vt AD, ad EG, ſed eſt <anchor type="note" xlink:href="" symbol="a"/> AD maior EG, quare & </s> <s xml:id="echoid-s3629" xml:space="preserve">DH erit <lb/>maior HE, eademq; </s> <s xml:id="echoid-s3630" xml:space="preserve">ratione EM <lb/>maior MF, vnde harum Ellipſiũ <lb/>centra cadent infra H, & </s> <s xml:id="echoid-s3631" xml:space="preserve">M, vt <lb/>in O, & </s> <s xml:id="echoid-s3632" xml:space="preserve">O, ex quibus applicatis <lb/>OP, QR Ellipſium ſemi-diame-<lb/>tris coniugatis, productaque QR <lb/>vſque ad ſectionem in S, cum in <lb/>Ellipſi DVE ſit OP <anchor type="note" xlink:href="" symbol="b"/> maior HV, <anchor type="note" xlink:label="note-0132-02a" xlink:href="note-0132-02"/> & </s> <s xml:id="echoid-s3633" xml:space="preserve">in angulo, vel ſectione ABC <lb/>ſit HV <anchor type="note" xlink:href="" symbol="c"/> maior QS, & </s> <s xml:id="echoid-s3634" xml:space="preserve">QS maior <anchor type="note" xlink:label="note-0132-03a" xlink:href="note-0132-03"/> QR, eò magis OP erit maior QR, & </s> <s xml:id="echoid-s3635" xml:space="preserve">duplum duplo maios, hoc eſt Ellipſis <lb/>DVE coniugata diameter, maior coniugata diametro Ellipſis ETF, ſed trãſ-<lb/>uerſa latera ED, EF ſunt æqualia, vnde & </s> <s xml:id="echoid-s3636" xml:space="preserve">latus rectum Ellipſis DVE maios <lb/>recto ETF, ſuntque huiuſmodi Ellipſes æqualiter inclinatæ cum eidem ſe-<lb/>ctioni ſint ſimul adſcriptæ: </s> <s xml:id="echoid-s3637" xml:space="preserve">quare Ellipſis DVE, maius habens rectum latus, <lb/>maior erit <anchor type="note" xlink:href="" symbol="d"/> ETF minoris recti lateris, quę dati anguli, vel ſectionis vertici <anchor type="note" xlink:label="note-0132-04a" xlink:href="note-0132-04"/> propior eſt. </s> <s xml:id="echoid-s3638" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s3639" xml:space="preserve"/> </p> <div xml:id="echoid-div348" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0132-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve">32. vel <lb/>63. huius.</note> <note symbol="b" position="left" xlink:label="note-0132-02" xlink:href="note-0132-02a" xml:space="preserve">63. h.</note> <note symbol="c" position="left" xlink:label="note-0132-03" xlink:href="note-0132-03a" xml:space="preserve">32. h.</note> <note symbol="d" position="left" xlink:label="note-0132-04" xlink:href="note-0132-04a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> </div> </div> <div xml:id="echoid-div350" type="section" level="1" n="150"> <head xml:id="echoid-head155" xml:space="preserve">PROBL. XXIV. PROP. LXV.</head> <p> <s xml:id="echoid-s3640" xml:space="preserve">Per datum punctum in axe dati anguli rectilinei MAXIMVM <lb/>circulum inſcribere & </s> <s xml:id="echoid-s3641" xml:space="preserve">è contra.</s> <s xml:id="echoid-s3642" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3643" xml:space="preserve">SIt datus angulus rectilineus ABC, cuius axis, ſiue linea ipſum bifariam <lb/>ſecans ſit BD, in quo datum ſit punctum E, per quod oporteat _MAXI_-<lb/>_MVM_ circulum inſcribere.</s> <s xml:id="echoid-s3644" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3645" xml:space="preserve">Ducatur ex E ſuper axim BD perpendicularis <lb/> <anchor type="figure" xlink:label="fig-0132-02a" xlink:href="fig-0132-02"/> EF, cui infra F ſumatur FA æqualis, & </s> <s xml:id="echoid-s3646" xml:space="preserve">ex A eri-<lb/>gatur AD perpendicularis ad BA, quæ axi oc-<lb/>curret in D (cum angulus ABD ſit omnino acu-<lb/>tus, & </s> <s xml:id="echoid-s3647" xml:space="preserve">BAD rectus, hoc eſt ſimul ſumpti minores <lb/>duobus rectis). </s> <s xml:id="echoid-s3648" xml:space="preserve">Dico punctum D eſſe centrum <lb/>quæſiti circuli. </s> <s xml:id="echoid-s3649" xml:space="preserve">Nam iuncta AE; </s> <s xml:id="echoid-s3650" xml:space="preserve">cum ſint FA, <lb/>FE inter ſe æquales, erunt anguli ad baſim AE æ-<lb/>quales, ſed toti FED, FAD æquales ſunt, cum <lb/>ſint recti, vnde reliqui DEA, DAE æquales erũt, <lb/>ſiue latus DE ipſi DA æqualle. </s> <s xml:id="echoid-s3651" xml:space="preserve">Ductaque DC <lb/>perpendiculari ad BC; </s> <s xml:id="echoid-s3652" xml:space="preserve">in triangulis DBA, DBC <lb/>ſunt anguli ad B, & </s> <s xml:id="echoid-s3653" xml:space="preserve">ad A, & </s> <s xml:id="echoid-s3654" xml:space="preserve">C æquales inter ſe, <lb/>& </s> <s xml:id="echoid-s3655" xml:space="preserve">latus BD commune, ergo, & </s> <s xml:id="echoid-s3656" xml:space="preserve">DC ipſi DA, ſiue DE, ęqualis erit: </s> <s xml:id="echoid-s3657" xml:space="preserve">quapro-<lb/>pter ſi cum centro D, interuallo DA circulus deſcribatur, ipſæ per puncta E, <lb/>& </s> <s xml:id="echoid-s3658" xml:space="preserve">C tranſibit, eritque angulo ABC inſcrintus. </s> <s xml:id="echoid-s3659" xml:space="preserve">cum obrectos angulos ad A, <pb o="109" file="0133" n="133" rhead=""/> Cipſius latera BA, BC eum contingant; </s> <s xml:id="echoid-s3660" xml:space="preserve">& </s> <s xml:id="echoid-s3661" xml:space="preserve">erit _MAXIMVS_: </s> <s xml:id="echoid-s3662" xml:space="preserve">Nam licet fa-<lb/>cta ſupra EF eadem penitùs, conſtructione nempe ſumpta FG æquali ad FE, <lb/>& </s> <s xml:id="echoid-s3663" xml:space="preserve">ducta GH perpendiculari ad GA, oſtendetur pariter H eſſe centrum alte-<lb/>rius circuli dato angulo inſcripti, ſed is erit minor circulo ex DE, cum ob <lb/>parallelas GH, AD, ſit AB ad BG, vt DA ad HG, ſed eſt AB maior BG, vn-<lb/>de radius DA erit maior radio HG, ſiue circulus AEC maior circulo ex HE. <lb/></s> <s xml:id="echoid-s3664" xml:space="preserve">Iam quilibet alius circulus per E, dato angulo adſcriptus, cuius diameter <lb/>minor ſit EI, minor eſt ipſo AEC, & </s> <s xml:id="echoid-s3665" xml:space="preserve">quilibet alius, cuius diameter ſit maior <lb/>ipſa EI, eſt quidem maior AEC, ſed omnino ſecat dati anguli latera, cum <lb/>hæc circulum contingant: </s> <s xml:id="echoid-s3666" xml:space="preserve">ex quo circulus ex DE erit _MAXIMVS_ inſcriptus <lb/>quæſitus. </s> <s xml:id="echoid-s3667" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s3668" xml:space="preserve">c.</s> <s xml:id="echoid-s3669" xml:space="preserve"/> </p> <div xml:id="echoid-div350" type="float" level="2" n="1"> <figure xlink:label="fig-0132-02" xlink:href="fig-0132-02a"> <image file="0132-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0132-02"/> </figure> </div> <p> <s xml:id="echoid-s3670" xml:space="preserve">SIverò ad datum punctum B extra circulum AEC, cuius ſit centrum D, ſit <lb/>ei circumſcribendus _MINIMVS_ angulus rectilineus; </s> <s xml:id="echoid-s3671" xml:space="preserve">iam per ſe patet <lb/>angulum ABC, à ductis contingentibus ex B, eſſe _MINIMVM_ quæſitum. <lb/></s> <s xml:id="echoid-s3672" xml:space="preserve">Quod vltimò faciendum, ac demonſtrandum erat.</s> <s xml:id="echoid-s3673" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div352" type="section" level="1" n="151"> <head xml:id="echoid-head156" xml:space="preserve">LEMMA VII. PROP. LXVI.</head> <p> <s xml:id="echoid-s3674" xml:space="preserve">In dato angulo, à recta linea per verticem vtcunque ſecto, lineas <lb/>applicare, quæ à prædicta diuidantur in data ratione.</s> <s xml:id="echoid-s3675" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3676" xml:space="preserve">SIt datus angulus ABC, vtcunque ſectus à re-<lb/> <anchor type="figure" xlink:label="fig-0133-01a" xlink:href="fig-0133-01"/> cta BD, punctum in eo ſit D, ex quo opor-<lb/>tet rectam, qualis eſt ADC, applicare ita vt ip-<lb/>ſius partes AD, DC ſint in data ratione, veluti <lb/>E ad F.</s> <s xml:id="echoid-s3677" xml:space="preserve"/> </p> <div xml:id="echoid-div352" type="float" level="2" n="1"> <figure xlink:label="fig-0133-01" xlink:href="fig-0133-01a"> <image file="0133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0133-01"/> </figure> </div> <p> <s xml:id="echoid-s3678" xml:space="preserve">Ducatur DG parallela ad alteram linearum, <lb/>angulum continentium, vt ad AB; </s> <s xml:id="echoid-s3679" xml:space="preserve">& </s> <s xml:id="echoid-s3680" xml:space="preserve">fiat vt E ad <lb/>F, ita BG ad GC, iungaturque CD, quæ cum <lb/>BA conueniat in A. </s> <s xml:id="echoid-s3681" xml:space="preserve">Dico factum eſſe, quod <lb/>proponebatur.</s> <s xml:id="echoid-s3682" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3683" xml:space="preserve">Et enim, ob parallelas, vt AD ad DC, ita BG <lb/>ad GC, vel E ad F. </s> <s xml:id="echoid-s3684" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s3685" xml:space="preserve">c.</s> <s xml:id="echoid-s3686" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div354" type="section" level="1" n="152"> <head xml:id="echoid-head157" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3687" xml:space="preserve">SI data ratio E ad F, fuerit ratio æqualitatis, tunc BD, licet præter morem, <lb/>vocetur dati anguli diameter, & </s> <s xml:id="echoid-s3688" xml:space="preserve">ſi bifariam, & </s> <s xml:id="echoid-s3689" xml:space="preserve">ad rectos angulos ipſas <lb/>applicatas ſecuerit, dicatur axis.</s> <s xml:id="echoid-s3690" xml:space="preserve"/> </p> <pb o="110" file="0134" n="134" rhead=""/> </div> <div xml:id="echoid-div355" type="section" level="1" n="153"> <head xml:id="echoid-head158" xml:space="preserve">PROBL. XXV. PROP. LXVII.</head> <p> <s xml:id="echoid-s3691" xml:space="preserve">Dato angulo rectilineo, per punctum intra ipſum datum MA-<lb/>XIMAM Parabolen inſcribere: </s> <s xml:id="echoid-s3692" xml:space="preserve">& </s> <s xml:id="echoid-s3693" xml:space="preserve">è contra.</s> <s xml:id="echoid-s3694" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3695" xml:space="preserve">SIt datus angulus rectilineus ABC, & </s> <s xml:id="echoid-s3696" xml:space="preserve">datum, intra ipſum, punctum ſit D. <lb/></s> <s xml:id="echoid-s3697" xml:space="preserve">Oportet per D _MAXIMAM_ Parabolen inſcribere.</s> <s xml:id="echoid-s3698" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3699" xml:space="preserve">Sumatur DE æqualis DB, & </s> <s xml:id="echoid-s3700" xml:space="preserve">per E in angulo ABC <anchor type="note" xlink:href="" symbol="a"/> applicetur, recta <anchor type="note" xlink:label="note-0134-01a" xlink:href="note-0134-01"/> AEC, quæ à diametro AE ſit bifariam ſecta in E, & </s> <s xml:id="echoid-s3701" xml:space="preserve">per verticem D circa <lb/> <anchor type="note" xlink:label="note-0134-02a" xlink:href="note-0134-02"/> diametrum ED, & </s> <s xml:id="echoid-s3702" xml:space="preserve">applicatam AC magnitudine, & </s> <s xml:id="echoid-s3703" xml:space="preserve">poſitione datam <anchor type="note" xlink:href="" symbol="b"/> de- ſcribatur Parabole ADC. </s> <s xml:id="echoid-s3704" xml:space="preserve">Dico ipſam eſſe quæſitam.</s> <s xml:id="echoid-s3705" xml:space="preserve"/> </p> <div xml:id="echoid-div355" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0134-01" xlink:href="note-0134-01a" xml:space="preserve">66. h.</note> <note symbol="b" position="left" xlink:label="note-0134-02" xlink:href="note-0134-02a" xml:space="preserve">57. h.</note> </div> <p> <s xml:id="echoid-s3706" xml:space="preserve">Quoniam cum ſint DE, DB æquales, rectæ <lb/> <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> <anchor type="note" xlink:label="note-0134-03a" xlink:href="note-0134-03"/> AB, CB ſectionem <anchor type="note" xlink:href="" symbol="c"/> contingent, vnde Parabo- le erit dato angulo inſcripta; </s> <s xml:id="echoid-s3707" xml:space="preserve">eritque _MAXIMA_ <lb/>quoniam quælibet Parabole per D ipſi ADC <lb/>adſcripta cum recto, quod eius recto ſit minus <lb/>ipſa ADC <anchor type="note" xlink:href="" symbol="d"/> minor eſt, quęlibet verò adſcripta <anchor type="note" xlink:label="note-0134-04a" xlink:href="note-0134-04"/> cum recto, quod prædictum excedat licet ea-<lb/>dem ſit <anchor type="note" xlink:href="" symbol="e"/> maior, ſecat tamen latera dati anguli.</s> <s xml:id="echoid-s3708" xml:space="preserve"> <anchor type="note" xlink:label="note-0134-05a" xlink:href="note-0134-05"/> Quare Parabole ADC eſt _MAXIMA_. </s> <s xml:id="echoid-s3709" xml:space="preserve">Quod <lb/>primò, &</s> <s xml:id="echoid-s3710" xml:space="preserve">c.</s> <s xml:id="echoid-s3711" xml:space="preserve"/> </p> <div xml:id="echoid-div356" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0134-01"/> </figure> <note symbol="c" position="left" xlink:label="note-0134-03" xlink:href="note-0134-03a" xml:space="preserve">2. huius.</note> <note symbol="d" position="left" xlink:label="note-0134-04" xlink:href="note-0134-04a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="e" position="left" xlink:label="note-0134-05" xlink:href="note-0134-05a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s3712" xml:space="preserve">SI verò data ſit Parabole ADC, & </s> <s xml:id="echoid-s3713" xml:space="preserve">extra ip-<lb/>ſam datum ſit pũctum B, per quod ei opor-<lb/>teat _MINIMVM_ angulum rectilineum circum-<lb/>ſcribere. </s> <s xml:id="echoid-s3714" xml:space="preserve">Ducta BE parabolæ diametro, & </s> <s xml:id="echoid-s3715" xml:space="preserve">ſumpta DE æquali DB applica-<lb/>taque AEC, iunctiſque BA, BC, Erit angulus ABC _MINIMVS_ quæſitus, <lb/> <anchor type="note" xlink:label="note-0134-06a" xlink:href="note-0134-06"/> vt ſatis perſpicuè patet. </s> <s xml:id="echoid-s3716" xml:space="preserve">Nam cum ipſæ BA, BC ſectionem <anchor type="note" xlink:href="" symbol="f"/> contingant om- nes aliæ ex B ductæ minorem angulum dato ABC adſcriptum conſtituentes, <lb/>ſectionem ſecabunt, quare, &</s> <s xml:id="echoid-s3717" xml:space="preserve">c. </s> <s xml:id="echoid-s3718" xml:space="preserve">Quod vltimò, &</s> <s xml:id="echoid-s3719" xml:space="preserve">c.</s> <s xml:id="echoid-s3720" xml:space="preserve"/> </p> <div xml:id="echoid-div357" type="float" level="2" n="3"> <note symbol="f" position="left" xlink:label="note-0134-06" xlink:href="note-0134-06a" xml:space="preserve">2. huius.</note> </div> </div> <div xml:id="echoid-div359" type="section" level="1" n="154"> <head xml:id="echoid-head159" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s3721" xml:space="preserve">MOnendus hìc Lector eſt, quod dum in hoc, & </s> <s xml:id="echoid-s3722" xml:space="preserve">in ſequentibus <lb/>problematibus; </s> <s xml:id="echoid-s3723" xml:space="preserve">dato angulo, per datum punctum adſcribitur, <lb/>velinſcribi, aut circumſcribi proponitur, quæſita coni-ſectio, <lb/>vel circulus; </s> <s xml:id="echoid-s3724" xml:space="preserve">& </s> <s xml:id="echoid-s3725" xml:space="preserve">è contra, dum datæ com-ſectioni, vel circulo <lb/>per datum punctum adſcribitur, velinſcribitur, aut circumſcribitur quæſi-<lb/>tus angulus; </s> <s xml:id="echoid-s3726" xml:space="preserve">id ſemper à nobis accipi intelligitur in eodem ſenſu quintæ ſe-<lb/>cundarum definitionum huius, qua in præcedentibus hactenus vſi ſumus; <lb/></s> <s xml:id="echoid-s3727" xml:space="preserve">nempe lineam, quæ per datum punctum educta diameter eſt datæ, vel quæſitæ <lb/>ſectionis, eſſe quoque diametrum dati, vel quæſiti anguli, ſiue eius verti-<lb/>ci occurrere; </s> <s xml:id="echoid-s3728" xml:space="preserve">ita vt quæ in angulo ducuntur æquidiſtantes ordinatim appli-<lb/>catis com-ſectionis, velcirculi, ſint quoque ab eadem ſectionis diametro per <pb o="111" file="0135" n="135" rhead=""/> datum punctum tranſeunte bifariam ſectæ, quod à lineis ad anguli verticem <lb/>non collimantibus conſequi minimè poſſet. </s> <s xml:id="echoid-s3729" xml:space="preserve">Si verò inſcriptio, ac circumſcri-<lb/>ptio alijs conditionibus confici iubeatur, aliæ item defintiones, & </s> <s xml:id="echoid-s3730" xml:space="preserve">conſtru-<lb/>ctiones diuerſæ ad problematum ſolutiones requirerentur, quas omnes, licet <lb/>nobis fortuitò datum ſit Geometriæ legibus ſubijcere, temporis tamen angu-<lb/>ſtijs obſequentes, hic <gap/> omittere neceſſe fuit; </s> <s xml:id="echoid-s3731" xml:space="preserve">ſed aliàs forſan, Deo dante, ſi <lb/>quid vnquam ocij nacti fuerimus, hanc ipſam de MAXIMIS, & </s> <s xml:id="echoid-s3732" xml:space="preserve">MI-<lb/>NIMIS doctrinam, & </s> <s xml:id="echoid-s3733" xml:space="preserve">duplò, & </s> <s xml:id="echoid-s3734" xml:space="preserve">triplò auctiorem denuò proferemus: </s> <s xml:id="echoid-s3735" xml:space="preserve">inte-<lb/>rim varijs ſtimulis, qui ad hæc edenda nos vrgent, obtemperantes, præſens <lb/>argumentum abſoluere properemus, vt citius (alteram huius tractationis <lb/>partem aggrediendo) ad noua pariter, & </s> <s xml:id="echoid-s3736" xml:space="preserve">apprimè iucunda in conicis acciden-<lb/>tia deueniamus, & </s> <s xml:id="echoid-s3737" xml:space="preserve">quod pluris eſt, præcipuè vtilitatis fundamenta iacien-<lb/>do, abſtruſionis doctrinæ myſteria perſpicacioribus ingenijs aperiamus.</s> <s xml:id="echoid-s3738" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div360" type="section" level="1" n="155"> <head xml:id="echoid-head160" xml:space="preserve">PROBL. XXVI. PROP. LXVIII.</head> <p> <s xml:id="echoid-s3739" xml:space="preserve">Dato angulo rectilineo, per punctum intra ipſum datum, cum <lb/>dato ſemi-tranſuerſo latere, MAXIMAM Hyperbolen inſcribere. <lb/></s> <s xml:id="echoid-s3740" xml:space="preserve">Item.</s> <s xml:id="echoid-s3741" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3742" xml:space="preserve">Datę Hyperbolæ, per punctum extra ipſam datum, MINIMVM <lb/>angulum rectilineum circumſcribere.</s> <s xml:id="echoid-s3743" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3744" xml:space="preserve">Oportet autem, ad hoc vt anguli circumſcriptio fiat iuxta alla-<lb/>tam definitionem, ac præcedens monitum, datum punctum, vel <lb/>eſſe in centro, vel intra angulos, ab aſymptotis conſtitutos.</s> <s xml:id="echoid-s3745" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3746" xml:space="preserve">SIt, in tribus primis figuris, datus angulus rectilineus ABC, & </s> <s xml:id="echoid-s3747" xml:space="preserve">datum in-<lb/>tra ipſum punctum ſit D: </s> <s xml:id="echoid-s3748" xml:space="preserve">oportet per D _MAXIMAM_ Hyperbolen inſcri-<lb/>bere, cuius ſemi-tranſuerſum latus æquale ſit dato E.</s> <s xml:id="echoid-s3749" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3750" xml:space="preserve">Iungatur DB, & </s> <s xml:id="echoid-s3751" xml:space="preserve">ſe-<lb/> <anchor type="figure" xlink:label="fig-0135-01a" xlink:href="fig-0135-01"/> cetur ex ipſa, DO ęqua <lb/>lis E. </s> <s xml:id="echoid-s3752" xml:space="preserve">Iam, vel DO æ-<lb/>qualis eſt DB, vt in pri-<lb/>ma figura, vel minor vt <lb/>in ſecunda, vel maior <lb/>vt in tertia. </s> <s xml:id="echoid-s3753" xml:space="preserve">Si primùm, <lb/>deſcribatur <anchor type="note" xlink:href="" symbol="a"/> per D, cũ <anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/> aſymptotis BA, BC <lb/>Hyperbole FDG: </s> <s xml:id="echoid-s3754" xml:space="preserve">& </s> <s xml:id="echoid-s3755" xml:space="preserve"><lb/>ipſa erit _MAXIMA_ <lb/>quæſita.</s> <s xml:id="echoid-s3756" xml:space="preserve"/> </p> <div xml:id="echoid-div360" type="float" level="2" n="1"> <figure xlink:label="fig-0135-01" xlink:href="fig-0135-01a"> <image file="0135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0135-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">4. ſec. <lb/>conic.</note> </div> <p> <s xml:id="echoid-s3757" xml:space="preserve">Nam, quæ cum eo-<lb/>dem tranſuerſo, eidem angulo per D adſcribitur, cum recto, quod minus ſit <pb o="112" file="0136" n="136" rhead=""/> recto FDG, minor eſt <anchor type="note" xlink:href="" symbol="a"/> ipſa FDG, quæ verò cum recto maiori, eſt quidem <anchor type="note" xlink:label="note-0136-01a" xlink:href="note-0136-01"/> maior <anchor type="note" xlink:href="" symbol="b"/> FDG, qualis eſt HDI, ſed omnino ſecat latera dati anguli ABC:</s> <s xml:id="echoid-s3758" xml:space="preserve"> <anchor type="note" xlink:label="note-0136-02a" xlink:href="note-0136-02"/> quoniam ducta BL aſymptoto ſectionis HDI, ipſa cadet <anchor type="note" xlink:href="" symbol="c"/> extra BA, ſed BH <anchor type="note" xlink:label="note-0136-03a" xlink:href="note-0136-03"/> eſt aſymptotos inſcriptæ FDG, quare ipſa BH producta ſecabit Hyperbolen <lb/>circumſcriptam DH, eadem ratione BC ſecabit DI: </s> <s xml:id="echoid-s3759" xml:space="preserve">quapropter Hyperbole <lb/>FDG eſt dato angulo _MAXIMA_ inſcripta quæſita. </s> <s xml:id="echoid-s3760" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s3761" xml:space="preserve">c.</s> <s xml:id="echoid-s3762" xml:space="preserve"/> </p> <div xml:id="echoid-div361" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0136-01" xlink:href="note-0136-01a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="b" position="left" xlink:label="note-0136-02" xlink:href="note-0136-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="left" xlink:label="note-0136-03" xlink:href="note-0136-03a" xml:space="preserve">37. h.</note> </div> <p> <s xml:id="echoid-s3763" xml:space="preserve">Siverò data magni-<lb/> <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> tudo E, vel ei æqualis <lb/>DO, minor fuerit di-<lb/>ſtantia DB inter datum <lb/>punctum, & </s> <s xml:id="echoid-s3764" xml:space="preserve">dati angu-<lb/>li ABC verticem, vt in <lb/>ſecunda figura; </s> <s xml:id="echoid-s3765" xml:space="preserve">ducan-<lb/>tur ex O, rectæ OP, OH, <lb/>aſymptotis BA, BC æ-<lb/>quidiſtantes, & </s> <s xml:id="echoid-s3766" xml:space="preserve">intra <lb/>aſymptotos OP, OH <lb/> <anchor type="note" xlink:label="note-0136-04a" xlink:href="note-0136-04"/> deſcribatur <anchor type="note" xlink:href="" symbol="d"/> per D Hy perbole FDG: </s> <s xml:id="echoid-s3767" xml:space="preserve">& </s> <s xml:id="echoid-s3768" xml:space="preserve">hæc <lb/>erit _MAXIMA_ inſcripta quæſita.</s> <s xml:id="echoid-s3769" xml:space="preserve"/> </p> <div xml:id="echoid-div362" type="float" level="2" n="3"> <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/QN4GHYBF/figures/0136-01"/> </figure> <note symbol="d" position="left" xlink:label="note-0136-04" xlink:href="note-0136-04a" xml:space="preserve">4. ſec. <lb/>conic.</note> </div> <p> <s xml:id="echoid-s3770" xml:space="preserve">Quoniam, quæ cum eodem tranſuerſo, ſed cum recto minori adſcribitur <lb/>per D, minor eſt <anchor type="note" xlink:href="" symbol="e"/> FDG, quæ verò cum recto maiori, qualis eſt IDL, eſt qui- <anchor type="note" xlink:label="note-0136-05a" xlink:href="note-0136-05"/> dem <anchor type="note" xlink:href="" symbol="f"/> maior, ſed omnino ſecat latera dati anguli BA, BC: </s> <s xml:id="echoid-s3771" xml:space="preserve">quoniam ducta <anchor type="note" xlink:label="note-0136-06a" xlink:href="note-0136-06"/> OM aſymptoto circumſcriptæ IDL, cadet <anchor type="note" xlink:href="" symbol="g"/> extra OP aſymptoton inſcriptæ <anchor type="note" xlink:label="note-0136-07a" xlink:href="note-0136-07"/> FDG, & </s> <s xml:id="echoid-s3772" xml:space="preserve">producta ſecabit BA, cum ſecet in O alteram parallelam OP; </s> <s xml:id="echoid-s3773" xml:space="preserve">qua-<lb/>re BA producta ſecabit <anchor type="note" xlink:href="" symbol="h"/> quidem Hyperbolen DIL: </s> <s xml:id="echoid-s3774" xml:space="preserve">vnde FDG eſt _MAXI-_ <anchor type="note" xlink:label="note-0136-08a" xlink:href="note-0136-08"/> _MA_ quæſita. </s> <s xml:id="echoid-s3775" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s3776" xml:space="preserve">c.</s> <s xml:id="echoid-s3777" xml:space="preserve"/> </p> <div xml:id="echoid-div363" type="float" level="2" n="4"> <note symbol="e" position="left" xlink:label="note-0136-05" xlink:href="note-0136-05a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="f" position="left" xlink:label="note-0136-06" xlink:href="note-0136-06a" xml:space="preserve">ibidem.</note> <note symbol="g" position="left" xlink:label="note-0136-07" xlink:href="note-0136-07a" xml:space="preserve">ex 37. h.</note> <note symbol="h" position="left" xlink:label="note-0136-08" xlink:href="note-0136-08a" xml:space="preserve">35. h.</note> </div> <p> <s xml:id="echoid-s3778" xml:space="preserve">Sitandem DO, quæ ipſi E æqualis eſt, excedat DB. </s> <s xml:id="echoid-s3779" xml:space="preserve">Fiat vt OB ad OD, <lb/>ita OD ad OF, & </s> <s xml:id="echoid-s3780" xml:space="preserve">per F applicetur <anchor type="note" xlink:href="" symbol="i"/> in angulo ABC ordinata AFC, & </s> <s xml:id="echoid-s3781" xml:space="preserve">cũ ſe- <anchor type="note" xlink:label="note-0136-09a" xlink:href="note-0136-09"/> mi-tranſuerſo OD, per puncta A,D,C, deſcribatur <anchor type="note" xlink:href="" symbol="l"/> Hyperbole ADC, cir- <anchor type="note" xlink:label="note-0136-10a" xlink:href="note-0136-10"/> ca diametri ſegmentum DF, & </s> <s xml:id="echoid-s3782" xml:space="preserve">applicatam AC. </s> <s xml:id="echoid-s3783" xml:space="preserve">Dico hanc eſſe _MAXIMAM_ <lb/>quęſitam.</s> <s xml:id="echoid-s3784" xml:space="preserve"/> </p> <div xml:id="echoid-div364" type="float" level="2" n="5"> <note symbol="i" position="left" xlink:label="note-0136-09" xlink:href="note-0136-09a" xml:space="preserve">Schol. <lb/>66. h.</note> <note symbol="l" position="left" xlink:label="note-0136-10" xlink:href="note-0136-10a" xml:space="preserve">57. h.</note> </div> <p> <s xml:id="echoid-s3785" xml:space="preserve">Quoniam, cum ſit FO ad OD, vt DO ad OB, erit rectangulum FOB æqua-<lb/>le quadrato OD, quare BA, BC Hyperbolen <anchor type="note" xlink:href="" symbol="m"/> contingent; </s> <s xml:id="echoid-s3786" xml:space="preserve">ſiue Hyperbo- <anchor type="note" xlink:label="note-0136-11a" xlink:href="note-0136-11"/> le ADC dato angulo ABC erit inſcripta; </s> <s xml:id="echoid-s3787" xml:space="preserve">eritque _MAXIMA_; </s> <s xml:id="echoid-s3788" xml:space="preserve">quoniam, quæ <lb/>cumrecto minori <anchor type="note" xlink:href="" symbol="n"/> cadit intra, quæ verò cum maiori cadit quidem <anchor type="note" xlink:href="" symbol="o"/> extra <anchor type="note" xlink:label="note-0136-12a" xlink:href="note-0136-12"/> ADC, ſed neceſſariò ſecat dati anguli latera BA, BC, cum ſectio Hyper-<lb/>bole in infinitum produci poſſit, & </s> <s xml:id="echoid-s3789" xml:space="preserve">ſpacium ABCDA ſit vndique clauſum: <lb/></s> <s xml:id="echoid-s3790" xml:space="preserve"> <anchor type="note" xlink:label="note-0136-13a" xlink:href="note-0136-13"/> quare ipſa ADC eſt _MAXIMA_ inſcripta quæſita, per datum punctum D. <lb/></s> <s xml:id="echoid-s3791" xml:space="preserve">Quod primò faciendum, ac demonſtrandum erat.</s> <s xml:id="echoid-s3792" xml:space="preserve"/> </p> <div xml:id="echoid-div365" type="float" level="2" n="6"> <note symbol="m" position="left" xlink:label="note-0136-11" xlink:href="note-0136-11a" xml:space="preserve">cõuerſ. <lb/>37. primi <lb/>conic. à <lb/>Comand.</note> <note symbol="n" position="left" xlink:label="note-0136-12" xlink:href="note-0136-12a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="o" position="left" xlink:label="note-0136-13" xlink:href="note-0136-13a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s3793" xml:space="preserve">IAM oporteat (in quarta figura) datæ Hyperbolæ ABC, cuius aſymptoti <lb/>ED, EF, per datum extra ipſam punctum G, _MINIMV M_ angulum circũ-<lb/>ſcribere.</s> <s xml:id="echoid-s3794" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3795" xml:space="preserve">Itaque, vel datum punctum G congruit cum centro E, vel cadit in angu-<lb/>lo aſymptotali, vel in eo, qui huic eſt ad verticem; </s> <s xml:id="echoid-s3796" xml:space="preserve">ſic enim ſemper, quę per <lb/>G, & </s> <s xml:id="echoid-s3797" xml:space="preserve">centrum E ducitur, tum Hyperbolæ, tum anguli eſt communis diame-<lb/>ter, non autem ſi datum punctum alibi cadat. </s> <s xml:id="echoid-s3798" xml:space="preserve">Si primùm; </s> <s xml:id="echoid-s3799" xml:space="preserve">ipſæ angulus aſym- <pb o="113" file="0137" n="137" rhead=""/> ptotalis DEF erit Hyperbolæ circumſcriptus, cum totus cadat extra, & </s> <s xml:id="echoid-s3800" xml:space="preserve">quę-<lb/>libet ſectionis diameter, eaſdem ipſi applicatas, ad latcra anguli productas, <lb/>bifariam <anchor type="note" xlink:href="" symbol="a"/> ſecet: </s> <s xml:id="echoid-s3801" xml:space="preserve">eritque _MINIMV S_, nam<unsure/> quælibet alia linea, quæ per G, <anchor type="note" xlink:label="note-0137-01a" xlink:href="note-0137-01"/> vel per E (quod idem eſt) intra ipſum ducitur, minorem quidem cum altera <lb/> <anchor type="note" xlink:label="note-0137-02a" xlink:href="note-0137-02"/> aſymptoto conſtituit angulum, ſed omnino ſecat <anchor type="note" xlink:href="" symbol="b"/> Hyperbolen. </s> <s xml:id="echoid-s3802" xml:space="preserve">Si ſecun- dum, duci poterunt <anchor type="note" xlink:href="" symbol="c"/> ex G Hyperbolen contingentes GA, GC, & </s> <s xml:id="echoid-s3803" xml:space="preserve">tunc an- <anchor type="note" xlink:label="note-0137-03a" xlink:href="note-0137-03"/> gulus AGC erit quæſitus circumſcriptus: </s> <s xml:id="echoid-s3804" xml:space="preserve">quoniam ſi iungatur AC, & </s> <s xml:id="echoid-s3805" xml:space="preserve">bifa-<lb/>riam ſecetur in N, iuncta GN <anchor type="note" xlink:href="" symbol="d"/> diameter eſt ſectionis, ſimulque anguli; </s> <s xml:id="echoid-s3806" xml:space="preserve">qui <anchor type="note" xlink:label="note-0137-04a" xlink:href="note-0137-04"/> erit _MINIMV S_, vt per ſe patet, cum quæ ex G ducitur intra angulum AGC <lb/>ſecet omnino Hyperbolen. </s> <s xml:id="echoid-s3807" xml:space="preserve">Sitertium: </s> <s xml:id="echoid-s3808" xml:space="preserve">ducantur GL, GM aſymptotis ęqui-<lb/>diſtantes, & </s> <s xml:id="echoid-s3809" xml:space="preserve">angulus LGM erit Hyperbolæ ABC circumſcriptus, cum cir-<lb/>cumſcriptus ſit angulo aſymptotali DEF: </s> <s xml:id="echoid-s3810" xml:space="preserve">nam ducta GEN ſectionis diame-<lb/>tro, applicataque quacunque LDANCFM; </s> <s xml:id="echoid-s3811" xml:space="preserve">in triangulis LGN, MGN eſt <lb/>ND ad DL, vt NE ad EG, vel vt NF ad FM, ſuntq; </s> <s xml:id="echoid-s3812" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> ND, NF inter ſe ęqua- <anchor type="note" xlink:label="note-0137-05a" xlink:href="note-0137-05"/> les, quare DL, FM ęquales erunt, & </s> <s xml:id="echoid-s3813" xml:space="preserve">totę NL, NM ęquales, ſiue GEN circũ-<lb/>ſcripti etiam anguli LGM diameter erit: </s> <s xml:id="echoid-s3814" xml:space="preserve">inſuper idem angulus LGM erit _MI-_ <lb/>_NIMVS_: </s> <s xml:id="echoid-s3815" xml:space="preserve">nam recta, quę ex G intra ipſum ducitur, minorem angulum cum al-<lb/>tera nunc ductarum conſtituens, ſi producatur, ſecat vnam aſymptoton (cum <lb/>ei æquidiſtanter ductam ſecet in G) quare vlterius producta ſecabit <anchor type="note" xlink:href="" symbol="f"/> ipſam <anchor type="note" xlink:label="note-0137-06a" xlink:href="note-0137-06"/> Hyperbolen. </s> <s xml:id="echoid-s3816" xml:space="preserve">Datę igitur Hyperbolę per datum extra ipſam pũctum in locis <lb/>poſſibilibus, circumſcriptus eſt _MINIMVS_ quæſitus angulus. </s> <s xml:id="echoid-s3817" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s3818" xml:space="preserve">c.</s> <s xml:id="echoid-s3819" xml:space="preserve"/> </p> <div xml:id="echoid-div366" type="float" level="2" n="7"> <note symbol="a" position="right" xlink:label="note-0137-01" xlink:href="note-0137-01a" xml:space="preserve">ex 8. 2. <lb/>conic.</note> <note symbol="b" position="right" xlink:label="note-0137-02" xlink:href="note-0137-02a" xml:space="preserve">8. huius.</note> <note symbol="c" position="right" xlink:label="note-0137-03" xlink:href="note-0137-03a" xml:space="preserve">49. ſec. <lb/>conic.</note> <note symbol="d" position="right" xlink:label="note-0137-04" xlink:href="note-0137-04a" xml:space="preserve">29. ſec. <lb/>conic.</note> <note symbol="e" position="right" xlink:label="note-0137-05" xlink:href="note-0137-05a" xml:space="preserve">ex 8. 2. <lb/>conic.</note> <note symbol="f" position="right" xlink:label="note-0137-06" xlink:href="note-0137-06a" xml:space="preserve">35. h.</note> </div> </div> <div xml:id="echoid-div368" type="section" level="1" n="156"> <head xml:id="echoid-head161" xml:space="preserve">PROBL. XXVII. PROP. LXIX.</head> <p> <s xml:id="echoid-s3820" xml:space="preserve">Datę Hyperbolę, per punctum intra ipſam datum, MAXIMVM <lb/>angulum inſcribere. </s> <s xml:id="echoid-s3821" xml:space="preserve">Item.</s> <s xml:id="echoid-s3822" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3823" xml:space="preserve">Dato angulo, per punctum extra ipſum datum, cum dato ſemi-<lb/>tranſuerſo latere, MINIMAM Hyperbolen circumſcribere.</s> <s xml:id="echoid-s3824" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3825" xml:space="preserve">Oportet autem datum punctum eſſe in angulo, qui eſt ad verti-<lb/>cem dato.</s> <s xml:id="echoid-s3826" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3827" xml:space="preserve">SIt data Hyperbole ABC, cuius aſymptoti ſint DE, DF, & </s> <s xml:id="echoid-s3828" xml:space="preserve">punctum intra <lb/>ipſam ſit G, per quod ei oporteat _MAXIMV M_ angulum inſcribere.</s> <s xml:id="echoid-s3829" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3830" xml:space="preserve">Ducantur ex G rectæ GH, GI aſymptotis æquidiſtantes. </s> <s xml:id="echoid-s3831" xml:space="preserve">Dico angulum <lb/>HGI eſſe _MAXIMVM_ quæſitum.</s> <s xml:id="echoid-s3832" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3833" xml:space="preserve">Nam iuncta DG, & </s> <s xml:id="echoid-s3834" xml:space="preserve">producta ad <lb/> <anchor type="figure" xlink:label="fig-0137-01a" xlink:href="fig-0137-01"/> L, ipſa GL neceſſariò diuidet angu-<lb/>lum HGI (vt ſatis patet) ſumptoque <lb/>in ea quolibet puncto L, & </s> <s xml:id="echoid-s3835" xml:space="preserve">applica-<lb/>ta in Hyperbola, ad diametrũ BL, or-<lb/>dinata ELF, Intera anguli HGI ſecã <lb/>in H, I; </s> <s xml:id="echoid-s3836" xml:space="preserve">erit ob triangulorum ſimili-<lb/>tudinem, DL ad LE, vt GL ad LH, <lb/>ſed DL ad LE eſt vt DL ad LF, cum <lb/>LE, LF ſint æquales, & </s> <s xml:id="echoid-s3837" xml:space="preserve">DL ad LF <lb/>eſt vt GL ad LI, quare GL ad LH erit vt GL ad LI, ſiue LH ęqualis LI: </s> <s xml:id="echoid-s3838" xml:space="preserve">vnde <pb o="114" file="0138" n="138" rhead=""/> BGL erit diameter, tum datæ Hyperbolæ, tum deſcripti anguli; </s> <s xml:id="echoid-s3839" xml:space="preserve">& </s> <s xml:id="echoid-s3840" xml:space="preserve">cum GH, <lb/>GI, per punctum G, in Hyperbola ſumptum, ductæ ſint aſymptotis æquidi-<lb/>ſtantes, ipſæ, ad partes B productæ, Hyperbolæ occurrent, cum aſymptotos <lb/>ſecent, ſed ad partes H, I, nunquam cum ſectione conuenient, <anchor type="note" xlink:href="" symbol="a"/> at quæcun- <anchor type="note" xlink:label="note-0138-01a" xlink:href="note-0138-01"/> que ducatur ex G extra angulum HGI, ſecabit producta alteram aſympto-<lb/>ton (cum ſecet in G ipſi parallelam GH) ac ideò priùs Hyperbolen datam: <lb/></s> <s xml:id="echoid-s3841" xml:space="preserve">eſt igitur HGI _MAXIMV S_ inſcriptus angulus, vti quærebatur. </s> <s xml:id="echoid-s3842" xml:space="preserve">Quod primò <lb/>erat, &</s> <s xml:id="echoid-s3843" xml:space="preserve">c.</s> <s xml:id="echoid-s3844" xml:space="preserve"/> </p> <div xml:id="echoid-div368" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0137-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0138-01" xlink:href="note-0138-01a" xml:space="preserve">Coroll. <lb/>11. huius.</note> </div> <p> <s xml:id="echoid-s3845" xml:space="preserve">IAM ſit datus angulus HGI, & </s> <s xml:id="echoid-s3846" xml:space="preserve">datum punctum ſit B, in angulo tamen, qui <lb/>ei eſt ad verticem: </s> <s xml:id="echoid-s3847" xml:space="preserve">oportet per B _MINIMAM_ Hyperbolen circumſcribe-<lb/>re, cum dato ſemi- tranſuerſo R.</s> <s xml:id="echoid-s3848" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3849" xml:space="preserve">Iungatur GB, & </s> <s xml:id="echoid-s3850" xml:space="preserve">producatur, ſu-<lb/> <anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/> maturque B D æqualis R; </s> <s xml:id="echoid-s3851" xml:space="preserve">& </s> <s xml:id="echoid-s3852" xml:space="preserve">per D <lb/>agantur DE, DF, ipſis GH, GI pa-<lb/>rallelæ, & </s> <s xml:id="echoid-s3853" xml:space="preserve">per B cum afymptotis <lb/>DE, DF deſcribatur <anchor type="note" xlink:href="" symbol="b"/> Hyperbole <anchor type="note" xlink:label="note-0138-02a" xlink:href="note-0138-02"/> ABC, Dico hanc eſſe _MINIMAM_ <lb/>circumſcriptam quæſitam.</s> <s xml:id="echoid-s3854" xml:space="preserve"/> </p> <div xml:id="echoid-div369" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0138-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0138-02" xlink:href="note-0138-02a" xml:space="preserve">4. ſec. <lb/>conic.</note> </div> <p> <s xml:id="echoid-s3855" xml:space="preserve">Nam eadem ratione, vt ſupra.</s> <s xml:id="echoid-s3856" xml:space="preserve">, <lb/>oſtendetur DBGL eſſe diametrum <lb/>ſectionis, & </s> <s xml:id="echoid-s3857" xml:space="preserve">dati anguli, & </s> <s xml:id="echoid-s3858" xml:space="preserve">rectas <lb/>GH, GI, ad partes H, I productas (cum aſymptotis æquidiſtent) nunquam <lb/>cum ſectione conuenire; </s> <s xml:id="echoid-s3859" xml:space="preserve">ideoque Hyperbolen ABC dato angulo eſſe circũ-<lb/>ſcriptam: </s> <s xml:id="echoid-s3860" xml:space="preserve">ſed eſt quoque _MINIMA_, quoniam, quæ cum eodem tranſuerſo <lb/>adſcribitur, ſed cum recto maiori, <anchor type="note" xlink:href="" symbol="c"/> maioreſt ipſa ABC; </s> <s xml:id="echoid-s3861" xml:space="preserve">quæ verò cum re- <anchor type="note" xlink:label="note-0138-03a" xlink:href="note-0138-03"/> cto minori, qualis eſt MBN, eſt quidem <anchor type="note" xlink:href="" symbol="d"/> minor ABC, ſed omnino ſecat la- <anchor type="note" xlink:label="note-0138-04a" xlink:href="note-0138-04"/> tera dati anguli: </s> <s xml:id="echoid-s3862" xml:space="preserve">quoniam ducta DO, quæ ſit aſymptotos inſcriptæ MBN, <lb/>ipſa cadet <anchor type="note" xlink:href="" symbol="e"/> infra DE, ſed eam ſecat in D, quare producta, alteram paralle- <anchor type="note" xlink:label="note-0138-05a" xlink:href="note-0138-05"/> lam ſecabit GH, ſed DO tota cadit extra BM, vnde occurſus DO cum GH <lb/>erit extra BM; </s> <s xml:id="echoid-s3863" xml:space="preserve">ſiue GH neceſſariò ſecabit priùs ſectionem BM. </s> <s xml:id="echoid-s3864" xml:space="preserve">Eſt igitur ſe-<lb/>ctio MBN _MINIMA_ circumſcripta quæſita dato angulo HGI, per datum <lb/>punctum B, & </s> <s xml:id="echoid-s3865" xml:space="preserve">cum dato ſemi-tranſuerſo R. </s> <s xml:id="echoid-s3866" xml:space="preserve">Quod vltimò faciendum erat.</s> <s xml:id="echoid-s3867" xml:space="preserve"/> </p> <div xml:id="echoid-div370" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0138-03" xlink:href="note-0138-03a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="d" position="left" xlink:label="note-0138-04" xlink:href="note-0138-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="left" xlink:label="note-0138-05" xlink:href="note-0138-05a" xml:space="preserve">ex 37. h.</note> </div> </div> <div xml:id="echoid-div372" type="section" level="1" n="157"> <head xml:id="echoid-head162" xml:space="preserve">PROBL. XXVIII. PROP. LXX.</head> <p> <s xml:id="echoid-s3868" xml:space="preserve">Dato angulo rectilineo, per punctum intra ipſum datum, cum <lb/>dato tranſuerſo latere, MAXIMAM Ellipſim inſcribere: </s> <s xml:id="echoid-s3869" xml:space="preserve">& </s> <s xml:id="echoid-s3870" xml:space="preserve">è con-<lb/>tra.</s> <s xml:id="echoid-s3871" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3872" xml:space="preserve">SIt datus angulus ABC, & </s> <s xml:id="echoid-s3873" xml:space="preserve">punctum intra ipſum ſit D, per quod ei opor-<lb/>teat, cum dato tranſuerſo R, _MAXIMAM_ Ellipſim inſcribere.</s> <s xml:id="echoid-s3874" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3875" xml:space="preserve">Sumatur DE ęqualis R, & </s> <s xml:id="echoid-s3876" xml:space="preserve">per D, & </s> <s xml:id="echoid-s3877" xml:space="preserve">E in angulo ABC applicentur <anchor type="note" xlink:href="" symbol="a"/> FDG, <anchor type="note" xlink:label="note-0138-06a" xlink:href="note-0138-06"/> HEL, & </s> <s xml:id="echoid-s3878" xml:space="preserve">iungatur HG diametrum ſecans in M, per quod applicetur AMD, <lb/>& </s> <s xml:id="echoid-s3879" xml:space="preserve">circa diametrum DE, ac per terminos applicatæ AC, deſcribatur <anchor type="note" xlink:href="" symbol="b"/> Ellipſis <anchor type="note" xlink:label="note-0138-07a" xlink:href="note-0138-07"/> DAEC. </s> <s xml:id="echoid-s3880" xml:space="preserve">Dico hanc eſſe _MAXIMAM_ quæſitam. </s> <s xml:id="echoid-s3881" xml:space="preserve">Quoniam, ipſa ADCE eſt <lb/>menſali HFGL, ſiue dato angulo <anchor type="note" xlink:href="" symbol="c"/> inſcripta, & </s> <s xml:id="echoid-s3882" xml:space="preserve">quælibet alia Ellipſis eidem <anchor type="note" xlink:label="note-0138-08a" xlink:href="note-0138-08"/> <pb o="115" file="0139" n="139" rhead=""/> angulo per D adſcripta, cum eodem tranſuerſo <lb/> <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> latere DE, ſed cum recto, quod minus ſit recto <lb/>adſcriptæ DAEC, eſt ipſa <anchor type="note" xlink:href="" symbol="a"/> minor, adſcripta <anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/> verò cum recto maiori, eſt quidem maior ea-<lb/>dem, ſed omnino ſecat anguli latera BA, BC, <lb/>vt ſatis conſtat. </s> <s xml:id="echoid-s3883" xml:space="preserve">Ampliùs, Ellipſis, quæ per <lb/>D ſupra applicatam FG eidem angulo contin-<lb/>genter inſcribitur, cum tranſuerſo latere ęqua-<lb/>li ipſo DE, vel dato R (ſi tamen interceptum <lb/>diametri ſegmentum DB maius fuerit DE) eſt <lb/>omnino <anchor type="note" xlink:href="" symbol="b"/> minor prædicta ADCE: </s> <s xml:id="echoid-s3884" xml:space="preserve">quare ipſa <anchor type="note" xlink:label="note-0139-02a" xlink:href="note-0139-02"/> eſt _MAXIMA_ inſcripta quæſita. </s> <s xml:id="echoid-s3885" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s3886" xml:space="preserve">c.</s> <s xml:id="echoid-s3887" xml:space="preserve"/> </p> <div xml:id="echoid-div372" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0138-06" xlink:href="note-0138-06a" xml:space="preserve">66. h.</note> <note symbol="b" position="left" xlink:label="note-0138-07" xlink:href="note-0138-07a" xml:space="preserve">Coroll. <lb/>57. h.</note> <note symbol="c" position="left" xlink:label="note-0138-08" xlink:href="note-0138-08a" xml:space="preserve">Schol. <lb/>62. h.</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/QN4GHYBF/figures/0139-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> <note symbol="b" position="right" xlink:label="note-0139-02" xlink:href="note-0139-02a" xml:space="preserve">64. h.</note> </div> <p> <s xml:id="echoid-s3888" xml:space="preserve">Notandum eſt autem, quod ſi ABC fuerit <lb/>quælibet coni- ſectio, vel circulus, eadem penitùs conſtructione, ac demon-<lb/>ſtratione inſcribetur ei _MAXIMA_ Ellipſis ADCE, cum dato tranſuerſo R, <lb/>quod tamen in Ellipſi, vel circulo, non excedat maius diametri ſegmentum.</s> <s xml:id="echoid-s3889" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3890" xml:space="preserve">SI verò data ſit Ellipſis ADCE, & </s> <s xml:id="echoid-s3891" xml:space="preserve">per punctum B extra ipſam datum cir-<lb/>cumſcribendus ſit ei _MINIMV S_ angulus rectilineus. </s> <s xml:id="echoid-s3892" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Ducantur ex B El- <anchor type="note" xlink:label="note-0139-03a" xlink:href="note-0139-03"/> lipſim contingentes BA, BC; </s> <s xml:id="echoid-s3893" xml:space="preserve">nam angulus ABC erit _MINIMV S_ circum-<lb/>ſcriptus quæſitus: </s> <s xml:id="echoid-s3894" xml:space="preserve">quoniam ducta AC, ac bifariam ſecta in M, iunctaque <lb/>BME, ipſa <anchor type="note" xlink:href="" symbol="d"/> erit Ellipſis diameter, ſimulque dati anguli ABC, cum omnes <anchor type="note" xlink:label="note-0139-04a" xlink:href="note-0139-04"/> ipſi AC æquidiſtanter ductæ ab eadem BM bifariam ſecentur: </s> <s xml:id="echoid-s3895" xml:space="preserve">vnde angulus <lb/>ABC erit datæ Ellipſi ADCE circumſcriptus: </s> <s xml:id="echoid-s3896" xml:space="preserve">eritque _MINIMV S_; </s> <s xml:id="echoid-s3897" xml:space="preserve">quoniam <lb/>quæcunque recta, quæ ex B intra angulum ABC ducitur, cum altera con-<lb/>tingentium minorem angulum conſtituens, neceſſariò ſecat datam Ellipſim <lb/>ADC: </s> <s xml:id="echoid-s3898" xml:space="preserve">quare angulus ABC eſt _MIMIMV S_ circũſcriptus quæſitus. </s> <s xml:id="echoid-s3899" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s3900" xml:space="preserve">c.</s> <s xml:id="echoid-s3901" xml:space="preserve"/> </p> <div xml:id="echoid-div373" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0139-03" xlink:href="note-0139-03a" xml:space="preserve">49. ſec. <lb/>conic.</note> <note symbol="d" position="right" xlink:label="note-0139-04" xlink:href="note-0139-04a" xml:space="preserve">29. ſec. <lb/>conic.</note> </div> </div> <div xml:id="echoid-div375" type="section" level="1" n="158"> <head xml:id="echoid-head163" xml:space="preserve">LEMMA VIII. PROP. LXXI.</head> <p> <s xml:id="echoid-s3902" xml:space="preserve">Si duæ rectæ AB, CD ſe mutuò ſecent in E, ſitque AE æqualis <lb/>EB, ſed CE maior ED, dico iunctas CA, BD, ſi producantur, con-<lb/>uenire ſimul ad partes A, D, vt in F, & </s> <s xml:id="echoid-s3903" xml:space="preserve">ſi per D ducatur DG pa-<lb/>rallela ad AE, eſſe FC ad CA, vt FG ad GA.</s> <s xml:id="echoid-s3904" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3905" xml:space="preserve">SVmpta enim EH æquali ipſi EC, erit EH ma-<lb/> <anchor type="figure" xlink:label="fig-0139-02a" xlink:href="fig-0139-02"/> ior ED, & </s> <s xml:id="echoid-s3906" xml:space="preserve">iuncta BH, in triangulis BEH, AE <lb/>C erunt latera circùm æquales angulos ad E, æ-<lb/>qualia: </s> <s xml:id="echoid-s3907" xml:space="preserve">quare reliqui anguli EBH, EAC ęquales, <lb/>vnde BH parallela ad CA, hoc eſt anguli BAG, <lb/>ABH duobus rectis æquales, ideoque duo BAG, <lb/>ABD minores duobus rectis: </s> <s xml:id="echoid-s3908" xml:space="preserve">occurrit ergo BD <lb/>cum CA producta ad partes D, A; </s> <s xml:id="echoid-s3909" xml:space="preserve">ſitque occur-<lb/>ſus in F, ex quo ducatur FI parallela ad DG, vel <lb/>ad AE.</s> <s xml:id="echoid-s3910" xml:space="preserve"/> </p> <div xml:id="echoid-div375" type="float" level="2" n="1"> <figure xlink:label="fig-0139-02" xlink:href="fig-0139-02a"> <image file="0139-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0139-02"/> </figure> </div> <p> <s xml:id="echoid-s3911" xml:space="preserve">Cum ſint ergo triangula FDI, BDE ſimilia, erit <lb/>FI ad EB, vel ad AE, hoc eſt FC ad CA, vt FD <lb/>ad DB, vel vt FG ad GA. </s> <s xml:id="echoid-s3912" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s3913" xml:space="preserve">c.</s> <s xml:id="echoid-s3914" xml:space="preserve"/> </p> <pb o="116" file="0140" n="140" rhead=""/> </div> <div xml:id="echoid-div377" type="section" level="1" n="159"> <head xml:id="echoid-head164" xml:space="preserve">LEMMA IX. PROP. LXXII.</head> <p> <s xml:id="echoid-s3915" xml:space="preserve">Dato angulo rectilineo ABC, cuius diameter BDE, & </s> <s xml:id="echoid-s3916" xml:space="preserve">applica-<lb/>ta ADC: </s> <s xml:id="echoid-s3917" xml:space="preserve">oportet ex C, infra ADC, ſecantem ducere CEF, ita vt <lb/>ſi ex E, & </s> <s xml:id="echoid-s3918" xml:space="preserve">F applicentur EH, FG ipſi ADC parallelæ, quadratum <lb/>HE ad rectangulum DEG, datam habeaa<unsure/>t rationem O ad P.</s> <s xml:id="echoid-s3919" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3920" xml:space="preserve">SVmatur Q media proportionalis inter O, & </s> <s xml:id="echoid-s3921" xml:space="preserve">P; </s> <s xml:id="echoid-s3922" xml:space="preserve">& </s> <s xml:id="echoid-s3923" xml:space="preserve">ex D, niſi DA ſit dia-<lb/>metro BD perpendicularis, erigatur DL, & </s> <s xml:id="echoid-s3924" xml:space="preserve">fiat vt O ad Q, ita DA ad <lb/>DL, iunctaque BLM, ſumatur LM æqualis LD, & </s> <s xml:id="echoid-s3925" xml:space="preserve">per M demittatur ME per-<lb/>pendicularis ipſi BDE: </s> <s xml:id="echoid-s3926" xml:space="preserve">dico per punctum E quæſitam ſecantem tranſire.</s> <s xml:id="echoid-s3927" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3928" xml:space="preserve">Nam iuncta MD, ductaque MN ipſi BM <lb/> <anchor type="figure" xlink:label="fig-0140-01a" xlink:href="fig-0140-01"/> perpendiculari cum ſint anguli L M N, <lb/>LDN recti, erũt anguli NMD, NDM duo-<lb/>bus rectis minores; </s> <s xml:id="echoid-s3929" xml:space="preserve">quare MN ipſi DE oc-<lb/>curret in N; </s> <s xml:id="echoid-s3930" xml:space="preserve">cumque angulus LMD, ęqua-<lb/>lis ſit angulo LDM, erunt reſidui ex rectis <lb/>DMN, MDN æquales, hoc eſt ND æqua-<lb/>lis NM, facto igitur centro N, interuallo <lb/>ND, deſcribatur circulus DMG, qui vtrã-<lb/>que LD, LM, continget in D, M, cum an-<lb/>guli ad D, M ſint recti: </s> <s xml:id="echoid-s3931" xml:space="preserve">ducatur tandem <lb/>EH parallela ad DA.</s> <s xml:id="echoid-s3932" xml:space="preserve"/> </p> <div xml:id="echoid-div377" type="float" level="2" n="1"> <figure xlink:label="fig-0140-01" xlink:href="fig-0140-01a"> <image file="0140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0140-01"/> </figure> </div> <p> <s xml:id="echoid-s3933" xml:space="preserve">Iam cum BM circulum DMG contingat <lb/>in M, ſitque ME diametro DG perpendi-<lb/>cularis <anchor type="note" xlink:href="" symbol="a"/> erit GB ad BD, vt GE ad ED, &</s> <s xml:id="echoid-s3934" xml:space="preserve"> <anchor type="note" xlink:label="note-0140-01a" xlink:href="note-0140-01"/> permutando BG ad GE, vt BD ad DE, ſed <lb/>eſt BG maior GE, ergo & </s> <s xml:id="echoid-s3935" xml:space="preserve">BD maior DE, <lb/>eſtque AD æqualis DC, & </s> <s xml:id="echoid-s3936" xml:space="preserve">anguli ad verticem D æquales, quare iunctæ <lb/>BA, CE, productę, conuenient ſimul ad partes A, E, vt in F <anchor type="note" xlink:href="" symbol="b"/> eritque FB ad <anchor type="note" xlink:label="note-0140-02a" xlink:href="note-0140-02"/> BA, vt FH ad HA, & </s> <s xml:id="echoid-s3937" xml:space="preserve">permutando BF ad FH, vt BE ad AH, vel vt BD ad <lb/>DE, vel vt BG ad GE, & </s> <s xml:id="echoid-s3938" xml:space="preserve">diuidendo BH ad HF, vt BE ad EG, quare iuncta <lb/>FG ipſis EH, DA æquidiſtabit. </s> <s xml:id="echoid-s3939" xml:space="preserve">Et quoniam eſt HE ad EB, vt AD ad DB, <lb/>& </s> <s xml:id="echoid-s3940" xml:space="preserve">BE ad EM, vt BD ad DL (ob triangulorum ſimilitudinem) erit ex æquo <lb/>HE ad EM, vt AD ad DL, & </s> <s xml:id="echoid-s3941" xml:space="preserve">quadratum HE ad quadratum EM, hoc eſt ad <lb/>rectangulum DEG, vt quadratum AD ad DL, vel vt quadratum O ad qua-<lb/>dratum Q, vel vt linea O ad P. </s> <s xml:id="echoid-s3942" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s3943" xml:space="preserve"/> </p> <div xml:id="echoid-div378" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0140-01" xlink:href="note-0140-01a" xml:space="preserve">36. pri-<lb/>mi conic.</note> <note symbol="b" position="left" xlink:label="note-0140-02" xlink:href="note-0140-02a" xml:space="preserve">71. h.</note> </div> <pb o="117" file="0141" n="141" rhead=""/> </div> <div xml:id="echoid-div380" type="section" level="1" n="160"> <head xml:id="echoid-head165" xml:space="preserve">PROBL. XXIX. PROP. LXXIII.</head> <p> <s xml:id="echoid-s3944" xml:space="preserve">Dato angulo rectilineo, per punctum in qualibet eius diametro <lb/>datum, MAXIMAM Ellipſim inſcribere, cuius latera datam ha-<lb/>beant rationem.</s> <s xml:id="echoid-s3945" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3946" xml:space="preserve">SIt datus angulus ABC, diameter BD, & </s> <s xml:id="echoid-s3947" xml:space="preserve">datum punctum D, per quod <lb/>oporteat Ellipſim inſcribere, cuius tranſuerſum latus ad rectum, datam <lb/>quamcunque habeat rationem E ad F, & </s> <s xml:id="echoid-s3948" xml:space="preserve">ſit _MAXIMA_.</s> <s xml:id="echoid-s3949" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3950" xml:space="preserve">Applicetur per D, <anchor type="note" xlink:href="" symbol="a"/> ordinatim GDH, &</s> <s xml:id="echoid-s3951" xml:space="preserve"> <anchor type="note" xlink:label="note-0141-01a" xlink:href="note-0141-01"/> <anchor type="figure" xlink:label="fig-0141-01a" xlink:href="fig-0141-01"/> per H ducatur HIL diametrum ſecans in I, & </s> <s xml:id="echoid-s3952" xml:space="preserve"><lb/>BA in L, ita vt ex I, & </s> <s xml:id="echoid-s3953" xml:space="preserve">L ductis A I, LM ipſi <lb/>DH parallelis, rectangulum DIM, ad quadra-<lb/>tum AI, rationem <anchor type="note" xlink:href="" symbol="b"/> habeat E ad F, & </s> <s xml:id="echoid-s3954" xml:space="preserve">cum trãſ- <anchor type="note" xlink:label="note-0141-02a" xlink:href="note-0141-02"/> uerſo DM, per extrema applicatæ AC, Elli-<lb/>pſis <anchor type="note" xlink:href="" symbol="c"/> deſcribatur DAMC. </s> <s xml:id="echoid-s3955" xml:space="preserve">Dico hanc eſſe <anchor type="note" xlink:label="note-0141-03a" xlink:href="note-0141-03"/> _MAXIMAM_ quæſitam.</s> <s xml:id="echoid-s3956" xml:space="preserve"/> </p> <div xml:id="echoid-div380" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve">66. h.</note> <figure xlink:label="fig-0141-01" xlink:href="fig-0141-01a"> <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0141-01"/> </figure> <note symbol="b" position="right" xlink:label="note-0141-02" xlink:href="note-0141-02a" xml:space="preserve">72. h.</note> <note symbol="c" position="right" xlink:label="note-0141-03" xlink:href="note-0141-03a" xml:space="preserve">Coroll. <lb/>57. h.</note> </div> <p> <s xml:id="echoid-s3957" xml:space="preserve">Eſt enim <anchor type="note" xlink:href="" symbol="d"/> LB ad BG, ſiue MB ad BD, vt <anchor type="note" xlink:label="note-0141-04a" xlink:href="note-0141-04"/> LA ad AG, ſiue vt MI ad ID, quare BA, BC <lb/>Ellipſim <anchor type="note" xlink:href="" symbol="e"/> contingent, ideoque ipſa erit angu- <anchor type="note" xlink:label="note-0141-05a" xlink:href="note-0141-05"/> lo inſcripta, eritque _MAXIMA_, vt in præce-<lb/>dentibus oſtenſum fuit. </s> <s xml:id="echoid-s3958" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s3959" xml:space="preserve">c.</s> <s xml:id="echoid-s3960" xml:space="preserve"/> </p> <div xml:id="echoid-div381" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-0141-04" xlink:href="note-0141-04a" xml:space="preserve">71. h.</note> <note symbol="e" position="right" xlink:label="note-0141-05" xlink:href="note-0141-05a" xml:space="preserve">34. pri. <lb/>conic.</note> </div> </div> <div xml:id="echoid-div383" type="section" level="1" n="161"> <head xml:id="echoid-head166" xml:space="preserve">LEMMA X. PROP. LXXIV.</head> <p> <s xml:id="echoid-s3961" xml:space="preserve">Datis medijs proportionalibus, Arithmetica nempe, & </s> <s xml:id="echoid-s3962" xml:space="preserve">Geo-<lb/>metrica inter eaſdem ignotas extremas; </s> <s xml:id="echoid-s3963" xml:space="preserve">ipſas extremas inuenire.</s> <s xml:id="echoid-s3964" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3965" xml:space="preserve">SIt AB media arithmetica, & </s> <s xml:id="echoid-s3966" xml:space="preserve">AC media <lb/> <anchor type="figure" xlink:label="fig-0141-02a" xlink:href="fig-0141-02"/> geometrica inter duas eaſdem ignotas <lb/>extremas, quarum idem ſit terminus A, & </s> <s xml:id="echoid-s3967" xml:space="preserve"><lb/>ſimul congruere intelligantur: </s> <s xml:id="echoid-s3968" xml:space="preserve">patet primò <lb/>AB ſuperare ipſam AC, cum media ari-<lb/>thmetica ſit maior media geometrica. </s> <s xml:id="echoid-s3969" xml:space="preserve">Iam <lb/>oporteat datis AC, AB ignotas extremas <lb/>proportionales inuenire.</s> <s xml:id="echoid-s3970" xml:space="preserve"/> </p> <div xml:id="echoid-div383" type="float" level="2" n="1"> <figure xlink:label="fig-0141-02" xlink:href="fig-0141-02a"> <image file="0141-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0141-02"/> </figure> </div> <p> <s xml:id="echoid-s3971" xml:space="preserve">Fiat centro A interuallo A C circulus <lb/>CF, cui ex puncto B contingens ducatur <lb/>BF, quæ eum radio FA rectum efficiet an-<lb/>gulum, vnde ſubtenſa BA erit maior ipſa <lb/>BF; </s> <s xml:id="echoid-s3972" xml:space="preserve">ſi ergo cum centro B, interuallo BF <lb/>deſcribatur ſemi- circulus DFE, ipſæ ſeca-<lb/>bit BA infra A, ſed tamen vltra C (cum ſit <lb/>BC minor BF, eo quod AC æquatur AF, <lb/>&</s> <s xml:id="echoid-s3973" xml:space="preserve">tota AB minor eſt duobus AF, FB) ſecabitque productam AB in E; </s> <s xml:id="echoid-s3974" xml:space="preserve">quem <pb o="118" file="0142" n="142" rhead=""/> circulum dico in punctis D, E, quæſitum ſoluere: </s> <s xml:id="echoid-s3975" xml:space="preserve">nempe AE, & </s> <s xml:id="echoid-s3976" xml:space="preserve">AD eſſe <lb/>quæſitas extremas. </s> <s xml:id="echoid-s3977" xml:space="preserve">Nam cum ſit BD æqualis BE, erit data AB media ari-<lb/>thmetica inter inuentas EA, AD. </s> <s xml:id="echoid-s3978" xml:space="preserve">Cumque ſit BF radius circuli EFD, & </s> <s xml:id="echoid-s3979" xml:space="preserve">an-<lb/>gulus BFA rectus, erit FA ipſi circulo contingens, quare rectangulum EAD <lb/>æquabitur quadrato AF, ſiue quadrato AC, vnde data AC erit media geo-<lb/>metrica inter eaſdem inuentas EA, AD. </s> <s xml:id="echoid-s3980" xml:space="preserve">Quare ignotæ extremæ, ſunt in-<lb/>uentæ, vti quærebantur. </s> <s xml:id="echoid-s3981" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s3982" xml:space="preserve">c.</s> <s xml:id="echoid-s3983" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div385" type="section" level="1" n="162"> <head xml:id="echoid-head167" xml:space="preserve">PROBL. XXX. PROP. LXXV.</head> <p> <s xml:id="echoid-s3984" xml:space="preserve">Datæ Parabolæ, per punctum intra ipſam datum, MAXIMAM <lb/>Ellipſim inſcribere, cuius latera datam habeant rationem: </s> <s xml:id="echoid-s3985" xml:space="preserve">& </s> <s xml:id="echoid-s3986" xml:space="preserve">è <lb/>contra.</s> <s xml:id="echoid-s3987" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3988" xml:space="preserve">Datæ Ellipſi, per punctum extra ipſam datum, MINIMAM <lb/>Parabolen circumſcribere.</s> <s xml:id="echoid-s3989" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3990" xml:space="preserve">ESto data Parabole ABC, & </s> <s xml:id="echoid-s3991" xml:space="preserve">datum intra ipſam punctum ſit E; </s> <s xml:id="echoid-s3992" xml:space="preserve">oportet per <lb/>E _MAXIMAM_ Ellipſim inſcribere, cuius rectum latus ad tranſuerſum <lb/>rationem habeat R ad S.</s> <s xml:id="echoid-s3993" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3994" xml:space="preserve">Ducatur ex E Parabolę diameter BED, <lb/> <anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/> & </s> <s xml:id="echoid-s3995" xml:space="preserve">applicetur EF, & </s> <s xml:id="echoid-s3996" xml:space="preserve">ſumpta V media pro-<lb/>portionali inter S, & </s> <s xml:id="echoid-s3997" xml:space="preserve">R; </s> <s xml:id="echoid-s3998" xml:space="preserve">fiat vt R ad V, ita <lb/>FE ad ED, iunctaque FD, quæ producta <lb/>ſectioni occurrat <anchor type="note" xlink:href="" symbol="a"/> in G, ex quo applicata <anchor type="note" xlink:label="note-0142-01a" xlink:href="note-0142-01"/> GHI, circa tranſuerſum latus EH, & </s> <s xml:id="echoid-s3999" xml:space="preserve">ter-<lb/>minos applicatæ AC deſcribatur <anchor type="note" xlink:href="" symbol="b"/> Ellipſis <anchor type="note" xlink:label="note-0142-02a" xlink:href="note-0142-02"/> AECH. </s> <s xml:id="echoid-s4000" xml:space="preserve">Hanc dico eſſe quæſitam.</s> <s xml:id="echoid-s4001" xml:space="preserve"/> </p> <div xml:id="echoid-div385" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0142-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0142-01" xlink:href="note-0142-01a" xml:space="preserve">27. pri-<lb/>mi conic.</note> <note symbol="b" position="left" xlink:label="note-0142-02" xlink:href="note-0142-02a" xml:space="preserve">Coroll. <lb/>57. h.</note> </div> <p> <s xml:id="echoid-s4002" xml:space="preserve">Cum enim in Parabola <anchor type="note" xlink:href="" symbol="c"/> ſint diametri ſe- <anchor type="note" xlink:label="note-0142-03a" xlink:href="note-0142-03"/> gmenta BH, BD, BE proportionalia, ſint-<lb/>que quadrata applicatarum IH, AD, FE <lb/>in eadem ratione <anchor type="note" xlink:href="" symbol="d"/> ipſorum ſegmentorum, <anchor type="note" xlink:label="note-0142-04a" xlink:href="note-0142-04"/> erunt quoq; </s> <s xml:id="echoid-s4003" xml:space="preserve">ipſæ applicatæ continuæ pro-<lb/>portionales, quapropter rectangulum ſub <lb/>IH, vel ſub HG, & </s> <s xml:id="echoid-s4004" xml:space="preserve">FE æquabitur quadrato AD, ac proinde quadratum <lb/>AD, ad rectangulum HDE, erit vt rectangulum ſub GH, EF, ad idem re-<lb/>ctangulum HDE, ſed rectangulum ſub GH, EF, ad ſibi ſimile rectangulum <lb/>HDE, (habent enim circa rectos angulos latera proportionalia, cum ſit GH <lb/>ad HD, vt FE ad ED, & </s> <s xml:id="echoid-s4005" xml:space="preserve">permutando GH ad FE, vt HD ad DE) eſt vt qua-<lb/>dratum FE ad ED (vtraque enim proportio, duplicata eſt proportionis linee <lb/>FE ad ED) quo circa, & </s> <s xml:id="echoid-s4006" xml:space="preserve">quadratum AD ad rectangulum HDE, hoc eſt in <lb/>Ellipſi, <anchor type="note" xlink:href="" symbol="e"/> @@ rectum latus ad tranſuerſum, erit vt quadratum FE ad ED, vel <anchor type="note" xlink:label="note-0142-05a" xlink:href="note-0142-05"/> vt quadratum R ad V, vel vt data linea R ad S. </s> <s xml:id="echoid-s4007" xml:space="preserve">Deſcripta eſt ergo Ellipſis <lb/>AECH, cuius latera habent datam rationem R ad S. </s> <s xml:id="echoid-s4008" xml:space="preserve">& </s> <s xml:id="echoid-s4009" xml:space="preserve">eſt <anchor type="note" xlink:href="" symbol="f"/> datæ Parabolæ <anchor type="note" xlink:label="note-0142-06a" xlink:href="note-0142-06"/> ABC inſcripta. </s> <s xml:id="echoid-s4010" xml:space="preserve">Amplius dico, ipſam eſſe _MAXIMAM_ Ellipſium quarum <lb/>latera ſint in ratione R ad S, ſiue eſſe _MAXIMAM_ ſibi ſimilium: </s> <s xml:id="echoid-s4011" xml:space="preserve">nam, quæ <lb/>cum minoribus lateribus datæ Parabolæ per E adſcribitur ad partes H, mi- <pb o="119" file="0143" n="143" rhead=""/> nor <anchor type="note" xlink:href="" symbol="a"/> eſt; </s> <s xml:id="echoid-s4012" xml:space="preserve">quæ verò cum maioribus eſt quidem <anchor type="note" xlink:href="" symbol="b"/> maior, ſed omnino ſecat Pa- <anchor type="note" xlink:label="note-0143-01a" xlink:href="note-0143-01"/> <anchor type="note" xlink:label="note-0143-02a" xlink:href="note-0143-02"/> rabolen ABC, vti oſtenſum fuit in præcedentibus. </s> <s xml:id="echoid-s4013" xml:space="preserve">Quamobrem Ellipſis <lb/>AECH, datæ Parabolæ per datum intra ipſam punctum E eſt _MAXIMA_ in-<lb/>ſcripta quæſita. </s> <s xml:id="echoid-s4014" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s4015" xml:space="preserve">c.</s> <s xml:id="echoid-s4016" xml:space="preserve"/> </p> <div xml:id="echoid-div386" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-0142-03" xlink:href="note-0142-03a" xml:space="preserve">Coroll. <lb/>1. 13. h.</note> <note symbol="d" position="left" xlink:label="note-0142-04" xlink:href="note-0142-04a" xml:space="preserve">20. pri-<lb/>mi conic.</note> <note symbol="e" position="left" xlink:label="note-0142-05" xlink:href="note-0142-05a" xml:space="preserve">22. pri-<lb/>mi conic.</note> <note symbol="f" position="left" xlink:label="note-0142-06" xlink:href="note-0142-06a" xml:space="preserve">Schol. <lb/>62. h.</note> <note symbol="a" position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">5. Co-<lb/>roll. 19. h.</note> <note symbol="b" position="right" xlink:label="note-0143-02" xlink:href="note-0143-02a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s4017" xml:space="preserve">IAM ſit data Ellipſis AECH, cuius centrum N, & </s> <s xml:id="echoid-s4018" xml:space="preserve">datum extra ipſam pun-<lb/>ctum ſit B, per quod oporteat _MINIMAM_ Parabolen circumſcribere.</s> <s xml:id="echoid-s4019" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4020" xml:space="preserve">Iungatur BN ſecans Ellipſim in E, & </s> <s xml:id="echoid-s4021" xml:space="preserve">poſita NE media geometrica, & </s> <s xml:id="echoid-s4022" xml:space="preserve">NB <lb/>media arithmetica inter eaſdem ignotas extremas, reperiantur <anchor type="note" xlink:href="" symbol="c"/> ipſæ extre- <anchor type="note" xlink:label="note-0143-03a" xlink:href="note-0143-03"/> mę, quę ſint ND, NL, & </s> <s xml:id="echoid-s4023" xml:space="preserve">per Dad Ellipſis diametrum EH applicetur ADC, <lb/>& </s> <s xml:id="echoid-s4024" xml:space="preserve">per verticem B, circa diametri ſegmentum BD, & </s> <s xml:id="echoid-s4025" xml:space="preserve">per terminos A, C de-<lb/>ſcribatur Parabole ABC. </s> <s xml:id="echoid-s4026" xml:space="preserve">Dico hanc eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s4027" xml:space="preserve"/> </p> <div xml:id="echoid-div387" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0143-03" xlink:href="note-0143-03a" xml:space="preserve">74. h.</note> </div> <p> <s xml:id="echoid-s4028" xml:space="preserve">Cum enim ſit NE media geometrica inter LN, ND, erit rectangulum <lb/>LND æquale quadrato NE; </s> <s xml:id="echoid-s4029" xml:space="preserve">& </s> <s xml:id="echoid-s4030" xml:space="preserve">per D applicata eſt in Ellipſi recta ADC, ſi <lb/>iungantur LA, LC ipſæ Ellipſim contingent <anchor type="note" xlink:href="" symbol="d"/> in A, C; </s> <s xml:id="echoid-s4031" xml:space="preserve">cumque ſit NB me- <anchor type="note" xlink:label="note-0143-04a" xlink:href="note-0143-04"/> dia arithmetica inter eaſdem LN, ND, erunt ipſarum differentiæ LB, BD <lb/>inter ſe æquales; </s> <s xml:id="echoid-s4032" xml:space="preserve">vnde eædem LA, LC Parabolen <anchor type="note" xlink:href="" symbol="e"/> contingent, quocirca <anchor type="note" xlink:label="note-0143-05a" xlink:href="note-0143-05"/> hæc datæ Ellipſi erit circumſcripta. </s> <s xml:id="echoid-s4033" xml:space="preserve">Eritque _MINIMA_: </s> <s xml:id="echoid-s4034" xml:space="preserve">quoniam quæ per B <lb/>eidem Ellipſi adſcribitur cum recto maiori, maior <anchor type="note" xlink:href="" symbol="f"/> eſt ABC, quæ verò cum <anchor type="note" xlink:label="note-0143-06a" xlink:href="note-0143-06"/> minori eſt quidem <anchor type="note" xlink:href="" symbol="g"/> minor, ſed omnino ſecat Ellipſim, vti ex præcedentibus, <anchor type="note" xlink:label="note-0143-07a" xlink:href="note-0143-07"/> & </s> <s xml:id="echoid-s4035" xml:space="preserve">per ſe ſatis conſtat. </s> <s xml:id="echoid-s4036" xml:space="preserve">Quapropter Parabole ABC eſt _MINIMA_ circumſcri-<lb/>pta quæſita. </s> <s xml:id="echoid-s4037" xml:space="preserve">Quod ſecundò faciendum, ac demonſtrandum erat.</s> <s xml:id="echoid-s4038" xml:space="preserve"/> </p> <div xml:id="echoid-div388" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-0143-04" xlink:href="note-0143-04a" xml:space="preserve">57. h.</note> <note symbol="e" position="right" xlink:label="note-0143-05" xlink:href="note-0143-05a" xml:space="preserve">conuer. <lb/>37. primi <lb/>conic. ex <lb/>Comand.</note> <note symbol="f" position="right" xlink:label="note-0143-06" xlink:href="note-0143-06a" xml:space="preserve">2. h.</note> <note symbol="g" position="right" xlink:label="note-0143-07" xlink:href="note-0143-07a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> </div> </div> <div xml:id="echoid-div390" type="section" level="1" n="163"> <head xml:id="echoid-head168" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s4039" xml:space="preserve">EX prima parte huius patet, quod ſi datum punctum D fuerit in axe Para-<lb/>bolæ, & </s> <s xml:id="echoid-s4040" xml:space="preserve">data ratio ſit æqualitatis, inſcribenda Ellipſis, idem erit, ac <lb/>circulus; </s> <s xml:id="echoid-s4041" xml:space="preserve">tunc enim applicata ADC erit axi perpendicularis, & </s> <s xml:id="echoid-s4042" xml:space="preserve">quadratum <lb/>AD æquabitur rectangulo HDE; </s> <s xml:id="echoid-s4043" xml:space="preserve">ideoque AECH erit circulus: </s> <s xml:id="echoid-s4044" xml:space="preserve">ex quo ha-<lb/>bebitur, quo pacto per punctum E in axe Parabolæ, _MAXIMVS_ circulus in-<lb/>ſcribatur: </s> <s xml:id="echoid-s4045" xml:space="preserve">applicata enim EF, cui ſumpta æquali ED, iunctaque FD, & </s> <s xml:id="echoid-s4046" xml:space="preserve">pro-<lb/>ducta in G, & </s> <s xml:id="echoid-s4047" xml:space="preserve">applicata GH, ipſa dabit EH diametrum quæſiti circuli.</s> <s xml:id="echoid-s4048" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div391" type="section" level="1" n="164"> <head xml:id="echoid-head169" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s4049" xml:space="preserve">PAtet etiam ſemi-applicatas in Parabola, ex terminis diametri _MAXIMI_ <lb/>inſcripti circuli, ęquari contiguis ſegmentis eiuſdem diametri, ab appli-<lb/>cata ex contactu circuli cum ſectione abſciſſis. </s> <s xml:id="echoid-s4050" xml:space="preserve">Sienim ſit FE æqualis ED, <lb/>ob ſimilitudinem triangulorum, crit etiam GH æqualis HD.</s> <s xml:id="echoid-s4051" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div392" type="section" level="1" n="165"> <head xml:id="echoid-head170" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s4052" xml:space="preserve">SI quis in vigeſimo nono, ac trigeſimo antecedenti Problemate, a <lb/>ſeueritate geometricæ demonſtrationis expeteret, nontantum El-<lb/>lipſes, per datum punctum ibi contingenter inſcriptas, ad par-<lb/>tes verticis, tum anguli, tum Parabolæ oppoſitas, MAXI-<lb/>MAS eſſe ſibi ipſis ſimilium per idem punctum, adeaſdem partes inſcri- <pb o="120" file="0144" n="144" rhead=""/> ptarum, ſed eſſe MAXIMAS quoque earum, quæ ad partes verticum in-<lb/>ſcribuntur; </s> <s xml:id="echoid-s4053" xml:space="preserve">id ſequenti Theoremate, in angulo, & </s> <s xml:id="echoid-s4054" xml:space="preserve">qualibet coni-ſectione, <lb/>vel circulo conſequetur, ſimulque dabitur Methodus ipſis inſcribendi ſimiles <lb/>Ellipſes, quæ ſucceſsiuè ſe mutuò, & </s> <s xml:id="echoid-s4055" xml:space="preserve">anguli, vel ſectionum latera contin-<lb/>gant.</s> <s xml:id="echoid-s4056" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div393" type="section" level="1" n="166"> <head xml:id="echoid-head171" xml:space="preserve">THEOR. XXXVI. PROP. LXXVI.</head> <p> <s xml:id="echoid-s4057" xml:space="preserve">Ellipſes inſcriptæ eidem angulo, vel Parabolæ, vel Hyperbolę, <lb/>aut portioni Ellipticæ, vel circulari, quæ non excedat Ellipſis, vel <lb/>circuli dimidium, ſe mutuò, & </s> <s xml:id="echoid-s4058" xml:space="preserve">anguli latera, vel ſectionem, vel <lb/>circulum contingentes, & </s> <s xml:id="echoid-s4059" xml:space="preserve">quarum diagonales menſalium, quibus <lb/>inſcribuntur, inter ſe æquidiſtent, ſunt ſimiles, & </s> <s xml:id="echoid-s4060" xml:space="preserve">quæ propior eſt <lb/>vertici, minor eſt remotiori.</s> <s xml:id="echoid-s4061" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4062" xml:space="preserve">SIt ABC, vel angulusrectilineus, vt in prima figura, vel Parabolæ, vel <lb/>Hyperbolæ, aut portio non maior ſemi-circuli, vel ſemi-Ellipſis dimi-<lb/>dio, vt in ſecunda, cuius vertex B, diameter BD, & </s> <s xml:id="echoid-s4063" xml:space="preserve">circa ipſius ſegmentum <lb/>DE, inter applicatas AC, IF, ducta diagonali AF, ſecan@ diametrum ED <lb/>in K, & </s> <s xml:id="echoid-s4064" xml:space="preserve">applicata per K recta GKH, per extrema G, E, H, D, <anchor type="note" xlink:href="" symbol="a"/> deſcribatur <anchor type="note" xlink:label="note-0144-01a" xlink:href="note-0144-01"/> Ellipſis GEHD, quæ per Scholium 62. </s> <s xml:id="echoid-s4065" xml:space="preserve">huius, menſali AIFC, hoc eſt dato <lb/>angulo, vel ſectioni crit inſcripta. </s> <s xml:id="echoid-s4066" xml:space="preserve">Et per I ducta IL parallela diagonali AF, <lb/>diametrum ſecan@ in O, conſimili conſtructione, ac ſupra, deſcribatur in <lb/>menſali INLF Ellipſis PMQE. </s> <s xml:id="echoid-s4067" xml:space="preserve">Dico primùm has Ellipſes inter ſe ſimiles <lb/>eſſe.</s> <s xml:id="echoid-s4068" xml:space="preserve"/> </p> <div xml:id="echoid-div393" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0144-01" xlink:href="note-0144-01a" xml:space="preserve">Coroll. <lb/>57. h.</note> </div> <p> <s xml:id="echoid-s4069" xml:space="preserve">Nam, in prima figura, proportio rectanguli GKH, ad rectangulum AKF, <lb/>componitur ex ratione GK ad KA, ſiue (per triangulorum ſimilitudinem) <lb/>PO ad OI, & </s> <s xml:id="echoid-s4070" xml:space="preserve">ex ratione HK ad KF, ſiue QO ad OL, ſed etiam proportio re-<lb/>ctanguli POQ, IOL, ex ijſdem rationibus componitur, quare in triangulo, <lb/>rectangulum GKH ad AKF, eſt vt rectangulum POQ ad IOL.</s> <s xml:id="echoid-s4071" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4072" xml:space="preserve">Iam, in ſecunda figura, eadem ratione, vt in 64. </s> <s xml:id="echoid-s4073" xml:space="preserve">huius, oſtendetur huiuſ-<lb/>modi Ellipſium centra cadere infra K, O, nempe in R, S, per quæ ſi appli-<lb/>centur RT, SV, ipſæ, diagonales ſecabunt in T, V; </s> <s xml:id="echoid-s4074" xml:space="preserve"><gap/> cum ſit DR ęqua-<lb/>lis RE, erit AT æqualis TF, ob parallelas; </s> <s xml:id="echoid-s4075" xml:space="preserve">item IV æqualis VL; </s> <s xml:id="echoid-s4076" xml:space="preserve">ſuntque <lb/>AF, IL æquidiſtanter ductæ in ſectione, vel circulo, quare iuncta TV erit <lb/>earundem æquidiſtantium diameter, quæ producta ad aliud punctum, præ-<lb/>ter B, ſectioni occurret, vt in Z, eritque <anchor type="note" xlink:href="" symbol="b"/> ſectionis, vel circuli diameter: </s> <s xml:id="echoid-s4077" xml:space="preserve">ſi <anchor type="note" xlink:label="note-0144-02a" xlink:href="note-0144-02"/> ergo ex verticibus B, Z, agantur BX, ZX ordinatim applicatis GH, AF æ-<lb/>quidiſtantes, hæ ſectionem <anchor type="note" xlink:href="" symbol="c"/> contingent, & </s> <s xml:id="echoid-s4078" xml:space="preserve">ſimul <anchor type="note" xlink:href="" symbol="d"/> conuenient in X; </s> <s xml:id="echoid-s4079" xml:space="preserve">eritque <anchor type="note" xlink:label="note-0144-03a" xlink:href="note-0144-03"/> <anchor type="note" xlink:label="note-0144-04a" xlink:href="note-0144-04"/> rectangulum GKH ad rectangulum AKF, vt quadratum BX ad quadratum <lb/>ZX; </s> <s xml:id="echoid-s4080" xml:space="preserve">item erit <anchor type="note" xlink:href="" symbol="e"/> rectangulum POQ ad IOL, vt idem quadratum BX ad idem <anchor type="note" xlink:label="note-0144-05a" xlink:href="note-0144-05"/> ZX, quapropter rectangulum GKH ad AKF, erit vt rectangulum POQ ad <lb/>IOL, quod etiam ſuperius in prima figura demonſtratum fuit. </s> <s xml:id="echoid-s4081" xml:space="preserve">Itaque, cum <lb/>ſit in vtraque, rectangulum GKH ad AKF, vt rectangulum POQ ad IOL, &</s> <s xml:id="echoid-s4082" xml:space="preserve"> <pb o="121" file="0145" n="145" rhead=""/> rectangulum AKF ad rectangulum DKE, vt quadratum AK ad KD (ob la-<lb/>terum proportionalitatem) vel vt quadratum IO ad OE (ob triangulorum <lb/>AKD, IOE ſimilitudinem) vel vt rectangulum IOL ad EOM (ob homolo-<lb/>gorum laterum proportionalitatem) erit, ex æquali, rectangulum GKH, <lb/>vel quadratum GK ad rectangulum DKE, ſiue vt rectũ latus Ellipſis GEHD <lb/>ad eiuſdem tranſuerſum, vt rectangulum POQ, ſiue quadratum PO, ad re-<lb/>ctangulum EOM, vel vt rectum Ellipſis PMQE ad ipſius tranſuerſum: </s> <s xml:id="echoid-s4083" xml:space="preserve">cum <lb/>ergo huiuſmodi Ellipſes habeant latera proportionalia, ſintque æqualiter <lb/> <anchor type="note" xlink:label="note-0145-01a" xlink:href="note-0145-01"/> inclinatæ, erunt <anchor type="note" xlink:href="" symbol="a"/> inter ſe ſimiles. </s> <s xml:id="echoid-s4084" xml:space="preserve">Quod erat primò demonſtrandum.</s> <s xml:id="echoid-s4085" xml:space="preserve"/> </p> <div xml:id="echoid-div394" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0144-02" xlink:href="note-0144-02a" xml:space="preserve">28. ſec. <lb/>conic.</note> <note symbol="c" position="left" xlink:label="note-0144-03" xlink:href="note-0144-03a" xml:space="preserve">17. pri-<lb/>mi conic.</note> <note symbol="d" position="left" xlink:label="note-0144-04" xlink:href="note-0144-04a" xml:space="preserve">59. h.</note> <note symbol="e" position="left" xlink:label="note-0144-05" xlink:href="note-0144-05a" xml:space="preserve">17. tertij <lb/>conic.</note> <note symbol="a" position="right" xlink:label="note-0145-01" xlink:href="note-0145-01a" xml:space="preserve">6. ſec. <lb/>defin. h.</note> </div> <figure> <image file="0145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0145-01"/> </figure> <p> <s xml:id="echoid-s4086" xml:space="preserve">Præterea, cum ſit AD´ ad DK, vt IE ad EO, ſitque AD maior <anchor type="note" xlink:href="" symbol="b"/> IE, erit <anchor type="note" xlink:label="note-0145-02a" xlink:href="note-0145-02"/> DK maior EO. </s> <s xml:id="echoid-s4087" xml:space="preserve">Item, cum ſit FE ad EK, vt LM ad MO, ſitque FE maior <lb/>LM, erit EK maior MO, ergo integra tranſuerſa diameter DE, maior toto <lb/>tranſuerſo latere EM; </s> <s xml:id="echoid-s4088" xml:space="preserve">ſed tranſuerſum DE ad tranſuerſum EM, eſt vt rectum <lb/>vnius ad rectum alterius, vt ſuperiùs demonſtrauimus, eſtq; </s> <s xml:id="echoid-s4089" xml:space="preserve">tranſuerſum DE <lb/>maios EM, quare rectum recto maios erit, ſiue Ellipſis GEHD maiorum la-<lb/>terum, maior <anchor type="note" xlink:href="" symbol="c"/> erit Ellipſi PMQE minorum laterum, quæ tùm in angulo, <anchor type="note" xlink:label="note-0145-03a" xlink:href="note-0145-03"/> tùm in ſectione, aut ſemi-Ellipſi, vel ſemi-circulo vertici B propior eſt. <lb/></s> <s xml:id="echoid-s4090" xml:space="preserve">Quod vltimò oſtendere propoſitum fuit.</s> <s xml:id="echoid-s4091" xml:space="preserve"/> </p> <div xml:id="echoid-div395" type="float" level="2" n="3"> <note symbol="b" position="right" xlink:label="note-0145-02" xlink:href="note-0145-02a" xml:space="preserve">32. vel <lb/>63. h.</note> <note symbol="c" position="right" xlink:label="note-0145-03" xlink:href="note-0145-03a" xml:space="preserve">5. Co-<lb/>roll. 19. h.</note> </div> </div> <div xml:id="echoid-div397" type="section" level="1" n="167"> <head xml:id="echoid-head172" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s4092" xml:space="preserve">SI ergo in figuris 29. </s> <s xml:id="echoid-s4093" xml:space="preserve">& </s> <s xml:id="echoid-s4094" xml:space="preserve">30. </s> <s xml:id="echoid-s4095" xml:space="preserve">Problematis, concipiantur methodo ſuperiùs <lb/>allata, per datum punctum inſcribi Ellipſes ad partes verticis, vel angu-<lb/>li, vel ſectionis, quæ ſimiles ſint alijs ad oppoſitas partes per idem punctum <lb/>inſcriptis (ſi tamen ſectionis, vel circuli portio, quæ ab applicata per datum <lb/>punctum terminatur, huius Ellipſis ſit capax, quod accidet, quando in ſe-<lb/>cunda præcedentium figurarum, diagonalis IL, quæ ex I ducitur diagonali <lb/>AF æquidiſtans, & </s> <s xml:id="echoid-s4096" xml:space="preserve">occurrens ſectioni in L, punctum L pertingat ad B, vel <pb o="122" file="0146" n="146" rhead=""/> cadat inter I, & </s> <s xml:id="echoid-s4097" xml:space="preserve">B; </s> <s xml:id="echoid-s4098" xml:space="preserve">tunc enim m<unsure/> portione<unsure/> IBF, per punctum E, Ellipſis alteri <lb/>GEHD ſimilis inſcribi nũquam poterit, qualis ſemper inſcribi poteſt m<unsure/> triã-<lb/>gulis IBF, MBL, &</s> <s xml:id="echoid-s4099" xml:space="preserve">c. </s> <s xml:id="echoid-s4100" xml:space="preserve">primæ figuræ; </s> <s xml:id="echoid-s4101" xml:space="preserve">quod omne, vel leuiter intuenti ſatis <lb/>patebit) illę omnino his minores erunt, cum verticibus ſint propiores; </s> <s xml:id="echoid-s4102" xml:space="preserve">& </s> <s xml:id="echoid-s4103" xml:space="preserve">ob <lb/>id, quæ ad oppoſitas partes ibi inſcribuntur, erunt quidem _MAXIMAE_ <lb/>quæſitæ.</s> <s xml:id="echoid-s4104" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div398" type="section" level="1" n="168"> <head xml:id="echoid-head173" xml:space="preserve">THEOR. XXXVII. PROP. LXXVII.</head> <p> <s xml:id="echoid-s4105" xml:space="preserve">MAXIMI circuli m<unsure/> Parabolæ inſcripti, & </s> <s xml:id="echoid-s4106" xml:space="preserve">à vertice ſucceſſiuè ſe <lb/>mutuò contingentes, ſunt inter ſe in ratione quadratorũ, diſparium <lb/>numerorum ab vnitate incipientium.</s> <s xml:id="echoid-s4107" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4108" xml:space="preserve">SIt Parabole ABC, cuius axis BH, vertex B; </s> <s xml:id="echoid-s4109" xml:space="preserve">& </s> <s xml:id="echoid-s4110" xml:space="preserve">_MAXIMI_ circuli m<unsure/> e@ in-<lb/>ſcripti, & </s> <s xml:id="echoid-s4111" xml:space="preserve">à vertice ſucceſſiuè ſe mutuò contingentes ſint, quorum dia-<lb/>metri BE, EF, FG, GH, &</s> <s xml:id="echoid-s4112" xml:space="preserve">c. </s> <s xml:id="echoid-s4113" xml:space="preserve">contactus veròſint, primi vertex B, ſecundi <lb/>punctum L, ter@ij O, quarti R, &</s> <s xml:id="echoid-s4114" xml:space="preserve">c. </s> <s xml:id="echoid-s4115" xml:space="preserve">dico huiuſmodi circulos, eſſe inter ſe, vt <lb/>quadrata numerorum diſparium ab vnitate incipientium, nempe 1. </s> <s xml:id="echoid-s4116" xml:space="preserve">9. </s> <s xml:id="echoid-s4117" xml:space="preserve">25. <lb/></s> <s xml:id="echoid-s4118" xml:space="preserve">49. </s> <s xml:id="echoid-s4119" xml:space="preserve">&</s> <s xml:id="echoid-s4120" xml:space="preserve">c.</s> <s xml:id="echoid-s4121" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4122" xml:space="preserve">Ducantur, tum ex diametrorum terminis E, F, G; </s> <s xml:id="echoid-s4123" xml:space="preserve">tum ex contactibus L, <lb/>O, R ordinatæ EI, FN, GQ, LM, OP, RS.</s> <s xml:id="echoid-s4124" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4125" xml:space="preserve">Iam cum circulus BE ſit _MAXIMVS_ in-<lb/> <anchor type="figure" xlink:label="fig-0146-01a" xlink:href="fig-0146-01"/> ſcriptibilium per verticẽ B, <anchor type="note" xlink:href="" symbol="a"/> erit BE ęqua- <anchor type="note" xlink:label="note-0146-01a" xlink:href="note-0146-01"/> lis recto lateri Parabolę, ſed EI eſt <anchor type="note" xlink:href="" symbol="b"/> media <anchor type="note" xlink:label="note-0146-02a" xlink:href="note-0146-02"/> proportionalis inter EB, & </s> <s xml:id="echoid-s4126" xml:space="preserve">rectum latus, <lb/>hoc eſt inter æquales lineas, quare EI æ-<lb/>qualis erit ipſi EB, quæ concipiatur, vt <lb/>vnum; </s> <s xml:id="echoid-s4127" xml:space="preserve">eſtque EI <anchor type="note" xlink:href="" symbol="c"/> æqualis EM, ergo EM, <anchor type="note" xlink:label="note-0146-03a" xlink:href="note-0146-03"/> eſt vt 1, & </s> <s xml:id="echoid-s4128" xml:space="preserve">tota BM, vt 2; </s> <s xml:id="echoid-s4129" xml:space="preserve">ſed eſt <anchor type="note" xlink:href="" symbol="d"/> vt BE ad <anchor type="note" xlink:label="note-0146-04a" xlink:href="note-0146-04"/> BM, ita BM ad BF, vel vt 1 ad 2, ita 2, ad <lb/>4; </s> <s xml:id="echoid-s4130" xml:space="preserve">erit ergo BF, 4: </s> <s xml:id="echoid-s4131" xml:space="preserve">eſtque BM, 2; </s> <s xml:id="echoid-s4132" xml:space="preserve">quare <lb/> <anchor type="note" xlink:label="note-0146-05a" xlink:href="note-0146-05"/> FM, ſiue <anchor type="note" xlink:href="" symbol="e"/> FN, ſiue <anchor type="note" xlink:href="" symbol="f"/> FP erit pariter 2; </s> <s xml:id="echoid-s4133" xml:space="preserve">vn- <anchor type="note" xlink:label="note-0146-06a" xlink:href="note-0146-06"/> detota BP erit 6; </s> <s xml:id="echoid-s4134" xml:space="preserve">eſtque BF ad BP, vel vt <lb/> <anchor type="note" xlink:label="note-0146-07a" xlink:href="note-0146-07"/> 4 ad 6, ita <anchor type="note" xlink:href="" symbol="g"/> BP ad BG, & </s> <s xml:id="echoid-s4135" xml:space="preserve">vt 4 ad 6, ita 6 ad 9, vnde BG erit 9, ſed eſt BP, 6, ergo GP, <lb/>ſiue GQ, vel GS erit 3; </s> <s xml:id="echoid-s4136" xml:space="preserve">quare tota BS, erit <lb/>12, ſed vt BG ad BS, vel vt 9 ad 12, ita BS <lb/>ad BH, & </s> <s xml:id="echoid-s4137" xml:space="preserve">vt 9 ad 12, ita 12 ad 16, quare <lb/>BH, erit 16. </s> <s xml:id="echoid-s4138" xml:space="preserve">Siergo dum BE eſt, vt I, BF <lb/>eſt 4, BG, 9, & </s> <s xml:id="echoid-s4139" xml:space="preserve">BH, 16; </s> <s xml:id="echoid-s4140" xml:space="preserve">ipſa BE cum ea-<lb/>rum differentijs EF, FG, GH, erunt, vt <lb/>ſunt numeri 1, 3, 5, 7, qui ſunt numeri im-<lb/>pares ab vnitate incipientes, ſed circuli <lb/>ſunt, vt quadrata ſuorum diametrorum, <lb/>ipſæque BF, EF, FG, GH ſunt inſcripto-<lb/>rum circulorum diametri, quare hi _MAXI-_ <pb o="123" file="0147" n="147" rhead=""/> _MI_ circuli, erunt, vt quadrata eorumdem numerorum diſparium ab vnita <lb/>te. </s> <s xml:id="echoid-s4141" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s4142" xml:space="preserve"/> </p> <div xml:id="echoid-div398" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0146-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0146-01" xlink:href="note-0146-01a" xml:space="preserve">1. Co-<lb/>roll. 20. h.</note> <note symbol="b" position="left" xlink:label="note-0146-02" xlink:href="note-0146-02a" xml:space="preserve">Coroll. <lb/>I. h.</note> <note symbol="c" position="left" xlink:label="note-0146-03" xlink:href="note-0146-03a" xml:space="preserve">2. Co-<lb/>roll. 75. h.</note> <note symbol="d" position="left" xlink:label="note-0146-04" xlink:href="note-0146-04a" xml:space="preserve">1. Co-<lb/>roll. 13. h.</note> <note symbol="e" position="left" xlink:label="note-0146-05" xlink:href="note-0146-05a" xml:space="preserve">2. Co-<lb/>roll. 75 h.</note> <note symbol="f" position="left" xlink:label="note-0146-06" xlink:href="note-0146-06a" xml:space="preserve">ibidem.</note> <note symbol="g" position="left" xlink:label="note-0146-07" xlink:href="note-0146-07a" xml:space="preserve">1. Co-<lb/>roll. 13. h.</note> </div> </div> <div xml:id="echoid-div400" type="section" level="1" n="169"> <head xml:id="echoid-head174" xml:space="preserve">PROBL. XXXI. PROP. LXXVIII.</head> <p> <s xml:id="echoid-s4143" xml:space="preserve">Datæ Hyperbolę, per punctum intra ipſam datum MAXIMAM <lb/>Parabolen inſcribere; </s> <s xml:id="echoid-s4144" xml:space="preserve">& </s> <s xml:id="echoid-s4145" xml:space="preserve">è contra.</s> <s xml:id="echoid-s4146" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4147" xml:space="preserve">Datæ Parabolæ, per punctum extra ipſam datum cum dato ſe-<lb/>mi- tranſuerſo latere MINIMAM Hyperbolen circumſcribere.</s> <s xml:id="echoid-s4148" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4149" xml:space="preserve">SIt data Hyperbole ABC, cuius centrum E, & </s> <s xml:id="echoid-s4150" xml:space="preserve">punctum intra ipſam da-<lb/>tum ſit G. </s> <s xml:id="echoid-s4151" xml:space="preserve">Oportet per G _MAXIMAM_ Parabolen inſcribere.</s> <s xml:id="echoid-s4152" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4153" xml:space="preserve">Iungatur EG ſecans Hyperbolen in B, & </s> <s xml:id="echoid-s4154" xml:space="preserve">concipiatur EG eſſe mediam <lb/>arithmeticam, EB verò mediam geometricam inter eaſdem ignotas extre-<lb/>mas, quæ reperiantur, <anchor type="note" xlink:href="" symbol="a"/> & </s> <s xml:id="echoid-s4155" xml:space="preserve">ſint EH, EF, & </s> <s xml:id="echoid-s4156" xml:space="preserve">per F applicetur AFC, & </s> <s xml:id="echoid-s4157" xml:space="preserve">circa <anchor type="note" xlink:label="note-0147-01a" xlink:href="note-0147-01"/> diametrum GF adſcribatur <anchor type="note" xlink:href="" symbol="b"/> ipſi Hyperbolæ ABC, Parabole DAGCM, <anchor type="note" xlink:label="note-0147-02a" xlink:href="note-0147-02"/> quarum communis applicata ſit AC. </s> <s xml:id="echoid-s4158" xml:space="preserve">Dico ipſam Parabolen eſſe quæſitam.</s> <s xml:id="echoid-s4159" xml:space="preserve"/> </p> <div xml:id="echoid-div400" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve">74. h.</note> <note symbol="b" position="right" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve">57. h.</note> </div> <p> <s xml:id="echoid-s4160" xml:space="preserve">Cum enim ſit FE ad EB, vt EB ad EH, erit <lb/> <anchor type="figure" xlink:label="fig-0147-01a" xlink:href="fig-0147-01"/> rectangulum FEH æquale quadrato EB, qua-<lb/>re AH Hyperbolen <anchor type="note" xlink:href="" symbol="c"/> continget. </s> <s xml:id="echoid-s4161" xml:space="preserve">Cumque ſit <anchor type="note" xlink:label="note-0147-03a" xlink:href="note-0147-03"/> EG media arithmetica inter FE, EH, erunt <lb/>ipſarum diſferentiæ FG, GH æquales, vnde <lb/> <anchor type="note" xlink:label="note-0147-04a" xlink:href="note-0147-04"/> eadem AH Parabolen quoque <anchor type="note" xlink:href="" symbol="d"/> continget:</s> <s xml:id="echoid-s4162" xml:space="preserve"> <anchor type="note" xlink:label="note-0147-05a" xlink:href="note-0147-05"/> quare Parabole D G M Hyperbolæ ABC <anchor type="note" xlink:href="" symbol="e"/> erit inſcripta. </s> <s xml:id="echoid-s4163" xml:space="preserve">Quod autem ſit _MAXIMA_, <lb/>patet; </s> <s xml:id="echoid-s4164" xml:space="preserve">cum quælibet alia per G adſcripta cum <lb/>recto minori, minor eſt AGC, quę verò cum <lb/>maiori, eſt quidem maior, ſed omninò ſecat <lb/>Hyperbolen ABC, cum ſectio Parabole in in-<lb/>finitum abeat, & </s> <s xml:id="echoid-s4165" xml:space="preserve">ſuperficies ABCGA vndi-<lb/>que ſit clauſa. </s> <s xml:id="echoid-s4166" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s4167" xml:space="preserve">c.</s> <s xml:id="echoid-s4168" xml:space="preserve"/> </p> <div xml:id="echoid-div401" 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/QN4GHYBF/figures/0147-01"/> </figure> <note symbol="c" position="right" xlink:label="note-0147-03" xlink:href="note-0147-03a" xml:space="preserve">conuerſ. <lb/>37. primi <lb/>conic. <lb/>Comand.</note> <note symbol="d" position="right" xlink:label="note-0147-04" xlink:href="note-0147-04a" xml:space="preserve">2. h.</note> <note symbol="e" position="right" xlink:label="note-0147-05" xlink:href="note-0147-05a" xml:space="preserve">61. h.</note> </div> <p> <s xml:id="echoid-s4169" xml:space="preserve">IAM ſit data Parabole AGC, & </s> <s xml:id="echoid-s4170" xml:space="preserve">datum extra ipſam pũctum ſit B, per quod <lb/>oporteat, cum dato quolibet ſemi-tranſuerſo D, _MINIMAM_ Hyperbo-<lb/>len circumſcribere.</s> <s xml:id="echoid-s4171" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4172" xml:space="preserve">Ducatur per B diameter Parabolæ DGF, quæ vltra B producatur, ſuma-<lb/>turque BE ipſi D æqualis, & </s> <s xml:id="echoid-s4173" xml:space="preserve">facta EB media geometrica, & </s> <s xml:id="echoid-s4174" xml:space="preserve">EG media ari-<lb/>thmetica inter eaſdem ignotas extremas, reperiantur <anchor type="note" xlink:href="" symbol="f"/> ipſæ extremæ, quæ <anchor type="note" xlink:label="note-0147-06a" xlink:href="note-0147-06"/> ſint EH, EF; </s> <s xml:id="echoid-s4175" xml:space="preserve">& </s> <s xml:id="echoid-s4176" xml:space="preserve">per F applicetur in Parabola AFC; </s> <s xml:id="echoid-s4177" xml:space="preserve">& </s> <s xml:id="echoid-s4178" xml:space="preserve">ſuper AC ad diame-<lb/>tri ſegmentum BF, cum ſemi- tranſuerſo BE deſcribatur <anchor type="note" xlink:href="" symbol="g"/> Hyperbole ABC.</s> <s xml:id="echoid-s4179" xml:space="preserve"> <anchor type="note" xlink:label="note-0147-07a" xlink:href="note-0147-07"/> Dico ipſam eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s4180" xml:space="preserve"/> </p> <div xml:id="echoid-div402" type="float" level="2" n="3"> <note symbol="f" position="right" xlink:label="note-0147-06" xlink:href="note-0147-06a" xml:space="preserve">74. h.</note> <note symbol="g" position="right" xlink:label="note-0147-07" xlink:href="note-0147-07a" xml:space="preserve">57. h.</note> </div> <p> <s xml:id="echoid-s4181" xml:space="preserve">Si quidem iuncta HA, ijſdem omnino argumentis, ac ſupra, demonſtra-<lb/>bitur ipſam HA, & </s> <s xml:id="echoid-s4182" xml:space="preserve">Parabolen, & </s> <s xml:id="echoid-s4183" xml:space="preserve">Hyperbolen contingere, vnde ſectiones <lb/>ſe mutuò contingent, <anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s4184" xml:space="preserve">Hyperbole ABC erit Parabolæ circumſcripta:</s> <s xml:id="echoid-s4185" xml:space="preserve"> <anchor type="note" xlink:label="note-0147-08a" xlink:href="note-0147-08"/> eritque _MINIMA_; </s> <s xml:id="echoid-s4186" xml:space="preserve">nam quælibet alia Hyperbole per B adſcripta cum eo-<lb/>dem tranſuerſo, ſed cum recto maiori, maior eſt ipſa ABC, quæ verò cum <lb/>minori, eſt quidem minor, ſed cum ipſi ABC ſit inſcripta, & </s> <s xml:id="echoid-s4187" xml:space="preserve">ad partes verti- <pb o="124" file="0148" n="148" rhead=""/> cioppoſitas, ſit infinitæ extenſionis, ſecaret omnino Parabolen AGC, vt <lb/>per ſe patet: </s> <s xml:id="echoid-s4188" xml:space="preserve">quapropter Hyperbole ABC erit _MINIMA_ circumſcripta quę-<lb/>ſita. </s> <s xml:id="echoid-s4189" xml:space="preserve">Quod ſecundò faciendum, ac demonſtrandum erat.</s> <s xml:id="echoid-s4190" xml:space="preserve"/> </p> <div xml:id="echoid-div403" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-0147-08" xlink:href="note-0147-08a" xml:space="preserve">61. h.</note> </div> </div> <div xml:id="echoid-div405" type="section" level="1" n="170"> <head xml:id="echoid-head175" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s4191" xml:space="preserve">HAEC de MAXIMARVM, & </s> <s xml:id="echoid-s4192" xml:space="preserve">MINIMARVM coni- ſe-<lb/>ctionum, circuli, & </s> <s xml:id="echoid-s4193" xml:space="preserve">anguli reciproca inſcriptione, ac circumſcri-<lb/>ptione, per punctum in ipſis, vel intra, vel extra datum, iuxta <lb/>ſæpius memoratam definitionem, hactenus pertractaſſe ſuffi-<lb/>iat, quæ ſi grata vobis fuiſſe perceperimus, multa his ſimilia, & </s> <s xml:id="echoid-s4194" xml:space="preserve">alia <lb/>quàm plurima ad aliud tempus proferemus. </s> <s xml:id="echoid-s4195" xml:space="preserve">Cæterum, in proximè ſequenti-<lb/>bus, quæ ad vberiorem doctrinam, & </s> <s xml:id="echoid-s4196" xml:space="preserve">alteri præſertim huius operis parti ma-<lb/>xime conducunt, hac omiſſa definitione, inſcriptio, & </s> <s xml:id="echoid-s4197" xml:space="preserve">circumſcriptio aliter fiet, <lb/>prout in ipſis propoſitionibus exponetur.</s> <s xml:id="echoid-s4198" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div406" type="section" level="1" n="171"> <head xml:id="echoid-head176" xml:space="preserve">LEMMA XI. PROP. LXXIX.</head> <p> <s xml:id="echoid-s4199" xml:space="preserve">Si recta AB ſecta fuerit in C, & </s> <s xml:id="echoid-s4200" xml:space="preserve">in D, ita vt AB ad BC, ſit vt <lb/>AD ad DC: </s> <s xml:id="echoid-s4201" xml:space="preserve">Dico ſi BD bifariam ſecetur in E, punctum E cadere <lb/>inter B, & </s> <s xml:id="echoid-s4202" xml:space="preserve">C, & </s> <s xml:id="echoid-s4203" xml:space="preserve">rectangulum AEC, æquari quadrato ED.</s> <s xml:id="echoid-s4204" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4205" xml:space="preserve">CVm ſit enim AB ad BC, vt AD ad DC, erit permu-<lb/> <anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/> tando BA ad AD, vt BC ad CD, ſed eſt BA ma-<lb/>ior AD, quare BC erit maior CD: </s> <s xml:id="echoid-s4206" xml:space="preserve">ex quo punctum E <lb/>bifariam ſecans BD cadit inter B, & </s> <s xml:id="echoid-s4207" xml:space="preserve">C.</s> <s xml:id="echoid-s4208" xml:space="preserve"/> </p> <div xml:id="echoid-div406" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0148-01"/> </figure> </div> <p> <s xml:id="echoid-s4209" xml:space="preserve">Ampliùs producatur BA ad F, & </s> <s xml:id="echoid-s4210" xml:space="preserve">ſecetur AF æqualis <lb/>AD.</s> <s xml:id="echoid-s4211" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4212" xml:space="preserve">Iam cum demonſtratum ſit eſſe BA ad AD, vt BC ad <lb/>CD, erit BA ad AF, vt BC ad CD; </s> <s xml:id="echoid-s4213" xml:space="preserve">& </s> <s xml:id="echoid-s4214" xml:space="preserve">componendo BF <lb/>ad FA, vt BD ad DC; </s> <s xml:id="echoid-s4215" xml:space="preserve">& </s> <s xml:id="echoid-s4216" xml:space="preserve">ſumptis antecedentium dimi-<lb/>dijs, EA ad AF, ſiue ad AD, vt ED ad DC, & </s> <s xml:id="echoid-s4217" xml:space="preserve">per con-<lb/>uerſionem rationis, AE ad ED, vt DE ad EC, vnde re-<lb/>ctãgulum AEC ęquabitur quadrato ED. </s> <s xml:id="echoid-s4218" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s4219" xml:space="preserve">c.</s> <s xml:id="echoid-s4220" xml:space="preserve"/> </p> <pb o="125" file="0149" n="149" rhead=""/> </div> <div xml:id="echoid-div408" type="section" level="1" n="172"> <head xml:id="echoid-head177" xml:space="preserve">LEMMA XII. PROP. LXXX.</head> <p> <s xml:id="echoid-s4221" xml:space="preserve">Si fuerit rectangulum ABC, ęquale rectangulo DEF, ſitque AC <lb/>maior DF, erit BC minor EF.</s> <s xml:id="echoid-s4222" xml:space="preserve"/> </p> <figure> <image file="0149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0149-01"/> </figure> <p> <s xml:id="echoid-s4223" xml:space="preserve">SI enim dicatur BC æqualem, vel maiorem <lb/>eſſe EF, cum ſit data AC maior DF, eſſet <lb/>omnino AB maior DE, & </s> <s xml:id="echoid-s4224" xml:space="preserve">BC dicitur æqualis, <lb/>vel maior EF, ergo rectangulum ABC eſſet om-<lb/>nino maius rectangulo DEF, ſed æquale poſi-<lb/>tum fuit. </s> <s xml:id="echoid-s4225" xml:space="preserve">Ergo patet propoſitum.</s> <s xml:id="echoid-s4226" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div409" type="section" level="1" n="173"> <head xml:id="echoid-head178" xml:space="preserve">THEOR. XXXVIII. PROP. LXXXI.</head> <p> <s xml:id="echoid-s4227" xml:space="preserve">Si recta linea ad alterũ terminũ cuiuſdam applicatæ coni-ſectio-<lb/>nem, vel circulũ cõtingat, ipſa omnino ſecabit ſibi adſcriptã eiuſdẽ <lb/>nominis ſectionẽ circa eandẽ applicatam, & </s> <s xml:id="echoid-s4228" xml:space="preserve">cum æquali tranſuerſa <lb/>diametro, ſi ſectiones fuerint Hyperbolæ, vel Ellipſes, aut circuli.</s> <s xml:id="echoid-s4229" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4230" xml:space="preserve">ESto quæcunque coni-ſectio, vel circulus ABC, cuius diameter BD, yo<unsure/> <lb/>vna applicatarum ſit AC. </s> <s xml:id="echoid-s4231" xml:space="preserve">& </s> <s xml:id="echoid-s4232" xml:space="preserve">ad ipſius terminum A, ſit recta contingens <lb/>EAF, quæ diametro occurret <anchor type="note" xlink:href="" symbol="a"/> in F; </s> <s xml:id="echoid-s4233" xml:space="preserve">& </s> <s xml:id="echoid-s4234" xml:space="preserve">ſumpto in diametri ſegmento BD <anchor type="note" xlink:label="note-0149-01a" xlink:href="note-0149-01"/> quolibet puncto G, cum diametro GD, & </s> <s xml:id="echoid-s4235" xml:space="preserve">applicata AC in qualibet figura, <lb/> <anchor type="figure" xlink:label="fig-0149-02a" xlink:href="fig-0149-02"/> ſed etiam, pro Hyperbola in ſecunda, & </s> <s xml:id="echoid-s4236" xml:space="preserve">pro Ellipſi, vel circulo, in tertia, <lb/>cum dato tranſuerſo latere GM, quod æquale ſit tranſuerſo BL datæ ſectio-<lb/>nis, deſcribatur <anchor type="note" xlink:href="" symbol="b"/> eiuſdem nomin@s ſectio AGC: </s> <s xml:id="echoid-s4237" xml:space="preserve">dico hanc omnino ſecarià <anchor type="note" xlink:label="note-0149-02a" xlink:href="note-0149-02"/> contingente EAF.</s> <s xml:id="echoid-s4238" xml:space="preserve"/> </p> <div xml:id="echoid-div409" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0149-01" xlink:href="note-0149-01a" xml:space="preserve">24. 25. <lb/>pr. conic.</note> <figure xlink:label="fig-0149-02" xlink:href="fig-0149-02a"> <image file="0149-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0149-02"/> </figure> <note symbol="b" position="right" xlink:label="note-0149-02" xlink:href="note-0149-02a" xml:space="preserve">57. h.</note> </div> <p> <s xml:id="echoid-s4239" xml:space="preserve">Ducatur enim per A recta IAH, quæ ſectionem continge<unsure/>t AGC, cum <lb/>eius diametro conueniat in H.</s> <s xml:id="echoid-s4240" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4241" xml:space="preserve">Iam in prima figura exhiben@ Parabolas, cum AF contingat ſectionem <pb o="126" file="0150" n="150" rhead=""/> ABC, erit DB æqualis <anchor type="note" xlink:href="" symbol="a"/> BF; </s> <s xml:id="echoid-s4242" xml:space="preserve">cumque AH contingat AGC erit DG <anchor type="note" xlink:href="" symbol="b"/> æqualis <anchor type="note" xlink:label="note-0150-01a" xlink:href="note-0150-01"/> GH, ſed eſt DB maior DG ex conſtructione, quare, & </s> <s xml:id="echoid-s4243" xml:space="preserve">BF erit maior GH, & </s> <s xml:id="echoid-s4244" xml:space="preserve"><lb/>GF eò maior GH: </s> <s xml:id="echoid-s4245" xml:space="preserve">quod memento.</s> <s xml:id="echoid-s4246" xml:space="preserve"/> </p> <div xml:id="echoid-div410" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0150-01" xlink:href="note-0150-01a" xml:space="preserve">35. pri. <lb/>conic.</note> </div> <note symbol="b" position="left" xml:space="preserve">ibidem.</note> <p> <s xml:id="echoid-s4247" xml:space="preserve">Præterea, in ſecunda figura, cum ſit LB, æqualis MG, & </s> <s xml:id="echoid-s4248" xml:space="preserve">BD maior GD, <lb/>ex conſtructione, habebit LB ad BD minorem rationem, quàm MG ad GD, <lb/>& </s> <s xml:id="echoid-s4249" xml:space="preserve">componendo LD ad DB, ſiue <anchor type="note" xlink:href="" symbol="c"/> LF ad FB minorem quàm MD ad DG, <anchor type="note" xlink:label="note-0150-03a" xlink:href="note-0150-03"/> ſiue <anchor type="note" xlink:href="" symbol="d"/> quàm MH ad HG, & </s> <s xml:id="echoid-s4250" xml:space="preserve">iterum componendo LB ad BF, minorem quàm <anchor type="note" xlink:label="note-0150-04a" xlink:href="note-0150-04"/> MG ad GH, ſed eſt LB æqualis MG, quare BF erit maior GH, & </s> <s xml:id="echoid-s4251" xml:space="preserve">eò magis <lb/>GF maior GH.</s> <s xml:id="echoid-s4252" xml:space="preserve"/> </p> <div xml:id="echoid-div411" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0150-03" xlink:href="note-0150-03a" xml:space="preserve">36. pri-<lb/>mi conic.</note> <note symbol="d" position="left" xlink:label="note-0150-04" xlink:href="note-0150-04a" xml:space="preserve">ibidem.</note> </div> <figure> <image file="0150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0150-01"/> </figure> <p> <s xml:id="echoid-s4253" xml:space="preserve">Intertia denique, cum ſit LB æqualis MG, & </s> <s xml:id="echoid-s4254" xml:space="preserve">DB maior DG, ex conſtru-<lb/>ctione, habebit LB ad BD minorem rationem, quàm MG ad GD, & </s> <s xml:id="echoid-s4255" xml:space="preserve">diui-<lb/>dendo LD ad DB ſiue <anchor type="note" xlink:href="" symbol="e"/> LF ad FB, minorem quàm MD ad DG, vel <anchor type="note" xlink:href="" symbol="f"/> quàm <anchor type="note" xlink:label="note-0150-05a" xlink:href="note-0150-05"/> MH ad HG, & </s> <s xml:id="echoid-s4256" xml:space="preserve">diuidendo iterum, LB ad BF minorem rationem, quàm MG <lb/> <anchor type="note" xlink:label="note-0150-06a" xlink:href="note-0150-06"/> ad GH, ſed eſt LB æqualis MG, ergo BF maior erit GH, & </s> <s xml:id="echoid-s4257" xml:space="preserve">eò magis GF <lb/>maior GH.</s> <s xml:id="echoid-s4258" xml:space="preserve"/> </p> <div xml:id="echoid-div412" type="float" level="2" n="4"> <note symbol="e" position="left" xlink:label="note-0150-05" xlink:href="note-0150-05a" xml:space="preserve">36. pri-<lb/>mi conic.</note> <note symbol="f" position="left" xlink:label="note-0150-06" xlink:href="note-0150-06a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s4259" xml:space="preserve">Itaque cum demonſtratum ſit in qualibet figura eſſe GF maiorem GH, <lb/>punctum H incidet infra F; </s> <s xml:id="echoid-s4260" xml:space="preserve">ſed HAI contingit ſectionem AGC in A, qua-<lb/>re FA, quæ contingit ABC, ſi producatur ad partes E, ſecabit ipſam ſectio-<lb/>nem ABC, cum inter ſectionem, & </s> <s xml:id="echoid-s4261" xml:space="preserve">contingentem, ex puncto contactus al-<lb/>tera recta linea non <anchor type="note" xlink:href="" symbol="g"/> cadat: </s> <s xml:id="echoid-s4262" xml:space="preserve">quare, &</s> <s xml:id="echoid-s4263" xml:space="preserve">c. </s> <s xml:id="echoid-s4264" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s4265" xml:space="preserve">c.</s> <s xml:id="echoid-s4266" xml:space="preserve"/> </p> <note symbol="g" position="left" xml:space="preserve">32. pri-<lb/>mi conic.</note> </div> <div xml:id="echoid-div414" type="section" level="1" n="174"> <head xml:id="echoid-head179" xml:space="preserve">PROBL. XXXII. PROP. LXXXII.</head> <p> <s xml:id="echoid-s4267" xml:space="preserve">Dato angulo rectilineo, vel coni-ſectione<unsure/>, vel circulo, perter-<lb/>minos, cuiuſcunque in ipſo applicatæ, MAXIMAM Ellipſim in-<lb/>ſcribere, cuius tranſuerſum latus æquale ſit dato.</s> <s xml:id="echoid-s4268" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4269" xml:space="preserve">Oportet autem, ſi data ſectio ſuerit Ellipſis, datum tranſuerſum <lb/>minus eſſe diametro datæ Ellipſis, ad quam data applicata ordi-<lb/>natim ducitur.</s> <s xml:id="echoid-s4270" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4271" xml:space="preserve">ESto ABC datus angulus, vt in prima figura, vel Parabole, vt in ſecunda; <lb/></s> <s xml:id="echoid-s4272" xml:space="preserve">vel Hyperbole, vt in tertia; </s> <s xml:id="echoid-s4273" xml:space="preserve">vel tandem Ellipſis, aut circulus, vt in <pb o="127" file="0151" n="151" rhead=""/> quarta, & </s> <s xml:id="echoid-s4274" xml:space="preserve">in ipſis concipiatur quædam AC ad diametrum BF ordinatim du-<lb/>cta; </s> <s xml:id="echoid-s4275" xml:space="preserve">oportet per eius terminos A, C, dato angulo, velſectioni, _MAXIMAM_ <lb/>Ellipſim inſcribere, cuius tranſuerſa diameter æqualis ſit datæ lineæ DE, <lb/>quæ tamen, pro Ellipſi ABCO, quartæ figuræ, minor ſit eius tranſuerſa dia-<lb/>metro BO.</s> <s xml:id="echoid-s4276" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4277" xml:space="preserve">Ducatur <anchor type="note" xlink:href="" symbol="a"/> ex A, ſectionem <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> <anchor type="note" xlink:label="note-0151-01a" xlink:href="note-0151-01"/> ABC contingens AK, quæ dia-<lb/>metro occurret <anchor type="note" xlink:href="" symbol="b"/> in K, & </s> <s xml:id="echoid-s4278" xml:space="preserve">KF, in <anchor type="note" xlink:label="note-0151-02a" xlink:href="note-0151-02"/> angulo etiam rectilineo, bifariam <lb/>ſecetur in puncto G, quod in Pa-<lb/>rabola cadet in ipſō B, cum (ob <lb/>tangentem AK) ſit KB <anchor type="note" xlink:href="" symbol="c"/> æqualis <anchor type="note" xlink:label="note-0151-03a" xlink:href="note-0151-03"/> BF, & </s> <s xml:id="echoid-s4279" xml:space="preserve">in Hyperbola cadet infra <lb/>B, cum ſit FB maior BK (ſumpta <lb/>enim eius tranſuerſa diametro <lb/> <anchor type="note" xlink:label="note-0151-04a" xlink:href="note-0151-04"/> BO, eſt OF ad FB, <anchor type="note" xlink:href="" symbol="d"/> vt OK ad KB, & </s> <s xml:id="echoid-s4280" xml:space="preserve">permutando OF ad OK, <lb/>vt FB ad BK, ſed eſt OF maior <lb/>OK, quare, & </s> <s xml:id="echoid-s4281" xml:space="preserve">FB erit maior BK) <lb/>in Ellipſi verò cadet ſupra B, cũ <lb/>ſit KB maior BF (nam eſt OK ad <lb/> <anchor type="note" xlink:label="note-0151-05a" xlink:href="note-0151-05"/> KB, <anchor type="note" xlink:href="" symbol="e"/> vt OF ad FB, & </s> <s xml:id="echoid-s4282" xml:space="preserve">KF bifa- riam ſecta eſt in G, ac ideo G ca-<lb/>det ſupra B.) </s> <s xml:id="echoid-s4283" xml:space="preserve">Præterea ad datam rectam DE applicetur parallelogrammum <lb/>æquale quadrato GF, excedens figura quadrata, idque ſit rectangulum <lb/>DHE; </s> <s xml:id="echoid-s4284" xml:space="preserve">ſumptaque HI media proportionali inter DH, HE, erit rectangulum <lb/>DHE, ſiue quadratum GF, æquale quadrato HI, ergo rectæ GF, HI æqua-<lb/>les inter ſe. </s> <s xml:id="echoid-s4285" xml:space="preserve">Inſuper ſumatur GL æqualis HE, & </s> <s xml:id="echoid-s4286" xml:space="preserve">erit reliqua LF æqualis re-<lb/>liquæ EI; </s> <s xml:id="echoid-s4287" xml:space="preserve">& </s> <s xml:id="echoid-s4288" xml:space="preserve">punctum L cadet omnino infra B, ſiue intra angulum, vel ſe-<lb/>ctionem, cum in angulo, & </s> <s xml:id="echoid-s4289" xml:space="preserve">Hyperbola cadat infra G, quod eſt intra angu-<lb/>lum, vel ſectionem, & </s> <s xml:id="echoid-s4290" xml:space="preserve">in Parabola cadat infra G, quod eſt in ipſa ſectione; <lb/></s> <s xml:id="echoid-s4291" xml:space="preserve">in Ellipſi verò, prædictum punctum L cadet infra B; </s> <s xml:id="echoid-s4292" xml:space="preserve">quoniam cum ſit OK <lb/>ad KB, vt OF ad FB, & </s> <s xml:id="echoid-s4293" xml:space="preserve">KF bifariam ſecta in G, per conſtructionem, erit re-<lb/>ctangulum OGB <anchor type="note" xlink:href="" symbol="f"/> æquale quadrato GF, (hic notatione dignum videtur, <anchor type="note" xlink:label="note-0151-06a" xlink:href="note-0151-06"/> hanc ipſam affectionem verificari etiam in Hyperbola, nempe rectangulo <lb/>OGB æquari quadrato GF, vel GK) ſiue quadrato HI, ſiue rectangulo <lb/>DHE; </s> <s xml:id="echoid-s4294" xml:space="preserve">ſed eſt OB maior DE, quare GB erit <anchor type="note" xlink:href="" symbol="g"/> minor HE, ſiue minor GL, <anchor type="note" xlink:label="note-0151-07a" xlink:href="note-0151-07"/> hoc eſt punctum L erit quoque intra Ellipſim A B C O. </s> <s xml:id="echoid-s4295" xml:space="preserve">Sumatur præte-<lb/>rea in quacumque figura FN æqualis ID, erit ergo LN æqualis datæ ED <lb/>(cum ſit quoque LF æqualis EI) & </s> <s xml:id="echoid-s4296" xml:space="preserve">punctum N in quarta figura cadet omni-<lb/>no intra Eilipſim ABCO: </s> <s xml:id="echoid-s4297" xml:space="preserve">quoniam cum ſit rectangulum DHE, ſiue NGL, <lb/>æquale quadrato HI, ſiue GE, & </s> <s xml:id="echoid-s4298" xml:space="preserve">ſit etiam rectangulum OGB æquale eidem <lb/>quadrato GF, vt ſuperiùs demonſtrauimus, erunt rectangulo<unsure/> OGB, NGL <lb/>inter ſe æquale<unsure/>@, & </s> <s xml:id="echoid-s4299" xml:space="preserve">ideo, vt OG ad GN, ita LG ad GB, ſed eſt LG maior <lb/>GB, vt paulò ante oſtendimus, quapropter, & </s> <s xml:id="echoid-s4300" xml:space="preserve">OG erit maior GN, ſiue pun-<lb/>ctum N cadet intra Ellipſim ABCO.</s> <s xml:id="echoid-s4301" xml:space="preserve"/> </p> <div xml:id="echoid-div414" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0151-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0151-01" xlink:href="note-0151-01a" xml:space="preserve">2.4. h.</note> <note symbol="b" position="right" xlink:label="note-0151-02" xlink:href="note-0151-02a" xml:space="preserve">24. 25. <lb/>pr. conic.</note> <note symbol="c" position="right" xlink:label="note-0151-03" xlink:href="note-0151-03a" xml:space="preserve">35. pri-<lb/>mi conic.</note> <note symbol="d" position="right" xlink:label="note-0151-04" xlink:href="note-0151-04a" xml:space="preserve">36. pri-<lb/>mi conic.</note> <note symbol="e" position="right" xlink:label="note-0151-05" xlink:href="note-0151-05a" xml:space="preserve">ibidem.</note> <note symbol="f" position="right" xlink:label="note-0151-06" xlink:href="note-0151-06a" xml:space="preserve">79. h.</note> <note symbol="g" position="right" xlink:label="note-0151-07" xlink:href="note-0151-07a" xml:space="preserve">80. h.</note> </div> <p> <s xml:id="echoid-s4302" xml:space="preserve">Tandem cum trãſuerſo LN, quod æquatur datę lineæ ED, circa applica- <pb o="128" file="0152" n="152" rhead=""/> tam AC deſcribatur <anchor type="note" xlink:href="" symbol="*"/> Ellipſis ALCN. </s> <s xml:id="echoid-s4303" xml:space="preserve">Dico hanc eſſe _MAXIMAM_ inſcriptã <anchor type="note" xlink:label="note-0152-01a" xlink:href="note-0152-01"/> quæſitam.</s> <s xml:id="echoid-s4304" xml:space="preserve"/> </p> <div xml:id="echoid-div415" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0152-01" xlink:href="note-0152-01a" xml:space="preserve">Coroll. <lb/>57. h.</note> </div> <p> <s xml:id="echoid-s4305" xml:space="preserve">Nam, in qualibet figura, cum <lb/>ſit rectangulum DHE, ſiue NGL <lb/> <anchor type="figure" xlink:label="fig-0152-01a" xlink:href="fig-0152-01"/> æquale quadrato HI, ſiue GF, <lb/>erit NG ad GF, vt GF ad GL, & </s> <s xml:id="echoid-s4306" xml:space="preserve"><lb/>componendo NG cum GF, hoc <lb/>eſt NK, erit ad GF, vt FG cum <lb/>GL, ſiue vt KL ad GL, & </s> <s xml:id="echoid-s4307" xml:space="preserve">permu-<lb/>tando NK ad KL, vt GF ad GL, <lb/>vel vt NG ad GF, vel vt <anchor type="note" xlink:href="" symbol="a"/> NF ad <anchor type="note" xlink:label="note-0152-02a" xlink:href="note-0152-02"/> FL, quæ ſunt differentiæ, trium <lb/>proportionalium NG, GF, GL; <lb/></s> <s xml:id="echoid-s4308" xml:space="preserve">ergo recta KAM tanget <anchor type="note" xlink:href="" symbol="b"/> Elli- <anchor type="note" xlink:label="note-0152-03a" xlink:href="note-0152-03"/> pſim ALCN, ſiue hæc angulo <lb/>ABC erit inſcripta, ſed in alijs fi-<lb/>guris, ipſa KAM tangit quoque <lb/>datã ei ſimul adſcriptam ſectio-<lb/>nem ABC, (ex conſtructione) <lb/>adūdem terminum cõmunis ap-<lb/>plicatæ AC, quapropter Ellipſis ALCN datæ ſectioni, vel circulo <anchor type="note" xlink:href="" symbol="c"/> erit <anchor type="note" xlink:label="note-0152-04a" xlink:href="note-0152-04"/> inſcripta.</s> <s xml:id="echoid-s4309" xml:space="preserve"/> </p> <div xml:id="echoid-div416" type="float" level="2" n="3"> <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a"> <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0152-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0152-02" xlink:href="note-0152-02a" xml:space="preserve">Coroll. <lb/>12. h.</note> <note symbol="b" position="left" xlink:label="note-0152-03" xlink:href="note-0152-03a" xml:space="preserve">4. h.</note> <note symbol="c" position="left" xlink:label="note-0152-04" xlink:href="note-0152-04a" xml:space="preserve">61. h.</note> </div> <p> <s xml:id="echoid-s4310" xml:space="preserve">Dico demum hanc eſſe _MAXIMAM_: </s> <s xml:id="echoid-s4311" xml:space="preserve">quoniam quæ ipſi adſcribitur per <lb/>eoſdem terminos applicatæ AC, & </s> <s xml:id="echoid-s4312" xml:space="preserve">cum tranſuerſa diametro æquali ipſi LN, <lb/>_licet minor fuerit eadem ALCN_, ſecat <anchor type="note" xlink:href="" symbol="d"/> contingentem KAM, in A, atque aliò <anchor type="note" xlink:label="note-0152-05a" xlink:href="note-0152-05"/> ad partes AK, ſi nempe vertex nouiter adſcriptæ cadat ſupra L; </s> <s xml:id="echoid-s4313" xml:space="preserve">vel ad par-<lb/>tes AM, ſi cadat infra, vt ex ipſa 81. </s> <s xml:id="echoid-s4314" xml:space="preserve">huius facilè elicitur: </s> <s xml:id="echoid-s4315" xml:space="preserve">cum ergo inouiter <lb/>adſcripta Ellipſis ſecet contingentem KAM, in ſe ipſam rediens, ſecabit om-<lb/>nino datam ſectionem ABC, quare Ellipſis ALCN, eſt _MAXIMA_ dato an-<lb/>gulo, vel ſectioni ABC inſcripta, circa datam applicatam, & </s> <s xml:id="echoid-s4316" xml:space="preserve">cum data <lb/>tranſuerſa diametro DE, immo potiùs ipſa ALCN eſt vnica huiuſmodi con-<lb/>ditionibus inſcriptibilius. </s> <s xml:id="echoid-s4317" xml:space="preserve">Quod faciendum, & </s> <s xml:id="echoid-s4318" xml:space="preserve">demonſtrandum erat.</s> <s xml:id="echoid-s4319" xml:space="preserve"/> </p> <div xml:id="echoid-div417" type="float" level="2" n="4"> <note symbol="d" position="left" xlink:label="note-0152-05" xlink:href="note-0152-05a" xml:space="preserve">81. h.</note> </div> </div> <div xml:id="echoid-div419" type="section" level="1" n="175"> <head xml:id="echoid-head180" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s4320" xml:space="preserve">EX hac conſtat interceptum diametri ſegmentum inter quamlibet appli-<lb/>tam, & </s> <s xml:id="echoid-s4321" xml:space="preserve">verticem, æquale eſſe (in Parabola) intercepto ſegmento eiuſ-<lb/>dem diametri inter verticem, & </s> <s xml:id="echoid-s4322" xml:space="preserve">occurſum contingentis, ductæ ex termino <lb/>applicatæ, cum diametro: </s> <s xml:id="echoid-s4323" xml:space="preserve">in Hyperbola, verò maius, ſed in Ellipſi, vel cir-<lb/>culo minus eſſe.</s> <s xml:id="echoid-s4324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4325" xml:space="preserve">Demonſtratum eſt enim, in ſecunda figura, KB æ qualem eſſe BF, in ter-<lb/>tia verò KB, minorem BF, & </s> <s xml:id="echoid-s4326" xml:space="preserve">in quarta KB maiorem BF.</s> <s xml:id="echoid-s4327" xml:space="preserve"/> </p> <pb o="129" file="0153" n="153" rhead=""/> </div> <div xml:id="echoid-div420" type="section" level="1" n="176"> <head xml:id="echoid-head181" xml:space="preserve">THEOR. XXXIX. PROP. LXXXIII.</head> <p> <s xml:id="echoid-s4328" xml:space="preserve">Si binarum Ellipſium ſimul adſcriptarum altera alteri fuerit in-<lb/>ſcripta, & </s> <s xml:id="echoid-s4329" xml:space="preserve">per terminos communis applicatæ ſe mutuò contingant; <lb/></s> <s xml:id="echoid-s4330" xml:space="preserve">quælibet alia Ellipſis datis adſcripta, cum eadem applicata, & </s> <s xml:id="echoid-s4331" xml:space="preserve"><lb/>cum æquali tranſuerſo latere, inſcriptam Ellipſim omnino ſecabit.</s> <s xml:id="echoid-s4332" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4333" xml:space="preserve">SInt duæ Ellipſes ABCD, AECF ſimul adſcriptæ, circa communem ap-<lb/>plicatam AC, ſitque ipſarum altera, nempe AECF, alteri inſcripta, ita <lb/>vt in extremis tantùm A, C, ſe mutuò contingant: </s> <s xml:id="echoid-s4334" xml:space="preserve">dico, ſi his alia adſcriba-<lb/>tur Ellipſis ALCI, circa eandem applicatam AC, & </s> <s xml:id="echoid-s4335" xml:space="preserve">cum tranſuerſo LI, <lb/>quod æquale ſit ipſo BD tranſuerſo circumſcriptæ, ipſam ALCI omnino ſe-<lb/>care inſcriptam AECF.</s> <s xml:id="echoid-s4336" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4337" xml:space="preserve">Nam ſi alterum extremum tranſuerſæ <lb/> <anchor type="figure" xlink:label="fig-0153-01a" xlink:href="fig-0153-01"/> diametri LI, quale eſt punctum L, cadat <lb/>intra inſcriptam, vt inter E, & </s> <s xml:id="echoid-s4338" xml:space="preserve">O, tunc aliud <lb/>extremum I neceſſariò cadet extra, infra F, <lb/>cum ſit LI, ſiue BD maior EF, ex quo ma-<lb/>nifeſtè patet, Ellipſim ALCI ſecare inſcri-<lb/>ptam AECF.</s> <s xml:id="echoid-s4339" xml:space="preserve"/> </p> <div xml:id="echoid-div420" type="float" level="2" n="1"> <figure xlink:label="fig-0153-01" xlink:href="fig-0153-01a"> <image file="0153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0153-01"/> </figure> </div> <p> <s xml:id="echoid-s4340" xml:space="preserve">Si verò vtrunque extremum L, I, cadit <lb/>extra inſcriptam, vti exhibetur ab hac figu-<lb/>ra; </s> <s xml:id="echoid-s4341" xml:space="preserve">tunc ducta AG, quæ circumſcriptam <lb/>ABCD contingat in A, ipſa, vtrinque pro-<lb/>ducta, ad alteram partium ſecabit <anchor type="note" xlink:href="" symbol="a"/> omni- <anchor type="note" xlink:label="note-0153-01a" xlink:href="note-0153-01"/> no ALCI; </s> <s xml:id="echoid-s4342" xml:space="preserve">vnde, quæ ex A cõtingit ALCI, <lb/>diuerſa erit ab AG, & </s> <s xml:id="echoid-s4343" xml:space="preserve">ſit ipſa AM.</s> <s xml:id="echoid-s4344" xml:space="preserve"/> </p> <div xml:id="echoid-div421" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0153-01" xlink:href="note-0153-01a" xml:space="preserve">81. h.</note> </div> <p> <s xml:id="echoid-s4345" xml:space="preserve">Iam ſi ALCI non ſecat inſcriptã AECF; <lb/></s> <s xml:id="echoid-s4346" xml:space="preserve">contingat in A, C, ſi poſſibile eſt. </s> <s xml:id="echoid-s4347" xml:space="preserve">Cum ergo recta AM contingat Ellipſim <lb/>ALCI, atque hæc contingat inſcriptam Ellipſim AECF, eadem recta AM, <lb/>in A quoque continget AECF, ſed etiam AG eandem AECF contingit in <lb/>A: </s> <s xml:id="echoid-s4348" xml:space="preserve">quare ex eodem puncto A ductæ erunt binæ rectę lineę eandem Ellipſim <lb/>contingentes; </s> <s xml:id="echoid-s4349" xml:space="preserve">quod eſt <anchor type="note" xlink:href="" symbol="b"/> impoſſibile. </s> <s xml:id="echoid-s4350" xml:space="preserve">Non igitur Ellipſis ALCI contingit <anchor type="note" xlink:label="note-0153-02a" xlink:href="note-0153-02"/> inſcriptam AECF, quapropter in occurſibus A, C neceſſariò eam ſecabit, <lb/>Quod oſtendere propoſitum fuit. </s> <s xml:id="echoid-s4351" xml:space="preserve">Sed hoc idem</s> </p> <div xml:id="echoid-div422" type="float" level="2" n="3"> <note symbol="b" position="right" xlink:label="note-0153-02" xlink:href="note-0153-02a" xml:space="preserve">ex 32. <lb/>pr. conic.</note> </div> </div> <div xml:id="echoid-div424" type="section" level="1" n="177"> <head xml:id="echoid-head182" xml:space="preserve">ALITER affirmatiuè.</head> <p> <s xml:id="echoid-s4352" xml:space="preserve">CVm recta AG contingat ad A circumſcriptam Ellipſim ABCD, atque <lb/>hæc ad idem A contingat inſcriptam AECF, ipſa AG omninò con-<lb/>tinget ad A inſcriptam AECF; </s> <s xml:id="echoid-s4353" xml:space="preserve">ſed MA (quę vt ſupra oſtendimus, diuerſa eſt <lb/>à GA) hanc ſecat in A; </s> <s xml:id="echoid-s4354" xml:space="preserve">quare MA producta, ad alteram partium omninò <lb/>ſecabit inſcriptam AECF, & </s> <s xml:id="echoid-s4355" xml:space="preserve">eò magis Ellipſis ALCI, quam contingit ad A <lb/>recta MA, ad eandem partem ſecabit inſcriptam AECF. </s> <s xml:id="echoid-s4356" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s4357" xml:space="preserve">c.</s> <s xml:id="echoid-s4358" xml:space="preserve"/> </p> <pb o="130" file="0154" n="154" rhead=""/> </div> <div xml:id="echoid-div425" type="section" level="1" n="178"> <head xml:id="echoid-head183" xml:space="preserve">PROBL. XXXIII. PROP. LXXXIV.</head> <p> <s xml:id="echoid-s4359" xml:space="preserve">Datæ Ellipſi, vel circulo, per terminos cuiuſcunque in ipſo ap-<lb/>plicatę MINIMAM Ellipſim circumſcribere, cuius tranſuerſum la-<lb/>tus æquale ſit dato, quod tamen maius ſit tranſuerſa diametro datæ <lb/>Ellipſis.</s> <s xml:id="echoid-s4360" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4361" xml:space="preserve">SIt data Ellipſis, vel circulus ABCO, cuius tranſuerſa diameter BO, & </s> <s xml:id="echoid-s4362" xml:space="preserve"><lb/>quædam ad eam applicata AC: </s> <s xml:id="echoid-s4363" xml:space="preserve">oportet per terminos A, C, cum tranſ-<lb/>uerſo DE, quod excedat BO _MINIMAM_ Ellipſim circumſcribere.</s> <s xml:id="echoid-s4364" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4365" xml:space="preserve">Ducatur ex A contingens AK productæ <lb/> <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> diametro occurrens in K, & </s> <s xml:id="echoid-s4366" xml:space="preserve">KF bifariam <lb/>ſecetur in puncto G, quod cadet iuter B, & </s> <s xml:id="echoid-s4367" xml:space="preserve"><lb/>K, vt in 83. </s> <s xml:id="echoid-s4368" xml:space="preserve">h. </s> <s xml:id="echoid-s4369" xml:space="preserve">oſtenſum fuit; </s> <s xml:id="echoid-s4370" xml:space="preserve">& </s> <s xml:id="echoid-s4371" xml:space="preserve">ad datam <lb/>lineam DE applicetur parallelogrammum, <lb/>æquale quadrato GF, excedens figura <lb/>quadrata, ſitque rectangulum DHE, & </s> <s xml:id="echoid-s4372" xml:space="preserve"><lb/>ſumpta HI media proportionali inter DH, <lb/>HE, erit rectangulum DHI, ſiue quadra-<lb/>tum GF, ęquale quadrato HI, hoc eſt linea <lb/>GF æqualis HI: </s> <s xml:id="echoid-s4373" xml:space="preserve">ſumpta ergo GL æquali <lb/>HE, erit reliqua LF æqualis EI, & </s> <s xml:id="echoid-s4374" xml:space="preserve">pũctum <lb/>L cadet extra B: </s> <s xml:id="echoid-s4375" xml:space="preserve">quoniam cum ſit OK ad <lb/>KB, <anchor type="note" xlink:href="" symbol="a"/> vt AF ad FB, ſitque KF bifariam ſe- <anchor type="note" xlink:label="note-0154-01a" xlink:href="note-0154-01"/> cta in G, erit rectangulum OGB æquale <anchor type="note" xlink:href="" symbol="b"/> quadrato GF, ſiue quadrato HI, <anchor type="note" xlink:label="note-0154-02a" xlink:href="note-0154-02"/> ſiue rectangulo DHE; </s> <s xml:id="echoid-s4376" xml:space="preserve">ſed eſt OB minor DE, ex conſtructione, quare GB <lb/>erit maior <anchor type="note" xlink:href="" symbol="c"/> HE, ſiue maior GL; </s> <s xml:id="echoid-s4377" xml:space="preserve">itaque punctum L cadet extra Ellipſim AB <anchor type="note" xlink:label="note-0154-03a" xlink:href="note-0154-03"/> CO. </s> <s xml:id="echoid-s4378" xml:space="preserve">Sumatur ampliùs FN æqualis ID, & </s> <s xml:id="echoid-s4379" xml:space="preserve">erit tota LN æqualis datæ ED, <lb/>itemque punctum N cadet extra Ellipſim ABCO: </s> <s xml:id="echoid-s4380" xml:space="preserve">Nam cum ſit rectangulum <lb/>DHE, ſiue NGL æquale quadrato HI, ſiue GF, ſitque rectangulum OGB, <lb/>æquale eidem quadrato GF, vt ſupra oſtendimus, erunt rectangula OGB, <lb/>NGL inter ſe æqualia, & </s> <s xml:id="echoid-s4381" xml:space="preserve">ideo vt OG ad GN, ita LG ad GB, ſed eſt LG mi-<lb/>nor GB, vt ſuperiùs demonſtrauimus, vnde, & </s> <s xml:id="echoid-s4382" xml:space="preserve">OG minor erit GN, nempe <lb/>punctum N cadet extra Ellipſim ABCO. </s> <s xml:id="echoid-s4383" xml:space="preserve">Poſtremò cum tranſuerſo latere <lb/>NL, quod æquale eſt datæ lineæ DE, circa applicatam AC <anchor type="note" xlink:href="" symbol="d"/> deſcribatur <anchor type="note" xlink:label="note-0154-04a" xlink:href="note-0154-04"/> Ellipſis ALCN. </s> <s xml:id="echoid-s4384" xml:space="preserve">Dico hanc eſſe _MINIMAM_ circumſcriptam quæſitam.</s> <s xml:id="echoid-s4385" xml:space="preserve"/> </p> <div xml:id="echoid-div425" type="float" level="2" n="1"> <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a"> <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0154-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0154-01" xlink:href="note-0154-01a" xml:space="preserve">36. pri-<lb/>mi conic.</note> <note symbol="b" position="left" xlink:label="note-0154-02" xlink:href="note-0154-02a" xml:space="preserve">79. h.</note> <note symbol="c" position="left" xlink:label="note-0154-03" xlink:href="note-0154-03a" xml:space="preserve">80. h.</note> <note symbol="d" position="left" xlink:label="note-0154-04" xlink:href="note-0154-04a" xml:space="preserve">Coroll. <lb/>57. h.</note> </div> <p> <s xml:id="echoid-s4386" xml:space="preserve">Quoniam cum ſit rectangulum NGL æquale quadrato GF, erit NG ad <lb/>GF, vt GF ad GL, & </s> <s xml:id="echoid-s4387" xml:space="preserve">componendo, NG cum GF, ſiue NK, erit ad GF, vt <lb/>FG cum GL, ſiue vt KL ad GL, & </s> <s xml:id="echoid-s4388" xml:space="preserve">permutando NK ad KL, vt GF ad GL, <lb/>vel vt NG ad GF, vel vt <anchor type="note" xlink:href="" symbol="e"/> NF ad FL, ergo recta KAM Ellipſim ALCN con- <anchor type="note" xlink:label="note-0154-05a" xlink:href="note-0154-05"/> tingit <anchor type="note" xlink:href="" symbol="f"/> in A, ſed eadem KAM contingit quoque ad idem punctum A Elli- <anchor type="note" xlink:label="note-0154-06a" xlink:href="note-0154-06"/> pſim ABCO: </s> <s xml:id="echoid-s4389" xml:space="preserve">quapropter Ellipſis ALCN datæ ABCO erit <anchor type="note" xlink:href="" symbol="g"/> circumſcripta.</s> <s xml:id="echoid-s4390" xml:space="preserve"> <anchor type="note" xlink:label="note-0154-07a" xlink:href="note-0154-07"/> At ipſa erit _MINIMA_: </s> <s xml:id="echoid-s4391" xml:space="preserve">nam quælibet alia, quæ ipſi adſcribitur per eoſdem <lb/>terminos communis applicatæ AC, & </s> <s xml:id="echoid-s4392" xml:space="preserve">cum tranſuerſa diametro æquali ipſi <lb/>LN, _licet maior ſuerit eadem ALCN_, inſcriptam ABCO omnino <anchor type="note" xlink:href="" symbol="b"/> ſecat; </s> <s xml:id="echoid-s4393" xml:space="preserve">ergo <anchor type="note" xlink:label="note-0154-08a" xlink:href="note-0154-08"/> <pb o="131" file="0155" n="155" rhead=""/> ALCO eſt _MINIMA_ circumſcripta datæ Ellipſi ABCO, per terminos ap-<lb/>plicatæ AC, cum dato tranſuerſo DE: </s> <s xml:id="echoid-s4394" xml:space="preserve">immo ipſa ALCN vnica eſt, his con-<lb/>ditionibus circumſcriptibilis. </s> <s xml:id="echoid-s4395" xml:space="preserve">Quod faciendum, & </s> <s xml:id="echoid-s4396" xml:space="preserve">demonſtrandum erat.</s> <s xml:id="echoid-s4397" xml:space="preserve"/> </p> <div xml:id="echoid-div426" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-0154-05" xlink:href="note-0154-05a" xml:space="preserve">Coroll. <lb/>12. h.</note> <note symbol="f" position="left" xlink:label="note-0154-06" xlink:href="note-0154-06a" xml:space="preserve">4. h.</note> <note symbol="g" position="left" xlink:label="note-0154-07" xlink:href="note-0154-07a" xml:space="preserve">61. h.</note> <note symbol="b" position="left" xlink:label="note-0154-08" xlink:href="note-0154-08a" xml:space="preserve">83. h.</note> </div> </div> <div xml:id="echoid-div428" type="section" level="1" n="179"> <head xml:id="echoid-head184" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s4398" xml:space="preserve">SIquæratur, qua nam ratione in prop. </s> <s xml:id="echoid-s4399" xml:space="preserve">82. </s> <s xml:id="echoid-s4400" xml:space="preserve">ad finem, dicatur _licet minor fue-_ <lb/>_rit eadem ALCN_ in hac verò, _licet maior fuerit eadem ALCN_ (perinde ac <lb/>ſi, per terminos A, C, cum diametro æquali ipſi LN alia in ea deſcribi poſſit <lb/>Ellipſis minor ALCN, in hac verò alia maior ALCN) vtrunq; </s> <s xml:id="echoid-s4401" xml:space="preserve">noshaud te-<lb/>merè dixiſſe ex ſequéti Theoremate manifeſtum fiet, à quo habebitur quam-<lb/>libet aliam Ellipſim per A, C, adſcriptam, cum tranſuerſo ęquali ipſi LN, ſed <lb/>cuius ſegmenta ab applicata AC abſciſſa, ſint magis inæqualia quàm ſint ſe-<lb/>gmenta NF, FL, minorem eſſe ipſa ALCN; </s> <s xml:id="echoid-s4402" xml:space="preserve">& </s> <s xml:id="echoid-s4403" xml:space="preserve">è contra, eam quę cum ſegmentis <lb/>minus inæqualibus, quàm ſint NF, FL, eadem ALCN maiorem eſſe.</s> <s xml:id="echoid-s4404" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div429" type="section" level="1" n="180"> <head xml:id="echoid-head185" xml:space="preserve">THEOR. XL. PROP. LXXXV.</head> <p> <s xml:id="echoid-s4405" xml:space="preserve">Ellipſium, perterminos communis applicatæ ſimul adſcripta-<lb/>rum, & </s> <s xml:id="echoid-s4406" xml:space="preserve">quarum tranſuerſa latera ſint æqualia, MINIMA eſt ea, <lb/>cuius communis ordinatim ducta ſit diameter coniugata: </s> <s xml:id="echoid-s4407" xml:space="preserve">aliarum <lb/>verò illa, cuius ſegmenta diametri ſunt minùs inæqualia, minor eſt <lb/>ea, cuius diametri ſegmenta ſunt magis inæqualia.</s> <s xml:id="echoid-s4408" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4409" xml:space="preserve">SInt duę Ellipſes ABCD, AECF, per terminos eiuſdem applicatæ AC <lb/>ſimul adſcriptæ, & </s> <s xml:id="echoid-s4410" xml:space="preserve">quarum tranſuerſa BD, EF ſint æqualia, ſitq; </s> <s xml:id="echoid-s4411" xml:space="preserve">AGC <lb/>coniugata diameter Ellipſis ABCD, ſiue G eius centrum. </s> <s xml:id="echoid-s4412" xml:space="preserve">Dico primùm <lb/>hanc minorem eſſe altera AECF, ſiue eſſe _MINIMAM_, &</s> <s xml:id="echoid-s4413" xml:space="preserve">c.</s> <s xml:id="echoid-s4414" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4415" xml:space="preserve">Etenim, cum ſit DB æqualis EF, & </s> <s xml:id="echoid-s4416" xml:space="preserve">DB <lb/> <anchor type="figure" xlink:label="fig-0155-01a" xlink:href="fig-0155-01"/> bifariam ſecta in G, erit EF in pũcto Ginæ-<lb/>qualiter ſecta, vnde rectangulum BGD ma-<lb/>ius erit rectangulo EGF, cum ſit <anchor type="note" xlink:href="" symbol="a"/> _MAXI-_ <anchor type="note" xlink:label="note-0155-01a" xlink:href="note-0155-01"/> _MVM_; </s> <s xml:id="echoid-s4417" xml:space="preserve">ideoque rectangulum BGD ad qua-<lb/>dratum AG, ſiue tranſuerſum <anchor type="note" xlink:href="" symbol="b"/> BD ad re- <anchor type="note" xlink:label="note-0155-02a" xlink:href="note-0155-02"/> ctum Ellipſis ABCD, maiorem habebit ra-<lb/>tionem quàm rectangulum EGF ad idem <lb/>quadratum AG, ſiue quàm <anchor type="note" xlink:href="" symbol="c"/> tranſuerſum <anchor type="note" xlink:label="note-0155-03a" xlink:href="note-0155-03"/> EF ad rectum Ellipſis AECF: </s> <s xml:id="echoid-s4418" xml:space="preserve">ſed tranſuerſa <lb/>BD, EF ſunt æqualia, ergo rectũ Ellipſis AB <lb/>CD, minus erit recto AECF:</s> <s xml:id="echoid-s4419" xml:space="preserve">ſi igitur Ellipſis <lb/>huiuſmodi Ellipſes (cum ſint ęqualiter incli-<lb/>natæ) concipiantur eſſe per eundem verticem ſimul adſcriptæ, ita vt tranſ-<lb/>uerſæ diametri ſimul congruant, ipſa ABCD, cuius rectum minus eſt, inſcri-<lb/>pta erit, <anchor type="note" xlink:href="" symbol="d"/> ſiue minor AECF, cuius rectum maius eſt, & </s> <s xml:id="echoid-s4420" xml:space="preserve">ſic minor quacũque <anchor type="note" xlink:label="note-0155-04a" xlink:href="note-0155-04"/> alia, cuius diametri ſegmenta ſint inæqualia: </s> <s xml:id="echoid-s4421" xml:space="preserve">quare ABCD erit _MINI-_ <lb/>_MA_, &</s> <s xml:id="echoid-s4422" xml:space="preserve">c.</s> <s xml:id="echoid-s4423" xml:space="preserve"/> </p> <div xml:id="echoid-div429" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0155-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0155-01" xlink:href="note-0155-01a" xml:space="preserve">60. h.</note> <note symbol="b" position="right" xlink:label="note-0155-02" xlink:href="note-0155-02a" xml:space="preserve">21. pri-<lb/>mi conic.</note> <note symbol="c" position="right" xlink:label="note-0155-03" xlink:href="note-0155-03a" xml:space="preserve">ibidem.</note> <note symbol="d" position="right" xlink:label="note-0155-04" xlink:href="note-0155-04a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> </div> <pb o="132" file="0156" n="156" rhead=""/> <p> <s xml:id="echoid-s4424" xml:space="preserve">Inſuper, ſit alia adſcripta Ellipſis AHCI, cuius ſegmenta diametri HG, <lb/>GI ſint adhuc magis inæqualia, quàm ſegmenta EG, GF: </s> <s xml:id="echoid-s4425" xml:space="preserve">dico AECF mi-<lb/>rcm eſſe Ellipſi AHCI. </s> <s xml:id="echoid-s4426" xml:space="preserve">Oſtendetur enim, vt ſupra, rectangulum EGF ma-<lb/>ius eſſe rectangulo HGI, & </s> <s xml:id="echoid-s4427" xml:space="preserve">rectum latus Ellipſis AECF, minus eſſe recto <lb/>AHCI, ſiue Ellipſim AECF inſcribi poſſe AHCI, hoc eſt ipſa minorem eſſe. <lb/></s> <s xml:id="echoid-s4428" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s4429" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div431" type="section" level="1" n="181"> <head xml:id="echoid-head186" xml:space="preserve">THEOR. XLI. PROP. LXXXVI.</head> <p> <s xml:id="echoid-s4430" xml:space="preserve">MAXIMA ſemi-diametrorum, à centro Ellipſeos eductarum, <lb/>eſt ſemi-axis maior, MINIMA verò ſemi-axis minor: </s> <s xml:id="echoid-s4431" xml:space="preserve">aliarum <lb/>autem, quæ cum MAXIMA minorem conſtituit angulum maior <lb/>eſt: </s> <s xml:id="echoid-s4432" xml:space="preserve">& </s> <s xml:id="echoid-s4433" xml:space="preserve">quatuor ſunt in Ellipſi æquales ſemi-diametri, quarum vna <lb/>tantùm cadit in vnoquoque Ellipſis quadrante, genito ex axium <lb/>interſectione.</s> <s xml:id="echoid-s4434" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4435" xml:space="preserve">SIt Ellipſis ABCD, cuius axis maior BD; </s> <s xml:id="echoid-s4436" xml:space="preserve">minor AC, centrum E. </s> <s xml:id="echoid-s4437" xml:space="preserve">Dico <lb/>primùm maiorem ſemi-axim EB eſſe omnium ſemi-diametrorum _MA-_ <lb/>_XIMAM_, & </s> <s xml:id="echoid-s4438" xml:space="preserve">ſemi-axim minorem EA omnium _MINIMAM_.</s> <s xml:id="echoid-s4439" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4440" xml:space="preserve">Cum centro enim E, & </s> <s xml:id="echoid-s4441" xml:space="preserve">interuallo EB <lb/> <anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/> deſcripto circulo BHD, ipſæ cadit totus <lb/>extra Ellipſim, cum eiſit <anchor type="note" xlink:href="" symbol="a"/> circumſcriptus, <anchor type="note" xlink:label="note-0156-01a" xlink:href="note-0156-01"/> vnde ſemi-diameter EB erit _MAXIMA_; <lb/></s> <s xml:id="echoid-s4442" xml:space="preserve">& </s> <s xml:id="echoid-s4443" xml:space="preserve">facto cétro E, cum radio EA deſcripto <lb/>circulo EOC, ipſæ totus cadet intra Elli-<lb/>pſim, cum ei ſit <anchor type="note" xlink:href="" symbol="b"/> inſcriptus, ex quo, E A <anchor type="note" xlink:label="note-0156-02a" xlink:href="note-0156-02"/> erit _MINIMA_. </s> <s xml:id="echoid-s4444" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s4445" xml:space="preserve">c.</s> <s xml:id="echoid-s4446" xml:space="preserve"/> </p> <div xml:id="echoid-div431" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0156-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0156-01" xlink:href="note-0156-01a" xml:space="preserve">ex26. h.</note> <note symbol="b" position="left" xlink:label="note-0156-02" xlink:href="note-0156-02a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s4447" xml:space="preserve">Ampliùs in quadrãte Ellipſeos AFCE <lb/>ductæ ſint quotcunque ſemi-diametri EF, <lb/>EG, & </s> <s xml:id="echoid-s4448" xml:space="preserve">ſit angulus BEF minor BEG: </s> <s xml:id="echoid-s4449" xml:space="preserve">dico <lb/>EF maiorem eſſe EG.</s> <s xml:id="echoid-s4450" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4451" xml:space="preserve">Applicentur enim per F, G, ad maio-<lb/>rem axim BE rectæ KF, LG, quæ produ-<lb/>ctæ, circuli peripheriæ BIH occurrant in I, <lb/>M, & </s> <s xml:id="echoid-s4452" xml:space="preserve">iungantur E I, EM.</s> <s xml:id="echoid-s4453" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4454" xml:space="preserve">Erit in ſemi-circulo BHD, quadratum ML ad IK, vt rectangulum DLB <lb/>ad DKB; </s> <s xml:id="echoid-s4455" xml:space="preserve">& </s> <s xml:id="echoid-s4456" xml:space="preserve">in ſemi-Ellipſi BCD, quadratum GL ad FK, <anchor type="note" xlink:href="" symbol="c"/> vt idem rectan- <anchor type="note" xlink:label="note-0156-03a" xlink:href="note-0156-03"/> gulum DLB ad idem DKB, ergo quadratum ML ad IK, erit vt quadratum <lb/>GL ad FK, ſiue linea ML ad IK, vt pars GL ad partem FK, & </s> <s xml:id="echoid-s4457" xml:space="preserve">vt reliqua MG <lb/>ad reliquam IF, ſed eſt GL <anchor type="note" xlink:href="" symbol="d"/> maior FK: </s> <s xml:id="echoid-s4458" xml:space="preserve">quare MG erit maior I F, ideoque <anchor type="note" xlink:label="note-0156-04a" xlink:href="note-0156-04"/> rectangulum MGL ſub maioribus lateribus contentum, maius erit rectan-<lb/>gulo IFK ſub minoribus, & </s> <s xml:id="echoid-s4459" xml:space="preserve">duplum vnius, alterius duplo maius.</s> <s xml:id="echoid-s4460" xml:space="preserve"/> </p> <div xml:id="echoid-div432" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-0156-03" xlink:href="note-0156-03a" xml:space="preserve">2 I. pri-<lb/>mi conic.</note> <note symbol="d" position="left" xlink:label="note-0156-04" xlink:href="note-0156-04a" xml:space="preserve">63. h.</note> </div> <p> <s xml:id="echoid-s4461" xml:space="preserve">Iam cum triangula EKI, ELM ſint rectangula ad K, L, erunt triangula <lb/>EFI, EGM obtuſiangula ad F, G, eſtque linea E I æqualis EM, ergo qua-<lb/>dradratum E I, hoc eſt duo ſimul quadrata EF, F I, cum duplo rectanguli <lb/>KEI, maiora erunt quadrato EM, ſiue duobus ſimul quadratis EG, GM, <pb o="133" file="0157" n="157" rhead=""/> cum duplo rectanguli LGM, ſed duplum rectanguli K F I, maius eſt duplo <lb/>rectanguli LGM, vt ſuperiùs oſtenſum fuit; </s> <s xml:id="echoid-s4462" xml:space="preserve">quare his demptis, erunt reli-<lb/>qua quadrata EF, FI ſimul, maiora reliquis ſimul EG, GM, ſed quadratum <lb/>FI minus eſt quadrato GM, cum ſit linea FI minor GM, ergo reliquum qua-<lb/>dratum EF maius erit reliquo EG, ſiue ſemi-diameter EF maior ſemi-dia-<lb/>metro EG. </s> <s xml:id="echoid-s4463" xml:space="preserve">Quod ſecundò erat, &</s> <s xml:id="echoid-s4464" xml:space="preserve">c.</s> <s xml:id="echoid-s4465" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4466" xml:space="preserve">Dico tandem, in vno quoque Ellipſeos rectangulo quadrante, nempe <lb/>in quadrante ABE, reperiri aliam ſemi-diametrum ipſi EG æqualem.</s> <s xml:id="echoid-s4467" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4468" xml:space="preserve">Producta enim applicata GL ad N, & </s> <s xml:id="echoid-s4469" xml:space="preserve">iuncta EN, in triangulis ELG, <lb/>ELN, eſt NL æqualis LG, & </s> <s xml:id="echoid-s4470" xml:space="preserve">LE communis, & </s> <s xml:id="echoid-s4471" xml:space="preserve">anguli ad L recti, quare ba-<lb/>ſes EG, GN æquales erunt, & </s> <s xml:id="echoid-s4472" xml:space="preserve">ſic in quolibet alio quadrante; </s> <s xml:id="echoid-s4473" xml:space="preserve">& </s> <s xml:id="echoid-s4474" xml:space="preserve">vnaquequę <lb/>ſemi-diametrorum vnica eſt in eodem quadrante, cum quę ad partes maio-<lb/>ris axis ducuntur, maiores ſint, & </s> <s xml:id="echoid-s4475" xml:space="preserve">quæ ad partes minoris, minores: </s> <s xml:id="echoid-s4476" xml:space="preserve">quapro-<lb/>pter à centro Ellipſeos quatuor tantùm (in rectangulis quadrantibus inter ſe-<lb/>mi-axes) ſemi-diametri æquales ad eius peripheriam duci poterunt. </s> <s xml:id="echoid-s4477" xml:space="preserve">Quod <lb/>vltimò demonſtrandum erat.</s> <s xml:id="echoid-s4478" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div434" type="section" level="1" n="182"> <head xml:id="echoid-head187" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s4479" xml:space="preserve">HInc eſt, quod _MAXIMA_ diametrorum, in Ellipſi, eſt axis maior, _MINI-_ <lb/>_MA_ verò axis minor; </s> <s xml:id="echoid-s4480" xml:space="preserve">eadem enim eſt ratio de duplis, ac de ſubduplis.</s> <s xml:id="echoid-s4481" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div435" type="section" level="1" n="183"> <head xml:id="echoid-head188" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s4482" xml:space="preserve">PAtet etiam, quod ſi ex Ellipſeos centro ad interuallum cuiuſcunque ſe-<lb/>mi-diametri, quæ non ſit axis, circulus deſcribatur, ipſum ad partes <lb/>maioris axis cadere intra, ad partes verò minoris cadere extra, & </s> <s xml:id="echoid-s4483" xml:space="preserve">in quatuor <lb/>tantùm punctis Ellipſim ſecare.</s> <s xml:id="echoid-s4484" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div436" type="section" level="1" n="184"> <head xml:id="echoid-head189" xml:space="preserve">THEOR. XLII. PROP. LXXXVII.</head> <p> <s xml:id="echoid-s4485" xml:space="preserve">Si ad extremum axis datæ coni-ſectionis ducta fuerit contingens <lb/>linea, quæ cum alia ad alterum ſectionis punctum contingente <lb/>conueniat; </s> <s xml:id="echoid-s4486" xml:space="preserve">ſemper ea, quæ inter occurſum, & </s> <s xml:id="echoid-s4487" xml:space="preserve">axem intercipitur <lb/>(qui tamen in ſectione Ellipſis, ſit axis maior) minor eſt altera con-<lb/>tingente: </s> <s xml:id="echoid-s4488" xml:space="preserve">& </s> <s xml:id="echoid-s4489" xml:space="preserve">in Ellipſi tantùm, contingens ex minori axe altera <lb/>contingente maior eſt.</s> <s xml:id="echoid-s4490" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4491" xml:space="preserve">SIt coni-ſectio AB, cuius axis BC (quitamen in Ellipſi ſit axis maior) & </s> <s xml:id="echoid-s4492" xml:space="preserve">in <lb/>Hyperbola, ac Ellipſi ſit centrum D, ſitque ex vertice B contingens li-<lb/>nea BE; </s> <s xml:id="echoid-s4493" xml:space="preserve">ſumptoque in ſectione quolibet alio puncto A (quod tamen in Elli-<lb/>pſi non ſit alterum axis extremum; </s> <s xml:id="echoid-s4494" xml:space="preserve">nam ipſæ contingentes, per 27. </s> <s xml:id="echoid-s4495" xml:space="preserve">ſecundi <lb/>conic. </s> <s xml:id="echoid-s4496" xml:space="preserve">inter ſe æquidiſtarent) ab eo ducatur contingens AE <anchor type="note" xlink:href="" symbol="a"/> quæ quidem <anchor type="note" xlink:label="note-0157-01a" xlink:href="note-0157-01"/> cum BE conueniet in E. </s> <s xml:id="echoid-s4497" xml:space="preserve">Dico tangentem BE ipſa AE minorem eſſe.</s> <s xml:id="echoid-s4498" xml:space="preserve"/> </p> <div xml:id="echoid-div436" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0157-01" xlink:href="note-0157-01a" xml:space="preserve">58. h.</note> </div> <p> <s xml:id="echoid-s4499" xml:space="preserve">Ducatur per occurſum E diameter DEF, iungaturque AB. </s> <s xml:id="echoid-s4500" xml:space="preserve">Patet ipſam <lb/> <anchor type="note" xlink:label="note-0157-02a" xlink:href="note-0157-02"/> diametrum, cumtranſeat per occurſum tangentium, ſecare AB tactus iun-<lb/>gentem <anchor type="note" xlink:href="" symbol="b"/> biſariam in F.</s> <s xml:id="echoid-s4501" xml:space="preserve"/> </p> <div xml:id="echoid-div437" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0157-02" xlink:href="note-0157-02a" xml:space="preserve">30. fecú-<lb/>di conic.</note> </div> <pb o="134" file="0158" n="158" rhead=""/> <p> <s xml:id="echoid-s4502" xml:space="preserve">Iam in Parabola, quam exhibet prima huius ſchematis ſigura, cum ſint <lb/>BC, EF diametri <anchor type="note" xlink:href="" symbol="a"/> ipſæ erunt inter ſe parallelæ, BA verò eas ſecat, quare <anchor type="note" xlink:label="note-0158-01a" xlink:href="note-0158-01"/> angulus GBF æquatur angulo EFA, ſed eſt GBF obtuſus, cum GBE ſit re-<lb/>ctus (nam eſt CB axis Parabolæ) ergo angulus quoque EFA obtuſus erit, <lb/>ſiue maior conſequenti BFE.</s> <s xml:id="echoid-s4503" xml:space="preserve"/> </p> <div xml:id="echoid-div438" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-0158-01" xlink:href="note-0158-01a" xml:space="preserve">cõuerſ. <lb/>46. pr. co-<lb/>nic.</note> </div> <p> <s xml:id="echoid-s4504" xml:space="preserve">In Hyperbola verò ſecundæ ſiguræ, cum angulus CBA externus triangu-<lb/>li DBF ſit acutus, (nam CBE rectus eſt) ſitque maior interno BFE, is quidem <lb/>acutus erit, & </s> <s xml:id="echoid-s4505" xml:space="preserve">qui ei deinceps EFA erit obtuſus, ſiue maior ipſo BFE.</s> <s xml:id="echoid-s4506" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4507" xml:space="preserve">In Ellipſi tandem tertiæ ſiguræ iuncta DA, cum in trianguiis DFB, DFA <lb/>ſit BF ęqualis AF, & </s> <s xml:id="echoid-s4508" xml:space="preserve">communis FD <anchor type="note" xlink:href="" symbol="b"/> baſis verò BD maior DA, erit angulus <anchor type="note" xlink:label="note-0158-02a" xlink:href="note-0158-02"/> BFD maior angulo DFA, & </s> <s xml:id="echoid-s4509" xml:space="preserve">eiad verticẽ EFA maior angulo ad verticẽ BFE.</s> <s xml:id="echoid-s4510" xml:space="preserve"/> </p> <div xml:id="echoid-div439" type="float" level="2" n="4"> <note symbol="b" position="left" xlink:label="note-0158-02" xlink:href="note-0158-02a" xml:space="preserve">86. h.</note> </div> <p> <s xml:id="echoid-s4511" xml:space="preserve">In triangulis itaq; </s> <s xml:id="echoid-s4512" xml:space="preserve">AFE, BFE, <lb/> <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> cuiuslibet harum ſigurarum, cum <lb/>ſit latus A F æqualis FB, & </s> <s xml:id="echoid-s4513" xml:space="preserve">FE <lb/>commune, augulus verò E F A <lb/>demonſtratus ſit maior angulo <lb/>BFE, erit baſis A F maior baſi <lb/>BE. </s> <s xml:id="echoid-s4514" xml:space="preserve">Quare contingens B E ex <lb/>termino maioris axis, minor eſt <lb/>altera contingente A E. </s> <s xml:id="echoid-s4515" xml:space="preserve">Quod <lb/>primò probandum erat.</s> <s xml:id="echoid-s4516" xml:space="preserve"/> </p> <div xml:id="echoid-div440" type="float" level="2" n="5"> <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/QN4GHYBF/figures/0158-01"/> </figure> </div> <p> <s xml:id="echoid-s4517" xml:space="preserve">Si verò, in Ellipſi ABC, quar-<lb/>tæ ſiguræ, axis BC fuerit minor. <lb/></s> <s xml:id="echoid-s4518" xml:space="preserve">Poſitis, & </s> <s xml:id="echoid-s4519" xml:space="preserve">conſtructis ijſdem. </s> <s xml:id="echoid-s4520" xml:space="preserve">Cum in triangulis AFD, BFD ſit latus AF æ-<lb/>qualle lateri BF, & </s> <s xml:id="echoid-s4521" xml:space="preserve">commune FD, baſis verò AD maior baſi DB (cum minor <lb/>ſemi-axis DB ſit _MINIMA_ <anchor type="note" xlink:href="" symbol="c"/> ſemi-diametrorum) erit angulus AFD, ſiue BFE <anchor type="note" xlink:label="note-0158-03a" xlink:href="note-0158-03"/> maior angulo BFD, hoc eſt AFE, ſuntque in triangulis BFE, AFE latera BF, <lb/>AF <anchor type="note" xlink:href="" symbol="d"/> inter ſe æqualia, & </s> <s xml:id="echoid-s4522" xml:space="preserve">latus FE commune: </s> <s xml:id="echoid-s4523" xml:space="preserve">quare baſis BE, erit maior baſi <anchor type="note" xlink:label="note-0158-04a" xlink:href="note-0158-04"/> AE Quod ſuit vltimò demonſtrandum.</s> <s xml:id="echoid-s4524" xml:space="preserve"/> </p> <div xml:id="echoid-div441" type="float" level="2" n="6"> <note symbol="c" position="left" xlink:label="note-0158-03" xlink:href="note-0158-03a" xml:space="preserve">ibidem.</note> <note symbol="d" position="left" xlink:label="note-0158-04" xlink:href="note-0158-04a" xml:space="preserve">30. ſec. <lb/>conic.</note> </div> </div> <div xml:id="echoid-div443" type="section" level="1" n="185"> <head xml:id="echoid-head190" xml:space="preserve">THEOR. XLIII. PROP. LXXXVIII.</head> <p> <s xml:id="echoid-s4525" xml:space="preserve">Si coni-ſectionem recta linea contingens cum axe conueniat, & </s> <s xml:id="echoid-s4526" xml:space="preserve"><lb/>à tactu erigatur contingenti perpendicularis, hæc neceſſariò cum <lb/>axe conueniet, in Ellipſi cum vtroque axe, ſed priùs cum maiori; <lb/></s> <s xml:id="echoid-s4527" xml:space="preserve">parſque ipſius intercepta inter contactum, & </s> <s xml:id="echoid-s4528" xml:space="preserve">occurſum cum axe, <lb/>qui tamen in Ellipſi ſit axis maior, ſemper minor erit eo axis ſe-<lb/>gmento, quod inter occurſum, & </s> <s xml:id="echoid-s4529" xml:space="preserve">verticem intercipitur.</s> <s xml:id="echoid-s4530" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4531" xml:space="preserve">Cum autem in Ellipſi, contingens linea minori axi occurret, <lb/>tunc prædicta perpendicularis inter contactum, & </s> <s xml:id="echoid-s4532" xml:space="preserve">minorem axem <lb/>intercepta, maior ſemper erit ſegmento minoris axis, quod inter <lb/>occurſum, & </s> <s xml:id="echoid-s4533" xml:space="preserve">verticem intercipitur.</s> <s xml:id="echoid-s4534" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4535" xml:space="preserve">SIt coni-ſectio ABC, cuius axis BD, & </s> <s xml:id="echoid-s4536" xml:space="preserve">prima ſigura Parabolen, aut Hy-<lb/>perbolen repræſentet, ſecunda verò Ellipſim, cuius axis maior, ſit BS, <pb o="135" file="0159" n="159" rhead=""/> & </s> <s xml:id="echoid-s4537" xml:space="preserve">ex puncto A in ſectione extra verticem ſumpto ipſam <anchor type="note" xlink:href="" symbol="a"/> contingat recta <anchor type="note" xlink:label="note-0159-01a" xlink:href="note-0159-01"/> AE, quæ cum axe SB, <anchor type="note" xlink:href="" symbol="b"/> conueniet, & </s> <s xml:id="echoid-s4538" xml:space="preserve">in Ellipſi cum vtraque axe SB, TH;</s> <s xml:id="echoid-s4539" xml:space="preserve"> <anchor type="note" xlink:label="note-0159-02a" xlink:href="note-0159-02"/> ſintque occurſus E, L, & </s> <s xml:id="echoid-s4540" xml:space="preserve">à contactu A erigatur ipſi perpendicularis AD. <lb/></s> <s xml:id="echoid-s4541" xml:space="preserve">Dico primùm hanc cum axe conuenire, & </s> <s xml:id="echoid-s4542" xml:space="preserve">in Ellipſi cum vtraque axe, ſed <lb/>priùs cum maiori.</s> <s xml:id="echoid-s4543" xml:space="preserve"/> </p> <div xml:id="echoid-div443" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0159-01" xlink:href="note-0159-01a" xml:space="preserve">2.4. h.</note> <note symbol="b" position="right" xlink:label="note-0159-02" xlink:href="note-0159-02a" xml:space="preserve">24. 25. <lb/>pr. conic.</note> </div> <p> <s xml:id="echoid-s4544" xml:space="preserve">Ducatur ex A recta <lb/> <anchor type="figure" xlink:label="fig-0159-01a" xlink:href="fig-0159-01"/> AF axi ordinatim appli-<lb/>cata, quæ cum axe re-<lb/>ctum angulum AFE cõ-<lb/>ſtituet, ac ideo angulus <lb/>AEF acutus erit, ſed eſt <lb/>rectus EAD, quare AD <lb/>conuenit cum EBD, vt-<lb/>puta in D. </s> <s xml:id="echoid-s4545" xml:space="preserve">Eadem ra-<lb/>tione in Ellipſi demon-<lb/>ſtrabitur ipſam AD con-<lb/>uenire quoque cum mi-<lb/>nori axe HT, ſi ex A or-<lb/>dinatè ei applicetur AR: </s> <s xml:id="echoid-s4546" xml:space="preserve">nam cum angulus ARL ſit rectus, angulus ALR <lb/>acutus erit, ſed LAD rectus ponitur, quare AD conuenit quoque cum axe <lb/>minori HT, vt in I. </s> <s xml:id="echoid-s4547" xml:space="preserve">Quod autem priùs cum maiori axe conueniat, ita oſten-<lb/>detur. </s> <s xml:id="echoid-s4548" xml:space="preserve">Etenim cum recta AF ſit ad axim applicata, & </s> <s xml:id="echoid-s4549" xml:space="preserve">contingens AE cum <lb/>axe in E conueniat, N verò ſit centrum Ellipſis, erit rectangulum EFN ad <lb/>quadratum AF, <anchor type="note" xlink:href="" symbol="c"/> vt tranſuerſum latus ad rectum, ſed quadratum AF æqua- <anchor type="note" xlink:label="note-0159-03a" xlink:href="note-0159-03"/> tur rectangulo EFD, ergo rectangulum EFN ad rectangulum EFD, ſiue li-<lb/>nea FN ad FD, erit vt tranſuerſum latus ad rectum, hoc eſt vt quadratum <lb/>BS ad quadratum HT (nam ſecunda diameter HT media proportionalis eſt <lb/>inter tranſuerſum BS, & </s> <s xml:id="echoid-s4550" xml:space="preserve">latus rectum) ſed quadratum BS maius eſt quadra-<lb/>to HT, cum ſit BS axis maior, ergo & </s> <s xml:id="echoid-s4551" xml:space="preserve">linea NF maior erit ipſa FD. </s> <s xml:id="echoid-s4552" xml:space="preserve">Perpen-<lb/>dicularis ergo AD ſecat priùs maiorem axem, quàm minorem.</s> <s xml:id="echoid-s4553" xml:space="preserve"/> </p> <div xml:id="echoid-div444" type="float" level="2" n="2"> <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a"> <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0159-01"/> </figure> <note symbol="c" position="right" xlink:label="note-0159-03" xlink:href="note-0159-03a" xml:space="preserve">37. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s4554" xml:space="preserve">Dico inſuper in vtraque ſigura interceptam DA minorem eſſe intercepto <lb/>axis ſegmento DB.</s> <s xml:id="echoid-s4555" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4556" xml:space="preserve">_Ducta enim ex B recta BG_ ordinatim ductæ FA æquidiſtant, ipſa quidem <lb/>ſectionem <anchor type="note" xlink:href="" symbol="d"/> continget, & </s> <s xml:id="echoid-s4557" xml:space="preserve">alteri contingenti AE <anchor type="note" xlink:href="" symbol="e"/> occurret, vt in G. </s> <s xml:id="echoid-s4558" xml:space="preserve">Iungatur <anchor type="note" xlink:label="note-0159-04a" xlink:href="note-0159-04"/> GD: </s> <s xml:id="echoid-s4559" xml:space="preserve">cumque anguli GAD, GBD ſint recti, erunt duo quadrata DA, AG <lb/> <anchor type="note" xlink:label="note-0159-05a" xlink:href="note-0159-05"/> quadrato DG itemque duo quadrata DB, BG eidem quodrato DG æqua-<lb/>lia, ergo duo ſimul DA, AG duobus ſimul DB, BG æqualia erunt, ſed AG <lb/>quadratum maius eſt quadrato BG cum ipſa tangens AG, ſit <anchor type="note" xlink:href="" symbol="f"/> maior tangen- <anchor type="note" xlink:label="note-0159-06a" xlink:href="note-0159-06"/> te BG, ergo quadtatum DA minus erit quadrato DB, ſiue perpendicularis <lb/>DA minor maioris axis ſegmento DB. </s> <s xml:id="echoid-s4560" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s4561" xml:space="preserve">c.</s> <s xml:id="echoid-s4562" xml:space="preserve"/> </p> <div xml:id="echoid-div445" type="float" level="2" n="3"> <note symbol="d" position="right" xlink:label="note-0159-04" xlink:href="note-0159-04a" xml:space="preserve">32. pri-<lb/>mi conic.</note> <note symbol="e" position="right" xlink:label="note-0159-05" xlink:href="note-0159-05a" xml:space="preserve">58. h.</note> <note symbol="f" position="right" xlink:label="note-0159-06" xlink:href="note-0159-06a" xml:space="preserve">87. h.</note> </div> <p> <s xml:id="echoid-s4563" xml:space="preserve">Cum verò in Ellipſi tangens AL occurret minori axi TH, vt in L. </s> <s xml:id="echoid-s4564" xml:space="preserve">Dico in-<lb/>terceptam perpendicularem AI maiorem eſſe axis ſegmento IH.</s> <s xml:id="echoid-s4565" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4566" xml:space="preserve">Si enim ex H ducatur HM ordinatim applicatæ NB æquidiſtans hæc Elli-<lb/>pſim <anchor type="note" xlink:href="" symbol="g"/> continget, & </s> <s xml:id="echoid-s4567" xml:space="preserve">alteritangenti AL <anchor type="note" xlink:href="" symbol="h"/> occurret vt in M; </s> <s xml:id="echoid-s4568" xml:space="preserve">iuncta ergo M I, <anchor type="note" xlink:label="note-0159-07a" xlink:href="note-0159-07"/> erunt duo triangula rectangula MAI, MHI, quorum anguli ad A, & </s> <s xml:id="echoid-s4569" xml:space="preserve">H recti <lb/> <anchor type="note" xlink:label="note-0159-08a" xlink:href="note-0159-08"/> ſunt; </s> <s xml:id="echoid-s4570" xml:space="preserve">quare duo quadrata MA, AI vnico MI, & </s> <s xml:id="echoid-s4571" xml:space="preserve">duo MH, HI eidem MI æ-<lb/>qualia erunt, ergo duo ſimul MA, AI duobus ſimul MH, HI ſunt æqualia, <pb o="136" file="0160" n="160" rhead=""/> ſed quadratum MA minus <anchor type="note" xlink:href="" symbol="a"/> eſt quadrato HM, ergo quadratum A I maius <anchor type="note" xlink:label="note-0160-01a" xlink:href="note-0160-01"/> erit quadrato HI, ſiue perpendicularis intercepta A I, maior intercepto mi-<lb/>noris axis ſegmento IH. </s> <s xml:id="echoid-s4572" xml:space="preserve">Quod tandem demonſtrare oportebat.</s> <s xml:id="echoid-s4573" xml:space="preserve"/> </p> <div xml:id="echoid-div446" type="float" level="2" n="4"> <note symbol="g" position="right" xlink:label="note-0159-07" xlink:href="note-0159-07a" xml:space="preserve">32. pri-<lb/>mi conic.</note> <note symbol="h" position="right" xlink:label="note-0159-08" xlink:href="note-0159-08a" xml:space="preserve">58. h.</note> <note symbol="a" position="left" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve">87. h.</note> </div> <p> <s xml:id="echoid-s4574" xml:space="preserve">ALITER abſque ope propoſitionis 87. </s> <s xml:id="echoid-s4575" xml:space="preserve">premiſso <lb/>tantum ſequenti lemmate pro Ellipſi.</s> <s xml:id="echoid-s4576" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div448" type="section" level="1" n="186"> <head xml:id="echoid-head191" xml:space="preserve">LEMMA XIII. PROP. XIC.</head> <p> <s xml:id="echoid-s4577" xml:space="preserve">Si ſuerit, in vtraque figura, rectangulum ſub extremis AB, BD <lb/>æquale quadrato mediæ BC, dico, in prima ſigura, ſi à tertia BD <lb/>dematur aliqua pars BE, rectangulum ſub AE, ED, minus eſſe <lb/>quadrato mediæ EC.</s> <s xml:id="echoid-s4578" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4579" xml:space="preserve">Cum ſit enim, vt totum AB ad totum BC, ita ablatum BC ad ablatũ BD, <lb/>erit reliquum AC ad reliquum CD, vt totum AB ad totum BC.</s> <s xml:id="echoid-s4580" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4581" xml:space="preserve">Et cum ſit CE minor C B, habebit <lb/> <anchor type="figure" xlink:label="fig-0160-01a" xlink:href="fig-0160-01"/> AC ad CE maiorem rationem quàm <lb/>AC ad CB, & </s> <s xml:id="echoid-s4582" xml:space="preserve">componendo AE ad <lb/>EC maiorem quàm AB ad BC, vel <lb/>quàm AC ad CD. </s> <s xml:id="echoid-s4583" xml:space="preserve">Siergo totum AE <lb/>ad totum EC maioré habet rationem <lb/>quàm ablatum AC ad ablatum CD, <lb/>habebit reliquum CE ad reliquũ ED <lb/>maiorem rationem, quàm totum AE <lb/> <anchor type="note" xlink:label="note-0160-02a" xlink:href="note-0160-02"/> ad totum EC, vel AE ad EC minorem <lb/>habebit rationem quàm CE ad ED; <lb/></s> <s xml:id="echoid-s4584" xml:space="preserve">ergo rectangulum ſub extremis A E, <lb/>ED minus <anchor type="note" xlink:href="" symbol="b"/> erit quadrato mediæ EC.</s> <s xml:id="echoid-s4585" xml:space="preserve"/> </p> <div xml:id="echoid-div448" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0160-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0160-02" xlink:href="note-0160-02a" xml:space="preserve">16. 7. <lb/>Pappi.</note> </div> <p> <s xml:id="echoid-s4586" xml:space="preserve">SI verò, ijſdem poſitis, in ſecunda ſigura, tertiæ proportionali BD recta <lb/>quædam BE adijciatur; </s> <s xml:id="echoid-s4587" xml:space="preserve">dico rectangulum ſub AE, ED maius eſſe qua-<lb/>drato EC; </s> <s xml:id="echoid-s4588" xml:space="preserve">quod licet in 9. </s> <s xml:id="echoid-s4589" xml:space="preserve">prop. </s> <s xml:id="echoid-s4590" xml:space="preserve">huius iam ſit oſtenſum, hic idem aliter nulla <lb/>facta conſtructione demonſtrabimus.</s> <s xml:id="echoid-s4591" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4592" xml:space="preserve">Quoniam enim CE maior eſt CB, habebit AC ad CE minorem rationem <lb/>quàm AC ad CB, & </s> <s xml:id="echoid-s4593" xml:space="preserve">componendo, tota AE ad totam EC, minorem quàm <lb/> <anchor type="note" xlink:label="note-0160-03a" xlink:href="note-0160-03"/> ablata AB ad ablatam BC, vel quàm AC ad CD, ergo reliqua CE ad re-<lb/>liquam ED, minorem quoque habebit rationem quàm tota AE ad EC, <lb/>hoc eſt AE ad EC maiorem quàm EC ad ED, ergo rectangulum ſub AE, <lb/>ED maius <anchor type="note" xlink:href="" symbol="c"/> quadrato mediæ EC. </s> <s xml:id="echoid-s4594" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s4595" xml:space="preserve">c.</s> <s xml:id="echoid-s4596" xml:space="preserve"/> </p> <div xml:id="echoid-div449" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-0160-03" xlink:href="note-0160-03a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s4597" xml:space="preserve">IAM, vt ad expeditiorem demonſtrationem præcedentis propoſitionis ac-<lb/>cedamus, ſuper eiſdem delineationibus, repetitis ijs omnibus, quæ ibi <lb/>(vſque ad ea verba excluſiuè _Ducta enim ex B recta BG, &</s> <s xml:id="echoid-s4598" xml:space="preserve">c.)</s> <s xml:id="echoid-s4599" xml:space="preserve">_ exponuntur, ac <lb/>demonſtrantur, ſic vlteriùs proſequemur. </s> <s xml:id="echoid-s4600" xml:space="preserve">Cum enim in ſingulis figuris triã-<lb/>gula DAE, LAI ſint rectangula ad A, ex quo baſibus ductæ ſunt perpendi-<lb/>culares AF, AR; </s> <s xml:id="echoid-s4601" xml:space="preserve">erit in triangulo DAE rectangulum EDF æquale quadrato <pb o="137" file="0161" n="161" rhead=""/> DA, & </s> <s xml:id="echoid-s4602" xml:space="preserve">in triangulo LAI rectangulum LIR æquale quadrato IA. </s> <s xml:id="echoid-s4603" xml:space="preserve">Quod <lb/>ſerua.</s> <s xml:id="echoid-s4604" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4605" xml:space="preserve">Iam ſi ſectio primæ figure ABC fuerit Parabole, cum AE ſit ei contingens <lb/>erit EB æqualis <anchor type="note" xlink:href="" symbol="*"/> BF, ergo rectangulum EDF cum quadrato FB æquabitur <anchor type="note" xlink:label="note-0161-01a" xlink:href="note-0161-01"/> quadrato BD, quare <lb/> <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> ſolum rectangulũ EDF, <lb/>ſiue quadratum DA mi-<lb/>nus erit quadrato DB, <lb/>ſiue linea D A minor <lb/>DB.</s> <s xml:id="echoid-s4606" xml:space="preserve"/> </p> <div xml:id="echoid-div450" type="float" level="2" n="3"> <note symbol="*" position="right" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve">20. pr. <lb/>conic.</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/QN4GHYBF/figures/0161-01"/> </figure> </div> <p> <s xml:id="echoid-s4607" xml:space="preserve">Siverò eadem figura <lb/>Hyperbolen reprefen-<lb/>tet, reperto eius centro <lb/>Q, erit rectangulum <lb/>FQE <anchor type="note" xlink:href="" symbol="a"/> ęquale quadrato <anchor type="note" xlink:label="note-0161-02a" xlink:href="note-0161-02"/> QB, ergo FQ ad QB, vt <lb/>QB ad QE, vel vt <anchor type="note" xlink:href="" symbol="b"/> FB <anchor type="note" xlink:label="note-0161-03a" xlink:href="note-0161-03"/> ad BE, ſed FQ maior eſt QB, ergo FB erit maior BE, ſiue pluſquam dimi-<lb/>dium ipſa FE, diuiſa ergo FE bifariam in V, erit FV minor FB, eritque re-<lb/>ctangulum EDF cum quadrato FV æquale quadrato DV, igitur ſolum re-<lb/>ctangulum EDF, hoc eſt quadratum DA minus quadrato DV, ſeu linea DA <lb/>minor DV, & </s> <s xml:id="echoid-s4608" xml:space="preserve">eò minor ipſa DB.</s> <s xml:id="echoid-s4609" xml:space="preserve"/> </p> <div xml:id="echoid-div451" type="float" level="2" n="4"> <note symbol="a" position="right" xlink:label="note-0161-02" xlink:href="note-0161-02a" xml:space="preserve">37. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0161-03" xlink:href="note-0161-03a" xml:space="preserve">Coroll. <lb/>12. h.</note> </div> <p> <s xml:id="echoid-s4610" xml:space="preserve">Amplius in Ellipſi ſecundæ figuræ, dum perpendicularis AD conuenit <lb/>cum axe maiori, eſt rectangulum ENF <anchor type="note" xlink:href="" symbol="c"/> æquale quadrato NB, & </s> <s xml:id="echoid-s4611" xml:space="preserve">à tertia <anchor type="note" xlink:label="note-0161-04a" xlink:href="note-0161-04"/> proportionali NF dempta eſt pars ND, ergo per Lemma præcedens erit re-<lb/>ctangulum EDF, ſiue quadratum DA minus quadrato DB, hoc eſt perpen-<lb/>dicularis DA maiori axi occurrens, minor eiuſdem axis portione DB.</s> <s xml:id="echoid-s4612" xml:space="preserve"/> </p> <div xml:id="echoid-div452" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-0161-04" xlink:href="note-0161-04a" xml:space="preserve">37. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s4613" xml:space="preserve">Tandem rectangulum LNR æquatur quadrato NH, & </s> <s xml:id="echoid-s4614" xml:space="preserve">tertiæ proportio-<lb/>nali NR addita eſt NI, ergo per idem Lemma erit rectangulum LIR, ſiue <lb/>quadratum IA maius quadrato IH, ſiue perpendicularis AI minori axi oc-<lb/>currens maior eiuſdem axis portione HI. </s> <s xml:id="echoid-s4615" xml:space="preserve">Quod fuit, &</s> <s xml:id="echoid-s4616" xml:space="preserve">c.</s> <s xml:id="echoid-s4617" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div454" type="section" level="1" n="187"> <head xml:id="echoid-head192" xml:space="preserve">THEOR. XLIV. PROP. XC.</head> <p> <s xml:id="echoid-s4618" xml:space="preserve">Si quamcunque coni-ſectionem recta linea contingat ad pun-<lb/>ctum, quod non ſit axis vertex, à quo ductæ ſint duæ rectæ lineæ, <lb/>altera contingenti, altera autem axi perpendicularis; </s> <s xml:id="echoid-s4619" xml:space="preserve">erit in Para-<lb/>bola ea axis portio inter perpendiculares inrercepta æqualis, in <lb/>Hyperbola verò maior, ſed in Ellipſi minor dimidio recti lateris <lb/>eius axis, cui perpendiculares occurrunt.</s> <s xml:id="echoid-s4620" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4621" xml:space="preserve">SIt quæcunque coni-ſectio ABC, cuius axis BD, vertex B, & </s> <s xml:id="echoid-s4622" xml:space="preserve">aliud in ea <lb/>punctum ſit A, à quo ducta ſit <anchor type="note" xlink:href="" symbol="d"/> contingens AE cum axe <anchor type="note" xlink:href="" symbol="e"/> conueniens <anchor type="note" xlink:label="note-0161-05a" xlink:href="note-0161-05"/> <anchor type="note" xlink:label="note-0161-06a" xlink:href="note-0161-06"/> in E, atque ex A erecta ſit AD ipſi AE perpendicularis (quæ cum axe con-<lb/>ueniet <anchor type="note" xlink:href="" symbol="f"/> in D) & </s> <s xml:id="echoid-s4623" xml:space="preserve">AF perpendicularis ad axem. </s> <s xml:id="echoid-s4624" xml:space="preserve">Dico primùm in Parabola <anchor type="note" xlink:label="note-0161-07a" xlink:href="note-0161-07"/> primæ figuræ, interceptam axis portionem DF dimidio recti lateris æqua-<lb/>lem eſſe.</s> <s xml:id="echoid-s4625" xml:space="preserve"/> </p> <div xml:id="echoid-div454" type="float" level="2" n="1"> <note symbol="d" position="right" xlink:label="note-0161-05" xlink:href="note-0161-05a" xml:space="preserve">2. 4. h.</note> <note symbol="e" position="right" xlink:label="note-0161-06" xlink:href="note-0161-06a" xml:space="preserve">24. 25. <lb/>pr. eonic.</note> <note symbol="f" position="right" xlink:label="note-0161-07" xlink:href="note-0161-07a" xml:space="preserve">88. h.</note> </div> <pb o="138" file="0162" n="162" rhead=""/> <p> <s xml:id="echoid-s4626" xml:space="preserve">Nam quadratum AF æquatur <anchor type="note" xlink:href="" symbol="a"/> rectangulo ſub FB, & </s> <s xml:id="echoid-s4627" xml:space="preserve">recto latere, vel <anchor type="note" xlink:label="note-0162-01a" xlink:href="note-0162-01"/> ſub dupla FB, ſiue ſub <anchor type="note" xlink:href="" symbol="b"/> EF, & </s> <s xml:id="echoid-s4628" xml:space="preserve">dimidio recti, ſed idem quadratum A F æ- quatur rectangulo ſub eadem EF, & </s> <s xml:id="echoid-s4629" xml:space="preserve">ſub FD; </s> <s xml:id="echoid-s4630" xml:space="preserve">quare FD erit dimidium recti. <lb/></s> <s xml:id="echoid-s4631" xml:space="preserve"> <anchor type="note" xlink:label="note-0162-02a" xlink:href="note-0162-02"/> Quod primò, &</s> <s xml:id="echoid-s4632" xml:space="preserve">c.</s> <s xml:id="echoid-s4633" xml:space="preserve"/> </p> <div xml:id="echoid-div455" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0162-01" xlink:href="note-0162-01a" xml:space="preserve">Coroll. <lb/>1. h.</note> <note symbol="b" position="left" xlink:label="note-0162-02" xlink:href="note-0162-02a" xml:space="preserve">35. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s4634" xml:space="preserve">Amplius in Hyperbola ſecundæ figuræ, dico interceptam portionem FD <lb/>eſſe pluſquam dimidium recti lateris.</s> <s xml:id="echoid-s4635" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4636" xml:space="preserve">Nam reperto eius centro G, erit rectangulum GFE ad quadratum AF, <lb/>vel ad rectangulum DFE, vt <anchor type="note" xlink:href="" symbol="c"/> tranſuerſum latus ad rectum, ſed rectangulum <anchor type="note" xlink:label="note-0162-03a" xlink:href="note-0162-03"/> GFE ad DFE, eſt vt linea GF ad FD, ergo GF ad FD eſt vt tranſuerſum la-<lb/>tus ad rectum, vel vt ſemi-tranſuerſum GB ad ſemi-rectum, & </s> <s xml:id="echoid-s4637" xml:space="preserve">permutando <lb/>GF ad GB, erit vt FD ad ſemirectum, ſed eſt GF maior GB, ergo FD erit <lb/>maior ſemi-recto latere. </s> <s xml:id="echoid-s4638" xml:space="preserve">Quod ſecundò erat, &</s> <s xml:id="echoid-s4639" xml:space="preserve">c.</s> <s xml:id="echoid-s4640" xml:space="preserve"/> </p> <div xml:id="echoid-div456" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0162-03" xlink:href="note-0162-03a" xml:space="preserve">37. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s4641" xml:space="preserve">Tandem in Ellipſi <lb/> <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> tertiæ figuræ, in qua <lb/>intercepta axis portio <lb/>DF eſt de maiori axe, <lb/>vel in quarta figura, in <lb/>qua prædicta portio <lb/>DF eſt de minori axe, <lb/>dico item ipſam DF <lb/>minorem eſſe dimidio <lb/>recti lateris eius axis, <lb/>cui ductæ perpendicu-<lb/>lares occurrunt.</s> <s xml:id="echoid-s4642" xml:space="preserve"/> </p> <div xml:id="echoid-div457" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0162-01"/> </figure> </div> <p> <s xml:id="echoid-s4643" xml:space="preserve">Sumpto enim Elli-<lb/>pſis centro G, eſt re-<lb/>ctangulũ EFG ad qua-<lb/>dratum AF, vel ad re-<lb/>ctangulum E F D, <anchor type="note" xlink:href="" symbol="d"/> vt <anchor type="note" xlink:label="note-0162-04a" xlink:href="note-0162-04"/> tranſuerſum latus ad <lb/>rectum, ſed idem rectangulum EFG ad EFD eſt vt linea GF ad FD quare <lb/>GF ad FD eſt vt tranſuerſum ad rectum, vel vt GB dimidium tranſuerſi ad <lb/>dimidium recti, & </s> <s xml:id="echoid-s4644" xml:space="preserve">permutando GF ad GB, vt FD ad dimidium recti, ſed eſt <lb/>GF minor GB, ergo & </s> <s xml:id="echoid-s4645" xml:space="preserve">FD erit minor quàm dimidium recti. </s> <s xml:id="echoid-s4646" xml:space="preserve">Quod vlti-<lb/>mò, &</s> <s xml:id="echoid-s4647" xml:space="preserve">c.</s> <s xml:id="echoid-s4648" xml:space="preserve"/> </p> <div xml:id="echoid-div458" type="float" level="2" n="5"> <note symbol="d" position="left" xlink:label="note-0162-04" xlink:href="note-0162-04a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div460" type="section" level="1" n="188"> <head xml:id="echoid-head193" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s4649" xml:space="preserve">HInc patet in Parabola, & </s> <s xml:id="echoid-s4650" xml:space="preserve">Hyperbola contingenti perpendicularem in-<lb/>ter contactum, & </s> <s xml:id="echoid-s4651" xml:space="preserve">axem, ſemper eſſe pluſquam dimidium recti lateris <lb/>ſectionis. </s> <s xml:id="echoid-s4652" xml:space="preserve">Nam in triangulo AFD recta AD recto angulo oppoſita maior eſt <lb/>latere DF, ſed DF, vel æqualis eſt (in Parabola) vel maior (in Hyperbola) <lb/>prædicto dimidio, quare perpẽdicularis AD erit omninò maior ipſo dimidio.</s> <s xml:id="echoid-s4653" xml:space="preserve"/> </p> <pb o="139" file="0163" n="163" rhead=""/> </div> <div xml:id="echoid-div461" type="section" level="1" n="189"> <head xml:id="echoid-head194" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s4654" xml:space="preserve">PAtet quoque in Parabola, & </s> <s xml:id="echoid-s4655" xml:space="preserve">Hyperbola interceptam axis portionem in-<lb/>ter verticem, & </s> <s xml:id="echoid-s4656" xml:space="preserve">contingenti perpendicularem ſemper item eſſe pluſ-<lb/>quam dimidium recti lateris propriæ ſectionis. </s> <s xml:id="echoid-s4657" xml:space="preserve">Quoniam cum demonſtra-<lb/>tum ſit DB maiorem <anchor type="note" xlink:href="" symbol="a"/> eſſe DA, & </s> <s xml:id="echoid-s4658" xml:space="preserve">DA in præcedenti Corollario ſit maior di- <anchor type="note" xlink:label="note-0163-01a" xlink:href="note-0163-01"/> midio rectilateris, eò magis DB erit maior prædicto dimidio.</s> <s xml:id="echoid-s4659" xml:space="preserve"/> </p> <div xml:id="echoid-div461" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve">88. h.</note> </div> </div> <div xml:id="echoid-div463" type="section" level="1" n="190"> <head xml:id="echoid-head195" xml:space="preserve">COROLL. III.</head> <p> <s xml:id="echoid-s4660" xml:space="preserve">MAnifeſtum eſt etiam in Hyperbola, & </s> <s xml:id="echoid-s4661" xml:space="preserve">Ellipſi ſemper eam axis portio-<lb/>nem, quæ eſt inter centrum ſectionis, & </s> <s xml:id="echoid-s4662" xml:space="preserve">ordinatim ductam ex con-<lb/>tactu, ad portionem eiuſdem axis inter ipſam ordinatam, & </s> <s xml:id="echoid-s4663" xml:space="preserve">contingenti <lb/>perpendicularem, eſſe vt ſemi-tranſuerſum ſectionis ad ſemi-rectum, vel vt <lb/>tranſuerſum ad rectum. </s> <s xml:id="echoid-s4664" xml:space="preserve">Demonſtratum eſt enim in ſecunda, tertia, & </s> <s xml:id="echoid-s4665" xml:space="preserve">quar-<lb/>ta figura rectam GF ad FD eſſe vt tranſuerſum latus ad rectum.</s> <s xml:id="echoid-s4666" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div464" type="section" level="1" n="191"> <head xml:id="echoid-head196" xml:space="preserve">THEOR. XLV. PROP. XCI.</head> <p> <s xml:id="echoid-s4667" xml:space="preserve">Si Ellipſim quædam recta linea contingat inter axium extrema, <lb/>cui à tactu ducta ſit perpendicularis cum vtroque axe conueniens, <lb/>ſemper ipſius portio inter contactum, & </s> <s xml:id="echoid-s4668" xml:space="preserve">minorem axim intercepta, <lb/>eſt maior ſemi-axe maiori; </s> <s xml:id="echoid-s4669" xml:space="preserve">portio verò inter contactum, & </s> <s xml:id="echoid-s4670" xml:space="preserve">maio-<lb/>rem axim, maior eſt ſemi-recto latere maioris axis; </s> <s xml:id="echoid-s4671" xml:space="preserve">& </s> <s xml:id="echoid-s4672" xml:space="preserve">eadem por-<lb/>tio eſt minor ſemi-axe minori; </s> <s xml:id="echoid-s4673" xml:space="preserve">ac demum portio inter contactum, <lb/>& </s> <s xml:id="echoid-s4674" xml:space="preserve">minorem axim minor eſt ſemi-recto latere minoris axis.</s> <s xml:id="echoid-s4675" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4676" xml:space="preserve">SIt Ellipſis ABC, cuius maior axis BC, minor IL, centrum G, & </s> <s xml:id="echoid-s4677" xml:space="preserve">quædam <lb/>contingens MAE inter axium extrema, quæ ipſis <anchor type="note" xlink:href="" symbol="b"/> occurret in E, M; </s> <s xml:id="echoid-s4678" xml:space="preserve">&</s> <s xml:id="echoid-s4679" xml:space="preserve"> <anchor type="note" xlink:label="note-0163-02a" xlink:href="note-0163-02"/> ex A ducta ſit ADH contingenti perpendicularis, quæ vtrique axi occurret, <lb/>ſed <anchor type="note" xlink:href="" symbol="c"/> priùs cum maiori in D, cum minori verò in H.</s> <s xml:id="echoid-s4680" xml:space="preserve"/> </p> <div xml:id="echoid-div464" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-0163-02" xlink:href="note-0163-02a" xml:space="preserve">25. pri-<lb/>mi conic.</note> </div> <note symbol="c" position="right" xml:space="preserve">88. h.</note> <p> <s xml:id="echoid-s4681" xml:space="preserve">Dico primùm interceptam AH ſemper maiorem eſſe maiori ſemi-axe <lb/>G B.</s> <s xml:id="echoid-s4682" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4683" xml:space="preserve">Agatur HP æquidiſtans ad GE, & </s> <s xml:id="echoid-s4684" xml:space="preserve">AFO ad NH. </s> <s xml:id="echoid-s4685" xml:space="preserve">Et quoniam eſt HP ma-<lb/>ior GE, & </s> <s xml:id="echoid-s4686" xml:space="preserve">HO æqualis GF, erit rectangulum PHO, ſiue quadratum HA (in <lb/>triangulo rectangulo PAH) maius rectangulo EGF, ſiue <anchor type="note" xlink:href="" symbol="d"/> quadrato GB, <anchor type="note" xlink:label="note-0163-04a" xlink:href="note-0163-04"/> hoc eſt linea AH maior ipſa GB. </s> <s xml:id="echoid-s4687" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s4688" xml:space="preserve">c.</s> <s xml:id="echoid-s4689" xml:space="preserve"/> </p> <div xml:id="echoid-div465" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-0163-04" xlink:href="note-0163-04a" xml:space="preserve">37. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s4690" xml:space="preserve">Ampliùs, dico AD eſſe pluſquam dimidium recti lateris axis BC.</s> <s xml:id="echoid-s4691" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4692" xml:space="preserve">Quoniam cum ſit GB minor AH, vt modò oſtendimus, habebit GD ad <lb/>AD minorem rationem quàm AH ad AD, vel quàm FG ad FD, vel quàm <lb/>eadem <anchor type="note" xlink:href="" symbol="e"/> GB ſemi-tranſuerſum, ad ſemi-rectum; </s> <s xml:id="echoid-s4693" xml:space="preserve">vnde AD erit maior quam <anchor type="note" xlink:label="note-0163-05a" xlink:href="note-0163-05"/> ſemi-rectum latus maioris axis. </s> <s xml:id="echoid-s4694" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s4695" xml:space="preserve">c.</s> <s xml:id="echoid-s4696" xml:space="preserve"/> </p> <div xml:id="echoid-div466" type="float" level="2" n="3"> <note symbol="e" position="right" xlink:label="note-0163-05" xlink:href="note-0163-05a" xml:space="preserve">3. Co-<lb/>roll. 90. h.</note> </div> <p> <s xml:id="echoid-s4697" xml:space="preserve">Dico præterea eandem portionem AD minorem eſſe quam IG dimidium <lb/>minoris axis.</s> <s xml:id="echoid-s4698" xml:space="preserve"/> </p> <pb o="140" file="0164" n="164" rhead=""/> <p> <s xml:id="echoid-s4699" xml:space="preserve">Quoniam ducta AQN parallela ad GF, <lb/> <anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/> & </s> <s xml:id="echoid-s4700" xml:space="preserve">DQR ad HM; </s> <s xml:id="echoid-s4701" xml:space="preserve">cum ſit RD minor MG, & </s> <s xml:id="echoid-s4702" xml:space="preserve"><lb/>DQ æqualis GN, erit rectangulum RDQ, <lb/>ſiue quadratum DA, (in triangulo rectan-<lb/>gulo RAD) minus rectangulo MGN <anchor type="note" xlink:href="" symbol="a"/> ſiue <anchor type="note" xlink:label="note-0164-01a" xlink:href="note-0164-01"/> quadrato GI; </s> <s xml:id="echoid-s4703" xml:space="preserve">hoc eſt intercepta linea D A <lb/>minor ſemi-axe minori GI. </s> <s xml:id="echoid-s4704" xml:space="preserve">Quod ter-<lb/>tiò, &</s> <s xml:id="echoid-s4705" xml:space="preserve">c.</s> <s xml:id="echoid-s4706" xml:space="preserve"/> </p> <div xml:id="echoid-div467" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0164-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" xml:space="preserve">37. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s4707" xml:space="preserve">Tandem, dico interceptam perpendi-<lb/>cularem AH minorẽ eſſe quàm dimidium <lb/>recti lateris minoris axis.</s> <s xml:id="echoid-s4708" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4709" xml:space="preserve">Etenim cum ſit IG maior AD, vt ſupra <lb/>oſtendimus; </s> <s xml:id="echoid-s4710" xml:space="preserve">habebit IG ad AH maiorem <lb/>rationem quàm AD ad AH, vel quàm NG <lb/>ad NH, vel quàm <anchor type="note" xlink:href="" symbol="b"/> eadem I G ſemi- tranſ- <anchor type="note" xlink:label="note-0164-02a" xlink:href="note-0164-02"/> uerſum ad ſemi- rectum; </s> <s xml:id="echoid-s4711" xml:space="preserve">quare intercepta <lb/>AH erit minor ſemi-recto minoris axis IL. <lb/></s> <s xml:id="echoid-s4712" xml:space="preserve">Quod vltimò oſtendere proponebatur.</s> <s xml:id="echoid-s4713" xml:space="preserve"/> </p> <div xml:id="echoid-div468" type="float" level="2" n="5"> <note symbol="b" position="left" xlink:label="note-0164-02" xlink:href="note-0164-02a" xml:space="preserve">3. Co-<lb/>roli. 90. h.</note> </div> </div> <div xml:id="echoid-div470" type="section" level="1" n="192"> <head xml:id="echoid-head197" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s4714" xml:space="preserve">HInc eſt, quod ſemper in Ellipſi intercepta maioris axis portio inter cõ-<lb/>tingenti perpendicularem, & </s> <s xml:id="echoid-s4715" xml:space="preserve">verticem, maior eſt dimidio recti late-<lb/>ris maioris axis. </s> <s xml:id="echoid-s4716" xml:space="preserve">Nam in figura huius, oſtenſa eſt AD ad numerum 2. </s> <s xml:id="echoid-s4717" xml:space="preserve">maior <lb/> <anchor type="note" xlink:label="note-0164-03a" xlink:href="note-0164-03"/> ſemi-recto maioris axis BC, ſed eſt DB <anchor type="note" xlink:href="" symbol="c"/> maior DA, quare DB eò maior erit prædicto ſemi-recto.</s> <s xml:id="echoid-s4718" xml:space="preserve"/> </p> <div xml:id="echoid-div470" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-0164-03" xlink:href="note-0164-03a" xml:space="preserve">88. h.</note> </div> </div> <div xml:id="echoid-div472" type="section" level="1" n="193"> <head xml:id="echoid-head198" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s4719" xml:space="preserve">PAtet etiam in Ellipſi, quod intercepta minoris axis portio inter contin-<lb/>genti perpendicularem, & </s> <s xml:id="echoid-s4720" xml:space="preserve">verticem, eſt minor dimidio recti lateris <lb/> <anchor type="note" xlink:label="note-0164-04a" xlink:href="note-0164-04"/> eiuſdem minoris axis. </s> <s xml:id="echoid-s4721" xml:space="preserve">Quoniam ſupra ad numerum 4. </s> <s xml:id="echoid-s4722" xml:space="preserve">demonſtrauimus AH <lb/>minorem eſſe ſemi-recto minoris axis IL, ſed eſt <anchor type="note" xlink:href="" symbol="d"/> IH minor AH, quare IH eò minor erit prædicto ſemi-recto.</s> <s xml:id="echoid-s4723" xml:space="preserve"/> </p> <div xml:id="echoid-div472" type="float" level="2" n="1"> <note symbol="d" position="left" xlink:label="note-0164-04" xlink:href="note-0164-04a" xml:space="preserve">ibidem.</note> </div> <pb o="141" file="0165" n="165" rhead=""/> </div> <div xml:id="echoid-div474" type="section" level="1" n="194"> <head xml:id="echoid-head199" xml:space="preserve">THEOR. XLVI. PROP. XCII.</head> <p> <s xml:id="echoid-s4724" xml:space="preserve">Si Parabolen, vel Hyperbolen, aut Ellipſim circa maiorem <lb/>axim recta linea, præter ad verticem contingat, cui à tactu ducta <lb/>ſit perpendicularis axi occurrens; </s> <s xml:id="echoid-s4725" xml:space="preserve">circulus, cuius centrum ſit idem <lb/>occurſus, radius verò ſit ipſa perpẽdicularis erit ſectioni inſcriptus.</s> <s xml:id="echoid-s4726" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4727" xml:space="preserve">Si autem Ellipſis fuerit circa minorem axim, cui prædicta per-<lb/>pendicularis occurrat, circulus ex ea tanquam radio, at centro fa-<lb/>cto ipſo occurſu, erit eidem Ellipſi circumſcriptus.</s> <s xml:id="echoid-s4728" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4729" xml:space="preserve">ESto ABC, Parabole, vel Hyperbole, in prima figura, aut Ellipſis in ſe-<lb/>cunda, circa maiorem axim BO; </s> <s xml:id="echoid-s4730" xml:space="preserve">vel circa minorẽ, vt in tertia, quarum <lb/>vertex B, & </s> <s xml:id="echoid-s4731" xml:space="preserve">ad aliud punctum quædam contingens EF, cui ducta ſit perpen-<lb/>dicularis ED, quæ axi occurret <anchor type="note" xlink:href="" symbol="a"/> in D, quo facto centro, & </s> <s xml:id="echoid-s4732" xml:space="preserve">interuallo DE <anchor type="note" xlink:label="note-0165-01a" xlink:href="note-0165-01"/> circulus EGHI deſcribatur. </s> <s xml:id="echoid-s4733" xml:space="preserve">Dico primùmhunc, in prima, & </s> <s xml:id="echoid-s4734" xml:space="preserve">ſecunda figu-<lb/>ra, datæ ſectioni eſſe inſcriptum.</s> <s xml:id="echoid-s4735" xml:space="preserve"/> </p> <div xml:id="echoid-div474" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0165-01" xlink:href="note-0165-01a" xml:space="preserve">88. h.</note> </div> <figure> <image file="0165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0165-01"/> </figure> <p> <s xml:id="echoid-s4736" xml:space="preserve">Applicata enim EH, ſecans<unsure/> axim in L, & </s> <s xml:id="echoid-s4737" xml:space="preserve">iuncta DH. </s> <s xml:id="echoid-s4738" xml:space="preserve">Cum in triangulis <lb/>ELD, HLD anguli ad L ſint recti, & </s> <s xml:id="echoid-s4739" xml:space="preserve">latera EL, LD æqualia lateribus HL, <lb/>LD, erit baſis DE æqualis DH, exquo circulus ex DE tranſibit omnino per <lb/>H, ideoque coni-ſectio, & </s> <s xml:id="echoid-s4740" xml:space="preserve">circulus, ſunt binæ ſectiones ſimul adſcriptæ <lb/>(cum earum diametri, & </s> <s xml:id="echoid-s4741" xml:space="preserve">applicatæ ſimul congruant) quæ in ijſdem extre-<lb/>mis communis applicatæ EH ſimul conueniunt, atque ad eorum alterum E, <lb/>eadem recta EF vtranque ſectionem contingit, nempe ſectionem ABC, ex <lb/>ſuppoſitione, & </s> <s xml:id="echoid-s4742" xml:space="preserve">circulum EGHI, cum EF ſit ad extremum ſemi-diametri <lb/>ED perpendicularis, atque vertex circuli G cadit infra B verticem ſectionis, <lb/>cum ſit DB <anchor type="note" xlink:href="" symbol="b"/> maior DE, ſiue maior DG, quare circulus ex DE erit <anchor type="note" xlink:href="" symbol="c"/> ſectioni <anchor type="note" xlink:label="note-0165-02a" xlink:href="note-0165-02"/> <anchor type="note" xlink:label="note-0165-03a" xlink:href="note-0165-03"/> inſcriptus. </s> <s xml:id="echoid-s4743" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s4744" xml:space="preserve">c.</s> <s xml:id="echoid-s4745" xml:space="preserve"/> </p> <div xml:id="echoid-div475" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0165-02" xlink:href="note-0165-02a" xml:space="preserve">ibideni.</note> <note symbol="c" position="right" xlink:label="note-0165-03" xlink:href="note-0165-03a" xml:space="preserve">@ 1. h.</note> </div> <p> <s xml:id="echoid-s4746" xml:space="preserve">AMpliùs, dico in tertia figura, prædictum circulum EGHI eſſe datæ El-<lb/>lipſi ABCO circumſcriptum.</s> <s xml:id="echoid-s4747" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4748" xml:space="preserve">Nam facta eadem conſtructione, ac ſupra oſtendetur pariter circulum <pb o="142" file="0166" n="166" rhead=""/> tranſire per H, adſcriptum eſſe Ellipſi ABCO, & </s> <s xml:id="echoid-s4749" xml:space="preserve">Ellipſeos contingentem <lb/>EF circulum quoque contingere, ſed huius verticem G, cadere vltra Elli-<lb/>pſeos verticem B, cum ſit DE, vel DG maior <anchor type="note" xlink:href="" symbol="a"/> DB, quare circulus ex DE <anchor type="note" xlink:label="note-0166-01a" xlink:href="note-0166-01"/> erit Ellipſi ABCO <anchor type="note" xlink:href="" symbol="b"/> circumſcriptus. </s> <s xml:id="echoid-s4750" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s4751" xml:space="preserve"/> </p> <div xml:id="echoid-div476" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-0166-01" xlink:href="note-0166-01a" xml:space="preserve">88. h.</note> </div> <note symbol="b" position="left" xml:space="preserve">61. h.</note> </div> <div xml:id="echoid-div478" type="section" level="1" n="195"> <head xml:id="echoid-head200" xml:space="preserve">THEOR. XLVIII. PROP. XCIII.</head> <p> <s xml:id="echoid-s4752" xml:space="preserve">Si Parabolen, vel Hyperbolen, aut Ellipſim circa maiorem axim <lb/>quotcunque rectæ lineæ ad eaſdem axis partes, præter verticem <lb/>contingant, quibus à tactibus ductæ ſint perpendiculares axi occur-<lb/>rentes: </s> <s xml:id="echoid-s4753" xml:space="preserve">ipſæ, quò magis contactuum puncta à maioris axis vertice <lb/>diſtabunt eò maiores erunt.</s> <s xml:id="echoid-s4754" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4755" xml:space="preserve">E contra: </s> <s xml:id="echoid-s4756" xml:space="preserve">ſi Ellipſis fuerit circa minorem axim, huiuſmodi per-<lb/>pendiculares ſemper decreſce<unsure/>nt, quò magis earum contactus à mi-<lb/>noris axis vertice remouentur.</s> <s xml:id="echoid-s4757" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4758" xml:space="preserve">SIt AB Parabole, vt in prima figura, vel Hyperbole, vt in ſecunda, aut <lb/>Ellipſis circa maiorem axim BR, vt in tertia, vel circa minorem, vt in <lb/>quarta, quas ſectiones duæ rectæ AE, DF ad eaſdem axis partes, & </s> <s xml:id="echoid-s4759" xml:space="preserve">in Elli-<lb/>pſi in eodem quadrante BLM ad duo quælibet puncta contingant, præter <lb/>verticem B, quibus erectæ ſint perpendiculares AC, DG axi occurrentes <lb/>in C, G. </s> <s xml:id="echoid-s4760" xml:space="preserve">Dico primùm in Parabola, & </s> <s xml:id="echoid-s4761" xml:space="preserve">Hyperbola, ac in Ellipſi tertiæ figu-<lb/>ræ interceptam perpendicularem AC ex puncto A, remotiori à vertice, <lb/>maiorem eſſe perpendiculari DG ex puncto D propinquiori.</s> <s xml:id="echoid-s4762" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4763" xml:space="preserve">1. </s> <s xml:id="echoid-s4764" xml:space="preserve">Nam, in ſingulis figuris, cum anguli CAE, GDF ſint duo recti, & </s> <s xml:id="echoid-s4765" xml:space="preserve">con-<lb/>tingentes AE, DF cadant extra ſectionem, & </s> <s xml:id="echoid-s4766" xml:space="preserve">ſi concipiatur iungi recta AD, <lb/>ipſa cadat tota intra ſectionem, anguli, <lb/>quos eadem A D conficiet cum perpen-<lb/>dicularibus A C, D G, minores erunt <lb/> <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> duobus rectis, quare ipſæ conuenient <lb/>ſimul ad partem axis, vel vltra, vel inter <lb/>contactus, & </s> <s xml:id="echoid-s4767" xml:space="preserve">axim.</s> <s xml:id="echoid-s4768" xml:space="preserve"/> </p> <div xml:id="echoid-div478" 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/QN4GHYBF/figures/0166-01"/> </figure> </div> <p> <s xml:id="echoid-s4769" xml:space="preserve">2. </s> <s xml:id="echoid-s4770" xml:space="preserve">Iam, ducta D I parallela ad AH, ſiue <lb/>axi perpendiculari, cum in Parabola pri-<lb/>mæ figuræ <anchor type="note" xlink:href="" symbol="c"/> ſit CH æqualis G I, vtraque <anchor type="note" xlink:label="note-0166-03a" xlink:href="note-0166-03"/> enim eſt <anchor type="note" xlink:href="" symbol="d"/> dimidium recti lateris, & </s> <s xml:id="echoid-s4771" xml:space="preserve">AH <anchor type="note" xlink:label="note-0166-04a" xlink:href="note-0166-04"/> maior D I, erunt quadrata ſimul C H, <lb/>AH, ſiue quadratum AC, maius qua-<lb/>dratis ſimul G I, D I, ſiue quadrato DG, <lb/>hoc eſt linea AC maior DG.</s> <s xml:id="echoid-s4772" xml:space="preserve"/> </p> <div xml:id="echoid-div479" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-0166-03" xlink:href="note-0166-03a" xml:space="preserve">35. pri-<lb/>mi conic.</note> <note symbol="d" position="left" xlink:label="note-0166-04" xlink:href="note-0166-04a" xml:space="preserve">90. h.</note> </div> <p> <s xml:id="echoid-s4773" xml:space="preserve">3. </s> <s xml:id="echoid-s4774" xml:space="preserve">In Hyperbola verò ſecundæ figuræ ſumpto eius centro L: </s> <s xml:id="echoid-s4775" xml:space="preserve">cum L H <lb/>ad H C, itemque L I ad I G, <anchor type="note" xlink:href="" symbol="e"/> vt tranſuerſum latus ad rectum, erit L H ad <anchor type="note" xlink:label="note-0166-05a" xlink:href="note-0166-05"/> H C, vt L I ad I G, & </s> <s xml:id="echoid-s4776" xml:space="preserve">permutando L H ad L I, vt H C ad I G, ſed eſt L H <lb/>maior L I, ergo, & </s> <s xml:id="echoid-s4777" xml:space="preserve">H C, maior I G, eſtque H A maior I D, quare duo <pb o="143" file="0167" n="167" rhead=""/> ſimul quadrata C H, H A, ſiue vnicum <lb/>quadratum A C, maius eſt duobus ſi-<lb/>mul quadratis G I, I D, ſiue vnico qua-<lb/>drato D G, hoc eſt linea A C maior <lb/> <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> D G.</s> <s xml:id="echoid-s4778" xml:space="preserve"/> </p> <div xml:id="echoid-div480" type="float" level="2" n="3"> <note symbol="e" position="left" xlink:label="note-0166-05" xlink:href="note-0166-05a" xml:space="preserve">3. Co-<lb/>roll. 90. h.</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/QN4GHYBF/figures/0167-01"/> </figure> </div> <p> <s xml:id="echoid-s4779" xml:space="preserve">4. </s> <s xml:id="echoid-s4780" xml:space="preserve">At in Ellipſi tertiæ figuræ cum licet <lb/>A H excedens ſemper D I, non tamen <lb/>ſit C H, vel æqualis, vel maior G I, ſed <lb/>omnino minor (eſt enim L H ad H C, <lb/>itemque L I, ad I G, vt <anchor type="note" xlink:href="" symbol="a"/> tranſuerſum <anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/> ad rectum, ideoque L H ad H C, eſt vt <lb/>L I ad I G, ſed permutando L H maior <lb/>eſt L I, ergo, & </s> <s xml:id="echoid-s4781" xml:space="preserve">H C maior I G) opor-<lb/>ruit hic aliam demonſtrationem inqui-<lb/>rere, quæ, tum Hyperbolæ, tum Elli-<lb/>pſi circa maiorem axim ſimul inſeruiet, <lb/>ſi concipiatur tertia figura vtriuſque <lb/>ſectionis ſpeciem exhibere.</s> <s xml:id="echoid-s4782" xml:space="preserve"/> </p> <div xml:id="echoid-div481" type="float" level="2" n="4"> <note symbol="a" position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">3. Co-<lb/>roll. 90. h.</note> </div> <p> <s xml:id="echoid-s4783" xml:space="preserve">Itaque, vel ordinata AH, quæ ex re-<lb/>motiori contactu à vertice B applicatur, <lb/>occurrit axi in puncto G, vel infra, vel <lb/>ſupra. </s> <s xml:id="echoid-s4784" xml:space="preserve">Si primum, vel ſecundum, patet <lb/>punctum C eò magis cadere infra G. </s> <s xml:id="echoid-s4785" xml:space="preserve">Si <lb/>tertium, hoc idem tamen demonſtrabi-<lb/>tur, videlicet punctum C cadere omnino <lb/>infra G. </s> <s xml:id="echoid-s4786" xml:space="preserve">Cum ſit enim G I maior G H <lb/>habebit L G ad G I minorem rationem <lb/>quàm L G ad GH, & </s> <s xml:id="echoid-s4787" xml:space="preserve">componendo L I ad <lb/>I G minorem item rationem quàm LH ad <lb/>HG, ſed vt L I ad I G, ita LH ad HC, vt <lb/>ſuperiùs oſtendimus, quare LH ad HC, <lb/>minorem habebit rationem quàm eadem <lb/>LH ad HG, vnde HC maior eſt HG, ſiue <lb/>punctum C cadit infra G; </s> <s xml:id="echoid-s4788" xml:space="preserve">quapropter in-<lb/>tercepta perpendicularis AC, ex A re-<lb/>motiori contactu à vertice B, occurrit axi <lb/>infra occurſum G interceptæ perpendi-<lb/>cularis DG, ex propiori contactu D.</s> <s xml:id="echoid-s4789" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4790" xml:space="preserve">5. </s> <s xml:id="echoid-s4791" xml:space="preserve">Iam AC, & </s> <s xml:id="echoid-s4792" xml:space="preserve">DG conueniunt ſimul ad partem axis BC, vt hic ad nume-<lb/>rum 1. </s> <s xml:id="echoid-s4793" xml:space="preserve">oſtenſum fuit, & </s> <s xml:id="echoid-s4794" xml:space="preserve">eſt punctum C infra G, quare ſi ex G ducatur GN, <lb/>parallela ad C A ipſa ſectionis peripheriam ſecabit inter A, & </s> <s xml:id="echoid-s4795" xml:space="preserve">D, vt in N. <lb/></s> <s xml:id="echoid-s4796" xml:space="preserve">Si igitur concipiantur puncta A, N, iungi recta linea, ipſa cadet tota intra <lb/>ſectionem, & </s> <s xml:id="echoid-s4797" xml:space="preserve">producta, axi occurret extra ad partes B, & </s> <s xml:id="echoid-s4798" xml:space="preserve">fiet triangulum, <lb/>in quo A C erit maior NG: </s> <s xml:id="echoid-s4799" xml:space="preserve">itaque ſi cum centro G, interuallo GD deſcriba-<lb/>tur circulus DO, cum <anchor type="note" xlink:href="" symbol="b"/> ſit ſectioni ſemper inſcriptus, ipſæ ſecabit rectam <anchor type="note" xlink:label="note-0167-02a" xlink:href="note-0167-02"/> GN, vt in O, eritque NG maior GO, ſiue maior GD, quare eò magis A C <lb/>maior erit DG. </s> <s xml:id="echoid-s4800" xml:space="preserve">Quod erat primò demonſtrandum.</s> <s xml:id="echoid-s4801" xml:space="preserve"/> </p> <div xml:id="echoid-div482" type="float" level="2" n="5"> <note symbol="b" position="right" xlink:label="note-0167-02" xlink:href="note-0167-02a" xml:space="preserve">92. h.</note> </div> <pb o="144" file="0168" n="168" rhead=""/> <p> <s xml:id="echoid-s4802" xml:space="preserve">6. </s> <s xml:id="echoid-s4803" xml:space="preserve">IN quarta autem figura Ellipſis <lb/>circa minorem axim BR, in <lb/> <anchor type="figure" xlink:label="fig-0168-01a" xlink:href="fig-0168-01"/> qua prædictæ contingentibus <lb/>perpendiculares ipſi BR occur-<lb/>runt: </s> <s xml:id="echoid-s4804" xml:space="preserve">dico AC, quæ à remotiori <lb/>contactu educitur minorem eſſe <lb/>DG, quæ à propinquiori. </s> <s xml:id="echoid-s4805" xml:space="preserve">Nam <lb/>cum ſit DO ad DG, vt I L ad IG, <lb/>vel vt <anchor type="note" xlink:href="" symbol="a"/> tranſuerſum latus ad re- <anchor type="note" xlink:label="note-0168-01a" xlink:href="note-0168-01"/> ctum, vel vt HL ad HC, vel vt <lb/>AN ad AC, erit DO ad DG, vt <lb/>AN ad AC, & </s> <s xml:id="echoid-s4806" xml:space="preserve">permutando, vt <lb/>DO ad AN, ita DG ad AC, ſed <lb/>eſt DO maior AN, vt ſupra ad numerum 5. </s> <s xml:id="echoid-s4807" xml:space="preserve">oſtenſum eſt, ergo, & </s> <s xml:id="echoid-s4808" xml:space="preserve">DG maior <lb/>erit ipſa AC. </s> <s xml:id="echoid-s4809" xml:space="preserve">Quod ſecundò oſtendere propoſitum fuit.</s> <s xml:id="echoid-s4810" xml:space="preserve"/> </p> <div xml:id="echoid-div483" type="float" level="2" n="6"> <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/QN4GHYBF/figures/0168-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0168-01" xlink:href="note-0168-01a" xml:space="preserve">3. Co-<lb/>roll. 90. h.</note> </div> </div> <div xml:id="echoid-div485" type="section" level="1" n="196"> <head xml:id="echoid-head201" xml:space="preserve">PROBL. XXXIV. PROP. XCIV.</head> <p> <s xml:id="echoid-s4811" xml:space="preserve">Dato angulo rectilineo, ad punctum in eius latere datum MA-<lb/>XIMVM circulum inſcribere.</s> <s xml:id="echoid-s4812" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4813" xml:space="preserve">SIt datus angulus rectilineus ABC, & </s> <s xml:id="echoid-s4814" xml:space="preserve">punctum in eius latere datum ſit A, <lb/>ad quod oporteat _MAXIMVM_ circulum inſcribere.</s> <s xml:id="echoid-s4815" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4816" xml:space="preserve">Bifariam ſecetur angulus à recta BD, & </s> <s xml:id="echoid-s4817" xml:space="preserve">ex A ipſi AB perpendicularis eri-<lb/>gatur AE, occurrens BD in E; </s> <s xml:id="echoid-s4818" xml:space="preserve">& </s> <s xml:id="echoid-s4819" xml:space="preserve">centro E, interuallo EA deſcribatur circu-<lb/>lus. </s> <s xml:id="echoid-s4820" xml:space="preserve">Dico hunc eſſe _MAXIMVM_ quæſitum.</s> <s xml:id="echoid-s4821" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4822" xml:space="preserve">Nam ſumpta BC ipſi BA æquali, iunctiſque AC, EC; </s> <s xml:id="echoid-s4823" xml:space="preserve">cum latera AB, <lb/>BE, æqualia ſint lateribus CB, BE, & </s> <s xml:id="echoid-s4824" xml:space="preserve">anguli ad B æquales, erit EA æqualis <lb/>EC. </s> <s xml:id="echoid-s4825" xml:space="preserve">Inſuper ſunt BA, AE, ipſis BC, CE æqualia, vtrunque vtrique, & </s> <s xml:id="echoid-s4826" xml:space="preserve">ba-<lb/>ſis BE communis, ergo angulus BAE angulo BCE æqualis, nempe rectus <lb/>quare circulus ex EA per C tranſibit, contigetque latera BA, BC, ſiue erit <lb/>angulo ABC inſcriptus. </s> <s xml:id="echoid-s4827" xml:space="preserve">Dico hunc eſſe _MAXIMVM_ quæſitum.</s> <s xml:id="echoid-s4828" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4829" xml:space="preserve">Nam ſi centra circulorum ad A pertinen-<lb/> <anchor type="figure" xlink:label="fig-0168-02a" xlink:href="fig-0168-02"/> tium, fuerint in portione perpendicularis <lb/>AE, inter A, & </s> <s xml:id="echoid-s4830" xml:space="preserve">E; </s> <s xml:id="echoid-s4831" xml:space="preserve">ipſi, vt ſatis conſtat, erũt <lb/>quidem angulo inſcripti, cum circulo quo-<lb/>que inſcripti ſint; </s> <s xml:id="echoid-s4832" xml:space="preserve">ſed minores erunt circulo <lb/>ADC cum ſint minoris radij; </s> <s xml:id="echoid-s4833" xml:space="preserve">illi verò quo-<lb/>rum centra ſunt in producta AE, vt in F, ſunt <lb/>quidem maiores, ſed latus BC omnino ſecát: <lb/></s> <s xml:id="echoid-s4834" xml:space="preserve">quoniam ducta F G parallela ad E C, quæ <lb/>productæ A C occurrat in G, cum ſit AF ad <lb/>FG, vt AE ad EC, ſitque AE ipſi EC æqua-<lb/>lis, erit quoque AF æqualis FG: </s> <s xml:id="echoid-s4835" xml:space="preserve">quare cir-<lb/>culus ex FA tranſibit per punctum G, quod <lb/>eſt extra angulum; </s> <s xml:id="echoid-s4836" xml:space="preserve">ideoque in ſe remeans ſecabit omnino latus BC, quod <lb/>eſt infinitæ extenſionis. </s> <s xml:id="echoid-s4837" xml:space="preserve">Si verò centrum ſumatur extra prædicta perpendi- <pb o="145" file="0169" n="169" rhead=""/> cularem AE, vt in H, patet iunctam HA, cum recta BAI inæquales angu-<lb/>los efficere, ac ideò peripheriam circuli ad partem acuti anguli cadere extra <lb/>datum angulum, & </s> <s xml:id="echoid-s4838" xml:space="preserve">ad partem obtuſi cadere intra, ſicque latus dati anguli <lb/>ſecare. </s> <s xml:id="echoid-s4839" xml:space="preserve">Quapropter circulus ACD erit _MAXIMVS_ inſcriptus ad datum <lb/>punctum A. </s> <s xml:id="echoid-s4840" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s4841" xml:space="preserve"/> </p> <div xml:id="echoid-div485" type="float" level="2" n="1"> <figure xlink:label="fig-0168-02" xlink:href="fig-0168-02a"> <image file="0168-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0168-02"/> </figure> </div> </div> <div xml:id="echoid-div487" type="section" level="1" n="197"> <head xml:id="echoid-head202" xml:space="preserve">PROBL. XXXV. PROP. XCV.</head> <p> <s xml:id="echoid-s4842" xml:space="preserve">Datæ Parabolæ, vel Hyperbolæ, ſiue Ellipſi circa maiorem <lb/>axim, ad datum punctum in eius peripheria, præter axis verticem, <lb/>MAXIMVM circulum inſcribere.</s> <s xml:id="echoid-s4843" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4844" xml:space="preserve">SIt ABC data Parabole, vel Hyperbole, in prima figura, vel Ellipſis circa <lb/>maiorem axim BO, in ſecunda, quarum vertex ſit B, & </s> <s xml:id="echoid-s4845" xml:space="preserve">punctum in ea <lb/>ſumptum præter B ſit E. </s> <s xml:id="echoid-s4846" xml:space="preserve">Oportet ad punctum E _MAXIMVM_ datæ ſectioni <lb/>circulum inſcribere.</s> <s xml:id="echoid-s4847" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4848" xml:space="preserve">Ducatur ex E ſectionem con-<lb/> <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> tingens EF, cui erigatur perpen-<lb/>dicularis ED axi <anchor type="note" xlink:href="" symbol="a"/> occurrens in <anchor type="note" xlink:label="note-0169-01a" xlink:href="note-0169-01"/> D. </s> <s xml:id="echoid-s4849" xml:space="preserve">Dico ſi cum centro D, inter-<lb/>uallo DE, circulus EGHI deſcri-<lb/>batur ipſum eſſe quæſitum: </s> <s xml:id="echoid-s4850" xml:space="preserve">nam <lb/>eſſe inſcriptum patet ex prima <lb/>parte 92. </s> <s xml:id="echoid-s4851" xml:space="preserve">huius; </s> <s xml:id="echoid-s4852" xml:space="preserve">quod autem ſit <lb/>_MAXIMVS_ conſtabit ſic: </s> <s xml:id="echoid-s4853" xml:space="preserve">appli-<lb/>cata enim ELH, & </s> <s xml:id="echoid-s4854" xml:space="preserve">producta EF <lb/>axi occurrens in F, iunctaque <lb/>FH, hæc pariter ſectionem <anchor type="note" xlink:href="" symbol="b"/> cõ- <anchor type="note" xlink:label="note-0169-02a" xlink:href="note-0169-02"/> tinget, & </s> <s xml:id="echoid-s4855" xml:space="preserve">fiet angulus E F H, & </s> <s xml:id="echoid-s4856" xml:space="preserve"><lb/>quilibet alius circulus, vel cadet intra AGHI, vel ſecabit latera anguli EFH, <lb/>vt in præcedenti oſtenſum fuit, ac ideò ſecabit priùs ſectionem. </s> <s xml:id="echoid-s4857" xml:space="preserve">Quare cir-<lb/>culus EGHI erit _MAXIMVS_ ſectioni inſcriptus ad punctum AE. </s> <s xml:id="echoid-s4858" xml:space="preserve">Quod erat <lb/>faciendum.</s> <s xml:id="echoid-s4859" xml:space="preserve"/> </p> <div xml:id="echoid-div487" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0169-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0169-01" xlink:href="note-0169-01a" xml:space="preserve">88. h.</note> <note symbol="b" position="right" xlink:label="note-0169-02" xlink:href="note-0169-02a" xml:space="preserve">59. h.</note> </div> </div> <div xml:id="echoid-div489" type="section" level="1" n="198"> <head xml:id="echoid-head203" xml:space="preserve">PROBL. XXXVI. PROP. XCVI.</head> <p> <s xml:id="echoid-s4860" xml:space="preserve">Datæ Ellipſi circa minorem axim, ad datum punctum in <lb/>eius peripheria, præter axis verticem, MINIMVM circulum <lb/>circumſcribere.</s> <s xml:id="echoid-s4861" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4862" xml:space="preserve">SIt data Ellipſis ABC, circa minorem axim BO, cuius vertex B, & </s> <s xml:id="echoid-s4863" xml:space="preserve">in pe-<lb/>pheria datum punctum, præter B, ſit E, per quod oporteat _MINIMVM_ <lb/>circulum circumſcribere.</s> <s xml:id="echoid-s4864" xml:space="preserve"/> </p> <pb o="146" file="0170" n="170" rhead=""/> <p> <s xml:id="echoid-s4865" xml:space="preserve">Ducatur EF Ellipſim contingens, cui ex E perpendicularis erigatur ED, <lb/>maiori axi occurrens in L, minori verò in D: </s> <s xml:id="echoid-s4866" xml:space="preserve">quo facto centro, & </s> <s xml:id="echoid-s4867" xml:space="preserve">interual-<lb/>lo DE circulus deſcribatur EGHI. </s> <s xml:id="echoid-s4868" xml:space="preserve">Dico hunc eſſe quæſitum.</s> <s xml:id="echoid-s4869" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4870" xml:space="preserve">Nam eſſe circumſcriptum, pater ex ſecunda parte 92. </s> <s xml:id="echoid-s4871" xml:space="preserve">huius. </s> <s xml:id="echoid-s4872" xml:space="preserve">Sed eſt quoq; <lb/></s> <s xml:id="echoid-s4873" xml:space="preserve">_MINIMVS_: </s> <s xml:id="echoid-s4874" xml:space="preserve">quoniam quilibet alius <lb/> <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> circulus, cuius radius, maior ſit ipſo <lb/>DE, eſt omnino maior circulo EG-<lb/>HI, & </s> <s xml:id="echoid-s4875" xml:space="preserve">cuius radius minor ſit D E, <lb/>eſt quidem minor, ſed vel totus ca-<lb/>dit intra Ellipſim, vel eius periphe-<lb/>riam neceſſariò ſecat. </s> <s xml:id="echoid-s4876" xml:space="preserve">Nam ſi cen-<lb/>trum fuerit in perpendiculari ED, <lb/>& </s> <s xml:id="echoid-s4877" xml:space="preserve">radius non maior diſtantia E L, <lb/>quæ cadit inter <anchor type="note" xlink:href="" symbol="a"/> contactum E, &</s> <s xml:id="echoid-s4878" xml:space="preserve"> <anchor type="note" xlink:label="note-0170-01a" xlink:href="note-0170-01"/> maiorem axim, circulus cadet totus <lb/>intra, & </s> <s xml:id="echoid-s4879" xml:space="preserve">ſi radius fuerit maior E L, <lb/>qualis eſt EP, tunc eius circulus ca-<lb/>det totus intra circulum EGHI, ſed <lb/>licet ipſius peripheria ad partes G, <lb/>B, ſtatim ac diſcedit ab E, cadat in-<lb/>ter peripheriam circuli AGH, & </s> <s xml:id="echoid-s4880" xml:space="preserve">perip heriam Ellipſis EBH, cum tamen in <lb/>ſe ipſum redeat, neceſſariò Ellipticam peripheriam EBH ſecabit, nam ſpa-<lb/>tium EGHB eſt vndique occluſum.</s> <s xml:id="echoid-s4881" xml:space="preserve"/> </p> <div xml:id="echoid-div489" 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/QN4GHYBF/figures/0170-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0170-01" xlink:href="note-0170-01a" xml:space="preserve">92. h.</note> </div> <p> <s xml:id="echoid-s4882" xml:space="preserve">Si verò centrum fuerit extra perpendicularem ED, vt in Q: </s> <s xml:id="echoid-s4883" xml:space="preserve">iuncta QE <lb/>cum contingente SEF inæquales angulos efficiet, quorum alterum, videli-<lb/>cet SEQ obtuſus erit, quare ſi ipſi EQ erigatur perpendicularis ER, hæc <lb/>omninò ſecabit <anchor type="note" xlink:href="" symbol="b"/> Ellipſim: </s> <s xml:id="echoid-s4884" xml:space="preserve">quare ſi cum centro Q, interuallo QE circulus <anchor type="note" xlink:label="note-0170-02a" xlink:href="note-0170-02"/> deſcribatur XEV, ipſæ ad partes ſecantis ER ſecabit omnino Ellipſis peri-<lb/>pheriam, vt per ſe patet. </s> <s xml:id="echoid-s4885" xml:space="preserve">Ergo circulus ex DE eſt _MINIMVS_ circumſcri-<lb/>ptus quæſitus. </s> <s xml:id="echoid-s4886" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s4887" xml:space="preserve"/> </p> <div xml:id="echoid-div490" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0170-02" xlink:href="note-0170-02a" xml:space="preserve">32. pri-<lb/>mi conic.</note> </div> </div> <div xml:id="echoid-div492" type="section" level="1" n="199"> <head xml:id="echoid-head204" xml:space="preserve">THEOR. XLVIII. PROP. XCVII.</head> <p> <s xml:id="echoid-s4888" xml:space="preserve">MAXIMI circuli angulo rectilineo inſcripti, & </s> <s xml:id="echoid-s4889" xml:space="preserve">ſucceſſiuè ſe <lb/>mutuò contingentes, ſunt inter ſe in continua, eademque ratione <lb/>geometrica, quæ progreditur iuxta quadrata tangentium, ex ver-<lb/>tice dati anguli ductarum.</s> <s xml:id="echoid-s4890" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4891" xml:space="preserve">ESto angulus ABC, cuius axis B D E F, in quo ſint centra D, E, F, &</s> <s xml:id="echoid-s4892" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s4893" xml:space="preserve">_MAXIMORVM_ circulorum dato angulo inſcriptorum, & </s> <s xml:id="echoid-s4894" xml:space="preserve">mutui ipſorum <lb/>contactus ſint G, H, &</s> <s xml:id="echoid-s4895" xml:space="preserve">c. </s> <s xml:id="echoid-s4896" xml:space="preserve">ad latus verò anguli, contactus ſint L, M, C, &</s> <s xml:id="echoid-s4897" xml:space="preserve">c. </s> <s xml:id="echoid-s4898" xml:space="preserve"><lb/>Dico hos circulos inter ſe eſſe in continua, eademque ratione geometrica, <lb/>ipſamque incedere iuxta quadrata contingentium BL, BM, BC, &</s> <s xml:id="echoid-s4899" xml:space="preserve">c.</s> <s xml:id="echoid-s4900" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4901" xml:space="preserve">Iunctis enim DL, EM, FC, & </s> <s xml:id="echoid-s4902" xml:space="preserve">GL, IC. </s> <s xml:id="echoid-s4903" xml:space="preserve">Cum in triangulis BLD, BCF, <lb/>anguli BLD, BCF ſint recti, & </s> <s xml:id="echoid-s4904" xml:space="preserve">angulus ad B communis, erit reliquus BDL, <pb o="147" file="0171" n="171" rhead=""/> reliquo BFC æqualis, qui ſunt anguli ad <lb/> <anchor type="figure" xlink:label="fig-0171-01a" xlink:href="fig-0171-01"/> centra D, F: </s> <s xml:id="echoid-s4905" xml:space="preserve">ergo ipſorum dimidia ad <lb/>circumferentias, hoc eſt anguli B G L, <lb/>B I C æquales erunt, vnde G L æquidi-<lb/>ſtabit I C: </s> <s xml:id="echoid-s4906" xml:space="preserve">quare, vt C B ad B L, ita I B, <lb/>ad BG, vel ſumpta communi altitudine <lb/>BH, ita rectangulum IBH, ſiue quadra-<lb/>tum B C, ad rectangulum H B G, vel ad <lb/>quadratum BM: </s> <s xml:id="echoid-s4907" xml:space="preserve">cum ergo ſit CB ad BL, <lb/>vt quadratum C B ad quadratum B M, <lb/>erunt tres contingentes BC, BM, BL, <lb/>in eadem ratione geometrica, ſed C B <lb/>ad B M, eſt vt C F ad M E, & </s> <s xml:id="echoid-s4908" xml:space="preserve">M B ad <lb/>B L, vt M E ad L D; </s> <s xml:id="echoid-s4909" xml:space="preserve">ergo C F, M E, <lb/>L D, vti etiam ipſarum quadrata, ſiue <lb/>_MAXIMI_ circuli ex FC, EM, DL erunt <lb/>in eadem ratione geometrica, quę pro-<lb/>cedit iuxta quadrata contingentium <lb/>B C, B M, B L. </s> <s xml:id="echoid-s4910" xml:space="preserve">Quod oſtendere pro-<lb/>ponebatur.</s> <s xml:id="echoid-s4911" xml:space="preserve"/> </p> <div xml:id="echoid-div492" type="float" level="2" n="1"> <figure xlink:label="fig-0171-01" xlink:href="fig-0171-01a"> <image file="0171-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0171-01"/> </figure> </div> </div> <div xml:id="echoid-div494" type="section" level="1" n="200"> <head xml:id="echoid-head205" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s4912" xml:space="preserve">HInc elicitur, quod ſi datus angulus fuerit angulus trianguli æquilateri, <lb/>ſiue duæ tertiæ vnius recti, prædicti _MAXIMI_ circuli erunt inter ſe <lb/>in continua progreſſione nonupla. </s> <s xml:id="echoid-s4913" xml:space="preserve">Tunc enim in triangulo ęquilatero BNO, <lb/>_MAXIMVS_ inſcriptus circulus ex DG ſingula latera ad puncta contactuum <lb/>bifariam ſecabit, quare BL æquabitur LN, ſiue NG, ſiue NM, (cum circu-<lb/>lum contingentes, ex eodem puncto ſint æquales) hoc eſt BM erit tripla <lb/>BL, & </s> <s xml:id="echoid-s4914" xml:space="preserve">quadratum BM nonuplum quadrati B L, vel circulus ex EM nonu-<lb/>plus circuli ex DL, itemque circulus ex F C nonuplus circuli ex E M, cum <lb/>ſint in eadem proportione geometrica, & </s> <s xml:id="echoid-s4915" xml:space="preserve">hoc ſemper, quotcunq; </s> <s xml:id="echoid-s4916" xml:space="preserve">ſint huiuſ-<lb/>modi circuli ſe mutuò, & </s> <s xml:id="echoid-s4917" xml:space="preserve">prædicti anguli latera contingentes.</s> <s xml:id="echoid-s4918" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4919" xml:space="preserve">Hic autem notandum eſt inter hos _MAXIMOS_ circulos non dari _MAXI-_ <lb/>_MVM_, cum infra circulum FC alij infiniti in eadem progreſſione dato angu-<lb/>lo inſcribi poſſint, eò quod ipſe ad partes L ſit infinitæ extenſionis.</s> <s xml:id="echoid-s4920" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4921" xml:space="preserve">Item inter eoſdem _MAXIMOS_ circulos non dari _MINIMVM_; </s> <s xml:id="echoid-s4922" xml:space="preserve">quoniam <lb/>ad partes verticis B, ſupra circulum DL, reſiduo trilineo, licet terminato, <lb/>alij infiniti circuli perpetuò decreſcentes inſcribi poſſunt.</s> <s xml:id="echoid-s4923" xml:space="preserve"/> </p> <pb o="148" file="0172" n="172" rhead=""/> </div> <div xml:id="echoid-div495" type="section" level="1" n="201"> <head xml:id="echoid-head206" xml:space="preserve">THEOR. IL. PROP. IIC.</head> <p> <s xml:id="echoid-s4924" xml:space="preserve">MAXIMORVM circulorum, ad puncta Parabolicę, aut Hy-<lb/>perbolicæ peripheriæ inſcriptorum, MINIMVS eſt, qui ad axis <lb/>verticem inſcribitur. </s> <s xml:id="echoid-s4925" xml:space="preserve">Aliorum verò is, cuius contactus magis <lb/>diſtat à vertice, maior eſt, neque datur MAXIMVS.</s> <s xml:id="echoid-s4926" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4927" xml:space="preserve">ESto Parabole, vel Hyperbole ABC, cuius axis B D, vertex B, & </s> <s xml:id="echoid-s4928" xml:space="preserve">in <lb/>eius peripheria ſumpta ſint quælibet puncta A, E extra verticem <lb/>B, à quo agantur contingentibus per-<lb/> <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> pendiculares AD, E G, & </s> <s xml:id="echoid-s4929" xml:space="preserve">ab axe <lb/>abſciſſa ſit B F, æqualis dimidio recti <lb/>datæ ſectionis. </s> <s xml:id="echoid-s4930" xml:space="preserve">Patet ſi cum centris <lb/>F, G, D, inueruallis verò FB, GE, <lb/>DA circuli deſcribantur, ipſos datæ <lb/>ſectioni ABC eſſe inſcriptos, atque <lb/>_MAXIMOS_ <anchor type="note" xlink:href="" symbol="a"/> ad puncta B, E, A in- <anchor type="note" xlink:label="note-0172-01a" xlink:href="note-0172-01"/> ſcriptibilium. </s> <s xml:id="echoid-s4931" xml:space="preserve">Dico iam inter hos _MA-_ <lb/>_XIMOS, MINIMVM_ eſſe eum, qui ad <lb/>verticem B inſcribitur. </s> <s xml:id="echoid-s4932" xml:space="preserve">Aliorum au-<lb/>tem illum, qui ad punctum E propin-<lb/>quius vertici, minorem eſſe eo, qui <lb/>ad A vertici remotius, inſcribitur.</s> <s xml:id="echoid-s4933" xml:space="preserve"/> </p> <div xml:id="echoid-div495" 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/QN4GHYBF/figures/0172-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0172-01" xlink:href="note-0172-01a" xml:space="preserve">1. Co-<lb/>roll. 20. h. <lb/>& 95. h.</note> </div> <p> <s xml:id="echoid-s4934" xml:space="preserve">Nam quælibet perpendicularis GE, DA, &</s> <s xml:id="echoid-s4935" xml:space="preserve">c. </s> <s xml:id="echoid-s4936" xml:space="preserve">maior <anchor type="note" xlink:href="" symbol="b"/> eſt dimidio re- <anchor type="note" xlink:label="note-0172-02a" xlink:href="note-0172-02"/> cti, ſiue maior FB: </s> <s xml:id="echoid-s4937" xml:space="preserve">quare circulus ex FB erit _MINIMVS_, &</s> <s xml:id="echoid-s4938" xml:space="preserve">c. </s> <s xml:id="echoid-s4939" xml:space="preserve">ſed G E, <lb/>quæ à contactu vertici propiori, minor <anchor type="note" xlink:href="" symbol="c"/> eſt D A, que à remotiori: </s> <s xml:id="echoid-s4940" xml:space="preserve">qua- <anchor type="note" xlink:label="note-0172-03a" xlink:href="note-0172-03"/> re circulus ex G E, erit minor circulo ex G A, &</s> <s xml:id="echoid-s4941" xml:space="preserve">c. </s> <s xml:id="echoid-s4942" xml:space="preserve">neque inter hos, <lb/>_MAXIMVS_ reperitur, cum ſectio Parabole, aut Hyperbole ad partes ver-<lb/>tici oppoſitas ſit infinitæ cxtenſionis, ac proinde vnquam ei inſcribi ne-<lb/>queat circulus tàm longi interualli, quin infra alij adhuc maioris inter-<lb/>ualli inſcribi poſſint. </s> <s xml:id="echoid-s4943" xml:space="preserve">Quod tandem erat demonſtrandum.</s> <s xml:id="echoid-s4944" xml:space="preserve"/> </p> <div xml:id="echoid-div496" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">1. Co-<lb/>roll. 90. h.</note> <note symbol="c" position="left" xlink:label="note-0172-03" xlink:href="note-0172-03a" xml:space="preserve">93. h.</note> </div> </div> <div xml:id="echoid-div498" type="section" level="1" n="202"> <head xml:id="echoid-head207" xml:space="preserve">THEOR. L. PROP. IC.</head> <p> <s xml:id="echoid-s4945" xml:space="preserve">MAXIMORVM circulorum, ad puncta Ellipticæ peri-<lb/>pheriæ inſcriptorum, MAXIMVS eſt qui ad verticem mino-<lb/>ris axis inſcribitur. </s> <s xml:id="echoid-s4946" xml:space="preserve">MINIMVS verò, qui ad verticem maio-<lb/>ris. </s> <s xml:id="echoid-s4947" xml:space="preserve">Aliorum autem is, cuius contactus à vertice maioris axis <lb/>magis remouetur, maior eſt.</s> <s xml:id="echoid-s4948" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4949" xml:space="preserve">ESto Ellipſis ABCD, cuius axis maior BD, minor A C, centrum E, <lb/>ſitq; </s> <s xml:id="echoid-s4950" xml:space="preserve">DF æqualis dimidio recti, cuius tranſuerſum latus eſt BD; </s> <s xml:id="echoid-s4951" xml:space="preserve">& </s> <s xml:id="echoid-s4952" xml:space="preserve">ex <pb o="149" file="0173" n="173" rhead=""/> punctis G, H, in Ellipſis peripheria vbi-<lb/> <anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/> cunque inter ſemi-axes aſſumptis, ſint <lb/>contingentibus perpẽdiculares GI, HL. <lb/></s> <s xml:id="echoid-s4953" xml:space="preserve">Conſtat, ſi cum centris E, L, I, F, inter-<lb/>uallis verò EA, LH, IG, FB, circuli de-<lb/>ſcribantur, ipſos Ellipſi ABCD inſcri-<lb/>ptos eſſe, ac _MAXIMOS_ <anchor type="note" xlink:href="" symbol="a"/> ad puncta <anchor type="note" xlink:label="note-0173-01a" xlink:href="note-0173-01"/> A, H, G, B inſcriptibilium. </s> <s xml:id="echoid-s4954" xml:space="preserve">Dico iam <lb/>inter hos _MAXIMOS, MAXIMV M_ <lb/>eſſe qui ad A, _MINIMVM_ verò, qui <lb/>ad B inſcribitur. </s> <s xml:id="echoid-s4955" xml:space="preserve">Aliorum autem inſcri-<lb/>ptum ad punctum H, quod à vertice <lb/>B maioris axis magis remouetur, maio-<lb/>rem eſſe inſcripto ad punctum G, quod <lb/>ipſi vertici propius eſt.</s> <s xml:id="echoid-s4956" xml:space="preserve"/> </p> <div xml:id="echoid-div498" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0173-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0173-01" xlink:href="note-0173-01a" xml:space="preserve">26. 92. h. <lb/>1. Coroll. <lb/>20. h.</note> </div> <p> <s xml:id="echoid-s4957" xml:space="preserve">Etenim quelibet perpendicularis LH, <lb/>IG inter ſemi-axes, minor eſt ſemi- axe maiori EA, ſed maior <anchor type="note" xlink:href="" symbol="b"/> ſemper ſe- <anchor type="note" xlink:label="note-0173-02a" xlink:href="note-0173-02"/> mi- recto F B: </s> <s xml:id="echoid-s4958" xml:space="preserve">vnde circulus ex E A erit _MAXIMVS_, & </s> <s xml:id="echoid-s4959" xml:space="preserve">ex F B _MINI-_ <lb/>_MVS_ inſcriptibilium: </s> <s xml:id="echoid-s4960" xml:space="preserve">ſed L H maior <anchor type="note" xlink:href="" symbol="c"/> eſt I G: </s> <s xml:id="echoid-s4961" xml:space="preserve">quapropter circulus ex <anchor type="note" xlink:label="note-0173-03a" xlink:href="note-0173-03"/> L A, erit maior circulo ex I G, Quod probandum erat.</s> <s xml:id="echoid-s4962" xml:space="preserve"/> </p> <div xml:id="echoid-div499" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0173-02" xlink:href="note-0173-02a" xml:space="preserve">91. h.</note> <note symbol="c" position="right" xlink:label="note-0173-03" xlink:href="note-0173-03a" xml:space="preserve">94. h.</note> </div> </div> <div xml:id="echoid-div501" type="section" level="1" n="203"> <head xml:id="echoid-head208" xml:space="preserve">THEOR. LI. PROP. C.</head> <p> <s xml:id="echoid-s4963" xml:space="preserve">MINIMORVM circulorum ad puncta Ellipticæ periphe-<lb/>riæ circumſcriptorum, MINIMVS eſt, qui ad verticem maio-<lb/>ris axis circumſcribitur. </s> <s xml:id="echoid-s4964" xml:space="preserve">MAXIMVS verò qui ad verticem <lb/>minoris. </s> <s xml:id="echoid-s4965" xml:space="preserve">Aliorum autem is, cuius contactus à vertice minoris <lb/>axis magis diſtat, minor eſt.</s> <s xml:id="echoid-s4966" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4967" xml:space="preserve">ESto Ellipſis ABCD, cuius axis maior A C, minor B D, centrum E, <lb/>& </s> <s xml:id="echoid-s4968" xml:space="preserve">ſumpta ſit BF æqualis dimidio recti, cuius tranſuerſum latus eſt <lb/>BD, & </s> <s xml:id="echoid-s4969" xml:space="preserve">ex punctis G, H, vbi-<lb/>cunque in Ellipſis peripheria <lb/> <anchor type="figure" xlink:label="fig-0173-02a" xlink:href="fig-0173-02"/> inter ſemi- axes aſſumptis, ſint <lb/>contingentibus perpendicula-<lb/>res GI, H L. </s> <s xml:id="echoid-s4970" xml:space="preserve">Conſtat iam, ſi <lb/>ex centris E, L, I, F, cum in-<lb/>teruallis EA, LH, I G, F B de-<lb/>ſcribantur circuli, ipſos Ellipſi <lb/>ABCD circumſcriptos eſſe, & </s> <s xml:id="echoid-s4971" xml:space="preserve"><lb/>_MINIMOS_ <anchor type="note" xlink:href="" symbol="d"/> ad puncta A, H, <anchor type="note" xlink:label="note-0173-04a" xlink:href="note-0173-04"/> G, B, circumſcriptibilium. </s> <s xml:id="echoid-s4972" xml:space="preserve">Di-<lb/>co tamen inter hos _MINI-_ <lb/>_MOS, MINIMVM_ eſſe, qui <lb/>ad A; </s> <s xml:id="echoid-s4973" xml:space="preserve">_MAXIMVM_, quiad B <pb o="150" file="0174" n="174" rhead=""/> circumſcribitur. </s> <s xml:id="echoid-s4974" xml:space="preserve">Aliorum verò, inſcriptum ad H, minorem eſſe inſcripto <lb/>ad punctum G, quod minoris axis vertici propinquius eſt.</s> <s xml:id="echoid-s4975" xml:space="preserve"/> </p> <div xml:id="echoid-div501" type="float" level="2" n="1"> <figure xlink:label="fig-0173-02" xlink:href="fig-0173-02a"> <image file="0173-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0173-02"/> </figure> <note symbol="d" position="right" xlink:label="note-0173-04" xlink:href="note-0173-04a" xml:space="preserve">26. 92. h. <lb/>1. Coroll. <lb/>20. h.</note> </div> <p> <s xml:id="echoid-s4976" xml:space="preserve">Quæuis enim perpendicularis LH, I G inter ſemi - axes, maior eſt <lb/>ſemi- axe maiori E A, ſed minor <anchor type="note" xlink:href="" symbol="a"/> ſemper ſemi- recto FB;</s> <s xml:id="echoid-s4977" xml:space="preserve"> <anchor type="note" xlink:label="note-0174-01a" xlink:href="note-0174-01"/> vnde circulus ex EA erit _MINIMVS_, & </s> <s xml:id="echoid-s4978" xml:space="preserve">ex FB <lb/>_MAXIMVS_ circumſcriptibilium; </s> <s xml:id="echoid-s4979" xml:space="preserve">ſed eſt <lb/>L H <anchor type="note" xlink:href="" symbol="b"/> minor I G: </s> <s xml:id="echoid-s4980" xml:space="preserve">quare circulus ex <anchor type="note" xlink:label="note-0174-02a" xlink:href="note-0174-02"/> L H erit minor circulo ex <lb/>IG. </s> <s xml:id="echoid-s4981" xml:space="preserve">Quod erat pro-<lb/>poſitum.</s> <s xml:id="echoid-s4982" xml:space="preserve"/> </p> <div xml:id="echoid-div502" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0174-01" xlink:href="note-0174-01a" xml:space="preserve">91. h.</note> <note symbol="b" position="left" xlink:label="note-0174-02" xlink:href="note-0174-02a" xml:space="preserve">94. h.</note> </div> <p style="it"> <s xml:id="echoid-s4983" xml:space="preserve">At rotundus hic Propoſitionum nnmerus, eſt quæſo</s> </p> </div> <div xml:id="echoid-div504" type="section" level="1" n="204"> <head xml:id="echoid-head209" xml:space="preserve">PRIMI LIBRI <lb/>FINIS.</head> <pb o="151" file="0175" n="175"/> </div> <div xml:id="echoid-div505" type="section" level="1" n="205"> <head xml:id="echoid-head210" xml:space="preserve">ADDENDA LIB. I.</head> <p style="it"> <s xml:id="echoid-s4984" xml:space="preserve">IN huius operis contextu, vel etiam in ipſa perſcriptione, quædam <lb/>ſunt, quæ aut mentem noſtram, aut Amanuenſis, quamuis accura-<lb/>tiſsimi, oculum effugerant: </s> <s xml:id="echoid-s4985" xml:space="preserve">itaque ſub calcem vniuſcuiuſque libri eadem ſic <lb/>addere liceat.</s> <s xml:id="echoid-s4986" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div506" type="section" level="1" n="206"> <head xml:id="echoid-head211" xml:space="preserve">Pag. 74. ad finem Prim. Coroll.</head> <p> <s xml:id="echoid-s4987" xml:space="preserve">Quapropter huiuſmodi Parabolæ iuxta has interceptas lineas diametro B <lb/>E parallelas, ſunt ſemper inter ſe ęquidiſtantes, licet iuxta intercepta appli-<lb/>catarum ſegmenta A E, I D, L M, & </s> <s xml:id="echoid-s4988" xml:space="preserve">ad eaſdem partes A I, E D ſint ſem-<lb/>per ſimul accedentes, nunquam verò coeuntes.</s> <s xml:id="echoid-s4989" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div507" type="section" level="1" n="207"> <head xml:id="echoid-head212" xml:space="preserve">Ad calcem Pag. 78. <lb/>COROLL. II.</head> <p> <s xml:id="echoid-s4990" xml:space="preserve">PAtet denique congruentes Hyperbolas per diuerſos vèrtices ſimul ad-<lb/>ſcriptas, & </s> <s xml:id="echoid-s4991" xml:space="preserve">ad eaſdem partes productas, eſſe inter ſe, & </s> <s xml:id="echoid-s4992" xml:space="preserve">ſimul ſemper ma-<lb/>gis accedentes, & </s> <s xml:id="echoid-s4993" xml:space="preserve">ſemper æquidiſtantes. </s> <s xml:id="echoid-s4994" xml:space="preserve">Nam iuxta intercepta applicata-<lb/>rum ſegmenta A E, S D, X Y, in præcedentibus figuris huiuſmodi Hyper-<lb/>bolæ ſemper fiunt propiores, licet nunquam ſimul conueniant; </s> <s xml:id="echoid-s4995" xml:space="preserve">iuxta autem <lb/>rectas B E, M D, Z Y, ad eaſdem partes A S, E D, perpetuam ſeruant ęqui-<lb/>diſtantiam, cum ipſæ B E, M D, Z Y inter ſe æquales ſint oſtenſæ, &</s> <s xml:id="echoid-s4996" xml:space="preserve">c.</s> <s xml:id="echoid-s4997" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div508" type="section" level="1" n="208"> <head xml:id="echoid-head213" xml:space="preserve">Pag. 87. ad finem Moniti.</head> <p style="it"> <s xml:id="echoid-s4998" xml:space="preserve">atque item congruentes Hyperbolæ, &</s> <s xml:id="echoid-s4999" xml:space="preserve">c. </s> <s xml:id="echoid-s5000" xml:space="preserve">prout in 2. </s> <s xml:id="echoid-s5001" xml:space="preserve">Coroll. </s> <s xml:id="echoid-s5002" xml:space="preserve">prop. </s> <s xml:id="echoid-s5003" xml:space="preserve">quadrageſi-<lb/>quartæ monuimus.</s> <s xml:id="echoid-s5004" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div509" type="section" level="1" n="209"> <head xml:id="echoid-head214" xml:space="preserve">Pag. 123. poſt Prop. 77. <lb/>Aliter idem, ac Vniuerſaliùs.</head> <p> <s xml:id="echoid-s5005" xml:space="preserve">MAXIMÆ ſimiles Ellipſes, Parabolæ inſcriptę, & </s> <s xml:id="echoid-s5006" xml:space="preserve">à vertice <lb/>ſucceſſiuè ſe mutuò contingentes, ſunt inter ſe in ratione quadra-<lb/>torum, diſparium numerorum ab vnitate incipientium.</s> <s xml:id="echoid-s5007" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5008" xml:space="preserve">ESto Parabole A B C, cuius diameter B D, latus rectum B E, & </s> <s xml:id="echoid-s5009" xml:space="preserve">circa <lb/>quodlibet diametri ſegmentum B F ſit ipſi Parabolæ per verticem E <lb/>inicripta _MAXIMA_ Ellipſis B F (quę erit <anchor type="note" xlink:href="" symbol="*"/> illa, cuius rectum latus idem ſit, <anchor type="note" xlink:label="note-0175-01a" xlink:href="note-0175-01"/> ac rectum B E) & </s> <s xml:id="echoid-s5010" xml:space="preserve">applicata ex F ad diametrum recta H F G, ſumptaque <lb/>F I æquali ipſi F B, ducatur diagonalis G I L, ex L applicetur L M N, <pb o="152" file="0176" n="176" rhead=""/> atque ex N agatur N P O ipſi G L parallela, ex O verò recta O Q R paral-<lb/>lela ad L N, & </s> <s xml:id="echoid-s5011" xml:space="preserve">R S A ad N O, atque A D C ipſi O R, & </s> <s xml:id="echoid-s5012" xml:space="preserve">hoc fiat quoties li-<lb/>buerit: </s> <s xml:id="echoid-s5013" xml:space="preserve">patet, ſi per puncta I, P, S, interſectionum ipſarum diagona-<lb/>lium cum diametro, agantur applicatæ T V, X Y, Z K, &</s> <s xml:id="echoid-s5014" xml:space="preserve">c. </s> <s xml:id="echoid-s5015" xml:space="preserve">& </s> <s xml:id="echoid-s5016" xml:space="preserve">circa diametri <lb/>ſegmenta F M, M Q, Q D, &</s> <s xml:id="echoid-s5017" xml:space="preserve">c. </s> <s xml:id="echoid-s5018" xml:space="preserve">& </s> <s xml:id="echoid-s5019" xml:space="preserve">per extrema prædictarum applicatarum <lb/>deſcribantur Ellipſes T F V M, X M Y Q, Z Q K D, has omnes eſſe Para-<lb/>bolæ A B C inſcriptas, <anchor type="note" xlink:href="" symbol="a"/> & </s> <s xml:id="echoid-s5020" xml:space="preserve">ſimiles inter ſe, ac ſe mutuò ſucceſſiuè contin- <anchor type="note" xlink:label="note-0176-01a" xlink:href="note-0176-01"/> gentes. </s> <s xml:id="echoid-s5021" xml:space="preserve">Iam dico eaſdem Ellipſes, primæ B F ſimiles eſſe, atq; </s> <s xml:id="echoid-s5022" xml:space="preserve">inter ſe eam <lb/>rationem habere, ac numeri quadrati diſparium numerorum ab vnitate: </s> <s xml:id="echoid-s5023" xml:space="preserve">ni-<lb/>mirum eſſe in progreſſione numerorum 1. </s> <s xml:id="echoid-s5024" xml:space="preserve">9. </s> <s xml:id="echoid-s5025" xml:space="preserve">25. </s> <s xml:id="echoid-s5026" xml:space="preserve">49. </s> <s xml:id="echoid-s5027" xml:space="preserve">&</s> <s xml:id="echoid-s5028" xml:space="preserve">c.</s> <s xml:id="echoid-s5029" xml:space="preserve"/> </p> <div xml:id="echoid-div509" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-0175-01" xlink:href="note-0175-01a" xml:space="preserve">20. h.</note> <note symbol="a" position="left" xlink:label="note-0176-01" xlink:href="note-0176-01a" xml:space="preserve">@. h.</note> </div> <p> <s xml:id="echoid-s5030" xml:space="preserve">Quoniam igitur eſt <anchor type="note" xlink:href="" symbol="b"/> M B ad B I, vt B I ad <anchor type="note" xlink:label="note-0176-02a" xlink:href="note-0176-02"/> <anchor type="figure" xlink:label="fig-0176-01a" xlink:href="fig-0176-01"/> B F, erit diuidédo M I ad I B, vt I F ad F B, <lb/>ſed eſt I F æqualis F B, ex conſtructione, <lb/>quare M I ipſi I B æqualis erit, ac ideo in <lb/>Ellipſi T F V M, erit quadratum T I ad re-<lb/>ctangulum M I F, hoc eſt rectum eius <anchor type="note" xlink:href="" symbol="c"/> latus <anchor type="note" xlink:label="note-0176-03a" xlink:href="note-0176-03"/> ad tranſuerſum, vt idem quadratum T I, vel <lb/>rectangulum <anchor type="note" xlink:href="" symbol="d"/> ſub I B, & </s> <s xml:id="echoid-s5031" xml:space="preserve">recto B E, ad rectá- <anchor type="note" xlink:label="note-0176-04a" xlink:href="note-0176-04"/> gulum ſub eadem I B, & </s> <s xml:id="echoid-s5032" xml:space="preserve">ſub I F, hoc eſt vt <lb/>linea B E ad I F, (cum ſit I B communis re-<lb/>ctangulorum altitudo) vel ad ei æqualem B <lb/>F, nempe vt rectum ad tranſuerſum Ellipſis <lb/>BF: </s> <s xml:id="echoid-s5033" xml:space="preserve">quapropter Ellipſis B F ipſi T F V M eri <lb/>ſimilis, ſed vnaqueque aliarum inſcriptarum <lb/>Ellipſium circ? </s> <s xml:id="echoid-s5034" xml:space="preserve">diametri ſegmenta M Q, Q <lb/>D, &</s> <s xml:id="echoid-s5035" xml:space="preserve">c. </s> <s xml:id="echoid-s5036" xml:space="preserve">eidem T F V M eſt ſimilis, vt ſupra <lb/>monuimus, quare omnes huiuſm odi inſcri <lb/>ptæ Ellipſes erunt ſimiles inter ſe. </s> <s xml:id="echoid-s5037" xml:space="preserve">Et cum ſit <lb/>M I ęqualis I B, & </s> <s xml:id="echoid-s5038" xml:space="preserve">I B dupla F B, erit to-<lb/>ta M B quadrupla B F. </s> <s xml:id="echoid-s5039" xml:space="preserve">Si ergo B F conci-<lb/>piatur vt vnum, erit B I vt 2, & </s> <s xml:id="echoid-s5040" xml:space="preserve">M B vt 4, <lb/>atque M I vt 2, & </s> <s xml:id="echoid-s5041" xml:space="preserve">M F vt 3.</s> <s xml:id="echoid-s5042" xml:space="preserve"/> </p> <div xml:id="echoid-div510" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0176-02" xlink:href="note-0176-02a" xml:space="preserve">1. Co-<lb/>roll. 13. h.</note> <figure xlink:label="fig-0176-01" xlink:href="fig-0176-01a"> <image file="0176-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0176-01"/> </figure> <note symbol="c" position="left" xlink:label="note-0176-03" xlink:href="note-0176-03a" xml:space="preserve">21. pri-<lb/>mi Conic.</note> <note symbol="d" position="left" xlink:label="note-0176-04" xlink:href="note-0176-04a" xml:space="preserve">Coroll. <lb/>1. huius.</note> </div> <p> <s xml:id="echoid-s5043" xml:space="preserve">Cumq; </s> <s xml:id="echoid-s5044" xml:space="preserve">in triangulis P M N, I F G ſint anguli ad M, F inter ſe æquales, <lb/>ob æquidiſtantes applicatas M N, F G; </s> <s xml:id="echoid-s5045" xml:space="preserve">atque anguli ad P, I item ęquales, <lb/>ob parallelas diagonales N P, G I, erunt reliqui ad N, G pariter æquales, <lb/>ſiue ipſa triangula inter ſe ſimilia, vnde latus N M ad M P erit, vt latus G F <lb/>ad F I, & </s> <s xml:id="echoid-s5046" xml:space="preserve">permutando N M ad G F, vt M P ad F I, vel quadratum N M ad <lb/>G F, ſiue <anchor type="note" xlink:href="" symbol="e"/> recta M B ad B F; </s> <s xml:id="echoid-s5047" xml:space="preserve">hoc eſt 4. </s> <s xml:id="echoid-s5048" xml:space="preserve">ad 1, vt quadratum M P ad qua- <anchor type="note" xlink:label="note-0176-05a" xlink:href="note-0176-05"/> dratum F I; </s> <s xml:id="echoid-s5049" xml:space="preserve">vnde quadratum M P quadruplum erit quadrati F I, ſiue linea <lb/>M P dupla F I, ſiue dupla ad B F, ſed B F ponitur vt vnum, ergo M P erit <lb/>2; </s> <s xml:id="echoid-s5050" xml:space="preserve">eſtque B M 4, ergo B P erit 6, eſtque B M ad B P, vt eſt <anchor type="note" xlink:href="" symbol="f"/> B P ad B <anchor type="note" xlink:label="note-0176-06a" xlink:href="note-0176-06"/> Q, quare B Q erit vt 9, ſed B M eſt vt 4, ergo M Q erit 5. </s> <s xml:id="echoid-s5051" xml:space="preserve">Præterea, ea-<lb/>dem ratione, ac ſupra, oſtendetur triangulum R Q S ſimile triangulo G F I, <lb/>& </s> <s xml:id="echoid-s5052" xml:space="preserve">quadratum R Q ad G F eſſe vt quadratum Q S ad F I, ſed eſt <anchor type="note" xlink:href="" symbol="g"/> quadra- <anchor type="note" xlink:label="note-0176-07a" xlink:href="note-0176-07"/> tum R Q nonuplum quadrati G F, cum ſit recta Q B nonupla B F, vt mo-<lb/>dò oſtendimus, ergo, & </s> <s xml:id="echoid-s5053" xml:space="preserve">quadratum Q S erit nonuplum quadrati F I, ſiue <lb/>quadrati B F, hoc eſt linea Q S tripla B F, quare tota B S erit vt 12; </s> <s xml:id="echoid-s5054" xml:space="preserve">eſtq;</s> <s xml:id="echoid-s5055" xml:space="preserve"> <pb o="153" file="0177" n="177" rhead=""/> B Q ad B S, <anchor type="note" xlink:href="" symbol="a"/> vt B S ad B D, quare cum B Q ſit 9, & </s> <s xml:id="echoid-s5056" xml:space="preserve">B S 12, erit B D 16, <anchor type="note" xlink:label="note-0177-01a" xlink:href="note-0177-01"/> & </s> <s xml:id="echoid-s5057" xml:space="preserve">Q D 7, & </s> <s xml:id="echoid-s5058" xml:space="preserve">ſic vlteriùs demonſtrabuntur diametri huiuſmodi ſimilium El-<lb/>lipſium Parabolæ inſcriptarum, &</s> <s xml:id="echoid-s5059" xml:space="preserve">c. </s> <s xml:id="echoid-s5060" xml:space="preserve">à vertice ſumptæ, augeri iuxta progreſ-<lb/>ſionem diſparium numerorum ab vnitate, nempe vt numeri 1. </s> <s xml:id="echoid-s5061" xml:space="preserve">3. </s> <s xml:id="echoid-s5062" xml:space="preserve">5. </s> <s xml:id="echoid-s5063" xml:space="preserve">7. </s> <s xml:id="echoid-s5064" xml:space="preserve">9. <lb/></s> <s xml:id="echoid-s5065" xml:space="preserve">11. </s> <s xml:id="echoid-s5066" xml:space="preserve">&</s> <s xml:id="echoid-s5067" xml:space="preserve">c. </s> <s xml:id="echoid-s5068" xml:space="preserve">Sed Ellipſes ſimiles ſunt inter ſe, vt quadrata homologarum diame-<lb/>trorum: </s> <s xml:id="echoid-s5069" xml:space="preserve">quare eædem _MAXIMAE_ Ellipſes Parabolæ A B C inſcriptæ, & </s> <s xml:id="echoid-s5070" xml:space="preserve"><lb/>à vertice ſucceſſiuè ſe mutuò contingentes, ſunt in ratione quadratorum diſ-<lb/>parium numerorum ab vnitate. </s> <s xml:id="echoid-s5071" xml:space="preserve">Quod probandum erat.</s> <s xml:id="echoid-s5072" xml:space="preserve"/> </p> <div xml:id="echoid-div511" type="float" level="2" n="3"> <note symbol="e" position="left" xlink:label="note-0176-05" xlink:href="note-0176-05a" xml:space="preserve">20. pri-<lb/>mi Conic.</note> <note symbol="f" position="left" xlink:label="note-0176-06" xlink:href="note-0176-06a" xml:space="preserve">1. Co-<lb/>roll. 13. h.</note> <note symbol="g" position="left" xlink:label="note-0176-07" xlink:href="note-0176-07a" xml:space="preserve">20. pri-<lb/>mi Conic.</note> <note symbol="a" position="right" xlink:label="note-0177-01" xlink:href="note-0177-01a" xml:space="preserve">1. Co-<lb/>roll. 13. h.</note> </div> </div> <div xml:id="echoid-div513" type="section" level="1" n="210"> <head xml:id="echoid-head215" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s5073" xml:space="preserve">HInc iterum apparet veritas Prop. </s> <s xml:id="echoid-s5074" xml:space="preserve">77. </s> <s xml:id="echoid-s5075" xml:space="preserve">huius. </s> <s xml:id="echoid-s5076" xml:space="preserve">Nam ſi B D diameter da-<lb/>tæ Parabolæ A B C, fuerit axis; </s> <s xml:id="echoid-s5077" xml:space="preserve">& </s> <s xml:id="echoid-s5078" xml:space="preserve">prima Ellipſis circa ſegmentum <lb/>B F fuerit circulus, reliquæ Ellipſes infra hanc ſucceſſiuè inſcriptæ, erunt <lb/>pariter Circuli, & </s> <s xml:id="echoid-s5079" xml:space="preserve">demonſtratio, ac concluſio omnino erit eadem, ac in <lb/>ſuperiori.</s> <s xml:id="echoid-s5080" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div514" type="section" level="1" n="211"> <head xml:id="echoid-head216" xml:space="preserve">Pag. 131. poſt Prop. 84.</head> <p> <s xml:id="echoid-s5081" xml:space="preserve">In hac, & </s> <s xml:id="echoid-s5082" xml:space="preserve">in proxima præcedenti 82. </s> <s xml:id="echoid-s5083" xml:space="preserve">propoſitione ex ipſamet conſtru-<lb/>ctione, ac demonſtratione elicitur, nos vtrobique aſſumpſiſſe datam appli-<lb/>catam A C ad diametrum datæ Ellipſis, nunquam per centrum tranſire: </s> <s xml:id="echoid-s5084" xml:space="preserve">in <lb/>hoc enim caſu vtriuſque Problematis ſolutio facillimè patebit, tunc nimirũ, <lb/>ſi hinc inde à centro ſuper diametrum ſumatur dimidium dati tranſuerſi la-<lb/>teris, atque circa ipſorum dimidiorum aggregatum, tanquam circa tranſ-<lb/>uerſum diametrum, & </s> <s xml:id="echoid-s5085" xml:space="preserve">per extrema ipſius applicatæ deſcribatur Ellipſis, quę <lb/>vel erit <anchor type="note" xlink:href="" symbol="b"/> _MAXIMA_ inſcripta, vel _MINIMA_ datæ Ellipſi circumſcripta, cum eadem applicata A C ſit tanquam communis ſecunda diameter, vel <lb/> <anchor type="note" xlink:label="note-0177-02a" xlink:href="note-0177-02"/> prout commune tranſuerſum latus vtriuſque Ellipſis, &</s> <s xml:id="echoid-s5086" xml:space="preserve">c.</s> <s xml:id="echoid-s5087" xml:space="preserve"/> </p> <div xml:id="echoid-div514" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-0177-02" xlink:href="note-0177-02a" xml:space="preserve">2. Co-<lb/>roll. 19. h.</note> </div> </div> <div xml:id="echoid-div516" type="section" level="1" n="212"> <head xml:id="echoid-head217" xml:space="preserve">Pag. 144. ad calcem Prop. 93.</head> <p style="it"> <s xml:id="echoid-s5088" xml:space="preserve">Lineæ, quæ ibi in figuris iungentes puncta C, D, manifeſtò indicant <lb/>in ipſa tranſcriptione omiſſum fuiſſe ſequens</s> </p> </div> <div xml:id="echoid-div517" type="section" level="1" n="213"> <head xml:id="echoid-head218" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s5089" xml:space="preserve">EX his aliàs manifeſtum fiet haud inutiliter animaduertiſſe in Parabola, <lb/>vel in Hyperbola, aut in quadrante Ellipſis circa maiorem axim, præ-<lb/>dictarum contingentibus perpendicularium ad eaſdem axis partes ductarũ, <lb/>quæ à contactu vertici remotiori ducitur occurrere axi infra occurſum ſupe-<lb/>rioris perpendicularis, ac ſimul vltra axim conuenire ad partes contactibus <lb/>oppoſitas: </s> <s xml:id="echoid-s5090" xml:space="preserve">ſed in quadrante Ellipſis circa minorem axim ſe mutuò ſecare in-<lb/>ter tangentium contactus, & </s> <s xml:id="echoid-s5091" xml:space="preserve">minorem axim in angulo quadrantis, qui dein-<lb/>ceps eſt ei, ad cuius peripheriam ductæ ſunt perpendiculares; </s> <s xml:id="echoid-s5092" xml:space="preserve">ac ideo oc-<lb/>curſum inferioris perpendicularis cum axe minori cadere ſupra occurſum <lb/>ſuperioris, quæ ducitur ex contactu vertici propiori.</s> <s xml:id="echoid-s5093" xml:space="preserve"/> </p> <pb o="154" file="0178" n="178" rhead=""/> <p> <s xml:id="echoid-s5094" xml:space="preserve">In ſingulis enim figuris iuncta recta C D: </s> <s xml:id="echoid-s5095" xml:space="preserve">erit in tribus primis circa maio-<lb/>rem axim, recta C D maior C A (cum circulus ex C A ſit ſectioni <anchor type="note" xlink:href="" symbol="a"/> inſcri- <anchor type="note" xlink:label="note-0178-01a" xlink:href="note-0178-01"/> ptus, ac propterea ſecet C D) ſed C A maior eſt G D, v thìc ad numeros <lb/>2, 3, & </s> <s xml:id="echoid-s5096" xml:space="preserve">5. </s> <s xml:id="echoid-s5097" xml:space="preserve">oſtenſum eſt, ergo C D eò ampliùs maior erit ipſa G D, ſiue <lb/>quadratum C D maius quadrato G D, vel duo ſimul C I, I D maiora <lb/>duobus ſimul G I, I D, quare dempto communi D I, erit quadratum C I <lb/>maius quadrato G I, vnde punctum C cadet infra G: </s> <s xml:id="echoid-s5098" xml:space="preserve">ſed A C, D G ſi-<lb/>mul conueniunt ad partes axis B R, vt ad num. </s> <s xml:id="echoid-s5099" xml:space="preserve">1. </s> <s xml:id="echoid-s5100" xml:space="preserve">oſtendimus, ergo ipſa-<lb/>rum occurſus erit vltra axim B R.</s> <s xml:id="echoid-s5101" xml:space="preserve"/> </p> <div xml:id="echoid-div517" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0178-01" xlink:href="note-0178-01a" xml:space="preserve">92. h</note> </div> <p> <s xml:id="echoid-s5102" xml:space="preserve">In quarta demum figura, eſt C D minor C A (cum circulus ex C A ſit <lb/>Ellipſi circumſcriptus <anchor type="note" xlink:href="" symbol="b"/>) & </s> <s xml:id="echoid-s5103" xml:space="preserve">C A minor G D, prout ad num. </s> <s xml:id="echoid-s5104" xml:space="preserve">6. </s> <s xml:id="echoid-s5105" xml:space="preserve">huius de- <anchor type="note" xlink:label="note-0178-02a" xlink:href="note-0178-02"/> monſtrauimus, quare C D erit omnino minor G D, ſiue quadratum C D <lb/>minus quadrato G D, vel duo ſimul C I, I D minora duobus ſimul G I, <lb/>I D; </s> <s xml:id="echoid-s5106" xml:space="preserve">quamobré dempto I D, erit C I minus G I, ſiue punctum C occurſus <lb/>inferioris perpendicularis A C cadet ſupra G occurſum ſuperioris D G; <lb/></s> <s xml:id="echoid-s5107" xml:space="preserve">ſed tales perpendiculares A C, D G ſe mutuò ſecant (vt ſuperiùs oſten-<lb/>dimus ad num. </s> <s xml:id="echoid-s5108" xml:space="preserve">1.) </s> <s xml:id="echoid-s5109" xml:space="preserve">ad partes axis B R, quare ipſarum occurſus erit inter <lb/>contactus, & </s> <s xml:id="echoid-s5110" xml:space="preserve">minorem axim, ſed reſpectu maiorem axim M L ſe mutuò <lb/>ſecant vltra M L, vti paulò ante demonſtrauimus. </s> <s xml:id="echoid-s5111" xml:space="preserve">Quare in Ellipſi oc-<lb/>curſus huiuſmodi perpendicularium A C, D G cadet in angulo quadran-<lb/>tis M L G, qui deinceps eſt quadranti M L B, ad cuius peripheriam M A <lb/>B ductæ ſunt perpendiculares A C, D G, &</s> <s xml:id="echoid-s5112" xml:space="preserve">c.</s> <s xml:id="echoid-s5113" xml:space="preserve"/> </p> <div xml:id="echoid-div518" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0178-02" xlink:href="note-0178-02a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div520" type="section" level="1" n="214"> <head xml:id="echoid-head219" xml:space="preserve">Pag. 147. ad finem Prop. 97.</head> <p> <s xml:id="echoid-s5114" xml:space="preserve">quodque de _MAXIMIS_ ſimilibus Ellipſibus angulo rectilineo inſcriptis <lb/>facillimùm eſt demonſtrare.</s> <s xml:id="echoid-s5115" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div521" type="section" level="1" n="215"> <head xml:id="echoid-head220" xml:space="preserve">FINIS.</head> <pb file="0179" n="179"/> </div> <div xml:id="echoid-div522" type="section" level="1" n="216"> <head xml:id="echoid-head221" xml:space="preserve"><emph style="red">DE MAXIMIS,</emph> <lb/>ET <lb/><emph style="red">MINIMIS</emph> <lb/>GEOMETRICA DIVINATIO <lb/><emph style="red"><emph style="sc">In</emph> <emph style="sc">Qvintvm</emph> <emph style="sc">Conicorvm</emph></emph> <lb/><emph style="red">APOLLONII PERGÆI</emph> <lb/>_IAMDIV DESIDERATVM._ <lb/>AD SER ENISSIMVM <lb/><emph style="red">PRINCIPEM LEOPOLDVM</emph> <lb/>AB ETRVRIA. <lb/><emph style="red">LIBER SECVNDVS.</emph> <lb/>_AVCTORE_ <lb/><emph style="red">VINCENTIO VIVIANI.</emph></head> <figure> <image file="0179-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0179-01"/> </figure> </div> <div xml:id="echoid-div523" type="section" level="1" n="217"> <head xml:id="echoid-head222" xml:space="preserve"><emph style="red">FLORENTIÆ MDCLIX.</emph> <lb/>Apud Ioſeph Cocchini, Typis Nouis, ſub Signo STELLÆ. <lb/><emph style="red">_SVPERIORVM PERMISSV._</emph></head> <pb file="0180" n="180"/> <pb file="0181" n="181"/> </div> <div xml:id="echoid-div524" type="section" level="1" n="218"> <head xml:id="echoid-head223" xml:space="preserve">SERENISSIMO <lb/>PRINCIPI LEOPOLODO <lb/>AB ETRVRIA.</head> <p> <s xml:id="echoid-s5116" xml:space="preserve">ABSVRDVM, aut inſolens minimè <lb/>quidem eſt SER ENISSIME <lb/>PRINCEPS, non tantùm aliena <lb/>largiri, verùm etiam muneris nomi-<lb/>ne animo libenti propria ſuſcipere. <lb/></s> <s xml:id="echoid-s5117" xml:space="preserve">Quid enim vnquam Deo Opt. </s> <s xml:id="echoid-s5118" xml:space="preserve">Max. </s> <s xml:id="echoid-s5119" xml:space="preserve"><lb/>mortales offerre poſsẽt, niſi ſuis quo-<lb/>que hoſtijs diuina benignitas oblectaretur? </s> <s xml:id="echoid-s5120" xml:space="preserve">Quid ego <lb/>Celſitudini tuæ, cuius patrocinio omnia debeo, niſi quę <lb/>tua ſunt tibi reddi magnanimè patereris? </s> <s xml:id="echoid-s5121" xml:space="preserve">Ab impuden-<lb/>tiæ nota me liberas, & </s> <s xml:id="echoid-s5122" xml:space="preserve">frontem meam rubori ſubtrahis <lb/>SERENISS. </s> <s xml:id="echoid-s5123" xml:space="preserve">LEOPOLDE. </s> <s xml:id="echoid-s5124" xml:space="preserve">Fidentiùs enim mentis <lb/>meæ tenuiſsimos partus tibi nunc exhibere audeo, Re-<lb/>gia namq; </s> <s xml:id="echoid-s5125" xml:space="preserve">manu obſtetrice, è tenebris in quibus delite-<lb/>ſcebant in lucem eductos, quos nuper vt proprios deſpi-<lb/>ciebam, modò à perſpicaciſsimo iudicio tuo in cliente-<lb/>lam, atque, vt ita dicam, in liberorum locum humaniſ-<lb/>ſimè ſuſceptos nonnihil æſtimare cogor. </s> <s xml:id="echoid-s5126" xml:space="preserve">Quid ergo <lb/>lucubrationes haſce meas, quæ tuæ iam ſunt, tibi am-<lb/>pliùs commendem? </s> <s xml:id="echoid-s5127" xml:space="preserve">Quod te iubente lucem aſpicerent, <lb/>tuæ magnanimitatis beneficium fuit; </s> <s xml:id="echoid-s5128" xml:space="preserve">quod tutæ à malo- <pb file="0182" n="182"/> rum inuidia te propugnante per Geometrarum eruditas <lb/>manus incedant, tui in literas amoris beneficium erit. <lb/></s> <s xml:id="echoid-s5129" xml:space="preserve">Hæc me alioquin, & </s> <s xml:id="echoid-s5130" xml:space="preserve">iure, formidoloſum bono eſſe ani-<lb/>mo eſſicaciter ſuadent. </s> <s xml:id="echoid-s5131" xml:space="preserve">Et verè, ſi mihi Genethliaco-<lb/>rum more diuinare liceret, non infelix futurum DIVI-<lb/>NATIONIS meę fatum ſperarem, quam naſcentem <lb/>fulgidiſsima lumina, luppiter, atque Apollo Etruriæ <lb/>tam benignè aſpexerunt. </s> <s xml:id="echoid-s5132" xml:space="preserve">Hoc ſi vnquam videre dabi-<lb/>tur, tuis auſpicijs SERENISSIME PRINCEPS, <lb/>non modò ingenium ad maiores conatus, ſed & </s> <s xml:id="echoid-s5133" xml:space="preserve">diu ia-<lb/>centẽ fortunam meam aliquando ſe ſe erecturam confi-<lb/>do. </s> <s xml:id="echoid-s5134" xml:space="preserve">Faxit Deus: </s> <s xml:id="echoid-s5135" xml:space="preserve">qui (vt enixè precor) te, literarum <lb/>præſidium, & </s> <s xml:id="echoid-s5136" xml:space="preserve">decus ſeruet incolumem, & </s> <s xml:id="echoid-s5137" xml:space="preserve">Heroicæ <lb/>virtutis tuæ incœptis faueat.</s> <s xml:id="echoid-s5138" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5139" xml:space="preserve">Florentiæ Tertio Cal. </s> <s xml:id="echoid-s5140" xml:space="preserve">Ian. </s> <s xml:id="echoid-s5141" xml:space="preserve">1658.</s> <s xml:id="echoid-s5142" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5143" xml:space="preserve">MÆ NIS</s> </p> <p> <s xml:id="echoid-s5144" xml:space="preserve">SER. </s> <s xml:id="echoid-s5145" xml:space="preserve">CELS. </s> <s xml:id="echoid-s5146" xml:space="preserve">TVÆ</s> </p> <p style="it"> <s xml:id="echoid-s5147" xml:space="preserve">Humillimus, Obſequentiſs.</s> <s xml:id="echoid-s5148" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5149" xml:space="preserve">Obſtrictiſs. </s> <s xml:id="echoid-s5150" xml:space="preserve">Seruus</s> </p> <p> <s xml:id="echoid-s5151" xml:space="preserve">Vincentius Viuiani.</s> <s xml:id="echoid-s5152" xml:space="preserve"/> </p> <pb o="1" file="0183" n="183"/> </div> <div xml:id="echoid-div525" type="section" level="1" n="219"> <head xml:id="echoid-head224" xml:space="preserve">VINCENTII VIVIANI <lb/>DE MAXIMIS, ET MINIMIS</head> <head xml:id="echoid-head225" xml:space="preserve">Geometrica diuinatio in V. conic. <lb/>Apoll. Pergæi.</head> <head xml:id="echoid-head226" style="it" xml:space="preserve">LIBER SECVNDVS.</head> <head xml:id="echoid-head227" xml:space="preserve">LEMMA I. PROP. I.</head> <p> <s xml:id="echoid-s5153" xml:space="preserve">Si recta linea vtcunque ſecta fuerit: </s> <s xml:id="echoid-s5154" xml:space="preserve">quadratum totius æqua-<lb/>bitur quadrato vnius partis, vnà cum rectangulo ſub tota, & </s> <s xml:id="echoid-s5155" xml:space="preserve">di-<lb/>cta parte, tanquam ab vna linea, & </s> <s xml:id="echoid-s5156" xml:space="preserve">ſub altera parte contento.</s> <s xml:id="echoid-s5157" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5158" xml:space="preserve">ESTO data recta A B vtcunque ſecta in C. </s> <s xml:id="echoid-s5159" xml:space="preserve">Dico quadratum <lb/>A B æquale eſſe quadrato alterius partis, nempe A C, vna <lb/>cum rectangulo ſub B A cum A C, tanquam vna linea, & </s> <s xml:id="echoid-s5160" xml:space="preserve"><lb/>ſub reliqua parte B C comprehenſo. </s> <s xml:id="echoid-s5161" xml:space="preserve">Nam producta B A ſu-<lb/>matur A D æqualis ipſi BC. </s> <s xml:id="echoid-s5162" xml:space="preserve">Quoniam igitur D C eſt bifa-<lb/>riam ſecta in A, ipſique adiecta C B, erit <lb/>quadratum A B æquale rectangulo ſub <lb/>D B, B C, vnà cum quadrato C A; </s> <s xml:id="echoid-s5163" xml:space="preserve">ſed <lb/>DB linea conficitur ex D A cum A B, vel <lb/> <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> ex A C cum A B; </s> <s xml:id="echoid-s5164" xml:space="preserve">ergo quadratum totius <lb/>A B æquatur quadrato partis C A, vna <lb/>cum rectangulo ſub B A cum A C, tan-<lb/>quam vna linea, & </s> <s xml:id="echoid-s5165" xml:space="preserve">ſub reliqua parte B C <lb/>comprehenſo. </s> <s xml:id="echoid-s5166" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s5167" xml:space="preserve">c.</s> <s xml:id="echoid-s5168" xml:space="preserve"/> </p> <div xml:id="echoid-div525" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0183-01"/> </figure> </div> </div> <div xml:id="echoid-div527" type="section" level="1" n="220"> <head xml:id="echoid-head228" xml:space="preserve">LEMMA II. PROP. II.</head> <p> <s xml:id="echoid-s5169" xml:space="preserve">Si quatuor quantitatum eiuſdem generis, prima ſuperet ſecun-<lb/>dam maiori exceſſu, quo tertia ſuperat quartam, aggregatum <lb/>extremarum maius erit aggregato mediarum.</s> <s xml:id="echoid-s5170" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5171" xml:space="preserve">SInt quatuor quantitates eiuſdem generis A, B, C, D, & </s> <s xml:id="echoid-s5172" xml:space="preserve">prima A ſu-<lb/>peret ſecundam B, maiori exceſſu, quo tertia C ſuperat quartam D. <lb/></s> <s xml:id="echoid-s5173" xml:space="preserve">Dico aggregatum extremarum A, D maius eſſe aggregato mediarum B, C.</s> <s xml:id="echoid-s5174" xml:space="preserve"/> </p> <pb o="2" file="0184" n="184" rhead=""/> <p> <s xml:id="echoid-s5175" xml:space="preserve">Nam intelligatur magnitudo E F æ-<lb/>qualis primæ A, FG verò æqualis ſecun-<lb/>dę B; </s> <s xml:id="echoid-s5176" xml:space="preserve">atque ipſis in directum magnitu-<lb/> <anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/> do F H æqualis tertiæ C, & </s> <s xml:id="echoid-s5177" xml:space="preserve">F I quar-<lb/>tæ D. </s> <s xml:id="echoid-s5178" xml:space="preserve">Erit exceſſus magnitudinis E F <lb/>ſupra F G, hoc est E G, maios exceſſu <lb/>quantitatis H F ſupra F I, ſiue maios-<lb/>ipſo H I, ex ſuppoſitione, quibus addi-<lb/>ta communi quantitate G I, proueniet <lb/>E I maior G H, ſiue aggregatum ex EF, <lb/>& </s> <s xml:id="echoid-s5179" xml:space="preserve">F I, nempe extremarum A, & </s> <s xml:id="echoid-s5180" xml:space="preserve">D, <lb/>maius aggregato ex G F, & </s> <s xml:id="echoid-s5181" xml:space="preserve">F H, velex medijs B, & </s> <s xml:id="echoid-s5182" xml:space="preserve">C. </s> <s xml:id="echoid-s5183" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s5184" xml:space="preserve">c</s> </p> <div xml:id="echoid-div527" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0184-01"/> </figure> </div> </div> <div xml:id="echoid-div529" type="section" level="1" n="221"> <head xml:id="echoid-head229" xml:space="preserve">THEOR. I. PROP. III.</head> <p> <s xml:id="echoid-s5185" xml:space="preserve">MINIMA linearum in Parabola ducibilium ad eius peri-<lb/>pheriam à puncto axis intra ſectionem ſumpto, quod diſtet à <lb/>vertice per interuallum non maius dimidio recti lateris, eſt ip-<lb/>ſum axis ſegmentum inter punctum, & </s> <s xml:id="echoid-s5186" xml:space="preserve">verticem interceptum. <lb/></s> <s xml:id="echoid-s5187" xml:space="preserve">Aliarum verò ea, quæ cum MINIMA minorem conſtituit an-<lb/>gulum, minor eſt.</s> <s xml:id="echoid-s5188" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5189" xml:space="preserve">ESto Parabole AB, cuius ſegmentum axis B D non excedat dimidium <lb/>recti lateris B C datæ Parabolæ. </s> <s xml:id="echoid-s5190" xml:space="preserve">Dico D B eſſe _MINIMAM_ du-<lb/>cibilium ex eodem puncto D ad Parabolæ peripheriam A B, &</s> <s xml:id="echoid-s5191" xml:space="preserve">c.</s> <s xml:id="echoid-s5192" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5193" xml:space="preserve">Applicetur axi ex D, recta D A, Erit <lb/>quadratum A D <anchor type="note" xlink:href="" symbol="a"/> æquale rectangulo <anchor type="figure" xlink:label="fig-0184-02a" xlink:href="fig-0184-02"/> ſub D B, & </s> <s xml:id="echoid-s5194" xml:space="preserve">recto B C; </s> <s xml:id="echoid-s5195" xml:space="preserve">ſed rectangulum <lb/> <anchor type="note" xlink:label="note-0184-01a" xlink:href="note-0184-01"/> D B C maius eſt quadrato D B (cum <lb/>latus rectum B C poſitum ſit, vel du-<lb/>plum, vel magis quàm duplum ipſius <lb/>B D) igitur quadratum A D maius erit <lb/>quadrato D B, ſiue linea D A maior <lb/>D B.</s> <s xml:id="echoid-s5196" xml:space="preserve"/> </p> <div xml:id="echoid-div529" 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/QN4GHYBF/figures/0184-02"/> </figure> <note symbol="a" position="left" xlink:label="note-0184-01" xlink:href="note-0184-01a" xml:space="preserve">Coroll. <lb/>primę pri <lb/>mi huius.</note> </div> <p> <s xml:id="echoid-s5197" xml:space="preserve">Rurſus ducatur infra D A ex D quę-<lb/>cunque alia D E ad peripheriam, & </s> <s xml:id="echoid-s5198" xml:space="preserve">ex <lb/>A recta A F parallela ad B D, quæ to-<lb/>ta ad partes F <anchor type="note" xlink:href="" symbol="b"/> cadet intra Parabolen;</s> <s xml:id="echoid-s5199" xml:space="preserve"> <anchor type="note" xlink:label="note-0184-02a" xlink:href="note-0184-02"/> nec ei ad alium punctum occurret quàm <lb/>ad A; </s> <s xml:id="echoid-s5200" xml:space="preserve">ideoque ſecabit eductam D E, vt in F, eritque E D maior D F, <lb/>ſed eſt D F maior D A (cum in triangulo D A F angulus ad A ſit rectus, <lb/>ſiue maior acuto ad F) & </s> <s xml:id="echoid-s5201" xml:space="preserve">D A maior ipſa D B, vt ſupra oſtendimus, qua-<lb/>re D E multò maior erit ipſa D B.</s> <s xml:id="echoid-s5202" xml:space="preserve"/> </p> <div xml:id="echoid-div530" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0184-02" xlink:href="note-0184-02a" xml:space="preserve">26. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s5203" xml:space="preserve">Ampliùs ſit quæcunque D G ducta ex D ſupra D A, & </s> <s xml:id="echoid-s5204" xml:space="preserve">ex G applice-<lb/>tur G H. </s> <s xml:id="echoid-s5205" xml:space="preserve">Cumque latus rectum B C ſit maios aggregato B D cum D H <lb/>(poſitum enim fuit B C non maius quàm duplum ſegmenti BD, eſtque <pb o="3" file="0185" n="185" rhead=""/> B D maior D H) erit rectangulum ſub recto C B in ſegmentum B H <lb/>ſiue <anchor type="note" xlink:href="" symbol="a"/> quadrarum G H, maius rectangulo ſub aggregato B D cum D H, in <anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/> idem ſegmentum B H, quibus addito communi quadrato D H, erit qua-<lb/>dratum G H cum H D quadrato, ſiue vnicum quadratum G D, maius re-<lb/>ctangulo ſub aggregato B D cum D H in B H. </s> <s xml:id="echoid-s5206" xml:space="preserve">vnà cum quadrato D H, <lb/>ſiue <anchor type="note" xlink:href="" symbol="b"/> maius vnico quadrato B D, hoc eſt linea D G maior erit D B. </s> <s xml:id="echoid-s5207" xml:space="preserve">Eſt <anchor type="note" xlink:label="note-0185-02a" xlink:href="note-0185-02"/> ergo D B _MINIMA_ ducibilium ad Parabolæ peripheriam ex axis puncto <lb/>D, quod abeſt à vertice per interuallum non maius dimidio recti lateris <lb/>B C. </s> <s xml:id="echoid-s5208" xml:space="preserve">Quod primò demonſtrandum erat.</s> <s xml:id="echoid-s5209" xml:space="preserve"/> </p> <div xml:id="echoid-div531" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve">Coroll. <lb/>primę pri. <lb/>mi huius.</note> <note symbol="b" position="right" xlink:label="note-0185-02" xlink:href="note-0185-02a" xml:space="preserve">1. h.</note> </div> <p> <s xml:id="echoid-s5210" xml:space="preserve">Præterea ſit quæpiam alia D M maiorem eſſiciens angulum cum _MI-_ <lb/>_NIMA_ D B, quàm D G, & </s> <s xml:id="echoid-s5211" xml:space="preserve">ex M applicetur M N. </s> <s xml:id="echoid-s5212" xml:space="preserve">Iam quadratum M N <lb/>ſuperat quadratum G H eo exceſſu, quo rectangulum C B N ſuperat re-<lb/>ctangulum C B H, (ob <anchor type="note" xlink:href="" symbol="c"/> æqualitatem) hoc eſt rectangulo ſub recto C B <anchor type="note" xlink:label="note-0185-03a" xlink:href="note-0185-03"/> in H N, ſed quadratum D H <anchor type="note" xlink:href="" symbol="d"/> ſuperat quadratum D N rectangulo ſub ea- dem H N, & </s> <s xml:id="echoid-s5213" xml:space="preserve">ſub aggregato H D cum D N, quod aggregatum, ex hypo-<lb/> <anchor type="note" xlink:label="note-0185-04a" xlink:href="note-0185-04"/> teſi, minus eſt ipſo recto B C, ergo exceſſus quadrati M N ſupra quadra-<lb/>tum G H, maior eſt exceſſu quadrati H D ſupra D N, vnde aggregatum <lb/>extremorum quadratorum M N, N D, ſiue vnicum quadratum MD, ma-<lb/>ius erit <anchor type="note" xlink:href="" symbol="e"/> aggregato quadratorum mediorum G H, H D, ſiue vnico qua- <anchor type="note" xlink:label="note-0185-05a" xlink:href="note-0185-05"/> drato G D, hoc eſt linea D M maior D G.</s> <s xml:id="echoid-s5214" xml:space="preserve"/> </p> <div xml:id="echoid-div532" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-0185-03" xlink:href="note-0185-03a" xml:space="preserve">Coroll. <lb/>primę pri <lb/>mi huius.</note> <note symbol="d" position="right" xlink:label="note-0185-04" xlink:href="note-0185-04a" xml:space="preserve">1. h.</note> <note symbol="e" position="right" xlink:label="note-0185-05" xlink:href="note-0185-05a" xml:space="preserve">2. h.</note> </div> <p> <s xml:id="echoid-s5215" xml:space="preserve">Vlteriùs, quadratum A D ſuperat quadratum M N rectangulo ſub D N, <lb/>& </s> <s xml:id="echoid-s5216" xml:space="preserve">recto B C, & </s> <s xml:id="echoid-s5217" xml:space="preserve">quadratum D M, ſuperat idem quadratum M N quadra-<lb/>to D N, quod eſt minus prædicto rectangulo ſub D N, & </s> <s xml:id="echoid-s5218" xml:space="preserve">recto C B, qua-<lb/>re exceſſus quadrati A D ſupra M N, maior eſt exceſſu quadrati D M, <lb/>ſupra idem quadratum M N; </s> <s xml:id="echoid-s5219" xml:space="preserve">quapropter A D quadratum maius eſt qua-<lb/>drato D M, ſiue linea A D maior ipſa D M.</s> <s xml:id="echoid-s5220" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5221" xml:space="preserve">Tandem ducta quacunque D O infra D E, agatur ex E recta E L ęqui-<lb/>diſtans ad B D. </s> <s xml:id="echoid-s5222" xml:space="preserve">Cum angulus B D E ſit obtuſus, erit quoque parallela-<lb/>rum alternus D E L obtuſus, ideoque in triangulo D E L angulus D L E <lb/>acutus, ſiue minor angulo D E L: </s> <s xml:id="echoid-s5223" xml:space="preserve">quare latus D E minus latere D L, & </s> <s xml:id="echoid-s5224" xml:space="preserve"><lb/>eò minus educta D O. </s> <s xml:id="echoid-s5225" xml:space="preserve">Vnde quæ minorem cum _MINIMA_ conſtituit an-<lb/>gulum minor eſt, &</s> <s xml:id="echoid-s5226" xml:space="preserve">c. </s> <s xml:id="echoid-s5227" xml:space="preserve">Quod omnino oſtendere propoſitum fuit.</s> <s xml:id="echoid-s5228" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div534" type="section" level="1" n="222"> <head xml:id="echoid-head230" xml:space="preserve">LEMMA III. PROP. IV.</head> <p> <s xml:id="echoid-s5229" xml:space="preserve">Si inter latera parallela AD, BC, menſalis ABCD rectangulę ad <lb/>B, ducta fuerit quædam linea EH ipſis lateribus æquidiſtans, ſitq; <lb/></s> <s xml:id="echoid-s5230" xml:space="preserve">AD minor BC. </s> <s xml:id="echoid-s5231" xml:space="preserve">Dico rectangulum ABC, ſuperare rectangulum <lb/>AEH maiori exceſſu, quàm ſit rectangulum EBC.</s> <s xml:id="echoid-s5232" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5233" xml:space="preserve">COmpletis enim rectãgulis EG, BF, EC; <lb/></s> <s xml:id="echoid-s5234" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0185-01a" xlink:href="fig-0185-01"/> patet rectangulum ABC ſuperare re-<lb/>ctangulum AEH gnomone ECG, ſed gno-<lb/>mon ECG maios eſt rectangulo EBC, vnde <lb/>rectangulum ABC, ſuperat rectangulum A <lb/>EH maiori quantitate quàm ſit rectãgulum <lb/>EBC. </s> <s xml:id="echoid-s5235" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s5236" xml:space="preserve">c.</s> <s xml:id="echoid-s5237" xml:space="preserve"/> </p> <div xml:id="echoid-div534" type="float" level="2" n="1"> <figure xlink:label="fig-0185-01" xlink:href="fig-0185-01a"> <image file="0185-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0185-01"/> </figure> </div> <pb o="4" file="0186" n="186" rhead=""/> </div> <div xml:id="echoid-div536" type="section" level="1" n="223"> <head xml:id="echoid-head231" xml:space="preserve">THEOR. II. PROP. V.</head> <p> <s xml:id="echoid-s5238" xml:space="preserve">MINIMA linearum in Hyperbola ducibilium ad ipſius pe-<lb/>ripheriam à puncto axis intra ſectionem ſumpto, quod diſtet à <lb/>vertice per interuallum, non maius quàm dimidium recti late-<lb/>ris, eſt idem axis ſegmentum inter punctum, & </s> <s xml:id="echoid-s5239" xml:space="preserve">verticem inter-<lb/>ceptum. </s> <s xml:id="echoid-s5240" xml:space="preserve">Aliarum autem, quæ cum MINIMA minorem con-<lb/>ſtituit angulum minor eſt.</s> <s xml:id="echoid-s5241" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5242" xml:space="preserve">ESto Hyperbole A B C, cuius ſegmentum axis B D non excedat dimi-<lb/>dium recti lateris B F (quod axi ordinatim applicetur, &</s> <s xml:id="echoid-s5243" xml:space="preserve">c.) </s> <s xml:id="echoid-s5244" xml:space="preserve">Dico <lb/>D B eſſe _MINIMAM_ ducibilium ex ipſo puncto D ad Hyperbolæ peri-<lb/>pheriam A B C, &</s> <s xml:id="echoid-s5245" xml:space="preserve">c.</s> <s xml:id="echoid-s5246" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5247" xml:space="preserve">Sumatur in directum axi, tranſuerſum latus B E, iungaturque regula E F, <lb/>& </s> <s xml:id="echoid-s5248" xml:space="preserve">producatur; </s> <s xml:id="echoid-s5249" xml:space="preserve">appliceturque per D ordinata A D C, regulæ occurrens <lb/>in G.</s> <s xml:id="echoid-s5250" xml:space="preserve"/> </p> <figure> <image file="0186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0186-01"/> </figure> <p> <s xml:id="echoid-s5251" xml:space="preserve">Iam, cum in triangulo E D G, ſit D G maior B F, & </s> <s xml:id="echoid-s5252" xml:space="preserve">B F maior ſegmen-<lb/>to B D (ex hypoteſi) erit D G eò maior ipſo ſegmento D B, quare re-<lb/>ctangulum G D B, <anchor type="note" xlink:href="" symbol="a"/> ſiue quadratum A D, maius erit quadrato D B; </s> <s xml:id="echoid-s5253" xml:space="preserve">hoc <anchor type="note" xlink:label="note-0186-01a" xlink:href="note-0186-01"/> eſt linea D A maior ipſa D B.</s> <s xml:id="echoid-s5254" xml:space="preserve"/> </p> <div xml:id="echoid-div536" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0186-01" xlink:href="note-0186-01a" xml:space="preserve">Coroll. <lb/>primę pri. <lb/>mi huius.</note> </div> <p> <s xml:id="echoid-s5255" xml:space="preserve">Eodem modò, ac in Parabola, oſtendetur D A minorem eſſe quacun-<lb/>que educta D H infra D A, & </s> <s xml:id="echoid-s5256" xml:space="preserve">D H adhuc minor D R, &</s> <s xml:id="echoid-s5257" xml:space="preserve">c.</s> <s xml:id="echoid-s5258" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5259" xml:space="preserve">Nunc verò ſit quælibet D L ducta ex D ſupra D A, & </s> <s xml:id="echoid-s5260" xml:space="preserve">perL applice-<lb/>tur L M, quę producatur, donec regulæ E F occurrat in N. </s> <s xml:id="echoid-s5261" xml:space="preserve">Erit in trian-<lb/>gulo E D G, recta M N maior B F, ſed B F maior eſt aggregato B D cum <pb o="5" file="0187" n="187" rhead=""/> D M (cum latus rectum B F, vel duplum ſit, vel plus quàm duplum ad <lb/>B D) ergo M N ipſo aggregato B D cum D M adhuc maior erit, vnde <lb/>rectangulum ſub N M in M B, <anchor type="note" xlink:href="" symbol="a"/> ſiue quadratum L M, maius erit rectan- <anchor type="note" xlink:label="note-0187-01a" xlink:href="note-0187-01"/> gulo ſub aggregato B D cum D M, in eadem M B, quibus communi ad-<lb/>dito quadrato M D, erit quadratum L M cum M D, ſiue vnicum qua-<lb/>dratum D L, maius rectangulo ſub B D cum D M in M B, vna cum qua-<lb/>drato D M, ſiue maius vnico quadrato D B, quod prædicto <anchor type="note" xlink:href="" symbol="b"/> rectãgulo <anchor type="note" xlink:label="note-0187-02a" xlink:href="note-0187-02"/> æquale eſt, ſiue linea D L maior D B. </s> <s xml:id="echoid-s5262" xml:space="preserve">Quare ſegmentum axis D B, non <lb/>excedens dimidium recti lateris B F, eſt _MINIMA_ linearum ducibilium <lb/>ex D ad Hyperbolæ peripheriam. </s> <s xml:id="echoid-s5263" xml:space="preserve">Quod primò oſtendere oportebat.</s> <s xml:id="echoid-s5264" xml:space="preserve"/> </p> <div xml:id="echoid-div537" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0187-01" xlink:href="note-0187-01a" xml:space="preserve">Coroll. <lb/>primæ pri <lb/>mi huius.</note> <note symbol="b" position="right" xlink:label="note-0187-02" xlink:href="note-0187-02a" xml:space="preserve">1. huius.</note> </div> <p> <s xml:id="echoid-s5265" xml:space="preserve">Præterea, quadratum A D ſuperat quadratum L M, eo exceſſu quo re-<lb/>ctangulum B D G ſuperat rectangulum B M N, (ſunt <anchor type="note" xlink:href="" symbol="c"/> enim ſingula ſingu- <anchor type="note" xlink:label="note-0187-03a" xlink:href="note-0187-03"/> lis æqualia) ſed exceſſus rectanguli B D G ſupra rectangulum B M N ma-<lb/>ius eſt <anchor type="note" xlink:href="" symbol="d"/> rectangulo M D G, ergo quadratum A D ſuperat quadratum <anchor type="note" xlink:label="note-0187-04a" xlink:href="note-0187-04"/> L M maiori rectangulo quàm M D G; </s> <s xml:id="echoid-s5266" xml:space="preserve">ſed quadratum D L ſuperat idem <lb/>quadratum L M quadrato D M, quod minus eſt rectangulo MDG (nam <lb/>eſt D G maior D M, cum ſuperiùs demonſtrata ſit maior ipſa D B) ergo <lb/>quadratum D A maius eſt quadrato D L, ſiue linea D A maior quacun-<lb/>que D L, intercepta inter applicatam D A, & </s> <s xml:id="echoid-s5267" xml:space="preserve">axem D B.</s> <s xml:id="echoid-s5268" xml:space="preserve"/> </p> <div xml:id="echoid-div538" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0187-03" xlink:href="note-0187-03a" xml:space="preserve">Coroll. <lb/>primæ pri <lb/>mi huius.</note> <note symbol="d" position="right" xlink:label="note-0187-04" xlink:href="note-0187-04a" xml:space="preserve">4. huius.</note> </div> <p> <s xml:id="echoid-s5269" xml:space="preserve">Ampliùs, ducatur alia quæpiam D O ſupra D A, ſed remotior à ſe-<lb/>gmento DB quàm D L, applicataque O P, producatur donec regulatrici <lb/>E F occurrat in Q. </s> <s xml:id="echoid-s5270" xml:space="preserve">Erit exceſſus quadrati O P ſupra quadratum L M, <lb/>idem ac exceſſus rectanguli B P Q ſupra B M N (nam <anchor type="note" xlink:href="" symbol="e"/> ſunt rectangula <anchor type="note" xlink:label="note-0187-05a" xlink:href="note-0187-05"/> quadratis æqualia, vtrumque vtrique) ſed exceſſus rectanguli B P Q ſu-<lb/>pra B M N maior <anchor type="note" xlink:href="" symbol="f"/> eſt rectangulo M P Q, ergo exceſſus quadrati O P, ſu- <anchor type="note" xlink:label="note-0187-06a" xlink:href="note-0187-06"/> pra quadratum L M, maior eſt rectangulo ſub M P, & </s> <s xml:id="echoid-s5271" xml:space="preserve">P Q; </s> <s xml:id="echoid-s5272" xml:space="preserve">at exceſſus <lb/>quadrati M D ſupra quadratum D P, minor eſt prædicto rectangulo (nam <lb/>quadratum M D <anchor type="note" xlink:href="" symbol="g"/> ſuperat quadratum D P, rectangulo ſub M D cum DP <anchor type="note" xlink:label="note-0187-07a" xlink:href="note-0187-07"/> in M P, quod eſt minus rectangulo ſub Q P in eadem M P, quoniam <lb/>M D cum D P minor eſt recto latere B F, & </s> <s xml:id="echoid-s5273" xml:space="preserve">eò minor ipſa QP, <lb/>quę maior eſt B F) quare exceſſus quadrati O P ſupra L M, <lb/>maior eſt exceſſu quadrati M D ſupra D P: </s> <s xml:id="echoid-s5274" xml:space="preserve">duo igi-<lb/> <anchor type="handwritten" xlink:label="hd-0187-1a" xlink:href="hd-0187-1"/> tur extrema ſimul quadrata O P, P D, ſiue vni-<lb/>cum quadratum D O, maius eſt duobus ſi-<lb/>mul quadratis medijs L M, M D, hoc <lb/>eſt vnico quadrato D L, ſiue li-<lb/>nea DO maior eſt linea DL. <lb/></s> <s xml:id="echoid-s5275" xml:space="preserve">Vnde quæ minorem <lb/>efficit angulum <lb/>cum _MINI_-<lb/>_M A_ <lb/>D B, minor eſt, &</s> <s xml:id="echoid-s5276" xml:space="preserve">c. </s> <s xml:id="echoid-s5277" xml:space="preserve">Quod <lb/>fuit vltimò demon-<lb/>ſtrandum. </s> <s xml:id="echoid-s5278" xml:space="preserve"><lb/>* * * <lb/>* * <lb/>*</s> </p> <div xml:id="echoid-div539" type="float" level="2" n="4"> <note symbol="e" position="right" xlink:label="note-0187-05" xlink:href="note-0187-05a" xml:space="preserve">Coroll. <lb/>primæ pri <lb/>mi huius.</note> <note symbol="f" position="right" xlink:label="note-0187-06" xlink:href="note-0187-06a" xml:space="preserve">4. huius.</note> <note symbol="g" position="right" xlink:label="note-0187-07" xlink:href="note-0187-07a" xml:space="preserve">1. huius.</note> <handwritten xlink:label="hd-0187-1" xlink:href="hd-0187-1a"/> </div> <pb o="6" file="0188" n="188" rhead=""/> </div> <div xml:id="echoid-div541" type="section" level="1" n="224"> <head xml:id="echoid-head232" xml:space="preserve">THEOR. III. PROP. VI.</head> <p> <s xml:id="echoid-s5279" xml:space="preserve">MAXIMA linearum ad vniuerſam Ellipſis peripheriam du-<lb/>cibilium, à puncto maioris axis, quod non ſit centrum, ea eſt, <lb/>in qua centrum. </s> <s xml:id="echoid-s5280" xml:space="preserve">Et eductarum ad peripheriam maioris Ellipti-<lb/>cæ portionis, cuius baſis, ſit recta ad axim ordinatim ducta, ex <lb/>prędicto puncto; </s> <s xml:id="echoid-s5281" xml:space="preserve">quę cum MAXIMA minorem conſtituit an-<lb/>gulum, maior eſt. </s> <s xml:id="echoid-s5282" xml:space="preserve">MINIMA verò in eadem portione, eſt ip-<lb/>ſa ſemi-applicata.</s> <s xml:id="echoid-s5283" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5284" xml:space="preserve">ESto Ellipſis A B C D, cuius axis maior B D, minor H I, centrum E, <lb/>& </s> <s xml:id="echoid-s5285" xml:space="preserve">quodlibet aliud punctum in maiori axe ſit F. </s> <s xml:id="echoid-s5286" xml:space="preserve">Dico _MAXIMAM_ <lb/>ducibilium ab F ad vniuerſam Ellipſis peripheriam eſſe F D, in qua cen-<lb/>trum.</s> <s xml:id="echoid-s5287" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5288" xml:space="preserve">Nam, quod D F ſit maior reliqua F B patet, cum F D, maior ſit axis <lb/>dimidio, F B verò minor.</s> <s xml:id="echoid-s5289" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5290" xml:space="preserve">Iam, ad quodcunque Ellipticæ peri-<lb/> <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> pheriæ punctum G, ſit quædam educta <lb/>F G, & </s> <s xml:id="echoid-s5291" xml:space="preserve">iungatnr E G. </s> <s xml:id="echoid-s5292" xml:space="preserve">Itaque cum ſe-<lb/>mi-axis maior E D, ſit <anchor type="note" xlink:href="" symbol="a"/> _MAXIMA_ ſe- <anchor type="note" xlink:label="note-0188-01a" xlink:href="note-0188-01"/> mi-diametrorum, ipſa maior erit E G, <lb/>quibus communi addita E F, erit tota <lb/>D F maior duobus G E, E F, & </s> <s xml:id="echoid-s5293" xml:space="preserve">eò ma-<lb/>ior vnica F G. </s> <s xml:id="echoid-s5294" xml:space="preserve">Quare F D eſt ad vni-<lb/>uerſam peripheriam ducibilium _MAXI_-<lb/>_MA_.</s> <s xml:id="echoid-s5295" xml:space="preserve"/> </p> <div xml:id="echoid-div541" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0188-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0188-01" xlink:href="note-0188-01a" xml:space="preserve">86. pri-<lb/>mi huius.</note> </div> <p> <s xml:id="echoid-s5296" xml:space="preserve">Inſuper applicetur ex F axi ordinata <lb/>A F C, & </s> <s xml:id="echoid-s5297" xml:space="preserve">ad peripheriam eiuſdem qua-<lb/>drantis H D E, ductæ ſint ex F duę quę-<lb/>libet F L, F M, & </s> <s xml:id="echoid-s5298" xml:space="preserve">F L minorem, F M <lb/>verò maiorem angulum efficiat cum <lb/>_MAXIMA_ F D. </s> <s xml:id="echoid-s5299" xml:space="preserve">Dico F L maiorem eſſe <lb/>F M. </s> <s xml:id="echoid-s5300" xml:space="preserve">Iunctis enim E L, E M; </s> <s xml:id="echoid-s5301" xml:space="preserve">erit <anchor type="note" xlink:href="" symbol="b"/> E L <anchor type="note" xlink:label="note-0188-02a" xlink:href="note-0188-02"/> maior E M, quæ producatur, & </s> <s xml:id="echoid-s5302" xml:space="preserve">fiat E O æqualis E L, & </s> <s xml:id="echoid-s5303" xml:space="preserve">iungatur F O: <lb/></s> <s xml:id="echoid-s5304" xml:space="preserve">erunt igitur duo latera F E, E L, duobus F E, E O æqualia, alterum al-<lb/>teri, ſed angulus F E L maior eſt angulo F E O, ergo baſis F L, maior eſt <lb/>F O, ſed F O maior eſt F M, (cum in triangulo F M O angulus ad M ob-<lb/>tuſus ſit, eò quod ſit maior obtuſo F E M) quare F L eò maior erit ipſa <lb/>F M, quę cum _MAXIMA_ maiorem efficit angulum: </s> <s xml:id="echoid-s5305" xml:space="preserve">ſimili modo oſtende-<lb/>tur F M maiorem eſſe educta F H.</s> <s xml:id="echoid-s5306" xml:space="preserve"/> </p> <div xml:id="echoid-div542" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0188-02" xlink:href="note-0188-02a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s5307" xml:space="preserve">De eductis verò ad portionem peripheriæ H A, ita ratiocinabimur. <lb/></s> <s xml:id="echoid-s5308" xml:space="preserve">Sit enim quælibet F P, & </s> <s xml:id="echoid-s5309" xml:space="preserve">per H ſit Ellipſim contingens H Q, quæ cum <lb/>æquidiſtet axi B F D, ſecabit omnino productam F P extra Ellipſim in Q; </s> <s xml:id="echoid-s5310" xml:space="preserve"><lb/>eritque in triangulo F Q H, latus F H maius latere F Q (cum angulus <pb o="7" file="0189" n="189" rhead=""/> F Q H ſit obtuſus, eò quod alterno Q F B obtuſo ſit æqualis) ſed eſt F Q <lb/>maius F P, quare educta F H eò maior erit educta F P. </s> <s xml:id="echoid-s5311" xml:space="preserve">Ampliùs ducta <lb/>qualibet alia F R, adhuc maiorem angulum facient<unsure/> cum _MAXIMA_ F D, <lb/>agatur per R recta R S axi F E parallela, quæ cadet intra Ellipſim, (cum <lb/>ſit ad minorem axim H I ordinatim ducta) ſecabitque F P in S, ac in <lb/>triangulo F R S obtuſiangulo ad R, erit latus F S maius latere F R, & </s> <s xml:id="echoid-s5312" xml:space="preserve"><lb/>educta F P eò maior educta F R; </s> <s xml:id="echoid-s5313" xml:space="preserve">eademque ratione oſtendetur quamli-<lb/>bet eductarum ad peripheriam H A, vtputa F R, maiorem eſſe ſemi-ap-<lb/>plicata F A, ſi ex A ducatur A V parallela ad E F, &</s> <s xml:id="echoid-s5314" xml:space="preserve">c. </s> <s xml:id="echoid-s5315" xml:space="preserve">quare eadem <lb/>ſemi-applicata F A omnium eductarum in portione maiori A D C erit <lb/>_MINIMA_. </s> <s xml:id="echoid-s5316" xml:space="preserve">Aliarum autem, quæ cum _MAXIMA_ F D maiorem angulum <lb/>conſtituit, maior eſt. </s> <s xml:id="echoid-s5317" xml:space="preserve">Quod omnino oſtendere opus fuerat.</s> <s xml:id="echoid-s5318" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div544" type="section" level="1" n="225"> <head xml:id="echoid-head233" xml:space="preserve">LEMMA IV. PROP. VII.</head> <p> <s xml:id="echoid-s5319" xml:space="preserve">Si in triangulo A B C, cuius rectus angulus ſit ad B, fuerit <lb/>latus A B maius altero B C, ſitque de maiori B A abſciſſa pars <lb/> <anchor type="note" xlink:label="note-0189-01a" xlink:href="note-0189-01"/> B E, quæ non excedat dimidium ipſius B C, & </s> <s xml:id="echoid-s5320" xml:space="preserve">ex quolibet eius <lb/>puncto G ducta ſit G H parallela ad B C. </s> <s xml:id="echoid-s5321" xml:space="preserve">Dico primùm ipſam <lb/>G H ſemper maiorem eſſe aggregato B E cum E G.</s> <s xml:id="echoid-s5322" xml:space="preserve"/> </p> <div xml:id="echoid-div544" type="float" level="2" n="1"> <note position="right" xlink:label="note-0189-01" xlink:href="note-0189-01a" xml:space="preserve">1.</note> </div> <p> <s xml:id="echoid-s5323" xml:space="preserve">DVcatur E F ęquidiſtans ad B C. </s> <s xml:id="echoid-s5324" xml:space="preserve">Et quoniam A B ponitur maior ip-<lb/>ſa B C; </s> <s xml:id="echoid-s5325" xml:space="preserve">B C verò dupla, vel plus quàm dupla ad B E, erit omni-<lb/>no A B plus quàm dupla ad B E, ſiue AE <lb/> <anchor type="figure" xlink:label="fig-0189-01a" xlink:href="fig-0189-01"/> plus quàm dimidium ipſius A B, quod <lb/>memento ſed, vt A E ad A B, ita E F <lb/>ad B C; </s> <s xml:id="echoid-s5326" xml:space="preserve">quare E F eſt maior dimidio <lb/>ipſius B C, hoc eſt maior ipſa B E. </s> <s xml:id="echoid-s5327" xml:space="preserve">Secta <lb/>igitur E S æquali ipſi B E, ducatur S K <lb/>D parallela ad B E, eritque B S paralle-<lb/>logrammum æquilaterum (cum E B, <lb/>E S ſint æquales) iungatur denique C S <lb/>rectam G H ſecans in T.</s> <s xml:id="echoid-s5328" xml:space="preserve"/> </p> <div xml:id="echoid-div545" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0189-01"/> </figure> </div> <p> <s xml:id="echoid-s5329" xml:space="preserve">Itaque cum E B, ſiue B D poſita ſit <lb/>æqualis, vel minor dimidio ipſius B C, <lb/>erit C D æqualis, vel maior ipſa D B, <lb/>vel D S. </s> <s xml:id="echoid-s5330" xml:space="preserve">Cumque ſit, vt C D ad D S, <lb/>ita T K ad K S, erit quoque T K ęqua-<lb/>lis, vel maior ipſa K S, ſiue G E, qui-<lb/>bus T K, & </s> <s xml:id="echoid-s5331" xml:space="preserve">G E additis ęqualibus K G, E B, proueniet tota T G æqua-<lb/>lis, vel maior aggregato G E cum E B, ſed eſt H G maior ipſa T G: </s> <s xml:id="echoid-s5332" xml:space="preserve">qua-<lb/>re H G erit omnino maior aggregato B E cnm E G. </s> <s xml:id="echoid-s5333" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s5334" xml:space="preserve">c.</s> <s xml:id="echoid-s5335" xml:space="preserve"/> </p> <pb o="8" file="0190" n="190" rhead=""/> <p> <s xml:id="echoid-s5336" xml:space="preserve">Præterea, ijſdem poſitis in eadem figura. </s> <s xml:id="echoid-s5337" xml:space="preserve">Dico rectangulum <lb/> <anchor type="note" xlink:label="note-0190-01a" xlink:href="note-0190-01"/> B E F ſuperare rectangulum B G H maiori exceſſu quàm ſit qua-<lb/>dratum G E.</s> <s xml:id="echoid-s5338" xml:space="preserve"/> </p> <div xml:id="echoid-div546" type="float" level="2" n="3"> <note position="left" xlink:label="note-0190-01" xlink:href="note-0190-01a" xml:space="preserve">2.</note> </div> <p> <s xml:id="echoid-s5339" xml:space="preserve">COmpletis enim rectangulis B E F I, B G H L, productiſque E F, L H <lb/>vſque ad occurſum in O; </s> <s xml:id="echoid-s5340" xml:space="preserve">cum ſit A E pluſquàm dimidium ipſius <lb/>A B, vt ſupra oſtendimus, erit A E maior E B; </s> <s xml:id="echoid-s5341" xml:space="preserve">cumque ſit B A ad A E, <lb/>ita B C ad E F, vel ad B I, erit diuidendo B E ad E A, vt C I ad I B, <lb/>ſed eſt B E minor ipſa E A, ergo, & </s> <s xml:id="echoid-s5342" xml:space="preserve">C I minor erit ipſa I B, quare ſum-<lb/>pta L M æquali ipſi C I punctum M non pertinget ad B.</s> <s xml:id="echoid-s5343" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5344" xml:space="preserve">Iam cum M L, I C ſint ęquales, erit <lb/> <anchor type="figure" xlink:label="fig-0190-01a" xlink:href="fig-0190-01"/> M L ad N H, vt I C ad N H, vel vt I F <lb/>ad F N, vel vt L O ad O H, quare pun-<lb/>cta M, N, O erunt in vna, eademque <lb/>recta M N O. </s> <s xml:id="echoid-s5345" xml:space="preserve">Poſtremò ducatur recta <lb/>M P Q parallela ad B E. </s> <s xml:id="echoid-s5346" xml:space="preserve">Erunt in re-<lb/>ctangulo Q L ſupplementa Q N, L N <lb/>inter ſe æqualia, quibus addito com-<lb/>muni rectangulo B N, fiet gnomon G I <lb/>Q æqualis rectangulo B H, ſed exceſ-<lb/>ſus rectanguli B F ſupra gnomonem G I <lb/>Q, eſt rectangulum G Q, quare exceſ-<lb/>ſus quoque rectanguli B F, ſupra B H, <lb/>erit idem rectangulum G Q. </s> <s xml:id="echoid-s5347" xml:space="preserve">Cumque <lb/>ſit C B minor B A, & </s> <s xml:id="echoid-s5348" xml:space="preserve">vt C B ad B A, <lb/>ita C L ad L H, erit quoque C L, vel M I, vel Q F minor L H, vel BG; <lb/></s> <s xml:id="echoid-s5349" xml:space="preserve">eſtque tota E F, maior tota E B, vt ſuperiùs oſtendimus, ergo reliqua <lb/>Q E maior erit reliqua E G, vnde rectangulum G E Q, quod eſt exceſ-<lb/>ſus rectanguli B E F ſupra B G H maius erit quadrato G E. </s> <s xml:id="echoid-s5350" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s5351" xml:space="preserve">c.</s> <s xml:id="echoid-s5352" xml:space="preserve"/> </p> <div xml:id="echoid-div547" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0190-01"/> </figure> </div> <p> <s xml:id="echoid-s5353" xml:space="preserve">Poſtremò ijſdem poſitis, & </s> <s xml:id="echoid-s5354" xml:space="preserve">conſtructis, concipiatur quoque <lb/> <anchor type="note" xlink:label="note-0190-02a" xlink:href="note-0190-02"/> alia B R maior quidem B E, ſed minor adhuc dimidio ipſius B <lb/>A, & </s> <s xml:id="echoid-s5355" xml:space="preserve">non maior dimidio ipſius B C. </s> <s xml:id="echoid-s5356" xml:space="preserve">Dico tandem exceſſum <lb/>rectanguli B E F ſupra rectangulum B G H, quod eſt G E Q, <lb/>maius eſſe exceſſu quadrati G R ſupra R E.</s> <s xml:id="echoid-s5357" xml:space="preserve"/> </p> <div xml:id="echoid-div548" type="float" level="2" n="5"> <note position="left" xlink:label="note-0190-02" xlink:href="note-0190-02a" xml:space="preserve">3.</note> </div> <p> <s xml:id="echoid-s5358" xml:space="preserve">NAm, vt primo loco ſuperiùs demonſtrauimus, erit tota linea E F <lb/>maior aggregato B R, cum R E, ſed pars Q F minor eſt parte BG <lb/>prædicti aggregati (nam eſt Q F æqualis M I, ſiue L C, & </s> <s xml:id="echoid-s5359" xml:space="preserve">B G æqualis <lb/>eſt L H, eſtque C L minor L H, cum ſit data C B minor quoque B A) er-<lb/>go reliqua E Q maior erit reliquo eiuſdem aggregati, quod eſt G R cum <lb/>R E; </s> <s xml:id="echoid-s5360" xml:space="preserve">vnde rectangulum ſub Q E, & </s> <s xml:id="echoid-s5361" xml:space="preserve">E G, quod eſt exceſſus rectanguli B <lb/>E F ſupra B G H, maius erit rectangulo ſub G E cum R E, in eadem G E: <lb/></s> <s xml:id="echoid-s5362" xml:space="preserve">ſed rectangulum ſub G R cum R E, in G E, <anchor type="note" xlink:href="" symbol="a"/> eſt exceſſus quadrati G R <anchor type="note" xlink:label="note-0190-03a" xlink:href="note-0190-03"/> ſupra R E, ideoque rectangulum B E F ſuperat rectangulum B G H maio-<lb/>ri exceſſu, quo quadratum G R ſuperat quadratum RE. </s> <s xml:id="echoid-s5363" xml:space="preserve">Quod tandem, &</s> <s xml:id="echoid-s5364" xml:space="preserve">c.</s> <s xml:id="echoid-s5365" xml:space="preserve"/> </p> <div xml:id="echoid-div549" type="float" level="2" n="6"> <note symbol="a" position="left" xlink:label="note-0190-03" xlink:href="note-0190-03a" xml:space="preserve">1. huius.</note> </div> <pb o="9" file="0191" n="191" rhead=""/> </div> <div xml:id="echoid-div551" type="section" level="1" n="226"> <head xml:id="echoid-head234" xml:space="preserve">THEOR. IV. PROP. VIII.</head> <p> <s xml:id="echoid-s5366" xml:space="preserve">MINIMA linearum ad vniuerſam Ellipſis peripheriam du-<lb/>cibilium, à puncto maioris axis, quod diſtet à vertice per in-<lb/>teruallum non maius dimidio recti lateris, eſt idem axis ſegmen-<lb/>tum, inter datum punctum, & </s> <s xml:id="echoid-s5367" xml:space="preserve">verticem interceptum.</s> <s xml:id="echoid-s5368" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5369" xml:space="preserve">Aliarum autem eductarum in minori portione Ellipſis, cuius <lb/>baſis, ſit applicata per datum punctum; </s> <s xml:id="echoid-s5370" xml:space="preserve">quæ cum MINIMA <lb/>minorem angulum conſtituit, minor eſt.</s> <s xml:id="echoid-s5371" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5372" xml:space="preserve">ESto Ellipſis A B C D, cuius axis maior A C, minor B D, centrum E, <lb/>& </s> <s xml:id="echoid-s5373" xml:space="preserve">latus rectum maioris axis C A ſit C F, & </s> <s xml:id="echoid-s5374" xml:space="preserve">regula A F: </s> <s xml:id="echoid-s5375" xml:space="preserve">ſegmentum <lb/>verò C G, ſit non mains<unsure/> dimidio C F. </s> <s xml:id="echoid-s5376" xml:space="preserve">Dico primùm G C eſſe _MINIMAM_ <lb/>ducibilium ex G ad vniuerſam Ellipſis peripheriam A B C D.</s> <s xml:id="echoid-s5377" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5378" xml:space="preserve">Quod enim G C, licet ponatur <lb/> <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> æqualis dimidio recti C E, ſit mi-<lb/>nor reliquo axis ſegmento G A, pa-<lb/>tet: </s> <s xml:id="echoid-s5379" xml:space="preserve">quoniam C A ad B D, eſt vt B D <lb/>ad C F, & </s> <s xml:id="echoid-s5380" xml:space="preserve">ſumptis ſubduplis, C E <lb/>ad E B, vt E B ad C G, eſtque C E <lb/>maior E B, quare E B quoque maior <lb/>eſt C G, & </s> <s xml:id="echoid-s5381" xml:space="preserve">eò magis A E, immò A <lb/>G maior G C.</s> <s xml:id="echoid-s5382" xml:space="preserve"/> </p> <div xml:id="echoid-div551" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0191-01"/> </figure> </div> <p> <s xml:id="echoid-s5383" xml:space="preserve">Iam applicetur per G recta H G S, <lb/>regulæ occurrens in I. </s> <s xml:id="echoid-s5384" xml:space="preserve">Erit A E ad <lb/>ad A C, vt E L ad C F, ſed eſt A E <lb/>dimidia A C, quare E L recti C F <lb/>dimidia erit; </s> <s xml:id="echoid-s5385" xml:space="preserve">eſtque G I maior E L, <lb/>ergo G I maior eſt dimidio recti C F, <lb/>& </s> <s xml:id="echoid-s5386" xml:space="preserve">poſita eſt G C non maior dimidio <lb/>recti; </s> <s xml:id="echoid-s5387" xml:space="preserve">ergo G C erit omnino minor <lb/>G I, ſiue quadratum G C minus re-<lb/>ctangulo C G I, ſiue <anchor type="note" xlink:href="" symbol="a"/> quadrato G H, hoc eſt linea G C minor ipſa G H, <anchor type="note" xlink:label="note-0191-01a" xlink:href="note-0191-01"/> ſed G H eſt <anchor type="note" xlink:href="" symbol="b"/> _MINIMA_ ducibilium ex G ad peripheriam H A S, ergo GC eò ampliùs _MINIMA_ erit ad eandem maioris portionis peripheriam H A S. <lb/></s> <s xml:id="echoid-s5388" xml:space="preserve"/> </p> <div xml:id="echoid-div552" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve">Coroll. <lb/>primę pri <lb/>mi huius.</note> </div> <note symbol="b" position="right" xml:space="preserve">6. h.</note> <p> <s xml:id="echoid-s5389" xml:space="preserve">Ampliùs, ad peripheriam minoris portionis H C S ducatur quęcunque <lb/>G M, & </s> <s xml:id="echoid-s5390" xml:space="preserve">per M applicetur M N O. </s> <s xml:id="echoid-s5391" xml:space="preserve">Cum in triangulo rectangulo A C F <lb/>oſtenſa ſit C G minor quàm dimidium C A, ſed poſita ſit non maior di-<lb/>midio C F, & </s> <s xml:id="echoid-s5392" xml:space="preserve">ex puncto N in C G ſumpto, ducta ſit N O parallela ad <lb/>C F, erit N O maior aggregato C G cum G N, per primam partem 7. </s> <s xml:id="echoid-s5393" xml:space="preserve">hu-<lb/>ius; </s> <s xml:id="echoid-s5394" xml:space="preserve">ergo ſumpta communi altitudine N C, erit rectangulum O N C, ſiue <lb/> <anchor type="note" xlink:label="note-0191-03a" xlink:href="note-0191-03"/> quadratum <anchor type="note" xlink:href="" symbol="c"/> M N maius rectangulo ſub C G cum G N in N C: </s> <s xml:id="echoid-s5395" xml:space="preserve">addito communi quadrato G N, erit quadratum M N cum quadrato N G, ſiue <lb/>vnicum quadratum G M, maius rectangulo ſub C G cum G N in N C, vnà <pb o="10" file="0192" n="192" rhead=""/> quadrato G N ſed rectangulum C G cum G N, in N C, vnà cum qua-<lb/>drato G N, <anchor type="note" xlink:href="" symbol="a"/> conficit quadratum vnicæ C G, ergo quadratum G M ma- <anchor type="note" xlink:label="note-0192-01a" xlink:href="note-0192-01"/> ius eſt quadrato G C, ſiue linea G M maior G C: </s> <s xml:id="echoid-s5396" xml:space="preserve">ex quò G C erit etiam <lb/>_MINIMA_ ductarum ex G ad peripheriam minoris portionis H C S. </s> <s xml:id="echoid-s5397" xml:space="preserve">Vn-<lb/>de ipſa G C erit _MINIMA_ ad totam peripheriam A B C D.</s> <s xml:id="echoid-s5398" xml:space="preserve"/> </p> <div xml:id="echoid-div553" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0191-03" xlink:href="note-0191-03a" xml:space="preserve">Coroll. <lb/>primę pri <lb/>mi huius.</note> <note symbol="a" position="left" xlink:label="note-0192-01" xlink:href="note-0192-01a" xml:space="preserve">1. h.</note> </div> <p> <s xml:id="echoid-s5399" xml:space="preserve">Inſuper rectangulum C G I ſuperat rectangulum C N O ſpatio minori, <lb/>quàm ſit quadratum N G, per ſecundam partem 7. </s> <s xml:id="echoid-s5400" xml:space="preserve">huius; </s> <s xml:id="echoid-s5401" xml:space="preserve">quare (alijs <lb/>ſumptis <anchor type="note" xlink:href="" symbol="b"/> æqualibus) quadratum G H <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"/> ſuperabit quadratum M N maiori ex-<lb/>ceſſu quadrati G N; </s> <s xml:id="echoid-s5402" xml:space="preserve">ſed quadratum <lb/>G M ſuperat idem quadratum M N <lb/>quadrato tantùm G N, ergo exceſſus <lb/>quadrati G H ſupra N M, maior eſt <lb/>exceſſu quadrati G M ſupra idem qua-<lb/>dratum M N, quare quadratum G H <lb/>maius eſt quadrato G M, ſiue linea <lb/>G H maior G M.</s> <s xml:id="echoid-s5403" xml:space="preserve"/> </p> <div xml:id="echoid-div554" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0192-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0192-02" xlink:href="note-0192-02a" xml:space="preserve">Coroll. <lb/>primę pri. <lb/>mi huius.</note> </div> <p> <s xml:id="echoid-s5404" xml:space="preserve">Tandem ducatur G P minorem có-<lb/>ſtituens angulum cum _MINIMA_ G C <lb/>quàm G M, appliceturque PQR. </s> <s xml:id="echoid-s5405" xml:space="preserve">Erit <lb/>exceſſus rectanguli C Q R ſupra CNO <lb/>maior exceſſu quadrati N G ſupra <lb/>G Q, per tertiam partem 7. </s> <s xml:id="echoid-s5406" xml:space="preserve">huius, <lb/>ergo (permutatis æqualibus, <anchor type="note" xlink:href="" symbol="c"/> &</s> <s xml:id="echoid-s5407" xml:space="preserve">c.)</s> <s xml:id="echoid-s5408" xml:space="preserve"> <anchor type="note" xlink:label="note-0192-03a" xlink:href="note-0192-03"/> quadratum P Q ſuperabit quadratum <lb/>M N maiori exceſſu, quàm quadrati N G ſupra G Q: </s> <s xml:id="echoid-s5409" xml:space="preserve">vnde aggregatum <lb/>extremorum quadratorum P Q, G Q, ſiue vnicum quadratum G P, ma-<lb/>ius erit <anchor type="note" xlink:href="" symbol="d"/> aggregato mediorum M N, N G, ſiue vnico quadrato GM; </s> <s xml:id="echoid-s5410" xml:space="preserve">hoc <anchor type="note" xlink:label="note-0192-04a" xlink:href="note-0192-04"/> eſt linea G P erit maior linea G M. </s> <s xml:id="echoid-s5411" xml:space="preserve">Quapropter linearum ex G ducibi-<lb/>lium ad minoris portionis peripheriam H C S, quæ minorem angulum <lb/>conſtituit cum _MINIMA_ minor eſt. </s> <s xml:id="echoid-s5412" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s5413" xml:space="preserve"/> </p> <div xml:id="echoid-div555" type="float" level="2" n="5"> <note symbol="c" position="left" xlink:label="note-0192-03" xlink:href="note-0192-03a" xml:space="preserve">ibidem.</note> <note symbol="d" position="left" xlink:label="note-0192-04" xlink:href="note-0192-04a" xml:space="preserve">2. h.</note> </div> <p style="it"> <s xml:id="echoid-s5414" xml:space="preserve">Verùm prætermiſſa hac methodo mihi, vt fateor, moleſiiori, quod <lb/>in quatuor præcedentibus theorematibus, quò ad MAXI-<lb/>MAS tantùm, & </s> <s xml:id="echoid-s5415" xml:space="preserve">MINIMAS attinet, hic ſi-<lb/>mul, & </s> <s xml:id="echoid-s5416" xml:space="preserve">aliquid vltra, aliter, & </s> <s xml:id="echoid-s5417" xml:space="preserve">expeditiùs <lb/>demonſtrabitur.</s> <s xml:id="echoid-s5418" xml:space="preserve"/> </p> <pb o="11" file="0193" n="193" rhead=""/> </div> <div xml:id="echoid-div557" type="section" level="1" n="227"> <head xml:id="echoid-head235" xml:space="preserve">THEOR. V. PROP. IX.</head> <p> <s xml:id="echoid-s5419" xml:space="preserve">MINIMA linearum, ad peripheriam cuiulibet coni - ſectio-<lb/>nis ducibilium à puncto axis (quod<unsure/> in Ellipſi ſit axis maior) di-<lb/>ſtante<unsure/> à vertice per interuallum non maius dimidio recti lateris, <lb/>eſt idem axis ſegmentum inter aſſignatum punctum, & </s> <s xml:id="echoid-s5420" xml:space="preserve">verticem <lb/>interceptum. </s> <s xml:id="echoid-s5421" xml:space="preserve">At in Ellipſi tantùm, MAXIMA eſt reliquum ma-<lb/>ioris axis ſegmentum, in quo centrum reperitur.</s> <s xml:id="echoid-s5422" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5423" xml:space="preserve">In Ellipſi verò circa minorem axim; </s> <s xml:id="echoid-s5424" xml:space="preserve">MAXIMA ducibilium <lb/>à puncto eiuſdem axis, quod diſtet à vertice per interuallum non <lb/>minus dimidio recti, eſt ipſum axis ſegmentum, inter aſſumptum <lb/>punctum, & </s> <s xml:id="echoid-s5425" xml:space="preserve">verticem interceptum. </s> <s xml:id="echoid-s5426" xml:space="preserve">MINIMA verò eſt reliquum <lb/>minoris axis ſegmentum, in quo centrum non reperitur.</s> <s xml:id="echoid-s5427" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5428" xml:space="preserve">1. </s> <s xml:id="echoid-s5429" xml:space="preserve">ESto A B C quæcunque coni-ſectio, vel Parabole, vel Hyperbole, vt <lb/>in prima figura, vel Ellipſis, vt in ſecunda, circa maiorem axim <lb/>B D, in quo ſumptum ſit pun-<lb/> <anchor type="figure" xlink:label="fig-0193-01a" xlink:href="fig-0193-01"/> ctum E, quod primò diſtet à <lb/>vertice B per interuallum ęqua-<lb/>le dimidio recti lateris axis BD, <lb/>quodq; </s> <s xml:id="echoid-s5430" xml:space="preserve">in Ellipſi omnino minus <lb/>erit ſemi - axe B H (eſt enim ſe-<lb/>mi - axis maior ad ſemi - axim <lb/>minorem, vt ſemi - axis minor <lb/>ad ſemi-rectum.) </s> <s xml:id="echoid-s5431" xml:space="preserve">Dico ſegmen-<lb/>tum axis E B eſſe _MINIMAM_ <lb/>linearum ex E ducibilium ad <lb/>ſectionis peripheriam ABC, & </s> <s xml:id="echoid-s5432" xml:space="preserve"><lb/>reliquam B D, in qua eſt cen-<lb/>trum, eſſe _MAXIMAM._</s> <s xml:id="echoid-s5433" xml:space="preserve"/> </p> <div xml:id="echoid-div557" type="float" level="2" n="1"> <figure xlink:label="fig-0193-01" xlink:href="fig-0193-01a"> <image file="0193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0193-01"/> </figure> </div> <p> <s xml:id="echoid-s5434" xml:space="preserve">Deſcripto enim cum centro <lb/>E, interuallo E B circulo B F, <lb/>ipſe cadet totus <anchor type="note" xlink:href="" symbol="a"/> intra ſectioné <anchor type="note" xlink:label="note-0193-01a" xlink:href="note-0193-01"/> A B C: </s> <s xml:id="echoid-s5435" xml:space="preserve">quare, quę ex centro E <lb/>ad ſectionis peripheriam ducẽ-<lb/>tur, præter ad B, omnino maio-<lb/>res erunt, quàm ductæ ex eo-<lb/>dem centro ad circuli periphe-<lb/>riam, quibus æqualis eſt E B. <lb/></s> <s xml:id="echoid-s5436" xml:space="preserve">Ergo ipſa E B erit _MINIMA_.</s> <s xml:id="echoid-s5437" xml:space="preserve"/> </p> <div xml:id="echoid-div558" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0193-01" xlink:href="note-0193-01a" xml:space="preserve">1. Co-<lb/>roll. 20. 1. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s5438" xml:space="preserve">Si verò, diſtantia à vertice B fuerit minor eodem recti dimidio qualis <lb/>eſt G B: </s> <s xml:id="echoid-s5439" xml:space="preserve">cum ad peripheriam circuli B F ipſa G B ſit _MINIMA_, eò magis <lb/>_MINIMA_ erit ad Ellipſis circumſcriptam peripheriam A B C D.</s> <s xml:id="echoid-s5440" xml:space="preserve"/> </p> <pb o="12" file="0194" n="194" rhead=""/> <p> <s xml:id="echoid-s5441" xml:space="preserve">2. </s> <s xml:id="echoid-s5442" xml:space="preserve">At in ſecunda tantùm figura, quod reliquum maioris axis ſegmentum <lb/>E D, vel G D ſit _MAXIMA_ ex E, vel G ducibililium, patet: </s> <s xml:id="echoid-s5443" xml:space="preserve">quoniam <lb/>circulus ex radio H D cadit totus <anchor type="note" xlink:href="" symbol="a"/> extra Ellipſim A B C D, ſed in circu- <anchor type="note" xlink:label="note-0194-01a" xlink:href="note-0194-01"/> lo, cuius radius H D, ipſa E D, vel G D eſt _MAXIMA_, cum in ea ſit cir-<lb/>culi centrum: </s> <s xml:id="echoid-s5444" xml:space="preserve">quapropter E D, vel G D eò magis erit _MAXIMA_ ad in-<lb/>ſcriptam Ellipſim A B C D. </s> <s xml:id="echoid-s5445" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s5446" xml:space="preserve">c.</s> <s xml:id="echoid-s5447" xml:space="preserve"/> </p> <div xml:id="echoid-div559" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-0194-01" xlink:href="note-0194-01a" xml:space="preserve">26. pri-<lb/>mi huius.</note> </div> <p> <s xml:id="echoid-s5448" xml:space="preserve">3. </s> <s xml:id="echoid-s5449" xml:space="preserve">Iam in tertia figura ſit A B C <lb/> <anchor type="figure" xlink:label="fig-0194-01a" xlink:href="fig-0194-01"/> D Ellipſis circa minorem axim <lb/>B D, in quo infra verticem B <lb/>ſumptum ſit punctum E, quod <lb/>à vertice diſtet per interuallum, <lb/>quod primò ſit æquale dimidio <lb/>recti lateris axis B D. </s> <s xml:id="echoid-s5450" xml:space="preserve">Dico E B <lb/>eſſe _MAXIMAM_ ex E ducibiliũ <lb/>ad Ellipſis peripheriam ABCD.</s> <s xml:id="echoid-s5451" xml:space="preserve"/> </p> <div xml:id="echoid-div560" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0194-01"/> </figure> </div> <p> <s xml:id="echoid-s5452" xml:space="preserve">Si enim facto centro E, cum <lb/>radio E B circulus deſcribatur <lb/>B F, ipſe cadet totus <anchor type="note" xlink:href="" symbol="b"/> extra El- <anchor type="note" xlink:label="note-0194-02a" xlink:href="note-0194-02"/> lipſim; </s> <s xml:id="echoid-s5453" xml:space="preserve">vnde, quæ ex E ad Elli-<lb/>pſis peripheriam ducẽtur, pręter <lb/>ad B, minores erunt, quàm quę <lb/>ex eodem E, ad circuli circum-<lb/>ſcriptam circumferentiam, hoc <lb/>eſt minores ipſa E B. </s> <s xml:id="echoid-s5454" xml:space="preserve">Quare <lb/>E B erit _MAXIMA_, &</s> <s xml:id="echoid-s5455" xml:space="preserve">c.</s> <s xml:id="echoid-s5456" xml:space="preserve"/> </p> <div xml:id="echoid-div561" type="float" level="2" n="5"> <note symbol="b" position="left" xlink:label="note-0194-02" xlink:href="note-0194-02a" xml:space="preserve">1. Co-<lb/>roll. 20. 1. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s5457" xml:space="preserve">Si verò diſtantia à vertice B, <lb/>maior fuerit eodem recti dimi-<lb/>dio, qualis eſt G B: </s> <s xml:id="echoid-s5458" xml:space="preserve">cum ſit in <lb/>circulo B F, ipſa G B, in qua <lb/>eſt circuli centrum, _MAXIMA_ ad eius peripheriam ducibilium, eò ma-<lb/>gis _MAXIMA_ erit ad inſcriptæ Ellipſis peripheriam A B C D.</s> <s xml:id="echoid-s5459" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5460" xml:space="preserve">4. </s> <s xml:id="echoid-s5461" xml:space="preserve">Quod autem in eadem tertia figura reliquum minoris axis ſegmentum <lb/>E D, vel G D, ſit _MINIMA_ ex E, vel G ducibilium ad Ellipſis periphe-<lb/>riam A B C D, ſic manifeſtum fiet.</s> <s xml:id="echoid-s5462" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5463" xml:space="preserve">Quoniam circulus ex radio H D cadit totus <anchor type="note" xlink:href="" symbol="c"/> intra Ellipſim A B C D, <anchor type="note" xlink:label="note-0194-03a" xlink:href="note-0194-03"/> ſed ad peripheriam circuli ex radio H D ipſa E D, vel G D eſt _MINIMA_, <lb/>cum in ea non ſit circuli centrum: </s> <s xml:id="echoid-s5464" xml:space="preserve">quare eadem E D, vel G D eò am-<lb/>pliùs erit _MINIMA_ ducibilium ad eidem circulo circumſcriptam Ellipſis <lb/>peripheriam A B C D. </s> <s xml:id="echoid-s5465" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s5466" xml:space="preserve"/> </p> <div xml:id="echoid-div562" type="float" level="2" n="6"> <note symbol="c" position="left" xlink:label="note-0194-03" xlink:href="note-0194-03a" xml:space="preserve">26. pri-<lb/>mi huius.</note> </div> </div> <div xml:id="echoid-div564" type="section" level="1" n="228"> <head xml:id="echoid-head236" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s5467" xml:space="preserve">HOC loco animaduertendum eſt, ſemper in Ellipſi circa minorem <lb/>axim, tertiæ figuræ, interuallum B E ſemi-rectis lateris, omnino <lb/>excedere minorem ſemi-axim B H, (cum integrum rectum latus excedat <lb/>integrum minorem axim; </s> <s xml:id="echoid-s5468" xml:space="preserve">vt in primo Coroll. </s> <s xml:id="echoid-s5469" xml:space="preserve">20. </s> <s xml:id="echoid-s5470" xml:space="preserve">primi huius monitum <lb/>fuit) ac idem punctum E cadere poſſe in quocunq; </s> <s xml:id="echoid-s5471" xml:space="preserve">puncto infra H, ha- <pb o="13" file="0195" n="195" rhead=""/> bita tamen ratione proportionis inter minorem axim, & </s> <s xml:id="echoid-s5472" xml:space="preserve">maiorem, quæ <lb/>proportio, quò minor fuerit, eò magis E, terminus ſemi - recti lateris, <lb/>remouebitur à centro H, vt vel modicè introſpicienti ſatit conſtat.</s> <s xml:id="echoid-s5473" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div565" type="section" level="1" n="229"> <head xml:id="echoid-head237" xml:space="preserve">THEOR. VI. PROP. X.</head> <p> <s xml:id="echoid-s5474" xml:space="preserve">Si quamcunque coni - fectionem recta linea contingat, cui à <lb/>tactu extra ſectionem perpendicularis erigatur, in qua ſumptum <lb/>ſit quodlibet punctum. </s> <s xml:id="echoid-s5475" xml:space="preserve">Linea intercepta inter aſſumptum pun-<lb/>ctum, & </s> <s xml:id="echoid-s5476" xml:space="preserve">contactum, erit MINIMA ducibilium ab eodem pun-<lb/>cto, ad conuexam coni-ſectionis peripheriam.</s> <s xml:id="echoid-s5477" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5478" xml:space="preserve">ESto coni-ſectio A B C, quam contingat recta D E in B, à quo ipſi <lb/>erecta ſit perpendicularis B F ad partes conuexæ peripheriæ ABC, <lb/>ſitque in ea aſſumptum quodlibet punctum F. </s> <s xml:id="echoid-s5479" xml:space="preserve">Dico rectam F B eſſe _MI-_ <lb/>_NIMAM_ rectarum ducibilium ab F ad conuexam peripheriam A B C.</s> <s xml:id="echoid-s5480" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5481" xml:space="preserve">Hoc enim per ſe ſatis patet: </s> <s xml:id="echoid-s5482" xml:space="preserve">nam cum <lb/> <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> F B ſit perpendicularis rectæ D E, erit <lb/>quoque _MINIMA_ <anchor type="note" xlink:href="" symbol="a"/> ducibilium ad ipſam <anchor type="note" xlink:label="note-0195-01a" xlink:href="note-0195-01"/> D E, quare F B eò magis erit _MINIMA_ <lb/>ducibilium ad conuexam A B C, quę ca-<lb/>dit infra D E. </s> <s xml:id="echoid-s5483" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s5484" xml:space="preserve">c.</s> <s xml:id="echoid-s5485" xml:space="preserve"/> </p> <div xml:id="echoid-div565" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0195-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0195-01" xlink:href="note-0195-01a" xml:space="preserve">ex ele-<lb/>mentis.</note> </div> <p> <s xml:id="echoid-s5486" xml:space="preserve">Quod autem de coni - ſectione hoc <lb/>loco oſtenditur, de quacunque etiam <lb/>curua linea verificari ex ipſa figura ſatis <lb/>patet, dummodo curua A B C ſit tota ad <lb/>alteram partem contingentis D E, per-<lb/>pendicularis verò B F ad aliam.</s> <s xml:id="echoid-s5487" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div567" type="section" level="1" n="230"> <head xml:id="echoid-head238" xml:space="preserve">THEOR. VII. PROP. XI.</head> <p> <s xml:id="echoid-s5488" xml:space="preserve">Si quamcunq, coni-ſectionem recta linea, pręter ad axis ver-<lb/>ticem contingat, cui à tactu intra ſectionem erigatur perpendi-<lb/>cularis, in qua ſumptum ſit punctum quodlibet, non tamen, quò <lb/>ad Ellipſim, vltra maiorem axim; </s> <s xml:id="echoid-s5489" xml:space="preserve">linea intercepta inter aſſum-<lb/>ptum punctum, & </s> <s xml:id="echoid-s5490" xml:space="preserve">contactum erit MINIMA ducibilium ex eo-<lb/>dem puncto, ad coni- ſectionis peripheriam.</s> <s xml:id="echoid-s5491" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5492" xml:space="preserve">Si verò in Ellipſi aſſumptum punctum in perpendiculari fue-<lb/>rit, vel in ipſo minori axe, vel vltra: </s> <s xml:id="echoid-s5493" xml:space="preserve">linea inter punctum, & </s> <s xml:id="echoid-s5494" xml:space="preserve"><lb/>contactum intercepta erit MAXIMA ducibilium ex ipſomet pun-<lb/>cto ad Ellipſis peripheriam.</s> <s xml:id="echoid-s5495" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5496" xml:space="preserve">ESto A B C Parabole, vel Hyperbole, vt in prima figura, vel Ellipſis, <lb/>vt in ſecunda, circa maiorem axim B D, quas in puncto A extra <pb o="14" file="0196" n="196" rhead=""/> axium vertices contingat recta A E, cui intra ſectionem ducta ſit perpen-<lb/>dicularis A D, quæ priùs maiori axi occurret, <anchor type="note" xlink:href="" symbol="a"/> vt in D. </s> <s xml:id="echoid-s5497" xml:space="preserve">Dico rectam <anchor type="note" xlink:label="note-0196-01a" xlink:href="note-0196-01"/> D A, & </s> <s xml:id="echoid-s5498" xml:space="preserve">quamlibet ipſa minorem F A, eſſe _MINIMAM_ ducibilium ad ſe-<lb/>ctionis peripheriam A B C, ex punctis D, vel F.</s> <s xml:id="echoid-s5499" xml:space="preserve"/> </p> <div xml:id="echoid-div567" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0196-01" xlink:href="note-0196-01a" xml:space="preserve">88. pri-<lb/>mih.</note> </div> <p> <s xml:id="echoid-s5500" xml:space="preserve">Nam facto centro D, inter-<lb/> <anchor type="figure" xlink:label="fig-0196-01a" xlink:href="fig-0196-01"/> uallo D A, ac circulo deſcripto <lb/>A C G, ipſe cadet totus <anchor type="note" xlink:href="" symbol="b"/> intra <anchor type="note" xlink:label="note-0196-02a" xlink:href="note-0196-02"/> fectionem A B C, in duobus tan-<lb/>tùm punctis A, C, eam contin-<lb/>gens: </s> <s xml:id="echoid-s5501" xml:space="preserve">quare quę ducentur ex D <lb/>ad ſectionis peripheriam, pręter <lb/>ad puncta A, C, interuallo D A <lb/>maiores erunt: </s> <s xml:id="echoid-s5502" xml:space="preserve">exquo ipſa D A, <lb/>vel D C erit _MINIMA_, &</s> <s xml:id="echoid-s5503" xml:space="preserve">c. </s> <s xml:id="echoid-s5504" xml:space="preserve">Si <lb/>verò interuallum F A minus ſit <lb/>ipſo D A. </s> <s xml:id="echoid-s5505" xml:space="preserve">Cum in circulo A C <lb/>G ipſum F A diametri ſegmen-<lb/>tum, in quo centrum non repe-<lb/>ritur, ſit rectarum _MINIMA_ ad <lb/>circuli peripheriam ducibilium, <lb/>eò magis eadem F A _MINIMA_ <lb/>erit ducibilium ex F, ad peri-<lb/>pheriam circumſcriptę ſectionis <lb/>A B C. </s> <s xml:id="echoid-s5506" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s5507" xml:space="preserve">c.</s> <s xml:id="echoid-s5508" xml:space="preserve"/> </p> <div xml:id="echoid-div568" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0196-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0196-02" xlink:href="note-0196-02a" xml:space="preserve">92. pri-<lb/>mi h.</note> </div> <p> <s xml:id="echoid-s5509" xml:space="preserve">2. </s> <s xml:id="echoid-s5510" xml:space="preserve">Iam, in tertia figura, ſit Elli-<lb/>pſis A B C, circa minorem axim B D, & </s> <s xml:id="echoid-s5511" xml:space="preserve">contingens linea ad punctum A, <lb/>quod non ſit axium vertex, ſit A E, cui ex contactu A, ducta ſit intra. <lb/></s> <s xml:id="echoid-s5512" xml:space="preserve">fectionem recta A D, quę poſt occurſum cum maiori axe, occurret quo-<lb/>que <anchor type="note" xlink:href="" symbol="c"/> minori, vt in D. </s> <s xml:id="echoid-s5513" xml:space="preserve">Dico rectam D A, & </s> <s xml:id="echoid-s5514" xml:space="preserve">quamlibet aliam F A ipſa.</s> <s xml:id="echoid-s5515" xml:space="preserve"> <anchor type="note" xlink:label="note-0196-03a" xlink:href="note-0196-03"/> D A maiorem, _MAXIMAM_ eſſe ducibilium ex D, vel F, ad Ellipſis peri-<lb/>pheriam A B C.</s> <s xml:id="echoid-s5516" xml:space="preserve"/> </p> <div xml:id="echoid-div569" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0196-03" xlink:href="note-0196-03a" xml:space="preserve">88. pri-<lb/>mi huius.</note> </div> <p> <s xml:id="echoid-s5517" xml:space="preserve">Deſcripto enim circulo A C G ex radio D A, ipſe cadet totus <anchor type="note" xlink:href="" symbol="d"/> extra <anchor type="note" xlink:label="note-0196-04a" xlink:href="note-0196-04"/> Ellipſim A B C hanc tantùm contingens in duobus punctis A, C; </s> <s xml:id="echoid-s5518" xml:space="preserve">qua-<lb/>propter, quæ ducentur ex D ad Ellipſis peripheriam, præter ad puncta <lb/>A, C, diſtantia D A minores erunt: </s> <s xml:id="echoid-s5519" xml:space="preserve">vnde D A, vel D C erit _MAXIMA_, &</s> <s xml:id="echoid-s5520" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s5521" xml:space="preserve">Si autem interuallum F A maius fuerit ipſo D A. </s> <s xml:id="echoid-s5522" xml:space="preserve">Cum in circulo <lb/>A C G in diametri ſegmento F A ſit circuli centrum, ipſam F <lb/>A, erit _MAXIMA_ ad circuli peripheriam A C G ducibi-<lb/>lium, & </s> <s xml:id="echoid-s5523" xml:space="preserve">eò magis eadem F A _MAXIMA_ ducibilium <lb/>ex F, ad peripheriam inſcriptæ Ellipſis A B C. </s> <s xml:id="echoid-s5524" xml:space="preserve"><lb/>Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s5525" xml:space="preserve"/> </p> <div xml:id="echoid-div570" type="float" level="2" n="4"> <note symbol="d" position="left" xlink:label="note-0196-04" xlink:href="note-0196-04a" xml:space="preserve">92. pri-<lb/>mihuius.</note> </div> <pb o="15" file="0197" n="197" rhead=""/> </div> <div xml:id="echoid-div572" type="section" level="1" n="231"> <head xml:id="echoid-head239" xml:space="preserve">THEOR. VIII. PROP. XII.</head> <p> <s xml:id="echoid-s5526" xml:space="preserve">Si per punctum quodlibet ſumptum in angulo à rectis lineis <lb/>comprehenſo, quarum altera ſit datæ Parabolæ, vel Hyperbo-<lb/>læ diameter, aut ipſi æquidiſtans, altera verò ſit quęlibet ſectio-<lb/>ni ordinatim ducta, vel huic parallela, deſcripta ſit ſectio Hy-<lb/>perbole, cuius aſymptoti ſint prædicti anguli latera; </s> <s xml:id="echoid-s5527" xml:space="preserve">huiuſmodi <lb/>Hyperbole datam ſectionem in vno tantùm puncto neceſſariò <lb/>ſecabit.</s> <s xml:id="echoid-s5528" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5529" xml:space="preserve">ESto Parabole, vel Hyperbole A B, cuius diameter, vel diametro æ-<lb/>quidiſtans ſit B C, quàm ad quemcunque angulum D E C ſecet D <lb/>E, quæ vel ſit vna applicatarum in ſectione, vel ipſis æquidiſtans, & </s> <s xml:id="echoid-s5530" xml:space="preserve">in <lb/>angulo D E C, per datum in eo punctum F, deſcribatur <anchor type="note" xlink:href="" symbol="a"/> Hyperbole G F <anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/> H, cuius aſymptoti ſint D E, E C. </s> <s xml:id="echoid-s5531" xml:space="preserve">Dico hancvltrò, citroque productam, <lb/>in vno tantùm puncto ſectionem ſecare.</s> <s xml:id="echoid-s5532" xml:space="preserve"/> </p> <div xml:id="echoid-div572" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">4. ſec. <lb/>conic.</note> </div> <figure> <image file="0197-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0197-01"/> </figure> <p> <s xml:id="echoid-s5533" xml:space="preserve">Ductis enim, in prima figura, per punctum F, quod eſt in ſectione <lb/>A B, rectis L F M, I F K aſymptotis D E, E C æquidiſtantibus, eiſque <lb/>occurrentibus in M, K. </s> <s xml:id="echoid-s5534" xml:space="preserve">Patet rectam M F L etiam ſi in infinitum produ-<lb/>ctam ad partes L, in ipſo tantùm puncto F ſectioni A B occurrere, cum <lb/>ſit vna applicatarum in data ſectione; </s> <s xml:id="echoid-s5535" xml:space="preserve">& </s> <s xml:id="echoid-s5536" xml:space="preserve">rectam I F K in eodem tantùm <lb/> <anchor type="note" xlink:label="note-0197-02a" xlink:href="note-0197-02"/> puncto F cum ſectione A B conuenire <anchor type="note" xlink:href="" symbol="b"/> cum ipſa rectæ B C, vel diame- tro datæ ſectionis æquidiſtet: </s> <s xml:id="echoid-s5537" xml:space="preserve">ſed Hyperbole G F H à puncto F ad par-<lb/>tes G, tota incedit in angulo K F L, & </s> <s xml:id="echoid-s5538" xml:space="preserve">inter æquidiſtantes F L, K D; </s> <s xml:id="echoid-s5539" xml:space="preserve">& </s> <s xml:id="echoid-s5540" xml:space="preserve"><lb/>à puncto F ad partes H, tota incedit in angulo M F I, ac inter paralle-<lb/>las F I, M C. </s> <s xml:id="echoid-s5541" xml:space="preserve">quare ipſa Hyperbole G F H in nullo alio puncto quàm F <lb/>ſectioni A B occurret.</s> <s xml:id="echoid-s5542" xml:space="preserve"/> </p> <div xml:id="echoid-div573" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0197-02" xlink:href="note-0197-02a" xml:space="preserve">26. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s5543" xml:space="preserve">In ſecunda verò, ac tertia figura, ductis ex dato puncto F (quod ibi <lb/>extra cadit, hic verò intra ſectionem) rectis F L, F I alteri aſymptoto, <pb o="16" file="0198" n="198" rhead=""/> & </s> <s xml:id="echoid-s5544" xml:space="preserve">ſectioni occurrentibus in L,I. </s> <s xml:id="echoid-s5545" xml:space="preserve">Conſtat Hyperbolen ex F ad partes H <lb/>omnino incedere intra angulum L F I, & </s> <s xml:id="echoid-s5546" xml:space="preserve">cum ipſa in infinitum extendi <lb/>poſſit, cumque in ſecunda figura ſpatium F I B ſit occluſum ad I, & </s> <s xml:id="echoid-s5547" xml:space="preserve">ad <lb/>rectam L B nunquam poſſit prouenire, eò quod ipſa L B ponatur Hyper-<lb/>bole G F H aſymptotos: </s> <s xml:id="echoid-s5548" xml:space="preserve">in tertia verò cum ſpatium F I N ſit vndique oc-<lb/>cluſum, neceſſariò, in vtraque figura, deſcripta Hyperbole G F H in ali-<lb/>quo puncto datam ſectionem ſecabit. </s> <s xml:id="echoid-s5549" xml:space="preserve">Sit ergo harum mutua interſectio <lb/>punctum M, per quod ductis, vt factum fuit in prima figura, rectis lineis <lb/>quæ aſymptotis E D, E C æquidiſtent, ijſdem penitus argumentis, ac in <lb/>primo caſu, demonſtrabitur ipſam Hyperbolen in nullo alio puncto quàm <lb/>M cum data ſectione A B conuenire. </s> <s xml:id="echoid-s5550" xml:space="preserve">Quare ſi per punctum in angulo, &</s> <s xml:id="echoid-s5551" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s5552" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s5553" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div575" type="section" level="1" n="232"> <head xml:id="echoid-head240" xml:space="preserve">THEOR. IX. PROP. XIII.</head> <p> <s xml:id="echoid-s5554" xml:space="preserve">Si in Hyperbola, ſumpta fuerint duo quælibet puncta, à qui-<lb/>bus ductæ ſint aſymptotis æquidiſtantes, eiſque occurrentes: </s> <s xml:id="echoid-s5555" xml:space="preserve">re-<lb/>cta linea iungens occurſus; </s> <s xml:id="echoid-s5556" xml:space="preserve">lineæ, data puncta iungenti, æqui-<lb/>diſtabit.</s> <s xml:id="echoid-s5557" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5558" xml:space="preserve">ESto Hyperbole A B, cuius aſymptoti C D, C E, ſumptaque ſint in <lb/>ſectione duo quælibet puncta A, B, à quibus ductæ ſint A F, B G, <lb/>aſymptotis æquidiſtantes. </s> <s xml:id="echoid-s5559" xml:space="preserve">Dico iunctas A B, F G, eſſe inter ſe paralle-<lb/>las.</s> <s xml:id="echoid-s5560" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5561" xml:space="preserve">Nam vtrinque producta A B vſque-<lb/> <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> ad aſymptotos in D, & </s> <s xml:id="echoid-s5562" xml:space="preserve">E. </s> <s xml:id="echoid-s5563" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Erit in pri- <anchor type="note" xlink:label="note-0198-01a" xlink:href="note-0198-01"/> ma figura, B D æqualis A E: </s> <s xml:id="echoid-s5564" xml:space="preserve">in ſecun-<lb/>da verò, cum ſit A D æqualis B E, ad-<lb/>dita communi A B, erit item D B æ-<lb/>qualis ipſi A E. </s> <s xml:id="echoid-s5565" xml:space="preserve">Sed in triangulis D B <lb/>G, E A F, anguli ad D, B, æquantur <lb/>angulis ad A, & </s> <s xml:id="echoid-s5566" xml:space="preserve">E, vterque vtrique, <lb/>ob paralellas D G, A F, & </s> <s xml:id="echoid-s5567" xml:space="preserve">B G, E F; <lb/></s> <s xml:id="echoid-s5568" xml:space="preserve">quare triangula D B G, A E F ſunt ſimi-<lb/>lia inter ſe, ac propterea vt D B ad B <lb/>G, ita A E ad E F, ſed antecedentes <lb/>D B, A E ſunt ęquales, vt modò oſten-<lb/>dimus, ergo, & </s> <s xml:id="echoid-s5569" xml:space="preserve">conſequentes B G, E F, <lb/>æquales erunt, at ſunt quoque inter ſe <lb/>parallelæ, quare, & </s> <s xml:id="echoid-s5570" xml:space="preserve">F G ipſi A B ęqui-<lb/>diſtabit. </s> <s xml:id="echoid-s5571" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s5572" xml:space="preserve">c.</s> <s xml:id="echoid-s5573" xml:space="preserve"/> </p> <div xml:id="echoid-div575" 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/QN4GHYBF/figures/0198-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0198-01" xlink:href="note-0198-01a" xml:space="preserve">8. ſec. <lb/>conic.</note> </div> <pb o="17" file="0199" n="199" rhead=""/> </div> <div xml:id="echoid-div577" type="section" level="1" n="233"> <head xml:id="echoid-head241" xml:space="preserve">THEOR. X. PROP. XIV.</head> <p> <s xml:id="echoid-s5574" xml:space="preserve">Si in Hyperbola ſumpta fuerint duo quælibet puncta, è quo-<lb/>rum vno ducta ſit recta linea, alteri aſymptoto æquidiſtans, <lb/>aliamque ſecans; </s> <s xml:id="echoid-s5575" xml:space="preserve">ex reliquo verò alia vtranque aſymptoton di-<lb/>uidens in angulo, qui aſymptotali deinceps eſt, à qua, producta <lb/>in angulo ad verticem aſymptotalis, ſumatur ęqualis ei, quę ex <lb/>ipſa inter prædictum punctum, & </s> <s xml:id="echoid-s5576" xml:space="preserve">alteram aſymptoton interci-<lb/>pitur, atque ex ſumptæ termino ducta ſit parallela ei aſympto-<lb/>to, cui prima eductarum occurrit, hanc ipſam ſecans: </s> <s xml:id="echoid-s5577" xml:space="preserve">recta li-<lb/>nea huiuſmodi interſectionem iungens cum puncto, in quo ſe-<lb/>cunda eductarum eam aſymptoton ſecat, cui prima æquidiſtat, <lb/>rectæ data puncta iungenti æquidiſtabit.</s> <s xml:id="echoid-s5578" xml:space="preserve"/> </p> <figure> <image file="0199-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0199-01"/> </figure> <p> <s xml:id="echoid-s5579" xml:space="preserve">SInt in Hyperbola A B, cuius aſymptoti C D, C E, ſumpta duo <lb/>quæcunque puncta A, B, è quorum altero A ducta ſit A E I alteri <lb/>aſymptoto C D æquidiſtans, ex B verò quælibet B G F vtranque ſecans <lb/>in G, & </s> <s xml:id="echoid-s5580" xml:space="preserve">F; </s> <s xml:id="echoid-s5581" xml:space="preserve">ſectaque G H in directum, & </s> <s xml:id="echoid-s5582" xml:space="preserve">æquali ipſi B F, ducatur ex H <lb/>recta H I parallela ad C E occurrens cum productis D C, A E in L, & </s> <s xml:id="echoid-s5583" xml:space="preserve"><lb/>I. </s> <s xml:id="echoid-s5584" xml:space="preserve">Dico iunctas A B, F I eſſe inter ſe parallelas.</s> <s xml:id="echoid-s5585" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5586" xml:space="preserve">Ducta enim B D parallela ad C E, iunctaque D E, cum ſit B F æqua-<lb/>lis G H, erit quoque D F æqualis C L, ob parallelas D B, G E, HI, ſed <lb/>eſt C L æqualis ipſi E I, quare D F, & </s> <s xml:id="echoid-s5587" xml:space="preserve">E I æquales erunt, ſuntque etiam <lb/>parallelæ, ergo F I æquidiſtat ipſi D E, ſed eſt A B <anchor type="note" xlink:href="" symbol="a"/> æquidiſtans eidem <anchor type="note" xlink:label="note-0199-01a" xlink:href="note-0199-01"/> D E, quare F I, & </s> <s xml:id="echoid-s5588" xml:space="preserve">A B ſunt quoque inter ſe parallelæ. </s> <s xml:id="echoid-s5589" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s5590" xml:space="preserve">c.</s> <s xml:id="echoid-s5591" xml:space="preserve"/> </p> <div xml:id="echoid-div577" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve">13. h.</note> </div> <pb o="18" file="0200" n="200" rhead=""/> </div> <div xml:id="echoid-div579" type="section" level="1" n="234"> <head xml:id="echoid-head242" xml:space="preserve">THEOR. XI. PROP. XV.</head> <p> <s xml:id="echoid-s5592" xml:space="preserve">Si à puncto, quod eſt intra Hyperbolen, ductæ ſint duæ re-<lb/>ctæ lineæ aſymptotis æquidiſtantes, & </s> <s xml:id="echoid-s5593" xml:space="preserve">Hyperbolæ in duobus <lb/>punctis occurrentes, è quorum altero ducta ſit recta linea vtran-<lb/>que aſymptoton ſecans, à qua, producta in angulo, qui aſym-<lb/>ptotalis eſt ad verticem, à puncto alteram aſymptoton ſecans <lb/>dematur æqualis ei, quę inter eductæ occurſum cum alia aſym-<lb/>ptoto intercipitur: </s> <s xml:id="echoid-s5594" xml:space="preserve">recta linea hoc idem occurſum iungens cum <lb/>dato puncto, æquidiſtabit rectæ, ſumptæ terminum iungenti, & </s> <s xml:id="echoid-s5595" xml:space="preserve"><lb/>ſectionis punctum, in quo conuenit recta alteri aſymptoto ęqui-<lb/>diſtanter ducta.</s> <s xml:id="echoid-s5596" xml:space="preserve"/> </p> <figure> <image file="0200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0200-01"/> </figure> <p> <s xml:id="echoid-s5597" xml:space="preserve">ESto intra Hyperbolen A B, cuius centrum C, & </s> <s xml:id="echoid-s5598" xml:space="preserve">aſymptoti C D, <lb/>C E vltra centrum productæ, ſumptum quodcunque punctum F, à <lb/>quo ductæ ſint F A D, F B E aſymptotis æquidiſtantes, quæ Hyperbolen <lb/>ſecent in punctis A, B, è quorum altero, vt ex B, ducta ſit quæcunque <lb/>B I aſymptoton C E ſecans in G, & </s> <s xml:id="echoid-s5599" xml:space="preserve">C D in H, ſumptaque H I æquali, <lb/>& </s> <s xml:id="echoid-s5600" xml:space="preserve">in directum ipſi B G, iungantur rectæ I A, G F. </s> <s xml:id="echoid-s5601" xml:space="preserve">Dico has inter ſe eſſe <lb/>parallelas.</s> <s xml:id="echoid-s5602" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5603" xml:space="preserve">Nam cum recta G H ſecet vtranque linearum C G, C H continentium <lb/>angulum H C G, qui deinceps eſt angulo D C E Hyperbolen A B conti-<lb/>nenti, ſitque ea (per conſtructionem) hinc inde æqualiter producta in. <lb/></s> <s xml:id="echoid-s5604" xml:space="preserve">B, I, & </s> <s xml:id="echoid-s5605" xml:space="preserve">punctum B ſit ad Hyperbolen A B, erit etiam punctum I ad ei <lb/>oppoſitam ſectionem. </s> <s xml:id="echoid-s5606" xml:space="preserve">Si enim oppoſita ſectio in alio puncto, pręter I, ſe- <pb o="19" file="0201" n="201" rhead=""/> caret rectam G I, vt in L, tunc G L <anchor type="note" xlink:href="" symbol="a"/> æquaretur ipſi H B, ideoque G I, <anchor type="note" xlink:label="note-0201-01a" xlink:href="note-0201-01"/> G L inter ſe æquales eſſent, totum, & </s> <s xml:id="echoid-s5607" xml:space="preserve">pars, quod eſt abſurdum.</s> <s xml:id="echoid-s5608" xml:space="preserve"/> </p> <div xml:id="echoid-div579" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve">16. ſec. <lb/>conic.</note> </div> <p> <s xml:id="echoid-s5609" xml:space="preserve">Cum ergo puncta I, A cadant in oppoſitas ſectiones, iunctaque ſit I <lb/>A ſecans rectas C O, C D continentes angulum O C D, qui deinceps eſt <lb/>angulo D C E ſectionem A B continenti <anchor type="note" xlink:href="" symbol="b"/> erunt ex ipſa abſciſſæ lineæ M <anchor type="note" xlink:label="note-0201-02a" xlink:href="note-0201-02"/> I, N A inter aſymptotos, & </s> <s xml:id="echoid-s5610" xml:space="preserve">ſectiones interiectæ inter ſe æquales. </s> <s xml:id="echoid-s5611" xml:space="preserve">Pro-<lb/>ducantur F A, F B vſque ad aſymptotos in D, E, agaturque ex I recta <lb/>I O æquidiſtans ad C D. </s> <s xml:id="echoid-s5612" xml:space="preserve">Cumque triangulorum I O M, N D A, baſes I <lb/>M, N A ſint in directum conſtitutæ ſintque latera I O, N D; </s> <s xml:id="echoid-s5613" xml:space="preserve">M O, A D <lb/>inter ſe parallela, ſingula ſingulis, erunt quoque anguli ad I, & </s> <s xml:id="echoid-s5614" xml:space="preserve">N; </s> <s xml:id="echoid-s5615" xml:space="preserve">vti <lb/>etiam ad M, & </s> <s xml:id="echoid-s5616" xml:space="preserve">A inter ſe æquales; </s> <s xml:id="echoid-s5617" xml:space="preserve">ſed & </s> <s xml:id="echoid-s5618" xml:space="preserve">baſes I M, N A inter ſe ſunt <lb/>æquales, vt ſuperiùs demonſtratum fuit, quare, & </s> <s xml:id="echoid-s5619" xml:space="preserve">reliqua latera M O, <lb/>A D æqualia erunt.</s> <s xml:id="echoid-s5620" xml:space="preserve"/> </p> <div xml:id="echoid-div580" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0201-02" xlink:href="note-0201-02a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s5621" xml:space="preserve">Præterea cum ſit linea B G æqualis H I, erunt quoque E G, C O inter <lb/>ſe æquales (ob æ quidiſtantiam linearum I O, H C, B E,) quibus addita <lb/>communi G C in prima, ſecunda, & </s> <s xml:id="echoid-s5622" xml:space="preserve">tertia figura, vel dempta in quin-<lb/>ta, proueniet E C, æqualis ipſi G O, ſed F D, E C ſunt æquales (nam <lb/>ſunt latera oppoſita in parallelogrammo C F,) quare F D ipſi G O ęqua-<lb/>lis erit; </s> <s xml:id="echoid-s5623" xml:space="preserve">ſi ergo ex his demantur æquales M O, A D; </s> <s xml:id="echoid-s5624" xml:space="preserve">reliquæ G M, F A <lb/>æquales erunt, at ſunt quoque parallelæ, vnde G F, I A inter ſe ęquidi <lb/>ſtabunt. </s> <s xml:id="echoid-s5625" xml:space="preserve">Quod demonſtrare oportebat.</s> <s xml:id="echoid-s5626" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div582" type="section" level="1" n="235"> <head xml:id="echoid-head243" xml:space="preserve">LEMMA V. PROP. XVI.</head> <p> <s xml:id="echoid-s5627" xml:space="preserve">Sint duæ rationes, A B nempe ad B C, & </s> <s xml:id="echoid-s5628" xml:space="preserve">D E ad F maioris <lb/>inæqualitatis, & </s> <s xml:id="echoid-s5629" xml:space="preserve">ſit ratio A B ad B C, minor ratione D E ad F. <lb/></s> <s xml:id="echoid-s5630" xml:space="preserve">Oportet B C, ita ſecare in G, ita vt A G ad G C ſit vt D E <lb/>ad F.</s> <s xml:id="echoid-s5631" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5632" xml:space="preserve">FIat E H æqualis F, & </s> <s xml:id="echoid-s5633" xml:space="preserve">vt D H ad H E, ita A C ad C G, & </s> <s xml:id="echoid-s5634" xml:space="preserve">punctum <lb/>G erit quæſitum. </s> <s xml:id="echoid-s5635" xml:space="preserve">Quoniam cum A C ad C G ſit vt D H ad H E, <lb/>erit componendo A G ad G C, <lb/> <anchor type="figure" xlink:label="fig-0201-01a" xlink:href="fig-0201-01"/> vt D E ad E H, velad F. </s> <s xml:id="echoid-s5636" xml:space="preserve">Quod <lb/>faciendum erat.</s> <s xml:id="echoid-s5637" xml:space="preserve"/> </p> <div xml:id="echoid-div582" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0201-01"/> </figure> </div> <p> <s xml:id="echoid-s5638" xml:space="preserve">Quod autem punctum G ca-<lb/>dat infra B, patet. </s> <s xml:id="echoid-s5639" xml:space="preserve">Nam ex hy-<lb/>poteſi, A B ad B C habet mi-<lb/>norem rationem quàm D E ad <lb/>F, vel ad E H, quare diuiden-<lb/>do A C ad C B habebit mino-<lb/>rem rationem, quàm D H ad H <lb/>E, vel quàm eadem A C ad C <lb/>G; </s> <s xml:id="echoid-s5640" xml:space="preserve">ergo C B eſt maior C G; </s> <s xml:id="echoid-s5641" xml:space="preserve">ſiue <lb/>punctum G cadit infra B. </s> <s xml:id="echoid-s5642" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s5643" xml:space="preserve"/> </p> <pb o="20" file="0202" n="202" rhead=""/> </div> <div xml:id="echoid-div584" type="section" level="1" n="236"> <head xml:id="echoid-head244" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s5644" xml:space="preserve">HInc, data ratione maioris inæqualitatis, hoc eſt D E, ad E H, & </s> <s xml:id="echoid-s5645" xml:space="preserve"><lb/>differentia A C inter duo s terminos ignotos A G, G C, qui de-<lb/>beant eſſe in data ratione, eruitur quomodo reperiantur ipſi termini A G, <lb/>G C. </s> <s xml:id="echoid-s5646" xml:space="preserve">Facta enim fuit vt D H differentia primorum, ad H E minorem ter-<lb/>minum, ita data differentia A C, ad aliam C G, & </s> <s xml:id="echoid-s5647" xml:space="preserve">reperti ſunt quæſiti <lb/>termini A G, G C, Nam ſtatim oſtenſum fuit eſſe A G ad G C, vt D E <lb/>ad E H.</s> <s xml:id="echoid-s5648" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div585" type="section" level="1" n="237"> <head xml:id="echoid-head245" xml:space="preserve">THEOR. XII. PROP. XVII.</head> <p> <s xml:id="echoid-s5649" xml:space="preserve">Si fuerit in angulo rectilineo quælibet applicata, à qua hinc <lb/>inde ab eius termino æqualia ſegmenta ſint abſciſſa, & </s> <s xml:id="echoid-s5650" xml:space="preserve">per v-<lb/>num diuiſionis punctum deſcribatur Hyperbole, cuius aſympto-<lb/>ti ſint latera dati anguli, ipſa per alterum punctum neceſſariò <lb/>tranſibit.</s> <s xml:id="echoid-s5651" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5652" xml:space="preserve">SIt in angulo A B C applicata quæcunque A C, quæ inæqualiter ſece-<lb/>tur in D, & </s> <s xml:id="echoid-s5653" xml:space="preserve">ſumatur C E æqualis A D. </s> <s xml:id="echoid-s5654" xml:space="preserve">Dico ſi per punctum D de-<lb/>ſcribatur Hyperbole, cuius aſymptoti ſint B A, B C, ipſam omnino tran-<lb/>ſire per E.</s> <s xml:id="echoid-s5655" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5656" xml:space="preserve">Quod huiuſmodi Hyper-<lb/> <anchor type="figure" xlink:label="fig-0202-01a" xlink:href="fig-0202-01"/> bole tranſiens per D, alibi <lb/>ſecet applicatam A C, pa-<lb/>tet. </s> <s xml:id="echoid-s5657" xml:space="preserve">Nam ſi eam continge-<lb/>ret in D, eſſet A C æquali-<lb/>ter <anchor type="note" xlink:href="" symbol="a"/> ſecta in D: </s> <s xml:id="echoid-s5658" xml:space="preserve">quod eſt <anchor type="note" xlink:label="note-0202-01a" xlink:href="note-0202-01"/> contra hypoteſim. </s> <s xml:id="echoid-s5659" xml:space="preserve">Secet er-<lb/>go in F; </s> <s xml:id="echoid-s5660" xml:space="preserve">& </s> <s xml:id="echoid-s5661" xml:space="preserve">erit F C <anchor type="note" xlink:href="" symbol="b"/> ęqua- <anchor type="note" xlink:label="note-0202-02a" xlink:href="note-0202-02"/> lis A D, ſed eſt quoque E <lb/>C eidem A D ęqualis, qua-<lb/>re F C, E C ęquales erunt; <lb/></s> <s xml:id="echoid-s5662" xml:space="preserve">hoc eſt punctum F congruet <lb/>cum ipſo E; </s> <s xml:id="echoid-s5663" xml:space="preserve">quare Hyper-<lb/>bole D F, quæ in angulo aſymptotali A B C deſcribitur per D, omnino <lb/>tranſit per E. </s> <s xml:id="echoid-s5664" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s5665" xml:space="preserve"/> </p> <div xml:id="echoid-div585" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0202-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0202-01" xlink:href="note-0202-01a" xml:space="preserve">3. ſecun-<lb/>di conic.</note> <note symbol="b" position="left" xlink:label="note-0202-02" xlink:href="note-0202-02a" xml:space="preserve">8. ibid.</note> </div> <pb o="21" file="0203" n="203" rhead=""/> </div> <div xml:id="echoid-div587" type="section" level="1" n="238"> <head xml:id="echoid-head246" xml:space="preserve">THEOR. XIII. PROP. XVIII.</head> <p> <s xml:id="echoid-s5666" xml:space="preserve">Si per centrum Ellipſis deſcribatur Hyperbole, cuius aſym-<lb/>ptoti coniugatis diametris æquidiſtent; </s> <s xml:id="echoid-s5667" xml:space="preserve">ipſa in duobus tantùm <lb/>punctis Ellipſis peripheriam ſecabit.</s> <s xml:id="echoid-s5668" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5669" xml:space="preserve">ESto Ellipſis A B C D, cuius centrum E, & </s> <s xml:id="echoid-s5670" xml:space="preserve">diametri coniugatæ ſint <lb/>A C, B D quibus ductæ ſint F G, H C ipſis diametris altera alteri ę-<lb/>quidiſtantes, & </s> <s xml:id="echoid-s5671" xml:space="preserve">ſimul occurrentes in G; </s> <s xml:id="echoid-s5672" xml:space="preserve">& </s> <s xml:id="echoid-s5673" xml:space="preserve">cum aſymptotis G F, G H, per <lb/>centrum E, deſcripta ſit Hyperbole I E L. </s> <s xml:id="echoid-s5674" xml:space="preserve">Dico hanc, Ellipſis periphe-<lb/>riam in duobus tantùm punctis ſecare.</s> <s xml:id="echoid-s5675" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5676" xml:space="preserve">Nam cum in Hyperbola I E L <lb/>ſumptum ſit punctum E, per <lb/> <anchor type="figure" xlink:label="fig-0203-01a" xlink:href="fig-0203-01"/> quod ductæ ſunt A E C, D E B <lb/>aſymptotis æquidiſtantes, ipſæ <lb/>in puncto tantùm E ſectioni oc-<lb/>current, & </s> <s xml:id="echoid-s5677" xml:space="preserve">Hyperbole in angu-<lb/>lo B E A, inter E A, & </s> <s xml:id="echoid-s5678" xml:space="preserve">G F <lb/>ſemper incedet, pariterque in <lb/>angulo C E D, inter E D, & </s> <s xml:id="echoid-s5679" xml:space="preserve"><lb/>G H; </s> <s xml:id="echoid-s5680" xml:space="preserve">ſed anguli B E A, C E D <lb/>terminantur à peripherijs B A, <lb/>C D, quare Hyperbole ex vtra-<lb/>que parte producta ipſas peri-<lb/>pherias omninò ſecabit, vt in <lb/>I, L. </s> <s xml:id="echoid-s5681" xml:space="preserve">Si ergo ex I ducantur M <lb/>I N, O I F diametris æquidiſtã-<lb/>tes, ob eandem rationem ſupe-<lb/>riùs allatam ſectio E I P, in nullo alio puncto, quàm I cum rectis N I M, <lb/>F I O conueniet, ſed ipſæ N I M, F I O nil aliud commune habent <lb/>cum peripheria quadrantis A B, quàm idem punctum I, quare <lb/>Hyperbole E I P in vno tantùm puncto I Ellipſis periphe-<lb/>riam ſecabit in quadrante A B. </s> <s xml:id="echoid-s5682" xml:space="preserve">Cõſimili conſtructione, <lb/>& </s> <s xml:id="echoid-s5683" xml:space="preserve">argumento, oſtendetur ſectionem E L Q in <lb/>alio puncto quàm L peripheriam D C non <lb/>ſecare: </s> <s xml:id="echoid-s5684" xml:space="preserve">quare huiuſmodi Hyperbole in <lb/>duobus tantùm punctis ſecat El-<lb/>lipſis peripheriam. </s> <s xml:id="echoid-s5685" xml:space="preserve">Quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s5686" xml:space="preserve"/> </p> <div xml:id="echoid-div587" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0203-01"/> </figure> </div> <pb o="22" file="0204" n="204" rhead=""/> </div> <div xml:id="echoid-div589" type="section" level="1" n="239"> <head xml:id="echoid-head247" xml:space="preserve">THEOR. XIV. PROP. XIX.</head> <p> <s xml:id="echoid-s5687" xml:space="preserve">Si à puncto, quod eſt in angulo aſymptotali, ductæ ſint re-<lb/>ctæ lineæ aſymptotis æquidiſtantes, & </s> <s xml:id="echoid-s5688" xml:space="preserve">Hyperbolæ occurrentes, <lb/>atque ex vnius eductarum occurſu agatur recta, quæ ſectionem, <lb/>vel in ipſo tangens puncto, vel alibi ſecans, producta ſecet <lb/>quoque eam aſymptoton, cui altera eductarum æqui diſtat; </s> <s xml:id="echoid-s5689" xml:space="preserve">re-<lb/>cta linea iungens hoc idem punctum cum puncto contactus, vel <lb/>interſectionis nouiter ductæ lineæ cum Hyperbola, æquidiſtabit <lb/>rectæ, quę ab occurſu eiuſdem lineæ cum prædicta aſymptoto <lb/>ad datum punctum educitur.</s> <s xml:id="echoid-s5690" xml:space="preserve"/> </p> <figure> <image file="0204-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0204-01"/> </figure> <p> <s xml:id="echoid-s5691" xml:space="preserve">SIt Hyperbole A B C, in cuius angulo aſymptotali E D F ſumptum ſit <lb/>quodlibet punctum G, vel extra Hyperbolen, vt in prima, ſecunda, <lb/>& </s> <s xml:id="echoid-s5692" xml:space="preserve">tertia; </s> <s xml:id="echoid-s5693" xml:space="preserve">vel intra, vt in quarta, quinta, & </s> <s xml:id="echoid-s5694" xml:space="preserve">ſexta figura, à quo ductæ <lb/>ſint aſymptotis æquidiſtantes G A, G C, ſectioni occurrentes in A, C; <lb/></s> <s xml:id="echoid-s5695" xml:space="preserve">& </s> <s xml:id="echoid-s5696" xml:space="preserve">ex altero occurſuum C ducta ſit quæcunque alia C B E, quæ, vel ſe-<lb/>ctionem contingat in C, vt in prima, & </s> <s xml:id="echoid-s5697" xml:space="preserve">quarta figura, vel alibi ſecet in <lb/>B, vt in reliquis, & </s> <s xml:id="echoid-s5698" xml:space="preserve">producta conueniat cum aſymptoto D E, quæ rectæ <lb/>G A ęquidiſtat. </s> <s xml:id="echoid-s5699" xml:space="preserve">Dico, ſi iungantur A B, E G ipſas inter ſe æquidiſtare.</s> <s xml:id="echoid-s5700" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5701" xml:space="preserve">Nam ducta B H parallela ad F D, productiſque A G, C G vſque ad <lb/>aſymptotos in F, L; </s> <s xml:id="echoid-s5702" xml:space="preserve">& </s> <s xml:id="echoid-s5703" xml:space="preserve">E B C ad aliam aſymptoton D F in I. </s> <s xml:id="echoid-s5704" xml:space="preserve">Erit iuncta <lb/>A B iunctæ H F <anchor type="note" xlink:href="" symbol="a"/> parallela, eſt autem E B æqualis C I; </s> <s xml:id="echoid-s5705" xml:space="preserve">quare, ob paral- <anchor type="note" xlink:label="note-0204-01a" xlink:href="note-0204-01"/> lelas B H, C L, I D, erit quoque E H æqualis ipſi L D, ſiue ęqualis G F;</s> <s xml:id="echoid-s5706" xml:space="preserve"> <pb o="23" file="0205" n="205" rhead=""/> ſed E H, G F ſunt etiam parallelæ, ergo, & </s> <s xml:id="echoid-s5707" xml:space="preserve">E G æquidiſtat H F, ſed A <lb/>B quoque ipſi H F æquidiſtat, vt modò oſtendimus: </s> <s xml:id="echoid-s5708" xml:space="preserve">quare A B, & </s> <s xml:id="echoid-s5709" xml:space="preserve">E G <lb/>ſunt inter ſe parallelæ. </s> <s xml:id="echoid-s5710" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s5711" xml:space="preserve">c.</s> <s xml:id="echoid-s5712" xml:space="preserve"/> </p> <div xml:id="echoid-div589" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve">13. h.</note> </div> </div> <div xml:id="echoid-div591" type="section" level="1" n="240"> <head xml:id="echoid-head248" xml:space="preserve">PROBL. I. PROP. XX.</head> <p> <s xml:id="echoid-s5713" xml:space="preserve">A dato puncto, ad datæ Parabolę peripheriam, MINIMAM <lb/>rectam lineam ducere.</s> <s xml:id="echoid-s5714" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5715" xml:space="preserve">SIt data Parabole A B C, cuius axis B D, vertex B, rectum latus B E, <lb/>& </s> <s xml:id="echoid-s5716" xml:space="preserve">datum vbicunque punctum ſit F. </s> <s xml:id="echoid-s5717" xml:space="preserve">Oportet ex F ad peripheriam <lb/>A B C, _MINIMAM_ rectam lineam ducere.</s> <s xml:id="echoid-s5718" xml:space="preserve"/> </p> <figure> <image file="0205-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0205-01"/> </figure> <p> <s xml:id="echoid-s5719" xml:space="preserve">Eſto primùm datum punctum F extra <lb/>Parabolen in axe producto, vt in prima <lb/>figura. </s> <s xml:id="echoid-s5720" xml:space="preserve">Dico ipſam F B eſſe _MINIMAM_.</s> <s xml:id="echoid-s5721" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5722" xml:space="preserve">Nam cum B D ſit axis Parabolæ, ſi ex <lb/>B ducatur B G ordinatis æquidiſtans, ipſa <lb/>cum F D rectos angulos efficiet, ac Para-<lb/> <anchor type="note" xlink:label="note-0205-01a" xlink:href="note-0205-01"/> bolen <anchor type="note" xlink:href="" symbol="a"/> continget. </s> <s xml:id="echoid-s5723" xml:space="preserve">Cum ergo B F perpen- dicularis ſit contingenti B G, erit F B _MI_-<lb/>_MIMA_ <anchor type="note" xlink:href="" symbol="b"/> omnium, quæ ex F ad periphe- <anchor type="note" xlink:label="note-0205-02a" xlink:href="note-0205-02"/> riam A B C educi poſſunt. </s> <s xml:id="echoid-s5724" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s5725" xml:space="preserve">c.</s> <s xml:id="echoid-s5726" xml:space="preserve"/> </p> <div xml:id="echoid-div591" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0205-01" xlink:href="note-0205-01a" xml:space="preserve">32. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0205-02" xlink:href="note-0205-02a" xml:space="preserve">10. h.</note> </div> <p> <s xml:id="echoid-s5727" xml:space="preserve">Si verò datum punctum F, in ſecunda <lb/>figura, fuerit in ipſo axe B D intra Para-<lb/>bolen A B C, quod diſtet à vertice B, per <lb/>interuallum non maius dimidio recti B E, idem axis ſegmentum F B erit <lb/>_MINIMA_ <anchor type="note" xlink:href="" symbol="c"/> recta quæſita.</s> <s xml:id="echoid-s5728" xml:space="preserve"/> </p> <note symbol="c" position="right" xml:space="preserve">9. hulus <lb/>ad nu. h.</note> <p> <s xml:id="echoid-s5729" xml:space="preserve">Si autem datum punctum F in eadem fi-<lb/> <anchor type="figure" xlink:label="fig-0205-02a" xlink:href="fig-0205-02"/> gura ſit in axe B D, ſed interuallum F B <lb/>maius ſit dimidio recti B E. </s> <s xml:id="echoid-s5730" xml:space="preserve">Secetur F G <lb/>æqualis eidem dimidio, & </s> <s xml:id="echoid-s5731" xml:space="preserve">applicetut G A <lb/>peripheriæ occurrens in A. </s> <s xml:id="echoid-s5732" xml:space="preserve">Dico iunctam <lb/>F A eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s5733" xml:space="preserve"/> </p> <div xml:id="echoid-div592" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0205-02"/> </figure> </div> <p> <s xml:id="echoid-s5734" xml:space="preserve">Ducta enim ex A <anchor type="note" xlink:href="" symbol="d"/> contingente A H, <anchor type="note" xlink:label="note-0205-04a" xlink:href="note-0205-04"/> ipſa cum axe producta <anchor type="note" xlink:href="" symbol="e"/> conueniet in H;</s> <s xml:id="echoid-s5735" xml:space="preserve"> <anchor type="note" xlink:label="note-0205-05a" xlink:href="note-0205-05"/> eritque H B ęqualis <anchor type="note" xlink:href="" symbol="f"/> B G, ſiue H G dupla <anchor type="note" xlink:label="note-0205-06a" xlink:href="note-0205-06"/> G B, eſtque E B dupla G F, ex conſtructio-<lb/>ne, ergo H G ad G B eſt vt E B ad G F; </s> <s xml:id="echoid-s5736" xml:space="preserve">ex <lb/>quo rectangulum H G F æquabitur rectan-<lb/>gulo E B G, ſiue <anchor type="note" xlink:href="" symbol="g"/> quadrato G A; </s> <s xml:id="echoid-s5737" xml:space="preserve">quare <anchor type="note" xlink:label="note-0205-07a" xlink:href="note-0205-07"/> angulus F A H rectus <anchor type="note" xlink:href="" symbol="h"/> erit. </s> <s xml:id="echoid-s5738" xml:space="preserve">Cumque A F ſit ex contactu A Contingenti A H perpendicularis, & </s> <s xml:id="echoid-s5739" xml:space="preserve">punctum F ſit in axe, erit F A _MINIMA_ <anchor type="note" xlink:href="" symbol="i"/> du- <anchor type="note" xlink:label="note-0205-08a" xlink:href="note-0205-08"/> @ibilium ad Parabolæ peripheriam A B C. </s> <s xml:id="echoid-s5740" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s5741" xml:space="preserve">c.</s> <s xml:id="echoid-s5742" xml:space="preserve"/> </p> <div xml:id="echoid-div593" type="float" level="2" n="3"> <note symbol="d" position="right" xlink:label="note-0205-04" xlink:href="note-0205-04a" xml:space="preserve">2. pr. h.</note> <note symbol="e" position="right" xlink:label="note-0205-05" xlink:href="note-0205-05a" xml:space="preserve">24. pri-<lb/>mi conic.</note> <note symbol="f" position="right" xlink:label="note-0205-06" xlink:href="note-0205-06a" xml:space="preserve">35. ibid.</note> <note symbol="g" position="right" xlink:label="note-0205-07" xlink:href="note-0205-07a" xml:space="preserve">Coroll. <lb/>pr. 1. h.</note> <note symbol="h" position="right" xlink:label="note-0205-08" xlink:href="note-0205-08a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> </div> <note symbol="i" position="right" xml:space="preserve">11. h. ad <lb/>num. 1.</note> <p> <s xml:id="echoid-s5743" xml:space="preserve">Si denique datum punctum F ſit extra Parabolen A B C, vt in tertia <lb/>figura, vel extra, vt in quarta, inter axem B D, & </s> <s xml:id="echoid-s5744" xml:space="preserve">peripheriam B A; <lb/></s> <s xml:id="echoid-s5745" xml:space="preserve">Applicetur ex recta F F G axi occurrens in G, dematurque de axe ſupra <pb o="24" file="0206" n="206" rhead=""/> F G recta G H ęqualis dimidio re-<lb/> <anchor type="figure" xlink:label="fig-0206-01a" xlink:href="fig-0206-01"/> cti B E, & </s> <s xml:id="echoid-s5746" xml:space="preserve">ex H agatur H I paral-<lb/> <anchor type="note" xlink:label="note-0206-01a" xlink:href="note-0206-01"/> lela ad G F, & </s> <s xml:id="echoid-s5747" xml:space="preserve">in angulo I H D per <lb/>punctum F deſcribatur <anchor type="note" xlink:href="" symbol="a"/> Hyperbo- le F A, quæ Parabolæ periphe-<lb/> <anchor type="note" xlink:label="note-0206-02a" xlink:href="note-0206-02"/> riam in vno tantùm puncto A ſe-<lb/>cabit, <anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s5748" xml:space="preserve">iungatur F A. </s> <s xml:id="echoid-s5749" xml:space="preserve">Dico hãc eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s5750" xml:space="preserve"/> </p> <div xml:id="echoid-div594" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0206-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0206-01" xlink:href="note-0206-01a" xml:space="preserve">4. ſec. <lb/>conic.</note> <note symbol="b" position="left" xlink:label="note-0206-02" xlink:href="note-0206-02a" xml:space="preserve">12. h.</note> </div> <p> <s xml:id="echoid-s5751" xml:space="preserve">Applicetur A L, & </s> <s xml:id="echoid-s5752" xml:space="preserve">ex A duca-<lb/>tur contingens A M, axi occur-<lb/> <anchor type="note" xlink:label="note-0206-03a" xlink:href="note-0206-03"/> rens in M, producaturque F A ad <lb/>vtranque partem, quæ cum aſym-<lb/>ptotis <anchor type="note" xlink:href="" symbol="c"/> conueniet in I, D, eruntq;</s> <s xml:id="echoid-s5753" xml:space="preserve"> in vtraque figura, interceptę A I, <lb/>F D inter ſe ęquales, ac ideo H L, <lb/>G D ęquales erunt, ob ęquidiſtan-<lb/>tes lineas I H, A L, F G, in trian-<lb/>gulo I H D; </s> <s xml:id="echoid-s5754" xml:space="preserve">ſi ergo, in tertia figu-<lb/>ra, dematur communis L G, & </s> <s xml:id="echoid-s5755" xml:space="preserve"><lb/>in quarta, addatur, fient H G, <lb/>L D inter ſe æquales; </s> <s xml:id="echoid-s5756" xml:space="preserve">ſed eſt G <lb/>H dimidia B E, quare, & </s> <s xml:id="echoid-s5757" xml:space="preserve">L D <lb/>ipſius B E dimidia erit. </s> <s xml:id="echoid-s5758" xml:space="preserve">Et quo-<lb/>niam quadratum A L æquatur re-<lb/> <anchor type="note" xlink:label="note-0206-04a" xlink:href="note-0206-04"/> ctangulo <anchor type="note" xlink:href="" symbol="d"/> L B E, & </s> <s xml:id="echoid-s5759" xml:space="preserve">rectangulum L B E, æquale eſt rectangulo ſub <lb/> <anchor type="note" xlink:label="note-0206-05a" xlink:href="note-0206-05"/> dupla L B, ſiue ſub <anchor type="note" xlink:href="" symbol="e"/> M L, & </s> <s xml:id="echoid-s5760" xml:space="preserve">ſub dimidia B E, hoc eſt ſub L D, ergo qua- dratum A L æquale erit rectangulo M L D, ac ideo angulus M A D re-<lb/> <anchor type="note" xlink:label="note-0206-06a" xlink:href="note-0206-06"/> ctus <anchor type="note" xlink:href="" symbol="f"/> erit, ſiue F A erit ex contactu A contingenti A M perpendicu- <anchor type="note" xlink:label="note-0206-07a" xlink:href="note-0206-07"/> laris: </s> <s xml:id="echoid-s5761" xml:space="preserve">quare F A, in vtraque figura, erit <anchor type="note" xlink:href="" symbol="g"/> _MINIMA_ quæſita. </s> <s xml:id="echoid-s5762" xml:space="preserve">Quod fa- ciendum erat.</s> <s xml:id="echoid-s5763" xml:space="preserve"/> </p> <div xml:id="echoid-div595" type="float" level="2" n="5"> <note symbol="c" position="left" xlink:label="note-0206-03" xlink:href="note-0206-03a" xml:space="preserve">8. ſecũ-<lb/>di conic.</note> <note symbol="d" position="left" xlink:label="note-0206-04" xlink:href="note-0206-04a" xml:space="preserve">Coroll. <lb/>primæ 1. <lb/>huius.</note> <note symbol="e" position="left" xlink:label="note-0206-05" xlink:href="note-0206-05a" xml:space="preserve">35. pri-<lb/>mi conic.</note> <note symbol="f" position="left" xlink:label="note-0206-06" xlink:href="note-0206-06a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> <note symbol="g" position="left" xlink:label="note-0206-07" xlink:href="note-0206-07a" xml:space="preserve">10. h. & <lb/>11. h. ad <lb/>num. 1.</note> </div> </div> <div xml:id="echoid-div597" type="section" level="1" n="241"> <head xml:id="echoid-head249" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s5764" xml:space="preserve">NOn te pigeat hoc loco, Lector humaniſsime, à ſuſcepta MA-<lb/>XIMARVM, MINIMARV MQVE linearum inueſti-<lb/>gatione circa reliquas coni- ſectiones, aliquantisper recedere, <lb/>dum elegantiſsimam quandam, ac vere admirabilem affe-<lb/>ctionem exhibere tibi decernimus, circa MINIMAS lineas, ad peri-<lb/>pherias infinitarum Parabolarum, per eundem verticem ſimul adſcripta-<lb/>rum, ex eodem communis axis puncto ducibiles, quarum veſtigia, dum <lb/>hoc ipſam<unsure/> propoſitio prælo ſubijcitur, neſcio qua parùm morata cura inſe-<lb/>qui voluimus. </s> <s xml:id="echoid-s5765" xml:space="preserve">Huius itaque itineris delineatio, eſt quæ conſequitur.</s> <s xml:id="echoid-s5766" xml:space="preserve"/> </p> <pb o="25" file="0207" n="207" rhead=""/> </div> <div xml:id="echoid-div598" type="section" level="1" n="242"> <head xml:id="echoid-head250" xml:space="preserve">THEOR. XV. PROP. XXI.</head> <p> <s xml:id="echoid-s5767" xml:space="preserve">Semita MINIMARVM linearum, ducibilium à puncto com-<lb/>munis axis infinitarum Parabolarum, per eundem verticem ſi-<lb/>mul adſcriptarum, ad earundem ſectionum peripherias, eſt cir-<lb/>cumferentia Ellipſis, cuius tranſuerſum latus ſit ipſum axis ſe-<lb/>gmentum, inter aſſumptum punctum, & </s> <s xml:id="echoid-s5768" xml:space="preserve">vertieem interceptum: <lb/></s> <s xml:id="echoid-s5769" xml:space="preserve">rectum verò eiuſdem tranſuerſi ſit duplum.</s> <s xml:id="echoid-s5770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5771" xml:space="preserve">ESto Parabole A B C, cuius axis B D, in quo ſumptum ſit punctum D <lb/>à vertice B diſtans per interuallum æquale dimidio ſui recti B E: </s> <s xml:id="echoid-s5772" xml:space="preserve">pa-<lb/>tet ipſam D B eſſe <anchor type="note" xlink:href="" symbol="a"/> _MINIMAM_ ad peripheriam A B C; </s> <s xml:id="echoid-s5773" xml:space="preserve">& </s> <s xml:id="echoid-s5774" xml:space="preserve">ſi aliæ Para- <anchor type="note" xlink:label="note-0207-01a" xlink:href="note-0207-01"/> bolæ concipiantur per B adſcriptæ, quarum recta latera excedant B E, <lb/>conſtat ipſas cadere <anchor type="note" xlink:href="" symbol="b"/> extra, qualis eſt M B N, & </s> <s xml:id="echoid-s5775" xml:space="preserve">eandem D B (quæ om- <anchor type="note" xlink:label="note-0207-02a" xlink:href="note-0207-02"/> nino erit minor dimidio ipſius rectilateris) ad eius peripheriam eſſe <anchor type="note" xlink:href="" symbol="c"/> _MI-_ _NIMAM_. </s> <s xml:id="echoid-s5776" xml:space="preserve">At ſi Parabolæ fuerint ipſi A B C per B verticem inſcriptæ, <lb/> <anchor type="note" xlink:label="note-0207-03a" xlink:href="note-0207-03"/> <anchor type="figure" xlink:label="fig-0207-01a" xlink:href="fig-0207-01"/> patet etiam ipſarum latera minora <anchor type="note" xlink:href="" symbol="d"/> eſſe recto B E, ac ideo D E quorun- <anchor type="note" xlink:label="note-0207-04a" xlink:href="note-0207-04"/> libet ipſorum laterum dimidium excedere, & </s> <s xml:id="echoid-s5777" xml:space="preserve">_MINIMAS_ ducibiles ex D, <lb/>ad harum Parabolarum peripherias pertingere, præter ad verticem B. </s> <s xml:id="echoid-s5778" xml:space="preserve">Si <lb/>ergo quæratur, quàm delineent ſemitam harum _MINIMARV M_ extrema <lb/>puncta. </s> <s xml:id="echoid-s5779" xml:space="preserve">Deſcribatur circa ſegmentum axis B D, tanquam circa tranſuer-<lb/>ſum latus, Ellipſis B F D G, cuius rectum ſit ipſum B E. </s> <s xml:id="echoid-s5780" xml:space="preserve">Conſtat hanc <lb/>eſſe _MAXIMAM_ Parabolæ A B C per B verticem <anchor type="note" xlink:href="" symbol="e"/> inſcriptibilem. </s> <s xml:id="echoid-s5781" xml:space="preserve">Dico <anchor type="note" xlink:label="note-0207-05a" xlink:href="note-0207-05"/> huius peripheriam B F D G prædictarum _MINIMARV M_ eſſe tramitem.</s> <s xml:id="echoid-s5782" xml:space="preserve"/> </p> <div xml:id="echoid-div598" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0207-01" xlink:href="note-0207-01a" xml:space="preserve">9. huius <lb/>ad nu. 1.</note> <note symbol="b" position="right" xlink:label="note-0207-02" xlink:href="note-0207-02a" xml:space="preserve">2. Co-<lb/>roll. 19. <lb/>pr. huius.</note> <note symbol="c" position="right" xlink:label="note-0207-03" xlink:href="note-0207-03a" xml:space="preserve">9. huius <lb/>ad nu. 1.</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/QN4GHYBF/figures/0207-01"/> </figure> <note symbol="d" position="right" xlink:label="note-0207-04" xlink:href="note-0207-04a" xml:space="preserve">ex 2. Co <lb/>roll. 19. <lb/>pr. huius.</note> <note symbol="e" position="right" xlink:label="note-0207-05" xlink:href="note-0207-05a" xml:space="preserve">ex 20. <lb/>pr. huius.</note> </div> <p> <s xml:id="echoid-s5783" xml:space="preserve">Iungatur Ellipſis regula D E: </s> <s xml:id="echoid-s5784" xml:space="preserve">& </s> <s xml:id="echoid-s5785" xml:space="preserve">Parabolę A B C inſcribatur quælibet <lb/>alia F B G, quæ Ellipſis peripheriam ad vtranq; </s> <s xml:id="echoid-s5786" xml:space="preserve">partem omnino ſecabit, vt <pb o="26" file="0208" n="208" rhead=""/> in F, G (nam Parabole A B C eſt <anchor type="note" xlink:href="" symbol="a"/> _MINIMA_ Ellipſi F B G circumſcri- <anchor type="note" xlink:label="note-0208-01a" xlink:href="note-0208-01"/> ptibilium) è quorum altero F ducta ſit ordinata F H I communem axem <lb/> <anchor type="note" xlink:label="note-0208-02a" xlink:href="note-0208-02"/> in H, regulam verò ſecante<unsure/> in I; </s> <s xml:id="echoid-s5787" xml:space="preserve">ſitque F L Parabolen <anchor type="note" xlink:href="" symbol="b"/> contingens ad F, axemque ſecans <anchor type="note" xlink:href="" symbol="c"/> in L.</s> <s xml:id="echoid-s5788" xml:space="preserve"/> </p> <div xml:id="echoid-div599" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0208-01" xlink:href="note-0208-01a" xml:space="preserve">ibidem.</note> <note symbol="b" position="left" xlink:label="note-0208-02" xlink:href="note-0208-02a" xml:space="preserve">2. primi <lb/>huius.</note> </div> <note symbol="c" position="left" xml:space="preserve">24. pri-<lb/>mi conic.</note> <figure> <image file="0208-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0208-01"/> </figure> <p> <s xml:id="echoid-s5789" xml:space="preserve">Iam, in triangulo E B D cum ſit E B dupla B D, erit I H dupla H D, <lb/>ſed eſt quoque L H dupla H B, quare vt L H ad H B, ita I H ad H D: <lb/></s> <s xml:id="echoid-s5790" xml:space="preserve">rectangulum ergo L H D æquale eſt rectangulo B H I, <anchor type="note" xlink:href="" symbol="d"/> ſiue quadrato F <anchor type="note" xlink:label="note-0208-04a" xlink:href="note-0208-04"/> H, eſtque F H ipſi L D perpendicularis, quare angulus D F L rectus <anchor type="note" xlink:href="" symbol="e"/> eſt, & </s> <s xml:id="echoid-s5791" xml:space="preserve">F L Parabolen contingit in F: </s> <s xml:id="echoid-s5792" xml:space="preserve">vnde D F eſt <anchor type="note" xlink:href="" symbol="f"/> _MINIMA_ ducibilium ex <anchor type="note" xlink:label="note-0208-05a" xlink:href="note-0208-05"/> dato puncto D ad peripheriam Parabolæ F B G. </s> <s xml:id="echoid-s5793" xml:space="preserve">Conſimili ratione oſten-<lb/>detur, quamlibet aliam inſcriptam P B R Ellipſis peripheriam B F G D <lb/> <anchor type="note" xlink:label="note-0208-06a" xlink:href="note-0208-06"/> ſecare, vt in P, R, & </s> <s xml:id="echoid-s5794" xml:space="preserve">iunctam D P, vel D R eſſe _MINIMAM_, &</s> <s xml:id="echoid-s5795" xml:space="preserve">c. </s> <s xml:id="echoid-s5796" xml:space="preserve">Qua-<lb/>re ſemita _MINIMARV M_ ex D ad huiuſmodi Parabolarum peripherias, eſt <lb/>prædictæ Ellipſis perimeter. </s> <s xml:id="echoid-s5797" xml:space="preserve">Quod oſtendere propoſitum fuit.</s> <s xml:id="echoid-s5798" xml:space="preserve"/> </p> <div xml:id="echoid-div600" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-0208-04" xlink:href="note-0208-04a" xml:space="preserve">Coroll. <lb/>primæ 1. <lb/>huius.</note> <note symbol="e" position="left" xlink:label="note-0208-05" xlink:href="note-0208-05a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> <note symbol="f" position="left" xlink:label="note-0208-06" xlink:href="note-0208-06a" xml:space="preserve">11. huius <lb/>ad nu. 1.</note> </div> </div> <div xml:id="echoid-div602" type="section" level="1" n="243"> <head xml:id="echoid-head251" xml:space="preserve">PROBL. II. PROP. XXII.</head> <p> <s xml:id="echoid-s5799" xml:space="preserve">A dato puncto, ad datę Hyperbolæ peripheriam, MINI-<lb/>MAM rectam lineam ducere.</s> <s xml:id="echoid-s5800" xml:space="preserve"/> </p> <figure> <image file="0208-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0208-02"/> </figure> <p> <s xml:id="echoid-s5801" xml:space="preserve">SIt data Hyperbole A B C, <lb/>cuius axis B D, rectum B E <lb/>tranſuerfum verò B G, centrum <lb/>H, & </s> <s xml:id="echoid-s5802" xml:space="preserve">datum vbicunque ſit pun-<lb/>ctum F. </s> <s xml:id="echoid-s5803" xml:space="preserve">Oportet ex F ad Hyper-<lb/>bolæ peripheriam A B C _MINI-_ <lb/>_MAM_ rectam lineam ducere.</s> <s xml:id="echoid-s5804" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5805" xml:space="preserve">Si primò datum punctum F, <lb/>in prima figura fuerit in axe pro-<lb/>ducto, extra Hyperbolen, ipſa <lb/>F B erit <anchor type="note" xlink:href="" symbol="g"/> _MINIMA_.</s> <s xml:id="echoid-s5806" xml:space="preserve"/> </p> <note symbol="g" position="left" xml:space="preserve">10. h.</note> <pb o="27" file="0209" n="209" rhead=""/> <p> <s xml:id="echoid-s5807" xml:space="preserve">Si datum punctum F ſit in axe intra ſectionem, vt in ſecunda figura, <lb/>quod tamen diſtet à vertice per interuallum non maius dimidio recti B E: <lb/></s> <s xml:id="echoid-s5808" xml:space="preserve">item F B erit <anchor type="note" xlink:href="" symbol="a"/> _MINIMA_.</s> <s xml:id="echoid-s5809" xml:space="preserve"/> </p> <note symbol="a" position="right" xml:space="preserve">9. huius <lb/>ad nu. 1.</note> <p> <s xml:id="echoid-s5810" xml:space="preserve">Cum verò, in eadem figura, <lb/> <anchor type="figure" xlink:label="fig-0209-01a" xlink:href="fig-0209-01"/> ſegmentũ F B excedet prædictum <lb/>recti dimidium: </s> <s xml:id="echoid-s5811" xml:space="preserve">dematur B I ęqua-<lb/>lis ſemi-recto B E, & </s> <s xml:id="echoid-s5812" xml:space="preserve">tunc habe-<lb/>bit H B ad B I maiorem rationem <lb/>quàm ad B F: </s> <s xml:id="echoid-s5813" xml:space="preserve">ſi ergo H F ſecetur <lb/>in L, ita vt H L ad L F, ſit vt H B <lb/>ad B I, punctum L omnino cadet <lb/>inter B & </s> <s xml:id="echoid-s5814" xml:space="preserve">F; </s> <s xml:id="echoid-s5815" xml:space="preserve">itaque ducta A L C <lb/>ordinatim axi applicata, iunctaq; <lb/></s> <s xml:id="echoid-s5816" xml:space="preserve">F A. </s> <s xml:id="echoid-s5817" xml:space="preserve">Dico ipſam F A eſſe _MINI-_ <lb/>_MAM_ quæſitam.</s> <s xml:id="echoid-s5818" xml:space="preserve"/> </p> <div xml:id="echoid-div602" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0209-01"/> </figure> </div> <p> <s xml:id="echoid-s5819" xml:space="preserve">Ducta enim ex A <anchor type="note" xlink:href="" symbol="b"/> contingente <anchor type="note" xlink:label="note-0209-02a" xlink:href="note-0209-02"/> A M, quæ axi occurret <anchor type="note" xlink:href="" symbol="c"/> in M. </s> <s xml:id="echoid-s5820" xml:space="preserve">Erit rectangulum H L M ad quadratum <anchor type="note" xlink:label="note-0209-03a" xlink:href="note-0209-03"/> L A, vt <anchor type="note" xlink:href="" symbol="d"/> tranſuerſum latus ad rectum, vel vt G B ad B E; </s> <s xml:id="echoid-s5821" xml:space="preserve">vel ſumptis ſubduplis, vt H B ad B I; </s> <s xml:id="echoid-s5822" xml:space="preserve">vel, ob conſtructionem, vt H L ad L F; </s> <s xml:id="echoid-s5823" xml:space="preserve">vel, <lb/> <anchor type="note" xlink:label="note-0209-04a" xlink:href="note-0209-04"/> ſumpta communi altitudine L M, vt idem rectangulum H L M ad rectan-<lb/>gulum F L M: </s> <s xml:id="echoid-s5824" xml:space="preserve">ergo quadratum L A æquabitur rectangulo F L M, ſed eſt <lb/>A L ipſi F M perpendicularis: </s> <s xml:id="echoid-s5825" xml:space="preserve">quare angulus F A M <anchor type="note" xlink:href="" symbol="e"/> rectus erit, ſed A M <anchor type="note" xlink:label="note-0209-05a" xlink:href="note-0209-05"/> ſectionem contingit in A: </s> <s xml:id="echoid-s5826" xml:space="preserve">ergo F A eſt _MINIMA_ ducibilium ex F ad <lb/>Hyperbolæ <anchor type="note" xlink:href="" symbol="f"/> peripheriam A B C, eſt autem F C ęqualis F A: </s> <s xml:id="echoid-s5827" xml:space="preserve">vnde in <anchor type="note" xlink:label="note-0209-06a" xlink:href="note-0209-06"/> hoc caſu duę erunt _MINIMAE_, &</s> <s xml:id="echoid-s5828" xml:space="preserve">c.</s> <s xml:id="echoid-s5829" xml:space="preserve"/> </p> <div xml:id="echoid-div603" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0209-02" xlink:href="note-0209-02a" xml:space="preserve">2. pr. h.</note> <note symbol="c" position="right" xlink:label="note-0209-03" xlink:href="note-0209-03a" xml:space="preserve">24. pri-<lb/>mi conic.</note> <note symbol="d" position="right" xlink:label="note-0209-04" xlink:href="note-0209-04a" xml:space="preserve">37. ibid.</note> <note symbol="e" position="right" xlink:label="note-0209-05" xlink:href="note-0209-05a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> <note symbol="f" position="right" xlink:label="note-0209-06" xlink:href="note-0209-06a" xml:space="preserve">11. h. ad <lb/>num. 1.</note> </div> <p> <s xml:id="echoid-s5830" xml:space="preserve">At ſi datum punctum F fuerit in axe coniugato H F, vt in tertia figu-<lb/>ra. </s> <s xml:id="echoid-s5831" xml:space="preserve">Diuidatur F H in I, ita vt F I ad I H ſit vt tranſuerſum G B ad rectũ <lb/>B E, & </s> <s xml:id="echoid-s5832" xml:space="preserve">per I agatur I A axi æquidiſtans, quæ in vno tantùm puncto A <lb/>Hyperbolæ <anchor type="note" xlink:href="" symbol="g"/> occurret. </s> <s xml:id="echoid-s5833" xml:space="preserve">Dico iunctam F A eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s5834" xml:space="preserve"/> </p> <note symbol="g" position="right" xml:space="preserve">26. pri-<lb/>mi conic.</note> <p> <s xml:id="echoid-s5835" xml:space="preserve">Producatur F A axi occurrens <lb/> <anchor type="figure" xlink:label="fig-0209-02a" xlink:href="fig-0209-02"/> in L, cui applicetur A M, duca-<lb/> <anchor type="note" xlink:label="note-0209-08a" xlink:href="note-0209-08"/> turque ex A <anchor type="note" xlink:href="" symbol="h"/> contingens A N, quę <anchor type="note" xlink:label="note-0209-09a" xlink:href="note-0209-09"/> axi occurret <anchor type="note" xlink:href="" symbol="i"/> in Q. </s> <s xml:id="echoid-s5836" xml:space="preserve">Erit in trian- gulo F L H, ob parallelas, H M ad <lb/>ad M L, vt F A ad A L, vel vt F I <lb/>ad I H; </s> <s xml:id="echoid-s5837" xml:space="preserve">vel vt tranſuerſum ad re-<lb/>ctum per conſtructionem; </s> <s xml:id="echoid-s5838" xml:space="preserve">vel vt re-<lb/> <anchor type="note" xlink:label="note-0209-10a" xlink:href="note-0209-10"/> ctangulum H M N <anchor type="note" xlink:href="" symbol="l"/> ad quadratum M A, ſed eadem H M ad M L, (ſum-<lb/>pta communi altitudine M N) eſt <lb/>vt idem rectangulum H M N ad re-<lb/>ctangulum L M N; </s> <s xml:id="echoid-s5839" xml:space="preserve">vnde quadratum <lb/>M A, æquabitur rectangulo N M L, & </s> <s xml:id="echoid-s5840" xml:space="preserve">eſt A M ipſi L N perpendicularis: <lb/></s> <s xml:id="echoid-s5841" xml:space="preserve">quare angulus L A N, & </s> <s xml:id="echoid-s5842" xml:space="preserve">qui ei deinceps eſt F A N rectus <anchor type="note" xlink:href="" symbol="m"/> erit, ſed A N <anchor type="note" xlink:label="note-0209-11a" xlink:href="note-0209-11"/> ſectionem contingit, ergo F A <anchor type="note" xlink:href="" symbol="n"/> eſt _MINIMA_ quæſita.</s> <s xml:id="echoid-s5843" xml:space="preserve"/> </p> <div xml:id="echoid-div604" type="float" level="2" n="3"> <figure xlink:label="fig-0209-02" xlink:href="fig-0209-02a"> <image file="0209-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0209-02"/> </figure> <note symbol="h" position="right" xlink:label="note-0209-08" xlink:href="note-0209-08a" xml:space="preserve">2. pr. h.</note> <note symbol="i" position="right" xlink:label="note-0209-09" xlink:href="note-0209-09a" xml:space="preserve">24. primi <lb/>conic.</note> <note symbol="l" position="right" xlink:label="note-0209-10" xlink:href="note-0209-10a" xml:space="preserve">37. ibid.</note> <note symbol="m" position="right" xlink:label="note-0209-11" xlink:href="note-0209-11a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> </div> <note symbol="n" position="right" xml:space="preserve">10. h.</note> <p> <s xml:id="echoid-s5844" xml:space="preserve">Si autem datum punctum F ſit extra Hyperbolen inter axem coniuga-<lb/>tum S H T, & </s> <s xml:id="echoid-s5845" xml:space="preserve">ſectionis peripheriam, vt in quarta, & </s> <s xml:id="echoid-s5846" xml:space="preserve">quinta figura, vel <pb o="28" file="0210" n="210" rhead=""/> intra Hyperbolen, inter axem, & </s> <s xml:id="echoid-s5847" xml:space="preserve">peripheriam, vt in ſexta, & </s> <s xml:id="echoid-s5848" xml:space="preserve">ſeptima <lb/>(nam ſi eſſet in ipſa peripheria, vt in A, tunc _MINIMA_ abiret in pun-<lb/>ctum.) </s> <s xml:id="echoid-s5849" xml:space="preserve">Iungatur H, centrum Hyperbolæ, cum dato puncto F recta linea <lb/>H F, quæ ita ſecetur in I, vt H I ad I F ſit vt tranſuerſum latus G B ad <lb/>rectum B E, ſumaturque H L æqualis F I, & </s> <s xml:id="echoid-s5850" xml:space="preserve">per L agatur L M axi B D <lb/>æquidiſtans, ac per I axi ordinata N I O, & </s> <s xml:id="echoid-s5851" xml:space="preserve">in angulo N O M per datum <lb/>in eo punctum A deſcribatur <anchor type="note" xlink:href="" symbol="a"/> Hyperbole F A, quæ in vno tantùm pun- <anchor type="note" xlink:label="note-0210-01a" xlink:href="note-0210-01"/> ctum A cum ſectione A B C <anchor type="note" xlink:href="" symbol="b"/> conueniet. </s> <s xml:id="echoid-s5852" xml:space="preserve">Dico iunctam F A eſſe _MINI-_ <anchor type="note" xlink:label="note-0210-02a" xlink:href="note-0210-02"/> _MAM_ quæſitam.</s> <s xml:id="echoid-s5853" xml:space="preserve"/> </p> <div xml:id="echoid-div605" type="float" level="2" n="4"> <note symbol="a" position="left" xlink:label="note-0210-01" xlink:href="note-0210-01a" xml:space="preserve">4. ſecun-<lb/>di conic.</note> <note symbol="b" position="left" xlink:label="note-0210-02" xlink:href="note-0210-02a" xml:space="preserve">12. h.</note> </div> <figure> <image file="0210-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0210-01"/> </figure> <p> <s xml:id="echoid-s5854" xml:space="preserve">Ducatur A P axi ordinata, & </s> <s xml:id="echoid-s5855" xml:space="preserve">A Q Hyperbolen contingens <anchor type="note" xlink:href="" symbol="c"/> in A, quę <anchor type="note" xlink:label="note-0210-03a" xlink:href="note-0210-03"/> ſecabit <anchor type="note" xlink:href="" symbol="d"/> axim in Q. </s> <s xml:id="echoid-s5856" xml:space="preserve">Et quoniam in Hyperbola A F ſumpta ſunt duo pun- <anchor type="note" xlink:label="note-0210-04a" xlink:href="note-0210-04"/> cta A, F, è quorum altero A ducta eſt A P alteri aſymptoto N O æqui-<lb/>diſtans, ex altero verò F, recta F I L H vtranque aſymptoton ſecans in <lb/>I, L; </s> <s xml:id="echoid-s5857" xml:space="preserve">eſtque L H in directum, & </s> <s xml:id="echoid-s5858" xml:space="preserve">æqualis poſita ipſi I F, & </s> <s xml:id="echoid-s5859" xml:space="preserve">ex H recta <lb/>H B P alteri aſymptoto O M æquidiſtans, cum alia A P conueniens in P, <lb/>erit iuncta I P <anchor type="note" xlink:href="" symbol="e"/> parallela ad F A; </s> <s xml:id="echoid-s5860" xml:space="preserve">ſed H P ſecat I P alteram parallelarum, <anchor type="note" xlink:label="note-0210-05a" xlink:href="note-0210-05"/> quare producta ſecabit quoque reliquam H F: </s> <s xml:id="echoid-s5861" xml:space="preserve">ſecet igitur in R. </s> <s xml:id="echoid-s5862" xml:space="preserve">Erit er-<lb/>go in triangulo H F R (ob parallelas) H P ad P R, vt H I ad I F, vel vt <pb o="29" file="0211" n="211" rhead=""/> tranſuerſum latus ad rectum, per conſtructionem, <anchor type="note" xlink:href="" symbol="a"/> vel vt rectangulum <anchor type="note" xlink:label="note-0211-01a" xlink:href="note-0211-01"/> H P Q ad quadratum P A; </s> <s xml:id="echoid-s5863" xml:space="preserve">ſed eadem H P ad P R (ſumpta communi al-<lb/>titudine P Q) eſt vt idem rectangulum H P Q ad rectangulum R P Q, <lb/>quare quadratum P A æquabitur rectangulo R P Q, eſtque A P ipſi R Q <lb/>perpendicularis, vnde angulus R A Q, & </s> <s xml:id="echoid-s5864" xml:space="preserve">in quarta, & </s> <s xml:id="echoid-s5865" xml:space="preserve">quinta figura, qui <lb/>ei deinceps Q A F <anchor type="note" xlink:href="" symbol="b"/> rectus erit, ſed A Q ſectionem contingit, quare per- <anchor type="note" xlink:label="note-0211-02a" xlink:href="note-0211-02"/> pendicularis F A <anchor type="note" xlink:href="" symbol="c"/> erit _MINIMA_ quæſita.</s> <s xml:id="echoid-s5866" xml:space="preserve"/> </p> <div xml:id="echoid-div606" type="float" level="2" n="5"> <note symbol="c" position="left" xlink:label="note-0210-03" xlink:href="note-0210-03a" xml:space="preserve">2. pr. h.</note> <note symbol="d" position="left" xlink:label="note-0210-04" xlink:href="note-0210-04a" xml:space="preserve">24. pri-<lb/>mi conic.</note> <note symbol="e" position="left" xlink:label="note-0210-05" xlink:href="note-0210-05a" xml:space="preserve">14. h.</note> <note symbol="a" position="right" xlink:label="note-0211-01" xlink:href="note-0211-01a" xml:space="preserve">37. primi <lb/>conic.</note> <note symbol="b" position="right" xlink:label="note-0211-02" xlink:href="note-0211-02a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> </div> <note symbol="c" position="right" xml:space="preserve">10. et 11. <lb/>huius ad <lb/>num. 1.</note> <p> <s xml:id="echoid-s5867" xml:space="preserve">Si denique datum punctum F ſit extra Hyperbolen, ſed ſupra axem <lb/>coniugatum H S, vt in octaua, & </s> <s xml:id="echoid-s5868" xml:space="preserve">nona figura. </s> <s xml:id="echoid-s5869" xml:space="preserve">A centro H date Hyper-<lb/>bolæ ad datum punctum F ducatur H F, quæ ita ſecetur in I, ita vt H I <lb/>ad I F, ſit vt tranſuerſum latus G B ad rectum B E, ſumptaque H L ęqua-<lb/>li ipſi I F, per I agatur I O N ordinatim ductis æquidiſtans, & </s> <s xml:id="echoid-s5870" xml:space="preserve">per L re-<lb/>cta L O M axi H B D parallela, & </s> <s xml:id="echoid-s5871" xml:space="preserve">in angulo N O M per datum punctum <lb/>H (quod eſt centrum Hyperbolæ) deſcribatur <anchor type="note" xlink:href="" symbol="d"/> alia Hyperbole H A, quæ <anchor type="note" xlink:label="note-0211-04a" xlink:href="note-0211-04"/> alteram A B C in vno tantùm <anchor type="note" xlink:href="" symbol="e"/> puncto A ſecabit. </s> <s xml:id="echoid-s5872" xml:space="preserve">Dico iunctam F A eſſe <anchor type="note" xlink:label="note-0211-05a" xlink:href="note-0211-05"/> _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s5873" xml:space="preserve"/> </p> <div xml:id="echoid-div607" type="float" level="2" n="6"> <note symbol="d" position="right" xlink:label="note-0211-04" xlink:href="note-0211-04a" xml:space="preserve">4. ſecun-<lb/>diconic.</note> <note symbol="e" position="right" xlink:label="note-0211-05" xlink:href="note-0211-05a" xml:space="preserve">12. h.</note> </div> <figure> <image file="0211-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0211-01"/> </figure> <p> <s xml:id="echoid-s5874" xml:space="preserve">Sit A P axi ordinatim applicata, & </s> <s xml:id="echoid-s5875" xml:space="preserve">A Q ex A ſectionem <anchor type="note" xlink:href="" symbol="f"/> contingens, <anchor type="note" xlink:label="note-0211-06a" xlink:href="note-0211-06"/> axemque ſecans <anchor type="note" xlink:href="" symbol="g"/> in Q, iungaturque IP. </s> <s xml:id="echoid-s5876" xml:space="preserve">Iam in Hyperbola A H, cuius <anchor type="note" xlink:label="note-0211-07a" xlink:href="note-0211-07"/> aſymptoti O N, O M, ſumptum eſt punctum P, à quo ductæ ſunt P A, <lb/>P H aſymptotis æquidiſtantes, & </s> <s xml:id="echoid-s5877" xml:space="preserve">Hyperbolæ occurrentes in A, H, & </s> <s xml:id="echoid-s5878" xml:space="preserve">ab <lb/>eorum altero H ducta eſt H L I vtranque aſymptoton ſecans in I, L, eſt-<lb/>que I F in directum, & </s> <s xml:id="echoid-s5879" xml:space="preserve">æqualis poſita ipſi H L, rectaque F A coniungit <lb/>extremum F cum altero datorum A; </s> <s xml:id="echoid-s5880" xml:space="preserve">ipſa F A <anchor type="note" xlink:href="" symbol="h"/>æquidiſtabit iungenti I <anchor type="note" xlink:label="note-0211-08a" xlink:href="note-0211-08"/> P; </s> <s xml:id="echoid-s5881" xml:space="preserve">ſed H P ſecat I P quare producta ſecabit quoque alteram parallela-<lb/>rum F A, ſi hæc vltra F A producatur. </s> <s xml:id="echoid-s5882" xml:space="preserve">Sit ergo harum occurſus R. </s> <s xml:id="echoid-s5883" xml:space="preserve">Erit <lb/>in triangulo F H R recta H P ad P R, vt H I ad I F, vel vt latus tranſuer-<lb/>ſum ad rectum, ex conſtructione, vel <anchor type="note" xlink:href="" symbol="i"/> vt rectangulum H P Q ad quadra- <anchor type="note" xlink:label="note-0211-09a" xlink:href="note-0211-09"/> tum P A, ſed eadem H P ad P R (ſumpta communi altitudine P Q) eſt <lb/>vt idem rectangulum H P Q ad rectangulum R P Q: </s> <s xml:id="echoid-s5884" xml:space="preserve">quare quadratum <pb o="30" file="0212" n="212" rhead=""/> P A, & </s> <s xml:id="echoid-s5885" xml:space="preserve">rectangulum R P Q inter ſe ſunt æqualia, ſed eſt A P ipſi Q R <lb/>perpendicularis, ergo angulus Q A R rectus <anchor type="note" xlink:href="" symbol="a"/> erit, pariterque is qui ei de- <anchor type="note" xlink:label="note-0212-01a" xlink:href="note-0212-01"/> inceps Q A F. </s> <s xml:id="echoid-s5886" xml:space="preserve">Quare perpendicularis F A <anchor type="note" xlink:href="" symbol="b"/> erit _MINIMA_ quæſita. </s> <s xml:id="echoid-s5887" xml:space="preserve">Quod <anchor type="note" xlink:label="note-0212-02a" xlink:href="note-0212-02"/> faciendum erat.</s> <s xml:id="echoid-s5888" xml:space="preserve"/> </p> <div xml:id="echoid-div608" type="float" level="2" n="7"> <note symbol="f" position="right" xlink:label="note-0211-06" xlink:href="note-0211-06a" xml:space="preserve">2. pr. h.</note> <note symbol="g" position="right" xlink:label="note-0211-07" xlink:href="note-0211-07a" xml:space="preserve">24. primi <lb/>conic.</note> <note symbol="h" position="right" xlink:label="note-0211-08" xlink:href="note-0211-08a" xml:space="preserve">15. h.</note> <note symbol="i" position="right" xlink:label="note-0211-09" xlink:href="note-0211-09a" xml:space="preserve">37. primi <lb/>conic.</note> <note symbol="a" position="left" xlink:label="note-0212-01" xlink:href="note-0212-01a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> <note symbol="b" position="left" xlink:label="note-0212-02" xlink:href="note-0212-02a" xml:space="preserve">10. h.</note> </div> </div> <div xml:id="echoid-div610" type="section" level="1" n="244"> <head xml:id="echoid-head252" xml:space="preserve">PROBL. III. PROP. XXIII.</head> <p> <s xml:id="echoid-s5889" xml:space="preserve">A dato puncto, ad datæ Ellipſis peripheriam, MAXIMAM, <lb/>& </s> <s xml:id="echoid-s5890" xml:space="preserve">MINIMAM rectam lineam ducere.</s> <s xml:id="echoid-s5891" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5892" xml:space="preserve">SIt data Ellipſis A B C D, cuius centrum E, axis minor A C, maior B <lb/>D, rectum latus B F, & </s> <s xml:id="echoid-s5893" xml:space="preserve">datum punctum ſit G. </s> <s xml:id="echoid-s5894" xml:space="preserve">Oportet ex G, ad <lb/>peripheriam A B C, _MAXIMAM_, & </s> <s xml:id="echoid-s5895" xml:space="preserve">_MINIMAM_ rectam lineam ducere.</s> <s xml:id="echoid-s5896" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5897" xml:space="preserve">1. </s> <s xml:id="echoid-s5898" xml:space="preserve">Si primò datum punctum congruit cum centro E: </s> <s xml:id="echoid-s5899" xml:space="preserve">duo maiores ſemi-<lb/>-axes E B, E D, erunt _MAXIMAE_, duo verò ſemi- axes minores E A, <lb/> <anchor type="note" xlink:label="note-0212-03a" xlink:href="note-0212-03"/> E C <anchor type="note" xlink:href="" symbol="c"/> erunt _MINIMAE_.</s> <s xml:id="echoid-s5900" xml:space="preserve"/> </p> <div xml:id="echoid-div610" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-0212-03" xlink:href="note-0212-03a" xml:space="preserve">86. primi <lb/>huius.</note> </div> <figure> <image file="0212-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0212-01"/> </figure> <p> <s xml:id="echoid-s5901" xml:space="preserve">2. </s> <s xml:id="echoid-s5902" xml:space="preserve">Si datum punctum fuerit in vertice B maioris axis; </s> <s xml:id="echoid-s5903" xml:space="preserve">ipſæ maior axis <lb/>B D erit _MAXIMA_ ducibilium ex B, &</s> <s xml:id="echoid-s5904" xml:space="preserve">c. </s> <s xml:id="echoid-s5905" xml:space="preserve">Nam ſi concipiatur deſcriptus <lb/>circulus B H D I ex radio E B, hoc eſt circa diametrum B D, eius peri-<lb/>pheria cadet tota extra <anchor type="note" xlink:href="" symbol="d"/> peripheriam Ellipſis A B C D; </s> <s xml:id="echoid-s5906" xml:space="preserve">& </s> <s xml:id="echoid-s5907" xml:space="preserve">cum B D ſit <anchor type="note" xlink:label="note-0212-04a" xlink:href="note-0212-04"/> _MAXIMA_ ad peripheriam circuli, eò ampliùs erit _MAXIMA_ ad inſcri-<lb/>ptam Ellipſis peripheriam. </s> <s xml:id="echoid-s5908" xml:space="preserve">Verùm non dabitur _MINIMA_ ex B, cum ipſa <lb/>in punctum abeat.</s> <s xml:id="echoid-s5909" xml:space="preserve"/> </p> <div xml:id="echoid-div611" type="float" level="2" n="2"> <note symbol="d" position="left" xlink:label="note-0212-04" xlink:href="note-0212-04a" xml:space="preserve">ex 26. <lb/>pr. huius.</note> </div> <p> <s xml:id="echoid-s5910" xml:space="preserve">3. </s> <s xml:id="echoid-s5911" xml:space="preserve">Si autem datum punctum G in eadem prima figura fuerit in axe maio-<lb/>ri, extra tamen Ellipſim: </s> <s xml:id="echoid-s5912" xml:space="preserve">tota G D erit _MAXIMA_: </s> <s xml:id="echoid-s5913" xml:space="preserve">eſt enim _MAXIMA_ <lb/>ad peripheriam circuli B H D I, cum in ea ſit centrum, ergo ad periphe-<lb/>riam inſcriptæ Ellipſis omnino _MAXIMA_ erit. </s> <s xml:id="echoid-s5914" xml:space="preserve">G B verò erit <anchor type="note" xlink:href="" symbol="e"/> _MINIMA_, <anchor type="note" xlink:label="note-0212-05a" xlink:href="note-0212-05"/> cum ipſa G B ſit extra Ellipſim perpendicularis ad rectum B F, quod ad <lb/>B contingit Ellipſim.</s> <s xml:id="echoid-s5915" xml:space="preserve"/> </p> <div xml:id="echoid-div612" type="float" level="2" n="3"> <note symbol="e" position="left" xlink:label="note-0212-05" xlink:href="note-0212-05a" xml:space="preserve">10. h.</note> </div> <pb o="31" file="0213" n="213" rhead=""/> <p> <s xml:id="echoid-s5916" xml:space="preserve">4. </s> <s xml:id="echoid-s5917" xml:space="preserve">Si verò, in ſecunda figura, datum punctum G fuerit in maiori ſemi-<lb/>axe, at diſtet à vertice B per interuallum G B non maius dimidio recti <lb/>B F, ipſa G D, in qua centrum, erit _MAXIMA_, <anchor type="note" xlink:href="" symbol="a"/> & </s> <s xml:id="echoid-s5918" xml:space="preserve">reliqua G B _MINIMA_.</s> <s xml:id="echoid-s5919" xml:space="preserve"/> </p> <note symbol="a" position="right" xml:space="preserve">9. huius <lb/>ad nu. 1. 2.</note> <p> <s xml:id="echoid-s5920" xml:space="preserve">5. </s> <s xml:id="echoid-s5921" xml:space="preserve">At ſi in eadem figura datum punctum G item fuerit, in maiori ſemi-<lb/>axe E B, ſed diſter à vertice B per interuallum maius dimidio recti B F <lb/>(nam ſemi-axis maior E B, eſt ſemper maior ſemi-recto B F, cum totus <lb/>axis B D ſit maior toto recto B F) _MAXIMA_ erit <anchor type="note" xlink:href="" symbol="b"/> GD, in qua centrum:</s> <s xml:id="echoid-s5922" xml:space="preserve"> <anchor type="note" xlink:label="note-0213-02a" xlink:href="note-0213-02"/> _MINIMA_ verò venabitur ſic.</s> <s xml:id="echoid-s5923" xml:space="preserve"/> </p> <div xml:id="echoid-div613" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-0213-02" xlink:href="note-0213-02a" xml:space="preserve">6. huius.</note> </div> <p> <s xml:id="echoid-s5924" xml:space="preserve">Cum ſit B G maior ſemi-recto B F, habebit E B ad B G minorem ra-<lb/>tionem, quàm E B ad ſemi-rectum B F, vel ſumptis duplis, quàm tranſ-<lb/>uerſum D B ad rectum B F, ſuntque hæ rationes maioris inæqualitatis: <lb/></s> <s xml:id="echoid-s5925" xml:space="preserve">Itaque diuidatur <anchor type="note" xlink:href="" symbol="c"/> B G in H, ita vt E H ad H G ſit vt D B ad B F, & </s> <s xml:id="echoid-s5926" xml:space="preserve">per <anchor type="note" xlink:label="note-0213-03a" xlink:href="note-0213-03"/> H applicetur I H K, & </s> <s xml:id="echoid-s5927" xml:space="preserve">iungantur G I, G K: </s> <s xml:id="echoid-s5928" xml:space="preserve">nam ipſæ, quæ ſunt ęquales, <lb/>erunt _MINIMAE_.</s> <s xml:id="echoid-s5929" xml:space="preserve"/> </p> <div xml:id="echoid-div614" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-0213-03" xlink:href="note-0213-03a" xml:space="preserve">16. h.</note> </div> <p> <s xml:id="echoid-s5930" xml:space="preserve">Quoniam ducta I L contingente, hæc axi occurret <anchor type="note" xlink:href="" symbol="d"/> in L: </s> <s xml:id="echoid-s5931" xml:space="preserve">& </s> <s xml:id="echoid-s5932" xml:space="preserve">cum ſit <anchor type="note" xlink:label="note-0213-04a" xlink:href="note-0213-04"/> E H ad H G, vt tranſuerſum D B ad rectum B F, ſumpta communi altitu-<lb/>dine H L, erit rectangulum E H L ad G H L, vt tranſuerſum ad rectum, <lb/>ſed eſt quoque rectangulum E H L ad quadratum H I, <anchor type="note" xlink:href="" symbol="e"/> vt tranſuerſum <anchor type="note" xlink:label="note-0213-05a" xlink:href="note-0213-05"/> ad rectum, ergo rectangulum E H L ad G H L, eſt vt idem E H L ad qua-<lb/>dratum H I, quare rectangulum G H L æquale eſt quadrato H I: </s> <s xml:id="echoid-s5933" xml:space="preserve">eſtque <lb/>H I ipſi G L perpendicularis, ergo angulus G I L rectus erit, & </s> <s xml:id="echoid-s5934" xml:space="preserve">I L ſectio-<lb/>nem contingit in I, à quo ducta eſt I G perpendicularis, & </s> <s xml:id="echoid-s5935" xml:space="preserve">maiori axi <lb/>occurrens, quapropter G I erit <anchor type="note" xlink:href="" symbol="f"/> _MINIMA_, eſtque G K æqualis G I. </s> <s xml:id="echoid-s5936" xml:space="preserve">Vn- <anchor type="note" xlink:label="note-0213-06a" xlink:href="note-0213-06"/> de in hoc caſu duæ erunt _MINIMAE_, & </s> <s xml:id="echoid-s5937" xml:space="preserve">vna tantùm _MAXIMA_.</s> <s xml:id="echoid-s5938" xml:space="preserve"/> </p> <div xml:id="echoid-div615" type="float" level="2" n="6"> <note symbol="d" position="right" xlink:label="note-0213-04" xlink:href="note-0213-04a" xml:space="preserve">25. pri-<lb/>mi conic.</note> <note symbol="e" position="right" xlink:label="note-0213-05" xlink:href="note-0213-05a" xml:space="preserve">37. ibid.</note> <note symbol="f" position="right" xlink:label="note-0213-06" xlink:href="note-0213-06a" xml:space="preserve">11. h.</note> </div> <p> <s xml:id="echoid-s5939" xml:space="preserve">6. </s> <s xml:id="echoid-s5940" xml:space="preserve">Si verò datum punctum G fuerit in axe minori, vt in tertia figura, & </s> <s xml:id="echoid-s5941" xml:space="preserve"><lb/>diſtantia G B ſit non minor dimidio recti lateris B E: </s> <s xml:id="echoid-s5942" xml:space="preserve">(quæ G B omnino <lb/>maior erit ſemi-axe B E, vt ad finem 9. </s> <s xml:id="echoid-s5943" xml:space="preserve">huius monuimus) tunc ipſa G B <lb/>erit _MAXIMA_, <anchor type="note" xlink:href="" symbol="g"/> & </s> <s xml:id="echoid-s5944" xml:space="preserve">G D _MINIMA_, vel punctum G cadat infra D; </s> <s xml:id="echoid-s5945" xml:space="preserve">vel ſu- <anchor type="note" xlink:label="note-0213-07a" xlink:href="note-0213-07"/> pra inter D, & </s> <s xml:id="echoid-s5946" xml:space="preserve">E. </s> <s xml:id="echoid-s5947" xml:space="preserve">Nam ſi caderet in ipſo puncto D (dummodo D B ſit <lb/>vt ponitur, nempe non minor dimidio recti) ipſa D B eſſet _MAXIMA_, <lb/>nec daretur _MINIMA_, cum hæc in punctum euaneſcat.</s> <s xml:id="echoid-s5948" xml:space="preserve"/> </p> <div xml:id="echoid-div616" type="float" level="2" n="7"> <note symbol="g" position="right" xlink:label="note-0213-07" xlink:href="note-0213-07a" xml:space="preserve">9. huius <lb/>ad num. 3.</note> </div> <p> <s xml:id="echoid-s5949" xml:space="preserve">7. </s> <s xml:id="echoid-s5950" xml:space="preserve">Verùm, ſi datum punctum G ſit in axe minori, ſed diſtet à vertice B <lb/>per interuallum minus dimidio recti B E, & </s> <s xml:id="echoid-s5951" xml:space="preserve">cadat infra centrum E, vel <lb/>inter E, & </s> <s xml:id="echoid-s5952" xml:space="preserve">D; </s> <s xml:id="echoid-s5953" xml:space="preserve">vt in quarta figura, aut infra D, vt in quinta. </s> <s xml:id="echoid-s5954" xml:space="preserve">Cum ſit <lb/>G B minor ſemi-recto, & </s> <s xml:id="echoid-s5955" xml:space="preserve">E B æqualis ſemi-tranſuerſo B D, habebit G B <lb/>ad B E minorem rationem, quàm ſemi-rectum ad ſemi-tranſuerſum, vel <lb/>quàm rectum F B ad tranſuerſum B D. </s> <s xml:id="echoid-s5956" xml:space="preserve">Diuidatur ergo B E in H, ita vt <lb/>G H ad H E, <anchor type="note" xlink:href="" symbol="h"/> ſit vt rectum F B ad B D tranſuerſum, & </s> <s xml:id="echoid-s5957" xml:space="preserve">per H agatur or- <anchor type="note" xlink:label="note-0213-08a" xlink:href="note-0213-08"/> dinata H I, & </s> <s xml:id="echoid-s5958" xml:space="preserve">I L ſectionem contingens, & </s> <s xml:id="echoid-s5959" xml:space="preserve">axi occurrens in L, iunga-<lb/>turque G I. </s> <s xml:id="echoid-s5960" xml:space="preserve">Dico G I eſſe _MAXIMAM_.</s> <s xml:id="echoid-s5961" xml:space="preserve"/> </p> <div xml:id="echoid-div617" type="float" level="2" n="8"> <note symbol="h" position="right" xlink:label="note-0213-08" xlink:href="note-0213-08a" xml:space="preserve">16. h.</note> </div> <p> <s xml:id="echoid-s5962" xml:space="preserve">Cum ſit enim G H ad H E, vt F B ad B D, ſumpta communi altitudi-<lb/>ne H L erit rectangulum G H L ad E H L, vt F B ad B D, vel vt <anchor type="note" xlink:href="" symbol="i"/> qua- <anchor type="note" xlink:label="note-0213-09a" xlink:href="note-0213-09"/> dratum G<unsure/> I<unsure/> H ad idem rectangulum E H L, quare rectangulum G H L æ-<lb/>quale eſt quadrato G H, eſtque H I perpendicularis ad G L; </s> <s xml:id="echoid-s5963" xml:space="preserve">ergo angu-<lb/>lus G I L rectus erit, eſtque I L ſectionem contingens in L, à quo ducta <lb/>eſt I G perpendicularis, & </s> <s xml:id="echoid-s5964" xml:space="preserve">minori axi in G, occurrens, quare ipſa G I <anchor type="note" xlink:href="" symbol="l"/> erit <anchor type="note" xlink:label="note-0213-10a" xlink:href="note-0213-10"/> _MAXIMA_, & </s> <s xml:id="echoid-s5965" xml:space="preserve">eſt G K æqualis G I: </s> <s xml:id="echoid-s5966" xml:space="preserve">ergo ex G duæ erunt _MAXIMAE. </s> <s xml:id="echoid-s5967" xml:space="preserve">MI-_ <pb o="32" file="0214" n="214" rhead=""/> _NIMA_ verò in hoc caſn, tum in quarta, tum in quinta figura eſt <anchor type="note" xlink:href="" symbol="a"/> ipſa <anchor type="note" xlink:label="note-0214-01a" xlink:href="note-0214-01"/> G D; </s> <s xml:id="echoid-s5968" xml:space="preserve">niſi punctum G cadat in ipſo D; </s> <s xml:id="echoid-s5969" xml:space="preserve">tunc enim _MINIMA_ abit in pun-<lb/>ctum.</s> <s xml:id="echoid-s5970" xml:space="preserve"/> </p> <div xml:id="echoid-div618" type="float" level="2" n="9"> <note symbol="i" position="right" xlink:label="note-0213-09" xlink:href="note-0213-09a" xml:space="preserve">37. primi <lb/>conic.</note> <note symbol="l" position="right" xlink:label="note-0213-10" xlink:href="note-0213-10a" xml:space="preserve">11. h.</note> <note symbol="a" position="left" xlink:label="note-0214-01" xlink:href="note-0214-01a" xml:space="preserve">9. huius <lb/>ad num. 4.</note> </div> <p> <s xml:id="echoid-s5971" xml:space="preserve">8. </s> <s xml:id="echoid-s5972" xml:space="preserve">At, ſi, vt in ſexta figura, quando interuallum G B minus eſt dimidio <lb/>B E, punctum G cadat inter B, & </s> <s xml:id="echoid-s5973" xml:space="preserve">E, tunc ſi concipiatur D eſſe Ellipſis <lb/>verticem, reliquum interuallum D G, vel erit non minus, vel minus di-<lb/>midio B F, quo in caſu duæ _MAXIMAE_ reperientur ad partem periphe-<lb/>riæ A D C: </s> <s xml:id="echoid-s5974" xml:space="preserve">eadem conſtructione, & </s> <s xml:id="echoid-s5975" xml:space="preserve">demonſtratione, ac ad num. </s> <s xml:id="echoid-s5976" xml:space="preserve">6. </s> <s xml:id="echoid-s5977" xml:space="preserve">& </s> <s xml:id="echoid-s5978" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s5979" xml:space="preserve">huius, & </s> <s xml:id="echoid-s5980" xml:space="preserve">reliqua G B erit _MINIMA_, &</s> <s xml:id="echoid-s5981" xml:space="preserve">c.</s> <s xml:id="echoid-s5982" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5983" xml:space="preserve">9. </s> <s xml:id="echoid-s5984" xml:space="preserve">Si denique datum pun-<lb/>ctum G fuerit inter ſemi-<lb/> <anchor type="figure" xlink:label="fig-0214-01a" xlink:href="fig-0214-01"/> axes, aut extra ſectioné, <lb/>vt in ſeptima figura; </s> <s xml:id="echoid-s5985" xml:space="preserve">vel <lb/>intra, vt in octaua; </s> <s xml:id="echoid-s5986" xml:space="preserve">vel in <lb/>ipſa ſectione, vt in nona. <lb/></s> <s xml:id="echoid-s5987" xml:space="preserve">Iungatur E G, quæ hinc <lb/>inde producatur, & </s> <s xml:id="echoid-s5988" xml:space="preserve">fiat, <lb/>vt tranſuerſum D B ad re-<lb/>ctum B F, <anchor type="note" xlink:href="" symbol="b"/> ita E H ad H <anchor type="note" xlink:label="note-0214-02a" xlink:href="note-0214-02"/> G, ac ita G I ad I E, & </s> <s xml:id="echoid-s5989" xml:space="preserve"><lb/>ex H, I, ducantur H L, <lb/>minori axi A C, & </s> <s xml:id="echoid-s5990" xml:space="preserve">I L <lb/>maiori D B parallelę, quę <lb/>ſimul occurrent in L, & </s> <s xml:id="echoid-s5991" xml:space="preserve"><lb/>in angulo H L I per pun-<lb/>ctum E (quod eſt cen-<lb/>trum Ellipſis) deſcriba-<lb/>tur <anchor type="note" xlink:href="" symbol="c"/> Hyperbole M G E <anchor type="note" xlink:label="note-0214-03a" xlink:href="note-0214-03"/> N, quæ neceſſariò <anchor type="note" xlink:href="" symbol="d"/> tran- <anchor type="note" xlink:label="note-0214-04a" xlink:href="note-0214-04"/> ſibit per G (cum ſegmen-<lb/>ta G H, E I rectæ H I ap-<lb/>plicatæ in angulo aſym-<lb/>ptotali H L I, ſint ęqua-<lb/>lia,) & </s> <s xml:id="echoid-s5992" xml:space="preserve">in duobus tantùm <lb/>punctis M, N, Ellipſis <lb/>peripheriam <anchor type="note" xlink:href="" symbol="e"/> ſecabit. </s> <s xml:id="echoid-s5993" xml:space="preserve">Di- <anchor type="note" xlink:label="note-0214-05a" xlink:href="note-0214-05"/> co has interſectiones da-<lb/>re puncta quæſita: </s> <s xml:id="echoid-s5994" xml:space="preserve">hoc <lb/>eſt iunctam G N in ſepti-<lb/>ma, octaua, & </s> <s xml:id="echoid-s5995" xml:space="preserve">nona fi-<lb/>gura eſſe _MAXIMAM_, & </s> <s xml:id="echoid-s5996" xml:space="preserve"><lb/>G M _MINIMAM_, in ſe-<lb/>ptima, & </s> <s xml:id="echoid-s5997" xml:space="preserve">octaua figura, tantùm, quoniam in nona ipſa _MINIMA_ abit in <lb/>punctum.</s> <s xml:id="echoid-s5998" xml:space="preserve"/> </p> <div xml:id="echoid-div619" type="float" level="2" n="10"> <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/QN4GHYBF/figures/0214-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0214-02" xlink:href="note-0214-02a" xml:space="preserve">Coroll. <lb/>16. h.</note> <note symbol="c" position="left" xlink:label="note-0214-03" xlink:href="note-0214-03a" xml:space="preserve">4. ſecun-<lb/>diconic.</note> <note symbol="d" position="left" xlink:label="note-0214-04" xlink:href="note-0214-04a" xml:space="preserve">17. h.</note> <note symbol="e" position="left" xlink:label="note-0214-05" xlink:href="note-0214-05a" xml:space="preserve">18. h.</note> </div> <p> <s xml:id="echoid-s5999" xml:space="preserve">Quò autem ad _MINIMAM_ oſtendendam. </s> <s xml:id="echoid-s6000" xml:space="preserve">Ducatur ex M, Ellipſim <lb/>contingens M P maiori axi occurrens <anchor type="note" xlink:href="" symbol="f"/> in P; </s> <s xml:id="echoid-s6001" xml:space="preserve">appliceturque M Q.</s> <s xml:id="echoid-s6002" xml:space="preserve"> </s> </p> <note symbol="f" position="left" xml:space="preserve">25. primi <lb/>conic.</note> <p> <s xml:id="echoid-s6003" xml:space="preserve">Et quoniam in angulo aſymptotali H L I ſumptum eſt punctum Q, ex-<unsure/> <pb o="33" file="0215" n="215" rhead=""/> tra ſectionem, à quo ductæ ſunt Q M, Q E aſymptotis parallelæ, & </s> <s xml:id="echoid-s6004" xml:space="preserve">Hy-<lb/>perbolæ occurrentes in M, E, & </s> <s xml:id="echoid-s6005" xml:space="preserve">ab altero occurſuum E, ducta eſt E G H, <lb/>ſecans Hyperbolen in G, & </s> <s xml:id="echoid-s6006" xml:space="preserve">aſymptoton H L in H, erunt iunctæ H Q, <lb/>M G O <anchor type="note" xlink:href="" symbol="a"/> inter ſe parallelæ; </s> <s xml:id="echoid-s6007" xml:space="preserve">quare in triangulo Q E H, recta G M, quæ <anchor type="note" xlink:label="note-0215-01a" xlink:href="note-0215-01"/> baſi H Q æquidiſtat, producta conueniet cum latere E Q, vt in O; </s> <s xml:id="echoid-s6008" xml:space="preserve">erit-<lb/>que E Q ad Q O, vt E H ad H G, hoc eſt vt tranſuerſum A B ad rectum <lb/>B F, ſed E Q ad Q O, ſumpta communi altitudine Q P, eſt vt rectangu-<lb/>lum E Q P ad rectangulum O Q P, ergo rectangulum E Q P ad O Q P erit <lb/>vt tranſuerſum ad rectum, vel vt <anchor type="note" xlink:href="" symbol="b"/> idem rectangulum E Q P ad quadra- <anchor type="note" xlink:label="note-0215-02a" xlink:href="note-0215-02"/> tum Q M; </s> <s xml:id="echoid-s6009" xml:space="preserve">vnde rectangulum O Q P, æquale eſt quadrato Q M, eſtque <lb/>QM ipſi O P perpendicularis, ergo angulus O M P <anchor type="note" xlink:href="" symbol="c"/> rectus eſt, & </s> <s xml:id="echoid-s6010" xml:space="preserve">in ſe- <anchor type="note" xlink:label="note-0215-03a" xlink:href="note-0215-03"/> ptima figura, qui ei deinceps eſt G M P rectus erit, ſed eſt G M extra <lb/>ſectionem, contingenti M P perpendicularis: </s> <s xml:id="echoid-s6011" xml:space="preserve">quare G M erit <anchor type="note" xlink:href="" symbol="d"/> _MINIMA_.</s> <s xml:id="echoid-s6012" xml:space="preserve"> <anchor type="note" xlink:label="note-0215-04a" xlink:href="note-0215-04"/> At, in octaua figura, M P Ellipſim contingit, & </s> <s xml:id="echoid-s6013" xml:space="preserve">ei perpendicularis M G <lb/>eſt intra Ellipſim, ſed non excedit interceptam M O inter contactum, & </s> <s xml:id="echoid-s6014" xml:space="preserve"><lb/>maiorem axim, quare G M erit <anchor type="note" xlink:href="" symbol="e"/> _MINIMA_.</s> <s xml:id="echoid-s6015" xml:space="preserve"/> </p> <div xml:id="echoid-div620" type="float" level="2" n="11"> <note symbol="a" position="right" xlink:label="note-0215-01" xlink:href="note-0215-01a" xml:space="preserve">19. h.</note> <note symbol="b" position="right" xlink:label="note-0215-02" xlink:href="note-0215-02a" xml:space="preserve">37. primi <lb/>conic.</note> <note symbol="c" position="right" xlink:label="note-0215-03" xlink:href="note-0215-03a" xml:space="preserve">203. Se-<lb/>pt. Pappi.</note> <note symbol="d" position="right" xlink:label="note-0215-04" xlink:href="note-0215-04a" xml:space="preserve">10. h.</note> </div> <note symbol="e" position="right" xml:space="preserve">11. h. ad <lb/>num. 1.</note> <p> <s xml:id="echoid-s6016" xml:space="preserve">Quod tandem in quouis prædictorum ſchematum, ducta G N ſit _MA-_ <lb/>_XIMA_, ita oſtendetur, ſed in nona tantùm figura, ne in reliquis noua li-<lb/>nearum, & </s> <s xml:id="echoid-s6017" xml:space="preserve">characterum appoſitio confuſionem pariat.</s> <s xml:id="echoid-s6018" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6019" xml:space="preserve">Secet ergo G N ſemi-axim minorem A E in K, & </s> <s xml:id="echoid-s6020" xml:space="preserve">maiorem E D in R, <lb/>applicetur N S, contingens agatur N T, iungaturque S H.</s> <s xml:id="echoid-s6021" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6022" xml:space="preserve">Et cum à puncto S, & </s> <s xml:id="echoid-s6023" xml:space="preserve">in angulo aſymptotali H L I intra ſectionem <lb/>ductæ ſint S E, S N aſymptotis parallelæ, Hyperbolæ occurrentes in E, <lb/>N, & </s> <s xml:id="echoid-s6024" xml:space="preserve">ab altero occurſuum E ducta ſit E G H, Hyperbolen ſecans in G, <lb/>& </s> <s xml:id="echoid-s6025" xml:space="preserve">aſymptoton in H, erunt iunctæ S H, N R G <anchor type="note" xlink:href="" symbol="f"/> inter ſe parallelæ quare <anchor type="note" xlink:label="note-0215-06a" xlink:href="note-0215-06"/> in triangulo H E S, erit E S ad S R, vt E H ad H G, vel vt tranſuerſum <lb/>D B <anchor type="note" xlink:href="" symbol="g"/> ad rectum B F, vel vt rectangulum E S T ad quadratum S N, ſed <anchor type="note" xlink:label="note-0215-07a" xlink:href="note-0215-07"/> E S ad S R, eſt vt idem rectangulum E S T ad rectangulum R S T, ergo <lb/>quadratum S N æquale eſt rectangulo R S T, ex quo angulus R N T re-<lb/>ctus erit, ſed T N Ellipſim contingit in N, eſtque N G maior intercepta <lb/>N K inter contactum, & </s> <s xml:id="echoid-s6026" xml:space="preserve">minorem axim, quare G N omnino erit <anchor type="note" xlink:href="" symbol="h"/> _MAXI-_ <anchor type="note" xlink:label="note-0215-08a" xlink:href="note-0215-08"/> _MA_ quæſita. </s> <s xml:id="echoid-s6027" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s6028" xml:space="preserve"/> </p> <div xml:id="echoid-div621" type="float" level="2" n="12"> <note symbol="f" position="right" xlink:label="note-0215-06" xlink:href="note-0215-06a" xml:space="preserve">19. h.</note> <note symbol="g" position="right" xlink:label="note-0215-07" xlink:href="note-0215-07a" xml:space="preserve">37. primi <lb/>conic.</note> <note symbol="h" position="right" xlink:label="note-0215-08" xlink:href="note-0215-08a" xml:space="preserve">11. h. ad <lb/>num. 2.</note> </div> </div> <div xml:id="echoid-div623" type="section" level="1" n="245"> <head xml:id="echoid-head253" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s6029" xml:space="preserve">DE inuentione MAXIMARVM à puncto dato ad univerſam <lb/>Parabolæ, vel Hyperbolæ peripheriam hactenus w<unsure/>ihil egimus, <lb/>cum manifeſtè pateat ad eas educi minimè poſſe lineas tantæ <lb/>longitudinis, quin ipſis maiores, & </s> <s xml:id="echoid-s6030" xml:space="preserve">maiores adhuc in infini-<lb/>tum reperiantur; </s> <s xml:id="echoid-s6031" xml:space="preserve">eò quod ſectiones ipſæ ſint infinitæ extenſionis: </s> <s xml:id="echoid-s6032" xml:space="preserve">itaque con-<lb/>ſultò de hac re demonſtrationem omiſimus, cum hæc in promptu ſatis ſit. <lb/></s> <s xml:id="echoid-s6033" xml:space="preserve">Verùm ſi quærantur MAXIMAE, ducibiles à puncto extra ſectionem da-<lb/>to, ad conuexas tantùm quarumlibet coni-ſectionum peripherias: </s> <s xml:id="echoid-s6034" xml:space="preserve">ſi punctum <lb/>fuerit in axe producto, ex eo ductæ lineæ contingentes æquales erunt, & </s> <s xml:id="echoid-s6035" xml:space="preserve">MA-<lb/>XIMAE ad ipſius ſectionis conuexam peripheriam. </s> <s xml:id="echoid-s6036" xml:space="preserve">Si autem punctum fue-<lb/>rit extra axim Parabolæ vel Hyperbolæ, ſed intra angulum ab aſymptotis <pb o="34" file="0216" n="216" rhead=""/> factum, tunc ex dictis binis contingentibus, quæ ad partem axis ducitur ſem-<lb/>per altera contingente ad oppofitam axis partem minor erit, atq; </s> <s xml:id="echoid-s6037" xml:space="preserve">hæc erit MA-<lb/>XIMA. </s> <s xml:id="echoid-s6038" xml:space="preserve">Si verò punctum fuerit extra Ellipſim inter axes, tunc contingens <lb/>ad partem maioris axis ducta, minor erit altera contingente ad partem mino-<lb/>ris, pariterque hæc erit MAXIMA ad conuexam Ellipſis peripheriã. </s> <s xml:id="echoid-s6039" xml:space="preserve">Quæ <lb/>omnia facili negotio demonſtrabuntur ſi animaduertatur, quod in quocunque <lb/>triangulo, cuius vnum latus altero ſit maius, hoc ipſum eſſe MAXIMIAM <lb/>linearum omnium à vertice anguli ab ipſis lateribus comprehenſi, ad puncta <lb/>baſis prædicti trianguli ducibilium, (tale enim triangulum eſt, quod a prædi-<lb/>ctis contingentibus tanquam lateribus, & </s> <s xml:id="echoid-s6040" xml:space="preserve">à recta puncta contactuum iungen-<lb/>te, tanquam baſi efficitur, in quo idem maius latus, ſiue contingentium ma-<lb/>ior eò magis erit MAXIMA ad incluſam ſectionis peripheriam.) </s> <s xml:id="echoid-s6041" xml:space="preserve">Si tandem <lb/>punctum fuerit in angulo ad verticem aſymptotalis, aut in aſymptotis eum <lb/>comprehendentibus, tunc vllam contingentium ducere imposſibile eſt, & </s> <s xml:id="echoid-s6042" xml:space="preserve">du-<lb/>cibiles lineæ ad conuexam Hyperbolæ peripheriam ſemper augentur, ideoque <lb/>non datur MAXIMA; </s> <s xml:id="echoid-s6043" xml:space="preserve">& </s> <s xml:id="echoid-s6044" xml:space="preserve">cum eſt in altero angulorum, qui deinceps ſunt <lb/>aſymptotali, vel in ipſis aſymptotis Hyperbolen continentilem<unsure/>, tunc vnica <lb/>tantùm contingens linea ab eo duci poteſt, & </s> <s xml:id="echoid-s6045" xml:space="preserve">hæc ad partem axis, quæ erit <lb/>MAXIMA ad eandem partem ducibilium; </s> <s xml:id="echoid-s6046" xml:space="preserve">ſed ad oppoſitam, ipſæ ducibiles <lb/>ad Hyperbolæ conuexam peripheriam perpetuò pariter augentur. </s> <s xml:id="echoid-s6047" xml:space="preserve">Sed in re <lb/>haud difficilis inueſtigationis ne ampliùs quæſo immoremur.</s> <s xml:id="echoid-s6048" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div624" type="section" level="1" n="246"> <head xml:id="echoid-head254" xml:space="preserve">THEOR. XVI. PROP. XXIV.</head> <p> <s xml:id="echoid-s6049" xml:space="preserve">Tranſuerſorũ laterũ in Hyperbola, MINIMVM eſt axis; </s> <s xml:id="echoid-s6050" xml:space="preserve">in Elli-<lb/>pſi autẽ, MAXIMVM eſt axis maior, MINIMVM verò axis minor.</s> <s xml:id="echoid-s6051" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6052" xml:space="preserve">SIt Hyperbole A B C, cuius axis tranſuerſus D B, centrum E. </s> <s xml:id="echoid-s6053" xml:space="preserve">Dico D <lb/>B omnium tranſuerſorum eſſe _MINIMVM._</s> <s xml:id="echoid-s6054" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6055" xml:space="preserve">Sit quodcunque aliud H <lb/> <anchor type="figure" xlink:label="fig-0216-01a" xlink:href="fig-0216-01"/> E A, & </s> <s xml:id="echoid-s6056" xml:space="preserve">per B axi applicetur <lb/>G B F, quę axi perpendicu-<lb/>laris erit, ac ſectionem con-<lb/>tinget in B. </s> <s xml:id="echoid-s6057" xml:space="preserve">Erit ergo per-<lb/>pendicularis E B _MINIMA_ <lb/>ad <anchor type="note" xlink:href="" symbol="a"/> peripheriam A B C:</s> <s xml:id="echoid-s6058" xml:space="preserve"> <anchor type="note" xlink:label="note-0216-01a" xlink:href="note-0216-01"/> quare E B minor erit E A, <lb/>& </s> <s xml:id="echoid-s6059" xml:space="preserve">duplum D B maius du-<lb/>plo H A: </s> <s xml:id="echoid-s6060" xml:space="preserve">ex quo D B erit <lb/>tranſuerſorum _MINIMVM._</s> <s xml:id="echoid-s6061" xml:space="preserve"/> </p> <div xml:id="echoid-div624" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0216-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0216-01" xlink:href="note-0216-01a" xml:space="preserve">10. h.</note> </div> <p> <s xml:id="echoid-s6062" xml:space="preserve">In Ellipſi verò A B C, <lb/>cuius centrum E, & </s> <s xml:id="echoid-s6063" xml:space="preserve">B D ſit <lb/>axis maior, & </s> <s xml:id="echoid-s6064" xml:space="preserve">A C minor: </s> <s xml:id="echoid-s6065" xml:space="preserve">patet B D eſſe tranſuerſorum _MAXIMVM_, & </s> <s xml:id="echoid-s6066" xml:space="preserve"><lb/>A C _MINIMVM_, ex primo Coroll. </s> <s xml:id="echoid-s6067" xml:space="preserve">86. </s> <s xml:id="echoid-s6068" xml:space="preserve">primihuius. </s> <s xml:id="echoid-s6069" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s6070" xml:space="preserve">c.</s> <s xml:id="echoid-s6071" xml:space="preserve"/> </p> <pb o="35" file="0217" n="217" rhead=""/> </div> <div xml:id="echoid-div626" type="section" level="1" n="247"> <head xml:id="echoid-head255" xml:space="preserve">THEOR. XVII. PROP. XXV.</head> <p> <s xml:id="echoid-s6072" xml:space="preserve">Rectorum laterum in Parabola, MINIMVM eſt rectum axis.</s> <s xml:id="echoid-s6073" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6074" xml:space="preserve">ESto Parabole A B C, cuius axis B D, rectum B E. </s> <s xml:id="echoid-s6075" xml:space="preserve">Dico ipſum B E <lb/>reliquorum rectorum eſſe _MINIMVM_. </s> <s xml:id="echoid-s6076" xml:space="preserve">Sit quælibet alia diameter <lb/>A F, quæ axi B D <anchor type="note" xlink:href="" symbol="a"/> æquidiſtabit, ſitque ad A contingens A G, & </s> <s xml:id="echoid-s6077" xml:space="preserve">B F <anchor type="note" xlink:label="note-0217-01a" xlink:href="note-0217-01"/> ipſi A G æquidiſtans, quæ diametro A F erit ordinatim applicata; </s> <s xml:id="echoid-s6078" xml:space="preserve">tan-<lb/>dem axi applicetur A H, ſumaturque A I æqualis recto diametri A F.</s> <s xml:id="echoid-s6079" xml:space="preserve"/> </p> <div xml:id="echoid-div626" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0217-01" xlink:href="note-0217-01a" xml:space="preserve">ex 46. <lb/>pr. conic.</note> </div> <p> <s xml:id="echoid-s6080" xml:space="preserve">Iam, ob contingentem A G, cum ſit <lb/>H B æqualis B G, & </s> <s xml:id="echoid-s6081" xml:space="preserve">F A eidem B G ę-<lb/> <anchor type="figure" xlink:label="fig-0217-01a" xlink:href="fig-0217-01"/> qualis, erit H B ęqualis F A: </s> <s xml:id="echoid-s6082" xml:space="preserve">rectan-<lb/>gulum ergo H B E ad F A I, vel qua-<lb/>dratum <anchor type="note" xlink:href="" symbol="b"/> H A, ad quadratum B F, <anchor type="note" xlink:label="note-0217-02a" xlink:href="note-0217-02"/> vel ad quadratum G A, erit vt B E <lb/>ad A I, ſed eſt quadratum A H minus <lb/>quadrato A G, ſiue recta A H minor <lb/>recta A G, cum acutus angulus A G B <lb/>minor ſit recto A H G, quare B E <lb/>rectum, minus erit recto A I: </s> <s xml:id="echoid-s6083" xml:space="preserve">eadem-<lb/>que ratione demonſtrabitur B E quo-<lb/>cunque alio recto minus eſſe: </s> <s xml:id="echoid-s6084" xml:space="preserve">quare <lb/>B E rectum axis, eſt _MINIMVM._ <lb/></s> <s xml:id="echoid-s6085" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s6086" xml:space="preserve"/> </p> <div xml:id="echoid-div627" 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/QN4GHYBF/figures/0217-01"/> </figure> <note symbol="b" position="right" xlink:label="note-0217-02" xlink:href="note-0217-02a" xml:space="preserve">Coroll. <lb/>primæ 1. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div629" type="section" level="1" n="248"> <head xml:id="echoid-head256" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s6087" xml:space="preserve">HInc patet, data quacunque Parabolæ diametro, ſi quæratur ratio <lb/>inter eius rectum, rectumque axis, hanc ipſam reperiri inter qua-<lb/>dratum contingentis interceptæ, à vertice datæ diametri vſque ad axim, <lb/>& </s> <s xml:id="echoid-s6088" xml:space="preserve">quadratum axi ſemi-applicatæ ab eodem vertice.</s> <s xml:id="echoid-s6089" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6090" xml:space="preserve">Verùm ſi omnium rectorum continuam proportionem, in lineis, & </s> <s xml:id="echoid-s6091" xml:space="preserve"><lb/>veluti ipſorum quandam propagationem ante oculos ponere expetemus, id <lb/>à proximo Theoremate addiſcere liceat.</s> <s xml:id="echoid-s6092" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div630" type="section" level="1" n="249"> <head xml:id="echoid-head257" xml:space="preserve">THEOR. XIIX. PROP. XXVI.</head> <p> <s xml:id="echoid-s6093" xml:space="preserve">Recta latera diametrorum in Parabola, ſunt inter ſe in ratio-<lb/>ne linearum ex puncto axis remoto à vertice per quadrantem <lb/>ſui recti, ad ipſarum diametrorum vertices eductarum.</s> <s xml:id="echoid-s6094" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6095" xml:space="preserve">ESto Parabole A B C, cuius axis B D rectum B I, ac eius quarta pars <lb/>ſit B D, & </s> <s xml:id="echoid-s6096" xml:space="preserve">quælibet aliæ diametri ſint A E, F G, &</s> <s xml:id="echoid-s6097" xml:space="preserve">c. </s> <s xml:id="echoid-s6098" xml:space="preserve">quarum ver-<lb/>tices iungantur rectis D B, D A, D F, &</s> <s xml:id="echoid-s6099" xml:space="preserve">c. </s> <s xml:id="echoid-s6100" xml:space="preserve">Dico, tùm axis, tùm prædi-<lb/>ctorum diametrorum latera eſſe inter ſe, vt ſunt ipſæ eductæ D B, D A, <lb/>D F, &</s> <s xml:id="echoid-s6101" xml:space="preserve">c.</s> <s xml:id="echoid-s6102" xml:space="preserve"/> </p> <pb o="36" file="0218" n="218" rhead=""/> <p> <s xml:id="echoid-s6103" xml:space="preserve">Erigatur ex A contingenti A G perpendicularis A L, quæ axi <anchor type="note" xlink:href="" symbol="a"/> occur- <anchor type="note" xlink:label="note-0218-01a" xlink:href="note-0218-01"/> ret in L, cui applicata A H, erit intercepta L H <anchor type="note" xlink:href="" symbol="b"/> æqualis dimidio recti <anchor type="note" xlink:label="note-0218-02a" xlink:href="note-0218-02"/> B I, hoc eſt dupla interuallo D B, (cum punctum D diſtet à vertice B <lb/>per quartam recti lateris partem ex hypoteſi) & </s> <s xml:id="echoid-s6104" xml:space="preserve">H G dupla <anchor type="note" xlink:href="" symbol="c"/> eſt quoq;</s> <s xml:id="echoid-s6105" xml:space="preserve"> <anchor type="note" xlink:label="note-0218-03a" xlink:href="note-0218-03"/> G B, quare, & </s> <s xml:id="echoid-s6106" xml:space="preserve">tota L G dupla eſt tota G D, ſiue L D æqualis D G, eſt-<lb/>que angulus L A G rectus, quare ſi <lb/>cum centro D, interuallo G, vel L <lb/> <anchor type="figure" xlink:label="fig-0218-01a" xlink:href="fig-0218-01"/> circulus deſcribatur, ipſe omnino <lb/>tranſibit per A; </s> <s xml:id="echoid-s6107" xml:space="preserve">vnde D A item æ-<lb/>qualis erit ipſis D G, D L, ſiue L G <lb/>erit dupla D A. </s> <s xml:id="echoid-s6108" xml:space="preserve">Et cum rectum axis <lb/>B D, ad rectum diametri A E, ſit vt <lb/>quadratum <anchor type="note" xlink:href="" symbol="d"/> A H ad A G, vel ob <anchor type="note" xlink:label="note-0218-04a" xlink:href="note-0218-04"/> triangulorum ſimilitudinem, vt qua-<lb/>dratum A L ad L G, vel vt recta <lb/>H L ad rectam L G (cum L A ſit <lb/>media proportionalis inter G L, L H) <lb/>ſumptis harum ſubduplis, erit rectũ <lb/>axis ad rectum diametri A E, vt D <lb/>B dimidium H L ad D A dimidium L G. </s> <s xml:id="echoid-s6109" xml:space="preserve">Quod erat demonſtrandum. <lb/></s> <s xml:id="echoid-s6110" xml:space="preserve">Vocatur autem punctum D, focus Parabolæ.</s> <s xml:id="echoid-s6111" xml:space="preserve"/> </p> <div xml:id="echoid-div630" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0218-01" xlink:href="note-0218-01a" xml:space="preserve">88. pri-<lb/>mi huius.</note> <note symbol="b" position="left" xlink:label="note-0218-02" xlink:href="note-0218-02a" xml:space="preserve">90. pri-<lb/>mi huius.</note> <note symbol="c" position="left" xlink:label="note-0218-03" xlink:href="note-0218-03a" xml:space="preserve">35. pri-<lb/>mi conic.</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/QN4GHYBF/figures/0218-01"/> </figure> <note symbol="d" position="left" xlink:label="note-0218-04" xlink:href="note-0218-04a" xml:space="preserve">Coroll. <lb/>24. huius.</note> </div> </div> <div xml:id="echoid-div632" type="section" level="1" n="250"> <head xml:id="echoid-head258" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s6112" xml:space="preserve">HInc cõſtat, omnes eductas à foco ad Parabolę peripheriam, ęqua-<lb/>ri quartæ parti rectorum, earum diametrorum, quarum vertices <lb/>ſint termini, quibus ipſæ eductæ ſectioni occurrunt: </s> <s xml:id="echoid-s6113" xml:space="preserve">rectum enim axis <lb/>B D ad rectum diametri A E, eſt vt D B ad D A, eſtque D B quarta pars <lb/>recti B I, quare, & </s> <s xml:id="echoid-s6114" xml:space="preserve">D A erit quarta pars recti lateris diametri A E, & </s> <s xml:id="echoid-s6115" xml:space="preserve">D F <lb/>quadrans recti, diametri F R. </s> <s xml:id="echoid-s6116" xml:space="preserve">Vnde quò diametri ab axe remotiores <lb/>fuerint, eò ipſarum recta maiora erunt. </s> <s xml:id="echoid-s6117" xml:space="preserve">nam eſt D F maior D A, &</s> <s xml:id="echoid-s6118" xml:space="preserve">c.</s> <s xml:id="echoid-s6119" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div633" type="section" level="1" n="251"> <head xml:id="echoid-head259" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s6120" xml:space="preserve">PAtet etiam, quamlibet eductam ex foco, ęquari aggregato ex inter-<lb/>uallo foci ab axis vertice, & </s> <s xml:id="echoid-s6121" xml:space="preserve">ſegmento axis inter verticem, & </s> <s xml:id="echoid-s6122" xml:space="preserve">ap-<lb/>plicatam ex occurſu eductæ cum ſectione. </s> <s xml:id="echoid-s6123" xml:space="preserve">Oſtenſa eſt enim D A æqua-<lb/>lis D G, quæ æqualis eſt aggregato G B, cum B D, vel H B cum B D.</s> <s xml:id="echoid-s6124" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div634" type="section" level="1" n="252"> <head xml:id="echoid-head260" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s6125" xml:space="preserve">CVm demonſtratum ſit D G æqualem eſſe D A, erit angulus D G A, <lb/>vel parallelarum externus E A M, æqualis angulo D A G, ſed M <lb/>A G Parabolen contingit in A, quare ex Opticæ legibus, ſi E A fuerit <lb/>radius <anchor type="note" xlink:href="" symbol="e"/> incidens ad concauam peripheriam A B C, ipſe A D erit refle- <anchor type="note" xlink:label="note-0218-05a" xlink:href="note-0218-05"/> xus, atque omnes radij axi Parabolę æquidiſtantes in punctum D coi-<lb/>bunt; </s> <s xml:id="echoid-s6126" xml:space="preserve">vnde ſi ipſi fuerint ſonori, aut lucidi, ſimulque calidi, ibi ſonus, <pb o="37" file="0219" n="219" rhead=""/> aut lux, & </s> <s xml:id="echoid-s6127" xml:space="preserve">calor, augebitur: </s> <s xml:id="echoid-s6128" xml:space="preserve">& </s> <s xml:id="echoid-s6129" xml:space="preserve">ſi à Solis corpore directè emanantes, ibi <lb/>fiet iccenſio<unsure/>, à qua punctum D foci nomen ademptum fuit.</s> <s xml:id="echoid-s6130" xml:space="preserve"/> </p> <div xml:id="echoid-div634" type="float" level="2" n="1"> <note symbol="e" position="left" xlink:label="note-0218-05" xlink:href="note-0218-05a" xml:space="preserve">Breuiùs, <lb/>& clariùs <lb/>quàm à <lb/>Vitellione <lb/>in 41. 9.</note> </div> <p> <s xml:id="echoid-s6131" xml:space="preserve">Si autem incidentes radij axi paralleli R F, O A à quadam recta N P <lb/>axi ordinatim ducta ſecentur, erunt aggregata incidentium cum earum <lb/>reflexis, ſimul æqualia.</s> <s xml:id="echoid-s6132" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6133" xml:space="preserve">Nam cum A D, ex præcedenti Coroll. </s> <s xml:id="echoid-s6134" xml:space="preserve">2. </s> <s xml:id="echoid-s6135" xml:space="preserve">ſit æqualis aggregato H B, <lb/>cum B D, additis hinc inde æqualibus A O, H P, proueniet aggregatum <lb/>O A, A D, æquale aggregato P B cum B D, itemque aggregatum R F, <lb/>cum F D oſtendetur ęquale eidem aggregato P B cum B D, quare aggre-<lb/>gata O A D, R F D æqualia erunt: </s> <s xml:id="echoid-s6136" xml:space="preserve">quod acutiſſimè quidem à perſpica-<lb/>ciſſimo Caualerio, in eius Speculo Vſtorio animaduerſum fuit.</s> <s xml:id="echoid-s6137" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div636" type="section" level="1" n="253"> <head xml:id="echoid-head261" xml:space="preserve">LEMMA VI. PROP. XXVII.</head> <p> <s xml:id="echoid-s6138" xml:space="preserve">Si in triangulo A B C, latus A C, ita ſectum fuerit in D, vt <lb/>rectangulum A C D ęquale ſit quadrato baſis B C. </s> <s xml:id="echoid-s6139" xml:space="preserve">Dico, iuncta <lb/>B D, angulum A B C ęqualem eſſe angulo B D C: </s> <s xml:id="echoid-s6140" xml:space="preserve">ſi verò A C D <lb/>rectangulum maius fuerit prædicto quadrato, & </s> <s xml:id="echoid-s6141" xml:space="preserve">angulus angulo <lb/>maior erit: </s> <s xml:id="echoid-s6142" xml:space="preserve">& </s> <s xml:id="echoid-s6143" xml:space="preserve">è contra.</s> <s xml:id="echoid-s6144" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6145" xml:space="preserve">NAm cum fuerit rectangulum A C D æquale quadrato C B, erit A C <lb/>ad C B, vt B C ad C D, quare triangula A B C, B D C, ad com-<lb/>munem angulum A conſtituta, ſimilia erunt, ob <lb/>idque angulus A B C æqualis angulo B D C.</s> <s xml:id="echoid-s6146" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6147" xml:space="preserve">At cum rectangulum A C D maius fuerit qua-<lb/> <anchor type="figure" xlink:label="fig-0219-01a" xlink:href="fig-0219-01"/> drato C B, facto rectangulo A C E æquali qua-<lb/>drato C B, erit C E minor C D, ergo iuncta B E, <lb/>erit angulus A B C, æqualis angulo B E C, ſed <lb/>B E C maior eſt angulo B D C, quare A B C om-<lb/>ninò maior erit angulo B D C.</s> <s xml:id="echoid-s6148" xml:space="preserve"/> </p> <div xml:id="echoid-div636" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0219-01"/> </figure> </div> <p> <s xml:id="echoid-s6149" xml:space="preserve">Si tandem rectangulum A C D minus fuerit <lb/>quadrato C B, non abſimili modo oſtendetur <lb/>angulum A B C minorem eſſe angulo A D C. <lb/></s> <s xml:id="echoid-s6150" xml:space="preserve">Quod vltimò erat, &</s> <s xml:id="echoid-s6151" xml:space="preserve">c.</s> <s xml:id="echoid-s6152" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div638" type="section" level="1" n="254"> <head xml:id="echoid-head262" xml:space="preserve">LEMMA VII. PROP. XXVIII.</head> <p> <s xml:id="echoid-s6153" xml:space="preserve">Si duæ rectæ lineæ A B, C D proportionaliter ſectæ fuerint <lb/>in E, F, & </s> <s xml:id="echoid-s6154" xml:space="preserve">homologis ſegmentis A E, C F æqualia ſumantur <lb/>A G, C H, & </s> <s xml:id="echoid-s6155" xml:space="preserve">ſuper A B, C D deſcripta ſint ſimilia triangula <lb/>I A B, L C D. </s> <s xml:id="echoid-s6156" xml:space="preserve">Dico vt rectangulum G B E, ad quadratum B I, <lb/>ita eſſe rectangulum H D F ad quadratum D L.</s> <s xml:id="echoid-s6157" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6158" xml:space="preserve">CVm ſit enim A E ad E B, vt C F ad F D, erit conuertendo, & </s> <s xml:id="echoid-s6159" xml:space="preserve">com-<lb/>ponendo B A ad A E, vt D C ad C F, vel quadratum B A ad A E, <pb o="38" file="0220" n="220" rhead=""/> vt quadratum D C ad C F, & </s> <s xml:id="echoid-s6160" xml:space="preserve">per con-<lb/>uerſionem rationis, quadratum A B ad <lb/> <anchor type="figure" xlink:label="fig-0220-01a" xlink:href="fig-0220-01"/> rectangulum G B E, vt quadratum C D <lb/>ad rectangulum H D F, & </s> <s xml:id="echoid-s6161" xml:space="preserve">conuertendo, <lb/>rectangulum G B E ad quadratum A B, <lb/>vt rectangulum H D F ad quadratum C <lb/>D, & </s> <s xml:id="echoid-s6162" xml:space="preserve">quadratum A B ad B I, eſt vt qua-<lb/>dratum C D ad D L, ob triangulorum <lb/>I A B, L C D ſimilitudinem; </s> <s xml:id="echoid-s6163" xml:space="preserve">quare ex <lb/>æquo rectangulum G B E ad quadratum <lb/>B I, erit vt rectangulum H D F ad quadra-<lb/>tum D L. </s> <s xml:id="echoid-s6164" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s6165" xml:space="preserve">c.</s> <s xml:id="echoid-s6166" xml:space="preserve"/> </p> <div xml:id="echoid-div638" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0220-01"/> </figure> </div> </div> <div xml:id="echoid-div640" type="section" level="1" n="255"> <head xml:id="echoid-head263" xml:space="preserve">LEMMA VIII. PROP. XXIX.</head> <p> <s xml:id="echoid-s6167" xml:space="preserve">Si quatuor magnitudinum eiuſdem generis, prima A ad ſe-<lb/>cundam B maiorem habuerit rationem, quàm tertia C ad quar-<lb/>tam D E, ſitque prima minor tertia, erit ſecunda minor quar-<lb/>ta.</s> <s xml:id="echoid-s6168" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6169" xml:space="preserve">FIat, vt A ad B, ita C ad D F, & </s> <s xml:id="echoid-s6170" xml:space="preserve">cum <lb/>A ad B habeat maiorem rationem, <lb/> <anchor type="figure" xlink:label="fig-0220-02a" xlink:href="fig-0220-02"/> quàm C ad D E, habebit quoque C ad D <lb/>F maiorem quàm ad D E, vnde D F erit <lb/>minor D E, & </s> <s xml:id="echoid-s6171" xml:space="preserve">eſt A ad B, vt C ad D F, <lb/>erit permutando A ad C, vt B ad D F, <lb/>eſtque A minor C, ergo B erit minor D <lb/>F, & </s> <s xml:id="echoid-s6172" xml:space="preserve">D F oſtenſa eſt minor D E, quare B <lb/>eò ampliùs erit minor D E. </s> <s xml:id="echoid-s6173" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s6174" xml:space="preserve">c.</s> <s xml:id="echoid-s6175" xml:space="preserve"/> </p> <div xml:id="echoid-div640" type="float" level="2" n="1"> <figure xlink:label="fig-0220-02" xlink:href="fig-0220-02a"> <image file="0220-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0220-02"/> </figure> </div> </div> <div xml:id="echoid-div642" type="section" level="1" n="256"> <head xml:id="echoid-head264" xml:space="preserve">THEOR. XIX. PROP. XXX.</head> <p> <s xml:id="echoid-s6176" xml:space="preserve">Rectorum laterum in Hyperbola, cuius axis tranſuerſus non <lb/>ſit minor eius recto latere, MINIMVM eſt rectum axis.</s> <s xml:id="echoid-s6177" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6178" xml:space="preserve">ESto Hyperbole A B C, cuius centrum D, axis tranſnerſus E B, qui <lb/>primò ſit minor recto B F. </s> <s xml:id="echoid-s6179" xml:space="preserve">Dico rectum B F eſſe rectorum laterum <lb/>_MINIMVM._</s> <s xml:id="echoid-s6180" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6181" xml:space="preserve">Sit quæcunque alia tranſuerſa diameter G D A, in ſectione producta <lb/>ad I, cuius rectum ſit A K ex A contingenter applicatum, & </s> <s xml:id="echoid-s6182" xml:space="preserve">axi occur-<lb/>rens in H; </s> <s xml:id="echoid-s6183" xml:space="preserve">& </s> <s xml:id="echoid-s6184" xml:space="preserve">ſit B I æquidiſtans A H, quæ ad diametrum G A I erit or-<lb/>dinatim ducta, atque ex I ſit I L ipſi D I perpendicularis, ex A verò A <lb/>M axi applicata, cui ex vertice B ſit parallela, vel contingens B O, ſe-<lb/>cans A H in P, iunganturque A B, O H.</s> <s xml:id="echoid-s6185" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6186" xml:space="preserve">Iam cum rectangulum D M H ad quadratum M A, ſit <anchor type="note" xlink:href="" symbol="a"/> vt E B ad B F, <anchor type="note" xlink:label="note-0220-01a" xlink:href="note-0220-01"/> ſitque E B maior B F, erit rectangulum D M H maius quadrato M A, <pb o="39" file="0221" n="221" rhead=""/> quare angulus D A M, ſiue in ſimili triangulo D L I, angulus D L I erit <lb/>maior <anchor type="note" xlink:href="" symbol="*"/> angulo A H M, ſiue angulo parallelarum externo I B L: </s> <s xml:id="echoid-s6187" xml:space="preserve">cum igi- <anchor type="note" xlink:label="note-0221-01a" xlink:href="note-0221-01"/> tur in triangulo I B L ſit angulus I B L minor I L B, erit latus I L minus <lb/>latere I B.</s> <s xml:id="echoid-s6188" xml:space="preserve"/> </p> <div xml:id="echoid-div642" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0220-01" xlink:href="note-0220-01a" xml:space="preserve">25. pri-<lb/>miconic.</note> <note symbol="*" position="right" xlink:label="note-0221-01" xlink:href="note-0221-01a" xml:space="preserve">27. h.</note> </div> <figure> <image file="0221-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0221-01"/> </figure> <p> <s xml:id="echoid-s6189" xml:space="preserve">Præterea, cum trian-<lb/>gula A P O, B P H <anchor type="note" xlink:href="" symbol="a"/> ſint <anchor type="note" xlink:label="note-0221-02a" xlink:href="note-0221-02"/> æqualia, addito commu-<lb/>ni triangulo A P B, erunt <lb/>triangula A O B, A H B <lb/>ſuper eadem baſi A B in-<lb/>ter ſe ęqualia, quare O H <lb/>æquidiſtabit A B, ideo-<lb/>que vt D O ad O A, vel <lb/>D B ad B M, ita D H ad <lb/>H B, vel D A ad A I. <lb/></s> <s xml:id="echoid-s6190" xml:space="preserve">Sunt ergo D M, D I pro-<lb/>portionaliter ſectæ in B, <lb/>A, quibus additæ ſunt D <lb/>E, D G, æquales ipſis D <lb/>B, D A, vtraq; </s> <s xml:id="echoid-s6191" xml:space="preserve">vtrique, <lb/>ſuntq; </s> <s xml:id="echoid-s6192" xml:space="preserve">rectangula triãgu-<lb/>la D M A, D I L ſimilia <lb/>inter ſe, quare recta ngu-<lb/>lum E M B ad quadratũ <lb/>M A, <anchor type="note" xlink:href="" symbol="b"/> ſiue E B ad B F, <anchor type="note" xlink:label="note-0221-03a" xlink:href="note-0221-03"/> eſt vt <anchor type="note" xlink:href="" symbol="c"/> rectãgulum G IA <anchor type="note" xlink:label="note-0221-04a" xlink:href="note-0221-04"/> ad quadratum I L, cumque ſit I L minor I B, erit quadratum I L minus <lb/>quadrato I B, ideoque rectangulum G I A ad quadratum I L, hoc eſt tranſ-<lb/>uerſum E B ad rectum B F, habebit maiorem rationem, quàm rectangu-<lb/>lum G I A ad quadratum I B, vel quàm <anchor type="note" xlink:href="" symbol="d"/> tranſuerſum G A ad rectum A K;</s> <s xml:id="echoid-s6193" xml:space="preserve"> <anchor type="note" xlink:label="note-0221-05a" xlink:href="note-0221-05"/> ergo prima E B, ad ſecundam B F, maiorem habet rationem quàm tertia G <lb/>A ad quartam A K, ſed eſt prima E B minor <anchor type="note" xlink:href="" symbol="e"/> tertia G A, ergo, & </s> <s xml:id="echoid-s6194" xml:space="preserve">ſecun- <anchor type="note" xlink:label="note-0221-06a" xlink:href="note-0221-06"/> da B F erit <anchor type="note" xlink:href="" symbol="f"/> minor quarta A K; </s> <s xml:id="echoid-s6195" xml:space="preserve">& </s> <s xml:id="echoid-s6196" xml:space="preserve">ſic de reliquis diametrorum rectis la- <anchor type="note" xlink:label="note-0221-07a" xlink:href="note-0221-07"/> teribus: </s> <s xml:id="echoid-s6197" xml:space="preserve">quare B F, rectum axis tranſuerſi, eſt _MINIMVM_, & </s> <s xml:id="echoid-s6198" xml:space="preserve">c.</s> <s xml:id="echoid-s6199" xml:space="preserve"/> </p> <div xml:id="echoid-div643" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0221-02" xlink:href="note-0221-02a" xml:space="preserve">I. tertij <lb/>conic.</note> <note symbol="b" position="right" xlink:label="note-0221-03" xlink:href="note-0221-03a" xml:space="preserve">21. primi <lb/>conic.</note> <note symbol="c" position="right" xlink:label="note-0221-04" xlink:href="note-0221-04a" xml:space="preserve">28. h.</note> <note symbol="d" position="right" xlink:label="note-0221-05" xlink:href="note-0221-05a" xml:space="preserve">21 pri-<lb/>mi conic.</note> <note symbol="e" position="right" xlink:label="note-0221-06" xlink:href="note-0221-06a" xml:space="preserve">24. h.</note> <note symbol="f" position="right" xlink:label="note-0221-07" xlink:href="note-0221-07a" xml:space="preserve">29. h.</note> </div> <p> <s xml:id="echoid-s6200" xml:space="preserve">Si autem axis E B æqualis fuerit eius recto B F; </s> <s xml:id="echoid-s6201" xml:space="preserve">cum demonſtratum ſit re-<lb/>ctangulum G I A ad quadratum I L eſſe vt tranſuerſus axis E B ad rectum <lb/>B F; </s> <s xml:id="echoid-s6202" xml:space="preserve">patet rectangulum quoque G I A æquari quadrato I L, ſed quando <lb/>E B æquatur B F, rectangulum etiam D M H æquatur <anchor type="note" xlink:href="" symbol="g"/> quadrato M A, &</s> <s xml:id="echoid-s6203" xml:space="preserve"> <anchor type="note" xlink:label="note-0221-08a" xlink:href="note-0221-08"/> tunc angulus D A M, ęqualis eſt <anchor type="note" xlink:href="" symbol="h"/> angulo A H M, ergo etiam angulus D L I <anchor type="note" xlink:label="note-0221-09a" xlink:href="note-0221-09"/> æquabitur angulo I B L, hoc eſt linea I B æqualis erit I L, ſed erat rectan-<lb/>gulum G I A æquale quadrato I L, ergo idem rectangulum G I A æqua-<lb/>bitur quadrato I B, ſiue tranſuerſa diameter A G, eius recto A K æqualis <lb/>erit, & </s> <s xml:id="echoid-s6204" xml:space="preserve">hoc ſemper, quæcunque ſit ducta tranſuerſa diameter præter axim.</s> <s xml:id="echoid-s6205" xml:space="preserve"/> </p> <div xml:id="echoid-div644" type="float" level="2" n="3"> <note symbol="g" position="right" xlink:label="note-0221-08" xlink:href="note-0221-08a" xml:space="preserve">37. primi <lb/>conic.</note> <note symbol="h" position="right" xlink:label="note-0221-09" xlink:href="note-0221-09a" xml:space="preserve">27. h.</note> </div> <p> <s xml:id="echoid-s6206" xml:space="preserve">Cum ergo Hyperbole fuerit rectangula æquilatera, ad aliam quoque <lb/>diametri applicationem æquilatera erit, ſed axis eſt tranſuerſorum <anchor type="note" xlink:href="" symbol="i"/> _MI-_ <anchor type="note" xlink:label="note-0221-10a" xlink:href="note-0221-10"/> _MIMVS_: </s> <s xml:id="echoid-s6207" xml:space="preserve">ergo in Hyperbola, cuius axis tranſuerſus eius rectum adæquet, <lb/>rectum axis aliorum rectorum eſt _MINIMVM_. </s> <s xml:id="echoid-s6208" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s6209" xml:space="preserve">c.</s> <s xml:id="echoid-s6210" xml:space="preserve"/> </p> <div xml:id="echoid-div645" type="float" level="2" n="4"> <note symbol="i" position="right" xlink:label="note-0221-10" xlink:href="note-0221-10a" xml:space="preserve">24. h.</note> </div> <pb o="40" file="0222" n="222" rhead=""/> </div> <div xml:id="echoid-div647" type="section" level="1" n="257"> <head xml:id="echoid-head265" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s6211" xml:space="preserve">CVm fuerit axis E B minor ſuo recto B F, ijſdem rationibus oſtende-<lb/>tur rectangulum D M H minus eſſe quadrato M A, & </s> <s xml:id="echoid-s6212" xml:space="preserve">angulum D A <lb/>M, ſiue D L I minorem eſſe angulo A H M, ſiue angulo I B L, ac propte-<lb/>rea latus I L maius eſſe latere I B, ideoque rectangulum G I A ad quadra-<lb/>tum I L, ſiue tranſuerſum axem E B ad rectum B F, minorem habere ratio-<lb/>nem, quàm idem rectangulum G I A ad quadratum I B, vel quàm tranſuer-<lb/>fa G A ad rectum A K.</s> <s xml:id="echoid-s6213" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div648" type="section" level="1" n="258"> <head xml:id="echoid-head266" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s6214" xml:space="preserve">EX his patet, in Hyperbola, cuius axis tranſuerſus ſit maior recto, maio-<lb/>rem eſſe rationem axis ad propriumrectum, quàm cuiuslibet aliæ trãſ-<lb/>uerſæ diametri ad proprium rectum.</s> <s xml:id="echoid-s6215" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6216" xml:space="preserve">Et ſi axis, ſuo recto æqualis fuerit, axem ad proprium rectum eandem ra-<lb/>tionem habere, quàm quęlibet alia tranſuerſa ad proprium rectum, ob ęqua-<lb/>litatem.</s> <s xml:id="echoid-s6217" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6218" xml:space="preserve">Si denique axis ſuo recto fuerit minor, minorem eſſe rationem inter axem, <lb/>ac proprium rectum, quàm inter quamcunque aliam diametrum propriumq; <lb/></s> <s xml:id="echoid-s6219" xml:space="preserve">rectum. </s> <s xml:id="echoid-s6220" xml:space="preserve">Sed hæc ſunt præter inſtitutum noſtrum, & </s> <s xml:id="echoid-s6221" xml:space="preserve">fuſim à præclariſſimo <lb/>Mydorgio pertractata.</s> <s xml:id="echoid-s6222" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6223" xml:space="preserve">Hinc, humanum eſſe errare deprehenditur, cum propoſitio 70. </s> <s xml:id="echoid-s6224" xml:space="preserve">de Hy-<lb/>perbola Gregorij à Sancto Vincentio, contrarium his falsò concludat cx <lb/>præcedenti 69. </s> <s xml:id="echoid-s6225" xml:space="preserve">in qua (pace tanti Viri dictum ſit) neſcio quo fato halluci-<lb/>natus eſt.</s> <s xml:id="echoid-s6226" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div649" type="section" level="1" n="259"> <head xml:id="echoid-head267" xml:space="preserve">LEMMA IX. PROP. XXXI.</head> <p> <s xml:id="echoid-s6227" xml:space="preserve">Si quatuor magnitudinum, prima A ad ſecundam B minorem <lb/>habuerit rationem, quàm tertia C D ad quartam E, ſitque prima <lb/>maior ſecunda, erit tertia maior quarta.</s> <s xml:id="echoid-s6228" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6229" xml:space="preserve">FIat enim vt A ad B, ita C F ad E; </s> <s xml:id="echoid-s6230" xml:space="preserve">cum <lb/> <anchor type="figure" xlink:label="fig-0222-01a" xlink:href="fig-0222-01"/> ergo A ad B minorem habeat ratio-<lb/>nem quàm C D ad E, habebit quoque C F <lb/>ad E, minorem quàm C D ad E; </s> <s xml:id="echoid-s6231" xml:space="preserve">quare C <lb/>F erit minor C D. </s> <s xml:id="echoid-s6232" xml:space="preserve">Et cum ſit A ad B vt C F <lb/>ad E, dataque ſit A, maior B, erit C F <lb/>maior E, & </s> <s xml:id="echoid-s6233" xml:space="preserve">eò magis C D maior eadem <lb/>E. </s> <s xml:id="echoid-s6234" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s6235" xml:space="preserve">c.</s> <s xml:id="echoid-s6236" xml:space="preserve"/> </p> <div xml:id="echoid-div649" type="float" level="2" n="1"> <figure xlink:label="fig-0222-01" xlink:href="fig-0222-01a"> <image file="0222-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0222-01"/> </figure> </div> <pb o="41" file="0223" n="223" rhead=""/> </div> <div xml:id="echoid-div651" type="section" level="1" n="260"> <head xml:id="echoid-head268" xml:space="preserve">THEOR. XX. PROP. XXXII</head> <p> <s xml:id="echoid-s6237" xml:space="preserve">Rectorum laterum in Ellipſi MAXIMVM eſt rectum minoris <lb/>axis, MINIMVM verò rectum maioris.</s> <s xml:id="echoid-s6238" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6239" xml:space="preserve">ESto Ellipſis A B C D, cuius centrum E, axis minor A C, rectum A <lb/>G, & </s> <s xml:id="echoid-s6240" xml:space="preserve">axis maior B D, rectum B F. </s> <s xml:id="echoid-s6241" xml:space="preserve">Dico A G rectorum omnium <lb/>eſſe _MAXIMVM_; </s> <s xml:id="echoid-s6242" xml:space="preserve">B F verò _MINIMVM_.</s> <s xml:id="echoid-s6243" xml:space="preserve"/> </p> <figure> <image file="0223-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0223-01"/> </figure> <p> <s xml:id="echoid-s6244" xml:space="preserve">Sit enim quælibet alia tranſuerſa diame-<lb/>ter H I, cuius rectum H L, ſitque diame-<lb/>ter M N ipſi H I coniugata, quæ media <lb/>proportionalis erit inter I H, & </s> <s xml:id="echoid-s6245" xml:space="preserve">H L; </s> <s xml:id="echoid-s6246" xml:space="preserve">vn-<lb/>de quadratum ipſius M N æquabitur re-<lb/>ctangulo I H L, vti etiam quadratum A C <lb/>æquatur rectangulo D B F, & </s> <s xml:id="echoid-s6247" xml:space="preserve">quadratum <lb/>B D rectangulo C A G; </s> <s xml:id="echoid-s6248" xml:space="preserve">ſed eſt quadratum <lb/>A C, minus quadrato M N, cum ſit tranſ-<lb/>uerſa A C minor <anchor type="note" xlink:href="" symbol="a"/> tranſuerſa M N, ergo <anchor type="note" xlink:label="note-0223-01a" xlink:href="note-0223-01"/> rectangulum D B F minus erit rectangulo <lb/>I H L, quare B D ad H I minorem habe-<lb/>bit rationem quàm H L ad B F, eſtque B <lb/>D maior <anchor type="note" xlink:href="" symbol="b"/> H I, ergo & </s> <s xml:id="echoid-s6249" xml:space="preserve">rectum H L erit <anchor type="note" xlink:label="note-0223-02a" xlink:href="note-0223-02"/> maior <anchor type="note" xlink:href="" symbol="c"/> recto B F.</s> <s xml:id="echoid-s6250" xml:space="preserve"/> </p> <div xml:id="echoid-div651" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0223-01" xlink:href="note-0223-01a" xml:space="preserve">24. h.</note> <note symbol="b" position="right" xlink:label="note-0223-02" xlink:href="note-0223-02a" xml:space="preserve">ibidem.</note> </div> <note symbol="c" position="right" xml:space="preserve">31. h.</note> <p> <s xml:id="echoid-s6251" xml:space="preserve">Præterea, cum ſit M N minor <anchor type="note" xlink:href="" symbol="d"/> D B, <anchor type="note" xlink:label="note-0223-04a" xlink:href="note-0223-04"/> erit quadratum M N minus quadrato D B, ſiue rectangulum I H L minus <lb/>rectangulo C A G, vnde I H ad C A minorem habebit rationem quàm <lb/>A G ad H L, ſed eſt I H maior <anchor type="note" xlink:href="" symbol="e"/> C A, ergo rectum A G erit maior <anchor type="note" xlink:href="" symbol="f"/> recto <anchor type="note" xlink:label="note-0223-05a" xlink:href="note-0223-05"/> H L. </s> <s xml:id="echoid-s6252" xml:space="preserve">Cum ſit ergo A G maior H L, & </s> <s xml:id="echoid-s6253" xml:space="preserve">H L maior B F erit A G adhuc <lb/> <anchor type="note" xlink:label="note-0223-06a" xlink:href="note-0223-06"/> maior B F. </s> <s xml:id="echoid-s6254" xml:space="preserve">Quare A G rectum minoris axis eſt _MAXIMVM_, B F verò <lb/>maioris axis rectum, eſt _MINIMVM_. </s> <s xml:id="echoid-s6255" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s6256" xml:space="preserve"/> </p> <div xml:id="echoid-div652" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-0223-04" xlink:href="note-0223-04a" xml:space="preserve">24. h.</note> <note symbol="e" position="right" xlink:label="note-0223-05" xlink:href="note-0223-05a" xml:space="preserve">ibidem.</note> <note symbol="f" position="right" xlink:label="note-0223-06" xlink:href="note-0223-06a" xml:space="preserve">31. h.</note> </div> </div> <div xml:id="echoid-div654" type="section" level="1" n="261"> <head xml:id="echoid-head269" xml:space="preserve">PROBL. IV. PROP. XXXIII.</head> <p> <s xml:id="echoid-s6257" xml:space="preserve">A puncto dato intra angulum rectilineum rectam applicare, <lb/>cuius rectangulum ſegmentorum ſit MINIMVM.</s> <s xml:id="echoid-s6258" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6259" xml:space="preserve">ESto ABC angulus rectilineus, in quo datum punctum ſit D. </s> <s xml:id="echoid-s6260" xml:space="preserve">Opor-<lb/>tet ex D rectam in angulo applicare, ita vt rectangulum ſub ipſius <lb/>ſegmentis ſit _MINIMVM_.</s> <s xml:id="echoid-s6261" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6262" xml:space="preserve">Ducatur B E angulum A B C bifariam ſecans, cui per D recta perpen-<lb/>dicularis applicetur A D C. </s> <s xml:id="echoid-s6263" xml:space="preserve">Dico hanc ipſam quæſitum ſoluere.</s> <s xml:id="echoid-s6264" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6265" xml:space="preserve">Cum enim in triangulis B E A, B E C anguli ad E ſint recti, & </s> <s xml:id="echoid-s6266" xml:space="preserve">ad B <lb/>facti æquales, erunt reliqui anguli B A E, B C E æquales, & </s> <s xml:id="echoid-s6267" xml:space="preserve">qui infra A <lb/>C, baſim trianguli æquicruris A B C, pariter æquales.</s> <s xml:id="echoid-s6268" xml:space="preserve"/> </p> <pb o="42" file="0224" n="224" rhead=""/> <p> <s xml:id="echoid-s6269" xml:space="preserve">Iam ducatur per D quælibet alia F D G. <lb/></s> <s xml:id="echoid-s6270" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0224-01a" xlink:href="fig-0224-01"/> Et cum in triangulo D G C ſit externus <lb/>angulus D C L maior interno D G C, fiat <lb/>angulus D G H ipſi D C L, ſiue D A F æ-<lb/>qualis, eſtque angulus G D C æqualis an-<lb/>gulo A D F, & </s> <s xml:id="echoid-s6271" xml:space="preserve">duo ſimul D A F, A D F <lb/>minores ſunt duobus rectis, ergo & </s> <s xml:id="echoid-s6272" xml:space="preserve">duo <lb/>D G H, G D C erunt duobus rectis mino-<lb/>res, ſiue G H cum D C producta conue-<lb/>niet, vt in H, eritque reliquus angulus H <lb/>in triangulo D H G æqualis reliquo F in <lb/>triangulo D F A: </s> <s xml:id="echoid-s6273" xml:space="preserve">quare huiuſmodi trian-<lb/>gula ſimilia erunt, & </s> <s xml:id="echoid-s6274" xml:space="preserve">circùm æquales an-<lb/>gulos ad D habebunt latera proportio-<lb/>nalia, ſiue vt A D ad D F, ita G D ad D <lb/>H, vnde rectangulum A D H æquale erit <lb/>rectangulo F D G, ideoque rectangulum <lb/>A D C minus erit rectangulo F D G, & </s> <s xml:id="echoid-s6275" xml:space="preserve">hoc ſemper vbicunque applicata <lb/>ſit per D, recta F D G præter A D C. </s> <s xml:id="echoid-s6276" xml:space="preserve">Quare rectangulum ſub ſegmentis <lb/>A D, D C eſt _MINIMV M_ quæſitum. </s> <s xml:id="echoid-s6277" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s6278" 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/QN4GHYBF/figures/0224-01"/> </figure> </div> </div> <div xml:id="echoid-div656" type="section" level="1" n="262"> <head xml:id="echoid-head270" xml:space="preserve">PROBL. V. PROP. XXXIV.</head> <p> <s xml:id="echoid-s6279" xml:space="preserve">A puncto intra coni-ſectionem dato rectam applicare, cuius <lb/>rectangulum ſegmentorum ſit MINIMVM. </s> <s xml:id="echoid-s6280" xml:space="preserve">In Ellipſi verò, & </s> <s xml:id="echoid-s6281" xml:space="preserve"><lb/>MAXIMVM rectangulum reperire.</s> <s xml:id="echoid-s6282" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6283" xml:space="preserve">ESto primùm A B C Parabole, vel Hyperbole, vt in prima figura, cu-<lb/>ius axis B D, & </s> <s xml:id="echoid-s6284" xml:space="preserve">datum intra ipſam punctum ſit E. </s> <s xml:id="echoid-s6285" xml:space="preserve">Oportet per E re-<lb/>ctam ſectioni applicare, ita vt rectangulum ſub eius ſegmentis ſit _MINI-_ <lb/>_MVM_.</s> <s xml:id="echoid-s6286" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6287" xml:space="preserve">Applicetur per E recta A E D C axi ordinatim ducta. </s> <s xml:id="echoid-s6288" xml:space="preserve">Dico hanc ip-<lb/>ſam quæſitum ſoluere: </s> <s xml:id="echoid-s6289" xml:space="preserve">ſiue rectangulum A E C eſſe _MINIMVM_.</s> <s xml:id="echoid-s6290" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6291" xml:space="preserve">Nam applicata per E qualibet alia inclinata F E G: </s> <s xml:id="echoid-s6292" xml:space="preserve">non abſimili mo-<lb/>do, ac in 26. </s> <s xml:id="echoid-s6293" xml:space="preserve">ſecundi conicorum, demonſtrabitur applicatas A C, F G in-<lb/>tra ſectionem ſe mutuò ſecantes in E, in ipſo E nunquam bifariam ſimul <lb/>ſecari, ex quo ipſarum applicatarum diametri diſiunctæ erunt inter ſe, <lb/>ideoque B vertex portionis A B C non erit vertex portionis F H G: </s> <s xml:id="echoid-s6294" xml:space="preserve">is er-<lb/>go ſit H; </s> <s xml:id="echoid-s6295" xml:space="preserve">ducaturque ex B ſectionem contingens B I, ſiue applicatę A C <lb/>æquidiſtans; </s> <s xml:id="echoid-s6296" xml:space="preserve">itemque ex H recta contingens H I, ſiue F G parallela, que <lb/>contingentes ſimul conuenient <anchor type="note" xlink:href="" symbol="a"/> in I. </s> <s xml:id="echoid-s6297" xml:space="preserve">Erit ergo rectangulum A E C, ad <anchor type="note" xlink:label="note-0224-01a" xlink:href="note-0224-01"/> rectangulum G E C, <anchor type="note" xlink:href="" symbol="b"/> vt quadratum B I ad quadratum H I; </s> <s xml:id="echoid-s6298" xml:space="preserve">ſed eſt con- <anchor type="note" xlink:label="note-0224-02a" xlink:href="note-0224-02"/> tingens B I, ad axis verticem, minor <anchor type="note" xlink:href="" symbol="c"/> contingente H I, ergo & </s> <s xml:id="echoid-s6299" xml:space="preserve">quadra- tum quadrato minus erit, ſiue rectangulum A E C minus rectangulo F E <lb/> <anchor type="note" xlink:label="note-0224-03a" xlink:href="note-0224-03"/> G, & </s> <s xml:id="echoid-s6300" xml:space="preserve">hoc ſemper quæcunque ſit quæ per E applicatur diuerſa ab appli-<lb/>cata A C, ergo rectangulum A E C eſt _MINIMVM_ quæſitum.</s> <s xml:id="echoid-s6301" xml:space="preserve"/> </p> <div xml:id="echoid-div656" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0224-01" xlink:href="note-0224-01a" xml:space="preserve">58. pri-<lb/>mih.</note> <note symbol="b" position="left" xlink:label="note-0224-02" xlink:href="note-0224-02a" xml:space="preserve">16. tertij <lb/>conic.</note> <note symbol="c" position="left" xlink:label="note-0224-03" xlink:href="note-0224-03a" xml:space="preserve">87. primi <lb/>huius.</note> </div> <pb o="43" file="0225" n="225" rhead=""/> <p> <s xml:id="echoid-s6302" xml:space="preserve">Sit verò A B C D in ſecunda figura Ellipſis, cuius axis maior B D, mi-<lb/>nor A C, centrum E, & </s> <s xml:id="echoid-s6303" xml:space="preserve">punctum intra datum ſit F. </s> <s xml:id="echoid-s6304" xml:space="preserve">Oportet per F re-<lb/>ctas in ſectione applicare quales inuenire propoſuimus.</s> <s xml:id="echoid-s6305" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6306" xml:space="preserve">Sit per F maiori axi B D ordinatim ducta G F H, minori verò ſit I F L. <lb/></s> <s xml:id="echoid-s6307" xml:space="preserve">Dico rectangulum G F H eſſe _MINIMVM, MAXIMVM_ verò I F L.</s> <s xml:id="echoid-s6308" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6309" xml:space="preserve">Sit quælibet alia per F ap-<lb/> <anchor type="figure" xlink:label="fig-0225-01a" xlink:href="fig-0225-01"/> plicata M F N, & </s> <s xml:id="echoid-s6310" xml:space="preserve">portionis <lb/>M O N ſit vertex O, atque ex <lb/>axium verticibus A, B, vti e-<lb/>tiam ex O agantur contingen-<lb/>tes A P, B Q, P O Q, quæ ſi-<lb/>mul occurrent <anchor type="note" xlink:href="" symbol="a"/> in R, P, Q.</s> <s xml:id="echoid-s6311" xml:space="preserve"> <anchor type="note" xlink:label="note-0225-01a" xlink:href="note-0225-01"/> Erit ergo rectangulum G F H <lb/>ad I F L, <anchor type="note" xlink:href="" symbol="b"/> vt quadratum B R <anchor type="note" xlink:label="note-0225-02a" xlink:href="note-0225-02"/> ad quadratum A R, ſed eſt <lb/>contingens B R, <anchor type="note" xlink:href="" symbol="c"/> minor A R, <anchor type="note" xlink:label="note-0225-03a" xlink:href="note-0225-03"/> ſiue quadratum B R minus quadrato A R, ergo, & </s> <s xml:id="echoid-s6312" xml:space="preserve">rectangulum G F H <lb/>minus erit rectangulo I F L.</s> <s xml:id="echoid-s6313" xml:space="preserve"/> </p> <div xml:id="echoid-div657" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0225-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0225-01" xlink:href="note-0225-01a" xml:space="preserve">58. pri-<lb/>mih.</note> <note symbol="b" position="right" xlink:label="note-0225-02" xlink:href="note-0225-02a" xml:space="preserve">16. tertij <lb/>conic.</note> <note symbol="c" position="right" xlink:label="note-0225-03" xlink:href="note-0225-03a" xml:space="preserve">87 primi <lb/>huius.</note> </div> <p> <s xml:id="echoid-s6314" xml:space="preserve">Præterea rectangulum G F H ad M F N eſt vt quadratum B Q ad qua-<lb/>dratum O Q, ſed eſt contingens B Q <anchor type="note" xlink:href="" symbol="d"/> minor contingente O Q, ſiue qua- <anchor type="note" xlink:label="note-0225-04a" xlink:href="note-0225-04"/> dratum B Q minus quadrato O Q, ergo rectangulum G F H minus eſt re-<lb/>ctangulo M F N, & </s> <s xml:id="echoid-s6315" xml:space="preserve">hoc ſemper vbicunque cadat applicata M F N: </s> <s xml:id="echoid-s6316" xml:space="preserve">qua-<lb/>re rectangulum G F H eſt _MINIMVM_ quæſitum.</s> <s xml:id="echoid-s6317" xml:space="preserve"/> </p> <div xml:id="echoid-div658" type="float" level="2" n="3"> <note symbol="d" position="right" xlink:label="note-0225-04" xlink:href="note-0225-04a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s6318" xml:space="preserve">Demùm cum rectangulum I F L ad N F M, ſit <anchor type="note" xlink:href="" symbol="e"/> vt quadratum A P ad <anchor type="note" xlink:label="note-0225-05a" xlink:href="note-0225-05"/> quadratum QP, ſitque contingens A P <anchor type="note" xlink:href="" symbol="f"/> maior contingente Q P erit qua- <anchor type="note" xlink:label="note-0225-06a" xlink:href="note-0225-06"/> dratum A P maius quadrato Q P, ergo rectangulum quoque I F L maius <lb/>erit rectangulo N F M, & </s> <s xml:id="echoid-s6319" xml:space="preserve">hoc ſemper vbicunque ſit ducta N F M inter <lb/>applicatas I F L, G F H quare rectangulum I F L eſt _MAXIMVM_ quæſi-<lb/>tum. </s> <s xml:id="echoid-s6320" xml:space="preserve">Quod vltimò inuenire propoſitum fuit.</s> <s xml:id="echoid-s6321" xml:space="preserve"/> </p> <div xml:id="echoid-div659" type="float" level="2" n="4"> <note symbol="e" position="right" xlink:label="note-0225-05" xlink:href="note-0225-05a" xml:space="preserve">16. tertij <lb/>huius.</note> <note symbol="f" position="right" xlink:label="note-0225-06" xlink:href="note-0225-06a" xml:space="preserve">87. primi <lb/>huius.</note> </div> </div> <div xml:id="echoid-div661" type="section" level="1" n="263"> <head xml:id="echoid-head271" xml:space="preserve">DEFINITIONES.</head> <head xml:id="echoid-head272" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s6322" xml:space="preserve">PLANVM ACVMINATVM REGVLARE, vel ACVMINATVM <lb/>tantùm voco omnem figuram planam, circa diametrum, in alteram par-<lb/>tem deficientem, & </s> <s xml:id="echoid-s6323" xml:space="preserve">cuius perimeter ſit in eaſdem partes cauus.</s> <s xml:id="echoid-s6324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6325" xml:space="preserve">Hoc eſt figura plana A B C, <lb/> <anchor type="figure" xlink:label="fig-0225-02a" xlink:href="fig-0225-02"/> in qua omnes rectæ lineæ A <lb/>C, E F, G H, &</s> <s xml:id="echoid-s6326" xml:space="preserve">c. </s> <s xml:id="echoid-s6327" xml:space="preserve">à figurę pe-<lb/>rimetro terminatæ, ac inter ſe <lb/>æquidiſtantes, à quadam re-<lb/>cta B D bifariam ſecentur, & </s> <s xml:id="echoid-s6328" xml:space="preserve"><lb/>in alteram partem, vt puta ad <lb/>B, continuò decreſcant, do-<lb/>nec abeant in punctum B, ſit-<lb/>que earum perimeter A G B H C ad eaſdem partes cauus vocetur PLA- <pb o="44" file="0226" n="226" rhead=""/> NVM ACVMINATVM REGVLARE, vel potius (breuitatis cauſa) <lb/>ACVMINATVM, cuius terminus B vocetur VERTEX; </s> <s xml:id="echoid-s6329" xml:space="preserve">& </s> <s xml:id="echoid-s6330" xml:space="preserve">æquidiſtan-<lb/>tes A C, E F, G H, &</s> <s xml:id="echoid-s6331" xml:space="preserve">c. </s> <s xml:id="echoid-s6332" xml:space="preserve">quæ à B D bifariam diuiduntur, dicantur AP-<lb/>PLICAT Æ ad ipſam B D, qnæ vocetur DIAMETER, vel AXIS quan-<lb/>do ipſa perpendiculariter ſecet eaſdem applicatas. </s> <s xml:id="echoid-s6333" xml:space="preserve">A C verò dicatur BA-<lb/>SIS ACVMINATI; </s> <s xml:id="echoid-s6334" xml:space="preserve">& </s> <s xml:id="echoid-s6335" xml:space="preserve">B I, quæ à vertice ſuper baſim ducitur perpen-<lb/>dicularis, ACVMINATI ALTITVDO nuncupetur.</s> <s xml:id="echoid-s6336" xml:space="preserve"/> </p> <div xml:id="echoid-div661" type="float" level="2" n="1"> <figure xlink:label="fig-0225-02" xlink:href="fig-0225-02a"> <image file="0225-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0225-02"/> </figure> </div> </div> <div xml:id="echoid-div663" type="section" level="1" n="264"> <head xml:id="echoid-head273" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s6337" xml:space="preserve">PLANA ACVMINATA REGVLARIA PROPORTIONALIA, vel <lb/>tantùm ACVMINATA PROPORTIONALIA dicantur illa, quorum <lb/>omnes applicatæ à punctis eorum diametros proportionaliter diuidenti-<lb/>bus, ſint quoque inter ſe proportionales.</s> <s xml:id="echoid-s6338" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6339" xml:space="preserve">Sint nempe duo Acu-<lb/> <anchor type="figure" xlink:label="fig-0226-01a" xlink:href="fig-0226-01"/> minata Regularia ABC, <lb/>E F G, ſuper baſes A C, <lb/>E G, qualia in præce-<lb/>denti definitione expli-<lb/>cauimus, quorum dia-<lb/>metri B D, F H propor-<lb/>tionaliter ſectæ ſint in <lb/>quotcunque punctis I, <lb/>M; </s> <s xml:id="echoid-s6340" xml:space="preserve">L, N, &</s> <s xml:id="echoid-s6341" xml:space="preserve">c. </s> <s xml:id="echoid-s6342" xml:space="preserve">ſiue ſit B I <lb/>ad I D, vt F M ad M H, <lb/>& </s> <s xml:id="echoid-s6343" xml:space="preserve">B L ad L D, vt F N <lb/>ad N H, &</s> <s xml:id="echoid-s6344" xml:space="preserve">c. </s> <s xml:id="echoid-s6345" xml:space="preserve">atque m <lb/>punctis inter ſectionum <lb/>applicatæ ſint O P, QR; <lb/></s> <s xml:id="echoid-s6346" xml:space="preserve">S T, V X, quę ex homo-<lb/>logis punctis ſint ad inuicem proportionales, hoc eſt vt A C ad E G, ita <lb/>O P ad S T, & </s> <s xml:id="echoid-s6347" xml:space="preserve">Q R ad V X, &</s> <s xml:id="echoid-s6348" xml:space="preserve">c. </s> <s xml:id="echoid-s6349" xml:space="preserve">huiuſmodi figuræ vocentur PLANA <lb/>ACVMINATA REGVLARIA PROPORTIONALIA, veltantùm ACV-<lb/>MINATA PROPORTIONALIA.</s> <s xml:id="echoid-s6350" xml:space="preserve"/> </p> <div xml:id="echoid-div663" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0226-01"/> </figure> </div> <pb o="45" file="0227" n="227" rhead=""/> </div> <div xml:id="echoid-div665" type="section" level="1" n="265"> <head xml:id="echoid-head274" xml:space="preserve">LEMMA X. PROP. XXXV.</head> <p> <s xml:id="echoid-s6351" xml:space="preserve">Si duæ rectæ lineæ terminatæ A B, C D bifariam ſectæ fue-<lb/>rint in E, F, & </s> <s xml:id="echoid-s6352" xml:space="preserve">proportionaliter producantur, vt in prima figu-<lb/>ra; </s> <s xml:id="echoid-s6353" xml:space="preserve">vel diuidantur, vt in ſecunda, in G, H, ita vt ſit A B ad B G, <lb/>vt C D ad D H, parteſq; </s> <s xml:id="echoid-s6354" xml:space="preserve">adiectæ, vel demptæ B G, D H iterum <lb/>proportionaliter ſecentur in I, L, ita vt B I ad I G, ſit vt D L ad <lb/>L H. </s> <s xml:id="echoid-s6355" xml:space="preserve">Dico rectangulum A G B ad rectangulum A I B, eſſe vt re-<lb/>ctangulum C H D ad rectangulum C L D.</s> <s xml:id="echoid-s6356" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6357" xml:space="preserve">NAm cum ſit A B ad B G, vt C D ad D H, erit in prima figura com-<lb/>ponendo, in ſecunda verò diuidendo A G ad G B, vt C H ad H D, <lb/> <anchor type="figure" xlink:label="fig-0227-01a" xlink:href="fig-0227-01"/> & </s> <s xml:id="echoid-s6358" xml:space="preserve">eſt B G ad G I, vt D H ad H L (cum diui-<lb/>dendo factum ſit B I ad I G, vt D L ad L H) <lb/>ergo ex æquo A G ad G I erit vt C H ad H L, <lb/>& </s> <s xml:id="echoid-s6359" xml:space="preserve">in prima figura per conuerſionem rationis, <lb/>in ſecunda verò, componendo, per conuer-<lb/>ſionem rationis, & </s> <s xml:id="echoid-s6360" xml:space="preserve">conuertendo, erit G A ad <lb/>A I, vt H C ad C L: </s> <s xml:id="echoid-s6361" xml:space="preserve">& </s> <s xml:id="echoid-s6362" xml:space="preserve">cum ſuperiùs demon-<lb/>ſtratum ſit eſſe B G ad G I, vt D H ad H L, <lb/>erit, per conuerſionem rationis, G B ad B I, <lb/>vt H D ad D L. </s> <s xml:id="echoid-s6363" xml:space="preserve">Iam rectangulum A G B ad <lb/>A I B habet rationem compoſitam ex ratione <lb/>G A ad A I, vel ex H C ad C L, & </s> <s xml:id="echoid-s6364" xml:space="preserve">ex ratio-<lb/>ne G B ad B I, vel ex H D ad D L, ſed ex ijſ-<lb/>dem rationibus H C ad C D, & </s> <s xml:id="echoid-s6365" xml:space="preserve">H D ad D L <lb/>componitur ratio rectanguli C H D ad rectan-<lb/>gulum C L D, quare vt rectangulum A G B ad <lb/>A I B, ita rectangulum C H D ad C L D. </s> <s xml:id="echoid-s6366" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s6367" xml:space="preserve">c.</s> <s xml:id="echoid-s6368" xml:space="preserve"/> </p> <div xml:id="echoid-div665" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0227-01"/> </figure> </div> </div> <div xml:id="echoid-div667" type="section" level="1" n="266"> <head xml:id="echoid-head275" xml:space="preserve">THEOR. XXI. PROP. XXXVI.</head> <p> <s xml:id="echoid-s6369" xml:space="preserve">Quælibet Portiones eiuſdem, vel diuerſarum Parabolarum <lb/>ſunt Acuminata Proportionalia.</s> <s xml:id="echoid-s6370" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6371" xml:space="preserve">Item, Portiones eiuſdem, vel diuerſarum Hyperbolarum, <lb/>Ellipſium, aut Circulorum; </s> <s xml:id="echoid-s6372" xml:space="preserve">quarum tamen ſegmenta diametro-<lb/>rum in ijſdem portionibus intercepta ad ſuas ſemi-diametros <lb/>eandem homologam habeant rationem, ſunt pariter inter ſe <lb/>Acuminata proportionalia.</s> <s xml:id="echoid-s6373" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6374" xml:space="preserve">SInt primò duę portiones A B C, D E F eiuſdem, vel diuerſarum Pa-<lb/>rabolarum in prima figura, quarum baſes ſint A C, D F. </s> <s xml:id="echoid-s6375" xml:space="preserve">Dico ip- <pb o="46" file="0228" n="228" rhead=""/> fas portiones eſſe Acuminata Proportionalia.</s> <s xml:id="echoid-s6376" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6377" xml:space="preserve">Repertis enim earum diametris B G, E H, diuidantur ipſæ proportio-<lb/>naliter in I, L, applicenturque M I N, O L P. </s> <s xml:id="echoid-s6378" xml:space="preserve">Cum ſit ergo G I ad I B, <lb/>vt H L ad L E, erit componendo G B ad B I, hoc eſt quadratum <anchor type="note" xlink:href="" symbol="a"/> A C <anchor type="note" xlink:label="note-0228-01a" xlink:href="note-0228-01"/> ad M N, vt H E ad E L, ſiue vt quadratum D F ad O P, ideoque & </s> <s xml:id="echoid-s6379" xml:space="preserve">ap-<lb/>plicata A C ad M N, vt applicata D F ad O P. </s> <s xml:id="echoid-s6380" xml:space="preserve">Quare, ex ſecunda præ-<lb/>cedentium definitionum, ipſæ portiones A B C, D E F erunt Acuminata <lb/>Proportionalia. </s> <s xml:id="echoid-s6381" xml:space="preserve">Quod primò, &</s> <s xml:id="echoid-s6382" xml:space="preserve">c.</s> <s xml:id="echoid-s6383" xml:space="preserve"/> </p> <div xml:id="echoid-div667" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0228-01" xlink:href="note-0228-01a" xml:space="preserve">20. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s6384" xml:space="preserve">Præterea ſint A B C, D E <lb/> <anchor type="figure" xlink:label="fig-0228-01a" xlink:href="fig-0228-01"/> F duæ portiones eiuſdem, vel <lb/>diuerſarum Hyperbolarum, vt <lb/>in ſecunda figura, vel Elli-<lb/>pſium, aut circulorum, vt in <lb/>tertia, quarum baſes A C, D <lb/>F, & </s> <s xml:id="echoid-s6385" xml:space="preserve">diametrorum ſegmenta <lb/>in ipſis intercepta ſint B G, E <lb/>H, quæ vſque ad ſectionum <lb/>centra Q, R producantur, & </s> <s xml:id="echoid-s6386" xml:space="preserve"><lb/>ſit vt G B ad BQ, ita H E ad <lb/>E R. </s> <s xml:id="echoid-s6387" xml:space="preserve">Dico item has portio-<lb/>nes A B C, D E F eſſe inter ſe <lb/>Acuminata Proportionalia.</s> <s xml:id="echoid-s6388" xml:space="preserve"/> </p> <div xml:id="echoid-div668" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0228-01"/> </figure> </div> <p> <s xml:id="echoid-s6389" xml:space="preserve">Nam diuiſis diametris B G, <lb/>E H proportionaliter in I, L, <lb/>per I, L applicentur M I N, <lb/>O L P, & </s> <s xml:id="echoid-s6390" xml:space="preserve">productis ſemi-dia-<lb/>metris B Q, E R ſumantur eis <lb/>æquales Q S, R T, ita vt S B, <lb/>T E ſint ſectionum diametri.</s> <s xml:id="echoid-s6391" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6392" xml:space="preserve">Iam cum ſit G B ad B Q vt <lb/>H E ad E R, erit conuerten-<lb/>do, & </s> <s xml:id="echoid-s6393" xml:space="preserve">ſumptis antecedentium duplis S B ad B G, vt T E ad E H, ſuntq; <lb/></s> <s xml:id="echoid-s6394" xml:space="preserve">S B, T E bifariam ſectæ in Q, R, & </s> <s xml:id="echoid-s6395" xml:space="preserve">partes adiectæ, in ſecunda figura, <lb/>vel demptæ in tertia B G, E H proportionaliter diuiſæ ſunt in I, L, ergo <lb/>rectangulum S G B ad S I B, ſiue <anchor type="note" xlink:href="" symbol="b"/> quadratum A C, ad M N, erit vt <anchor type="note" xlink:href="" symbol="c"/> re- <anchor type="note" xlink:label="note-0228-02a" xlink:href="note-0228-02"/> <anchor type="note" xlink:label="note-0228-03a" xlink:href="note-0228-03"/> ctangulum T H E, ad T L E, vel vt <anchor type="note" xlink:href="" symbol="d"/> quadratum D F ad O P, nempe ap- <anchor type="note" xlink:label="note-0228-04a" xlink:href="note-0228-04"/> plicata A C ad M N erit vt applicata D F ad O P, & </s> <s xml:id="echoid-s6396" xml:space="preserve">permutando A C <lb/>ad D F, vt M N ad O P, & </s> <s xml:id="echoid-s6397" xml:space="preserve">hoc ſemper de quibuslibet applicatis per <lb/>pnncta diametrorum B G, E H ipſas proportionaliter ſecantia, quare, ex <lb/>definitione ſecunda, ipſæ portiones A B C, D E F erunt Acuminata pro-<lb/>portionalia. </s> <s xml:id="echoid-s6398" xml:space="preserve">Quod vltimò demonſtrandum erat.</s> <s xml:id="echoid-s6399" xml:space="preserve"/> </p> <div xml:id="echoid-div669" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-0228-02" xlink:href="note-0228-02a" xml:space="preserve">21. primi <lb/>conic.</note> <note symbol="c" position="left" xlink:label="note-0228-03" xlink:href="note-0228-03a" xml:space="preserve">35. h.</note> <note symbol="d" position="left" xlink:label="note-0228-04" xlink:href="note-0228-04a" xml:space="preserve">21. primi <lb/>conic.</note> </div> <pb o="47" file="0229" n="229" rhead=""/> </div> <div xml:id="echoid-div671" type="section" level="1" n="267"> <head xml:id="echoid-head276" xml:space="preserve">THEOR. XXII. PROP. XXXVII.</head> <p> <s xml:id="echoid-s6400" xml:space="preserve">Proportionalia Acuminata, quorum baſes eorum altitudini-<lb/>bus ſint reciprocè proportionales, ſunt inter ſe æqualia.</s> <s xml:id="echoid-s6401" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6402" xml:space="preserve">SInt duo proportionalia Acuminata A B C, D E F, quorum diametri <lb/>ſint B G, E H, altitudines verò B I, E L, quæ inter ſe reciprocam <lb/>habeant rationem baſium A C, D F; </s> <s xml:id="echoid-s6403" xml:space="preserve">ſiue ſit vt A C ad D F, ita E L ad <lb/>B I. </s> <s xml:id="echoid-s6404" xml:space="preserve">Dico huiuſmodi Acuminata inter ſe æqualia eſſe.</s> <s xml:id="echoid-s6405" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6406" xml:space="preserve">Si enim poſſibile eſt, <lb/> <anchor type="figure" xlink:label="fig-0229-01a" xlink:href="fig-0229-01"/> ſit alterum ipſorum, <lb/>nempe A B C reliquo D <lb/>E F minus, & </s> <s xml:id="echoid-s6407" xml:space="preserve">per con-<lb/>tinuam diametri B G <lb/>biſectionem, iuxta vul-<lb/>gatam methodum, cir-<lb/>cumſcribatur ipſi A B <lb/>C, figura exparallelo-<lb/>grammis conſtans æ-<lb/>qualium altitudinum A <lb/>L, M N, &</s> <s xml:id="echoid-s6408" xml:space="preserve">c. </s> <s xml:id="echoid-s6409" xml:space="preserve">quorum <lb/>altitudines I T, T V, &</s> <s xml:id="echoid-s6410" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s6411" xml:space="preserve">æquales erunt (cum <lb/>altitudo B I in tot æ-<lb/>quales partes diuidatur <lb/>ab æquidiſtantibus parallelogrammorum baſibus A C, M O, &</s> <s xml:id="echoid-s6412" xml:space="preserve">c. </s> <s xml:id="echoid-s6413" xml:space="preserve">in quot <lb/>partes diameter B G ſecta fuit) huiuſmodi autem circumſcripta figura ex <lb/>parallelogrammis, acuminatum A B C ſuperet minori exceſſu, quò acu-<lb/>minatum D E F ponitur excedere idem acuminatum A B C, adeo vt ipſa <lb/>circumſcripta A B N L C ſit adhuc minor acuminato D E F, cui circum-<lb/>ſcribatur item figura D E R P F ex totidem parallelogrammis D P, Q R &</s> <s xml:id="echoid-s6414" xml:space="preserve">c. </s> <s xml:id="echoid-s6415" xml:space="preserve"><lb/>æqualium altitudinum K X, X Y, &</s> <s xml:id="echoid-s6416" xml:space="preserve">c.</s> <s xml:id="echoid-s6417" xml:space="preserve"/> </p> <div xml:id="echoid-div671" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0229-01"/> </figure> </div> <p> <s xml:id="echoid-s6418" xml:space="preserve">Iam, cum ſit baſis A C ad D F, vt altitudo E K ad B I, vel vt ſubmul-<lb/>tiplex K X ad æque-ſubmultiplicem I T, erit parallelogrammum A L, æ-<lb/>quale parallelogrammo D P. </s> <s xml:id="echoid-s6419" xml:space="preserve">Et cum, ex conſtructione, ſit G B ad B Z, <lb/>vt H E ad E 3, erit, ex definitione proportionalium acuminatorum, A C <lb/>ad D F, vt M O ad Q S, ſed A C ad D F eſt vt E K ad B I, ergo, & </s> <s xml:id="echoid-s6420" xml:space="preserve">M <lb/>O ad Q S erit vt E K ad B I, vel vt ſubmultiplex X Y ad æque-ſubmul-<lb/>tiplicem T V: </s> <s xml:id="echoid-s6421" xml:space="preserve">parallelogrammum igitur M N æquatur parallelogrammo <lb/>Q R; </s> <s xml:id="echoid-s6422" xml:space="preserve">& </s> <s xml:id="echoid-s6423" xml:space="preserve">ſic de reliquis, ſingula ſingulis: </s> <s xml:id="echoid-s6424" xml:space="preserve">ergo vniuerſa figura A B N L C <lb/>æqualis erit vniuerſæ D E R P F, ſed figura A B N L C facta eſt minor <lb/>acuminato D E F, quare figura D E R P F erit quoque minor eodem ſibi <lb/>inſcripto acuminato D E F: </s> <s xml:id="echoid-s6425" xml:space="preserve">totum parte, quod eſt abſurdum. </s> <s xml:id="echoid-s6426" xml:space="preserve">Nullum <lb/>ergo horum acuminatorum eſt reliquo minus, quapropter æqualia eſſe <lb/>inter ſe neceſſe eſt. </s> <s xml:id="echoid-s6427" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s6428" xml:space="preserve"/> </p> <pb o="48" file="0230" n="230" rhead=""/> </div> <div xml:id="echoid-div673" type="section" level="1" n="268"> <head xml:id="echoid-head277" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s6429" xml:space="preserve">EX hac facilè elicietur methodus, qua precipuas quorumlibet Acumi-<lb/>natorum paſſiones oſtendi poſſint, nempe: </s> <s xml:id="echoid-s6430" xml:space="preserve">ipſa Acuminata à diame-<lb/>tris bifariam ſecari: </s> <s xml:id="echoid-s6431" xml:space="preserve">& </s> <s xml:id="echoid-s6432" xml:space="preserve">Proportionalia Acuminata ęqualium altitudinum <lb/>inter ſe eſſe vt baſes: </s> <s xml:id="echoid-s6433" xml:space="preserve">& </s> <s xml:id="echoid-s6434" xml:space="preserve">(tanquam Corollarium) Acuminata proportio-<lb/>nalia æqualium baſium eſſe inter ſe vt altitudines: </s> <s xml:id="echoid-s6435" xml:space="preserve">item duo quæcunque <lb/>Acuminata proportionalia habere inter ſe rationem compoſitam ex ratio-<lb/>ne baſium, & </s> <s xml:id="echoid-s6436" xml:space="preserve">ex ratione altitudinum: </s> <s xml:id="echoid-s6437" xml:space="preserve">& </s> <s xml:id="echoid-s6438" xml:space="preserve">ad inſcripta triangula, vel cir-<lb/>cumſcripta parallelogramma eandem retinere rationem, aliaque his ſimi-<lb/>lia: </s> <s xml:id="echoid-s6439" xml:space="preserve">& </s> <s xml:id="echoid-s6440" xml:space="preserve">quod de proportionalibus Acuminatis, idem penitus euenire de ſi-<lb/>milibus menſalibus proportionalium Acuminatorum, præmiſſa priùs ha-<lb/>rum menſalium definitione, &</s> <s xml:id="echoid-s6441" xml:space="preserve">c. </s> <s xml:id="echoid-s6442" xml:space="preserve">quæ omnia infinitas figurarum ſpecies <lb/><gap/>, ne dum hactenus tractatis Parabolis, Hyperbolis, &</s> <s xml:id="echoid-s6443" xml:space="preserve">c. </s> <s xml:id="echoid-s6444" xml:space="preserve">maximè con-<lb/>ducunt. </s> <s xml:id="echoid-s6445" xml:space="preserve">Sed hæc aliàs, quę tamen cum ſint haud obſcurę indagationis, <lb/>& </s> <s xml:id="echoid-s6446" xml:space="preserve">huic noſtro inſtituto prorſus aliena, erudito Lectori ſic præmonſtraſſe <lb/>ſuſſiciat.</s> <s xml:id="echoid-s6447" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div674" type="section" level="1" n="269"> <head xml:id="echoid-head278" xml:space="preserve">LEMMA XI. PROP. XXXVIII.</head> <p> <s xml:id="echoid-s6448" xml:space="preserve">Si duæ rectæ lineæ inter ſe æquales fuerint, & </s> <s xml:id="echoid-s6449" xml:space="preserve">parallelæ, & </s> <s xml:id="echoid-s6450" xml:space="preserve"><lb/>ab earum extremis terminis ducantur lineæ quemlibet angulum <lb/>efficientes, ab alteris autem terminis aliæ ipſis æquidiſtantes; <lb/></s> <s xml:id="echoid-s6451" xml:space="preserve">hæ quoque angulum inter datas conſtituent, & </s> <s xml:id="echoid-s6452" xml:space="preserve">recta angulo-<lb/>rum vertices coniungens erit vtrique datarum æqualis, & </s> <s xml:id="echoid-s6453" xml:space="preserve">pa-<lb/>rallela.</s> <s xml:id="echoid-s6454" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6455" xml:space="preserve">Si verò datæ rectæ lineæ terminatæ ad quemcunque angulum <lb/>applicatæ fuerint, & </s> <s xml:id="echoid-s6456" xml:space="preserve">vbicunque proportionaliter ſectę, aut pro-<lb/>ductæ, atque ab homologis earum punctis, hoc eſt, vel ab ex-<lb/>tremis terminis, vel ab inter-ſectionum, aut productionum pun-<lb/>ctis, ductæ fuerint intra datum angulum aliæ rectæ lineæ, quæ <lb/>item angulum quemlibet conſtituant, à reliquis verò punctis aliæ <lb/>ipſis æquidiſtanter ducantur, hæ pariter tertium angulum effi-<lb/>cient intra datum, & </s> <s xml:id="echoid-s6457" xml:space="preserve">horum trium angulorum vertices in vna <lb/>eademque recta linea reperientur.</s> <s xml:id="echoid-s6458" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6459" xml:space="preserve">SIt, in prima figura, recta A B æqualis, & </s> <s xml:id="echoid-s6460" xml:space="preserve">parallela ad C D, & </s> <s xml:id="echoid-s6461" xml:space="preserve">exter-<lb/>minis A, C inter eas conſtituatur angulus quicunque A E C ducta-<lb/>que B F parallela ad A E, D F verò ad C E. </s> <s xml:id="echoid-s6462" xml:space="preserve">Dico B F, D F inter datas <lb/>æquidiſtantes conuenire, & </s> <s xml:id="echoid-s6463" xml:space="preserve">E F iungentem angulorum vertices, alteri A <lb/>B, vel C D eſſe æqualem, & </s> <s xml:id="echoid-s6464" xml:space="preserve">parallelam.</s> <s xml:id="echoid-s6465" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6466" xml:space="preserve">Iungantur A C, B D: </s> <s xml:id="echoid-s6467" xml:space="preserve">& </s> <s xml:id="echoid-s6468" xml:space="preserve">quoniam B F eſt parallela ad A E, erit angu-<lb/>lus G B F æqualis angulo B A E; </s> <s xml:id="echoid-s6469" xml:space="preserve">cumque A B ſit æqualis, & </s> <s xml:id="echoid-s6470" xml:space="preserve">parallela ad <pb o="49" file="0231" n="231" rhead=""/> C B, erunt A C, B D æquales, & </s> <s xml:id="echoid-s6471" xml:space="preserve">parallelæ, idcirco angulus G B D æ-<lb/>qualis angulo B A C, ergo reliquus angulus D B F, æqualis erit reliquo <lb/>C A E; </s> <s xml:id="echoid-s6472" xml:space="preserve">eadem ratione oſtendetur angulum B D F æquari angulo A C E, <lb/>& </s> <s xml:id="echoid-s6473" xml:space="preserve">B D demonſtrata eſt æqualis ipſi A C, ergo in triangulis B F D, A E <lb/>C, cum æqualia latera B D, A C æqualibus angulis adiaceant, erit ter-<lb/>tius angulus B F D, tertio A E C æqualis, & </s> <s xml:id="echoid-s6474" xml:space="preserve">reliqua latera B F, A E, <lb/>itemque D F, C E, inter ſe æqualia, ſed ſunt quoque parallela, ob hy-<lb/>poteſim, ergo E F angulorum vertices iungens, erit æqualis, & </s> <s xml:id="echoid-s6475" xml:space="preserve">paralle-<lb/>la ad A B, vel ad C D; </s> <s xml:id="echoid-s6476" xml:space="preserve">cadetque inter datas A B, C D, cum punctum <lb/>E, ex quo ducitur ſit inter eas, ſicque angulus B F D cadet intra datas <lb/>æquidiſtantes A B, C D. </s> <s xml:id="echoid-s6477" xml:space="preserve">Quod primò oſtendere opus erat.</s> <s xml:id="echoid-s6478" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6479" xml:space="preserve">Sint verò, in reli-<lb/>quis tribus figuris, <lb/>datæ rectæ A B, A C <lb/> <anchor type="figure" xlink:label="fig-0231-01a" xlink:href="fig-0231-01"/> terminatæ angulum <lb/>B A C conſtituentes, <lb/>quę proportionaliter <lb/>ſecentur, vel produ-<lb/>cantur in D, E; </s> <s xml:id="echoid-s6480" xml:space="preserve">& </s> <s xml:id="echoid-s6481" xml:space="preserve"><lb/>ex punctis D, E du-<lb/>ctæ ſint D F, E F ſibi <lb/>ipſis occurrentes in-<lb/>tra datum angulum <lb/>in F, ex reliquis ve-<lb/>rò punctis B C, alię <lb/>ipſis æquidiſtantes B <lb/>G, C H. </s> <s xml:id="echoid-s6482" xml:space="preserve">Dico item <lb/>has intra angulum B <lb/>A C inter ſe conue-<lb/>nire, vt in G, ac tres <lb/>angulorum occurſus <lb/>A, F, G in eadem re-<lb/>cta linea reperiri.</s> <s xml:id="echoid-s6483" xml:space="preserve"/> </p> <div xml:id="echoid-div674" type="float" level="2" n="1"> <figure xlink:label="fig-0231-01" xlink:href="fig-0231-01a"> <image file="0231-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0231-01"/> </figure> </div> <p> <s xml:id="echoid-s6484" xml:space="preserve">Nam iuncta A F, & </s> <s xml:id="echoid-s6485" xml:space="preserve">producta cum B G ipſi D F ęquidiſtet, & </s> <s xml:id="echoid-s6486" xml:space="preserve">A F cum <lb/>D F conueniat, conueniet quoque producta cum B G, ſit ergo G punctum <lb/>occurſus. </s> <s xml:id="echoid-s6487" xml:space="preserve">Item cum C H æquidiſtet ipſi E F, & </s> <s xml:id="echoid-s6488" xml:space="preserve">A F ſecet E F, producta <lb/>ſecabit quoque C H: </s> <s xml:id="echoid-s6489" xml:space="preserve">ſecet in H. </s> <s xml:id="echoid-s6490" xml:space="preserve">Oſtendam puncta G, H, quæ iam in <lb/>recta A F reperiri demonſtratum eſt, eſſe vnum idemque punctum rectæ <lb/>A F: </s> <s xml:id="echoid-s6491" xml:space="preserve">eſt enim in triangulo A G B vt G A ad A F, ita B A ad A D, vel, <lb/>ob hypoteſim, vt C A ad A E, vel vt H A ad A F, ergo G A, & </s> <s xml:id="echoid-s6492" xml:space="preserve">H A <lb/>ſunt æquales, hoc eſt puncta G, & </s> <s xml:id="echoid-s6493" xml:space="preserve">H non ſunt duo, ſed vnum tantùm, <lb/>& </s> <s xml:id="echoid-s6494" xml:space="preserve">in eadem recta linea in qua ſunt puncta A, F. </s> <s xml:id="echoid-s6495" xml:space="preserve">Ergo B G, C H inter <lb/>ſe conueniunt intra datum angulum, ac trium angulorum vertices ſunt <lb/>in directum poſiti. </s> <s xml:id="echoid-s6496" xml:space="preserve">Quod vltimò oſtendere propoſitum fuit.</s> <s xml:id="echoid-s6497" xml:space="preserve"/> </p> <pb o="50" file="0232" n="232" rhead=""/> </div> <div xml:id="echoid-div676" type="section" level="1" n="270"> <head xml:id="echoid-head279" xml:space="preserve">LEMMA XII. PROP. XXXIX.</head> <p> <s xml:id="echoid-s6498" xml:space="preserve">Si fuerit quodcunque quadrilaterum rectilineum A B C D, cu-<lb/>ius oppoſita latera A D, B C bifariam ſecta ſint in punctis F, <lb/>E, iunctaque ſit recta F E, in qua ſumptum ſit quodlibet pun-<lb/>ctum G, vel intra, vel extra quadrilaterum à quo ad terminos <lb/>alterius ęquidiſtantium veluti ad A, D, ductæ ſint G A, G D, <lb/>ac in triangulo A G D, ſit quædam H I ipſis A D, B C æquidi-<lb/>ſtans, & </s> <s xml:id="echoid-s6499" xml:space="preserve">E F ſecans in L. </s> <s xml:id="echoid-s6500" xml:space="preserve">Dico, ſi iungantur B H, C I, trian-<lb/>gula A B H, D C I inter ſe æqualia eſſe.</s> <s xml:id="echoid-s6501" xml:space="preserve"/> </p> <figure> <image file="0232-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0232-01"/> </figure> <p> <s xml:id="echoid-s6502" xml:space="preserve">NAm totum quadrilaterum A B E F, æquale eſt integro quadrilatero <lb/>D C E F (vtrunque enim diuiditur per diagonales A E, D E, in <lb/>duo triangula alterum alteri æquale, eò quod ſint ſuper æqualibus baſi-<lb/>bus, ac inter eaſdem parallelas) eadem ratione quadrilaterum A H L F <lb/>æquale eſt quadrilatero D I L F, & </s> <s xml:id="echoid-s6503" xml:space="preserve">quadrilaterum B E L H æquale qua-<lb/>drilatero C E L I, ergo, & </s> <s xml:id="echoid-s6504" xml:space="preserve">reliquum triangulum A B H reliquo triangulo <lb/>D C I eſt æquale. </s> <s xml:id="echoid-s6505" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s6506" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6507" xml:space="preserve">His itaque præoſtenſis, ad inueſtigationem MAXIMARVM, MI-<lb/>NIMARV MQVE portionum per idem datum punctum ex qualibet coni-<lb/>ſectione abſciſſarum accedamus, præmiſſo tamen, ſuper figurastertij Sche-<lb/>matiſmi, ſequenti Theoremate, vniuerſalem, ſimulque facilem methodum <lb/>exhibente, qua æquales portiones de eadem coni-ſectione abſcindi poſſunt.</s> <s xml:id="echoid-s6508" xml:space="preserve"/> </p> <pb file="0233" n="233"/> <pb file="0234" n="234"/> <figure> <image file="0234-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0234-01"/> </figure> <pb o="51" file="0235" n="235" rhead=""/> </div> <div xml:id="echoid-div677" type="section" level="1" n="271"> <head xml:id="echoid-head280" xml:space="preserve">THEOR. XXIII. PROP. XXXX.</head> <p> <s xml:id="echoid-s6509" xml:space="preserve">Si in Parabola, ex binis ipſius diametris duo æqualia ſegmen-<lb/>ta ſint abſciſſa: </s> <s xml:id="echoid-s6510" xml:space="preserve">in Hyperbola verò, Ellipſi, vel circulo duæ ſe-<lb/>mi-diametri proportionaliter intra ſectionem ſectę fuerint, & </s> <s xml:id="echoid-s6511" xml:space="preserve">ex <lb/> <anchor type="note" xlink:label="note-0235-01a" xlink:href="note-0235-01"/> terminis æqualium diametrorum in Parabola, vel ex punctis di-<lb/>uiſionum, in reliquis ſectionibus, ordinatim applicentur lineæ ad <lb/>fuas diametros, & </s> <s xml:id="echoid-s6512" xml:space="preserve">producantur, donec ad vtranque partem ſe-<lb/>ctioni occurrant: </s> <s xml:id="echoid-s6513" xml:space="preserve">coni- ſectionum portiones; </s> <s xml:id="echoid-s6514" xml:space="preserve">at in Ellipſi, vel <lb/>circulo, minores portiones ijſdem applicatis, tanquam baſibus <lb/>inſiſtentes, inter ſe æquales erunt.</s> <s xml:id="echoid-s6515" xml:space="preserve"/> </p> <div xml:id="echoid-div677" type="float" level="2" n="1"> <note position="right" xlink:label="note-0235-01" xlink:href="note-0235-01a" xml:space="preserve">Schema-<lb/>tiſmus 3.</note> </div> <p> <s xml:id="echoid-s6516" xml:space="preserve">ESto A B C Parabole, in prima, ſecunda, & </s> <s xml:id="echoid-s6517" xml:space="preserve">tertia figura, vel Hyper-<lb/>bole in quarta, quinta, & </s> <s xml:id="echoid-s6518" xml:space="preserve">ſexta, aut Ellipſis in ſeptima, octaua, & </s> <s xml:id="echoid-s6519" xml:space="preserve"><lb/>nona, aut circulus, in reliquis, quarum ſectionum binæ diametri in Pa-<lb/>rabola ſint D B, D E, à quibus dempta ſint æqualia ſegmenta B F, E G, <lb/>& </s> <s xml:id="echoid-s6520" xml:space="preserve">in reliquis binæ ſemi-diametri D B, D E (quæ primò in Ellipſi, vel <lb/>circulo omnino conſtituant angulum B D E) ita intra ſectiones ſectæ ſint <lb/>in F, G, vt D B ad B F, ſit vt D E ad E G, & </s> <s xml:id="echoid-s6521" xml:space="preserve">per puncta F, G, in ſin-<lb/>gulis figuris ſint ad diametros D B, D E ordinatim ductæ A F C, H G I, <lb/>quæ ad vtranque partem ſectioni occurrent <anchor type="note" xlink:href="" symbol="a"/> in punctis A, C; </s> <s xml:id="echoid-s6522" xml:space="preserve">H, I, &</s> <s xml:id="echoid-s6523" xml:space="preserve"> <anchor type="note" xlink:label="note-0235-02a" xlink:href="note-0235-02"/> bifariam in F, G ſecabuntur, cum D B, D G, ſint ipſarum diametri. </s> <s xml:id="echoid-s6524" xml:space="preserve">Di-<lb/>co portiones A B C, H E I ſuper ijſdem applicatis, tanquam baſibus inſi-<lb/>ſtentes, inter ſe æquales eſſe.</s> <s xml:id="echoid-s6525" xml:space="preserve"/> </p> <div xml:id="echoid-div678" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0235-02" xlink:href="note-0235-02a" xml:space="preserve">19. primi <lb/>conic.</note> </div> <p> <s xml:id="echoid-s6526" xml:space="preserve">Nam, ductis ex B, E rectis B N, E N ſectionem contingentibus in B, <lb/>E; </s> <s xml:id="echoid-s6527" xml:space="preserve">ipſæ occurrent <anchor type="note" xlink:href="" symbol="b"/> ſimul in N inter diametros D B, D E, & </s> <s xml:id="echoid-s6528" xml:space="preserve">applicatis <anchor type="note" xlink:label="note-0235-03a" xlink:href="note-0235-03"/> H I, A C æquidiſtabunt. </s> <s xml:id="echoid-s6529" xml:space="preserve">Iungantur præterea E B, G F.</s> <s xml:id="echoid-s6530" xml:space="preserve"/> </p> <div xml:id="echoid-div679" type="float" level="2" n="3"> <note symbol="b" position="right" xlink:label="note-0235-03" xlink:href="note-0235-03a" xml:space="preserve">58. primi <lb/>huius.</note> </div> <p> <s xml:id="echoid-s6531" xml:space="preserve">Iam in Parabolis, cum ſint E G, B F inter ſe æquales, & </s> <s xml:id="echoid-s6532" xml:space="preserve">parallelę, iun-<lb/>ctæ quoq; </s> <s xml:id="echoid-s6533" xml:space="preserve">E B, G F inter ſe æquidiſtabunt, & </s> <s xml:id="echoid-s6534" xml:space="preserve">cum ex illarum terminis E, <lb/>B, ductæ ſint rectę E N, B N angulum E N B inter eas conſtituentes, atq; <lb/></s> <s xml:id="echoid-s6535" xml:space="preserve">ex reliquis terminis G, F, ſint G I, F A, ipſis E N, B N æqurdiſtantes; </s> <s xml:id="echoid-s6536" xml:space="preserve"><lb/>ipſæ G I, F A inter eaſdem E G, B F ſimul conuenient, vt in M, & </s> <s xml:id="echoid-s6537" xml:space="preserve">iuncta <lb/>N M ijſdem E G, B F æquidiſtabit, ſiue erit altera Parabolæ <anchor type="note" xlink:href="" symbol="c"/> diameter.</s> <s xml:id="echoid-s6538" xml:space="preserve"> <anchor type="note" xlink:label="note-0235-04a" xlink:href="note-0235-04"/> Cum ergo ſit E G parallela ad <anchor type="note" xlink:href="" symbol="d"/> N M, & </s> <s xml:id="echoid-s6539" xml:space="preserve">E N ad G M, erit E N æqualis <anchor type="note" xlink:label="note-0235-05a" xlink:href="note-0235-05"/> G M; </s> <s xml:id="echoid-s6540" xml:space="preserve">eademque ratione B N æqualis F M, quare vt E N ad N B, ita G <lb/>M ad M F.</s> <s xml:id="echoid-s6541" xml:space="preserve"/> </p> <div xml:id="echoid-div680" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-0235-04" xlink:href="note-0235-04a" xml:space="preserve">38. h.</note> <note symbol="d" position="right" xlink:label="note-0235-05" xlink:href="note-0235-05a" xml:space="preserve">46. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s6542" xml:space="preserve">In reliquis verò figuris, cum rectæ D B, D E angulum E D B efficien-<lb/>tes, proportionaliter ſectæ, aut productæ ſint in G, F, ſintque ex earum <lb/>homologis terminis E, B ductæ E N, B N angulum inter ipſas conſti-<lb/>tuentes E N B, & </s> <s xml:id="echoid-s6543" xml:space="preserve">ex reliquis diuiſionum punctis G, F, ſint G I, F A ijſdem <lb/>E N, B N parallelę, hæ intra datum angulum E D B ſimul conuenient, vt <lb/>in M; </s> <s xml:id="echoid-s6544" xml:space="preserve">& </s> <s xml:id="echoid-s6545" xml:space="preserve">recta iungens puncta D, M, per occurſum M omnino tranſibit, <lb/>ſiue <anchor type="note" xlink:href="" symbol="e"/> erit alia ſectionis diameter. </s> <s xml:id="echoid-s6546" xml:space="preserve">Cumque ob parallelas G M, E N ſit G <anchor type="note" xlink:label="note-0235-06a" xlink:href="note-0235-06"/> M ad E N, vt M D ad D N, & </s> <s xml:id="echoid-s6547" xml:space="preserve">ob parallelas M F, N B ſit M F ad N B, <lb/> <anchor type="note" xlink:label="note-0235-07a" xlink:href="note-0235-07"/> <pb o="52" file="0236" n="236" rhead=""/> vt eadem M D ad D N, erit G M ad E N, vt M F ad N B, & </s> <s xml:id="echoid-s6548" xml:space="preserve">permu-<lb/>tando G M ad M F, vt E N ad N B.</s> <s xml:id="echoid-s6549" xml:space="preserve"/> </p> <div xml:id="echoid-div681" type="float" level="2" n="5"> <note symbol="e" position="right" xlink:label="note-0235-06" xlink:href="note-0235-06a" xml:space="preserve">38. h.</note> <note position="right" xlink:label="note-0235-07" xlink:href="note-0235-07a" xml:space="preserve">_f_ 47. primi <lb/>conic.</note> </div> <p> <s xml:id="echoid-s6550" xml:space="preserve">Cum ergo, in figuris prima, ſecunda, quarta, quinta, ſeptima, octaua, <lb/>decima, ac decimaprima ſit G M ad M F, vt E N ad N B, erit quoq; </s> <s xml:id="echoid-s6551" xml:space="preserve">qua-<lb/>dratum G M ad M F, vt quadratum E N ad N B, vel vt <anchor type="note" xlink:href="" symbol="a"/> rectangulum H <anchor type="note" xlink:label="note-0236-01a" xlink:href="note-0236-01"/> M I ad rectangulum C M A, & </s> <s xml:id="echoid-s6552" xml:space="preserve">permutando quadratum G M ad rectan-<lb/>gulum H M I, vt quadratum F M ad rectangulum C M A, & </s> <s xml:id="echoid-s6553" xml:space="preserve">couertendo <lb/>in prima, quarta, ſeptima, & </s> <s xml:id="echoid-s6554" xml:space="preserve">decima figura (in quibus applicatæ H I, C A <lb/>ſecant ſe mutuò intra ſectionem in puncto M) rectangulum H M I ad qua-<lb/>dratum G M, vtrectangulum C M A ad quadratum F M, & </s> <s xml:id="echoid-s6555" xml:space="preserve">componendo <lb/>rectangulum H M I cum quadrato G M, ſiue vnicum quadratum H G, (nam <lb/>eſt A C bifariam ſecta in G, & </s> <s xml:id="echoid-s6556" xml:space="preserve">non bifariam in M) ad quadratum G M, vt <lb/>rectangulum C M A cum quadrato F M, ſiue vt vnicum quadratum C F <lb/>(cum A C quoque ſecta ſit bifariam in F, & </s> <s xml:id="echoid-s6557" xml:space="preserve">non bifariam in M) ad quadra-<lb/>tum F M. </s> <s xml:id="echoid-s6558" xml:space="preserve">In figuris verò ſecunda, quinta, octaua, & </s> <s xml:id="echoid-s6559" xml:space="preserve">vndecima, in quibus <lb/>applicatæ H I, C A ſe mutuò ſecant extra ſectionem in puncto M, cum ſit <lb/>G M quadratum ad rectangulum H M I, vt quadratum F M ad rectangu-<lb/>lum C M A, erit per conuerſionem rationis quadratum M G ad quadratum <lb/>G H (eſt enim rectangulum H M I cum quadrato G H æquale quadrato G <lb/>M, cum ſit H I bifariam ſecta in G, & </s> <s xml:id="echoid-s6560" xml:space="preserve">ei adiecta ſit I M) vt quadratum M <lb/>F ad quadratum F C, ob eandem rationem, (nam C A quoq; </s> <s xml:id="echoid-s6561" xml:space="preserve">bifariam ſecta <lb/>eſt in C, eiq; </s> <s xml:id="echoid-s6562" xml:space="preserve">addita eſt in directum A M) & </s> <s xml:id="echoid-s6563" xml:space="preserve">conuertendo quadratum H G ad <lb/>G M quadratum, erit vt quadratum C F ad F M. </s> <s xml:id="echoid-s6564" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s6565" xml:space="preserve">in ſingulis prædictis <lb/>figuris, déptis tertia, ſexta, nona, & </s> <s xml:id="echoid-s6566" xml:space="preserve">duodecima, cum demonſtratum ſit qua-<lb/>dratum H G ad G M eſſe vt quadratum C F ad F M, erit quoque linea H G <lb/>G M, vt linea C F ad F M. </s> <s xml:id="echoid-s6567" xml:space="preserve">In figuris deniq; </s> <s xml:id="echoid-s6568" xml:space="preserve">tertia, ſexta, nona, & </s> <s xml:id="echoid-s6569" xml:space="preserve">duodeci-<lb/>ma, in quibus applicatę H I, C A conueniunt ſimul cum ipſa ſectione in pun-<lb/>cto M, patet quoque eſſe H G ad G M, vt C F ad F M, cum ipſæ H I, C <lb/>A, vel H M, C M bifariam ſecentur in G, F ab earum diametris E G, B F. <lb/></s> <s xml:id="echoid-s6570" xml:space="preserve">Eſt igitur in qualibet datarum figurarum huius ſchematiſmi, H G ad G M, vt <lb/>C F ad F M, quare iuncta H C æquidiſtabit iunctæ G F; </s> <s xml:id="echoid-s6571" xml:space="preserve">ſed eſt I G æqua-<lb/>lis H G, & </s> <s xml:id="echoid-s6572" xml:space="preserve">A F æqualis C F, ergo etiam I G ad G M erit vt A F ad F M, <lb/>ideoque iuncta A I æquidiſtabit eidem G F, ſed E B quoque ipſi G F ęqui-<lb/>diſtat (vt iam ſupra oſtendimus in Parabolis, & </s> <s xml:id="echoid-s6573" xml:space="preserve">cum in reliquis ſectionibus <lb/>ſit D E ad E G, vt D B ad B F ex hypoteſi) ergo quatuor iunctæ rectæ lineæ <lb/>E B, A I, G F, H C ſunt inter ſe parallelæ; </s> <s xml:id="echoid-s6574" xml:space="preserve">ſed N M, quàm ſuperiùs oſten-<lb/>dimus eſſe ſectionis diametrum, tranſit per N occurſum contingentium E <lb/>N, B N, ergo recta E B puncta contactuum iungens, ab eadem diametro N <lb/>M D bifariam ſecabitur, <anchor type="note" xlink:href="" symbol="b"/> vt in O, ac ideò omnes aliæ in ſectione applicatæ <anchor type="note" xlink:label="note-0236-02a" xlink:href="note-0236-02"/> ipſi E B ęquidiſtantes, nempe A I, G F, H C, ab eadem D N M bifariam <lb/>ſecabuntur, vt H C in P.</s> <s xml:id="echoid-s6575" xml:space="preserve"/> </p> <div xml:id="echoid-div682" type="float" level="2" n="6"> <note symbol="a" position="left" xlink:label="note-0236-01" xlink:href="note-0236-01a" xml:space="preserve">17. tertij <lb/>conic.</note> <note symbol="b" position="left" xlink:label="note-0236-02" xlink:href="note-0236-02a" xml:space="preserve">30.ſecũ-<lb/>di conic.</note> </div> <p> <s xml:id="echoid-s6576" xml:space="preserve">Denique iungantur rectæ H E, C B, & </s> <s xml:id="echoid-s6577" xml:space="preserve">fiet quadrilaterum H E B C, cu-<lb/>ius oppoſita latera H C, E B ſunt parallela, & </s> <s xml:id="echoid-s6578" xml:space="preserve">bifariam ſecta à recta P O, in <lb/>qua ſumptum eſt punctum M, & </s> <s xml:id="echoid-s6579" xml:space="preserve">ab ipſo ad terminos alterius ęquidiſtantium <lb/>nempe ad H, C ductę ſunt rectæ M H, M C, ac in triangulo H M C eſt G F <lb/>ipſi H C parallela, quare iunctę E G, B F auferent triangula E G H, B F C <lb/>inter ſe <anchor type="note" xlink:href="" symbol="c"/> æqualia; </s> <s xml:id="echoid-s6580" xml:space="preserve">quapropter baſis H G ad baſim C F erit reciprocè, vt al- <anchor type="note" xlink:label="note-0236-03a" xlink:href="note-0236-03"/> <pb o="53" file="0237" n="237" rhead=""/> titudo trianguli C B F ad altitudinem trianguli H E G, ſed horum triangu-<lb/>lorum altitudines eædem ſunt, ac altitudines portionum A B C, H E I, cum <lb/>puncta B, E ſint earundem portionum vertices; </s> <s xml:id="echoid-s6581" xml:space="preserve">quare vt baſis H G ad ba-<lb/>ſim C F, vel ſumptis duplis, vt H I baſis portionis H E I, ad A C baſim <lb/>portionis A B C, ita reciprocè altitudo portionis A B C ad altitudinem por-<lb/>tionis H E I, ſuntque huiuſmodi portiones <anchor type="note" xlink:href="" symbol="a"/> Acuminata regularia, & </s> <s xml:id="echoid-s6582" xml:space="preserve">pro- <anchor type="note" xlink:label="note-0237-01a" xlink:href="note-0237-01"/> portionalia, & </s> <s xml:id="echoid-s6583" xml:space="preserve">eorum baſes altitudinibus reciprocantur, quare ipſa Acumi-<lb/>nata, ſeu portiones H E I, A B C inter ſe ſunt <anchor type="note" xlink:href="" symbol="b"/> æquales. </s> <s xml:id="echoid-s6584" xml:space="preserve">Quod oſtendere <anchor type="note" xlink:label="note-0237-02a" xlink:href="note-0237-02"/> propoſitum fuit, quodque de ſola Parabola demonſtrauit Geometrarum <lb/>Princeps in 4. </s> <s xml:id="echoid-s6585" xml:space="preserve">Prop. </s> <s xml:id="echoid-s6586" xml:space="preserve">de Conoid. </s> <s xml:id="echoid-s6587" xml:space="preserve">ac Sphæroid. </s> <s xml:id="echoid-s6588" xml:space="preserve">ſuppoſita tamen eiuſdem Pa-<lb/>rabolę quadratura.</s> <s xml:id="echoid-s6589" xml:space="preserve"/> </p> <div xml:id="echoid-div683" type="float" level="2" n="7"> <note symbol="c" position="left" xlink:label="note-0236-03" xlink:href="note-0236-03a" xml:space="preserve">39. h.</note> <note symbol="a" position="right" xlink:label="note-0237-01" xlink:href="note-0237-01a" xml:space="preserve">36. h.</note> <note symbol="b" position="right" xlink:label="note-0237-02" xlink:href="note-0237-02a" xml:space="preserve">37. h.</note> </div> </div> <div xml:id="echoid-div685" type="section" level="1" n="272"> <head xml:id="echoid-head281" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s6590" xml:space="preserve">HInc eſt, quod applicatæ ex terminis æqualium diametrorum in Para-<lb/>bola, vel ex punctis, in reliquis ſectionibus, proportionaliter diuidẽ-<lb/>tibus ſemi-diametros ad angulum conſtitutas, omnino ſe mutuò ſecant; </s> <s xml:id="echoid-s6591" xml:space="preserve">& </s> <s xml:id="echoid-s6592" xml:space="preserve"><lb/>quod rectæ lineę, tùm harum applicatarum puncta media, tùm extrema iun-<lb/>gentes, rectæ ſemi-diametrorum terminos iungenti æquidiſtant. </s> <s xml:id="echoid-s6593" xml:space="preserve">Demon-<lb/>ſtratum eſt enim H I, A C ſecare ſe mutuò in M, & </s> <s xml:id="echoid-s6594" xml:space="preserve">iunctas H C, G F, A I <lb/>ipſi E B eſſe parallelas.</s> <s xml:id="echoid-s6595" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div686" type="section" level="1" n="273"> <head xml:id="echoid-head282" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s6596" xml:space="preserve">PAtet quoq; </s> <s xml:id="echoid-s6597" xml:space="preserve">in quarta, quinta, ſeptima, & </s> <s xml:id="echoid-s6598" xml:space="preserve">octaua figura, portiones eiuſ-<lb/>dem Ellipſis, vel circuli, quarum baſes tranſeant per puncta earum ſe-<lb/>mi-diametros proportionaliter ſecantia, etiam ſi ipſæ ſemi-diametri ſint in <lb/>directum poſitæ, hoc eſt applicatæ inter ſe æquidiſtent, eſſe quoque inter ſe <lb/>æquales. </s> <s xml:id="echoid-s6599" xml:space="preserve">Vtra enim talium portionum æqualis demonſtratur, (vt in ſupe-<lb/>riori propoſitione) ei portioni, cuius baſis ſit applicata per punctum propor-<lb/>tionaliter ſecans aliam ſemi-diametrum, quæ cum prædictis angulum con-<lb/>ſtituat.</s> <s xml:id="echoid-s6600" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div687" type="section" level="1" n="274"> <head xml:id="echoid-head283" xml:space="preserve">COROLL. III.</head> <p> <s xml:id="echoid-s6601" xml:space="preserve">EX ijſdem conſtat, quod ſi quotcunque applicatæ in eadem Ellipſi, vel <lb/>circulo integras diametros proportionaliter ſecent, abſciſſæ portiones <lb/>viciſſim æquales erunt, hoc eſt minor minori, & </s> <s xml:id="echoid-s6602" xml:space="preserve">maior maiori.</s> <s xml:id="echoid-s6603" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6604" xml:space="preserve">Si enim in prædictis figuris ſint duæ diametri B R E L, ita ſectæ in F, G; <lb/></s> <s xml:id="echoid-s6605" xml:space="preserve">vt R F ad F B ſit vt L G ad G E, erit componendo, & </s> <s xml:id="echoid-s6606" xml:space="preserve">ſumptis antece-<lb/>dentium ſubduplis D B ad B F, vt D E ad E G; </s> <s xml:id="echoid-s6607" xml:space="preserve">applicatis ergo A F C, <lb/>H G I erunt portiones A B C, H E I inter ſe æquales, & </s> <s xml:id="echoid-s6608" xml:space="preserve">reliqua portio <lb/>A R C reliquæ portioni H R I æqualis erit.</s> <s xml:id="echoid-s6609" xml:space="preserve"/> </p> <pb o="54" file="0238" n="238" rhead=""/> </div> <div xml:id="echoid-div688" type="section" level="1" n="275"> <head xml:id="echoid-head284" xml:space="preserve">PROBL. VI. PROP. XXXXI.</head> <p> <s xml:id="echoid-s6610" xml:space="preserve">Per datum punctum in angulo rectilineo, rectam applicare, <lb/>quæ de angulo abſcindat triangulum MINIMVM.</s> <s xml:id="echoid-s6611" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6612" xml:space="preserve">ESto angulus rectilineus A B C, in quo datum ſit punctum D. </s> <s xml:id="echoid-s6613" xml:space="preserve">Oportet ex <lb/>D rectam applicare, quæ ab angulo auferat triangulum _MINIMVM_.</s> <s xml:id="echoid-s6614" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6615" xml:space="preserve">Iungatur diameter B D, ad quàm applicetur per D recta A D C, quæ in <lb/>dato puncto D <anchor type="note" xlink:href="" symbol="a"/> bifariam ſecetur. </s> <s xml:id="echoid-s6616" xml:space="preserve">Dico hanc ipſam quæſitum ſoluere, hoc <anchor type="note" xlink:label="note-0238-01a" xlink:href="note-0238-01"/> eſt triangulum A B C eſſe _MINIMVM_.</s> <s xml:id="echoid-s6617" xml:space="preserve"/> </p> <div xml:id="echoid-div688" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0238-01" xlink:href="note-0238-01a" xml:space="preserve">ex 66. 1. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s6618" xml:space="preserve">Ducatur quælibet alia E D F, & </s> <s xml:id="echoid-s6619" xml:space="preserve">ab extremo <lb/> <anchor type="figure" xlink:label="fig-0238-01a" xlink:href="fig-0238-01"/> applicatæ A C, quod cadit ſupra E F, ſiue ex <lb/>puncto C agatur C G ipſi E A ęquidiſtans. </s> <s xml:id="echoid-s6620" xml:space="preserve">Et <lb/>cum ſit A D æqualis D C, ob conſtructionem, <lb/>erit quoque E D ęqualis D G, & </s> <s xml:id="echoid-s6621" xml:space="preserve">angulus A D E <lb/>ęquatur angulo C D G, ergo triangulum A D E, <lb/>triangulo C D G ęquale erit, ac ideò A D E mi-<lb/>nus triangulo C D F; </s> <s xml:id="echoid-s6622" xml:space="preserve">ſi ergo addatur commune <lb/>trapetium B E D C, erit triangulum A B C mi-<lb/>nus triangulo E B F, & </s> <s xml:id="echoid-s6623" xml:space="preserve">hoc ſemper: </s> <s xml:id="echoid-s6624" xml:space="preserve">quare trian-<lb/>gulum A B C eſt _MINIMVM_. </s> <s xml:id="echoid-s6625" xml:space="preserve">Quod reperien-<lb/>dum erat.</s> <s xml:id="echoid-s6626" xml:space="preserve"/> </p> <div xml:id="echoid-div689" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0238-01"/> </figure> </div> </div> <div xml:id="echoid-div691" type="section" level="1" n="276"> <head xml:id="echoid-head285" xml:space="preserve">PROBL. VII. PROP. XXXXII.</head> <p> <s xml:id="echoid-s6627" xml:space="preserve">Per datum punctum intra coni-ſectionem, vel circulum rectam <lb/>applicare, quæ de ipſa auferat portionem MINIMAM.</s> <s xml:id="echoid-s6628" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6629" xml:space="preserve">ESto A B C data Parabole, vt in prima figura, vel Hyperbole, vt in ſe-<lb/>cunda, aut Ellipſis, vel circulus, vt in tertia, quarum centrum H, & </s> <s xml:id="echoid-s6630" xml:space="preserve"><lb/>punctum intra datum ſit D. </s> <s xml:id="echoid-s6631" xml:space="preserve">Oportet per D rectam applicare, quæ de ſe-<lb/>ctione abſcindat portionem _MINIMAM_.</s> <s xml:id="echoid-s6632" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6633" xml:space="preserve">Ducatur H B D ſectionis diameter tranſiens per datum punctum D, per <lb/>quod ei ordinatim applicetur recta A D C. </s> <s xml:id="echoid-s6634" xml:space="preserve">Dico portionem A B C eſſe _MI-_ <lb/>_NIMAM_ quæſitam.</s> <s xml:id="echoid-s6635" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6636" xml:space="preserve">Nam applicata per D in ſectione qualibet alia E D F, cum ipſa E F alte-<lb/>ram applicatam A C in ſectione bifariam ſecet in D, ipſæ ſe mutuò bifariam <lb/>non ſecabunt, per 6. </s> <s xml:id="echoid-s6637" xml:space="preserve">ſecundi conicorum, quæ licet de ſola Ellipſi, vel circu-<lb/>lo agat, verificatur quoque de quacunque data coni-ſectione. </s> <s xml:id="echoid-s6638" xml:space="preserve">Secetur er-<lb/>go E F bifariam in G, per quod ducatur eius diameter G I H ſectioni oc-<lb/>currens in I, per quod agatur ſectionem contingens IL, quæ ipſi E G F æ-<lb/>quidiſtabit, <anchor type="note" xlink:href="" symbol="b"/> quare ſi iungatur I B, cum ipſa tota cadat <anchor type="note" xlink:href="" symbol="c"/> intra ſectionem, &</s> <s xml:id="echoid-s6639" xml:space="preserve"> <anchor type="note" xlink:label="note-0238-02a" xlink:href="note-0238-02"/> <anchor type="note" xlink:label="note-0238-03a" xlink:href="note-0238-03"/> alteram parallelarum L I ſecet in I, producta ad partes B, conueniet cum <lb/>reliqua producta F D E ad partes E, ac ideò D M, quæ ex D ducitur ipſi <lb/>B I æquidiſtans cadet ſupra D F, ſecabitque diametrum I G, vt in M, cui <pb o="55" file="0239" n="239" rhead=""/> per M applicetur recta N M O, quæ applicatæ E G F æquidiſtabit.</s> <s xml:id="echoid-s6640" xml:space="preserve"/> </p> <div xml:id="echoid-div691" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-0238-02" xlink:href="note-0238-02a" xml:space="preserve">5. ſecun-<lb/>di conic.</note> <note symbol="c" position="left" xlink:label="note-0238-03" xlink:href="note-0238-03a" xml:space="preserve">10. primi <lb/>conic.</note> </div> <p> <s xml:id="echoid-s6641" xml:space="preserve">Iam, in prima figura, cum ſit B D parallela ad I M, & </s> <s xml:id="echoid-s6642" xml:space="preserve">B I ad D M, erit <lb/>diametri ſegmentum B D æquale diametri ſegmento I M; </s> <s xml:id="echoid-s6643" xml:space="preserve">ſuntque ex D, M <lb/>applicatæ diametris rectæ A D C, N M O, vnde portiones A B C, N I O <lb/>æquales <anchor type="note" xlink:href="" symbol="a"/> erunt.</s> <s xml:id="echoid-s6644" xml:space="preserve"/> </p> <note symbol="a" position="right" xml:space="preserve">40. h.</note> <figure> <image file="0239-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0239-01"/> </figure> <p> <s xml:id="echoid-s6645" xml:space="preserve">In reliquis verò, cum in triangulo D H M ſit B I parallela ad D M, erit <lb/>H B ad B D, vt H I ad I M, ſuntque ex D, M applicatæ diametris rectæ <lb/>A D C, N M O, quare portiones A B C, N I O æquales <anchor type="note" xlink:href="" symbol="b"/> erunt. </s> <s xml:id="echoid-s6646" xml:space="preserve">Cum er- <anchor type="note" xlink:label="note-0239-02a" xlink:href="note-0239-02"/> go in ſingulis figuris portio A B C demonſtrata ſit æqualis portioni N I O, <lb/>& </s> <s xml:id="echoid-s6647" xml:space="preserve">ſit portio N I O minor portione E I F, pars toto, ergo portio A B C erit <lb/>quoque minor portione E I F, & </s> <s xml:id="echoid-s6648" xml:space="preserve">ſic quacunque alia portione, ab applicata <lb/>per D abſciſſa, minor demonſtrabitur. </s> <s xml:id="echoid-s6649" xml:space="preserve">Vnde portio A B C eſt _MINIMA_ <lb/>quæſita. </s> <s xml:id="echoid-s6650" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s6651" xml:space="preserve"/> </p> <div xml:id="echoid-div692" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0239-02" xlink:href="note-0239-02a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div694" type="section" level="1" n="277"> <head xml:id="echoid-head286" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s6652" xml:space="preserve">HInc eſt, quod dum per datum punctum D intra Ellipſim, ducitur appli-<lb/>cata A D C _MINIMAM_ portionem abſcindes, habetur ſimul _MA_-<lb/>_XIMA_ portio, quæ eſt reliqua A P C, vt per ſe ſatis conſtat.</s> <s xml:id="echoid-s6653" xml:space="preserve"/> </p> <pb o="56" file="0240" n="240" rhead=""/> </div> <div xml:id="echoid-div695" type="section" level="1" n="278"> <head xml:id="echoid-head287" xml:space="preserve">THEOR. XXIV. PROP. XXXXIII.</head> <p> <s xml:id="echoid-s6654" xml:space="preserve">In congruentibus Parabolis per diuerſos vertices ſimul adſcri-<lb/>ptis, intercepta communium diametrorum ſegmenta inter ſe ſunt <lb/>æqualia, & </s> <s xml:id="echoid-s6655" xml:space="preserve">huiuſmodi Parabolæ dicantur ęquidiſtantes. </s> <s xml:id="echoid-s6656" xml:space="preserve">Contin-<lb/>gentes verò vtranq; </s> <s xml:id="echoid-s6657" xml:space="preserve">ſectionem ad terminos eiuſdem diametri inter <lb/>ſe æquidiſtant.</s> <s xml:id="echoid-s6658" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6659" xml:space="preserve">SInt duæ congruentes Parabolæ A B C, D E F per diuerſos vertices B, E <lb/>ſimul adſcriptæ circa communem diametrum B E H, & </s> <s xml:id="echoid-s6660" xml:space="preserve">inter ipſas du-<lb/>cta ſit quæcunque alia A D ipſi B E parallela, (quæ vtrique Parabolæ con-<lb/>ueniet <anchor type="note" xlink:href="" symbol="a"/> in A, D eritque earum communis <anchor type="note" xlink:href="" symbol="b"/> diameter) atque ex terminis A, <anchor type="note" xlink:label="note-0240-01a" xlink:href="note-0240-01"/> <anchor type="note" xlink:label="note-0240-02a" xlink:href="note-0240-02"/> D, agantur A I, D L Parabolas contingentes in A, D, & </s> <s xml:id="echoid-s6661" xml:space="preserve">communi diame-<lb/>tro B E occurrentes <anchor type="note" xlink:href="" symbol="c"/> in I, L. </s> <s xml:id="echoid-s6662" xml:space="preserve">Dico diametrorum intercepta ſegmenta B E, <anchor type="note" xlink:label="note-0240-03a" xlink:href="note-0240-03"/> A D æqualia eſſe, & </s> <s xml:id="echoid-s6663" xml:space="preserve">contingentes A I, D L inter ſe æquidiſtare.</s> <s xml:id="echoid-s6664" xml:space="preserve"/> </p> <div xml:id="echoid-div695" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0240-01" xlink:href="note-0240-01a" xml:space="preserve">26. primi <lb/>conic.</note> <note symbol="b" position="left" xlink:label="note-0240-02" xlink:href="note-0240-02a" xml:space="preserve">46. ibid.</note> <note symbol="c" position="left" xlink:label="note-0240-03" xlink:href="note-0240-03a" xml:space="preserve">24. ibid.</note> </div> <p> <s xml:id="echoid-s6665" xml:space="preserve">Nam primum patet ex primo Coroll. </s> <s xml:id="echoid-s6666" xml:space="preserve">42. <lb/></s> <s xml:id="echoid-s6667" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0240-01a" xlink:href="fig-0240-01"/> primi huius: </s> <s xml:id="echoid-s6668" xml:space="preserve">cumq; </s> <s xml:id="echoid-s6669" xml:space="preserve">omnes interceptæ B E, <lb/>A D, &</s> <s xml:id="echoid-s6670" xml:space="preserve">c. </s> <s xml:id="echoid-s6671" xml:space="preserve">ſint æquales vocentur, huiuſmodi <lb/>Parabolæ inter ſe ęquidiſtantes. </s> <s xml:id="echoid-s6672" xml:space="preserve">Secundum <lb/>verò, ita oſtenditur.</s> <s xml:id="echoid-s6673" xml:space="preserve"/> </p> <div xml:id="echoid-div696" type="float" level="2" n="2"> <figure xlink:label="fig-0240-01" xlink:href="fig-0240-01a"> <image file="0240-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0240-01"/> </figure> </div> <p> <s xml:id="echoid-s6674" xml:space="preserve">Applicentur ex A, D ad diametrum B H <lb/>rectæ A G, D H; </s> <s xml:id="echoid-s6675" xml:space="preserve">erit A H parallelogram-<lb/>mum, ex quo G H æqualis erit A D, ſiue <lb/>ipſi B E, quare dempta, vel addita, vti opus <lb/>fuerit, communi G E, proueniet B G ęqua-<lb/>lis H E, & </s> <s xml:id="echoid-s6676" xml:space="preserve">dupla <anchor type="note" xlink:href="" symbol="d"/> I G duplæ L H æqualis <anchor type="note" xlink:label="note-0240-04a" xlink:href="note-0240-04"/> erit, & </s> <s xml:id="echoid-s6677" xml:space="preserve">eſt G A æqualis H D, & </s> <s xml:id="echoid-s6678" xml:space="preserve">angulus <lb/>I G A angulo L H D æqualis, ergo angu-<lb/>lus quoque G I A angulo H L D æqualis erit. </s> <s xml:id="echoid-s6679" xml:space="preserve">Quare contingentes A I, D <lb/>L inter ſe æquidiſtant. </s> <s xml:id="echoid-s6680" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s6681" xml:space="preserve">c.</s> <s xml:id="echoid-s6682" xml:space="preserve"/> </p> <div xml:id="echoid-div697" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-0240-04" xlink:href="note-0240-04a" xml:space="preserve">35. ibid.</note> </div> </div> <div xml:id="echoid-div699" type="section" level="1" n="279"> <head xml:id="echoid-head288" xml:space="preserve">THEOR. XXV. PROP. XXXXIV.</head> <p> <s xml:id="echoid-s6683" xml:space="preserve">In Hyperbolis, aut Ellipſibus ſimilibus, & </s> <s xml:id="echoid-s6684" xml:space="preserve">concentricis, per <lb/>diuerſos vertices ſimul adſcriptis, intercepta communium dia-<lb/>metrorum ſegmenta ad proprias ſemi-diametros vnam eandem-<lb/>que habent rationem, & </s> <s xml:id="echoid-s6685" xml:space="preserve">quæ ſectiones contingunt ad terminos <lb/>eiuſdem diametri inter ſe æquidiſtant.</s> <s xml:id="echoid-s6686" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6687" xml:space="preserve">SInt duæ Hyperbolæ ſimiles in prima figura, vel duæ ſimiles Ellipſes in ſe-<lb/>cunda, quarum commune centrum ſit G, & </s> <s xml:id="echoid-s6688" xml:space="preserve">communis ſemi- diameter <lb/>G B E, ſitque ducta quæcumque alia G A D, (quæ tamen in Ellipſi cadat in-<lb/>ter coniugatas ſemi-diametros G E, G N) eritque G A D, <anchor type="note" xlink:href="" symbol="e"/> item commu- <anchor type="note" xlink:label="note-0240-05a" xlink:href="note-0240-05"/> nis ſectionum ſemi-diameter, ducãturque A L, D M ad terminos A, D ſe- <pb o="57" file="0241" n="241" rhead=""/> ctiones contingentes, quæ productæ, communi diametro G B E <anchor type="note" xlink:href="" symbol="a"/> occurent <anchor type="note" xlink:label="note-0241-01a" xlink:href="note-0241-01"/> in L, M. </s> <s xml:id="echoid-s6689" xml:space="preserve">Dico primùm G A ad A D eſſe vt G B ad B E, & </s> <s xml:id="echoid-s6690" xml:space="preserve">contingentes <lb/>A L, D L inter ſe æquidiſtare.</s> <s xml:id="echoid-s6691" xml:space="preserve"/> </p> <div xml:id="echoid-div699" type="float" level="2" n="1"> <note symbol="e" position="left" xlink:label="note-0240-05" xlink:href="note-0240-05a" xml:space="preserve">47. ibid.</note> <note symbol="a" position="right" xlink:label="note-0241-01" xlink:href="note-0241-01a" xml:space="preserve">24. 25. <lb/>primi co-<lb/>nic.</note> </div> <p> <s xml:id="echoid-s6692" xml:space="preserve">Applicentur ex A, D ad diametrum communem G B M rectę A I, D H. <lb/></s> <s xml:id="echoid-s6693" xml:space="preserve">Erit iam in ſectione D E F, rectangulum G H M ad quadratum H D, <anchor type="note" xlink:href="" symbol="b"/> vt <anchor type="note" xlink:label="note-0241-02a" xlink:href="note-0241-02"/> tranſuerſum ad rectum, vel, ob ſectionum ſimilitudinem, vt tranſuerſum ſe-<lb/>ctionis A B C ad eius rectum, vel vt rectangulum G I L ad quadratum I A, <lb/>& </s> <s xml:id="echoid-s6694" xml:space="preserve">quadratum D H ad H G, eſt vt quadratum A I ad I G, ergo ex æquo <lb/>rectangulum G H M ad quadratum G H, erit vt rectangulum G I L ad qua-<lb/>dratum I G, & </s> <s xml:id="echoid-s6695" xml:space="preserve">conuertendo quadratum G H ad rectangulum G H M, vt <lb/>quadratum I G ad rectangulum G I L, & </s> <s xml:id="echoid-s6696" xml:space="preserve">per conuerſionem rationis in pri-<lb/>ma figura, & </s> <s xml:id="echoid-s6697" xml:space="preserve">componendo in ſecunda, quadratum G H ad rectangulum. <lb/></s> <s xml:id="echoid-s6698" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0241-01a" xlink:href="fig-0241-01"/> H G M, vt quadratum I G ad rectangulum I G L, & </s> <s xml:id="echoid-s6699" xml:space="preserve">permutando quadra-<lb/>tum H G ad G I, vel quadratum D G ad G A, erit vt rectangulum H G M <lb/>ad rectangulum I G L, vel permutatis æqualibus, <anchor type="note" xlink:href="" symbol="c"/> vt quadratum E G ad <anchor type="note" xlink:label="note-0241-03a" xlink:href="note-0241-03"/> quadratum G B, ſeulinea D G ad G A, vt linea E G ad G B, & </s> <s xml:id="echoid-s6700" xml:space="preserve">diuiden-<lb/>do, & </s> <s xml:id="echoid-s6701" xml:space="preserve">conuertendo G A ad A D, vt G B ad B E. </s> <s xml:id="echoid-s6702" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s6703" xml:space="preserve">c.</s> <s xml:id="echoid-s6704" xml:space="preserve"/> </p> <div xml:id="echoid-div700" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0241-02" xlink:href="note-0241-02a" xml:space="preserve">37. ibid.</note> <figure xlink:label="fig-0241-01" xlink:href="fig-0241-01a"> <image file="0241-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0241-01"/> </figure> <note symbol="c" position="right" xlink:label="note-0241-03" xlink:href="note-0241-03a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s6705" xml:space="preserve">Præterea, cum ſuperiùs demonſtratum ſit eſſe rectangulum G H M ad <lb/>quadratum H D, vt rectangulum G I L ad quadratum I A, erit permutan-<lb/>do rectangulum G H M ad G I L, vt quadratum H D ad I A; </s> <s xml:id="echoid-s6706" xml:space="preserve">ſed propor-<lb/>tio quadrati H D ad I A componitur ex du@bus rationibus H D ad I A, <lb/>vel ex duobus rationibus H G ad G I, & </s> <s xml:id="echoid-s6707" xml:space="preserve">proportio rectanguli G H M ad <lb/>G I L componitur ex duobus rationibus, nempe ex G H ad G I, & </s> <s xml:id="echoid-s6708" xml:space="preserve">ex H M <lb/>ad I L; </s> <s xml:id="echoid-s6709" xml:space="preserve">ergo proportio G H ad G I, hoc eſt H D ad I A, æqualis eſt pro-<lb/>portioni H M ad I I.</s> <s xml:id="echoid-s6710" xml:space="preserve">, & </s> <s xml:id="echoid-s6711" xml:space="preserve">permutando D H ad H M, erit vt A I ad I L, &</s> <s xml:id="echoid-s6712" xml:space="preserve"> <pb o="58" file="0242" n="242" rhead=""/> anguli ad H, I ſunt æquales, ergo triangula D H M, A I L ſunt æquiangu-<lb/>la, hoc eſt angulus D M H æqualis erit angulo A L I, ac ideo D M, A L <lb/>inter ſe æquidiſtant. </s> <s xml:id="echoid-s6713" xml:space="preserve">Quod vltimò demonſtrandum erat.</s> <s xml:id="echoid-s6714" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div702" type="section" level="1" n="280"> <head xml:id="echoid-head289" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s6715" xml:space="preserve">PRoportionalitas, quàm primo loco ſuperioris theorematis inter ſemi-<lb/>diametros concentricorum quadrantum N G E, O G B ſimilium Elli-<lb/>pſium inuenimus, eadem penitùs reperietur in alijs deinceps quadrantibus, <lb/>& </s> <s xml:id="echoid-s6716" xml:space="preserve">ad verticem, vt per ſe ſatis patet.</s> <s xml:id="echoid-s6717" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div703" type="section" level="1" n="281"> <head xml:id="echoid-head290" xml:space="preserve">THEOR. XXVI. PROP. XLV.</head> <p> <s xml:id="echoid-s6718" xml:space="preserve">In Hyperbola intra angulum aſymptotalem; </s> <s xml:id="echoid-s6719" xml:space="preserve">vel in Parabolis <lb/>parallelis, ſiue in Hyperbolis, aut Ellipſibus ſimilibus, & </s> <s xml:id="echoid-s6720" xml:space="preserve">concen-<lb/>tricis circa eandem diametrum per diuerſos vertices ſimul adſcri-<lb/>ptis, portiones omnes anguli, vel exterioris ſectionis, quarum ba-<lb/>ſes interiorem ſectionem contingant, inter ſe ſunt æquales.</s> <s xml:id="echoid-s6721" xml:space="preserve"/> </p> <figure> <image file="0242-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0242-01"/> </figure> <p> <s xml:id="echoid-s6722" xml:space="preserve">SIt intra angulum aſymptotalem A B C deſcripta Hyperbole D E F, vt <lb/>in prima figura, vel duæ æquidiſtantes Parabolæ A B C, D E F, vt in <lb/>ſecunda; </s> <s xml:id="echoid-s6723" xml:space="preserve">vel ſimiles concentricæ Hyperbolæ, vt in tertia, aut Ellipſes, vt in <lb/>quarta, quarum commune centrum ſit G, ac omnes per diuerſos vertices <lb/>B, E ſint ſimul adſcriptæ circa eandem diametrum G B E, & </s> <s xml:id="echoid-s6724" xml:space="preserve">ad verticem E <lb/>interiorem ſectionem contingat recta A E C, & </s> <s xml:id="echoid-s6725" xml:space="preserve">ad quodcunque aliud pun- <pb o="59" file="0243" n="243" rhead=""/> ctum D contingat eandem recta H D I. </s> <s xml:id="echoid-s6726" xml:space="preserve">Dico ipſas contingentes exteriori <lb/>ſectioni ad vtranque partem occurrere, ac de ea æquales portiones abſcin-<lb/>dere.</s> <s xml:id="echoid-s6727" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6728" xml:space="preserve">Nam ductis diametris G B E, G M D; </s> <s xml:id="echoid-s6729" xml:space="preserve">cum in prima figura rectæ A E C, <lb/>H D I Hyperbolen contingant in E, D, ipſæ productæ cum vtraque aſym-<lb/>ptoto conuenient in A, C, & </s> <s xml:id="echoid-s6730" xml:space="preserve">in H, I, atque <anchor type="note" xlink:href="" symbol="a"/> bifariam ſecabuntur in E, <anchor type="note" xlink:label="note-0243-01a" xlink:href="note-0243-01"/> D, à quibus ſi ducantur aſymptotis æquidiſtantes E N, E O, & </s> <s xml:id="echoid-s6731" xml:space="preserve">D P, D Q, <lb/>erit rectangulum N E O ęquale <anchor type="note" xlink:href="" symbol="b"/> rectangulo P D Q, ſiue parallelogrammum <anchor type="note" xlink:label="note-0243-02a" xlink:href="note-0243-02"/> N O ęquale ſibi æquiangulo parallelogrammo P Q, & </s> <s xml:id="echoid-s6732" xml:space="preserve">duplum duplo ęqua-<lb/>le erit, hoc eſt triangulum A B C, triangulo H B I (cum A C, H I ſint bi-<lb/>fariam ſectæ in E, D.)</s> <s xml:id="echoid-s6733" xml:space="preserve"/> </p> <div xml:id="echoid-div703" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0243-01" xlink:href="note-0243-01a" xml:space="preserve">3 <lb/>dic</note> <note symbol="b" position="right" xlink:label="note-0243-02" xlink:href="note-0243-02a" xml:space="preserve">12. ibid.</note> </div> <p> <s xml:id="echoid-s6734" xml:space="preserve">In reliquis verò figuris cum A E C contingat in E interiorem ſectionem <lb/>D E F, ipſa æquidiſtabit <anchor type="note" xlink:href="" symbol="c"/> contingenti ex B exteriorem, ac ideo erit vna <anchor type="note" xlink:label="note-0243-03a" xlink:href="note-0243-03"/> applicatarum ad diametrum G B E in exteriori ſectione A B C, & </s> <s xml:id="echoid-s6735" xml:space="preserve">bifariam <lb/>ſecabitur in E. </s> <s xml:id="echoid-s6736" xml:space="preserve">Eadem ratione contingens H D I erit vna applicatarum ad <lb/>diametrum G M D in exteriori, & </s> <s xml:id="echoid-s6737" xml:space="preserve">bifariam ſecabitur in D, eritque in ſe-<lb/>cunda figura ſegmentum diametri B E æquale ſegmento M D, & </s> <s xml:id="echoid-s6738" xml:space="preserve">in tertia <lb/>habebit <anchor type="note" xlink:href="" symbol="d"/> G B ad B E eandem rationem, ac G M ad M D, in quarta de- <anchor type="note" xlink:label="note-0243-04a" xlink:href="note-0243-04"/> nique G E ad E B eandem, ac G D ad D M: </s> <s xml:id="echoid-s6739" xml:space="preserve">quare portiones A B C, H <lb/>M I exterioris ſectionis A B C, quarum baſes contingunt interiorem D E F <lb/>inter ſe ſunt <anchor type="note" xlink:href="" symbol="e"/> æquales. </s> <s xml:id="echoid-s6740" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s6741" xml:space="preserve"/> </p> <div xml:id="echoid-div704" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0243-03" xlink:href="note-0243-03a" xml:space="preserve">43. 44. h.</note> <note symbol="d" position="right" xlink:label="note-0243-04" xlink:href="note-0243-04a" xml:space="preserve">ibidem.</note> </div> <note symbol="e" position="right" xml:space="preserve">40. h.</note> </div> <div xml:id="echoid-div706" type="section" level="1" n="282"> <head xml:id="echoid-head291" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s6742" xml:space="preserve">HInc eſt, quod contingentes ad puncta interioris concentricæ ſectio-<lb/>nis, exteriori ſemper ad vtranque partem occurrunt, & </s> <s xml:id="echoid-s6743" xml:space="preserve">à tactibus <lb/>bifariam ſecantur.</s> <s xml:id="echoid-s6744" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div707" type="section" level="1" n="283"> <head xml:id="echoid-head292" xml:space="preserve">THEOR. XXVII. PROP. XLVI.</head> <p> <s xml:id="echoid-s6745" xml:space="preserve">Si in Parabolis parallelis, vel in Hyperbolis, aut circulis, ſiue in <lb/>Ellipſibus ſimilibus, & </s> <s xml:id="echoid-s6746" xml:space="preserve">concentricis ad punctum quodlibet interio-<lb/>ris ſectionis, quædam recta linea contingat, cui ducta ſit quęcunq; <lb/></s> <s xml:id="echoid-s6747" xml:space="preserve">alia æquidiſtans, vtranque ſectionem ſecans, erit rectangulum <lb/>ſub ſegmentis huiuſmodi applicatę inter vtranque ſectionem in-<lb/>terceptis, æquale quadrato ſemi-tangentis.</s> <s xml:id="echoid-s6748" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6749" xml:space="preserve">SInt due Parabolę ęquidiſtãtes, vt in prima figura, vel ſimiles, & </s> <s xml:id="echoid-s6750" xml:space="preserve">concẽtricę <lb/>Hyperbolę, vt in ſecũda, aut Ellipſes, vel circuli, vt in tertia, A B C, D <lb/>E F, quarũ centrum, reſpectiuè ſit R, & </s> <s xml:id="echoid-s6751" xml:space="preserve">ad quodcunq; </s> <s xml:id="echoid-s6752" xml:space="preserve">punctum E interioris <lb/>ſit contingens recta A E C, (quæ ad vtranque partem exteriori <anchor type="note" xlink:href="" symbol="f"/> occurret <anchor type="note" xlink:label="note-0243-06a" xlink:href="note-0243-06"/> in A, C, & </s> <s xml:id="echoid-s6753" xml:space="preserve">à tactu E bifariam ſecabitur) eique ſit æquidiſtanter ducta <lb/>quælibet alia G D H, (quæ item ad vtranque partem exterioris occurret in <pb o="60" file="0244" n="244" rhead=""/> G, H cum ſit vna applicatarum, &</s> <s xml:id="echoid-s6754" xml:space="preserve">c.) </s> <s xml:id="echoid-s6755" xml:space="preserve">interiorem ſecans in D, F. </s> <s xml:id="echoid-s6756" xml:space="preserve">Dico re-<lb/>ctangulum ſub ſegmentis G D, D H æquari quadrato ſemi-tangentis A E.</s> <s xml:id="echoid-s6757" xml:space="preserve"/> </p> <div xml:id="echoid-div707" type="float" level="2" n="1"> <note symbol="f" position="right" xlink:label="note-0243-06" xlink:href="note-0243-06a" xml:space="preserve">Coroll. <lb/>45. h.</note> </div> <p> <s xml:id="echoid-s6758" xml:space="preserve">Nam iuncta D E, & </s> <s xml:id="echoid-s6759" xml:space="preserve">bifariam ſecta in N, ducatur eius diameter N O P, <lb/>quæ erit vtriuſque ſectionis diameter (cum ipſę ponantur parallelę, vel con-<lb/>centricæ) eas ſecans in O, P. </s> <s xml:id="echoid-s6760" xml:space="preserve">Patet, ſi ex O, P concipiantur contingentes <lb/>ſectiones O V, P Q has inter ſe <anchor type="note" xlink:href="" symbol="a"/> æquidiſtare, ſed O V ipſi D E ęquidiſtat, <anchor type="note" xlink:label="note-0244-01a" xlink:href="note-0244-01"/> cum hæc ſit vna applicatarum in ſectione D E F ad diametrum R N O, qua-<lb/>re, & </s> <s xml:id="echoid-s6761" xml:space="preserve">P Q ipſi D E æquidiſtabit, hoc eſt D E erit vna applicatarum in ſe-<lb/>ctione A B C ad diametrum R N P; </s> <s xml:id="echoid-s6762" xml:space="preserve">ex quo N E producta ad vtranque par-<lb/>tem exteriori ſectioni A B C occurret, vt in L, M, & </s> <s xml:id="echoid-s6763" xml:space="preserve">à diametro P N bifa-<lb/>riam ſecabitur in N, ſed D E quoque bifariam ſecta fuit in N, quare inter-<lb/>ceptæ L D, E M inter ſe ſunt æquales, hoc eſt rectangulum L D M æquale <lb/>eſt rectangulo L E M.</s> <s xml:id="echoid-s6764" xml:space="preserve"/> </p> <div xml:id="echoid-div708" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0244-01" xlink:href="note-0244-01a" xml:space="preserve">43. 44. <lb/>huius.</note> </div> <figure> <image file="0244-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0244-01"/> </figure> <p> <s xml:id="echoid-s6765" xml:space="preserve">Iam cum ſit applicata A C bifariam ſecta in E, ducta eius diametro B <lb/>E, hæc quoque bifariam ſecabit aliam applicatam G H, vt in I, eritque <lb/>etiam diameter ſectionis parallelæ, vel concentricæ D E F; </s> <s xml:id="echoid-s6766" xml:space="preserve">& </s> <s xml:id="echoid-s6767" xml:space="preserve">cum A C <lb/>contingat ſectionem D E F in E, ſitque D F ei æquidiſtans, hæc item bi-<lb/>fariam ſecabitur à diametro E I, vt in I. </s> <s xml:id="echoid-s6768" xml:space="preserve">Cum ſit ergo G I æqualis I H, & </s> <s xml:id="echoid-s6769" xml:space="preserve"><lb/>ablata D I æqualis ablatæ I F, erit reliqua G D reliquæ F H æqualis, ſiue <lb/>rectangulum G D H æquale rectangulo G F H.</s> <s xml:id="echoid-s6770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6771" xml:space="preserve">Tandem ex B ducatur contingens B Q alteri contingenti P Q conue-<lb/>niens in Q. </s> <s xml:id="echoid-s6772" xml:space="preserve">Erit ergo rectangulum G D H ad L D M, <anchor type="note" xlink:href="" symbol="b"/> vt quadratum B Q <anchor type="note" xlink:label="note-0244-02a" xlink:href="note-0244-02"/> ad P Q; </s> <s xml:id="echoid-s6773" xml:space="preserve">eademque ratione rectangulum A E C ad L E M, vt quadratum <lb/>B Q ad P Q: </s> <s xml:id="echoid-s6774" xml:space="preserve">quapropter rectangulum G D H ad L D M, erit vt A E C ad <lb/>L E M, & </s> <s xml:id="echoid-s6775" xml:space="preserve">permutando G D H ad A E C, vel ad quadratum A C, (cum <lb/>A E, E C ſint æquales) vt rectangulum L D M ad L E M, ſed L D M ipſi <lb/>L E M æquale oſtenſum fuit, quare, & </s> <s xml:id="echoid-s6776" xml:space="preserve">rectangulum G D H, vel G F H <lb/>æquale erit quadrato ſemi-tangentis A E. </s> <s xml:id="echoid-s6777" xml:space="preserve">Quod erat demonſtrandum:</s> <s xml:id="echoid-s6778" xml:space="preserve"> <pb o="61" file="0245" n="245" rhead=""/> quodque in parallelis Parabolis, ac ſimilibus concentricis Hyperbolis in 42. <lb/></s> <s xml:id="echoid-s6779" xml:space="preserve">& </s> <s xml:id="echoid-s6780" xml:space="preserve">47. </s> <s xml:id="echoid-s6781" xml:space="preserve">primi huius, ſed alijs aggreſſionibus oſtenſum fuit.</s> <s xml:id="echoid-s6782" xml:space="preserve"/> </p> <div xml:id="echoid-div709" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-0244-02" xlink:href="note-0244-02a" xml:space="preserve">17. tertij <lb/>conic.</note> </div> </div> <div xml:id="echoid-div711" type="section" level="1" n="284"> <head xml:id="echoid-head293" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s6783" xml:space="preserve">HInc eſt, quod in parallelis Parabolis, vel concentricis, ac ſimilibus <lb/>Hyperbolis, aut Ellipſibus, applicata in interiori ſectione hinc inde <lb/>producta exteriori neceſſariò occurrit, totaque ab illius diametro bifariam <lb/>ſecatur, & </s> <s xml:id="echoid-s6784" xml:space="preserve">quod huius applicatæ intercepta ſegmenta inter ſe ſunt æqualia.</s> <s xml:id="echoid-s6785" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6786" xml:space="preserve">Demonſtratum eſt enim applicatas D E, D F in interiori ſectioni D E F <lb/>exteriori A B C occurrere in L, M, & </s> <s xml:id="echoid-s6787" xml:space="preserve">in G, H, & </s> <s xml:id="echoid-s6788" xml:space="preserve">diametros O N, F I, <lb/>quæ bifariam ſecant D E, D F in N, I, bifariam quoque diuidere totas L <lb/>M, G H, atque interceptas portiones L D, E M inter ſe æquales eſſe, item-<lb/>que G D, F H æquales.</s> <s xml:id="echoid-s6789" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div712" type="section" level="1" n="285"> <head xml:id="echoid-head294" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s6790" xml:space="preserve">COnſtat etiam ex vltima parte huius Theorematis, quod, ſi in quacunq; <lb/></s> <s xml:id="echoid-s6791" xml:space="preserve">coni-ſectione, vel circulo duæ rectæ lineæ applicatæ fuerint inter ſe <lb/>æquidiſtantes, ad vtranque partem ſectioni occurrentes, quæ à tertia qua-<lb/>dam applicata vtcunque ſecentur, rectangula ſub ſegmentis æquidiſtantium <lb/>eandem inter ſe habere rationem, quam rectangula ſub ſegmentis tertiæ ſe-<lb/>cantis homologè ſumpta.</s> <s xml:id="echoid-s6792" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6793" xml:space="preserve">Ibi enim oſtenſum fuit tùm in Parabola, tùm in Hyperbola, aut Ellipſi, <lb/>vel circulo A B C, in quibus duę æquidiſtanter applicatæ A C, G H ſecan-<lb/>tur à tertia applicata L M in punctis E, D, rectangulum G D H ad A E C, <lb/>eſſe vt rectangulum L D M ad L E M.</s> <s xml:id="echoid-s6794" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div713" type="section" level="1" n="286"> <head xml:id="echoid-head295" xml:space="preserve">THEOR. XXVIII. PROP. XLVII.</head> <p> <s xml:id="echoid-s6795" xml:space="preserve">In Hyperbola intra angulum aſymptotalem deſcripta, vel in <lb/>æquidiſtantibus Parabolis, aut ſimilibus concentricis Hyperbolis, <lb/>aut Ellipſibus, rectarum in exteriori applicatarum, ac interiorem <lb/>ſectionem contingentium, MINIMA eſt ea, quæ ad verticem <lb/>maioris axis ducitur. </s> <s xml:id="echoid-s6796" xml:space="preserve">At in Ellipſibus, MAXIMA eſt quæ ad ver-<lb/>ticem minoris axis.</s> <s xml:id="echoid-s6797" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6798" xml:space="preserve">ESto, in prima figura, in angulo aſymptotali A B C deſcripta Hyperbole <lb/>D E F, cuius axis B E G, vel in ſecunda, ſint duæ Parabolæ æquidi-<lb/>ſtantes, vel duæ ſimiles concentricæ Hyperbolæ A B C, D E F circa axim <lb/>B E; </s> <s xml:id="echoid-s6799" xml:space="preserve">aut in tertia, duæ ſimiles concentricæ Ellipſes A B C, D E F, ſitque <lb/>exterioris ſectionis axis maior B P N, minor O P Q, & </s> <s xml:id="echoid-s6800" xml:space="preserve">in interiori ſit maior <pb o="62" file="0246" n="246" rhead=""/> E P K, minor S P Y, & </s> <s xml:id="echoid-s6801" xml:space="preserve">in quauis figura ad E verticem màioris axis interio-<lb/>rem ſectionem contingat recta A E C, quæ ad vtranque partem exterioris <lb/>pertinget, <anchor type="note" xlink:href="" symbol="a"/> ac bifariam ſecabitur in E. </s> <s xml:id="echoid-s6802" xml:space="preserve">Dico ipſam A E C eſſe _MINIMAM_ <anchor type="note" xlink:label="note-0246-01a" xlink:href="note-0246-01"/> exteriori ſectioni applicatarum, atque interiorem contingentium. </s> <s xml:id="echoid-s6803" xml:space="preserve">Et in El-<lb/>lipſibus contingentem R S T ad verticem minoris axis eſſe _MAXIMAM_.</s> <s xml:id="echoid-s6804" xml:space="preserve"/> </p> <div xml:id="echoid-div713" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0246-01" xlink:href="note-0246-01a" xml:space="preserve">Coroll. <lb/>45. huius.</note> </div> <p> <s xml:id="echoid-s6805" xml:space="preserve">Sit quæcunque alia contingens L D M ad punctum D, quæ item exte-<lb/>riori ſectioui occurret in L, M, <anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s6806" xml:space="preserve">bifariam ſecabitur in D, & </s> <s xml:id="echoid-s6807" xml:space="preserve">per D aga- <anchor type="note" xlink:label="note-0246-02a" xlink:href="note-0246-02"/> tur H D I ipſi A E C æquidiſtans, exteriori occurrens in H, I. </s> <s xml:id="echoid-s6808" xml:space="preserve">Et cum in <lb/>ſectione A B C per punctum D intra ipſam ſumptum, ſint duæ H D I, L D <lb/>M, quarum prima maiori axi B G eſt perpendicularis, altera verò inclina-<lb/>ta, erit rectangulum H D I minus rectangulo L D M, (cum ipſum H D I ſit <lb/>_MINIMVM_ <anchor type="note" xlink:href="" symbol="c"/>) ſed H D I æquatur <anchor type="note" xlink:href="" symbol="d"/> quadrato A E, ergo quadratum A E <anchor type="note" xlink:label="note-0246-03a" xlink:href="note-0246-03"/> <anchor type="note" xlink:label="note-0246-04a" xlink:href="note-0246-04"/> <anchor type="figure" xlink:label="fig-0246-01a" xlink:href="fig-0246-01"/> minus erit rectangulo L D M, ſiue quadrato L D, & </s> <s xml:id="echoid-s6809" xml:space="preserve">quadruplum quadru-<lb/>plo minus, hoc eſt quadratum A C minus quadrato L M, ſiue contingens <lb/>linea A C minor contingente A M, & </s> <s xml:id="echoid-s6810" xml:space="preserve">hoc ſemper, vbicunque contingat <lb/>obliqua A M: </s> <s xml:id="echoid-s6811" xml:space="preserve">quare A E C erit _MINIMA_ interiorem ſectionem contin-<lb/>gentium. </s> <s xml:id="echoid-s6812" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s6813" xml:space="preserve">c.</s> <s xml:id="echoid-s6814" xml:space="preserve"/> </p> <div xml:id="echoid-div714" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0246-02" xlink:href="note-0246-02a" xml:space="preserve">ibidem.</note> <note symbol="c" position="left" xlink:label="note-0246-03" xlink:href="note-0246-03a" xml:space="preserve">33. 34. h.</note> <note symbol="d" position="left" xlink:label="note-0246-04" xlink:href="note-0246-04a" xml:space="preserve">46. h.</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/QN4GHYBF/figures/0246-01"/> </figure> </div> <p> <s xml:id="echoid-s6815" xml:space="preserve">Iam, ducta ſit per D, recta V D X æquidiſtans contingenti R S T. </s> <s xml:id="echoid-s6816" xml:space="preserve">Et cum <lb/>in Ellipſi A B C ſit per punctum D recta V D X minori axi O Q perpendi-<lb/>cularis, ſitq; </s> <s xml:id="echoid-s6817" xml:space="preserve">alia obliqua L D M; </s> <s xml:id="echoid-s6818" xml:space="preserve">erit rectangulum V D X maius rectangulo <lb/>L D M (cum V D X ſit <anchor type="note" xlink:href="" symbol="e"/> _MAXIMVM_) ſed V D X æquatur <anchor type="note" xlink:href="" symbol="f"/> quadrato R S, <anchor type="note" xlink:label="note-0246-05a" xlink:href="note-0246-05"/> <anchor type="note" xlink:label="note-0246-06a" xlink:href="note-0246-06"/> quare quadratum R S maius erit rectangulo L D M, ſiue quadrato L D, & </s> <s xml:id="echoid-s6819" xml:space="preserve"><lb/>quadruplum quadruplo maius, hoc eſt quadratum R T maius quadrato L <lb/>M, hoc eſt linea R T maior linea R M, & </s> <s xml:id="echoid-s6820" xml:space="preserve">hoc ſemper de qualibet contin-<lb/>gente inter S, & </s> <s xml:id="echoid-s6821" xml:space="preserve">E, quare ipſa R T erit _MAXIMA_ interiorem Ellipſim <lb/>contingentium. </s> <s xml:id="echoid-s6822" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s6823" xml:space="preserve"/> </p> <div xml:id="echoid-div715" type="float" level="2" n="3"> <note symbol="e" position="left" xlink:label="note-0246-05" xlink:href="note-0246-05a" xml:space="preserve">34. h.</note> <note symbol="f" position="left" xlink:label="note-0246-06" xlink:href="note-0246-06a" xml:space="preserve">46. h.</note> </div> <pb o="63" file="0247" n="247" rhead=""/> </div> <div xml:id="echoid-div717" type="section" level="1" n="287"> <head xml:id="echoid-head296" xml:space="preserve">THEOR. XXIX. PROP. XLVIII.</head> <p> <s xml:id="echoid-s6824" xml:space="preserve">MAXIMA portionum eiuſdem anguli rectilinei, vel Hyperbo-<lb/>le, & </s> <s xml:id="echoid-s6825" xml:space="preserve">quarum diametri ſint æquales, eſt ea, cuius diameter ſit axis <lb/>dati anguli, vel Hyperbolæ.</s> <s xml:id="echoid-s6826" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6827" xml:space="preserve">ESto primùm, in prima figura, A B C angulus rectilineus, circa axim B <lb/>D, cui applicata ſit perpendiculariter quæcunque A E C, eum ſecans <lb/>in E. </s> <s xml:id="echoid-s6828" xml:space="preserve">Dico portionum, ſiue triangulorum ex dato angulo abſciſſorum, & </s> <s xml:id="echoid-s6829" xml:space="preserve"><lb/>quorum diametri ſint æquales ipſi B E, _MAXIMVM_ eſſe A B C.</s> <s xml:id="echoid-s6830" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6831" xml:space="preserve">Nam cum B E ſit perpendicu-<lb/> <anchor type="figure" xlink:label="fig-0247-01a" xlink:href="fig-0247-01"/> laris ad A C, facto centro B in-<lb/>teruallo B D, ac circulo deſcri-<lb/>pto, eius peripheria continget re-<lb/>ctam A C in D, anguli latera ſe-<lb/>cans in F, K; </s> <s xml:id="echoid-s6832" xml:space="preserve">quare diametri æ-<lb/>quales abſciſſorum triangulorum <lb/>ad peripheriam F E K pertingẽt: <lb/></s> <s xml:id="echoid-s6833" xml:space="preserve">ſumpto igitur in ipſa quocunque <lb/>puncto G, iungatur B G, & </s> <s xml:id="echoid-s6834" xml:space="preserve">du-<lb/>catur per G recta L G M ipſi A C <lb/>æquidiſtans, axim ſecans in N, <lb/>& </s> <s xml:id="echoid-s6835" xml:space="preserve">erit L N æqualis N M, vnde <lb/>L G minor G M; </s> <s xml:id="echoid-s6836" xml:space="preserve">ſecetur ergo G <lb/>O ipſi L G ęqualis, & </s> <s xml:id="echoid-s6837" xml:space="preserve">agatur O I <lb/>parallela ad B A, iungaturque <lb/>I G, & </s> <s xml:id="echoid-s6838" xml:space="preserve">producatur, quæ cum O I <lb/>ſecet in I, alteram quoque paral-<lb/>lelam B A ſecabit in H, eritque I G æqualis G H, ſed anguli ad verticem <lb/>I G O, H G L ſunt æquales, ergo, & </s> <s xml:id="echoid-s6839" xml:space="preserve">triangulum I G O triangulo H G L æ-<lb/>quale erit, & </s> <s xml:id="echoid-s6840" xml:space="preserve">communi addito trapetio B L G I, erit quadrilaterum B L O I <lb/>æquale triangulo H B I, ſed triangulum A B C maius eſt quadrilatero B L <lb/>O I, totum ſua parte, quare triangulum A B C erit quoque maius triangulo <lb/>H B I, cuius diameter B G æqualis eſt axi B E trianguli A B C, & </s> <s xml:id="echoid-s6841" xml:space="preserve">hoc ſem-<lb/>per de quolibet alio triangulo circa diametrum ipſi B E ęqualem; </s> <s xml:id="echoid-s6842" xml:space="preserve">quare <lb/>triangulum A B C eſt _MAXIMVM_. </s> <s xml:id="echoid-s6843" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s6844" xml:space="preserve">c.</s> <s xml:id="echoid-s6845" xml:space="preserve"/> </p> <div xml:id="echoid-div717" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0247-01"/> </figure> </div> <p> <s xml:id="echoid-s6846" xml:space="preserve">Sit præterea, in ſecunda figura, Hyperbole A B C, cuius centrum D, <lb/>axis D B E, ex quo dempta ſit B E, eique per E applicata A E C, & </s> <s xml:id="echoid-s6847" xml:space="preserve">ſit <lb/>quælibet alia diameter D F G, ex qua ſumatur F G ipſi B E æqualis, appli-<lb/>ceturque H G I. </s> <s xml:id="echoid-s6848" xml:space="preserve">Dico portionem A B C portione H F I maiorem eſſe.</s> <s xml:id="echoid-s6849" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6850" xml:space="preserve">Nam cum ſit ſemi-axis D B ſemi-diametrorum <anchor type="note" xlink:href="" symbol="a"/> _MINIMA_, hæc erit ma- <anchor type="note" xlink:label="note-0247-01a" xlink:href="note-0247-01"/> ior D F, eſtque B E æqualis F G, quare D B ad B E minorem habebit ra-<lb/>tionem quàm D F ad F G: </s> <s xml:id="echoid-s6851" xml:space="preserve">fiat ergo D F ad F L, vt D B ad B E, & </s> <s xml:id="echoid-s6852" xml:space="preserve">habe-<lb/>bit D F ad F L minorem rationem quàm D F ad F G, ideoque F L maior <lb/>erit F G, ſi ergo per L applicetur M L N, quæ ipſi H G I æquidiſtet, erit <pb o="64" file="0248" n="248" rhead=""/> portio M F N maior portione H F I (totum ſua parte) ſed portio M F N æ-<lb/>qualis <anchor type="note" xlink:href="" symbol="a"/> eſt portioni A B C (cum ſit D F ad F L, vt D B ad B E) quare <anchor type="note" xlink:label="note-0248-01a" xlink:href="note-0248-01"/> portio A B C erit maior H F I, & </s> <s xml:id="echoid-s6853" xml:space="preserve">hoc ſemper de qualibet alia portione, cu-<lb/>ius diameter æqualis ſit axi B E: </s> <s xml:id="echoid-s6854" xml:space="preserve">ergo portio A B C eſt _MAXIMA_ portio-<lb/>num æqualium diametrorum. </s> <s xml:id="echoid-s6855" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s6856" xml:space="preserve"/> </p> <div xml:id="echoid-div718" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0247-01" xlink:href="note-0247-01a" xml:space="preserve">24. h.</note> <note symbol="a" position="left" xlink:label="note-0248-01" xlink:href="note-0248-01a" xml:space="preserve">40. h.</note> </div> </div> <div xml:id="echoid-div720" type="section" level="1" n="288"> <head xml:id="echoid-head297" xml:space="preserve">THEOR. XXX. PROP. XLIX.</head> <p> <s xml:id="echoid-s6857" xml:space="preserve">MAXIMA portionum ſemi- Ellipſi minorum, & </s> <s xml:id="echoid-s6858" xml:space="preserve">æqualium dia-<lb/>metrorum eſt ea, cuius diameter ſit minoris ſemi-axis ſegmentum. <lb/></s> <s xml:id="echoid-s6859" xml:space="preserve">MINIMA verò, cuius diameter ſit ſegmentum maioris ſemi-axis.</s> <s xml:id="echoid-s6860" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6861" xml:space="preserve">ESto A B C D Ellipſis, cuius axis maior ſit B D, minor A C, centrum <lb/>E, ſitque ex minori ſemi-axe A E demptum ſegmentum A G, & </s> <s xml:id="echoid-s6862" xml:space="preserve">ex <lb/>maiori B E ipſi A G ſit æquale B F perque puncta G, F applicatæ ſint <lb/>axibus rectæ L G M, H F I. </s> <s xml:id="echoid-s6863" xml:space="preserve">Dico portionem L A M eſſe _MAXIMAM_, & </s> <s xml:id="echoid-s6864" xml:space="preserve"><lb/>H B I _MINIMAM_ aliarum portionum eiuſdem Ellipſis circa diametros ipſis <lb/>A G, B F æquales.</s> <s xml:id="echoid-s6865" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6866" xml:space="preserve">Quod L A M ſit maior H B I patet ſic. <lb/></s> <s xml:id="echoid-s6867" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0248-01a" xlink:href="fig-0248-01"/> Nam cum ſit E A minor E B, A G verò <lb/>æqualis B F, habebit. </s> <s xml:id="echoid-s6868" xml:space="preserve">E A ad A G mi-<lb/>norem rationem quàm E B ad B F: </s> <s xml:id="echoid-s6869" xml:space="preserve">fiat <lb/>ergo E B ad B N, vt E A ad A G, & </s> <s xml:id="echoid-s6870" xml:space="preserve">ha-<lb/>bebit E B ad B N minorem rationem <lb/>quàm E B ad B F, ſiue B N erit maior <lb/>B F; </s> <s xml:id="echoid-s6871" xml:space="preserve">quare applicata O N P cadet infra <lb/>H I: </s> <s xml:id="echoid-s6872" xml:space="preserve">& </s> <s xml:id="echoid-s6873" xml:space="preserve">cum ſit vt E A ad A G, ita E B <lb/>ad B N, erit portio L A M <anchor type="note" xlink:href="" symbol="b"/> ęqualis por- <anchor type="note" xlink:label="note-0248-02a" xlink:href="note-0248-02"/> tioni O B P, ſed hæc maior eſt portione <lb/>H B I, totum parte, ergo L A M maior <lb/>eſt H B I.</s> <s xml:id="echoid-s6874" xml:space="preserve"/> </p> <div xml:id="echoid-div720" type="float" level="2" n="1"> <figure xlink:label="fig-0248-01" xlink:href="fig-0248-01a"> <image file="0248-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0248-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0248-02" xlink:href="note-0248-02a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s6875" xml:space="preserve">Præterea, ducta inter ſemi-axes qua-<lb/>cunque ſemi-diametro E Q, ex ipſa, quę <lb/>maior eſt E A (eo quod hæc ſit ſemi-dia-<lb/>metrorum _MINIMA_ <anchor type="note" xlink:href="" symbol="c"/>) & </s> <s xml:id="echoid-s6876" xml:space="preserve">eò maior ipſa <anchor type="note" xlink:label="note-0248-03a" xlink:href="note-0248-03"/> A G, dematur Q R æqualis ipſi A G, vel B F, appliceturque S R T. </s> <s xml:id="echoid-s6877" xml:space="preserve">Iam <lb/>cum ſit E A minor E Q, & </s> <s xml:id="echoid-s6878" xml:space="preserve">A G æqualis Q R, habebit E A ad A G mi-<lb/>norem rationem, quàm E Q ad Q R, ac ideò vti ſuperiùs oſtendimus, por-<lb/>tio L A M erit maior portione S Q T. </s> <s xml:id="echoid-s6879" xml:space="preserve">Eadem ratione, cum ſit E Q minor <lb/>E B, (eò quod hæc ſit <anchor type="note" xlink:href="" symbol="d"/> ſemi-diametrorum _MAXIMA_) & </s> <s xml:id="echoid-s6880" xml:space="preserve">Q R ęqualis B F, <anchor type="note" xlink:label="note-0248-04a" xlink:href="note-0248-04"/> habebit E Q ad Q R minorem rationem quàm E B ad B F, quapropter <lb/>portio S Q T maior erit portione H B I, & </s> <s xml:id="echoid-s6881" xml:space="preserve">hoc ſemper de qualibet portio-<lb/>ne, cuius diameter ſit inter ſemi- axes; </s> <s xml:id="echoid-s6882" xml:space="preserve">quare portio L A M erit _MAXIMA_, <lb/>& </s> <s xml:id="echoid-s6883" xml:space="preserve">H B I _MINIMA_ portionum æqualium diametrorum. </s> <s xml:id="echoid-s6884" xml:space="preserve">Quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s6885" xml:space="preserve"/> </p> <div xml:id="echoid-div721" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-0248-03" xlink:href="note-0248-03a" xml:space="preserve">86. pri-<lb/>mi huius.</note> <note symbol="d" position="left" xlink:label="note-0248-04" xlink:href="note-0248-04a" xml:space="preserve">ibidem.</note> </div> <pb o="65" file="0249" n="249" rhead=""/> </div> <div xml:id="echoid-div723" type="section" level="1" n="289"> <head xml:id="echoid-head298" xml:space="preserve">THEOR. XXXI. PROP. L.</head> <p> <s xml:id="echoid-s6886" xml:space="preserve">MAXIMA portionum eiuſdem anguli rectilinei, vel cuiuſcunq; <lb/></s> <s xml:id="echoid-s6887" xml:space="preserve">coni-ſectionis, quarum baſes ſint æquales, eſt ea, cuius diameter <lb/>ſit ſegmentum axis, vel maioris ſemi- axis (reſpectiuè ad Ellipſim) <lb/>datæ ſectionis. </s> <s xml:id="echoid-s6888" xml:space="preserve">MINIMA verò in Ellipſi eſt, cuius diameter ſit ſe-<lb/>gmentum minoris ſemi- axis.</s> <s xml:id="echoid-s6889" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6890" xml:space="preserve">ESto A B C angulus rectilineus, vt in prima figura; </s> <s xml:id="echoid-s6891" xml:space="preserve">vel Parabole, aut <lb/>Hyperbole, vt in ſecunda; </s> <s xml:id="echoid-s6892" xml:space="preserve">vel Ellipſis, vt in tertia, quarum axes ſint B <lb/>D, & </s> <s xml:id="echoid-s6893" xml:space="preserve">in Ellipſi axis maior ſit B D N, minor L K M, centrum E, atque ma-<lb/>iori axi in quauis figura applicata ſit quęcunque A D C. </s> <s xml:id="echoid-s6894" xml:space="preserve">Dico primùm por-<lb/>tionem A B C, quæ tamen in tertia figura ſit minor ſemi-Ellipſi L B M, eſſe <lb/>_MAXIMAM_ omnium portionum eiuſdem anguli, vel coni-ſectionis, qua-<lb/>rum baſes æquales ſint baſi A C.</s> <s xml:id="echoid-s6895" xml:space="preserve"/> </p> <figure> <image file="0249-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0249-01"/> </figure> <p> <s xml:id="echoid-s6896" xml:space="preserve">Nam, in prima figura, deſcribatur per D in angulo aſymptotali A B C <lb/>Hyperbole F D G, in ſecunda verò, ſi A B C fuerit Parabole, deſcribatur per <lb/>D congruens Parabole F D G, vel ſi fuerit Hyperbole, deſcribatur item per <lb/>D, vti etiam in tertia, eiuſdem nominis ſectio F D G ſimilis, & </s> <s xml:id="echoid-s6897" xml:space="preserve">concentri-<lb/>ca ipſi A B C, & </s> <s xml:id="echoid-s6898" xml:space="preserve">tunc recta A D C continget omnino ſectionem F D G in <lb/>D; </s> <s xml:id="echoid-s6899" xml:space="preserve">ſumptoque in interiori ſectione F D G quolibet puncto F, per ipſum <lb/>ducatur ſectionem contingens H F I exteriori occurrens in H I, deque ipſa <lb/>abſcindens portionem H O I, cuius diameter ſit O F.</s> <s xml:id="echoid-s6900" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6901" xml:space="preserve">Iam, in ſingulis ſiguris, baſis A C minor eſt baſi H I, cum ſit <anchor type="note" xlink:href="" symbol="a"/> _MINIMA_ <anchor type="note" xlink:label="note-0249-01a" xlink:href="note-0249-01"/> contingentium ſectionem F D G, quare, & </s> <s xml:id="echoid-s6902" xml:space="preserve">dimidium D C dimidio F I mi-<lb/>nus erit. </s> <s xml:id="echoid-s6903" xml:space="preserve">Fiat ergo F P æqualis D C, & </s> <s xml:id="echoid-s6904" xml:space="preserve">ex P agatur P R diametro F O <lb/>æquidiſtans, cui ex R applicetur R Q S: </s> <s xml:id="echoid-s6905" xml:space="preserve">patet ipſam R Q S ęquari baſi A C, <pb o="66" file="0250" n="250" rhead=""/> hoc eſt portiones A B C, S O R eſſe æqualium baſium, ſed H O I maior eſt <lb/>S O R, totum parte, ergo, & </s> <s xml:id="echoid-s6906" xml:space="preserve">A B C, quæ ipſi H O I <anchor type="note" xlink:href="" symbol="a"/> eſt æqualis, erit maior <anchor type="note" xlink:label="note-0250-01a" xlink:href="note-0250-01"/> eadem S O R, & </s> <s xml:id="echoid-s6907" xml:space="preserve">hoc ſemper, &</s> <s xml:id="echoid-s6908" xml:space="preserve">c. </s> <s xml:id="echoid-s6909" xml:space="preserve">vnde portio A B C eſt _MAXIMA_ portio-<lb/>num æqualium baſium. </s> <s xml:id="echoid-s6910" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s6911" xml:space="preserve">c.</s> <s xml:id="echoid-s6912" xml:space="preserve"/> </p> <div xml:id="echoid-div723" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0249-01" xlink:href="note-0249-01a" xml:space="preserve">47. h.</note> <note symbol="a" position="left" xlink:label="note-0250-01" xlink:href="note-0250-01a" xml:space="preserve">45. h.</note> </div> <figure> <image file="0250-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0250-01"/> </figure> <p> <s xml:id="echoid-s6913" xml:space="preserve">Pręterea, cũ in tertia figura, quæ ex K ducitur interiorem Ellipſim F D G <lb/>contingens ſit <anchor type="note" xlink:href="" symbol="b"/> _MAXIMA_ eandem Ellipſim contingentium, ipſa erit omni- <anchor type="note" xlink:label="note-0250-02a" xlink:href="note-0250-02"/> no maior A C; </s> <s xml:id="echoid-s6914" xml:space="preserve">quare eidem axi applicata, quæ ipſi A C ſit æqualis, mino-<lb/>rem axim ſecabit inter L, & </s> <s xml:id="echoid-s6915" xml:space="preserve">K, & </s> <s xml:id="echoid-s6916" xml:space="preserve">ſit ea T V X. </s> <s xml:id="echoid-s6917" xml:space="preserve">Si ergo concipiatur per V <lb/>deſcripta Ellipſis, datis A B C, F D G ſimilis, & </s> <s xml:id="echoid-s6918" xml:space="preserve">concentrica, recta T V X <lb/>hanc Ellipſim continget, eritque <anchor type="note" xlink:href="" symbol="c"/> _MAXIMA_ eandem Ellipſim contingen- <anchor type="note" xlink:label="note-0250-03a" xlink:href="note-0250-03"/> tium, quapropter portiones, quarum baſes ſint æquales baſi T V X, hanc <lb/>mediam Ellipſim omnino ſecabunt, ac ideo maiores erunt portione T L X, <lb/>cum portiones ab ijſdem contingentibus abſciſſæ ſint <anchor type="note" xlink:href="" symbol="d"/> omnes portioni TLX <anchor type="note" xlink:label="note-0250-04a" xlink:href="note-0250-04"/> æquales. </s> <s xml:id="echoid-s6919" xml:space="preserve">Quare portio T L X eſt _MINIMA_ portionum æqualium baſium, <lb/>ex eadem Ellipſi A B C abſciſſarum. </s> <s xml:id="echoid-s6920" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s6921" xml:space="preserve"/> </p> <div xml:id="echoid-div724" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0250-02" xlink:href="note-0250-02a" xml:space="preserve">47. h.</note> <note symbol="c" position="left" xlink:label="note-0250-03" xlink:href="note-0250-03a" xml:space="preserve">ibidem.</note> <note symbol="d" position="left" xlink:label="note-0250-04" xlink:href="note-0250-04a" xml:space="preserve">45. h.</note> </div> </div> <div xml:id="echoid-div726" type="section" level="1" n="290"> <head xml:id="echoid-head299" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s6922" xml:space="preserve">EX his conſtat _MINIMAM_ portionum ſemi-Ellipſi maiorum, quarum <lb/>baſes ſint ęquales eam eſſe, cuius diameter ſit ſegmentum maioris axis, <lb/>_MAXIMAM_ verò, cuius diameter ſit ſegmentum minoris.</s> <s xml:id="echoid-s6923" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6924" xml:space="preserve">Nam in tertia figura, cum portionum A B C, S O R, T L X, &</s> <s xml:id="echoid-s6925" xml:space="preserve">c. </s> <s xml:id="echoid-s6926" xml:space="preserve">ſemi-El-<lb/>lipſi minorum, & </s> <s xml:id="echoid-s6927" xml:space="preserve">ſuper æqualibus baſibus, ipſa A B C ſit _MAXIMA_, & </s> <s xml:id="echoid-s6928" xml:space="preserve">TLX <lb/>_MINIMA_, ac ipſæ ſint portiones eiuſdem terminatæ magnitudinis, ſiue Elli-<lb/>pſis eiuſdem A B C N, patet reliquarum portionum ſemi-Ellipſi maiorum <lb/>A N C, S N R, X M T, &</s> <s xml:id="echoid-s6929" xml:space="preserve">c. </s> <s xml:id="echoid-s6930" xml:space="preserve">quæ item ſunt ſuper æquales baſes A C, S R, <lb/>T X, portionem A N C eſſe _MAXIMAM_, & </s> <s xml:id="echoid-s6931" xml:space="preserve">X M T _MINIMAM_.</s> <s xml:id="echoid-s6932" xml:space="preserve"/> </p> <pb o="67" file="0251" n="251" rhead=""/> </div> <div xml:id="echoid-div727" type="section" level="1" n="291"> <head xml:id="echoid-head300" xml:space="preserve">THEOR. XXXII. PROP. LI.</head> <p> <s xml:id="echoid-s6933" xml:space="preserve">MINIMA portionum eiuſdem anguli, vel cuiuslibet coni-ſectio-<lb/>nis, quarum altitudines ſint equales, eſt ea, cuius diameter ſit ſegmẽ-<lb/>tum maioris axis: </s> <s xml:id="echoid-s6934" xml:space="preserve">in Ellipſi verò MAXIMA eſt, cuius diameter ſit <lb/>ſegmentum minoris axis.</s> <s xml:id="echoid-s6935" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6936" xml:space="preserve">ESto A B C, in prima figura, angulus rectilineus, vel in ſecunda, Parabole, <lb/>aut Hyperbole, ſiue in tertia Ellipſis, quarum axes ſint B D, at in Ellipſi <lb/>axis maior ſit B D N, minor L K; </s> <s xml:id="echoid-s6937" xml:space="preserve">centrum E, atque axi B D in quauis figura <lb/>applicata ſit quælibet A D C. </s> <s xml:id="echoid-s6938" xml:space="preserve">Dico portionem A B C, quæ in tertia figura <lb/>ſit, vel maior, vel minor ſemi-Ellipſi, eſſe _MINIMAM_ omnium portionum <lb/>eiuſdem anguli, vel coni-ſectionis, quarum altitudines ſint æquales ipſi B D.</s> <s xml:id="echoid-s6939" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6940" xml:space="preserve">Deſcripta. </s> <s xml:id="echoid-s6941" xml:space="preserve">n. </s> <s xml:id="echoid-s6942" xml:space="preserve">per D, vel Hyperbola <lb/> <anchor type="figure" xlink:label="fig-0251-01a" xlink:href="fig-0251-01"/> in prima figura, cuius aſymptoti ſint B <lb/>A, B C; </s> <s xml:id="echoid-s6943" xml:space="preserve">vel in reliquis figuris, deſcri-<lb/>pta eiuſdem nominis coni-ſectione ſi-<lb/>mili, & </s> <s xml:id="echoid-s6944" xml:space="preserve">concentrica F D G, que rectam <lb/>A D C continget in D; </s> <s xml:id="echoid-s6945" xml:space="preserve">ſumatur in in-<lb/>teriori ſectione quodlibet aliud punctũ <lb/>F, ad quod ſit contingens H F I exte-<lb/>riori occurrens in H, I, atque portionẽ <lb/>abſcindens H O I, cuius diameter ſit <lb/>O F, altitudo verò ſit O P.</s> <s xml:id="echoid-s6946" xml:space="preserve"/> </p> <div xml:id="echoid-div727" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0251-01"/> </figure> </div> <p> <s xml:id="echoid-s6947" xml:space="preserve">Itaque cum portio H O I equalis <anchor type="note" xlink:href="" symbol="a"/> ſit <anchor type="note" xlink:label="note-0251-01a" xlink:href="note-0251-01"/> portioni A B C eiuſdem ſectionis, erit <lb/>reciprocè baſis H I ad baſim A C, vt <lb/>altitudo B D ad altitudinem O P, ſed <lb/>eſt H I maior A C, cum A C ſit om-<lb/>nium <anchor type="note" xlink:href="" symbol="b"/> contingentium _MINIMA_, ergo, <anchor type="note" xlink:label="note-0251-02a" xlink:href="note-0251-02"/> & </s> <s xml:id="echoid-s6948" xml:space="preserve">B D erit maior O P: </s> <s xml:id="echoid-s6949" xml:space="preserve">producatur ergo <lb/>O P, & </s> <s xml:id="echoid-s6950" xml:space="preserve">ſumatur O Q ipſi B D æqualis, <lb/>appliceturque S Q R contingenti H I <lb/>æquidiſtans: </s> <s xml:id="echoid-s6951" xml:space="preserve">eruntque portiones S O R, A B C æqualium altitudinum, ſed eſt <lb/>portio H O I minor S O R, pars ſuo toto, ergo, & </s> <s xml:id="echoid-s6952" xml:space="preserve">portio A B C, quæ ipſi H O I <lb/>eſt æqualis, minor erit portione S O R, & </s> <s xml:id="echoid-s6953" xml:space="preserve">hoc ſemper, &</s> <s xml:id="echoid-s6954" xml:space="preserve">c. </s> <s xml:id="echoid-s6955" xml:space="preserve">Vnde portio A B <lb/>C eſt _MINIMA_ portionum eiuſdem anguli, vel coni-ſectionis, & </s> <s xml:id="echoid-s6956" xml:space="preserve">æqualium <lb/>altitudinum. </s> <s xml:id="echoid-s6957" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s6958" xml:space="preserve">c.</s> <s xml:id="echoid-s6959" xml:space="preserve"/> </p> <div xml:id="echoid-div728" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0251-01" xlink:href="note-0251-01a" xml:space="preserve">45. h.</note> <note symbol="b" position="right" xlink:label="note-0251-02" xlink:href="note-0251-02a" xml:space="preserve">47. h.</note> </div> <p> <s xml:id="echoid-s6960" xml:space="preserve">Ampliùs in tertia figura eſto recta V K T minori axi L M ordinatim appli-<lb/>cata. </s> <s xml:id="echoid-s6961" xml:space="preserve">Dico portionem V M T (quæ ſit vel maior, vel minor ſemi-Ellipſi) cu-<lb/>ius diameter, vel altitudo eſt M K, eſſe _MAXIMAM_ portionum omnium, qua-<lb/>rum altitudines ipſi M K ſint æquales.</s> <s xml:id="echoid-s6962" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6963" xml:space="preserve">Deſcripta enim per K Ellipſi K D G ſimili, & </s> <s xml:id="echoid-s6964" xml:space="preserve">concentrica datæ A B C N, <lb/>quæ rectam V K T continget in K, ſumptoque in eius peripheria quocunque <lb/>puncto F, ducatur contingens H F I exteriori ſectioni occurrens in H, I, de-<lb/>que ipſa abſcindens portionem I X H, cuius diameter ſit F X, altitudo verò <lb/>ſit X Z.</s> <s xml:id="echoid-s6965" xml:space="preserve"/> </p> <pb o="68" file="0252" n="252" rhead=""/> <p> <s xml:id="echoid-s6966" xml:space="preserve">Iam cum portio V M T æqualis ſit <anchor type="note" xlink:href="" symbol="a"/> portioni I X H, erit baſis V T ad baſim <anchor type="note" xlink:label="note-0252-01a" xlink:href="note-0252-01"/> I H reciprocè vt altitudo X Z ad altitudinem M K, ſed eſt V T maior I H, <lb/>cum ipſa V T ſit contingentium <anchor type="note" xlink:href="" symbol="b"/> _MAXIMA_, ergo, & </s> <s xml:id="echoid-s6967" xml:space="preserve">X Z erit maior M K;</s> <s xml:id="echoid-s6968" xml:space="preserve"> <anchor type="note" xlink:label="note-0252-02a" xlink:href="note-0252-02"/> facta igitur X Y æquali ipſi M K, applicataque S Y R, erunt portiones V M <lb/>T, R X S æqualium altitudinum, ſed eſt portio R X S minor portione I X H, <lb/>pars ſuo toto, ergo ipſa R X S minor quoque erit portione V M T, & </s> <s xml:id="echoid-s6969" xml:space="preserve">hoc ſem-<lb/>per, &</s> <s xml:id="echoid-s6970" xml:space="preserve">c. </s> <s xml:id="echoid-s6971" xml:space="preserve">Quare portio V M T eſt _MAXIMA_ portionum eiuſdem Ellipſis, & </s> <s xml:id="echoid-s6972" xml:space="preserve"><lb/>æqualium altitudinum. </s> <s xml:id="echoid-s6973" xml:space="preserve">Quod erat vltimò demonſtrandum.</s> <s xml:id="echoid-s6974" xml:space="preserve"/> </p> <div xml:id="echoid-div729" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-0252-01" xlink:href="note-0252-01a" xml:space="preserve">45. h.</note> <note symbol="b" position="left" xlink:label="note-0252-02" xlink:href="note-0252-02a" xml:space="preserve">47. h.</note> </div> </div> <div xml:id="echoid-div731" type="section" level="1" n="292"> <head xml:id="echoid-head301" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s6975" xml:space="preserve">PRoxima quatuor præcedentia Theoremata, ſuper hoc ipſo Diagrammate, <lb/>facilè ſimul, tanquam Conſectaria demonſtrabuntur, ſi tamen hæ tres <lb/>concluſiones notatu dignæ præmittantur, à quibus ipſa ortum ducant. </s> <s xml:id="echoid-s6976" xml:space="preserve">Nimirũ.</s> <s xml:id="echoid-s6977" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6978" xml:space="preserve">1. </s> <s xml:id="echoid-s6979" xml:space="preserve">INter diametros æqualium portionum eiuſdem anguli, vel Hyperbolæ, aut <lb/>Ellipſis, _MINIMA_ eſt ea illius portionis, cuius diameter ſimul ſit ſegmentũ <lb/>axis dati anguli, vel Hyperbolæ: </s> <s xml:id="echoid-s6980" xml:space="preserve">ſed in Ellipſi, quæ ſit ſegmentum minoris <lb/>axis, & </s> <s xml:id="echoid-s6981" xml:space="preserve">_MAXIMA_, quæ ſit ſegmentum maioris.</s> <s xml:id="echoid-s6982" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6983" xml:space="preserve">Etenim in prima figura angulum ex-<lb/> <anchor type="figure" xlink:label="fig-0252-01a" xlink:href="fig-0252-01"/> hibente, in portionibus A B C, H O I, <lb/>quæ ſunt <anchor type="note" xlink:href="" symbol="c"/> æquales, (eò quod ipſarum <anchor type="note" xlink:label="note-0252-03a" xlink:href="note-0252-03"/> baſes contingant eandem ſimilem con-<lb/>cẽtricam Hyperbolen interiorem) dia-<lb/>meter B D, quæ eſt axis dati anguli, <lb/>minor eſt diametro O F, cum ſit B D <lb/>ſemi-tranſuerſorum <anchor type="note" xlink:href="" symbol="d"/> _MINIMA_. </s> <s xml:id="echoid-s6984" xml:space="preserve">Et in <anchor type="note" xlink:label="note-0252-04a" xlink:href="note-0252-04"/> ſecunda, Hyperbolen repræſentante, <lb/>in portionibus item A B C, H O I, quę <lb/>ob eandem rationem æquales ſunt, dia-<lb/>meter B D, quæ eſt ſegmentum axis <lb/>Hyperbolæ, minor eſt diametro O F, <lb/>cum ſit B D ad O F, vt ſemi - axis per-<lb/>tingens ad B ex centro exterioris Hy-<lb/>perbole, A B C, ad ſemi-tranſuerſum <lb/>pertingens ad O ex eodem centro, vt <lb/>ſatis conſtat ex 44. </s> <s xml:id="echoid-s6985" xml:space="preserve">huius, at ſemi-axis, <lb/>minor eſt ſemi-tranſuerſo, quare pa-<lb/>tet, &</s> <s xml:id="echoid-s6986" xml:space="preserve">c. </s> <s xml:id="echoid-s6987" xml:space="preserve">In tertia denique in portioni-<lb/>bus T L V, H O I, A B C interſe pariter æqualibus, diameter L K portionis <lb/>T L V, quæ eſt ex minori axe datæ Ellipſis, minor eſt diametro O F portionis <lb/>H O I, atque minor diametro B D portionis A B C, & </s> <s xml:id="echoid-s6988" xml:space="preserve">ſic de ſingulis, quoniam <lb/>E K ad K L eſt vt E F ad F O, & </s> <s xml:id="echoid-s6989" xml:space="preserve">vt E D ad D B, eſtque antecedens E K minor <lb/>qualibet alia antecedentium, cum ea ſit <anchor type="note" xlink:href="" symbol="e"/> ſemi-tranſuerſorum _MINIMA_, & </s> <s xml:id="echoid-s6990" xml:space="preserve">E <anchor type="note" xlink:label="note-0252-05a" xlink:href="note-0252-05"/> D maior eſt ipſarum antecedentiũ, cum ſit ſemi-trãſuerſorum _MAXIMA_, qua-<lb/>re & </s> <s xml:id="echoid-s6991" xml:space="preserve">K L erit _MINIMA_, & </s> <s xml:id="echoid-s6992" xml:space="preserve">D B _MAXIMA_, &</s> <s xml:id="echoid-s6993" xml:space="preserve">c. </s> <s xml:id="echoid-s6994" xml:space="preserve">idemque dicetur de æqualibus <lb/>portionibus ſemi-Ellipſi maioribus. </s> <s xml:id="echoid-s6995" xml:space="preserve">Verùm inter diametros æqualium por-<lb/>tionum eiuſdem Parabolæ non datur _MAXIMA_, cum omnes æquales ſint.</s> <s xml:id="echoid-s6996" xml:space="preserve"/> </p> <div xml:id="echoid-div731" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0252-01"/> </figure> <note symbol="c" position="left" xlink:label="note-0252-03" xlink:href="note-0252-03a" xml:space="preserve">45. h.</note> <note symbol="d" position="left" xlink:label="note-0252-04" xlink:href="note-0252-04a" xml:space="preserve">24. h.</note> <note symbol="e" position="left" xlink:label="note-0252-05" xlink:href="note-0252-05a" xml:space="preserve">ibidem.</note> </div> <pb o="69" file="0253" n="253" rhead=""/> <p> <s xml:id="echoid-s6997" xml:space="preserve">2. </s> <s xml:id="echoid-s6998" xml:space="preserve">INter baſes æqualiũ portionum eiuſdem anguli, vel coni-ſectionis _MINIMA_ <lb/>eſt ea illius portionis, cuius diameter ſit ſegmentum maioris axis, reſpectiuè <lb/>ad Ellipſim: </s> <s xml:id="echoid-s6999" xml:space="preserve">& </s> <s xml:id="echoid-s7000" xml:space="preserve">_MAXIMA_ eius, cuius diameter ſit ſegmentum minoris.</s> <s xml:id="echoid-s7001" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7002" xml:space="preserve">In qualibet enim ſigura, baſis A C portionis A B C, circa maiorem axim, <lb/>_MINIMA_ <anchor type="note" xlink:href="" symbol="a"/> eſt baſium, aliarum æqualium portionum; </s> <s xml:id="echoid-s7003" xml:space="preserve">& </s> <s xml:id="echoid-s7004" xml:space="preserve">in Ellipſi baſis V T <anchor type="note" xlink:label="note-0253-01a" xlink:href="note-0253-01"/> portionis V L T circa minorem, _MAXIMA_ eſt baſium, reliquarum æqualium <lb/>portionum, vel ipſæ ſimul ſint ſemi-Ellipſi minores, vel ſimul maiores, &</s> <s xml:id="echoid-s7005" xml:space="preserve">c.</s> <s xml:id="echoid-s7006" xml:space="preserve"/> </p> <div xml:id="echoid-div732" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0253-01" xlink:href="note-0253-01a" xml:space="preserve">47. h.</note> </div> <p> <s xml:id="echoid-s7007" xml:space="preserve">3. </s> <s xml:id="echoid-s7008" xml:space="preserve">INter altitudines æqualium portionum de eodem angulo, vel coni-ſectione <lb/>_MAXIMA_ eſt ea illius portionis, cuius diameter ſit ſegmentum maioris axis <lb/>reſpectiuè ad Ellipſim, & </s> <s xml:id="echoid-s7009" xml:space="preserve">_MINIMA_ eius, cuius diameter ſit ſegmétum minoris.</s> <s xml:id="echoid-s7010" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7011" xml:space="preserve">Id autem in ſuperiori propoſitione oſtenſum fuit: </s> <s xml:id="echoid-s7012" xml:space="preserve">nempe B D, quæ eſt alti-<lb/>tudo portionis A B C, circa maiorem axim, maiorem eſſe O P altitudine ęqua-<lb/>lis portionis H O I, atque ampliùs, in Ellipſi, altitudinem M K portionis T M <lb/>V circa minorẽ axim, minorem eſſe altitudine X Z æqualis portionis HXI, &</s> <s xml:id="echoid-s7013" xml:space="preserve">c.</s> <s xml:id="echoid-s7014" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7015" xml:space="preserve">E´ prima itaque harum concluſionum, elicitur veritas prop. </s> <s xml:id="echoid-s7016" xml:space="preserve">48. </s> <s xml:id="echoid-s7017" xml:space="preserve">& </s> <s xml:id="echoid-s7018" xml:space="preserve">49. </s> <s xml:id="echoid-s7019" xml:space="preserve">h. </s> <s xml:id="echoid-s7020" xml:space="preserve">ex <lb/>altera verò prop. </s> <s xml:id="echoid-s7021" xml:space="preserve">50. </s> <s xml:id="echoid-s7022" xml:space="preserve">è tertia denique prop. </s> <s xml:id="echoid-s7023" xml:space="preserve">51. </s> <s xml:id="echoid-s7024" xml:space="preserve">quæ omnia per ſe ſatis patent.</s> <s xml:id="echoid-s7025" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7026" xml:space="preserve">Sed hæc de planis, pro hac vice, dixiſſe ſufſiciat. </s> <s xml:id="echoid-s7027" xml:space="preserve">Nonnulla ſequuntur quæ <lb/>iam diù pariter circa ſolida à coni-ſectionibus genita excogitauimus. </s> <s xml:id="echoid-s7028" xml:space="preserve">Noua <lb/>omnia, ni fallor, omnia ſaltem geometrica: </s> <s xml:id="echoid-s7029" xml:space="preserve">quæ ſi apertæ iucunditatis referta <lb/>comperies amice Lector, reconditæ vtilitatis haud expertia eße aliquando te <lb/>certiorem factum non dubito.</s> <s xml:id="echoid-s7030" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div734" type="section" level="1" n="293"> <head xml:id="echoid-head302" xml:space="preserve">THEOR. XXXIII. PROP. LII.</head> <p> <s xml:id="echoid-s7031" xml:space="preserve">Recta linea, quę à puncto extra planũ dato ſit ipſi plano perpẽdicu-<lb/>laris, MINIMA eſt rectarũ ab eodem pũcto ad idem planũ ducibiliũ.</s> <s xml:id="echoid-s7032" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7033" xml:space="preserve">SIt extra planum A B, punctum C, à quo ducta ſit ipſi <lb/> <anchor type="figure" xlink:label="fig-0253-01a" xlink:href="fig-0253-01"/> plano perpendicularis C D. </s> <s xml:id="echoid-s7034" xml:space="preserve">Dico hanc eſſe _MINI_-<lb/>_MAM_ ducibilium ex C ad alia puncta plani A B.</s> <s xml:id="echoid-s7035" xml:space="preserve"/> </p> <div xml:id="echoid-div734" type="float" level="2" n="1"> <figure xlink:label="fig-0253-01" xlink:href="fig-0253-01a"> <image file="0253-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0253-01"/> </figure> </div> <p> <s xml:id="echoid-s7036" xml:space="preserve">Sumatur vbicunque in dato plano aliud punctum E, <lb/>iunganturque D E, C E. </s> <s xml:id="echoid-s7037" xml:space="preserve">Et cum C D recta ſit ad pla-<lb/>num A B, erit <anchor type="note" xlink:href="" symbol="b"/> angulus C D E rectus, ideoque C E D <anchor type="note" xlink:label="note-0253-02a" xlink:href="note-0253-02"/> acutus, ſiue minor C D E: </s> <s xml:id="echoid-s7038" xml:space="preserve">quare C D minor erit C E, <lb/>& </s> <s xml:id="echoid-s7039" xml:space="preserve">hoc ſemper. </s> <s xml:id="echoid-s7040" xml:space="preserve">Vnde C D eſt _MINIMA_, &</s> <s xml:id="echoid-s7041" xml:space="preserve">c. </s> <s xml:id="echoid-s7042" xml:space="preserve">Quod &</s> <s xml:id="echoid-s7043" xml:space="preserve">c.</s> <s xml:id="echoid-s7044" xml:space="preserve"/> </p> <div xml:id="echoid-div735" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0253-02" xlink:href="note-0253-02a" xml:space="preserve">3. deſ. <lb/>vnd. Ele.</note> </div> </div> <div xml:id="echoid-div737" type="section" level="1" n="294"> <head xml:id="echoid-head303" xml:space="preserve">THEOR. XXXIV. PROP. LIII.</head> <p> <s xml:id="echoid-s7045" xml:space="preserve">Si in Cono, vel Cylindro recto planum ductum per vnum laterum <lb/>trianguli, vel rectanguli per axem eidem triangulo, vel rectangulo <lb/>rectum fuerit, idem planum in ipſo tantùm latere conicam, vel cy-<lb/>lindricam ſuperficiem continget, quæ tota cadet ad alteram partem <lb/>plani contingentis.</s> <s xml:id="echoid-s7046" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7047" xml:space="preserve">ESto in figura, (que & </s> <s xml:id="echoid-s7048" xml:space="preserve">Conum, & </s> <s xml:id="echoid-s7049" xml:space="preserve">Cylindrum rectum exhibeat) planum per <lb/>axẽ A B C, cui rectũ ſit aliud planũ G D K H tranſiens per latus A B, cum <pb o="70" file="0254" n="254" rhead=""/> plano baſis Coni, vel Cylindri A C efficiens communem ſectionem G A D. <lb/></s> <s xml:id="echoid-s7050" xml:space="preserve">Dico ipſum planum G D K H, licet in infinitum extendatur, in vnico tantùm <lb/>latere B A ſuperficiem Conicam, vel Cylindricam contingere, ac propterea <lb/>hanc totam cadere infra planum contingens.</s> <s xml:id="echoid-s7051" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7052" xml:space="preserve">Quoniam cum axis Coni, vel Cylindrirecti ſit perpendicularis plano baſis, <lb/>erit planum per axem B A C rectum baſi A C, ſiue planum baſis A C rectum <lb/>plano per axem A B C, cui rectum quoque poſitum fuit planum per B A, A D <lb/>ductum, quare G A D communis planorum ſectio eidem plano per axem erit <lb/>perpendicularis, <anchor type="note" xlink:href="" symbol="a"/> vnde angulus D A C rectus erit, ſed eſt C A diameter cir- <anchor type="note" xlink:label="note-0254-01a" xlink:href="note-0254-01"/> culi A C, quare G A D circuli peripheriam continget, ac tota cadet extra co-<lb/>nicam, vel cylindricam ſuperſiciem.</s> <s xml:id="echoid-s7053" xml:space="preserve"/> </p> <div xml:id="echoid-div737" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0254-01" xlink:href="note-0254-01a" xml:space="preserve">19. vnd. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s7054" xml:space="preserve">Iam per B axis verticem concipiatur ductum pla-<lb/> <anchor type="figure" xlink:label="fig-0254-01a" xlink:href="fig-0254-01"/> num S T baſi A C ęquidiſtans, quod communem <lb/>ſectionem faciet cum plano G K rectam Q B R ipſi <lb/>G A D <anchor type="note" xlink:href="" symbol="b"/> parallelam, abſcindetque de plano G K <anchor type="note" xlink:label="note-0254-02a" xlink:href="note-0254-02"/> vtrinque in infinitum extenſo, partem Q R K H, <lb/>quæ tota cadet ſupra planum S T ad oppoſitas par-<lb/>tes conicæ, vel cylindricæ ſuperficiei B A C (cum <lb/>hæc tota cadat inter æquidiſtantia plana S T, A C, <lb/>vt ſatis conſtat,) & </s> <s xml:id="echoid-s7055" xml:space="preserve">partem Q R D G, quæ tota <lb/>erit ad partes eiuſdem ſuperſiciei. </s> <s xml:id="echoid-s7056" xml:space="preserve">Sumatur ergo in <lb/>plano Q R D G extra lineam B A, inter æquidiſtã-<lb/>tes Q R, G D quodlibet punctum E, & </s> <s xml:id="echoid-s7057" xml:space="preserve">iuncta B E <lb/>producatur: </s> <s xml:id="echoid-s7058" xml:space="preserve">patet ipſam cum A D conuenire: </s> <s xml:id="echoid-s7059" xml:space="preserve">(cumrecta B E ſit in eodem <lb/>plano in quo ſunt B A, & </s> <s xml:id="echoid-s7060" xml:space="preserve">A D, & </s> <s xml:id="echoid-s7061" xml:space="preserve">alteram parallelarum ſecet in B) conueniat <lb/>in F, & </s> <s xml:id="echoid-s7062" xml:space="preserve">cum punctum F ſit extra ſolidi ſuperſiciem, ipſa quoque B F cadet to-<lb/>ta extra <anchor type="note" xlink:href="" symbol="c"/> eandem, quare punctum E erit extra ipſam ſuperficiem, & </s> <s xml:id="echoid-s7063" xml:space="preserve">ſic de <anchor type="note" xlink:label="note-0254-03a" xlink:href="note-0254-03"/> quolibet alio puncto plani G R, quod ſit extra latus B A, quapropter planum <lb/>G R ſuperficiem dati ſolidi contingit per rectam B A, ac ideo ipſa ſuperficies <lb/>cadit tota ad alteram partem plani G R. </s> <s xml:id="echoid-s7064" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s7065" xml:space="preserve">c.</s> <s xml:id="echoid-s7066" xml:space="preserve"/> </p> <div xml:id="echoid-div738" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0254-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0254-02" xlink:href="note-0254-02a" xml:space="preserve">16. ibid.</note> <note symbol="c" position="left" xlink:label="note-0254-03" xlink:href="note-0254-03a" xml:space="preserve">Coroll. <lb/>primę pri-<lb/>mi Conic.</note> </div> </div> <div xml:id="echoid-div740" type="section" level="1" n="295"> <head xml:id="echoid-head304" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s7067" xml:space="preserve">SI per quodcunque aliud punctum N lateris B A concipiatur duci planum <lb/>ſecans Conum, vel Cylindrum, quod ſit baſi A C parallelum, ipſum in <lb/>ſolidi ſuperficie circuli peripheriam <anchor type="note" xlink:href="" symbol="d"/> deſcribet, & </s> <s xml:id="echoid-s7068" xml:space="preserve">in plano per axem rectam, <anchor type="note" xlink:label="note-0254-04a" xlink:href="note-0254-04"/> ſeu diametrum N P, quæ ipſi A C <anchor type="note" xlink:href="" symbol="e"/> æ quidiſtabit, in plano verò Q D rectam <anchor type="note" xlink:label="note-0254-05a" xlink:href="note-0254-05"/> M N O, quæ item rectæ G A D erit parallela (cum ſint communes ſectiones <lb/>æquidiſtantium planorum cum altero plano) eritque angulus O N P <anchor type="note" xlink:href="" symbol="f"/> æqualis <anchor type="note" xlink:label="note-0254-06a" xlink:href="note-0254-06"/> angulo D A C, ſiue rectus, (cum ſuperiùs demonſtratum ſit ipſum D A C <lb/>rectum eſſe) hoc eſt recta M N O peripheriam N P continget in N, & </s> <s xml:id="echoid-s7069" xml:space="preserve">ex vtra-<lb/>que parte cadet extra ſolidi ſuperficiem, & </s> <s xml:id="echoid-s7070" xml:space="preserve">hoc ſemper de qualibet alia ducta <lb/>in plano B D ipſi G D æquidiſtante: </s> <s xml:id="echoid-s7071" xml:space="preserve">quare totum planum G R, quod per la-<lb/>tus B A ductum fuit rectum ad planum A B C per axem ductum, ſolidi ſuper-<lb/>ficiem contingit tantùm per latus B A: </s> <s xml:id="echoid-s7072" xml:space="preserve">vnde ipſa ſuperficies cadit tota ad alte-<lb/>ram partem plani G R. </s> <s xml:id="echoid-s7073" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s7074" xml:space="preserve">c.</s> <s xml:id="echoid-s7075" xml:space="preserve"/> </p> <div xml:id="echoid-div740" type="float" level="2" n="1"> <note symbol="d" position="left" xlink:label="note-0254-04" xlink:href="note-0254-04a" xml:space="preserve">4. primi <lb/>Conic.</note> <note symbol="e" position="left" xlink:label="note-0254-05" xlink:href="note-0254-05a" xml:space="preserve">16. vnd. <lb/>Elem.</note> <note symbol="f" position="left" xlink:label="note-0254-06" xlink:href="note-0254-06a" xml:space="preserve">10. ibid.</note> </div> <pb o="71" file="0255" n="255" rhead=""/> </div> <div xml:id="echoid-div742" type="section" level="1" n="296"> <head xml:id="echoid-head305" xml:space="preserve">THEOR. XXXV. PROP. LIV.</head> <p> <s xml:id="echoid-s7076" xml:space="preserve">Si Conus rectus plano per axem ſecetur, per in quo verticem du-<lb/>cta ſit quędam linea, quę non in directum ſit poſita cum aliquo late-<lb/>rum trianguli per axem perque ipſam agatur planum, quod rectum <lb/>ſit ad idem planum, per axem ductum: </s> <s xml:id="echoid-s7077" xml:space="preserve">Huiuſmodi planum in ipſo <lb/>tantùm vertice coni ſuperficiem continget, quæ tota cadet ad alte-<lb/>ram partem ducti plani.</s> <s xml:id="echoid-s7078" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7079" xml:space="preserve">SIt conus rectus A B C plano per axem B D ſectus efficiente triangulum <lb/>A B C, in cuius plano, & </s> <s xml:id="echoid-s7080" xml:space="preserve">per verticem B ſit quælibet linea E B F, non <lb/>tamen cum aliquo laterum B A, B C ſit in directũ poſita, per quam tranſeat <lb/>planum G H I K, quod ad planum per axem A B C ſit rectum. </s> <s xml:id="echoid-s7081" xml:space="preserve">Dico tale <lb/>planum G I in nullo alio puncto, quàm in vertice B conicam ſuperficiem <lb/>contingere, &</s> <s xml:id="echoid-s7082" xml:space="preserve">c.</s> <s xml:id="echoid-s7083" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7084" xml:space="preserve">Quoniam ſi recta E B F ęquidiſtat <lb/> <anchor type="figure" xlink:label="fig-0255-01a" xlink:href="fig-0255-01"/> ipſi A C baſi trianguli per axem, an-<lb/>guli interiores E B D, A D B duobus <lb/>rectis æquales erunt, ſed A D B re-<lb/>ctus eſt, cum ſit axis B D plano baſis <lb/>A C perpendicularis, quare, & </s> <s xml:id="echoid-s7085" xml:space="preserve">an-<lb/>gulus E B D rectus erit, ſed planum <lb/>A B C ponitur rectum ad planum G <lb/>I, & </s> <s xml:id="echoid-s7086" xml:space="preserve">in eo ad communem horum ſe-<lb/>ctionem E B F ducta eſt perpendi-<lb/>cularis D B, ergo ipſa D B erit <anchor type="note" xlink:href="" symbol="a"/> re- <anchor type="note" xlink:label="note-0255-01a" xlink:href="note-0255-01"/> cta ad planum G I, eſtque eadem B <lb/>D recta ad planum baſis A C, quare <lb/>duo plana G I, A C inter ſe <anchor type="note" xlink:href="" symbol="b"/> æquidiſtant, atque eſt punctum B in vno pla- <anchor type="note" xlink:label="note-0255-02a" xlink:href="note-0255-02"/> no G I, & </s> <s xml:id="echoid-s7087" xml:space="preserve">circuli peripheria A C in altero A C, ergo recta B A, quæ ma-<lb/>nente puncto B circa peripheriam C A circumducitur conicam ſuperficiem <lb/>deſcribens, hoc eſt ipſa conica ſuperficies tota cadet inter plana ęquidiſtan-<lb/>tia (vbicunque enim ducatur planum per axem, habentur communes æqui-<lb/>diſtantium planorum fectiones inter ſe parallelę, inter quas cadit communis <lb/>ſectio ſecantis plani cum ſuperficie) ac ideò planum G I in ipſo tantùm ver-<lb/>tice B, coni ſuperficiem continget.</s> <s xml:id="echoid-s7088" xml:space="preserve"/> </p> <div xml:id="echoid-div742" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0255-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0255-01" xlink:href="note-0255-01a" xml:space="preserve">4. defin. <lb/>vndec. E-<lb/>lem.</note> <note symbol="b" position="right" xlink:label="note-0255-02" xlink:href="note-0255-02a" xml:space="preserve">14. vnd. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s7089" xml:space="preserve">Si verò recta F B E conueniet cum C A, vt in E; </s> <s xml:id="echoid-s7090" xml:space="preserve">patet, dum triangulum <lb/>B E D circa axim B D conuerti concipitur, rectam B E coni B E L ſuperfi-<lb/>ciem deſcribere, cuius triangulum per axem eſt B E L idem cum plano A B <lb/>C, cui rectum eſt planum G I ductum per latus B E, quare idem planum G <lb/>I continget conicam B E L in ipſo tantùm <anchor type="note" xlink:href="" symbol="c"/> latere B E, ſed latus B E con- <anchor type="note" xlink:label="note-0255-03a" xlink:href="note-0255-03"/> tingit conicam B C in vnico tantùm vertice B, ergo planum G I conicam <lb/>A B C in ipſo tantùm vertice B contingit, ac propterea ipſa coni ſuperficies <lb/>cadit tota infra planum G I. </s> <s xml:id="echoid-s7091" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s7092" xml:space="preserve"/> </p> <div xml:id="echoid-div743" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0255-03" xlink:href="note-0255-03a" xml:space="preserve">53. h.</note> </div> <pb o="72" file="0256" n="256" rhead=""/> </div> <div xml:id="echoid-div745" type="section" level="1" n="297"> <head xml:id="echoid-head306" xml:space="preserve">THEOR. XXXIV. PROP. LV.</head> <p> <s xml:id="echoid-s7093" xml:space="preserve">Si rectum Conoides Parabolicum, vel Hyperbolicum, aut Sphę-<lb/>ra, aut Sphæroides rectum plano per axem ſecetur, & </s> <s xml:id="echoid-s7094" xml:space="preserve">communem <lb/>ſectionem plani ſecantis cum ſolidi ſuperficie quædam recta linea <lb/>in puncto contingat, per quam ductum ſit aliud planum, quod re-<lb/>ctum ſit ei per axem ducto: </s> <s xml:id="echoid-s7095" xml:space="preserve">huiuſmodi planum in prædicto tantùm <lb/>puncto ſolidi ſuperficiem continget, ipſaque ſuperficies cadet tota <lb/>ad alteram partem plani contingentis.</s> <s xml:id="echoid-s7096" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7097" xml:space="preserve">ESto rectum Conoides Parabolicum, vel Hyperbolicum, vt in prima fi-<lb/>gura; </s> <s xml:id="echoid-s7098" xml:space="preserve">vel Sphæra, aut Sphæroides rectum, vt in ſecunda, plano per <lb/>axem B D ſectum efficiente in ſolidi ſuperficie ſectionem A B C, (quæ erit <lb/>genitrix <anchor type="note" xlink:href="" symbol="a"/> datiſolidi) & </s> <s xml:id="echoid-s7099" xml:space="preserve">per punctum E in ipſa ſumptum, ſit ei contingens li- <anchor type="note" xlink:label="note-0256-01a" xlink:href="note-0256-01"/> nea F E G, per quam concipiatur duci planum H I, quod ſit rectum plano <lb/>per axem A B C: </s> <s xml:id="echoid-s7100" xml:space="preserve">dico huiuſmodi planum H I in ipſo tantùm puncto E con-<lb/>uexam ſolidi ſuperficiem contingere, atque hanc totam cadere infra pla-<lb/>num H I.</s> <s xml:id="echoid-s7101" xml:space="preserve"/> </p> <div xml:id="echoid-div745" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0256-01" xlink:href="note-0256-01a" xml:space="preserve">ex com-<lb/>ment. Cõ. <lb/>mand. in <lb/>lib. Arch. <lb/>de Conoi. <lb/>& Sphær.</note> </div> <p> <s xml:id="echoid-s7102" xml:space="preserve">Cum enim recta F E G ſe-<lb/> <anchor type="figure" xlink:label="fig-0256-01a" xlink:href="fig-0256-01"/> ctionem A B C cõtingat, pro-<lb/>ducta conueniet <anchor type="note" xlink:href="" symbol="b"/> cum axe ſe- <anchor type="note" xlink:label="note-0256-02a" xlink:href="note-0256-02"/> ctionis B D ad partes verticis <lb/>B; </s> <s xml:id="echoid-s7103" xml:space="preserve">qua propter ſi concipiatur <lb/>planum A B C denuò conuer-<lb/>ti circa axim B D, patet ſectio-<lb/>nem A B C, dati ſolidi, & </s> <s xml:id="echoid-s7104" xml:space="preserve">cõ-<lb/>tingentem F E G, coni ſuper-<lb/>ficiem deſcribere, quæ conue-<lb/>xam ſolidi ſuperficiem per cir-<lb/>culi tantùm peripheriam à pũ-<lb/>cto E deſcriptam continget <lb/>(cum punctum E ſit tum in contingente, tum in ipſa ſectione, & </s> <s xml:id="echoid-s7105" xml:space="preserve">in reuolu-<lb/>tione peripheriam circuli deſignet, ac reliqua puncta rectæ F G ſint extra <lb/>ſectionem A B C.) </s> <s xml:id="echoid-s7106" xml:space="preserve">Et quoniam planum H I per contingentem F G du-<lb/>ctum, poſitum fuit rectum ad planum per axem A B C, quod eſt idem, ac <lb/>planum per axem coni â latere F G deſcripti, ergo planum H I ſecundùm <lb/>latus tantùm F G conicam ſuperficiem continget, <anchor type="note" xlink:href="" symbol="c"/> ſed latus F G conuexam <anchor type="note" xlink:label="note-0256-03a" xlink:href="note-0256-03"/> ſolidi ſuperficiem contingit tantùm in puncto E, quare planum H I in vnico <lb/>puncto E ſolidi ſuperficiem contingit, ac ideò hæc cadit tota infra planum <lb/>H I. </s> <s xml:id="echoid-s7107" xml:space="preserve">Quod probandum erat.</s> <s xml:id="echoid-s7108" xml:space="preserve"/> </p> <div xml:id="echoid-div746" type="float" level="2" n="2"> <figure xlink:label="fig-0256-01" xlink:href="fig-0256-01a"> <image file="0256-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0256-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0256-02" xlink:href="note-0256-02a" xml:space="preserve">24. 25. <lb/>pr. conic.</note> <note symbol="c" position="left" xlink:label="note-0256-03" xlink:href="note-0256-03a" xml:space="preserve">53. h.</note> </div> <pb o="73" file="0257" n="257" rhead=""/> </div> <div xml:id="echoid-div748" type="section" level="1" n="298"> <head xml:id="echoid-head307" xml:space="preserve">THEOR. XXXVII. PROP. LVI.</head> <p> <s xml:id="echoid-s7109" xml:space="preserve">Si coni-ſectio, vel circulus coni-ſectionem, vel circulum intus, <lb/>vel extra, in vno, aut in duobus punctis contingat, & </s> <s xml:id="echoid-s7110" xml:space="preserve">harum ſectio-<lb/>num axes, vel ſibi mutuò congruant, vel æquidiſtent, vtraque au-<lb/>tem figura, altera immota, circa proprium axem conuertatur. </s> <s xml:id="echoid-s7111" xml:space="preserve">So-<lb/>lidum factum ab vna ſectionum nunquam ſecabit ſolidum ab altera <lb/>genitum, ſed omnino ſe mutuò contingent, velin vnico puncto, ſi <lb/>figurarum planarum contactus fuerit tantùm in puncto, ſiue axes <lb/>congruant, ſiue æquidiſtent; </s> <s xml:id="echoid-s7112" xml:space="preserve">vel in duobus tantùm, ſi ad duo pun-<lb/>cta ſe mutuò contingant, dum axes ſint paralleli; </s> <s xml:id="echoid-s7113" xml:space="preserve">vel denique ad <lb/>integram circuli peripheriam à contactibus genitam, ſi ad duo <lb/>puncta ſectiones ſimul occurrant, dum axes ſimul congruant.</s> <s xml:id="echoid-s7114" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7115" xml:space="preserve">SInt duæ coni-ſectiones A B C, D E F, quarum axes ſint B G, E H, & </s> <s xml:id="echoid-s7116" xml:space="preserve"><lb/>vel ſimul congruant, vt in prima, quinta, & </s> <s xml:id="echoid-s7117" xml:space="preserve">ſexta figura, vel inter ſe æ-<lb/>quidiſtent, vt in ſecunda, tertia, & </s> <s xml:id="echoid-s7118" xml:space="preserve">quarta, atque ſe mutuò contingant, vel <lb/> <anchor type="figure" xlink:label="fig-0257-01a" xlink:href="fig-0257-01"/> in vnico puncto I, vt in prima, ſecunda, & </s> <s xml:id="echoid-s7119" xml:space="preserve">tertia, vel in duobus tantùm I <lb/>L, vt in quarta, quinta, & </s> <s xml:id="echoid-s7120" xml:space="preserve">ſexta; </s> <s xml:id="echoid-s7121" xml:space="preserve">& </s> <s xml:id="echoid-s7122" xml:space="preserve">concipiatur modò figura A B C, ma-<lb/>nente alia D E F, circa axim B G conuerti; </s> <s xml:id="echoid-s7123" xml:space="preserve">modò figura D E F, manente <lb/>altera, ita vt ab ipſis ſolida conoidalia, ſphærica, aut ſphæroidalia deſcri-<lb/>bantur. </s> <s xml:id="echoid-s7124" xml:space="preserve">Dico talia ſolida nunquam ſimul ſecari, ſed vel in vnico puncto I, <lb/>in quo plana ſe contingunt, ſe quoque mutuò contingere in prima, ſecun-<lb/>da, & </s> <s xml:id="echoid-s7125" xml:space="preserve">tertia, vel in duobus tantùm I, L, in quarta vbi axes B G, E H inter <pb o="74" file="0258" n="258" rhead=""/> ſe æquidiſtant: </s> <s xml:id="echoid-s7126" xml:space="preserve">vel tandem ad integram circuli peripheriam à contactibus I, <lb/>L in figurarum reuolutione deſcriptam in quinta, & </s> <s xml:id="echoid-s7127" xml:space="preserve">ſexta vbi axes ſimul <lb/>congruunt.</s> <s xml:id="echoid-s7128" xml:space="preserve"/> </p> <div xml:id="echoid-div748" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0257-01"/> </figure> </div> <p> <s xml:id="echoid-s7129" xml:space="preserve">Cum enim harum ſectionum axes, vel congruant ſimul, vel æquidiſtent, <lb/>quæ ad vnum ipſorum plana ducentur erecta, alteri quoque erecta erunt, <lb/>deſcribentque circulos in proprijs ſolidis, quorum centra in ipſ@s ax<gap/>s ca-<lb/>dent: </s> <s xml:id="echoid-s7130" xml:space="preserve">vnde cum axes ſimul congruent, vt in prima, quinta, & </s> <s xml:id="echoid-s7131" xml:space="preserve">ſexta, huiuſ-<lb/>modi circuli erunt concentrici; </s> <s xml:id="echoid-s7132" xml:space="preserve">at ſi æquidiſtent, vt in reliquis, circuli erunt <lb/>eccentrici, & </s> <s xml:id="echoid-s7133" xml:space="preserve">communes ſectiones horum planorum cum ipſis ſectionibus <lb/>A B C, D E F erunt eorundem circulorum diametri: </s> <s xml:id="echoid-s7134" xml:space="preserve">quare ducto quocun-<lb/>que plano A D F C ad axes erecto, non per contactus I, vel L tranſeunte, <lb/>efficiente verò in ſectione A B C diametrum A C, in ſectione autem D E F <lb/>diametrum D F: </s> <s xml:id="echoid-s7135" xml:space="preserve">patet in prima, ſecunda, quarta, quinta, & </s> <s xml:id="echoid-s7136" xml:space="preserve">ſexta figura, <lb/>in quibus ſectio D E F inſcripta eſt ſectioni A B C diametrum D F totam <lb/> <anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/> cadere intra diametrum A C, ac ideo circulum ex D F inſolido D E F <lb/>diſiunctum eſſe à circulo ex A C in ſolido A B C, vel per armillam A D <lb/>C, vt in prima, ſecunda, & </s> <s xml:id="echoid-s7137" xml:space="preserve">ſexta, ob circulorum concentricitatem, vel <lb/>per armillam excentricam A D C, in ſecunda, & </s> <s xml:id="echoid-s7138" xml:space="preserve">quarta ob ipſorum cir-<lb/>culorum excentricitatem. </s> <s xml:id="echoid-s7139" xml:space="preserve">Rurſus in tertia figura in qua ſectio D E F to-<lb/>ta cadit extra A B C, prædicta diameter D F tota cadet extra diametrum <lb/>A C, ideoque circulus ex D F in ſolido D E F totus cadet extra circulum <lb/>ex A C in ſolido A B C, & </s> <s xml:id="echoid-s7140" xml:space="preserve">hoc ſemper: </s> <s xml:id="echoid-s7141" xml:space="preserve">quare in ſingulis figuris vbicun-<lb/>que ductum ſit planum A D F C, præter ad contactus, huiuſmodi ſolida <lb/>erunt in totum diſiuncta, ex quo nullibi ſe mutuò ſecabunt.</s> <s xml:id="echoid-s7142" xml:space="preserve"/> </p> <div xml:id="echoid-div749" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0258-01"/> </figure> </div> <p> <s xml:id="echoid-s7143" xml:space="preserve">Præterea cum in prima figura ſectionum contactus ſit in ipſo axium verti-<lb/>ce, patet, & </s> <s xml:id="echoid-s7144" xml:space="preserve">ſolida circa communem axim ab ipſis ſectionibus genita in eo-<lb/>dem puncto ſe mutuò contingere. </s> <s xml:id="echoid-s7145" xml:space="preserve">In ſecunda verò tertia, & </s> <s xml:id="echoid-s7146" xml:space="preserve">quarta ducto <lb/>plano ad axes erecto per punctum contactus I, in ſolido A B C efficientæ<unsure/> <lb/>circulum, cuius diameter ſit I M, at in ſolido D E F circulum, cuius diame- <pb o="75" file="0259" n="259" rhead=""/> ter ſit I N; </s> <s xml:id="echoid-s7147" xml:space="preserve">patet tales circulos in ipſo puncto I ſe mutuò contingere, ideo-<lb/>que, & </s> <s xml:id="echoid-s7148" xml:space="preserve">ſolida in eodem contactus puncto I ſe tantùm contingere, & </s> <s xml:id="echoid-s7149" xml:space="preserve">ob ean-<lb/>demrationẽ in quarta figura in altero contactus puncto L ſe contingent, &</s> <s xml:id="echoid-s7150" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s7151" xml:space="preserve">At in quinta, & </s> <s xml:id="echoid-s7152" xml:space="preserve">ſexta, in quibus ſectiones ſunt circa communem axim B G, <lb/>& </s> <s xml:id="echoid-s7153" xml:space="preserve">in duobus punctis I, L ſe contingunt, ſi ex contactu I ducatur communis <lb/>applicata I M, & </s> <s xml:id="echoid-s7154" xml:space="preserve">producatur, ipſa ad alterum contactus punctum L omni-<lb/>no pertinget; </s> <s xml:id="echoid-s7155" xml:space="preserve">quoniam producta I M vtranque ſectionem ſecan@ in O, P, eſt <lb/>ſemi- applicata I M, in ſectione A B C, æqualis ſemi- applicatæ I M, in <lb/>ſectione D E F, ſed eſt M O in ſectione A B C æqualis I M, & </s> <s xml:id="echoid-s7156" xml:space="preserve">M P in ſe-<lb/>ctione D E F æqualis eidem I M, ergo M O, M P ſunt æquales, hoc eſt <lb/>puncta O, P vnum, ac idem ſunt; </s> <s xml:id="echoid-s7157" xml:space="preserve">quare ſectiones in puncto P ſimul con-<lb/>ueniunt, ſed conueniunt quoque in I, & </s> <s xml:id="echoid-s7158" xml:space="preserve">in duobus tantùm punctis I, & </s> <s xml:id="echoid-s7159" xml:space="preserve">L <lb/>poſitum fuit eas ſimul occurrere, ergo punctum P idem eſt, ac punctum <lb/>contactus L: </s> <s xml:id="echoid-s7160" xml:space="preserve">quare I M L eſt communis ſectionum applicata, per quam ſi <lb/>ducatur planum ad axem erectum, efficiet in vtroque ſolido circulum, cuius <lb/>diameter <anchor type="note" xlink:href="" symbol="a"/> erit eadem I L; </s> <s xml:id="echoid-s7161" xml:space="preserve">itaque per huius circuli peripheriam à puncto I <anchor type="note" xlink:label="note-0259-01a" xlink:href="note-0259-01"/> ex ſectionum reuolutione deſcriptam, huiuſmodiſolida ſe cõtingent. </s> <s xml:id="echoid-s7162" xml:space="preserve">Quod <lb/>erat vltimò demonſtrandum.</s> <s xml:id="echoid-s7163" xml:space="preserve"/> </p> <div xml:id="echoid-div750" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0259-01" xlink:href="note-0259-01a" xml:space="preserve">ex Com <lb/>mand. cõ-<lb/>ment. in <lb/>lib. Arch. <lb/>de Co-<lb/>noid.</note> </div> </div> <div xml:id="echoid-div752" type="section" level="1" n="299"> <head xml:id="echoid-head308" xml:space="preserve">PROBL. VIII. PROP. LVII.</head> <p> <s xml:id="echoid-s7164" xml:space="preserve">A puncto extra conum rectum dato ad eius conuexam ſuperfi-<lb/>ciem, MINIMAM rectam lineam ducere.</s> <s xml:id="echoid-s7165" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7166" xml:space="preserve">ESto conus rectus, cuius axis A B. </s> <s xml:id="echoid-s7167" xml:space="preserve">Oportet per punctum G datum extra <lb/>conum ad eius conuexam ſuperficiem _MINIMAM_ rectam lineam du-<lb/>cere.</s> <s xml:id="echoid-s7168" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7169" xml:space="preserve">Secetur conus, in <lb/> <anchor type="figure" xlink:label="fig-0259-01a" xlink:href="fig-0259-01"/> vtraque figura, plano <lb/>per axem A B, ac per <lb/>datum punctum G <lb/>tranſeunte, quod ef-<lb/>ficiat in ſuperficie <lb/>triangulum C A D: <lb/></s> <s xml:id="echoid-s7170" xml:space="preserve">producatur axis B A <lb/>in K; </s> <s xml:id="echoid-s7171" xml:space="preserve">& </s> <s xml:id="echoid-s7172" xml:space="preserve">cum anguli <lb/>C A B, D A B ſint ę-<lb/>quales, & </s> <s xml:id="echoid-s7173" xml:space="preserve">acuti, qui <lb/>ipſis deinceps ſunt C <lb/>A K, D A K erunt ę-<lb/>quales, & </s> <s xml:id="echoid-s7174" xml:space="preserve">obtuſi. </s> <s xml:id="echoid-s7175" xml:space="preserve">Fiant igitur ex vertice A anguli C A E, D A F recti, & </s> <s xml:id="echoid-s7176" xml:space="preserve"><lb/>primò ſit datum punctum G in prima figura in altero rectorum angulorum, <lb/>vt puta in ipſo C A E, demittaturque ex G recta G H perpendicularis late-<lb/>ri A C (quæ, vt patet _MINIMA_ eſt ad anguli latera, &</s> <s xml:id="echoid-s7177" xml:space="preserve">c.) </s> <s xml:id="echoid-s7178" xml:space="preserve">Dico ipſam G <lb/>H eſſe _MINIMAM_ quæſitam.</s> <s xml:id="echoid-s7179" xml:space="preserve"/> </p> <div xml:id="echoid-div752" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0259-01"/> </figure> </div> <p> <s xml:id="echoid-s7180" xml:space="preserve">Concipiatur enim per rectam A C duci planum N I L M, quod rectum <lb/>ſit ad planum per axem D A C, in quo eſt recta G H.</s> <s xml:id="echoid-s7181" xml:space="preserve"/> </p> <pb o="76" file="0260" n="260" rhead=""/> <p> <s xml:id="echoid-s7182" xml:space="preserve">Iam cum planum N L rectum ſit ad planum D A C, cumque in plano N <lb/>L ſit G H communi planorum ſectioni A C perpendicularis, erit ipſa G H <lb/>ad idem planum N L recta <anchor type="note" xlink:href="" symbol="a"/> hoc eſt _MINIMA_ <anchor type="note" xlink:href="" symbol="b"/> ducibilium à puncto G ad <anchor type="note" xlink:label="note-0260-01a" xlink:href="note-0260-01"/> <anchor type="note" xlink:label="note-0260-02a" xlink:href="note-0260-02"/> quodcunque aliud punctum eiuſdem plani N L, ſed conuexa coni ſuperfi-<lb/>cies tota eſt infra planum N L, ipſum tantùm contingens <anchor type="note" xlink:href="" symbol="c"/> per rectam A C, <anchor type="note" xlink:label="note-0260-03a" xlink:href="note-0260-03"/> quare eadem G H eò ampliùs _MINIMA_ erit ad conuexam dati conirecti <lb/>C A B ſuperficiem.</s> <s xml:id="echoid-s7183" xml:space="preserve"/> </p> <div xml:id="echoid-div753" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0260-01" xlink:href="note-0260-01a" xml:space="preserve">4. def. <lb/>11. Elem.</note> <note symbol="b" position="left" xlink:label="note-0260-02" xlink:href="note-0260-02a" xml:space="preserve">52. h.</note> <note symbol="c" position="left" xlink:label="note-0260-03" xlink:href="note-0260-03a" xml:space="preserve">53. h.</note> </div> <p> <s xml:id="echoid-s7184" xml:space="preserve">Si autem datum <lb/> <anchor type="figure" xlink:label="fig-0260-01a" xlink:href="fig-0260-01"/> punctum fuerit in ip-<lb/>ſa perpendiculari E <lb/>A, vt in E, eodem <lb/>modo demonſtrabi-<lb/>tur E A rectam eſſe <lb/>ad planũ N L, ideo-<lb/>que ad ipſum _MINI_-<lb/>_MAM_, & </s> <s xml:id="echoid-s7185" xml:space="preserve">eò magis <lb/>ad coni ſuperficiem.</s> <s xml:id="echoid-s7186" xml:space="preserve"/> </p> <div xml:id="echoid-div754" type="float" level="2" n="3"> <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/QN4GHYBF/figures/0260-01"/> </figure> </div> <p> <s xml:id="echoid-s7187" xml:space="preserve">Si denique datum <lb/>punctum G fuerit in-<lb/>tra angulum E A F, <lb/>vt in ſecunda figura. </s> <s xml:id="echoid-s7188" xml:space="preserve">Iungatur G A, & </s> <s xml:id="echoid-s7189" xml:space="preserve">hæc erit _MINIMA_ quæſita.</s> <s xml:id="echoid-s7190" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7191" xml:space="preserve">Nam cũ angulus C A G ſit maior recto, in plano per axem D A C, in quo <lb/>eſt A G, fiat rectus angulus O A G, & </s> <s xml:id="echoid-s7192" xml:space="preserve">linea O A producatur ad P: </s> <s xml:id="echoid-s7193" xml:space="preserve">patet A P <lb/>cadere inter A G, & </s> <s xml:id="echoid-s7194" xml:space="preserve">A D cum angulus G A P ſit rectus, & </s> <s xml:id="echoid-s7195" xml:space="preserve">duo ſimul G A F, <lb/>F A D recto ſint maiores: </s> <s xml:id="echoid-s7196" xml:space="preserve">(eſt. </s> <s xml:id="echoid-s7197" xml:space="preserve">n. </s> <s xml:id="echoid-s7198" xml:space="preserve">vnicus D A F rectus, ex conſtructione) ita-<lb/>que ſi per rectam O P concipiatur planum Q R, quod rectum ſit ad planum <lb/>D A C, in quo eſt A G, ob rationem ſuperiùs allatam, ipſa G A recta erit <lb/>ad planum Q R, hoc eſt _MINIMA_, <anchor type="note" xlink:href="" symbol="d"/> ſed planum Q R in ipſo tantùm verti- <anchor type="note" xlink:label="note-0260-04a" xlink:href="note-0260-04"/> ce A coni ſuperficiem contingit, <anchor type="note" xlink:href="" symbol="e"/> quæ tota cadit ad inferiorem partem pla- <anchor type="note" xlink:label="note-0260-05a" xlink:href="note-0260-05"/> ni Q R, quare eadem G A erit _MINIMA_ ducibilium ex G ad conuexam <lb/>coni ſuperficiem. </s> <s xml:id="echoid-s7199" xml:space="preserve">Ducta eſt ergo à puncto G extra conum rectum dato, &</s> <s xml:id="echoid-s7200" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s7201" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s7202" xml:space="preserve"/> </p> <div xml:id="echoid-div755" type="float" level="2" n="4"> <note symbol="d" position="left" xlink:label="note-0260-04" xlink:href="note-0260-04a" xml:space="preserve">52. h.</note> <note symbol="e" position="left" xlink:label="note-0260-05" xlink:href="note-0260-05a" xml:space="preserve">54. h,</note> </div> </div> <div xml:id="echoid-div757" type="section" level="1" n="300"> <head xml:id="echoid-head309" xml:space="preserve">PROBL. IX. PROP. LVIII.</head> <p> <s xml:id="echoid-s7203" xml:space="preserve">A puncto extra Conoides Parabolicum, aut Hyperbolicum, <lb/>vel Sphæram, aut Sphæroides dato, ad eius conuexam ſuperficiem <lb/>MINIMAM rectam lineam ducere.</s> <s xml:id="echoid-s7204" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7205" xml:space="preserve">ESto Conoides Parabolicũ, aut Hyperbolicũ, in 1. </s> <s xml:id="echoid-s7206" xml:space="preserve">figura, vel Sphęra, aut <lb/>Sphęroides in ſecunda, cuius axis A B, & </s> <s xml:id="echoid-s7207" xml:space="preserve">oporteat per pũctum C extra <lb/>datum ad conuexam ſolidi ſuperficiem _MINIMAM_ rectam lineam ducere.</s> <s xml:id="echoid-s7208" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7209" xml:space="preserve">Secetur datum ſolidum plano per axem A B, ac per datum punctum C, <lb/>efficiente in ſuperficie genitricem ſolidi ſectionem D A E, ad cuius peri-<lb/>pheriam ex puncto C ducatur <anchor type="note" xlink:href="" symbol="f"/> _MINIMA_ linea C F. </s> <s xml:id="echoid-s7210" xml:space="preserve">Dico hanc quoque <anchor type="note" xlink:label="note-0260-06a" xlink:href="note-0260-06"/> eſſe _MINIMAM_ ad conuexam dati ſolidi ſuperficiem.</s> <s xml:id="echoid-s7211" xml:space="preserve"/> </p> <div xml:id="echoid-div757" type="float" level="2" n="1"> <note symbol="f" position="left" xlink:label="note-0260-06" xlink:href="note-0260-06a" xml:space="preserve">20. 22. <lb/>23. h.</note> </div> <pb o="77" file="0261" n="261" rhead=""/> <p> <s xml:id="echoid-s7212" xml:space="preserve">Ducatur enim in plano ſecante D A E, per punctum F ſectionem con-<lb/>tingens G F H, quæ, (vtielicitur ex propoſitionibus 20. </s> <s xml:id="echoid-s7213" xml:space="preserve">22. </s> <s xml:id="echoid-s7214" xml:space="preserve">ac 23. </s> <s xml:id="echoid-s7215" xml:space="preserve">huius) <lb/>cum _MINIMA_ C F rectos an-<lb/>gulos efficiet. </s> <s xml:id="echoid-s7216" xml:space="preserve">Concipiatur <lb/> <anchor type="figure" xlink:label="fig-0261-01a" xlink:href="fig-0261-01"/> denique per contingentem G <lb/>H, ductum planũ L M, quod <lb/>ad planum D A E, in quo iam <lb/>ponitur eſſe C F, rectum ſit. <lb/></s> <s xml:id="echoid-s7217" xml:space="preserve">Cum ergo plana L M, D A E, <lb/>ſe mutuò ſecent per rectam G <lb/>H, cui in plano D A E ducta <lb/>eſt perpendicularis C F, erit <lb/>ipſa C F, <anchor type="note" xlink:href="" symbol="a"/> recta quoque ad <anchor type="note" xlink:label="note-0261-01a" xlink:href="note-0261-01"/> planum L M, ſiue ad idem planum ex puncto C erit <anchor type="note" xlink:href="" symbol="b"/> _MINIMA_; </s> <s xml:id="echoid-s7218" xml:space="preserve">ſed pla- <anchor type="note" xlink:label="note-0261-02a" xlink:href="note-0261-02"/> num L M conuexam ſolidi ſuperficiem contingit in puncto tantùm F, quę <lb/>cadit <anchor type="note" xlink:href="" symbol="c"/> tota infra idem planum, ergo recta C F eò magis eſt _MINIMA_ ad <anchor type="note" xlink:label="note-0261-03a" xlink:href="note-0261-03"/> conuexam ſolidi ſuperficiem D A E. </s> <s xml:id="echoid-s7219" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s7220" xml:space="preserve">c.</s> <s xml:id="echoid-s7221" xml:space="preserve"/> </p> <div xml:id="echoid-div758" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0261-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0261-01" xlink:href="note-0261-01a" xml:space="preserve">4. def. <lb/>II. Elem.</note> <note symbol="b" position="right" xlink:label="note-0261-02" xlink:href="note-0261-02a" xml:space="preserve">52. h.</note> <note symbol="c" position="right" xlink:label="note-0261-03" xlink:href="note-0261-03a" xml:space="preserve">55. h.</note> </div> </div> <div xml:id="echoid-div760" type="section" level="1" n="301"> <head xml:id="echoid-head310" xml:space="preserve">PROBL. X. PROP. LIX.</head> <p> <s xml:id="echoid-s7222" xml:space="preserve">A puncto non intra ſphæram dato, ad eius ſuperficiem, MA-<lb/>XIMAM rectam lineam ducere.</s> <s xml:id="echoid-s7223" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7224" xml:space="preserve">SIt data ſphæra, cuius centrum A, & </s> <s xml:id="echoid-s7225" xml:space="preserve">oporteat per punctum B non intra <lb/>ſphæram datum, ad eius ſuperficiẽ, _MAXIMAM_ rectam lineam ducere. <lb/></s> <s xml:id="echoid-s7226" xml:space="preserve">Iungatur B A, & </s> <s xml:id="echoid-s7227" xml:space="preserve">producatur, donec <lb/> <anchor type="figure" xlink:label="fig-0261-02a" xlink:href="fig-0261-02"/> ſphæricæ ſuperficiei occurrat in D, & </s> <s xml:id="echoid-s7228" xml:space="preserve">E. <lb/></s> <s xml:id="echoid-s7229" xml:space="preserve">Dico B E, in qua eſt centrum, eſſe _MAXI_-<lb/>_MAM._</s> <s xml:id="echoid-s7230" xml:space="preserve"/> </p> <div xml:id="echoid-div760" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0261-02"/> </figure> </div> <p> <s xml:id="echoid-s7231" xml:space="preserve">Concipiatur per B E ductum planum, <lb/>quod in ſpæræ ſuperficie maximum circu-<lb/>lum deſignabit D F E, ad cuius periphe-<lb/>riam eſt recta B E _MAXIMA._</s> <s xml:id="echoid-s7232" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7233" xml:space="preserve">Iam in plano circuli D F E, cum radio <lb/>B E deſcripto circulo G E H, & </s> <s xml:id="echoid-s7234" xml:space="preserve">circa im-<lb/>motum axim B E reuoluto, ab ipſo deſcri-<lb/>betur ſphæra G E H, quæ datam ſphæram <lb/>D F E circa eundem axim deſcriptam comprehendet, ac ſe ſimul contingét <lb/>in ipſo <anchor type="note" xlink:href="" symbol="d"/> circulorum contactu E, ſed quæ à centro B ad ſphæricam ſuperfi- <anchor type="note" xlink:label="note-0261-04a" xlink:href="note-0261-04"/> ciem G E H ducuntur omnes ſunt æquales rectæ B E, ergo quæ ab eodem <lb/>puncto B ad interioris ſphæræ D F E ſuperficiem ducentur ipſa B E mino-<lb/>res erunt. </s> <s xml:id="echoid-s7235" xml:space="preserve">Vnde B E eſt _MAXIMA_ quæſita, &</s> <s xml:id="echoid-s7236" xml:space="preserve">c. </s> <s xml:id="echoid-s7237" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s7238" xml:space="preserve">c.</s> <s xml:id="echoid-s7239" xml:space="preserve"/> </p> <div xml:id="echoid-div761" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-0261-04" xlink:href="note-0261-04a" xml:space="preserve">56. h.</note> </div> <pb o="78" file="0262" n="262" rhead=""/> </div> <div xml:id="echoid-div763" type="section" level="1" n="302"> <head xml:id="echoid-head311" xml:space="preserve">PROBL. XI. PROP. LX.</head> <p> <s xml:id="echoid-s7240" xml:space="preserve">A puncto intra ſphæram dato, ad eius concauam ſuperficiem, <lb/>_MAXIMAM, & </s> <s xml:id="echoid-s7241" xml:space="preserve">MINIMAM rectam lineam ducere._</s> <s xml:id="echoid-s7242" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7243" xml:space="preserve">ESto ſphæra, cuius centrum A, & </s> <s xml:id="echoid-s7244" xml:space="preserve">oporteat per datum intra ipſam pun-<lb/>ctum B ad concauam ſphæræ ſuperficiem _MAXIMAM_, & </s> <s xml:id="echoid-s7245" xml:space="preserve">_MINIMAM_ <lb/>rectam lineam ducere.</s> <s xml:id="echoid-s7246" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7247" xml:space="preserve">Si punctum B fuerit in centro ſphæræ, patet tunc neque _MAXIMAM,_ <lb/>neque _MINIMAM_ dari, cum omnes eductæ à centro ad ſphærę ſuperficiem <lb/>ſint æquales.</s> <s xml:id="echoid-s7248" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7249" xml:space="preserve">Si autem datum punctum fuerit præter cen-<lb/>trum: </s> <s xml:id="echoid-s7250" xml:space="preserve">iungatur cum centro A recta B A, quæ <lb/> <anchor type="figure" xlink:label="fig-0262-01a" xlink:href="fig-0262-01"/> hinc inde producta occurrat ſphęricæ ſuperficiei <lb/>in punctis C, D. </s> <s xml:id="echoid-s7251" xml:space="preserve">Dico B D, in quà eſt centrum, <lb/>eſſe _MAXIMAM_, reliquam B C _MINIMAM_.</s> <s xml:id="echoid-s7252" xml:space="preserve"/> </p> <div xml:id="echoid-div763" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0262-01"/> </figure> </div> <p> <s xml:id="echoid-s7253" xml:space="preserve">Si enim circà axim C D intelligatur quicun-<lb/>que _MAXIMVS_ ſphæræ circulus C D F: </s> <s xml:id="echoid-s7254" xml:space="preserve">patet <lb/>linearum ex B ad peripheriam C D F ducibi-<lb/>lium, B D in qua centrum A, eſſe _MAXIMAM_, <lb/>& </s> <s xml:id="echoid-s7255" xml:space="preserve">B C _MINIMAM_.</s> <s xml:id="echoid-s7256" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7257" xml:space="preserve">Siverò ducta ſit quælibet alia B E extra peri-<lb/>pheriam C D F, ſphæricæ ſuperficiei occurrens <lb/>in E; </s> <s xml:id="echoid-s7258" xml:space="preserve">per rectas C D, & </s> <s xml:id="echoid-s7259" xml:space="preserve">B E intelligatur pla-<lb/>num, cuius communis ſectio cum ſphæræ ſuperficie erit cuiuſdam _MAXIMI_ <lb/>circuli peripheria C E D, & </s> <s xml:id="echoid-s7260" xml:space="preserve">eius diameter C D: </s> <s xml:id="echoid-s7261" xml:space="preserve">quare B D, in qua eſt <lb/>centrum, cum ſit _MAXIMA_, erit maior B E; </s> <s xml:id="echoid-s7262" xml:space="preserve">& </s> <s xml:id="echoid-s7263" xml:space="preserve">B C, cum ſit _MINIMA_ <lb/>minor erit eadem B E, & </s> <s xml:id="echoid-s7264" xml:space="preserve">hoc ſemper vbicunque pertingat ducta B E: </s> <s xml:id="echoid-s7265" xml:space="preserve">ideo-<lb/>que B D eſt _MAXIMA_ ad vniuerſam ſphæræ ſuperficiem ducibilium ex da-<lb/>to puncto B, & </s> <s xml:id="echoid-s7266" xml:space="preserve">B C _MINIMA_. </s> <s xml:id="echoid-s7267" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s7268" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div765" type="section" level="1" n="303"> <head xml:id="echoid-head312" xml:space="preserve">PROBL. XII. PROP. LXI.</head> <p> <s xml:id="echoid-s7269" xml:space="preserve">A puncto intra Conum rectum, vel Conoides Parabolicum, <lb/>aut Hyperbolicum dato, ad eius concauam ſuperficiem, MI-<lb/>NIMAM rectam lineam ducere.</s> <s xml:id="echoid-s7270" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7271" xml:space="preserve">ESto Conus rectus; </s> <s xml:id="echoid-s7272" xml:space="preserve">vt in prima ſigura, vel Conoides Parabolicum, aut <lb/>Hyperbolicum, vt in ſecunda, cuius axis A B, & </s> <s xml:id="echoid-s7273" xml:space="preserve">oporteat per punctum <lb/>intra ipſum datum ad concauam ſolidi ſuperficiem _MINIMAM_ rectam li-<lb/>neam ducere.</s> <s xml:id="echoid-s7274" xml:space="preserve"/> </p> <note symbol="a" position="left" xml:space="preserve">ex Com <lb/>ment. Có-<lb/>mand. in <lb/>12. Arch. <lb/>de Co. <lb/>noid. & <lb/>Spheroid,</note> <p> <s xml:id="echoid-s7275" xml:space="preserve">Secetur ſolidum plano per axem A B, ac per datum punctum ducto effi-<lb/>ciente in ſolidi ſuperficie ſectionem D A E, quæ eadem erit, <anchor type="note" xlink:href="" symbol="a"/> ac ipſius ſo- lidi genitrix ſectio, & </s> <s xml:id="echoid-s7276" xml:space="preserve">in Cono angulum rectilineum conſtituet.</s> <s xml:id="echoid-s7277" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7278" xml:space="preserve">Iam ſi datum punctum fuerit in axe; </s> <s xml:id="echoid-s7279" xml:space="preserve">vt in H; </s> <s xml:id="echoid-s7280" xml:space="preserve">ducta H D, quæ in ſe- <pb file="0263" n="263"/> <pb file="0264" n="264"/> <anchor type="figure" xlink:label="fig-0264-01a" xlink:href="fig-0264-01"/> <pb o="79" file="0265" n="265" rhead=""/> ctione D A E ſit _MINIMA_, (ſed quæ in angulo, primæ figuræ, erit perpen-<lb/>dicularis ad A D) ipſa H D erit quoque _MINIMA_ in ſolido.</s> <s xml:id="echoid-s7281" xml:space="preserve"/> </p> <div xml:id="echoid-div765" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0264-01"/> </figure> </div> <p> <s xml:id="echoid-s7282" xml:space="preserve">Nam ſi H D eſt _MINIMA_ <lb/>ad peripheriam D A E patet <lb/> <anchor type="figure" xlink:label="fig-0265-01a" xlink:href="fig-0265-01"/> ex 20. </s> <s xml:id="echoid-s7283" xml:space="preserve">22. </s> <s xml:id="echoid-s7284" xml:space="preserve">ac 23. </s> <s xml:id="echoid-s7285" xml:space="preserve">huius, ipſam <lb/>H D perpendicularem eſſe <lb/>rectæ F D G, quæ ad pun-<lb/>ctum D ſectionem contingat. <lb/></s> <s xml:id="echoid-s7286" xml:space="preserve">Si ergo centro H, interuallo <lb/>H D circulus deſcribatur <anchor type="note" xlink:href="" symbol="a"/> D <anchor type="note" xlink:label="note-0265-01a" xlink:href="note-0265-01"/> E B, ipſe cadet totus intra ſe-<lb/>ctionem, eam contingens tan-<lb/>tùm in duobus punctis D E: <lb/></s> <s xml:id="echoid-s7287" xml:space="preserve">quare in reuolutione ſectio-<lb/>nis D A E circa axim A B <lb/>deſcribetur datum ſolidum, & </s> <s xml:id="echoid-s7288" xml:space="preserve">à circulo ſphæra, quæ tota cadet intra ſoli-<lb/>dum, eius concauam ſuperficiem contingens <anchor type="note" xlink:href="" symbol="b"/> tantùm per peripheriam D I <anchor type="note" xlink:label="note-0265-02a" xlink:href="note-0265-02"/> E eius circuli, qui in reuolutione deſcribitur à puncto D; </s> <s xml:id="echoid-s7289" xml:space="preserve">& </s> <s xml:id="echoid-s7290" xml:space="preserve">ipſa H D, vna <lb/>cum qualibet alia eductarum ab H ad prædictam peripheriam D I E, erit <lb/>_MINIMA_ in ſolido quæſita; </s> <s xml:id="echoid-s7291" xml:space="preserve">cum hæ omnes ſint æquales inter ſe, eò quod <lb/>ſint latera Conirecti, cuius baſis eſt circulus D I E, vertex H; </s> <s xml:id="echoid-s7292" xml:space="preserve">cumque om-<lb/>nes alię eductæ ab H ad ſolidi ſuperficiem, occurrant priùs ſphęricæ ſuper-<lb/>ficiei (quæ cadit tota intra ſolidi ſuperficiem) quàm ſuperficiei conicæ, aut <lb/>dati ſolidi conoidalis.</s> <s xml:id="echoid-s7293" xml:space="preserve"/> </p> <div xml:id="echoid-div766" type="float" level="2" n="2"> <figure xlink:label="fig-0265-01" xlink:href="fig-0265-01a"> <image file="0265-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0265-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0265-01" xlink:href="note-0265-01a" xml:space="preserve">92. pri-<lb/>mihuius.</note> <note symbol="b" position="right" xlink:label="note-0265-02" xlink:href="note-0265-02a" xml:space="preserve">56. h.</note> </div> <p> <s xml:id="echoid-s7294" xml:space="preserve">Siverò datum punctum ſit C inter axem, & </s> <s xml:id="echoid-s7295" xml:space="preserve">ſectionem: </s> <s xml:id="echoid-s7296" xml:space="preserve">ducta item C D, <lb/>quæ in ſectione ſit <anchor type="note" xlink:href="" symbol="c"/> _MINIMA_. </s> <s xml:id="echoid-s7297" xml:space="preserve">Dico ipſam quoque eſſe _MINIMAM_ in ſo- <anchor type="note" xlink:label="note-0265-03a" xlink:href="note-0265-03"/> lido.</s> <s xml:id="echoid-s7298" xml:space="preserve"/> </p> <div xml:id="echoid-div767" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0265-03" xlink:href="note-0265-03a" xml:space="preserve">20. 22. <lb/>23. h.</note> </div> <p> <s xml:id="echoid-s7299" xml:space="preserve">Cum enim C D ſit _MINIMA_ ad ſectionis peripheriam D A E, ipſa C D <lb/>erit contingenti F D G perpendicularis, quare, & </s> <s xml:id="echoid-s7300" xml:space="preserve">producta axi <anchor type="note" xlink:href="" symbol="d"/> occurret, <anchor type="note" xlink:label="note-0265-04a" xlink:href="note-0265-04"/> vt in H: </s> <s xml:id="echoid-s7301" xml:space="preserve">quo facto centro, ac interuallo H D deſcripto circulo D E B, & </s> <s xml:id="echoid-s7302" xml:space="preserve"><lb/>facta reuolutione circa axim A B, procreabitur denuo datum ſolidum, & </s> <s xml:id="echoid-s7303" xml:space="preserve"><lb/>ſphæra, cuius ſuperficies cadet tota intra <anchor type="note" xlink:href="" symbol="e"/> ſolidi ſuperficiem, ſed recta C D <anchor type="note" xlink:label="note-0265-05a" xlink:href="note-0265-05"/> eſt _MINIMA_ <anchor type="note" xlink:href="" symbol="f"/> à puncto C ad ſphæræ ſuperficiem eductarú quare ipſa C <anchor type="note" xlink:label="note-0265-06a" xlink:href="note-0265-06"/> D eſt omnino _MINIMA_ ex C ducibilium ad concauam, & </s> <s xml:id="echoid-s7304" xml:space="preserve">exteriorem ſo-<lb/>lidi ſuperficiem. </s> <s xml:id="echoid-s7305" xml:space="preserve">Quod facere oportebat.</s> <s xml:id="echoid-s7306" xml:space="preserve"/> </p> <div xml:id="echoid-div768" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-0265-04" xlink:href="note-0265-04a" xml:space="preserve">88. pr. h.</note> <note symbol="e" position="right" xlink:label="note-0265-05" xlink:href="note-0265-05a" xml:space="preserve">56. h.</note> <note symbol="f" position="right" xlink:label="note-0265-06" xlink:href="note-0265-06a" xml:space="preserve">ex 60. h.</note> </div> </div> <div xml:id="echoid-div770" type="section" level="1" n="304"> <head xml:id="echoid-head313" xml:space="preserve">PROBL. XIII. PROP. LXII.</head> <p> <s xml:id="echoid-s7307" xml:space="preserve">A puncto vbicunque dato, ad Sphæroidis ſuperficiem, MAXI-<lb/> <anchor type="note" xlink:label="note-0265-07a" xlink:href="note-0265-07"/> MAM, & </s> <s xml:id="echoid-s7308" xml:space="preserve">MINIMAM rectam lineam ducere.</s> <s xml:id="echoid-s7309" xml:space="preserve"/> </p> <div xml:id="echoid-div770" type="float" level="2" n="1"> <note position="right" xlink:label="note-0265-07" xlink:href="note-0265-07a" xml:space="preserve">Schema-<lb/>tiſmus 4.</note> </div> <p> <s xml:id="echoid-s7310" xml:space="preserve">ESto datum Sphæroides A B C D, cuius axis reuolutionis ſit B D, cen-<lb/>trum E, & </s> <s xml:id="echoid-s7311" xml:space="preserve">punctum datum ſit F. </s> <s xml:id="echoid-s7312" xml:space="preserve">Oportet primò ex F ad Sphæroidis <lb/>fuperficiem _MAXIMAM_ rectam lineam ducere.</s> <s xml:id="echoid-s7313" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7314" xml:space="preserve">Pro huius lineæ indagatione, generalis conſtructio in ſingulis figuris <lb/>quarti Schematiſmi, talis eſt.</s> <s xml:id="echoid-s7315" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7316" xml:space="preserve">Secetur Sphæroides A B C D plano per axem B D, ac per datum pun- <pb o="80" file="0266" n="266" rhead=""/> ctum F ducto, ſectionem eſſicient in ſolido ſiguram A B C D, quæ ſemper <lb/>eſt eadem, ac Ellipſis quæ ſolidum genuit; </s> <s xml:id="echoid-s7317" xml:space="preserve">& </s> <s xml:id="echoid-s7318" xml:space="preserve">à dato puncto F ad huius ſe-<lb/>ctionis peripheriam ducatur <anchor type="note" xlink:href="" symbol="a"/> _MAXIMA_ linea. </s> <s xml:id="echoid-s7319" xml:space="preserve">Dico ipſam quoque eſſe <anchor type="note" xlink:label="note-0266-01a" xlink:href="note-0266-01"/> _MAXIMAM_ ad ſolidi ſuperficiem.</s> <s xml:id="echoid-s7320" xml:space="preserve"/> </p> <div xml:id="echoid-div771" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">23. h.</note> </div> <p> <s xml:id="echoid-s7321" xml:space="preserve">Iam, vel datum Sphæroides eſt Oblongum, vt in 9. </s> <s xml:id="echoid-s7322" xml:space="preserve">primis figuris; </s> <s xml:id="echoid-s7323" xml:space="preserve">vel <lb/>Prolatum, vt in totidem proximè ſequentibus.</s> <s xml:id="echoid-s7324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7325" xml:space="preserve">Si primum: </s> <s xml:id="echoid-s7326" xml:space="preserve">vel datum punctum F idem eſt cum centro E, vt in prima fi-<lb/>gura, & </s> <s xml:id="echoid-s7327" xml:space="preserve">tunc duo ſemi- axes maiores F B, F D erunt _MAXIMAE_ ad Ellipſis <lb/>peripheriam per 23. </s> <s xml:id="echoid-s7328" xml:space="preserve">huius ad num. </s> <s xml:id="echoid-s7329" xml:space="preserve">1. </s> <s xml:id="echoid-s7330" xml:space="preserve">Vel eſt in maiori axe B D, hoc eſt in-<lb/>ter verticem, & </s> <s xml:id="echoid-s7331" xml:space="preserve">centrum, vt in ſecunda, & </s> <s xml:id="echoid-s7332" xml:space="preserve">tunc F D tantùm, in qua eſt <lb/>centrum eſt _MAXIMA_, vt ad num. </s> <s xml:id="echoid-s7333" xml:space="preserve">4. </s> <s xml:id="echoid-s7334" xml:space="preserve">& </s> <s xml:id="echoid-s7335" xml:space="preserve">5. </s> <s xml:id="echoid-s7336" xml:space="preserve">Aut in ipſo vertice B, vt in ter-<lb/>tia, quo in caſu F B, item eſt _MAXIMA_, vt ad num. </s> <s xml:id="echoid-s7337" xml:space="preserve">2. </s> <s xml:id="echoid-s7338" xml:space="preserve">Vel in ipſo maiori <lb/>axe, extra tamen ſectionem, vt in quarta, & </s> <s xml:id="echoid-s7339" xml:space="preserve">tunc ipſa F D, in qua eſt cen-<lb/>trum pariter eſt _MAXIMA_, vt ad num. </s> <s xml:id="echoid-s7340" xml:space="preserve">3. </s> <s xml:id="echoid-s7341" xml:space="preserve">Vel eſt in minori axe A C, hŏc eſt <lb/>vel diſtans à vertice A per interuallum F A non minus dimidio recti, cuius <lb/>tranſuerſum eſt A C, vt in quinta figura, & </s> <s xml:id="echoid-s7342" xml:space="preserve">tunc ipſa F A eſt _MAXIMA_, vt <lb/>ad num. </s> <s xml:id="echoid-s7343" xml:space="preserve">6. </s> <s xml:id="echoid-s7344" xml:space="preserve">Aut diſtat ab A per interuallum minus prædicto dimidio, vt in <lb/>ſexta figura, & </s> <s xml:id="echoid-s7345" xml:space="preserve">ſic duæ tantùm F H, F G ſunt _MAXIMAE_, vt ad num. </s> <s xml:id="echoid-s7346" xml:space="preserve">7. </s> <s xml:id="echoid-s7347" xml:space="preserve">Vel <lb/>denique datum punctum F eſt inter axes, & </s> <s xml:id="echoid-s7348" xml:space="preserve">hoc, vel in ipſa peripheria, vt <lb/>in ſeptima figura, vel intra, vt in octaua, vel extra, vt in nona, atque in his <lb/>caſibus vna tantùm duci poteſt ex F _MAXIMA_, vt ad num. </s> <s xml:id="echoid-s7349" xml:space="preserve">9. </s> <s xml:id="echoid-s7350" xml:space="preserve">quæ ſit F G.</s> <s xml:id="echoid-s7351" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7352" xml:space="preserve">Si autem Sphæroides fuerit Prolatum, vt in nouem proximis figuris eiuſ-<lb/>dem Schematiſmi, vel datum punctũ eſt idem cum centro F, vt in 10. </s> <s xml:id="echoid-s7353" xml:space="preserve">figura, <lb/>& </s> <s xml:id="echoid-s7354" xml:space="preserve">tunc duo ſemi - axes maiores F A, F C erunt _MAXIMAE_ ad Ellipſis pe-<lb/>ripheriam. </s> <s xml:id="echoid-s7355" xml:space="preserve">Vel eſt in maiori axe, & </s> <s xml:id="echoid-s7356" xml:space="preserve">hoc vel inter centrum, & </s> <s xml:id="echoid-s7357" xml:space="preserve">verticem C, <lb/>vt in vndecima, vel in ipſo vertice, vt in duodecima, vel extra verticem vt <lb/>in decimatertia, quibus in caſibus F A, in qua centrum, eſt _MAXIMA_. </s> <s xml:id="echoid-s7358" xml:space="preserve">Vel <lb/>eſt in minori axe diſtans à vertice B per interuallum non minus dimidio re-<lb/>cti, cuius tranſuerſum ſit B D, vt in decima quarta figura, & </s> <s xml:id="echoid-s7359" xml:space="preserve">tunc F B, vel <lb/>F G eſt _MAXIMA_, vel diſtans à vertice B per interuallum minus prædicto <lb/>dimidio, vt in decimaquinta, & </s> <s xml:id="echoid-s7360" xml:space="preserve">tunc duæ ſunt _MAXIMAE_ F G, F H. </s> <s xml:id="echoid-s7361" xml:space="preserve">Vel <lb/>eſt inter axes, & </s> <s xml:id="echoid-s7362" xml:space="preserve">hoc aut in ipſa peripheria, aut intra, aut extra, vt in 16. </s> <s xml:id="echoid-s7363" xml:space="preserve">17. <lb/></s> <s xml:id="echoid-s7364" xml:space="preserve">& </s> <s xml:id="echoid-s7365" xml:space="preserve">18. </s> <s xml:id="echoid-s7366" xml:space="preserve">in quibus vna tantùm F G eſt _MAXIMA_, quæ omnia ad præcitatos <lb/>numeros propoſ. </s> <s xml:id="echoid-s7367" xml:space="preserve">23. </s> <s xml:id="echoid-s7368" xml:space="preserve">huius ſunt demonſtrata. </s> <s xml:id="echoid-s7369" xml:space="preserve">Si ergo in ſingulis figuris ad <lb/>interuallum _MAXIMAE_ repertæ F D, vel F G reſpectiuè, cum centro dati <lb/>puncti F circulus deſcribatur, ipſe cadet totus extra Ellipſim, hanc tantùm <lb/>contingens in eò puncto, vel in ijs duobus ad quę _MAXIMA_, vel _MAXIMAE_ <lb/>perueniunt; </s> <s xml:id="echoid-s7370" xml:space="preserve">nam ſi circulus alibi cum Ellipſi conueniret _MAXIMAE_ quoq; </s> <s xml:id="echoid-s7371" xml:space="preserve"><lb/>plures eſſent quàm vna, vel duæ reſpectiuè, quod eſt contra oſtenſa in 23. </s> <s xml:id="echoid-s7372" xml:space="preserve"><lb/>huius.</s> <s xml:id="echoid-s7373" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7374" xml:space="preserve">Præterea, vbi F centrum deſcripti circuli G H non eſt in B D axe reuo-<lb/>lutionis Ellipſis A B C D, vti reperitur in 1. </s> <s xml:id="echoid-s7375" xml:space="preserve">2. </s> <s xml:id="echoid-s7376" xml:space="preserve">3. </s> <s xml:id="echoid-s7377" xml:space="preserve">4. </s> <s xml:id="echoid-s7378" xml:space="preserve">10. </s> <s xml:id="echoid-s7379" xml:space="preserve">14. </s> <s xml:id="echoid-s7380" xml:space="preserve">ac 15. </s> <s xml:id="echoid-s7381" xml:space="preserve">figura, in <lb/>quibus eadem B D eſt circuli diameter, ducatur I F L diameter circuli G H, <lb/>atque axi B D ęquidiſtans; </s> <s xml:id="echoid-s7382" xml:space="preserve">& </s> <s xml:id="echoid-s7383" xml:space="preserve">concipiatur, modo circulum circa diametrum <lb/>I L, tanquam circa axim conuerti, interea manente Ellipſi, & </s> <s xml:id="echoid-s7384" xml:space="preserve">fiet ſphæra G <lb/>H, modò Ellipſim circa axim B D, manente tamen circulo, & </s> <s xml:id="echoid-s7385" xml:space="preserve">procreabitur <lb/>Sphæroides A B C D, quod cadet totum intra ſphæram, hanc <anchor type="note" xlink:href="" symbol="b"/> tantùm con- <anchor type="note" xlink:label="note-0266-02a" xlink:href="note-0266-02"/> <pb o="81" file="0267" n="267" rhead=""/> ad vnicum punctum D, aut G, vt in 2. </s> <s xml:id="echoid-s7386" xml:space="preserve">3. </s> <s xml:id="echoid-s7387" xml:space="preserve">4. </s> <s xml:id="echoid-s7388" xml:space="preserve">5. </s> <s xml:id="echoid-s7389" xml:space="preserve">7. </s> <s xml:id="echoid-s7390" xml:space="preserve">8. </s> <s xml:id="echoid-s7391" xml:space="preserve">9. </s> <s xml:id="echoid-s7392" xml:space="preserve">11. </s> <s xml:id="echoid-s7393" xml:space="preserve">12. </s> <s xml:id="echoid-s7394" xml:space="preserve">13. </s> <s xml:id="echoid-s7395" xml:space="preserve">14. </s> <s xml:id="echoid-s7396" xml:space="preserve">16. <lb/></s> <s xml:id="echoid-s7397" xml:space="preserve">17. </s> <s xml:id="echoid-s7398" xml:space="preserve">ac 18. </s> <s xml:id="echoid-s7399" xml:space="preserve">figura, quoniam in his quoque vnicus eſt contactus inter circulũ, <lb/>& </s> <s xml:id="echoid-s7400" xml:space="preserve">Ellipſim; </s> <s xml:id="echoid-s7401" xml:space="preserve">vel ad duo tantùm puncta B, D, vt in prima, aut G, H, vt in <lb/>ſexta, in quot circulus Ellipſim contingit, & </s> <s xml:id="echoid-s7402" xml:space="preserve">quæ non ſunt extrema eiuſdem <lb/>applicatæ in vtraq; </s> <s xml:id="echoid-s7403" xml:space="preserve">ſectione ad communem axim; </s> <s xml:id="echoid-s7404" xml:space="preserve">vel tandem ad integram <lb/>circuli peripheriam à puncto A in decima figura, vel à puncto G in 15. </s> <s xml:id="echoid-s7405" xml:space="preserve">ex <lb/>figurarum reuolutione circa communem axim B D deſcriptam. </s> <s xml:id="echoid-s7406" xml:space="preserve">Cum ergo <lb/>Sphæra G H claudat Sphæroides A B C D, atque ipſum contingat tantùm, <lb/>vel in vno, vel in duobus punctis, vel ad integram circuli peripheriam, cũq; </s> <s xml:id="echoid-s7407" xml:space="preserve"><lb/>omnes rectæ, quæ à centro F ad punctum ſphæricæ ſuperficiei duci poſſunt <lb/>ſint æquales ijs, quæ ad prædicta contactuum pũcta, vel peripherias ducun-<lb/>tur, ideò quæ ab eodem centro ad incluſam Sphæroidis ſuperficiem, præter <lb/>ad prædicta puncta, vel peripherias ducentur minores erunt, ac propterea <lb/>ipſæ eductæ à centro F, ſiue à puncto dato ad prędicta puncta, vel periphe-<lb/>rias in Sphæroidis ſuperficie erunt _MAXIMAE_ quæſitæ. </s> <s xml:id="echoid-s7408" xml:space="preserve">Quod erat pri-<lb/>mò faciendum.</s> <s xml:id="echoid-s7409" xml:space="preserve"/> </p> <div xml:id="echoid-div772" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-0266-02" xlink:href="note-0266-02a" xml:space="preserve">56. h.</note> </div> <p> <s xml:id="echoid-s7410" xml:space="preserve">SIverò ad Sphæroidis ſuperficiem A B C D ducenda ſit _MINIMA_ linea à <lb/>puncto dato F. </s> <s xml:id="echoid-s7411" xml:space="preserve">Vel datum punctum eſt in ipſa ſuperficie, & </s> <s xml:id="echoid-s7412" xml:space="preserve">tunc _MI-_ <lb/>_NIMA_ in punctum abit. </s> <s xml:id="echoid-s7413" xml:space="preserve">Vel cadit extra, & </s> <s xml:id="echoid-s7414" xml:space="preserve">tunc _MINIMA_ reperitur, vt in <lb/>58. </s> <s xml:id="echoid-s7415" xml:space="preserve">huius. </s> <s xml:id="echoid-s7416" xml:space="preserve">Vel tandem eſt intra Sphæroides, & </s> <s xml:id="echoid-s7417" xml:space="preserve">tunc ad _MINIMAM_ venan-<lb/>dam generalis conſtructio eſt huiuſmodi.</s> <s xml:id="echoid-s7418" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7419" xml:space="preserve">Secetur Sphæroides plano per axem B D, & </s> <s xml:id="echoid-s7420" xml:space="preserve">per datum punctum F, geni-<lb/>tricem Ellipſim efficiente A B C D, ductaque ex F ad Ellipſis peripheriam <lb/>_MINIMA_<anchor type="note" xlink:href="" symbol="*"/> recta linea, ipſa quoque erit _MINIMA_ ad Sphæroidis ſuperficiẽ.</s> <s xml:id="echoid-s7421" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">23. h.</note> <p> <s xml:id="echoid-s7422" xml:space="preserve">Iam, vel datum Sphæroides eſt Oblongum, vel Prolatum. </s> <s xml:id="echoid-s7423" xml:space="preserve">Sit primò <lb/>Oblongum, vt in figuris 19. </s> <s xml:id="echoid-s7424" xml:space="preserve">20. </s> <s xml:id="echoid-s7425" xml:space="preserve">21. </s> <s xml:id="echoid-s7426" xml:space="preserve">22. </s> <s xml:id="echoid-s7427" xml:space="preserve">23. </s> <s xml:id="echoid-s7428" xml:space="preserve">Itaque datum punctum F, vel <lb/>eſt in centro, vt in 19. </s> <s xml:id="echoid-s7429" xml:space="preserve">& </s> <s xml:id="echoid-s7430" xml:space="preserve">tunc duæ F A, F C, ſunt _MINIMAE_, vel in ma-<lb/>iori axe A B diſtans à vertice B per interuallum maius dimidio recti, &</s> <s xml:id="echoid-s7431" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s7432" xml:space="preserve">itemque duæ F G, F H ſunt _MINIMAE_, vt in 20. </s> <s xml:id="echoid-s7433" xml:space="preserve">vel per interuallum non <lb/>maius prædicto dimidio, vt in 21. </s> <s xml:id="echoid-s7434" xml:space="preserve">& </s> <s xml:id="echoid-s7435" xml:space="preserve">tunc vnica F B, in qua non eſt centrũ, <lb/>eſt _MINIMA_; </s> <s xml:id="echoid-s7436" xml:space="preserve">vel eſt in minori axe, vt in 22. </s> <s xml:id="echoid-s7437" xml:space="preserve">in qua F C vbi centrum non <lb/>reperitur eſt _MINIMA_; </s> <s xml:id="echoid-s7438" xml:space="preserve">vel tandem eſt inter axes, vt in 23. </s> <s xml:id="echoid-s7439" xml:space="preserve">& </s> <s xml:id="echoid-s7440" xml:space="preserve">tunc vnica F <lb/>G eſt _MINIMA_, &</s> <s xml:id="echoid-s7441" xml:space="preserve">c.</s> <s xml:id="echoid-s7442" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7443" xml:space="preserve">Sit denique Sphæroides Prolatum, vt in poſtremis figuris huius quarti <lb/>Schematiſmi. </s> <s xml:id="echoid-s7444" xml:space="preserve">Si punctum F congruit cum centro E, vt in 24. </s> <s xml:id="echoid-s7445" xml:space="preserve">figura duæ F <lb/>D, F B ſunt _MINIMAE_; </s> <s xml:id="echoid-s7446" xml:space="preserve">ſi eſt in ſemi- axe maiori E C, diſtans à C per in-<lb/>teruallum maius recti dimidio, &</s> <s xml:id="echoid-s7447" xml:space="preserve">c. </s> <s xml:id="echoid-s7448" xml:space="preserve">vt in 25. </s> <s xml:id="echoid-s7449" xml:space="preserve">duo item F G, F H ſunt _MINI-_ <lb/>_MAE_; </s> <s xml:id="echoid-s7450" xml:space="preserve">ſi per interuallum non maius prædicto dimidio, vt in 26. </s> <s xml:id="echoid-s7451" xml:space="preserve">vnica F C <lb/>eſt _MINIMA_; </s> <s xml:id="echoid-s7452" xml:space="preserve">ſi in ſemi- axe minori E B, vt in 27. </s> <s xml:id="echoid-s7453" xml:space="preserve">ipſa F B, in qua non eſt <lb/>centrum eſt _MINIMA_; </s> <s xml:id="echoid-s7454" xml:space="preserve">ſi tandem inter axes, vt in 28. </s> <s xml:id="echoid-s7455" xml:space="preserve">vnica F G eſt _MINI-_ <lb/>_MA_, quæ omnia in prop. </s> <s xml:id="echoid-s7456" xml:space="preserve">23. </s> <s xml:id="echoid-s7457" xml:space="preserve">huius ſunt demonſtrata.</s> <s xml:id="echoid-s7458" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7459" xml:space="preserve">Siergo in his omnibus figuris cum centro F, ad interuallum nuper inuen-<lb/>tæ _MINIMAE_ deſcribatur circulus G H, ipſe circumſcriptus erit Ellipſi, <lb/>hanc tantùm contingens in eo, vel in ijs punctis, ad quæ _MINIMA_, vel <lb/>_MINIMAE_ perueniunt; </s> <s xml:id="echoid-s7460" xml:space="preserve">nam ſi alibi cum Ellipſi conuenirent, _MINIMAE_ <lb/>plures eſſent, quàm eſſe poſſint. </s> <s xml:id="echoid-s7461" xml:space="preserve">Itaque in circulis figurarum 22. </s> <s xml:id="echoid-s7462" xml:space="preserve">23. </s> <s xml:id="echoid-s7463" xml:space="preserve">25. </s> <s xml:id="echoid-s7464" xml:space="preserve">26.</s> <s xml:id="echoid-s7465" xml:space="preserve"> <pb o="82" file="0268" n="268" rhead=""/> 28. </s> <s xml:id="echoid-s7466" xml:space="preserve">in quibus eorum centra non ſunt in B D axe reuolutionis Ellipſis, prout <lb/>ſunt in reliquis, ducatur per centrum F diameter I L eidem axi B D æqui-<lb/>diſtans, & </s> <s xml:id="echoid-s7467" xml:space="preserve">concipiatur, tum circulum, tum Ellipſim conuerti eadem arte, <lb/>qua ſuperiùs vſi ſumus, non abſimili ratiocinatione, atque ope 56. </s> <s xml:id="echoid-s7468" xml:space="preserve">huius, <lb/>oſtendetur incluſam Sphæram Sphæroides contingere, vel in vnico puncto, <lb/>vt euenit in 21. </s> <s xml:id="echoid-s7469" xml:space="preserve">22. </s> <s xml:id="echoid-s7470" xml:space="preserve">23. </s> <s xml:id="echoid-s7471" xml:space="preserve">26. </s> <s xml:id="echoid-s7472" xml:space="preserve">27. </s> <s xml:id="echoid-s7473" xml:space="preserve">ac 28. </s> <s xml:id="echoid-s7474" xml:space="preserve">vel in duobus tantùm, vt in 24. </s> <s xml:id="echoid-s7475" xml:space="preserve">& </s> <s xml:id="echoid-s7476" xml:space="preserve">25. <lb/></s> <s xml:id="echoid-s7477" xml:space="preserve">vel ad integram circuli peripheriam, vt in 19. </s> <s xml:id="echoid-s7478" xml:space="preserve">& </s> <s xml:id="echoid-s7479" xml:space="preserve">20. </s> <s xml:id="echoid-s7480" xml:space="preserve">ideoque omnes rectas, <lb/>quæ à centro F ad puncta Sphæricæ ſuperficiei ducuntur, æquales eſſe ijs, <lb/>quæ ad prædicta contactuum puncta, vel ad peripherias ducuntur, ac pro-<lb/>pterea, quæ ad circumſcriptam Sphæroidis ſuperſiciem, præter ad eadem <lb/>puncta, vel peripherias ducentur, maiores erunt. </s> <s xml:id="echoid-s7481" xml:space="preserve">Vnde ipſæ eductæ, à <lb/>dato puncto F ad reperta contactuum puncta, vel ad peripherias ſuper dati <lb/>Sphæroidis ſuperficiem erunt _MINIMAE_. </s> <s xml:id="echoid-s7482" xml:space="preserve">Quod vltimò faciendum erat.</s> <s xml:id="echoid-s7483" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div774" type="section" level="1" n="305"> <head xml:id="echoid-head314" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s7484" xml:space="preserve">Iraberis fortaſſe, ac non immeritò, proximas haſce quinque <lb/>propoſitiones circa planas portiones verſantes, & </s> <s xml:id="echoid-s7485" xml:space="preserve">immediatè <lb/>poſt quadrageſimam quintam huius aptè apponendas, locum <lb/>hunc inter ſolida ſortitas fuiſſe: </s> <s xml:id="echoid-s7486" xml:space="preserve">ſed inuitam, vel fortuitam <lb/>potiùs huius tranſmisſionis cauſam, hic tibi enarrare ſuperuacaneum <lb/>puto. </s> <s xml:id="echoid-s7487" xml:space="preserve">His itaque vtaris prout ſuo loco inſertis; </s> <s xml:id="echoid-s7488" xml:space="preserve">nulla namque ipſarum <lb/>indiget aliqua præcedentium vſque ad num. </s> <s xml:id="echoid-s7489" xml:space="preserve">46. </s> <s xml:id="echoid-s7490" xml:space="preserve">incluſiuè, licet ſola quin-<lb/>quageſima prima nonnullarum ſequentium notionem aſſumat.</s> <s xml:id="echoid-s7491" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div775" type="section" level="1" n="306"> <head xml:id="echoid-head315" xml:space="preserve">THEOR. XXXVIII. PROP. LXIII.</head> <p> <s xml:id="echoid-s7492" xml:space="preserve">Æquales portiones eiuſdem coni - ſectionis, vel circuli, ſi <lb/>fuerint de eadem Parabola habebunt intercepta diametrorum <lb/> <anchor type="note" xlink:label="note-0268-01a" xlink:href="note-0268-01"/> ſegmenta inter ſe æqualia. </s> <s xml:id="echoid-s7493" xml:space="preserve">Si de eadem Hyperbola, vel Ellipſi, <lb/>vel circulo, prædicta diametrorum ſegmenta erunt proprijs ſe-<lb/>mi- diametris proportionalia.</s> <s xml:id="echoid-s7494" xml:space="preserve"/> </p> <div xml:id="echoid-div775" type="float" level="2" n="1"> <note position="left" xlink:label="note-0268-01" xlink:href="note-0268-01a" xml:space="preserve">Conuer-<lb/>ſum Pro-<lb/>p. 40. h.</note> </div> <p> <s xml:id="echoid-s7495" xml:space="preserve">SInt, in quacunque harum figurarum, duæ portiones A B C, D E F inter <lb/>ſe æquales, quæ in ſectione Ellipſis tertiæ figuræ ſint primò minores ſe-<lb/>mi- Ellipſi, & </s> <s xml:id="echoid-s7496" xml:space="preserve">harum omnium ſegmenta diametrorum ſint B G, E H, tùm <lb/>in Parabola primæ figuræ, tùm in reliquis, quarum centrum ſit O. </s> <s xml:id="echoid-s7497" xml:space="preserve">Dico, <lb/>in prima, ſegmenta E H, B G inter ſe æqualia eſſe, in reliquis verò, eſſe vt <lb/>H E ad E O, ita G B ad B O.</s> <s xml:id="echoid-s7498" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7499" xml:space="preserve">Ex altera diametrorum, vtputa ex E H, ſecetur, in prima figura, E I <lb/>æqualis ſegmento B G; </s> <s xml:id="echoid-s7500" xml:space="preserve">& </s> <s xml:id="echoid-s7501" xml:space="preserve">in reliquis, fiat O E ad E I, vt O B ad B G, atq; <lb/></s> <s xml:id="echoid-s7502" xml:space="preserve">in omnibus ordinatim applicetur per I ipſi diametro E I recta L I M, quæ <pb o="83" file="0269" n="269" rhead=""/> rectæ D H F æquidiſtabit, cum & </s> <s xml:id="echoid-s7503" xml:space="preserve">hæc quoque ſit eidem diametro ordina-<lb/>tim ducta.</s> <s xml:id="echoid-s7504" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7505" xml:space="preserve">Iam in ſingulis figuris erit portio L E M æqualis <anchor type="note" xlink:href="" symbol="a"/> portioni A B C, ſed <anchor type="note" xlink:label="note-0269-01a" xlink:href="note-0269-01"/> eſt quoque portio D E F eidem portioni A B C æqualis, ex hypotheſi, <lb/>quare duæ portiones L E M, D E F inter ſe æquales erunt, ſed vtraque eſt <lb/>de eadem ſectione, & </s> <s xml:id="echoid-s7506" xml:space="preserve">circa communem diametrum E H I, & </s> <s xml:id="echoid-s7507" xml:space="preserve">ſuper baſes <lb/>parallelas, ergo baſis L I M tota congruet cum baſi D H F, vnde & </s> <s xml:id="echoid-s7508" xml:space="preserve">pun-<lb/>ctum I cum puncto H; </s> <s xml:id="echoid-s7509" xml:space="preserve">quare ſegmenta E I, E H inter ſe æqualia erunt, <lb/>ac propterea erit, in prima, ſegmentum quoque E H æquale B G, & </s> <s xml:id="echoid-s7510" xml:space="preserve">in re-<lb/>liquis erit H E ad E O, vt G B ad B O. </s> <s xml:id="echoid-s7511" xml:space="preserve">Quod primò oſtendere propone-<lb/>batur.</s> <s xml:id="echoid-s7512" xml:space="preserve"/> </p> <div xml:id="echoid-div776" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-0269-01" xlink:href="note-0269-01a" xml:space="preserve">40. h.</note> </div> <figure> <image file="0269-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0269-01"/> </figure> <p> <s xml:id="echoid-s7513" xml:space="preserve">Sint iam in tertia figura duæ portiones æquales A N C, D P F ſemi- El-<lb/>lipſi maiores, quarum ſegmenta diametrorum ſint G N, H P, & </s> <s xml:id="echoid-s7514" xml:space="preserve">commune <lb/>centrum O. </s> <s xml:id="echoid-s7515" xml:space="preserve">Dico item eſſe G N ad N O, vt H P ad P O.</s> <s xml:id="echoid-s7516" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7517" xml:space="preserve">Producantur diametri N G, P H, ad B, E.</s> <s xml:id="echoid-s7518" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7519" xml:space="preserve">Et cum portiones A N C, D P F ſint æquales, & </s> <s xml:id="echoid-s7520" xml:space="preserve">ſemi- Ellipſi maiores <lb/>erunt quoque reliquæ A B C, D E F de eadem Ellipſi inter ſe æquales, <lb/>ſed ſemi- Ellipſi minores; </s> <s xml:id="echoid-s7521" xml:space="preserve">quare erit, vt ſupra oſtendimus, G B ad B O, <lb/>vt H E ad E O, & </s> <s xml:id="echoid-s7522" xml:space="preserve">conuertendo, & </s> <s xml:id="echoid-s7523" xml:space="preserve">diuidendo O G ad G B, vt O H ad <lb/>H E, & </s> <s xml:id="echoid-s7524" xml:space="preserve">eſt G B ad B O, vel ad O N, vt H E ad E O, vel ad O P, ergo, <lb/>ex æquali G O ad O N, vt H O ad O P, & </s> <s xml:id="echoid-s7525" xml:space="preserve">componendo, G N ad N O, <lb/>vt H P ad P O. </s> <s xml:id="echoid-s7526" xml:space="preserve">Quod vltimò erat, &</s> <s xml:id="echoid-s7527" xml:space="preserve">c.</s> <s xml:id="echoid-s7528" xml:space="preserve"/> </p> <pb o="84" file="0270" n="270" rhead=""/> </div> <div xml:id="echoid-div778" type="section" level="1" n="307"> <head xml:id="echoid-head316" xml:space="preserve">THEOR. XXXIX. PROP. LXIV.</head> <p> <s xml:id="echoid-s7529" xml:space="preserve">Portiones eiuſdem coni-ſectionis, vel circuli, aut etiam an-<lb/>guli rectilinei, quarum intercepta diametrorum ſegmenta in <lb/>Parabola ſint æqualia, vel in Hyperbola, aut in Ellipſi, vel <lb/>circulo, ad proprias ſemi- diametros eandem ſimul habeant ra-<lb/>tionem, vel in angulo pertingant ad eandem inſcriptam con-<lb/>centricam Hyperbolen, habent baſes altitudinibus reciprocè <lb/>proportionales.</s> <s xml:id="echoid-s7530" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7531" xml:space="preserve">NAm, quo ad primùm, reiterata inſpectione figurarum tertij Schemati-<lb/>ſmi pro propoſitione 40. </s> <s xml:id="echoid-s7532" xml:space="preserve">huius; </s> <s xml:id="echoid-s7533" xml:space="preserve">ibi in portionibus A B C, H E I, tùm <lb/>quandò, in Parabola, diametri B F, E G ſint æquales; </s> <s xml:id="echoid-s7534" xml:space="preserve">tùm quandò, in <lb/> <anchor type="note" xlink:label="note-0270-01a" xlink:href="note-0270-01"/> reliquis ſectionibus, ſit ſemi- diameter D B ad B F diametrum portionis A <lb/>B C, vt ſemi- diameter D E, ad E G diametrum portionis H E I, demon-<lb/>ſtratum ſuit, propè finem, baſim H I portionis H E I, ad baſim A C portio-<lb/>nis A B C, eſſe reciprocè, vt altitudo portionis A B C ad altitudinem por-<lb/>tionis H E I. </s> <s xml:id="echoid-s7535" xml:space="preserve">Quod tanquam Coroll. </s> <s xml:id="echoid-s7536" xml:space="preserve">Prop. </s> <s xml:id="echoid-s7537" xml:space="preserve">40. </s> <s xml:id="echoid-s7538" xml:space="preserve">huius elici poterat. </s> <s xml:id="echoid-s7539" xml:space="preserve">At cum <lb/>ibi tantùm loquatur de portionibus Ellipticis, quæ ſint ſemi- Ellipſi mino-<lb/>res, hoc idem verificari etiam de portionibus ſemi - Ellipſi maioribus, vel <lb/>etiam de ijſdem ſemi-Ellipſibus, ita demonſtrabitur.</s> <s xml:id="echoid-s7540" xml:space="preserve"/> </p> <div xml:id="echoid-div778" type="float" level="2" n="1"> <note position="left" xlink:label="note-0270-01" xlink:href="note-0270-01a" xml:space="preserve">Schema-<lb/>tiſmus 3.</note> </div> <p> <s xml:id="echoid-s7541" xml:space="preserve">Sint duæ portiones A B C, D E F de ea-<lb/> <anchor type="figure" xlink:label="fig-0270-01a" xlink:href="fig-0270-01"/> dem Ellipſi, cuius centrum O; </s> <s xml:id="echoid-s7542" xml:space="preserve">vtraque ve-<lb/>rò ſit ſemi- Ellipſi maior, quarum diametri <lb/>G B, H E ad proprias ſemi - diametros B <lb/>O, E O ſint in eadem ratione. </s> <s xml:id="echoid-s7543" xml:space="preserve">Dico, baſim <lb/>A C vnius, ad D F baſim alterius, eſſe vt <lb/>huius altitudo E M, ad illius altitudinem <lb/>B N.</s> <s xml:id="echoid-s7544" xml:space="preserve"/> </p> <div xml:id="echoid-div779" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0270-01"/> </figure> </div> <p> <s xml:id="echoid-s7545" xml:space="preserve">Productis enim diametris B G, E H vſq; <lb/></s> <s xml:id="echoid-s7546" xml:space="preserve">ad Ellipſis peripheriam in punctis I, L, è <lb/>quibus ductis I P, L R, baſibus A C, D F <lb/>perpendicularibus, hæ erunt altitudines <lb/>portionum A I C, D L F, & </s> <s xml:id="echoid-s7547" xml:space="preserve">reliquarum <lb/>portionum altitudinibus, B N, E M æqui-<lb/>diſtabunt.</s> <s xml:id="echoid-s7548" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7549" xml:space="preserve">Et cum, ex hypotheſi, ſit G B ad B O, vt H E ad E O, ſumptis conſe-<lb/>quentium duplis, conuertendo, & </s> <s xml:id="echoid-s7550" xml:space="preserve">per conuerſionem rationis B I ad I G, <lb/>erit vt E L ad L H; </s> <s xml:id="echoid-s7551" xml:space="preserve">& </s> <s xml:id="echoid-s7552" xml:space="preserve">ſumptis antecedentium ſubduplis, O I ad I G, vt O <lb/>L ad L H: </s> <s xml:id="echoid-s7553" xml:space="preserve">quare, per ſuperiùs oſtenſa, in portionibus A I C, D L F, ſemi-<lb/>Ellipſi minoribus, erit baſis A C ad D F, vt altitudo L R ad altitudinem <lb/>I P, fed L R ad I P eſt, vt E M ad B N, vt mox demonſtrabitur, ergo A <lb/>C ad D F erit quoque, vt E M ad B N.</s> <s xml:id="echoid-s7554" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7555" xml:space="preserve">Quod autem ſit L R ad I P, vt E M ad B N. </s> <s xml:id="echoid-s7556" xml:space="preserve">Cum demonſtratum ſit <pb o="85" file="0271" n="271" rhead=""/> eſſe E L ad L H, vt B I ad I G, erit diuidendo, & </s> <s xml:id="echoid-s7557" xml:space="preserve">conuertendo L H ad <lb/>H E, vel L R ad E M (ob triangulorum L H R, E H M ſimilitudinem) <lb/>vt I G ad G B, vel ita I P ad B N (ob ſimilitudinem triangulorum I G P, <lb/>B G N) & </s> <s xml:id="echoid-s7558" xml:space="preserve">permutando L R ad I P, vt E M ad B N. </s> <s xml:id="echoid-s7559" xml:space="preserve">Quod reliquum erat <lb/>oſtendere de portionibus ſemi-Ellipſi maioribus.</s> <s xml:id="echoid-s7560" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7561" xml:space="preserve">Tandem intelligantur duæ ſemi- Ellipſes I E B, E B L de eadem Ellipſi. <lb/></s> <s xml:id="echoid-s7562" xml:space="preserve">Dicobaſim I B ad baſim L E eſſe reciprocè, vt altitudo portionis E B L ad <lb/>altitudinem portionis I E B.</s> <s xml:id="echoid-s7563" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7564" xml:space="preserve">Iunctis enim E I, E B; </s> <s xml:id="echoid-s7565" xml:space="preserve">cum in triangulis I E O, B E O, quorum com-<lb/>munis vertex E, ſit baſis I O æqualis baſi B O, erit triangulum I E O, <lb/>triangulo B E O æquale; </s> <s xml:id="echoid-s7566" xml:space="preserve">& </s> <s xml:id="echoid-s7567" xml:space="preserve">ſi concipiatur baſis trianguli B E O permutari, <lb/>ita vt ipſa ſit O E, & </s> <s xml:id="echoid-s7568" xml:space="preserve">vertex B: </s> <s xml:id="echoid-s7569" xml:space="preserve">cum huiuſmodi triangula ſint æqualia, erit <lb/>baſis I O, vnius I E O, ad baſim O E, alterius B E O, ita reciprocè altitu-<lb/>do trianguli B E O, cuius vertex B, ad altitudinem trianguli I E O, cuius <lb/>vertex E; </s> <s xml:id="echoid-s7570" xml:space="preserve">ſed horum triangulorum altitudines ſunt eædem, ac ſemi-Elli-<lb/>pſium E B L, I E B, ergo I O ad O E, vel ſumptis duplis, baſis I B ad ba-<lb/>ſim L E, erit reciprocè, vt altitudo ſemi- Ellipſis E B L ad altitudinem <lb/>ſemi- Ellipſis I E B.</s> <s xml:id="echoid-s7571" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7572" xml:space="preserve">Quò autem ad portiones eiuſdem anguli, ſuper figuram primam Propoſ. <lb/></s> <s xml:id="echoid-s7573" xml:space="preserve">45. </s> <s xml:id="echoid-s7574" xml:space="preserve">huius, in qua diametri B E, M D portionum, ſiue triangulorum A B C, <lb/>H M I pertingunt ad eandem Hyperbolen D E concentricam, cum ibi de-<lb/>monſtratum ſit ipſa triangula inter ſe eſſe æqualia, erit baſis A C vnius, ad <lb/>H I baſim alterius, vt altitudo trianguli H M I ad altitudinem trianguli A <lb/>B C: </s> <s xml:id="echoid-s7575" xml:space="preserve">hoc enim elicitur ex elementis, nam triangula æqualia habent baſes <lb/>altitudinibus reciprocè proportionales. </s> <s xml:id="echoid-s7576" xml:space="preserve">Quare portiones eiuſdem coni- ſe-<lb/>ctionis, &</s> <s xml:id="echoid-s7577" xml:space="preserve">c. </s> <s xml:id="echoid-s7578" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s7579" xml:space="preserve">c.</s> <s xml:id="echoid-s7580" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div781" type="section" level="1" n="308"> <head xml:id="echoid-head317" xml:space="preserve">THEOR. XL. PROP. LXV.</head> <p> <s xml:id="echoid-s7581" xml:space="preserve">Æquales portiones eiuſdem coni - ſectionis, vel circuli, aut <lb/>etiam anguli, habent baſes altitudinibus reciprocè proportiona-<lb/>les. </s> <s xml:id="echoid-s7582" xml:space="preserve">Et è conuerſo.</s> <s xml:id="echoid-s7583" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7584" xml:space="preserve">Si portiones de eadem coni - ſectione, vel circulo, aut etiam <lb/>angulo habuerint baſes altitudinibus reciprocè proportionales, <lb/>ipſæ portiones æquales erunt.</s> <s xml:id="echoid-s7585" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7586" xml:space="preserve">1. </s> <s xml:id="echoid-s7587" xml:space="preserve">ETenim, quò ad primùm, quandò portiones de eadem coni- ſectione, <lb/>vel circulo, aut etiam angulo ſunt æquales, ſi fuerint de cadem Para-<lb/>bola, habent intercepta diametrorum ſegmenta inter ſe æqualia, & </s> <s xml:id="echoid-s7588" xml:space="preserve">ſi de ea-<lb/>dem Hyperbola, vel Ellipſi, vel circulo habent ſegmenta proprijs ſemi- dia-<lb/>metris <anchor type="note" xlink:href="" symbol="a"/> proportionalia (nam ſi fuerint de eodem angulo propoſitum ſatis <anchor type="note" xlink:label="note-0271-01a" xlink:href="note-0271-01"/> conſtat, ex Elementis;) </s> <s xml:id="echoid-s7589" xml:space="preserve">ſed quandò huiuſmodi portionibus inſunt condi-<lb/>tiones prædictæ, ipſæ habent <anchor type="note" xlink:href="" symbol="b"/> baſes altitudinibus reciprocè proportiona- <anchor type="note" xlink:label="note-0271-02a" xlink:href="note-0271-02"/> les, ergo, & </s> <s xml:id="echoid-s7590" xml:space="preserve">cum fuerint equales, ipſarum baſes altitudinibus erunt recipro- <pb o="86" file="0272" n="272" rhead=""/> cæ. </s> <s xml:id="echoid-s7591" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s7592" xml:space="preserve">c. </s> <s xml:id="echoid-s7593" xml:space="preserve">quodque tanquam præoſtenſum bis aſſumpſi-<lb/>mus in 5 1. </s> <s xml:id="echoid-s7594" xml:space="preserve">h.</s> <s xml:id="echoid-s7595" xml:space="preserve"/> </p> <div xml:id="echoid-div781" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0271-01" xlink:href="note-0271-01a" xml:space="preserve">63. h.</note> <note symbol="b" position="right" xlink:label="note-0271-02" xlink:href="note-0271-02a" xml:space="preserve">64. h.</note> </div> <p> <s xml:id="echoid-s7596" xml:space="preserve">2. </s> <s xml:id="echoid-s7597" xml:space="preserve">PRo demóſtratione <lb/> <anchor type="figure" xlink:label="fig-0272-01a" xlink:href="fig-0272-01"/> auté cóuerſi huius, <lb/>ponantur portiones A <lb/>B C, D E F de eadem <lb/>coni-ſectione, in pri-<lb/>mis tribus figuris, (quę <lb/>tamen in tertia ſint ſe-<lb/>mi-Ellipſi minores) vel <lb/>de eodem angulo, vt in <lb/>quarta, quarum omniũ <lb/>diametri ſint G B, H E, <lb/>baſes A C, D F, alti-<lb/>tudines verò B K, E I, <lb/>centrum autem in ſe-<lb/>cunda, & </s> <s xml:id="echoid-s7598" xml:space="preserve">tertia ſit R: <lb/></s> <s xml:id="echoid-s7599" xml:space="preserve">ſitque baſis A C ad ba-<lb/>ſim D F, reciprocè, vt <lb/>altitudo E I ad altitu-<lb/>dinem B K. </s> <s xml:id="echoid-s7600" xml:space="preserve">Dico ip-<lb/>ſas portiones inter ſe <lb/>æquales eſſe.</s> <s xml:id="echoid-s7601" xml:space="preserve"/> </p> <div xml:id="echoid-div782" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0272-01"/> </figure> </div> <p> <s xml:id="echoid-s7602" xml:space="preserve">Nam ſi ſegmenta <lb/>diametrorum B G, E <lb/>H, in prima exhibente <lb/>Parabolen, fuerint æ-<lb/>qualia; </s> <s xml:id="echoid-s7603" xml:space="preserve">& </s> <s xml:id="echoid-s7604" xml:space="preserve">in ſecunda, ac tertia exhibentibus Hyperbolen, & </s> <s xml:id="echoid-s7605" xml:space="preserve">Ellipſim, ha-<lb/>buerint ad proprias ſemi- diametros B R, E R eandem rationem; </s> <s xml:id="echoid-s7606" xml:space="preserve">iam patet, <lb/>per 40. </s> <s xml:id="echoid-s7607" xml:space="preserve">huius, ipſas portiones inter ſe æquales eſſe.</s> <s xml:id="echoid-s7608" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7609" xml:space="preserve">At ſi inter hæc diametrorum ſegmẽta non eſt prædicta æqualitas in prima <lb/>figura; </s> <s xml:id="echoid-s7610" xml:space="preserve">vel proportionalitas, in ſecunda, & </s> <s xml:id="echoid-s7611" xml:space="preserve">tertia, alterum ipſorum ſegmẽ-<lb/>torum erit æquo maius. </s> <s xml:id="echoid-s7612" xml:space="preserve">Sit ipſum B G, & </s> <s xml:id="echoid-s7613" xml:space="preserve">ad æquum reducatur in L: </s> <s xml:id="echoid-s7614" xml:space="preserve">erit <lb/>ergo B L minus B G, cui per L ordinatim applicetur N L O (quæ baſi A <lb/>C æquidiſtabit) altitudinem B K ſecans in M; </s> <s xml:id="echoid-s7615" xml:space="preserve">& </s> <s xml:id="echoid-s7616" xml:space="preserve">erit B M altitudo por-<lb/>tionis N B O.</s> <s xml:id="echoid-s7617" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7618" xml:space="preserve">Iam, diameter L B, in prima, facta eſt æqualis diametro H E; </s> <s xml:id="echoid-s7619" xml:space="preserve">in ſecunda <lb/>verò, & </s> <s xml:id="echoid-s7620" xml:space="preserve">tertia nũc ponitur L B ad B R habere eandem rationem quàm H E <lb/>ad E R, ergo per primam partem huius, erit baſis N O ad D F, vt altitudo <lb/>E I ad B M; </s> <s xml:id="echoid-s7621" xml:space="preserve">vnde rectangulum ſub N O, B M æquabitur rectangulo ſub <lb/>D F, E I; </s> <s xml:id="echoid-s7622" xml:space="preserve">ſed eſt, ex hypotheſi, baſis A C ad D F, vt altitudo E I ad B K, <lb/>ergo, & </s> <s xml:id="echoid-s7623" xml:space="preserve">rectangulum ſub A C, & </s> <s xml:id="echoid-s7624" xml:space="preserve">B K, æquabitur eidem rectangulo ſub D <lb/>F, & </s> <s xml:id="echoid-s7625" xml:space="preserve">E I; </s> <s xml:id="echoid-s7626" xml:space="preserve">quare duo rectangula ſub N O, & </s> <s xml:id="echoid-s7627" xml:space="preserve">B M, & </s> <s xml:id="echoid-s7628" xml:space="preserve">ſub A C, & </s> <s xml:id="echoid-s7629" xml:space="preserve">B K ſunt <lb/>æqualia, quod eſt falſum. </s> <s xml:id="echoid-s7630" xml:space="preserve">Rectangulum enim ſub N O, B M minus eſt re-<lb/>ctangulo ſub A C, B K, eò quod ſub minoribus lateribus contineatur, cum <lb/>ſit applicata N O minor applicata A C, & </s> <s xml:id="echoid-s7631" xml:space="preserve">altitudo B M minor altitudine <lb/>B K: </s> <s xml:id="echoid-s7632" xml:space="preserve">quapropter ipſa diametrorum ſegmenta, in prima, æqualia erunt; </s> <s xml:id="echoid-s7633" xml:space="preserve">&</s> <s xml:id="echoid-s7634" xml:space="preserve"> <pb o="87" file="0273" n="273" rhead=""/> & </s> <s xml:id="echoid-s7635" xml:space="preserve">in reliquis, erunt proprijs ſemi- diametris proportionalia, hoc eſt ipſæ <lb/>portiones æquales <anchor type="note" xlink:href="" symbol="a"/> erunt. </s> <s xml:id="echoid-s7636" xml:space="preserve">De portionibus tandem eiuſdem anguli, quæ <anchor type="note" xlink:label="note-0273-01a" xlink:href="note-0273-01"/> ſunt triangula, iam notum eſt, quandò baſes ipſorum altitudinibus ſint reci-<lb/>procè proportionales, ipſa triangula eſſe æqualia. </s> <s xml:id="echoid-s7637" xml:space="preserve">Quare, &</s> <s xml:id="echoid-s7638" xml:space="preserve">c. </s> <s xml:id="echoid-s7639" xml:space="preserve">quod ſecun-<lb/>dò probandum erat.</s> <s xml:id="echoid-s7640" xml:space="preserve"/> </p> <div xml:id="echoid-div783" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0273-01" xlink:href="note-0273-01a" xml:space="preserve">40. h.</note> </div> <p style="it"> <s xml:id="echoid-s7641" xml:space="preserve">Haud incongruum, neque inutile duximus hic adnotaſſe Theorema <lb/>huiuſmodi.</s> <s xml:id="echoid-s7642" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div785" type="section" level="1" n="309"> <head xml:id="echoid-head318" xml:space="preserve">THEOR. XLI. PROP. LXVI.</head> <p> <s xml:id="echoid-s7643" xml:space="preserve">Æquales portiones eiuſdem coni-ſectionis, vel circuli (quæ <lb/>tamen in Ellipſi ſint, vel vnà æquales, vel vnà maiores, vel vnà <lb/>minores ſemi- Ellipſi) ad inſcripta ſibi triangula, (nempè ad ea, <lb/>quorum baſes eædem ſunt, ac portionum, eædemque altitudi-<lb/>nes, ſiuè ijdem vertices) vel ad circumſcripta parallelogram-<lb/>ma, ſunt inter ſe in vnà eademque ratione.</s> <s xml:id="echoid-s7644" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7645" xml:space="preserve">NAm cum baſes æqualium portionum eiuſdem coni- ſectionis, vel cir-<lb/>culi earum altitudinibus ſint <anchor type="note" xlink:href="" symbol="b"/> reciprocæ, baſes quoque inſcriptorum <anchor type="note" xlink:label="note-0273-02a" xlink:href="note-0273-02"/> triangulorum, eorum altitudinibus reciprocabuntur, cum vtrobique altitu-<lb/>dines, & </s> <s xml:id="echoid-s7646" xml:space="preserve">baſes ponantur eædem; </s> <s xml:id="echoid-s7647" xml:space="preserve">ac propterea ipſa triangula æqualia erunt. <lb/></s> <s xml:id="echoid-s7648" xml:space="preserve">Quare, vt portio ad portionem, ita triangulum ad triangulum, ob æquali-<lb/>tatem tùm portionum, tùm triangulorum; </s> <s xml:id="echoid-s7649" xml:space="preserve">& </s> <s xml:id="echoid-s7650" xml:space="preserve">permutando, portio ad ſibi in-<lb/>ſcriptum triangulum, vt altera æqualis portio de eadem coni- ſectione, vel <lb/>circulo ad ſibi inſcriptum triangulum. </s> <s xml:id="echoid-s7651" xml:space="preserve">Et ſumptis conſequentium duplis, <lb/>portio ad circumſcriptum parallelogrammum, erit vt altera portio ad cir-<lb/>cumſcriptum parallelogrammum. </s> <s xml:id="echoid-s7652" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s7653" xml:space="preserve">c.</s> <s xml:id="echoid-s7654" xml:space="preserve"/> </p> <div xml:id="echoid-div785" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-0273-02" xlink:href="note-0273-02a" xml:space="preserve">65. h. ad <lb/>num. 1.</note> </div> <p> <s xml:id="echoid-s7655" xml:space="preserve">Hoc de ſolis Parabolæ portionibus, etiam ſi inæqualibus, nec de <lb/>eadem Parabola, manifeſtum iam erat ex Archimede (omnis <lb/>enim Parabolæ portio ad ſibi inſcriptum triangulum ha-<lb/>bet <anchor type="note" xlink:href="" symbol="c"/> rationem ſeſquitertiam.) </s> <s xml:id="echoid-s7656" xml:space="preserve">De reliquarum autem <anchor type="note" xlink:label="note-0273-03a" xlink:href="note-0273-03"/> coni- ſectionum æqualibus portionibus, <lb/>non dum.</s> <s xml:id="echoid-s7657" xml:space="preserve"/> </p> <div xml:id="echoid-div786" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0273-03" xlink:href="note-0273-03a" xml:space="preserve">17. pr. h.</note> </div> <pb o="88" file="0274" n="274" rhead=""/> </div> <div xml:id="echoid-div788" type="section" level="1" n="310"> <head xml:id="echoid-head319" xml:space="preserve">LEMMA XIII. PROP. LXVII.</head> <p> <s xml:id="echoid-s7658" xml:space="preserve">Si in angulo A B C applicatæ ſint duæ rectæ lineæ D E, A <lb/>C, quæ ab eadem recta B G per verticem B ducta proportio-<lb/>naliter ſecentur, ita vt ſit A G ad G C, homologè, vt D F ad <lb/>F E. </s> <s xml:id="echoid-s7659" xml:space="preserve">Dico ipſas A C, D F inter ſe æquidiſtare.</s> <s xml:id="echoid-s7660" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7661" xml:space="preserve">SI enim A C non eſt ipſi D E parallela, ſit alia <lb/> <anchor type="figure" xlink:label="fig-0274-01a" xlink:href="fig-0274-01"/> applicata A H, ſecans B G in I: </s> <s xml:id="echoid-s7662" xml:space="preserve">erit ergo <lb/>(ob parallelas) A I ad I H, vt D F ad F E; </s> <s xml:id="echoid-s7663" xml:space="preserve">vel <lb/>ob hypotheſim, vt A G ad G C, ergo in trian-<lb/>gulo A C H erit I G parallela ad H C, ſed ipſæ <lb/>conueniunt in B. </s> <s xml:id="echoid-s7664" xml:space="preserve">Quare non erit alia ex A ipſi <lb/>D E parallela, quàm A C. </s> <s xml:id="echoid-s7665" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s7666" xml:space="preserve">c.</s> <s xml:id="echoid-s7667" xml:space="preserve"/> </p> <div xml:id="echoid-div788" type="float" level="2" n="1"> <figure xlink:label="fig-0274-01" xlink:href="fig-0274-01a"> <image file="0274-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0274-01"/> </figure> </div> </div> <div xml:id="echoid-div790" type="section" level="1" n="311"> <head xml:id="echoid-head320" xml:space="preserve">THEOR. XLII. PROP. LXVIII.</head> <p> <s xml:id="echoid-s7668" xml:space="preserve">Baſes æqualium portionum, ex eodem angulo, ſiue ex eadem <lb/> <anchor type="note" xlink:label="note-0274-01a" xlink:href="note-0274-01"/> quacunque coni- ſectione, vel circulo abſciſſarum, eandem in-<lb/>ſcriptam eiuſdem nominis ſectionem ſimilem, & </s> <s xml:id="echoid-s7669" xml:space="preserve">concentricam <lb/>ad puncta media contingunt.</s> <s xml:id="echoid-s7670" xml:space="preserve"/> </p> <div xml:id="echoid-div790" type="float" level="2" n="1"> <note position="left" xlink:label="note-0274-01" xlink:href="note-0274-01a" xml:space="preserve">Conuer-<lb/>ſum Pro-<lb/>p. 45. h.</note> </div> <p> <s xml:id="echoid-s7671" xml:space="preserve">SInt de angulo rectilineo, vt in prima figura, vel de qualibet alia coni- ſe-<lb/>ctione, vel circulo, vt in ſecunda, abſciſſæ duæ æquales portiones A <lb/>B C, D E F, quarum baſes A C, D F ſint bifariam ſectæ in G, H, & </s> <s xml:id="echoid-s7672" xml:space="preserve">per G <lb/>inſcribatur <anchor type="note" xlink:href="" symbol="a"/> eiuſdem nominis ſectio ſimilis, & </s> <s xml:id="echoid-s7673" xml:space="preserve">concentrica exteriori A B F, <anchor type="note" xlink:label="note-0274-02a" xlink:href="note-0274-02"/> quæ ſit I G H. </s> <s xml:id="echoid-s7674" xml:space="preserve">Dico baſim A C ſectionem I G H contingere in G, & </s> <s xml:id="echoid-s7675" xml:space="preserve">ba-<lb/>ſim quoque D F eandem ſectionem contingere in H.</s> <s xml:id="echoid-s7676" xml:space="preserve"/> </p> <div xml:id="echoid-div791" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0274-02" xlink:href="note-0274-02a" xml:space="preserve">4. ſec. <lb/>conic & <lb/>5 6.7. p. h.</note> </div> <p> <s xml:id="echoid-s7677" xml:space="preserve">Iungatur, in prima, B G, & </s> <s xml:id="echoid-s7678" xml:space="preserve">producatur, nam ipſa erit diameter Hyper-<lb/>bolæ I G H (cum ſit B eius centrum) bifariam ſecans omnes in ea applica-<lb/>tas, quæ ſi vſque ad aſymptotos producantur, erunt, & </s> <s xml:id="echoid-s7679" xml:space="preserve">ipſarum ſegmenta <lb/>inter aſymptotos, & </s> <s xml:id="echoid-s7680" xml:space="preserve">ſectionem æqualia <anchor type="note" xlink:href="" symbol="b"/> inter ſe, quare ſi ipſa ſegmenta <anchor type="note" xlink:label="note-0274-03a" xlink:href="note-0274-03"/> concipiantur addita æqualibus ſemi- applicatis in ſectione eis in directum <lb/>poſitis, prouenient totæ applicatæ in angulo A B E biſariam ſectæ à dia-<lb/>metro B G producta, ſed ponitur quoque applicata A C bifariam ſecta in <lb/>G, quare A C ipſis applicatis in ſectione <anchor type="note" xlink:href="" symbol="c"/> æquidiſtabit, ac ideò ſectionem <anchor type="note" xlink:label="note-0274-04a" xlink:href="note-0274-04"/> I G H continget <anchor type="note" xlink:href="" symbol="d"/> in G.</s> <s xml:id="echoid-s7681" xml:space="preserve"/> </p> <div xml:id="echoid-div792" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-0274-03" xlink:href="note-0274-03a" xml:space="preserve">8. ſecũd. <lb/>conic.</note> <note symbol="c" position="left" xlink:label="note-0274-04" xlink:href="note-0274-04a" xml:space="preserve">67. h.</note> </div> <note symbol="d" position="left" xml:space="preserve">32. pri-<lb/>mi conic.</note> <p> <s xml:id="echoid-s7682" xml:space="preserve">In ſecunda autem figura quaſcunque coni- ſectiones exhibente ducatur <lb/>ex G diameter G B, quæ vtriuſque ſectionis A B E, I G H erit communis <lb/>diameter (cumipſæ ponantur ſectiones concentricæ, &</s> <s xml:id="echoid-s7683" xml:space="preserve">c.) </s> <s xml:id="echoid-s7684" xml:space="preserve">ad applicatas in <pb o="89" file="0275" n="275" rhead=""/> ipſis æqualiter inclinata; </s> <s xml:id="echoid-s7685" xml:space="preserve">quare applicatæ in ſectione I G H ad diametrum <lb/>B G æquidiſtabunt applicatis in ſectione A B C ad eandem diametrum, <lb/>quarum vna eſt A C per verticé G ducta, cum in G ſit bifariam ſecta; </s> <s xml:id="echoid-s7686" xml:space="preserve">ergo <lb/>ipſa A C continget <anchor type="note" xlink:href="" symbol="a"/> in G ſectionem I G H.</s> <s xml:id="echoid-s7687" xml:space="preserve"/> </p> <note symbol="a" position="right" xml:space="preserve">ibidem.</note> <p style="it"> <s xml:id="echoid-s7688" xml:space="preserve">Sed hoc idem breuiùs, tùm in angulo, tùm in qualibet coni-ſectione, <lb/>omiſſo precedenti Lemmate.</s> <s xml:id="echoid-s7689" xml:space="preserve"/> </p> <figure> <image file="0275-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0275-01"/> </figure> <p> <s xml:id="echoid-s7690" xml:space="preserve">COncedatur ſectionem I G H occurrere rectæ A C in alio puncto quàm <lb/>G, quod ſit K. </s> <s xml:id="echoid-s7691" xml:space="preserve">Dico tamen punctum K idem eſſe ac G.</s> <s xml:id="echoid-s7692" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7693" xml:space="preserve">Quoniam erit A K <anchor type="note" xlink:href="" symbol="b"/> æqualis G C, ſed eſt quoque A G æqualis eidem <anchor type="note" xlink:label="note-0275-02a" xlink:href="note-0275-02"/> G C, ergo A K, & </s> <s xml:id="echoid-s7694" xml:space="preserve">A G ſunt æquales, ſed hæ habent communes terminos <lb/>ad A, ergo, & </s> <s xml:id="echoid-s7695" xml:space="preserve">punctum K congruet cum G. </s> <s xml:id="echoid-s7696" xml:space="preserve">Quare ipſa baſis A C con-<lb/>tingit omnino ſectionem I G H in G.</s> <s xml:id="echoid-s7697" xml:space="preserve"/> </p> <div xml:id="echoid-div793" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">8. ſec. <lb/>conic. & <lb/>ex 1. Co-<lb/>roll 46. h.</note> </div> <p> <s xml:id="echoid-s7698" xml:space="preserve">Ampliùs, in prima figura, iungatur E H, quæ eſt <anchor type="note" xlink:href="" symbol="c"/> diameter inſcriptæ <anchor type="note" xlink:label="note-0275-03a" xlink:href="note-0275-03"/> Hyperbolæ I G H, & </s> <s xml:id="echoid-s7699" xml:space="preserve">in ſecunda ex H ducatur vnius ſectionis diameter H <lb/>E, quæ erit quoque diameter alterius (cum ponantur concentricæ, &</s> <s xml:id="echoid-s7700" xml:space="preserve">c.) </s> <s xml:id="echoid-s7701" xml:space="preserve">Si <lb/>ergo hæc diameter E H producatur, ipſa ſecabit interiorem ſectionem I G <lb/>H in aliquo puncto, vt in L, ex quo ducatur in ſectione A B F recta M L N <lb/>ipſi D F æquidiſtans.</s> <s xml:id="echoid-s7702" xml:space="preserve"/> </p> <div xml:id="echoid-div794" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-0275-03" xlink:href="note-0275-03a" xml:space="preserve">8. pr. h.</note> </div> <p> <s xml:id="echoid-s7703" xml:space="preserve">Et quoniam, in ſingulis, figuris D F eſt bifariam ſecta in H, erit quoque <lb/>M N bifariam ſecta in L (cum M N ex conſtructione æquidiſter ordinatim <lb/>ductæ D F in eadem ſectione A B F) ſed ſectio I G tranſit per L, quare <lb/>ſectio ipſa I G continget omnino rectam M N in L (quod ijſdem rationi-<lb/>bus, ac ſupra de A C oſtenſum fuit, demonſtrabitur) ergo portio M E N <lb/>æquabitur <anchor type="note" xlink:href="" symbol="d"/> portioni A B C, ſed portio quoque D E F æquatur eidem por- <anchor type="note" xlink:label="note-0275-04a" xlink:href="note-0275-04"/> tioni A B C, ex hypotheſi, quare portiones M E N, D E F inter ſe æqua-<lb/>les erunt, ſuntque de eodem angulo, vel de eadem coni- ſectione, vel cir-<lb/>culo, & </s> <s xml:id="echoid-s7704" xml:space="preserve">circa communem diametrum E H L, & </s> <s xml:id="echoid-s7705" xml:space="preserve">ipſarum baſes ſimul æqui-<lb/>diſtant, qua propter, & </s> <s xml:id="echoid-s7706" xml:space="preserve">baſes quoque ſimul in totum congruent, nempe M <lb/>N cum D F, ac ideò punctum L cum puncto H. </s> <s xml:id="echoid-s7707" xml:space="preserve">Recta igitur D F, quæ <lb/>eadem eſt cum M N, contingit ſectionem I G in H. </s> <s xml:id="echoid-s7708" xml:space="preserve">Quod tandem erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s7709" xml:space="preserve"/> </p> <div xml:id="echoid-div795" type="float" level="2" n="6"> <note symbol="d" position="right" xlink:label="note-0275-04" xlink:href="note-0275-04a" xml:space="preserve">45. h.</note> </div> <pb o="90" file="0276" n="276" rhead=""/> </div> <div xml:id="echoid-div797" type="section" level="1" n="312"> <head xml:id="echoid-head321" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s7710" xml:space="preserve">HInc elicitur, quod baſis angularis portionis, vel baſis cuiuslibet coni-<lb/>ſectionis, vel circuli ad punctum medium contingit eiuſdem nominis <lb/>ſectionem ſimilem, & </s> <s xml:id="echoid-s7711" xml:space="preserve">concentricam peripſum punctum dato angulo, vel <lb/>ſectioni, aut circulo inſcriptam.</s> <s xml:id="echoid-s7712" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7713" xml:space="preserve">Nam primò loco ſuperiùs demonſtratum fuit, in vtraque figura, baſim <lb/>A C ad eius punctum medium G omnino contingere ſectionem I G H per <lb/>punctum G concentricè inſcriptam, &</s> <s xml:id="echoid-s7714" xml:space="preserve">c.</s> <s xml:id="echoid-s7715" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div798" type="section" level="1" n="313"> <head xml:id="echoid-head322" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s7716" xml:space="preserve">SEquitur etiam, quod ſegmenta diametrorum, omnium æqualium por-<lb/>tionum ex eodem angulo, aut ex eadem coni- ſectione, vel circulo ab-<lb/>ſciſſarum, cum earum extremis terminis ad baſim, perueniunt ad eandem <lb/>eiuſdem nominis, ſimilem, & </s> <s xml:id="echoid-s7717" xml:space="preserve">inſcriptam concentricam ſectionem.</s> <s xml:id="echoid-s7718" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7719" xml:space="preserve">Etenim puncta media baſium ipſarum portionum, quæ iam eandem ſimi-<lb/>lem inſcriptam concentricam ſectionem contingunt, eadem ſunt, ac prædi-<lb/>cta diametrorum extrema puncta, &</s> <s xml:id="echoid-s7720" xml:space="preserve">c. </s> <s xml:id="echoid-s7721" xml:space="preserve">vt ſatis conſtat.</s> <s xml:id="echoid-s7722" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div799" type="section" level="1" n="314"> <head xml:id="echoid-head323" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s7723" xml:space="preserve">OPportunè monendus hic Lector eſt, nos ſuperiùs, & </s> <s xml:id="echoid-s7724" xml:space="preserve">in ſe-<lb/>quentibus, Hyperbolen intra angulum aſymptotalem deſcri-<lb/>ptam, & </s> <s xml:id="echoid-s7725" xml:space="preserve">Parabolen Parabolæ æquidiſtantem, interdum <lb/>nuncupaſſe ſimiles, & </s> <s xml:id="echoid-s7726" xml:space="preserve">concentricas ſectiones, perindè ac ſi <lb/>angulus rectilineus aſymptotalis, ſectio eſſet ſimilis, & </s> <s xml:id="echoid-s7727" xml:space="preserve">concentrica Hy-<lb/>perbolæ, & </s> <s xml:id="echoid-s7728" xml:space="preserve">quaſi Parabole æquidiſtanti Parabolæ concentrica eſſet. </s> <s xml:id="echoid-s7729" xml:space="preserve">Ve-<lb/>rum ſi id accuratius perpendamus, quo ad angulum rectilineum, ani-<lb/>maduertere licebit ipſum non abs re haberi poſſe tanquam vnam Hyper-<lb/>bolarum, quarum centrum ſit vertex eiuſdem anguli, & </s> <s xml:id="echoid-s7730" xml:space="preserve">aſymptoti ſint <lb/>eadem anguli latera: </s> <s xml:id="echoid-s7731" xml:space="preserve">Omnes enim Hyperbolæ cum ijſdem aſymptotis, <lb/>ſiue cum eodem centro deſcriptæ, ſed cum diuerſis ſemi-axibus, inter ſe <lb/>ſimiles ſunt, vti ex doctrina primi huius iam ſatis patuit; </s> <s xml:id="echoid-s7732" xml:space="preserve">& </s> <s xml:id="echoid-s7733" xml:space="preserve">quò ſe-<lb/>mi- axes ſunt minores, eò tales Hyperbolæ fiunt anguſtiores (nempe in-<lb/>ſcriptibiles per vertices ijs, quarum ſemi-axes ſint maiores) ſed tantò <lb/>magis accedunt ad latera eiuſdem anguli, nunquam tamen eis occur-<lb/>runt, & </s> <s xml:id="echoid-s7734" xml:space="preserve">in hoc ſemi-axium decremento, peruenitur tandem ad MI-<lb/>NIMV M, nempe ad punctum, ſeu verticem anguli, qui eſt centrum <lb/>omnium ſimilium Hyperbolarum, & </s> <s xml:id="echoid-s7735" xml:space="preserve">ad MINIMAM Hyperbolen, <pb o="91" file="0277" n="277" rhead=""/> hoc eſt ad omnium ſimilium, & </s> <s xml:id="echoid-s7736" xml:space="preserve">concentricarum anguſtisſimam, cum <lb/>ipſis anguli lateribus, ſeu cum aſymptotis in totum congruentem. </s> <s xml:id="echoid-s7737" xml:space="preserve">Itaque <lb/>angulus rectilineus vocari quodammodo poteſt prima, & </s> <s xml:id="echoid-s7738" xml:space="preserve">MINIMA <lb/>ſimilium Hyperbolarum concentricarum, quarum angulus aſymptotalis <lb/>ſit æqualis dato, & </s> <s xml:id="echoid-s7739" xml:space="preserve">quælibet prædictarum ſimilium Hyperbolarum in-<lb/>ſcriptarum dici poteſt ſectio eiuſdem nominis cum angulo ſimilis, & </s> <s xml:id="echoid-s7740" xml:space="preserve"><lb/>concentrica, & </s> <s xml:id="echoid-s7741" xml:space="preserve">c. </s> <s xml:id="echoid-s7742" xml:space="preserve">quales meritò appellantur duæ Hyperbolæ, vel duæ El-<lb/>lipſes inter ſe ſimiles, & </s> <s xml:id="echoid-s7743" xml:space="preserve">concentricæ.</s> <s xml:id="echoid-s7744" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7745" xml:space="preserve">Quò autem ad congruentes Parabolas, vel etiam non congruentes, <lb/>(omnes enim Parabolæ ſunt ſimiles inter ſe) ſed per diuerſos vertices <lb/>ſimul adſcriptas, quas alibi æquidiſtantes diximus, liceat etiam, quam-<lb/>uis impropriè, concentricas appellare. </s> <s xml:id="echoid-s7746" xml:space="preserve">Etenim, & </s> <s xml:id="echoid-s7747" xml:space="preserve">Parabole ſuum ha-<lb/>bet centrum à quo procedunt eius diametri, ſed cum id poſitum ſit in infi-<lb/>nitam diſtantiam extra ſectionem, ideò ipſæ diametri ab eodem centro <lb/>emanantes inter ſe æquidiſtant, & </s> <s xml:id="echoid-s7748" xml:space="preserve">c.</s> <s xml:id="echoid-s7749" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7750" xml:space="preserve">Ob eaſdem quoque rationes, ſi concipiantur Hyperbolæ intra angulos <lb/>aſymptotales, vel Parabolæ æquidiſtantes, vel Hyperbolæ, aut Elli-<lb/>pſes, vel circuli ſimiles, & </s> <s xml:id="echoid-s7751" xml:space="preserve">concentrici circa communes axes in gyrum <lb/>conuersi, ſolida ab ipſis genita vocabuntur in poſterum ſolida eiuſdem <lb/>nominis ſimilia, & </s> <s xml:id="echoid-s7752" xml:space="preserve">concentrica. </s> <s xml:id="echoid-s7753" xml:space="preserve">Conus enim ab angulo procreatus ha-<lb/>bebitur pro primo, & </s> <s xml:id="echoid-s7754" xml:space="preserve">MINIMO Conoidorum Hyperbolicorum ſimilium, <lb/>& </s> <s xml:id="echoid-s7755" xml:space="preserve">concentricorum, &</s> <s xml:id="echoid-s7756" xml:space="preserve">c. </s> <s xml:id="echoid-s7757" xml:space="preserve">& </s> <s xml:id="echoid-s7758" xml:space="preserve">Conoidalia Parabolica tanquam ſimul con-<lb/>centrica, quarum commune centrum abeat in infinitam diſtantiam. </s> <s xml:id="echoid-s7759" xml:space="preserve">De <lb/>ſimilibus verò, & </s> <s xml:id="echoid-s7760" xml:space="preserve">concentricis Conoidibus Hyperbolicis, aut Sphæroidi-<lb/>bus, vel Sphæris, à ſimilibus, & </s> <s xml:id="echoid-s7761" xml:space="preserve">concentricis ſectionibus genitis, nihil <lb/>eſi quod ad nominum declar ationem addamus, cum eadem defi-<lb/>nitio ipſi definito perquàm rectè conueniat. </s> <s xml:id="echoid-s7762" xml:space="preserve">Verumenim-<lb/>uerò iam ſuſcepta, ac nuper interciſa ſolidorum tra-<lb/>ctatio, antequam reſumatùr, nouarum quarun-<lb/>dam vocum explicationem requirit, quam <lb/>ideò in ſequentibus ita exhibemus.</s> <s xml:id="echoid-s7763" xml:space="preserve"/> </p> <pb o="92" file="0278" n="278" rhead=""/> </div> <div xml:id="echoid-div800" type="section" level="1" n="315"> <head xml:id="echoid-head324" xml:space="preserve">DEFINITIONES. <lb/>I.</head> <p> <s xml:id="echoid-s7764" xml:space="preserve">PLANA ACVMINATA SIMILIA vocentur illa, quæ inter ſe ſint <lb/>proportionalia, & </s> <s xml:id="echoid-s7765" xml:space="preserve">quorum diametri ſuper baſes ſint æqualiter inclinatæ, ac <lb/>ijſdem baſibus proportionales.</s> <s xml:id="echoid-s7766" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7767" xml:space="preserve">Hoc eſt ſi ſint duo quælibet plana Acuminata proportionalia A B C, D <lb/>E F, quorum diametri B G, E H cum baſibus A C, D F æquales angulos <lb/>alterum alteri conſtituant, nempe A G B <lb/>ipſi D H E, & </s> <s xml:id="echoid-s7768" xml:space="preserve">qui ei eſt deinceps C G B <lb/> <anchor type="figure" xlink:label="fig-0278-01a" xlink:href="fig-0278-01"/> reliquo F H E ſit æqualis, ſitque diame-<lb/>ter B G ad baſim A C, vt diameter E H <lb/>ad baſim D F; </s> <s xml:id="echoid-s7769" xml:space="preserve">huiuſmodi plana inter ſe <lb/>vocentur SIMILIA ACVMINATA. <lb/></s> <s xml:id="echoid-s7770" xml:space="preserve">Vnde, & </s> <s xml:id="echoid-s7771" xml:space="preserve">duæ ſimiles Ellipſes vocari pote-<lb/>runt ſimilia Acuminata, cum vtraque ex <lb/>duobus proportionalibus Acuminatis con-<lb/>ſtet, ſiue ex dua<unsure/>bus ſemi-Ellipſibus, per diametros æqualiter inclinatas diſ-<lb/>ſectis, quarum diametri ſunt baſibus proportionales, &</s> <s xml:id="echoid-s7772" xml:space="preserve">c. </s> <s xml:id="echoid-s7773" xml:space="preserve">Idemque de duo-<lb/>bus circulis, &</s> <s xml:id="echoid-s7774" xml:space="preserve">c.</s> <s xml:id="echoid-s7775" xml:space="preserve"/> </p> <div xml:id="echoid-div800" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0278-01"/> </figure> </div> </div> <div xml:id="echoid-div802" type="section" level="1" n="316"> <head xml:id="echoid-head325" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s7776" xml:space="preserve">SOLIDVM ACVMINATVM REGVLARE, vel tantùm SOLIDVM <lb/>ACVMINATVM, voco omnem figuram ſolidam ad alteram partem defi-<lb/>cientem, circa planum Acuminatum deſcriptam, cuius omnia plana baſi ſo-<lb/>lidi æquidiſtantia per Acuminati applicatas ducta, ſint quoque plana Acu-<lb/>minata, eidem baſi, ac inter ſe ſimilia, & </s> <s xml:id="echoid-s7777" xml:space="preserve">ſimiliter poſita, & </s> <s xml:id="echoid-s7778" xml:space="preserve">quorum homo-<lb/>logæ diametri ſint ipſæ applicatæ prædicti Acuminati, &</s> <s xml:id="echoid-s7779" xml:space="preserve">c.</s> <s xml:id="echoid-s7780" xml:space="preserve"/> </p> <figure> <image file="0278-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0278-02"/> </figure> <p> <s xml:id="echoid-s7781" xml:space="preserve">Eſto planum quodcunque Acuminatum A B C, cuius baſis A C, dia-<lb/>meter B D, vertex B, & </s> <s xml:id="echoid-s7782" xml:space="preserve">ipſa A C, ſit vel diameter circuli, aut Ellipſis, <lb/>vel cuiuſcun que ipſarum figurarum portionis, aut diameter Parabolæ, <lb/>vel Hyperbolæ, vel cuiuslibet alij plani Acuminati A E C F, quod tan-<lb/>quam baſis, ad quemlibet inclinationis angulum cum plano A B C ſit <pb o="93" file="0279" n="279" rhead=""/> diſpoſitum, ſintque omnia plana G M H, I N L, &</s> <s xml:id="echoid-s7783" xml:space="preserve">c. </s> <s xml:id="echoid-s7784" xml:space="preserve">quæ baſi A E C F <lb/>æquidiſtanter ducuntur per Acuminati A B C applicatas G H, I L, &</s> <s xml:id="echoid-s7785" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s7786" xml:space="preserve">ipſi baſi, ac inter ſe, ſimilia Acuminata, & </s> <s xml:id="echoid-s7787" xml:space="preserve">ſimiliter poſita, atque ipſæ <lb/>applicatæ G H, I L ſint eorundem Acuminatorum homologæ diametri: </s> <s xml:id="echoid-s7788" xml:space="preserve"><lb/>huiuſmodi figura SOLIDVM REGVLARE ACVMINATVM vocetur, <lb/>vel tantùm ACVMINATVM SOLIDVM; </s> <s xml:id="echoid-s7789" xml:space="preserve">A E C F verò BASIS ſoli-<lb/>di Acuminati; </s> <s xml:id="echoid-s7790" xml:space="preserve">ſed portionem A B C Acuminati plani intra Acuminatum <lb/>ſolidum interceptam (eò quod ipſa ſit tanquam Regula, vel Modulus, <lb/>aut Canon homologarum diametrorum ſimilium planorum ęquidiſtantium, <lb/>ac ſolidum procreantium) nuncupare liceat CANONEM ſolidi Acumina-<lb/>ti, qui ſi ad planum ba<unsure/>ſis A E C F rectus fuerit, dicatur CANON RECTVS <lb/>ſolidi Acuminati, & </s> <s xml:id="echoid-s7791" xml:space="preserve">B D diameter Canonis, nuncupetur quoque AXIS <lb/>ſolidi, & </s> <s xml:id="echoid-s7792" xml:space="preserve">eius VERTEX punctum B, in quod abit ſolidum, atque eiuſdem <lb/>ſolidi ALTITVDO dicatur recta B O, quæ à vertice B ſuper baſim A E C <lb/>F recta ducitur. </s> <s xml:id="echoid-s7793" xml:space="preserve">Plana verò A C, G H, I L, &</s> <s xml:id="echoid-s7794" xml:space="preserve">c. </s> <s xml:id="echoid-s7795" xml:space="preserve">dicantur PLANA OR-<lb/>DINATIM DVCTA ad axim ſolidi Acuminati.</s> <s xml:id="echoid-s7796" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div803" type="section" level="1" n="317"> <head xml:id="echoid-head326" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s7797" xml:space="preserve">SOLIDA ACVMINATA PROPORTIONALIA dicantur illa, quo-<lb/>rum omnia plana ordinatim applicata per puncta, eorum axes proportio-<lb/>naliter diuidentia, ſint quoque inter ſe, & </s> <s xml:id="echoid-s7798" xml:space="preserve">baſibus proportionalia.</s> <s xml:id="echoid-s7799" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7800" xml:space="preserve">Videlicet ſi duo ſolida Acumi-<lb/> <anchor type="figure" xlink:label="fig-0279-01a" xlink:href="fig-0279-01"/> nata A B C, D E F, quorum baſes <lb/>ſint A G C I, L F H D axes verò <lb/>ſint B K, E O proportionaliter ſe-<lb/>cti in M, P; </s> <s xml:id="echoid-s7801" xml:space="preserve">& </s> <s xml:id="echoid-s7802" xml:space="preserve">in N, Q; </s> <s xml:id="echoid-s7803" xml:space="preserve">ita vt K <lb/>M, ad M B ſit vt O P, ad P E; </s> <s xml:id="echoid-s7804" xml:space="preserve">& </s> <s xml:id="echoid-s7805" xml:space="preserve"><lb/>K N ad N B, vt O Q ad Q E, &</s> <s xml:id="echoid-s7806" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s7807" xml:space="preserve">ſitque baſis A G C ad baſim L F H, <lb/>vt planum ordinatim applicatum <lb/>per M ad applicatum per P, & </s> <s xml:id="echoid-s7808" xml:space="preserve">vt <lb/>applicatum per N ad applicatum <lb/>per Q, &</s> <s xml:id="echoid-s7809" xml:space="preserve">c. </s> <s xml:id="echoid-s7810" xml:space="preserve">talia ſolida, dicentur <lb/>SOLIDA ACVMINATA PROPORTIONALIA.</s> <s xml:id="echoid-s7811" xml:space="preserve"/> </p> <div xml:id="echoid-div803" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0279-01"/> </figure> </div> </div> <div xml:id="echoid-div805" type="section" level="1" n="318"> <head xml:id="echoid-head327" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s7812" xml:space="preserve">Si ſuper diametrum Acuminati plani deſcriptum ſit parallelogrammum <lb/>quodlibet ſuper ipſum planum quomodocunque eleuatum, idem que Acu-<lb/>minatum concipiatur ſibi ipſi æquidiſtanter moueri, ita vt eius diameter ſuo <lb/>motu parallelo prædictum parallelogrammum deſcribat: </s> <s xml:id="echoid-s7813" xml:space="preserve">ſolidum occluſum <lb/>à duobus oppoſitis Acuminatis congruentibus, ac parallelis, atque à ſuper-<lb/>ficie, quæ à perimetro figuræ motæ deſcribitur CYLINDRICVS vocetur. <lb/></s> <s xml:id="echoid-s7814" xml:space="preserve">Acuminatum verò ſolidum procreans, dicatur BASIS, & </s> <s xml:id="echoid-s7815" xml:space="preserve">parallelogram-<lb/>mum, per quod fit æquidiſtans latio Acuminati plani Cylindricum pro-<lb/>creantis, CANON DIAMETRALIS nuncupetur.</s> <s xml:id="echoid-s7816" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7817" xml:space="preserve">Nimirum, ſit Acuminatum planum A B C, cuius diameter B D, cui in-<lb/>ſiſtat parallelogrammum quodcumq; </s> <s xml:id="echoid-s7818" xml:space="preserve">B D E F ſuper planum figuræ A B C <pb o="94" file="0280" n="280" rhead=""/> vtcunque eleuatum, concipiaturque Acu-<lb/> <anchor type="figure" xlink:label="fig-0280-01a" xlink:href="fig-0280-01"/> minatum A B C moueri motu ſibi ipſi pa-<lb/>rallelo, ſed ita vt recta B D æquidiſtanter <lb/>incedat ſuper parallelogrammum B E, do-<lb/>nec congruat cum oppoſito latere E F. <lb/></s> <s xml:id="echoid-s7819" xml:space="preserve">Huiuſmodi ſolidum occluſum à parallelis, <lb/>& </s> <s xml:id="echoid-s7820" xml:space="preserve">congruentibus Acuminatis A B C, G F <lb/>H, atque à ſuperficie, quæ à perimetro A <lb/>B C A in ſua latione deſcribitur, vocetur <lb/>CYLINDRICVS, Acuminatum verò A B C eius BASIS, & </s> <s xml:id="echoid-s7821" xml:space="preserve">parallelo-<lb/>grammum B E CANON DIAMETRALIS prædicti Cylindrici, cuius <lb/>altitudo metietur per rectam ad vtrunque oppoſitorum planorum perpen-<lb/>dicularem.</s> <s xml:id="echoid-s7822" xml:space="preserve"/> </p> <div xml:id="echoid-div805" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0280-01"/> </figure> </div> <p> <s xml:id="echoid-s7823" xml:space="preserve">Itaque CYLINDRICVS dicetur omne ſolidum circa parallelogrãmum <lb/>quodcunque deſcriptum, & </s> <s xml:id="echoid-s7824" xml:space="preserve">cuius omnia plana baſi ſolidi æquidiſtantia, ac <lb/>per applicatas in parallelogrammo ducta, ſint plana Acuminata, eidem <lb/>baſi, ac inter ſe æqualia, & </s> <s xml:id="echoid-s7825" xml:space="preserve">ſimilia, & </s> <s xml:id="echoid-s7826" xml:space="preserve">ſimiliter poſita, & </s> <s xml:id="echoid-s7827" xml:space="preserve">quorum homologę <lb/>diametri ſint ipſæ applicatæ in prædicto parallelogrammo; </s> <s xml:id="echoid-s7828" xml:space="preserve">quod CANON <lb/>DIAMETRALIS Cylindrici vocabitur.</s> <s xml:id="echoid-s7829" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7830" xml:space="preserve">Omittimus vniuerſaliores Solidorum Acuminatorũ, ac Cylindricorum <lb/>definitiones, cum hoc loco de ijs ſermo minimè habendus ſit.</s> <s xml:id="echoid-s7831" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div807" type="section" level="1" n="319"> <head xml:id="echoid-head328" xml:space="preserve">PROBL. XIV. PROP. LXIX.</head> <p> <s xml:id="echoid-s7832" xml:space="preserve">Si Conoides quodcunque, vel Sphæra, aut Sphæroides ob-<lb/>longum, vel prolatum plano ſecetur ex dato ſolido portionem <lb/>abſcindent: </s> <s xml:id="echoid-s7833" xml:space="preserve">poſſibile eſt per axem ſolidi, planum ducere, quod <lb/>ad baſim abſciſſæ portionis ſit erectum. </s> <s xml:id="echoid-s7834" xml:space="preserve">Item.</s> <s xml:id="echoid-s7835" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7836" xml:space="preserve">Poſſibile eſt baſi portionis aliud planum æquidiſtans ducere, <lb/>quod conuexam ſolidæ portionis ſnperficiem contingat.</s> <s xml:id="echoid-s7837" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7838" xml:space="preserve">ESto quodcunque ex prædictis ſolidis A B C, cuius axis reuolutionis ſit <lb/>B D, atque ex eo per planum E H G I ſit abſciſſa portio ſolida E F G, <lb/>cuius baſis E H G I (quæ, vel erit <anchor type="note" xlink:href="" symbol="a"/> Ellipſis, vel circulus.) </s> <s xml:id="echoid-s7839" xml:space="preserve">Dico poſſibile <anchor type="note" xlink:label="note-0280-01a" xlink:href="note-0280-01"/> eſſe baſi E H G I planum ducere per ſolidi axem B D, quod ad baſim E H <lb/>G I rectum ſit. </s> <s xml:id="echoid-s7840" xml:space="preserve">Præterea poſſibile eſſe eidem baſi aliud planum æquidiſtans <lb/>ducere, quod ſolidæ portionis ſuperficiem contingat.</s> <s xml:id="echoid-s7841" xml:space="preserve"/> </p> <div xml:id="echoid-div807" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0280-01" xlink:href="note-0280-01a" xml:space="preserve">ex 13. 14 <lb/>15. Arch. <lb/>de Conoi. <lb/>&c.</note> </div> <p> <s xml:id="echoid-s7842" xml:space="preserve">Si enim planum ſecans E I G fuerit ad axem B D erectum, hunc ſecans <lb/>in K, ſectio circulus erit, <anchor type="note" xlink:href="" symbol="b"/> cuius centrum K; </s> <s xml:id="echoid-s7843" xml:space="preserve">& </s> <s xml:id="echoid-s7844" xml:space="preserve">ſi per axim B K agatur <anchor type="note" xlink:label="note-0280-02a" xlink:href="note-0280-02"/> quodcunque planum E B G baſim portionis E H G I ſecans per rectam E <lb/>G, ſectionis portio plana E B G erit <anchor type="note" xlink:href="" symbol="c"/> ea, quæ ſolidum genuit, cuius baſis <anchor type="note" xlink:label="note-0280-03a" xlink:href="note-0280-03"/> eadem E G, axis verò ipſe B K, & </s> <s xml:id="echoid-s7845" xml:space="preserve">ad baſim E H G I recta <anchor type="note" xlink:href="" symbol="d"/> erit. </s> <s xml:id="echoid-s7846" xml:space="preserve">Quod <anchor type="note" xlink:label="note-0280-04a" xlink:href="note-0280-04"/> primò, &</s> <s xml:id="echoid-s7847" xml:space="preserve">c.</s> <s xml:id="echoid-s7848" xml:space="preserve"/> </p> <div xml:id="echoid-div808" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0280-02" xlink:href="note-0280-02a" xml:space="preserve">12. Ar-<lb/>chim. ib. <lb/>à Comãd. <lb/>reſtit.</note> <note symbol="c" position="left" xlink:label="note-0280-03" xlink:href="note-0280-03a" xml:space="preserve">ibidem.</note> <note symbol="d" position="left" xlink:label="note-0280-04" xlink:href="note-0280-04a" xml:space="preserve">18. vnd. <lb/>Elem.</note> </div> <pb o="95" file="0281" n="281" rhead=""/> <p> <s xml:id="echoid-s7849" xml:space="preserve">Iam ſi per verticem B ducatur in plano portionis E B G recta B L, ipſam <lb/> <anchor type="note" xlink:label="note-0281-01a" xlink:href="note-0281-01"/> portionem contingens, hæc baſi E G <anchor type="note" xlink:href="" symbol="a"/> æquidiſtabit: </s> <s xml:id="echoid-s7850" xml:space="preserve">& </s> <s xml:id="echoid-s7851" xml:space="preserve">ſi per B L concipia- tur planum duci, quod plano per axem E B G ſit erectum, id ſolidæ por-<lb/> <anchor type="note" xlink:label="note-0281-02a" xlink:href="note-0281-02"/> tionis ſuperficiem continget <anchor type="note" xlink:href="" symbol="b"/> in B, atque baſi E H G I erit parallelum <anchor type="note" xlink:href="" symbol="c"/> cum <anchor type="note" xlink:label="note-0281-03a" xlink:href="note-0281-03"/> vtrunque planorum ſit eidem E B G rectum, & </s> <s xml:id="echoid-s7852" xml:space="preserve">communes ſectiones B L, <lb/>E G ſint parallelæ. </s> <s xml:id="echoid-s7853" xml:space="preserve">Quod ſecundò, &</s> <s xml:id="echoid-s7854" xml:space="preserve">c.</s> <s xml:id="echoid-s7855" xml:space="preserve"/> </p> <div xml:id="echoid-div809" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0281-01" xlink:href="note-0281-01a" xml:space="preserve">32. pri-<lb/>mi conic.</note> <note symbol="b" position="right" xlink:label="note-0281-02" xlink:href="note-0281-02a" xml:space="preserve">55. h.</note> <note symbol="c" position="right" xlink:label="note-0281-03" xlink:href="note-0281-03a" xml:space="preserve">per Sch. <lb/>Clauijpoſt <lb/>18. vndec. <lb/>elem.</note> </div> <p> <s xml:id="echoid-s7856" xml:space="preserve">Siverò planum ſecans E H G I rectum non fuerit ad axem B D; </s> <s xml:id="echoid-s7857" xml:space="preserve">(& </s> <s xml:id="echoid-s7858" xml:space="preserve">tunc <lb/>ſectio <anchor type="note" xlink:href="" symbol="d"/> erit Ellipſis) ſecetur denuò datum ſolidum quocunque alio plano A <anchor type="note" xlink:label="note-0281-04a" xlink:href="note-0281-04"/> H C I ad axem recto: </s> <s xml:id="echoid-s7859" xml:space="preserve">(quod tamen non tranſeat per interſectionem axis B <lb/>D cum plano E H G I, ſi hoc axem ſecuerit intra ſolidum) id in ſolido ſe-<lb/>ctionem faciet <anchor type="note" xlink:href="" symbol="e"/> circulum, centrum habentem in axe B D, vti in D, omninò <anchor type="note" xlink:label="note-0281-05a" xlink:href="note-0281-05"/> autem ſecabit baſim E H G I per communem rectam H I tùm in Ellipſi, tùm <lb/>in circulo applicatam, cui ex D, circuli centro, ducta perpendiculari D M; <lb/></s> <s xml:id="echoid-s7860" xml:space="preserve">per axem B D, ac rectam D M agatur planum in ſolido efficiens genitricem <lb/>ſectionem E A B G C, cuius communis ſectio cum circulo erit diameter A <lb/>C, & </s> <s xml:id="echoid-s7861" xml:space="preserve">cum Ellipſi erit recta E G.</s> <s xml:id="echoid-s7862" xml:space="preserve"/> </p> <div xml:id="echoid-div810" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-0281-04" xlink:href="note-0281-04a" xml:space="preserve">13. 14. <lb/>15. Arch. <lb/>de Conoi. <lb/>&c.</note> <note symbol="e" position="right" xlink:label="note-0281-05" xlink:href="note-0281-05a" xml:space="preserve">12. Arch. <lb/>ib. à Co-<lb/>mãd. reſt.</note> </div> <p> <s xml:id="echoid-s7863" xml:space="preserve">Iam priùs oſtendam ſectio-<lb/> <anchor type="figure" xlink:label="fig-0281-01a" xlink:href="fig-0281-01"/> nem hanc per B D axem du-<lb/>ctam ad ſecans planum E H <lb/>G I, ſiue ad baſim ſolidę por-<lb/>tionis E F G rectam eſſe. <lb/></s> <s xml:id="echoid-s7864" xml:space="preserve">Quoniam cum planum circu-<lb/>li E H C I rectum ſit ad pla-<lb/>nũ per axem E A B C, cumq; </s> <s xml:id="echoid-s7865" xml:space="preserve"><lb/>linea I M in circulo perpen-<lb/>dicularis ſit ad A C horum <lb/>planorum communem ſectio-<lb/>nem, erit eadem linea I M <lb/>recta <anchor type="note" xlink:href="" symbol="f"/> ad planum per axem <anchor type="note" xlink:label="note-0281-06a" xlink:href="note-0281-06"/> E A B C: </s> <s xml:id="echoid-s7866" xml:space="preserve">quare omnia plana, quæ per ipſam ducentur ad idem planum E A <lb/>B C recta erunt, <anchor type="note" xlink:href="" symbol="g"/> ſed E H G I baſis ſolidæ portionis tranſit per I M, ergo <anchor type="note" xlink:label="note-0281-07a" xlink:href="note-0281-07"/> baſis E H G I, ſiue planum ſecans rectum erit ad planum per axem E A B C, <lb/>ſiue id rectum ad planum ſecans, hoc eſt ad baſim ſolidæ portionis. </s> <s xml:id="echoid-s7867" xml:space="preserve">Quod <lb/>primò, &</s> <s xml:id="echoid-s7868" xml:space="preserve">c.</s> <s xml:id="echoid-s7869" xml:space="preserve"/> </p> <div xml:id="echoid-div811" type="float" level="2" n="5"> <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/QN4GHYBF/figures/0281-01"/> </figure> <note symbol="f" position="right" xlink:label="note-0281-06" xlink:href="note-0281-06a" xml:space="preserve">4. def. <lb/>vnd. Ele.</note> <note symbol="g" position="right" xlink:label="note-0281-07" xlink:href="note-0281-07a" xml:space="preserve">18. vnd. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s7870" xml:space="preserve">Cum ergo E G ſit communis ſectio planorum, eius ſcilicet, quod ſolidũ <lb/>ſecat, & </s> <s xml:id="echoid-s7871" xml:space="preserve">cius, quod per axem ducitur erectum ſuper planum ſecans, ipſa E <lb/>G erit <anchor type="note" xlink:href="" symbol="h"/> axis Ellipſis E H G I, qua bifariam ſecta in N, erit N Ellipſis cen- <anchor type="note" xlink:label="note-0281-08a" xlink:href="note-0281-08"/> trum, ex quo, in plana portione E F G ſectionis per axem à recta E G ab-<lb/>ſciſſæ, & </s> <s xml:id="echoid-s7872" xml:space="preserve">ſuper baſim ſolidæ portionis erectæ, ducta diametro N F, & </s> <s xml:id="echoid-s7873" xml:space="preserve">per F <lb/>ſectionem <anchor type="note" xlink:href="" symbol="i"/> contingente F O, per ipſam F O agatur planum, quod ad idem <anchor type="note" xlink:label="note-0281-09a" xlink:href="note-0281-09"/> planum per axem E B G rectum ſit, id ſolidæ portionis E F G ſuperficiem <lb/> <anchor type="note" xlink:label="note-0281-10a" xlink:href="note-0281-10"/> continget <anchor type="note" xlink:href="" symbol="l"/> in F, & </s> <s xml:id="echoid-s7874" xml:space="preserve">baſi E H G I æquidiſtabit. </s> <s xml:id="echoid-s7875" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> Quod ſecundò, &</s> <s xml:id="echoid-s7876" xml:space="preserve">c. </s> <s xml:id="echoid-s7877" xml:space="preserve">Si <anchor type="note" xlink:label="note-0281-11a" xlink:href="note-0281-11"/> fuerit ergo Conoides quodcunque, vel Sphæra, &</s> <s xml:id="echoid-s7878" xml:space="preserve">c. </s> <s xml:id="echoid-s7879" xml:space="preserve">poſſibile eſt, &</s> <s xml:id="echoid-s7880" xml:space="preserve">c. </s> <s xml:id="echoid-s7881" xml:space="preserve">Quod <lb/>erat faciendum, ac demondrandum.</s> <s xml:id="echoid-s7882" xml:space="preserve"/> </p> <div xml:id="echoid-div812" type="float" level="2" n="6"> <note symbol="h" position="right" xlink:label="note-0281-08" xlink:href="note-0281-08a" xml:space="preserve">13. 14. <lb/>15. Arch. <lb/>de Conoi. <lb/>&c.</note> <note symbol="i" position="right" xlink:label="note-0281-09" xlink:href="note-0281-09a" xml:space="preserve">2. & 4. <lb/>pr. h.</note> <note symbol="l" position="right" xlink:label="note-0281-10" xlink:href="note-0281-10a" xml:space="preserve">55. h.</note> <note symbol="m" position="right" xlink:label="note-0281-11" xlink:href="note-0281-11a" xml:space="preserve">Schol. <lb/>Clauijpoſt <lb/>18. vndec. <lb/>Elem.</note> </div> <pb o="96" file="0282" n="282" rhead=""/> </div> <div xml:id="echoid-div814" type="section" level="1" n="320"> <head xml:id="echoid-head329" xml:space="preserve">SCHOLIVM I.</head> <p> <s xml:id="echoid-s7883" xml:space="preserve">CVm huiuſmodi ſolida portio E F G de quolibet prędictorum ſolidorum <lb/>abſciſſa, ſit ſolidum ad alteram partem F deficiens, circa Acuminatũ <lb/>planum E F G deſcriptum, cumque omnia plana eius baſi E H G I æquidi-<lb/>ſtantia, ſint plana Acuminata, vt in prima proximè præcedentium definitio-<lb/> <anchor type="note" xlink:label="note-0282-01a" xlink:href="note-0282-01"/> num monuimus, ſintque omnia inter ſe <anchor type="note" xlink:href="" symbol="a"/> ſimilia, ac ſimiliter poſita, eò quod vel ſint circuli, vel Ellipſes, quarum homologi axes ſunt <anchor type="note" xlink:href="" symbol="b"/> eædem applicatæ in Acuminato E F G, idcircò per ſecundam prædictarum definit. </s> <s xml:id="echoid-s7884" xml:space="preserve">talis ſoli-<lb/> <anchor type="note" xlink:label="note-0282-02a" xlink:href="note-0282-02"/> da portio in poſterum vocari poterit aliquandò ſolidum Acuminatum; </s> <s xml:id="echoid-s7885" xml:space="preserve">& </s> <s xml:id="echoid-s7886" xml:space="preserve"><lb/>planum Acuminatum, ſeu portio plana E F G, cum ſit recta ad baſim E H <lb/>G I, dicetur Canon rectus ſolidæ portionis.</s> <s xml:id="echoid-s7887" xml:space="preserve"/> </p> <div xml:id="echoid-div814" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0282-01" xlink:href="note-0282-01a" xml:space="preserve">Coroll. <lb/>15. Arch. <lb/>de Conoi. <lb/>&c.</note> <note symbol="b" position="left" xlink:label="note-0282-02" xlink:href="note-0282-02a" xml:space="preserve">13. 14. <lb/>15. ibid.</note> </div> </div> <div xml:id="echoid-div816" type="section" level="1" n="321"> <head xml:id="echoid-head330" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s7888" xml:space="preserve">EX hac elicitur, qua methodo per axem cuiuslibet Conoidis, aut Sphæ-<lb/>roidis, vel Sphæræ, aut etiam Coni recti duci poſſit planum, quod ad <lb/>datum quodcunque planum non per axem ductum, & </s> <s xml:id="echoid-s7889" xml:space="preserve">ſolidum ſecans, re-<lb/>ctum ſit, etiam ſi ſecans planum in Conoide Parabolico, aut Hyperbolico, <lb/>vel Cono non ſit circulus, neque Ellipſis: </s> <s xml:id="echoid-s7890" xml:space="preserve">ſimulque patet, quod prædictum <lb/>planum per axem, aliud non per axem ductum omnino ſecat intra ſolidum: <lb/></s> <s xml:id="echoid-s7891" xml:space="preserve">quæ omnia, velleuiter perpendenti manifeſta ſunt ex dictis, quæque ab ip-<lb/>ſo Archimede tanquam poſſibilia, & </s> <s xml:id="echoid-s7892" xml:space="preserve">iam nota paſſim ſupponuntur in libro <lb/>de Conoid. </s> <s xml:id="echoid-s7893" xml:space="preserve">&</s> <s xml:id="echoid-s7894" xml:space="preserve">c.</s> <s xml:id="echoid-s7895" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div817" type="section" level="1" n="322"> <head xml:id="echoid-head331" xml:space="preserve">SCHOLIVM II.</head> <p> <s xml:id="echoid-s7896" xml:space="preserve">POterat quidem prima pars huius Problematis breuiùs perſolui. </s> <s xml:id="echoid-s7897" xml:space="preserve">Nam <lb/>ex vertice B, vel ex quolibet alio axis puncto, ſuper planum ſe-<lb/>cans E H G I ducta perpẽdiculari, per quàm, & </s> <s xml:id="echoid-s7898" xml:space="preserve">per axem B D ducto plano; <lb/></s> <s xml:id="echoid-s7899" xml:space="preserve">conſtat hoc idem ſuper planum ſecans rectum <anchor type="note" xlink:href="" symbol="c"/> eſſe. </s> <s xml:id="echoid-s7900" xml:space="preserve">Verùm cum ſæpe eue- <anchor type="note" xlink:label="note-0282-03a" xlink:href="note-0282-03"/> niat, quod ipſa perpendicularis occurrat ſecanti plano non intra ſolidum, <lb/>ſed vel in eius ſuperficie, vel extra, cumq; </s> <s xml:id="echoid-s7901" xml:space="preserve">omnino oſtendere opus ſit, quod <lb/>huiuſmodi planum per axem, rectum ad planum ſecans, hoc idem planum <lb/>ſecat ſemper intra ſolidum, idcircò prò huius Problematis ſolutione ſupe-<lb/>riorem viam elegimus, quæ ad vtrunq; </s> <s xml:id="echoid-s7902" xml:space="preserve">ſimul nos perduceret vnica conſtru-<lb/>ctione.</s> <s xml:id="echoid-s7903" xml:space="preserve"/> </p> <div xml:id="echoid-div817" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-0282-03" xlink:href="note-0282-03a" xml:space="preserve">18. vnd. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div819" type="section" level="1" n="323"> <head xml:id="echoid-head332" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s7904" xml:space="preserve">COlligitur quoque planum, quod baſi portionis cuiuslibet prædicto-<lb/>rum ſolidorum æquidiſtat, atque eius conuexam ſuperficiem con-<lb/>tingit, eam contingere ad verticem diametri recti Canonis; </s> <s xml:id="echoid-s7905" xml:space="preserve">hoc eſt tan-<lb/>gere ad verticem axis portionis ſolidæ.</s> <s xml:id="echoid-s7906" xml:space="preserve"/> </p> <pb o="97" file="0283" n="283" rhead=""/> <p> <s xml:id="echoid-s7907" xml:space="preserve">Nam, ad finem propoſitionis oſtenſum fuit, planum contingens portio-<lb/>nem ſolidam E F G, & </s> <s xml:id="echoid-s7908" xml:space="preserve">baſi E H G I parallelum, eam contingere ad pun-<lb/>ctum F, quod eſt vertex diametri N F Canonis recti E F G, atque inſuper <lb/>idem punctum contactus F, iuxta Archim. </s> <s xml:id="echoid-s7909" xml:space="preserve">definitiones præmiſſas ad librum <lb/>de Conoid. </s> <s xml:id="echoid-s7910" xml:space="preserve">&</s> <s xml:id="echoid-s7911" xml:space="preserve">c. </s> <s xml:id="echoid-s7912" xml:space="preserve">iam notum eſt verticem vocari axis portionis ſolidæ E F G.</s> <s xml:id="echoid-s7913" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div820" type="section" level="1" n="324"> <head xml:id="echoid-head333" xml:space="preserve">SCHOLIVM III.</head> <p> <s xml:id="echoid-s7914" xml:space="preserve">EX his itaque notandum eſt, axim ſolidæ portionis eundem eſſe cum dia-<lb/>metro prædicti Canonis recti, & </s> <s xml:id="echoid-s7915" xml:space="preserve">altitudinem, eandem cum altitudine.</s> <s xml:id="echoid-s7916" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7917" xml:space="preserve">Nam eadem recta F N <lb/> <anchor type="figure" xlink:label="fig-0283-01a" xlink:href="fig-0283-01"/> quæ ex conſtructione diame-<lb/>ter eſt planæ portionis E F <lb/>G, eſt quoque axis ſolidæ, <lb/>cum ab F eius vertice, ad N <lb/>centrum baſis E H G I ince-<lb/>dat. </s> <s xml:id="echoid-s7918" xml:space="preserve">Præterea ducta ex ha-<lb/>rum portionum cómuni ver-<lb/>tice F recta F P ad baſim E <lb/>G planæ portionis, ſeu recti <lb/>Canonis E F G perpendicu-<lb/>lari. </s> <s xml:id="echoid-s7919" xml:space="preserve">Patet hanc eſſe Canonis <lb/>altitudinem, ſed Canon E F <lb/>G rectus ponitur ad baſim E H G I; </s> <s xml:id="echoid-s7920" xml:space="preserve">quare F P, quæ ad communem horum <lb/>planorum ſectionem E G eſt perpendicularis, recta erit ad planum baſis <lb/>E H G I, ac propterea ipſa erit quoque altitudo portionis ſolidæ E F G, <lb/>cum perpendiculariter cadat ex eius vertice F ſuper baſim E H G I, &</s> <s xml:id="echoid-s7921" xml:space="preserve">c.</s> <s xml:id="echoid-s7922" xml:space="preserve"/> </p> <div xml:id="echoid-div820" type="float" level="2" n="1"> <figure xlink:label="fig-0283-01" xlink:href="fig-0283-01a"> <image file="0283-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0283-01"/> </figure> </div> </div> <div xml:id="echoid-div822" type="section" level="1" n="325"> <head xml:id="echoid-head334" xml:space="preserve">COROLL. III.</head> <p> <s xml:id="echoid-s7923" xml:space="preserve">PAtet denique axim portionis cuiuſcunque prædictorum ſolidorum, & </s> <s xml:id="echoid-s7924" xml:space="preserve"><lb/>axim ſolidi, cuius eſt portio, eſſe in vno eodemque plano, quod per <lb/>axem eiuſdem ſolidi ad baſim portionis rectum ducitur, ſiue eſſe in plano <lb/>Canonis recti.</s> <s xml:id="echoid-s7925" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7926" xml:space="preserve">Etenim, & </s> <s xml:id="echoid-s7927" xml:space="preserve">B D axis dati ſolidi, & </s> <s xml:id="echoid-s7928" xml:space="preserve">F N axis ſolidæ portionis E F G ſunt <lb/>in plano E B C ducto per axem B D, ſed erecto ſuper baſim E I G H por-<lb/>tionis ſolidę E F G, quod planum E B C idem eſt, ac planum recti Canonis <lb/>E F G intra ſolidam portionem intercepti.</s> <s xml:id="echoid-s7929" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7930" xml:space="preserve">Siergo per axim datæ ſolidæ portionis, & </s> <s xml:id="echoid-s7931" xml:space="preserve">per axim ſolidi, cuius eſt por-<lb/>tio ducatur planum, hoc erit ad planum baſis portionis erectum, atque in <lb/>ſolida portione rectum Canonem exhibebit.</s> <s xml:id="echoid-s7932" xml:space="preserve"/> </p> <pb o="98" file="0284" n="284" rhead=""/> </div> <div xml:id="echoid-div823" type="section" level="1" n="326"> <head xml:id="echoid-head335" xml:space="preserve">THEOR. XLIII. PROP. LXX.</head> <p> <s xml:id="echoid-s7933" xml:space="preserve">Portiones eiuſdem, vel diuerſorum Conorum, aut Conoidum <lb/>Parabolicorum, ſunt ſolida Acuminata proportionalia. </s> <s xml:id="echoid-s7934" xml:space="preserve">Item.</s> <s xml:id="echoid-s7935" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7936" xml:space="preserve">Portiones eiuſdem, vel diuerſorum Conoidum Hyperbolico-<lb/>rum, vel Sphærarum, aut Sphæroidum, quarum ſegmenta diame-<lb/>trorum in portionibus genitricium earum ſectionum ad baſes ere-<lb/>ctis intercepta, ad ſuas ſemi-diametros eandem homologam ha-<lb/>beant rationem, ſunt pariter ſolida Acuminata proportionalia.</s> <s xml:id="echoid-s7937" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7938" xml:space="preserve">SInt primò duæ quæcunque portiones A B C, D E F eiuſdem, vel diuer-<lb/>ſorum Conorum, vt in prima figura, vel eiuſdem, aut diu erſorum Co-<lb/>noidum Parabolicorum, vt in ſecunda, quarum axes ſint B G, E H, baſes <lb/>verò circuli, aut Ellipſes A C, D F, ipſæque portiones ſolidæ, (quæ iam <lb/> <anchor type="note" xlink:label="note-0284-01a" xlink:href="note-0284-01"/> per primum Scholium precedentis ſunt ſolida Acuminata) planis per eorum <lb/> <anchor type="note" xlink:label="note-0284-02a" xlink:href="note-0284-02"/> ſolidorum axes ductis ad baſes rectis <anchor type="note" xlink:href="" symbol="a"/> ſecentur, & </s> <s xml:id="echoid-s7939" xml:space="preserve">ſient <anchor type="note" xlink:href="" symbol="b"/> in ſolidis recti Ca- <anchor type="note" xlink:label="note-0284-03a" xlink:href="note-0284-03"/> nones A B C, D E F, qui erunt <anchor type="note" xlink:href="" symbol="c"/> portiones ſectionum ſolida genitricium, & </s> <s xml:id="echoid-s7940" xml:space="preserve">communes ſectiones ipſorum cum baſibus erunt <anchor type="note" xlink:href="" symbol="d"/> rectæ A C, D F, quæ circulorum, aut Ellipſium <anchor type="note" xlink:href="" symbol="e"/> erunt axes. </s> <s xml:id="echoid-s7941" xml:space="preserve">Dico in vtraque ſigura ſolidas por- tiones A B C, D E F eſſe Acuminata ſolida proportionalia.</s> <s xml:id="echoid-s7942" xml:space="preserve"/> </p> <div xml:id="echoid-div823" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0284-01" xlink:href="note-0284-01a" xml:space="preserve">69. h.</note> <note symbol="b" position="left" xlink:label="note-0284-02" xlink:href="note-0284-02a" xml:space="preserve">ibid. 1. <lb/>Schol.</note> <note symbol="c" position="left" xlink:label="note-0284-03" xlink:href="note-0284-03a" xml:space="preserve">ex 12. <lb/>Archim. <lb/>de Co-<lb/>noid. & <lb/>Comand. <lb/>ſuppleta.</note> </div> <p> <s xml:id="echoid-s7943" xml:space="preserve">Etenim horum Acuminato-<lb/> <anchor type="note" xlink:label="note-0284-04a" xlink:href="note-0284-04"/> <anchor type="figure" xlink:label="fig-0284-01a" xlink:href="fig-0284-01"/> rum ſolidorum axibus B G, E <lb/>H proportionaliter vtcunque <lb/> <anchor type="note" xlink:label="note-0284-05a" xlink:href="note-0284-05"/> ſectis in I, L, ducantur per I, <lb/>L plana M N, O P baſibus A <lb/>C, D F æquidiſtantia, quæ in <lb/>ſolidis efficient ſectiones ipſa-<lb/>rum baſibus ſimiles <anchor type="note" xlink:href="" symbol="f"/> earumq;</s> <s xml:id="echoid-s7944" xml:space="preserve"> <anchor type="note" xlink:label="note-0284-06a" xlink:href="note-0284-06"/> communes ſectiones cum pla-<lb/> <anchor type="note" xlink:label="note-0284-07a" xlink:href="note-0284-07"/> nis A B C, D E F <anchor type="note" xlink:href="" symbol="g"/> erunt re- ctæ M N, O P ipſis A C, D <lb/> <anchor type="note" xlink:label="note-0284-08a" xlink:href="note-0284-08"/> F <anchor type="note" xlink:href="" symbol="h"/> parallelæ, & </s> <s xml:id="echoid-s7945" xml:space="preserve">earundem ſi- milium ſectionum homologæ <lb/>diametri.</s> <s xml:id="echoid-s7946" xml:space="preserve"/> </p> <div xml:id="echoid-div824" type="float" level="2" n="2"> <note symbol="d" position="left" xlink:label="note-0284-04" xlink:href="note-0284-04a" xml:space="preserve">3. vnd. <lb/>Elem.</note> <figure xlink:label="fig-0284-01" xlink:href="fig-0284-01a"> <image file="0284-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0284-01"/> </figure> <note symbol="e" position="left" xlink:label="note-0284-05" xlink:href="note-0284-05a" xml:space="preserve">ex 13. <lb/>Archim. <lb/>ibidem.</note> <note symbol="f" position="left" xlink:label="note-0284-06" xlink:href="note-0284-06a" xml:space="preserve">ex Co-<lb/>roll. 15. ib.</note> <note symbol="g" position="left" xlink:label="note-0284-07" xlink:href="note-0284-07a" xml:space="preserve">3. vnd. <lb/>Elem.</note> <note symbol="h" position="left" xlink:label="note-0284-08" xlink:href="note-0284-08a" xml:space="preserve">16. ib.</note> </div> <p> <s xml:id="echoid-s7947" xml:space="preserve">Iam cum ſit G B ad B I, vt <lb/>H E ad E L, ob conſtructio-<lb/>nem, ſitque in prima figura A <lb/>C ad M N, vt G B ad B I; </s> <s xml:id="echoid-s7948" xml:space="preserve">& </s> <s xml:id="echoid-s7949" xml:space="preserve"><lb/>D F ad O P, vt H E ad E L (cum Canones A B C, D E F ſint triangula) <lb/>erit A C ad M N vt D F ad O P, & </s> <s xml:id="echoid-s7950" xml:space="preserve">quadratum A C ad M N, vt <lb/>quadratum D F ad O P. </s> <s xml:id="echoid-s7951" xml:space="preserve">In ſecunda verò eſt quadratum A C ad M N, vt <lb/>re cta G B ad B I (cum Canon A B C ſit portio Parabolæ) vel vt recta H <lb/> <anchor type="note" xlink:label="note-0284-09a" xlink:href="note-0284-09"/> E ad E L, per conſtructionem, vel vt quadratum D E ad O P: </s> <s xml:id="echoid-s7952" xml:space="preserve">eſt ergo in <lb/>vtraque ſigura, vt quadratum A C ad M N, vel vt circulus, <anchor type="note" xlink:href="" symbol="i"/> aut Ellipſis <pb o="99" file="0285" n="285" rhead=""/> A C ad ſibi ſimilem M N, ita quadratum D F ad O P, vel ita circulus, aut <lb/>Ellipſis D F ad ſibi ſimilem O P, & </s> <s xml:id="echoid-s7953" xml:space="preserve">permutando, ſectio A C ad D F erit <lb/>vt ſectio M N ad O P, & </s> <s xml:id="echoid-s7954" xml:space="preserve">hoc ſemper vbicunque ſolidorum Acuminatorum <lb/>axes ſint proportionaliter ſecti: </s> <s xml:id="echoid-s7955" xml:space="preserve">quare, ex tertia præmiſſarum definitionum, <lb/>Acuminata ſolida A B C, D E F erunt ſolida Acuminata proportionalia. <lb/></s> <s xml:id="echoid-s7956" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s7957" xml:space="preserve">c.</s> <s xml:id="echoid-s7958" xml:space="preserve"/> </p> <div xml:id="echoid-div825" type="float" level="2" n="3"> <note symbol="i" position="left" xlink:label="note-0284-09" xlink:href="note-0284-09a" xml:space="preserve">Coroll. <lb/>7. Arch. <lb/>ibid.</note> </div> <p> <s xml:id="echoid-s7959" xml:space="preserve">PRæterea ſint A B C, D E F <lb/> <anchor type="figure" xlink:label="fig-0285-01a" xlink:href="fig-0285-01"/> duæ portiones eiuſdem, vel <lb/>diuerſorum Conoidum Hyper-<lb/>bolicorum, vt in tertia figura, vel <lb/>eiuſdem, aut diuerſorum Sphæ-<lb/>roidum, vel Sphærarum, vt in <lb/>quarta, (quæ portiones ſunt pa-<lb/>riter ſolida Acuminata per 1. <lb/></s> <s xml:id="echoid-s7960" xml:space="preserve">Schol. </s> <s xml:id="echoid-s7961" xml:space="preserve">69. </s> <s xml:id="echoid-s7962" xml:space="preserve">h.) </s> <s xml:id="echoid-s7963" xml:space="preserve">quarum baſes ſint <lb/>circuli, aut Ellipſes A C, D F. </s> <s xml:id="echoid-s7964" xml:space="preserve"><lb/>Patet quod ſi per axes ſolidorũ, <lb/>quorum ſunt portiones ducantur <lb/>plana, <anchor type="note" xlink:href="" symbol="a"/> quæ portionum baſibus ſint erecta, fient in ſolidis portio-<lb/> <anchor type="note" xlink:label="note-0285-01a" xlink:href="note-0285-01"/> nes genitricium <anchor type="note" xlink:href="" symbol="b"/> ſectionum A B C, D E F, hoc eſt in tertia por-<lb/> <anchor type="note" xlink:label="note-0285-02a" xlink:href="note-0285-02"/> nes Hyperbolarum, & </s> <s xml:id="echoid-s7965" xml:space="preserve">in quarta portiones Ellipſium, quas vocamus <anchor type="note" xlink:href="" symbol="c"/> Ca- nones, & </s> <s xml:id="echoid-s7966" xml:space="preserve">communes horum Canonum ſectiones cum baſibus erunt <anchor type="note" xlink:href="" symbol="d"/> rectæ <anchor type="note" xlink:label="note-0285-03a" xlink:href="note-0285-03"/> A C, D F, quæ ipſarum baſium erunt <anchor type="note" xlink:href="" symbol="e"/> axes. </s> <s xml:id="echoid-s7967" xml:space="preserve">Sint iam Canonum A B C, <anchor type="note" xlink:label="note-0285-04a" xlink:href="note-0285-04"/> D E F intercepta diametrorum ſegmenta B G, E H, (quæ & </s> <s xml:id="echoid-s7968" xml:space="preserve">ſolidarum <lb/>portionum axes vocantur ab Archimede) quibus productis vſque ad earum <lb/> <anchor type="note" xlink:label="note-0285-05a" xlink:href="note-0285-05"/> centra Q, R, habeat ſegmentum G B ad ſemi-diametrum B Q, eandem <lb/>rationem, ac ſegmentum H E ad ſemi - diametrum E R. </s> <s xml:id="echoid-s7969" xml:space="preserve">Dico in vtraque <lb/>harum figurarum, portiones ſolidas, vel ſolida Acuminata A B C, D E F <lb/>eſſe Acuminata ſolida proportionalia.</s> <s xml:id="echoid-s7970" xml:space="preserve"/> </p> <div xml:id="echoid-div826" type="float" level="2" n="4"> <figure xlink:label="fig-0285-01" xlink:href="fig-0285-01a"> <image file="0285-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0285-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0285-01" xlink:href="note-0285-01a" xml:space="preserve">69. h.</note> <note symbol="b" position="right" xlink:label="note-0285-02" xlink:href="note-0285-02a" xml:space="preserve">ex 12. <lb/>Arch. de <lb/>Conoid.</note> <note symbol="c" position="right" xlink:label="note-0285-03" xlink:href="note-0285-03a" xml:space="preserve">1. Schol. <lb/>69. h.</note> <note symbol="d" position="right" xlink:label="note-0285-04" xlink:href="note-0285-04a" xml:space="preserve">3. vnd. <lb/>Elem.</note> <note symbol="e" position="right" xlink:label="note-0285-05" xlink:href="note-0285-05a" xml:space="preserve">ex 14. <lb/>& 15. Ar-<lb/>chim. ib.</note> </div> <p> <s xml:id="echoid-s7971" xml:space="preserve">Diuiſis enim ipſorum axibus B G, E H proportionaliter vtcunque in I, <lb/>L, ductiſque per I, L planis M N, O P ipſis baſibus A C, D F æquidi-<lb/>ſtantibus, erit ſectio M N in ſolido A B C ſimilis <anchor type="note" xlink:href="" symbol="f"/> baſi A C, & </s> <s xml:id="echoid-s7972" xml:space="preserve">ſectio O P <anchor type="note" xlink:label="note-0285-06a" xlink:href="note-0285-06"/> in-ſolido D E F ſimilis baſi D F, & </s> <s xml:id="echoid-s7973" xml:space="preserve">earum communes ſectiones cum planis <lb/>Acuminatis A B C, D E F erunt rectæ M N, O P ipſis A C, D F paralle-<lb/>læ <anchor type="note" xlink:href="" symbol="g"/> vtraque vtrique, eruntque homologæ diametri earundem ſimilium ſe- <anchor type="note" xlink:label="note-0285-07a" xlink:href="note-0285-07"/> ctionum.</s> <s xml:id="echoid-s7974" xml:space="preserve"/> </p> <div xml:id="echoid-div827" type="float" level="2" n="5"> <note symbol="f" position="right" xlink:label="note-0285-06" xlink:href="note-0285-06a" xml:space="preserve">ex Co-<lb/>roll. 15. <lb/>eiuſdem.</note> <note symbol="g" position="right" xlink:label="note-0285-07" xlink:href="note-0285-07a" xml:space="preserve">3. & 16. <lb/>vnd. El.</note> </div> <p> <s xml:id="echoid-s7975" xml:space="preserve">Et quoniam, per conſtructionem, in Acuminatis planis A B C, D E F, <lb/>Hyperbolarum, vt in tertia figura, aut Ellipſium, vt in quarta, ſeginenta <lb/>diametrorum G B, E H ad proprias ſemi-diametros B Q, E R eandem ha-<lb/>bent rationem, erunt <anchor type="note" xlink:href="" symbol="h"/> ipſa Acuminata, plana Acuminata proportionalia;</s> <s xml:id="echoid-s7976" xml:space="preserve"> <anchor type="note" xlink:label="note-0285-08a" xlink:href="note-0285-08"/> ſuntque B G, E H proportionaliter ſectæ in I, L, ex conſtructione, quare <lb/>vt recta A C ad D F, ita recta M N ad O P (ex definitione planorum Acu-<lb/>minatorum proportionalium) & </s> <s xml:id="echoid-s7977" xml:space="preserve">quadratum A C ad D F, hoc eſt circulus, <lb/> <anchor type="note" xlink:label="note-0285-09a" xlink:href="note-0285-09"/> vel <anchor type="note" xlink:href="" symbol="i"/> Ellipſis A C ad ſibi ſimilem D F, vt quadratum M N ad O P, vel vt circulus, aut Ellipſis M N ad ſibi ſimilem O P, & </s> <s xml:id="echoid-s7978" xml:space="preserve">hoc ſemper vbicunque <pb o="100" file="0286" n="286" rhead=""/> axes B G, E H ſolidarum portionum ſint proportionaliter ſecti: </s> <s xml:id="echoid-s7979" xml:space="preserve">quare, ex <lb/>definitione, ipſæ ſolidæ portiones ABC, DEF erunt ſolida Acuminata <lb/>proportionalia. </s> <s xml:id="echoid-s7980" xml:space="preserve">Quod vltimò demonſtrandum erat.</s> <s xml:id="echoid-s7981" xml:space="preserve"/> </p> <div xml:id="echoid-div828" type="float" level="2" n="6"> <note symbol="h" position="right" xlink:label="note-0285-08" xlink:href="note-0285-08a" xml:space="preserve">36. h.</note> <note symbol="i" position="right" xlink:label="note-0285-09" xlink:href="note-0285-09a" xml:space="preserve">ex co-<lb/>roll. ſept. <lb/>Arch. de <lb/>Conoid.</note> </div> </div> <div xml:id="echoid-div830" type="section" level="1" n="327"> <head xml:id="echoid-head336" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s7982" xml:space="preserve">HInc manifeſtum fit ſolidas portiones eiuſdem Conirecti, vel Conoidis <lb/>Parabolici, aut Hyperbolici, ſiue Sphæræ, aut Sphæroidis oblongi, <lb/>vel prolati, quarum recti Canones ſint æquales, inter ſe eſſe Acuminata ſo-<lb/>lida proportionalia.</s> <s xml:id="echoid-s7983" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7984" xml:space="preserve">Nam quò ad portiones eiuſdem Conirecti, vel Conoidis Parabolici, iam <lb/>in prima parte huius propoſitionis oſtenſum eſt eas omnes, quæcunque ſint, <lb/>eſſe ſolida Acuminata proportionalia, ac ideò, & </s> <s xml:id="echoid-s7985" xml:space="preserve">illæ quarum recti Ca-<lb/>nones ſint æquales, erunt pariter ſolida Acuminata proportionalia.</s> <s xml:id="echoid-s7986" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7987" xml:space="preserve">Quò autem ad ſolidas portiones eiuſdem Conoidis Hyperbolici, ſiue <lb/>Sphæræ, aut Sphæroidis oblongi, vel prolati; </s> <s xml:id="echoid-s7988" xml:space="preserve">quandò earum portiones ge-<lb/>nitricium ſectionum ad plana baſium rectæ(quæ eædem ſunt, ac recti Cano-<lb/>nes) fuerint æquales: </s> <s xml:id="echoid-s7989" xml:space="preserve">patet ex prop. </s> <s xml:id="echoid-s7990" xml:space="preserve">63. </s> <s xml:id="echoid-s7991" xml:space="preserve">huius, ſegmenta diametrorum ipſarum <lb/>ad proprias ſemi - diametros, vnam, eandemque ſimul rationem habere, ac <lb/>propterea ex ijs, quæ in hac vitimò loco demonſtrauimus, huiuſmodi ſolidæ <lb/>portiones erunt Acuminata ſolida proportionalia.</s> <s xml:id="echoid-s7992" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div831" type="section" level="1" n="328"> <head xml:id="echoid-head337" xml:space="preserve">THEOR. XLIV. PROP. LXXI.</head> <p> <s xml:id="echoid-s7993" xml:space="preserve">Cylindrici æqualium altitudinum, inter ſe ſunt vt baſes.</s> <s xml:id="echoid-s7994" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7995" xml:space="preserve">SInt duo Cylindrici A, B, quorum baſes ſint plana Acuminata C D E, <lb/>F G H, altitudines verò, ſint æquales cuidam rectæ I. </s> <s xml:id="echoid-s7996" xml:space="preserve">Dico Cylindri-<lb/>cum A ad Cylindricum B, eſſe vt baſis C D E ad baſim F G H.</s> <s xml:id="echoid-s7997" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7998" xml:space="preserve">Concipiatur alius quicunque Cylindricus L, cuius baſis ſit parallelo-<lb/>grammum K T, altitudo verò ſit eadem I: </s> <s xml:id="echoid-s7999" xml:space="preserve">quod erit parallepipedum. <lb/></s> <s xml:id="echoid-s8000" xml:space="preserve">Oſtendam priùs Cylindricum A ad parallelepipedum, vel Cylindricum <lb/>L eſſe vt baſis C D E ad baſim K T.</s> <s xml:id="echoid-s8001" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8002" xml:space="preserve">Nam ſi non eſt ita, erit baſis C D E, vel maior, vel minor quàm ſit opus, <lb/>ad hoc vt ad baſim K T habeat eandem rationem, ac Cylindricus A ad L. <lb/></s> <s xml:id="echoid-s8003" xml:space="preserve">Eſto primùm maior, ſitque exceſſus O. </s> <s xml:id="echoid-s8004" xml:space="preserve">Et cum Acuminatum C D E ſit ſi-<lb/>gura circa diametrum D M ad partem D deſiciens, & </s> <s xml:id="echoid-s8005" xml:space="preserve">cuius perimeter eſt <lb/>ad eandem partem cauus, poterit, vſitata methodo, per continuam diame-<lb/>tri D M biſectionem, inſcribi Acuminato C D E figura ex parallelogram-<lb/>mis, ita vt ipſum Acuminatum ſuperet inſcriptam minori exceſſu, quàm ſit <lb/>O; </s> <s xml:id="echoid-s8006" xml:space="preserve">ſit ergo hæc inſcripta P Q, R S, &</s> <s xml:id="echoid-s8007" xml:space="preserve">c. </s> <s xml:id="echoid-s8008" xml:space="preserve">Itaque cum Acuminatum C D E <lb/>ſuperet inſcriptam P Q, R S, &</s> <s xml:id="echoid-s8009" xml:space="preserve">c. </s> <s xml:id="echoid-s8010" xml:space="preserve">minori quantitate O, erit inſcripta adhuc <lb/>maior, quam opus eſt ad hoc, vt ad baſim K T ſit vt Cylindricus A ad Cy-<lb/>lindricum L.</s> <s xml:id="echoid-s8011" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8012" xml:space="preserve">Iam intra Cylindricum A ſuper omnia inſcriptæ figuræ parallelogram- <pb o="101" file="0287" n="287" rhead=""/> ma P Q, R S, &</s> <s xml:id="echoid-s8013" xml:space="preserve">c, concipiantur deſcripta ſolida parallelepipeda æqualium <lb/>altitudinum cum Cylindrico A, vel L; </s> <s xml:id="echoid-s8014" xml:space="preserve">& </s> <s xml:id="echoid-s8015" xml:space="preserve">quorum inſiſtentes lineæ ſint <lb/>æquidiſtantes inſiſtentibus Cylindrici A, &</s> <s xml:id="echoid-s8016" xml:space="preserve">c. </s> <s xml:id="echoid-s8017" xml:space="preserve">erit ergo vnumquodque pa-<lb/>rallelepipedorum inſcriptorum, ad parallelepipedum L, <anchor type="note" xlink:href="" symbol="a"/> vt propria baſis ad <anchor type="note" xlink:label="note-0287-01a" xlink:href="note-0287-01"/> baſim, ac ideò omnia ſimul inſcripta ſuper P Q, R S, &</s> <s xml:id="echoid-s8018" xml:space="preserve">c. </s> <s xml:id="echoid-s8019" xml:space="preserve">ad vnicum paralle-<lb/>lepipedum, vel Cylindricum L, erunt vt omnes baſes P Q, R S, &</s> <s xml:id="echoid-s8020" xml:space="preserve">c. </s> <s xml:id="echoid-s8021" xml:space="preserve">hoc eſt <lb/>vt figura inſcripta ad baſim R T; </s> <s xml:id="echoid-s8022" xml:space="preserve">ſed inſcripta ad K T maiorem habet ratio-<lb/>nem quàm Cylindricus A ad L, ergo, & </s> <s xml:id="echoid-s8023" xml:space="preserve">omnia ſimul parallelepipeda in-<lb/>ſcripta, ad Cylindricum L maiorem habebunt rationem, quàm Cylindri-<lb/>cus A circumſcriptus ad eundem Cylindricum L, ergo inſcripta ſimul pa-<lb/>rallelepipeda maiora erunt Cylindrico A, pars ſuo toto, quod eſt abſurdũ: <lb/></s> <s xml:id="echoid-s8024" xml:space="preserve">non eſt ergo baſis C D E maior quàm opus eſt ad hoc vt ad baſim K T ſit vt <lb/>Cylindricus A ad L.</s> <s xml:id="echoid-s8025" xml:space="preserve"/> </p> <div xml:id="echoid-div831" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0287-01" xlink:href="note-0287-01a" xml:space="preserve">32. vnd. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s8026" xml:space="preserve">Si verò ponatur baſim <lb/> <anchor type="figure" xlink:label="fig-0287-01a" xlink:href="fig-0287-01"/> C D E ad K T hab ere mi-<lb/>norem rationem quàm <lb/>Cylindricus A ad L, erit <lb/>baſis C D E minor quàm <lb/>opus eſt ad hoc vt huiuſ-<lb/>modi magnitudines ſint <lb/>proportionales, inuento <lb/>igitur defectu, &</s> <s xml:id="echoid-s8027" xml:space="preserve">c. </s> <s xml:id="echoid-s8028" xml:space="preserve">& </s> <s xml:id="echoid-s8029" xml:space="preserve">facta <lb/>baſi C D E circumſcri-<lb/>ptione figuræ ex paralle-<lb/>logrammis, &</s> <s xml:id="echoid-s8030" xml:space="preserve">c. </s> <s xml:id="echoid-s8031" xml:space="preserve">quæ ad <lb/>baſim K T adhuc minorẽ <lb/>habeat rationem quàm <lb/>Cylindricus A ad L, & </s> <s xml:id="echoid-s8032" xml:space="preserve"><lb/>circumſcriptis parallele-<lb/>pipedis vt ſupra, oſtendetur aggregatum circumſcriptorum parallelepipe-<lb/>dorum ad Cylindricum L eſſe vt figura circumſcripta ab baſim K T, hoc eſt <lb/>habere minorem rationem quàm Cylindricus A ad eundem Cylindricum <lb/>L, ideoque prædictum aggregatum parallelepipedorum minùs eſſe Cylin-<lb/>drico A, totum ſua parte, quod eſt abſurdum. </s> <s xml:id="echoid-s8033" xml:space="preserve">Non ergo baſis C ad K T <lb/>habet maiorem, nec minorem rationem quàm Cylindricus A ad L, ergo <lb/>erit baſis C D E ad baſim K T, vt Cylindricus A ad L. </s> <s xml:id="echoid-s8034" xml:space="preserve">Eadem ratione <lb/>demonſtrabitur, baſim K T ad Acuminatum F G H, ſiue ad baſim Cylin-<lb/>drici B, eſſe vt Cylindricus L ad Cylindricum B; </s> <s xml:id="echoid-s8035" xml:space="preserve">quare, ex æquo, erit vt <lb/>baſis C D E ad baſim F G H, ita Cylindricus A ad Cylindricum B. </s> <s xml:id="echoid-s8036" xml:space="preserve">Quod <lb/>erat, &</s> <s xml:id="echoid-s8037" xml:space="preserve">c.</s> <s xml:id="echoid-s8038" xml:space="preserve"/> </p> <div xml:id="echoid-div832" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0287-01"/> </figure> </div> </div> <div xml:id="echoid-div834" type="section" level="1" n="329"> <head xml:id="echoid-head338" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s8039" xml:space="preserve">PErſpicuum hinc eſt, quod ſi huiuſmodi Cylindrici æqualiũ altitudinum <lb/>æquales baſes habuerint inter ſe æquales erunt.</s> <s xml:id="echoid-s8040" xml:space="preserve"/> </p> <pb o="102" file="0288" n="288" rhead=""/> </div> <div xml:id="echoid-div835" type="section" level="1" n="330"> <head xml:id="echoid-head339" xml:space="preserve">THEOR. XLV. PROP. LXXII.</head> <p> <s xml:id="echoid-s8041" xml:space="preserve">Si Cylindricus plano ſecetur baſi æquidiſtante, erit Cylin-<lb/>dricus ad Cylindricum, vt altitudo ad altitudinem.</s> <s xml:id="echoid-s8042" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8043" xml:space="preserve">HOc eadem penitus conſtructione, ijſdemque argumentis demonſtrabi-<lb/>tur, ac 13. </s> <s xml:id="echoid-s8044" xml:space="preserve">duodecimi Elem. </s> <s xml:id="echoid-s8045" xml:space="preserve">opetamen præcedentis Corollarij; </s> <s xml:id="echoid-s8046" xml:space="preserve">ani-<lb/>maduertendo ſimul, quod dum Cylindricus plano ſecatur baſi æquidiſtante, <lb/>in ipſa ſectione oritur figura ſimilis, & </s> <s xml:id="echoid-s8047" xml:space="preserve">æqualis, ſiue in totum congruens baſi <lb/>Cylindrici: </s> <s xml:id="echoid-s8048" xml:space="preserve">nam ipſæ Cylindricus, ex motu parallelo ſuæ baſis procreari <lb/>concipitur, ex definitione, &</s> <s xml:id="echoid-s8049" xml:space="preserve">c.</s> <s xml:id="echoid-s8050" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div836" type="section" level="1" n="331"> <head xml:id="echoid-head340" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s8051" xml:space="preserve">EX hac pendet huius concluſionis demonſtratio, quod eſt conuerſum <lb/>prop. </s> <s xml:id="echoid-s8052" xml:space="preserve">71. </s> <s xml:id="echoid-s8053" xml:space="preserve">huius; </s> <s xml:id="echoid-s8054" xml:space="preserve">nempe.</s> <s xml:id="echoid-s8055" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8056" xml:space="preserve">Cylindrici æqualium baſium ſunt inter ſe, vt altitudines; </s> <s xml:id="echoid-s8057" xml:space="preserve">quod oſtenditur <lb/>vt in 14. </s> <s xml:id="echoid-s8058" xml:space="preserve">duodecimi Elementorum.</s> <s xml:id="echoid-s8059" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div837" type="section" level="1" n="332"> <head xml:id="echoid-head341" xml:space="preserve">THEOR. XLVI. PROP. LXXIII.</head> <p> <s xml:id="echoid-s8060" xml:space="preserve">Cylindrici, quorum baſes altitudinibus reciprocantur, inter ſe <lb/>ſunt æquales: </s> <s xml:id="echoid-s8061" xml:space="preserve">& </s> <s xml:id="echoid-s8062" xml:space="preserve">è conuerſo.</s> <s xml:id="echoid-s8063" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8064" xml:space="preserve">HVius Theorematis demonſtratio elicitur ex præcèdenti, eſtque omni-<lb/>nò ſimilis 15. </s> <s xml:id="echoid-s8065" xml:space="preserve">duodec. </s> <s xml:id="echoid-s8066" xml:space="preserve">Element. </s> <s xml:id="echoid-s8067" xml:space="preserve">itaque breuitatis gratia, hanc ipſam <lb/>o mittimus, ſimulque nonnullas alias Cylindricorum affectiones, cum hìc <lb/>de ijs diſſerere non ſit opus.</s> <s xml:id="echoid-s8068" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div838" type="section" level="1" n="333"> <head xml:id="echoid-head342" xml:space="preserve">THEOR. XLVII. PROP. LXXIV.</head> <p> <s xml:id="echoid-s8069" xml:space="preserve">Solida Acuminata proportionalia, quorum baſes altitudinibus <lb/>ſint reciprocè proportionales inter ſe ſunt æqualia.</s> <s xml:id="echoid-s8070" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8071" xml:space="preserve">SInt duo Acuminata ſolida proportionalia, quorum Canones A B C, D <lb/>E F ſint ſuper baſes A C, D F, & </s> <s xml:id="echoid-s8072" xml:space="preserve">circa diametros B G, E H; </s> <s xml:id="echoid-s8073" xml:space="preserve">baſes <lb/>verò horum ſolidorum ſint Acuminata plana A L C, N F O circa diametros <lb/>A C, D F, ſitque vnius ſolidi altitudo B I, ad alterius altitudinem E Q re-<lb/>ciprocè, vt baſis N F O ad baſim A L C. </s> <s xml:id="echoid-s8074" xml:space="preserve">Dico huiuſmodi ſolida inter ſe <lb/>æqualia eſſe.</s> <s xml:id="echoid-s8075" xml:space="preserve"/> </p> <pb o="103" file="0289" n="289" rhead=""/> <p> <s xml:id="echoid-s8076" xml:space="preserve">Si enim fuerint inæqualia, alterum ipſorum minus erit: </s> <s xml:id="echoid-s8077" xml:space="preserve">ſit ipſum A B C, <lb/>quod cum ſit ad partem B deficiens, patet ei circumſcribi poſſe per conti-<lb/>nuam axis B G biſectionem, &</s> <s xml:id="echoid-s8078" xml:space="preserve">c. </s> <s xml:id="echoid-s8079" xml:space="preserve">figuram ex Cylindris æque-altis, quæ <lb/>inſcriptum ſolidum Acuminatum A B C ſuperet minori exceſſu, quo ſoli-<lb/>dum D E F dicitur excedere idem ſolidum Acuminatum A B C: </s> <s xml:id="echoid-s8080" xml:space="preserve">(ſuſſi-<lb/>cit enim vt circumſcripto Canoni A B C parallelogrammo A R, eius <lb/>ope, tan quam circa diametralem Canonem, ad æquidiſtantem motum <lb/>baſis A L C deſcribatur Cylindricus A R, vt vides, circumſeribens Acu-<lb/>minatum ſolidum A B C, ſic enim plano per punctum medium axis B G <lb/>applicato, bifariam <anchor type="note" xlink:href="" symbol="a"/> ſecabitur Cylindricus, quod ſi iterum axis dimidium <anchor type="note" xlink:label="note-0289-01a" xlink:href="note-0289-01"/> biſariam ſecetur, &</s> <s xml:id="echoid-s8081" xml:space="preserve">c. </s> <s xml:id="echoid-s8082" xml:space="preserve">relinquetur tandem Cylindricus A S, qui prædicto <lb/>exceſſu minor erit: </s> <s xml:id="echoid-s8083" xml:space="preserve">vnde hac vltima diametri B G diuiſione per æqualia <lb/>ſegmenta, completa circumſcriptione Cylindricorum T M, X V, Z Y <lb/>æqualium altitudinum, quorum diametrales Canones ſint A S; </s> <s xml:id="echoid-s8084" xml:space="preserve">T M; </s> <s xml:id="echoid-s8085" xml:space="preserve">X <lb/>V; </s> <s xml:id="echoid-s8086" xml:space="preserve">X Y; </s> <s xml:id="echoid-s8087" xml:space="preserve">aggregatum ipſorum excedet ſolidum A B C minori quantitate, <lb/>quàm ſit primus Cylindricus A S, cum A S ſit ſemper exceſſus circum-<lb/>ſcriptæ figuræ ex Cylindricis ſuper inſcriptam ex æque - altis Cylindri-<lb/>cis, &</s> <s xml:id="echoid-s8088" xml:space="preserve">c. </s> <s xml:id="echoid-s8089" xml:space="preserve">ſed Cylindricus A S ponitur minor exceſſu ſolidi D E F ſuper A B <lb/>C, ergo circumſcripta figura A S M Y ex Cylindricis, ſuperat inſcriptum <lb/>ſolidum A B C minori exceſſu ipſius ſolidi D E F ſupra A B C) ſit ergo quę-<lb/>ſita figura circumſcripta, ex Cylindricis A S, T M, X V, Z Y, &</s> <s xml:id="echoid-s8090" xml:space="preserve">c. </s> <s xml:id="echoid-s8091" xml:space="preserve">quæ <lb/>ideò adhuc minor erit ſolido D E F, cuieadem arte circumſcribatur figura <lb/>ex totidem æque - altis Cylindricis D K; </s> <s xml:id="echoid-s8092" xml:space="preserve">2 3; </s> <s xml:id="echoid-s8093" xml:space="preserve">4 5; </s> <s xml:id="echoid-s8094" xml:space="preserve">6 7; </s> <s xml:id="echoid-s8095" xml:space="preserve">quorum maximi, <lb/>diametralis Canon ſit D K ſuper baſim O F N; </s> <s xml:id="echoid-s8096" xml:space="preserve">proximi verò diametralis <lb/>Canon ſit 2 3, &</s> <s xml:id="echoid-s8097" xml:space="preserve">c.</s> <s xml:id="echoid-s8098" xml:space="preserve"/> </p> <div xml:id="echoid-div838" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0289-01" xlink:href="note-0289-01a" xml:space="preserve">72. h.</note> </div> <p> <s xml:id="echoid-s8099" xml:space="preserve">Iam patet horum ſoli-<lb/> <anchor type="figure" xlink:label="fig-0289-01a" xlink:href="fig-0289-01"/> dorum altitudines BI, E<unsure/> <lb/>Q in tot æquas partes ſe-<lb/>cari à parallelis baſibus <lb/>circumſcriptorum Cylin-<lb/>dricorum, in quot <anchor type="note" xlink:href="" symbol="b"/> ſe- <anchor type="note" xlink:label="note-0289-02a" xlink:href="note-0289-02"/> cãtur diametri B G, E H, <lb/>Canonum A B C, D E F. <lb/></s> <s xml:id="echoid-s8100" xml:space="preserve">Sit igitur primi Cylindri-<lb/>ci A S altitudo 8 I, & </s> <s xml:id="echoid-s8101" xml:space="preserve"><lb/>primi D K altitudo 9 Q: </s> <s xml:id="echoid-s8102" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s8103" xml:space="preserve">cum ſit baſis A L C, ad <lb/>baſim O F N, ita recipro-<lb/>cè altitudo E Q ad altitudinem B I, ſumptis conſequentium æque-ſubmul-<lb/>tiplicibus 9 Q, 8 I; </s> <s xml:id="echoid-s8104" xml:space="preserve">erit baſis A L C, ad O F N, vt altitudo 9 Q, ad 8 I; </s> <s xml:id="echoid-s8105" xml:space="preserve"><lb/>quare Cylindricus A S æqualis erit <anchor type="note" xlink:href="" symbol="c"/> Cylindrico D K. </s> <s xml:id="echoid-s8106" xml:space="preserve">Eadem ratione de- <anchor type="note" xlink:label="note-0289-03a" xlink:href="note-0289-03"/> monſtrabuntur reliqu@ Cylindric@ T M, X V, Z Y, reliquis 23, 45, 67, <lb/>æqual@a eſſe, ſingul@ ſingulis, quapropter vniuerſa figura ex Cylindricis, <lb/>circumſcripta ſolido A B C, æqualis erit vniuerſæ circumſcriptæ ſolido D <lb/>E F, ſed circumſcripta ipſi A B C demonſtrata eſt minor ſolido D E F, er-<lb/>go, & </s> <s xml:id="echoid-s8107" xml:space="preserve">circumſcripta ſolido D E F, ipſo ſibi inſcripto ſolido minor erit, to-<lb/>tum ſua parte, quod eſt abſurdum. </s> <s xml:id="echoid-s8108" xml:space="preserve">Non eſt ergo vllum horum Acuminato- <pb o="104" file="0290" n="290" rhead=""/> rum altero maius: </s> <s xml:id="echoid-s8109" xml:space="preserve">quare omnino inter ſe ſunt æqualia. </s> <s xml:id="echoid-s8110" xml:space="preserve">Quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s8111" xml:space="preserve"/> </p> <div xml:id="echoid-div839" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0289-01"/> </figure> <note symbol="b" position="right" xlink:label="note-0289-02" xlink:href="note-0289-02a" xml:space="preserve">17. vnd. <lb/>Elem.</note> <note symbol="c" position="right" xlink:label="note-0289-03" xlink:href="note-0289-03a" xml:space="preserve">73. h.</note> </div> </div> <div xml:id="echoid-div841" type="section" level="1" n="334"> <head xml:id="echoid-head343" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s8112" xml:space="preserve">NOn paucas alias ſolidorum Acuminatorum, eorumque trunco-<lb/>rum proprietates (quales nimirum attigimus de planis, & </s> <s xml:id="echoid-s8113" xml:space="preserve"><lb/>menſalibus Acuminatis in Scholio propoſ. </s> <s xml:id="echoid-s8114" xml:space="preserve">37. </s> <s xml:id="echoid-s8115" xml:space="preserve">huius) facilè <lb/>huc eſſet, ſi locus requireret, ex ſuperioribus afferre: </s> <s xml:id="echoid-s8116" xml:space="preserve">ve-<lb/>rùm ad opportuniorem occaſionem hæc omnia, aliaque fuſius forſan per-<lb/>tractabimus, ſi Deo nobis valetudinem cum vita, vel ſaltem mitio-<lb/>rem ægritudinem præſtare placuerit. </s> <s xml:id="echoid-s8117" xml:space="preserve">Modò ad inuentionem MAXI-<lb/>MARVM, MINIMARVMQVE ſolidarum portionum acce-<lb/>damus, nonnullis adhuc præoſtenſis.</s> <s xml:id="echoid-s8118" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div842" type="section" level="1" n="335"> <head xml:id="echoid-head344" xml:space="preserve">LEMMA XIV. PROP. LXXV.</head> <p> <s xml:id="echoid-s8119" xml:space="preserve">Datæ portioni anguli rectilinei, circa diuerſam diametrum <lb/>datam, æqualem portionem conſtituere.</s> <s xml:id="echoid-s8120" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8121" xml:space="preserve">ESto ex angulo rectilineo abſciſſa portio A B C, cuius baſis A C, dia-<lb/>meter verò B D; </s> <s xml:id="echoid-s8122" xml:space="preserve">ſitque data alia diameter B E, circa quam oporteat <lb/>portionem ipſi A B C æqualem conſtituere.</s> <s xml:id="echoid-s8123" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8124" xml:space="preserve">Latus A B ſecetur bifariam in F, & </s> <s xml:id="echoid-s8125" xml:space="preserve">per F agatur F G parallela ad B C <lb/>cum B E occurrens in G, per G verò ducatur A G H ipſam B C ſecans <lb/>in H, atque inter C B, B H ſumatur media proportionalis B I agaturque <lb/>per I recta IL baſim A C ſecans in M, & </s> <s xml:id="echoid-s8126" xml:space="preserve">datam diametrum B E in N, & </s> <s xml:id="echoid-s8127" xml:space="preserve"><lb/>B A productam, in L. </s> <s xml:id="echoid-s8128" xml:space="preserve">Dico ipſam I L abſcindere L B I portionem quæ-<lb/>ſitam.</s> <s xml:id="echoid-s8129" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8130" xml:space="preserve">Triangulum enim A B C ad triangulum A B H eſt vt baſis C B ad B H, <lb/>vel vt quadratum mediæ proportionalis I B ad quadratum tertiæ B H, vel <lb/>vt triangulum L B I ad idem triangulum A B H. </s> <s xml:id="echoid-s8131" xml:space="preserve">(ob ſimilitudinem) quare <lb/>triangula A B C, L B I ſunt æqualia.</s> <s xml:id="echoid-s8132" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8133" xml:space="preserve">Et cum rectæ L I, A H ſimul æquidiſtent, ſecenturque ab eadem B N ex <lb/>vertice B trianguli L B I ducta, erit L N ad N I, vt A G ad G H, vel vt <lb/>A F ad F B (ob parallelas F G, B H) ſed eſt A F ipſi F B æqualis (per con-<lb/>ſtructionem) ergo, & </s> <s xml:id="echoid-s8134" xml:space="preserve">L N ipſi N I æqualis erit. </s> <s xml:id="echoid-s8135" xml:space="preserve">Itaque ad datam diame-<lb/>trum B N, in angulo A B C ordinatim applicata eſt L I abſcindens trian-<lb/>gulum, vel portionem L B I alteri datæ portioni A B C æqualem. </s> <s xml:id="echoid-s8136" xml:space="preserve">Quod <lb/>faciendum erat.</s> <s xml:id="echoid-s8137" xml:space="preserve"/> </p> <pb o="105" file="0291" n="291" rhead=""/> </div> <div xml:id="echoid-div843" type="section" level="1" n="336"> <head xml:id="echoid-head345" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s8138" xml:space="preserve">HIs peractis, patet baſes A C, I L æqualium portionum de eodem an-<lb/>gulo A B C neceſſariò ſe mutuò ſecare intra angulum. </s> <s xml:id="echoid-s8139" xml:space="preserve">Nam I M, <lb/>quæ ex puncto I inter H, & </s> <s xml:id="echoid-s8140" xml:space="preserve">C ſumpto æquidiſtans ducitur rectæ A H <lb/>neceſſariò occurrit cum A C, vt in M.</s> <s xml:id="echoid-s8141" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8142" xml:space="preserve">Dico ampliùs earum baſium occurſum M cadere omninò inter diame-<lb/>tros B D, B E; </s> <s xml:id="echoid-s8143" xml:space="preserve">hoc eſt inter puncta E, D; </s> <s xml:id="echoid-s8144" xml:space="preserve">atque rectas N D, A I, L C <lb/>harũ baſim tùm puncta media, tùm extrema iungẽtes eſſe inter ſe parallelas.</s> <s xml:id="echoid-s8145" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8146" xml:space="preserve">Si enim per E agatur O <lb/> <anchor type="figure" xlink:label="fig-0291-01a" xlink:href="fig-0291-01"/> E P ipſis A H, L I æquidi-<lb/>ſtans, & </s> <s xml:id="echoid-s8147" xml:space="preserve">per P recta P Q <lb/>parallela ad A B, erit (ob <lb/>ipſarum æquidiſtantiam) O <lb/>E æqualis E P, itemque A <lb/>E ęqualis E Q (ob triangu-<lb/>lorum ſimilitudinem A E <lb/>O, Q E P) atque anguli ad <lb/>E ſunt æquales, quare & </s> <s xml:id="echoid-s8148" xml:space="preserve"><lb/>ipſa triangula ęqualia erunt, <lb/>quibus communi addito tra-<lb/>petio A B P E, fiet triangu-<lb/>lum O B P æquale menſali <lb/>A B P Q, hoc eſt minus triangulo A B C, vel triangulo L B I, quare L I <lb/>eſt infra æquidiſtantem baſim O P, ſiue baſis L I ſecat baſim A C vltra E, <lb/>verſus D. </s> <s xml:id="echoid-s8149" xml:space="preserve">Præterea cum ſit C B ad B I, <anchor type="note" xlink:href="" symbol="a"/> vt C I ad I H, vel vt C M ad <anchor type="note" xlink:label="note-0291-01a" xlink:href="note-0291-01"/> ad M A, ſitque C B maior B I erit C M maior M A, hoc eſt punctum M <lb/>cadet vltra D, verſus E. </s> <s xml:id="echoid-s8150" xml:space="preserve">Itaque harum baſium occurſus eſt inter diame-<lb/>tros B N, B D. </s> <s xml:id="echoid-s8151" xml:space="preserve">Quod idem eſt, ac ſi dicatur nullam ipſarum baſium tranſi-<lb/>re per medium punctum alterius.</s> <s xml:id="echoid-s8152" xml:space="preserve"/> </p> <div xml:id="echoid-div843" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0291-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0291-01" xlink:href="note-0291-01a" xml:space="preserve">Coroll. <lb/>12. primi <lb/>huius.</note> </div> <p> <s xml:id="echoid-s8153" xml:space="preserve">Tandem cum triangula A B C, L B I ſint æqualia, dempto com-<lb/>munitriangulo A B I, remanebit triangulum A C I ęquale trian-<lb/>gulo A L I, ſuntque ſuper eadem baſi A I, quare A I ipſi <lb/>L C æquidiſtabit; </s> <s xml:id="echoid-s8154" xml:space="preserve">& </s> <s xml:id="echoid-s8155" xml:space="preserve">cum inter parallelas A I, L C <lb/>interceptæ ſint duæ C A, L I proportionaliter <lb/>ſectæ in N, D, (ibi enim bifariam ſectæ <lb/>ſunt ex hypotheſi) erit quoque iun-<lb/>cta N D ipſi L C, vel A I æqui-<lb/>diſtans; </s> <s xml:id="echoid-s8156" xml:space="preserve">vt patet ex Ele-<lb/>mentis.</s> <s xml:id="echoid-s8157" xml:space="preserve"/> </p> <pb o="106" file="0292" n="292" rhead=""/> </div> <div xml:id="echoid-div845" type="section" level="1" n="337"> <head xml:id="echoid-head346" xml:space="preserve">LEMMA XV. PROP. LXXVI.</head> <p> <s xml:id="echoid-s8158" xml:space="preserve">Si in angulo A B C applicatæ fuerint quotcunque rectæ lineæ <lb/>A C, D E, &</s> <s xml:id="echoid-s8159" xml:space="preserve">c. </s> <s xml:id="echoid-s8160" xml:space="preserve">inter ſe parallelæ; </s> <s xml:id="echoid-s8161" xml:space="preserve">quæ a quacunque alia recta F <lb/>G vtrique lateri dati anguli occurrente in F, G, ſecentur in <lb/>H, I, &</s> <s xml:id="echoid-s8162" xml:space="preserve">c. </s> <s xml:id="echoid-s8163" xml:space="preserve">Dico vt rectangulum F H G ad D H E, ita eſſe rectan-<lb/>gulum FIG, ad AIC.</s> <s xml:id="echoid-s8164" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8165" xml:space="preserve">ETenim ratio rectanguli F <lb/> <anchor type="figure" xlink:label="fig-0292-01a" xlink:href="fig-0292-01"/> H G ad D H E, com-<lb/>ponitur ex ratione F H ad H <lb/>D, ſiue ex F I ad I A; </s> <s xml:id="echoid-s8166" xml:space="preserve">& </s> <s xml:id="echoid-s8167" xml:space="preserve">ex <lb/>ratione G H ad H E, ſiue ex <lb/>G I ad I C; </s> <s xml:id="echoid-s8168" xml:space="preserve">ſed & </s> <s xml:id="echoid-s8169" xml:space="preserve">ratio re-<lb/>ctanguli F I G ad A I C ex <lb/>ijſdem rationibus componi-<lb/>tur, quapropter rectangulum <lb/>F H G ad D H E, eſt vt re-<lb/>ctangulum F I G ad A I C. <lb/></s> <s xml:id="echoid-s8170" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s8171" xml:space="preserve">c. </s> <s xml:id="echoid-s8172" xml:space="preserve">Et permutan-<lb/>do, rectangulum F H G ad FIG, vt rectangulum D H E ad AIC.</s> <s xml:id="echoid-s8173" xml:space="preserve"/> </p> <div xml:id="echoid-div845" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0292-01"/> </figure> </div> </div> <div xml:id="echoid-div847" type="section" level="1" n="338"> <head xml:id="echoid-head347" xml:space="preserve">THEOR. XLVIII. PROP. LXXVII.</head> <p> <s xml:id="echoid-s8174" xml:space="preserve">Si fuerint duæ æquales portiones de eodem angulo, vel de <lb/>eadem coni - ſectione, vel circulo, & </s> <s xml:id="echoid-s8175" xml:space="preserve">ex puncto medio baſis <lb/>vnius portionis applicata ſit in angulo, vel ſectione quædam <lb/>recta linea baſi alterius portionis æquidiſtans: </s> <s xml:id="echoid-s8176" xml:space="preserve">rectangulum ſub <lb/>ſegmentis huius applicatæ æquabitur quadrato ſemi - baſis eiuſ-<lb/>dem portionis, cui hæc ipſa applicata æquidiſtanter ducta fuit.</s> <s xml:id="echoid-s8177" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8178" xml:space="preserve">SInt de eodem angulo, vt in præſenti figura, vel de quacunque coni-ſe-<lb/>ctione, vt in figuris tertij Schematiſmi, abſciſſæ duæ æquales portio-<lb/>nes A B C, H E I (vti docuimus in præcedenti, atque ex quadrageſima <lb/>huius abſolui poſſe elicitur) quarum diametri ſint B F, E G baſes verò <lb/>ſint A C, H I ab ipſis diametris bifariam ſectæ in F, G. </s> <s xml:id="echoid-s8179" xml:space="preserve">Iam <anchor type="note" xlink:href="" symbol="a"/> patet has <anchor type="note" xlink:label="note-0292-01a" xlink:href="note-0292-01"/> baſes omninò ſe mutuò ſecare, atque inter portionum diametros vt in <lb/>M, ſiue, punctum earum occurſus M differre à punctis medijs F, G. <lb/></s> <s xml:id="echoid-s8180" xml:space="preserve">Itaque ſi per alterum ipſorum, vtputa per G puncto medio baſis H I, ap-<lb/>plicetur in angulo, vel ſectione recta S G T parallela alteri baſi A C, hæc <lb/>omnino ad vtranque partem cum anguli lateribus, vel cum ſectione conue-<lb/>niet, cumipſa ſit vna applicatarum ad diametrum B F. </s> <s xml:id="echoid-s8181" xml:space="preserve">Occurrat ergo in S, <lb/>T: </s> <s xml:id="echoid-s8182" xml:space="preserve">Dico rectangulum S G T quadrato dimidiæ baſis A C, ſiue quadrato F <lb/>C æquale eſſe.</s> <s xml:id="echoid-s8183" xml:space="preserve"/> </p> <div xml:id="echoid-div847" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0292-01" xlink:href="note-0292-01a" xml:space="preserve">Schol. <lb/>75. h. & <lb/>per 1. Co-<lb/>roll. 40. h.</note> </div> <pb o="107" file="0293" n="293" rhead=""/> <p> <s xml:id="echoid-s8184" xml:space="preserve">Iunctis enim rectis A I, G F, H C: <lb/></s> <s xml:id="echoid-s8185" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0293-01a" xlink:href="fig-0293-01"/> patet has inter ſe <anchor type="note" xlink:href="" symbol="a"/> æquidiſtare, ac ideò <anchor type="note" xlink:label="note-0293-01a" xlink:href="note-0293-01"/> ipſas proportionaliter diuidere rectas H <lb/>G M I, C F M A inter eas interceptas in <lb/>punctis G, F, M. </s> <s xml:id="echoid-s8186" xml:space="preserve">Erit ergo, in ſingulis <lb/>figuris, quadratum H G ad quadratum <lb/>G M, vt quadratum C F ad quadratum <lb/>F M, & </s> <s xml:id="echoid-s8187" xml:space="preserve">permutando quadratum H G <lb/>ad quadratum C F, vt quadratum G M <lb/>ad F M: </s> <s xml:id="echoid-s8188" xml:space="preserve">ſed, in præſenti figura, eſt qua-<lb/>dratum H G æquale rectangulo H M I <lb/>vnà cum quadrato G M, & </s> <s xml:id="echoid-s8189" xml:space="preserve">quadratum <lb/>C F æquatur rectangulo C M A vnà <lb/>cum quadrato F M, (cumrectæ H I, A C bifariam ſectæ ſint in G & </s> <s xml:id="echoid-s8190" xml:space="preserve">F, & </s> <s xml:id="echoid-s8191" xml:space="preserve"><lb/>non bifariam in M) atque eſt totum quadratum H G ad totum C F vt pars <lb/>ad partem, vel vt quadratum G M ad F M, ergo reliquum ad reliquum, <lb/>nempe rectangulum H M I ad C M A, vel <anchor type="note" xlink:href="" symbol="b"/> rectangulum H G I ad T G S <anchor type="note" xlink:label="note-0293-02a" xlink:href="note-0293-02"/> erit vt totum ad totum, ſiue vt quadratum H G ad quadratum C F, ſed an-<lb/>tecedentia ſunt æqualia, hoc eſt rectangulum H G I, & </s> <s xml:id="echoid-s8192" xml:space="preserve">quadratum H G, <lb/>cum ſit recta H G æqualis G I, ergo, & </s> <s xml:id="echoid-s8193" xml:space="preserve">conſequentia æqualia erunt, nem-<lb/>pe rectangulum T G S, & </s> <s xml:id="echoid-s8194" xml:space="preserve">quadratum C F. </s> <s xml:id="echoid-s8195" xml:space="preserve">Quod in anguli portionibus <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s8196" xml:space="preserve"/> </p> <div xml:id="echoid-div848" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0293-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0293-01" xlink:href="note-0293-01a" xml:space="preserve">ibidem.</note> <note symbol="b" position="right" xlink:label="note-0293-02" xlink:href="note-0293-02a" xml:space="preserve">76. h.</note> </div> <p> <s xml:id="echoid-s8197" xml:space="preserve">In reliquis autem figuris iam dicti tertij Schematiſmi, eſt rectangulum H <lb/>G I ad rectangulum T G S, vt quadratum <anchor type="note" xlink:href="" symbol="c"/> contingentis E N ipſi H I pa- <anchor type="note" xlink:label="note-0293-03a" xlink:href="note-0293-03"/> rallelæ ad quadratum contingentis B N alteri A C æquidiſtantis, vel vt <lb/>quadratum G M ad M F (nam ibi primo loco oſtenſum fuit in ſingulis eſſe <lb/>E N ad N B, vt G M ad M F (vel ob parallelas A I, F G, C H, vt qua-<lb/>dratum H G ad quadratum C F, atque antecedentia ſunt æqualia, nempe <lb/>rectangulum H G I quadrato H G, cum recta H G ſit æqualis rectæ G I, <lb/>ergo, & </s> <s xml:id="echoid-s8198" xml:space="preserve">conſequentia, ſiue rectangulum T G S quadrato C F æquale erit. <lb/></s> <s xml:id="echoid-s8199" xml:space="preserve">Quod omnino oſtendere propoſitum fuit.</s> <s xml:id="echoid-s8200" xml:space="preserve"/> </p> <div xml:id="echoid-div849" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0293-03" xlink:href="note-0293-03a" xml:space="preserve">17. tertij <lb/>Conic.</note> </div> </div> <div xml:id="echoid-div851" type="section" level="1" n="339"> <head xml:id="echoid-head348" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s8201" xml:space="preserve">CVM ad abſcindendas MAXIMAS, & </s> <s xml:id="echoid-s8202" xml:space="preserve">MINIMAS co-<lb/>ni - ſectionum portiones per punctum in ijs datum, animad-<lb/>uertiſsemus olim præmittendam eſſe inueſtigationem æqua-<lb/>lium portionum eiuſdem coni - ſectionis, quas deinde pro <lb/>quacunque coni - ſectione reperimus, atque vnica demonſtratione confir-<lb/>mauimus, (vt viſum eſt in 40. </s> <s xml:id="echoid-s8203" xml:space="preserve">huius, ac ſimul vt in 45. </s> <s xml:id="echoid-s8204" xml:space="preserve">eas om-<lb/>nes proprijs baſibus ſimilem, & </s> <s xml:id="echoid-s8205" xml:space="preserve">concentricam eiuſdem nominis ſectio-<lb/>nem contingere) ita dum MAXIMAS, ac MINIMAS Conorum, aut <lb/>Conoidalium, vel Sphæroidalium ſolidorum portiones nobis duximus in-<lb/>quirendum, neceſſe fuit prius contemplari, quæ nam eiuſdem Coni recti, <pb o="108" file="0294" n="294" rhead=""/> vel Conoidis, ſiue Sphær<unsure/>æ, aut Sphæroidis portiones inter ſe æquales eſ-<lb/>ſent: </s> <s xml:id="echoid-s8206" xml:space="preserve">vnde mox venit nobis in animum perpendendi, an illæ inter ſe <lb/>æqualitatem ſortirentur, quarum portiones planæ genitricium ſectionum <lb/>ad plana baſium erectæ, nempe quarum recti Canones inter ſe pariter <lb/>æquales eſſent, prout æquales inſpexeramus in Conoide Paraboli co, ex <lb/>25. </s> <s xml:id="echoid-s8207" xml:space="preserve">Archimedis in libro de Conoid. </s> <s xml:id="echoid-s8208" xml:space="preserve">& </s> <s xml:id="echoid-s8209" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s8210" xml:space="preserve">Res quidem ex cogi-<lb/>tatione ſuccesſit, tunc enim in ſequentem vniuerſalem demonſtr ationem <lb/>incidimus, cuius, atque ſuperioris quadrageſimæ propoſitionis, ſolæ enun-<lb/>ciationes, cum præſtantisſimis Geometris, Galileo, ac Torricellio com-<lb/>municatæ, tantos Viros, meruerunt habere laudatores.</s> <s xml:id="echoid-s8211" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div852" type="section" level="1" n="340"> <head xml:id="echoid-head349" xml:space="preserve">THEOR. IL. PROP. LXXVIII.</head> <p> <s xml:id="echoid-s8212" xml:space="preserve">Solidæ portiones eiuſdem Coni recti, vel Conoidis Parabo-<lb/>lici, aut Hyperbolici, ſiue Sphæræ, aut Sphæroidis oblongi, <lb/>vel prolati, quarum recti Canones ſint æquales, inter ſe quoq; <lb/></s> <s xml:id="echoid-s8213" xml:space="preserve">æquales ſunt.</s> <s xml:id="echoid-s8214" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8215" xml:space="preserve">ESto Coni recti, vt in prima figura, vel Conoidis Parabolici, aut Hy-<lb/>perbolici, ſiue Sphæræ, aut Sphæroidis oblongi, vel prolati, vt in ſe-<lb/>cunda, quorum axis B D, quælibet ſectio per axem A B C, quæ erit <anchor type="note" xlink:href="" symbol="a"/> geni- <anchor type="note" xlink:label="note-0294-01a" xlink:href="note-0294-01"/> trix dati ſolidi, à qua demantur duæ æquales portiones planæ A B C, E F <lb/>G (hoc autem fieri poſſe manifeſtum iam <anchor type="note" xlink:href="" symbol="b"/> eſt) quarum baſes ſint A C, E <anchor type="note" xlink:label="note-0294-02a" xlink:href="note-0294-02"/> G bifariam ſectæ in H, I, & </s> <s xml:id="echoid-s8216" xml:space="preserve">ipſarum altera A C ſit axi perpendicularis, <lb/>altera verò vtcunque inclinata; </s> <s xml:id="echoid-s8217" xml:space="preserve">& </s> <s xml:id="echoid-s8218" xml:space="preserve">per eas concipiantur duci plana A L C, <lb/>E M G ad planum per axem A B C erecta, auferentia portiones ſolidas A <lb/>B C, E F G, quarum recti Canones erunt ipſæ portiones planæ A B C, E <lb/>F G: </s> <s xml:id="echoid-s8219" xml:space="preserve">patet ſectionem A L C circulum eſſe, <anchor type="note" xlink:href="" symbol="c"/> cuius diameter A C, centrum <anchor type="note" xlink:label="note-0294-03a" xlink:href="note-0294-03"/> H, atque E M G eſſe Ellipſim, cuius axis maior, in Cono, vel in Conoide <lb/>Parabolico, aut Hyperbolico, atque in Sphæroide oblongo, erit ipſa baſis <lb/>E G, ſed in prolato erit <anchor type="note" xlink:href="" symbol="d"/> minor axis, vbique autem centrum I. </s> <s xml:id="echoid-s8220" xml:space="preserve">Dico hu- <anchor type="note" xlink:label="note-0294-04a" xlink:href="note-0294-04"/> iuſmodi ſolidas portiones A B C, E F G inter ſe æquales eſſe.</s> <s xml:id="echoid-s8221" xml:space="preserve"/> </p> <div xml:id="echoid-div852" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0294-01" xlink:href="note-0294-01a" xml:space="preserve">12. Ar-<lb/>chim. de <lb/>Conoid.</note> <note symbol="b" position="left" xlink:label="note-0294-02" xlink:href="note-0294-02a" xml:space="preserve">ex 40. & <lb/>ex 75. h.</note> <note symbol="c" position="left" xlink:label="note-0294-03" xlink:href="note-0294-03a" xml:space="preserve">ex 1. pri-<lb/>mihuius, <lb/>& ex 12. <lb/>13. 14. 15. <lb/>Archim. <lb/>de Co-<lb/>noid. &c.</note> <note symbol="d" position="left" xlink:label="note-0294-04" xlink:href="note-0294-04a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s8222" xml:space="preserve">Secetur iterum datum ſolidum A B C, plano per punctum I tranſeunte, <lb/>& </s> <s xml:id="echoid-s8223" xml:space="preserve">ad axem B D erecto, ſiue plano A L C æquidiſtanti, quod in ſolido <lb/>efficiet <anchor type="note" xlink:href="" symbol="e"/> pariter circulum N M O, cuius centrum P in axe, & </s> <s xml:id="echoid-s8224" xml:space="preserve">diameter N <anchor type="note" xlink:label="note-0294-05a" xlink:href="note-0294-05"/> O, quæ ipſi A C <anchor type="note" xlink:href="" symbol="f"/> æquidiſtabit, communis autem ſectio recti plani N M <anchor type="note" xlink:label="note-0294-06a" xlink:href="note-0294-06"/> O, cum alio plano E M G, erit <anchor type="note" xlink:href="" symbol="g"/> recta M I, quæ quidem recta erit <anchor type="note" xlink:href="" symbol="h"/> ad pla- <anchor type="note" xlink:label="note-0294-07a" xlink:href="note-0294-07"/> <anchor type="note" xlink:label="note-0294-08a" xlink:href="note-0294-08"/> num per axem A B C (cum ea ſit communis ſectio duorum planorum ad <lb/>idem planum per axem erectorum) ideoque tùm ad circuli diametrum N <lb/>O, tum ad E G axem Ellipſis, erit perpendicularis, & </s> <s xml:id="echoid-s8225" xml:space="preserve">in Cono, aut Co-<lb/>noide, vel Sphæroide oblongo erit ſemi- axis minor, in prolato verò ſemi-<lb/>axis maior. </s> <s xml:id="echoid-s8226" xml:space="preserve">Et quoniam M I ad diametrum N O ſemi - circuli N M O eſt <lb/>perpendicularis, erit quadratum M I ęquale rectangulo NIO, ſed & </s> <s xml:id="echoid-s8227" xml:space="preserve">qua- <pb o="109" file="0295" n="295" rhead=""/> dratum A H eidem rectangulo NIO eſt æquale, cum ſit NIO <anchor type="note" xlink:href="" symbol="a"/> parallela <anchor type="note" xlink:label="note-0295-01a" xlink:href="note-0295-01"/> ad A C, & </s> <s xml:id="echoid-s8228" xml:space="preserve">per I punctum medium baſis E G ducta, &</s> <s xml:id="echoid-s8229" xml:space="preserve">c. </s> <s xml:id="echoid-s8230" xml:space="preserve">ergo, & </s> <s xml:id="echoid-s8231" xml:space="preserve">quadra-<lb/>tum M I ipſi A H, ſeu linea M I lineæ A H æqualis erit, ſed Ellipſis E M <lb/>G ad circulum A L C eſt vt <anchor type="note" xlink:href="" symbol="b"/> rectangulum ſub G I, & </s> <s xml:id="echoid-s8232" xml:space="preserve">I M ad quadratum <anchor type="note" xlink:label="note-0295-02a" xlink:href="note-0295-02"/> ex A H, vel vt linea G I ad A H (ob communem altitudinem M I) vel <lb/>ſumptis duplis, vt E G ad A C, ergo baſis portionis ſolidę E F G, ad baſim <lb/>portionis ſolidę A B C, eſt vt E G baſis Canonis E F G, ad A C baſim Ca-<lb/>nonis A B C; </s> <s xml:id="echoid-s8233" xml:space="preserve">verùm vt E G ad A C, ita <anchor type="note" xlink:href="" symbol="c"/> eſt reciprocè altitudo Canonis <anchor type="note" xlink:label="note-0295-03a" xlink:href="note-0295-03"/> <anchor type="figure" xlink:label="fig-0295-01a" xlink:href="fig-0295-01"/> A B C ad altitudinem Canonis E F G (cum ipſi Canones ęquales facti ſint) <lb/>atque Canonum altitudines eædem ſunt <anchor type="note" xlink:href="" symbol="d"/> cum altitudinibus ſolidarum por- <anchor type="note" xlink:label="note-0295-04a" xlink:href="note-0295-04"/> tionum, vnde baſis E M G ad baſim A L C erit reciprocè, vt altitudo ſoli-<lb/>dæ portionis A B C ad altitudinem ſolidæ E F G: </s> <s xml:id="echoid-s8234" xml:space="preserve">hæ autem portiones ſunt <lb/>ſolida <anchor type="note" xlink:href="" symbol="e"/> Acuminata proportionalia, eò quod ipſarum Canones ſint æquales, <anchor type="note" xlink:label="note-0295-05a" xlink:href="note-0295-05"/> atque baſes altitudinibus ſunt reciprocæ, ergo huiuſmodi portiones ſolidæ <lb/>A B C, E F G ſunt <anchor type="note" xlink:href="" symbol="f"/> æquales. </s> <s xml:id="echoid-s8235" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s8236" xml:space="preserve"/> </p> <div xml:id="echoid-div853" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-0294-05" xlink:href="note-0294-05a" xml:space="preserve">12. ibid.</note> <note symbol="f" position="left" xlink:label="note-0294-06" xlink:href="note-0294-06a" xml:space="preserve">16. Vnd. <lb/>Elem.</note> <note symbol="g" position="left" xlink:label="note-0294-07" xlink:href="note-0294-07a" xml:space="preserve">3. ibid.</note> <note symbol="h" position="left" xlink:label="note-0294-08" xlink:href="note-0294-08a" xml:space="preserve">19. ibid.</note> <note symbol="a" position="right" xlink:label="note-0295-01" xlink:href="note-0295-01a" xml:space="preserve">77. h.</note> <note symbol="b" position="right" xlink:label="note-0295-02" xlink:href="note-0295-02a" xml:space="preserve">ex 6. Ar-<lb/>chim. de <lb/>Conoid.</note> <note symbol="c" position="right" xlink:label="note-0295-03" xlink:href="note-0295-03a" xml:space="preserve">65. h.</note> <figure xlink:label="fig-0295-01" xlink:href="fig-0295-01a"> <image file="0295-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0295-01"/> </figure> <note symbol="d" position="right" xlink:label="note-0295-04" xlink:href="note-0295-04a" xml:space="preserve">3. Schol. <lb/>69. h.</note> <note symbol="e" position="right" xlink:label="note-0295-05" xlink:href="note-0295-05a" xml:space="preserve">Coroll. <lb/>70 h.</note> </div> <note symbol="f" position="right" xml:space="preserve">74. h.</note> <p style="it"> <s xml:id="echoid-s8237" xml:space="preserve">Sed hoc idem, tribus proximè præcedentibus propoſitionibus omisſis, <lb/>ſuper nouo diagrammate ſic oſtendere conabimur</s> </p> </div> <div xml:id="echoid-div855" type="section" level="1" n="341"> <head xml:id="echoid-head350" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s8238" xml:space="preserve">SIt Conus rectus, vt in prima figura, vel aliud quodcunque prædictorum <lb/>ſolidorum, vt in ſecunda, circa axim A B, & </s> <s xml:id="echoid-s8239" xml:space="preserve">ſectio per axim ſit E A <lb/>D, quæ genitrix erit <anchor type="note" xlink:href="" symbol="g"/> dati ſolidi, à qua demptæ ſint duæ quælibet portio- <anchor type="note" xlink:label="note-0295-07a" xlink:href="note-0295-07"/> nes planæ æquales C A D, E A F, quarum baſes ſint C D, E F, & </s> <s xml:id="echoid-s8240" xml:space="preserve">per ip-<lb/>ſas ducantur piana ſecantia data ſolida, & </s> <s xml:id="echoid-s8241" xml:space="preserve">ad ipſum planum per axem E A <lb/>D erecta, circulos, vel <anchor type="note" xlink:href="" symbol="h"/> Ellipſes E O F, C P D deſcribentia (quarum ma- <anchor type="note" xlink:label="note-0295-08a" xlink:href="note-0295-08"/> iores axes in Cono, Conoide Parabolico, Hyperbolico, & </s> <s xml:id="echoid-s8242" xml:space="preserve">Sphæroide ob- <pb o="110" file="0296" n="296" rhead=""/> longo erunt <anchor type="note" xlink:href="" symbol="a"/> ipſę C D, E F, in prolato verò erunt axes minores) auferen- <anchor type="note" xlink:label="note-0296-01a" xlink:href="note-0296-01"/> tiaque ſolidas portiones C A D, E A F, quarum recti Canones erunt ipſæ <lb/>æquales portiones planæ C A D, E A F. </s> <s xml:id="echoid-s8243" xml:space="preserve">Dico tales portiones ſolidas inter <lb/>ſe æquales eſſe.</s> <s xml:id="echoid-s8244" xml:space="preserve"/> </p> <div xml:id="echoid-div855" type="float" level="2" n="1"> <note symbol="g" position="right" xlink:label="note-0295-07" xlink:href="note-0295-07a" xml:space="preserve">ex 12. <lb/>Archim. <lb/>de Co-<lb/>noid. &c.</note> <note symbol="h" position="right" xlink:label="note-0295-08" xlink:href="note-0295-08a" xml:space="preserve">ex pri-<lb/>ma primi <lb/>huius, & <lb/>ex 13. 14. <lb/>15. Arch. <lb/>de Co-<lb/>noid. &c.</note> <note symbol="a" position="left" xlink:label="note-0296-01" xlink:href="note-0296-01a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s8245" xml:space="preserve">Nam bifariam ſectis E F in G, & </s> <s xml:id="echoid-s8246" xml:space="preserve">C D in H, patet puncta G, H eſſe cen-<lb/>tra circulorum, ſiue Ellipſium E O F, C P D; </s> <s xml:id="echoid-s8247" xml:space="preserve">& </s> <s xml:id="echoid-s8248" xml:space="preserve">ſi per punctum G deſcri-<lb/> <anchor type="note" xlink:label="note-0296-02a" xlink:href="note-0296-02"/> batur <anchor type="note" xlink:href="" symbol="b"/> in vtraque ſigura eiuſdem nominis Coni-ſectio G H, que ſimilis ſit, & </s> <s xml:id="echoid-s8249" xml:space="preserve">concentrica ſectioni E A D, & </s> <s xml:id="echoid-s8250" xml:space="preserve">qualis in Monito poſt 68. </s> <s xml:id="echoid-s8251" xml:space="preserve">h. </s> <s xml:id="echoid-s8252" xml:space="preserve">definiuimus, <lb/>patet inquam ipſam ſectionem G H omnino tranſire per H, ſimulque E F, <lb/>& </s> <s xml:id="echoid-s8253" xml:space="preserve">C D in punctis medijs G, H <anchor type="note" xlink:href="" symbol="c"/> contingere.</s> <s xml:id="echoid-s8254" xml:space="preserve"/> </p> <div xml:id="echoid-div856" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0296-02" xlink:href="note-0296-02a" xml:space="preserve">4. ſec. <lb/>Conic & <lb/>5. 6. 7. pri-<lb/>mi huius.</note> </div> <note symbol="c" position="left" xml:space="preserve">68. h.</note> <p> <s xml:id="echoid-s8255" xml:space="preserve">Iam ductis per G, H, rectis I G L, M H N ad axem A B perpendicula-<lb/>ribus, concipiantur per ipſas duci plana ad planum per axem E A D erecta, <lb/> <anchor type="note" xlink:label="note-0296-04a" xlink:href="note-0296-04"/> quæ efficient in exteriori ſolido circulos <anchor type="note" xlink:href="" symbol="d"/> circa diametros I L, M N, &</s> <s xml:id="echoid-s8256" xml:space="preserve"> communes eorum ſectiones cum planis per E F, C D ductis, erunt <anchor type="note" xlink:href="" symbol="e"/> rectæ G O, H P, quæ ad planum E A D rectæ erunt <anchor type="note" xlink:href="" symbol="f"/> (ſunt enim communes ſe- ctiones duorum planorum ad idem planum erectorum) hoc eſt, tùm O G <lb/> <anchor type="note" xlink:label="note-0296-05a" xlink:href="note-0296-05"/> cum vtriſque E F, I L, tùm P H cum vtriſque C D, M N rectos eſſiciet <lb/> <anchor type="note" xlink:label="note-0296-06a" xlink:href="note-0296-06"/> angulos; </s> <s xml:id="echoid-s8257" xml:space="preserve">vnde in circulis tranſeuntibus per I L, M N, rectangulum I G L <lb/>æquabitur quadrato G O, & </s> <s xml:id="echoid-s8258" xml:space="preserve">re-<lb/> <anchor type="figure" xlink:label="fig-0296-01a" xlink:href="fig-0296-01"/> ctangulum M H N quadrato H <lb/>P, atque ipſæ G O, H P erunt <lb/>circulorum, aut Ellipſium E O <lb/>F, C P D minores ſemi-axes, <lb/>in Cono tamen, vel Conoide <lb/>Parabolico, aut Hyperbolico, <lb/>vel in Sphæroide oblongo; </s> <s xml:id="echoid-s8259" xml:space="preserve">nam <lb/>in prolato, erunt maiores ſemi-<lb/>axes: </s> <s xml:id="echoid-s8260" xml:space="preserve">ſed rectangula I G L, M <lb/>H N ſunt <anchor type="note" xlink:href="" symbol="g"/> æqualia, vtrunque <anchor type="note" xlink:label="note-0296-07a" xlink:href="note-0296-07"/> enim æquatur quadrato ſemi-tangentis per verticem interioris ſectionis, &</s> <s xml:id="echoid-s8261" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s8262" xml:space="preserve">ergo, & </s> <s xml:id="echoid-s8263" xml:space="preserve">quadrato G O, H P æqualia erunt, ſiue ſemi-axis G O æqualis ſe-<lb/>mi- axi H P; </s> <s xml:id="echoid-s8264" xml:space="preserve">ſed circulus, aut Ellipſis E O F ad C P D, eſt vt <anchor type="note" xlink:href="" symbol="h"/> rectangu- <anchor type="note" xlink:label="note-0296-08a" xlink:href="note-0296-08"/> lum ſub E F, G O, ad rectangulum ſub C D, H P, & </s> <s xml:id="echoid-s8265" xml:space="preserve">rectangulum ſub E F, <lb/>G O ad rectangulum ſub C D, H P eſt vt E F ad C D (cum eorum latitu-<lb/>dines G O, H P ſint æquales) ergo circulus, vel Ellipſis E O F ad C P D <lb/>erit in ſolido Parabolico, vel Hyperbolico, aut Sphæroide oblongo, vt ma-<lb/>ior axis E F ad maiorem axim C D, vel in Sphæroide prolato, vt minor <lb/>axis E F ad minorem C D: </s> <s xml:id="echoid-s8266" xml:space="preserve">ſed E F ad C D eſt <anchor type="note" xlink:href="" symbol="i"/> vt altitudo Canonis C A <anchor type="note" xlink:label="note-0296-09a" xlink:href="note-0296-09"/> D, ad altitudinem Canonis E A F, cum ipſi ſint æquales portiones eiuſ-<lb/>dem coni-ſectionis, & </s> <s xml:id="echoid-s8267" xml:space="preserve">horum Canonum altitudines ſunt <anchor type="note" xlink:href="" symbol="l"/> eædem, ac alti- <anchor type="note" xlink:label="note-0296-10a" xlink:href="note-0296-10"/> tudines ſolidarum portionum C A D, E A F, quare circulus, vel Ellipſis E <lb/>O F ad C P D, erit reciprocè vt altitudo ſolidæ portionis C A D, ad alti-<lb/>tucinem ſolidæ E A F: </s> <s xml:id="echoid-s8268" xml:space="preserve">at huiuſmodi portiones ſunt <anchor type="note" xlink:href="" symbol="m"/> ſolida Acuminata <anchor type="note" xlink:label="note-0296-11a" xlink:href="note-0296-11"/> proportionalia, & </s> <s xml:id="echoid-s8269" xml:space="preserve">ipſorum baſes altitudinibus reciprocantur, ergo ipſæ ſo-<lb/>lidæ portiones inter ſe ſunt <anchor type="note" xlink:href="" symbol="n"/> æquales. </s> <s xml:id="echoid-s8270" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s8271" xml:space="preserve"/> </p> <div xml:id="echoid-div857" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-0296-04" xlink:href="note-0296-04a" xml:space="preserve">4. primi <lb/>Conic. & <lb/>12. Arch. <lb/>de Co-<lb/>noid. &c.</note> <note symbol="e" position="left" xlink:label="note-0296-05" xlink:href="note-0296-05a" xml:space="preserve">3. vnd. <lb/>Elem.</note> <note symbol="f" position="left" xlink:label="note-0296-06" xlink:href="note-0296-06a" xml:space="preserve">19. ibid.</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/QN4GHYBF/figures/0296-01"/> </figure> <note symbol="g" position="left" xlink:label="note-0296-07" xlink:href="note-0296-07a" xml:space="preserve">3. Co-<lb/>roll. 46. h.</note> <note symbol="h" position="left" xlink:label="note-0296-08" xlink:href="note-0296-08a" xml:space="preserve">7. Arch. <lb/>de Co-<lb/>noid. &c.</note> <note symbol="i" position="left" xlink:label="note-0296-09" xlink:href="note-0296-09a" xml:space="preserve">65. h.</note> <note symbol="l" position="left" xlink:label="note-0296-10" xlink:href="note-0296-10a" xml:space="preserve">3. Schol. <lb/>69. h.</note> <note symbol="m" position="left" xlink:label="note-0296-11" xlink:href="note-0296-11a" xml:space="preserve">Coroll. <lb/>70. h.</note> </div> <note symbol="n" position="left" xml:space="preserve">74. h.</note> <pb o="111" file="0297" n="297" rhead=""/> </div> <div xml:id="echoid-div859" type="section" level="1" n="342"> <head xml:id="echoid-head351" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s8272" xml:space="preserve">HInc colligitur, quod puncta media rectarum quarumlibet applicata-<lb/>rum in ſectione per axem ducta, cuiuſcunque prædictorum ſolidorũ, <lb/>ſunt centra baſium earum portionum ſolidarum à planis per eaſdem rectas <lb/>ductis, atque ad eandem ſectionem per axem erectis abſciſſarum.</s> <s xml:id="echoid-s8273" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8274" xml:space="preserve">Nam puncta media G, H, applicatarum E F, C D demonſtrata ſunt <lb/>eſſe centra prædictarum baſium E O F, C P D, &</s> <s xml:id="echoid-s8275" xml:space="preserve">c.</s> <s xml:id="echoid-s8276" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div860" type="section" level="1" n="343"> <head xml:id="echoid-head352" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s8277" xml:space="preserve">PErſpicuum eſt quoque, baſes ſolidarum portionum inter ſe æqualium <lb/>eiuſdem Coni recti, vel Conoidis Parabolici, aut Hyperbolici, Sphe-<lb/>ræ, aut Sphæroidis oblongi, habere inter ſe axes minores æquales, ſiue eſſe <lb/>æqualium latitudinum, ac ideò eſſe inter ſe, vt axes maiores, vel vt baſes <lb/>rectorum Canonum. </s> <s xml:id="echoid-s8278" xml:space="preserve">Baſes verò æqualium portionum eiuſdem Sphæroidis <lb/>prolati habere maiores axes æquales, ac propterea eſſe inter ſe vt axes mi-<lb/>nores, vel vt baſes eorundem rectorum Canonum.</s> <s xml:id="echoid-s8279" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8280" xml:space="preserve">Etenim, in pręcedentibus figuris, de baſibus E O F, C P D (vel ſint Cir-<lb/>culi, vel Ellipſes) portionum ſolidarum E A F, C A D, quas æquales eſſe <lb/>demonſtrauimus, oſtenſum priùs fuit ſemi- axes minores G O, H P in Co-<lb/>no recto, vel Conoide, aut Sphæroide oblongo eſſe æquales, ac ideò, & </s> <s xml:id="echoid-s8281" xml:space="preserve"><lb/>eorum duplos, hoc eſt integros minores axes æquales eſſe; </s> <s xml:id="echoid-s8282" xml:space="preserve">& </s> <s xml:id="echoid-s8283" xml:space="preserve">paulò poſt <lb/>circulum, vel Ellipſim E O F ad C P D eſſe vt maior axis E F ad maiorem <lb/>C D, vel vt baſis recti Canonis E A F, ad baſim recti Canonis C A D. </s> <s xml:id="echoid-s8284" xml:space="preserve">In <lb/>Sphæroide autem prolato demonſtratum eſt ipſas G O, H P maiores ſemi-<lb/>axes, item æquales eſſe, ſiue integros maiores axes æquales, & </s> <s xml:id="echoid-s8285" xml:space="preserve">poſtea cir-<lb/>culos, vel Ellipſes E O F, C B D habere inter ſe eandem rationem, ac ipſi <lb/>minores axes E F, C D; </s> <s xml:id="echoid-s8286" xml:space="preserve">nimirum eſſe inter ſe, vt ſunt baſes rectorum <lb/>Canonum E A F, C A D.</s> <s xml:id="echoid-s8287" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div861" type="section" level="1" n="344"> <head xml:id="echoid-head353" xml:space="preserve">THEOR. L. PROP. LXXIX.</head> <p> <s xml:id="echoid-s8288" xml:space="preserve">Solidæ portiones eiuſdem Coni recti, vel cuiuſcunque Conoi-<lb/>dis, vel Sphæræ, aut cuiuslibet Sphæroidis tunc æquales ſunt, <lb/>qnando, in Cono, portionum axes pertingant ad idem Conoides <lb/>Hyperbolicum concentricum, &</s> <s xml:id="echoid-s8289" xml:space="preserve">c. </s> <s xml:id="echoid-s8290" xml:space="preserve">In Conoide verò Parabolico, <lb/>quando portionum axes ſint æquales. </s> <s xml:id="echoid-s8291" xml:space="preserve">At in Conoide Hyperboli-<lb/>co, Sphæra, aut quocunque Sphæroide, quando portionum axes, <lb/>ad proprias ſemi- diametros ijſdem axibus in directum poſitas, ſint <lb/>in vna eademque ratione.</s> <s xml:id="echoid-s8292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8293" xml:space="preserve">ETenim quandò in portionibus eiuſdem cuiuſcunque prædictorum ſoli-<lb/>dorum diametri rectorum Canonum habuerint relatiuè conditiones <pb o="112" file="0298" n="298" rhead=""/> ſuperiùs allatas, ipſi Canones recti, qui iam ſunt portiones, vel anguli, vel <lb/>coni-ſectionis, aut circuli, æquales iam ſunt oſtenſi, vti de anguli portioni-<lb/>bus patet ex prima parte 45. </s> <s xml:id="echoid-s8294" xml:space="preserve">huius, pro reliquis autem Coni-ſectionibus, &</s> <s xml:id="echoid-s8295" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s8296" xml:space="preserve">ex 40. </s> <s xml:id="echoid-s8297" xml:space="preserve">ſed dum huiuſmodi Canones recti ſunt æquales, & </s> <s xml:id="echoid-s8298" xml:space="preserve">portiones ſolidæ <lb/>demonſtrantur æquales, ex ſuperiori Theoremate, ſuntque rectorum Ca-<lb/>nonum diametri <anchor type="note" xlink:href="" symbol="a"/> eædem, ac axes ſolidarum, quare, & </s> <s xml:id="echoid-s8299" xml:space="preserve">dum diametri re- <anchor type="note" xlink:label="note-0298-01a" xlink:href="note-0298-01"/> ctorum Canonum, ſiue dum axes ſolidarum portionum reſpectiuè ſerua-<lb/>bunt, quod modò expoſuimus, ipſæ portiones ſolidæ æquales erunt. </s> <s xml:id="echoid-s8300" xml:space="preserve">Quod <lb/>erat, &</s> <s xml:id="echoid-s8301" xml:space="preserve">c.</s> <s xml:id="echoid-s8302" xml:space="preserve"/> </p> <div xml:id="echoid-div861" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0298-01" xlink:href="note-0298-01a" xml:space="preserve">3. Schol. <lb/>69. h.</note> </div> <p style="it"> <s xml:id="echoid-s8303" xml:space="preserve">Itaque prop. </s> <s xml:id="echoid-s8304" xml:space="preserve">25. </s> <s xml:id="echoid-s8305" xml:space="preserve">præcitati libri Archimedis, quæ ſolùm de portionibus <lb/>Conoidis Parabolici diſſerit, ſuppoſita etiam proportione Conoidis ad ſibi <lb/>inſcriptum Conum, nobis hic eſt præſens Theorema, quod generaliter <lb/>proponit ea, quæ ad cognitionem faciunt æqualium portionum, cuiuslibet <lb/>ſimul prædictorum ſolidorum, atque ipſa diuerſa ratiocinatione confirmat, <lb/>nulla habita ratione proportionis, quæ cadit inter ſolidas portiones, & </s> <s xml:id="echoid-s8306" xml:space="preserve"><lb/>inſcriptos Conos, aut circumſcriptos Cylindros.</s> <s xml:id="echoid-s8307" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div863" type="section" level="1" n="345"> <head xml:id="echoid-head354" xml:space="preserve">THEOR. LI. PROP. LXXX.</head> <p> <s xml:id="echoid-s8308" xml:space="preserve">Omnes ſolidæ portiones eiuſdem Coni recti, vel Conoidis <lb/>Parabolici, aut Hyperbolici, ſiue Sphæræ, aut Sphæroidis ob-<lb/>longi, vel prolati, quarum baſes contingant eandem ſimilis, & </s> <s xml:id="echoid-s8309" xml:space="preserve"><lb/>inſcripti concentrici ſolidi ſuperficiem, inter ſe ſunt æquales, & </s> <s xml:id="echoid-s8310" xml:space="preserve"><lb/>ad centra baſium eandem ſuperficiem contingunt.</s> <s xml:id="echoid-s8311" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8312" xml:space="preserve">REpetito ſecundo Schemate præcedentis 78. </s> <s xml:id="echoid-s8313" xml:space="preserve">ijſdemque poſitis, quæ <lb/>ibi, ſi concipiantur figuræ conuerti circa axim A B, procreabitur de-<lb/>nuò à ſectione E A F datum ſo-<lb/> <anchor type="figure" xlink:label="fig-0298-01a" xlink:href="fig-0298-01"/> lidum, & </s> <s xml:id="echoid-s8314" xml:space="preserve">à ſectione G H inſcri-<lb/>ptum ſimile ſolidum concentri-<lb/>cum. </s> <s xml:id="echoid-s8315" xml:space="preserve">Ampliùs ſi fuerint quot-<lb/>cunque rectæ E F, C D, &</s> <s xml:id="echoid-s8316" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s8317" xml:space="preserve">interiorem ſectionem G H con-<lb/>tingentes, per quas agantur <lb/>plana E O F, C P D ad ipſas <lb/>ſectiones recta, hæc abſcin-<lb/>dent de exteriori ſolido por-<lb/>tiones ſolidas E A F, C A D, <lb/>atque erunt earundem portionum baſes, quæ concentrici ſolidi G H ſuper-<lb/>ficiem contingent <anchor type="note" xlink:href="" symbol="b"/> in ijſdem punctis G, H, in quibus rectæ E F, C D ſe- <anchor type="note" xlink:label="note-0298-02a" xlink:href="note-0298-02"/> ctionem G H contingunt, quæ puncta, per iam demonſtrata, ſunt <anchor type="note" xlink:href="" symbol="c"/> centra <anchor type="note" xlink:label="note-0298-03a" xlink:href="note-0298-03"/> ipſarum baſium, ſed huiuſmodi portionum ſolidarum E A F, C A D, Ca- <pb o="113" file="0299" n="299" rhead=""/> nones E A F, C A D (qui, ex conſtructione, ſunt ad plana baſium recti) <lb/>ſunt <anchor type="note" xlink:href="" symbol="a"/> æquales, ergo & </s> <s xml:id="echoid-s8318" xml:space="preserve">ipſæ ſolidæ portiones æquales <anchor type="note" xlink:href="" symbol="b"/> erunt. </s> <s xml:id="echoid-s8319" xml:space="preserve">Vnde om- <anchor type="note" xlink:label="note-0299-01a" xlink:href="note-0299-01"/> <anchor type="note" xlink:label="note-0299-02a" xlink:href="note-0299-02"/> nes ſolidæ portiones eiuſdem Coni recti, vel cuiuslibet prædictorum ſolido-<lb/>rum, quarum baſes contingant eiuſdem ſimilis, & </s> <s xml:id="echoid-s8320" xml:space="preserve">concentrici ſolidi ſuper-<lb/>ficiem inter ſe ſunt æquales, & </s> <s xml:id="echoid-s8321" xml:space="preserve">ad centra baſium eandem ſuperficiem con-<lb/>tingunt. </s> <s xml:id="echoid-s8322" xml:space="preserve">Quod oſtendere propoſitum fuerat; </s> <s xml:id="echoid-s8323" xml:space="preserve">quodque Cl. </s> <s xml:id="echoid-s8324" xml:space="preserve">Tor. </s> <s xml:id="echoid-s8325" xml:space="preserve">inter pro-<lb/>prios pugillares geometricos regerere non eſt dedignatus: </s> <s xml:id="echoid-s8326" xml:space="preserve">animo, vt opina-<lb/>ri libet, huiuſce haud iniucundi Theorematis, a me ipſi tantummodo expo-<lb/>ſiti demonſtrationem inquirendi, quam poſtea ſolùm de Coni portionibus <lb/>nactus fuit, vel potiùs circa ipſas tantùm placuit ei meditari: </s> <s xml:id="echoid-s8327" xml:space="preserve">eminentiſſimi <lb/>enim, ac propè diuini ingenij Vir, & </s> <s xml:id="echoid-s8328" xml:space="preserve">de aliorum ſolidorum portionibus fe-<lb/>liciùs quàm à nobis ſuperiùs factum ſit, hoc idem reperiſſet, ſi tantillùm ex-<lb/>cogitaſſet: </s> <s xml:id="echoid-s8329" xml:space="preserve">verùm proprias, ac ideò ſublimiores contem plationes affectans, <lb/>ab his nugis meis fortaſſe ſe abſtinuit.</s> <s xml:id="echoid-s8330" xml:space="preserve"/> </p> <div xml:id="echoid-div863" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0298-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0298-02" xlink:href="note-0298-02a" xml:space="preserve">55. h.</note> <note symbol="c" position="left" xlink:label="note-0298-03" xlink:href="note-0298-03a" xml:space="preserve">Coroll. <lb/>1. 78. h.</note> <note symbol="a" position="right" xlink:label="note-0299-01" xlink:href="note-0299-01a" xml:space="preserve">45. h.</note> <note symbol="b" position="right" xlink:label="note-0299-02" xlink:href="note-0299-02a" xml:space="preserve">78. h.</note> </div> </div> <div xml:id="echoid-div865" type="section" level="1" n="346"> <head xml:id="echoid-head355" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s8331" xml:space="preserve">Hlc autem animaduertendum eſt, quod nihil refert vtrùm baſes, huiuſ-<lb/>modi portionum ſolidarum inſcriptum ſolidum concentricum con-<lb/>tingant ad puncta eiuſdem ſectionis ſolidum genitricis, vel diuerſarum: </s> <s xml:id="echoid-s8332" xml:space="preserve">nam <lb/>omnes genitrices ſectiones eiuſdem ſolidi concentrici, ſe mutuò ſecant in <lb/>eodem vertice axis reuolutionis prædicti ſolidi; </s> <s xml:id="echoid-s8333" xml:space="preserve">ſed omnes portiones ſolidæ <lb/>exterioris, quæ quamlibet ſolidi interioris genitricem ſectionem per centra <lb/>earum baſium contingunt, æquales oſtendi poſſunt per ſuperiorem prop. </s> <s xml:id="echoid-s8334" xml:space="preserve">78. <lb/></s> <s xml:id="echoid-s8335" xml:space="preserve">eidem tertiæ portioni ſolidæ ab ipſo exteriori ſolido abſciſſæ, ei nempe, <lb/>cuius baſis tranſiens per axis verticem ad eundem axim ſit recta, circulum <lb/>in ſectione efficiens; </s> <s xml:id="echoid-s8336" xml:space="preserve">ergo omnes prædictæ portiones ſolidæ, vbicunque ea-<lb/>rum baſes contingant ſuperficiem ſimilis, & </s> <s xml:id="echoid-s8337" xml:space="preserve">concentrici inſcripti ſolidi, in-<lb/>ter ſe æquales erunt, cum tertiæ cuidam portioni ſint æquales, &</s> <s xml:id="echoid-s8338" xml:space="preserve">c.</s> <s xml:id="echoid-s8339" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div866" type="section" level="1" n="347"> <head xml:id="echoid-head356" xml:space="preserve">THEOR. LII. PROP. LXXXI.</head> <p> <s xml:id="echoid-s8340" xml:space="preserve">Si planum ductum per axem Coni recti, vel Conoidis Parabo-<lb/>lici, aut Hyperbolici, Sphæræ, aut Sphæroidis oblongi, vel pro-<lb/>lati à quadam recta linea ſecetur, per quam ductum ſit planum, <lb/>quod ad planum per axem rectum ſit: </s> <s xml:id="echoid-s8341" xml:space="preserve">ſolidi portio, quæ per hoc <lb/>planum abſcinditur, MINIMA eſt omnium portionum à quibuſ-<lb/>libet alijs planis per eandem rectam ductis abſciſſarum.</s> <s xml:id="echoid-s8342" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8343" xml:space="preserve">ESto quodlibet prædictorum ſolidorum A B C, cuius axis reuolutionis <lb/>ſit B D, & </s> <s xml:id="echoid-s8344" xml:space="preserve">planum per axem ductum ſit A B C vbicunque ſectum à <lb/>quadam recta A F, ad vtranque partem ſectioni occurrente, per quam con-<lb/>cipiatur duci planum A E F ad ipſum A B C rectum, portionem ex ſolido <pb o="114" file="0300" n="300" rhead=""/> abſcindens A B F, cuius baſis ſit A E F, & </s> <s xml:id="echoid-s8345" xml:space="preserve">Canon rectus A B F. </s> <s xml:id="echoid-s8346" xml:space="preserve">Dico <lb/>hanc ſolidam portionem _MINIMAM_ eſſe earum, quæ à quocunque alio <lb/>plano per eandem A F ducto ex dato ſolido abſcindi poſſunt.</s> <s xml:id="echoid-s8347" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8348" xml:space="preserve">Diuidatur A F bifariam in G, & </s> <s xml:id="echoid-s8349" xml:space="preserve">per G in plano per axem A B C de-<lb/>ſcribatur <anchor type="note" xlink:href="" symbol="a"/> in prima figura (exhibente Conum) Hyperbole G H I, cuius <anchor type="note" xlink:label="note-0300-01a" xlink:href="note-0300-01"/> aſymptoti ſint B A, B C; </s> <s xml:id="echoid-s8350" xml:space="preserve">in ſecunda verò (quodcunque aliorum ſolidorum <lb/>repræſentante) deſciibatur <anchor type="note" xlink:href="" symbol="b"/> coni-ſectio G H I ſimilis, & </s> <s xml:id="echoid-s8351" xml:space="preserve">concentrica ſe- <anchor type="note" xlink:label="note-0300-02a" xlink:href="note-0300-02"/> ctioni A B C, quæ in vtraque figura omnino continget rectam A F in G, <lb/>(nam ſi alia eſſet contingens per G ſectionem G H I, ipſa producta ad <lb/>vtranque partem exteriori ſectioni A B C occurreret, ac bifariam ſecare-<lb/>tur <anchor type="note" xlink:href="" symbol="c"/> in G: </s> <s xml:id="echoid-s8352" xml:space="preserve">vnde duæ applicate per G in ſectione A B C ſe mutuò bifariam <anchor type="note" xlink:label="note-0300-03a" xlink:href="note-0300-03"/> ſecarent, quod eſſet contra 26. </s> <s xml:id="echoid-s8353" xml:space="preserve">ſec. </s> <s xml:id="echoid-s8354" xml:space="preserve">Conic.</s> <s xml:id="echoid-s8355" xml:space="preserve">, quæ vnicuique coni-ſectioni <lb/>inſeruit, licet de ſola Ellipſi, vel Circulo agat, ſed hoc idem, & </s> <s xml:id="echoid-s8356" xml:space="preserve">pro angulo <lb/>ſimul, aliter patet, ex primo Coroll. </s> <s xml:id="echoid-s8357" xml:space="preserve">68. </s> <s xml:id="echoid-s8358" xml:space="preserve">h.) </s> <s xml:id="echoid-s8359" xml:space="preserve">& </s> <s xml:id="echoid-s8360" xml:space="preserve">concipiatur circa commu-<lb/>nem axim B D H ſectio G H I in gyrum conuersá: </s> <s xml:id="echoid-s8361" xml:space="preserve">patet hanc deſcribere <lb/>ſolidum G H I ſimile, & </s> <s xml:id="echoid-s8362" xml:space="preserve">concentricum exteriori A B C; </s> <s xml:id="echoid-s8363" xml:space="preserve">& </s> <s xml:id="echoid-s8364" xml:space="preserve">cum recta A F <lb/>contingat ſectionem G H I in G, & </s> <s xml:id="echoid-s8365" xml:space="preserve">per A F ductum ſit planum A E F ipſi <lb/>plano per axem G H I perpendiculare, hoc ipſum continget <anchor type="note" xlink:href="" symbol="d"/> concentrici <anchor type="note" xlink:label="note-0300-04a" xlink:href="note-0300-04"/> ſolidi ſuperficiem in puncto G.</s> <s xml:id="echoid-s8366" xml:space="preserve"/> </p> <div xml:id="echoid-div866" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0300-01" xlink:href="note-0300-01a" xml:space="preserve">4. ſec. <lb/>Conic.</note> <note symbol="b" position="left" xlink:label="note-0300-02" xlink:href="note-0300-02a" xml:space="preserve">5. 6. 7. <lb/>primi h.</note> <note symbol="c" position="left" xlink:label="note-0300-03" xlink:href="note-0300-03a" xml:space="preserve">Coroll. <lb/>45. h.</note> <note symbol="d" position="left" xlink:label="note-0300-04" xlink:href="note-0300-04a" xml:space="preserve">55. h.</note> </div> <figure> <image file="0300-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0300-01"/> </figure> <p> <s xml:id="echoid-s8367" xml:space="preserve">Ponatur primò punctum G eſſe extra axis verticem H, & </s> <s xml:id="echoid-s8368" xml:space="preserve">per G intelli-<lb/>gatur duci planum G L I ad axem erectum, quod in ſolido G H I circulum <lb/>efficiet <anchor type="note" xlink:href="" symbol="e"/> centrum habentem in axe, vt in D, & </s> <s xml:id="echoid-s8369" xml:space="preserve">cuius communis ſectio cum <anchor type="note" xlink:label="note-0300-05a" xlink:href="note-0300-05"/> plano per axem erit diameter G D I, cum alio autem plano A E F erit recta <lb/>E G, quæ cum ſit communis ſectio duorum planorum ad planum A B C <lb/>erectorum, erit ad idem planum <anchor type="note" xlink:href="" symbol="f"/> recta, ac ideo cum diametro G D I re- <anchor type="note" xlink:label="note-0300-06a" xlink:href="note-0300-06"/> ctum conſtituet angulum E G I, ſiue ipſa E G in puncto tantùm G circu-<lb/>lum continget.</s> <s xml:id="echoid-s8370" xml:space="preserve"/> </p> <div xml:id="echoid-div867" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-0300-05" xlink:href="note-0300-05a" xml:space="preserve">4. primi <lb/>Conic. & <lb/>12. Arch. <lb/>de Co-<lb/>noid. &c.</note> <note symbol="f" position="left" xlink:label="note-0300-06" xlink:href="note-0300-06a" xml:space="preserve">19. vnd. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s8371" xml:space="preserve">Iam intelligatur per A F aliud planum duci ad planum per axem A B C <lb/>non erectum (ſed tale quod de exteriori ſolido aliam terminatam ſectionem <lb/>abſcindat) cuius communis ſectio cum circuli plano diuerſa erit à linea G E <lb/>(planum enim nunc ductum conuenit cum plano A E F per rectam tantùm <lb/>A F.) </s> <s xml:id="echoid-s8372" xml:space="preserve">Sit ipſa G L. </s> <s xml:id="echoid-s8373" xml:space="preserve">Et quoniam G E rectos facit angulos cum G I, ipſa.</s> <s xml:id="echoid-s8374" xml:space="preserve"> <pb o="115" file="0301" n="301" rhead=""/> G L cum eadem G I haud rectos efficiet, vnde producta hinc inde ad alte-<lb/>ram partem cadet intra circulum G L I, eius peripheriæ occurrens in L. <lb/></s> <s xml:id="echoid-s8375" xml:space="preserve">Cum ergo G L ſit tota intra circulum, circulus verò totus intra ſolidum, <lb/>erit quoquè G L tota intra ſolidum: </s> <s xml:id="echoid-s8376" xml:space="preserve">quare planum, quod per A F, & </s> <s xml:id="echoid-s8377" xml:space="preserve">G L <lb/>ductum fuit, fecabit omnino interius ſolidum G H I, de quo aliquam ter-<lb/>minatam portionem abſcindet (cum idem planum vndique productum de <lb/>exteriori ſolido ponatur quoque portionem quandam auferre) cuius con-<lb/>uexa ſuperficies tota erit intra portionem exterioris ſolidi ab eodem plano <lb/>abſciſſam.</s> <s xml:id="echoid-s8378" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8379" xml:space="preserve">Si verò punctum G (quod nuper oſtẽſum fuit eſſe cõtactum plani per A F <lb/>ducti, ad planum per axem A B C recti, cum interioris ſolidi G H I ſuper-<lb/>ficie) fuerit in ipſo axis vertice H, vt in hac tertia figura, oſtendetur etiam <lb/>quodlibet aliud planum A L F per rectam A F ductum, ſed ad planum per <lb/>axem A B C inclinatum, quodque de exteriori ſolido aliquam portionem <lb/>abſcindat, omnino ſecare interius ſolidum, ideoque de ipſo quandam por-<lb/>tionem terminatam auferre.</s> <s xml:id="echoid-s8380" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8381" xml:space="preserve">Nam, in prædicto contingente plano A <lb/> <anchor type="figure" xlink:label="fig-0301-01a" xlink:href="fig-0301-01"/> E F, ducta per G quacumq; </s> <s xml:id="echoid-s8382" xml:space="preserve">recta G E cũ <lb/>G A quemlibet angulum conſtituente, & </s> <s xml:id="echoid-s8383" xml:space="preserve"><lb/>per rectam G E, ac per axim G D ducto <lb/>alio plano, id in interiori ſolido deſcribet <lb/>genitricem <anchor type="note" xlink:href="" symbol="a"/> ſectionem L G M, quam cõ- <anchor type="note" xlink:label="note-0301-01a" xlink:href="note-0301-01"/> tinget in G recta G E eorundem plano-<lb/>rum communis ſectio, cum hæc ponatur <lb/>eſſe in plano contingente vniuerſam ſolid <lb/>ſuperficiem, ſed planum inclinatum A L <lb/>F vndiq; </s> <s xml:id="echoid-s8384" xml:space="preserve">productum ad alteram partium, <lb/>vtputa ad E, cadit infra contingens pla-<lb/>num, cum eo commune habens tantùm <lb/>rectam A F, ergo & </s> <s xml:id="echoid-s8385" xml:space="preserve">communis ſectio ipſius plani inclinati cum ſectione L <lb/>G M, nempe recta G L cadet infra idem planum contingens, ac ideo infra <lb/>rectam G E; </s> <s xml:id="echoid-s8386" xml:space="preserve">ſed G L, & </s> <s xml:id="echoid-s8387" xml:space="preserve">G E ſunt in plano L G M, atque G E ipſam ſe-<lb/>ctionem contingit, vt modò oſtendimus, quare G L, quæ cadit infra G E <lb/>cadet omnino <anchor type="note" xlink:href="" symbol="b"/> intra ſectionem L G M, ſiue intra ſolidum, ac propterea <anchor type="note" xlink:label="note-0301-02a" xlink:href="note-0301-02"/> planum inclinatum, quod per A F, & </s> <s xml:id="echoid-s8388" xml:space="preserve">G L ducitur, ſecabit omnino interius <lb/>ſolidum, ac de ipſo quandam terminatam portionem auferet, cum idem <lb/>planum inclinatum ponatur de exteriori terminatam portionem abſcindere.</s> <s xml:id="echoid-s8389" xml:space="preserve"/> </p> <div xml:id="echoid-div868" type="float" level="2" n="3"> <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/QN4GHYBF/figures/0301-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">12. Ar-<lb/>chim. de <lb/>Conoid. <lb/>&c.</note> <note symbol="b" position="right" xlink:label="note-0301-02" xlink:href="note-0301-02a" xml:space="preserve">32. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s8390" xml:space="preserve">Itaque, cum in vtroque caſu demonſtratum ſit, planum inclinatum tran-<lb/>ſiens per A F, & </s> <s xml:id="echoid-s8391" xml:space="preserve">G L, de interiori ſolido G H I aliquam portionem ſecare, <lb/>poſſibile <anchor type="note" xlink:href="" symbol="c"/> erit ipſi plano, hoc eſt baſibus vtriuſque portionis, aliud planum <anchor type="note" xlink:label="note-0301-03a" xlink:href="note-0301-03"/> æquidiſtans ducere, quod interioris portionis ſuperficiem contingat: </s> <s xml:id="echoid-s8392" xml:space="preserve">quare <lb/>ſi mente concipiatur iam hoc ductum eſſe, ac vndique productum, patet hoc <lb/>ipſum planum contingens, de prædicta exteriori portione dempta à plano <lb/>per A F, & </s> <s xml:id="echoid-s8393" xml:space="preserve">G L ducto, aliam portionem abſcindere, ſed illa omninò mi-<lb/>norem (pars enim ſuo toto minor eſt) at hęc minor portio æqualis eſt <anchor type="note" xlink:href="" symbol="d"/> por- <anchor type="note" xlink:label="note-0301-04a" xlink:href="note-0301-04"/> tioni A B F abſciſſæ à plano, quod per A F ductum fuit ad planum per axem <lb/>A B C rectum (vtraque enim talium portionum terminatur à planis baſium, <pb o="116" file="0302" n="302" rhead=""/> eiuſdem ſimilis concentrici ſolidi ſuperſiciem contingentium) ergo, & </s> <s xml:id="echoid-s8394" xml:space="preserve">por-<lb/>tio A B F à prædicto plano recto abſciſſa, erit minor eadem portione, quæ <lb/>dempta fuit à plano per A F, & </s> <s xml:id="echoid-s8395" xml:space="preserve">G L ducto, ſiue à plano, quod in conſtru-<lb/>ctione per A F obliquè ductum fuit ſuper planum per axem A B C: </s> <s xml:id="echoid-s8396" xml:space="preserve">& </s> <s xml:id="echoid-s8397" xml:space="preserve">hoc <lb/>ſemper verum eſſe demonſtrabitur, quodcunque ſit planum inclinatum <lb/>tranſiens per A F; </s> <s xml:id="echoid-s8398" xml:space="preserve">ergo portio ſolida A B F, quæ ex dato ſolido à plano <lb/>per A F ducto, & </s> <s xml:id="echoid-s8399" xml:space="preserve">ad planum per axem A B C erecto abſcinditur, _M I N I-_ <lb/>_M A_ eſt omnium portionum à quibuslibet alijs planis per eandem A F du-<lb/>ctis abſciſſarum. </s> <s xml:id="echoid-s8400" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s8401" xml:space="preserve"/> </p> <div xml:id="echoid-div869" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-0301-03" xlink:href="note-0301-03a" xml:space="preserve">69. h.</note> <note symbol="d" position="right" xlink:label="note-0301-04" xlink:href="note-0301-04a" xml:space="preserve">ex Sch. <lb/>Prop. 80. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div871" type="section" level="1" n="348"> <head xml:id="echoid-head357" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s8402" xml:space="preserve">EX eo, quod prope finem huius demonſtratum eſt, elicitur, omnem por-<lb/>tionem cuiuſcunque prædictorum ſolidorum, cuius baſis ſecet ſimile <lb/>inſcriptum ſolidum concentricum, maiorem eſſe qualibet alia portione de <lb/>eodem exteriori ſolido, cuius baſis contingat idem inſeriptum ſolidum.</s> <s xml:id="echoid-s8403" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8404" xml:space="preserve">Ibienim oſtendimus prædictam exterioris ſolidi portionem, cuius baſis <lb/>ſecet inſcriptum ſolidum, maiorem eſſe ea, cuius baſis contingens idem in-<lb/>ſcriptum, ſimul ſit parallela ſecanti baſi; </s> <s xml:id="echoid-s8405" xml:space="preserve">ſed omnes portiones de eodem ſo-<lb/>lido, quarum baſes contingant idem ſimile inſcriptum concentricum, inter <lb/>ſe ſunt <anchor type="note" xlink:href="" symbol="a"/> æquales: </s> <s xml:id="echoid-s8406" xml:space="preserve">ergo patet propoſitum, &</s> <s xml:id="echoid-s8407" xml:space="preserve">c.</s> <s xml:id="echoid-s8408" xml:space="preserve"/> </p> <note symbol="a" position="left" xml:space="preserve">Propoſ. <lb/>80. h.</note> </div> <div xml:id="echoid-div872" type="section" level="1" n="349"> <head xml:id="echoid-head358" xml:space="preserve">PROBL. XV. PROP. LXXXII.</head> <p> <s xml:id="echoid-s8409" xml:space="preserve">Per datum punctum intra Conum rectum, vel Conoides Para-<lb/>bolicum, aut Hyperbolicum, ſiue Sphæram, aut Sphæroides ob-<lb/>longum, vel prolatum, planum ducere, quod de ſolido abſcindat <lb/>portionem MINIMAM; </s> <s xml:id="echoid-s8410" xml:space="preserve">atque in Sphæroide, vel Sphæra portio-<lb/>nem MAXIMAM.</s> <s xml:id="echoid-s8411" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8412" xml:space="preserve">ESto quodlibet prædictorum ſolidorum A B C, cuius axis reuolutionis <lb/>ſit B D, ac datum vbicunque intra ſolidum ſit punctum E: </s> <s xml:id="echoid-s8413" xml:space="preserve">oportet per <lb/>E planum ducere, quod ex dato ſolido abſcindat portionem _MINIMAM_, <lb/>atque ampliùs in Sphæroide, vel Sphæra, portionem _MAXIMAM_. </s> <s xml:id="echoid-s8414" xml:space="preserve">Opor-<lb/>tet autem ſi ſolidum fuerit Sphæroides, vel Sphæra, quod datum punctum <lb/>non ſit idem, ac centrum, tune enim neque _MAXIMA_, neque _MINIMA_ <lb/>portio exhiberi poſſet, cum omnia plana per centra eorum ſolidorum ducta <lb/>in duas æquas portiones diuidant ipſa ſolida; </s> <s xml:id="echoid-s8415" xml:space="preserve">veluti in Ellipſi, vel circulo <lb/>dum quærebatur _MAXIMA_, & </s> <s xml:id="echoid-s8416" xml:space="preserve">_MINIMA_ portio, neceſſe fuit datum pun-<lb/>ctum non eſſe in centro, cum rectæ omnes per ipſum ductæ, huiuſmodi ſu-<lb/>perficies bifariam ſecent, vt iam ſatis conſtat.</s> <s xml:id="echoid-s8417" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8418" xml:space="preserve">Secetur folidum plano per axem B D, ac per datum punctum E tran-<lb/>ſeunte, efficienteque in ſolido <anchor type="note" xlink:href="" symbol="b"/> genitricem ſectionem A B C, quæ indefi- <anchor type="note" xlink:label="note-0302-02a" xlink:href="note-0302-02"/> nitè producatur, ac de ipſa per idem punctum E, cum recta F E G abſcin- <pb o="117" file="0303" n="303" rhead=""/> datur <anchor type="note" xlink:href="" symbol="a"/> _MINIMA_ portio plana F B G, & </s> <s xml:id="echoid-s8419" xml:space="preserve">per eandem F E G agatur planum <anchor type="note" xlink:label="note-0303-01a" xlink:href="note-0303-01"/> F H G I, quod ad ductum per axem A B C rectum ſit. </s> <s xml:id="echoid-s8420" xml:space="preserve">Dico tale planum <lb/>F H G quæſitum ſoluere, ſiue de dato ſolido auferre portionem ſolidam F <lb/>B G _MINIMAM_ omnium, quæ ex eodem ſolido à quibuslibet alijs planis, <lb/>per idem punctum E ducibilibus, abſcindi poſſunt.</s> <s xml:id="echoid-s8421" xml:space="preserve"/> </p> <div xml:id="echoid-div872" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-0302-02" xlink:href="note-0302-02a" xml:space="preserve">12. Ar-<lb/>chim. de <lb/>Conoid. <lb/>&c.</note> <note symbol="a" position="right" xlink:label="note-0303-01" xlink:href="note-0303-01a" xml:space="preserve">41. 42. h.</note> </div> <p> <s xml:id="echoid-s8422" xml:space="preserve">Iam patet primò portionem F B G _MINIMAM_ eſſe <anchor type="note" xlink:href="" symbol="b"/> aliarum portionum <anchor type="note" xlink:label="note-0303-02a" xlink:href="note-0303-02"/> abſciſſarum à planis, tranſeuntibus quidem per rectam F G, ac ideo per <lb/>datum punctum E, non autem rectis ſuper planum per axem A B C. </s> <s xml:id="echoid-s8423" xml:space="preserve">Ve-<lb/>rùm quod ſit quoque _MINIMA_ abſcindendarum ab alijs planis non per re-<lb/>ctam F G, ſed omnino per punctum E ducibilibus, ſic demonſtrabitur.</s> <s xml:id="echoid-s8424" xml:space="preserve"/> </p> <div xml:id="echoid-div873" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0303-02" xlink:href="note-0303-02a" xml:space="preserve">81. h.</note> </div> <p> <s xml:id="echoid-s8425" xml:space="preserve">In plano enim per axem A B C deſcripta per punctum E (quod bifariam <lb/>ſecat applicatam F G, vti elicitur ex 41. </s> <s xml:id="echoid-s8426" xml:space="preserve">& </s> <s xml:id="echoid-s8427" xml:space="preserve">42. </s> <s xml:id="echoid-s8428" xml:space="preserve">huius) ſimili, & </s> <s xml:id="echoid-s8429" xml:space="preserve">concentri-<lb/>ca ſectione E L M; </s> <s xml:id="echoid-s8430" xml:space="preserve">ipſa rectam F G continget <anchor type="note" xlink:href="" symbol="c"/> in E: </s> <s xml:id="echoid-s8431" xml:space="preserve">& </s> <s xml:id="echoid-s8432" xml:space="preserve">facta reuolutione <anchor type="note" xlink:label="note-0303-03a" xlink:href="note-0303-03"/> ipſius ſectionis E L M circa eundem axim B D, deſcribetur ſimile concen-<lb/>tricum ſolidum, quod continget <anchor type="note" xlink:href="" symbol="d"/> planum F H G I in E; </s> <s xml:id="echoid-s8433" xml:space="preserve">itaque ducto per <anchor type="note" xlink:label="note-0303-04a" xlink:href="note-0303-04"/> datum punctum E quolibet alio plano non per F G tranſeunte, ſed neque <lb/>per axim B D; </s> <s xml:id="echoid-s8434" xml:space="preserve">(tunc enim planum hoc, datum ſolidum in duas partes diui-<lb/>deret, quarum vtra eſſet quidem maior portione F B G, quoniam vel eſſet <lb/>infinitæ magnitudinis, ſi datum ſolidum <lb/>fuerit Conus, vel Conoides, vel eſſet <lb/> <anchor type="figure" xlink:label="fig-0303-01a" xlink:href="fig-0303-01"/> ſolidi dimidium, ſi fuerit Sphæroides, <lb/>vel Sphæra, ac propterea omnino eſſet <lb/>maior portione F B G, quæ dimidio <lb/>occluſi ſolidi minor eſt, cum extra ip-<lb/>ſam ſit centrum; </s> <s xml:id="echoid-s8435" xml:space="preserve">nam centrum _MINI-_ <lb/>_MAE_ portionis planæ F B G, quod <lb/>idem eſt, ac centrum ſolidi, iam con-<lb/>ſtat eſſe extra ipſam portionem, quando <lb/>datum punctum E in ſectione ſit extra <lb/>centrum, vt ponitur) patet id iuxta <lb/>quandam rectam N E M C neceſſariò <lb/>ſecare planum per axem A B C, in quo <lb/>eſt punctum E. </s> <s xml:id="echoid-s8436" xml:space="preserve">Et quoniam F G ſectionem E L M contingit in E, recta <lb/>N C, quæ per E ponitur tranſire, omninò ſecabit interiorem ſectionem E <lb/>L M, ſiue per aliquam ſui partem, vt puta per E M, tota cadet intra ſe-<lb/>ctionem E L M; </s> <s xml:id="echoid-s8437" xml:space="preserve">ſed ſectio E L M tota eſt intra concentricum inſcriptum <lb/>ſolidum, cum ſit ducta per axem, quare, & </s> <s xml:id="echoid-s8438" xml:space="preserve">ipſa recta E M tota erit intra ſo-<lb/>lidum inſcriptum, vnde planum, quod modò per ipſam duximus, quodque <lb/>de exteriori aufert ſolidam portionem N B C, cuius baſis eſt N O C P, ſe-<lb/>cabit prorſus interius ſolidum, deque ipſo quandam ſolidam portionem <lb/>abſcindet, nimirum E L M, cuius baſis ſit E Q M R: </s> <s xml:id="echoid-s8439" xml:space="preserve">portio igitur N B C, <lb/>cuius baſis eſt N O C P interius ſolidum ſecans, maior erit <anchor type="note" xlink:href="" symbol="e"/> portione F B <anchor type="note" xlink:label="note-0303-05a" xlink:href="note-0303-05"/> G, cuius baſis eſt F H G I idem interius ſolidum contingens, & </s> <s xml:id="echoid-s8440" xml:space="preserve">hoc ſem-<lb/>per, quodcunque ſit planum tranſiens per datum punctum E præter pla-<lb/>num F H G I. </s> <s xml:id="echoid-s8441" xml:space="preserve">Quare ex dato ſolido A B C per datum punctum E abſciſſa <lb/>eſt _MINIMA_ portio F B G. </s> <s xml:id="echoid-s8442" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s8443" xml:space="preserve"/> </p> <div xml:id="echoid-div874" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0303-03" xlink:href="note-0303-03a" xml:space="preserve">1. Corol. <lb/>68. h.</note> <note symbol="d" position="right" xlink:label="note-0303-04" xlink:href="note-0303-04a" xml:space="preserve">55. h.</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/QN4GHYBF/figures/0303-01"/> </figure> <note symbol="e" position="right" xlink:label="note-0303-05" xlink:href="note-0303-05a" xml:space="preserve">Schol. <lb/>81. h.</note> </div> <pb o="118" file="0304" n="304" rhead=""/> </div> <div xml:id="echoid-div876" type="section" level="1" n="350"> <head xml:id="echoid-head359" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s8444" xml:space="preserve">SI datum ſolidum fuerit quodcunque Sphæroides, vel Sphæra; </s> <s xml:id="echoid-s8445" xml:space="preserve">patet re-<lb/>liquam portionem ſolidam, dempta _MINIMA_ nuper inuenta, eſſe <lb/>_MAXIMAM_ quæſitam.</s> <s xml:id="echoid-s8446" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div877" type="section" level="1" n="351"> <head xml:id="echoid-head360" xml:space="preserve">THEOR. LIII. PROP. LXXXIII.</head> <p> <s xml:id="echoid-s8447" xml:space="preserve">Æquales portiones ſolidæ eiſdem Conoidis, vel Sphæræ, aut <lb/>cuiuslibet Sphæroidis, ſi fuerint de eodem Conoide Parabolico <lb/> <anchor type="note" xlink:label="note-0304-01a" xlink:href="note-0304-01"/> habebunt axes æquales. </s> <s xml:id="echoid-s8448" xml:space="preserve">Si de eodem Hyperbolico, vel de Sphæ-<lb/>ra, aut Sphæroide quocunque, erunt axes proprijs ſemi- diametris <lb/>proportionales. </s> <s xml:id="echoid-s8449" xml:space="preserve">At ſi fuerint de eodem Cono recto, extrema ip-<lb/>ſorum axium pertingent ad idem inſcriptum ſolidum ſimile, & </s> <s xml:id="echoid-s8450" xml:space="preserve"><lb/>concentricum.</s> <s xml:id="echoid-s8451" xml:space="preserve"/> </p> <div xml:id="echoid-div877" type="float" level="2" n="1"> <note position="left" xlink:label="note-0304-01" xlink:href="note-0304-01a" xml:space="preserve">Conuer-<lb/>ſum Pro-<lb/>p. 79. h.</note> </div> <p> <s xml:id="echoid-s8452" xml:space="preserve">SInt duæ de eodem quocunque prædictorum ſolidorum portiones æqua-<lb/>les, quarum recti Canones concipiantur transferri ſuper eadem ſectio-<lb/>ne A B F per ſolidi axem ducta (hoc enim fieri poſſe manifeſtum eſt, cum <lb/>ipſi recti Canones intra ſolidas portiones intercepti, ſint portiones eiuſdem <lb/>ſectionis, quæ in reuolutione circa axim ſolidum genuit) & </s> <s xml:id="echoid-s8453" xml:space="preserve">ſint A B C, D <lb/>E F, quarum baſes ſint A C, D F, & </s> <s xml:id="echoid-s8454" xml:space="preserve">diametri B G, E H, quæ ſimul ſunt <lb/> <anchor type="figure" xlink:label="fig-0304-01a" xlink:href="fig-0304-01"/> axes ſolidarum <anchor type="note" xlink:href="" symbol="a"/> portionum. </s> <s xml:id="echoid-s8455" xml:space="preserve">Dico, in prima figura exhibente Conoides <anchor type="note" xlink:label="note-0304-02a" xlink:href="note-0304-02"/> Parabolicum, axes B G, E H eſſe inter ſe æquales, & </s> <s xml:id="echoid-s8456" xml:space="preserve">in ſecunda exhibente <lb/>Hyperbolicum, atque in tertia Sphæram, vel Sphæroides, quarum centra <lb/>ſint O, eſſe axim H E ad ſemi-diametrum E O, vt axis G B ad ſemi-dia-<lb/>metrum B O.</s> <s xml:id="echoid-s8457" xml:space="preserve"/> </p> <div xml:id="echoid-div878" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0304-01"/> </figure> <note symbol="a" position="left" xlink:label="note-0304-02" xlink:href="note-0304-02a" xml:space="preserve">3. Schol. <lb/>69. h.</note> </div> <p> <s xml:id="echoid-s8458" xml:space="preserve">Ex altero axium, videlicet ex E H, ſecetur in prima figura ſegmentum <lb/>E I ipſi B G æquale; </s> <s xml:id="echoid-s8459" xml:space="preserve">& </s> <s xml:id="echoid-s8460" xml:space="preserve">in reliquis, fiat O E ad E I, vt O B ad B G, in <pb o="119" file="0305" n="305" rhead=""/> omnibus verò per I applicetur ordinatim ad E I recta L I M, quæ rectæ D <lb/>F æquidiſtabit, & </s> <s xml:id="echoid-s8461" xml:space="preserve">per ipſam L I M concipiatur duci planum, quod plano <lb/>per D F tranſeunti, ſiue baſi portionis ſolidæ D E F æquidiftet, aliam por-<lb/>tionem ſolidam abſcindens L E M, quæ portioni ſolidæ A B C <anchor type="note" xlink:href="" symbol="a"/> æqualis <anchor type="note" xlink:label="note-0305-01a" xlink:href="note-0305-01"/> erit; </s> <s xml:id="echoid-s8462" xml:space="preserve">ſed ponitur etiam D E F eidem A B C æqualis; </s> <s xml:id="echoid-s8463" xml:space="preserve">ergo duæ L E M, D <lb/>E F inter ſe æquales erunt, ſed vtraque eſt de eodem ſolido, circa commu-<lb/>nem axim E H I, & </s> <s xml:id="echoid-s8464" xml:space="preserve">ſuper baſes parallelas, quare planum baſis ductum per <lb/>L M, congruet cum plano baſis, quod tranſit per D F, vnde, & </s> <s xml:id="echoid-s8465" xml:space="preserve">axis termi-<lb/>nus I, cum termino axis H. </s> <s xml:id="echoid-s8466" xml:space="preserve">Erit ergo axis E I æqualis axi E H. </s> <s xml:id="echoid-s8467" xml:space="preserve">Sed in <lb/>prima, factus fuit E I æqualis B G, & </s> <s xml:id="echoid-s8468" xml:space="preserve">in reliquis O E ad E I, vt O B ad <lb/>B G, quare axis quoque E H, in prima, æquabitur axi B G, in alijs verò <lb/>erit O E ad E H, vt O B ad B G, & </s> <s xml:id="echoid-s8469" xml:space="preserve">conuertendo H E ad E O, vt G B <lb/>ad B O.</s> <s xml:id="echoid-s8470" xml:space="preserve"/> </p> <div xml:id="echoid-div879" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0305-01" xlink:href="note-0305-01a" xml:space="preserve">79. h.</note> </div> <p> <s xml:id="echoid-s8471" xml:space="preserve">Sint tandem duæ æquales portiones de eodem Cono recto A B C, D B <lb/>E, quarum recti Canones concipiantur coaptari ſuper eadem ſectione A B <lb/>E per ſolidi axem ducta, & </s> <s xml:id="echoid-s8472" xml:space="preserve">ſint A B C, D B E, quarum baſes A C, D E, <lb/>& </s> <s xml:id="echoid-s8473" xml:space="preserve">diametri B F, B G, (quæ iam ſunt <anchor type="note" xlink:href="" symbol="b"/> axes ſolidarum portionum.) </s> <s xml:id="echoid-s8474" xml:space="preserve">Et per <anchor type="note" xlink:label="note-0305-02a" xlink:href="note-0305-02"/> F cum aſymptotis B A, B C deſcribatur Hyperbole F G; </s> <s xml:id="echoid-s8475" xml:space="preserve">quæ omnino <lb/>continget <anchor type="note" xlink:href="" symbol="c"/> A C in F, termino axis B F. </s> <s xml:id="echoid-s8476" xml:space="preserve">Dico iam extremum G axis B <anchor type="note" xlink:label="note-0305-03a" xlink:href="note-0305-03"/> G, ad eandem quoque ſectionem pertin-<lb/>gere: </s> <s xml:id="echoid-s8477" xml:space="preserve">hoc eſt ſectionem F G ſecare dia-<lb/> <anchor type="figure" xlink:label="fig-0305-01a" xlink:href="fig-0305-01"/> metrum B G in puncto G. </s> <s xml:id="echoid-s8478" xml:space="preserve">Si poffibile <lb/>eſt ſectio F G alibi ſecet axim B G, vt in-<lb/>fra G in puncto H, & </s> <s xml:id="echoid-s8479" xml:space="preserve">per H ducatur L <lb/>H M ipſi D E æquidiſtans: </s> <s xml:id="echoid-s8480" xml:space="preserve">erit D G ad <lb/>G E, vt L H ad H M, eſtque D G ęqua-<lb/>lis G E, quare L M quoque bifariam ſe-<lb/>cta erit in H: </s> <s xml:id="echoid-s8481" xml:space="preserve">ſed dicitur per H tranſire <lb/>ſectionem, ergo L M ipfam <anchor type="note" xlink:href="" symbol="d"/> continget <anchor type="note" xlink:label="note-0305-04a" xlink:href="note-0305-04"/> in H, quapropter portio plana L B M <lb/>æquabitur <anchor type="note" xlink:href="" symbol="e"/> portioni A B C, & </s> <s xml:id="echoid-s8482" xml:space="preserve">ſi per L <anchor type="note" xlink:label="note-0305-05a" xlink:href="note-0305-05"/> M agatur planum ſecans Conum, & </s> <s xml:id="echoid-s8483" xml:space="preserve">ad planum L B M rectum, quod & </s> <s xml:id="echoid-s8484" xml:space="preserve"><lb/>plano datæ portionis ſolidæ D B E per D E ductum æquidiſtabit, cum hoc <lb/>ad idem planum L B M ponatur rectum eſſe; </s> <s xml:id="echoid-s8485" xml:space="preserve">erit ſolida portio L B M ęqua-<lb/>lis <anchor type="note" xlink:href="" symbol="f"/> portioni A B C, cum earum recti Canones L B M, A B C æquales <anchor type="note" xlink:label="note-0305-06a" xlink:href="note-0305-06"/> ſint oſtenſi; </s> <s xml:id="echoid-s8486" xml:space="preserve">ſed D B E quoque eidem A B C data eſt æqualis, ergo duæ <lb/>portiones L B M, D B E ſimul æquales erunt, totum ſuæ parti, quod eſt <lb/>abſurdum: </s> <s xml:id="echoid-s8487" xml:space="preserve">non ergo ſectio F G ſecat axim B G infra H; </s> <s xml:id="echoid-s8488" xml:space="preserve">& </s> <s xml:id="echoid-s8489" xml:space="preserve">ob eandem ra-<lb/>tionem neque ſupra; </s> <s xml:id="echoid-s8490" xml:space="preserve">ergo ſectio F G omnino tranſibit per G extremum <lb/>axis B G: </s> <s xml:id="echoid-s8491" xml:space="preserve">ſed facta reuolutione anguli, ac ſectionis circa communem axim <lb/>procreatur Conus, & </s> <s xml:id="echoid-s8492" xml:space="preserve">Conoides Hyperbolicum ſimile, ac concentricum: <lb/></s> <s xml:id="echoid-s8493" xml:space="preserve">ergo F, G, extrema puncta axium æqualium portionum ſolidarum A B C, <lb/>D B E, ex eodem Cono recto, pertingunt ad idem Conoides Hyperboli-<lb/>cum ſimile, & </s> <s xml:id="echoid-s8494" xml:space="preserve">concentricum inſcriptum. </s> <s xml:id="echoid-s8495" xml:space="preserve">Quod vltimò demonſtrandum <lb/>erat.</s> <s xml:id="echoid-s8496" xml:space="preserve"/> </p> <div xml:id="echoid-div880" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-0305-02" xlink:href="note-0305-02a" xml:space="preserve">3. Schol. <lb/>69. h.</note> <note symbol="c" position="right" xlink:label="note-0305-03" xlink:href="note-0305-03a" xml:space="preserve">1. Co-<lb/>roll. 68. h.</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/QN4GHYBF/figures/0305-01"/> </figure> <note symbol="d" position="right" xlink:label="note-0305-04" xlink:href="note-0305-04a" xml:space="preserve">ibidem.</note> <note symbol="e" position="right" xlink:label="note-0305-05" xlink:href="note-0305-05a" xml:space="preserve">45. h.</note> <note symbol="f" position="right" xlink:label="note-0305-06" xlink:href="note-0305-06a" xml:space="preserve">78. h.</note> </div> <pb o="120" file="0306" n="306" rhead=""/> </div> <div xml:id="echoid-div882" type="section" level="1" n="352"> <head xml:id="echoid-head361" xml:space="preserve">THEOR. LIV. PROP. LXXXIV.</head> <p> <s xml:id="echoid-s8497" xml:space="preserve">Æquales portiones de eodem ſolido, quodcunque ſit ex ſæ-<lb/> <anchor type="note" xlink:label="note-0306-01a" xlink:href="note-0306-01"/> pius memoratis, habent Canones rectos, in ipſis interceptos, <lb/>inter ſe æquales.</s> <s xml:id="echoid-s8498" xml:space="preserve"/> </p> <div xml:id="echoid-div882" type="float" level="2" n="1"> <note position="left" xlink:label="note-0306-01" xlink:href="note-0306-01a" xml:space="preserve">Conuerſ. <lb/>Prop. 78. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s8499" xml:space="preserve">ETenim huiuſmodi portiones ſolidæ æquales, habent axes, <anchor type="note" xlink:href="" symbol="a"/> vel inter <anchor type="note" xlink:label="note-0306-02a" xlink:href="note-0306-02"/> ſe æquales, vel proprijs ſemi-diametris proportionales, vel ad idem <lb/>Conoides ſimile concentricum, & </s> <s xml:id="echoid-s8500" xml:space="preserve">inſcriptum pertingentes, ſed ijdem <lb/>axes ſunt quoque <anchor type="note" xlink:href="" symbol="b"/> diametri prædictorum Canonum, & </s> <s xml:id="echoid-s8501" xml:space="preserve">quando hæ dia- <anchor type="note" xlink:label="note-0306-03a" xlink:href="note-0306-03"/> metri habuerint conditiones huiuſmodi, ipſi Canones recti ſunt <anchor type="note" xlink:href="" symbol="c"/> æqua- les, ergo ſolidæ portiones æquales, habebunt rectos Canones inter ſe <lb/> <anchor type="note" xlink:label="note-0306-04a" xlink:href="note-0306-04"/> æquales. </s> <s xml:id="echoid-s8502" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s8503" xml:space="preserve">c.</s> <s xml:id="echoid-s8504" xml:space="preserve"/> </p> <div xml:id="echoid-div883" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-0306-02" xlink:href="note-0306-02a" xml:space="preserve">83. h.</note> <note symbol="b" position="left" xlink:label="note-0306-03" xlink:href="note-0306-03a" xml:space="preserve">3. Schol. <lb/>prop. 69. <lb/>huius.</note> <note symbol="c" position="left" xlink:label="note-0306-04" xlink:href="note-0306-04a" xml:space="preserve">40. h. & <lb/>ex 45. h.</note> </div> </div> <div xml:id="echoid-div885" type="section" level="1" n="353"> <head xml:id="echoid-head362" xml:space="preserve">THEOR. LV. PROP. LXXXV.</head> <p> <s xml:id="echoid-s8505" xml:space="preserve">Baſes æqualium portionum ex eodem quocunque prædicto-<lb/>rum ſolidorum, ſuperſiciem eiuſdem ſimilis inſcripti ſolidi con-<lb/> <anchor type="note" xlink:label="note-0306-05a" xlink:href="note-0306-05"/> centrici ad earum centra contingunt.</s> <s xml:id="echoid-s8506" xml:space="preserve"/> </p> <div xml:id="echoid-div885" type="float" level="2" n="1"> <note position="left" xlink:label="note-0306-05" xlink:href="note-0306-05a" xml:space="preserve">Conuerſ. <lb/>Prop. 80. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s8507" xml:space="preserve">POrtiones enim æquales eiuſdem ſolidi habent rectos Canones in ipſis <lb/>interceptos inter ſe <anchor type="note" xlink:href="" symbol="d"/> æquales, ſed quando huiuſmodi Canones ſunt æquales (ſi concipiantur translati ſuper eandem ſectionem ſolidi geni-<lb/> <anchor type="note" xlink:label="note-0306-06a" xlink:href="note-0306-06"/> tricem) ipſorum baſes ad puncta media, eandem concentricam, inſcri-<lb/>ptam, & </s> <s xml:id="echoid-s8508" xml:space="preserve">ſimilem ſectionem <anchor type="note" xlink:href="" symbol="e"/> contingunt, & </s> <s xml:id="echoid-s8509" xml:space="preserve">baſes ſolidarum portionum <anchor type="note" xlink:label="note-0306-07a" xlink:href="note-0306-07"/> tranſeunt per has baſes rectorum Canonum, atque ad eos ſunt erectæ, nem-<lb/>pe ad planum per axem dati ſolidi, quare eędem baſes ſolidarum portionum <lb/>contingent ſuperſiciem interioris ſolidi concentrici ab inſcripta concentri-<lb/>ca ſectioni geniti (dum hæc circa axim conuertatur) ad eadem puncta, <anchor type="note" xlink:href="" symbol="f"/> in <anchor type="note" xlink:label="note-0306-08a" xlink:href="note-0306-08"/> quibus baſes planarum, ſectionem interiorem contingunt, quę puncta ſunt <lb/>centra axium, vel baſium ſolidarum portionum ex Archimede, & </s> <s xml:id="echoid-s8510" xml:space="preserve">ex iam à <lb/>nobis animaduerſis.</s> <s xml:id="echoid-s8511" xml:space="preserve"/> </p> <div xml:id="echoid-div886" type="float" level="2" n="2"> <note symbol="d" position="left" xlink:label="note-0306-06" xlink:href="note-0306-06a" xml:space="preserve">84. h.</note> <note symbol="e" position="left" xlink:label="note-0306-07" xlink:href="note-0306-07a" xml:space="preserve">68. h.</note> <note symbol="f" position="left" xlink:label="note-0306-08" xlink:href="note-0306-08a" xml:space="preserve">55. h.</note> </div> </div> <div xml:id="echoid-div888" type="section" level="1" n="354"> <head xml:id="echoid-head363" xml:space="preserve">THEOR. LVI. PROP. LXXXVI.</head> <p> <s xml:id="echoid-s8512" xml:space="preserve">Solidæ portiones eiuſdem Coni recti, vel Conoidis, ſiue Sphæ-<lb/>ræ, aut Sphæroidis, quarum axes (pro Cono recto) pertingant ad <lb/>idem inſcriptum concentricum Conoides Hyperbolicum (vel pro <lb/>Conoide Parabolico) ſint æquales (ſiue pro reliquis) ad proprias <lb/>ſemi - diametros eandem habeant rationem, habent baſes altitu-<lb/>dinibus reciprocè proportionales.</s> <s xml:id="echoid-s8513" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8514" xml:space="preserve">Eſto vt ponitur dico, &</s> <s xml:id="echoid-s8515" xml:space="preserve">c.</s> <s xml:id="echoid-s8516" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8517" xml:space="preserve">ETenim cum axibus huiuſmodi ſolidarum portionum inſint prædictæ <lb/>conditiones, ipſæ portiones ſolidæ æquales <anchor type="note" xlink:href="" symbol="g"/> erunt, pariterque earum <anchor type="note" xlink:label="note-0306-09a" xlink:href="note-0306-09"/> <pb o="121" file="0307" n="307" rhead=""/> recti Canones erunt <anchor type="note" xlink:href="" symbol="a"/> æquales (eo quod ijdem ſint <anchor type="note" xlink:href="" symbol="b"/> axes ſolidarum, &</s> <s xml:id="echoid-s8518" xml:space="preserve"> <anchor type="note" xlink:label="note-0307-01a" xlink:href="note-0307-01"/> diametri Canonum) ac propterea ipſorum baſes altitudinibus erunt <anchor type="note" xlink:href="" symbol="c"/> reci- <anchor type="note" xlink:label="note-0307-02a" xlink:href="note-0307-02"/> procè proportionales, ſed in æqualibus portionibus de eodem ſolido, vt <lb/> <anchor type="note" xlink:label="note-0307-03a" xlink:href="note-0307-03"/> ſunt baſes rectorum Canonum ita ſunt <anchor type="note" xlink:href="" symbol="d"/> baſes ſolidarum portionum, & </s> <s xml:id="echoid-s8519" xml:space="preserve">al- <anchor type="note" xlink:label="note-0307-04a" xlink:href="note-0307-04"/> titudines tùm portionum, tùm Canonum ſunt <anchor type="note" xlink:href="" symbol="e"/> eædem, ergo in datis por- <anchor type="note" xlink:label="note-0307-05a" xlink:href="note-0307-05"/> tionibus, quibus inſunt prædictæ conditiones, erunt quoque baſes altitudi-<lb/>nibus reciprocè proportionales. </s> <s xml:id="echoid-s8520" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s8521" xml:space="preserve">c.</s> <s xml:id="echoid-s8522" xml:space="preserve"/> </p> <div xml:id="echoid-div888" type="float" level="2" n="1"> <note symbol="g" position="left" xlink:label="note-0306-09" xlink:href="note-0306-09a" xml:space="preserve">Prop. 79. <lb/>huius.</note> <note symbol="a" position="right" xlink:label="note-0307-01" xlink:href="note-0307-01a" xml:space="preserve">84. h.</note> <note symbol="b" position="right" xlink:label="note-0307-02" xlink:href="note-0307-02a" xml:space="preserve">3. Schol. <lb/>69. h.</note> <note symbol="c" position="right" xlink:label="note-0307-03" xlink:href="note-0307-03a" xml:space="preserve">65. h.</note> <note symbol="d" position="right" xlink:label="note-0307-04" xlink:href="note-0307-04a" xml:space="preserve">2. Co-<lb/>roll. 78. h.</note> <note symbol="e" position="right" xlink:label="note-0307-05" xlink:href="note-0307-05a" xml:space="preserve">3. Schol. <lb/>69. h.</note> </div> </div> <div xml:id="echoid-div890" type="section" level="1" n="355"> <head xml:id="echoid-head364" xml:space="preserve">THEOR. LVII. PROP. LXXXVII.</head> <p> <s xml:id="echoid-s8523" xml:space="preserve">Æquales portiones ſolidæ de eodem Conoide, vel Sphæra, aut <lb/>quocunque Sphæroide, vel etiam de Cono recto, habent baſes al-<lb/>titudinibus reciprocè proportionales: </s> <s xml:id="echoid-s8524" xml:space="preserve">& </s> <s xml:id="echoid-s8525" xml:space="preserve">è conuerſo.</s> <s xml:id="echoid-s8526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8527" xml:space="preserve">Si baſes portionum de eodem ſolido fuerint altitudinibus reci-<lb/>procè proportionales, ipſæ portiones æquales erunt.</s> <s xml:id="echoid-s8528" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8529" xml:space="preserve">QVando enim huiuſmodi portiones ſolidæ ſunt æquales, neceſſariò ea-<lb/>rum axes (ſi portiones fuerint de eodem Conoide Parabolico) erunt <lb/>æquales (ſi de eodem Hyperbolico, aut Sphæra, aut Sphæ-<lb/>roide) erunt <anchor type="note" xlink:href="" symbol="f"/> proprijs ſemi - diametris proportionales; </s> <s xml:id="echoid-s8530" xml:space="preserve">ſed in his caſibus <anchor type="note" xlink:label="note-0307-06a" xlink:href="note-0307-06"/> eædem portiones ſolidæ habent <anchor type="note" xlink:href="" symbol="g"/> baſes altitudinibus proportionales, ergo, <anchor type="note" xlink:label="note-0307-07a" xlink:href="note-0307-07"/> & </s> <s xml:id="echoid-s8531" xml:space="preserve">cum portiones de eodem quocunque prædictorum ſolidorum fuerint <lb/>æquales, ipſarum baſes altitudinibus reciprocabuntur.</s> <s xml:id="echoid-s8532" xml:space="preserve"/> </p> <div xml:id="echoid-div890" type="float" level="2" n="1"> <note symbol="f" position="right" xlink:label="note-0307-06" xlink:href="note-0307-06a" xml:space="preserve">83. h.</note> <note symbol="g" position="right" xlink:label="note-0307-07" xlink:href="note-0307-07a" xml:space="preserve">86. h.</note> </div> <p> <s xml:id="echoid-s8533" xml:space="preserve">De portionibus autem æqualibus eiuſdem, vel etiam diuerſi Coni recti, <lb/>aut obliqui, iam id oſtenſum fuit à Commandino in Comment. </s> <s xml:id="echoid-s8534" xml:space="preserve">ſuper Ar-<lb/>chim. </s> <s xml:id="echoid-s8535" xml:space="preserve">de Conoid. </s> <s xml:id="echoid-s8536" xml:space="preserve">&</s> <s xml:id="echoid-s8537" xml:space="preserve">c. </s> <s xml:id="echoid-s8538" xml:space="preserve">Quod erat primò, &</s> <s xml:id="echoid-s8539" xml:space="preserve">c.</s> <s xml:id="echoid-s8540" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8541" xml:space="preserve">PRæterea ſint duæ ſolidæ portiones A B C, D E F de eodem ſolido, <lb/>quodcunque ſit ex prædictis (quæ tamen in Sphæroide non excedant <lb/>eius dimidium) quarum axes ſint B G, E H, & </s> <s xml:id="echoid-s8542" xml:space="preserve">baſes A I C, D K F, alti-<lb/>tudines verò B L, E M, & </s> <s xml:id="echoid-s8543" xml:space="preserve">ſit <lb/>baſis A I C ad D K F reci-<lb/> <anchor type="figure" xlink:label="fig-0307-01a" xlink:href="fig-0307-01"/> procè, vt altitudo E M ad B <lb/>L. </s> <s xml:id="echoid-s8544" xml:space="preserve">Dico has portiones inter <lb/>ſe æquales eſſe.</s> <s xml:id="echoid-s8545" xml:space="preserve"/> </p> <div xml:id="echoid-div891" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0307-01"/> </figure> </div> <p> <s xml:id="echoid-s8546" xml:space="preserve">Concipiantur ipſarum ſoli-<lb/>darum portionum recti Ca-<lb/>nones A B C, D E F, quo-<lb/>rum diametri, & </s> <s xml:id="echoid-s8547" xml:space="preserve">altitudines <lb/>eædem <anchor type="note" xlink:href="" symbol="h"/> erunt atque axes, &</s> <s xml:id="echoid-s8548" xml:space="preserve"> <anchor type="note" xlink:label="note-0307-08a" xlink:href="note-0307-08"/> altitudines ſolidarum portio-<lb/>num.</s> <s xml:id="echoid-s8549" xml:space="preserve"/> </p> <div xml:id="echoid-div892" type="float" level="2" n="3"> <note symbol="h" position="right" xlink:label="note-0307-08" xlink:href="note-0307-08a" xml:space="preserve">3. Schol. <lb/>69. h.</note> </div> <p> <s xml:id="echoid-s8550" xml:space="preserve">Iam, ſi huiuſmodi Cano-<lb/>nes ſunt æquales, & </s> <s xml:id="echoid-s8551" xml:space="preserve">portiones ſolidæ æquales <anchor type="note" xlink:href="" symbol="i"/> erunt. </s> <s xml:id="echoid-s8552" xml:space="preserve">At ſi dicatur eos <anchor type="note" xlink:label="note-0307-09a" xlink:href="note-0307-09"/> inæquales eſſet alter ipſorum, vt puta A B C, altero D E F maior erit: </s> <s xml:id="echoid-s8553" xml:space="preserve">vnde <pb o="122" file="0308" n="308" rhead=""/> & </s> <s xml:id="echoid-s8554" xml:space="preserve">diameter B G erit æquo maior: </s> <s xml:id="echoid-s8555" xml:space="preserve">ſi igitur ipſa ad æquum reducatur in N, <lb/>ita vt, vel B N ſit æqualis ipſi E H, (dum ſolidum ſuerit Conoides Parabo-<lb/>licum,) vel ita vt B N, & </s> <s xml:id="echoid-s8556" xml:space="preserve">E H ad proprias ſemi- diametros ſint in eadem ra-<lb/>tione,) dum ſolidum ſuerit Hyperbolicum, vel Sphæra, aut Sphæroides;) </s> <s xml:id="echoid-s8557" xml:space="preserve">vel <lb/>ita vt eędem pertingant ad eandẽ ſimilem concentricam ſectionem inſcriptã; <lb/></s> <s xml:id="echoid-s8558" xml:space="preserve">erit B N omnino minor B G, & </s> <s xml:id="echoid-s8559" xml:space="preserve">ſi per N agatur ipſi A C ęquidiſtans O N P, <lb/>quę ad eandem diametrum B G erit ordinatim ducta, atq; </s> <s xml:id="echoid-s8560" xml:space="preserve">minor ipſa A C, <lb/>ſiet portio, ſeu Canon O B P æqualis <anchor type="note" xlink:href="" symbol="a"/> portioni, ſiue Canoni D E F, & </s> <s xml:id="echoid-s8561" xml:space="preserve">ipſa <anchor type="note" xlink:label="note-0308-01a" xlink:href="note-0308-01"/> O P ſecabit B L in R, eritque B R altitudo Canonis O B P, cum ob paral-<lb/>lelas ſit angulus B R N rectus: </s> <s xml:id="echoid-s8562" xml:space="preserve">& </s> <s xml:id="echoid-s8563" xml:space="preserve">ſi per rectam O P ducatur planum O Q P, <lb/>quod baſi A I C ſit parallelum, ſiue rectum ad Canonem A B C, id abſcin-<lb/>det ex dato ſolido portionem <lb/>O B P, cuius altitudo erit B <lb/> <anchor type="figure" xlink:label="fig-0308-01a" xlink:href="fig-0308-01"/> R eadem atque Canonis O B <lb/>P. </s> <s xml:id="echoid-s8564" xml:space="preserve">Cumque Canon O B P <lb/>æqualis ſit Canoni D E F, <lb/>erit ſolida portio O B P <anchor type="note" xlink:href="" symbol="b"/> æ- <anchor type="note" xlink:label="note-0308-02a" xlink:href="note-0308-02"/> qualis ſolidæ portioni D E F, <lb/>ac ideo vt baſis O Q P ad ba-<lb/>ſim D K F, ita <anchor type="note" xlink:href="" symbol="c"/> reciprocè al- <anchor type="note" xlink:label="note-0308-03a" xlink:href="note-0308-03"/> titudo E M ad altitudinem B <lb/>R, eſtque baſis D K F ad ba-<lb/>ſim A I C, ex hypotheſi, vt <lb/>altitudo B L ad altitudinem <lb/>E M, quare, ex æquali in ratione perturbata, erit baſis O Q P ad baſim A <lb/>I C, vt altitudo B L ad altitudinem B R, ſed eſt B L maior B R, ergo & </s> <s xml:id="echoid-s8565" xml:space="preserve"><lb/>baſis O Q P maior erit baſi A I C, quod eſt falſum, cum ſit minor, eò quod <lb/>O P diameter Ellipſis, aut circuli O Q P minor ſit homologa diametro A C <lb/>ſimilis <anchor type="note" xlink:href="" symbol="d"/> Ellipſis, vel circuli A I C. </s> <s xml:id="echoid-s8566" xml:space="preserve">Non erit ergo Canonum A B C, D E <anchor type="note" xlink:label="note-0308-04a" xlink:href="note-0308-04"/> F alter altero maior, quare inter ſe æquales eſſe neceſſe eſt: </s> <s xml:id="echoid-s8567" xml:space="preserve">ideoque, & </s> <s xml:id="echoid-s8568" xml:space="preserve"><lb/>portiones ſolidæ A B C, D E F ęquales <anchor type="note" xlink:href="" symbol="e"/> erunt. </s> <s xml:id="echoid-s8569" xml:space="preserve">Quod ſecundò oſtendere <anchor type="note" xlink:label="note-0308-05a" xlink:href="note-0308-05"/> propoſitum ſuit.</s> <s xml:id="echoid-s8570" xml:space="preserve"/> </p> <div xml:id="echoid-div893" type="float" level="2" n="4"> <note symbol="i" position="right" xlink:label="note-0307-09" xlink:href="note-0307-09a" xml:space="preserve">78. h.</note> <note symbol="a" position="left" xlink:label="note-0308-01" xlink:href="note-0308-01a" xml:space="preserve">40. h. & <lb/>ex 45. h.</note> <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/QN4GHYBF/figures/0308-01"/> </figure> <note symbol="b" position="left" xlink:label="note-0308-02" xlink:href="note-0308-02a" xml:space="preserve">78. h.</note> <note symbol="c" position="left" xlink:label="note-0308-03" xlink:href="note-0308-03a" xml:space="preserve">ex pri-<lb/>ma parte <lb/>huius.</note> <note symbol="d" position="left" xlink:label="note-0308-04" xlink:href="note-0308-04a" xml:space="preserve">Coroll. <lb/>15. Arch. <lb/>de Co-<lb/>noid. &c.</note> <note symbol="e" position="left" xlink:label="note-0308-05" xlink:href="note-0308-05a" xml:space="preserve">75. h.</note> </div> </div> <div xml:id="echoid-div895" type="section" level="1" n="356"> <head xml:id="echoid-head365" xml:space="preserve">THEOR. LVIII. PROP. LXXXVIII.</head> <p> <s xml:id="echoid-s8571" xml:space="preserve">Æquales portiones ſolidæ de eodem quocunque Conoide, aut <lb/>Sphæra, aut Sphæroide ad ſibi inſcriptam Coni portionem, vel ad <lb/>circumſcriptum Cylindricum, vnam, eandemque ſimul habent <lb/>rationem.</s> <s xml:id="echoid-s8572" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8573" xml:space="preserve">ETenim huiuſmodi portiones habent baſes altitudinibus reciprocè pro-<lb/>portionales, vt in præcedenti, primo loco demonſtratum eſt, ſed ba-<lb/>ſes, & </s> <s xml:id="echoid-s8574" xml:space="preserve">altitudines portionum eædem ſunt, ac baſes, & </s> <s xml:id="echoid-s8575" xml:space="preserve">altitudines inſcripta-<lb/>rum Coniportionum, quare, & </s> <s xml:id="echoid-s8576" xml:space="preserve">Coni portionum baſes ipſarum altitudini-<lb/>bus erunt reciprocè proportionales, ſed eædem portiones Conorum ſunt <pb o="123" file="0309" n="309" rhead=""/> inter ſe <anchor type="note" xlink:href="" symbol="a"/> ſolida Acuminata proportionalia, & </s> <s xml:id="echoid-s8577" xml:space="preserve">baſes altitudinibus recipro- <anchor type="note" xlink:label="note-0309-01a" xlink:href="note-0309-01"/> cantur, vnde Coni portiones inſcriptæ inter ſe æquales <anchor type="note" xlink:href="" symbol="b"/> erunt; </s> <s xml:id="echoid-s8578" xml:space="preserve">erit ergo <anchor type="note" xlink:label="note-0309-02a" xlink:href="note-0309-02"/> ſolida portio ad portionem æqualem de eodem ſolido, vt inſcripta Coni <lb/>portio ad inſcriptam Coni portionem (ob æqualitatem) & </s> <s xml:id="echoid-s8579" xml:space="preserve">permutando <lb/>ſolida portio ad ſibi inſcriptam Coni portionem, vt altera æqualis portio ad <lb/>ſibi inſcriptam Coni portionem, & </s> <s xml:id="echoid-s8580" xml:space="preserve">ſumptis conſequentium <anchor type="note" xlink:href="" symbol="c"/> triplis, ſolida <anchor type="note" xlink:label="note-0309-03a" xlink:href="note-0309-03"/> portio ad circumſcriptum Cylindricum, vt reliqua portio ad ſibi circum-<lb/>ſcriptum Cylindricum, &</s> <s xml:id="echoid-s8581" xml:space="preserve">c. </s> <s xml:id="echoid-s8582" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s8583" xml:space="preserve">c.</s> <s xml:id="echoid-s8584" xml:space="preserve"/> </p> <div xml:id="echoid-div895" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0309-01" xlink:href="note-0309-01a" xml:space="preserve">70. h.</note> <note symbol="b" position="right" xlink:label="note-0309-02" xlink:href="note-0309-02a" xml:space="preserve">74. h.</note> <note symbol="c" position="right" xlink:label="note-0309-03" xlink:href="note-0309-03a" xml:space="preserve">ex Com <lb/>mand. in <lb/>lib. de Co <lb/>noid. & <lb/>Sphęroid. <lb/>Archim.</note> </div> </div> <div xml:id="echoid-div897" type="section" level="1" n="357"> <head xml:id="echoid-head366" xml:space="preserve">THEOR. LIX. PROP. LXXXIX.</head> <p> <s xml:id="echoid-s8585" xml:space="preserve">MAXIMA portionum eiuſdem Coni recti, aut Conoidis Hy-<lb/>perbolici, ſiue Sphæroidis oblongi, vel prolati, & </s> <s xml:id="echoid-s8586" xml:space="preserve">quarum axes <lb/>ſint æquales, ea eſt, cuius axis congruat cum axe ſectionis, quæ <lb/>ſolidum genuit; </s> <s xml:id="echoid-s8587" xml:space="preserve">& </s> <s xml:id="echoid-s8588" xml:space="preserve">reſpectiue ad Sphæroides, cum minori axe El-<lb/>lipſis genitricis.</s> <s xml:id="echoid-s8589" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8590" xml:space="preserve">MINIMA verò, cuius axis congruat cum maiori axe eiuſdem <lb/>Ellipſis.</s> <s xml:id="echoid-s8591" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8592" xml:space="preserve">ETenim quando portiones eiuſdem Conirecti, aut Conoidis Hyperboli-<lb/>ci, ſiue Sphæroidis cuiuslibet ſunt æquales, & </s> <s xml:id="echoid-s8593" xml:space="preserve">eorum recti Canones <lb/>ſunt <anchor type="note" xlink:href="" symbol="d"/> æquales, & </s> <s xml:id="echoid-s8594" xml:space="preserve">quando recti Canones ſiue portiones de eodem angulo, <anchor type="note" xlink:label="note-0309-04a" xlink:href="note-0309-04"/> vel Hyperbola, aut Ellipſi æquales ſunt, inter ipſorum diametros _MINIMA_ <lb/>eſt <anchor type="note" xlink:href="" symbol="e"/> ea, quæ ſimul ſit axis anguli, vel Hyperbolæ, & </s> <s xml:id="echoid-s8595" xml:space="preserve">in Ellipſi, quæ ſit axis <anchor type="note" xlink:label="note-0309-05a" xlink:href="note-0309-05"/> minor, & </s> <s xml:id="echoid-s8596" xml:space="preserve">_MAXIMA_, quæ ſit axis maior, ergo, & </s> <s xml:id="echoid-s8597" xml:space="preserve">dum portiones eiuſdem <lb/>Coni recti, aut Conoidis Hyperbolici, vel Sphæroidis fuerint æquales, in-<lb/>ter ipſorum axes (qui ijſdem ſunt, <anchor type="note" xlink:href="" symbol="f"/> ac diametri rectorum Canonum) _MI_- <anchor type="note" xlink:label="note-0309-06a" xlink:href="note-0309-06"/> _NIMVS_ erit is, qui congruet cum axe Coni, vel Conoidis Hyperbolici, <lb/>aut cum minori axe Ellipſis Sphæroidis, & </s> <s xml:id="echoid-s8598" xml:space="preserve">_MAXIMVS_, qui congruat cum <lb/>maiori: </s> <s xml:id="echoid-s8599" xml:space="preserve">quare ſi primùm axes harum omnium equalium portionum, dempta <lb/>ea circa _MINIMV M_ axem, huic _MINIMO_ axi æquales ſecentur, atque ex <lb/>interſectionibus ducantur plana baſibus portionum æquidiſtantia, auferen-<lb/>tur ab ipſis portiones ſolidæ æqualium axium, ſed vnaquæque erit minor <lb/>quacunque æqualium portionum, (cum ſit pars ſuo toto minor) ac propte-<lb/>rea minor ea, è cuius, axe, ſiue à qua portione nihil ablatũ ſuit, quę quidem <lb/>ea eſt, cuius axis congruit cum axe Coni recti, vel Conoidis Hyperbolici, <lb/>& </s> <s xml:id="echoid-s8600" xml:space="preserve">in Sphæroide cum minori axe Ellipſis genìtricis. </s> <s xml:id="echoid-s8601" xml:space="preserve">Si ergo omnes aliæ por-<lb/>tiones æqualium axium ſunt hac portione minores, erit è contra hæc ipſa <lb/>portio, cuius axis congruit cum axe dati Coni, vel Conoidis Hyperbolici, <lb/>& </s> <s xml:id="echoid-s8602" xml:space="preserve">pro Sphæroide, cum minori axe genitricis Ellipſis, earundem omnium <lb/>portionum, æqualium axium, _MAXIMA_. </s> <s xml:id="echoid-s8603" xml:space="preserve">Quod primò erat, &</s> <s xml:id="echoid-s8604" xml:space="preserve">c.</s> <s xml:id="echoid-s8605" xml:space="preserve"/> </p> <div xml:id="echoid-div897" type="float" level="2" n="1"> <note symbol="d" position="right" xlink:label="note-0309-04" xlink:href="note-0309-04a" xml:space="preserve">84. h.</note> <note symbol="e" position="right" xlink:label="note-0309-05" xlink:href="note-0309-05a" xml:space="preserve">Schol. <lb/>poſt 5 1. h. <lb/>ad nu. 1.</note> <note symbol="f" position="right" xlink:label="note-0309-06" xlink:href="note-0309-06a" xml:space="preserve">3. Schol. <lb/>69. h.</note> </div> <p> <s xml:id="echoid-s8606" xml:space="preserve">PRæterea ſi axes omnium æqualium portionum eiuſdem Sphæroidis pro-<lb/>ducantur, ac prædicto _MAXIMO_ axi (qui iam, vt ſuperiùs diximus, <pb o="124" file="0310" n="310" rhead=""/> congruit cum maiori axe Sphæroidis) æquales ſecentur, atque ex interſe-<lb/>ctionum punctis plana ducantur portionum baſibus æquidiſtantia, abſcin-<lb/>dentur portiones ſolidæ æqualium axium, & </s> <s xml:id="echoid-s8607" xml:space="preserve">vnaquæque erit maior quali-<lb/>bet æqualium (totum enim ſua parte maius eſt) ac ideò maior ea portione, <lb/>cuius axi, vel cui portioni nihil additum fuit, quæ quidem eſt ea, cuius axis <lb/>congruit cum maiori axe Sphæroidis. </s> <s xml:id="echoid-s8608" xml:space="preserve">Itaque ſi omnes planæ portiones <lb/>æqualium axium ſunt hac portione maiores, erit è contra hæc ipſa portio, <lb/>cuius axis conuenit cum maiori axe Sphæroidis, _MINIMA_ earundem om-<lb/>nium portionum æqualium axium, in caſibus tamen poſſibilibus. </s> <s xml:id="echoid-s8609" xml:space="preserve">Quod vl-<lb/>timò demonſtrandum erat.</s> <s xml:id="echoid-s8610" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div899" type="section" level="1" n="358"> <head xml:id="echoid-head367" xml:space="preserve">THEOR. LX. PROP. LXXXX.</head> <p> <s xml:id="echoid-s8611" xml:space="preserve">MAXIMA portionum de codem Cono recto, vel de quocun-<lb/>que Conoide, aut Sphæroide, & </s> <s xml:id="echoid-s8612" xml:space="preserve">quarum baſes ſint æquales, ea eſt, <lb/>cuius axis ſit ſegmentum maioris ſemi- axis genitricis ſectionis dati <lb/>ſolidi, reſpectiuè ad Sphæroides.</s> <s xml:id="echoid-s8613" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8614" xml:space="preserve">In Sphæroide autem, MINIMA, cuius axis ſit ſegmentum mi-<lb/>noris ſemi- axis Ellipſis, quæ ſolidum procreat.</s> <s xml:id="echoid-s8615" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8616" xml:space="preserve">QVando enim portiones eiuſdem Coni recti, vel cuiuslibet Conoidis, <lb/>aut Sphæroidis ſunt æquales, & </s> <s xml:id="echoid-s8617" xml:space="preserve">recti earum Canones ſunt <anchor type="note" xlink:href="" symbol="a"/> æquales, <anchor type="note" xlink:label="note-0310-01a" xlink:href="note-0310-01"/> & </s> <s xml:id="echoid-s8618" xml:space="preserve">cum recti Canones, vel portiones de eodem angulo, vel de <lb/>eadem coni- ſectione, quæ ſolidum genuit æquales ſunt, inter ipſorum ba-<lb/> <anchor type="note" xlink:label="note-0310-02a" xlink:href="note-0310-02"/> ſes, _MINIMA_ eſt <anchor type="note" xlink:href="" symbol="b"/> ea illius portionis, cuius diameter ſit ſegmentum maio- ris axis reſpectiuè ad Ellipſim, & </s> <s xml:id="echoid-s8619" xml:space="preserve">_MAXIMA_ eius, cuius diameter ſit ſegmen-<lb/>tum minoris, atque vt ſunt baſes æqualium planarum portionum de eodem <lb/>angulo, vel coni-ſectione, ita ſunt <anchor type="note" xlink:href="" symbol="c"/> baſes ſolidarum portionum, quarum <anchor type="note" xlink:label="note-0310-03a" xlink:href="note-0310-03"/> ipſæ planæ portiones ſint recti Canones, ergo & </s> <s xml:id="echoid-s8620" xml:space="preserve">inter baſes æqualium por-<lb/>tionum de eodem Cono recto, vel Conoide, aut Sphæroide quocunque, <lb/>_MINIMA_ erit ea illius portionis, cuius axis (qui idem eſt <anchor type="note" xlink:href="" symbol="d"/> cum diametro <anchor type="note" xlink:label="note-0310-04a" xlink:href="note-0310-04"/> recti Canonis) congruat cum maiori axe genitricis ſectionis ſolidi, cuius <lb/>eſt portio, & </s> <s xml:id="echoid-s8621" xml:space="preserve">_MAXIMA_, in Sphæroide, erit baſis illius portionis, cuius axis <lb/>ſit ſegmentum minoris axis Ellipſis genitricis eiuſdem Sphæroidis; </s> <s xml:id="echoid-s8622" xml:space="preserve">quare ſi <lb/>primò intra has æquales portiones, dempta ea ſuper _MINIMA_ baſi, ducan-<lb/>tur plana baſibus æquidiſtantia, quorum vnumquodque efficiat in portione <lb/>ſectionem prædictæ _MINIMAE_ baſi æqualem (hoc autem ſieri poſſe, & </s> <s xml:id="echoid-s8623" xml:space="preserve"><lb/>quomodò infra docebimus) per huiuſmodi plana abſcindentur portiones <lb/>ſolidæ æqualium baſium, ſed harum quælibet minor erit quacunque æqua-<lb/>lium portionum (cum ſit pars minor ſuo toto) ideoque minor ea, à qua ni-<lb/>hil ablatum fuit, ſiue minor ea, cuius axis conuenit cum maiori axe dati ſo-<lb/>lidi. </s> <s xml:id="echoid-s8624" xml:space="preserve">Si ergo omnes aliæ portiones æqualium baſium hac portione ſunt mi-<lb/>nores, erit è contra hæc ipſa portio, cuius axis eſt ſegmentum maioris ſemi-<lb/>axis ſectionis genitricis dati ſolidi earundem portionum æqualium baſium, <lb/>ac de eodem ſolido _MAXIMA_, &</s> <s xml:id="echoid-s8625" xml:space="preserve">c.</s> <s xml:id="echoid-s8626" xml:space="preserve"/> </p> <div xml:id="echoid-div899" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0310-01" xlink:href="note-0310-01a" xml:space="preserve">84. h.</note> <note symbol="b" position="left" xlink:label="note-0310-02" xlink:href="note-0310-02a" xml:space="preserve">Schol. <lb/>poſt 5 1. h. <lb/>ad nu. 2.</note> <note symbol="c" position="left" xlink:label="note-0310-03" xlink:href="note-0310-03a" xml:space="preserve">2. Co-<lb/>roll. 78. h.</note> <note symbol="d" position="left" xlink:label="note-0310-04" xlink:href="note-0310-04a" xml:space="preserve">3. Schol. <lb/>69. h.</note> </div> <pb o="125" file="0311" n="311" rhead=""/> <p> <s xml:id="echoid-s8627" xml:space="preserve">QVod autem in quolibet Sphæroide, inter portiones eius dimidio mi-<lb/>nores, & </s> <s xml:id="echoid-s8628" xml:space="preserve">æqualium baſium, _MINIMA_ ſit ea, cuius axis ſit ſegmen-<lb/>tum minoris axis Ellipſis datum Sphæroides procreantis, id con-<lb/>ſimili conſtructione, atque argumentis oſtendetur, vti factum fuit in ſecun-<lb/>da parte Prop. </s> <s xml:id="echoid-s8629" xml:space="preserve">50. </s> <s xml:id="echoid-s8630" xml:space="preserve">huius, ſi tamen ſuper tertia figura lineæ rectæ, & </s> <s xml:id="echoid-s8631" xml:space="preserve">Ellipſes <lb/>ibi animaduerſæ, concipiantur tanquam baſes ſolidarum portionum, & </s> <s xml:id="echoid-s8632" xml:space="preserve">ve-<lb/>luti Sphæroidalia ſolida, &</s> <s xml:id="echoid-s8633" xml:space="preserve">c. </s> <s xml:id="echoid-s8634" xml:space="preserve">Quod fuit, &</s> <s xml:id="echoid-s8635" xml:space="preserve">c.</s> <s xml:id="echoid-s8636" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div901" type="section" level="1" n="359"> <head xml:id="echoid-head368" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s8637" xml:space="preserve">HInc conſtat _MINIM AM_ portionum ſemi- Sphæroide maiorum, & </s> <s xml:id="echoid-s8638" xml:space="preserve"><lb/>quarum baſes ſint æquales, eam eſſe, cuius axis ſit ſegmentum maio-<lb/>ris axis Ellipſis genitrics; </s> <s xml:id="echoid-s8639" xml:space="preserve">_MAXIM AM_ autem, cuius axis ſit ſegmentum <lb/>minoris.</s> <s xml:id="echoid-s8640" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div902" type="section" level="1" n="360"> <head xml:id="echoid-head369" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s8641" xml:space="preserve">QVod ſuperius promiſſimus abſoluetur ſic, ſuper figuras prædictæ 50. </s> <s xml:id="echoid-s8642" xml:space="preserve">h. <lb/></s> <s xml:id="echoid-s8643" xml:space="preserve">Cum ibi ſit A C minor H I, erit quoque dimidium D C minus di-<lb/>midio F I. </s> <s xml:id="echoid-s8644" xml:space="preserve">Detrahatur ergo F P, quę ſit media proportionalis <lb/>inter F I, D C; </s> <s xml:id="echoid-s8645" xml:space="preserve">agatur P R diametro F O æquidiſtans, & </s> <s xml:id="echoid-s8646" xml:space="preserve">ſectioni occur-<lb/>rensin R, atque ex R applicetur R Q S, & </s> <s xml:id="echoid-s8647" xml:space="preserve">facta figurarum reuolutione <lb/>circa axim B D, concipiantur deſcribiſolida, &</s> <s xml:id="echoid-s8648" xml:space="preserve">c. </s> <s xml:id="echoid-s8649" xml:space="preserve">èquibus cum planis per <lb/>rectas A C, H I, S R ductis, & </s> <s xml:id="echoid-s8650" xml:space="preserve">ad eaſdem genitrices ſectiones erectis, ab-<lb/>ſcindentur portiones ſolidæ A B C, H O I inter ſe <anchor type="note" xlink:href="" symbol="a"/> æquales, & </s> <s xml:id="echoid-s8651" xml:space="preserve">portio S O <anchor type="note" xlink:label="note-0311-01a" xlink:href="note-0311-01"/> R. </s> <s xml:id="echoid-s8652" xml:space="preserve">Dico huius baſim per S R ductam, æqualem eſſe baſi per A C.</s> <s xml:id="echoid-s8653" xml:space="preserve"/> </p> <div xml:id="echoid-div902" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0311-01" xlink:href="note-0311-01a" xml:space="preserve">80. h.</note> </div> <p> <s xml:id="echoid-s8654" xml:space="preserve">Nam baſis per H I ad baſim per A C, eſt <anchor type="note" xlink:href="" symbol="b"/> vt recta H I ad rectam A C, <anchor type="note" xlink:label="note-0311-02a" xlink:href="note-0311-02"/> vel ſumptis dimidijs, vt F I ad D C, vel vt quadratum F I, ad quadratum <lb/>F P, ſiue ad quadratum Q R, vel ſumptis quadruplis, vt quadratum H I ad <lb/>quadratum S R, ſed etiam baſis per H I ad baſim per S R, eſt vt quadra-<lb/>tum H I ad quadratum S R, cum ob planorum æquidiſtantiam ſint <anchor type="note" xlink:href="" symbol="c"/> ſectio- <anchor type="note" xlink:label="note-0311-03a" xlink:href="note-0311-03"/> nes ſimiles, ergo baſis per H I ad baſim per A C, erit vt eadem baſis per H <lb/>I ad baſim per S R: </s> <s xml:id="echoid-s8655" xml:space="preserve">vnde baſis per S R æqualis eſt baſi per A C, &</s> <s xml:id="echoid-s8656" xml:space="preserve">c. </s> <s xml:id="echoid-s8657" xml:space="preserve">Quod <lb/>facere oportebat.</s> <s xml:id="echoid-s8658" xml:space="preserve"/> </p> <div xml:id="echoid-div903" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-0311-02" xlink:href="note-0311-02a" xml:space="preserve">2. Co-<lb/>78. h.</note> <note symbol="c" position="right" xlink:label="note-0311-03" xlink:href="note-0311-03a" xml:space="preserve">Coroll. <lb/>15. Arch. <lb/>de Co-<lb/>noid.</note> </div> </div> <div xml:id="echoid-div905" type="section" level="1" n="361"> <head xml:id="echoid-head370" xml:space="preserve">THEOR. LXI. PROP. LXXXXI.</head> <p> <s xml:id="echoid-s8659" xml:space="preserve">MINIMA portionum de eodem Cono recto, vel de quocunque <lb/>Conoide, aut Sphæroide, & </s> <s xml:id="echoid-s8660" xml:space="preserve">quarum altitudines ſint æquales ea <lb/>eſt, cuius axis congruat cum maiori axe genitricis ſectionis dati <lb/>ſolidi.</s> <s xml:id="echoid-s8661" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8662" xml:space="preserve">In Sphæroide, MAXIMA eſt, cuius axis cum minori axe eiuſ-<lb/>dem genitricis ſectionis conueniat.</s> <s xml:id="echoid-s8663" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8664" xml:space="preserve">NAm quando portiones de eodem Cono recto, vel Conoide, aut Sphę-<lb/>roide quocunque ſunt æquales, & </s> <s xml:id="echoid-s8665" xml:space="preserve">ipſarum recti Canones inter ſe <pb o="126" file="0312" n="312" rhead=""/> ſunt <anchor type="note" xlink:href="" symbol="a"/> æquales, quando verò recti Canones, ſiue portiones de eodem angu- <anchor type="note" xlink:label="note-0312-01a" xlink:href="note-0312-01"/> lo, vel de eadem coni-ſectione, quæ ſolidum procreat æquales ſunt, inter <lb/>ipſarum altitudines _MAXIM A_ eſt <anchor type="note" xlink:href="" symbol="b"/> ea illius portionis, cuius diameter ſit <anchor type="note" xlink:label="note-0312-02a" xlink:href="note-0312-02"/> ſegmentum maioris axis, & </s> <s xml:id="echoid-s8666" xml:space="preserve">_MINIMA_, cuius diameter ſit ſegmentum mi-<lb/>noris; </s> <s xml:id="echoid-s8667" xml:space="preserve">atque altitudines, & </s> <s xml:id="echoid-s8668" xml:space="preserve">diametri rectorum Canonum, ſiue planarum <lb/>portionum eædem ſunt, <anchor type="note" xlink:href="" symbol="c"/> ac altitudines, & </s> <s xml:id="echoid-s8669" xml:space="preserve">axes ſolidarum, ergo, & </s> <s xml:id="echoid-s8670" xml:space="preserve">dum <anchor type="note" xlink:label="note-0312-03a" xlink:href="note-0312-03"/> portiones eiuſdem Coni recti, vel Conoidis, aut Sphæroidis ſunt æquales, <lb/>inter earum altitudines _MAXIM A_ erit ea illius portionis, cuius axis ſit ſe-<lb/>gmentum maioris axis genitricis ſolidi, cuius eſt portio, & </s> <s xml:id="echoid-s8671" xml:space="preserve">_MINIM A_ eius, <lb/>cuius axis ſit ſegmentum minoris. </s> <s xml:id="echoid-s8672" xml:space="preserve">Itaque ſi primò altitudines omnium ha-<lb/>rum æqualium portionum, (dempta ea circa _MAXIM AM_ altitudinem) <lb/>producantur, & </s> <s xml:id="echoid-s8673" xml:space="preserve">huic _MINIM AE_ altitudini æquales fiant, atque ex interſe-<lb/>ctionum punctis ducantur plana portionum baſibus æquidiſtantia, abſcin-<lb/>dentur ab ipſis portiones ſolidæ æqualium altitudinum, & </s> <s xml:id="echoid-s8674" xml:space="preserve">vnaquæque ma-<lb/>ior erit quacunque æqualium portionum (nam totum ſua parte maius eſt) <lb/>vnde, & </s> <s xml:id="echoid-s8675" xml:space="preserve">maior ea portione, cuius altitudini, vel cui portioni nihil additum <lb/>fuit, quæ ea eſt, cuius axis conuenit cum maiori axe genitricis ſectionis dati <lb/>ſolidi. </s> <s xml:id="echoid-s8676" xml:space="preserve">Si ergo omnes aliæ portiones æqualium altitudinum hane portio-<lb/>nem excedunt, erit è contra hæc ipſa portio, cuius axis congruit cum maio-<lb/>ri axe genitricis ſectionis dati ſolidi aliarum portionum æqualium altitudi-<lb/>num _MINIM A_.</s> <s xml:id="echoid-s8677" xml:space="preserve"/> </p> <div xml:id="echoid-div905" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0312-01" xlink:href="note-0312-01a" xml:space="preserve">84. h.</note> <note symbol="b" position="left" xlink:label="note-0312-02" xlink:href="note-0312-02a" xml:space="preserve">Schol. <lb/>poſt 51. h. <lb/>ad nu. 3.</note> <note symbol="c" position="left" xlink:label="note-0312-03" xlink:href="note-0312-03a" xml:space="preserve">3. Schol. <lb/>69. h.</note> </div> <p> <s xml:id="echoid-s8678" xml:space="preserve">PRo Sphæroide autem, ſi altitudines omnium prædictarum æqualium <lb/>portionum (dempta ea circa _MINIM AM_ altitudinem, quæ iam ea eſt <lb/>circa minorem axem Ellipſis Sphæroidis genitricis) ę quales ſecentur eidem <lb/>_MINIM AE_ altitudini, atque per puncta ſectionum, plana ſolidarum por-<lb/>tionum baſibus æquidiſtantia ducantur, hæc à portionibus auferent portio-<lb/>nes ſolidas æqualium altitudinum, ſed vnaquæque ipſarum minor erit <lb/>quacunque æqualium portionum (eò quod pars ſuo toto ſit minor) quapro-<lb/>pter & </s> <s xml:id="echoid-s8679" xml:space="preserve">minor ea portione a cuius altitudine, vel à qua portione nihil dem-<lb/>ptum fuit, quæ quidem eſt ea, cuius axis congruit cum minori axe Ellipſis <lb/>datum Sphæroides procreantis: </s> <s xml:id="echoid-s8680" xml:space="preserve">ſi igitur omnes portiones æqualium altitu-<lb/>dinum hac portione ſunt minores, erit ex aduerſo hæc eadem portio, cuius <lb/>axis conuenit cum minori axe genitricis Ellipſis dati Sphæroidis earundem <lb/>omnium portionum, æqualium altitudinum, _MAXIMA_. </s> <s xml:id="echoid-s8681" xml:space="preserve">Quod tandem ſu-<lb/>pererat demonſtrandum.</s> <s xml:id="echoid-s8682" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div907" type="section" level="1" n="362"> <head xml:id="echoid-head371" xml:space="preserve">SCHOLIV M.</head> <p> <s xml:id="echoid-s8683" xml:space="preserve">HVc etiam, prout expoſuimus in Scholio poſt 51. </s> <s xml:id="echoid-s8684" xml:space="preserve">huius, hæc tria ſunt <lb/>animaduertenda. </s> <s xml:id="echoid-s8685" xml:space="preserve">Videlicet.</s> <s xml:id="echoid-s8686" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8687" xml:space="preserve">1. </s> <s xml:id="echoid-s8688" xml:space="preserve">I Nter axes æqualium portionum eiuſdem Coni recti, vel Conoidis Hy-<lb/>perbolici, aut cuiuſcunque Sphæroidis, _MINIMV S_ eſt is eius portionis, <lb/>cuius axis congruat cum axe, & </s> <s xml:id="echoid-s8689" xml:space="preserve">pro Sphæroide, cum minori axe genitricis <lb/>ſectionis dati ſolidi, & </s> <s xml:id="echoid-s8690" xml:space="preserve">in Sphæroide _MAXIMV S_ eius portionis, cuius axis <lb/>congruat cum maiori axe eiuſdem genitricis ſectionis.</s> <s xml:id="echoid-s8691" xml:space="preserve"/> </p> <pb o="127" file="0313" n="313" rhead=""/> <p> <s xml:id="echoid-s8692" xml:space="preserve">2. </s> <s xml:id="echoid-s8693" xml:space="preserve">INter baſes æqualium portionum de eodem Cono recto, aut de quocun-<lb/>que Conoide, aut Sphæroide _MINIM A_ eſt ea illius portionis, cuius axis <lb/>ſit ſegmentum axis, & </s> <s xml:id="echoid-s8694" xml:space="preserve">pro Sphæroide ſit ſegmen tum maioris axis genitricis <lb/>ſectionis dati ſolidi. </s> <s xml:id="echoid-s8695" xml:space="preserve">_MAXIM A_ verò eius, cuius axis ſit ſegmentum mino-<lb/>ris axis eiuſdem ſectionis genitricis.</s> <s xml:id="echoid-s8696" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8697" xml:space="preserve">3. </s> <s xml:id="echoid-s8698" xml:space="preserve">INter altitudines æqualium portionum de eodem Cono recto, ſiue de quo-<lb/>libet Conoide, aut Sphæroide, _MAXIMA_ eſt ea illius portionis, cuius <lb/>axis congruat cum maiori axe genitricis ſectionis dati ſolidi, & </s> <s xml:id="echoid-s8699" xml:space="preserve">in Sphæroi-<lb/>de _MINIM A_ eius, cuius axis cum minori axe eiuſdem genitricis ſectionis <lb/>conueniat.</s> <s xml:id="echoid-s8700" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8701" xml:space="preserve">Quæ omnia, ex hucuſque demonſtratis, paucis oſtendentur (vti factum <lb/>fuit in præfato Scholio, & </s> <s xml:id="echoid-s8702" xml:space="preserve">ſuper eaſdem figuras 51. </s> <s xml:id="echoid-s8703" xml:space="preserve">h.) </s> <s xml:id="echoid-s8704" xml:space="preserve">conſimilibus, ac ibi <lb/>argumentis, veruntamen circa ſolidas portiones verſantibus, è quibus de-<lb/>nique vniuſcuiuſque trium proximè præcedentium propoſitionum veritas <lb/>iterum eluceſcet. </s> <s xml:id="echoid-s8705" xml:space="preserve">Sed de his hactenus.</s> <s xml:id="echoid-s8706" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div908" type="section" level="1" n="363"> <head xml:id="echoid-head372" xml:space="preserve">MONIT V M.</head> <p style="it"> <s xml:id="echoid-s8707" xml:space="preserve">PLacuit SERENO, Antinſ enſi Philoſopho, in quibuslibet Conis <lb/>terminatis MAXIMV M, & </s> <s xml:id="echoid-s8708" xml:space="preserve">MINIMV M triangulum <lb/>per verticem ductum inquirere, liceat nobis tanti Geometræ <lb/>veſtigia inſequentibus in Cono pariter terminato quocunque <lb/>MAXIMAM, & </s> <s xml:id="echoid-s8709" xml:space="preserve">MINIMAM Paraboæ portionem aſsignare, pro <lb/>cuius indigatione nonnulla circa plana, nec præter ſuſceptam materiam, <lb/>nec ſcitu iniucunda occurrunt afferenda.</s> <s xml:id="echoid-s8710" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div909" type="section" level="1" n="364"> <head xml:id="echoid-head373" xml:space="preserve">LEMMA XVI. PROP. XCII.</head> <p> <s xml:id="echoid-s8711" xml:space="preserve">Si duo triangula habuerint latus lateri æquale, atque alterum <lb/>adiacentium angulorum in vno triangulo, alteri adiacentium in <lb/>reliquo æqualem, ſitque reliquus angulus adiacentium in primo, <lb/>maior reliquo adiacentium in altero, & </s> <s xml:id="echoid-s8712" xml:space="preserve">latus illi oppoſitum, late-<lb/>re huic oppoſito maius erit.</s> <s xml:id="echoid-s8713" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8714" xml:space="preserve">SInt duo triangula A B C, D E F, quo-<lb/> <anchor type="figure" xlink:label="fig-0313-01a" xlink:href="fig-0313-01"/> rum latera B C, E F ſint æqualia, & </s> <s xml:id="echoid-s8715" xml:space="preserve"><lb/>anguli pariter A B C, D E F æquales, an-<lb/>gulus verò A C B maior ſit angulo D F E. <lb/></s> <s xml:id="echoid-s8716" xml:space="preserve">Dico, & </s> <s xml:id="echoid-s8717" xml:space="preserve">latus A B maiori angulo oppoſitũ, <lb/>maius eſſe latere D E oppoſitum minori.</s> <s xml:id="echoid-s8718" xml:space="preserve"/> </p> <div xml:id="echoid-div909" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0313-01"/> </figure> </div> <p> <s xml:id="echoid-s8719" xml:space="preserve">Fiat angulus B C G æqualis ipſi E F D.</s> <s xml:id="echoid-s8720" xml:space="preserve"> <pb o="128" file="0314" n="314" rhead=""/> Et quoniam angulus quoque G B C ponitur æqualis angulo D E F, & </s> <s xml:id="echoid-s8721" xml:space="preserve">latus <lb/>B C lateri E F æquale, erunt in triangulis G C B, D F E reliqua latera G <lb/>B, D E æqualibus angulis oppoſita, inter ſe æqualia, ſed eſt latus A B ma-<lb/>ius latere B G, cum recta C G ſecet angulum A C B, ergo latus A B erit <lb/>quoque maius latere D E. </s> <s xml:id="echoid-s8722" xml:space="preserve">Quod erat probandum.</s> <s xml:id="echoid-s8723" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div911" type="section" level="1" n="365"> <head xml:id="echoid-head374" xml:space="preserve">PROBL. XVI. PROP. XCIII.</head> <p> <s xml:id="echoid-s8724" xml:space="preserve">A data circuli peripheria arcum abſcindere, ita vt rectangulum <lb/>ſub eius chorda in ſagittam ſit MINIMVM.</s> <s xml:id="echoid-s8725" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8726" xml:space="preserve">1. </s> <s xml:id="echoid-s8727" xml:space="preserve">ESto circulus, cuius diameter A B, centrum C, & </s> <s xml:id="echoid-s8728" xml:space="preserve">exequi oporteat, <lb/>quod imperatum eſt.</s> <s xml:id="echoid-s8729" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8730" xml:space="preserve">Sumantur in peripheria, hinc inde à puncto A, duo trientes A D, A E, <lb/>& </s> <s xml:id="echoid-s8731" xml:space="preserve">iungatur chorda D E ſecans diametrum A B in F. </s> <s xml:id="echoid-s8732" xml:space="preserve">Dico arcum D A E <lb/>eſſe quæſitum; </s> <s xml:id="echoid-s8733" xml:space="preserve">hoc eſt rectangulum ſub eius chorda D E in ſagittam A F <lb/>eſſe _MAXIMV M_.</s> <s xml:id="echoid-s8734" xml:space="preserve"/> </p> <figure> <image file="0314-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0314-01"/> </figure> <p> <s xml:id="echoid-s8735" xml:space="preserve">Secta enim ſemi - peripheria A K B bifariam in K, iunctaque K C, ac <lb/>ſumpto in arcu D K quolibet puncto G, quod vel in ipſum K, vel inter <lb/>K, & </s> <s xml:id="echoid-s8736" xml:space="preserve">D vbicunque cadat, demiſſaque ex G ſuper diametrum A B per-<lb/>pendiculari G H, quæ producta occurrat peripheriæ in I, iungatur G D.</s> <s xml:id="echoid-s8737" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8738" xml:space="preserve">Et cum arcus A G ſit non minor quadrante A K, erit duplus G A I <lb/>non minor ſemi - circulo, atque arcus D A I omnino maior ſemi - circu-<lb/>lo; </s> <s xml:id="echoid-s8739" xml:space="preserve">vnde iuncta G D, angulus I G D erit acutus, eſtque G H B rectus, <lb/>quare duo ſimul D G H, G H B duobus rectis minores erunt, ex quo G <lb/>D producta conueniet cum diametro ad partes D, vt in L. </s> <s xml:id="echoid-s8740" xml:space="preserve">Et cum ar-<lb/>cus A K D, A I E ſint trientes totius peripheriæ, erit D B E, quod ſupe-<lb/>reſt de aſſe, eiuſdem peripheriæ triens, ſiue æqualis arcui A I E, itaque <lb/>arcus D B I erit maior arcu A I E: </s> <s xml:id="echoid-s8741" xml:space="preserve">ſi ergo iungatur A D, erit angulus A <lb/>D E, ſiue A D F minor angulo I G D, ſiue parallelarum externo F D L, <pb o="129" file="0315" n="315" rhead=""/> ſuntque in triangulis D F A, D F L anguli ad F æquales, cum ſint recti, <lb/>& </s> <s xml:id="echoid-s8742" xml:space="preserve">latus F D commune, atque angulus A D F minor eſt angulo L D F, <lb/>quare & </s> <s xml:id="echoid-s8743" xml:space="preserve">latus A F minus <anchor type="note" xlink:href="" symbol="a"/> erit latere F L, & </s> <s xml:id="echoid-s8744" xml:space="preserve">A H eò minus F L; </s> <s xml:id="echoid-s8745" xml:space="preserve">habe- <anchor type="note" xlink:label="note-0315-01a" xlink:href="note-0315-01"/> bit ergo H F ad F L minorem rationem, quàm eadem F H ad H A, & </s> <s xml:id="echoid-s8746" xml:space="preserve"><lb/>componendo H L ad L F, ſiue G H ad D F, minorem quàm F A ad A <lb/>H, vnde rectangulum G H A ſub extremis minus <anchor type="note" xlink:href="" symbol="b"/> erit rectangulo D F A <anchor type="note" xlink:label="note-0315-02a" xlink:href="note-0315-02"/> ſub medijs, & </s> <s xml:id="echoid-s8747" xml:space="preserve">hoc ſemper, vbicunque ſumptum ſit punctum G, vel in-<lb/>ter D, & </s> <s xml:id="echoid-s8748" xml:space="preserve">K, vel in ipſo K, nempe rectangulum ad G, vel K, pertin-<lb/>gens, minus eſſe rectangulo D F A, ſiue D F A maius eſſe quocunque <lb/>prædictorum rectangulorum G H A, vel K C A, &</s> <s xml:id="echoid-s8749" xml:space="preserve">c.</s> <s xml:id="echoid-s8750" xml:space="preserve"/> </p> <div xml:id="echoid-div911" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0315-01" xlink:href="note-0315-01a" xml:space="preserve">92. h.</note> <note symbol="b" position="right" xlink:label="note-0315-02" xlink:href="note-0315-02a" xml:space="preserve">16. ſept. <lb/>Pappi.</note> </div> <p> <s xml:id="echoid-s8751" xml:space="preserve">Si autem punctum ſumatur in quadrante A K, vt in O; </s> <s xml:id="echoid-s8752" xml:space="preserve">demiſſa per-<lb/>pendiculari O P. </s> <s xml:id="echoid-s8753" xml:space="preserve">Cum ſit K C maior O P, & </s> <s xml:id="echoid-s8754" xml:space="preserve">C A maior A P, erit re-<lb/>ctangulum K C A maius rectangulo O P C, ſed rectangulum D F A oſtẽ-<lb/>ſum eſt maius rectangulo K C A, ergo rectangulum D F A eò amplius <lb/>maius erit rectangulo O P A.</s> <s xml:id="echoid-s8755" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8756" xml:space="preserve">Si denique punctum ſumatur in peripheriæ ſextante D B, veluti in Q, <lb/>demiſſa perpendiculari Q R, & </s> <s xml:id="echoid-s8757" xml:space="preserve">iuncta D Q, & </s> <s xml:id="echoid-s8758" xml:space="preserve">producta, ipſa conueniet <lb/>o mnino cum diametro A B ad partes B, vt in S, quoniam angulus E D <lb/>Q eſt in portione E A Q ſemi - circulo maiori, ac propterea acutus, & </s> <s xml:id="echoid-s8759" xml:space="preserve"><lb/>angulus D F S rectus eſt, &</s> <s xml:id="echoid-s8760" xml:space="preserve">c. </s> <s xml:id="echoid-s8761" xml:space="preserve">Et cum arcus A I E æqualis ſit arcui D B <lb/>E, vterque enim eſt triens peripheriæ, erit arcus A I E maior arcu Q B <lb/>E, ac ideo angulus A D E, vel A D F maior angulo Q D E, vel S D F, <lb/>ſed in triangulis A F D, S F D latus F D eſt commune, & </s> <s xml:id="echoid-s8762" xml:space="preserve">anguli ad F <lb/>ſunt æquales, eò quod ſint recti, & </s> <s xml:id="echoid-s8763" xml:space="preserve">angulus A D F maior eſt angulo S <lb/>D F, vnde latus A F maius eſt <anchor type="note" xlink:href="" symbol="c"/> latere F S, & </s> <s xml:id="echoid-s8764" xml:space="preserve">adhuc maius latere R S, <anchor type="note" xlink:label="note-0315-03a" xlink:href="note-0315-03"/> habebit ergo F R ad R S maiorem rationem quàm eadem R F ad F A, & </s> <s xml:id="echoid-s8765" xml:space="preserve"><lb/>componendo F S ad S R, vel D F ad Q R, maiorem quàm R A ad A F; <lb/></s> <s xml:id="echoid-s8766" xml:space="preserve">vnde rectangulum D F A ſub extremis, maius <anchor type="note" xlink:href="" symbol="d"/> erit rectangulo Q R A <anchor type="note" xlink:label="note-0315-04a" xlink:href="note-0315-04"/> ſub medijs, & </s> <s xml:id="echoid-s8767" xml:space="preserve">hoc ſemper vbicunque aſſumptum ſit punctum Q in ſex-<lb/>tante D B. </s> <s xml:id="echoid-s8768" xml:space="preserve">Quare cum rectangulum A F D demonſtratum ſit maius om-<lb/>nium applicatosum, tum in triente A D, tum in ſextante D B, ipſum A <lb/>F D erit _MAXIMV M_, & </s> <s xml:id="echoid-s8769" xml:space="preserve">ſumptis duplis, rectangulum ſub ſagitta A F in <lb/>chordam D E, erit _MAXIMV M_ rectangulum ſub qualibet alia ſagitta in <lb/>ſuam chordam. </s> <s xml:id="echoid-s8770" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s8771" xml:space="preserve">c. </s> <s xml:id="echoid-s8772" xml:space="preserve">Quodque alibi aliter enodabimus.</s> <s xml:id="echoid-s8773" xml:space="preserve"/> </p> <div xml:id="echoid-div912" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-0315-03" xlink:href="note-0315-03a" xml:space="preserve">92. h.</note> <note symbol="d" position="right" xlink:label="note-0315-04" xlink:href="note-0315-04a" xml:space="preserve">16. ſept. <lb/>Pappi.</note> </div> <p> <s xml:id="echoid-s8774" xml:space="preserve">2. </s> <s xml:id="echoid-s8775" xml:space="preserve">AD pleniorem autem doctrinã, in proxima ſequenti ſecunda figura, ma-<lb/>nentibus poſitione ijſdem punctis K, D, E, dico talium rectang lo-<lb/>rum id, quod puncto D propinquius eſt, ſemper maius eſſe remotiori.</s> <s xml:id="echoid-s8776" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8777" xml:space="preserve">Nam de ijs, quæ ad arcum quadrantis A K pertingunt, vtputa de re-<lb/>ctangulis A C K, A F R, A H G, &</s> <s xml:id="echoid-s8778" xml:space="preserve">c. </s> <s xml:id="echoid-s8779" xml:space="preserve">patet A C K propinquius puncto D <lb/>maius eſſe rectangulo A F R, quod ab ipſo D magis remouetur, & </s> <s xml:id="echoid-s8780" xml:space="preserve">A F <lb/>R maius eſſe A H G, &</s> <s xml:id="echoid-s8781" xml:space="preserve">c. </s> <s xml:id="echoid-s8782" xml:space="preserve">cum, tum altitudines K C, R F, G H, tum ba-<lb/>ſes C A, F A, H A continuè decreſcant.</s> <s xml:id="echoid-s8783" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8784" xml:space="preserve">De ijs verò, quæ perueniunt ad arcum K D, videlicet in punctis I, <lb/>L, ita ratiocinabimur. </s> <s xml:id="echoid-s8785" xml:space="preserve">Demittantur ex I, L ad diametrum perpendicu-<lb/>lares I M N, L O P, & </s> <s xml:id="echoid-s8786" xml:space="preserve">iungatur I L, quæ producta conueniet ad partes <lb/>L cum diametro in Q (nam arcus N A L maior eſt ſemi-peripheria, ex <pb o="130" file="0316" n="316" rhead=""/> quo angulus N I L eſt acutus, atque I M O rectus eſt, ideoque duo ſimul <lb/>N I L, I M B duobus rectis minores.) </s> <s xml:id="echoid-s8787" xml:space="preserve">Et cum arcus A E æqualis ſit arcui <lb/>D E, erit arcus A P minor arcu D E, & </s> <s xml:id="echoid-s8788" xml:space="preserve">multò minor arcu L B N: </s> <s xml:id="echoid-s8789" xml:space="preserve">vnde <lb/>iuncta A L, erit angulus A L P, ſiue A L O minor angulo L I N, ſiue L I <lb/>M, ſiue angulo Q L O parallelarum externo, eſtque in triangulis A L O, <lb/>Q L O latus O L commune, & </s> <s xml:id="echoid-s8790" xml:space="preserve">anguli ad O ſunt æquales, cum ſint recti, <lb/>ergo latus A O erit minus <anchor type="note" xlink:href="" symbol="a"/> latere O Q, & </s> <s xml:id="echoid-s8791" xml:space="preserve">A M eò minus O Q; </s> <s xml:id="echoid-s8792" xml:space="preserve">habebit <anchor type="note" xlink:label="note-0316-01a" xlink:href="note-0316-01"/> igitur O M ad M A maiorem rationem, quàm M O ad O Q, & </s> <s xml:id="echoid-s8793" xml:space="preserve">compo-<lb/>nendo O A ad A M maiorem quàm M Q ad Q O, vel quàm I M ad L O, <lb/>vnde rectangulum A O L ſub extremis, quod propinquius eſt puncto D, <lb/>maius <anchor type="note" xlink:href="" symbol="b"/> erit rectangulo A M I ſub medijs, quod à puncto D magis diſtat.</s> <s xml:id="echoid-s8794" xml:space="preserve"/> </p> <div xml:id="echoid-div913" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-0316-01" xlink:href="note-0316-01a" xml:space="preserve">92. h.</note> </div> <note symbol="b" position="left" xml:space="preserve">16. ſept. <lb/>Pappi.</note> <figure> <image file="0316-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0316-01"/> </figure> <p> <s xml:id="echoid-s8795" xml:space="preserve">De rectangulis denique pertingentibus ad puncta in ſextante D B, nimi-<lb/>rum ad S, T, idem ſic demonſtrabitur. </s> <s xml:id="echoid-s8796" xml:space="preserve">Ductis enim S V Y, T X diametro <lb/>perpendicularibus, & </s> <s xml:id="echoid-s8797" xml:space="preserve">iunctis A S, & </s> <s xml:id="echoid-s8798" xml:space="preserve">S T, hæc producta conueniet cum <lb/>A B in Z, quoniam angulus T S Y eſt in portione T A Y ſemi- circulo ma-<lb/>iori, nempe acutus, & </s> <s xml:id="echoid-s8799" xml:space="preserve">angulus S V B rectus eſt, &</s> <s xml:id="echoid-s8800" xml:space="preserve">c. </s> <s xml:id="echoid-s8801" xml:space="preserve">Et cum arcus A E Y <lb/>ſit triente maior, & </s> <s xml:id="echoid-s8802" xml:space="preserve">arcus Y B T minor E B D, ſiue minor triente, erit an-<lb/>gulus A S Y, ſiue A S V maior angulo Y S T, ſiue V S Z, & </s> <s xml:id="echoid-s8803" xml:space="preserve">in triangulis <lb/>A S V, Z S V ſunt anguli ad V æquales, cum ſint recti, & </s> <s xml:id="echoid-s8804" xml:space="preserve">latus S V com-<lb/>mune, ergo latus A V erit <anchor type="note" xlink:href="" symbol="c"/> maius latere V Z, & </s> <s xml:id="echoid-s8805" xml:space="preserve">eò maius latere X Z: </s> <s xml:id="echoid-s8806" xml:space="preserve">ha- <anchor type="note" xlink:label="note-0316-03a" xlink:href="note-0316-03"/> bebit ergo V X ad X Z maiorem rationem quàm ad V A, & </s> <s xml:id="echoid-s8807" xml:space="preserve">componen-<lb/>do, V Z ad Z X, ſiue S V ad T X maiorem rationem quàm X A ad A <lb/>V: </s> <s xml:id="echoid-s8808" xml:space="preserve">quapropter rectangulum S V A ſub extremis, quod propius eſt puncto <lb/>D maius erit <anchor type="note" xlink:href="" symbol="d"/> rectangulo T X A ſub medijs, quod à puncto D magis di- <anchor type="note" xlink:label="note-0316-04a" xlink:href="note-0316-04"/> ſtat. </s> <s xml:id="echoid-s8809" xml:space="preserve">Qnod ex abundanti oſtendere propoſitum fuit.</s> <s xml:id="echoid-s8810" xml:space="preserve"/> </p> <div xml:id="echoid-div914" type="float" level="2" n="4"> <note symbol="c" position="left" xlink:label="note-0316-03" xlink:href="note-0316-03a" xml:space="preserve">92. h.</note> <note symbol="d" position="left" xlink:label="note-0316-04" xlink:href="note-0316-04a" xml:space="preserve">16. ſept. <lb/>Pappi.</note> </div> </div> <div xml:id="echoid-div916" type="section" level="1" n="366"> <head xml:id="echoid-head375" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s8811" xml:space="preserve">EX eo, quod ad num. </s> <s xml:id="echoid-s8812" xml:space="preserve">1. </s> <s xml:id="echoid-s8813" xml:space="preserve">ſuperiùs oſtenſum fuit; </s> <s xml:id="echoid-s8814" xml:space="preserve">facilè conſtat, in prima <lb/>figura, quæſitam chordam D E ſecare circuli diametrum A B in F, <lb/>in 3. </s> <s xml:id="echoid-s8815" xml:space="preserve">ratione ad 1.</s> <s xml:id="echoid-s8816" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8817" xml:space="preserve">Nam iunctis C B, B D. </s> <s xml:id="echoid-s8818" xml:space="preserve">Cum ſit arcus B D circuli ſextans, ipſius chor- <pb o="131" file="0317" n="317" rhead=""/> da B D erit æqualis radio B C, ſiue C D, vnde in triangulo æquilatero C <lb/>D B anguli ad C, B, æquales erunt, & </s> <s xml:id="echoid-s8819" xml:space="preserve">in triangulis C F D, B F D cum <lb/>anguli ad C, B, ſint æquales, atque etiam æquales ad F, cum ſint recti, <lb/>& </s> <s xml:id="echoid-s8820" xml:space="preserve">latus D F commune, erit reliquum latus C F, reliquo F B æquale, eſtq; <lb/></s> <s xml:id="echoid-s8821" xml:space="preserve">A C æqualis C B, ergo A F erit tripla F B.</s> <s xml:id="echoid-s8822" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s8823" xml:space="preserve">Verum hæc omnia conſimili ratione perſolui, ac verificari de rectan-<lb/>gulis in Ellipſi applicatis, &</s> <s xml:id="echoid-s8824" xml:space="preserve">c. </s> <s xml:id="echoid-s8825" xml:space="preserve">ita ſequenti Problemate demonſtrabitur.</s> <s xml:id="echoid-s8826" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div917" type="section" level="1" n="367"> <head xml:id="echoid-head376" xml:space="preserve">PROBL. XVII. PROP. XCIV.</head> <p> <s xml:id="echoid-s8827" xml:space="preserve">Ad diametrum datæ ſemi - Ellipſis rectam applicare, cuius <lb/>rectangulum in alterum diametri ſegmentum ſit MAXIMVM.</s> <s xml:id="echoid-s8828" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8829" xml:space="preserve">1. </s> <s xml:id="echoid-s8830" xml:space="preserve">ESto ſemi - Ellipſis A D B, cuius centrum C, & </s> <s xml:id="echoid-s8831" xml:space="preserve">diameter A B, ad quam <lb/>applicare oporteat D E, ita vt rectangulum A E D ſit _MAXIMVM._ <lb/></s> <s xml:id="echoid-s8832" xml:space="preserve">Secetur B C bifariam in E, appliceturque E D, quæ erit quæſita.</s> <s xml:id="echoid-s8833" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8834" xml:space="preserve">Nam deſcripto ſuper A B ſemi - circulo A F B, erigatur ex E ipſi A B <lb/>perpendicularis E F. </s> <s xml:id="echoid-s8835" xml:space="preserve">Patet ex præcedenti Scholio, rectangulum A E F <lb/>eſſe _MAXIMVM_ in ſemi - circulo, &</s> <s xml:id="echoid-s8836" xml:space="preserve">c. </s> <s xml:id="echoid-s8837" xml:space="preserve">cum A E ſit tripla E B.</s> <s xml:id="echoid-s8838" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8839" xml:space="preserve">Sumatur ampliùs quodlibet aliud <lb/>punctum G, præter E, applicenturq; <lb/></s> <s xml:id="echoid-s8840" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/> tum in ſemi - circulo, tum in ſemi - El-<lb/>lipſi rectæ G H, G I. </s> <s xml:id="echoid-s8841" xml:space="preserve">Et cum ſit qua-<lb/>dratum E F ad G H vt rectangulum <lb/>A E B ad A G B, vel vt <anchor type="note" xlink:href="" symbol="*"/> quadratum <anchor type="note" xlink:label="note-0317-01a" xlink:href="note-0317-01"/> E D ad G I, erit & </s> <s xml:id="echoid-s8842" xml:space="preserve">linea E F ad G H, <lb/>vt E D ad G I, ſed ratio rectanguli A <lb/>E F ad rectangulum A G H compo-<lb/>nitur ex ratione E F, ad G H, ſiue ex <lb/>ratione E D ad G I, & </s> <s xml:id="echoid-s8843" xml:space="preserve">ex ratione E <lb/>A ad A G, atque rectangulum A E D <lb/>ad A G I ex ijſdem componitur ratio-<lb/>nibus, vnde rectangulum A E F ad A <lb/>G H erit vt rectangulum A E D ad A G I, & </s> <s xml:id="echoid-s8844" xml:space="preserve">hoc ſemper, ſed eſt rectangu-<lb/>lum A E F _MAXIMVM_ in ſemi - circulo, ergo, & </s> <s xml:id="echoid-s8845" xml:space="preserve">A E D erit _MAXIMVM_ <lb/>in ſemi - Ellipſi. </s> <s xml:id="echoid-s8846" xml:space="preserve">Applicatum eſt ergo, &</s> <s xml:id="echoid-s8847" xml:space="preserve">c. </s> <s xml:id="echoid-s8848" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s8849" xml:space="preserve"/> </p> <div xml:id="echoid-div917" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0317-01"/> </figure> <note symbol="*" position="right" xlink:label="note-0317-01" xlink:href="note-0317-01a" xml:space="preserve">21. pri-<lb/>mi conic.</note> </div> <p> <s xml:id="echoid-s8850" xml:space="preserve">2. </s> <s xml:id="echoid-s8851" xml:space="preserve">QVod autem eorum, quæ hinc inde à puncto D applicantur, nempe de <lb/>rectangulis A L M, A G I id, quod _MAXIMO_ propius eſt maius ſit <lb/>remotiori, eadem penitus arte nuper adhibita oſtendetur, ſi ex L <lb/>in ſemi - circulo applicetur L N. </s> <s xml:id="echoid-s8852" xml:space="preserve">Nam eodem argumento demonſtrabitur <lb/>rectangulum A L M ad A G I, eſſe vt A L N ad A G H, ſed A L N maius <lb/>eſt A G H, prout in præcedenti ad num. </s> <s xml:id="echoid-s8853" xml:space="preserve">2. </s> <s xml:id="echoid-s8854" xml:space="preserve">concluſum fuit, ergo & </s> <s xml:id="echoid-s8855" xml:space="preserve">rectan-<lb/>gulum A L M maius erit rectangulo A G I, & </s> <s xml:id="echoid-s8856" xml:space="preserve">hoc ſemper verum eſt, tum <pb o="132" file="0318" n="318" rhead=""/> de applicatis ad puncta arcus A I D, tum de ijs, quæ pertingunt ad puncta <lb/>reliqui arcus D B, hoc eſt prædicta rectangula hinc inde à puncto D, con-<lb/>tinuè decreſcere, quò magis diſtant à _MAXIMO_ rectangulo A E D.</s> <s xml:id="echoid-s8857" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s8858" xml:space="preserve">Hinc ſoluendum fit obuiam Problema huiuſmodi.</s> <s xml:id="echoid-s8859" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div919" type="section" level="1" n="368"> <head xml:id="echoid-head377" xml:space="preserve">PROBL. XVIII. PROP. XCV.</head> <p> <s xml:id="echoid-s8860" xml:space="preserve">In dato ſemi - circulo, vel ſemi - Ellipſi, hinc inde à MA-<lb/>XIMO rectangulo nuper inuento, bina æqualia rectangula re-<lb/>perire.</s> <s xml:id="echoid-s8861" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8862" xml:space="preserve">SIt datus ſemi- circulus, vel ſemi-Ellipſis, cuius diameter A B, centrum <lb/>C, & </s> <s xml:id="echoid-s8863" xml:space="preserve">punctum, ad quod peruenit _MAXIMVM_ rectangulum, ſit D, <lb/>(quod habebitur ſi diameter A B ſecetur in L, ita vt A L ſit <anchor type="note" xlink:href="" symbol="a"/> tripla L B, <anchor type="note" xlink:label="note-0318-01a" xlink:href="note-0318-01"/> & </s> <s xml:id="echoid-s8864" xml:space="preserve">applicetur L D,) ſitque exempli gratia è quolibet puncto E arcus A E <lb/>D, applicata E F ad diametrum A B, & </s> <s xml:id="echoid-s8865" xml:space="preserve">oporteat in reliquo arcu D B pun-<lb/>ctum G reperire, ita vt ducta G H ipſi E F parallela, rectangula A F E, A <lb/>H G inter ſe ſint æqualia.</s> <s xml:id="echoid-s8866" xml:space="preserve"/> </p> <div xml:id="echoid-div919" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0318-01" xlink:href="note-0318-01a" xml:space="preserve">Schol. <lb/>93. h. & <lb/>ex 94. h.</note> </div> <p> <s xml:id="echoid-s8867" xml:space="preserve">Ducatur ex A ſectionem contingens A I, quę ipſis applicatis æquidiſta-<lb/>bit, atque in angulo aſymptotali I A B per punctum E deſcribatur <anchor type="note" xlink:href="" symbol="b"/> Hy- <anchor type="note" xlink:label="note-0318-02a" xlink:href="note-0318-02"/> perbole E G. </s> <s xml:id="echoid-s8868" xml:space="preserve">Dico hanc neceſſariò in aliquo puncto circuli arcum D B ſe-<lb/>care, vt in G, & </s> <s xml:id="echoid-s8869" xml:space="preserve">hoc eſſe quæſitum, atque vnicum.</s> <s xml:id="echoid-s8870" xml:space="preserve"/> </p> <div xml:id="echoid-div920" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0318-02" xlink:href="note-0318-02a" xml:space="preserve">4. ſec. <lb/>Conic.</note> </div> <p> <s xml:id="echoid-s8871" xml:space="preserve">Etenim demiſſa ordinata D L, cum hæc aſymptoto A I æquidiſtet, ipſa <lb/>neceſſariò Hyperbolen E G ſecabit, <anchor type="note" xlink:href="" symbol="c"/> at in <anchor type="note" xlink:label="note-0318-03a" xlink:href="note-0318-03"/> vno tantùm puncto, veluti in M, & </s> <s xml:id="echoid-s8872" xml:space="preserve">ob Hy-<lb/> <anchor type="figure" xlink:label="fig-0318-01a" xlink:href="fig-0318-01"/> perbolen, erit rectangulum A L M <anchor type="note" xlink:href="" symbol="d"/> æquale <anchor type="note" xlink:label="note-0318-04a" xlink:href="note-0318-04"/> rectangulo A F E, ſed eſt rectangulùm A L <lb/>D maius eodem rectangulo A F E, cum ſit <lb/>_MAXIMVM_, ex hypotheſi, ergo idem rectan-<lb/>gulum A L D maius erit rectangulo A L M, <lb/>atq; </s> <s xml:id="echoid-s8873" xml:space="preserve">eſt A L communis eorum altitudo, qua-<lb/>re L D maior erit L M. </s> <s xml:id="echoid-s8874" xml:space="preserve">Hyperbole igitur E <lb/>G ſecat omnino D L inter D, & </s> <s xml:id="echoid-s8875" xml:space="preserve">L, vnde & </s> <s xml:id="echoid-s8876" xml:space="preserve"><lb/>producta neceſſariò ſecabit peripheriam arcus <lb/>D B, cum ſpatium L D B ſit vndique clau-<lb/>ſum, & </s> <s xml:id="echoid-s8877" xml:space="preserve">Hyperbole ſit infinitæ productionis: <lb/></s> <s xml:id="echoid-s8878" xml:space="preserve">ſecet igitur in G. </s> <s xml:id="echoid-s8879" xml:space="preserve">Dico punctum G quæſitum ſoluere, vt ſatis patet, cùm <lb/>rectangulum G H A, ob Hyperbolen, ſit <anchor type="note" xlink:href="" symbol="e"/> æquale rectangulo E F A.</s> <s xml:id="echoid-s8880" xml:space="preserve"/> </p> <div xml:id="echoid-div921" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0318-03" xlink:href="note-0318-03a" xml:space="preserve">Coroll. <lb/>11. primi <lb/>huius.</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/QN4GHYBF/figures/0318-01"/> </figure> <note symbol="d" position="left" xlink:label="note-0318-04" xlink:href="note-0318-04a" xml:space="preserve">12. ſec. <lb/>Conic.</note> </div> <note symbol="e" position="left" xml:space="preserve">ibidem.</note> <p> <s xml:id="echoid-s8881" xml:space="preserve">Quod autem in nullo alio puncto, præter in E, & </s> <s xml:id="echoid-s8882" xml:space="preserve">G, huiuſmodi Hyper-<lb/>bole arcui A D, vel arcui D B occurrat, manifeſtum eſt: </s> <s xml:id="echoid-s8883" xml:space="preserve">nam ſi alibi oc-<lb/>curreret, vt in N; </s> <s xml:id="echoid-s8884" xml:space="preserve">eſſet ob Hyperbolen, rectangulum pertingens ad N <lb/>æquale rectangulo A F E, quod eſt falſum, quoniam ob circulum, vel El-<lb/>lipſim, quando punctum N eſt inter E, & </s> <s xml:id="echoid-s8885" xml:space="preserve">D, rectangulum ad N maius eſt <lb/>quàm rectangulum ad E, & </s> <s xml:id="echoid-s8886" xml:space="preserve">ſi fuerit inter A, & </s> <s xml:id="echoid-s8887" xml:space="preserve">E, ipſo rectangulo ad E <pb o="123" file="0319" n="319" rhead=""/> minus eſt, <anchor type="note" xlink:href="" symbol="*"/> prout in præcedenti demonſtratum fuit: </s> <s xml:id="echoid-s8888" xml:space="preserve">idemque ſequetur, ſi <anchor type="note" xlink:label="note-0319-01a" xlink:href="note-0319-01"/> dicatur Hyperbolen alibi quàm in G arcui D B occurrere. </s> <s xml:id="echoid-s8889" xml:space="preserve">Itaque inuenta <lb/>ſunt in ſemi - circulo, vel ſemi - Ellipſi vltrò citròque à _MAXIMO_ rectangu-<lb/>lo, duo rectangula inter ſe æqualia. </s> <s xml:id="echoid-s8890" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s8891" xml:space="preserve"/> </p> <div xml:id="echoid-div922" type="float" level="2" n="4"> <note symbol="*" position="right" xlink:label="note-0319-01" xlink:href="note-0319-01a" xml:space="preserve">94. h.</note> </div> </div> <div xml:id="echoid-div924" type="section" level="1" n="369"> <head xml:id="echoid-head378" xml:space="preserve">PROBL. XIX. PROP. XCVI.</head> <p> <s xml:id="echoid-s8892" xml:space="preserve">In quocunque Cono terminato, ex infinitis Parabolæ portioni-<lb/>bus, quæ à planis inter ſe æquidiſtantibus, iuxta quodlibet Coni <lb/>latus, tanquam regulam ductis, in ipſo Cono procreantur, MA-<lb/>XIMAM aſſignare.</s> <s xml:id="echoid-s8893" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8894" xml:space="preserve">ESto Conus quicunque terminatus A B C, cuius vertex B, baſis circu-<lb/>lus A C, & </s> <s xml:id="echoid-s8895" xml:space="preserve">quodcunque triangulum per axem ductum ſit A B C. <lb/></s> <s xml:id="echoid-s8896" xml:space="preserve">Patet, ſi huinſmodi Conus, & </s> <s xml:id="echoid-s8897" xml:space="preserve">triangulum per axem alio plano ſecetur, quo-<lb/>rum communis ſectio D E æquidiſtet alterutri laterum trianguli per axem, <lb/>nempe B C, & </s> <s xml:id="echoid-s8898" xml:space="preserve">communis ſectio plani ſecantis per D E cum baſi A C, quę <lb/>ſit F G, ſit ad baſim A C trianguli per axem perpendicularis, patet inquam <lb/>ſectionem in Cono genitam G E F (quam vocò factam iuxta latus B C, <lb/>quod communi ſectioni E D æquidiſtat) ſemper eſſe <anchor type="note" xlink:href="" symbol="a"/> quandam Parabolæ <anchor type="note" xlink:label="note-0319-02a" xlink:href="note-0319-02"/> portionem: </s> <s xml:id="echoid-s8899" xml:space="preserve">quæritur modò, quæ ſit _MAXIMA_ harum æquidiſtantium infi-<lb/>nitarum Parabolæ portionum in Cono, iuxta latus B C, tanquam regulam, <lb/>progenitarum.</s> <s xml:id="echoid-s8900" xml:space="preserve"/> </p> <div xml:id="echoid-div924" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-0319-02" xlink:href="note-0319-02a" xml:space="preserve">1. primi <lb/>huius.</note> </div> <p> <s xml:id="echoid-s8901" xml:space="preserve">Secetur diameter A C in D, ita vt A <lb/> <anchor type="figure" xlink:label="fig-0319-01a" xlink:href="fig-0319-01"/> D ſit tripla ad D C, & </s> <s xml:id="echoid-s8902" xml:space="preserve">per D agatur pla-<lb/>num iuxta regulam B C, vti dictum eſt, <lb/>ſectionem faciens Parabolen G E F. </s> <s xml:id="echoid-s8903" xml:space="preserve">Di-<lb/>co hanc eſſe _MAXIMAM_ quæſitam.</s> <s xml:id="echoid-s8904" xml:space="preserve"/> </p> <div xml:id="echoid-div925" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0319-01"/> </figure> </div> <p> <s xml:id="echoid-s8905" xml:space="preserve">Secto enim Cono, quocunque alio <lb/>plano iuxta eandem regulam B C, quod <lb/>ſectionem faciat Parabolen H I K, cuius <lb/>communis ſectio cum triangulo per axem <lb/>ſit I L, cum circulo verò ſit K L H, erit <lb/>D E ipſi L I, & </s> <s xml:id="echoid-s8906" xml:space="preserve">F D ipſi K L <anchor type="note" xlink:href="" symbol="b"/> parallela, <anchor type="note" xlink:label="note-0319-03a" xlink:href="note-0319-03"/> quare angulus F D E angulo K L I æqua-<lb/>lis <anchor type="note" xlink:href="" symbol="c"/> erit, vnde, ſi concipiantur iungi re- <anchor type="note" xlink:label="note-0319-04a" xlink:href="note-0319-04"/> ctæ F E, K I, triangula F D E, K L I cum <lb/>ſint æquiangula ad D, L, habebunt rationem compoſitam ex latere E D <lb/>ad I L, ſiue ex D A ad A L, & </s> <s xml:id="echoid-s8907" xml:space="preserve">ex D F ad L K, ſed rectangulum quoque <lb/>A D F, ad rectangulum A L K habet rationem ex ijſdem rationibus com-<lb/>poſitam, ergo triangulum E D F ad I L H erit vt rectangulum A D F ad A <lb/>L K, ſed rectangulum A D F maius eſt ipſo A L K, cum ſit <anchor type="note" xlink:href="" symbol="d"/> _MAXIMVM_, <anchor type="note" xlink:label="note-0319-05a" xlink:href="note-0319-05"/> ergo & </s> <s xml:id="echoid-s8908" xml:space="preserve">triangulum E D F ipſo I L K maius erit, & </s> <s xml:id="echoid-s8909" xml:space="preserve">ſumptis duplis <anchor type="note" xlink:href="" symbol="e"/> ſuperbi- <anchor type="note" xlink:label="note-0319-06a" xlink:href="note-0319-06"/> partibus tertijs, erit Parabolæ portio G E F maior Parabolæ portione H I <lb/>K, & </s> <s xml:id="echoid-s8910" xml:space="preserve">hoc ſemper, vbicunque æquidiſtans planum ducatur extra G E F <pb o="134" file="0320" n="320" rhead=""/> iuxta regulam B C: </s> <s xml:id="echoid-s8911" xml:space="preserve">quare Parabolica portio G E F, aliarum, iuxta ean-<lb/>dem regulam B C progenitarum, eſt _MAXIMA._ </s> <s xml:id="echoid-s8912" xml:space="preserve">Quod inuenire propoſi-<lb/>tum fuerat.</s> <s xml:id="echoid-s8913" xml:space="preserve"/> </p> <div xml:id="echoid-div926" type="float" level="2" n="3"> <note symbol="b" position="right" xlink:label="note-0319-03" xlink:href="note-0319-03a" xml:space="preserve">16. vnd. <lb/>Elem.</note> <note symbol="c" position="right" xlink:label="note-0319-04" xlink:href="note-0319-04a" xml:space="preserve">10. ibid.</note> <note symbol="d" position="right" xlink:label="note-0319-05" xlink:href="note-0319-05a" xml:space="preserve">93 h.</note> <note symbol="e" position="right" xlink:label="note-0319-06" xlink:href="note-0319-06a" xml:space="preserve">17. pri-<lb/>mi h.</note> </div> </div> <div xml:id="echoid-div928" type="section" level="1" n="370"> <head xml:id="echoid-head379" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s8914" xml:space="preserve">HInc eſt, quod _MAXIMAE_ Parabolæ iuxta quæuis Coni latera genitæ, <lb/>habent baſes æquales: </s> <s xml:id="echoid-s8915" xml:space="preserve">nam ipſæ baſes, vti conſtat ex ſuperiori con-<lb/>ſtructione æqualiter diſtant à centro circuli (qui eſt baſis Coni) ſiue per <lb/>quadrantem ſui ipſius diametri, ac propterea inter ſe ſunt æquales.</s> <s xml:id="echoid-s8916" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div929" type="section" level="1" n="371"> <head xml:id="echoid-head380" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s8917" xml:space="preserve">SI hinc inde à _MAXIMA_ inuenta Parabolica ſectione, quærantur binæ <lb/>æquales, id facili negotio conſequetur, & </s> <s xml:id="echoid-s8918" xml:space="preserve">conſimilibus argumentis, ac <lb/>ſupra demonſtrabitur, eas nimirum æquales eſſe inter ſe, quæ ductæ ſint ex <lb/>punctis in circuli diametro A C, hinc inde à puncto D æqualia rectangula <lb/>præſtantibus.</s> <s xml:id="echoid-s8919" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8920" xml:space="preserve">Si autem quæratur inter has _MAXIMAS_ Parabolicas ſectiones, iuxta in-<lb/>finita Conilatera genitas, quæ ſit _MAXIMA_, quæue _MINIMA_, hoc, non-<lb/>nullis præmiſſis, proximo Problemate venabimur, ſed tantummodò in Co-<lb/>no Scaleno, nam in recto, ſatis ſuperque patet, omnes huiuſmodi _MAXI-_ <lb/>_MAS_ inter ſe æquales eſſe, cùm omnia triangula per axem Coni recti, ſint <lb/>ad baſim erecta, æqualia, æquicruria, & </s> <s xml:id="echoid-s8921" xml:space="preserve">æqualium laterum, &</s> <s xml:id="echoid-s8922" xml:space="preserve">c.</s> <s xml:id="echoid-s8923" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div930" type="section" level="1" n="372"> <head xml:id="echoid-head381" xml:space="preserve">THEOR. LXII. PROP. XCVII.</head> <p> <s xml:id="echoid-s8924" xml:space="preserve">In plano dati circuli, perpendicularium à puncto dato, quod <lb/>non ſit centrum, ſuper rectas eiuſdem circuli peripheriam contin-<lb/>gentes ducibilium, MAXIMA eſt ea, in qua centrum, MINIMA <lb/>verò, ſi punctum fuerit intra circulum, eſt reliquum diametri ſe-<lb/>gmentum; </s> <s xml:id="echoid-s8925" xml:space="preserve">ſi autem datum punctum fuerit in ipſa peripheria, vel <lb/>extra, tunc non datur MINIMA.</s> <s xml:id="echoid-s8926" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8927" xml:space="preserve">ESto circulus A B, cuius centrum C, & </s> <s xml:id="echoid-s8928" xml:space="preserve">datum punctum vbicunque ſit <lb/>D præter in centro, & </s> <s xml:id="echoid-s8929" xml:space="preserve">iuncta D C, ac producta vſque ad peripheriam <lb/>in A, B punctis, è quibus ductis contingentibus A E, B L (quæ diametro <lb/>A B perpendiculares erunt) & </s> <s xml:id="echoid-s8930" xml:space="preserve">ex quolibet alio peripheriæ puncto F, ducta <lb/>item contingente F H, ſuper qua ex dato puncto D demiſſa ſit perpendicu-<lb/>laris D H, &</s> <s xml:id="echoid-s8931" xml:space="preserve">c. </s> <s xml:id="echoid-s8932" xml:space="preserve">Dico huiuſmodi perpendicularium _MAXIMAM_ eſſe D A, <lb/>in qua eſt centrum C, & </s> <s xml:id="echoid-s8933" xml:space="preserve">in prima figura, in qua punctum cadit intra, _MI-_ <lb/>_NIMAM_ eſſe D B: </s> <s xml:id="echoid-s8934" xml:space="preserve">ſi verò datum punctum D cadat in ipſam peripheriam, <lb/>vt in B, vel extra, vt in ſecunda figura, tunc dico non dari _MINIMAM._</s> <s xml:id="echoid-s8935" xml:space="preserve"/> </p> <pb o="135" file="0321" n="321" rhead=""/> <p> <s xml:id="echoid-s8936" xml:space="preserve">Ex centro C ad punctum contactus F ducatur radius C F; </s> <s xml:id="echoid-s8937" xml:space="preserve">patet ipſum <lb/>cum contingente F H rectum angulum efficere, ſed angulus quoque D H F, <lb/>rectus eſt ex hypotheſi, quare D H ipſi C F eſt parallela, vnde perpendi-<lb/>cularis D H, occurrit tangenti extra punctum contactus F. </s> <s xml:id="echoid-s8938" xml:space="preserve">Iungatur de-<lb/>nique D F, &</s> <s xml:id="echoid-s8939" xml:space="preserve">c.</s> <s xml:id="echoid-s8940" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8941" xml:space="preserve">Cum enim ex puncto D in circuli peripheriam cadant rectæ D A, D F, <lb/>D B, &</s> <s xml:id="echoid-s8942" xml:space="preserve">c. </s> <s xml:id="echoid-s8943" xml:space="preserve">patet, ex elementis, D A, in qua eſt centrum, _MAXIMAM_ eſſe, <lb/>nempe maiorem D F, ſed eſt obliqua D F maior perpendiculari D H, er-<lb/>go D A eò magis maior erit D H. </s> <s xml:id="echoid-s8944" xml:space="preserve">Quod D A quoque ſit maior D B, pa-<lb/>tet cum ipſa ſit diametri ſegmentum, in quo eſt centrum, & </s> <s xml:id="echoid-s8945" xml:space="preserve">hoc ſemper <lb/>oſtendetur de quibuslibet alijs perpendicularibus ad contingentes; </s> <s xml:id="echoid-s8946" xml:space="preserve">ergo D <lb/>A, in qua centrum reperitur, eſt _MAXIMA_ in vtraque figura, etiam ſi da-<lb/>tum punctum cadat in ipſam peripheriam.</s> <s xml:id="echoid-s8947" xml:space="preserve"/> </p> <figure> <image file="0321-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0321-01"/> </figure> <p> <s xml:id="echoid-s8948" xml:space="preserve">In prima verò, iam eſt D B minor D A; </s> <s xml:id="echoid-s8949" xml:space="preserve">item eſt D B minor D G, eſtq; <lb/></s> <s xml:id="echoid-s8950" xml:space="preserve">D G minor D H, ergo D B eò ampliùs eſt minor D H, & </s> <s xml:id="echoid-s8951" xml:space="preserve">hoc ſemper de <lb/>qualibet perpendiculari ad quamcunque contingentem, pręter ad punctum <lb/>D; </s> <s xml:id="echoid-s8952" xml:space="preserve">quare, dum datum punctum D cadit intra circulum, _MINIMA_ eſt D <lb/>B reliquum diametri ſe gmentum, dempta _MAXIMA_.</s> <s xml:id="echoid-s8953" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8954" xml:space="preserve">Si autem datum punctum incidat in ipſam peripheriam, vt in B: </s> <s xml:id="echoid-s8955" xml:space="preserve">patet <lb/>perpendicularem ex B, ſuper contingentem ex eodem B ductam, pun-<lb/>ctum euadere, ac propterea non dari _MINIMAM_, niſi dicatur illud idem <lb/>punctum eſſe _MINIMAM_.</s> <s xml:id="echoid-s8956" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8957" xml:space="preserve">Si tandem punctum D cadat extra, vt in ſecunda figura: </s> <s xml:id="echoid-s8958" xml:space="preserve">ducta ex D <lb/>circulum contingente D I, conſtat pariter perpendicularem ductam ex D <lb/>ſuper ipſam D I in punctum abire, ac ideo in hoc etiam caſu non dari _MINI-_ <lb/>_MAM_, &</s> <s xml:id="echoid-s8959" xml:space="preserve">c. </s> <s xml:id="echoid-s8960" xml:space="preserve">Quod vltimò probandum erat.</s> <s xml:id="echoid-s8961" xml:space="preserve"/> </p> <pb o="136" file="0322" n="322" rhead=""/> </div> <div xml:id="echoid-div931" type="section" level="1" n="373"> <head xml:id="echoid-head382" xml:space="preserve">THEOR. LXIII. PROP. XCVIII.</head> <p> <s xml:id="echoid-s8962" xml:space="preserve">Perpendicularium à vertice Coniſcaleni ſuper rectas baſis peri-<lb/>pheriam contingentes ducibilium, MAXIMA eſt, quæ ſuper con-<lb/>tingentẽ extermino MAXIMI lateris Coni ducitur, ſiue eſt ipſum <lb/>MAXIMVM Coni latus: </s> <s xml:id="echoid-s8963" xml:space="preserve">& </s> <s xml:id="echoid-s8964" xml:space="preserve">dum veſtigium verticis cadit intra ba-<lb/>ſim, vel in ipſius peripheriam, MINIMA eſt, quæ ſuper contin-<lb/>gentem ex termino MINIMI lateris, ſiue eſt idem latus MINI-<lb/>MVM: </s> <s xml:id="echoid-s8965" xml:space="preserve">dum autem cadit extra, MINIMA eſt, quæ cadit ſuper <lb/>contingentem ductam à puncto veſtigij verticis ad eandem baſis <lb/>peripheriam, ſiue MINIMA eſt ipſa Coni altitudo.</s> <s xml:id="echoid-s8966" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8967" xml:space="preserve">ESto Conus ſcalenus A B C, cuius vertex B, baſis A C, centrum D, & </s> <s xml:id="echoid-s8968" xml:space="preserve"><lb/>altitudo B E baſi occurrens in puncto E (quod verticis veſtigium vo-<lb/>co,) quod vel cadat intra baſim, vt in prima figura, vel in ipſam peripheriã, <lb/>vt in ſecunda, vel extra, vt in tertia, per quàm B E, & </s> <s xml:id="echoid-s8969" xml:space="preserve">per centrum D con-<lb/>cipiatur ductum planum efficiens in Cono triangulum A B C, quod rectum <lb/>erit <anchor type="note" xlink:href="" symbol="a"/> ad planum circuli A C, eritque triangulum ſcalenum, cuius maius la- <anchor type="note" xlink:label="note-0322-01a" xlink:href="note-0322-01"/> tus, nempe B A erit <anchor type="note" xlink:href="" symbol="b"/> _MAXIMVM_, minus verò B C _MINIMVM_ laterum, <anchor type="note" xlink:label="note-0322-02a" xlink:href="note-0322-02"/> à vertice B ad baſis circumferentiam ducibilium.</s> <s xml:id="echoid-s8970" xml:space="preserve"/> </p> <div xml:id="echoid-div931" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0322-01" xlink:href="note-0322-01a" xml:space="preserve">14. ſe-<lb/>cundi Se-<lb/>reni.</note> <note symbol="b" position="left" xlink:label="note-0322-02" xlink:href="note-0322-02a" xml:space="preserve">15. ibid.</note> </div> <p> <s xml:id="echoid-s8971" xml:space="preserve">Præterea ex terminis diametri A, C, contingant peripheriam rectæ A <lb/>F, H C, & </s> <s xml:id="echoid-s8972" xml:space="preserve">ducto per axem quolibet alio plano efficiente triangulum I B L <lb/>obliquũ ad planum baſis A C, ex terminis I, L alterius diametri I D L, agan-<lb/>tur contingentes I M, L N, & </s> <s xml:id="echoid-s8973" xml:space="preserve">hoc fiat vt contingit, &</s> <s xml:id="echoid-s8974" xml:space="preserve">c. </s> <s xml:id="echoid-s8975" xml:space="preserve">Dico perpendicula-<lb/>rium, quæ à vertice B ad ipſas contingentes A F, C H, I M, L N, &</s> <s xml:id="echoid-s8976" xml:space="preserve">c. </s> <s xml:id="echoid-s8977" xml:space="preserve">du-<lb/>ci poſſunt, in ſigulis caſibus, _MAXIMAM_ eſſe, quæ ſuper A F, atque eam <lb/>eſſe ipſum _MAXIMVM_ latus B A: </s> <s xml:id="echoid-s8978" xml:space="preserve">in primò autem, & </s> <s xml:id="echoid-s8979" xml:space="preserve">ſecundò caſu _MINI-_ <lb/>_MAM_ eſſe, quæ ſuper C H, atque hanc eſſe, ipſum _MINIMVM_ latus B C: <lb/></s> <s xml:id="echoid-s8980" xml:space="preserve">in tertio denique ſi ex puncto veſtigij E ducatur E G peripheriam baſis <lb/>contingens. </s> <s xml:id="echoid-s8981" xml:space="preserve">Dico earundem perpendicularium _MINIMAM_ eſſe, quæ ſu-<lb/>per E G ducitur, & </s> <s xml:id="echoid-s8982" xml:space="preserve">hanc eſſe ipſam altitudinem B E.</s> <s xml:id="echoid-s8983" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8984" xml:space="preserve">Etenim, in ſingulis figuris, cum triangulum A B C ſit, ex hypotheſi re-<lb/>ctum ad planum baſis A C, & </s> <s xml:id="echoid-s8985" xml:space="preserve">ad communem eorum ſectionem A C ſit F A <lb/>perpendicularis (nam eſt A F contingens circulum, & </s> <s xml:id="echoid-s8986" xml:space="preserve">A D centrum iun-<lb/>gens) erit eadem F A recta ad planum A B C, ac propterea recta erit quo-<lb/>que ad A B, quæ eſt in eodem plano A B C, in quo eſt A C, hoc eſt B A <lb/>perpendicularis erit ſuper contingentem A F; </s> <s xml:id="echoid-s8987" xml:space="preserve">eadem ratione oſtendetur B <lb/>C perpendicularem eſſe ad contingentem C H.</s> <s xml:id="echoid-s8988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8989" xml:space="preserve">Præterea ducta ex E recta M E N parallela ad I L, cum anguli D I M, D <lb/>L N ſint recti, à contingentibus cum radijs conſtituti, erunt quoque reliqui <lb/>parallelarum interni I M E, L N E recti. </s> <s xml:id="echoid-s8990" xml:space="preserve">Iungantur denique B M, B N. <lb/></s> <s xml:id="echoid-s8991" xml:space="preserve">Et cum B E ſit recta ad planum baſis A C, erit etiam planum trianguli <lb/>M B N, quod per eam ducitur, rectum <anchor type="note" xlink:href="" symbol="c"/> ad ipſam baſim, ſiue baſis recta ad <anchor type="note" xlink:label="note-0322-03a" xlink:href="note-0322-03"/> triangulum M B N, eſtque I M perpendicularis ad eorum communem ſe- <pb o="137" file="0323" n="323" rhead=""/> ctionem M N, vt modò oſtendimus, ergo, & </s> <s xml:id="echoid-s8992" xml:space="preserve">ad rectam M B, quæ eſt in <lb/>eodem trianguli plano perpendicularis erit, ſiue B M perpendicularis ſuper <lb/>I M: </s> <s xml:id="echoid-s8993" xml:space="preserve">eodem modo oſtendetur B N perpendicularem eſſe ad L N.</s> <s xml:id="echoid-s8994" xml:space="preserve"/> </p> <div xml:id="echoid-div932" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-0322-03" xlink:href="note-0322-03a" xml:space="preserve">18. vnd. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s8995" xml:space="preserve">1. </s> <s xml:id="echoid-s8996" xml:space="preserve"><emph style="sc">Iam</emph> perpendicularis B A maior eſt B C, cum B A ſit _MAXIMVM_ Coni <lb/>latus, & </s> <s xml:id="echoid-s8997" xml:space="preserve">B C _MINIMVM_, vt ſupra monuimus; </s> <s xml:id="echoid-s8998" xml:space="preserve">ob eandem rationem eſt <lb/>B A maior B I, ſed B I maior eſt B M, cum B M ſit perpendicularis ad I <lb/>M, ac ideo _MINIMA_ ad ipſam I M, ergo B A eò magis maior erit per-<lb/>pendiculari B M: </s> <s xml:id="echoid-s8999" xml:space="preserve">eodem modo demonſtrabitur B A maiorem eſſe perpen-<lb/>diculari B N, & </s> <s xml:id="echoid-s9000" xml:space="preserve">hoc ſemper, &</s> <s xml:id="echoid-s9001" xml:space="preserve">c. </s> <s xml:id="echoid-s9002" xml:space="preserve">quare in ſingulis caſibus _MAXIMVM_ <lb/>Conilatus B A eſt _MAXIMA_ prædictarum perpendicularium.</s> <s xml:id="echoid-s9003" xml:space="preserve"/> </p> <figure> <image file="0323-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0323-01"/> </figure> <p> <s xml:id="echoid-s9004" xml:space="preserve">2. </s> <s xml:id="echoid-s9005" xml:space="preserve">QVo autem ad _MINIMAM_ in prima figura. </s> <s xml:id="echoid-s9006" xml:space="preserve">Eſt B C minor B A, cum ea <lb/>ſit _MINIMVM_ Coni latus. </s> <s xml:id="echoid-s9007" xml:space="preserve">Ampliùs eſt <anchor type="note" xlink:href="" symbol="a"/> perpendicularis E C minor <anchor type="note" xlink:label="note-0323-01a" xlink:href="note-0323-01"/> perpendiculari E M, vnde, & </s> <s xml:id="echoid-s9008" xml:space="preserve">quadratum E C minus eſt quadra-<lb/>to E M, & </s> <s xml:id="echoid-s9009" xml:space="preserve">communi addito quadrato E B, erunt duo ſimul quadrata C E, <lb/>E B, ſiue vnicum quadratum B C, minus duobus ſimul quadratis M E, E B, <lb/>ſiue vnico quadrato B M (ponitur enim B E recta ad baſim, ac ideo cum om-<lb/>nibus E C, E M, &</s> <s xml:id="echoid-s9010" xml:space="preserve">c. </s> <s xml:id="echoid-s9011" xml:space="preserve">rectos efficit angulos) hoc eſt recta B C, quæ perpen-<lb/>dicularis eſt ad contingentem C H, minor erit recta B M, quæ eſt perpen-<lb/>dicularis ad contingentem I M; </s> <s xml:id="echoid-s9012" xml:space="preserve">eadem ratione oſtendetur B C minorem <lb/>eſſe perpendiculari B N, vel quacunque alia ex B ad quamlibet contingen-<lb/>tium ducta: </s> <s xml:id="echoid-s9013" xml:space="preserve">quare B C eſt ipſarum perpendicularium _MINIMA_.</s> <s xml:id="echoid-s9014" xml:space="preserve"/> </p> <div xml:id="echoid-div933" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-0323-01" xlink:href="note-0323-01a" xml:space="preserve">97. h.</note> </div> <p> <s xml:id="echoid-s9015" xml:space="preserve">In ſecunda verò cum altitudo B E congruat cum perpendiculari B C ad <lb/>contingentem C H, cumque eadem B E ſit <anchor type="note" xlink:href="" symbol="b"/> _MINIMA_ ad planum baſis A <anchor type="note" xlink:label="note-0323-02a" xlink:href="note-0323-02"/> C, erit etiam perpendicularis B C _MINIMA_ ad idem planum, hoc eſt _MI-_ <lb/>_NIMA_ quarumlibet perpendicularium. </s> <s xml:id="echoid-s9016" xml:space="preserve">In primo igitur, ac ſecundo caſu <lb/>recta B C, quæ eſt _MINIMVM_ Coni latus, perpendicularium ad prædi-<lb/>ctas contingentes eſt _MINIMA_.</s> <s xml:id="echoid-s9017" xml:space="preserve"/> </p> <div xml:id="echoid-div934" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-0323-02" xlink:href="note-0323-02a" xml:space="preserve">52. h.</note> </div> <p> <s xml:id="echoid-s9018" xml:space="preserve">3. </s> <s xml:id="echoid-s9019" xml:space="preserve">IN tertia denique, cum ſit recta B E ad planum baſis perpendicularis, ipſa <lb/>cum contingente E G rectos efficiet <anchor type="note" xlink:href="" symbol="c"/> angulos, ſed ipſa B E eſt <anchor type="note" xlink:href="" symbol="d"/> _MINI-_ <anchor type="note" xlink:label="note-0323-03a" xlink:href="note-0323-03"/> <anchor type="note" xlink:label="note-0323-04a" xlink:href="note-0323-04"/> _MA_ ad ipſum baſis planum, quare, & </s> <s xml:id="echoid-s9020" xml:space="preserve">_MINIMA_ quoque erit prædictarum <lb/>quarumlibet perpendicularium. </s> <s xml:id="echoid-s9021" xml:space="preserve">Quod vltimò oſtendere proponebatur.</s> <s xml:id="echoid-s9022" xml:space="preserve"/> </p> <div xml:id="echoid-div935" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-0323-03" xlink:href="note-0323-03a" xml:space="preserve">3. def. 11 <lb/>Elem.</note> <note symbol="d" position="right" xlink:label="note-0323-04" xlink:href="note-0323-04a" xml:space="preserve">52. h.</note> </div> <pb o="138" file="0324" n="324" rhead=""/> </div> <div xml:id="echoid-div937" type="section" level="1" n="374"> <head xml:id="echoid-head383" xml:space="preserve">COROLL. I.</head> <p> <s xml:id="echoid-s9023" xml:space="preserve">EX hac igitur conſtat in Cono ſcaleno, tum _MAXIMVM_, tum _MINI-_ <lb/>_MVM_ latus perpendiculare eſſe ad rectas ex eorum extremis terminis <lb/>baſis peripheriam contingentes.</s> <s xml:id="echoid-s9024" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9025" xml:space="preserve">Nam ſuperiùs primo loco demonſtrauimus rectam B A, quæ eſt _MAXI-_ <lb/>_MVM_ Coni latus, rectum angulum efficere cum contingente A F, & </s> <s xml:id="echoid-s9026" xml:space="preserve">rectam <lb/>B C, quæ eſt latus _MINIMVM_, cum contingente C H rectum pariter an-<lb/>gulum conſtituere.</s> <s xml:id="echoid-s9027" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div938" type="section" level="1" n="375"> <head xml:id="echoid-head384" xml:space="preserve">COROLL. II.</head> <p> <s xml:id="echoid-s9028" xml:space="preserve">PAtet quoque in eodem Cono ſcaleno, perpendicularem ex vertice du-<lb/>ctam ſuper aliam contingentem ad extrema baſis cuiuſcunque trian-<lb/>guli per axem non recti ad baſim Coni, eam eſſe, quæ iungit eundem verti-<lb/>cem cum interſectione ipſius tangentis cum ea recta linea, quæ à veſtigio <lb/>verticis ipſi baſi prædictitrianguli per axem æquidiſtans ducitur.</s> <s xml:id="echoid-s9029" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9030" xml:space="preserve">In triangulo enim I B L per axem ducto, ſed ſuper baſim A I C L obli-<lb/>quo, ibi demonſtratum fuit rectas B M, & </s> <s xml:id="echoid-s9031" xml:space="preserve">B N perpendiculares eſſe <lb/>ſuper contingentes I M, & </s> <s xml:id="echoid-s9032" xml:space="preserve">L N, ductas ex terminis I, & </s> <s xml:id="echoid-s9033" xml:space="preserve">L baſis I L eiuſ-<lb/>dem trianguli, atque iam puncta M, & </s> <s xml:id="echoid-s9034" xml:space="preserve">N ſunt interſectiones ipſarum tan-<lb/>gentium cum recta M E N, quæ per verticis veſtigium E æquidiſtans duci-<lb/>tur ad I L baſim trianguli.</s> <s xml:id="echoid-s9035" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div939" type="section" level="1" n="376"> <head xml:id="echoid-head385" xml:space="preserve">THEOR. LXIV. PROP. IC.</head> <p> <s xml:id="echoid-s9036" xml:space="preserve">In quocunque Cono ſcaleno, Parabolæ portiones iuxta quæli-<lb/>bet Coni latera genitæ, & </s> <s xml:id="echoid-s9037" xml:space="preserve">quarum diametri, in earum triangulis <lb/>per axem ab ijſdem lateribus proportionaliter diſtent, vel qua rum <lb/>baſes ſint æquales, habent altitudines proportionales perpendicu-<lb/>laribus, quę ducuntur à Coni vertice ſuper rectas baſis peripheriam <lb/>contingentes ad puncta, quibus eadem latera occurrunt.</s> <s xml:id="echoid-s9038" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9039" xml:space="preserve">ESto Conus ſcalenus A B C, cuius vertex B, baſis circulus A C, cen-<lb/>trum D, & </s> <s xml:id="echoid-s9040" xml:space="preserve">Coni altitudo ſit B E, per quam, & </s> <s xml:id="echoid-s9041" xml:space="preserve">per axim ductum ſit <lb/>planum ad baſim erectum, efficiens in Cono triangulum A B C: </s> <s xml:id="echoid-s9042" xml:space="preserve">& </s> <s xml:id="echoid-s9043" xml:space="preserve">iterum <lb/>ſectus ſit Conus quocunque alio plano per axem efficiente triangulum ſuper <lb/>baſim obliquum G B H, atque iuxta vtriuſque horum triangulorum latera <lb/>B A, B G tanquam regulas, cõcipiantur duci - plana, parabolicas portiones <lb/>efficientia, ita vt communis ſectio Parabolæ genitæ iuxta latus B A cum <lb/>triangulo A B C ſit recta P I, (quæ in triangulo A B C æquidiſtabit lateri <lb/>B A <anchor type="note" xlink:href="" symbol="a"/> eritque Parabolæ diameter) & </s> <s xml:id="echoid-s9044" xml:space="preserve">cum baſi A C ſit recta L I M (quæ <anchor type="note" xlink:label="note-0324-01a" xlink:href="note-0324-01"/> rectæ A D C erit perpendicularis, atque eiuſdem Parabolæ baſis) commu-<lb/>nis autem ſectio Parabolæ genitæ iuxta latus B G cum triangulo G B H, ſit <lb/>recta Q S, (quæ parallela erit ipſi B G, ac item erit <anchor type="note" xlink:href="" symbol="b"/> diameter Parabolæ) <anchor type="note" xlink:label="note-0324-02a" xlink:href="note-0324-02"/> <pb o="139" file="0325" n="325" rhead=""/> & </s> <s xml:id="echoid-s9045" xml:space="preserve">cum baſi A C erit recta N S O, (quæ ad rectam G D H erit perpendi-<lb/>cularis, & </s> <s xml:id="echoid-s9046" xml:space="preserve">ipſius Parabolæ baſis) quæ baſes inter ſe æquales erunt, cum ſint <lb/>rectæ in circulo A C à centro D æqualiter diſtantes, atque huiuſmodi Pa-<lb/>rabolarum diametri P I, Q S proportionaliter diſtent à lateribus, ſeu ab ip-<lb/>ſarum regulis B A, B G, ita vt ſit B P ad P C, vel A I ad I C, vt B Q ad <lb/>Q H, vel G S ad S H. </s> <s xml:id="echoid-s9047" xml:space="preserve">Dico altitudinem Parabolæ per P I ad altitudinem <lb/>Parabolæ per Q S (quæ ſunt Parabolæ æqualium baſium) habere eandem <lb/>rationem, ac perpendicularis ex vertice B ſuper contingentem ex A, ter-<lb/>mino lateris B A, ad perpendicularem ex B ſuper contingentem ex G, <lb/>termino lateris B G. </s> <s xml:id="echoid-s9048" xml:space="preserve">Et è conuerſo, &</s> <s xml:id="echoid-s9049" xml:space="preserve">c.</s> <s xml:id="echoid-s9050" xml:space="preserve"/> </p> <div xml:id="echoid-div939" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0324-01" xlink:href="note-0324-01a" xml:space="preserve">1. primi <lb/>buius.</note> <note symbol="b" position="left" xlink:label="note-0324-02" xlink:href="note-0324-02a" xml:space="preserve">ibidem.</note> </div> <p> <s xml:id="echoid-s9051" xml:space="preserve">Nam ſit A F baſim contingens <lb/> <anchor type="figure" xlink:label="fig-0325-01a" xlink:href="fig-0325-01"/> ad A, ſiue perpendicularis ad <lb/>diametrum A C, quę erit <anchor type="note" xlink:href="" symbol="a"/> quoq;</s> <s xml:id="echoid-s9052" xml:space="preserve"> <anchor type="note" xlink:label="note-0325-01a" xlink:href="note-0325-01"/> cum A B perpendicularis: </s> <s xml:id="echoid-s9053" xml:space="preserve">ſitque <lb/>G R contingens ad G, quæ item <lb/>cum diametro G D H rectos an-<lb/>gulos efficiet; </s> <s xml:id="echoid-s9054" xml:space="preserve">atque ex E Coni <lb/>verticis veſtigio, ducatur E R pa-<lb/>rallela ad H D G, iungaturque B <lb/>R, quæ ſuper contingentem G R <lb/>erit <anchor type="note" xlink:href="" symbol="b"/> perpendicularis, iunctaque <anchor type="note" xlink:label="note-0325-02a" xlink:href="note-0325-02"/> H R, quæ rectam G S N ſecet in <lb/>T, agatur recta Q T.</s> <s xml:id="echoid-s9055" xml:space="preserve"/> </p> <div xml:id="echoid-div940" type="float" level="2" n="2"> <figure xlink:label="fig-0325-01" xlink:href="fig-0325-01a"> <image file="0325-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0325-01"/> </figure> <note symbol="a" position="right" xlink:label="note-0325-01" xlink:href="note-0325-01a" xml:space="preserve">1. Co-<lb/>roll. 98. h.</note> <note symbol="b" position="right" xlink:label="note-0325-02" xlink:href="note-0325-02a" xml:space="preserve">2. Co-<lb/>roll. ibid.</note> </div> <p> <s xml:id="echoid-s9056" xml:space="preserve">Iam cum ſit I M parallela ad A <lb/>F, (vtraque enim perpendicularis <lb/>eſt ad A C) & </s> <s xml:id="echoid-s9057" xml:space="preserve">I P ad A B, erit <lb/>angulus P I M <anchor type="note" xlink:href="" symbol="c"/> æqualis angulo B <anchor type="note" xlink:label="note-0325-03a" xlink:href="note-0325-03"/> A F, nempe rectus, quare ipſa P I erit altitudo Parabolicæ portionis, quæ <lb/>ducitur per P I iuxta latus B A, cum ſit M I L eius baſis. </s> <s xml:id="echoid-s9058" xml:space="preserve">Præterea cum ſit <lb/>R H ad H T, vt G H ad H S, (ob parallelas R G, T S in triangulo G H R) <lb/>vel vt B H ad H Q (ob æquidiſtantes G B, S Q in triangulo G H B) erit <lb/>in triangulo R H B recta B R parallela ad Q T, eſtque R G parallela ad T <lb/>S, ergo angulus Q T S æquabitur <anchor type="note" xlink:href="" symbol="d"/> angulo B R G, ſiue rectus erit, ex quo <anchor type="note" xlink:label="note-0325-04a" xlink:href="note-0325-04"/> ipſa Q T erit altitudo Parabolicæ portionis ductæ per Q S iuxta latus <lb/>B G, cum N S O ſit baſis ipſius Parabolæ. </s> <s xml:id="echoid-s9059" xml:space="preserve">Et quoniam demonſtrata eſt B <lb/>R parallela ad Q T, erit B R ad Q T, vt B H ad H Q in triangulo B H R, <lb/>vel vt B C ad C P, ex hypotheſi, vel vt B A ad P I, ob parallelas in trian-<lb/>gulo A B C, & </s> <s xml:id="echoid-s9060" xml:space="preserve">permutando B R, quæ eſt perpendicularis ex vertice B ſu-<lb/>per contingentem G R, ad B A, quæ eſt perpendicularis ex B ſuper con-<lb/>tingentem A F, ita Q T, quæ eſt altitudo Parabolæ per Q S, ad P I, quæ <lb/>eſt altitudo Parabolæ per P I, & </s> <s xml:id="echoid-s9061" xml:space="preserve">hoc ſemper; </s> <s xml:id="echoid-s9062" xml:space="preserve">quare patet propoſitum.</s> <s xml:id="echoid-s9063" xml:space="preserve"/> </p> <div xml:id="echoid-div941" type="float" level="2" n="3"> <note symbol="c" position="right" xlink:label="note-0325-03" xlink:href="note-0325-03a" xml:space="preserve">10. vnd. <lb/>Elem.</note> <note symbol="d" position="right" xlink:label="note-0325-04" xlink:href="note-0325-04a" xml:space="preserve">ibidem.</note> </div> </div> <div xml:id="echoid-div943" type="section" level="1" n="377"> <head xml:id="echoid-head386" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s9064" xml:space="preserve">HInc eſt, quod Parabolarum in Cono genitarum, iuxta quodlibet latus <lb/>trianguli per axem ad baſem recti, eędẽ ſunt diametri, ac altitudines. <lb/></s> <s xml:id="echoid-s9065" xml:space="preserve">Superiùs enim oſtendimus diametrum Parabolæ per P I in triangulo per <lb/>axem A B C iuxta latus B A, eſſe quoque altitudinem eiuſdem Parabolæ.</s> <s xml:id="echoid-s9066" xml:space="preserve"/> </p> <pb o="140" file="0326" n="326" rhead=""/> </div> <div xml:id="echoid-div944" type="section" level="1" n="378"> <head xml:id="echoid-head387" xml:space="preserve">PROBL. XX. PROP. C.</head> <p> <s xml:id="echoid-s9067" xml:space="preserve">In dato quocunque Cono ſcaleno, MAXIMAM MAXIMA-<lb/>RVM, & </s> <s xml:id="echoid-s9068" xml:space="preserve">MAXIMARVM MINIMAM Parabolæ portionem <lb/>aſſignare.</s> <s xml:id="echoid-s9069" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9070" xml:space="preserve">ESto Conus ſcalenus A B C, cuius vertex B, baſis B C, centrum D. <lb/></s> <s xml:id="echoid-s9071" xml:space="preserve">Oportet inter _MAXIMAS._ </s> <s xml:id="echoid-s9072" xml:space="preserve">Parabolas, & </s> <s xml:id="echoid-s9073" xml:space="preserve">_MAXIMAM_, & </s> <s xml:id="echoid-s9074" xml:space="preserve">_MINIMAM_ <lb/>aſſignare.</s> <s xml:id="echoid-s9075" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9076" xml:space="preserve">Secetur Conus plano per axem, & </s> <s xml:id="echoid-s9077" xml:space="preserve">ad baſim erecto, efficiente triangulum <lb/>A B C. </s> <s xml:id="echoid-s9078" xml:space="preserve">Patet alterum ipſius laterum, vt puta B A eſſe <anchor type="note" xlink:href="" symbol="a"/> _MAXIMVM_, alte- <anchor type="note" xlink:label="note-0326-01a" xlink:href="note-0326-01"/> rum verò B C _MINIMVM._</s> <s xml:id="echoid-s9079" xml:space="preserve"/> </p> <div xml:id="echoid-div944" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-0326-01" xlink:href="note-0326-01a" xml:space="preserve">15. ſec. <lb/>Sereni.</note> </div> <p> <s xml:id="echoid-s9080" xml:space="preserve">Radius D A ad partes _MAXIMI_ lateris ſecetur bifariam in E, ita vt C <lb/>E ſit tripla ad E A; </s> <s xml:id="echoid-s9081" xml:space="preserve">& </s> <s xml:id="echoid-s9082" xml:space="preserve">per E iuxta regulam _MAXIMI_ lateris B A concipia-<lb/> <anchor type="note" xlink:label="note-0326-02a" xlink:href="note-0326-02"/> tur ductum planum efficiens Parabolen: </s> <s xml:id="echoid-s9083" xml:space="preserve">patet hanc eſſe <anchor type="note" xlink:href="" symbol="b"/> _MAXIMAM_ iuxta idem latus B A, quam dico eſſe quoque _MAXIMARVM MAXIMAM,_ vbi-<lb/>cunque cadat punctum H veſtigium verticis.</s> <s xml:id="echoid-s9084" xml:space="preserve"/> </p> <div xml:id="echoid-div945" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-0326-02" xlink:href="note-0326-02a" xml:space="preserve">96. h.</note> </div> <p> <s xml:id="echoid-s9085" xml:space="preserve">Nam _MAXIMA_ Parabole, ducta per E iuxta latus B A, ad quamlibet <lb/>aliam _MAXIMAM_ Parabolen iuxta aliud quodcunque latus, nempe iuxta <lb/>B F (cum ipſæ ſint <anchor type="note" xlink:href="" symbol="c"/> æqualium baſium) eſt homologè, vt altitudo <anchor type="note" xlink:href="" symbol="d"/> vnius ad <anchor type="note" xlink:label="note-0326-03a" xlink:href="note-0326-03"/> altitudinem alterius, ſed altitudo ad altitudinem eſt vt <anchor type="note" xlink:href="" symbol="e"/> perpendicularis ex <anchor type="note" xlink:label="note-0326-04a" xlink:href="note-0326-04"/> <anchor type="note" xlink:label="note-0326-05a" xlink:href="note-0326-05"/> B ſuper contingentem circuli B C peripheriam ad punctum A, <anchor type="note" xlink:href="" symbol="f"/> quæ eſt <anchor type="note" xlink:label="note-0326-06a" xlink:href="note-0326-06"/> ipſum latus B A, ad perpendicularem ex B ſuper contingentem ad pun-<lb/>ctum F, atque perpendicularis B A maior eſt perpendiculari ex B ſuper <lb/>contingentem ad F, cum ipſa B A ſit <anchor type="note" xlink:href="" symbol="g"/> earundem perpendicularium _MAXI_- <anchor type="note" xlink:label="note-0326-07a" xlink:href="note-0326-07"/> _MA,_ ergo, & </s> <s xml:id="echoid-s9086" xml:space="preserve">_MAXIMA_ Parabole ducta per E iuxta latus B A erit maior <lb/>_MAXIMA_ Parabola ducta iuxta latus B F, & </s> <s xml:id="echoid-s9087" xml:space="preserve">hoc ſemper, vnde ipſa ducta <lb/>per E iuxta _MAXIMVM_ Coni latus B A, erit _MAXIMARVM MAXIMA:_ <lb/></s> <s xml:id="echoid-s9088" xml:space="preserve">quod primò erat, &</s> <s xml:id="echoid-s9089" xml:space="preserve">c.</s> <s xml:id="echoid-s9090" xml:space="preserve"/> </p> <div xml:id="echoid-div946" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-0326-03" xlink:href="note-0326-03a" xml:space="preserve">Coroll. <lb/>96. h.</note> <note symbol="d" position="left" xlink:label="note-0326-04" xlink:href="note-0326-04a" xml:space="preserve">15. pri-<lb/>mi h.</note> <note symbol="e" position="left" xlink:label="note-0326-05" xlink:href="note-0326-05a" xml:space="preserve">99. h.</note> <note symbol="f" position="left" xlink:label="note-0326-06" xlink:href="note-0326-06a" xml:space="preserve">1. Co-<lb/>roll. 98. h.</note> <note symbol="g" position="left" xlink:label="note-0326-07" xlink:href="note-0326-07a" xml:space="preserve">98. h. ad <lb/>num. 1.</note> </div> <p> <s xml:id="echoid-s9091" xml:space="preserve">Præterea ſi H veſtigium verticis B ce-<lb/> <anchor type="figure" xlink:label="fig-0326-01a" xlink:href="fig-0326-01"/> ciderit, vel intra circulum B C, vel in ip-<lb/>ſius peripheria: </s> <s xml:id="echoid-s9092" xml:space="preserve">ſecto radio D C, (qui eſt <lb/>ad partem _MINIMI_ lateris B C Coni A <lb/>B C) bifariam in G, & </s> <s xml:id="echoid-s9093" xml:space="preserve">per ipſum ducto <lb/>plano iuxta regulam lateris B C efficiente <lb/> <anchor type="note" xlink:label="note-0326-08a" xlink:href="note-0326-08"/> _MAXIMA_ <anchor type="note" xlink:href="" symbol="h"/> Parabola. </s> <s xml:id="echoid-s9094" xml:space="preserve">Dico hanc eſſe _MAXIMARVM, MINIMAM_ quæſitam.</s> <s xml:id="echoid-s9095" xml:space="preserve"/> </p> <div xml:id="echoid-div947" type="float" level="2" n="4"> <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/QN4GHYBF/figures/0326-01"/> </figure> <note symbol="h" position="left" xlink:label="note-0326-08" xlink:href="note-0326-08a" xml:space="preserve">96. h.</note> </div> <p> <s xml:id="echoid-s9096" xml:space="preserve">Etenim _MAXIMA_ Parabole per G iux-<lb/>ta latus B C, ad quamcumque aliam _MA_-<lb/>_XIMAM_ iuxta quodcunque aliud latus B <lb/>F, eſt homologè <anchor type="note" xlink:href="" symbol="i"/> vt altitudo vnius ad al- <anchor type="note" xlink:label="note-0326-09a" xlink:href="note-0326-09"/> titudinem alterius, cum ipſæ ſint <anchor type="note" xlink:href="" symbol="l"/> æqua- <anchor type="note" xlink:label="note-0326-10a" xlink:href="note-0326-10"/> lium baſium; </s> <s xml:id="echoid-s9097" xml:space="preserve">ſed altitudo ad altitudinem <lb/>eſt vt <anchor type="note" xlink:href="" symbol="m"/> perpendicularis, ex B ſuper contingentem ad C, quæ <anchor type="note" xlink:href="" symbol="n"/> eſt ipſum <anchor type="note" xlink:label="note-0326-11a" xlink:href="note-0326-11"/> <anchor type="note" xlink:label="note-0326-12a" xlink:href="note-0326-12"/> _MINIMVM_ latus B C, ad perpendicularem ex B ſuper contingentem ad F, <pb o="141" file="0327" n="327" rhead=""/> & </s> <s xml:id="echoid-s9098" xml:space="preserve">perpendicularis B C minor eſt perpendiculari ex B ſuper contingentem <lb/>ad F, cum ea B C ſit ipſarum perpendicularium <anchor type="note" xlink:href="" symbol="a"/> _MINIMA_, ergo, & </s> <s xml:id="echoid-s9099" xml:space="preserve">_MA_- <anchor type="note" xlink:label="note-0327-01a" xlink:href="note-0327-01"/> _XIMA_ Parabole per G ducta iuxta Coni latus B C, erit minor _MAXIMA_ <lb/>Parabola genita iuxta latus B F, & </s> <s xml:id="echoid-s9100" xml:space="preserve">hoc ſemper; </s> <s xml:id="echoid-s9101" xml:space="preserve">quapropter ipſa _MAXI_-<lb/>_MA_ Parabole, ducta per G iuxta _MINIMVM_ Coni latus B C, in his caſi-<lb/>bus, erit _MAXIMARVM MINIMA._ </s> <s xml:id="echoid-s9102" xml:space="preserve">Quod ſecundò erat, &</s> <s xml:id="echoid-s9103" xml:space="preserve">c.</s> <s xml:id="echoid-s9104" xml:space="preserve"/> </p> <div xml:id="echoid-div948" type="float" level="2" n="5"> <note symbol="i" position="left" xlink:label="note-0326-09" xlink:href="note-0326-09a" xml:space="preserve">15. primi <lb/>huius.</note> <note symbol="l" position="left" xlink:label="note-0326-10" xlink:href="note-0326-10a" xml:space="preserve">Coroll. <lb/>96. h.</note> <note symbol="m" position="left" xlink:label="note-0326-11" xlink:href="note-0326-11a" xml:space="preserve">99. h.</note> <note symbol="n" position="left" xlink:label="note-0326-12" xlink:href="note-0326-12a" xml:space="preserve">1. Co-<lb/>roll. 98. h.</note> <note symbol="a" position="right" xlink:label="note-0327-01" xlink:href="note-0327-01a" xml:space="preserve">98. h. ad <lb/>num. 2.</note> </div> <p> <s xml:id="echoid-s9105" xml:space="preserve">Sitandem veſtigium verticis H ceciderit extra baſim Coni, vti apparet <lb/>in hac ſigura. </s> <s xml:id="echoid-s9106" xml:space="preserve">Ducta contingente H I, atque iuncta B I, ſi radius D I biſa-<lb/>riam ſecetur in puncto L, per quod iuxta latus B I ducatur planum Para-<lb/>bolen efficiens, quæ erit _MAXIMA._ </s> <s xml:id="echoid-s9107" xml:space="preserve">Dico hanc eſſe _MAXIMARVM MI_-<lb/>_NIMAM._</s> <s xml:id="echoid-s9108" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9109" xml:space="preserve">Quoniam _MAXIMA_ per L iuxta latus B I ad _MAXIMAM_ iuxta aliud <lb/>quodcunque latus B F, eſt vt <anchor type="note" xlink:href="" symbol="b"/> altitudo ad altitudinem, cum ipſæ Parabolæ <anchor type="note" xlink:label="note-0327-02a" xlink:href="note-0327-02"/> ſint <anchor type="note" xlink:href="" symbol="c"/> æqualium baſium, ſed altitudo ad altitudinem eſt vt <anchor type="note" xlink:href="" symbol="d"/> perpendicularis <anchor type="note" xlink:label="note-0327-03a" xlink:href="note-0327-03"/> ex B ſuper contingentem ad I, quæ eſt ipſa B H Coni altitudo (quæ ad <lb/>omnes rectas in plano baſis Coni ad punctum H pertingentes eſt <anchor type="note" xlink:href="" symbol="e"/> perpen- <anchor type="note" xlink:label="note-0327-04a" xlink:href="note-0327-04"/> <anchor type="note" xlink:label="note-0327-05a" xlink:href="note-0327-05"/> dicularis) ad perpendicularem ex B ſuper contingentem ad F, & </s> <s xml:id="echoid-s9110" xml:space="preserve">perpen-<lb/>dicularis B H minor eſt perpendiculari ex B ſuper contingentem ad F, <lb/>cum ipſa ſit <anchor type="note" xlink:href="" symbol="f"/> huiuſmodi perpendicularium _MINIMA,_ quare, & </s> <s xml:id="echoid-s9111" xml:space="preserve">_MAXIMA_ <anchor type="note" xlink:label="note-0327-06a" xlink:href="note-0327-06"/> Parabole iuxta latus B I, iungens Coni verticem, & </s> <s xml:id="echoid-s9112" xml:space="preserve">contactum rectæ <lb/>lineæ H I, quæ à veſtigio H ad peripheriam baſis ducitur, mi-<lb/>nor erit _MAXIMA_ Parabola iuxta latus B F, & </s> <s xml:id="echoid-s9113" xml:space="preserve">hoc ſem-<lb/>per, vnde ipſa _MAXIMA_ Parabole per L iuxta la-<lb/>tus B I, erit, in hoc caſu, _MAXIMARVM MI_-<lb/>_NIMA._ </s> <s xml:id="echoid-s9114" xml:space="preserve">Quod vltimò faciendum erat, <lb/>quodque eſto DIVINATIO-<lb/>NIS, ac</s> </p> <div xml:id="echoid-div949" type="float" level="2" n="6"> <note symbol="b" position="right" xlink:label="note-0327-02" xlink:href="note-0327-02a" xml:space="preserve">15. pri-<lb/>mi h.</note> <note symbol="c" position="right" xlink:label="note-0327-03" xlink:href="note-0327-03a" xml:space="preserve">Coroll. <lb/>96. h.</note> <note symbol="d" position="right" xlink:label="note-0327-04" xlink:href="note-0327-04a" xml:space="preserve">99. h.</note> <note symbol="e" position="right" xlink:label="note-0327-05" xlink:href="note-0327-05a" xml:space="preserve">ex def. 3. <lb/>vnd. Ele.</note> <note symbol="f" position="right" xlink:label="note-0327-06" xlink:href="note-0327-06a" xml:space="preserve">98. h. ad <lb/>num. 3.</note> </div> </div> <div xml:id="echoid-div951" type="section" level="1" n="379"> <head xml:id="echoid-head388" xml:space="preserve">LIBRISECVNDI <lb/>FINIS.</head> <pb o="142" file="0328" n="328" rhead=""/> </div> <div xml:id="echoid-div952" type="section" level="1" n="380"> <head xml:id="echoid-head389" xml:space="preserve">Pag. 53. Coroll. I. ita reſtituendum.</head> <p> <s xml:id="echoid-s9115" xml:space="preserve">HInc eſt, quod applicatæ ex terminis ęqualium diametrorum in Parabo-<lb/>la, vel (in reliquis ſectionibus) ex punctis proportionaliter diuiden-<lb/>tibus ſemi-diametros ad quemlibet angulum conſtitutas; </s> <s xml:id="echoid-s9116" xml:space="preserve">nempe quod baſes <lb/>equalium portionum de eadem coni-ſectione, vel circulo, omnino ſemu-<lb/>tuò ſecant inter diametros; </s> <s xml:id="echoid-s9117" xml:space="preserve">& </s> <s xml:id="echoid-s9118" xml:space="preserve">quod rectæ lineæ, tum harum applicatarum, <lb/>vel baſium portionum puncta media, tum extrema iungentes, rectæ ſemi-<lb/>diametrorum terminos iungentiæquidiſtant.</s> <s xml:id="echoid-s9119" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9120" xml:space="preserve">Demonſtratum eſt enim rectas H I, E C, quæ ſunt baſes æqualiũ portio-<lb/>num H E I, A B C, ſecare ſe mutuò in M inter diametros E D, B D; </s> <s xml:id="echoid-s9121" xml:space="preserve">& </s> <s xml:id="echoid-s9122" xml:space="preserve">iun-<lb/>ctas H C, G F, A I ipſi E B eſſe parallelas.</s> <s xml:id="echoid-s9123" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div953" type="section" level="1" n="381"> <head xml:id="echoid-head390" xml:space="preserve">Pag. 59. poſt Coroll. adde ſequens</head> <head xml:id="echoid-head391" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s9124" xml:space="preserve">QVod in Ellipſi demonſtratum fuit de portionibus A B C, H M I, ſemi-<lb/>Ellipſi minoribus, idem ſequitur de maioribus A H C, H C I, qua-<lb/>rum baſes A C, H I ſimilem concentricam interiorem Ellipſim. <lb/></s> <s xml:id="echoid-s9125" xml:space="preserve">contingunt; </s> <s xml:id="echoid-s9126" xml:space="preserve">nempe has quoque inter ſe æquales eſſe. </s> <s xml:id="echoid-s9127" xml:space="preserve">Nam ipſæ portiones <lb/>A H C, H C I ſunt partes ſuperſtites de eadem Ellipſi A B C H, demptis <lb/>æqualibus portionibus A B C, H M I.</s> <s xml:id="echoid-s9128" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div954" type="section" level="1" n="382"> <head xml:id="echoid-head392" xml:space="preserve">Pag. 61. poſt Coroll. II.</head> <head xml:id="echoid-head393" xml:space="preserve">COROLL. III.</head> <p> <s xml:id="echoid-s9129" xml:space="preserve">PAtet denique in Parabolis parallelis, vel in ſimilibus concentricis Hy-<lb/>perbolis, aut Ellipſibus, vel Circulis A B C, D E F, omnia rectangu-<lb/>la ſub ſegmentis applicatarum, interſe, & </s> <s xml:id="echoid-s9130" xml:space="preserve">prædictæ contingenti A E C <lb/>æquidiſtantium (quorum vnum eſt rectangulum G D H, vel G F H) eſſe. <lb/></s> <s xml:id="echoid-s9131" xml:space="preserve">inter ſe æqualia, cum quodlibet ipſorum æquale ſit eidem quadrato ſemi-<lb/>tangentis A E.</s> <s xml:id="echoid-s9132" xml:space="preserve"/> </p> <pb o="143" file="0329" n="329"/> <figure> <image file="0329-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0329-01"/> </figure> </div> <div xml:id="echoid-div955" type="section" level="1" n="383"> <head xml:id="echoid-head394" xml:space="preserve">VINCENTII VIVIANI <lb/>AD LIB DE MAX. ET MIN. <lb/>APPENDIX.</head> <figure> <image file="0329-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0329-02"/> </figure> </div> <div xml:id="echoid-div956" type="section" level="1" n="384"> <head xml:id="echoid-head395" xml:space="preserve">MONITVM.</head> <p style="it"> <s xml:id="echoid-s9133" xml:space="preserve">_H_ACT ENVS babes Amice Lector plurima eorum, quæ iam-<lb/>diu occaſione Diuinationis in V. </s> <s xml:id="echoid-s9134" xml:space="preserve">Conicor. </s> <s xml:id="echoid-s9135" xml:space="preserve">excogitauimus, <lb/>dum ex tribus illis faſciculis SERENISS. </s> <s xml:id="echoid-s9136" xml:space="preserve">LEOPOLDI <lb/>inuicto teſtimonio comprobatis, de quibus latius in Proæmio, <lb/>priorem exinaniuimus, alterum extenuauimus. </s> <s xml:id="echoid-s9137" xml:space="preserve">Ex eorum reliquijs ter-<lb/>tium ſaltem librum efformare ſtatueramus, circa MAXIMAS pariter, <lb/>ac MINIMAS magnitudinis verſantem, atque ampliùs illas eiuſdem <lb/>nominis, quæ à MAXIMIS, & </s> <s xml:id="echoid-s9138" xml:space="preserve">MINIMIS plus minuſue rece-<lb/>dunt excutientem; </s> <s xml:id="echoid-s9139" xml:space="preserve">quod rarò bucuſque, ac tantùm neceſsitate cogente <lb/>de monſtrauimus, quodque de induſtria omiſimus, tum ne à ſuſcepta <lb/>materia longiùs diſcederemus, tum vt ipſam expeditiùs perſolueremus. <lb/></s> <s xml:id="echoid-s9140" xml:space="preserve">Verùm graues, ac diuturnæ egritudines, quæ nos, huic editioni in-<lb/>cumbentes, exagitarunt, ita ipſimet remoram fecere, totque è contra <lb/>ſunt ſtimuli ad hoc in vulgus manandum, vt cætera ad aliud tempus <lb/>proferre cogamur, ſi hæc tibi grata comperiamus. </s> <s xml:id="echoid-s9141" xml:space="preserve">Liceat tamen ex tertio <lb/>libro quaſdam Propoſitiones aliunde receptas deſumere, atque Appendicis <lb/>nomine huc apponere, ad id præſertim impulſi, tum quod noſtræ harum <lb/>Propoſitionum demonſtrationes huic tertio libro ſint penitus inutiles, tum <lb/>quia pollicitam quorundam fidem, ſolidam, incorruptamque prorſus non <lb/>inuenerimus.</s> <s xml:id="echoid-s9142" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9143" xml:space="preserve">Duo potiſsimùm ſunt Problemata, quibus bæc Appendicula conftatur.</s> <s xml:id="echoid-s9144" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9145" xml:space="preserve">Primum (vti conſtat ex quadam variarum Propoſitionum narratio-<lb/>ne, quæ inter ſummum Geometram Torricellium, præſtantioreſque Gal- <pb o="144" file="0330" n="330" rhead=""/> liæ, ne dicam Europæ Mathematicos interceſſere, quales, inter hos D. <lb/></s> <s xml:id="echoid-s9146" xml:space="preserve">Fermat Senator Tholoſanus, D. </s> <s xml:id="echoid-s9147" xml:space="preserve">Roberuallius in Pariſienſi Academia Re-<lb/>gius Mathematum Profeſſor, ac D. </s> <s xml:id="echoid-s9148" xml:space="preserve">de Verdus) præfatus Cl. </s> <s xml:id="echoid-s9149" xml:space="preserve">Vir de <lb/>Fermat ipſi Torricellio olim propoſuerat, qui licet ſtatim in ipſius ſolu-<lb/>tionem non incidiſſet, inde mox animaduertens Problema determinatum <lb/>eſſe, illud demum triplici via, altera nimirum per locos planos, reliquis <lb/>per ſolidos demonſtrauit, nobiſque poſtmodum exercitationis gratia in. </s> <s xml:id="echoid-s9150" xml:space="preserve"><lb/>bunc, qui ſequitur modum enodandum tradidit.</s> <s xml:id="echoid-s9151" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9152" xml:space="preserve">Dato triangulo, cuius vnuſquiſq; </s> <s xml:id="echoid-s9153" xml:space="preserve">angulorum minor ſit graduum <lb/>120. </s> <s xml:id="echoid-s9154" xml:space="preserve">punctum reperire, à quo ſi ad angulos tres rectæ educantur <lb/>ipſarum aggregatum ſit MINIMVM.</s> <s xml:id="echoid-s9155" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9156" xml:space="preserve">Quod, vt vera fatear, non niſi iteratis oppugnationibus tunc nobis <lb/>vincere datum fuit, ſed aggreſsione omnino ab alijs diſcrepante, ac, <lb/>ni decipimur, ſatis iucunda, & </s> <s xml:id="echoid-s9157" xml:space="preserve">ad ipſiuſmet Problematis propagatio-<lb/>nem valde accommoda, dum non tantum ad tria data puncta, (qualia <lb/>ſunt vertices angulorum propoſiti trianguli) verùm etiam ad quotquot li-<lb/>buerit, ex alio quæſito puncto, MINIMVM eductarum aggregatum <lb/>reperiri queat, manente tamen determinata eorum poſitione, prout deter-<lb/>minatum eſt prædictum triangulum.</s> <s xml:id="echoid-s9158" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9159" xml:space="preserve">Alterum Problema præclariſsimum Virum, & </s> <s xml:id="echoid-s9160" xml:space="preserve">Auorum ſplendore, & </s> <s xml:id="echoid-s9161" xml:space="preserve"><lb/>morum integritate conſpicuum agnoſcit Auctorem: </s> <s xml:id="echoid-s9162" xml:space="preserve">P. </s> <s xml:id="echoid-s9163" xml:space="preserve">Honoratum F abbri, <lb/>natione Gallum, in Ieſuitarum celeberrima Societate magni nominis Theo-<lb/>logum, omnigena hiſtoriarum, humaniorumque literarum eruditione de-<lb/>coratum, Mathematicum præſtantiſsimum, Philoſophum acutiſsimum, <lb/>qui olim Lugduni apud Gallos Philoſophiam publicè edocens, ſummam <lb/>egregij acuminis famam ſibi peperit, quod manifeſtò teſtantur (ita nobis <lb/>aſſerente alibi iam, ſed parum commendato nobiliſsimo Adoleſcente Lau-<lb/>rentio Magalotti tanti Viri amantiſsimo, & </s> <s xml:id="echoid-s9164" xml:space="preserve">obſequentiſsimo) quædam <lb/>ipſius PROPOSITIONES PHYSICAE, CVM BREVISSIMIS <lb/>RATIONVM MOMENTIS, tunc ibidem publici iuris factæ, & </s> <s xml:id="echoid-s9165" xml:space="preserve"><lb/>prout fuſiùs, Deo dante, patebit ex nouis eiuſdem geometricis, ac phyſi-<lb/>comathematicis contemplationibus, quibus Literatorum Reſpublica ali-<lb/>quando ſe locupletaturam expectat.</s> <s xml:id="echoid-s9166" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9167" xml:space="preserve">Hoc igitur Problema, anno 1656. </s> <s xml:id="echoid-s9168" xml:space="preserve">idem Cl. </s> <s xml:id="echoid-s9169" xml:space="preserve">Adoleſcens Laurentius <lb/>Magalotis, (dum in Piſano Lyceo Iuriſprudentiam excoleret) à prædicto <lb/>P. </s> <s xml:id="echoid-s9170" xml:space="preserve">F abbri, tunc Romæ immorante receperat, nobiſque per epiſtolam, Pi-<lb/>ſis, ſub 27. </s> <s xml:id="echoid-s9171" xml:space="preserve">Decembris datam communicarat, cui poſt triduum reſcri-<lb/>bentes, vniuerſaliorem quæſiti propoſitionem, ita expoſuimus;</s> <s xml:id="echoid-s9172" xml:space="preserve"/> </p> <pb o="145" file="0331" n="331" rhead=""/> <p> <s xml:id="echoid-s9173" xml:space="preserve">Duabus datis rectis lineis terminatis, non modò ad rectum, ſed <lb/>ad quemlibet angulum conſtitutis, & </s> <s xml:id="echoid-s9174" xml:space="preserve">per vnius ipſarum terminum <lb/>alia alteri ipſarum æquidiſtanter ducta, ad contrarias tamen par-<lb/>tes, & </s> <s xml:id="echoid-s9175" xml:space="preserve">in infinitum producta: </s> <s xml:id="echoid-s9176" xml:space="preserve">oportet per extremum terminum al-<lb/>terius, rectam ducere æquidiſtanti occurrentem, quæ cum bina <lb/>ſimilia triangula ad verticem conſtituat, ipſorum aggregatum ſit <lb/>MINIMA quantitas.</s> <s xml:id="echoid-s9177" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9178" xml:space="preserve">ſimulque noſtram Problematis enodationem his verbis enunciauimus;</s> <s xml:id="echoid-s9179" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9180" xml:space="preserve">Diuidatur ſecanda linea, ita vt ſegmentum ipſius propè termi-<lb/>natam parallelam, ad ſegmentum reliquum ſit in ratione diametri <lb/>cuiuslibet quadrati ad exceſſum diametri ſuper latus: </s> <s xml:id="echoid-s9181" xml:space="preserve">nam pũctum <lb/>interſectionis erit quæſitum.</s> <s xml:id="echoid-s9182" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9183" xml:space="preserve">ac demum de inuentione binorum æqualium ex triangulis aggregatorum, <lb/>tam ſupra, quàm infra punctum MINIMI aggregati eundem Cl. </s> <s xml:id="echoid-s9184" xml:space="preserve">Ado-<lb/>leſcentem commonefecimus. </s> <s xml:id="echoid-s9185" xml:space="preserve">Sed iam Appendicem aggrediamur.</s> <s xml:id="echoid-s9186" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div957" type="section" level="1" n="385"> <head xml:id="echoid-head396" xml:space="preserve">LEMMA I. PROP. I.</head> <p> <s xml:id="echoid-s9187" xml:space="preserve">Si fuerint duo ordines quotcunque triangulorum æqualem al-<lb/>titudinem habentium; </s> <s xml:id="echoid-s9188" xml:space="preserve">erit aggregatum baſium triangulorum pri-<lb/>mi ordinis, ad aggregatum baſium triangulorum ſecundi, vt ag-<lb/>gregatum triangulorum primi, ad aggregatum triangulorum ſe-<lb/>cundi ordinis.</s> <s xml:id="echoid-s9189" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9190" xml:space="preserve">SIt vnus ordo triangulorum A B C, C D E, E F G, G H I, alter verò <lb/>triangulorum ordo L M N, N O P, P Q R, & </s> <s xml:id="echoid-s9191" xml:space="preserve">omnia ſint æqualis alti-<lb/>tudinis, vtriuſque autem ordinis triangula ſint ad eaſdem partes, & </s> <s xml:id="echoid-s9192" xml:space="preserve">ipſorum <lb/>baſes in directum diſponãtur, quarum baſium aggregatum, in primo ſit A I, <lb/>& </s> <s xml:id="echoid-s9193" xml:space="preserve">in ſecundo ſit L R. </s> <s xml:id="echoid-s9194" xml:space="preserve">Dico aggregatum A I, ad aggregatum L R eſſe vt <lb/>aggregatum triangulorum primi ordinis ad aggregatum tr iangulorũ ſecũdi.</s> <s xml:id="echoid-s9195" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9196" xml:space="preserve">Quoniam iunctis rectis A H, <lb/> <anchor type="figure" xlink:label="fig-0331-01a" xlink:href="fig-0331-01"/> C H, E H; </s> <s xml:id="echoid-s9197" xml:space="preserve">& </s> <s xml:id="echoid-s9198" xml:space="preserve">L Q, N Q: </s> <s xml:id="echoid-s9199" xml:space="preserve">erit <lb/>triangulum A B C ęquale trian-<lb/>gulo A H C, (cum ſint ſuper ea-<lb/>dembaſi A C, & </s> <s xml:id="echoid-s9200" xml:space="preserve">habeant ex hy-<lb/>potheſi eandem altitudinem) & </s> <s xml:id="echoid-s9201" xml:space="preserve"><lb/>C D E ęquale C H E, ac E F G <lb/>æquale E H G; </s> <s xml:id="echoid-s9202" xml:space="preserve">vnde communi <lb/>addito G H I, erunt omnia ſimul <lb/>primi ordinis æqualia vnico A <lb/>H I: </s> <s xml:id="echoid-s9203" xml:space="preserve">item oſtẽdetur omnia ſimul <lb/>ſecundi ordinis æqualia eſſe vni-<lb/>co L Q R; </s> <s xml:id="echoid-s9204" xml:space="preserve">ſed triangulum A H I ad L Q R eſt vt baſis A I ad L R, cum po- <pb o="146" file="0332" n="332" rhead=""/> nantur æqualium altitudinum, quare aggregatum triangulorum primi, ad <lb/>aggregatum triangulorum ſecundi ordinis erit, vt A I ad L R, vel vt aggre-<lb/>gatum baſium primi ordinis ad aggregatum baſium ſecundi. </s> <s xml:id="echoid-s9205" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s9206" xml:space="preserve">c.</s> <s xml:id="echoid-s9207" xml:space="preserve"/> </p> <div xml:id="echoid-div957" type="float" level="2" n="1"> <figure xlink:label="fig-0331-01" xlink:href="fig-0331-01a"> <image file="0331-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0331-01"/> </figure> </div> </div> <div xml:id="echoid-div959" type="section" level="1" n="386"> <head xml:id="echoid-head397" xml:space="preserve">LEMMA II. PROP. II.</head> <p> <s xml:id="echoid-s9208" xml:space="preserve">In quocunque polygono regulari, aggregata perpendicularium <lb/>ex quibuſcunque punctis, (quæ tamen non ſint extra perimetrum <lb/>polygoni) ſuper omnia eius latera eductarum, inter ſe ſunt æqua-<lb/>lia. </s> <s xml:id="echoid-s9209" xml:space="preserve">Si verò alterum punctorum fuerit extra perimetrum, aggrega-<lb/>tum perpendicularium ex eo eductarum, maius ſemper erit quoli-<lb/>bet prædictorum aggregatorum ex puncto, quod non ſit extra.</s> <s xml:id="echoid-s9210" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9211" xml:space="preserve">ESto polygonum regulare A B C D E, & </s> <s xml:id="echoid-s9212" xml:space="preserve">duo quælibet puncta F, G, in <lb/>prima figura, vel intra, vel in ipſius perimetro, à quibus ſuper eius late-<lb/>ra eductæ ſint perpendiculares F N, F H, F I, F L, F M; </s> <s xml:id="echoid-s9213" xml:space="preserve">& </s> <s xml:id="echoid-s9214" xml:space="preserve">G O, G P, G <lb/>Q, G R, G S. </s> <s xml:id="echoid-s9215" xml:space="preserve">Dico talium perpendicularium aggregata inter ſe æqualia <lb/>eſſe. </s> <s xml:id="echoid-s9216" xml:space="preserve">Si verò alterum punctorum G, cadat extra, vt in ſecunda ſigura, dico <lb/>aggregatum perpendicularium ex G maius eſſe quolibet prædictorum ag-<lb/>gregatorum, vtputa perpendicularium ex F.</s> <s xml:id="echoid-s9217" xml:space="preserve"/> </p> <figure> <image file="0332-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0332-01"/> </figure> <p> <s xml:id="echoid-s9218" xml:space="preserve">Ductis enim rectis ex G, F ad omnes àngulos polygoni, vt in ſiguris: <lb/></s> <s xml:id="echoid-s9219" xml:space="preserve">Patet ipſum polygonum vtrinque diuiſum eſſe in duos triangulorum ordines <lb/>æquales altitudineshabentium, quæ ſunt ipſa polygonilatera, ſuper quæ ca-<lb/>dunt perpendiculares, (ſinempe hæ accipiantur tanquam baſes) erit ergo <lb/>aggregatum baſiun triangulorum, quæ ſimul conueniunt in F, ad aggre-<lb/>gatum baſium triangulorum, quæ conueniunt in G, <anchor type="note" xlink:href="" symbol="*"/> vt aggregatum trian- <anchor type="note" xlink:label="note-0332-01a" xlink:href="note-0332-01"/> gulorum, primiordinisex F, ad aggregatum triangulornm ſecundi ex G, <lb/>ſed hęc triangulorumaggregata in prima figura ſunt æqualia (namipſa idem <lb/>polygonum complent) ergo, & </s> <s xml:id="echoid-s9220" xml:space="preserve">aggregata baſium eorundem, hoc eſt ag-<lb/>gregata perpendicularium ex F, & </s> <s xml:id="echoid-s9221" xml:space="preserve">G, ſuper polygoni latera eductarum <pb o="147" file="0333" n="333" rhead=""/> ſunt æqualia. </s> <s xml:id="echoid-s9222" xml:space="preserve">In ſecunda verò figura, aggregatum triangulorum ex G ma-<lb/>ius eſt aggregato triangulorum ex F, vt ſatis patet (cum illud, ipſum poly-<lb/>gonum excedat) quare, & </s> <s xml:id="echoid-s9223" xml:space="preserve">aggregatum baſium triangulorum ex G, (quæ <lb/>ſuntipſæ perpendiculares ex G) maius eſt aggregato baſium triangulorum <lb/>ex F, (quæ ſunt perpendiculares ex F.) </s> <s xml:id="echoid-s9224" xml:space="preserve">Quapropter, &</s> <s xml:id="echoid-s9225" xml:space="preserve">c. </s> <s xml:id="echoid-s9226" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s9227" xml:space="preserve">c.</s> <s xml:id="echoid-s9228" xml:space="preserve"/> </p> <div xml:id="echoid-div959" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-0332-01" xlink:href="note-0332-01a" xml:space="preserve">per pri-<lb/>mam Ap-<lb/>pend.</note> </div> </div> <div xml:id="echoid-div961" type="section" level="1" n="387"> <head xml:id="echoid-head398" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s9229" xml:space="preserve">HInc eſt, quod aggregatum perpendicularium ex centro dati polygoni <lb/>ſuper eius latera eductarum, ſemper eſt non maius quolibet ex alio <lb/>puncto perpendicularium aggregato, vbicunque aſſumptum ſit punctum <lb/>hoc, velintra, vel in perimetro, vel extra perimetrum dati polygoni.</s> <s xml:id="echoid-s9230" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div962" type="section" level="1" n="388"> <head xml:id="echoid-head399" xml:space="preserve">THEOR. I. PROP. III.</head> <p> <s xml:id="echoid-s9231" xml:space="preserve">In quocunque polygono regulari, aggregatorum linearum ex <lb/>punctis vbicunque aſſumptis ad ipſius angulos eductarum, MINI-<lb/>MVM eſt, quod ex centro.</s> <s xml:id="echoid-s9232" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9233" xml:space="preserve">SIt polygonum regulare A B C D E, cuius centrum P, à quo ad angulos <lb/>eductæ ſint rectę P A, P B, P C, P D, P E, ſumptoq; </s> <s xml:id="echoid-s9234" xml:space="preserve">vbicunque alio <lb/>puncto O, vei intra polygonum A B C D E, vel in eius perimetro, vel ex-<lb/>tra, iungantur item O A, O B, O C, O D, O E. </s> <s xml:id="echoid-s9235" xml:space="preserve">Dico aggregatum edu-<lb/>ctarum ex centro P, minus eſſe aggregato ductarum ex O.</s> <s xml:id="echoid-s9236" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9237" xml:space="preserve">Ex punctis enim A, B, C, D, E, erigan-<lb/> <anchor type="figure" xlink:label="fig-0333-01a" xlink:href="fig-0333-01"/> turipſis P A, P B, P C, P D, P E perpen-<lb/>diculares L I, I H, H G, G F, F L vtrinq; <lb/></s> <s xml:id="echoid-s9238" xml:space="preserve">productæ. </s> <s xml:id="echoid-s9239" xml:space="preserve">Patet has ſimul conuenire, & </s> <s xml:id="echoid-s9240" xml:space="preserve"><lb/>polygonum L I H G F dato ſimile conſti-<lb/>tuere circa idem centrum P, ad cuius late-<lb/>ra ex puncto O ducantur perpendiculares <lb/>O R, O Q, O N, O M, O S.</s> <s xml:id="echoid-s9241" xml:space="preserve"/> </p> <div xml:id="echoid-div962" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0333-01"/> </figure> </div> <p> <s xml:id="echoid-s9242" xml:space="preserve">Iam per Coroll. </s> <s xml:id="echoid-s9243" xml:space="preserve">præcedentis Lemmatis <lb/>in polygono I H G F L aggregatum per-<lb/>pendicularium, quæ ex centro P eſt non <lb/>maius aggregato perpendicularium, quæ <lb/>ex puncto O vbicunq; </s> <s xml:id="echoid-s9244" xml:space="preserve">aſſumpto, ſed aggregatum perpendicularium ex O, <lb/>minus eſt aggregato obliquarum O A, O B, O C, O D, O E, ſuper ijſdem <lb/>lateribus circumſcripti polygoni eductarum, (eſt enim perpendicularis O <lb/>R, minor obliqua O A, & </s> <s xml:id="echoid-s9245" xml:space="preserve">O Q minor O B; </s> <s xml:id="echoid-s9246" xml:space="preserve">O N minor O C; </s> <s xml:id="echoid-s9247" xml:space="preserve">O M minor <lb/>O D, & </s> <s xml:id="echoid-s9248" xml:space="preserve">O S minor O E) ergo aggregatum perpendicularium ex P, hoc eſt <lb/>ad angulos dati polygoni A B C D E eductarum, eſt omnino minus aggre-<lb/>gato obliquarum ex O, nempe eductarum ad eoſdem angulos dati poly-<lb/>goni à puncto O, vbicunque ſit ipſum O. </s> <s xml:id="echoid-s9249" xml:space="preserve">Quare aggregatum ductarum ex <lb/>centro ad angulos polygoni regularis _MINIMVM_ eſt. </s> <s xml:id="echoid-s9250" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s9251" xml:space="preserve">c.</s> <s xml:id="echoid-s9252" xml:space="preserve"/> </p> <pb o="148" file="0334" n="334" rhead=""/> </div> <div xml:id="echoid-div964" type="section" level="1" n="389"> <head xml:id="echoid-head400" xml:space="preserve">THEOR. II. PROP. IV.</head> <p> <s xml:id="echoid-s9253" xml:space="preserve">Si quotcunque rectæ lineæ terminatæ (non minus verò quam <lb/>tres) cuiuslibet longitudinis, ad vnum idemque punctum occur-<lb/>rant, totidem angulos inter ſe æquales conſtituentes, & </s> <s xml:id="echoid-s9254" xml:space="preserve">quatuor <lb/>rectos complentes. </s> <s xml:id="echoid-s9255" xml:space="preserve">Erit aggregatum harum ſimul omnium occur-<lb/>rentium, MINIMVM aggregatorum rectarum, à quibuſcunque <lb/>alijs aſſumptis punctis, ad eoſdem datarum terminos eductarum.</s> <s xml:id="echoid-s9256" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9257" xml:space="preserve">SInt quotcunque rectę A B, A C, A D, A E, A F, A G terminatę, quę <lb/>ad punctum A ſimul occurrant, conſtituantque angulos B A C, C A <lb/>D, D A E, E A F, F A G, G A B inter ſe æquales, & </s> <s xml:id="echoid-s9258" xml:space="preserve">ſimul ſumpti ęquales <lb/>quatuor rectis: </s> <s xml:id="echoid-s9259" xml:space="preserve">dico aggregatum harum omniũ minus eſſe aggregato linea-<lb/>rum, quæ ex quolibet alio puncto I ad eoſdem terminos B, C, D, E, F, G, <lb/>educi poſſunt, quales ſunt I B, I C, I D, I E, I F, I G.</s> <s xml:id="echoid-s9260" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9261" xml:space="preserve">Sit enim A G _MAXIMA_ ductarum ex A, ſuper qua ſumatur A P ipſa A <lb/>G non minor, cui demantur æquales A H, A L, A M, A N, A O, & </s> <s xml:id="echoid-s9262" xml:space="preserve">com-<lb/>pleatur polygonum H L M N O P, quod erit æquilaterum, & </s> <s xml:id="echoid-s9263" xml:space="preserve">æquiangulũ, <lb/>ſiue regulare, cum anguli ad A ſint æquales, eiuſque centrum erit A; </s> <s xml:id="echoid-s9264" xml:space="preserve">deni-<lb/>que iungantur I H, I L, I M, I N, I O, I P.</s> <s xml:id="echoid-s9265" xml:space="preserve"/> </p> <figure> <image file="0334-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0334-01"/> </figure> <p> <s xml:id="echoid-s9266" xml:space="preserve">Iam aggregatum ductarum A H, A L, A M, A N, A O, A P ex centro <lb/>A ad angulos polygoni, cum ſit <anchor type="note" xlink:href="" symbol="*"/> _MINIMVM_, erit minus aggregato ducta- <anchor type="note" xlink:label="note-0334-01a" xlink:href="note-0334-01"/> rum I H, I L, I M, I N, I O, I P ex puncto I, ſed harum aggregatum mi-<lb/>nus eſt aggregato binarum I B, B H; </s> <s xml:id="echoid-s9267" xml:space="preserve">I C, C L; </s> <s xml:id="echoid-s9268" xml:space="preserve">I D, D M; </s> <s xml:id="echoid-s9269" xml:space="preserve">I E, E N; </s> <s xml:id="echoid-s9270" xml:space="preserve">I F, <lb/>F O; </s> <s xml:id="echoid-s9271" xml:space="preserve">I G, G P; </s> <s xml:id="echoid-s9272" xml:space="preserve">nam I B, B H maiores ſunt I H, & </s> <s xml:id="echoid-s9273" xml:space="preserve">I C, C L maiores I <lb/>L, &</s> <s xml:id="echoid-s9274" xml:space="preserve">c. </s> <s xml:id="echoid-s9275" xml:space="preserve">quare eò magis aggregatum, ex A ductarum, A H, A L, A M, A <lb/>N, A O, A P minus erit aggregato binarum I B, B H; </s> <s xml:id="echoid-s9276" xml:space="preserve">I C, C L; </s> <s xml:id="echoid-s9277" xml:space="preserve">I D, D <lb/>M; </s> <s xml:id="echoid-s9278" xml:space="preserve">I E, E N; </s> <s xml:id="echoid-s9279" xml:space="preserve">I F, F O; </s> <s xml:id="echoid-s9280" xml:space="preserve">I G, G P; </s> <s xml:id="echoid-s9281" xml:space="preserve">demptis ergo communibus ſegmentis B <lb/>H, C L, D M, E N, F O, G P, erit reliquum aggregatum datarum A B, <lb/>A C, A D, A E, A F, A G minus reliquo aggregato ductarum I B, I C, <pb o="149" file="0335" n="335" rhead=""/> I D, I E, I F, I G ex aſſumpto puncto I ad datarum terminos B, C, D, E, <lb/>F, G; </s> <s xml:id="echoid-s9282" xml:space="preserve">itaque aggregatum ductarum ex A æquales angulos inter ſe efficien-<lb/>tes, & </s> <s xml:id="echoid-s9283" xml:space="preserve">quatuor rectos ſimul complentes eſt _MINIMVM_. </s> <s xml:id="echoid-s9284" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s9285" xml:space="preserve">c.</s> <s xml:id="echoid-s9286" xml:space="preserve"/> </p> <div xml:id="echoid-div964" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-0334-01" xlink:href="note-0334-01a" xml:space="preserve">per 3. <lb/>Append.</note> </div> <p style="it"> <s xml:id="echoid-s9287" xml:space="preserve">Hinc ſolutio Gallici Problematis, ſequenti Lemmate præoſtenſo.</s> <s xml:id="echoid-s9288" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div966" type="section" level="1" n="390"> <head xml:id="echoid-head401" xml:space="preserve">LEMMA III. PROP. V.</head> <p> <s xml:id="echoid-s9289" xml:space="preserve">Si in triangulo A B C fuerit angulus A B C, minor grad. </s> <s xml:id="echoid-s9290" xml:space="preserve">120. <lb/></s> <s xml:id="echoid-s9291" xml:space="preserve">& </s> <s xml:id="echoid-s9292" xml:space="preserve">ſuper latera B A, B C deſcribantur ad partes baſis A C ſimiles <lb/>circuli portiones A E B, C D B capientes angulos graduum 120. </s> <s xml:id="echoid-s9293" xml:space="preserve"><lb/>Dico ipſarum peripherias ſe mutuò ſecare, atque omnino intra <lb/>triangulum A B C.</s> <s xml:id="echoid-s9294" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9295" xml:space="preserve">NOn enim ſe contingunt in B: </s> <s xml:id="echoid-s9296" xml:space="preserve">quoniam ducta ex B recta F B G vnam <lb/>harum portionum peripheriam contingente, ipſa, & </s> <s xml:id="echoid-s9297" xml:space="preserve">alteram quoque <lb/>continget: </s> <s xml:id="echoid-s9298" xml:space="preserve">quare angulas G B A à contingente, & </s> <s xml:id="echoid-s9299" xml:space="preserve">ſecante confectus equa-<lb/>lis erit ei, qui ſit in alterna portione A E B, nempe erit gr. </s> <s xml:id="echoid-s9300" xml:space="preserve">120. </s> <s xml:id="echoid-s9301" xml:space="preserve">& </s> <s xml:id="echoid-s9302" xml:space="preserve">ob eandem <lb/>rationem angulus G B C erit grad. </s> <s xml:id="echoid-s9303" xml:space="preserve">120. </s> <s xml:id="echoid-s9304" xml:space="preserve">vnde reliquus A B C, è quatuor <lb/>rectis, erit pariter gr. </s> <s xml:id="echoid-s9305" xml:space="preserve">120. </s> <s xml:id="echoid-s9306" xml:space="preserve">quod eſt contra hypotheſim, cum ſit minor.</s> <s xml:id="echoid-s9307" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9308" xml:space="preserve">Nec autem ſe ſecant extra trian-<lb/> <anchor type="figure" xlink:label="fig-0335-01a" xlink:href="fig-0335-01"/> gulum ad partes G, vt in G: </s> <s xml:id="echoid-s9309" xml:space="preserve">nam <lb/>ducta G B, eſſet angulus G B A mi-<lb/>noreo, qui fit à contingente ex B cũ <lb/>ſecante B A, ſiue minor facto in al-<lb/>terna portione A E B, qui eſt grad. <lb/></s> <s xml:id="echoid-s9310" xml:space="preserve">120. </s> <s xml:id="echoid-s9311" xml:space="preserve">itemque G B C minor eſſet gr. </s> <s xml:id="echoid-s9312" xml:space="preserve"><lb/>120. </s> <s xml:id="echoid-s9313" xml:space="preserve">quare reliquus A B C è grad. </s> <s xml:id="echoid-s9314" xml:space="preserve"><lb/>360. </s> <s xml:id="echoid-s9315" xml:space="preserve">maior eſſet omnino 120. </s> <s xml:id="echoid-s9316" xml:space="preserve">quod <lb/>item eſt contra hypotheſim, cum ſit <lb/>minor; </s> <s xml:id="echoid-s9317" xml:space="preserve">quapropter huiuſmodi peri-<lb/>pherias ſe mutuò ſecare infra B ad <lb/>partes baſis A C neceſſe eſt.</s> <s xml:id="echoid-s9318" xml:space="preserve"/> </p> <div xml:id="echoid-div966" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0335-01"/> </figure> </div> <p> <s xml:id="echoid-s9319" xml:space="preserve">Verùm ipſarum interſectio haud fiet in baſi A C, nec infra, quoniam <lb/>ſi in ipſa baſi A C, vt in F, eſſet, ex conſtructione, angulus A F B grad. <lb/></s> <s xml:id="echoid-s9320" xml:space="preserve">120. </s> <s xml:id="echoid-s9321" xml:space="preserve">ſiue maior recto, & </s> <s xml:id="echoid-s9322" xml:space="preserve">C F B pariter maior recto; </s> <s xml:id="echoid-s9323" xml:space="preserve">ex quo duo ſimul A F <lb/>B, C F B eſſent duobus rectis maiores; </s> <s xml:id="echoid-s9324" xml:space="preserve">quod eſt abſurdum, cum duos re-<lb/>ctos adæquent.</s> <s xml:id="echoid-s9325" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9326" xml:space="preserve">Sitandem eædem peripheriæ ſe mutuò ſecarent infra baſim A C, vt in H; <lb/></s> <s xml:id="echoid-s9327" xml:space="preserve">iunctis A H, C H, eſſent pariter, ex conſtructione, duo ſimul anguli A H B, <lb/>C H B, ſiue vnicus A H C maior duobus rectis; </s> <s xml:id="echoid-s9328" xml:space="preserve">quod eſt falſum cum ipſe à <lb/>duobus rectis deficiat per aggregatum duorum angulorum A C H, C A H. </s> <s xml:id="echoid-s9329" xml:space="preserve"><lb/>Quamobrem huiuſmodi ſimilium portionum peripherię neceſſariò ſe mutuò <lb/>ſecabunt, atque intra triangulum A B C|. </s> <s xml:id="echoid-s9330" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s9331" xml:space="preserve"/> </p> <pb o="150" file="0336" n="336" rhead=""/> </div> <div xml:id="echoid-div968" type="section" level="1" n="391"> <head xml:id="echoid-head402" xml:space="preserve">PROBL. I. PROP. VI.</head> <p> <s xml:id="echoid-s9332" xml:space="preserve">Dato triangulo, cuius vnuſquiſq; </s> <s xml:id="echoid-s9333" xml:space="preserve">angulorum minor ſit gr. </s> <s xml:id="echoid-s9334" xml:space="preserve">120. <lb/></s> <s xml:id="echoid-s9335" xml:space="preserve">punctum reperire, à quo ſi ad angulos tres rectę educantur, ipſarum <lb/>aggregatum ſit MINIMVM.</s> <s xml:id="echoid-s9336" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9337" xml:space="preserve">ESto triangulum A B C vt ponitur, & </s> <s xml:id="echoid-s9338" xml:space="preserve">inuenire oporteat punctum quale <lb/>imperatum eſt.</s> <s xml:id="echoid-s9339" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9340" xml:space="preserve">Super latera B A, B C ad partes baſis A C deſcribantur circuli portio-<lb/>nes A D B, C D B capientes angulos grad. </s> <s xml:id="echoid-s9341" xml:space="preserve">120. </s> <s xml:id="echoid-s9342" xml:space="preserve">ſiue æquales externo cuiuſ-<lb/>libet trianguli æquilateri, quarum portionum arcus omnino ſe mutuò <anchor type="note" xlink:href="" symbol="*"/> ſe- <anchor type="note" xlink:label="note-0336-01a" xlink:href="note-0336-01"/> cabunt intra triangulum A B C, ſitque eorum interſectio punctum D. </s> <s xml:id="echoid-s9343" xml:space="preserve">Di-<lb/>co ipſum eſſe quæſitum.</s> <s xml:id="echoid-s9344" xml:space="preserve"/> </p> <div xml:id="echoid-div968" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-0336-01" xlink:href="note-0336-01a" xml:space="preserve">5. App.</note> </div> <p> <s xml:id="echoid-s9345" xml:space="preserve">Nam iunctis D A, D B, D C, erunt an-<lb/> <anchor type="figure" xlink:label="fig-0336-01a" xlink:href="fig-0336-01"/> guli A D B, C D B graduum 120. </s> <s xml:id="echoid-s9346" xml:space="preserve">vnde reli-<lb/>quus A D C, vſque ad quatuor rectorum cõ-<lb/>plementum item erit gr. </s> <s xml:id="echoid-s9347" xml:space="preserve">120. </s> <s xml:id="echoid-s9348" xml:space="preserve">Cum ergo tres <lb/>rectę D A, D B, D C ad punctum D coeun-<lb/>tes tres æquales angulos efficiant, cumque hi <lb/>ſimul ſumpti æquales ſint quatuor rectis, erit <lb/>ipſarum D A, D B, D C aggregatum _MINIMA_ <anchor type="note" xlink:href="" symbol="*"/> quantitas. </s> <s xml:id="echoid-s9349" xml:space="preserve">Quare in- <anchor type="note" xlink:label="note-0336-02a" xlink:href="note-0336-02"/> uentum eſt punctum D, vti quærebatur. </s> <s xml:id="echoid-s9350" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s9351" xml:space="preserve"/> </p> <div xml:id="echoid-div969" type="float" level="2" n="2"> <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/QN4GHYBF/figures/0336-01"/> </figure> <note symbol="*" position="left" xlink:label="note-0336-02" xlink:href="note-0336-02a" xml:space="preserve">4. App.</note> </div> </div> <div xml:id="echoid-div971" type="section" level="1" n="392"> <head xml:id="echoid-head403" xml:space="preserve">PROBL. II. PROP. VII.</head> <p> <s xml:id="echoid-s9352" xml:space="preserve">Datam rectam lineam terminatam ita diuidere, vt ſumpta par-<lb/>tium ipſius tertia proportionali, aggregatum extremarum ſit MI-<lb/>NIMA quantitas.</s> <s xml:id="echoid-s9353" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9354" xml:space="preserve">ESto data linea A B, quam ſecare oporteat, vt imperatum eſt.</s> <s xml:id="echoid-s9355" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9356" xml:space="preserve">Erigatur ex A ipſi A B perpendicularis, & </s> <s xml:id="echoid-s9357" xml:space="preserve">æqualis A D, iunctaq; </s> <s xml:id="echoid-s9358" xml:space="preserve">D <lb/>B ſecetur D E æqualis D A, & </s> <s xml:id="echoid-s9359" xml:space="preserve">ex E ſuper A B perpendicularis demitta-<lb/>tur E C. </s> <s xml:id="echoid-s9360" xml:space="preserve">Dico punctum C quæſitum ſoluere.</s> <s xml:id="echoid-s9361" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9362" xml:space="preserve">Nam bifariam ſecto angulo A D E per rectam D F ſecante A B in F, & </s> <s xml:id="echoid-s9363" xml:space="preserve"><lb/>iuncta F E: </s> <s xml:id="echoid-s9364" xml:space="preserve">cum ſit latus D A æquale D E, & </s> <s xml:id="echoid-s9365" xml:space="preserve">D F commune, & </s> <s xml:id="echoid-s9366" xml:space="preserve">anguli <lb/>A D F, E D F æquales, erunt baſes F A, F E æquales, & </s> <s xml:id="echoid-s9367" xml:space="preserve">reliquus angulus <lb/>F E D reliquo F A D æqualis ſiue rectus: </s> <s xml:id="echoid-s9368" xml:space="preserve">quare ſi cum centro F interuallo <lb/>F A circulus deſcribatur A E G, is tranſibit quoque per E, & </s> <s xml:id="echoid-s9369" xml:space="preserve">vtramque D. <lb/></s> <s xml:id="echoid-s9370" xml:space="preserve">A, D B continget in A, E.</s> <s xml:id="echoid-s9371" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9372" xml:space="preserve">Iam cum in ſemi-circulo ſit A C ad C E, vt C E ad C G, ſitque C B <lb/>æqualis C E (cum etiam A D ſit æqualis A B) erit A C ad C B, vt C B <lb/>ad C G. </s> <s xml:id="echoid-s9373" xml:space="preserve">Vnde aggregatum extremarum poſt ſegmenta A C, C B erit A <lb/>G; </s> <s xml:id="echoid-s9374" xml:space="preserve">quod eſſe _MINIMVM_ ſic demonſtrabitur.</s> <s xml:id="echoid-s9375" xml:space="preserve"/> </p> <pb o="151" file="0337" n="337" rhead=""/> <p> <s xml:id="echoid-s9376" xml:space="preserve">Sumpto enim in data recta A B quocunque alio puncto H, vel in ipſius <lb/>parte producta vltra B, vt in prima figura, vel in ipſa A B, vt in ſecunda, <lb/>& </s> <s xml:id="echoid-s9377" xml:space="preserve">ex H ducta H I perpendiculari ad A B, ſecante diagonalem D B in I, <lb/>ductaque A I ſecante circuli peripheriam in L, iunctiſque G L, G I: </s> <s xml:id="echoid-s9378" xml:space="preserve">erit <lb/>angulus A L G rectus, atque externus trianguli L I G; </s> <s xml:id="echoid-s9379" xml:space="preserve">quare internus L I <lb/>G acutus erit, ac ideo recta I M, quæ ex I erigitur perpendicularis ad I A, <lb/>hoc eſt, quæ ipſi L G æquidiſtat, ſecabit A B vltra punctum G, vt in M, ac <lb/>ideo erit A G minor A M. </s> <s xml:id="echoid-s9380" xml:space="preserve">Et cum in triangulo rectangulo A I M, ſit vt A <lb/>H ad H I, ita H I ad H M, ſitque H I æqualis H B, erit A H ad H B, vt <lb/>H B ad H M, ergo A M eſt aggregatum extremarum proportionalium poſt <lb/>partes A H, H B, ſed eſt A G minor A M, vt modò oſtendimus: </s> <s xml:id="echoid-s9381" xml:space="preserve">ergo ag-<lb/>gregatum A G minus eſt aggregato A M: </s> <s xml:id="echoid-s9382" xml:space="preserve">& </s> <s xml:id="echoid-s9383" xml:space="preserve">hoc ſemper vbicunque aſſum-<lb/>ptum fuerit punctum H extra C: </s> <s xml:id="echoid-s9384" xml:space="preserve">ergo aggregatum A G minus eſt aggrega-<lb/>to A M: </s> <s xml:id="echoid-s9385" xml:space="preserve">& </s> <s xml:id="echoid-s9386" xml:space="preserve">hoc ſemper vbicunque aſſumptum fuerit punctum H extra C: <lb/></s> <s xml:id="echoid-s9387" xml:space="preserve">quare A G eſt _MINIMVM_ aggregatum quæſitum; </s> <s xml:id="echoid-s9388" xml:space="preserve">& </s> <s xml:id="echoid-s9389" xml:space="preserve">recta A B ſecta eſt in <lb/>C, vt imperatum fuit. </s> <s xml:id="echoid-s9390" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s9391" xml:space="preserve"/> </p> <figure> <image file="0337-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0337-01"/> </figure> </div> <div xml:id="echoid-div972" type="section" level="1" n="393"> <head xml:id="echoid-head404" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s9392" xml:space="preserve">SI quæratur iuxta quam rationem repertum punctum C diuidat datam A <lb/>B; </s> <s xml:id="echoid-s9393" xml:space="preserve">id ex ipſa Theorematis conſtructione elicietur. </s> <s xml:id="echoid-s9394" xml:space="preserve">Nam cum triangu-<lb/>la D A B, B E F ſint ſimilia inter ſe, erit B D ad D A, ſiue diameter qua-<lb/>drati ad latus, vt B F ad F E, vel ad F A, & </s> <s xml:id="echoid-s9395" xml:space="preserve">cum ſit B C ad C E, vt C E <lb/>ad C F, ſitque B C æqualis C E (cum & </s> <s xml:id="echoid-s9396" xml:space="preserve">B A æqualis ſit A D) erit etiam <lb/>C E ſiue C B æqualis C F. </s> <s xml:id="echoid-s9397" xml:space="preserve">Quare ſi data recta B A diuidatur, ita vt pars <lb/>B F ad reliquam partem F A, ſit vt diameter cuiuſdam quadrati ad eius la-<lb/>tus, & </s> <s xml:id="echoid-s9398" xml:space="preserve">maior pars B F ſecetur bifariam in C, hoc ipſum punctum erit quæ-<lb/>ſitum.</s> <s xml:id="echoid-s9399" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9400" xml:space="preserve">Vel. </s> <s xml:id="echoid-s9401" xml:space="preserve">Cum rectæ A B, A D ſint æquales, & </s> <s xml:id="echoid-s9402" xml:space="preserve">perpendiculariter conſtitu-<lb/>tæ, erit A D, ſiue D E latus quadrati, & </s> <s xml:id="echoid-s9403" xml:space="preserve">D B diameter, & </s> <s xml:id="echoid-s9404" xml:space="preserve">E B exceſſus <lb/>diametri ſuper latus, ſed eſt A C ad C B, vt D E ad E B: </s> <s xml:id="echoid-s9405" xml:space="preserve">ergo quæſitum <lb/>punctum C ſecat datam rectam A B, ita vt maior pars A C ad minorem C <lb/>B, ſit vt latus cuiuſdam quadrati ad exceſſum diametri ſuper latus, quæ ra-<lb/>tio, vt iam conſtat, cadit inter terminos incommenſurabiles.</s> <s xml:id="echoid-s9406" xml:space="preserve"/> </p> <pb o="152" file="0338" n="338" rhead=""/> </div> <div xml:id="echoid-div973" type="section" level="1" n="394"> <head xml:id="echoid-head405" xml:space="preserve">LEMMA IV. PROP. VIII.</head> <p> <s xml:id="echoid-s9407" xml:space="preserve">Si in triangulo A B C, cuius baſis A B, ex vertice C ducta ſit <lb/>C E ipſi B A parallela, vel ad eaſdem, vel ad oppoſitas partes, & </s> <s xml:id="echoid-s9408" xml:space="preserve"><lb/>ducatur quælibet A D E vtranque B C, C E ſecans in D, & </s> <s xml:id="echoid-s9409" xml:space="preserve">E: <lb/></s> <s xml:id="echoid-s9410" xml:space="preserve">dico aggregatum triangulorum A D B, D C E ad triangulum A <lb/>C B eſſe vt aggregatum extremarum poſt B D, D C, ad B C.</s> <s xml:id="echoid-s9411" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9412" xml:space="preserve">SVmatur D F tertia proportionalis poſt B D, D C.</s> <s xml:id="echoid-s9413" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9414" xml:space="preserve">Iam triangulum D C E ad A D C eſt vt E D ad D A, vel vt C D ad <lb/> <anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/> D B, vel vt D F ad D C; </s> <s xml:id="echoid-s9415" xml:space="preserve">& </s> <s xml:id="echoid-s9416" xml:space="preserve"><lb/>triangulum A D C ad trian-<lb/>gulum A B C, eſt vt D C <lb/>ad C B, ergo ex æquali triã-<lb/>gulum D C E ad A B C, erit <lb/>vt D F ad C B; </s> <s xml:id="echoid-s9417" xml:space="preserve">ſed triangu-<lb/>lum A D B ad idem A B C <lb/>eſt vt B D ad B C, quare <lb/>duo ſimul triangula D C E, <lb/>A D B, ad triangulum A C <lb/>B, erunt vt duæ ſimul lineæ <lb/>D F, D B, hoc eſt tota B F, <lb/>aggregatum extremarum poſt B D, D C, ad B C. </s> <s xml:id="echoid-s9418" xml:space="preserve">Quod erat, &</s> <s xml:id="echoid-s9419" xml:space="preserve">c.</s> <s xml:id="echoid-s9420" xml:space="preserve"/> </p> <div xml:id="echoid-div973" type="float" level="2" n="1"> <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/QN4GHYBF/figures/0338-01"/> </figure> </div> </div> <div xml:id="echoid-div975" type="section" level="1" n="395"> <head xml:id="echoid-head406" xml:space="preserve">PROBL. III. PROP. IX.</head> <p> <s xml:id="echoid-s9421" xml:space="preserve">Duabus datis rectis lineis terminatis ad quemlibet angulum <lb/>conſtitutis, & </s> <s xml:id="echoid-s9422" xml:space="preserve">per vnius ipſarum terminum alia alteri datarum <lb/>æquidiſtanter ducta, ad contrarias tamen partes, & </s> <s xml:id="echoid-s9423" xml:space="preserve">in infinitum <lb/>producta: </s> <s xml:id="echoid-s9424" xml:space="preserve">oportet per extremum terminum alterius, rectam duce-<lb/>re ęquidiſtanti occurrentem, ita vt, cum ipſa bina ſimilia triangula <lb/>ad verticem conſtituat, horũ aggregatum ſit MINIMA quantitas.</s> <s xml:id="echoid-s9425" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9426" xml:space="preserve">SInt A B, B C rectæ lineæ terminatæ ad quemcunque angulum A B C <lb/>compoſitæ, ſitque C D in infinitum producta ipſi B A parallela, ſed ad <lb/>oppoſit as partes rectæ C B: </s> <s xml:id="echoid-s9427" xml:space="preserve">oportet ex A rectam ducere, qualis eſt A D, <lb/>ita vt aggregatum ſimilium triangulorum A E B, C E D ad verticem E <lb/>ſit _MINIMVM_.</s> <s xml:id="echoid-s9428" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9429" xml:space="preserve">Diuidatur B C in E, ita vt B E ad E C ſit vt latus cuiuſdam quadrati ad <lb/>exceſſum diametri ſuper latus: </s> <s xml:id="echoid-s9430" xml:space="preserve">dico punctum E eſſe quæſitum.</s> <s xml:id="echoid-s9431" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9432" xml:space="preserve">Nam ducta qualibet alia A F G; </s> <s xml:id="echoid-s9433" xml:space="preserve">iunctaque A C: </s> <s xml:id="echoid-s9434" xml:space="preserve">cum aggregatum ex-<lb/>tremarum proportionalium poſt B E, E C ſit _MINIMVM_ (per Scholium <pb o="153" file="0339" n="339" rhead=""/> prop. </s> <s xml:id="echoid-s9435" xml:space="preserve">7. </s> <s xml:id="echoid-s9436" xml:space="preserve">huius) ipſum erit minus aggregato extremarum poſt B F, F C; </s> <s xml:id="echoid-s9437" xml:space="preserve">qua-<lb/>re primum aggregatum, ad rectam B C minorem habebit rationem, quam <lb/>ſecundum aggregatum ad eandem B C, ſed primum ad B C eſt <anchor type="note" xlink:href="" symbol="*"/> vt aggre- <anchor type="note" xlink:label="note-0339-01a" xlink:href="note-0339-01"/> <anchor type="figure" xlink:label="fig-0339-01a" xlink:href="fig-0339-01"/> gatum triangulorum A E B, D E C ad triã-<lb/>gulum A C B, & </s> <s xml:id="echoid-s9438" xml:space="preserve">ſecundum ad eandem B <lb/>C eſt vt aggregatum triangulorum A F B, <lb/>G F C ad idem trian gulum A C B, quare <lb/>aggregatum A E B, D E C ad triangulum <lb/>A C B minorem habebit rationem quàm <lb/>aggregatum A F B, G F C ad idem trian-<lb/>gulum A C B, vnde aggregatum ex A E <lb/>B, D E C minus erit aggregato ex A F B, <lb/>G F C, ac propterea aggregatum triangu-<lb/>lorum ad punctum E erit _MINIMVM_. <lb/></s> <s xml:id="echoid-s9439" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s9440" xml:space="preserve"/> </p> <div xml:id="echoid-div975" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-0339-01" xlink:href="note-0339-01a" xml:space="preserve">8. App.</note> <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/QN4GHYBF/figures/0339-01"/> </figure> </div> </div> <div xml:id="echoid-div977" type="section" level="1" n="396"> <head xml:id="echoid-head407" xml:space="preserve">COROLL.</head> <p> <s xml:id="echoid-s9441" xml:space="preserve">HInc, cum ſit vt ſubduplum ad ſubduplum, ita duplum ad duplum, ſi <lb/>compleantur parallelogramma B H, C I, ipſorum aggregatum erit <lb/>_MINIMVM_, &</s> <s xml:id="echoid-s9442" xml:space="preserve">c.</s> <s xml:id="echoid-s9443" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div978" type="section" level="1" n="397"> <head xml:id="echoid-head408" xml:space="preserve">PROBL. IV. PROP. X.</head> <p> <s xml:id="echoid-s9444" xml:space="preserve">Ijſdem poſitis, ac in præcedenti. </s> <s xml:id="echoid-s9445" xml:space="preserve">Si datum ſit in linea B C, <lb/>quodlibet aliud punctum F inter inuentum punctum E, & </s> <s xml:id="echoid-s9446" xml:space="preserve">extre-<lb/>mum B, & </s> <s xml:id="echoid-s9447" xml:space="preserve">oporteat aliud in ipſa punctum aſſignare, quæ ſimul <lb/>exhibeant aggregata triangulorum ad verticem inter ſe æqualia.</s> <s xml:id="echoid-s9448" xml:space="preserve"/> </p> <figure> <image file="0339-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0339-02"/> </figure> <p> <s xml:id="echoid-s9449" xml:space="preserve">ERigatur B G perpendicularis, & </s> <s xml:id="echoid-s9450" xml:space="preserve">æqualis ipſi B C, iungatur G C, & </s> <s xml:id="echoid-s9451" xml:space="preserve"><lb/>per F agatur F H æquidiſtans B G, & </s> <s xml:id="echoid-s9452" xml:space="preserve">fiat vt B F ad F H, ita F H ad <lb/>aliam F I, & </s> <s xml:id="echoid-s9453" xml:space="preserve">circa diametrum B I circulus deſcribatur rectam G C ſecans <lb/>in H, & </s> <s xml:id="echoid-s9454" xml:space="preserve">L, & </s> <s xml:id="echoid-s9455" xml:space="preserve">ex L ducatur L M parallela ad G B; </s> <s xml:id="echoid-s9456" xml:space="preserve">dico punctum M eſſe <lb/>quæſitum, hoc eſt ſi producantur A F, A M rectam C D ſecantes in D, <pb o="154" file="0340" n="340" rhead=""/> & </s> <s xml:id="echoid-s9457" xml:space="preserve">N; </s> <s xml:id="echoid-s9458" xml:space="preserve">aggregatum triangulorum A F B, D F C, æquale eſſe aggregato <lb/>triangulorum A M B, N M C.</s> <s xml:id="echoid-s9459" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9460" xml:space="preserve">Quoniam cum ſit vt B F ad F H, ita F H, vel F C ad F I, erit B I aggre-<lb/>gatum extremarum B F, F I, poſt B F, F C. </s> <s xml:id="echoid-s9461" xml:space="preserve">Item cum ſit B M ad M L, vt <lb/>M L, vel M C ad M I, erit idem BI aggregatum extremarum B M, M I, <lb/>poſt B M, M C; </s> <s xml:id="echoid-s9462" xml:space="preserve">ſed aggregatum triangulorum ad F ad triangulum A B C <lb/> <anchor type="figure" xlink:label="fig-0340-01a" xlink:href="fig-0340-01"/> (iuncta A C) eſt <anchor type="note" xlink:href="" symbol="*"/> vt aggregatum extremarum poſt B F, F C ad B C, &</s> <s xml:id="echoid-s9463" xml:space="preserve"> <anchor type="note" xlink:label="note-0340-01a" xlink:href="note-0340-01"/> aggregatum triangulorum ad M ad idem triangulum A B C eſt vt aggrega-<lb/>tum extremarum poſt B M, M C ad eandem B C, ſuntque prædicta extre-<lb/>marum aggregata inter ſe æqualia, cum vtrinque conficiant eandem B I, <lb/>quare, & </s> <s xml:id="echoid-s9464" xml:space="preserve">aggregatum triangulorum A F B, D F C, æquale erit aggregato <lb/>triangulorum A M B, N M C. </s> <s xml:id="echoid-s9465" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s9466" xml:space="preserve"/> </p> <div xml:id="echoid-div978" 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/QN4GHYBF/figures/0340-01"/> </figure> <note symbol="*" position="left" xlink:label="note-0340-01" xlink:href="note-0340-01a" xml:space="preserve">8. App.</note> </div> </div> <div xml:id="echoid-div980" type="section" level="1" n="398"> <head xml:id="echoid-head409" xml:space="preserve">APPENDICIS <lb/>FINIS.</head> <pb file="0341" n="341" rhead=""/> <p style="it"> <s xml:id="echoid-s9467" xml:space="preserve">ERrata, quæ non niſi peracta Operis impreſſione pacato animo adnotare potuimus, & </s> <s xml:id="echoid-s9468" xml:space="preserve">quæ parsim ob <lb/>noſtrum authogr apbum multis lituris, & </s> <s xml:id="echoid-s9469" xml:space="preserve">contr actionibus conſperſum, in Amanuenſis tranſcriptione <lb/>exciderunt, partim ex Typothetæ incuria irrepſerunt, quæque ipſo calamo reſtitui nequeunt, antequam <lb/>ad lectionem accedas, ita ſuis locis corrigere te rogatum volumus. </s> <s xml:id="echoid-s9470" xml:space="preserve">Reliqua minutiora ad orthogr aphiam <lb/>præſertim pertinentia, veluti, & </s> <s xml:id="echoid-s9471" xml:space="preserve">quaſdam paucas citationes turbatas, vel omiſſas æquiſſimo iudicio tuo <lb/>relinquimus emendandas.</s> <s xml:id="echoid-s9472" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9473" xml:space="preserve">Pag. </s> <s xml:id="echoid-s9474" xml:space="preserve">17. </s> <s xml:id="echoid-s9475" xml:space="preserve">verſ. </s> <s xml:id="echoid-s9476" xml:space="preserve">12. </s> <s xml:id="echoid-s9477" xml:space="preserve">ductæ diametro BD - ductæ ad diametrum BD | p. </s> <s xml:id="echoid-s9478" xml:space="preserve">18. </s> <s xml:id="echoid-s9479" xml:space="preserve">v. </s> <s xml:id="echoid-s9480" xml:space="preserve">10. </s> <s xml:id="echoid-s9481" xml:space="preserve">& </s> <s xml:id="echoid-s9482" xml:space="preserve">diametro - & </s> <s xml:id="echoid-s9483" xml:space="preserve">ad diametrum <lb/>v. </s> <s xml:id="echoid-s9484" xml:space="preserve">14. </s> <s xml:id="echoid-s9485" xml:space="preserve">& </s> <s xml:id="echoid-s9486" xml:space="preserve">diametro - & </s> <s xml:id="echoid-s9487" xml:space="preserve">ad diametrum | p. </s> <s xml:id="echoid-s9488" xml:space="preserve">20. </s> <s xml:id="echoid-s9489" xml:space="preserve">v. </s> <s xml:id="echoid-s9490" xml:space="preserve">6. </s> <s xml:id="echoid-s9491" xml:space="preserve">& </s> <s xml:id="echoid-s9492" xml:space="preserve">ab ipſa ex - & </s> <s xml:id="echoid-s9493" xml:space="preserve">ex ipſa à | p. </s> <s xml:id="echoid-s9494" xml:space="preserve">31. </s> <s xml:id="echoid-s9495" xml:space="preserve">v. </s> <s xml:id="echoid-s9496" xml:space="preserve">9. </s> <s xml:id="echoid-s9497" xml:space="preserve">A B C, ABC - A B C, <lb/>ADC | v. </s> <s xml:id="echoid-s9498" xml:space="preserve">11. </s> <s xml:id="echoid-s9499" xml:space="preserve">& </s> <s xml:id="echoid-s9500" xml:space="preserve">quod de - & </s> <s xml:id="echoid-s9501" xml:space="preserve">quod priùs de | v. </s> <s xml:id="echoid-s9502" xml:space="preserve">38. </s> <s xml:id="echoid-s9503" xml:space="preserve">40. </s> <s xml:id="echoid-s9504" xml:space="preserve">41. </s> <s xml:id="echoid-s9505" xml:space="preserve">ALCE - DLCE | p. </s> <s xml:id="echoid-s9506" xml:space="preserve">38. </s> <s xml:id="echoid-s9507" xml:space="preserve">v. </s> <s xml:id="echoid-s9508" xml:space="preserve">5. </s> <s xml:id="echoid-s9509" xml:space="preserve">à quadam ſectionis --<lb/>à quadam in ſectione | p. </s> <s xml:id="echoid-s9510" xml:space="preserve">47. </s> <s xml:id="echoid-s9511" xml:space="preserve">v. </s> <s xml:id="echoid-s9512" xml:space="preserve">7. </s> <s xml:id="echoid-s9513" xml:space="preserve">eſt MINIMA ſibi - eſt MAXIMA ſibi | p. </s> <s xml:id="echoid-s9514" xml:space="preserve">49. </s> <s xml:id="echoid-s9515" xml:space="preserve">v. </s> <s xml:id="echoid-s9516" xml:space="preserve">4. </s> <s xml:id="echoid-s9517" xml:space="preserve">Hyperbolen AEC, - Hyper-<lb/>bolen HBI | v. </s> <s xml:id="echoid-s9518" xml:space="preserve">17. </s> <s xml:id="echoid-s9519" xml:space="preserve">MAXIMAM - MINIMAM | v. </s> <s xml:id="echoid-s9520" xml:space="preserve">35. </s> <s xml:id="echoid-s9521" xml:space="preserve">regula LE - regula LF | p. </s> <s xml:id="echoid-s9522" xml:space="preserve">51. </s> <s xml:id="echoid-s9523" xml:space="preserve">v. </s> <s xml:id="echoid-s9524" xml:space="preserve">1. </s> <s xml:id="echoid-s9525" xml:space="preserve">in margine deeſt cita-<lb/>tio - _a._ </s> <s xml:id="echoid-s9526" xml:space="preserve">1. </s> <s xml:id="echoid-s9527" xml:space="preserve">coroll. </s> <s xml:id="echoid-s9528" xml:space="preserve">19. </s> <s xml:id="echoid-s9529" xml:space="preserve">h. </s> <s xml:id="echoid-s9530" xml:space="preserve">| p. </s> <s xml:id="echoid-s9531" xml:space="preserve">55. </s> <s xml:id="echoid-s9532" xml:space="preserve">v. </s> <s xml:id="echoid-s9533" xml:space="preserve">2. </s> <s xml:id="echoid-s9534" xml:space="preserve">regula IG - regula ſit ducta I G | v. </s> <s xml:id="echoid-s9535" xml:space="preserve">14. </s> <s xml:id="echoid-s9536" xml:space="preserve">C I, & </s> <s xml:id="echoid-s9537" xml:space="preserve">quidem - C I, eſt quidem | <lb/>p. </s> <s xml:id="echoid-s9538" xml:space="preserve">57. </s> <s xml:id="echoid-s9539" xml:space="preserve">v. </s> <s xml:id="echoid-s9540" xml:space="preserve">5. </s> <s xml:id="echoid-s9541" xml:space="preserve">circũſcripta: </s> <s xml:id="echoid-s9542" xml:space="preserve">- circumſcripta: </s> <s xml:id="echoid-s9543" xml:space="preserve">quam dico eſſe MINIMAM | p. </s> <s xml:id="echoid-s9544" xml:space="preserve">58. </s> <s xml:id="echoid-s9545" xml:space="preserve">v. </s> <s xml:id="echoid-s9546" xml:space="preserve">8. </s> <s xml:id="echoid-s9547" xml:space="preserve">conueniret - conueniret ſupra C <lb/>v. </s> <s xml:id="echoid-s9548" xml:space="preserve">43. </s> <s xml:id="echoid-s9549" xml:space="preserve">GI cum- GL cum | p. </s> <s xml:id="echoid-s9550" xml:space="preserve">59. </s> <s xml:id="echoid-s9551" xml:space="preserve">v. </s> <s xml:id="echoid-s9552" xml:space="preserve">17. </s> <s xml:id="echoid-s9553" xml:space="preserve">verſum AF - verſum CF | p. </s> <s xml:id="echoid-s9554" xml:space="preserve">68. </s> <s xml:id="echoid-s9555" xml:space="preserve">v. </s> <s xml:id="echoid-s9556" xml:space="preserve">34. </s> <s xml:id="echoid-s9557" xml:space="preserve">& </s> <s xml:id="echoid-s9558" xml:space="preserve">Hyperbole - & </s> <s xml:id="echoid-s9559" xml:space="preserve">ſimilis Hyperbole <lb/>v. </s> <s xml:id="echoid-s9560" xml:space="preserve">35. </s> <s xml:id="echoid-s9561" xml:space="preserve">M G, rectum G N, aſymptotos O P, & </s> <s xml:id="echoid-s9562" xml:space="preserve">ipſarum - MI, rectum I N, aſymptotos OP, & </s> <s xml:id="echoid-s9563" xml:space="preserve">ex ipſarum | <lb/>p. </s> <s xml:id="echoid-s9564" xml:space="preserve">74. </s> <s xml:id="echoid-s9565" xml:space="preserve">v. </s> <s xml:id="echoid-s9566" xml:space="preserve">3. </s> <s xml:id="echoid-s9567" xml:space="preserve">Parabolis hactenus - Parabolis in hac | p. </s> <s xml:id="echoid-s9568" xml:space="preserve">75. </s> <s xml:id="echoid-s9569" xml:space="preserve">v. </s> <s xml:id="echoid-s9570" xml:space="preserve">3. </s> <s xml:id="echoid-s9571" xml:space="preserve">AB, BE, item altera - AB, D E, tem latera | p. </s> <s xml:id="echoid-s9572" xml:space="preserve">77. <lb/></s> <s xml:id="echoid-s9573" xml:space="preserve">v. </s> <s xml:id="echoid-s9574" xml:space="preserve">35. </s> <s xml:id="echoid-s9575" xml:space="preserve">ex vettice BG; </s> <s xml:id="echoid-s9576" xml:space="preserve">- ex vertice ſit B G; </s> <s xml:id="echoid-s9577" xml:space="preserve">| p. </s> <s xml:id="echoid-s9578" xml:space="preserve">78. </s> <s xml:id="echoid-s9579" xml:space="preserve">v. </s> <s xml:id="echoid-s9580" xml:space="preserve">13. </s> <s xml:id="echoid-s9581" xml:space="preserve">adſcriptarum aſymptotos - adſcriptarum regulas, & </s> <s xml:id="echoid-s9582" xml:space="preserve"><lb/>aſymptotos | p. </s> <s xml:id="echoid-s9583" xml:space="preserve">79. </s> <s xml:id="echoid-s9584" xml:space="preserve">v. </s> <s xml:id="echoid-s9585" xml:space="preserve">21. </s> <s xml:id="echoid-s9586" xml:space="preserve">Hyperbolæ, per - Hyperbolæ ABC, per | p. </s> <s xml:id="echoid-s9587" xml:space="preserve">83. </s> <s xml:id="echoid-s9588" xml:space="preserve">v. </s> <s xml:id="echoid-s9589" xml:space="preserve">7. </s> <s xml:id="echoid-s9590" xml:space="preserve">Iam, cum rectangulum GE 3 <lb/>ſit - Iam, rectangulũ GE 3 eſt | p. </s> <s xml:id="echoid-s9591" xml:space="preserve">92. </s> <s xml:id="echoid-s9592" xml:space="preserve">v. </s> <s xml:id="echoid-s9593" xml:space="preserve">10. </s> <s xml:id="echoid-s9594" xml:space="preserve">in puncto Q, - in puncto P, | v. </s> <s xml:id="echoid-s9595" xml:space="preserve">32. </s> <s xml:id="echoid-s9596" xml:space="preserve">tamen eas eligemus, que appor-<lb/>tunæ - tamen eas in reliquis eligemus, quæ opportunæ | p. </s> <s xml:id="echoid-s9597" xml:space="preserve">96. </s> <s xml:id="echoid-s9598" xml:space="preserve">v. </s> <s xml:id="echoid-s9599" xml:space="preserve">14. </s> <s xml:id="echoid-s9600" xml:space="preserve">Hyperbolen circumſcribere - Hyperbolen <lb/>concentricam circumſcribere | v. </s> <s xml:id="echoid-s9601" xml:space="preserve">22. </s> <s xml:id="echoid-s9602" xml:space="preserve">eſſe MAXIMAM - eſſe MINIMAM | p. </s> <s xml:id="echoid-s9603" xml:space="preserve">99. </s> <s xml:id="echoid-s9604" xml:space="preserve">in prima ſigura, Hyperbole N <lb/>E concipiatur punctata, & </s> <s xml:id="echoid-s9605" xml:space="preserve">HEK continuata | p. </s> <s xml:id="echoid-s9606" xml:space="preserve">100. </s> <s xml:id="echoid-s9607" xml:space="preserve">v. </s> <s xml:id="echoid-s9608" xml:space="preserve">9. </s> <s xml:id="echoid-s9609" xml:space="preserve">latere MINIMAM - latere BR MINIMAM | p. </s> <s xml:id="echoid-s9610" xml:space="preserve">101. </s> <s xml:id="echoid-s9611" xml:space="preserve"><lb/>v. </s> <s xml:id="echoid-s9612" xml:space="preserve">22. </s> <s xml:id="echoid-s9613" xml:space="preserve">rectus latus - tranſuerſum latus |. </s> <s xml:id="echoid-s9614" xml:space="preserve">p. </s> <s xml:id="echoid-s9615" xml:space="preserve">107. </s> <s xml:id="echoid-s9616" xml:space="preserve">v. </s> <s xml:id="echoid-s9617" xml:space="preserve">10. </s> <s xml:id="echoid-s9618" xml:space="preserve">remotiori. </s> <s xml:id="echoid-s9619" xml:space="preserve">- remotiori GH. </s> <s xml:id="echoid-s9620" xml:space="preserve">| v. </s> <s xml:id="echoid-s9621" xml:space="preserve">vlt. </s> <s xml:id="echoid-s9622" xml:space="preserve">LEG - IEG | p. </s> <s xml:id="echoid-s9623" xml:space="preserve">109. </s> <s xml:id="echoid-s9624" xml:space="preserve"><lb/>v. </s> <s xml:id="echoid-s9625" xml:space="preserve">29. </s> <s xml:id="echoid-s9626" xml:space="preserve">Et enim - Eſt enim | p. </s> <s xml:id="echoid-s9627" xml:space="preserve">110. </s> <s xml:id="echoid-s9628" xml:space="preserve">v. </s> <s xml:id="echoid-s9629" xml:space="preserve">6. </s> <s xml:id="echoid-s9630" xml:space="preserve">Sumatur D E - Iungarur B D, & </s> <s xml:id="echoid-s9631" xml:space="preserve">producatur, & </s> <s xml:id="echoid-s9632" xml:space="preserve">ſumatur DE | v. </s> <s xml:id="echoid-s9633" xml:space="preserve">7. </s> <s xml:id="echoid-s9634" xml:space="preserve">dia-<lb/>metro AE - diametro BE. </s> <s xml:id="echoid-s9635" xml:space="preserve">| p. </s> <s xml:id="echoid-s9636" xml:space="preserve">111. </s> <s xml:id="echoid-s9637" xml:space="preserve">v. </s> <s xml:id="echoid-s9638" xml:space="preserve">vlt. </s> <s xml:id="echoid-s9639" xml:space="preserve">adſcribitur, cum recto - adſcribitur, ſed cum recto | p. </s> <s xml:id="echoid-s9640" xml:space="preserve">112. </s> <s xml:id="echoid-s9641" xml:space="preserve">v. </s> <s xml:id="echoid-s9642" xml:space="preserve">3. </s> <s xml:id="echoid-s9643" xml:space="preserve">ſed <lb/>BH - ſed BA | v. </s> <s xml:id="echoid-s9644" xml:space="preserve">4. </s> <s xml:id="echoid-s9645" xml:space="preserve">ipſa BH - ipſa BA | v. </s> <s xml:id="echoid-s9646" xml:space="preserve">24. </s> <s xml:id="echoid-s9647" xml:space="preserve">OM aſymptoto - OM aſymptotos | v. </s> <s xml:id="echoid-s9648" xml:space="preserve">39. </s> <s xml:id="echoid-s9649" xml:space="preserve">punctum D - punctum <lb/>D, & </s> <s xml:id="echoid-s9650" xml:space="preserve">cum dato ſemi-tranſuerſo E. </s> <s xml:id="echoid-s9651" xml:space="preserve">| p. </s> <s xml:id="echoid-s9652" xml:space="preserve">114. </s> <s xml:id="echoid-s9653" xml:space="preserve">v. </s> <s xml:id="echoid-s9654" xml:space="preserve">30. </s> <s xml:id="echoid-s9655" xml:space="preserve">MBN - ABC | p. </s> <s xml:id="echoid-s9656" xml:space="preserve">120. </s> <s xml:id="echoid-s9657" xml:space="preserve">v. </s> <s xml:id="echoid-s9658" xml:space="preserve">30. </s> <s xml:id="echoid-s9659" xml:space="preserve">in qua cum - & </s> <s xml:id="echoid-s9660" xml:space="preserve">cum. </s> <s xml:id="echoid-s9661" xml:space="preserve">| p. </s> <s xml:id="echoid-s9662" xml:space="preserve">122. </s> <s xml:id="echoid-s9663" xml:space="preserve">v. </s> <s xml:id="echoid-s9664" xml:space="preserve">1. </s> <s xml:id="echoid-s9665" xml:space="preserve"><lb/>in portione - portioni | v. </s> <s xml:id="echoid-s9666" xml:space="preserve">2. </s> <s xml:id="echoid-s9667" xml:space="preserve">in triangulis - triangulis | v. </s> <s xml:id="echoid-s9668" xml:space="preserve">8. </s> <s xml:id="echoid-s9669" xml:space="preserve">in Parabola - Parabolæ | v. </s> <s xml:id="echoid-s9670" xml:space="preserve">11. </s> <s xml:id="echoid-s9671" xml:space="preserve">in ea inſcripti - ei in-<lb/>ſcripti | p. </s> <s xml:id="echoid-s9672" xml:space="preserve">123. </s> <s xml:id="echoid-s9673" xml:space="preserve">v. </s> <s xml:id="echoid-s9674" xml:space="preserve">23. </s> <s xml:id="echoid-s9675" xml:space="preserve">cum quælibet - nam quælibet | v. </s> <s xml:id="echoid-s9676" xml:space="preserve">32. </s> <s xml:id="echoid-s9677" xml:space="preserve">Parabolæ DGF - Parabolæ BGF | p. </s> <s xml:id="echoid-s9678" xml:space="preserve">125. </s> <s xml:id="echoid-s9679" xml:space="preserve">v. </s> <s xml:id="echoid-s9680" xml:space="preserve">2. </s> <s xml:id="echoid-s9681" xml:space="preserve">equa-<lb/>le rectangulo - æquale, vel minus rectangulo | v. </s> <s xml:id="echoid-s9682" xml:space="preserve">8. </s> <s xml:id="echoid-s9683" xml:space="preserve">æquale poſitum - æquale, vel minus poſitum | v. </s> <s xml:id="echoid-s9684" xml:space="preserve">12. </s> <s xml:id="echoid-s9685" xml:space="preserve">ſeca-<lb/>bit ſibi - ſecabit aliam ſibi | p. </s> <s xml:id="echoid-s9686" xml:space="preserve">127. </s> <s xml:id="echoid-s9687" xml:space="preserve">v. </s> <s xml:id="echoid-s9688" xml:space="preserve">22. </s> <s xml:id="echoid-s9689" xml:space="preserve">vt OF ad FB, & </s> <s xml:id="echoid-s9690" xml:space="preserve">KF - vt OF ad FB, & </s> <s xml:id="echoid-s9691" xml:space="preserve">permutando OK ad OF, vt KB <lb/>ad BF, & </s> <s xml:id="echoid-s9692" xml:space="preserve">eſt OK maior OF, ergo, & </s> <s xml:id="echoid-s9693" xml:space="preserve">KB maior eſt FB, & </s> <s xml:id="echoid-s9694" xml:space="preserve">KF | p. </s> <s xml:id="echoid-s9695" xml:space="preserve">129. </s> <s xml:id="echoid-s9696" xml:space="preserve">v. </s> <s xml:id="echoid-s9697" xml:space="preserve">5. </s> <s xml:id="echoid-s9698" xml:space="preserve">cum æquali - cum circumſcriptæ <lb/>æquali | v. </s> <s xml:id="echoid-s9699" xml:space="preserve">10. </s> <s xml:id="echoid-s9700" xml:space="preserve">ſit ipſo - ſit ipſi | p. </s> <s xml:id="echoid-s9701" xml:space="preserve">130. </s> <s xml:id="echoid-s9702" xml:space="preserve">v. </s> <s xml:id="echoid-s9703" xml:space="preserve">12. </s> <s xml:id="echoid-s9704" xml:space="preserve">vt in 83 - vtin 82. </s> <s xml:id="echoid-s9705" xml:space="preserve">| v. </s> <s xml:id="echoid-s9706" xml:space="preserve">22. </s> <s xml:id="echoid-s9707" xml:space="preserve">vt AF - vt OF | p. </s> <s xml:id="echoid-s9708" xml:space="preserve">131. </s> <s xml:id="echoid-s9709" xml:space="preserve">v. </s> <s xml:id="echoid-s9710" xml:space="preserve">1. </s> <s xml:id="echoid-s9711" xml:space="preserve">ALCO - A <lb/>LCN | v. </s> <s xml:id="echoid-s9712" xml:space="preserve">12. </s> <s xml:id="echoid-s9713" xml:space="preserve">& </s> <s xml:id="echoid-s9714" xml:space="preserve">è contra, quæ - & </s> <s xml:id="echoid-s9715" xml:space="preserve">è contra, eam, quę | v. </s> <s xml:id="echoid-s9716" xml:space="preserve">35. </s> <s xml:id="echoid-s9717" xml:space="preserve">ſi igitur ellipſis - ſi igitur | p. </s> <s xml:id="echoid-s9718" xml:space="preserve">132. </s> <s xml:id="echoid-s9719" xml:space="preserve">v. </s> <s xml:id="echoid-s9720" xml:space="preserve">vlt. </s> <s xml:id="echoid-s9721" xml:space="preserve">KEI <lb/>maiora - KFI æqualia | p. </s> <s xml:id="echoid-s9722" xml:space="preserve">133. </s> <s xml:id="echoid-s9723" xml:space="preserve">v. </s> <s xml:id="echoid-s9724" xml:space="preserve">11. </s> <s xml:id="echoid-s9725" xml:space="preserve">EG, GN - EG, EN | p. </s> <s xml:id="echoid-s9726" xml:space="preserve">135. </s> <s xml:id="echoid-s9727" xml:space="preserve">v. </s> <s xml:id="echoid-s9728" xml:space="preserve">42. </s> <s xml:id="echoid-s9729" xml:space="preserve">eſſe axis - eſſe minoris axis | p. </s> <s xml:id="echoid-s9730" xml:space="preserve">142. </s> <s xml:id="echoid-s9731" xml:space="preserve">v. </s> <s xml:id="echoid-s9732" xml:space="preserve">37. </s> <s xml:id="echoid-s9733" xml:space="preserve"><lb/>cum LH - cum ſit LH | p. </s> <s xml:id="echoid-s9734" xml:space="preserve">143. </s> <s xml:id="echoid-s9735" xml:space="preserve">v. </s> <s xml:id="echoid-s9736" xml:space="preserve">12. </s> <s xml:id="echoid-s9737" xml:space="preserve">LH maior - LH minor | v. </s> <s xml:id="echoid-s9738" xml:space="preserve">13. </s> <s xml:id="echoid-s9739" xml:space="preserve">& </s> <s xml:id="echoid-s9740" xml:space="preserve">HC maior - & </s> <s xml:id="echoid-s9741" xml:space="preserve">HC minor | p. </s> <s xml:id="echoid-s9742" xml:space="preserve">144. </s> <s xml:id="echoid-s9743" xml:space="preserve">v. </s> <s xml:id="echoid-s9744" xml:space="preserve">30. </s> <s xml:id="echoid-s9745" xml:space="preserve"><lb/>pertinentium - pertingentium | p. </s> <s xml:id="echoid-s9746" xml:space="preserve">146. </s> <s xml:id="echoid-s9747" xml:space="preserve">v. </s> <s xml:id="echoid-s9748" xml:space="preserve">14. </s> <s xml:id="echoid-s9749" xml:space="preserve">inter _a_ contactum - inter contactum | v. </s> <s xml:id="echoid-s9750" xml:space="preserve">15. </s> <s xml:id="echoid-s9751" xml:space="preserve">cadet totus intra, & </s> <s xml:id="echoid-s9752" xml:space="preserve"><lb/>ſi - cadet _a_ totus intra Ellipſim, & </s> <s xml:id="echoid-s9753" xml:space="preserve">ſi | p. </s> <s xml:id="echoid-s9754" xml:space="preserve">148. </s> <s xml:id="echoid-s9755" xml:space="preserve">v. </s> <s xml:id="echoid-s9756" xml:space="preserve">vlt. </s> <s xml:id="echoid-s9757" xml:space="preserve">ſitque DF - ſitque BF | pag. </s> <s xml:id="echoid-s9758" xml:space="preserve">149. </s> <s xml:id="echoid-s9759" xml:space="preserve">v. </s> <s xml:id="echoid-s9760" xml:space="preserve">20. </s> <s xml:id="echoid-s9761" xml:space="preserve">LA - LH.</s> <s xml:id="echoid-s9762" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div981" type="section" level="1" n="399"> <head xml:id="echoid-head410" style="it" xml:space="preserve">Lib. II. errata ſic reſtituenda.</head> <p> <s xml:id="echoid-s9763" xml:space="preserve">Pag. </s> <s xml:id="echoid-s9764" xml:space="preserve">1. </s> <s xml:id="echoid-s9765" xml:space="preserve">verſ. </s> <s xml:id="echoid-s9766" xml:space="preserve">14. </s> <s xml:id="echoid-s9767" xml:space="preserve">ipſi BC. </s> <s xml:id="echoid-s9768" xml:space="preserve">- ipſi AC. </s> <s xml:id="echoid-s9769" xml:space="preserve">| p. </s> <s xml:id="echoid-s9770" xml:space="preserve">5. </s> <s xml:id="echoid-s9771" xml:space="preserve">v. </s> <s xml:id="echoid-s9772" xml:space="preserve">7. </s> <s xml:id="echoid-s9773" xml:space="preserve">rectangulo æquale eſt - rectangulo cum quadrato DM æquale <lb/>eſt | p. </s> <s xml:id="echoid-s9774" xml:space="preserve">8. </s> <s xml:id="echoid-s9775" xml:space="preserve">v. </s> <s xml:id="echoid-s9776" xml:space="preserve">5. </s> <s xml:id="echoid-s9777" xml:space="preserve">in O; </s> <s xml:id="echoid-s9778" xml:space="preserve">cum - in O; </s> <s xml:id="echoid-s9779" xml:space="preserve">FI ſecet GH in N. </s> <s xml:id="echoid-s9780" xml:space="preserve">Et cum | p. </s> <s xml:id="echoid-s9781" xml:space="preserve">9. </s> <s xml:id="echoid-s9782" xml:space="preserve">v. </s> <s xml:id="echoid-s9783" xml:space="preserve">33. </s> <s xml:id="echoid-s9784" xml:space="preserve">MINIMA erit - minor erit quacunque <lb/>ducibilium | p. </s> <s xml:id="echoid-s9785" xml:space="preserve">11. </s> <s xml:id="echoid-s9786" xml:space="preserve">v. </s> <s xml:id="echoid-s9787" xml:space="preserve">25. </s> <s xml:id="echoid-s9788" xml:space="preserve">& </s> <s xml:id="echoid-s9789" xml:space="preserve">reliquam B D - & </s> <s xml:id="echoid-s9790" xml:space="preserve">in ſecunda figura, reliquam E D | p. </s> <s xml:id="echoid-s9791" xml:space="preserve">12. </s> <s xml:id="echoid-s9792" xml:space="preserve">v. </s> <s xml:id="echoid-s9793" xml:space="preserve">11. </s> <s xml:id="echoid-s9794" xml:space="preserve">à vertice diſtet - à <lb/>vertice B diſtet | p. </s> <s xml:id="echoid-s9795" xml:space="preserve">16. </s> <s xml:id="echoid-s9796" xml:space="preserve">v. </s> <s xml:id="echoid-s9797" xml:space="preserve">4. </s> <s xml:id="echoid-s9798" xml:space="preserve">prouenire - peruenire | p. </s> <s xml:id="echoid-s9799" xml:space="preserve">21. </s> <s xml:id="echoid-s9800" xml:space="preserve">v. </s> <s xml:id="echoid-s9801" xml:space="preserve">6. </s> <s xml:id="echoid-s9802" xml:space="preserve">BD quibus - BD, atque | p. </s> <s xml:id="echoid-s9803" xml:space="preserve">23. </s> <s xml:id="echoid-s9804" xml:space="preserve">v. </s> <s xml:id="echoid-s9805" xml:space="preserve">40. </s> <s xml:id="echoid-s9806" xml:space="preserve">Quod, &</s> <s xml:id="echoid-s9807" xml:space="preserve">c. </s> <s xml:id="echoid-s9808" xml:space="preserve">-<lb/>Quod, &</s> <s xml:id="echoid-s9809" xml:space="preserve">c. </s> <s xml:id="echoid-s9810" xml:space="preserve">Sed FC æqualis eſt ipſi FA: </s> <s xml:id="echoid-s9811" xml:space="preserve">ergo in hoc caſu duæ ſunt MINIMAE. </s> <s xml:id="echoid-s9812" xml:space="preserve">| v. </s> <s xml:id="echoid-s9813" xml:space="preserve">42. </s> <s xml:id="echoid-s9814" xml:space="preserve">vel extra - vel intra | <lb/>v. </s> <s xml:id="echoid-s9815" xml:space="preserve">43. </s> <s xml:id="echoid-s9816" xml:space="preserve">ex recta F FG - ex F recta FG | p. </s> <s xml:id="echoid-s9817" xml:space="preserve">32. </s> <s xml:id="echoid-s9818" xml:space="preserve">v. </s> <s xml:id="echoid-s9819" xml:space="preserve">9. </s> <s xml:id="echoid-s9820" xml:space="preserve">erit - eſſet | p. </s> <s xml:id="echoid-s9821" xml:space="preserve">34. </s> <s xml:id="echoid-s9822" xml:space="preserve">v. </s> <s xml:id="echoid-s9823" xml:space="preserve">12. </s> <s xml:id="echoid-s9824" xml:space="preserve">MAXIMA ad incluſam - MAXIMA du-<lb/>cibilium ad incluſam | p. </s> <s xml:id="echoid-s9825" xml:space="preserve">37. </s> <s xml:id="echoid-s9826" xml:space="preserve">v. </s> <s xml:id="echoid-s9827" xml:space="preserve">2. </s> <s xml:id="echoid-s9828" xml:space="preserve">ademptum - adeptum | v. </s> <s xml:id="echoid-s9829" xml:space="preserve">34. </s> <s xml:id="echoid-s9830" xml:space="preserve">ſumantur - addantur | p. </s> <s xml:id="echoid-s9831" xml:space="preserve">38. </s> <s xml:id="echoid-s9832" xml:space="preserve">v. </s> <s xml:id="echoid-s9833" xml:space="preserve">31. </s> <s xml:id="echoid-s9834" xml:space="preserve">ſit minor - ſit <lb/>maior | citat. </s> <s xml:id="echoid-s9835" xml:space="preserve">25. </s> <s xml:id="echoid-s9836" xml:space="preserve">pr. </s> <s xml:id="echoid-s9837" xml:space="preserve">conic. </s> <s xml:id="echoid-s9838" xml:space="preserve">- 27. </s> <s xml:id="echoid-s9839" xml:space="preserve">pr. </s> <s xml:id="echoid-s9840" xml:space="preserve">conic. </s> <s xml:id="echoid-s9841" xml:space="preserve">| p. </s> <s xml:id="echoid-s9842" xml:space="preserve">42. </s> <s xml:id="echoid-s9843" xml:space="preserve">v. </s> <s xml:id="echoid-s9844" xml:space="preserve">39. </s> <s xml:id="echoid-s9845" xml:space="preserve">rectangulum GEC - rectangulum GEF | p. </s> <s xml:id="echoid-s9846" xml:space="preserve">43. </s> <s xml:id="echoid-s9847" xml:space="preserve">v. </s> <s xml:id="echoid-s9848" xml:space="preserve">25. <lb/></s> <s xml:id="echoid-s9849" xml:space="preserve">quadratum Q P - quadratum OP | v. </s> <s xml:id="echoid-s9850" xml:space="preserve">25. </s> <s xml:id="echoid-s9851" xml:space="preserve">contingente Q P - contingente OP | v 26. </s> <s xml:id="echoid-s9852" xml:space="preserve">quadrato Q P - quadrato <lb/>OP | p. </s> <s xml:id="echoid-s9853" xml:space="preserve">50. </s> <s xml:id="echoid-s9854" xml:space="preserve">v. </s> <s xml:id="echoid-s9855" xml:space="preserve">3. </s> <s xml:id="echoid-s9856" xml:space="preserve">latera AD - latera parallela AD | p. </s> <s xml:id="echoid-s9857" xml:space="preserve">51. </s> <s xml:id="echoid-s9858" xml:space="preserve">v. </s> <s xml:id="echoid-s9859" xml:space="preserve">40. </s> <s xml:id="echoid-s9860" xml:space="preserve">D, M, - D, N, | v. </s> <s xml:id="echoid-s9861" xml:space="preserve">41. </s> <s xml:id="echoid-s9862" xml:space="preserve">ſiue erit - atque erit | <lb/>p. </s> <s xml:id="echoid-s9863" xml:space="preserve">53. </s> <s xml:id="echoid-s9864" xml:space="preserve">In Coroll. </s> <s xml:id="echoid-s9865" xml:space="preserve">II. </s> <s xml:id="echoid-s9866" xml:space="preserve">dele ea verba in 4. </s> <s xml:id="echoid-s9867" xml:space="preserve">5. </s> <s xml:id="echoid-s9868" xml:space="preserve">6. </s> <s xml:id="echoid-s9869" xml:space="preserve">7. </s> <s xml:id="echoid-s9870" xml:space="preserve">& </s> <s xml:id="echoid-s9871" xml:space="preserve">8. </s> <s xml:id="echoid-s9872" xml:space="preserve">figura | p. </s> <s xml:id="echoid-s9873" xml:space="preserve">56. </s> <s xml:id="echoid-s9874" xml:space="preserve">v. </s> <s xml:id="echoid-s9875" xml:space="preserve">23. </s> <s xml:id="echoid-s9876" xml:space="preserve">comm uni B E - communi G E | p. </s> <s xml:id="echoid-s9877" xml:space="preserve">60. </s> <s xml:id="echoid-s9878" xml:space="preserve"><lb/>v. </s> <s xml:id="echoid-s9879" xml:space="preserve">9. </s> <s xml:id="echoid-s9880" xml:space="preserve">ex quo NE - ex quo DE | p. </s> <s xml:id="echoid-s9881" xml:space="preserve">61. </s> <s xml:id="echoid-s9882" xml:space="preserve">v. </s> <s xml:id="echoid-s9883" xml:space="preserve">5. </s> <s xml:id="echoid-s9884" xml:space="preserve">Hyperbolis, aut Ellipſibus - Hyperbolis, vel circulis, aut Ellipſibus | <lb/>p. </s> <s xml:id="echoid-s9885" xml:space="preserve">62. </s> <s xml:id="echoid-s9886" xml:space="preserve">citat. </s> <s xml:id="echoid-s9887" xml:space="preserve">46. </s> <s xml:id="echoid-s9888" xml:space="preserve">h. </s> <s xml:id="echoid-s9889" xml:space="preserve">- 10. </s> <s xml:id="echoid-s9890" xml:space="preserve">ſec. </s> <s xml:id="echoid-s9891" xml:space="preserve">conic. </s> <s xml:id="echoid-s9892" xml:space="preserve">& </s> <s xml:id="echoid-s9893" xml:space="preserve">46. </s> <s xml:id="echoid-s9894" xml:space="preserve">h. </s> <s xml:id="echoid-s9895" xml:space="preserve">| v. </s> <s xml:id="echoid-s9896" xml:space="preserve">18. </s> <s xml:id="echoid-s9897" xml:space="preserve">Iam, ducta - Iam, in prima figura, ducta | p. </s> <s xml:id="echoid-s9898" xml:space="preserve">63. </s> <s xml:id="echoid-s9899" xml:space="preserve">v. </s> <s xml:id="echoid-s9900" xml:space="preserve">2. </s> <s xml:id="echoid-s9901" xml:space="preserve">Hy-<lb/>perbole - Hyperbolæ | v. </s> <s xml:id="echoid-s9902" xml:space="preserve">11. </s> <s xml:id="echoid-s9903" xml:space="preserve">BD - BE | v. </s> <s xml:id="echoid-s9904" xml:space="preserve">13. </s> <s xml:id="echoid-s9905" xml:space="preserve">in D - in E | p. </s> <s xml:id="echoid-s9906" xml:space="preserve">72. </s> <s xml:id="echoid-s9907" xml:space="preserve">v. </s> <s xml:id="echoid-s9908" xml:space="preserve">18. </s> <s xml:id="echoid-s9909" xml:space="preserve">producta conueniet - producta, vel con-<lb/>ueniet | v. </s> <s xml:id="echoid-s9910" xml:space="preserve">20. </s> <s xml:id="echoid-s9911" xml:space="preserve">verticis B; </s> <s xml:id="echoid-s9912" xml:space="preserve">qua propter - verticis B; </s> <s xml:id="echoid-s9913" xml:space="preserve">velin ſecunda figura aliquando axi æquidiſtabit, quapro-<lb/>pter | v. </s> <s xml:id="echoid-s9914" xml:space="preserve">24. </s> <s xml:id="echoid-s9915" xml:space="preserve">Coni ſuperficiem - Coni, vel Cylindri ſuperficiem | v. </s> <s xml:id="echoid-s9916" xml:space="preserve">33. </s> <s xml:id="echoid-s9917" xml:space="preserve">Coni à latere - Coni, vel Cylindri à la-<lb/>tere | v. </s> <s xml:id="echoid-s9918" xml:space="preserve">34. </s> <s xml:id="echoid-s9919" xml:space="preserve">Conicam ſuperficiem - Conicam, vel Cylindricam ſuperficiem | p. </s> <s xml:id="echoid-s9920" xml:space="preserve">75. </s> <s xml:id="echoid-s9921" xml:space="preserve">v. </s> <s xml:id="echoid-s9922" xml:space="preserve">41. </s> <s xml:id="echoid-s9923" xml:space="preserve">latera, &</s> <s xml:id="echoid-s9924" xml:space="preserve">c. </s> <s xml:id="echoid-s9925" xml:space="preserve">- latus A <lb/>C, &</s> <s xml:id="echoid-s9926" xml:space="preserve">c. </s> <s xml:id="echoid-s9927" xml:space="preserve">| p. </s> <s xml:id="echoid-s9928" xml:space="preserve">76. </s> <s xml:id="echoid-s9929" xml:space="preserve">v. </s> <s xml:id="echoid-s9930" xml:space="preserve">1. </s> <s xml:id="echoid-s9931" xml:space="preserve">in plano NL - in plano DAC | p. </s> <s xml:id="echoid-s9932" xml:space="preserve">79. </s> <s xml:id="echoid-s9933" xml:space="preserve">v. </s> <s xml:id="echoid-s9934" xml:space="preserve">5. </s> <s xml:id="echoid-s9935" xml:space="preserve">ex 20. </s> <s xml:id="echoid-s9936" xml:space="preserve">22. </s> <s xml:id="echoid-s9937" xml:space="preserve">ac 23. </s> <s xml:id="echoid-s9938" xml:space="preserve">huius - ex 20. </s> <s xml:id="echoid-s9939" xml:space="preserve">ac 22. </s> <s xml:id="echoid-s9940" xml:space="preserve">huius | v. </s> <s xml:id="echoid-s9941" xml:space="preserve">11. </s> <s xml:id="echoid-s9942" xml:space="preserve"><lb/>DEB - DE | v. </s> <s xml:id="echoid-s9943" xml:space="preserve">30. </s> <s xml:id="echoid-s9944" xml:space="preserve">DEB - DE | p. </s> <s xml:id="echoid-s9945" xml:space="preserve">121. </s> <s xml:id="echoid-s9946" xml:space="preserve">v. </s> <s xml:id="echoid-s9947" xml:space="preserve">vlt. </s> <s xml:id="echoid-s9948" xml:space="preserve">eſſet alter - eſſe, alter | p. </s> <s xml:id="echoid-s9949" xml:space="preserve">129. </s> <s xml:id="echoid-s9950" xml:space="preserve">v. </s> <s xml:id="echoid-s9951" xml:space="preserve">33. </s> <s xml:id="echoid-s9952" xml:space="preserve">rectangulum - rectangulo-<lb/>rum | p. </s> <s xml:id="echoid-s9953" xml:space="preserve">130. </s> <s xml:id="echoid-s9954" xml:space="preserve">v. </s> <s xml:id="echoid-s9955" xml:space="preserve">penult. </s> <s xml:id="echoid-s9956" xml:space="preserve">in 3. </s> <s xml:id="echoid-s9957" xml:space="preserve">ratione ad 1. </s> <s xml:id="echoid-s9958" xml:space="preserve">- in ratione 3. </s> <s xml:id="echoid-s9959" xml:space="preserve">ad 1.</s> <s xml:id="echoid-s9960" xml:space="preserve"/> </p> <pb file="0342" n="342"/> <p> <s xml:id="echoid-s9961" xml:space="preserve">Imprimatur ſeruatis ſeruandis 18. </s> <s xml:id="echoid-s9962" xml:space="preserve">Martij 1658.</s> <s xml:id="echoid-s9963" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9964" xml:space="preserve">Vinc. </s> <s xml:id="echoid-s9965" xml:space="preserve">de Bardis Vic. </s> <s xml:id="echoid-s9966" xml:space="preserve">Gen. </s> <s xml:id="echoid-s9967" xml:space="preserve">Florentiæ.</s> <s xml:id="echoid-s9968" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9969" xml:space="preserve">Excellentiſsimus Dominus Auguſtinus Coltellinus Aduocatus, & </s> <s xml:id="echoid-s9970" xml:space="preserve"><lb/>S. </s> <s xml:id="echoid-s9971" xml:space="preserve">Officij Conſultor, videat hoc Opus inſcriptum DE MAXI-<lb/>MIS, & </s> <s xml:id="echoid-s9972" xml:space="preserve">MINIMIS, &</s> <s xml:id="echoid-s9973" xml:space="preserve">c. </s> <s xml:id="echoid-s9974" xml:space="preserve">& </s> <s xml:id="echoid-s9975" xml:space="preserve">referat, die 9. </s> <s xml:id="echoid-s9976" xml:space="preserve">Aprilis 1659.</s> <s xml:id="echoid-s9977" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9978" xml:space="preserve">F. </s> <s xml:id="echoid-s9979" xml:space="preserve">Gabriel Pierotius Florentinus S. </s> <s xml:id="echoid-s9980" xml:space="preserve">Officij <lb/>Flor. </s> <s xml:id="echoid-s9981" xml:space="preserve">Cancell. </s> <s xml:id="echoid-s9982" xml:space="preserve">&</s> <s xml:id="echoid-s9983" xml:space="preserve">c.</s> <s xml:id="echoid-s9984" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9985" xml:space="preserve">Sic diuinare licet Reuerendiſs. </s> <s xml:id="echoid-s9986" xml:space="preserve">Pater, nec malè de arte ſua au-<lb/>diet Mathematicus, dum per retortos linearum tramites itur <lb/>ad rectam geometricæ veritatis; </s> <s xml:id="echoid-s9987" xml:space="preserve">bonis interim lætantibus, <lb/>cum nihil obliquum ab orthodoxa fide inueniatur S. </s> <s xml:id="echoid-s9988" xml:space="preserve">R. </s> <s xml:id="echoid-s9989" xml:space="preserve">E. </s> <s xml:id="echoid-s9990" xml:space="preserve">in-<lb/>uiſum, prout refero. </s> <s xml:id="echoid-s9991" xml:space="preserve">Die x v j. </s> <s xml:id="echoid-s9992" xml:space="preserve">April. </s> <s xml:id="echoid-s9993" xml:space="preserve">MDCLIX.</s> <s xml:id="echoid-s9994" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9995" xml:space="preserve">Auguſtinus Coltellini manu propria.</s> <s xml:id="echoid-s9996" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9997" xml:space="preserve">Stante prędicta atteſtatione imprimatur. </s> <s xml:id="echoid-s9998" xml:space="preserve">Hac die 19. </s> <s xml:id="echoid-s9999" xml:space="preserve">April. </s> <s xml:id="echoid-s10000" xml:space="preserve">1659.</s> <s xml:id="echoid-s10001" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10002" xml:space="preserve">F. </s> <s xml:id="echoid-s10003" xml:space="preserve">Gabriel Pierotius S. </s> <s xml:id="echoid-s10004" xml:space="preserve">Officij Flor. <lb/></s> <s xml:id="echoid-s10005" xml:space="preserve">Cancell. </s> <s xml:id="echoid-s10006" xml:space="preserve">de mandato.</s> <s xml:id="echoid-s10007" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10008" xml:space="preserve">Alexander Victorius Sereniſs. </s> <s xml:id="echoid-s10009" xml:space="preserve">Magni Ducis Auditor.</s> <s xml:id="echoid-s10010" xml:space="preserve"/> </p> <pb file="0343" n="343"/> <pb file="0344" n="344"/> <pb file="0345" n="345"/> <handwritten/> <handwritten/> <pb file="0346" n="346"/> </div></text> </echo>