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DESpecs 2.0 Autumn 2009
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:14:40 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> <metadata> <dcterms:identifier>ECHO:Y97ESDAP.xml</dcterms:identifier> <dcterms:creator identifier="GND:118639749">Huygens, Christiaan</dcterms:creator> <dcterms:title xml:lang="la">Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica.</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1724</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> <log>curly braces also left around greek ligatures red tag is not yet recognized</log> </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"/> <pb file="0003" n="3"/> <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"><emph style="red">CHRISTIANI HUGENII</emph></head> <head xml:id="echoid-head2" xml:space="preserve">AZULICHEM,</head> <head xml:id="echoid-head3" style="it" xml:space="preserve">Dum viveret Zelhemi Toparchæ,</head> <head xml:id="echoid-head4" xml:space="preserve"><emph style="red">OPERA VARIA.</emph></head> <head xml:id="echoid-head5" xml:space="preserve"><emph style="sc">Volumen</emph> <emph style="sc">Secundum</emph>.</head> <figure> <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0005-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head6" xml:space="preserve"><emph style="sc">Lugduni</emph> <emph style="sc">Batavorum</emph>, <lb/>Apud <emph style="red">JANSSONIOS VANDER A@,</emph> <lb/>Bibliopolas. <emph style="sc">MDCCXXIV</emph>.</head> <handwritten/> <pb file="0006" n="6"/> <handwritten/> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head7" xml:space="preserve">MAX-PLANCK-INSTITUT <lb/>FOR WISSENSCHAFTSGESCHICHTE <lb/>Bibliothek</head> <pb file="0007" n="7"/> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head8" xml:space="preserve"><emph style="bf">CHRISTIANI HUGENII <lb/>OPERA <lb/>GEOMETRICA.</emph></head> <head xml:id="echoid-head9" xml:space="preserve"><emph style="sc">Tomus</emph> <emph style="sc">Secundus</emph>.</head> <pb file="0008" n="8"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head10" xml:space="preserve">Tomi ſecundi contenta.</head> <note position="right" xml:space="preserve"> <lb/><emph style="sc">Theoremata de quadratura</emph> <emph style="sc">Hyperboles</emph>, <emph style="sc">Ellipsis et</emph> <lb/><emph style="sc">Circuli</emph>, ex dato portionum gravitatis centro. Quibus ſub-<lb/>juncta eſt E’ξετα{σι}ς Cyclometriæ Cl. Viri <emph style="sc">Gregorii</emph> à S. <emph style="sc">Vin-<lb/>centio</emph>, editæ Anno <emph style="sc">CIƆIƆCXLVII</emph>. # 309. <lb/><emph style="sc">Epistola ad</emph> C. V. <emph style="sc">Franc</emph>. <emph style="sc">Xaverium</emph> <emph style="sc">Ainscom</emph>. S. I. qua <lb/>diluuntur ea quibus Ἐξέτα{σι}ς Cyclometriæ <emph style="sc">Gregor ii</emph> à S. <emph style="sc">Vin-<lb/>centio</emph> impugnata fuit. # 341. <lb/><emph style="sc">De</emph> <emph style="sc">Circuli magnitudine inventa</emph>. Accedunt ejuſdem Pro-<lb/>blematum quorundam illuſtrium Conſtructiones. # 351. <lb/><emph style="sc">De</emph> <emph style="sc">Circuli et</emph> <emph style="sc">Hyperbolæ quadratura controver-<lb/>sia</emph>. # 405. <lb/><emph style="sc">Geometrica varia</emph>. # 483. <lb/></note> <pb file="0009" n="9"/> </div> <div xml:id="echoid-div7" type="section" level="1" n="7"> <head xml:id="echoid-head11" xml:space="preserve"><emph style="bf">CHRISTIANI HUGENII,</emph> <lb/><emph style="sc">Const</emph>. F. <lb/>THEOREMATA <lb/>DE <lb/><emph style="bf">QUADRATURA</emph></head> <head xml:id="echoid-head12" xml:space="preserve"><emph style="bf">HYPERBOLES, ELLIPSIS</emph> <lb/><emph style="bf">ET CIRCULI,</emph> <lb/>EX DATO <lb/><emph style="bf">PORTIONUM GRAVITATIS CENTRO.</emph></head> <head xml:id="echoid-head13" style="it" xml:space="preserve">Quibus ſubjuncta eſt</head> <head xml:id="echoid-head14" xml:space="preserve">Ε’ξέ{τα}{σι}ς Cyclometriæ Cl. Viri <emph style="sc">Gregorii</emph> à S. <emph style="sc">Vincentio</emph>, <lb/>editæ Anno <emph style="sc">C</emph><emph style="sc">Iɔ</emph> <emph style="sc">Iɔcxlvii</emph>.</head> <pb file="0010" n="10"/> <pb file="0011" n="11"/> <figure> <image file="0011-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0011-01"/> </figure> </div> <div xml:id="echoid-div8" type="section" level="1" n="8"> <head xml:id="echoid-head15" xml:space="preserve"><emph style="bf">AD LECTOREM.</emph></head> <p> <s xml:id="echoid-s1" xml:space="preserve">DE Conicis Sectionibus & </s> <s xml:id="echoid-s2" xml:space="preserve">Cir-<lb/>culo novi quid adferimus, <lb/>Amice Lector, ſi tamen ita vo-<lb/>cari queat, quod æternâ le-<lb/>ge definitum conſtitutum que, <lb/>quale nunc eſt, perpetuò fuit. <lb/></s> <s xml:id="echoid-s3" xml:space="preserve">Sic quod recens effoditur, no-<lb/>vum quis aurum dicat. </s> <s xml:id="echoid-s4" xml:space="preserve">Sic ſtellas in cœlo novas, <lb/>quæ ſuperioribus ſæculis incognitæ, noſtrorum <lb/>artificio deteguntur. </s> <s xml:id="echoid-s5" xml:space="preserve">Neque verò his Geometri-<lb/>cum Theorema ullum vetuſtate cedit, ſed nos <lb/>novitatem ſingulis tribuimus, prout quæque ſe-<lb/>ſe nobis offerunt fiuntque manifeſta. </s> <s xml:id="echoid-s6" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s7" xml:space="preserve"><lb/>inſcripti trigoni Parabola ante Archimedem ſeſ-<lb/>quitertia fuiſſe dicenda eſt; </s> <s xml:id="echoid-s8" xml:space="preserve">neque illa minùs <lb/>immutabili veritate reliquarum Sectionum Cir-<lb/>culique portionibus ſemper inerant, quæ nunc <lb/>circa eas prodimus, licet antehac nemini, de <lb/>quo quidem ad nos pervenerit, comperta fuiſſe <lb/>conſtet vel determinata. </s> <s xml:id="echoid-s9" xml:space="preserve">Damus autem non iſti, <lb/>quam retulimus, Archimedeæ ſimilem determi-<lb/>nationem, neque vel ipſa rerum natura, poſt <lb/>tot ſubtiliſſimorum hominum deluſos conatus, <lb/>ſpem reliquiſſe videtur tale quid unquam de fi-<lb/>guris, quas tractandas ſumpſimus, expectandi: </s> <s xml:id="echoid-s10" xml:space="preserve"><lb/>Verùm id præſtitiſſe profitemur, quod in ipſa <pb file="0012" n="12" rhead="PRÆFATIO."/> quoque in ſcriptione expreſſum eſt, & </s> <s xml:id="echoid-s11" xml:space="preserve">paulò quid <lb/>amplius, ſi Lectoris æqui judicium prævenire per-<lb/>mittitur. </s> <s xml:id="echoid-s12" xml:space="preserve">Namque ex datis gravitatum centris <lb/>Hyperbolas, Ellipſes & </s> <s xml:id="echoid-s13" xml:space="preserve">Circulos ad quadrata <lb/>redigere non finis eſt horum Theorematum, ſed <lb/>conſequentia duntaxat; </s> <s xml:id="echoid-s14" xml:space="preserve">eoque nomine potiſſi-<lb/>mum placere debent, quòd aliquatenus certam <lb/>trium Portionum ad inſcripta triangula rationem <lb/>dcmonſtrant. </s> <s xml:id="echoid-s15" xml:space="preserve">Eam in Hyperbola primum mihi <lb/>deprehendere contigit, viâ deſtinatâ planè, ſed <lb/>impeditâ difficilique; </s> <s xml:id="echoid-s16" xml:space="preserve">quâ deinde breviorem ex-<lb/>quirens, in hanc incidi quæ ad Ellipſin & </s> <s xml:id="echoid-s17" xml:space="preserve">Cir-<lb/>culum quoque pertineret, commodumque ob-<lb/>venit conſtans illa in cognatis figuris mirabiliſ-<lb/>que convenientia. </s> <s xml:id="echoid-s18" xml:space="preserve">Quæ quidem univerſim in o-<lb/>mnibus hiſce quo minus locum habeat, ſola fa-<lb/>cit Propoſitionum noviſſima, ſupernumeraria illa <lb/>quaſi, atque ultra propoſitum adſcita, cujuſque <lb/>hoc ſolum mihi tribui par eſt, quod ex præce-<lb/>dentibus ipſam non ineleganti ratione compro-<lb/>bari poſſe oſtendi. </s> <s xml:id="echoid-s19" xml:space="preserve">Dudum enim hîc nos præ-<lb/>venit, egregiumque Theorema ante annos un-<lb/>deviginti demonſtratum dedit acutiſſimus Geo-<lb/>metra I. </s> <s xml:id="echoid-s20" xml:space="preserve">Della Faille, felix, meâ quidem ſenten-<lb/>tiâ, quod ante alios perſpexerit, quomodo à ſe-<lb/>ctorum gravitatis centro Quadratura depende-<lb/>ret: </s> <s xml:id="echoid-s21" xml:space="preserve">cumque illum in Circulo præcipuam laudem <lb/>promeruiſſe agnoſcam, non æquè gaviſus ſum de-<lb/>tectâ in reliquis hujus ſegmentis ſimili connexio- <pb file="0013" n="13" rhead="PRÆFATIO."/> ne, quàm cum eandem in Hyperboles portioni-<lb/>bus obſervaſſem, illudque inveniſſem de quo <lb/>tantus Vir non potuit non & </s> <s xml:id="echoid-s22" xml:space="preserve">ipſe cogitaſſe. </s> <s xml:id="echoid-s23" xml:space="preserve">Ra-<lb/>ros alioqui ſemper hæc figura, ſi cum Circulo <lb/>conferatur, ſui contemplatores nacta eſt; </s> <s xml:id="echoid-s24" xml:space="preserve">ejuſ-<lb/>que rei vel effectum vel indicium habemus, quod <lb/>cum varia ſint inſpecta quibus datis Quadratu-<lb/>ram quoque Circuli dari neceſſe ſit; </s> <s xml:id="echoid-s25" xml:space="preserve">cujuſmodi <lb/>ſunt exacta perimetri longitudo, Helicis Archi-<lb/>medeæ contingens, Quadratricis Dinoſtrati ter-<lb/>minus, vel tangens quoque ejuſdem Quadratri-<lb/>cis ad terminum alterum, (ſicut aliquando me <lb/>demonſtraſſe memini) & </s> <s xml:id="echoid-s26" xml:space="preserve">alia nonnulla quæ recen-<lb/>tioribus debentur; </s> <s xml:id="echoid-s27" xml:space="preserve">nihil interea à quoquam de-<lb/>finitum extet, quo vel ſub ulla conditione Hy-<lb/>perbole cum ſpatio rectis comprehenſo lineis <lb/>comparari poſſit. </s> <s xml:id="echoid-s28" xml:space="preserve">Noſtrâ ſanè ætate, pauciſque <lb/>abhinc annis Vir Clariſſ. </s> <s xml:id="echoid-s29" xml:space="preserve">D. </s> <s xml:id="echoid-s30" xml:space="preserve">Gregorius à S. </s> <s xml:id="echoid-s31" xml:space="preserve">Vin-<lb/>centio, de quo mihi deinceps dicendum reſtat, <lb/>exquiſitâ prorſus novâque methodo utramque <lb/>Quadraturam aggreſſus eſt, & </s> <s xml:id="echoid-s32" xml:space="preserve">credidit eâdem <lb/>ſe propemodum demonſtratione abſolviſſe. </s> <s xml:id="echoid-s33" xml:space="preserve">At <lb/>ego cum ampliſſima quæ de hiſce volumina emi-<lb/>ſit, perſcriptis jam Theorematis meis, diligentius <lb/>evolverem, (certus, ſi quod intenderat obtineret, <lb/>ſaltem gravitatis me centra exhibiturum,) in-<lb/>tellexi tandem, majori ſubtilitate quàm ſucceſ-<lb/>ſu rem arduam tentatam fuiſſe, ratione quoque <lb/>repertâ quâ id clariſſime oſtendi poſſe confido.</s> <s xml:id="echoid-s34" xml:space="preserve"> <pb file="0014" n="14" rhead="PRÆFATIO."/> Et quando inter tot eximios hoc ævo Geome-<lb/>tras nondum licuit animadvertere qui ſibi hanc <lb/>provinciam delegerit, ac proinde fieri poſſet <lb/>ut longa porro dubitatio maneret circa demon-<lb/>ſtrationes, quas certiſſimas eſſe oportet; </s> <s xml:id="echoid-s35" xml:space="preserve">arbi-<lb/>tratus ſum me facturum quod & </s> <s xml:id="echoid-s36" xml:space="preserve">in publicum <lb/>utile eſſet & </s> <s xml:id="echoid-s37" xml:space="preserve">à propoſiti argumenti ratione non <lb/>alienum, ſi ſimul hîc prodire ſinerem, quæ no-<lb/>vam in re obſcura lucem allatura videbantur. <lb/></s> <s xml:id="echoid-s38" xml:space="preserve">Nulla autem temeritate ad elevandam Viri gra-<lb/>vis & </s> <s xml:id="echoid-s39" xml:space="preserve">eruditi auctoritatem acceſſi, ſed cauſæ <lb/>bonitate adductus, putavi quæ compereram li-<lb/>berè citraque offenſam proponi poſſe. </s> <s xml:id="echoid-s40" xml:space="preserve">Majori <lb/>quoque fiduciâ, poſteaquam is ſeſe ipſum per <lb/>literas, quarum aliquod inter nos commercium <lb/>eſt, auctorem hortatoremque candidè præbuit, <lb/>ut ſiqua commentatus eſſem, ea cum univerſis <lb/>communicarem. </s> <s xml:id="echoid-s41" xml:space="preserve">Ingenuitatem hanc inſignem <lb/>lubenti gratoque uti meretur animo accepi, & </s> <s xml:id="echoid-s42" xml:space="preserve"><lb/>ſpero modeſtâ reprehenſione ſatis me decla-<lb/>raſſe, quanti æſtimem Doctiſſ. </s> <s xml:id="echoid-s43" xml:space="preserve">Viro haberi ami-<lb/>cus. </s> <s xml:id="echoid-s44" xml:space="preserve">Cujus invicem ſumma, quâ me uſque ad-<lb/>huc excepit, humanitas facit ne, quid aliud ex-<lb/>ſpectem, niſi ut vel moderatè & </s> <s xml:id="echoid-s45" xml:space="preserve">ſine ulla acer-<lb/>bitate ad mea reſpondeat, ſi quid iis reponen-<lb/>dum eſſe duxerit, vel rationibus evidentiſſimis <lb/>perſuaſus, æquè lubens noſtra quam alterius <lb/>poſthac opera veriora ſentiat & </s> <s xml:id="echoid-s46" xml:space="preserve">amplectatur.</s> <s xml:id="echoid-s47" xml:space="preserve"/> </p> <pb file="0015" n="15"/> <figure> <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0015-01"/> </figure> </div> <div xml:id="echoid-div9" type="section" level="1" n="9"> <head xml:id="echoid-head16" xml:space="preserve"><emph style="bf">CHRISTIANI HUGENII,</emph> <lb/><emph style="sc">Const</emph>. F.</head> <head xml:id="echoid-head17" xml:space="preserve">THEOREMATA <lb/>DE</head> <head xml:id="echoid-head18" xml:space="preserve"><emph style="bf">QUADRATURA</emph></head> <head xml:id="echoid-head19" xml:space="preserve">HYPERBOLES, ELLIPSIS, <lb/>ET CIRCULI, <lb/>EX DATO</head> <head xml:id="echoid-head20" xml:space="preserve">PORTIONUM GRAVITATIS CENTRO</head> <head xml:id="echoid-head21" xml:space="preserve"><emph style="sc">Theorema</emph> I.</head> <p style="it"> <s xml:id="echoid-s48" xml:space="preserve">POrtioni hyperboles, vel ellipſis vel <lb/>circuli portioni, dimidiâ ellipſi di-<lb/>midiove circulo non majori, circum-<lb/>ſcribi poteſt figura ex parallelo-<lb/>grammis æqualem latitudinem ha-<lb/>bentibus, quæ portionem excedat ſpatio quod minus <lb/>ſit quovis dato.</s> <s xml:id="echoid-s49" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s50" xml:space="preserve"><emph style="sc">DAta</emph> ſit portio A B C, cujus diameter B D. <lb/></s> <s xml:id="echoid-s51" xml:space="preserve"> <anchor type="note" xlink:label="note-0015-01a" xlink:href="note-0015-01"/> Super baſin A C conſtituatur parallelogrammum <lb/>A E, latera duo habens diametro B D parallela <lb/>& </s> <s xml:id="echoid-s52" xml:space="preserve">æqualia, quo fiet ut latus reliquum portionem <lb/>in vertice contingat. </s> <s xml:id="echoid-s53" xml:space="preserve">Hoc parallelogrammo con-<lb/>tinuè in duo æqualia ſecto, relinquetur tandem pars quæ <pb o="316" file="0016" n="16" rhead="THEOR. DE QUADRAT."/> minor erit dato ſpatio; </s> <s xml:id="echoid-s54" xml:space="preserve">ſit ea parallelogrammum B F, & </s> <s xml:id="echoid-s55" xml:space="preserve">di-<lb/>vidatur baſis A C in partes æquales ipſi D F, punctis <lb/>G, H, K &</s> <s xml:id="echoid-s56" xml:space="preserve">c. </s> <s xml:id="echoid-s57" xml:space="preserve">atque inde ducantur ad ſectionem rectæ <lb/>G L, H M, K N &</s> <s xml:id="echoid-s58" xml:space="preserve">c. </s> <s xml:id="echoid-s59" xml:space="preserve">diametro B D parallelæ, & </s> <s xml:id="echoid-s60" xml:space="preserve">perfi-<lb/>ciantur parallelogramma D O, G P, H Q, K R &</s> <s xml:id="echoid-s61" xml:space="preserve">c. </s> <s xml:id="echoid-s62" xml:space="preserve">Di-<lb/>co figuram ex omnibus iſtis parallelogrammis compoſitam <lb/>(quæ impoſterum ordinatè circumſcripta vocabitur) ſupera-<lb/>re portionem A B C minori quàm datum ſit ſpatio.</s> <s xml:id="echoid-s63" xml:space="preserve"/> </p> <div xml:id="echoid-div9" type="float" level="2" n="1"> <note position="right" xlink:label="note-0015-01" xlink:href="note-0015-01a" xml:space="preserve">TAB. XXXIV. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s64" xml:space="preserve">Jungantur enim A N, N M, M L, L B, B S, &</s> <s xml:id="echoid-s65" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s66" xml:space="preserve">eritque hac ratione inſcripta quoque portioni figura quædam <lb/>rectilinea; </s> <s xml:id="echoid-s67" xml:space="preserve">majorque erit exceſſus figuræ circumſcriptæ quæ <lb/>ex parallelogrammis compoſita eſt, ſuper inſcriptam, quàm <lb/>ſupra portionem A B C. </s> <s xml:id="echoid-s68" xml:space="preserve">Exceſſus autem circumſcriptæ ſuper <lb/>inſcriptam ex triangulis conſtat, quorum quæ ſunt ab una <lb/>diametri parte, ut A R N, N Q M, M P L, L O B, <lb/>æquantur dimidio parallelogrammi O D vel B F, quia ſin-<lb/>gulorum baſes baſi D F æquales ſunt, & </s> <s xml:id="echoid-s69" xml:space="preserve">omnium ſimul al-<lb/>titudo, parallelogrammi B F altitudini. </s> <s xml:id="echoid-s70" xml:space="preserve">Eâdem ratione trian-<lb/>gula qu& </s> <s xml:id="echoid-s71" xml:space="preserve">ſunt ab altera diametri parte, æquantur dimidio <lb/>parallelogrammi B F: </s> <s xml:id="echoid-s72" xml:space="preserve">Ergo omnia ſimul triangula ſive di-<lb/>ctus exceſſus æqualis eſt parallelogrammo B F, eóque mi-<lb/>nor ſpatio dato. </s> <s xml:id="echoid-s73" xml:space="preserve">Sed eodem exceſſu adhuc minor erat ex-<lb/>ceſſus figuræ circumſcriptæ ſupra portionem A B C: </s> <s xml:id="echoid-s74" xml:space="preserve">igitur <lb/>hic exceſſus dato ſpatio multo minor eſt. </s> <s xml:id="echoid-s75" xml:space="preserve">Et apparet fieri <lb/>poſſe quod proponebatur.</s> <s xml:id="echoid-s76" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div11" type="section" level="1" n="10"> <head xml:id="echoid-head22" xml:space="preserve"><emph style="sc">Theorema</emph> II.</head> <p style="it"> <s xml:id="echoid-s77" xml:space="preserve">DAtâ portione hyperboles, vel ellipſis vel circuli <lb/>portione, dimidiâ ellipſi dimidiove circulo non <lb/>majore, & </s> <s xml:id="echoid-s78" xml:space="preserve">dato triangulo qui baſin habeat baſi por-<lb/>tionis æqualem; </s> <s xml:id="echoid-s79" xml:space="preserve">poteſt utrique figura circumſcribi ex <lb/>parallelogrammis quorum ſit omnium eadem latitu-<lb/>do, ita ut uterque ſimulexceſſus quo figuræ circum-<lb/>ſcriptæ portionem & </s> <s xml:id="echoid-s80" xml:space="preserve">triangulum ſuperant, ſit minor <lb/>ſpatio quovis dato.</s> <s xml:id="echoid-s81" xml:space="preserve"/> </p> <pb o="317" file="0017" n="17" rhead="HYPERB. ELLIPS. ET CIRC."/> <p> <s xml:id="echoid-s82" xml:space="preserve">Data ſit portio A B C & </s> <s xml:id="echoid-s83" xml:space="preserve">triangulus D E F, baſibus A C, <lb/> <anchor type="note" xlink:label="note-0017-01a" xlink:href="note-0017-01"/> D F æqualibus; </s> <s xml:id="echoid-s84" xml:space="preserve">& </s> <s xml:id="echoid-s85" xml:space="preserve">portionis diameter ſit B G, in trian-<lb/>gulo verò ducta à vertice in mediam baſin linea E H. </s> <s xml:id="echoid-s86" xml:space="preserve">Sint <lb/>autem utræque B G, E H vel ad baſes rectæ vel æqualiter <lb/>inclinatæ; </s> <s xml:id="echoid-s87" xml:space="preserve">& </s> <s xml:id="echoid-s88" xml:space="preserve">quam rationem habet B G ad E H, in eandem <lb/>dividatur ſpatium datum, ſintque partes K & </s> <s xml:id="echoid-s89" xml:space="preserve">L. </s> <s xml:id="echoid-s90" xml:space="preserve">Circumſcri-<lb/>batur jam ſicut antea portioni A B C figura ordinatè, quæ <lb/>portionem ſuperet exceſſu minore quàm ſit ſpatium K. </s> <s xml:id="echoid-s91" xml:space="preserve">Et <lb/>triangulo D E F circumſcribatur figura quæ totidem paral-<lb/>lelogrammis conſtet, quot ſunt in figura portioni A B C cir-<lb/>cumſcripta.</s> <s xml:id="echoid-s92" xml:space="preserve"/> </p> <div xml:id="echoid-div11" type="float" level="2" n="1"> <note position="right" xlink:label="note-0017-01" xlink:href="note-0017-01a" xml:space="preserve">TAB. XXXIV <lb/>Fig, 2.</note> </div> <p> <s xml:id="echoid-s93" xml:space="preserve">Quoniam igitur baſes portionis & </s> <s xml:id="echoid-s94" xml:space="preserve">trianguli æquales ſunt, <lb/>apparet quidem omnium parallelogrammorum eandem fore <lb/>latitudinem. </s> <s xml:id="echoid-s95" xml:space="preserve">Hinc quum parallelogrammum B M ſit ad E R <lb/>ut B G ad E H, id eſt ut K ad L, ſitque B M minus quam <lb/>K <anchor type="note" xlink:href="" symbol="1"/>, erit quoque E R minus quam L <anchor type="note" xlink:href="" symbol="2"/>. </s> <s xml:id="echoid-s96" xml:space="preserve">Verùm omnia trian- <anchor type="note" xlink:label="note-0017-02a" xlink:href="note-0017-02"/> <anchor type="note" xlink:label="note-0017-03a" xlink:href="note-0017-03"/> gula quibus conſtat exceſſus figuræ circumſcriptæ ſupra trian-<lb/>gulum D E F, æqualia ſunt parallelogrammo E R, ergo <lb/>minor eſt idem exceſſus ſpatio L. </s> <s xml:id="echoid-s97" xml:space="preserve">Sed & </s> <s xml:id="echoid-s98" xml:space="preserve">exceſſus quo figu-<lb/>ra circumſcripta portionem A B C ſuperat, minor eſt ſpa-<lb/>tio K. </s> <s xml:id="echoid-s99" xml:space="preserve">Ergo utergue ſimul exceſſus minor erit ſpatio K L <lb/>dato. </s> <s xml:id="echoid-s100" xml:space="preserve">Et conſtat fieri poſſe quod proponebatur.</s> <s xml:id="echoid-s101" xml:space="preserve"/> </p> <div xml:id="echoid-div12" type="float" level="2" n="2"> <note symbol="1" position="right" xlink:label="note-0017-02" xlink:href="note-0017-02a" xml:space="preserve">Ex conſit<unsure/>.</note> <note symbol="2" position="right" xlink:label="note-0017-03" xlink:href="note-0017-03a" xml:space="preserve">14. 5. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div14" type="section" level="1" n="11"> <head xml:id="echoid-head23" xml:space="preserve"><emph style="sc">Theorema</emph> III.</head> <p style="it"> <s xml:id="echoid-s102" xml:space="preserve">SI portioni hyperboles, vel ellipſis vel circuli por-<lb/>tioni, dimidiâ ellipſi dimidiove circulo non majori, <lb/>circumſcribatur figur a or dinatè; </s> <s xml:id="echoid-s103" xml:space="preserve">ejus figuræ centrum <lb/>gravitatis erit in portionis diametro.</s> <s xml:id="echoid-s104" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s105" xml:space="preserve">Sit portio quælibet iſtarum A B C, diameter ejus B D; <lb/></s> <s xml:id="echoid-s106" xml:space="preserve"> <anchor type="note" xlink:label="note-0017-04a" xlink:href="note-0017-04"/> & </s> <s xml:id="echoid-s107" xml:space="preserve">circumſcribatur ei ut ſupra figura ordinatè. </s> <s xml:id="echoid-s108" xml:space="preserve">Oſtenden-<lb/>dum eſt ejus figuræ centrum gravitatis fore in B D diametro. <lb/></s> <s xml:id="echoid-s109" xml:space="preserve">Ducantur lineæ H K, N R, P S, conjungentes ſuprema <lb/>latera parallelogrammorum quæ à diametro portionis æqua-<lb/>liter utrinque diſtant.</s> <s xml:id="echoid-s110" xml:space="preserve"/> </p> <div xml:id="echoid-div14" type="float" level="2" n="1"> <note position="right" xlink:label="note-0017-04" xlink:href="note-0017-04a" xml:space="preserve">TAB. XXXVI. <lb/>Fig. 3.</note> </div> <pb o="318" file="0018" n="18" rhead="THEOR. DE QUADRAT."/> <p> <s xml:id="echoid-s111" xml:space="preserve">Quoniam igitur F H, L K ſunt diametro B D parallelæ, <lb/>ſuntque D F, D L æquales, oportet lineam H K, quæ duas <lb/>F H, L K conjungit, à diametro B D bifariam ſecari; </s> <s xml:id="echoid-s112" xml:space="preserve">qua-<lb/>re eadem H K parallela erit baſi A C <anchor type="note" xlink:href="" symbol="1"/>, & </s> <s xml:id="echoid-s113" xml:space="preserve">E H K G recta <anchor type="note" xlink:label="note-0018-01a" xlink:href="note-0018-01"/> linea. </s> <s xml:id="echoid-s114" xml:space="preserve">Itaque E C parallelogrammum eſt; </s> <s xml:id="echoid-s115" xml:space="preserve">cujus oppoſita la-<lb/>tera quum bifariam dividat diameter B D, erit in ea paral-<lb/>lelogrammi centrum gravitatis <anchor type="note" xlink:href="" symbol="2"/>. </s> <s xml:id="echoid-s116" xml:space="preserve">Eâdem ratione parallelo- <anchor type="note" xlink:label="note-0018-02a" xlink:href="note-0018-02"/> gramma erunt H M, N O, P Q, & </s> <s xml:id="echoid-s117" xml:space="preserve">ſingulorum centra gra-<lb/>vitatis in linea B D. </s> <s xml:id="echoid-s118" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s119" xml:space="preserve">figuræ ex omnibus dictis pa-<lb/>rallelogrammis compoſitæ centrum gravitatis in eadem B D <lb/>reperiri neceſſe eſt. </s> <s xml:id="echoid-s120" xml:space="preserve">Iſta autem figura eadem eſt quæ portio-<lb/>ni ordinatè fuerat circumſcripta. </s> <s xml:id="echoid-s121" xml:space="preserve">Ergo figuræ portioni ordi-<lb/>natè circumſcriptæ centrum gravitatis conſtat eſſe in B D por-<lb/>tionis diametro. </s> <s xml:id="echoid-s122" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s123" xml:space="preserve"/> </p> <div xml:id="echoid-div15" type="float" level="2" n="2"> <note symbol="1" position="left" xlink:label="note-0018-01" xlink:href="note-0018-01a" xml:space="preserve">5. lib. 2. <lb/>con.</note> <note symbol="2" position="left" xlink:label="note-0018-02" xlink:href="note-0018-02a" xml:space="preserve">9. lib. 1. <lb/>Arch. de <lb/>Æquipond.</note> </div> </div> <div xml:id="echoid-div17" type="section" level="1" n="12"> <head xml:id="echoid-head24" xml:space="preserve"><emph style="sc">Theorema</emph> IV.</head> <p style="it"> <s xml:id="echoid-s124" xml:space="preserve">POrtionis hyperboles, ellipſis & </s> <s xml:id="echoid-s125" xml:space="preserve">circuli, centrum <lb/>gravitatis eſt in portionis diametro.</s> <s xml:id="echoid-s126" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s127" xml:space="preserve">Eſto portio hyperboles, vel ellipſis vel circuli dimidiâ pri-<lb/> <anchor type="note" xlink:label="note-0018-03a" xlink:href="note-0018-03"/> mum figurâ non major, A B C; </s> <s xml:id="echoid-s128" xml:space="preserve">diameter ejus B D. </s> <s xml:id="echoid-s129" xml:space="preserve">O-<lb/>ſtendendum eſt, in B D reperiri portionis A B C gravitatis <lb/>centrum.</s> <s xml:id="echoid-s130" xml:space="preserve"/> </p> <div xml:id="echoid-div17" type="float" level="2" n="1"> <note position="left" xlink:label="note-0018-03" xlink:href="note-0018-03a" xml:space="preserve">TAB. XXXIV. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s131" xml:space="preserve">Si enim fieri poteſt, ſit extra diametrum in E, & </s> <s xml:id="echoid-s132" xml:space="preserve">ducatur <lb/>E H diametro B D parallela. </s> <s xml:id="echoid-s133" xml:space="preserve">Dividendo itaque D C conti-<lb/>nuè bifariam, relinquetur tandem linea minor quam D H; <lb/></s> <s xml:id="echoid-s134" xml:space="preserve">ſit ea D F, & </s> <s xml:id="echoid-s135" xml:space="preserve">circumſcribatur portioni figura ordinatè ex <lb/>parallelogrammis quorum baſes æquales ſint lineæ D F, & </s> <s xml:id="echoid-s136" xml:space="preserve"><lb/>jungantur B A, B C. </s> <s xml:id="echoid-s137" xml:space="preserve">Figuræ itaque portioni circumſcri-<lb/>ptæ centrum gravitatis eſt in B D portionis diametro. </s> <s xml:id="echoid-s138" xml:space="preserve">Sit hoc <lb/>K, & </s> <s xml:id="echoid-s139" xml:space="preserve">jungatur E K, producaturque, & </s> <s xml:id="echoid-s140" xml:space="preserve">occurrat ei A L <lb/>parallela B D. </s> <s xml:id="echoid-s141" xml:space="preserve">Quia autem portio major eſt triangulo A B C, <lb/>& </s> <s xml:id="echoid-s142" xml:space="preserve">exceſſus quo figura circumſcripta portionem ſuperat, mi-<lb/>nor parallelogrammo B F, uti ſupra demonſtratum fuit <anchor type="note" xlink:href="" symbol="*"/>;</s> <s xml:id="echoid-s143" xml:space="preserve"> <anchor type="note" xlink:label="note-0018-04a" xlink:href="note-0018-04"/> erit major ratio portionis A B C ad dictum exceſſum, quàm <lb/>trianguli A B C ad B F parallelogrammum, id eſt quàm <pb o="319" file="0019" n="19" rhead="HYPERB. ELLIPS. ET CIRC."/> A D ad D F; </s> <s xml:id="echoid-s144" xml:space="preserve">multoque major quam A D ad D H, vel <lb/>quàm L K ad K E. </s> <s xml:id="echoid-s145" xml:space="preserve">Sit itaque M K ad K E ſicut portio <lb/>A B C ad exceſſum quo ipſa ſuperatur à figura ordinatè cir-<lb/>cumſcripta. </s> <s xml:id="echoid-s146" xml:space="preserve">Itaque cum K ſit centrum grav. </s> <s xml:id="echoid-s147" xml:space="preserve">figuræ portio-<lb/>ni circumſcriptæ, & </s> <s xml:id="echoid-s148" xml:space="preserve">E centrum grav. </s> <s xml:id="echoid-s149" xml:space="preserve">ipſius portionis; </s> <s xml:id="echoid-s150" xml:space="preserve">erit <lb/>M centrum gravitatis omnium ſpatiorum quæ eundem exceſ-<lb/>ſum conſtituunt <anchor type="note" xlink:href="" symbol="1"/>. </s> <s xml:id="echoid-s151" xml:space="preserve">Quod eſſe non poteſt; </s> <s xml:id="echoid-s152" xml:space="preserve">Nam ſi per M <anchor type="note" xlink:label="note-0019-01a" xlink:href="note-0019-01"/> linea ducatur diametro B D parallela, erunt ab una parte <lb/>omnia quæ diximus ſpatia. </s> <s xml:id="echoid-s153" xml:space="preserve">Manifeſtum eſt igitur, portio-<lb/>nis A B C centrum grav. </s> <s xml:id="echoid-s154" xml:space="preserve">eſſe in B D portionis diametro.</s> <s xml:id="echoid-s155" xml:space="preserve"/> </p> <div xml:id="echoid-div18" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0018-04" xlink:href="note-0018-04a" xml:space="preserve">Theor. 1.</note> <note symbol="1" position="right" xlink:label="note-0019-01" xlink:href="note-0019-01a" xml:space="preserve">8. lib. 1. <lb/>Arch. de <lb/>Æquipond.</note> </div> <p> <s xml:id="echoid-s156" xml:space="preserve">Eſto nunc A B C portio ellipſis vel circuli, dimidiâ fi-<lb/> <anchor type="note" xlink:label="note-0019-02a" xlink:href="note-0019-02"/> gurá major. </s> <s xml:id="echoid-s157" xml:space="preserve">Abſolvatur figura, & </s> <s xml:id="echoid-s158" xml:space="preserve">producatur B D uſque <lb/>dum ſectioni occurrat in E; </s> <s xml:id="echoid-s159" xml:space="preserve">erit igitur portionis A E C dia-<lb/>meter E D, & </s> <s xml:id="echoid-s160" xml:space="preserve">B D E diameter totius figuræ. </s> <s xml:id="echoid-s161" xml:space="preserve">Et quoniam <lb/>in B D E diametro eſt figuræ totius centrum gravitatis, (hoc <lb/>enim ex prædemonſtratis conſtabit, ſi in duo æqualia tota <lb/>figura dividatur diametro quæ ipſi A C ſit parallela,) & </s> <s xml:id="echoid-s162" xml:space="preserve">in <lb/>eadem centr. </s> <s xml:id="echoid-s163" xml:space="preserve">gravitatis A E C portionis minoris, ſicut mo-<lb/>dò oſtenſum fuit; </s> <s xml:id="echoid-s164" xml:space="preserve">erit quoque centr. </s> <s xml:id="echoid-s165" xml:space="preserve">gravitatis portionis re-<lb/>liquæ A B C in B D E <anchor type="note" xlink:href="" symbol="2"/>; </s> <s xml:id="echoid-s166" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s167" xml:space="preserve"/> </p> <div xml:id="echoid-div19" type="float" level="2" n="3"> <note position="right" xlink:label="note-0019-02" xlink:href="note-0019-02a" xml:space="preserve">TAB. XXXIV. <lb/>Fig. 5.</note> </div> <note symbol="2" position="right" xml:space="preserve">8. lib. 1. <lb/>Archim. dc <lb/>Æqu@pond.</note> </div> <div xml:id="echoid-div21" type="section" level="1" n="13"> <head xml:id="echoid-head25" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/><emph style="sc">Lemma</emph>.</head> <p> <s xml:id="echoid-s168" xml:space="preserve">Eſto linea E B, cui ad utrumque terminum adjiciantur æ-<lb/> <anchor type="note" xlink:label="note-0019-04a" xlink:href="note-0019-04"/> quales duæ E S, B P, & </s> <s xml:id="echoid-s169" xml:space="preserve">inſuper alia P D. </s> <s xml:id="echoid-s170" xml:space="preserve">Dico id quo <lb/>rectangulum E D B excedit E P B, æquari rectangulo S D P. <lb/></s> <s xml:id="echoid-s171" xml:space="preserve">Eſt enim rectangulum E D B æquale iſtis duobus, rectangulo <lb/>E D P & </s> <s xml:id="echoid-s172" xml:space="preserve">rectangulo ſub E D, P B: </s> <s xml:id="echoid-s173" xml:space="preserve">quorum ultimum ſuperat <lb/>rectangulum E P B rectangulo D P B. </s> <s xml:id="echoid-s174" xml:space="preserve">Igitur exceſſus rectan-<lb/>guli E D B ſupra rectangulum E P B æqualis eſt duobus iſtis, <lb/>rectangulo E D P, & </s> <s xml:id="echoid-s175" xml:space="preserve">D P B. </s> <s xml:id="echoid-s176" xml:space="preserve">Sed rectangulum E D P addito <lb/>rectangulo D P B, id eſt rectangulo ſub E S, D P, æquale <lb/>fit rectangulo S D P. </s> <s xml:id="echoid-s177" xml:space="preserve">Manifeſtum eſt igitur, exceſſum re-<lb/>ctanguli E D B ſupra E P B, æquari rectangulo S D P.</s> <s xml:id="echoid-s178" xml:space="preserve"/> </p> <div xml:id="echoid-div21" type="float" level="2" n="1"> <note position="right" xlink:label="note-0019-04" xlink:href="note-0019-04a" xml:space="preserve">TAB. XXXIV. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s179" xml:space="preserve"><anchor type="note" xlink:href="" symbol="†"/>Eſto rurſus linea E B, cui ad utrumque terminum aufe- <anchor type="note" xlink:label="note-0019-05a" xlink:href="note-0019-05"/> <anchor type="note" xlink:label="note-0019-06a" xlink:href="note-0019-06"/> <anchor type="note" xlink:label="note-0019-07a" xlink:href="note-0019-07"/> <pb o="320" file="0020" n="20" rhead="THEOR. DE QUADRAT."/> rantur duæ æquales E S, B P, & </s> <s xml:id="echoid-s180" xml:space="preserve">inſuper alia P D. </s> <s xml:id="echoid-s181" xml:space="preserve">Dico <lb/>iterum, id quo rectangulum E D B excedit E P B, æquari <lb/>rectangulo S D P. </s> <s xml:id="echoid-s182" xml:space="preserve">Rectangulum enim E D B æquale eſt iſtis <lb/>duobus, rectangulo E D P, & </s> <s xml:id="echoid-s183" xml:space="preserve">rectangulo ſub E D, P B; <lb/></s> <s xml:id="echoid-s184" xml:space="preserve">horum autem E D P rurſus æquale eſt duobus, rectangulo ni-<lb/>mirum S D P, & </s> <s xml:id="echoid-s185" xml:space="preserve">ei quod continetur ſub E S, D P, ſive <lb/>rectangulo D P B. </s> <s xml:id="echoid-s186" xml:space="preserve">Igitur rectangulum E D B iſtis tribus æ-<lb/>quale eſt rectangulis, S D P, D P B, & </s> <s xml:id="echoid-s187" xml:space="preserve">rectangulo ſub <lb/>E D, P B; </s> <s xml:id="echoid-s188" xml:space="preserve">horum vero duo poſtrema æquantur rectangu-<lb/>lo E P B; </s> <s xml:id="echoid-s189" xml:space="preserve">ergo rectangulum E D B æquale eſt duobus, re-<lb/>ctangulo nimirum S D P & </s> <s xml:id="echoid-s190" xml:space="preserve">E P B, unde apparet exceſ-<lb/>ſum rectanguli E D B ſupra rectangulum E P B æquari re-<lb/>ctangulo S D P.</s> <s xml:id="echoid-s191" xml:space="preserve"/> </p> <div xml:id="echoid-div22" type="float" level="2" n="2"> <note position="right" xlink:label="note-0019-05" xlink:href="note-0019-05a" xml:space="preserve">TAB. XXXIV. <lb/>Fig. 7.</note> <note symbol="*" position="foot" xlink:label="note-0019-06" xlink:href="note-0019-06a" xml:space="preserve">Idem hoc aliter demonſtratum reperi apud Pappum, lib. 7. Prop. 24.</note> <note symbol="†" position="foot" xlink:label="note-0019-07" xlink:href="note-0019-07a" xml:space="preserve">Vide eundem, lib. 7. Prop. 57.</note> </div> </div> <div xml:id="echoid-div24" type="section" level="1" n="14"> <head xml:id="echoid-head26" xml:space="preserve"><emph style="sc">Theorema</emph> V.</head> <p style="it"> <s xml:id="echoid-s192" xml:space="preserve">DAtâ portione hyperboles, vel ellipſis vel cir-<lb/>culi portione, dimidiâ figurâ non majore; </s> <s xml:id="echoid-s193" xml:space="preserve">ſi ad <lb/>diametrum conſtituatur triangulus hujuſmodi, qui <lb/>verticem habeat in centro figuræ, & </s> <s xml:id="echoid-s194" xml:space="preserve">baſin portio-<lb/>nis baſi æqualem & </s> <s xml:id="echoid-s195" xml:space="preserve">parallelam; </s> <s xml:id="echoid-s196" xml:space="preserve">eam verò quæ de-<lb/>inceps à vertice ad mediam baſin pertingit tantam, <lb/>ut poſſit ipſa rectangulum comprehenſum lineis, quæ <lb/>inter portionis baſin & </s> <s xml:id="echoid-s197" xml:space="preserve">terminos diametri figuræ in-<lb/>terjiciuntur. </s> <s xml:id="echoid-s198" xml:space="preserve">Erit magnitudinis, quæ ex portione & </s> <s xml:id="echoid-s199" xml:space="preserve"><lb/>præſcripto triangulo componitur, centrum gravita-<lb/>tis punctum idem quod eſt trianguli vertex, cen-<lb/>trum nimirum figuræ.</s> <s xml:id="echoid-s200" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s201" xml:space="preserve">Data ſit portio hyberboles, vel ellipſis vel circuli portio <lb/> <anchor type="note" xlink:label="note-0020-01a" xlink:href="note-0020-01"/> dimidiâ figurâ non major, A B C. </s> <s xml:id="echoid-s202" xml:space="preserve">Diameter ejus ſit B D, <lb/>& </s> <s xml:id="echoid-s203" xml:space="preserve">figuræ diameter B E, in cujus medio centrum figuræ F. <lb/></s> <s xml:id="echoid-s204" xml:space="preserve">Et ſumatur F G quæ poſſit rectangulum B D E, ductâque <lb/>K G H æquali & </s> <s xml:id="echoid-s205" xml:space="preserve">parallelâ baſi A C, quæque ad G biſa- <pb o="321" file="0021" n="21" rhead="HYPERB. ELLIPS. ET CIRC."/> riam dividatur, jungantur K F, F H. </s> <s xml:id="echoid-s206" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> Demonſtrandum eſt, quòd magnitudinis compoſitæ ex portione A B C & </s> <s xml:id="echoid-s207" xml:space="preserve">trian-<lb/>gulo K F H, centrum gravitatis eſt punctum F.</s> <s xml:id="echoid-s208" xml:space="preserve"/> </p> <div xml:id="echoid-div24" type="float" level="2" n="1"> <note position="left" xlink:label="note-0020-01" xlink:href="note-0020-01a" xml:space="preserve">TAB. XXXV. <lb/>Fig. 1. 2. 3.</note> </div> <p> <s xml:id="echoid-s209" xml:space="preserve">Si non eſt in F, ſit ſi fieri poteſt primùm ab ea parte pun-<lb/>cti F quæ eſt verſus A B C portionem, atque eſto pun-<lb/>ctum L; </s> <s xml:id="echoid-s210" xml:space="preserve">conſtat autem futurum in recta B D G, quum in <lb/>hac ſint utraque centra gravitatis portionis & </s> <s xml:id="echoid-s211" xml:space="preserve">trianguli K F H. <lb/></s> <s xml:id="echoid-s212" xml:space="preserve">Jungantur A B, B C, & </s> <s xml:id="echoid-s213" xml:space="preserve">quam rationem habet G F ad F L, <lb/>eam habeat magnitudo compoſita ex triangulis A B C, K F H <lb/>ad ſpatium quoddam M; </s> <s xml:id="echoid-s214" xml:space="preserve">& </s> <s xml:id="echoid-s215" xml:space="preserve">circumſcribantur portioni & </s> <s xml:id="echoid-s216" xml:space="preserve">tri-<lb/>angulo K F H figuræ ordinatè, ex parallelogrammis quo-<lb/>rum omnium ſit eadem latitudo, ita ut duo ſimul exceſſus <lb/>quibus iſtæ figuræ ſuperant portionem A B C & </s> <s xml:id="echoid-s217" xml:space="preserve">triangu-<lb/>lum K F H, minores ſint ſpatio M <anchor type="note" xlink:href="" symbol="1"/>. </s> <s xml:id="echoid-s218" xml:space="preserve">Igitur duorum ſimul <anchor type="note" xlink:label="note-0021-01a" xlink:href="note-0021-01"/> triangulorum A B C, K F H ad dictos duos exceſſus ſive <lb/>reſidua major erit ratio quàm ad M, id eſt quàm G F ad <lb/>F L; </s> <s xml:id="echoid-s219" xml:space="preserve">ac proinde longè major ratio portionis A B C unà cum <lb/>K F H triangulo ad eadem reſidua quam G F ad F L. </s> <s xml:id="echoid-s220" xml:space="preserve">Sit <lb/>itaque N F ad F L, ſicut portio A B C ſimul cum trian-<lb/>gulo K F H ad duo reſidua, & </s> <s xml:id="echoid-s221" xml:space="preserve">cadet terminus N ultra tri-<lb/>anguli baſin K H. </s> <s xml:id="echoid-s222" xml:space="preserve">Jam per F ducatur O Ξ parallela baſi <lb/>A C vel K H; </s> <s xml:id="echoid-s223" xml:space="preserve">& </s> <s xml:id="echoid-s224" xml:space="preserve">duorum quorumcunque parallelogram-<lb/>morum, quæ in portione & </s> <s xml:id="echoid-s225" xml:space="preserve">in triangulo K F H æqualiter <lb/>à diametro diſtabunt, ut ſunt R Q, Σ T, ſint centra gra-<lb/>vitatis V & </s> <s xml:id="echoid-s226" xml:space="preserve">X; </s> <s xml:id="echoid-s227" xml:space="preserve">per quæ ducatur recta Z Λ Δ Ω, ſecans li-<lb/>neam Ο Ξ in Y; </s> <s xml:id="echoid-s228" xml:space="preserve">& </s> <s xml:id="echoid-s229" xml:space="preserve">ducatur R P baſi A C parallela, abſciſ-<lb/>ſæque ad verticem lineæ P B ſumatur æqualis, ex altero <lb/>diametri figuræ termino, E S.</s> <s xml:id="echoid-s230" xml:space="preserve"/> </p> <div xml:id="echoid-div25" type="float" level="2" n="2"> <note symbol="1" position="right" xlink:label="note-0021-01" xlink:href="note-0021-01a" xml:space="preserve">Theor. 2. h.</note> </div> <p> <s xml:id="echoid-s231" xml:space="preserve">Quoniam igitur ad diametrum figuræ ordinatim ſunt ap-<lb/>plicatæ C D & </s> <s xml:id="echoid-s232" xml:space="preserve">R P, erit ut rectangulum B D E ad rectan-<lb/>gulum B P E, ita quadratum C D ad R P quadratum <anchor type="note" xlink:href="" symbol="2"/>;</s> <s xml:id="echoid-s233" xml:space="preserve"> <anchor type="note" xlink:label="note-0021-02a" xlink:href="note-0021-02"/> verùm ut C D ad R P, hoc eſt, ut H G ad Ψ G, ita eſt H F <lb/>ad Σ F, & </s> <s xml:id="echoid-s234" xml:space="preserve">ita Z Y ad Λ Y igitur ut C D quadratum ad <lb/>quadratum R P, id eſt ut rectangulum B D E ad B P E, <lb/> <anchor type="note" xlink:label="note-0021-03a" xlink:href="note-0021-03"/> <pb o="322" file="0022" n="22" rhead="THEOR. DE QUADRAT."/> ita eſt quadratum Z Y ad Λ Y quadratum. </s> <s xml:id="echoid-s235" xml:space="preserve">Quare & </s> <s xml:id="echoid-s236" xml:space="preserve">per con-<lb/>verſionem rationis, ſicut rectangulum B D E ad differenti-<lb/>am rectangulorum B D E, B P E, ita quadratum Z Y ad <lb/>differentiam quadratorum Z Y, Λ Y. </s> <s xml:id="echoid-s237" xml:space="preserve">Eſt autem differentia <lb/>rectangulorum B D E, B P E, æqualis rectangulo S D P, <lb/>ſicut lemmate præmiſſo demonſtratum eſt; </s> <s xml:id="echoid-s238" xml:space="preserve">differentia verò <lb/>quadratorum Z Y, Λ Y, æqualis quadrato Z Λ & </s> <s xml:id="echoid-s239" xml:space="preserve">duobus <lb/>rectangulis Z Λ Y <anchor type="note" xlink:href="" symbol="3"/>, ſive quod idem eſt, rectangulis Z Λ X, <anchor type="note" xlink:label="note-0022-01a" xlink:href="note-0022-01"/> Z Λ Y bis ſumptis, hoc eſt, duplo rectangulo ſub Z Λ, <lb/>X Y. </s> <s xml:id="echoid-s240" xml:space="preserve">Itaque ſicut eſt rectangulum B D E ad rectangulum <lb/>S D P, ita quadratum Z Y ad duplum rectangulum ſub <lb/>X Y, Z Λ. </s> <s xml:id="echoid-s241" xml:space="preserve">quare cum rectangulum B D E quadrato F G <lb/>æquale ſit <anchor type="note" xlink:href="" symbol="4"/>, ideoque & </s> <s xml:id="echoid-s242" xml:space="preserve">quadrato Z Y, erit quoque rectan- <anchor type="note" xlink:label="note-0022-02a" xlink:href="note-0022-02"/> gulum S D P æquale duplo rectangulo ſub X Y, Z Λ <anchor type="note" xlink:href="" symbol="5"/>.</s> <s xml:id="echoid-s243" xml:space="preserve"> <anchor type="note" xlink:label="note-0022-03a" xlink:href="note-0022-03"/> Quia verò F punctum dividit B E per medium, ſuntque <lb/>æquales B P, E S, etiam F P, F S æquales erunt, unde <lb/>additi utrique F D, erit S D æqualis toti P F D id eſt <lb/>Δ Y Ω: </s> <s xml:id="echoid-s244" xml:space="preserve">ſed Δ Y Ω dupla eſt lineæ V Y, quia bis continet <lb/>utramque Y Δ, Δ V in hyperbole, in ellipſi verò & </s> <s xml:id="echoid-s245" xml:space="preserve">circulo <lb/>bis utramque V Ω & </s> <s xml:id="echoid-s246" xml:space="preserve">Ω Y; </s> <s xml:id="echoid-s247" xml:space="preserve">ergo & </s> <s xml:id="echoid-s248" xml:space="preserve">S D dupla V Y, ideo-<lb/>que rectangulum S D P æquale duplo rectangulo ſub Y V, <lb/>Ω Δ. </s> <s xml:id="echoid-s249" xml:space="preserve">Sed idem rectangulum S D P æquale oſtenſum fuit <lb/>duplo rectangulo ſub X Y, Z Λ; </s> <s xml:id="echoid-s250" xml:space="preserve">ergo æquale eſt rectangu-<lb/>lum ſub Y V, Ω Δ, rectangulo ſub X Y, Z Λ. </s> <s xml:id="echoid-s251" xml:space="preserve">Eſt itaque <lb/>Y V ad Y X, ut Λ Z ad Ω Δ <anchor type="note" xlink:href="" symbol="6"/>; </s> <s xml:id="echoid-s252" xml:space="preserve">verùm ut Λ Z ad Ω Δ, ita <anchor type="note" xlink:label="note-0022-04a" xlink:href="note-0022-04"/> eſt parallelogrammum Σ T ad R Q; </s> <s xml:id="echoid-s253" xml:space="preserve">itaque & </s> <s xml:id="echoid-s254" xml:space="preserve">Y V eſt ad <lb/>Y Χ ut parallelogrammum Σ T ad R Q parallelogr. </s> <s xml:id="echoid-s255" xml:space="preserve">Sunt <lb/>autem puncta X & </s> <s xml:id="echoid-s256" xml:space="preserve">V centra gravitatis dictorum parallelo-<lb/>grammorum; </s> <s xml:id="echoid-s257" xml:space="preserve">ergo magnitudinis ex utroque parallelogram-<lb/>mo compoſitæ centrum gravitatis eſt punctum Y <anchor type="note" xlink:href="" symbol="7"/>. </s> <s xml:id="echoid-s258" xml:space="preserve">Eâdem <anchor type="note" xlink:label="note-0022-05a" xlink:href="note-0022-05"/> ratione oſtendi poteſt de reliquis omnibus parallelogrammis, <lb/>quod duorum quorumlibet oppoſitorum centrum gravitatis <lb/>eſt in linea O Ξ. </s> <s xml:id="echoid-s259" xml:space="preserve">Ergo totius magnitudinis quæ ex duabus <lb/>ſiguris utrimque ordinatè circumſoriptis componitur, centr. <lb/></s> <s xml:id="echoid-s260" xml:space="preserve">gravitatis in eadem O Ξ reper@ri neceſſe eſt. </s> <s xml:id="echoid-s261" xml:space="preserve">Sed ejuſdem com-<lb/>poſitæ magnitudinis centrum gravit. </s> <s xml:id="echoid-s262" xml:space="preserve">eſt quoque in recta <pb o="323" file="0023" n="23" rhead="HYPERB. ELLIPS. ET CIRC."/> B D G, quoniam in ea ſunt centra gravitatis utriusque fi-<lb/>guræ circumſcriptæ <anchor type="note" xlink:href="" symbol="8"/>; </s> <s xml:id="echoid-s263" xml:space="preserve">igitur magnitudinis ex dictis figuris <anchor type="note" xlink:label="note-0023-01a" xlink:href="note-0023-01"/> compoſitæ centrum grav. </s> <s xml:id="echoid-s264" xml:space="preserve">eſt ipſum punctum F. </s> <s xml:id="echoid-s265" xml:space="preserve">Poſitum au-<lb/>tem fuit L punctum centrum gravitatis ejus magnitudinis quæ <lb/>ex portione A B C & </s> <s xml:id="echoid-s266" xml:space="preserve">K F H triangulo componitur; </s> <s xml:id="echoid-s267" xml:space="preserve">igi-<lb/>tur magnitudinis reliquæ, compoſitæ ex duobus reſiduis, <lb/>quæ in figuris circumſcriptis remanent, erit centr. </s> <s xml:id="echoid-s268" xml:space="preserve">grav. </s> <s xml:id="echoid-s269" xml:space="preserve">in <lb/>producta L F, ubi ea ſic terminatur, ut pars adjecta habeat <lb/>ad F L eandem rationem quam portio A B C ſimul cum <lb/>K F H triangulo ad dicta duo reſidua <anchor type="note" xlink:href="" symbol="9"/>: </s> <s xml:id="echoid-s270" xml:space="preserve">is autem termi- <anchor type="note" xlink:label="note-0023-02a" xlink:href="note-0023-02"/> nus eſt N; </s> <s xml:id="echoid-s271" xml:space="preserve">itaque N punctum eſt centrum gravitatis duo-<lb/>rum reſiduorum. </s> <s xml:id="echoid-s272" xml:space="preserve">Quod fieri nequit; </s> <s xml:id="echoid-s273" xml:space="preserve">Nam ſi per N ducatur <lb/>recta baſi K H parallela, erunt ab una parte ſpatia omnia è <lb/>quibus utrumque reſiduum conſtat. </s> <s xml:id="echoid-s274" xml:space="preserve">Non eſt igitur L pun-<lb/>ctum centrum gravitatis magnitudinis ex portione A B C & </s> <s xml:id="echoid-s275" xml:space="preserve"><lb/>K F H triangulo compoſitæ. </s> <s xml:id="echoid-s276" xml:space="preserve">Sed neque erit ab altera parte <lb/>puncti F. </s> <s xml:id="echoid-s277" xml:space="preserve">Namque hoc ſi dicatur, planè ſimili demonſtratio-<lb/>ne eò devenietur ut duorum reſiduorum quæ demptâ portio-<lb/>ne A B C & </s> <s xml:id="echoid-s278" xml:space="preserve">K F H triangulo, in circumſcriptis figuris ſu-<lb/>pererunt, centrum gravitatis ſit ultra portionem A B C; <lb/></s> <s xml:id="echoid-s279" xml:space="preserve">quod eſt æquè abſurdum. </s> <s xml:id="echoid-s280" xml:space="preserve">Reliquum eſt igitur ut ſit ipſum pun-<lb/>ctum F; </s> <s xml:id="echoid-s281" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s282" xml:space="preserve"/> </p> <div xml:id="echoid-div26" type="float" level="2" n="3"> <note symbol="2" position="right" xlink:label="note-0021-02" xlink:href="note-0021-02a" xml:space="preserve">21. lib. 1. <lb/>Con.</note> <note symbol="*" position="foot" xlink:label="note-0021-03" xlink:href="note-0021-03a" xml:space="preserve">Notatu dignum quod K F, F H in hyperbole ſunt aſymptoti.</note> <note symbol="3" position="left" xlink:label="note-0022-01" xlink:href="note-0022-01a" xml:space="preserve">4. lib. 2. <lb/>Elem.</note> <note symbol="4" position="left" xlink:label="note-0022-02" xlink:href="note-0022-02a" xml:space="preserve">Ex conſtr.</note> <note symbol="5" position="left" xlink:label="note-0022-03" xlink:href="note-0022-03a" xml:space="preserve">14. 5. E-<lb/>lem.</note> <note symbol="6" position="left" xlink:label="note-0022-04" xlink:href="note-0022-04a" xml:space="preserve">16. l. 6. 6. <lb/>Elem.</note> <note symbol="7" position="left" xlink:label="note-0022-05" xlink:href="note-0022-05a" xml:space="preserve">7. lib. 1. <lb/>A@chim. de <lb/>Æquip.</note> <note symbol="8" position="right" xlink:label="note-0023-01" xlink:href="note-0023-01a" xml:space="preserve">Theor. 3. h.</note> <note symbol="9" position="right" xlink:label="note-0023-02" xlink:href="note-0023-02a" xml:space="preserve">8<unsure/>. lib. 1. <lb/>Archine. d e <lb/>Æquipond</note> </div> </div> <div xml:id="echoid-div28" type="section" level="1" n="15"> <head xml:id="echoid-head27" xml:space="preserve"><emph style="sc">Theorema</emph> VI.</head> <p style="it"> <s xml:id="echoid-s283" xml:space="preserve">OMnis hyperboles portio ad triangulum inſcri-<lb/>ptum, eandem cum ipſa baſin habentem ean-<lb/>demque altitudinem, hanc habet rationem; </s> <s xml:id="echoid-s284" xml:space="preserve">quam <lb/>ſubſeſquialtera duarum, lateris tranſverſi & </s> <s xml:id="echoid-s285" xml:space="preserve">dia-<lb/>metri portionis, ad eam quæ ex centro ſectionis <lb/>ducitur ad portionis centrum gravitatis.</s> <s xml:id="echoid-s286" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s287" xml:space="preserve">Eſto hyperboles portio, & </s> <s xml:id="echoid-s288" xml:space="preserve">inſcriptus ei, qualem diximus, <lb/> <anchor type="note" xlink:label="note-0023-03a" xlink:href="note-0023-03"/> triangulus A B C; </s> <s xml:id="echoid-s289" xml:space="preserve">diameter autem portionis ſit B D, & </s> <s xml:id="echoid-s290" xml:space="preserve"><lb/>latus tranſverſum ſive diameter ſectionis B E, in cujus me-<lb/>dio centrum ſectionis F. </s> <s xml:id="echoid-s291" xml:space="preserve">Et ponatur centrum gravitatis in <pb o="324" file="0024" n="24" rhead="THEOR. DE QUADRAT."/> portione punctum L. </s> <s xml:id="echoid-s292" xml:space="preserve">Dico portionem ad inſcriptum trian-<lb/>gulum A B C eam habere rationem, quam duæ tertiæ to-<lb/>tius E D ad F L.</s> <s xml:id="echoid-s293" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="2" n="1"> <note position="right" xlink:label="note-0023-03" xlink:href="note-0023-03a" xml:space="preserve">TAB. XXXIV. <lb/>Fig. 8.</note> </div> <p> <s xml:id="echoid-s294" xml:space="preserve">Conſtituatur enim ad diametrum, ut in præcedentibus, <lb/>triangulus K F H; </s> <s xml:id="echoid-s295" xml:space="preserve">ſcilicet ut quadratum F G æquetur re-<lb/>ctangulo E D B, & </s> <s xml:id="echoid-s296" xml:space="preserve">ut baſis K H ſit baſi A C æqualis & </s> <s xml:id="echoid-s297" xml:space="preserve"><lb/>parallela: </s> <s xml:id="echoid-s298" xml:space="preserve">ejuſque trianguli ſit centrum gravitatis M, ſum-<lb/>ptâ nimirum F M æquali duabus tertiis lineæ F G <anchor type="note" xlink:href="" symbol="1"/>.</s> <s xml:id="echoid-s299" xml:space="preserve"/> </p> <note symbol="1" position="left" xml:space="preserve">14. lib. 1. <lb/>Arch. de <lb/>Æquip.</note> <p> <s xml:id="echoid-s300" xml:space="preserve">Eſt itaque triangulus K F H ad A B C triangulum, ut <lb/>F G ad B D: </s> <s xml:id="echoid-s301" xml:space="preserve">verùm ut F G ad B D, ita eſt E D ad F G, <lb/>quia quadratum F G æquale eſt rectangulo E D B; </s> <s xml:id="echoid-s302" xml:space="preserve">& </s> <s xml:id="echoid-s303" xml:space="preserve">ut <lb/>E D ad F G, ita ſunt duæ tertiæ E D ad duas tertias F G, <lb/>id eſt F M; </s> <s xml:id="echoid-s304" xml:space="preserve">ergo triangulus K F H ad triangulum A B C, <lb/>ut duæ tertiæ E D ad F M. </s> <s xml:id="echoid-s305" xml:space="preserve">Eſt autem portio hyperboles <lb/>ad triangulum K F H, ut F M ad F L <anchor type="note" xlink:href="" symbol="2"/>, quoniam æqui- <anchor type="note" xlink:label="note-0024-02a" xlink:href="note-0024-02"/> librium portionis & </s> <s xml:id="echoid-s306" xml:space="preserve">trianguli K F H eſt in puncto F <anchor type="note" xlink:href="" symbol="3"/>, &</s> <s xml:id="echoid-s307" xml:space="preserve"> centra gravitatis ſingulorum puncta L & </s> <s xml:id="echoid-s308" xml:space="preserve">M; </s> <s xml:id="echoid-s309" xml:space="preserve">ex æquali igi-<lb/> <anchor type="note" xlink:label="note-0024-03a" xlink:href="note-0024-03"/> tur in proportione perturbata, erit portio ad triangulum <lb/>A B C, ut duæ tertiæ lineæ E D ad F L <anchor type="note" xlink:href="" symbol="4"/>: </s> <s xml:id="echoid-s310" xml:space="preserve">quod erat de- <anchor type="note" xlink:label="note-0024-04a" xlink:href="note-0024-04"/> monſtrandum.</s> <s xml:id="echoid-s311" xml:space="preserve"/> </p> <div xml:id="echoid-div29" type="float" level="2" n="2"> <note symbol="2" position="left" xlink:label="note-0024-02" xlink:href="note-0024-02a" xml:space="preserve">7. lib. 1. <lb/>Archim. de <lb/>Æquipond.</note> <note symbol="3" position="left" xlink:label="note-0024-03" xlink:href="note-0024-03a" xml:space="preserve">Thcor. 5. h.</note> <note symbol="4" position="left" xlink:label="note-0024-04" xlink:href="note-0024-04a" xml:space="preserve">23. lib. 5. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div31" type="section" level="1" n="16"> <head xml:id="echoid-head28" xml:space="preserve"><emph style="sc">Theorema</emph> VII.</head> <p style="it"> <s xml:id="echoid-s312" xml:space="preserve">OMnis ellipſis vel circuli portio ad triangulum <lb/>inſcriptum, eandem cum ipſa baſin habentem <lb/>eandemque altitudinem, hanc habet rationem; </s> <s xml:id="echoid-s313" xml:space="preserve">quam <lb/>ſubſeſquialtera diametri portionis reliquæ, ad eam <lb/>quæ ex figuræ centro ducitur ad centrum gravitatis <lb/>in portione.</s> <s xml:id="echoid-s314" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s315" xml:space="preserve">Eſto ellipſis vel circuli portio primùm dimidiâ figurâ non <lb/> <anchor type="note" xlink:label="note-0024-05a" xlink:href="note-0024-05"/> major, & </s> <s xml:id="echoid-s316" xml:space="preserve">inſcriptus ei triangulus A B C, eandem cum <lb/>portione baſin habens, eandemque altitudinem; </s> <s xml:id="echoid-s317" xml:space="preserve">diameter au-<lb/>tem portionis ſit B D, quæ producatur, & </s> <s xml:id="echoid-s318" xml:space="preserve">manifeſtum eſt <lb/>quod tranſibit per centrum figuræ; </s> <s xml:id="echoid-s319" xml:space="preserve">ſit hoc F, & </s> <s xml:id="echoid-s320" xml:space="preserve">diameter <lb/>portionis reliquæ D E. </s> <s xml:id="echoid-s321" xml:space="preserve">Et ponatur centrum gravitatis in por- <pb file="0025" n="25"/> <pb file="0025a" n="26"/> <anchor type="figure" xlink:label="fig-0025a-01a" xlink:href="fig-0025a-01"/> <anchor type="figure" xlink:label="fig-0025a-02a" xlink:href="fig-0025a-02"/> <anchor type="figure" xlink:label="fig-0025a-03a" xlink:href="fig-0025a-03"/> <anchor type="figure" xlink:label="fig-0025a-04a" xlink:href="fig-0025a-04"/> <anchor type="figure" xlink:label="fig-0025a-05a" xlink:href="fig-0025a-05"/> <anchor type="figure" xlink:label="fig-0025a-06a" xlink:href="fig-0025a-06"/> <anchor type="figure" xlink:label="fig-0025a-07a" xlink:href="fig-0025a-07"/> <anchor type="figure" xlink:label="fig-0025a-08a" xlink:href="fig-0025a-08"/> <pb file="0026" n="27"/> <pb o="325" file="0027" n="28" rhead="HYPERB. ELLIPS. ET CIRC."/> tione A B C punctum L. </s> <s xml:id="echoid-s322" xml:space="preserve">Dico igitur portionem ad inſcri-<lb/>ptum ſibi triangulum A B C eam habere rationem quam duæ <lb/>tertiæ E D ad F L. </s> <s xml:id="echoid-s323" xml:space="preserve">Conſtituatur enim ut ſupra triangulus <lb/>K F H, cujus nimirum baſis K H ſit baſi A C æqualis & </s> <s xml:id="echoid-s324" xml:space="preserve"><lb/>parallela, & </s> <s xml:id="echoid-s325" xml:space="preserve">F G quæ à vertice ad mediam baſin pertingit <lb/>poſſit rectangulum B D E: </s> <s xml:id="echoid-s326" xml:space="preserve">& </s> <s xml:id="echoid-s327" xml:space="preserve">centrum gravitatis trianguli <lb/>K F H ſit M punctum, ſumptâ ſcilicet F M æquali duabus <lb/> <anchor type="note" xlink:label="note-0027-01a" xlink:href="note-0027-01"/> tertiis lineæ F G <anchor type="note" xlink:href="" symbol="1"/>.</s> <s xml:id="echoid-s328" xml:space="preserve"/> </p> <div xml:id="echoid-div31" type="float" level="2" n="1"> <note position="left" xlink:label="note-0024-05" xlink:href="note-0024-05a" xml:space="preserve">TAB. XXXV. <lb/>Fig. 4. 5.</note> <figure xlink:label="fig-0025a-01" xlink:href="fig-0025a-01a"> <caption xml:id="echoid-caption1" style="it" xml:space="preserve">Pag. 324.<lb/>TAB. XXXIV.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables1" xml:space="preserve">O B E P L S Q M R N A K H G D F C</variables> </figure> <figure xlink:label="fig-0025a-02" xlink:href="fig-0025a-02a"> <caption xml:id="echoid-caption2" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables2" xml:space="preserve">B Q P S O N R M E H K G A F D L C</variables> </figure> <figure xlink:label="fig-0025a-03" xlink:href="fig-0025a-03a"> <caption xml:id="echoid-caption3" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables3" xml:space="preserve">B E A G M C D H R F K L</variables> </figure> <figure xlink:label="fig-0025a-04" xlink:href="fig-0025a-04a"> <caption xml:id="echoid-caption4" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables4" xml:space="preserve">B M L K E A D F H C</variables> </figure> <figure xlink:label="fig-0025a-05" xlink:href="fig-0025a-05a"> <caption xml:id="echoid-caption5" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables5" xml:space="preserve">B B A D C A D C E E</variables> </figure> <figure xlink:label="fig-0025a-06" xlink:href="fig-0025a-06a"> <caption xml:id="echoid-caption6" style="it" xml:space="preserve">Fig. 8.</caption> <variables xml:id="echoid-variables6" xml:space="preserve">K G H M E F B L A D C</variables> </figure> <figure xlink:label="fig-0025a-07" xlink:href="fig-0025a-07a"> <caption xml:id="echoid-caption7" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables7" xml:space="preserve">S E B P D</variables> </figure> <figure xlink:label="fig-0025a-08" xlink:href="fig-0025a-08a"> <caption xml:id="echoid-caption8" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables8" xml:space="preserve">E S D P B</variables> </figure> <note symbol="1" position="right" xlink:label="note-0027-01" xlink:href="note-0027-01a" xml:space="preserve">14. lib. 1. <lb/>Arch. de <lb/>Æquip.</note> </div> <p> <s xml:id="echoid-s329" xml:space="preserve">Triangulus igitur K F H eſt ad triangulum A B C, ut <lb/>F G ad B D; </s> <s xml:id="echoid-s330" xml:space="preserve">ut autem F G ad B D, ſic eſt E D ad F G, <lb/>quia quadratum F G æquale eſt B D E rectangulo; </s> <s xml:id="echoid-s331" xml:space="preserve">& </s> <s xml:id="echoid-s332" xml:space="preserve">ut <lb/>E D ad F G, ſic ſunt duæ tertiæ E D ad duas tertias F G, <lb/>id eſt, ad F M. </s> <s xml:id="echoid-s333" xml:space="preserve">Ergo triangulus K F H ad triangulum A B C, <lb/>ſicut duæ tertiæ E D ad F M. </s> <s xml:id="echoid-s334" xml:space="preserve">Portio autem A B C eſt ad <lb/>triangulum K F H, ut F M ad F L <anchor type="note" xlink:href="" symbol="2"/>, quoniam æquili- <anchor type="note" xlink:label="note-0027-02a" xlink:href="note-0027-02"/> brium eorum eſt in F <anchor type="note" xlink:href="" symbol="3"/>, & </s> <s xml:id="echoid-s335" xml:space="preserve">centra gravitatis ſingulorum pun- <anchor type="note" xlink:label="note-0027-03a" xlink:href="note-0027-03"/> cta L & </s> <s xml:id="echoid-s336" xml:space="preserve">M; </s> <s xml:id="echoid-s337" xml:space="preserve">Ergo ex æquali in proportione perturbata, <lb/>erit portio A B C ad A B C triangulum, ſicut duæ tertiæ <lb/>E D ad F L <anchor type="note" xlink:href="" symbol="4"/>.</s> <s xml:id="echoid-s338" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="2"> <note symbol="2" position="right" xlink:label="note-0027-02" xlink:href="note-0027-02a" xml:space="preserve">7. lib. 1. <lb/>Archim. de <lb/>Æquipond.</note> <note symbol="3" position="right" xlink:label="note-0027-03" xlink:href="note-0027-03a" xml:space="preserve">Theor. 5. h.</note> </div> <note symbol="4" position="right" xml:space="preserve">23. lib. 5. <lb/>Elem.</note> <p> <s xml:id="echoid-s339" xml:space="preserve">Sit nunc portio A B C dimidiâ figurâ major. </s> <s xml:id="echoid-s340" xml:space="preserve">Dico eam <lb/> <anchor type="note" xlink:label="note-0027-05a" xlink:href="note-0027-05"/> rurſus ad inſcriptum triangulum eam habere rationem, quam <lb/>duæ tertiæ E D ad F L.</s> <s xml:id="echoid-s341" xml:space="preserve"/> </p> <div xml:id="echoid-div33" type="float" level="2" n="3"> <note position="right" xlink:label="note-0027-05" xlink:href="note-0027-05a" xml:space="preserve">TAB. XXXVI. <lb/>Fig. 1. 2.</note> </div> <p> <s xml:id="echoid-s342" xml:space="preserve">Ponatur enim portionis reliquæ A E C centrum gravitatis <lb/>H punctum, & </s> <s xml:id="echoid-s343" xml:space="preserve">jungantur A E, E C. </s> <s xml:id="echoid-s344" xml:space="preserve">Igitur per ea quæ jam <lb/>oſtendimus, erit portio A E C ad A E C triangulum, ut <lb/>duæ tertiæ B D ad F H; </s> <s xml:id="echoid-s345" xml:space="preserve">verùm ut triangulus A E C ad <lb/>triangulum A B C, ſic eſt E D ad B D, ſive duæ tertiæ <lb/>E D ad duas tertias B D; </s> <s xml:id="echoid-s346" xml:space="preserve">ex æquali igitur in proportione <lb/>perturbata, erit ſicut portio A E C ad triangulum A B C, <lb/>ita duæ tertiæ E D ad F H <anchor type="note" xlink:href="" symbol="5"/>. </s> <s xml:id="echoid-s347" xml:space="preserve">Sed ut portio A B C ad <anchor type="note" xlink:label="note-0027-06a" xlink:href="note-0027-06"/> A E C portionem, ita eſt F H ad F L <anchor type="note" xlink:href="" symbol="6"/>, quoniam totius <anchor type="note" xlink:label="note-0027-07a" xlink:href="note-0027-07"/> figuræ centrum gravitatis eſt F, centraque dictarum portio-<lb/>num L & </s> <s xml:id="echoid-s348" xml:space="preserve">H; </s> <s xml:id="echoid-s349" xml:space="preserve">Ergo iterum ex æquali in proportione per-<lb/>turbata, erit portio A B C ad A B C triangulum, ut duæ <lb/>tertiæ E D ad F L. </s> <s xml:id="echoid-s350" xml:space="preserve">Omnis igitur Ellipſis vel circuli portio <lb/>&</s> <s xml:id="echoid-s351" xml:space="preserve">c. </s> <s xml:id="echoid-s352" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s353" xml:space="preserve"/> </p> <div xml:id="echoid-div34" type="float" level="2" n="4"> <note symbol="5" position="right" xlink:label="note-0027-06" xlink:href="note-0027-06a" xml:space="preserve">23. lib. 5. <lb/>Elem.</note> <note symbol="6" position="right" xlink:label="note-0027-07" xlink:href="note-0027-07a" xml:space="preserve">8. lib. 1. <lb/>Arch. de <lb/>Æquipond.</note> </div> <pb o="326" file="0028" n="29" rhead="THEOR. DE QUADRAT."/> </div> <div xml:id="echoid-div36" type="section" level="1" n="17"> <head xml:id="echoid-head29" xml:space="preserve"><emph style="sc">Theorema</emph> VIII.</head> <p style="it"> <s xml:id="echoid-s354" xml:space="preserve">IN ſemicirculo & </s> <s xml:id="echoid-s355" xml:space="preserve">quolibet circuli ſectore, habet <lb/>arcus ad duas tertias rectæ ſibi ſubtenſæ hanc ra-<lb/>tionem, quam ſemidiameter ad eam, quæ ex centro <lb/>ducitur ad ſectoris centrum gravitatis.</s> <s xml:id="echoid-s356" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s357" xml:space="preserve">Eſto primùm ſemicirculus A B C, deſcriptus centro D, <lb/> <anchor type="note" xlink:label="note-0028-01a" xlink:href="note-0028-01"/> ſectuſque bifariam rectâ B D, in qua centrum gravitatis <lb/>ſemicirculi ſit E <anchor type="note" xlink:href="" symbol="1"/>. </s> <s xml:id="echoid-s358" xml:space="preserve">Dico arcum A B C eſſe ad duas tertias <anchor type="note" xlink:label="note-0028-02a" xlink:href="note-0028-02"/> A C, ſicut B D ad D E. </s> <s xml:id="echoid-s359" xml:space="preserve">Jungantur enim A B, B C. </s> <s xml:id="echoid-s360" xml:space="preserve">Igi-<lb/>tur, ut ſemicirculus ad triangulum A B C, ſic ſunt duæ ter-<lb/>tiæ B D ad D E <anchor type="note" xlink:href="" symbol="2"/>, eſt enim B D æqualis diametro portio- <anchor type="note" xlink:label="note-0028-03a" xlink:href="note-0028-03"/> nis reliquæ. </s> <s xml:id="echoid-s361" xml:space="preserve">Verùm etiam ut ſemicirculus, id eſt, ut trian-<lb/>gulus habens baſin æqualem arcui A B C & </s> <s xml:id="echoid-s362" xml:space="preserve">altitudinem B D, <lb/>ad A B C triangulum, ita eſt arcus A B C ad A C re-<lb/>ctam; </s> <s xml:id="echoid-s363" xml:space="preserve">ergo ut arcus A B C ad A C, ita ſunt duæ tertiæ <lb/>B D ad D E, & </s> <s xml:id="echoid-s364" xml:space="preserve">permutando, ut arcus A B C ad duas tertias <lb/>B D, ita A C ad D E, ſive ita {2/3} A C ad {2/3} D E, unde rur-<lb/>ſus permutando, ut arcus A B C ad {2/3} A C, ita {2/3} B D ad {2/3} <lb/>D E, ſive ita, B D ad D E.</s> <s xml:id="echoid-s365" xml:space="preserve"/> </p> <div xml:id="echoid-div36" type="float" level="2" n="1"> <note position="left" xlink:label="note-0028-01" xlink:href="note-0028-01a" xml:space="preserve">TAB. XXXVI. <lb/>Fig. 3.</note> <note symbol="1" position="left" xlink:label="note-0028-02" xlink:href="note-0028-02a" xml:space="preserve">Theor. 4. h.</note> <note symbol="2" position="left" xlink:label="note-0028-03" xlink:href="note-0028-03a" xml:space="preserve">Theor. 7. h.</note> </div> <note position="left" xml:space="preserve">TAB. XXXVI. <lb/>Fig. 4.</note> <p> <s xml:id="echoid-s366" xml:space="preserve">Sit deinde ſector D A B C, ſemicirculo minor, bifariam <lb/>ſectus rectâ D B, in qua ſectoris centrum gravitatis ponatur <lb/>E punctum, & </s> <s xml:id="echoid-s367" xml:space="preserve">ducatur ſubtenſa A C. </s> <s xml:id="echoid-s368" xml:space="preserve">Dico rurſus, arcum <lb/>A B C ad duas tertias rectæ A C eam habere rationem, <lb/>quam B D ad D E. </s> <s xml:id="echoid-s369" xml:space="preserve">Jungantur enim A B, B C, & </s> <s xml:id="echoid-s370" xml:space="preserve">totius <lb/>circuli ſit diameter K D B, quæ producatur in Q, ut fiat <lb/>Q F, ad B F, ſicut portio A C B ad A B C triangulum, <lb/>& </s> <s xml:id="echoid-s371" xml:space="preserve">jungantur A Q, Q C; </s> <s xml:id="echoid-s372" xml:space="preserve">erit jam triangulus A Q C portio-<lb/>ni A C B æqualis. </s> <s xml:id="echoid-s373" xml:space="preserve">Ponantur deinde centra gravitatis, G tri-<lb/>anguli A C D, & </s> <s xml:id="echoid-s374" xml:space="preserve">H portionis A C B; </s> <s xml:id="echoid-s375" xml:space="preserve">& </s> <s xml:id="echoid-s376" xml:space="preserve">ſicut D Q ad <lb/>Q F, ita ſit H D ad D R.</s> <s xml:id="echoid-s377" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s378" xml:space="preserve">Quia igitur ſicut portio A C B ſive triangulus A Q C ad <lb/>triangulum A B C, id eſt, ut Q F ad B F, ita {2/3} K F ad <lb/>D H <anchor type="note" xlink:href="" symbol="3"/>, erit rectangulum ſub Q F, D H, æquale duabus <anchor type="note" xlink:label="note-0028-05a" xlink:href="note-0028-05"/> <pb file="0029" n="30"/> <pb file="0029a" n="31"/> <anchor type="figure" xlink:label="fig-0029a-01a" xlink:href="fig-0029a-01"/> <anchor type="figure" xlink:label="fig-0029a-02a" xlink:href="fig-0029a-02"/> <anchor type="figure" xlink:label="fig-0029a-03a" xlink:href="fig-0029a-03"/> <anchor type="figure" xlink:label="fig-0029a-04a" xlink:href="fig-0029a-04"/> <anchor type="figure" xlink:label="fig-0029a-05a" xlink:href="fig-0029a-05"/> <pb file="0030" n="32"/> <pb o="327" file="0031" n="33" rhead="HYPERB. ELLIPS. ET CIRC."/> tertiis rectanguli K F B <anchor type="note" xlink:href="" symbol="4"/>, id eſt, duabus tertiis quadrati <anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/> A F; </s> <s xml:id="echoid-s379" xml:space="preserve">ſed idem rectangulum ſub Q F, D H, æquale eſt re-<lb/>ctangulo Q D R, quia ut Q D ad Q F, ita fecimus eſſe <lb/>D H ad D R; </s> <s xml:id="echoid-s380" xml:space="preserve">ergo rectangulum Q D R æquale duabus ter-<lb/>tiis quadrati A F, ideoque ut Q D ad A F ita {2/3} A F ad D R <lb/> <anchor type="note" xlink:href="" symbol="5"/>: </s> <s xml:id="echoid-s381" xml:space="preserve">ſed ut Q D ad A F, ſic quoque eſt rectangulum ſub Q D, <anchor type="note" xlink:label="note-0031-02a" xlink:href="note-0031-02"/> A F, cui æquale quadrilaterum D A Q C, id eſt, ſector <lb/>D A B C ad A F quadratum; </s> <s xml:id="echoid-s382" xml:space="preserve">ergo & </s> <s xml:id="echoid-s383" xml:space="preserve">ſector D A B C ad <lb/>quadratum A F, ut {2/3} A F ad D R. </s> <s xml:id="echoid-s384" xml:space="preserve">Porro quoniam E cen-<lb/>trum gravitatis eſt totius ſectoris, & </s> <s xml:id="echoid-s385" xml:space="preserve">H centrum grav. </s> <s xml:id="echoid-s386" xml:space="preserve">por-<lb/>tionis A C B, G vero trianguli A C D, conſtat eſſe, ſicut <lb/>triangulus A C D ad A C B portionem ſive ad triangulum <lb/>A Q C, id eſt, ut D F ad F Q, ita H E ad E G <anchor type="note" xlink:href="" symbol="6"/>; </s> <s xml:id="echoid-s387" xml:space="preserve">quare <anchor type="note" xlink:label="note-0031-03a" xlink:href="note-0031-03"/> convertendo & </s> <s xml:id="echoid-s388" xml:space="preserve">per compoſitionem rationis erit ut D Q ad <lb/>D F, ita G H, ad H E. </s> <s xml:id="echoid-s389" xml:space="preserve">Sed quia fecimus ut D Q ad Q F, <lb/>ita H D ad D R, erit quoque per converſionem rationis, <lb/>ut D Q ad D F, ita H D ad H R; </s> <s xml:id="echoid-s390" xml:space="preserve">ergo H D ad H R ut <lb/>G H ad H E; </s> <s xml:id="echoid-s391" xml:space="preserve">quare & </s> <s xml:id="echoid-s392" xml:space="preserve">reliqua G D ad reliquam E R, ut <lb/> <anchor type="note" xlink:label="note-0031-04a" xlink:href="note-0031-04"/> H D ad H R <anchor type="note" xlink:href="" symbol="7"/>, hoc eſt, ut D Q ad D F. </s> <s xml:id="echoid-s393" xml:space="preserve">Sicut autem D Q ad D F, ita eſt quadrilaterum D A Q C, cui æqualis ſector <lb/>D A B C ad A C D triangulum; </s> <s xml:id="echoid-s394" xml:space="preserve">igitur ſector D A B C <lb/>ad A C D triangulum ut G D ad E R: </s> <s xml:id="echoid-s395" xml:space="preserve">Eſt autem A C D <lb/>triangulus ad D F quadratum, ut A F ad D F, ſive ut {2/3} A F <lb/>ad {2/3} D F id eſt D G. </s> <s xml:id="echoid-s396" xml:space="preserve">Igitur ex æquali in proportione perturbata, <lb/>ſicut ſector D A B C ad quadratum D F, ita {2/3} A F ad E R <lb/> <anchor type="note" xlink:href="" symbol="8"/>, & </s> <s xml:id="echoid-s397" xml:space="preserve">convertendo, quadratum D F ad ſectorem D A B C, <anchor type="note" xlink:label="note-0031-05a" xlink:href="note-0031-05"/> ut E R ad {2/3} A F. </s> <s xml:id="echoid-s398" xml:space="preserve">Fuit autem ante oſtenſum, quadratum <lb/>A F eſſe ad ſectorem D A B C, ut D R ad {2/3} A F; </s> <s xml:id="echoid-s399" xml:space="preserve">igitur <lb/>duo ſimul quadrata, D F & </s> <s xml:id="echoid-s400" xml:space="preserve">A F, ſive unum quadratum <lb/>D A ad ſectorem D A B C ut duæ ſimul E R & </s> <s xml:id="echoid-s401" xml:space="preserve">R D, id <lb/>eſt ut tota E D ad {2/3} A F <anchor type="note" xlink:href="" symbol="9"/>. </s> <s xml:id="echoid-s402" xml:space="preserve">Eſt verò etiam quadratum D A <anchor type="note" xlink:label="note-0031-06a" xlink:href="note-0031-06"/> ad D A B C ſectorem, ſicut linea D A ad arcum A B, quia <lb/>nimirum ſector D A B C æqualis eſt rectangulo, baſin ha-<lb/>benti æqualem arcui A B & </s> <s xml:id="echoid-s403" xml:space="preserve">altitudinem D A; </s> <s xml:id="echoid-s404" xml:space="preserve">ergo ſicut <lb/>D A ad arcum A B, ita E D ad {2/3} A F; </s> <s xml:id="echoid-s405" xml:space="preserve">& </s> <s xml:id="echoid-s406" xml:space="preserve">permutando, <lb/>arcus A B ad {2/3} A F, ſive arcus A B C ad {2/3} A C, ut D A <lb/>vel B D ad D E.</s> <s xml:id="echoid-s407" xml:space="preserve"/> </p> <div xml:id="echoid-div37" type="float" level="2" n="2"> <note symbol="3" position="left" xlink:label="note-0028-05" xlink:href="note-0028-05a" xml:space="preserve">Theor. 7. h.</note> <figure xlink:label="fig-0029a-01" xlink:href="fig-0029a-01a"> <caption xml:id="echoid-caption9" style="it" xml:space="preserve">Pag. 326.<lb/>TAB. XXXV.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables9" xml:space="preserve">N H T Z Ψ G K X S Σ Α E Ξ Y F O L B Δ R P V C Q Ω D M</variables> </figure> <figure xlink:label="fig-0029a-02" xlink:href="fig-0029a-02a"> <caption xml:id="echoid-caption10" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables10" xml:space="preserve">B L A C D F M G K E H</variables> </figure> <figure xlink:label="fig-0029a-03" xlink:href="fig-0029a-03a"> <caption xml:id="echoid-caption11" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables11" xml:space="preserve">B L A C D F M G K H E</variables> </figure> <figure xlink:label="fig-0029a-04" xlink:href="fig-0029a-04a"> <caption xml:id="echoid-caption12" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables12" xml:space="preserve">B Δ P R V C Q Ω D A L F O Y Ξ Α Σ X S G K Ψ Z T H E N M</variables> </figure> <figure xlink:label="fig-0029a-05" xlink:href="fig-0029a-05a"> <caption xml:id="echoid-caption13" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables13" xml:space="preserve">B Δ P R V A D Ω Q C L F O Y Ξ Α Σ X S G K E Ψ Z T H E N M</variables> </figure> <note symbol="4" position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">16. lib. 6. <lb/>Elem.</note> <note symbol="5" position="right" xlink:label="note-0031-02" xlink:href="note-0031-02a" xml:space="preserve">16. lib. 6. <lb/>Elem.</note> <note symbol="6" position="right" xlink:label="note-0031-03" xlink:href="note-0031-03a" xml:space="preserve">8. lib. 1. <lb/>Arch. de <lb/>Æquipond.</note> <note symbol="7" position="right" xlink:label="note-0031-04" xlink:href="note-0031-04a" xml:space="preserve">19. lib. 5. <lb/>Elem.</note> <note symbol="8" position="right" xlink:label="note-0031-05" xlink:href="note-0031-05a" xml:space="preserve">23. lib. 5. <lb/>Elem.</note> <note symbol="9" position="right" xlink:label="note-0031-06" xlink:href="note-0031-06a" xml:space="preserve">24 lib. 5. <lb/>Elem.</note> </div> <pb o="328" file="0032" n="34" rhead="THEOR. DE QUADRAT. HYPERB. &c."/> <p> <s xml:id="echoid-s408" xml:space="preserve">Eſto jam denique ſector D A B C ſemicirculo major, & </s> <s xml:id="echoid-s409" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0032-01a" xlink:href="note-0032-01"/> ponantur ea quæ priùs, & </s> <s xml:id="echoid-s410" xml:space="preserve">perficiatur circulus B A F C, <lb/>& </s> <s xml:id="echoid-s411" xml:space="preserve">totius diameter ſit B D F, in qua erit quoque centrum <lb/>grav. </s> <s xml:id="echoid-s412" xml:space="preserve">ſectoris reliqui D A F C <anchor type="note" xlink:href="" symbol="1"/>, quot ſit G.</s> <s xml:id="echoid-s413" xml:space="preserve"/> </p> <div xml:id="echoid-div38" type="float" level="2" n="3"> <note position="left" xlink:label="note-0032-01" xlink:href="note-0032-01a" xml:space="preserve">TAB. XXXVI. <lb/>Fig. 5.</note> </div> <note symbol="1" position="left" xml:space="preserve">8. lib. 1. <lb/>Arch. de <lb/>Æquip.</note> <p> <s xml:id="echoid-s414" xml:space="preserve">Quia igitur circuli totius centrum gravitatis eſt D, & </s> <s xml:id="echoid-s415" xml:space="preserve">duorum <lb/>ſectorum centra grav. </s> <s xml:id="echoid-s416" xml:space="preserve">E & </s> <s xml:id="echoid-s417" xml:space="preserve">G, erit ſicut ſector D A B C, ad ſecto-<lb/>rem D A F C, id eſt, ſicut arcus A B C ad arcum A F C, ita <lb/>G D ad D E <anchor type="note" xlink:href="" symbol="2"/>: </s> <s xml:id="echoid-s418" xml:space="preserve">verum ut arcus A F C ad {2/3} A C, ita eſt D F <anchor type="note" xlink:label="note-0032-03a" xlink:href="note-0032-03"/> ad D G, ſicuti modò oſtenſum eſt; </s> <s xml:id="echoid-s419" xml:space="preserve">ergo ex æquali in pro-<lb/>portione perturbata, ſicut arcus A B C ad {2/3} A C, ita erit <lb/>D F vel B D ad D E <anchor type="note" xlink:href="" symbol="3"/>. </s> <s xml:id="echoid-s420" xml:space="preserve">Conſtat itaque quod in ſemicircu- <anchor type="note" xlink:label="note-0032-04a" xlink:href="note-0032-04"/> lo & </s> <s xml:id="echoid-s421" xml:space="preserve">quolibet circuli ſectore &</s> <s xml:id="echoid-s422" xml:space="preserve">c. </s> <s xml:id="echoid-s423" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s424" xml:space="preserve"/> </p> <div xml:id="echoid-div39" type="float" level="2" n="4"> <note symbol="2" position="left" xlink:label="note-0032-03" xlink:href="note-0032-03a" xml:space="preserve">8. lib. 1. <lb/>Arch. de <lb/>Æquip.</note> <note symbol="3" position="left" xlink:label="note-0032-04" xlink:href="note-0032-04a" xml:space="preserve">23. lib. 5. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div41" type="section" level="1" n="18"> <head xml:id="echoid-head30" xml:space="preserve">ἘΞἘΤΑΣΙΣ <lb/>CYCLOMETRIÆ <lb/>CLARISSIMI VIRI, <lb/>GREGORII à S. VINCENTIO, S. J.</head> <head xml:id="echoid-head31" xml:space="preserve">Editæ Anno D. cIↄ Iↄc <emph style="sc">XLVII</emph>.</head> <p> <s xml:id="echoid-s425" xml:space="preserve">ANte quinquennium circiter Vir eruditiſſimus & </s> <s xml:id="echoid-s426" xml:space="preserve"><lb/>Geometriâ celeberrimus, Gregorius à S. </s> <s xml:id="echoid-s427" xml:space="preserve">Vin-<lb/>centio, quatuor modos propoſuit quadrandi Cir-<lb/>culum, unum vero eorum etiam Quadraturæ Hy-<lb/>perboles applicavit: </s> <s xml:id="echoid-s428" xml:space="preserve">quem cæteris potiorem ab <lb/>ipſo exiſtimari ex multis indiciis colligere licet. </s> <s xml:id="echoid-s429" xml:space="preserve">Unum eſt <lb/>hoc ipſum quod duas diverſarum figurarum quadraturas per <lb/>eundem hunc demonſtravit; </s> <s xml:id="echoid-s430" xml:space="preserve">alterum quod evidentior multò <lb/>ſit hic modus quam reliqui tres, ideoque minus errori obno-<lb/>xius videri debuerit; </s> <s xml:id="echoid-s431" xml:space="preserve">nonnullum etiam quod primo eum lo-<lb/>co produxit; </s> <s xml:id="echoid-s432" xml:space="preserve">Et denique hoc maximum eſt, quod in iis quæ <lb/>@d lectorem in principio totius operis præfatur, ubi ſuæ in- <pb file="0033" n="35"/> <pb file="0033a" n="36"/> <anchor type="figure" xlink:label="fig-0033a-01a" xlink:href="fig-0033a-01"/> <anchor type="figure" xlink:label="fig-0033a-02a" xlink:href="fig-0033a-02"/> <anchor type="figure" xlink:label="fig-0033a-03a" xlink:href="fig-0033a-03"/> <anchor type="figure" xlink:label="fig-0033a-04a" xlink:href="fig-0033a-04"/> <anchor type="figure" xlink:label="fig-0033a-05a" xlink:href="fig-0033a-05"/> <pb file="0034" n="37"/> <pb o="329" file="0035" n="38" rhead="ΕΞΕΤΑΣΙΣ CYCLOM. GR. à S. VINC."/> ventionis hiſtoriam & </s> <s xml:id="echoid-s433" xml:space="preserve">progreſſum paucis expoſuit, nullius <lb/>modi præter hunc unum meminerit. </s> <s xml:id="echoid-s434" xml:space="preserve">Potuit autem & </s> <s xml:id="echoid-s435" xml:space="preserve">aliam <lb/>rationem habuiſſe tres poſteriores quadraturas illic ſilentio <lb/>prætereundi, eam videlicet, quod quatuor omnes ſciret ex <lb/>iiſdem principiis deductas & </s> <s xml:id="echoid-s436" xml:space="preserve">demonſtratas eſſe. </s> <s xml:id="echoid-s437" xml:space="preserve">Sed mihi vel <lb/>alterutra harum conſiderationum ſufficere viſa eſt, ut per-<lb/>ſuaderet unam pro omnibus fore diſcuſſionem quæ quadra-<lb/>turam primariam infirmatura eſſet, reliquarum agmen du-<lb/>centem. </s> <s xml:id="echoid-s438" xml:space="preserve">Si enim erratum in ea oſtenderimus quæ minus <lb/>obſcuritatis habet, non video quâ ratione melior ſuc-<lb/>ceſſus expectandus ſit in tribus ſequentibus, quæ maximâ <lb/>caligine involvuntur, quaſque Auctor ipſe vel uni illi poſt-<lb/>habere videtur. </s> <s xml:id="echoid-s439" xml:space="preserve">Principia quæ communia eſſe omnibus qua-<lb/>draturis dixi, ea ſunt nova inventa de Proportionalitatibus <lb/>Geometricis ſive de Proportionum proportionibus, & </s> <s xml:id="echoid-s440" xml:space="preserve">de <lb/>Ductibus plani in planum. </s> <s xml:id="echoid-s441" xml:space="preserve">Quæ quidem prorſus non impu-<lb/>gnabo, nam & </s> <s xml:id="echoid-s442" xml:space="preserve">ſolida corpora quæcunque in Geometria con-<lb/>ſiderare licere exiſtimo, & </s> <s xml:id="echoid-s443" xml:space="preserve">alia omnia adhiberi poſſe, quæ <lb/>modò ullo auxilio fore credimus ad inveſtigationem veri. </s> <s xml:id="echoid-s444" xml:space="preserve">U-<lb/>num tamen prætermittere nequeo quin dicam, Clar. </s> <s xml:id="echoid-s445" xml:space="preserve">Virum <lb/>non ſatis feliciter quædam inventa in materia Proportiona-<lb/>litatum ad Quadraturam applicaſſe, atque hinc, meâ opi-<lb/>nione, ipſi extitiſſe erroris cauſam. </s> <s xml:id="echoid-s446" xml:space="preserve">Primùm omnium id in <lb/>propoſ. </s> <s xml:id="echoid-s447" xml:space="preserve">39. </s> <s xml:id="echoid-s448" xml:space="preserve">lib. </s> <s xml:id="echoid-s449" xml:space="preserve">10. </s> <s xml:id="echoid-s450" xml:space="preserve">Oper. </s> <s xml:id="echoid-s451" xml:space="preserve">Geom. </s> <s xml:id="echoid-s452" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> obſervaveram. </s> <s xml:id="echoid-s453" xml:space="preserve">Poſitis enim numeris fortuitò aſſumptis, 2, 3, 4, 5; </s> <s xml:id="echoid-s454" xml:space="preserve">deinde horum <lb/>quadratis 4, 9, 16, 25; </s> <s xml:id="echoid-s455" xml:space="preserve">& </s> <s xml:id="echoid-s456" xml:space="preserve">quadratorum, quadratis, 16, <lb/>81, 256, 625; </s> <s xml:id="echoid-s457" xml:space="preserve">videbam duodecim hiſce eandem demonſtra-<lb/>tionem convenire, quæ in dicta prop. </s> <s xml:id="echoid-s458" xml:space="preserve">39. </s> <s xml:id="echoid-s459" xml:space="preserve">ſcripta eſt de to-<lb/>tidem parallelepipedis. </s> <s xml:id="echoid-s460" xml:space="preserve">Et quum tamen concluſio nullam ido-<lb/>neam admitteret interpretationem, non dubitabam quin æ-<lb/>què ipſius ac mea argumentatio, quam iiſdem verbis forma-<lb/>veram, aliquid abſurdi contineret. </s> <s xml:id="echoid-s461" xml:space="preserve">Poſterior autem demon-<lb/>ſtrationis pars rectè ſe habebat, ideoque arguebat peccatum <lb/> <anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/> <pb o="330" file="0036" n="39" rhead="ΕΞΕΤΑΣΙΣ CYCLOM."/> in priori. </s> <s xml:id="echoid-s462" xml:space="preserve">Sed veritus ne intricata & </s> <s xml:id="echoid-s463" xml:space="preserve">prolixa hinc nobis diſ-<lb/>putatio oriretur, ad alias inventiones me converti, & </s> <s xml:id="echoid-s464" xml:space="preserve">tan-<lb/>dem ea ſeſe obtulerunt, quæ paucis hic perſcribere conſtitui. <lb/></s> <s xml:id="echoid-s465" xml:space="preserve">Nulla per hæc propoſitionum Cl. </s> <s xml:id="echoid-s466" xml:space="preserve">Viri in controverſiam vo-<lb/>cabitur; </s> <s xml:id="echoid-s467" xml:space="preserve">ſed contrà multis earum probatis, atque in uſum <lb/>meum converſis, eò rem deducam, ut ſi quidem non impoſ-<lb/>ſibile dicet quadraturam ſuam ad exitum perducere, & </s> <s xml:id="echoid-s468" xml:space="preserve">per <lb/>eam reapſe invenire rectilineum circulo æquale, oſtendam <lb/>qui id facillimè impoſterum aſſequatur. </s> <s xml:id="echoid-s469" xml:space="preserve">Deinde veſtigia i-<lb/>pſius inſiſtens demonſtrabo, quibus hactenus nobis præceſſit, <lb/>iis nequaquam ad optatum finem perveniri poſſe, ſed eſſe <lb/>ſubſiſtendum ad concluſiones perquam abſurdas. </s> <s xml:id="echoid-s470" xml:space="preserve">Atque ut <lb/>ad rem veniamus.</s> <s xml:id="echoid-s471" xml:space="preserve"/> </p> <div xml:id="echoid-div41" type="float" level="2" n="1"> <figure xlink:label="fig-0033a-01" xlink:href="fig-0033a-01a"> <caption xml:id="echoid-caption14" style="it" xml:space="preserve">Pag. 328.<lb/>Fig. 2.</caption> <variables xml:id="echoid-variables14" xml:space="preserve">B L F A D C H E</variables> </figure> <figure xlink:label="fig-0033a-02" xlink:href="fig-0033a-02a"> <caption xml:id="echoid-caption15" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables15" xml:space="preserve">B L F A D C H E</variables> </figure> <figure xlink:label="fig-0033a-03" xlink:href="fig-0033a-03a"> <caption xml:id="echoid-caption16" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables16" xml:space="preserve">B E A D C</variables> </figure> <figure xlink:label="fig-0033a-04" xlink:href="fig-0033a-04a"> <caption xml:id="echoid-caption17" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables17" xml:space="preserve">Q B H A F C E G R D K</variables> </figure> <figure xlink:label="fig-0033a-05" xlink:href="fig-0033a-05a"> <caption xml:id="echoid-caption18" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables18" xml:space="preserve">B E D A C G F</variables> </figure> <note symbol="*" position="foot" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">In hac Propoſ. 39. etiam Carteſius paralogiſmum oſtendit Epiſt. ad <lb/>Schotenium.</note> </div> <p> <s xml:id="echoid-s472" xml:space="preserve">Eſto circulus cujus centrum F, diameter C D. </s> <s xml:id="echoid-s473" xml:space="preserve">Et diviſo <lb/> <anchor type="note" xlink:label="note-0036-01a" xlink:href="note-0036-01"/> radio F C bifariam in G, ducantur ipſi ad angulos rectos <lb/>F E, G H. </s> <s xml:id="echoid-s474" xml:space="preserve">Dico, datâ ratione ſegmenti C H G ad G H E F <lb/>ſegmentum, dari quoque rationem circuli ad inſcriptum ſibi <lb/>hexagonum regulare. </s> <s xml:id="echoid-s475" xml:space="preserve">Jungantur enim F H, H C, & </s> <s xml:id="echoid-s476" xml:space="preserve">mani-<lb/>feſtum eſt triangulum F H C fore æquilaterum; </s> <s xml:id="echoid-s477" xml:space="preserve">item qua-<lb/>drantis arcum C E triplum fore arcus H E. </s> <s xml:id="echoid-s478" xml:space="preserve">Si ergo data ſit <lb/>ratio ſegmenti C H G ad G H E F ſegmentum, componen-<lb/>do quoque, data erit ratio quadrantis F E C ad ſegmentum <lb/>G H E F. </s> <s xml:id="echoid-s479" xml:space="preserve">Sed data quoque eſt ratio ſectoris F H E ad qua-<lb/>drantem F E C, ergo datur quoque ratio ſegmenti G H E F <lb/>ad ſectorem F H E; </s> <s xml:id="echoid-s480" xml:space="preserve">ac proinde dabitur quoque ratio ſe-<lb/>gmenti G H E F ad triangulum F H G; </s> <s xml:id="echoid-s481" xml:space="preserve">quare & </s> <s xml:id="echoid-s482" xml:space="preserve">ratio ſe-<lb/>ctoris F H E ad triangulum F H G data erit. </s> <s xml:id="echoid-s483" xml:space="preserve">Sed huic ra-<lb/>tioni eadem eſt ratio ſectoris F H C ad triangulum F C H, <lb/>(quoniam hæc utriuſque præcedentium dupla ſunt;) </s> <s xml:id="echoid-s484" xml:space="preserve">eadem-<lb/>que eſt circuli ratio ad hexagonum regulare ſibi inſcriptum. <lb/></s> <s xml:id="echoid-s485" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s486" xml:space="preserve">hanc datam eſſe apparet: </s> <s xml:id="echoid-s487" xml:space="preserve">quod erat demonſtran-<lb/>dum.</s> <s xml:id="echoid-s488" xml:space="preserve"/> </p> <div xml:id="echoid-div42" type="float" level="2" n="2"> <note position="left" xlink:label="note-0036-01" xlink:href="note-0036-01a" xml:space="preserve">TAB. XXXVII. <lb/>Fig. 1.</note> </div> <note position="left" xml:space="preserve">TAB. XXXVII. <lb/>Fig. 1. 2.</note> <p> <s xml:id="echoid-s489" xml:space="preserve">Sunto nunc lineæ A B, C D, E F, ſingulæ æquales dia-<lb/>metro circuli C D: </s> <s xml:id="echoid-s490" xml:space="preserve">& </s> <s xml:id="echoid-s491" xml:space="preserve">ſuper unaquaque harum conſtruantur <lb/>bina quadrata. </s> <s xml:id="echoid-s492" xml:space="preserve">Deinde verticibus A & </s> <s xml:id="echoid-s493" xml:space="preserve">B deſcribantur ſemi-<lb/>parabolæ A V G, B T H, quarum baſes ſint quadratorum <pb o="331" file="0037" n="40" rhead="GREGORII à S. VINCENTIO."/> latera B G, A H. </s> <s xml:id="echoid-s494" xml:space="preserve">In duobus ſequentibus quadratis duean-<lb/>tur diagonii C I, D K. </s> <s xml:id="echoid-s495" xml:space="preserve">Sed in poſtremis rurſus ſemipara-<lb/>bolæ deſcribantur E Σ L, F Π M, quarum vertices E & </s> <s xml:id="echoid-s496" xml:space="preserve"><lb/>F, axes vero ſint quadratorum latera E Ψ, F Ω, & </s> <s xml:id="echoid-s497" xml:space="preserve">baſes <lb/>Ψ L, Ω M. </s> <s xml:id="echoid-s498" xml:space="preserve">Porro diviſis bifariam ſingulis lineis quæ ab <lb/>initio poſitæ fuerunt, in N, O, P, & </s> <s xml:id="echoid-s499" xml:space="preserve">medietatibus rurſus <lb/>bifariam in Q, R, S, ducantur per diviſionum puncta, qua-<lb/>dratorum lateribus parallelæ, T V, X Y; </s> <s xml:id="echoid-s500" xml:space="preserve">Ζ Γ, Δ Θ, <lb/>Π Σ, Λ Ξ.</s> <s xml:id="echoid-s501" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s502" xml:space="preserve">Oſtendit itaque Cl. </s> <s xml:id="echoid-s503" xml:space="preserve">V. </s> <s xml:id="echoid-s504" xml:space="preserve">in demonſtr. </s> <s xml:id="echoid-s505" xml:space="preserve">prop. </s> <s xml:id="echoid-s506" xml:space="preserve">52. </s> <s xml:id="echoid-s507" xml:space="preserve">lib. </s> <s xml:id="echoid-s508" xml:space="preserve">10. <lb/></s> <s xml:id="echoid-s509" xml:space="preserve">Oper. </s> <s xml:id="echoid-s510" xml:space="preserve">Geom. </s> <s xml:id="echoid-s511" xml:space="preserve">& </s> <s xml:id="echoid-s512" xml:space="preserve">veriſſimum eſt, in circulo ſuperiori ſe-<lb/>gmentum C H G ad ſegmentum G H E F, eandem habere <lb/>rationem quam habet hîc ſolidum quod fit ex ductu plani <lb/>A Y Q in planum A H X Q, ad ſolidum ortum ex ductu <lb/>plani Q Y V N in planum Q X T N; </s> <s xml:id="echoid-s513" xml:space="preserve">ſicut enim ille in <lb/>ſuo ſchemate ſumit æquales lineas h i, k l, ita nobis æqua-<lb/>les ſunt ſumptæ in circulo, C G, G F, & </s> <s xml:id="echoid-s514" xml:space="preserve">hiſce pares A Q, <lb/>Q N.</s> <s xml:id="echoid-s515" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s516" xml:space="preserve">Atque ut ipſa demonſtrandi methodus quoque noſcatur, <lb/>ea hujuſmodi eſt. </s> <s xml:id="echoid-s517" xml:space="preserve">In prop. </s> <s xml:id="echoid-s518" xml:space="preserve">51. </s> <s xml:id="echoid-s519" xml:space="preserve">lib. </s> <s xml:id="echoid-s520" xml:space="preserve">10. </s> <s xml:id="echoid-s521" xml:space="preserve">oſtenditur, ſolidum <lb/>quod fit ex ductu ſemiparabolæ A B G in ſemipar. </s> <s xml:id="echoid-s522" xml:space="preserve">A B H, <lb/>æquari ſemicylindro, baſin habenti ſemicirculum C E D & </s> <s xml:id="echoid-s523" xml:space="preserve"><lb/>altitudinem C D. </s> <s xml:id="echoid-s524" xml:space="preserve">Deinde in Corollario ejuſdem prop. </s> <s xml:id="echoid-s525" xml:space="preserve">idem <lb/>quoque ſingulis partibus quod totis ſolidis convenire doce-<lb/>tur. </s> <s xml:id="echoid-s526" xml:space="preserve">Nimirum id ſolidum quod fit ex ductu plani Q Y V N <lb/>in planum Q X T N, æquatur quoque parti dicti ſemicy-<lb/>lindri quæ inſiſtit ſegmento G H E F; </s> <s xml:id="echoid-s527" xml:space="preserve">Itemque ſolidum ex <lb/>ductu plani A Y Q in pl. </s> <s xml:id="echoid-s528" xml:space="preserve">A H X Q, æquatur ejuſdem ſemicy-<lb/>lindri parti quæ inſiſtit ſegmento C H G. </s> <s xml:id="echoid-s529" xml:space="preserve">Quorum hoc vel <lb/>ex eo conſtat, quod alioqui duo iſta ſolida ſimul ſumpta, <lb/>hoc eſt, ſolidum ex ductu plani A V N in pl. </s> <s xml:id="echoid-s530" xml:space="preserve">A H T N, <lb/>æquale non eſſet dimidio ejus quem diximus, ſemicylin-<lb/>dri; </s> <s xml:id="echoid-s531" xml:space="preserve">& </s> <s xml:id="echoid-s532" xml:space="preserve">conſequenter falſum quoque eſſet quod in confeſ-<lb/>ſo eſt, nimirum ſolidum ex ductu ſemiparab. </s> <s xml:id="echoid-s533" xml:space="preserve">A B G in ſe-<lb/>miparab. </s> <s xml:id="echoid-s534" xml:space="preserve">A B H æquari toti ſemicylindro. </s> <s xml:id="echoid-s535" xml:space="preserve">Apparet igitur, <lb/>quoniam dictæ ſemicylindri partes eandem inter ſe rationem <pb o="332" file="0038" n="41" rhead="ΕΞΕΤΑΣΙΣ CYCLOM."/> habent quam baſes quibus inſiſtunt, certum eſſe quod dixi-<lb/>mus, ſegmentum circuli C H G ad G H E F, eſſe ut ſoli-<lb/>dum ex ductu plani A Y Q in pl. </s> <s xml:id="echoid-s536" xml:space="preserve">A H X Q ad ſolidum ex <lb/>ductu plani Q Y V N in pl. </s> <s xml:id="echoid-s537" xml:space="preserve">Q X T N.</s> <s xml:id="echoid-s538" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s539" xml:space="preserve">Hæc ita enucleatè ſcribere volui, ne cui ignaro fortaſſe na-<lb/>turæ demonſtrationum quibus Cl. </s> <s xml:id="echoid-s540" xml:space="preserve">V. </s> <s xml:id="echoid-s541" xml:space="preserve">utitur, ſcrupulus re-<lb/>ſtare poſſet, quod ubi ille in d. </s> <s xml:id="echoid-s542" xml:space="preserve">prop. </s> <s xml:id="echoid-s543" xml:space="preserve">52, lib. </s> <s xml:id="echoid-s544" xml:space="preserve">10, duo cir-<lb/>culi ſegmenta conſiderat, quale ferè eſt G H E F, ego pro <lb/>altero eorum ſumpſerim ſegmentum C H G: </s> <s xml:id="echoid-s545" xml:space="preserve">Quodque in <lb/>linea A B ab ipſo termino A æquales partes capiam A Q, <lb/>Q N. </s> <s xml:id="echoid-s546" xml:space="preserve">Ipſum autem Cl. </s> <s xml:id="echoid-s547" xml:space="preserve">Virum hæc remorari non poſſunt, <lb/>neque hîc, neque in ſequentibus; </s> <s xml:id="echoid-s548" xml:space="preserve">quia cum in d. </s> <s xml:id="echoid-s549" xml:space="preserve">prop. </s> <s xml:id="echoid-s550" xml:space="preserve">52. <lb/></s> <s xml:id="echoid-s551" xml:space="preserve">& </s> <s xml:id="echoid-s552" xml:space="preserve">44, lib. </s> <s xml:id="echoid-s553" xml:space="preserve">10. </s> <s xml:id="echoid-s554" xml:space="preserve">præcipit in linea a b æquales inter ſe ſumi <lb/>h i, k l, ſcit hoc nullam limitationem admittere; </s> <s xml:id="echoid-s555" xml:space="preserve">ſicut & </s> <s xml:id="echoid-s556" xml:space="preserve"><lb/>in ſchemate communi prop. </s> <s xml:id="echoid-s557" xml:space="preserve">39, lib. </s> <s xml:id="echoid-s558" xml:space="preserve">10, ubivis in linea a b <lb/>ſumitur i k, quæ dividitur in duas æquales i m, m k. </s> <s xml:id="echoid-s559" xml:space="preserve">Idem <lb/>contingit in prop. </s> <s xml:id="echoid-s560" xml:space="preserve">ſequenti 40.</s> <s xml:id="echoid-s561" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s562" xml:space="preserve">Revertor autem ad propoſitum, & </s> <s xml:id="echoid-s563" xml:space="preserve">conſtat nunc quidem, <lb/>ſi detur Ratio ſolidi quod fit ex ductu plani A Y Q in pl. <lb/></s> <s xml:id="echoid-s564" xml:space="preserve">A H X Q, ad ſolidum ex ductu plani Q Y V N in pl. </s> <s xml:id="echoid-s565" xml:space="preserve"><lb/>Q X T N, eo ipſo dari quoque rationem ſegmenti C H G <lb/>ad ſegmentum G H E F, ac proinde continuò tunc inveni-<lb/>ri poſſe quam rationem circulus habeat ad inſcriptum hexa-<lb/>gonum regulare.</s> <s xml:id="echoid-s566" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s567" xml:space="preserve">Vocemus autem brevitatis gratia, id quod fieri diximus <lb/>ex ductu plani A Y Q in planum A H X Q, ſolidum H Y. <lb/></s> <s xml:id="echoid-s568" xml:space="preserve">Item quod ſit ex ductu plani Q Y V N in planum Q X T N, <lb/>ſolidum X V. </s> <s xml:id="echoid-s569" xml:space="preserve">Similiter quod oritur ex ductu plani C Θ R <lb/>in planum C K Δ R, vocemus ſolidum K Θ; </s> <s xml:id="echoid-s570" xml:space="preserve">eâdemque <lb/>brevitate dicamus ſolida Δ Γ, Μ Ξ, Λ Σ, quibus quæ de-<lb/>notentur jam ſatis intelligitur.</s> <s xml:id="echoid-s571" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s572" xml:space="preserve">His ſic conſtitutis, ſciendum eſt, omnem ſpem & </s> <s xml:id="echoid-s573" xml:space="preserve">fundamen-<lb/>tum perficiendæ Quadraturæ Cl. </s> <s xml:id="echoid-s574" xml:space="preserve">Viro in eo poſitum eſſe, <lb/>quod exiſtimet rationem ſolidi H Y ad ſolidum X V (quam <lb/>unicam tantum deſiderari jam admonui) facile inveniri poſ-<lb/>ſe, ſi cognitæ ſint duæ rationes hæ, nimirum ratio ſolidi <pb o="333" file="0039" n="42" rhead="GREGORII à S. VINCENTIO."/> Μ Ξ ad ſol. </s> <s xml:id="echoid-s575" xml:space="preserve">Λ Σ, & </s> <s xml:id="echoid-s576" xml:space="preserve">ratio ſolidi Κ Θ ad ſol. </s> <s xml:id="echoid-s577" xml:space="preserve">Δ Γ. </s> <s xml:id="echoid-s578" xml:space="preserve">Sic enim <lb/>tunc argumentabitur; </s> <s xml:id="echoid-s579" xml:space="preserve">Nota eſt ratio ſolidi Μ Ξ ad ſol. </s> <s xml:id="echoid-s580" xml:space="preserve">Λ Σ, <lb/>item ratio ſolidi Κ Θ ad ſol. </s> <s xml:id="echoid-s581" xml:space="preserve">Δ Γ, ergo notum quoque quo-<lb/>ties illa ratio hanc contineat; </s> <s xml:id="echoid-s582" xml:space="preserve">Quoties autem illa hanc con-<lb/>tinet toties hæc ipſa, ſcilicet ratio ſolidi Κ Θ ad ſol. </s> <s xml:id="echoid-s583" xml:space="preserve">Δ Γ, <lb/>continet rationem ſolidi H Y ad X V; </s> <s xml:id="echoid-s584" xml:space="preserve">ergo & </s> <s xml:id="echoid-s585" xml:space="preserve">hæc ratio <lb/>nota erit. </s> <s xml:id="echoid-s586" xml:space="preserve">Quomodo hæc intelligenda ſint paulò inferiùs me-<lb/>lius patebit, ubi eandem argumentationem repetemus. </s> <s xml:id="echoid-s587" xml:space="preserve">Inter-<lb/>ea certo ſcio, nihil horum quæ dixi mihi à Cl. </s> <s xml:id="echoid-s588" xml:space="preserve">V. </s> <s xml:id="echoid-s589" xml:space="preserve">nega-<lb/>tum iri, modò conſideret in linea A B, ſumptas eſſe æqua-<lb/>les inter ſe partes A Q, Q N, & </s> <s xml:id="echoid-s590" xml:space="preserve">hiſce pares C R, R O; <lb/></s> <s xml:id="echoid-s591" xml:space="preserve">E S, S P.</s> <s xml:id="echoid-s592" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s593" xml:space="preserve">Si igitur indicavero ipſi quæ ſit ratio ſolidi Μ Ξ ad ſol. <lb/></s> <s xml:id="echoid-s594" xml:space="preserve">Λ Σ, item quæ ſit ratio ſolidi Κ Θ ad ſol. </s> <s xml:id="echoid-s595" xml:space="preserve">Δ Γ, & </s> <s xml:id="echoid-s596" xml:space="preserve">ne tum <lb/>quidem dicere poſſit quam rationem habeat ſolidum H Y <lb/>ad ſol. </s> <s xml:id="echoid-s597" xml:space="preserve">X V, fateatur ſane ſe fruſtra utramque Quadraturam <lb/>tentaſſe, tam Circuli quam Hyperboles. </s> <s xml:id="echoid-s598" xml:space="preserve">Circuli; </s> <s xml:id="echoid-s599" xml:space="preserve">quoniam <lb/>tunc videbit nequaquam procedere Propoſitionem 44. </s> <s xml:id="echoid-s600" xml:space="preserve">lib. </s> <s xml:id="echoid-s601" xml:space="preserve"><lb/>10. </s> <s xml:id="echoid-s602" xml:space="preserve">Oper. </s> <s xml:id="echoid-s603" xml:space="preserve">Geom. </s> <s xml:id="echoid-s604" xml:space="preserve">quæ vana & </s> <s xml:id="echoid-s605" xml:space="preserve">inanis erit, niſi ex notis ra-<lb/>tionibus ſolidi Μ Ξ ad ſol. </s> <s xml:id="echoid-s606" xml:space="preserve">Λ Σ, & </s> <s xml:id="echoid-s607" xml:space="preserve">ſolidi Κ Θ ad ſol. </s> <s xml:id="echoid-s608" xml:space="preserve">Δ Γ, <lb/>innoteſcat ratio ſolidi H Y ad ſol. </s> <s xml:id="echoid-s609" xml:space="preserve">X V. </s> <s xml:id="echoid-s610" xml:space="preserve">Hyperboles ve-<lb/>ro; </s> <s xml:id="echoid-s611" xml:space="preserve">quoniam prop. </s> <s xml:id="echoid-s612" xml:space="preserve">146. </s> <s xml:id="echoid-s613" xml:space="preserve">ejusd. </s> <s xml:id="echoid-s614" xml:space="preserve">lib. </s> <s xml:id="echoid-s615" xml:space="preserve">10. </s> <s xml:id="echoid-s616" xml:space="preserve">cui hæc quadratura <lb/>innititur, eadem eſt cum dicta prop. </s> <s xml:id="echoid-s617" xml:space="preserve">44. </s> <s xml:id="echoid-s618" xml:space="preserve">& </s> <s xml:id="echoid-s619" xml:space="preserve">iiſdem verbis <lb/>Hyperbolæ applicatur.</s> <s xml:id="echoid-s620" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s621" xml:space="preserve">Sin vero datis iſtis duabus rationibus invenire poſthac po-<lb/>tuerit rationem ſolidi H Y ad ſol. </s> <s xml:id="echoid-s622" xml:space="preserve">X V, tum ſe credat Cir-<lb/>culum reverâ quadraviſſe. </s> <s xml:id="echoid-s623" xml:space="preserve">Nota enim ſic erit ratio ſegmenti <lb/>C H G in circulo ad ſegmentum G H E F, & </s> <s xml:id="echoid-s624" xml:space="preserve">reliqua facile <lb/>perficientur.</s> <s xml:id="echoid-s625" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s626" xml:space="preserve">Dicam autem nunc ipſas Rationes. </s> <s xml:id="echoid-s627" xml:space="preserve">Et primam quidem, <lb/>hoc eſt rationem ſolidi Μ Ξ ad ſol. </s> <s xml:id="echoid-s628" xml:space="preserve">Λ Σ, ajo eſſe eandem <lb/>quæ numeri 53 ad 203. </s> <s xml:id="echoid-s629" xml:space="preserve">Alteram vero, rationem ſolidi Κ Θ <lb/>ad ſol. </s> <s xml:id="echoid-s630" xml:space="preserve">Δ Γ, eam quæ 5 ad 11. </s> <s xml:id="echoid-s631" xml:space="preserve">atque horum utrumque in-<lb/>frà ſum demonſtraturus.</s> <s xml:id="echoid-s632" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s633" xml:space="preserve">Priùs autem quod ab initio promiſi etiam oſtendam, hiſce <lb/>Rationibus cognitis, tamen rationem ſol. </s> <s xml:id="echoid-s634" xml:space="preserve">H Y ad ſol. </s> <s xml:id="echoid-s635" xml:space="preserve">X V, <pb o="334" file="0040" n="43" rhead="ΕΞΕΤΑΣΙΣ CYCLOM."/> per ea quidem quæ nos adhuc docuit V. </s> <s xml:id="echoid-s636" xml:space="preserve">Cl. </s> <s xml:id="echoid-s637" xml:space="preserve">inveniri non <lb/>poſſe. </s> <s xml:id="echoid-s638" xml:space="preserve">Etenim inventurus ex datis rationibus, ſol. </s> <s xml:id="echoid-s639" xml:space="preserve">Μ Ξ ad <lb/>ſol. </s> <s xml:id="echoid-s640" xml:space="preserve">Λ Σ, & </s> <s xml:id="echoid-s641" xml:space="preserve">ſolidi Κ Θ ad ſol. </s> <s xml:id="echoid-s642" xml:space="preserve">Δ Γ, rationem tertiam ſo-<lb/>lidi H Y ad ſol. </s> <s xml:id="echoid-s643" xml:space="preserve">X V, in hunc modum rationcinabitur, ut <lb/>videre eſt ex demonſtratione prop. </s> <s xml:id="echoid-s644" xml:space="preserve">44. </s> <s xml:id="echoid-s645" xml:space="preserve">ſuprà citatæ, cui hunc <lb/>caſum convenire liquidò conſtat. </s> <s xml:id="echoid-s646" xml:space="preserve">Dicet enim, Notæ ſunt <lb/>prima & </s> <s xml:id="echoid-s647" xml:space="preserve">ſecunda ratio, (iſtæ enim ſunt 53 ad 203, & </s> <s xml:id="echoid-s648" xml:space="preserve">5 ad <lb/>11,) ergo notum quoque quoties prima ſecundam contineat. <lb/></s> <s xml:id="echoid-s649" xml:space="preserve">Sed quoties prima continet ſecundam, toties ſecunda continet <lb/>tertiam, (hoc aſſerit prop. </s> <s xml:id="echoid-s650" xml:space="preserve">40. </s> <s xml:id="echoid-s651" xml:space="preserve">lib. </s> <s xml:id="echoid-s652" xml:space="preserve">10. </s> <s xml:id="echoid-s653" xml:space="preserve">oper. </s> <s xml:id="echoid-s654" xml:space="preserve">Geom.) </s> <s xml:id="echoid-s655" xml:space="preserve">Ergo <lb/>notum quoque quoties ſecunda tertiam contineat. </s> <s xml:id="echoid-s656" xml:space="preserve">Quare cum <lb/>nota ſit ſecunda, etiam tertia nota erit, ea nimirum quam <lb/>habet ſolidum H Y ad ſol. </s> <s xml:id="echoid-s657" xml:space="preserve">X V.</s> <s xml:id="echoid-s658" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s659" xml:space="preserve">Conſequenter hoc nunc definiendum ei incumbet, Quo-<lb/>ties Ratio harum prima ſecundam contineat; </s> <s xml:id="echoid-s660" xml:space="preserve">hoc eſt, quo-<lb/>ties ratio 53 ad 203, contineat rationem 5 ad 11. </s> <s xml:id="echoid-s661" xml:space="preserve">Sed enim <lb/>quo ſenſu verbum continere hîc explicaturus eſt? </s> <s xml:id="echoid-s662" xml:space="preserve">Num eo, <lb/>ut idem ſignificet quod alibi continere per multiplicationem? <lb/></s> <s xml:id="echoid-s663" xml:space="preserve">utque ratio 53 ad 203 rationem 5 ad 11. </s> <s xml:id="echoid-s664" xml:space="preserve">multiplicare dica-<lb/>tur vel bis (hoc eſt ut illa hujus ſit duplicata, ita enim con-<lb/>tinere intelligendum videtur in propoſitione 40. </s> <s xml:id="echoid-s665" xml:space="preserve">lib. </s> <s xml:id="echoid-s666" xml:space="preserve">10, <lb/>modo citata) vel ter, vel quater, vel ſæpiùs etiam. </s> <s xml:id="echoid-s667" xml:space="preserve">Et hoc <lb/>quidem eſſe non poteſt; </s> <s xml:id="echoid-s668" xml:space="preserve">nam ratio 53 ad 203, rationis 5 <lb/>ad 11, neque duplicata eſt neque triplicata vel ulterius mul-<lb/>tiplex, quum demum ratio 53 ad 256 {13/25} ſit duplicata rationis <lb/>5 ad 11.</s> <s xml:id="echoid-s669" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s670" xml:space="preserve">An igitur verbum Continere in eum ſenſum trahet, quem <lb/>habet in propoſitione 125. </s> <s xml:id="echoid-s671" xml:space="preserve">lib. </s> <s xml:id="echoid-s672" xml:space="preserve">8. </s> <s xml:id="echoid-s673" xml:space="preserve">Oper. </s> <s xml:id="echoid-s674" xml:space="preserve">Geom.</s> <s xml:id="echoid-s675" xml:space="preserve">? Vix qui-<lb/>dem illud ſuſpicari poſſum; </s> <s xml:id="echoid-s676" xml:space="preserve">ſed etiamſi vellet rurſus inde <lb/>abſurdum conſequetur. </s> <s xml:id="echoid-s677" xml:space="preserve">Nam ſecundum interpretationem iſtam; <lb/></s> <s xml:id="echoid-s678" xml:space="preserve">quoties ratio 53 ad 203 continet rationem 5 ad 11, toties <lb/>hæc ipſa continebit rationem 5075 ad 6413; </s> <s xml:id="echoid-s679" xml:space="preserve">hoc autem pa-<lb/>tebit horum numerorum inter ſe rationis examinanti ſecun-<lb/>dùm regulam dictæ propoſitionis 125. </s> <s xml:id="echoid-s680" xml:space="preserve">Eſſet igitur ratio ſoli-<lb/>di H Y ad ſol. </s> <s xml:id="echoid-s681" xml:space="preserve">X V, hoc eſt, ratio ſegmenti circuli ab ini-<lb/>tio propoſiti C H G, ad ſegmentum G H E F, eadem quæ <pb o="335" file="0041" n="44" rhead="GREGORII à S. VINCENTIO."/> 5075 ad 6413. </s> <s xml:id="echoid-s682" xml:space="preserve">Quare qualium partium ſegmentum C H G <lb/>eſſet 5075, talium ſegmentum G H E F eſſet 6413; </s> <s xml:id="echoid-s683" xml:space="preserve">& </s> <s xml:id="echoid-s684" xml:space="preserve">pro-<lb/>inde quadrans F C E 11488; </s> <s xml:id="echoid-s685" xml:space="preserve">& </s> <s xml:id="echoid-s686" xml:space="preserve">ſector F H E (qui qua-<lb/>drantis tertia pars eſt) 3829 {1/3}: </s> <s xml:id="echoid-s687" xml:space="preserve">& </s> <s xml:id="echoid-s688" xml:space="preserve">triangulum G H F 2583 {@/3}. <lb/></s> <s xml:id="echoid-s689" xml:space="preserve">Sicut autem ſector F H E ad triang. </s> <s xml:id="echoid-s690" xml:space="preserve">G H F, ita eſt ſector <lb/>F H C ad triangul. </s> <s xml:id="echoid-s691" xml:space="preserve">F C H, & </s> <s xml:id="echoid-s692" xml:space="preserve">ita circulus C D ad in-<lb/>ſcriptum ſibi hexagonum regulare. </s> <s xml:id="echoid-s693" xml:space="preserve">Ergo quoque qualium par-<lb/>tium circulus C D eſſet 3829 {1/3} talium hexagonum inſcri-<lb/>ptum foret 2583 {2/3}. </s> <s xml:id="echoid-s694" xml:space="preserve">Qualium autem hexagonum inſcriptum eſt <lb/>2583 {2/3}, talium hexagonum regulare circumſcriptum eſt 3444 {8/9}; </s> <s xml:id="echoid-s695" xml:space="preserve"><lb/>quoniam hoc inſcripti eſt ſeſquitertium: </s> <s xml:id="echoid-s696" xml:space="preserve">Ergo qualium <lb/>partium circulus C D eſſet 3829 {1/3}, talium hexagonum cir-<lb/>cumſcriptum eſſet 3444 {8/9}, atque ita eſſet ipſo circulo mi-<lb/>nus, quod eſt abſurdum.</s> <s xml:id="echoid-s697" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s698" xml:space="preserve">Manifeſtum igitur fecimus, ex duabus interpretationibus <lb/>verbi Continere, neutram caſui noſtro accommodari poſſe. <lb/></s> <s xml:id="echoid-s699" xml:space="preserve">Aliam autem præter illas nullam in ſuo opere attulit; </s> <s xml:id="echoid-s700" xml:space="preserve">non <lb/>docuit igitur modum determinandi, quoties ratio ſol. </s> <s xml:id="echoid-s701" xml:space="preserve">Μ Ξ <lb/>ad ſol. </s> <s xml:id="echoid-s702" xml:space="preserve">Λ Σ contineat rationem ſol. </s> <s xml:id="echoid-s703" xml:space="preserve">Κ Θ ad ſolid. </s> <s xml:id="echoid-s704" xml:space="preserve">Δ Γ, ac <lb/>proinde nec determinari poterit quoties hæc ratio contineat <lb/>rationem ſolidi H Y ad ſolid. </s> <s xml:id="echoid-s705" xml:space="preserve">X V. </s> <s xml:id="echoid-s706" xml:space="preserve">Quare liquet, hanc <lb/>rationem, ne duabus quidem prioribus iſtis datis, per in-<lb/>venta Clariſſ. </s> <s xml:id="echoid-s707" xml:space="preserve">Viri cognoſci poſſe: </s> <s xml:id="echoid-s708" xml:space="preserve">ideoque fruſtra ipſum <lb/>ſperaſſe hoc modo perficere Circuli quadraturam.</s> <s xml:id="echoid-s709" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s710" xml:space="preserve">Reſtat nunc tantùm ut manifeſta faciam quæ in præce-<lb/>dentibus poſita fuere, dixi enim me demonſtraturm, quod <lb/>ſolidum Μ Ξ eſſet ad ſolid. </s> <s xml:id="echoid-s711" xml:space="preserve">Λ Σ, ut 53 ad 203: </s> <s xml:id="echoid-s712" xml:space="preserve">item quod <lb/>ſolidum Κ Θ rationem haberet ad ſolidum Δ Γ, quam 5 ad <lb/>11.</s> <s xml:id="echoid-s713" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s714" xml:space="preserve">Quoniam autem ad horum primi demonſtrationem neceſſa-<lb/>rium eſt, ut notum habeamus, quæ ſit Ratio ungulæ Paraboli-<lb/>cæ ad Cylindrum ſuum, qui baſi inſiſtit eidem, & </s> <s xml:id="echoid-s715" xml:space="preserve">eandem <lb/>habet altitudinem; </s> <s xml:id="echoid-s716" xml:space="preserve">idcirco hanc Rationem declarantes, Tra-<lb/>ctatum Clariſſ. </s> <s xml:id="echoid-s717" xml:space="preserve">Viri, quem de eadem Ungula, Parte 5. </s> <s xml:id="echoid-s718" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s719" xml:space="preserve">9. </s> <s xml:id="echoid-s720" xml:space="preserve">propoſuit, uno egregrio Theoremate auctiorem reddemus, <lb/>quod miror ipſum non inveniſſe, quum ex iis quæ jam <pb o="336" file="0042" n="45" rhead="ΕΞΕΤΑΣΙΣ CYCLOM."/> oſtenderat facili negotio deducatur, ut jam ſtatim appa-<lb/>rebit.</s> <s xml:id="echoid-s721" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">TAB. XXXVII. <lb/>Fig. 3.</note> <p> <s xml:id="echoid-s722" xml:space="preserve">Repetitâ enim quatenus hîc neceſſe erit figurâ ipſius, quæ <lb/>eſt in propoſitione 99. </s> <s xml:id="echoid-s723" xml:space="preserve">lib. </s> <s xml:id="echoid-s724" xml:space="preserve">9. </s> <s xml:id="echoid-s725" xml:space="preserve">Eſto Cylindrus Parabolicus, <lb/>baſes oppoſitas habens parabolas A B D, V C E; </s> <s xml:id="echoid-s726" xml:space="preserve">à quo ſit <lb/>abſciſſa Ungula A B C D, eâdem baſi & </s> <s xml:id="echoid-s727" xml:space="preserve">altitudine. </s> <s xml:id="echoid-s728" xml:space="preserve">Dico <lb/>Cylindrum ad hanc Ungulam habere rationem duplam ſeſ-<lb/>quialteram, ſive quam 5 ad 2.</s> <s xml:id="echoid-s729" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s730" xml:space="preserve">Tranſcriptis enim reliquis ex figura eadem, eſt F B dia-<lb/>meter parabolæ A B D: </s> <s xml:id="echoid-s731" xml:space="preserve">& </s> <s xml:id="echoid-s732" xml:space="preserve">lineæ rectæ A B, B D. </s> <s xml:id="echoid-s733" xml:space="preserve">Ductâ <lb/>porrò B C rectâ in ſuperficie cylindri, ſumptâque ejus quar-<lb/>tâ parte C Q, abſcinditur plano P Q N ungula P Q C N <lb/>& </s> <s xml:id="echoid-s734" xml:space="preserve">junguntur C A, C D. </s> <s xml:id="echoid-s735" xml:space="preserve">Denique toti cylindro adjuncta eſt <lb/>pyramis A D γ C æqualis parti B X D E C, quæ à cylin-<lb/>dro abſciſſa eſt plano B D E C. </s> <s xml:id="echoid-s736" xml:space="preserve">Et hactenus quidem ſuffi-<lb/>ciet nobis conſtructionem Cl. </s> <s xml:id="echoid-s737" xml:space="preserve">V. </s> <s xml:id="echoid-s738" xml:space="preserve">repetiiſſe. </s> <s xml:id="echoid-s739" xml:space="preserve">Demonſtravit <lb/>autem hæc duo quæ ſequuntur, ſicut videre eſt in dicta prop. <lb/></s> <s xml:id="echoid-s740" xml:space="preserve">99. </s> <s xml:id="echoid-s741" xml:space="preserve">lib. </s> <s xml:id="echoid-s742" xml:space="preserve">9. </s> <s xml:id="echoid-s743" xml:space="preserve">Nimirum quod ungula A B C D eſt ad ungulam <lb/>P Q C N, ſicut 32 ad 1. </s> <s xml:id="echoid-s744" xml:space="preserve">Item quod hæc ungula P Q C N <lb/>eſt ad pyramidem totam A γ D B C, (quæ compoſita eſt <lb/>ex duabus pyramidibus A D B C & </s> <s xml:id="echoid-s745" xml:space="preserve">A D γ C) ut 1 ad <lb/>30.</s> <s xml:id="echoid-s746" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s747" xml:space="preserve">Erit igitur ex æquo ungula A B C D ad pyramidem <lb/>A γ D B C ut 32 ad 30, hoc eſt, ut 16 ad 15. </s> <s xml:id="echoid-s748" xml:space="preserve">Porrò cùm <lb/>parabolæ A B D octava pars ſit ſegmentum B D X, erit <lb/>quoque ſegmentum ſolidum B X D E C vel huic æqualis <lb/>pyramis A D γ C, octava pars cylindri totius parabolici <lb/>A V C E D B: </s> <s xml:id="echoid-s749" xml:space="preserve">ſed pyramis altera A D B C æquatur dua-<lb/>bus octavis ſive uni quartæ ejuſdem parabolici cylindri; </s> <s xml:id="echoid-s750" xml:space="preserve">(eſt <lb/>enim ipſa tertia pars ſui priſmatis, quod æquale eſt tribus <lb/>quartis cylindri iſtius, ut ex quadratura parabolæ conſtat) <lb/>ergo tota pyramis A γ D B C tribus octavis æquatur cylin-<lb/>dri parab. </s> <s xml:id="echoid-s751" xml:space="preserve">A V C E D B. </s> <s xml:id="echoid-s752" xml:space="preserve">Cylindrus igitur parabolicus <lb/>A V C E D B erit ad pyramidem A γ D B C, ut 8 ad 3, <lb/>hoc eſt, ut 40 ad 15; </s> <s xml:id="echoid-s753" xml:space="preserve">ſed oſtenſum eſt eandem pyramidem <lb/>A γ D B C eſſe ad ungulam A B C D ut 15 ad 16. </s> <s xml:id="echoid-s754" xml:space="preserve">Igitur <pb o="337" file="0043" n="46" rhead="GREGORII à S. VINCENTIO."/> ex æquo erit cylindrus parabolicus A V C E D B ad ungu-<lb/>lam A B C D ut 40 ad 16, hoc eſt, ut 5 ad 2; </s> <s xml:id="echoid-s755" xml:space="preserve">quod fuit <lb/>demonſtrandum.</s> <s xml:id="echoid-s756" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s757" xml:space="preserve">Quæ hîc dixi à Cl. </s> <s xml:id="echoid-s758" xml:space="preserve">Viro oſtenſa fuiſſe, veriſſima ſunt, <lb/>ac proinde non eſt quod de veritate hujus Theorematis du-<lb/>bitemus: </s> <s xml:id="echoid-s759" xml:space="preserve">Cujus aliam quoque demonſtr. </s> <s xml:id="echoid-s760" xml:space="preserve">adferre poſſem, lon-<lb/>ge ab iſta diverſam, niſi ad ſequentia properarem.</s> <s xml:id="echoid-s761" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s762" xml:space="preserve">Repetitâ igitur parte ultimâ ſchematis, quod ſuprà de-<lb/> <anchor type="note" xlink:label="note-0043-01a" xlink:href="note-0043-01"/> ſcripſimus, ſit oſtendendum, quod ſolidum Μ Ξ, id eſt, <lb/>quod oritur ex ductu plani E Ξ S in planum E M Λ S ad <lb/>ſolidum Λ Σ, id eſt, quod oritur ex ductu plani S Ξ Σ P <lb/>in planum S Λ Π P, eam habet rationem quam 53 ad 203. <lb/></s> <s xml:id="echoid-s763" xml:space="preserve">Deſcribatur ſuper E F parabola E Π F, axem habens P Π, <lb/>quam conſtat eſſe quartam partem ipſius E F ſive M E. </s> <s xml:id="echoid-s764" xml:space="preserve">Erit <lb/>igitur parabola E Π F eadem quam V. </s> <s xml:id="echoid-s765" xml:space="preserve">Cl. </s> <s xml:id="echoid-s766" xml:space="preserve">in prop. </s> <s xml:id="echoid-s767" xml:space="preserve">41. </s> <s xml:id="echoid-s768" xml:space="preserve">& </s> <s xml:id="echoid-s769" xml:space="preserve"><lb/>42. </s> <s xml:id="echoid-s770" xml:space="preserve">lib. </s> <s xml:id="echoid-s771" xml:space="preserve">10. </s> <s xml:id="echoid-s772" xml:space="preserve">notat literis A R B. </s> <s xml:id="echoid-s773" xml:space="preserve">Ait autem in dicta prop. </s> <s xml:id="echoid-s774" xml:space="preserve"><lb/>42. </s> <s xml:id="echoid-s775" xml:space="preserve">& </s> <s xml:id="echoid-s776" xml:space="preserve">veriſſimum eſt, ſolidum quod producitur ex ductu <lb/>plani E Σ L F in planum F Π M E æquari ſolido quod fit <lb/>ex parabola E Π F ducta in ſe ipſam: </s> <s xml:id="echoid-s777" xml:space="preserve">ſicut illud quoque <lb/>quod ſubjungit in Coroll. </s> <s xml:id="echoid-s778" xml:space="preserve">1. </s> <s xml:id="echoid-s779" xml:space="preserve">nimirum quod ſolidum ex pla-<lb/>no S Ξ Σ P in planum S Λ Π P, æquatur ſolido ex ductu <lb/>plani S Φ Π P in ſe ipſum; </s> <s xml:id="echoid-s780" xml:space="preserve">unde ſimiliter ſolidum ex pla-<lb/>no E Ξ S in planum E M Λ S æquabitur ſolido ex plano <lb/>E Φ S in ſe ipſum ducto.</s> <s xml:id="echoid-s781" xml:space="preserve"/> </p> <div xml:id="echoid-div43" type="float" level="2" n="3"> <note position="right" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">TAB. XXXVII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s782" xml:space="preserve">Oportet itaque oſtendere ſolidum ortum ex plano E Φ S <lb/>ad ſolidum ex plano S Φ Π P, utroque in ſe ipſum ducto, <lb/>eſſe ut 53 ad 203.</s> <s xml:id="echoid-s783" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s784" xml:space="preserve">Eſto ungula parabolica A E F Π, cujus baſis parabola <lb/> <anchor type="note" xlink:label="note-0043-02a" xlink:href="note-0043-02"/> E Π F repetita ſit ex figura præcedenti, eodemque modo <lb/>ut iſtic diviſa lineis Π P, Φ S. </s> <s xml:id="echoid-s785" xml:space="preserve">Sit autem altitudo ungulæ <lb/>A Π dupla diametri baſis Π P. </s> <s xml:id="echoid-s786" xml:space="preserve">Erit igitur hæc ea ungula, <lb/>quam intelligit in prop. </s> <s xml:id="echoid-s787" xml:space="preserve">dicta 42. </s> <s xml:id="echoid-s788" xml:space="preserve">lib. </s> <s xml:id="echoid-s789" xml:space="preserve">10. </s> <s xml:id="echoid-s790" xml:space="preserve">ejuſdemque coroll. <lb/></s> <s xml:id="echoid-s791" xml:space="preserve">2. </s> <s xml:id="echoid-s792" xml:space="preserve">fieri ex ductu parabolæ E Π F in ſe ipſam. </s> <s xml:id="echoid-s793" xml:space="preserve">Eâdem nimi-<lb/>rum conſideratione uſus quæ eſt in Scholio propoſ. </s> <s xml:id="echoid-s794" xml:space="preserve">19. </s> <s xml:id="echoid-s795" xml:space="preserve">lib. </s> <s xml:id="echoid-s796" xml:space="preserve"><lb/>9. </s> <s xml:id="echoid-s797" xml:space="preserve">Nam alioqui ex ejuſmodi ductu potius dicendum eſſet ge-<lb/>minas ungulas produci, ſingulas altitudine æquales diame- <pb o="338" file="0044" n="47" rhead="ΕΞΕΤΑΣΙΣ CYCLOM."/> tro Π P. </s> <s xml:id="echoid-s798" xml:space="preserve">Ducto deinde plano per A Π P, & </s> <s xml:id="echoid-s799" xml:space="preserve">alio huic pa-<lb/>rallelo D Φ S ſecundum lineam Φ S, erit jam pars ungulæ <lb/>hiſce duobus planis terminata, æqualis ſolido quod fit ex du-<lb/>ctu plani S Φ Π P in ſe ipſum; </s> <s xml:id="echoid-s800" xml:space="preserve">& </s> <s xml:id="echoid-s801" xml:space="preserve">pars ungulæ E D S Φ, æ-<lb/>qualis ei ſolido quod fit ex ductu plani E Φ S in ſe ipſum. <lb/></s> <s xml:id="echoid-s802" xml:space="preserve">Quare nunc demonſtrandum erit duntaxat, partem E D S Φ <lb/>eſſe ad partem Φ A P ut 53 ad 203, Sit Φ N parallela E P, <lb/>& </s> <s xml:id="echoid-s803" xml:space="preserve">N C parallela Π A. </s> <s xml:id="echoid-s804" xml:space="preserve">Ergo quoniam ex proprietate Para-<lb/>boles, P N eſt {3/4} Π P, erit quoque P C {3/4} A P. </s> <s xml:id="echoid-s805" xml:space="preserve">Verùm & </s> <s xml:id="echoid-s806" xml:space="preserve"><lb/>S D æquatur {3/4} A P, quum ſit huic parallela <anchor type="note" xlink:href="" symbol="*"/>, ſitque pa- <anchor type="note" xlink:label="note-0044-01a" xlink:href="note-0044-01"/> rabola E A F: </s> <s xml:id="echoid-s807" xml:space="preserve">Itaque junctâ C D, ea parallela & </s> <s xml:id="echoid-s808" xml:space="preserve">æqualis <lb/>erit lineis P S, N Φ. </s> <s xml:id="echoid-s809" xml:space="preserve">Ducatur ſecundùm D C planum <lb/>D B C parallelum baſi E Π F, fietque ſemiparabola B D C <lb/>æqualis & </s> <s xml:id="echoid-s810" xml:space="preserve">ſimilis ſemiparabolæ Π Φ N; </s> <s xml:id="echoid-s811" xml:space="preserve">& </s> <s xml:id="echoid-s812" xml:space="preserve">erit Φ B N di-<lb/>midius cylindrus parabolicus: </s> <s xml:id="echoid-s813" xml:space="preserve">D A C B verò dimidiata un-<lb/>gula. </s> <s xml:id="echoid-s814" xml:space="preserve">Hæc autem æquatur ſicut antea oſtendimus, duabus <lb/>quintis cylindri dimidiati, baſin habentis D B C & </s> <s xml:id="echoid-s815" xml:space="preserve">altitudi-<lb/>nem B A. </s> <s xml:id="echoid-s816" xml:space="preserve">Ergo quum ſemicylindrus Φ B N habeat altitu-<lb/>dinem B Π triplam ipſius B A, erit ungula dimid. </s> <s xml:id="echoid-s817" xml:space="preserve">D A C B <lb/>ad ſemicyl. </s> <s xml:id="echoid-s818" xml:space="preserve">Φ B N, ut 2 ad 15, hoc eſt, ut 8 ad 60.</s> <s xml:id="echoid-s819" xml:space="preserve"/> </p> <div xml:id="echoid-div44" type="float" level="2" n="4"> <note position="right" xlink:label="note-0043-02" xlink:href="note-0043-02a" xml:space="preserve">TAB. XXXVII. <lb/>Fig. 4.</note> <note symbol="*" position="left" xlink:label="note-0044-01" xlink:href="note-0044-01a" xml:space="preserve">11. 16. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s820" xml:space="preserve">Junctâ porro Φ Π, conſtat ſemiparabolam Π Φ N ad trian-<lb/>gulum Π Φ N eſſe ut 4 ad@ 3; </s> <s xml:id="echoid-s821" xml:space="preserve">ſed triangulus Π Φ N eſt ad <lb/>rectangulum Φ P ut 1 ad 6, (eſt enim baſis Π N tertia pars <lb/>ipſius N P) hoc eſt, ut 3 ad 18. </s> <s xml:id="echoid-s822" xml:space="preserve">Ergo ex æquo erit ſemi-<lb/>parab. </s> <s xml:id="echoid-s823" xml:space="preserve">Π Φ N ad rectang. </s> <s xml:id="echoid-s824" xml:space="preserve">Φ P ut 4 ad 18. </s> <s xml:id="echoid-s825" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s826" xml:space="preserve">ſemi-<lb/>cylindrus Φ B N eſt ad parallelepipedum ejuſdem altitudinis <lb/>ſuper baſi Φ P, ut 4 ad 18. </s> <s xml:id="echoid-s827" xml:space="preserve">Dictiautem parallelepipedi di-<lb/>midium eſt priſma D N S; </s> <s xml:id="echoid-s828" xml:space="preserve">ergo ſemicylindrus Φ B N eſt ad <lb/>priſma D N S, ut 4 ad 9, hoc eſt, ut 60 ut 135. </s> <s xml:id="echoid-s829" xml:space="preserve">Qua-<lb/>lium igitur partium dimidiata ungula D A C B erat 8, ta-<lb/>lium ſemicylindrus parab. </s> <s xml:id="echoid-s830" xml:space="preserve">Φ B N erat 60, (ut ſuprà oſten-<lb/>ſum eſt) taliumque priſma D N S erit 135. </s> <s xml:id="echoid-s831" xml:space="preserve">Ac proinde ſo-<lb/>lidum A Π S D, quod ex iſtis tribus componitur, erit 203. <lb/></s> <s xml:id="echoid-s832" xml:space="preserve">Eſt autem ungula dimidiata A D C B ad dimidiatam un-<lb/>gulam E A P Π, ut 1 ad 32, ſicut Cl. </s> <s xml:id="echoid-s833" xml:space="preserve">Vir. </s> <s xml:id="echoid-s834" xml:space="preserve">demonſtravit in <lb/>prop. </s> <s xml:id="echoid-s835" xml:space="preserve">95. </s> <s xml:id="echoid-s836" xml:space="preserve">lib. </s> <s xml:id="echoid-s837" xml:space="preserve">9. </s> <s xml:id="echoid-s838" xml:space="preserve">Ergo qualium partium ungula dimid.</s> <s xml:id="echoid-s839" xml:space="preserve"> <pb o="339" file="0045" n="48" rhead="GREGORII à S. VINCENTIO."/> A D C B eſt 8, talium erit dimid. </s> <s xml:id="echoid-s840" xml:space="preserve">ungula E A P Π 256, <lb/>quoniam ut 1 ad 32, ita eſt 8 ad 256. </s> <s xml:id="echoid-s841" xml:space="preserve">Diximus autem par-<lb/>tem ſol. </s> <s xml:id="echoid-s842" xml:space="preserve">A Π S D eſſe talium 203. </s> <s xml:id="echoid-s843" xml:space="preserve">Igitur dim. </s> <s xml:id="echoid-s844" xml:space="preserve">ungula <lb/>E A P Π eſt ad partem A Π S D ut 256 ad 203; </s> <s xml:id="echoid-s845" xml:space="preserve">& </s> <s xml:id="echoid-s846" xml:space="preserve">divi-<lb/>dendo pars reliqua E D S Φ ad partem A Π S D, ut 53 ad <lb/>203; </s> <s xml:id="echoid-s847" xml:space="preserve">quod erat demonſtr. </s> <s xml:id="echoid-s848" xml:space="preserve">Oſtendimus igitur illud quoque <lb/>ſolidum, quod ſuprà diximus fieri ex ductu plani E Ξ S in <lb/>planum E M Λ S, eam habere rationem ad ſolidum ortum ex <lb/>ductu plani S Ξ Σ P in planum S Λ Π P, quam 53 ad 203.</s> <s xml:id="echoid-s849" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s850" xml:space="preserve">Tandem ad alterum eorum quæ demonſtrare promiſimus <lb/> <anchor type="note" xlink:label="note-0045-01a" xlink:href="note-0045-01"/> accedamus, repetitâque parte mediâ ſchematis triplicis <lb/>quod ſuprà deſcriptum fuit, oſtendendum ſit; </s> <s xml:id="echoid-s851" xml:space="preserve">ſolidum or-<lb/>tum ex ductu plani C Θ R in planum C K Δ R, ad ſoli-<lb/>dum ex ductu plani R Θ Γ O in planum R Δ Z O eam ha-<lb/> <anchor type="note" xlink:label="note-0045-02a" xlink:href="note-0045-02"/> bere rationem, quam 5 ad 11. </s> <s xml:id="echoid-s852" xml:space="preserve">Supra latus C D trianguli <lb/>C D I, erigatur ad perpendiculum triangulum C K D, & </s> <s xml:id="echoid-s853" xml:space="preserve"><lb/>jungatur K I. </s> <s xml:id="echoid-s854" xml:space="preserve">Erit jam pyramis C D I K illud ſolidum quod <lb/>intelligitur fieri ex ductu trianguli C D I in triangulum C D K. <lb/></s> <s xml:id="echoid-s855" xml:space="preserve">Etenim ſectâ pyramide plano A Z O Γ ſecundum O Γ, <lb/>quod rectum ſit ad baſin C D I, erit ſectio quadratum, id <lb/>eſt, rectangul. </s> <s xml:id="echoid-s856" xml:space="preserve">quod fit ex lineis Γ O, O Z; </s> <s xml:id="echoid-s857" xml:space="preserve">eademque ſe-<lb/>ctio dividet pyramidem bifariam. </s> <s xml:id="echoid-s858" xml:space="preserve">Secta item plano E Δ R Θ <lb/>priori parallelo, ſecundùm lineam R Θ, exiſtet inde re-<lb/>ctangulum E R, quale continetur lineis Θ R, R Δ. </s> <s xml:id="echoid-s859" xml:space="preserve">Opor-<lb/>tet itaque oſtendere, quòd ſolidum K C R E Δ eſt ad ſo-<lb/>lidum Δ Λ Ο Θ Δ, ut 5 ad 11.</s> <s xml:id="echoid-s860" xml:space="preserve"/> </p> <div xml:id="echoid-div45" type="float" level="2" n="5"> <note position="right" xlink:label="note-0045-01" xlink:href="note-0045-01a" xml:space="preserve">TAB. XXXVII. <lb/>Fig. 2.</note> <note position="right" xlink:label="note-0045-02" xlink:href="note-0045-02a" xml:space="preserve">Fig. 5.</note> </div> <p> <s xml:id="echoid-s861" xml:space="preserve">Ducatur ſecundùm E Δ planum Δ E B parallelum baſi <lb/>C D I; </s> <s xml:id="echoid-s862" xml:space="preserve">abſcindet illud pyramidem B E Δ K ſimilem toti <lb/>pyramidi C I D K, quæque proinde erit ad hanc in tripli-<lb/>cata ratione laterum homologorum B Δ ad C D. </s> <s xml:id="echoid-s863" xml:space="preserve">Sed B Δ, <lb/>cum ſit æqualis ipſi C R, quarta pars eſt lateris C D. </s> <s xml:id="echoid-s864" xml:space="preserve">Ita-<lb/>que qualium partium pyramis B E Δ K eſt unius, talium <lb/>pyramis C I D K erit 64: </s> <s xml:id="echoid-s865" xml:space="preserve">& </s> <s xml:id="echoid-s866" xml:space="preserve">dimidium hujus, hoc eſt, ſo-<lb/>lidum K A O C erit 32. </s> <s xml:id="echoid-s867" xml:space="preserve">Qualium autem pyramis B E Δ K <lb/>eſt unius talium quoque priſma B E R eſt 9; </s> <s xml:id="echoid-s868" xml:space="preserve">quoniam ba-<lb/>ſin habent communem B E Δ, & </s> <s xml:id="echoid-s869" xml:space="preserve">priſmatis altitudo B C tri- <pb o="340" file="0046" n="49" rhead="ΕΞΕΤΑΣΙΣ CYCLOM. GR. à S. VINC."/> pla eſt ad altitudinem pyramidis B K. </s> <s xml:id="echoid-s870" xml:space="preserve">Ergo ſolidum K C R E Δ <lb/>quod ex hiſce duobus componitur, erit partium 10, qua-<lb/>lium ſolidum K A O C eſt 32. </s> <s xml:id="echoid-s871" xml:space="preserve">Apparet igitur hoc eſſe ad <lb/>illud ut 16 ad 5; </s> <s xml:id="echoid-s872" xml:space="preserve">ideoque dividendo & </s> <s xml:id="echoid-s873" xml:space="preserve">convertendo ſoli-<lb/>dum K C R E Δ eſſe ad ſolidum Δ Α Ο Θ Δ, ut 5 ad 11: <lb/></s> <s xml:id="echoid-s874" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s875" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div47" type="section" level="1" n="19"> <head xml:id="echoid-head32" xml:space="preserve">FINIS.</head> <figure> <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0046-01"/> </figure> <pb file="0047" n="50"/> <pb file="0047a" n="51"/> <figure> <caption xml:id="echoid-caption19" style="it" xml:space="preserve">Pag. 340.<lb/>TAB. XXXVII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables19" xml:space="preserve">C G H F E D<lb/>H A X Q Y T N V B G</variables> </figure> <figure> <caption xml:id="echoid-caption20" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables20" xml:space="preserve">γ A F D X B P N V E Q C</variables> </figure> <figure> <caption xml:id="echoid-caption21" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables21" xml:space="preserve">K C Δ R Θ Z O Γ D I</variables> </figure> <figure> <caption xml:id="echoid-caption22" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables22" xml:space="preserve">A B D C Π Φ N E S P F</variables> </figure> <figure> <caption xml:id="echoid-caption23" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables23" xml:space="preserve">M E Ψ Λ Φ S Ξ Π Ρ Σ Ω F L</variables> </figure> <figure> <caption xml:id="echoid-caption24" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables24" xml:space="preserve">K B Δ E Z A C R O D Θ Γ I</variables> </figure> <pb file="0048" n="52"/> <pb file="0049" n="53"/> </div> <div xml:id="echoid-div48" type="section" level="1" n="20"> <head xml:id="echoid-head33" xml:space="preserve">CHRISTIANI HUGENII, <lb/><emph style="sc">Const</emph>. F. <lb/><emph style="sc">AD</emph> <lb/>C. V. FRAN. XAVERIUM <lb/>AINSCOM. S.I.</head> <head xml:id="echoid-head34" xml:space="preserve">EPISTOLA,</head> <head xml:id="echoid-head35" xml:space="preserve">Qua diluuntur ea quibus Ε’ξε{τα}{σι}ς Cyclometriæ <lb/>Gregorii à S<emph style="super">to</emph>. Vincentio impugnata fuit.</head> <pb file="0050" n="54"/> <pb file="0051" n="55"/> <figure> <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0051-01"/> </figure> </div> <div xml:id="echoid-div49" type="section" level="1" n="21"> <head xml:id="echoid-head36" xml:space="preserve">CHRISTIANI HUGENII, <lb/><emph style="sc">Const</emph>. F. <lb/><emph style="sc">AD</emph> <lb/>C. V. FRAN. XAVERIUM <lb/>AINSCOM. S. I.</head> <head xml:id="echoid-head37" xml:space="preserve">EPISTOLA.</head> <head xml:id="echoid-head38" xml:space="preserve"><emph style="sc">Cl</emph>. <emph style="sc">Viro</emph> D°. XAVERIO AINSCOM <lb/>CHRISTIANUS HUGENIUS S. D.</head> <p> <s xml:id="echoid-s876" xml:space="preserve">LIber ille quem non ita pridem tuo no-<lb/>mine huc miſit Apelles veſter Segerus, <lb/>tam mihi acceptus fuit, Vir Clariſſ. <lb/></s> <s xml:id="echoid-s877" xml:space="preserve">quam ſolent eſſe ea quorum diutina ex-<lb/>pectatio deſiderium auget. </s> <s xml:id="echoid-s878" xml:space="preserve">Jam diu enim <lb/>intellexeram te Quadraturæ Vincentia-<lb/>næ patrocinium ſuſcepiſſe, noviſſiméque <lb/>& </s> <s xml:id="echoid-s879" xml:space="preserve">Lovanio & </s> <s xml:id="echoid-s880" xml:space="preserve">Româ ſignificatum fuerat <lb/>opus illud jam penè à te ad umbilicum <lb/>perductum, in quo pars etiam quædam noſtræ Exetaſi dica-<lb/>ta eſſet. </s> <s xml:id="echoid-s881" xml:space="preserve">Itaque cum avide totum commentarium tuum evol-<lb/>vi, tum accuratius reliquis illa expendi quæ propius ad me <lb/>pertinebant. </s> <s xml:id="echoid-s882" xml:space="preserve">De quibus quid viſum fuerit breviter tibi perſcri-<lb/>bere conſtitui. </s> <s xml:id="echoid-s883" xml:space="preserve">Equidem miratus ſum, cum me non ultimum <lb/>inter eos recenſeas qui cæteris ſolidius in examinanda Qua-<lb/>dratura veſtra verſati ſint, poſtea tamen adeo nihili animad-<lb/>verſiones omnes meas, omniáque argumenta prædicare, ut <lb/>quod convellere nituntur, id ne attingant quidem. </s> <s xml:id="echoid-s884" xml:space="preserve">Nempe <lb/>ego totâ viâ, totoque, quod ajunt, cœlo erravi, quemque <lb/>refutare volui, ejus mentem minime ſum aſſecutus. </s> <s xml:id="echoid-s885" xml:space="preserve">Verun-<lb/>tamen Viri Doctiſſimi funditus evertiſſe me commenta veſtra <lb/>pronunciavere, quorum judiciis, etſi vos fortaſſe non ſtatis, <lb/>apud intelligentes tamen multo pluris futura reor quam eo- <pb o="344" file="0052" n="56" rhead="CHRIST. HUGENII AD XAVERIUM"/> rum qui vobis de reperta Quadratura gratulantur. </s> <s xml:id="echoid-s886" xml:space="preserve">E ſocie-<lb/>tate veſtra Vir eximius A. </s> <s xml:id="echoid-s887" xml:space="preserve">Tacquetus, accurate ſibi lectam <lb/>eſſe m@ltumque prohari Exetaſin noſtram reſcripſit, & </s> <s xml:id="echoid-s888" xml:space="preserve">recte <lb/>me urgere autorem Quadraturæ, ut exhibeat, quoties ratio <lb/>prima contineat ſecundam & </s> <s xml:id="echoid-s889" xml:space="preserve">ſecunda tertiam, idque niſipræ-<lb/>ſtet, tertiam incognitam explicaturum nunquam, ac proinde <lb/>non daturum quadraturam, quæ à notitia tertiæ illius ratio-<lb/>nis dependet. </s> <s xml:id="echoid-s890" xml:space="preserve">Alter item apud vos eſt Clariſſimus Gutſcho-<lb/>vius, quem paſſim profiteri ſcio magnos P. </s> <s xml:id="echoid-s891" xml:space="preserve">Gregorii cona-<lb/>tus noſtrâ operâ penitus concidiſſe. </s> <s xml:id="echoid-s892" xml:space="preserve">Neque aliter ſentit Vir <lb/>undiquaque Doctiſſimus & </s> <s xml:id="echoid-s893" xml:space="preserve">in Academia Oxonienſi Mathe-<lb/>matum Profeſſor J. </s> <s xml:id="echoid-s894" xml:space="preserve">Walliſius, idque publice teſtatum fecit <lb/>in edito nuper ſubtiliſſimo opere de Infinitorum Arithmeti-<lb/>ca. </s> <s xml:id="echoid-s895" xml:space="preserve">Poſſemque & </s> <s xml:id="echoid-s896" xml:space="preserve">alios complures referre quorum pro me fa-<lb/>cit calculus, ni perſuaſum haberem in re Geometrica ratio-<lb/>nibus magis quam autoritate agendum. </s> <s xml:id="echoid-s897" xml:space="preserve">Neque enim dubito <lb/>quin dicturus ſis, eodem mecum errore ductos qui mihi ap-<lb/>plaudunt, ipſos quoque nihilo rectius penetraſſe ſenſa au-<lb/>toris tui. </s> <s xml:id="echoid-s898" xml:space="preserve">Quare id agam potius, ut procul à me ſimul atque <lb/>illis hanc, ſive inſcitiæ, ſive oſcitantiæ culpam amoliar. </s> <s xml:id="echoid-s899" xml:space="preserve">Prius <lb/>autem ad alia quoque nonnulla quæ mihi objicis reſponden-<lb/>dum opinor. </s> <s xml:id="echoid-s900" xml:space="preserve">Variis allatis conjecturis veriſimile reddere co-<lb/>natus eram, ex quatuor quadraturis eam à vobis præferri quæ <lb/>prima ponitur. </s> <s xml:id="echoid-s901" xml:space="preserve">Hoc ita refutas, ut, quod ego præcipuum <lb/>argumentum dixeram, diſſimules prætereáſque. </s> <s xml:id="echoid-s902" xml:space="preserve">Verum per <lb/>me licet ut quo loco vobis viſum erit primam quadraturam <lb/>habeatis. </s> <s xml:id="echoid-s903" xml:space="preserve">Ego me abunde præſtitiſſe arbitrabor ſi hancabſur-<lb/>dam eſſe evincam: </s> <s xml:id="echoid-s904" xml:space="preserve">cuique hoc planum fecero, eum non pu-<lb/>to reliquarum trium confutationem expetiturum, imo, ſi <lb/>offeratur, ne lecturum quidem. </s> <s xml:id="echoid-s905" xml:space="preserve">Etenim quod iiſdem omnes <lb/>principiis innitantur, Proportionalitatum nimirum doctrinæ <lb/>atque ei quæ eſt de ductibus plani in planum, tam certum eſt, <lb/>ut negari nulla ratione poſſit. </s> <s xml:id="echoid-s906" xml:space="preserve">Negas tu tamen hoc, crebro-<lb/>que inculcas, in prima hac quadratura, proportionalitatum <lb/>conſideratione, non uti autorem tuum. </s> <s xml:id="echoid-s907" xml:space="preserve">Sed miror qua fron-<lb/>te; </s> <s xml:id="echoid-s908" xml:space="preserve">cum non ignores utique propoſitionem 12. </s> <s xml:id="echoid-s909" xml:space="preserve">39. </s> <s xml:id="echoid-s910" xml:space="preserve">& </s> <s xml:id="echoid-s911" xml:space="preserve">40. <lb/></s> <s xml:id="echoid-s912" xml:space="preserve">libri 10. </s> <s xml:id="echoid-s913" xml:space="preserve">ex 8<emph style="super">va</emph>. </s> <s xml:id="echoid-s914" xml:space="preserve">ejuſdem libri demonſtrari, hanc vero per 114.</s> <s xml:id="echoid-s915" xml:space="preserve"> <pb o="345" file="0053" n="57" rhead="EPISTOLA."/> lib. </s> <s xml:id="echoid-s916" xml:space="preserve">8. </s> <s xml:id="echoid-s917" xml:space="preserve">qui totus eſt de Proportionalitatibus.</s> <s xml:id="echoid-s918" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s919" xml:space="preserve">Porro ſuperfluam me ais operam ſumpſiſſe, cum priores <lb/>duas corporum rationes numero exhibui, ex quibus tertia <lb/>vobis definienda erat; </s> <s xml:id="echoid-s920" xml:space="preserve">illas enim autor operis Geometrici, ſi <lb/>credimus, multo antè quàm ego edidiſſem, imo quàm ipſe <lb/>editus eſſem, perſpectas habuit aliiſque demonſtravit. </s> <s xml:id="echoid-s921" xml:space="preserve">Quæ-<lb/>ſo curnon explicuit igitur, noſque ea levavit moleſtia? </s> <s xml:id="echoid-s922" xml:space="preserve">Nam <lb/>certum erat plurimum ad abſolvendam quadraturam, ſi mo-<lb/>do abſolvi poſſet, eorum notitiam conferre debere, planéque <lb/>eſſe neceſſariam. </s> <s xml:id="echoid-s923" xml:space="preserve">Sed vobis cuncta perinde nota dici video <lb/>quæ cognoſci poſſe aliquâ ſaltem ratione imaginamini, at-<lb/>que ea quæ liquido comperta fuerint. </s> <s xml:id="echoid-s924" xml:space="preserve">Itaque ad propoſ. </s> <s xml:id="echoid-s925" xml:space="preserve">43. <lb/></s> <s xml:id="echoid-s926" xml:space="preserve">lib. </s> <s xml:id="echoid-s927" xml:space="preserve">10. </s> <s xml:id="echoid-s928" xml:space="preserve">me remittis, in qua utramque rationem notam fieri <lb/>aſſeris. </s> <s xml:id="echoid-s929" xml:space="preserve">Illa vero non magis ipſas expedit quam propoſitio <lb/>poſtrema ejuſdem libri, rationem quæ ſit inter circulum & </s> <s xml:id="echoid-s930" xml:space="preserve"><lb/>quadratum diametri. </s> <s xml:id="echoid-s931" xml:space="preserve">Prorsus huic ſimile eſt quod de Para-<lb/>bolica ungula reſpondes. </s> <s xml:id="echoid-s932" xml:space="preserve">Videlicet jam à triginta annis ex-<lb/>ploratum habuiſſe autorem tuum, quænam ſit illius ad cy-<lb/>lindrum ſuum proportio. </s> <s xml:id="echoid-s933" xml:space="preserve">Equidem ex iis quæ jam tradide-<lb/>rat, erui illam poſſe faſſus ſum; </s> <s xml:id="echoid-s934" xml:space="preserve">ipſum vero adhuc cujuſmo-<lb/>di foret neſciviſſe, ſatis evidens argumentum videbatur, <lb/>quod eam non expromeret. </s> <s xml:id="echoid-s935" xml:space="preserve">Neque enim credibile, cujus theo-<lb/>rematis gratia duodeviginti propoſitiones elucubraſſet, id <lb/>tanquam ſuperfluum non eſſe adſcripturum, ſi tam nullo ne-<lb/>gotio inveniri poſſe ſperaret. </s> <s xml:id="echoid-s936" xml:space="preserve">Parum intererat utrum propo-<lb/>ſitione illud dignatus fuiſſet, (quod noluiſſe eum dicis) an <lb/>corollario tantum. </s> <s xml:id="echoid-s937" xml:space="preserve">Sed nec in corollario ratio illa uſpiam ex-<lb/>preſſa eſt. </s> <s xml:id="echoid-s938" xml:space="preserve">Nam in eo quod adducis, hoc ſolum legitur, <lb/>methodum traditam eſſe qua ratio ungulæ ad cylindrum quo <lb/>continetur, inveſtigari queat, eamque notam fore, ſi quo-<lb/>rundam inter ſe corporum rationes inventæ fuerint. </s> <s xml:id="echoid-s939" xml:space="preserve">Atqui & </s> <s xml:id="echoid-s940" xml:space="preserve"><lb/>horum corporum rationes, & </s> <s xml:id="echoid-s941" xml:space="preserve">ex iis quæ ſit inter ungulam <lb/>cylindrumque ſuum analogia, lectoribus diſquirenda relin-<lb/>quuntur: </s> <s xml:id="echoid-s942" xml:space="preserve">idque ipſe non neſcis. </s> <s xml:id="echoid-s943" xml:space="preserve">Quare non ſatis ingenue hic <lb/>me diſſimulationis arguis, ubi ipſe contra quam ſentias, ſcri-<lb/>bere videaris.</s> <s xml:id="echoid-s944" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s945" xml:space="preserve">Jam vero de palmari errore quem mihi impingis videa- <pb o="346" file="0054" n="58" rhead="CHRIST. HUGENII AD XAVERIUM"/> mus. </s> <s xml:id="echoid-s946" xml:space="preserve">Is circa verbum continere commiſſus eſt, ex quo non <lb/>recte percepto factum eſt ſcilicet, ut, cum Quadraturam ve-<lb/>ſtram oppugnare me crederem, nihil minus egerim, omnes-<lb/>que item, qui me labefeciſſe eam judicarunt, cæcutierint. <lb/></s> <s xml:id="echoid-s947" xml:space="preserve">Ego ſignificationem duplicem ejus verbi quam in opere Ge-<lb/>ometrico inveneram, adduxi, tuam, quæ & </s> <s xml:id="echoid-s948" xml:space="preserve">P. </s> <s xml:id="echoid-s949" xml:space="preserve">Sarraſæ eſt, <lb/>interpretationem, quoniam adhuc ignorabam, præterii. </s> <s xml:id="echoid-s950" xml:space="preserve">Igi-<lb/>tur hic palmaris eſt error meus, quod nec P. </s> <s xml:id="echoid-s951" xml:space="preserve">Sarraſæ librum, <lb/>nec tuum Corollarium tum temporis videram. </s> <s xml:id="echoid-s952" xml:space="preserve">Sed nec for-<lb/>taſſe ſi ſciviſſem explicationem veſtram, propterea memo-<lb/>randam duxiſſem, cum parum adeo ad rem faciat, ſitque <lb/>monſtroſa plane atque abſona, uti ex adjecto ſpecimine li-<lb/>quebit: </s> <s xml:id="echoid-s953" xml:space="preserve">quantum vero ea promoveritis deinde exponam. </s> <s xml:id="echoid-s954" xml:space="preserve"><lb/>Propoſitio 40. </s> <s xml:id="echoid-s955" xml:space="preserve">lib. </s> <s xml:id="echoid-s956" xml:space="preserve">10. </s> <s xml:id="echoid-s957" xml:space="preserve">eſt hujuſmodi. </s> <s xml:id="echoid-s958" xml:space="preserve">Iiſdempoſitis, dicora-<lb/>tionem ſolidiex R S in X Y ad ſolidum ex T V in Z &</s> <s xml:id="echoid-s959" xml:space="preserve">, to-<lb/>ties continere rationem ſolidiex I K in N O ad ſolidum ex L M <lb/>in P Q, quoties hæc ipſa ratio continet rationem ſolidi ex A B <lb/>in E F ad ſolidum ex C D in G H. </s> <s xml:id="echoid-s960" xml:space="preserve">Quam propoſitionem jux-<lb/>ta mentem, ut ais, autoris, (variatâ tantùm phraſi ſcilicet) <lb/>ſic nobis enarras. </s> <s xml:id="echoid-s961" xml:space="preserve">Iiſdem poſitis, dico rationem ſolidi ex R S <lb/>in X Y ad ſolidum ex T V in Z &</s> <s xml:id="echoid-s962" xml:space="preserve">, conſtitui ex iis rationibus <lb/>quæ toties multiplicatæ ſunt illarum rationum ex quibus con-<lb/>ſtituitur ratio ſolidi ex I K in N O ad ſolidum ex L M in <lb/>P Q, quoties hæ ipſæ rationes multiplicatæ ſunt earum ex qui-<lb/>bus conſtituitur ratio ſolidi ex A B in E F ad ſolidum ex C D <lb/>in G H.</s> <s xml:id="echoid-s963" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s964" xml:space="preserve">Pulchra vero explanatio! quam quia ego pervidere non <lb/>valui, ſenſum convenientem ratiociniis veſtris non percepi. <lb/></s> <s xml:id="echoid-s965" xml:space="preserve">At cui hoc in mentem veniret, Mathematicum longe aliud <lb/>ſcribere quam intelligi poſtulet? </s> <s xml:id="echoid-s966" xml:space="preserve">quiſve magis adhuc intri-<lb/>catum ſenſum theorematibus jam nunc nimium obſcuris af-<lb/>fingere vellet? </s> <s xml:id="echoid-s967" xml:space="preserve">Omnes profecto qui vobis controverſiam mo-<lb/>verunt, haud aliter atque ego, verbum continere accepiſſe <lb/>noſti, neque ulli hoc incidiſſe, ut cum de ratione inter duas <lb/>magnitudines legeret, id ad partiales referret, ex quibus to-<lb/>tales conſtituerentur. </s> <s xml:id="echoid-s968" xml:space="preserve">Ecce vero ut præter eos quorum ani-<lb/>madverſiones ad manus veſtras pervenere, eadem plane quæ <pb o="347" file="0055" n="59" rhead="EPISTOLA."/> nobis, circa has propoſitiones & </s> <s xml:id="echoid-s969" xml:space="preserve">ſignificationem verbi continere, <lb/>opinio fuit Incomparabili Carteſio, quem ſi minus inſignem <lb/>Geometram quam Algebriſtam fuiſſe arbitraris, parum ex ve-<lb/>ro judicas. </s> <s xml:id="echoid-s970" xml:space="preserve">Ejus ad amicum epiſtolæ copia mihi facta eſt, cum <lb/>jam diu exetaſis noſtra prodiiſſet, quâ quoniam non tantum id <lb/>quod dixi comprobatur, ſed & </s> <s xml:id="echoid-s971" xml:space="preserve">tota inſuper ad opus Geometri-<lb/>cum P. </s> <s xml:id="echoid-s972" xml:space="preserve">à S<emph style="super">to</emph>. </s> <s xml:id="echoid-s973" xml:space="preserve">Vincentio pertinet, integram hic adſcribere vi-<lb/>ſum eſt. </s> <s xml:id="echoid-s974" xml:space="preserve">Gallicè ſic habet.</s> <s xml:id="echoid-s975" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s976" xml:space="preserve"><emph style="sc">Monsieur</emph>.</s> <s xml:id="echoid-s977" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s978" xml:space="preserve">J’Aygardé vos livres un peulong temps, pource que je deſirois <lb/>en vous les renvoyant, vous rendre compte de la Quadrature <lb/>du cercle pretendue, & </s> <s xml:id="echoid-s979" xml:space="preserve">j’ avois bien de la peine à me reſoudre de <lb/>feuilleter tout le gros volume qui en traite. </s> <s xml:id="echoid-s980" xml:space="preserve">En fin j’ en ay veu <lb/>quelque choſe & </s> <s xml:id="echoid-s981" xml:space="preserve">aſſez ce me ſemble pour pouvoir dire qu’ilne conti-<lb/>ent rien de bon qui ne ſoit facile, & </s> <s xml:id="echoid-s982" xml:space="preserve">qu’on ne puſt eſcrire tout en <lb/>une ou deux pages. </s> <s xml:id="echoid-s983" xml:space="preserve">Lereſte n’ eſt qu’un paralogiſme touchant la <lb/>Quadrature du cercle, enveloppé en quantité de propoſitions qui <lb/>ne ſervent qu’ à embroüiller la matiere, & </s> <s xml:id="echoid-s984" xml:space="preserve">ſont tresſimples & </s> <s xml:id="echoid-s985" xml:space="preserve"><lb/>facilles pour la pluspart, bien que la façon dont il les traite, les <lb/>face paroiſtre un peu obſcures. </s> <s xml:id="echoid-s986" xml:space="preserve">Pour trouver ſon paralogiſme, j’ai <lb/>commencé par la 1134<emph style="super">e</emph> page, ou il dit: </s> <s xml:id="echoid-s987" xml:space="preserve">Nota autem eſt propor-<lb/>tio ſegmenti L M N K ad ſegmentum E G H F, ce qui eſt faux, <lb/>& </s> <s xml:id="echoid-s988" xml:space="preserve">la preuve qu’il en donne eſt fondée ſur la 39<emph style="super">e</emph> propoſition en la <lb/>page 1121. </s> <s xml:id="echoid-s989" xml:space="preserve">du meſme livre, ou ily a une erreur tresmanifeſte, <lb/>qui conſiſte en ce qu’il veut appliquer à pluſieurs quantitez conjo-<lb/>intes ce qu’il a prouvé auparavant des meſmes quantitez divisées. <lb/></s> <s xml:id="echoid-s990" xml:space="preserve">Carpar exemple, ayant les 4. </s> <s xml:id="echoid-s991" xml:space="preserve">ordres de proportionelles <lb/> <anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/> bien qu’il ſoit vray que 8. </s> <s xml:id="echoid-s992" xml:space="preserve">eſt à 32. </s> <s xml:id="echoid-s993" xml:space="preserve">en raiſon duoblée de ce que 4. <lb/></s> <s xml:id="echoid-s994" xml:space="preserve">eſt à 8. </s> <s xml:id="echoid-s995" xml:space="preserve">Et que 18. </s> <s xml:id="echoid-s996" xml:space="preserve">eſt auſſi à 50. </s> <s xml:id="echoid-s997" xml:space="preserve">en raiſon doublée de ce que 6. </s> <s xml:id="echoid-s998" xml:space="preserve"><lb/>eſt à 10. </s> <s xml:id="echoid-s999" xml:space="preserve">il n’ eſt pas vray pour cela que 8 + 18. </s> <s xml:id="echoid-s1000" xml:space="preserve">c’ eſt à dire 26. </s> <s xml:id="echoid-s1001" xml:space="preserve">ſo-<lb/>it à 32 + 50. </s> <s xml:id="echoid-s1002" xml:space="preserve">c’ eſt à dire 82. </s> <s xml:id="echoid-s1003" xml:space="preserve">en raiſon double de celle qui eſt entre <lb/>4 + 6. </s> <s xml:id="echoid-s1004" xml:space="preserve">c’ eſt à dire 10, & </s> <s xml:id="echoid-s1005" xml:space="preserve">8 + 10, c’ eſt à dire 18. </s> <s xml:id="echoid-s1006" xml:space="preserve">Tous ſesraiſon-<lb/>nements ne ſont fondez que ſur cettefaute, & </s> <s xml:id="echoid-s1007" xml:space="preserve">ce qu’il eſcrit de <pb o="348" file="0056" n="60" rhead="CHRIST. HUGENII AD XAVERIUM"/> Proportionalitatibus & </s> <s xml:id="echoid-s1008" xml:space="preserve">de Ductibus, ne ſert qu’ à l’embaraſſer, <lb/>& </s> <s xml:id="echoid-s1009" xml:space="preserve">ne me ſemble d’ aucun uſage, pour ce que fruſtra fit per plur a <lb/>quod poteſt fieri per pauciora.</s> <s xml:id="echoid-s1010" xml:space="preserve"/> </p> <div xml:id="echoid-div49" type="float" level="2" n="1"> <note position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve"> <lb/>2, # 4, # 8, # 2, # 8, # 32. <lb/>###### & <lb/>2, # 6, # 18, # 2, # 10, # 50. <lb/></note> </div> <p> <s xml:id="echoid-s1011" xml:space="preserve">Quorum latinè hæc eſt ſententia.</s> <s xml:id="echoid-s1012" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1013" xml:space="preserve">Libros tuos retinui diutius, quod remittere eos nolebam quin <lb/>ſimul opinionem meam tibi exponerem de nova iſta quam vendi-<lb/>tant circuli Quadratura; </s> <s xml:id="echoid-s1014" xml:space="preserve">vix autem à me ipſo impetr are poteram, <lb/>ut ingentia quibus tractatur volumina evolverem. </s> <s xml:id="echoid-s1015" xml:space="preserve">Tandem ta-<lb/>men nonnulla in iis delibavi, è quibus ſatis tutò mihi pronunciare <lb/>poſſe videor, nihil ibi boni inveniri, quod non captu facile ſit; <lb/></s> <s xml:id="echoid-s1016" xml:space="preserve">unâque aut alter â paginâ explicari potuerit. </s> <s xml:id="echoid-s1017" xml:space="preserve">Cætera merum pa-<lb/>ralogiſmum de quadratura circuli continent, multis propoſitio-<lb/>nibus implicitum, quæque hoc tantum efficiunt, ut omnia eva-<lb/>dant intricatior a. </s> <s xml:id="echoid-s1018" xml:space="preserve">Pler æque vero ſimpliciſſimæ ſunt & </s> <s xml:id="echoid-s1019" xml:space="preserve">facili ratio-<lb/>ne conſtant, licet tractandi methodus obſcuriores reddiderit. </s> <s xml:id="echoid-s1020" xml:space="preserve">Pa-<lb/>ralogiſmum quærere inſtitui, initio facto ad paginam 1134. </s> <s xml:id="echoid-s1021" xml:space="preserve">ubi <lb/>hoc ait: </s> <s xml:id="echoid-s1022" xml:space="preserve">Nota autem eſt proportio ſegmenti L M N K ad ſegmen-<lb/>tum E G H F; </s> <s xml:id="echoid-s1023" xml:space="preserve">quod falſum eſt, pendet enim hujus demonſtratio <lb/>à propoſitione 39, pag. </s> <s xml:id="echoid-s1024" xml:space="preserve">1121. </s> <s xml:id="echoid-s1025" xml:space="preserve">ejuſdem libri, ubi manifeſtus er-<lb/>ror occurrit, dum pluribus quantitatibus conjunctim applicatur, <lb/>quod de ſingulis ſeorſim fuerat oſtenſum. </s> <s xml:id="echoid-s1026" xml:space="preserve">Etenim ex. </s> <s xml:id="echoid-s1027" xml:space="preserve">gr. </s> <s xml:id="echoid-s1028" xml:space="preserve">poſitis <lb/>quatuor proportionalium ordinibus <lb/> <anchor type="note" xlink:label="note-0056-01a" xlink:href="note-0056-01"/> licet verum ſit rationem 8. </s> <s xml:id="echoid-s1029" xml:space="preserve">ad 32. </s> <s xml:id="echoid-s1030" xml:space="preserve">duplicatam eſſe ejus quæ 4. </s> <s xml:id="echoid-s1031" xml:space="preserve">ad <lb/>8. </s> <s xml:id="echoid-s1032" xml:space="preserve">itemque rationem 18. </s> <s xml:id="echoid-s1033" xml:space="preserve">ad 50. </s> <s xml:id="echoid-s1034" xml:space="preserve">duplicatam eſse ejus quæ 6. </s> <s xml:id="echoid-s1035" xml:space="preserve">ad 10. <lb/></s> <s xml:id="echoid-s1036" xml:space="preserve">non tamen idcirco verum eſt 8 + 18. </s> <s xml:id="echoid-s1037" xml:space="preserve">hoc eſt, 26. </s> <s xml:id="echoid-s1038" xml:space="preserve">eſſe ad 32 + 50. </s> <s xml:id="echoid-s1039" xml:space="preserve"><lb/>hoc eſt ad 82. </s> <s xml:id="echoid-s1040" xml:space="preserve">inratione duplicata ejus quæ 4 + 6. </s> <s xml:id="echoid-s1041" xml:space="preserve">hoc eſt, 10. </s> <s xml:id="echoid-s1042" xml:space="preserve">ad <lb/>8 + 10. </s> <s xml:id="echoid-s1043" xml:space="preserve">hoc eſt, 18. </s> <s xml:id="echoid-s1044" xml:space="preserve">Unicum ei fundamentum hæc vitioſa argu-<lb/>mentatio; </s> <s xml:id="echoid-s1045" xml:space="preserve">quæque de Proportionalitatibus ſcribit & </s> <s xml:id="echoid-s1046" xml:space="preserve">de Ducti-<lb/>bus, tantum majoribus ipſum diſſicultatibus involvunt, neque <lb/>alicujus uſus videntur, ſiquidem fruſtra fit per plura quod poteſt <lb/>fieri per pauciora.</s> <s xml:id="echoid-s1047" xml:space="preserve"/> </p> <div xml:id="echoid-div50" type="float" level="2" n="2"> <note position="right" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve"> <lb/>2, 4, 8, # 2, 8, 32, <lb/>## & <lb/>2, 6, 18, # 2, 10, 50, <lb/></note> </div> <p> <s xml:id="echoid-s1048" xml:space="preserve">Vides, Vir Egregie, neque Carteſium, veſtrum illud Hoc <lb/>eſt juxta mentem autoris, agniturum fuiſſe, ſed potius, quod <lb/>res eſt, dicturum, deſperatâ causâ hoc vobis effugium quæſi-<lb/>tum, ut quadratura veſtra ad inſtar Protei cujuſdem aliâ atque <pb o="349" file="0057" n="61" rhead="EPISTOLA."/> aliâ aſſumptâ formâ quantum libet arctè ſeſe conſtringentibus e-<lb/>laberetur. </s> <s xml:id="echoid-s1049" xml:space="preserve">Verum age, inſpiciamus jam quo rem deducas, poſt-<lb/>eaquam verbi continere novam ſignificationem elicuiſti, eâque <lb/>vetera theoremata tam ſcitè interpolaſti. </s> <s xml:id="echoid-s1050" xml:space="preserve">In Corollario propo-<lb/>ſitionis 40. </s> <s xml:id="echoid-s1051" xml:space="preserve">lib. </s> <s xml:id="echoid-s1052" xml:space="preserve">10. </s> <s xml:id="echoid-s1053" xml:space="preserve">quò tam ſæpe provocas, id unum egiſſe vi-<lb/>deris, unas ex aliis difficultates nectendo, ut ſi quis argumen-<lb/>tationis tuæ tenorem conſectari cupiat, is defeſſus abſiſtat pri-<lb/>uſquam ad finem pervenerit. </s> <s xml:id="echoid-s1054" xml:space="preserve">Ego ad eum uſque locum te ſecu-<lb/>tus ſum, ubi ſpatia Y & </s> <s xml:id="echoid-s1055" xml:space="preserve">Z aſſumi jubes: </s> <s xml:id="echoid-s1056" xml:space="preserve">Inde non ulterius pro-<lb/>cedendum putavi. </s> <s xml:id="echoid-s1057" xml:space="preserve">Adeo enim manifeſto vitio atque ἀγεωμετρησίἀ <lb/>ibi laborat conſtructio tua, ut tibimet ipſi exploratum id eſſe du-<lb/>bitare nequeam: </s> <s xml:id="echoid-s1058" xml:space="preserve">ſed quoniam alia evadendi ratio non occurre-<lb/>bat, ſperaſti, credo, in tanta obſcuritate nemini illud facile <lb/>animadverſum iri. </s> <s xml:id="echoid-s1059" xml:space="preserve">Dein, inquis, aſſumantur duo plana Hyper-<lb/>bolica Y & </s> <s xml:id="echoid-s1060" xml:space="preserve">Z, rectis alteri aſymptotorum parallelis incluſa. </s> <s xml:id="echoid-s1061" xml:space="preserve">Nullâ <lb/>aliâ præcautione aſſumuntur quam quod rectis alteri aſympto-<lb/>torum parallelis includi ea neceſſe ſit. </s> <s xml:id="echoid-s1062" xml:space="preserve">De magnitudine utriuſ-<lb/>que aut ratione quam inter ſe ſervare debeant nihil præcipis. </s> <s xml:id="echoid-s1063" xml:space="preserve">I-<lb/>gitur quamlibet magnum aut parvum unum quodque eorum <lb/>abſcindi poterit. </s> <s xml:id="echoid-s1064" xml:space="preserve">Mox tamen rationem ſpatii Y ad Z cum aliis <lb/>rationibus comparare inſtituis, quas prius ſecundum certam de-<lb/>terminationem aſſumpſiſti, tibique hoc demonſtrandum propo-<lb/>nis, Rationem totalem planorum X ad T tam eſſe multiplicem ra-<lb/>tionis totalis planorum Y ad Z, quàm ratio totalis ſolidorum G H <lb/>ad I K multiplicata eſt rationis totalis ſolidi L M ad N O. </s> <s xml:id="echoid-s1065" xml:space="preserve">Quid-<lb/>nam, quæſo, abſurdius, quam de quantitate ejus rationis ali-<lb/>quid enunciare, quæ prorſus in certa ſit ac vaga? </s> <s xml:id="echoid-s1066" xml:space="preserve">Equidem ex <lb/>hoc ſolo ſatis liquere puto, quam fruſtra primæ Quadraturæ <lb/>ſuppetias ferre tentaveris, cum in eo quod præcipuè tibi ex-<lb/>plicandum erat, tam inſigniter delinquas. </s> <s xml:id="echoid-s1067" xml:space="preserve">In tribus reliquis an <lb/>meliore fortunâ uſus ſis, ſi me inquirere oporteat, talentum <lb/>non meream. </s> <s xml:id="echoid-s1068" xml:space="preserve">Id tamen ſcito perpetuum adverſus vos argumen-<lb/>tum fore, quod rationem peripheriæ ad diametrum quam ſin-<lb/>gulis quadraturis datam eſſe profitemini, ipſi tamen exhibere <lb/>non poteſtis; </s> <s xml:id="echoid-s1069" xml:space="preserve">non autor ipſe Quadraturæ, non tot ejus diſci-<lb/>puli, qui tot jam annis in id incumbunt, ut paucioribus Ilium <lb/>expugnatum ſit. </s> <s xml:id="echoid-s1070" xml:space="preserve">Datam eſſe rationem, Euclides definivit, cui <pb o="350" file="0058" n="62" rhead="CHR. HUGENII AD XAVER. EPIST."/> poſſumus æqualem invenire. </s> <s xml:id="echoid-s1071" xml:space="preserve">Quis autem ad veſtram illam hoc <lb/>pertinere credet, quæ irrito labore toto decennio quæſita eſt? <lb/></s> <s xml:id="echoid-s1072" xml:space="preserve">Nam quod ſufficere exiſtimatis ſi modo viam commonſtraveri-<lb/>tis quâ emensâ ad quæſitum perveniatur, obſtacula vero, at-<lb/>que innumeras difficultates quibus præſepta eſt, non remove-<lb/>tis, videte cui perſuadere poſſitis, eâratione tetragoniſmi ne-<lb/>gotium à vobis confectum eſſe. </s> <s xml:id="echoid-s1073" xml:space="preserve">Illud ſanè vos conſequi apparet, <lb/>ut, dum ultra non proceditis, minus expoſiti ſitis ad promi-<lb/>ſcuos omnium inſultus, difficilius etiam à peritioribus oppu-<lb/>gnemini, paratioremque habeatis receptum. </s> <s xml:id="echoid-s1074" xml:space="preserve">Facile enim acrius <lb/>inſtantes proportionum & </s> <s xml:id="echoid-s1075" xml:space="preserve">proportionalitatum veſtrarum tene-<lb/>bris involvere poteſtis, atque eſſicere ut tandem veluti nox præ-<lb/>lium dirimat. </s> <s xml:id="echoid-s1076" xml:space="preserve">Hoc ipſum ne mihi eveniret, cum exetaſin Qua-<lb/>draturæ conſcriberem, metuebam, atque ut caverem operam <lb/>dedi; </s> <s xml:id="echoid-s1077" xml:space="preserve">id unum conatus, ut, quatenus fieri poſſet, autorem ad <lb/>abſurdum compellerem, nimirum ut vel nolle ſe vel non poſſe <lb/>Quadraturam ſuam abſolvere fateretur. </s> <s xml:id="echoid-s1078" xml:space="preserve">Eo fine ignota prius at-<lb/>que informia corpora dimenſus ſum, exhibitiſque prioribus <lb/>duabus ſolidorum proportionibus, petii ut inde tertiam elice-<lb/>ret, utpote quam cognitis illis notam dixiſſet. </s> <s xml:id="echoid-s1079" xml:space="preserve">Ad quas angu-<lb/>ſtias redactum non aliâ ratione defendis, quam expoſtulando <lb/>mecum quod autori tuo modum præſcribere præſumam qua-<lb/>drandi circulum, ac jubendo denique ut meminerim quid & </s> <s xml:id="echoid-s1080" xml:space="preserve">cui <lb/>ſcribam. </s> <s xml:id="echoid-s1081" xml:space="preserve">Ego verò quomodo quadratus fiat circulus, nec di-<lb/>dici, nec præſcribo; </s> <s xml:id="echoid-s1082" xml:space="preserve">ſed hoc urgeo; </s> <s xml:id="echoid-s1083" xml:space="preserve">ut quem ille modum ſe <lb/>inveniſſe contendit, eum reapſe utilem & </s> <s xml:id="echoid-s1084" xml:space="preserve">efficacem eſſe demon-<lb/>ſtret. </s> <s xml:id="echoid-s1085" xml:space="preserve">Atque ita, quid ſcripſerim & </s> <s xml:id="echoid-s1086" xml:space="preserve">in quem finem, me non <lb/>neſciviſſe, ſatis jam tibi conſtare arbitror. </s> <s xml:id="echoid-s1087" xml:space="preserve">Cui vero ſcripſerim, <lb/>ne hoc quidem puto me oblitum fuiſſe. </s> <s xml:id="echoid-s1088" xml:space="preserve">Vides autem quam hac <lb/>in parte longe diverſum ſonent Carteſii literæ atque Elogia ve-<lb/>ſtra: </s> <s xml:id="echoid-s1089" xml:space="preserve">quorum utris potius ſubſcribendum ſit aliorum judicio de-<lb/>cerni malim quam meum interponere. </s> <s xml:id="echoid-s1090" xml:space="preserve">Hoc tamen autorem <lb/>Quadraturæ ſcire velim, tanto majori eruditionis & </s> <s xml:id="echoid-s1091" xml:space="preserve">candoris <lb/>opinione apud me futurum, quanto maturius ab errore ſuo reſi-<lb/>piſcet. </s> <s xml:id="echoid-s1092" xml:space="preserve">Vale.</s> <s xml:id="echoid-s1093" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1094" xml:space="preserve">Dat. </s> <s xml:id="echoid-s1095" xml:space="preserve">Hagæ Com. </s> <s xml:id="echoid-s1096" xml:space="preserve">2. </s> <s xml:id="echoid-s1097" xml:space="preserve">Oct. <lb/></s> <s xml:id="echoid-s1098" xml:space="preserve">1656.</s> <s xml:id="echoid-s1099" xml:space="preserve"/> </p> <pb file="0059" n="63"/> </div> <div xml:id="echoid-div52" type="section" level="1" n="22"> <head xml:id="echoid-head39" xml:space="preserve">CHRISTIANI HUGENII, <lb/><emph style="sc">Const.</emph> F. <lb/><emph style="sc">DE</emph> <lb/>CIRCULI <lb/>MAGNITUDINE <lb/>INVENTA.</head> <head xml:id="echoid-head40" style="it" xml:space="preserve">ACCEDUNT EJUSDEM</head> <head xml:id="echoid-head41" xml:space="preserve">Problematum quorundam illuſtrium <lb/>Conſtructiones.</head> <pb file="0060" n="64"/> <pb file="0061" n="65"/> <figure> <image file="0061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0061-01"/> </figure> </div> <div xml:id="echoid-div53" type="section" level="1" n="23"> <head xml:id="echoid-head42" xml:space="preserve">PRÆFATIO.</head> <p style="it"> <s xml:id="echoid-s1100" xml:space="preserve">CIrca antiquum Tetragoniſmiproble-<lb/>ma, quo vel apud Mathematum ig-<lb/>naros nihil eſt celebrius, recens ope-<lb/>ræ pretium nos feciſſe rati, & </s> <s xml:id="echoid-s1101" xml:space="preserve">quæ-<lb/>dam hactenus compertis meliora ut <lb/>putamus conſecuti, Geometris ea de-<lb/>monſtrata impertiri volumus. </s> <s xml:id="echoid-s1102" xml:space="preserve">Nam-<lb/>que & </s> <s xml:id="echoid-s1103" xml:space="preserve">ſtudiis eorum profutura arbitramur, & </s> <s xml:id="echoid-s1104" xml:space="preserve">no-<lb/>vitate ipſâ ad rerum abditarum inveſtigationem <lb/>incitamento futura, reputantibus in eo quoque ar-<lb/>gumento, ubi omnes pridem ſummâ contentione ver-<lb/>ſati ſint, aliqua ſuperfuiſſe haud indigna diligen-<lb/>tiæ præmia. </s> <s xml:id="echoid-s1105" xml:space="preserve">Plurimi quidem antehac inventæ Qua-<lb/>draturæ gloriam ſibi aſſerere conati ſunt, varia-<lb/>que ſubinde commenta protulere, falſis vera mi-<lb/>ſcentes. </s> <s xml:id="echoid-s1106" xml:space="preserve">Verum à peritioribus omnia veleverſa fuiſ-<lb/>ſe vel contempta ſcimus, neque aliud adhuc rece-<lb/>ptum, quo omnis circuli dimenſio niteretur, præ-<lb/>ter unum illud, majorem eſſe eum inſcripto ſibi <lb/>polygono, circumſcripto minorem. </s> <s xml:id="echoid-s1107" xml:space="preserve">Nos autem pro-<lb/>piorem determinationem nunc exhibemus oſtendi-<lb/>muſque, quod duobus ſumptis polygonis propor-<lb/>tione mediis inter inſcriptum circumſcriptum-<lb/>que ipſis ſimile, minoris eorum perimeter cir-<lb/>cumferentia circuli major exiſtit, reliquum ve- <pb file="0062" n="66" rhead="PRÆFATIO."/> ro polygonum eâdem proportione circuli are-<lb/>am exuperat. </s> <s xml:id="echoid-s1108" xml:space="preserve">Et hoc quidem ut inter ea quæ de-<lb/>monſtraturi ſumus & </s> <s xml:id="echoid-s1109" xml:space="preserve">difficillimum & </s> <s xml:id="echoid-s1110" xml:space="preserve">contempla-<lb/>tione præcipue dignum videatur, alia tamen ſunt <lb/>non accuratiora modò, ſed quæ & </s> <s xml:id="echoid-s1111" xml:space="preserve">uſu magis pro-<lb/>bentur; </s> <s xml:id="echoid-s1112" xml:space="preserve">quæ ſane hic in anteceſſum non recenſebi-<lb/>mus, quippe in ſequentibus rectius percipienda. </s> <s xml:id="echoid-s1113" xml:space="preserve">Bre-<lb/>viter tamen quid ſtudiis Geometriæ conferant ex-<lb/>poſuiſſe proderit, cum non minimam habeant utili-<lb/>tatis commendationem. </s> <s xml:id="echoid-s1114" xml:space="preserve">Cum igitur duplicem propo-<lb/>ſiti tractationem inſtituerimus, primum eatraden-<lb/>do quorum demonſtratio conſuetis Geometriæ ele-<lb/>mentis contenta eſt, deinde centrorum gravitatis <lb/>quoque conſiderationem adhibendo: </s> <s xml:id="echoid-s1115" xml:space="preserve">in prioribus <lb/>quidem illud explicatum reperietur, quomodo non <lb/>tantum circumferentiæ toti, ſed & </s> <s xml:id="echoid-s1116" xml:space="preserve">arcui cuilibet <lb/>dato recta linea æqualis invenienda ſit; </s> <s xml:id="echoid-s1117" xml:space="preserve">expeditâ <lb/>ratione ad Mechanicas conſtructiones, quæque vel <lb/>ſubtiliſſimas earum minime fruſtretur. </s> <s xml:id="echoid-s1118" xml:space="preserve">Quomodo <lb/>item numeros exercentibus peripheriæ ad diametrum <lb/>ratio, quam Archimedes ex polygonis laterum 96 <lb/>eruit, per dodecagona ſola comprobari queat. </s> <s xml:id="echoid-s1119" xml:space="preserve">Ex <lb/>polygonis autem laterum 10800, cum iis qui vete-<lb/>rem inſiſtunt viam vix hi peripheriæ termini exi-<lb/>ſtant 62831852 & </s> <s xml:id="echoid-s1120" xml:space="preserve">62831855, ad diametrum par-<lb/>tium 20000000, noſtrâ Methodo iſti prodiiſſe cer-<lb/>nentur, 6283185307179584, 6283185307179589; <lb/></s> <s xml:id="echoid-s1121" xml:space="preserve">ſemperque duplicem obtineri verorum characterum <lb/>numerum, quacunque laterum multitudine poly- <pb file="0063" n="67" rhead="PRÆFATIO."/> gona adhibeantur. </s> <s xml:id="echoid-s1122" xml:space="preserve">Quod quidem certâ ratione con-<lb/>tingere perſpeximus ſicuti & </s> <s xml:id="echoid-s1123" xml:space="preserve">Quadratum cujuſque <lb/>numeri bis totidem quot latus characteribus ple-<lb/>rumque conſtituitur. </s> <s xml:id="echoid-s1124" xml:space="preserve">At majora etiam compendia <lb/>centrorum gravitatis proprietas ſubminiſtrat, & </s> <s xml:id="echoid-s1125" xml:space="preserve"><lb/>propius quodammodo ad perfectionem inſuperabilis <lb/>problematis per hæc acceſſiſſe videmur. </s> <s xml:id="echoid-s1126" xml:space="preserve">Certe ad <lb/>Archimedeos peripheriæ limites conſtituendos, ſo-<lb/>lo nunc inſcripti trigoni cognito latere indigemus. <lb/></s> <s xml:id="echoid-s1127" xml:space="preserve">E ſexagintangulo autem inter hoſce eam contineri <lb/>probamus 31415926538 & </s> <s xml:id="echoid-s1128" xml:space="preserve">31415926533, poſitâ <lb/>diametro partium 10000000000, cum ſolitâ me-<lb/>thodo vix iſti producantur 3145, 3140. </s> <s xml:id="echoid-s1129" xml:space="preserve">Adeo ut <lb/>triplus jam & </s> <s xml:id="echoid-s1130" xml:space="preserve">ultra ſit verarum hîc notarum nume-<lb/>rus, ſicut per præcedentia duplus; </s> <s xml:id="echoid-s1131" xml:space="preserve">& </s> <s xml:id="echoid-s1132" xml:space="preserve">perpetuo qui-<lb/>dem ſucceſſu, haud aliter quam in majoribus nu-<lb/>meris cubum ſui lateris triplum eſſe animadverti-<lb/>tur. </s> <s xml:id="echoid-s1133" xml:space="preserve">Ergo poſthac ſi qui falsò circumferentiæ ma-<lb/>gnitudinem definient, pernumeroſapolygona non re-<lb/>futabuntur, ſed calculo brevi miniméque intrica-<lb/>to, quemque erroris inſimulare, quod hactenus fe-<lb/>rè ſoliti ſunt, haud facile poſſint. </s> <s xml:id="echoid-s1134" xml:space="preserve">Ad hæc ſi quid <lb/>in ſubienſarum Canone, quem emendatum haberi <lb/>quantum referat omnes ſciunt, in eo contexendo ſi <lb/>quid erit admiſſum aut aliunde perverſum irrepſe-<lb/>rit, non dif@icile erit horum ope reſtituere, cum <lb/>aliâ nunc ratione ex inſcriptis in circulo longitu-<lb/>dinem arcuum quibus ſubtenduntur invenire lice-<lb/>at. </s> <s xml:id="echoid-s1135" xml:space="preserve">Quinimo & </s> <s xml:id="echoid-s1136" xml:space="preserve">omni Canonum auxilio deſtitutis <pb file="0064" n="68" rhead="PRÆFATIO."/> oſtendimus, quo pacto ex lateribus triangulorum <lb/>datis angulos eorum inveſtigare queant, ut nun-<lb/>quam duorum ſecundorum ſcrupulorum ſit à vero <lb/>diſſenſus, ſæpe ne unius quidem tertii. </s> <s xml:id="echoid-s1137" xml:space="preserve">Et hæc qui-<lb/>dem non levia commoda viſum iri confidimus. </s> <s xml:id="echoid-s1138" xml:space="preserve">Com-<lb/>perimus autem & </s> <s xml:id="echoid-s1139" xml:space="preserve">Renatum Carteſium, cujus viri <lb/>inventis cum Philoſophia univerſa tum Matheſis <lb/>plurimùm illuſtrata eſt, nonnulla quæ huc ſpectent <lb/>ſcriptis mandaſſe. </s> <s xml:id="echoid-s1140" xml:space="preserve">Ea vero defuncto ipſo in commen-<lb/>tariis reperta feruntur, neque adhuc reſcire potui-<lb/>mus quâ induſtriâ aut eventu hiſce manum admo-<lb/>verit. </s> <s xml:id="echoid-s1141" xml:space="preserve">Willebrordi autem Snellii geometræ eruditi <lb/>Cyclometricus extat, multo labore conſcriptus, qui-<lb/>que omnis in his eſt. </s> <s xml:id="echoid-s1142" xml:space="preserve">Atque ille non exiguam lau-<lb/>dem promeritus videretur, ſi præcipua duo theore-<lb/>mata, quibus omne id opus velut fundamentis ſu-<lb/>perſtructum eſt, demonſtrare potuiſſet. </s> <s xml:id="echoid-s1143" xml:space="preserve">Sed quas <lb/>ibi pro demonſtrationibus haberi poſtulat, propoſi-<lb/>tum minime comprobant: </s> <s xml:id="echoid-s1144" xml:space="preserve">ipſa vero theoremata, ſic-<lb/>ut in utroque evidenti ratione nos oſtendimus, <lb/>præclaram continent veritatem. </s> <s xml:id="echoid-s1145" xml:space="preserve">Et ea quidem ſe-<lb/>quentibus meritò inſerenda putavimus, quod cau-<lb/>ſæ eorum à noſtris pendeant inventis.</s> <s xml:id="echoid-s1146" xml:space="preserve"/> </p> <pb file="0065" n="69"/> <figure> <image file="0065-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0065-01"/> </figure> </div> <div xml:id="echoid-div54" type="section" level="1" n="24"> <head xml:id="echoid-head43" xml:space="preserve">CHRISTIANI HUGENII, <lb/><emph style="sc">Const. f</emph>. <lb/><emph style="sc">DE</emph></head> <head xml:id="echoid-head44" xml:space="preserve">CIRCULI MAGNITUDINE <lb/>INVENTA.</head> <head xml:id="echoid-head45" xml:space="preserve"><emph style="sc">Theorema</emph> I. <emph style="sc">Propositio</emph> I.</head> <p style="it"> <s xml:id="echoid-s1147" xml:space="preserve">SI Circuli portioni, ſemicirculo minori, trian-<lb/>gulum maximum inſcribatur, & </s> <s xml:id="echoid-s1148" xml:space="preserve">portioni-<lb/>bus reliquis triangula ſimiliter inſcribantur, <lb/>erit triangulum primo deſcriptum duorum ſimul <lb/>quæ in portionibus reliquis deſcripta ſunt minus <lb/>quam quadruplum.</s> <s xml:id="echoid-s1149" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1150" xml:space="preserve">Eſto circuli portio A B C, ſemicirculo minor, cujus diameter <lb/> <anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/> B D; </s> <s xml:id="echoid-s1151" xml:space="preserve">maximum autem inſcriptum ſit triangulum A B C, <lb/>hoc eſt, quod baſin & </s> <s xml:id="echoid-s1152" xml:space="preserve">altitudinem habeat cum portione eandem. <lb/></s> <s xml:id="echoid-s1153" xml:space="preserve">Et reliquis duabus portionibus inſcribantur triangula item ma-<lb/>xima A E B, B F C. </s> <s xml:id="echoid-s1154" xml:space="preserve">Dico triangulum A B C minus eſſe quam <lb/>quadruplum triangulorum A E B, B F C ſimul ſumpto-<lb/>rum. </s> <s xml:id="echoid-s1155" xml:space="preserve">Jungatur enim E F, quæ ſecet diametrum portionis <lb/>in puncto G. </s> <s xml:id="echoid-s1156" xml:space="preserve">Quoniam igitur arcus A B bifariam dividitur <lb/>in E puncto, erit utraque harum E A, E B, major dimi-<lb/>diâ A B. </s> <s xml:id="echoid-s1157" xml:space="preserve">Quamobrem quadratum A B minus erit quam qua-<lb/>druplum quadrati E B vel E A. </s> <s xml:id="echoid-s1158" xml:space="preserve">Sicut autem quadratum A B <lb/>ad quadr. </s> <s xml:id="echoid-s1159" xml:space="preserve">E B, ita eſt D B ad B G longitudine; </s> <s xml:id="echoid-s1160" xml:space="preserve">quia qua-<lb/>dratum quidem A B æquale eſt rectangulo quod à D B & </s> <s xml:id="echoid-s1161" xml:space="preserve"><lb/>circuli totius diametro continetur, quadratum vero E B æ-<lb/>quale rectangulo ſub eadem diametro & </s> <s xml:id="echoid-s1162" xml:space="preserve">recta B G. </s> <s xml:id="echoid-s1163" xml:space="preserve">Minor <lb/>igitur eſt B D quam quadrupla B G. </s> <s xml:id="echoid-s1164" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1165" xml:space="preserve">A C minor <lb/>eſt quam dupla E F, quoniam hæc ipſi A B æquatur. </s> <s xml:id="echoid-s1166" xml:space="preserve">Er-<lb/>go patet triangulum A B C minus eſſe quam octuplum tri- <pb o="358" file="0066" n="70" rhead="CHRISTIANI HUGENII"/> anguli E B F. </s> <s xml:id="echoid-s1167" xml:space="preserve">Huic autem triangulo æquantur ſingula A E B, <lb/>B F C. </s> <s xml:id="echoid-s1168" xml:space="preserve">Ergo utriuſque ſimul triangulum A B C minus erit <lb/>quam quadruplum. </s> <s xml:id="echoid-s1169" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1170" xml:space="preserve"/> </p> <div xml:id="echoid-div54" type="float" level="2" n="1"> <note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">TAB. XXXVI@ <lb/>Fig. 1.</note> </div> </div> <div xml:id="echoid-div56" type="section" level="1" n="25"> <head xml:id="echoid-head46" xml:space="preserve"><emph style="sc">Theor</emph>. II. <emph style="sc">Prop</emph>. II.</head> <p style="it"> <s xml:id="echoid-s1171" xml:space="preserve">Si fuerit circuli portio, ſemicirculo minor, & </s> <s xml:id="echoid-s1172" xml:space="preserve">ſu-<lb/>per eadem baſi triangulum, cujus latera portio-<lb/>nem contingant; </s> <s xml:id="echoid-s1173" xml:space="preserve">ducatur autem quæ contingat por-<lb/>tionem in vertice: </s> <s xml:id="echoid-s1174" xml:space="preserve">Hæc à triangulo dicto triangu-<lb/>lum abſcindet majus dimidio maximi trianguli in-<lb/>tra portionem deſcripti.</s> <s xml:id="echoid-s1175" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1176" xml:space="preserve">Eſto circuli portio ſemicirculo minor A B C, cujus vertex <lb/> <anchor type="note" xlink:label="note-0066-01a" xlink:href="note-0066-01"/> B. </s> <s xml:id="echoid-s1177" xml:space="preserve">Et contingant portionem ad terminos baſis rectæ A E, <lb/>C E, quæ conveniant in E: </s> <s xml:id="echoid-s1178" xml:space="preserve">convenient enim quia portio ſe-<lb/>micirculo minor eſt. </s> <s xml:id="echoid-s1179" xml:space="preserve">Porro ducatur F G, quæ contingati-<lb/>pſam in vertice B; </s> <s xml:id="echoid-s1180" xml:space="preserve">& </s> <s xml:id="echoid-s1181" xml:space="preserve">jungantur A B, B C. </s> <s xml:id="echoid-s1182" xml:space="preserve">Oſtendendum eſt <lb/>itaque, triangulum F E G majus eſſe dimidio trianguli <lb/>A B C. </s> <s xml:id="echoid-s1183" xml:space="preserve">Conſtat triangula A E C, F E G, item A F B, <lb/>B G C æquicruria eſſe, dividique F G ad B bifariam. </s> <s xml:id="echoid-s1184" xml:space="preserve">Utra-<lb/>que autem ſimul F E, E G, major eſt quam F G; </s> <s xml:id="echoid-s1185" xml:space="preserve">ergo <lb/>E F major quam F B, vel quam F A. </s> <s xml:id="echoid-s1186" xml:space="preserve">Tota igitur A E minor <lb/>quam dupla F E. </s> <s xml:id="echoid-s1187" xml:space="preserve">Quare triangulum F E G majus erit quarta <lb/>parte trianguli A E C. </s> <s xml:id="echoid-s1188" xml:space="preserve">Sicut autem F A ad A E, ita eſt al-<lb/>titudo trianguli A B C ad altitudinem trianguli A E C, & </s> <s xml:id="echoid-s1189" xml:space="preserve"><lb/>baſis utrique eadem A C. </s> <s xml:id="echoid-s1190" xml:space="preserve">Ergo, quum F A ſit minor quam <lb/>ſubdupla totius A E, erit triangulum A B C minus dimi-<lb/>dio triangulo A E C. </s> <s xml:id="echoid-s1191" xml:space="preserve">Hujus vero quarta parte majus erat <lb/>triangulum F E G. </s> <s xml:id="echoid-s1192" xml:space="preserve">Ergo triangulum F E G majus dimidio <lb/>trianguli A B C. </s> <s xml:id="echoid-s1193" xml:space="preserve">Quod oſtendendum fuit.</s> <s xml:id="echoid-s1194" xml:space="preserve"/> </p> <div xml:id="echoid-div56" type="float" level="2" n="1"> <note position="left" xlink:label="note-0066-01" xlink:href="note-0066-01a" xml:space="preserve">TAB. XXXVIII. <lb/>Fig. 2.</note> </div> </div> <div xml:id="echoid-div58" type="section" level="1" n="26"> <head xml:id="echoid-head47" xml:space="preserve"><emph style="sc">Theor</emph>. III. <emph style="sc">Prop</emph>. III.</head> <p style="it"> <s xml:id="echoid-s1195" xml:space="preserve">OMnis circuli portio, ſemicirculo minor, ad ma-<lb/>ximum triangulum inſcriptum majorem ratio-<lb/>nem habet quam ſeſquitertiam.</s> <s xml:id="echoid-s1196" xml:space="preserve"/> </p> <pb o="359" file="0067" n="71" rhead="DE CIRCULI MAGNIT. INVENTA."/> <p> <s xml:id="echoid-s1197" xml:space="preserve">Eſto Circuli portio, ſemicirculo minor, cui maximum ſit in-<lb/> <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> ſcriptum triangulum A B C. </s> <s xml:id="echoid-s1198" xml:space="preserve">Dico portionem ad dictum tri-<lb/>angulum majorem rationem habere quam quatuor ad tria. </s> <s xml:id="echoid-s1199" xml:space="preserve">In-<lb/>ſcribantur enim & </s> <s xml:id="echoid-s1200" xml:space="preserve">reliquis portionibus duabus maxima triangu-<lb/>la A D B, B E C. </s> <s xml:id="echoid-s1201" xml:space="preserve">Itaque minus eſt triangulum A B C quam illo <lb/>rum ſimul quadruplum <anchor type="note" xlink:href="" symbol="*"/>: </s> <s xml:id="echoid-s1202" xml:space="preserve">ac proinde ſpatium aliquod adjungi <anchor type="note" xlink:label="note-0067-02a" xlink:href="note-0067-02"/> poteſt triangulo A B C, quod una cum ipſo minus etiam ſit <lb/>quam quadruplum dictorum ſimul triangulorum A D B, B E C. <lb/></s> <s xml:id="echoid-s1203" xml:space="preserve">Eſto itaque eâ ratione adjectum triangulum A F C, ut to-<lb/>tum ſpatium A B C F minus ſit quam quadruplum triangu-<lb/>lorum A D B, B E C. </s> <s xml:id="echoid-s1204" xml:space="preserve">Et porro in reſiduis portionibus ma-<lb/>xima triangula inſcribi intelligantur; </s> <s xml:id="echoid-s1205" xml:space="preserve">itemque in reſiduis ſem-<lb/>per, donec portiones quibus poſtremùm inſcribentur ſimul <lb/>minores ſint triangulo A C F, hoc enim fieri poteſt. </s> <s xml:id="echoid-s1206" xml:space="preserve">Ita-<lb/>que & </s> <s xml:id="echoid-s1207" xml:space="preserve">triangula poſtremùm inſcripta ſimul triangulo A C F <lb/>minora erunt. </s> <s xml:id="echoid-s1208" xml:space="preserve">Quia autem ſpatii A B C F quarta parte ma-<lb/>jora ſunt duo ſimul triangula A D B, B E C. </s> <s xml:id="echoid-s1209" xml:space="preserve">Rurſuſque <lb/>quarta horum parte majora triangula quatuor, quæ portio-<lb/>nibus reliquis inſcribuntur. </s> <s xml:id="echoid-s1210" xml:space="preserve">Et horum quartâ majora ſimili-<lb/>ter, quæ deinceps: </s> <s xml:id="echoid-s1211" xml:space="preserve">atque ita continue, ſi plura fuerint de-<lb/>ſcripta. </s> <s xml:id="echoid-s1212" xml:space="preserve">Erit propterea ſpatium ex quadrilatero A B C F & </s> <s xml:id="echoid-s1213" xml:space="preserve"><lb/>cæteris inſcriptis triangulis, & </s> <s xml:id="echoid-s1214" xml:space="preserve">triente eorum, quæ poſtremò <lb/>inſcripta erunt, compoſitum, majus quam ſeſquitertium i-<lb/>pſius quadrilateri A B C F. </s> <s xml:id="echoid-s1215" xml:space="preserve">Hoc enim ab Archimede demon-<lb/>ſtratum eſt, quod ſi fuerint ſpatia quotcunque in ratione qua-<lb/>drupla, ea omnia ſimul cum triente minimi ad maximum ra-<lb/>tionem habebunt ſeſquitertiam. </s> <s xml:id="echoid-s1216" xml:space="preserve">Dividendo itaque, triangula <lb/>omnia intra portiones A D B, B E C deſcripta cum trien-<lb/>te poſtremo diſcriptorum majora erunt tertia parte ſpatii <lb/>A B C F. </s> <s xml:id="echoid-s1217" xml:space="preserve">Sed triens dictus minor eſt triente trianguli A C F. </s> <s xml:id="echoid-s1218" xml:space="preserve"><lb/>Igitur dempto illinc triente poſtremò inſcriptorum; </s> <s xml:id="echoid-s1219" xml:space="preserve">à ſpatio <lb/>autem A B C F ablato triangulo A F C, erunt triangula <lb/>omnia intra portiones A D B, B E C deſcripta, majora <lb/>triente trianguli A B C <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1220" xml:space="preserve">Quare componendo, tota figura <anchor type="note" xlink:label="note-0067-03a" xlink:href="note-0067-03"/> rectilinea portioni A B C inſcripta major quam ſeſquitertia <lb/>trianguli A B C, multoque magis portio ipſa. </s> <s xml:id="echoid-s1221" xml:space="preserve">Quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s1222" xml:space="preserve"/> </p> <div xml:id="echoid-div58" type="float" level="2" n="1"> <note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">TAB. XXXVIII. <lb/>Fig. 3.</note> <note symbol="*" position="right" xlink:label="note-0067-02" xlink:href="note-0067-02a" xml:space="preserve">per. 1. huj.</note> <note symbol="*" position="right" xlink:label="note-0067-03" xlink:href="note-0067-03a" xml:space="preserve">33. 5. Elem.</note> </div> <pb o="360" file="0068" n="72" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div60" type="section" level="1" n="27"> <head xml:id="echoid-head48" xml:space="preserve"><emph style="sc">Theor</emph>. IV. <emph style="sc">Prop</emph>. IV.</head> <p style="it"> <s xml:id="echoid-s1223" xml:space="preserve">Omnis circuli portio, ſemicirculo minor, minor <lb/>eſt duabus tertiis trianguli eandem cum ipſa <lb/>baſin babentis, & </s> <s xml:id="echoid-s1224" xml:space="preserve">latera portionem contingentia.</s> <s xml:id="echoid-s1225" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1226" xml:space="preserve">Eſto circuli portio, ſemicirculo minor, A B C, & </s> <s xml:id="echoid-s1227" xml:space="preserve">contin-<lb/> <anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/> gant ipſam ad terminos baſis rectæ A D, C D, quæ con-<lb/>veniant in puncto D. </s> <s xml:id="echoid-s1228" xml:space="preserve">Dico Portionem A B C minorem eſſe <lb/>duabus tertiis trianguli A D C. </s> <s xml:id="echoid-s1229" xml:space="preserve">Ducatur enim E F quæ por-<lb/>tionem contingat in vertice B, & </s> <s xml:id="echoid-s1230" xml:space="preserve">inſcribatur ipſi triangu-<lb/>lum maximum A B C. </s> <s xml:id="echoid-s1231" xml:space="preserve">Quum igitur triangulum E D F ma-<lb/>jus ſit dimidio trianguli A B C <anchor type="note" xlink:href="" symbol="*"/>, manifeſtum eſt ab illo <anchor type="note" xlink:label="note-0068-02a" xlink:href="note-0068-02"/> partem abſcindi poſſe, ita ut reliquum tamen majus ſit di-<lb/>midio dicti A B C trianguli. </s> <s xml:id="echoid-s1232" xml:space="preserve">Sit igitur hoc pacto abſciſſum <lb/>triangulum E D G. </s> <s xml:id="echoid-s1233" xml:space="preserve">Et ducantur porro rectæ H I, K L, <lb/>quæ portiones reliquas A M B, B N C in verticibus ſuis <lb/>contingant, ipſiſque portionibus triangula maxima inſcri-<lb/>bantur. </s> <s xml:id="echoid-s1234" xml:space="preserve">Idemque prorſus circa reliquas portiones fieri intel-<lb/>ligatur, donec tandem portiones reſiduæ ſimul minores ſint <lb/>quam duplum trianguli E D G. </s> <s xml:id="echoid-s1235" xml:space="preserve">Erit igitur inſcripta portio-<lb/>ni figura quædam rectilinea, atque alia circumſcripta. </s> <s xml:id="echoid-s1236" xml:space="preserve">Et <lb/>quoniam triangulum E G F majus eſt dimidio trianguli <lb/>A B C; </s> <s xml:id="echoid-s1237" xml:space="preserve">& </s> <s xml:id="echoid-s1238" xml:space="preserve">rurſus triangula H E I, K F L, majora quam <lb/>dimidia triangulorum A M B, B N C; </s> <s xml:id="echoid-s1239" xml:space="preserve">idque eadem ſem-<lb/>per ratione in reliquis locum habet, ut triangula ſuper por-<lb/>tionum verticibus conſtituta, eorum quæ intra portiones i-<lb/>pſas deſcripta ſunt, majora ſint quam ſubdupla: </s> <s xml:id="echoid-s1240" xml:space="preserve">apparet tri-<lb/>angula omnia extra portionem poſita etiam abſque triangu-<lb/>lo E G D majora ſimul eſſe quam dimidia triangulorum o-<lb/>mnium intra portionem deſcriptorum. </s> <s xml:id="echoid-s1241" xml:space="preserve">Atqui ſegmentorum in <lb/>portione reliquorum triangulum quoque E G D majus eſt <lb/>quam ſubduplum. </s> <s xml:id="echoid-s1242" xml:space="preserve">Ergo triangulum E D F ſimul cum reli-<lb/>quis triangulis, quæ ſunt extra portionem, majus erit dimi-<lb/>dio portionis totius A B C. </s> <s xml:id="echoid-s1243" xml:space="preserve">Quare multo magis ſpatium à <pb o="361" file="0069" n="73" rhead="DE CIRCULI MAGNIT. INVENTA."/> rectis A D, D C & </s> <s xml:id="echoid-s1244" xml:space="preserve">arcu A B C comprehenſum majus erit <lb/>portionis A B C dimidio. </s> <s xml:id="echoid-s1245" xml:space="preserve">Ac proinde triangulum A D C <lb/>majus quam portionis A B C ſeſquialterum. </s> <s xml:id="echoid-s1246" xml:space="preserve">Quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s1247" xml:space="preserve"/> </p> <div xml:id="echoid-div60" type="float" level="2" n="1"> <note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">TAB. XXXVIII. <lb/>Fig. 4.</note> <note symbol="*" position="left" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">per. 2. huj.</note> </div> </div> <div xml:id="echoid-div62" type="section" level="1" n="28"> <head xml:id="echoid-head49" xml:space="preserve"><emph style="sc">Theor</emph>. V. <emph style="sc">Prop</emph>. V.</head> <p style="it"> <s xml:id="echoid-s1248" xml:space="preserve">OMnis circulus major eſt pylogono æqualium <lb/>laterum ſibi inſcripto & </s> <s xml:id="echoid-s1249" xml:space="preserve">triente exceſſus quo <lb/>id polygonum ſuperat aliud inſcriptum ſubduplo la-<lb/>terum numero.</s> <s xml:id="echoid-s1250" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1251" xml:space="preserve">Eſto circulus centro C; </s> <s xml:id="echoid-s1252" xml:space="preserve">ſitque ipſi inſcriptum polygonum <lb/> <anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> æqualium laterum, quorum unum ſit A B. </s> <s xml:id="echoid-s1253" xml:space="preserve">Atque alterum <lb/>item polygonum ſit inſcriptum, cujus bina latera A D, D B, <lb/>ſubtendat A B. </s> <s xml:id="echoid-s1254" xml:space="preserve">Hoc igitur priore polygono majus eſt. </s> <s xml:id="echoid-s1255" xml:space="preserve">Sit <lb/>autem exceſſus trienti æquale H ſpatium. </s> <s xml:id="echoid-s1256" xml:space="preserve">Dico circulum ma-<lb/>jorem eſſe polygono A D B una cum ſpatio H. </s> <s xml:id="echoid-s1257" xml:space="preserve">Ducantur <lb/>enim ex centro rectæ C A, C B. </s> <s xml:id="echoid-s1258" xml:space="preserve">Quoniam igitur portio <lb/>circuli A D B major eſt quam ſeſquitertia trianguli A D B ſibi <lb/>inſcripti <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s1259" xml:space="preserve">erunt portiones A D, D B, ſimul majores tri- <anchor type="note" xlink:label="note-0069-02a" xlink:href="note-0069-02"/> ente trianguli A D B. </s> <s xml:id="echoid-s1260" xml:space="preserve">Quamobrem & </s> <s xml:id="echoid-s1261" xml:space="preserve">ſector C A B major <lb/>erit utriſque ſimul quadrilatero C A D B & </s> <s xml:id="echoid-s1262" xml:space="preserve">triente trian-<lb/>guli A D B. </s> <s xml:id="echoid-s1263" xml:space="preserve">Sicut autem ſector C A B ad circulum totum, <lb/>ita eſt quadrilaterum C D B A ad polygonum A D B, & </s> <s xml:id="echoid-s1264" xml:space="preserve"><lb/>ita quoque triens trianguli A D B ad trientem exceſſus po-<lb/>lygoni A D B ſupra polygonum A B. </s> <s xml:id="echoid-s1265" xml:space="preserve">Ergo manifeſtum eſt <lb/>circulum quoque totum majorem fore polygono A D B una <lb/>cum triente exceſſus quo polygonum A D B ſuperat poly-<lb/>gonum A B, hoc eſt, unà cum ſpatio H. </s> <s xml:id="echoid-s1266" xml:space="preserve">Quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s1267" xml:space="preserve"/> </p> <div xml:id="echoid-div62" type="float" level="2" n="1"> <note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">TAB. XXXVIII. <lb/>Fig. 5.</note> <note symbol="*" position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">per. 3. huj.</note> </div> </div> <div xml:id="echoid-div64" type="section" level="1" n="29"> <head xml:id="echoid-head50" xml:space="preserve"><emph style="sc">Theor</emph>. VI. <emph style="sc">Prop</emph>. VI.</head> <p style="it"> <s xml:id="echoid-s1268" xml:space="preserve">Omnis circulus minor eſt duabus tertiis polygo-<lb/>ni æqualium laterum ſibi circumſcripti & </s> <s xml:id="echoid-s1269" xml:space="preserve">tri-<lb/>ente polygoni ſimilis inſcripti.</s> <s xml:id="echoid-s1270" xml:space="preserve"/> </p> <pb o="362" file="0070" n="74" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1271" xml:space="preserve">Eſto circulus cujus centrum A, & </s> <s xml:id="echoid-s1272" xml:space="preserve">inſcribatur ipſi polygo-<lb/> <anchor type="note" xlink:label="note-0070-01a" xlink:href="note-0070-01"/> num lateribus æqualibus, quorum unum ſit B C; </s> <s xml:id="echoid-s1273" xml:space="preserve">& </s> <s xml:id="echoid-s1274" xml:space="preserve">ali-<lb/>ud ſimile circumſcribatur F E G, cujus latera circulum con-<lb/>tingant ad occurſum angulorum polygoni prioris. </s> <s xml:id="echoid-s1275" xml:space="preserve">Dico cir-<lb/>culum minorem eſſe duabus tertiis polygoni F E G ſimul <lb/>cum triente polygoni B C. </s> <s xml:id="echoid-s1276" xml:space="preserve">Ducantur namque ex centro re-<lb/>ctæ A B, A C. </s> <s xml:id="echoid-s1277" xml:space="preserve">Igitur quoniam ſuper baſi portionis B D C <lb/>conſiſtit triangulum B E C, cujus latera portionem contin-<lb/>gunt, erit ipſa minor duabus tertiis trianguli B E C <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1278" xml:space="preserve">I- <anchor type="note" xlink:label="note-0070-02a" xlink:href="note-0070-02"/> taque ſi triangulo A B C addantur duæ tertiæ trianguli B E C, <lb/>hoc eſt, duæ tertiæ exceſſus quadrilateri A B E C ſupra tri-<lb/>angulum A B C, ex utriſque compoſitum ſpatium majus <lb/>erit ſectore circuli A B C. </s> <s xml:id="echoid-s1279" xml:space="preserve">Idem eſt autem, ſive triangulo <lb/>A B C addantur duæ tertiæ exceſſus dicti, ſive addantur duæ <lb/>tertiæ quadrilateri A B E C, contraque auferantur duæ ter-<lb/>tiæ trianguli A B C: </s> <s xml:id="echoid-s1280" xml:space="preserve">hinc autem fiunt duæ tertiæ quadri-<lb/>lateri A B E C cum triente trianguli A B C. </s> <s xml:id="echoid-s1281" xml:space="preserve">Ergo apparet <lb/>ſectorem A B C minorem eſſe duabus tertiis quadrilateri <lb/>A B E C & </s> <s xml:id="echoid-s1282" xml:space="preserve">triente trianguli A B C. </s> <s xml:id="echoid-s1283" xml:space="preserve">Quare ſumptis omni-<lb/>bus quoties ſector A B C circulo continetur, totus quoque <lb/>circulus minor erit duabus tertiis polygoni circumſcripti <lb/>F E G & </s> <s xml:id="echoid-s1284" xml:space="preserve">triente inſcripti B C. </s> <s xml:id="echoid-s1285" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1286" xml:space="preserve"/> </p> <div xml:id="echoid-div64" type="float" level="2" n="1"> <note position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">TAB. XXXVIII. <lb/>Fig. 6.</note> <note symbol="*" position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">per. 4. huj.</note> </div> </div> <div xml:id="echoid-div66" type="section" level="1" n="30"> <head xml:id="echoid-head51" xml:space="preserve"><emph style="sc">Theor</emph>. VII. <emph style="sc">Prop</emph>. VII.</head> <p style="it"> <s xml:id="echoid-s1287" xml:space="preserve">OMnis circuli circumferentia major eſt perime-<lb/>tro polygoni æqualium laterum ſibi inſcripti, <lb/>& </s> <s xml:id="echoid-s1288" xml:space="preserve">triente exceſſus quo perimeter eadem ſuperat pe-<lb/>rimetrum alterius polygoni inſcripti ſubduplo late-<lb/>terum numero.</s> <s xml:id="echoid-s1289" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1290" xml:space="preserve">Eſto circulus A B, centro O, cui inſcribatur polygonum <lb/> <anchor type="note" xlink:label="note-0070-03a" xlink:href="note-0070-03"/> æquilaterum A C D, atque alterum duplo laterum nume-<lb/>ro A E C B D F. </s> <s xml:id="echoid-s1291" xml:space="preserve">Sitque recta G I æqualis perimetro po-<lb/>lygoni A E C B D F, G H vero æqualis perimetro poly- <pb o="363" file="0071" n="75" rhead="DE CIRCULI MAGNIT. INVENTA."/> goni A C D. </s> <s xml:id="echoid-s1292" xml:space="preserve">Exceſſus igitur perimetrorum eſt H I; </s> <s xml:id="echoid-s1293" xml:space="preserve">cujus <lb/>triens I K adjiciatur ipſi G I. </s> <s xml:id="echoid-s1294" xml:space="preserve">Dico totâ G K majorem eſſe <lb/>circuli A B circumferentiam. </s> <s xml:id="echoid-s1295" xml:space="preserve">Inſcribatur enim circulo tertium <lb/>polygonum æquilaterum A L E M C, quod ſit duplo nu-<lb/>mero laterum polygoni A E C B D F. </s> <s xml:id="echoid-s1296" xml:space="preserve">Et ſuper lineis G H, <lb/>H I, I K, triangula conſtituantur quorum communis vertex <lb/>N, altitudo autem æqualis ſemidiametro circuli A B. </s> <s xml:id="echoid-s1297" xml:space="preserve">Igi-<lb/>tur quoniam G H baſis æqualis eſt perimetro polygoni <lb/>A C D, erit triangulum G N H æquale polygono, cui bis <lb/>totidem ſunt latera, hoc eſt, polygono A E C B D F. </s> <s xml:id="echoid-s1298" xml:space="preserve">Hoc <lb/>enim patet, ductis ex centro rectis O A & </s> <s xml:id="echoid-s1299" xml:space="preserve">O E, quarum <lb/>hæc ſecet A C in P. </s> <s xml:id="echoid-s1300" xml:space="preserve">Nam triangulum quidem A E O æ-<lb/>quale eſt triangulo baſin habenti A P & </s> <s xml:id="echoid-s1301" xml:space="preserve">altitudinem radii <lb/>O E. </s> <s xml:id="echoid-s1302" xml:space="preserve">Quanta autem pars eſt triangulum A E O polygo-<lb/>ni A E C B D F, eadem eſt recta A P perimetri A C D. <lb/></s> <s xml:id="echoid-s1303" xml:space="preserve">Itaque polygonum A E C B D F æquabitur triangulo cu-<lb/>jus baſis æqualis perimetro A C D, altitudo autem radio <lb/>E O: </s> <s xml:id="echoid-s1304" xml:space="preserve">hoc eſt, triangulo G N H. </s> <s xml:id="echoid-s1305" xml:space="preserve">Eâdem ratione, quo-<lb/>niam baſis G I eſt æqualis polygoni A E C B D F <lb/>perimetro, & </s> <s xml:id="echoid-s1306" xml:space="preserve">altitudo trianguli G N I æqualis radio circu-<lb/>li, erit triangulum G N I æquale polygono A L E M C. </s> <s xml:id="echoid-s1307" xml:space="preserve"><lb/>Itaque triangulum H N I æquale exceſſui polygoni <lb/>A L E M C ſupra polygonum A E C B D F. </s> <s xml:id="echoid-s1308" xml:space="preserve">Trianguli <lb/>autem H N I ſubtriplum eſt ex conſtr triangulum I N K. </s> <s xml:id="echoid-s1309" xml:space="preserve"><lb/>Ergo hoc æquale erit dicti exceſſus trienti. </s> <s xml:id="echoid-s1310" xml:space="preserve">Quare totum tri-<lb/>angulum G N K minus erit circulo A B <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1311" xml:space="preserve">Altitudo autem <anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/> trianguli æqualis eſt circuli ſemidiametro. </s> <s xml:id="echoid-s1312" xml:space="preserve">Ergo evidens eſt <lb/>rectam G K totâ circuli circumferentiâ minorem eſſe. </s> <s xml:id="echoid-s1313" xml:space="preserve">Quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s1314" xml:space="preserve"/> </p> <div xml:id="echoid-div66" type="float" level="2" n="1"> <note position="left" xlink:label="note-0070-03" xlink:href="note-0070-03a" xml:space="preserve">TAB. XXXVIII. <lb/>Fig. 7.</note> <note symbol="*" position="right" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">per 5. huj.</note> </div> <p> <s xml:id="echoid-s1315" xml:space="preserve">Hinc manifeſtum eſt, ſi à feſquitertio laterum polygoni <lb/>circulo inſcripti auferatur triens laterum polygoni alterius <lb/>inſcripti ſubduplo laterum numero, reliquum circumferen-<lb/>tiâ minus eſſe. </s> <s xml:id="echoid-s1316" xml:space="preserve">Idem enim eſt, ſive perimetro majori adda-<lb/>tur {1/3} exceſſus quo ipſa ſuperat perimetrum minorem, ſive <lb/>addatur {1/3} perimetri majoris contraque auferatur {1/3} perimetri <lb/>minoris. </s> <s xml:id="echoid-s1317" xml:space="preserve">Hinc autem fit ſeſquitertium majoris perimetri mi- <pb o="364" file="0072" n="76" rhead="CHRISTIANI HUGENII"/> nus triente minoris. </s> <s xml:id="echoid-s1318" xml:space="preserve">Quare ſi à ſexdecim inſcripti dodecago-<lb/>ni lateribus duo latera inſcripti hexagoni, hoc eſt, diameter <lb/>circuli deducatur, reliqua circuli circumferentiâ minor erit, <lb/>aut ſi ab octo dodecagoni lateribus radius deducatur, reliqua <lb/>minor erit circumferentiæ ſemiſſe. </s> <s xml:id="echoid-s1319" xml:space="preserve">Hoc autem ad conſtructio-<lb/>nem mechanicam utile eſt, quoniam exigua eſt differentia, <lb/>ſicut poſtea oſtendetur.</s> <s xml:id="echoid-s1320" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1321" xml:space="preserve">Manifeſtum etiam, in omni arcu qui ſemicircumferen-<lb/>tiâ minor ſit, ſi ad ſubtenſam addatur triens exceſſus quo <lb/>ſubtenſa ſinum ſuperat, compoſitam arcu minorem eſſe.</s> <s xml:id="echoid-s1322" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div68" type="section" level="1" n="31"> <head xml:id="echoid-head52" xml:space="preserve"><emph style="sc">Theor</emph>. VIII. <emph style="sc">Prop</emph>. VIII.</head> <p style="it"> <s xml:id="echoid-s1323" xml:space="preserve">CIrculo dato, ſi ad diametriterminum contingens <lb/>ducatur, ducatur autem & </s> <s xml:id="echoid-s1324" xml:space="preserve">ab oppoſito diametri <lb/>termino quæ circumferentiam ſecet occurratque tan-<lb/>genti ductæ: </s> <s xml:id="echoid-s1325" xml:space="preserve">erunt interceptæ tangentis duæ tertiæ <lb/>cum triente ejus quæ ab interſectionis puncto dia-<lb/>metro ad angulos rectos incidet, ſimul arcu abſciſ-<lb/>ſo adjacente majores.</s> <s xml:id="echoid-s1326" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1327" xml:space="preserve">Eſto circulus centro A, diametro B C; </s> <s xml:id="echoid-s1328" xml:space="preserve">& </s> <s xml:id="echoid-s1329" xml:space="preserve">ducatur ex C <lb/> <anchor type="note" xlink:label="note-0072-01a" xlink:href="note-0072-01"/> recta quæ circulum contingat C D: </s> <s xml:id="echoid-s1330" xml:space="preserve">huic autem occurrat <lb/>ducta ab altero diametri termino recta B D, quæ circumfe-<lb/>rentiam ſecet in E: </s> <s xml:id="echoid-s1331" xml:space="preserve">ſitque E F diametro B C ad angulos re-<lb/>ctos. </s> <s xml:id="echoid-s1332" xml:space="preserve">Dico tangentis interceptæ C D duas tertias ſimul cum <lb/>triente ipſius E F, arcu E C majores eſſe. </s> <s xml:id="echoid-s1333" xml:space="preserve">Jungantur enim <lb/>A E, E C; </s> <s xml:id="echoid-s1334" xml:space="preserve">& </s> <s xml:id="echoid-s1335" xml:space="preserve">ducatur tangens circulum in E puncto, quæ <lb/>tangenti C D occurrat in G. </s> <s xml:id="echoid-s1336" xml:space="preserve">Erit igitur G E ipſi G C æqua-<lb/>lis, itemque D G; </s> <s xml:id="echoid-s1337" xml:space="preserve">nam ſi centro G circumferentia deſcriba-<lb/>tur quæ tranſeat per puncta C, E, eadem tranſibit quoque <lb/>per D punctum, quoniam angulus C E D rectus eſt. </s> <s xml:id="echoid-s1338" xml:space="preserve">Oſten-<lb/>ſum autem fuit ſuprà, duas tertias quadrilateri A E G C una <lb/>cum triente trianguli A E C ſimul majores eſſe ſectore A E C <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s1339" xml:space="preserve"> <anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/> Eſtque quadrilaterum A E G C æquale triangulo baſin ha- <pb o="365" file="0073" n="77" rhead="DE CIRCULI MAGNIT. INVENTA."/> benti duplam C G, hoc eſt, C D, & </s> <s xml:id="echoid-s1340" xml:space="preserve">altitudinem C A: </s> <s xml:id="echoid-s1341" xml:space="preserve">tri-<lb/>angulum vero A E C æquale triangulo baſin ipſi E F æqua-<lb/>lem habenti & </s> <s xml:id="echoid-s1342" xml:space="preserve">altitudinem dictam A C. </s> <s xml:id="echoid-s1343" xml:space="preserve">Itaque apparet duas <lb/>tertias quadrilateri A E G C ſimul cum triente trianguli A E C <lb/>æquari triangulo qui baſin habeat compoſitam ex duabus ter-<lb/>tiis C D & </s> <s xml:id="echoid-s1344" xml:space="preserve">triente E F, altitudinem vero radii A C. </s> <s xml:id="echoid-s1345" xml:space="preserve">Qua-<lb/>re ejuſmodi quoque triangulum majus erit ſectore A E C. <lb/></s> <s xml:id="echoid-s1346" xml:space="preserve">Unde liquet baſin ipſius, hoc eſt, compoſitam ex duabus <lb/>tertiis ipſius C D & </s> <s xml:id="echoid-s1347" xml:space="preserve">triente ipſius E F, majorem eſſe arcu <lb/>C E. </s> <s xml:id="echoid-s1348" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1349" xml:space="preserve"/> </p> <div xml:id="echoid-div68" type="float" level="2" n="1"> <note position="left" xlink:label="note-0072-01" xlink:href="note-0072-01a" xml:space="preserve">TAB. XXXVIII. <lb/>Fig. 8.</note> <note symbol="*" position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">per 6. huj.</note> </div> </div> <div xml:id="echoid-div70" type="section" level="1" n="32"> <head xml:id="echoid-head53" xml:space="preserve"><emph style="sc">Theor</emph>. IX. <emph style="sc">Prop</emph>. IX.</head> <p style="it"> <s xml:id="echoid-s1350" xml:space="preserve">OMnis circuli circumferentia minor eſt duabus <lb/>tertiis perimetri polygoni æqualium laterum ſibi <lb/>inſcripti & </s> <s xml:id="echoid-s1351" xml:space="preserve">triente perimetri polygoni ſimilis circum-<lb/>ſcripti.</s> <s xml:id="echoid-s1352" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1353" xml:space="preserve">Eſto Circulus cujus A centrum; </s> <s xml:id="echoid-s1354" xml:space="preserve">& </s> <s xml:id="echoid-s1355" xml:space="preserve">inſcribatur ei polygo-<lb/> <anchor type="note" xlink:label="note-0073-01a" xlink:href="note-0073-01"/> num æquilaterum, cujus latus C D: </s> <s xml:id="echoid-s1356" xml:space="preserve">ſimileque aliud cir-<lb/>cumſcribatur lateribus ad priora parallelis, quorum unum ſit <lb/>E F. </s> <s xml:id="echoid-s1357" xml:space="preserve">Dico circuli totius circumferentiam minorem eſſe dua-<lb/>bus tertiis ambitus polygoni C D & </s> <s xml:id="echoid-s1358" xml:space="preserve">triente ambitus polygo-<lb/>ni E F. </s> <s xml:id="echoid-s1359" xml:space="preserve">Ducatur namque diameter circuli B G, quæ ſimul <lb/>inſcripti polygoni latus C D medium dividat in H, & </s> <s xml:id="echoid-s1360" xml:space="preserve">cir-<lb/>cumſcripti latus E F in G, (conſtat autem G fore punctum <lb/>contactus lateris E F,) Et ponatur H L æqualis ipſi H G, <lb/>& </s> <s xml:id="echoid-s1361" xml:space="preserve">jungantur A C, B C & </s> <s xml:id="echoid-s1362" xml:space="preserve">producantur, occurrátque B C <lb/>lateri E F in K, producta autem A C incidet in E angu-<lb/>lum polygoni circumſcripti. </s> <s xml:id="echoid-s1363" xml:space="preserve">Quoniam igitur H L æqualis <lb/>H G, erit B L dupla ipſius A H: </s> <s xml:id="echoid-s1364" xml:space="preserve">Ideoque ut G A ad A H, <lb/>ita G B ad B L. </s> <s xml:id="echoid-s1365" xml:space="preserve">Major autem eſt ratio H B ad B L, quam <lb/>G B ad B H; </s> <s xml:id="echoid-s1366" xml:space="preserve">quoniam hætres ſeſe æqualiter excedunt G B, <lb/>H B, L B. </s> <s xml:id="echoid-s1367" xml:space="preserve">Itaque major erit ratio G B ad B L, hoc eſt, <lb/>G A ad A H, quam duplicata rationis G B ad B H. </s> <s xml:id="echoid-s1368" xml:space="preserve">Sicut <lb/>autem G A ad A H, ita eſt E G ad C H; </s> <s xml:id="echoid-s1369" xml:space="preserve">& </s> <s xml:id="echoid-s1370" xml:space="preserve">ſicut G B <pb o="366" file="0074" n="78" rhead="CHRISTIANI HUGENII"/> ad B H, ita K G ad C H. </s> <s xml:id="echoid-s1371" xml:space="preserve">Ergo major erit ratio E G ad <lb/>C H, quam duplicata ejus, quam habet K G ad C H. </s> <s xml:id="echoid-s1372" xml:space="preserve">Qua-<lb/>re major ratio E G ad K G, quam K G ad C H. </s> <s xml:id="echoid-s1373" xml:space="preserve">Ideoque <lb/>duæ ſimul E G, C H omnino majores duplâ K G. </s> <s xml:id="echoid-s1374" xml:space="preserve">Et ſumptis <lb/>omnium trientibus, erunt trientes utriuſque E G & </s> <s xml:id="echoid-s1375" xml:space="preserve">C H ſi-<lb/>mul majores duabus tertiis K G. </s> <s xml:id="echoid-s1376" xml:space="preserve">Quamobrem addito utrim-<lb/>que ipſius C H triente, erit triens E G cum duabus tertiis <lb/>C H, major duabus tertiis K G cum triente C H. </s> <s xml:id="echoid-s1377" xml:space="preserve">Hiſce <lb/>vero minor etiam eſt arcus C G <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1378" xml:space="preserve">Igitur duæ tertiæ C H <anchor type="note" xlink:label="note-0074-01a" xlink:href="note-0074-01"/> ſimul cum triente ipſius E G majores omnino ſunt eodem ar-<lb/>cu C G. </s> <s xml:id="echoid-s1379" xml:space="preserve">Unde ſumptis omnibus toties quoties arcus C G <lb/>circumferentiâ totâ continetur, erunt quoque duæ tertiæ pe-<lb/>rimetri polygoni C D, cum triente perimetri polygoni E F, <lb/>majores circuli totius circumferentiâ. </s> <s xml:id="echoid-s1380" xml:space="preserve">Quod fuerat oſtenden-<lb/>dum.</s> <s xml:id="echoid-s1381" xml:space="preserve"/> </p> <div xml:id="echoid-div70" type="float" level="2" n="1"> <note position="right" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">TAB. XXXIX. <lb/>Fig. 1.</note> <note symbol="*" position="left" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">per pra<unsure/>ced.</note> </div> <p> <s xml:id="echoid-s1382" xml:space="preserve">Omnis igitur circumferentiæ arcus quadrante minor, mi-<lb/>nor eſt ſinus ſui beſſe & </s> <s xml:id="echoid-s1383" xml:space="preserve">tangentis triente.</s> <s xml:id="echoid-s1384" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div72" type="section" level="1" n="33"> <head xml:id="echoid-head54" xml:space="preserve"><emph style="sc">Problema</emph> I. <emph style="sc">Prop</emph>. X.</head> <head xml:id="echoid-head55" style="it" xml:space="preserve">Peripheriæ ad diametrum rationem invenire <lb/>quamlibet veræ propinquam.</head> <p> <s xml:id="echoid-s1385" xml:space="preserve">MInorem eſſe peripheriæ ad diametrum rationem quam tri-<lb/>plam ſeſquiſeptimam: </s> <s xml:id="echoid-s1386" xml:space="preserve">majorem vero quam 3 {10/71}, Archi-<lb/>medes oſtendit inſcripto circumſcriptoque 96 laterum po-<lb/>lygono. </s> <s xml:id="echoid-s1387" xml:space="preserve">Idem verò hic per dodecagona demonſtrabimus.</s> <s xml:id="echoid-s1388" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1389" xml:space="preserve">Quia enim latus inſcripti circulo dodecagoni majus eſt par-<lb/>tibus 5176 {3/8}, qualium radius continet 10000: </s> <s xml:id="echoid-s1390" xml:space="preserve">duodecim la-<lb/>tera proinde, hoc eſt, perimeter inſcripti dodecagoni major <lb/>erit quam 62116 {1/2}: </s> <s xml:id="echoid-s1391" xml:space="preserve">perimeter autem hexagoni inſcripti eſt <lb/>radii ſextupla, ideoque partium 60000. </s> <s xml:id="echoid-s1392" xml:space="preserve">Igitur dodecagoni <lb/>perimeter perimetrum hexagoni excedit amplius quam par-<lb/>tibus 2116 {1/2}. </s> <s xml:id="echoid-s1393" xml:space="preserve">Quare triens exceſſus major erit quam 705 {1/2}. </s> <s xml:id="echoid-s1394" xml:space="preserve">Igi-<lb/>tur dodecagoni perimeter unà cum triente exceſſus, quo pe-<lb/>rimetrum hexagoni ſuperat, major erit aggregato partium <pb file="0075" n="79"/> <pb file="0075a" n="80"/> <anchor type="figure" xlink:label="fig-0075a-01a" xlink:href="fig-0075a-01"/> <anchor type="figure" xlink:label="fig-0075a-02a" xlink:href="fig-0075a-02"/> <anchor type="figure" xlink:label="fig-0075a-03a" xlink:href="fig-0075a-03"/> <anchor type="figure" xlink:label="fig-0075a-04a" xlink:href="fig-0075a-04"/> <anchor type="figure" xlink:label="fig-0075a-05a" xlink:href="fig-0075a-05"/> <anchor type="figure" xlink:label="fig-0075a-06a" xlink:href="fig-0075a-06"/> <anchor type="figure" xlink:label="fig-0075a-07a" xlink:href="fig-0075a-07"/> <anchor type="figure" xlink:label="fig-0075a-08a" xlink:href="fig-0075a-08"/> <pb file="0076" n="81"/> <pb o="367" file="0077" n="82" rhead="DE CIRCULI MAGNIT. INVENTA."/> 62116 {1/2} & </s> <s xml:id="echoid-s1395" xml:space="preserve">705 {1/2}, hoc eſt, partibus 62822. </s> <s xml:id="echoid-s1396" xml:space="preserve">Atque hiſce pro-<lb/>inde omnino major erit circuli peripheria <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1397" xml:space="preserve">Eſt autem ma- <anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/> jor ratio 62822 ad 20000, longitudinem diametri, quam 3{10/71} <lb/>ad 1. </s> <s xml:id="echoid-s1398" xml:space="preserve">Ergo omnino etiam peripheriæ ad diametrum ratio ma-<lb/>jor erit.</s> <s xml:id="echoid-s1399" xml:space="preserve"/> </p> <div xml:id="echoid-div72" type="float" level="2" n="1"> <figure xlink:label="fig-0075a-01" xlink:href="fig-0075a-01a"> <caption xml:id="echoid-caption25" style="it" xml:space="preserve">Pag. 366.<lb/>TAB.XXXVIII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables25" xml:space="preserve">B E F G A D C</variables> </figure> <figure xlink:label="fig-0075a-02" xlink:href="fig-0075a-02a"> <caption xml:id="echoid-caption26" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables26" xml:space="preserve">E F G B A C</variables> </figure> <figure xlink:label="fig-0075a-03" xlink:href="fig-0075a-03a"> <caption xml:id="echoid-caption27" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables27" xml:space="preserve">B E D C A F</variables> </figure> <figure xlink:label="fig-0075a-04" xlink:href="fig-0075a-04a"> <caption xml:id="echoid-caption28" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables28" xml:space="preserve">D G E F I B K M N H L A C</variables> </figure> <figure xlink:label="fig-0075a-05" xlink:href="fig-0075a-05a"> <caption xml:id="echoid-caption29" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables29" xml:space="preserve">H<lb/>D A B C</variables> </figure> <figure xlink:label="fig-0075a-06" xlink:href="fig-0075a-06a"> <caption xml:id="echoid-caption30" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables30" xml:space="preserve">E D C B F G A</variables> </figure> <figure xlink:label="fig-0075a-07" xlink:href="fig-0075a-07a"> <caption xml:id="echoid-caption31" style="it" xml:space="preserve">Fig. 8.</caption> <variables xml:id="echoid-variables31" xml:space="preserve">D E G B A F C</variables> </figure> <figure xlink:label="fig-0075a-08" xlink:href="fig-0075a-08a"> <caption xml:id="echoid-caption32" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables32" xml:space="preserve">N G H I K<lb/>E L M A P C O F B D</variables> </figure> <note symbol="*" position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">per 7. huj.</note> </div> <p> <s xml:id="echoid-s1400" xml:space="preserve">Rurſus quoniam latus dodecagoni inſcripti minus eſt par-<lb/>tibus 5176 {2/5}. </s> <s xml:id="echoid-s1401" xml:space="preserve">Erunt octo latera, hoc eſt, {2/3} perimetri minora <lb/>quam 41411 {1/5}. </s> <s xml:id="echoid-s1402" xml:space="preserve">Item quia latus dodecagoni circumſcripti mi-<lb/>nus eſt quam 5359, erunt quatuor latera, hoc eſt, triens <lb/>perimetri minor quam 21436. </s> <s xml:id="echoid-s1403" xml:space="preserve">Quamobrem {2/3} perimetri dode-<lb/>cagoni inſcripti cum triente perimetri circumſcripti minores <lb/>erunt quam 62847 {1/5}. </s> <s xml:id="echoid-s1404" xml:space="preserve">Sed iſtis ſimul minor etiam eſt circuli <lb/>circumferentia <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1405" xml:space="preserve">Ergo hæc ad diametrum omnino minorem ha- <anchor type="note" xlink:label="note-0077-02a" xlink:href="note-0077-02"/> bebit rationem, quam 62847 {1/5} ad 20000; </s> <s xml:id="echoid-s1406" xml:space="preserve">& </s> <s xml:id="echoid-s1407" xml:space="preserve">multo minorem <lb/>proinde, quam 62857 {1/7} ad 20000, hoc eſt, quam triplam ſeſ-<lb/>quiſeptimam. </s> <s xml:id="echoid-s1408" xml:space="preserve">Demonſtrati itaque ſunt termini Rationis pe-<lb/>ripheriæ ad diametrum, quos Archimedes ſtatuit. </s> <s xml:id="echoid-s1409" xml:space="preserve">Eoſdem <lb/>verò poſtmodum ſolius inſcripti trigoni æquilateri latere in-<lb/>dagato comprobabimus. </s> <s xml:id="echoid-s1410" xml:space="preserve">Porrò ut propinquior inveniatur ra-<lb/>tio plurium laterum polygona conſideranda ſunt. </s> <s xml:id="echoid-s1411" xml:space="preserve">Intelliga-<lb/>tur igitur circumſcriptum circulo polygonum aliudque inſcri-<lb/>ptum laterum 60. </s> <s xml:id="echoid-s1412" xml:space="preserve">Et præter hæc ſubduplo numero laterum <lb/>inſcriptum, nempe trigintangulum.</s> <s xml:id="echoid-s1413" xml:space="preserve"/> </p> <div xml:id="echoid-div73" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0077-02" xlink:href="note-0077-02a" xml:space="preserve">per 9. huj.</note> </div> <p> <s xml:id="echoid-s1414" xml:space="preserve">Et invenitur quidem latus inſcripti ſexagintanguli majus <lb/>partibus 10467191, qualium radius 100000000 & </s> <s xml:id="echoid-s1415" xml:space="preserve">latus tri-<lb/>gintanguli minus quam 20905693: </s> <s xml:id="echoid-s1416" xml:space="preserve">cujus dimidium 10452846 {1/2} <lb/>eſt ſinus arcus æquantis {1/60} circumferentiæ. </s> <s xml:id="echoid-s1417" xml:space="preserve">Subtenſa autem <lb/>erat 10467191. </s> <s xml:id="echoid-s1418" xml:space="preserve">Ergo differentia 14344 {1/2} minor verâ: </s> <s xml:id="echoid-s1419" xml:space="preserve">& </s> <s xml:id="echoid-s1420" xml:space="preserve">triens <lb/>differentiæ 4781 {1/2}, qui additus ad ſubtenſam 10467191 facit <lb/>10471972 {1/2}. </s> <s xml:id="echoid-s1421" xml:space="preserve">Quibus itaque major eſt arcus {1/60} circumferentiæ. <lb/></s> <s xml:id="echoid-s1422" xml:space="preserve">Ductis autem 10471972 {1/2} ſexagies fiunt 628318250. </s> <s xml:id="echoid-s1423" xml:space="preserve">Hiſce <lb/>igitur partibus omnino major eſt circumferentia tota.</s> <s xml:id="echoid-s1424" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1425" xml:space="preserve">Rurſus quoniam latus inſcripti 60 anguli minus eſt quam <lb/>10467192, erunt duæ tertiæ ipſius minores quam 6978128. <lb/></s> <s xml:id="echoid-s1426" xml:space="preserve">Circumſcripti autem 60 anguli latus cum ſit minus quam <lb/>10481556, erit triens ipſius minor quam 3493852. </s> <s xml:id="echoid-s1427" xml:space="preserve">Quibus <pb o="368" file="0078" n="83" rhead="CHRISTIANI HUGENII"/> additis ad 6978128, fiunt 10471980. </s> <s xml:id="echoid-s1428" xml:space="preserve">Hæigitur omnino ex-<lb/>cedunt {1/60} circumferentiæ, & </s> <s xml:id="echoid-s1429" xml:space="preserve">ſexagecuplum ipſarum, hoc <lb/>eſt 628318800 majus erit circumferentiâ totâ. </s> <s xml:id="echoid-s1430" xml:space="preserve">Quod ſi verò <lb/>polygona adhibeamus laterum 10800, quorum inſcripti qui-<lb/>dem latus, calculo Ludolphi Colonienſis, Arithmetici nobi-<lb/>lis, inventum eſt partium 58177640912684919 non unâ am-<lb/>plius, ſubtenditurque duobus ſcrup. </s> <s xml:id="echoid-s1431" xml:space="preserve">primis; </s> <s xml:id="echoid-s1432" xml:space="preserve">circumſcripti <lb/>autem latus 58177643374063182 non unâ minus. </s> <s xml:id="echoid-s1433" xml:space="preserve">Prætereaque <lb/>latus polygoni ſubduplo laterum numero inſcripti, quod eſt <lb/>116355276902613523 non unâ minus. </s> <s xml:id="echoid-s1434" xml:space="preserve">Hinc peripheriæ lon-<lb/>gitudo invenitur major quam partium 6283185307179584; <lb/></s> <s xml:id="echoid-s1435" xml:space="preserve">minor autem quam 6283185307179589, qualium radius <lb/>1000000000000000. </s> <s xml:id="echoid-s1436" xml:space="preserve">Solitâ autem methodo ex additis inſcri-<lb/>pti circumſcriptique polygoni iſtius lateribus, invenietur <lb/>tantum majorem eſſe peripheriam partibus 62831852, & </s> <s xml:id="echoid-s1437" xml:space="preserve">mi-<lb/>norem 62831855. </s> <s xml:id="echoid-s1438" xml:space="preserve">Patet igitur notarum verarum amplius quam <lb/>duplum numerum eſſe à nobis inventum. </s> <s xml:id="echoid-s1439" xml:space="preserve">Hoc autem & </s> <s xml:id="echoid-s1440" xml:space="preserve">in <lb/>præcedentibus ita ſe habet, ſemperque evenire neceſſe eſt quot-<lb/>cunque laterum polygonis utamur. </s> <s xml:id="echoid-s1441" xml:space="preserve">At per ea, quæ poſtea <lb/>trademus, triplum numerum notarum facile obtineri appare-<lb/>bit.</s> <s xml:id="echoid-s1442" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div75" type="section" level="1" n="34"> <head xml:id="echoid-head56" xml:space="preserve"><emph style="sc">Problema</emph> II. <emph style="sc">Prop</emph>. XI.</head> <p style="it"> <s xml:id="echoid-s1443" xml:space="preserve">Rectam ſumere peripheriæ dati circuli æqualem.</s> <s xml:id="echoid-s1444" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1445" xml:space="preserve">Oſtenſum eſt ſuperius, quod octo inſcripti dodecagoni la-<lb/>tera dempto circuli radio minora ſunt peripheriâ dimidiâ. <lb/></s> <s xml:id="echoid-s1446" xml:space="preserve">In conſtructione autem ut plurimum defectus animadverti ne-<lb/>quit. </s> <s xml:id="echoid-s1447" xml:space="preserve">Nam ſi quatermilleſima diametri pars accedat longitu-<lb/>dini ſic inventæ, jam dimidiam peripheriam excedet. </s> <s xml:id="echoid-s1448" xml:space="preserve">Quod <lb/>ſic fiet manifeſtum. </s> <s xml:id="echoid-s1449" xml:space="preserve">Quarum partium radius eſt 10000, earum <lb/>latus dodecagoni inſcripti circulo eſt amplius quam 5176 {3/8}. </s> <s xml:id="echoid-s1450" xml:space="preserve"><lb/>Unde latera octo majora quam 41411. </s> <s xml:id="echoid-s1451" xml:space="preserve">& </s> <s xml:id="echoid-s1452" xml:space="preserve">dempto radio 10000, <lb/>erit reliqua major quam 31411. </s> <s xml:id="echoid-s1453" xml:space="preserve">cui ſi addantur partes 5, <lb/>hoc eſt, {1/4000} diametri, fient jam 31416; </s> <s xml:id="echoid-s1454" xml:space="preserve">quibus minorem eſſe <lb/>circumferentiam dimidiam liquet ex præcedentibus. </s> <s xml:id="echoid-s1455" xml:space="preserve">Latus au- <pb o="369" file="0079" n="84" rhead="DE CIRCULI MAGNIT. INVENTA."/> tem dodecagoni inſcripti facile invenitur, quia radius pe-<lb/>ripheriæ ſextantem ſubtendit. </s> <s xml:id="echoid-s1456" xml:space="preserve">Eſtque hæc ratio accuratior <lb/>quam ſi triplâ ſeſquiſeptimâ utamur. </s> <s xml:id="echoid-s1457" xml:space="preserve">Nam ſecundum eam <lb/>excedetur {1/2} peripheriæ longitudo amplius quam {1/100} diame-<lb/>tri.</s> <s xml:id="echoid-s1458" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div76" type="section" level="1" n="35"> <head xml:id="echoid-head57" xml:space="preserve"><emph style="sc">Aliter</emph>.</head> <p> <s xml:id="echoid-s1459" xml:space="preserve">Eſto datus circulus cujus B C diameter. </s> <s xml:id="echoid-s1460" xml:space="preserve">Dividatur ſemi-<lb/> <anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/> circumferentia B C bifariam in D. </s> <s xml:id="echoid-s1461" xml:space="preserve">reliqua vero trifariam <lb/>in E & </s> <s xml:id="echoid-s1462" xml:space="preserve">F. </s> <s xml:id="echoid-s1463" xml:space="preserve">Et jungantur D E, D F, quæ ſecent diametrum <lb/>in G & </s> <s xml:id="echoid-s1464" xml:space="preserve">H. </s> <s xml:id="echoid-s1465" xml:space="preserve">Erit trianguli G D H latus alterum una cum <lb/>baſi G H quadrante B D exiguo majus, neque enim exce-<lb/>det {1/5000} diametri B C. </s> <s xml:id="echoid-s1466" xml:space="preserve">Sciendum eſt enim fieri D G vel D H <lb/>duobus inſcripti dodecagoni lateribus æquales. </s> <s xml:id="echoid-s1467" xml:space="preserve">G H autem <lb/>lateri dodecagoni circumſcripti. </s> <s xml:id="echoid-s1468" xml:space="preserve">Unde quidem junctas D G <lb/>& </s> <s xml:id="echoid-s1469" xml:space="preserve">G H majores eſſe conſtat quadrante B D. </s> <s xml:id="echoid-s1470" xml:space="preserve">Nam quia per <lb/>8. </s> <s xml:id="echoid-s1471" xml:space="preserve">huj. </s> <s xml:id="echoid-s1472" xml:space="preserve">octo latera dodecagoni circulo inſcripti cum quatuor <lb/>lateribus circumſcripti majora ſunt peripheriâ totâ, ideo ſum-<lb/>ptâ omnium quartâ parte erunt quoque duo latera inſcripti <lb/>cum latere uno circumſcripti majora peripheriæ quadrante. <lb/></s> <s xml:id="echoid-s1473" xml:space="preserve">Porro quoniam latus inſcripti dodecagoni minus eſt quam <lb/>partium 51764 qualium B C 200000: </s> <s xml:id="echoid-s1474" xml:space="preserve">erunt latera duo, hoc <lb/>eſt, G D, minor quam 103528. </s> <s xml:id="echoid-s1475" xml:space="preserve">Circumſcripti autem dode-<lb/>cagoni latus minus eſt partibus 53590, ipſa nimirum G H. </s> <s xml:id="echoid-s1476" xml:space="preserve"><lb/>Itaque junctæ una D G, G H efficiunt minus quam 157118. </s> <s xml:id="echoid-s1477" xml:space="preserve"><lb/>At quadrantem B D conſtat ex præcedentibus majorem <lb/>eſſe quam 157079. </s> <s xml:id="echoid-s1478" xml:space="preserve">Ergo differentia minor eſt quam partium <lb/>39, cum 40 demum efficiant {1/3000} diametri B C.</s> <s xml:id="echoid-s1479" xml:space="preserve"/> </p> <div xml:id="echoid-div76" type="float" level="2" n="1"> <note position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">TAB. XXXIX. <lb/>Fig. 2.</note> </div> </div> <div xml:id="echoid-div78" type="section" level="1" n="36"> <head xml:id="echoid-head58" xml:space="preserve"><emph style="sc">Aliter</emph>.</head> <p> <s xml:id="echoid-s1480" xml:space="preserve">Tribus ſemidiametris addatur {1/10} lateris inſcripti quadrati; <lb/></s> <s xml:id="echoid-s1481" xml:space="preserve">compoſita ſemicircumferentiæ æquabitur tam propè, ut <lb/>non {1/18000} diametri brevior ſit. </s> <s xml:id="echoid-s1482" xml:space="preserve">Latus quadrati eſt majus quam <lb/>partium 141421 qualium radius 100000, unde quod dictum eſt <lb/>facile demonſtratur.</s> <s xml:id="echoid-s1483" xml:space="preserve"/> </p> <pb o="370" file="0080" n="85" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1484" xml:space="preserve">Vel potius ſex ſe midiameſtris addatur {1/5} dicti lateris qua-<lb/>drati inſcripti ut habeatur recta æqualis peripheriæ toti.</s> <s xml:id="echoid-s1485" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div79" type="section" level="1" n="37"> <head xml:id="echoid-head59" xml:space="preserve"><emph style="sc">Problbma</emph> III. <emph style="sc">Prop</emph>. XII.</head> <head xml:id="echoid-head60" style="it" xml:space="preserve">Dato arcui cuicunque rectam æqualem <lb/>ſumere.</head> <p> <s xml:id="echoid-s1486" xml:space="preserve">Eſto datus circumferentiæ arcus C D, primum quadrante mi-<lb/> <anchor type="note" xlink:label="note-0080-01a" xlink:href="note-0080-01"/> nor, cui rectam æqualem ſumere oporteat. </s> <s xml:id="echoid-s1487" xml:space="preserve">Dividatur ar-<lb/>cus C D bifariam in E, ſitque ſubtenſæ C D æqualis recta F G. <lb/></s> <s xml:id="echoid-s1488" xml:space="preserve">Duabus vero C E, E D, quæ ſubtendunt arcus dimidios, <lb/>æqualis F H. </s> <s xml:id="echoid-s1489" xml:space="preserve">Et ipſi F H jungatur H I triens exceſſus G H. </s> <s xml:id="echoid-s1490" xml:space="preserve"><lb/>Erit tota F I arcui C D æqualis ferè: </s> <s xml:id="echoid-s1491" xml:space="preserve">adeo ut unâ ſui particulâ, <lb/>qualium 1200 continet, aucta, major futura ſit, etiamſi arcus <lb/>C D quadranti æqualis detur. </s> <s xml:id="echoid-s1492" xml:space="preserve">In minoribus autem arcubus <lb/>minor erit differentia. </s> <s xml:id="echoid-s1493" xml:space="preserve">Nam ſi fuerit datus non major peri-<lb/>pheriæ ſextante, linea inventa minus quam {1/6000} ſui parte à ve-<lb/>ra arcus longitudine deficiet. </s> <s xml:id="echoid-s1494" xml:space="preserve">Et minores quidem eſſe arcu-<lb/>bus rectas eo modo inventas conſtat ex Theoremate 7. </s> <s xml:id="echoid-s1495" xml:space="preserve">huj. </s> <s xml:id="echoid-s1496" xml:space="preserve"><lb/>De quantitate autem differentiæ eſt oſtendendum.</s> <s xml:id="echoid-s1497" xml:space="preserve"/> </p> <div xml:id="echoid-div79" type="float" level="2" n="1"> <note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">TAB. XXXIX. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s1498" xml:space="preserve">Primum itaque ponendo arcum C D quadranti peripheriæ <lb/>æqualem, erit C D recta, hoc eſt, F G, latus quadrati <lb/>circulo inſcripti, & </s> <s xml:id="echoid-s1499" xml:space="preserve">minor proinde quam partium 141422, <lb/>qualium radius circuli 100000. </s> <s xml:id="echoid-s1500" xml:space="preserve">C E autem vel E D latus in-<lb/>ſcripti octogoni, ideoque major quam 76536. </s> <s xml:id="echoid-s1501" xml:space="preserve">Eſt autem du-<lb/>plæ E D æqualis F H. </s> <s xml:id="echoid-s1502" xml:space="preserve">Ergo hæc major quam 153072. </s> <s xml:id="echoid-s1503" xml:space="preserve">Qua-<lb/>re exceſſus G H major quam 11650: </s> <s xml:id="echoid-s1504" xml:space="preserve">Et hujus triens H I <lb/>major quam 3883. </s> <s xml:id="echoid-s1505" xml:space="preserve">Ideoque tota F I major quam 156955. <lb/></s> <s xml:id="echoid-s1506" xml:space="preserve">Arcus autem C D cum quadranti æqualis ponitur, minor eſt <lb/>quam 157080. </s> <s xml:id="echoid-s1507" xml:space="preserve">Itaque minus ab hoc diſcrepat recta F I quam <lb/>partibus 125, qualium ipſa eſt 156955. </s> <s xml:id="echoid-s1508" xml:space="preserve">Quæ utique minus <lb/>efficiunt quam {1/1200} ipſius F I.</s> <s xml:id="echoid-s1509" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1510" xml:space="preserve">Si vero ſextans peripheriæ ſit arcus C D, erit recta C D, <lb/>hoc eſt, F G, latus hexagoni inſcripti, ideoque partium <lb/>10000, & </s> <s xml:id="echoid-s1511" xml:space="preserve">C E vel E D latus dodecagoni, ac proinde ma- <pb o="371" file="0081" n="86" rhead="DE CIRCULI MAGNIT. INVENTA."/> jor quam 5176 {3/8}. </s> <s xml:id="echoid-s1512" xml:space="preserve">cujus dupla F H major quam 10352 {3/4}. </s> <s xml:id="echoid-s1513" xml:space="preserve">unde <lb/>G H major quam 352 {3/4}; </s> <s xml:id="echoid-s1514" xml:space="preserve">& </s> <s xml:id="echoid-s1515" xml:space="preserve">H I major quam 117 {7/12}. </s> <s xml:id="echoid-s1516" xml:space="preserve">Tota igi-<lb/>tur F I major quam 10470 {1/3}. </s> <s xml:id="echoid-s1517" xml:space="preserve">Arcus autem C D, ſextans pe-<lb/>ripheriæ, minor eſt quam 10472. </s> <s xml:id="echoid-s1518" xml:space="preserve">Ergo deficiunt lineæ F I <lb/>partium earundem pauciores quam 1 {2/3}. </s> <s xml:id="echoid-s1519" xml:space="preserve">Quæ non æquant {1/6000} <lb/>F I. </s> <s xml:id="echoid-s1520" xml:space="preserve">Porro cum arcus quadrante major datus erit, dividen-<lb/>dus eſt in partes æquales 4 vel 6 vel plures, prout accura-<lb/>tiori dimenſione uti voluerimus; </s> <s xml:id="echoid-s1521" xml:space="preserve">ſed numero pares: </s> <s xml:id="echoid-s1522" xml:space="preserve">Earum-<lb/>que partium ſubtenſis ſimul ſumptis adjungendus eſt triens <lb/>exceſſus quo ipſæ ſuperant aggregatum earum quæ arcubus <lb/>duplis ſubtenduntur. </s> <s xml:id="echoid-s1523" xml:space="preserve">Ita namque componetur longitudo ar-<lb/>cus totius. </s> <s xml:id="echoid-s1524" xml:space="preserve">Vel hac etiam ratione eadem habebitur, ſi arcus <lb/>reliqui ad ſemicircumferentiam longitudo inveniatur aut ſu-<lb/>pra eandem exceſſus, aut reliqui ad circumferentiam totam, <lb/>ſi dodrante major fuerit datus; </s> <s xml:id="echoid-s1525" xml:space="preserve">eaque longitudo adjungatur <lb/>vel auferatur à dimidiæ vel totius circumferentiæ longitudi-<lb/>ne, quam antea invenire docuimus.</s> <s xml:id="echoid-s1526" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div81" type="section" level="1" n="38"> <head xml:id="echoid-head61" xml:space="preserve"><emph style="sc">Theor</emph>. X. <emph style="sc">Prop</emph>. XIII.</head> <p style="it"> <s xml:id="echoid-s1527" xml:space="preserve"><emph style="bf">L</emph>Atus Polygoni æquilateri circulo inſcripti, pro-<lb/>portione medium eſt inter latus polygoni ſimi-<lb/>lis circumſcripti, & </s> <s xml:id="echoid-s1528" xml:space="preserve">dimidium latus polygoni in-<lb/>ſcriptiſub duplo laterum numero.</s> <s xml:id="echoid-s1529" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1530" xml:space="preserve">IN circulo cujus centrum A, radius A B, ſit latus inſcri-<lb/> <anchor type="note" xlink:label="note-0081-01a" xlink:href="note-0081-01"/> pti polygoni æquilateri B C; </s> <s xml:id="echoid-s1531" xml:space="preserve">& </s> <s xml:id="echoid-s1532" xml:space="preserve">latus circumſcripti ſimilis <lb/>polygoni D E ipſi B C parallelum. </s> <s xml:id="echoid-s1533" xml:space="preserve">Ergo producta A B trans-<lb/>ibit per D, & </s> <s xml:id="echoid-s1534" xml:space="preserve">A C per E. </s> <s xml:id="echoid-s1535" xml:space="preserve">Et ſi ducatur C F ipſi A B ad <lb/>angulos rectos, ea erit dimidium latus polygoni inſcripti ſub-<lb/>duplo numero laterum. </s> <s xml:id="echoid-s1536" xml:space="preserve">Itaque oſtendendum eſt, B C me-<lb/>diam eſſe proportione inter E D & </s> <s xml:id="echoid-s1537" xml:space="preserve">C F. </s> <s xml:id="echoid-s1538" xml:space="preserve">Ducatur A G, quæ <lb/>dividat E D bifariam, itaque erit ipſa quoque circuli ſemi-<lb/>diameter & </s> <s xml:id="echoid-s1539" xml:space="preserve">æqualis A B. </s> <s xml:id="echoid-s1540" xml:space="preserve">Et quoniam eſt ut E D ad C B, <lb/>ſic D A ad A B, hoc eſt, D A ad A G; </s> <s xml:id="echoid-s1541" xml:space="preserve">ſicut autem D A <pb o="372" file="0082" n="87" rhead="CHRISTIANI HUGENII"/> ad A G, ita B C ad C F, propter triangulos ſimiles D A G, <lb/>B C F. </s> <s xml:id="echoid-s1542" xml:space="preserve">Erit proinde ut E D ad C B, ita quoque C B ad C F. <lb/></s> <s xml:id="echoid-s1543" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1544" xml:space="preserve"/> </p> <div xml:id="echoid-div81" type="float" level="2" n="1"> <note position="right" xlink:label="note-0081-01" xlink:href="note-0081-01a" xml:space="preserve">TAB. XXXIX. <lb/>Fig. 4.</note> </div> </div> <div xml:id="echoid-div83" type="section" level="1" n="39"> <head xml:id="echoid-head62" xml:space="preserve"><emph style="sc">Lemma</emph>.</head> <p> <s xml:id="echoid-s1545" xml:space="preserve">Eſto linea B C diviſa æqualiter in R; </s> <s xml:id="echoid-s1546" xml:space="preserve">& </s> <s xml:id="echoid-s1547" xml:space="preserve">inæqualiter in F, <lb/> <anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/> ſitque ſegmentum majus F C; </s> <s xml:id="echoid-s1548" xml:space="preserve">& </s> <s xml:id="echoid-s1549" xml:space="preserve">fiat B O æqualis utrique <lb/>ſimul B C, C F; </s> <s xml:id="echoid-s1550" xml:space="preserve">B M vero utrique B C, C R. </s> <s xml:id="echoid-s1551" xml:space="preserve">Dico ma-<lb/>jorem eſſe rationem R B ad B F, quam triplicatam ejus, <lb/>quam habet O B ad B M. </s> <s xml:id="echoid-s1552" xml:space="preserve">Sumatur enim ipſi O M æqualis <lb/>utraque harum M L, L P. </s> <s xml:id="echoid-s1553" xml:space="preserve">Quoniam igitur M O ipſi R F <lb/>æqualis eſt, (nam hoc ex conſtructione intelligitur) erit P O <lb/>tripla ipſius F R. </s> <s xml:id="echoid-s1554" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1555" xml:space="preserve">B M tripla eſt B R. </s> <s xml:id="echoid-s1556" xml:space="preserve">Ergo ut B R <lb/>ad B M, ita F R ad P O. </s> <s xml:id="echoid-s1557" xml:space="preserve">Et permutando ut B R ad F R, <lb/>ſic B M ad P O. </s> <s xml:id="echoid-s1558" xml:space="preserve">Major autem eſt B O quam B M. </s> <s xml:id="echoid-s1559" xml:space="preserve">Ergo <lb/>major erit ratio B O ad O P, quam B R ad R F: </s> <s xml:id="echoid-s1560" xml:space="preserve">& </s> <s xml:id="echoid-s1561" xml:space="preserve">per <lb/>converſionem rationis minor O B ad B P, quam R B ad <lb/>B F. </s> <s xml:id="echoid-s1562" xml:space="preserve">Porro quoniam æquales ſunt O M, M L, major erit <lb/>ratio B O ad O M, quam B M ad M L: </s> <s xml:id="echoid-s1563" xml:space="preserve">& </s> <s xml:id="echoid-s1564" xml:space="preserve">per converſio-<lb/>nem rationis minor O B ad B M, quam M B ad B L. </s> <s xml:id="echoid-s1565" xml:space="preserve">Eo-<lb/>dem modo minor adhuc oſtendetur ratio M B ad B L, quam <lb/>L B ad B P. </s> <s xml:id="echoid-s1566" xml:space="preserve">Itaque omnino ratio triplicata ejus quam ha-<lb/>bet O B ad B M minor erit quam compoſita ex rationibus <lb/>O B ad B M, B M ad B L, & </s> <s xml:id="echoid-s1567" xml:space="preserve">B L ad B P, hoc eſt, <lb/>quam ratio O B ad B P. </s> <s xml:id="echoid-s1568" xml:space="preserve">Major autem erat R B ad B F, <lb/>quam O B ad B P. </s> <s xml:id="echoid-s1569" xml:space="preserve">Ergo omnino major erit ratio R B ad <lb/>B F, quam triplicata rationis O B ad B M. </s> <s xml:id="echoid-s1570" xml:space="preserve">Quod erat pro-<lb/>poſitum.</s> <s xml:id="echoid-s1571" xml:space="preserve"/> </p> <div xml:id="echoid-div83" type="float" level="2" n="1"> <note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">TAB. XXXIX. <lb/>Fig. 5.</note> </div> </div> <div xml:id="echoid-div85" type="section" level="1" n="40"> <head xml:id="echoid-head63" xml:space="preserve"><emph style="sc">Theor</emph>. XI. <emph style="sc">Prop</emph>. XIV.</head> <p style="it"> <s xml:id="echoid-s1572" xml:space="preserve"><emph style="bf">O</emph>Mnis circuli circumferentia minor eſt minore <lb/>duarum mediarum proportionalium inter peri-<lb/>metros polygonorum ſimilium, quorum alterum or-<lb/>dinate circulo inſcriptum ſit, alterum circumſcri- <pb o="373" file="0083" n="88" rhead="DE CIRCULI MAGNIT. INVENTA."/> ptum. </s> <s xml:id="echoid-s1573" xml:space="preserve">Et circulus minor eſt polygono iſtis ſimili cu-<lb/>jus ambitus majori mediarum æquetur.</s> <s xml:id="echoid-s1574" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1575" xml:space="preserve">Eſto circulus B D, cujus centrum A. </s> <s xml:id="echoid-s1576" xml:space="preserve">Et inſcribatur ei po-<lb/> <anchor type="note" xlink:label="note-0083-01a" xlink:href="note-0083-01"/> lygonum æquilaterum B C D L, ſimileque circumſcri-<lb/>batur lateribus parallelis H K M N. </s> <s xml:id="echoid-s1577" xml:space="preserve">Sitque perimetro po-<lb/>lygoni H K M N æqualis recta T, perimetro autem B C D L <lb/>æqualis Z. </s> <s xml:id="echoid-s1578" xml:space="preserve">Et inter Z & </s> <s xml:id="echoid-s1579" xml:space="preserve">T duæ ſint mediæ proportionales <lb/>X & </s> <s xml:id="echoid-s1580" xml:space="preserve">V, quarum X minor. </s> <s xml:id="echoid-s1581" xml:space="preserve">Dico circumferentiam circuli <lb/>B D minorem eſſe rectâ X. </s> <s xml:id="echoid-s1582" xml:space="preserve">Et ſi fiat polygonum in quo Y, <lb/>cujus perimeter æquetur rectæ V, ſimile autem ſit polygono <lb/>B C D L aut H K M N; </s> <s xml:id="echoid-s1583" xml:space="preserve">Dico circulum B N minorem <lb/>haberi polygono Y. </s> <s xml:id="echoid-s1584" xml:space="preserve">Ducatur enim diameter circuli P E, <lb/>quæ dividat bifariam latera parallela B C, H K, inſcripti <lb/>circumſcriptique polygoni in R & </s> <s xml:id="echoid-s1585" xml:space="preserve">E; </s> <s xml:id="echoid-s1586" xml:space="preserve">erit autem E punctum <lb/>contactus lateris H K, & </s> <s xml:id="echoid-s1587" xml:space="preserve">B C ſecabitur in R ad angulos <lb/>rectos. </s> <s xml:id="echoid-s1588" xml:space="preserve">Ducatur etiam ex centro recta A C K, quæ utriuſ-<lb/>que polygoni angulos C & </s> <s xml:id="echoid-s1589" xml:space="preserve">K bifariam ſecet, nam hoc ab <lb/>eadem recta fieri conſtat; </s> <s xml:id="echoid-s1590" xml:space="preserve">& </s> <s xml:id="echoid-s1591" xml:space="preserve">jungatur C E. </s> <s xml:id="echoid-s1592" xml:space="preserve">Ipſi autem C E <lb/>ponatur æqualis C F; </s> <s xml:id="echoid-s1593" xml:space="preserve">ſitque duabus his C R, C F tertia <lb/>proportionalis C G. </s> <s xml:id="echoid-s1594" xml:space="preserve">Ergo qualis polygoni inſcripti latus eſt <lb/>C E ſive C F, talis circumſcripti latus erit C G <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1595" xml:space="preserve">Ideoque <anchor type="note" xlink:label="note-0083-02a" xlink:href="note-0083-02"/> duæ tertiæ C F cum triente C G ſimul majores erunt arcu <lb/>E C <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1596" xml:space="preserve">Sit autem duabus tertiis C F cum triente C G æqua- <anchor type="note" xlink:label="note-0083-03a" xlink:href="note-0083-03"/> lis recta S. </s> <s xml:id="echoid-s1597" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1598" xml:space="preserve">hæc major erit arcu E C.</s> <s xml:id="echoid-s1599" xml:space="preserve"/> </p> <div xml:id="echoid-div85" type="float" level="2" n="1"> <note position="right" xlink:label="note-0083-01" xlink:href="note-0083-01a" xml:space="preserve">TAB. XXXIX. <lb/>Fig. 6.</note> <note symbol="*" position="right" xlink:label="note-0083-02" xlink:href="note-0083-02a" xml:space="preserve">per 13. huj.</note> <note symbol="*" position="right" xlink:label="note-0083-03" xlink:href="note-0083-03a" xml:space="preserve">per 9. huj.</note> </div> <p> <s xml:id="echoid-s1600" xml:space="preserve">Et quoniam ſe habet C R ad C F, ut C F ad C G; <lb/></s> <s xml:id="echoid-s1601" xml:space="preserve">erit quoque dupla C R una cum C F ad triplam C R, <lb/>hoc eſt, utraque ſimul B C, C F ad utramque B C, C R, <lb/>ut dupla C F una cum C G ad triplam C F: </s> <s xml:id="echoid-s1602" xml:space="preserve">vel ſumptis <lb/>horum trientibus, ut {2/3} C F una cum {1/3} C G ad C F, hoc <lb/>eſt, ut S ad C F. </s> <s xml:id="echoid-s1603" xml:space="preserve">Quare etiam triplicata ratio ejus quam ha-<lb/>bet utraque ſimul B C, C F ad utramque B C, C R ea-<lb/>dem erit triplicatæ rationi S ad C F. </s> <s xml:id="echoid-s1604" xml:space="preserve">Major autem eſt ratio <lb/>R B ad B F quam triplicata ejus, quam habet utraque ſi-<lb/>mul B C, C F ad utramque B C, C R <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1605" xml:space="preserve">Ergo major ea- <anchor type="note" xlink:label="note-0083-04a" xlink:href="note-0083-04"/> dem ratio R B ad B F quam triplicata ejus quam habet S ad <pb o="374" file="0084" n="89" rhead="CHRISTIANI HUGENII"/> C F, hoc eſt, quam cubi S ad cubum C F. </s> <s xml:id="echoid-s1606" xml:space="preserve">Sicut autem <lb/>R B ad B F, ita eſt cubus R B ad id quod fit ex quadra-<lb/>to R B in B F. </s> <s xml:id="echoid-s1607" xml:space="preserve">Ergo major quoque ratio cubi R B ad qua-<lb/>dratum R B in B F, quam cubi S ad cubum C F. </s> <s xml:id="echoid-s1608" xml:space="preserve">Qua-<lb/>drato autem R B in B F minus eſt rectangulum ſub R B, <lb/>B G, in F C; </s> <s xml:id="echoid-s1609" xml:space="preserve">quod ſic oſtenditur. </s> <s xml:id="echoid-s1610" xml:space="preserve">Quia enim proportiona-<lb/>les ſunt R C, C F, C G, Erit id quo major mediam exce-<lb/>dit, hoc eſt F G, major quam quo media minimam, hoc <lb/>eſt, quam F R. </s> <s xml:id="echoid-s1611" xml:space="preserve">Major autem eſt F C quam F B. </s> <s xml:id="echoid-s1612" xml:space="preserve">Ergo o-<lb/>mnino major erit ratio C F ad F R, quam B F ad F G. </s> <s xml:id="echoid-s1613" xml:space="preserve">Et <lb/>per converſionem rationis, minor ratio F C ad C R, quam <lb/>F B ad B G. </s> <s xml:id="echoid-s1614" xml:space="preserve">Et permutando minor F C ad F B, quam <lb/>C R ſeu R B ad B G: </s> <s xml:id="echoid-s1615" xml:space="preserve">hoc eſt, (ſumptâ communi altitu-<lb/>dine B R) quam quadrati R B ad rectangulum R B G. </s> <s xml:id="echoid-s1616" xml:space="preserve">Un-<lb/>de quod fit ex rectangulo R B G in F C minus erit quam <lb/>quod ex quadrato R B in F B, uti dictum fuit. </s> <s xml:id="echoid-s1617" xml:space="preserve">Quum ita-<lb/>que major oſtenſa fuerit ratio cubi R B ad quadratum R B <lb/>in B F, quam cubi S ad cubum C F; </s> <s xml:id="echoid-s1618" xml:space="preserve">omnino quoque major <lb/>erit ratio cubi R B ad ſolidum ſub rectangulo R B G in <lb/>F C, quam cubi S ad cubum C F. </s> <s xml:id="echoid-s1619" xml:space="preserve">Et permutando major <lb/>ratio cubi R B ad cubum S, quam rectanguli R B G in <lb/>F C ad cubum C F; </s> <s xml:id="echoid-s1620" xml:space="preserve">hoc eſt, quam rectanguli R B G ad <lb/>quadratum C F. </s> <s xml:id="echoid-s1621" xml:space="preserve">Eſt autem quadrato C F æquale rectan-<lb/>gulum G C R, hoc eſt rectangulum ſub G C, R B, quia <lb/>proportionales ſunt C R, C F, C G. </s> <s xml:id="echoid-s1622" xml:space="preserve">Itaque major erit ra-<lb/>tio cubi R B ad cubum S, quam rectanguli R B G ad re-<lb/>ctangulum ſub G C, R B, hoc eſt, quam B G ad G C. <lb/></s> <s xml:id="echoid-s1623" xml:space="preserve">Sicut autem B G ad G C, ita R C ad E K. </s> <s xml:id="echoid-s1624" xml:space="preserve">Quia enim <lb/>eſt C R ad C G, ut quadratum C R ad quadratum C F ſeu <lb/>quadratum C E: </s> <s xml:id="echoid-s1625" xml:space="preserve">ut autem quadratum C R ad quadratum <lb/>C E, ita eſt P R ad P E diametrum: </s> <s xml:id="echoid-s1626" xml:space="preserve">Erit idcirco C R ad <lb/>C G, ut P R ad P E. </s> <s xml:id="echoid-s1627" xml:space="preserve">Unde dupla C R, hoc eſt, C B ad <lb/>C G, ut dupla P R ad P E, hoc eſt, ut P R ad P A. </s> <s xml:id="echoid-s1628" xml:space="preserve">Et <lb/>dividendo, B G ad G C, ut R A ad A P, ſeu A E, hoc <lb/>eſt, ut R C ad E K, quod dicebamus. </s> <s xml:id="echoid-s1629" xml:space="preserve">Itaque major quo-<lb/>que ratio cubi R B ad cubum S, hoc eſt, ratio triplicata <pb o="375" file="0085" n="90" rhead="DE CIRCULI MAGNIT. INVENTA."/> R B ad S, quam R C ad E K. </s> <s xml:id="echoid-s1630" xml:space="preserve">Eſt autem S major oſtenſa <lb/>arcu E C. </s> <s xml:id="echoid-s1631" xml:space="preserve">Ergo omnino major erit ratio triplicata R B ſeu <lb/>R C ad æqualem arcui E C, quam R C ad E K. </s> <s xml:id="echoid-s1632" xml:space="preserve">Sicut au-<lb/>tem R C ad arcum E C, ita eſt perimeter polygoni B C D L, <lb/>hoc eſt, linea Z ad circumferentiam circuli B D; </s> <s xml:id="echoid-s1633" xml:space="preserve">Et ſicut <lb/>R C ad E K, ita perimeter polygoni B C D L ad perime-<lb/>trum polygoni H K M N, hoc eſt, ita Z ad T. </s> <s xml:id="echoid-s1634" xml:space="preserve">Ergo ma-<lb/>jor quoque triplicata ratio Z ad circumferentiam totam B D, <lb/>quam Z ad T. </s> <s xml:id="echoid-s1635" xml:space="preserve">Ratio autem triplicata Z ad X eadem eſt <lb/>rationi Z ad T. </s> <s xml:id="echoid-s1636" xml:space="preserve">Itaque major eſt ratio ipſius Z ad dictam <lb/>circumferentiam, quam Z ad X. </s> <s xml:id="echoid-s1637" xml:space="preserve">Ac proinde circumferentia <lb/>minor quam recta X. </s> <s xml:id="echoid-s1638" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1639" xml:space="preserve"/> </p> <div xml:id="echoid-div86" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0083-04" xlink:href="note-0083-04a" xml:space="preserve">per lemm@ <lb/>præ.</note> </div> <p> <s xml:id="echoid-s1640" xml:space="preserve">Sciendum eſt autem ipſam X minorem eſſe duabus tertiis <lb/>Z & </s> <s xml:id="echoid-s1641" xml:space="preserve">triente T: </s> <s xml:id="echoid-s1642" xml:space="preserve">hoc eſt, duabus tertiis perimetri polygoni <lb/>inſcripti & </s> <s xml:id="echoid-s1643" xml:space="preserve">triente circumſcripti, quibus alioqui minorem eſſe <lb/>circuli circumferentiam conſtat ex præcedentibus. </s> <s xml:id="echoid-s1644" xml:space="preserve">Nam {2/3} Z <lb/>cum {1/3} T æquantur minori duarum mediarum ſecundum Ari-<lb/>thmeticam proportionem, quæ major eſt minore mediarum <lb/>ſecundum proportionem Geometricam.</s> <s xml:id="echoid-s1645" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1646" xml:space="preserve">Jam vero & </s> <s xml:id="echoid-s1647" xml:space="preserve">de polygono Y demonſtrabimus, ipſum videlicet <lb/>circulo B D majus eſſe. </s> <s xml:id="echoid-s1648" xml:space="preserve">Quia enim polygonum Y habet ad po-<lb/>lygonum ſimile H K M N rationem duplicatam ejus quam peri-<lb/>meter ad perimetrum: </s> <s xml:id="echoid-s1649" xml:space="preserve">perimeter autem polygoni Y æquatur <lb/>rectæ V, & </s> <s xml:id="echoid-s1650" xml:space="preserve">perim. </s> <s xml:id="echoid-s1651" xml:space="preserve">H K M N ipſi T. </s> <s xml:id="echoid-s1652" xml:space="preserve">habebit proinde polygon. </s> <s xml:id="echoid-s1653" xml:space="preserve">Y <lb/>ad polyg. </s> <s xml:id="echoid-s1654" xml:space="preserve">H K M N rationem duplicatam ejus quam V ad <lb/>T, hoc eſt, eam quam X ad T. </s> <s xml:id="echoid-s1655" xml:space="preserve">Sicut autem polygonum <lb/>H K M N ad circulum B D, ita eſt perimeter ipſius poly-<lb/>goni, hoc eſt, linea T ad circuli B D circumferentiam; </s> <s xml:id="echoid-s1656" xml:space="preserve">quo-<lb/>niam polygonum æquale eſt triangulo baſin habenti perime-<lb/>tro ſuæ æqualem & </s> <s xml:id="echoid-s1657" xml:space="preserve">altitudinem radii A E, circulus autem <lb/>æqualis ejuſdem altitudinis triangulo cujus baſis circumferen-<lb/>tiæ æquetur. </s> <s xml:id="echoid-s1658" xml:space="preserve">Ex æquali igitur, erit polygonum Y ad circu-<lb/>lum B D ſicut X ad circumferentiam B D. </s> <s xml:id="echoid-s1659" xml:space="preserve">Eſt autem X <lb/>major oſtenſa quam B D circumferentia. </s> <s xml:id="echoid-s1660" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1661" xml:space="preserve">polygo-<lb/>num Y majus erit circulo B D. </s> <s xml:id="echoid-s1662" xml:space="preserve">Quod erat demonſtran-<lb/>dum.</s> <s xml:id="echoid-s1663" xml:space="preserve"/> </p> <pb o="376" file="0086" n="91" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1664" xml:space="preserve">Ex his manifeſtus eſt Orontii Finei error, qui circumfe-<lb/>rentiæ quadrantem æqualem minori duarum proportione me-<lb/>diarum inter inſcripti & </s> <s xml:id="echoid-s1665" xml:space="preserve">circumſcripti quadrati latera prodi-<lb/>dit, circulum vero æqualem quadrato quod fieret à majori.</s> <s xml:id="echoid-s1666" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div88" type="section" level="1" n="41"> <head xml:id="echoid-head64" xml:space="preserve"><emph style="sc">Theor</emph>. XII. <emph style="sc">Prop</emph>. XV.</head> <p style="it"> <s xml:id="echoid-s1667" xml:space="preserve"><emph style="bf">S</emph>I inter productam circuli diametrum & </s> <s xml:id="echoid-s1668" xml:space="preserve">circum-<lb/>ferentiam recta aptetur radio æqualis, & </s> <s xml:id="echoid-s1669" xml:space="preserve">pro-<lb/>ducta circulum ſecet, occurr atque tangenti circulum <lb/>ad alterum diametri terminum: </s> <s xml:id="echoid-s1670" xml:space="preserve">Intercipiet eapar-<lb/>tem tangentis arcu adjacente abſciſſo majorem.</s> <s xml:id="echoid-s1671" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1672" xml:space="preserve">Eſto deſcriptus circulus centro C, cujus diameter A B. <lb/></s> <s xml:id="echoid-s1673" xml:space="preserve"> <anchor type="note" xlink:label="note-0086-01a" xlink:href="note-0086-01"/> Hæc autem producatur verſus A, interque ipſam & </s> <s xml:id="echoid-s1674" xml:space="preserve">cir-<lb/>cumferentiam ponatur E D recta radio A C æqualis. </s> <s xml:id="echoid-s1675" xml:space="preserve">Quæ <lb/>producta ſecet circumferentiam in F, occurratque tangenti <lb/>in G, ei nimirum quæ circulum contingit ad diametri ter-<lb/>minum B. </s> <s xml:id="echoid-s1676" xml:space="preserve">Dico tangentem B G majorem eſſe arcu B F. <lb/></s> <s xml:id="echoid-s1677" xml:space="preserve">Ducatur enim per centrum recta H L parallela E G, quæ <lb/>circumferentiæ occurrat in punctis H, M: </s> <s xml:id="echoid-s1678" xml:space="preserve">tangenti vero B G <lb/>in L Et jungatur D H, quæ diametrum ſecet in K. </s> <s xml:id="echoid-s1679" xml:space="preserve">Simi-<lb/>les itaque ſunt trianguli E D K, C H K, quoniam angu-<lb/>los ad K æquales habent, & </s> <s xml:id="echoid-s1680" xml:space="preserve">angulum E æqualem angulo <lb/>C. </s> <s xml:id="echoid-s1681" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1682" xml:space="preserve">latus E D æquale eſt lateri H C, ſuntque hæc <lb/>latera æqualibus angulis ſubtenſa. </s> <s xml:id="echoid-s1683" xml:space="preserve">Ergo æquale etiam latus <lb/>D K lateri K H. </s> <s xml:id="echoid-s1684" xml:space="preserve">Itaque C A ſecat bifariam ipſam D H, <lb/>itemque arcum D A H. </s> <s xml:id="echoid-s1685" xml:space="preserve">Arcus igitur D H ſive huic æqua-<lb/>lis F M duplus eſt ad arcum A H. </s> <s xml:id="echoid-s1686" xml:space="preserve">Ipſi autem A H æqua-<lb/>lis eſt arcus M B. </s> <s xml:id="echoid-s1687" xml:space="preserve">Igitur arcus F B triplus erit ad arcum <lb/>A H. </s> <s xml:id="echoid-s1688" xml:space="preserve">Porro quoniam H K ſinus eſt arcus H A, ejuſdem-<lb/>que tangenti æquatur L B, erunt duæ tertiæ H K & </s> <s xml:id="echoid-s1689" xml:space="preserve">triens <lb/>L B ſimul majores arcu A H <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1690" xml:space="preserve">Quare ſumptis omnium tri- <anchor type="note" xlink:label="note-0086-02a" xlink:href="note-0086-02"/> plis erit dupla H K, hoc eſt, H D ſive G L una cum L B <lb/>major arcu A H triplo, hoceſt, arcu F B. </s> <s xml:id="echoid-s1691" xml:space="preserve">Apparet igitur <lb/>totam G B arcu F B majorem eſſe.</s> <s xml:id="echoid-s1692" xml:space="preserve"/> </p> <div xml:id="echoid-div88" type="float" level="2" n="1"> <note position="left" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve">TAB. XXXIX. <lb/>Fig. 7.</note> <note symbol="*" position="left" xlink:label="note-0086-02" xlink:href="note-0086-02a" xml:space="preserve">per 9. huj.</note> </div> <pb file="0087" n="92"/> <pb file="0087a" n="93"/> <figure> <caption xml:id="echoid-caption33" style="it" xml:space="preserve">Pag. 376.<lb/>TAB. XXXIX<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables33" xml:space="preserve">E K C B A L H G D F</variables> </figure> <figure> <caption xml:id="echoid-caption34" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables34" xml:space="preserve">D B G H C E F</variables> </figure> <figure> <caption xml:id="echoid-caption35" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables35" xml:space="preserve">E C G A F B D</variables> </figure> <figure> <caption xml:id="echoid-caption36" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables36" xml:space="preserve">E C D F G H I</variables> </figure> <figure> <caption xml:id="echoid-caption37" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables37" xml:space="preserve">B F R C P L M O</variables> </figure> <figure> <caption xml:id="echoid-caption38" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables38" xml:space="preserve">Y S H E K B C G F R A L D N P M Z X V T</variables> </figure> <figure> <caption xml:id="echoid-caption39" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables39" xml:space="preserve">G F D M L E A K C B H</variables> </figure> <pb file="0088" n="94"/> <pb o="377" file="0089" n="95" rhead="DE CIRCULI MAGNIT. INVENTA."/> <p> <s xml:id="echoid-s1693" xml:space="preserve">Hoc Theorema alterum eſt ex iis quibus Cyclometria <lb/>Willebrordi Snellii tota innititur, quæque demonſtraſſe ipſe <lb/>videri voluit, argumentatione uſus quæ meram quæſiti pe-<lb/>titionem continet. </s> <s xml:id="echoid-s1694" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1695" xml:space="preserve">alterum ſubjungemus, quod utile <lb/>eſt imprimis & </s> <s xml:id="echoid-s1696" xml:space="preserve">contemplatione digniſſimum.</s> <s xml:id="echoid-s1697" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div90" type="section" level="1" n="42"> <head xml:id="echoid-head65" xml:space="preserve"><emph style="sc">Theor</emph>. XIII. <emph style="sc">Prop</emph>. XVI.</head> <p style="it"> <s xml:id="echoid-s1698" xml:space="preserve"><emph style="bf">S</emph>I diametro circuli ſemidiameter in directum adji-<lb/>ciatur, & </s> <s xml:id="echoid-s1699" xml:space="preserve">ab adjectæ termino recta ducatur quæ <lb/>circulum ſecet, occurr atque tangenti circulum ad ter-<lb/>minum diametri oppoſitum: </s> <s xml:id="echoid-s1700" xml:space="preserve">Intercipiet eapartem tan-<lb/>gentis arcu adjacente abſciſſo minorem.</s> <s xml:id="echoid-s1701" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1702" xml:space="preserve">Eſto circulus, cujus diameter A B; </s> <s xml:id="echoid-s1703" xml:space="preserve">quæ producatur, & </s> <s xml:id="echoid-s1704" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/> ſit A C ſemidiametro æqualis. </s> <s xml:id="echoid-s1705" xml:space="preserve">Et ducatur recta C L, <lb/>quæ circumferentiam ſecundò ſecet in E; </s> <s xml:id="echoid-s1706" xml:space="preserve">occurratque tan-<lb/>genti in L, ei nimirum quæ circulum contingit in termino <lb/>diametri B. </s> <s xml:id="echoid-s1707" xml:space="preserve">Dico interceptam B L arcu B E minorem eſſe. <lb/></s> <s xml:id="echoid-s1708" xml:space="preserve">Jungantur enim A E, E B, poſitâque A H ipſi A E æqua-<lb/>li ducatur H E & </s> <s xml:id="echoid-s1709" xml:space="preserve">producatur, occurratque tangenti in K. </s> <s xml:id="echoid-s1710" xml:space="preserve"><lb/>Denique ſit E G diametro A B ad angulos rectos, E D ve-<lb/>ro tangenti B L. </s> <s xml:id="echoid-s1711" xml:space="preserve">Quoniam igitur iſoſceles eſt triangulus <lb/>H A E, erunt anguli inter ſe æquales H & </s> <s xml:id="echoid-s1712" xml:space="preserve">H E A. </s> <s xml:id="echoid-s1713" xml:space="preserve">Quia <lb/>autem angulus A E B rectus eſt, etiam recto æquales erunt <lb/>duo ſimul H E A, K E B. </s> <s xml:id="echoid-s1714" xml:space="preserve">Verùm duo quoque iſti H & </s> <s xml:id="echoid-s1715" xml:space="preserve"><lb/>H K B uni recto æquantur, quoniam in triangulo H K B <lb/>rectus eſt angulus B. </s> <s xml:id="echoid-s1716" xml:space="preserve">Ergo demptis utrimque æqualibus, <lb/>hinc nimirum angulo H, inde angulo H E A, relinquen-<lb/>tur inter ſe æquales anguli K E B, H K B. </s> <s xml:id="echoid-s1717" xml:space="preserve">Triangulus <lb/>igitur iſoſceles eſt K B E, ejuſque latera æqualia E B, B K. </s> <s xml:id="echoid-s1718" xml:space="preserve"><lb/>Eſt autem B D æqualis E G. </s> <s xml:id="echoid-s1719" xml:space="preserve">Ergo D K differentia eſt quâ <lb/>B E excedit E G. </s> <s xml:id="echoid-s1720" xml:space="preserve">Porro quoniam eſt A G ad A E, ut A E <lb/>ad A B, erunt duæ ſimul A G, A B majores duplâ A E <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s1721" xml:space="preserve"> <anchor type="note" xlink:label="note-0089-02a" xlink:href="note-0089-02"/> Ideoque A E, hoc eſt, A H minor quam dimidia utriuſque <pb o="378" file="0090" n="96" rhead="CHRISTIANI HUGENII"/> ſimul A G, A B; </s> <s xml:id="echoid-s1722" xml:space="preserve">hoc eſt, minor quam C A cum dimidia <lb/>A G. </s> <s xml:id="echoid-s1723" xml:space="preserve">Quare ablatâ utrimque C A, erit C H minor dimi-<lb/>diâ A G. </s> <s xml:id="echoid-s1724" xml:space="preserve">C A vero dimidiâ A G major eſt. </s> <s xml:id="echoid-s1725" xml:space="preserve">Ergo ſi adda-<lb/>tur A C ad A G, erit tota C G major quam tripla ipſius <lb/>C H. </s> <s xml:id="echoid-s1726" xml:space="preserve">Quia autem ut H G ad G E, ita eſt E D ad D K; <lb/></s> <s xml:id="echoid-s1727" xml:space="preserve">ut autem G E ad G C, ita L D ad D E: </s> <s xml:id="echoid-s1728" xml:space="preserve">Erit ex æquo in <lb/>proportione turbata ut H G ad G C, ita L D ad D K. </s> <s xml:id="echoid-s1729" xml:space="preserve">Et <lb/>per converſionem rationis & </s> <s xml:id="echoid-s1730" xml:space="preserve">dividendo, ut G C ad C H, <lb/>ita D K ad K L. </s> <s xml:id="echoid-s1731" xml:space="preserve">Ergo etiam D K major quam tripla K L. </s> <s xml:id="echoid-s1732" xml:space="preserve"><lb/>Erat autem D K exceſſus ipſius E B ſupra E G. </s> <s xml:id="echoid-s1733" xml:space="preserve">Ergo K L <lb/>minor eſt triente dicti exceſſus. </s> <s xml:id="echoid-s1734" xml:space="preserve">K B autem æqualis eſt ipſi <lb/>E B ſubtenſæ. </s> <s xml:id="echoid-s1735" xml:space="preserve">Ergo K B unà cum K L, hoc eſt, tota <lb/>L B omnino minor erit arcu B E <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1736" xml:space="preserve">Quod erat demonſtran- <anchor type="note" xlink:label="note-0090-01a" xlink:href="note-0090-01"/> dum.</s> <s xml:id="echoid-s1737" xml:space="preserve"/> </p> <div xml:id="echoid-div90" type="float" level="2" n="1"> <note position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">TAB. XL. <lb/>Fig. 1.</note> <note symbol="*" position="right" xlink:label="note-0089-02" xlink:href="note-0089-02a" xml:space="preserve">25.5. Elem.</note> <note symbol="*" position="left" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve">per 7. huj.</note> </div> <p> <s xml:id="echoid-s1738" xml:space="preserve">Perpenſo autem Theoremate præcedenti, liquet non poſſe <lb/>ſumi punctum aliud in producta B A diametro, quod minus <lb/>à circulo diſtet quam punctum C, eandemque ſervet proprie-<lb/>tatem, ut nimirum ductâ C L fiat tangens intercepta B L <lb/>ſemper minor arcu abſciſſo B E.</s> <s xml:id="echoid-s1739" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1740" xml:space="preserve">Porro uſus hujus Theorematis multiplex eſt, cum in inve-<lb/>niendis triangulorum angulis quorum cognita ſint latera, id-<lb/>que citra tabularum opem, tum ut latera ex angulis datis <lb/>inveniantur, vel cuilibet peripheriæ arcui ſubtenſa aſſigne-<lb/>tur. </s> <s xml:id="echoid-s1741" xml:space="preserve">Quæ omnia à Snellio in Cyclometricis diligenter pertra-<lb/>ctata ſunt.</s> <s xml:id="echoid-s1742" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div92" type="section" level="1" n="43"> <head xml:id="echoid-head66" xml:space="preserve"><emph style="sc">Theorema</emph> XIV. <emph style="sc">Propos</emph>. XVII.</head> <p style="it"> <s xml:id="echoid-s1743" xml:space="preserve"><emph style="bf">P</emph>Ortionis circuli centrum gravitatis diametrum <lb/>portionis ita dividit, ut pars quæ ad verticem <lb/>reliquâ major ſit, minor autem quam ejuſdem ſeſ-<lb/>quialtera.</s> <s xml:id="echoid-s1744" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1745" xml:space="preserve">Eſto circuli portio A B C, (ponatur autem ſemicirculo <lb/> <anchor type="note" xlink:label="note-0090-02a" xlink:href="note-0090-02"/> minor, quoniam cæteræ ad propoſitum non faciunt) & </s> <s xml:id="echoid-s1746" xml:space="preserve">dia-<lb/>meter portionis ſit B D, quæ bifariam ſecetur in E. </s> <s xml:id="echoid-s1747" xml:space="preserve">Itaque <pb o="379" file="0091" n="97" rhead="DE CIRCULI MAGNIT. INVENTA."/> oſtendendum eſt primò centrum gravitatis portionis A B di-<lb/>ſtare à vertice B ultra punctum E; </s> <s xml:id="echoid-s1748" xml:space="preserve">nam, quod in diametro <lb/>ſitum ſit, alibi oſtendimus. </s> <s xml:id="echoid-s1749" xml:space="preserve">Ducatur per E recta baſi paral-<lb/>lela, quæ utrimque circumferentiæ occurrat in punctis F & </s> <s xml:id="echoid-s1750" xml:space="preserve"><lb/>G. </s> <s xml:id="echoid-s1751" xml:space="preserve">Per quæ ducantur K I, H L baſi A C ad angulos re-<lb/>ctos, atque hæ cum ea, quæ portionem in vertice contingit, <lb/>conſtituant rectangulum K L. </s> <s xml:id="echoid-s1752" xml:space="preserve">Quoniam igitur portio ſemi-<lb/>circulo minor eſt, conſtat rectanguli dicti dimidium F L con-<lb/>tineri intra ſegmentum A F G C, atque inſuper ſpatia quæ-<lb/>dam A F I, L G C. </s> <s xml:id="echoid-s1753" xml:space="preserve">Alterum vero rectanguli K L ſemiſſem <lb/>K G complecti ſegmentum F B G unà cum ſpatiis F B K, <lb/>B G H. </s> <s xml:id="echoid-s1754" xml:space="preserve">Quæ ſpatia quum ſint tota ſupra rectam F G, et-<lb/>iam centrum commune gravitatis eorum ſupra eandem ſitum <lb/>erit. </s> <s xml:id="echoid-s1755" xml:space="preserve">Eſt autem E punctum in ipſa F G centrum grav. </s> <s xml:id="echoid-s1756" xml:space="preserve">to-<lb/>tius rectanguli K L. </s> <s xml:id="echoid-s1757" xml:space="preserve">Igitur ſpatii reliqui B F I L G B cen-<lb/>trum grav. </s> <s xml:id="echoid-s1758" xml:space="preserve">erit infra rectam F G. </s> <s xml:id="echoid-s1759" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1760" xml:space="preserve">ſpatiorum A F I, <lb/>L G C commune gravitatis centrum eſt infra eandem F G. <lb/></s> <s xml:id="echoid-s1761" xml:space="preserve">Ergo magnitudinis ex ſpatiis hiſce & </s> <s xml:id="echoid-s1762" xml:space="preserve">dicto ſpatio B F I L G B <lb/>compoſitæ, quæ eſt portio ipſa A B C, centrum gravitatis <lb/>infra lineam F G reperiri neceſſe eſt, ideoque infra E pun-<lb/>ctum.</s> <s xml:id="echoid-s1763" xml:space="preserve"/> </p> <div xml:id="echoid-div92" type="float" level="2" n="1"> <note position="left" xlink:label="note-0090-02" xlink:href="note-0090-02a" xml:space="preserve">TAB. XL. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s1764" xml:space="preserve">Eadem verò diameter B D ſecetur nunc in S, ita ut B S <lb/> <anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/> ſit ſeſquialtera reliquæ S D. </s> <s xml:id="echoid-s1765" xml:space="preserve">Dico centrum grav. </s> <s xml:id="echoid-s1766" xml:space="preserve">portionis <lb/>A B C minus diſtare à vertice B quam punctum S. </s> <s xml:id="echoid-s1767" xml:space="preserve">Sit enim <lb/>B D P totius circuli diameter. </s> <s xml:id="echoid-s1768" xml:space="preserve">& </s> <s xml:id="echoid-s1769" xml:space="preserve">ducatur per S recta baſi <lb/>parallela quæ circumferentiæ occurrat in F & </s> <s xml:id="echoid-s1770" xml:space="preserve">G. </s> <s xml:id="echoid-s1771" xml:space="preserve">Et parabo-<lb/>le intelligatur cujus vertex B, axis B D, rectum vero latus <lb/>æquale S P. </s> <s xml:id="echoid-s1772" xml:space="preserve">Et occurat baſi portionis in H & </s> <s xml:id="echoid-s1773" xml:space="preserve">K. </s> <s xml:id="echoid-s1774" xml:space="preserve">Quoniam <lb/>igitur quadratum F S æquale eſt rectangulo B S P, hoc eſt, <lb/>ei quod ſub B S & </s> <s xml:id="echoid-s1775" xml:space="preserve">latere recto parabolæ continetur, tranſibit <lb/>ea per F punctum, itemque per G. </s> <s xml:id="echoid-s1776" xml:space="preserve">Partes autem lineæ pa-<lb/>rabolicæ B F, B G intra circumferentiam cadent, ſed reli-<lb/>quæ F H, G K erunt exteriores. </s> <s xml:id="echoid-s1777" xml:space="preserve">Hoc enim oſtenditur du-<lb/>ctâ inter B & </s> <s xml:id="echoid-s1778" xml:space="preserve">S ordinatim applicatâ N L, quæ circumfe-<lb/>rentiæ occurrat in N, parabolæ autem in M. </s> <s xml:id="echoid-s1779" xml:space="preserve">Nam quia qua-<lb/>dratum N L æquale eſt rectangulo B L P, quadratum vero <pb o="380" file="0092" n="98" rhead="CHRISTIANI HUGENII"/> M L rectangulo contento lineis B L, S P: </s> <s xml:id="echoid-s1780" xml:space="preserve">rectangulum au-<lb/>tem B L P majus eo quod ſub B L, S P continetur: </s> <s xml:id="echoid-s1781" xml:space="preserve">erit <lb/>quadratum N L majus quadrato M L, & </s> <s xml:id="echoid-s1782" xml:space="preserve">N L linea major <lb/>quam M L. </s> <s xml:id="echoid-s1783" xml:space="preserve">Idem autem continget ubicunque inter B & </s> <s xml:id="echoid-s1784" xml:space="preserve">S <lb/>aliqua ordinatim applicabitur. </s> <s xml:id="echoid-s1785" xml:space="preserve">Igitur partem circumferentiæ <lb/>B F totam extra parabolam ferri neceſſe eſt, eâdemque ra-<lb/>tione partem B G. </s> <s xml:id="echoid-s1786" xml:space="preserve">Rurſus quia rectangulum B D P æquale <lb/>eſt quadrato D A; </s> <s xml:id="echoid-s1787" xml:space="preserve">rectangulum verò ſub B D, S P conten-<lb/>tum quadrato D H; </s> <s xml:id="echoid-s1788" xml:space="preserve">erit H D major quam A D potentiâ, <lb/>ideoque & </s> <s xml:id="echoid-s1789" xml:space="preserve">longitudine. </s> <s xml:id="echoid-s1790" xml:space="preserve">Idemque eveniet ubicunque inter <lb/>S, D, ordinatim aliqua applicabitur. </s> <s xml:id="echoid-s1791" xml:space="preserve">Quare partes circum-<lb/>ferentiæ F A, itemque G C intra parabolam cadent. </s> <s xml:id="echoid-s1792" xml:space="preserve">Fiunt <lb/>igitur ſpatia quædam F N B M, & </s> <s xml:id="echoid-s1793" xml:space="preserve">B Q G, itemque alia <lb/>H F A, G C K. </s> <s xml:id="echoid-s1794" xml:space="preserve">Quorum hæc cum tota ſint infra lineam <lb/>F G, etiam centrum commune gravitatis eorum infra eandem <lb/>erit. </s> <s xml:id="echoid-s1795" xml:space="preserve">At parabolicæ portionis H B K centrum grav eſt in <lb/>ipſa F G, nimirum S punctum <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1796" xml:space="preserve">Ergo partis reliquæ <anchor type="note" xlink:label="note-0092-01a" xlink:href="note-0092-01"/> A F M B Q G C centrum grav. </s> <s xml:id="echoid-s1797" xml:space="preserve">erit ſupra rectam F G. <lb/></s> <s xml:id="echoid-s1798" xml:space="preserve">Sed ſupra hanc ſitum quoque apparet centrum grav. </s> <s xml:id="echoid-s1799" xml:space="preserve">ſpatio-<lb/>rum F M B N, B Q G, quum tota ſint ſupra ipſam <lb/>F G. </s> <s xml:id="echoid-s1800" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1801" xml:space="preserve">ſpatii ex hiſce duobus & </s> <s xml:id="echoid-s1802" xml:space="preserve">A F M B Q G C <lb/>compoſiti, hoc eſt, portionis circuli A B C centrum grav. </s> <s xml:id="echoid-s1803" xml:space="preserve"><lb/>ſupra lineam F G reperietur: </s> <s xml:id="echoid-s1804" xml:space="preserve">quumque ſit in B D diametro, <lb/>minus aberit à vertice B quam punctum S. </s> <s xml:id="echoid-s1805" xml:space="preserve">Quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s1806" xml:space="preserve"/> </p> <div xml:id="echoid-div93" type="float" level="2" n="2"> <note position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve">TAB. XL. <lb/>Fig. 3.</note> <note symbol="*" position="left" xlink:label="note-0092-01" xlink:href="note-0092-01a" xml:space="preserve">8. lib. 2. <lb/>Archim. de <lb/>Æquipond.</note> </div> </div> <div xml:id="echoid-div95" type="section" level="1" n="44"> <head xml:id="echoid-head67" xml:space="preserve"><emph style="sc">Theor</emph>. XV. <emph style="sc">Propos</emph>. XVIII.</head> <p style="it"> <s xml:id="echoid-s1807" xml:space="preserve">CIrculi portio ſemicirculo minor ad inſcriptum <lb/>triangulum maximum majorem rationem habet <lb/>quam ſeſquitertiam, minorem vero quam diameter <lb/>portionis reliquæ tripla ſeſquitertia ad circuli diame-<lb/>trum cum tripla ea, quæ à centro circuli pertingit <lb/>ad portionis baſin.</s> <s xml:id="echoid-s1808" xml:space="preserve"/> </p> <pb o="381" file="0093" n="99" rhead="DE CIRCULI MAGNIT. INVENTA."/> <p> <s xml:id="echoid-s1809" xml:space="preserve">SIt portio ſemicirculo minor, cui inſcriptum triangulum <lb/> <anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/> maximum A B C. </s> <s xml:id="echoid-s1810" xml:space="preserve">Diameter autem portionis ſit B D; </s> <s xml:id="echoid-s1811" xml:space="preserve">& </s> <s xml:id="echoid-s1812" xml:space="preserve"><lb/>diameter circuli à quo portio reſecta eſt, B F, centrum E. <lb/></s> <s xml:id="echoid-s1813" xml:space="preserve">Oſtendendum eſt primo, portionis A B C ad triangulum in-<lb/>ſcriptum majorem eſſe rationem quam ſeſquitertiam. </s> <s xml:id="echoid-s1814" xml:space="preserve">Eſto <lb/>portionis A B C centrum grav. </s> <s xml:id="echoid-s1815" xml:space="preserve">punctum G, & </s> <s xml:id="echoid-s1816" xml:space="preserve">ſecetur D F <lb/>in H, ut ſit H D dupla reliquæ H F.</s> <s xml:id="echoid-s1817" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="1"> <note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">TAB. XL. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s1818" xml:space="preserve">Quoniam igitur F B eſt dupla E B; </s> <s xml:id="echoid-s1819" xml:space="preserve">D B autem minor <lb/>quam dupla G B. </s> <s xml:id="echoid-s1820" xml:space="preserve">Erit major ratio F B ad B D, quam E B <lb/>ad B G. </s> <s xml:id="echoid-s1821" xml:space="preserve">Et per converſionem rationis, minor B F ad F D, <lb/>quam B E ad E G. </s> <s xml:id="echoid-s1822" xml:space="preserve">Et permutando minor B F ad B E, <lb/>(quæ proportio dupla eſt) quam F D ad E G. </s> <s xml:id="echoid-s1823" xml:space="preserve">Igitur F D <lb/>major eſt quam dupla E G. </s> <s xml:id="echoid-s1824" xml:space="preserve">Ipſius autem F D duas tertias <lb/>continet H D. </s> <s xml:id="echoid-s1825" xml:space="preserve">Ergo H D major eſt quam ſeſquitertia E G. <lb/></s> <s xml:id="echoid-s1826" xml:space="preserve">Sicut autem H D ad E G, ita eſt portio A B C ad inſcri-<lb/>ptum ſibi triangulum: </s> <s xml:id="echoid-s1827" xml:space="preserve">hoc enim antehac demonſtravimus in <lb/>Theorematis de Hyperboles Ellipſis & </s> <s xml:id="echoid-s1828" xml:space="preserve">Circuli quadratura <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s1829" xml:space="preserve"> <anchor type="note" xlink:label="note-0093-02a" xlink:href="note-0093-02"/> Itaque major eſt ratio portionis ad inſcriptum triangulum <lb/>A B C quam ſeſquitertia.</s> <s xml:id="echoid-s1830" xml:space="preserve"/> </p> <div xml:id="echoid-div96" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">Vide ſupra <lb/>p. 324.</note> </div> <p> <s xml:id="echoid-s1831" xml:space="preserve">Quod autem ad triangulum A B C portio minorem ha-<lb/>beat rationem quam tripla ſeſquitertia ipſius D F ad diame-<lb/>trum circuli B F unà cum tripla E D, id nunc oſtendemus. <lb/></s> <s xml:id="echoid-s1832" xml:space="preserve">Secetur diameter portionis in R, ut B R ſit ſeſquialtera re-<lb/>liquæ R D. </s> <s xml:id="echoid-s1833" xml:space="preserve">Ergo cadit R punctum inter G & </s> <s xml:id="echoid-s1834" xml:space="preserve">D <anchor type="note" xlink:href="" symbol="*"/> quo- <anchor type="note" xlink:label="note-0093-03a" xlink:href="note-0093-03"/> niam poſitum fuit G centrum gravitatis in portione A B C. <lb/></s> <s xml:id="echoid-s1835" xml:space="preserve">Quumque portionis ad inſcriptum triangulum eadem ſit ra-<lb/>tio, quæ H D ad E G, ut modo dictum fuit; </s> <s xml:id="echoid-s1836" xml:space="preserve">minor au-<lb/>tem ſit ratio H D ad E G, quam H D ad E R: </s> <s xml:id="echoid-s1837" xml:space="preserve">Erit <lb/>propterea minor quoque portionis ad inſcriptum triangulum <lb/>ratio quam H D ad E R, ſive quam H D quinquies ſum-<lb/>pta ad quintuplam E R. </s> <s xml:id="echoid-s1838" xml:space="preserve">Atqui H D, (cum ſit æqualis <lb/>duabus tertiis D F) quinquies ſumpta æquabitur decem ter-<lb/>tiis, hoc eſt, triplæ ſeſquitertiæ D F. </s> <s xml:id="echoid-s1839" xml:space="preserve">E R verò quæ con-<lb/>tinet E D & </s> <s xml:id="echoid-s1840" xml:space="preserve">duas quintas ipſius D B, ſi quinquies ſuma-<lb/>tur, æquabitur duplæ B D & </s> <s xml:id="echoid-s1841" xml:space="preserve">quintuplæ E D; </s> <s xml:id="echoid-s1842" xml:space="preserve">hoc eſt, <lb/>duplæ totius E B atque inſuper triplæ E D. </s> <s xml:id="echoid-s1843" xml:space="preserve">Igitur appa- <pb o="382" file="0094" n="100" rhead="CHRISTIANI HUGENII"/> ret portionem A B C ad inſcriptum triangulum minorem ha-<lb/>bere rationem quam triplam ſeſquitertiam D F ad duplam <lb/>E B, hoc eſt, diametrum B F, unà cum tripla E D. </s> <s xml:id="echoid-s1844" xml:space="preserve">Quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s1845" xml:space="preserve"/> </p> <div xml:id="echoid-div97" type="float" level="2" n="3"> <note symbol="*" position="right" xlink:label="note-0093-03" xlink:href="note-0093-03a" xml:space="preserve">per præced.</note> </div> </div> <div xml:id="echoid-div99" type="section" level="1" n="45"> <head xml:id="echoid-head68" xml:space="preserve"><emph style="sc">Theor</emph>. XVI. <emph style="sc">Propos</emph>. XIX.</head> <p style="it"> <s xml:id="echoid-s1846" xml:space="preserve">ARcus quilibet ſemicirumferentiâ minor, ma-<lb/>jor eſt ſuâ ſubtenſâ ſimul & </s> <s xml:id="echoid-s1847" xml:space="preserve">triente differen-<lb/>tiæ quâ ſubtenſa ſinum excedit. </s> <s xml:id="echoid-s1848" xml:space="preserve">Idem verò minor <lb/>quam ſubtenſa ſimul cum ea quæ ad dictum trien-<lb/>tem ſeſe habeat, ut quadrupla ſubtenſa juncta ſi-<lb/>nui ad ſubtenſam duplam cum ſinu triplo.</s> <s xml:id="echoid-s1849" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1850" xml:space="preserve">Eſto circulus cujus D centrum, diameter F B. </s> <s xml:id="echoid-s1851" xml:space="preserve">Et ſit ar-<lb/> <anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/> cus B A ſemicircumferentiâ minor, cui ſubtenſa ducatur <lb/>B A, ſinus autem A M: </s> <s xml:id="echoid-s1852" xml:space="preserve">quæ nimirum diametro F B ſit ad <lb/>angulos rectos. </s> <s xml:id="echoid-s1853" xml:space="preserve">Porro ipſi A M ſit æqualis recta G H, & </s> <s xml:id="echoid-s1854" xml:space="preserve"><lb/>G I æqualis ſubtenſæ A B. </s> <s xml:id="echoid-s1855" xml:space="preserve">Exceſſus igitur eſt H I; </s> <s xml:id="echoid-s1856" xml:space="preserve">cujus <lb/>triens I K ipſi G I adjiciatur. </s> <s xml:id="echoid-s1857" xml:space="preserve">Oſtendendum eſt primo, ar-<lb/>cum A B totâ G K majorem eſſe. </s> <s xml:id="echoid-s1858" xml:space="preserve">Hoc autem ex Theore-<lb/>mate 7. </s> <s xml:id="echoid-s1859" xml:space="preserve">eſt manifeſtum. </s> <s xml:id="echoid-s1860" xml:space="preserve">At cum ipſi G I additur IO quæ <lb/>ad I K trientem ipſius H I rationem habeat, quam quadru-<lb/>pla G I una cum G H ad duplam G I cum tripla G H. <lb/></s> <s xml:id="echoid-s1861" xml:space="preserve">Dico totam G O arcu A B majorem eſſe. </s> <s xml:id="echoid-s1862" xml:space="preserve">Conſtituantur enim <lb/>ſuper lineis G H, H I, IO, triangula quorum communis <lb/>vertex ſit L, altitudo autem æqualis radio D B. </s> <s xml:id="echoid-s1863" xml:space="preserve">Et junga-<lb/>tur D A, ducaturque diameter circuli C E quæ rectam <lb/>A B bifariam dividat in N, arcum vero A B in E. </s> <s xml:id="echoid-s1864" xml:space="preserve">Et jun-<lb/>gantur A E, E B.</s> <s xml:id="echoid-s1865" xml:space="preserve"/> </p> <div xml:id="echoid-div99" type="float" level="2" n="1"> <note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">TAB. XL. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s1866" xml:space="preserve">Quoniam igitur O I eſt ad I K ut quadrupla G I unà <lb/>cum G H ad duplam G I cum tripla G H; </s> <s xml:id="echoid-s1867" xml:space="preserve">ſumptis conſequen-<lb/>tium triplis erit O I ad I H (hæc enim tripla eſt I K,) ut <lb/>quadrupla G I unà cum G H ad ſexcuplam G I cum non-<lb/>cupla G H. </s> <s xml:id="echoid-s1868" xml:space="preserve">Et componendo, O H ad H I, ut decupla <pb o="383" file="0095" n="101" rhead="DE CIRCULI MAGNIT. INVENTA."/> utriuſque I G, G H ad ſexcuplam I G cum noncupla G H: <lb/></s> <s xml:id="echoid-s1869" xml:space="preserve">vel ſumptis horum trientibus ut decem tertiæ duarum ſimul <lb/>G I, G H ad duplam G I cum tripla G H. </s> <s xml:id="echoid-s1870" xml:space="preserve">Eſt autem ea-<lb/>dem ratio linearum G I ad G H, hoc eſt, B A ad A M, <lb/>quæ B D ad D N, propter ſimiles triangulos B A M, <lb/>B D N. </s> <s xml:id="echoid-s1871" xml:space="preserve">Ergo etiam O H ad H I, ut {10/3} utriuſque ſimul <lb/>B D, D N ad duplam B D cum tripla D N; </s> <s xml:id="echoid-s1872" xml:space="preserve">hoc eſt, ut <lb/>{10/3} N C ad diametrum E C cum tripla D N. </s> <s xml:id="echoid-s1873" xml:space="preserve">Hac autem ra-<lb/>tione minor eſt ratio portionis A E B ad A E B triangulum <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s1874" xml:space="preserve"> <anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/> Ergo dictæ portionis ad dictum triang. </s> <s xml:id="echoid-s1875" xml:space="preserve">minor quoque ratio <lb/>quam O H ad H I, hoc eſt, quam trianguli O H L ad <lb/>triangulum I H L. </s> <s xml:id="echoid-s1876" xml:space="preserve">Triangulum autem I H L æquale eſt tri-<lb/>angulo A E B. </s> <s xml:id="echoid-s1877" xml:space="preserve">Quod ſic oſtenditur. </s> <s xml:id="echoid-s1878" xml:space="preserve">Triangulum enim <lb/>G H L æquale eſt triangulo D A B, quoniam baſes & </s> <s xml:id="echoid-s1879" xml:space="preserve">alti-<lb/>tudines reciprocè æquales habent. </s> <s xml:id="echoid-s1880" xml:space="preserve">Similique ratione quo-<lb/>niam G I æqualis eſt rectæ A B, erit triangulum G I L æ-<lb/>quale duobus ſimul triangulis D A E, D B E, hoc eſt, <lb/>quadrilatero D A E B. </s> <s xml:id="echoid-s1881" xml:space="preserve">Itaque triangulum H I L triangulo <lb/>A E B æquari neceſſe eſt, quod dicebamus. </s> <s xml:id="echoid-s1882" xml:space="preserve">Habebit itaque <lb/>portio A E B ad triangulum ſibi inſcriptum A E B mino-<lb/>rem quoque rationem quam triangulum O H L ad idem tri-<lb/>angulum A E B. </s> <s xml:id="echoid-s1883" xml:space="preserve">Quamobrem triangulum O H L portione <lb/>A E B majus erit. </s> <s xml:id="echoid-s1884" xml:space="preserve">Et totum proinde triangulum O G L <lb/>majus ſectore D A E B. </s> <s xml:id="echoid-s1885" xml:space="preserve">Altitudo autem trianguli G L O <lb/>æqualis eſt radio D B. </s> <s xml:id="echoid-s1886" xml:space="preserve">Ergo baſis G O major erit arcu A B. <lb/></s> <s xml:id="echoid-s1887" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1888" xml:space="preserve"/> </p> <div xml:id="echoid-div100" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">per præced.</note> </div> <p style="it"> <s xml:id="echoid-s1889" xml:space="preserve">Ex his autem manifeſtum eſt de tota quoque circumferen-<lb/>tia pronunciari poſſe, quod, Si circulo inſcribantur polygona <lb/>duo æquilatera, quorum alterum alterius ſit duplo laterum nu-<lb/>mero, & </s> <s xml:id="echoid-s1890" xml:space="preserve">differentiæ perimetrorum triens perimetro polygoni <lb/>majoris adjungatur, compoſita ex his circuli circumferentiâ mi-<lb/>nor erit. </s> <s xml:id="echoid-s1891" xml:space="preserve">Eidem vero majori perimetro ſi linea addatur quæ <lb/>ad dictum differentiæ trientem ſeſe habeat, ſicut quadrupla <lb/>perimetri majoris juncta perimetro minori, ad duplam ma-<lb/>joris cum tripla minoris, compoſita circumferentiam circuli ex-<lb/>cedet.</s> <s xml:id="echoid-s1892" xml:space="preserve"/> </p> <pb o="384" file="0096" n="102" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div102" type="section" level="1" n="46"> <head xml:id="echoid-head69" xml:space="preserve"><emph style="sc">Problema</emph> IV. <emph style="sc">Propos</emph>. XX.</head> <p style="it"> <s xml:id="echoid-s1893" xml:space="preserve">CIrcumferentiæ ad diametrum rationem inve-<lb/>ſtigare; </s> <s xml:id="echoid-s1894" xml:space="preserve">& </s> <s xml:id="echoid-s1895" xml:space="preserve">ex datis inſcriptis in dato circulo <lb/>invenire longitudinem arcuum quibus illæ ſubtendun-<lb/>tur.</s> <s xml:id="echoid-s1896" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1897" xml:space="preserve">Eſto circulus centro D, cujus diameter C B, & </s> <s xml:id="echoid-s1898" xml:space="preserve">ſit arcus <lb/> <anchor type="note" xlink:label="note-0096-01a" xlink:href="note-0096-01"/> B A ſextans circumferentiæ, cui ſubtenſa ducatur A B, <lb/>itemque ſinus A M. </s> <s xml:id="echoid-s1899" xml:space="preserve">Poſitâ igitur D B ſemidiametro par-<lb/>tium 100000, totidem quoque erit ſubtenſa B A. </s> <s xml:id="echoid-s1900" xml:space="preserve">A M ve-<lb/>rò partium 86603 non unâ minus, hoc eſt, ſi una pars ſi-<lb/>ve unitas auferatur ab 86603 fiet minor debito. </s> <s xml:id="echoid-s1901" xml:space="preserve">quippe ſe-<lb/>miſſis lateris trianguli æquilateri circulo inſcripti.</s> <s xml:id="echoid-s1902" xml:space="preserve"/> </p> <div xml:id="echoid-div102" type="float" level="2" n="1"> <note position="left" xlink:label="note-0096-01" xlink:href="note-0096-01a" xml:space="preserve">TAB. XL. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s1903" xml:space="preserve">Hinc exceſſus A B ſupra A M fit 13397 vero minor. <lb/></s> <s xml:id="echoid-s1904" xml:space="preserve">Cujus triens 4465 {2/3} additus ipſi A B 100000, fiunt partes <lb/>104465 {2/3} minores arcu A B. </s> <s xml:id="echoid-s1905" xml:space="preserve">Et hic primus eſt minor termi-<lb/>nus, quo poſtea alium vero propiorem inveniemus. </s> <s xml:id="echoid-s1906" xml:space="preserve">Prius <lb/>autem major quoque terminus ſecundum Theorema præce-<lb/>dens inquirendus eſt.</s> <s xml:id="echoid-s1907" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1908" xml:space="preserve">Tres nimirum ſunt numeri quibus quartum proportiona-<lb/>lem invenire oportet. </s> <s xml:id="echoid-s1909" xml:space="preserve">Primus eſt partium duplæ A B & </s> <s xml:id="echoid-s1910" xml:space="preserve">tri-<lb/>plæ A M qui erit 459807, vero minor, (nam hoc quoque <lb/>obſervandum ut minor ſit, idemque in cæteris prout dicetur) <lb/>ſecundus quadruplæ A B & </s> <s xml:id="echoid-s1911" xml:space="preserve">ſimplæ A M qui 486603 vero <lb/>maj. </s> <s xml:id="echoid-s1912" xml:space="preserve">Et tertius triens exceſſus A B ſupra A M, 4466 vero <lb/>major. </s> <s xml:id="echoid-s1913" xml:space="preserve">Itaque quartus proportionalis erit 4727 vero maj. <lb/></s> <s xml:id="echoid-s1914" xml:space="preserve">quo addito ad A B 100000 fit 104727, major numero par-<lb/>tium, quas continet arcus A B, peripheriæ ſextans. </s> <s xml:id="echoid-s1915" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> Jam <anchor type="note" xlink:label="note-0096-02a" xlink:href="note-0096-02"/> igitur invenimus longitudinem arcus A B ſecundum mino-<lb/>rem majoremque terminum, quorum hic quidem longe pro-<lb/>pior vero eſt, cum vero proximus ſit 104719.</s> <s xml:id="echoid-s1916" xml:space="preserve"/> </p> <div xml:id="echoid-div103" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0096-02" xlink:href="note-0096-02a" xml:space="preserve">per præced.</note> </div> <p> <s xml:id="echoid-s1917" xml:space="preserve">Sed ex utroque iſtorum alius minor terminus habebi-<lb/>tur priore accuratior ſi@ utamur præcepto ſequenti, quod <lb/>à diligentiori centrorum gravitatis inſpectione dependet.</s> <s xml:id="echoid-s1918" xml:space="preserve"/> </p> <pb o="385" file="0097" n="103" rhead="DE CIRCULI MAGNIT. INVENTA."/> <p style="it"> <s xml:id="echoid-s1919" xml:space="preserve">Inventorum terminorum differentia ſeſquitertia jungatur du-<lb/>plæ ſubtenſæ & </s> <s xml:id="echoid-s1920" xml:space="preserve">ſinui triplo, & </s> <s xml:id="echoid-s1921" xml:space="preserve">quam rationem habet ex his <lb/>compoſita ad triplam ſeſquitertiam ſeu {10/3} utriuſque ſimul, ſi-<lb/>nus, ſubtenſæque, eandem habeat ſubtenſæ ſupra ſinum exceſ-<lb/>ſus ad aliam quandam; </s> <s xml:id="echoid-s1922" xml:space="preserve">Hæc ad ſinum addita rectam conſtituet <lb/>arcu minorem.</s> <s xml:id="echoid-s1923" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1924" xml:space="preserve">Minor terminus erat 104465 {2/3}. </s> <s xml:id="echoid-s1925" xml:space="preserve">Major 104727. </s> <s xml:id="echoid-s1926" xml:space="preserve">differentia <lb/>horum eſt 261 {1/3}. </s> <s xml:id="echoid-s1927" xml:space="preserve">Eſtque rurſus tribus numeris inveniendus <lb/>quartus proportionalis. </s> <s xml:id="echoid-s1928" xml:space="preserve">Primus eſt partium duplæ A B & </s> <s xml:id="echoid-s1929" xml:space="preserve"><lb/>triplæ A M & </s> <s xml:id="echoid-s1930" xml:space="preserve">ſeſquitertiæ terminorum differentiæ, 460158 <lb/>vero major. </s> <s xml:id="echoid-s1931" xml:space="preserve">Secundus {10/3} utriuſque ſimul A B, A M, 622008 <lb/>vero minor. </s> <s xml:id="echoid-s1932" xml:space="preserve">Tertius denique exceſſus A B ſupra A M, 13397 <lb/>vero min. </s> <s xml:id="echoid-s1933" xml:space="preserve">Quibus quartus proportionalis eſt 18109 vero min. <lb/></s> <s xml:id="echoid-s1934" xml:space="preserve">Hic igitur additus numero partium A M 86602 {1/2} vero min. </s> <s xml:id="echoid-s1935" xml:space="preserve"><lb/>fiunt 104711 {1/2} minores arcu A B. </s> <s xml:id="echoid-s1936" xml:space="preserve">Quare ſexcuplum earum, <lb/>628269 minus erit circumferentiâ totâ. </s> <s xml:id="echoid-s1937" xml:space="preserve">At quoniam 104727 <lb/>majores inventæ ſunt arcu A B, earum ſexcuplum 628362 <lb/>circumferentiâ majus erit. </s> <s xml:id="echoid-s1938" xml:space="preserve">Itaque circumferentiæ ad diame-<lb/>trum ratio minor eſt quam 628362, major autem quam 628269 <lb/>ad 200000. </s> <s xml:id="echoid-s1939" xml:space="preserve">Sive minor quam 314181, major autem quam <lb/>314135 ad 100000. </s> <s xml:id="echoid-s1940" xml:space="preserve">Unde conſtat minorem utique eſſe quam <lb/>triplam ſeſquiſeptimam, & </s> <s xml:id="echoid-s1941" xml:space="preserve">majorem quam 3 {10/@}. </s> <s xml:id="echoid-s1942" xml:space="preserve">Quin etiam <lb/>Longomontani error per hæc refutatur, qui ſcripſit peri-<lb/>pheriam majorem eſſe partibus 314185 qualium rad. </s> <s xml:id="echoid-s1943" xml:space="preserve"><lb/>100000.</s> <s xml:id="echoid-s1944" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1945" xml:space="preserve">Eſto nunc arcus A B {1/@} circumferentiæ, & </s> <s xml:id="echoid-s1946" xml:space="preserve">erit A M, ſe-<lb/>miſſis lateris quadrati circulo inſcripti, partium 7071068, <lb/>non unâ minus, qualium radius D B 10000000. </s> <s xml:id="echoid-s1947" xml:space="preserve">A B vero <lb/>latus octanguli partium 7653668 non unâ majus. </s> <s xml:id="echoid-s1948" xml:space="preserve">Quibus da-<lb/>tis ad ſimilitudinem præcedentium invenietur primus minor <lb/>terminus longitudinis arcus A B 7847868. </s> <s xml:id="echoid-s1949" xml:space="preserve">Deinde major <lb/>terminus 7854066. </s> <s xml:id="echoid-s1950" xml:space="preserve">Et ex utroque rurſus terminus minor ac-<lb/>curatior 7853885. </s> <s xml:id="echoid-s1951" xml:space="preserve">Unde conſtat peripheriæ ad diametrum <lb/>rationem minorem haberi quam 31416 {1/3}, majorem autem quam <lb/>31415 ad 10000.</s> <s xml:id="echoid-s1952" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1953" xml:space="preserve">Et quum terminus major 7854066 à vera arcus A B lon- <pb o="386" file="0098" n="104" rhead="CHRISTIANI HUGENII"/> gitudine minus diſtet quam partibus 85; </s> <s xml:id="echoid-s1954" xml:space="preserve">(Eſt enim arcus <lb/>A B, per ea quæ ſupra oſtendimus, major quam 7853981) <lb/>partes autem 85 efficiant minus quam duos ſcrupulos ſecun-<lb/>dos, hoc eſt, quam {2/1296600} circumferentiæ, nam tota earun-<lb/>dem plures habet quam 60000000: </s> <s xml:id="echoid-s1955" xml:space="preserve">Hinc manifeſtum eſt, ſi <lb/>trianguli rectanguli angulos quæramus ex datis lateribus, eo <lb/>modo quo majorem iſtum terminum paulò antè, nunquam <lb/>duobus ſcrupulis ſecundis aberraturos; </s> <s xml:id="echoid-s1956" xml:space="preserve">etiamſi æqualia inter <lb/>ſe fuerint latera circa angulum rectum, veluti hic erant in <lb/>triangulo D A M.</s> <s xml:id="echoid-s1957" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1958" xml:space="preserve">Si vero ea ſit ratio lateris D M ad M A, ut angulus A D M <lb/>non excedat {1/4} recti; </s> <s xml:id="echoid-s1959" xml:space="preserve">non unius tertii ſcrupuli error erit. </s> <s xml:id="echoid-s1960" xml:space="preserve">Po-<lb/>ſito enim arcu A B {1/16} circumferentiæ, erit A M ſemiſſis la-<lb/>teris octanguli æquilateri circulo inſcripti partium 382683433, <lb/>non unâ minus. </s> <s xml:id="echoid-s1961" xml:space="preserve">A B vero latus ſexdecanguli 390180644 <lb/>non unâ amplius, qualium radius D B 1000000000. </s> <s xml:id="echoid-s1962" xml:space="preserve">Unde <lb/>primus minor terminus longitudinis arcus A B invenitur par-<lb/>tium 392679714. </s> <s xml:id="echoid-s1963" xml:space="preserve">Terminus autem major 392699148. </s> <s xml:id="echoid-s1964" xml:space="preserve">Et ex <lb/>his minor rurſus 392699010. </s> <s xml:id="echoid-s1965" xml:space="preserve">Conſtat autem ex ſupra demon-<lb/>ſtratis arcum A B {1/16} peripheriæ, majorem eſſe quam <lb/>392699081, quas terminus major ſuperat partibus 67. </s> <s xml:id="echoid-s1966" xml:space="preserve">Hæ <lb/>autem minus efficiunt uno ſcrupulo tertio, hoc eſt, {1/77760000} to-<lb/>tius circumferentiæ, quoniam ea major eſt utique quam <lb/>6000000000.</s> <s xml:id="echoid-s1967" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1968" xml:space="preserve">Porro ex noviſſimis terminis inventis orietur ratio circum-<lb/>ferentiæ ad diametrum minor quam 3141593 {1/5}, major autem <lb/>quam 3141592 ad 1000000.</s> <s xml:id="echoid-s1969" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1970" xml:space="preserve">Quod ſi {1/60} circumferentiæ ponatur arcus A B, ſeu par-<lb/>tium 6 qualium tota 360: </s> <s xml:id="echoid-s1971" xml:space="preserve">Erit A M ſemiſſis lateris trigin-<lb/>tanguli inſcripti partium 10452846326766, non unâ minus, <lb/>qualium radius 100000000000000. </s> <s xml:id="echoid-s1972" xml:space="preserve">Et A B latus ſexagintan-<lb/>guli inſcripti 10467191248588 non unâ amplius. </s> <s xml:id="echoid-s1973" xml:space="preserve">Invenietur-<lb/>que ex his arcus A B ſecundum primum minorem terminum <lb/>10471972889195. </s> <s xml:id="echoid-s1974" xml:space="preserve">Secundum majorem 10471975512584. <lb/></s> <s xml:id="echoid-s1975" xml:space="preserve">Et ex his minor alter terminus 10471975511302. </s> <s xml:id="echoid-s1976" xml:space="preserve">Unde <lb/>efficitur peripheriæ ad diametrum ratio minor quam <pb file="0099" n="105"/> <pb o="386" file="0099a" n="106"/> <anchor type="figure" xlink:label="fig-0099a-01a" xlink:href="fig-0099a-01"/> <anchor type="figure" xlink:label="fig-0099a-02a" xlink:href="fig-0099a-02"/> <anchor type="figure" xlink:label="fig-0099a-03a" xlink:href="fig-0099a-03"/> <anchor type="figure" xlink:label="fig-0099a-04a" xlink:href="fig-0099a-04"/> <anchor type="figure" xlink:label="fig-0099a-05a" xlink:href="fig-0099a-05"/> <anchor type="figure" xlink:label="fig-0099a-06a" xlink:href="fig-0099a-06"/> <pb file="0100" n="107"/> <pb o="387" file="0101" n="108" rhead="DE CIRCULI MAGNIT. INVENTA."/> 31415926538, major autem quam 31415926533 ad <lb/>10000000000.</s> <s xml:id="echoid-s1977" xml:space="preserve"/> </p> <div xml:id="echoid-div104" type="float" level="2" n="3"> <figure xlink:label="fig-0099a-01" xlink:href="fig-0099a-01a"> <caption xml:id="echoid-caption40" style="it" xml:space="preserve">Pag. 386.<lb/>TAB. XL.<lb/>Fig. 2.</caption> <variables xml:id="echoid-variables40" xml:space="preserve">K B H F G E A I D L C</variables> </figure> <figure xlink:label="fig-0099a-02" xlink:href="fig-0099a-02a"> <caption xml:id="echoid-caption41" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables41" xml:space="preserve">L K E D H C A G B</variables> </figure> <figure xlink:label="fig-0099a-03" xlink:href="fig-0099a-03a"> <caption xml:id="echoid-caption42" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables42" xml:space="preserve">B Q N L M F G S H K A D C P</variables> </figure> <figure xlink:label="fig-0099a-04" xlink:href="fig-0099a-04a"> <caption xml:id="echoid-caption43" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables43" xml:space="preserve">B G R A C D E H F</variables> </figure> <figure xlink:label="fig-0099a-05" xlink:href="fig-0099a-05a"> <caption xml:id="echoid-caption44" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables44" xml:space="preserve">A C D M B</variables> </figure> <figure xlink:label="fig-0099a-06" xlink:href="fig-0099a-06a"> <caption xml:id="echoid-caption45" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables45" xml:space="preserve">A E N F B L D M C G H I K O</variables> </figure> </div> <p> <s xml:id="echoid-s1978" xml:space="preserve">Quos terminos ſi ex additis inſcriptorum & </s> <s xml:id="echoid-s1979" xml:space="preserve">circumſcripto-<lb/>rum polygonorum lateribus inquirendum eſſet ferè ad late-<lb/>rum quadringenta millia deveniendum. </s> <s xml:id="echoid-s1980" xml:space="preserve">Nam ex ſexagintan-<lb/>gulo inſcripto circumſcriptoque hoc tantum probatur, mi-<lb/>norem eſſe rationem peripheriæ ad diametrum quam 3145 ad <lb/>1000, majorem autem quam 3140. </s> <s xml:id="echoid-s1981" xml:space="preserve">Adeo ut triplum & </s> <s xml:id="echoid-s1982" xml:space="preserve">am-<lb/>plius verarum notarum numerum noſtro ratiocinio produ-<lb/>ctum appareat. </s> <s xml:id="echoid-s1983" xml:space="preserve">Idem vero in ulterioribus polygonis ſi quis <lb/>experiatur ſemper evenire cernet: </s> <s xml:id="echoid-s1984" xml:space="preserve">non ignota nobis ratione, <lb/>ſed quæ longiori explicatione indigeret.</s> <s xml:id="echoid-s1985" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1986" xml:space="preserve">Porro autem quomodo, datis quibuſcunque aliis inſcriptis, <lb/>arcuum quibus ſubtenduntur longitudo per hæc inveniri que-<lb/>at ſatis puto manifeſtum. </s> <s xml:id="echoid-s1987" xml:space="preserve">Si enim quadrati inſcripti latere <lb/>majores ſunt, longitudo arcus ad ſemicircumferentiam reli-<lb/>qui inquirenda eſt, cujus tum quoque ſubtenſa datur. </s> <s xml:id="echoid-s1988" xml:space="preserve">Sci-<lb/>endum autem & </s> <s xml:id="echoid-s1989" xml:space="preserve">dimidiorum arcuum ſubtenſas inveniri cum <lb/>totius arcus ſubtenſa data eſt. </s> <s xml:id="echoid-s1990" xml:space="preserve">Atque hâc ratione ſi biſectioni-<lb/>bus uti placebit, poterimus ad omnem ſubtenſam, arcus i-<lb/>pſius longitudinem quamlibet veræ propinquam non difficul-<lb/>ter cognoſcere. </s> <s xml:id="echoid-s1991" xml:space="preserve">Utile hoc ad ſinuum tabulas examinandas. <lb/></s> <s xml:id="echoid-s1992" xml:space="preserve">Imo ad componendas quoque: </s> <s xml:id="echoid-s1993" xml:space="preserve">quia cognitâ arcus alicujus <lb/>ſubtenſâ, etiam ejus qui paulò major minorve ſit ſatis accu-<lb/>ratè definiri poteſt.</s> <s xml:id="echoid-s1994" xml:space="preserve"/> </p> <figure> <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0101-01"/> </figure> <pb file="0102" n="109"/> <figure> <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0102-01"/> </figure> </div> <div xml:id="echoid-div106" type="section" level="1" n="47"> <head xml:id="echoid-head70" xml:space="preserve"><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> C. F.</head> <head xml:id="echoid-head71" xml:space="preserve">ILLVSTRIVM QVORVNDAM</head> <head xml:id="echoid-head72" xml:space="preserve">PROBLEMATVM CONSTRVCTIONES.</head> <head xml:id="echoid-head73" xml:space="preserve"><emph style="sc">Probl</emph>. I.</head> <head xml:id="echoid-head74" style="it" xml:space="preserve">Datam ſphæram plano ſecare, ut portiones <lb/>inter ſe rationem habeant datam.</head> <p> <s xml:id="echoid-s1995" xml:space="preserve">HOc Archimedes problema reſolvit lib. </s> <s xml:id="echoid-s1996" xml:space="preserve">2. </s> <s xml:id="echoid-s1997" xml:space="preserve">de <lb/>Sphæra & </s> <s xml:id="echoid-s1998" xml:space="preserve">Cylind. </s> <s xml:id="echoid-s1999" xml:space="preserve">Compoſitionem autem pro-<lb/>miſſam non videtur explicuiſſe, niſi ipſius eſt il-<lb/>la quam Eutocius in vetuſto quodam libro reper-<lb/>tam commentariis ſuis inferuit. </s> <s xml:id="echoid-s2000" xml:space="preserve">Ea vero parabo-<lb/>les & </s> <s xml:id="echoid-s2001" xml:space="preserve">hyperboles interſectione perficitur, uti & </s> <s xml:id="echoid-s2002" xml:space="preserve">illa cujus <lb/>Dionyſidorus autor eſt, quæ tamen à priori differt. </s> <s xml:id="echoid-s2003" xml:space="preserve">Præter <lb/>has tertiam quoque adfert Eutocius è Dioclis de Pyriis li-<lb/>bro, quæ hyperboles & </s> <s xml:id="echoid-s2004" xml:space="preserve">ellipſis deſcriptionem requirit. </s> <s xml:id="echoid-s2005" xml:space="preserve">No-<lb/>ſtra autem quam hic conſcribemus anguli triſectionem poſtu-<lb/>lat; </s> <s xml:id="echoid-s2006" xml:space="preserve">Et hæc conſtruendi ratio in ſolidis problematibus quo-<lb/>dammodo ſimpliciſſima videtur, atque ad uſum maxime ac-<lb/>commodata.</s> <s xml:id="echoid-s2007" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2008" xml:space="preserve">Eſto igitur data ſphæra cujus centrum M, diameter C A. <lb/></s> <s xml:id="echoid-s2009" xml:space="preserve"> <anchor type="note" xlink:label="note-0102-01a" xlink:href="note-0102-01"/> Et data ſit proportio lineæ S ad T majoris ad minorem. <lb/></s> <s xml:id="echoid-s2010" xml:space="preserve">Intelligatur ſecari ſphæra plano ſecundum A C diametrum, <lb/>ſitque maximus in ea circulus C B A D. </s> <s xml:id="echoid-s2011" xml:space="preserve">Et producatur <lb/>utrimque diameter C A, & </s> <s xml:id="echoid-s2012" xml:space="preserve">ponatur ſemidiametro æqua-<lb/>lis utraque harum C H, A E. </s> <s xml:id="echoid-s2013" xml:space="preserve">Et dividatur tota H E in <lb/>Q, ut ſit E Q ad Q H ſicut S ad T. </s> <s xml:id="echoid-s2014" xml:space="preserve">Ipſi autem M Q æqua-<lb/>lis ponatur ad circumferentiam recta A R. </s> <s xml:id="echoid-s2015" xml:space="preserve">Et ei quæ ter-<lb/>tiam partem ſubtendit arcus A R, æqualis ſumatur M N.</s> <s xml:id="echoid-s2016" xml:space="preserve"> <pb o="389" file="0103" n="110" rhead="CHR.HUGENII ILL.QUOR.PROB.CONSTR."/> Et per N punctum ducatur planum K L quod diametro <lb/>C A ſit ad angulos rectos. </s> <s xml:id="echoid-s2017" xml:space="preserve">Dico hoc ſphæram ſic ſecare, ut <lb/>portio cujus A vertex eſt ad eam cujus vertex C rationem ha-<lb/>beat quam S ad T.</s> <s xml:id="echoid-s2018" xml:space="preserve"/> </p> <div xml:id="echoid-div106" type="float" level="2" n="1"> <note position="left" xlink:label="note-0102-01" xlink:href="note-0102-01a" xml:space="preserve">TAB. XLI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s2019" xml:space="preserve">Secetur enim ſphæra per centrum M plano B D ipſi K L <lb/>parallelo, & </s> <s xml:id="echoid-s2020" xml:space="preserve">jungantur K M, M L; </s> <s xml:id="echoid-s2021" xml:space="preserve">& </s> <s xml:id="echoid-s2022" xml:space="preserve">intelligatur conus <lb/>baſin habens circulum factum ſectione K L, verticem vero <lb/>M. </s> <s xml:id="echoid-s2023" xml:space="preserve">Et ſicut quadratum C M ad quadratum M N, ita ſit <lb/>M N ad N O longitudine. </s> <s xml:id="echoid-s2024" xml:space="preserve">Erit igitur per converſionem ra-<lb/>tionis ut quadratum C M ſive quadr. </s> <s xml:id="echoid-s2025" xml:space="preserve">K M ad quadratum <lb/>K N (eſt enim quadr. </s> <s xml:id="echoid-s2026" xml:space="preserve">K N exceſſus quadrati K M ſupra <lb/>quadr. </s> <s xml:id="echoid-s2027" xml:space="preserve">M N) ita linea N M ad M O. </s> <s xml:id="echoid-s2028" xml:space="preserve">Sicut autem quadr. <lb/></s> <s xml:id="echoid-s2029" xml:space="preserve">K M, hoc eſt, quadr. </s> <s xml:id="echoid-s2030" xml:space="preserve">B M ad quadr. </s> <s xml:id="echoid-s2031" xml:space="preserve">K N, ita eſt circu-<lb/>lus circa diametrum B D ad eum qui circa diametrum K L. </s> <s xml:id="echoid-s2032" xml:space="preserve"><lb/>Ergo quoque ille circulus ad hunc ſeſe habebit ut N M ad <lb/>M O. </s> <s xml:id="echoid-s2033" xml:space="preserve">Ac proinde conus K M L æqualis erit cono cujus <lb/>baſis circulus circa diametrum B D, altitudo M O <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2034" xml:space="preserve">Hic <anchor type="note" xlink:label="note-0103-01a" xlink:href="note-0103-01"/> autem conus ad hemiſphæram B C D, hoc eſt, ad conum <lb/>qui baſin habeat eundem circulum circa B D diametrum, <lb/>& </s> <s xml:id="echoid-s2035" xml:space="preserve">altitudinem M H <anchor type="note" xlink:href="" symbol="*"/>, eam habet rationem quam M O ad <anchor type="note" xlink:label="note-0103-02a" xlink:href="note-0103-02"/> M H. </s> <s xml:id="echoid-s2036" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s2037" xml:space="preserve">conus K M L erit ad hemiſphæram B C D <lb/>ſicut M O ad M H. </s> <s xml:id="echoid-s2038" xml:space="preserve">Et invertendo.</s> <s xml:id="echoid-s2039" xml:space="preserve"/> </p> <div xml:id="echoid-div107" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0103-01" xlink:href="note-0103-01a" xml:space="preserve">15. 12. E. <lb/>lem.</note> <note symbol="*" position="right" xlink:label="note-0103-02" xlink:href="note-0103-02a" xml:space="preserve">32. 1. Ar-<lb/>chim. de <lb/>Sphær. & <lb/>Cylin.</note> </div> <p> <s xml:id="echoid-s2040" xml:space="preserve">Porro autem quoniam hemiſphæra B C D eſt ad ſectorem <lb/>ſolidum M K C L ſicut ſuperficies illius ſphærica ad ſphæ-<lb/>ricam hujus ſuperficiem <anchor type="note" xlink:href="" symbol="*"/>, hoc eſt, ut M C ad C N <anchor type="note" xlink:href="" symbol="†"/>.</s> <s xml:id="echoid-s2041" xml:space="preserve"> <anchor type="note" xlink:label="note-0103-03a" xlink:href="note-0103-03"/> Erit per converſionem rationis hemiſphæra B C D ad par <lb/>tem ſui quæ remanet dempto ſectore M K C L, ſicut C M <lb/> <anchor type="note" xlink:label="note-0103-04a" xlink:href="note-0103-04"/> ad M N: </s> <s xml:id="echoid-s2042" xml:space="preserve">vel ſumptis horum duplis ut H M ad O Q. </s> <s xml:id="echoid-s2043" xml:space="preserve">Quod <lb/>enim O Q dupla ſit ipſius M N poſtea oſtendemus. </s> <s xml:id="echoid-s2044" xml:space="preserve">Fuit <lb/>autem oſtenſum, quod hemiſphæra B C D ad conum K M L <lb/>ſicut H M ad M O. </s> <s xml:id="echoid-s2045" xml:space="preserve">Ergo jam hemiſphæra B C D ad to-<lb/>tam portionem inter plana B D, K L contentam erit ut H M <lb/>ad utramque ſimul Q O, O M <anchor type="note" xlink:href="" symbol="*"/>, hoc eſt, ad M Q. </s> <s xml:id="echoid-s2046" xml:space="preserve">Qua- <anchor type="note" xlink:label="note-0103-05a" xlink:href="note-0103-05"/> re & </s> <s xml:id="echoid-s2047" xml:space="preserve">per converſionem rationis, erit hemiſphæra B C D ad <lb/>portionem K C L, ut M H ad H Q. </s> <s xml:id="echoid-s2048" xml:space="preserve">Et ſumptis antece-<lb/>dentium duplis, ſphæra tota ad portionem K C L ut E H <pb o="390" file="0104" n="111" rhead="CHRISTIANI HUGENII"/> ad H Q. </s> <s xml:id="echoid-s2049" xml:space="preserve">Et dividendo, portio K A L ad portionem <lb/>K C L ut E Q ad Q H, hoc eſt ut S ad T. </s> <s xml:id="echoid-s2050" xml:space="preserve">Quod erat fa-<lb/>ciendum. </s> <s xml:id="echoid-s2051" xml:space="preserve">Quod autem dictum fuit O Q duplam eſſe ipſius <lb/>M N, ſic fiet manifeſtum. </s> <s xml:id="echoid-s2052" xml:space="preserve">Quia enim ut quadratum C M <lb/>ad quadr. </s> <s xml:id="echoid-s2053" xml:space="preserve">M N, ita eſt M N ad N O longitudine: </s> <s xml:id="echoid-s2054" xml:space="preserve">Eſt au-<lb/>tem Q M æqualis ſubtenſæ arcus A R cujus trienti ſubten-<lb/>ditur M N. </s> <s xml:id="echoid-s2055" xml:space="preserve">Erunt propterea duæ ſimul Q M & </s> <s xml:id="echoid-s2056" xml:space="preserve">N O æqua-<lb/>les triplæ M N, uti ſequenti lemmate demonſtratur. </s> <s xml:id="echoid-s2057" xml:space="preserve">Quam-<lb/>obrem ablata communi O N, erit ſola Q M æqualis du-<lb/>plæ N M & </s> <s xml:id="echoid-s2058" xml:space="preserve">ipſi M O. </s> <s xml:id="echoid-s2059" xml:space="preserve">Sed eadem Q M æqualis eſt dua-<lb/>bus ſimul his Q O, O M, ergo apparet duplam M N æqua-<lb/>ri ipſi O Q.</s> <s xml:id="echoid-s2060" xml:space="preserve"/> </p> <div xml:id="echoid-div108" type="float" level="2" n="3"> <note symbol="*" position="right" xlink:label="note-0103-03" xlink:href="note-0103-03a" xml:space="preserve">42. 1. Ar-<lb/>chim. de <lb/>Sphær. & <lb/>Cyl.</note> <note symbol="†" position="right" xlink:label="note-0103-04" xlink:href="note-0103-04a" xml:space="preserve">3. 2. Ar-<lb/>chim. de <lb/>Sphær. <lb/>& Cyl.</note> <note symbol="*" position="right" xlink:label="note-0103-05" xlink:href="note-0103-05a" xml:space="preserve">24. 5 Ele@.</note> </div> </div> <div xml:id="echoid-div110" type="section" level="1" n="48"> <head xml:id="echoid-head75" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s2061" xml:space="preserve">SI Circumferentiæ arcus in tria æqualia ſecetur, tres ſimul <lb/> <anchor type="note" xlink:label="note-0104-01a" xlink:href="note-0104-01"/> rectæ quæ æqualibus partibus ſubtenduntur, æquantur ſub-<lb/>tenſæ arcus totius & </s> <s xml:id="echoid-s2062" xml:space="preserve">ei quæ ad ſubtenſam trientis ſeſe habe-<lb/>at, ſicut hujus quadratum ad quadratum ſemidiametri. </s> <s xml:id="echoid-s2063" xml:space="preserve">Ar-<lb/>cus ſectoris A B C in tria æqualia diviſus ſit punctis D, E. <lb/></s> <s xml:id="echoid-s2064" xml:space="preserve">Et ſubtendantur partibus rectæ B D, D E, E C; </s> <s xml:id="echoid-s2065" xml:space="preserve">& </s> <s xml:id="echoid-s2066" xml:space="preserve">toti <lb/>arcui linea B C. </s> <s xml:id="echoid-s2067" xml:space="preserve">Porro jungantur D A, E A, atque inter-<lb/>ſecent ſubtenſam B C in punctis G & </s> <s xml:id="echoid-s2068" xml:space="preserve">H. </s> <s xml:id="echoid-s2069" xml:space="preserve">Sitque H F pa-<lb/>rallela G D.</s> <s xml:id="echoid-s2070" xml:space="preserve"/> </p> <div xml:id="echoid-div110" type="float" level="2" n="1"> <note position="left" xlink:label="note-0104-01" xlink:href="note-0104-01a" xml:space="preserve">TAB. XLI. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s2071" xml:space="preserve">Quoniam igitur circumferentia B D E dupla eſt circum-<lb/>ferentiæ E C, angulus autem huic inſiſtens E A C ad cen-<lb/>trum conſtitutus, qui vero illi inſiſtit angulus B C E ad <lb/>circumferentiam. </s> <s xml:id="echoid-s2072" xml:space="preserve">Erit propterea angulus B C E, hoc eſt, <lb/>angulus H C E in triangulo H C E æqualis angulo C A E <lb/>in triangulo C A E. </s> <s xml:id="echoid-s2073" xml:space="preserve">Sed angulus ad E utrique eſt commu-<lb/>nis; </s> <s xml:id="echoid-s2074" xml:space="preserve">itaque ſimiles inter ſe ſunt dicti trianguli: </s> <s xml:id="echoid-s2075" xml:space="preserve">Eritque ut <lb/>A E ad E C ita E C ad E H. </s> <s xml:id="echoid-s2076" xml:space="preserve">Ratio igitur A E ad E H <lb/>hoc eſt D E ad E F, duplicata eſt rationis A E ad E C, <lb/>ac proinde cadem quæ quadrati A E ad quadr. </s> <s xml:id="echoid-s2077" xml:space="preserve">E C ſeu <lb/>quadr. </s> <s xml:id="echoid-s2078" xml:space="preserve">E D. </s> <s xml:id="echoid-s2079" xml:space="preserve">Erit igitur invertendo F E ad E D ſicut <lb/>quadr. </s> <s xml:id="echoid-s2080" xml:space="preserve">E D ad quadr. </s> <s xml:id="echoid-s2081" xml:space="preserve">E A. </s> <s xml:id="echoid-s2082" xml:space="preserve">Quamobrem oſtendendum <lb/>eſt, tres ſimul B D, D E, E C æquari ſubtenſæ B C <pb o="391" file="0105" n="112" rhead="ILLUST.QUORUND.PROB.CONSTRUCT."/> atque ipſi E F. </s> <s xml:id="echoid-s2083" xml:space="preserve">quod ſane manifeſtum eſt; </s> <s xml:id="echoid-s2084" xml:space="preserve">nam C E eſt æ-<lb/>qualis C H; </s> <s xml:id="echoid-s2085" xml:space="preserve">B D æqualis B G; </s> <s xml:id="echoid-s2086" xml:space="preserve">D E vero utriſque ſimul <lb/>G H, & </s> <s xml:id="echoid-s2087" xml:space="preserve">F E. </s> <s xml:id="echoid-s2088" xml:space="preserve">Ergo conſtat propoſitum.</s> <s xml:id="echoid-s2089" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2090" xml:space="preserve">Sumpſimus autem arcum B C ſemicircumferentiâ mino-<lb/>rem quoniam in conſtructione problematis ejuſmodi ſemper <lb/>invenitur. </s> <s xml:id="echoid-s2091" xml:space="preserve">Nam lemma ad quoſvis arcus pertinet, eſtque <lb/>in ſemicircumferentiâ majoribus demonſtratio parum diverſa.</s> <s xml:id="echoid-s2092" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div112" type="section" level="1" n="49"> <head xml:id="echoid-head76" xml:space="preserve"><emph style="sc">Probl</emph>. II.</head> <head xml:id="echoid-head77" style="it" xml:space="preserve">Cubum invenire dati cubi duplum.</head> <p> <s xml:id="echoid-s2093" xml:space="preserve">AD hoc imperfectam primò conſtructionem proponemus <lb/>ad mechanicen utilem; </s> <s xml:id="echoid-s2094" xml:space="preserve">deinde accuratam ſubjiciemus, <lb/>quæ tamen non niſi ſæpius tentando perficiatur. </s> <s xml:id="echoid-s2095" xml:space="preserve">Etenim ſoli-<lb/>da problemata omnia vel hoc exigunt vel ſectionum conica-<lb/>rum deſcriptionem.</s> <s xml:id="echoid-s2096" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2097" xml:space="preserve">Sit itaque datus cubus cujus latus A B, oporteatque in-<lb/> <anchor type="note" xlink:label="note-0105-01a" xlink:href="note-0105-01"/> venire latus cubi dupli.</s> <s xml:id="echoid-s2098" xml:space="preserve"/> </p> <div xml:id="echoid-div112" type="float" level="2" n="1"> <note position="right" xlink:label="note-0105-01" xlink:href="note-0105-01a" xml:space="preserve">TAB. XLI. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s2099" xml:space="preserve">Radio B A ſemicirculus deſcribatur A F C. </s> <s xml:id="echoid-s2100" xml:space="preserve">Sitque arcus <lb/>A F ſemicircumferentiæ triens, C D vero quadrans; </s> <s xml:id="echoid-s2101" xml:space="preserve">& </s> <s xml:id="echoid-s2102" xml:space="preserve">du-<lb/>cantur C F, A D, quarum interſectio ad E punctum. </s> <s xml:id="echoid-s2103" xml:space="preserve">Erit <lb/>A E latus cubi quæſiti; </s> <s xml:id="echoid-s2104" xml:space="preserve">exiguo tamen excedens, quodque <lb/>minus ſit {1/2000} ſui parte, ut facile numeris explorari poteſt. <lb/></s> <s xml:id="echoid-s2105" xml:space="preserve">Fit enim A E ſecans anguli p. </s> <s xml:id="echoid-s2106" xml:space="preserve">37. </s> <s xml:id="echoid-s2107" xml:space="preserve">ſcr. </s> <s xml:id="echoid-s2108" xml:space="preserve">30. </s> <s xml:id="echoid-s2109" xml:space="preserve">quæ proinde ma-<lb/>jor eſt partibus 12600 qualium A F vel A B 10000. </s> <s xml:id="echoid-s2110" xml:space="preserve">Minor <lb/>autem quam 12605. </s> <s xml:id="echoid-s2111" xml:space="preserve">Itaque cum cubus ex 12600 ſit major <lb/>quam duplus ejus qui ex A B 10000, erit & </s> <s xml:id="echoid-s2112" xml:space="preserve">cubus A E <lb/>major duplo cubo ex A B. </s> <s xml:id="echoid-s2113" xml:space="preserve">Rurſus quia A E major eſt par-<lb/>tibus 12600 erit {1/2000} A E major part. </s> <s xml:id="echoid-s2114" xml:space="preserve">6. </s> <s xml:id="echoid-s2115" xml:space="preserve">Tota vero A E mi-<lb/>nor eſt quam 12605. </s> <s xml:id="echoid-s2116" xml:space="preserve">Ergo auferendo ab A E partem bis-<lb/>milleſimam ſui ipſius reliqua minor erit partibus 12599, tot <lb/>enim ſuperſunt cum ex 12605 deducuntur 6. </s> <s xml:id="echoid-s2117" xml:space="preserve">Atqui cubus <lb/>ex 12599 minor eſt duplo cubo ex 10000. </s> <s xml:id="echoid-s2118" xml:space="preserve">Ergo omnino <lb/>quoque A E diminuta parte ſui biſmilleſima cubum mino-<lb/>rem producet quam ſit duplus cubus à latere A B.</s> <s xml:id="echoid-s2119" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2120" xml:space="preserve">Porro ad perfectam conſtructionem, C F quidem uti pri- <pb o="392" file="0106" n="113" rhead="CHRISTIANI HUGENII"/> us ducenda eſt: </s> <s xml:id="echoid-s2121" xml:space="preserve">A D vero ſic, ut ſubtenſa C D æqualis ſit <lb/>abſciſſæ E F. </s> <s xml:id="echoid-s2122" xml:space="preserve">Etenim his poſitis dico cubum A E ejus qui <lb/>ex A B duplum exiſtere.</s> <s xml:id="echoid-s2123" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2124" xml:space="preserve">Producatur enim C A, & </s> <s xml:id="echoid-s2125" xml:space="preserve">ſit ipſi A E æqualis A G. <lb/></s> <s xml:id="echoid-s2126" xml:space="preserve">Propter triangulos ſimiles igitur eſt E C ad C D, hoc eſt, <lb/>E F ut E A ad A F, hoc eſt, ut G A ad A B. </s> <s xml:id="echoid-s2127" xml:space="preserve">Et com-<lb/>ponendo C F ad F E ut G B ad B A ſive A F. </s> <s xml:id="echoid-s2128" xml:space="preserve">Et permu-<lb/>tando C F ad G B ut E F ad F A. </s> <s xml:id="echoid-s2129" xml:space="preserve">Quare ut C F qua-<lb/>dratum ad quadr. </s> <s xml:id="echoid-s2130" xml:space="preserve">G B, ita quadr. </s> <s xml:id="echoid-s2131" xml:space="preserve">E F ad quadr. </s> <s xml:id="echoid-s2132" xml:space="preserve">F A. </s> <s xml:id="echoid-s2133" xml:space="preserve"><lb/>Et componendo ut quadr. </s> <s xml:id="echoid-s2134" xml:space="preserve">C F & </s> <s xml:id="echoid-s2135" xml:space="preserve">G B ad quadr. </s> <s xml:id="echoid-s2136" xml:space="preserve">G B, ita <lb/>quadr. </s> <s xml:id="echoid-s2137" xml:space="preserve">E F & </s> <s xml:id="echoid-s2138" xml:space="preserve">F A ſimul, hoc eſt, quadr. </s> <s xml:id="echoid-s2139" xml:space="preserve">E A ad quadr. </s> <s xml:id="echoid-s2140" xml:space="preserve"><lb/>A F. </s> <s xml:id="echoid-s2141" xml:space="preserve">Quadr. </s> <s xml:id="echoid-s2142" xml:space="preserve">autem C F & </s> <s xml:id="echoid-s2143" xml:space="preserve">G B ſimul æquantur rectangu-<lb/>lo G C A cum quadr. </s> <s xml:id="echoid-s2144" xml:space="preserve">A G, quod ſic oſtenditur. </s> <s xml:id="echoid-s2145" xml:space="preserve">Quadra-<lb/>tum enim G B æquale eſt rectangulo C G A & </s> <s xml:id="echoid-s2146" xml:space="preserve">quadrato <lb/>A B ſeu A F <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2147" xml:space="preserve">Quare addito utrimque quadrato F C, <anchor type="note" xlink:label="note-0106-01a" xlink:href="note-0106-01"/> erunt quadrata G B, F C ſimul æqualia rectangulo C G A <lb/>& </s> <s xml:id="echoid-s2148" xml:space="preserve">quadrato A C. </s> <s xml:id="echoid-s2149" xml:space="preserve">Rectangulum autem C G A cum quadra-<lb/>to A C æquatur rectangulo G C A cum quadrato G A. </s> <s xml:id="echoid-s2150" xml:space="preserve">Ita-<lb/>que & </s> <s xml:id="echoid-s2151" xml:space="preserve">quadrata C F, G B ſimul æqualia ſunt rectangulo <lb/>G C A cum quadrato A G, ſicut diximus. </s> <s xml:id="echoid-s2152" xml:space="preserve">Sicut igitur re-<lb/>ctangulum G C A cum quadrato A G ad quadr. </s> <s xml:id="echoid-s2153" xml:space="preserve">G B, ita <lb/>eſt quadr. </s> <s xml:id="echoid-s2154" xml:space="preserve">E A ad quadr. </s> <s xml:id="echoid-s2155" xml:space="preserve">A F, hoc eſt, ita quadratum <lb/>G A ad quadr. </s> <s xml:id="echoid-s2156" xml:space="preserve">A B. </s> <s xml:id="echoid-s2157" xml:space="preserve">Et permutando, ut rectangulum <lb/>G C A cum quadrato G A ad quadratum G A ita quadr. <lb/></s> <s xml:id="echoid-s2158" xml:space="preserve">G B ad quadr. </s> <s xml:id="echoid-s2159" xml:space="preserve">A B. </s> <s xml:id="echoid-s2160" xml:space="preserve">Dividendo igitur, erit ut rectang. </s> <s xml:id="echoid-s2161" xml:space="preserve"><lb/>G C A ad quadr. </s> <s xml:id="echoid-s2162" xml:space="preserve">G A, ita quadr. </s> <s xml:id="echoid-s2163" xml:space="preserve">G B dempto quadrato <lb/>A B, hoc eſt, rectangulum C G A ad quadr. </s> <s xml:id="echoid-s2164" xml:space="preserve">A B. </s> <s xml:id="echoid-s2165" xml:space="preserve">Et per-<lb/>mutando rurſus, ut rectang. </s> <s xml:id="echoid-s2166" xml:space="preserve">G C A ad rectang. </s> <s xml:id="echoid-s2167" xml:space="preserve">C G A, <lb/>hoc eſt, ut C A ad A G ita quadratum G A ad quadr. </s> <s xml:id="echoid-s2168" xml:space="preserve"><lb/>A B. </s> <s xml:id="echoid-s2169" xml:space="preserve">Quamobrem quod fit ex quadrato G A in ipſam <lb/>G A, hoc eſt, cubus G A æquabitur ei quod fit ex quadra-<lb/>to A B in A C, hoc eſt, duplo cubo ex A B. </s> <s xml:id="echoid-s2170" xml:space="preserve">Quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s2171" xml:space="preserve"/> </p> <div xml:id="echoid-div113" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0106-01" xlink:href="note-0106-01a" xml:space="preserve">6.2. Elem.</note> </div> <pb o="393" file="0107" n="114" rhead="ILLUST. QUORUND. PROB. CONSTRUCT."/> </div> <div xml:id="echoid-div115" type="section" level="1" n="50"> <head xml:id="echoid-head78" xml:space="preserve"><emph style="sc">Probl</emph>. III.</head> <head xml:id="echoid-head79" style="it" xml:space="preserve">Datis duabus rectis duas medias propor-<lb/>tionales invenire.</head> <p> <s xml:id="echoid-s2172" xml:space="preserve">VEterum Geometrarum ad hoc Problema conſtructiones <lb/>complures retulit Eutocius ad lib. </s> <s xml:id="echoid-s2173" xml:space="preserve">2. </s> <s xml:id="echoid-s2174" xml:space="preserve">Archimedis de <lb/>Sphæra & </s> <s xml:id="echoid-s2175" xml:space="preserve">Cylindro, at non omnes inventione diverſas, uti <lb/>recte quoque ipſe animadvertit. </s> <s xml:id="echoid-s2176" xml:space="preserve">Heronis enim inventionem <lb/>ſecuti videntur Apollonius & </s> <s xml:id="echoid-s2177" xml:space="preserve">Philo Byzantius: </s> <s xml:id="echoid-s2178" xml:space="preserve">quanquam <lb/>Heronem Apollonio ætate poſteriorem nonnulli exiſtiment. <lb/></s> <s xml:id="echoid-s2179" xml:space="preserve">Dioclis modum Pappus & </s> <s xml:id="echoid-s2180" xml:space="preserve">Sporus. </s> <s xml:id="echoid-s2181" xml:space="preserve">Nicomedea autem con-<lb/>ſtructio præ cæteris ſubtilis ibidem extat, quam Fr. </s> <s xml:id="echoid-s2182" xml:space="preserve">Viëta <lb/>paulò aliter concinnatam ſuo Geometriæ ſupplemento inſe-<lb/>ruit. </s> <s xml:id="echoid-s2183" xml:space="preserve">R. </s> <s xml:id="echoid-s2184" xml:space="preserve">Carteſii egregia eſt & </s> <s xml:id="echoid-s2185" xml:space="preserve">nova per paraboles & </s> <s xml:id="echoid-s2186" xml:space="preserve">cir-<lb/>cumferentiæ interſectionem, cujus demonſtratio legitur in <lb/>libris Harmonicôn M. </s> <s xml:id="echoid-s2187" xml:space="preserve">Merſenni. </s> <s xml:id="echoid-s2188" xml:space="preserve">Noſtræ autem ſequen-<lb/>tes.</s> <s xml:id="echoid-s2189" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2190" xml:space="preserve">Sit datarum linearum major A C, quæ bifariam ſecetur <lb/> <anchor type="note" xlink:label="note-0107-01a" xlink:href="note-0107-01"/> in E. </s> <s xml:id="echoid-s2191" xml:space="preserve">Minor autem ſit A B, quæ ſic conſtituatur ut trian-<lb/>gulus E A B habeat crura æqualia A E, E B. </s> <s xml:id="echoid-s2192" xml:space="preserve">Et perficia-<lb/>tur parallelogrammum C A B D. </s> <s xml:id="echoid-s2193" xml:space="preserve">Et producantur A C, <lb/>A B. </s> <s xml:id="echoid-s2194" xml:space="preserve">Porro applicetur regula ad punctum D, & </s> <s xml:id="echoid-s2195" xml:space="preserve">moveatur <lb/>quousque poſitionem habeat G F, abſcindens nimirum E F <lb/>æqualem rectæ E G; </s> <s xml:id="echoid-s2196" xml:space="preserve">(Hoc autem vel ſæpius tentando aſſe-<lb/>quemur, vel deſcriptâ hyperbole, uti poſtea oſtendetur) <lb/>Dico inter A C, A B medias duas inventas eſſe B G, <lb/>C F.</s> <s xml:id="echoid-s2197" xml:space="preserve"/> </p> <div xml:id="echoid-div115" type="float" level="2" n="1"> <note position="right" xlink:label="note-0107-01" xlink:href="note-0107-01a" xml:space="preserve">TAB. XLI. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s2198" xml:space="preserve">Sit enim E K ipſi A B ad angulos rectos. </s> <s xml:id="echoid-s2199" xml:space="preserve">Quia igitur <lb/>B E æqualis E A, dividetur A B in K bifariam: </s> <s xml:id="echoid-s2200" xml:space="preserve">adjecta au-<lb/>tem eſt linea B G. </s> <s xml:id="echoid-s2201" xml:space="preserve">Ergo rectangulum A G B cum quadrato <lb/>ex K B, æquabitur quadrato K G. </s> <s xml:id="echoid-s2202" xml:space="preserve">Et addito utrimque qua-<lb/>drato K E, erit rectangulum A G B unà cum quadratis <lb/>B K, K E, hoc eſt unà cum quadrato B E, æquale qua-<lb/>drato E G. </s> <s xml:id="echoid-s2203" xml:space="preserve">Similiter quia A C bifariam dividitur in E, & </s> <s xml:id="echoid-s2204" xml:space="preserve">ad- <pb o="394" file="0108" n="115" rhead="CHRISTIANI HUGENII"/> jecta eſt linea C F, erit rectangulum A F C cum quadrato <lb/>E C æquale quadrato E F. </s> <s xml:id="echoid-s2205" xml:space="preserve">Quadratum autem E F æquale <lb/>eſt quadrato E G. </s> <s xml:id="echoid-s2206" xml:space="preserve">Erit igitur rectangulum A F C cum qua-<lb/>drato C E, æquale rectangulo A G B cum quadrato B E. <lb/></s> <s xml:id="echoid-s2207" xml:space="preserve">Atqui quadratum C E ſeu E A æquale eſt quadrato E B. </s> <s xml:id="echoid-s2208" xml:space="preserve"><lb/>Ergo & </s> <s xml:id="echoid-s2209" xml:space="preserve">reliquum rectangulum A F C æquale rectangulo <lb/>A G B. </s> <s xml:id="echoid-s2210" xml:space="preserve">Quare ſicut F A ad A G ita B G ad C F. </s> <s xml:id="echoid-s2211" xml:space="preserve">Ut au-<lb/>tem F A ad A G ita eſt D B ad B G, & </s> <s xml:id="echoid-s2212" xml:space="preserve">ita quoque F C ad <lb/>C D. </s> <s xml:id="echoid-s2213" xml:space="preserve">Igitur ut D B, hoc eſt, A C ad B G ita B G ad <lb/>F C, & </s> <s xml:id="echoid-s2214" xml:space="preserve">F C ad C D, hoc eſt, A B. </s> <s xml:id="echoid-s2215" xml:space="preserve">Quod erat dem. </s> <s xml:id="echoid-s2216" xml:space="preserve"><lb/>Quod autem dictum eſt, etiam deſcriptâ hyperbole inveni-<lb/>ri quomodo linea F D G ducenda ſit, hinc conſtabit: </s> <s xml:id="echoid-s2217" xml:space="preserve">Fa-<lb/>ctum enim ſit, ut E F, E G ſint æquales, & </s> <s xml:id="echoid-s2218" xml:space="preserve">ſumatur G N <lb/>æqualis D F. </s> <s xml:id="echoid-s2219" xml:space="preserve">Itaque punctum N eſt ad hyperbolem quæ <lb/>deſcribetur per D punctum circa aſymptotos F A, A G <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s2220" xml:space="preserve"> <anchor type="note" xlink:label="note-0108-01a" xlink:href="note-0108-01"/> Sed idem punctum N eſt quoque ad circuli circumferentiam <lb/>cujus centrum E radius E D: </s> <s xml:id="echoid-s2221" xml:space="preserve">(Hoc enim facile intelligitur <lb/>quia triangulus F E G eſt æquicruris, & </s> <s xml:id="echoid-s2222" xml:space="preserve">N G æqualis D F) <lb/>Itaque datum eſt punctum N ad interſectionem hyperboles <lb/>& </s> <s xml:id="echoid-s2223" xml:space="preserve">circumferentiæ dictæ. </s> <s xml:id="echoid-s2224" xml:space="preserve">Sed & </s> <s xml:id="echoid-s2225" xml:space="preserve">D datum eſt. </s> <s xml:id="echoid-s2226" xml:space="preserve">Datur igitur <lb/>poſitione linea F G ducenda per puncta N, D. </s> <s xml:id="echoid-s2227" xml:space="preserve">Et compo-<lb/>ſitio manifeſta eſt.</s> <s xml:id="echoid-s2228" xml:space="preserve"/> </p> <div xml:id="echoid-div116" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve">8. 2. Conic.</note> </div> </div> <div xml:id="echoid-div118" type="section" level="1" n="51"> <head xml:id="echoid-head80" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2229" xml:space="preserve">CIrca diametrum A C majori datarum linearum æqualem <lb/> <anchor type="note" xlink:label="note-0108-02a" xlink:href="note-0108-02"/> circulus deſcribatur & </s> <s xml:id="echoid-s2230" xml:space="preserve">ponatur A B minori datarum æqua-<lb/>lis, & </s> <s xml:id="echoid-s2231" xml:space="preserve">perficiatur parallelogrammum A D: </s> <s xml:id="echoid-s2232" xml:space="preserve">productâque A B, <lb/>ducatur ex centro E recta E H G eâ ratione ut H D, H G <lb/>ſint inter ſe æquales. </s> <s xml:id="echoid-s2233" xml:space="preserve">Secet autem circumferentiam in L. <lb/></s> <s xml:id="echoid-s2234" xml:space="preserve">Dico duabus A C, A B duas medias inventas eſſe B G, <lb/>G L.</s> <s xml:id="echoid-s2235" xml:space="preserve"/> </p> <div xml:id="echoid-div118" type="float" level="2" n="1"> <note position="left" xlink:label="note-0108-02" xlink:href="note-0108-02a" xml:space="preserve">TAB. XLI. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s2236" xml:space="preserve">Producatur enim G E uſque ad circumferentiam in K, & </s> <s xml:id="echoid-s2237" xml:space="preserve"><lb/>jungatur A K, eique parrallela ducatur B O. </s> <s xml:id="echoid-s2238" xml:space="preserve">Similes ita-<lb/>que ſunt trianguli A E K, B H O; </s> <s xml:id="echoid-s2239" xml:space="preserve">& </s> <s xml:id="echoid-s2240" xml:space="preserve">quia A E æqualis <lb/>E K, etiam B H, H O æquales erunt. </s> <s xml:id="echoid-s2241" xml:space="preserve">Sed & </s> <s xml:id="echoid-s2242" xml:space="preserve">H G, H D <lb/>inter ſe æquales ſunt. </s> <s xml:id="echoid-s2243" xml:space="preserve">Igitur tota O G æqualis B D, hoc <pb o="395" file="0109" n="116" rhead="ILLUST. QUORUND. PROB. CONSTRUCT."/> eſt, diametro A C vel L K; </s> <s xml:id="echoid-s2244" xml:space="preserve">& </s> <s xml:id="echoid-s2245" xml:space="preserve">ablatâ communi L O, re-<lb/>linquentur æquales L G, O K. </s> <s xml:id="echoid-s2246" xml:space="preserve">Eſt autem rectangulum <lb/>K G L æquale rectangulo A G B. </s> <s xml:id="echoid-s2247" xml:space="preserve">Ergo ut K G ad G A <lb/>ita B G ad G L. </s> <s xml:id="echoid-s2248" xml:space="preserve">Sed ut K G ad G A ita eſt O G ad G B <lb/>& </s> <s xml:id="echoid-s2249" xml:space="preserve">ita reliqua O K, hoc eſt, L G ad B A. </s> <s xml:id="echoid-s2250" xml:space="preserve">Ergo ut O G, <lb/>hoc eſt, A C ad G B ita B G ad G L & </s> <s xml:id="echoid-s2251" xml:space="preserve">G L ad A B. <lb/></s> <s xml:id="echoid-s2252" xml:space="preserve">Quod erat demonſtr. </s> <s xml:id="echoid-s2253" xml:space="preserve">Hujus autem conſtructionis inventio <lb/>eandem cum præcedenti originem habet.</s> <s xml:id="echoid-s2254" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div120" type="section" level="1" n="52"> <head xml:id="echoid-head81" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2255" xml:space="preserve">SInt datæ A B & </s> <s xml:id="echoid-s2256" xml:space="preserve">Q quibus duas medias proportionales in-<lb/> <anchor type="note" xlink:label="note-0109-01a" xlink:href="note-0109-01"/> venire opus ſit; </s> <s xml:id="echoid-s2257" xml:space="preserve">A B autem quam Q major.</s> <s xml:id="echoid-s2258" xml:space="preserve"/> </p> <div xml:id="echoid-div120" type="float" level="2" n="1"> <note position="right" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">TAB. XLI. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s2259" xml:space="preserve">Dimidiæ Q ſumatur æqualis A F, & </s> <s xml:id="echoid-s2260" xml:space="preserve">productâ A B u-<lb/>trimque, ſit ipſi æqualis B R. </s> <s xml:id="echoid-s2261" xml:space="preserve">Erigatur autem ad A B perpen-<lb/>dicularis F C, & </s> <s xml:id="echoid-s2262" xml:space="preserve">ipſi R A æqualis ponatur R C: </s> <s xml:id="echoid-s2263" xml:space="preserve">& </s> <s xml:id="echoid-s2264" xml:space="preserve">junga-<lb/>tur B C, & </s> <s xml:id="echoid-s2265" xml:space="preserve">huic parallela ducatur A E. </s> <s xml:id="echoid-s2266" xml:space="preserve">Denique applica-<lb/>tâ regulâ ad punctum C, moveatur ea quouſque poſitionem <lb/>habeat C D, faciens C E æqualem A D. </s> <s xml:id="echoid-s2267" xml:space="preserve">Dico inter A B & </s> <s xml:id="echoid-s2268" xml:space="preserve"><lb/>Q duas medias eſſe C E, E D.</s> <s xml:id="echoid-s2269" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2270" xml:space="preserve">Jungatur enim C A. </s> <s xml:id="echoid-s2271" xml:space="preserve">Igitur quia æquales ſunt R A, R C <lb/>& </s> <s xml:id="echoid-s2272" xml:space="preserve">angulus C F A rectus, erit R A ad A C ut A C ad du-<lb/>plam A F, hoc eſt, Q: </s> <s xml:id="echoid-s2273" xml:space="preserve">ac proinde quadratum A C æqua-<lb/>le rectangulo ſub R A & </s> <s xml:id="echoid-s2274" xml:space="preserve">Q. </s> <s xml:id="echoid-s2275" xml:space="preserve">Quadratum autem A C cum <lb/>quadrato A D & </s> <s xml:id="echoid-s2276" xml:space="preserve">duplo rectangulo D A F, hoc eſt, ſub <lb/>D A & </s> <s xml:id="echoid-s2277" xml:space="preserve">Q contento, æquatur quadrato D C<anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2278" xml:space="preserve">Igitur qua- <anchor type="note" xlink:label="note-0109-02a" xlink:href="note-0109-02"/> dratum D C æquabitur quadrato D A unà cum rectangulis <lb/>ſub D A, Q, & </s> <s xml:id="echoid-s2279" xml:space="preserve">ſub R A, Q, hoc eſt, unà cum rectan-<lb/>gulo ſub D R & </s> <s xml:id="echoid-s2280" xml:space="preserve">Q. </s> <s xml:id="echoid-s2281" xml:space="preserve">Quadratum autem D B æquale rectan-<lb/>gulo R D A & </s> <s xml:id="echoid-s2282" xml:space="preserve">quadrato A B <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2283" xml:space="preserve">Igitur ut quadratum <anchor type="note" xlink:label="note-0109-03a" xlink:href="note-0109-03"/> D B ad quadratum D C, hoc eſt, ut quadr. </s> <s xml:id="echoid-s2284" xml:space="preserve">A B ad quadr. <lb/></s> <s xml:id="echoid-s2285" xml:space="preserve">A D, (eſt enim ut D B ad D C ſic A B ad E C ſive A D) <lb/>ita erit rectangulum R D A cum quadrato A B ad rectan-<lb/>gulum ſub R D, Q, cum quadrato A D. </s> <s xml:id="echoid-s2286" xml:space="preserve">Quamobrem & </s> <s xml:id="echoid-s2287" xml:space="preserve"><lb/>rectangulum R D A ad rectangulum ſub R D, Q, ſicut quadr. </s> <s xml:id="echoid-s2288" xml:space="preserve"><lb/>A B ad quadr. </s> <s xml:id="echoid-s2289" xml:space="preserve">A D <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2290" xml:space="preserve">Eſt autem ut quadratum A B ad <anchor type="note" xlink:label="note-0109-04a" xlink:href="note-0109-04"/> quadr. </s> <s xml:id="echoid-s2291" xml:space="preserve">A D, ita A B ad E D longitudine: </s> <s xml:id="echoid-s2292" xml:space="preserve">nam ut B A ad A D <pb o="396" file="0110" n="117" rhead="CHRISTIANI HUGENII"/> ſic C E, hoc eſt, ipſa A D ad E D. </s> <s xml:id="echoid-s2293" xml:space="preserve">Ergo rectangulum <lb/>R D A ad rectangulum ſub R D, Q, hoc eſt, A D ad Q <lb/>ſicut A B ad E D. </s> <s xml:id="echoid-s2294" xml:space="preserve">Et permutando & </s> <s xml:id="echoid-s2295" xml:space="preserve">invertendo, B A ad <lb/>A D ut E D ad Q. </s> <s xml:id="echoid-s2296" xml:space="preserve">Atqui ut B A ad A D, hoc eſt, C E <lb/>ita C E ad E D. </s> <s xml:id="echoid-s2297" xml:space="preserve">Ergo ut A B ad C E ita C E ad E D, & </s> <s xml:id="echoid-s2298" xml:space="preserve"><lb/>E D ad Q. </s> <s xml:id="echoid-s2299" xml:space="preserve">Itaque inter A B & </s> <s xml:id="echoid-s2300" xml:space="preserve">Q mediæ proportionales <lb/>ſunt C E, E D. </s> <s xml:id="echoid-s2301" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s2302" xml:space="preserve"/> </p> <div xml:id="echoid-div121" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0109-02" xlink:href="note-0109-02a" xml:space="preserve">12. 2. E-<lb/>lem.</note> <note symbol="*" position="right" xlink:label="note-0109-03" xlink:href="note-0109-03a" xml:space="preserve">6.2. Elem.</note> <note symbol="*" position="right" xlink:label="note-0109-04" xlink:href="note-0109-04a" xml:space="preserve">19. 5. E-<lb/>lem.</note> </div> </div> <div xml:id="echoid-div123" type="section" level="1" n="53"> <head xml:id="echoid-head82" xml:space="preserve"><emph style="sc">Probl</emph>. IV.</head> <p style="it"> <s xml:id="echoid-s2303" xml:space="preserve">QUadrato dato & </s> <s xml:id="echoid-s2304" xml:space="preserve">uno latere producto, aptare <lb/>ſub angulo exteriori rectam magnitudine da-<lb/>tam quæ ad angulum oppoſitum pertineat.</s> <s xml:id="echoid-s2305" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2306" xml:space="preserve">Eſto quadratum B A cujus productum ſit latus F A. </s> <s xml:id="echoid-s2307" xml:space="preserve">Data <lb/> <anchor type="note" xlink:label="note-0110-01a" xlink:href="note-0110-01"/> verò linea K. </s> <s xml:id="echoid-s2308" xml:space="preserve">Et oporteat ducere rectam B D C, ita ut <lb/>pars intercepta D C ſit datæ K æqualis.</s> <s xml:id="echoid-s2309" xml:space="preserve"/> </p> <div xml:id="echoid-div123" type="float" level="2" n="1"> <note position="left" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">TAB. XLI. <lb/>Fig. 7.</note> </div> <p> <s xml:id="echoid-s2310" xml:space="preserve">Quadratis ex K & </s> <s xml:id="echoid-s2311" xml:space="preserve">E B ſit æquale quadratum E G; </s> <s xml:id="echoid-s2312" xml:space="preserve">& </s> <s xml:id="echoid-s2313" xml:space="preserve"><lb/>ſuper B G diametro deſcribatur ſemicirculus B C G ſecans <lb/>rectam F A productam in C, & </s> <s xml:id="echoid-s2314" xml:space="preserve">ducatur B D C. </s> <s xml:id="echoid-s2315" xml:space="preserve">Dico D C <lb/>æqualem eſſe ipſi K. </s> <s xml:id="echoid-s2316" xml:space="preserve">Jungantur enim C G, G D; </s> <s xml:id="echoid-s2317" xml:space="preserve">ſitque <lb/>C H ipſi B G ad angulos rectos.</s> <s xml:id="echoid-s2318" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2319" xml:space="preserve">Quia igitur ſimiles ſunt trianguli B E D, C H G, & </s> <s xml:id="echoid-s2320" xml:space="preserve">la-<lb/>tera B E, C H circa angulos rectos inter ſe æqualia, erit <lb/>& </s> <s xml:id="echoid-s2321" xml:space="preserve">latus D B æquale lateri G C, & </s> <s xml:id="echoid-s2322" xml:space="preserve">D E ipſi G H. </s> <s xml:id="echoid-s2323" xml:space="preserve">Sunt <lb/>autem quadrata D C, C G, hoc eſt, quadrata D C, C H, <lb/>& </s> <s xml:id="echoid-s2324" xml:space="preserve">H G æqualia quadrato D G <anchor type="note" xlink:href="" symbol="*"/>, hoc eſt, quadratis G E, <anchor type="note" xlink:label="note-0110-02a" xlink:href="note-0110-02"/> E D. </s> <s xml:id="echoid-s2325" xml:space="preserve">Ergo dempto hinc quadrato E D, inde vero quadrato <lb/>H G; </s> <s xml:id="echoid-s2326" xml:space="preserve">erunt duo quadrata D C & </s> <s xml:id="echoid-s2327" xml:space="preserve">C H æqualia quadrato <lb/>E G, hoc eſt, quadratis ex K & </s> <s xml:id="echoid-s2328" xml:space="preserve">E B <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2329" xml:space="preserve">Quadratum autem <anchor type="note" xlink:label="note-0110-03a" xlink:href="note-0110-03"/> E B æquale eſt quadrato C H. </s> <s xml:id="echoid-s2330" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2331" xml:space="preserve">reliquum quadra-<lb/>tum D C æquabitur K quadrato; </s> <s xml:id="echoid-s2332" xml:space="preserve">& </s> <s xml:id="echoid-s2333" xml:space="preserve">recta D C ipſi K. </s> <s xml:id="echoid-s2334" xml:space="preserve">Quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s2335" xml:space="preserve"/> </p> <div xml:id="echoid-div124" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0110-02" xlink:href="note-0110-02a" xml:space="preserve">47.1. Elem.</note> <note symbol="*" position="left" xlink:label="note-0110-03" xlink:href="note-0110-03a" xml:space="preserve">Ex con-<lb/>ſtruct.</note> </div> <p> <s xml:id="echoid-s2336" xml:space="preserve">Demonſtratio hæc ab ea diverſa eſt quæ apud Pappum A-<lb/>lex. </s> <s xml:id="echoid-s2337" xml:space="preserve">legitur lib 7. </s> <s xml:id="echoid-s2338" xml:space="preserve">prop. </s> <s xml:id="echoid-s2339" xml:space="preserve">72. </s> <s xml:id="echoid-s2340" xml:space="preserve">Conſtructio verò non differt. </s> <s xml:id="echoid-s2341" xml:space="preserve">Cæ-<lb/>terum eandem ad caſum quoque ſequentem pertinere inveni-<lb/>mus.</s> <s xml:id="echoid-s2342" xml:space="preserve"/> </p> <pb o="397" file="0111" n="118" rhead="ILLUST. QUORUND. PROB. CONSTRUCT."/> </div> <div xml:id="echoid-div126" type="section" level="1" n="54"> <head xml:id="echoid-head83" xml:space="preserve"><emph style="sc">Probl</emph>. V.</head> <p style="it"> <s xml:id="echoid-s2343" xml:space="preserve">DAto quadrato, & </s> <s xml:id="echoid-s2344" xml:space="preserve">duobus contiguis lateribus <lb/>productis, aptare ſub angulo interiori rectam <lb/>magnitudine datam quæ per angulum oppoſitum <lb/>tranſeat. </s> <s xml:id="echoid-s2345" xml:space="preserve">Oportet autem non minorem eſſe datam <lb/>quam ſit quadrati diameter dupla.</s> <s xml:id="echoid-s2346" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2347" xml:space="preserve">Datum ſit quadratum A B, productaque latera A F, A E.</s> <s xml:id="echoid-s2348" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">TAB. XLI. <lb/>Fig. 8.</note> <p> <s xml:id="echoid-s2349" xml:space="preserve">Data verò linea K, non minor duplâ A B diametro. </s> <s xml:id="echoid-s2350" xml:space="preserve">Et <lb/>oporteat per angulum B ponere rectam D C ipſi K æqua-<lb/>lem.</s> <s xml:id="echoid-s2351" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2352" xml:space="preserve">Si K æqualis fuerit duplæ A B, conſtructio manifeſta eſt; <lb/></s> <s xml:id="echoid-s2353" xml:space="preserve">tunc enim per B ducenda eſt quæ ſit ipſi A B ad angulos re-<lb/>ctos, eaque propoſitum efficiet. </s> <s xml:id="echoid-s2354" xml:space="preserve">Cum verò major erit K quam <lb/>dupla A B, conſtructio iiſdem verbis præcipitur atque in Pro-<lb/>blemate præcedenti: </s> <s xml:id="echoid-s2355" xml:space="preserve">& </s> <s xml:id="echoid-s2356" xml:space="preserve">demonſtratio quoque eſt eadem.</s> <s xml:id="echoid-s2357" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2358" xml:space="preserve">Quod autem circumferentia B C G ſecabit productam A F <lb/>hinc manifeſtum eſt. </s> <s xml:id="echoid-s2359" xml:space="preserve">Etenim quia K major eſt duplâ diame-<lb/>tro A B, erit K quadratum majus octo quadratis E B. </s> <s xml:id="echoid-s2360" xml:space="preserve">Itaque <lb/>quadratum E G majus quam quadrata novem ex E B; </s> <s xml:id="echoid-s2361" xml:space="preserve">ac pro-<lb/>inde E G major quam tripla E B. </s> <s xml:id="echoid-s2362" xml:space="preserve">Dimidia igitur B G, hoc <lb/>eſt, radius deſcripti ſemicirculi major quam E B vel B F; <lb/></s> <s xml:id="echoid-s2363" xml:space="preserve">unde neceſſe eſt rectam A C ſecari à circumferentia B C G.</s> <s xml:id="echoid-s2364" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div127" type="section" level="1" n="55"> <head xml:id="echoid-head84" xml:space="preserve"><emph style="sc">Probl.</emph> VI.</head> <p style="it"> <s xml:id="echoid-s2365" xml:space="preserve">RHombo dato, & </s> <s xml:id="echoid-s2366" xml:space="preserve">uno latere producto, aptare <lb/>ſub angulo exteriori lineam magnitudine da-<lb/>tam quæ ad oppoſitum angulum pertineat.</s> <s xml:id="echoid-s2367" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2368" xml:space="preserve">Sit datus rhombus A B, cujus productum latus F A. </s> <s xml:id="echoid-s2369" xml:space="preserve">Data <lb/> <anchor type="note" xlink:label="note-0111-02a" xlink:href="note-0111-02"/> autem linea K. </s> <s xml:id="echoid-s2370" xml:space="preserve">Et oporteat ducere rectam B D C, ita ut <lb/>pars intercepta D C ſit ipſi K æqualis.</s> <s xml:id="echoid-s2371" xml:space="preserve"/> </p> <div xml:id="echoid-div127" type="float" level="2" n="1"> <note position="right" xlink:label="note-0111-02" xlink:href="note-0111-02a" xml:space="preserve">TAB. XLII. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s2372" xml:space="preserve">Ducatur diameter A B, & </s> <s xml:id="echoid-s2373" xml:space="preserve">latus B E producatur; </s> <s xml:id="echoid-s2374" xml:space="preserve">& </s> <s xml:id="echoid-s2375" xml:space="preserve">qua-<lb/>dratis ex K & </s> <s xml:id="echoid-s2376" xml:space="preserve">A B ſit æquale quadratum A G. </s> <s xml:id="echoid-s2377" xml:space="preserve">Et ſuper B G <pb o="398" file="0112" n="119" rhead="CHRISTIANI HUGENII"/> circumferentiæ portio deſcribatur quæ capiat angulum ipſi <lb/>B F A æqualem. </s> <s xml:id="echoid-s2378" xml:space="preserve">Secabit ea productum latus F A, ut mo-<lb/>do oſtendetur. </s> <s xml:id="echoid-s2379" xml:space="preserve">Itaque ad interſectionis punctum C ducatur <lb/>B C. </s> <s xml:id="echoid-s2380" xml:space="preserve">Dico hujus partem interceptam D C lineæ datæ K æ-<lb/>qualem eſſe. </s> <s xml:id="echoid-s2381" xml:space="preserve">Quod autem circumferentia deſcripta latus F A <lb/>productum ſecabit, ſic primùm oſtenditur. </s> <s xml:id="echoid-s2382" xml:space="preserve">Ducatur A N <lb/>ita ut ſit angulus B A N angulo B F A vel B E A æqualis. <lb/></s> <s xml:id="echoid-s2383" xml:space="preserve">Itaque triangulus B A N triangulo B E A ſimilis eſt, ac <lb/>proinde iſoſceles quoque. </s> <s xml:id="echoid-s2384" xml:space="preserve">Quare ſi ſuper B N circumferen-<lb/>tia deſcribatur quæ capiat angulum B F A, ea continget <lb/>latus F A in A puncto. </s> <s xml:id="echoid-s2385" xml:space="preserve">Sed B G major eſt quam B N: </s> <s xml:id="echoid-s2386" xml:space="preserve"><lb/>nam quadratum A G majus eſt quadrato A N vel A B, cum <lb/>ſit æquale quadratis ex K & </s> <s xml:id="echoid-s2387" xml:space="preserve">A B. </s> <s xml:id="echoid-s2388" xml:space="preserve">Quare A G cadet extra <lb/>triangulum iſoſcelem B A N. </s> <s xml:id="echoid-s2389" xml:space="preserve">Itaque manifeſtum eſt cir-<lb/>cumferentiam ſuper B G deſcriptam capientemque angulum <lb/>ipſi B F A vel B A N æqualem ſecare lineam F A C. </s> <s xml:id="echoid-s2390" xml:space="preserve">Eſto <lb/>alterum interſectionis punctum M & </s> <s xml:id="echoid-s2391" xml:space="preserve">jungantur B M, G C, <lb/>& </s> <s xml:id="echoid-s2392" xml:space="preserve">cadat in B E ex A perpendicularis A L.</s> <s xml:id="echoid-s2393" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2394" xml:space="preserve">Quia igitur quadratum A G æquale eſt quadratis ex K & </s> <s xml:id="echoid-s2395" xml:space="preserve"><lb/>A B: </s> <s xml:id="echoid-s2396" xml:space="preserve">atque idem quadratum A G æquale quadratis A B & </s> <s xml:id="echoid-s2397" xml:space="preserve"><lb/>B G minus duplo rectangulo G B L, hoc eſt, minus rectan-<lb/>gulo G B N; </s> <s xml:id="echoid-s2398" xml:space="preserve">erit K quadratum æquale quadrato B G mi-<lb/>nùs rectangulo G B N, hoc eſt, rectangulo B G N. </s> <s xml:id="echoid-s2399" xml:space="preserve">Eſt <lb/>autem ut rectangulum B G N ad rectang. </s> <s xml:id="echoid-s2400" xml:space="preserve">B E, G N, ita <lb/>B G ad B E. </s> <s xml:id="echoid-s2401" xml:space="preserve">Ergo ut B G ad B E ita quoque quadratum <lb/>K ad rectangulum G N, B E, hoc eſt, rectangulum G B E <lb/>minùs rectangulo N B E. </s> <s xml:id="echoid-s2402" xml:space="preserve">Eſt autem rectangulo G B E æ-<lb/>quale rectang. </s> <s xml:id="echoid-s2403" xml:space="preserve">C B D, quoniam G B ad B C ut D B ad B E <lb/>propter triangulos ſimiles G B C, D B E; </s> <s xml:id="echoid-s2404" xml:space="preserve">habent enim an-<lb/>gulum ad B communem, & </s> <s xml:id="echoid-s2405" xml:space="preserve">angulus B C G ipſi B E D eſt <lb/>æqualis. </s> <s xml:id="echoid-s2406" xml:space="preserve">Item rectangulo N B E æquale eſt quadratum A B <lb/>quia propter triangulos ſimiles eſt N B ad B A ut A B ad <lb/>B E. </s> <s xml:id="echoid-s2407" xml:space="preserve">Ergo erit G B ad B E ut quadratum K ad rectangu-<lb/>lum C B D minùs quadrato A B. </s> <s xml:id="echoid-s2408" xml:space="preserve">Eſt autem rectangulo <lb/>C B D minùs quadr. </s> <s xml:id="echoid-s2409" xml:space="preserve">A B æquale rectangulum D A, A C; <lb/></s> <s xml:id="echoid-s2410" xml:space="preserve">quod ſic oſtenditur. </s> <s xml:id="echoid-s2411" xml:space="preserve">Etenim quia quadrilaterum C G B M <pb file="0113" n="120"/> <pb file="0113a" n="121"/> <anchor type="figure" xlink:label="fig-0113a-01a" xlink:href="fig-0113a-01"/> <anchor type="figure" xlink:label="fig-0113a-02a" xlink:href="fig-0113a-02"/> <anchor type="figure" xlink:label="fig-0113a-03a" xlink:href="fig-0113a-03"/> <anchor type="figure" xlink:label="fig-0113a-04a" xlink:href="fig-0113a-04"/> <anchor type="figure" xlink:label="fig-0113a-05a" xlink:href="fig-0113a-05"/> <anchor type="figure" xlink:label="fig-0113a-06a" xlink:href="fig-0113a-06"/> <anchor type="figure" xlink:label="fig-0113a-07a" xlink:href="fig-0113a-07"/> <anchor type="figure" xlink:label="fig-0113a-08a" xlink:href="fig-0113a-08"/> <pb file="0114" n="122"/> <pb o="399" file="0115" n="123" rhead="ILLUST. QUORUND. PROB. CONSTRUCT."/> eſt in circulo, ſunt anguli C G B & </s> <s xml:id="echoid-s2412" xml:space="preserve">B M C ſimul duobus <lb/>rectis æquales. </s> <s xml:id="echoid-s2413" xml:space="preserve">Sed & </s> <s xml:id="echoid-s2414" xml:space="preserve">anguli E D B, A D B. </s> <s xml:id="echoid-s2415" xml:space="preserve">Quorum E D B <lb/>æqualis angulo C G B propter ſimilitudinem triangulorum <lb/>G B C, D B E. </s> <s xml:id="echoid-s2416" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2417" xml:space="preserve">angulus B M C æqualis erit an-<lb/>gulo A D B. </s> <s xml:id="echoid-s2418" xml:space="preserve">Trianguli igitur A B M, A B D angulos M <lb/>& </s> <s xml:id="echoid-s2419" xml:space="preserve">D inter ſe æquales habent. </s> <s xml:id="echoid-s2420" xml:space="preserve">Verum & </s> <s xml:id="echoid-s2421" xml:space="preserve">angulos ad A, & </s> <s xml:id="echoid-s2422" xml:space="preserve"><lb/>latus A B commune. </s> <s xml:id="echoid-s2423" xml:space="preserve">Itaque dicti trianguli ſimiles ſunt & </s> <s xml:id="echoid-s2424" xml:space="preserve"><lb/>æquales. </s> <s xml:id="echoid-s2425" xml:space="preserve">Quare A M æqualis A D, & </s> <s xml:id="echoid-s2426" xml:space="preserve">M B æqualis B D, <lb/>& </s> <s xml:id="echoid-s2427" xml:space="preserve">angulus M B A æqualis A B D. </s> <s xml:id="echoid-s2428" xml:space="preserve">In triangulo igitur M B C <lb/>angulus B in duo æqualia dividitur à recta B A, ideoque <lb/>rectang. </s> <s xml:id="echoid-s2429" xml:space="preserve">M B C minus quadrato B A æquatur rectangulo <lb/>M A C. </s> <s xml:id="echoid-s2430" xml:space="preserve">Sed rectangulo C B M æquale eſt rectangulum <lb/>C B D; </s> <s xml:id="echoid-s2431" xml:space="preserve">& </s> <s xml:id="echoid-s2432" xml:space="preserve">rectangulo M A C æquale rectang. </s> <s xml:id="echoid-s2433" xml:space="preserve">D A C. </s> <s xml:id="echoid-s2434" xml:space="preserve">Igi-<lb/>tur rectang. </s> <s xml:id="echoid-s2435" xml:space="preserve">C B D minus quadrato B A æquale rectangulo <lb/>C A D, uti dictum fuit. </s> <s xml:id="echoid-s2436" xml:space="preserve">Eſt itaque G B ad B E ut quadr. <lb/></s> <s xml:id="echoid-s2437" xml:space="preserve">K ad rectangulum D A C. </s> <s xml:id="echoid-s2438" xml:space="preserve">Sicut autem G B ad B E ita eſt <lb/>rectang. </s> <s xml:id="echoid-s2439" xml:space="preserve">G B E, hoc eſt, rectang. </s> <s xml:id="echoid-s2440" xml:space="preserve">C B D ad quadratum B E. </s> <s xml:id="echoid-s2441" xml:space="preserve"><lb/>Ergo ut quadratum K ad rectang. </s> <s xml:id="echoid-s2442" xml:space="preserve">D A C ita rectang. </s> <s xml:id="echoid-s2443" xml:space="preserve">C B D <lb/>ad quadratum B E. </s> <s xml:id="echoid-s2444" xml:space="preserve">Ratio autem rectanguli C B D ad quadr. </s> <s xml:id="echoid-s2445" xml:space="preserve"><lb/>B E compoſita eſt ex ratione D B ad B E, hoc eſt, D C <lb/>ad C A, & </s> <s xml:id="echoid-s2446" xml:space="preserve">ex ratione C B ad B E ſive B F, hoc eſt, C D <lb/>ad D A. </s> <s xml:id="echoid-s2447" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2448" xml:space="preserve">quadr. </s> <s xml:id="echoid-s2449" xml:space="preserve">K ad rectang. </s> <s xml:id="echoid-s2450" xml:space="preserve">D A C eam habet <lb/>rationem quæ componitur ex ratione D C ad C A & </s> <s xml:id="echoid-s2451" xml:space="preserve">D C <lb/>ad D A. </s> <s xml:id="echoid-s2452" xml:space="preserve">hoc eſt, eam quam quadratum D C ad rectang. </s> <s xml:id="echoid-s2453" xml:space="preserve"><lb/>D A C. </s> <s xml:id="echoid-s2454" xml:space="preserve">Quamobrem quadr. </s> <s xml:id="echoid-s2455" xml:space="preserve">K. </s> <s xml:id="echoid-s2456" xml:space="preserve">quadrato D C æquale eſt: </s> <s xml:id="echoid-s2457" xml:space="preserve"><lb/>Et D C ipſi K longitudine. </s> <s xml:id="echoid-s2458" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s2459" xml:space="preserve"/> </p> <div xml:id="echoid-div128" type="float" level="2" n="2"> <figure xlink:label="fig-0113a-01" xlink:href="fig-0113a-01a"> <caption xml:id="echoid-caption46" style="it" xml:space="preserve">Pag. 398.<lb/>TAB. XLI.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables46" xml:space="preserve">S T B R K H Q C N O M A E L D</variables> </figure> <figure xlink:label="fig-0113a-02" xlink:href="fig-0113a-02a"> <caption xml:id="echoid-caption47" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables47" xml:space="preserve">D E F B G H C A</variables> </figure> <figure xlink:label="fig-0113a-03" xlink:href="fig-0113a-03a"> <caption xml:id="echoid-caption48" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables48" xml:space="preserve">F D E G A B C</variables> </figure> <figure xlink:label="fig-0113a-04" xlink:href="fig-0113a-04a"> <caption xml:id="echoid-caption49" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables49" xml:space="preserve">G N B H D K A E C F</variables> </figure> <figure xlink:label="fig-0113a-05" xlink:href="fig-0113a-05a"> <caption xml:id="echoid-caption50" style="it" xml:space="preserve">Fig. 8</caption> <variables xml:id="echoid-variables50" xml:space="preserve">K A F c C E B h H G D d</variables> </figure> <figure xlink:label="fig-0113a-06" xlink:href="fig-0113a-06a"> <caption xml:id="echoid-caption51" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables51" xml:space="preserve">C E D A F B R Q</variables> </figure> <figure xlink:label="fig-0113a-07" xlink:href="fig-0113a-07a"> <caption xml:id="echoid-caption52" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables52" xml:space="preserve">G L B H D O A E C K</variables> </figure> <figure xlink:label="fig-0113a-08" xlink:href="fig-0113a-08a"> <caption xml:id="echoid-caption53" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables53" xml:space="preserve">K F A C D B E H G</variables> </figure> </div> </div> <div xml:id="echoid-div130" type="section" level="1" n="56"> <head xml:id="echoid-head85" xml:space="preserve"><emph style="sc">Probl.</emph> VII.</head> <p style="it"> <s xml:id="echoid-s2460" xml:space="preserve">RHombo dato & </s> <s xml:id="echoid-s2461" xml:space="preserve">duobus contiguis lateribus pro-<lb/>ductis, aptare ſub angulo interiorirectam ma-<lb/>gnitudine datam quæ per oppoſitum angulum tranſ-<lb/>eat. </s> <s xml:id="echoid-s2462" xml:space="preserve">Oportet autem datam non minorem eſſe quam <lb/>duplam diametri quæ reliquos duos rhombi angulos <lb/>conjungit.</s> <s xml:id="echoid-s2463" xml:space="preserve"/> </p> <pb o="400" file="0116" n="124" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s2464" xml:space="preserve">Sit datus rhombus A B cujus producantur latera A F, A E; <lb/></s> <s xml:id="echoid-s2465" xml:space="preserve"> <anchor type="note" xlink:label="note-0116-01a" xlink:href="note-0116-01"/> data autem ſit recta K cui æqualem ponere oporteat C D, <lb/>per angulum B tranſeuntem. </s> <s xml:id="echoid-s2466" xml:space="preserve">Ducatur diameter A B, eique <lb/>ad angulos rectos linea S B R, quæ quidem æqualis erit <lb/>duplæ diametro F E. </s> <s xml:id="echoid-s2467" xml:space="preserve">Igitur K non minor debet eſſe quam <lb/>S R. </s> <s xml:id="echoid-s2468" xml:space="preserve">Si vero æqualis, factum eſt quod proponebatur. </s> <s xml:id="echoid-s2469" xml:space="preserve">Sed <lb/>ponatur K major data eſſe quam S R. </s> <s xml:id="echoid-s2470" xml:space="preserve">Erit jam in ſchemate <lb/>hoc prout propoſitum eſt conſtructio eadem, quæ in Pro-<lb/>blemate præcedenti. </s> <s xml:id="echoid-s2471" xml:space="preserve">Demonſtratio autem nonnihil diverſa. <lb/></s> <s xml:id="echoid-s2472" xml:space="preserve">Etenim hoc primò aliter oſtenditur quod circumferentia ſu-<lb/>per B G deſcripta ſecat productam A F. </s> <s xml:id="echoid-s2473" xml:space="preserve">Sit A L ad E B <lb/>perpendicularis & </s> <s xml:id="echoid-s2474" xml:space="preserve">ducatur S T ut ſit angulus B S T æqua-<lb/>lis angulo E A F vel B F S. </s> <s xml:id="echoid-s2475" xml:space="preserve">Eſt itaque triangulus B S T <lb/>triangulo B F S ſimilis; </s> <s xml:id="echoid-s2476" xml:space="preserve">(nam & </s> <s xml:id="echoid-s2477" xml:space="preserve">angulos ad B æquales ha-<lb/>bent:) </s> <s xml:id="echoid-s2478" xml:space="preserve">ac proinde æquicruris etiam triangulus B S T. </s> <s xml:id="echoid-s2479" xml:space="preserve">Ap-<lb/>paret igitur lineam A S æquari ipſi L B cum dimidia B T. </s> <s xml:id="echoid-s2480" xml:space="preserve"><lb/>Quare dupla A S æquabitur duplæ L B & </s> <s xml:id="echoid-s2481" xml:space="preserve">toti B T. </s> <s xml:id="echoid-s2482" xml:space="preserve">Sed <lb/>dupla A S eſt quadrupla A F vel E B. </s> <s xml:id="echoid-s2483" xml:space="preserve">Ergo quadrupla E B <lb/>æqualis duplæ L B & </s> <s xml:id="echoid-s2484" xml:space="preserve">B T. </s> <s xml:id="echoid-s2485" xml:space="preserve">Sumptâque communî altitudi-<lb/>ne B T, erit rectangulum ſub quadrupla E B & </s> <s xml:id="echoid-s2486" xml:space="preserve">B T con-<lb/>tentum, æquale duplo rectangulo L B T & </s> <s xml:id="echoid-s2487" xml:space="preserve">quadrato B T. </s> <s xml:id="echoid-s2488" xml:space="preserve"><lb/>Et addito utrimque quadrato B L, erit rectangulum E B T <lb/>quater cum quadrato L B æquale rectangulo L B T bis cum <lb/>quadratis B T, B L, hoc eſt quadrato L T. </s> <s xml:id="echoid-s2489" xml:space="preserve">Quia vero <lb/>propter triangulos ſimiles eſt T B ad B S ut B S ad B F <lb/>ſive B E, æquale erit rectang. </s> <s xml:id="echoid-s2490" xml:space="preserve">E B T quadrato B S; </s> <s xml:id="echoid-s2491" xml:space="preserve">& </s> <s xml:id="echoid-s2492" xml:space="preserve"><lb/>quater ſumptum quadrato R S. </s> <s xml:id="echoid-s2493" xml:space="preserve">Itaque quadr. </s> <s xml:id="echoid-s2494" xml:space="preserve">S R cum qua-<lb/>drato L B æquale quadrato L T. </s> <s xml:id="echoid-s2495" xml:space="preserve">Quadratum vero K (quod <lb/>majus eſt quam R S quadr.) </s> <s xml:id="echoid-s2496" xml:space="preserve">unà cum eodem quadrato L B <lb/>æquale eſt quadrato L G, uti ex conſtructione manifeſtum <lb/>eſt, quia ſcilicet quadr. </s> <s xml:id="echoid-s2497" xml:space="preserve">A G æquale poſitum fuit quadratis <lb/>ex K & </s> <s xml:id="echoid-s2498" xml:space="preserve">A B. </s> <s xml:id="echoid-s2499" xml:space="preserve">Itaque majus eſt quadr. </s> <s xml:id="echoid-s2500" xml:space="preserve">L G quam L T, & </s> <s xml:id="echoid-s2501" xml:space="preserve"><lb/>L G major quam L T, & </s> <s xml:id="echoid-s2502" xml:space="preserve">B G quam B T. </s> <s xml:id="echoid-s2503" xml:space="preserve">Quamobrem <lb/>circumferentia ſuper B G deſcripta capax anguli E A F ſe-<lb/>cabit rectam A S; </s> <s xml:id="echoid-s2504" xml:space="preserve">nam ſimilis circumferentia, ſi ſuper B T <lb/>deſcribatur, ea continget ipſam in S puncto, quoniam an- <pb o="401" file="0117" n="125" rhead="ILLUST. QUORUND. PROB. CONSTRUCT."/> gulus B S T ipſi E A F æqualis eſt trianguluſque B S T æ-<lb/>quicruris.</s> <s xml:id="echoid-s2505" xml:space="preserve"/> </p> <div xml:id="echoid-div130" type="float" level="2" n="1"> <note position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">TAB. XLII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s2506" xml:space="preserve">Porrò quod C D ipſi K æqualis eſt, ſic demonſtrabitur. <lb/></s> <s xml:id="echoid-s2507" xml:space="preserve">Quia quadratum A G æquale eſt quadratis ex K & </s> <s xml:id="echoid-s2508" xml:space="preserve">A B: </s> <s xml:id="echoid-s2509" xml:space="preserve"><lb/>idemque quadratum A G æquale quadratis A B, B G cum <lb/>duplo rectangulo G B L. </s> <s xml:id="echoid-s2510" xml:space="preserve">Erit propterea quadratum K æ-<lb/>quale quadrato B G cum duplo rectangulo G B L. </s> <s xml:id="echoid-s2511" xml:space="preserve">Sicut <lb/>autem B G ad B E ita eſt quadratum B G cum duplo re-<lb/>ctangulo G B L ad rectangulum G B E cum duplo rectan-<lb/>ctulo E B L; </s> <s xml:id="echoid-s2512" xml:space="preserve">ſingula enim ad ſingula eam habent rationem. </s> <s xml:id="echoid-s2513" xml:space="preserve"><lb/>Ergo & </s> <s xml:id="echoid-s2514" xml:space="preserve">quadratum K ad rectangulum G B E cum duplo re-<lb/>ctangulo E B L ut B G ad B E. </s> <s xml:id="echoid-s2515" xml:space="preserve">Eſt autem rectangulo G B E <lb/>æquale rectangulum C B D, quoniam C B ad B G ut E B <lb/>ad B D, propter triangulos ſimiles C B G, E B D; </s> <s xml:id="echoid-s2516" xml:space="preserve">habent <lb/>enim angulos ad B æquales & </s> <s xml:id="echoid-s2517" xml:space="preserve">angulum B C G angulo B E D. </s> <s xml:id="echoid-s2518" xml:space="preserve"><lb/>Item duplo rectangulo E B L æquale eſt quadratum A B, <lb/>quia propter triangulos ſimiles ut S A, hoc eſt, dupla B E <lb/>ad A B ita A B ad B L. </s> <s xml:id="echoid-s2519" xml:space="preserve">Igitur ut B G ad B E ita erit <lb/>quadratum K ad rectangulum C B D cum quadrato A B. </s> <s xml:id="echoid-s2520" xml:space="preserve"><lb/>Sed hiſce duobus æquale eſt rectangulum C A D; </s> <s xml:id="echoid-s2521" xml:space="preserve">quoniam <lb/>in triangulo C A D angulus A bifariam dividitur à linea A B. </s> <s xml:id="echoid-s2522" xml:space="preserve"><lb/>Ergo ut B G ad B E ita eſt quadr. </s> <s xml:id="echoid-s2523" xml:space="preserve">K ad rectangulum C A D. </s> <s xml:id="echoid-s2524" xml:space="preserve"><lb/>Atque hinc porrò eodem modo ut in caſu præcedenti con-<lb/>cludemus lineam D C ipſi K æqualem eſſe, repetendo iſta: </s> <s xml:id="echoid-s2525" xml:space="preserve"><lb/>Sicut autem G B ad B E, &</s> <s xml:id="echoid-s2526" xml:space="preserve">c.</s> <s xml:id="echoid-s2527" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div132" type="section" level="1" n="57"> <head xml:id="echoid-head86" style="it" xml:space="preserve">Utrumque præcedentium Aliter.</head> <p> <s xml:id="echoid-s2528" xml:space="preserve">SIt datus rhombus A D B C cujus productum latus <lb/> <anchor type="note" xlink:label="note-0117-01a" xlink:href="note-0117-01"/> D B. </s> <s xml:id="echoid-s2529" xml:space="preserve">Et data ſit linea G. </s> <s xml:id="echoid-s2530" xml:space="preserve">Oportet ducere rectam A N F, <lb/>ut pars intercepta N F ſit datæ G æqualis.</s> <s xml:id="echoid-s2531" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="1"> <note position="right" xlink:label="note-0117-01" xlink:href="note-0117-01a" xml:space="preserve">TAB. XLII. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s2532" xml:space="preserve">Ducatur diameter A B, & </s> <s xml:id="echoid-s2533" xml:space="preserve">quadratis ex G & </s> <s xml:id="echoid-s2534" xml:space="preserve">A B ſit æ-<lb/>quale quadratum A H, & </s> <s xml:id="echoid-s2535" xml:space="preserve">ducatur H E ipſi B A parallela. <lb/></s> <s xml:id="echoid-s2536" xml:space="preserve">Et A E ipſi G ponatur æqualis, eademque producatur ad <lb/>F. </s> <s xml:id="echoid-s2537" xml:space="preserve">Dico N F ipſi G æqualem eſſe.</s> <s xml:id="echoid-s2538" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2539" xml:space="preserve">Quod autem ad H E poni poteſt A E ipſi G æqualis, <pb o="402" file="0118" n="126" rhead="CHRISTIANI HUGENII"/> hinc manifeſtum eſt. </s> <s xml:id="echoid-s2540" xml:space="preserve">Etenim quadratum A H majus eſt <lb/>quadratis A X & </s> <s xml:id="echoid-s2541" xml:space="preserve">X H, quum ſit angulus A X H obtuſus. <lb/></s> <s xml:id="echoid-s2542" xml:space="preserve">Sed idem quadratum A H æquale ponitur quadratis A B ſeu <lb/>H X & </s> <s xml:id="echoid-s2543" xml:space="preserve">G. </s> <s xml:id="echoid-s2544" xml:space="preserve">Itaque quadratum G ſeu A E majus eſt quadrato <lb/>A X. </s> <s xml:id="echoid-s2545" xml:space="preserve">Unde apparet interſectionem E accidere inter puncta <lb/>H & </s> <s xml:id="echoid-s2546" xml:space="preserve">X.</s> <s xml:id="echoid-s2547" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2548" xml:space="preserve">Producatur B D & </s> <s xml:id="echoid-s2549" xml:space="preserve">ponatur ipſi æqualis D R. </s> <s xml:id="echoid-s2550" xml:space="preserve">& </s> <s xml:id="echoid-s2551" xml:space="preserve">ſit R K <lb/>parallela D A vel B C, eique occurrant productæ F A, <lb/>B A, H E, in punctis M, Q, K: </s> <s xml:id="echoid-s2552" xml:space="preserve">& </s> <s xml:id="echoid-s2553" xml:space="preserve">jungatur R A, & </s> <s xml:id="echoid-s2554" xml:space="preserve"><lb/>producatur ad P.</s> <s xml:id="echoid-s2555" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2556" xml:space="preserve">Quoniam igitur D R æqualis eſt D B, & </s> <s xml:id="echoid-s2557" xml:space="preserve">R Q K paral-<lb/>lela D A, erit & </s> <s xml:id="echoid-s2558" xml:space="preserve">M A æqualis A N, & </s> <s xml:id="echoid-s2559" xml:space="preserve">Q A æqualis A B; <lb/></s> <s xml:id="echoid-s2560" xml:space="preserve">angulus autem B A R rectus, quum ſit in ſemicirculo, <lb/>nam tres hæ æquales ſunt D B, D A, D R. </s> <s xml:id="echoid-s2561" xml:space="preserve">Parallelæ au-<lb/>tem ſunt B Q, H E K, ergo & </s> <s xml:id="echoid-s2562" xml:space="preserve">anguli ad P recti, & </s> <s xml:id="echoid-s2563" xml:space="preserve">erit <lb/>H P æqualis P K. </s> <s xml:id="echoid-s2564" xml:space="preserve">Eſt itaque quadratum A H æquale qua-<lb/>drato A E unà cum rectangulo H E K <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2565" xml:space="preserve">Sed idem quadra- <anchor type="note" xlink:label="note-0118-01a" xlink:href="note-0118-01"/> tum A H æquale eſt etiam quadratis ex G ſeu A E, & </s> <s xml:id="echoid-s2566" xml:space="preserve">ex <lb/>A B. </s> <s xml:id="echoid-s2567" xml:space="preserve">Itaque quadr. </s> <s xml:id="echoid-s2568" xml:space="preserve">A B æquale erit rectangulo K E H. <lb/></s> <s xml:id="echoid-s2569" xml:space="preserve">Ac propterea K E ad A B ut A B ad E H. </s> <s xml:id="echoid-s2570" xml:space="preserve">Verum ut K E <lb/>ad A B ſeu Q A ita eſt E M ad M A: </s> <s xml:id="echoid-s2571" xml:space="preserve">& </s> <s xml:id="echoid-s2572" xml:space="preserve">ut A B ad E H <lb/>ita A F ad F E. </s> <s xml:id="echoid-s2573" xml:space="preserve">Igitur E M ad M A ut A F ad F E: </s> <s xml:id="echoid-s2574" xml:space="preserve">Et <lb/>proinde E A ad A M ut E A ad E F. </s> <s xml:id="echoid-s2575" xml:space="preserve">Æqualis eſt igi-<lb/>tur E F ipſi A M; </s> <s xml:id="echoid-s2576" xml:space="preserve">quare & </s> <s xml:id="echoid-s2577" xml:space="preserve">ipſi A N. </s> <s xml:id="echoid-s2578" xml:space="preserve">Ideoque & </s> <s xml:id="echoid-s2579" xml:space="preserve">F N <lb/>ipſi A E, hoc eſt, datæ G. </s> <s xml:id="echoid-s2580" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s2581" xml:space="preserve"/> </p> <div xml:id="echoid-div133" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0118-01" xlink:href="note-0118-01a" xml:space="preserve">12.2 Elem.</note> </div> <p> <s xml:id="echoid-s2582" xml:space="preserve">Sit denuo datus rhomdus A D B C, cujus producta la-<lb/> <anchor type="note" xlink:label="note-0118-02a" xlink:href="note-0118-02"/> tera B D, B C; </s> <s xml:id="echoid-s2583" xml:space="preserve">& </s> <s xml:id="echoid-s2584" xml:space="preserve">data ſit linea G. </s> <s xml:id="echoid-s2585" xml:space="preserve">Oportet ducere re-<lb/>ctam N F tranſeuntem per angulum A, quæque æqualis ſit <lb/>ipſi G.</s> <s xml:id="echoid-s2586" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="3"> <note position="left" xlink:label="note-0118-02" xlink:href="note-0118-02a" xml:space="preserve">TAB. XLII. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s2587" xml:space="preserve">Ducatur diameter B A, eique ad angulos rectos R A L. <lb/></s> <s xml:id="echoid-s2588" xml:space="preserve">Si igitur G minor detur quam R L, problema conſtrui ne-<lb/>quit, uti ſupra quoque dictum fuit. </s> <s xml:id="echoid-s2589" xml:space="preserve">Si vero æqualis, jam fa-<lb/>ctum eſt quod quærebatur. </s> <s xml:id="echoid-s2590" xml:space="preserve">Sit igitur G major quam R L. </s> <s xml:id="echoid-s2591" xml:space="preserve"><lb/>Erit in ſchemate adjecto, ſicut propoſitum eſt, conſtru-<lb/>ctio & </s> <s xml:id="echoid-s2592" xml:space="preserve">demonſtratio eadem quæ in caſu præcedenti.</s> <s xml:id="echoid-s2593" xml:space="preserve"/> </p> <pb o="403" file="0119" n="127" rhead="ILLUST. QUORUND. PROB. CONSTRUCT."/> <p> <s xml:id="echoid-s2594" xml:space="preserve">Illud autem hic aliter eſt oſtendendum, quod ad lineam <lb/>H E poni poteſt A E ipſi G æqualis. </s> <s xml:id="echoid-s2595" xml:space="preserve">Sit R S æqualis R B, <lb/>& </s> <s xml:id="echoid-s2596" xml:space="preserve">jungatur A S. </s> <s xml:id="echoid-s2597" xml:space="preserve">Quoniam igitur in triangulo B A S à ver-<lb/>tice ad mediam baſin ducta eſt A R, erunt quadrata B R <lb/>& </s> <s xml:id="echoid-s2598" xml:space="preserve">R A ſimul ſumpta, hoc eſt, quadratum B A cum duplo <lb/> <anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/> quadrato A R, ſubdupla quadratorum B A, A S <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2599" xml:space="preserve">Itaque quadratum A B duplum cum quadruplo quadrato A R, hoc <lb/>eſt, cum quadrato R L, æquabitur quadratis B A, A S. <lb/></s> <s xml:id="echoid-s2600" xml:space="preserve">Quare ablato utrimque quadrato B A, erit quadratum A S <lb/>æquale quadratis B A & </s> <s xml:id="echoid-s2601" xml:space="preserve">R L, ac proinde minus quam quadr. </s> <s xml:id="echoid-s2602" xml:space="preserve"><lb/>A H; </s> <s xml:id="echoid-s2603" xml:space="preserve">nam hoc æquale eſt quadratis A B & </s> <s xml:id="echoid-s2604" xml:space="preserve">G. </s> <s xml:id="echoid-s2605" xml:space="preserve">Eſt igitur <lb/>A S minor quam A H. </s> <s xml:id="echoid-s2606" xml:space="preserve">Sed major eſt quam A R. </s> <s xml:id="echoid-s2607" xml:space="preserve">Ergo pun-<lb/>ctum S cadit inter R & </s> <s xml:id="echoid-s2608" xml:space="preserve">H; </s> <s xml:id="echoid-s2609" xml:space="preserve">angulus enim A R H obtuſus <lb/>eſt. </s> <s xml:id="echoid-s2610" xml:space="preserve">Major itaque eſt R H quam R S vel R B. </s> <s xml:id="echoid-s2611" xml:space="preserve">Et quum <lb/>propter triangulos ſimiles ſit R H ad H P ut R B ad B A, <lb/>erit quoque H P major quam B A; </s> <s xml:id="echoid-s2612" xml:space="preserve">& </s> <s xml:id="echoid-s2613" xml:space="preserve">quadratum H P ma-<lb/>jus quadrato A B. </s> <s xml:id="echoid-s2614" xml:space="preserve">At quadratum H P cum quadrato P A <lb/>æquatur quadrato A H, hoc eſt, quadratis B A & </s> <s xml:id="echoid-s2615" xml:space="preserve">G. </s> <s xml:id="echoid-s2616" xml:space="preserve">Er-<lb/>go cum quadratum H P ſit majus quadrato A B, erit invi-<lb/>cem quadr. </s> <s xml:id="echoid-s2617" xml:space="preserve">P A minus quam quadr. </s> <s xml:id="echoid-s2618" xml:space="preserve">G. </s> <s xml:id="echoid-s2619" xml:space="preserve">Patet igitur quod <lb/>ſi centro A circumferentia deſcribatur radio A E ipſi G æ-<lb/>quali, ea lineam H E ſecabit.</s> <s xml:id="echoid-s2620" xml:space="preserve"/> </p> <div xml:id="echoid-div135" type="float" level="2" n="4"> <note symbol="*" position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">per 122. <lb/>lib.7. Pappi.</note> </div> </div> <div xml:id="echoid-div137" type="section" level="1" n="58"> <head xml:id="echoid-head87" xml:space="preserve"><emph style="sc">Probl.</emph> VIII.</head> <head xml:id="echoid-head88" style="it" xml:space="preserve">In Conchoide linea invenire confinia <lb/>flexus contrarii.</head> <p> <s xml:id="echoid-s2621" xml:space="preserve">Conchoidem intelligimus quam Nicomedes excogitavit; <lb/></s> <s xml:id="echoid-s2622" xml:space="preserve"> <anchor type="note" xlink:label="note-0119-02a" xlink:href="note-0119-02"/> quâ & </s> <s xml:id="echoid-s2623" xml:space="preserve">angulum diviſit trifariam, & </s> <s xml:id="echoid-s2624" xml:space="preserve">duas medias invenit <lb/>proportionales: </s> <s xml:id="echoid-s2625" xml:space="preserve">Eſto ea C Q D, polus G, regula autem <lb/>A B cujus ope deſcripta eſt; </s> <s xml:id="echoid-s2626" xml:space="preserve">quam ſecet G Q ad angulos <lb/>rectos. </s> <s xml:id="echoid-s2627" xml:space="preserve">Hæc igitur lineæ proprietas eſt, ut ductâ ad ipſam <lb/>rectâ qualibet ex G puncto, pars hujus inter conchoidem & </s> <s xml:id="echoid-s2628" xml:space="preserve"><lb/>rectam A B intercepta ſit ipſi A Q æqualis.</s> <s xml:id="echoid-s2629" xml:space="preserve"/> </p> <div xml:id="echoid-div137" type="float" level="2" n="1"> <note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">TAB. XLII. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s2630" xml:space="preserve">Quum autem appareat partem quandam Conchoidis ut in <lb/>ſchemate ſubjecto C Q D verſus polum G cavam eſſe, lineam <pb o="404" file="0120" n="128" rhead="CHR. HUG. ILL. QUOR. PROB. CONSTR."/> verò reliquam in infinitum licet utrimque productam in diver-<lb/>ſum curvari; </s> <s xml:id="echoid-s2631" xml:space="preserve">quæſitum eſt qua ratione puncta ea determinari <lb/>poſſent ubi contraria flexio initium capit. </s> <s xml:id="echoid-s2632" xml:space="preserve">Et nos quidem ad <lb/>hoc ſequentem invenimus conſtructionem.</s> <s xml:id="echoid-s2633" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2634" xml:space="preserve">Sit duabus A G, A Q tertia proportionalis A E, ſumenda <lb/>verſus G. </s> <s xml:id="echoid-s2635" xml:space="preserve">Et ponatur G F æqualis G E. </s> <s xml:id="echoid-s2636" xml:space="preserve">Porro ſit G R ipſi G Q <lb/>ad angulos rectos, & </s> <s xml:id="echoid-s2637" xml:space="preserve">æqualis duplæ G A. </s> <s xml:id="echoid-s2638" xml:space="preserve">Et deſcribatur pa-<lb/>rabole R O, cujus vertex ſit R axis R G, latus rectum ipſi A G <lb/>æquale. </s> <s xml:id="echoid-s2639" xml:space="preserve">Centro autem F radio F R circumferentia deſcriba-<lb/>tur, quæ parabolen ſecet in O; </s> <s xml:id="echoid-s2640" xml:space="preserve">& </s> <s xml:id="echoid-s2641" xml:space="preserve">ducatur O C parallela A B <lb/>occurratque conchoidi in punctis C, D. </s> <s xml:id="echoid-s2642" xml:space="preserve">Hæc erunt puncta <lb/>quæſita in confinio flexionis contrariæ.</s> <s xml:id="echoid-s2643" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2644" xml:space="preserve">Iſta autem Univerſalis eſt conſtructio. </s> <s xml:id="echoid-s2645" xml:space="preserve">At quando quadra-<lb/> <anchor type="note" xlink:label="note-0120-01a" xlink:href="note-0120-01"/> tum ex A Q non majus eſt quam duplum quadrati A G, ar-<lb/>cus triſectione propoſitum quoque efficiemus. </s> <s xml:id="echoid-s2646" xml:space="preserve">Et diversè qui-<lb/>dem prout A Q major vel minor erit quam A G. </s> <s xml:id="echoid-s2647" xml:space="preserve">Etenim ſi <lb/>minor, deſcribenda eſt circumferentia centro A radio A G, <lb/>in eaque ponenda G K æqualis duplæ G E, inventæ ut priùs. <lb/></s> <s xml:id="echoid-s2648" xml:space="preserve">Et rectæ G H quæ ſubtendit trientem circumferentiæ K H G <lb/>æqualis ſumenda G M, & </s> <s xml:id="echoid-s2649" xml:space="preserve">per M ducenda ut ante D C ipſi <lb/> <anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/> A B parallela. </s> <s xml:id="echoid-s2650" xml:space="preserve">Cum verò A Q major eſt quam A G, cæte-<lb/>ris ad eundem modum compoſitis, hæc tantum differentia <lb/>erit quod arcum K P, qui unà cum arcu G K ſemicircum-<lb/>ferentiam explet, in tria æqualia dividere oportet, & </s> <s xml:id="echoid-s2651" xml:space="preserve">partium <lb/>unam conſtituere P H, & </s> <s xml:id="echoid-s2652" xml:space="preserve">ſubtenſæ G H æqualem ſumere G M.</s> <s xml:id="echoid-s2653" xml:space="preserve"/> </p> <div xml:id="echoid-div138" type="float" level="2" n="2"> <note position="left" xlink:label="note-0120-01" xlink:href="note-0120-01a" xml:space="preserve">TAB. XLII. <lb/>Fig. 6.</note> <note position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">TAB. XLII. <lb/>Fig. 7.</note> </div> <p> <s xml:id="echoid-s2654" xml:space="preserve">Porro planum eſt Problema cum A G æqualis A Q. </s> <s xml:id="echoid-s2655" xml:space="preserve">Tunc <lb/>enim G M fit æqualis lateri trigoni ordinati in circulo in-<lb/>ſcripti. </s> <s xml:id="echoid-s2656" xml:space="preserve">Item cum quadratum A Q duplum eſt quadrati A G: <lb/></s> <s xml:id="echoid-s2657" xml:space="preserve">fit enim G M dupla ipſius G A.</s> <s xml:id="echoid-s2658" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2659" xml:space="preserve">Sed & </s> <s xml:id="echoid-s2660" xml:space="preserve">aliis caſibus innumeris planum erit, quorum ii qui-<lb/>dem facilè diſcerni poterunt, qui ad anguli triſectionem re-<lb/>ducuntur.</s> <s xml:id="echoid-s2661" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div140" type="section" level="1" n="59"> <head xml:id="echoid-head89" style="it" xml:space="preserve">FINIS.</head> <pb file="0121" n="129"/> <pb file="0121a" n="130"/> <figure> <caption xml:id="echoid-caption54" style="it" xml:space="preserve">Pag. 404.<lb/>TAB. XLII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables54" xml:space="preserve">K F M A C D B L E N G</variables> </figure> <figure> <caption xml:id="echoid-caption55" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables55" xml:space="preserve">G R D B H F E N A X C M P Q K</variables> </figure> <figure> <caption xml:id="echoid-caption56" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables56" xml:space="preserve">K A F c S C L E B T G D R d</variables> </figure> <figure> <caption xml:id="echoid-caption57" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables57" xml:space="preserve">K e G P E m B D f R F S H M C A N L Q n</variables> </figure> <figure> <caption xml:id="echoid-caption58" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables58" xml:space="preserve">B C R E G A F M Q D O</variables> </figure> <figure> <caption xml:id="echoid-caption59" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables59" xml:space="preserve">B C H G E A M Q P K D</variables> </figure> <figure> <caption xml:id="echoid-caption60" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables60" xml:space="preserve">B C E G A M P Q K H D</variables> </figure> <pb file="0122" n="131"/> <pb file="0123" n="132"/> </div> <div xml:id="echoid-div141" type="section" level="1" n="60"> <head xml:id="echoid-head90" xml:space="preserve">DE <lb/>CIRCULI <lb/>ET HYPERBOLÆ <lb/>QUADRATURA <lb/>CONTROVERSIA.</head> <pb file="0124" n="133"/> <pb file="0125" n="134"/> <figure> <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0125-01"/> </figure> </div> <div xml:id="echoid-div142" type="section" level="1" n="61"> <head xml:id="echoid-head91" xml:space="preserve">VERA <lb/>CIRCULI ET HYPERBOLÆ <lb/>QUADRATURA <lb/>AUTHORE <lb/>JACOBO GREGORIO.</head> <head xml:id="echoid-head92" xml:space="preserve">LECTORI GEOMETRÆ <lb/>SALUTEM.</head> <p style="it"> <s xml:id="echoid-s2662" xml:space="preserve">MEcum aliquando cogitabam, ami-<lb/>ce Lector, num Analytica cum <lb/>ſuis quinque operationibus eſſet <lb/>ſufficiens, & </s> <s xml:id="echoid-s2663" xml:space="preserve">generalis methodus <lb/>inveſtigandi omnes quantitatum <lb/>proportiones, ut in initio ſuæ Ge-<lb/>ometriæ affirmare videtur Car-<lb/>teſius; </s> <s xml:id="echoid-s2664" xml:space="preserve">ſi enim ita eſſet, poſſibile foret ejus ope to-<lb/>ties decantatam circuli quadraturam exhibere: <lb/></s> <s xml:id="echoid-s2665" xml:space="preserve">cumque hæc mente revolverem, facile percepi ex <lb/>hactenus repertis circuli proprietatibus nullam poſſe <lb/>analyſin inſtitui tali ſtructuræ inſervientem: </s> <s xml:id="echoid-s2666" xml:space="preserve">dein-<lb/>de mihi alias quærenti incidit in mentem hujus ſe-<lb/>cunda, prima enim in circulo vulgo eſt cognita: </s> <s xml:id="echoid-s2667" xml:space="preserve"><lb/>ex hiſce percepi ſeriem polygonorum convergentem, <lb/>cujus terminatio eſt circuli ſector; </s> <s xml:id="echoid-s2668" xml:space="preserve">ubi ſtatim vidi <pb file="0126" n="135" rhead="PRÆFATIO"/> aliquod analyſios veſtigium. </s> <s xml:id="echoid-s2669" xml:space="preserve">deinde ſerierum con- vergentium naturis non ſolum in facilioribus qui- busdam caſibus, ſed etiam in genere conſideratis, & </s> <s xml:id="echoid-s2670" xml:space="preserve">prædictis circuli proprietatibus ad ellipſim &</s> <s xml:id="echoid-s2671" xml:space="preserve"> hyperbolam nullo negotio reductis, infallibilis mihi videbatur omnium ſectionum conicarum quadratu- ra: </s> <s xml:id="echoid-s2672" xml:space="preserve">dum autem me illuc converti ut polygonorum ſeriem convergentem terminarem, inſuperabilem difficultatem in ejus terminatione invenienda poſt omnes artis & </s> <s xml:id="echoid-s2673" xml:space="preserve">aleæ conatus deprehendi: </s> <s xml:id="echoid-s2674" xml:space="preserve">Sed ani- mo revolvens analyſios oſſicium eſſe ſicut algebræ communis, non ſolum problemata reſolvere, ſed etiam eorum impoſſibilitatem (ſi opus ſit) demon- ſtrare; </s> <s xml:id="echoid-s2675" xml:space="preserve">cumque in primo difficultatem indicibilem expertus eſſem, ad ſecundum me converti, quod certe ſupra votum ſucceſſit; </s> <s xml:id="echoid-s2676" xml:space="preserve">non enim ſolius circuli (quam mihi ab initio propoſueram) ſed omnium ſe- ctionum conicarum veram & </s> <s xml:id="echoid-s2677" xml:space="preserve">legitimam in ſua pro- portionum ſpecie quadraturam, & </s> <s xml:id="echoid-s2678" xml:space="preserve">integram pro- portionis ſpeciem ante incognitam orbi Geometrico patefacio, quam etiam proportionem ſaltem in re- latione ad dimenſionem ſectionum conicarum ad commenſurabilem veræ quam proximam reduco, praxi facili, demonſtrabili, & </s> <s xml:id="echoid-s2679" xml:space="preserve">extractione radicis ſurde ſolidæ (ni fallor) multo breviore; </s> <s xml:id="echoid-s2680" xml:space="preserve">in omni enim proportione incommenſurabili ad tales approxi- mationes recurrunt Mathematici: </s> <s xml:id="echoid-s2681" xml:space="preserve">ut autem melius concipiamus hujus proportionis naturam, loquamur de proportione quatenus ortum habet à quinque <pb file="0127" n="136" rhead="AD LECTOREM."/> operationibus analyticis, ſeu arithmeticæ vulgaris, proportio enim nobis notior eſt in numeris ſeu in quantitate diſcreta, quam in continua, neque ve- reor poſt Carteſium has operationes in geometriam adducere. </s> <s xml:id="echoid-s2682" xml:space="preserve">Primo itaque ſciendum eſt nos ſemper nobis proponere quantitates commenſurabiles, ſeu quæ inter ſe ſunt ut numerus ad numerum; </s> <s xml:id="echoid-s2683" xml:space="preserve">pro- portionem enim incommenſur abilem niſi relativè ad commenſurabilem nullo modo percipimus, habet enim in ſe neſcio quid infiniti, mentem noſtram obtun- dens & </s> <s xml:id="echoid-s2684" xml:space="preserve">ſimplicem perceptionem impediens: </s> <s xml:id="echoid-s2685" xml:space="preserve">deinde ex illis quinque operationibus arithmeticis, duæ ſunt tantum ſimplices, additio & </s> <s xml:id="echoid-s2686" xml:space="preserve">ſubſtractio; </s> <s xml:id="echoid-s2687" xml:space="preserve">mul- tiplicatio enim eſt compoſita ex additione, & </s> <s xml:id="echoid-s2688" xml:space="preserve">divi- ſio ex ſubſtractione; </s> <s xml:id="echoid-s2689" xml:space="preserve">& </s> <s xml:id="echoid-s2690" xml:space="preserve">extractio radicum, quæ in genere nihil aliud eſt quam inventio proportionis commenſurabilis, quæ quam proximè accedit ad proportionem analyticam incommenſur abilem, com- ponitur ex præcedentibus quatuor; </s> <s xml:id="echoid-s2691" xml:space="preserve">& </s> <s xml:id="echoid-s2692" xml:space="preserve">noſtra ſexta operatio quæ in genere nihil aliud eſt quam inven- tio proportionis commenſurahilis, quam proximè accedentis ad noſtram proportionem non analyticam, componitur ex prioribus quinque. </s> <s xml:id="echoid-s2693" xml:space="preserve">Advertendum quoque eſt ſicut numeri fracti nunquam procedunt ex integrorum additione, ſubductione, multiplica- tione, ſed tantum ex diviſione; </s> <s xml:id="echoid-s2694" xml:space="preserve">& </s> <s xml:id="echoid-s2695" xml:space="preserve">numeri incom- menſurabiles nunquam procedunt ex commenſura- bilium additione, ſubductione, multiplicatione, diviſione, ſed tantum ex radicum extractione;</s> <s xml:id="echoid-s2696" xml:space="preserve"> <pb file="0128" n="137" rhead="PRÆFATIO"/> ita numeros, vel quantitates non analyticas nun- quam provenire ex analyticarum additione, ſub- ductione, multiplicatione, diviſione, radicum ex- tractione, ſed ex ſexta hac operatione; </s> <s xml:id="echoid-s2697" xml:space="preserve">ita ut hæc noſtra inventio addat arithmeticæ aliam operatio- nem, & </s> <s xml:id="echoid-s2698" xml:space="preserve">geometriæ aliam rationis ſpeciem, idem enim eſt (ſicut in hoc tractatulo demonſtro) ratio- nem circuli ad diametri quadratum in analytica ſeu illa rationis ſpecie hactenus cognita exhibere, ac rationem inter quadrati latus & </s> <s xml:id="echoid-s2699" xml:space="preserve">ejuſdem diame- trum in commenſur abilibus: </s> <s xml:id="echoid-s2700" xml:space="preserve">verum certè eſt me hanc demonſtrationem integram ad phraſem geo- metricam non reduxiſſe, nam ut hoc perficiatur, opus eſt non parvo volumine de quantitatum analy- ticarum mutuo inter ſe relatione & </s> <s xml:id="echoid-s2701" xml:space="preserve">incommenſura- bilitate in genere, quod miror nullum unquam ſcripſiſſe, cum in his tam latè pateant inventio- num campi; </s> <s xml:id="echoid-s2702" xml:space="preserve">nam ex his petenda eſt demonſtratio, quod meſolabium non poſſit perfici ope regulæ & </s> <s xml:id="echoid-s2703" xml:space="preserve">circi- ni, item quod non ſemper & </s> <s xml:id="echoid-s2704" xml:space="preserve">quando æquationes affectæ poſſunt reduci ad puras, item quod neceſſaria ſit ad minimum talis generis curva ad mechanicam talium æquationum reſolutionem, cum talibus innumeris, quæ à præſtantioribus geometris impoſ- ſibilia eſſe deprehenduntur ex analyſi, & </s> <s xml:id="echoid-s2705" xml:space="preserve">à rudio- ribus quotidie & </s> <s xml:id="echoid-s2706" xml:space="preserve">fruſtra quæruntur. </s> <s xml:id="echoid-s2707" xml:space="preserve">Scripſit Eu- clides decimum ſuum librum ſolummodo (niſi in pau- cis quibuſdam propoſitionibus generalibus) de incom- @nenſurabilitate facta ab extractione radicis quadra- <pb file="0129" n="138" rhead="AD LECTOREM."/> tæ; </s> <s xml:id="echoid-s2708" xml:space="preserve">neque quantum ſcio ab ullo alio tractata eſt hæc materia, etiamſi geometriæ ſpeculativæ non ſo- lum utiliſſima ſit, ſed etiam maxime admirabilis;</s> <s xml:id="echoid-s2709" xml:space="preserve"> in ipſo enim limine admiranda occurrunt theorema- ta; </s> <s xml:id="echoid-s2710" xml:space="preserve">e.</s> <s xml:id="echoid-s2711" xml:space="preserve">g. </s> <s xml:id="echoid-s2712" xml:space="preserve">Si fuerit progreſſio geometrica cujus unus terminus fuerit propoſitæ quantitati commenſurabi- lis longitudine vel poteſtate quacunque, & </s> <s xml:id="echoid-s2713" xml:space="preserve">alius quilibet, binomium, trinomium, &</s> <s xml:id="echoid-s2714" xml:space="preserve">c. </s> <s xml:id="echoid-s2715" xml:space="preserve">quodcunque, impoſsibile eſt duos totius progreſſionis terminos in infinitum continuatæ eſſe inter ſe commenſurabiles longitudine vel poteſtate quacunque: </s> <s xml:id="echoid-s2716" xml:space="preserve">alia multa poſſem afferre, ſed pro commodiore fortaſſis tempo- re hæc reſervo, ſatis exiſtimans pro præſenti hæc analyticè demonſtraſſe; </s> <s xml:id="echoid-s2717" xml:space="preserve">etſi enim analyſis aſſenſum adeo violenter non cogat ac geometria, nunquam ta- men reſpuit nec reſpuere poteſt geometria, quodpro- bavit ſemel analyſis geometrica. </s> <s xml:id="echoid-s2718" xml:space="preserve">Ex hac inventione deduco quoque novam ſectionem angularium & </s> <s xml:id="echoid-s2719" xml:space="preserve">lo- gorithmorum doctrinam, facilem quidem, in praxi expeditiſsimam & </s> <s xml:id="echoid-s2720" xml:space="preserve">geometrica demonſtratione muni- tam; </s> <s xml:id="echoid-s2721" xml:space="preserve">hactenus enim logorithmorum conſtructio pro- lixiſſima, conjectura potius quam ſcientia videba- tur, & </s> <s xml:id="echoid-s2722" xml:space="preserve">diviſio anguli in partes æquales ultra quin- que numero primo numeratas in praxim vix ad- mitti poterat. </s> <s xml:id="echoid-s2723" xml:space="preserve">hæc omnia ſumma (qua poſſum) bre- vitate & </s> <s xml:id="echoid-s2724" xml:space="preserve">perſpicuitate demonſtro; </s> <s xml:id="echoid-s2725" xml:space="preserve">neque ſcrupulo- ſus ſum in citationibus, utpote peregrinus, & </s> <s xml:id="echoid-s2726" xml:space="preserve">li- bris ad tale opus deſtitutus, te enim ſuppono in geometricis non mediocriter verſatum, alioquin nul- <pb file="0130" n="139" rhead="PRÆFATIO AD LECTOREM."/> lum hinc fructum colliges.</s> <s xml:id="echoid-s2727" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div143" type="section" level="1" n="62"> <head xml:id="echoid-head93" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s2728" xml:space="preserve">1 Si in circulo, ellipſe vel hyperbola ducantur è centro <lb/>in ejus perimetrum duæ rectæ, appellamus planum <lb/>ab illis rectis & </s> <s xml:id="echoid-s2729" xml:space="preserve">perimetri ſegmento comprehenſum, <lb/>ſectorem.</s> <s xml:id="echoid-s2730" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2731" xml:space="preserve">2 Si perimetri ſegmentum inter illas rectas comprehenſum à <lb/>rectis quotcumque ſubtendatur, ita ut triangula rectili-<lb/>nea (quorum communis vertex eſt ſectionis centrum & </s> <s xml:id="echoid-s2732" xml:space="preserve"><lb/>baſes rectæ ſubtendentes) ſint æqualia; </s> <s xml:id="echoid-s2733" xml:space="preserve">vocamus rectili-<lb/>neum illud ab iſtis triangulis conflatum, polygonum re-<lb/>gulare inſcriptum, ſi ſectio conica fuerit circulus vel el-<lb/>lipſis; </s> <s xml:id="echoid-s2734" xml:space="preserve">quod ſi fuerit hyperbola, vocamus illud rectili-<lb/>neum polygonum regulare circumſcriptum.</s> <s xml:id="echoid-s2735" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2736" xml:space="preserve">3 Si perimetri ſegmentum inter illas rectas comprehenſum à <lb/>rectis quotcunque tangatur & </s> <s xml:id="echoid-s2737" xml:space="preserve">à tactibus ad ſectionis cen-<lb/>trum ducantur rectæ; </s> <s xml:id="echoid-s2738" xml:space="preserve">ſi inquam omnia trapezia, a tan-<lb/>gentibus proximis & </s> <s xml:id="echoid-s2739" xml:space="preserve">rectis ad centrum comprehenſa, fue-<lb/>rint æqualia; </s> <s xml:id="echoid-s2740" xml:space="preserve">appello rectilineum ab illis conflatum, poly-<lb/>gonum regulare circumſcriptum, ſi ſectio conica ſit elli-<lb/>pſis vel circulus, & </s> <s xml:id="echoid-s2741" xml:space="preserve">polygonum regulare inſcriptum ſi <lb/>fuerit hyperbola.</s> <s xml:id="echoid-s2742" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2743" xml:space="preserve">4 Si omnes anguli (excepto illo ad ſectionis centrum) po-<lb/>lygoni regularis à ſubtendentibus comprehenſi inſiſtant <pb o="413" file="0131" n="140" rhead="DEFINITIONES."/> omnibus contactum punctis polygoni regularis à tangen-<lb/>tibus comprehenſi, appello hæc polygona complicata.</s> <s xml:id="echoid-s2744" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2745" xml:space="preserve">5 Quantitatem dicimus à quantitatibus eſſe compoſitam; <lb/></s> <s xml:id="echoid-s2746" xml:space="preserve">cum à quantitatum additione, ſubductione, multiplica-<lb/>tione, diviſione, radicum extractione, vel quacunque alia <lb/>imaginabili operatione, fit alia quantitas.</s> <s xml:id="echoid-s2747" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2748" xml:space="preserve">6 Quando quantitas componitur ex quantitatum additione, <lb/>ſubductione, multiplicatione, diviſione, radicum extra-<lb/>ctione; </s> <s xml:id="echoid-s2749" xml:space="preserve">dicimus illam componi analyticè.</s> <s xml:id="echoid-s2750" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2751" xml:space="preserve">7 Quando quantitates à quantitatibus inter ſe commenſura-<lb/>bilibus analyticè componi poſſunt, dicimus illas eſſe inter <lb/>ſe analyticas.</s> <s xml:id="echoid-s2752" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2753" xml:space="preserve">8 Si à quantitatibus quotcunque A, B, C, D, E, com-<lb/>ponatur quantitas X, & </s> <s xml:id="echoid-s2754" xml:space="preserve">à quantitatibus F, G, C, D, E, <lb/>componatur quantitas Z, eâdem omnino methodo & </s> <s xml:id="echoid-s2755" xml:space="preserve">iis-<lb/>dem omnino operationibus quibus antè componebatur X, <lb/>poſitis quantitatibus F, G, loco quantitatum A, B, ſi <lb/>inquam hoc fiat. </s> <s xml:id="echoid-s2756" xml:space="preserve">dicimus quantitatem X eodem modo <lb/>componi à quantitatibus A, B, quo Z componitur à <lb/>quantitatibus F, G.</s> <s xml:id="echoid-s2757" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2758" xml:space="preserve">9 Sint duæ quantitates A, B, à quibus compo-<lb/> <anchor type="note" xlink:label="note-0131-01a" xlink:href="note-0131-01"/> nantur duæ aliæ quantitates C, D, quarum <lb/>differentia ſit minor differentia quantitatum <lb/>A, B, & </s> <s xml:id="echoid-s2759" xml:space="preserve">eodem modo quo C, componitur à <lb/>quantitatibus A, B, componatur E à quantita-<lb/>tibus C, D; </s> <s xml:id="echoid-s2760" xml:space="preserve">& </s> <s xml:id="echoid-s2761" xml:space="preserve">eodem modo quo D componitur à quan-<lb/>titatibus A, B, componatur F, à quantitatibus C, D; </s> <s xml:id="echoid-s2762" xml:space="preserve">&</s> <s xml:id="echoid-s2763" xml:space="preserve"> <pb o="414" file="0132" n="141" rhead="DEFINITIONES."/> eodem modo quo E componitur à quantitatibus C, D, <lb/>vel C à quantitaiibus A, B, componatur G à quantita-<lb/>tibus E, F; </s> <s xml:id="echoid-s2764" xml:space="preserve">& </s> <s xml:id="echoid-s2765" xml:space="preserve">eodem modo quo F componitur à quan-<lb/>titatibus C, D, vel D à quantitatibus A, B, componatur <lb/>H à quantitatibus E, F; </s> <s xml:id="echoid-s2766" xml:space="preserve">atque ita continuetur ſeries: <lb/></s> <s xml:id="echoid-s2767" xml:space="preserve">appello hanc ſeriem, ſeriem convergentem.</s> <s xml:id="echoid-s2768" xml:space="preserve"/> </p> <div xml:id="echoid-div143" type="float" level="2" n="1"> <note position="right" xlink:label="note-0131-01" xlink:href="note-0131-01a" xml:space="preserve"> <lb/>A # B <lb/>C # D <lb/>E # F <lb/>G # H <lb/></note> </div> <p> <s xml:id="echoid-s2769" xml:space="preserve">10 Ejus termini juxta ſe poſiti nempe A, B, vel C, D, vel <lb/>E, F, vel G, H, dicuntur termini convergentes.</s> <s xml:id="echoid-s2770" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div145" type="section" level="1" n="63"> <head xml:id="echoid-head94" xml:space="preserve">PETITIONES.</head> <p> <s xml:id="echoid-s2771" xml:space="preserve">1 Petimus quantitates, à quantitatibus datis inter ſe ana-<lb/>lyticis analyticè compoſitas, eſſe inter ſe & </s> <s xml:id="echoid-s2772" xml:space="preserve">cum <lb/>quantitatibus datis analyticas.</s> <s xml:id="echoid-s2773" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2774" xml:space="preserve">2 Item quantitates, quæ à quantitatibus datis inter ſe ana-<lb/>lyticis non poſſunt analyticè componi, non eſſe cum quan-<lb/>titatibus datis analyticas.</s> <s xml:id="echoid-s2775" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2776" xml:space="preserve">Præcedentes petitiones quibusdam fortaſſis obſcuræ videbun-<lb/>tur, ſed tamen ex analyſeos elementis ſunt ſatis mani-<lb/>feſtæ.</s> <s xml:id="echoid-s2777" xml:space="preserve"/> </p> <pb o="415" file="0133" n="142"/> </div> <div xml:id="echoid-div146" type="section" level="1" n="64"> <head xml:id="echoid-head95" xml:space="preserve">VERA <lb/>CIRCULI ET HYPERBOLÆ <lb/>QUADRATURA.</head> <p> <s xml:id="echoid-s2778" xml:space="preserve">Sit circuli, ellipſeos vel hyperbolæ ſegmentum B I P <lb/> <anchor type="note" xlink:label="note-0133-01a" xlink:href="note-0133-01"/> cujus centrum A: </s> <s xml:id="echoid-s2779" xml:space="preserve">compleatur triangulum A B P, & </s> <s xml:id="echoid-s2780" xml:space="preserve"><lb/>ſegmentum in punctis, B, P, tangentes ducantur re-<lb/>ctæ B F, P F, ſe invicem ſecantes in puncto F; </s> <s xml:id="echoid-s2781" xml:space="preserve">pro-<lb/>ducatur (ſi opus ſit) recta A F ſegmentum interſecans in <lb/>puncto I & </s> <s xml:id="echoid-s2782" xml:space="preserve">rectam B P in puncto Q; </s> <s xml:id="echoid-s2783" xml:space="preserve">deinde jungantur re-<lb/>ctæ B I, P I.</s> <s xml:id="echoid-s2784" xml:space="preserve"/> </p> <div xml:id="echoid-div146" type="float" level="2" n="1"> <note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> </div> </div> <div xml:id="echoid-div148" type="section" level="1" n="65"> <head xml:id="echoid-head96" xml:space="preserve">PROP. I. THEOREMA.</head> <head xml:id="echoid-head97" style="it" xml:space="preserve">Dico trapezium B A P I eſſe medium propor-<lb/>tionale inter trapezium B A P F, & <lb/>triangulum B A P.</head> <p> <s xml:id="echoid-s2785" xml:space="preserve">Quoniam recta A Q ducitur per F concurſum duarum re-<lb/>ctarum F B, F P, ſegmentum in punctis B, P, tan-<lb/>gentium; </s> <s xml:id="echoid-s2786" xml:space="preserve">igitur recta A Q rectam B P contactuum <lb/>puncta jungentem bifariam ſecabit in puncto Q; </s> <s xml:id="echoid-s2787" xml:space="preserve">& </s> <s xml:id="echoid-s2788" xml:space="preserve">proinde <lb/>triangulum A B Q eſt æquale triangulo A Q P, & </s> <s xml:id="echoid-s2789" xml:space="preserve">trian-<lb/>gnlum F B Q triangulo F Q P; </s> <s xml:id="echoid-s2790" xml:space="preserve">& </s> <s xml:id="echoid-s2791" xml:space="preserve">igitur triangulum A B F <lb/>æquale eſt triangulo A P F; </s> <s xml:id="echoid-s2792" xml:space="preserve">eſt ergo triangulum A B F di-<lb/>midium trapezii A B F P: </s> <s xml:id="echoid-s2793" xml:space="preserve">eodem modo probatur triangu-<lb/>lum A B I eſſe dimidium trapezii A B I P; </s> <s xml:id="echoid-s2794" xml:space="preserve">& </s> <s xml:id="echoid-s2795" xml:space="preserve">triangulum <lb/>A B Q eſt dimidium trianguli A B P: </s> <s xml:id="echoid-s2796" xml:space="preserve">cumque triangula <lb/>A B F, A B I, A B Q, eandem habeant altitudinem, in-<lb/>ter ſe ſunt ut baſes, ſed eorum baſes nempe A F, A I, A Q, <lb/>ſunt continuè proportionales; </s> <s xml:id="echoid-s2797" xml:space="preserve">& </s> <s xml:id="echoid-s2798" xml:space="preserve">igitur ipſa quoque triangu-<lb/>la ſunt continuè proportionalia; </s> <s xml:id="echoid-s2799" xml:space="preserve">& </s> <s xml:id="echoid-s2800" xml:space="preserve">proinde eorum dupla ni-<lb/>mirum trapezia A B F P, A B I P, & </s> <s xml:id="echoid-s2801" xml:space="preserve">triangulum A B P <lb/>ſunt continuè proportionalia in ratione A F ad A I, quod <lb/>demonſtrare oportuit.</s> <s xml:id="echoid-s2802" xml:space="preserve"/> </p> <pb o="416" file="0134" n="143" rhead="VERA CIRCULI"/> <p> <s xml:id="echoid-s2803" xml:space="preserve">Ducatur recta D L ſegmentum tangens in puncto I, & </s> <s xml:id="echoid-s2804" xml:space="preserve"><lb/>rectis B F, F P, occurrens in punctis D, L, ita ut com-<lb/>pleatur polygonum A B D L P.</s> <s xml:id="echoid-s2805" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div149" type="section" level="1" n="66"> <head xml:id="echoid-head98" xml:space="preserve">PROP. II. THEOREMA.</head> <head xml:id="echoid-head99" style="it" xml:space="preserve">Dico trapezia A B F P, A B I P ſimul, eſſe ad du-<lb/> plum trapezii A B I P, ſicut trapezium A B F P ad polygonum A B D L P.</head> <note position="left" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> <p> <s xml:id="echoid-s2806" xml:space="preserve">Quoniam recta A F, ducta per contactum rectæ D L cum <lb/>ſegmento, ducitur etiam per concurſum duarum recta-<lb/>rum F B, F P, rectam D L terminantium & </s> <s xml:id="echoid-s2807" xml:space="preserve">ſegmen-<lb/>tum in duobus punctis tangentium; </s> <s xml:id="echoid-s2808" xml:space="preserve">igitur recta D L bifa-<lb/>riam ſecatur in puncto I; </s> <s xml:id="echoid-s2809" xml:space="preserve">& </s> <s xml:id="echoid-s2810" xml:space="preserve">proinde triangulum F D I æ-<lb/>quale eſt triangulo F I L, at triangulum A B F æquale eſt <lb/>triangulo A P F; </s> <s xml:id="echoid-s2811" xml:space="preserve">& </s> <s xml:id="echoid-s2812" xml:space="preserve">igitur trapezium A B D I æquale eſt <lb/>trapezio A P L I; </s> <s xml:id="echoid-s2813" xml:space="preserve">trapezium ergo A P L I dimidium eſt <lb/>polygoni A B D L P. </s> <s xml:id="echoid-s2814" xml:space="preserve">ducatur recta A L: </s> <s xml:id="echoid-s2815" xml:space="preserve">manifeſtum eſt <lb/>ex præcedentis demonſtratione triangulum A I L eſſe æqua-<lb/>le triangulo A L P; </s> <s xml:id="echoid-s2816" xml:space="preserve">ſed ut triangulum A L F ad triangu-<lb/>lum A L I ita F A ad A I, & </s> <s xml:id="echoid-s2817" xml:space="preserve">ut F A ad A I ita trapezium <lb/>A B F P ad trapezium A B I P; </s> <s xml:id="echoid-s2818" xml:space="preserve">& </s> <s xml:id="echoid-s2819" xml:space="preserve">igitur ut trapezium <lb/>A B F P ad trapezium A B I P; </s> <s xml:id="echoid-s2820" xml:space="preserve">ita triangulum A L F ad <lb/>triangulum A L I; </s> <s xml:id="echoid-s2821" xml:space="preserve">& </s> <s xml:id="echoid-s2822" xml:space="preserve">componendo, ut trapezia A B F P, <lb/>A B I P ſimul, ad trapezium A B I P, ita triangulum A F L <lb/>& </s> <s xml:id="echoid-s2823" xml:space="preserve">triangulum A I L ſimul, hoc eſt triangulum A F P, ad <lb/>triangulum A I L: </s> <s xml:id="echoid-s2824" xml:space="preserve">& </s> <s xml:id="echoid-s2825" xml:space="preserve">conſequentes duplicando, ut trape-<lb/>zia A B F P, A B I P ſimul, ad duplum trapezii A B I P, <lb/>ita triangulum A F P, ad trapezium A I L P: </s> <s xml:id="echoid-s2826" xml:space="preserve">at triangu-<lb/>lum A F P eſt dimidium trapezii A B F P, & </s> <s xml:id="echoid-s2827" xml:space="preserve">trapezium <lb/>A I L P eſt dimidium polygoni A B D L P; </s> <s xml:id="echoid-s2828" xml:space="preserve">& </s> <s xml:id="echoid-s2829" xml:space="preserve">igitur ut <lb/>trapezia A B F P, A B I P ſimul, ad duplum trapezii <lb/>A B I P, ita trapezium A B F P ad polygonum A B D L P, <lb/>quod demonſtrare oportuit.</s> <s xml:id="echoid-s2830" xml:space="preserve"/> </p> <pb o="417" file="0135" n="144" rhead="ET HYPERBOLÆ QUADRATURA."/> </div> <div xml:id="echoid-div150" type="section" level="1" n="67"> <head xml:id="echoid-head100" xml:space="preserve">PROP. III. THEOREMA.</head> <head xml:id="echoid-head101" style="it" xml:space="preserve">Dico triangulum B A P, & trapezium A B I P ſimul, <lb/> eſſe ad trapezium A B I P, ut duplum trapezii A B I P ad polygonum A B D L P.</head> <note position="right" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> <p> <s xml:id="echoid-s2831" xml:space="preserve">In antecedente demonſtratum eſt trapezia A B F P, A B I P <lb/>ſimul, eſſe ad duplum trapezii A B I P, ſicut trapezium <lb/>A B F P ad polygonum A B D L P: </s> <s xml:id="echoid-s2832" xml:space="preserve">& </s> <s xml:id="echoid-s2833" xml:space="preserve">permutando tra-<lb/>pezia A B F P, A B I P ſimul, ſunt ad trapezium A B F P, <lb/>ut duplum trapezii A B I P ad polygonum A B D L P. </s> <s xml:id="echoid-s2834" xml:space="preserve">& </s> <s xml:id="echoid-s2835" xml:space="preserve"><lb/>quoniam trapezium A B F P, trapezium A B I P & </s> <s xml:id="echoid-s2836" xml:space="preserve">trian-<lb/>gulum A B P, ſunt continuè proportionalia; </s> <s xml:id="echoid-s2837" xml:space="preserve">erit trape-<lb/>zium A B I P ad trapezium A B F P, ut triangulum A B P <lb/>ad trapezium A B I P; </s> <s xml:id="echoid-s2838" xml:space="preserve">& </s> <s xml:id="echoid-s2839" xml:space="preserve">componendo, ut trapezia A B I P, <lb/>A B F P ſimul, ad trapezium A B F P, ita triangulum <lb/>A B P & </s> <s xml:id="echoid-s2840" xml:space="preserve">trapezium A B I P ſimul, ad trapezium A B I P: <lb/></s> <s xml:id="echoid-s2841" xml:space="preserve">erat autem, ut trapezia A B I P, A B F P, ſimul, ad tra-<lb/>pezium A B F P, ita duplum trapezii A B I P ad polygo-<lb/>num A B D L P; </s> <s xml:id="echoid-s2842" xml:space="preserve">& </s> <s xml:id="echoid-s2843" xml:space="preserve">igitur ut triangulum A B P & </s> <s xml:id="echoid-s2844" xml:space="preserve">trape-<lb/>zium A B I P ſimul, ad trapezium A B I P, ita duplum <lb/>trapezii A B I P ad polygonum A B D L P, quod demon-<lb/>ſtrare oportuit.</s> <s xml:id="echoid-s2845" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2846" xml:space="preserve">Producantur (ſi opus ſit) rectæ A D, A L, ſegmentum <lb/>ſecantes in punctis E & </s> <s xml:id="echoid-s2847" xml:space="preserve">O, & </s> <s xml:id="echoid-s2848" xml:space="preserve">rectas B I, I P, in H & </s> <s xml:id="echoid-s2849" xml:space="preserve">M: <lb/></s> <s xml:id="echoid-s2850" xml:space="preserve">deinde jungantur rectæ B E, E I, I O, O P, ut complea-<lb/>tur polygonum A B E I O P.</s> <s xml:id="echoid-s2851" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div151" type="section" level="1" n="68"> <head xml:id="echoid-head102" xml:space="preserve">PROP. IV. THEOREMA.</head> <head xml:id="echoid-head103" style="it" xml:space="preserve">Dico polygonum A B E I O P eſſe medium pro-<lb/> portionale inter polygonum A B D L P & trapezium A B I P.</head> <note position="right" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> <p> <s xml:id="echoid-s2852" xml:space="preserve">Ex hujus prima manifeſtum eſt trapezium A I L P, tra-<lb/>pezium A I O P & </s> <s xml:id="echoid-s2853" xml:space="preserve">triangulum A I P eſſe continuè <pb o="418" file="0136" n="145" rhead="VERA CIRCULI"/> proportionalia, & </s> <s xml:id="echoid-s2854" xml:space="preserve">ex prædictis ſatis facile colligi poteſt <lb/>trapezium A I L P eſſe dimidium polygoni A B D L P & </s> <s xml:id="echoid-s2855" xml:space="preserve"><lb/>trapezium A I O P eſſe dimidium polygoni A B E I O P <lb/>& </s> <s xml:id="echoid-s2856" xml:space="preserve">triangulum A I P eſſe dimidium trapezii A B I P: </s> <s xml:id="echoid-s2857" xml:space="preserve">& </s> <s xml:id="echoid-s2858" xml:space="preserve"><lb/>proinde terminos duplicando, polygonum A B D L P, po-<lb/>lygonum A B E I O P & </s> <s xml:id="echoid-s2859" xml:space="preserve">trapezium A B I P ſunt continuè <lb/>proportionalia, quod demonſtrare oportuit.</s> <s xml:id="echoid-s2860" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2861" xml:space="preserve">Ducantur rectæ C G, K N, ſegmentum tangentes in <lb/>punctis E, O, & </s> <s xml:id="echoid-s2862" xml:space="preserve">rectis D L, D B, L P, occurrentes in <lb/>punctis C, G, K, N, ut compleatur polygonum <lb/>A B C G K N P.</s> <s xml:id="echoid-s2863" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div152" type="section" level="1" n="69"> <head xml:id="echoid-head104" xml:space="preserve">PROP. V. THEOREMA.</head> <p style="it"> <s xml:id="echoid-s2864" xml:space="preserve">Dico trapezium A B I P & </s> <s xml:id="echoid-s2865" xml:space="preserve">polygonum A B E I O P <lb/> <anchor type="note" xlink:label="note-0136-01a" xlink:href="note-0136-01"/> ſimul, eße ad polygonum A B E I O P, ut <lb/>duplum polygoni A B E I O P ad poly-<lb/>gonum A B C G K N P.</s> <s xml:id="echoid-s2866" xml:space="preserve"/> </p> <div xml:id="echoid-div152" type="float" level="2" n="1"> <note position="left" xlink:label="note-0136-01" xlink:href="note-0136-01a" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> </div> <p> <s xml:id="echoid-s2867" xml:space="preserve">Ex hujus tertia manifeſtum eſt triangulum A B I & </s> <s xml:id="echoid-s2868" xml:space="preserve">tra-<lb/>pezium A B E I ſimul, eſſe ad trapezium A B E I, <lb/>ut duplum trapezii A B E Iad polygonum A B C G I: </s> <s xml:id="echoid-s2869" xml:space="preserve">& </s> <s xml:id="echoid-s2870" xml:space="preserve"><lb/>ex prædictis facile concludi poteſt triangulum A B I eſſe di-<lb/>midium trapezii A B I P, & </s> <s xml:id="echoid-s2871" xml:space="preserve">trapezium A B E I eſſe dimi-<lb/>dium polygoni A B E I O P, & </s> <s xml:id="echoid-s2872" xml:space="preserve">polygonum A B C G I <lb/>eſſe dimidium polygoni A B C G K N P; </s> <s xml:id="echoid-s2873" xml:space="preserve">& </s> <s xml:id="echoid-s2874" xml:space="preserve">proinde termi-<lb/>nos duplicando, trapezium A B I P & </s> <s xml:id="echoid-s2875" xml:space="preserve">polygonum A B E I O P <lb/>ſimul, erunt ad polygonum A B E I O P ut duplum polygo-<lb/>ni A B E I O P ad polygonum A B C G K N P, quod <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s2876" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2877" xml:space="preserve">Hinc facile colligi poteſt polygonum A B C G K N P <lb/>eſſe medium harmonicum inter polygona A B E I O P, <lb/>A B D L P, quod hic admonuiſſe ſufficiat, in ſequentibus <lb/>enim demonſtrabitur.</s> <s xml:id="echoid-s2878" xml:space="preserve"/> </p> <pb o="419" file="0137" n="146" rhead="ET HYPERBOLÆ QUADRATURA."/> </div> <div xml:id="echoid-div154" type="section" level="1" n="70"> <head xml:id="echoid-head105" xml:space="preserve">SCHOLIUM.</head> <p> <s xml:id="echoid-s2879" xml:space="preserve">Duæ præcedentes propoſitiones eodem modo demon-<lb/>ſtrari poſſunt de duobus quibuſcunque polygonis <lb/>complicatis loco polygonorum complicatorum ABIP, <lb/>A B D L P; </s> <s xml:id="echoid-s2880" xml:space="preserve">polygonum enim à tangentibus comprehenſum <lb/>tot continet æqualia trapezia, quot continet polygonum à <lb/>ſubtendentibus comprehenſum æqualia triangula: </s> <s xml:id="echoid-s2881" xml:space="preserve">atque hinc <lb/>evidens eſt has polygonorum analogias ita ſe habere in infi-<lb/>nitum, ducendo nimirum rectas AN, AK, AG, AC, per <lb/>puncta R, T, S, V, & </s> <s xml:id="echoid-s2882" xml:space="preserve">adhuc alia & </s> <s xml:id="echoid-s2883" xml:space="preserve">alia polygona intra & </s> <s xml:id="echoid-s2884" xml:space="preserve"><lb/>extra ſemper ſcribendo: </s> <s xml:id="echoid-s2885" xml:space="preserve">notandum nos appellare hanc poly-<lb/>gonorum inſcriptionem & </s> <s xml:id="echoid-s2886" xml:space="preserve">circumſcriptionem, inſcriptionem <lb/>& </s> <s xml:id="echoid-s2887" xml:space="preserve">circumſcriptionem ſubduplam, ex prædictis patet (ſi po-<lb/>natur triangulum A B P = <emph style="super">a</emph>, & </s> <s xml:id="echoid-s2888" xml:space="preserve">trapezium A B F P = <emph style="super">b</emph>) tra-<lb/>pezium A B I P eſſe vqab & </s> <s xml:id="echoid-s2889" xml:space="preserve">polygonum A B D L P {2ab/a + vqab}: <lb/></s> <s xml:id="echoid-s2890" xml:space="preserve">eodem modo poſito trapezio A B I P = <emph style="super">c</emph>, & </s> <s xml:id="echoid-s2891" xml:space="preserve">polygono <lb/>A B D L P = <emph style="super">d</emph>, erit polygonum A B E I O P = vqcd & </s> <s xml:id="echoid-s2892" xml:space="preserve">po-<lb/>lygonum A B C G K N P = {2cd/c + vqcd,}, ita ut evidens ſit hanc <lb/>polygonorum ſeriem eſſe convergentem; </s> <s xml:id="echoid-s2893" xml:space="preserve">atque in infinitum <lb/>illam continuando, manifeſtum eſt tandem exhiberi quanti-<lb/>tatem ſectori circulari, elliptico vel hyperbolico A B E I O P <lb/>æqualem; </s> <s xml:id="echoid-s2894" xml:space="preserve">differentia enim polygonorum complicatorum in <lb/>ſeriei continuatione ſemper diminuitur, ita ut omni exhibita <lb/>quantitate fieri poſſit minor, ut in ſequentis theorematis <lb/>Scholio demonſtrabimus: </s> <s xml:id="echoid-s2895" xml:space="preserve">ſi igitur prædicta polygonorum ſe-<lb/>ries terminari poſſet, hoc eſt, ſi inveniretur ultimum illud <lb/>polygonum inſcriptum (ſi ita loqui liceat) æquale ultimo <lb/>illi polygono circumſcripto, daretur infallibiliter circuli & </s> <s xml:id="echoid-s2896" xml:space="preserve"><lb/>hyperbolæ quadratura: </s> <s xml:id="echoid-s2897" xml:space="preserve">ſed quoniam difficile eſt, & </s> <s xml:id="echoid-s2898" xml:space="preserve">in geo-<lb/>metria omnino fortaſſe inauditum tales ſeries terminare; </s> <s xml:id="echoid-s2899" xml:space="preserve">præ-<lb/>mittendæ ſunt quædam propoſitiones è quibus inveniri poſ-<lb/>ſint hujuſmodi aliquot ſerierum terminationes, & </s> <s xml:id="echoid-s2900" xml:space="preserve">tandem (ſi <pb o="420" file="0138" n="147" rhead="VERA CIRCULI"/> fieri poſſit) generalis methodus inveniendi omnium ſerierum <lb/>convergentium terminationes.</s> <s xml:id="echoid-s2901" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div155" type="section" level="1" n="71"> <head xml:id="echoid-head106" xml:space="preserve">PROP. VI. THEOREMATA.</head> <p style="it"> <s xml:id="echoid-s2902" xml:space="preserve">Dico differentiam inter triangulum A B P & </s> <s xml:id="echoid-s2903" xml:space="preserve">trape-<lb/> <anchor type="note" xlink:label="note-0138-01a" xlink:href="note-0138-01"/> zium A B F P majorem eſſe duplo differentiæ inter <lb/>trapezium A B I P & </s> <s xml:id="echoid-s2904" xml:space="preserve">polygonum A B D L P.</s> <s xml:id="echoid-s2905" xml:space="preserve"/> </p> <div xml:id="echoid-div155" type="float" level="2" n="1"> <note position="left" xlink:label="note-0138-01" xlink:href="note-0138-01a" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> </div> <p> <s xml:id="echoid-s2906" xml:space="preserve">Sit triangulum A B P, A; </s> <s xml:id="echoid-s2907" xml:space="preserve">trapezium A B F P, B; </s> <s xml:id="echoid-s2908" xml:space="preserve">tra-<lb/>pezium A B I P, C, & </s> <s xml:id="echoid-s2909" xml:space="preserve">polygonum A B D L P, D: </s> <s xml:id="echoid-s2910" xml:space="preserve">quo-<lb/>niam A eſt ad C ut C ad B, igitur ut differentia inter A <lb/>& </s> <s xml:id="echoid-s2911" xml:space="preserve">C ad A, ita differentia inter C & </s> <s xml:id="echoid-s2912" xml:space="preserve">B ad C; </s> <s xml:id="echoid-s2913" xml:space="preserve">& </s> <s xml:id="echoid-s2914" xml:space="preserve">permutan-<lb/>do, ut differentia inter A & </s> <s xml:id="echoid-s2915" xml:space="preserve">C ad differentiam inter C & </s> <s xml:id="echoid-s2916" xml:space="preserve">B <lb/>ita A ad C; </s> <s xml:id="echoid-s2917" xml:space="preserve">& </s> <s xml:id="echoid-s2918" xml:space="preserve">componendo, ut differentia inter A & </s> <s xml:id="echoid-s2919" xml:space="preserve">C & </s> <s xml:id="echoid-s2920" xml:space="preserve"><lb/>differentia inter C & </s> <s xml:id="echoid-s2921" xml:space="preserve">B ſimul, hoc eſt differentia inter A & </s> <s xml:id="echoid-s2922" xml:space="preserve"><lb/>B, ad differentiam inter C & </s> <s xml:id="echoid-s2923" xml:space="preserve">B, ita A + C ad C; </s> <s xml:id="echoid-s2924" xml:space="preserve">ſed ut <lb/>A + C ad C ita 2 C ad D, & </s> <s xml:id="echoid-s2925" xml:space="preserve">ideo differentia inter A & </s> <s xml:id="echoid-s2926" xml:space="preserve"><lb/>B eſt ad differentiam inter C & </s> <s xml:id="echoid-s2927" xml:space="preserve">B ut 2 C ad D. </s> <s xml:id="echoid-s2928" xml:space="preserve">quoniam <lb/>A + C eſt ad C ut 2 C ad D, erit permutando ut A + C <lb/>ad 2 C ita C ad D; </s> <s xml:id="echoid-s2929" xml:space="preserve">& </s> <s xml:id="echoid-s2930" xml:space="preserve">dividendo, ut differentia inter A & </s> <s xml:id="echoid-s2931" xml:space="preserve"><lb/>C ad 2 C ita differentia inter C & </s> <s xml:id="echoid-s2932" xml:space="preserve">D ad D; </s> <s xml:id="echoid-s2933" xml:space="preserve">& </s> <s xml:id="echoid-s2934" xml:space="preserve">permutan-<lb/>do, differentia inter A & </s> <s xml:id="echoid-s2935" xml:space="preserve">C eſt ad differentiam inter C & </s> <s xml:id="echoid-s2936" xml:space="preserve">D <lb/>ut 2 C ad D; </s> <s xml:id="echoid-s2937" xml:space="preserve">ſed demonſtratum eſt differentiam inter A & </s> <s xml:id="echoid-s2938" xml:space="preserve"><lb/>B eſſe ad differentiam inter C & </s> <s xml:id="echoid-s2939" xml:space="preserve">B ut 2 C ad D; </s> <s xml:id="echoid-s2940" xml:space="preserve">& </s> <s xml:id="echoid-s2941" xml:space="preserve">proin-<lb/>de differentia inter A & </s> <s xml:id="echoid-s2942" xml:space="preserve">B eſt ad differentiam inter C & </s> <s xml:id="echoid-s2943" xml:space="preserve">B, <lb/>ut differentia inter A & </s> <s xml:id="echoid-s2944" xml:space="preserve">C ad differentiam inter C & </s> <s xml:id="echoid-s2945" xml:space="preserve">D; </s> <s xml:id="echoid-s2946" xml:space="preserve">ſed <lb/>differentia inter A & </s> <s xml:id="echoid-s2947" xml:space="preserve">B eſt major differentia inter C & </s> <s xml:id="echoid-s2948" xml:space="preserve">B, <lb/>& </s> <s xml:id="echoid-s2949" xml:space="preserve">ideo differentia inter A & </s> <s xml:id="echoid-s2950" xml:space="preserve">C eſt major differentia inter C <lb/>& </s> <s xml:id="echoid-s2951" xml:space="preserve">D: </s> <s xml:id="echoid-s2952" xml:space="preserve">& </s> <s xml:id="echoid-s2953" xml:space="preserve">prædictam analogiam permutando, differentia in-<lb/>ter A & </s> <s xml:id="echoid-s2954" xml:space="preserve">B eſt ad differentiam inter A & </s> <s xml:id="echoid-s2955" xml:space="preserve">C, ut differentia <lb/>inter C & </s> <s xml:id="echoid-s2956" xml:space="preserve">B ad differentiam inter C & </s> <s xml:id="echoid-s2957" xml:space="preserve">D; </s> <s xml:id="echoid-s2958" xml:space="preserve">at differentia in-<lb/>ter A & </s> <s xml:id="echoid-s2959" xml:space="preserve">B eſt major differentia inter A & </s> <s xml:id="echoid-s2960" xml:space="preserve">C, & </s> <s xml:id="echoid-s2961" xml:space="preserve">ideo diffe-<lb/>rentia inter C & </s> <s xml:id="echoid-s2962" xml:space="preserve">B eſt major differentia inter C & </s> <s xml:id="echoid-s2963" xml:space="preserve">D; </s> <s xml:id="echoid-s2964" xml:space="preserve">atque <lb/>differentia inter A & </s> <s xml:id="echoid-s2965" xml:space="preserve">B æqualis eſt differentiis, inter A & </s> <s xml:id="echoid-s2966" xml:space="preserve">C, <lb/>inter C & </s> <s xml:id="echoid-s2967" xml:space="preserve">B; </s> <s xml:id="echoid-s2968" xml:space="preserve">cumque earum utravis ſit major differentia in- <pb o="421" file="0139" n="148" rhead="ET HYPERBOLÆ QUADRATURA."/> ter C & </s> <s xml:id="echoid-s2969" xml:space="preserve">D, manifeſtum eſt differentiam inter A & </s> <s xml:id="echoid-s2970" xml:space="preserve">B majo-<lb/>rem eſſe duplo differentiæ inter C & </s> <s xml:id="echoid-s2971" xml:space="preserve">D, hoc eſt differen-<lb/>tiam inter triangulum A B P & </s> <s xml:id="echoid-s2972" xml:space="preserve">trapezium A B F P majo-<lb/>rem eſſe duplo differentiæ inter trapezium A B I P, & </s> <s xml:id="echoid-s2973" xml:space="preserve">po-<lb/>lygonum A B D L P, quod demonſtrare oportuit.</s> <s xml:id="echoid-s2974" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div157" type="section" level="1" n="72"> <head xml:id="echoid-head107" xml:space="preserve">SCHOLIUM.</head> <p> <s xml:id="echoid-s2975" xml:space="preserve">Eodem prorſus modo demonſtratur differentiam inter tra-<lb/>pezium A B I P & </s> <s xml:id="echoid-s2976" xml:space="preserve">polygonum A B D L P majorem <lb/>eſſe duplo differentiæ inter polygonum A B E I O P & </s> <s xml:id="echoid-s2977" xml:space="preserve">poly-<lb/>gonum A B C G K N P. </s> <s xml:id="echoid-s2978" xml:space="preserve">denique eodem modo demonſtra-<lb/>ri poteſt hic differentiarum exceſſus in ſubdupla noſtra poly-<lb/>gonorum complicatorum deſcriptione in infinitum; </s> <s xml:id="echoid-s2979" xml:space="preserve">differen-<lb/>tia enim priorum nempe inſcripti & </s> <s xml:id="echoid-s2980" xml:space="preserve">circumſcripti major ſem-<lb/>per erit duplo differentiæ immediatè ſequentium nimirum in-<lb/>ſcripti quoque & </s> <s xml:id="echoid-s2981" xml:space="preserve">circumſcripti, & </s> <s xml:id="echoid-s2982" xml:space="preserve">proinde aufertur majus <lb/>quam dimidium a priorum differentia ut fiat differentia im-<lb/>mediatè ſequentium; </s> <s xml:id="echoid-s2983" xml:space="preserve">& </s> <s xml:id="echoid-s2984" xml:space="preserve">igitur continuando ſubduplam po-<lb/>lygonorum deſcriptionem, inveniri poſſunt duo polygona <lb/>complicata, quorum differentia ſit minor qualibet exhibita <lb/>quantitate, ut in præcedentis Scholio aſſumpſimus.</s> <s xml:id="echoid-s2985" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2986" xml:space="preserve">Sint duæ quantitates indefinitæ <emph style="super">a</emph> minor <emph style="sub">b</emph> major, ſintque <lb/>datæ duæ rationes majoris inæqualitatis <emph style="super">c</emph> ad <emph style="super">d</emph>, & </s> <s xml:id="echoid-s2987" xml:space="preserve"><emph style="super">c</emph> ad <emph style="super">e</emph>; </s> <s xml:id="echoid-s2988" xml:space="preserve">de-<lb/>inde ſit ut <emph style="super">c</emph> ad <emph style="super">d</emph> ita <emph style="super">b - a</emph> ad {bd - ad/c} cui addatur quantitas <emph style="sub">a</emph> ut <lb/>fiat {ca + bd - ad/c}, quæ quantitas ponatur immediate ſub a: </s> <s xml:id="echoid-s2989" xml:space="preserve">fiatque <lb/>ut <emph style="super">c</emph> ad <emph style="sub">e</emph> ita <emph style="super">b</emph> - <emph style="sub">a</emph> ad {be - ae/c}, quæ quantitas ſubſtrahatur ex <emph style="super">b</emph> & </s> <s xml:id="echoid-s2990" xml:space="preserve"><lb/>relictum nempe {bc - be + ae/c} ponatur ſub <emph style="sub">b.</emph> continuetur de-<lb/>inde ſeries convergens cujus pri-<lb/> <anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/> mi termini convergentes ſunt <emph style="super">a, b</emph>, <lb/>& </s> <s xml:id="echoid-s2991" xml:space="preserve">ſecundi termini convergentes <lb/>{ca + bd - ad/c}, {bc - be + ae/c}. </s> <s xml:id="echoid-s2992" xml:space="preserve">manifeſtum eſt terminum {ca + bd - ad/c} majorem <pb o="422" file="0140" n="149" rhead="VERA CIRCULI"/> eſſe termino <emph style="sub">a</emph>, quoniam termino <emph style="super">a</emph> additur {bd - ad/c} ut fiat termi-<lb/>nus {ca + bd - ad/c}: </s> <s xml:id="echoid-s2993" xml:space="preserve">manifeſtum quoque eſt terminum {ca + bd - ad/c} mi-<lb/>norem eſſe termino b, quoniam differentia inter <emph style="super">a</emph> & </s> <s xml:id="echoid-s2994" xml:space="preserve"><emph style="super">b</emph> eſt ad <lb/>differentiam inter a & </s> <s xml:id="echoid-s2995" xml:space="preserve">{ca + bd - ad/c} in ratione majoris inæqualita-<lb/>tis: </s> <s xml:id="echoid-s2996" xml:space="preserve">evidens quoque eſt terminum {bc - be + ae/c} minorem eſſe ter-<lb/>mino b. </s> <s xml:id="echoid-s2997" xml:space="preserve">quoniam ex <emph style="sub">b</emph> ſubſtrahitur {be - ae/c} ut fiat {bc - be + ae/c}; </s> <s xml:id="echoid-s2998" xml:space="preserve">ma-<lb/>nifeſtum etiam eſt terminum {bc - be + ae/c} majorem eſſe termino <emph style="super">a</emph>, <lb/>quoniam differentia inter <emph style="super">a</emph> & </s> <s xml:id="echoid-s2999" xml:space="preserve"><emph style="sub">b</emph> eſt ad differentiam inter <lb/>{bc - be + ae/c} & </s> <s xml:id="echoid-s3000" xml:space="preserve"><emph style="super">b</emph> in ratione majoris inæqualitatis: </s> <s xml:id="echoid-s3001" xml:space="preserve">evidens igitur <lb/>eſt differentiam inter terminos convergentes <emph style="super">a</emph> & </s> <s xml:id="echoid-s3002" xml:space="preserve"><emph style="super">b</emph> majorem <lb/>eſſe differentiâ inter terminos convergentes {ca + bd - ad/c} & </s> <s xml:id="echoid-s3003" xml:space="preserve">{bc - be + ae/c}. <lb/></s> <s xml:id="echoid-s3004" xml:space="preserve">fed quoniam termini convergentes <emph style="super">a</emph> & </s> <s xml:id="echoid-s3005" xml:space="preserve"><emph style="super">b</emph> ponuntur indefiniti, <lb/>poſſunt <emph style="super">a</emph> & </s> <s xml:id="echoid-s3006" xml:space="preserve"><emph style="super">b</emph> ſumi loco quorumlibet terminorum convergen-<lb/>tium totius hujus ſeriei; </s> <s xml:id="echoid-s3007" xml:space="preserve">& </s> <s xml:id="echoid-s3008" xml:space="preserve">poſitis <emph style="super">a</emph> & </s> <s xml:id="echoid-s3009" xml:space="preserve"><emph style="super">b</emph> pro terminis hujus <lb/>ſeriei convergentibus quibuſcunque, ſequitur neceſſario ex <lb/>ſeriei compoſitione {ca + bd - ad.</s> <s xml:id="echoid-s3010" xml:space="preserve">/c}, {bc - be + ae/c} eſſe terminos conver-<lb/>gentes immediatè ſequentes: </s> <s xml:id="echoid-s3011" xml:space="preserve">cumque differentia terminorum <lb/><emph style="super">a</emph> & </s> <s xml:id="echoid-s3012" xml:space="preserve"><emph style="super">b</emph> major ſit differentia terminorum {ca + bd - ad/c} & </s> <s xml:id="echoid-s3013" xml:space="preserve">{bc - be + ae/c}, <lb/>evidens eſt differentiam terminorum convergentium priorum <lb/>ſemper eſſe majorem differentia terminorum convergentium <lb/>immediatè ſequentium; </s> <s xml:id="echoid-s3014" xml:space="preserve">& </s> <s xml:id="echoid-s3015" xml:space="preserve">igitur quò magis continuatur hæc <lb/>ſeries convergens eò minor fit differentia terminorum con-<lb/>vergentium: </s> <s xml:id="echoid-s3016" xml:space="preserve">& </s> <s xml:id="echoid-s3017" xml:space="preserve">quoniam hæc differentiarum diminutio ſem-<lb/>per fit proportionaliter nempe in ratione <emph style="super">b-a</emph> ad {bc - be + ae - ca - bd + ad;</s> <s xml:id="echoid-s3018" xml:space="preserve">/c} <lb/>igitur poſſunt inveniri hujus ſeriei termini convergentes quo-<lb/>rum differentia ſit omni exhibita quantitate minor; </s> <s xml:id="echoid-s3019" xml:space="preserve">& </s> <s xml:id="echoid-s3020" xml:space="preserve">igitur <lb/>imaginando hanc ſeriem in infinitum continuari, poſſumus <lb/>imaginari ultimos terminos convergentes eſſe æquales, <pb o="423" file="0141" n="150" rhead="ET HYPERBOLÆ QUADRATURA."/> quos terminos æquales appellamus ſeriei terminatio-<lb/>nem.</s> <s xml:id="echoid-s3021" xml:space="preserve"/> </p> <div xml:id="echoid-div157" type="float" level="2" n="1"> <note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve"> <lb/># d <lb/>c # e # a # b <lb/>## {ca + bd - ad/c} # {bc - be + ae/ c} <lb/></note> </div> </div> <div xml:id="echoid-div159" type="section" level="1" n="73"> <head xml:id="echoid-head108" xml:space="preserve">PROP. VII. PROBLEMA.</head> <head xml:id="echoid-head109" style="it" xml:space="preserve">Oportet prædictæ ſeriei terminationem <lb/>invenire.</head> <p> <s xml:id="echoid-s3022" xml:space="preserve">Ut huic problemati ſatisfiat, oportet primò invenire <lb/>quantitatem quæ eodem modo componitur ex termi-<lb/>nis convergentibus <emph style="super">a</emph>, <emph style="super">b</emph>, quo ex terminis convergentibus <lb/>{ca + bd - ad/c}, {bc - bc + ae/c}, hoc autem facile fit hoc modo: </s> <s xml:id="echoid-s3023" xml:space="preserve">inveniatur <lb/>quantitas quæ multiplicata in <emph style="sub">a</emph> & </s> <s xml:id="echoid-s3024" xml:space="preserve">addita <emph style="super">b</emph> multiplicata in <lb/>quantitatem data m, eandem quantitatem facit ac ſi multi-<lb/>plicaretur in {ca + bd - ad/c} & </s> <s xml:id="echoid-s3025" xml:space="preserve">adderetur {bc - be + ae/c} multiplicata etiam <lb/>in eandem quantitatem data\m m. </s> <s xml:id="echoid-s3026" xml:space="preserve">ſit quantitas illa <emph style="sub">z</emph>, & </s> <s xml:id="echoid-s3027" xml:space="preserve">pro-<lb/>inde za + bm æquatur {zca + zbd - zad + mbe - mbe + mae/c}, & </s> <s xml:id="echoid-s3028" xml:space="preserve">æquatione <lb/>reducta invenitur {z = mac - mbe/ad - bd}; </s> <s xml:id="echoid-s3029" xml:space="preserve">quæ quantitas ſive multiplica-<lb/>ta in <emph style="super">a</emph> & </s> <s xml:id="echoid-s3030" xml:space="preserve">addita <emph style="super">m</emph>, ſive multiplicata in {ca + bd - ad/c} & </s> <s xml:id="echoid-s3031" xml:space="preserve">addita <lb/>{mbe - mbe + mae/c} efficit eandem in utroque caſu quantitatem nempe <lb/>{maae - mbae + mbad - mbbd/cd - bd}: </s> <s xml:id="echoid-s3032" xml:space="preserve">& </s> <s xml:id="echoid-s3033" xml:space="preserve">proinde prædicta quantitas eodem mo-<lb/>do componitur ex terminis convergentibus <emph style="super">a</emph>, <emph style="super">b</emph>, quo compo-<lb/>nitur ex terminis convergentibus {ca + bd - ad/c}, {bc - be + ae/c}. </s> <s xml:id="echoid-s3034" xml:space="preserve">atque <emph style="super">a</emph> <lb/>& </s> <s xml:id="echoid-s3035" xml:space="preserve"><emph style="sub">b</emph> quoniam ſunt quantitates indefinitæ poſſunt eſſe quili-<lb/>bet totius ſeriei termini convergentes, modò termini con-<lb/>vergentes immediatè ſequentes ſint {ca + bd - ad/c} & </s> <s xml:id="echoid-s3036" xml:space="preserve">{bc - be + ae/c}, & </s> <s xml:id="echoid-s3037" xml:space="preserve"><lb/>proinde quantitas {maae - mbae + mbad - mbbd/cd - bd} eodem modo componi-<lb/>tur ex quibuslibet totius ſeriei terminis convergentibus quo <lb/>componitur ex terminis convergentibus <emph style="super">a</emph>, <emph style="super">b</emph>; </s> <s xml:id="echoid-s3038" xml:space="preserve">& </s> <s xml:id="echoid-s3039" xml:space="preserve">igitur præ- <pb o="424" file="0142" n="151" rhead="VERA CIRCULI"/> dicta quantitas eodem etiam modo componitur ex ul-<lb/>timis ejus terminis convergentibus, qui æquales ſunt: <lb/></s> <s xml:id="echoid-s3040" xml:space="preserve">ſit ultimus ille terminus <emph style="super">x</emph> qui multiplicatus in {mae-mbe/ad-bd} & </s> <s xml:id="echoid-s3041" xml:space="preserve"><lb/>in <emph style="super">m</emph> efficit xm & </s> <s xml:id="echoid-s3042" xml:space="preserve">{xmae - xmbe/ad - bd}, quorum factorum ſumma nempe <lb/>{xmae - xmbe + xmad - xmbd/ad - bd} æquatur {maae - mbae + mbad - mbbd/ad - bd} & </s> <s xml:id="echoid-s3043" xml:space="preserve">æquatio-<lb/>ne reducta invenitur x ſeu ſeriei terminatio {aae - bae + bad - bbd/ae - be + ad - bd}, <lb/>quam invenire oportuit.</s> <s xml:id="echoid-s3044" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3045" xml:space="preserve">Ne minus exercitatis obſcurum videatur hoc problema, <lb/>illud in numeris illuſtrabimus: </s> <s xml:id="echoid-s3046" xml:space="preserve">ſit <emph style="super">c 7, d 2, e 3, a 28, b 42,</emph> erunt ſe-<lb/>cundi termini convergentes 32, 36, tertii 33 {1/7}, 34 {2/7}, & </s> <s xml:id="echoid-s3047" xml:space="preserve">ejus ter-<lb/>minatio 33 {3/5}.</s> <s xml:id="echoid-s3048" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3049" xml:space="preserve">Neminem moveat, quod (etiamſi <emph style="super">a</emph> ſit minor quam <emph style="super">b</emph>) <lb/>{ca + bd - ad/c} poſſit eſſe major quam {bc - be + ae/c}, analyticè enim major <lb/>à minore poteſt ſubſtrahi, cnjus tamen exemplum non grava-<lb/>bimus exhibere, ſit <emph style="super">c 7, d 5, e 4, a 28, b 42;</emph> erunt ſecundi termini <lb/>convergentes 38, 34, & </s> <s xml:id="echoid-s3050" xml:space="preserve">tertii 35 {1/7}, 36 {2/7}, ejuſque terminatio <lb/>35 {7/9}.</s> <s xml:id="echoid-s3051" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3052" xml:space="preserve">Animadvertendum eſt hujus problematis ſolutionem eo-<lb/>dem modo ſe habere, etiamſi loco <emph style="super">a</emph> ponatur cyphra ſeu me-<lb/>rum nihil, Ex. </s> <s xml:id="echoid-s3053" xml:space="preserve">Gr; </s> <s xml:id="echoid-s3054" xml:space="preserve">ſit <emph style="super">c 8, d 3, e 4, a 0, b 24;</emph> erunt ſecundi ter-<lb/>mini convergentes 9, 12, & </s> <s xml:id="echoid-s3055" xml:space="preserve">tertii 10 {1/8}, 10 {1/2}, & </s> <s xml:id="echoid-s3056" xml:space="preserve">ſeriei termina-<lb/>tio 10 {2/7},</s> </p> <p> <s xml:id="echoid-s3057" xml:space="preserve">Harum etiam ſerierum terminationes poſſunt inveniri ex <lb/>Gregorii à S. </s> <s xml:id="echoid-s3058" xml:space="preserve">Vincentio lib. </s> <s xml:id="echoid-s3059" xml:space="preserve">de progreſſ. </s> <s xml:id="echoid-s3060" xml:space="preserve">geometrica, etiamſi <lb/>methodo longe ab hac diverſa.</s> <s xml:id="echoid-s3061" xml:space="preserve"/> </p> <pb o="425" file="0143" n="152" rhead="ET HYPERBOLÆ QUADRATURA."/> </div> <div xml:id="echoid-div160" type="section" level="1" n="74"> <head xml:id="echoid-head110" xml:space="preserve">PROP. VIII. PROBLEMA.</head> <p style="it"> <s xml:id="echoid-s3062" xml:space="preserve">Sint duæ quantitates datæ A, B, & </s> <s xml:id="echoid-s3063" xml:space="preserve">ratio quæli-<lb/>libet data C ad D: </s> <s xml:id="echoid-s3064" xml:space="preserve">oportet invenire aliam <lb/>quantitatem, ut ratio ejus ad A ſit multipli-<lb/>cata rationis B ad A in ratione C ad D.</s> <s xml:id="echoid-s3065" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3066" xml:space="preserve">Sit primò ratio C ad D commen-<lb/> <anchor type="note" xlink:label="note-0143-01a" xlink:href="note-0143-01"/> ſurabilis, ſitque inter C & </s> <s xml:id="echoid-s3067" xml:space="preserve">D com-<lb/>munis menſura E; </s> <s xml:id="echoid-s3068" xml:space="preserve">& </s> <s xml:id="echoid-s3069" xml:space="preserve">quoties E continetur in D toties ſit <lb/>ratio F ad A ſubmultiplicata rationis B ad A; </s> <s xml:id="echoid-s3070" xml:space="preserve">& </s> <s xml:id="echoid-s3071" xml:space="preserve">quoties E <lb/>continetur in C toties ſit ratio G ad A multiplicata rationis <lb/>F ad A: </s> <s xml:id="echoid-s3072" xml:space="preserve">dico G eſſe quantitatem illam quæſitam. </s> <s xml:id="echoid-s3073" xml:space="preserve">ratio G ad <lb/>A eſt multiplicata rationis F ad A in ratione C ad E, & </s> <s xml:id="echoid-s3074" xml:space="preserve">ra-<lb/>tio F ad A eſt multiplicata rationis B ad A in ratione E ad D; </s> <s xml:id="echoid-s3075" xml:space="preserve">& </s> <s xml:id="echoid-s3076" xml:space="preserve"><lb/>igitur ex æqualitate, ratio G ad A eſt multiplicata rationis <lb/>B ad A in ratione C ad D, quod demonſtrare oportuit.</s> <s xml:id="echoid-s3077" xml:space="preserve"/> </p> <div xml:id="echoid-div160" type="float" level="2" n="1"> <note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve"> <lb/>EDC # AFBG <lb/></note> </div> <p> <s xml:id="echoid-s3078" xml:space="preserve">Quod ſi ratio C ad D ſit incommenſurabilis, geometricam <lb/>hujus problematis praxim eſſe impoſſibilem mihi perſuadeo; <lb/></s> <s xml:id="echoid-s3079" xml:space="preserve">approximatione tamen fieri poteſt, aſſumendo rationem com-<lb/>menſurabilem ejus loco, quæ quàm proximè ad illam acce-<lb/>dat.</s> <s xml:id="echoid-s3080" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3081" xml:space="preserve">Sit ſeries convergens, cujus primi <lb/> <anchor type="note" xlink:label="note-0143-02a" xlink:href="note-0143-02"/> termini convergentes ſint A, B, ſe-<lb/>cundi C, D, tertii E, F; </s> <s xml:id="echoid-s3082" xml:space="preserve">ſintque <lb/>ſecundi termini ita facti à primis, ut <lb/>ratio B majoris ad A minorem ſit <lb/>multiplicata rationis C ad A in ra-<lb/>tione data mojoris inæqualitatis M ad N, & </s> <s xml:id="echoid-s3083" xml:space="preserve">ut ratio B ad <lb/>A ſit multiplicata rationis D ad A in ratione data majoris <lb/>inæqualitatis M ad O: </s> <s xml:id="echoid-s3084" xml:space="preserve">ſintque tertii termini eodem modo <lb/>facti ex ſecundis quo ſecundi facti ſunt ex primis; </s> <s xml:id="echoid-s3085" xml:space="preserve">atque ita <lb/>continuetur ſeries.</s> <s xml:id="echoid-s3086" xml:space="preserve"/> </p> <div xml:id="echoid-div161" type="float" level="2" n="2"> <note position="right" xlink:label="note-0143-02" xlink:href="note-0143-02a" xml:space="preserve"> <lb/>## G # H # A # B <lb/># N # I # K # C # D <lb/>M ## R # S # E # F <lb/># O # T # V # X # Y <lb/>### L ## Z <lb/></note> </div> <pb o="426" file="0144" n="153" rhead="VERA CIRCULI"/> </div> <div xml:id="echoid-div163" type="section" level="1" n="75"> <head xml:id="echoid-head111" xml:space="preserve">PROP. IX. PROBLEMA.</head> <p style="it"> <s xml:id="echoid-s3087" xml:space="preserve">Oportet prædictæ ſeriei terminationem invenire.</s> <s xml:id="echoid-s3088" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3089" xml:space="preserve">Ponatur G cyphra ſeu nihil hoc eſt exponens rationis æ-<lb/>qualitatis, ſeu rationis A ad A; </s> <s xml:id="echoid-s3090" xml:space="preserve">ſitque H ad libitum ex-<lb/>ponens rationis B ad A: </s> <s xml:id="echoid-s3091" xml:space="preserve">ſit ut M ad N ita differentia inter <lb/>G & </s> <s xml:id="echoid-s3092" xml:space="preserve">H, hoc eſt ipſa H vel exponens rationis B ad A ad ex-<lb/>ceſſum quo I ſuperat G hoc eſt ipſam I, ſed ut M ad N <lb/>ita ratio B ad A eſt multiplicata rationis C ad A; </s> <s xml:id="echoid-s3093" xml:space="preserve">& </s> <s xml:id="echoid-s3094" xml:space="preserve">igitur <lb/>Exceſſus quo I ſuperat G hoc eſt ipſa I eſt exponens ratio-<lb/>nis C ad A. </s> <s xml:id="echoid-s3095" xml:space="preserve">ſit ut M ad O ita differentia inter G & </s> <s xml:id="echoid-s3096" xml:space="preserve">H hoc <lb/>eſt H ad exceſſum quo K ſuperat G hoc eſt ipſam K, ſed <lb/>ut M ad O ita ratio B ad A eſt multiplicata rationis D ad <lb/>A, cumque H ſit exponens rationis B ad A, erit K expo-<lb/>nens rationis D ad A: </s> <s xml:id="echoid-s3097" xml:space="preserve">ſi igitur I ſit exponens rationis C ad <lb/>A & </s> <s xml:id="echoid-s3098" xml:space="preserve">K exponens rationis D ad A; </s> <s xml:id="echoid-s3099" xml:space="preserve">erit exceſſus quo K ſu-<lb/>perat I exponens rationis D ad C. </s> <s xml:id="echoid-s3100" xml:space="preserve">deinde ſit ut M ad N <lb/>ita exceſſus quo K ſuperat I ſeu exponens rationis D ad C <lb/>ad exceſſum quo R ſuperat I, ſed ut M ad N ita ex ſeriei <lb/>compoſitione ratio D ad C eſt multiplicata rationis E ad <lb/>C, atque exceſſus quo K ſuperat I eſt exponens rationis <lb/>D ad C; </s> <s xml:id="echoid-s3101" xml:space="preserve">& </s> <s xml:id="echoid-s3102" xml:space="preserve">proinde exceſſus quo R ſuperat I eſt exponens <lb/>rationis E ad C, atque I eſt exponens rationis C ad A, & </s> <s xml:id="echoid-s3103" xml:space="preserve">pro-<lb/>inde R eſt exponens rationis E ad A. </s> <s xml:id="echoid-s3104" xml:space="preserve">deinde ſit ut M ad <lb/>O ita exceſſus quo K ſuperat I ad exceſſum quo S ſuperat <lb/>I, ſed ut M ad O ita ex ſeriei compoſitione ratio D ad <lb/>C eſt multiplicata rationis F ad C, cumque exceſſus quo <lb/>K ſuperat I ſit exponens rationis D ad C; </s> <s xml:id="echoid-s3105" xml:space="preserve">erit exceſſus quo <lb/>S ſuperat I exponens rationis F ad C, atque I eſt expo-<lb/>nens rationis C ad A, & </s> <s xml:id="echoid-s3106" xml:space="preserve">proinde S eſt exponens rationis F <lb/>ad A: </s> <s xml:id="echoid-s3107" xml:space="preserve">cum igitur R ſit exponens E ad A & </s> <s xml:id="echoid-s3108" xml:space="preserve">S exponens ra-<lb/>tionis F ad A; </s> <s xml:id="echoid-s3109" xml:space="preserve">erit exceſſus quo S ſuperat R exponens ra-<lb/>tionis F ad E: </s> <s xml:id="echoid-s3110" xml:space="preserve">& </s> <s xml:id="echoid-s3111" xml:space="preserve">utramque ſeriem continuando, demonſtra-<lb/>tur ut antè T eſſe exponentem rationis X ad A, & </s> <s xml:id="echoid-s3112" xml:space="preserve">V eſſe <pb o="427" file="0145" n="154" rhead="ET HYPERBOLÆ QUADRATURA."/> exponentem rationis Y ad A; </s> <s xml:id="echoid-s3113" xml:space="preserve">denique ſemper demonſtrabi-<lb/>tur terminos convergentes ſeriei exponentium eſſe exponen-<lb/>tes rationum, terminorum convergentium ſeriei propoſitæ <lb/>ad primam ſeriei quantitatem A, modò utriuſque ſeriei ter-<lb/>mini convergentes ſint in eodem ab initio ordine: </s> <s xml:id="echoid-s3114" xml:space="preserve">& </s> <s xml:id="echoid-s3115" xml:space="preserve">proin-<lb/>de terminatio ſeriei exponentium per hujus 7 inventa, quæ <lb/>Ex: </s> <s xml:id="echoid-s3116" xml:space="preserve">Gr: </s> <s xml:id="echoid-s3117" xml:space="preserve">ſit L, erit exponens rationis, terminationis ſeriei <lb/>propoſitæ ad primum terminum A: </s> <s xml:id="echoid-s3118" xml:space="preserve">inveniatur igitur ratio <lb/>Z ad A quæ ſit multiplicata rationis datæ B ad A in ratio-<lb/>ne data L ad H; </s> <s xml:id="echoid-s3119" xml:space="preserve">eritque Z terminatio quæſita, quam in-<lb/>venire oportuit.</s> <s xml:id="echoid-s3120" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3121" xml:space="preserve">Ad hoc problema in numeris illuſtrandum ſit M 4, N 2, <lb/>O I, A <emph style="super">6</emph>, B <emph style="super">10</emph>; </s> <s xml:id="echoid-s3122" xml:space="preserve">erunt ſecundi termini convergentes v<emph style="sub">960</emph>, <lb/><emph style="super">V992160,</emph> tertii termini convergentes <emph style="super">V9997776000, V9999100776960000000.</emph> <lb/>& </s> <s xml:id="echoid-s3123" xml:space="preserve">ſeriei terminatio <emph style="super">Vc360.</emph></s> </p> <p> <s xml:id="echoid-s3124" xml:space="preserve">Aliud exemplum, ſit M 6, N 2, O 3, A 5, B 10; <lb/></s> <s xml:id="echoid-s3125" xml:space="preserve">erunt ſecundi termini convergentes <emph style="super">Vc250, Vq50,</emph> tertii termini <lb/>convergentes <emph style="super">Vqcc488281250000000, Vqqc7812500000,</emph> & </s> <s xml:id="echoid-s3126" xml:space="preserve">ſeriei terminatio <lb/><emph style="super">Vſ12500.</emph> hactenus terminavimus omnes ſeries convergentes quæ <lb/>fieri poſſunt vel à ſola proportione arithmetica vel a ſola pro-<lb/>portione geometrica, nunc vero methodum aggredimur, cu-<lb/>jus ope omnium ſerierum convergentium terminationes (ſi <lb/>modò ſint in rerum natura) inveniri poſſunt.</s> <s xml:id="echoid-s3127" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div164" type="section" level="1" n="76"> <head xml:id="echoid-head112" xml:space="preserve">PROP. X. PROBLEMA.</head> <p style="it"> <s xml:id="echoid-s3128" xml:space="preserve">Ex data quantitate, eodem modo compoſita à duo-<lb/>bus terminis convergentibus cujuſcunque ſeriei <lb/>convergentis, quo componitur ex terminis con-<lb/>vergentibus ejuſdem ſeriei immediatè ſe-<lb/>quentibus; </s> <s xml:id="echoid-s3129" xml:space="preserve">ſeriei propoſitæ terminationem <lb/>invenire.</s> <s xml:id="echoid-s3130" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3131" xml:space="preserve">Sit ſeries convergens, cujus duo termini convergentes <lb/>quicunque ſint <emph style="super">a, b,</emph> & </s> <s xml:id="echoid-s3132" xml:space="preserve">termini convergentes immediatè <pb o="428" file="0146" n="155" rhead="VERA CIRCULI"/> ſequentes Vqab, {aa.</s> <s xml:id="echoid-s3133" xml:space="preserve">/Vqab} ſumma terminorum convergentium <emph style="super">a + b</emph> <lb/>multiplicata in terminum convergentem primum <emph style="sub">a</emph> efficit <lb/>aa + ab: </s> <s xml:id="echoid-s3134" xml:space="preserve">& </s> <s xml:id="echoid-s3135" xml:space="preserve">ſumma terminorum convergentium immediate ſe-<lb/>quentium nempe Vqab + {aa/Vqab} multiplicata in primum terminum <lb/>convergentem <emph style="super">Vqab</emph> efficit etiam aa + ab; </s> <s xml:id="echoid-s3136" xml:space="preserve">ex his invenienda ſit ſe-<lb/>riei propoſitæ terminatio. </s> <s xml:id="echoid-s3137" xml:space="preserve">manifeſtum eſt quantitatem aa + ab <lb/>eodem modo fieri à terminis convergentibus <emph style="super">a, b,</emph> quo à termi-<lb/>nis convergentibus immediatè ſequentibus Vqab, {aa/Vqab:</s> <s xml:id="echoid-s3138" xml:space="preserve">} & </s> <s xml:id="echoid-s3139" xml:space="preserve">quo-<lb/>niam quantitates <emph style="super">a, b,</emph> indefinitæ ponuntur pro quibuslibet to-<lb/>tius ſeriei terminis convergentibus, evidens eſt ſummam quo-<lb/>rumcunque terminorum convergentium propoſitæ ſeriei mul-<lb/>tiplicatam in primum terminum convergentem efficere quan-<lb/>titatem æqualem illi, quæ fit à ſumma terminorum conver-<lb/>gentium immediatè ſequentium multiplicata etiam in primum <lb/>ſuum terminum convergentem; </s> <s xml:id="echoid-s3140" xml:space="preserve">cumque duo termini conver-<lb/>gentes duos terminos convergentes ſemper immediatè ſe-<lb/>quuntur, manifeſtum eſt ſummam duorum quorumlibet ter-<lb/>minorum convergentium multiplicatam in primum ſemper <lb/>efficere eandem quantitatem nempe aa + ab, atque ultimi ter-<lb/>mini convergentes ſunt æquales, & </s> <s xml:id="echoid-s3141" xml:space="preserve">proinde ſit ultimus ille <lb/>terminus ſeu ſeriei terminatio <emph style="super">z</emph>, quæ ſibi addita & </s> <s xml:id="echoid-s3142" xml:space="preserve">in ſum-<lb/>mam multiplicata efficit <emph style="super">2 zz</emph>, quæ quantitas debet eſſe æqua-<lb/>lis quantitati aa + ab, & </s> <s xml:id="echoid-s3143" xml:space="preserve">æquatione reſoluta dabitur <emph style="super">z</emph> ſeu ſeriei <lb/>terminatio {Vq aa + ab,/2} quam invenire oportuit.</s> <s xml:id="echoid-s3144" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3145" xml:space="preserve">Et proinde ad inveniendam cujuscunque ſeriei convergen-<lb/>tis terminationem; </s> <s xml:id="echoid-s3146" xml:space="preserve">opus eſt ſolummodo invenire quantitatem <lb/>eodem modo compoſitam ex terminis convergentibus primis, <lb/>quo componitur eadem quantitas ex terminis convergentibus <lb/>ſecundis.</s> <s xml:id="echoid-s3147" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div165" type="section" level="1" n="77"> <head xml:id="echoid-head113" xml:space="preserve">CONSECTARIUM.</head> <p> <s xml:id="echoid-s3148" xml:space="preserve">Quoniam non refert in problemate ſive termini conver-<lb/>gentes a, b, ſint primi, ſecundi, vel tertii &</s> <s xml:id="echoid-s3149" xml:space="preserve">c; </s> <s xml:id="echoid-s3150" xml:space="preserve">manifeſtum eſt <pb o="429" file="0147" n="156" rhead="ET HYPERBOLÆ QUADRATURA."/> omnes ſeriei convergentis terminationem eodem modo eſſe <lb/>compoſitam ex terminis convergentibus primis quo ex termi-<lb/>nis convergentibus ſecundis, tertiis, vel quartis, &</s> <s xml:id="echoid-s3151" xml:space="preserve">c.</s> <s xml:id="echoid-s3152" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div166" type="section" level="1" n="78"> <head xml:id="echoid-head114" xml:space="preserve">PROP. XI. THEOREMA.</head> <p style="it"> <s xml:id="echoid-s3153" xml:space="preserve">Dico ſectorem circuli, ellipſeos vel hyperbolæ A B I P <lb/> <anchor type="note" xlink:label="note-0147-01a" xlink:href="note-0147-01"/> non eſſe compoſitum analyticè à triangulo <lb/>A B P & </s> <s xml:id="echoid-s3154" xml:space="preserve">trapezio A B F P.</s> <s xml:id="echoid-s3155" xml:space="preserve"/> </p> <div xml:id="echoid-div166" type="float" level="2" n="1"> <note position="right" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> </div> <p> <s xml:id="echoid-s3156" xml:space="preserve">Ponatur triangulum A B P <emph style="super">a</emph> & </s> <s xml:id="echoid-s3157" xml:space="preserve">trapezium A B F P <emph style="super">b</emph>: </s> <s xml:id="echoid-s3158" xml:space="preserve">ma-<lb/>nifeſtum eſt ex prædictis trapezium A B I P eſſe Vqab & </s> <s xml:id="echoid-s3159" xml:space="preserve"><lb/>polygonum A B D L P {2 ab/a + Vqab}, item ſectorem A B I P eſſe <lb/>hujus ſeriei convergentis terminationem. </s> <s xml:id="echoid-s3160" xml:space="preserve">ut ex ſeriei termi-<lb/>nis auferantur ſigna radicis & </s> <s xml:id="echoid-s3161" xml:space="preserve">fractionis, pro <emph style="super">a</emph> & </s> <s xml:id="echoid-s3162" xml:space="preserve"><emph style="super">b</emph> primis <lb/>ſeriei terminis convergentibus, hoc eſt pro triangulo A B P <lb/>& </s> <s xml:id="echoid-s3163" xml:space="preserve">trapezio A B F P ponantur a<emph style="super">3</emph> + a<emph style="super">2</emph> b & </s> <s xml:id="echoid-s3164" xml:space="preserve">ab<emph style="super">2</emph> + b<emph style="super">3</emph>; </s> <s xml:id="echoid-s3165" xml:space="preserve">erunt-<lb/>que ſecundi ſeriei termini convergentes, hoc eſt trapezium <lb/>A B I P & </s> <s xml:id="echoid-s3166" xml:space="preserve">polygonum A B D L P, ba<emph style="super">2</emph> + b<emph style="super">2</emph> a & </s> <s xml:id="echoid-s3167" xml:space="preserve">2 b <emph style="super">2</emph> a, di-<lb/>co ſeriei convergentis (cujus primi termini convergentes ſunt <lb/>a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ab<emph style="super">2</emph> + b<emph style="super">3</emph> & </s> <s xml:id="echoid-s3168" xml:space="preserve">ſecundi ſunt ba<emph style="super">2</emph> + b<emph style="super">2</emph> a, 2 b<emph style="super">2</emph> a) termina-<lb/>tionem non eſſe compoſitam analyticè a terminis a<emph style="super">3</emph> + a<emph style="super">2</emph> b, <lb/>ab<emph style="super">2</emph> + b<emph style="super">3</emph>: </s> <s xml:id="echoid-s3169" xml:space="preserve">ſi enim componatur prædicta terminatio analyticè a <lb/>terminis convergentibus a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ab<emph style="super">2</emph> + b<emph style="super">3</emph>; </s> <s xml:id="echoid-s3170" xml:space="preserve">componetur etiam <lb/>eadem terminatio analyticè & </s> <s xml:id="echoid-s3171" xml:space="preserve">eodem omnino modo à termi-<lb/>nis convergentibus ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, 2b<emph style="super">2</emph>a; </s> <s xml:id="echoid-s3172" xml:space="preserve">& </s> <s xml:id="echoid-s3173" xml:space="preserve">proinde eadem quan-<lb/>titas, nempe prædicta terminatio, eodem modo componitur <lb/>analyticè ex terminis a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ab<emph style="super">2</emph> + b<emph style="super">3</emph>, quo componitur ex <lb/>terminis ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, 2b<emph style="super">2</emph>a, <lb/> <anchor type="note" xlink:label="note-0147-02a" xlink:href="note-0147-02"/> ſed nulla quantitas poteſt <lb/>eodem modo analyticè com-<lb/>poni ex terminis a<emph style="super">3</emph> + a<emph style="super">2</emph>b, <lb/>ab<emph style="super">2</emph> + b<emph style="super">3</emph>, quo componitur <lb/>ex terminis ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, 2b<emph style="super">2</emph>a, quod ſic demonſtro. </s> <s xml:id="echoid-s3174" xml:space="preserve">ſi analy-<lb/>ticè componeretur quantitas ex terminis a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ab<emph style="super">2</emph> + b<emph style="super">3</emph>, <pb o="430" file="0148" n="157" rhead="VERA CIRCULI"/> eodem modo, quo analyticè componitur eadem quantitas ex <lb/>terminis ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, 2b<emph style="super">2</emph>a; </s> <s xml:id="echoid-s3175" xml:space="preserve">addendo, ſubtrahendo, multi-<lb/>plicando & </s> <s xml:id="echoid-s3176" xml:space="preserve">dividendo terminos a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ab<emph style="super">2</emph> + b<emph style="super">3</emph> & </s> <s xml:id="echoid-s3177" xml:space="preserve">radices <lb/>ex factis extrahendo, eadem fieret quantitas ac ſi eodem mo-<lb/>do adderentur, ſubducerentur, multiplicarentur & </s> <s xml:id="echoid-s3178" xml:space="preserve">divide-<lb/>rentur termini ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, 2b<emph style="sub">2</emph>a, & </s> <s xml:id="echoid-s3179" xml:space="preserve">radices eædem ex factis <lb/>extraherentur, ſed poſterius fieri non poteſt, ergo nec prius; <lb/></s> <s xml:id="echoid-s3180" xml:space="preserve">minorem ſic probo, ſi eadem fieret quantitas ex additione, <lb/>ſubductione, multiplicatione, diviſione & </s> <s xml:id="echoid-s3181" xml:space="preserve">radicum extractio-<lb/>ne terminorum a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ab<emph style="super">2</emph> + b<emph style="super">3</emph>, quæ fieret ex eadem ad-<lb/>ditione, ſubductione, multiplicatione, diviſione & </s> <s xml:id="echoid-s3182" xml:space="preserve">radicum <lb/>extractione terminorum ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, 2b<emph style="super">2</emph>a; </s> <s xml:id="echoid-s3183" xml:space="preserve">tunc addendo æ-<lb/>quales quantitates terminis a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, vel ab illis <lb/>ſive ipſorum factis æquales quantitates ſubducendo, vel il-<lb/>los ſive ipſorum factos æqualibus quantitatibus multiplican-<lb/>do vel dividendo, vel denique illos ſive ipſorum factos eo-<lb/>dem modo in ſe multiplicando, vel ex iisdem easdem radi-<lb/>ces extrahendo, hasce analyticas operationes aliquo modo <lb/>mutando, reiterando vel utrumque vel neutrum faciendo, <lb/>fieri poſſent duo ultima producta, nempe unum à termino <lb/>ab<emph style="super">2</emph> + b<emph style="sub">3</emph> & </s> <s xml:id="echoid-s3184" xml:space="preserve">alterum à termino 2b<emph style="super">2</emph> a; </s> <s xml:id="echoid-s3185" xml:space="preserve">ita ut ultimum produ-<lb/>ctum ex termino a<emph style="super">3</emph> + a<emph style="super">2</emph> b cum ultimo producto ex termino <lb/>ab<emph style="super">2</emph> + b<emph style="sub">3</emph> additum, ſubductum, multiplicatum, diviſum, & </s> <s xml:id="echoid-s3186" xml:space="preserve"><lb/>ex facto radice aliqua extracta (haſce analyticas operationes <lb/>aliquo modo mutando, reiterando vel utrumque vel neu-<lb/>trum faciendo) eandem efficiat quantitatem quam efficit ul-<lb/>timum productum ex termino ba<emph style="super">2</emph> + b<emph style="super">2</emph>a cum ultimo produ-<lb/>cto ex termino 2b<emph style="super">2</emph>a eodem omnino modo additum, ſubdu-<lb/>ctum, multiplicatum, diviſum, & </s> <s xml:id="echoid-s3187" xml:space="preserve">ex facto eadem etiam ra-<lb/>dice extracta, haſce analyticas operationes eodem omnino <lb/>modo mutando, reiterando vel utrumque vel neutrum fa-<lb/>ciendo; </s> <s xml:id="echoid-s3188" xml:space="preserve">ſed poſterius eſt abſurdum ergò & </s> <s xml:id="echoid-s3189" xml:space="preserve">prius: </s> <s xml:id="echoid-s3190" xml:space="preserve">ſequela <lb/>majoris patet ex octava definitione hujus, minorem ſic pro-<lb/>bo, in termino a<emph style="super">3</emph> + a<emph style="super">2</emph> b reperitur poteſtas ipſius a nempe <lb/>a<emph style="super">3</emph>, quæ eſt altior ulla poteſtate ejusdem a in termino <lb/>ba<emph style="super">2</emph> + b<emph style="super">2</emph> a; </s> <s xml:id="echoid-s3191" xml:space="preserve">& </s> <s xml:id="echoid-s3192" xml:space="preserve">proinde terminos a<emph style="super">3</emph> + a<emph style="super">2</emph> b, ba<emph style="sub">2</emph> + b<emph style="super">2</emph>a, cum <pb o="431" file="0149" n="158" rhead="ET HYPERBOLÆ QUADRATURA."/> æqualibus quantitatibus addendo, ſubtrahendo, multiplican-<lb/>do, dividendo, &</s> <s xml:id="echoid-s3193" xml:space="preserve">c: </s> <s xml:id="echoid-s3194" xml:space="preserve">ſicut ſuperius dictum eſt, ſemper ma-<lb/>net poteſtas ipſius a in ultimo producto ex termino a<emph style="super">3</emph> + a<emph style="super">2</emph> b <lb/>altior poteſtate ulla ipſius a in ultimo producto termini <lb/>ba<emph style="super">2</emph> + b<emph style="super">2</emph> a, quoniam illos cum æqualibus quantitatibus ad-<lb/>dendo, ſubſtrahendo, &</s> <s xml:id="echoid-s3195" xml:space="preserve">c, ſemper manent in factis eædem <lb/>poteſtates, & </s> <s xml:id="echoid-s3196" xml:space="preserve">illos in ſe eodem modo multiplicando ſemper <lb/>magis elevatur altior poteſtas in termino a<emph style="super">3</emph> + a<emph style="super">2</emph> b quam ele-<lb/>vatur depreſſior poteſtas in termino ba<emph style="super">2</emph> + b<emph style="super">2</emph> a, & </s> <s xml:id="echoid-s3197" xml:space="preserve">ex illis <lb/>eaſdem radices extrahendo, ubi a fuerat elevatior in poteſta-<lb/>te erit etiam elevatior in radice: </s> <s xml:id="echoid-s3198" xml:space="preserve">& </s> <s xml:id="echoid-s3199" xml:space="preserve">quoniam eadem reperitur <lb/>altiſſima poteſtas ipſius a in termino ab<emph style="super">2</emph> + b<emph style="super">3</emph> quæ reperitur <lb/>in termino 2b<emph style="super">2</emph> a, demonſtratur ut antè altiſſimam poteſta-<lb/>tem ipſius a in ultimo producto ex termino ab<emph style="super">2</emph> + b<emph style="super">3</emph> eandem <lb/>eſſe cum altiſſima poteſtate ipſius a in ultimo producto ex <lb/>termino 2b<emph style="super">2</emph> a: </s> <s xml:id="echoid-s3200" xml:space="preserve">in ultimo igitur producto termini a<emph style="super">3</emph> + a<emph style="super">2</emph> b <lb/>reperitur altior poteſtas ipſius a quam in ultimo producto <lb/>termini ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, & </s> <s xml:id="echoid-s3201" xml:space="preserve">in ultimo producto termini ab<emph style="super">2</emph> + b<emph style="super">3</emph> <lb/>altiſſima poteſtas ipſius a eadem eſt cum altiſſima poteſtate <lb/>ipſius a in ultimo producto termini 2b<emph style="super">2</emph> a; </s> <s xml:id="echoid-s3202" xml:space="preserve">& </s> <s xml:id="echoid-s3203" xml:space="preserve">igitur ultima <lb/>producta ex terminis a<emph style="super">3</emph> + a<emph style="super">2</emph>b, ab<emph style="super">2</emph> + b<emph style="super">3</emph>, eodem modo inter <lb/>ſe addita, ſubducta, multiplicata, diviſa, &</s> <s xml:id="echoid-s3204" xml:space="preserve">c. </s> <s xml:id="echoid-s3205" xml:space="preserve">ſemper effi-<lb/>cient quantitatem, in qua reperitur altior poteſtas ipſius a <lb/>quam ulla quæ reperiri poteſt in quantitate facta ex eadem <lb/>prorſus additione, ſubductione, multiplicatione, diviſione, <lb/>&</s> <s xml:id="echoid-s3206" xml:space="preserve">c, productorum à terminis ba<emph style="super">2</emph> + b<emph style="super">2</emph> a, 2b<emph style="super">2</emph> a; </s> <s xml:id="echoid-s3207" xml:space="preserve">quoniam al-<lb/>tior poteſtas cum altera poteſtate ſemper altiorem facit po-<lb/>teſtatem quam depreſſior poteſtas cum eadem altera poteſta-<lb/>te; </s> <s xml:id="echoid-s3208" xml:space="preserve">& </s> <s xml:id="echoid-s3209" xml:space="preserve">igitur iſtæ duæ quantitates non poſſunt eſſe indefini-<lb/>tè æquales, cum reperiatur altior poteſtas ipſius a in una <lb/>quam in altera: </s> <s xml:id="echoid-s3210" xml:space="preserve">atque hinc evidens eſt quod ſector circuli, <lb/>ellipſeos vel hyperbolæ A B I P non poſſit componi analy-<lb/>ticè à triangulo A B P & </s> <s xml:id="echoid-s3211" xml:space="preserve">trapezio A B F P, quod demon-<lb/>ſtrandum erat.</s> <s xml:id="echoid-s3212" xml:space="preserve"/> </p> <div xml:id="echoid-div167" type="float" level="2" n="2"> <note position="right" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve"> <lb/>a<emph style="super">3</emph> + a<emph style="super">2</emph>b # ab<emph style="super">2</emph> + b<emph style="super">3</emph>. <lb/>ba<emph style="super">2</emph> + b<emph style="super">2</emph> a # 2b<emph style="super">2</emph>a <lb/></note> </div> <p> <s xml:id="echoid-s3213" xml:space="preserve">Ut autem evidentius fiat propoſitum, aliam demonſtratio-<lb/>nem treviorem faciliorem & </s> <s xml:id="echoid-s3214" xml:space="preserve">ex alio medio petitam hic ſub- <pb o="432" file="0150" n="159" rhead="VERA CIRCULI"/> jungo: </s> <s xml:id="echoid-s3215" xml:space="preserve">quantitas non poteſt componi analyticè ex terminis <lb/>a<emph style="super">3</emph> + a<emph style="super">2</emph> b, ab<emph style="super">2</emph> + b<emph style="super">3</emph>, eodem modo quo componitur eadem <lb/>quantitas ex terminis ba<emph style="super">2</emph> + b<emph style="super">2</emph> a, 2b<emph style="super">2</emph> a; </s> <s xml:id="echoid-s3216" xml:space="preserve">quoniam adden-<lb/>do, ſubſtrahendo, multiplicando, dividendo duo binomia <lb/>a<emph style="super">3</emph> + a<emph style="super">2</emph> b, ab<emph style="super">2</emph> + b<emph style="super">3</emph> & </s> <s xml:id="echoid-s3217" xml:space="preserve">radices ex ultimo facto extrahendo, <lb/>plura fiunt nomina in ultimo producto, quam ſi eodem mo-<lb/>do adderentur, ſubducerentur, multiplicarentur, divideren-<lb/>tur, binomium ba<emph style="super">2</emph> + b<emph style="super">2</emph>a & </s> <s xml:id="echoid-s3218" xml:space="preserve">ſimplex quantitas 2b<emph style="super">2</emph> a, & </s> <s xml:id="echoid-s3219" xml:space="preserve">eæ-<lb/>dem quoque radices ex facto extraherentur; </s> <s xml:id="echoid-s3220" xml:space="preserve">& </s> <s xml:id="echoid-s3221" xml:space="preserve">ſi plura ſint <lb/>nomina in uno producto quam in altero, impoſſibile eſt ut <lb/>ſint indefinitè æqualia, quod eſt propoſitum, reliqua enim <lb/>ex priore demonſtratione haberi poſſunt.</s> <s xml:id="echoid-s3222" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div169" type="section" level="1" n="79"> <head xml:id="echoid-head115" xml:space="preserve">SCHOLIUM.</head> <p> <s xml:id="echoid-s3223" xml:space="preserve">Obſcurum fortaſſis videbitur hoc theorema ob multas in-<lb/>uſitatas in geometria voces quas hic adhibere oportet, <lb/>& </s> <s xml:id="echoid-s3224" xml:space="preserve">ob multa ſuppoſita lemmata, quæ demonſtrare pigebat, <lb/>quoniam cuivis analyſtæ vel prima lectione ſunt obvia, ex <lb/>natura enim operationum analyticarum omnino dependent.</s> <s xml:id="echoid-s3225" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3226" xml:space="preserve">Locus hic requirit ut aliquid dicam de proportione inter <lb/>triangulum A B P & </s> <s xml:id="echoid-s3227" xml:space="preserve">ſectorem A B I P; </s> <s xml:id="echoid-s3228" xml:space="preserve">quod ut fiat, adver-<lb/>tendum eſt veriſſimum philoſophorum axioma, nempe omnem <lb/>noſtram cognitionem à ſenſu ortum habere: </s> <s xml:id="echoid-s3229" xml:space="preserve">inter proportio-<lb/>nes enim, ſola commenſurabilis ſenſu attingitur & </s> <s xml:id="echoid-s3230" xml:space="preserve">perfectè <lb/>ab humana mente intelligitur; </s> <s xml:id="echoid-s3231" xml:space="preserve">incommenſurabilis enim à ma-<lb/>thematicis ſolummodo adhuc contemplatur, quatenus com-<lb/>menſurabilis cujusdam rationis eſt ſubduplicata, ſubtriplica-<lb/>ta, &</s> <s xml:id="echoid-s3232" xml:space="preserve">c. </s> <s xml:id="echoid-s3233" xml:space="preserve">vel ex talium additione, ſubductione, &</s> <s xml:id="echoid-s3234" xml:space="preserve">c. </s> <s xml:id="echoid-s3235" xml:space="preserve">genita: <lb/></s> <s xml:id="echoid-s3236" xml:space="preserve">hoc eſt, quantitas quæ quantitati propoſitæ eſt incommen-<lb/>ſurabilis ex eo ſolummodo ab humana mente contemplatur, <lb/>quod ex aliquot quantitatum cognitarum & </s> <s xml:id="echoid-s3237" xml:space="preserve">propoſitæ quan-<lb/>titati commenſurabilium additione, ſubductione, multiplica-<lb/>tione, diviſione & </s> <s xml:id="echoid-s3238" xml:space="preserve">radicum extractione componi poſſit: </s> <s xml:id="echoid-s3239" xml:space="preserve">at <lb/>ex hactenus demonſtratis manifeſtum eſt ſectorem A B I P <lb/>non poſſe componi ex additione, ſubductione, multiplica- <pb o="433" file="0151" n="160" rhead="ET HYPERBOLÆ QUADRATURA."/> tione, diviſione, & </s> <s xml:id="echoid-s3240" xml:space="preserve">radicum extractione trianguli A B P & </s> <s xml:id="echoid-s3241" xml:space="preserve"><lb/>trapezii A B F P: </s> <s xml:id="echoid-s3242" xml:space="preserve">triangulum autem A B P & </s> <s xml:id="echoid-s3243" xml:space="preserve">trapezium <lb/>A B F P ſupponimus eſſe quantitates inter ſe analyticas; </s> <s xml:id="echoid-s3244" xml:space="preserve">& </s> <s xml:id="echoid-s3245" xml:space="preserve"><lb/>proinde ſector A B I P illis analytica eſſe non poteſt, hoc <lb/>eſt ex quantitatum ipſis A B P, A B F P analyticarum addi-<lb/>tione, ſubductione, multiplicatione, diviſione & </s> <s xml:id="echoid-s3246" xml:space="preserve">radicum <lb/>extractione componi non poteſt; </s> <s xml:id="echoid-s3247" xml:space="preserve">& </s> <s xml:id="echoid-s3248" xml:space="preserve">proinde ex hoc capite <lb/>nulla poteſt exhiberi ratio inter triangulum A B P & </s> <s xml:id="echoid-s3249" xml:space="preserve">ſecto-<lb/>rem A B I P, cum evidens ſit illam non eſſe analyticam. </s> <s xml:id="echoid-s3250" xml:space="preserve">ſed <lb/>dicet fortè aliquis rationem inter triangulum A B P & </s> <s xml:id="echoid-s3251" xml:space="preserve">ſecto-<lb/>rem A B I P omnifariam variari poſſe; </s> <s xml:id="echoid-s3252" xml:space="preserve">& </s> <s xml:id="echoid-s3253" xml:space="preserve">proinde poſſe eſſe <lb/>inter ſe in ratione qualibet data ſive analytica ſive etiam com-<lb/>menſurabili: </s> <s xml:id="echoid-s3254" xml:space="preserve">reſpondeo hoc eſſe veriſſimum, ſed in hoc ca-<lb/>ſu ratio inter triangulum A B P & </s> <s xml:id="echoid-s3255" xml:space="preserve">trapezium A B F P non <lb/>erit analytica; </s> <s xml:id="echoid-s3256" xml:space="preserve">& </s> <s xml:id="echoid-s3257" xml:space="preserve">proinde ex dato circulo ellipſe vel hyper-<lb/>bola nunquam dabitur in analyticis triangulum A B P, quod <lb/>ex prædictis clariſſimè patet. </s> <s xml:id="echoid-s3258" xml:space="preserve">etiamſi ex prædicto capite non <lb/>poſſimus comprehendere rationem inter triangulum A B P & </s> <s xml:id="echoid-s3259" xml:space="preserve"><lb/>ſectorem A B I P, poſſumus tamen ejus aliquam habere cogni-<lb/>tionem, ex eo quod ſector A B I P ſit terminatio ſeriei con-<lb/>vergentis datæ; </s> <s xml:id="echoid-s3260" xml:space="preserve">& </s> <s xml:id="echoid-s3261" xml:space="preserve">ex hac conſideratione poſſibile eſt inve-<lb/>nire quantitatem datæ commenſurabilem cujus differentia à <lb/>ſectore A B I P minor fuerit quacunque quantitate propoſi-<lb/>ta, ad hoc enim ſemper recurrendum eſt, cum de quantita-<lb/>tibus quibuscunque incommenſurabilibus tractant practici, <lb/>& </s> <s xml:id="echoid-s3262" xml:space="preserve">in hac noſtra approximatione praxis non erit operoſior <lb/>quam in multis aliis etiam quantitatum analyticarum appro-<lb/>ximationibus, immo multo brevior, facilior & </s> <s xml:id="echoid-s3263" xml:space="preserve">paratior erit <lb/>illis Vietæ ſectionibus angularibus, quæ tamen ſummæ ma-<lb/>theſeos utilitati in praxem reducuntur. </s> <s xml:id="echoid-s3264" xml:space="preserve">non video ergo qua-<lb/>re circuli quadratura diutius æſtimetur ignorari: </s> <s xml:id="echoid-s3265" xml:space="preserve">cum enim <lb/>demonſtratum ſit rationem circuli ad diametri quadratum <lb/>non eſſe analyticam, vanum certè erit & </s> <s xml:id="echoid-s3266" xml:space="preserve">ineptum illam ſicut <lb/>talem impoſterum quærere: </s> <s xml:id="echoid-s3267" xml:space="preserve">at rejectis quantitatibus analyti-<lb/>cis, vix credo ullam poſſe eſſe notiorem hisce noſtrarum ſe-<lb/>rierum convergentium terminationibus, ſicut ex ſequentibus <lb/>pleniſſimè apparebit.</s> <s xml:id="echoid-s3268" xml:space="preserve"/> </p> <pb o="434" file="0152" n="161" rhead="VERA CIRCULI"/> </div> <div xml:id="echoid-div170" type="section" level="1" n="80"> <head xml:id="echoid-head116" xml:space="preserve">PROP. XII. THEOREMA.</head> <p> <s xml:id="echoid-s3269" xml:space="preserve">Sit trapezium A B I P, A; </s> <s xml:id="echoid-s3270" xml:space="preserve">polygonum <lb/> <anchor type="note" xlink:label="note-0152-01a" xlink:href="note-0152-01"/> <anchor type="note" xlink:label="note-0152-02a" xlink:href="note-0152-02"/> A B E I O P, C; </s> <s xml:id="echoid-s3271" xml:space="preserve">polygonum A B C G <lb/>K N P, D; </s> <s xml:id="echoid-s3272" xml:space="preserve">& </s> <s xml:id="echoid-s3273" xml:space="preserve">polygonum A B D L P, B. </s> <s xml:id="echoid-s3274" xml:space="preserve">di-<lb/>co D eſſe medium harmonicum inter C & </s> <s xml:id="echoid-s3275" xml:space="preserve">B. </s> <s xml:id="echoid-s3276" xml:space="preserve">ex hujus 4, <lb/>A: </s> <s xml:id="echoid-s3277" xml:space="preserve">C:</s> <s xml:id="echoid-s3278" xml:space="preserve">: C: </s> <s xml:id="echoid-s3279" xml:space="preserve">B, & </s> <s xml:id="echoid-s3280" xml:space="preserve">componendo A + C: </s> <s xml:id="echoid-s3281" xml:space="preserve">C:</s> <s xml:id="echoid-s3282" xml:space="preserve">: C + B: </s> <s xml:id="echoid-s3283" xml:space="preserve">B, ſed ex <lb/>hujus 5, A + C: </s> <s xml:id="echoid-s3284" xml:space="preserve">C:</s> <s xml:id="echoid-s3285" xml:space="preserve">: 2 C: </s> <s xml:id="echoid-s3286" xml:space="preserve">D; </s> <s xml:id="echoid-s3287" xml:space="preserve">& </s> <s xml:id="echoid-s3288" xml:space="preserve">ideo C + B: </s> <s xml:id="echoid-s3289" xml:space="preserve">B:</s> <s xml:id="echoid-s3290" xml:space="preserve">: 2 C: </s> <s xml:id="echoid-s3291" xml:space="preserve">D, & </s> <s xml:id="echoid-s3292" xml:space="preserve"><lb/>permutando B + C: </s> <s xml:id="echoid-s3293" xml:space="preserve">2 C:</s> <s xml:id="echoid-s3294" xml:space="preserve">: B: </s> <s xml:id="echoid-s3295" xml:space="preserve">D, & </s> <s xml:id="echoid-s3296" xml:space="preserve">dividendo, differentia in-<lb/>ter B & </s> <s xml:id="echoid-s3297" xml:space="preserve">C eſt ad 2 C, ut differentia inter B & </s> <s xml:id="echoid-s3298" xml:space="preserve">D ad D, & </s> <s xml:id="echoid-s3299" xml:space="preserve">per-<lb/>mutando differentia inter B & </s> <s xml:id="echoid-s3300" xml:space="preserve">C eſt ad differentiam inter <lb/>B & </s> <s xml:id="echoid-s3301" xml:space="preserve">D ut 2 C ad D, hoc eſt, ut C + B ad B, & </s> <s xml:id="echoid-s3302" xml:space="preserve">dividendo, <lb/>differentia inter D & </s> <s xml:id="echoid-s3303" xml:space="preserve">C eſt ad differentiam inter B & </s> <s xml:id="echoid-s3304" xml:space="preserve">D ut <lb/>C ad B; </s> <s xml:id="echoid-s3305" xml:space="preserve">& </s> <s xml:id="echoid-s3306" xml:space="preserve">proinde D eſt medium harmonicum inter C & </s> <s xml:id="echoid-s3307" xml:space="preserve">B, <lb/>quod demonſtrare oportuit.</s> <s xml:id="echoid-s3308" xml:space="preserve"/> </p> <div xml:id="echoid-div170" type="float" level="2" n="1"> <note position="left" xlink:label="note-0152-01" xlink:href="note-0152-01a" xml:space="preserve">TAB. XLIII. <lb/>Fig. 1. 2. 3.</note> <note position="right" xlink:label="note-0152-02" xlink:href="note-0152-02a" xml:space="preserve"> <lb/>A # C # D # B <lb/></note> </div> <p> <s xml:id="echoid-s3309" xml:space="preserve">Hæc propoſitio eodem modo locum habet in omnibus po-<lb/>lygonis complicatis, ut patet ex ſcholio 5 hujus.</s> <s xml:id="echoid-s3310" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div172" type="section" level="1" n="81"> <head xml:id="echoid-head117" xml:space="preserve">PROP. XIII. THEOREMA.</head> <p> <s xml:id="echoid-s3311" xml:space="preserve">Inter duas quantitates A, B, ſit media a-<lb/> <anchor type="note" xlink:label="note-0152-03a" xlink:href="note-0152-03"/> rithmetica C, media geometrica D & </s> <s xml:id="echoid-s3312" xml:space="preserve">me-<lb/>dia harmonica E. </s> <s xml:id="echoid-s3313" xml:space="preserve">dico C, D, E, eſſe con-<lb/>tinuè proportionales. </s> <s xml:id="echoid-s3314" xml:space="preserve">quoniam A, E, B, <lb/>ſunt in ratione harmonica; </s> <s xml:id="echoid-s3315" xml:space="preserve">erit differentia inter A & </s> <s xml:id="echoid-s3316" xml:space="preserve">E ad <lb/>differentiam inter E & </s> <s xml:id="echoid-s3317" xml:space="preserve">B ut A ad B; </s> <s xml:id="echoid-s3318" xml:space="preserve">& </s> <s xml:id="echoid-s3319" xml:space="preserve">componendo erit <lb/>differentia inter A & </s> <s xml:id="echoid-s3320" xml:space="preserve">B ad differentiam inter E & </s> <s xml:id="echoid-s3321" xml:space="preserve">B, ut <lb/>A + B ad B; </s> <s xml:id="echoid-s3322" xml:space="preserve">deinde permutando & </s> <s xml:id="echoid-s3323" xml:space="preserve">componendo 2 A: </s> <s xml:id="echoid-s3324" xml:space="preserve">A + B:</s> <s xml:id="echoid-s3325" xml:space="preserve">: <lb/>E: </s> <s xml:id="echoid-s3326" xml:space="preserve">B, ſed 2A eſt duplum ipſius A & </s> <s xml:id="echoid-s3327" xml:space="preserve">A + B duplum ipſius C; <lb/></s> <s xml:id="echoid-s3328" xml:space="preserve">& </s> <s xml:id="echoid-s3329" xml:space="preserve">ideo A: </s> <s xml:id="echoid-s3330" xml:space="preserve">C:</s> <s xml:id="echoid-s3331" xml:space="preserve">: E: </s> <s xml:id="echoid-s3332" xml:space="preserve">B; </s> <s xml:id="echoid-s3333" xml:space="preserve">& </s> <s xml:id="echoid-s3334" xml:space="preserve">proinde CE = AB, & </s> <s xml:id="echoid-s3335" xml:space="preserve">AB = DD, ideo-<lb/>que CE = DD; </s> <s xml:id="echoid-s3336" xml:space="preserve">& </s> <s xml:id="echoid-s3337" xml:space="preserve">igitur C: </s> <s xml:id="echoid-s3338" xml:space="preserve">D:</s> <s xml:id="echoid-s3339" xml:space="preserve">: D: </s> <s xml:id="echoid-s3340" xml:space="preserve">E, quod demonſtrare o-<lb/>portuit.</s> <s xml:id="echoid-s3341" xml:space="preserve"/> </p> <div xml:id="echoid-div172" type="float" level="2" n="1"> <note position="right" xlink:label="note-0152-03" xlink:href="note-0152-03a" xml:space="preserve"> <lb/>A ## B <lb/>C # D # E <lb/></note> </div> <pb o="435" file="0153" n="162" rhead="ET HYPERBOLÆ QUADRATURA."/> </div> <div xml:id="echoid-div174" type="section" level="1" n="82"> <head xml:id="echoid-head118" xml:space="preserve">PROP. XIV. THEOREMA.</head> <p> <s xml:id="echoid-s3342" xml:space="preserve">Sint duo polygona complicata A, B, nem-<lb/> <anchor type="note" xlink:label="note-0153-01a" xlink:href="note-0153-01"/> pe A intra circuli vel ellipſeos ſectorem & </s> <s xml:id="echoid-s3343" xml:space="preserve"><lb/>B extra: </s> <s xml:id="echoid-s3344" xml:space="preserve">continuetur ſeries convergens horum <lb/>polygonorum complicatorum ſecundum no-<lb/>ſtram methodum ſubduplam deſcriptorum, ita <lb/>ut polygona intra circulum ſint A, C, E, &</s> <s xml:id="echoid-s3345" xml:space="preserve">c, & </s> <s xml:id="echoid-s3346" xml:space="preserve">extra cir-<lb/>culum B, D, F, &</s> <s xml:id="echoid-s3347" xml:space="preserve">c; </s> <s xml:id="echoid-s3348" xml:space="preserve">dico A + E minorem eſſe quam 2 C: <lb/></s> <s xml:id="echoid-s3349" xml:space="preserve">ex prædictis manifeſtæ ſunt ſequentes analogiæ; </s> <s xml:id="echoid-s3350" xml:space="preserve">prima quo-<lb/>niam A, C, B, ſunt continue pro-<lb/> <anchor type="note" xlink:label="note-0153-02a" xlink:href="note-0153-02"/> portionales; </s> <s xml:id="echoid-s3351" xml:space="preserve">& </s> <s xml:id="echoid-s3352" xml:space="preserve">ſecunda quoniam <lb/>C, D, B, ſunt harmonice pro-<lb/>portionales: </s> <s xml:id="echoid-s3353" xml:space="preserve">& </s> <s xml:id="echoid-s3354" xml:space="preserve">proinde exceſſus <lb/>C ſupra A, hoc eſt C — A, eſt ad exceſſum D ſupra C ſeu <lb/>D - C in ratione compoſita ex proportione A ad C & </s> <s xml:id="echoid-s3355" xml:space="preserve">ex <lb/>proportione A + C ad A, hoc eſt in ratione A + C ad C; <lb/></s> <s xml:id="echoid-s3356" xml:space="preserve">at A + C eſt major quam C, & </s> <s xml:id="echoid-s3357" xml:space="preserve">ideo exceſſus C ſupra A eſt <lb/>major quam exceſſus D ſupra C, eſt autem D major quam <lb/>E, & </s> <s xml:id="echoid-s3358" xml:space="preserve">proinde exceſſus C ſupra A multò major eſt quam <lb/>exceſſus E ſupra C; </s> <s xml:id="echoid-s3359" xml:space="preserve">eſt igitur A + E minor quam 2 C; </s> <s xml:id="echoid-s3360" xml:space="preserve"><lb/>quod demonſtrare oportuit.</s> <s xml:id="echoid-s3361" xml:space="preserve"/> </p> <div xml:id="echoid-div174" type="float" level="2" n="1"> <note position="right" xlink:label="note-0153-01" xlink:href="note-0153-01a" xml:space="preserve"> <lb/>A # B <lb/>C # D <lb/>E # F <lb/></note> <note position="right" xlink:label="note-0153-02" xlink:href="note-0153-02a" xml:space="preserve"> <lb/>C - A:B - C::A:C <lb/>B - C:D - C::A + C:A <lb/></note> </div> </div> <div xml:id="echoid-div176" type="section" level="1" n="83"> <head xml:id="echoid-head119" xml:space="preserve">PROP. XV. THEOREMA.</head> <p> <s xml:id="echoid-s3362" xml:space="preserve">Iiſdem poſitis: </s> <s xml:id="echoid-s3363" xml:space="preserve">dico exceſſum C ſupra A minorem eſſe qua-<lb/>druplo exceſſus E ſupra C. </s> <s xml:id="echoid-s3364" xml:space="preserve">ex prædictis manifeſtæ ſunt <lb/>ſequentes tres analogiæ, prima quoniam A, C, B, ſunt con-<lb/>tinuè proportionales; </s> <s xml:id="echoid-s3365" xml:space="preserve">ſecunda, quoniam C, D, B, ſunt har-<lb/>monicè proportionales; </s> <s xml:id="echoid-s3366" xml:space="preserve">& </s> <s xml:id="echoid-s3367" xml:space="preserve">tertia, quoniam C, E, D, ſunt con-<lb/>tinuè proportionales; </s> <s xml:id="echoid-s3368" xml:space="preserve">& </s> <s xml:id="echoid-s3369" xml:space="preserve">ideo <lb/> <anchor type="note" xlink:label="note-0153-03a" xlink:href="note-0153-03"/> exceſſus C ſupra A (hoc eſt) <lb/>C - A eſt ad exceſſum E ſu-<lb/>pra C ſeu E - C, ut A C + E C <lb/>+ AE + CC ad CC; </s> <s xml:id="echoid-s3370" xml:space="preserve">at B ma- <pb o="436" file="0154" n="163" rhead="VERA CIRCULI"/> jor eſt quam E, & </s> <s xml:id="echoid-s3371" xml:space="preserve">ideo A B ſeu C C major eſt quam A E, <lb/>& </s> <s xml:id="echoid-s3372" xml:space="preserve">igitur AE + CC minor eſt quam 2 CC: </s> <s xml:id="echoid-s3373" xml:space="preserve">atque AC + E C <lb/>eſt ad 2 CC ut A + E ad 2 C, ſed A + E minor eſt quam <lb/>2 C; </s> <s xml:id="echoid-s3374" xml:space="preserve">& </s> <s xml:id="echoid-s3375" xml:space="preserve">ideo A C + E C minor eſt quam 2 CC; </s> <s xml:id="echoid-s3376" xml:space="preserve">proinde <lb/>A C + E C + A E + CC minor eſt quam 4 CC; </s> <s xml:id="echoid-s3377" xml:space="preserve">& </s> <s xml:id="echoid-s3378" xml:space="preserve">igitur <lb/>C - A minor eſt quadruplo ipſius E - C, quod demon-<lb/>ſtrare oportuit.</s> <s xml:id="echoid-s3379" xml:space="preserve"/> </p> <div xml:id="echoid-div176" type="float" level="2" n="1"> <note position="right" xlink:label="note-0153-03" xlink:href="note-0153-03a" xml:space="preserve"> <lb/>C - A:B - C::A:C <lb/>B - C:D - C::A + C:A <lb/>D - C:E - C::E + C:C <lb/></note> </div> </div> <div xml:id="echoid-div178" type="section" level="1" n="84"> <head xml:id="echoid-head120" xml:space="preserve">PROP. XVI. THEOREMA.</head> <p> <s xml:id="echoid-s3380" xml:space="preserve">SInt duo Polygona complicata A, B; <lb/></s> <s xml:id="echoid-s3381" xml:space="preserve"> <anchor type="note" xlink:label="note-0154-01a" xlink:href="note-0154-01"/> nempe A extra hyperbolæ ſecto-<lb/>rem & </s> <s xml:id="echoid-s3382" xml:space="preserve">B intra: </s> <s xml:id="echoid-s3383" xml:space="preserve">Continuetur ſeries con-<lb/>vergens horum polygonorum complica-<lb/>torum ſecundum noſtram methodum <lb/>ſubduplam deſcriptorum, ita ut polygona extra hyperbolem <lb/>ſint A, C, E, &</s> <s xml:id="echoid-s3384" xml:space="preserve">c. </s> <s xml:id="echoid-s3385" xml:space="preserve">& </s> <s xml:id="echoid-s3386" xml:space="preserve">intra hyperbolem B, D, F &</s> <s xml:id="echoid-s3387" xml:space="preserve">c; </s> <s xml:id="echoid-s3388" xml:space="preserve">Dico A <lb/>+ E majorem eſſe quam 2 C. </s> <s xml:id="echoid-s3389" xml:space="preserve">ex prædictis manifeſtæ ſunt ſe-<lb/>quentes duæ Analogiæ, prima quoniam A, C, B, ſunt con-<lb/>tinue proportionales; </s> <s xml:id="echoid-s3390" xml:space="preserve">& </s> <s xml:id="echoid-s3391" xml:space="preserve">ſecunda, quoniam C, D, B, ſunt <lb/>harmonicè proportionales; </s> <s xml:id="echoid-s3392" xml:space="preserve">& </s> <s xml:id="echoid-s3393" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0154-02a" xlink:href="note-0154-02"/> proinde exceſſus A ſupra C, <lb/>hoc eſt A - C, eſt ad ex ceſſum <lb/>C ſupra D ſeu C - D; </s> <s xml:id="echoid-s3394" xml:space="preserve">In ratione <lb/>compoſita ex proportione A ad C & </s> <s xml:id="echoid-s3395" xml:space="preserve">ex proportione A + C <lb/>ad A hoc eſt in ratione A + C ad C, at A + C eſt ma-<lb/>jor quam C & </s> <s xml:id="echoid-s3396" xml:space="preserve">ideo exceſſus A ſupra C eſt major exceſſu C <lb/>ſupra D, eſt autem E major quam D; </s> <s xml:id="echoid-s3397" xml:space="preserve">& </s> <s xml:id="echoid-s3398" xml:space="preserve">proinde exceſſus A <lb/>ſupra C multo major eſt exceſſu C ſupra E; </s> <s xml:id="echoid-s3399" xml:space="preserve">manifeſtum eſt <lb/>igitur A + E majorem eſſe quam 2 C, quod demonſtrare <lb/>oportuit.</s> <s xml:id="echoid-s3400" xml:space="preserve"/> </p> <div xml:id="echoid-div178" type="float" level="2" n="1"> <note position="right" xlink:label="note-0154-01" xlink:href="note-0154-01a" xml:space="preserve"> <lb/>A # B <lb/>C # D <lb/>E # F <lb/></note> <note position="right" xlink:label="note-0154-02" xlink:href="note-0154-02a" xml:space="preserve"> <lb/>A - C:C - B::A:C <lb/>C - B:C - D::A + C:A <lb/></note> </div> <pb o="437" file="0155" n="164" rhead="ET HYPERBOLÆ QUADRATURA."/> </div> <div xml:id="echoid-div180" type="section" level="1" n="85"> <head xml:id="echoid-head121" xml:space="preserve">PROP. XVII. THEOREMA.</head> <p> <s xml:id="echoid-s3401" xml:space="preserve">IIsdem poſitis: </s> <s xml:id="echoid-s3402" xml:space="preserve">dico exceſſum A ſupra C majorem eſſe <lb/>quadruplo exceſſus C ſupra E. </s> <s xml:id="echoid-s3403" xml:space="preserve">ex prædictis manifeſtæ <lb/>ſunt ſequentes tres analogiæ; </s> <s xml:id="echoid-s3404" xml:space="preserve">prima, quoniam A, C, B, ſunt <lb/>continuè proportionales; </s> <s xml:id="echoid-s3405" xml:space="preserve">ſecunda, quoniam C, D, B, ſunt <lb/>harmonicè proportionales; </s> <s xml:id="echoid-s3406" xml:space="preserve">& </s> <s xml:id="echoid-s3407" xml:space="preserve">tertia, quoniam C, E, D, ſunt <lb/>continuè proportionales; </s> <s xml:id="echoid-s3408" xml:space="preserve">& </s> <s xml:id="echoid-s3409" xml:space="preserve">ideo exceſſus A ſupra C, hoc eſt <lb/>A - C, eſt ad exceſſum C ſupra E, hoc eſt C - E in ratio-<lb/>ne compoſita ex proportionibus <lb/> <anchor type="note" xlink:label="note-0155-01a" xlink:href="note-0155-01"/> A ad C, A + C ad A & </s> <s xml:id="echoid-s3410" xml:space="preserve">E + C ad C; <lb/></s> <s xml:id="echoid-s3411" xml:space="preserve">& </s> <s xml:id="echoid-s3412" xml:space="preserve">ideo A - C eſt ad C - E, <lb/>ut A C + E C + A E + CC ad <lb/>CC; </s> <s xml:id="echoid-s3413" xml:space="preserve">at B minor eſt quam E, <lb/>& </s> <s xml:id="echoid-s3414" xml:space="preserve">ideo AB, ſeu CC minor eſt quam A E; </s> <s xml:id="echoid-s3415" xml:space="preserve">& </s> <s xml:id="echoid-s3416" xml:space="preserve">igitur AE + <lb/>CC major eſt quam 2 CC. </s> <s xml:id="echoid-s3417" xml:space="preserve">atque A C + E C eſt ad 2 CC ut <lb/>A + E ad 2 C; </s> <s xml:id="echoid-s3418" xml:space="preserve">ſed A + E major eſt quam duo C, & </s> <s xml:id="echoid-s3419" xml:space="preserve">ideo <lb/>A C + E C major eſt quam 2 CC; </s> <s xml:id="echoid-s3420" xml:space="preserve">& </s> <s xml:id="echoid-s3421" xml:space="preserve">proinde A C + E C + A E <lb/>+ C C major eſt quam 4 CC; </s> <s xml:id="echoid-s3422" xml:space="preserve">& </s> <s xml:id="echoid-s3423" xml:space="preserve">igitur A - C major eſt qua-<lb/>druplo ipſius C - E, quod demonſtrare oportuit.</s> <s xml:id="echoid-s3424" xml:space="preserve"/> </p> <div xml:id="echoid-div180" type="float" level="2" n="1"> <note position="right" xlink:label="note-0155-01" xlink:href="note-0155-01a" xml:space="preserve"> <lb/>A - C:C: - B::A:C <lb/>C - B:C - D::A + :CA <lb/>C - D:C - E::E + C:C <lb/></note> </div> </div> <div xml:id="echoid-div182" type="section" level="1" n="86"> <head xml:id="echoid-head122" xml:space="preserve">PROP. XVIII. THEOREMA.</head> <p> <s xml:id="echoid-s3425" xml:space="preserve">SInt duæ quantitates inæquales; </s> <s xml:id="echoid-s3426" xml:space="preserve">A <lb/> <anchor type="note" xlink:label="note-0155-02a" xlink:href="note-0155-02"/> minor, B major, C media geome-<lb/>trica, D media arithmetica. </s> <s xml:id="echoid-s3427" xml:space="preserve">dico D <lb/>majorem eſſe quam C, quoniam B, C, <lb/>A, ſunt continuè proportionales; </s> <s xml:id="echoid-s3428" xml:space="preserve">erit divi-<lb/>dendo, permutando, & </s> <s xml:id="echoid-s3429" xml:space="preserve">componendo; </s> <s xml:id="echoid-s3430" xml:space="preserve">ut exceſſus B ſupra A <lb/>ad exceſſum C ſupra A, ita A + C ad A, atque A + C major eſt <lb/>duplo ipſius A; </s> <s xml:id="echoid-s3431" xml:space="preserve">& </s> <s xml:id="echoid-s3432" xml:space="preserve">proinde exceſſus B ſupra A major eſt duplo <lb/>exceſſus C ſupra; </s> <s xml:id="echoid-s3433" xml:space="preserve">ſed exceſſus B ſupra A duplus eſt exceſſus <lb/>D ſupra A, & </s> <s xml:id="echoid-s3434" xml:space="preserve">ideo exceſſus D ſupra A major eſt exceſſu C <pb o="438" file="0156" n="165" rhead="VERA CIRCULI"/> ſupra A; </s> <s xml:id="echoid-s3435" xml:space="preserve">eſt igitur D major quam C, quod demonſtrare <lb/>oportuit.</s> <s xml:id="echoid-s3436" xml:space="preserve"/> </p> <div xml:id="echoid-div182" type="float" level="2" n="1"> <note position="right" xlink:label="note-0155-02" xlink:href="note-0155-02a" xml:space="preserve"> <lb/># E <lb/>A # C # B <lb/># D <lb/></note> </div> </div> <div xml:id="echoid-div184" type="section" level="1" n="87"> <head xml:id="echoid-head123" xml:space="preserve">PROP. XIX. THEOREMA.</head> <p> <s xml:id="echoid-s3437" xml:space="preserve">IIsdem poſitis; </s> <s xml:id="echoid-s3438" xml:space="preserve">ſit inter A & </s> <s xml:id="echoid-s3439" xml:space="preserve">B media harmonica E. </s> <s xml:id="echoid-s3440" xml:space="preserve">dico <lb/>C majorem eſſe quam E. </s> <s xml:id="echoid-s3441" xml:space="preserve">ex hujus 13, D eſt ad C ut C ad <lb/>E, ſed D major eſt quam C; </s> <s xml:id="echoid-s3442" xml:space="preserve">& </s> <s xml:id="echoid-s3443" xml:space="preserve">ideo C major eſt quam E, <lb/>quod demonſtrare oportuit.</s> <s xml:id="echoid-s3444" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div185" type="section" level="1" n="88"> <head xml:id="echoid-head124" xml:space="preserve">CONSECTARIUM.</head> <p> <s xml:id="echoid-s3445" xml:space="preserve">Ex duabus præcedentibus manifeſtum eſt D majorem eſſe <lb/>quam E, hoc eſt mediam arithmeticam inter duas quantita-<lb/>tes inæquales majorem eſſe media harmonica inter eaſdem.</s> <s xml:id="echoid-s3446" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div186" type="section" level="1" n="89"> <head xml:id="echoid-head125" xml:space="preserve">PROP. XX. THEOREMA.</head> <p> <s xml:id="echoid-s3447" xml:space="preserve">SInt duo polygona complicata <lb/> <anchor type="note" xlink:label="note-0156-01a" xlink:href="note-0156-01"/> A, B, nempe A intra circuli <lb/>vel ellipſeos ſectorem, B extra. <lb/></s> <s xml:id="echoid-s3448" xml:space="preserve">continuetur ſeries convergens ho-<lb/>rum polygonorum complicato-<lb/>rum ſecundum methodum no-<lb/>ſtram ſubduplam deſcriptorum; </s> <s xml:id="echoid-s3449" xml:space="preserve"><lb/>ita ut polygona intra circulum ſint A, C, E, K, &</s> <s xml:id="echoid-s3450" xml:space="preserve">c, & </s> <s xml:id="echoid-s3451" xml:space="preserve">ex-<lb/>tra circulum B, D, F, L, &</s> <s xml:id="echoid-s3452" xml:space="preserve">c; </s> <s xml:id="echoid-s3453" xml:space="preserve">ſitque ſeriei convergentis ter-<lb/>minatio ſeu circuli vel ellipſeos ſector Z. </s> <s xml:id="echoid-s3454" xml:space="preserve">dico Z majorem <lb/>eſſe quam C una cum triente exceſſus C ſupra A. </s> <s xml:id="echoid-s3455" xml:space="preserve">ſit exceſſus <lb/>G ſupra C quarta pars exceſſus C ſupra A, & </s> <s xml:id="echoid-s3456" xml:space="preserve">exceſſus H <lb/>ſupra G quarta pars exceſſus G ſupra C; </s> <s xml:id="echoid-s3457" xml:space="preserve">continueturque <lb/>hæc ſeries in infinitum, ut ejus terminatio ſit X. </s> <s xml:id="echoid-s3458" xml:space="preserve">Exceſſus <lb/>C ſupra A minor eſt quadruplo exceſſus E ſupra C; </s> <s xml:id="echoid-s3459" xml:space="preserve">& </s> <s xml:id="echoid-s3460" xml:space="preserve">ideo <lb/>exceſſus E ſupra C major eſt exceſſu G ſupra C, eſt ergo <lb/>E major quam G. </s> <s xml:id="echoid-s3461" xml:space="preserve">deinde exceſſus E ſupra C minor eſt qua-<lb/>druplo exceſſus K ſupra E, & </s> <s xml:id="echoid-s3462" xml:space="preserve">ideo exceſſus G ſupra C mul-<lb/>to minor eſt quadruplo exceſſus K ſupra E, eſt igitur ex- <pb o="439" file="0157" n="166" rhead="ET HYPERBOLÆ QUADRATURA."/> ceſſus K ſupra E major exceſſus H ſupra G; </s> <s xml:id="echoid-s3463" xml:space="preserve">cumque E ma-<lb/>jor ſit quam G, manifeſtum eſt K majorem eſſe quam H: <lb/></s> <s xml:id="echoid-s3464" xml:space="preserve">eodem prorſus modo demonſtratur in omni ſerierum A, C, <lb/>E; </s> <s xml:id="echoid-s3465" xml:space="preserve">A, C, G, continuatione, terminum quemcunque ſeriei <lb/>A, C, E, majorem eſſe quam idem numero terminus ſeriei <lb/>A, C, G; </s> <s xml:id="echoid-s3466" xml:space="preserve">& </s> <s xml:id="echoid-s3467" xml:space="preserve">ideo terminatio ſeriei A, C, E, nempe Z ma-<lb/>jor erit terminatione ſeriei A, C, G, nempe X, at ex Archi-<lb/>medis quadratura parabloæ conſtat X æqualem eſſe ipſi C <lb/>una cum triente exceſſus C ſupra A, & </s> <s xml:id="echoid-s3468" xml:space="preserve">proinde Z eadem <lb/>major eſt, quod demonſtrare oportuit.</s> <s xml:id="echoid-s3469" xml:space="preserve"/> </p> <div xml:id="echoid-div186" type="float" level="2" n="1"> <note position="right" xlink:label="note-0156-01" xlink:href="note-0156-01a" xml:space="preserve"> <lb/>A B # A <lb/>C D # C <lb/>E F # G <lb/>K L # H <lb/>Z # X <lb/></note> </div> </div> <div xml:id="echoid-div188" type="section" level="1" n="90"> <head xml:id="echoid-head126" xml:space="preserve">PROP. XXI. THEOREMA.</head> <p> <s xml:id="echoid-s3470" xml:space="preserve">IIsdem poſitis quæ ſupra; </s> <s xml:id="echoid-s3471" xml:space="preserve">dico Z <lb/> <anchor type="note" xlink:label="note-0157-01a" xlink:href="note-0157-01"/> ſeu ſectorem circuli vel ellipſeos <lb/>minorem eſſe quam major duarum <lb/>mediarum continuè proportionalium <lb/>arithmeticè inter A & </s> <s xml:id="echoid-s3472" xml:space="preserve">B. </s> <s xml:id="echoid-s3473" xml:space="preserve">inter A & </s> <s xml:id="echoid-s3474" xml:space="preserve"><lb/>B ſit media Arithmetica G, & </s> <s xml:id="echoid-s3475" xml:space="preserve">inter <lb/>G & </s> <s xml:id="echoid-s3476" xml:space="preserve">B ſit media Arithmetica H; </s> <s xml:id="echoid-s3477" xml:space="preserve">item <lb/>inter G & </s> <s xml:id="echoid-s3478" xml:space="preserve">H ſit media Arithmetica M, & </s> <s xml:id="echoid-s3479" xml:space="preserve">inter M & </s> <s xml:id="echoid-s3480" xml:space="preserve">H ſed me-<lb/>dia Arithmetica N; </s> <s xml:id="echoid-s3481" xml:space="preserve">continueturque hæc ſeries convergens A B, <lb/>G H, M N, O P, in infinitum, ut fiat ejus terminatio X. </s> <s xml:id="echoid-s3482" xml:space="preserve">ſatis <lb/>pater ex prædictis G majorem eſſe quam C, atque H media <lb/>Arithmetica inter G & </s> <s xml:id="echoid-s3483" xml:space="preserve">B major eſt media harmonica inter eas-<lb/>dem G, B; </s> <s xml:id="echoid-s3484" xml:space="preserve">media autem harmonica inter G & </s> <s xml:id="echoid-s3485" xml:space="preserve">B major eſt quam <lb/>D media harmonica inter C & </s> <s xml:id="echoid-s3486" xml:space="preserve">B, quoniam G major eſt quam <lb/>C; </s> <s xml:id="echoid-s3487" xml:space="preserve">& </s> <s xml:id="echoid-s3488" xml:space="preserve">ideo media Arithmetica inter G & </s> <s xml:id="echoid-s3489" xml:space="preserve">B, hoc eſt H, major <lb/>eſt quam D media harmonica inter C & </s> <s xml:id="echoid-s3490" xml:space="preserve">B. </s> <s xml:id="echoid-s3491" xml:space="preserve">eodem modo M <lb/>media arithmetica inter G & </s> <s xml:id="echoid-s3492" xml:space="preserve">H major eſt media geometrica <lb/>inter easdem G & </s> <s xml:id="echoid-s3493" xml:space="preserve">H: </s> <s xml:id="echoid-s3494" xml:space="preserve">& </s> <s xml:id="echoid-s3495" xml:space="preserve">quoniam G eſt major quam C, & </s> <s xml:id="echoid-s3496" xml:space="preserve"><lb/>H quam D; </s> <s xml:id="echoid-s3497" xml:space="preserve">media geometrica inter G & </s> <s xml:id="echoid-s3498" xml:space="preserve">H major eſt quam <lb/>E media geometrica inter C & </s> <s xml:id="echoid-s3499" xml:space="preserve">D; </s> <s xml:id="echoid-s3500" xml:space="preserve">& </s> <s xml:id="echoid-s3501" xml:space="preserve">proinde M major <lb/>eſt quam E. </s> <s xml:id="echoid-s3502" xml:space="preserve">deinde N media arithmetica inter M & </s> <s xml:id="echoid-s3503" xml:space="preserve">H ma-<lb/>jor eſt media harmonica inter eaſdem, & </s> <s xml:id="echoid-s3504" xml:space="preserve">quoniam H major <lb/>eſt quam D & </s> <s xml:id="echoid-s3505" xml:space="preserve">M quam E, media harmonica inter M & </s> <s xml:id="echoid-s3506" xml:space="preserve">H <lb/>major eſt quam F media harmonica inter E & </s> <s xml:id="echoid-s3507" xml:space="preserve">D; </s> <s xml:id="echoid-s3508" xml:space="preserve">& </s> <s xml:id="echoid-s3509" xml:space="preserve">ideo N <pb o="440" file="0158" n="167" rhead="VERA CIRCULI"/> eadem F major eſt. </s> <s xml:id="echoid-s3510" xml:space="preserve">eadem modo utramque ſeriem in infini-<lb/>tum continuando, ſemper demonſtratur terminum quemlibet <lb/>ſeriei A B, C D, minorem eſſe quam idem numero terminus <lb/>ſeriei. </s> <s xml:id="echoid-s3511" xml:space="preserve">A B, G H; </s> <s xml:id="echoid-s3512" xml:space="preserve">& </s> <s xml:id="echoid-s3513" xml:space="preserve">igitur terminatio ſeriei A B, C D, nem-<lb/>pe Z minor erit terminatione ſeriei A B, G H, nempe X; <lb/></s> <s xml:id="echoid-s3514" xml:space="preserve">atque ex hujus 7, terminatio ſeriei A B, G H, ſeu X æqua-<lb/>lis eſt majori duarum mediarum arithmeticè continuè propor-<lb/>tionalium inter A & </s> <s xml:id="echoid-s3515" xml:space="preserve">B, & </s> <s xml:id="echoid-s3516" xml:space="preserve">ideo Z eadem minor eſt, quod <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s3517" xml:space="preserve"/> </p> <div xml:id="echoid-div188" type="float" level="2" n="1"> <note position="right" xlink:label="note-0157-01" xlink:href="note-0157-01a" xml:space="preserve"> <lb/>A B # A B <lb/>C D # G H <lb/>E F # M N <lb/>K L # O P <lb/>Z # X <lb/></note> </div> </div> <div xml:id="echoid-div190" type="section" level="1" n="91"> <head xml:id="echoid-head127" xml:space="preserve">PROP. XXII. THEOREMA.</head> <p> <s xml:id="echoid-s3518" xml:space="preserve">IIsdem poſitis quæ ſupra; </s> <s xml:id="echoid-s3519" xml:space="preserve">dico Z <lb/> <anchor type="note" xlink:label="note-0158-01a" xlink:href="note-0158-01"/> ſeu ſectorem circuli vel ellipſeos <lb/>minorem eſſe quam major duarum <lb/>mediarum geometricè continuè pro-<lb/>portionalium inter A & </s> <s xml:id="echoid-s3520" xml:space="preserve">B. </s> <s xml:id="echoid-s3521" xml:space="preserve">inter A <lb/>& </s> <s xml:id="echoid-s3522" xml:space="preserve">B ſit media geometrica G, & </s> <s xml:id="echoid-s3523" xml:space="preserve">inter <lb/>G & </s> <s xml:id="echoid-s3524" xml:space="preserve">B ſit media geometrica H; </s> <s xml:id="echoid-s3525" xml:space="preserve">Item <lb/>inter G & </s> <s xml:id="echoid-s3526" xml:space="preserve">H media Geometrica M, & </s> <s xml:id="echoid-s3527" xml:space="preserve">inter M & </s> <s xml:id="echoid-s3528" xml:space="preserve">H media Geo-<lb/>metrica N; </s> <s xml:id="echoid-s3529" xml:space="preserve">continuetúrque hæc ſeries convergens A B, G H, <lb/>M N, O P, &</s> <s xml:id="echoid-s3530" xml:space="preserve">c, in infinitum, ut fiat ejus terminatio X. </s> <s xml:id="echoid-s3531" xml:space="preserve">ſatis <lb/>patet ex prædictis C & </s> <s xml:id="echoid-s3532" xml:space="preserve">G eſſe inter ſe æquales, item H majorem <lb/>eſſe quam D; </s> <s xml:id="echoid-s3533" xml:space="preserve">atque ob hanc rationem M media Geometrica in-<lb/>ter G & </s> <s xml:id="echoid-s3534" xml:space="preserve">H major eſt quam E media geometrica inter G & </s> <s xml:id="echoid-s3535" xml:space="preserve">D. <lb/></s> <s xml:id="echoid-s3536" xml:space="preserve">deinde N media Geometrica inter M & </s> <s xml:id="echoid-s3537" xml:space="preserve">H major eſt media har-<lb/>monica inter easdem; </s> <s xml:id="echoid-s3538" xml:space="preserve">& </s> <s xml:id="echoid-s3539" xml:space="preserve">quoniam M major eſt quam E & </s> <s xml:id="echoid-s3540" xml:space="preserve">H <lb/>major quam D, erit media harmonica inter M & </s> <s xml:id="echoid-s3541" xml:space="preserve">H major quam <lb/>F media harmonica inter E & </s> <s xml:id="echoid-s3542" xml:space="preserve">D; </s> <s xml:id="echoid-s3543" xml:space="preserve">& </s> <s xml:id="echoid-s3544" xml:space="preserve">ideo N media Geometrica <lb/>inter M & </s> <s xml:id="echoid-s3545" xml:space="preserve">H major erit quam F. </s> <s xml:id="echoid-s3546" xml:space="preserve">eadem methodo utramque <lb/>ſeriem in infinitum continuando ſemper demonſtratur termi-<lb/>num quemlibet ſeriei A B, C D, minorem eſſe quam idem <lb/>numero terminus ſeriei A B, G H; </s> <s xml:id="echoid-s3547" xml:space="preserve">& </s> <s xml:id="echoid-s3548" xml:space="preserve">igitur terminatio ſeriei <lb/>A B, C D, nempe Z minor erit terminatione ſeriei A B, <lb/>G H, nempè X; </s> <s xml:id="echoid-s3549" xml:space="preserve">atque ex hujus 9 terminatio ſeriei A B, <lb/>G H, ſeu X, æqualis eſt majori duarum mediarum Geometri-<lb/>cè continuè proportionalium inter A & </s> <s xml:id="echoid-s3550" xml:space="preserve">B; </s> <s xml:id="echoid-s3551" xml:space="preserve">& </s> <s xml:id="echoid-s3552" xml:space="preserve">ideo Z eadem <lb/>minor eſt, quod demonſtrare oportuit.</s> <s xml:id="echoid-s3553" xml:space="preserve"/> </p> <div xml:id="echoid-div190" type="float" level="2" n="1"> <note position="right" xlink:label="note-0158-01" xlink:href="note-0158-01a" xml:space="preserve"> <lb/>A B # A B <lb/>C D # G H <lb/>E F # M N <lb/>K L # O P <lb/>Z # X <lb/></note> </div> <pb o="441" file="0159" n="168" rhead="ET HYPERBOLÆ QUADRATURA."/> </div> <div xml:id="echoid-div192" type="section" level="1" n="92"> <head xml:id="echoid-head128" xml:space="preserve">SCHOLIUM.</head> <p> <s xml:id="echoid-s3554" xml:space="preserve">NOn opus eſt ut hic demonſtrem majorem duarum me-<lb/>diarum arithmeticè continuè proportionalium inter duas <lb/>inæquales quantitates majorem eſſe quam major duarum me-<lb/>diarum Geometricè continuè proportionalium inter eaſdem, <lb/>& </s> <s xml:id="echoid-s3555" xml:space="preserve">igitur hujus propoſitionis approximationem præcedentis <lb/>eſſe exactiorem, quod etſi fiat; </s> <s xml:id="echoid-s3556" xml:space="preserve">præcedente tamen ob facilita-<lb/>tem potius utimur.</s> <s xml:id="echoid-s3557" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div193" type="section" level="1" n="93"> <head xml:id="echoid-head129" xml:space="preserve">PROP. XXIII. THEOREMA.</head> <p> <s xml:id="echoid-s3558" xml:space="preserve">Sint duo polygona complicata <lb/> <anchor type="note" xlink:label="note-0159-01a" xlink:href="note-0159-01"/> A, B, nempè A extra hyperbolæ <lb/>ſectorem, B intra. </s> <s xml:id="echoid-s3559" xml:space="preserve">continuetur ſe-<lb/>ries convergens horum polygono-<lb/>rum complicatorum ſecundum me-<lb/>thodum noſtram ſubduplam de-<lb/>ſcriptorum, ita ut polygona extra <lb/>hyperbolam ſint A, C, E, K, &</s> <s xml:id="echoid-s3560" xml:space="preserve">c, & </s> <s xml:id="echoid-s3561" xml:space="preserve">intra hyperbolam B, <lb/>D, F, L, &</s> <s xml:id="echoid-s3562" xml:space="preserve">c; </s> <s xml:id="echoid-s3563" xml:space="preserve">Sitque ſeriei convergentis terminatio ſeu hy-<lb/>perbolæ ſector Z. </s> <s xml:id="echoid-s3564" xml:space="preserve">dico Z majorem eſſe quam C dempto tri-<lb/>ente exceſſus A ſupra C. </s> <s xml:id="echoid-s3565" xml:space="preserve">ſit exceſſus C ſupra G quarta pars <lb/>exceſſus A ſupra C, & </s> <s xml:id="echoid-s3566" xml:space="preserve">exceſſus G ſupra H quarta pars ex-<lb/>ceſſus C ſupra G, continueturque hæc ſeries in infinitum ut <lb/>ejus terminatio ſit X. </s> <s xml:id="echoid-s3567" xml:space="preserve">exceſſus A ſupra C major eſt quadru-<lb/>plo exceſſus C ſupra E, & </s> <s xml:id="echoid-s3568" xml:space="preserve">ideo exceſſus C ſupra E minor <lb/>eſt exceſſus C ſupra G, eſt ergo E major quam G. </s> <s xml:id="echoid-s3569" xml:space="preserve">Deinde <lb/>exceſſus C ſupra E major eſt quadruplo exceſſus E ſupra K, <lb/>& </s> <s xml:id="echoid-s3570" xml:space="preserve">ideo exceſſus C ſupra G multò major eſt quadruplo ex-<lb/>ceſſu E ſupra K, & </s> <s xml:id="echoid-s3571" xml:space="preserve">igitur exceſſus G ſupra H major eſt <lb/>exceſſu E ſupra K; </s> <s xml:id="echoid-s3572" xml:space="preserve">cumque E major ſit quam G, ma-<lb/>nifeſtum eſt K etiam majorem eſſe quam H: </s> <s xml:id="echoid-s3573" xml:space="preserve">eodem pror-<lb/>ſus modo demonſtratur in omni ſerierum A, C, E, K; </s> <s xml:id="echoid-s3574" xml:space="preserve">A, <lb/>C, G, H, continuatione, terminum quemcumque ſeriei A, <lb/>C, E, majorem eſſe quam idem numero terminus ſeriei A, <pb o="442" file="0160" n="169" rhead="VERA CIRCULI"/> C, G; </s> <s xml:id="echoid-s3575" xml:space="preserve">& </s> <s xml:id="echoid-s3576" xml:space="preserve">ideo terminatio ſeriei A, C, E, nempe Z, major <lb/>erit terminatione ſeriei A, C, G, nempè X; </s> <s xml:id="echoid-s3577" xml:space="preserve">at ex Archime-<lb/>dis quadratura parabolæ conſtat X æqualem eſſe ipſi C dem-<lb/>pto triente exceſſus A ſupra C, & </s> <s xml:id="echoid-s3578" xml:space="preserve">proinde Z eadem major <lb/>eſt, quod demonſtrare oportuit.</s> <s xml:id="echoid-s3579" xml:space="preserve"/> </p> <div xml:id="echoid-div193" type="float" level="2" n="1"> <note position="right" xlink:label="note-0159-01" xlink:href="note-0159-01a" xml:space="preserve"> <lb/>A B # A <lb/>C D # C <lb/>E F # G <lb/>K L # H <lb/>Z # X <lb/></note> </div> </div> <div xml:id="echoid-div195" type="section" level="1" n="94"> <head xml:id="echoid-head130" xml:space="preserve">PROP. XXIV. THEOREMA.</head> <p> <s xml:id="echoid-s3580" xml:space="preserve">IIsdem poſitis; </s> <s xml:id="echoid-s3581" xml:space="preserve">dico Z ſeu ſe-<lb/> <anchor type="note" xlink:label="note-0160-01a" xlink:href="note-0160-01"/> ctorem hyperbolæ minorem eſ-<lb/>ſe quam minor duarum mediarum <lb/>arithmeticè continuè proportio-<lb/>nalium inter A & </s> <s xml:id="echoid-s3582" xml:space="preserve">B. </s> <s xml:id="echoid-s3583" xml:space="preserve">Inter A & </s> <s xml:id="echoid-s3584" xml:space="preserve"><lb/>B ſit media arithmetica G, & </s> <s xml:id="echoid-s3585" xml:space="preserve">in-<lb/>ter G & </s> <s xml:id="echoid-s3586" xml:space="preserve">B ſit media Arithmetica <lb/>H, Item inter G & </s> <s xml:id="echoid-s3587" xml:space="preserve">H ſit media Arithmetica M, & </s> <s xml:id="echoid-s3588" xml:space="preserve">inter M <lb/>& </s> <s xml:id="echoid-s3589" xml:space="preserve">H ſit media Arithmetica N: </s> <s xml:id="echoid-s3590" xml:space="preserve">continueturque hæc ſeries con-<lb/>vergens A B, G H, M N, O P, in infinitum, ut fiat ejus termi-<lb/>natio X. </s> <s xml:id="echoid-s3591" xml:space="preserve">ſatis patet ex prædictis G majorem eſſe quam C; <lb/></s> <s xml:id="echoid-s3592" xml:space="preserve">atque H media arithmetica inter G & </s> <s xml:id="echoid-s3593" xml:space="preserve">B major eſt media har-<lb/>monica inter easdem G & </s> <s xml:id="echoid-s3594" xml:space="preserve">B; </s> <s xml:id="echoid-s3595" xml:space="preserve">media autem harmonica inter <lb/>G & </s> <s xml:id="echoid-s3596" xml:space="preserve">B; </s> <s xml:id="echoid-s3597" xml:space="preserve">major eſt media harmonica inter C & </s> <s xml:id="echoid-s3598" xml:space="preserve">B, nempe D, quo-<lb/>niam G major eſt quam C; </s> <s xml:id="echoid-s3599" xml:space="preserve">& </s> <s xml:id="echoid-s3600" xml:space="preserve">ideo media Arithmetica inter G <lb/>& </s> <s xml:id="echoid-s3601" xml:space="preserve">B nempe H major eſt quam D media harmonica inter C & </s> <s xml:id="echoid-s3602" xml:space="preserve">B <lb/>eodem modo M media Arithmetica inter G & </s> <s xml:id="echoid-s3603" xml:space="preserve">H major eſt me-<lb/>dia geometrica inter eaſdem G & </s> <s xml:id="echoid-s3604" xml:space="preserve">H; </s> <s xml:id="echoid-s3605" xml:space="preserve">& </s> <s xml:id="echoid-s3606" xml:space="preserve">quoniam G eſt ma-<lb/>jor quam C & </s> <s xml:id="echoid-s3607" xml:space="preserve">H quam D, media geometrica inter G & </s> <s xml:id="echoid-s3608" xml:space="preserve">H <lb/>major eſt quam E media geometrica inter C & </s> <s xml:id="echoid-s3609" xml:space="preserve">D; </s> <s xml:id="echoid-s3610" xml:space="preserve">& </s> <s xml:id="echoid-s3611" xml:space="preserve">proin-<lb/>de M major eſt quam E. </s> <s xml:id="echoid-s3612" xml:space="preserve">Deinde N media Arithmetica in-<lb/>ter M & </s> <s xml:id="echoid-s3613" xml:space="preserve">H major eſt media harmonica inter easdem; </s> <s xml:id="echoid-s3614" xml:space="preserve">& </s> <s xml:id="echoid-s3615" xml:space="preserve">quo-<lb/>niam H major eſt quam D & </s> <s xml:id="echoid-s3616" xml:space="preserve">M quam E, media harmonica <lb/>inter M & </s> <s xml:id="echoid-s3617" xml:space="preserve">H major eſt quam F media harmonica inter E & </s> <s xml:id="echoid-s3618" xml:space="preserve"><lb/>D; </s> <s xml:id="echoid-s3619" xml:space="preserve">& </s> <s xml:id="echoid-s3620" xml:space="preserve">ideo N eadem F major eſt. </s> <s xml:id="echoid-s3621" xml:space="preserve">eodem modo utramque <lb/>ſeriem in infinitum continuando, ſemper demonſtratur ter-<lb/>minum quemlibet ſeriei A B, C D, minorem eſſe quam idem <lb/>numero terminum ſeriei A B, G H; </s> <s xml:id="echoid-s3622" xml:space="preserve">& </s> <s xml:id="echoid-s3623" xml:space="preserve">igitur terminatio ſe-<lb/>riei A B, C D, nempe Z, minor erit terminatione ſeriei A B, <pb o="443" file="0161" n="170" rhead="ET HYPERBOLÆ QUADRATURA."/> G H, nempe X; </s> <s xml:id="echoid-s3624" xml:space="preserve">atque ex hujus 7 terminatio ſeriei A B, <lb/>G H, nempe X, æqualis eſt minori duarum mediarum arith-<lb/>meticè continuè proportionalium inter A & </s> <s xml:id="echoid-s3625" xml:space="preserve">B, & </s> <s xml:id="echoid-s3626" xml:space="preserve">ideo Z ea-<lb/>dem minor eſt, quod demonſtrare oportuit.</s> <s xml:id="echoid-s3627" xml:space="preserve"/> </p> <div xml:id="echoid-div195" type="float" level="2" n="1"> <note position="right" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve"> <lb/>A B # A B <lb/>C D # G H <lb/>E F # M N <lb/>K L # O P <lb/>Z # X <lb/></note> </div> </div> <div xml:id="echoid-div197" type="section" level="1" n="95"> <head xml:id="echoid-head131" xml:space="preserve">PROP. XXV. THEOREMA.</head> <p> <s xml:id="echoid-s3628" xml:space="preserve">Iisdem poſitis; </s> <s xml:id="echoid-s3629" xml:space="preserve">dico Z ſeu ſectorem <lb/> <anchor type="note" xlink:label="note-0161-01a" xlink:href="note-0161-01"/> hyperbolæ minorem eſſe quam mi-<lb/>nor duarum mediarum geometricè con-<lb/>tinuè proportionalium inter A & </s> <s xml:id="echoid-s3630" xml:space="preserve">B. <lb/></s> <s xml:id="echoid-s3631" xml:space="preserve">Inter A & </s> <s xml:id="echoid-s3632" xml:space="preserve">B ſit media geometrica G, <lb/>& </s> <s xml:id="echoid-s3633" xml:space="preserve">inter G & </s> <s xml:id="echoid-s3634" xml:space="preserve">B media geometrica H; </s> <s xml:id="echoid-s3635" xml:space="preserve"><lb/>Item inter G & </s> <s xml:id="echoid-s3636" xml:space="preserve">H media geometrica M, & </s> <s xml:id="echoid-s3637" xml:space="preserve">inter M & </s> <s xml:id="echoid-s3638" xml:space="preserve">H media <lb/>geometriea N; </s> <s xml:id="echoid-s3639" xml:space="preserve">continueturque hæc ſeries convergens AB, GH, <lb/>MN, OP, &</s> <s xml:id="echoid-s3640" xml:space="preserve">c. </s> <s xml:id="echoid-s3641" xml:space="preserve">in infinitum ut fiat ejus terminatio X. </s> <s xml:id="echoid-s3642" xml:space="preserve">ſatis patet <lb/>ex prædictis C & </s> <s xml:id="echoid-s3643" xml:space="preserve">G eſſe inter ſe æquales, & </s> <s xml:id="echoid-s3644" xml:space="preserve">H majorem eſſe <lb/>quam D; </s> <s xml:id="echoid-s3645" xml:space="preserve">atque ob hanc rationem M media geometrica inter G <lb/>& </s> <s xml:id="echoid-s3646" xml:space="preserve">H major eſt quam E media geometrica inter C & </s> <s xml:id="echoid-s3647" xml:space="preserve">D. </s> <s xml:id="echoid-s3648" xml:space="preserve">Deinde <lb/>N media geometrica inter M & </s> <s xml:id="echoid-s3649" xml:space="preserve">H major eſt media harmonica <lb/>inter eaſdem; </s> <s xml:id="echoid-s3650" xml:space="preserve">& </s> <s xml:id="echoid-s3651" xml:space="preserve">quoniam M major eſt quam E & </s> <s xml:id="echoid-s3652" xml:space="preserve">H quam D, erit <lb/>media harmonica inter M & </s> <s xml:id="echoid-s3653" xml:space="preserve">H major quam F media harmo-<lb/>nica inter E & </s> <s xml:id="echoid-s3654" xml:space="preserve">D; </s> <s xml:id="echoid-s3655" xml:space="preserve">proinde N media geometrica inter M & </s> <s xml:id="echoid-s3656" xml:space="preserve">H <lb/>major eritquam F. </s> <s xml:id="echoid-s3657" xml:space="preserve">eadem methodo utramque ſeriem in in-<lb/>finitum continuando, ſemper demonſtratur terminum quem-<lb/>libet ſeriei A B, C D, minorem eſſe quam idem numero ter-<lb/>minus ſeriei A B, G H; </s> <s xml:id="echoid-s3658" xml:space="preserve">& </s> <s xml:id="echoid-s3659" xml:space="preserve">igitur terminatio ſeriei A B, C D, <lb/>nempe Z minor erit quam terminatio ſeriei A B, G H, nem-<lb/>pe X; </s> <s xml:id="echoid-s3660" xml:space="preserve">atque ex hujus 9 terminatio ſeriei A B, G H, ſeu X, æqua-<lb/>lis eſt minori duarum mediarum geometricè continuè propor-<lb/>tionalium inter A & </s> <s xml:id="echoid-s3661" xml:space="preserve">B; </s> <s xml:id="echoid-s3662" xml:space="preserve">& </s> <s xml:id="echoid-s3663" xml:space="preserve">ideo Z eadem minor eſt, quod <lb/>demonſtrare oportuit.</s> <s xml:id="echoid-s3664" xml:space="preserve"/> </p> <div xml:id="echoid-div197" type="float" level="2" n="1"> <note position="right" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve"> <lb/>A B # A B <lb/>C D # G H <lb/>E F # M N <lb/>K L # O P <lb/>Z # X <lb/></note> </div> <p> <s xml:id="echoid-s3665" xml:space="preserve">Ex dictis manifeſtum eſt hanc approximationem exactio-<lb/>rem eſſe illa, in antecedenti propoſitione, demonſtrata, et-<lb/>iamſi hæc ſit paulò laborioſior. </s> <s xml:id="echoid-s3666" xml:space="preserve">ſed non diſſimulandum <lb/>eſt duas poſſe eſſe ſeries æquales terminationes habentes, ita <pb o="444" file="0162" n="171" rhead="VERA CIRCULI"/> ut ſemper quilibet terminus unius ſeriei ſit major quam idem <lb/>numero terminus alterius ſeriei; </s> <s xml:id="echoid-s3667" xml:space="preserve">ſed in talibus ſeriebus quò <lb/>longius producuntur, eò minor eſt eorundem numero termi-<lb/>norum differentia: </s> <s xml:id="echoid-s3668" xml:space="preserve">ſed è contra noſtræ ſeries quò longius <lb/>producuntur, eò magis differunt iidem numero termini, ſicut <lb/>facillimè demonſtrari poteſt.</s> <s xml:id="echoid-s3669" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3670" xml:space="preserve">Experientia obſervo differentiam inter ſecundam duarum <lb/>mediarum arithmetice proportionalium & </s> <s xml:id="echoid-s3671" xml:space="preserve">ſecundam duarum <lb/>mediarum geometricè proportionalium ſemper eſſe multò <lb/>majorem differentia inter ſecundam duarum mediarum geo-<lb/>metricè proportionalium & </s> <s xml:id="echoid-s3672" xml:space="preserve">ſectorem circuli, ellipſeos vel <lb/>hyperbolæ; </s> <s xml:id="echoid-s3673" xml:space="preserve">quod notatu dignum exiſtimo, hinc enim col-<lb/>ligitur ſectorem differre vix ultra unitatem à ſecunda duarum <lb/>mediarum arithmeticè continuè proportionalium, quando <lb/>medium arithmeticum non excedit medium geometricum ul-<lb/>tra unitatem, quod ſummopere notandum, nam ex hoc evi-<lb/>dens eſt approximationem audacter eſſe adhibendam, quan-<lb/>do ita continuatur ſeries ut medietas prima notarum ſit <lb/>eadem in utroque termino convergente, quod experientia <lb/>etiam evincit; </s> <s xml:id="echoid-s3674" xml:space="preserve">nunquam enim in hoc caſu differt ſector <lb/>unitate à ſecunda duarum mediarum arithmeticè continuè <lb/>proportionalium.</s> <s xml:id="echoid-s3675" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3676" xml:space="preserve">Eſt etiam alia approximatio omnium breviſſima & </s> <s xml:id="echoid-s3677" xml:space="preserve">maximè <lb/>admiranda, etiamſi mihi non contingat illam demonſtratio-<lb/>ne geometrica munire; </s> <s xml:id="echoid-s3678" xml:space="preserve">nempe ſi primus notarum triens in <lb/>utroque termino convergente ſit eadem, ſector circuli, el-<lb/>lipſeos vel hyperbolæ ſemper differt infra unitatem à maxi-<lb/>mo quatuor arithmeticè continuè proportionalium inter ter-<lb/>minos noſtræ approximationis.</s> <s xml:id="echoid-s3679" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div199" type="section" level="1" n="96"> <head xml:id="echoid-head132" xml:space="preserve">PROP. XXVI. THEOREMA.</head> <p> <s xml:id="echoid-s3680" xml:space="preserve">Sit hyperbola quæcunque C F N cujus centrum A, aſym-<lb/> <anchor type="note" xlink:label="note-0162-01a" xlink:href="note-0162-01"/> ptota A B, A O; </s> <s xml:id="echoid-s3681" xml:space="preserve">ſitque ejus ſector A F G L cum triangulo <lb/>circum ſcripto A F L: </s> <s xml:id="echoid-s3682" xml:space="preserve">aſymptotorum uni A B parallellæ du-<lb/>cantur rectæ F D, I M; </s> <s xml:id="echoid-s3683" xml:space="preserve">& </s> <s xml:id="echoid-s3684" xml:space="preserve">compleantur parallelogramma <pb o="445" file="0163" n="172" rhead="ET HYPERBOLÆ QUADRATURA."/> F D M K, P L M D. </s> <s xml:id="echoid-s3685" xml:space="preserve">dico triangulum A F L eſſe medium arith-<lb/>meticum inter parallelogramma F D M K, P L M D. </s> <s xml:id="echoid-s3686" xml:space="preserve">Grego-<lb/>rius à S. </s> <s xml:id="echoid-s3687" xml:space="preserve">Vincentio in Lib. </s> <s xml:id="echoid-s3688" xml:space="preserve">de Hyperbola demonſtrat triangu-<lb/>lum A F L eſſe æquale trapezio D F L M, ſed manifeſtum eſt <lb/>trapezium D F L M eſſe medium arithmeticum inter paral-<lb/>lelogramma F D M K, P L M D; </s> <s xml:id="echoid-s3689" xml:space="preserve">& </s> <s xml:id="echoid-s3690" xml:space="preserve">ideo patet propo-<lb/>ſitum.</s> <s xml:id="echoid-s3691" xml:space="preserve"/> </p> <div xml:id="echoid-div199" type="float" level="2" n="1"> <note position="left" xlink:label="note-0162-01" xlink:href="note-0162-01a" xml:space="preserve">TAB. XLIII. <lb/>fig. 4.</note> </div> </div> <div xml:id="echoid-div201" type="section" level="1" n="97"> <head xml:id="echoid-head133" xml:space="preserve">PROP. XXVII. THEOREMA.</head> <p> <s xml:id="echoid-s3692" xml:space="preserve">Iisdem poſitis: </s> <s xml:id="echoid-s3693" xml:space="preserve">ducatur A I rectam F L bifariam dividens in <lb/> <anchor type="note" xlink:label="note-0163-01a" xlink:href="note-0163-01"/> I & </s> <s xml:id="echoid-s3694" xml:space="preserve">hyperbolam interſecans in puncto G, fiatque trape-<lb/>zium ſectori circumſcriptum A F G L, quod dico eſſe me-<lb/>dium geometricum inter parallellogramma F D M K, P L M D. <lb/></s> <s xml:id="echoid-s3695" xml:space="preserve">ex demonſtratis Gregorii à S. </s> <s xml:id="echoid-s3696" xml:space="preserve">Vincentio evidens eſt trape-<lb/>zium A F G L æquale eſſe rectilineo D F G L M. </s> <s xml:id="echoid-s3697" xml:space="preserve">& </s> <s xml:id="echoid-s3698" xml:space="preserve">quoniam <lb/>A G I recta ſecat rectam F L bifariam in I, ex ejuſdem Gre-<lb/>gorii à S. </s> <s xml:id="echoid-s3699" xml:space="preserve">Vincentio Lib. </s> <s xml:id="echoid-s3700" xml:space="preserve">de hyperbola, manifeſtum eſt rectas <lb/>L M, G H, FD, eſſe continuè proportionales in eadem ratione <lb/>cum tribus continuè proportionalibus A D, A H, A M. </s> <s xml:id="echoid-s3701" xml:space="preserve">aſym-<lb/>ptoto A O per punctum G ducatur parallela recta R G S <lb/>rectis F D, M K, occurrens in punctis R, S. </s> <s xml:id="echoid-s3702" xml:space="preserve">quoniam rectæ <lb/>F D, G H, L M, ſunt continuè proportionales, erit dividen-<lb/>do & </s> <s xml:id="echoid-s3703" xml:space="preserve">permutando F R ad S L ut G H ad L M: </s> <s xml:id="echoid-s3704" xml:space="preserve">& </s> <s xml:id="echoid-s3705" xml:space="preserve">quoniam <lb/>rectæ M A, H A, D A, ſunt continuè proportionales, erit <lb/>etiam dividendo & </s> <s xml:id="echoid-s3706" xml:space="preserve">permutando M H ad H D hoc eſt S G ad <lb/>G R ut H A ad D A, vel ut G H ad L M; </s> <s xml:id="echoid-s3707" xml:space="preserve">& </s> <s xml:id="echoid-s3708" xml:space="preserve">proinde F R <lb/>eſt ad S L ut S G ad G R, cumque anguli F R G, G S L, ſint <lb/>æquales ob parallelas F R, S L, erunt triangula F R G, G L S, <lb/>æqualia; </s> <s xml:id="echoid-s3709" xml:space="preserve">& </s> <s xml:id="echoid-s3710" xml:space="preserve">proinde parallelogrammum R D M S æquale eſt <lb/>rectilineo D F G L M ſeu trapezio A F G L; </s> <s xml:id="echoid-s3711" xml:space="preserve">ſed parallelo-<lb/>grammum R D M S eſt medium geometricum inter parale-<lb/>logramma P D M L, F D M K, quoniam eandem habentia alti-<lb/>tudinem eorum baſes nempe L M, S M, K M, ſunt continuè <lb/>proportionales; </s> <s xml:id="echoid-s3712" xml:space="preserve">& </s> <s xml:id="echoid-s3713" xml:space="preserve">ideo trapezium A F G L eſt medium geo- <pb o="446" file="0164" n="173" rhead="VERA CIRCULI"/> metricum inter parallelogramma P D M L, F D M K, quod de-<lb/>monſtrare oportuit.</s> <s xml:id="echoid-s3714" xml:space="preserve"/> </p> <div xml:id="echoid-div201" type="float" level="2" n="1"> <note position="right" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve">TAB. <lb/>XLVIII. <lb/>fig. 4.</note> </div> </div> <div xml:id="echoid-div203" type="section" level="1" n="98"> <head xml:id="echoid-head134" xml:space="preserve">PROP. XXVIII. THEOREMA.</head> <p> <s xml:id="echoid-s3715" xml:space="preserve">Iisdem poſitis ducantur rectæ F E, L E, hyperbolam tan-<lb/> <anchor type="note" xlink:label="note-0164-01a" xlink:href="note-0164-01"/> gentes in punctis F, L, ut compleatur trapezium A F E L, <lb/>quod dico eſſe medium harmonicum inter parallelogramma <lb/>P D M L, F D M K. </s> <s xml:id="echoid-s3716" xml:space="preserve">triangulum A F L, trapezium A F G L, <lb/>& </s> <s xml:id="echoid-s3717" xml:space="preserve">medium harmonicum inter parallelogramma P D M L, <lb/>F D M K, ſunt continuè proportionalia, quoniam triangu-<lb/>lum A F L eſt medium arithmeticum & </s> <s xml:id="echoid-s3718" xml:space="preserve">trapezium A F G L <lb/>medium geometricum inter eadem parallelogramma, ut pa-<lb/>tet ex hujus 13; </s> <s xml:id="echoid-s3719" xml:space="preserve">ſed triangulum A F L, trapezium A F G L, <lb/>& </s> <s xml:id="echoid-s3720" xml:space="preserve">trapezium A F E L ſunt continuè proportionalia ex hujus <lb/>1; </s> <s xml:id="echoid-s3721" xml:space="preserve">& </s> <s xml:id="echoid-s3722" xml:space="preserve">proinde trapezium A F E L eſt medium harmonicum <lb/>inter parallelogramma P D M L, F D M K, quod demon-<lb/>ftrare oportuit.</s> <s xml:id="echoid-s3723" xml:space="preserve"/> </p> <div xml:id="echoid-div203" type="float" level="2" n="1"> <note position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" xml:space="preserve">TAB. XLIII. <lb/>fig. 4.</note> </div> </div> <div xml:id="echoid-div205" type="section" level="1" n="99"> <head xml:id="echoid-head135" xml:space="preserve">PROP. XXIX. PROBLEMA. <lb/>Dato circulo æquale invenire quadratum.</head> <p> <s xml:id="echoid-s3724" xml:space="preserve">Sit quadratum circulo circumſcriptum 4000000000000000; <lb/></s> <s xml:id="echoid-s3725" xml:space="preserve">erit ergo eidem inſcriptum 2000000000000000, inter quæ <lb/>quadrata ſit medium geometricum 2828427124746190 <lb/>octagonum nempe intra circulum: </s> <s xml:id="echoid-s3726" xml:space="preserve">deinde inter octago-<lb/>num intra circulum & </s> <s xml:id="echoid-s3727" xml:space="preserve">quadratum extra ſit medium har-<lb/>monicum, quod levi labore invenitur dividendoduplum <lb/>quadrati octagoni intra circulum ſeu duplum rectanguli à <lb/>quadratis intra & </s> <s xml:id="echoid-s3728" xml:space="preserve">extra circulum per ſummam quadrati & </s> <s xml:id="echoid-s3729" xml:space="preserve"><lb/>octagoni intra; </s> <s xml:id="echoid-s3730" xml:space="preserve">eritque inventum medium harmonicum, octa-<lb/>gonum circumſcriptum, nempe 3313708498984760. </s> <s xml:id="echoid-s3731" xml:space="preserve">conti-<lb/>nuetur hæc ſeries convergens polygonorum complicatorum <lb/>donec prima medietas notarum ſit eadem in utroque termino <pb o="447" file="0165" n="174" rhead="ET HYPERBOLÆ QUADRATURA."/> convergente, nimirum uſque ad polygona laterum 16384; <lb/></s> <s xml:id="echoid-s3732" xml:space="preserve">inſcriptum enim eſt 3141592576586860 & </s> <s xml:id="echoid-s3733" xml:space="preserve">circumſcriptum <lb/>3141592692091258; </s> <s xml:id="echoid-s3734" xml:space="preserve">non conſideratur nota ultima, quoniam <lb/>in diviſionibus & </s> <s xml:id="echoid-s3735" xml:space="preserve">radicum extractionibus ſemper à vera quan-<lb/>titate paululum aberramus, quod ultimam notam imperfectam <lb/>plerumque reddit. </s> <s xml:id="echoid-s3736" xml:space="preserve">adhibeatur deinde approximatio in hujus <lb/>20 & </s> <s xml:id="echoid-s3737" xml:space="preserve">21 demonſtrata, & </s> <s xml:id="echoid-s3738" xml:space="preserve">invenientur termini intra quos eſt vera <lb/>circuli menſura, poſito diametri quadrato 4000000000000000, <lb/>3141592653589789 minor circulo & </s> <s xml:id="echoid-s3739" xml:space="preserve">3141592653589792 eo-<lb/>dem major, & </s> <s xml:id="echoid-s3740" xml:space="preserve">proinde non latet circuli menſura, quam <lb/>invenire oportuit. </s> <s xml:id="echoid-s3741" xml:space="preserve">polygonorum ſeriem hic appono.</s> <s xml:id="echoid-s3742" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/># Intra circulum # extra circulum <lb/>4 # 2000000000000000 # 4000000000000000 <lb/>8 # 2828427124746190 # 3313708498984760 <lb/>16 # 3061467458920718 # 3182597878074527 <lb/>32 # 3121445152258051 # 3151724907429255 <lb/>64 # 3136548490545938 # 3144118385245904 <lb/>128 # 3140331156954752 # 3142223629942456 <lb/>256 # 3141277250932772 # 3141750369168965 <lb/>512 # 3141513801144299 # 3141632080703181 <lb/>1024 # 3141572940367090 # 3141602510256808 <lb/>2048 # 3141587725277158 # 3141595117749588 <lb/>4096 # 3141591421543029 # 3141593269613390 <lb/>8192 # 3141592345578073 # 3141592807595664 <lb/>16384 # 3141592576586860 # 3141592692091258 <lb/>### Circulus intra ſequentes terminos conſiſtit <lb/># 3141592653589789 # 3141592653589792 <lb/></note> <p> <s xml:id="echoid-s3743" xml:space="preserve">@odem omnino modo reperitur rectilineum æquale cuicun-<lb/>que ſectori circulari vel elliptico ex cognito triangulo inſcri-<lb/>pr<unsure/>o & </s> <s xml:id="echoid-s3744" xml:space="preserve">trapezio circumſcripto.</s> <s xml:id="echoid-s3745" xml:space="preserve"/> </p> <pb o="448" file="0166" n="175" rhead="VERA CIRCULI."/> </div> <div xml:id="echoid-div206" type="section" level="1" n="100"> <head xml:id="echoid-head136" xml:space="preserve">PROP. XXX. PROBLEMA.</head> <head xml:id="echoid-head137" style="it" xml:space="preserve">Ex dato ſinu invenire arcum.</head> <p> <s xml:id="echoid-s3746" xml:space="preserve">Sit arcus circuli A E deſcriptus ex centro B. </s> <s xml:id="echoid-s3747" xml:space="preserve">hujus arcus da-<lb/> <anchor type="note" xlink:label="note-0166-01a" xlink:href="note-0166-01"/> tur radius nempe A B & </s> <s xml:id="echoid-s3748" xml:space="preserve">ſinus nempe A D: </s> <s xml:id="echoid-s3749" xml:space="preserve">oportet in-<lb/>venire quam proportionem habetipſe arcusad integram circuli <lb/>circunferentiam. </s> <s xml:id="echoid-s3750" xml:space="preserve">ſit arcus chorda A E & </s> <s xml:id="echoid-s3751" xml:space="preserve">ejus ſemiſſis tan-<lb/>gens A C vel C E. </s> <s xml:id="echoid-s3752" xml:space="preserve">ex quadrato radii A B auferatur quadra-<lb/>tum ſinus A D & </s> <s xml:id="echoid-s3753" xml:space="preserve">relinquetur quadratum ſinus complementi <lb/>B D, datur igitur B D; </s> <s xml:id="echoid-s3754" xml:space="preserve">& </s> <s xml:id="echoid-s3755" xml:space="preserve">ideo datur area trianguli rectanguli <lb/>A B D; </s> <s xml:id="echoid-s3756" xml:space="preserve">datur quoque area trianguli A B E nempe rectangu-<lb/>lum ſinus dati A D in ſemiſſem radii B E. </s> <s xml:id="echoid-s3757" xml:space="preserve">deinde fiat ut ſumma <lb/>triangulorum A B D, A B E, ad triangulum A B E ita duplum <lb/>trianguli A B E ad trapezium A B E C, ut conſtat ex hujus <lb/>5. </s> <s xml:id="echoid-s3758" xml:space="preserve">& </s> <s xml:id="echoid-s3759" xml:space="preserve">ex dato triangulo inſcripto A B E & </s> <s xml:id="echoid-s3760" xml:space="preserve">trapezio circum-<lb/>ſcripto A B E C, inveniatur per præcedentem ſector ipſe <lb/>A B E, qui ad circulum integrum datum quæſitam habet pro-<lb/>portionem arcus A E ad totam circumferentiam, quam inve-<lb/>nire oportuit.</s> <s xml:id="echoid-s3761" xml:space="preserve"/> </p> <div xml:id="echoid-div206" type="float" level="2" n="1"> <note position="left" xlink:label="note-0166-01" xlink:href="note-0166-01a" xml:space="preserve">TAB. XLIII. <lb/>fig. 5.</note> </div> </div> <div xml:id="echoid-div208" type="section" level="1" n="101"> <head xml:id="echoid-head138" xml:space="preserve">PROP. XXXI. PROBLEMA.</head> <head xml:id="echoid-head139" style="it" xml:space="preserve">Ex dato arcu invenire ſinum.</head> <p> <s xml:id="echoid-s3762" xml:space="preserve">Ex dato arcu manifeſtum eſt dari ſectoris aream; </s> <s xml:id="echoid-s3763" xml:space="preserve">hoc igitur <lb/> <anchor type="note" xlink:label="note-0166-02a" xlink:href="note-0166-02"/> ſectore dato conſideretur ex quot notis arithmeticis con-<lb/>ſtet ſinus totus: </s> <s xml:id="echoid-s3764" xml:space="preserve">deinde ſumatur ſectoris dati talis pars ali-<lb/>quota nempe ſector A B E, ut trianguli inſcripti A B E & </s> <s xml:id="echoid-s3765" xml:space="preserve"><lb/>trapezii circumſcripti A B E C toties multiplicia, quoties ſe-<lb/>ctor datus multiplex eſt ſectoris A B E, concordent in totno-<lb/>tis arithmeticis quot continet radix quadrata ſinus totius; </s> <s xml:id="echoid-s3766" xml:space="preserve">hoc <lb/>enim facile fieri poteſt ex ſpeculatione tabellæ hujus 29: </s> <s xml:id="echoid-s3767" xml:space="preserve">non <lb/>enim requiritur in hoc praxis præciſa, nam nihil refertetiam-<lb/>ſi in magnis radiis diſcrepantia ſit notarum aliquot. </s> <s xml:id="echoid-s3768" xml:space="preserve">Sectoris <pb o="449" file="0167" n="176" rhead="ET HYPERBOLÆ QUADRATURA."/> cogniti A B E, cujus arcus A E etiam innoteſcit, datur radius <lb/>B A: </s> <s xml:id="echoid-s3769" xml:space="preserve">ſit ejus ſinus A D, z; </s> <s xml:id="echoid-s3770" xml:space="preserve">& </s> <s xml:id="echoid-s3771" xml:space="preserve">proinde è dato ſinu & </s> <s xml:id="echoid-s3772" xml:space="preserve">radio, da-<lb/>bitur ut in antecedente triangulum ſectori inſcriptum A B E & </s> <s xml:id="echoid-s3773" xml:space="preserve"><lb/>eidem trapezium circumſcriptum A B E C; </s> <s xml:id="echoid-s3774" xml:space="preserve">atque ipſe ſector <lb/>datus, eſt ſecunda duarum mediarum arithmeticè continuè <lb/>proportionalium inter triangulum ſibi inſcriptum & </s> <s xml:id="echoid-s3775" xml:space="preserve">trapezium <lb/>circumſcriptum; </s> <s xml:id="echoid-s3776" xml:space="preserve">& </s> <s xml:id="echoid-s3777" xml:space="preserve">proinde datur æquatio inter duplum tra-<lb/>pezii A B E C unà cum triangulo A B E & </s> <s xml:id="echoid-s3778" xml:space="preserve">triplum ſectoris <lb/>cogniti A B E, cujus reſolutio manifeſtat valorem quantita-<lb/>tis ignotæ z, hoc eſt ſinum A D: </s> <s xml:id="echoid-s3779" xml:space="preserve">at dato arcu A E & </s> <s xml:id="echoid-s3780" xml:space="preserve">ejus <lb/>ſinu, dantur etiam ex vulgata ſectionum angularium doctri-<lb/>na ſinus omnium multiplicium ejuſdem arcus A E; </s> <s xml:id="echoid-s3781" xml:space="preserve">& </s> <s xml:id="echoid-s3782" xml:space="preserve">proin-<lb/>de non latet ſinus arcus in initio propoſiti, cum ſit ad arcum <lb/>A E in data ratione multiplici, quem invenire oportuit.</s> <s xml:id="echoid-s3783" xml:space="preserve"/> </p> <div xml:id="echoid-div208" type="float" level="2" n="1"> <note position="left" xlink:label="note-0166-02" xlink:href="note-0166-02a" xml:space="preserve">TAB. XLIII. <lb/>fig. 5.</note> </div> </div> <div xml:id="echoid-div210" type="section" level="1" n="102"> <head xml:id="echoid-head140" xml:space="preserve">PROP. XXXII. PROBLEMA.</head> <head xml:id="echoid-head141" style="it" xml:space="preserve">Invenire quadratum æquale ſpatio hyperbolico con-<lb/>tento à curva hyperbolica, uno aſymptoto & dua-<lb/>bus rectis alteri aſymptoto parallelis; quod <lb/>ſpatium æquale eſt ſectori hyperbolico <lb/>cujus baſis eſt eadem curva.</head> <p> <s xml:id="echoid-s3784" xml:space="preserve">Sit hyperpola D I L, cujus aſymptota A O, A K, ſibioc-<lb/> <anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/> currunt in angulo recto O A K. </s> <s xml:id="echoid-s3785" xml:space="preserve">proponitur hujus hyper-<lb/>bolæ ſpatium I L M K, contentum à curva hyperbolica I L, <lb/>aſymptoto K M & </s> <s xml:id="echoid-s3786" xml:space="preserve">duabus rectis I K, L M, alteri aſymptoto <lb/>A O parallelis. </s> <s xml:id="echoid-s3787" xml:space="preserve">oportet è datis recti I K 10 000000000000, <lb/>L M 10000000000000, A M 1000000000000, & </s> <s xml:id="echoid-s3788" xml:space="preserve">proinde <lb/>recta quoque K M 9000000000000, invenire menſuram ſpa-<lb/>tii I L M K. </s> <s xml:id="echoid-s3789" xml:space="preserve">Producantur rectæ I K, O L, & </s> <s xml:id="echoid-s3790" xml:space="preserve">ducatur recta <lb/>I P, ut compleantur rectangula L N K M, Q I K M; </s> <s xml:id="echoid-s3791" xml:space="preserve">manife-<lb/>ſtum eſt rectangulum L N K M eſſe 9000000000000000000-<lb/>0000000 & </s> <s xml:id="echoid-s3792" xml:space="preserve">Q I K M 9000000000000000000000000 & </s> <s xml:id="echoid-s3793" xml:space="preserve">tra- <pb o="450" file="0168" n="177" rhead="VERA CIRCULI."/> pezium L I K M eſſe medium arithmeticum inter prædicta re-<lb/>ctangula, hoc eſt 49500000000000000000000000. </s> <s xml:id="echoid-s3794" xml:space="preserve">invenia-<lb/>tur inter rectangula L N K M, Q I K M, medium geometri-<lb/>cum 28460498941515413987990042 quod erit pentagonum <lb/>ſpatio hyperbolico L I K M regulariter circumſcriptum. </s> <s xml:id="echoid-s3795" xml:space="preserve">Sit-<lb/>que ut trapezium LIKM unà cum dicto pentagono circumſcri-<lb/>pto ad dictum pentagonum, ita duplum dicti pentagoni ad <lb/>hexagonum ſpatio hyperbolico L I K M regulariter inſcriptum, <lb/>nempe 20779754131836628160009835; </s> <s xml:id="echoid-s3796" xml:space="preserve">quod hexagonum <lb/>erit polygonum complicatum cum prædicto pentagono; </s> <s xml:id="echoid-s3797" xml:space="preserve">quæ <lb/>duo rectilinea efficiant primos ſeriei terminos convergentes: <lb/></s> <s xml:id="echoid-s3798" xml:space="preserve">inter quæ ſit medium geometricum cujus quadrati duplum di-<lb/>vidatur per idem medium geometricum unà cum majori ter-<lb/>mino ſeu pentagono circumſcripto; </s> <s xml:id="echoid-s3799" xml:space="preserve">eruntque medium geo-<lb/>metricum & </s> <s xml:id="echoid-s3800" xml:space="preserve">quotus, ſecundi terminiconvergentes: </s> <s xml:id="echoid-s3801" xml:space="preserve">atque ita <lb/>continuetur ſeries hæc convergens polygonorum complica-<lb/>torum, donec medietas prima notarum eadem ſit in utroque <lb/>termino convergente, nempe ad terminum vigeſimum; </s> <s xml:id="echoid-s3802" xml:space="preserve">po-<lb/>lygonum enim circumſcriptum eſt 23025850929958961534-<lb/>173864, & </s> <s xml:id="echoid-s3803" xml:space="preserve">inſcriptum 23025850929931203593181124: </s> <s xml:id="echoid-s3804" xml:space="preserve">ad-<lb/>hibeatur deinde approximatio in hujus 23 & </s> <s xml:id="echoid-s3805" xml:space="preserve">24 demonſtra-<lb/>ta, & </s> <s xml:id="echoid-s3806" xml:space="preserve">invenientur termini intra quos conſiſtit vera ſpatii <lb/>hyperbolici L I K M menſura, nempe 230258509299404562-<lb/>40178681, minor ſpatio, & </s> <s xml:id="echoid-s3807" xml:space="preserve">23025850929940456240178704 <lb/>eodem major, & </s> <s xml:id="echoid-s3808" xml:space="preserve">ideo non latet ſpatii menſura, quam in-<lb/>venire oportuit: </s> <s xml:id="echoid-s3809" xml:space="preserve">totam polygonorum ſeriem hic appono unà <lb/>cum numero rectarum curvam hyperbolicam ſubtendentium <lb/>in unoquoque polygono circumſcripto.</s> <s xml:id="echoid-s3810" xml:space="preserve"/> </p> <div xml:id="echoid-div210" type="float" level="2" n="1"> <note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">TAB. XLIV. <lb/>fig. 1.</note> </div> <pb file="0169" n="178"/> <pb file="0169a" n="179"/> <figure> <caption xml:id="echoid-caption61" style="it" xml:space="preserve">Pag. 450.<lb/>TAB.XLIII.<lb/>Fig. 4.</caption> <variables xml:id="echoid-variables61" xml:space="preserve">B A F R P C D E G H I K S L M N O</variables> </figure> <figure> <caption xml:id="echoid-caption62" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables62" xml:space="preserve">F G I K D L E S T O C N H M V R B Q P A</variables> </figure> <figure> <caption xml:id="echoid-caption63" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables63" xml:space="preserve">F G I K D L E S T O C N V R B Q P A</variables> </figure> <figure> <caption xml:id="echoid-caption64" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables64" xml:space="preserve">A C B D E</variables> </figure> <figure> <caption xml:id="echoid-caption65" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables65" xml:space="preserve">A F G I K D L S T E O C N H M V R B Q P</variables> </figure> <pb file="0170" n="180"/> <pb o="451" file="0171" n="181" rhead="ET HYPERBOLÆ QUADRATURA."/> <note position="right" xml:space="preserve"> <lb/>## Extra hyperbolam # Intra hyperbolam <lb/>2 # 28460498941515413987990042 # 20779754131836628160009835 <lb/>4 # 24318761696971474416609403 # 22410399968461612921314879 <lb/>8 # 23345088913234727934949897 # 22868197570682058351436953 <lb/>16 # 23105412906351426185065096 # 22986193244865462241217428 <lb/>32 # 23045725982658962868047234 # 23015921117139340153267671 <lb/>64 # 23030818728479610745741910 # 23023367512879647736902891 <lb/>128 # 23027092819292183214705676 # 23025230015404383009313933 <lb/>256 # 23026161398510805910921810 # 23025695697539046352276636 <lb/>512 # 23025928546847571901068394 # 23025812121604634087915779 <lb/>1024 # 23025870334152518169052273 # 23025841227841783762272302 <lb/>2048 # 23025855780992551911165543 # 23025848504414868310197241 <lb/>4096 # 23025852142703422669729927 # 23025850323559001769499206 <lb/>8192 # 23025851233131194254554390 # 23025850778345089029496888 <lb/>16384 # 23025851005738140519209367 # 23025850892041614212944994 <lb/>32768 # 23025850948889877295901163 # 23025850920465745719335070 <lb/>65536 # 23025850934677811503232115 # 23025850927571778609090592 <lb/>131072 # 23025850931124795055887228 # 23025850929348286832351848 <lb/>262144 # 23025850930236540944102405 # 23025850929792413888218560 <lb/>524288 # 23025850930014477416159412 # 23025850929903445652188450 <lb/>1048576 # 23025850929958961534173864 # 23025850929931203593181124 <lb/># ## hyperbolæ ſector intra ſequentes terminos conſiſtit <lb/>## 23025850929940456240178681 # 23025850929940456240178704 <lb/></note> <pb o="452" file="0172" n="182" rhead="VERA CIRCULI"/> <p> <s xml:id="echoid-s3811" xml:space="preserve">Poteſtigitur abſque periculo erroris ſumi ſequens numerus <lb/>pro hyperbolæ ſectore, cujus numeri multiplices uſque ad <lb/>decem, diviſionis facilitandæ gratia in compoſitione logorith-<lb/>morum, hic ſubjicimus; </s> <s xml:id="echoid-s3812" xml:space="preserve">in magnis namque diviſionibus præ-<lb/>ſtat uti repetita diviſoris multiplicium ſubſtractione quam or-<lb/>dinaria diviſione, ut conſtat expertis arithmeticis.</s> <s xml:id="echoid-s3813" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3814" xml:space="preserve">Manifeſtum eſt hoc <lb/> <anchor type="note" xlink:label="note-0172-01a" xlink:href="note-0172-01"/> problema eodem <lb/>ferè modo poſſe reſol-<lb/>vi, etiamſi aſymptota <lb/>A O, A K, non ſint <lb/>conſtiruta ad angulum <lb/>rectum: </s> <s xml:id="echoid-s3815" xml:space="preserve">nos autem ita <lb/>fuppoſuimus, ut faci-<lb/>lior & </s> <s xml:id="echoid-s3816" xml:space="preserve">paratior eſſet <lb/>problematis uſus in doctrina logorithmica, quam primò <lb/>invenit nobiliſſimus noſter Neperus, & </s> <s xml:id="echoid-s3817" xml:space="preserve">quam nos (ni fal-<lb/>lor) ad ſummum perfectionis faſtigium nunc elevamus.</s> <s xml:id="echoid-s3818" xml:space="preserve"/> </p> <div xml:id="echoid-div211" type="float" level="2" n="2"> <note position="right" xlink:label="note-0172-01" xlink:href="note-0172-01a" xml:space="preserve"> <lb/>1 # 23025850929940456240178700 <lb/>2 # 46051701859880912480357400 <lb/>3 # 69077552789821368720536100 <lb/>4 # 92103403719761824960714800 <lb/>5 # 115129254649702281200893500 <lb/>6 # 138155105579642737441072200 <lb/>7 # 161180956509583193681250900 <lb/>8 # 184206807439523649921429600 <lb/>9 # 207232658369464106161608300 <lb/>10 # 230258509299404562401787000 <lb/></note> </div> </div> <div xml:id="echoid-div213" type="section" level="1" n="103"> <head xml:id="echoid-head142" xml:space="preserve">PROP. XXXIII. PROBLEMA.</head> <head xml:id="echoid-head143" style="it" xml:space="preserve">Propoſiti cujuscunque numeri logorithmum invenire.</head> <p> <s xml:id="echoid-s3819" xml:space="preserve">Eiſdem poſitis quæ in antecedente, manifeſtum eſt, poſita <lb/> <anchor type="note" xlink:label="note-0172-02a" xlink:href="note-0172-02"/> I K unitate, M L eſſe decem: </s> <s xml:id="echoid-s3820" xml:space="preserve">poſita ergo I K unitate ſit <lb/>G H aſymptoto quoque A O parallela numerus propoſitus <lb/>cujus deſideratur logorithmus@ manifeſtum eſt ex data rectà <lb/>G H dari K F, & </s> <s xml:id="echoid-s3821" xml:space="preserve">ex præcedenti dari etiam ſpatium hyper-<lb/>bolicum G I K H, quod ſpatium hyperbolicum dico eſſe lo-<lb/>gorithmum numeri propoſiti G H, poſito ſpatio hyperbo-<lb/>lico L I K M logorithmo numeri denarii: </s> <s xml:id="echoid-s3822" xml:space="preserve">eſt enim (ex Gre-<lb/>gorio à S. </s> <s xml:id="echoid-s3823" xml:space="preserve">Vincentio) ſpatium G H K I in eadem ratione ad <lb/>ſpatium L M K I, in qua ratio G H ad I K eſt multiplicata <lb/>rationis L M ad I K; </s> <s xml:id="echoid-s3824" xml:space="preserve">ſed ratio G H ad I K eſt multiplicata <pb o="453" file="0173" n="183" rhead="ET HYPERBOLÆ QUADRATURA."/> rationis L M ad I K in eadem ratione qua numerus G H eſt <lb/>multiplicatus numeri L M, quoniam idem eſt conſequens in <lb/>utraque ratione; </s> <s xml:id="echoid-s3825" xml:space="preserve">& </s> <s xml:id="echoid-s3826" xml:space="preserve">proinde ſpatium G I K H eſt in eadem <lb/>ratione ad ſpatium L I K M, in qua numerus G H eſt mul-<lb/>tiplicatus numeri L M; </s> <s xml:id="echoid-s3827" xml:space="preserve">& </s> <s xml:id="echoid-s3828" xml:space="preserve">ideo (quoniam ex hypotheſi ſpa-<lb/>tium L I K M eſt logorithmus numeri L M ſeu denarii) erit <lb/>ſpatium G I K H logorithmus numeri propoſiti G H, quo-<lb/>niam hæc eſt logorithmorum eſſentialis proprietas, ut ſint <lb/>inter ſe in eadem directa ratione, in qua eorum numeri ſunt <lb/>unus alterius multiplicati: </s> <s xml:id="echoid-s3829" xml:space="preserve">at ponitur communiter logo@ith-<lb/>mus numeri denarii ad arbitrium unitas cum numero quodam <lb/>cyphrarum: </s> <s xml:id="echoid-s3830" xml:space="preserve">ſi igitur fiat, ut ſpatium L I K M ad ſpatium <lb/>G I K H, ita arbitrarius denarii logorithmus ad alium nume-<lb/>rum; </s> <s xml:id="echoid-s3831" xml:space="preserve">erit inventus ille numerus, logorithmus numeri propo-<lb/>ſiti G H, quem invenire oportuit.</s> <s xml:id="echoid-s3832" xml:space="preserve"/> </p> <div xml:id="echoid-div213" type="float" level="2" n="1"> <note position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">TAB. XLIV. <lb/>fig. 1.</note> </div> </div> <div xml:id="echoid-div215" type="section" level="1" n="104"> <head xml:id="echoid-head144" xml:space="preserve">SCHOLIUM.</head> <p> <s xml:id="echoid-s3833" xml:space="preserve">PRaxis prædicti problematis prolixa eſt & </s> <s xml:id="echoid-s3834" xml:space="preserve">laborioſa; </s> <s xml:id="echoid-s3835" xml:space="preserve">& </s> <s xml:id="echoid-s3836" xml:space="preserve">pro-<lb/>inde, ut abbrevietur labor noſter in compoſitione tabulæ <lb/>logorithmorum; </s> <s xml:id="echoid-s3837" xml:space="preserve">ſciendum eſt nos ſolummodo laborare in <lb/>inventione logorithmorum, numerorum primorum; </s> <s xml:id="echoid-s3838" xml:space="preserve">nume-<lb/>rorum enim compoſitorum logorithmi ex primorum additio-<lb/>ne & </s> <s xml:id="echoid-s3839" xml:space="preserve">ſubductione nullo negotio invenientur. </s> <s xml:id="echoid-s3840" xml:space="preserve">ſed ut nu-<lb/>merorum primorum logorithmi facilius inveniantur, ordine <lb/>progrediendum eſt à prioribus ad poſteriores, nempe à 10 <lb/>cujus logorithmus eſt arbitrarius ad 2 numerum omnium pri-<lb/>mum, & </s> <s xml:id="echoid-s3841" xml:space="preserve">à 10 & </s> <s xml:id="echoid-s3842" xml:space="preserve">2 ad 3, item à 10, 2 & </s> <s xml:id="echoid-s3843" xml:space="preserve">3 ad 7, item à 10, <lb/>2, 3 & </s> <s xml:id="echoid-s3844" xml:space="preserve">7 ad 11, & </s> <s xml:id="echoid-s3845" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s3846" xml:space="preserve">deinde inveniendi ſunt duo <lb/>numeri compoſiti parum inter ſe differentes, quorum unus <lb/>compoſitus eſt ex numeris logorithmos cognitos habentibus, <lb/>& </s> <s xml:id="echoid-s3847" xml:space="preserve">ideo logorithmum datum habens, alter autem numerus <lb/>compoſitus eſt ex ſolo numero primo (cujus quæritur logo-<lb/>rithmus) vel ex illo unà cum aliis numeris logorithmos co-<lb/>gnitos habentibus. </s> <s xml:id="echoid-s3848" xml:space="preserve">deinde applicentur hi numeri compoſiti <pb o="454" file="0174" n="184" rhead="VERA CIRCULI"/> (qui exempli gratia ſint G H, E F) in hyperbola aſymptoto <lb/>O A paralleli; </s> <s xml:id="echoid-s3849" xml:space="preserve">& </s> <s xml:id="echoid-s3850" xml:space="preserve">inveniatur ſpatium hyperbolicum EGHF <lb/>per hujus 32, quod breviter, fit quoniam G H, E F, parum <lb/>inter ſe differunt. </s> <s xml:id="echoid-s3851" xml:space="preserve">ex ſuppoſitione, unius numeri, exempli <lb/>gratia G H, datur logorithmus, & </s> <s xml:id="echoid-s3852" xml:space="preserve">proinde datur ejus logo-<lb/>rithmi ratio ad logorithmum denarii arbitrarium, quæ ea-<lb/>dem eſt (ex hactenus demonſtratis) cum ratione ſpatii hyper-<lb/>bolici G I K H ad ſpatium hyperbolicum L I K M, datur au-<lb/>tem ex hujus 32 ſpatium L I K M; </s> <s xml:id="echoid-s3853" xml:space="preserve">& </s> <s xml:id="echoid-s3854" xml:space="preserve">ideo innoteſcit quoque <lb/>ſpatium I K H G, cumque detur ſpatium E G H F, innote-<lb/>ſcit quoque E I K F; </s> <s xml:id="echoid-s3855" xml:space="preserve">& </s> <s xml:id="echoid-s3856" xml:space="preserve">proinde datur logorithmus numeri <lb/>compoſiti E F; </s> <s xml:id="echoid-s3857" xml:space="preserve">cumque ex ſuppoſitione dentur logorithmi <lb/>omnium numerorum numerum E F componentium, exce-<lb/>pto numero illo primo cujus logorithmus deſideratur, dabi-<lb/>tur quoque illius numeri primi logorithmus, quem invenire <lb/>oportuit. </s> <s xml:id="echoid-s3858" xml:space="preserve">Exempli gratia, propoſitum ſit invenire logorith-<lb/>mum numeri binarii, ſuppoſito arbitrario numeri denariilo-<lb/>gorithmo, unitate cum 25 cyphris. </s> <s xml:id="echoid-s3859" xml:space="preserve">duo numeri compoſiti <lb/>parum inter ſe differentes ſunt 1000 & </s> <s xml:id="echoid-s3860" xml:space="preserve">1024; </s> <s xml:id="echoid-s3861" xml:space="preserve">numeri 1000 <lb/>datur logorithmus, nempe triplum ſpatii 23025850929940-<lb/>456240178700 in antecedente inventi, poſito ſcilicet illo <lb/>ſpatio numeri denarii logorithmo arbitrario; </s> <s xml:id="echoid-s3862" xml:space="preserve">numeri 1024 <lb/>ignoratur logorithmus, eſt enim compoſitus ex ſolo numero <lb/>primo 2, nempe ejus decies multiplicatus eſt. </s> <s xml:id="echoid-s3863" xml:space="preserve">applicentur hi <lb/>numeri compoſiti in hyperbola, ut dictum eſt; </s> <s xml:id="echoid-s3864" xml:space="preserve">ſitque G H <lb/>1000, E F 1024: </s> <s xml:id="echoid-s3865" xml:space="preserve">ſed quoniam I K unitas eſt 1000000000000, <lb/>erit G H 1000000000000000 & </s> <s xml:id="echoid-s3866" xml:space="preserve">E F 1024000000000000, <lb/>& </s> <s xml:id="echoid-s3867" xml:space="preserve">per hujus 32 inveniatur ſpatium E G H F 237165266173-<lb/>160421183067 (ſeriem convergentem hic appono,</s> </p> <pb o="455" file="0175" n="185" rhead="ET HYPERBOLÆ QUADRATURA."/> <note position="right" xml:space="preserve"> <lb/># Extra hyperbolam # Intra hyperbolam <lb/>2 # 237170824512628449899917 # 237162487062045867846886 <lb/>4 # 237166655750699903737556 # 237164571388054419219371 <lb/>8 # 237165613567087322970403 # 237165092476425954356426 <lb/>16 # 237165353021613523599438 # 237165222748948181485250 <lb/>32 # 237165287885271907848389 # 237165255317105572320456 <lb/>64 # 237165271601188181041012 # 237165263459146597159038 <lb/># ## hyperbolæ ſector intra ſequentes terminos conſiſtit <lb/># 237165266173160272103220 # 237165266173160458453029 <lb/></note> <p> <s xml:id="echoid-s3868" xml:space="preserve">Sit inter hos terminos maximus quatuor continuè arithmeticè <lb/>proportionalium 237165266173160421183067; </s> <s xml:id="echoid-s3869" xml:space="preserve">qui proinde erit <lb/>verus hyperbolæ ſector in notarum numero propoſito, quoniam pri-<lb/>mus triens notarum idem eſt in utroque termino convergente.</s> <s xml:id="echoid-s3870" xml:space="preserve"> <pb o="456" file="0176" n="186" rhead="VERA CIRCULI"/> ut lectori compendium patefiat) ſeu logorithmus numeri 1{24/1000} <lb/>poſito logorithmo denarii arbitrario 23025850929940456-<lb/>240178700; </s> <s xml:id="echoid-s3871" xml:space="preserve">deinde eodem ſuppoſito logorithmo arbitrario <lb/>denarii, addatur logorithmus numeri 1000, ſeu triplus logo-<lb/>rithmi denarii, logorithmo numeri 1{24/1000}, eritque ſumma <lb/>logorithmus numeri 1024, cujus pars decima erit logorithmus <lb/>numeri binarii, pro eodem logorithmo denarii arbitrario, <lb/>nempe 6931471805599452914171917: </s> <s xml:id="echoid-s3872" xml:space="preserve">fiatque ut logorith-<lb/>mus numerii denarii 23025850929940456240178700 ad lo-<lb/>gorithmum numeri binarii correſpondentem 69314718055-<lb/>99452914171917, ita logorithmus numeri denarii arbitrarius <lb/>propoſitus nempe 100000000000000000000000000 ad lo-<lb/>gorithmum numeri binarii quæſitum 3010299956639811952-<lb/>405804, quem invenire oportuit: </s> <s xml:id="echoid-s3873" xml:space="preserve">eodem modo invenitur lo-<lb/>gorithmus ternarii 4771212547196624373502993, &</s> <s xml:id="echoid-s3874" xml:space="preserve">c.</s> <s xml:id="echoid-s3875" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3876" xml:space="preserve">Ut in promptu habeantur numeri illi compoſiti parum in-<lb/>ter ſe differentes pro unoquoque numero primo, hic tabel-<lb/>lam exhibeo pro numeris primis uſque ad 100, & </s> <s xml:id="echoid-s3877" xml:space="preserve">unam re-<lb/>gulam pro numeris primis inter 100 & </s> <s xml:id="echoid-s3878" xml:space="preserve">1000 & </s> <s xml:id="echoid-s3879" xml:space="preserve">alteram pro <lb/>numeris primis ſupra 1000; </s> <s xml:id="echoid-s3880" xml:space="preserve">quæ omnia ita excogitata ſunt, <lb/>ut verus cujuscunque numeri primi logorithmus inveniri <lb/>poſſit correſpondens logorithmo arbitrario denarii 100000-<lb/>00000000000000000000 ex una ſola multiplicatione, dua-<lb/>bus diviſionibus & </s> <s xml:id="echoid-s3881" xml:space="preserve">una radicis quadratæ extractione, ultra <lb/>inconſiderabiles aliquot operatiunculas.</s> <s xml:id="echoid-s3882" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>2 # 1000(<emph style="super">3</emph>)10 <lb/>" # 1024(<emph style="super">10</emph>)2 <lb/>3 # 32805 factus ex 5 & 6561(<emph style="super">8</emph>)3 <lb/>" # 32768(<emph style="super">15</emph>) 2 <lb/>7 # 2400 factus ex 3 & 32(<emph style="super">5</emph>)2 & 25(<emph style="super">2</emph>)5 <lb/>" # 2401(<emph style="super">4</emph>)7 <lb/>11 # 9800 factus ex 2, 49(<emph style="super">2</emph>)7 & 100(<emph style="super">2</emph>)10 <lb/>" # 9801 factus ex 121(<emph style="super">2</emph>)11 & 81(<emph style="super">4</emph>)3 <lb/></note> <pb o="457" file="0177" n="187" rhead="ET HYPERBOLÆ QUADRATURA."/> <note position="right" xml:space="preserve"> <lb/>13 # 123200 factus ex 7, 11, 25(<emph style="super">2</emph>)5, 64(<emph style="super">6</emph>)2 <lb/>" # 123201 factus ex 169(<emph style="super">2</emph>)13 & 729(<emph style="super">6</emph>)3 <lb/>17 # 2600 factus ex 13, 8(<emph style="super">3</emph>)2 & 25(<emph style="super">2</emph>)5 <lb/>" # 2601 factus ex 9(<emph style="super">2</emph>)3 & 289(<emph style="super">2</emph>)17 <lb/>19 # 28899 factus ex 169(<emph style="super">2</emph>)13, 9(<emph style="super">2</emph>)3 & 19 <lb/>" # 28900 factus ex 100(<emph style="super">2</emph>)10 & 289(<emph style="super">2</emph>)17 <lb/>23 # 25920 factus ex 10, 32(<emph style="super">5</emph>)2 & 81(<emph style="super">4</emph>)3 <lb/>" # 25921 factus ex 49(<emph style="super">2</emph>)7 & 529(<emph style="super">2</emph>)23 <lb/>29 # 613088 factus ex 17, 23, 32(<emph style="super">5</emph>)2, & 49(<emph style="super">2</emph>)7 <lb/>" # 613089 factus ex 729(<emph style="super">6</emph>)3 & 841(<emph style="super">2</emph>)29 <lb/>31 # 116280 factus ex 10, 17, 19, 4(<emph style="super">2</emph>)2, & 9(<emph style="super">2</emph>)3 <lb/>" # 116281 factus ex 121(<emph style="super">2</emph>)11 & 961(<emph style="super">2</emph>)31 <lb/>37 # 165648 factus ex 3, 7, 17, 29, & 16(<emph style="super">4</emph>)2 <lb/>" # 165649 factus ex 121(<emph style="super">2</emph>)11 & 1369(<emph style="super">2</emph>)37 <lb/>41 # 1413720 factus ex 7, 10, 11, 17, 4(<emph style="super">2</emph>)2 & 27(<emph style="super">3</emph>)3 <lb/>" # 1413721 factus ex 1681(<emph style="super">2</emph>)41 & 841(<emph style="super">2</emph>)29 <lb/>43 # 978120 factus ex 10, 11, 13, 19, 4(<emph style="super">2</emph>)2, 9(<emph style="super">2</emph>)3 <lb/>" # 978121 factus ex 529(<emph style="super">2</emph>)23 & 1849(<emph style="super">2</emph>)43 <lb/>47 # 664848 factus ex 7, 13, 32(<emph style="super">5</emph>)2 & 729(<emph style="super">6</emph>)3 <lb/>" # 664849 factus ex 961(<emph style="super">2</emph>)31 & 2209(<emph style="super">2</emph>)47 <lb/>53 # 3059000 factus ex 7, 19, 23, 8,(<emph style="super">3</emph>)2, & 125(<emph style="super">3</emph>)5 <lb/>" # 3059001 factus ex, 9(<emph style="super">2</emph>)3, 121,(<emph style="super">2</emph>)11 & 2809(<emph style="super">2</emph>)53 <lb/>57 # 5851560 factus ex 3, 5, 13, 31, & 121(<emph style="super">2</emph>)11 <lb/>" # 5851561 factus ex 1681(<emph style="super">2</emph>)41 & 3481(<emph style="super">2</emph>)59 <lb/>61 # 3575880 factus ex 5, 7, 11, 43, 8(<emph style="super">3</emph>)2 & 27(<emph style="super">3</emph>)3 <lb/>" # 3575881 factus ex 961(<emph style="super">2</emph>)31 & 3721(<emph style="super">2</emph>)61 <lb/>67 # 1620528 factus ex 3, 13, 16(<emph style="super">4</emph>)2 & 49(<emph style="super">2</emph>)7 <lb/>" # 1620529 factus ex 361(<emph style="super">2</emph>)19 & 4489(<emph style="super">2</emph>)67 <lb/></note> <pb o="458" file="0178" n="188" rhead="VERA CIRCULI"/> <note position="right" xml:space="preserve"> <lb/>71 # 2016399 factus ex 3, 11, 29, 43 & 49(<emph style="super">2</emph>)7 <lb/>" # 2016400 factus ex 16(<emph style="super">4</emph>)2, 25(<emph style="super">2</emph>)5 & 5041(<emph style="super">2</emph>)71 <lb/>73 # 5116644 factus ex 4(<emph style="super">2</emph>)2, 9(<emph style="super">2</emph>)3, 169(<emph style="super">2</emph>)13, & 841(<emph style="super">2</emph>)29 <lb/>" # 5116645 factus ex 7, 17, 19, 31, 73 <lb/>79 # 5997600 factus ex 17, 32(<emph style="super">5</emph>)2, 9(<emph style="super">2</emph>)3, 25(<emph style="super">2</emph>)5 & 49(<emph style="super">2</emph>)7 <lb/>" # 5997601 factus ex 961(<emph style="super">2</emph>)31 & 6241(<emph style="super">2</emph>)79 <lb/>83 # 1164240 factus ex 5, 11, 49(<emph style="super">2</emph>)7, 16(<emph style="super">4</emph>)2, 27(<emph style="super">3</emph>)3 <lb/>" # 1164241 factus ex 169(<emph style="super">2</emph>)13 & 6889(<emph style="super">2</emph>)83 <lb/>89 # 2859480 factus ex 5, 47, 8(<emph style="super">3</emph>)2, 169(<emph style="super">2</emph>)13 & 9(<emph style="super">2</emph>)3 <lb/>" # 2859481 factus ex 361(<emph style="super">2</emph>)19 & 7921(<emph style="super">2</emph>)89 <lb/>97 # 1138488 factus ex 3, 13, 41, 89, & 8(<emph style="super">3</emph>)2 <lb/>" # 1138489 factus ex 121(<emph style="super">2</emph>)11 & 9409(<emph style="super">2</emph>)97 <lb/></note> <p> <s xml:id="echoid-s3883" xml:space="preserve">Pro numeris primis inter 100 & </s> <s xml:id="echoid-s3884" xml:space="preserve">1000 ſit hæc regula: </s> <s xml:id="echoid-s3885" xml:space="preserve">ante <lb/>numerum primum cujus logorithmus quæritur, ſumantur im-<lb/>mediatè duo numeri proximi, & </s> <s xml:id="echoid-s3886" xml:space="preserve">poſt eum numerus imme-<lb/>diatè ſequens, qui tres numeri cum illo primo ſunt quatuor <lb/>numeri in ſuo naturali ordine ſe invicem ſequentes; </s> <s xml:id="echoid-s3887" xml:space="preserve">deinde <lb/>multiplicetur primus numerus in cubum tertii & </s> <s xml:id="echoid-s3888" xml:space="preserve">quartus in <lb/>cubum ſecundi, eritque factorum differentia æqualis ſummæ <lb/>primi & </s> <s xml:id="echoid-s3889" xml:space="preserve">quarti vel ſecundi & </s> <s xml:id="echoid-s3890" xml:space="preserve">tertii, ut facile demonſtrari po-<lb/>teſt; </s> <s xml:id="echoid-s3891" xml:space="preserve">iſtique numeri facti habent ad minimum ſex notas pri-<lb/>mas omnino eaſdem, & </s> <s xml:id="echoid-s3892" xml:space="preserve">proinde parum inter ſe differunt; </s> <s xml:id="echoid-s3893" xml:space="preserve">at-<lb/>que omnium horum quatuor numerorum (excepto tertii) lo-<lb/>gorithmi cognoſcuntur ex ipſa progrediendi methodo, & </s> <s xml:id="echoid-s3894" xml:space="preserve"><lb/>ideo ad noſtram abbreviationem ſunt idonei. </s> <s xml:id="echoid-s3895" xml:space="preserve">in numeris ul-<lb/>tra 1000 non opus eſt tanto apparatu, quoniam rectangulum <lb/>numerorum, inter quos immediatè comprehenditur numerus <lb/>primus cujus quæritur logorithmus, unitate ſolummodo defi-<lb/>cit à quadrato numeri primi; </s> <s xml:id="echoid-s3896" xml:space="preserve">eorumque ideo primæ ſex no-<lb/>tæ ad minimum ſunt eædem; </s> <s xml:id="echoid-s3897" xml:space="preserve">atque primi & </s> <s xml:id="echoid-s3898" xml:space="preserve">tertii dantur lo-<lb/>gorithmi, & </s> <s xml:id="echoid-s3899" xml:space="preserve">ideo ad noſtrum inſtitutum ſunt idonei.</s> <s xml:id="echoid-s3900" xml:space="preserve"/> </p> <pb o="459" file="0179" n="189" rhead="ET HYTERBOLÆ QUADRATURA."/> </div> <div xml:id="echoid-div216" type="section" level="1" n="105"> <head xml:id="echoid-head145" xml:space="preserve">PROP. XXXIV. PROBLEMA.</head> <head xml:id="echoid-head146" style="it" xml:space="preserve">Ex dato logorithmo invenire ejus <lb/>numerum.</head> <p> <s xml:id="echoid-s3901" xml:space="preserve">Ex demonſtratis manifeſtum eſt hoc problema idem eſſe <lb/> <anchor type="note" xlink:label="note-0179-01a" xlink:href="note-0179-01"/> ac ſi quis proponeret; </s> <s xml:id="echoid-s3902" xml:space="preserve">ex dato ſpatio hyperbolico, & </s> <s xml:id="echoid-s3903" xml:space="preserve"><lb/>una recta uni aſymptotorum parallela illud comprehenden-<lb/>te, alteram invenire idem ſpatium comprehendentem, & </s> <s xml:id="echoid-s3904" xml:space="preserve"><lb/>eidem aſymptoto parallelam. </s> <s xml:id="echoid-s3905" xml:space="preserve">Conſideretur ex quot notis <lb/>arithmeticis conſtet logorithmus denarii arbitrarius; </s> <s xml:id="echoid-s3906" xml:space="preserve">& </s> <s xml:id="echoid-s3907" xml:space="preserve">ſuma-<lb/>tur logorithmi vel ſpatii dati talis pars aliquota nempe ſpa-<lb/>tium L I K M, ut pentagoni ſpatio L I K M regulariter cir-<lb/>cumſcripti, & </s> <s xml:id="echoid-s3908" xml:space="preserve">hexagoni eidem regulariter inſcripti toties <lb/>multiplicia, quoties ſpatium datum multiplex eſt ſpatii <lb/>L I K M, concordent in tot notis arithmeticis, quot conti-<lb/>net radix quadrata logorithmi arbitrarii; </s> <s xml:id="echoid-s3909" xml:space="preserve">hoc enim facile fie-<lb/>ri poteſt ex inſpectione tabellæ 32 hujus: </s> <s xml:id="echoid-s3910" xml:space="preserve">datur ergo ſpatii <lb/>L I K M menſura & </s> <s xml:id="echoid-s3911" xml:space="preserve">recta I K unitas ex ſuppoſitione. </s> <s xml:id="echoid-s3912" xml:space="preserve">Sit <lb/>L M, z; </s> <s xml:id="echoid-s3913" xml:space="preserve">ſicut in hujus 32 datur pentagonum ſpatio L I K M <lb/>regulariter circumſcriptum & </s> <s xml:id="echoid-s3914" xml:space="preserve">hexagonum eidem regulariter <lb/>inſcriptum, inter quæ ſpatium datum L I K M eſt ſecunda <lb/>duarum mediarum arithmeticè continuè proportionalium; <lb/></s> <s xml:id="echoid-s3915" xml:space="preserve">& </s> <s xml:id="echoid-s3916" xml:space="preserve">ideo duplum haxagoni una cum pentagono æquatur triplo <lb/>ſpatii, cujus æquationis reſolutio manifeſtat ignotam z ſeu <lb/>numerum L M, cujus toties multiplicatus, quoties ſpatium <lb/>L I K M eſt ſubmultiplex ſpatii vel logorithmi dati, eſt nu-<lb/>merus quæſitus, quem invenire oportuit.</s> <s xml:id="echoid-s3917" xml:space="preserve"/> </p> <div xml:id="echoid-div216" type="float" level="2" n="1"> <note position="right" xlink:label="note-0179-01" xlink:href="note-0179-01a" xml:space="preserve">TAB. XLIV. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s3918" xml:space="preserve">Hoc problema idem eſt cum hujus 8, ſed aliter genera-<lb/>lius & </s> <s xml:id="echoid-s3919" xml:space="preserve">methodo plerumque minus operoſa hic reſolu-<lb/>tum.</s> <s xml:id="echoid-s3920" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div218" type="section" level="1" n="106"> <head xml:id="echoid-head147" style="it" xml:space="preserve">Tom. II. Mmm</head> <pb o="460" file="0180" n="190" rhead="VERA CIRCULI"/> </div> <div xml:id="echoid-div219" type="section" level="1" n="107"> <head xml:id="echoid-head148" xml:space="preserve">PROP. XXXV. PROBLEMA.</head> <head xml:id="echoid-head149" style="it" xml:space="preserve">Rectâ per datum punctum in diametro ductâ, <lb/>ſemicirculum in ratione data dividere.</head> <p> <s xml:id="echoid-s3921" xml:space="preserve">Sit ſemicirculus A D G, cujus diameter A G, centrum E, <lb/> <anchor type="note" xlink:label="note-0180-01a" xlink:href="note-0180-01"/> punctum in diametro datum B. </s> <s xml:id="echoid-s3922" xml:space="preserve">ſupponatur factum quod <lb/>jubetur; </s> <s xml:id="echoid-s3923" xml:space="preserve">ſitque recta B D ſemicirculum dividens in ratione <lb/>data: </s> <s xml:id="echoid-s3924" xml:space="preserve">quoniam datur ſemicirculi menſura & </s> <s xml:id="echoid-s3925" xml:space="preserve">ratio in qua <lb/>dividitur, igitur datur ejus portio nempe D B G. </s> <s xml:id="echoid-s3926" xml:space="preserve">Sit recta <lb/>B D, z: </s> <s xml:id="echoid-s3927" xml:space="preserve">ex datis rectis B D, B E, E D, innoteſcunt tri-<lb/>angula D E B, D E F, D E G: </s> <s xml:id="echoid-s3928" xml:space="preserve">deinde ſit ut D E F una cum <lb/>D E G ad D E G ità duplum D E G ad trapezium circum-<lb/>ſcriptum D E G H: </s> <s xml:id="echoid-s3929" xml:space="preserve">& </s> <s xml:id="echoid-s3930" xml:space="preserve">poſitis primis terminis convergenti-<lb/>bus D E G, D E G H continuetur ſeries convergens poly-<lb/>gonorum complicatorum, ſecundum circuli proprietates ſæ-<lb/>pius repetitas, donec conveniens fuerit approximationem ad-<lb/>hibere ita ut exhibeatur ſector D E G, qui una cum trian-<lb/>gulo D B E æquatur portioni D B G cognitæ, cujus æquatio-<lb/>nis reſolutio manifeſtat ignotam quantitatem z ſeu rectam <lb/>B D: </s> <s xml:id="echoid-s3931" xml:space="preserve">reliqua patent.</s> <s xml:id="echoid-s3932" xml:space="preserve"/> </p> <div xml:id="echoid-div219" type="float" level="2" n="1"> <note position="left" xlink:label="note-0180-01" xlink:href="note-0180-01a" xml:space="preserve">TAB. XLIV. <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s3933" xml:space="preserve">Idem problema eodem omnino modo reſolvitur in ellipſe, <lb/>hyperbola vel earum ſectore dato.</s> <s xml:id="echoid-s3934" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div221" type="section" level="1" n="108"> <head xml:id="echoid-head150" xml:space="preserve">SCHOLIUM.</head> <p> <s xml:id="echoid-s3935" xml:space="preserve">Si quis prædictorum problematum mechanicam deſideret <lb/>praxim; </s> <s xml:id="echoid-s3936" xml:space="preserve">non difficile erit calculum, approximationem, <lb/>& </s> <s xml:id="echoid-s3937" xml:space="preserve">æquationis reſolutionem ſecundum vulgatas Geome-<lb/>triæ practicæ regulas quodammodo imitari, multa talia <lb/>problemata poſſem hic reſolvere ope analyſios & </s> <s xml:id="echoid-s3938" xml:space="preserve">noſtræ ſe-<lb/>ricrum convergentium doctrinæ, quæ antea impoſſibilia æſti-<lb/>mabantur: </s> <s xml:id="echoid-s3939" xml:space="preserve">ſed dicet fortè aliquis has reſolutiones non eſſe geo-<lb/>metricas; </s> <s xml:id="echoid-s3940" xml:space="preserve">reſpondeo, ſi per geometricum intelligatur praxis <pb o="461" file="0181" n="191" rhead="ET HYPERBOLÆ QUADRATURA."/> ope ſolius regulæ & </s> <s xml:id="echoid-s3941" xml:space="preserve">circini peracta, hanc in his non ſolum <lb/>eſſe impoſſibilem ſed etiam in omnibus problematis quæ ad <lb/>æquationem quadratic<unsure/>am reduci non poſſunt, ſicut facile <lb/>demonſtrari poſſet; </s> <s xml:id="echoid-s3942" xml:space="preserve">& </s> <s xml:id="echoid-s3943" xml:space="preserve">ſi per geometricum intelligatur redu-<lb/>ctio problematis ad æquationem analyticam, omnia hæc <lb/>problemata ſunt geometrice impoſſibilia, cum ex hic demon-<lb/>ſtratis, manifeſtum ſit talem reductionem fieri non poſſe: <lb/></s> <s xml:id="echoid-s3944" xml:space="preserve">ſi verò per geometricum intelligatur methodus omnium poſ-<lb/>ſibilium ſimpliciſſima; </s> <s xml:id="echoid-s3945" xml:space="preserve">invenietur fortaſſe poſt maturam con-<lb/>ſiderationem omnia prædicta problemata eſſe geometriciſſi-<lb/>mè reſoluta, diligenter animadvertendum totam ſerierum <lb/>convergentium doctrinam poſſe etiam nullo negotio applicari <lb/>ſeriebus ſimplicibus. </s> <s xml:id="echoid-s3946" xml:space="preserve">Sit enim ſeries A, B, C, D, E, &</s> <s xml:id="echoid-s3947" xml:space="preserve">c, <lb/>talis naturæ ut tertius terminus C eodem modo <lb/> <anchor type="note" xlink:label="note-0181-01a" xlink:href="note-0181-01"/> componatur ex primo & </s> <s xml:id="echoid-s3948" xml:space="preserve">ſecundo A, B, quo <lb/>quartus D componitur ex ſecundo & </s> <s xml:id="echoid-s3949" xml:space="preserve">tertio B, C, <lb/>& </s> <s xml:id="echoid-s3950" xml:space="preserve">quintus E ex tertio & </s> <s xml:id="echoid-s3951" xml:space="preserve">quarto C, D, & </s> <s xml:id="echoid-s3952" xml:space="preserve">ſic dein-<lb/>ceps in infinitum; </s> <s xml:id="echoid-s3953" xml:space="preserve">ſitque differentia anteceden-<lb/>tium A, B, major ſemper differentia immediatè <lb/>ſequentium B, C; </s> <s xml:id="echoid-s3954" xml:space="preserve">ſupponamus hanc ſeriem ita in infinitum <lb/>continuari donec duorum terminorum immediate ſe invicem <lb/>ſequentium nulla ſit differentia, ſitque unus ex illis terminis <lb/>z, quem ſeriei terminationem appellamus: </s> <s xml:id="echoid-s3955" xml:space="preserve">dico z eodem <lb/>modo componi ex A & </s> <s xml:id="echoid-s3956" xml:space="preserve">B quo ex B & </s> <s xml:id="echoid-s3957" xml:space="preserve">C vel C & </s> <s xml:id="echoid-s3958" xml:space="preserve">D; </s> <s xml:id="echoid-s3959" xml:space="preserve">de-<lb/>monſtratio vix differt ab hujus 10 & </s> <s xml:id="echoid-s3960" xml:space="preserve">ejus conſectario: </s> <s xml:id="echoid-s3961" xml:space="preserve">hac <lb/>ratione ſi ponatur triangulum, ſectori circulari vel elliptico <lb/>inſcriptum, vel ſectori hyperbolico circumſcriptum a, & </s> <s xml:id="echoid-s3962" xml:space="preserve"><lb/>trapezium, ſectori circulari vel elliptico regulariter in-<lb/>ſcriptum vel hyperbolico regulariter circumſcriptum b; <lb/></s> <s xml:id="echoid-s3963" xml:space="preserve">erit hexagonum ſectori circulari vel elliptico regulari-<lb/>ter inſcriptum vel hyperbolico regulariter circumſcri-<lb/>ptum Vq {2 b3/a + b;</s> <s xml:id="echoid-s3964" xml:space="preserve">} & </s> <s xml:id="echoid-s3965" xml:space="preserve">proinde ſector circuli, ellipſeos vel hyperbo-<lb/>læ eodem modo componitur ex a & </s> <s xml:id="echoid-s3966" xml:space="preserve">b quo ex b & </s> <s xml:id="echoid-s3967" xml:space="preserve">Vq {2 b3/a + b;</s> <s xml:id="echoid-s3968" xml:space="preserve">} <lb/>atque hinc etiam demonſtrari poteſt, quod ratio inter ſecto-<lb/>rem & </s> <s xml:id="echoid-s3969" xml:space="preserve">ejus triangulum datum non ſit analytica, ſecundum <pb o="462" file="0182" n="192" rhead="VERA CIRC. ET HYPERB. QUADR."/> tenorem hujus 11: </s> <s xml:id="echoid-s3970" xml:space="preserve">poſſem quoque adhuc alia methodo par-<lb/>ticulari demonſtrare arcum circularem non habere rationem <lb/>analyticam ad ſuam chordam datam: </s> <s xml:id="echoid-s3971" xml:space="preserve">ſed plura non addo, <lb/>geometras interim pro ſcientiæ incremento admonens, me <lb/>reperiſſe in quibusdam figuris (quas Carteſius ſecundi gene-<lb/>ris appellat) tres focos, ſeu tria puncta, à quibus ductarum <lb/>in quodlibet curvæ punctum rectarum ſumma vel differen-<lb/>tia ſemper eſt eadem: </s> <s xml:id="echoid-s3972" xml:space="preserve">unde mihi veriſimile videtur ſicut omnis <lb/>curvæ primi generis duos habent focos vel reales vel imagi-<lb/>narios, ita omnes ſecundi generis habere tres, omnes ter-<lb/>tii quatuor, & </s> <s xml:id="echoid-s3973" xml:space="preserve">ſic in infinitum: </s> <s xml:id="echoid-s3974" xml:space="preserve">quæ certè ſpeculatio ſcru-<lb/>tatu digniſſima eſt, eſſet enim admiranda figurarum geome-<lb/>tricarum proprietas, & </s> <s xml:id="echoid-s3975" xml:space="preserve">mechanicæ omnium æquationum pra-<lb/>xi utiliſſima.</s> <s xml:id="echoid-s3976" xml:space="preserve"/> </p> <div xml:id="echoid-div221" type="float" level="2" n="1"> <note position="right" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve"> <lb/>A <lb/>B <lb/>C <lb/>D <lb/>E <lb/>Z <lb/></note> </div> </div> <div xml:id="echoid-div223" type="section" level="1" n="109"> <head xml:id="echoid-head151" xml:space="preserve">FINIS.</head> <figure> <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0182-01"/> </figure> <pb o="463" file="0183" n="193"/> <figure> <image file="0183-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0183-01"/> </figure> </div> <div xml:id="echoid-div224" type="section" level="1" n="110"> <head xml:id="echoid-head152" xml:space="preserve">II. <lb/>HUGENII <lb/>OBSERVATIONES <lb/>IN LIBRUM <lb/>JACOBI GREGORII, <lb/>DE VERA CIRCULI ET HYPERBOLÆ <lb/>QUADRATURA.</head> <p> <s xml:id="echoid-s3977" xml:space="preserve">D<emph style="super">nus</emph> Hugenius rogatus à Societate quæ Pariſiis ſuos <lb/>habebat conventus (1668) in Bibliotheca Regia, ut <lb/>vellet hunc librum examinare, ſocietati retu-<lb/>lit ſe multos errores obſervaſſe in demonſtratio-<lb/>ne, qua Auctor contendit impoſſibilitatem quadra-<lb/>turæ Circuli & </s> <s xml:id="echoid-s3978" xml:space="preserve">Hyperboles probare.</s> <s xml:id="echoid-s3979" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3980" xml:space="preserve">I. </s> <s xml:id="echoid-s3981" xml:space="preserve">in 11<emph style="super">a</emph>. </s> <s xml:id="echoid-s3982" xml:space="preserve">Propoſitione, in qua auctor contendit Circuli <lb/>quadraturam analytice impoſſibilem demonſtrare, hæc ha-<lb/>bentur Si terminatio magnitudinum propoſitarum analyti-<lb/>ce componatur a primis terminis a<emph style="super">3</emph> + a<emph style="super">2</emph>b & </s> <s xml:id="echoid-s3983" xml:space="preserve">ab<emph style="super">2</emph> + b<emph style="super">3</emph>, com-<lb/>ponetur etiam analyticê & </s> <s xml:id="echoid-s3984" xml:space="preserve">eodem omnino modo a ſecundis <lb/>terminis ba<emph style="super">2</emph> + bba & </s> <s xml:id="echoid-s3985" xml:space="preserve">2 b<emph style="super">2</emph> a. </s> <s xml:id="echoid-s3986" xml:space="preserve">ſed licet id verum ſit quando <lb/>terminatio inventa eſt per Methodum, quam indicat, non <lb/>poteſt generalis inde deduci concluſio. </s> <s xml:id="echoid-s3987" xml:space="preserve">niſi ponamus, non <lb/>poſſe inveniri, niſi per ejus Methodum, terminationem ſeriei <lb/>Magnitudinum, quas appellat convergentes, vel ſi aliâ me-<lb/>thodo inveniatur illam etiam per ſuam methodum inveniri <lb/>poſſe quod non demonſtravit.</s> <s xml:id="echoid-s3988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3989" xml:space="preserve">II. </s> <s xml:id="echoid-s3990" xml:space="preserve">Autor immediate poſt addit; </s> <s xml:id="echoid-s3991" xml:space="preserve">nullam quantitatem poſſe <pb o="464" file="0184" n="194" rhead="HUGENII OBSERVATIONES"/> eodem modo analytice componi e terminis a<emph style="super">3</emph> + aab, abb <lb/>+ b<emph style="super">3</emph> quo componitur e terminis aab + bba, 2 bba. </s> <s xml:id="echoid-s3992" xml:space="preserve">Hæc ni-<lb/>hilominus invenitur per eandem methodum quam indi-<lb/>cat in 7. </s> <s xml:id="echoid-s3993" xml:space="preserve">Propoſitione. </s> <s xml:id="echoid-s3994" xml:space="preserve">Methodus autem hæc eſt. </s> <s xml:id="echoid-s3995" xml:space="preserve">Pri-<lb/>mo quærenda eſt quantitas perquam ſi multiplices a<emph style="super">3</emph> + aab, <lb/>& </s> <s xml:id="echoid-s3996" xml:space="preserve">producto addas productum abb + b<emph style="super">3</emph> multiplicati per da-<lb/>tam quantitatem m, ſumma æqualis ſit ſummæ duorum alio-<lb/>rum productorum, unius aab + bba multiplicati per eandem <lb/>quantitatem quæſitam, alterius 2 bba multiplicati per datam <lb/>quantitatem m. </s> <s xml:id="echoid-s3997" xml:space="preserve">Ponamus igitur quantitatem illam æqualem <lb/>z, erit a<emph style="super">3</emph>z + aabz + abbm + b<emph style="super">3</emph>m = aabz + bbaz + 2 bbam; <lb/></s> <s xml:id="echoid-s3998" xml:space="preserve">& </s> <s xml:id="echoid-s3999" xml:space="preserve">z = {bbm/aa + ab}: </s> <s xml:id="echoid-s4000" xml:space="preserve">Et certum eſt, ſive multiplicetur {bbm/aa + ab} per a<emph style="super">3</emph> + aab <lb/>& </s> <s xml:id="echoid-s4001" xml:space="preserve">addatur abb + b<emph style="super">3</emph> multiplicatum per m, ſive eadem illa <lb/>quantitas multiplicetur per aab + bba & </s> <s xml:id="echoid-s4002" xml:space="preserve">addatur 2 bbam, <lb/>ſemper prodire eandem quantitatem 2 abbm + b<emph style="super">3</emph>m, & </s> <s xml:id="echoid-s4003" xml:space="preserve">conſe-<lb/>quenter ultimam hanc quantitatem componi eodem modo e <lb/>primis & </s> <s xml:id="echoid-s4004" xml:space="preserve">ſecundis terminis progreſſionis convergentis pro-<lb/>poſitæ, quod Autor fieri poſſe negavit.</s> <s xml:id="echoid-s4005" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4006" xml:space="preserve">III°. </s> <s xml:id="echoid-s4007" xml:space="preserve">Datâ autem hac quantitate 2 abbm + bbbm, ſi hac <lb/>utamur ad quærendam terminationem progreſſionis propo-<lb/>ſitæ, juxta methodum ab auctore indicatam in 7 propoſi-<lb/>tione, & </s> <s xml:id="echoid-s4008" xml:space="preserve">in 10, reperietur = {3 aab3 + ab4 + 2 a3 bb/bb + ab + aa}; </s> <s xml:id="echoid-s4009" xml:space="preserve">& </s> <s xml:id="echoid-s4010" xml:space="preserve">poſito a = 1 <lb/>& </s> <s xml:id="echoid-s4011" xml:space="preserve">b = 2, illa terminatio, quæ deſignat in eo caſu ſectorem <lb/>circuli continentem {1/3} totius circuli, erit = {48/7}; </s> <s xml:id="echoid-s4012" xml:space="preserve">& </s> <s xml:id="echoid-s4013" xml:space="preserve">primus ter-<lb/>minus progreſſionis a<emph style="super">3</emph> + aab, qui deſignat {1/3} trianguli æquila-<lb/>teri inſcripti in eodem Circulo, erit æqualis 3; </s> <s xml:id="echoid-s4014" xml:space="preserve">ita ut propor-<lb/>tio circuli ad triangulum æquilaterum inſcriptum ſit ut {48/7} ad <lb/>3@ id eſt 16 ad 7. </s> <s xml:id="echoid-s4015" xml:space="preserve">a vero nihilominus omnes has proportiones ab-<lb/>errare facile patet.</s> <s xml:id="echoid-s4016" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4017" xml:space="preserve">IV°. </s> <s xml:id="echoid-s4018" xml:space="preserve">Si examinemus, cur terminatio aliquando occurrat ve-<lb/>ra per methodum Autoris, ut in 7 Propoſitione, interdum <lb/>verò non; </s> <s xml:id="echoid-s4019" xml:space="preserve">reperiemus id ex eo oriri quod problema 10<emph style="super">mæ</emph> <pb o="465" file="0185" n="195" rhead="IN LIB JAC. GREG."/> Propoſitionis non bene ſit ſolutum; </s> <s xml:id="echoid-s4020" xml:space="preserve">Non enim ſufficit, ut <lb/>inveniatur terminatio progreſſionis convergentis, inveniſſe <lb/>quantitatem compoſitam eodem modo e primis & </s> <s xml:id="echoid-s4021" xml:space="preserve">ſecundis <lb/>terminis; </s> <s xml:id="echoid-s4022" xml:space="preserve">nam talis ſolutio tantum locum habet quando <lb/>quantitas illa detegitur, non inveſtigatâ quantitate quæ di-<lb/>citur z in 7 Propoſitione; </s> <s xml:id="echoid-s4023" xml:space="preserve">vel quum eadem quantitas non <lb/>componitur ex ulla quantitate quæ datur in terminis progreſ-<lb/>ſionis, ut in 7<emph style="super">a</emph>. </s> <s xml:id="echoid-s4024" xml:space="preserve">Propoſitione, ubi z = {me/d}; </s> <s xml:id="echoid-s4025" xml:space="preserve">nam Autor ſu-<lb/>mendo z = {mae - mbe/ad - bd} non videtur obſervaſſe, diviſionem poſ-<lb/>ſe fieri per a-b.</s> <s xml:id="echoid-s4026" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4027" xml:space="preserve">In exemplo propoſitionis 10, non quæritur quantitas z, <lb/>ſed z appellatur ipſa terminatio; </s> <s xml:id="echoid-s4028" xml:space="preserve">& </s> <s xml:id="echoid-s4029" xml:space="preserve">in tranſitu notandum eſt <lb/>exemplum illud allatum eſſe citra propoſitum nam progreſ-<lb/>ſio, cujus primi termini ſunt a, b, & </s> <s xml:id="echoid-s4030" xml:space="preserve">ſecundi & </s> <s xml:id="echoid-s4031" xml:space="preserve">Vab, <lb/>{aa/Vab} non eſt progreſſio conyergens, neque habet terminatio-<lb/>nem, quamvis autor illam inveniat.</s> <s xml:id="echoid-s4032" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4033" xml:space="preserve">Quod attinet ad methodum, quam Autor propoſuit ad ap-<lb/>propinquandum per numeros ad dimenſionem Circuli; </s> <s xml:id="echoid-s4034" xml:space="preserve">@<emph style="super">us</emph> <lb/>Hugenius dixit ſibi videri ſe quid magis accuratum dediſſe <lb/>in libro cui titulus eſt de Circuli magnitudine, edito 1654. <lb/></s> <s xml:id="echoid-s4035" xml:space="preserve">Addidit illa, quæ Gregorius habet de dimenſione Hy-<lb/>perbolæ & </s> <s xml:id="echoid-s4036" xml:space="preserve">ratione quam habet cum Logarithmis, bona <lb/>quidem eſſe ſed minime nova membris ſocietatis, cum poſ-<lb/>ſint recordari, ſe ipſis idem jam propoſuiſſe, regulamque ad in-<lb/>veniendos logarithmos, jamdudum actis eorum inſertam eſſe. </s> <s xml:id="echoid-s4037" xml:space="preserve"><lb/>Se etiam non credere illud novum videri regiæ ſocietati An-<lb/>glicanæ, cum qua jam ante plures annos communicavit me-<lb/>thodum ad invenienda pondera aëris in diverſis altitudini-<lb/>bus ſupra terram, quæ fundamentum in hac ipsâ Hyperbolæ <lb/>dimenſione habet.</s> <s xml:id="echoid-s4038" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4039" xml:space="preserve">Hæc autem eſt methodus, quam D<emph style="super">us</emph> Hugenius propoſuit <lb/>ad inveniendum per logarithmos dimenſionem Spatii Hy-<lb/>perbolici contenti inter Curvam & </s> <s xml:id="echoid-s4040" xml:space="preserve">unam ex Aſymptotis & </s> <s xml:id="echoid-s4041" xml:space="preserve"><lb/>duas lineas paralelas ad alteram Aſymptoton, data ratione quam <pb o="466" file="0186" n="196" rhead="HUGENII OBSERVATIONES"/> inter ſe habent duæ hæ lineæ. </s> <s xml:id="echoid-s4042" xml:space="preserve">Si logarithmus differentiæ lo-<lb/>garithmorum horum numerorum ſemper addatur ad 0, <lb/>36221, 56868, ſumma erit logarithmus numeri partium, <lb/>quas continet Spatium Hyperbolicum, illarum quarum <lb/>1.</s> <s xml:id="echoid-s4043" xml:space="preserve">0000000000 conv<unsure/>inet parallelogr ammum Hyperbolæ, id <lb/>eſt illud, quod eſt contentum a duabus lineis ductis ex uno <lb/>Hyperbolæ puncto, & </s> <s xml:id="echoid-s4044" xml:space="preserve">quarum quævis parallela eſt uni ex <lb/>Aſymptotis; </s> <s xml:id="echoid-s4045" xml:space="preserve">Exempli gratia ſi parallelæ, quæ includunt ſpa-<lb/>tium Hyperbolicum ſint inter ſe ut 36 ad 5 operatio hoc pa-<lb/>cto fiet.</s> <s xml:id="echoid-s4046" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>1, 5563025008 # Logarithmns 36. <lb/>0, 6989700043 # Logar. 5. <lb/>0, 8573324965. # Differentia. <lb/>9, 9331492856 # Logar. Differentiæ. <lb/>0, 36221 56868 # Logar. qui ſemper additur. <lb/>10, 2953649724; # Logar. cujus numerus facit contentum \\ in Spatio Hyperbolico, quod hic eſt 1.9740810180. <lb/></note> <figure> <image file="0186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0186-01"/> </figure> </div> <div xml:id="echoid-div225" type="section" level="1" n="111"> <head xml:id="echoid-head153" xml:space="preserve">III. <lb/>DOMINI GREGORII RESPONSUM <lb/>AD <lb/>ANIMADVERSIONES <lb/>DOMINI HUGENII, <lb/>IN EJUS LIBRUM, DE VERA CIRCULI <lb/>ET HYPERBOLÆ QUADRATURA.</head> <p> <s xml:id="echoid-s4047" xml:space="preserve">AD ea quæ dicit D. </s> <s xml:id="echoid-s4048" xml:space="preserve">Hugenius contra meam Circ. <lb/></s> <s xml:id="echoid-s4049" xml:space="preserve">& </s> <s xml:id="echoid-s4050" xml:space="preserve">Hyperb. </s> <s xml:id="echoid-s4051" xml:space="preserve">Quadraturam, ingenuè fateor (cum <lb/>illa ſcriberem) me non animadvertiſſe exemplum <lb/>in prop. </s> <s xml:id="echoid-s4052" xml:space="preserve">10. </s> <s xml:id="echoid-s4053" xml:space="preserve">non eſſe ſeriem convergentem ex- <pb o="467" file="0187" n="197" rhead="JAC. GREG. RESPONS."/> perientiam enim feci ſolummodo de primis & </s> <s xml:id="echoid-s4054" xml:space="preserve">ſecundis ter-<lb/>minis, non conſiderando tertios cum primis coincidere, nam <lb/>ratiociniis inſiſtebam, de exemplis parum ſolicitus. </s> <s xml:id="echoid-s4055" xml:space="preserve">Ut au-<lb/>tem appareat in hoc nihil contineri contra noſtram Doctri-<lb/>nam, agedum hoc loco 10: </s> <s xml:id="echoid-s4056" xml:space="preserve">prop. </s> <s xml:id="echoid-s4057" xml:space="preserve">totidem verbis, ſed cum <lb/>legitimo exemplo repetamus.</s> <s xml:id="echoid-s4058" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div226" type="section" level="1" n="112"> <head xml:id="echoid-head154" xml:space="preserve">PROP. X. PROBLEMA.</head> <p style="it"> <s xml:id="echoid-s4059" xml:space="preserve">Ex data quantitate eodem modo compoſita à duobus <lb/>terminis convergentibus cujuſcunque ſeriei convergen-<lb/>tis, quo componitur ex terminis convergentibus ejuſ-<lb/>dem ſeriei immediate ſequentibus; </s> <s xml:id="echoid-s4060" xml:space="preserve">ſeriei propoſitæ <lb/>terminationem invenire.</s> <s xml:id="echoid-s4061" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4062" xml:space="preserve">Sit ſeries convergens, cujus duo termini convergentes qui-<lb/>cunque ſint a, b, & </s> <s xml:id="echoid-s4063" xml:space="preserve">termini convergentes immediatè ſe-<lb/>quentes {2 a b/a + b}, {a + b/2}, termini priores inter ſe multiplicati effi-<lb/>ciunt eandem a b, item ſequentes inter ſe multiplicati effi-<lb/>ciunt eandem a b; </s> <s xml:id="echoid-s4064" xml:space="preserve">ex his invenienda ſit propoſitæ ſeriei ter-<lb/>minatio. </s> <s xml:id="echoid-s4065" xml:space="preserve">Manifeſtum eſt, quantitatem a b eodem modo <lb/>fieri à terminis convergentibus a, b, quo à terminis conver-<lb/>gentibus immediatè ſequentibus {2 a b/a + b}, {a + b/z}: </s> <s xml:id="echoid-s4066" xml:space="preserve">& </s> <s xml:id="echoid-s4067" xml:space="preserve">quoniam quan-<lb/>titates a, b, indefinitè ponuntur pro quibuslibet totius ſe-<lb/>riei terminis convergentibus, evidens eſt, duos quoſcunque <lb/>terminos convergentes propoſitæ ſeriei inter ſe multiplica-<lb/>tos idem efficere productum, quod faciunt termini imme-<lb/>diatè ſequentes etiam inter ſe multiplicati; </s> <s xml:id="echoid-s4068" xml:space="preserve">cumque duo ter-<lb/>mini convergentes duos terminos convergentes ſemper im-<lb/>mediatè ſequantur, manifeſtum eſt, duos quoſcunque ter-<lb/>minos convergentes inter ſe multiplicatos idem ſemper effi-<lb/>cere productum, nempe a b; </s> <s xml:id="echoid-s4069" xml:space="preserve">atque ultimi termini conver-<lb/>gentes ſunt æquales, & </s> </p> </div> <div xml:id="echoid-div227" type="section" level="1" n="113"> <head xml:id="echoid-head155" style="it" xml:space="preserve">Tom. II. Nnn</head> <p> <s xml:id="echoid-s4070" xml:space="preserve">proinde ſit ultimus ille terminus, ſeu <lb/>ſeriei terminatio Z, quæ in ſe ipſam multiplicata facit <lb/> <pb o="468" file="0188" n="198" rhead="JAC. GREG. RESPONSUM"/> Z<emph style="super">2</emph> = a b; </s> <s xml:id="echoid-s4071" xml:space="preserve">eſt igitur Z. </s> <s xml:id="echoid-s4072" xml:space="preserve">ſeu ſeriei terminatio r<emph style="super">2</emph> a b, quam invenire oportuit: </s> <s xml:id="echoid-s4073" xml:space="preserve">& </s> <s xml:id="echoid-s4074" xml:space="preserve">proinde ad inveniendam cujuſcunque ſeriei convergentis terminationem opus eſt ſolummodo inve- nire quantitatem eodem modo compoſitam ex terminis con- vergentibus primis, quo componitur eadem quantitas ex ter- minis convergentibus ſecundis.</s> <s xml:id="echoid-s4075" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div228" type="section" level="1" n="114"> <head xml:id="echoid-head156" xml:space="preserve">CONSECTARIUM.</head> <p style="it"> <s xml:id="echoid-s4076" xml:space="preserve">Quoniam non refert in Problemate, ſive termini con-<lb/>vergentes, a, b, ſint primi, ſecundi vel tertii, <lb/>&</s> <s xml:id="echoid-s4077" xml:space="preserve">c. </s> <s xml:id="echoid-s4078" xml:space="preserve">manifeſtum eſt, omnis ſeriei convergentis ter-<lb/>minationem eodem modo eſſe compoſitam ex termi-<lb/>nis convergentibus primis, quo ex ſecundis, ter-<lb/>tiis, &</s> <s xml:id="echoid-s4079" xml:space="preserve">c.</s> <s xml:id="echoid-s4080" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4081" xml:space="preserve">Si quis aliud Exemplum deſideret, ſint primi termini a, b, <lb/>ſecundi r<emph style="super">7</emph> a<emph style="super">5</emph> b<emph style="super">2</emph>, r2 a b, quantitas eodem modo compo-<lb/>ſita &</s> <s xml:id="echoid-s4082" xml:space="preserve">c. </s> <s xml:id="echoid-s4083" xml:space="preserve">eſt a<emph style="super">7</emph> b<emph style="super">4</emph> & </s> <s xml:id="echoid-s4084" xml:space="preserve">ſeriei terminatio r<emph style="super">11</emph> a<emph style="super">7</emph> b<emph style="super">4</emph>: </s> <s xml:id="echoid-s4085" xml:space="preserve">videat Huge-<lb/>nius, duo exempla legitima hic adducta inquiſitionem Septi-<lb/>mæ non admittere; </s> <s xml:id="echoid-s4086" xml:space="preserve">ope tamen prop. </s> <s xml:id="echoid-s4087" xml:space="preserve">decimæ (ſuppoſita ter-<lb/>tia illa quantitate) facilè reſolvuntur, neque ullo modo con-<lb/>ſectarium reſpuunt, quod ſolummodo eſſe momenti ſatis <lb/>ſit indicaſſe; </s> <s xml:id="echoid-s4088" xml:space="preserve">plura autem exempla deſideranti millena affe-<lb/>ram.</s> <s xml:id="echoid-s4089" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4090" xml:space="preserve">Ad primam Hugenii objectionem quod ſpectat, miror <lb/>eum non conſiderâſſe præcedens conſectarium, ubi illa, <lb/>quæ deſiderat, evidenter deduco ex prop. </s> <s xml:id="echoid-s4091" xml:space="preserve">10. </s> <s xml:id="echoid-s4092" xml:space="preserve">At agnoſcit <lb/>hoc verum eſſe in illis ſeriebus, quæ ope noſtræ methodi <lb/>terminantur: </s> <s xml:id="echoid-s4093" xml:space="preserve">velim certè ut aſſignet mihi Nobiliſſimus Vir <lb/>ſeriem aliquam convergentem cum ſua terminatione, quæ <lb/>conſectarium noſtrum reſpuat; </s> <s xml:id="echoid-s4094" xml:space="preserve">vel ſi eam aſſignare non poſ-<lb/>ſit, ſolidam dubitandi rationem tantum deſidero. </s> <s xml:id="echoid-s4095" xml:space="preserve">Ut au-<lb/>tem funditus evertatur hæc objectio, ſequentem exhibeo de-<lb/>monſtrationem Geometricam.</s> <s xml:id="echoid-s4096" xml:space="preserve"/> </p> <pb o="469" file="0189" n="199" rhead="AD ANIM. HUGENII."/> <p> <s xml:id="echoid-s4097" xml:space="preserve">Sit A. </s> <s xml:id="echoid-s4098" xml:space="preserve">Polygonum regulare ſectori inſcriptum. </s> <s xml:id="echoid-s4099" xml:space="preserve">B eidem <lb/>ſimile circumſcriptum; </s> <s xml:id="echoid-s4100" xml:space="preserve">continetur ſeries convergens poly-<lb/>gonorum &</s> <s xml:id="echoid-s4101" xml:space="preserve">c. </s> <s xml:id="echoid-s4102" xml:space="preserve">ut ſit ejus terminatio ſeu circuli ſector Z: </s> <s xml:id="echoid-s4103" xml:space="preserve">ſit <lb/>X eodem modo compoſita à terminis C, D, quo Z à ter-<lb/>minis A, B; </s> <s xml:id="echoid-s4104" xml:space="preserve">dico Z & </s> <s xml:id="echoid-s4105" xml:space="preserve">X eſſe indefinitè æquales; </s> <s xml:id="echoid-s4106" xml:space="preserve">ſi non ſint <lb/>indefinitè æquales, ſit inter illas indefinita differentia a, & </s> <s xml:id="echoid-s4107" xml:space="preserve"><lb/>continuetur ſeries convergens in terminos convergentes I, K, <lb/>ita ut eorum differentia ſit minor quam a; </s> <s xml:id="echoid-s4108" xml:space="preserve">hoc <lb/> <anchor type="note" xlink:label="note-0189-01a" xlink:href="note-0189-01"/> enim abſque dubio concipi poteſt, etiamſi hic <lb/>omnes quantitates ſint indefinitæ, quoniam <lb/>definitis quantitatibus A, B, definitur etiam a, <lb/>ſed adhuc reſtat K-1 quantitas indeterminata <lb/>in infinitum decreſcens. </s> <s xml:id="echoid-s4109" xml:space="preserve">Manifeſtum eſt, ſe-<lb/>ctorem Z eſſe indefinitè minorem quam K, & </s> <s xml:id="echoid-s4110" xml:space="preserve"><lb/>majorem quam I: </s> <s xml:id="echoid-s4111" xml:space="preserve">item quoniam Zeodem mo-<lb/>do componitur ex quantitatibus A, B, quo X. </s> <s xml:id="echoid-s4112" xml:space="preserve">è quantita-<lb/>tibus C, D, & </s> <s xml:id="echoid-s4113" xml:space="preserve">Z indefinitè minor eſt quam K & </s> <s xml:id="echoid-s4114" xml:space="preserve">major <lb/>quam I, patet ex Proprietatibus ſerierum convergentium, <lb/>X etiam eſſe indefinitè majorem quàm I, & </s> <s xml:id="echoid-s4115" xml:space="preserve">minorem quàm <lb/>K (eſt enim revera indefinitè major quàm L & </s> <s xml:id="echoid-s4116" xml:space="preserve">minor quam <lb/>M) & </s> <s xml:id="echoid-s4117" xml:space="preserve">proinde ſunt quatuor quantitates indefinitæ, quarum <lb/>maxima & </s> <s xml:id="echoid-s4118" xml:space="preserve">minima ſunt I, K, intermediæ autem Z & </s> <s xml:id="echoid-s4119" xml:space="preserve">X, <lb/>& </s> <s xml:id="echoid-s4120" xml:space="preserve">ideo differentia extremarum K-I major eſt quàm a diffe-<lb/>rentia mediarum, quod eſt abſurdum, ponitur enim minor: <lb/></s> <s xml:id="echoid-s4121" xml:space="preserve">quantitates ergò Z & </s> <s xml:id="echoid-s4122" xml:space="preserve">X non ſunt indefinitè inæquales, & </s> <s xml:id="echoid-s4123" xml:space="preserve"><lb/>ideo ſunt indefinitè æquales, quod demonſtrandum erat. </s> <s xml:id="echoid-s4124" xml:space="preserve"><lb/>Manifeſtum eſt hanc demonſtrationem eodem modo appli-<lb/>cabilem eſſe omni ſeriei convergenti.</s> <s xml:id="echoid-s4125" xml:space="preserve"/> </p> <div xml:id="echoid-div228" type="float" level="2" n="1"> <note position="right" xlink:label="note-0189-01" xlink:href="note-0189-01a" xml:space="preserve"> <lb/>A # B <lb/>C # D <lb/>E # F <lb/>G # H a <lb/>I # K <lb/>L # M <lb/># Z <lb/># X <lb/></note> </div> <p> <s xml:id="echoid-s4126" xml:space="preserve">In objectionibus 2, 3, & </s> <s xml:id="echoid-s4127" xml:space="preserve">4, contra ſuas ipſius imaginatio-<lb/>nes argumentatur Hugenius: </s> <s xml:id="echoid-s4128" xml:space="preserve">Ego enim ſatis dilucidè affir-<lb/>mo in Scholio propoſit. </s> <s xml:id="echoid-s4129" xml:space="preserve">5. </s> <s xml:id="echoid-s4130" xml:space="preserve">& </s> <s xml:id="echoid-s4131" xml:space="preserve">in fine prop. </s> <s xml:id="echoid-s4132" xml:space="preserve">9. </s> <s xml:id="echoid-s4133" xml:space="preserve">Septimam & </s> <s xml:id="echoid-s4134" xml:space="preserve">no-<lb/>nam propoſitionem eſſe Particularem, unamquamque ſuo ca-<lb/>ſui; </s> <s xml:id="echoid-s4135" xml:space="preserve">item in Prop. </s> <s xml:id="echoid-s4136" xml:space="preserve">decima (quàm ergo pro generali ſubſti-<lb/>tuo) evidenter ſuppono, & </s> <s xml:id="echoid-s4137" xml:space="preserve">non quæro, illam quantitatem <lb/>eo modo compoſitam ex primis, quo ex ſecundis terminis <lb/>convergentibus; </s> <s xml:id="echoid-s4138" xml:space="preserve">ſatis enim ſcio, talem methodum genera- <pb o="470" file="0190" n="200" rhead="JAC. GREG. RESP."/> lem eſſe impoſſibilem. </s> <s xml:id="echoid-s4139" xml:space="preserve">Sed omnium maxime admiror, Cla-<lb/>riſſimum Virum non animadvertiſſe in 8. </s> <s xml:id="echoid-s4140" xml:space="preserve">definitione, Quan-<lb/>titates C, D, E, compoſitionem ingredientes, ſemper eſſe <lb/>eaſdem, nempe definitas & </s> <s xml:id="echoid-s4141" xml:space="preserve">invariabiles, ipſos autem termi-<lb/>nos A, B, eſſe indefinitos & </s> <s xml:id="echoid-s4142" xml:space="preserve">variabiles, nimirum in F, <lb/>G, & </s> <s xml:id="echoid-s4143" xml:space="preserve">infinitos alios: </s> <s xml:id="echoid-s4144" xml:space="preserve">at quis eſt qui non videt Hugenii <lb/>{b2 m/a2 + ba} non minus eſſe indefinitam, quam ſunt ipſi termini? <lb/></s> <s xml:id="echoid-s4145" xml:space="preserve">Deinde in Proœmio noſtræ Geometriæ Partis univerſalis, <lb/>ſic dico. </s> <s xml:id="echoid-s4146" xml:space="preserve">Alii objiciunt contra prop. </s> <s xml:id="echoid-s4147" xml:space="preserve">11. </s> <s xml:id="echoid-s4148" xml:space="preserve">ita: </s> <s xml:id="echoid-s4149" xml:space="preserve">ſi addatur <lb/>a<emph style="super">3</emph> termino a<emph style="super">3</emph> + a<emph style="super">2</emph>b & </s> <s xml:id="echoid-s4150" xml:space="preserve">temino ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, enervetur vis u-<lb/>triuſque demonſtrationis. </s> <s xml:id="echoid-s4151" xml:space="preserve">Reſpondeo, a<emph style="super">3</emph> eſſe quantitatem <lb/>indefinitam, & </s> <s xml:id="echoid-s4152" xml:space="preserve">alias quantitates indefinitas præter ipſos <lb/>terminos convergentes compoſitionem non poſſe ingredi, quod <lb/>analyſtam latere non poteſt: </s> <s xml:id="echoid-s4153" xml:space="preserve">Eodem modo reſpondeo Huge-<lb/>nio {bbm/a2 + ab} eſſe quantitatem indefinitam, & </s> <s xml:id="echoid-s4154" xml:space="preserve">ideo compoſitio-<lb/>nem non poſſe ingredi: </s> <s xml:id="echoid-s4155" xml:space="preserve">Si autem mihi objiciat, in ſeptima <lb/>me credidiſſe, {mae - mbe/ad - bd} fuiſſe quantitatem indefinitam; </s> <s xml:id="echoid-s4156" xml:space="preserve">Re-<lb/>ſpondeo, etiamſi diviſio per a-b à me ſatis inconſideratè ne-<lb/>glecta ſit, apertè tamen conſtat, me hoc cognoviſſe, ex di-<lb/>verſitate methodorum, quibus utor in ſeptima & </s> <s xml:id="echoid-s4157" xml:space="preserve">decima, <lb/>quippe iſta particulari, in qua quantitatem illam quæro, & </s> <s xml:id="echoid-s4158" xml:space="preserve"><lb/>hac generali, in qua illam ſuppono; </s> <s xml:id="echoid-s4159" xml:space="preserve">nulla enim alia ratio <lb/>hujus diverſitatis excogitari poteſt; </s> <s xml:id="echoid-s4160" xml:space="preserve">quod etiam ex ipſis ſepti-<lb/>ma & </s> <s xml:id="echoid-s4161" xml:space="preserve">decima eſt manifeſtum, cum appellem ſemper termi-<lb/>nos convergentes quantitates indefinitas, hoc ipſo ſatis ſi-<lb/>gnificans, nullas alias quantitates indefinitas calculo in-<lb/>eſſe.</s> <s xml:id="echoid-s4162" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4163" xml:space="preserve">Semper credidi in rebus ſcientificis verba ita candidè eſſe <lb/>explicanda (ſi modo poſſibile ſit) ut diſcurſus nullum inclu-<lb/>dat abſurdum; </s> <s xml:id="echoid-s4164" xml:space="preserve">at Hugenius fatis percipit, diſcurſum nihil <lb/>continere abſurdi, modò nulla quantitas indefinita præter i-<lb/>pſos terminos compoſitionem ingrediatur; </s> <s xml:id="echoid-s4165" xml:space="preserve">judicat tamen abſ-<lb/>que omni ratione, me contrarium exiſtimaſſe; </s> <s xml:id="echoid-s4166" xml:space="preserve">libenter enim <lb/>optarem Hugenium aſſignaſſe locum, ubi aſſero, illam in- <pb o="471" file="0191" n="201" rhead="AD ANIMADV. HUGENII."/> quiſitionem Septimæ eſſe univerſalem. </s> <s xml:id="echoid-s4167" xml:space="preserve">Dico igitur & </s> <s xml:id="echoid-s4168" xml:space="preserve">de-<lb/>claro me intelligere, nullam quantitatem indefinitam præter <lb/>ipſos terminos convergentes compoſitionem poſſe ingredi. <lb/></s> <s xml:id="echoid-s4169" xml:space="preserve">Atque ita corruunt tres ultimæ Hugenii ſive diverſæ obje-<lb/>ctiones, ſive ejuſdem portiones; </s> <s xml:id="echoid-s4170" xml:space="preserve">neſcio enim, quare in tot <lb/>partes dividatur.</s> <s xml:id="echoid-s4171" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4172" xml:space="preserve">Præcedentibus perceptis, evidentiſſimum eſt, Circuli, El-<lb/>lipſeos, vel Hyperbolæ ſectorem eſſe terminationem ſeriei <lb/>convergentis, cujus primi termini a<emph style="super">3</emph> + a<emph style="super">2</emph> b, ab<emph style="super">2</emph> + b<emph style="super">3</emph>, & </s> <s xml:id="echoid-s4173" xml:space="preserve"><lb/>ſecundi ba<emph style="super">2</emph> + b<emph style="super">2</emph>a, <emph style="sub">2</emph>b<emph style="super">2</emph>a <anchor type="note" xlink:href="" symbol="*"/> & </s> <s xml:id="echoid-s4174" xml:space="preserve">proinde Sectorem eodem mo- <anchor type="note" xlink:label="note-0191-01a" xlink:href="note-0191-01"/> <anchor type="note" xlink:label="note-0191-02a" xlink:href="note-0191-02"/> do componi ex primis terminis quo ex ſecundis <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s4175" xml:space="preserve">atque evidens eſt, nullam dari quantitatem eodem modo ana-<lb/>lyticè compoſitam ex primis terminis quo ex ſecundis, quo-<lb/>niam primos eodem modo analyticè tractando quo ſe-<lb/>cundos, ſemper reſtat altior poteſtas ipſius a in primorum <lb/>producto, quam in producto ſecundorum; </s> <s xml:id="echoid-s4176" xml:space="preserve">de hoc (ſi non cre-<lb/>datur) fiat experientia, & </s> <s xml:id="echoid-s4177" xml:space="preserve">conſtabit non ſolum aſſertionis veri-<lb/>tas, ſed etiam ejuſdem demonſtratio; </s> <s xml:id="echoid-s4178" xml:space="preserve">quando autem altior eſt <lb/>ejuſdem poteſtas in una quantitate quàm in altera, nulla datur <lb/>indefinita æquatio, de qua hic tantum loquimur, hoc eſt, <lb/>ut (poſitis a, b, ad libitum,) æqualitas ſemper rite proce-<lb/>dat. </s> <s xml:id="echoid-s4179" xml:space="preserve">Atque hæc eſt ſumma non ſolum propoſitionis unde-<lb/>cimæ ſed etiam totius noſtræ Circ. </s> <s xml:id="echoid-s4180" xml:space="preserve">& </s> <s xml:id="echoid-s4181" xml:space="preserve">Hyp. </s> <s xml:id="echoid-s4182" xml:space="preserve">Quadraturæ, <lb/>ab Hugenio adhuc intactæ. </s> <s xml:id="echoid-s4183" xml:space="preserve">Gratias tamen ago nobiliſſimo <lb/>viro, quod meas qualeſcunque lucubrationes examinare di-<lb/>gnatus eſt, hinc enim mihi data eſt occaſio illas fuſius ex-<lb/>plicandi & </s> <s xml:id="echoid-s4184" xml:space="preserve">confirmandi. </s> <s xml:id="echoid-s4185" xml:space="preserve">Num Hugeniana methodus cir-<lb/>culum menſurandi mea ſit præciſior, experientiæ relinquo <lb/>judicandum; </s> <s xml:id="echoid-s4186" xml:space="preserve">quod autem noſtra, Hyperbolam quadrandi, <lb/>illi etiam innotuerat, de hoc nihil habeo quod dicam, niſi <lb/>quod mihi gratuler, inventa mea ipſo Hugenio non æſtima-<lb/>@i indigna.</s> <s xml:id="echoid-s4187" xml:space="preserve"/> </p> <div xml:id="echoid-div229" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve">Schol. 5.</note> <note symbol="*" position="right" xlink:label="note-0191-02" xlink:href="note-0191-02a" xml:space="preserve">Conſ. 10.</note> </div> <pb o="472" file="0192" n="202" rhead="HUGENII EXCEPSIO"/> <figure> <image file="0192-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0192-01"/> </figure> </div> <div xml:id="echoid-div231" type="section" level="1" n="115"> <head xml:id="echoid-head157" xml:space="preserve">IV. <lb/>EXCERPTA EX LITERIS <lb/>D<emph style="super">ni</emph>. HUGENII <lb/>DE RESPONSO, QUOD <lb/>D<emph style="super">nus</emph>. GREGORIUS <lb/>DEDIT AD EXAMEN LIBRI, CUI TITULUS EST, <lb/>VERA CIRCULI ET HYPERBOLÆ QUADRATURA.</head> <p> <s xml:id="echoid-s4188" xml:space="preserve">In circuli quadraturam inquirentes Geometræ, va-<lb/>ria & </s> <s xml:id="echoid-s4189" xml:space="preserve">quidem pulcherrima detexere, ideo ne u-<lb/>tili exercitio deſtituantur adverſus Gregorium de-<lb/>fendere conſtitui, quadraturam hanc detegi poſ-<lb/>ſe; </s> <s xml:id="echoid-s4190" xml:space="preserve">neque huc uſque reſponſionem hanc diſtuliſſem, niſi oc-<lb/>caſione controverſiæ noſtræ examen ſuſcepiſſem exactiſſimum <lb/>de proxima Circuli & </s> <s xml:id="echoid-s4191" xml:space="preserve">Hyperboles Quadratura: </s> <s xml:id="echoid-s4192" xml:space="preserve">in quo me <lb/>interpellarunt ſæpe alia impedimenta.</s> <s xml:id="echoid-s4193" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4194" xml:space="preserve">Dico nunc primo; </s> <s xml:id="echoid-s4195" xml:space="preserve">Quod ſpectat impoſſibilitatem analy-<lb/>ticam quadraturæ illarum figurarum, tantum abeſſe etiam poſt <lb/>ſupplementum demonſtrationum, quod D<emph style="super">nus</emph> Gregorius dedit, <lb/>impoſſibilitatem hanc bene probatam eſſe, ut adhuc dum in-<lb/>certum maneat, num Circulus & </s> <s xml:id="echoid-s4196" xml:space="preserve">quadratum diametri ejus ſint <lb/>incommenſurabilia, id eſt an non horum ratio per numeros <lb/>poſſit exprimi; </s> <s xml:id="echoid-s4197" xml:space="preserve">quod etiam applicari poteſt portioni deter-<lb/>minatæ Hyperboles & </s> <s xml:id="echoid-s4198" xml:space="preserve">figuræ huic inſcriptæ. </s> <s xml:id="echoid-s4199" xml:space="preserve">Quod ut demon-<lb/>ſtremus ſufficit obſervaſſe; </s> <s xml:id="echoid-s4200" xml:space="preserve">quod XI ejus Propoſitio & </s> <s xml:id="echoid-s4201" xml:space="preserve">ſup-<lb/>plementum nil probent quando in ejus ſerie convergen-<lb/>te determinantur quantitates a & </s> <s xml:id="echoid-s4202" xml:space="preserve">b per numeros ra-<lb/>tionales vel ſurdos; </s> <s xml:id="echoid-s4203" xml:space="preserve">quoniam tunc terminatio et-<lb/>iam poterit eſſe numerus quidam ſimilis; </s> <s xml:id="echoid-s4204" xml:space="preserve">& </s> <s xml:id="echoid-s4205" xml:space="preserve">con-<lb/>trarium per illam propoſitionem demonſtrari non po- <pb o="473" file="0193" n="203" rhead="AD RESP. JAC. GREG."/> terit, quoniam tunc ignotum erit, quomodo terminatio com-<lb/>poſita ſit ex primis & </s> <s xml:id="echoid-s4206" xml:space="preserve">ſecundis terminis. </s> <s xml:id="echoid-s4207" xml:space="preserve">E. </s> <s xml:id="echoid-s4208" xml:space="preserve">G. </s> <s xml:id="echoid-s4209" xml:space="preserve">Si a ſit 1 & </s> <s xml:id="echoid-s4210" xml:space="preserve">b, 2 <lb/>quomodo probabitur per <emph style="sc">XI</emph> ejus Propoſitionem, termina-<lb/>tionem non eſſe {7/3}. </s> <s xml:id="echoid-s4211" xml:space="preserve">Ut ergo concludatur, rationem Circu-<lb/>li ad quadratum diametri ejus non eſſe analyticam, demon-<lb/>ftrari non ſolum debet, quod ſector Circuli non ſit ana-<lb/>lyticus indefinite ad figuram ſuam inſcriptam, licet illa de-<lb/>monſtratio non careat ſuâ pulchritudine, ſed quod illud etiam <lb/>verum ſit in omni caſu definito.</s> <s xml:id="echoid-s4212" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4213" xml:space="preserve">Dico præterea, ſi quantitates a & </s> <s xml:id="echoid-s4214" xml:space="preserve">b maneant indetermi-<lb/>natæ, terminationem forte reduci ad æquationem ex iis, qua-<lb/>rum radices dari nequeunt; </s> <s xml:id="echoid-s4215" xml:space="preserve">& </s> <s xml:id="echoid-s4216" xml:space="preserve">contrarium probari per <emph style="sc">XI</emph> e-<lb/>jus Propoſitionem, neque per Supplementum, non poterit, ni-<lb/>hilominus ſi illa terminatio reducta ſit ad quandam æquatio-<lb/>nem ejus naturæ, crederem Quadraturam Geometricè eſſe in-<lb/>ventam per interſectionem quarundam linearum curvarum, <lb/>quæ in Geometriâ admittuntur.</s> <s xml:id="echoid-s4217" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4218" xml:space="preserve">Non inhærebo in aliis objectionibus, quas propoſui; </s> <s xml:id="echoid-s4219" xml:space="preserve">Di-<lb/>cam tantum, has, ſicuti locum amplius non habent poſt cor-<lb/>rectionem Dn. </s> <s xml:id="echoid-s4220" xml:space="preserve">Gregorii, ita antea firmo fundamento nixas <lb/>fuiſſe; </s> <s xml:id="echoid-s4221" xml:space="preserve">quoniam omiſſa diviſione neceſſariâ per a@ b in tot <lb/>locis propoſitionis <emph style="sc">VII</emph>. </s> <s xml:id="echoid-s4222" xml:space="preserve">poterat præſumi illum non noviſſe <lb/>talem diviſionem eſſe poſſibilem, & </s> <s xml:id="echoid-s4223" xml:space="preserve">conſequenter illum <lb/>credidiſſe, in compoſitione de qua agitur indefinitas ad-<lb/>mitti debere quantitates.</s> <s xml:id="echoid-s4224" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4225" xml:space="preserve">Tranſeo igitur ad comparationem noſtrarum methodorum <lb/>pro dimenſione Circuli proxima: </s> <s xml:id="echoid-s4226" xml:space="preserve">Certum eſt, primas ejus <lb/>approximationes quarum fundamentum ſunt ipſius propo-<lb/>fitiones <emph style="sc">XX</emph>, & </s> <s xml:id="echoid-s4227" xml:space="preserve"><emph style="sc">XXI</emph>. </s> <s xml:id="echoid-s4228" xml:space="preserve">eaſdem eſſe cum iis, quas dedi in tra-<lb/>ctatu de Circuli magnitudine; </s> <s xml:id="echoid-s4229" xml:space="preserve">ubi demonſtravi eadem illa <lb/>Theoremata, ſcil. </s> <s xml:id="echoid-s4230" xml:space="preserve">ſi Polygonum in Circulo inſcriptum ſit <lb/>a; </s> <s xml:id="echoid-s4231" xml:space="preserve">& </s> <s xml:id="echoid-s4232" xml:space="preserve">Polygonum ſimul circumſcriptum ſit d: </s> <s xml:id="echoid-s4233" xml:space="preserve">contentum <lb/>Circuli minus eſſe quam {1/3}a + {2/3} d <anchor type="note" xlink:href="" symbol="*"/>, ſed majus quam {4/3} c- <anchor type="note" xlink:label="note-0193-01a" xlink:href="note-0193-01"/> {1/3} a <anchor type="note" xlink:href="" symbol="*"/>, poſito c pro Polygono inſcripto quod duplo majorem <anchor type="note" xlink:label="note-0193-02a" xlink:href="note-0193-02"/> habeat laterum numerum quam a vel d. </s> <s xml:id="echoid-s4234" xml:space="preserve">Quidquid hic dico <pb o="474" file="0194" n="204" rhead="HUGENII EXCEPTIO"/> & </s> <s xml:id="echoid-s4235" xml:space="preserve">in ſequentibus dicam de Circulo debet intelligi pariter de <lb/>ſectore Circuli.</s> <s xml:id="echoid-s4236" xml:space="preserve"/> </p> <div xml:id="echoid-div231" type="float" level="2" n="1"> <note symbol="* it" position="right" xlink:label="note-0193-01" xlink:href="note-0193-01a" xml:space="preserve">Vide ſupræ <lb/>pag. 361. <lb/>pr. <emph style="sc">VI</emph>.</note> <note symbol="* it" position="right" xlink:label="note-0193-02" xlink:href="note-0193-02a" xml:space="preserve">Vide ibid. <lb/>prop. <emph style="sc">V</emph>.</note> </div> <p> <s xml:id="echoid-s4237" xml:space="preserve">Præter hanc approximationem D. </s> <s xml:id="echoid-s4238" xml:space="preserve">Gregorius aliam pro-<lb/>ponit in fine ſuæ <emph style="sc">XXV</emph> Propoſitionis, quam admirandam di-<lb/>cit, cujus demonſtrationem ſe ignorare fatetur; </s> <s xml:id="echoid-s4239" xml:space="preserve">hæc eſt, in-<lb/>ter duos terminos, ſtatim memoratos {1/3} a + {2/3} d & </s> <s xml:id="echoid-s4240" xml:space="preserve">{4/3} c - {1/3} a, <lb/>inventis quatuor mediis quantitatibus in proportione Arith-<lb/>metica, aſſerit maximam harum quantitatum adeo Circuli <lb/>magnitudini vicinam eſſe, ut, ſi in numeris, qui deſignant <lb/>Polygona ſimilia a & </s> <s xml:id="echoid-s4241" xml:space="preserve">d, prima notarum triens ſit eadem, <lb/>error ad unitatem non pertingat.</s> <s xml:id="echoid-s4242" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4243" xml:space="preserve">Sed invenio hanc approximationem in Circulo veram non <lb/>eſſe licet in Hyperbola locum habeat, &</s> <s xml:id="echoid-s4244" xml:space="preserve">, dum in hac utimur <lb/>maxima quatuor mediarum Arithmeticarum proportiona-<lb/>lium, minimam pro approximatione Circuli adhibendam <lb/>eſſe.</s> <s xml:id="echoid-s4245" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4246" xml:space="preserve">Ita minima quatuor mediarum proportionalium inter <lb/>terminos dictos primæ approximationis erit {16 c + 2 d - 3 a/15}, uti <lb/>facile eſt videre per Calculum; </s> <s xml:id="echoid-s4247" xml:space="preserve">& </s> <s xml:id="echoid-s4248" xml:space="preserve">probare poſſum non ſo-<lb/>lum experientiâ, ſed & </s> <s xml:id="echoid-s4249" xml:space="preserve">per demonſtrationem quantitatem <lb/>hanc, poſitis numeris quorum prima notarum triens eadem <lb/>eſt, ex primentibus polygonis a & </s> <s xml:id="echoid-s4250" xml:space="preserve">d, a vera Circuli ma-<lb/>gnitudine non aberrare niſi in duabus ultimis notis, & </s> <s xml:id="echoid-s4251" xml:space="preserve">ple-<lb/>rumque in omnibus notis & </s> <s xml:id="echoid-s4252" xml:space="preserve">ulterius cum vera magnitudi-<lb/>ne conincidere, quam tamen ſemper ſuperat, cum e contra-<lb/>rio maxima quatuor Mediarum, qua utitur D<emph style="super">us</emph> Gregorius in <lb/>Hyperbola deficiat.</s> <s xml:id="echoid-s4253" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4254" xml:space="preserve">Inveni præterea, approximationem hanc pro Circulo non <lb/>æque accuratam eſſe, ac eſt illa quam dedi in Tractatu deCircu-<lb/>li magnitudine, juxta quam, quando a, c & </s> <s xml:id="echoid-s4255" xml:space="preserve">d deſignant eadem <lb/>Polygona, ac ſupra, terminus excedens contentum Circuli <lb/>eſt a + {10cc - 10aa <anchor type="note" xlink:href="" symbol="*"/>/6c + 9a} Neque demonſtratio difficilis eſt, quoniam <anchor type="note" xlink:label="note-0194-01a" xlink:href="note-0194-01"/> ſi neges illum terminum eſſe minorem ideoque magis exa- <pb o="475" file="0195" n="205" rhead="AD RESP. JAC. GREG."/> ctum quam præcedentem {16c + 2d - 3a/15}, ſequetur, cubum c - a non<unsure/> <lb/>fore majorem nihilo, & </s> <s xml:id="echoid-s4256" xml:space="preserve">c non majorem quam a contra hypo-<lb/>theſin, uti facile videre eſt per calculum analyticum, obſervan-<lb/>do quod d ſit = {cc/a}.</s> <s xml:id="echoid-s4257" xml:space="preserve"/> </p> <div xml:id="echoid-div232" type="float" level="2" n="2"> <note symbol="* it" position="left" xlink:label="note-0194-01" xlink:href="note-0194-01a" xml:space="preserve">vide ſupra <lb/>p. 383. in <lb/>ſ<unsure/>ine.</note> </div> <p> <s xml:id="echoid-s4258" xml:space="preserve">Poſſunt etiam a & </s> <s xml:id="echoid-s4259" xml:space="preserve">c ſumi pro circumferentiis Polygono-<lb/>rum inſcriptorum, quorum unum ſubduplum laterum ha-<lb/>bet numerum; </s> <s xml:id="echoid-s4260" xml:space="preserve">Et tum terminus a + {10cc - 10aa/6c + 9a} eſt longitu-<lb/>do circumferentiæ Circuli, aut arcus ſectoris parum admo-<lb/>dum excedens, ut ſi notarum triens in a & </s> <s xml:id="echoid-s4261" xml:space="preserve">c ſit eadem, er-<lb/>ror dari non poterit niſi in ultima nota, & </s> <s xml:id="echoid-s4262" xml:space="preserve">ſæpius nequi-<lb/>dem in quatuor vel quinque notis ſequentibus ultra illas quæ <lb/>dantur in numeris a vel c.</s> <s xml:id="echoid-s4263" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4264" xml:space="preserve">Sed & </s> <s xml:id="echoid-s4265" xml:space="preserve">ut illi, qui contem plationes has negligunt, tamen <lb/>quid commodi ex noſtra controverſia capiant, addam hic <lb/>Geometricam conſtructionem ex ultimâ approximatione <lb/>deductam qua invenitur longitudo arcus Circuli dati adeo <lb/>acurata quam ad uſum deſideratur.</s> <s xml:id="echoid-s4266" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4267" xml:space="preserve">Sit arcus Circuli, qui non excedat ſemicircumferentiam <lb/> <anchor type="note" xlink:label="note-0195-01a" xlink:href="note-0195-01"/> A B C, cujus ſubtenſa ſit A C; </s> <s xml:id="echoid-s4268" xml:space="preserve">dividat hanc ut & </s> <s xml:id="echoid-s4269" xml:space="preserve">arcum in duas <lb/>partes æquales linea B D.</s> <s xml:id="echoid-s4270" xml:space="preserve"/> </p> <div xml:id="echoid-div233" type="float" level="2" n="3"> <note position="right" xlink:label="note-0195-01" xlink:href="note-0195-01a" xml:space="preserve">TAB. XLIV. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s4271" xml:space="preserve">Ducta ſubtenſa A B ſumantur ejus {2/3} & </s> <s xml:id="echoid-s4272" xml:space="preserve">ponantur ab A ad E <lb/>in producta lineâ C D; </s> <s xml:id="echoid-s4273" xml:space="preserve">tum diminuta D E decimâ parte E F, <lb/>ducatur F B, & </s> <s xml:id="echoid-s4274" xml:space="preserve">tandem ipſi perpendicularis B G. </s> <s xml:id="echoid-s4275" xml:space="preserve">Erit linea <lb/>A G æqualis arcui A B, vel dupla linea æqualis arcui A B C, <lb/>quæ tam parum excedet, ut tunc etiam, quum arcus erit æ-<lb/>qualis ſemicircumferentiæ Circuli exceſſus non ſit {1/1400}. </s> <s xml:id="echoid-s4276" xml:space="preserve">lon-<lb/>gitudinis; </s> <s xml:id="echoid-s4277" xml:space="preserve">ſed ſi tantum ſit {1/3} circumferentiæ, differentia non erit <lb/>{1/13000}, ſi autem arcus tantum ſit pars quarta circumferentiæ er-<lb/>ror non erit {1/90000} longitudinis.</s> <s xml:id="echoid-s4278" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4279" xml:space="preserve">Poſſem hic addere approximationem & </s> <s xml:id="echoid-s4280" xml:space="preserve">conſtructionem <lb/>omnino ſimilem pro quadratura Hyperboles, paulo magis <lb/>accedentem ad veram quam media Arithmetica D<emph style="super">ni</emph>. </s> <s xml:id="echoid-s4281" xml:space="preserve">Grego- <pb o="476" file="0196" n="206" rhead="JAC. GREG. CONSID."/> rii, de qua ſuperius dixi, ſed vererer, ne nimium hoc ſcriptum <lb/>producerem, & </s> <s xml:id="echoid-s4282" xml:space="preserve">certus ſum præterea, poſt illa, quæ D<emph style="super">us</emph>. </s> <s xml:id="echoid-s4283" xml:space="preserve">Mer-<lb/>cator nuper feliciter invenit de illa quadratura, & </s> <s xml:id="echoid-s4284" xml:space="preserve">correctionem <lb/>D<emph style="super">ni</emph>. </s> <s xml:id="echoid-s4285" xml:space="preserve">Walliſii, inventa noſtra hac de re parum eſſe conſi-<lb/>deranda.</s> <s xml:id="echoid-s4286" xml:space="preserve"/> </p> <figure> <image file="0196-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0196-01"/> </figure> </div> <div xml:id="echoid-div235" type="section" level="1" n="116"> <head xml:id="echoid-head158" xml:space="preserve">V.</head> <head xml:id="echoid-head159" xml:space="preserve">EXCERPTA EX EPISTOLA <lb/>D. JACOBI GREGORII, <lb/>CONTINENTE QUASDAM EJUS CONSIDERATIO-<lb/>NES, SUPER EPISTOLA</head> <head xml:id="echoid-head160" xml:space="preserve">D. HUGENII, <lb/>IMPRESSA IN VINDICATIONEM EXAMINIS <lb/>SUI LIBRI, DE VERA CIRCULI ET HY-<lb/>PERBOLÆ QUADRATURA.</head> <p> <s xml:id="echoid-s4287" xml:space="preserve">Ex duobus argumentis, quibus conatur Nob. </s> <s xml:id="echoid-s4288" xml:space="preserve">D. <lb/></s> <s xml:id="echoid-s4289" xml:space="preserve">Hugenius doctrinam meam evertere, primo qui-<lb/>dem, reſponſionis fundamentum dedi in Proœm. </s> <s xml:id="echoid-s4290" xml:space="preserve"><lb/>ad Geometr. </s> <s xml:id="echoid-s4291" xml:space="preserve">partem univer ſalem; </s> <s xml:id="echoid-s4292" xml:space="preserve">alterum au-<lb/>tem provenit ſolummodo à Prop. </s> <s xml:id="echoid-s4293" xml:space="preserve">11. </s> <s xml:id="echoid-s4294" xml:space="preserve">non re-<lb/>cte, opinor, ab Hugenio intellecta, quam tandem admit-<lb/>tit poſt correctiones (ut inquit) à me factas. </s> <s xml:id="echoid-s4295" xml:space="preserve">Ut autem, <lb/>ſimul cum reſolutione Objectionum, omnem evertam du-<lb/>bitandi rationem, ex admiſſa Prop. </s> <s xml:id="echoid-s4296" xml:space="preserve">11. </s> <s xml:id="echoid-s4297" xml:space="preserve">in forma proba-<lb/>re conabor Syllogiſtica, Nullam eſſe rationem analyticam <lb/>inter Circulum & </s> <s xml:id="echoid-s4298" xml:space="preserve">Diametri Quadratum: </s> <s xml:id="echoid-s4299" xml:space="preserve">Præter Modum <lb/>quippe & </s> <s xml:id="echoid-s4300" xml:space="preserve">figuram nil deeſt in hactenus à me publicatis, <pb o="477" file="0197" n="207" rhead="SUPER HUGEN. EXCEPT."/> quin id integrè demonſtretur; </s> <s xml:id="echoid-s4301" xml:space="preserve">quæ interim forma rarò à <lb/>Geometris exigitur. </s> <s xml:id="echoid-s4302" xml:space="preserve">Dico itaque;</s> <s xml:id="echoid-s4303" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4304" xml:space="preserve">Si daretur ratio Analytica (ſeu ratio notis Analyticis ex-<lb/>primenda) inter Circulum & </s> <s xml:id="echoid-s4305" xml:space="preserve">Diametri Quadratum, tunc Cir-<lb/>culus analyticè componeretur ex Quadratis, inſcripto & </s> <s xml:id="echoid-s4306" xml:space="preserve">cir-<lb/>cumſcripto. </s> <s xml:id="echoid-s4307" xml:space="preserve">Sed poſterius eſt abſurdum E. </s> <s xml:id="echoid-s4308" xml:space="preserve">Sequela Majoris <lb/>ſic probatur;</s> <s xml:id="echoid-s4309" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4310" xml:space="preserve">Quantitas quæſita & </s> <s xml:id="echoid-s4311" xml:space="preserve">determinata invenitur ex quantitati-<lb/>bus quibuſcunque eam determinantibus, in ea ratione, ſeu <lb/>relatione, quam habet quantitas determinata ad dictas quan-<lb/>titates determinantes. </s> <s xml:id="echoid-s4312" xml:space="preserve">Sed Quadratum inſcriptum & </s> <s xml:id="echoid-s4313" xml:space="preserve">cir-<lb/>cumſcriptum Circulum determinat, idemque ex illis Circu-<lb/>lus daretur in ea relatione, quam habet ad Diametri Quadra-<lb/>tum vel ejus ſemiſſem, h. </s> <s xml:id="echoid-s4314" xml:space="preserve">e. </s> <s xml:id="echoid-s4315" xml:space="preserve">ſi eſſet ratio analytica inter Cir-<lb/>culum & </s> <s xml:id="echoid-s4316" xml:space="preserve">diametri Quadratum; </s> <s xml:id="echoid-s4317" xml:space="preserve">Ex dictis quantitatibus de-<lb/>terminantibus analyticè componeretur Circulus. </s> <s xml:id="echoid-s4318" xml:space="preserve">Ex dictis <lb/>enim quantitatibus omnia analyticè componi poſſunt, quæ <lb/>ad ea rationem habent analyticam.</s> <s xml:id="echoid-s4319" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4320" xml:space="preserve">Secundi Syllogiſmi Minor eſt evidentiſſima. </s> <s xml:id="echoid-s4321" xml:space="preserve">Major autem <lb/>eſt axioma ab omnibus Geometris tacitè admiſſum.</s> <s xml:id="echoid-s4322" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4323" xml:space="preserve">Minor Syllogiſmi prioris ſic probatur.</s> <s xml:id="echoid-s4324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4325" xml:space="preserve">Eodem modo componitur Circulus ex Quadrato inſcripto <lb/>& </s> <s xml:id="echoid-s4326" xml:space="preserve">circumſcripto, quo componitur Quadrans Circuli ex <lb/>Triangulo inſcripto & </s> <s xml:id="echoid-s4327" xml:space="preserve">Trapezio vel potius Quadrato cir-<lb/>cumſcripto. </s> <s xml:id="echoid-s4328" xml:space="preserve">Sed ex 11. </s> <s xml:id="echoid-s4329" xml:space="preserve">Prop. </s> <s xml:id="echoid-s4330" xml:space="preserve">Quadrans Circuli ſeu ſector <lb/>non poteſt componi analyticè ex Triangulo inſcripto & </s> <s xml:id="echoid-s4331" xml:space="preserve">Qua-<lb/>drilatero circumſcripto. </s> <s xml:id="echoid-s4332" xml:space="preserve">E.</s> <s xml:id="echoid-s4333" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4334" xml:space="preserve">Major eſt evidens. </s> <s xml:id="echoid-s4335" xml:space="preserve">At poterit fortaſſe diſtingui Minor, <lb/>dicendo; </s> <s xml:id="echoid-s4336" xml:space="preserve">Prop. </s> <s xml:id="echoid-s4337" xml:space="preserve">11. </s> <s xml:id="echoid-s4338" xml:space="preserve">veram eſſe in Methodo indefinita; </s> <s xml:id="echoid-s4339" xml:space="preserve">ſed <lb/>poſſe eſſe falſam in methodis Particularibus. </s> <s xml:id="echoid-s4340" xml:space="preserve">At inſto. </s> <s xml:id="echoid-s4341" xml:space="preserve">Omnis <lb/>methodus indefinita in methodos ſeu caſus Particulares eſt re-<lb/>ſolubilis. </s> <s xml:id="echoid-s4342" xml:space="preserve">Sed hæc methodus indefinita, nempe quod ſe-<lb/>ctor ſit terminatio datæ ſerie convergentis, in nullam parti-<lb/>culararem reſolvi poteſt. </s> <s xml:id="echoid-s4343" xml:space="preserve">Nulla igitur datur hic methodus <lb/>particul@ris. </s> <s xml:id="echoid-s4344" xml:space="preserve">Major patet, quia quantitates æquales in ſe mu-<lb/>tuò ſunt reſolubiles. </s> <s xml:id="echoid-s4345" xml:space="preserve">Minorem ita probo; </s> <s xml:id="echoid-s4346" xml:space="preserve">Si hæc Methodus <pb o="478" file="0198" n="208" rhead="JAC. GREG. CONSID."/> indefinita reſolvetur in aliquam particularem, reſolutio fie-<lb/>ret vel ab Analyſi ſpecioſa vel numeroſa. </s> <s xml:id="echoid-s4347" xml:space="preserve">Sed neutrum dici <lb/>poteſt. </s> <s xml:id="echoid-s4348" xml:space="preserve">E. </s> <s xml:id="echoid-s4349" xml:space="preserve">Major patet ex ſufficienti enumeratione. </s> <s xml:id="echoid-s4350" xml:space="preserve">Minor <lb/>ſic probatur: </s> <s xml:id="echoid-s4351" xml:space="preserve">Non ab Analyſi ſpecioſa, quoniam hæc Me-<lb/>thodus indefinita ad eam eſt irreducibilis, ut patet ex Prop. <lb/></s> <s xml:id="echoid-s4352" xml:space="preserve">11. </s> <s xml:id="echoid-s4353" xml:space="preserve">Non à Numeroſa, quæ hic eſt interminabilis, proinde-<lb/>que invariabilis.</s> <s xml:id="echoid-s4354" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4355" xml:space="preserve">In hanc ultimam diſtinctionem reſolvitur 1. </s> <s xml:id="echoid-s4356" xml:space="preserve">objectio Hu-<lb/>genii. </s> <s xml:id="echoid-s4357" xml:space="preserve">Velim enim Nobiliſſ. </s> <s xml:id="echoid-s4358" xml:space="preserve">Virum conſiderare, omnem <lb/>plenam Problematis ſolutionem eſſe indefinitam. </s> <s xml:id="echoid-s4359" xml:space="preserve">Nam Me-<lb/>thodi particulares, cum ſint infinitæ, exhiberi omnes ne-<lb/>queunt; </s> <s xml:id="echoid-s4360" xml:space="preserve">neque dirigi poſſunt à tenore Problematis quippe <lb/>illis omnibus communi: </s> <s xml:id="echoid-s4361" xml:space="preserve">Ideoque requiritur Methodus Ge-<lb/>neralis ſeu Indefinita, Particularium directrix. </s> <s xml:id="echoid-s4362" xml:space="preserve">Agnoſco u-<lb/>tique Methodos Particulares caſu ſæpe inveniri abſque ope <lb/>Generalis; </s> <s xml:id="echoid-s4363" xml:space="preserve">attamen fatendum eſt Geometris, nullam eſſe, <lb/>nec poſſe fieri Mothodum Particularem, in quam reſolubi-<lb/>lis non ſit Methodus indefinita. </s> <s xml:id="echoid-s4364" xml:space="preserve">Si igitur Methodus Indefi-<lb/>nita omni reſolutioni ſit impervia (ut in Prop. </s> <s xml:id="echoid-s4365" xml:space="preserve">11. </s> <s xml:id="echoid-s4366" xml:space="preserve">eſt de-<lb/>monſtratum) eodem modo omnes Particulares reſolutionem <lb/>@tiam reſpuent; </s> <s xml:id="echoid-s4367" xml:space="preserve">proindeque tam Definita, quam Indefinita <lb/>nullam compoſitionem agnoſcit. </s> <s xml:id="echoid-s4368" xml:space="preserve">Talis enim Compoſitio, <lb/>qualis Reſolutio.</s> <s xml:id="echoid-s4369" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4370" xml:space="preserve">Etiamſi prædicta, meo quidem judicio, adundè ſufficiant, <lb/>ne tamen ullus relinquatur cavillationi locus, 11<emph style="super">mam</emph> noſtram <lb/>Prop. </s> <s xml:id="echoid-s4371" xml:space="preserve">etiam in Definitis hic demonſtrabimus. </s> <s xml:id="echoid-s4372" xml:space="preserve">Sit ergò B <lb/>Polygonum intra Circuli ſectorem, 2 B Polygonum circum-<lb/>ſcriptum & </s> <s xml:id="echoid-s4373" xml:space="preserve">priori ſimile; </s> <s xml:id="echoid-s4374" xml:space="preserve">ſufficit enim Polygonorum propor-<lb/>tionem definire, ut Theorema definitè demonſtretur. </s> <s xml:id="echoid-s4375" xml:space="preserve">Con-<lb/>tinuetur ſeries convergens ut ſit ejus teminatio <lb/> <anchor type="note" xlink:label="note-0198-01a" xlink:href="note-0198-01"/> feu Circuli Sector Z. </s> <s xml:id="echoid-s4376" xml:space="preserve">Dico, Z non poſſe com-<lb/>poni analyticè ex Polygonis definitis B, 2 B. </s> <s xml:id="echoid-s4377" xml:space="preserve">Si <lb/>fieri poteſt, componatur Z Analytice ex Poly-<lb/>gonis Definitis B, 2 B. </s> <s xml:id="echoid-s4378" xml:space="preserve">ſintque duæ quantitates <lb/>Indefinitæ a & </s> <s xml:id="echoid-s4379" xml:space="preserve">x, è quibus componatur m eo- <pb o="479" file="0199" n="209" rhead="SUPER HUGENII EXCEPT."/> dem modo, quo Z. </s> <s xml:id="echoid-s4380" xml:space="preserve">componitur à quantitatibus <lb/> <anchor type="note" xlink:label="note-0199-01a" xlink:href="note-0199-01"/> B, 2 B; </s> <s xml:id="echoid-s4381" xml:space="preserve">Item eodem modo componatur n ex <lb/>quantitatibus Vax, {2ax/a + Vax}: </s> <s xml:id="echoid-s4382" xml:space="preserve">quantitates m, n, non <lb/>ſunt indefinite æquales ex prop. </s> <s xml:id="echoid-s4383" xml:space="preserve">11 Si igitur in-<lb/>ter m & </s> <s xml:id="echoid-s4384" xml:space="preserve">n fingatur æquatio; </s> <s xml:id="echoid-s4385" xml:space="preserve">a manente quantitate indefinita, <lb/>æquatio inter m & </s> <s xml:id="echoid-s4386" xml:space="preserve">n tot habebit radices ſeu quantitates, in <lb/>quas reſolvitur x, quot quantitatum, inter ſe diverſas ra-<lb/>tiones habentium, binarii ſunt in rerum natura, quæ vices <lb/>quantitatum a, x, ſubire poſſunt, h. </s> <s xml:id="echoid-s4387" xml:space="preserve">e. </s> <s xml:id="echoid-s4388" xml:space="preserve">quæ eandem quan-<lb/>titatem Analyticè ex ſe ipſis componit eodem modo, quo <lb/>eandem quantitas componitur ex ipſarum media Geometri-<lb/>ca Vax, & </s> <s xml:id="echoid-s4389" xml:space="preserve">ex media Harmonica inter dictam mediam <lb/>Geometricam & </s> <s xml:id="echoid-s4390" xml:space="preserve">x, nempe {2ax/a + Vax,} ita ut compoſitio ſit eo-<lb/>dem modo quo Z componitur ex B & </s> <s xml:id="echoid-s4391" xml:space="preserve">2B. </s> <s xml:id="echoid-s4392" xml:space="preserve">atque ex Conſe-<lb/>ctario Prop. </s> <s xml:id="echoid-s4393" xml:space="preserve">10. </s> <s xml:id="echoid-s4394" xml:space="preserve">omnes quantitatum binarii, rationes quoque <lb/>diverſas inter ſe habentium, B 2 B, C D, E F, G H, &</s> <s xml:id="echoid-s4395" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s4396" xml:space="preserve">in infinitum, poſſunt ſupplere vices quantitatum a, x, quo-<lb/>niam Z eodem modo componitur ex B 2 B, quo ex C D, E F, <lb/>vel G H, &</s> <s xml:id="echoid-s4397" xml:space="preserve">c. </s> <s xml:id="echoid-s4398" xml:space="preserve">& </s> <s xml:id="echoid-s4399" xml:space="preserve">proinde æquatio inter m & </s> <s xml:id="echoid-s4400" xml:space="preserve">n radices habet <lb/>numero infinitas. </s> <s xml:id="echoid-s4401" xml:space="preserve">Sed omnis æquatio habet ad ſummum tot <lb/>radices, quot habet dimenſiones; </s> <s xml:id="echoid-s4402" xml:space="preserve">& </s> <s xml:id="echoid-s4403" xml:space="preserve">proinde æquatio inter <lb/>m & </s> <s xml:id="echoid-s4404" xml:space="preserve">n dimenſiones habet numero infinitas, quod eſt abſur-<lb/>dum; </s> <s xml:id="echoid-s4405" xml:space="preserve">ideoque Z ſeu Circuli Sector non poteſt analyticè <lb/>componi ex Polygonis definitis B, 2B. </s> <s xml:id="echoid-s4406" xml:space="preserve">quod demonſtran-<lb/>dum erat. </s> <s xml:id="echoid-s4407" xml:space="preserve">Hinc manifeſtum eſt, Terminationem cujuſlibet <lb/>ſeriei convergentis, ſi non poſſit componi ex terminis con-<lb/>vergentibus indefinitè, nec poſſe componi definitè; </s> <s xml:id="echoid-s4408" xml:space="preserve">adeo-<lb/>que evaneſcit ſimul cum noſtra diſtinctione Objectio Huge-<lb/>nii prima.</s> <s xml:id="echoid-s4409" xml:space="preserve"/> </p> <div xml:id="echoid-div235" type="float" level="2" n="1"> <note position="right" xlink:label="note-0198-01" xlink:href="note-0198-01a" xml:space="preserve"> <lb/>B # 2 B <lb/>C # D <lb/>E # F <lb/>G # H <lb/>## Z <lb/>a # x <lb/></note> <note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve"> <lb/>Vax {2ax/a + Vax} <lb/>m <lb/>n <lb/></note> </div> <p> <s xml:id="echoid-s4410" xml:space="preserve">Idem in Objectione ſua ſecunda non videtur advertiſſe, <lb/>me non ſolum in Prop. </s> <s xml:id="echoid-s4411" xml:space="preserve">11. </s> <s xml:id="echoid-s4412" xml:space="preserve">ſed etiam in toto meo Tracta-<lb/>tulo intelligere per Extractionem radicum, Reſolutionem <lb/>omnium poteſtatum ſive purarum ſive affectarum; </s> <s xml:id="echoid-s4413" xml:space="preserve">omnium <lb/>quippe eadem eſt ratio, neque ulla imaginabilis eſt in de-<lb/>monſtratione diverſitas, ſive Sector ſupponatur Radix ali- <pb o="480" file="0200" n="210" rhead="JAC. GREG. CONSID."/> cujus poteſtatis puræ, ſive affectæ ad puram irreducibilis. <lb/></s> <s xml:id="echoid-s4414" xml:space="preserve">Nam ſi Sector eodem modo fiat cx primis terminis conver-<lb/>gentibus, quo ex ſecundis (ut in Conſect. </s> <s xml:id="echoid-s4415" xml:space="preserve">Prop. </s> <s xml:id="echoid-s4416" xml:space="preserve">10. </s> <s xml:id="echoid-s4417" xml:space="preserve">eſt de-<lb/>monſtratum) etiam omnes ejus poteſtates ſive puræ, ſive <lb/>quocunque modo affectæ eodem modo componitur è pri-<lb/>mis, quo è ſecundis terminis convergentibus, quæ (in A-<lb/>nalyticis exhibitæ) erunt æquales quantitates eodem modo <lb/>Analyticè compoſitæ ex primis, quo ex ſecundis terminis <lb/>convergentibus; </s> <s xml:id="echoid-s4418" xml:space="preserve">quod eſt abſurdum, nempe contra Prop. </s> <s xml:id="echoid-s4419" xml:space="preserve"><lb/>11. </s> <s xml:id="echoid-s4420" xml:space="preserve">admiſlam. </s> <s xml:id="echoid-s4421" xml:space="preserve">Senſus igitur integer Prop. </s> <s xml:id="echoid-s4422" xml:space="preserve">11. </s> <s xml:id="echoid-s4423" xml:space="preserve">eſt Hoc Pro-<lb/>blema (E datis duobus polygonis complicatis, invenire Se-<lb/>ctorem ſive Circularem ſive Hyperbolicum ab illis determi-<lb/>natum) non poteſt reduci ad ullam æquationem Analyticam.</s> <s xml:id="echoid-s4424" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4425" xml:space="preserve">In comparatione Hugeniana inter noſtras methodos, a-<lb/>gnoſco, meas approximationes prop. </s> <s xml:id="echoid-s4426" xml:space="preserve">20. </s> <s xml:id="echoid-s4427" xml:space="preserve">& </s> <s xml:id="echoid-s4428" xml:space="preserve">21. </s> <s xml:id="echoid-s4429" xml:space="preserve">eaſdem eſſe <lb/>cum Hugenianis, ſed methodo mihi peculiari demonſtratas. <lb/></s> <s xml:id="echoid-s4430" xml:space="preserve">At meam approximationem in fine prop. </s> <s xml:id="echoid-s4431" xml:space="preserve">25. </s> <s xml:id="echoid-s4432" xml:space="preserve">non percipere <lb/>videtur Hugenius; </s> <s xml:id="echoid-s4433" xml:space="preserve">aliam interim ſibi fingit: </s> <s xml:id="echoid-s4434" xml:space="preserve">hanc primò <lb/>meam non eſſe probat, deinde tamen eam cum ſua compa-<lb/>rat, victoriaque potitur. </s> <s xml:id="echoid-s4435" xml:space="preserve">Sed lentè hic feſtinandum.</s> <s xml:id="echoid-s4436" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4437" xml:space="preserve">Sit a Polygonum, Circulo vel ſectori inſcriptum, c Po-<lb/>lygonum inſcriptum duplo plura habens latera, d autem ſit <lb/>Polygonum circumſcriptum ſimile ipſi c. </s> <s xml:id="echoid-s4438" xml:space="preserve">Ex 20. </s> <s xml:id="echoid-s4439" xml:space="preserve">prop. </s> <s xml:id="echoid-s4440" xml:space="preserve">Sector <lb/>eſt major quam {4c - a;</s> <s xml:id="echoid-s4441" xml:space="preserve">/3} & </s> <s xml:id="echoid-s4442" xml:space="preserve">ex 21. </s> <s xml:id="echoid-s4443" xml:space="preserve">Sector eſt minor quam <lb/>{2d + c/3}, inter quos terminos ſit maximus quatuor continuè <lb/>proportionalium {8d + 8c-a/15}, nempe noſtra approximatio@ <lb/>quam rigidiſſimis Hugenii cenſuris ſubjicio. </s> <s xml:id="echoid-s4444" xml:space="preserve">Hallucinatur au-<lb/>tem Hugenius, quod Polygona a & </s> <s xml:id="echoid-s4445" xml:space="preserve">d ſimilia ſumeret, cum <lb/>debeant eſſe c & </s> <s xml:id="echoid-s4446" xml:space="preserve">d, quæ duplò plura habent latera. </s> <s xml:id="echoid-s4447" xml:space="preserve">Ne <lb/>autem dicat, factam eſſe à me correctionem, conſideret <lb/>hanc approximationem non ſolum verbis prop. </s> <s xml:id="echoid-s4448" xml:space="preserve">25. </s> <s xml:id="echoid-s4449" xml:space="preserve">ſed & </s> <s xml:id="echoid-s4450" xml:space="preserve"><lb/>praxi prop. </s> <s xml:id="echoid-s4451" xml:space="preserve">30. </s> <s xml:id="echoid-s4452" xml:space="preserve">eſſe conſonam, ubi approximationem prop. <lb/></s> <s xml:id="echoid-s4453" xml:space="preserve">21. </s> <s xml:id="echoid-s4454" xml:space="preserve">ex ultimis ſimilibus Polygonis conſtruo: </s> <s xml:id="echoid-s4455" xml:space="preserve">ridiculum enim <lb/>eſſet, illam è penultimis minus præciſam dare, cum eadem <pb o="481" file="0201" n="211" rhead="SUPER HUGENII EXCEPT."/> opera detur magis præciſa ex ultimis. </s> <s xml:id="echoid-s4456" xml:space="preserve">At miror, cum Hu-<lb/>genius incidiſſet in meam Hyperbolæ approximationem, quod <lb/>eam non potuerit Circulo applicare; </s> <s xml:id="echoid-s4457" xml:space="preserve">Nam in Hyperbola abſ-<lb/>que dubio 24 prop. </s> <s xml:id="echoid-s4458" xml:space="preserve">approximationem ex ultimis ſimilibus <lb/>polygonis conſtruxit: </s> <s xml:id="echoid-s4459" xml:space="preserve">Omnis enim ad Circulum approxi-<lb/>matio ex polygonis deducta, Hyperbolæ eſt etiam applica-<lb/>bilis, & </s> <s xml:id="echoid-s4460" xml:space="preserve">vice verſa. </s> <s xml:id="echoid-s4461" xml:space="preserve">Sed hoc non videtur animadvertiſſe Hu-<lb/>genius; </s> <s xml:id="echoid-s4462" xml:space="preserve">alioqui in fine fuarum Animadverſionum non pro-<lb/>mitteret talem Hyperbolicam approximationem, de cujus <lb/>applicatione ad Circulum nihil dicit. </s> <s xml:id="echoid-s4463" xml:space="preserve">Quæ autem illic affir-<lb/>mat (ſi de ſemet loquitur in plurali) tranſeant; </s> <s xml:id="echoid-s4464" xml:space="preserve">ſi verò etiam <lb/>de me adeo fidenter ſibi perſuadeat, falli ipſum putem, cum <lb/>hæc eadem quadratura, de qua loquitur, antequam ab eo <lb/>videretur, ad laboris dimidium à me ſit reducta.</s> <s xml:id="echoid-s4465" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4466" xml:space="preserve">Ne autem Hugenii praxis geometrica minus peritis videa-<lb/>tur noſtram ſuperaſſe, ex noſtra approximatione, ab Hu-<lb/>genio rejecta, ſequentem praxin exhibebo.</s> <s xml:id="echoid-s4467" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4468" xml:space="preserve">In fig. </s> <s xml:id="echoid-s4469" xml:space="preserve">Hugeniana ſit A C = A, A B C = B, ſitque <lb/> <anchor type="note" xlink:label="note-0201-01a" xlink:href="note-0201-01"/> A + B: </s> <s xml:id="echoid-s4470" xml:space="preserve">B :</s> <s xml:id="echoid-s4471" xml:space="preserve">: 2B: </s> <s xml:id="echoid-s4472" xml:space="preserve">C; </s> <s xml:id="echoid-s4473" xml:space="preserve">eritque {8C + 8B - A/15} major, quam <lb/>arcus A B C; </s> <s xml:id="echoid-s4474" xml:space="preserve">differentia autem, in ſemi-circumferen-<lb/>tia minor erit quàm ipſius {1/3500}, in triente minor quam <lb/>ipſius {1/40000}, & </s> <s xml:id="echoid-s4475" xml:space="preserve">in quadrante minor quam ipſius {1/300000}. </s> <s xml:id="echoid-s4476" xml:space="preserve">Sed quo-<lb/>niam præcedens approximatio major eſt quàm arcus, aliam <lb/>addamus eodem minorem. </s> <s xml:id="echoid-s4477" xml:space="preserve">Sit A:</s> <s xml:id="echoid-s4478" xml:space="preserve">B:</s> <s xml:id="echoid-s4479" xml:space="preserve">:B:</s> <s xml:id="echoid-s4480" xml:space="preserve">D; </s> <s xml:id="echoid-s4481" xml:space="preserve">{12 C + 4 B - D/15} <lb/>minor erit quam arcus A B C; </s> <s xml:id="echoid-s4482" xml:space="preserve">differentia autem in ſemi-cir-<lb/>cumferentia minor erit quam ipſius {1/1000}, & </s> <s xml:id="echoid-s4483" xml:space="preserve">in quadrante <lb/>minor quam ipſius {1/60000}, inter has approximationes ſit maxi-<lb/>ma, penultima ſex continuè Arithmeticè proportionalium, <lb/>quæ minor erit quàm arcus, differentia autem in ſemi-cir-<lb/>cumferentia minor erit quam ejuſdem {1/23000}, & </s> <s xml:id="echoid-s4484" xml:space="preserve">in quadrante <pb o="482" file="0202" n="212" rhead="J. GREG. CONS. SUPER HUG. EXCEPT."/> minor quam ejuſdem {1/3000000}. </s> <s xml:id="echoid-s4485" xml:space="preserve">Sed hæc levia mihi videntur, <lb/>cum poſſim Approximationes exhibere, quæ ab ipſa <lb/>ſemi-circumferentia differant minori intervallo, quam quæ-<lb/>libet ejus pars aſſignata, neque nobis amplius apparent hæc <lb/>mirabilia, cum demonſtratio ſolida innoteſcat. </s> <s xml:id="echoid-s4486" xml:space="preserve">Ad reliqua <lb/>ab Hugenio publicata, cum à meo inſtituto ſint aliena, nihil <lb/>dico, niſi quod ipſa Hugenii dicta (non obſtante exactiſſi-<lb/>ma ſua, ut ait, materiæ hujus examinatione) à meæ Ap-<lb/>pendiculæ factis ni fallor, longè ſuperentur. </s> <s xml:id="echoid-s4487" xml:space="preserve">Vale. </s> <s xml:id="echoid-s4488" xml:space="preserve">Decemb. <lb/></s> <s xml:id="echoid-s4489" xml:space="preserve">15. </s> <s xml:id="echoid-s4490" xml:space="preserve">1668.</s> <s xml:id="echoid-s4491" xml:space="preserve"/> </p> <div xml:id="echoid-div236" type="float" level="2" n="2"> <note position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve">TAB. XLIV. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s4492" xml:space="preserve">Figura Hugenii hæc eſt, quam ipſe hoc ſenſu, licet Gal-<lb/>licè, ſic explicat. </s> <s xml:id="echoid-s4493" xml:space="preserve">Sit Arcus Circuli, qui non excedat ſemicir-<lb/>cumferentiam, A B C, cujus ſubtenſa ſit A C, & </s> <s xml:id="echoid-s4494" xml:space="preserve">dividan-<lb/>tur ambo in partes æquales per lineam B D. </s> <s xml:id="echoid-s4495" xml:space="preserve">Ducta ſubtenſa <lb/>A B, capias inde {2/3}, eaſque jungas inde ab A ad E in linea <lb/>C A protracta. </s> <s xml:id="echoid-s4496" xml:space="preserve">Dein, reſecta lineæ D E parte decima E F, <lb/>ducas F B, & </s> <s xml:id="echoid-s4497" xml:space="preserve">tandem B G, ipſi perpendicularem: </s> <s xml:id="echoid-s4498" xml:space="preserve">& </s> <s xml:id="echoid-s4499" xml:space="preserve">habe-<lb/>bis lineam A G æqualem Arcui A B C, cujus exceſſus tantil-<lb/>lus erit, ut etiam tunc, quando hic arcus æqualis erit ſemi-<lb/>circumferentiæ Circuli, futura non ſit differentia {1/3400} ſuæ <lb/>longitudinis; </s> <s xml:id="echoid-s4500" xml:space="preserve">at quando non eſt niſi tertiæ partis circumfe-<lb/>ferentiæ, differentia non erit {1/13000}; </s> <s xml:id="echoid-s4501" xml:space="preserve">& </s> <s xml:id="echoid-s4502" xml:space="preserve">ſi non ſit niſi quar-<lb/>tæ partis, non differet niſi {1/90000}, ſuæ longitudinis.</s> <s xml:id="echoid-s4503" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div238" type="section" level="1" n="117"> <head xml:id="echoid-head161" xml:space="preserve">FINIS.</head> <pb file="0203" n="213"/> </div> <div xml:id="echoid-div239" type="section" level="1" n="118"> <head xml:id="echoid-head162" xml:space="preserve">CHRISTIANI HUGENII <lb/>GEOMETRICA <lb/>VARIA.</head> <head xml:id="echoid-head163" style="it" xml:space="preserve">Tom. II. Ppp</head> <pb file="0204" n="214"/> <pb o="485" file="0205" n="215"/> <figure> <image file="0205-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0205-01"/> </figure> </div> <div xml:id="echoid-div240" type="section" level="1" n="119"> <head xml:id="echoid-head164" xml:space="preserve">I. <lb/>CONSTRUCTIO LOCI <lb/>AD HYPERBOLAM <lb/>PER ASYMPTOTOS.</head> <p> <s xml:id="echoid-s4504" xml:space="preserve">In æquatione loci ad hyperbolam, ſi neutra indeter-<lb/> <anchor type="note" xlink:label="note-0205-01a" xlink:href="note-0205-01"/> minatarum linearum in ſeipſam ducta inveniatur, <lb/>velut ſi ſit xy = bb; </s> <s xml:id="echoid-s4505" xml:space="preserve">vel xy = cx. </s> <s xml:id="echoid-s4506" xml:space="preserve">bb; </s> <s xml:id="echoid-s4507" xml:space="preserve">(literis x & </s> <s xml:id="echoid-s4508" xml:space="preserve">y <lb/>lineas indeterminatas A B, B C ſignificantibus, <lb/>quæ in dato angulo ſibi mutuò ſint applicatæ, quarumque al-<lb/>tera, ut A B, poſitione data intelligitur, & </s> <s xml:id="echoid-s4509" xml:space="preserve">in ea datum pun-<lb/>ctum A) conſtructio per aſymptotorum inventionem facilè <lb/>abſolvitur, ut oſtenſum eſt à Fl. </s> <s xml:id="echoid-s4510" xml:space="preserve">de Beaune in Notis ad Geo-<lb/>metriam Carteſii. </s> <s xml:id="echoid-s4511" xml:space="preserve">Cum verò habetur x x vel y y in æquatio-<lb/>ne, vel utrumque nihilominus ad aſymptotos rem deduci <lb/>poſſe, & </s> <s xml:id="echoid-s4512" xml:space="preserve">quidem brevius quàm ad diametri laterumque re-<lb/>cti & </s> <s xml:id="echoid-s4513" xml:space="preserve">transverſi inventionem, oſtendemus hoc modo.</s> <s xml:id="echoid-s4514" xml:space="preserve"/> </p> <div xml:id="echoid-div240" type="float" level="2" n="1"> <note position="right" xlink:label="note-0205-01" xlink:href="note-0205-01a" xml:space="preserve">TAB. XLIV. <lb/>fig. 4. 5. 6. 7.</note> </div> <p> <s xml:id="echoid-s4515" xml:space="preserve">Sit æquatio ejuſmodi reducta, y = l. </s> <s xml:id="echoid-s4516" xml:space="preserve">{nx/z} √mm.</s> <s xml:id="echoid-s4517" xml:space="preserve">ox + {ppxx;</s> <s xml:id="echoid-s4518" xml:space="preserve">/gg} <lb/>ſemper enim ad hos terminos reduci poteſt, nempe ut y al-<lb/>tera linearum indeterminatarum, quæ applicata eſt ad poſi-<lb/>tionem datam, ſola ab una parte æquationis habeatur, ab alte-<lb/>ra verò non plures termini quàm hîc inveniantur; </s> <s xml:id="echoid-s4519" xml:space="preserve">nam ſæ-<lb/>pe pauciores etiam eſſe poſſunt, cum ſoli neceſſarii ſint <lb/>+ {ppxx/gg} cum alterutro horum mm vel ox.</s> <s xml:id="echoid-s4520" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4521" xml:space="preserve">Quum angulus A B C datus ſit, ducatur per A punctum <lb/>linea X Y quæ ſit rectæ B C parallela, & </s> <s xml:id="echoid-s4522" xml:space="preserve">in ea accipiatur AI <lb/>æqualis l, idque ad partes B C, ſi habeatur + l in æquatio-<lb/>ne, in contrarias verò ſi habeatur — l, & </s> <s xml:id="echoid-s4523" xml:space="preserve">agatur I K pa- <pb o="486" file="0206" n="216" rhead="CHRIST. HUGENII"/> rallela A B. </s> <s xml:id="echoid-s4524" xml:space="preserve">Si verò non habeatur omnino l, recta I K in <lb/>A B incidere intelligenda eſt.</s> <s xml:id="echoid-s4525" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4526" xml:space="preserve">Deinde ſicut z ad n, quæ ratio data, ita ſit I K ad libi-<lb/>tum ſumpta, ad K L; </s> <s xml:id="echoid-s4527" xml:space="preserve">quæ ipſi A I parallela ducendaeſt, ſu-<lb/>mendaque hoc pacto, ut puncta K L ſita ſint quo ordinc <lb/>A I, ſi habeatur + {nx/z}, at contrà ſi habeatur - {nx/z}, & </s> <s xml:id="echoid-s4528" xml:space="preserve">du-<lb/>catur recta per IL; </s> <s xml:id="echoid-s4529" xml:space="preserve">ſi verò deſit {nx/z}, eadem eſt I L & </s> <s xml:id="echoid-s4530" xml:space="preserve">I K.</s> <s xml:id="echoid-s4531" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4532" xml:space="preserve">Porro ut p ad g, ita ſit {1/2}o ad ſingulas IX, I Y ſumendas <lb/>in recta A I; </s> <s xml:id="echoid-s4533" xml:space="preserve">atque ita quoque I X ad I V ſumendam in I K <lb/>ad partes A B ſi habeatur - o x, aut in contrarias ſi habea-<lb/>tur + ox; </s> <s xml:id="echoid-s4534" xml:space="preserve">& </s> <s xml:id="echoid-s4535" xml:space="preserve">ſit V M parallela A I, occurratque rectæ I L <lb/>in M: </s> <s xml:id="echoid-s4536" xml:space="preserve">erit jam M centrum hyperbolæ quæſitæ aſymptoti <lb/>vero, rectæ per M X, M Y ductæ.</s> <s xml:id="echoid-s4537" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4538" xml:space="preserve">Si vero non habeatur o x in æquatione, erit I centrum hy-<lb/>perbolæ; </s> <s xml:id="echoid-s4539" xml:space="preserve">ſumptisque I X, I Y ad libitum ſed inter ſe æqua-<lb/>libus, inventiſque inde punctis V & </s> <s xml:id="echoid-s4540" xml:space="preserve">M, ut ante, ducentur <lb/>aſymptoti per I parallelæ ipſis M X, M Y.</s> <s xml:id="echoid-s4541" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4542" xml:space="preserve">Jam porro ſi habeatur + mm, puncta S & </s> <s xml:id="echoid-s4543" xml:space="preserve">R, per quæ <lb/>hyperbola vel oppoſitæ ſectiones tranſire debent, invenien-<lb/>tur ſumendo in recta A I à puncto I, ſingulas I S, I R æqua-<lb/>les m: </s> <s xml:id="echoid-s4544" xml:space="preserve">unde jam hyperbola data erit ac deſcribi poterit, in <lb/>qua B C erit ordinatim applicata ad diametrum, ſi {{1/2}og/z} ma-<lb/>jor quam m; </s> <s xml:id="echoid-s4545" xml:space="preserve">ſin verò {{1/2}og/p} minor quam m, erit B C paralle-<lb/>la diametro hyperbolæ ad quam eſt C punctum, ut hic caſu <lb/>ſecundo. </s> <s xml:id="echoid-s4546" xml:space="preserve">Quod ſi forte punctum S incidat in X, locus <lb/>puncti C, erunt ipſæ aſymptoti. </s> <s xml:id="echoid-s4547" xml:space="preserve">Si verò non habeatur mm, <lb/>erit ipſum I punctum in hyperbola quæſita.</s> <s xml:id="echoid-s4548" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4549" xml:space="preserve">At ſi habeatur — mm, accommodanda eſt intra angu-<lb/>lum X M I recta G N parallela I X, quæque poſſit quadrata <lb/>@b I X & </s> <s xml:id="echoid-s4550" xml:space="preserve">I S, vel tantum ipſi I S æqualis, ſi non habeatur <pb o="487" file="0207" n="217" rhead="GEOMET. VARIA."/> o x; </s> <s xml:id="echoid-s4551" xml:space="preserve">eritque punctum N in hyperbola quæſita, quæ proin-<lb/>de rurſus data erit.</s> <s xml:id="echoid-s4552" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4553" xml:space="preserve">Sumpta enim in caſu primo A B = x ad arbitrium, eique <lb/> <anchor type="note" xlink:label="note-0207-01a" xlink:href="note-0207-01"/> applicata B C = y in angulo dato, quæ ad hyperbolam in-<lb/>ventam terminetur, oſtendendum ſit quod</s> </p> <div xml:id="echoid-div241" type="float" level="2" n="2"> <note position="right" xlink:label="note-0207-01" xlink:href="note-0207-01a" xml:space="preserve">fig. 4.</note> </div> <p> <s xml:id="echoid-s4554" xml:space="preserve">y = l - {nx/z} + √mm - ox + {ppxx.</s> <s xml:id="echoid-s4555" xml:space="preserve">/gg} </s> </p> </div> <div xml:id="echoid-div243" type="section" level="1" n="120"> <head xml:id="echoid-head165" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s4556" xml:space="preserve">Occurrat B C utrinque ſi opus ſit producta, aſymptotis <lb/>in O & </s> <s xml:id="echoid-s4557" xml:space="preserve">Q. </s> <s xml:id="echoid-s4558" xml:space="preserve">Ex conſtructione eſt I X vel I Y = {{1/2}og/p}, <lb/>I V = {{1/2}ogg/pp}, Ratio verò data I K ad K L, eadem nempe <lb/>quæ z ad n. </s> <s xml:id="echoid-s4559" xml:space="preserve">Sed & </s> <s xml:id="echoid-s4560" xml:space="preserve">angulus I K L datus eſt. </s> <s xml:id="echoid-s4561" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s4562" xml:space="preserve">ra-<lb/>tio I K ad I L, quæ ſit ea quæ z ad a. </s> <s xml:id="echoid-s4563" xml:space="preserve">Ergo quia ut I K <lb/>ad I L ita I V ad I M, erit I M = {{1/2}aogg/zpp}. </s> <s xml:id="echoid-s4564" xml:space="preserve">Ut autem I M ad <lb/>IX, hoc eſt ut {{1/2}aogg/zpp} ad {{1/2}go/p}, ſive ut ag ad pz, ita M L, <lb/>ſive M I minus I L, hoc eſt, {{1/2}aogg}zpp - {ax/z} ad L O vel L Q; <lb/></s> <s xml:id="echoid-s4565" xml:space="preserve">quæ itaque erit {{1/2}og/p} - {px/g}. </s> <s xml:id="echoid-s4566" xml:space="preserve">Porro quia B K = l, & </s> <s xml:id="echoid-s4567" xml:space="preserve">L K = {nx/z}, <lb/>erit B L = l - {nx/z}, quà ablatâ à B C=y, fit L C = y - l + {nx/z}. </s> <s xml:id="echoid-s4568" xml:space="preserve"><lb/>Propter hyperbolam verò erit rectangulum Q C O æquale <lb/>rectangulo Y S X. </s> <s xml:id="echoid-s4569" xml:space="preserve">Sed rectangulum Q C O æquale eſt <lb/>quadrato L O minus quadrato L C, hoc eſt quadrato ab <lb/>{{1/2}go/p} - {px/g} minus quadrato ab y - l + {nx/z}: </s> <s xml:id="echoid-s4570" xml:space="preserve">quorum quadra- <pb o="488" file="0208" n="218" rhead="CHRIST. HUGENII"/> torum differentia eſt {{1/4}ggoo/pp} - ox + {ppxx/gg} - yy + 2 ly - ll <lb/>+ {2nxy/z} + {2lnx/z} - {nnxx/zz}. </s> <s xml:id="echoid-s4571" xml:space="preserve">Ergo hæc æquatur rectangulo <lb/>Y S X, hoc eſt quadrato I X minus quadrato I S, hoc eſt <lb/>{{1/4}ggoo/pp} - mm; </s> <s xml:id="echoid-s4572" xml:space="preserve">quia I X = {{1/2}go/p} & </s> <s xml:id="echoid-s4573" xml:space="preserve">I S = m. </s> <s xml:id="echoid-s4574" xml:space="preserve">In qua æqua-<lb/>tione deleto utrinque {{1/4}ggoo/pp}, invenietur y = l - {nx/z} + <lb/>√mm - ox + {ppxx/gg} , ut oportebat.</s> <s xml:id="echoid-s4575" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4576" xml:space="preserve">In Secundo calu rectangulum Q C O æquatur quadrato <lb/> <anchor type="note" xlink:label="note-0208-01a" xlink:href="note-0208-01"/> L C minus quadrato L O; </s> <s xml:id="echoid-s4577" xml:space="preserve">& </s> <s xml:id="echoid-s4578" xml:space="preserve">rectangulum Y S X quadrato <lb/>I S minus quadrato I X. </s> <s xml:id="echoid-s4579" xml:space="preserve">Unde rurſus valor Y idem qui ca-<lb/>ſu primo invenietur.</s> <s xml:id="echoid-s4580" xml:space="preserve"/> </p> <div xml:id="echoid-div243" type="float" level="2" n="1"> <note position="left" xlink:label="note-0208-01" xlink:href="note-0208-01a" xml:space="preserve">fig. 5.</note> </div> <p> <s xml:id="echoid-s4581" xml:space="preserve">Sit tertius caſus quo habeatur - mm, ſitque æquatio y = l <lb/>- {nx/z} + √- mm + ox + {ppxx/gg} , producta GN occurrat al-<lb/>teri aſymptoto in D. </s> <s xml:id="echoid-s4582" xml:space="preserve">Hîc jam eadem ratione qua prius, ap-<lb/>parebit L O vel LQ eſſe {{1/2}go/p} + {px/g}, & </s> <s xml:id="echoid-s4583" xml:space="preserve">L C = y + {nx/z} - l. <lb/></s> <s xml:id="echoid-s4584" xml:space="preserve">Et propter hyperbolam erit rectangulum Q C O = rectan-<lb/>gulo D N G ſeu quadrato N G, hoc eſt {{1/4}ggoo/pp} + mm, quia <lb/>X I = {{1/2}go/p}, & </s> <s xml:id="echoid-s4585" xml:space="preserve">I S = m, quorum quadratis æquale fecimus <lb/>quadratum G N. </s> <s xml:id="echoid-s4586" xml:space="preserve">Rectangulum autem Q C O æquatur qua-<lb/>drato L O minus quadrato L C, hoceſt {{1/4}ggoo/pp} + ox + {ppxx/gg} <lb/>- yy - {2nxy/z} — {nnxx/zz} + 2 ly + {2nlx/z} - ll. </s> <s xml:id="echoid-s4587" xml:space="preserve">Ergo hoc <lb/>æquale {{1/4}ggoo/pp} + mm. </s> <s xml:id="echoid-s4588" xml:space="preserve">In qua æquatione deleto rurſus utrin- <pb o="489" file="0209" n="219" rhead="GEOMET. VARIA."/> que {{1/4}ggoo/pp}, invenitur y = l - {nx/z} + √- mm + ox + {ppxx/gg} <lb/>Eademque eſt demonſtrandi ratio in caſu quarto, & </s> <s xml:id="echoid-s4589" xml:space="preserve">aliis <lb/> <anchor type="note" xlink:label="note-0209-01a" xlink:href="note-0209-01"/> quibusvis, habita ratione ſignorum + & </s> <s xml:id="echoid-s4590" xml:space="preserve">-.</s> <s xml:id="echoid-s4591" xml:space="preserve"/> </p> <div xml:id="echoid-div244" type="float" level="2" n="2"> <note position="right" xlink:label="note-0209-01" xlink:href="note-0209-01a" xml:space="preserve">fig. 7.</note> </div> <p> <s xml:id="echoid-s4592" xml:space="preserve">Cum non habetur {nx/z} in æquatione, puncta M & </s> <s xml:id="echoid-s4593" xml:space="preserve">V unum <lb/>ſunt, tunc vero ſi p = g, hoc eſt ſi habeatur + xx pro <lb/>{ppxx/gg}, erunt ſemper aſymptoti ſibi mutuò ad angulos rectos, <lb/>quia ut p ad g, ita fecimus {1/2}o ad I X & </s> <s xml:id="echoid-s4594" xml:space="preserve">ad I Y, & </s> <s xml:id="echoid-s4595" xml:space="preserve">ita IX <lb/>ad I V; </s> <s xml:id="echoid-s4596" xml:space="preserve">fiunt enim jam æquales I X, I Y, I V, & </s> <s xml:id="echoid-s4597" xml:space="preserve">ſingulæ = <lb/>{1/2}o, unde punctum V eſt in ſemicirculo ſuper X Y & </s> <s xml:id="echoid-s4598" xml:space="preserve">proin-<lb/>de angulus X V Y rectus. </s> <s xml:id="echoid-s4599" xml:space="preserve">Item quia I M = {{1/2}aogg/zpp}, patet <lb/>quod ſi ag = zp, hoc eſt ſi g ad p ut z ad a, tunc erit <lb/>I M = {{1/2}og/p}, ac proinde æqualis ipſi IX & </s> <s xml:id="echoid-s4600" xml:space="preserve">IY quæ etiam erant <lb/>{{1/2}og/p}. </s> <s xml:id="echoid-s4601" xml:space="preserve">Adeoque hoc caſu erunt aſymptoti ſibi mutuo ad angu-<lb/>los rectos; </s> <s xml:id="echoid-s4602" xml:space="preserve">cum rurſus punctum M ſit futurum in circumfe-<lb/>rentia circuli deſcripti ſuper X Y centro I.</s> <s xml:id="echoid-s4603" xml:space="preserve"/> </p> <pb o="490" file="0210" n="220" rhead="CHRIST. HUGENII"/> <figure> <image file="0210-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0210-01"/> </figure> </div> <div xml:id="echoid-div246" type="section" level="1" n="121"> <head xml:id="echoid-head166" xml:space="preserve">II. <lb/>DEMONSTRATIO <lb/>REGULÆ <lb/>DE <lb/>MAXIMIS ET MINIMIS.</head> <p> <s xml:id="echoid-s4604" xml:space="preserve">Ad inveſtiganda Maxima & </s> <s xml:id="echoid-s4605" xml:space="preserve">Minima in Geometricis quæ-<lb/>ſtionibus, regulam certam primus, quod ſciam, Fer-<lb/>matius adhibuit: </s> <s xml:id="echoid-s4606" xml:space="preserve">cujus originem ab ipſo non traditam cum <lb/>exquirerem, inveni ſimul quo pacto ea ipſa regula ad mira-<lb/>bilem brevitatem perduci poſſet, utque inde eadem illa exiſte-<lb/>ret quam poſtea vir ampliſſimus Joh. </s> <s xml:id="echoid-s4607" xml:space="preserve">Huddenius dederat, tan-<lb/>quam partem regulæ ſuæ generalioris atque elegantiſſimæ, <lb/>quæ ab alio prorſus principio pendet. </s> <s xml:id="echoid-s4608" xml:space="preserve">Hæc à Fr. </s> <s xml:id="echoid-s4609" xml:space="preserve">Schote-<lb/>nio edita eſt unà cum Carteſianis de Geometria libris. </s> <s xml:id="echoid-s4610" xml:space="preserve">Fer-<lb/>matianæ autem regulæ examen quod inſtitui eſt hujuſ-<lb/>modi.</s> <s xml:id="echoid-s4611" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4612" xml:space="preserve">Quoties Maximum aut Minimum in problemate aliquo de-<lb/> <anchor type="note" xlink:label="note-0210-01a" xlink:href="note-0210-01"/> terminandum proponitur, certum eſt utrinque æqualitatis <lb/>caſum exiſtere: </s> <s xml:id="echoid-s4613" xml:space="preserve">ut ſi data ſit poſitione recta E D & </s> <s xml:id="echoid-s4614" xml:space="preserve">puncta A, <lb/>B, oporteatque invenire in E D punctum C, unde ductis C A, <lb/>C B, quadrata earum ſimul ſumpta, ſint minima quæ eſſe poſ-<lb/>ſint; </s> <s xml:id="echoid-s4615" xml:space="preserve">neceſſe eſt ab utraque parte puncti C, eſſe puncta G & </s> <s xml:id="echoid-s4616" xml:space="preserve"><lb/>F, à quibus ducendo rectas G A, G B; </s> <s xml:id="echoid-s4617" xml:space="preserve">F A, F B oriatur ſum-<lb/>ma quadratorum G A, G B æqualis ſummæ quadratorum F A, <lb/>F B, & </s> <s xml:id="echoid-s4618" xml:space="preserve">utraque ſumma major quadratis C A, C B ſimul <lb/>ſumptis.</s> <s xml:id="echoid-s4619" xml:space="preserve"/> </p> <div xml:id="echoid-div246" type="float" level="2" n="1"> <note position="left" xlink:label="note-0210-01" xlink:href="note-0210-01a" xml:space="preserve">TAB. XLV. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s4620" xml:space="preserve">Ut igitur inveniam punctum C, unde ductis C A, C B <lb/>fiat ſumma quadratorum ab ipſis omnium minima; </s> <s xml:id="echoid-s4621" xml:space="preserve">ductis A E, <lb/>B D perpendicularibus in E D, quarum A E dicatur a; </s> <s xml:id="echoid-s4622" xml:space="preserve">B D, <lb/>b; </s> <s xml:id="echoid-s4623" xml:space="preserve">intervallum verò E, D, c: </s> <s xml:id="echoid-s4624" xml:space="preserve">fingo primùm G F, differentiam <pb file="0211" n="221"/> <pb file="0211a" n="222"/> <anchor type="figure" xlink:label="fig-0211a-01a" xlink:href="fig-0211a-01"/> <anchor type="figure" xlink:label="fig-0211a-02a" xlink:href="fig-0211a-02"/> <anchor type="figure" xlink:label="fig-0211a-03a" xlink:href="fig-0211a-03"/> <anchor type="figure" xlink:label="fig-0211a-04a" xlink:href="fig-0211a-04"/> <anchor type="figure" xlink:label="fig-0211a-05a" xlink:href="fig-0211a-05"/> <anchor type="figure" xlink:label="fig-0211a-06a" xlink:href="fig-0211a-06"/> <anchor type="figure" xlink:label="fig-0211a-07a" xlink:href="fig-0211a-07"/> <pb file="0212" n="223"/> <pb o="491" file="0213" n="224" rhead="GEOMET. VARIA."/> duarum E G, E F æqualem datælineæ quæ vocetur e; </s> <s xml:id="echoid-s4625" xml:space="preserve">& </s> <s xml:id="echoid-s4626" xml:space="preserve">quæ-<lb/>ro quanta futura ſit E G, quam appello x, ut quadrata G A, <lb/>G B ſimul ſumpta æquentur quadratis F A, F B.</s> <s xml:id="echoid-s4627" xml:space="preserve"/> </p> <div xml:id="echoid-div247" type="float" level="2" n="2"> <figure xlink:label="fig-0211a-01" xlink:href="fig-0211a-01a"> <caption xml:id="echoid-caption66" style="it" xml:space="preserve">TAB. XLIV.<lb/>Fig. 2.</caption> <variables xml:id="echoid-variables66" xml:space="preserve">D H A B E F G</variables> </figure> <figure xlink:label="fig-0211a-02" xlink:href="fig-0211a-02a"> <caption xml:id="echoid-caption67" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables67" xml:space="preserve">E G N L O I Q P D K M H F A</variables> </figure> <figure xlink:label="fig-0211a-03" xlink:href="fig-0211a-03a"> <caption xml:id="echoid-caption68" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables68" xml:space="preserve">B E F A D G C</variables> </figure> <figure xlink:label="fig-0211a-04" xlink:href="fig-0211a-04a"> <caption xml:id="echoid-caption69" style="it" xml:space="preserve">I. Casus<lb/>Fig. 4.</caption> <variables xml:id="echoid-variables69" xml:space="preserve">Y Q R C A B M L I K V C O S X</variables> </figure> <figure xlink:label="fig-0211a-05" xlink:href="fig-0211a-05a"> <caption xml:id="echoid-caption70" style="it" xml:space="preserve">II. Casus<lb/>Fig. 5.</caption> <variables xml:id="echoid-variables70" xml:space="preserve">R C Y Q A B I L M K V O X S C</variables> </figure> <figure xlink:label="fig-0211a-06" xlink:href="fig-0211a-06a"> <caption xml:id="echoid-caption71" style="it" xml:space="preserve">III. Casus<lb/>Fig. 6.</caption> <variables xml:id="echoid-variables71" xml:space="preserve">Q C D Y K L I N M S V B X C A G O</variables> </figure> <figure xlink:label="fig-0211a-07" xlink:href="fig-0211a-07a"> <caption xml:id="echoid-caption72" style="it" xml:space="preserve">Fig. 7.<lb/>IV. Casus</caption> <variables xml:id="echoid-variables72" xml:space="preserve">Q D C A B S L N X M I V Y K C G O</variables> </figure> </div> <p> <s xml:id="echoid-s4628" xml:space="preserve">Itaque quia A E = a, & </s> <s xml:id="echoid-s4629" xml:space="preserve">E G = x, erit quadratum A G = <lb/>aa + xx. </s> <s xml:id="echoid-s4630" xml:space="preserve">Et quia G D = c - x, & </s> <s xml:id="echoid-s4631" xml:space="preserve">D B = b, erit quadra-<lb/>tum G B = bb + cc - 2cx + xx, unde quadrata A G, <lb/>G B ſimul ſumpta fient = aa + bb + cc - 2cx + 2xx, <lb/>qui dicantur termini priores; </s> <s xml:id="echoid-s4632" xml:space="preserve">idque ſimiliter in quovis alio pro-<lb/>blemate intelligendum, ubi maximum aut minimum inquiri-<lb/>tur. </s> <s xml:id="echoid-s4633" xml:space="preserve">Rurfus autem quia E F = x + e, ſi ubique in ſumma <lb/>quadratorum inventa ſubſtituam x + e pro x, & </s> <s xml:id="echoid-s4634" xml:space="preserve">quadratum ab <lb/>x + e pro xx, adque ita deinceps ſi altior poteſtas ipſius x repc-<lb/>riatur, certum eſt exorituram ſummam quadratorum F A, F B; <lb/></s> <s xml:id="echoid-s4635" xml:space="preserve">quæ quidem erit aa + bb + cc - 2cx - 2ce + 2xx + 4ex + 2ee, <lb/>æquanda ſummæ quadratorum A G, G B; </s> <s xml:id="echoid-s4636" xml:space="preserve">dicantur autem hi ter-<lb/>mini poſteriores.</s> <s xml:id="echoid-s4637" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4638" xml:space="preserve">Itaque erit aa + bb + cc - 2cx + 2xx = aa + bb + cc - 2cx <lb/>- 2ce + 2xx + 4ex + 2ee. </s> <s xml:id="echoid-s4639" xml:space="preserve">Ex qua æquatione prodibit valor E G <lb/>ſive x, quando G F ſive e certæ magnitudinis lineam <lb/>refert.</s> <s xml:id="echoid-s4640" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4641" xml:space="preserve">Ponendo autem e infinitè parvam, apparebit ex eadem æ-<lb/>quatione quanta futura ſit E G, cum ipſi E F æqualis eſt, ad-<lb/>eoque habebitur determinatio quæſita puncti C, unde du-<lb/>ctæ C A, C B faciant ſummam quadratorum minimam; <lb/></s> <s xml:id="echoid-s4642" xml:space="preserve">nempe ſublatis primùm, ſi quæ ſunt, fractionibus, (quæ <lb/>in hoc exemplo nullæ ſunt) delentur termini qui utrin-<lb/>que iidem habentur, quales ſunt neceſſariò omnes quibus <lb/>litera e admixta non eſt; </s> <s xml:id="echoid-s4643" xml:space="preserve">idque facile eſt intelligere, cum <lb/>dixerimus poſteriores terminos ex prioribus deſcribi, po-<lb/>nendo x + e vel poteſtatem ejus, quoties invenitur x vel po-<lb/>teſtas ejus aliqua in prioribus. </s> <s xml:id="echoid-s4644" xml:space="preserve">Deinde omnes termini per <lb/>e dividuntur, quibuſque poſt eam diviſionem adhuc unum <lb/>e aut plura ineſſe inveniuntur, 11 delentur, quippe cum <lb/>quantitates infinitè parvas contineant reſpectu cæterorum <lb/>terminorum quibus nullum amplius ineſt e. </s> </p> </div> <div xml:id="echoid-div249" type="section" level="1" n="122"> <head xml:id="echoid-head167" style="it" xml:space="preserve">Tom. II. Qqq</head> <p> <s xml:id="echoid-s4645" xml:space="preserve">Ex quibus de-<lb/>nique ſolis invenitur quantitas x quæſita in caſu determina-<lb/> <pb o="492" file="0214" n="225" rhead="CHRIST. HUGENII"/> tionis propoſito; </s> <s xml:id="echoid-s4646" xml:space="preserve">& </s> <s xml:id="echoid-s4647" xml:space="preserve">hæc eſt ratio methodi Fermatianæ, quâ in compendium redactâ hanc aliam inveni, cujus partes duæ ſunt. </s> <s xml:id="echoid-s4648" xml:space="preserve">Nam primò,</s> </p> <p> <s xml:id="echoid-s4649" xml:space="preserve">Quando termini, quosmaximum aut minimum deſigna-<lb/>re volumus, nullam fractionem habent, in cujus deno-<lb/>minatore quantitas incognita quæſita continetur; </s> <s xml:id="echoid-s4650" xml:space="preserve">multi-<lb/>plicandus eſt terminus quiſque per numerum dimenſionum, <lb/>quem in illo habet quantitas incognita, omiſſis terminis iis <lb/>in quibus incognita quantitas non reperitur; </s> <s xml:id="echoid-s4651" xml:space="preserve">omniaque illa <lb/>producta æquanda nihilo.</s> <s xml:id="echoid-s4652" xml:space="preserve">Ita in exemplo propoſito, ubi termini priores in-<lb/>venti ſunt aa + bb + cc - 2cx + 2xx, ſummam <lb/>duorum quadratorum continentes, quam volo eſſe mi-<lb/>nimam; </s> <s xml:id="echoid-s4653" xml:space="preserve">tantummodo hujuſmodi inſtituenda erit multipli-<lb/>catio, <lb/>{aa + bb + cc - 2cx + 2xx <lb/>1 2/ <lb/>Ex qua orientur termini æquandi nihilo - 2cx + 4xx = 0};</s> <s xml:id="echoid-s4654" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>Unde fit # {1/2} c = x. <lb/></note> <p> <s xml:id="echoid-s4655" xml:space="preserve">Ita quoque ſi priores termini ſint 3ax<emph style="super">3</emph> - bx<emph style="super">3</emph> - {2bba<emph style="super">2</emph>/2c} x + ab<emph style="super">2</emph>, <lb/> <anchor type="note" xlink:label="note-0214-02a" xlink:href="note-0214-02"/> Unde termini æquandi nihilo {9ax<emph style="super">3</emph> - 3bx<emph style="super">3</emph> - {2bba<emph style="super">2</emph>/3c} x = 0/ <lb/>9axx - 3bxx - {2bba<emph style="super">2</emph>/3c} = 0}.</s> <s xml:id="echoid-s4656" xml:space="preserve"/> </p> <div xml:id="echoid-div249" type="float" level="2" n="1"> <note position="right" xlink:label="note-0214-02" xlink:href="note-0214-02a" xml:space="preserve"> <lb/>multiplicatio erit hujuſmodi # 3 # 3 # 1 <lb/></note> </div> <p> <s xml:id="echoid-s4657" xml:space="preserve">Hujus compendii ratio ut intelligatur, ſciendum primò, <lb/>quoniam termini poſter ores ex prioribus deſcribuntur, po-<lb/>nendo tantum ubique x + e pro x, neceſſariò omnes ter-<lb/>minos priores etiam in poſterioribus reperiri; </s> <s xml:id="echoid-s4658" xml:space="preserve">ideoque illos <pb o="493" file="0215" n="226" rhead="GEOMET. VARIA."/> nihil opus eſſe deſcribi, cum utrobique mox delendi forent, <lb/>atque adeo illos tantum ſcribendos in quibus unum e vel plu-<lb/>ra inſunt, ut in exemplo noſtro - 2ce + 4ex + 2ee; </s> <s xml:id="echoid-s4659" xml:space="preserve">eoſ-<lb/>que æquandos nihilo. </s> <s xml:id="echoid-s4660" xml:space="preserve">Sed etiam illos quibus plura quam u-<lb/>num e inerunt, ſcribi ſruſtra apparet, cum diviſione facta <lb/>per e delendos poſtea conſtet, ut paulò ante diximus. </s> <s xml:id="echoid-s4661" xml:space="preserve">Ita-<lb/>que nulli præterea ab initio deſcribendi inter terminos poſte-<lb/>riores quam quibus inerit e ſimplex.</s> <s xml:id="echoid-s4662" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4663" xml:space="preserve">Hi autem termini ex terminis prioribus facilè deducuntur, <lb/>cum conſtet nihil aliud eſſe quam ſecundos terminos poteſta-<lb/>tum ab x + e, quia cæteri omnes plura quam unum e vel nullum <lb/>habent. </s> <s xml:id="echoid-s4664" xml:space="preserve">Adeo ut ubicunque in prioribus terminis habe-<lb/>tur x, ſcribendum ſit in poſterioribus e; </s> <s xml:id="echoid-s4665" xml:space="preserve">& </s> <s xml:id="echoid-s4666" xml:space="preserve">ubi habe-<lb/>tur xx in prioribus, ponendum 2ex in poſterioribus; </s> <s xml:id="echoid-s4667" xml:space="preserve">& </s> <s xml:id="echoid-s4668" xml:space="preserve">ubi <lb/>x<emph style="super">3</emph> in prioribus, in poſterioribus 3exx, atque ita deinceps. <lb/></s> <s xml:id="echoid-s4669" xml:space="preserve">Dicti autem termini ſecundi cujuſque poteſtatis x + e exipſa <lb/>poteſtate x facilè deſcribuntur mutando unum x in e, & </s> <s xml:id="echoid-s4670" xml:space="preserve"><lb/>præponendo numerum dimenſionum ipſius x, ita enim ab <lb/>xx fit 2ex, & </s> <s xml:id="echoid-s4671" xml:space="preserve">ab x<emph style="super">3</emph>, 3exx; </s> <s xml:id="echoid-s4672" xml:space="preserve">atque in cæteris pari modo. </s> <s xml:id="echoid-s4673" xml:space="preserve"><lb/>Itaque ex terminis prioribus in quibus x, quos ſolos conſi-<lb/>derandos eſſe patuit, facilè etiam termini poſteriores, ii <lb/>quos nihilo adæquandos diximus, deſcribuntur; </s> <s xml:id="echoid-s4674" xml:space="preserve">multipli-<lb/>cando tantum ſingulos in numerum dimenſionum quas in ipſis <lb/>habet x. </s> <s xml:id="echoid-s4675" xml:space="preserve">Nam mutare unum x in e ne quidem opus eſt, cum <lb/>eodem redeat, ſive omnes poſtea per e ſive per x dividan-<lb/>tur, & </s> <s xml:id="echoid-s4676" xml:space="preserve">ex his quidem aperta eſt ratio compendii ad primam <lb/>partem regulæ pertinentis: </s> <s xml:id="echoid-s4677" xml:space="preserve">nunc ad alteram veniamus quæ <lb/>eſt hujuſmodi.</s> <s xml:id="echoid-s4678" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4679" xml:space="preserve">Si termini quos maximum aut minimum deſignare volu-<lb/>mus fractiones habeant in quarum denominatore occurrat <lb/>quantitas incognita, delendæ primùm ſunt quantitates co-<lb/>gnitæ ſi quæ adſint; </s> <s xml:id="echoid-s4680" xml:space="preserve">deinde ſi reliquæ quantitates non ha-<lb/>beant eundem denominatorem, eò reducendæ ſunt. </s> <s xml:id="echoid-s4681" xml:space="preserve">Tunc <lb/>termini ſinguli numeratorem fractionis conſtituentes, du-<lb/>cendi in terminos ſingulos denominatoris, productaque <lb/>ſingula multipla ſumenda ſecundum numerum quo dimen- <pb o="494" file="0216" n="227" rhead="CHRIST. HUGENII"/> ſiones quantitatis incognitæ in termino numeratoris diffe-<lb/>runt à dimenſionibus ejuſdem incognitæ quantitatis in ter-<lb/>mino denominatoris. </s> <s xml:id="echoid-s4682" xml:space="preserve">Signa autem affectionis productis <lb/>ſingulis præponenda qualia lex multiplicationis exigit, quo-<lb/>ties dimenſiones quantitatis incognitæ plures ſunt in termi-<lb/>no numeratoris quam in termino denominatoris: </s> <s xml:id="echoid-s4683" xml:space="preserve">at quo-<lb/>ties contra evenit, contraria quoque ſigna productis præ-<lb/>ponenda; </s> <s xml:id="echoid-s4684" xml:space="preserve">quæ denique omnia æquanda nihilo.</s> <s xml:id="echoid-s4685" xml:space="preserve">Sint, exempli gratiâ, inventi termini priores, quos maxi-<lb/>mum deſignare velimus, iſti {bx<emph style="super">3</emph> - ccxx - 2bccx/bcc + x<emph style="super">3</emph>}, ubi nul-<lb/>la eſt quantitas cognita. </s> <s xml:id="echoid-s4686" xml:space="preserve">Hîc ergo, ſecundum regulam, multi-<lb/>plico terminos omnes numeratoris primum per bcc, prioris-<lb/>que producti ex bx<emph style="super">3</emph> in bcc, ſcribo triplum, quia bx<emph style="super">3</emph> ha-<lb/>bet tres dimenſiones quantitatis incognitæ x, bcc verò <lb/>nullam. </s> <s xml:id="echoid-s4687" xml:space="preserve">Secundi producti ex - ccxx in bcc ſcribo duplum, <lb/>propterea quod in - ccxx duæ ſunt dimenſiones x, & </s> <s xml:id="echoid-s4688" xml:space="preserve">in <lb/>bcc nulla. </s> <s xml:id="echoid-s4689" xml:space="preserve">Tertium verò productum ex - 2bccx in bcc ſcri-<lb/>bo ſimplex, quia in - 2bccx & </s> <s xml:id="echoid-s4690" xml:space="preserve">bcc differentia dimenſionum <lb/>x eſt unitas. </s> <s xml:id="echoid-s4691" xml:space="preserve">Tribus autem hiſce productis vera ſigna af-<lb/>fectionis adſcribo, quoniam dimenſiones x in terminis nu-<lb/>meratoris excedunt eas quæ in termino bcc, quippe quæ <lb/>nullæ ſunt, ita ut tria hæc producta ſint <lb/>3bbccx<emph style="super">3</emph> - 2bc<emph style="super">4</emph>xx - 2bbc<emph style="super">4</emph>x.</s> <s xml:id="echoid-s4692" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4693" xml:space="preserve">Jam porrò terminos omnes eoſdem numeratoris duco in x<emph style="super">3</emph>, <lb/>terminum alterum denominatoris, primumque productum ex <lb/>bx<emph style="super">3</emph> in x<emph style="super">3</emph> ſcribere omitto, ſive per 0 multiplico, quoniam <lb/>eædem dimenſiones utrobique ſunt ipſius x, ideoque diffe-<lb/>rentia nulla. </s> <s xml:id="echoid-s4694" xml:space="preserve">Secundum autem productum ex - ccxx in <lb/>x<emph style="super">3</emph> ſcribo ſimplex, quia in his terminis differentia dimenſio-<lb/>num x eſt unitas. </s> <s xml:id="echoid-s4695" xml:space="preserve">At tertium productum ex - 2bccx in <lb/>x<emph style="super">3</emph> ſcribo duplum, quia differentia dimenſionum x in his eſt <lb/>2. </s> <s xml:id="echoid-s4696" xml:space="preserve">Signa verò affectionis productis hiſce duobus adſcribo <lb/>contraria iis quæ requireret lex multiplicationis, eo quod di- <pb o="495" file="0217" n="228" rhead="GEOMET. VARIA."/> menſiones x pauciores ſunt utrobique in terminis numerato-<lb/>ris quam in x<emph style="super">3</emph>, termino denominatoris.</s> <s xml:id="echoid-s4697" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4698" xml:space="preserve">Itaque producta bina erunt hæc + ccx<emph style="super">5</emph> + 4bccx<emph style="super">4</emph>; </s> <s xml:id="echoid-s4699" xml:space="preserve">quæ ad-<lb/>dita tribus præcedentibus <lb/>+ 3bbccx<emph style="super">3</emph> - 2bc<emph style="super">4</emph>xx - 2bbcx<emph style="super">4</emph>, <lb/>faciunt ſummam æquandam nihilo <lb/>ccx<emph style="super">5</emph> + 4bccx<emph style="super">4</emph> + 3bbccx<emph style="super">3</emph> - 2bc<emph style="super">4</emph>xx - 2bbc<emph style="super">4</emph>x = 0; <lb/></s> <s xml:id="echoid-s4700" xml:space="preserve">qua æquatione diviſa per ccbx + ccxx, fit x<emph style="super">3</emph> + 3bxx - 2bcc = 0.</s> <s xml:id="echoid-s4701" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4702" xml:space="preserve">Quomodo autem ad hæc perventum ſit uno exemplo rur-<lb/>ſus explicabimus, ex quo eandem in omnibus cæteris ratio-<lb/>nem eſſe intelligetur. </s> <s xml:id="echoid-s4703" xml:space="preserve">Videamus igitur priores terminos quos <lb/>modò propoſueram, nempe {bx<emph style="super">3</emph> - ccxx - 2bccx/bcc + x<emph style="super">3</emph>}; </s> <s xml:id="echoid-s4704" xml:space="preserve">ex qui-<lb/>bus ſi alios quibuſcum eos comparem, ut initio factum eſt, <lb/>deſcribere velim, ponendo ubique x + e ubi eſt x; </s> <s xml:id="echoid-s4705" xml:space="preserve">video <lb/>quidem primò omnes illos in poſterioribus terminis poſſe ne-<lb/>gligi in quibus plura quam unum e inerit, quia ſemper ex iis <lb/>quantitates orientur in quibus plura uno e inerunt, quæque <lb/>proinde delendæ tandem erunt, ob cauſam in ſuperioribus <lb/>traditam.</s> <s xml:id="echoid-s4706" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4707" xml:space="preserve">Itaque erunt termini priores æquandi poſterioribus <lb/>{bx<emph style="super">3</emph> - ccxx - 2bccx, <lb/>{bx<emph style="super">3</emph> - ccxx - 2bccx/bcc + x<emph style="super">3</emph>} = {+ 3bexx - 2ccex - 2bcce,/bcc + x<emph style="super">3</emph> + 3exx} <lb/>qui nempe ex prioribus hac lege deſcripti ſunt, ut ubicun-<lb/>que eſt x vel poteſtas ejus in prioribus, ibi ponatur x + e <lb/>vel poteſtatis x + e duo priores termini; </s> <s xml:id="echoid-s4708" xml:space="preserve">quoniam ſcimus <lb/>in cæteris plura quam unum e contineri.</s> <s xml:id="echoid-s4709" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4710" xml:space="preserve">Jam verò porrò, quia termini in quibus nullum e in nu-<lb/>meratore ac denominatore priorum ac poſteriorum termi-<lb/>norum, iidem planè reperiuntur, patet multiplicationes al-<lb/>ternas eorum terminorum denominatoris in terminos nu- <pb o="496" file="0218" n="229" rhead="CHRIST. HUGENII"/> meratoris partis alterius e carentes, omitti poſſe, cum quan-<lb/>titates inde ortæ eædem utrinque eſſent futuræ ideoque de-<lb/>lendæ. </s> <s xml:id="echoid-s4711" xml:space="preserve">Quare in terminis poſterioribus ii tantum ab ini-<lb/>tio ſcribendi erant in quibus unum e, omiſſis omnibus reli-<lb/>quis, ut æquatio hîc futura ſit iſta <lb/>{bx<emph style="super">3</emph> - ccxx - 2bccx/bcc + x<emph style="super">3</emph>} = {3bexx - 2ccex - 2bcce/3exx}</s> </p> <p> <s xml:id="echoid-s4712" xml:space="preserve">Hîc jam multiplicationes alternæ per denominatores in-<lb/>ſtituendæ eſſent ad tollendas fractiones. </s> <s xml:id="echoid-s4713" xml:space="preserve">Verum examinan-<lb/>do diligentius quænam futura ſint harum multiplicationum <lb/>producta, aliud adhuc compendium inveniemus, & </s> <s xml:id="echoid-s4714" xml:space="preserve">nec <lb/>ſcribendos quidem omnino eſſe terminos poſteriores: </s> <s xml:id="echoid-s4715" xml:space="preserve">quia <lb/>enim deſcribuntur ex prioribus mutato x in e, præpoſito-<lb/>que numero dimenſionum ipſius x, non difficile eſt col-<lb/>ligere ex ſolis terminis prioribus quænam futura ſint iſta <lb/>omnia producta.</s> <s xml:id="echoid-s4716" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4717" xml:space="preserve">Ita quoniam propter - ccxx in prioribus, habetur -<lb/>2ccex in poſterioribus; </s> <s xml:id="echoid-s4718" xml:space="preserve">& </s> <s xml:id="echoid-s4719" xml:space="preserve">propter x<emph style="super">3</emph> in denominatore prio-<lb/>rum, in poſteriorum denominatore eſt 3exx; </s> <s xml:id="echoid-s4720" xml:space="preserve">facile per-<lb/>ſpicitur utraque producta ex - ccxx in 3exx & </s> <s xml:id="echoid-s4721" xml:space="preserve">ex -<lb/>2ccex in x<emph style="super">3</emph>, quæ ſunt - 3ccex<emph style="super">4</emph> & </s> <s xml:id="echoid-s4722" xml:space="preserve">- 2ccex<emph style="super">4</emph>, eaſdem <lb/>literas habitura, ſed diverſos numeros præpoſitos 3 & </s> <s xml:id="echoid-s4723" xml:space="preserve">2, <lb/>idque inde fieri quòd in termino ccxx unam dimenſio-<lb/>nem minus habeat x quam in termino x<emph style="super">3</emph>. </s> <s xml:id="echoid-s4724" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s4725" xml:space="preserve">au-<lb/>ferendo poſtea ex utraque parte æquationis, - 2ccex<emph style="super">4</emph>, ap-<lb/>paret ſuperfuturum - ccex<emph style="super">4</emph> à parte terminorum priorum. <lb/></s> <s xml:id="echoid-s4726" xml:space="preserve">Quare ab initio hoc ſciri poteſt, multiplicando tantum in <lb/>terminis prioribus - ccxx numeratoris in x<emph style="super">3</emph> denomina-<lb/>toris, unumque x in e mutando, ac productum ſimplex ſcri-<lb/>bendo; </s> <s xml:id="echoid-s4727" xml:space="preserve">quia differentia dimenſionum x in iſtis duobus ter-<lb/>minis eſt unitas.</s> <s xml:id="echoid-s4728" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4729" xml:space="preserve">Eadem ratione producta ex - 2bccx in 3exx, & </s> <s xml:id="echoid-s4730" xml:space="preserve">ex -<lb/>2bcce in x<emph style="super">3</emph>, quæ eaſdem literas habent, ſunt enim -<lb/>6bccex<emph style="super">3</emph> & </s> <s xml:id="echoid-s4731" xml:space="preserve">- 2bccex<emph style="super">3</emph>, habebunt numeros præpoſitos di- <pb o="497" file="0219" n="230" rhead="CEOMET. VARIA"/> verſos, propterea quod in - 2bccx una tantum eſt dimen-<lb/>ſio x; </s> <s xml:id="echoid-s4732" xml:space="preserve">at in x<emph style="super">3</emph> tres, unde ablato ex utraque parte æqua-<lb/>tionis - 2bccex<emph style="super">3</emph>, ſcio ſuperfuturum à parte terminorum <lb/>priorum - 4bccex<emph style="super">3</emph>: </s> <s xml:id="echoid-s4733" xml:space="preserve">quod rurſus ab initio cognoſci po-<lb/>tuit, quia eadem quantitas oritur, multiplicando - 2bccx <lb/>numeratoris terminorum priorum, in x<emph style="super">3</emph> denominatoris, <lb/>mutandoque unum x in e, & </s> <s xml:id="echoid-s4734" xml:space="preserve">productum multiplicando per <lb/>2, quæ eſt differentia dimenſionum x in terminis - 2bccx <lb/>& </s> <s xml:id="echoid-s4735" xml:space="preserve">x<emph style="super">3</emph>.</s> <s xml:id="echoid-s4736" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4737" xml:space="preserve">At quoniam in bx<emph style="super">3</emph> & </s> <s xml:id="echoid-s4738" xml:space="preserve">in x<emph style="super">3</emph> eadem eſt dimenſio x, <lb/>ſequetur producta ex bx<emph style="super">3</emph> in 3exx, & </s> <s xml:id="echoid-s4739" xml:space="preserve">ex 3bexx in x<emph style="super">3</emph>, <lb/>tum literas eaſdem, tum eoſdem numeros præpoſitos ha-<lb/>bitura, ideoque ſeſe mutuo ſublatura, ut proinde multi-<lb/>plicatio illa omitti poſſit.</s> <s xml:id="echoid-s4740" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4741" xml:space="preserve">Atque hujuſmodi animadverſionibus inventum eſt quod in <lb/>regula præcipitur, terminos ſingulos numeratoris in ſingu-<lb/>los denominatoris terminos eſſe ducendos, productaque quæ-<lb/>libet multipla ſumenda ſecundum differentiam dimenſionum <lb/>quantitatis incognitæ in terminis binis qui in ſe mutuò ducun-<lb/>tur. </s> <s xml:id="echoid-s4742" xml:space="preserve">Nam quod non præcipitur unum x in e mutandum, id <lb/>hanc rationem habet, quod non referat utrum poſtea per e an <lb/>per x omnes termini dividantur.</s> <s xml:id="echoid-s4743" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4744" xml:space="preserve">Quod vero ſi ſigna affectionis vera productis ſingulis præ-<lb/>ponenda dicuntur, quoties dimenſiones x plures ſunt in nu-<lb/>meratore quam in denominatore, id quoque ex jam dictis in-<lb/>telligetur; </s> <s xml:id="echoid-s4745" xml:space="preserve">uti conſequenter etiam hoc quod contraria ſigna <lb/>ſunt adponenda, quoties dimenſionum numerus contra ſe <lb/>habet. </s> <s xml:id="echoid-s4746" xml:space="preserve">Velut hîc, productum ex bx<emph style="super">3</emph> in bcc ſcribendum <lb/>eſt cum ſigno - præpoſito numero 3, ut fiat - 3bbccx<emph style="super">3</emph>, <lb/>quia nempe propter bx<emph style="super">3</emph> ſcimus in poſterioribus terminis <lb/>fore 3bexx; </s> <s xml:id="echoid-s4747" xml:space="preserve">quod ductum in bcc faciet + 3bbccexx, ſed <lb/>tranſlatum in partem priorem æquationis, fiet - 3bbccexx; <lb/></s> <s xml:id="echoid-s4748" xml:space="preserve">ſive, non mutato x in e, - 3bbccx<emph style="super">3</emph>.</s> <s xml:id="echoid-s4749" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4750" xml:space="preserve">Quod denique in regula habetur, quoties in prioribus ter-<lb/>minis priusquam ad eundem denominatorem reducantur, <lb/>quantitates cognitæ occurrunt eas primum omnium delendas, <pb o="498" file="0220" n="231" rhead="CHRIST. HUGENII"/> id ex hoc ſequenti exemplo intelligetur rectè præcipi. </s> <s xml:id="echoid-s4751" xml:space="preserve">Sint <lb/>enim reperti termini priores, quos maximum aut mini-<lb/>mum deſignare oporteat, iſti {x<emph style="super">3</emph>/2a - x} - 2vx + xx + vv; <lb/></s> <s xml:id="echoid-s4752" xml:space="preserve">ubi vv quantitatem cognitam ſignificet: </s> <s xml:id="echoid-s4753" xml:space="preserve">id igitur delendum <lb/>eſſe ut appareat, videamus quid futurum ſit ſi non delea-<lb/>tur. </s> <s xml:id="echoid-s4754" xml:space="preserve">Nempe ut ad eundem denominatorem cum cæteris <lb/>omnibus reducatur, ducendum erit vv in 2a - x, fietque <lb/>inde {2avv - xvv/2a - x} in terminis prioribus. </s> <s xml:id="echoid-s4755" xml:space="preserve">Propter quos in <lb/>terminis poſterioribus, ſecundum ſuperius, explicata ſcribetur <lb/>{- evv/- e}, adeoque multiplicatione alternatim utrinque per <lb/>denominatores inſtituta, ducendum erit hinc 2a - x in <lb/>- evv; </s> <s xml:id="echoid-s4756" xml:space="preserve">inde - e in 2avv - xvv. </s> <s xml:id="echoid-s4757" xml:space="preserve">Ex quibus multiplica-<lb/>tionibus eoſdem utrinque terminos oriri neceſſe eſt, cum <lb/>utrobique eadem hæc tria in ſe mutuo ducantur 2a - x in <lb/>- e in vv, qui proinde termini ſe ſe mutuo ſublaturi eſſent, <lb/>eoque fruſtra ſcriberentur; </s> <s xml:id="echoid-s4758" xml:space="preserve">ac proinde liquet tuto deleri <lb/>poſſe ab initio quantitatem vv, idemque quod in hoc exem-<lb/>plo accidit, neceſſario quoque in quibuslibet aliis continge-<lb/>re, diligenter intuenti manifeſtum erit.</s> <s xml:id="echoid-s4759" xml:space="preserve"/> </p> <figure> <image file="0220-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0220-01"/> </figure> </div> <div xml:id="echoid-div251" type="section" level="1" n="123"> <head xml:id="echoid-head168" xml:space="preserve">III.</head> <head xml:id="echoid-head169" xml:space="preserve">REGULA</head> <head xml:id="echoid-head170" style="it" xml:space="preserve">Ad inveniendas Tangentes linearum curvarum.</head> <p> <s xml:id="echoid-s4760" xml:space="preserve">IDem Fermatius linearum curvarum Tangentes regula ſibi <lb/>peculiari inquirebat, quam Carteſius ſuſpicabatur non ſa-<lb/>tis ipſum intelligere quo fundamento niteretur, ut ex epiſto-<lb/>lis ejus hac de re ſcriptis apparet. </s> <s xml:id="echoid-s4761" xml:space="preserve">Sanè in Fermatii operi-<lb/>bus poſt mortem editis, ncc bene expoſitus eſt regulæ uſus, <lb/>nec demonſtrationem ullam adjectam habet. </s> <s xml:id="echoid-s4762" xml:space="preserve">Carteſium ve-<lb/>ro in his quas dixi literis, rationem ejus aliquatenus aſſecu-<lb/>tum invenio, nec tamen tam perſpicuè eam explicuiſſe quam <pb o="499" file="0221" n="232" rhead="GEOMET. VARIA."/> per hæc quæ nunc trademus fiet, quæ jam olim, multò an-<lb/>te iſtas literas vulgatas conſcripſimus.</s> <s xml:id="echoid-s4763" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4764" xml:space="preserve">Præcipuum verò operæ pretium tunc fuit compendioſa hu-<lb/>juſce regulæ contractio, quam, quoad potui, proſecutus, <lb/>tandem in ipſas illas inſignes Huddenii, Sluſiique regulas <lb/>deſinere inveni, quas mihi Viri hi Clariſſimi uterque ferè eo-<lb/>dem tempore exhibuerant: </s> <s xml:id="echoid-s4765" xml:space="preserve">an vero hac eadem viâ an aliâ in <lb/>illas inciderint nondum mihi compertum.</s> <s xml:id="echoid-s4766" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4767" xml:space="preserve">Sit data linea curva ut B C, quæ cognitam relationem ha-<lb/> <anchor type="note" xlink:label="note-0221-01a" xlink:href="note-0221-01"/> beat ad rectam aliquam poſitione datam A F; </s> <s xml:id="echoid-s4768" xml:space="preserve">ac proinde ap-<lb/>plicatâ è puncto quolibet curvæ, ut B, rectâ B F, in dato <lb/>angulo B F A, datoque in recta A F puncto A, certa æqua-<lb/>tione relatio quæ eſt inter A F & </s> <s xml:id="echoid-s4769" xml:space="preserve">F B expreſſa habeatur. </s> <s xml:id="echoid-s4770" xml:space="preserve">Ex-<lb/>empli gratiâ, appellando A F, x; </s> <s xml:id="echoid-s4771" xml:space="preserve">F B, y, ſit æquatio x<emph style="super">3</emph> <lb/>= xya - y<emph style="super">3</emph>, ubi a lineam quandam datam ſignificare cen-<lb/>ſenda eſt.</s> <s xml:id="echoid-s4772" xml:space="preserve"/> </p> <div xml:id="echoid-div251" type="float" level="2" n="1"> <note position="right" xlink:label="note-0221-01" xlink:href="note-0221-01a" xml:space="preserve">TAB. XLV. <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s4773" xml:space="preserve">Quod ſi jam ad punctum B tangens ducenda ſit B E, quæ <lb/>occurrat rectæ A F in E, voceturque F E, z, ejus longitu-<lb/>do per hanc regulam Fermatianæ regulæ compendiariam, in-<lb/>venietur, ex ſola æquatione data.</s> <s xml:id="echoid-s4774" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4775" xml:space="preserve">Tranſlatis terminis omnibus æquationis datæ ad unam æqua-<lb/>tionis partem, qui proinde æquales fiunt nihilo, multiplicen-<lb/>tur primò termini ſinguli, in quibus reperitur y, per nume-<lb/>rum dimenſionum quas in ipſis habet y, atque ea erit quan-<lb/>titas dividenda. </s> <s xml:id="echoid-s4776" xml:space="preserve">Deinde ſimiliter termini ſinguli in quibus x, <lb/>multiplicentur per numerum dimenſionum quas in ipſis habet <lb/>x, & </s> <s xml:id="echoid-s4777" xml:space="preserve">è ſingulis unum x tollatur; </s> <s xml:id="echoid-s4778" xml:space="preserve">atque hæc quantitas pro diviſo-<lb/>re erit ſubſcribenda quantitati dividendæ jam inventæ. </s> <s xml:id="echoid-s4779" xml:space="preserve">Quo fa-<lb/>cto habebitur quantitas æqualis z ſive F E. </s> <s xml:id="echoid-s4780" xml:space="preserve">Signa autem + & </s> <s xml:id="echoid-s4781" xml:space="preserve">-<lb/>eadem ubique retinenda ſunt; </s> <s xml:id="echoid-s4782" xml:space="preserve">atque etiam ſi forte quantitas di-<lb/>viſoris, vel dividenda, vel utraque minor nihilo ſive negata ſit, <lb/>tamen tanquam adfirmatæ ſunt conſiderandæ: </s> <s xml:id="echoid-s4783" xml:space="preserve">hoc tantum <lb/>obſervando, ut cum altera adfirmata eſt, altera negata, tunc <lb/>F E ſumatur verſus punctum A, cum verò utraque vel affir-<lb/>mata eſt vel negata, ut tunc ſumatur F E in partem contra-<lb/>riam.</s> <s xml:id="echoid-s4784" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div253" type="section" level="1" n="124"> <head xml:id="echoid-head171" style="it" xml:space="preserve">Tom. II. Rrr</head> <pb o="500" file="0222" n="233" rhead="CHRIST. HUGENII"/> <p> <s xml:id="echoid-s4785" xml:space="preserve">In curvâ propoſita cujus æquatio x<emph style="super">3</emph> + y<emph style="super">3</emph> - axy = 0, <lb/>fiet ſecundum hanc regulam dividenda quantitas 3y<emph style="super">3</emph> - axy, <lb/>diviſor verò 3xx - ay; </s> <s xml:id="echoid-s4786" xml:space="preserve">ideoque z = {3y<emph style="super">3</emph> - axy/3xx - ay}, quæ <lb/>eſt longitudo cognita, cum dentur x, y & </s> <s xml:id="echoid-s4787" xml:space="preserve">a.</s> <s xml:id="echoid-s4788" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4789" xml:space="preserve">Eſto item alia curva A B H, cujus æquatio axx - x<emph style="super">3</emph> -<lb/> <anchor type="note" xlink:label="note-0222-01a" xlink:href="note-0222-01"/> qqy = 0, poſito ſcilicet a & </s> <s xml:id="echoid-s4790" xml:space="preserve">q eſſe lineas datas, A F vero <lb/>= x, F B = y. </s> <s xml:id="echoid-s4791" xml:space="preserve">Sit B E tangens, & </s> <s xml:id="echoid-s4792" xml:space="preserve">F E dicatur ut ante, <lb/>z. </s> <s xml:id="echoid-s4793" xml:space="preserve">Hîc fiet ſecundum regulam, dividenda quantitas - qqy; <lb/></s> <s xml:id="echoid-s4794" xml:space="preserve">diviſor autem 2ax - 3xx; </s> <s xml:id="echoid-s4795" xml:space="preserve">unde z = {- qqy/2ax - 3xx} Ubi cum <lb/>dividenda quantitas ſit negata, ſi fuerit etiam diviſor minor <lb/>nihilo, hoc eſt ſi 2a minor quam 3x, erit z ſive fe ſumen-<lb/>da in partem ab A averſam. </s> <s xml:id="echoid-s4796" xml:space="preserve">Si vero 2a major quam 3x, ſu-<lb/>menda erit F E verſus A, ex præcepto regulæ.</s> <s xml:id="echoid-s4797" xml:space="preserve"/> </p> <div xml:id="echoid-div253" type="float" level="2" n="1"> <note position="left" xlink:label="note-0222-01" xlink:href="note-0222-01a" xml:space="preserve">TAB. XLV. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s4798" xml:space="preserve">Horum vero rationem, ipſiuſque regulæ & </s> <s xml:id="echoid-s4799" xml:space="preserve">compendii quò <lb/> <anchor type="note" xlink:label="note-0222-02a" xlink:href="note-0222-02"/> reducta eſt, originem ut explicemus, proponatur ut ante <lb/>curva B C, ad cujus punctum B tangens ducenda ſit.</s> <s xml:id="echoid-s4800" xml:space="preserve"/> </p> <div xml:id="echoid-div254" type="float" level="2" n="2"> <note position="left" xlink:label="note-0222-02" xlink:href="note-0222-02a" xml:space="preserve">TAB. XLV. <lb/>fig. 4.</note> </div> <p> <s xml:id="echoid-s4801" xml:space="preserve">Intelligatur primum recta E B D, quæ non tangat curvam <lb/>ſed eam ſecet in B, atque item in alio puncto D, ipſi B <lb/>proximo; </s> <s xml:id="echoid-s4802" xml:space="preserve">rectæ autem A G occurrat in E; </s> <s xml:id="echoid-s4803" xml:space="preserve">& </s> <s xml:id="echoid-s4804" xml:space="preserve">ab utriſque <lb/>punctis B, D ducantur ad rectam A G, iiſdem angulis in-<lb/>clinatæ B F, D G; </s> <s xml:id="echoid-s4805" xml:space="preserve">& </s> <s xml:id="echoid-s4806" xml:space="preserve">ſit A F = x, F B = y, ſicut antea; <lb/></s> <s xml:id="echoid-s4807" xml:space="preserve">ponatur que etiam F G data eſſe, quæ ſit e, quæraturque F E <lb/>= z.</s> <s xml:id="echoid-s4808" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4809" xml:space="preserve">Eſt itaque ſicut E F ad F B, hoc eſt, ſicut z ad y, ita E G, <lb/>hoc eſt, z + e ad G D; </s> <s xml:id="echoid-s4810" xml:space="preserve">quæ erit y + {ey/z}; </s> <s xml:id="echoid-s4811" xml:space="preserve">& </s> <s xml:id="echoid-s4812" xml:space="preserve">hoc quidem <lb/>in qualibet curva ita ſe habere manifeſtum eſt.</s> <s xml:id="echoid-s4813" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4814" xml:space="preserve">Nunc porrò conſideretur æquatio naturam curvæ conti-<lb/>nens, ex. </s> <s xml:id="echoid-s4815" xml:space="preserve">gr. </s> <s xml:id="echoid-s4816" xml:space="preserve">illa ſuperius propoſita x<emph style="super">3</emph> + y<emph style="super">3</emph> - xya = 0, <lb/>ubi a rectam longitudine datam, velut A H ſignificabat; </s> <s xml:id="echoid-s4817" xml:space="preserve">& </s> <s xml:id="echoid-s4818" xml:space="preserve"><lb/>patet, cum punctum D in curva ponatur, debere eodem mo-<lb/>do duas A G, G D, hoc eſt x + e & </s> <s xml:id="echoid-s4819" xml:space="preserve">y + {ey/z} ad ſe mutuo referri <pb o="501" file="0223" n="234" rhead="GEOMET. VARIA."/> atque A F, F B, hoc eſt x & </s> <s xml:id="echoid-s4820" xml:space="preserve">y. </s> <s xml:id="echoid-s4821" xml:space="preserve">Nempe ſi in æquatione <lb/>propoſita pro x ſubſtituatur ubique x + e, & </s> <s xml:id="echoid-s4822" xml:space="preserve">pro y, ubique <lb/>y + {ey/z}, debebit æquatio hinc formata terminos omnes ha-<lb/>bere æquales nihilo; </s> <s xml:id="echoid-s4823" xml:space="preserve">hoc eſt <lb/>x<emph style="super">3</emph> + [3exx] + 3eex + e<emph style="super">3</emph>, + y<emph style="super">3</emph> + [{3ey<emph style="super">3</emph>/z}] + {3eey<emph style="super">3</emph>/zz} + {e<emph style="super">3</emph>y<emph style="super">3</emph>/z<emph style="super">3</emph>} = 0. <lb/></s> <s xml:id="echoid-s4824" xml:space="preserve">- axy - [aey] - [{aeyx/z}] - {aeey/z}</s> </p> <p> <s xml:id="echoid-s4825" xml:space="preserve">In hac autem æquatione conſtat neceſſario terminos prio-<lb/>ris æquationis, ex qua formata eſt, contineri debere, nem-<lb/>pe x<emph style="super">3</emph> + y<emph style="super">3</emph> - axy: </s> <s xml:id="echoid-s4826" xml:space="preserve">qui cum ſint æquales nihilo ex proprie-<lb/>tate curvæ, idcirco his in æquatione deletis, neceſſe eſt etiam <lb/>reliquos nihilo æquari, in quibus ſingulis manifeſtum quoque <lb/>eſt vel unum e vel plura reperiri, ideoque omnes per e divi-<lb/>di poſſe. </s> <s xml:id="echoid-s4827" xml:space="preserve">Qui autem poſt hanc diviſionem non amplius ha-<lb/>bebunt e, eos, neglectis reliquis, ſcio nihilo æquari debe-<lb/>re, quantitatemque lineæ z ſive F E oſtenſuros; </s> <s xml:id="echoid-s4828" xml:space="preserve">ſi nempe <lb/>B E jam tanquam tangens conſideretur, ideoque F G, ſeu <lb/>e, infinitè parva. </s> <s xml:id="echoid-s4829" xml:space="preserve">Nam termini in quibus adhuc e ſupereſt, <lb/>etiam quantitates infinite parvas ſive omnino evaneſcentes <lb/>continebunt. </s> <s xml:id="echoid-s4830" xml:space="preserve">Et his quidem hactenus Fermatianæ regulæ <lb/>origo ac ratio declaratur: </s> <s xml:id="echoid-s4831" xml:space="preserve">nunc porro oſtendemus quomodo <lb/>eadem ad tantam brevitatem perducta ſit. </s> <s xml:id="echoid-s4832" xml:space="preserve">Video itaque ex <lb/>æquatione totâ noviſſimâ, tantum eos terminos ſeribi neceſ-<lb/>ſe eſſe quibus ineſt e ſimplex, velut hic 3exx + {3ey<emph style="super">3</emph>/z} - aey -<lb/>{aeyx/z} = 0. </s> <s xml:id="echoid-s4833" xml:space="preserve">Qui termini quomodo facili negotio ex datis æqua-<lb/>tionis terminis x<emph style="super">3</emph> + y<emph style="super">3</emph> - axy = 0, deſcribi poſſint, dein-<lb/>ceps explicandum. </s> <s xml:id="echoid-s4834" xml:space="preserve">Et primò quidem apparet 3exx + <lb/>{3ey<emph style="super">3</emph>/z} nihil aliud eſſe quam ſecundos terminos cuborum ab <pb o="502" file="0224" n="235" rhead="CHRIST. HUGENII"/> x + e & </s> <s xml:id="echoid-s4835" xml:space="preserve">aby + {e/z} ideo ſcriptos quia in æquatione habentur <lb/>cubi ab x & </s> <s xml:id="echoid-s4836" xml:space="preserve">y. </s> <s xml:id="echoid-s4837" xml:space="preserve">Nam reliqui omnes termini cuborum, ut & </s> <s xml:id="echoid-s4838" xml:space="preserve"><lb/>quarumvis aliarum poteſtatem ab x + e, & </s> <s xml:id="echoid-s4839" xml:space="preserve">ab y + {ey/z}, vel <lb/>plura quam unum e habent, vel nullum; </s> <s xml:id="echoid-s4840" xml:space="preserve">ideoque, ut jam <lb/>diximus, fruſtra ſcriberentur. </s> <s xml:id="echoid-s4841" xml:space="preserve">Eâdem itaque ratione, ſi aliæ <lb/>poteſtates ab x vel y eſſent in æquatione propoſitæ, ſcriben-<lb/>di forent in æquatione alterâ termini ſecundi tantùm ſimi-<lb/>lium poteſtatum ab x + e & </s> <s xml:id="echoid-s4842" xml:space="preserve">aby + {ey/z}. </s> <s xml:id="echoid-s4843" xml:space="preserve">Notandumque ſe-<lb/>cundos hoſce terminos, ex ipſis datis poteſtatibus ab x & </s> <s xml:id="echoid-s4844" xml:space="preserve">y, <lb/>certa ratione confici; </s> <s xml:id="echoid-s4845" xml:space="preserve">nempe ex poteſtate quavis x, velut <lb/>x<emph style="super">3</emph>, mutando unum x in e, & </s> <s xml:id="echoid-s4846" xml:space="preserve">præponendo numerum di-<lb/>menſionum ipſius x: </s> <s xml:id="echoid-s4847" xml:space="preserve">ita hîc fit 3exx. </s> <s xml:id="echoid-s4848" xml:space="preserve">Ex poteſtate y verò <lb/>ducendo eam in {e/z} præponendoque ſimiliter numerum di-<lb/>menſionum ipſius y: </s> <s xml:id="echoid-s4849" xml:space="preserve">ita hîc ab y<emph style="super">3</emph> fit {3y<emph style="super">3</emph>e/z}. </s> <s xml:id="echoid-s4850" xml:space="preserve">Quorum qui-<lb/>dem rationem ex poteſtatum formatione intelligere facilli-<lb/>mum eſt.</s> <s xml:id="echoid-s4851" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4852" xml:space="preserve">Porrò propter xy in termino æquationis - axy, facile <lb/>quoque apparet quid in æquatione ſecunda ſcribendum ſit. <lb/></s> <s xml:id="echoid-s4853" xml:space="preserve">Cum enim ſubſtituendum ſit pro xy productum ab x + e in y + <lb/>{ey/z}, ſed ea tantum ſcribenda in quibus unum e, ideo de duo-<lb/>bus x + e tantùm e ducemus in y, & </s> <s xml:id="echoid-s4854" xml:space="preserve">tantùm x in {ey/z}; </s> <s xml:id="echoid-s4855" xml:space="preserve">ad-<lb/>eoque fient ey + {exy/z}; </s> <s xml:id="echoid-s4856" xml:space="preserve">quibus in a ductis, præpoſitoque ſi-<lb/>gno -, quia habetur - axy, exiſtet - aey - {aexy/z}, ſicut <lb/>ſuprà.</s> <s xml:id="echoid-s4857" xml:space="preserve"/> </p> <pb o="503" file="0225" n="236" rhead="GEOMET. VARIA."/> <p> <s xml:id="echoid-s4858" xml:space="preserve">Sic quoque ſi in æquatione propoſita haberetur xxy<emph style="super">3</emph>; </s> <s xml:id="echoid-s4859" xml:space="preserve">ſu-<lb/>merem propter xx duos priores terminos quadrati ab x + e, <lb/>nempe xx + 2ex; </s> <s xml:id="echoid-s4860" xml:space="preserve">& </s> <s xml:id="echoid-s4861" xml:space="preserve">propter y<emph style="super">3</emph> duos priores terminos cu-<lb/>bi ab y + {ey/z}, nempe y<emph style="super">3</emph> + {3ey<emph style="super">3</emph>/z}; </s> <s xml:id="echoid-s4862" xml:space="preserve">quorum productum pro <lb/>xxy<emph style="super">3</emph> ſurrogandum. </s> <s xml:id="echoid-s4863" xml:space="preserve">Sed etiam hîc de duobus xx + 2ex tan-<lb/>tùm xx ducendum in {3ey<emph style="super">3</emph>/z}, tantumque 2ex in y<emph style="super">3</emph> (nam cæ-<lb/>tera vel plura quàm unum e vel nullum haberent) adeo ut <lb/>fiat {3exxy<emph style="super">3</emph>/z} + 2exy<emph style="super">3</emph>.</s> <s xml:id="echoid-s4864" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4865" xml:space="preserve">Atque ex his animadvertere licet, ſemper utrumque ho-<lb/>rum terminorum deſcribi poſſe ex dato termino, qui hic <lb/>xxy<emph style="super">3</emph>, alterum quidem mutato uno x in e, & </s> <s xml:id="echoid-s4866" xml:space="preserve">præponendo <lb/>numerum dimenſionum ipſius; </s> <s xml:id="echoid-s4867" xml:space="preserve">ita enim fit 2exy<emph style="super">3</emph>: </s> <s xml:id="echoid-s4868" xml:space="preserve">alterum ve-<lb/>rò ducendo datum terminum in {e/z}, præponendoque ſimili-<lb/>ter numerum dimenſionum ipſius y; </s> <s xml:id="echoid-s4869" xml:space="preserve">ita enim fit {3exxy<emph style="super">3</emph>/z}. <lb/></s> <s xml:id="echoid-s4870" xml:space="preserve">Cumque hac eadem immutatione, paulo ante, etiam ſecun-<lb/>dos terminos poteſtatum ab x + e & </s> <s xml:id="echoid-s4871" xml:space="preserve">ab y + {ey/z} ex pote-<lb/>ſtatibus x & </s> <s xml:id="echoid-s4872" xml:space="preserve">y æquationis datæ deſcribi oſtenſum ſit, mani-<lb/>feſtum jam eſt à ſingulis terminis æquationis datæ, in quibus <lb/>x vel poteſtas ejus, deſcribi prædicta methodo in ſecunda <lb/>æquatione totidem terminos in quibus non eſt z; </s> <s xml:id="echoid-s4873" xml:space="preserve">à ſingulis <lb/>verò in quibus y vel poteſtas ejus, deſcribitotidem terminos, <lb/>dicta etiam methodo, quarum fractionis denominator ſit z; </s> <s xml:id="echoid-s4874" xml:space="preserve"><lb/>nec alibi hanc literam in ſecunda æquatione repertum <lb/>iri.</s> <s xml:id="echoid-s4875" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4876" xml:space="preserve">Hoc igitur cognito, quo pacto ex æquatione quavis pro-<lb/>poſita, velut hîc x<emph style="super">3</emph> + y<emph style="super">3</emph> - axy = 0, alia deſcribenda ſit, ut hîc <lb/>3exx + {3ey<emph style="super">3</emph>/z} - aey - {aeyx/z} = 0, animadverto porro, ſi termi- <pb o="504" file="0226" n="237" rhead="CHRIST. HUGENII"/> ni diviſi per z ad alteram partem æquationis transferantur, <lb/>ductisque omnibus in z, diviſio deinde fiat per terminos in <lb/>quibus initio non erat z, exiſtere tunc ipſam quantitatem z <lb/>ab una æquationis parte; </s> <s xml:id="echoid-s4877" xml:space="preserve">uti hîc fiet z = - {3ey<emph style="super">3</emph> + aeyx/3exx - aey}. <lb/></s> <s xml:id="echoid-s4878" xml:space="preserve">Atque hinc intelligo ad conſequendam quantitatem z, po-<lb/>nendos tantum eos terminos æquationis ſecundæ, qui deſcri-<lb/>pti ſunt ex terminis æquationis primæ in quibus y, ſublato <lb/>tantum denominatore z, mutatiſque ſignis + & </s> <s xml:id="echoid-s4879" xml:space="preserve">-. </s> <s xml:id="echoid-s4880" xml:space="preserve">Dein-<lb/>de dividendo iſtos terminos per eos qui deſcripti ſunt ex ter-<lb/>minis æquationis primæ in quibus x. </s> <s xml:id="echoid-s4881" xml:space="preserve">Porro ex omnibus, tam <lb/>diviſis quàm dividentibus, patet rejici poſſe e, adeo ut in hoc <lb/>exemplo fiat z = - {y<emph style="super">3</emph>3 + ayx/3xx - ay}. </s> <s xml:id="echoid-s4882" xml:space="preserve">Itaque rejicitur {e/z} ex <lb/>terminis qui deſcripti ſunt ab iis qui habent y. </s> <s xml:id="echoid-s4883" xml:space="preserve">Sic autem <lb/>deſcriptos eos ſuperius diximus ut ducerentur in idem <lb/>{e/z}, præponereturque numerus dimenſionum y. </s> <s xml:id="echoid-s4884" xml:space="preserve">Itaque ni-<lb/>hil requiri apparet ad terminos hoſce (quatenus ad definien-<lb/>dam quantitatem z hic adhibentur) ex terminis æquationis <lb/>primæ, in quibus y, deſcribendos, quam ut præponamus <lb/>tantum iis numerum dimenſionum quas in ipſis habet y, ſigna-<lb/>que + & </s> <s xml:id="echoid-s4885" xml:space="preserve">- invertamus. </s> <s xml:id="echoid-s4886" xml:space="preserve">Sic nempe ab y<emph style="super">3</emph> - axy, deſcribetur <lb/>- 3y<emph style="super">3</emph> + axy. </s> <s xml:id="echoid-s4887" xml:space="preserve">A terminis verò qui deſcripti ſunt à terminis æ-<lb/>quationis primæ in quibus x, cum tantum e hîc rejiciendum <lb/>patuerit, cumque hos ita prius deſcriptos dixerimus ut unum <lb/>x mutaretur in e, præponereturque numerus dimenſionum <lb/>ipſius x; </s> <s xml:id="echoid-s4888" xml:space="preserve">apparet eos, quatenus h@c ad conſtituendum diviſo-<lb/>rum adhibentur, ſic tantum deſcribi opus eſſe ex terminis pro-<lb/>poſitæ æquationis in quibus x, ut præponatur iis numerus di-<lb/>menſionum ipſius x, ac deinde unum x auferatur. </s> <s xml:id="echoid-s4889" xml:space="preserve">Sic nem-<lb/>pe ab x<emph style="super">3</emph> - axy deſcribetur 3x<emph style="super">3</emph> - axy; </s> <s xml:id="echoid-s4890" xml:space="preserve">& </s> <s xml:id="echoid-s4891" xml:space="preserve">dempto ubique x <lb/>uno, fiet 3xx - ay. </s> <s xml:id="echoid-s4892" xml:space="preserve">Atque ex his ratio regulæ ab initio poſitæ <lb/>manifeſta eſt. </s> <s xml:id="echoid-s4893" xml:space="preserve">Nam quod ſigna + & </s> <s xml:id="echoid-s4894" xml:space="preserve">- in terminis qui deſcri-<lb/>buntur ab iis in quibusy, hîc immutanda diximus, in regulâ <pb o="505" file="0227" n="238" rhead="CEOMET. VARIA"/> verò nulla omnino immutanda, id eodem redire liquet cùm <lb/>quantitatem negatam, ſive minorem nihilo, tanquam affir-<lb/>matam conſiderandam ibi dixerimus. </s> <s xml:id="echoid-s4895" xml:space="preserve">Ut autem ratio obſer-<lb/>vationis ibidem adjectæ, in utram partem linea F E accipien-<lb/>da ſit, intelligatur, repetemus figuram in principio poſitam, <lb/> <anchor type="note" xlink:label="note-0227-01a" xlink:href="note-0227-01"/> ubi vidimus A G eſſe x + e, E G vero z + e; </s> <s xml:id="echoid-s4896" xml:space="preserve">unde fiebat GD <lb/>+ y + {ey/z}. </s> <s xml:id="echoid-s4897" xml:space="preserve">Si autem tangens ab altera parte lineæ B F cadere <lb/>intelligatur, velut be, atque hæc primùm curvam ſecare fin-<lb/>gatur, ut ibi factum eſt in d, ducaturque dg parallela bf; </s> <s xml:id="echoid-s4898" xml:space="preserve">fiet <lb/>ponendo rurſus fg = e, fe = z, ut A g quidem fiat x + e, <lb/>ſed eg erit z - e, unde gd = y - {ey/z}. </s> <s xml:id="echoid-s4899" xml:space="preserve">Atque hinc porrò fa-<lb/>cile eſt perſpicere æquationem ſecundam, quæ ex propoſita <lb/>æquatione, x<emph style="super">3</emph> + y<emph style="super">3</emph> - axy = o deſcribitur, hoc caſu fore <lb/>3exx - {3ey3/z} - aey + {aeyx/z} = o, termini ut nempe qui per <lb/>z dividuntur, habeant ſigna contraria iis quæ habebant <lb/>in æquatione deſcripta caſu priori, quæ erat 3exx + <lb/>{3ey<emph style="super">3</emph>/z} - aey - {aeyx/z}. </s> <s xml:id="echoid-s4900" xml:space="preserve">Ex hac verò priori ſequitur, quan-<lb/>do quantitas 3exx - aey, ſive quando 3xx - ay (quæ diviſo-<lb/>rem conſtituit ſecundum regulam) fuerit minor nihilo, ſive ne-<lb/>gata, tunc quantitatem reliquam {3ey<emph style="super">3</emph>/z} - {aeyx/z}; </s> <s xml:id="echoid-s4901" xml:space="preserve">ſive etiam <lb/>3y<emph style="super">3</emph> - ayx (quæ quantitatem dividendam ſecundùm regulam <lb/>conſtituit) eſſe affirmatam, aut cum illa eſt affirmata, hanc eſ-<lb/>ſenegatam; </s> <s xml:id="echoid-s4902" xml:space="preserve">quia omnes ſimul æquationis termini æquantur ni-<lb/>hilo. </s> <s xml:id="echoid-s4903" xml:space="preserve">At contra ex illa æquatione 3exx - {3ey<emph style="super">3</emph>/z} - aey + <lb/>{aeyx/z} = o, ſequitur, quando quantitas 3exx - aey, ſive <lb/>3xx - ay, fuerit negata, tunc reliquam - {3ey<emph style="super">3</emph>/z} + {aeyx/z}, <pb o="506" file="0228" n="239" rhead="CHRIST. HUGENII"/> ſive etiam - 3y<emph style="super">3</emph> + ayx eſſe affirmatam, ac proinde 3y<emph style="super">3</emph> - ayx <lb/>eſſe negatam: </s> <s xml:id="echoid-s4904" xml:space="preserve">aut quando 3xx - ay fuerit affirmata, tunc <lb/>- 3y<emph style="super">3</emph> + ayx eſſe negatam; </s> <s xml:id="echoid-s4905" xml:space="preserve">ac proinde 3y<emph style="super">3</emph> - ayx eſſe affir-<lb/>matam. </s> <s xml:id="echoid-s4906" xml:space="preserve">Per hæc itaque apparet ex quantitatibus per regulam in-<lb/>ventis, quæ erant {3y<emph style="super">3</emph> - ayx/3xx - ay} = z judiciari poſſe ad utrum ca-<lb/>ſum conſtructio tangentis pertineat; </s> <s xml:id="echoid-s4907" xml:space="preserve">nempe excomperta diſſi-<lb/>militudine affectionis in diviſore & </s> <s xml:id="echoid-s4908" xml:space="preserve">dividendo, ſequi ad prio-<lb/>rem caſum eam pertinere, hoc eſt z, ſive F E, accipiendam <lb/>eſſe verſus A: </s> <s xml:id="echoid-s4909" xml:space="preserve">ex ſimilitudine vero eorum affectionis ſequi ad <lb/>contrariam partem ſumendam.</s> <s xml:id="echoid-s4910" xml:space="preserve"/> </p> <div xml:id="echoid-div255" type="float" level="2" n="3"> <note position="right" xlink:label="note-0227-01" xlink:href="note-0227-01a" xml:space="preserve">TAB. XLV. <lb/>fig. 5.</note> </div> <note position="left" xml:space="preserve">TAB. XLV. <lb/>fig. 6.</note> <p> <s xml:id="echoid-s4911" xml:space="preserve">Poteſt autem quantitas z ſive FE per regulam inventa, non-<lb/>nunquam ad ſimpliciores terminos reduci ope æquationis datæ, <lb/>quæ naturam curvæ continet: </s> <s xml:id="echoid-s4912" xml:space="preserve">velut in hac curva A C, axem <lb/>habente A D, verticem A, cujuſque ea eſt proprietas ut, ſi à <lb/>puncto C in eâ ſumpta, applicetur ordinatim C D, fiat pro-<lb/>ductum ex cubo B D (eſt autem B punctum in axe extra cur-<lb/>vam datam) in quadratum D A æquale cubo quadrato DC. </s> <s xml:id="echoid-s4913" xml:space="preserve">Si-<lb/>ve ponendo B A = a, B D = x, D C = y, fiat æquatio cur-<lb/>væ naturam continens, iſta x<emph style="super">5</emph> - 2ax<emph style="super">4</emph> + aax<emph style="super">3</emph> - y<emph style="super">5</emph> = o. </s> <s xml:id="echoid-s4914" xml:space="preserve">Hîc <lb/>ponendo CG eſſe tangentem, quæ occurrat axi in G, vocan-<lb/>doque GD, z, fit ſecundum regulam z = {- 5y<emph style="super">5</emph>/5x<emph style="super">4</emph> - 8ax<emph style="super">3</emph> + 3aaxx}. <lb/></s> <s xml:id="echoid-s4915" xml:space="preserve">Quia autem ex datâ æquatione eſt y<emph style="super">5</emph> = x<emph style="super">5</emph> - 2ax<emph style="super">4</emph> + aax<emph style="super">3</emph>, re-<lb/>ſtituendo pro 5y id quod ipſi æquale eſt, fiet z = <lb/>{- 5x<emph style="super">5</emph> + 10ax<emph style="super">4</emph> - 5aax<emph style="super">3</emph>/5x<emph style="super">4</emph> - 8ax<emph style="super">3</emph> + 3aaxx}; </s> <s xml:id="echoid-s4916" xml:space="preserve">ſive dividendo per xx, erit z = <lb/>{- 5x<emph style="super">3</emph> + 10axx - 5aax/5xx - 8ax + 3aa,}. </s> <s xml:id="echoid-s4917" xml:space="preserve">Et rurſus, dividendo hanc fractionem <lb/>per x - a, habebitur z = {-5xx + 5ax/5x - 3a}. </s> <s xml:id="echoid-s4918" xml:space="preserve">Quod ſignificat <lb/>faciendum ut ſicut B D quinquies ſumpta minus B A ter, ſive <lb/>ut B A bis unà cum A D quinquies ad AD quinquies, ita BD <lb/>ad D G; </s> <s xml:id="echoid-s4919" xml:space="preserve">atque ita G C tacturam in C curvam A C.</s> <s xml:id="echoid-s4920" xml:space="preserve"/> </p> <pb file="0229" n="240"/> <pb file="0229a" n="241"/> <figure> <caption xml:id="echoid-caption73" style="it" xml:space="preserve">Pag. 506.<lb/>TAB. XLV.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables73" xml:space="preserve">C F D B</variables> </figure> <figure> <caption xml:id="echoid-caption74" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables74" xml:space="preserve">C B A E F</variables> </figure> <figure> <caption xml:id="echoid-caption75" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables75" xml:space="preserve">B b F f H c</variables> </figure> <figure> <caption xml:id="echoid-caption76" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables76" xml:space="preserve">C D B A E F G H</variables> </figure> <figure> <caption xml:id="echoid-caption77" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables77" xml:space="preserve">C b d D B E F G f g e</variables> </figure> <figure> <caption xml:id="echoid-caption78" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables78" xml:space="preserve">B G A C D</variables> </figure> <pb file="0230" n="242"/> <pb o="507" file="0231" n="243"/> <figure> <image file="0231-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0231-01"/> </figure> </div> <div xml:id="echoid-div257" type="section" level="1" n="125"> <head xml:id="echoid-head172" xml:space="preserve"><emph style="bf">IV.</emph> <lb/><emph style="bf">CHRISTIANI HUGENII</emph> <lb/><emph style="bf">EPISTOLA</emph> <lb/>DE <lb/>CURVIS QUIBUSDAM PECULIARIBUS.</head> <p> <s xml:id="echoid-s4921" xml:space="preserve">MItto tibi conſtructionem Problematis, quod ſine <lb/>dubio Geometris placebit, ſi id cum ipſis com-<lb/>municare velis, cum pulcherrimum ſit, & </s> <s xml:id="echoid-s4922" xml:space="preserve">ſingula-<lb/>ria quædam contineat: </s> <s xml:id="echoid-s4923" xml:space="preserve">accepi hoc a Marchione <lb/>de l’Hoſpital, quem ex ſpecimine hoc & </s> <s xml:id="echoid-s4924" xml:space="preserve">variis aliis, in-<lb/>ter ſummos noſtri ævi Geometras referendum puto. </s> <s xml:id="echoid-s4925" xml:space="preserve">Hac <lb/>etiam occaſione ad te mitto quaſdam ex poſterioribus, ſuper <lb/>rebus fere ſimilibus, contemplationibus noſtris.</s> <s xml:id="echoid-s4926" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4927" xml:space="preserve">Problema March. </s> <s xml:id="echoid-s4928" xml:space="preserve">eſt; </s> <s xml:id="echoid-s4929" xml:space="preserve">Invenire lineam rectam æqualem <lb/>datæ portioni lineæ logarithmicæ. </s> <s xml:id="echoid-s4930" xml:space="preserve">Ut hujus inveniat ſolu-<lb/>tionem, ſagaciter utitur Calculo differentiali celeberrimi Lei-<lb/>bnitii, & </s> <s xml:id="echoid-s4931" xml:space="preserve">reducit problema ad quadraturam curvæ, cujus æ-<lb/>quatio eſt, a<emph style="super">6</emph> = aaxxyy + y<emph style="super">4</emph>xx; </s> <s xml:id="echoid-s4932" xml:space="preserve">poſitis x & </s> <s xml:id="echoid-s4933" xml:space="preserve">y indeter <lb/>minatis, quæ angulum rectum efficiunt; </s> <s xml:id="echoid-s4934" xml:space="preserve">Hanc quadra-<lb/>turam oſtendit dependere à quadraturâ Hyperboles; </s> <s xml:id="echoid-s4935" xml:space="preserve">quæ da-<lb/>tur, ut notum eſt, poſitâ logarithmicâ deſcriptâ; </s> <s xml:id="echoid-s4936" xml:space="preserve">conſtru-<lb/>ctio autem huc redit. </s> <s xml:id="echoid-s4937" xml:space="preserve">Sit logarithmica indefinita A C D, a-<lb/> <anchor type="note" xlink:label="note-0231-01a" xlink:href="note-0231-01"/> ſymptos L O, ſubtangens conſtans data a, & </s> <s xml:id="echoid-s4938" xml:space="preserve">portio curvæ <lb/>ſit C D, cui æqualem rectam invenire oportet.</s> <s xml:id="echoid-s4939" xml:space="preserve"/> </p> <div xml:id="echoid-div257" type="float" level="2" n="1"> <note position="right" xlink:label="note-0231-01" xlink:href="note-0231-01a" xml:space="preserve">TAB. XLVI. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s4940" xml:space="preserve">Duc D L, C O perpendiculares ad aſymptoton, CE perpen-<lb/>dicularem ad D L, & </s> <s xml:id="echoid-s4941" xml:space="preserve">fac L T in aſymptoto æqualem ſubtan-<lb/>genti a, & </s> <s xml:id="echoid-s4942" xml:space="preserve">ductis rectis T D, T E, fiat T V = T D & </s> <s xml:id="echoid-s4943" xml:space="preserve">T I <pb o="508" file="0232" n="244" rhead="CHRIST. HUGENII"/> = T E: </s> <s xml:id="echoid-s4944" xml:space="preserve">tunc junge VD, ductâque ipſi parallelâ I K, e pun-<lb/>cto K, ubi occurrit ipſi D L, duc parallelam aſymptoto <lb/>K A, ſecantem D V in F, C O in X, & </s> <s xml:id="echoid-s4945" xml:space="preserve">Logarithmicam in <lb/>A. </s> <s xml:id="echoid-s4946" xml:space="preserve">Tum rectæ A X & </s> <s xml:id="echoid-s4947" xml:space="preserve">F K ſimul ſumtæ erunt æquales cur-<lb/>væ C D.</s> <s xml:id="echoid-s4948" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4949" xml:space="preserve">Solutio hujus Problematis, prout ego invenio, poteſt et-<lb/>iam reduci ad quadraturam curvæ, cujus Æquatio eſt <lb/>a<emph style="super">4</emph> = xxyy - aayy, quæ, ut & </s> <s xml:id="echoid-s4950" xml:space="preserve">altera, dependet à quadratu-<lb/>ra Hyperboles, uti poſſem ſatis facile demonſtrare; </s> <s xml:id="echoid-s4951" xml:space="preserve">ſed <lb/>conſtructio à modo deſcripta non differt.</s> <s xml:id="echoid-s4952" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4953" xml:space="preserve">Neſcio, an multæ lineæ curvæ hanc habeant proprietatem <lb/>ut ipſarum longitudines per ipſas curvas menſurari queant; <lb/></s> <s xml:id="echoid-s4954" xml:space="preserve">interim ecce unam, quam haud ita pridem inveni, dignam, ut <lb/>videbis, quæ & </s> <s xml:id="echoid-s4955" xml:space="preserve">ob alia etiam notetur; </s> <s xml:id="echoid-s4956" xml:space="preserve">Eſt curva A X K O <lb/> <anchor type="note" xlink:label="note-0232-01a" xlink:href="note-0232-01"/> extenſa in infinitum ſecundum rectam D N, quæ eſt ejus a-<lb/>ſymptos, ad quam A D, tangens ad verticem A, inſiſtit <lb/>perpendicularis; </s> <s xml:id="echoid-s4957" xml:space="preserve">curvæ princeps & </s> <s xml:id="echoid-s4958" xml:space="preserve">ſimpliciſſima pro-<lb/>prietas eſt, ut omnis tangens inter punctum contactus & </s> <s xml:id="echoid-s4959" xml:space="preserve"><lb/>aſymptoton, ut K N, ſit æqualis lineæ A D; </s> <s xml:id="echoid-s4960" xml:space="preserve">curva pariter ex-<lb/>tenditur ad alteram partem hujus perpendicularis A D. </s> <s xml:id="echoid-s4961" xml:space="preserve">Ut <lb/>invenias rectam lineam æqualem portioni hujus curvæ datæ a <lb/>vertice A, ut A K (ſic enim invenies alias portiones quaſcun-<lb/>que) duc K P perpendicularem ad A D, & </s> <s xml:id="echoid-s4962" xml:space="preserve">deſcripto arcu <lb/>circuli P Q, qui habeat centrum D & </s> <s xml:id="echoid-s4963" xml:space="preserve">radium D P, quæ-<lb/>re in A B parallelâ Aſymptoto punctum B, quod ſit centrum <lb/>circumferentiæ circuli, quæ tranſit per A & </s> <s xml:id="echoid-s4964" xml:space="preserve">tangit arcum PQ, <lb/>quod facile eſt; </s> <s xml:id="echoid-s4965" xml:space="preserve">porro ductâ rectâ B D, ſume in illâ DY = DA, <lb/>& </s> <s xml:id="echoid-s4966" xml:space="preserve">e puncto Y duc parallelam Aſymptoto uſque ad curvam in <lb/>X, tunc Y X erit æqualis curvæ A K; </s> <s xml:id="echoid-s4967" xml:space="preserve">Et natura hujus lineæ <lb/>talis eſt, ut ſi ſumas tot proportionales quot volueris, in re-<lb/>cta A D, incipiendo a D, ut DS, DI, DP & </s> <s xml:id="echoid-s4968" xml:space="preserve">ducas applica-<lb/>tas SR, IO, PK: </s> <s xml:id="echoid-s4969" xml:space="preserve">partes interceptæ curvæ, ut R O, O K, <lb/>omnes ſint æquales.</s> <s xml:id="echoid-s4970" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="2" n="2"> <note position="left" xlink:label="note-0232-01" xlink:href="note-0232-01a" xml:space="preserve">TAB. XLVI. <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s4971" xml:space="preserve">Ad quadraturam Hyperboles quoque inſervit curva hæc; </s> <s xml:id="echoid-s4972" xml:space="preserve">nam <lb/>eadem recta Y X facit cum A D rectangulum æquale ſpatio <lb/>Hyperbolico A D E V, terminato lineis A D, E V perpen- <pb o="509" file="0233" n="245" rhead="GEOMET. VARIA."/> dicularibus ad F D E, unam ex Aſymptotis, & </s> <s xml:id="echoid-s4973" xml:space="preserve">quæ ſunt in-<lb/>ter ſe in ratione A D ad D P, ſi Hyperbola A V ſit æquila-<lb/>tera & </s> <s xml:id="echoid-s4974" xml:space="preserve">quadratum ejus ad angulum Aſymptῶtωn ſit A D F H; <lb/></s> <s xml:id="echoid-s4975" xml:space="preserve">unde reciproce patet, quomodo queant & </s> <s xml:id="echoid-s4976" xml:space="preserve">inveniri puncta hu-<lb/>jus curvæ poſitâ quadraturâ Hyperboles.</s> <s xml:id="echoid-s4977" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4978" xml:space="preserve">Habet illa adhuc alias notabiles affectiones, quales ſunt, <lb/>quod ſpatium infinitum, inter curvam, Aſymptoton, & </s> <s xml:id="echoid-s4979" xml:space="preserve">rectam <lb/>A D, ſit æquale {1/4} circuli cujus radius eſt A D: </s> <s xml:id="echoid-s4980" xml:space="preserve">quod ſoli-<lb/>dum infinitum, quod producit hoc ſpatium rotando cir-<lb/>ca Aſymptoton, ſit æquale {1/4} ſphæræ ejuſdem radii; </s> <s xml:id="echoid-s4981" xml:space="preserve">quod <lb/>ſuperficies ejus ſolidi infiniti, ſine baſi, ſit æqualis circulo, cujus <lb/>radius eſt diagonalis quadrati ex A D.</s> <s xml:id="echoid-s4982" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4983" xml:space="preserve">Non autem propter ea qua huc uſque propoſui de hac cur-<lb/>va hic egi, ſed quia ſimplici Machinâ deſcribi poteſt, quo <lb/>reducitur Hyperbola ad quadratum, quod mihi viſum eſt Geo-<lb/>metrarum conſideratione dignum.</s> <s xml:id="echoid-s4984" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4985" xml:space="preserve">Conſtructio Machinæ nititur in dictâ Tangentis proprieta-<lb/>te & </s> <s xml:id="echoid-s4986" xml:space="preserve">principio vel lege motus; </s> <s xml:id="echoid-s4987" xml:space="preserve">ſcilicet, ſi in plano horizon-<lb/>tali detur punctum, quod ſuo pondere vel alio modo ali-<lb/>quantulum reſiſtit, junctum extremitati fili vel vectis inflexi-<lb/>lis, cujus altera extremitas movetur, punctum illud deſcri-<lb/>bet curvam, cujus Tangens ſemper erit filum vel vectis. </s> <s xml:id="echoid-s4988" xml:space="preserve">In <lb/>inſtrumento vel machinâ, de qua dixi, movenda eſt extremitas <lb/>D fili vel vectis D A juxta lineam rectam D N, & </s> <s xml:id="echoid-s4989" xml:space="preserve">cavendum ut <lb/>cuſpis in extremitate altera A hærens erecta maneat dum in-<lb/>terim premetur in planum horizontale, potius elaterio quam <lb/>pondere, quoniam ſic curva A K deſcribitur ſine errore ſen-<lb/>ſibili, licet planum non ſit exacte horizontale; </s> <s xml:id="echoid-s4990" xml:space="preserve">Et detegi-<lb/>tur, an habeat veram figuram reducendo extremitatem vectis <lb/>N per eandem rectam N D; </s> <s xml:id="echoid-s4991" xml:space="preserve">quoniam requiritur, ut cuſpis <lb/>regrediatur ex K in A, per eandem viam.</s> <s xml:id="echoid-s4992" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4993" xml:space="preserve">Si hæc deſcriptio, quæ per leges Mechanicæ eſt <lb/>acurata poſſet haberi pro Geometricâ, eodem modo ut de- <pb o="510" file="0234" n="246" rhead="CHRISTIANI HUGENII"/> ſcriptiones ſectionum Conicarum, quæ fiunt per inſtrumen-<lb/>ta, haberemus inde & </s> <s xml:id="echoid-s4994" xml:space="preserve">quadraturam Hyperboles & </s> <s xml:id="echoid-s4995" xml:space="preserve">perfe-<lb/>ctam conſtructionem omnium Problematum, quæ ad hanc <lb/>quadraturam reducuntur; </s> <s xml:id="echoid-s4996" xml:space="preserve">ut inter alia ſunt, determinatio <lb/>punctorum Catenariæ, & </s> <s xml:id="echoid-s4997" xml:space="preserve">logarithmi. </s> <s xml:id="echoid-s4998" xml:space="preserve">Si enim B Y ſit = <lb/>A C, quæ ſumitur in axe Catenariæ, id eſt D B = DC ap-<lb/>plicata ejus C G erit = Y X; </s> <s xml:id="echoid-s4999" xml:space="preserve">& </s> <s xml:id="echoid-s5000" xml:space="preserve">eadem quoque Y X eſt lo-<lb/>garithmus rationis quam habet A D ad P D; </s> <s xml:id="echoid-s5001" xml:space="preserve">id eſt, æqualis <lb/>eſt diſtantiæ duarum linearum A D, P D, vel aliarum duarum qua-<lb/>rumcunque, quæ eandem habent rationem, ordinatarum per-<lb/>pendicularium ad Aſymptoton lineæ logarithmicæ, quæ habet <lb/>D A pro Subtangente univerſali; </s> <s xml:id="echoid-s5002" xml:space="preserve">unde poſſunt inveniri logarith-<lb/>mi tabularum, prout demonſtravi in additione ad diſſertationem <lb/>de cauſa gravitatis. </s> <s xml:id="echoid-s5003" xml:space="preserve">Leibnitius, qui primus initium fecit redu-<lb/>ctionis curvæ Catenariæ ad leges Geometriæ, ipſam illam <lb/>lineam ope veræ Catenæ tenuiſſimæ formatam, dixit inſer-<lb/>vire poſſe inventioni logarithmorum, vel quadraturæ Hy-<lb/>perboles; </s> <s xml:id="echoid-s5004" xml:space="preserve">licet ad id cognita requiratur (ut quidem ipſe no-<lb/>verat) longitudo rectæ, quam vocat curvæ Parametrum, <lb/>cujus inventionem non demonſtrat. </s> <s xml:id="echoid-s5005" xml:space="preserve">Ita ut noſtra qua-<lb/>dratrix in his uſibus præferenda videatur, quia poſt de-<lb/>ſcriptionem Parameter ejus, quæ eſt univerſalis ejus Tan-<lb/>gens, datur.</s> <s xml:id="echoid-s5006" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5007" xml:space="preserve">Sed quoniam hæc materia me perduxit ad conſideratio-<lb/>nem Catenariæ quæ elegantiſſimis hujus temporis Geome-<lb/>trarum inquiſitionibus occaſionem præbuit, libet hic addere <lb/>quam inveni peculiarem ſatis methodum qua hæc delincatur cur-<lb/>va, quod eſt omnium difficillimum inter ea quæ de hac ſibi inqui-<lb/>renda propoſuere Mathematici. </s> <s xml:id="echoid-s5008" xml:space="preserve">Inter illa, quæ inſerenda dedi in <lb/>actis Lipſienſibus cum pulcris & </s> <s xml:id="echoid-s5009" xml:space="preserve">eruditis Leibnitii & </s> <s xml:id="echoid-s5010" xml:space="preserve">Ber-<lb/>noullii inventis, dixi, me reduxiſſe conſtructionem vel <lb/>inventionem punctorum hujus lineæ ad quadra@uram curvæ, <lb/>cujus æquatio eſt a<emph style="super">4</emph> = aaxx + yyxx <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s5011" xml:space="preserve">& </s> <s xml:id="echoid-s5012" xml:space="preserve">me cognoviſ- <anchor type="note" xlink:label="note-0234-01a" xlink:href="note-0234-01"/> ſe, hanc quadraturam dependere à cognitione ſummæ ſe-<lb/>cantium arcuum circuli, quæ æqualiter creſcerent per mi-<lb/>nima; </s> <s xml:id="echoid-s5013" xml:space="preserve">quæ ſumma jam dudum reducta fuerat ad quadratu- <pb o="511" file="0235" n="247" rhead="GEOMETRICA VARIA."/> ram Hyperboles per Jac. </s> <s xml:id="echoid-s5014" xml:space="preserve">Gregorium in exercitationibus ſuis <lb/>Geometricis, ubi inde deducit ſolutionem problematis lon-<lb/>gitudinum, datis vento & </s> <s xml:id="echoid-s5015" xml:space="preserve">latitudinum differentiâ, quod novum <lb/>credidit Leibnitius, & </s> <s xml:id="echoid-s5016" xml:space="preserve">quod à Gregorio traditum tunc tem-<lb/>poris non recordabar. </s> <s xml:id="echoid-s5017" xml:space="preserve">Leibnitius & </s> <s xml:id="echoid-s5018" xml:space="preserve">Bernoullius, ut cenſeo, <lb/>pervenerunt ad Catenariæ Conſtructionem ope Curvæ, <lb/>quam poſterior illorum habet in 1<emph style="super">a</emph>. </s> <s xml:id="echoid-s5019" xml:space="preserve">Figurarum quas exhi-<lb/>bet ad ſolvendum hoc Problema; </s> <s xml:id="echoid-s5020" xml:space="preserve">nam Leibnitius mihi ſcri-<lb/>pſit, ſe etiam ad eandem perveniſſe; </s> <s xml:id="echoid-s5021" xml:space="preserve">Et invenio eandem <lb/>cum illâ de qua ante, cujus æquatio eſt a<emph style="super">4</emph> = xxyy -<lb/>aayy, cujus quadratura, ut dixi, dependet à quadraturâ Hy-<lb/>perboles: </s> <s xml:id="echoid-s5022" xml:space="preserve">licet nondum concipere potuerim, quomodo cal-<lb/>culus illos perduxerit ad hanc lineam. </s> <s xml:id="echoid-s5023" xml:space="preserve">Sed tranſeo ad meam <lb/>conſtructionem, quæ abſque conſideratione aliûs lineæ <lb/>curvæ, dat puncta Catenariæ per dimenſionem lineæ Pa-<lb/>rabolicæ.</s> <s xml:id="echoid-s5024" xml:space="preserve"/> </p> <div xml:id="echoid-div259" type="float" level="2" n="3"> <note symbol="*" position="left" xlink:label="note-0234-01" xlink:href="note-0234-01a" xml:space="preserve">Vide ſupra <lb/>pag. 29@.</note> </div> <p> <s xml:id="echoid-s5025" xml:space="preserve">Primum fundamentum totius inquiſitionis reſpectu hujus <lb/> <anchor type="note" xlink:label="note-0235-01a" xlink:href="note-0235-01"/> lineæ eſt hoc; </s> <s xml:id="echoid-s5026" xml:space="preserve">Si habeas catenam compoſitam ex variis pon-<lb/>deribus æqualibus filo appenſis, ut BCDEF ſemper trium <lb/>interſtitiorum ſe mutuo ſequentium duæ lineæ extremæ, ut CD, <lb/>F E continuatæ ſibi mutuo occurrunt in linea IH per-<lb/>pendiculari ad Horizontem, quæ dividit interſtitium me-<lb/>dium in duas partes æquales. </s> <s xml:id="echoid-s5027" xml:space="preserve">Conſiderando porro catenam ita <lb/>compoſitam à ponderibus connexis ad æquales diſtantias, <lb/>quas ponimus infinite exiguas, & </s> <s xml:id="echoid-s5028" xml:space="preserve">diſpoſitis, ita, ut inter-<lb/>ſtitium infimum BC ſit horizonti parallelum, ſi ſuper quo-<lb/>vis alio interſtitio concipiamus triangula rectangula CDK, <lb/>D E L, quorum unum latus ſit horizontale, videbimus, <lb/>quod ab infimo initium faciendo anguli DCK, EDL, FEM, <lb/>tales ſint, ut illorum Tangentes æqualiter creſcant, ut nu-<lb/>meri 1, 2, 3, 4, id quod demonſtratu facile eſt ex dicto <lb/>principio, licet forſitan eo non perveniſſemus ſine calculo Al-<lb/>gebraico.</s> <s xml:id="echoid-s5029" xml:space="preserve"/> </p> <div xml:id="echoid-div260" type="float" level="2" n="4"> <note position="right" xlink:label="note-0235-01" xlink:href="note-0235-01a" xml:space="preserve">TAB. XLVI. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s5030" xml:space="preserve">Si porro concipiamus partes æquales catenæ CDEFG ex-<lb/>tenſas in recta horizontali in C O P Q R, & </s> <s xml:id="echoid-s5031" xml:space="preserve">ex prima divi-<lb/>ſione O ductam O S, quæ concurrat cum perpendiculari C S, <pb o="512" file="0236" n="248" rhead="CHRIST. HUGENII"/> ita ut angulus C O S ſit æqualis angulo C D O, ſi deinde <lb/>ducantur aliæ rectæ S P, S Q, S R; </s> <s xml:id="echoid-s5032" xml:space="preserve">triangula S C O, S C P, <lb/>S C Q, S C R erunt neceſſario ſimilia C O D, D L E, E M F, <lb/>F N G, quoniam S C O eſt ſimile ipſi C O D per conſtru-<lb/>ctionem, & </s> <s xml:id="echoid-s5033" xml:space="preserve">aliorum C S P, C S Q &</s> <s xml:id="echoid-s5034" xml:space="preserve">c. </s> <s xml:id="echoid-s5035" xml:space="preserve">tangentes æqualiter <lb/>creſcunt.</s> <s xml:id="echoid-s5036" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5037" xml:space="preserve">Si porro ducas C T, O V, P X &</s> <s xml:id="echoid-s5038" xml:space="preserve">c. </s> <s xml:id="echoid-s5039" xml:space="preserve">perpendiculares ad SO, <lb/>S P, S Q evidens eſt, triangula C T O, O V P, P X Q &</s> <s xml:id="echoid-s5040" xml:space="preserve">c. </s> <s xml:id="echoid-s5041" xml:space="preserve">æ-<lb/>qualia eſſe & </s> <s xml:id="echoid-s5042" xml:space="preserve">ſimilia triangulis C O D, D L E, E M F &</s> <s xml:id="echoid-s5043" xml:space="preserve">c. </s> <s xml:id="echoid-s5044" xml:space="preserve">eo-<lb/>dem ordine ſumendo: </s> <s xml:id="echoid-s5045" xml:space="preserve">unde concluditur, ſi ſpatia C D, <lb/>D E &</s> <s xml:id="echoid-s5046" xml:space="preserve">c. </s> <s xml:id="echoid-s5047" xml:space="preserve">ſint infinite parva, ut & </s> <s xml:id="echoid-s5048" xml:space="preserve">partes C O, O P &</s> <s xml:id="echoid-s5049" xml:space="preserve">c.</s> <s xml:id="echoid-s5050" xml:space="preserve">, id eſt <lb/>ſi C G ſit curva catenæ, & </s> <s xml:id="echoid-s5051" xml:space="preserve">C R æqualis ejus longitudini, <lb/>tum ſumma T O, V P, X Q &</s> <s xml:id="echoid-s5052" xml:space="preserve">c. </s> <s xml:id="echoid-s5053" xml:space="preserve">erit æqualis ſummæ perpen-<lb/>dicularium K D, L E, M F &</s> <s xml:id="echoid-s5054" xml:space="preserve">c. </s> <s xml:id="echoid-s5055" xml:space="preserve">id eſt rectæ G Σ vel axi Φ C <lb/>(nam ſpatium B C tum pro nihilo habetur) & </s> <s xml:id="echoid-s5056" xml:space="preserve">ſumma C T, <lb/>O V, P X erit æqualis ſummæ C K, D L, E M &</s> <s xml:id="echoid-s5057" xml:space="preserve">c. </s> <s xml:id="echoid-s5058" xml:space="preserve">ideſt appli-<lb/>catæ G Φ.</s> <s xml:id="echoid-s5059" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5060" xml:space="preserve">Deſcribendo autem centro S arcum CZ uſque ad ulti-<lb/>mam ſecantium S R facile patet, ſummam infinite parva-<lb/>rum T O, V P, X Q æqualem eſſe rectæ Z R; </s> <s xml:id="echoid-s5061" xml:space="preserve">conſequenter, <lb/>ſi ponamus quod S C Φ ſit axis Catenæ, & </s> <s xml:id="echoid-s5062" xml:space="preserve">linea C S certæ <lb/>longitudinis, & </s> <s xml:id="echoid-s5063" xml:space="preserve">quod C Φ ſit æqualis Z R exceſſui ſecantis <lb/>cujuſvis S R ſupra radium S C, & </s> <s xml:id="echoid-s5064" xml:space="preserve">quod applicata Φ G ſit <lb/>æqualis ſummæ omnium C T, O V, P X &</s> <s xml:id="echoid-s5065" xml:space="preserve">c. </s> <s xml:id="echoid-s5066" xml:space="preserve">uſque ad illam, <lb/>quæ cadit in S R, punctum G erit in curva catenæ, cujus <lb/>longitudo C G erit æqualis rectæ C R: </s> <s xml:id="echoid-s5067" xml:space="preserve">ſed quæritur ſumma <lb/>infinitarum C T, O V, P X &</s> <s xml:id="echoid-s5068" xml:space="preserve">c. </s> <s xml:id="echoid-s5069" xml:space="preserve">quam obtineo hâc conſidera-<lb/>tione, quod anguli S O V, S P X, S Q Y poſſint haberi pro <lb/>rectis, utpote quorum differentia cum recto eſt infinite exi-<lb/>gua, & </s> <s xml:id="echoid-s5070" xml:space="preserve">quod tum lineæ O V, P X, productæ utrinque, ut & </s> <s xml:id="echoid-s5071" xml:space="preserve"><lb/>R Ω perpendicularis ad S R, fiant tangentes Parabolæ C Ω, <lb/>cujus vertex eſt C, axis C S, focus S, & </s> <s xml:id="echoid-s5072" xml:space="preserve">in qua S C eſt pars <lb/>quarta Parametri; </s> <s xml:id="echoid-s5073" xml:space="preserve">quarum tangentium quævis ſecatur in duas <lb/>partes æqualiter per C R, ita ut una dimidia pars pertingat <lb/>ad axem, altera ad punctum contactus, ſic ΔΩ ſecta eſt in <lb/>R; </s> <s xml:id="echoid-s5074" xml:space="preserve">quæ facile demonſtrantur. </s> <s xml:id="echoid-s5075" xml:space="preserve">Hinc porro intelligo ex E- <pb o="513" file="0237" n="249" rhead="CEOMET. VARIA"/> volutione linearum curvarum, de qua locutus ſum in Horo-<lb/>logio oſcillatorio, quod ſumma omnium Q Y, P X, O V, C T, <lb/>debeat eſſe æqualis exceſſui curvæ Parabolicæ Ω C ſupra re-<lb/>ctam Ω R; </s> <s xml:id="echoid-s5076" xml:space="preserve">id quod Geometræ ſatis facile intelligent, licet in <lb/>demonſtrando non inhæream, quum non conſtituerim hic <lb/>demonſtrationes ſcribere, ſed tantum indicare viam, quâ ad <lb/>inventionem pervenis.</s> <s xml:id="echoid-s5077" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5078" xml:space="preserve">Datâ ergo Catenariæ Parametro SC, ſi ſumas in axe punctum <lb/>quodcunque Φ, & </s> <s xml:id="echoid-s5079" xml:space="preserve">centro S radio S Φ deſcribas arcum circuli, <lb/>qui ſecat C R tangentem ad verticem in R; </s> <s xml:id="echoid-s5080" xml:space="preserve">tangente R Ω du-<lb/>ctâ ad Parabolam dictame puncto R, & </s> <s xml:id="echoid-s5081" xml:space="preserve">ſubtractâ hac ex lon-<lb/>gitudine curvæ ſuæ CΩ quam ponimus poſſe menſurari, quod <lb/>reliquum eſt, erit recta applicata Φ G, & </s> <s xml:id="echoid-s5082" xml:space="preserve">ſic per eandem Para-<lb/>bolam invenies tot puncta in illâ curvâ, quot volueris; </s> <s xml:id="echoid-s5083" xml:space="preserve">miſi hanc <lb/>conſtructionem ad Leibnitium initio Septembris 1691.</s> <s xml:id="echoid-s5084" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5085" xml:space="preserve">Poteſt porro in tranſitu notari, quod curva G C (ſu-<lb/>mendo ſemper numerum interſtitiorum infinitum, & </s> <s xml:id="echoid-s5086" xml:space="preserve">ideo <lb/>punctum C ac ſi foret in axe & </s> <s xml:id="echoid-s5087" xml:space="preserve">vertex) erit æqualis rectæ <lb/>C R; </s> <s xml:id="echoid-s5088" xml:space="preserve">& </s> <s xml:id="echoid-s5089" xml:space="preserve">quod dimenſio ſpatii curvi quoque facile de-<lb/>monſtretur, perficiendo rectangulum R C S Θ & </s> <s xml:id="echoid-s5090" xml:space="preserve">producen-<lb/>do perpendiculares G N, F M &</s> <s xml:id="echoid-s5091" xml:space="preserve">c. </s> <s xml:id="echoid-s5092" xml:space="preserve">uſque ad S Θ in Λ, Γ <lb/>&</s> <s xml:id="echoid-s5093" xml:space="preserve">c.</s> <s xml:id="echoid-s5094" xml:space="preserve">: patet enim triangulum S Q Y eſſe ſemiſſem rectanguli F Λ, <lb/>baſin & </s> <s xml:id="echoid-s5095" xml:space="preserve">altitudinem eandem habentis, & </s> <s xml:id="echoid-s5096" xml:space="preserve">pariter triangulum <lb/>S P X ſemiſſem rectanguli E Γ & </s> <s xml:id="echoid-s5097" xml:space="preserve">ſic porro de aliis. </s> <s xml:id="echoid-s5098" xml:space="preserve">Et con-<lb/>ſequenter triangulum S C R æquale dimidio ſpatii S C G Λ. <lb/></s> <s xml:id="echoid-s5099" xml:space="preserve">Poſſem ita paucis demonſtrare fundamenta omnium quæ de <lb/>hac lineâ curvâ detecta ſunt; </s> <s xml:id="echoid-s5100" xml:space="preserve">ſed hæc ſpectant ad Leibnitium & </s> <s xml:id="echoid-s5101" xml:space="preserve"><lb/>Bernoullium, utpote hujus inventionis magis participes, quos <lb/>rogatos oportet, ut in bonum publicum laborem hunc in ſe <lb/>ſuſcipiant.</s> <s xml:id="echoid-s5102" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5103" xml:space="preserve">Finem ſcripto huic impoſuiſſem, niſi nuperrime literas à <lb/>Marchione de l’Hoſpital accepiſſem, ubi quum duo notabi-<lb/>lia hac de materia memorat, non poſſum quin paucis de iis <lb/>loquar; </s> <s xml:id="echoid-s5104" xml:space="preserve">primum ſpectat ad conſtructionem & </s> <s xml:id="echoid-s5105" xml:space="preserve">plures proprie-<lb/>tates lineæ curvæ D<emph style="super">ni</emph> de Beaune, quam Carteſius in Ep. <lb/></s> <s xml:id="echoid-s5106" xml:space="preserve">79. </s> <s xml:id="echoid-s5107" xml:space="preserve">3 vol. </s> <s xml:id="echoid-s5108" xml:space="preserve">dicit ipſi inveniendam propoſitam per datam <pb o="514" file="0238" n="250" rhead="CHRISTIANI HUGENII"/> Tangentis proprietatem; </s> <s xml:id="echoid-s5109" xml:space="preserve">Problema, quod mihi apparuit <lb/>difficillimum. </s> <s xml:id="echoid-s5110" xml:space="preserve">Solutio Marchionis exſtat in 34. </s> <s xml:id="echoid-s5111" xml:space="preserve">Diario Pari-<lb/>ſienſi anni ultimi, quare hanc hic non trado.</s> <s xml:id="echoid-s5112" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5113" xml:space="preserve">Alterum eſt ejus reſponſum de aliâ curvâ valde cognitâ, & </s> <s xml:id="echoid-s5114" xml:space="preserve"><lb/>quam Carteſius aliter adhuc conſideravit, ut & </s> <s xml:id="echoid-s5115" xml:space="preserve">Huddenius <lb/>eo tempore, quo negotia Reip. </s> <s xml:id="echoid-s5116" xml:space="preserve">non impediebant, quo-<lb/>minus vacaret ſtudiis. </s> <s xml:id="echoid-s5117" xml:space="preserve">Curva in fig. </s> <s xml:id="echoid-s5118" xml:space="preserve">4. </s> <s xml:id="echoid-s5119" xml:space="preserve">exhibetur, fo-<lb/> <anchor type="note" xlink:label="note-0238-01a" xlink:href="note-0238-01"/> lium A B C H circumſcribit & </s> <s xml:id="echoid-s5120" xml:space="preserve">utrimque ſeſe extendit juxta <lb/>Aſymptoton E F G. </s> <s xml:id="echoid-s5121" xml:space="preserve">Æquatio ejus eſt x<emph style="super">3</emph> + y<emph style="super">3</emph> = xyn ſi A D <lb/>ponatur x in recta, quæ cum diametro C A format angulum <lb/>45 graduum; </s> <s xml:id="echoid-s5122" xml:space="preserve">perpendicularis DB, vel D H, vel D K, y; </s> <s xml:id="echoid-s5123" xml:space="preserve">& </s> <s xml:id="echoid-s5124" xml:space="preserve"><lb/>n recta data. </s> <s xml:id="echoid-s5125" xml:space="preserve">Cum ipſi indicaſſem, me inveniſſe qua-<lb/>draturam hujus curvæ, & </s> <s xml:id="echoid-s5126" xml:space="preserve">quod contentum folii A B C H <lb/>eſſet = {1/6} nn id eſt {1/3} quadrati diametri A C; </s> <s xml:id="echoid-s5127" xml:space="preserve">quod ſpa-<lb/>tium infinitum inter Aſymptoton & </s> <s xml:id="echoid-s5128" xml:space="preserve">ambo curvæ brachia <lb/>etiam ejuſdem magnitudinis eſſent; </s> <s xml:id="echoid-s5129" xml:space="preserve">& </s> <s xml:id="echoid-s5130" xml:space="preserve">quod quadratura ge-<lb/>neralis ſegmentorum exprimeretur per unicum terminum; <lb/></s> <s xml:id="echoid-s5131" xml:space="preserve">ille invenit veram illam quadraturam generalem. </s> <s xml:id="echoid-s5132" xml:space="preserve">Scilicet quod <lb/>contentum ſegmentorum A H vel A K exprimatur per {nxx/6y} <lb/>& </s> <s xml:id="echoid-s5133" xml:space="preserve">ſegmenti A B per {nyy/6x}; </s> <s xml:id="echoid-s5134" xml:space="preserve">ſed ulterius mihi affirmat ſe eo <lb/>perveniſſe per tres diverſas vias; </s> <s xml:id="echoid-s5135" xml:space="preserve">quod miror, cum per-<lb/>ſuaſum habeam me non parum profeciſſe unicam dete-<lb/>gendo.</s> <s xml:id="echoid-s5136" xml:space="preserve"/> </p> <div xml:id="echoid-div261" type="float" level="2" n="5"> <note position="left" xlink:label="note-0238-01" xlink:href="note-0238-01a" xml:space="preserve">TAB. XLVI. <lb/>fig. 4.</note> </div> <figure> <image file="0238-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0238-01"/> </figure> <pb o="515" file="0239" n="251" rhead="GEOMET. VARIA."/> <figure> <image file="0239-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0239-01"/> </figure> </div> <div xml:id="echoid-div263" type="section" level="1" n="126"> <head xml:id="echoid-head173" xml:space="preserve">V. <lb/>PROBLEMA AB ERUDITIS SOLVENDUM: <lb/>A <lb/>JOHANNE BERNOULLIO <lb/>IN ACTIS LIPSIENSIBUS <lb/>ANNI MDCXCIII. <lb/>PROPOSITUM.</head> <p> <s xml:id="echoid-s5137" xml:space="preserve">QUæritur, qualis ſit curva A B C, quæ hanc habet <lb/> <anchor type="note" xlink:label="note-0239-01a" xlink:href="note-0239-01"/> proprietatem, ut, ducta ubicunque tangente <lb/>B D terminata ab axe A E, portio ejus abſciſſa <lb/>A D ſit ad tangentem B D in ratione conſtan-<lb/>te M ad N.</s> <s xml:id="echoid-s5138" xml:space="preserve"/> </p> <div xml:id="echoid-div263" type="float" level="2" n="1"> <note position="right" xlink:label="note-0239-01" xlink:href="note-0239-01a" xml:space="preserve">TAB. XLVI. <lb/>fig. 5.</note> </div> <p> <s xml:id="echoid-s5139" xml:space="preserve">Problema hoc ſolutu dignum eſt, & </s> <s xml:id="echoid-s5140" xml:space="preserve">facile Mathemati-<lb/>corum applicationem meretur. </s> <s xml:id="echoid-s5141" xml:space="preserve">In quacunque enim ratio-<lb/>ne ſit M ad N, curva A B C ſemper eadem facilitate mo-<lb/>tu quodam continuo deſcribi poteſt, non obſtante, quod <lb/>curva pro ratione M ad N magis vel minus compoſita <lb/>evadat; </s> <s xml:id="echoid-s5142" xml:space="preserve">in caſu quippe rationis æqualitatis illico patet, <lb/>curvam A B C eſſe circulum: </s> <s xml:id="echoid-s5143" xml:space="preserve">in reliquis ſi M ad N eſt ut <lb/>numerus ad numerum, erit quidem curva geometrica, ſe-<lb/>cus autem tranſcendentalis eſt. </s> <s xml:id="echoid-s5144" xml:space="preserve">Quæritur generalis deter-<lb/>matio puncti in curva.</s> <s xml:id="echoid-s5145" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div265" type="section" level="1" n="127"> <head xml:id="echoid-head174" style="it" xml:space="preserve">Tom. II. Ttt</head> <pb o="516" file="0240" n="252" rhead="CHRIST. HUGENII"/> <figure> <image file="0240-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0240-01"/> </figure> </div> <div xml:id="echoid-div266" type="section" level="1" n="128"> <head xml:id="echoid-head175" xml:space="preserve">VI. <lb/>C. H. Z. DE PROBLEMATE <lb/>BERNOULLIANO <lb/>IN <lb/>ACTIS LIPSIENSIBUS <lb/>PROPOSITO.</head> <p> <s xml:id="echoid-s5146" xml:space="preserve">ELegans inprimis eſſe hoc Problema, cum ex iis <lb/>quæ Clariſſimus inventor de eo prodidit, tum <lb/>ex ſolutione & </s> <s xml:id="echoid-s5147" xml:space="preserve">commentatione fraterna mani-<lb/>feſtum eſt. </s> <s xml:id="echoid-s5148" xml:space="preserve">A quo inveſtigando cum propter <lb/>inſignem difficultatem, quæ ſtatim ſeſe offerebat, abſtinere <lb/>ſtatuerim (neque enim omnibus perquirendis, quæ à Viris <lb/>eruditis exercitii gratia proponuntur, incumbere neceſſe exi-<lb/>ſtimo, aut aſſequendis parem me profiteor) non deſiit <lb/>tamen quaſi invitum compellere recurrens identidem quæ-<lb/>ſiti non vulgaris idea, donec tandem quod deſiderabam <lb/>obtinui. </s> <s xml:id="echoid-s5149" xml:space="preserve">Inventa nimirum æquatione differentiali, in qua <lb/>ex altera parte erat elementum trapezii hyperbolici, ab <lb/>aſymptoto perpendicularibus intercepti; </s> <s xml:id="echoid-s5150" xml:space="preserve">ab altera elemen-<lb/>tum ſpatii curvilinei, quod itidem ad trapezium hyper-<lb/>bolicum reduci poſſet. </s> <s xml:id="echoid-s5151" xml:space="preserve">Quod apertius exponerem, niſi <lb/>relinquendam etiamnum aliis putarem inquirendi volupta-<lb/>tem. </s> <s xml:id="echoid-s5152" xml:space="preserve">Inde eo rem deducebam, ut trapezium ejuſmodi <lb/>hyperbolicum ſecandum eſſet aut augendum ſecundum ra-<lb/>tionem datam. </s> <s xml:id="echoid-s5153" xml:space="preserve">Quod cum per medias aut continue pro- <pb o="517" file="0241" n="253" rhead="GEOMET. VARIA."/> portionales fieri poſſit, ubi ratio tangentis ad abſciſ-<lb/>ſam eſt ea quæ numeri ad numerum, hinc apparuit curvam <lb/>quæſitam tunc iis accenſendam quæ geometricæ vocantur, <lb/>alias eſſe ex heterogeneis; </s> <s xml:id="echoid-s5154" xml:space="preserve">ac tamen conſtructionem dari <lb/>poſita lineæ logarithmicæ deſcriptione, quam quidem hic ad-<lb/>ducerem, niſi viderem haud difficulter ex ipſa Jacobi Ber-<lb/>noullii doctiſſima ſimul breviſſimaque ſolutione omnia erui <lb/>poſſe, ut jam ab aliis occupatam dubitem.</s> <s xml:id="echoid-s5155" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5156" xml:space="preserve">Colligitur vero ex his illud animadverſione dignum, <lb/>nempe quandocunque in inveſtigatione curvarum ex tan-<lb/>gentibus aut ſubtangentibus ejus, ad ſimiles ei, quam dixi, <lb/>æquationes pervenietur, aut in quibus habeatur utrinque <lb/>elementum ſpatii ad trapezium hyperbolicum reductibilis; <lb/></s> <s xml:id="echoid-s5157" xml:space="preserve">tunc idem hoc, quod mirabile hic accidit, eventurum, ut <lb/>curvæ geometricæ diverſorum generum graduumque exi-<lb/>ſtant, ſi hyperbolarum ad quas devenitur rectangula <lb/>quæ in aſymptotis, ſint commenſurabilia. </s> <s xml:id="echoid-s5158" xml:space="preserve">Præterea <lb/>obſervanda venit in hoc problemate inuſitata ac ſin-<lb/>gularis analyſis via, quæ ad alia multa in hac Tan-<lb/>gentium doctrina aditum aperit, ut egregie jam ani-<lb/>madvertit Vir Celeberrimus calculi differentialis inven-<lb/>tor, ſine quo vix eſſet, ut ad haſce geometriæ ſubti-<lb/>litates admitteremur. </s> <s xml:id="echoid-s5159" xml:space="preserve">Porro quod ad curvarum, de qui-<lb/>bus agitur, deſignationem in plano attinet, poſſem, ſi <lb/>operæ pretium eſſet, alios modos ac fortaſſe com-<lb/>modiores indicare quam qui a Cl. </s> <s xml:id="echoid-s5160" xml:space="preserve">Bernoullio præſcri-<lb/>bitur, atque etiam docere qua ratione optime peraga-<lb/>tur deſcriptio noſtræ quadratricis hyperbolæ, quæ in-<lb/>ter Tractorias (ita enim vocari poſſunt) ſimpliciſſi-<lb/>ma cenſenda eſt, cum ad eam filis nihil opus ſit, <lb/>ſed bacillo tantum utrimque cuſpidem lateri infixam <lb/>habente, quo fit ut & </s> <s xml:id="echoid-s5161" xml:space="preserve">regreſſu explorari poſſit quam <lb/>recte exarata ſit. </s> <s xml:id="echoid-s5162" xml:space="preserve">Sed his ſuperſedendum arbitror, do-<lb/>nec inſignis uſus aliquis harum linearum in lucem <lb/>proferatur. </s> <s xml:id="echoid-s5163" xml:space="preserve">Interim aliam quandam utiliſſimam cur- <pb o="518" file="0242" n="254" rhead="CHRIST. HUGENII"/> vam nuper mihi repertam Geometræ ſciant, cujus o-<lb/>pera horologiis æqualis motus conciliatur, atque ejuſ-<lb/>modi ut maris agitatione nequaquam turbari aut immi-<lb/>nui queat; </s> <s xml:id="echoid-s5164" xml:space="preserve">quod in pendulis noſtris hactenus uſurpatis <lb/>non ſatis caveri potuit. </s> <s xml:id="echoid-s5165" xml:space="preserve">Adeo ut nova ac certior ſpes <lb/>nunc affulgeat perficiendi Longitudinum inventi. </s> <s xml:id="echoid-s5166" xml:space="preserve">Curva <lb/>hæc formatur,</s> </p> <p> <s xml:id="echoid-s5167" xml:space="preserve">a a b b c d e e e e e f i i i i i l l l m m m m n o r r ſ ſ t t u u x.</s> <s xml:id="echoid-s5168" xml:space="preserve"/> </p> <figure> <image file="0242-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0242-01"/> </figure> </div> <div xml:id="echoid-div267" type="section" level="1" n="129"> <head xml:id="echoid-head176" xml:space="preserve">VII. <lb/>C. H. Z. CONSTRUCTIO <lb/>UNIVERSALIS PROBLEMATIS <lb/>A <lb/>CLARISSIMO VIRO <lb/>JOH. BERNOULLIO <lb/>PROPOSITI.</head> <p> <s xml:id="echoid-s5169" xml:space="preserve">CUm in actis Lipſienſibus Conſtructionem <lb/> <anchor type="note" xlink:label="note-0242-01a" xlink:href="note-0242-01"/> hanc me reperiſſe ſignificarem, menſe O-<lb/>ctobri anno 1693, edenda tamen ea ſuper-<lb/>ſedi quod futurum putabam, ut vel ab <lb/>Auctore ipſo, vel Clariſſimo Viro fratre <lb/>ejus, vel alio quopiam, non multum ab- <pb o="519" file="0243" n="255" rhead="GEOMET. VARIA."/> ſimilis brevi in lucem mitteretur; </s> <s xml:id="echoid-s5170" xml:space="preserve">ac ſubverebar etiam, <lb/>ne actum agerem. </s> <s xml:id="echoid-s5171" xml:space="preserve">Quoniam vero nuſquam adhuc com-<lb/>paruit, & </s> <s xml:id="echoid-s5172" xml:space="preserve">eſt inter eas quæ dari poſſint quodammo-<lb/>do ſimpliciſſima, non videtur abſque ea diutius relin-<lb/>quendum tam eximium problema. </s> <s xml:id="echoid-s5173" xml:space="preserve">Eſt autem hujuſ-<lb/>modi. </s> <s xml:id="echoid-s5174" xml:space="preserve">In recta A B ſit datum punctum A, & </s> <s xml:id="echoid-s5175" xml:space="preserve">opor-<lb/>teat invenire curvam A F C talem, ut tangens ejus-<lb/>quævis C D abſcindat a recta A B partem A D, quæ <lb/>ad ipſam C D habeat rationem datam lineæ C ad <lb/>L.</s> <s xml:id="echoid-s5176" xml:space="preserve"/> </p> <div xml:id="echoid-div267" type="float" level="2" n="1"> <note position="left" xlink:label="note-0242-01" xlink:href="note-0242-01a" xml:space="preserve">TAB. XLVI. <lb/>fig. 6.</note> </div> <p> <s xml:id="echoid-s5177" xml:space="preserve">Conſtructio: </s> <s xml:id="echoid-s5178" xml:space="preserve">ſicut C ad L, ita quælibet A E, in <lb/>recta A B aſſumta ad E F ipſi perpendicularem: </s> <s xml:id="echoid-s5179" xml:space="preserve">& </s> <s xml:id="echoid-s5180" xml:space="preserve"><lb/>per F punctum ponatur ducta Logarithmica quæcun-<lb/>que cujus aſymptotos ſit A B, ad quam illa accedat <lb/>verſus A. </s> <s xml:id="echoid-s5181" xml:space="preserve">Deinde ab A verſus E accepta diſtan-<lb/>tia qualibet A D, ſit ut C ad L, ſive ut A E ad <lb/>E F, ita A D ad aliam D H; </s> <s xml:id="echoid-s5182" xml:space="preserve">qua tanquam radio, <lb/>centroque D, deſcribatur Circuli circumferentia H C; <lb/></s> <s xml:id="echoid-s5183" xml:space="preserve">ac præterea applicetur ad Logarithmicam recta I G <lb/>aſymptoto perpendicularis, ipſique D H æqualis. </s> <s xml:id="echoid-s5184" xml:space="preserve">Jam <lb/>ſicut L ad duplum C, ita fiat I E ad E K, ſumen-<lb/>dam in aſymptoto in partem alterutram, nihil enim <lb/>refert, & </s> <s xml:id="echoid-s5185" xml:space="preserve">applicetur rurſus ad Logarithmicam recta <lb/>K L. </s> <s xml:id="echoid-s5186" xml:space="preserve">Utque duæ ſimul K L, E F ad earum differen-<lb/>tiam, ita ſit D H ad D B; </s> <s xml:id="echoid-s5187" xml:space="preserve">quæ ſumenda verſus A <lb/>punctum, ſi A D major ſit quam A E; </s> <s xml:id="echoid-s5188" xml:space="preserve">at in con-<lb/>trariam, ſi minor. </s> <s xml:id="echoid-s5189" xml:space="preserve">Denique erigatur ad aſymptoton <lb/>perpendicularis B C; </s> <s xml:id="echoid-s5190" xml:space="preserve">ea ſecabit circumferentiam H C in <lb/>puncto C, quod erit in curva quæſita A F C. </s> <s xml:id="echoid-s5191" xml:space="preserve">Tangit <lb/>autem hanc recta E F in F.</s> <s xml:id="echoid-s5192" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5193" xml:space="preserve">Porro animadverſione dignum eſt, non ſimpli-<lb/>cem eſſe curvaturam lineæ hujus cum C major eſt <lb/>quam L, ſed ex duabus eam tunc componi, ex uno <lb/>quodam puncto exeuntibus, ut C F A, C M; </s> <s xml:id="echoid-s5194" xml:space="preserve">quarum <pb o="520" file="0244" n="256" rhead="CHR. HUGENII GEOMET. VARIA."/> hæc in infinitum progreditur. </s> <s xml:id="echoid-s5195" xml:space="preserve">In puncto autem ex-<lb/>tremo C, recta ex A educta occurrit curvæ utrique <lb/>ad angulos rectos, ac proportionales ſunt D A, D C, <lb/>D B.</s> <s xml:id="echoid-s5196" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div269" type="section" level="1" n="130"> <head xml:id="echoid-head177" xml:space="preserve">FINIS.</head> <figure> <image file="0244-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0244-01"/> </figure> <pb file="0245" n="257"/> <pb file="0245a" n="258"/> <figure> <caption xml:id="echoid-caption79" style="it" xml:space="preserve">Pag. 520.<lb/>TAB. XLVI.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables79" xml:space="preserve">D C E A X F K V O I L T α M N</variables> </figure> <figure> <caption xml:id="echoid-caption80" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables80" xml:space="preserve">Δ A Φ G F N E M I D H L B C K O P Q Σ R T V X Y Z S Γ Δ Θ @</variables> </figure> <figure> <caption xml:id="echoid-caption81" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables81" xml:space="preserve">C B A D E</variables> </figure> <figure> <caption xml:id="echoid-caption82" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables82" xml:space="preserve">H C L E B A D F K G</variables> </figure> <figure> <caption xml:id="echoid-caption83" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables83" xml:space="preserve">L G C F M A H B E I D K</variables> </figure> <figure> <caption xml:id="echoid-caption84" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables84" xml:space="preserve">G C H B A Y L X P K V Q I O S R F D E N</variables> </figure> <pb file="0246" n="259"/> <pb file="0247" n="260"/> </div> <div xml:id="echoid-div270" type="section" level="1" n="131"> <head xml:id="echoid-head178" xml:space="preserve">CHRISTIANI HUGENII <lb/>OPERA <lb/>ASTRONOMICA.</head> <head xml:id="echoid-head179" style="it" xml:space="preserve"><emph style="sc">Tomus</emph> <emph style="sc">Tertius</emph>.</head> <pb file="0248" n="261"/> </div> <div xml:id="echoid-div271" type="section" level="1" n="132"> <head xml:id="echoid-head180" xml:space="preserve">Tomi tertii contenta.</head> <note position="right" xml:space="preserve"> <lb/><emph style="sc">De</emph> <emph style="sc">Saturni</emph> <emph style="sc">Luna observatio nova</emph>. # 521. <lb/><emph style="sc">Systema</emph> <emph style="sc">Saturnium</emph>, ſive de cauſis mirandorum <emph style="sc">Saturni</emph> <lb/># Phænomenon; & Comite ejus Planeta novo. # 527. <lb/><emph style="sc">Eustachii de</emph> <emph style="sc">Divinis</emph> <emph style="sc">Septempedani brevis annotatio</emph> <lb/># in Syſtema Saturnium Chriſtiani Hugenii. # 597. <lb/><emph style="sc">Brevis assertio</emph> <emph style="sc">Systematis</emph> <emph style="sc">Saturnii sui</emph>. # 619. <lb/><emph style="sc">De</emph> <emph style="sc">Saturni</emph> <emph style="sc">Annulo observationes</emph>. # 635. <lb/><emph style="sc">ΚΟΣΜΟΘΕΩΡ@Σ</emph>, ſive de Terris Cœleſtibus, earumque ornatu, <lb/># <emph style="sc">Conjecturæ</emph>. # 641. <lb/></note> <pb file="0249" n="262"/> </div> <div xml:id="echoid-div272" type="section" level="1" n="133"> <head xml:id="echoid-head181" xml:space="preserve">CHRISTIANI HUGENII <lb/>DE <lb/>SATURNILUNA <lb/>OBSERVATIO NOVA.</head> <head xml:id="echoid-head182" style="it" xml:space="preserve">Tom. III. Ttt</head> <pb file="0250" n="263"/> <pb o="523" file="0251" n="264"/> <figure> <image file="0251-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0251-01"/> </figure> </div> <div xml:id="echoid-div273" type="section" level="1" n="134"> <head xml:id="echoid-head183" xml:space="preserve">CHRISTIANI HUGENII <lb/>DE <lb/>SATURNI <lb/>LUNA <lb/>OBSERVATIO NOVA.</head> <p> <s xml:id="echoid-s5197" xml:space="preserve">ANNO milleſimo ſexcenteſimo quinquage-<lb/>ſimo quinto, Menſis Martii die quinto & </s> <s xml:id="echoid-s5198" xml:space="preserve"><lb/>viceſimo, Saturni planetam per tubum <lb/>dioptricum aſpectans, animadverti præter <lb/>anſas ſive brachia quæ utrimque illi cohæ-<lb/>rent, ſtellulam quandam ab occaſu adſtan-<lb/>tem, tribus circiter ſcrupulis remotam, eratque diſpo-<lb/>ſita ſecundum eam quæ per utraque brachia ducta fuiſ-<lb/>ſet rectam. </s> </p> </div> <div xml:id="echoid-div274" type="section" level="1" n="135"> <head xml:id="echoid-head184" style="it" xml:space="preserve">Tom. III. Vvv. </head> <p> <s xml:id="echoid-s5199" xml:space="preserve">Et cum ſubdubitarem nunquid fortaſſe <lb/>planeta eſſet ejus generis, quales circa Jovem quatuor <lb/> <pb o="524" file="0252" n="265" rhead="CHRIST. HUGENIUS"/> circumferuntur, locum Saturni ſtellulæque & </s> <s xml:id="echoid-s5200" xml:space="preserve">poſitum utriuſque ad aliam quandam quæ tantumdem fere, ſed in contrarias partes, à Saturno diſſita erat annotavi;</s> <s xml:id="echoid-s5201" xml:space="preserve"> ex inerrantium numero hanc potiùs quam illam fore ratus, quod ab ea quam dixi recta linea deflectebat.</s> <s xml:id="echoid-s5202" xml:space="preserve"> Neque me fefellit opinio. </s> <s xml:id="echoid-s5203" xml:space="preserve">Poſtridie enim repetita ob- ſervatione, eam quæ ad occaſum ſpectabat ſtellam eo- dem ad Saturnum ſitu, eodemque quo prius interval- 10 ſejunctam deprehendi, alteram verò ad duplam fe- rè prioris diſtantiam receſſiſſe. </s> <s xml:id="echoid-s5204" xml:space="preserve">Unde hanc quidem è fixis unam eſſe, atque à Saturno tum temporis retro- gradiente longius relictam, illam autem ſimul cum eo progreſſam, comitem ipſi adeſſe intelligere mihi viſus ſum. </s> <s xml:id="echoid-s5205" xml:space="preserve">Sequentium verò dierum obſervationibus omnis dubitatio ſublata eſt. </s> <s xml:id="echoid-s5206" xml:space="preserve">Namque ab eo tempore per tres continuos menſes, quoties ſerenitas aëris patiebatur, novum planetam notavi, oſtendique amicis, nunc dex- trum Saturno nunc ſiniſtrum; </s> <s xml:id="echoid-s5207" xml:space="preserve">redactiſque in commen- tarios obſervationibus, ſexto decimo die periodum ex- plere cognovi. </s> <s xml:id="echoid-s5208" xml:space="preserve">Digreſſio ſumma tribus ſcrupulis pau- lo minor viſa eſt, ad quam ubi pervenit maximè fit conſpicuus: </s> <s xml:id="echoid-s5209" xml:space="preserve">at cum Saturno appropinquat ante aut po- nè tranſiens biduo propter ſplendorem ejus deliteſcit.</s> <s xml:id="echoid-s5210" xml:space="preserve"> Tempus verò ſexdecim dierum tam exactè circuitum planetæ metitur, ut cum annus jam & </s> <s xml:id="echoid-s5211" xml:space="preserve">amplius à pri- mis obſervationibus effluxerit, nihil adhuc aut abunda- re aut deficere deprehendatur, quoque loco prædici- mus ibi ſeſe in cœlo ſiſtat. </s> <s xml:id="echoid-s5212" xml:space="preserve">Scio ante annos complu- res Ant. </s> <s xml:id="echoid-s5213" xml:space="preserve">Mariam de Rheita non unum, ſed ſenos jam Saturno errones attribuiſſe. </s> <s xml:id="echoid-s5214" xml:space="preserve">Verum æquè circa hos, quàm circa alios illos quinque, quos præter Mediceos Jovi circumpoſuerat, deceptum fuiſſe, inde perſpici- tur, quod cum meliori Teleſcopio ſeſe uti Clar. </s> <s xml:id="echoid-s5215" xml:space="preserve">Vir Joh. </s> <s xml:id="echoid-s5216" xml:space="preserve">Hevelius demonſtret, nullum tamen Saturno ut- ut diligentiſſimè ſæpiſſimeque inſpecto comitatum adeſ- <pb o="525" file="0253" n="266" rhead="DE SATURNI LUNA."/> ſe ſenſerit. </s> <s xml:id="echoid-s5217" xml:space="preserve">Hoc enim ultrò fatetur. </s> <s xml:id="echoid-s5218" xml:space="preserve">Præter Rheitam verò nemo quod ſciam ſimile quid de Saturno prodi- dit. </s> <s xml:id="echoid-s5219" xml:space="preserve">Nam gemini illi quos Galilæus detexerat latero- nes longè aliud eſſe deprehenſi ſunt quàm prima ſpe- cie videbantur. </s> <s xml:id="echoid-s5220" xml:space="preserve">Quid tamen ſint in incerto eſt, ne- que adhuc pronunciare audent Aſtronomi. </s> <s xml:id="echoid-s5221" xml:space="preserve">Cæterùm mihi novum Saturniæ lunæ phænomenon ad hæc quo- que viam aperuit, tandemque cauſam reſcivimus, cur interdum inter binas veluti anſas Saturnus medius te- neatur, alias recta quaſi brachia protendat, tum non- nunquam omnibus amiſſis rotundus inveniatur, qualis anno 1642. </s> <s xml:id="echoid-s5222" xml:space="preserve">ſpectatus fuit, jamque rurſus trimeſtri ſpa- tio perſtitit. </s> <s xml:id="echoid-s5223" xml:space="preserve">Et harum quidem viciſſitudinum tempo- ra in futurum definire non erit difficile ſi duorum ad- huc menſium obſervationibus attendere licuerit, quæ vi- dendum an hypotheſi noſtræ conſentiant. </s> <s xml:id="echoid-s5224" xml:space="preserve">Expectamus enim ut ſub finem Aprilis, ſi non ante, brachia Sa- turno renaſcantur, non curva illa, cujuſmodi à Franc.</s> <s xml:id="echoid-s5225" xml:space="preserve"> Fontana & </s> <s xml:id="echoid-s5226" xml:space="preserve">Hevelio depicta cernuntur, ſed ſecundùm lineam rectam utrimque prominentia, ſi quis melioris notæ perſpicillo intueatur. </s> <s xml:id="echoid-s5227" xml:space="preserve">Nam vulgaria ſi adhibeat bi- nos orbiculos referent, ſicuti Galilæo primum ſeſe ob- tulere. </s> <s xml:id="echoid-s5228" xml:space="preserve">Noſtrum, quo Saturni aſſeclam reperimus, quin- quagies diametrum rei viſæ multiplicat, duodenos pe- des æquans: </s> <s xml:id="echoid-s5229" xml:space="preserve">cui poſtea duplum longitudine conſtruxi- mus, multiplicatione centupla. </s> <s xml:id="echoid-s5230" xml:space="preserve">Cum autem longiora etiam hiſce Teleſcopia, utpote triginta & </s> <s xml:id="echoid-s5231" xml:space="preserve">quadraginta pedum ab aliis fabricari dicantur, aliquid aut vitris vi- tii ineſſe, aut hæc eadem non debita proportione mu- tuò reſpondere credibile eſt. </s> <s xml:id="echoid-s5232" xml:space="preserve">Neque enim alias huc- uſque aciem eorum effugiſſet novus Saturni ſatelles.</s> <s xml:id="echoid-s5233" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5234" xml:space="preserve">Obſervationes præterito præſentique anno collectas, <lb/>quibus periodus ipſius demonſtratur, tunc unà edituri <lb/>ſumus cum integrum Saturni ſyſtema perfecerimus. </s> <s xml:id="echoid-s5235" xml:space="preserve">Cu-<lb/>jus interea ſummam ſequenti grypho conſignare viſum <pb o="526" file="0254" n="267" rhead="CHR. HUGEN. DE SATURNI LUNA."/> eſt, ut ſi quis fortaſſe idem ſe inveniſſe exiſtimet, <lb/>ſpatium habeat ad expromendum, neque à nobis ille <lb/>aut nos ab illo mutuati dicamur.</s> <s xml:id="echoid-s5236" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5237" xml:space="preserve">a a a a a a a c c c c c d e e e e e g h i i i i i i i l l l l m m n n n n n n n <lb/>n n o o o o p p q r r s t t t t t u u u u u.</s> <s xml:id="echoid-s5238" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5239" xml:space="preserve">Hagæ-Com. </s> <s xml:id="echoid-s5240" xml:space="preserve">5. </s> <s xml:id="echoid-s5241" xml:space="preserve">Mart. </s> <s xml:id="echoid-s5242" xml:space="preserve">1656.</s> <s xml:id="echoid-s5243" xml:space="preserve"/> </p> <figure> <image file="0254-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0254-01"/> </figure> <pb file="0255" n="268"/> </div> <div xml:id="echoid-div275" type="section" level="1" n="136"> <head xml:id="echoid-head185" xml:space="preserve">CHRISTIANI HUGENII <lb/>ZULICHEMII, CONST. F. <lb/>SYSTEMA <lb/>SATURNIUM, <lb/>SIVE <lb/>DE CAUSIS MIRANDORUM SATURNI <lb/>PHÆNOMENON; <lb/>ET <lb/>COMITE EJUS <lb/>PLANETA NOVO.</head> <pb file="0256" n="269"/> <pb o="529" file="0257" n="270"/> <figure> <image file="0257-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0257-01"/> </figure> </div> <div xml:id="echoid-div276" type="section" level="1" n="137"> <head xml:id="echoid-head186" xml:space="preserve">SERENISSIMO PRINCIPI <lb/>LEOPOLDO <lb/>AB HETRURIA</head> <head xml:id="echoid-head187" style="it" xml:space="preserve">Chriſtianus Hugenius S.D.</head> <p> <s xml:id="echoid-s5244" xml:space="preserve">REs in cælo remotiſſimas, extraque ho-<lb/>minum conſpectum poſitas, niſi cum <lb/>ab arte ſibi auxilium adſciſcunt, hoc <lb/>opuſculo perſequor, Princeps Sere-<lb/>niſſime, nec dubito quin multis ni-<lb/>mia diligentia verſatus dicar in his quæ tam parum <lb/>ad nos attinere arbitrantur, cum eorum quæ hîc <lb/>coram & </s> <s xml:id="echoid-s5245" xml:space="preserve">in propinquo nobis ſita ſunt, plurima in-<lb/>veſtigatu digna ſuperſint. </s> <s xml:id="echoid-s5246" xml:space="preserve">Verum hi parum at-<lb/>tendere videntur, quanto præſtet cæteris omni-<lb/>bus ſublimium rerum conſideratio, quamque hoc <lb/>ipſum præclarum ſit, ad tam longe diſſitas na-<lb/>turæ partes contemplationem mitti; </s> <s xml:id="echoid-s5247" xml:space="preserve">quæ licet <lb/>viſu obſcuræ & </s> <s xml:id="echoid-s5248" xml:space="preserve">exiles, reipſa tamen illuſtres mul-<lb/>toque maximæ exiſtunt. </s> <s xml:id="echoid-s5249" xml:space="preserve">Nam ſi quod procul <lb/>abſunt, ideo parum ad nos pertinere illas putemus, <lb/>indigni profecto ſimus mente rationis participe, <lb/>qua facile immenſa cæli ſpatia tranſcendimus, in-<lb/>digni etiam egregio illo, nec unquam ſatis lauda-<lb/>to, propagandæ viſionis invento, quo ad aſtrorum <lb/>regionem ipſo quoque oculorum ſenſu pertingimus.</s> <s xml:id="echoid-s5250" xml:space="preserve"> <pb o="530" file="0258" n="271" rhead="DEDICATIO."/> Cujus quidem inventi beneficio ad longinqua Sa-<lb/>turni regna propius nunc quam antehac quiſquam <lb/>adivi, & </s> <s xml:id="echoid-s5251" xml:space="preserve">uſque eò progreſſus ſum, ut vaſti adeo <lb/>itineris, pars una centeſima tantummodo reliqua <lb/>fuerit: </s> <s xml:id="echoid-s5252" xml:space="preserve">quam ſi quo pacto ſuperare potuiſſem, quot <lb/>qualiaque, dii boni, narranda haberem! Nunc au-<lb/>tem ea perſcribo quæ ex intervallo iſto notare ocu-<lb/>lis valui, quæque & </s> <s xml:id="echoid-s5253" xml:space="preserve">ipſa miranda eſſe & </s> <s xml:id="echoid-s5254" xml:space="preserve">relatu di-<lb/>gniſſima nemo diffitebitur. </s> <s xml:id="echoid-s5255" xml:space="preserve">Quem enim non ad-<lb/>miratio capiet, ubi Saturnum annulo circumdatum <lb/>ac velut corona redimitum viderit? </s> <s xml:id="echoid-s5256" xml:space="preserve">atque hanc eam <lb/>formam eſſe, quæ, cum perpetuo eadem ſit, di-<lb/>verſas tamen facies induat, & </s> <s xml:id="echoid-s5257" xml:space="preserve">pertinaciter hacte-<lb/>nus conjecturas Aſtronomorum fruſtretur. </s> <s xml:id="echoid-s5258" xml:space="preserve">At ne-<lb/>que hoc minus novum atque inopinabile omnium <lb/>auribus accidiſſet, Planetam aliquem non antea vi-<lb/>ſum in cælo repertum eſſe, niſi novitatis gratiam <lb/>ſtellæ Mediceæ abſtuliſſent. </s> <s xml:id="echoid-s5259" xml:space="preserve">Verum hic noſter Sa-<lb/>turni accola, quo diutius latuit, majorique moli-<lb/>mine ad terram deducendus fuit, eo magis depre-<lb/>henſo gaudendum eſt: </s> <s xml:id="echoid-s5260" xml:space="preserve">quodque unus hactenus de-<lb/>ſideratus, cumulum nunc tandem errantium ſtel-<lb/>larum explet, numerumque earum duodenarium; <lb/></s> <s xml:id="echoid-s5261" xml:space="preserve">quo majorem poſthac repertum non iri, prope <lb/>eſt ut confirmare audeam. </s> <s xml:id="echoid-s5262" xml:space="preserve">Certè jam majoribus <lb/>illis ac primariis, inter quos Tellus hæc reponen-<lb/>da eſt, æquales multitudine minores exiſtunt, & </s> <s xml:id="echoid-s5263" xml:space="preserve"><lb/>utrique illo, quem perfectum dicimus, numero <lb/>continentur, ut conſilio ſummi opificis modus hic <pb o="531" file="0259" n="272" rhead="DEDICATIO."/> præfinitus videri poſſit. </s> <s xml:id="echoid-s5264" xml:space="preserve">Cæterum multiplicem <lb/>contemplandi materiam, Philoſophorum ingeniis <lb/>circa hæc cæleſtia corpora exorituram, quibus ve-<lb/>luti nova acceſſione mundum auximus, non per-<lb/>ſequar. </s> <s xml:id="echoid-s5265" xml:space="preserve">Unum hoc inanimadverſum eos præterire <lb/>nolim; </s> <s xml:id="echoid-s5266" xml:space="preserve">nempe quam non leve argumentum ad a-<lb/>ſtruendum pulcherrimum illum mundi univerſi or-<lb/>dinem, qui à Copernico nomen habet, Satur-<lb/>nius hic mundus adferat: </s> <s xml:id="echoid-s5267" xml:space="preserve">ſi enim gravatè olimiſti <lb/>ſyſtemati aſſentientibus, ſcrupulum demere potue-<lb/>runt quaternæ circa Jovem repertæ Lunæ; </s> <s xml:id="echoid-s5268" xml:space="preserve">mani-<lb/>feſtius utique nunc eos convincet unica illa circa <lb/>Saturnum oberrans, atque ob hoc ipſum quod uni-<lb/>ca eſt, noſtratis Lunæ ſimilitudinem magis ex-<lb/>primens: </s> <s xml:id="echoid-s5269" xml:space="preserve">ut omittam nunc aliam quoque Saturnii <lb/>globi cum hoc noſtro cognationem, quam in ſi-<lb/>mili axium utriuſque inclinatione invenient Aſtro-<lb/>nomiæ periti. </s> <s xml:id="echoid-s5270" xml:space="preserve">Quæ ſanè cum mecum reputo, <lb/>fieri non poſſe videtur, ut veritatem hiſce in rebus <lb/>tam feliciter repertam, tamque manifeſtis indiciis <lb/>fultam, ulla ætas obliterare valeat, quamdiu mo-<lb/>do obſervationum Galilæi aut noſtrarum aliqua me-<lb/>moria ſupererit. </s> <s xml:id="echoid-s5271" xml:space="preserve">Haſce igitur ut ab oblivione vindi-<lb/>carem, utque deficientibus forſan olim organis qui-<lb/>bus eaſdem repetere liceat, eſſet tamen unde aliquan-<lb/>do extitiſſe probari poſſet, hac qualicunque ſcriptio-<lb/>ne publicandas cenſui, omnibuſque impertiendas. <lb/></s> <s xml:id="echoid-s5272" xml:space="preserve">Quod autem Celſitudini Tuæ commentarium hunc <lb/>inſcripſerim, feci id non una ratione. </s> </p> </div> <div xml:id="echoid-div277" type="section" level="1" n="138"> <head xml:id="echoid-head188" style="it" xml:space="preserve">Tom. III. Xxx</head> <p><s xml:id="echoid-s5273" xml:space="preserve">Namque in <lb/> <pb o="532" file="0260" n="273" rhead="DEDICATIO."/> primis celebritatem claritatemque ei non exiguam ab lluſtriſſimo nomine tuo acquiri poſſe credidi;</s> <s xml:id="echoid-s5274" xml:space="preserve"> cujus cum per orbem univerſum, quà modo aliquis virtuti aut humanitati locus eſt, latè fama pervaſe- rit, librum hunc tibi nuncupare, hoc eſt velut in edito cunctiſque conſpicuo loco eum deponere.</s> <s xml:id="echoid-s5275" xml:space="preserve"> Deinde nec ignorabam quantum momenti acceſſu- rum eſſet invento illi noſtro, quo perplexa Saturni myſteria exponere conatus ſum, ſi exactiſſimo tuo judicio illud probari contingeret: </s> <s xml:id="echoid-s5276" xml:space="preserve">quod utinam non fruſtra ſperaverim. </s> <s xml:id="echoid-s5277" xml:space="preserve">Sed ante omnia occaſionem aliquam me inveniſſe gaviſus ſum, neque omit- tendam duxi, qua palam commemorarem quantum tibi, Princeps Celſiſſime, artes diſciplinæque opti- mæ, & </s> <s xml:id="echoid-s5278" xml:space="preserve">in his Mathematicæ præſertim debeant, quod contra invaleſcentem indies barbariem patronum iis ac defenſorem te præſtas, quodque familiariter eas colendo, ac velut in contubernium tuum admitten- do, plurimum dignitatis ipſis concilias: </s> <s xml:id="echoid-s5279" xml:space="preserve">quod de- nique præſtantiſſimorum ex omni antiquitate Au- torum ſcriptis in vitam revocatis eaſdem promoves ac locupletas. </s> <s xml:id="echoid-s5280" xml:space="preserve">Nempe ad hæc facienda, & </s> <s xml:id="echoid-s5281" xml:space="preserve">illu- ſtria majorum tuorum exempla & </s> <s xml:id="echoid-s5282" xml:space="preserve">innata virtus &</s> <s xml:id="echoid-s5283" xml:space="preserve"> egregia animi tui propenſio te impellunt. </s> <s xml:id="echoid-s5284" xml:space="preserve">nos au- tem ad quos optimæ hujus tuæ voluntatis curæque utilitas pervenit, grato animo illa agnoſcere &</s> <s xml:id="echoid-s5285" xml:space="preserve"> prædicare æquum eſt. </s> <s xml:id="echoid-s5286" xml:space="preserve">Hagæ Comitis. </s> <s xml:id="echoid-s5287" xml:space="preserve">5. </s> <s xml:id="echoid-s5288" xml:space="preserve">Julii.</s> <s xml:id="echoid-s5289" xml:space="preserve"> Anno 1659.</s> <s xml:id="echoid-s5290" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div278" type="section" level="1" n="139"> <pb o="533" file="0261" n="274"/> <head xml:id="echoid-head189" xml:space="preserve">NICOLAUS HEINSIUS, D. F. <lb/>AD AUCTOREM SYSTEMATIS.</head> <p style="it"> <s xml:id="echoid-s5291" xml:space="preserve">LAudibus Hugeni pars addite magna paternis, <lb/>Quem totum Vranie vindicat una ſibi.</s> <s xml:id="echoid-s5292" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5293" xml:space="preserve">In cunis placiti reptare per avia cæli, <lb/>Aſtra tibi puero volvere ludus erat:</s> <s xml:id="echoid-s5294" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5295" xml:space="preserve">Aſtra minus patriis non trita penatibus olim; <lb/></s> <s xml:id="echoid-s5296" xml:space="preserve">Cognita natali non minus aſtra ſolo.</s> <s xml:id="echoid-s5297" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5298" xml:space="preserve">Jamque eadem populis mirantibus aſtra recludis, <lb/>Perſpicua ingenii lumine facta tui.</s> <s xml:id="echoid-s5299" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5300" xml:space="preserve">Qualis ſidereo radiatus in æthere Titan <lb/>Oppoſitam nocti ſpargit ubique facem.</s> <s xml:id="echoid-s5301" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5302" xml:space="preserve">Ardue ſtellantis ſalve metator Olympi, <lb/>Qui ſuperâ nobis das regione frui:</s> <s xml:id="echoid-s5303" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5304" xml:space="preserve">Per quem, diſcuſſâ dubiæ caligine mentis, <lb/>Inſerimur liquido cominus ora polo.</s> <s xml:id="echoid-s5305" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5306" xml:space="preserve">Devocat in terras, magico ſine carmine, Lunam <lb/>Æqua Syracoſio cui manus arte ſeni.</s> <s xml:id="echoid-s5307" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5308" xml:space="preserve">Ecce Jovis genitor tenebroſo carcere per te <lb/>Et fugit, & </s> <s xml:id="echoid-s5309" xml:space="preserve">lætam rurſus oberrat humum.</s> <s xml:id="echoid-s5310" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5311" xml:space="preserve">Macte auſis ſtudioque; </s> <s xml:id="echoid-s5312" xml:space="preserve">Deos qui vindice chartâ <lb/>Aſſeris: </s> <s xml:id="echoid-s5313" xml:space="preserve">inventi qui facis aſtra tui.</s> <s xml:id="echoid-s5314" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5315" xml:space="preserve">Nunc ſua Saturno cum vincula demſeris ipſi; <lb/></s> <s xml:id="echoid-s5316" xml:space="preserve">Sæcla tuum terris aurea munus eunt.</s> <s xml:id="echoid-s5317" xml:space="preserve"/> </p> <pb o="534" file="0262" n="275" rhead="IN IDEM SATURNI SYSTEMA."/> <p style="it"> <s xml:id="echoid-s5318" xml:space="preserve">OMnia qui magni diſpexit ſidera Mundi, <lb/>Viderat hæc oculo debiliore Conon.</s> <s xml:id="echoid-s5319" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5320" xml:space="preserve">Attigit illa Conon miris adjutus ab alis, <lb/>Attigit, & </s> <s xml:id="echoid-s5321" xml:space="preserve">viſu nobiliore, meus.</s> <s xml:id="echoid-s5322" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5323" xml:space="preserve">Perque vias Lunæ, per, qua Cyllenius errat, <lb/>Volvitur & </s> <s xml:id="echoid-s5324" xml:space="preserve">Veneri Martia flamma comes;</s> <s xml:id="echoid-s5325" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5326" xml:space="preserve">Quique nitet famulos inter tot Jupiter ignes; <lb/></s> <s xml:id="echoid-s5327" xml:space="preserve">Lumina falcigeri miſit ad aſtra Dei:</s> <s xml:id="echoid-s5328" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5329" xml:space="preserve">Et didicit vario quare mutabilis ore, <lb/>Ludat in obſcurâ mobilitate Senex:</s> <s xml:id="echoid-s5330" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5331" xml:space="preserve">Quod frontem diadema premat, quo, Circulus illi <lb/>Aureus, infauſtum cingat honore caput:</s> <s xml:id="echoid-s5332" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5333" xml:space="preserve">Quæ noctes ibi Luna regat, quæque, æmula noſtræ, <lb/>Expleat amiſſum Cynthia luce diem.</s> <s xml:id="echoid-s5334" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5335" xml:space="preserve">Nec ſatis hæc vidiſſe ſibi miracula, teſtes <lb/>Convocat, & </s> <s xml:id="echoid-s5336" xml:space="preserve">viſis quærit ubique novis.</s> <s xml:id="echoid-s5337" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5338" xml:space="preserve">Noluit hæc noſtros fugerent arcana nepotes, <lb/>Ignaros cœli nec ſinit eſſe ſui.</s> <s xml:id="echoid-s5339" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5340" xml:space="preserve">Ampla ſatis Juveni eſt, ut debita gloria, merces, <lb/>Vocibus innumeris quam ſua fama ſonet;</s> <s xml:id="echoid-s5341" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5342" xml:space="preserve">Gloria ſideribus quam convenit eſſe coævam, <lb/>Et tantum Cœlo commoriente mori.</s> <s xml:id="echoid-s5343" xml:space="preserve"/> </p> <pb o="535" file="0263" n="276"/> <figure> <image file="0263-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0263-01"/> </figure> </div> <div xml:id="echoid-div279" type="section" level="1" n="140"> <head xml:id="echoid-head190" xml:space="preserve">CHRISTIANI HUGENII <lb/><emph style="sc">Zulichemii</emph>, <emph style="sc">Cθnst</emph>. F. <lb/>SYSTEMA SATURNIUM.</head> <p> <s xml:id="echoid-s5344" xml:space="preserve">CUM ad cæleſtium contemplationem tubos <lb/>opticos, nobiliſſimum Belgicæ noſtræ inven-<lb/>tum, Galileus admoviſſet, celeberrimaque il-<lb/>la Planetarum phænomena mortalibus primus <lb/>aperuiſſet; </s> <s xml:id="echoid-s5345" xml:space="preserve">in his, ea quæ de Saturni ſtella <lb/>prodidit, vel præcipuè admiratione digna <lb/>fuiſſe videntur. </s> <s xml:id="echoid-s5346" xml:space="preserve">Nam cætera quidem, etſi ſuſpicienda meri-<lb/>tò ac magni facienda, non tamen ejuſmodi erant, ut, qui-<lb/>bus de cauſis talia cernerentur, enixè quærendum eſſet. </s> <s xml:id="echoid-s5347" xml:space="preserve">Sa-<lb/>turni vero mutabiles figuræ novum quoddam & </s> <s xml:id="echoid-s5348" xml:space="preserve">reconditum <lb/>naturæ artificium præferebant, cujus certè rationem neque <lb/>Galileo ipſi, neque tanto poſt tempore Aſtronomorum cui-<lb/>quam (pace eorum dixerim) divinare contigit. </s> <s xml:id="echoid-s5349" xml:space="preserve">Hunc primò <lb/>non ſimplici orbe lucentem, ſed veluti tergeminum conſpe-<lb/>xerat, binis ſtellis minoribus mediæ majori proximè utrin-<lb/>que adjectis. </s> <s xml:id="echoid-s5350" xml:space="preserve">Eâque formâ triennio fere abſque ulla muta-<lb/>tione perſeverante, certò ſibi perſuaſerat, quales Jovi qua-<lb/>tuor, tales duos comites Saturno obtigiſſe, nullo tamen mo-<lb/>tu præditos, eóque ſimili poſitu ſemper lateribus hæſuros. <lb/></s> <s xml:id="echoid-s5351" xml:space="preserve">Verùm ſententiam mutare coactus eſt, ſolitario Saturno pro-<lb/>deunte, ac priore ſatellitio penitus deſtituto. </s> <s xml:id="echoid-s5352" xml:space="preserve">Quod cum <lb/>admirabundus vidiſſet & </s> <s xml:id="echoid-s5353" xml:space="preserve">cauſam rei conjectura aſſequi tenta-<lb/>ret, de reditus tempore, quo prior illa phaſis reſtitui debe-<lb/>ret, nonnulla vaticinatus eſt. </s> <s xml:id="echoid-s5354" xml:space="preserve">Sed neque hæc ita tunc ſuc-<lb/>ceſſiſſe quemadmodum ſperaverat deprehenſum eſt, nec ge-<lb/>mina modò aſpectus diverſitate Saturnum contentum eſſe. </s> <s xml:id="echoid-s5355" xml:space="preserve"><lb/>Etenim aliæ deinceps mirabiles ac prodigioſæ formæ appa- <pb o="536" file="0264" n="277" rhead="CHRIST. HUGENII"/> ruerunt, quas primùm à Joſepho Blancano & </s> <s xml:id="echoid-s5356" xml:space="preserve">Franciſco <lb/>Fontana deſcriptas novimus; </s> <s xml:id="echoid-s5357" xml:space="preserve">adeo quidem inſolita ſpecie, <lb/>ut multis oculorum ludibria cenſerentur, imagineſque vitris <lb/>potius quam cælo hærentes: </s> <s xml:id="echoid-s5358" xml:space="preserve">donec pluribus eadem videnti-<lb/>bus, haud vano indicio proditas fuiſſe conſtitit.</s> <s xml:id="echoid-s5359" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5360" xml:space="preserve">Igitur ipſe quoque ad hæc cæli miracula conſpicienda ma-<lb/>gno deſiderio actus; </s> <s xml:id="echoid-s5361" xml:space="preserve">cum non niſi vulgaria ſuppeterent per-<lb/>ſpicilla, quinum aut ſenum pedum longitudine; </s> <s xml:id="echoid-s5362" xml:space="preserve">artem eam <lb/>qua vitra in hoſce uſus figurantur quanta potui cum cura di-<lb/>ligentiaque excolere aggreſſus ſum, nec piguit ipſummet o-<lb/>peri manus admoviſſe; </s> <s xml:id="echoid-s5363" xml:space="preserve">quoad multis ſuperatis difficultatibus <lb/>(nam plures in receſſu hæc arshabet quam prima fronte præ-<lb/>ferre videatur) ea denique vitra mihi effeci, per quæ ad hæc <lb/>ſcribenda præbitum eſt argumentum. </s> <s xml:id="echoid-s5364" xml:space="preserve">Continuo enim ad Sa-<lb/>turnum teleſcopia dirigens, aliam ibi rerum faciem reperi, <lb/>quàm pleriſque antehac fuerat credita. </s> <s xml:id="echoid-s5365" xml:space="preserve">Nam quæ vicinæ il-<lb/>li hærebant appendices, eas non ſane geminos planetas, ſed <lb/>quidvis potius aliud eſſe, diverſum vero ab his unumque nu-<lb/>mero planetam, majori intervallo à Saturno remotum, die-<lb/>bus ſexdecim circa eum ambire apparuit; </s> <s xml:id="echoid-s5366" xml:space="preserve">& </s> <s xml:id="echoid-s5367" xml:space="preserve">hunc quidem <lb/>omnibus antehac ſæculis ignoratum. </s> <s xml:id="echoid-s5368" xml:space="preserve">De qua nova noſtra ob-<lb/>ſervatione tribusabhinc annis Aſtronomos certiores feci, pru-<lb/>denti conſilio obſecutus viri Illuſtris, ingenioque juxta ac vir-<lb/>tute conſpicui, Joh. </s> <s xml:id="echoid-s5369" xml:space="preserve">Capelani. </s> <s xml:id="echoid-s5370" xml:space="preserve">Huic enim, uti & </s> <s xml:id="echoid-s5371" xml:space="preserve">Gaſſen-<lb/>do aliiſque, cum Lutetiæ Pariſiorum agens, de Saturni co-<lb/>mite à me viſo narraſſem, multas ob cauſas cenſuit, non re-<lb/>ticendum tam gratum omnibus futurum nuncium, quoad il-<lb/>lud quod meditabar integrum Saturni Syſtema perſcripſiſſem. <lb/></s> <s xml:id="echoid-s5372" xml:space="preserve">Itaque anno 1656 die 5 Martii, de Saturni Luna (ita enim <lb/>novam ſtellam nec immerito appellavimus) obſervationem <lb/>emiſi <anchor type="note" xlink:href="" symbol="*"/>, atque unà hypotheſin quæ cauſam reliquorum Satur- <anchor type="note" xlink:label="note-0264-01a" xlink:href="note-0264-01"/> ni phænomenôn contineret; </s> <s xml:id="echoid-s5373" xml:space="preserve">ſed hanc confuſo elementorum <lb/>quibus ſcribebatur ordine, quo tantùm, non ignoraſſe nos <lb/>eam illo jam tempore, teſtatum eſſet, aliique etiam ad vul-<lb/>ganda quæ commenti eſſent hoc pacto invitarentur, neque ſi-<lb/>bi præreptam quererentur inventionis gloriam. </s> <s xml:id="echoid-s5374" xml:space="preserve">Deinceps <pb o="537" file="0265" n="278" rhead="SYSTEMA SATURNIUM."/> vero rogatu ejuſdem Viri eximii, ſolvi quoque hunc litera-<lb/>rium gryphum, totamque hypotheſin ſummatim illi expoſui: <lb/></s> <s xml:id="echoid-s5375" xml:space="preserve">unde jam ad plures forſitan noſtra de Saturni phaſibus ſen-<lb/>tentia manaverit. </s> <s xml:id="echoid-s5376" xml:space="preserve">Sed pleniorem utique tractationem poſtu-<lb/>lat mira & </s> <s xml:id="echoid-s5377" xml:space="preserve">inſolens circa hunc planetam naturæ fabrica, ne-<lb/>que expectare debemus ut vel à nobis relatis, vel ad expli-<lb/>canda phænomena adſumtis, fidem omnes habeant, niſi & </s> <s xml:id="echoid-s5378" xml:space="preserve"><lb/>hæc rationum momentis, & </s> <s xml:id="echoid-s5379" xml:space="preserve">illa obſervationum teſtimonio <lb/>adſtructa viderint. </s> <s xml:id="echoid-s5380" xml:space="preserve">Quamobrem horum utrumque nunc præ-<lb/>ſtare nobis propoſitum eſt. </s> <s xml:id="echoid-s5381" xml:space="preserve">Ac primò quæ ad comitis pla-<lb/>netæ motum periodumque ſpectant accuratè, quantum fieri <lb/>poterit, ex obſervatis definiemus, motuſque ejus tabulas con-<lb/>demus. </s> <s xml:id="echoid-s5382" xml:space="preserve">Deinde ipſius Saturni phaſes ſingulas ſuis cauſis aſ-<lb/>ſignabimus, ita ut futuras quoque inde prænoſcere in prom-<lb/>ptu ſit.</s> <s xml:id="echoid-s5383" xml:space="preserve"/> </p> <div xml:id="echoid-div279" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-0264-01" xlink:href="note-0264-01a" xml:space="preserve">Vide ſupra <lb/>pag. 523.</note> </div> <p> <s xml:id="echoid-s5384" xml:space="preserve">Sed antequam obſervationes exhibeamus, de teleſcopiis <lb/>noſtris quibus cælo eas deduximus, pauca referre expediet, <lb/>ut ſciant hinc, qui comitem Saturni, mirabileſque Planetæ <lb/>ipſius figuras intueri cupiunt, qualibus ad hoc turbis vitriſ-<lb/>que indigeant; </s> <s xml:id="echoid-s5385" xml:space="preserve">útque ſuos ſi quos habent, poſſint cum no-<lb/>ſtris contendere. </s> <s xml:id="echoid-s5386" xml:space="preserve">Primus quem adhibuimus duodenos pe-<lb/> <anchor type="note" xlink:label="note-0265-01a" xlink:href="note-0265-01"/> des non excedebat, duobus convexis vitris inſtructus, quo-<lb/>rum id quod oculo vicinum erat, radios parallelos cogebat <lb/>ad trium paulo minus pollicum, ſive unciarum pedis Rhe-<lb/>nolandici diſtantiam. </s> <s xml:id="echoid-s5387" xml:space="preserve">Eo planetam novum & </s> <s xml:id="echoid-s5388" xml:space="preserve">deteximus pri-<lb/>mùm, & </s> <s xml:id="echoid-s5389" xml:space="preserve">per aliquot menſes obſervavimus, nec non formam <lb/>eam Saturni, quæ à nemine adhuc percepta fuerat, quám-<lb/>que poſtea deſcribemus, licet non prorſus erroris expertem. <lb/></s> <s xml:id="echoid-s5390" xml:space="preserve">Inde verò duplicata priori longitudine, ſimul duplo propio-<lb/>res ſideribus facti ſumus, multóque meliùs faciliúſque phœ-<lb/>nomena omnia adnotavimus. </s> <s xml:id="echoid-s5391" xml:space="preserve">Et hi quidem tubi 23 pedum, <lb/>è ferri bractea conſtructi ſunt, habentque ab altera parte vi-<lb/>trum inſertum, cujus latitudo ad quatuor pollices, ſed in <lb/>quo non major pateat circulus quam diametro duorum pol-<lb/>licum cum triente. </s> <s xml:id="echoid-s5392" xml:space="preserve">Ab altera parte, quæ nimirum oculo <lb/>admovetur, bina ſunt vitra minora, 1 {1/2} pollicem diametro <pb o="538" file="0266" n="279" rhead="CHRIST. HUGENII"/> æquantia, jucta invicem, quæque hoc pacto æquipollent <lb/>convexo colligenti radios parallelos ad intervallum unciarum <lb/>3, aut paulo etiam anguſtius. </s> <s xml:id="echoid-s5393" xml:space="preserve">Ex quo ſanè majoris vitri ex-<lb/>cellentia æſtimanda eſt, tam breve convexum perferre valen-<lb/>tis: </s> <s xml:id="echoid-s5394" xml:space="preserve">quoniam quanto minori de ſphæra id fuerit, tanto res <lb/>viſæ magis ampliantur. </s> <s xml:id="echoid-s5395" xml:space="preserve">Illud enim in Dioptrici@ noſtri@ de-<lb/>monſtratum invenietur, ſpeciei per tubum viſæ ad eam quæ <lb/>nudo oculo percipitur, hanc ſecundum diametrum eſſe ratio-<lb/>nem, quæ diſtantiæ foci in exteriori vitro ad illam, quæ in <lb/>interiori ſive oculari vitro eſt, foci diſtantiam. </s> <s xml:id="echoid-s5396" xml:space="preserve">Centuplam <lb/> <anchor type="note" xlink:label="note-0266-01a" xlink:href="note-0266-01"/> itaque fere rationem hanc in perſpicillis noſtris eſſe conſtat, <lb/>cum Galileana non ultra trigecuplam proceſſerint. </s> <s xml:id="echoid-s5397" xml:space="preserve">Nam quan-<lb/>titatem incrementi eodem modo nos atque ille æſtimamus; <lb/></s> <s xml:id="echoid-s5398" xml:space="preserve">nempe ut tanto major res quæque per tubos quam nudo vi-<lb/>ſu conſpici dicatur, quanto majori angulo ad oculum extre-<lb/>ma ejus deferuntur, ſive quanto latior ejus imago in fundo <lb/>oculi depingitur.</s> <s xml:id="echoid-s5399" xml:space="preserve"/> </p> <div xml:id="echoid-div280" type="float" level="2" n="2"> <note position="right" xlink:label="note-0265-01" xlink:href="note-0265-01a" xml:space="preserve">Teleſcopio-<lb/>rum noſtro-<lb/>rum deſcri-<lb/>ptio.</note> <note position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">Quantum <lb/>iis res viſæ <lb/>æmplientur. <lb/>Augmentum <lb/>teleſcopio per-<lb/>ceptum quo-<lb/>modo æſtime-<lb/>tur.</note> </div> <p> <s xml:id="echoid-s5400" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s5401" xml:space="preserve">alia æſtimandi augmenti ratio, ſed parum ex <lb/> <anchor type="note" xlink:label="note-0266-02a" xlink:href="note-0266-02"/> vero, cùm abſque ulla anguli conſideratione apparentem per-<lb/>ſpicillo alicujus rei magnitudinem determinamus; </s> <s xml:id="echoid-s5402" xml:space="preserve">velut cum <lb/>Jovis orbem circello duorum aut trium digitorum latitudine <lb/>æqualem nobis cerni putamus. </s> <s xml:id="echoid-s5403" xml:space="preserve">At enim cum idem circulus, <lb/>trium puta digitorum diametro, major minorve neceſſario <lb/>appareat, pro diverſa ſui ab oculo diſtantia, nonne etiam <lb/>adjiciendum ſit, quanto ex intervallo conſpectus circulus di-<lb/>ſco Jovis æqualis cernatur? </s> <s xml:id="echoid-s5404" xml:space="preserve">Profectò niſi hoc addatur, nihil <lb/>certi ea comparatione deſignari videtur. </s> <s xml:id="echoid-s5405" xml:space="preserve">Et tamen ratio ſub-<lb/>eſt cur magis una quæpiam quam alia magnitudo imagini vi-<lb/>ſæ tribuatur, & </s> <s xml:id="echoid-s5406" xml:space="preserve">quidem à pluribus ſæpe ſpectatoribus eadem. <lb/></s> <s xml:id="echoid-s5407" xml:space="preserve">Verùm de his aliàs fortaſſe. </s> <s xml:id="echoid-s5408" xml:space="preserve">In præſentia illud oſtendiſſe ſuf-<lb/>fecerit, fallacem omnino eſſe hoc modo initam æſtimatio-<lb/>nem. </s> <s xml:id="echoid-s5409" xml:space="preserve">Idque primùm inde liquet, quod Lunâ aut ſigno ali-<lb/>quo cæleſti, velut Orione, prope horizontem conſpecto, <lb/>idem longè majus viſus judicet, quam ubi altè jam ac ſupra <lb/>verticem penè adſtiterit, cum tamen hîc nihilo minori angu-<lb/>lo illud comprehendi certum ſit. </s> <s xml:id="echoid-s5410" xml:space="preserve">In his autem quæ teleſcopio <pb o="539" file="0267" n="280" rhead="SYSTEMA SATURNIUM."/> intuemur major adhuc error contingit. </s> <s xml:id="echoid-s5411" xml:space="preserve">nam cum, exempli <lb/>gratia, vel triplo latior ſecundum diametrum appareat Jupi-<lb/>ter, oculo altero per teleſcopium noſtrum ſpectatus, quam <lb/>Luna oculo altero vacuo, atque adeo utrâque hac ſpecie, in <lb/>unum convenire juſsâ, latè à JoveLuna contegatur; </s> <s xml:id="echoid-s5412" xml:space="preserve">nihilo-<lb/>minus cum ſeorſim Jupiter inſpicitur, trium circiter digito-<lb/>rum latitudinem pleriſque ſpectatoribus æquare tantummodo <lb/>exiſtimatur. </s> <s xml:id="echoid-s5413" xml:space="preserve">quanquam aliquos repererim qui diſco Lunari æ-<lb/>qualem faciebant, atque ita tertiam partem ſaltem tribuebant <lb/>ejus quæ reipſa apparet amplitudinis. </s> <s xml:id="echoid-s5414" xml:space="preserve">Quamobrem de mul-<lb/>tiplicatione teleſcopii malè hoc modo inquiri certum eſt. </s> <s xml:id="echoid-s5415" xml:space="preserve">Fiet <lb/>autem rectè Galilei methodo, quam in Sidereo nuntio tradi-<lb/>dit; </s> <s xml:id="echoid-s5416" xml:space="preserve">vel, quia hæc in prægrandibus teleſcopiis difficultatem <lb/>habet, inquirendo foci diſtantias in ſingulis vitris, eaſque in-<lb/>ter ſe comparando. </s> <s xml:id="echoid-s5417" xml:space="preserve">Qua ratione diximus centuplum fere au-<lb/>gmentum tuborum noſtrorum reperiri.</s> <s xml:id="echoid-s5418" xml:space="preserve"/> </p> <div xml:id="echoid-div281" type="float" level="2" n="3"> <note position="left" xlink:label="note-0266-02" xlink:href="note-0266-02a" xml:space="preserve">Alia ejuſdem <lb/>falſa æſtima-<lb/>tio.</note> </div> <p> <s xml:id="echoid-s5419" xml:space="preserve">Cæterùm libenter intellecturos credo qui hæc legent, qua-<lb/> <anchor type="note" xlink:label="note-0267-01a" xlink:href="note-0267-01"/> lia etiam eorum ope de reliquis præter Saturnum planetis fi-<lb/>xiſque ſtellis obſervaverimus: </s> <s xml:id="echoid-s5420" xml:space="preserve">de quibus breviter ergo hæc <lb/>habeant. </s> <s xml:id="echoid-s5421" xml:space="preserve">In primis ſæpe illud quæſivimus, num aliqui etiam <lb/>apud Venerem, Martem aut Mercurium comites circumfer-<lb/>rentur, ubi tamen nullos unquam reperimus. </s> <s xml:id="echoid-s5422" xml:space="preserve">Apud Jovem <lb/>autem quatuor neque amplius. </s> <s xml:id="echoid-s5423" xml:space="preserve">Qui quidem ſemper ac facilè <lb/>teleſcopio noſtro patent, niſi cum diſco ſuo aut umbra Jupi-<lb/>ter aliquem abſcondit. </s> <s xml:id="echoid-s5424" xml:space="preserve">Inde verò quamprimum rurſus emer-<lb/>gere incipiunt fiunt conſpicui, imò priuſquam toti exierint, <lb/>ut non ſemel me vidiſſe memini. </s> <s xml:id="echoid-s5425" xml:space="preserve">Porrò quæ in Jove zonæ <lb/> <anchor type="note" xlink:label="note-0267-02a" xlink:href="note-0267-02"/> ſeu faſciæ quibuſdam animadverſæ ſunt, non ſemper eâdem <lb/>formâ præditæ; </s> <s xml:id="echoid-s5426" xml:space="preserve">has ego & </s> <s xml:id="echoid-s5427" xml:space="preserve">qui mecum obſervarunt perſpi-<lb/>cuè ſæpe animadvertimus reliquo Jovis corpore magis luci-<lb/>das, cum tamen alii obſcuriores aſſerant; </s> <s xml:id="echoid-s5428" xml:space="preserve">quibus forſitan in-<lb/>terjectum ſpatium inter binas zonas lucidiores pro una ob-<lb/>ſcuriore fuerit. </s> <s xml:id="echoid-s5429" xml:space="preserve">Atque anno quidem 1656, multo majori in-<lb/> <anchor type="note" xlink:label="note-0267-03a" xlink:href="note-0267-03"/> tervallo, quam ſequentibus tribus, illas à ſe mutuo diſtare <lb/>comperimus, ſicut in adjunctis delineationibus videre eſt. <lb/></s> <s xml:id="echoid-s5430" xml:space="preserve">Qua ex inſtabilitate non malè forſan colligemus, ad inſtat <pb o="540" file="0268" n="281" rhead="CHRIST. HUGENII"/> nubium noſtrarum, vapores quoſdam vicinum Jovi ætherem <lb/>inſidere, qui nunc his nunc illis climatis crebri magis con-<lb/>fertique exoriantur.</s> <s xml:id="echoid-s5431" xml:space="preserve"/> </p> <div xml:id="echoid-div282" type="float" level="2" n="4"> <note position="right" xlink:label="note-0267-01" xlink:href="note-0267-01a" xml:space="preserve">Quæna@ <lb/>circa plan@-<lb/>tas cæteros & <lb/>fixas obſer-<lb/>vata.</note> <note position="right" xlink:label="note-0267-02" xlink:href="note-0267-02a" xml:space="preserve">In Jovis <lb/>diſco zonæ <lb/>candicantes.</note> <note position="right" xlink:label="note-0267-03" xlink:href="note-0267-03a" xml:space="preserve">TAB. <lb/>XLVII. <lb/>Fig. 1. 2.</note> </div> <p> <s xml:id="echoid-s5432" xml:space="preserve">In Marte quoque cingulum ejuſmodi unicum anno 1656 <lb/> <anchor type="note" xlink:label="note-0268-01a" xlink:href="note-0268-01"/> deprehendi, latum admodum, mediámque diſci partem of-<lb/>fuſcans, quemadmodum figura adjecta demonſtrat. </s> <s xml:id="echoid-s5433" xml:space="preserve">Inſu-<lb/> <anchor type="note" xlink:label="note-0268-02a" xlink:href="note-0268-02"/> per diſcum planetæ hujus parte aliqua deficientem vidi ali-<lb/>quoties, & </s> <s xml:id="echoid-s5434" xml:space="preserve">in Venere phaſes omnes quales Lunæ. </s> <s xml:id="echoid-s5435" xml:space="preserve">Verùm <lb/>hæc minoribus etiam teleſcopiis alii notarunt.</s> <s xml:id="echoid-s5436" xml:space="preserve"/> </p> <div xml:id="echoid-div283" type="float" level="2" n="5"> <note position="left" xlink:label="note-0268-01" xlink:href="note-0268-01a" xml:space="preserve">In Marte <lb/>zona ob-<lb/>ſcura.</note> <note position="left" xlink:label="note-0268-02" xlink:href="note-0268-02a" xml:space="preserve">TAB. <lb/>XLVII. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s5437" xml:space="preserve">Fixarum autem diametros etiam maximè ſplendidarum nul-<lb/> <anchor type="note" xlink:label="note-0268-03a" xlink:href="note-0268-03"/> la unquam latitudine cernere potui, ſed tantum minimi pun-<lb/>cti inſtar, quoties vitris uſus ſum fuligine leviter infectis ad <lb/>auferendos radios. </s> <s xml:id="echoid-s5438" xml:space="preserve">At ex Hevelii conſilio, quod in egre-<lb/>gio ejus extat opere Selenographico, exterius vitrum conte-<lb/>gens, ita ut exiguum tantum foramen relinquatur, ali-<lb/>quam magnitudinem præ ſe ferre illas vidi; </s> <s xml:id="echoid-s5439" xml:space="preserve">quam proinde <lb/>non ſtellarum propriam eſſe, ſed ex aliqua viſus fallacia <lb/>naſci arbitror. </s> <s xml:id="echoid-s5440" xml:space="preserve">Nam noſtra quidem illa methodus, trans <lb/>fumum, quo lens proxima oculo tincta eſt, ſtellas inſpicien-<lb/>di, certa eſt omnique erroris ſuſpicione carens; </s> <s xml:id="echoid-s5441" xml:space="preserve">atque ita <lb/>planetas quoque nimia luce radiantes, ſolemque ipſum in-<lb/>tueri ſolemus. </s> <s xml:id="echoid-s5442" xml:space="preserve">Foramine autem exiguo majorem lucis par-<lb/>tem excludendo, non tolli penitus circumfuſos ſideribus ra-<lb/>dios, ſed in orbem minorem ſatiſque perfectè circinatum <lb/>eos cogi opinor, qui imprudentibus pro ipſius ſtellæ corpo-<lb/>re imponat.</s> <s xml:id="echoid-s5443" xml:space="preserve"/> </p> <div xml:id="echoid-div284" type="float" level="2" n="6"> <note position="left" xlink:label="note-0268-03" xlink:href="note-0268-03a" xml:space="preserve">Fixarum <lb/>diametri <lb/>nulla la-<lb/>titudin@.</note> </div> <p> <s xml:id="echoid-s5444" xml:space="preserve">Unum verò circa fixas phænomenon relatu dignum oc-<lb/> <anchor type="note" xlink:label="note-0268-04a" xlink:href="note-0268-04"/> currit, à nemine hucuſque, quod ſciam, animadverſum, <lb/>nec quidem niſi grandibus hiſce teleſcopiis rectè obſer-<lb/>vandum. </s> <s xml:id="echoid-s5445" xml:space="preserve">In Orionis enſe tres ſtellæ ab Aſtronomis repo-<lb/>nuntur inter ſe proximæ. </s> <s xml:id="echoid-s5446" xml:space="preserve">Harum mediam Anno 1656 for-<lb/>tè per tubum inſpicientimihi, pro ſtella una duodecim (quod <lb/>quidem minimè novum) ſeſe obtulerunt; </s> <s xml:id="echoid-s5447" xml:space="preserve">eo poſitu quem <lb/> <anchor type="note" xlink:label="note-0268-05a" xlink:href="note-0268-05"/> ſubjecta figura expreſſimus.</s> <s xml:id="echoid-s5448" xml:space="preserve"/> </p> <div xml:id="echoid-div285" type="float" level="2" n="7"> <note position="left" xlink:label="note-0268-04" xlink:href="note-0268-04a" xml:space="preserve">Phænom@-<lb/>non in <lb/>Orione no-<lb/>vum.</note> <note position="left" xlink:label="note-0268-05" xlink:href="note-0268-05a" xml:space="preserve">TAB. <lb/>XLVII. <lb/>fig. 4.</note> </div> <p> <s xml:id="echoid-s5449" xml:space="preserve">Ex his autem tres illæ pene inter ſe contiguæ, cumque <lb/>his aliæ quatuor, velut trans nebulam lucebant, ita ut ſpa- <pb o="541" file="0269" n="282" rhead="SYSTEMA SATURNIUM."/> tium circa ipſas, qua forma hîcconſpicitur, multo illuſtrius <lb/>appareret reliquo omni cælo; </s> <s xml:id="echoid-s5450" xml:space="preserve">quod cum apprime ſerenum <lb/>eſſet ac cerneretur nigerrimum, velut hiatu quodam inter-<lb/>ruptum videbatur, per quem in plagam magis lucidam eſſet <lb/>proſpectus. </s> <s xml:id="echoid-s5451" xml:space="preserve">Idem verò in hanc uſque diem nihil immutata <lb/>facie ſæpiùs atque eodem loco conſpexi; </s> <s xml:id="echoid-s5452" xml:space="preserve">adeo ut perpetuam <lb/>illic ſedem habere credibile ſit hoc quidquid eſt portenti; </s> <s xml:id="echoid-s5453" xml:space="preserve">cui <lb/>certè ſimile aliud nuſquam apud reliquas fixas potui animad-<lb/>vertere. </s> <s xml:id="echoid-s5454" xml:space="preserve">Nam cæteræ nebuloſæ olim exiſtimatæ, atque ipſa <lb/>via lactea, perſpicillo inſpectæ, nullas nebulas habere com-<lb/>periuntur, neque aliud eſſe quam plurium ſtellarum conge-<lb/>ries & </s> <s xml:id="echoid-s5455" xml:space="preserve">frequentia.</s> <s xml:id="echoid-s5456" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5457" xml:space="preserve">In Lunæ facie autem quàm multa, diligentiſſimis quibus-<lb/>que obſervatoribus præterita, tubis noſtris detegantur, non <lb/>referam; </s> <s xml:id="echoid-s5458" xml:space="preserve">quandoquidem ſchemate ad hoc deſcripto opus eſ-<lb/>ſet, eoque ampliſſimo. </s> <s xml:id="echoid-s5459" xml:space="preserve">Quem laborem hactenus non ſuſ-<lb/>cepimus, credimuſque exiturum in immenſum, ſi montium <lb/>omnium eminentias & </s> <s xml:id="echoid-s5460" xml:space="preserve">anfractus, qua multitudine nobis vi-<lb/>dentur, depingere conemur. </s> <s xml:id="echoid-s5461" xml:space="preserve">Itaque ad Saturni obſervatio-<lb/>nes pergo, de quibus ſciendum eſt, priores omnes, uſque <lb/>ad eam quam 19 Febr. </s> <s xml:id="echoid-s5462" xml:space="preserve">Anno 1656 annotavimus, tubo 12 <lb/>pedum peractas eſſe, reliquas pedum 23. </s> <s xml:id="echoid-s5463" xml:space="preserve">Uterque autem <lb/>everſo ſitu viſibile referebant, ideoque ſchemata omnia, non <lb/>ut primùm deſcripta fuerant, hîc expreſſimus, ſed ſupera <lb/>inferis, dextra ſiniſtris permutavimus, ut vera pateret diſ-<lb/>poſitio.</s> <s xml:id="echoid-s5464" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5465" xml:space="preserve">Die igitur, ſecundum Calendarium Gregorianum, 25 <lb/> <anchor type="note" xlink:label="note-0269-01a" xlink:href="note-0269-01"/> Martii, Anno 1655, circa horam 8 veſpertinam, Saturnum <lb/>conſpexi cum brachiis utrinque ſecundum rectam lineam ex-<lb/> <anchor type="note" xlink:label="note-0269-02a" xlink:href="note-0269-02"/> tentis; </s> <s xml:id="echoid-s5466" xml:space="preserve">tribuſque fere ſcrupulis ab eo diſtantem occaſum <lb/>verſus ſtellulam quandam exiguam a, ſic ſitam, ut ſi per bra-<lb/>chium u@rumque recta duceretur, ea in illam incurreret, aut <lb/>certè pauxillo tantum inferior tranſiret. </s> <s xml:id="echoid-s5467" xml:space="preserve">Alteraque item <lb/>verſus orientem ſtellula b aderat, paulo longius à Saturno re-<lb/>mota, & </s> <s xml:id="echoid-s5468" xml:space="preserve">brachiorum lineâ multo inferior. </s> <s xml:id="echoid-s5469" xml:space="preserve">Et hac quidem <lb/>prima vice ſuſpicatus ſum ſtellam a Saturnum comitari, quo- <pb o="542" file="0270" n="283" rhead="CHRIST. HUGENII"/> niam alias quoque vicinam illi animadverteram, ſimilique fe-<lb/>re poſitu.</s> <s xml:id="echoid-s5470" xml:space="preserve"/> </p> <div xml:id="echoid-div286" type="float" level="2" n="8"> <note position="right" xlink:label="note-0269-01" xlink:href="note-0269-01a" xml:space="preserve">Circa Sa-<lb/>turnum ob-<lb/>ſervationes.</note> <note position="right" xlink:label="note-0269-02" xlink:href="note-0269-02a" xml:space="preserve">TAB, XLVII. <lb/>fig. 5.</note> </div> <p> <s xml:id="echoid-s5471" xml:space="preserve">Porrò brachia Saturni recta quidem utrinque extenſa cer-<lb/> <anchor type="note" xlink:label="note-0270-01a" xlink:href="note-0270-01"/> nebantur, ſed verſus extremas cuſpides craſſiora paulo quam <lb/>qua parte Saturni diſco cohærebant, qualia ſequens ſchema <lb/>exhibet.</s> <s xml:id="echoid-s5472" xml:space="preserve"/> </p> <div xml:id="echoid-div287" type="float" level="2" n="9"> <note position="left" xlink:label="note-0270-01" xlink:href="note-0270-01a" xml:space="preserve">TAB. XLVII. <lb/>fig. 6.</note> </div> <p> <s xml:id="echoid-s5473" xml:space="preserve">Eâque formâ uſque ad occaſum Heliacum Saturnus perſtitit. <lb/></s> <s xml:id="echoid-s5474" xml:space="preserve">Cæterum cum poſt phaſin rotundam Anni 1656 brachia <lb/>denuo recepiſſet, eadem quidem illa forma reverſa eſt <lb/>duodecempedali tubo ſpectanda: </s> <s xml:id="echoid-s5475" xml:space="preserve">ſed tunc majori tubo 23 <lb/> <anchor type="note" xlink:label="note-0270-02a" xlink:href="note-0270-02"/> pedum adhibito alteram hanc figuram veriorem eſſe patuit; <lb/></s> <s xml:id="echoid-s5476" xml:space="preserve">unde antea quoque talem extitiſſe credibile fiebat, quæ ta-<lb/>men minori teleſcopio perfectè conſpici nequiiſſet. </s> <s xml:id="echoid-s5477" xml:space="preserve">At li-<lb/>neam illam obſcuram, brachia utraque conjungentem, ac to-<lb/>ta tamen infra ſe relinquentem, etiam 12 pedum teleſcopio <lb/>notavimus.</s> <s xml:id="echoid-s5478" xml:space="preserve"/> </p> <div xml:id="echoid-div288" type="float" level="2" n="10"> <note position="left" xlink:label="note-0270-02" xlink:href="note-0270-02a" xml:space="preserve">TAB. XLIX. <lb/>fig. 6.</note> </div> <p> <s xml:id="echoid-s5479" xml:space="preserve">Die ſequenti, nempe 26 Martii ſtella a eodem modo ea-<lb/> <anchor type="note" xlink:label="note-0270-03a" xlink:href="note-0270-03"/> demque qua prius diſtantia juxta Saturnum collocata erat, <lb/>b verò duplo quam ante remotior. </s> <s xml:id="echoid-s5480" xml:space="preserve">Unde, quum diſtantia <lb/>inter ſe ſtellularum a & </s> <s xml:id="echoid-s5481" xml:space="preserve">b major eſſet effecta, ſequebatur vel <lb/>utramque vel alteram ſaltem erraticam eſſe. </s> <s xml:id="echoid-s5482" xml:space="preserve">Et ſtellam qui-<lb/>dem a neceſſario talem judicavi, quoniam Saturnum eo tem-<lb/>pore retrogradum eſſe noveram; </s> <s xml:id="echoid-s5483" xml:space="preserve">itaque oportebat cum Sa-<lb/>turno illam in eandem plagam delatam eſſe, quum alioqui <lb/>propinquior multo fieri debuiſſet. </s> <s xml:id="echoid-s5484" xml:space="preserve">Altera vero b, quomi-<lb/>nus fixa cenſeretur nihil obſtabat, imo prorſus ita exiſtima-<lb/>ri conſentaneum erat, cum una die tantum ab illa Saturnus <lb/>receſſiſſet, quantum motus ejus poſtulabat. </s> <s xml:id="echoid-s5485" xml:space="preserve">Neque verò a-<lb/>liter ſe rem habuiſſe ſequentes obſervationes oſtendunt.</s> <s xml:id="echoid-s5486" xml:space="preserve"/> </p> <div xml:id="echoid-div289" type="float" level="2" n="11"> <note position="left" xlink:label="note-0270-03" xlink:href="note-0270-03a" xml:space="preserve">TAB. XLVII. <lb/>fig. 7.</note> </div> <p> <s xml:id="echoid-s5487" xml:space="preserve">Martii 27. </s> <s xml:id="echoid-s5488" xml:space="preserve">ſtella a Saturno propior facta erat: </s> <s xml:id="echoid-s5489" xml:space="preserve">b verò <lb/> <anchor type="note" xlink:label="note-0270-04a" xlink:href="note-0270-04"/> adhuc longius receſſerat.</s> <s xml:id="echoid-s5490" xml:space="preserve"/> </p> <div xml:id="echoid-div290" type="float" level="2" n="12"> <note position="left" xlink:label="note-0270-04" xlink:href="note-0270-04a" xml:space="preserve">TAB. <lb/>XLVIII. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s5491" xml:space="preserve">Inde nubili dies interceſſerunt uſque ad 3 Apr. </s> <s xml:id="echoid-s5492" xml:space="preserve">quo die <lb/>fixam b non amplius annotavi, ſed ſtellula a migraverat ad Sa-<lb/>turni latus Orientale, rurſuſque fere 3 ſcrupulis diſtabat. </s> <s xml:id="echoid-s5493" xml:space="preserve">fig. </s> <s xml:id="echoid-s5494" xml:space="preserve">2.</s> <s xml:id="echoid-s5495" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5496" xml:space="preserve">Quinque diebus ſequentibus, rurſus ut ante impeditæ ob-<lb/>ſervationes.</s> <s xml:id="echoid-s5497" xml:space="preserve"/> </p> <pb file="0271" n="284"/> <pb file="0271a" n="285"/> <figure> <caption xml:id="echoid-caption85" style="it" xml:space="preserve">Pag. 542.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables85" xml:space="preserve">♃</variables> </figure> <figure> <caption xml:id="echoid-caption86" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables86" xml:space="preserve">♃</variables> </figure> <figure> <caption xml:id="echoid-caption87" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables87" xml:space="preserve">♂</variables> </figure> <figure> <caption xml:id="echoid-caption88" style="it" xml:space="preserve">Fig. 5.</caption> <description xml:id="echoid-description1" xml:space="preserve">25 Mart. 1655.</description> <description xml:id="echoid-description2" xml:space="preserve">* a</description> <description xml:id="echoid-description3" xml:space="preserve">b *</description> </figure> <figure> <caption xml:id="echoid-caption89" style="it" xml:space="preserve">Fig. 7.</caption> <description xml:id="echoid-description4" xml:space="preserve">26 Mart.</description> <description xml:id="echoid-description5" xml:space="preserve">* a b *</description> </figure> <figure> <caption xml:id="echoid-caption90" style="it" xml:space="preserve">Fig. 4.</caption> </figure> <figure> <caption xml:id="echoid-caption91" style="it" xml:space="preserve">Fig. 6.</caption> </figure> <pb file="0272" n="286"/> <pb o="543" file="0273" n="287" rhead="SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s5498" xml:space="preserve">9 Apr. </s> <s xml:id="echoid-s5499" xml:space="preserve">denuo ad occidentem ſitus erat comes a ſicut 27 <lb/> <anchor type="note" xlink:label="note-0273-01a" xlink:href="note-0273-01"/> Martii, fixaque altera c à parte orientali Saturno admodum <lb/>propinqua animadvertebatur, uno circiter ſcrup. </s> <s xml:id="echoid-s5500" xml:space="preserve">diſtans, <lb/>lineaque brachiorum, uti præcedens, inferior.</s> <s xml:id="echoid-s5501" xml:space="preserve"/> </p> <div xml:id="echoid-div291" type="float" level="2" n="13"> <note position="right" xlink:label="note-0273-01" xlink:href="note-0273-01a" xml:space="preserve">TAB. <lb/>XLVIII. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s5502" xml:space="preserve">10 Apr. </s> <s xml:id="echoid-s5503" xml:space="preserve">Saturnus à fixa c longius abſceſſerat, & </s> <s xml:id="echoid-s5504" xml:space="preserve">comes <lb/>à Saturno. </s> <s xml:id="echoid-s5505" xml:space="preserve">fig. </s> <s xml:id="echoid-s5506" xml:space="preserve">4.</s> <s xml:id="echoid-s5507" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5508" xml:space="preserve">Diebus duobus ſequentibus aucta fuit continuè diſtantia <lb/>ſtellæ c, comitis a eadem fere manſit. </s> <s xml:id="echoid-s5509" xml:space="preserve">Nempe</s> </p> <p> <s xml:id="echoid-s5510" xml:space="preserve">11 Apr. </s> <s xml:id="echoid-s5511" xml:space="preserve">hic poſitus fuit. </s> <s xml:id="echoid-s5512" xml:space="preserve">fig. </s> <s xml:id="echoid-s5513" xml:space="preserve">5.</s> <s xml:id="echoid-s5514" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5515" xml:space="preserve">12 Apr. </s> <s xml:id="echoid-s5516" xml:space="preserve">talis. </s> <s xml:id="echoid-s5517" xml:space="preserve">fig. </s> <s xml:id="echoid-s5518" xml:space="preserve">6.</s> <s xml:id="echoid-s5519" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5520" xml:space="preserve">13 Apr. </s> <s xml:id="echoid-s5521" xml:space="preserve">comes videri nequiit, quoniam & </s> <s xml:id="echoid-s5522" xml:space="preserve">Saturno pro-<lb/>pior factus erat, & </s> <s xml:id="echoid-s5523" xml:space="preserve">Luna adventabat. </s> <s xml:id="echoid-s5524" xml:space="preserve">Fixa c ulterius ſem-<lb/>per recedebat, ſed lentiore paſſu. </s> <s xml:id="echoid-s5525" xml:space="preserve">fig. </s> <s xml:id="echoid-s5526" xml:space="preserve">7.</s> <s xml:id="echoid-s5527" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5528" xml:space="preserve">14 & </s> <s xml:id="echoid-s5529" xml:space="preserve">15 Apr. </s> <s xml:id="echoid-s5530" xml:space="preserve">comes non apparuit ob viciniam Saturni.</s> <s xml:id="echoid-s5531" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5532" xml:space="preserve">16. </s> <s xml:id="echoid-s5533" xml:space="preserve">Cælum nubilum.</s> <s xml:id="echoid-s5534" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5535" xml:space="preserve">17. </s> <s xml:id="echoid-s5536" xml:space="preserve">Comes ad ortum ſitus erat, diſtans fere 3 ſcrup. </s> <s xml:id="echoid-s5537" xml:space="preserve">ſtel-<lb/>la vero c longius abierat, ſemper tamen oculis notata; </s> <s xml:id="echoid-s5538" xml:space="preserve">Ac <lb/>poſtea quidem eadem ad Saturnum reverſa eſt, ut ſequentes <lb/>obſervationes docebunt. </s> <s xml:id="echoid-s5539" xml:space="preserve">fig. </s> <s xml:id="echoid-s5540" xml:space="preserve">8.</s> <s xml:id="echoid-s5541" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5542" xml:space="preserve">18. </s> <s xml:id="echoid-s5543" xml:space="preserve">Apr. </s> <s xml:id="echoid-s5544" xml:space="preserve">comes ſitum mutaſſe non videbatur.</s> <s xml:id="echoid-s5545" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5546" xml:space="preserve">19. </s> <s xml:id="echoid-s5547" xml:space="preserve">Apr. </s> <s xml:id="echoid-s5548" xml:space="preserve">paulo propius ad Saturnum acceſſerat. </s> <s xml:id="echoid-s5549" xml:space="preserve">fig. </s> <s xml:id="echoid-s5550" xml:space="preserve">9.</s> <s xml:id="echoid-s5551" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5552" xml:space="preserve">20. </s> <s xml:id="echoid-s5553" xml:space="preserve">Apr. </s> <s xml:id="echoid-s5554" xml:space="preserve">magis appropinquaverat. </s> <s xml:id="echoid-s5555" xml:space="preserve">fig. </s> <s xml:id="echoid-s5556" xml:space="preserve">10.</s> <s xml:id="echoid-s5557" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5558" xml:space="preserve">21. </s> <s xml:id="echoid-s5559" xml:space="preserve">Apr. </s> <s xml:id="echoid-s5560" xml:space="preserve">Adhuc magis; </s> <s xml:id="echoid-s5561" xml:space="preserve">cernebaturque anſarum lineâ pau-<lb/>lo ſuperior. </s> <s xml:id="echoid-s5562" xml:space="preserve">fig. </s> <s xml:id="echoid-s5563" xml:space="preserve">11.</s> <s xml:id="echoid-s5564" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5565" xml:space="preserve">29 Apr. </s> <s xml:id="echoid-s5566" xml:space="preserve">Occidentalior Saturno comes factus erat, ac tan-<lb/>tundem diſtabat quantum 21. </s> <s xml:id="echoid-s5567" xml:space="preserve">Apr. </s> <s xml:id="echoid-s5568" xml:space="preserve">fig. </s> <s xml:id="echoid-s5569" xml:space="preserve">12.</s> <s xml:id="echoid-s5570" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5571" xml:space="preserve">Ultima Apr. </s> <s xml:id="echoid-s5572" xml:space="preserve">Comes prope Saturnum delituit.</s> <s xml:id="echoid-s5573" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5574" xml:space="preserve">3 Maji, in quem diem inſtitio Saturni incidit, comes in <lb/>maxima ab eo diſtantia videbatur, orientem verſus. </s> <s xml:id="echoid-s5575" xml:space="preserve">fig. </s> <s xml:id="echoid-s5576" xml:space="preserve">13.</s> <s xml:id="echoid-s5577" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5578" xml:space="preserve">Diebus inſequentibus uſque ad 27 Maii, ejuſmodi poſi-<lb/>tus fuit qualem exhibent ſchemata ſubjecta.</s> <s xml:id="echoid-s5579" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5580" xml:space="preserve">6. </s> <s xml:id="echoid-s5581" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5582" xml:space="preserve">fig. </s> <s xml:id="echoid-s5583" xml:space="preserve">14.</s> <s xml:id="echoid-s5584" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5585" xml:space="preserve">7. </s> <s xml:id="echoid-s5586" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5587" xml:space="preserve">fig. </s> <s xml:id="echoid-s5588" xml:space="preserve">15.</s> <s xml:id="echoid-s5589" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5590" xml:space="preserve">10. </s> <s xml:id="echoid-s5591" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5592" xml:space="preserve">fig. </s> <s xml:id="echoid-s5593" xml:space="preserve">16.</s> <s xml:id="echoid-s5594" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5595" xml:space="preserve">11. </s> <s xml:id="echoid-s5596" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5597" xml:space="preserve">fig 17.</s> <s xml:id="echoid-s5598" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5599" xml:space="preserve">12. </s> <s xml:id="echoid-s5600" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5601" xml:space="preserve">fig. </s> <s xml:id="echoid-s5602" xml:space="preserve">18.</s> <s xml:id="echoid-s5603" xml:space="preserve"/> </p> <pb o="544" file="0274" n="288" rhead="CHRIST. HUGENII"/> <p> <s xml:id="echoid-s5604" xml:space="preserve">14. </s> <s xml:id="echoid-s5605" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5606" xml:space="preserve">fig. </s> <s xml:id="echoid-s5607" xml:space="preserve">19.</s> <s xml:id="echoid-s5608" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5609" xml:space="preserve">15. </s> <s xml:id="echoid-s5610" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5611" xml:space="preserve">ægre conſpectus comes propter inſtans pleni-<lb/>lunium. </s> <s xml:id="echoid-s5612" xml:space="preserve">fig. </s> <s xml:id="echoid-s5613" xml:space="preserve">20.</s> <s xml:id="echoid-s5614" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5615" xml:space="preserve">17. </s> <s xml:id="echoid-s5616" xml:space="preserve">Maji, Comes non apparuit.</s> <s xml:id="echoid-s5617" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5618" xml:space="preserve">18. </s> <s xml:id="echoid-s5619" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5620" xml:space="preserve">fig. </s> <s xml:id="echoid-s5621" xml:space="preserve">21.</s> <s xml:id="echoid-s5622" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5623" xml:space="preserve">19. </s> <s xml:id="echoid-s5624" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5625" xml:space="preserve">fig. </s> <s xml:id="echoid-s5626" xml:space="preserve">22.</s> <s xml:id="echoid-s5627" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5628" xml:space="preserve">20. </s> <s xml:id="echoid-s5629" xml:space="preserve">Maji. </s> <s xml:id="echoid-s5630" xml:space="preserve">fig. </s> <s xml:id="echoid-s5631" xml:space="preserve">23.</s> <s xml:id="echoid-s5632" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5633" xml:space="preserve">27. </s> <s xml:id="echoid-s5634" xml:space="preserve">Maji, Comes occidentem verſus in maxima à Satur-<lb/>no diſtantia reperiebatur. </s> <s xml:id="echoid-s5635" xml:space="preserve">Saturnus autem jam rurſus ad <lb/>ſtellam c ſuperius notatam appropinquare cœperat, remotus <lb/>circiter 10 ſcrup. </s> <s xml:id="echoid-s5636" xml:space="preserve">Ita verò nunc poſita hæc erat, ut linea <lb/>brachiorum Saturni continuata ſubter eam ferretur, cum die <lb/>10 Apr. </s> <s xml:id="echoid-s5637" xml:space="preserve">ſupra tranſiiſſet. </s> <s xml:id="echoid-s5638" xml:space="preserve">Sed & </s> <s xml:id="echoid-s5639" xml:space="preserve">Saturni ipſius ſemita, uti <lb/>ex ſequenti obſervatione liquebit, ſtellâ c inferior hac vice <lb/>fuit, quæ priori illius applicatione ſuperior contigerat. </s> <s xml:id="echoid-s5640" xml:space="preserve">fig. </s> <s xml:id="echoid-s5641" xml:space="preserve">24.</s> <s xml:id="echoid-s5642" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5643" xml:space="preserve">Ultima Maji, Saturnus ſtellam c jam præterierat; </s> <s xml:id="echoid-s5644" xml:space="preserve">comes ad <lb/>occaſum ſitus erat, ſed aliquanto propior quam die 27. </s> <s xml:id="echoid-s5645" xml:space="preserve">fig. </s> <s xml:id="echoid-s5646" xml:space="preserve">25.</s> <s xml:id="echoid-s5647" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5648" xml:space="preserve">13 Jun. </s> <s xml:id="echoid-s5649" xml:space="preserve">Ultima fuit ante occaſum Heliacum ob@ervatio, <lb/>comes in maxima diſtantia cernebatur, Saturno occidenta-<lb/>lior. </s> <s xml:id="echoid-s5650" xml:space="preserve">fig. </s> <s xml:id="echoid-s5651" xml:space="preserve">26.</s> <s xml:id="echoid-s5652" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5653" xml:space="preserve">Poſtquam ex radiis Solaribus Saturnus emerſiſſet Anno 1656. <lb/></s> <s xml:id="echoid-s5654" xml:space="preserve"> <anchor type="note" xlink:label="note-0274-01a" xlink:href="note-0274-01"/> Jan. </s> <s xml:id="echoid-s5655" xml:space="preserve">16, hora 12, primùm à me obſervatus eſt, cum ad hoc <lb/>uſque tempus peregre abfuiſſem. </s> <s xml:id="echoid-s5656" xml:space="preserve">Inveni autem brachiis ſuis <lb/>ſpoliatum penitus, & </s> <s xml:id="echoid-s5657" xml:space="preserve">comitem ad orientem ſitum, in ma-<lb/>xima fere diſtantia. </s> <s xml:id="echoid-s5658" xml:space="preserve">fig. </s> <s xml:id="echoid-s5659" xml:space="preserve">27.</s> <s xml:id="echoid-s5660" xml:space="preserve"/> </p> <div xml:id="echoid-div292" type="float" level="2" n="14"> <note position="left" xlink:label="note-0274-01" xlink:href="note-0274-01a" xml:space="preserve">Saturnus <lb/>brachiorum <lb/>expers in-<lb/>ventus.</note> </div> <p> <s xml:id="echoid-s5661" xml:space="preserve">Rotundum autem Saturnum, ſub finem Novembris, ali-<lb/> <anchor type="note" xlink:label="note-0274-02a" xlink:href="note-0274-02"/> qui jam obſervaverant; </s> <s xml:id="echoid-s5662" xml:space="preserve">eâque formâ perſtitit uſque dum rur-<lb/>ſum radios Solis ſubiret.</s> <s xml:id="echoid-s5663" xml:space="preserve"/> </p> <div xml:id="echoid-div293" type="float" level="2" n="15"> <note position="left" xlink:label="note-0274-02" xlink:href="note-0274-02a" xml:space="preserve">TAB. LI. <lb/>fig. 4.</note> </div> <p> <s xml:id="echoid-s5664" xml:space="preserve">19 Febr. </s> <s xml:id="echoid-s5665" xml:space="preserve">tubo 23 pedum primùm uſus ſum, comitemque <lb/> <anchor type="note" xlink:label="note-0274-03a" xlink:href="note-0274-03"/> Saturno propinquum reperi, ubi priori teleſcopio ægre po-<lb/>tuiſſet animadverti.</s> <s xml:id="echoid-s5666" xml:space="preserve"/> </p> <div xml:id="echoid-div294" type="float" level="2" n="16"> <note position="left" xlink:label="note-0274-03" xlink:href="note-0274-03a" xml:space="preserve">TAB. <lb/>XLVIII. <lb/>fig. 28.</note> </div> <p> <s xml:id="echoid-s5667" xml:space="preserve">16 Martii, circa octavam adhuc propiorem Saturno co-<lb/>mitem vidi, ad orientem ſpectantem, uti obſervatione ſu-<lb/>periori. </s> <s xml:id="echoid-s5668" xml:space="preserve">fig. </s> <s xml:id="echoid-s5669" xml:space="preserve">29.</s> <s xml:id="echoid-s5670" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5671" xml:space="preserve">30 Martii, hora 8, pari propinquitate, ſed ad alteram <lb/>partem comes ſtabat. </s> <s xml:id="echoid-s5672" xml:space="preserve">fig. </s> <s xml:id="echoid-s5673" xml:space="preserve">30.</s> <s xml:id="echoid-s5674" xml:space="preserve"/> </p> <pb o="545" file="0275" n="289" rhead="SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s5675" xml:space="preserve">18 Apr. </s> <s xml:id="echoid-s5676" xml:space="preserve">ad ortum ſitus erat comes in maxima diſtantia. </s> <s xml:id="echoid-s5677" xml:space="preserve">fig. </s> <s xml:id="echoid-s5678" xml:space="preserve">31.</s> <s xml:id="echoid-s5679" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5680" xml:space="preserve">17 Jun. </s> <s xml:id="echoid-s5681" xml:space="preserve">hora 9{1/2} ultimo ea vice obſervatus fuit Saturnus, <lb/>ſatellite verſus occidentem adſtante, & </s> <s xml:id="echoid-s5682" xml:space="preserve">mediocriter re-<lb/>moto. </s> <s xml:id="echoid-s5683" xml:space="preserve">fig. </s> <s xml:id="echoid-s5684" xml:space="preserve">32.</s> <s xml:id="echoid-s5685" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5686" xml:space="preserve">Hi@ce autem obſervationibus omnibus quandiu Saturnus <lb/>rotundus apparuit, tranſverſa illa linea, cæteris diſci parti-<lb/>bus paulo obſcurior, ex æquo medium ejus diſcum ſecabat, <lb/>eratque ad ſatellitem directa. </s> <s xml:id="echoid-s5687" xml:space="preserve">Et hac quidem obſervatione <lb/>17 Junii habita, primùm animadverti motum Saturni, eum <lb/>ſcilicet quo propter telluris vertiginem cum cælo pariter quo-<lb/>tidie circumferri putatur, ſecundum eandem illam incedere <lb/>lineam.</s> <s xml:id="echoid-s5688" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5689" xml:space="preserve">Eodem Anno 1656, Octobris die 13, manè hora 6, rur-<lb/> <anchor type="note" xlink:label="note-0275-01a" xlink:href="note-0275-01"/> ſus Saturnus videri cœpit, cui jam brachia erant renata; </s> <s xml:id="echoid-s5690" xml:space="preserve">ſe-<lb/>rius quidem quàm in obſervatione edita prædixeram, verùm <lb/> <anchor type="note" xlink:label="note-0275-02a" xlink:href="note-0275-02"/> haud aliâ formâ, quæ nimirum eadem planè fuit atque anno <lb/>præcedenti: </s> <s xml:id="echoid-s5691" xml:space="preserve">licet melius nunc cujuſmodi eſſet diſcerneretur, <lb/>ob adhibitos tubos præſtantiores.</s> <s xml:id="echoid-s5692" xml:space="preserve"/> </p> <div xml:id="echoid-div295" type="float" level="2" n="17"> <note position="right" xlink:label="note-0275-01" xlink:href="note-0275-01a" xml:space="preserve">Brachia Sa-<lb/>turno rena-<lb/>ta.</note> <note position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">TAB. LI. <lb/>fig. 5.</note> </div> <p> <s xml:id="echoid-s5693" xml:space="preserve">Faſcia autem illa ſeu zona obſcurior, paulò inferior bra-<lb/>chiorum linea nunc apparebat, cum anno 1655 ſuperior fu-<lb/>iſſet. </s> <s xml:id="echoid-s5694" xml:space="preserve">Eratque Saturni motus, quo cum cælo corripi vide-<lb/>tur, ſecundum hujus zonæ ductum, ac proinde ſecundum <lb/>rectam quoque lineam per utraque brachia protenſam, ac <lb/>ſemper poſtea quotquot obſervationibus idem inquirere li-<lb/>buit, eodem modo rem ſeſe habere comperi. </s> <s xml:id="echoid-s5695" xml:space="preserve">Comes con-<lb/>ſpici hac vice nequiit, forte an ob ingrue@tem Solis exortum, <lb/>aëremque craſſiorem prope horizontem.</s> <s xml:id="echoid-s5696" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5697" xml:space="preserve">Die autem 19 Oct. </s> <s xml:id="echoid-s5698" xml:space="preserve">hora 6 mat. </s> <s xml:id="echoid-s5699" xml:space="preserve">apparuit comes Saturno <lb/> <anchor type="note" xlink:label="note-0275-03a" xlink:href="note-0275-03"/> occidentalior, vix mediocri diſtantia abſiſtens, quæ diebus <lb/>ſequentibus duobus aucta eſt continuè.</s> <s xml:id="echoid-s5700" xml:space="preserve"/> </p> <div xml:id="echoid-div296" type="float" level="2" n="18"> <note position="right" xlink:label="note-0275-03" xlink:href="note-0275-03a" xml:space="preserve">TAB. <lb/>XLVIII. <lb/>fig. 33.</note> </div> <p> <s xml:id="echoid-s5701" xml:space="preserve">21 Oct. </s> <s xml:id="echoid-s5702" xml:space="preserve">hora eâdem, erat comes in diſtantia maxima oc-<lb/>cidentali. </s> <s xml:id="echoid-s5703" xml:space="preserve">fig. </s> <s xml:id="echoid-s5704" xml:space="preserve">34.</s> <s xml:id="echoid-s5705" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5706" xml:space="preserve">25 Oct. </s> <s xml:id="echoid-s5707" xml:space="preserve">hora 6. </s> <s xml:id="echoid-s5708" xml:space="preserve">mat. </s> <s xml:id="echoid-s5709" xml:space="preserve">non apparuit comes.</s> <s xml:id="echoid-s5710" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5711" xml:space="preserve">9 Nov. </s> <s xml:id="echoid-s5712" xml:space="preserve">hora 5{1/2} mat propinquus Saturno comes exiſtebat, <lb/>occidentalis rurſum, ac lineâ anſarum paulo ſuperior. </s> <s xml:id="echoid-s5713" xml:space="preserve">fig. </s> <s xml:id="echoid-s5714" xml:space="preserve">35.</s> <s xml:id="echoid-s5715" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5716" xml:space="preserve">26 Nov. </s> <s xml:id="echoid-s5717" xml:space="preserve">hora 6{1/2} mat. </s> <s xml:id="echoid-s5718" xml:space="preserve">comes latuit. </s> <s xml:id="echoid-s5719" xml:space="preserve">Brachia vero paulo <pb o="546" file="0276" n="290" rhead="CHRIST. HUGENII"/> latiora evaſerant, & </s> <s xml:id="echoid-s5720" xml:space="preserve">quà Saturno junguntur, minus intenſa <lb/>luce quàm verſus extremas cuſpides lucebant, & </s> <s xml:id="echoid-s5721" xml:space="preserve">hac fere ſpe-<lb/>cie ad occaſum uſque Heliacum Saturnus permanſit.</s> <s xml:id="echoid-s5722" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5723" xml:space="preserve">27 Nov. </s> <s xml:id="echoid-s5724" xml:space="preserve">hora 6 mat. </s> <s xml:id="echoid-s5725" xml:space="preserve">ſatelles ægre conſpiciebatur, ad o-<lb/>rientem ſitus, & </s> <s xml:id="echoid-s5726" xml:space="preserve">brachiorum linea ſuperior. </s> <s xml:id="echoid-s5727" xml:space="preserve">fig. </s> <s xml:id="echoid-s5728" xml:space="preserve">36.</s> <s xml:id="echoid-s5729" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5730" xml:space="preserve">16 Dec. </s> <s xml:id="echoid-s5731" xml:space="preserve">6 mat. </s> <s xml:id="echoid-s5732" xml:space="preserve">videbatur ſatelles in maxima diſtantia o-<lb/>rientali. </s> <s xml:id="echoid-s5733" xml:space="preserve">fig. </s> <s xml:id="echoid-s5734" xml:space="preserve">37.</s> <s xml:id="echoid-s5735" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5736" xml:space="preserve">Anno 1657, 5 Jan. </s> <s xml:id="echoid-s5737" xml:space="preserve">hora 12{1/2} comes latebat.</s> <s xml:id="echoid-s5738" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5739" xml:space="preserve">18 Jan. </s> <s xml:id="echoid-s5740" xml:space="preserve">hora 12, erat in mediocri diſtantia, orientem ſpe-<lb/>ctans. </s> <s xml:id="echoid-s5741" xml:space="preserve">fig. </s> <s xml:id="echoid-s5742" xml:space="preserve">38.</s> <s xml:id="echoid-s5743" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5744" xml:space="preserve">A Saturno autem polum verſus dimidio circiter gradu di-<lb/>ſtabat fixa 3 magnitudinis, quæ eſt in ventre Leonis, longi-<lb/>tudine reſpondens Virginis gr. </s> <s xml:id="echoid-s5745" xml:space="preserve">4. </s> <s xml:id="echoid-s5746" xml:space="preserve">5′. </s> <s xml:id="echoid-s5747" xml:space="preserve">cum latit. </s> <s xml:id="echoid-s5748" xml:space="preserve">borea gr. </s> <s xml:id="echoid-s5749" xml:space="preserve">2. </s> <s xml:id="echoid-s5750" xml:space="preserve">49′.</s> <s xml:id="echoid-s5751" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5752" xml:space="preserve">22 Martii hor. </s> <s xml:id="echoid-s5753" xml:space="preserve">7{1/2} veſp. </s> <s xml:id="echoid-s5754" xml:space="preserve">comes in maxima diſtantia orien-<lb/>tali, & </s> <s xml:id="echoid-s5755" xml:space="preserve">anſarum linea paulo ſuperior; </s> <s xml:id="echoid-s5756" xml:space="preserve">linea obſcura non ſa-<lb/>tis erat conſpicua.</s> <s xml:id="echoid-s5757" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5758" xml:space="preserve">Scripſit mihi Hevelius ſe quoque pridie hujus diei comi-<lb/>tem Saturni obſervaſſe ad orientem ſitum in maxima diſtan-<lb/>tia, quod ſatis bene cum noſtra hac obſervatione convenit.</s> <s xml:id="echoid-s5759" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5760" xml:space="preserve">29 Martii hora 7{1/2} comes erat in mediocri diſtantia occi-<lb/>dentem verſus, & </s> <s xml:id="echoid-s5761" xml:space="preserve">in eadem cum brachiis recta. </s> <s xml:id="echoid-s5762" xml:space="preserve">fig. </s> <s xml:id="echoid-s5763" xml:space="preserve">39.</s> <s xml:id="echoid-s5764" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5765" xml:space="preserve">30 Martii, comes in maxima diſtantia occidentali. </s> <s xml:id="echoid-s5766" xml:space="preserve">fig. </s> <s xml:id="echoid-s5767" xml:space="preserve">40.</s> <s xml:id="echoid-s5768" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5769" xml:space="preserve">Qua veſpera Hevelius quoque ad eandem partem ſibi con-<lb/>ſpectum aſſerit, ſed difficulter, unde fortaſſe de diſtantia mi-<lb/>nus recte judicaverit, nam 1{1/2} ſcrupulo tantum abfuiſſe à Sa-<lb/>turno ſcribit.</s> <s xml:id="echoid-s5770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5771" xml:space="preserve">18 Maji, comitem mecum obſervavit Bullialdus occiden-<lb/>taliorem Saturno, & </s> <s xml:id="echoid-s5772" xml:space="preserve">in mediocri diſtantia. </s> <s xml:id="echoid-s5773" xml:space="preserve">fig. </s> <s xml:id="echoid-s5774" xml:space="preserve">41.</s> <s xml:id="echoid-s5775" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5776" xml:space="preserve">19 Maji, proximus Saturno adſtabat comes occidentem <lb/>verſus, vix anſarum linea ſuperior. </s> <s xml:id="echoid-s5777" xml:space="preserve">fig. </s> <s xml:id="echoid-s5778" xml:space="preserve">42.</s> <s xml:id="echoid-s5779" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5780" xml:space="preserve">Anno eodem 1657. </s> <s xml:id="echoid-s5781" xml:space="preserve">17 Dec. </s> <s xml:id="echoid-s5782" xml:space="preserve">hora 5{1/2} manè, quo die pri-<lb/> <anchor type="note" xlink:label="note-0276-01a" xlink:href="note-0276-01"/> mùm poſt ortum Heliacum Saturnum obſervavi, comes me-<lb/>diocriter diſtabat orientem verſus, eratque anſarum lineâ ſu-<lb/>perior. </s> <s xml:id="echoid-s5783" xml:space="preserve">fig 43. </s> <s xml:id="echoid-s5784" xml:space="preserve">Brachia verò prope Saturni diſcum adaper-<lb/>ta ac bifida inveniebam, qualia ante non videram, lineâ quo-<lb/>que obſcurâ verſus inferiora ulterius promotâ.</s> <s xml:id="echoid-s5785" xml:space="preserve"/> </p> <div xml:id="echoid-div297" type="float" level="2" n="19"> <note position="left" xlink:label="note-0276-01" xlink:href="note-0276-01a" xml:space="preserve">Brachia <lb/>Saturni in <lb/>anſas muta-<lb/>ri cœpta. <lb/>TAB. XLIX. <lb/>fig. 7.</note> </div> <pb o="547" file="0277" n="291" rhead="SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s5786" xml:space="preserve">Et hac quidem figurâ permanſit, donec rurſus radiis ſolis <lb/>occultaretur.</s> <s xml:id="echoid-s5787" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5788" xml:space="preserve">18 Dec. </s> <s xml:id="echoid-s5789" xml:space="preserve">comes erat in maxima diſtantia, orientem ſpe-<lb/>ctans, & </s> <s xml:id="echoid-s5790" xml:space="preserve">in ipſa anſarum linea. </s> <s xml:id="echoid-s5791" xml:space="preserve">Tab. </s> <s xml:id="echoid-s5792" xml:space="preserve">XLVIII. </s> <s xml:id="echoid-s5793" xml:space="preserve">fig. </s> <s xml:id="echoid-s5794" xml:space="preserve">44.</s> <s xml:id="echoid-s5795" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5796" xml:space="preserve">22 Dec. </s> <s xml:id="echoid-s5797" xml:space="preserve">h. </s> <s xml:id="echoid-s5798" xml:space="preserve">6{1/2} mat. </s> <s xml:id="echoid-s5799" xml:space="preserve">comes non apparuit.</s> <s xml:id="echoid-s5800" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5801" xml:space="preserve">27 Dec. </s> <s xml:id="echoid-s5802" xml:space="preserve">h. </s> <s xml:id="echoid-s5803" xml:space="preserve">6{1/2} mat. </s> <s xml:id="echoid-s5804" xml:space="preserve">comes occidentem verſus in maxima <lb/>fere diſtantia ſitus erat, & </s> <s xml:id="echoid-s5805" xml:space="preserve">anſarum linea paulo altior. </s> <s xml:id="echoid-s5806" xml:space="preserve">fig 45.</s> <s xml:id="echoid-s5807" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5808" xml:space="preserve">Anno 1658. </s> <s xml:id="echoid-s5809" xml:space="preserve">24 Febr. </s> <s xml:id="echoid-s5810" xml:space="preserve">h. </s> <s xml:id="echoid-s5811" xml:space="preserve">10, comes videri nequiit.</s> <s xml:id="echoid-s5812" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5813" xml:space="preserve">1 Mart. </s> <s xml:id="echoid-s5814" xml:space="preserve">h. </s> <s xml:id="echoid-s5815" xml:space="preserve">10, idem comitis ſitus erat qui 27 Dec.</s> <s xml:id="echoid-s5816" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5817" xml:space="preserve">11 Martii, h. </s> <s xml:id="echoid-s5818" xml:space="preserve">10, comes difficile conſpiciebatur, quippe <lb/>propinquus admodum Saturno. </s> <s xml:id="echoid-s5819" xml:space="preserve">Orientem ſpectabat, erat-<lb/>que anſarum linea aliquanto inferior, & </s> <s xml:id="echoid-s5820" xml:space="preserve">quaſi ſub Saturno <lb/>tranſiturus. </s> <s xml:id="echoid-s5821" xml:space="preserve">fig. </s> <s xml:id="echoid-s5822" xml:space="preserve">46.</s> <s xml:id="echoid-s5823" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5824" xml:space="preserve">16 Mart. </s> <s xml:id="echoid-s5825" xml:space="preserve">h. </s> <s xml:id="echoid-s5826" xml:space="preserve">10 quantum poterat à Saturno comes receſſerat <lb/>occidentem verſus, vixque erat anſarum linea ſuperior. </s> <s xml:id="echoid-s5827" xml:space="preserve">fig. </s> <s xml:id="echoid-s5828" xml:space="preserve">47.</s> <s xml:id="echoid-s5829" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5830" xml:space="preserve">23 Martii, in contrariam partem pene tantundem diſta-<lb/>bat, lineâ anſarum rurſus paulo ſuperior. </s> <s xml:id="echoid-s5831" xml:space="preserve">fig. </s> <s xml:id="echoid-s5832" xml:space="preserve">48.</s> <s xml:id="echoid-s5833" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5834" xml:space="preserve">3 Apr. </s> <s xml:id="echoid-s5835" xml:space="preserve">paulo remotior erat à Saturno comes, quam 11 <lb/>Martii, & </s> <s xml:id="echoid-s5836" xml:space="preserve">occiduum latus tenebat, lineaque anſarum ſubli-<lb/>mior cernebatur. </s> <s xml:id="echoid-s5837" xml:space="preserve">fig. </s> <s xml:id="echoid-s5838" xml:space="preserve">49.</s> <s xml:id="echoid-s5839" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5840" xml:space="preserve">Anno eodem 1658. </s> <s xml:id="echoid-s5841" xml:space="preserve">10 Nov. </s> <s xml:id="echoid-s5842" xml:space="preserve">hor. </s> <s xml:id="echoid-s5843" xml:space="preserve">6{1/2} mat. </s> <s xml:id="echoid-s5844" xml:space="preserve">poſtquam He-<lb/> <anchor type="note" xlink:label="note-0277-01a" xlink:href="note-0277-01"/> liace ortus eſſet Saturnus, jam latius aperiri anſæ videban-<lb/>tur, quanquam ob humilitatem ſideris, ſurgenteſque vapo-<lb/>res, & </s> <s xml:id="echoid-s5845" xml:space="preserve">auroræ claritatem non admodum diſtincte poterant <lb/>diſcerni: </s> <s xml:id="echoid-s5846" xml:space="preserve">comes occidentem verſus adſtabat, remotus ut cum <lb/>maximè, anſarum linea nonnihil tamen ſuperior. </s> <s xml:id="echoid-s5847" xml:space="preserve">fig. </s> <s xml:id="echoid-s5848" xml:space="preserve">50.</s> <s xml:id="echoid-s5849" xml:space="preserve"/> </p> <div xml:id="echoid-div298" type="float" level="2" n="20"> <note position="right" xlink:label="note-0277-01" xlink:href="note-0277-01a" xml:space="preserve">Anſæ Satu@ <lb/>ni amplius <lb/>pateſactæ.</note> </div> <p> <s xml:id="echoid-s5850" xml:space="preserve">16 Jan. </s> <s xml:id="echoid-s5851" xml:space="preserve">anno 1659. </s> <s xml:id="echoid-s5852" xml:space="preserve">hora 5{1/2} mane, comes ad occidentem <lb/>denuo ſitus erat, non longe à Saturno diſtans, linea autem <lb/>anſarum integra fere Saturni diametro ſuperior. </s> <s xml:id="echoid-s5853" xml:space="preserve">fig. </s> <s xml:id="echoid-s5854" xml:space="preserve">51.</s> <s xml:id="echoid-s5855" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5856" xml:space="preserve">12 Febr. </s> <s xml:id="echoid-s5857" xml:space="preserve">6 mat. </s> <s xml:id="echoid-s5858" xml:space="preserve">tantundem infra lineam anſarum deſcen-<lb/>derat, occidentalis rurſus. </s> <s xml:id="echoid-s5859" xml:space="preserve">fig. </s> <s xml:id="echoid-s5860" xml:space="preserve">52.</s> <s xml:id="echoid-s5861" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5862" xml:space="preserve">Forma verò anſarum diſtincte hac vice percipi potuit, quam <lb/> <anchor type="note" xlink:label="note-0277-02a" xlink:href="note-0277-02"/> figura hæc exhibet; </s> <s xml:id="echoid-s5863" xml:space="preserve">atque ea ad ultimam uſque harum obſer-<lb/>vationum talis extitit.</s> <s xml:id="echoid-s5864" xml:space="preserve"/> </p> <div xml:id="echoid-div299" type="float" level="2" n="21"> <note position="right" xlink:label="note-0277-02" xlink:href="note-0277-02a" xml:space="preserve">TAB. LI. <lb/>fig. 6.</note> </div> <p> <s xml:id="echoid-s5865" xml:space="preserve">24 Febr. </s> <s xml:id="echoid-s5866" xml:space="preserve">hora dimidia poſt mediam noctem, comes erat <lb/>in mediocri diſtantia, orientem verſus, rectâ anſarum pau- <pb o="548" file="0278" n="292" rhead="CHRIST. HUGENII"/> lo inferior. </s> <s xml:id="echoid-s5867" xml:space="preserve">Tab. </s> <s xml:id="echoid-s5868" xml:space="preserve">XLVIII. </s> <s xml:id="echoid-s5869" xml:space="preserve">fig. </s> <s xml:id="echoid-s5870" xml:space="preserve">53.</s> <s xml:id="echoid-s5871" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5872" xml:space="preserve">25 Febr. </s> <s xml:id="echoid-s5873" xml:space="preserve">horâ eadem orientalis denuo comes cernebatur, <lb/>una Saturni diametro ab ipſo remotus. </s> <s xml:id="echoid-s5874" xml:space="preserve">fig. </s> <s xml:id="echoid-s5875" xml:space="preserve">54.</s> <s xml:id="echoid-s5876" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5877" xml:space="preserve">14 Martii, hora 12, comes recta fere infra Saturnum ob-<lb/> <anchor type="note" xlink:label="note-0278-01a" xlink:href="note-0278-01"/> ſervatus, unius circiter diametri longitudine diſtans; </s> <s xml:id="echoid-s5878" xml:space="preserve">paulum <lb/>tamen verſus occidentem declinabat. </s> <s xml:id="echoid-s5879" xml:space="preserve">fig. </s> <s xml:id="echoid-s5880" xml:space="preserve">55.</s> <s xml:id="echoid-s5881" xml:space="preserve"/> </p> <div xml:id="echoid-div300" type="float" level="2" n="22"> <note position="left" xlink:label="note-0278-01" xlink:href="note-0278-01a" xml:space="preserve">Comes inſra <lb/>Saturnum <lb/>tranſire vi-<lb/>ſus.</note> </div> <p> <s xml:id="echoid-s5882" xml:space="preserve">16 Martii, hora 11, ad latus occiduum poſitus erat, fere <lb/>in maxima diſtantia, inferiorque paulo eâ quæ per anſas du-<lb/>citur. </s> <s xml:id="echoid-s5883" xml:space="preserve">fig. </s> <s xml:id="echoid-s5884" xml:space="preserve">56.</s> <s xml:id="echoid-s5885" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5886" xml:space="preserve">21 Martii, hora 11, rurſus ad eandem partem conſiſtebat <lb/> <anchor type="note" xlink:label="note-0278-02a" xlink:href="note-0278-02"/> comes, motu latitudinis integra Saturni diametro ſupra an-<lb/>ſarum lineam elatus, longitudinis motu tantum dimidia dia-<lb/>metro diſtans. </s> <s xml:id="echoid-s5887" xml:space="preserve">fig. </s> <s xml:id="echoid-s5888" xml:space="preserve">57.</s> <s xml:id="echoid-s5889" xml:space="preserve"/> </p> <div xml:id="echoid-div301" type="float" level="2" n="23"> <note position="left" xlink:label="note-0278-02" xlink:href="note-0278-02a" xml:space="preserve">Idem ſupra <lb/>Saturnum <lb/>t<unsure/>ranſiens.</note> </div> <p> <s xml:id="echoid-s5890" xml:space="preserve">22 Martii, horæ quadrante ante undecimam; </s> <s xml:id="echoid-s5891" xml:space="preserve">rurſus in-<lb/>tegra diametro ſuperior erat rectâ anſarum, ac fere ſupra <lb/>orientalis anſæ extremam cuſpidem collocatus. </s> <s xml:id="echoid-s5892" xml:space="preserve">fig. </s> <s xml:id="echoid-s5893" xml:space="preserve">58.</s> <s xml:id="echoid-s5894" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5895" xml:space="preserve">26 Martii, hora 10{1/2} comes in maxima diſtantia videba-<lb/>tur; </s> <s xml:id="echoid-s5896" xml:space="preserve">quam accuratè hac vice dimenſus, inveni inter comi-<lb/>tem centrumque Saturni intervallum trium ſcrupulorum pri-<lb/>morum, 16 ſecundorum. </s> <s xml:id="echoid-s5897" xml:space="preserve">fig. </s> <s xml:id="echoid-s5898" xml:space="preserve">59.</s> <s xml:id="echoid-s5899" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5900" xml:space="preserve">Hucuſque obſervationes, & </s> <s xml:id="echoid-s5901" xml:space="preserve">plures quidem quam neceſſe <lb/>fuerat, recenſui; </s> <s xml:id="echoid-s5902" xml:space="preserve">rem gratam tamen iis me facturum ratus, <lb/>qui triennio iſto ſimul forſitan mecum novo Planetæ obſer-<lb/>vando vacaverint; </s> <s xml:id="echoid-s5903" xml:space="preserve">quibus procul dubio jucundum erit con-<lb/>ſenſum mutuum ſuarum cum noſtris obſervationibus depre-<lb/>hendere. </s> <s xml:id="echoid-s5904" xml:space="preserve">Jam enim & </s> <s xml:id="echoid-s5905" xml:space="preserve">Hevelius Gedani eum conſpicere an-<lb/>te biennium cœpit, ut ſupra quoque retuli, & </s> <s xml:id="echoid-s5906" xml:space="preserve">in Anglia D. <lb/></s> <s xml:id="echoid-s5907" xml:space="preserve">Paulus Nelius eques cum Clariſſimo Wrennio ipſo jam An-<lb/>no 1655, ſibi animadverſum quandoque aſſerunt, nec tamen <lb/>Planetam eſſe cognoviſſe donec à nobis eſſent admoniti. </s> <s xml:id="echoid-s5908" xml:space="preserve"><lb/>Nunc quo pacto periodum ejus inveſtigaverim, quæque por-<lb/>ro ad illam pertinent expediam.</s> <s xml:id="echoid-s5909" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5910" xml:space="preserve">Penſitatis priorum aliquot menſium obſervationibus, cum <lb/> <anchor type="note" xlink:label="note-0278-03a" xlink:href="note-0278-03"/> circiter 16 dierum ſpatio Saturnum à Luna ſua ambiri com-<lb/>periſſem; </s> <s xml:id="echoid-s5911" xml:space="preserve">nam quo loco animadverſa fuerat 25 Mart. </s> <s xml:id="echoid-s5912" xml:space="preserve">1655, <lb/>ad eum ſexto decimo inde die rediiſſe viſa eſt, 10 nimirum <pb o="549" file="0279" n="293" rhead="SYSTEMA SATURNIUM."/> Apr. </s> <s xml:id="echoid-s5913" xml:space="preserve">Itemque eodem anno die 3 & </s> <s xml:id="echoid-s5914" xml:space="preserve">19 Apr. </s> <s xml:id="echoid-s5915" xml:space="preserve">idem ſitus fue-<lb/> <anchor type="note" xlink:label="note-0279-01a" xlink:href="note-0279-01"/> rat deprehenſus; </s> <s xml:id="echoid-s5916" xml:space="preserve">nec non 13 & </s> <s xml:id="echoid-s5917" xml:space="preserve">29 ejuſdem menſis. </s> <s xml:id="echoid-s5918" xml:space="preserve">Hiſce <lb/>igitur animadverſis, circulum deſcripſi orbitam comitis refe-<lb/>rentem, in cujus centro Saturnus eſſet, atque in partes 16 <lb/>diſtribui, uti Schema ſubjectum exhibet. </s> <s xml:id="echoid-s5919" xml:space="preserve">In eo comitem, <lb/>ſecundum ſignorum ordinem, circumduxi; </s> <s xml:id="echoid-s5920" xml:space="preserve">nulla tum qui-<lb/>dem obſervatione ut ita ſtatuerem cogente, ſed quod in <lb/>eam partem Luna quoque noſtra & </s> <s xml:id="echoid-s5921" xml:space="preserve">Jovis comites deferren-<lb/>tur. </s> <s xml:id="echoid-s5922" xml:space="preserve">Poſtmodum autem ſtabilita hypotheſi, qua phæno-<lb/>mean anſarum explicantur, patuit recte ita hunc morum me <lb/>conſtituiſſe. </s> <s xml:id="echoid-s5923" xml:space="preserve">Porro in hoc circulo locum comitis quærendo <lb/>quo in prima noſtra obſervatione ſtetiſſet, ſæpiuſque eun-<lb/>dem retractando, ut obſervationibus per id tempus habitis <lb/>congruerent etiam reliqua loca in circulo comiti aſſignata; </s> <s xml:id="echoid-s5924" xml:space="preserve">ita <lb/>demum commodiſſimè repreſentari omnia viſum eſt, ſi pri-<lb/>ma obſervatione, nempe 25 Mart. </s> <s xml:id="echoid-s5925" xml:space="preserve">1655, ad numerum 12 <lb/>comes reponeretur, 3 partibus cum dimidia, qualium 16 cir-<lb/>cumferentia continet à loco perigæo B remotus; </s> <s xml:id="echoid-s5926" xml:space="preserve">nam dia-<lb/>metrum AB ad viſum noſtrum vergere ponimus, & </s> <s xml:id="echoid-s5927" xml:space="preserve">utraſque <lb/>hujus epicyclii apſides determinare. </s> <s xml:id="echoid-s5928" xml:space="preserve">Fuerit igitur 26 Mart. <lb/></s> <s xml:id="echoid-s5929" xml:space="preserve">ad num. </s> <s xml:id="echoid-s5930" xml:space="preserve">13, Saturni comes: </s> <s xml:id="echoid-s5931" xml:space="preserve">& </s> <s xml:id="echoid-s5932" xml:space="preserve">27 Mart. </s> <s xml:id="echoid-s5933" xml:space="preserve">ad num. </s> <s xml:id="echoid-s5934" xml:space="preserve">14, & </s> <s xml:id="echoid-s5935" xml:space="preserve">3 <lb/>Apr. </s> <s xml:id="echoid-s5936" xml:space="preserve">ad num. </s> <s xml:id="echoid-s5937" xml:space="preserve">5, atque ita deinceps iis orbitæ locis quæ ob-<lb/>ſervatis primi anni exactè ſatis conveniunt, quanquam ali-<lb/>quid ſubinde addendum auferendumve 16 dierum periodo <lb/>exiſtimaverim. </s> <s xml:id="echoid-s5938" xml:space="preserve">Cum autem ſcirem etiam orbitæ terreſtris, <lb/>qua nos circa Solem ferimur, ipſiuſque Saturni motus ratio-<lb/>nem habendam eſſe, ſi accuratè comitis periodum definire <lb/>vellem, proinde ſequenti methodo eam deinceps inveſtigavi.</s> <s xml:id="echoid-s5939" xml:space="preserve"/> </p> <div xml:id="echoid-div302" type="float" level="2" n="24"> <note position="left" xlink:label="note-0278-03" xlink:href="note-0278-03a" xml:space="preserve">Luna Satur-<lb/>nia periodus.</note> <note position="right" xlink:label="note-0279-01" xlink:href="note-0279-01a" xml:space="preserve">TAB. LI. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s5940" xml:space="preserve">Bina tempora quæſivi quibus in apogæo vel perigæo co-<lb/> <anchor type="note" xlink:label="note-0279-02a" xlink:href="note-0279-02"/> mes verſatus eſſet; </s> <s xml:id="echoid-s5941" xml:space="preserve">quorum alterum inveni 14 Martii, anno <lb/>1659, veſperi circa horam octavam. </s> <s xml:id="echoid-s5942" xml:space="preserve">Quia enim hora noctis <lb/>12 tantum prætergreſſus erat locum perigæi quantum ex ob-<lb/>ſervatione illo tempore habita apparet, oportet eum circiter <lb/>octavam in perigæo ipſo fuiſſe. </s> <s xml:id="echoid-s5943" xml:space="preserve">Alterum ſimilem comitis ſi-<lb/>tum colligo contigiſſe die 23 Martii, anno 1656, hora iti-<lb/>dem octava poſt meridiem. </s> <s xml:id="echoid-s5944" xml:space="preserve">Etenim quia die 16 & </s> <s xml:id="echoid-s5945" xml:space="preserve">30 ejuſ- <pb o="550" file="0280" n="294" rhead="CHRIST. HUGENII"/> dem menſis hora 8 p. </s> <s xml:id="echoid-s5946" xml:space="preserve">m. </s> <s xml:id="echoid-s5947" xml:space="preserve">æquali ſpatio à Saturno remotus <lb/>apparuit, quibus diebus circa partem ſuæ orbitæ ſuperiorem <lb/>verſatum conſtat, ſequitur 16 Martii medium locum eum <lb/>tenuiſſe inter puncta 1 & </s> <s xml:id="echoid-s5948" xml:space="preserve">2 circuli modò deſcripti; </s> <s xml:id="echoid-s5949" xml:space="preserve">30 verò <lb/>Martii medium fuiſſe inter puncta 15 & </s> <s xml:id="echoid-s5950" xml:space="preserve">16; </s> <s xml:id="echoid-s5951" xml:space="preserve">quoniam 14 die-<lb/>rum intervallum eſt. </s> <s xml:id="echoid-s5952" xml:space="preserve">Ac proinde neceſſario perigæus fuit <lb/>die 23 Mart. </s> <s xml:id="echoid-s5953" xml:space="preserve">poſt meridiem circa horam octavam.</s> <s xml:id="echoid-s5954" xml:space="preserve"/> </p> <div xml:id="echoid-div303" type="float" level="2" n="25"> <note position="right" xlink:label="note-0279-02" xlink:href="note-0279-02a" xml:space="preserve">Eadem peri@-<lb/>dus accura-<lb/>tius ſupputa-<lb/>ta.</note> </div> <p> <s xml:id="echoid-s5955" xml:space="preserve">Cognitis hiſce temporibus deſcribo circulum A B C, qui <lb/> <anchor type="note" xlink:label="note-0280-01a" xlink:href="note-0280-01"/> Saturni orbitam deſignet, itemque G F orbitam Telluris, in <lb/>cujus centro Sol. </s> <s xml:id="echoid-s5956" xml:space="preserve">Saturni locum ſumo ad diem 23 Martii <lb/>1656, hora 8 poſt mer. </s> <s xml:id="echoid-s5957" xml:space="preserve">fuiſſe in A; </s> <s xml:id="echoid-s5958" xml:space="preserve">quo tanquam centro <lb/>deſcribo comitis orbitam D E L; </s> <s xml:id="echoid-s5959" xml:space="preserve">Tellurem vero eo tempore <lb/>fuiſſe in G. </s> <s xml:id="echoid-s5960" xml:space="preserve">Locus itaque comitis erat in D, ubi recta G A <lb/>circulum comitis interſecat, ſiquidem perigæum fuiſſe con-<lb/>ftat. </s> <s xml:id="echoid-s5961" xml:space="preserve">Rurſus poſito ad diem 14 Martii 1659. </s> <s xml:id="echoid-s5962" xml:space="preserve">hora 8. </s> <s xml:id="echoid-s5963" xml:space="preserve">poſt <lb/>mer. </s> <s xml:id="echoid-s5964" xml:space="preserve">loco Saturni in B, tellure vero in F; </s> <s xml:id="echoid-s5965" xml:space="preserve">neceſſe eſt comi-<lb/>tem fuiſſe in H, ubi recta F B ſecat circulum ejus N H K. <lb/></s> <s xml:id="echoid-s5966" xml:space="preserve">Eſt autem temporis intervallum inter 23 Martii 1656, & </s> <s xml:id="echoid-s5967" xml:space="preserve">14 <lb/>Martii 1659, dierum 1086, quibus Saturnus ab A progreſ-<lb/>ſus eſt ad B: </s> <s xml:id="echoid-s5968" xml:space="preserve">Comitem vero ſpatio dierum 16 circuitum u-<lb/>num abſolvere ſcio, atque inſuper exiguum quid, quod in <lb/>annis tribus circuitum integrum conficere nequeat. </s> <s xml:id="echoid-s5969" xml:space="preserve">Quum <lb/>igitur diviſis diebus 1086 per 16 fiant 67, atque aliquot dies <lb/>abundent, apparet 68 circuitus integros noſtri reſpectu co-<lb/>mitem peregiſſe, quia in H rurſus perigæus fuit. </s> <s xml:id="echoid-s5970" xml:space="preserve">Sit B K <lb/>parallela A G. </s> <s xml:id="echoid-s5971" xml:space="preserve">Si igitur 14 Mart. </s> <s xml:id="echoid-s5972" xml:space="preserve">1659, comes non in H <lb/>ſed in K poſitus fuiſſet; </s> <s xml:id="echoid-s5973" xml:space="preserve">ſequeretur eum hiſce 1086 diebus <lb/>ſexagies octies orbitam ſuam decurriſſe fixarum reſpectu, <lb/>hoc eſt, totidem menſes periodicos, ſive ſidereos potius, <lb/>Saturni incolis præbuiſſe. </s> <s xml:id="echoid-s5974" xml:space="preserve">Nam quando rectæ A D, B K <lb/>fecundum quas ex Saturno comes proſpicitur, inter ſe paral-<lb/>lelæ ſunt, eundem inter fixas locum illis obtinere cernitur. </s> <s xml:id="echoid-s5975" xml:space="preserve"><lb/>Nunc verò inſuper arcum K H emenſus eſt, qui totidem eſt <lb/>graduum quot apparenti motu Saturnus inter prædicta duo <lb/>tempora tranſivit; </s> <s xml:id="echoid-s5976" xml:space="preserve">quoniam angulus H B K æqualis eſt ei <lb/>quem conſtituunt rectæ F B, G A, motus apparentis indi- <pb file="0281" n="295"/> <pb file="0281a" n="296"/> <anchor type="figure" xlink:label="fig-0281a-01a" xlink:href="fig-0281a-01"/> <anchor type="figure" xlink:label="fig-0281a-02a" xlink:href="fig-0281a-02"/> <anchor type="figure" xlink:label="fig-0281a-03a" xlink:href="fig-0281a-03"/> <anchor type="figure" xlink:label="fig-0281a-04a" xlink:href="fig-0281a-04"/> <anchor type="figure" xlink:label="fig-0281a-05a" xlink:href="fig-0281a-05"/> <anchor type="figure" xlink:label="fig-0281a-06a" xlink:href="fig-0281a-06"/> <anchor type="figure" xlink:label="fig-0281a-07a" xlink:href="fig-0281a-07"/> <anchor type="figure" xlink:label="fig-0281a-08a" xlink:href="fig-0281a-08"/> <anchor type="figure" xlink:label="fig-0281a-09a" xlink:href="fig-0281a-09"/> <anchor type="figure" xlink:label="fig-0281a-10a" xlink:href="fig-0281a-10"/> <anchor type="figure" xlink:label="fig-0281a-11a" xlink:href="fig-0281a-11"/> <anchor type="figure" xlink:label="fig-0281a-12a" xlink:href="fig-0281a-12"/> <anchor type="figure" xlink:label="fig-0281a-13a" xlink:href="fig-0281a-13"/> <anchor type="figure" xlink:label="fig-0281a-14a" xlink:href="fig-0281a-14"/> <anchor type="figure" xlink:label="fig-0281a-15a" xlink:href="fig-0281a-15"/> <anchor type="figure" xlink:label="fig-0281a-16a" xlink:href="fig-0281a-16"/> <anchor type="figure" xlink:label="fig-0281a-17a" xlink:href="fig-0281a-17"/> <anchor type="figure" xlink:label="fig-0281a-18a" xlink:href="fig-0281a-18"/> <anchor type="figure" xlink:label="fig-0281a-19a" xlink:href="fig-0281a-19"/> <anchor type="figure" xlink:label="fig-0281a-20a" xlink:href="fig-0281a-20"/> <anchor type="figure" xlink:label="fig-0281a-21a" xlink:href="fig-0281a-21"/> <anchor type="figure" xlink:label="fig-0281a-22a" xlink:href="fig-0281a-22"/> <anchor type="figure" xlink:label="fig-0281a-23a" xlink:href="fig-0281a-23"/> <anchor type="figure" xlink:label="fig-0281a-24a" xlink:href="fig-0281a-24"/> <anchor type="figure" xlink:label="fig-0281a-25a" xlink:href="fig-0281a-25"/> <anchor type="figure" xlink:label="fig-0281a-26a" xlink:href="fig-0281a-26"/> <anchor type="figure" xlink:label="fig-0281a-27a" xlink:href="fig-0281a-27"/> <anchor type="figure" xlink:label="fig-0281a-28a" xlink:href="fig-0281a-28"/> <anchor type="figure" xlink:label="fig-0281a-29a" xlink:href="fig-0281a-29"/> <anchor type="figure" xlink:label="fig-0281a-30a" xlink:href="fig-0281a-30"/> <anchor type="figure" xlink:label="fig-0281a-31a" xlink:href="fig-0281a-31"/> <anchor type="figure" xlink:label="fig-0281a-32a" xlink:href="fig-0281a-32"/> <anchor type="figure" xlink:label="fig-0281a-33a" xlink:href="fig-0281a-33"/> <anchor type="figure" xlink:label="fig-0281a-34a" xlink:href="fig-0281a-34"/> <anchor type="figure" xlink:label="fig-0281a-35a" xlink:href="fig-0281a-35"/> <anchor type="figure" xlink:label="fig-0281a-36a" xlink:href="fig-0281a-36"/> <anchor type="figure" xlink:label="fig-0281a-37a" xlink:href="fig-0281a-37"/> <anchor type="figure" xlink:label="fig-0281a-38a" xlink:href="fig-0281a-38"/> <anchor type="figure" xlink:label="fig-0281a-39a" xlink:href="fig-0281a-39"/> <anchor type="figure" xlink:label="fig-0281a-40a" xlink:href="fig-0281a-40"/> <anchor type="figure" xlink:label="fig-0281a-41a" xlink:href="fig-0281a-41"/> <anchor type="figure" xlink:label="fig-0281a-42a" xlink:href="fig-0281a-42"/> <anchor type="figure" xlink:label="fig-0281a-43a" xlink:href="fig-0281a-43"/> <anchor type="figure" xlink:label="fig-0281a-44a" xlink:href="fig-0281a-44"/> <anchor type="figure" xlink:label="fig-0281a-45a" xlink:href="fig-0281a-45"/> <anchor type="figure" xlink:label="fig-0281a-46a" xlink:href="fig-0281a-46"/> <anchor type="figure" xlink:label="fig-0281a-47a" xlink:href="fig-0281a-47"/> <anchor type="figure" xlink:label="fig-0281a-48a" xlink:href="fig-0281a-48"/> <anchor type="figure" xlink:label="fig-0281a-49a" xlink:href="fig-0281a-49"/> <anchor type="figure" xlink:label="fig-0281a-50a" xlink:href="fig-0281a-50"/> <anchor type="figure" xlink:label="fig-0281a-51a" xlink:href="fig-0281a-51"/> <anchor type="figure" xlink:label="fig-0281a-52a" xlink:href="fig-0281a-52"/> <anchor type="figure" xlink:label="fig-0281a-53a" xlink:href="fig-0281a-53"/> <anchor type="figure" xlink:label="fig-0281a-54a" xlink:href="fig-0281a-54"/> <anchor type="figure" xlink:label="fig-0281a-55a" xlink:href="fig-0281a-55"/> <anchor type="figure" xlink:label="fig-0281a-56a" xlink:href="fig-0281a-56"/> <anchor type="figure" xlink:label="fig-0281a-57a" xlink:href="fig-0281a-57"/> <anchor type="figure" xlink:label="fig-0281a-58a" xlink:href="fig-0281a-58"/> <anchor type="figure" xlink:label="fig-0281a-59a" xlink:href="fig-0281a-59"/> <pb file="0282" n="297"/> <pb o="551" file="0283" n="298" rhead="SYSTEMA SATURNIUM."/> ces; </s> <s xml:id="echoid-s5977" xml:space="preserve">iſque motus ex Ephemeridibus invenitur fuiſſe gr. </s> <s xml:id="echoid-s5978" xml:space="preserve">40, <lb/>48′. </s> <s xml:id="echoid-s5979" xml:space="preserve">Sic itaque colligo; </s> <s xml:id="echoid-s5980" xml:space="preserve">ſi diebus 1086 abſolvit periodos 68, <lb/>atque inſuper gr. </s> <s xml:id="echoid-s5981" xml:space="preserve">40, 48@, hoc eſt gr. </s> <s xml:id="echoid-s5982" xml:space="preserve">24520, 48′, quantum <lb/>ergo die una? </s> <s xml:id="echoid-s5983" xml:space="preserve">Prodeuntque gr. </s> <s xml:id="echoid-s5984" xml:space="preserve">22, 34′, 44″, qui motus co-<lb/>mitis diurnus eſt reſpectu fixarum. </s> <s xml:id="echoid-s5985" xml:space="preserve">Ad menſis ſiderei lon-<lb/>gitudinem inveniendam ita calculum inſtituo; </s> <s xml:id="echoid-s5986" xml:space="preserve">ſi gr. </s> <s xml:id="echoid-s5987" xml:space="preserve">24520, <lb/>48′ percurrit diebus 1086, quot dies impendet gradibus 360? <lb/></s> <s xml:id="echoid-s5988" xml:space="preserve">fiunt dies 15, horæ 22, ſcr. </s> <s xml:id="echoid-s5989" xml:space="preserve">39. </s> <s xml:id="echoid-s5990" xml:space="preserve">ſpatium quo ad eandem fi-<lb/>xam Saturni incolis Luna ſua revertitur.</s> <s xml:id="echoid-s5991" xml:space="preserve"/> </p> <div xml:id="echoid-div304" type="float" level="2" n="26"> <note position="left" xlink:label="note-0280-01" xlink:href="note-0280-01a" xml:space="preserve">TAB. XLIX. <lb/>fig. 1.</note> <figure xlink:label="fig-0281a-01" xlink:href="fig-0281a-01a"> <caption xml:id="echoid-caption92" style="it" xml:space="preserve">Pag. 550.<lb/>TAB. XLV III.<lb/>Fig. 1.</caption> <description xml:id="echoid-description6" xml:space="preserve">* a</description> <description xml:id="echoid-description7" xml:space="preserve">* b</description> <description xml:id="echoid-description8" xml:space="preserve">27. Mart. 1655.</description> </figure> <figure xlink:label="fig-0281a-02" xlink:href="fig-0281a-02a"> <caption xml:id="echoid-caption93" style="it" xml:space="preserve">Fig. 2.</caption> <description xml:id="echoid-description9" xml:space="preserve">a *</description> <description xml:id="echoid-description10" xml:space="preserve">3. Apr.</description> </figure> <figure xlink:label="fig-0281a-03" xlink:href="fig-0281a-03a"> <caption xml:id="echoid-caption94" style="it" xml:space="preserve">Fig. 3.</caption> <description xml:id="echoid-description11" xml:space="preserve">* a</description> <description xml:id="echoid-description12" xml:space="preserve">c *</description> <description xml:id="echoid-description13" xml:space="preserve">9. Apr.</description> </figure> <figure xlink:label="fig-0281a-04" xlink:href="fig-0281a-04a"> <caption xml:id="echoid-caption95" style="it" xml:space="preserve">Fig. 4.</caption> <description xml:id="echoid-description14" xml:space="preserve">* a</description> <description xml:id="echoid-description15" xml:space="preserve">* c</description> <description xml:id="echoid-description16" xml:space="preserve">10. Apr.</description> </figure> <figure xlink:label="fig-0281a-05" xlink:href="fig-0281a-05a"> <caption xml:id="echoid-caption96" style="it" xml:space="preserve">Fig. 5.</caption> <description xml:id="echoid-description17" xml:space="preserve">* a</description> <description xml:id="echoid-description18" xml:space="preserve">c *</description> <description xml:id="echoid-description19" xml:space="preserve">11. Apr.</description> </figure> <figure xlink:label="fig-0281a-06" xlink:href="fig-0281a-06a"> <caption xml:id="echoid-caption97" style="it" xml:space="preserve">Fig. 6.</caption> <description xml:id="echoid-description20" xml:space="preserve">* a</description> <description xml:id="echoid-description21" xml:space="preserve">c *</description> <description xml:id="echoid-description22" xml:space="preserve">12. Apr.</description> </figure> <figure xlink:label="fig-0281a-07" xlink:href="fig-0281a-07a"> <caption xml:id="echoid-caption98" style="it" xml:space="preserve">Fig. 7.</caption> <description xml:id="echoid-description23" xml:space="preserve">* c</description> <description xml:id="echoid-description24" xml:space="preserve">13. Apr.</description> </figure> <figure xlink:label="fig-0281a-08" xlink:href="fig-0281a-08a"> <caption xml:id="echoid-caption99" style="it" xml:space="preserve">Fig. 8.</caption> <description xml:id="echoid-description25" xml:space="preserve">a *</description> <description xml:id="echoid-description26" xml:space="preserve">17. Apr.</description> </figure> <figure xlink:label="fig-0281a-09" xlink:href="fig-0281a-09a"> <caption xml:id="echoid-caption100" style="it" xml:space="preserve">Fig. 9.</caption> <description xml:id="echoid-description27" xml:space="preserve">*</description> <description xml:id="echoid-description28" xml:space="preserve">19. Apr.</description> </figure> <figure xlink:label="fig-0281a-10" xlink:href="fig-0281a-10a"> <caption xml:id="echoid-caption101" style="it" xml:space="preserve">Fig. 10.</caption> <description xml:id="echoid-description29" xml:space="preserve">*</description> <description xml:id="echoid-description30" xml:space="preserve">20. Apr.</description> </figure> <figure xlink:label="fig-0281a-11" xlink:href="fig-0281a-11a"> <caption xml:id="echoid-caption102" style="it" xml:space="preserve">Fig. 11.</caption> <description xml:id="echoid-description31" xml:space="preserve">*</description> <description xml:id="echoid-description32" xml:space="preserve">21. Apr.</description> </figure> <figure xlink:label="fig-0281a-12" xlink:href="fig-0281a-12a"> <caption xml:id="echoid-caption103" style="it" xml:space="preserve">Fig. 12.</caption> <description xml:id="echoid-description33" xml:space="preserve">*</description> <description xml:id="echoid-description34" xml:space="preserve">29. Apr.</description> </figure> <figure xlink:label="fig-0281a-13" xlink:href="fig-0281a-13a"> <caption xml:id="echoid-caption104" style="it" xml:space="preserve">Fig. 13.</caption> <description xml:id="echoid-description35" xml:space="preserve">*</description> <description xml:id="echoid-description36" xml:space="preserve">3. Maii.</description> </figure> <figure xlink:label="fig-0281a-14" xlink:href="fig-0281a-14a"> <caption xml:id="echoid-caption105" style="it" xml:space="preserve">Fig. 14.</caption> <description xml:id="echoid-description37" xml:space="preserve">*</description> <description xml:id="echoid-description38" xml:space="preserve">6. Maii.</description> </figure> <figure xlink:label="fig-0281a-15" xlink:href="fig-0281a-15a"> <caption xml:id="echoid-caption106" style="it" xml:space="preserve">Fig. 15.</caption> <description xml:id="echoid-description39" xml:space="preserve">*</description> <description xml:id="echoid-description40" xml:space="preserve">7. Maii.</description> </figure> <figure xlink:label="fig-0281a-16" xlink:href="fig-0281a-16a"> <caption xml:id="echoid-caption107" style="it" xml:space="preserve">Fig. 16.</caption> <description xml:id="echoid-description41" xml:space="preserve">*</description> <description xml:id="echoid-description42" xml:space="preserve">10. Maii.</description> </figure> <figure xlink:label="fig-0281a-17" xlink:href="fig-0281a-17a"> <caption xml:id="echoid-caption108" style="it" xml:space="preserve">Fig. 17.</caption> <description xml:id="echoid-description43" xml:space="preserve">*</description> <description xml:id="echoid-description44" xml:space="preserve">11. Maii.</description> </figure> <figure xlink:label="fig-0281a-18" xlink:href="fig-0281a-18a"> <caption xml:id="echoid-caption109" style="it" xml:space="preserve">Fig. 18.</caption> <description xml:id="echoid-description45" xml:space="preserve">*</description> <description xml:id="echoid-description46" xml:space="preserve">12. Maii.</description> </figure> <figure xlink:label="fig-0281a-19" xlink:href="fig-0281a-19a"> <caption xml:id="echoid-caption110" style="it" xml:space="preserve">Fig. 19.</caption> <description xml:id="echoid-description47" xml:space="preserve">*</description> <description xml:id="echoid-description48" xml:space="preserve">14. Maii.</description> </figure> <figure xlink:label="fig-0281a-20" xlink:href="fig-0281a-20a"> <caption xml:id="echoid-caption111" style="it" xml:space="preserve">Fig. 20.</caption> <description xml:id="echoid-description49" xml:space="preserve">*</description> <description xml:id="echoid-description50" xml:space="preserve">15. Maii.</description> </figure> <figure xlink:label="fig-0281a-21" xlink:href="fig-0281a-21a"> <caption xml:id="echoid-caption112" style="it" xml:space="preserve">Fig. 21.</caption> <description xml:id="echoid-description51" xml:space="preserve">*</description> <description xml:id="echoid-description52" xml:space="preserve">18. Maii.</description> </figure> <figure xlink:label="fig-0281a-22" xlink:href="fig-0281a-22a"> <caption xml:id="echoid-caption113" style="it" xml:space="preserve">Fig. 22.</caption> <description xml:id="echoid-description53" xml:space="preserve">*</description> <description xml:id="echoid-description54" xml:space="preserve">19. Maii.</description> </figure> <figure xlink:label="fig-0281a-23" xlink:href="fig-0281a-23a"> <caption xml:id="echoid-caption114" style="it" xml:space="preserve">Fig. 23.</caption> <description xml:id="echoid-description55" xml:space="preserve">*</description> <description xml:id="echoid-description56" xml:space="preserve">20. Maii.</description> </figure> <figure xlink:label="fig-0281a-24" xlink:href="fig-0281a-24a"> <caption xml:id="echoid-caption115" style="it" xml:space="preserve">Fig. 24.</caption> <description xml:id="echoid-description57" xml:space="preserve">* c</description> <description xml:id="echoid-description58" xml:space="preserve">a *</description> <description xml:id="echoid-description59" xml:space="preserve">27. Maii.</description> </figure> <figure xlink:label="fig-0281a-25" xlink:href="fig-0281a-25a"> <caption xml:id="echoid-caption116" style="it" xml:space="preserve">Fig. 25.</caption> <description xml:id="echoid-description60" xml:space="preserve">c *</description> <description xml:id="echoid-description61" xml:space="preserve">31. Maii.</description> <description xml:id="echoid-description62" xml:space="preserve">a *</description> </figure> <figure xlink:label="fig-0281a-26" xlink:href="fig-0281a-26a"> <caption xml:id="echoid-caption117" style="it" xml:space="preserve">Fig. 26.</caption> <description xml:id="echoid-description63" xml:space="preserve">*</description> <description xml:id="echoid-description64" xml:space="preserve">13. Iun.</description> </figure> <figure xlink:label="fig-0281a-27" xlink:href="fig-0281a-27a"> <caption xml:id="echoid-caption118" style="it" xml:space="preserve">Fig. 27.</caption> <description xml:id="echoid-description65" xml:space="preserve">*</description> <description xml:id="echoid-description66" xml:space="preserve">16. Ian. 1656.</description> </figure> <figure xlink:label="fig-0281a-28" xlink:href="fig-0281a-28a"> <caption xml:id="echoid-caption119" style="it" xml:space="preserve">Fig. 28.</caption> <description xml:id="echoid-description67" xml:space="preserve">*</description> <description xml:id="echoid-description68" xml:space="preserve">19. Febr.</description> </figure> <figure xlink:label="fig-0281a-29" xlink:href="fig-0281a-29a"> <caption xml:id="echoid-caption120" style="it" xml:space="preserve">Fig. 29.</caption> <description xml:id="echoid-description69" xml:space="preserve">*</description> <description xml:id="echoid-description70" xml:space="preserve">16. Mart.</description> </figure> <figure xlink:label="fig-0281a-30" xlink:href="fig-0281a-30a"> <caption xml:id="echoid-caption121" style="it" xml:space="preserve">Fig. 30.</caption> <description xml:id="echoid-description71" xml:space="preserve">*</description> <description xml:id="echoid-description72" xml:space="preserve">30. Mart.</description> </figure> <figure xlink:label="fig-0281a-31" xlink:href="fig-0281a-31a"> <caption xml:id="echoid-caption122" style="it" xml:space="preserve">Fig. 31.</caption> <description xml:id="echoid-description73" xml:space="preserve">*</description> <description xml:id="echoid-description74" xml:space="preserve">18. Apr.</description> </figure> <figure xlink:label="fig-0281a-32" xlink:href="fig-0281a-32a"> <caption xml:id="echoid-caption123" style="it" xml:space="preserve">Fig. 32.</caption> <description xml:id="echoid-description75" xml:space="preserve">*</description> <description xml:id="echoid-description76" xml:space="preserve">17. Iun.</description> </figure> <figure xlink:label="fig-0281a-33" xlink:href="fig-0281a-33a"> <caption xml:id="echoid-caption124" style="it" xml:space="preserve">Fig. 33.</caption> <description xml:id="echoid-description77" xml:space="preserve">*</description> <description xml:id="echoid-description78" xml:space="preserve">19. Oct.</description> </figure> <figure xlink:label="fig-0281a-34" xlink:href="fig-0281a-34a"> <caption xml:id="echoid-caption125" style="it" xml:space="preserve">Fig. 34.</caption> <description xml:id="echoid-description79" xml:space="preserve">*</description> <description xml:id="echoid-description80" xml:space="preserve">21. Oct.</description> </figure> <figure xlink:label="fig-0281a-35" xlink:href="fig-0281a-35a"> <caption xml:id="echoid-caption126" style="it" xml:space="preserve">Fig. 35.</caption> <description xml:id="echoid-description81" xml:space="preserve">*</description> <description xml:id="echoid-description82" xml:space="preserve">9. Nov.</description> </figure> <figure xlink:label="fig-0281a-36" xlink:href="fig-0281a-36a"> <caption xml:id="echoid-caption127" style="it" xml:space="preserve">Fig. 36.</caption> <description xml:id="echoid-description83" xml:space="preserve">*</description> <description xml:id="echoid-description84" xml:space="preserve">27. Nov.</description> </figure> <figure xlink:label="fig-0281a-37" xlink:href="fig-0281a-37a"> <caption xml:id="echoid-caption128" style="it" xml:space="preserve">Fig. 37.</caption> <description xml:id="echoid-description85" xml:space="preserve">*</description> <description xml:id="echoid-description86" xml:space="preserve">16. Dec.</description> </figure> <figure xlink:label="fig-0281a-38" xlink:href="fig-0281a-38a"> <caption xml:id="echoid-caption129" style="it" xml:space="preserve">Fig. 38.</caption> <description xml:id="echoid-description87" xml:space="preserve">*</description> <description xml:id="echoid-description88" xml:space="preserve">18. Ian. 1657.</description> </figure> <figure xlink:label="fig-0281a-39" xlink:href="fig-0281a-39a"> <caption xml:id="echoid-caption130" style="it" xml:space="preserve">Fig. 39.</caption> <description xml:id="echoid-description89" xml:space="preserve">*</description> <description xml:id="echoid-description90" xml:space="preserve">29. Mart.</description> </figure> <figure xlink:label="fig-0281a-40" xlink:href="fig-0281a-40a"> <caption xml:id="echoid-caption131" style="it" xml:space="preserve">Fig. 40.</caption> <description xml:id="echoid-description91" xml:space="preserve">*</description> <description xml:id="echoid-description92" xml:space="preserve">30. Mart.</description> </figure> <figure xlink:label="fig-0281a-41" xlink:href="fig-0281a-41a"> <caption xml:id="echoid-caption132" style="it" xml:space="preserve">Fig. 41.</caption> <description xml:id="echoid-description93" xml:space="preserve">*</description> <description xml:id="echoid-description94" xml:space="preserve">18. Maii.</description> </figure> <figure xlink:label="fig-0281a-42" xlink:href="fig-0281a-42a"> <caption xml:id="echoid-caption133" style="it" xml:space="preserve">Fig. 42.</caption> <description xml:id="echoid-description95" xml:space="preserve">*</description> <description xml:id="echoid-description96" xml:space="preserve">19. Maii.</description> </figure> <figure xlink:label="fig-0281a-43" xlink:href="fig-0281a-43a"> <caption xml:id="echoid-caption134" style="it" xml:space="preserve">Fig. 43.</caption> <description xml:id="echoid-description97" xml:space="preserve">*</description> <description xml:id="echoid-description98" xml:space="preserve">17. Dec.</description> </figure> <figure xlink:label="fig-0281a-44" xlink:href="fig-0281a-44a"> <caption xml:id="echoid-caption135" style="it" xml:space="preserve">Fig. 44.</caption> <description xml:id="echoid-description99" xml:space="preserve">*</description> <description xml:id="echoid-description100" xml:space="preserve">18. Dec.</description> </figure> <figure xlink:label="fig-0281a-45" xlink:href="fig-0281a-45a"> <caption xml:id="echoid-caption136" style="it" xml:space="preserve">Fig. 45.</caption> <description xml:id="echoid-description101" xml:space="preserve">*</description> <description xml:id="echoid-description102" xml:space="preserve">27. Dec.</description> </figure> <figure xlink:label="fig-0281a-46" xlink:href="fig-0281a-46a"> <caption xml:id="echoid-caption137" style="it" xml:space="preserve">Fig. 46.</caption> <description xml:id="echoid-description103" xml:space="preserve">*</description> <description xml:id="echoid-description104" xml:space="preserve">11. Mart 1658.</description> </figure> <figure xlink:label="fig-0281a-47" xlink:href="fig-0281a-47a"> <caption xml:id="echoid-caption138" style="it" xml:space="preserve">Fig. 47.</caption> <description xml:id="echoid-description105" xml:space="preserve">*</description> <description xml:id="echoid-description106" xml:space="preserve">16. Mart.</description> </figure> <figure xlink:label="fig-0281a-48" xlink:href="fig-0281a-48a"> <caption xml:id="echoid-caption139" style="it" xml:space="preserve">Fig. 48.</caption> <description xml:id="echoid-description107" xml:space="preserve">*</description> <description xml:id="echoid-description108" xml:space="preserve">23. Mart.</description> </figure> <figure xlink:label="fig-0281a-49" xlink:href="fig-0281a-49a"> <caption xml:id="echoid-caption140" style="it" xml:space="preserve">Fig. 49.</caption> <description xml:id="echoid-description109" xml:space="preserve">*</description> <description xml:id="echoid-description110" xml:space="preserve">3. Apr.</description> </figure> <figure xlink:label="fig-0281a-50" xlink:href="fig-0281a-50a"> <caption xml:id="echoid-caption141" style="it" xml:space="preserve">Fig. 50.</caption> <description xml:id="echoid-description111" xml:space="preserve">*</description> <description xml:id="echoid-description112" xml:space="preserve">10. Nov.</description> </figure> <figure xlink:label="fig-0281a-51" xlink:href="fig-0281a-51a"> <caption xml:id="echoid-caption142" style="it" xml:space="preserve">Fig. 51.</caption> <description xml:id="echoid-description113" xml:space="preserve">*</description> <description xml:id="echoid-description114" xml:space="preserve">16. Ian. 1659.</description> </figure> <figure xlink:label="fig-0281a-52" xlink:href="fig-0281a-52a"> <caption xml:id="echoid-caption143" style="it" xml:space="preserve">Fig. 52.</caption> <description xml:id="echoid-description115" xml:space="preserve">12. Febr.</description> <description xml:id="echoid-description116" xml:space="preserve">*</description> </figure> <figure xlink:label="fig-0281a-53" xlink:href="fig-0281a-53a"> <caption xml:id="echoid-caption144" style="it" xml:space="preserve">Fig. 53.</caption> <description xml:id="echoid-description117" xml:space="preserve">*</description> <description xml:id="echoid-description118" xml:space="preserve">24. Febr.</description> </figure> <figure xlink:label="fig-0281a-54" xlink:href="fig-0281a-54a"> <caption xml:id="echoid-caption145" style="it" xml:space="preserve">Fig. 54.</caption> <description xml:id="echoid-description119" xml:space="preserve">25. Febr.</description> <description xml:id="echoid-description120" xml:space="preserve">*</description> </figure> <figure xlink:label="fig-0281a-55" xlink:href="fig-0281a-55a"> <caption xml:id="echoid-caption146" style="it" xml:space="preserve">Fig. 55.</caption> <description xml:id="echoid-description121" xml:space="preserve">14. Mart.</description> <description xml:id="echoid-description122" xml:space="preserve">*</description> </figure> <figure xlink:label="fig-0281a-56" xlink:href="fig-0281a-56a"> <caption xml:id="echoid-caption147" style="it" xml:space="preserve">Fig. 56.</caption> <description xml:id="echoid-description123" xml:space="preserve">16. Mart.</description> <description xml:id="echoid-description124" xml:space="preserve">*</description> </figure> <figure xlink:label="fig-0281a-57" xlink:href="fig-0281a-57a"> <caption xml:id="echoid-caption148" style="it" xml:space="preserve">Fig. 57.</caption> <description xml:id="echoid-description125" xml:space="preserve">*</description> <description xml:id="echoid-description126" xml:space="preserve">21. Mart.</description> </figure> <figure xlink:label="fig-0281a-58" xlink:href="fig-0281a-58a"> <caption xml:id="echoid-caption149" style="it" xml:space="preserve">Fig. 58.</caption> <description xml:id="echoid-description127" xml:space="preserve">*</description> <description xml:id="echoid-description128" xml:space="preserve">22. Mart.</description> </figure> <figure xlink:label="fig-0281a-59" xlink:href="fig-0281a-59a"> <caption xml:id="echoid-caption150" style="it" xml:space="preserve">Fig. 59.</caption> <description xml:id="echoid-description129" xml:space="preserve">26. Mart.</description> <description xml:id="echoid-description130" xml:space="preserve">*</description> </figure> </div> <p> <s xml:id="echoid-s5992" xml:space="preserve">Deinde medium motum diurnum à ſole (qui minor eſt <lb/>motu quo reſpectu fixarum comes progreditur, ut in noſtra <lb/>quoque Luna evenire novimus) ita reperio; </s> <s xml:id="echoid-s5993" xml:space="preserve">Saturni motum <lb/>medium diurnum qui 2 eſt minutorum, aufero ab invento <lb/>motu diurno reſpectu fixarum gr. </s> <s xml:id="echoid-s5994" xml:space="preserve">22, 34′, 44″. </s> <s xml:id="echoid-s5995" xml:space="preserve">unde ſuper-<lb/>ſunt gr. </s> <s xml:id="echoid-s5996" xml:space="preserve">22, 32′, 44″, diurnus motus à Sole. </s> <s xml:id="echoid-s5997" xml:space="preserve">Atque hinc <lb/> <anchor type="note" xlink:label="note-0283-01a" xlink:href="note-0283-01"/> facile quoque menſis Synodici Saturnicolarum media longi-<lb/>tudo computatur. </s> <s xml:id="echoid-s5998" xml:space="preserve">Nempe ſi gr. </s> <s xml:id="echoid-s5999" xml:space="preserve">22, 32′, 44″, uno die per-<lb/>curruntur, quot igitur diebus gr. </s> <s xml:id="echoid-s6000" xml:space="preserve">360? </s> <s xml:id="echoid-s6001" xml:space="preserve">Fiuntque dies 15, <lb/>horæ 23, ſcr. </s> <s xml:id="echoid-s6002" xml:space="preserve">13. </s> <s xml:id="echoid-s6003" xml:space="preserve">Reliquis duntaxat 47 ſcrupulis ad dies 16. <lb/></s> <s xml:id="echoid-s6004" xml:space="preserve">Atque illud etiam medium tempus eſt quo noſtri reſpectu ad <lb/>apogæum ſuum Saturni comes revertitur, ſive intra quod <lb/>cum Saturno bis conjungitur.</s> <s xml:id="echoid-s6005" xml:space="preserve"/> </p> <div xml:id="echoid-div305" type="float" level="2" n="27"> <note position="right" xlink:label="note-0283-01" xlink:href="note-0283-01a" xml:space="preserve">Menſis Sa-<lb/>turnicola-<lb/>rum vera <lb/>longitude.</note> </div> <p> <s xml:id="echoid-s6006" xml:space="preserve">Cæterum quia ex motu comitis illud præcipue inveſtigari <lb/>meretur, quo pacto ad datum quodvis tempus ſitus ejus a-<lb/>pud Saturnum exhiberi poſſit; </s> <s xml:id="echoid-s6007" xml:space="preserve">oſtendemus nunc breviſſimam <lb/>ad hoc calculi rationem, ſequentium tabularum ope abſol-<lb/>vendam; </s> <s xml:id="echoid-s6008" xml:space="preserve">in quibus motum comitis æqualem qualis ex Satur-<lb/>no fixarum reſpectu appareret, proponimus. </s> <s xml:id="echoid-s6009" xml:space="preserve">Hunc autem <lb/>diurnum adſumſimus gr. </s> <s xml:id="echoid-s6010" xml:space="preserve">22, 34′, 44″, ſicut modo inventus <lb/>fuit. </s> <s xml:id="echoid-s6011" xml:space="preserve">Etſi enim expertus ſum, iterato eodem examine adhi-<lb/>bitiſque aliis duobus temporibus quibus comes fuit apogæus <lb/>aut perigæus, aliquot ſecundis ſcrupulis majorem interdum <lb/>minoremve eundem motum reperiri, iſtum tamen medium quo-<lb/>dammodo inter alios retineri poſſe ratus ſum, poſt plures ab-<lb/>hinc elapſos annos facile emendandum. </s> <s xml:id="echoid-s6012" xml:space="preserve">Nam & </s> <s xml:id="echoid-s6013" xml:space="preserve">inæqualitas <lb/>puto aliqua, & </s> <s xml:id="echoid-s6014" xml:space="preserve">eccentricitas, quemadmodum in Luna noſtra, <lb/>ita circa hanc quoque, diligenti obſervatione olim deprehen-<lb/>di poterit.</s> <s xml:id="echoid-s6015" xml:space="preserve"/> </p> <pb o="552" file="0284" n="299" rhead="CHRIST. HUGENII"/> </div> <div xml:id="echoid-div307" type="section" level="1" n="141"> <head xml:id="echoid-head191" style="it" xml:space="preserve">Tabul@ motus æqualis Lunæ Saturniæ in orbita ſua reſpectu fixarum.</head> <note position="right" xml:space="preserve"> <lb/>## In Annis Fulianis. <lb/># Gr. # Min. <lb/>1 # 321 # 18 <lb/>2 # 282 # 35 <lb/>3 # 243 # 53 <lb/>4 # 227 # 45 <lb/>5 # 189 # 3 <lb/>6 # 150 # 21 <lb/>7 # 111 # 38 <lb/>8 # 95 # 31 <lb/>9 # 56 # 48 <lb/>10 # 18 # 6 <lb/>11 # 339 # 24 <lb/>12 # 323 # 16 <lb/>13 # 284 # 34 <lb/>14 # 245 # 51 <lb/>15 # 207 # 9 <lb/>16 # 191 # 2 <lb/>17 # 152 # 19 <lb/>18 # 113 # 37 <lb/>19 # 74 # 55 <lb/>20 # 58 # 47 <lb/></note> </div> <div xml:id="echoid-div308" type="section" level="1" n="142"> <head xml:id="echoid-head192" xml:space="preserve">In Menſibus anni @uli@-<lb/>ni ineuntibus.</head> <note position="right" xml:space="preserve"> <lb/>Janu. # 0 # 0 <lb/>Febr. # 339 # 57 <lb/>Mart. # 252 # 9 <lb/>April # 232 # 6 <lb/>Maji # 189 # 28 <lb/>Junii # 169 # 25 <lb/>Julii # 126 # 47 <lb/>Aug. # 106 # 43 <lb/>Sept. # 86 # 40 <lb/>Octo. # 44 # 2 <lb/>Nov. # 23 # 59 <lb/>Dec. # 341 # 21 <lb/></note> <p style="it"> <s xml:id="echoid-s6016" xml:space="preserve">Si fuerit datus annus interc@la-<lb/>ris, poſt Februarium @@us dies, <lb/>eoque & </s> <s xml:id="echoid-s6017" xml:space="preserve">unius diei m@tus ad@ <lb/>dendus eſt.</s> <s xml:id="echoid-s6018" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>## In Diebus. <lb/># Gr. # Min. <lb/>1 # 22 # 35 <lb/>2 # 45 # 9 <lb/>3 # 67 # 44 <lb/>4 # 90 # 19 <lb/>5 # 112 # 54 <lb/>6 # 135 # 28 <lb/>7 # 158 # 3 <lb/>8 # 180 # 38 <lb/>9 # 203 # 13 <lb/>10 # 225 # 47 <lb/>11 # 248 # 22 <lb/>12 # 270 # 57 <lb/>13 # 293 # 32 <lb/>14 # 316 # 6 <lb/>15 # 338 # 41 <lb/>16 # 1 # 16 <lb/>17 # 23 # 50 <lb/>18 # 46 # 25 <lb/>19 # 69 # 0 <lb/>20 # 91 # 35 <lb/>21 # 114 # 9 <lb/>22 # 136 # 4 <lb/>23 # 159 # 19 <lb/>24 # 181 # 54 <lb/>25 # 204 # 28 <lb/>26 # 227 # 3 <lb/>27 # 249 # 38 <lb/>28 # 272 # 13 <lb/>29 # 294 # 47 <lb/>30 # 317 # 22 <lb/></note> <note position="right" xml:space="preserve"> <lb/>## In Horis. <lb/># Gr. Mi. <lb/>1 # 0 # 56 <lb/>2 # 1 # 53 <lb/>3 # 2 # 49 <lb/>4 # 3 # 46 <lb/>5 # 4 # 42 <lb/>6 # 5 # 39 <lb/>7 # 6 # 35 <lb/>8 # 7 # 32 <lb/>9 # 8 # 28 <lb/>10 # 9 # 24 <lb/>11 # 10 # 21 <lb/>12 # 11 # 17 <lb/>13 # 12 # 14 <lb/>14 # 13 # 10 <lb/>15 # 14 # 7 <lb/>16 # 15 # 3 <lb/>17 # 16 # 0 <lb/>18 # 16 # 56 <lb/>19 # 17 # 52 <lb/>20 # 18 # 49 <lb/>21 # 19 # 45 <lb/>22 # 20 # 42 <lb/>23 # 21 # 38 <lb/>24 # 22 # 35 <lb/></note> <p style="it"> <s xml:id="echoid-s6019" xml:space="preserve">Primo @an. </s> <s xml:id="echoid-s6020" xml:space="preserve">anno 1653. <lb/></s> <s xml:id="echoid-s6021" xml:space="preserve">meridie, diſtantia Lu-<lb/>næ Saturniæ ab apogæo <lb/>erat gr. </s> <s xml:id="echoid-s6022" xml:space="preserve">274. </s> <s xml:id="echoid-s6023" xml:space="preserve">21′. </s> <s xml:id="echoid-s6024" xml:space="preserve">Satur-<lb/>ni locus apparens, eode@ <lb/>tempore in ♌ gr. </s> <s xml:id="echoid-s6025" xml:space="preserve">11. </s> <s xml:id="echoid-s6026" xml:space="preserve"><lb/>41.</s> <s xml:id="echoid-s6027" xml:space="preserve"/> </p> <pb o="553" file="0285" n="300" rhead="SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s6028" xml:space="preserve">Harum tabularum auxilio primùm Epocham, quæ <lb/>præcederet omnes obſervationes noſtras, conſtitui diem <lb/>1 Jan. </s> <s xml:id="echoid-s6029" xml:space="preserve">meridie, anno 1653. </s> <s xml:id="echoid-s6030" xml:space="preserve">Nempe ex eo, quod 14 <lb/>Martii, 1659, hora 8 pom. </s> <s xml:id="echoid-s6031" xml:space="preserve">in perigæo Saturni Luna <lb/>verſaretur, collegi retrorſum numerando, ad prædictam <lb/>diem 1 Jan. </s> <s xml:id="echoid-s6032" xml:space="preserve">1653, abfuiſſe illam ab apogæi loco gr. <lb/></s> <s xml:id="echoid-s6033" xml:space="preserve"> <anchor type="note" xlink:label="note-0285-01a" xlink:href="note-0285-01"/> 274, 21′. </s> <s xml:id="echoid-s6034" xml:space="preserve">Jam vero ut ad quodlibet datum tempus, e-<lb/>pocha poſterius, inveniatur ejus ab apogæo diſtantia, <lb/>(namque hinc facile deinde perſpicere eſt quam pro-<lb/>pinqua Saturno apparitura ſit) addendus eſt ad mo-<lb/>tum Epochæ, motus comitis uſque ad tempus datum, <lb/>ex tabulis collectus; </s> <s xml:id="echoid-s6035" xml:space="preserve">hinc auferendus Saturni motus <lb/>apparens per idem temporis intervallum, qui ex Ephe-<lb/>meridibus cognoſcitur; </s> <s xml:id="echoid-s6036" xml:space="preserve">reliquum erit diſtantia comitis <lb/>ab apogæo quæſita. </s> <s xml:id="echoid-s6037" xml:space="preserve">Ut ſi datum fuerit tempus dies <lb/>25 Martii, anno 1655, hora 8 pom. </s> <s xml:id="echoid-s6038" xml:space="preserve">quæ prima no-<lb/>ſtra fuit obſervatio: </s> <s xml:id="echoid-s6039" xml:space="preserve">Motus Epochæ, ſive diſtantia Lu-<lb/>næ Saturniæ ab apogæo tempore Epochæ, quæ eſt <lb/>gr. </s> <s xml:id="echoid-s6040" xml:space="preserve">274, 21′, addita ad motum ejuſdem Lunæ inde ab <lb/>Epocha ad tempus datum, colligit gr. </s> <s xml:id="echoid-s6041" xml:space="preserve">278, 31′; </s> <s xml:id="echoid-s6042" xml:space="preserve">Hinc <lb/>ablato Saturni motu apparente per idem tempus, qui <lb/>invenitur gr. </s> <s xml:id="echoid-s6043" xml:space="preserve">22, 58′, relinquuntur gr. </s> <s xml:id="echoid-s6044" xml:space="preserve">255, 33′, diſtan-<lb/>tia Lunæ Saturni ab apogæo, quæ invenienda erat. <lb/></s> <s xml:id="echoid-s6045" xml:space="preserve">Unde apparet in circulo (Tab. </s> <s xml:id="echoid-s6046" xml:space="preserve">LI. </s> <s xml:id="echoid-s6047" xml:space="preserve">fig. </s> <s xml:id="echoid-s6048" xml:space="preserve">3.)</s> <s xml:id="echoid-s6049" xml:space="preserve">, ubi A apogæi <lb/>locum referebat, fere ad numerum 12 illam conſtitiſſe, <lb/>eoque in maxima propemodum diſtantia viſam, uti re-<lb/>vera contigit. </s> <s xml:id="echoid-s6050" xml:space="preserve">Calculum autem ſubjicimus qui ſic or-<lb/>dinatur.</s> <s xml:id="echoid-s6051" xml:space="preserve"/> </p> <div xml:id="echoid-div308" type="float" level="2" n="1"> <note position="right" xlink:label="note-0285-01" xlink:href="note-0285-01a" xml:space="preserve">Locus Lun@ <lb/>Saturniæ <lb/>quomod@ <lb/>ſupputetur@.</note> </div> <pb o="554" file="0286" n="301" rhead="CHRIST. HUGENII"/> <note position="right" xml:space="preserve"> <lb/>## Gra. # Mi. # # G. # M. <lb/>Motus # Epochæ, # 274 # 21 # b # Locus app. 25 <lb/># Anni @ # 282 # 35 # Mart. 1655, # 4 # 39 # ♍ <lb/># Martii 1 # 252 # 9 # b # Locusapp. tem- \\ pore epochæ, # 11 # 41 # Ω <lb/># Dies 24 # 181 # 54 <lb/># Horæ 8 # 7 # 32 # Reſtat b motus \\ apparens, # 22 # 58 <lb/># Summa # 278 # 31 <lb/>Motus Saturni apparens # 22 # 58 <lb/># Reliquum # 255 # 33 ##### diſtantia Lunæ Saturni \\ ab apogæo. <lb/></note> <p> <s xml:id="echoid-s6052" xml:space="preserve">Ut verò conſtet methodi ratio, intelligatur in ſuperiori <lb/> <anchor type="note" xlink:label="note-0286-02a" xlink:href="note-0286-02"/> diagrammate, Saturnus die 1 Jan. </s> <s xml:id="echoid-s6053" xml:space="preserve">1653 poſitus fuiſſe in A, <lb/>terra in G, Luna Saturni in E, gradibus 274, 21′ ab apo-<lb/>gæo L. </s> <s xml:id="echoid-s6054" xml:space="preserve">Rurſumque 25 Martii 1655, Saturnus ponatur in <lb/>B, terra in F, Luna Saturni in M: </s> <s xml:id="echoid-s6055" xml:space="preserve">ſitque B N parallela <lb/>A L. </s> <s xml:id="echoid-s6056" xml:space="preserve">Quoniam igitur Lunæ motus periodicus inter bina <lb/>illa tempora, additus motui Epochæ, hoc eſt, arcui L D E <lb/>graduum 274, 21′, efficit gr. </s> <s xml:id="echoid-s6057" xml:space="preserve">278, 31′; </s> <s xml:id="echoid-s6058" xml:space="preserve">erit idcirco arcus <lb/>N K M hoc graduum numero. </s> <s xml:id="echoid-s6059" xml:space="preserve">Diſtantia autem Lunæ Sa-<lb/>turniæ ab apogæo eſt arcus O K M, qui ut cognoſcatur, <lb/>auferendus eſt ab arcu N K M arcus N O. </s> <s xml:id="echoid-s6060" xml:space="preserve">Ergo cum arcus <lb/>N O totidem ſit graduum atque angulus quem conſtituunt <lb/>rectæ O F, L G; </s> <s xml:id="echoid-s6061" xml:space="preserve">hujus autem quantitas definiatur apparen-<lb/>ti Saturni motu inter duo prædicta tempora; </s> <s xml:id="echoid-s6062" xml:space="preserve">apparet nos <lb/>rectè ab inventis gr. </s> <s xml:id="echoid-s6063" xml:space="preserve">278, 31′, hoc eſt ab arcu N K M, ſub-<lb/>traxiſſe Saturni motum apparentem gr. </s> <s xml:id="echoid-s6064" xml:space="preserve">22, 58′ (quippe <lb/>æqualem arcui N O) ad conſequendum arcum O K M, gr. <lb/></s> <s xml:id="echoid-s6065" xml:space="preserve">255, 33′, diſtantiam nempe Lunæ Saturni ab apogæo.</s> <s xml:id="echoid-s6066" xml:space="preserve"/> </p> <div xml:id="echoid-div309" type="float" level="2" n="2"> <note position="left" xlink:label="note-0286-02" xlink:href="note-0286-02a" xml:space="preserve">Ejus calculi <lb/>comprobatio. <lb/>TAB. XLIX. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s6067" xml:space="preserve">Nondum hic locus eſt explicandi alium quendam hujus <lb/>lunulæ motum in latitudinem, obſervationibus aliquot præ-<lb/>cedentibus jam ſeſe prodere incipientem; </s> <s xml:id="echoid-s6068" xml:space="preserve">quo nempe ab li-<lb/>nea recta per anſarum extrema tranſeunte plerumque exorbi-<lb/>tat, apparetque circa Saturnum ellipſin percurerre, inter-<lb/>dum quidem ſatis latam, alias verò anguſtiorem, & </s> <s xml:id="echoid-s6069" xml:space="preserve">rotun- <pb o="555" file="0287" n="302" rhead="SYSTEMA SATURNIUM."/> do Saturno lucente, in rectam lineam abeuntem. </s> <s xml:id="echoid-s6070" xml:space="preserve">Etſi e-<lb/>nim, ad verum lunulæ ſitum reſpectu Saturni determinan-<lb/>dum, hujus quoque motus ratio eſt habenda, cum tamen <lb/>cauſæ ejus cum cæterorum phænomenôn cauſis prorſus con-<lb/>junctæ ſint, ſimul cum illis atque una opera exponendum <lb/>cenſeo.</s> <s xml:id="echoid-s6071" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6072" xml:space="preserve">Illam igitur Syſtematis hujus partem alteram nunc aggre-<lb/>dior, in qua formæ Saturni inſtabilis atque à ſe ipſa diſcre-<lb/>pantis ratio reddenda, tum qua periodo ſingulæ mutationes <lb/>contingant dicendum eſt. </s> <s xml:id="echoid-s6073" xml:space="preserve">Harum aliquas quæ nobis ſeſe <lb/>obtulere ſupra exhibui; </s> <s xml:id="echoid-s6074" xml:space="preserve">ſed eæ partem tantummodo periodi <lb/>complectuntur. </s> <s xml:id="echoid-s6075" xml:space="preserve">Ideoque ut omnem phænomenôn diverſi-<lb/>tatem ab iis quas dicemus cauſis pendere conſtet, aliorum <lb/>quoque temporum obſervationes examinare neceſſe erit, <lb/>quas à 40 atque amplius annis complures ſcriptis prodide-<lb/>runt. </s> <s xml:id="echoid-s6076" xml:space="preserve">At verò cum Saturni figuras omnes quas nobis deli-<lb/>nearunt oculis luſtro, eas multiplices adeo ac prodigioſas <lb/>invenio, ut ſi qua hypotheſis comminiſcenda foret quâ cun-<lb/>ctis fieret ſatis, nemo non, ut opinor, in ea excogitanda <lb/>operam luſurus ſit: </s> <s xml:id="echoid-s6077" xml:space="preserve">cum nulla tam crebræ atque enormis<unsure/> <lb/>transformationis cauſa eſſe poſſit, niſi ponatur ipſam Satur-<lb/>nii corporis molem identidem aliam atque aliam faciem in-<lb/>duere, quod ab omni veriſimilitudine eſt alienum. </s> <s xml:id="echoid-s6078" xml:space="preserve">Quam-<lb/>obrem delectus adhibendus eſt obſervatis illorum, & </s> <s xml:id="echoid-s6079" xml:space="preserve">inqui-<lb/>rendum quænam ex iis fidem mereantur, quæve contra ut <lb/>ſuſpecta rejicienda ſint. </s> <s xml:id="echoid-s6080" xml:space="preserve">Quo in examine illud concedi no-<lb/>bis poſtulamus, ut quoniam Saturni comitem primi perſpi-<lb/>cillis noſtris deteximus, ac quoties libet clarè intuemur, præ-<lb/>ferantur propterea noſtra hæc illis quibus alii ad ſtellam ean-<lb/>dem nequaquam pertingere potuere, licet quotidie Saturno <lb/>obſervando intenti: </s> <s xml:id="echoid-s6081" xml:space="preserve">eoque noſtræ etiam circa formam pla-<lb/>netæ obſervationes veriores habeantur, quoties eodem tem-<lb/>pore nobis atque illis diverſæ phaſes animadverſæ fuerint. <lb/></s> <s xml:id="echoid-s6082" xml:space="preserve">Adjuncta itaque tabella omnes eas exhibet quas ex variis au-<lb/> <anchor type="note" xlink:label="note-0287-01a" xlink:href="note-0287-01"/> toribus deſcripſimus.</s> <s xml:id="echoid-s6083" xml:space="preserve"/> </p> <div xml:id="echoid-div310" type="float" level="2" n="3"> <note position="right" xlink:label="note-0287-01" xlink:href="note-0287-01a" xml:space="preserve">TAB. LII.</note> </div> <p> <s xml:id="echoid-s6084" xml:space="preserve">Ac prima quidem harum formarum eſt quam Galilæus ad-<lb/> <anchor type="note" xlink:label="note-0287-02a" xlink:href="note-0287-02"/> <pb o="556" file="0288" n="303" rhead="CHRIST. HUGENII"/> notavit anno 1610, in qua triceps Saturnus ſpectatur, mi-<lb/> <anchor type="note" xlink:label="note-0288-01a" xlink:href="note-0288-01"/> noribus duobus orbiculis majori utrinque adjacentibus. <lb/></s> <s xml:id="echoid-s6085" xml:space="preserve">Hanc alii quoque permulti viderunt, aut certè ſe vidiſſe cre-<lb/>diderunt. </s> <s xml:id="echoid-s6086" xml:space="preserve">Nam ſi grandiores tubos adhibuiſſent atque o-<lb/>ptimis vitris inſtructos, haud dubiè pro triplici hac globulo-<lb/>rum facie eadem ſeſe illis obtuliſſet quam nobis diximus an-<lb/>no 1655, ac rurſus anno inſequenti, die 13 Oct. </s> <s xml:id="echoid-s6087" xml:space="preserve">viſam. </s> <s xml:id="echoid-s6088" xml:space="preserve">Hoc <lb/>enim inde colligimus quod dum illis bini ad latera globuli <lb/>apparent, nobis porrecta in longitudinem brachia tubi no-<lb/>ſtri referunt; </s> <s xml:id="echoid-s6089" xml:space="preserve">uti contigit eo ipſo anno 1655, menſe Aprili <lb/>ac Majo, quo tempore triſphærica illa figura Ricciolo He-<lb/>velioque obſervata fuit. </s> <s xml:id="echoid-s6090" xml:space="preserve">Etenim quo certius conſtet ob per-<lb/>ſpicillorum parvitatem talem hanc cerni, experimur nobis <lb/>quoque, quoties breviori perſpicillo, quinûm puta aut ſe-<lb/>nûm pedum, Saturnum aſpicimus, binos globulos dictorum <lb/>loco brachiorum apparere; </s> <s xml:id="echoid-s6091" xml:space="preserve">etiam illâ anni 1658 jam exiſten-<lb/>te phaſi.</s> <s xml:id="echoid-s6092" xml:space="preserve"/> </p> <div xml:id="echoid-div311" type="float" level="2" n="4"> <note position="right" xlink:label="note-0287-02" xlink:href="note-0287-02a" xml:space="preserve">Ha<unsure/> ſingalæ <lb/>examina@ <lb/>tur.</note> <note position="left" xlink:label="note-0288-01" xlink:href="note-0288-01a" xml:space="preserve">TAB. LII. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s6093" xml:space="preserve">Quænam autem fallacis imaginis cauſa ſit facile perſpici-<lb/>tur. </s> <s xml:id="echoid-s6094" xml:space="preserve">Quippe enim quum circa extremas cuſpides amplio-<lb/>rem lucem brachia hæc ſive alæ emittant, quam qua parte <lb/>medio Saturni diſco adhærent, ubi ſemper umbræ aliquid <lb/>intercedit, non modo cum manifeſto jam bifida evaſerunt, <lb/>velut anno 1658, ſed antea quoque ut anno 1657; </s> <s xml:id="echoid-s6095" xml:space="preserve">non mi-<lb/>rum eſt lucem illam intenſiorem, debiliori interjecta, peni-<lb/>tus à medio orbe ſeparatam videri. </s> <s xml:id="echoid-s6096" xml:space="preserve">Neque item rotundam <lb/>ex oblonga fieri mirandum eſt, cum idem accidat omni fi-<lb/>guræ eminus, nec ſatis diſtinctè ob exilitatem, perceptæ, <lb/>atque eo magis quo fuerit lucidior. </s> <s xml:id="echoid-s6097" xml:space="preserve">Itaque plane conſtat te-<lb/>leſcopiorum culpa phaſin hanc vitiatam eſſe, licet obſerva-<lb/>tores bona fide eam tradiderint.</s> <s xml:id="echoid-s6098" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6099" xml:space="preserve">Quæ ſequitur hanc nihilo melius ſe habet, à Scheinero <lb/> <anchor type="note" xlink:label="note-0288-02a" xlink:href="note-0288-02"/> obſervata anno 1614. </s> <s xml:id="echoid-s6100" xml:space="preserve">Atque equidem dubito, perfectiori-<lb/>buſne an deterioribus perſpicillis, quam qui præcedentem <lb/>dederunt, hanc deprehenderit. </s> <s xml:id="echoid-s6101" xml:space="preserve">Quoniam hactenus quidem <lb/>ad veram magis accedit, quod affixas Saturno auriculas ex-<lb/>hibet; </s> <s xml:id="echoid-s6102" xml:space="preserve">at contra in eo aberrat, quod plus juſto earum con- <pb o="557" file="0289" n="304" rhead="SYSTEMA SATURNIUM."/> trahat longitudinem. </s> <s xml:id="echoid-s6103" xml:space="preserve">Hæc autem phaſis tum ſibi ipſi tum <lb/>ſuperiori fidem derogat, alterutram certe haud veram fuiſſe <lb/>arguit; </s> <s xml:id="echoid-s6104" xml:space="preserve">quoniam eodem tempore, anno nimirum 1614, al-<lb/>tera à Scheinero, altera à Galilæo aliisque obſervata perhibe-<lb/>tur. </s> <s xml:id="echoid-s6105" xml:space="preserve">Ut proinde dubitandum non ſit, quin & </s> <s xml:id="echoid-s6106" xml:space="preserve">hæc ſimilis ex-<lb/>titerit ei quam nos anno 1657 vel 1658 in commentaria re-<lb/>tulimus. </s> <s xml:id="echoid-s6107" xml:space="preserve">Neque aliud de tertia hujus tabellæ exiſtimandum <lb/> <anchor type="note" xlink:label="note-0289-01a" xlink:href="note-0289-01"/> eſt, quæ Ricciolo ſecunda ponitur, quamque anno 1640 & </s> <s xml:id="echoid-s6108" xml:space="preserve"><lb/>1643 obſervatam ſcribit. </s> <s xml:id="echoid-s6109" xml:space="preserve">Veritati tamen propior hæc vide-<lb/>ri poteſt, quod pro orbiculis oblongas atque olivæ ſimiles <lb/>figuras habeat.</s> <s xml:id="echoid-s6110" xml:space="preserve"/> </p> <div xml:id="echoid-div312" type="float" level="2" n="5"> <note position="left" xlink:label="note-0288-02" xlink:href="note-0288-02a" xml:space="preserve">TAB. LII. <lb/>fig. 2.</note> <note position="right" xlink:label="note-0289-01" xlink:href="note-0289-01a" xml:space="preserve">TAB. XLII. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s6111" xml:space="preserve">Quarta eſt quam in locum triſphæricæ formæ Hevelius ad-<lb/> <anchor type="note" xlink:label="note-0289-02a" xlink:href="note-0289-02"/> ſumſit in libro de Saturni nativa facie; </s> <s xml:id="echoid-s6112" xml:space="preserve">ubi ſecundum leges <lb/>hypotheſeos ſuæ aſſerit talem quandoque cerni debere, quæ <lb/>tamen ex viſus hallucinatione in triſphæricam degeneret. <lb/></s> <s xml:id="echoid-s6113" xml:space="preserve">Quanquam mihi ne iſtiuſmodi quidem forma ſatis convenire <lb/>hypotheſi illius videatur, ut poſtea oſtendemus.</s> <s xml:id="echoid-s6114" xml:space="preserve"/> </p> <div xml:id="echoid-div313" type="float" level="2" n="6"> <note position="right" xlink:label="note-0289-02" xlink:href="note-0289-02a" xml:space="preserve">fig. 4.</note> </div> <p> <s xml:id="echoid-s6115" xml:space="preserve">Ejuſdem Hevelii etiam quinta eſt, cui ſimilem Gaſſendus <lb/> <anchor type="note" xlink:label="note-0289-03a" xlink:href="note-0289-03"/> edidit. </s> <s xml:id="echoid-s6116" xml:space="preserve">Et hæc quidem ſatis prope cum noſtra illa anni 1658 <lb/>conſentit, niſi quod partes brachiorum tenuiores, qua me-<lb/>dio diſco adnectuntur utriuſque tubi non ſint aſſecuti.</s> <s xml:id="echoid-s6117" xml:space="preserve"/> </p> <div xml:id="echoid-div314" type="float" level="2" n="7"> <note position="right" xlink:label="note-0289-03" xlink:href="note-0289-03a" xml:space="preserve">fig. 5.</note> </div> <p> <s xml:id="echoid-s6118" xml:space="preserve">Idemque in 6 & </s> <s xml:id="echoid-s6119" xml:space="preserve">7 contigit, quæ ambæ ab Hevelio quo-<lb/> <anchor type="note" xlink:label="note-0289-04a" xlink:href="note-0289-04"/> que traduntur, ſimiles iis quas circa eadem tempora Riccio-<lb/>lus obſervavit, nimirum anno 1648, 1649 & </s> <s xml:id="echoid-s6120" xml:space="preserve">1650, quæ hic <lb/>octavo nonoque loco exhibentur. </s> <s xml:id="echoid-s6121" xml:space="preserve">Neque alia re differunt, <lb/> <anchor type="note" xlink:label="note-0289-05a" xlink:href="note-0289-05"/> quàm quod medius orbis Hevelio nonnihil oblongus appa-<lb/>ruit, cum Ricciolo rotundus fuerit: </s> <s xml:id="echoid-s6122" xml:space="preserve">quodque hic connexas <lb/>anſas cum inter ſe tum Saturno ipſi ſpectaverit, quæ Heve-<lb/>lio pauxillo à contactu abeſſe viſæ ſint. </s> <s xml:id="echoid-s6123" xml:space="preserve">Verum ipſe Heve-<lb/>lius cauſam cur ſeparatæ videantur viſus imbecillitati adſcri-<lb/>bit, cum alioqui reipſa cohærere eas Saturno ſtatuat.</s> <s xml:id="echoid-s6124" xml:space="preserve"/> </p> <div xml:id="echoid-div315" type="float" level="2" n="8"> <note position="right" xlink:label="note-0289-04" xlink:href="note-0289-04a" xml:space="preserve">fig. 6. 7.</note> <note position="right" xlink:label="note-0289-05" xlink:href="note-0289-05a" xml:space="preserve">fig. 8. 9.</note> </div> <p> <s xml:id="echoid-s6125" xml:space="preserve">Tales autem & </s> <s xml:id="echoid-s6126" xml:space="preserve">Euſtachius de Divinis notavit anno 1646, <lb/> <anchor type="note" xlink:label="note-0289-06a" xlink:href="note-0289-06"/> 1647 & </s> <s xml:id="echoid-s6127" xml:space="preserve">1648, à quo editum ſchema ad num. </s> <s xml:id="echoid-s6128" xml:space="preserve">10 hic exhi-<lb/>buimus. </s> <s xml:id="echoid-s6129" xml:space="preserve">lsque cum præſtantiſſimus perſpicillorum artifex <lb/>habeatur, credibile eſt omnium emendatiſſimè nativam Sa-<lb/>turni faciem nobis deſcripſiſſe, niſi quod umbras illas quæ in <pb o="558" file="0290" n="305" rhead="CHRIST. HUGENII"/> ſchemate apparent, de ſuo, ut opinor, adjecit.</s> <s xml:id="echoid-s6130" xml:space="preserve"/> </p> <div xml:id="echoid-div316" type="float" level="2" n="9"> <note position="right" xlink:label="note-0289-06" xlink:href="note-0289-06a" xml:space="preserve">fig. 10.</note> </div> <p> <s xml:id="echoid-s6131" xml:space="preserve">Porro ab hiſce figuris non multum recedit ea quæ à Fr. <lb/></s> <s xml:id="echoid-s6132" xml:space="preserve"> <anchor type="note" xlink:label="note-0290-01a" xlink:href="note-0290-01"/> Fontana vulgata fuit, undecima tabellæ noſtræ. </s> <s xml:id="echoid-s6133" xml:space="preserve">Quam qui-<lb/>dem & </s> <s xml:id="echoid-s6134" xml:space="preserve">Ricciolus anno 1646 ſibi viſam ſcribit. </s> <s xml:id="echoid-s6135" xml:space="preserve">Sed minus <lb/>bonis teleſcopiis tunc uſum crediderim, quam quibus modo <lb/>dictas octavam nonamque detexit. </s> <s xml:id="echoid-s6136" xml:space="preserve">Siquidem eodem anno <lb/>1646 ſeptimam formam ſe vidiſſe teſtatur Hevelius, cui po-<lb/>tius hic ſtandum eſt. </s> <s xml:id="echoid-s6137" xml:space="preserve">Nam Fontanæ obſervationes quomi-<lb/>nus in dubium vocare verear facit, quod etiam olim magis <lb/>monſtroſas formas Martis publicavit, veluti trilateræ cujuſ-<lb/>dam rupis, ac rurſus aliter cum nigra in medio orbe macula; <lb/></s> <s xml:id="echoid-s6138" xml:space="preserve">quæ nos cum aliis multis fabuloſa comperimus. </s> <s xml:id="echoid-s6139" xml:space="preserve">Hæc ta-<lb/>men, quam in Saturno prodidit formæ diverſitatem, neque <lb/>magna eſt, ut dixi, neque miranda.</s> <s xml:id="echoid-s6140" xml:space="preserve"/> </p> <div xml:id="echoid-div317" type="float" level="2" n="10"> <note position="left" xlink:label="note-0290-01" xlink:href="note-0290-01a" xml:space="preserve">TAB. LII. <lb/>fig. 11.</note> </div> <p> <s xml:id="echoid-s6141" xml:space="preserve">Plus negotii poſteriores duæ, duodecima decimatertiaque, <lb/> <anchor type="note" xlink:label="note-0290-02a" xlink:href="note-0290-02"/> exhibituræ videntur: </s> <s xml:id="echoid-s6142" xml:space="preserve">quarum priorem præter Blancanum e-<lb/>iam Gaſſendus prodidit; </s> <s xml:id="echoid-s6143" xml:space="preserve">reliquam Ricciolus, aliunde tamen <lb/>acceptam, nobis impertiit. </s> <s xml:id="echoid-s6144" xml:space="preserve">Mirabilis præſertim illa Gaſſen-<lb/>di apparet. </s> <s xml:id="echoid-s6145" xml:space="preserve">Verumtamen ſi bene perpendatur, facile eſt <lb/>intelligere, quo pacto ab nona figura hæc defluxerit. </s> <s xml:id="echoid-s6146" xml:space="preserve">Nam <lb/>ſi tantum in locum rotundarum lacunarum, lunatæ ſubſti-<lb/>tuantur, cornibus ſeſe mutuo reſpicientibus, jam profecto <lb/>nona illa exiſtet, quam Ricciolus adnotavit. </s> <s xml:id="echoid-s6147" xml:space="preserve">Nihil mi-<lb/>rum autem Gaſſendo ac Blancano, cum non magnis perſpi-<lb/>cillis uterentur, rotundas potius eas maculas quam lunula-<lb/>rum forma apparuiſſe, ſiquidem partes harum acuminatas di-<lb/>ſtinctè percipere illis negatum fuit. </s> <s xml:id="echoid-s6148" xml:space="preserve">Tertiadecima denique <lb/>quam Ricciolus à Fontana, itemque ab aliis Romæ viſam <lb/>memorat, anno 1644 & </s> <s xml:id="echoid-s6149" xml:space="preserve">1645: </s> <s xml:id="echoid-s6150" xml:space="preserve">eam quoque pro octava ac <lb/>nona ſuppoſitam eſſe certum eſt, vel etiam pro ſeptima quam <lb/>Hevelius prodidit. </s> <s xml:id="echoid-s6151" xml:space="preserve">Non ſolum enim Ricciolus hanc ſibi <lb/>unquam oblatam negat, ſolas octavam, nonamque cum an-<lb/>ſis ſe vidiſſe affirmans: </s> <s xml:id="echoid-s6152" xml:space="preserve">ſed & </s> <s xml:id="echoid-s6153" xml:space="preserve">eodem anno 1645 Heveliano <lb/>teleſcopio ſeptima illa conſpecta fuit. </s> <s xml:id="echoid-s6154" xml:space="preserve">Nempe anſas Satur-<lb/>no conjungi rectè hîc Fontana animadvertit; </s> <s xml:id="echoid-s6155" xml:space="preserve">ſed cum præci-<lb/>puus earum ſplendor à parte gibba procederet, orbicularem <lb/>ibi figuram conſtituere viſus eſt.</s> <s xml:id="echoid-s6156" xml:space="preserve"/> </p> <div xml:id="echoid-div318" type="float" level="2" n="11"> <note position="left" xlink:label="note-0290-02" xlink:href="note-0290-02a" xml:space="preserve">Fig. 12. 13.</note> </div> <pb o="559" file="0291" n="306" rhead="SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s6157" xml:space="preserve">Non adſcripſi phaſibus hiſce eam qua Saturnus ellipſis for-<lb/>mâ conſpectus dicitur, abſque illis comitibus brachiiſve; </s> <s xml:id="echoid-s6158" xml:space="preserve">quo-<lb/>niam ſatis conſtat ob exilitatem teleſcopiorum, in ipſo in-<lb/>venti hujus exortu, talem apparuiſſe; </s> <s xml:id="echoid-s6159" xml:space="preserve">poſtquam verò ad ma-<lb/>jorem perfectionem eadem adducta ſint, neminem amplius <lb/>ſolitarium Saturnum vidiſſe niſi ſimul & </s> <s xml:id="echoid-s6160" xml:space="preserve">rotundus fuerit.</s> <s xml:id="echoid-s6161" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6162" xml:space="preserve">Omiſi etiam phænomenon aliud à nonnullis quidem rela-<lb/> <anchor type="note" xlink:label="note-0291-01a" xlink:href="note-0291-01"/> tum, ſed vanum proculdubio atque à ſola imaginatione pro-<lb/>fectum: </s> <s xml:id="echoid-s6163" xml:space="preserve">quo nempe alterum quandoque Saturni ſive comi-<lb/>tem ſive anſam altera minorem deprehendi aſſerunt. </s> <s xml:id="echoid-s6164" xml:space="preserve">Ego <lb/>vero exiſtimo non tam ab indiſcreta perceptione phaſin hanc <lb/>enaſci (cur enim hanc anſam potius quam illam minorem <lb/>dicerent?) </s> <s xml:id="echoid-s6165" xml:space="preserve">quam quod ubi hypotheſin quiſpiam commentus <lb/>fuerit, ex qua talem prodire neceſſe ſit, facile ſibi ipſi im-<lb/>ponat, quodque cupit evenire credat. </s> <s xml:id="echoid-s6166" xml:space="preserve">Itàque viro eximio <lb/>J. </s> <s xml:id="echoid-s6167" xml:space="preserve">Hodiernæ qui ad nos è Sicilia Syſtema ſuum Saturnium, <lb/>de quo pluribus mox agemus, miſit, accidiſſe reor. </s> <s xml:id="echoid-s6168" xml:space="preserve">Hic e-<lb/>nim anno 1655 & </s> <s xml:id="echoid-s6169" xml:space="preserve">ſub finem anni inſequentis, orientalem <lb/>globulum reliquo minorem ſibi apparuiſſe ſcribit, quum ta-<lb/>men eodem tempore, nobis inſpectantibus, eadem utrique <lb/>magnitudo, claritas ac figura adfuerit, non quidem orbicula-<lb/>ris, ſed rectà in longitudinem utrinque à Saturni diſco pro-<lb/>cedens. </s> <s xml:id="echoid-s6170" xml:space="preserve">Quin etiam fruſtra cauſam hujuſce rei ex hypotheſi <lb/>ſua adducere Hevelius laborat, quum planè contrarium ex <lb/>ea ſequatur, perpetua videlicet utriuſque anſæ æqualitas. <lb/></s> <s xml:id="echoid-s6171" xml:space="preserve">Nam cum faciem Saturni nativam ejuſmodi fingat, qualis ſu-<lb/> <anchor type="note" xlink:label="note-0291-02a" xlink:href="note-0291-02"/> periori tabella ad numerum ſeptimum exhibetur, utrâque <lb/>ſcilicet ansâ pari formâ ac magnitudine corpori medio ad-<lb/>nexâ, non ſinunt ullæ opticorum leges, ut qualicunque hu-<lb/>jus corporis converſione, aliter una atque alia anſa ſeſe vi-<lb/>dendam nobis præbeat. </s> <s xml:id="echoid-s6172" xml:space="preserve">Nobis, inquam, in tam immenſa <lb/>poſitis diſtantia. </s> <s xml:id="echoid-s6173" xml:space="preserve">Cum enim 3000 fere diametris Saturni <lb/>maximis ab eo remoti ſimus, quo pacto exiſtimat vir Clar. <lb/></s> <s xml:id="echoid-s6174" xml:space="preserve">nos percepturos diſcrimen angulorum quibus propior remo-<lb/>tiorque anſa ſpectari debeat?</s> <s xml:id="echoid-s6175" xml:space="preserve"/> </p> <div xml:id="echoid-div319" type="float" level="2" n="12"> <note position="right" xlink:label="note-0291-01" xlink:href="note-0291-01a" xml:space="preserve">Non eſſe al-<lb/>teram anſa-<lb/>rum quando-<lb/>que altera <lb/>minorem.</note> <note position="right" xlink:label="note-0291-02" xlink:href="note-0291-02a" xml:space="preserve">TAB. LII. <lb/>fig. 7.</note> </div> <p> <s xml:id="echoid-s6176" xml:space="preserve">Æquè parum rationi conſentaneum eſt quod, ob eandem <lb/> <anchor type="note" xlink:label="note-0291-03a" xlink:href="note-0291-03"/> <pb o="560" file="0292" n="307" rhead="CHRIST. HUGENII"/> illam diſtantiæ differentiam, unam anſarum citius quam alte-<lb/> <anchor type="note" xlink:label="note-0292-01a" xlink:href="note-0292-01"/> ram cum medio diſco coaleſcere poſſe exiſtimat. </s> <s xml:id="echoid-s6177" xml:space="preserve">Hoc enim, <lb/>admiſſa licet Heveliana hypotheſi, atque etiamſi centuplo <lb/>præſtantiores tubos haberemus, nequaquam tamen nobis vi-<lb/>ſu deprehendere liceret. </s> <s xml:id="echoid-s6178" xml:space="preserve">Cæterum & </s> <s xml:id="echoid-s6179" xml:space="preserve">hujus ſententiam viri <lb/>ſolertiſſimi, quo certè nemo hac tempeſtate majori animo <lb/>atque induſtria rem promovet Aſtronomicam, pluribus ex-<lb/>ponere, & </s> <s xml:id="echoid-s6180" xml:space="preserve">aliorum inſuper de propria Saturni forma opinio-<lb/>nes, priuſquam noſtram adferamus, recenſere placet, quas <lb/>poſt editam de Luna Saturni obſervationem omnes accepi-<lb/>mus. </s> <s xml:id="echoid-s6181" xml:space="preserve">Hevelius igitur, in eo libro quem peculiarem huic <lb/>argumento dicavit, cauſas phænomenôn redditurus, ſphæ-<lb/> <anchor type="note" xlink:label="note-0292-02a" xlink:href="note-0292-02"/> roidis oblongi figuram medio corpori Saturni tribuit, cui ab <lb/>utroque latere appendices iſtæ, ut jam dixi, brachiorum ſi-<lb/>ve anſarum forma, firmiter adhæreant, quemadmodum ſu-<lb/>pra 7 loco expreſſimus. </s> <s xml:id="echoid-s6182" xml:space="preserve">Porro ſimul cum Saturno haſce an-<lb/> <anchor type="note" xlink:label="note-0292-03a" xlink:href="note-0292-03"/> ſas, ſpatio 30 circiter annorum, circa minorem ſphæroidis <lb/>axem converti facit, qui quidem axis plano orbitæ Saturni <lb/>ſit ad angulos rectos. </s> <s xml:id="echoid-s6183" xml:space="preserve">Enimverò his poſitis ſeptimæ qui-<lb/>dem ſuperioris figuræ phaſin nec non rotundam quoque re-<lb/>præſentari certum eſt: </s> <s xml:id="echoid-s6184" xml:space="preserve">ut nimirum in duobus eclipticæ locis <lb/>oppoſitis anſata hac facie Saturnus appariturus ſit, aliiſque <lb/>rurſus duobus rotundus anſiſque prorſus exutus. </s> <s xml:id="echoid-s6185" xml:space="preserve">Quinimo <lb/>& </s> <s xml:id="echoid-s6186" xml:space="preserve">ſexta ac quodammodo quinta quoque exhiberi poſſent, <lb/>niſi quod conjunctæ cum medio diſco anſæ videri debebant. <lb/></s> <s xml:id="echoid-s6187" xml:space="preserve">Sed quarta phaſis nequaquam ab eadem forma proficiſci po-<lb/>terit. </s> <s xml:id="echoid-s6188" xml:space="preserve">Nam cum ponatur Saturnus cum annexis ſibi anſis, <lb/>qualem 7 figura oſtendit, rectus conſiſtere ad planum ec-<lb/>centrici ſui, atque ita perpetuo manere, licet circa axem <lb/>proprium vertatur; </s> <s xml:id="echoid-s6189" xml:space="preserve">eveniet quidem ea converſione, ut pau-<lb/>latim arctius ad medium orbem anſæ applicentur, verunta-<lb/>men ſemper geminas lunas referent ejuſdem cum dicto orbe <lb/>altitudinis, minimeque in tam compreſſas formas abibunt. </s> <s xml:id="echoid-s6190" xml:space="preserve"><lb/>Nam quod hoc efficere poſſe declinationem Saturni orbitæ <lb/>ab ecliptica Vir Cl. </s> <s xml:id="echoid-s6191" xml:space="preserve">cenſet (ita enim mihi reſpondit cum <lb/>difficultatem hanc ei moviſſem) ſi diligentius rem expendat <pb o="561" file="0293" n="308" rhead="SYSTEMA SATURNIUM."/> intelliget fieri non poſſe, cum Saturni orbita tantum 3 gra-<lb/>dibus ab eclipticæ plano recedat, ut inde anſarum figuris <lb/>ulla nobis percipienda mutatio adveniat. </s> <s xml:id="echoid-s6192" xml:space="preserve">Cæterum nec ve-<lb/>ram quidem eſſe hanc quartam phaſin ſuperius indicavimus, <lb/>ſed cum hæc, ſive triſphærica potius, minoribus teleſcopiis <lb/>cernitur, noſtris prægrandibus illam comparere quam adno-<lb/>tavimus anno 1655 & </s> <s xml:id="echoid-s6193" xml:space="preserve">1657. </s> <s xml:id="echoid-s6194" xml:space="preserve">Quæ cum adhuc minus Heve-<lb/>lianæ hypotheſi conveniat, clarius demonſtrat illam ſcopum <lb/>non attigiſſe. </s> <s xml:id="echoid-s6195" xml:space="preserve">Neque ſanè vel rotundæ phaſis phænomena <lb/>ſatis congruunt præſtitutis ab Hevelio limitibus; </s> <s xml:id="echoid-s6196" xml:space="preserve">ut patet <lb/>non tantum ex prædictione ejus, eventu refutata, quæ fe-<lb/>rebat rotundam phaſin anni 1656 continuatum iri uique in <lb/>Menſem Septembrem anni 1657, cum tamen jam inde à 13 <lb/>Oct. </s> <s xml:id="echoid-s6197" xml:space="preserve">1656, anſas Saturnus recuperaverit, neque poſtea a-<lb/>miſerit; </s> <s xml:id="echoid-s6198" xml:space="preserve">ſed & </s> <s xml:id="echoid-s6199" xml:space="preserve">ex Galilæi obſervatis circa rotundam phaſin <lb/>anni 1612. </s> <s xml:id="echoid-s6200" xml:space="preserve">Solſtitio enim hujus anni, Saturnus gr. </s> <s xml:id="echoid-s6201" xml:space="preserve">18, 22′. <lb/></s> <s xml:id="echoid-s6202" xml:space="preserve">♓ obtinens, tricorpor adhuc Galilæo apparuit, quum per <lb/>tabulas Hevelii debuiſſet rotundus obſervari, ac viciſſim 1 <lb/>Dec. </s> <s xml:id="echoid-s6203" xml:space="preserve">ejuſdem anni 1612, cum verſaretur in gr. </s> <s xml:id="echoid-s6204" xml:space="preserve">11, 27′. </s> <s xml:id="echoid-s6205" xml:space="preserve">♓ <lb/>rotundus Galilæo repertus eſt, quo loco Hevelius triſphæri-<lb/>cum expectaſſet.</s> <s xml:id="echoid-s6206" xml:space="preserve"/> </p> <div xml:id="echoid-div320" type="float" level="2" n="13"> <note position="right" xlink:label="note-0291-03" xlink:href="note-0291-03a" xml:space="preserve">Non etiam <lb/>unamquam.</note> <note position="left" xlink:label="note-0292-01" xlink:href="note-0292-01a" xml:space="preserve">alteram ci-<lb/>tius ad me-<lb/>dium Satur-<lb/>ni corpus ap-<lb/>plicari.</note> <note position="left" xlink:label="note-0292-02" xlink:href="note-0292-02a" xml:space="preserve">Hevelii hy-<lb/>potheſis circa <lb/>anſas Satur-<lb/>ni examina-<lb/>tur.</note> <note position="left" xlink:label="note-0292-03" xlink:href="note-0292-03a" xml:space="preserve">TAB. LII. <lb/>fig. 7.</note> </div> <p> <s xml:id="echoid-s6207" xml:space="preserve">Alteram hypotheſin, ingenio ſuo acutiſſimo dignam, <lb/> <anchor type="note" xlink:label="note-0293-01a" xlink:href="note-0293-01"/> ſummus Geometra E. </s> <s xml:id="echoid-s6208" xml:space="preserve">Robervallius nobis expoſuit. </s> <s xml:id="echoid-s6209" xml:space="preserve">Qua <lb/>quidem Saturnum perinde ut cæteros planetas rotundum ſta-<lb/>tuit; </s> <s xml:id="echoid-s6210" xml:space="preserve">egredi autem è Zona ejus torrida, hoc eſt, quæ re-<lb/>ctiores ſolis radios excipit, vapores quoſdam non admo-<lb/>dum ſpiſſos, qui procul à ſuperficie ejus in ſublime evolan-<lb/>tes, undique illum ambiant, præterquam polos verſus, ubi <lb/>fortaſſis intenſum frigus eos à ſole attrahi prohibeat. </s> <s xml:id="echoid-s6211" xml:space="preserve">Hi ſi <lb/>quando omne ſpatium complent à Saturno ad uſque extre-<lb/>mam ipſorum regionem, elliptica forma, inquit, eum vi-<lb/>deri faciunt. </s> <s xml:id="echoid-s6212" xml:space="preserve">Cum vero minus denſi exoriuntur, interque <lb/>eos ac Saturnum locus ingens medius relinquitur, ob tenui-<lb/>tatem ſolares radios non reflectunt, niſi inde ubi magis con-<lb/>ferti iiſdem opponuntur; </s> <s xml:id="echoid-s6213" xml:space="preserve">quod noſtri reſpectu neceſſe eſt <lb/>fieri in partibus à medio diſco Saturni remotioribus. </s> <s xml:id="echoid-s6214" xml:space="preserve">Ubi <pb o="562" file="0294" n="309" rhead="CHRIST. HUGENII"/> proinde anſarum formam ex ea reflexione produci oportet, <lb/>illasque inter ac Saturnum ſpatia utrinque obſcura vel certè <lb/>minus lucida intercedere. </s> <s xml:id="echoid-s6215" xml:space="preserve">At quoties nullæ prorſus hujuſ-<lb/>modi exhalationes aſcendunt, rotundus planeta ſpectabi-<lb/>tur.</s> <s xml:id="echoid-s6216" xml:space="preserve"/> </p> <div xml:id="echoid-div321" type="float" level="2" n="14"> <note position="right" xlink:label="note-0293-01" xlink:href="note-0293-01a" xml:space="preserve">Cauſæ phæ-<lb/>nomen@n Sa-<lb/>turni à Ro-<lb/>bervallio ex-<lb/>cogitatæ.</note> </div> <p> <s xml:id="echoid-s6217" xml:space="preserve">Equidem rectè hoc conſideravit vir ſagaciſſimus eſſe ali-<lb/>quid quod æqualiter undique Saturnum circumdaret; </s> <s xml:id="echoid-s6218" xml:space="preserve">ſua-<lb/>dente ſcilicet periodo Lunæ Saturniæ dierum 16, etiam ipſum <lb/>circa axem ſuum, & </s> <s xml:id="echoid-s6219" xml:space="preserve">breviori tempore, converſiones facere; <lb/></s> <s xml:id="echoid-s6220" xml:space="preserve">ex quibus converſionibus intra paucos dies diverſas phaſes <lb/>naſci oporteret, niſi undique eodem modo materia quædam <lb/>circumpoſita eſſet. </s> <s xml:id="echoid-s6221" xml:space="preserve">Nec diſſimili collectione uſi fueramus <lb/>dum noſtrum Syſtema effingeremus, ut poſtea dicetur. </s> <s xml:id="echoid-s6222" xml:space="preserve">Ve-<lb/>rumtamen ex illa ſua hypotheſi neque phaſis noſtra anni <lb/>1655 & </s> <s xml:id="echoid-s6223" xml:space="preserve">1657, ſatis commodè exponitur, neque certis Sa-<lb/>turni orbitæ locis phaſes quæque peculiares ſunt, quod ta-<lb/>men ita ſe habere obſervationes omnes evincunt, & </s> <s xml:id="echoid-s6224" xml:space="preserve">rotundi <lb/>Saturni inprimis. </s> <s xml:id="echoid-s6225" xml:space="preserve">Hæ enim in duobus oppoſitis Eclipticæ <lb/>locis accidere animadverſæ ſunt, contraque in iis qui qua-<lb/>drante hinc diſtant, anſæ quam maximè expanſæ: </s> <s xml:id="echoid-s6226" xml:space="preserve">cum ta-<lb/>men cauſa non appareat cur aliis quidem orbitæ partibus <lb/>nullos vapores Sol è Saturno, aliis maximam eorum copiam <lb/>extracturus ſit. </s> <s xml:id="echoid-s6227" xml:space="preserve">Deinde vapores iſti, neſcio quam bene, at <lb/>multum certè diſſimiles noſtris hiſce qui terram ambiunt po-<lb/>nuntur, tum quod immenſo ſpatio, prout hi ſolent, à Sa-<lb/>rurno aſcendant, tum quod verſus polos pauciſſimi aut nul-<lb/>li ſint, cum circa terræ polos plures contra atque altiores <lb/>exiſtant quam circa Zonam torridam. </s> <s xml:id="echoid-s6228" xml:space="preserve">De elliptica autem <lb/>forma Saturni diximus ſupra, eam revera nunquam talem <lb/>cerni, ut proinde cauſam ejus adferre ſuperfluum fuerit.</s> <s xml:id="echoid-s6229" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6230" xml:space="preserve">Nunc verò tuam quoque Hodierna doctiſſime ſententiam <lb/> <anchor type="note" xlink:label="note-0294-01a" xlink:href="note-0294-01"/> excutiamus, quam novo nuncio meo de Saturni Luna & </s> <s xml:id="echoid-s6231" xml:space="preserve"><lb/>promiſſo Syſtemate excitatus, ſubito publicam feciſti. </s> <s xml:id="echoid-s6232" xml:space="preserve">Me-<lb/>retur tuus ille in hæc ſtudia non vulgaris amor, ut melioris <lb/>notæ teleſcopia tibi ſuppeditentur, qui qualibuſcunque et-<lb/>iam inſtructus, non ceſſas in cælum quà licet eniti, ſum- <pb o="563" file="0295" n="310" rhead="SYSTEMA SATURNIUM."/> mumque illum planetarum, formarum varietate omnes fru-<lb/>ſtrantem, certis legibus aſtringere aggreſſus es. </s> <s xml:id="echoid-s6233" xml:space="preserve">At ille te <lb/>quoque ut opinor deluſit. </s> <s xml:id="echoid-s6234" xml:space="preserve">Nam ſi poſthac ea facie ſe tibi <lb/>offerat, qua mihi anno 1655 & </s> <s xml:id="echoid-s6235" xml:space="preserve">1657 apparuit, vel ea quo-<lb/> <anchor type="note" xlink:label="note-0295-01a" xlink:href="note-0295-01"/> que quæ ſucceſſit anno 1658, cognoſces utique has non qua-<lb/>drare illi quod tibi finxiſti corpori. </s> <s xml:id="echoid-s6236" xml:space="preserve">Sphæroidi nimirum, cu-<lb/>juſmodi aut ovum aut prunum eſt, ſed magis etiam oblongo <lb/>quam Hevelianum illud, ſimilem Saturnum imaginaris; </s> <s xml:id="echoid-s6237" xml:space="preserve">in <lb/>quo binæ utrinque ſint maculæ lucis expertes, quales in ta-<lb/>bula ſuperiori phaſis octava exhibet, quæ nobis interſtitia <lb/> <anchor type="note" xlink:label="note-0295-02a" xlink:href="note-0295-02"/> illa inter anſas Saturni mediumque ejus diſcum referant. </s> <s xml:id="echoid-s6238" xml:space="preserve">In-<lb/>de converſiones hujus corporis circa axem, eadem periodo <lb/>qua & </s> <s xml:id="echoid-s6239" xml:space="preserve">Hevelius, definis; </s> <s xml:id="echoid-s6240" xml:space="preserve">& </s> <s xml:id="echoid-s6241" xml:space="preserve">rotundum tunc videri aſſeris, <lb/>cum longior ſphæroidis axis ad nos dirigitur, ideoque bis <lb/>hoc accidere annorum 30 decurſu. </s> <s xml:id="echoid-s6242" xml:space="preserve">Verùm enim cum pror-<lb/>ſus rotundus ac orbe integro lucens appareat Saturnus, quo-<lb/>ties brachiis ſuis nudatus eſt, expendendum tibi amplius re-<lb/>linquo, qua ratione lacunas quaſdam expleas quas à maculis <lb/>illis nigris ſuperfuturas rectè ipſe prævidiſti. </s> <s xml:id="echoid-s6243" xml:space="preserve">Præterea & </s> <s xml:id="echoid-s6244" xml:space="preserve"><lb/>phaſes illas te conſiderare velim quas hæc tua hypotheſis <lb/>non poteſt non producere, nec à quoquam tamen obſerva-<lb/>tæ perhibentur. </s> <s xml:id="echoid-s6245" xml:space="preserve">Quas ut facilius omnes coram inſpicias, <lb/>ovum aut aliud quod eam formam habeat, iſtis maculis or-<lb/>natum, tibi proponito, atque ita ut Saturnum converti vis <lb/>circumducito; </s> <s xml:id="echoid-s6246" xml:space="preserve">videbis non pauca in tuo hoc Saturno phæ-<lb/>nomena quæ cæleſtis Saturnus nunquam exhibuit, atque a-<lb/>lia rurſus, quæ ineſſe huic certo tibi affirmare audeo, non <lb/>videbis, ipſéque de Syſtemate tuo ſtatues.</s> <s xml:id="echoid-s6247" xml:space="preserve"/> </p> <div xml:id="echoid-div322" type="float" level="2" n="15"> <note position="left" xlink:label="note-0294-01" xlink:href="note-0294-01a" xml:space="preserve">Hodiernæ Si-<lb/>culi circa ea-<lb/>dem phæno-<lb/>mena opinio.</note> <note position="right" xlink:label="note-0295-01" xlink:href="note-0295-01a" xml:space="preserve">TAB. XLIX. <lb/>fig. 6. 7.</note> <note position="right" xlink:label="note-0295-02" xlink:href="note-0295-02a" xml:space="preserve">TAB. LII. <lb/>fig. 8.</note> </div> <p> <s xml:id="echoid-s6248" xml:space="preserve">Quod autem rei veritatem neque tu neque illi viri egregii, <lb/>quorum antea opiniones recenſui, aſſecuti ſitis, minimè mi-<lb/>randum eſt, aut vobis imputandum, utpote ad quos falſa <lb/>plurima pro veris phænomenis delata ſint; </s> <s xml:id="echoid-s6249" xml:space="preserve">alia vero quæ de <lb/>Saturno citra viſus fallaciam obſervantur, non pervenerint. <lb/></s> <s xml:id="echoid-s6250" xml:space="preserve">Quæ ſi nobiſcum aſpicere vobis contigiſſet, eadem inde quæ <lb/>nos de nativa Planetæ forma collecturos fuiſſe credibile eſt. </s> <s xml:id="echoid-s6251" xml:space="preserve"><lb/>Me vero præter phaſes illas ſynceriores, etiam Lunæ Satur- <pb o="564" file="0296" n="311" rhead="CHRIST. HUGENII"/> niæ motus jam inde à principio animadverſus, non parum <lb/>hîc adjuvit; </s> <s xml:id="echoid-s6252" xml:space="preserve">ſiquidem ex hujus circa Saturnum gyratione, <lb/>prima mihi ſpes de conſtituenda hypotheſi affulſit. </s> <s xml:id="echoid-s6253" xml:space="preserve">Quæ <lb/>cujuſmodi ſit deinceps explicare aggredior.</s> <s xml:id="echoid-s6254" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6255" xml:space="preserve">Quum ergo 16 dierum periodo circumferri Saturno pla-<lb/> <anchor type="note" xlink:label="note-0296-01a" xlink:href="note-0296-01"/> netam novum comperiſſem, haud dubiè minori etiam tem-<lb/>poris ſpatio Saturnum in ſeſe ſuper axem ſuum revolvi ar-<lb/>bitratus ſum. </s> <s xml:id="echoid-s6256" xml:space="preserve">Namque antea quoque, in eo cum Tellure <lb/>hac noſtra cæteris primariis Planetis convenire, ſemper cre-<lb/>didi, quod ſinguli in ſe ipſis rotentur, atque ita tota eorum <lb/>ſuperficies lumine ſolis per vices gaudeat. </s> <s xml:id="echoid-s6257" xml:space="preserve">Quin imo in u-<lb/>niverſum ita cum magnis Mundi corporibus comparatum eſ-<lb/>ſe, ut illa quibus alia minora circumferuntur, ipſa quoque <lb/>in medio poſita minori tempore circumeant. </s> <s xml:id="echoid-s6258" xml:space="preserve">Ita enim So-<lb/>lem diebus 26 circiter in ſe redire maculæ ejus declarant: <lb/></s> <s xml:id="echoid-s6259" xml:space="preserve">circa Solem vero Planetarum ſinguli, inter quos Tellus quo-<lb/>que reponenda eſt, prout quiſque remotior eſt ita tardius <lb/>curſum conficiunt. </s> <s xml:id="echoid-s6260" xml:space="preserve">Rurſus Tellus hæc diurno ſpatio gyra-<lb/>tur, quam Luna menſtruo motu ambit. </s> <s xml:id="echoid-s6261" xml:space="preserve">Jovis autem Plane-<lb/>tam quatuor minores, hoc eſt, totidem Lunæ circumſtant, <lb/>eâdem hac lege, ut propiores quæ ſunt, celeriore curſu fe-<lb/>rantur. </s> <s xml:id="echoid-s6262" xml:space="preserve">Unde Jupiter quidem breviori forſitan tempore <lb/>quam 24 horarum converti cenſendus eſt, cum citima ei lu-<lb/>nularum m@nus biduo impendat. </s> <s xml:id="echoid-s6263" xml:space="preserve">Quæ omnia cum pridem <lb/>cognoviſſem, Saturno quoque jam tum ſimilem motum in-<lb/>eſſe judicabam. </s> <s xml:id="echoid-s6264" xml:space="preserve">De celeritate autem periodi, comitis ſui ob-<lb/>ſervatio me certiorem fecit. </s> <s xml:id="echoid-s6265" xml:space="preserve">Qui cum 16 diebus orbitam <lb/>expleat, Saturnum in centro orbitæ ſitum, multo frequen-<lb/>tius circumagi arguit. </s> <s xml:id="echoid-s6266" xml:space="preserve">Jam verò & </s> <s xml:id="echoid-s6267" xml:space="preserve">hoc credibile videbatur, <lb/>omnem cæleſtem materiam, Saturnum inter comitemque e-<lb/>jus interjectam, eidem motui obnoxiam eſſe, hoc pacto ut <lb/>quo Saturno propinquior eſt, eo magis ad ipſiuſmet celeri-<lb/>tatem accedat. </s> <s xml:id="echoid-s6268" xml:space="preserve">Unde illud ſequebatur denique, etiam ap-<lb/>pendices ſive brachia Saturni, vel medio globoſo corpori <lb/>conjuncta atque affixa, ſimul cum eo volvi, vel, interval-<lb/>lo aliquo diſcreta, non multo tamen lentiorem periodum <lb/>ſortita eſſe.</s> <s xml:id="echoid-s6269" xml:space="preserve"/> </p> <div xml:id="echoid-div323" type="float" level="2" n="16"> <note position="left" xlink:label="note-0296-01" xlink:href="note-0296-01a" xml:space="preserve">Noſtra hy-<lb/>potheſis qua <lb/>ratione exco-<lb/>gitata.</note> </div> <pb o="565" file="0297" n="312" rhead="SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s6270" xml:space="preserve">Figura porro brachiorum, dum hæc circa motum eorum <lb/>mente agitabam, ejuſmodi apparebat, qualis in ſuperioribus <lb/>obſervationibus anni 1655 expreſſa eſt. </s> <s xml:id="echoid-s6271" xml:space="preserve">Nempe medium <lb/>Saturni corpus omnino rotundum erat, brachia vero utrin-<lb/>que ſecundum eandem rectam lineam protendebantur, velut <lb/>ſi axe quodam medius planeta trajiceretur, quanquam tubo <lb/>illo 12 pedum quo tunc utebar, utraque verſus extremas <lb/> <anchor type="note" xlink:label="note-0297-01a" xlink:href="note-0297-01"/> cuſpides paulo craſſiora clarioraque videbantur, quam ubi <lb/>mediæ ſphæræ cohærebant, ut indicat figura. </s> <s xml:id="echoid-s6272" xml:space="preserve">Quum ita-<lb/>que quotidie eandem hanc ſpeciem præ ſe ferret, intel-<lb/>lexi id alia ratione fieri non poſſe, ſiquidem tam brevis <lb/>eſſet Saturni eorumque quæ illi cohærent circuitus, niſi ut <lb/>globus Saturni à corpore alio æqualiter undique cinctus po-<lb/> <anchor type="note" xlink:label="note-0297-02a" xlink:href="note-0297-02"/> neretur, atque ita annulus quidam medium eum ambiret. <lb/></s> <s xml:id="echoid-s6273" xml:space="preserve">Hinc enim, quacunque celeritate circumvolveretur, ean-<lb/>dem ſemper faciem nobis oblatum iri, ſi nimirum axis ad i-<lb/>ſtius annuli planum erectus eſſet.</s> <s xml:id="echoid-s6274" xml:space="preserve"/> </p> <div xml:id="echoid-div324" type="float" level="2" n="17"> <note position="right" xlink:label="note-0297-01" xlink:href="note-0297-01a" xml:space="preserve">TAB. XLVII. <lb/>fig. 6.</note> <note position="right" xlink:label="note-0297-02" xlink:href="note-0297-02a" xml:space="preserve">Annulo sæ@ <lb/>turnum ci@-<lb/>gi.</note> </div> <p> <s xml:id="echoid-s6275" xml:space="preserve">Et ſic quidem ei quæ per id tempus aderat phaſi cauſa ſua <lb/>conſtabat. </s> <s xml:id="echoid-s6276" xml:space="preserve">Ergo deinceps expendere cœpi anne reliquæ et-<lb/>iam, quæ de Saturno ferebantur, eidem annulo imputari <lb/>poſſent. </s> <s xml:id="echoid-s6277" xml:space="preserve">Hoc autem non tardè ſucceſſit ex animadverſa, <lb/>per frequentes obſervationes, brachiorum Saturni ad eclipti-<lb/>cam obliquitate. </s> <s xml:id="echoid-s6278" xml:space="preserve">Cum enim lineam rectam, ſecundum quam <lb/>utrinque ea extabant, non ſequi ductum eclipticæ, ſed in-<lb/>terſecare eam angulo 20 partibus majore comperiſſem, ſta-<lb/>tui proinde planum annuli quem imaginatus eram tali circi-<lb/> <anchor type="note" xlink:label="note-0297-03a" xlink:href="note-0297-03"/> ter angulo ad eclipticæ planum inclinari. </s> <s xml:id="echoid-s6279" xml:space="preserve">Perpetua videli-<lb/>cet conſtantique inclinatione, quemadmodum in tellure hac <lb/>noſtra plano æquatoris contingere notum eſt. </s> <s xml:id="echoid-s6280" xml:space="preserve">Hinc autem <lb/>neceſſario illud ſequebatur, ut diverſis aſpectibus nunc el-<lb/>lipſin ſatis latam, nunc eandem ſtrictiorem, nonnunquam <lb/>denique & </s> <s xml:id="echoid-s6281" xml:space="preserve">rectam lineam idem annulus nobis exhiberet. <lb/></s> <s xml:id="echoid-s6282" xml:space="preserve">Quod autem anſæ effingerentur, intellexi id inde fieri, quod <lb/>non arctè Saturni globo annulus cohæreat, ſed pari inter-<lb/>ſtitio undique ab eo removeatur. </s> <s xml:id="echoid-s6283" xml:space="preserve">Quibus proinde ſic ordi-<lb/>natis, ac præterea adſumta ea quam dixi annuli inclinatio- <pb o="566" file="0298" n="313" rhead="CHRIST. HUGENII"/> ne, omnes mirabiles Saturni facies ſicut mox demonſtrabi-<lb/>tur, eo referri poſſe inveni. </s> <s xml:id="echoid-s6284" xml:space="preserve">Et hæc ea ipſa hypotheſis eſt <lb/>quam anno 1656 die 25 Martii permixtis literis una cum <lb/> <anchor type="note" xlink:label="note-0298-01a" xlink:href="note-0298-01"/> obſervatione Saturniæ Lunæ edidimus <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s6285" xml:space="preserve"/> </p> <div xml:id="echoid-div325" type="float" level="2" n="18"> <note position="right" xlink:label="note-0297-03" xlink:href="note-0297-03a" xml:space="preserve">Eum ad Ec-<lb/>lipticæ pla-<lb/>num obliquè <lb/>poſitum eſſe.</note> <note symbol="*" position="left" xlink:label="note-0298-01" xlink:href="note-0298-01a" xml:space="preserve">vide ſupra <lb/>pag. 523.</note> </div> <p> <s xml:id="echoid-s6286" xml:space="preserve">Erant enim Literæ aaaaaaa ccccc d eeeee g hiiiiiii <lb/>llllmmnnnnnnnnnooooppqrrstttttuuuuu <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s6287" xml:space="preserve">quæ <anchor type="note" xlink:label="note-0298-02a" xlink:href="note-0298-02"/> ſuis locis repoſitæ hoc ſignificant, Annulo cingitur, tenui, <lb/>plano, nuſquam cohærente, ad eclipticam inclinato. </s> <s xml:id="echoid-s6288" xml:space="preserve">La-<lb/>titudinem vero ſpatii inter annulum globumque Saturni in-<lb/>terjecti, æquare ipſius annuli latitudinem vel excedere etiam, <lb/>figura Saturni ab aliis obſervata, certiuſque deinde quæ mi-<lb/>hi ipſi conſpecta fuit, edocuit: </s> <s xml:id="echoid-s6289" xml:space="preserve">maximamque item annuli <lb/>diametrum eam circiter rationem habere ad diametrum Sa-<lb/> <anchor type="note" xlink:label="note-0298-03a" xlink:href="note-0298-03"/> turni quæ eſt 9 ad 4. </s> <s xml:id="echoid-s6290" xml:space="preserve">Ut vera proinde forma ſit ejuſmodi <lb/>qualem appoſito ſchemate adumbravimus.</s> <s xml:id="echoid-s6291" xml:space="preserve"/> </p> <div xml:id="echoid-div326" type="float" level="2" n="19"> <note symbol="*" position="left" xlink:label="note-0298-02" xlink:href="note-0298-02a" xml:space="preserve">vide pag. <lb/>526.</note> <note position="left" xlink:label="note-0298-03" xlink:href="note-0298-03a" xml:space="preserve">TAB. XLIX. <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s6292" xml:space="preserve">Cæterum obiter hic iis reſpondendum cenſeo, quibus <lb/> <anchor type="note" xlink:label="note-0298-04a" xlink:href="note-0298-04"/> novum nimis ac fortaſſe abſonum videbitur, quod non tan-<lb/>tum alicui cæleſtium corporum figuram ejuſmodi tribuam, <lb/>cui ſimilis in nullo hactenus corum deprehenſa eſt, cum con-<lb/>tra pro certo creditum fuerit, ac veluti naturali ratione con-<lb/>ſtitutum, ſolam iis ſphæricam convenire, ſed & </s> <s xml:id="echoid-s6293" xml:space="preserve">quod an-<lb/>nulum hunc ſolidum ac permanentem (talem enim arbitror) <lb/>Saturno ita circumponam, ut nullis compagibus retinaculiſ-<lb/>ve ei cohæreat, ac nihilominus æqualem ab omni parte di-<lb/>ſtantiam ſervet, unáque cum Saturno motu velociſſimo <lb/>transferatur. </s> <s xml:id="echoid-s6294" xml:space="preserve">Hos autem reputare illud oportet, non ex <lb/>mera inventione atque arbitrio meo hanc me fingere hypo-<lb/>theſin, ſicut Aſtronomi ſuos epicyclos, nuſquam in cælo <lb/>apparentes; </s> <s xml:id="echoid-s6295" xml:space="preserve">ſed oculorum ſenſu, quo nempe reliquarum re-<lb/>rum omnium figuras dignoſcimus, hunc quoque annulum <lb/>ſatis evidenter me percipere. </s> <s xml:id="echoid-s6296" xml:space="preserve">Neque verò cauſam eſſe cur <lb/>corpus aliquod inter cæ eſtia hac forma præditum exiſtere <lb/>nequeat, quæ ſi non ſphærica at ſaltem rotunda eſt, atque <lb/>ejuſmodi ut motum circa centrum æquè commodè atque <lb/>ipſi ſphærica ſuſcipiat. </s> <s xml:id="echoid-s6297" xml:space="preserve">Nam minus utique mirandum, hu-<lb/>juſmodi figuram tali corpori tributam, quam inconcinnam <pb o="567" file="0299" n="314" rhead="SYSTEMA SATURNIUM."/> quampiam miniméque tornatilem. </s> <s xml:id="echoid-s6298" xml:space="preserve">Porrò quum certo ſatis <lb/>colligi poſſe videatur, ob ſimilitudinem ac cognationem <lb/>magnam quæ Saturno cum Tellure noſtra intercedit, illum <lb/>perinde ut hæc in medio ſui vorticis ſitum eſſe, centrumque <lb/>ejus verſus omnia naturâ ſuâ tendere quæ illic gravia haben-<lb/>tur; </s> <s xml:id="echoid-s6299" xml:space="preserve">inde neceſſario quoque efficitur, annulum iſtum omni-<lb/>bus ſui partibus æqualivi ad centrum nitentem, hoc ipſo ita <lb/>conſiſtere, ut undiquaque pari intervallo à centro abſit. </s> <s xml:id="echoid-s6300" xml:space="preserve">Pla-<lb/>nè ſicuti quidam contemplati ſunt, quod ſi continuum for-<lb/>nicem per totum terrarum ambitum exſtrui poſſibile eſſet, is <lb/>abſque ullo fulcimine ſemet ipſum eſſet ſuſtentaturus. </s> <s xml:id="echoid-s6301" xml:space="preserve">Ergo <lb/>tale quid in Saturno reipſa effectum eſſe ne protinus abſur-<lb/>dum credant, ſed ſuſpiciant potius infinitam naturæ poten-<lb/>tiam & </s> <s xml:id="echoid-s6302" xml:space="preserve">majeſtatem, quæ ſubinde nova ſuorum operum ſpe-<lb/>cimina in lucem promens, plura etiam ſupereſſe admonet. <lb/></s> <s xml:id="echoid-s6303" xml:space="preserve">Verum ad propoſitum jam revertamur.</s> <s xml:id="echoid-s6304" xml:space="preserve"/> </p> <div xml:id="echoid-div327" type="float" level="2" n="20"> <note position="left" xlink:label="note-0298-04" xlink:href="note-0298-04a" xml:space="preserve">Occurritur iis <lb/>quæ de an-<lb/>nulo objici <lb/>poſſent.</note> </div> <p> <s xml:id="echoid-s6305" xml:space="preserve">Diximus brachiorum lineam anno 1655 eclipticæ occur-<lb/> <anchor type="note" xlink:label="note-0299-01a" xlink:href="note-0299-01"/> rere viſam angulo partium plus minus 20. </s> <s xml:id="echoid-s6306" xml:space="preserve">Quod cum præ-<lb/>cipuè faciat ad phænomena ex hypotheſi noſtra deducenda, <lb/>antequam ad illa accedamus, ita ſe habere ex obſervationi-<lb/>bus oſtendendum eſt.</s> <s xml:id="echoid-s6307" xml:space="preserve"/> </p> <div xml:id="echoid-div328" type="float" level="2" n="21"> <note position="right" xlink:label="note-0299-01" xlink:href="note-0299-01a" xml:space="preserve">Saturni ma-<lb/>jor diameter <lb/>Æquatori <lb/>parallela <lb/>oſtenditur.</note> </div> <p> <s xml:id="echoid-s6308" xml:space="preserve">Cum igitur 25 Martii, anno 1655, Saturni in A poſiti ad <lb/> <anchor type="note" xlink:label="note-0299-02a" xlink:href="note-0299-02"/> ſtellam fixam B ea fuerit conſtitutio quæ hîc adnotata eſt, <lb/>ut conſtat ex ſuperioribus obſervationibus; </s> <s xml:id="echoid-s6309" xml:space="preserve">die autem ſe-<lb/>quenti ſicut in D & </s> <s xml:id="echoid-s6310" xml:space="preserve">B, tertioque item die ſicut in C & </s> <s xml:id="echoid-s6311" xml:space="preserve">B, <lb/>ita ut quotidie linea anſarum producta altius ſupra ſtellam B <lb/>aſcenderet; </s> <s xml:id="echoid-s6312" xml:space="preserve">hinc perſpicuum fit viam Planetæ A C nequa-<lb/>quam congruere directioni anſarum, ſed diverſam ferri an-<lb/>gulo A C E, quem 20. </s> <s xml:id="echoid-s6313" xml:space="preserve">gr. </s> <s xml:id="echoid-s6314" xml:space="preserve">majorem invenio. </s> <s xml:id="echoid-s6315" xml:space="preserve">Eo autem <lb/>tempore nihil admodum mutabatur latitudo Saturni appa-<lb/>rens, unde conſtat viam A C eclipticæ parallelam fuiſſe; </s> <s xml:id="echoid-s6316" xml:space="preserve">ac <lb/>proinde eodem illo 20 gr. </s> <s xml:id="echoid-s6317" xml:space="preserve">angulo etiam ab ecliptica lineam <lb/>anſarum deflexiſſe.</s> <s xml:id="echoid-s6318" xml:space="preserve"/> </p> <div xml:id="echoid-div329" type="float" level="2" n="22"> <note position="right" xlink:label="note-0299-02" xlink:href="note-0299-02a" xml:space="preserve">TAB. XLIX. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s6319" xml:space="preserve">Rurſus 9 Apr. </s> <s xml:id="echoid-s6320" xml:space="preserve">& </s> <s xml:id="echoid-s6321" xml:space="preserve">tribus ſequentibus continuè diebus ad <lb/> <anchor type="note" xlink:label="note-0299-03a" xlink:href="note-0299-03"/> aliam fixam C talis annotatus fuit Saturni poſitus qualis in <lb/>L, M, N, O cernitur. </s> <s xml:id="echoid-s6322" xml:space="preserve">Ubi ſimiliter apparet, lineam an- <pb o="568" file="0300" n="315" rhead="CHRIST. HUGENII"/> ſarum O P, Saturni tramitem L O ſecare angulo L O P, qui <lb/>20 plus minus graduum deprehenditur. </s> <s xml:id="echoid-s6323" xml:space="preserve">Eclipticam verò <lb/>vel eclipticæ parallelam lineam R Q, angulo P Q R, paulo <lb/>etiam majore quam L O P, quoniam decreſcebat per eos dies <lb/>Saturni latitudo, quæ borea erat.</s> <s xml:id="echoid-s6324" xml:space="preserve"/> </p> <div xml:id="echoid-div330" type="float" level="2" n="23"> <note position="right" xlink:label="note-0299-03" xlink:href="note-0299-03a" xml:space="preserve">TAB. XLIX@. <lb/>fig. 4.</note> </div> <p> <s xml:id="echoid-s6325" xml:space="preserve">Denique cum poſt ſtationem Saturnus ad fixam eandem <lb/> <anchor type="note" xlink:label="note-0300-01a" xlink:href="note-0300-01"/> C revertiſſet, factuſque eſſet auſtralior, die 27 Maji, eo <lb/>poſitu juxta hanc adſtabat quo hîc ad S deſignatus eſt. </s> <s xml:id="echoid-s6326" xml:space="preserve">Die <lb/>vero Maji poſtrema, eo qui ad T. </s> <s xml:id="echoid-s6327" xml:space="preserve">Ita ut hic quoque linea <lb/>S V, quæ ſecundum brachia ducitur, interſecet S T, quam <lb/>quatriduo illo Saturnus permeaverat, angulo V S T, 20 gra-<lb/>dus exiguo ſuperante. </s> <s xml:id="echoid-s6328" xml:space="preserve">Eclipticam vero vel eclipticæ paralle-<lb/>lam X Y, paulo minore angulo S X Y, quoniam minuebatur <lb/>adhuc continuè Planetæ latitudo, etſi directus jam incedebat.</s> <s xml:id="echoid-s6329" xml:space="preserve"/> </p> <div xml:id="echoid-div331" type="float" level="2" n="24"> <note position="left" xlink:label="note-0300-01" xlink:href="note-0300-01a" xml:space="preserve">TAB. XLIX. <lb/>fig. 5.</note> </div> <p> <s xml:id="echoid-s6330" xml:space="preserve">Et his quidem intellectis, inclinatio brachiorum Saturni <lb/>ad eclipticam, in dubium vocari amplius nequit, etſi, fa-<lb/>teor, non admodum accuratè quanta ſit hoc modo definiri <lb/>poſſit. </s> <s xml:id="echoid-s6331" xml:space="preserve">Si tamen 21 graduum per id tempus apparuiſſe po-<lb/>natur, quantam ferè adductæ obſervationes comprobant, ac <lb/>porro ad locum Saturni, qui tunc fuit, attendamus, inve-<lb/>niemus æquatori planè parallela brachia extitiſſe. </s> <s xml:id="echoid-s6332" xml:space="preserve">Cum e-<lb/>nim circa 3 gr. </s> <s xml:id="echoid-s6333" xml:space="preserve">Virginis Saturnus verſaretur, ubi parallelus <lb/>æquatoris eclipticam interſecat angulo gr. </s> <s xml:id="echoid-s6334" xml:space="preserve">21. </s> <s xml:id="echoid-s6335" xml:space="preserve">quo angulo <lb/>etiam brachiorum linea illam interſecabat; </s> <s xml:id="echoid-s6336" xml:space="preserve">manifeſtum eſt <lb/>hanc ſecundùm dictum parallelum atque adeo ſecundùm <lb/>ipſum æquatorem incedere debuiſſe.</s> <s xml:id="echoid-s6337" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6338" xml:space="preserve">Cæterum alio nunc irrefragabili argumento eandem hanc <lb/>brachiorum inclinationem, & </s> <s xml:id="echoid-s6339" xml:space="preserve">cum æquatore circulo paralle-<lb/>liſmum confirmabimus. </s> <s xml:id="echoid-s6340" xml:space="preserve">Suprà in obſervationum Hiſtoria <lb/>ad diem 13 Octob. </s> <s xml:id="echoid-s6341" xml:space="preserve">anni 1656, adnotatum eſt, motum Sa-<lb/>turni, quo cum cælo quotidie defertur, ſecundum ipſam an-<lb/>ſarum lineam procedere nobis viſum, non ea die tantum, <lb/>ſed quotieſcunque exinde idem inquirere libuit: </s> <s xml:id="echoid-s6342" xml:space="preserve">inquiritur <lb/>autem hoc modo. </s> <s xml:id="echoid-s6343" xml:space="preserve">Teleſcopio ad Saturnum obverſo, eo-<lb/>que intra tubi aperturam, quam hic refert circulus A B, re-<lb/>cepto, ita ut primùm in extremo margine conſiſtat, vergat- <pb o="569" file="0301" n="316" rhead="SYSTEMA SATURNIUM."/> que linea brachiorum ad aperturæ centrum, immotum inde <lb/>tubum ſiſtimus, quo ita manente, mox totam tubi capaci-<lb/>tatem A B percurrere Saturnus conſpicitur, motu qui cælo <lb/>tribuitur abreptus, atque exire ad partem oppoſitam B, quo <lb/>tendebat brachiorum linea. </s> <s xml:id="echoid-s6344" xml:space="preserve">Quoniam ergo Saturnus, hoc <lb/>motu ſuo, circulum æquatori parallelum percurrit, cujus <lb/>circuli pars eſt linea A B, manifeſto patet brachiorum li-<lb/>neam æquatori quoque parallelam extendi. </s> <s xml:id="echoid-s6345" xml:space="preserve">Hujus vero rei <lb/>periculum facere licet quoties Saturni obſervandi facultas <lb/>datur, neque alia certiori ratione directionem anſarum in-<lb/>veſtigari poſſe credimus. </s> <s xml:id="echoid-s6346" xml:space="preserve">Non defuere tamen qui diverſi-<lb/>mode idem examen inſtituerunt, quorum obſervationes ſen-<lb/>tentiam hanc noſtram comprobant. </s> <s xml:id="echoid-s6347" xml:space="preserve">Nam Galilæus quidem <lb/>ipſe, qui primus phænomeni hujus indicium fecit, non <lb/>omnino eclipticæ parallelam lineam, in qua Saturni comi-<lb/>tes poſiti eſſent, ſibi viſam ſcribit, ſed evidenter ab ea de-<lb/>flectere; </s> <s xml:id="echoid-s6348" xml:space="preserve">& </s> <s xml:id="echoid-s6349" xml:space="preserve">fortaſſe, inquit, æquatori parallela eſt.</s> <s xml:id="echoid-s6350" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6351" xml:space="preserve">Poſt hunc vero quicunque rem examinaverunt, omnes <lb/>hanc ejus conjecturam nequaquam aberraſſe oſtendunt, & </s> <s xml:id="echoid-s6352" xml:space="preserve"><lb/>in his Aſtronomi inſignes Gaſſendus, Bullialdus & </s> <s xml:id="echoid-s6353" xml:space="preserve">Riccio-<lb/>lus; </s> <s xml:id="echoid-s6354" xml:space="preserve">quorum hic ſtellarum fixarum ope (alia tamen metho-<lb/>do quam qua modo nos uſi ſumus) non ſemel illum anſa-<lb/>rum cum æquatore paralleliſmum liquido ſibi compertum <lb/>demonſtrat.</s> <s xml:id="echoid-s6355" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6356" xml:space="preserve">Clariſſimus tamen Hevelius, ſuis obſervationibus confi-<lb/> <anchor type="note" xlink:label="note-0301-01a" xlink:href="note-0301-01"/> ſus, quæ cujuſmodi fuerint non indicat, diverſam hîc opi-<lb/>nionem tuetur, contenditque excentrico Saturni parallelam <lb/>eſſe brachiorum inclinationem, ut proinde eclipticæ quoque <lb/>ferè conveniat, non amplius unqnam quam 2{1/2} gr. </s> <s xml:id="echoid-s6357" xml:space="preserve">defle-<lb/>ctens: </s> <s xml:id="echoid-s6358" xml:space="preserve">& </s> <s xml:id="echoid-s6359" xml:space="preserve">hoc quidem cum circa nodos interſectionis Satur-<lb/>nus verſatur. </s> <s xml:id="echoid-s6360" xml:space="preserve">Quo fiet ut cum in Libræ & </s> <s xml:id="echoid-s6361" xml:space="preserve">Arietis gr. </s> <s xml:id="echoid-s6362" xml:space="preserve">20, <lb/>ubi limites ſtatuuntur, poſitus fuerit, prorſus eclipticam ſe-<lb/>quatur brachiorum diſpoſitio. </s> <s xml:id="echoid-s6363" xml:space="preserve">At circa hæc loca maximam <lb/>omnium inclinationem noſtræ & </s> <s xml:id="echoid-s6364" xml:space="preserve">aliorum obſervationes pro-<lb/>dunt, nam noſtræ quidem, Saturno prope libræ ſignum at-<lb/>que in ipſo ſigno verſante, habitæ ſunt; </s> <s xml:id="echoid-s6365" xml:space="preserve">aliorum vero quædam <pb o="570" file="0302" n="317" rhead="CHRIST. HUGENII"/> cum non procul ab oppoſito limite diſtaret. </s> <s xml:id="echoid-s6366" xml:space="preserve">Ergo quod & </s> <s xml:id="echoid-s6367" xml:space="preserve"><lb/>plurium autoritas & </s> <s xml:id="echoid-s6368" xml:space="preserve">propria experientia conſirmat, haud <lb/>cunctanter hac in re amplectemur: </s> <s xml:id="echoid-s6369" xml:space="preserve">Neque vero dubito, quum <lb/>certam ſemperque obviam obſervandi methodum tradiderim, <lb/>quin ea adhibita vir veri amantiſſimus priorem ſententiam <lb/>volens abdicaturus ſit. </s> <s xml:id="echoid-s6370" xml:space="preserve">Illud utique nobiſcum novit, fruſtra <lb/>ſive hunc ſive alium quemlibet obſervandi modum, ad diri-<lb/>mendam inter nos hanc litem, uſurpari, quando circa ſigna <lb/>Cancri vel Capricorni Saturnus commoratur, quoniam ibi <lb/>diſcerni nequit utrum æquatori an eclipticæ parallela ſit quæ <lb/>per anſas ducitur, eò quod in his locis parallelus æquatoris <lb/>eclipticam non interſecet ſed tangat. </s> <s xml:id="echoid-s6371" xml:space="preserve">Quamobrem miror <lb/> <anchor type="note" xlink:label="note-0302-01a" xlink:href="note-0302-01"/> etiam quo pacto Ricciolus, obſervationem Grimaldi adfe-<lb/>rens anno 1650, 18 Martii habitam, quæ probat eo tempo-<lb/>re hanc anſarum lineam eclipticæ obliquam incidiſſe, non a-<lb/>nimadverterit contrarium ejus quod intenderat illa obſerva-<lb/>tione evinci: </s> <s xml:id="echoid-s6372" xml:space="preserve">ſi enim tunc, Saturno prope initium Cancri <lb/>agente, non erat eclipticæ parallela anſarum linea, ne qui-<lb/>dem æquatori parallela fuerit, ut conſtat ex modo dictis. <lb/></s> <s xml:id="echoid-s6373" xml:space="preserve">Semper autem æquatori parallelam cerni, & </s> <s xml:id="echoid-s6374" xml:space="preserve">aliis multis ob-<lb/>ſervationibus docuit, & </s> <s xml:id="echoid-s6375" xml:space="preserve">hac ipſa oſtendere voluit. </s> <s xml:id="echoid-s6376" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s6377" xml:space="preserve"><lb/>experientiæ propriæ & </s> <s xml:id="echoid-s6378" xml:space="preserve">ſententiæ ſuæ, quam veram eſſe de-<lb/>monſtravimus, planè hîc adverſatur. </s> <s xml:id="echoid-s6379" xml:space="preserve">Grimaldi autem ob-<lb/>ſervatio, quanquam alioqui in hoc negotio verſatiſſimi at-<lb/>que experientiſſimi, vitio aliquo laborare cenſenda eſt; </s> <s xml:id="echoid-s6380" xml:space="preserve">in <lb/>qua forſitan priore fixa à Saturno contecta, vel ob fulgorem <lb/>cerni prohibita, aliam deinde teleſcopio detexerit prius non <lb/>animadverſam, proque eadem illa habuerit. </s> <s xml:id="echoid-s6381" xml:space="preserve">Nos verò, cum <lb/>parallelam æquatori brachiorum lineam eſſe indubitatis ratio-<lb/>nibus ſatis jam adſtruxerimus, deinceps ad annuli noſtri hy-<lb/>potheſin hunc cjus ſitum applicemus, indeque phænomenôn <lb/>cauſas ſingulas derivari doceamus.</s> <s xml:id="echoid-s6382" xml:space="preserve"/> </p> <div xml:id="echoid-div332" type="float" level="2" n="25"> <note position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">Hevelii de <lb/>inclinatione <lb/>anſarum <lb/>contraria <lb/>opinio.</note> <note position="left" xlink:label="note-0302-01" xlink:href="note-0302-01a" xml:space="preserve">Ricciolus ſibi <lb/>contrarius. <lb/>lib. 7. Al-<lb/>mag. novi.</note> </div> <p> <s xml:id="echoid-s6383" xml:space="preserve">Eſto itaque Saturni orbita, quam 30 circiter annis ille <lb/> <anchor type="note" xlink:label="note-0302-02a" xlink:href="note-0302-02"/> emetitur, A B C D; </s> <s xml:id="echoid-s6384" xml:space="preserve">& </s> <s xml:id="echoid-s6385" xml:space="preserve">in eodem plano (nam exiguæ de-<lb/>clinationis gr. </s> <s xml:id="echoid-s6386" xml:space="preserve">2 {1/2} non hic rationem habebimus) circulus <lb/>F E, orbem magnum ſive telluris orbitam referens, quam <lb/> <anchor type="note" xlink:label="note-0302-03a" xlink:href="note-0302-03"/> <pb file="0303" n="318"/> <pb file="0303a" n="319"/> <anchor type="figure" xlink:label="fig-0303a-01a" xlink:href="fig-0303a-01"/> <anchor type="figure" xlink:label="fig-0303a-02a" xlink:href="fig-0303a-02"/> <anchor type="figure" xlink:label="fig-0303a-03a" xlink:href="fig-0303a-03"/> <anchor type="figure" xlink:label="fig-0303a-04a" xlink:href="fig-0303a-04"/> <anchor type="figure" xlink:label="fig-0303a-05a" xlink:href="fig-0303a-05"/> <anchor type="figure" xlink:label="fig-0303a-06a" xlink:href="fig-0303a-06"/> <anchor type="figure" xlink:label="fig-0303a-07a" xlink:href="fig-0303a-07"/> <pb file="0304" n="320"/> <pb o="571" file="0305" n="321" rhead="SYSTEMA SATURNIUM."/> annuo ſpatio cum illa nos obimus: </s> <s xml:id="echoid-s6387" xml:space="preserve">inque hujus centro G <lb/>conſiſtat Sol. </s> <s xml:id="echoid-s6388" xml:space="preserve">Jam ſicuti axem terra habet ſemper ſibi paral-<lb/>lelum, circa quem in ſeſe volvitur, ita Saturnus quoque ha-<lb/>bere ponatur, qui ſit ad annuli ſibi circumdati planum ere-<lb/>ctus. </s> <s xml:id="echoid-s6389" xml:space="preserve">Adeo ut circa unum eundemque axem gyrentur & </s> <s xml:id="echoid-s6390" xml:space="preserve">cor-<lb/>pus Saturni ſphæricum, & </s> <s xml:id="echoid-s6391" xml:space="preserve">annulus, & </s> <s xml:id="echoid-s6392" xml:space="preserve">in eodem annuli pla-<lb/>no poſitus Saturni comes ſive Luna. </s> <s xml:id="echoid-s6393" xml:space="preserve">Intelligatur autem Sa-<lb/>turni hic axis axi terræ circiter æquidiſtans. </s> <s xml:id="echoid-s6394" xml:space="preserve">Unde & </s> <s xml:id="echoid-s6395" xml:space="preserve">pla-<lb/>num annuli plano æquatoris noſtri parallelum erit, ac pro-<lb/>inde angulo partium 23 {1/2} ad planum eclipticæ inclinabitur. <lb/></s> <s xml:id="echoid-s6396" xml:space="preserve">Nam ſic conſtituendum eſſe inde intelleximus quod anſarum <lb/>Saturni inclinatio æquatori parallela deprehendatur, ut modo <lb/>demonſtratum ſuit.</s> <s xml:id="echoid-s6397" xml:space="preserve"/> </p> <div xml:id="echoid-div333" type="float" level="2" n="26"> <note position="left" xlink:label="note-0302-02" xlink:href="note-0302-02a" xml:space="preserve">Ulterior hy-<lb/>potheſeos no-<lb/>ſtræ expla-<lb/>natio.</note> <note position="left" xlink:label="note-0302-03" xlink:href="note-0302-03a" xml:space="preserve">TAB. L. <lb/>fig. 2.</note> <figure xlink:label="fig-0303a-01" xlink:href="fig-0303a-01a"> <caption xml:id="echoid-caption151" style="it" xml:space="preserve">Pag. 574.<lb/>TAB. XLIX.<lb/>Fig. 2.</caption> </figure> <figure xlink:label="fig-0303a-02" xlink:href="fig-0303a-02a"> <caption xml:id="echoid-caption152" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables88" xml:space="preserve">C K O B H N G M S * F D A L E</variables> </figure> <figure xlink:label="fig-0303a-03" xlink:href="fig-0303a-03a"> <caption xml:id="echoid-caption153" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables89" xml:space="preserve">E C D A * B</variables> </figure> <figure xlink:label="fig-0303a-04" xlink:href="fig-0303a-04a"> <caption xml:id="echoid-caption154" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables90" xml:space="preserve">P Q O N M L * C R</variables> </figure> <figure xlink:label="fig-0303a-05" xlink:href="fig-0303a-05a"> <caption xml:id="echoid-caption155" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables91" xml:space="preserve">C * V S X T Y</variables> </figure> <figure xlink:label="fig-0303a-06" xlink:href="fig-0303a-06a"> <caption xml:id="echoid-caption156" style="it" xml:space="preserve">Fig. 6.</caption> </figure> <figure xlink:label="fig-0303a-07" xlink:href="fig-0303a-07a"> <caption xml:id="echoid-caption157" style="it" xml:space="preserve">Fig. 7.</caption> </figure> </div> <p> <s xml:id="echoid-s6398" xml:space="preserve">Nobis itaque in circulo E F circumlatis, varias annuli <lb/>hujus, à Sole pariter cum Saturno illuſtrati figuras cerni de-<lb/>bere perſpicuum eſt, prout nimirum nunc has nunc illas or-<lb/>bitæ ſuæ partes Saturnus peragrabit. </s> <s xml:id="echoid-s6399" xml:space="preserve">Nam eodem quidem <lb/>orbitæ loco, quacunque tandem celeritate ſuper axem ſuum <lb/>converti dicatur, diverſas phaſes non exhibebit, quoniam <lb/>ea converſione ſitum annuli, noſtri reſpectu, nihil immu-<lb/>tat, ſed ita demum ſi in eccentrico ſuo ſive orbita ſpatio <lb/>aliquo progreſſus fuerit. </s> <s xml:id="echoid-s6400" xml:space="preserve">Cumque triginta fere annis ad <lb/>loca eadem revertatur, hinc neceſſario intra id tempus <lb/>omnium phaſium viciſſitudinem conſpici neceſſe erit, at-<lb/>que alias quidem bis, alias quater: </s> <s xml:id="echoid-s6401" xml:space="preserve">quas ſingulas nunc ex-<lb/>pendemus.</s> <s xml:id="echoid-s6402" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6403" xml:space="preserve">Primò itaque duos orbitæ locos eſſe liquet è diametro <lb/> <anchor type="note" xlink:label="note-0305-01a" xlink:href="note-0305-01"/> oppoſitos, velut A & </s> <s xml:id="echoid-s6404" xml:space="preserve">C, in quibus Saturnus conſtitutus <lb/>latiſſimas, maximeque omnium diductas anſas exhibeat; <lb/></s> <s xml:id="echoid-s6405" xml:space="preserve">cum nimirum à plano per centra Telluris & </s> <s xml:id="echoid-s6406" xml:space="preserve">Saturni acto, <lb/>atque ad orbitæ Saturniæ planum erecto, annuli quoque <lb/>planum ad angulos rectos ſecatur, eoque viginti trium circi-<lb/>ter partium angulo ſupra planum annuli viſus noſter attolli-<lb/>tur.</s> <s xml:id="echoid-s6407" xml:space="preserve"/> </p> <div xml:id="echoid-div334" type="float" level="2" n="27"> <note position="right" xlink:label="note-0305-01" xlink:href="note-0305-01a" xml:space="preserve">Cauſa pha-<lb/>ſcos anſarum <lb/>latiſſima-<lb/>rum.</note> </div> <p> <s xml:id="echoid-s6408" xml:space="preserve">Cujus phaſeos vera proinde forma, ſecundum ea quæ ſu-<lb/> <anchor type="note" xlink:label="note-0305-02a" xlink:href="note-0305-02"/> pra circa annulum deſinivimus, ejuſmodi erit qualis hîc de- <pb o="572" file="0306" n="322" rhead="CHRIST. HUGENII"/> lineata cernitur, majori ellipſis diametro ad minorem ſe ha-<lb/>bente fere ut 5 ad 2. </s> <s xml:id="echoid-s6409" xml:space="preserve">Atque hæc ea figura eſt quam ab He-<lb/>velio, Ricciolo, & </s> <s xml:id="echoid-s6410" xml:space="preserve">Euſtachio de Divinis, anno 1648, 1649, <lb/>& </s> <s xml:id="echoid-s6411" xml:space="preserve">1650. </s> <s xml:id="echoid-s6412" xml:space="preserve">ſpectatam fuiſſe diximus. </s> <s xml:id="echoid-s6413" xml:space="preserve">Cujus itaque locum, hinc <lb/>inter ſigna ♊ & </s> <s xml:id="echoid-s6414" xml:space="preserve">♋, inde inter ♐ & </s> <s xml:id="echoid-s6415" xml:space="preserve">♑ conſtitui oportere cer-<lb/>tum eſt, quandoquidem iſtis annis ſignum ♊ Saturnus pera-<lb/>grabat & </s> <s xml:id="echoid-s6416" xml:space="preserve">in ♋ ſignum tranſibat. </s> <s xml:id="echoid-s6417" xml:space="preserve">Poſtea verò accuratius hunc <lb/>locum latiſſimè anſarum phaſeos inquirentes oſtendemus eum <lb/>cadere in 20{1/2} gr. </s> <s xml:id="echoid-s6418" xml:space="preserve">♊ & </s> <s xml:id="echoid-s6419" xml:space="preserve">♐.</s> <s xml:id="echoid-s6420" xml:space="preserve"/> </p> <div xml:id="echoid-div335" type="float" level="2" n="28"> <note position="right" xlink:label="note-0305-02" xlink:href="note-0305-02a" xml:space="preserve">TAB. L. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s6421" xml:space="preserve">Illud vero manifeſtum eſt, Saturnum, quoties hanc præ-<lb/> <anchor type="note" xlink:label="note-0306-01a" xlink:href="note-0306-01"/> fert ſpeciem, multo lucidiorem ſeſe intuentibus oſtendere <lb/>debere, quàm dum nullis brachiis auctus eſt: </s> <s xml:id="echoid-s6422" xml:space="preserve">quum tantun-<lb/>dem luminis fere ab hiſce emanet, quantum ab toto interio-<lb/>re diſco. </s> <s xml:id="echoid-s6423" xml:space="preserve">Atque ita ſemper quo propius verſus Cancri & </s> <s xml:id="echoid-s6424" xml:space="preserve"><lb/>Capricorni ſigna acceſſerit, eo majorem, aut certè ſplendi-<lb/>diorem, etiam abſque teleſcopio appariturum; </s> <s xml:id="echoid-s6425" xml:space="preserve">quippe an-<lb/>nuli ellipſis ſemper latius ſe pandente, ut in ſequentibus de-<lb/>clarabitur.</s> <s xml:id="echoid-s6426" xml:space="preserve"/> </p> <div xml:id="echoid-div336" type="float" level="2" n="29"> <note position="left" xlink:label="note-0306-01" xlink:href="note-0306-01a" xml:space="preserve">Clarion Sa-<lb/>turnus pro-<lb/>pter anſas <lb/>cernitur.</note> </div> <p> <s xml:id="echoid-s6427" xml:space="preserve">Sed & </s> <s xml:id="echoid-s6428" xml:space="preserve">aliud circa hanc phaſin obſervandum occurrit, mo-<lb/> <anchor type="note" xlink:label="note-0306-02a" xlink:href="note-0306-02"/> tus nimirum inſolens Saturnii comitis, qui quidem motus <lb/>obſervationibus anni 1659 adnotari jam cæpit, verùm hac <lb/>latiſſimâ anſarum phaſi exiſtente, omnium evidentiſſimè ut <lb/>ſeſe prodat neceſſe eſt. </s> <s xml:id="echoid-s6429" xml:space="preserve">Etenim quum eodem plano & </s> <s xml:id="echoid-s6430" xml:space="preserve">an-<lb/>nulus Saturni & </s> <s xml:id="echoid-s6431" xml:space="preserve">comitis orbita contineantur, aut certè pa-<lb/>rum diverſis; </s> <s xml:id="echoid-s6432" xml:space="preserve">conſtat ſimul cum annulo etiam orbitam hanc <lb/>latiſſimam omnium ellipſin nobis explicaturam: </s> <s xml:id="echoid-s6433" xml:space="preserve">eoque futu-<lb/>rum ut comes apparente motu altè ſupra Saturnum atque in-<lb/>fra tranſire conſpiciatur, ita ut corpore ejus vel radiis ni-<lb/>mium propinquis nequaquam nobis occultetur, quemadmo-<lb/>dum aliàs accidere ſolet. </s> <s xml:id="echoid-s6434" xml:space="preserve">Atque ea quidem ellipſis quam ta-<lb/>li motu deſcribere hiſce in locis cernetur, longitudinem la-<lb/>titudinis ſuæ duplam ſeſqui alteram habere invenietur, ſicut <lb/>& </s> <s xml:id="echoid-s6435" xml:space="preserve">annuli, quam diximus, ellipſis latiſſima. </s> <s xml:id="echoid-s6436" xml:space="preserve">Verum receden-<lb/>te hinc Saturno ſenſim anguſtior ipſa quoque evadet, quip-<lb/>pe annuli ellipſi ſemper ſimilis, adeo ut primum fulgore pla-<lb/>netæ tranſitus comitis conſpici prohibeatur, deinde corpo- <pb o="573" file="0307" n="323" rhead="SYSTEMA SATURNIUM."/> ris ipſius Saturni ante vel pone comitem objectu. </s> <s xml:id="echoid-s6437" xml:space="preserve">Ita nam-<lb/>que obſervationes anni 1655 & </s> <s xml:id="echoid-s6438" xml:space="preserve">1657, quo tempore exilia & </s> <s xml:id="echoid-s6439" xml:space="preserve"><lb/>quam maximè compreſſa brachia erant, oſtendunt comitem <lb/>etiam in ea recta tunc perpetuo apparuiſſe quæ per utrum-<lb/>que illorum extendebatur.</s> <s xml:id="echoid-s6440" xml:space="preserve"/> </p> <div xml:id="echoid-div337" type="float" level="2" n="30"> <note position="left" xlink:label="note-0306-02" xlink:href="note-0306-02a" xml:space="preserve">Lunæ Satur-<lb/>niæ motus <lb/>apparens <lb/>ellipticus.</note> </div> <p> <s xml:id="echoid-s6441" xml:space="preserve">Et hac quidem in re diſſidet Saturniæ Lunæ curſus ab eo <lb/>qui in noſtrate Luna animadvertitur; </s> <s xml:id="echoid-s6442" xml:space="preserve">ſiquidem hæc non æ-<lb/>quatoris noſtri ſed magis eclipticæ plano obnoxia eſt, à quo <lb/>quinque tantummodo partibus exorbitat.</s> <s xml:id="echoid-s6443" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6444" xml:space="preserve">Verum ad Saturni phaſes revertamur: </s> <s xml:id="echoid-s6445" xml:space="preserve">quo poſito ad H <lb/> <anchor type="note" xlink:label="note-0307-01a" xlink:href="note-0307-01"/> vel I, vel ex adverſo ad O vel P, paulo anguſtiorem jam <lb/>annuli ellipſin, eadem tamen qua prius longitudine nobis <lb/>aſpici oportet, quum ſupra planum annuli minus altè hoc <lb/>ſitu viſus noſter efferatur. </s> <s xml:id="echoid-s6446" xml:space="preserve">Unde illa phaſis exorietur quæ <lb/>in tabellæ LII. </s> <s xml:id="echoid-s6447" xml:space="preserve">ſerie octava poſita eſt; </s> <s xml:id="echoid-s6448" xml:space="preserve">vel, minus accuratè <lb/>perſpicere valentibus, ea quæ ſexta eſt efficietur.</s> <s xml:id="echoid-s6449" xml:space="preserve"/> </p> <div xml:id="echoid-div338" type="float" level="2" n="31"> <note position="right" xlink:label="note-0307-01" xlink:href="note-0307-01a" xml:space="preserve">Quomod@ <lb/>anſarum @@-<lb/>titudo ſenſim <lb/>contrah@uc. <lb/>TAB. L. <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s6450" xml:space="preserve">Rurſus autem eum ad K aut L Saturnus pervenit, vel ad <lb/>loca hiſce oppoſita N & </s> <s xml:id="echoid-s6451" xml:space="preserve">M, magis adhuc contrahi apparet <lb/>minorem ellipſis diametrum, quippe magis ex obliquo in-<lb/>ſpectam. </s> <s xml:id="echoid-s6452" xml:space="preserve">Ut jam ea Saturni facies proditura ſit quam nos <lb/>anno 1658 obſervavimus (T. </s> <s xml:id="echoid-s6453" xml:space="preserve">XLIX. </s> <s xml:id="echoid-s6454" xml:space="preserve">fig. </s> <s xml:id="echoid-s6455" xml:space="preserve">7.)</s> <s xml:id="echoid-s6456" xml:space="preserve">; </s> <s xml:id="echoid-s6457" xml:space="preserve">quæ minus bo-<lb/>nis teleſcopiis excepta degeneravit in illam quæ quinta in ta-<lb/>bella LII. </s> <s xml:id="echoid-s6458" xml:space="preserve">recenſetur; </s> <s xml:id="echoid-s6459" xml:space="preserve">pejoribus etiam in triſphæricam.</s> <s xml:id="echoid-s6460" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6461" xml:space="preserve">Deinde cum penè quadrante circuli à latiſſimæ phaſeos <lb/> <anchor type="note" xlink:label="note-0307-02a" xlink:href="note-0307-02"/> loco Saturnus remotus invenitur, velut in Q & </s> <s xml:id="echoid-s6462" xml:space="preserve">R, vel S <lb/>& </s> <s xml:id="echoid-s6463" xml:space="preserve">T; </s> <s xml:id="echoid-s6464" xml:space="preserve">uſque eo ellipſis annuli conſtringitur, ut ſi qua adhuc <lb/>rima ſuperſit viſui pervia, ea tamen propter exilitatem & </s> <s xml:id="echoid-s6465" xml:space="preserve">re-<lb/>liqui corporis ſplendorem conſpici nequeat; </s> <s xml:id="echoid-s6466" xml:space="preserve">ac proinde Sa-<lb/>turni forma ejuſmodi appareat qualem anno 1655 & </s> <s xml:id="echoid-s6467" xml:space="preserve">1657 <lb/>nos ſpectaſſe ſupra meminimus (T. </s> <s xml:id="echoid-s6468" xml:space="preserve">XLIX. </s> <s xml:id="echoid-s6469" xml:space="preserve">fig. </s> <s xml:id="echoid-s6470" xml:space="preserve">6. </s> <s xml:id="echoid-s6471" xml:space="preserve">T.</s> <s xml:id="echoid-s6472" xml:space="preserve">LI. </s> <s xml:id="echoid-s6473" xml:space="preserve">fig 5), <lb/>quæ quomodo aliis omnibus hactenus teleſcopiorum culpâ <lb/>triſphærica exiſtimata ſit, antea quoque oſtenſum eſt.</s> <s xml:id="echoid-s6474" xml:space="preserve"/> </p> <div xml:id="echoid-div339" type="float" level="2" n="32"> <note position="right" xlink:label="note-0307-02" xlink:href="note-0307-02a" xml:space="preserve">Quomodo <lb/>patulæ vide-<lb/>ri anſæ de-<lb/>ſinant.</note> </div> <p> <s xml:id="echoid-s6475" xml:space="preserve">Sola dehinc rotunda phaſis explicanda reſtat; </s> <s xml:id="echoid-s6476" xml:space="preserve">ad quam <lb/>priuſquam tranſeam, de inclinatione magnæ Saturni diame-<lb/>tri, ſive anſarum lineæ, ut ſupra eam vocavi, pauca adno-<lb/>tanda ſunt.</s> <s xml:id="echoid-s6477" xml:space="preserve"/> </p> <pb o="574" file="0308" n="324" rhead="CHRIST. HUGENII"/> <p> <s xml:id="echoid-s6478" xml:space="preserve">Ajo itaque in omnibus prædictarum phaſium, poſito, ut <lb/> <anchor type="note" xlink:label="note-0308-01a" xlink:href="note-0308-01"/> hactenus ſecimus, parallelum eſſe annuli planum plano æ-<lb/>quatoris, neceſſario fieri ut major diameter Saturni, ſive el-<lb/>lipſeos ejus cui annulus aſſimilatur, ſemper æquatori circulo <lb/>parallela ſpectetur, planè quemadmodum revera contingere <lb/>noſtris aliorumque obſervationibus ſupra probavimus. </s> <s xml:id="echoid-s6479" xml:space="preserve">Quum <lb/>enim viſus noſter reſpectu cæleſtium in centro æquatoris cir-<lb/>culi, atque adeo in plano ejus collocatus ſit, cui plano pla-<lb/>num annuli parallelum ponitur, neceſſe eſt maximam longi-<lb/>tudinem annularis ellipſeos eidem æquatori circulo paral-<lb/>lelam extendi: </s> <s xml:id="echoid-s6480" xml:space="preserve">eâdem prorſus ratione, quâ, ſi in aëre pla-<lb/>no terræ parallelum circulum ſuſpendamus, oculo noſtro <lb/>altiorem, indeque paſſibus aliquot retro abſiſtamus, major <lb/>diameter ellipſis quam ſuſpenſus circulus viſui offert Ho-<lb/>rizontis circulo æqui diſtare comperietur. </s> <s xml:id="echoid-s6481" xml:space="preserve">Hoc enim pro-<lb/>culdubio eventurum, cuivis vel citra experimentum intel-<lb/>ligitur.</s> <s xml:id="echoid-s6482" xml:space="preserve"/> </p> <div xml:id="echoid-div340" type="float" level="2" n="33"> <note position="left" xlink:label="note-0308-01" xlink:href="note-0308-01a" xml:space="preserve">Majorem <lb/>Saturni dia-<lb/>metrum ne-<lb/>ceſſario Æ-<lb/>quatori pa-<lb/>rallelam <lb/>ſemper vi-<lb/>deri.</note> </div> <p> <s xml:id="echoid-s6483" xml:space="preserve">Eſſe autem annuli planum plano æquatoris omnimodis <lb/> <anchor type="note" xlink:label="note-0308-02a" xlink:href="note-0308-02"/> parallelum, etſi obſervationibus, quas quidem expendimus <lb/>hactenus, minimè repugnantibus ſtatuere licuit; </s> <s xml:id="echoid-s6484" xml:space="preserve">exiguum ta-<lb/>men quid deeſſe, quo minus omnino perfectus ſit eorum pa-<lb/>ralleliſmus, alia ratione deprehenditur. </s> <s xml:id="echoid-s6485" xml:space="preserve">Etenim ſi prorſus <lb/>inter ſe parallela forent utraque plana, ſequeretur ut cum <lb/>in pr. </s> <s xml:id="echoid-s6486" xml:space="preserve">♈ & </s> <s xml:id="echoid-s6487" xml:space="preserve">♎ Saturnus ſpectaretur, omnium arctiſſima annu-<lb/>li ellipſis, & </s> <s xml:id="echoid-s6488" xml:space="preserve">quaſi recta linea exiſteret, latiſſima verò in pr. <lb/></s> <s xml:id="echoid-s6489" xml:space="preserve">♋ & </s> <s xml:id="echoid-s6490" xml:space="preserve">♑. </s> <s xml:id="echoid-s6491" xml:space="preserve">Atqui arctiſſimam annuli phaſium in gr. </s> <s xml:id="echoid-s6492" xml:space="preserve">20{1/2} ♍ & </s> <s xml:id="echoid-s6493" xml:space="preserve">♓ <lb/>cadere invenimus, ut in ſequentibus demonſtrabitur, ac pro-<lb/>inde latiſſimam in gr. </s> <s xml:id="echoid-s6494" xml:space="preserve">20{1/2} ♊ & </s> <s xml:id="echoid-s6495" xml:space="preserve">♐. </s> <s xml:id="echoid-s6496" xml:space="preserve">Non igitur exactè paral-<lb/>lela eſſe plana annuli & </s> <s xml:id="echoid-s6497" xml:space="preserve">æquatoris hinc patuit, ideoque nec <lb/>ſemper revera parallelam, licet ita videatur, magnam Satur-<lb/>ni diametrum æquatori circulo fore. </s> <s xml:id="echoid-s6498" xml:space="preserve">Quæſitoque per ſphæ-<lb/>rica triangula, quibus in locis & </s> <s xml:id="echoid-s6499" xml:space="preserve">quanta ſit maxima inclina-<lb/>tio, inveni eam gr. </s> <s xml:id="echoid-s6500" xml:space="preserve">4 8′. </s> <s xml:id="echoid-s6501" xml:space="preserve">quando Saturnus in gr. </s> <s xml:id="echoid-s6502" xml:space="preserve">25, 15′. </s> <s xml:id="echoid-s6503" xml:space="preserve">♊ <lb/>& </s> <s xml:id="echoid-s6504" xml:space="preserve">♐ verſatur. </s> <s xml:id="echoid-s6505" xml:space="preserve">Tali nimirum angulo, nec unquam majori, <lb/>anſarum linea circulum æquatori parallelum per Saturnum <lb/>tranſeuntem iis in locis interſecabit; </s> <s xml:id="echoid-s6506" xml:space="preserve">qui cum ſit adeo exi- <pb o="575" file="0309" n="325" rhead="SYSTEMA SATURNIUM."/> guus, vix puto obſervabilis erit: </s> <s xml:id="echoid-s6507" xml:space="preserve">aliis autem locis multo mi-<lb/>nus. </s> <s xml:id="echoid-s6508" xml:space="preserve">Nam iis quidem quæ quadrante Zodiaci inde abſunt, <lb/>nempe cum gr. </s> <s xml:id="echoid-s6509" xml:space="preserve">25. </s> <s xml:id="echoid-s6510" xml:space="preserve">15′. </s> <s xml:id="echoid-s6511" xml:space="preserve">♍ aut ♓ Saturnus obtinebit, prorſus <lb/>parallela æquatori eadem anſarum linea conſpicictur. </s> <s xml:id="echoid-s6512" xml:space="preserve">Eclipti-<lb/>cæ vero parallela fiet Saturno gr. </s> <s xml:id="echoid-s6513" xml:space="preserve">20{1/2} ♊ aut ♐ tenente: </s> <s xml:id="echoid-s6514" xml:space="preserve">quip-<lb/>pe quam in locis per quadrantem circuli inde diſtantibus, <lb/>nempe in gr. </s> <s xml:id="echoid-s6515" xml:space="preserve">20{1/2} ♍ & </s> <s xml:id="echoid-s6516" xml:space="preserve">♓, maximo angulo interſecat. </s> <s xml:id="echoid-s6517" xml:space="preserve">Hunc <lb/>autem maximum angulum quo linea anſarum eclipticam in-<lb/>terſecat eſſe circiter gr. </s> <s xml:id="echoid-s6518" xml:space="preserve">23{1/2} obſervationibus anni 1655, 1656 <lb/>& </s> <s xml:id="echoid-s6519" xml:space="preserve">1657 apparuit, talemque ad iſta ſupputanda adſumſimus. <lb/></s> <s xml:id="echoid-s6520" xml:space="preserve">Fortaſſe autem, longo ſæculorum lapſu, ſenſim omnia hæc <lb/>loca mutari continget, ſimili quodam motu Saturni globum <lb/>inclinante atque in Tellure noſtra eſt is qui præceſſiones æ-<lb/>quinoctiorum efficit: </s> <s xml:id="echoid-s6521" xml:space="preserve">atque ita phaſium quoque omnium lo-<lb/>ca transferri neceſſe erit. </s> <s xml:id="echoid-s6522" xml:space="preserve">Hoc verò non tam facile in cæ-<lb/>teris phaſibus quam in rotunda pateſcet, ad cujus conſide-<lb/>rationem deinceps veniendum.</s> <s xml:id="echoid-s6523" xml:space="preserve"/> </p> <div xml:id="echoid-div341" type="float" level="2" n="34"> <note position="left" xlink:label="note-0308-02" xlink:href="note-0308-02a" xml:space="preserve">Revera ta-<lb/>men mini-<lb/>mum quid <lb/>nonnun-<lb/>quam aber-<lb/>@are.</note> </div> <p> <s xml:id="echoid-s6524" xml:space="preserve">Schema itaque repetenti manifeſtum eſt Saturnum in-<lb/> <anchor type="note" xlink:label="note-0309-01a" xlink:href="note-0309-01"/> tegro orbitæ ſuæ circuitu non poſſe non ad eum locum bis <lb/>pervenire, (eſto in B & </s> <s xml:id="echoid-s6525" xml:space="preserve">D) ubi planum annuli recta ad nos <lb/>dirigatur, productumque in oculos noſtros incurrat. </s> <s xml:id="echoid-s6526" xml:space="preserve">Quod <lb/>quidem in locis è diametro oppoſitis contingeret, inque iiſ-<lb/>dem perpetuò, ſi ex Sole Saturni motum proſpiceremus; </s> <s xml:id="echoid-s6527" xml:space="preserve">at <lb/>nunc propter motum Terræ in ſua orbita annuum, nonnul-<lb/>la oritur inæqualitas. </s> <s xml:id="echoid-s6528" xml:space="preserve">Annulo igitur ſecundum latus inſpe-<lb/>cto, nec niſi rectam lineam referente, quum neutra plana-<lb/>rum ejus ſuperficierum appareat, ad nihilum rediguntur bra-<lb/>chia ac prorſus intereunt, ſolo rotundo Saturni corpore <lb/>quod intueamur reliquo. </s> <s xml:id="echoid-s6529" xml:space="preserve">Cujus phænomeni primum teſtem <lb/>Galilæum ſupra commemoravimus, qui anno 1612 ſimpli-<lb/>cem hanc Saturni formam obſervaverit. </s> <s xml:id="echoid-s6530" xml:space="preserve">Inde verò poſt an-<lb/>nos 30 Gaſſendum alioſque complures; </s> <s xml:id="echoid-s6531" xml:space="preserve">ac nos etiam deni-<lb/>que anno 1656. </s> <s xml:id="echoid-s6532" xml:space="preserve">Verùm enim ſi hôc tantum annuli poſitu <lb/>anſæ evaneſcerent, nequaquam tanto tempore perſtare ro-<lb/>tunda phaſis poſſet atque obſervationes teſtantur, toto enim <lb/>ſemeſtri rotundus Saturnus ſpectatus fuit. </s> <s xml:id="echoid-s6533" xml:space="preserve">Quamobrem por- <pb o="576" file="0310" n="326" rhead="CHRIST. HUGENII"/> ro quærendum eſt quæ cauſa, præter eam quam jam attuli-<lb/>mus, anſas quandoque cerni prohibeat: </s> <s xml:id="echoid-s6534" xml:space="preserve">eſt autem quæ ſe-<lb/>quitur manifeſtiſſima.</s> <s xml:id="echoid-s6535" xml:space="preserve"/> </p> <div xml:id="echoid-div342" type="float" level="2" n="35"> <note position="right" xlink:label="note-0309-01" xlink:href="note-0309-01a" xml:space="preserve">Rotundæ <lb/>phaſeos cau-<lb/>ſæ.</note> </div> <p> <s xml:id="echoid-s6536" xml:space="preserve">Frequenter nimirum evenire conſtat Saturno circa locos <lb/>modo dictos B vel D commorante, ut tum hujuſce tum So-<lb/>lis reſpectu ita poſiti ſimus, ut ſi produci planum annuli in-<lb/>telligatur, id inter nos ac Solem tranſiturum ſit: </s> <s xml:id="echoid-s6537" xml:space="preserve">velut in <lb/>ſchemate, cum Saturnus prope D conſiſtit, nos au-<lb/>tem cum Tellure in E. </s> <s xml:id="echoid-s6538" xml:space="preserve">Quo fit ut illam annuli ſuperſi-<lb/>ciem quæ ſolis radiis illuminatur conſpicere nequeamus, ſed <lb/>alteram tantummodo quæ tunc umbræ vices patitur. </s> <s xml:id="echoid-s6539" xml:space="preserve">Nulla <lb/>itaque ne hîc quidem brachia Saturno annulus præſtat, ſed <lb/>oculis noſtris ereptus orbum ac ſolitarium relinquit. </s> <s xml:id="echoid-s6540" xml:space="preserve">Atque <lb/>hæc cauſa ad continuos quinque aut ſex menſes rotundæ <lb/>phaſi interdum ſufficit, uti poſtea accuratius docebimus.</s> <s xml:id="echoid-s6541" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6542" xml:space="preserve">At non immerito dubitari poſſit, cur & </s> <s xml:id="echoid-s6543" xml:space="preserve">hôc & </s> <s xml:id="echoid-s6544" xml:space="preserve">præceden-<lb/>ti quoque annuli poſitu cum planum ejus rectà ad nos ver-<lb/>git, non ſaltem exterior ejus limbus à Sole illuſtratus appa-<lb/>reat: </s> <s xml:id="echoid-s6545" xml:space="preserve">quid enim dicemus? </s> <s xml:id="echoid-s6546" xml:space="preserve">anne tam tenue eſſe totum annu-<lb/>li corpus, ut licet revera ſplendeat extrema ejus margo, exi-<lb/>lior tamen ipſa ſit quam quæ noſtris teleſcopiis percipi poſ-<lb/>ſit? </s> <s xml:id="echoid-s6547" xml:space="preserve">Nequaquam: </s> <s xml:id="echoid-s6548" xml:space="preserve">verùm res eadem quæ cauſam hanc præ-<lb/> <anchor type="note" xlink:label="note-0310-01a" xlink:href="note-0310-01"/> texi vetat, eadem veram cauſam quoque haud dubiè ſuppe-<lb/>ditat: </s> <s xml:id="echoid-s6549" xml:space="preserve">faſcia nimirum illa reliquo Saturni diſco obſcurior, <lb/>quam & </s> <s xml:id="echoid-s6550" xml:space="preserve">rotundo illo apparente, & </s> <s xml:id="echoid-s6551" xml:space="preserve">rurſus brachiis aucto, <lb/>nobis viſam narravimus. </s> <s xml:id="echoid-s6552" xml:space="preserve">Hæc ita comparatum eſſe exte-<lb/>riorem illum annuli ambitum evincit, ut aliqua quidem craſ-<lb/>ſitudine præditus ſit, verum ejus naturæ, ut ſolis lumen, <lb/>vel nihil prorſus, vel certè leviter admodum reflectat. </s> <s xml:id="echoid-s6553" xml:space="preserve">Quia <lb/>enim in Saturno etiam brachiis prædito tractus iſte nigricans <lb/>animadvertitur, nempe cum ſuperſiciem annuli eandem quæ <lb/>à Sole illuſtratur deſpicimus; </s> <s xml:id="echoid-s6554" xml:space="preserve">quo poſitu nulla ejus regio <lb/>obumbrata oculis noſtris obverſa eſt; </s> <s xml:id="echoid-s6555" xml:space="preserve">ſequitur nigredinem <lb/>illam ex alia cauſa manare non poſſe, niſi quod ejuſcemodi <lb/>quadam materia annuli margo cooperta ſit, quæ non perin-<lb/>de ut reliqua ejus ſuperficies repercutiendo lumini ſit ido- <pb o="577" file="0311" n="327" rhead="SYSTEMA SATURNIUM."/> nea. </s> <s xml:id="echoid-s6556" xml:space="preserve">Sic in lunari quoque diſco maculas aliquas, cæteris <lb/>partibus multo obſcuriores, cernimus; </s> <s xml:id="echoid-s6557" xml:space="preserve">quæ quidem non <lb/>planè omni luce defectæ apparent, verùm ſi æquè procul <lb/>ac Saturnus à Sole diſtarent, ubi tantum centeſimam partem <lb/>ejus, quod nunc accipiunt, luminis ab illo mutuarentur, <lb/>credibile eſt penitus nobis inviſibiles fore, niſi quatenus lu-<lb/>cidioribus undique terminantur; </s> <s xml:id="echoid-s6558" xml:space="preserve">præſertim ſi tenuem modo <lb/>lineam, ut Saturnii annuli margo, conſtituerent. </s> <s xml:id="echoid-s6559" xml:space="preserve">Alioqui <lb/>vel illud forſitan dici poſſit, materiam quandam aquæ ſimi-<lb/>lem, aut certè lævi & </s> <s xml:id="echoid-s6560" xml:space="preserve">ſplendida ſuperſicie præditam, extre-<lb/>ma annuli præcingere, quæ unico tantum veluti puncto Solis <lb/>radios reflectens, nequaquam nobis conſpicua erit, ut ra-<lb/>tionibus opticis clarum eſt.</s> <s xml:id="echoid-s6561" xml:space="preserve"/> </p> <div xml:id="echoid-div343" type="float" level="2" n="36"> <note position="left" xlink:label="note-0310-01" xlink:href="note-0310-01a" xml:space="preserve">Quid ſit in <lb/>Saturni diſco <lb/>Zona nigri-<lb/>cans.</note> </div> <p> <s xml:id="echoid-s6562" xml:space="preserve">Sed quæ ad rotundam phaſin attinent ulterius etiam ex-<lb/>pendamus, in qua plura animadvertenda ſuperſunt. </s> <s xml:id="echoid-s6563" xml:space="preserve">Dixi-<lb/>mus paulò ante, tunc eam potiſſimum exiſtere, cum produ-<lb/>ctum annuli planum inter nos Solemque medium tranſit. <lb/></s> <s xml:id="echoid-s6564" xml:space="preserve">Hoc verò quando contingat, quo pacto cognoſcere poſſi-<lb/>mus, atque ex Aſtronomicis tabulis deſinire, deinceps ex-<lb/>plicandum eſt. </s> <s xml:id="echoid-s6565" xml:space="preserve">Sit itaque denuo Saturni orbita A B C, Tel-<lb/> <anchor type="note" xlink:label="note-0311-01a" xlink:href="note-0311-01"/> luris annua D E F, locus ſolis L.</s> <s xml:id="echoid-s6566" xml:space="preserve"/> </p> <div xml:id="echoid-div344" type="float" level="2" n="37"> <note position="right" xlink:label="note-0311-01" xlink:href="note-0311-01a" xml:space="preserve">TAB. @. <lb/>fig. 4.</note> </div> <p> <s xml:id="echoid-s6567" xml:space="preserve">Jam quia poſitum fuit, Saturni axem, qui ad annuli pla-<lb/>num erectus eſt, ſemper ſibi parallelum ferri, ſequitur com-<lb/>munem quoque planorum annuli & </s> <s xml:id="echoid-s6568" xml:space="preserve">orbitæ interſectionem <lb/>uni cuidam lineæ ſemper fore parallelam. </s> <s xml:id="echoid-s6569" xml:space="preserve">Eſto ea linea per <lb/>Solem ducta A C, quæ proinde in cælo locum Saturni æ-<lb/>quinoctiorum deſignabit. </s> <s xml:id="echoid-s6570" xml:space="preserve">Poſito igitur Saturno in H, <lb/>Tellure verò in D, utroque ad partem eandem rectæ A C, <lb/>verum ita ut minus ab ea Saturnus quam Tellus diſtet: </s> <s xml:id="echoid-s6571" xml:space="preserve">ne-<lb/>ceſſario interſectionis linea planorum annuli & </s> <s xml:id="echoid-s6572" xml:space="preserve">orbitæ Satur-<lb/>ni H M, quæ nimirum ex H parallela ducitur A C inter So-<lb/>lem L, Terramque in D poſitam excurret, ac propterea an-<lb/>nuli quoque planum manifeſtò inter utrumque medium in-<lb/>cedet. </s> <s xml:id="echoid-s6573" xml:space="preserve">Quod ubi contingit, rotundæ phaſi locum eſſe <lb/>demonſtravimus. </s> <s xml:id="echoid-s6574" xml:space="preserve">Contra verò, cum Saturnus quidem ac <lb/>Tellus ad eandem partem rectæ A C conſiſtunt, verum ita <pb o="578" file="0312" n="328" rhead="CHRIST. HUGENII"/> ut illa propius ad eam accedat, ut cum ille eſt in N, hæc in <lb/>F: </s> <s xml:id="echoid-s6575" xml:space="preserve">vel cum diverſas partes reſpectu A C obtinent, velut cum <lb/>Saturnus eſt in N, Tellus in f; </s> <s xml:id="echoid-s6576" xml:space="preserve">utroque caſu linea interſe-<lb/>ctionis annuli atque orbitæ Saturni, ut hîc N Q, quæ ni-<lb/>mirum ipſi A C æquidiſtans acta eſt, Terram Solemque ab <lb/>eadem parte habebit. </s> <s xml:id="echoid-s6577" xml:space="preserve">Unde intelligitur, utrovis poſitu <lb/>eandem annuli ſuperficiem à Sole & </s> <s xml:id="echoid-s6578" xml:space="preserve">nobis aſpici: </s> <s xml:id="echoid-s6579" xml:space="preserve">quod re-<lb/>quiri jam ſupra advertimus, ut brachia Saturni aut anſæ ap-<lb/>parere poſſint. </s> <s xml:id="echoid-s6580" xml:space="preserve">Quanquam ne ſic quidem ſemper cernuntur, <lb/>uti poſtea manifeſtum fiet.</s> <s xml:id="echoid-s6581" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6582" xml:space="preserve">Hi autem diverſi ſitus ut ex tabulis Aſtronomicis parvo <lb/> <anchor type="note" xlink:label="note-0312-01a" xlink:href="note-0312-01"/> negotio cognoſcantur, indeque rotunda phaſis, quatenus <lb/>ab hac cauſa pendet, prædici poſſit, ſciendum eſt, quan-<lb/>docunque Saturni locus apparens locuſque ex Sole ſive ec-<lb/>centricus, diverſas partes obtinent ejus loci quem inter fi-<lb/>xas deſignat recta A C, (quam in gr. </s> <s xml:id="echoid-s6583" xml:space="preserve">20 {1/2} ♍ & </s> <s xml:id="echoid-s6584" xml:space="preserve">♓ incidere <lb/>poſt hæc oſtendemus) id ejuſmodi ſitum Saturni ac Telluris <lb/>indicare qualis in H & </s> <s xml:id="echoid-s6585" xml:space="preserve">D, ut nimirum ab eadem parte am-<lb/>bo reperiantur rectæ A C, atque propior huic Saturnus con-<lb/>ſiſtat; </s> <s xml:id="echoid-s6586" xml:space="preserve">eoque tunc rotundam phaſin conſpiciendam dari. </s> <s xml:id="echoid-s6587" xml:space="preserve">Il-<lb/>lo enim poſito, ſit Saturnus in H. </s> <s xml:id="echoid-s6588" xml:space="preserve">Primùm itaque dico ab <lb/>hac parte lineæ A C etiam Tellurem conſiſtere. </s> <s xml:id="echoid-s6589" xml:space="preserve">Etenim ſi <lb/>ad alteram partem ſita eſſe dicatur, velut in P; </s> <s xml:id="echoid-s6590" xml:space="preserve">liquet rectam <lb/>ex P ad H ductam & </s> <s xml:id="echoid-s6591" xml:space="preserve">verſus ſixas porro continuatam, quæ <lb/>locum Saturni apparentem, itemque rectam L H, quæ pro-<lb/>ducta locum ejus ex Sole demonſtrat, ab eadem parte pun-<lb/>cti æquinoctiorum, quod L A producta inter fixas determi-<lb/>nat, utraque ea loca exhibituras; </s> <s xml:id="echoid-s6592" xml:space="preserve">contra quàm poſitum fue-<lb/>rat. </s> <s xml:id="echoid-s6593" xml:space="preserve">Eſt ergo Tellus neceſſario ab eadem parte lineæ A C, <lb/>qua Saturnus H. </s> <s xml:id="echoid-s6594" xml:space="preserve">Eſto jam ea alicubi puta in D. </s> <s xml:id="echoid-s6595" xml:space="preserve">Quoniam <lb/>igitur D H ad fixas protracta locum Saturni apparentem de-<lb/>ſignat, L H vero, uti diximus, locum ejus eccentricum, <lb/>inter quæ loca intercedere ponitur locus æquinoctiorum quò <lb/>tendit L A, neceſſe eſt rectam D H tandem ipſam L A in-<lb/>terſecare, ideoque punctum D, locum videlicet Telluris, <lb/>amplius diſtare à linea A C quam punctum H in quo Satur- <pb o="579" file="0313" n="329" rhead="SYSTEMA SATURNIUM."/> nus. </s> <s xml:id="echoid-s6596" xml:space="preserve">Conſtat itaque quando locus Saturni apparens & </s> <s xml:id="echoid-s6597" xml:space="preserve">ec-<lb/>centricus ad partes diverſas cadent loci æquinoctiorum, <lb/>neceſſario tunc Tellurem cum Saturno ab eadem parte rectæ <lb/>A C inventum iri, atque ita ut minus ab ea Saturnus remo-<lb/>veatur. </s> <s xml:id="echoid-s6598" xml:space="preserve">At vero quoties apparens locus Saturni itemque ec-<lb/>centricus ab eadem parte habentur rectæ A C, ſive loci æqui-<lb/>noctiorum Saturni, inde certo colligi ajo, Tellurem ac Sa-<lb/>turnum ita poſitos eſſe uti in F & </s> <s xml:id="echoid-s6599" xml:space="preserve">N. </s> <s xml:id="echoid-s6600" xml:space="preserve">Nempe ut vel diver-<lb/>ſas ad partes ipſius A C collocati ſint, vel ad eaſdem qui-<lb/>dem, ſed ita ut Saturnus amplius ab ea quam Tellus diſtet. <lb/></s> <s xml:id="echoid-s6601" xml:space="preserve">Ideoque, ſecundum ante demonſtrata, eandem annuli ſu-<lb/>perficiem quæ à Sole illuſtratur oculis quoque noſtris obver-<lb/>ſam eſſe. </s> <s xml:id="echoid-s6602" xml:space="preserve">Nam cum L N producta oſtendat inter fixas lo-<lb/>cum Saturni excentricum, F N vero locum ejus apparentem, <lb/>ac uterque cadere ponatur in partem eandem loci illius quem <lb/>inter fixas exhibet L C: </s> <s xml:id="echoid-s6603" xml:space="preserve">manifeſtum eſt rectam F N ab N <lb/>porrò productam neque ſecare debere neque parallelam eſſe <lb/>rectæ L C, ſed ſemper ab ea magis recedere. </s> <s xml:id="echoid-s6604" xml:space="preserve">Quare neceſ-<lb/>ſario punctum F vel propinquius erit rectæ A C quam pun-<lb/>ctum N, ſi ſint ambo ad eandem partem dictæ lineæ; </s> <s xml:id="echoid-s6605" xml:space="preserve">vel <lb/>N ad iſtam, F ad illam partem ſitum erit. </s> <s xml:id="echoid-s6606" xml:space="preserve">Quod proban-<lb/>dum erat.</s> <s xml:id="echoid-s6607" xml:space="preserve"/> </p> <div xml:id="echoid-div345" type="float" level="2" n="38"> <note position="left" xlink:label="note-0312-01" xlink:href="note-0312-01a" xml:space="preserve">Saturnus <lb/>quando ne-<lb/>ceſſario ro-<lb/>tundus ſpe-<lb/>ctari debeat.</note> </div> <p> <s xml:id="echoid-s6608" xml:space="preserve">Denique advertendum etiam, quando vel Saturni locus <lb/>ex Sole, vel apparens locus incidit in alterutrum æqui-<lb/>noctiorum, quæ determinat producta A C, nempe in gr. <lb/></s> <s xml:id="echoid-s6609" xml:space="preserve">20{1/2} ♍ vel ♓, neceſſario quoque rotundam phaſin exoriri. </s> <s xml:id="echoid-s6610" xml:space="preserve">Si <lb/>enim locus ex Sole inibi reperiatur, hoc eſt, ſi Saturnus oc-<lb/>cupet punctum A vel C, tunc planum annuli protractum <lb/>per Solem tranſire liquet: </s> <s xml:id="echoid-s6611" xml:space="preserve">unde ſequitur neutram annuli ſu-<lb/>perficiem tunc luce aliqua perfundi. </s> <s xml:id="echoid-s6612" xml:space="preserve">Rurſus vero cum locus <lb/>ejus apparens incidit in dictos æquinoctii locos, hoc eſt, <lb/>quando recta à Tellure ad Saturnum extenſa parallela ince-<lb/>dit ipſi A C; </s> <s xml:id="echoid-s6613" xml:space="preserve">velut cum Saturnus ſitus eſt in O, tellus in f, <lb/>eo poſitu productum annuli planum oculo noſtro occurrere <lb/>conſtat, quoniam communis ejus interſectio cum plano or-<lb/>bitæ Saturni ipſa eſt O f recta: </s> <s xml:id="echoid-s6614" xml:space="preserve">Unde ſit ut neutram ſuperſi- <pb o="580" file="0314" n="330" rhead="CHRIST. HUGENII"/> ciem annuli, etiamſi al@era jam illarum Solis radiis ſplen-<lb/>deat, conſpicere poſſimus.</s> <s xml:id="echoid-s6615" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6616" xml:space="preserve">Nunc illud videamus, quemnam Eclipticæ locum teneat <lb/>recta A C Saturni æquinoctiorum linea: </s> <s xml:id="echoid-s6617" xml:space="preserve">quam quidem in <lb/> <anchor type="note" xlink:label="note-0314-01a" xlink:href="note-0314-01"/> ♍ & </s> <s xml:id="echoid-s6618" xml:space="preserve">♓ gr. </s> <s xml:id="echoid-s6619" xml:space="preserve">20{1/2} cadere diximus, verum qua ratione id ex ob-<lb/>ſervationibus collegerimus declarandum eſt.</s> <s xml:id="echoid-s6620" xml:space="preserve"/> </p> <div xml:id="echoid-div346" type="float" level="2" n="39"> <note position="left" xlink:label="note-0314-01" xlink:href="note-0314-01a" xml:space="preserve">Quanam sa-<lb/>@urnicolis <lb/>ſint loca <lb/>aquinoctio-<lb/>rum.</note> </div> <p> <s xml:id="echoid-s6621" xml:space="preserve">Saturnum brachiis carentem ac prorſus rotundum ſpecta-<lb/>vimus à Menſe Decembri anni 1655, uſque in Jun. </s> <s xml:id="echoid-s6622" xml:space="preserve">1656. <lb/></s> <s xml:id="echoid-s6623" xml:space="preserve">Quæ phaſis ex ea cauſa, quam modo retulimus, tanto tem-<lb/>pore durare potuit, ſi linea A C eum ipſum locum obtinere <lb/>ponatur quem & </s> <s xml:id="echoid-s6624" xml:space="preserve">Saturnus tenebat tunc cum Soli eſſet oppo-<lb/>ſitus, hoc eſt, ♍ gr. </s> <s xml:id="echoid-s6625" xml:space="preserve">20. </s> <s xml:id="echoid-s6626" xml:space="preserve">Hinc enim fieri debuit ut tot il-<lb/>lo tempore Saturnus ac Tellus ab eadem parte lineæ A C ſi-<lb/>mul conſiſterent, atque ita quidem ut Saturnus ei vicinior <lb/>eſſet; </s> <s xml:id="echoid-s6627" xml:space="preserve">ſemel autem eodem momento, nempe oppoſitionis <lb/>tempore, in ipſa A C reperirentur: </s> <s xml:id="echoid-s6628" xml:space="preserve">unde non niſi rotunda <lb/>phaſis potuit exiſtere, ſicut ex prædictis intelligitur.</s> <s xml:id="echoid-s6629" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6630" xml:space="preserve">Cæterum uti non admodum verſimile eſt (etſi fieri po-<lb/>tuerit) eo ipſo tempore oppoſitionem illam contigiſſe cum <lb/>Saturnus in æquinocti ſui linea A C ſitus eſſet, ita non ve-<lb/>rebimur hanc lineam dimidio gradu emovere loco illo, at-<lb/>que in gr. </s> <s xml:id="echoid-s6631" xml:space="preserve">20{1/2} ♍ eam transferre; </s> <s xml:id="echoid-s6632" xml:space="preserve">quum hoc poſitu melius <lb/>ſatisſiat aliis rotundæ figuræ obſervationibus, anni 1642 & </s> <s xml:id="echoid-s6633" xml:space="preserve"><lb/>1612, ac nihilo ſecius noſtra illa anni 1656 conciliari quæat, <lb/>ut ex ſequentibus liquebit. </s> <s xml:id="echoid-s6634" xml:space="preserve">Poſita itaque A C in gr. </s> <s xml:id="echoid-s6635" xml:space="preserve">20{1/2} ♍ <lb/>& </s> <s xml:id="echoid-s6636" xml:space="preserve">♓ ſequitur rotundam phaſin, ab uſque principio Auguſti <lb/>anno 1642, ad Febr. </s> <s xml:id="echoid-s6637" xml:space="preserve">1643, continuè perſeveraſſe, prorſus <lb/>quemadmodum à Gaſſendo aliisque fuit adnotatum. </s> <s xml:id="echoid-s6638" xml:space="preserve">Erat <lb/>enim ineunte Auguſto locus Saturni eccentricus in 20 gr. </s> <s xml:id="echoid-s6639" xml:space="preserve">♓, <lb/>hoc eſt, dimidio gradu præcedebat locum æquinoctii: </s> <s xml:id="echoid-s6640" xml:space="preserve">at lo-<lb/>cus apparens in 24 gr. </s> <s xml:id="echoid-s6641" xml:space="preserve">♓, atque ita gr. </s> <s xml:id="echoid-s6642" xml:space="preserve">3{1/2} eundem æquinoctii <lb/>locum tranſierat. </s> <s xml:id="echoid-s6643" xml:space="preserve">Quare ſuperficiem annuli iliuminatam <lb/>Gaſſendus nequaquam conſpicere potuit, ſed eam quæ te-<lb/>nebris tegebatur ſibi obverſam habuit. </s> <s xml:id="echoid-s6644" xml:space="preserve">Et ſic quidem locus <lb/>Saturni apparens eccentricuſque perpetuo diverſis in parti-<lb/>bus loci æquinoctiorum perſtiterunt uſque ad initium fere <pb file="0315" n="331"/> <pb file="0315a" n="332"/> <anchor type="figure" xlink:label="fig-0315a-01a" xlink:href="fig-0315a-01"/> <anchor type="figure" xlink:label="fig-0315a-02a" xlink:href="fig-0315a-02"/> <anchor type="figure" xlink:label="fig-0315a-03a" xlink:href="fig-0315a-03"/> <anchor type="figure" xlink:label="fig-0315a-04a" xlink:href="fig-0315a-04"/> <pb file="0316" n="333"/> <pb o="581" file="0317" n="334" rhead="SYSTEMA SATURNIUM."/> Febr. </s> <s xml:id="echoid-s6645" xml:space="preserve">1643, unde non niſi rotunda forma prodire potuit, <lb/>ut antea demonſtratum eſt.</s> <s xml:id="echoid-s6646" xml:space="preserve"/> </p> <div xml:id="echoid-div347" type="float" level="2" n="40"> <figure xlink:label="fig-0315a-01" xlink:href="fig-0315a-01a"> <caption xml:id="echoid-caption158" style="it" xml:space="preserve">Pag. 580.<lb/>TAB. L.<lb/>Fig. 2.</caption> <variables xml:id="echoid-variables92" xml:space="preserve">R ♈ L D I T A N ♋ H G E P F K C Q O B M S</variables> </figure> <figure xlink:label="fig-0315a-02" xlink:href="fig-0315a-02a"> <caption xml:id="echoid-caption159" style="it" xml:space="preserve">Fig. 3.</caption> </figure> <figure xlink:label="fig-0315a-03" xlink:href="fig-0315a-03a"> <caption xml:id="echoid-caption160" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables93" xml:space="preserve">N Q F C P L E A M H O D f</variables> </figure> <figure xlink:label="fig-0315a-04" xlink:href="fig-0315a-04a"> <caption xml:id="echoid-caption161" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables94" xml:space="preserve">B A</variables> </figure> </div> <p> <s xml:id="echoid-s6647" xml:space="preserve">Idem quoque phænomenon anno 1627, proculdubio ſeſe <lb/>obtulit, ſed obſervatorum penuria an ſupinitate præteriit in-<lb/>obſervatum; </s> <s xml:id="echoid-s6648" xml:space="preserve">cum nemo, quod ſciam, prodiderit, utrum <lb/>Saturnus eo anno cum globulis apparuerit an ſecus.</s> <s xml:id="echoid-s6649" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6650" xml:space="preserve">Anno autem 1612, à Galilæo primo omnium rotunda fa-<lb/>cie animadverſus eſt. </s> <s xml:id="echoid-s6651" xml:space="preserve">Scribit is ad Marc. </s> <s xml:id="echoid-s6652" xml:space="preserve">Velſerum Epiſto-<lb/>la de Solaribus maculis 3. </s> <s xml:id="echoid-s6653" xml:space="preserve">circa ſolſtitium quidem illius an-<lb/>ni, tricorporem adhuc Saturnum ſibi viſum, inde verò, cum <lb/>duobus menſibus atque amplius teleſcopio eum inſpicere o-<lb/>miſiſſet, ut qui nullam formæ mutationem animo præſagi-<lb/>ret, poſtea nihil minus cogitanti ſolitarium oblatum, talem-<lb/>que deinceps in illam uſque diem, quæ prima Decembris <lb/>erat; </s> <s xml:id="echoid-s6654" xml:space="preserve">permanſiſle. </s> <s xml:id="echoid-s6655" xml:space="preserve">Quæ quidem obſervatio hactenus hypo-<lb/>theſi noſtræ conſentit, ut ſi eo tranſitu lineæ A C rotundus <lb/>apparere Saturnus debuerit, id hoc anno neque alio accide-<lb/>re fuerit neceſſe. </s> <s xml:id="echoid-s6656" xml:space="preserve">Sed cum per totum hunc annum Saturni <lb/>locus, quà apparens, quà eccentricus, ab eadem parte man-<lb/>ſerit loci æquinoctiorum, ſive ♓ gradus 20{1/2}; </s> <s xml:id="echoid-s6657" xml:space="preserve">videbatur ex <lb/>præcedentibus colligendum, ſalvis brachiis ſuis eum præter-<lb/>labi potuiſſe, quod tamen contra evenit. </s> <s xml:id="echoid-s6658" xml:space="preserve">Et ſexto quidem <lb/>Septembr. </s> <s xml:id="echoid-s6659" xml:space="preserve">dicti anni 1612, quo tempore jam rotundum Ga-<lb/>lilæus reperit, erat ejus locus ex Sole ſimul & </s> <s xml:id="echoid-s6660" xml:space="preserve">apparens, in <lb/>gr. </s> <s xml:id="echoid-s6661" xml:space="preserve">14, 44′ ♓. </s> <s xml:id="echoid-s6662" xml:space="preserve">Hoc eſt g.</s> <s xml:id="echoid-s6663" xml:space="preserve">5. </s> <s xml:id="echoid-s6664" xml:space="preserve">46′ citra locum æquinoctii ſui. <lb/></s> <s xml:id="echoid-s6665" xml:space="preserve">Adeo ut ſolares radii atque ii qui à viſu terricolarum flue-<lb/>bant in eandem annuli Saturnii ſuperficiem inciderent: </s> <s xml:id="echoid-s6666" xml:space="preserve">non <lb/>tamen angulo g. </s> <s xml:id="echoid-s6667" xml:space="preserve">5. </s> <s xml:id="echoid-s6668" xml:space="preserve">46′ deſuper in planum ejus directi, ſed <lb/>gr. </s> <s xml:id="echoid-s6669" xml:space="preserve">2. </s> <s xml:id="echoid-s6670" xml:space="preserve">15′ duntaxat. </s> <s xml:id="echoid-s6671" xml:space="preserve">Hic enim elevationis angulus, ut hoc <lb/>obiter adnotemus, ſic ſe habet ad illum gr. </s> <s xml:id="echoid-s6672" xml:space="preserve">5. </s> <s xml:id="echoid-s6673" xml:space="preserve">46′, quo Pla-<lb/>netæ locus diſtat ab æquinoctio ſuo, quemadmodum apud <lb/>nos declinatio Solis ad ejuſdem ab æquinoctio diſtantiam. </s> <s xml:id="echoid-s6674" xml:space="preserve"><lb/>Quandoquidem non aliud reſpectu Saturni eſt planum annu-<lb/>li ſui, atque nobis eſt planum æquatoris, ſimilique etiam <lb/>angulo ad eclipticæ planum inclinatur. </s> <s xml:id="echoid-s6675" xml:space="preserve">Proinde cum cog-<lb/>nita eſt diſtantia loci Saturni eccentrici ab æquinoctio ſuo, <pb o="582" file="0318" n="335" rhead="CHRIST. HUGENII"/> ſi ſcire libeat quali angulo Solis radii in ſuperficiem annuli <lb/>deferantur; </s> <s xml:id="echoid-s6676" xml:space="preserve">quærendum in tabula declinationis Eclipticæ, <lb/>qualis apud Aſtronomos in uſu eſt, quanta ſit declinatio <lb/>loci alicujus qui tantundem à princ. </s> <s xml:id="echoid-s6677" xml:space="preserve">Arietis abſit. </s> <s xml:id="echoid-s6678" xml:space="preserve">Ea enim <lb/>eſt ipſe angulus quæſitus. </s> <s xml:id="echoid-s6679" xml:space="preserve">Ac ſimili quoque ratione cum <lb/>data erit diſtantia loci Saturni apparentis ab æquinoctio ſuo, <lb/>hoc eſt, à gr. </s> <s xml:id="echoid-s6680" xml:space="preserve">20{1/2} ♍ aut ♓, invenire licebit quonam angu-<lb/>lo deſuper planum annuli nos inſpectemus; </s> <s xml:id="echoid-s6681" xml:space="preserve">qualemque pro-<lb/>pterea ellipſin ille nobis exhibere debeat. </s> <s xml:id="echoid-s6682" xml:space="preserve">Ex eadem nimi-<lb/>rum tabula capiendo declinationem huic diſtantiæ à pr. </s> <s xml:id="echoid-s6683" xml:space="preserve">A-<lb/>rietis reſpondentem, quæ declinatio rurſus hîc quæſitum <lb/>angulum æquat. </s> <s xml:id="echoid-s6684" xml:space="preserve">Quorum demonſtrationem adferre operæ <lb/>pretium non eſt, quippe quam Aſtronimiæ periti facile per-<lb/>ſpicient.</s> <s xml:id="echoid-s6685" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6686" xml:space="preserve">Illud verò videndum, qui evenerit ut viſu noſtro ac Sole <lb/>ſimul ſupra planum annuli plus quam 2 gr. </s> <s xml:id="echoid-s6687" xml:space="preserve">altis, nullum ta-<lb/>men brachiorum apud Saturnum veſtigium apparuerit. </s> <s xml:id="echoid-s6688" xml:space="preserve">An-<lb/>ne hæc fortaſſe cauſa fuit quod ellipſis annuli, tam obliquè <lb/>inſpecta ut magis tenuiſſimæ lineæ ſimilitudinem præferret, <lb/>exiliorem lucem emittebat quam quæ Galilæi tubo percipi <lb/>poſſet. </s> <s xml:id="echoid-s6689" xml:space="preserve">Hoc quidem non improbabile videri queat, niſi <lb/>eodem tubo, circa ſolſtitium ejuſdem anni 1612, brachia <lb/>Saturni ſive bini illi, ut tum putabant, ſatellites Galilæo a-<lb/>nimadverſi fuiſſent, quo tempore Saturnus obſervabatur <lb/>circa gr. </s> <s xml:id="echoid-s6690" xml:space="preserve">18, ♓. </s> <s xml:id="echoid-s6691" xml:space="preserve">hoc eſt gr. </s> <s xml:id="echoid-s6692" xml:space="preserve">2{1/2} ab æquinoctio ſuo remotus, <lb/>ideoque viſus noſter vix uno gradu plano annuli ſuperior. <lb/></s> <s xml:id="echoid-s6693" xml:space="preserve">Etenim ſi hoc poſitu brachia cerni potuere, multo clarius <lb/>cerni debuiſſent tunc cum gr. </s> <s xml:id="echoid-s6694" xml:space="preserve">2. </s> <s xml:id="echoid-s6695" xml:space="preserve">15′ ſupra idem planum vi-<lb/>ſus extolleretur, initio nimirum Septembr. </s> <s xml:id="echoid-s6696" xml:space="preserve">magiſque etiam <lb/>14 Nov. </s> <s xml:id="echoid-s6697" xml:space="preserve">ejuſdem anni 1612, tempore ſtationis Saturni; </s> <s xml:id="echoid-s6698" xml:space="preserve"><lb/>quia tunc gr. </s> <s xml:id="echoid-s6699" xml:space="preserve">11. </s> <s xml:id="echoid-s6700" xml:space="preserve">10′ ♓ tenebat, eoque gr. </s> <s xml:id="echoid-s6701" xml:space="preserve">9. </s> <s xml:id="echoid-s6702" xml:space="preserve">20′ ab æqui-<lb/>noctio ſuo aberat; </s> <s xml:id="echoid-s6703" xml:space="preserve">adeo ut angulo gr. </s> <s xml:id="echoid-s6704" xml:space="preserve">3. </s> <s xml:id="echoid-s6705" xml:space="preserve">42′ radii viſus in <lb/>planum annuli deſcenderent. </s> <s xml:id="echoid-s6706" xml:space="preserve">Atqui hîc, vel certe circa <lb/>hæc loca brachiorum expers manſit, quum ſatis utique dili-<lb/>gentem ac intentum obſervationibus Galilæum fuiſſe per eos <lb/>dies credibile ſit, qui rotundæ phaſeos miraculo nuper eſſet <lb/>excitatus.</s> <s xml:id="echoid-s6707" xml:space="preserve"/> </p> <pb o="583" file="0319" n="336" rhead="SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s6708" xml:space="preserve">Itaque in tenuitatem brachiorum, cur ea penitus latuerint <lb/>cauſa conferri non poteſt. </s> <s xml:id="echoid-s6709" xml:space="preserve">At neque transponendo amplius <lb/>Saturni æquinoctiorum locum, quicquam proficimus; </s> <s xml:id="echoid-s6710" xml:space="preserve">ne <lb/>quidem ſi mobilem ſtatuamus lento quodam progreſſu, ad <lb/>ſimilitudinem noſtrorum æquinoctiorum, quæ ſenſim in <lb/>præcedentia referuntur. </s> <s xml:id="echoid-s6711" xml:space="preserve">Quamobrem non aliam hujus rei <lb/>cauſam eſſe crediderim quam quæ nunc dicetur.</s> <s xml:id="echoid-s6712" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6713" xml:space="preserve">Certum eſt cognitæque experientiæ, ſuperficiem quamli-<lb/>bet ab eodem lumine plus minuſve illuſtrari, prout directis <lb/>vel obliquis radiis expoſita eſt: </s> <s xml:id="echoid-s6714" xml:space="preserve">quod & </s> <s xml:id="echoid-s6715" xml:space="preserve">à Galilæo, dialogo <lb/>1. </s> <s xml:id="echoid-s6716" xml:space="preserve">de Mundi ſyſtemate, optimè eſt demonſtratum. </s> <s xml:id="echoid-s6717" xml:space="preserve">Quam-<lb/>obrem cum exiguo admodum angulo ſupra annuli Saturnii <lb/>planum Sol attollitur, puta gradus unius vel duorum vel <lb/>etiam duorum cum dimidio, hoc eſt, cum Saturni locus ec-<lb/>centricus non ultra gr. </s> <s xml:id="echoid-s6718" xml:space="preserve">6. </s> <s xml:id="echoid-s6719" xml:space="preserve">ab æquinoctio ſuo aberit, etiam <lb/>prætenui tantum luce annuli ſuperficies à Sole impertietur. <lb/></s> <s xml:id="echoid-s6720" xml:space="preserve">Hanc autem, ita leviter ſplendentem, veriſimile eſt à meris <lb/>tenebris nos diſcernere non poſſe, præſertim Saturnii globi <lb/>vicino fulgore præpeditos; </s> <s xml:id="echoid-s6721" xml:space="preserve">eoque anſas tunc nullas animad-<lb/>verti. </s> <s xml:id="echoid-s6722" xml:space="preserve">Qua tamen in re illud ante omnia ſtatuere neceſſe <lb/>eſt, ſuperficiem annuli non eſſe aſperam montibuſque obſi-<lb/>tam, veluti maxima ex parte Lunæ noſtræ eſt ſuperficies: </s> <s xml:id="echoid-s6723" xml:space="preserve"><lb/>ſed æqualem planamque velut in iis Lunæ regionibus, quas <lb/>nonnulli maria eſſe ob inſignem planiciem arbitrati ſunt.</s> <s xml:id="echoid-s6724" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6725" xml:space="preserve">Alioqui enim, ſicut Luna plena circa diſci ſui extrema, <lb/>ubi tamen obliquos Solis radios excipit, nihilo languidiori <lb/>lumine quam verſus medias partes cernitur, ſcilicet quia illic <lb/>tota montoſa atque aſpera eſt, ita annuli quoque ſuperficies <lb/>ſi ſimili natura prædita foret, non ſecus obliquè incidenti-<lb/>bus radiis quam directis ſplendeſceret. </s> <s xml:id="echoid-s6726" xml:space="preserve">Quamobrem neceſ-<lb/>ſario plana, uti diximus, cenſenda eſt. </s> <s xml:id="echoid-s6727" xml:space="preserve">Atque hinc jam <lb/>intelligere licet cur Galilæo à Menſe Septembri anni 1612 <lb/>uſque in Febr. </s> <s xml:id="echoid-s6728" xml:space="preserve">1613, quoad nempe Heliacè Saturnus occi-<lb/>deret, nullæ circa eum anſæ conſpicerentur. </s> <s xml:id="echoid-s6729" xml:space="preserve">Quia nempe <lb/>inde à Septembri continuè locus Saturni eccentricus ad æqui-<lb/>noctii locum appropinquavit, eoque altitudo Solis ſupra an- <pb o="584" file="0320" n="337" rhead="CHRIST. HUGENII"/> nuli planum, quam initio Sept. </s> <s xml:id="echoid-s6730" xml:space="preserve">tantum gr. </s> <s xml:id="echoid-s6731" xml:space="preserve">2. </s> <s xml:id="echoid-s6732" xml:space="preserve">15′, fuiſſe <lb/>diximus, ſimul imminuta eſt. </s> <s xml:id="echoid-s6733" xml:space="preserve">Pari ratione nec Gaſſendus <lb/>brachia ulla percipere potuit menſe Febr. </s> <s xml:id="echoid-s6734" xml:space="preserve">anni 1643. </s> <s xml:id="echoid-s6735" xml:space="preserve">etſi <lb/>tunc quoque Sol oculuſque obſervantis eandem annuli ſuper-<lb/>ficiem intuebantur; </s> <s xml:id="echoid-s6736" xml:space="preserve">ſcilicet quia Saturni locus eccentricus in <lb/>gr. </s> <s xml:id="echoid-s6737" xml:space="preserve">26. </s> <s xml:id="echoid-s6738" xml:space="preserve">♓ inveniebatur, hoc eſt 5{1/2} gr. </s> <s xml:id="echoid-s6739" xml:space="preserve">à loco æquinoctii, <lb/>eoque altitudo Solis ſupra annuli ſuperſiciem tantum g. </s> <s xml:id="echoid-s6740" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s6741" xml:space="preserve">11′. </s> <s xml:id="echoid-s6742" xml:space="preserve">Denique & </s> <s xml:id="echoid-s6743" xml:space="preserve">anno 1656, menſis Martii diebus aliquot, <lb/>quibus ſimiliter eadem annuli ſuperficies ad nos Solemque <lb/>ſpectabat, multo minus brachiorum ullum veſtigium appa-<lb/>rere debuit, quod nunquam dimidio gr. </s> <s xml:id="echoid-s6744" xml:space="preserve">ſupra planum an-<lb/>nuli Sol attolleretur. </s> <s xml:id="echoid-s6745" xml:space="preserve">Nam reliquo tempore fulſionis illius, <lb/>diverſas partes loci æquinoctiorum locus Saturni ex Sole at-<lb/>que obſervatus locus obtinebant, unde ex ſuperius allata <lb/>cauſa brachiorum expertem cerni oportuit.</s> <s xml:id="echoid-s6746" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6747" xml:space="preserve">Neque verò magis conſpicuum nobis annulum fieri cre-<lb/>dendum eſt, ſi altius ſupra planum ejus oculus attollatur, <lb/>Sole tamen ex obliquo tantum radiis ſuis eum perſtringente, <lb/>quemadmodum contigit dicta Saturni ſtatione ad 14 Nov. <lb/></s> <s xml:id="echoid-s6748" xml:space="preserve">1612; </s> <s xml:id="echoid-s6749" xml:space="preserve">quippe quo tempore Sol gradu 1. </s> <s xml:id="echoid-s6750" xml:space="preserve">36′ ſupra planum <lb/>annuli aſcenderat, viſus autem noſter gr. </s> <s xml:id="echoid-s6751" xml:space="preserve">3. </s> <s xml:id="echoid-s6752" xml:space="preserve">53′. </s> <s xml:id="echoid-s6753" xml:space="preserve">Idem enim <lb/>in quacunqne plana ſuperficie experiri licet, in quam ſi à la-<lb/>tere radii luminis incidant, non apparcbit illuſtrior quocun-<lb/>que in loco viſus ſtatuatur, ſed tum demum, ſi radios à lu-<lb/>mine rectiores accipiat. </s> <s xml:id="echoid-s6754" xml:space="preserve">Uti contra quoque ſi amplius ad <lb/>lumen obverſa directiuſque illuminata fuerit, non refert <lb/>quàm obliquè in eam radii viſus incidant, ſed undecunque <lb/>ſpectata æque lucida apparebit. </s> <s xml:id="echoid-s6755" xml:space="preserve">Atque ita fit ut cum Sol <lb/>paulo altius ſupra planum annuli ſeſe extulit, tribus puta <lb/>gradibus aut paulo amplius; </s> <s xml:id="echoid-s6756" xml:space="preserve">viſu licet noſtro non niſi uno <lb/>gradu extante; </s> <s xml:id="echoid-s6757" xml:space="preserve">ſplendere jam nunc annulus incipiat, Satur-<lb/>noque brachia adnaſcantur. </s> <s xml:id="echoid-s6758" xml:space="preserve">Sicut anno 1612. </s> <s xml:id="echoid-s6759" xml:space="preserve">circa ſolſti-<lb/>tium accidit, Saturno ex Sole circa gr. </s> <s xml:id="echoid-s6760" xml:space="preserve">12. </s> <s xml:id="echoid-s6761" xml:space="preserve">♓, hoc eſt, gr. </s> <s xml:id="echoid-s6762" xml:space="preserve"><lb/>8{1/2} ab æquinoctio ſuo agente, ac proinde Solis altitudine ſu-<lb/>pra planum annuli graduum 3. </s> <s xml:id="echoid-s6763" xml:space="preserve">23′: </s> <s xml:id="echoid-s6764" xml:space="preserve">Galilæi verò oculo vix <lb/>uno gr. </s> <s xml:id="echoid-s6765" xml:space="preserve">ſupra idem planum elevato. </s> <s xml:id="echoid-s6766" xml:space="preserve">Videbatur enim Satur- <pb o="585" file="0321" n="338" rhead="SYSTEMA SATURNIUM."/> nus in gr. </s> <s xml:id="echoid-s6767" xml:space="preserve">18. </s> <s xml:id="echoid-s6768" xml:space="preserve">♓, hoc eſt 2{1/2} gr. </s> <s xml:id="echoid-s6769" xml:space="preserve">à loco æquinoctii, quæ di-<lb/>ſtantia dat declinationem gr. </s> <s xml:id="echoid-s6770" xml:space="preserve">unius.</s> <s xml:id="echoid-s6771" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6772" xml:space="preserve">Ut igitur ſecundum hæc de rotunda phaſi judicium ſera-<lb/> <anchor type="note" xlink:label="note-0321-01a" xlink:href="note-0321-01"/> tur, illud in primis reſpicere oportet, quod gradibus ab æ-<lb/>quinoctio ſuo, hoc eſt, à gr. </s> <s xml:id="echoid-s6773" xml:space="preserve">20{1/2}. </s> <s xml:id="echoid-s6774" xml:space="preserve">♍ vel ♓, Saturni locus ec-<lb/>centricus abſit. </s> <s xml:id="echoid-s6775" xml:space="preserve">Ac mihi quidem phænomena antecedentia <lb/>expendenti, videtur ita ſtatuendum, ut quoties non amplius <lb/>quam 6 gr. </s> <s xml:id="echoid-s6776" xml:space="preserve">circiter in alterutram partem, à dictis locis Sa-<lb/>turni locus ex Sole diſtabit, nunquam brachia ejus conſpici <lb/>queant, quicunque demum oculi noſtri ſitus fuerit; </s> <s xml:id="echoid-s6777" xml:space="preserve">hoc eſt, <lb/>nulla ducta ratione Saturni loci apparentis. </s> <s xml:id="echoid-s6778" xml:space="preserve">Quod tamen <lb/>ita accipiendum eſt, ſi perſpicillis non melioribus quam qui-<lb/>bus Galilæus & </s> <s xml:id="echoid-s6779" xml:space="preserve">Gaſſendus uſi ſunt Saturnus inſpiciatur. </s> <s xml:id="echoid-s6780" xml:space="preserve">Nam <lb/>ſi noſtris ſimilia adhibeantur, forſitan jam in illa 6 graduum <lb/>diſtantia tenue quoddam brachiorum exordium percipi poſ-<lb/>ſit. </s> <s xml:id="echoid-s6781" xml:space="preserve">Certè anno 1656, 13. </s> <s xml:id="echoid-s6782" xml:space="preserve">Oct. </s> <s xml:id="echoid-s6783" xml:space="preserve">renata illa vidimus, cum <lb/>Saturni locus ex Sole tantum gradibus 6. </s> <s xml:id="echoid-s6784" xml:space="preserve">46′. </s> <s xml:id="echoid-s6785" xml:space="preserve">æquinoctii ſui <lb/>punctum prætergreſſus eſſet. </s> <s xml:id="echoid-s6786" xml:space="preserve">Erant autem æquè conſpicua <lb/>ferè atque ante phaſin rotundam, anno 1655. </s> <s xml:id="echoid-s6787" xml:space="preserve">At ii qui mi-<lb/>noribus teleſcopiis eum tunc obſervitabant, ut Joh. </s> <s xml:id="echoid-s6788" xml:space="preserve">Hodier-<lb/>na, penè adhuc rotundum repererunt: </s> <s xml:id="echoid-s6789" xml:space="preserve">ait enim is, tantum <lb/>tenuiſſimos quoſdam veluti radios utrinque è lateribus Sa-<lb/>turnum vibrare viſum. </s> <s xml:id="echoid-s6790" xml:space="preserve">Quamobrem noſtris teleſcopiis for-<lb/>taſſe arctiores paulò prædicti limites ſumendi ſint; </s> <s xml:id="echoid-s6791" xml:space="preserve">quod ta-<lb/>men minimum quid erit, ſequentiumque annorum expe-<lb/>rientiâ definiendum.</s> <s xml:id="echoid-s6792" xml:space="preserve"/> </p> <div xml:id="echoid-div348" type="float" level="2" n="41"> <note position="right" xlink:label="note-0321-01" xlink:href="note-0321-01a" xml:space="preserve">Saturnus <lb/>brachiorum <lb/>expers futu-<lb/>rus quo mo-<lb/>do cognoſca-<lb/>tur.</note> </div> <p> <s xml:id="echoid-s6793" xml:space="preserve">Rurſus cum amplius 9. </s> <s xml:id="echoid-s6794" xml:space="preserve">gradib. </s> <s xml:id="echoid-s6795" xml:space="preserve">vel forſan ſupra 8. </s> <s xml:id="echoid-s6796" xml:space="preserve">gr. </s> <s xml:id="echoid-s6797" xml:space="preserve">ſo-<lb/>lum, ab æquinoctiis ſuis ſive 20{1/2} gr. </s> <s xml:id="echoid-s6798" xml:space="preserve">♍ aut ♓ locus Saturni <lb/>eccentricus diſtare invenietur, jam inde brachiis anſiſve præ-<lb/>ditum dicemus: </s> <s xml:id="echoid-s6799" xml:space="preserve">nullâ ne hic quidem apparentis loci conſi-<lb/>deratione, quoniam hinc jam apparens locus & </s> <s xml:id="echoid-s6800" xml:space="preserve">eccentricus <lb/>ab eadem parte loci æquinoctialis habentur; </s> <s xml:id="echoid-s6801" xml:space="preserve">ac propterea ea-<lb/>dem annuli ſuperficies, per ea quæ ſupra demonſtrata ſunt, <lb/>Soli & </s> <s xml:id="echoid-s6802" xml:space="preserve">viſui noſtro obverſa eſt, idque ita ut Sol plus tribus <lb/>gr. </s> <s xml:id="echoid-s6803" xml:space="preserve">ſupra ipſam elevatus ſit, viſus vero noſter ut minimum <lb/>gr. </s> <s xml:id="echoid-s6804" xml:space="preserve">uno: </s> <s xml:id="echoid-s6805" xml:space="preserve">quo fit ut brachia ſaltem tenuia animadverti de-<lb/>beant.</s> <s xml:id="echoid-s6806" xml:space="preserve"/> </p> <pb o="586" file="0322" n="339" rhead="CHRIST. HUGENII"/> <p> <s xml:id="echoid-s6807" xml:space="preserve">At cum nondum gr. </s> <s xml:id="echoid-s6808" xml:space="preserve">8 aut 9, plus verò quam ſenis, ec-<lb/>centricus locus Saturni à 20{1/2} gr. </s> <s xml:id="echoid-s6809" xml:space="preserve">♍ aut ♓ aberit, videtur <lb/>etiam apparens locus expendendus eſſe, nempe circa quadra-<lb/>tum quem vocant Saturni cum Sole aſpectum. </s> <s xml:id="echoid-s6810" xml:space="preserve">Fieri enim <lb/>poterit ut non ultra gradus unius ſemiſſem locus apparens <lb/>ab æquinoctii loco remotus ſit, ideoque viſus noſter tantum <lb/>12′ ſuper annuli ſuperficiem exurgat; </s> <s xml:id="echoid-s6811" xml:space="preserve">quæ cum à Sole ſimul <lb/>debiliter illuminetur, vix puto ſe videndam præbebit. </s> <s xml:id="echoid-s6812" xml:space="preserve">Quum <lb/>autem duo contingant quadrati aſpectus ſingulis Saturni ful-<lb/>ſionibus, priori eorum ita demum illud quod diximus per-<lb/>pendere opus erit, ſi locus planetæ ex Sole præcedat gr. <lb/></s> <s xml:id="echoid-s6813" xml:space="preserve">@0{1/2} ♍ aut ♓, poſteriori verò non niſi ſequatur. </s> <s xml:id="echoid-s6814" xml:space="preserve">Nam ſi ſe-<lb/>cus fuerit jam certo brachiis auctus cernetur.</s> <s xml:id="echoid-s6815" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6816" xml:space="preserve">Et ſic quidem quod maximè perplexum in toto hoc nego-<lb/>tio inerat, nulla diſſimulata difficultate, pertractavimus. </s> <s xml:id="echoid-s6817" xml:space="preserve">Il-<lb/>lud enim inquirere voluimus an certis locis rotunda phaſis <lb/>alligari poſſet, ita ut omnibus huc uſque obſervatis fieret <lb/>ſatis. </s> <s xml:id="echoid-s6818" xml:space="preserve">Quod & </s> <s xml:id="echoid-s6819" xml:space="preserve">præſtitimus tandem, etſi arctis adeo limiti-<lb/>bus circumſcripti, ut ne minimum quidem ultra citrave ab <lb/>iis diſcedere ſalvis apparentiis licuerit. </s> <s xml:id="echoid-s6820" xml:space="preserve">Simul verò hoc ope-<lb/>ræ pretium fecimus, quod de futuris Saturni phaſibus haud <lb/>dubia prænuntiare in poſterum licebit. </s> <s xml:id="echoid-s6821" xml:space="preserve">Etenim ſi modo ta-<lb/>bularum Aſtronomicarum ope inveniatur, quo tempore Sa-<lb/>turni locus ex ſole ſive eccentricus, cadat in gr. </s> <s xml:id="echoid-s6822" xml:space="preserve">20, 30′ ♍ <lb/>vel ♓; </s> <s xml:id="echoid-s6823" xml:space="preserve">ſenis menſibus qui id tempus præcedunt totidemque <lb/>ſequentibus brachia nulla percipi poſſe maniſeſtum erit; </s> <s xml:id="echoid-s6824" xml:space="preserve">quo-<lb/>niam ſcilicet in menſes ſingulos gradu uno Planetæ locus ex <lb/>Sole promovetur. </s> <s xml:id="echoid-s6825" xml:space="preserve">Præterquam quod aliquando uno etiam <lb/>atque altero menſe citius rotunda phaſis exiſtet, aliquando <lb/>tanto diutius videri perſeverabit; </s> <s xml:id="echoid-s6826" xml:space="preserve">ſi nempe quadrante cæli <lb/>per id tempus Saturnum à Sole diſtare contingat, locuſque <lb/>Planetæ eccentricus, ea qua diximus lege, præcedat vel ſe-<lb/>quatur ♍ aut ♓ gr. </s> <s xml:id="echoid-s6827" xml:space="preserve">20. </s> <s xml:id="echoid-s6828" xml:space="preserve">30′.</s> <s xml:id="echoid-s6829" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6830" xml:space="preserve">Secundùm hæc igitur, quoniam in gr. </s> <s xml:id="echoid-s6831" xml:space="preserve">20, 30′ ♓ incidit <lb/>locus eccentricus die 12. </s> <s xml:id="echoid-s6832" xml:space="preserve">Jan. </s> <s xml:id="echoid-s6833" xml:space="preserve">1672″, exiſtimo anno 1671 & </s> <s xml:id="echoid-s6834" xml:space="preserve"><lb/>1672 Saturnum brachiis cariturum; </s> <s xml:id="echoid-s6835" xml:space="preserve">non toto utriuſque anni <pb o="587" file="0323" n="340" rhead="SYSTEMA SATURNIUM."/> tempore, ſed hoc pacto, ut menſibus quidem Aprili, Ma-<lb/> <anchor type="note" xlink:label="note-0323-01a" xlink:href="note-0323-01"/> jo, Junioque anni 1671, nondum iis ſpoliatus cernatur, ſed <lb/>tenuia etiamnum utrinque extent qualia ſub finem anni 1656 <lb/>obſervavimus. </s> <s xml:id="echoid-s6836" xml:space="preserve">Julio autem aut Auguſto gracileſcant pror-<lb/>ſus tandemque diſpareant & </s> <s xml:id="echoid-s6837" xml:space="preserve">rotundum Saturnum relinquant. <lb/></s> <s xml:id="echoid-s6838" xml:space="preserve">Quo vultu non tantum ad Heliacum occaſum uſque conſpi-<lb/>ciendus erit, hoc eſt uſque ad finem Februarii anni 1672; </s> <s xml:id="echoid-s6839" xml:space="preserve"><lb/>ſed eundem quoque exoriens rurſus menſe Aprili referet, nec <lb/>amittet niſi Julio menſe aut Auguſto. </s> <s xml:id="echoid-s6840" xml:space="preserve">Circa hoc enim tem-<lb/>pus adcreſcere denuo brachia cernentur, ac paulatim eviden-<lb/>tiora fieri; </s> <s xml:id="echoid-s6841" xml:space="preserve">tandemque etiam ſingula prope medium Saturni <lb/>orbem biſida evadent, quemadmodum ab anno 1656 dein-<lb/>ceps augeri ea vidimus.</s> <s xml:id="echoid-s6842" xml:space="preserve"/> </p> <div xml:id="echoid-div349" type="float" level="2" n="42"> <note position="right" xlink:label="note-0323-01" xlink:href="note-0323-01a" xml:space="preserve">Quibus tem-<lb/>poribus ro-<lb/>tunda phaſis <lb/>reverſura ſit <lb/>prædicitur.</note> </div> <p> <s xml:id="echoid-s6843" xml:space="preserve">Manebunt autem ad annum uſque 1685, quando rurſus <lb/>evaneſcentia ſpectare continget, nempe circa Martii ini-<lb/>tium. </s> <s xml:id="echoid-s6844" xml:space="preserve">Inde rotundus planeta obſervabitur per annum inte-<lb/>grum, excepto eo tempore quo propter Solem deliteſcit. <lb/></s> <s xml:id="echoid-s6845" xml:space="preserve">Adeo ut menſe Martio ſequentis anni 1686 brachia demum <lb/>recuperaturus ſit, ſed tenui tantum incremento quandiu illa <lb/>vice viſendus erit. </s> <s xml:id="echoid-s6846" xml:space="preserve">Locum Saturni ex Sole reperio in gr. </s> <s xml:id="echoid-s6847" xml:space="preserve"><lb/>20{1/2} ♍ die ſecunda Sept. </s> <s xml:id="echoid-s6848" xml:space="preserve">anni 1685, unde prædictas viciſſitu-<lb/>dines conſequi neceſſe erat.</s> <s xml:id="echoid-s6849" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6850" xml:space="preserve">Rurſus verò anno 1701, 15 Jun. </s> <s xml:id="echoid-s6851" xml:space="preserve">redit Saturni locus ex <lb/>Sole ad ♓ gr. </s> <s xml:id="echoid-s6852" xml:space="preserve">20, 30′. </s> <s xml:id="echoid-s6853" xml:space="preserve">Unde circa finem anni 1700, princi-<lb/>piumque 1701, priuſquam Heliace occidat, rotundus luce-<lb/>bit, ac porro quoque ubi jam ex Solis radiis emerſerit, ea <lb/>facie permanebit ad initium uſque anni 1702: </s> <s xml:id="echoid-s6854" xml:space="preserve">à quo tempo-<lb/>re ad occaſum uſque Heliacum tenuia fortaſſe brachia cerni <lb/>poterunt. </s> <s xml:id="echoid-s6855" xml:space="preserve">Abſque dubio autem, menſe Majo ejuſdem an-<lb/>ni, cum denuo obſervari cœperit, reſtituta invenientur.</s> <s xml:id="echoid-s6856" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6857" xml:space="preserve">Atque ita porro ſingulis quatuordecim aut quindecim an-<lb/>nis, nimirum bis ad ſingulas Saturni in ſua orbita revolutio-<lb/>nes, rotunda forma conſpicienda dabitur; </s> <s xml:id="echoid-s6858" xml:space="preserve">neque enim un-<lb/>quam æquinoctii ſui locos tranſire quin brachia amittat po-<lb/>teſt. </s> <s xml:id="echoid-s6859" xml:space="preserve">Ac facile quidem accurata mutationum ejuſmodi tem-<lb/>pora veſtigiis noſtris inſiſtendo, quilibet inpoſterum præft- <pb o="588" file="0324" n="341" rhead="CHRIST. HUGENII"/> niet, certiuſque etiam ubi renaſcentes Saturni anſas conſpe-<lb/>xerit.</s> <s xml:id="echoid-s6860" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6861" xml:space="preserve">Cæterum ad eos obſervandos, quos hic adnotavimus, ala-<lb/>tæ formæ ad rotundam tranſitus, Aſtronomos omnes inten-<lb/>tos eſſe cupimus. </s> <s xml:id="echoid-s6862" xml:space="preserve">In quorum prædictione ſi à veritate aut ni-<lb/>hil aut pauxillum tantum aberraſſe nos invenient, tum procul <lb/>dubio cauſas quoque horum phænomenôn germanas qualeſ-<lb/>que revera ſunt ſibi explicitas credant. </s> <s xml:id="echoid-s6863" xml:space="preserve">Sin longè halluciua-<lb/>ti fuerimus, adeo ut brachiis præditus planeta cernatur, quo <lb/>tempore ex ſententia noſtra vel maximè iis carere deberet; <lb/></s> <s xml:id="echoid-s6864" xml:space="preserve">indicio id erit, quædam circa rotundam phaſin accidere no-<lb/>bis nondum ſatis perſpecta, nec ulli mortalium forſitan per-<lb/>videnda. </s> <s xml:id="echoid-s6865" xml:space="preserve">Nec tamen annuli propterea hypotheſin rejicien-<lb/>dam exiſtiment, quandiu reliquis quæ circa anſas animad-<lb/>vertentur ad amuſſim, uti hactenus, conſentiet.</s> <s xml:id="echoid-s6866" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6867" xml:space="preserve">Porro in hiſce anſatis Saturni phaſibus, cum nulla adeo <lb/>ſubita mutatio locum habeat, ideo nec tempora ſingularum <lb/>tam accuratè ut in rotunda diſtinguere neceſſe fuit; </s> <s xml:id="echoid-s6868" xml:space="preserve">ſed in <lb/>univerſum ſcire ſufficit, phaſin anſatarum latiſſimam medio <lb/> <anchor type="note" xlink:label="note-0324-01a" xlink:href="note-0324-01"/> tempore inter duas rotundas incidere; </s> <s xml:id="echoid-s6869" xml:space="preserve">ut proinde circa an-<lb/>num 1663 & </s> <s xml:id="echoid-s6870" xml:space="preserve">1664 illa reverſura ſit; </s> <s xml:id="echoid-s6871" xml:space="preserve">iterumque anno 1678 <lb/>& </s> <s xml:id="echoid-s6872" xml:space="preserve">1679; </s> <s xml:id="echoid-s6873" xml:space="preserve">ac poſtea anno 1693. </s> <s xml:id="echoid-s6874" xml:space="preserve">Reliquis enim inter iſtos in-<lb/>tercedentibus annis, tanto anguſtior annuli ellipſis cerne-<lb/>tur, quanto magis rotundæ phaſeos tempus imminebit. </s> <s xml:id="echoid-s6875" xml:space="preserve">Et <lb/>hactenus quidem eorum quæ circa Saturnum obſervantur <lb/>cauſas ac tempora digeſſimus.</s> <s xml:id="echoid-s6876" xml:space="preserve"/> </p> <div xml:id="echoid-div350" type="float" level="2" n="43"> <note position="left" xlink:label="note-0324-01" xlink:href="note-0324-01a" xml:space="preserve">Quando la-<lb/>tiſſimæ o-<lb/>mnium anſæ <lb/>apparituræ <lb/>ſint.</note> </div> <p> <s xml:id="echoid-s6877" xml:space="preserve">Nunc fortaſſe haud alienum propoſito videatur, ſi quem-<lb/>admodum ex noſtra hac ſtatione hucuſque ſyſtema ejus con-<lb/>templati ſumus, ita ad ipſius Saturni globum deinceps cogi-<lb/>tationem transferamus, atque illud diſpiciamus, qualis inde <lb/>univerſi facies, quænam futura ſint intervalla annorum men-<lb/>ſium ac dierum, quæve æſtatis hyemiſque viciſſitudo, ac <lb/>præſertim qualia ob annulum planetæ circundatum contin-<lb/>gere eum inhabitantibus neceſſe ſit: </s> <s xml:id="echoid-s6878" xml:space="preserve">ut nonnunquam longo <lb/>tempore aliquos conſpectu Solis privet, aliàs rurſum no-<lb/>cturnas tenebras imminuat, arcus lucidi ſpecie horizonti il- <pb o="589" file="0325" n="342" rhead="SYSTEMA SATURNIUM."/> lorum ſuperſtans. </s> <s xml:id="echoid-s6879" xml:space="preserve">Verum eo labore ſuperſedere rectius ar-<lb/>bitror, tum quod Aſtronomiæ gnaris ſingula hæc expende-<lb/>re, ſibique ob oculos ponere, non ſit difficilè futurum; </s> <s xml:id="echoid-s6880" xml:space="preserve">tum <lb/>quod in his ipſis multi ſint, (ut alios mittam) quibus otio-<lb/>ſa & </s> <s xml:id="echoid-s6881" xml:space="preserve">inanis ejuſmodi diſquiſitio videretur; </s> <s xml:id="echoid-s6882" xml:space="preserve">ac tanto quidem <lb/>magis, quanto abſurdius putant ut ammantia aliqua ratione <lb/>prædita Saturnum ac reliquos Planetarum incolere creda-<lb/>mus.</s> <s xml:id="echoid-s6883" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6884" xml:space="preserve">De magnitudine autem Saturni ejuſque à terris diſtantia <lb/> <anchor type="note" xlink:label="note-0325-01a" xlink:href="note-0325-01"/> operæ pretium videtur ut ſententiam noſtram his ſubjicia-<lb/>mus, ſiquidem & </s> <s xml:id="echoid-s6885" xml:space="preserve">alio modo illas inquiſivimus quam quo <lb/>fieri ſolet, & </s> <s xml:id="echoid-s6886" xml:space="preserve">aliquanto aliter de utraque ſtatuimus. </s> <s xml:id="echoid-s6887" xml:space="preserve">Nem-<lb/>pe ſphærulam illam annulo cinctam quam ſub exiguis modo <lb/>lineamentis ſpectavimus, diametrum quindecies circiter ma-<lb/>jorem habere, quam noſtra hæc in qua degimus terra, pu-<lb/>tandum eſt; </s> <s xml:id="echoid-s6888" xml:space="preserve">abeſſe autem à nobis, cum poxima eſt, terræ <lb/>diametris 100344; </s> <s xml:id="echoid-s6889" xml:space="preserve">cum longiſſimè diſtat, 122000. </s> <s xml:id="echoid-s6890" xml:space="preserve">Quæ <lb/>quidem ſequenti ratiocinio nituntur.</s> <s xml:id="echoid-s6891" xml:space="preserve"/> </p> <div xml:id="echoid-div351" type="float" level="2" n="44"> <note position="right" xlink:label="note-0325-01" xlink:href="note-0325-01a" xml:space="preserve">De Saturni <lb/>magnitudine <lb/>& à terris <lb/>diſtantia.</note> </div> <p> <s xml:id="echoid-s6892" xml:space="preserve">Docuit nos novo ſuo ac divinitus invento ſyſtemate Co-<lb/>pernicus, quamnam inter ſe proportionem ſervent ſingulo-<lb/>rum à Sole Planetarum diſtantiæ, apparentes vero eorundem <lb/>diametri quanto aliæ aliis majores ſint teleſcopii ope innote-<lb/>ſcit. </s> <s xml:id="echoid-s6893" xml:space="preserve">Collatis ergo invicem rationibus utriſque, tum diſtan-<lb/>tiæ tum magnitudinis apparentis, vera inde planetarum ad <lb/>ſe mutuo, nec non ad Solem magnitudo cognoſcitur. </s> <s xml:id="echoid-s6894" xml:space="preserve">Et <lb/> <anchor type="note" xlink:label="note-0325-02a" xlink:href="note-0325-02"/> ad Saturnum quod attinet, primùm annuli ejus diameter. <lb/></s> <s xml:id="echoid-s6895" xml:space="preserve">quum in minima à nobis diſtantia comprehendatur angulo <lb/>ſexaginta & </s> <s xml:id="echoid-s6896" xml:space="preserve">octo ſcrupulorum ſecundorum; </s> <s xml:id="echoid-s6897" xml:space="preserve">talem enim ad <lb/>ſummum reperimus; </s> <s xml:id="echoid-s6898" xml:space="preserve">cumque minima hæc Saturni diſtantia <lb/>ad mediocrem Solis diſtantiam ſit fere octupla; </s> <s xml:id="echoid-s6899" xml:space="preserve">ſequitur, ſi <lb/>tam propinquus nobis fieret Saturnus quam Sol in diſtantia <lb/>mediocri, apparituram tunc annuli diametrum octuplam ejus <lb/>quæ nunc apparet, hoc eſt 9′, 4″. </s> <s xml:id="echoid-s6900" xml:space="preserve">Solis autem diameter in <lb/>media diſtantia eſt 30, 30″. </s> <s xml:id="echoid-s6901" xml:space="preserve">Ergo revera ea erit proportio <lb/>diametri ann@@i Saturnii ad diametrum Solis quæ 9′, 4″, ad <lb/>30′, 30″, hoc eſt, fere quæ 11 ad 37. </s> <s xml:id="echoid-s6902" xml:space="preserve">Diameter vero Sa- <pb o="590" file="0326" n="343" rhead="CHRIST. HUGENII"/> turni ipſius, quam ſuperius diximus ad annuli diametrum ſe <lb/>habere ut 4 ad 9, hoc eſt fere ut 5 ad 11, ad diametrum <lb/>Solis erit paulo minor quam 5 ad 37.</s> <s xml:id="echoid-s6903" xml:space="preserve"/> </p> <div xml:id="echoid-div352" type="float" level="2" n="45"> <note position="right" xlink:label="note-0325-02" xlink:href="note-0325-02a" xml:space="preserve">Quanta ap-<lb/>pareat major <lb/>Saturni dia-<lb/>meter.</note> </div> <p> <s xml:id="echoid-s6904" xml:space="preserve">Quanta vero ſit Saturni diameter ad Telluris diametrum <lb/>collata haud æque certo definiri poteſt. </s> <s xml:id="echoid-s6905" xml:space="preserve">Aſtronomi ita hoc <lb/>inveſtigant, ut primò intervallum inter terram ac Solem ad <lb/>certum terreſtrium diametrorum numerum revocent, inde <lb/>quæſitam magnitudinum rationem eliciant. </s> <s xml:id="echoid-s6906" xml:space="preserve">At in illo Solis <lb/>terræque taxando intervallo nimium quantum inter ſe diſ-<lb/>ſentiunt; </s> <s xml:id="echoid-s6907" xml:space="preserve">nec mirum, quum nulla adhuc tolerabilis metho-<lb/>dus ad dimetiendum hoc ſpatium reperta ſit. </s> <s xml:id="echoid-s6908" xml:space="preserve">Nam ſive per <lb/>Eclipſes ſive per Lunæ dichotomias id deprehendere conen-<lb/>tur, facile oſtendi queat inanem operam ſumi. </s> <s xml:id="echoid-s6909" xml:space="preserve">Quare mihi <lb/>quidem unica illa, quam dicam, ratio reliqua eſſe videtur, <lb/>qua ſaltem veriſimiliter de Planetarum omnium ad terram <lb/>magnitudine ac diſtantia ſtatui poſſit. </s> <s xml:id="echoid-s6910" xml:space="preserve">Teleſcopio diametri <lb/>Planetarum apparentes explorentur; </s> <s xml:id="echoid-s6911" xml:space="preserve">ex his ſingulorum dein-<lb/>ceps ad Solem comparata magnitudo inveſtigetur, ut de Sa-<lb/>turno modo exemplum dedimus; </s> <s xml:id="echoid-s6912" xml:space="preserve">omnibuſque perpenſis, ea <lb/>Telluris ad cæteros aſſumatur magnitudo, quæ totius ſyſte-<lb/>matis ordini aptæque diſpoſitioni quàm maxime congruere <lb/>videbitur. </s> <s xml:id="echoid-s6913" xml:space="preserve">Ita cum proportio diametrorum Telluris ac reli-<lb/>quorum. </s> <s xml:id="echoid-s6914" xml:space="preserve">Planetarum ad Solis diametrum conſtituta fuerit, <lb/>conſtetque inſuper quot ſuis diametris Sol à nobis diſtet, ex <lb/>angulo videlicet quem ſubtendit diameter ejus apparens, jam <lb/>terræ quoque ad Solem magnitudo nota erit; </s> <s xml:id="echoid-s6915" xml:space="preserve">atque unà So-<lb/>lis diſtantia, tum à terra tum à cæteris Planetis, Terræ dia-<lb/>metris æſtimabitur. </s> <s xml:id="echoid-s6916" xml:space="preserve">Hanc itaque nunc viam ingredi placet, <lb/>ideoque ſicut Saturnum cum Sole modo comparavimus, ita <lb/>de reliquis quoque ſimile examen inſtituemus.</s> <s xml:id="echoid-s6917" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6918" xml:space="preserve">Jovis diameter, cum proximè nobis adeſt, ſexaginta qua-<lb/> <anchor type="note" xlink:label="note-0326-01a" xlink:href="note-0326-01"/> tuor ſecunda ſcrupula comprehendere mihi videtur; </s> <s xml:id="echoid-s6919" xml:space="preserve">quum-<lb/>que hæc ejus diſtantia ad mediam Solis diſtantiam ſit ut 26 <lb/>ad 5, hinc ſi fiat ut 5 ad 26 ita 64″ ad aliud, invenientur <lb/>5′, 35″, amplitudo anguli quem obtineret Jovis diameter ſi <lb/>tam propinquus nobis fieri intelligatur atque Sol in diſtantia <pb o="591" file="0327" n="344" rhead="SYSTEMA SATURNIUM."/> mediocri. </s> <s xml:id="echoid-s6920" xml:space="preserve">Sol autem hîc apparet diametro 30′, 30″. </s> <s xml:id="echoid-s6921" xml:space="preserve">Ergo <lb/>Jovialis diametri ad Solarem hæc proportio erit quæ 5′, <lb/>35″ ad 30′, 30″, hoc eſt paulo major quam 1 ad 5{1/2}.</s> <s xml:id="echoid-s6922" xml:space="preserve"/> </p> <div xml:id="echoid-div353" type="float" level="2" n="46"> <note position="left" xlink:label="note-0326-01" xlink:href="note-0326-01a" xml:space="preserve">Jovis terris <lb/>proximi dia-<lb/>meter appa-<lb/>rens quanta <lb/>obſervetur.</note> </div> <p> <s xml:id="echoid-s6923" xml:space="preserve">Accuratè etiam diametrum Veneris dimenſus ſum, ea <lb/> <anchor type="note" xlink:label="note-0327-01a" xlink:href="note-0327-01"/> quam poſtmodum exponam methodo, invenique cum ter-<lb/>ris proxima eſt, non majorem fore quam octoginta quinque <lb/>ſecundorum ſcrupulorum. </s> <s xml:id="echoid-s6924" xml:space="preserve">Eſt autem diſtantia hæc Veneris <lb/>perigææ ad mediam Solis à Tellure diſtantiam circiter ut 21 <lb/>ad 82. </s> <s xml:id="echoid-s6925" xml:space="preserve">Ergo ſi apud Solem Venus conſiſteret appareret ejus <lb/>diameter duntaxat 21″, 46″′. </s> <s xml:id="echoid-s6926" xml:space="preserve">Unde conſtat ita eſſe diame-<lb/>trum Veneris ad Solis diametrum ut 21″, 46″′ ad 30′, 30″, <lb/>hoc eſt ut 1 ad 84.</s> <s xml:id="echoid-s6927" xml:space="preserve"/> </p> <div xml:id="echoid-div354" type="float" level="2" n="47"> <note position="right" xlink:label="note-0327-01" xlink:href="note-0327-01a" xml:space="preserve">Veneris dia-<lb/>meter appa-<lb/>rens.</note> </div> <p> <s xml:id="echoid-s6928" xml:space="preserve">At Martis diametrum terris proximi non excedere 30″ de-<lb/> <anchor type="note" xlink:label="note-0327-02a" xlink:href="note-0327-02"/> prehendi, etſi obſervatione non tam exacta quam qua in Sa-<lb/>turno, Jove & </s> <s xml:id="echoid-s6929" xml:space="preserve">Venere uſus ſum, quippe cujus rationem no-<lb/>viſſimo Martis ad Tellurem acceſſu nondum inveneram. </s> <s xml:id="echoid-s6930" xml:space="preserve">Un-<lb/>de quum diſtantia Martis minima ſit ad mediocrem Solis ut <lb/>15 ad 41; </s> <s xml:id="echoid-s6931" xml:space="preserve">colligitur ratio diametri Martis ad diametrum So-<lb/>lis ea circiter quæ 1 ad 166. </s> <s xml:id="echoid-s6932" xml:space="preserve">Mars itaque duplo minor Ve-<lb/>nere ſecundum diametrum hac ratione efficitur. </s> <s xml:id="echoid-s6933" xml:space="preserve">Atque adeo <lb/>manifeſtum eſt in Planetis non ubique eum ſervari ordinem, <lb/>ut qui remotiores à Sole ſunt iidem quoque majori ſint mo-<lb/>le: </s> <s xml:id="echoid-s6934" xml:space="preserve">nam & </s> <s xml:id="echoid-s6935" xml:space="preserve">Jovis ſphæra Saturno ſine annulo major inventa <lb/>eſt. </s> <s xml:id="echoid-s6936" xml:space="preserve">Quo fit ut minus liquido de Terræ ad cæteros Plane-<lb/>tas proportione æſtimatio iniri poſſit. </s> <s xml:id="echoid-s6937" xml:space="preserve">Nam ſi pro ratione <lb/>ordinis magnitudines eſſent attributæ, ut Saturnus Jove <lb/>major eſſet, Jupiter Marte, hic Venere, hæc Mercurio; <lb/></s> <s xml:id="echoid-s6938" xml:space="preserve">inde quidem penè certo colligere liceret, Telluris magnitu-<lb/>dinem eſſe inter Martem Veneremque mediam. </s> <s xml:id="echoid-s6939" xml:space="preserve">Cum verò <lb/>in aliquibus contrarium deprehendatur, non æquè quid ſe-<lb/>quendum ſit apparet. </s> <s xml:id="echoid-s6940" xml:space="preserve">Veruntamen ut quatenus fieri poteſt <lb/>totius ſyſtematis concinnitas obſervetur, id nunc quoque <lb/>maximè conſentaneum videtur, ut ſicut loco media Terra <lb/>eſt inter Martem & </s> <s xml:id="echoid-s6941" xml:space="preserve">Venerem, ita quoque ſit magnitudine. </s> <s xml:id="echoid-s6942" xml:space="preserve"><lb/>Martis diametrum diximus diametri Solis eſſe {1/100}; </s> <s xml:id="echoid-s6943" xml:space="preserve">Veneris <lb/>vero diametrum {1/14}. </s> <s xml:id="echoid-s6944" xml:space="preserve">Inter utramque mediam igitur terræ dia- <pb o="592" file="0328" n="345" rhead="CHRIST. HUGENII"/> metrum ponendo fiet ea {1/111} diametri Solis. </s> <s xml:id="echoid-s6945" xml:space="preserve">Hujus autem <lb/>{5/37} diametro Saturni æquales repertæ ſunt; </s> <s xml:id="echoid-s6946" xml:space="preserve">ergo Saturni dia-<lb/>meter Telluris diametrum continebit quindecies, diameter <lb/>vero annuli Saturnii eandem Telluris diametrum circiter tri-<lb/>geſies & </s> <s xml:id="echoid-s6947" xml:space="preserve">quater. </s> <s xml:id="echoid-s6948" xml:space="preserve">Unde eximia horum corporum magnitudo <lb/>cognoſcitur; </s> <s xml:id="echoid-s6949" xml:space="preserve">quæ ſane omnem ab aliis hactenus traditam fa-<lb/>cile exuperat.</s> <s xml:id="echoid-s6950" xml:space="preserve"/> </p> <div xml:id="echoid-div355" type="float" level="2" n="48"> <note position="right" xlink:label="note-0327-02" xlink:href="note-0327-02a" xml:space="preserve">Martis diæ<unsure/>-<lb/>meter.</note> </div> <p> <s xml:id="echoid-s6951" xml:space="preserve">Hinc verò & </s> <s xml:id="echoid-s6952" xml:space="preserve">intervallum inter Terram ac Solem neceſſa-<lb/>rio omnium exiſtimatione majus conflabitur. </s> <s xml:id="echoid-s6953" xml:space="preserve">Si enim dia-<lb/>meter terræ diametri Solis {1/111} continet; </s> <s xml:id="echoid-s6954" xml:space="preserve">Solis autem diame-<lb/>ter ſuæ à nobis mediæ diſtantiæ {1/113} æquat, uti ſequitur ex eo <lb/>quod diameter ejus obſervetur 30′, 30″; </s> <s xml:id="echoid-s6955" xml:space="preserve">erit certè Terræ dia-<lb/>meter {1/12543} diſtantiæ quæ eſt inter ipſam ac Solem. </s> <s xml:id="echoid-s6956" xml:space="preserve">Deinde <lb/>cum Saturni minima diſtantia ſit ad mediam Solis diſtantiam <lb/>ferme octupla, hinc Saturni, cum erræ proximus eſt, di-<lb/>ſtantia habebitur diametrorum terreſtrium 100344; </s> <s xml:id="echoid-s6957" xml:space="preserve">cum verò <lb/>plurimum aberit, circiter 122000.</s> <s xml:id="echoid-s6958" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6959" xml:space="preserve">Fateor equidem lubrica eatenus ratione hæc niti, quate-<lb/>nus nimirum Terræ magnitudinem inter Martem Venerm-<lb/>que mediam adſumſimus, nullo niſi veriſimilitudinis argu-<lb/>mento: </s> <s xml:id="echoid-s6960" xml:space="preserve">adeoque vel millenis aliquot Terræ diametris facile <lb/>à veritate aberrari potuiſſe Verùm ut jam duplo majora <lb/>minorave quam reipſa ſunt intervalla iſta definiverimus, aut <lb/>triplo etiam; </s> <s xml:id="echoid-s6961" xml:space="preserve">non tamen parum videri debet hactenus ſaltem <lb/>menſuram eorum comprehendiſſe, quum alia nulla ratio ſup-<lb/>petat qua non vel decuplo major error timendus ſit. </s> <s xml:id="echoid-s6962" xml:space="preserve">Ita e-<lb/>nim omnino exiſtimo. </s> <s xml:id="echoid-s6963" xml:space="preserve">Ad reliquum verò calculum quod at-<lb/>tinet, quo Planetarum diametros Solis diametro comparavi-<lb/>mus, ſciendum eſt nihil in eo conjecturæ tribui, ſed ex iis <lb/>quæ data ſunt certa ratione illum procedere. </s> <s xml:id="echoid-s6964" xml:space="preserve">Atque adeo <lb/>poſitis iis Planetarum diametris apparentibus quas a nobis <lb/>obſervatas diximus, non poſle non Solis diametrum ad dia-<lb/> <anchor type="note" xlink:label="note-0328-01a" xlink:href="note-0328-01"/> metrum annuli Saturno circumdati ſeſe habere ut<unsure/> 37 ad 11; <lb/></s> <s xml:id="echoid-s6965" xml:space="preserve">ad diametrum vero Saturni ipſius, ut 37 ad 5; </s> <s xml:id="echoid-s6966" xml:space="preserve">ad Jovis dia-<lb/>metrum ut 11 ad 2; </s> <s xml:id="echoid-s6967" xml:space="preserve">ad Martis ut 166 ad 1; </s> <s xml:id="echoid-s6968" xml:space="preserve">ad Veneris ut <lb/>84 ad 1. </s> <s xml:id="echoid-s6969" xml:space="preserve">De Mercurio non definiam priuſquam ritè eum di- <pb o="593" file="0329" n="346" rhead="SYSTEMA SATURNIUM."/> menſus fuero; </s> <s xml:id="echoid-s6970" xml:space="preserve">quod hactenus non ſucceſſit, cum ob exilita-<lb/>tem ſideris, tum quod horizonti plerumque vicinum inve-<lb/>niatur, ubi vapores è Terra ſurgentes tremula quadam refra-<lb/>ctione figuram ejus præciſo ambitu terminari non ſinunt. </s> <s xml:id="echoid-s6971" xml:space="preserve">Pa-<lb/>tet autem & </s> <s xml:id="echoid-s6972" xml:space="preserve">Saturnii globi diametrum ad diametrum Jovis ra-<lb/>tionem habere quam 55 ad 74: </s> <s xml:id="echoid-s6973" xml:space="preserve">diametrum vero Martis conti-<lb/>nere amplius quam vicies & </s> <s xml:id="echoid-s6974" xml:space="preserve">bis, ad Veneris diametrum eſſe <lb/>ut 34 ad 3. </s> <s xml:id="echoid-s6975" xml:space="preserve">Jovis item diametrum diametri Martis amplius <lb/>quam trigecuplam eſſe; </s> <s xml:id="echoid-s6976" xml:space="preserve">diametri vero Veneris amplius <lb/>quam quindecuplam; </s> <s xml:id="echoid-s6977" xml:space="preserve">ac denique Martis diam. </s> <s xml:id="echoid-s6978" xml:space="preserve">ad diametrum <lb/>Veneris circiter ſubduplam. </s> <s xml:id="echoid-s6979" xml:space="preserve">Quæ omnes proportiones ratæ <lb/>fixæque permanent, quantacunque diſtantia Solem inter ac <lb/>Tellurem ſtatuatur, ſi modo apparentes diametri quales tra-<lb/>didimus retineantur. </s> <s xml:id="echoid-s6980" xml:space="preserve">Haſce autem quo modo obſervaveri-<lb/>mus denique dicendum eſt, atque eo magis quod longè ab <lb/>aliorum ſententia alicubi recedamus: </s> <s xml:id="echoid-s6981" xml:space="preserve">veluti cum Veneris dia-<lb/>metrum triplo minorem aſſerimus quam à Ricciolo definita <lb/>eſt, qui tamen ſumma cura circa hæc verſatus videtur. </s> <s xml:id="echoid-s6982" xml:space="preserve">No-<lb/>ſtrum igitur artificium eſt hujuſmodi.</s> <s xml:id="echoid-s6983" xml:space="preserve"/> </p> <div xml:id="echoid-div356" type="float" level="2" n="49"> <note position="left" xlink:label="note-0328-01" xlink:href="note-0328-01a" xml:space="preserve">Diametri So-<lb/>lis ad Plane-<lb/>tarum dia-<lb/>metre<unsure/>s ratio.</note> </div> <p> <s xml:id="echoid-s6984" xml:space="preserve">Locus quidam eſt intra tubos qui Solis convexis vitris in-<lb/> <anchor type="note" xlink:label="note-0329-01a" xlink:href="note-0329-01"/> ſtructi ſunt, circiter altero tanto amplius quàm convexum <lb/>oculare ab oculo diſtans; </s> <s xml:id="echoid-s6985" xml:space="preserve">quo in loco ſi quid intra tubi cavi-<lb/>tatem viſui objiciatur, quantumvis ſubtile aut exiguum, id di-<lb/>ſtinctè prorſus ambituque exquiſitè terminato conſpicitur, <lb/>atque ita pro ratione latitudinis ſuæ partem aliquam rei luci-<lb/>dæ, velut Lunæ per teleſcopium ſpectatæ, viſui ſubducit. <lb/></s> <s xml:id="echoid-s6986" xml:space="preserve">Exacta loci determinatio, his quibus nullo vitio viſus labo-<lb/>rat, in focum convexi ocularis cadit; </s> <s xml:id="echoid-s6987" xml:space="preserve">myopi aliquanto pro-<lb/>pinquius punctum accipiendum eſt, contraque, qui tantum <lb/>à longinquo clare vident, paulo remotius; </s> <s xml:id="echoid-s6988" xml:space="preserve">quod experientia <lb/>protinus docere poteſt. </s> <s xml:id="echoid-s6989" xml:space="preserve">Hîc igitur ſi primò annulus ſtatua-<lb/>tur cum foramine paulo anguſtiore quam ſit vitrum ipſum o-<lb/>culo proximum, eo tota tubi apertura, ſive ſpatium circulare <lb/>quod uno obtutu in cælo detegitur, præcisâ circumſerentiâ <lb/>deſcriptum habetur. </s> <s xml:id="echoid-s6990" xml:space="preserve">Cujus ſpatii diameter, quot ſcrupula <lb/>comprehendat, aliquo pacto inquirendum eſt, atque optimè <pb o="594" file="0330" n="347" rhead="CHRIST. HUGENII"/> quidem ex tranſitu ſideris alicujus, cujus tempus numeretur <lb/>vibrationibus perpendiculi, vel ope Horologii noſtri oſcil-<lb/>latorii nuper inventi, teleſcopio interim immoto manente. <lb/></s> <s xml:id="echoid-s6991" xml:space="preserve">Scimus enim 4 ſcrupulis horariis unum cæli gradum & </s> <s xml:id="echoid-s6992" xml:space="preserve">exi-<lb/>guum quid amplius tranſire: </s> <s xml:id="echoid-s6993" xml:space="preserve">ideoque ſi verbi gratia numeren-<lb/>tur ſcrupula ſecunda 69 interea dum ſtella quædam fixa to-<lb/>tam teleſcopii capacitatem emetitur, argumento id erit 17{1/4} <lb/>ſcrupula prima, teleſcopii hujuſmodi apertura comprehendi, <lb/>ſicut noſtro evenit. </s> <s xml:id="echoid-s6994" xml:space="preserve">Quo invento virgulam unam atque al-<lb/>teram ex ære aliave materia parare oportet, decreſcente pau-<lb/>latim latitudine, tubumque perforare utrinque circa locum <lb/>illum paulo ante memoratum, quo poſſint in ipſo ejus pun-<lb/>cto virgulæ tranſverſæ ante oculum obtendi. </s> <s xml:id="echoid-s6995" xml:space="preserve">Cum igitur <lb/>Planetæ alicujus diametrum metiri cupimus, adhibitâ eo quo <lb/>diximus loco virgulâ, notandum eſt quænam hujus latitudo <lb/>totum Planetam contegere poſſit. </s> <s xml:id="echoid-s6996" xml:space="preserve">Eâ enim latitudine acuto <lb/>deinde circino acceptâ, atque ad totius foraminis amplitudi-<lb/>nem collatâ, Planetæ diameter apparens facili ratiocinio in-<lb/> <anchor type="note" xlink:label="note-0330-01a" xlink:href="note-0330-01"/> noteſcet. </s> <s xml:id="echoid-s6997" xml:space="preserve">Sic die 29 Dec. </s> <s xml:id="echoid-s6998" xml:space="preserve">1658 diametro Veneris inveni-<lb/>mus convenire virgulam cujus latitudo æquabat {1/20} totius fo-<lb/>raminis: </s> <s xml:id="echoid-s6999" xml:space="preserve">eſt autem hoc, uti diximus, 17′, 15″. </s> <s xml:id="echoid-s7000" xml:space="preserve">Ergo Vene-<lb/>ris diameter erat 51″, 45″′. </s> <s xml:id="echoid-s7001" xml:space="preserve">Diſtantia autem Planetæ ad mini-<lb/>mam ſuam à Terris diſtantiam ſe habebat circiter ut 27 ad 16, <lb/>ergo diameter ejus terris proximæ efficitur 87′, 20″′. </s> <s xml:id="echoid-s7002" xml:space="preserve">Rurſus <lb/>anno 1659, 8 Mart. </s> <s xml:id="echoid-s7003" xml:space="preserve">hora 6 mat. </s> <s xml:id="echoid-s7004" xml:space="preserve">Veneris diametrum obſer-<lb/>vavimus quæ æquabat {4/67} aperturæ teleſcopii. </s> <s xml:id="echoid-s7005" xml:space="preserve">Ideoque erat <lb/>61″, 30″′. </s> <s xml:id="echoid-s7006" xml:space="preserve">Diſtantia autem eo tempore ad minimam Veneris <lb/>à terris diſtantiam ſe habebat ut 430 ad 316, ergo diameter <lb/>ejus maxima fit 83″, 40″′. </s> <s xml:id="echoid-s7007" xml:space="preserve">Sed & </s> <s xml:id="echoid-s7008" xml:space="preserve">alias eadem methodo ſem-<lb/>per tantum paulo majorem minoremve invenimus, nam ſæ-<lb/>pius examen hoc repetivimus, atque ex omnibus medium <lb/>quid ſumentes, 85″ pro maxima Veneris diametro ſupra ſta-<lb/>tuimus. </s> <s xml:id="echoid-s7009" xml:space="preserve">Hanc autem Ricciolus 4″, 8″, taxavit, atque adeo <lb/>triplo quam nos majorem exiſtimat, ſed procul dubio nuda <lb/>illa oculi æſtimatione, qua hîc uſus eſt, in tantum deceptus <lb/>fuit. </s> <s xml:id="echoid-s7010" xml:space="preserve">Nam Saturni Joviſque diametros, quas methodo cer- <pb o="595" file="0331" n="348" rhead="SYSTEMA SATURNIUM."/> tiori, appulſu nimirum ad fixas inveſtigavit, eadem fere qua <lb/>& </s> <s xml:id="echoid-s7011" xml:space="preserve">nos amplitudine definiit, paullum tantum excedens: </s> <s xml:id="echoid-s7012" xml:space="preserve">Sa-<lb/>turni enim maximam 72″ continere aſſerit, Jovis item maxi-<lb/>mam 68″, quæ mihi ſunt 68″ & </s> <s xml:id="echoid-s7013" xml:space="preserve">64″. </s> <s xml:id="echoid-s7014" xml:space="preserve">Certum autem eſt non <lb/>magis in Venere, quam in Jove aut Saturno dimetiendis, er-<lb/>rori obnoxiam eſſe rationem noſtram. </s> <s xml:id="echoid-s7015" xml:space="preserve">Et Veneris quidem <lb/>diametrum Hevelius quoque, ad Lunæ maculas inſtituta com-<lb/>paratione, tantum 82″ ſe inveniſſe teſtatur cum circa peri-<lb/>gæum verſaretur, unde diameter ejus maxima non multum ſa-<lb/>nè noſtram ſuperabit. </s> <s xml:id="echoid-s7016" xml:space="preserve">Porro illud in methodo à nobis tra-<lb/>dita commodiſſimum accidit, quod nec Lunæ, nec ſideris cu-<lb/>juſquam conjunctionem, cum eo quem metiri volumus pla-<lb/>neta, opperiri neceſſe eſt, ſed omni tempore ejus uſus con-<lb/>ceditur. </s> <s xml:id="echoid-s7017" xml:space="preserve">Nec ad planetarum tantum diametros pertinet, ſed <lb/>ad lunares quoque maculas accuratè deſcribendas, comitum-<lb/>que Jovialium diſtantias accipiendas rectiſſimè adhibebitur. <lb/></s> <s xml:id="echoid-s7018" xml:space="preserve">Ad planetarum autem diametros longis atque optimæ notæ <lb/>teleſcopiis opus eſſe ſciendum eſt. </s> <s xml:id="echoid-s7019" xml:space="preserve">Nec negligendum in Ve-<lb/>nere ac Mercurio, ut fuligine leviter inficiatur lens oculo pro-<lb/>xima, quo perfectius planetæ ambitus circùm terminetur.</s> <s xml:id="echoid-s7020" xml:space="preserve"/> </p> <div xml:id="echoid-div357" type="float" level="2" n="50"> <note position="right" xlink:label="note-0329-01" xlink:href="note-0329-01a" xml:space="preserve">Ratio obſer-<lb/>vandi Pla-<lb/>netarum dia-<lb/>metros appa-<lb/>rentes accu-<lb/>ratiſſima.</note> <note position="left" xlink:label="note-0330-01" xlink:href="note-0330-01a" xml:space="preserve">Diameter <lb/>Veneris ob-<lb/>ſervata.</note> </div> </div> <div xml:id="echoid-div359" type="section" level="1" n="143"> <head xml:id="echoid-head193" xml:space="preserve">FINIS.</head> <pb file="0332" n="349"/> <pb file="0333" n="350"/> </div> <div xml:id="echoid-div360" type="section" level="1" n="144"> <head xml:id="echoid-head194" xml:space="preserve"><emph style="sc">Eustachii</emph> <emph style="sc">De</emph> <emph style="sc">Divinis</emph> <emph style="sc">Septempedani</emph></head> <head xml:id="echoid-head195" xml:space="preserve">BREVIS ANNOTATIO <lb/>IN <lb/>SYSTEMA SATURNIUM <lb/>CHRISTIANI HUGENII. <lb/>A D <lb/>SERENISSIMUM PRINCIPEM <lb/>LEOPOLDUM <lb/>Magni Ducis HETRVRIÆ Fratrem.</head> <pb file="0334" n="351"/> <pb o="599" file="0335" n="352"/> <figure> <image file="0335-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0335-01"/> </figure> </div> <div xml:id="echoid-div361" type="section" level="1" n="145"> <head xml:id="echoid-head196" xml:space="preserve"><emph style="sc">Eustachii</emph> <emph style="sc">De</emph> <emph style="sc">Divinis</emph> <emph style="sc">Septempedani</emph></head> <head xml:id="echoid-head197" xml:space="preserve">BREVIS ANNOTATIO <lb/>IN <lb/>SYSTEMA SATURNIUM <lb/>CRISTIANI HUGENII.</head> <head xml:id="echoid-head198" style="it" xml:space="preserve">SERENISSIME PRINCEPS</head> <p> <s xml:id="echoid-s7021" xml:space="preserve">NOn ita pridem Chriſtiani Hugenii liber de Sa-<lb/>turnio ſyſtemate Celſitudini tuæ Sereniſſimæ <lb/>inſcriptus in meas manus incidit, in quo, & </s> <s xml:id="echoid-s7022" xml:space="preserve"><lb/>de me unum quidpiam, & </s> <s xml:id="echoid-s7023" xml:space="preserve">de reipſa nonnul-<lb/>la eſſe comperi, quæ cum veritate minùs con-<lb/>ſentiant. </s> <s xml:id="echoid-s7024" xml:space="preserve">Nollem literatos homines, quibus <lb/>teleſcopii majoris copia non eſt, in errorem induci; </s> <s xml:id="echoid-s7025" xml:space="preserve">cum <lb/>jam alioquin ſatis proclives ſint ad excipiendam credulis au-<lb/>ribus, vel oculis eam rerum cœleſtium novitatem, quæ ha-<lb/>ctenus incompertæ orbi fuerunt: </s> <s xml:id="echoid-s7026" xml:space="preserve">Accedit, quod libelli auctor <lb/>eam diligentiam à ſe adhibitam eſſe atteſtatur, qua certè quiſ-<lb/>piam majorem in illo non deſideret; </s> <s xml:id="echoid-s7027" xml:space="preserve">quod revera non par-<lb/>vi ponderis eſt, ad captandam hominum credulitatem, erro-<lb/>remque in eorum animos inſinuandum. </s> <s xml:id="echoid-s7028" xml:space="preserve">Itaque meum eſſe <lb/>duxi, tum ea, quæ mihi afficta ſunt, tum ea, quæ cœteris <lb/>omnibus errandi occaſionem facilè præbere poſſent, brevi <lb/>hoc & </s> <s xml:id="echoid-s7029" xml:space="preserve">publico ſcripto refellere; </s> <s xml:id="echoid-s7030" xml:space="preserve">brevi quidem, ne plus æ-<lb/>quo legentibus moleſtum ſit; </s> <s xml:id="echoid-s7031" xml:space="preserve">publico verò, ne privati Huge-<lb/>nii errores publici fiant; </s> <s xml:id="echoid-s7032" xml:space="preserve">aut ſi jam publici facti ſunt, ne quid <pb o="600" file="0336" n="353" rhead="EUSTACHIUS DE DIVINIS"/> inde reſpublica literaria detrimenti capiat, ut publicè à me <lb/>configantur. </s> <s xml:id="echoid-s7033" xml:space="preserve">Quia vero Hugenius tuis auſpiciis libellum <lb/>ſuum vulgavit, iiſdem ego potiori jure brevem hanc diſcuſ-<lb/>ſionem in publicam lucem prodire volui, qui tot nominibus <lb/>Celſitudini tuæ Sereniſſimæ, totique pluſquam regiæ Medi-<lb/>ceorum familiæ obſtrictus ſum.</s> <s xml:id="echoid-s7034" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7035" xml:space="preserve">Tria igitur in hac ſcriptione præſtabo; </s> <s xml:id="echoid-s7036" xml:space="preserve">primò enim affi-<lb/>ctam mihi fraudem amoliar; </s> <s xml:id="echoid-s7037" xml:space="preserve">aliquot deinde Auctoris errores <lb/>indicabo, ea tamen ſtyli moderatione, quæ & </s> <s xml:id="echoid-s7038" xml:space="preserve">Chriſtianum, <lb/>& </s> <s xml:id="echoid-s7039" xml:space="preserve">ingenuum, ne dicam, EUGENIUM, hominem deceat: <lb/></s> <s xml:id="echoid-s7040" xml:space="preserve">cum demum Auctor ſcripſerit pag. </s> <s xml:id="echoid-s7041" xml:space="preserve">1. </s> <s xml:id="echoid-s7042" xml:space="preserve">phænomeni Saturni ra-<lb/>tionem & </s> <s xml:id="echoid-s7043" xml:space="preserve">cauſam nec Galilæo, nec Aſtronomorum cuiquam <lb/>compertam fuiſſe, eamque dumtaxat explicari poſſe, inſerta <lb/>Saturnio globo ſplendentis annuli corona, ne Jovi, quem <lb/>aſtra Medicea jure coronant, detrahi corona & </s> <s xml:id="echoid-s7044" xml:space="preserve">jam olim ex-<lb/>auctorato Saturno reſtitui videatur, Fabriani ſyſtematis ru-<lb/>dimentum quoddam breviter exponam, in quo, adducta Sa-<lb/>turni phænomena ea facilitate explicantur, qua, neſcio, an <lb/>major excogitari poſſit; </s> <s xml:id="echoid-s7045" xml:space="preserve">in ea ſcilicet terræ quieſcentis hypo-<lb/>theſi; </s> <s xml:id="echoid-s7046" xml:space="preserve">quam catholici Aſtronomi amplecti dumtaxat & </s> <s xml:id="echoid-s7047" xml:space="preserve">con-<lb/>tra Heterodoxos Ariſtarchos tueri debent. </s> <s xml:id="echoid-s7048" xml:space="preserve">Cæterùm illa <lb/>omnia, quæ à me ſcribentur, æquiſſimæ tuæ cenſuræ, ac re-<lb/>cto judicio ſubjecta eſſe velim; </s> <s xml:id="echoid-s7049" xml:space="preserve">nec abs re, opinor, cùm te <lb/>literati omnes, ut Mecænatem; </s> <s xml:id="echoid-s7050" xml:space="preserve">& </s> <s xml:id="echoid-s7051" xml:space="preserve">politiorum operum arti-<lb/>fices, ut ſingularem patronum colant; </s> <s xml:id="echoid-s7052" xml:space="preserve">ſed ad rem venio.</s> <s xml:id="echoid-s7053" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7054" xml:space="preserve">Primo igitur loco Hugenius, antequam ſyſtema Saturnium <lb/>aggrediatur, præmittit ſui teleſcopii deſcriptionem; </s> <s xml:id="echoid-s7055" xml:space="preserve">cum e-<lb/>nim, ut ipſe ait, alia perſpicilla non eſſent, niſi quinque aut <lb/>ſex pedes longa, ſeriò & </s> <s xml:id="echoid-s7056" xml:space="preserve">maximo ſtudio ad longiora fabri-<lb/>canda animum & </s> <s xml:id="echoid-s7057" xml:space="preserve">manum applicuit: </s> <s xml:id="echoid-s7058" xml:space="preserve">ſed advertere potuit ac <lb/>debuit, in illa obſervatione mea Saturni, quam ipſe refert, <lb/>mentionem à me factam fuiſſe teleſcopiorum variæ longitudi-<lb/>nis, quæ ad prædictas obſervationes adhibueram, ſcilicet 15. <lb/></s> <s xml:id="echoid-s7059" xml:space="preserve">24. </s> <s xml:id="echoid-s7060" xml:space="preserve">& </s> <s xml:id="echoid-s7061" xml:space="preserve">36. </s> <s xml:id="echoid-s7062" xml:space="preserve">palmorum; </s> <s xml:id="echoid-s7063" xml:space="preserve">multa autem hujuſmodi fabricavi; </s> <s xml:id="echoid-s7064" xml:space="preserve">Illu-<lb/>ſtriſſimus Eques Dighby, vir ut nobili genere, ita ingenio <lb/>perſpicaciſſimo & </s> <s xml:id="echoid-s7065" xml:space="preserve">omni doctrinæ genere conſpicuus ſex, cum <pb o="601" file="0337" n="354" rhead="IN SYSTEMA SATURNIUM."/> Roma diſceſſit, ſecum detulit; </s> <s xml:id="echoid-s7066" xml:space="preserve">unum Thomas Paggi nobilis <lb/>Anglus; </s> <s xml:id="echoid-s7067" xml:space="preserve">Taurinum alterum miſſum eſt; </s> <s xml:id="echoid-s7068" xml:space="preserve">Fabritius Guaſtafer-<lb/>rus Romanus, vir conſuetudine & </s> <s xml:id="echoid-s7069" xml:space="preserve">amicitia mihi conjunctiſ-<lb/>ſimus, rerumqùe optimarum ſludioſiſſimus, unum habuit; <lb/></s> <s xml:id="echoid-s7070" xml:space="preserve">Bononiam unum miſi ad Illuſtriſſimum Comitem Carolum <lb/>Antonium Manzinum, virum ſanè rerum iſtarum peritiſſi-<lb/>mum, doctorem eximium, cujus Dioptricam in dies expe-<lb/>ctamus, in qua haud dubiè ea legemus, quæ multi hactenus <lb/>deſiderarunt: </s> <s xml:id="echoid-s7071" xml:space="preserve">Unum etiam fabricavi Illuſtriſſimo D. </s> <s xml:id="echoid-s7072" xml:space="preserve">Caramuel <lb/>Campaniæ Epiſcopo; </s> <s xml:id="echoid-s7073" xml:space="preserve">omitto alia; </s> <s xml:id="echoid-s7074" xml:space="preserve">nihil enim aliud jam à <lb/>multis annis ago.</s> <s xml:id="echoid-s7075" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7076" xml:space="preserve">Tum deſcribit teleſcopium à ſe fabricatum, longum pedes <lb/>Rhenolandicos 23. </s> <s xml:id="echoid-s7077" xml:space="preserve">qui ſi reducantur ad palmos Romanos, <lb/>quos adhibent architecti, faciunt palmos 31. </s> <s xml:id="echoid-s7078" xml:space="preserve">uncias 4. </s> <s xml:id="echoid-s7079" xml:space="preserve">Vi-<lb/>tri objectivi (ſic vulgò vocant) orbis apertus diametrum ha-<lb/>bet pollicum 2. </s> <s xml:id="echoid-s7080" xml:space="preserve">cum triente: </s> <s xml:id="echoid-s7081" xml:space="preserve">ſeu palmi Romani unc. </s> <s xml:id="echoid-s7082" xml:space="preserve">3. </s> <s xml:id="echoid-s7083" xml:space="preserve">ſcru-<lb/>pul. </s> <s xml:id="echoid-s7084" xml:space="preserve">1. </s> <s xml:id="echoid-s7085" xml:space="preserve">Vitrum oculare ex duobus invicem unitis conſtans, <lb/>lenti æquivalet, & </s> <s xml:id="echoid-s7086" xml:space="preserve">radios, parallelos colligit, ac projicit in <lb/>focum, diſtantem pollic. </s> <s xml:id="echoid-s7087" xml:space="preserve">3. </s> <s xml:id="echoid-s7088" xml:space="preserve">prædicti pedis, id eſt, uncias 4. <lb/></s> <s xml:id="echoid-s7089" xml:space="preserve">ſcrup. </s> <s xml:id="echoid-s7090" xml:space="preserve">2. </s> <s xml:id="echoid-s7091" xml:space="preserve">palmi Romani: </s> <s xml:id="echoid-s7092" xml:space="preserve">inde porrò deducit Hugenius, vi-<lb/>trum ſuum objectivum exquiſitum eſſe, cum ſit patiens len-<lb/>tis adeo acutæ, ut vocant, & </s> <s xml:id="echoid-s7093" xml:space="preserve">ad veras Saturni obſervationes, <lb/>ejuſmodi tubos, & </s> <s xml:id="echoid-s7094" xml:space="preserve">vitra adhiberi vult.</s> <s xml:id="echoid-s7095" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7096" xml:space="preserve">Hæc facilè omittere Hugenius potuiſſet; </s> <s xml:id="echoid-s7097" xml:space="preserve">nec enim his re-<lb/>gulis opus erat, cum longè ante fabricata fuerint teleſcopia, <lb/>quæ tum longitudine, tum vitrorum figura, tum operis per-<lb/>fectione Hugenianis minimè cedunt; </s> <s xml:id="echoid-s7098" xml:space="preserve">adhibui aliquando tu-<lb/>bum 36. </s> <s xml:id="echoid-s7099" xml:space="preserve">palmos longum, ac proinde longiorem Hugeniano <lb/>palmis 4. </s> <s xml:id="echoid-s7100" xml:space="preserve">unc. </s> <s xml:id="echoid-s7101" xml:space="preserve">8. </s> <s xml:id="echoid-s7102" xml:space="preserve">orbis autem aperti diameter fuit unc. </s> <s xml:id="echoid-s7103" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s7104" xml:space="preserve">ſcrup. </s> <s xml:id="echoid-s7105" xml:space="preserve">2. </s> <s xml:id="echoid-s7106" xml:space="preserve">ſed ſublatis 2. </s> <s xml:id="echoid-s7107" xml:space="preserve">illis ſcrup. </s> <s xml:id="echoid-s7108" xml:space="preserve">propter longitudinis dif-<lb/>ferentiam 4. </s> <s xml:id="echoid-s7109" xml:space="preserve">ſcilicet palm. </s> <s xml:id="echoid-s7110" xml:space="preserve">unc. </s> <s xml:id="echoid-s7111" xml:space="preserve">8. </s> <s xml:id="echoid-s7112" xml:space="preserve">reſtat diameter orbis mei <lb/>vitri, major diametro orbis vitri Hugeniani unc. </s> <s xml:id="echoid-s7113" xml:space="preserve">1. </s> <s xml:id="echoid-s7114" xml:space="preserve">ac pro-<lb/>inde vel hoc nomine, meum teleſcopium Hugeniano præ-<lb/>ferendum, ex vulgari & </s> <s xml:id="echoid-s7115" xml:space="preserve">ab omnibus accepta regula; </s> <s xml:id="echoid-s7116" xml:space="preserve">cum <lb/>ſcilicet majoris aperturæ, ut ſic loquar, vitrum objectivum <lb/>patiens ſit.</s> <s xml:id="echoid-s7117" xml:space="preserve"/> </p> <pb o="602" file="0338" n="355" rhead="EUSTACHIUS DE DIVINIS"/> <p> <s xml:id="echoid-s7118" xml:space="preserve">Præterea fruſtra omnino duo vitra in vitrum oculare u-<lb/>trinque convexum conjunxit; </s> <s xml:id="echoid-s7119" xml:space="preserve">cùm unica lens ſufficiat: </s> <s xml:id="echoid-s7120" xml:space="preserve">Ego <lb/>lentem in prædictis obſervationibus adhibui, radios paralle-<lb/>los colligentem, & </s> <s xml:id="echoid-s7121" xml:space="preserve">projicientem in focum diſtantem unc. </s> <s xml:id="echoid-s7122" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s7123" xml:space="preserve">ſcrup. </s> <s xml:id="echoid-s7124" xml:space="preserve">2. </s> <s xml:id="echoid-s7125" xml:space="preserve">ſed hæc diſtantia minor eſt diſtantia projectionis <lb/>Hugenianæ unc. </s> <s xml:id="echoid-s7126" xml:space="preserve">1. </s> <s xml:id="echoid-s7127" xml:space="preserve">igitur lens mea acutior: </s> <s xml:id="echoid-s7128" xml:space="preserve">licet tubi mei <lb/>longitudo major ſit; </s> <s xml:id="echoid-s7129" xml:space="preserve">& </s> <s xml:id="echoid-s7130" xml:space="preserve">cum ex hoc teleſcopia cenſeantur, <lb/>ut optimè ipſe obſervat; </s> <s xml:id="echoid-s7131" xml:space="preserve">inde certè pari jure deduco, meum <lb/>teleſcopium præſtantius eſſe, idque triplici nomine. </s> <s xml:id="echoid-s7132" xml:space="preserve">Primò, <lb/>quia longius eſt. </s> <s xml:id="echoid-s7133" xml:space="preserve">Secundò, quia vitrum objectivum habet <lb/>majoris aperturæ patiens. </s> <s xml:id="echoid-s7134" xml:space="preserve">Tertiò quia lentem acutiorem ſuſ-<lb/>tinet.</s> <s xml:id="echoid-s7135" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7136" xml:space="preserve">Hugenius pagina libelli 37. </s> <s xml:id="echoid-s7137" xml:space="preserve">de me ita ſcribit. </s> <s xml:id="echoid-s7138" xml:space="preserve">Tales au-<lb/>tem (Saturni ſcilicet anſas) Euſtachius de Divinis notavit <lb/>annis 1646 1647. </s> <s xml:id="echoid-s7139" xml:space="preserve">1648. </s> <s xml:id="echoid-s7140" xml:space="preserve">à quo editum Schema ad numerum <lb/>10. </s> <s xml:id="echoid-s7141" xml:space="preserve">hic exhibuimus; </s> <s xml:id="echoid-s7142" xml:space="preserve">iſque cum præſtantiſſimus perſpicillo-<lb/> <anchor type="note" xlink:label="note-0338-01a" xlink:href="note-0338-01"/> rum artifex habeatur, credibile eſt, omnium emendatiſſi-<lb/>mè Saturni faciem deſcripſiſſe, niſi quod umbras illas, quæ <lb/>in ſchemate apparent, de ſuo, ut opinor, adjecit. </s> <s xml:id="echoid-s7143" xml:space="preserve">Eſt ſa-<lb/>nè, quod multas illi gratias habeam, pro iis laudibus, quas <lb/>mihi ignoto homini ac minimè cogitanti aſperſit; </s> <s xml:id="echoid-s7144" xml:space="preserve">ſed quid <lb/>hoc eſt, de meo adjeciſſe, niſi hominibus impoſuiſſe? </s> <s xml:id="echoid-s7145" xml:space="preserve">Si <lb/>enim de meo eſt, fictum eſt, nec reipſa talem Saturnum ex-<lb/>hibui, qualem obſervavi; </s> <s xml:id="echoid-s7146" xml:space="preserve">quia de meo umbras illas adjeci, <lb/>quas in Saturno non vidi, nec videre potui, cùm revera in <lb/>Saturno non ſint; </s> <s xml:id="echoid-s7147" xml:space="preserve">fabula igitur eſt, fictio eſt; </s> <s xml:id="echoid-s7148" xml:space="preserve">immo in re <lb/>tanta pro luculenta fraude haberi debet, cum inde homines <lb/>in errorem facilè inducantur. </s> <s xml:id="echoid-s7149" xml:space="preserve">At ſi umbras de meo adjeci, <lb/>Hugenius ejuſdem criminis reus eſt, qui paſſim, ut ſuum <lb/>annulum Saturno injectum meliùs exprimeret, longe majo-<lb/>res umbras adjecit: </s> <s xml:id="echoid-s7150" xml:space="preserve">ſed profecto neuter adjecit de ſuo; </s> <s xml:id="echoid-s7151" xml:space="preserve">ut <lb/>enim globus, corpus ſcilicet ſolidum, in plano deſcribatur, <lb/>aliquid umbræ accerſendum eſt, quo figura ſolida exprima-<lb/>tur & </s> <s xml:id="echoid-s7152" xml:space="preserve">appareat, alioquin merus circulus evadet: </s> <s xml:id="echoid-s7153" xml:space="preserve">Quis vel <lb/>tyro in arte graphica hoc neſcit? </s> <s xml:id="echoid-s7154" xml:space="preserve">Umbras ſcilicet accerſen-<lb/>das eſſe ad figuras ſolidas deſignandas. </s> <s xml:id="echoid-s7155" xml:space="preserve">Umbram igitur de <pb o="603" file="0339" n="356" rhead="IN SYSTEMA SATURNIUM."/> meo non adjeci, ſed globum in plano deſcripſi, qui tamen <lb/>ſine umbræ beneficio deſcribi non potuit; </s> <s xml:id="echoid-s7156" xml:space="preserve">Umbra enim fa-<lb/>cit, ut reliquum figuræ quaſi extare, ac prominere videa-<lb/>tur; </s> <s xml:id="echoid-s7157" xml:space="preserve">in quo uno ſtereometria ab ichnographia diſcrepat: </s> <s xml:id="echoid-s7158" xml:space="preserve">aut <lb/>fortè dicere voluit, cavitatem anſulæ Saturniæ nigriore, <lb/>quàm par ſit, umbra, à me fuiſſe fœdatam: </s> <s xml:id="echoid-s7159" xml:space="preserve">cum tamen ejuſ-<lb/>dem cum reliquo cælo coloris eſſe videatur; </s> <s xml:id="echoid-s7160" xml:space="preserve">ſed pace illius <lb/>dixerim; </s> <s xml:id="echoid-s7161" xml:space="preserve">quotquot anſarum cavitates vidimus, nigriores re-<lb/>liquo cælo, & </s> <s xml:id="echoid-s7162" xml:space="preserve">magis atras aſpeximus, neque hoc teleſcopii <lb/>vitio quiſquam tribuat; </s> <s xml:id="echoid-s7163" xml:space="preserve">nempe ſi revera anſæ illæ vacuæ ſunt, <lb/>& </s> <s xml:id="echoid-s7164" xml:space="preserve">ad ipſum cælum radii viſorii per illas trajecti terminantur, <lb/>ejuſdem cum cælo coloris erunt, quocumque tandem tele-<lb/>ſcopio aſpiciantur. </s> <s xml:id="echoid-s7165" xml:space="preserve">Itaque, quod mihi Hugenius pro ſua <lb/>humanitate, præſtantiſſimi artificis elogium attribuit, ut me <lb/>præſtantiſſimum eſſe non admitto, ita pro iis laudibus, quæ <lb/>in me non quadrant, gratias, ut jam dixi, habeo; </s> <s xml:id="echoid-s7166" xml:space="preserve">quia tamen <lb/>unde altiùs extollit, inde profundiùs deprimit, dum fabrica-<lb/>cati à me teleſcopii fidelem uſum deſiderat, utpote quod Sa-<lb/>turnios ſatellites ea forma, eoque vultu non exhibeat, quo <lb/>revera exhibendi eſſent, ad alterum caput venio, in quo lu-<lb/>culenter oſtendam, Hugenium in nonnullis hallucinatum eſ-<lb/>ſe, quod an perſpicillis ab eo fabrefactis tribuendum ſit, an <lb/>verò auctoris fictioni, alius fortè dubitaret; </s> <s xml:id="echoid-s7167" xml:space="preserve">ego ſanè doctiſ-<lb/>ſimi hominis candori nihil detractum velim, quare haud du-<lb/>biè teleſcopii vitio, quidquid erroris admiſit, tribuendum <lb/>eſſe judico: </s> <s xml:id="echoid-s7168" xml:space="preserve">leges porro, quas vitris ſuis pag. </s> <s xml:id="echoid-s7169" xml:space="preserve">3. </s> <s xml:id="echoid-s7170" xml:space="preserve">& </s> <s xml:id="echoid-s7171" xml:space="preserve">4. </s> <s xml:id="echoid-s7172" xml:space="preserve">præ-<lb/>ſcripſit, ſufficiat ſupra indicaſſe.</s> <s xml:id="echoid-s7173" xml:space="preserve"/> </p> <div xml:id="echoid-div361" type="float" level="2" n="1"> <note position="left" xlink:label="note-0338-01" xlink:href="note-0338-01a" xml:space="preserve">TAB. LII. <lb/>fig. 10.</note> </div> <p> <s xml:id="echoid-s7174" xml:space="preserve">Conſtat verò auctorem duobus teleſcopiis uſum fuiſſe, in <lb/>traditis obſervationibus; </s> <s xml:id="echoid-s7175" xml:space="preserve">primum fuit 12. </s> <s xml:id="echoid-s7176" xml:space="preserve">pedum, vel pal-<lb/>morum 16. </s> <s xml:id="echoid-s7177" xml:space="preserve">more Romano paulò plus, duobus convexis in-<lb/>ſtructum; </s> <s xml:id="echoid-s7178" xml:space="preserve">alterum 23. </s> <s xml:id="echoid-s7179" xml:space="preserve">pedum, de quo ſupra; </s> <s xml:id="echoid-s7180" xml:space="preserve">In hoc, ejus <lb/>dictis ſtandum eſſe puto: </s> <s xml:id="echoid-s7181" xml:space="preserve">Ego tamen, ut dixi, longius tele-<lb/>ſcopium adhibui, ſcilicet triginta ſex palmorum; </s> <s xml:id="echoid-s7182" xml:space="preserve">quare non <lb/>dubitarim, quin Saturnum mihi æquè perſpicuè exhibeat; <lb/></s> <s xml:id="echoid-s7183" xml:space="preserve">præſertim cum vitra ſatis accuratè, & </s> <s xml:id="echoid-s7184" xml:space="preserve">diligenter elaborari, <lb/>tornari, formari, lævigari à me ſoleant: </s> <s xml:id="echoid-s7185" xml:space="preserve">ſi Hugeniani tele- <pb o="604" file="0340" n="357" rhead="EUSTACHIUS DE DIVINIS"/> ſcopii mihi copia fieret, facilis comparatio inſtitui poſſet, <lb/>quam certè ab ipſo Hugenio inſtitui poſſe, nihil vetat; </s> <s xml:id="echoid-s7186" xml:space="preserve">cum <lb/>enim per totam Europam perſpicilla à me fabrefacta, diſtra-<lb/>cta jam & </s> <s xml:id="echoid-s7187" xml:space="preserve">quaſi diſſeminata fuerint, in ejus manus facilè ve-<lb/>nire potuerunt: </s> <s xml:id="echoid-s7188" xml:space="preserve">rem narro, quæ mihi ante aliquot dies, ac-<lb/>cidit; </s> <s xml:id="echoid-s7189" xml:space="preserve">Jam unus, vel alter annus elapſus eſt, ex quo R. </s> <s xml:id="echoid-s7190" xml:space="preserve">P. <lb/></s> <s xml:id="echoid-s7191" xml:space="preserve">Antonius Maria de Reitha Bononiam veniens, teleſcopium, <lb/>quod ſecum habebat, 24. </s> <s xml:id="echoid-s7192" xml:space="preserve">palmorum, mirificè prædicabat, <lb/>aſſerebatque, nullum ejuſdem longitudinis in Italia eſſe, quod <lb/>ſuo non cederet; </s> <s xml:id="echoid-s7193" xml:space="preserve">hujus rei certior factus ab Illuſtriſſimo Co-<lb/>mite Car. </s> <s xml:id="echoid-s7194" xml:space="preserve">Ant. </s> <s xml:id="echoid-s7195" xml:space="preserve">Manzino viro, ut dicam iterum, rerum iſta-<lb/>rum peritiſſimo, & </s> <s xml:id="echoid-s7196" xml:space="preserve">curioſiſſimo, ad eum miſi, ut ipſe petie-<lb/>rat, teleſcopium ejuſdem longitudinis, utrumque æqualium <lb/>virium, & </s> <s xml:id="echoid-s7197" xml:space="preserve">ejuſdem bonitatis repertum eſt; </s> <s xml:id="echoid-s7198" xml:space="preserve">hæc jam penè mi-<lb/>hi exciderant, cùm ecce Illuſtriſſimus Dominus de Monco-<lb/>nis, nobilis Lugdunenſis, vir, tum ingenio, tum genere ple-<lb/>riſque notus, incogitanti mihi, poſtridie cinerum adfuit; </s> <s xml:id="echoid-s7199" xml:space="preserve">poſt <lb/>ſolita urbanitatis officia, cùm inter colloquendum, de per-<lb/>ſpicillis mentio facta eſſet, non habeo ampliùs, inquit, tuum <lb/>illud teleſcopum 24. </s> <s xml:id="echoid-s7200" xml:space="preserve">palmorum, illudque ſanè optimum, <lb/>quod ante aliquot annos Lugdunum ad me ignotum licèt ho-<lb/>minem miſeras; </s> <s xml:id="echoid-s7201" xml:space="preserve">illud enim bonus P. </s> <s xml:id="echoid-s7202" xml:space="preserve">Reitha ſuis precibus à me <lb/>extorſit; </s> <s xml:id="echoid-s7203" xml:space="preserve">mecum igitur, inquam, pugnavi, ejuſdemque pa-<lb/>rentis liberos quaſi hoſtes inter ſe commiſi; </s> <s xml:id="echoid-s7204" xml:space="preserve">in riſum proru-<lb/>pimus ambo; </s> <s xml:id="echoid-s7205" xml:space="preserve">non tamen eſt, quod aliquis hoc vitio vertat <lb/>Patri Reithæ, qui pro ſuo candore ultro fatebatur, præfa-<lb/>tum teleſcopium non à ſe, ſed ab alio elaboratum fuiſſe. </s> <s xml:id="echoid-s7206" xml:space="preserve">Scri-<lb/>pſi hæc, ut vel inde conſtet, fieri potuiſſe, ut aliquod ex meis <lb/>teleſcopiis in manus Hugenii veniret, ad comparationem in-<lb/>ſtituendam; </s> <s xml:id="echoid-s7207" xml:space="preserve">dum id fiat, facilè mihi perſuadebo, nihil eſſe, <lb/>cur mea Hugenianis primas deferant; </s> <s xml:id="echoid-s7208" xml:space="preserve">ac proinde ſuppoſita <lb/>eadem longitudine, æquè clare, & </s> <s xml:id="echoid-s7209" xml:space="preserve">fideliter Saturnios ſatel-<lb/>lites exhiberi à meis, ac ab Hugenianis.</s> <s xml:id="echoid-s7210" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7211" xml:space="preserve">His præmiſſis; </s> <s xml:id="echoid-s7212" xml:space="preserve">conſtanter aſſerit Hugenius, pag. </s> <s xml:id="echoid-s7213" xml:space="preserve">547. </s> <s xml:id="echoid-s7214" xml:space="preserve">& </s> <s xml:id="echoid-s7215" xml:space="preserve"><lb/>556. </s> <s xml:id="echoid-s7216" xml:space="preserve">Saturni ſatellites nunquam ab eo ſejunctos figura ſphæ-<lb/>rica videri, quales prima figura ab eo excuſa, & </s> <s xml:id="echoid-s7217" xml:space="preserve">à me recu- <pb o="605" file="0341" n="358" rhead="IN SYSTEMA SATURNIUM."/> ſa exhibet; </s> <s xml:id="echoid-s7218" xml:space="preserve">quod tamen falſum eſſe, omni aſſeveratione affir-<lb/> <anchor type="note" xlink:label="note-0341-01a" xlink:href="note-0341-01"/> mo, cùm anno 1657. </s> <s xml:id="echoid-s7219" xml:space="preserve">cœptis obſervationibus die 30 Junii, <lb/>& </s> <s xml:id="echoid-s7220" xml:space="preserve">ſepties, vel octies, ad vigeſimum uſque Julii repetitis, ad-<lb/>hibito ad hunc finem teleſcopio triginta ſex palmorum, ac-<lb/>curatiſſimè fabrefacto, prædictos ſatellites figura ſphærica <lb/>tornatos à Saturni globo omnino ſejunctos obſervarim, qua-<lb/>les prima figura Hugenii illos repræſentat, & </s> <s xml:id="echoid-s7221" xml:space="preserve">quales jam an-<lb/>tè Galilæus viderat, idem obſervavi anno proxime ſequenti, <lb/>teleſcopio 24. </s> <s xml:id="echoid-s7222" xml:space="preserve">palmorum uſus; </s> <s xml:id="echoid-s7223" xml:space="preserve">adfuerunt teſtes omni exce-<lb/>ptione majores; </s> <s xml:id="echoid-s7224" xml:space="preserve">præ cæteris, vir clariſſimus Alphonſus Bor-<lb/>relli, Magni Etruriæ Ducis inſignis Mathematicus, & </s> <s xml:id="echoid-s7225" xml:space="preserve">Geo-<lb/>metra, typorum gloria orbi ſatis notus: </s> <s xml:id="echoid-s7226" xml:space="preserve">neque hæc contra-<lb/>dicendi ſtudio à me adſtrui, quiſquam ſibi perſuadeat; </s> <s xml:id="echoid-s7227" xml:space="preserve">quo <lb/>enim tempore, hos ſatellites ſphæricos à Saturni globo, mo-<lb/>dico quidem, ſed quod ſenſu, adhibita teleſcopii opera, per-<lb/>cipi poterat, intervallo disjunctos vidi, nunquam mihi per-<lb/>ſuadere potuiſſem, aliquem fore, qui ſejunctos unquam vi-<lb/>deri poſſe negaret; </s> <s xml:id="echoid-s7228" xml:space="preserve">Dico igitur, disjunctos à me viſos fuiſ-<lb/>ſe; </s> <s xml:id="echoid-s7229" xml:space="preserve">negat Hugenius disjunctos videri poſſe; </s> <s xml:id="echoid-s7230" xml:space="preserve">produco teſtes <lb/>omni exceptione majores, qui una mecum viderunt; </s> <s xml:id="echoid-s7231" xml:space="preserve">Utri <lb/>quæſo habenda fides? </s> <s xml:id="echoid-s7232" xml:space="preserve">Sejunctos videri poſſe contendo, cùm <lb/>reipſa ſejunctos viderim, adductis teſtibus omni exceptione <lb/>majoribus; </s> <s xml:id="echoid-s7233" xml:space="preserve">negat Hugenius, videri poſſe; </s> <s xml:id="echoid-s7234" xml:space="preserve">quia ſcilicet nun-<lb/>quam ipſe vidit; </s> <s xml:id="echoid-s7235" xml:space="preserve">ſed alii viderunt; </s> <s xml:id="echoid-s7236" xml:space="preserve">negat tamen videri poſſe; <lb/></s> <s xml:id="echoid-s7237" xml:space="preserve">adſunt & </s> <s xml:id="echoid-s7238" xml:space="preserve">jurant teſtes; </s> <s xml:id="echoid-s7239" xml:space="preserve">illos ejurat, & </s> <s xml:id="echoid-s7240" xml:space="preserve">fidem alienis teleſco-<lb/>piis detrahit; </s> <s xml:id="echoid-s7241" xml:space="preserve">perperam quidem, meo judicio; </s> <s xml:id="echoid-s7242" xml:space="preserve">nempe bre-<lb/>viore teleſcopio diſcerni non poterant, licèt revera disjun-<lb/>cti eſſent: </s> <s xml:id="echoid-s7243" xml:space="preserve">igitur cùm eos ſejunctos viderim meo teleſcopio, <lb/>ideo neceſſaria conſecutionis vi deduco, hoc perfectius, & </s> <s xml:id="echoid-s7244" xml:space="preserve"><lb/>melius eſſe; </s> <s xml:id="echoid-s7245" xml:space="preserve">nec enim longitudo æqualis, immo, nec major <lb/>defuit; </s> <s xml:id="echoid-s7246" xml:space="preserve">nec accurata operantis manus vel artificis diligentia <lb/>deſiderata eſt. </s> <s xml:id="echoid-s7247" xml:space="preserve">Teleſcopium igitur Hugenii, longiùs licet, <lb/>breviorum vitio laboravit, cum illius opera, ſejunctos diſtin-<lb/>ctè videre non potuerit, quo vitio, ut dixi, breviora per-<lb/>ſpicilla laborant; </s> <s xml:id="echoid-s7248" xml:space="preserve">falſum enim eſt, quod habet pag. </s> <s xml:id="echoid-s7249" xml:space="preserve">35. </s> <s xml:id="echoid-s7250" xml:space="preserve">bre-<lb/>viore teleſcopio 8. </s> <s xml:id="echoid-s7251" xml:space="preserve">circiter palmorum ſejunctos, longiore <pb o="606" file="0342" n="359" rhead="EUSTACHIUS DE DIVINIS"/> verò 16. </s> <s xml:id="echoid-s7252" xml:space="preserve">palmorum conjunctos videri, nempe, ut dixi, po-<lb/>tiore jure, ſejuncti, majore teleſcopio, conjuncti verò, mi-<lb/>nore viderentur; </s> <s xml:id="echoid-s7253" xml:space="preserve">cùm verò ſuo minore ſejunctos viderit, <lb/>majore verò ad inſtar brachiorum, hinc inde adnatorum ex-<lb/>tantes, perſpicuum majoris teleſcopii vitium eſt, cujus ope-<lb/>râ radii minùs accuratè projiciuntur, & </s> <s xml:id="echoid-s7254" xml:space="preserve">ut vulgò dicimus, <lb/>inepta fit terminatio. </s> <s xml:id="echoid-s7255" xml:space="preserve">Teleſcopium igitur Hugenianum meo <lb/>primas deferat, oportet; </s> <s xml:id="echoid-s7256" xml:space="preserve">quo ſcilicet ea diſtinctè vidi, quæ <lb/>ille ſuo diſtinctè videre non potuit.</s> <s xml:id="echoid-s7257" xml:space="preserve"/> </p> <div xml:id="echoid-div362" type="float" level="2" n="2"> <note position="right" xlink:label="note-0341-01" xlink:href="note-0341-01a" xml:space="preserve">TAB. LII. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s7258" xml:space="preserve">At non me latet cauſa qua fiat, ut Hugenius obſtinatè ne-<lb/>get, ſejunctos videri poſſe; </s> <s xml:id="echoid-s7259" xml:space="preserve">commentus eſt ipſe hypotheſim <lb/>annuli Saturni globum cingentis, cujus operâ, illas phaſes <lb/>explicare conatur, quæ in hoc planeta obſervatæ fuere: </s> <s xml:id="echoid-s7260" xml:space="preserve">ut <lb/>autem ſuæ opinioni, vel ſuo commento conſulat, illud pro-<lb/>fectò ipſi negandum fuit, quo ſemel admiſſo commentitia il-<lb/>la hypotheſis penitùs corruebat; </s> <s xml:id="echoid-s7261" xml:space="preserve">ſi enim annulus Saturnum <lb/>ambit, nunquam ſejuncti ſatellites videri queunt: </s> <s xml:id="echoid-s7262" xml:space="preserve">pari modo <lb/>fingit, præfatas anſarum vacuitates ejuſdem cum reliquo cæ-<lb/>lo coloris eſſe, ut ſuæ hypotheſi conſulat; </s> <s xml:id="echoid-s7263" xml:space="preserve">ſuppoſito enim <lb/>illo annulo in iis vacuitatibus, nihil niſi cælum videtur; </s> <s xml:id="echoid-s7264" xml:space="preserve">at <lb/>revera, ut jam ſuprà dixi, nigriores illæ apparent, ex quo <lb/>Hugeniana hypotheſis corruit. </s> <s xml:id="echoid-s7265" xml:space="preserve">Ut igitur ſuam illam opinio-<lb/>nem indemnem ſervaret, ea removenda eſſe putavit, quæ ſi <lb/>adeſſent, illius opinionis falſitas manifeſta evaderet: </s> <s xml:id="echoid-s7266" xml:space="preserve">& </s> <s xml:id="echoid-s7267" xml:space="preserve">hæc <lb/>ſibi perſuaſit, nobis tamen minimè perſuaſurus, qui rem ſe-<lb/>cus eſſe, oculari demonſtratione prorſus evicimus.</s> <s xml:id="echoid-s7268" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7269" xml:space="preserve">Alterum eſt quod Hugenius vel finxit, vel leviter ſibi per-<lb/>ſuaſit, quod tamen ejuſdem furfuris eſt; </s> <s xml:id="echoid-s7270" xml:space="preserve">dicit enim, faſciam <lb/>quandam nigricantem in Saturni globo videri, quæ quidem <lb/>ex æquo medium diſcum ſecat, quando ſolus Saturni globus <lb/>ſuo ſatellitio deſtitutus apparet; </s> <s xml:id="echoid-s7271" xml:space="preserve">ubi verò ſatellites adſunt, <lb/>modò ſuprà, modò infra brachiorum lineam faſcia videtur; <lb/></s> <s xml:id="echoid-s7272" xml:space="preserve">ſed hoc etiam merum figmentum eſt ab auctore excogitatum, <lb/>ut annulo ſuo conſuleret; </s> <s xml:id="echoid-s7273" xml:space="preserve">nam cùm ipſi opponeretur, ad-<lb/>miſſo ſemel hoc annulo nunquam Saturnum videri poſſe, vel <lb/>anſulis, vel brachiis deſtitutum, quod tamen omnibus ob- <pb o="607" file="0343" n="360" rhead="IN SYSTEMA SATURNIUM."/> ſervationibus repugnat; </s> <s xml:id="echoid-s7274" xml:space="preserve">nempe ubi planum annuli directè <lb/>in oculos incidit, annulus appareret ad inſtar zonæ rectæ, <lb/>æqualis diametro annuli, ac proinde majoris diametro Satur-<lb/>ni; </s> <s xml:id="echoid-s7275" xml:space="preserve">igitur ad inſtar brachiorum duæ appendices hinc inde <lb/>Saturnio globo adhærerent: </s> <s xml:id="echoid-s7276" xml:space="preserve">cum igitur hac difficultate, & </s> <s xml:id="echoid-s7277" xml:space="preserve"><lb/>hoc nodo inſolubili annularis illa hypotheſis laboraret; </s> <s xml:id="echoid-s7278" xml:space="preserve">fin-<lb/>xit illius auctor, extimam annuli ſuperficiem ineptam eſſe <lb/>ad luminis repercuſſionem, ac proinde faſciam illam nigri-<lb/>cantem inde ortum ducere, & </s> <s xml:id="echoid-s7279" xml:space="preserve">ob penuriam luminis reper-<lb/>cuſſi, extantes illas appendices ſub aſpectum noſtrum non <lb/>cadere: </s> <s xml:id="echoid-s7280" xml:space="preserve">Hæc habet pag. </s> <s xml:id="echoid-s7281" xml:space="preserve">542. </s> <s xml:id="echoid-s7282" xml:space="preserve">545. </s> <s xml:id="echoid-s7283" xml:space="preserve">576. </s> <s xml:id="echoid-s7284" xml:space="preserve">Aliam cauſam ſubnectit <lb/>pag. </s> <s xml:id="echoid-s7285" xml:space="preserve">585. </s> <s xml:id="echoid-s7286" xml:space="preserve">ab obliquitate radiorum illapſorum petitam, qua <lb/>fiat ut pars annuli ad nos obverſa obliquè dumtaxat illumi-<lb/>nata non videatur; </s> <s xml:id="echoid-s7287" xml:space="preserve">immo hoc alterum addit, ſuperficiem <lb/>annuli ad nos obverſam minimè à Sole illuminari; </s> <s xml:id="echoid-s7288" xml:space="preserve">ac proin-<lb/>de mirum non eſſe, quod minimè videri poſſit; </s> <s xml:id="echoid-s7289" xml:space="preserve">omiſeram, <lb/>quæ habet pag. </s> <s xml:id="echoid-s7290" xml:space="preserve">551. </s> <s xml:id="echoid-s7291" xml:space="preserve">de Saturnicolis.</s> <s xml:id="echoid-s7292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7293" xml:space="preserve">Cuncta hæc facilè difflantur; </s> <s xml:id="echoid-s7294" xml:space="preserve">nunquam enim faſciam il-<lb/>lam nigricantem, quamvis exquiſito teleſcopio uſus videre <lb/>potui; </s> <s xml:id="echoid-s7295" xml:space="preserve">nemo unquam præter illum obſervavit; </s> <s xml:id="echoid-s7296" xml:space="preserve">nec tamen <lb/>teleſcopia noſtra Hugenianis deteriora eſſe crediderim; </s> <s xml:id="echoid-s7297" xml:space="preserve">fi-<lb/>ctitia igitur cenſenda eſt illa faſcia, aut certè vitrorum vitio <lb/>tribuenda: </s> <s xml:id="echoid-s7298" xml:space="preserve">quod autem dicit, extimam illam annuli ſuper-<lb/>ficiem ad lumen reflectendum ineptam eſſe, vel ex eo falſum <lb/>eſſe conſtat, quòd in Saturni diſco partes omnes æquali lu-<lb/>mine perfuſæ videantur: </s> <s xml:id="echoid-s7299" xml:space="preserve">Nulla igitur pars ad nos obverſa <lb/>ad luminis reflexionem inepta eſt. </s> <s xml:id="echoid-s7300" xml:space="preserve">Præterea demus ita eſſe, <lb/>& </s> <s xml:id="echoid-s7301" xml:space="preserve">faſciam illam paulò obſcuriorem in Saturni diſco videri; <lb/></s> <s xml:id="echoid-s7302" xml:space="preserve">nihil tamen obſtaret, quin brachia eodem lumine perſpicua <lb/>eſſent; </s> <s xml:id="echoid-s7303" xml:space="preserve">id eſt extrema ſegmenta apparentis zonæ, in quam <lb/>annulus hoc ſitu aſpectus projicitur, extantia, & </s> <s xml:id="echoid-s7304" xml:space="preserve">excurren-<lb/>tia hinc inde extra Saturnium globum, viderentur, eodem <lb/>licèt nigricante colore diluta. </s> <s xml:id="echoid-s7305" xml:space="preserve">Ad obliquitatem illapſorum <lb/>radiorum perperam confugit; </s> <s xml:id="echoid-s7306" xml:space="preserve">alioquin in cæteris planetis <lb/>eadem ratio militaret; </s> <s xml:id="echoid-s7307" xml:space="preserve">nempe fieri non poteſt, quin major <lb/>pars ſuperficiei convexæ, obliquum radiorum ſolarium illa- <pb o="608" file="0344" n="361" rhead="EUSTACHIUS DE DIVINIS"/> pſum excipiat. </s> <s xml:id="echoid-s7308" xml:space="preserve">Id demum, quod ſubnectit, ſuperficiem <lb/>annuli ad nos obverſam à ſole aliquando non illuminari, ex <lb/>regulis opticis omnino refellitur; </s> <s xml:id="echoid-s7309" xml:space="preserve">ſuppoſita ſcilicet maxima <lb/>illa Saturni diſtantia, nec non ſolis diametro longè majore, <lb/>quo fiat, ut ſuperficies Saturni ad nos obverſa ſemper illu-<lb/>minetur; </s> <s xml:id="echoid-s7310" xml:space="preserve">hinc totus Saturni diſcus ſemper apparet. </s> <s xml:id="echoid-s7311" xml:space="preserve">De Sa-<lb/>turnicolis nihil eſt, quod dicam, riſu potiùs, quàm argu-<lb/>mentis hoc commentum refellitur; </s> <s xml:id="echoid-s7312" xml:space="preserve">accedit, quòd catholicis <lb/>dogmatis adverſatur, quæ homines tantùm illos agnoſcunt, <lb/>qui ab Adamo duxerunt ortum.</s> <s xml:id="echoid-s7313" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7314" xml:space="preserve">Nonnulla etiam falſa admiſcuit, ut pag. </s> <s xml:id="echoid-s7315" xml:space="preserve">539. </s> <s xml:id="echoid-s7316" xml:space="preserve">ubi dicit fa-<lb/>ſcias Jovis lucidiores eſſe reliquo diſco; </s> <s xml:id="echoid-s7317" xml:space="preserve">cùm tamen obſcu-<lb/>riores certò videantur: </s> <s xml:id="echoid-s7318" xml:space="preserve">Item globo Martis latiorem faſciam <lb/>ſub obſcuram obduci; </s> <s xml:id="echoid-s7319" xml:space="preserve">nihil tale unquam obſervatum eſt; <lb/></s> <s xml:id="echoid-s7320" xml:space="preserve">deinde nunquam Saturnum vidimus ea forma, quam illi tri-<lb/>buit pag. </s> <s xml:id="echoid-s7321" xml:space="preserve">542. </s> <s xml:id="echoid-s7322" xml:space="preserve">545. </s> <s xml:id="echoid-s7323" xml:space="preserve">tractum illum cæli circa Oriona, illu-<lb/>ſtriorem eſſe vidimus, cum iis ſtellis ſplendentibus, quas <lb/>ipſe recenſet. </s> <s xml:id="echoid-s7324" xml:space="preserve">Multa etiam de Saturni phaſibus ſtatuit, ex <lb/>paucorum annorum obſervationibus, quæ pro certis habere <lb/>non poſſum, donec ſubſequentium annorum obſervationibus <lb/>congruant; </s> <s xml:id="echoid-s7325" xml:space="preserve">nempe ipſe obſervationes ſuas primum cœpit 25. </s> <s xml:id="echoid-s7326" xml:space="preserve"><lb/>Martii anno 1655. </s> <s xml:id="echoid-s7327" xml:space="preserve">eaſque ad 19. </s> <s xml:id="echoid-s7328" xml:space="preserve">Februarii anni 1656. </s> <s xml:id="echoid-s7329" xml:space="preserve">bre-<lb/>viore tamen teleſcopio uſus, 12. </s> <s xml:id="echoid-s7330" xml:space="preserve">pedes longo perduxit; </s> <s xml:id="echoid-s7331" xml:space="preserve">ac <lb/>proinde minùs fideles cenſendas, ut ipſe ultrò fatetur. </s> <s xml:id="echoid-s7332" xml:space="preserve">Stan-<lb/>dum igitur obſervationibus deinceps ab eo factis, ac repeti-<lb/>tis identidem, adhibito longiore tubo, 23. </s> <s xml:id="echoid-s7333" xml:space="preserve">circiter pedum, <lb/>perductiſque ad 26. </s> <s xml:id="echoid-s7334" xml:space="preserve">Martii anno 1659. </s> <s xml:id="echoid-s7335" xml:space="preserve">Si aliquæ eſſent ab <lb/>eo notatæ à 30. </s> <s xml:id="echoid-s7336" xml:space="preserve">Junii ad 20. </s> <s xml:id="echoid-s7337" xml:space="preserve">Julii anno 1657. </s> <s xml:id="echoid-s7338" xml:space="preserve">viderem u-<lb/>trùm meis conſentirent; </s> <s xml:id="echoid-s7339" xml:space="preserve">ſed nullas toto illo tempore factas <lb/>invenio; </s> <s xml:id="echoid-s7340" xml:space="preserve">an fortè paucæ illæ obſervationes ſufficient ad cer-<lb/>tum quid ſtatuendum? </s> <s xml:id="echoid-s7341" xml:space="preserve">præſertim cum toto illo, quo factæ <lb/>ſunt, temporis intervallo, vix Saturnus 38. </s> <s xml:id="echoid-s7342" xml:space="preserve">gradus confe-<lb/>cerit, traductus ſcilicet à 20 grad. </s> <s xml:id="echoid-s7343" xml:space="preserve">Virginis ad 28. </s> <s xml:id="echoid-s7344" xml:space="preserve">Libræ; </s> <s xml:id="echoid-s7345" xml:space="preserve"><lb/>an fortè inquam modicus ille arcus decurſus ſufficiet ad cer-<lb/>tam & </s> <s xml:id="echoid-s7346" xml:space="preserve">indubitatam hypotheſim ſtatuendam? </s> <s xml:id="echoid-s7347" xml:space="preserve">Statuit autem, <lb/>anſas magis apertas, & </s> <s xml:id="echoid-s7348" xml:space="preserve">patentes videri, Saturno exiſtente <pb o="609" file="0345" n="362" rhead="IN SYSTEMA SATURNIUM."/> in grad. </s> <s xml:id="echoid-s7349" xml:space="preserve">20, 30′. </s> <s xml:id="echoid-s7350" xml:space="preserve">Gemin. </s> <s xml:id="echoid-s7351" xml:space="preserve">& </s> <s xml:id="echoid-s7352" xml:space="preserve">in oppoſito 20. </s> <s xml:id="echoid-s7353" xml:space="preserve">30′. </s> <s xml:id="echoid-s7354" xml:space="preserve">Sagitt. <lb/></s> <s xml:id="echoid-s7355" xml:space="preserve">Sphæricum verò ſine anſis, & </s> <s xml:id="echoid-s7356" xml:space="preserve">ſatellitio in grad. </s> <s xml:id="echoid-s7357" xml:space="preserve">20. </s> <s xml:id="echoid-s7358" xml:space="preserve">30′. </s> <s xml:id="echoid-s7359" xml:space="preserve">Piſc. </s> <s xml:id="echoid-s7360" xml:space="preserve"><lb/>Itemque in 20. </s> <s xml:id="echoid-s7361" xml:space="preserve">30′. </s> <s xml:id="echoid-s7362" xml:space="preserve">Virg. </s> <s xml:id="echoid-s7363" xml:space="preserve">quod ſanè utrùm futurum ſit, ſe-<lb/>quentes anni docebunt. </s> <s xml:id="echoid-s7364" xml:space="preserve">Ipſe hactenus in 20. </s> <s xml:id="echoid-s7365" xml:space="preserve">Virg. </s> <s xml:id="echoid-s7366" xml:space="preserve">obſer-<lb/>vavit, in aliis punctis non vidit, nec eſt, quod aliorum ob-<lb/>ſervationes adducat, quibus fidem omnem detraxit, ac pro-<lb/>inde illi ſuffragari non debent. </s> <s xml:id="echoid-s7367" xml:space="preserve">Unum, quod habet, omit-<lb/>tere ſtatueram; </s> <s xml:id="echoid-s7368" xml:space="preserve">ſed quandoquidem in mentem venit, ſaltem <lb/>indicabo; </s> <s xml:id="echoid-s7369" xml:space="preserve">dicit enim, Saturni globum circumagi quodam <lb/>vertiginis motu; </s> <s xml:id="echoid-s7370" xml:space="preserve">ſed hoc tantummodo divinat, non probat; </s> <s xml:id="echoid-s7371" xml:space="preserve"><lb/>licèt enim Sol hoc vertiginis motu gaudeat, quod probari <lb/>ſolet ex maculis: </s> <s xml:id="echoid-s7372" xml:space="preserve">inde tamen nemo rectè deducat, hunc mo-<lb/>tum reliquis planetis competere. </s> <s xml:id="echoid-s7373" xml:space="preserve">Hoc ii præſertim inſinua-<lb/>re & </s> <s xml:id="echoid-s7374" xml:space="preserve">inculcare non ceſſant, qui globo terraqueo motum ad-<lb/>ſtruunt; </s> <s xml:id="echoid-s7375" xml:space="preserve">ſed fruſtra, cùm Lunam hoc vertiginis motu non <lb/>agi, certiſſimum ſit: </s> <s xml:id="echoid-s7376" xml:space="preserve">quod certe non dico, quaſi hic verti-<lb/>ginis motus in Saturno alicujus momenti ſit, quod ad rem <lb/>hanc pertineat, ultrò enim fateor fieri poſſe ut prædicto mo-<lb/>tu agatur: </s> <s xml:id="echoid-s7377" xml:space="preserve">dico tamen, ex obſervationibus huc uſque factis <lb/>haberi non poſſe.</s> <s xml:id="echoid-s7378" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7379" xml:space="preserve">Quia verò nemini ſuam laudem ereptam velim, admitto <lb/>& </s> <s xml:id="echoid-s7380" xml:space="preserve">oſculor alia multa ab Hugenio tradita, cum veritati appri-<lb/>mè conſentiant. </s> <s xml:id="echoid-s7381" xml:space="preserve">1. </s> <s xml:id="echoid-s7382" xml:space="preserve">planum illud, in quo videntur eſſe ſatel-<lb/>lites, æquatoris plano parallelum rectè judicat: </s> <s xml:id="echoid-s7383" xml:space="preserve">hoc jam in-<lb/>dicarat cum aliis Galilæus; </s> <s xml:id="echoid-s7384" xml:space="preserve">& </s> <s xml:id="echoid-s7385" xml:space="preserve">facilè crediderim, licèt autor <lb/>nullam hujus rei rationem adducat. </s> <s xml:id="echoid-s7386" xml:space="preserve">2. </s> <s xml:id="echoid-s7387" xml:space="preserve">Anſas illas hinc inde <lb/>ſemper æquales eſſe contendit; </s> <s xml:id="echoid-s7388" xml:space="preserve">jure quidem: </s> <s xml:id="echoid-s7389" xml:space="preserve">licèt enim for-<lb/>tè aliquando altera longior viſa ſit, alterius tamen cujuſpiam <lb/>per accidens adjuncti extrinſecus, acceſſio illa facta eſt. </s> <s xml:id="echoid-s7390" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s7391" xml:space="preserve">vult, Lunam quandam eſſe Saturniam, (ſic enim vocat, li-<lb/>cèt propriè loquendo Luna non ſit, cùm circa Saturnum <lb/>ſuum orbem non agat) ſupremum ſcilicet omnium ſatellitem, <lb/>ad inſtar ſtellulæ ſæpe à ſe obſervatum, modo ad ortum, <lb/>modò ad occaſum, modò conjunctum Saturno, cujus pri-<lb/>mus ille inſpector fuit, in quo ſanè maximam laudem mere-<lb/>tur: </s> <s xml:id="echoid-s7392" xml:space="preserve">ego quidem cùm jam anno 1657. </s> <s xml:id="echoid-s7393" xml:space="preserve">de hoc aliquid inau- <pb o="610" file="0346" n="363" rhead="EUSTACHIUS DE DIVINIS"/> diiſſem à clariſſimo viro Michaele Angelo Riccio, eodemque <lb/>nobiliſſimo hujus temporis Geometra: </s> <s xml:id="echoid-s7394" xml:space="preserve">Eundem ſæpe obſer-<lb/>vavi à 30. </s> <s xml:id="echoid-s7395" xml:space="preserve">Junii ad 20. </s> <s xml:id="echoid-s7396" xml:space="preserve">Julii, ſcilicet die 30. </s> <s xml:id="echoid-s7397" xml:space="preserve">Junii hora 2. <lb/></s> <s xml:id="echoid-s7398" xml:space="preserve">noctis juxta horologium Italicum, id eſt 2. </s> <s xml:id="echoid-s7399" xml:space="preserve">horis cum dimi-<lb/>dia poſt ſolis occaſum, illum obſervavi ad ortum Saturni; </s> <s xml:id="echoid-s7400" xml:space="preserve"><lb/>item 2. </s> <s xml:id="echoid-s7401" xml:space="preserve">Julii, 4. </s> <s xml:id="echoid-s7402" xml:space="preserve">9. </s> <s xml:id="echoid-s7403" xml:space="preserve">12. </s> <s xml:id="echoid-s7404" xml:space="preserve">eadem hora, die 14. </s> <s xml:id="echoid-s7405" xml:space="preserve">non apparuit; </s> <s xml:id="echoid-s7406" xml:space="preserve"><lb/>die 20 ad occaſum viſus eſt. </s> <s xml:id="echoid-s7407" xml:space="preserve">Ad prædictas obſervationes <lb/>adhibui teleſcopium 36. </s> <s xml:id="echoid-s7408" xml:space="preserve">palmorum. </s> <s xml:id="echoid-s7409" xml:space="preserve">Cur verò ab aliis Luna <lb/>illa Saturnia obſervata non fuerit ante Hugenium, non qui-<lb/>dem teleſcopiorum vitio, ſed vel inadvertentiæ, vel nimiæ <lb/>ejuſdem à Saturno diſtantiæ, vel parvitati, vel conjunctioni <lb/>tribuendum eſt. </s> <s xml:id="echoid-s7410" xml:space="preserve">4. </s> <s xml:id="echoid-s7411" xml:space="preserve">Anſæ Saturni nunquam mihi viſæ ſunt <lb/>una vice acutiores, alia obtuſiores; </s> <s xml:id="echoid-s7412" xml:space="preserve">majores tamen ſeu latio-<lb/>res aliquando eſſe comperi. </s> <s xml:id="echoid-s7413" xml:space="preserve">Quod autem ſæpè ac ſæpius in-<lb/>culcat, prædicta omnia explicari non poſſe, niſi annulus ab <lb/>eo excogitatus Saturni globum circumdet, pace illius dixerim, <lb/>à veritate alienum eſt; </s> <s xml:id="echoid-s7414" xml:space="preserve">non deſunt alii modi, quibus res iſta <lb/>facili deſignatione exponatur: </s> <s xml:id="echoid-s7415" xml:space="preserve">præ cæteris, mihi ſatis arridet <lb/>illa ſyſtematis Saturnii ratio, quam P. </s> <s xml:id="echoid-s7416" xml:space="preserve">Honoratus Fabri So-<lb/>cietatis Jeſu vir mihi maxima familiaritate conjunctus exco-<lb/>gitavit, cujus breve ſchema ſubjicio.</s> <s xml:id="echoid-s7417" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7418" xml:space="preserve">Supponit. </s> <s xml:id="echoid-s7419" xml:space="preserve">I. </s> <s xml:id="echoid-s7420" xml:space="preserve">terram immobilem in centro mundi, circa <lb/>quam cæleſtes globi ſuos gyros agunt; </s> <s xml:id="echoid-s7421" xml:space="preserve">hanc enim ſententiam <lb/>mordicus defendit, ut catholicis decretis, ſacris literis, ob-<lb/>ſervatis phænomenis, ac rectæ rationi probè conſentientem. <lb/></s> <s xml:id="echoid-s7422" xml:space="preserve">II. </s> <s xml:id="echoid-s7423" xml:space="preserve">Motum realem Saturni æqualem eſſe, apparentem verò <lb/>inæqualem; </s> <s xml:id="echoid-s7424" xml:space="preserve">abhorret enim hominis intellectus, ut vel ipſe <lb/>Copernicus prudenter monet, à reali motuum inæqualitate. </s> <s xml:id="echoid-s7425" xml:space="preserve">III. </s> <s xml:id="echoid-s7426" xml:space="preserve"><lb/>Supponit, motu æquali, æqualibus temporibus, æqualia ſpatia <lb/>decurri, reali ſcilicet, ſeu recta, ſeu curva; </s> <s xml:id="echoid-s7427" xml:space="preserve">ac proinde ma-<lb/>jorem orbem tardiùs, minorem citius, eodem motu deſcri-<lb/>bi. </s> <s xml:id="echoid-s7428" xml:space="preserve">IV. </s> <s xml:id="echoid-s7429" xml:space="preserve">Supponit, motum circularem diurnum ab ortu ad oc-<lb/>caſum. </s> <s xml:id="echoid-s7430" xml:space="preserve">V. </s> <s xml:id="echoid-s7431" xml:space="preserve">motum rectum ab apogæo verſus perigæum, & </s> <s xml:id="echoid-s7432" xml:space="preserve">vi-<lb/>ciſſim; </s> <s xml:id="echoid-s7433" xml:space="preserve">acceleratum quidem, ab utroque termino ad medio-<lb/>crem diſtantiam, quæ lineam apogæum, & </s> <s xml:id="echoid-s7434" xml:space="preserve">perigæum conne-<lb/>ctentem bifariam ſecat; </s> <s xml:id="echoid-s7435" xml:space="preserve">retardatum verò à mediocri diſtan- <pb o="611" file="0347" n="364" rhead="IN SYSTEMA SATURNIUM."/> tia ad utrumque terminum; </s> <s xml:id="echoid-s7436" xml:space="preserve">hunc autem primum motum re-<lb/>ctum vocat. </s> <s xml:id="echoid-s7437" xml:space="preserve">VI. </s> <s xml:id="echoid-s7438" xml:space="preserve">Supponit alium motum rectum, ſecundum <lb/>ſcilicet, ac declinationis ab Auſtro ad Boream, & </s> <s xml:id="echoid-s7439" xml:space="preserve">viciſſim; <lb/></s> <s xml:id="echoid-s7440" xml:space="preserve">fines porro diverſi ſunt in diverſis planetis, pro diverſa de-<lb/>clinatione maxima cujuſlibet planetæ; </s> <s xml:id="echoid-s7441" xml:space="preserve">eſt etiam hic motus <lb/>eodem modo acceleratus, & </s> <s xml:id="echoid-s7442" xml:space="preserve">retardatus, quo prior; </s> <s xml:id="echoid-s7443" xml:space="preserve">& </s> <s xml:id="echoid-s7444" xml:space="preserve">me-<lb/>diocris diſtantia eſt in æquatore: </s> <s xml:id="echoid-s7445" xml:space="preserve">ex his tribus motibus, Sa-<lb/>turni motus componitur; </s> <s xml:id="echoid-s7446" xml:space="preserve">vel potiùs eſt motus quidam ſim-<lb/>plex circulari affinis, prædictis duabus determinationibus in-<lb/>ſtructus; </s> <s xml:id="echoid-s7447" xml:space="preserve">vel, ut ajunt Scholaſtici, modificatus; </s> <s xml:id="echoid-s7448" xml:space="preserve">primus mo-<lb/>tus brevior eſt; </s> <s xml:id="echoid-s7449" xml:space="preserve">ſecundus diuturnior; </s> <s xml:id="echoid-s7450" xml:space="preserve">cujus ſanè diſcriminis <lb/>cauſam affert. </s> <s xml:id="echoid-s7451" xml:space="preserve">Sed ut rude aliquod ſchema exhibeam, omiſ-<lb/>ſo ſecundo motu recto, ſit centrum mundi B. </s> <s xml:id="echoid-s7452" xml:space="preserve">hic terram col-<lb/> <anchor type="note" xlink:label="note-0347-01a" xlink:href="note-0347-01"/> locat immobilem. </s> <s xml:id="echoid-s7453" xml:space="preserve">Sit G B diſtantia Saturni apogæi; </s> <s xml:id="echoid-s7454" xml:space="preserve">B O di-<lb/>ſtantia perigæi; </s> <s xml:id="echoid-s7455" xml:space="preserve">B P diſtantia mediocris, Saturnus circulum, <lb/>vel orbem radio G B tardius percurrit, quàm Stella vel pun-<lb/>ctum primi mobilis, cui reſpondet; </s> <s xml:id="echoid-s7456" xml:space="preserve">citius verò percurrit or-<lb/>bem radio B O, & </s> <s xml:id="echoid-s7457" xml:space="preserve">æquè citò radio B A minore mediocri <lb/>diſtantia B P. </s> <s xml:id="echoid-s7458" xml:space="preserve">His præmiſſis, ſit Saturnus in apogæo G, re-<lb/>ſpondens puncto C primi mobilis, vel initio arietis v. </s> <s xml:id="echoid-s7459" xml:space="preserve">g. </s> <s xml:id="echoid-s7460" xml:space="preserve">& </s> <s xml:id="echoid-s7461" xml:space="preserve"><lb/>ſupponatur punctum C per S deſcribere ſuum circulum, mo-<lb/>tu diurno; </s> <s xml:id="echoid-s7462" xml:space="preserve">item Saturnus in G ſuum, per R, ſcilicet versùs <lb/>occaſum; </s> <s xml:id="echoid-s7463" xml:space="preserve">punctum C citiùs abſolvit ſuum gyrum, quàm Sa-<lb/>turnus; </s> <s xml:id="echoid-s7464" xml:space="preserve">igitur Saturnum à tergo relinquit, versùs ortum, ſci-<lb/>licet in aliquo puncto curvæ G H; </s> <s xml:id="echoid-s7465" xml:space="preserve">nempe primo motu re-<lb/>cto, à circulo apogæi L R ſenſim deſcendit ad circulum pe-<lb/>rigæi Z X, donec tandem multis orbibus actis, perveniat in <lb/>H, & </s> <s xml:id="echoid-s7466" xml:space="preserve">reſpondeat puncto E primi mobilis, v. </s> <s xml:id="echoid-s7467" xml:space="preserve">g. </s> <s xml:id="echoid-s7468" xml:space="preserve">20. </s> <s xml:id="echoid-s7469" xml:space="preserve">gradui <lb/>Arietis: </s> <s xml:id="echoid-s7470" xml:space="preserve">toto illo tempore, moveri dicitur in conſequentia; <lb/></s> <s xml:id="echoid-s7471" xml:space="preserve">id eſt, juxta ſeriem ſignorum, initio quidem, motu accele-<lb/>rato; </s> <s xml:id="echoid-s7472" xml:space="preserve">& </s> <s xml:id="echoid-s7473" xml:space="preserve">ſub finem, retardato; </s> <s xml:id="echoid-s7474" xml:space="preserve">in H ſtare videtur: </s> <s xml:id="echoid-s7475" xml:space="preserve">hîc enim <lb/>æquè citò ac punctum E ſuum gyrum agit: </s> <s xml:id="echoid-s7476" xml:space="preserve">Quia verô adhuc <lb/>deſcendit versùs perigæum, decurrit deinceps ſuos orbes ci-<lb/>tiùs, quàm puncta primi mobilis, quibus reſpondet; </s> <s xml:id="echoid-s7477" xml:space="preserve">ac pro-<lb/>inde illa relinquit à tergo, versùs ortum ab H quidem, ad <lb/>perigæum I: </s> <s xml:id="echoid-s7478" xml:space="preserve">ab I verò, ad K; </s> <s xml:id="echoid-s7479" xml:space="preserve">hinc videtur regredi ab E ad <pb o="612" file="0348" n="365" rhead="EUSTACHIUS DE DIVINIS"/> D, quæ ſunt puncta primi mobilis; </s> <s xml:id="echoid-s7480" xml:space="preserve">idque motu retardato, <lb/>circa H & </s> <s xml:id="echoid-s7481" xml:space="preserve">K, id eſt, circa puncta ſtationum, accelerato <lb/>verò, hinc inde circa I. </s> <s xml:id="echoid-s7482" xml:space="preserve">hinc duplex ſtatio, in punctis ſcili-<lb/>cet E & </s> <s xml:id="echoid-s7483" xml:space="preserve">D. </s> <s xml:id="echoid-s7484" xml:space="preserve">hinc regreſſio, vel motus retrogradus brevior <lb/>directo; </s> <s xml:id="echoid-s7485" xml:space="preserve">quia radius circuli æquè diuturni (ſic eum vocat, <lb/>quem planeta æquè citò conficit, ac punctum primi mobi-<lb/>lis ſuum) eſt brevior, radio mediocris diſtantiæ B P. </s> <s xml:id="echoid-s7486" xml:space="preserve">ex <lb/>his ſequitur latitudinis mutatio ſingulis ferè diebus: </s> <s xml:id="echoid-s7487" xml:space="preserve">ſed hæc <lb/>rudi Minerva deſcripſiſſe ſufficiat; </s> <s xml:id="echoid-s7488" xml:space="preserve">ex quibus omnia Saturni <lb/>phænomena, nullo ferè negotio, per cauſas phyſicas expli-<lb/>cat. </s> <s xml:id="echoid-s7489" xml:space="preserve">Jam ad ſatellites Saturni orationem converto. </s> <s xml:id="echoid-s7490" xml:space="preserve">Quin-<lb/>que Fabrius agnoſcit; </s> <s xml:id="echoid-s7491" xml:space="preserve">minimus & </s> <s xml:id="echoid-s7492" xml:space="preserve">ſupremus omnium eſt <lb/>quem Hugenius, primus opinor, ſub aſpectuna<unsure/> hominum <lb/>adduxit, quemque, ut dixi, ab illuſtriſſimo, & </s> <s xml:id="echoid-s7493" xml:space="preserve">doctiſſimo <lb/>Riccio certior factus, adhibito 36. </s> <s xml:id="echoid-s7494" xml:space="preserve">palmorum teleſcopio, <lb/>an. </s> <s xml:id="echoid-s7495" xml:space="preserve">1657. </s> <s xml:id="echoid-s7496" xml:space="preserve">per multos dies, menſe Julio conſpexi, ortivum <lb/>modò, mox occiduum; </s> <s xml:id="echoid-s7497" xml:space="preserve">dicta autem die non comparuit, <lb/>quia conjunctus Saturno erat. </s> <s xml:id="echoid-s7498" xml:space="preserve">Hic ergo ſupra Saturnum eſt, <lb/>& </s> <s xml:id="echoid-s7499" xml:space="preserve">dum in perigæo verſatur, citiùs ſuum orbem abſolvit, quàm <lb/>Saturnus ſuum, quando eſt in apogæo, tardiùs; </s> <s xml:id="echoid-s7500" xml:space="preserve">& </s> <s xml:id="echoid-s7501" xml:space="preserve">æquè ci-<lb/>tò in mediocri diſtantia, æquè ab utroque termino diſtante: <lb/></s> <s xml:id="echoid-s7502" xml:space="preserve">hinc aliquando Saturnus illum à tergo relinquit, aliquando <lb/>ille Saturnum; </s> <s xml:id="echoid-s7503" xml:space="preserve">hinc aliquando ortivus, aliquando occiduus, <lb/>aliquando latet in conjunctione; </s> <s xml:id="echoid-s7504" xml:space="preserve">à conjunctione apogæi, <lb/>ſemper ortivus; </s> <s xml:id="echoid-s7505" xml:space="preserve">à conjunctione perigæi, ſemper occiduus; </s> <s xml:id="echoid-s7506" xml:space="preserve"><lb/>quid plura? </s> <s xml:id="echoid-s7507" xml:space="preserve">in Mediceis appoſitam analogiam habemus.</s> <s xml:id="echoid-s7508" xml:space="preserve"/> </p> <div xml:id="echoid-div363" type="float" level="2" n="3"> <note position="right" xlink:label="note-0347-01" xlink:href="note-0347-01a" xml:space="preserve">TAB. LI. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s7509" xml:space="preserve">Alii quatuor inferiores ſunt, Saturno licèt ſuperiores, <lb/> <anchor type="note" xlink:label="note-0348-01a" xlink:href="note-0348-01"/> quorum ſyſtema ſic Fabrius explicat: </s> <s xml:id="echoid-s7510" xml:space="preserve">Sit centrum terræ, & </s> <s xml:id="echoid-s7511" xml:space="preserve"><lb/>mundi A, Saturni diameter G C, centrum B, ducta cenſea-<lb/>tur recta A B P; </s> <s xml:id="echoid-s7512" xml:space="preserve">ſint duo N H, S D, in maxima digreſſio-<lb/>ne N H ortiva; </s> <s xml:id="echoid-s7513" xml:space="preserve">S D occidua, hinc inde æquali; </s> <s xml:id="echoid-s7514" xml:space="preserve">in qua ſit <lb/>illorum ſtatio; </s> <s xml:id="echoid-s7515" xml:space="preserve">& </s> <s xml:id="echoid-s7516" xml:space="preserve">mediocris diſtantia A I; </s> <s xml:id="echoid-s7517" xml:space="preserve">hoc loco, dia-<lb/>metros dumtaxat pro globis exhibeo: </s> <s xml:id="echoid-s7518" xml:space="preserve">tendat autem N H <lb/>versùs perigæum; </s> <s xml:id="echoid-s7519" xml:space="preserve">& </s> <s xml:id="echoid-s7520" xml:space="preserve">S D, versùs apogæum, pari greſſu, & </s> <s xml:id="echoid-s7521" xml:space="preserve"><lb/>paribus intervallis, ita ut hic apogæum, & </s> <s xml:id="echoid-s7522" xml:space="preserve">ille perigæum ſi-<lb/>mul attingant; </s> <s xml:id="echoid-s7523" xml:space="preserve">iſtud eſt inter B I, illud verò inter I O. </s> <s xml:id="echoid-s7524" xml:space="preserve">tunc <pb o="613" file="0349" n="366" rhead="IN SYSTEMA SATURNIUM."/> uterque ſimul erit in cunjunctione, alter perigæi, alter apo-<lb/>gæi, ſed paulò poſt N H à perigæo aſcendit ad maximam di-<lb/>greſſionem occiduam S D, & </s> <s xml:id="echoid-s7525" xml:space="preserve">S D ab apogæo deſcendit ad <lb/>maximam digreſſionem ortivam N H; </s> <s xml:id="echoid-s7526" xml:space="preserve">atque ita deinceps, re-<lb/>petitis ac reciprocis recurſibus: </s> <s xml:id="echoid-s7527" xml:space="preserve">idem prorſus dictum ſit de <lb/>duobus K L, K E, quorum mediocris diſtantia communis <lb/>eſt in O; </s> <s xml:id="echoid-s7528" xml:space="preserve">perigæum commune inter O I; </s> <s xml:id="echoid-s7529" xml:space="preserve">apogæum ſupra O, <lb/>æquè diſtans; </s> <s xml:id="echoid-s7530" xml:space="preserve">maxima digreſſio ortiva K L, occidua R E; <lb/></s> <s xml:id="echoid-s7531" xml:space="preserve">eſt autem digreſſio iſtorum, qui ſuperiores ſunt, major di-<lb/>greſſione illorum, quos inferiores eſſe ſupponit; </s> <s xml:id="echoid-s7532" xml:space="preserve">ſuperiores <lb/>appellat lucidos, inferiores obſcuros; </s> <s xml:id="echoid-s7533" xml:space="preserve">lucidi conſtant ex ma-<lb/>teria aptiſſima ad repercuſſionem luminis; </s> <s xml:id="echoid-s7534" xml:space="preserve">obſcuri verò, ex <lb/>ineptiſſima, ad eundem finem; </s> <s xml:id="echoid-s7535" xml:space="preserve">nec difficile fuit auctori na-<lb/>turæ, hujuſmodi materiam invenire, cùm etiam in hoc ſub-<lb/>lunari tractu hujuſmodi habeamus: </s> <s xml:id="echoid-s7536" xml:space="preserve">mediocris diſtantia in <lb/>utriſque ab apogæo, & </s> <s xml:id="echoid-s7537" xml:space="preserve">perigæo æquè diſtat: </s> <s xml:id="echoid-s7538" xml:space="preserve">vult etiam idem <lb/>Fabrius, quinque ſatellites ejuſdem eſſe cum Saturno decli-<lb/>nationis, ac proinde ſemper in eodem plano eſſe æquatori <lb/>parallelo: </s> <s xml:id="echoid-s7539" xml:space="preserve">quod facilè obtinetur, modò ab initio eandem <lb/>declinationem habuerint, eundemque ſimul inierint declina-<lb/>tionis motum, in eodem ſemper plano æquatori parallelo ſi-<lb/>ti: </s> <s xml:id="echoid-s7540" xml:space="preserve">nec eſt, quod quiſquam miretur, quinque ſatellites Sa-<lb/>turno adeſſe, cum Jovi quatuor adſint, Soli duo, Terræ <lb/>unus, Marti fortè tres, quis ſcit, nondum explorati; </s> <s xml:id="echoid-s7541" xml:space="preserve">quia <lb/>à Martis globo longius digrediuntur, quàm Medicei à Jo-<lb/>ve; </s> <s xml:id="echoid-s7542" xml:space="preserve">non tamen tam longè, quàm Venus aut Mercurius à <lb/>Sole; </s> <s xml:id="echoid-s7543" xml:space="preserve">vult demum, obſcuros citiùs ſuam revolutionem abſol-<lb/>vere, & </s> <s xml:id="echoid-s7544" xml:space="preserve">lucidos tardiùs; </s> <s xml:id="echoid-s7545" xml:space="preserve">cujus rei analogiam habemus in Ve-<lb/>nere, & </s> <s xml:id="echoid-s7546" xml:space="preserve">Mercurio.</s> <s xml:id="echoid-s7547" xml:space="preserve"/> </p> <div xml:id="echoid-div364" type="float" level="2" n="4"> <note position="left" xlink:label="note-0348-01" xlink:href="note-0348-01a" xml:space="preserve">TAB. LI. <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s7548" xml:space="preserve">His poſitis, omnes Saturni phaſes, omnia phænomena <lb/>facilè omnino explicantur. </s> <s xml:id="echoid-s7549" xml:space="preserve">Solus Saturni globus ſuis ſatelli-<lb/>tibus aliquando deſtitutus apparet, ſcilicet figura ſphærica; <lb/></s> <s xml:id="echoid-s7550" xml:space="preserve">nec enim alia cæleſtibus corporibus indita eſt: </s> <s xml:id="echoid-s7551" xml:space="preserve">hujus phæno-<lb/>meni ratio ex eo petitur, quòd duo lucidi (de primo enim <lb/>omnium ſupremo hoc loco non agitur) vel ſint conjuncti, id <pb o="614" file="0350" n="367" rhead="EUSTACHIUS DE DIVINIS"/> eſt, alter in apogæo, alter in perigæo, vel ab obſcuris peni-<lb/>tùs tecti, etiam extra conjunctionem:</s> <s xml:id="echoid-s7552" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7553" xml:space="preserve">II. </s> <s xml:id="echoid-s7554" xml:space="preserve">Duo lucidi aliquando videntur à globo Saturni omni-<lb/>no ſejuncti, & </s> <s xml:id="echoid-s7555" xml:space="preserve">figuramſ phæricam præferentes; </s> <s xml:id="echoid-s7556" xml:space="preserve">hoc ipſum <lb/>toties accidit, quoties duo obſcuri in conjunctione verſan-<lb/>tur, alter ſcilicet in apogæo, alter in perigæo, & </s> <s xml:id="echoid-s7557" xml:space="preserve">lucidi hinc <lb/>inde digreſſi.</s> <s xml:id="echoid-s7558" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7559" xml:space="preserve">III. </s> <s xml:id="echoid-s7560" xml:space="preserve">Aliquando lucidi apparent, quaſi lucidæ anſulæ eſ-<lb/>ſent, vel Lunæ falcatæ; </s> <s xml:id="echoid-s7561" xml:space="preserve">hoc evenit quando obſcuri ſunt qui-<lb/>dem interpoſiti, non tamen integrum lucidorum diſcum te-<lb/>gunt.</s> <s xml:id="echoid-s7562" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7563" xml:space="preserve">IV. </s> <s xml:id="echoid-s7564" xml:space="preserve">Hinc modò plus, modò minùs diſci lucidi ab obſcu-<lb/> <anchor type="note" xlink:label="note-0350-01a" xlink:href="note-0350-01"/> ris tegitur; </s> <s xml:id="echoid-s7565" xml:space="preserve">unde diverſa phaſis, & </s> <s xml:id="echoid-s7566" xml:space="preserve">diverſa figura; </s> <s xml:id="echoid-s7567" xml:space="preserve">neque <lb/>hoc teleſcopii vitio Hugenius tribuere debuit, cùm alioqui <lb/>optima ab authoribus, quos citat, adhibita fuerint; </s> <s xml:id="echoid-s7568" xml:space="preserve">Galilæo <lb/>ſcilicet, Scheinero, Ricciolo, Hevelio, Gaſſendo, quo-<lb/>rum figuræ ab Hugenio excuſæ, in hac hypotheſi, nullo-<lb/>negotio demonſtrari poſſunt; </s> <s xml:id="echoid-s7569" xml:space="preserve">undecima tamen inter alias <lb/>Fontanæ fabulas referenda eſt; </s> <s xml:id="echoid-s7570" xml:space="preserve">decima tertia, quæ caret <lb/>auctore, modò tantulùm reformetur, fortè ſuo modo ex-<lb/>plicari poſſet, in aliquo caſu; </s> <s xml:id="echoid-s7571" xml:space="preserve">ſeptima Hevelii fictitia eſt; <lb/></s> <s xml:id="echoid-s7572" xml:space="preserve">ſphæroidem eſſe putavit, ſed hæc figura cæleſtibus globis <lb/>minimè competit; </s> <s xml:id="echoid-s7573" xml:space="preserve">niſi fortè dicamus Saturnum ellipticum <lb/>videri, quando obſcuris in conjunctione ſtatutis, lucidi <lb/>tantulum à conjunctione recedunt, tunc autem anſulæ non <lb/>apparent.</s> <s xml:id="echoid-s7574" xml:space="preserve"/> </p> <div xml:id="echoid-div365" type="float" level="2" n="5"> <note position="left" xlink:label="note-0350-01" xlink:href="note-0350-01a" xml:space="preserve">TAB. LII.</note> </div> <p> <s xml:id="echoid-s7575" xml:space="preserve">V. </s> <s xml:id="echoid-s7576" xml:space="preserve">Veriſimile eſt, obſcuros æquales eſſe lucidis; </s> <s xml:id="echoid-s7577" xml:space="preserve">nihil <lb/>enim probat, vel majores, vel minores eſſe, æquales igitur <lb/>cenſendi ſunt, dum aliquid probet inæqualitatem; </s> <s xml:id="echoid-s7578" xml:space="preserve">immò <lb/>potiùs æqualitas ex ipſis phaſibus perſuadetur, quartæ ſci-<lb/>licet figuræ, quintæ, ſextæ, octavæ &</s> <s xml:id="echoid-s7579" xml:space="preserve">c. </s> <s xml:id="echoid-s7580" xml:space="preserve">Arcus enim falcis <lb/>lucidæ, cavus ſcilicet, & </s> <s xml:id="echoid-s7581" xml:space="preserve">convexus æqualium circulorum eſ-<lb/>ſe videntur.</s> <s xml:id="echoid-s7582" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7583" xml:space="preserve">VI. </s> <s xml:id="echoid-s7584" xml:space="preserve">Linea, vel diameter anſas lucidas connectens, quæ <lb/>per medium globi Saturnii centrum duci videtur, eſt ſem- <pb o="615" file="0351" n="368" rhead="IN SYSTEMA SATURNIUM."/> per in plano parallelo plano æquatoris; </s> <s xml:id="echoid-s7585" xml:space="preserve">quia cùm ſatelli-<lb/>tes Saturni ejuſdem ſemper ſint cum eo declinationis & </s> <s xml:id="echoid-s7586" xml:space="preserve"><lb/>motu ſemper æquali in aliquo parallelo moveantur, mo-<lb/>tus enim diurnus ab ortu ad occaſum in aliquo paralle-<lb/>lo fieri cenſetur, quid mirum, ſi linea eorum centra con-<lb/>nectens in eo plano ſit, quod plano æquatoris parallelum <lb/>eſt; </s> <s xml:id="echoid-s7587" xml:space="preserve">dixi, quæ per centrum Saturni duci videtur, quia <lb/>revera licèt Saturno ſuperiores ſint, oculus tamen judicat, <lb/>ejuſdem altitudinis eſſe; </s> <s xml:id="echoid-s7588" xml:space="preserve">ac proinde in eadem linea cum <lb/>Saturno.</s> <s xml:id="echoid-s7589" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7590" xml:space="preserve">VII. </s> <s xml:id="echoid-s7591" xml:space="preserve">Licèt ſupremi ſatellitis motus ab Hugenio defini-<lb/>tus ſit, utpote qui 16. </s> <s xml:id="echoid-s7592" xml:space="preserve">circiter dierum ſpatio abſolvatur, <lb/>quanquam obſervationes à me factæ tantulùm refragantur; <lb/></s> <s xml:id="echoid-s7593" xml:space="preserve">non tamen huc uſque quatuor inferiorum motus, vel ab <lb/>eo, vel ab alio quopiam definitus eſt, qui tamen ex fu-<lb/>turis obſervationibus haberi poſſet.</s> <s xml:id="echoid-s7594" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7595" xml:space="preserve">VIII. </s> <s xml:id="echoid-s7596" xml:space="preserve">Turbinatio Saturni, vel illius annuli fictitii nulla <lb/>obſervatione probatur, ut jam indicavi; </s> <s xml:id="echoid-s7597" xml:space="preserve">licèt enim ſol hoc <lb/>vertiginis motu agatur circa ſuum centrum, ut evincitur ex <lb/>illius maculis; </s> <s xml:id="echoid-s7598" xml:space="preserve">aliis tamen planetis nulla hujuſmodi, vel alia <lb/>quæpiam probatio ſuffragatur.</s> <s xml:id="echoid-s7599" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7600" xml:space="preserve">IX. </s> <s xml:id="echoid-s7601" xml:space="preserve">Illa brachia oblonga quæ Hugenius Saturno anne-<lb/>xa obſervavit annis 1655. </s> <s xml:id="echoid-s7602" xml:space="preserve">& </s> <s xml:id="echoid-s7603" xml:space="preserve">1656. </s> <s xml:id="echoid-s7604" xml:space="preserve">ſunt omnino fictitia, & </s> <s xml:id="echoid-s7605" xml:space="preserve"><lb/>vitio teleſcopii tribuenda: </s> <s xml:id="echoid-s7606" xml:space="preserve">cur enim alii non obſervaſſent? <lb/></s> <s xml:id="echoid-s7607" xml:space="preserve">præſertim cùm teleſcopium ad hunc finem ab eo adhibi-<lb/>tum brevius eſſet, ac proinde non melioris conditionis quàm <lb/>alia.</s> <s xml:id="echoid-s7608" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7609" xml:space="preserve">X. </s> <s xml:id="echoid-s7610" xml:space="preserve">Figuræ ab illo excuſæ, & </s> <s xml:id="echoid-s7611" xml:space="preserve">rejectæ facilè explicari <lb/>poſſunt in hac Fabriana hypotheſi: </s> <s xml:id="echoid-s7612" xml:space="preserve">1<emph style="super">a</emph> quidem lucidis in <lb/>maxima digreſſione, & </s> <s xml:id="echoid-s7613" xml:space="preserve">obſcuris in conjunctione exiſten-<lb/>tibus; </s> <s xml:id="echoid-s7614" xml:space="preserve">2<emph style="super">a</emph> verò, lucidis tangentibus Saturni globum, & </s> <s xml:id="echoid-s7615" xml:space="preserve">ob-<lb/>ſcuris in conjunctione poſitis; </s> <s xml:id="echoid-s7616" xml:space="preserve">4<emph style="super">a</emph> lucidis in maxima di-<lb/>greſſione poſitis, & </s> <s xml:id="echoid-s7617" xml:space="preserve">obſcuris tantulùm extantibus; </s> <s xml:id="echoid-s7618" xml:space="preserve">ita ut <lb/>modicam lucidorum portionem tegant; </s> <s xml:id="echoid-s7619" xml:space="preserve">3<emph style="super">a</emph>. </s> <s xml:id="echoid-s7620" xml:space="preserve">eadem eſt cum <lb/>4<emph style="super">a</emph>. </s> <s xml:id="echoid-s7621" xml:space="preserve">ſed teleſcopii vitio, in mucronem ivit, cùm in con- <pb o="616" file="0352" n="369" rhead="EUSTACHIUS DE DIVINIS"/> vexum ire debuiſſet; </s> <s xml:id="echoid-s7622" xml:space="preserve">5<emph style="super">a</emph>, & </s> <s xml:id="echoid-s7623" xml:space="preserve">6<emph style="super">a</emph>. </s> <s xml:id="echoid-s7624" xml:space="preserve">lucidis non totis extantibus, <lb/>& </s> <s xml:id="echoid-s7625" xml:space="preserve">majori ex parte tectis ab obſcuris, tunc enim reſtat <lb/>quaſi falcula, differt autem 5<emph style="super">a</emph> à 6<emph style="super">a</emph> ſecundùm plùs, & </s> <s xml:id="echoid-s7626" xml:space="preserve">mi-<lb/>nùs 7<emph style="super">a</emph>, ut dixi, fictitia eſt; </s> <s xml:id="echoid-s7627" xml:space="preserve">cùm Saturnus ſphærois non <lb/>ſit, niſi fortè eo modo, quem ſupra num. </s> <s xml:id="echoid-s7628" xml:space="preserve">4. </s> <s xml:id="echoid-s7629" xml:space="preserve">expoſui; </s> <s xml:id="echoid-s7630" xml:space="preserve">8<emph style="super">a</emph> <lb/>lucidis inter maximam digreſſionem, & </s> <s xml:id="echoid-s7631" xml:space="preserve">Saturni globum <lb/>exiſtentibus, tectiſque ex parte, à minore portione ob-<lb/>ſcurorum; </s> <s xml:id="echoid-s7632" xml:space="preserve">tunc enim falces majores ſunt; </s> <s xml:id="echoid-s7633" xml:space="preserve">latiores qui-<lb/>dem, quò propiùs ad maximam digreſſionem accedunt; <lb/></s> <s xml:id="echoid-s7634" xml:space="preserve">anguſtiores verò, quo Saturnio globo propiores ſiunt; </s> <s xml:id="echoid-s7635" xml:space="preserve">9<emph style="super">a</emph> <lb/>& </s> <s xml:id="echoid-s7636" xml:space="preserve">10<emph style="super">a</emph> ab octava tantummodo differunt, ſecundùm plus, <lb/>& </s> <s xml:id="echoid-s7637" xml:space="preserve">minùs; </s> <s xml:id="echoid-s7638" xml:space="preserve">ſunt enim multæ combinationes; </s> <s xml:id="echoid-s7639" xml:space="preserve">quas habes in <lb/>duobus circulis ſeſe invicem ſecantibus; </s> <s xml:id="echoid-s7640" xml:space="preserve">tres ultimas ficti-<lb/>tias reputo; </s> <s xml:id="echoid-s7641" xml:space="preserve">niſi fortè 12<emph style="super">a</emph> ad 10, reducatur itemque 13<emph style="super">a</emph> de <lb/>qua infra.</s> <s xml:id="echoid-s7642" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7643" xml:space="preserve">XI. </s> <s xml:id="echoid-s7644" xml:space="preserve">Anſa una non longiùs extenditur, quàm alia; </s> <s xml:id="echoid-s7645" xml:space="preserve">hinc <lb/>æqualis eſt utriuſque lucidi digreſſio; </s> <s xml:id="echoid-s7646" xml:space="preserve">quia uterque com-<lb/>muni apogæo & </s> <s xml:id="echoid-s7647" xml:space="preserve">perigæo gaudet, & </s> <s xml:id="echoid-s7648" xml:space="preserve">æquali motu move-<lb/>tur; </s> <s xml:id="echoid-s7649" xml:space="preserve">item obſcuri eodem apogæo & </s> <s xml:id="echoid-s7650" xml:space="preserve">perigæo gaudent; </s> <s xml:id="echoid-s7651" xml:space="preserve">ex <lb/>his, perfecta anſarum & </s> <s xml:id="echoid-s7652" xml:space="preserve">brachiorum æqualitas neceſſariò <lb/>ſequitur.</s> <s xml:id="echoid-s7653" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7654" xml:space="preserve">XII. </s> <s xml:id="echoid-s7655" xml:space="preserve">Anſulæ videntur, quando portio lucidorum ab ob-<lb/>ſcuris tegitur; </s> <s xml:id="echoid-s7656" xml:space="preserve">nempe ſunt quædam quaſi aperturæ ni-<lb/>græ, quas lucida corona cingit; </s> <s xml:id="echoid-s7657" xml:space="preserve">hæ ſunt anſulæ; </s> <s xml:id="echoid-s7658" xml:space="preserve">brachia <lb/>verò, quando lucidi tangunt globum Saturni, obſcuris in <lb/>conjunctione exiſtentibus.</s> <s xml:id="echoid-s7659" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7660" xml:space="preserve">XIII. </s> <s xml:id="echoid-s7661" xml:space="preserve">Anſulæ eò longiùs porriguntur quo propiùs lu-<lb/>cidi ad maximam digreſſionem accedunt; </s> <s xml:id="echoid-s7662" xml:space="preserve">obſcuris ſcilicet <lb/>lucidorum partem tegentibus.</s> <s xml:id="echoid-s7663" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7664" xml:space="preserve">XIV. </s> <s xml:id="echoid-s7665" xml:space="preserve">Anſulæ eò patentiores ſunt, quò major obſcuro-<lb/>rum portio extra Saturnum extat, lucidis in maxima di-<lb/>greſſione conſtitutis; </s> <s xml:id="echoid-s7666" xml:space="preserve">anguſtiores verò, cum minor obſcu-<lb/>rorum portio extat.</s> <s xml:id="echoid-s7667" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7668" xml:space="preserve">XV. </s> <s xml:id="echoid-s7669" xml:space="preserve">Anſulæ latiores ſunt, quando minor obſcurorum <lb/>portio tegit lucidos, in maxima digreſſione poſitos.</s> <s xml:id="echoid-s7670" xml:space="preserve"/> </p> <pb o="617" file="0353" n="370" rhead="IN SYSTEMA SATURNIUM."/> <p> <s xml:id="echoid-s7671" xml:space="preserve">XVI. </s> <s xml:id="echoid-s7672" xml:space="preserve">Anſularum brachia ad ipſum uſque Saturni glo-<lb/>bum excurrunt, quando minor obſcurorum portio lucidos <lb/>tegit.</s> <s xml:id="echoid-s7673" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7674" xml:space="preserve">XVII. </s> <s xml:id="echoid-s7675" xml:space="preserve">Anſulæ à Saturni globo ſunt prorſus avulſæ at-<lb/>que ſejunctæ, quando major obſcurorum portio lucidos <lb/>tegit.</s> <s xml:id="echoid-s7676" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7677" xml:space="preserve">XVIII. </s> <s xml:id="echoid-s7678" xml:space="preserve">Anſulæ ſenſim contrahuntur, quando à digreſ-<lb/>ſione lucidi recedunt, & </s> <s xml:id="echoid-s7679" xml:space="preserve">obſcuri ad digreſſionem acce-<lb/>dunt.</s> <s xml:id="echoid-s7680" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7681" xml:space="preserve">XIX. </s> <s xml:id="echoid-s7682" xml:space="preserve">Quando anſulæ latiores & </s> <s xml:id="echoid-s7683" xml:space="preserve">majores ſunt, ipſe <lb/>Saturni globus illuſtrior apparet, quia lumina quaſi com-<lb/>muni quodam niſu & </s> <s xml:id="echoid-s7684" xml:space="preserve">illapſu oculum feriunt; </s> <s xml:id="echoid-s7685" xml:space="preserve">ſic multi <lb/>ignes ſeorſim poſiti, actione quadam communi, plus calefa-<lb/>ciunt, ac proinde majoris lucis vim diffundunt.</s> <s xml:id="echoid-s7686" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7687" xml:space="preserve">XX. </s> <s xml:id="echoid-s7688" xml:space="preserve">Anſulæ ſenſim dilatantur & </s> <s xml:id="echoid-s7689" xml:space="preserve">augentur, quando luci-<lb/>di ad maximam digreſſionem accedunt, & </s> <s xml:id="echoid-s7690" xml:space="preserve">ab illa obſcuri <lb/>recedunt.</s> <s xml:id="echoid-s7691" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7692" xml:space="preserve">XXI. </s> <s xml:id="echoid-s7693" xml:space="preserve">Primus ille Saturni comes ſub Saturno aliquan-<lb/>do ponitur ab Hugenio, qui vult, illum circa Satur-<lb/>num moveri; </s> <s xml:id="echoid-s7694" xml:space="preserve">hoc tamen veritati repugnat; </s> <s xml:id="echoid-s7695" xml:space="preserve">ſi enim ſub <lb/>Saturno eſſet, prædictas anſularum phaſes omnino turba-<lb/>ret: </s> <s xml:id="echoid-s7696" xml:space="preserve">Idem dico de Mediceis, ſeu Jovialibus; </s> <s xml:id="echoid-s7697" xml:space="preserve">igitur & </s> <s xml:id="echoid-s7698" xml:space="preserve">Jo-<lb/>vialia ſupra Jovem, & </s> <s xml:id="echoid-s7699" xml:space="preserve">Saturnia ſupra Saturnum collocata <lb/>fuere.</s> <s xml:id="echoid-s7700" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7701" xml:space="preserve">XXII. </s> <s xml:id="echoid-s7702" xml:space="preserve">Si quando accidat, ut primus ille Saturni comes <lb/>extremitati alterius anſulæ adhærere videatur; </s> <s xml:id="echoid-s7703" xml:space="preserve">nihil enim ob-<lb/>ſtat, talem illius poſitionem eſſe poſſe; </s> <s xml:id="echoid-s7704" xml:space="preserve">hæc anſula paulò al-<lb/>terâ longior eſſe videbitur.</s> <s xml:id="echoid-s7705" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7706" xml:space="preserve">XXIII. </s> <s xml:id="echoid-s7707" xml:space="preserve">Immo ſi caſu aliquo ſtellula quæpiam alterius an-<lb/>ſæ extremitati adhæreat. </s> <s xml:id="echoid-s7708" xml:space="preserve">Figuræ 13<emph style="super">æ</emph>. </s> <s xml:id="echoid-s7709" xml:space="preserve">phaſis aliquo modo <lb/>ſalvabitur, modò anſarum forma emendetur; </s> <s xml:id="echoid-s7710" xml:space="preserve">hæc enim tri-<lb/>gona non eſt, ſed convexa; </s> <s xml:id="echoid-s7711" xml:space="preserve">ut jam dixi, de 3<emph style="super">a</emph>. </s> <s xml:id="echoid-s7712" xml:space="preserve">figura: </s> <s xml:id="echoid-s7713" xml:space="preserve">o-<lb/>mitto alia, quæ ex his facilè deducentur.</s> <s xml:id="echoid-s7714" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7715" xml:space="preserve">HÆC ſunt, Sereniſſime Princeps, quæ de Hugeniano <lb/>Syſtemate tibi inſcripto ad te ſcribenda, & </s> <s xml:id="echoid-s7716" xml:space="preserve">ne probandi <pb o="618" file="0354" n="371" rhead="EUST. DE DIVINIS IN SYST. SAT."/> modum deſiderares, teleſcopium triginta ſex palmos lon-<lb/>gum, unâ mittendum eſſe putavi; </s> <s xml:id="echoid-s7717" xml:space="preserve">ut & </s> <s xml:id="echoid-s7718" xml:space="preserve">rationes noſtras, <lb/>ingenii, quo ad ſtuporem polles, & </s> <s xml:id="echoid-s7719" xml:space="preserve">obſervationes, ocu-<lb/>lorum acie, accedente teleſcopii operâ, explorare valeas: <lb/></s> <s xml:id="echoid-s7720" xml:space="preserve">facilè, ni fallor, judicabis, utri potiùs habenda fides ſit; </s> <s xml:id="echoid-s7721" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s7722" xml:space="preserve">an noſtra teleſcopia Hugenianis deteriora ſint: </s> <s xml:id="echoid-s7723" xml:space="preserve">quidquid <lb/>ſit, ſi hæc opella mea tibi grata acciderit, faciam aliquan-<lb/>do ut aliquid majoris momenti ad perennem obſervan-<lb/>tiæ meæ teſtificationem Sereniſſimo tuo nomini inſcriba-<lb/>tur.</s> <s xml:id="echoid-s7724" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div367" type="section" level="1" n="146"> <head xml:id="echoid-head199" xml:space="preserve">FINIS.</head> <figure> <image file="0354-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0354-01"/> </figure> <pb file="0355" n="372"/> </div> <div xml:id="echoid-div368" type="section" level="1" n="147"> <head xml:id="echoid-head200" xml:space="preserve"><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> <emph style="sc">Zulichemii</emph></head> <head xml:id="echoid-head201" xml:space="preserve">BREVIS ASSERTIO</head> <head xml:id="echoid-head202" xml:space="preserve">SYSTEMATIS <lb/>SATURNII <lb/>S U I, <lb/><emph style="sc">Ad</emph> <emph style="sc">Serenissimum</emph> <emph style="sc">Principem</emph> <lb/>LEOPOLDUM AB HETRURIA.</head> <pb file="0356" n="373"/> <pb o="621" file="0357" n="374"/> </div> <div xml:id="echoid-div369" type="section" level="1" n="148"> <head xml:id="echoid-head203" xml:space="preserve"><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> <emph style="sc">Zulichemii</emph></head> <head xml:id="echoid-head204" xml:space="preserve">BREVIS ASSERTIO <lb/>SYSTEMATIS <lb/>SATURNII <lb/>S U I, <lb/><emph style="sc">Ad</emph> <emph style="sc">Serenissimum</emph> <emph style="sc">Principem</emph> <lb/>LEOPOLDUM AB HETRURIA.</head> <head xml:id="echoid-head205" style="it" xml:space="preserve">SERENISSIME PRINCEPS,</head> <p> <s xml:id="echoid-s7725" xml:space="preserve">VIdiſti quæ adverſus ſyſtema meum Saturnium <lb/>Romæ ſunt edita, utque ſub uno nomine ge-<lb/>mini mihi adverſarii exorti ſint. </s> <s xml:id="echoid-s7726" xml:space="preserve">Etſi enim <lb/>Euſtachius de Divinis libelli iſtius Annotatio-<lb/>num autor inſcribitur, eum tamen adjutum <lb/>fuiſſe opera P. </s> <s xml:id="echoid-s7727" xml:space="preserve">Honorati Fabrii nec Celſ. </s> <s xml:id="echoid-s7728" xml:space="preserve">T. <lb/></s> <s xml:id="echoid-s7729" xml:space="preserve">ignorat, & </s> <s xml:id="echoid-s7730" xml:space="preserve">ego, hoc agi, jam pridem ab amicis fueram ad-<lb/>monitus. </s> <s xml:id="echoid-s7731" xml:space="preserve">Credideram hoc ipſo futurum ut ſubtiliora quæ-<lb/>dam, non mihi præviſa, éque profundiore Aſtronomiæ ſi-<lb/>nu depromta, objicerentur; </s> <s xml:id="echoid-s7732" xml:space="preserve">tum vero ea civilitate & </s> <s xml:id="echoid-s7733" xml:space="preserve">vere-<lb/>cundia, quæ viro humanioribus ſtudiis dedito conveniret. </s> <s xml:id="echoid-s7734" xml:space="preserve"><lb/>Sed omnino ſpe deceptus ſum, cum nihil aliud quam obſer-<lb/>vationes meas temere impugnari videam, quas pleraſque in <lb/>dubium vocant, præterque veritatem mihi confictas eſſe ſa- <pb o="622" file="0358" n="375" rhead="CHRIST. HUGENII ASSERTIO"/> tis apcrtè criminantur. </s> <s xml:id="echoid-s7735" xml:space="preserve">Facile autem, ut ſpero, tam indi-<lb/>gnam à me ſuſpicionem repellam, nec multis ad hoc opus eſ-<lb/>ſe, coram C. </s> <s xml:id="echoid-s7736" xml:space="preserve">T. </s> <s xml:id="echoid-s7737" xml:space="preserve">cauſam agenti, exiſtimo, cujus ſumma æ-<lb/>quitas cum pari judicii perſpicacitate conjuncta eſt.</s> <s xml:id="echoid-s7738" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7739" xml:space="preserve">Primus præcipuuſque adverſariorum conatus eſt, ut vitra <lb/>ſua tuboſque opticos nihilo meis inferiores eſſe demonſtrent; <lb/></s> <s xml:id="echoid-s7740" xml:space="preserve">quod ubi egregiè ſe confeciſſe arbitrantur, inde porro, ni-<lb/>hil me in cælo deprehendere potuiſſe, quod non & </s> <s xml:id="echoid-s7741" xml:space="preserve">ipſi vi-<lb/>derint, concludunt. </s> <s xml:id="echoid-s7742" xml:space="preserve">Quod ſi autem phænomena ipſa Sa-<lb/>turni, quæ adduxi, vera non ſunt, vana utique & </s> <s xml:id="echoid-s7743" xml:space="preserve">Hypo-<lb/>theſis erit, qua viſorum eorum quæ nuſquam ſunt cauſas ex-<lb/>ponere ſuſcepi.</s> <s xml:id="echoid-s7744" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7745" xml:space="preserve">Teleſcopiorum ſuorum comparationem cum noſtris inſti-<lb/>tuere aliquatenus ut poſſent effeci, tradita accuratiſſima meo-<lb/>rum deſcriptione; </s> <s xml:id="echoid-s7746" xml:space="preserve">veruntamen quod præcipuum eſt omnium, <lb/>præſtantia vitrorum quæ ab exacta formatione proficiſcitur, <lb/>hanc ſcire debent nulla deſcriptione, ſed effectu demum ipſo <lb/>probari poſſe: </s> <s xml:id="echoid-s7747" xml:space="preserve">Ideoque etſi & </s> <s xml:id="echoid-s7748" xml:space="preserve">longiores paulo noſtris tubos, <lb/>& </s> <s xml:id="echoid-s7749" xml:space="preserve">vitra amplius patentia fabricarint, non idcirco meliora ex <lb/>his teleſcopia ſeſe compoſuiſſe exiſtiment. </s> <s xml:id="echoid-s7750" xml:space="preserve">Oportet nempe, <lb/>quæ pari bonitate cum noſtris futura ſunt, ut & </s> <s xml:id="echoid-s7751" xml:space="preserve">longitudine <lb/>illis æquentur, & </s> <s xml:id="echoid-s7752" xml:space="preserve">vitrorum non minore apertura gaudeant; <lb/></s> <s xml:id="echoid-s7753" xml:space="preserve">at non ideo quæcunque ſic ſe habuerint, eâdem virtute pol-<lb/>lent, aut illis æquiparanda ſunt.</s> <s xml:id="echoid-s7754" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7755" xml:space="preserve">Ut igitur reipſa & </s> <s xml:id="echoid-s7756" xml:space="preserve">effectu concedere noſtris Diviniana te-<lb/>leſcopia comprobem, certiſſimum argumentum mihi petere <lb/>poſſe videor à Saturnio illo comite ſive luna, quæ ſexdecim <lb/>dierum ſpatio circa eum revolvitur. </s> <s xml:id="echoid-s7757" xml:space="preserve">Hanc primum omnium <lb/>mihi conſpectam eſſe liquet, neque id Euſtachius cum ſibi <lb/>fuccenturiato P. </s> <s xml:id="echoid-s7758" xml:space="preserve">Fabrio denegat. </s> <s xml:id="echoid-s7759" xml:space="preserve">Quod autem nec aliis nec <lb/>ſibi antea animad verſa fuerit, in cauſa fuiſſe ait vel inadver-<lb/>tentiam, vel nimiam comitis à Saturno diſtantiam, vel <lb/>parvitatem, vel conjunctionem. </s> <s xml:id="echoid-s7760" xml:space="preserve">Sed vana effugia hæc eſſe <lb/>quis non videt? </s> <s xml:id="echoid-s7761" xml:space="preserve">Cum enim jam ab anno 1646 & </s> <s xml:id="echoid-s7762" xml:space="preserve">ultra fortè <lb/>Saturni phaſes deligenter reſpicere cœperit, teleſcopio è con-<lb/>vexis vitris compoſito, quo ſemper Saturnum ſimul & </s> <s xml:id="echoid-s7763" xml:space="preserve">co- <pb o="623" file="0359" n="376" rhead="SYSTEMATIS SATURNII."/> mitem hunc, licet triplo longius recederet comprehendi ne-<lb/>ceſſe erat, quænam illa fuit inadvertentia qua fiebat ut nun-<lb/>quam ei in oculos incurreret? </s> <s xml:id="echoid-s7764" xml:space="preserve">cur non eadem mihi quoque <lb/>illum ſubducebat, nihilo magis admonito? </s> <s xml:id="echoid-s7765" xml:space="preserve">Quin etiam hinc <lb/>patet non bene prætexi magnam à Saturno diſtantiam, cum <lb/>quoties ad hunc reſpiceret, ſimul ante oculos ſeſe ſiſteret co-<lb/>mes ejus. </s> <s xml:id="echoid-s7766" xml:space="preserve">Et ſanè diſtantia illa vix tria prima ſcrupula ex-<lb/>cedit, cum Medicei planetæ omnes longius à Jove evagen-<lb/>tur, extimus etiam ſcrupulis quatuordecim. </s> <s xml:id="echoid-s7767" xml:space="preserve">Parvitas ergo <lb/>illum occuluit. </s> <s xml:id="echoid-s7768" xml:space="preserve">At hoc idem eſt quaſi fateatur ſuis illum or-<lb/>ganis detegi nequiiſſe: </s> <s xml:id="echoid-s7769" xml:space="preserve">certe enim nimium exilis erat, at mi-<lb/>hi nequaquam.</s> <s xml:id="echoid-s7770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7771" xml:space="preserve">Poſtea tamen edita obſervatione mea, cum à Clariſſimo <lb/>Viro Mich. </s> <s xml:id="echoid-s7772" xml:space="preserve">Angelo Riccio eſſet admonitus, vidiſſe ſe ſtel-<lb/>lulam hanc affirmat, indicans etiam quo ad Saturnum ſitu, <lb/>& </s> <s xml:id="echoid-s7773" xml:space="preserve">quoties illam obſervaverit. </s> <s xml:id="echoid-s7774" xml:space="preserve">Verum cum diligentius haſce <lb/>obſervationes examino, invenio profecto non optima fide <lb/>eas commemorari; </s> <s xml:id="echoid-s7775" xml:space="preserve">ſed adeo miſere vereri Euſtachium ne bo-<lb/>nitatem noſtrorum teleſcopiorum non aſſecutus putetur, ut <lb/>quæ nunquam in cælo vidit, aut per rerum naturam videre <lb/>potuit, tamen ſibi viſa incogitanter aſſerat. </s> <s xml:id="echoid-s7776" xml:space="preserve">Quod ut appareat, <lb/>periodus primò Saturnii comitis expendatur, quam 16. <lb/></s> <s xml:id="echoid-s7777" xml:space="preserve">circiter dierum eſſe, ex obſervationibus meis, toto triennio <lb/>continuatis (niſi & </s> <s xml:id="echoid-s7778" xml:space="preserve">has confixiſſe dicar) manifeſtum feci: </s> <s xml:id="echoid-s7779" xml:space="preserve"><lb/>ſed eſto tantùm ea veræ proxima ſit, quoniam ſuas obſerva-<lb/>tiones tantulum refragari Euſtachius ſcribit. </s> <s xml:id="echoid-s7780" xml:space="preserve">Die igitur 30. </s> <s xml:id="echoid-s7781" xml:space="preserve"><lb/>Junii, anno 1657. </s> <s xml:id="echoid-s7782" xml:space="preserve">horis 2{1/2}, poſt ſolis occaſum, item ſe-<lb/>quentibus diebus 2, 4, 9, 12, Julii, ad partem orientalem Sa-<lb/>turni animadverſum ſibi comitem ait; </s> <s xml:id="echoid-s7783" xml:space="preserve">14° non apparuiſſe. </s> <s xml:id="echoid-s7784" xml:space="preserve"><lb/>Die 20 ad occaſum fuiſſe ſitum; </s> <s xml:id="echoid-s7785" xml:space="preserve">utrum in maxima, medio-<lb/>cri, an minima diſtantia apparuerit nuſquam addit. </s> <s xml:id="echoid-s7786" xml:space="preserve">Certe <lb/>equidem tribus prioribus harum obſervationum, ad occa-<lb/>ſum revera poſitus erat, ut ex tabulis colligo. </s> <s xml:id="echoid-s7787" xml:space="preserve">Sed putemus <lb/>inverſionis teleſcopii rationem non habuiſſe Euſtachium, at-<lb/>que ita ſicut ei videbatur, ſi tamen viſus eſt, orientem ſpe-<lb/>ctaſſe comitem rectè dixerit. </s> <s xml:id="echoid-s7788" xml:space="preserve">At die 9 & </s> <s xml:id="echoid-s7789" xml:space="preserve">12 Julii, quibus <pb o="624" file="0360" n="377" rhead="CHRIST. HUGENII ASSERTIO"/> ſibi orientalem itidem apparuiſſe aſſerit, omnino ad occaſum <lb/>tranſierat, ac rurſus 20. </s> <s xml:id="echoid-s7790" xml:space="preserve">Jul. </s> <s xml:id="echoid-s7791" xml:space="preserve">ad latus orientale, cum Euſta-<lb/>chius ab occiduo ſibi repertum ſcribat.</s> <s xml:id="echoid-s7792" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7793" xml:space="preserve">In figura appoſita clarius hoc liquebit; </s> <s xml:id="echoid-s7794" xml:space="preserve">in qua circulus <lb/> <anchor type="note" xlink:label="note-0360-01a" xlink:href="note-0360-01"/> A B orbitam Lunæ Saturniæ exhibet, in 16 partes diviſam, <lb/>ipſaque A B rectam ad nos obverſam. </s> <s xml:id="echoid-s7795" xml:space="preserve">Die igitur 30 Jun. <lb/></s> <s xml:id="echoid-s7796" xml:space="preserve">1657, hora 10 veſp. </s> <s xml:id="echoid-s7797" xml:space="preserve">invenio locum comitis grad. </s> <s xml:id="echoid-s7798" xml:space="preserve">207. </s> <s xml:id="echoid-s7799" xml:space="preserve">16′ <lb/>ab apogæo A, methodo ſcilicet ea quam in Syſtemate Satur-<lb/>nio tradidi. </s> <s xml:id="echoid-s7800" xml:space="preserve">unde prope num. </s> <s xml:id="echoid-s7801" xml:space="preserve">10 conſtitiſſe debuit, paulo <lb/>tantum citerior. </s> <s xml:id="echoid-s7802" xml:space="preserve">Atque hinc porro in ſingulos dies ſingulas <lb/>circuli partes numerando; </s> <s xml:id="echoid-s7803" xml:space="preserve">ſequitur die 2 Jul. </s> <s xml:id="echoid-s7804" xml:space="preserve">ad num. </s> <s xml:id="echoid-s7805" xml:space="preserve">12. </s> <s xml:id="echoid-s7806" xml:space="preserve"><lb/>progreſſum eſſe. </s> <s xml:id="echoid-s7807" xml:space="preserve">Item 4. </s> <s xml:id="echoid-s7808" xml:space="preserve">Jul. </s> <s xml:id="echoid-s7809" xml:space="preserve">ad num. </s> <s xml:id="echoid-s7810" xml:space="preserve">14. </s> <s xml:id="echoid-s7811" xml:space="preserve">Inde poſt dies <lb/>5, nempe 9 Jul. </s> <s xml:id="echoid-s7812" xml:space="preserve">ad num. </s> <s xml:id="echoid-s7813" xml:space="preserve">3. </s> <s xml:id="echoid-s7814" xml:space="preserve">Ac porro 12 Jul. </s> <s xml:id="echoid-s7815" xml:space="preserve">ad num. </s> <s xml:id="echoid-s7816" xml:space="preserve">6. </s> <s xml:id="echoid-s7817" xml:space="preserve"><lb/>Et denique 14 Jul. </s> <s xml:id="echoid-s7818" xml:space="preserve">ad num. </s> <s xml:id="echoid-s7819" xml:space="preserve">8. </s> <s xml:id="echoid-s7820" xml:space="preserve">qua die recte divinavit con-<lb/>junctum Saturno fuiſſe. </s> <s xml:id="echoid-s7821" xml:space="preserve">Reliquis verò plane infeliciter. </s> <s xml:id="echoid-s7822" xml:space="preserve">Si <lb/>enim ad num. </s> <s xml:id="echoid-s7823" xml:space="preserve">10, 12, 14, orientalis ſpectatus fuit, neceſſa-<lb/>rio ad num. </s> <s xml:id="echoid-s7824" xml:space="preserve">3 & </s> <s xml:id="echoid-s7825" xml:space="preserve">6 occidentalis factus eſt. </s> <s xml:id="echoid-s7826" xml:space="preserve">Nec quidem ul-<lb/>la ratione ad num. </s> <s xml:id="echoid-s7827" xml:space="preserve">14 & </s> <s xml:id="echoid-s7828" xml:space="preserve">6, qui è diametro ſibi oppoſiti ſunt, <lb/>idem latus Saturni obtinere poterat, uti Euſtachii obſerva-<lb/>tiones poſtulant. </s> <s xml:id="echoid-s7829" xml:space="preserve">Denique cum 20 Jul. </s> <s xml:id="echoid-s7830" xml:space="preserve">ad num. </s> <s xml:id="echoid-s7831" xml:space="preserve">14 rever-<lb/>terit, ubi & </s> <s xml:id="echoid-s7832" xml:space="preserve">die 4 Jul. </s> <s xml:id="echoid-s7833" xml:space="preserve">inveniebatur, quomodo evenit quæ-<lb/>ſo, ut, qui tunc ad orientem, idem nunc, quamvis eodem <lb/>loco, ad occidentem comparuerit? </s> <s xml:id="echoid-s7834" xml:space="preserve">Etiamſi mea periodus <lb/>vera non eſſet, quænam tamen illa Euſtachii fuerit oſtendi <lb/>mihi velim, paululum diſcrepans, qua ſua hæc viſa tueatur. </s> <s xml:id="echoid-s7835" xml:space="preserve"><lb/>Debuerat ſane hîc, ſi uſquam, Fabrius illi opem tuliſſe; </s> <s xml:id="echoid-s7836" xml:space="preserve">ſal-<lb/>temque perpendiſſe an ſtare ullo modo iſtæ obſervationes <lb/>poſſent: </s> <s xml:id="echoid-s7837" xml:space="preserve">aut, ſi nobis imponere vellet, quotlibet duntaxat <lb/>novas è tabulis meis concinnaſſe. </s> <s xml:id="echoid-s7838" xml:space="preserve">Nam quas nunc adferunt, <lb/>facile nimis deprehenditur, ideo tantum productas fuiſſe, ne <lb/>nihil vidiſſe exiſtimentur.</s> <s xml:id="echoid-s7839" xml:space="preserve"/> </p> <div xml:id="echoid-div369" type="float" level="2" n="1"> <note position="left" xlink:label="note-0360-01" xlink:href="note-0360-01a" xml:space="preserve">TAB. LI. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s7840" xml:space="preserve">Credo & </s> <s xml:id="echoid-s7841" xml:space="preserve">faſciam nigricantem in Saturni diſco, liquido <lb/>ſibi conſpici dixiſſet Euſtachius, ni Fabrio viſum fuiſſet eam <lb/>nimium hypotheſi meæ annulari favere. </s> <s xml:id="echoid-s7842" xml:space="preserve">Cum autem ne <lb/>optimis quidem ſuis perſpicillis eam cerni affirmet, hinc quo-<lb/>que quanto illa meis deteriora ſint perſpicuum fit. </s> <s xml:id="echoid-s7843" xml:space="preserve">Nam ne <pb o="625" file="0361" n="378" rhead="SYSTEMATIS SATURNII."/> mihi phænomenon illud confictum credatur, idem & </s> <s xml:id="echoid-s7844" xml:space="preserve">in An-<lb/>glia pridem obſervari cœpiſſe ſciendum eſt; </s> <s xml:id="echoid-s7845" xml:space="preserve">ut liquet ex lite-<lb/>ris Viri Clar. </s> <s xml:id="echoid-s7846" xml:space="preserve">Joh. </s> <s xml:id="echoid-s7847" xml:space="preserve">Walliſii, Oxonia ad me datis 22. </s> <s xml:id="echoid-s7848" xml:space="preserve">Dec. <lb/></s> <s xml:id="echoid-s7849" xml:space="preserve">1658. </s> <s xml:id="echoid-s7850" xml:space="preserve">quibus inter alia hæc ſcribit. </s> <s xml:id="echoid-s7851" xml:space="preserve">Monebam etiam eiſdem <lb/>literis (nempe datis 29 Maji, 1656) de Saturni faſcia <lb/>quam jam ante obſervaverat D. </s> <s xml:id="echoid-s7852" xml:space="preserve">Ball, & </s> <s xml:id="echoid-s7853" xml:space="preserve">ſciſcitabar num <lb/>tu eandem conſpexeras, &</s> <s xml:id="echoid-s7854" xml:space="preserve">c. </s> <s xml:id="echoid-s7855" xml:space="preserve">Eam porro faſciam à 5 Febr. </s> <s xml:id="echoid-s7856" xml:space="preserve"><lb/>1656 ad 2 Jul. </s> <s xml:id="echoid-s7857" xml:space="preserve">quo tempore rotundus Saturnus abſque anſis <lb/>apparuit, medium Planetæ diſcum ſecare D. </s> <s xml:id="echoid-s7858" xml:space="preserve">Ball. </s> <s xml:id="echoid-s7859" xml:space="preserve">adnota-<lb/>vit, ut in ſchemate ad me miſſo expreſſa eſt. </s> <s xml:id="echoid-s7860" xml:space="preserve">Atque ita mi-<lb/>hi quoque fuerat eo tempore obſervata, ut cernitur pag. </s> <s xml:id="echoid-s7861" xml:space="preserve">544 <lb/>ſyſtematis Saturnii, quam figuram hîc repeto. </s> <s xml:id="echoid-s7862" xml:space="preserve">Poſtmodum <lb/> <anchor type="note" xlink:label="note-0361-01a" xlink:href="note-0361-01"/> tamen renatis Saturni anſis cum difficillimè conſpici eadem <lb/>faſcia cœpiſſet, minus rectè quoque à D. </s> <s xml:id="echoid-s7863" xml:space="preserve">Ball, quantum ad <lb/>ſitum attinet, depicta eſt. </s> <s xml:id="echoid-s7864" xml:space="preserve">At in mearum obſervationum <lb/>adverſariis, die 26 Nov. </s> <s xml:id="echoid-s7865" xml:space="preserve">1656, & </s> <s xml:id="echoid-s7866" xml:space="preserve">alias adſcriptum invenio, <lb/>lineam obſcuram fuiſſe evidentiſſimam, eo nempe poſitu qui <lb/>pag. </s> <s xml:id="echoid-s7867" xml:space="preserve">545, Syſtem. </s> <s xml:id="echoid-s7868" xml:space="preserve">Saturnii memoratur. </s> <s xml:id="echoid-s7869" xml:space="preserve">Neque id monere <lb/>neglexiſſem, obſervationes eas recenſens, ſi quicquam profu-<lb/>turum putaſſem apud illos qui me data opera lectores fallere <lb/>voluiſſe ſuſpicarentur. </s> <s xml:id="echoid-s7870" xml:space="preserve">Difficillime autem, ut dixi, tractus <lb/>hic obſcurus deinceps cerni cœpit, imo vix jam amplius a-<lb/>nimadverti poteſt; </s> <s xml:id="echoid-s7871" xml:space="preserve">quod & </s> <s xml:id="echoid-s7872" xml:space="preserve">ſyſtemati meo conſentaneum eſt, <lb/>quippe magnâ jam annuli Saturnii inclinatione; </s> <s xml:id="echoid-s7873" xml:space="preserve">ad quam <lb/>accedit, quod & </s> <s xml:id="echoid-s7874" xml:space="preserve">anſarum ſplendor, duplo quam alias luci-<lb/>diorem Saturnum efficiens, oculorum aciem nunc magis im-<lb/>pedit.</s> <s xml:id="echoid-s7875" xml:space="preserve"/> </p> <div xml:id="echoid-div370" type="float" level="2" n="2"> <note position="right" xlink:label="note-0361-01" xlink:href="note-0361-01a" xml:space="preserve">TAB. LI. <lb/>fig. 4.</note> </div> <p> <s xml:id="echoid-s7876" xml:space="preserve">Qualia igitur ſint Diviniana teleſcopia cum noſtris collata, <lb/>ex his quæ dicta ſunt, cuilibet manifeſtum opinor. </s> <s xml:id="echoid-s7877" xml:space="preserve">Eadem <lb/>verò & </s> <s xml:id="echoid-s7878" xml:space="preserve">Anglicanis viliora eſſe alia præterea ratione oſten-<lb/>dam, quo minus dubitetur phænomena ea vera eſſe, quæ <lb/>pariter cum Anglis ego obſervavi. </s> <s xml:id="echoid-s7879" xml:space="preserve">Nam & </s> <s xml:id="echoid-s7880" xml:space="preserve">illa quæ ad an-<lb/>ſatas Saturni phaſes attinent, teſtimonio illorum confirma-<lb/>turus ſum. </s> <s xml:id="echoid-s7881" xml:space="preserve">Vir quidam nobilis ex Gallia, eruditus, acerri-<lb/>mique ingenii, qui Romæ teleſcopia apud artificem iſtum vi-<lb/>derat, hoc ipſo anno 1660, ſuper iis hæc ad me ſcribit. </s> <s xml:id="echoid-s7882" xml:space="preserve">Il <pb o="626" file="0362" n="379" rhead="CHRIST. HUGENII ASSERTIO"/> me monſtra les plus beaux de ſes teleſcopes, qui paſſent au <lb/>de la ue 30 pieds; </s> <s xml:id="echoid-s7883" xml:space="preserve">& </s> <s xml:id="echoid-s7884" xml:space="preserve">nous les comparames avec un de ceux <lb/>de la methode du Chevalier Neal, qu’ on a envoyé au Car-<lb/>dinal Ghiſi; </s> <s xml:id="echoid-s7885" xml:space="preserve">iln’ a garde de ne tirer l’ avantage de ſon coſté, <lb/>mais ſans mentir il ſe trompe lourdement.</s> <s xml:id="echoid-s7886" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7887" xml:space="preserve">Et meliora igitur Romanis eſſe, illa quæ ex Anglia depor-<lb/>tata erant, judicat is qui coram comparationem utrorumque <lb/>inſtituit, & </s> <s xml:id="echoid-s7888" xml:space="preserve">Euſtachium tamen obſtinate hoc pernegare teſta-<lb/>tur, adeo ut ſi jam mea Romam deferam, nihil apud illum <lb/>quidem ſim profecturus. </s> <s xml:id="echoid-s7889" xml:space="preserve">Quid autem huic homini facias, <lb/>aut quis hæc videns non merito exiſtimet, uſque adeo priva-<lb/>ti compendii cura eum duci, ut quid verum ſit diſcernere aut <lb/>non poſſit aut nolit?</s> <s xml:id="echoid-s7890" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7891" xml:space="preserve">Non ægre nunc fidem habitum iri ſpero, tum mihi tum <lb/>Anglis ſimul obſervatoribus, qui anno 1657 oblonga Satur-<lb/>ni brachia diſco utrinque conjuncta ſpectavimus, qualia ex-<lb/>hibet figura Syſtematis mei pag. </s> <s xml:id="echoid-s7892" xml:space="preserve">545. </s> <s xml:id="echoid-s7893" xml:space="preserve">quam hîc repono; </s> <s xml:id="echoid-s7894" xml:space="preserve">non <lb/> <anchor type="note" xlink:label="note-0362-01a" xlink:href="note-0362-01"/> autem binorum orbiculorum formâ à medio diſco disjuncto-<lb/>rum, ut Euſtachius ſe illa eodem tempore vidiſſe dejerat.</s> <s xml:id="echoid-s7895" xml:space="preserve"/> </p> <div xml:id="echoid-div371" type="float" level="2" n="3"> <note position="left" xlink:label="note-0362-01" xlink:href="note-0362-01a" xml:space="preserve">TAB. LI. <lb/>fig. 5.</note> </div> <p> <s xml:id="echoid-s7896" xml:space="preserve">Adderem hîc ſchema quod mihi à D. </s> <s xml:id="echoid-s7897" xml:space="preserve">Ball, ſuprà memora-<lb/>to, advenit, niſi planè ſimile eſſet huic noſtro, hoc uno tan-<lb/>tillum duntaxat abludens, quod brachia illa ubique paulo craſ-<lb/>ſiora ille referat.</s> <s xml:id="echoid-s7898" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7899" xml:space="preserve">Eam vero formam à 5 Nov. </s> <s xml:id="echoid-s7900" xml:space="preserve">1656, ad 9 Jul. </s> <s xml:id="echoid-s7901" xml:space="preserve">1657 ſibi ap-<lb/>paruiſſe ſcribit. </s> <s xml:id="echoid-s7902" xml:space="preserve">Apertis autem brachiis, qualis pag. </s> <s xml:id="echoid-s7903" xml:space="preserve">547. </s> <s xml:id="echoid-s7904" xml:space="preserve">Sy-<lb/>ſtematis mei & </s> <s xml:id="echoid-s7905" xml:space="preserve">hîc repræſentatur, talem à 9. </s> <s xml:id="echoid-s7906" xml:space="preserve">Nov. </s> <s xml:id="echoid-s7907" xml:space="preserve">1657 ad <lb/> <anchor type="note" xlink:label="note-0362-02a" xlink:href="note-0362-02"/> 7 Jun. </s> <s xml:id="echoid-s7908" xml:space="preserve">1658, idem obſervator depingit, ſimillima prorſus <lb/>figura, niſi quod ad poſitum zonæ obſcuræ attinet, de quo <lb/>dixi ſuprà. </s> <s xml:id="echoid-s7909" xml:space="preserve">Ac denique à 3 Jan. </s> <s xml:id="echoid-s7910" xml:space="preserve">1659 ad 17 Jun. </s> <s xml:id="echoid-s7911" xml:space="preserve">ejuſdem <lb/>anni, anſis paulo latius adhuc apertis. </s> <s xml:id="echoid-s7912" xml:space="preserve">Et hæc quidem il-<lb/>le, ignarus adhuc meæ hypotheſeos, ne ob præconceptam <lb/>opinionem aliquid indulſiſſe ſibi exiſtimetur. </s> <s xml:id="echoid-s7913" xml:space="preserve">Neque ego <lb/>aliter iſta quam ſe revera habent referre auderem, cum redar-<lb/>guere me, ſi fallam, autori obſervationum in promptu ſit.</s> <s xml:id="echoid-s7914" xml:space="preserve"/> </p> <div xml:id="echoid-div372" type="float" level="2" n="4"> <note position="left" xlink:label="note-0362-02" xlink:href="note-0362-02a" xml:space="preserve">TAB. LI. <lb/>fig. 6.</note> </div> <p> <s xml:id="echoid-s7915" xml:space="preserve">Quænam igitur illa ſunt Euſtachii perſpicilla palmorum 24, <lb/>quibus rotundi globuli rectorum loco brachiorum cerneban- <pb file="0363" n="380"/> <pb file="0363a" n="381"/> <anchor type="figure" xlink:label="fig-0363a-01a" xlink:href="fig-0363a-01"/> <anchor type="figure" xlink:label="fig-0363a-02a" xlink:href="fig-0363a-02"/> <anchor type="figure" xlink:label="fig-0363a-03a" xlink:href="fig-0363a-03"/> <anchor type="figure" xlink:label="fig-0363a-04a" xlink:href="fig-0363a-04"/> <anchor type="figure" xlink:label="fig-0363a-05a" xlink:href="fig-0363a-05"/> <anchor type="figure" xlink:label="fig-0363a-06a" xlink:href="fig-0363a-06"/> <pb file="0364" n="382"/> <pb o="627" file="0365" n="383" rhead="SYSTEMATIS SATURNII."/> tur? </s> <s xml:id="echoid-s7916" xml:space="preserve">Imo etiam loco anſarum jam adapertarum, quales in ſy-<lb/>ſtemate meo pag. </s> <s xml:id="echoid-s7917" xml:space="preserve">546. </s> <s xml:id="echoid-s7918" xml:space="preserve">Etenim & </s> <s xml:id="echoid-s7919" xml:space="preserve">anno 1658 globulos tantum <lb/>ſibi viſos ait, quales in Fig. </s> <s xml:id="echoid-s7920" xml:space="preserve">1. </s> <s xml:id="echoid-s7921" xml:space="preserve">iſtarum tredecim tabellæ <lb/>LII. </s> <s xml:id="echoid-s7922" xml:space="preserve">Quid teſtes citat qui ſecum iſta obſervaverint? <lb/></s> <s xml:id="echoid-s7923" xml:space="preserve">Quos equidem fide digniſſimos eſſe cenſeo, quæque vi-<lb/>derint ingenuè faſſuros, ſed eo ipſo mihi videntur de tele-<lb/>ſcopiorum illius vitio & </s> <s xml:id="echoid-s7924" xml:space="preserve">ineptitudine teſtimonium perhibere. </s> <s xml:id="echoid-s7925" xml:space="preserve"><lb/>Mallem aliquos adduxiſſet qui ſecum Saturni comitem obſer-<lb/>vaſſent.</s> <s xml:id="echoid-s7926" xml:space="preserve"/> </p> <div xml:id="echoid-div373" type="float" level="2" n="5"> <figure xlink:label="fig-0363a-01" xlink:href="fig-0363a-01a"> <caption xml:id="echoid-caption162" style="it" xml:space="preserve">Pag. 626.<lb/>TAB. LI.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables95" xml:space="preserve">F E D V S 30 20 10 C L G R H K P A M Z I O X B</variables> </figure> <figure xlink:label="fig-0363a-02" xlink:href="fig-0363a-02a"> <caption xml:id="echoid-caption163" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables96" xml:space="preserve">L K O R E H N I S D G B C</variables> </figure> <figure xlink:label="fig-0363a-03" xlink:href="fig-0363a-03a"> <caption xml:id="echoid-caption164" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables97" xml:space="preserve">A 16 15 14 13 12 11 10 9 B 8 7 6 5 4 3 2 1</variables> </figure> <figure xlink:label="fig-0363a-04" xlink:href="fig-0363a-04a"> <caption xml:id="echoid-caption165" style="it" xml:space="preserve">Fig. 4</caption> </figure> <figure xlink:label="fig-0363a-05" xlink:href="fig-0363a-05a"> <caption xml:id="echoid-caption166" style="it" xml:space="preserve">Fig. 5.</caption> </figure> <figure xlink:label="fig-0363a-06" xlink:href="fig-0363a-06a"> <caption xml:id="echoid-caption167" style="it" xml:space="preserve">Fig. 6.</caption> </figure> </div> <p> <s xml:id="echoid-s7927" xml:space="preserve">Globulorum formâ mihi quoque brachia apparuiſſe dixe-<lb/>ram, quoties 5 aut 6 pedum tubos adhibuiſſem. </s> <s xml:id="echoid-s7928" xml:space="preserve">Hiſce igi-<lb/>tur rectiſſime ea, atque uti debent, mihi cerni arbitratur; </s> <s xml:id="echoid-s7929" xml:space="preserve">cum <lb/>vero oblonga ac Saturni diſco affixa, tubo 23 pedum referun-<lb/>tur, falſa imagine me deludi, idque teleſcopii culpa contin-<lb/>gere. </s> <s xml:id="echoid-s7930" xml:space="preserve">Nimirum perſuadere mihi vult exigua illa omnium ma-<lb/>ximis præferenda eſſe, quaſi ne hoc quidem diſcernere adhuc <lb/>didicerim. </s> <s xml:id="echoid-s7931" xml:space="preserve">Sed jam ſatis ſuperque, quid de meis teleſcopiis <lb/>obſervationibuſque, quidque de Euſtachianis exiſtimandum <lb/>ſit, me oſtendiſſe arbitror. </s> <s xml:id="echoid-s7932" xml:space="preserve">Nec tamen quæ dixi in damnum <lb/>illius cedere velim, ſed potius ſtimulos addere, quò magis <lb/>magiſque incumbens, ſua ipſius primo, deinde & </s> <s xml:id="echoid-s7933" xml:space="preserve">noſtra te-<lb/>leſcopia ſuperet. </s> <s xml:id="echoid-s7934" xml:space="preserve">Adeo enim non invideo his qui artem ad-<lb/>eo egregiam promovere nituntur, ut decreverim etiam cun-<lb/>cta quæ circa eam mihi comperta ſunt, ſed & </s> <s xml:id="echoid-s7935" xml:space="preserve">præcipue quæ <lb/>ad theoriam Dioptrices ſpectant, propediem in lucem emit-<lb/>tere: </s> <s xml:id="echoid-s7936" xml:space="preserve">quod vel ideo mihi faciendum video, ut de veritate ob-<lb/>ſervatorum quæ in Saturni ſyſtemate protuli, plures inquire-<lb/>re poſſint, monſtrata arte qua ſibi paria noſtris perſpicilla pa-<lb/>rent.</s> <s xml:id="echoid-s7937" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7938" xml:space="preserve">Cæterum unicum etiamnum diſcutiendum reſtat Euſtachii <lb/>phænomenon, quo vel ſolo Syſtema meum univerſum cor-<lb/>ruere dicitur. </s> <s xml:id="echoid-s7939" xml:space="preserve">Eſt autem rurſus de genere eorum quæ non <lb/>tantum non extant, ſed ne quidem per rerum naturam appa-<lb/>rere ullo modo poſſunt: </s> <s xml:id="echoid-s7940" xml:space="preserve">quod, ſi non vitrarius artifex, at <lb/>P. </s> <s xml:id="echoid-s7941" xml:space="preserve">Fabrius certè animadvertere debuerat. </s> <s xml:id="echoid-s7942" xml:space="preserve">Ajunt ſpatia illa <lb/>bina, cavitate anſarum intercepta, nigriora reliquo cælo in- <pb o="628" file="0366" n="384" rhead="CHRIST. HUGENII ASSERTIO"/> veniri; </s> <s xml:id="echoid-s7943" xml:space="preserve">cum contra, ex hypotheſi mea, cælum ipſum ſit quod <lb/>trans aperturas eas conſpicitur. </s> <s xml:id="echoid-s7944" xml:space="preserve">Ego vero quæram ex Fabrio, <lb/>qui fiat ut cælum omne, cum vel interdiu vel noctu aſpici-<lb/>tur, non planè tenebroſum nigerrimumque appareat. </s> <s xml:id="echoid-s7945" xml:space="preserve">Fate-<lb/>ri cogetur in cauſſa eſſe vapores illos ſive atmoſphæram Ter-<lb/>ræ circumfuſam, quæ interdiu quidem à Sole, noctu verò à <lb/>Luna aut ſtellis illuſtretur. </s> <s xml:id="echoid-s7946" xml:space="preserve">Atque adeo ſi vapores ii auferan-<lb/>tur, planè nigrum appariturum eſſe cælum, æquè ac ſpatia <lb/>illa, Saturni anſis incluſa. </s> <s xml:id="echoid-s7947" xml:space="preserve">Atqui tota atmoſphæræ illuſtratæ <lb/>craſſitudo tam inter nosac Saturnum, quam inter nos nigrum-<lb/>que cæli convexum interjecta eſt; </s> <s xml:id="echoid-s7948" xml:space="preserve">ergo illius interpoſitu æque <lb/>multum de Saturniarum macularum nigredine, atque de illa <lb/>quæ cælum obtinet decedere neceſſe eſt; </s> <s xml:id="echoid-s7949" xml:space="preserve">ac proinde eæ ma-<lb/>culæ nihilo obſcuriores reliquo æthere apparere poſſunt. <lb/></s> <s xml:id="echoid-s7950" xml:space="preserve">Quod ſi igitur nihilominus hoc ſibi videri pertendant, fa-<lb/>teantur oportet viſus quadam fallacia id contingere, vicino <lb/>forſitan ſplendore Saturni, anſarumque ſuarum, paulo ob-<lb/>ſcuriora ſpatia illa reddente, quam abſque eo apparitura eſ-<lb/>ſent. </s> <s xml:id="echoid-s7951" xml:space="preserve">Quanquam mihi nunquam id evenit: </s> <s xml:id="echoid-s7952" xml:space="preserve">quinimo ejus con-<lb/>trarium quodammodo fieri animadverti. </s> <s xml:id="echoid-s7953" xml:space="preserve">Dum enim anſæ <lb/>exigua tantum adhuc rima paterent, veluti circa 26. </s> <s xml:id="echoid-s7954" xml:space="preserve">Nov. </s> <s xml:id="echoid-s7955" xml:space="preserve"><lb/>1656, contigiſſe nullus dubito, nondum nigræ lacunæ diſtin-<lb/>cte cerni poterant, ſed tantum tenuiori luce, illa brachiorum <lb/>pars, quæ diſco Saturni proxima erat, perfuſa videbatur. </s> <s xml:id="echoid-s7956" xml:space="preserve">Qua <lb/>eadem ratione & </s> <s xml:id="echoid-s7957" xml:space="preserve">faſcia in Saturno, de qua ſuprà, paulò tan-<lb/>tum obſcurior reliquo ejus diſco cernitur, quia nempe licèt <lb/>revera nigra ſatis vel etiam nigerrima ſit, eſt tamen tenuiſſi-<lb/>ma, unde & </s> <s xml:id="echoid-s7958" xml:space="preserve">latior ſimul juſto & </s> <s xml:id="echoid-s7959" xml:space="preserve">dilutior ſpectatur.</s> <s xml:id="echoid-s7960" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7961" xml:space="preserve">Atque hinc apparet, fruſtra etiam adverſus hypotheſin <lb/>meam objici, quod per eam nunquam Saturnus rotundus ſi-<lb/>ne brachiis videri poſſit; </s> <s xml:id="echoid-s7962" xml:space="preserve">quia nempe exiguam ſaltem lucem <lb/>in extrema annuli ora reliquiſſe videbar, qua futurum ſit ut <lb/>non penitus viſum effugiat annulus, quanquam à latere inſpe-<lb/>ctus nec brachia proinde in totum aboleantur. </s> <s xml:id="echoid-s7963" xml:space="preserve">Nunc enim <lb/>ſciant nihil obſtare quo minus omnem lucem margini annula-<lb/>ri adimam, neque hoc abſurdum in rerum natura ſtatui Fa- <pb o="629" file="0367" n="385" rhead="SYSTEMATIS SATURNII."/> brius contendet, eſſe nimirum materiam aliquam quæ radios <lb/>ſolares omnino non reflectat, quippe qui de quatuor novis, <lb/>quos fingit, Saturni ſatellitibus, duos atros, ac per ſe invi-<lb/>ſibiles, nulliſque ſolis radiis illuſtrandos, reliquos vero ſplen-<lb/>didos eſſe imaginetur. </s> <s xml:id="echoid-s7964" xml:space="preserve">Verum hypotheſim illam jucundiſſi-<lb/>mam poſtea videbimus; </s> <s xml:id="echoid-s7965" xml:space="preserve">ſunt enim alia etiam prius ventilan-<lb/>da quibus noſtra oppugnatur.</s> <s xml:id="echoid-s7966" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7967" xml:space="preserve">Dixi planam annuli ſuperficiem, eam quæ nobis obverſa <lb/>eſt, aliquando à Sole averſam non illuminari, unde nimirum <lb/>nec cerni nobis tunc poſſit, ac proinde nullas Saturno anſu-<lb/>las præſtet. </s> <s xml:id="echoid-s7968" xml:space="preserve">Quod cum certiſſima ratione fieri demonſtrave-<lb/>rim, adverſarii mei breviter atque uno verbo negant, opti-<lb/>ciſque regulis refelli dicunt. </s> <s xml:id="echoid-s7969" xml:space="preserve">Quomodo tamen? </s> <s xml:id="echoid-s7970" xml:space="preserve">Ponendo ſci-<lb/>licet & </s> <s xml:id="echoid-s7971" xml:space="preserve">Saturni diſtantiam & </s> <s xml:id="echoid-s7972" xml:space="preserve">Solis diametrum longe quam ego <lb/>majores, quo ſiatut ſuperficies dicti annuli, ad nos ſpectans, <lb/>ſemper quoque à Sole lumen accipiat. </s> <s xml:id="echoid-s7973" xml:space="preserve">Atqui ego & </s> <s xml:id="echoid-s7974" xml:space="preserve">diſtan-<lb/>tiam illam & </s> <s xml:id="echoid-s7975" xml:space="preserve">diametrum (nam una alia expendet) multo ma-<lb/>jores exhibui in Syſtemate meo quam alius quiſquam omnium <lb/>Aſtronomorum. </s> <s xml:id="echoid-s7976" xml:space="preserve">Nam cum Ptolemæus & </s> <s xml:id="echoid-s7977" xml:space="preserve">Copernicus dia-<lb/>metrum Solis, tantum 5 Terræ diametris taxent; </s> <s xml:id="echoid-s7978" xml:space="preserve">Ariſtarchus <lb/>& </s> <s xml:id="echoid-s7979" xml:space="preserve">Bullialdus 7; </s> <s xml:id="echoid-s7980" xml:space="preserve">Ricciolus 33; </s> <s xml:id="echoid-s7981" xml:space="preserve">Wendelinus, quo pluribus ne-<lb/>mo, 64; </s> <s xml:id="echoid-s7982" xml:space="preserve">ipſe 111 dedi, nec ſine ratione ut puto. </s> <s xml:id="echoid-s7983" xml:space="preserve">Et Fabrio <lb/>tamen parcus fuiſſe videor, qui Solem longe majorem etiam <lb/>fieri vult, ne ſtare hypotheſis mea poſſit, utque merito hanc <lb/>Optices ignorantiam mihi impegiſſe videatur. </s> <s xml:id="echoid-s7984" xml:space="preserve">Cum ipſe in-<lb/>terim vel Optices vel Aſtronomiæ uſque adeo imperitus ſit, <lb/>ut non advertat, licet centies millies majorem quam ego So-<lb/>lis diametrum ſtatuat, ac ſimul planetariorum orbium ampli-<lb/>tudinem eadem proportione adaugeat, tamen diametrum il-<lb/>lam, ex Saturno conſpectam, non niſi 3 circiter ſcrupulorum <lb/>fore; </s> <s xml:id="echoid-s7985" xml:space="preserve">atque etiamſi tanta illinc appareret quanta nobis in Ter-<lb/>ra poſitis, tamen illa quam dedi demonſtratione evinci, ſu-<lb/>perficiem annuli ad nos verſam, aliquando Solis lumine non <lb/>illuſtrari, propterea quod planum annuli productum, inter <lb/>nos Solemque tranſeat. </s> <s xml:id="echoid-s7986" xml:space="preserve">Nam ſi hoc fieri quandoque neget, <lb/>oportet ſtatuat diametrum Solis, (quod ridiculum eſt) ipſius <pb o="630" file="0368" n="386" rhead="CHRIST. HUGENII ASSERTIO"/> magni orbis diametri, in quo circa Solem Tellus defertur, ſal-<lb/>rem duabus quintis partibus æqualem eſſe. </s> <s xml:id="echoid-s7987" xml:space="preserve">Hoc enim ita eſ-<lb/>ſe facile oſtendere poſſem, ſed Aſtronomi ex demonſtratio-<lb/>ne mea pag. </s> <s xml:id="echoid-s7988" xml:space="preserve">570 haud ægre deducent. </s> <s xml:id="echoid-s7989" xml:space="preserve">Nam ſi planum annu-<lb/>li plano eclipticæ foret ad rectos angulos, non duas quintas <lb/>ſed integram orbis magni diametrum dicerem; </s> <s xml:id="echoid-s7990" xml:space="preserve">duæ quintæ <lb/>autem oriuntur ex inclinatione graduum 23 {1/2} quanta eſt di-<lb/>ctorum planorum.</s> <s xml:id="echoid-s7991" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7992" xml:space="preserve">Invenio vero & </s> <s xml:id="echoid-s7993" xml:space="preserve">Graphices inſcitiam mihi objici, quia <lb/>nempe umbras qualdam, quas in ſchemate Saturni num. </s> <s xml:id="echoid-s7994" xml:space="preserve">10. <lb/></s> <s xml:id="echoid-s7995" xml:space="preserve">Euſtachius depinxit, alioqui non fuerim reprehenſurus. </s> <s xml:id="echoid-s7996" xml:space="preserve"><lb/>Eum enim accerſiviſſe quidem illas, ſed neceſſario, conten-<lb/>dunt, ut ſphæræ formam exprimeret; </s> <s xml:id="echoid-s7997" xml:space="preserve">quia videlicet incertum <lb/>abſque iis futurum fuerit, an convexus Saturnus, an planus <lb/>eſſet. </s> <s xml:id="echoid-s7998" xml:space="preserve">Atque ego proinde ineptus, qui de ſuo eas umbras <lb/>Euſtachium addidiſſe dixerim, quas ille ex arte, atque ita <lb/>flagitante rei natura deſcripſerit. </s> <s xml:id="echoid-s7999" xml:space="preserve">Sed enim jocari hîc libet. </s> <s xml:id="echoid-s8000" xml:space="preserve"><lb/>Nempe cum temere citra ullam demonſtrationem me ſtatuiſ-<lb/>ſe arguant, Saturnum circa axem ſuum converti, rogo nunc <lb/>illos unde tam certo compererint, Saturni medium corpus <lb/>globoſum eſſe? </s> <s xml:id="echoid-s8001" xml:space="preserve">Non enim obſervatio ulla puro hoc docuit, <lb/>ſed colligunt tantum ex analogia quadam inter hunc & </s> <s xml:id="echoid-s8002" xml:space="preserve">alia <lb/>quædam cæleſtia corpora, ſicut & </s> <s xml:id="echoid-s8003" xml:space="preserve">ego de converſione circa <lb/>axem.</s> <s xml:id="echoid-s8004" xml:space="preserve"><unsure/> Quid igitur neceſſario umbras ad globum repræſen-<lb/>tandum accerſere opus erat? </s> <s xml:id="echoid-s8005" xml:space="preserve">Verum diſſimulare mihi viden-<lb/>tur, quaſi non advertant, non adeo me de medii corporis <lb/>umbra ſenſiſſe, quàm de illa quam ellipticæ figuræ, binas <lb/>anſas efficienti, tribuit. </s> <s xml:id="echoid-s8006" xml:space="preserve">Huic enim ea lege umbras adjecit, <lb/>ut annulum ellipticum, non planum ſed rotundum, ſimilem-<lb/>que ſerpenti caudam devoranti, exprimat; </s> <s xml:id="echoid-s8007" xml:space="preserve">neſcio enim quo-<lb/>modo melius formam eam deſignem. </s> <s xml:id="echoid-s8008" xml:space="preserve">Deinde & </s> <s xml:id="echoid-s8009" xml:space="preserve">hoc umbræ <lb/>ſuæ artificio præſtitit, ut totus iſte annulus poſt globum Sa-<lb/>turni poſitus videatur. </s> <s xml:id="echoid-s8010" xml:space="preserve">Qua ex re ſibi ipſi forte Euſtachius <lb/>abſurdam aliquam Saturni formam commentus eſt, vel aliis <lb/>certe commioiſcendi anſam præbere potuit. </s> <s xml:id="echoid-s8011" xml:space="preserve">Mihi autem, <lb/>de umbra illa temere addita, neceſſario admonendi erant le- <pb o="631" file="0369" n="387" rhead="SYSTEMATIS SATURNII."/> ctores; </s> <s xml:id="echoid-s8012" xml:space="preserve">quia ſi vera eſſet, refellebt<unsure/> hypotheſin meam, cum <lb/>alioqui ſchema Euſtachianum, ſi ſine umbris conſideraretur, <lb/>plurimum illam confirmaret. </s> <s xml:id="echoid-s8013" xml:space="preserve">Sed & </s> <s xml:id="echoid-s8014" xml:space="preserve">me ejuſdem criminis te-<lb/>neri ajunt, quod paſſim umbras majores quam revera ſint <lb/>adjiciam. </s> <s xml:id="echoid-s8015" xml:space="preserve">Neſcio equidem quas dicant; </s> <s xml:id="echoid-s8016" xml:space="preserve">ſemel enim Satur-<lb/>ni globo annuloque umbras addidi, at non tanquam ita ob-<lb/>ſervaſſem, ſed explicandæ hypotheſeos cauſa, ut nempe ap-<lb/>pareret quo pacto annulum Saturno circumdediſſem.</s> <s xml:id="echoid-s8017" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8018" xml:space="preserve">Quod ſi faſciam in diſco ejus obſervatam, uti & </s> <s xml:id="echoid-s8019" xml:space="preserve">illas quas <lb/>in Jove & </s> <s xml:id="echoid-s8020" xml:space="preserve">Marte exhibui, nigriores quàm in cælo ſpectan-<lb/>tur, ob oculos poni dixiſſent, non iviſſem inficias. </s> <s xml:id="echoid-s8021" xml:space="preserve">Fateor <lb/>enim ultro cælatoris culpa hoc accidiſſe; </s> <s xml:id="echoid-s8022" xml:space="preserve">ac in Marte præ-<lb/>ſertim zonam longe dilutiorem, imprimis circa margines, <lb/>pigendam fuiſſe. </s> <s xml:id="echoid-s8023" xml:space="preserve">Quam neque perpetuo ibi ſpectari opi-<lb/>nor, ſed, ut Joviales faſciæ, mutabilem formam habere, ſic-<lb/>ut poſtremis in Marte obſervationibus didici; </s> <s xml:id="echoid-s8024" xml:space="preserve">de quibus a-<lb/>lias fortaſſe plura. </s> <s xml:id="echoid-s8025" xml:space="preserve">Nihil autem ejuſmodi mihi viſum in u-<lb/>troque hoc planeta, Euſtachius non alio argumento probat <lb/>quàm cætera omnia, nempe quia ipſe non viderit.</s> <s xml:id="echoid-s8026" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8027" xml:space="preserve">Quod Syſtemati Copernicano Syſtema meum Saturnium <lb/>adaptaverim, nemo, ut opinor, jure me reprehenderit. </s> <s xml:id="echoid-s8028" xml:space="preserve">At-<lb/>tamen cum Catholicis omnibus Fabrius illo interdicat, mi-<lb/>ror quod non vel hoc ſolo nomine rejicienda eſſe omnia <lb/>commenta mea pronuntiet. </s> <s xml:id="echoid-s8029" xml:space="preserve">Sed videbat, credo, facile in <lb/>locum Copernicani Syſtematis Tychonicum me ſubſtitue-<lb/>ere poſſe. </s> <s xml:id="echoid-s8030" xml:space="preserve">Utrum enim adhibeam parum admodum intereſt <lb/>ad phænomena quod attinet. </s> <s xml:id="echoid-s8031" xml:space="preserve">Sed rei veritas haud aliter <lb/>quam Copernicum ſequendo explicatur; </s> <s xml:id="echoid-s8032" xml:space="preserve">cujus ſententiam <lb/>non parum quoque noſtrum Saturni Syſtema commendat.</s> <s xml:id="echoid-s8033" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8034" xml:space="preserve">Non intelligo autem qui tam confidenter, hanc de Terra <lb/>mota opinionem, tantùm ab hetorodoxis Ariſtarchis tenen-<lb/>dam Fabrius aſſeveret. </s> <s xml:id="echoid-s8035" xml:space="preserve">Quoties enim de ea cum Catholicis <lb/>(Romanis nempe) ſermones confero, profitentur illi ſe ne-<lb/>quaquam decretis in contrarium latis teneri, ſive ea à Car-<lb/>dinalibus, ſive ab ipſo ſummo Pontifice profecta fuerint. <lb/></s> <s xml:id="echoid-s8036" xml:space="preserve">Quibus videlicet non tantum tribuunt in explicando ſacra- <pb o="632" file="0370" n="388" rhead="CHRIST. HUGENII ASSERTIO"/> rum literarum ſenſu, ut etiam de controverſiis quæ, ut vo-<lb/>cant, facti ſunt, neceſſario iis ſtandum ſit: </s> <s xml:id="echoid-s8037" xml:space="preserve">ac plane, quie-<lb/>tem Telluris rationibus potius adſtruendam, quàm diploma-<lb/>tis ſanciendam, exiſtimant. </s> <s xml:id="echoid-s8038" xml:space="preserve">Quin etiam in Gallia paſſim <lb/>Syſtema Copernici non tanquam hypotheſin, ſed ut liqui-<lb/>dam veritatem propugnari certum eſt, idque ab ipſis Eccle-<lb/>ſiaſticis, & </s> <s xml:id="echoid-s8039" xml:space="preserve">Sacerdotibus, qui voluminibus totis publice do-<lb/>ctrinam eam tradunt, nihil, quod ſciam, Roma contradi-<lb/>cente. </s> <s xml:id="echoid-s8040" xml:space="preserve">Quæ omnia perpendens, pridem credidi præter A-<lb/>ſtronomiæ ignaros, imperitamque multitudinem, tantùm <lb/>adhuc Cleanthes aliquos, in quibus & </s> <s xml:id="echoid-s8041" xml:space="preserve">Fabrius, antiquo er-<lb/>rori adhæreſcere, vique irrita Telluris motui obniti.</s> <s xml:id="echoid-s8042" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8043" xml:space="preserve">Cæterum cum mordicus, ut ait, hoc propoſitum teneat, <lb/>ac planetas reliquos proinde nullo modo Terræ aſſimilandos <lb/>putet, nihil mirum eſt, nec ferre eum potuiſſe Saturnicola-<lb/>rum ullam mentionem fieri. </s> <s xml:id="echoid-s8044" xml:space="preserve">Ubi tamen injuria me culpat. <lb/></s> <s xml:id="echoid-s8045" xml:space="preserve">Nam non ita de illis diſſerui, ut eſſe aliquos affirmarem, aut, <lb/>rationibus adductis, veriſimile id eſſe evincerem. </s> <s xml:id="echoid-s8046" xml:space="preserve">Quin imo <lb/>abſtinere me dixi plura ſcribere de Aſtronomia, qualis inco-<lb/>lentibus Saturnum futura eſſet, eo quod abſurdum nimis ple-<lb/>rique arbitrentur homines in planetis degere; </s> <s xml:id="echoid-s8047" xml:space="preserve">qui proinde <lb/>fruſtra me inveſtigare dicturi eſſent, quid ii obſervent, qui <lb/>in rerum natura non ſunt. </s> <s xml:id="echoid-s8048" xml:space="preserve">Cum autem, in periodo Lunæ <lb/>Saturniæ computanda, menſes Saturnicolarum nominavi, <lb/>nihil novum aut inſolitum Aſtronomis feci, quibus nihil fre-<lb/>quentius eſt, quam ut in Soleaut Luna aliquem exiſtere imagi-<lb/>nentur, qui inde aſtrorum motus ſpeculetur. </s> <s xml:id="echoid-s8049" xml:space="preserve">Non erat itaque <lb/>quod commentum hoc de Saturnicolis me ibi propoſuiſſe Fa-<lb/>brius culparet. </s> <s xml:id="echoid-s8050" xml:space="preserve">Quanquam etſi ſecus foret, neque primus <lb/>ego hoc prodidiſſem, neque ridiculum adeo, apud Philoſo-<lb/>phos quidem, quam ille exiſtimat. </s> <s xml:id="echoid-s8051" xml:space="preserve">Verum præter inſtitu-<lb/>tum ad iſta non digrediar. </s> <s xml:id="echoid-s8052" xml:space="preserve">Vocat me Fabriani Syſtematis <lb/>contemplatio, illius nempe quo, poſtquam meum diruit, <lb/>phænomena circa Saturnum commodè exponi poſſeconfidit. </s> <s xml:id="echoid-s8053" xml:space="preserve"><lb/>Nam priora illa, quibus Saturni ipſius motum novis legibus <lb/>ordinare aggreditur, nihil huc pertinent, neque examinare <pb o="633" file="0371" n="389" rhead="SYSTEMATIS SATURNII."/> ea operæ pretium fuerit. </s> <s xml:id="echoid-s8054" xml:space="preserve">Miror tamen quid in Tychonica <lb/>hypotheſi illi diſplicuerit, ut novam à fundamentis extruere <lb/>in animum induceret: </s> <s xml:id="echoid-s8055" xml:space="preserve">quam quidem vereor ut Aſtronomi <lb/>ſatis percipere, nedum probare queant.</s> <s xml:id="echoid-s8056" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8057" xml:space="preserve">At nec illa quæ ad phaſes Saturni pertinet, ullo modo <lb/>clarior eſt, prætereaque ab omni ratione & </s> <s xml:id="echoid-s8058" xml:space="preserve">veriſimilitudine <lb/>tam longè remota, ut neſciam an refutari à me opus habeat. <lb/></s> <s xml:id="echoid-s8059" xml:space="preserve">Cui enim imponet quæſo bellum iſtud commentum de qua-<lb/>tuor globulis Saturno proximis, quorum binilumine ſolis ſplen-<lb/>dent, binialii radios prorſus non reflectunt, ſed natura ſua ſunt <lb/>obſcurriſſimi. </s> <s xml:id="echoid-s8060" xml:space="preserve">Videor mihi circulatorium quendam calculo-<lb/>rum ludum videre, alios ibi albos, alios nigros eſſe, nunc <lb/>hos, nunc illos oſtendi abſcondique viciſſim. </s> <s xml:id="echoid-s8061" xml:space="preserve">Tale quid enim <lb/>præ ſe fert mirabilis illa hypotheſis. </s> <s xml:id="echoid-s8062" xml:space="preserve">Quidni verò circulos ſal-<lb/>tem in ſchemate ſuo expreſſit, quos nova hæc Planetarum ſo-<lb/>boles obeunt? </s> <s xml:id="echoid-s8063" xml:space="preserve">Quos circulos non Saturnum in centro habe-<lb/>re cerneremus, nec quidem minorem à majore includi, ſed <lb/>poſt tergum Planetæ binos ex ordine jacere, in quorum utro-<lb/>que duo iſtorum ſatellitum ita currerent, ut nunquam alter <lb/>alterum aſſequeretur, ſed è diametro ſemper diſtarent. </s> <s xml:id="echoid-s8064" xml:space="preserve">Re-<lb/>ctè, ni fallor, mentem Fabrii intellexi, ſin minus, admone-<lb/>ri cupio, fuit enim nonnihil divinandum. </s> <s xml:id="echoid-s8065" xml:space="preserve">Quum autem, his <lb/>poſitis, phaſin 1, 4, 8, 9, tabellæ ſuæ, facile explicari dicat, <lb/> <anchor type="note" xlink:label="note-0371-01a" xlink:href="note-0371-01"/> ſcire velim quo pacto Planetæ novi ex minimis maximi fiant. <lb/></s> <s xml:id="echoid-s8066" xml:space="preserve">Nam in 1 & </s> <s xml:id="echoid-s8067" xml:space="preserve">4 Figura, diametrum ne quidem dimidiam ha-<lb/>bent medii Saturnii corporis, cum in Figura 8, & </s> <s xml:id="echoid-s8068" xml:space="preserve">9, vel æ-<lb/>quales huic, vel majores etiam eſſe neceſſe ſit. </s> <s xml:id="echoid-s8069" xml:space="preserve">Deinde ipſius <lb/>Euſtachii obſervata phaſis in Figura 10, quomodo ex hac hy-<lb/>potheſi deducetur? </s> <s xml:id="echoid-s8070" xml:space="preserve">Cum neque rotunda corpora ellipticos ar-<lb/>cus facere poſſint, & </s> <s xml:id="echoid-s8071" xml:space="preserve">nigri Planetæ minores candidis hîc eſ-<lb/>ſe debeant, contra quam poſitum fuit: </s> <s xml:id="echoid-s8072" xml:space="preserve">æquales enim inter ſe, <lb/>poſt aliquantam deliberationem, ſtatuuntur: </s> <s xml:id="echoid-s8073" xml:space="preserve">Quænam porro <lb/>ſit novorum Planetarum periodus non jam exigam, nondum <lb/>enim repertam eſſe ait. </s> <s xml:id="echoid-s8074" xml:space="preserve">Sed vereor ut aliqua eſſe poſſit. </s> <s xml:id="echoid-s8075" xml:space="preserve">Nam <lb/>trium quidem annorum ſpatio, quorum primus 1646, unam <lb/>illam Figuram 10 ſibi. </s> <s xml:id="echoid-s8076" xml:space="preserve">apparuiſſe Euſtachius affirmat, in ta- <pb o="634" file="0372" n="390" rhead="CHRST. HUG. ASSERT. SYST. SAT."/> bella quadam, Sereniſſimo Magno Duci Hetruriæ, Fratri T. <lb/></s> <s xml:id="echoid-s8077" xml:space="preserve">C. </s> <s xml:id="echoid-s8078" xml:space="preserve">dicata; </s> <s xml:id="echoid-s8079" xml:space="preserve">unde oportet nihil motos fuiſſe interea ſatellites, <lb/>quod quidem animadverti poſſet. </s> <s xml:id="echoid-s8080" xml:space="preserve">Poſtea vero alüs tribusan-<lb/>nis, 1655 nempe, cum duobus inſequentibus, in tantum pro-<lb/>greſſi fuere, ut medio horum annorum rotundus Saturnus ſpe-<lb/>ctatus ſit, extremis vero utriſque cum binisad latera orbibus, <lb/>interſtitio aliquo à medio corpore disjunctis. </s> <s xml:id="echoid-s8081" xml:space="preserve">Utinam ridi-<lb/>culi Syſtematis inventori pœna conſtituta ſit, ut motuum iſto-<lb/>rum anomalias inveſtigare teneatur. </s> <s xml:id="echoid-s8082" xml:space="preserve">Piget vero me ultra in <lb/>his immorari: </s> <s xml:id="echoid-s8083" xml:space="preserve">ſed tamen pauca pro meo illo Saturni comite <lb/>dicere cogor. </s> <s xml:id="echoid-s8084" xml:space="preserve">Hunc, quia videbatur aliquid in Syſtemate ſuo <lb/>turbaturus, ſupra Saturnum Fabrius relegavit, ita nimirum ut <lb/>totus quem percurrit circulus, Planetâ ſuperior jaceat. </s> <s xml:id="echoid-s8085" xml:space="preserve">Sed <lb/>immerito prorſus; </s> <s xml:id="echoid-s8086" xml:space="preserve">ego enim cautionem me dabo, nihil eum <lb/>anſatis illis phaſibus occurſu ſuo nociturum, cum puſillus ad-<lb/>eo ſit, ut ne quidem, cum prope ad Saturnum accedit, con-<lb/>ſpici queat: </s> <s xml:id="echoid-s8087" xml:space="preserve">raro etiam, nec niſi cum rotundus eſt Saturnus, <lb/>inter diſcum ejus & </s> <s xml:id="echoid-s8088" xml:space="preserve">nos tranſeat, idque ſemidiurno ſpatio. </s> <s xml:id="echoid-s8089" xml:space="preserve"><lb/>Sinat itaque in ſua illum manere orbita, unáque Mediceos <lb/>Planetas in vias ſuas, à Magno olim Galilæo circa Jovem at-<lb/>tributas, reſtituat, quos nullam omnino ob cauſam, quam <lb/>quidem aut ipſe adducit, aut ego animadvertere poſſum, lo-<lb/>co ſuo ſimiliter expulit.</s> <s xml:id="echoid-s8090" xml:space="preserve"/> </p> <div xml:id="echoid-div374" type="float" level="2" n="6"> <note position="right" xlink:label="note-0371-01" xlink:href="note-0371-01a" xml:space="preserve">TAB. LII.</note> </div> <p> <s xml:id="echoid-s8091" xml:space="preserve">Jamque percenſui, ni fallor, Princeps Sereniſſime, omnia <lb/>quæ adverſus me Euſtachius de Divinis, ſeu potius P. </s> <s xml:id="echoid-s8092" xml:space="preserve">Fabrius <lb/>conquiſivit, quæve ad ſuam hypotheſin ſtabiliendam commen-<lb/>tus eſt, quæ quum animo reputo, nulla ſane invenio, quibus <lb/>non pro me ſibi eruditi reſponſuri fuerint, aut venturi tem-<lb/>poris experientia. </s> <s xml:id="echoid-s8093" xml:space="preserve">Interim pauca hæc repoſuiſſe non me pœ-<lb/>nitet, quia, licet artificem illum merito neglexiſſe tacendo vi-<lb/>deri poteram, non idem fortaſſe contra alterum, quem ferunt <lb/>alicujus nominis eſſe adverſarium, valiturum erat ſilentium; <lb/></s> <s xml:id="echoid-s8094" xml:space="preserve">utique quum ſatis omnibus conſtet pleraque ab hoc fuiſſe ſup-<lb/>peditata, multique, non tam qul<unsure/>d, quam à quo ſit objectum, <lb/>reſpiciant. </s> <s xml:id="echoid-s8095" xml:space="preserve">De cætero rogo C. </s> <s xml:id="echoid-s8096" xml:space="preserve">T. </s> <s xml:id="echoid-s8097" xml:space="preserve">ut, qua uſus ſum in diſ-<lb/>putando, libertatem boni conſulat: </s> <s xml:id="echoid-s8098" xml:space="preserve">Idque ita, ſi & </s> <s xml:id="echoid-s8099" xml:space="preserve">provo-<lb/>catus in arenam hanc deſcendi, & </s> <s xml:id="echoid-s8100" xml:space="preserve">juſtæ defenſionis terminos <lb/>nuſquam tamen exceſſi, FINIS.</s> <s xml:id="echoid-s8101" xml:space="preserve"/> </p> <pb file="0373" n="391"/> <pb file="0373a" n="392"/> <figure> <caption xml:id="echoid-caption168" style="it" xml:space="preserve">Fig. 1.</caption> </figure> <figure> <caption xml:id="echoid-caption169" style="it" xml:space="preserve">Fig. 2.</caption> </figure> <figure> <caption xml:id="echoid-caption170" style="it" xml:space="preserve">Fig. 3.</caption> </figure> <figure> <caption xml:id="echoid-caption171" style="it" xml:space="preserve">Fig. 4.</caption> </figure> <figure> <caption xml:id="echoid-caption172" style="it" xml:space="preserve">Fig. 5.</caption> </figure> <figure> <caption xml:id="echoid-caption173" style="it" xml:space="preserve">Fig. 6.</caption> </figure> <figure> <caption xml:id="echoid-caption174" style="it" xml:space="preserve">Fig. 7.</caption> </figure> <figure> <caption xml:id="echoid-caption175" style="it" xml:space="preserve">Fig. 8.</caption> </figure> <figure> <caption xml:id="echoid-caption176" style="it" xml:space="preserve">Fig. 9.</caption> </figure> <figure> <caption xml:id="echoid-caption177" style="it" xml:space="preserve">Fig. 10.</caption> </figure> <figure> <caption xml:id="echoid-caption178" style="it" xml:space="preserve">Fig. 11.</caption> </figure> <figure> <caption xml:id="echoid-caption179" style="it" xml:space="preserve">Fig. 12.</caption> </figure> <figure> <caption xml:id="echoid-caption180" style="it" xml:space="preserve">Fig. 13.</caption> </figure> <pb file="0374" n="393"/> <pb file="0375" n="394"/> </div> <div xml:id="echoid-div376" type="section" level="1" n="149"> <head xml:id="echoid-head206" xml:space="preserve">CHRISTIANI HUGENII <lb/>DE <lb/>SATURNI ANNULO <lb/>OBSERVATIONES.</head> <pb file="0376" n="395"/> <pb o="637" file="0377" n="396"/> <figure> <image file="0377-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0377-01"/> </figure> </div> <div xml:id="echoid-div377" type="section" level="1" n="150"> <head xml:id="echoid-head207" xml:space="preserve">CHRISTIANI HUGENII <lb/>DE <lb/>SATURNI ANNULO <lb/>OBSERVATIONES.</head> <head xml:id="echoid-head208" xml:space="preserve">I.</head> <head xml:id="echoid-head209" style="it" xml:space="preserve">Obſervationes in Saturnum Pariſiis habitæ in Bi-<lb/>bliotheca Regia.</head> <p> <s xml:id="echoid-s8102" xml:space="preserve">ANno 1668, 17 Aug.</s> <s xml:id="echoid-s8103" xml:space="preserve">, hora 11 veſpertinâ D<emph style="super">ni</emph> Hu-<lb/>genius & </s> <s xml:id="echoid-s8104" xml:space="preserve">Picart obſervarunt Planetam Satur-<lb/> <anchor type="note" xlink:label="note-0377-01a" xlink:href="note-0377-01"/> num ope teleſcopiorum 21 pedum & </s> <s xml:id="echoid-s8105" xml:space="preserve">invene-<lb/>runt figuram ejus talem qualem hic exhibemus, <lb/>globo in medio manifeſte excedente ſupra & </s> <s xml:id="echoid-s8106" xml:space="preserve"><lb/>infra ovalem quod vix dum anno præceden-<lb/>ti erat conſpicuum.</s> <s xml:id="echoid-s8107" xml:space="preserve"/> </p> <div xml:id="echoid-div377" type="float" level="2" n="1"> <note position="right" xlink:label="note-0377-01" xlink:href="note-0377-01a" xml:space="preserve">TAB. LL <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s8108" xml:space="preserve">Menſurarunt diverſimode inclinationem magnæ diametri <lb/>Ovalis ad æquatorem, quæ inventa eſt circiter 9 graduum, li-<lb/>cet eo empore tantum debuerit eſſe 4 graduum, ſecundum illa, <lb/>quæ D. </s> <s xml:id="echoid-s8109" xml:space="preserve">Hugenius dixit in Syſtemate Saturni, planum annu-<lb/>li, qui circumdat globum Planetæ cum plano Eclipticæ tan-<lb/>tum efficere angulum 23. </s> <s xml:id="echoid-s8110" xml:space="preserve">gr. </s> <s xml:id="echoid-s8111" xml:space="preserve">3ó; </s> <s xml:id="echoid-s8112" xml:space="preserve">ſed cum ultima hæc obſer-<lb/>vatio, & </s> <s xml:id="echoid-s8113" xml:space="preserve">ſimiles aliæ hujus anni & </s> <s xml:id="echoid-s8114" xml:space="preserve">præcedentis, magis exa-<lb/>ctæ ſint & </s> <s xml:id="echoid-s8115" xml:space="preserve">habitæ tempore magis opportuno ad menſuran-<lb/>dam obliquitatem, illis, quæ fundamentum fuere memo-<lb/>ratæ detereminationis, D<emph style="super">us</emph> Hugenius invenit, loco 23 <lb/>gr 3ó angulum plani Annularis & </s> <s xml:id="echoid-s8116" xml:space="preserve">Eclipticæ eſſe circiter 31 <lb/> <anchor type="note" xlink:label="note-0377-02a" xlink:href="note-0377-02"/> graduum; </s> <s xml:id="echoid-s8117" xml:space="preserve">& </s> <s xml:id="echoid-s8118" xml:space="preserve">hoc poſito, non ſolum formam, quam jam ha- <pb o="638" file="0378" n="397" rhead="CHRISTIANUS HUGENIUS"/> bet Saturnus, ſed & </s> <s xml:id="echoid-s8119" xml:space="preserve">omnes illas, quas obſervarunt Aſtrono-<lb/>mi ab illo tempore a quo veras obſervare potuere, congruere <lb/>perfecte cum hypotheſi annuli, & </s> <s xml:id="echoid-s8120" xml:space="preserve">peculiariter obſervationem <lb/>D<emph style="super">ni</emph> Campani, Anni 1664 initio Julii, quam ipſe publici ju-<lb/>ris fecit, in quo magna Ovalis diameter duplicatæ minori dia-<lb/>metro æqualis eſt.</s> <s xml:id="echoid-s8121" xml:space="preserve"/> </p> <div xml:id="echoid-div378" type="float" level="2" n="2"> <note position="right" xlink:label="note-0377-02" xlink:href="note-0377-02a" xml:space="preserve">Vera annuii <lb/>inclinatio.</note> </div> <p> <s xml:id="echoid-s8122" xml:space="preserve">Quod autem ad rotundam Saturni Phaſin attinet, non poteſt <lb/> <anchor type="note" xlink:label="note-0378-01a" xlink:href="note-0378-01"/> niſi parum admodum tempus hujus mutari, ex memorata mu-<lb/>tata inclinatione ita ut D<emph style="super">us</emph> Hugenius adhuc dum phaſin hanc <lb/>rotundam expectet anno 1671, in quo æſtate anſas amittere <lb/>inchoabit, quas niſi poſt annum circiter recuperabit, juxta illa <lb/>quæ tradidit in Syſtemate Saturni.</s> <s xml:id="echoid-s8123" xml:space="preserve"/> </p> <div xml:id="echoid-div379" type="float" level="2" n="3"> <note position="left" xlink:label="note-0378-01" xlink:href="note-0378-01a" xml:space="preserve">Phaſis ro-<lb/>@unda.</note> </div> <figure> <image file="0378-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0378-01"/> </figure> </div> <div xml:id="echoid-div381" type="section" level="1" n="151"> <head xml:id="echoid-head210" xml:space="preserve">II.</head> <head xml:id="echoid-head211" style="it" xml:space="preserve">Excerpta ex literis D. Hugenii, Academiæ regiæ ſcien-<lb/>tiarum ſocii, ad auctorem Diarii Eruditorum <lb/>de figura Planetæ Saturni.</head> <p> <s xml:id="echoid-s8124" xml:space="preserve">CUm Saturnus abdicaverit figuram ſuam rotundam poſt-<lb/> <anchor type="note" xlink:label="note-0378-02a" xlink:href="note-0378-02"/> quam hoc anno 1672 ex radiis ſolaribus emerſit, uti <lb/>anno præcedenti prædixeram, & </s> <s xml:id="echoid-s8125" xml:space="preserve">cum illi brachia <lb/>ſint renata; </s> <s xml:id="echoid-s8126" xml:space="preserve">ultima mutatio non minus meretur, ut <lb/>in Diariis veſtris notetur, quam quædam aliæ præcedentes, qua-<lb/>rum ibidem mentionem feciſti.</s> <s xml:id="echoid-s8127" xml:space="preserve"/> </p> <div xml:id="echoid-div381" type="float" level="2" n="1"> <note position="left" xlink:label="note-0378-02" xlink:href="note-0378-02a" xml:space="preserve">Brachia <lb/>Saturno re-<lb/>@ata.</note> </div> <p> <s xml:id="echoid-s8128" xml:space="preserve">Ultimum in conjunctione cum Sole fuit Planeta hic 12 Mar-<lb/>tii præterito, in 22 gr. </s> <s xml:id="echoid-s8129" xml:space="preserve">35 min. </s> <s xml:id="echoid-s8130" xml:space="preserve">Piſcium; </s> <s xml:id="echoid-s8131" xml:space="preserve">magna autem obliqui-<lb/>tas hujus loci Zodiaci ad noſtrum horizontem, quando oritur, <lb/>in cauſa fuit quo minus tribus menſibus, aurora prohibende, <lb/>videretur; </s> <s xml:id="echoid-s8132" xml:space="preserve">nam 5 Junii tantum D<emph style="super">US</emph> Caſſini primum illum po-<lb/>tuit obſervare, cum brachia Saturni jam facta eſſent adeo <lb/>clara & </s> <s xml:id="echoid-s8133" xml:space="preserve">ampla, ut inde colligi debuerit, jam à longo tem-<lb/>pore fuiſſe inſtaurata; </s> <s xml:id="echoid-s8134" xml:space="preserve">Obſervabat etiam in diſco Saturni <lb/> <anchor type="note" xlink:label="note-0378-03a" xlink:href="note-0378-03"/> parvam umbram ad latus ſeptentrionale brachiorum, pa- <pb o="639" file="0379" n="398" rhead="DE SATURNI ANNULO."/> riter ac exhibui in Syſtemate Saturni Tab. </s> <s xml:id="echoid-s8135" xml:space="preserve">XLIX. </s> <s xml:id="echoid-s8136" xml:space="preserve">fig. </s> <s xml:id="echoid-s8137" xml:space="preserve">6.</s> <s xml:id="echoid-s8138" xml:space="preserve">; <lb/>quod, æque ac inſtaurata brachia, congruit cum iis quæ ſtatui <lb/>in illo Syſtemate de annulo, quo cingitur Saturnus. </s> <s xml:id="echoid-s8139" xml:space="preserve">Sed <lb/>quoniam hæc hypotheſis præcipue confirmatur obſervationi-<lb/>bus, quas ultimo anno fecimus, quarum quædam nondum cum <lb/>publico communicatæ fuere, ut mihi veniam concedas rogo has <lb/>nunc memorare, cum annotationibus, quas ad eas feci.</s> <s xml:id="echoid-s8140" xml:space="preserve"/> </p> <div xml:id="echoid-div382" type="float" level="2" n="2"> <note position="left" xlink:label="note-0378-03" xlink:href="note-0378-03a" xml:space="preserve">@mbra in <lb/>@iſco Saturni.</note> </div> <p> <s xml:id="echoid-s8141" xml:space="preserve">Anno 1671 Saturnus apparuit rotundus ſine brachiis aut <lb/>anſis uti prædixi ante annos 14, quum publici juris feci meum <lb/>Syſtema, licet id acciderit duobus menſibus prius quam ex-<lb/>pectaveram, ſcilicet in fine menſis Maji. </s> <s xml:id="echoid-s8142" xml:space="preserve">Percepta deinde eſt <lb/>quædam interruptio figuræ rotundæ, quam non prævidi, & </s> <s xml:id="echoid-s8143" xml:space="preserve"><lb/>quam difficulter admodum prævidere potuiſſem, quia Satur-<lb/>num tantum obſervaram per unicum annum, cum hæc va-<lb/>ticinarer. </s> <s xml:id="echoid-s8144" xml:space="preserve">Sed noſti, quam primum intellexerim, brachia <lb/>rediiſle, quod D<emph style="super">US</emph> Caſſini obſervavit 11 & </s> <s xml:id="echoid-s8145" xml:space="preserve">14 Auguſti, <lb/>me affirmaſſe illa certo certius brevi iterum peritura, <lb/>quod quoque verum repertum eſt; </s> <s xml:id="echoid-s8146" xml:space="preserve">nam 4 Novembris bra-<lb/>chia Saturni erant adeo obſcura, ut dubitarem an qui-<lb/>dem iterum apparerent, licet D<emph style="super">US</emph> Caſſini affirmet ſe illa per-<lb/>cepiſſe ultimâ vice 13 menſis ſequentis Decembris, poſt quod <lb/>tempus figura rotunda duravit, donec Saturnus in radios <lb/>Solares ſe abſcondit.</s> <s xml:id="echoid-s8147" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8148" xml:space="preserve">Hæc ultima brachiorum Eclipſis præcipue veritatem meæ <lb/> <anchor type="note" xlink:label="note-0379-01a" xlink:href="note-0379-01"/> hypotheſeos probat; </s> <s xml:id="echoid-s8149" xml:space="preserve">facile enim patet quam difficile fuiſſet ſe-<lb/>cundam prævidere mutationem, priori adeo vicinam, niſi <lb/>illius veram cauſam perſpectam habuiſſem; </s> <s xml:id="echoid-s8150" xml:space="preserve">præterquam quod <lb/>modus, quo brachia ſecunda vice ablata fuere isipſe ſit, quem <lb/>ſtatui in meo Syſtemate. </s> <s xml:id="echoid-s8151" xml:space="preserve">Vidimus enim illa pedetentim de-<lb/>ficere à claritate ſuâ, licet ſemper manſerint ſatis ampla; <lb/></s> <s xml:id="echoid-s8152" xml:space="preserve">quod erat certum indicium, radios ſolis valde oblique il-<lb/>luſtrare ſuperficiem annuli Saturni, quæ ad nos converſa <lb/>erat, quam tandem non amplius illuſtrarunt ſed quidem op-<lb/>poſitam. </s> <s xml:id="echoid-s8153" xml:space="preserve">In præcedenti apparitione figuræ rotundæ, à fine <lb/>Maji uſque ad 14 Auguſti, brachia latebant, non quia <lb/>non illuſtrabantur, ſed quia viſus noſter, parum admodum <pb o="640" file="0380" n="399" rhead="CHRIST. HUG. DE SAT. ANNULO."/> vel omnino nihil elevatus erat ſupra ſuperficiem annulí, <lb/>quem Sol intuebatur.</s> <s xml:id="echoid-s8154" xml:space="preserve"/> </p> <div xml:id="echoid-div383" type="float" level="2" n="3"> <note position="right" xlink:label="note-0379-01" xlink:href="note-0379-01a" xml:space="preserve">Saturnus <lb/>rotundus.</note> </div> <p> <s xml:id="echoid-s8155" xml:space="preserve">Hæc omnia tantum illis ſcripta ſunt, qui cum cura exami-<lb/>narunt, quæ tradidi in Syſtemate Saturni: </s> <s xml:id="echoid-s8156" xml:space="preserve">& </s> <s xml:id="echoid-s8157" xml:space="preserve">pro iis adhuc <lb/>addo, quantum attinet ad lineam æquinoctialem vel rotundæ <lb/> <anchor type="note" xlink:label="note-0380-01a" xlink:href="note-0380-01"/> apparitionis Saturni, quæ linea formatur interſectione an-<lb/>nuli & </s> <s xml:id="echoid-s8158" xml:space="preserve">plani orbitæ planetæ, me nullis cogi obſervationibus, <lb/>ad ſitum illius mutandum quem in Syſtemate Saturni præſcri-<lb/>pſi in 20{1/2} gr. </s> <s xml:id="echoid-s8159" xml:space="preserve">Piſcium & </s> <s xml:id="echoid-s8160" xml:space="preserve">Libræ. </s> <s xml:id="echoid-s8161" xml:space="preserve">Quoties Saturnus è Sole viſus <lb/>pervenietad illa Zodiaci loca, rotundus apparebit, etiam quan-<lb/>do ab hiſce tantum duobus gradibus removetur; </s> <s xml:id="echoid-s8162" xml:space="preserve">nam ex obſer-<lb/>vationibus anni ultimi percepi ita reſtringendos eſſe limites <lb/>quos 6 graduum ſtatueram, ut ſatis facerem quibuſdam Gali-<lb/>læi & </s> <s xml:id="echoid-s8163" xml:space="preserve">Gaſſendi obſervationibus, quorum Teleſcopia mino-<lb/>ris notæ fuerant, quam credere auſus eram. </s> <s xml:id="echoid-s8164" xml:space="preserve">Juxta ultimas <lb/>haſce limitationes minus apparitiones formæ rotundæ Satur-<lb/>ni durabunt quam ipſe olim prædixi; </s> <s xml:id="echoid-s8165" xml:space="preserve">ita ut anno 1685 non <lb/>initio menſis Martii, ſed menſe Julio, brachia amittet Satur-<lb/>nus prope finem apparitionis ſuæ, & </s> <s xml:id="echoid-s8166" xml:space="preserve">illa recuperabit ſequenti <lb/>menſe Novembri; </s> <s xml:id="echoid-s8167" xml:space="preserve">pariter 1701 non poterit videri rotun-<lb/>dus, niſi menſe Junio, initio ſuæ apparitionis, & </s> <s xml:id="echoid-s8168" xml:space="preserve">menſe Au-<lb/>guſto brachia ejus incipient renaſci.</s> <s xml:id="echoid-s8169" xml:space="preserve"/> </p> <div xml:id="echoid-div384" type="float" level="2" n="4"> <note position="left" xlink:label="note-0380-01" xlink:href="note-0380-01a" xml:space="preserve">Linea æqui-<lb/>noctialis an-<lb/>nuli.</note> </div> <p> <s xml:id="echoid-s8170" xml:space="preserve">Antequam finem huic ſcripto imponam, hic addam, Ta-<lb/>bulam, quam dedi pro motu parvæ Lunæ, vel ſtellæ, quæ <lb/>comitatur Saturnum, & </s> <s xml:id="echoid-s8171" xml:space="preserve">circa illum volvitur 47 minutis mi-<lb/>nus quam 16 diebus, huc uſque repertam eſſe adeo obſer-<lb/>vationibus congruam, ut nondum videre queam, ſitne quid-<lb/>piam addendum vel detrahendum.</s> <s xml:id="echoid-s8172" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div386" type="section" level="1" n="152"> <head xml:id="echoid-head212" xml:space="preserve">FINIS.</head> <pb file="0381" n="400"/> </div> <div xml:id="echoid-div387" type="section" level="1" n="153"> <head xml:id="echoid-head213" xml:space="preserve">CHRISTIANI HUGENII <lb/>ΚΟΣΜΟΘΕΩΡΟΣ, <lb/>SIVE <lb/>De Terris Cœleſtibus, earumque ornatu, <lb/>CONJECTURÆ <lb/>AD <lb/>CONTANTINUM HUGENIUM, <lb/>Fratrem: <lb/>CULIELMO III. MAGNÆ BRITANNIÆ REGI, <lb/>A SECRETIS.</head> <pb file="0382" n="401"/> </div> <div xml:id="echoid-div388" type="section" level="1" n="154"> <head xml:id="echoid-head214" xml:space="preserve">Horat. Epiſt. 6. lib. 1.</head> <p style="it"> <s xml:id="echoid-s8173" xml:space="preserve">Hunc ſolem, & </s> <s xml:id="echoid-s8174" xml:space="preserve">ſtellas, & </s> <s xml:id="echoid-s8175" xml:space="preserve">decedentia certis <lb/>Tempora momentis, ſunt qui formidine nulla <lb/>Imbuti ſpectent: </s> <s xml:id="echoid-s8176" xml:space="preserve">quid cenſes munera terræ <lb/>Quid maris extremos Arabas ditantis & </s> <s xml:id="echoid-s8177" xml:space="preserve">Indos? <lb/></s> <s xml:id="echoid-s8178" xml:space="preserve">Ludicra quid, plauſus, & </s> <s xml:id="echoid-s8179" xml:space="preserve">amici dona Quiritis, <lb/>Quo ſpectanda modo, quo ſenſu credis & </s> <s xml:id="echoid-s8180" xml:space="preserve">ore?</s> <s xml:id="echoid-s8181" xml:space="preserve"/> </p> <pb o="643" file="0383" n="402" rhead="PRÆFATIO AD LECTOREM."/> <figure> <image file="0383-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0383-01"/> </figure> </div> <div xml:id="echoid-div389" type="section" level="1" n="155"> <head xml:id="echoid-head215" xml:space="preserve">BENEVOLO LECTORI <lb/>SALUTEM.</head> <p style="it"> <s xml:id="echoid-s8182" xml:space="preserve">LIbellus hicce jam ad umbilicum dedu-<lb/>ctus, & </s> <s xml:id="echoid-s8183" xml:space="preserve">prælo deſtinatus erat, cum <lb/>maximo rei literariæ damno Illuſtrem <lb/>ejus Auctorem primum morbus, dein <lb/>mors occupavit. </s> <s xml:id="echoid-s8184" xml:space="preserve">Qui tamen ut in lu-<lb/>cem prodiret, cavit, ultima voluntate fratrem, ad <lb/>quem ſcriptus eſt, rogitans, hujus ut edendi curam <lb/>ſuſcipere vellet. </s> <s xml:id="echoid-s8185" xml:space="preserve">Cui rei Nobiliſſimo Viro innume-<lb/>ris occupationibus & </s> <s xml:id="echoid-s8186" xml:space="preserve">peregrinationibus, utpote qui <lb/>Magnæ Britanniæ Regi ad res Batavas à ſecretis <lb/>eſſet, diſtracto vacare non licuit, niſi anno ferme <lb/>poſt Auctoris obitum. </s> <s xml:id="echoid-s8187" xml:space="preserve">Qua re, intercedente dein-<lb/>de etiam Typothetarum mora, factum eſt, ut cum <lb/>editioni jam omnia pararentur, & </s> <s xml:id="echoid-s8188" xml:space="preserve">hic Vir fato ceſ-<lb/>ſerit, adeoque & </s> <s xml:id="echoid-s8189" xml:space="preserve">Parente & </s> <s xml:id="echoid-s8190" xml:space="preserve">eo, qui poſt parentis <lb/>obitum ejus vicem gerebat, & </s> <s xml:id="echoid-s8191" xml:space="preserve">ad quem deſtinatus <lb/>erat, deſtitutus fuerit hic Libellus. </s> <s xml:id="echoid-s8192" xml:space="preserve">Eadem tamen, <lb/>qua ab Auctore conſcriptus erat, ratione, eademque <lb/>ad fratrem, licet jam defunctum, inſcriptione, (Re- <pb o="644" file="0384" n="403" rhead="PRÆFATIO AD LECTOREM."/> ligio enim fuit quidquam immutare) prodit in publi-<lb/>cum, non dubia ſpe, fore, ut eruditi, ſicut reliqua <lb/>omnia Auctoris, ſic & </s> <s xml:id="echoid-s8193" xml:space="preserve">ultimum hunc ejus fœtum be-<lb/>nigne accipiant. </s> <s xml:id="echoid-s8194" xml:space="preserve">Demonſtrationes equidem Mathe-<lb/>maticas non invenient ubique, neque enim res pati-<lb/>tur, ſed, quo in his rebus nihil ultra deſiderari jure <lb/>poſſe videtur, veriſimiles & </s> <s xml:id="echoid-s8195" xml:space="preserve">ingenioſas conjecturas. <lb/></s> <s xml:id="echoid-s8196" xml:space="preserve">Quæ ex cœlorum notitia depromi potuerunt, ea hic <lb/>videbunt ratione demonſtrata; </s> <s xml:id="echoid-s8197" xml:space="preserve">quæ ex iis non pa-<lb/>tent, ex cœleſtium corporum cum tellure noſtra affini-<lb/>tate ſolerter conjecta. </s> <s xml:id="echoid-s8198" xml:space="preserve">Verum hujus quid ſit, ex ipſo <lb/>Auctore commodius perſpicies. </s> <s xml:id="echoid-s8199" xml:space="preserve">Vale.</s> <s xml:id="echoid-s8200" xml:space="preserve"/> </p> <pb o="645" file="0385" n="404"/> <figure> <image file="0385-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0385-01"/> </figure> </div> <div xml:id="echoid-div390" type="section" level="1" n="156"> <head xml:id="echoid-head216" xml:space="preserve">CHRISTIANI HUGENII <lb/>COSMOTHEOROS, <lb/>SIVE <lb/>De Terris Cœleſtibus, earumque ornatu, <lb/>Conjecturæ. <lb/>AD <lb/>CONSTANTINUM HUGENIUM, <lb/>Fratrem.</head> <head xml:id="echoid-head217" xml:space="preserve">LIBER I.</head> <p> <s xml:id="echoid-s8201" xml:space="preserve">FIeri vix poteſt, Frater optime, ſi quis cum <lb/>Copernico ſentiat, Terramque, quam inco-<lb/>limus, è Planetarum numero unum eſſe exi-<lb/>ſtimet, qui circa Solem circumferantur, ab <lb/>eoque lucem omnem accipiant; </s> <s xml:id="echoid-s8202" xml:space="preserve">quin inter-<lb/>dum cogitet haud a ratione alienum eſſe ut, <lb/>quemadmodum noſter hic Globus, ita cæteri quoque iſti, <lb/>cultu ornatuque, ac fortaſſe habitatoribus non vacent. </s> <s xml:id="echoid-s8203" xml:space="preserve">Præ-<lb/>ſertim ſi ad ea quoque reſpiciat quæ poſt Copernici tempo-<lb/>ra in cœlo deprehenſa ſunt; </s> <s xml:id="echoid-s8204" xml:space="preserve">Comites nempe ſtellarum Jovis <lb/>& </s> <s xml:id="echoid-s8205" xml:space="preserve">Saturni, Lunæ montes campoſque, & </s> <s xml:id="echoid-s8206" xml:space="preserve">alia multa; </s> <s xml:id="echoid-s8207" xml:space="preserve">quibus <lb/>non ſolum veritas inventi ab illo ſyſtematis, ſed & </s> <s xml:id="echoid-s8208" xml:space="preserve">ſimilitu-<lb/>do ac cognatio, Terram inter & </s> <s xml:id="echoid-s8209" xml:space="preserve">Planetarum corpora, ma-<lb/>gnopere confirmatur. </s> <s xml:id="echoid-s8210" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s8211" xml:space="preserve">nobis, cum prælongis Te-<lb/>leſcopiis ſidera unà ſpecularemur; </s> <s xml:id="echoid-s8212" xml:space="preserve">quod jam per multos an-<lb/>nos, propter occupationes tuas & </s> <s xml:id="echoid-s8213" xml:space="preserve">continuam fere abſen- <pb o="646" file="0386" n="405" rhead="CHRISTIANI HUGENII"/> tiam, non licuit; </s> <s xml:id="echoid-s8214" xml:space="preserve">ſæpius ea de re ſermones habitos memini. <lb/></s> <s xml:id="echoid-s8215" xml:space="preserve">Qualia vero eſſent, quæ in iſtis regionibus extarent Naturæ <lb/>opera, id ne ſperandum quidem eſſe ut unquam ſciri poſ-<lb/>ſit, fruſtraque proinde quæri, certo credebamus. </s> <s xml:id="echoid-s8216" xml:space="preserve">Neque <lb/>vero aut a priſcis Philoſophis, aut a recentioribus quidquam <lb/>ejuſmodi tentatum fuiſſe comperi. </s> <s xml:id="echoid-s8217" xml:space="preserve">Nam inter illos quidem, <lb/>jam ab ipſo Aſtronomiæ exortu, cum primum Sphæricam <lb/>eſſe Terræ formam intellectum eſt, eamque undique æthere <lb/>cingi, fuere qui auderent alios eſſe in ſideribus mundos, imo <lb/>innumerabiles dicere. </s> <s xml:id="echoid-s8218" xml:space="preserve">Poſteriores vero, ut Cardinalis Cu-<lb/> <anchor type="note" xlink:label="note-0386-01a" xlink:href="note-0386-01"/> ſanus, Brunus, Keplerus, qui & </s> <s xml:id="echoid-s8219" xml:space="preserve">Tychonem Braheum idem <lb/>ſenſiſſe ſcribit, Planetis quidem incolas ſuos tribuerunt; </s> <s xml:id="echoid-s8220" xml:space="preserve">Cu-<lb/>ſanus & </s> <s xml:id="echoid-s8221" xml:space="preserve">Brunus etiam Soli, & </s> <s xml:id="echoid-s8222" xml:space="preserve">ſtellis inerrantibus: </s> <s xml:id="echoid-s8223" xml:space="preserve">nihil ta-<lb/>men ulterius aut hi aut illi quæſiviſſe inveniuntur; </s> <s xml:id="echoid-s8224" xml:space="preserve">neque <lb/>etiam nuperus auctor Gallicus dialogi ingenioſi de Mundo-<lb/>rum multitudine. </s> <s xml:id="echoid-s8225" xml:space="preserve">Tantum fabulas quaſdam de Lunæ popu-<lb/>lis nonnulli contexuerunt, animi causâ, Lucianicis, quas <lb/>noſti, haud multo veriſimiliores. </s> <s xml:id="echoid-s8226" xml:space="preserve">Nam & </s> <s xml:id="echoid-s8227" xml:space="preserve">Keplerianas his <lb/>annumero, quibus ille in ſomnio Aſtronomico ludere voluit. <lb/></s> <s xml:id="echoid-s8228" xml:space="preserve">Mihi vero, qui tot viris egregiis nequaquam me perſpica-<lb/>ciorem eſſe exiſtimo, ſed eo feliciorem, quòd poſt illos. </s> <s xml:id="echoid-s8229" xml:space="preserve"><lb/>tantùm non omnes, natus ſim; </s> <s xml:id="echoid-s8230" xml:space="preserve">cum ab aliquo tempore di-<lb/>ligentius iſta meditari cœpiſſem, viſum eſt non prorſus ob-<lb/>ſeptas eſſe, de rebus tam procul diſſitis, inquirendi vias, ſed <lb/>veriſimilibus conjecturis abunde materiam præberi. </s> <s xml:id="echoid-s8231" xml:space="preserve">Quas <lb/>conjecturas meas, prout ſeſe ſubinde obtulerunt, in adver-<lb/>ſariis annotatas, nunc in ordinem redigere, tibique expo-<lb/>nere volui; </s> <s xml:id="echoid-s8232" xml:space="preserve">atque aliquid etiam adjicere de Sole, Stelliſque <lb/>inerrantibus, & </s> <s xml:id="echoid-s8233" xml:space="preserve">mundi magnitudine, cujus particula quæ-<lb/>dam minima eſt totius Syſtematis noſtri complexus. </s> <s xml:id="echoid-s8234" xml:space="preserve">Et hæc <lb/>quidem, pro ſolito tuo res ſuperas cognoſcendi ſtudio, li-<lb/>benter te lecturum arbitror. </s> <s xml:id="echoid-s8235" xml:space="preserve">Mihi certe ſcribere ea jucun-<lb/>dum fuit; </s> <s xml:id="echoid-s8236" xml:space="preserve">utque ſæpe aliàs, ita nunc, velut in re ipſa, ve-<lb/>rum eſſe expertus ſum illud Archytæ; </s> <s xml:id="echoid-s8237" xml:space="preserve">Si quis in cœlum a-<lb/>ſcendiſſet, naturamque mundi & </s> <s xml:id="echoid-s8238" xml:space="preserve">pluchritudinem ſiderum <lb/>perſpexiſſet, inſuavem illam admirationem ei futuram, <pb o="647" file="0387" n="406" rhead="COSMOTHEOROS, LIB. I."/> (quæ alioqui jucundiſſima fuiſſet) niſi aliquem cui narra-<lb/>ret habuiſſet. </s> <s xml:id="echoid-s8239" xml:space="preserve">Utinam vero hæc noſtra narrare non omni-<lb/>bus poſſem, ſed præter te lectores arbitratu meo deligere li-<lb/>ceret, qui nec Aſtronomicæ ſcientiæ, nec Philoſophiæ me-<lb/>lioris rudes eſſent; </s> <s xml:id="echoid-s8240" xml:space="preserve">quibus facile conatus hoſce probatum iri, <lb/>nec, propter novitatem, defenſione opus habituros confi-<lb/>derem. </s> <s xml:id="echoid-s8241" xml:space="preserve">Quia vero & </s> <s xml:id="echoid-s8242" xml:space="preserve">in imperitiorum manus venturos pro-<lb/>video, & </s> <s xml:id="echoid-s8243" xml:space="preserve">fortaſſe quorumdam ſeveriora judicia ſubituros, <lb/>puto non abs re fore ut utrorumque reprehenſiones jam <lb/>hinc repellere coner.</s> <s xml:id="echoid-s8244" xml:space="preserve"/> </p> <div xml:id="echoid-div390" type="float" level="2" n="1"> <note position="left" xlink:label="note-0386-01" xlink:href="note-0386-01a" xml:space="preserve">Fuiſſe qui <lb/>Planetis in-<lb/>colas tribue-<lb/>rint, ſed ni-<lb/>hil præterea <lb/>de iis inqui-<lb/>ſiviſſe.</note> </div> <p> <s xml:id="echoid-s8245" xml:space="preserve">Atque erunt quidem, qui cum Geometriam aut Mathe-<lb/> <anchor type="note" xlink:label="note-0387-01a" xlink:href="note-0387-01"/> maticas nunquam attigerint, omnino vanum ac ridiculum <lb/>hoc inceptum noſtrum cenſebunt. </s> <s xml:id="echoid-s8246" xml:space="preserve">Incredibile enim iis vi-<lb/>detur, ut Siderum diſtantias, aut quæ ſit magnitudo eorum, <lb/>metiri poſſimus. </s> <s xml:id="echoid-s8247" xml:space="preserve">Tum vero motum huic Terræ aut falſo <lb/>adſcribi exiſtimant, aut nequaquam adhuc probatum eſſe. <lb/></s> <s xml:id="echoid-s8248" xml:space="preserve">Quare nihil mirum, ſi, quæ talibus fundamentis exſtruun-<lb/>tur, pro ſomniis nugiſque ſint habituri. </s> <s xml:id="echoid-s8249" xml:space="preserve">Quid vero his di-<lb/>cemus, niſi aliter ſenſuros ſi diſciplinis iſtis, naturæque re-<lb/>rum contemplandæ, operam dediſſent. </s> <s xml:id="echoid-s8250" xml:space="preserve">Hoc vero longe <lb/>plurimis non licuiſſe ſcimus, vel quod ad ea parum ingenio <lb/>comparati eſſent, vel quod unde diſcerent non haberent, <lb/>vel denique quod ſuis, aut reipublicæ curandis negotiis, alio <lb/>vocarentur. </s> <s xml:id="echoid-s8251" xml:space="preserve">Itaque nihil eos reprehendimus; </s> <s xml:id="echoid-s8252" xml:space="preserve">ſed, ſi dili-<lb/>gentiam in his rebus noſtram condemnandam putabunt, ad <lb/>magis idoneos judices provocamus.</s> <s xml:id="echoid-s8253" xml:space="preserve"/> </p> <div xml:id="echoid-div391" type="float" level="2" n="2"> <note position="right" xlink:label="note-0387-01" xlink:href="note-0387-01a" xml:space="preserve">Occurritur <lb/>objectionibus <lb/>imperito-<lb/>rum.</note> </div> <p> <s xml:id="echoid-s8254" xml:space="preserve">Erunt alii qui ea, quæ veriſimilia eſſe oſtendere conati <lb/> <anchor type="note" xlink:label="note-0387-02a" xlink:href="note-0387-02"/> ſumus, Sacris Literis adverſari prædicent, cum de Terris <lb/>animalibuſque, atque etiam ratione præditis, nos diſſerere <lb/>animadvertent; </s> <s xml:id="echoid-s8255" xml:space="preserve">de quorum origine, aut quod omnino in <lb/>rerum natura extent, nihil illic traditum ſit, ſed ea potius <lb/>ex quibus contrarium ſequatur. </s> <s xml:id="echoid-s8256" xml:space="preserve">Tantum enim de Tellure <lb/>hac, cum ſuis animantibus, herbiſque, & </s> <s xml:id="echoid-s8257" xml:space="preserve">homine omnium <lb/>domino commemorari. </s> <s xml:id="echoid-s8258" xml:space="preserve">Quibus reſpondeo, quod & </s> <s xml:id="echoid-s8259" xml:space="preserve">ante <lb/>me alii, ſatis apparere non de omnibus iis, quæ Deus crea-<lb/>vit, particulatim nos edoceri eum voluiſſe. </s> <s xml:id="echoid-s8260" xml:space="preserve">Itaque cum vel <pb o="648" file="0388" n="407" rhead="CHRISTIANI HUGENII"/> Siderum vel Terræ nomine, in prima Geneſi, etiam Plane-<lb/>tæ, qui præter Solem Lunamque ſunt, comprehendantur; <lb/></s> <s xml:id="echoid-s8261" xml:space="preserve">atque etiam Jovis & </s> <s xml:id="echoid-s8262" xml:space="preserve">Saturni Comites; </s> <s xml:id="echoid-s8263" xml:space="preserve">poſſe non tantum <lb/>plures alios utriuſque generis includi, ſed & </s> <s xml:id="echoid-s8264" xml:space="preserve">res innumeras <lb/>quas in ſuperficie eorum ſummo opifici collocare placuerit. </s> <s xml:id="echoid-s8265" xml:space="preserve"><lb/>Porro non neſcire eos quo pacto interpretandum ſit, quod <lb/>dicitur omnia propter hominem condita eſſe; </s> <s xml:id="echoid-s8266" xml:space="preserve">neque eo ſi-<lb/>gnificari, ut a pluribus jam eſt animadverſum, tot ingentia <lb/>corpora ſtellarum, quas partim videmus, partim nec vidiſ-<lb/>ſemus quidem unquam, ſi Teleſcopiorum auxilium defuiſſet, <lb/>noſtræ utilitatis aut contemplationis gratia fuiſſe condita; </s> <s xml:id="echoid-s8267" xml:space="preserve"><lb/>quia id abſurde diceretur. </s> <s xml:id="echoid-s8268" xml:space="preserve">Quare cum operum Dei magna <lb/>pars extra conſpectum hominum ſit poſita, neque ad eos <lb/>pertinere videatur, haud alienum eſſe opinari, aliquos exta-<lb/>re, qui illa propius aſpiciant & </s> <s xml:id="echoid-s8269" xml:space="preserve">admirentur.</s> <s xml:id="echoid-s8270" xml:space="preserve"/> </p> <div xml:id="echoid-div392" type="float" level="2" n="3"> <note position="right" xlink:label="note-0387-02" xlink:href="note-0387-02a" xml:space="preserve">Conjecturas <lb/>haſce S. Scri-<lb/>pturis non <lb/>adverſari.</note> </div> <p> <s xml:id="echoid-s8271" xml:space="preserve">Sed dicent fortaſſe, cum de his ipſe ſupremus auctor nihil <lb/> <anchor type="note" xlink:label="note-0388-01a" xlink:href="note-0388-01"/> amplius docuerit aut revelarit, credendum eſſe ſibi ſcientiam <lb/>eorum reſervaſſe, ac proinde temere, & </s> <s xml:id="echoid-s8272" xml:space="preserve">curioſe nimis de iis <lb/>inquiri. </s> <s xml:id="echoid-s8273" xml:space="preserve">At nimium ipſos ſibi ſumere ajo, ſi definire velint, <lb/>quouſque homines inveſtigando progredi debeant, diligen-<lb/>tiæque eorum modum ſtatuere; </s> <s xml:id="echoid-s8274" xml:space="preserve">ac ſi terminos, quos hic <lb/>Deus præſcripſit, certo cognitos haberent; </s> <s xml:id="echoid-s8275" xml:space="preserve">aut in hominum <lb/>poteſtate eſſet illos prætergredi. </s> <s xml:id="echoid-s8276" xml:space="preserve">Et ſane, ſi talibus ſcrupu-<lb/>lis retenti fuiſſent qui ante nos vixerunt, adhuc ignorari po-<lb/>tuiſſet quænam Telluris eſſet figura, aut quæ magnitudo, & </s> <s xml:id="echoid-s8277" xml:space="preserve"><lb/>num aliqua Americæ regio. </s> <s xml:id="echoid-s8278" xml:space="preserve">Item an Solis radiis Luna illu-<lb/>ſtraretur, quibuſve ex cauſis aut hæc aut ille deficerent; </s> <s xml:id="echoid-s8279" xml:space="preserve">ac <lb/>pleraque alia, quæ Aſtronomorum laboribus repertiſque ac-<lb/>cepta referimus. </s> <s xml:id="echoid-s8280" xml:space="preserve">Quid enim tam abſconditum & </s> <s xml:id="echoid-s8281" xml:space="preserve">inacceſſum <lb/>videbatur, quam quæ de rebus cœleſtibus in aperta luce nunc <lb/>poſita ſunt? </s> <s xml:id="echoid-s8282" xml:space="preserve">Ex quo intelligitur induſtriam mentiſque acu-<lb/>men hominibus data eſſe, quibus paulatim rerum naturalium <lb/>cognitionem conſequerentur, neque eſſe cur conari deſina-<lb/>mus & </s> <s xml:id="echoid-s8283" xml:space="preserve">ulteriora inquirere. </s> <s xml:id="echoid-s8284" xml:space="preserve">Attamen reconditiora illa, qui-<lb/>bus hic præcipue inſiſtimus, ſcimus non eſſe ejuſmodi, ut <lb/>quærendo penitus inveſtigari poſſint. </s> <s xml:id="echoid-s8285" xml:space="preserve">Itaque nihil veluti <pb o="649" file="0389" n="408" rhead="COSMOTHEOROS, LIB. I."/> certum affirmamus, (quî poſſimus enim?) </s> <s xml:id="echoid-s8286" xml:space="preserve">ſed conjecturis <lb/>tantum agimus, quarum de veriſimilitudine ſuo cuique arbi-<lb/>tratu judicare liberum ſit. </s> <s xml:id="echoid-s8287" xml:space="preserve">Quod ſi quis irritam igitur, & </s> <s xml:id="echoid-s8288" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0389-01a" xlink:href="note-0389-01"/> inanem in his operam nos ponere dicat, de rebus iis conje-<lb/>cturas prodendo, de quibus ipſi fateamur nihil certi unquam <lb/>comprehendi poſſe: </s> <s xml:id="echoid-s8289" xml:space="preserve">reſpondebo totum Phyſices ſtudium, <lb/>quatenus in cauſis rerum eruendis verſatur, eadem ratione <lb/>fore improbandum; </s> <s xml:id="echoid-s8290" xml:space="preserve">ubi veriſimilia inveniſſe laus ſumma <lb/>eſt, & </s> <s xml:id="echoid-s8291" xml:space="preserve">indagatio ipſa rerum, tum maximarum, tum oc-<lb/>cultisſimarum, habet oblectationem. </s> <s xml:id="echoid-s8292" xml:space="preserve">Sed veriſimilium <lb/>multi ſunt gradus, alii aliis veritati propiores, in quo dili-<lb/>genter æſtimando præcipuus judicii uſus vertitur.</s> <s xml:id="echoid-s8293" xml:space="preserve"/> </p> <div xml:id="echoid-div393" type="float" level="2" n="4"> <note position="left" xlink:label="note-0388-01" xlink:href="note-0388-01a" xml:space="preserve">Inquiſitio-<lb/>nem horum <lb/>ut nimis cu-<lb/>rioſam repre-<lb/>hendi non <lb/>debere.</note> <note position="right" xlink:label="note-0389-01" xlink:href="note-0389-01a" xml:space="preserve">Conjecturas <lb/>non eſſe va-<lb/>nas, quia <lb/>non plane <lb/>certæ.</note> </div> <p> <s xml:id="echoid-s8294" xml:space="preserve">Ut vero mihi videtur, non tantum res ad cognitionem <lb/> <anchor type="note" xlink:label="note-0389-02a" xlink:href="note-0389-02"/> maximas hic indagamus, ſed quarum contemplatio ſtudiis <lb/>quoque ſapientiæ multum conducat. </s> <s xml:id="echoid-s8295" xml:space="preserve">Expedit nimirum ut, <lb/>velut extra Tellurem hanc poſiti, procul eam intueamur, <lb/>quæramuſque, an ſola ſit in quam omnem ornatum natura <lb/>contulerit. </s> <s xml:id="echoid-s8296" xml:space="preserve">Ita enim rectius quid ſit, quoque loco haben-<lb/>da, intelligere poterimus: </s> <s xml:id="echoid-s8297" xml:space="preserve">quemadmodum qui longinquas <lb/>regiones obeunt, de patriæ ſuæ rebus verius judicare ſolent, <lb/>quam qui nunquam inde ſe moverunt. </s> <s xml:id="echoid-s8298" xml:space="preserve">Nec ſane ille ma-<lb/>gnopere admirabitur quæcunque hic vulgo maxima haben-<lb/>tur, qui, rationibus noſtris aliquid tribuens, multitudinem <lb/>Terrarum noſtræ ſimilium, ſimiliterque incolis ſuis frequen-<lb/>tatarum, ſibi propoſuerit. </s> <s xml:id="echoid-s8299" xml:space="preserve">Deum vero, tantarum rerum <lb/>effectorem, quî poterit idem non valde ſuſpicere & </s> <s xml:id="echoid-s8300" xml:space="preserve">venera-<lb/>ri? </s> <s xml:id="echoid-s8301" xml:space="preserve">cujus providentiam, ſapientiamque mirabilem, paſſim <lb/>hic aſſertam inveniet, contra falſas opiniones eorum, qui <lb/>vel ex fortuito corpuſculorum concurſu ortam eſſe Terram, <lb/>vel omni principio eam carere dixerunt. </s> <s xml:id="echoid-s8302" xml:space="preserve">Sed jam ad pro-<lb/>poſitum.</s> <s xml:id="echoid-s8303" xml:space="preserve"/> </p> <div xml:id="echoid-div394" type="float" level="2" n="5"> <note position="right" xlink:label="note-0389-02" xlink:href="note-0389-02a" xml:space="preserve">Ad ſapien-<lb/>tiam & pie-<lb/>tatem facere <lb/>quæ hic tra-<lb/>ctantur.</note> </div> <p> <s xml:id="echoid-s8304" xml:space="preserve">Et quoniam maximum ſumetur argumentum, ad ea quæ <lb/>inſtituimus probanda, ex ordinatione Planetarum Coperni-<lb/>cea, quodque inter eos Tellus hæc haud dubie numeratur; <lb/></s> <s xml:id="echoid-s8305" xml:space="preserve">bina ſchemata hic initio deſcribo, quorum alterum orbes <lb/> <anchor type="note" xlink:label="note-0389-03a" xlink:href="note-0389-03"/> eorum, circa Solem diſpoſitos, continet, veris proportio- <pb o="650" file="0390" n="409" rhead="CHRISTIANI HUGENII"/> nibus expreſſos; </s> <s xml:id="echoid-s8306" xml:space="preserve">ſimile illi quod in Automato noſtro ſæpius <lb/>conſpexiſti: </s> <s xml:id="echoid-s8307" xml:space="preserve">alterum rationes magnitudinum oſtendit, qui-<lb/> <anchor type="note" xlink:label="note-0390-01a" xlink:href="note-0390-01"/> bus corpora Planetarum inter ſe, & </s> <s xml:id="echoid-s8308" xml:space="preserve">ad Solem, comparan-<lb/>tur; </s> <s xml:id="echoid-s8309" xml:space="preserve">quod in eodem Automato adjectum eſt. </s> <s xml:id="echoid-s8310" xml:space="preserve">In priore <lb/> <anchor type="note" xlink:label="note-0390-02a" xlink:href="note-0390-02"/> punctum medium Sol eſt; </s> <s xml:id="echoid-s8311" xml:space="preserve">à quo deinceps, noto omnibus <lb/>ordine, ſunt orbitæ Mercurii, Veneris, Telluris, cum ſu-<lb/>peraddita via Lunæ; </s> <s xml:id="echoid-s8312" xml:space="preserve">tum Martis, Jovis, & </s> <s xml:id="echoid-s8313" xml:space="preserve">Saturni: </s> <s xml:id="echoid-s8314" xml:space="preserve">ac <lb/> <anchor type="note" xlink:label="note-0390-03a" xlink:href="note-0390-03"/> circa Jovem, Saturnumque circelli Comitum; </s> <s xml:id="echoid-s8315" xml:space="preserve">illius quatuor, <lb/>hujus quinque. </s> <s xml:id="echoid-s8316" xml:space="preserve">Quos circellos, cum eo, qui Lunæ noſtræ <lb/>dicatus eſt, longe majores hic poni ſciendum, quam pro <lb/>ratione ad Planetarum primariorum orbitas; </s> <s xml:id="echoid-s8317" xml:space="preserve">ne, ob parvi-<lb/>tatem, penitus viſum effugerent. </s> <s xml:id="echoid-s8318" xml:space="preserve">Orbitarum vero quanta <lb/>reipſa ſit vaſtitas inde intelligere licet, quod diſtantia a Sole <lb/>ad Terram, decem vel duodecim millia Terræ diametrorum <lb/>continet: </s> <s xml:id="echoid-s8319" xml:space="preserve">de qua menſura pluribus poſtea agetur. </s> <s xml:id="echoid-s8320" xml:space="preserve">Omnes <lb/>porro in eodem fere plano ſitæ ſunt; </s> <s xml:id="echoid-s8321" xml:space="preserve">ut proinde non mul-<lb/>tum diſcedant ab eo in quo Tellus circumit, quod Eclipticæ <lb/>planum vocatur. </s> <s xml:id="echoid-s8322" xml:space="preserve">Hoc vero oblique ſecatur ab axe Telluris, <lb/>in quo illa volvitur horis viginti quatuor, reſpectu Solis: </s> <s xml:id="echoid-s8323" xml:space="preserve">iſ-<lb/>que axis, niſi quod mutationem lentiſſimam ſubit, quam <lb/>norunt Aſtronomi, ſibi ipſi parallelus manet, dum ipſa cir-<lb/>ca Solem defertur; </s> <s xml:id="echoid-s8324" xml:space="preserve">ex quo dierum noctiumque oriuntur vi-<lb/>ces, itemque temporum anni commutationes, ut paſſim do-<lb/>cent eorum libri. </s> <s xml:id="echoid-s8325" xml:space="preserve">Unde & </s> <s xml:id="echoid-s8326" xml:space="preserve">tempora periodorum, quibus <lb/>circuitus ſuos Planeta quiſque peragit, huc tranſcribo. </s> <s xml:id="echoid-s8327" xml:space="preserve">Nem-<lb/>pe Saturni, annorum 29, dierum 174, horarum 5. </s> <s xml:id="echoid-s8328" xml:space="preserve">Jovis <lb/>annorum 11, dierum 317, horarum 15. </s> <s xml:id="echoid-s8329" xml:space="preserve">Martis proxime <lb/>dierum 687. </s> <s xml:id="echoid-s8330" xml:space="preserve">Telluris dierum 365{1/4}. </s> <s xml:id="echoid-s8331" xml:space="preserve">Veneris dierum 224, hor. <lb/></s> <s xml:id="echoid-s8332" xml:space="preserve">18. </s> <s xml:id="echoid-s8333" xml:space="preserve">Mercurii dierum 88.</s> <s xml:id="echoid-s8334" xml:space="preserve"/> </p> <div xml:id="echoid-div395" type="float" level="2" n="6"> <note position="right" xlink:label="note-0389-03" xlink:href="note-0389-03a" xml:space="preserve">TAB. LIV. <lb/>fig. 1.</note> <note position="left" xlink:label="note-0390-01" xlink:href="note-0390-01a" xml:space="preserve">TAB. LV. <lb/>fig. 1.</note> <note position="left" xlink:label="note-0390-02" xlink:href="note-0390-02a" xml:space="preserve">Copernici ſy-<lb/>ſtema expo-<lb/>nitur.</note> <note position="left" xlink:label="note-0390-03" xlink:href="note-0390-03a" xml:space="preserve">TAB. LIV. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s8335" xml:space="preserve">Hic eſt ille, notiſſimus jam, cæleſtium corporum ordo, <lb/>a Copernico repertus, idemque naturæ ſimplicitati conve-<lb/>nientiſſimus. </s> <s xml:id="echoid-s8336" xml:space="preserve">Hunc ſi quis convellere aut improbare con-<lb/> <anchor type="note" xlink:label="note-0390-04a" xlink:href="note-0390-04"/> tendat; </s> <s xml:id="echoid-s8337" xml:space="preserve">is diſcat primum, ex demonſtrationibus Aſtronomo-<lb/>rum, quanto in hac deſcriptione melius faciliuſque omnium <lb/>eorum, quæ circa motum ſiderum animadvertuntur, ratio red-<lb/>datur, quam in P@olemaico aut Tychonis ſyſtemate. </s> <s xml:id="echoid-s8338" xml:space="preserve">Cog- <pb o="651" file="0391" n="410" rhead="COSMOTHEOROS, LIB. I."/> noſcat etiam, ex ſingulari Kepleri obſervatione, quomodo <lb/>Planetarum, interque eos Telluris, a Sole diſtantiæ tempo-<lb/>ribus periodorum, quas retuli, certa quadam proportione <lb/>reſpondeant; </s> <s xml:id="echoid-s8339" xml:space="preserve">quam poſtea Jovis quoque & </s> <s xml:id="echoid-s8340" xml:space="preserve">Saturni Comites, <lb/>horum reſpectu, ſervare deprehenſum eſt. </s> <s xml:id="echoid-s8341" xml:space="preserve">Intelligat quam <lb/>contra motus naturam quiddam comminiſcendum ſit, quo <lb/>demonſtretur cur ſtella Polaris, in extrema cauda minoris Ur-<lb/>ſæ, exiguo nunc circulo moveatur, duobus gradibus & </s> <s xml:id="echoid-s8342" xml:space="preserve">ter-<lb/>tia parte à Polo diſtans; </s> <s xml:id="echoid-s8343" xml:space="preserve">cum ante annos mille octingentos <lb/>viginti, ætate nempe Hipparchi, duodecim gradibus, 24 <lb/>ſcrupulis, ab eodem Polo abfuerit: </s> <s xml:id="echoid-s8344" xml:space="preserve">poſt aliquot vero ſæcu-<lb/>la, ad 45 gradus inde receſſura ſit, & </s> <s xml:id="echoid-s8345" xml:space="preserve">poſt annorum vigin-<lb/>ti quinque millia, eodem quo nunc eſt, reverſura. </s> <s xml:id="echoid-s8346" xml:space="preserve">Ut pro-<lb/>inde cælum totum, ſi circumrotari dicatur, ſuper alio atque <lb/>alio axe id faciat neceſſe ſit, quod eſt abſurdiſſimum; </s> <s xml:id="echoid-s8347" xml:space="preserve">cum in <lb/>Copernici hypotheſi nihil ſit explicatu facilius. </s> <s xml:id="echoid-s8348" xml:space="preserve">Denique ex-<lb/>pendat omnia illa, quibus, ad argumenta Copernico objici <lb/>ſolita, Galilæus, Gaſſendus, Keplerus, aliique plurimi re-<lb/>ſponderunt. </s> <s xml:id="echoid-s8349" xml:space="preserve">Quorum rationibus ita ſublati ſunt qui ſupere-<lb/>rant ſcrupuli, ut omnes nunc Aſtronomi, niſi tardiore ſint <lb/>ingenio, aut hominum imperio obnoxiam credulitatem ha-<lb/>beant, motum Telluri, locumque inter Planetas, abſque du-<lb/>bitatione decernant.</s> <s xml:id="echoid-s8350" xml:space="preserve"/> </p> <div xml:id="echoid-div396" type="float" level="2" n="7"> <note position="left" xlink:label="note-0390-04" xlink:href="note-0390-04a" xml:space="preserve">Copernici <lb/>doctrinam <lb/>quæ rationes <lb/>confirment.</note> </div> <p> <s xml:id="echoid-s8351" xml:space="preserve">In altero, quod dixi, ſchemate, ita horum globi cum So-<lb/> <anchor type="note" xlink:label="note-0391-01a" xlink:href="note-0391-01"/> le oculis ſubjiciuntur, ac ſi juxta ſe poſiti eſſent. </s> <s xml:id="echoid-s8352" xml:space="preserve">Atque hic <lb/>rationem diametrorum, ad Solis diametrum, eam ſecutus <lb/>ſum, quam tradidi in libro de Saturni Phænomenis. </s> <s xml:id="echoid-s8353" xml:space="preserve">Nem-<lb/> <anchor type="note" xlink:label="note-0391-02a" xlink:href="note-0391-02"/> pe Annuli Saturnii eam quæ 11. </s> <s xml:id="echoid-s8354" xml:space="preserve">ad 37; </s> <s xml:id="echoid-s8355" xml:space="preserve">Globi incluſi, ad <lb/>eandem Solis diametrum fere, quæ 5 ad 37; </s> <s xml:id="echoid-s8356" xml:space="preserve">Jovis, quæ 2 <lb/>ad 11; </s> <s xml:id="echoid-s8357" xml:space="preserve">Martis, quæ 1 ad 166; </s> <s xml:id="echoid-s8358" xml:space="preserve">Telluris, quæ 1 ad 111; </s> <s xml:id="echoid-s8359" xml:space="preserve">Ve-<lb/>neris, quæ 1 ad 84; </s> <s xml:id="echoid-s8360" xml:space="preserve">quibus nunc addo Mercurii, quæ eſt 1 <lb/>ad 290 ex Hevelii obſervatione Anno 1661 habita; </s> <s xml:id="echoid-s8361" xml:space="preserve">cum in <lb/>Solis diſco Mercurius conſpiceretur, noſtro tamen, non il-<lb/>lius calculo.</s> <s xml:id="echoid-s8362" xml:space="preserve"/> </p> <div xml:id="echoid-div397" type="float" level="2" n="8"> <note position="right" xlink:label="note-0391-01" xlink:href="note-0391-01a" xml:space="preserve">Planetarum <lb/>magnitudi-<lb/>nis inter ſe <lb/>& ad Solem <lb/>ratio.</note> <note position="right" xlink:label="note-0391-02" xlink:href="note-0391-02a" xml:space="preserve">TAB. LV. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s8363" xml:space="preserve">Quomodo autem hæ noſtræ magnitudinum rationes inven-<lb/>tæ ſint, tum ex cognita proportione diſtantiarum à Sole, tum <pb o="652" file="0392" n="411" rhead="CRISTIANI HUGENII"/> ex menſura Diametrorum, Teleſcopiis capta, eo, quem di-<lb/>xi, libro oſtendi: </s> <s xml:id="echoid-s8364" xml:space="preserve">neque adhuc video cur multum, ab iis quas <lb/>tunc definivi, recedam; </s> <s xml:id="echoid-s8365" xml:space="preserve">etſi nihil eis deeſſe non contende-<lb/>rim. </s> <s xml:id="echoid-s8366" xml:space="preserve">Nam quod multi exiſtimant, in metiendis apparenti-<lb/>bus diametris, præſtare lamellis noſtris uſum Micrometro-<lb/> <anchor type="note" xlink:label="note-0392-01a" xlink:href="note-0392-01"/> rum quæ vocant, quibus fila tenuiſſima in foco Lentis majo-<lb/>ris prætenduntur, nondum iis aſſentiri poſſum, ſed aptiores <lb/>eſſe lamellas virgulaſve tenues arbitror, quas eo loco objicien-<lb/>das docueram. </s> <s xml:id="echoid-s8367" xml:space="preserve">Ex quo iſtud Micrometrorum inventum, <lb/>itemque Teleſcopii ad organa Aſtronomica adaptatio, non <lb/>multo poſt emanavit: </s> <s xml:id="echoid-s8368" xml:space="preserve">non ſine laude tamen eorum, qui in <lb/>perficiendo tam utili invento elaborarunt.</s> <s xml:id="echoid-s8369" xml:space="preserve"/> </p> <div xml:id="echoid-div398" type="float" level="2" n="9"> <note position="left" xlink:label="note-0392-01" xlink:href="note-0392-01a" xml:space="preserve">Microme-<lb/>tris præſtare <lb/>lamellas vir-<lb/>gulaſve te-<lb/>nues.</note> </div> <p> <s xml:id="echoid-s8370" xml:space="preserve">Cæterum, in hac planetarum comparatione, notanda eſt <lb/> <anchor type="note" xlink:label="note-0392-02a" xlink:href="note-0392-02"/> ingens Solis magnitudo, cum interioribus quatuor Planetis <lb/>collata; </s> <s xml:id="echoid-s8371" xml:space="preserve">utque hi Saturno quoque, ac Jove, longe longeque <lb/>minores ſint. </s> <s xml:id="echoid-s8372" xml:space="preserve">Nam conſiderandum, non ordine creſcere eo-<lb/>rum corpora cum diſtantiis a Sole; </s> <s xml:id="echoid-s8373" xml:space="preserve">quippe cum multo major <lb/>ſit Veneris, quam Martis, globus.</s> <s xml:id="echoid-s8374" xml:space="preserve"/> </p> <div xml:id="echoid-div399" type="float" level="2" n="10"> <note position="left" xlink:label="note-0392-02" xlink:href="note-0392-02a" xml:space="preserve">Solem Pla-<lb/>netis multo <lb/>majorem <lb/>eſſe.</note> </div> <p> <s xml:id="echoid-s8375" xml:space="preserve">His de utroque Diagrammate expoſitis, nemo, ut puto, <lb/> <anchor type="note" xlink:label="note-0392-03a" xlink:href="note-0392-03"/> jam non videt, quam clare ex priore, in quo ſyſtematis eſt <lb/>typus, ſequatur, eodem genere, cum cæteris quinque Pla-<lb/>netis, Tellurem hanc noſtram contineri. </s> <s xml:id="echoid-s8376" xml:space="preserve">Nam vel ipſi cir-<lb/>culorum poſitus hoc teſtantur. </s> <s xml:id="echoid-s8377" xml:space="preserve">Atqui præterea conſtat, te-<lb/>leſcopiorum obſervationibus, & </s> <s xml:id="echoid-s8378" xml:space="preserve">globoſa eſſe omnium cor-<lb/>pora, itidem ut Telluris, & </s> <s xml:id="echoid-s8379" xml:space="preserve">à Sole ſplendorem ſimiliter eos <lb/>mutuari. </s> <s xml:id="echoid-s8380" xml:space="preserve">Ac denique in hoc quoque ei ſimiles eſſe, quod <lb/>in ſe ipſis circum proprios axes volvantur: </s> <s xml:id="echoid-s8381" xml:space="preserve">quis enim de cæ-<lb/>teris dubitet, cum in Jove & </s> <s xml:id="echoid-s8382" xml:space="preserve">Marte hoc certo compertum <lb/>ſit? </s> <s xml:id="echoid-s8383" xml:space="preserve">Sicut autem Tellus Lunam comitem habet, ita Jupiter <lb/>& </s> <s xml:id="echoid-s8384" xml:space="preserve">Saturnus ſuas. </s> <s xml:id="echoid-s8385" xml:space="preserve">Quid igitur tam probabile eſt, cum in his <lb/>tot rebus Telluri cum Planetis illis primariis intercedat ſimi-<lb/>litudo, quam non minori quoque dignitate & </s> <s xml:id="echoid-s8386" xml:space="preserve">pulchritudi-<lb/>ne eos eſſe, nihiloque minus ornatos cultoſque: </s> <s xml:id="echoid-s8387" xml:space="preserve">aut quænam <lb/>cur hoc aliter ſe habeat ratio excogitari poteſt?</s> <s xml:id="echoid-s8388" xml:space="preserve"/> </p> <div xml:id="echoid-div400" type="float" level="2" n="11"> <note position="left" xlink:label="note-0392-03" xlink:href="note-0392-03a" xml:space="preserve">Tellurem <lb/>Planetis, & <lb/>hos Telluri <lb/>recte aſſimi-<lb/>lari.</note> </div> <p> <s xml:id="echoid-s8389" xml:space="preserve">Sane ſi cui, in diſſecti canis corpore, viſcera oſtenderen-<lb/> <anchor type="note" xlink:label="note-0392-04a" xlink:href="note-0392-04"/> tur, cor, ſtomachus, pulmones, inteſtina; </s> <s xml:id="echoid-s8390" xml:space="preserve">tum venæ, ar- <pb o="653" file="0393" n="412" rhead="COSMOTHEOROS, LIB. I."/> teriæ, nervi; </s> <s xml:id="echoid-s8391" xml:space="preserve">etiamſi nunquam animalis corpus apertum con-<lb/>ſpexiſſet; </s> <s xml:id="echoid-s8392" xml:space="preserve">vix dubitaret, quin ſimilis quædam fabrica, ac <lb/>partium varietas, in bove, porco, cæteriſque beſtiis ineſ@et. <lb/></s> <s xml:id="echoid-s8393" xml:space="preserve">Nec ſi unius, ex Saturni aut Jovis Comitibus, naturam cog-<lb/>nitam haberemus, non eadem fere quæ in illo, in cæteris <lb/>quoque reperiri putaremus? </s> <s xml:id="echoid-s8394" xml:space="preserve">Similiterque ex uno quopiam <lb/>Cometa, ſi quidnam eſſet perſpici poſſet, eandem omnium <lb/>rationem eſſe ſtatueremus. </s> <s xml:id="echoid-s8395" xml:space="preserve">Itaque plurimum ponderis ha-<lb/>bet illa ex ſimilitudine petita, & </s> <s xml:id="echoid-s8396" xml:space="preserve">à rebus viſis ad non viſas <lb/>producta ratio. </s> <s xml:id="echoid-s8397" xml:space="preserve">Quam proinde ſequentes, ex Planeta uno, <lb/>quem coram aſpicimus, de reliquis ejuſdem generis rectè <lb/>conjecturam faciemus.</s> <s xml:id="echoid-s8398" xml:space="preserve"/> </p> <div xml:id="echoid-div401" type="float" level="2" n="12"> <note position="left" xlink:label="note-0392-04" xlink:href="note-0392-04a" xml:space="preserve">Ex ſimilitu-<lb/>dine in hiſce <lb/>recte argu-<lb/>m<unsure/>enta p@ti.</note> </div> <p> <s xml:id="echoid-s8399" xml:space="preserve">Ac primùm quidem, non aliter quam Tellus noſtra, ſoli-<lb/> <anchor type="note" xlink:label="note-0393-01a" xlink:href="note-0393-01"/> do corpore eos conſtare exiſtimabimus. </s> <s xml:id="echoid-s8400" xml:space="preserve">Deinde prorſus <lb/>etiam veriſimile cenſebimus, adeſſe globis eorum id quod <lb/>gravitatem appellamus; </s> <s xml:id="echoid-s8401" xml:space="preserve">cujus vi corpora quæque, in ſuper-<lb/>ficie eorum hærentia, premant eam; </s> <s xml:id="echoid-s8402" xml:space="preserve">aut, ſi dimoveantur, <lb/>ex omni parte velut attracta recidant. </s> <s xml:id="echoid-s8403" xml:space="preserve">Quod ex ipſa quoque <lb/>globi forma liquet, cum hæc ex conatu corporum, ad cen-<lb/>trum unum tendentium, generetur. </s> <s xml:id="echoid-s8404" xml:space="preserve">Imo jam, certo quo-<lb/>dam ratiocinio, colligere didicimus, quanto majus minuſve <lb/>in Jove ac Saturno, quam apud nos, gravitatis momentum <lb/>eſſe debeat. </s> <s xml:id="echoid-s8405" xml:space="preserve">Qua de re, deque auctore ejus, in Diatriba de <lb/>Cauſis gravium diximus.</s> <s xml:id="echoid-s8406" xml:space="preserve"/> </p> <div xml:id="echoid-div402" type="float" level="2" n="13"> <note position="right" xlink:label="note-0393-01" xlink:href="note-0393-01a" xml:space="preserve">Planet as <lb/>ſolidos eſſe & <lb/>gravitate <lb/>pollere.</note> </div> <p> <s xml:id="echoid-s8407" xml:space="preserve">Nunc vero ulterius quærere pergamus, quibus gradibus <lb/>ad penitiora quædam, de ſtatu ornatuque Terrarum iſtarum, <lb/>cognoſcenda perveniri poſſit. </s> <s xml:id="echoid-s8408" xml:space="preserve">Ac primùm quam veriſimile <lb/>ſit herbas, & </s> <s xml:id="echoid-s8409" xml:space="preserve">animalia in earum ſuperficie exiſtere, æque ac <lb/>in Tellure noſtra. </s> <s xml:id="echoid-s8410" xml:space="preserve">Nemo negabit puto, & </s> <s xml:id="echoid-s8411" xml:space="preserve">formam & </s> <s xml:id="echoid-s8412" xml:space="preserve">vi-<lb/> <anchor type="note" xlink:label="note-0393-02a" xlink:href="note-0393-02"/> tam, & </s> <s xml:id="echoid-s8413" xml:space="preserve">creſcendi generandique rationem, in ſtirpibus ani-<lb/>mantibuſque majus quid eſſe, magiſque mirandum quàm <lb/>corpora vitâ carentia, quantumvis mole conſpicua ſint; </s> <s xml:id="echoid-s8414" xml:space="preserve">ve-<lb/>lut montes, rupes, maria. </s> <s xml:id="echoid-s8415" xml:space="preserve">Patet etiam in utroque illo vi-<lb/>ventium genere, multo aliter longeque expreſſius, cerni <lb/>Divinæ providentiæ intelligentiæque præſtantiam. </s> <s xml:id="echoid-s8416" xml:space="preserve">Cum <pb o="654" file="0394" n="413" rhead="CHRISTIANI HUGENII"/> enim quæ in Terra, imo quæ in Cælo quoque aſpicimus, <lb/>aliquis Democriti, aut etiam Carteſii ſectator, ita ſe expla-<lb/>naturum profiteri poſſit, ut tantum atomis & </s> <s xml:id="echoid-s8417" xml:space="preserve">motu horum <lb/>indigeat; </s> <s xml:id="echoid-s8418" xml:space="preserve">in herbis tamen & </s> <s xml:id="echoid-s8419" xml:space="preserve">animalibus fruſtra erit, nec de <lb/>primo eorum exortu quidquam veriſimile adferet; </s> <s xml:id="echoid-s8420" xml:space="preserve">cum ni-<lb/>mis maniſeſto appareat, nunquam vago, ac fortuito corpu-<lb/>ſculorum motu, talia quædam prodire potuiſſe: </s> <s xml:id="echoid-s8421" xml:space="preserve">quippe in <lb/>quibus omnia ad certum finem egregie apta accommodata-<lb/>que cernantur; </s> <s xml:id="echoid-s8422" xml:space="preserve">cum ſumma prudentia, & </s> <s xml:id="echoid-s8423" xml:space="preserve">legum naturæ, <lb/>ipſiuſque Geometriæ, cognitione exquiſita, quemadmodum <lb/>in ſequentibus ſæpius oſtendetur: </s> <s xml:id="echoid-s8424" xml:space="preserve">ut jam omittamus illa in <lb/>progignendo miracula. </s> <s xml:id="echoid-s8425" xml:space="preserve">Quod ſi igitur in Planetis nihil aliud <lb/>quàm vaſtæ ſolitudines, corporaque inertia, & </s> <s xml:id="echoid-s8426" xml:space="preserve">vita caren-<lb/>tia reperiantur; </s> <s xml:id="echoid-s8427" xml:space="preserve">atque abſint ea in quibus clariſſime certiſſi-<lb/>meque Architecti ſupremi ſapientia eluceſcit; </s> <s xml:id="echoid-s8428" xml:space="preserve">haud dubiè <lb/>multum dignitate & </s> <s xml:id="echoid-s8429" xml:space="preserve">pulchritudine concedent Telluri noſtræ: <lb/></s> <s xml:id="echoid-s8430" xml:space="preserve">quod, ut jam dixi, rationi adverſatur.</s> <s xml:id="echoid-s8431" xml:space="preserve"/> </p> <div xml:id="echoid-div403" type="float" level="2" n="14"> <note position="right" xlink:label="note-0393-02" xlink:href="note-0393-02a" xml:space="preserve">Nec deeſſe <lb/>illis anim@ <lb/>lia.</note> </div> <p> <s xml:id="echoid-s8432" xml:space="preserve">Non igitur ſic; </s> <s xml:id="echoid-s8433" xml:space="preserve">ſed erunt & </s> <s xml:id="echoid-s8434" xml:space="preserve">ibi corpora quædam motu <lb/>prædita, ſeſeque ipſa moventia, neque his quæ in Terra <lb/>ſunt ignobiliora; </s> <s xml:id="echoid-s8435" xml:space="preserve">adeoque erunt animantia. </s> <s xml:id="echoid-s8436" xml:space="preserve">His autem po-<lb/> <anchor type="note" xlink:label="note-0394-01a" xlink:href="note-0394-01"/> ſitis, jam de herbis quoque fere neceſſario concedendum eſt; <lb/></s> <s xml:id="echoid-s8437" xml:space="preserve">ut ſit aliquid quo illa alantur. </s> <s xml:id="echoid-s8438" xml:space="preserve">Omnia verò hæc non aliter <lb/>quam in ſuperficie Planetariorum globorum exiſtere, dubita-<lb/>ri non poteſt; </s> <s xml:id="echoid-s8439" xml:space="preserve">cum calore Solis gaudere ac foveri debeant; </s> <s xml:id="echoid-s8440" xml:space="preserve"><lb/>cujus radiis, non ſecus quam Tellus noſtra, expoſiti ſint.</s> <s xml:id="echoid-s8441" xml:space="preserve"/> </p> <div xml:id="echoid-div404" type="float" level="2" n="15"> <note position="left" xlink:label="note-0394-01" xlink:href="note-0394-01a" xml:space="preserve">Ut nee plan-<lb/>tas.</note> </div> <p> <s xml:id="echoid-s8442" xml:space="preserve">Sed dicet aliquis, celerius quàm par eſt, hic nos progre-<lb/>di. </s> <s xml:id="echoid-s8443" xml:space="preserve">Nam, ut non negetur res aliquas in Planetarum ſuper-<lb/>ficie reperiri, quæ ibi creſcant & </s> <s xml:id="echoid-s8444" xml:space="preserve">moveantur, Deoque au-<lb/>ctore, non minus quam noſtra hæc, dignæ ſint; </s> <s xml:id="echoid-s8445" xml:space="preserve">longè di-<lb/>verſam tamen earum poſſe eſſe naturam, ut nec materia, nec <lb/>creſcendi more, nec extrinſeca forma, aut internis partibus, <lb/>quidquam iis, quæ apud nos ſunt, ſimile habeant: </s> <s xml:id="echoid-s8446" xml:space="preserve">ac talia <lb/>ſint denique, ut nihil ejuſmodi in mentem homini venire <lb/>poſſit. </s> <s xml:id="echoid-s8447" xml:space="preserve">Hoc igitur jam quæramus quam ſit veriſimile; </s> <s xml:id="echoid-s8448" xml:space="preserve">& </s> <s xml:id="echoid-s8449" xml:space="preserve">an <lb/>non potius credendum ſit, non tantam eſſe diverſitatem <pb o="655" file="0395" n="414" rhead="COSMOTHEOROS, LIB. I."/> quanta exiſtimetur. </s> <s xml:id="echoid-s8450" xml:space="preserve">Favet eorum ſententiæ, qui omnia alia <lb/>illic imaginantur, quod Natura videatur varietatem plerum-<lb/>que, & </s> <s xml:id="echoid-s8451" xml:space="preserve">plurimis in rebus, ſectari; </s> <s xml:id="echoid-s8452" xml:space="preserve">quodque Conditoris po-<lb/>tentia hoc ipſo magis declaretur. </s> <s xml:id="echoid-s8453" xml:space="preserve">Sed cogitare debent, non <lb/>eſſe hominum arbitrio definiendum quàm magna iſta ſit va-<lb/> <anchor type="note" xlink:label="note-0395-01a" xlink:href="note-0395-01"/> rietas ac diſſimilitudo. </s> <s xml:id="echoid-s8454" xml:space="preserve">Neque, quia poſſit eſſe immenſa, <lb/>reſque illæ ab intellectu, & </s> <s xml:id="echoid-s8455" xml:space="preserve">comprehenſione noſtra penitus <lb/>remotæ, idcirco neceſſe eſſe, ut reipſa tales exiſtant. </s> <s xml:id="echoid-s8456" xml:space="preserve">Quam-<lb/>vis enim ſimilia omnia iis quæ apud nos ſunt, finxiſſet Deus <lb/>in cæteris Planetis; </s> <s xml:id="echoid-s8457" xml:space="preserve">nihilo minor eſſet ſpectatoribus eorum, <lb/>ſi qui ſunt, admiratio, quam ſi plurimum diſtarent: </s> <s xml:id="echoid-s8458" xml:space="preserve">cum, <lb/>quid in aliis effectum ſit, nullo modo poſſiut cognoſcere. <lb/></s> <s xml:id="echoid-s8459" xml:space="preserve">Potuiſſet in terris Americæ, aliiſque longè remotis, aliqua <lb/>creaſſe viventia, quæ his noſtris nihil ſimile haberent; </s> <s xml:id="echoid-s8460" xml:space="preserve">neque <lb/>id fecit tamen. </s> <s xml:id="echoid-s8461" xml:space="preserve">Nam formarum quidem diverſitatem aliquam <lb/>eſſe voluit, quibus animalia herbæque noſtræ à tranſmarinis <lb/>illis, diſſiderent, ſed & </s> <s xml:id="echoid-s8462" xml:space="preserve">in his ipſis formis, inque creſcendi <lb/>& </s> <s xml:id="echoid-s8463" xml:space="preserve">generandi modis, multa utriſque convenire fecit. </s> <s xml:id="echoid-s8464" xml:space="preserve">Habent <lb/>enim & </s> <s xml:id="echoid-s8465" xml:space="preserve">illic animalia pedes, alas; </s> <s xml:id="echoid-s8466" xml:space="preserve">atque intus cor, pulmo-<lb/>nes, inteſtina, vulvas; </s> <s xml:id="echoid-s8467" xml:space="preserve">cum hæc omnia in unoquoque gene-<lb/>re illorum, ac noſtratium quoque, planè diverſa ratione or-<lb/>dinari potuerint, ab infinitæ ſolertiæ opifice. </s> <s xml:id="echoid-s8468" xml:space="preserve">Non igitur <lb/>omnem varietatem quam poterat in rebus creatis, earum au-<lb/>ctor exhibuit, nec proinde argumento illi, quod a Naturæ <lb/>novandi ſtudio petitur, tantum tribuendum eſt, ut omnem, <lb/>qui in cæteris Planetis eſt, ornatum ab eo, qui in Terra no-<lb/>ſtra conſpicitur, prorſus alienum putemus. </s> <s xml:id="echoid-s8469" xml:space="preserve">At contra cre-<lb/>dibile eſt, inter ea quæ in ſuperficie iſtorum globorum gene-<lb/>rantur, quæque apud nos ſunt, præcipuam eſſe differentiam, <lb/>quæ ex majori, minorive, eorum a Sole, caloris vitæque fon-<lb/>te, diſtantia oriatur. </s> <s xml:id="echoid-s8470" xml:space="preserve">Propter quam tamen magis materiam, <lb/>quam formam rerum, variari neceſſe ſit.</s> <s xml:id="echoid-s8471" xml:space="preserve"/> </p> <div xml:id="echoid-div405" type="float" level="2" n="16"> <note position="right" xlink:label="note-0395-01" xlink:href="note-0395-01a" xml:space="preserve">Non nimian<unsure/> <lb/>in hiſce ſin-<lb/>gendam diſ-<lb/>ſimilitudi-<lb/>nem.</note> </div> <p> <s xml:id="echoid-s8472" xml:space="preserve">Ad materiam vero quod attinet qualiumcunque ſtirpium, <lb/>atque animantium, quæ Planetas exornant, etſi qualis ſit co-<lb/>gitatione aſſequi nequeamus, illud tamen vix dubitari poteſt, <lb/>quin ex elemento humido, uti noſtra omnia, creſcant &</s> <s xml:id="echoid-s8473" xml:space="preserve"> <pb o="656" file="0396" n="415" rhead="CHISTIANI HUGENII"/> alantur. </s> <s xml:id="echoid-s8474" xml:space="preserve">Nihil enim aliter gigni poſſe omnes fere Philoſo-<lb/>phiarbitrantur; </s> <s xml:id="echoid-s8475" xml:space="preserve">& </s> <s xml:id="echoid-s8476" xml:space="preserve">fuere inter præcipuos, qui ex aqua omnium <lb/>rerum originem eſſe dicerent. </s> <s xml:id="echoid-s8477" xml:space="preserve">Etenim, ſicca & </s> <s xml:id="echoid-s8478" xml:space="preserve">arida quæ <lb/>ſunt, motu carent: </s> <s xml:id="echoid-s8479" xml:space="preserve">abſque motu vero nihil corporibus, quo <lb/>augeantur, accedere poſſe manifeſtum eſt. </s> <s xml:id="echoid-s8480" xml:space="preserve">At liquidorum <lb/>particulæ, & </s> <s xml:id="echoid-s8481" xml:space="preserve">inter ſe continue moventur, & </s> <s xml:id="echoid-s8482" xml:space="preserve">facile ſeſe ubi-<lb/>que inſinuant; </s> <s xml:id="echoid-s8483" xml:space="preserve">quo fit, ut non tantum ſeipſas, ſed & </s> <s xml:id="echoid-s8484" xml:space="preserve">alias <lb/>diverſæ naturæ, quas ſecum vehunt, creſcentibus apponere <lb/>aptæ ſint. </s> <s xml:id="echoid-s8485" xml:space="preserve">Ita enim, aquæ affluxu, & </s> <s xml:id="echoid-s8486" xml:space="preserve">herbas adoleſcere, fo-<lb/>liiſque & </s> <s xml:id="echoid-s8487" xml:space="preserve">fructibus augeri, & </s> <s xml:id="echoid-s8488" xml:space="preserve">lapides ex arena concreſcere cer-<lb/>nimus. </s> <s xml:id="echoid-s8489" xml:space="preserve">Itemque metalla & </s> <s xml:id="echoid-s8490" xml:space="preserve">cryſtallos, gemmaſque incre-<lb/>menta inde capere ſatis conſtat, etſi in his obſcurius id ani-<lb/>madvertitur, propter lentiſſimos progreſſus; </s> <s xml:id="echoid-s8491" xml:space="preserve">quodque ſæpe <lb/>non in iis, quibus enatæ ſint, locis cavitatibuſque reperian-<lb/>tur; </s> <s xml:id="echoid-s8492" xml:space="preserve">pervetuſtis, ut videtur, Terræ ruinis convulſionibuſ-<lb/> <anchor type="note" xlink:label="note-0396-01a" xlink:href="note-0396-01"/> que disjectæ. </s> <s xml:id="echoid-s8493" xml:space="preserve">Sed aquæ elementum a Planetis non abeſſe, <lb/>veriſimiles quoque conjecturæ ſuppetunt, ex teleſcopiorum <lb/>obſervationibus. </s> <s xml:id="echoid-s8494" xml:space="preserve">Apparent enim in Jove tractus quidam re-<lb/>liquo diſco obſcuriores, iique non eadem ſemper forma per-<lb/>manentes, quod nubium proprium eſt. </s> <s xml:id="echoid-s8495" xml:space="preserve">Maculæ vero, quæ <lb/>immutabiliter globo ejus inhærere conſpiciuntur, ſæpe lon-<lb/>go tempore obtectæ manent, nubibus videlicet illis interce-<lb/>ptæ, è quibus deinde rurſus emergant. </s> <s xml:id="echoid-s8496" xml:space="preserve">Atque etiam nubes <lb/>in medio Jovis diſco exoriri quandoque annotatum fuit, & </s> <s xml:id="echoid-s8497" xml:space="preserve"><lb/>maculas quaſdam minores exiſtere, reliquo corpore magis <lb/>lucidas, neque eas diu ſupereſſe; </s> <s xml:id="echoid-s8498" xml:space="preserve">quas Caſſinus ex nivibus <lb/>eſſe conjectabat, cacumina montium inſidentibus. </s> <s xml:id="echoid-s8499" xml:space="preserve">Mihi non <lb/>improbabile videtur, terræ regiones candidiores eſſe, ſuper-<lb/>fuſis nubibus plerumque occultatas, ac nonnunquam ab iis <lb/>liberas.</s> <s xml:id="echoid-s8500" xml:space="preserve"/> </p> <div xml:id="echoid-div406" type="float" level="2" n="17"> <note position="left" xlink:label="note-0396-01" xlink:href="note-0396-01a" xml:space="preserve">Aquas a <lb/>Planetis non <lb/>abeſſe.</note> </div> <p> <s xml:id="echoid-s8501" xml:space="preserve">Apparent, etiam in Marte, lucis & </s> <s xml:id="echoid-s8502" xml:space="preserve">obſcuritatis diſcrimi-<lb/>na, ex quibus converſio ejus ad Solem, viginti quatuor ho-<lb/>ris cum 40 ſcrupulis primis, abſolvi reperta eſt; </s> <s xml:id="echoid-s8503" xml:space="preserve">nubes tamen <lb/>nondum fuerunt animadverſæ, idcirco quod multò minor cer-<lb/>nitur quam Jupiter; </s> <s xml:id="echoid-s8504" xml:space="preserve">etiam cum maximè ad Tellurem appro-<lb/>pinquat. </s> <s xml:id="echoid-s8505" xml:space="preserve">Præterquam quod & </s> <s xml:id="echoid-s8506" xml:space="preserve">intenſior Martis lux, utpote <pb o="657" file="0397" n="416" rhead="COSMOTHEOROS, LIB. I."/> a propiore Sole accepta, intuentibus impedimento eſt. </s> <s xml:id="echoid-s8507" xml:space="preserve">Ea-<lb/>demque lux magis etiam obſtat in Venere. </s> <s xml:id="echoid-s8508" xml:space="preserve">Sed ſi Tellus ac <lb/>Jupiter nubes aquaſque habent, vix dubitandum eſt quin & </s> <s xml:id="echoid-s8509" xml:space="preserve"><lb/>in cæteris inveniantur Planetis. </s> <s xml:id="echoid-s8510" xml:space="preserve">Nec tamen noſtræ prorſus <lb/> <anchor type="note" xlink:label="note-0397-01a" xlink:href="note-0397-01"/> ſimiles eſſe aquas iſtas dixerim; </s> <s xml:id="echoid-s8511" xml:space="preserve">etſi liquidæ ut ſint, ad uſus <lb/>quos præſtare debent, requiritur; </s> <s xml:id="echoid-s8512" xml:space="preserve">ut verò perſpicuæ, ad pul-<lb/>chritudinem. </s> <s xml:id="echoid-s8513" xml:space="preserve">Noſtra enim hæc, in Jove & </s> <s xml:id="echoid-s8514" xml:space="preserve">Saturno, con-<lb/>tinuo gelu aſtringeretur, propter magnam Solis diſtantiam. <lb/></s> <s xml:id="echoid-s8515" xml:space="preserve">Itaque putandum eſt naturam earum, quæ in Planetis ſunt, <lb/>ad ſuam quamque regionem attemperatam eſſe; </s> <s xml:id="echoid-s8516" xml:space="preserve">ut in Jove <lb/>quidem ac Saturno difficilius in glaciem vertantur, in Vene-<lb/>re verò, ac Mercurio, minus facile in vapores abeant. </s> <s xml:id="echoid-s8517" xml:space="preserve">In <lb/>omnibus autem attractum a Sole humorem, ſubſidere rurſus, <lb/>& </s> <s xml:id="echoid-s8518" xml:space="preserve">unde venit reverti, neceſſe eſt, ne penitus aridum Solum <lb/>relinquatur. </s> <s xml:id="echoid-s8519" xml:space="preserve">Non cadet autem niſi in guttas denſatus; </s> <s xml:id="echoid-s8520" xml:space="preserve">quod <lb/>eveniet, ſicuti apud nos, cum in frigidiorem locum aſcende-<lb/>rit ex inferiore calidioreque ob terræ viciniam.</s> <s xml:id="echoid-s8521" xml:space="preserve"/> </p> <div xml:id="echoid-div407" type="float" level="2" n="18"> <note position="right" xlink:label="note-0397-01" xlink:href="note-0397-01a" xml:space="preserve">Noſtræta-<lb/>men non <lb/>prorſus ſimi-<lb/>les.</note> </div> <p> <s xml:id="echoid-s8522" xml:space="preserve">Habemus igitur in globis illis campos Solis radiis expoſitos, <lb/>pluviiſque aut rore irrigatos, in quibus ſi quid enaſcatur; </s> <s xml:id="echoid-s8523" xml:space="preserve">ut <lb/>fieri debere utilitatis & </s> <s xml:id="echoid-s8524" xml:space="preserve">ornatus gratia diximus; </s> <s xml:id="echoid-s8525" xml:space="preserve">id eodem quo <lb/>apud nos modo fieri veriſimile eſt: </s> <s xml:id="echoid-s8526" xml:space="preserve">cum nec aliter fere, nec <lb/>melius poſſit. </s> <s xml:id="echoid-s8527" xml:space="preserve">Ut nempe radicibus ſuis ſolo adhæreat, ſimul-<lb/>que harum fibris humorem inde combibat. </s> <s xml:id="echoid-s8528" xml:space="preserve">Neque vero ſa-<lb/> <anchor type="note" xlink:label="note-0397-02a" xlink:href="note-0397-02"/> tis ornatæ mihi eſſe terræ iſtæ videbuntur, niſi ſtirpes quaſ-<lb/>dam habeant alte excreſcentes, quæque adeo arbores, aut <lb/>arborum inſtar, fiant: </s> <s xml:id="echoid-s8529" xml:space="preserve">quandoquidem hæ maximum, ac, <lb/>præter aquas, unicum ſunt ornamentum, quod Natura ter-<lb/>ris largiri poſſit. </s> <s xml:id="echoid-s8530" xml:space="preserve">Quæ quantum amœnitatis & </s> <s xml:id="echoid-s8531" xml:space="preserve">gratiæ affe-<lb/>rant facile unuſquiſque ſecum exiſtimat. </s> <s xml:id="echoid-s8532" xml:space="preserve">Ut omittam jam <lb/>materiæ ex arboribus oportuniſſimum ad omnia uſum. </s> <s xml:id="echoid-s8533" xml:space="preserve">Por-<lb/>ro vix aliter quoque propagari ſtirpes, aut perennare poſſe <lb/>exiſtimo, quam producendis ſeminibus. </s> <s xml:id="echoid-s8534" xml:space="preserve">Cum unica fere <lb/>hæc ratio videatur, eademque tam mirabilis, ut non ſolius <lb/>Telluris noſtræ gratia inventa ſit. </s> <s xml:id="echoid-s8535" xml:space="preserve">Denique nihil vetat, ut, <lb/>quemadmodum in diverſis hujus terræ regionibus, ita in iſtis <lb/>quoque longè remotis, idem in iis quæ ad ſtirpes attinent, <lb/>Natura ſecuta ſit.</s> <s xml:id="echoid-s8536" xml:space="preserve"/> </p> <div xml:id="echoid-div408" type="float" level="2" n="19"> <note position="right" xlink:label="note-0397-02" xlink:href="note-0397-02a" xml:space="preserve">Nec alia <lb/>ratione illic <lb/>naſci & pro-<lb/>pagari ſtirpes <lb/>quam ap@d <lb/>nos.</note> </div> <pb o="658" file="0398" n="417" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s8537" xml:space="preserve">Neque vero diſpar ratio eſt in animalibus; </s> <s xml:id="echoid-s8538" xml:space="preserve">cur non & </s> <s xml:id="echoid-s8539" xml:space="preserve">pa-<lb/> <anchor type="note" xlink:label="note-0398-01a" xlink:href="note-0398-01"/> ſcendi, & </s> <s xml:id="echoid-s8540" xml:space="preserve">generandi, modus ſimilis putetur in Planetis ei <lb/>qui eſt apud nos. </s> <s xml:id="echoid-s8541" xml:space="preserve">Quia nempe univerſa terræ hujus anima-<lb/>lia, ſive quadrupedum generis, aut volucrum, aut natantia, <lb/>aut reptilia, ipſaque inſecta, idem naturæ præſcriptum ſe-<lb/>quuntur. </s> <s xml:id="echoid-s8542" xml:space="preserve">Veſcuntur enim vel herbis, fructibuſque, vel ipſis <lb/>animantibus, quæ inde nutrita fuere: </s> <s xml:id="echoid-s8543" xml:space="preserve">omniumque generatio <lb/>per conjunctionem maris & </s> <s xml:id="echoid-s8544" xml:space="preserve">fœminæ, perque fæcunditatem <lb/>ovorum (nam & </s> <s xml:id="echoid-s8545" xml:space="preserve">hæc ubique animadvertitur) peragitur. <lb/></s> <s xml:id="echoid-s8546" xml:space="preserve">Nam hoc quidem certum eſt, fieri non poſſe ut, vel her-<lb/>bæ, vel animantia quæ illic ſunt, ſine propagatione generis <lb/>ſui eſſe perſeverent; </s> <s xml:id="echoid-s8547" xml:space="preserve">quia vel fortuitis caſibus interire ea ac de-<lb/>ficere contingeret; </s> <s xml:id="echoid-s8548" xml:space="preserve">cum herbæ ſtirpeſque humida materia con-<lb/>ſtent, eoque etiam exareſcere debeant; </s> <s xml:id="echoid-s8549" xml:space="preserve">animalia mollibus fle-<lb/>xilibuſque membris, nec, ut ſilices, duris. </s> <s xml:id="echoid-s8550" xml:space="preserve">Quod ſi in his <lb/>alias naſcendi vias comminiſcamur, velut ex arboribus; </s> <s xml:id="echoid-s8551" xml:space="preserve"><lb/>quemadmodum diu creditum eſt, ex harum genere quodam <lb/>in Britannia anates naſci, apparet quàm id à ratione abhor-<lb/>reat, propter ſummam, quæ lignum inter carneſque eſt, dif-<lb/>ferentiam. </s> <s xml:id="echoid-s8552" xml:space="preserve">Vel ſi animalia ex limo terræ exiſtere putemus, <lb/>velut de muribus in Ægypto multi prodiderunt, quis, natu-<lb/>ræ paulo intelligentior, non videt hoc alienum eſſe inſtitutis <lb/>ejus? </s> <s xml:id="echoid-s8553" xml:space="preserve">aut quis non exiſtimet multò magis convenire Dei <lb/>magnitudini ac ſapientiæ, ut ſemel omnis generis animantia <lb/>creaverit, inque Terrarum orbem certo modo, (quem ne-<lb/>mo hominum adhuc divinare potuit) impoſuerit, quam ut <lb/>perpetuo novis ex terra producendis vacare neceſſe habeat? </s> <s xml:id="echoid-s8554" xml:space="preserve"><lb/>Quibus alendis, educandiſque, abeſſet quoque prorſus pa-<lb/>rentum cura ac charitas, quam neceſſaria quadam ratione, <lb/>omni animalium noſtrorum generi, inſitam, ingenitamque <lb/>novimus. </s> <s xml:id="echoid-s8555" xml:space="preserve">Sed hæc quæ ad propagationem attinent, etſi <lb/>fortaſſe aliter ſeſe habeant, hoc tamen rationibus ſuperius <lb/>adductis ſatis probatum eſt, & </s> <s xml:id="echoid-s8556" xml:space="preserve">ſtirpes & </s> <s xml:id="echoid-s8557" xml:space="preserve">animalia in Plane-<lb/>tarum terris inveniri, ne ſcilicet ſint hac noſtra viliores. </s> <s xml:id="echoid-s8558" xml:space="preserve"><lb/>Quod cum ita ſit, tum quoque, ne minus, quam noſtra <lb/>Tellus, iſtæ aliæ ornatæ ſint, neceſſe eſt, ut non minor ſit, <pb o="659" file="0399" n="418" rhead="COSMOTHEOROS, LIB. I."/> in utroque genere illo, quàm apud nos varietas. </s> <s xml:id="echoid-s8559" xml:space="preserve">Quænam <lb/>vero hæc eſſe poteſt? </s> <s xml:id="echoid-s8560" xml:space="preserve">Equidem cum, in omni animantium <lb/>noſtrorum genere, cogito quibus modis moveantur; </s> <s xml:id="echoid-s8561" xml:space="preserve">omnia <lb/>video eo reduci, ut vel pedibus ingrediantur binis, quater-<lb/>niſve; </s> <s xml:id="echoid-s8562" xml:space="preserve">inſecta ſenis, vel etiam centenis; </s> <s xml:id="echoid-s8563" xml:space="preserve">vel ut in aëre vo-<lb/>lent, alarum mirabili vi & </s> <s xml:id="echoid-s8564" xml:space="preserve">moderamine; </s> <s xml:id="echoid-s8565" xml:space="preserve">vel ſine pedibus <lb/>reptent; </s> <s xml:id="echoid-s8566" xml:space="preserve">vel flexu corporum vehementi, aut etiam pedum <lb/>percuſſu, in aqua ſibi viam aperiant. </s> <s xml:id="echoid-s8567" xml:space="preserve">Præter hos incedendi <lb/>modos, vix videtur alius dari, aut omnino mente concipi <lb/>poſſe. </s> <s xml:id="echoid-s8568" xml:space="preserve">Ergo quæ in Planetis extant animantia, uno aliquo <lb/>ex his utentur; </s> <s xml:id="echoid-s8569" xml:space="preserve">aut quædam pluribus etiam, quemadmodum <lb/>apud nos aves amphibiæ; </s> <s xml:id="echoid-s8570" xml:space="preserve">quæ & </s> <s xml:id="echoid-s8571" xml:space="preserve">pedibus incedunt, & </s> <s xml:id="echoid-s8572" xml:space="preserve">na-<lb/>tant in aquis, & </s> <s xml:id="echoid-s8573" xml:space="preserve">in aëre volitant: </s> <s xml:id="echoid-s8574" xml:space="preserve">& </s> <s xml:id="echoid-s8575" xml:space="preserve">crocodili & </s> <s xml:id="echoid-s8576" xml:space="preserve">hippopota-<lb/>mi, inter terreſtria, & </s> <s xml:id="echoid-s8577" xml:space="preserve">aquatica, medii generis. </s> <s xml:id="echoid-s8578" xml:space="preserve">Nulla au-<lb/>tem præter haſce vita cogitari poſſe videtur. </s> <s xml:id="echoid-s8579" xml:space="preserve">Quid enim eſ-<lb/>ſe poſſit, in quo animantia exiſtant, præter tellurem ſoli-<lb/>dam, aut Elementum liquidum, quale aquæ noſtræ, aut mul-<lb/>to liquidius, quale aër; </s> <s xml:id="echoid-s8580" xml:space="preserve">aut illis ſimilia. </s> <s xml:id="echoid-s8581" xml:space="preserve">Poſſet enim eſſe aër <lb/>multò, quàm apud nos denſior, graviorque; </s> <s xml:id="echoid-s8582" xml:space="preserve">eoque ad vo-<lb/>landum accommodatior, neque tamen minus perſpicuus. <lb/></s> <s xml:id="echoid-s8583" xml:space="preserve">Poſſent etiam liquidorum plura genera, alia aliis ſuperindu-<lb/>cta eſſe. </s> <s xml:id="echoid-s8584" xml:space="preserve">Velut ſi, ſuper mare, incumbere cogitetur alia <lb/>quæpiam materia, quæ decuplo levior ſit aqua, centuplo <lb/>gravior aëre; </s> <s xml:id="echoid-s8585" xml:space="preserve">ac ſua quidem ſuperficie extrinſecus termina-<lb/>ta, ſed ut extra eam, terræ partes ſolidæ emineant. </s> <s xml:id="echoid-s8586" xml:space="preserve">Sed <lb/>non eſt, cur plura hujuſmodi in cæteris Planetis, quàm in <lb/>noſtro, inveniri putemus, & </s> <s xml:id="echoid-s8587" xml:space="preserve">ſi inveniantur, non tamen aliis <lb/>modis ibi animalia moveri poterunt. </s> <s xml:id="echoid-s8588" xml:space="preserve">Cæterùm quod ad va-<lb/>rias eorum formas attinet; </s> <s xml:id="echoid-s8589" xml:space="preserve">cum videamus in variis terræ re-<lb/>gionibus miram adeo ac multiplicem diverſitatem; </s> <s xml:id="echoid-s8590" xml:space="preserve">inveniri-<lb/>que in America quæ fruſtra alibi quæras; </s> <s xml:id="echoid-s8591" xml:space="preserve">magna ratio eſt ut <lb/>nullam earum formarum, quæ in Planetis exſtant, imaginando <lb/>aſſequi nos poſſe credamus. </s> <s xml:id="echoid-s8592" xml:space="preserve">Quanquam ſi omnes iſtos mo-<lb/>vendi modos cogitemus, quos hic recenſui, nihil mirum eſ-<lb/>ſet non magis differre aliquod iſtorum animalium à noſtra- <pb o="660" file="0400" n="419" rhead="CHRISTIANI HUGENII"/> te quopiam, quam noſtra diſcrepant inter ſe. </s> <s xml:id="echoid-s8593" xml:space="preserve">Ea dico quibus <lb/>minimum eſt ſimilitudinis.</s> <s xml:id="echoid-s8594" xml:space="preserve"/> </p> <div xml:id="echoid-div409" type="float" level="2" n="20"> <note position="left" xlink:label="note-0398-01" xlink:href="note-0398-01a" xml:space="preserve">Idem & de <lb/>animalibus <lb/>verum eſſe.</note> </div> <p> <s xml:id="echoid-s8595" xml:space="preserve">Quam varia porro ſint genera eorum in Planetis ita opti-<lb/>mè colligemus, ſi ad ea quæ apud nos ſunt, miramque in <lb/>iis formarum diverſitatem, animum advertamus. </s> <s xml:id="echoid-s8596" xml:space="preserve">Planè <lb/>enim veriſimile eſt, non minori numero occurſuras, ſi quis <lb/>ad Jovis, aut Veneris globum cominus ſpectandum admitte-<lb/>retur. </s> <s xml:id="echoid-s8597" xml:space="preserve">Percurramus vero (nam de omnibus dicere longum <lb/>eſſet) majores noſtrorum animalium differentias, vel for-<lb/>mâ, vel proprietate aliqua ſingulari notabiles; </s> <s xml:id="echoid-s8598" xml:space="preserve">idque in ter-<lb/> <anchor type="note" xlink:label="note-0400-01a" xlink:href="note-0400-01"/> reſtribus, aquatilibus, volucribus. </s> <s xml:id="echoid-s8599" xml:space="preserve">Cogitemus quæ ſit inter <lb/>equum, elephantum, leonem, cervum, camelum, porcum, <lb/>ſimiam, hiſtricum, teſtudinem, chamæleontem, diſſimili-<lb/>tudo; </s> <s xml:id="echoid-s8600" xml:space="preserve">quanta in aquaticis, cetum inter & </s> <s xml:id="echoid-s8601" xml:space="preserve">phocam, raiam, <lb/>lucium, anguillam, ſepiam, polypum, crocodilum, piſcem <lb/>volantem, torpedinem, cancrum, oſtream, muricem. </s> <s xml:id="echoid-s8602" xml:space="preserve">In <lb/>avium genere quantum diſcrimen, aquilæ, ſtruthiocameli, <lb/>pavonis, cygni, noctuæ, veſpertilioni<unsure/>s. </s> <s xml:id="echoid-s8603" xml:space="preserve">Reptilia pro uno <lb/>tantum genere cenſeamus. </s> <s xml:id="echoid-s8604" xml:space="preserve">At in inſectis formicas ſpecte-<lb/>mus, araneos, muſcas, papiliones; </s> <s xml:id="echoid-s8605" xml:space="preserve">& </s> <s xml:id="echoid-s8606" xml:space="preserve">miram horum natu-<lb/>ram, quod ex vermibus volatilia evadant. </s> <s xml:id="echoid-s8607" xml:space="preserve">In omnibus ve-<lb/>ro his, ſcimus quàm magnus præterea ſit numerus minus diſ-<lb/>ſidentium.</s> <s xml:id="echoid-s8608" xml:space="preserve"/> </p> <div xml:id="echoid-div410" type="float" level="2" n="21"> <note position="left" xlink:label="note-0400-01" xlink:href="note-0400-01a" xml:space="preserve">Summam <lb/>animalium <lb/>apud nos va-<lb/>@jetatem eſſe.</note> </div> <p> <s xml:id="echoid-s8609" xml:space="preserve">At quantuſcunque ſit, nihilo minorem eſſe in unoquoque <lb/> <anchor type="note" xlink:label="note-0400-02a" xlink:href="note-0400-02"/> reliquorum Planetarum putandum eſt. </s> <s xml:id="echoid-s8610" xml:space="preserve">Quamvis vero de fi-<lb/>gura iſtorum animalium fruſtra per conjecturas quæratur, <lb/>tamen de vita eorum generatim jam aliquid aſſecuti vide-<lb/>mur; </s> <s xml:id="echoid-s8611" xml:space="preserve">& </s> <s xml:id="echoid-s8612" xml:space="preserve">de ſenſibus erit in ſequentibus quod dicamus.</s> <s xml:id="echoid-s8613" xml:space="preserve"/> </p> <div xml:id="echoid-div411" type="float" level="2" n="22"> <note position="left" xlink:label="note-0400-02" xlink:href="note-0400-02a" xml:space="preserve">Nec mino-<lb/>rem in Pla-<lb/>@@tis.</note> </div> <p> <s xml:id="echoid-s8614" xml:space="preserve">Sicuti verò animantium, ita ſtirpium quoque & </s> <s xml:id="echoid-s8615" xml:space="preserve">arborum <lb/> <anchor type="note" xlink:label="note-0400-03a" xlink:href="note-0400-03"/> noſtrarum præcipuæ differentiæ expendi poſſunt. </s> <s xml:id="echoid-s8616" xml:space="preserve">Velut quæ <lb/>in abiete, quercu, palma, vite, ficu; </s> <s xml:id="echoid-s8617" xml:space="preserve">tum ea quæ nuces, <lb/>Cocos dictas, generat arbore; </s> <s xml:id="echoid-s8618" xml:space="preserve">itemque alia apud Indos, è <lb/>cujus ramis radices novæ pullulant, inque terram demittun-<lb/>tur. </s> <s xml:id="echoid-s8619" xml:space="preserve">Item, in herbis, gramen, papaver, braſſica, hedera, <lb/>pepones, ficus Indica foliis craſſis, ſine caule, ſuccreſcenti- <pb o="661" file="0401" n="420" rhead="COSMOTHEOROS, LIB. I."/> bus, aloë. </s> <s xml:id="echoid-s8620" xml:space="preserve">In quibus rurſus ea quam ſcimus, minus diſſi-<lb/>milium eſt copia. </s> <s xml:id="echoid-s8621" xml:space="preserve">Ad hæc propagandi viæ variæ inſpician-<lb/>tur; </s> <s xml:id="echoid-s8622" xml:space="preserve">velut ex ſeminibus, nucleis, taleis, inſitione, bulbis. <lb/></s> <s xml:id="echoid-s8623" xml:space="preserve">Quibus omnibus nihilo pauciora, aut minus miranda, in <lb/>Planetarum terris reperiri, exiſtimandum ſit.</s> <s xml:id="echoid-s8624" xml:space="preserve"/> </p> <div xml:id="echoid-div412" type="float" level="2" n="23"> <note position="left" xlink:label="note-0400-03" xlink:href="note-0400-03a" xml:space="preserve">@lem in ſtir-<lb/>pibus locum <lb/>babere.</note> </div> <p> <s xml:id="echoid-s8625" xml:space="preserve">Sed quod in hac diſquiſitione præcipuum eſt, plurimam-<lb/>que jucunditatem habet, nondum attigiſſe mihi videor; <lb/></s> <s xml:id="echoid-s8626" xml:space="preserve">quamdiu nullos in terris illis ſpectatores poſui, qui tot rebus <lb/> <anchor type="note" xlink:label="note-0401-01a" xlink:href="note-0401-01"/> creatis fruantur, pulchritudinemque, & </s> <s xml:id="echoid-s8627" xml:space="preserve">varietatem earum, <lb/>admirentur. </s> <s xml:id="echoid-s8628" xml:space="preserve">Et video quidem, neminem fere eorum, qui-<lb/>bus vel leviter hæc meditari contigit, dubitaſſe quin ſpecta-<lb/>tores aliqui in Planetis collocandi ſint: </s> <s xml:id="echoid-s8629" xml:space="preserve">non quidem homi-<lb/>nes nobis ſimiles, ſed animantia tamen ratione utentia. </s> <s xml:id="echoid-s8630" xml:space="preserve">Nem-<lb/>pe iis viſum eſt, qualemcunque terrarum iſtarum ornatum, <lb/>velut fruſtra, nulloque fine aut conſilio, fore procreatum, <lb/>ſi non hoc propoſitum fuiſſet, ut ab aliquo cerneretur, qui <lb/>intelligere ejus elegantiam poſſet, fructumque ſimul percipe-<lb/>re, & </s> <s xml:id="echoid-s8631" xml:space="preserve">ſummi opificis admirari ſapientiam. </s> <s xml:id="echoid-s8632" xml:space="preserve">Ego vero non <lb/>hoc præcipuum argumentum habeo, cur animal rationis par-<lb/>ticeps Planetas incolere exiſtimem. </s> <s xml:id="echoid-s8633" xml:space="preserve">Quid ſi enim dicamus <lb/>ipſum Deum ſpectare quæ effecit; </s> <s xml:id="echoid-s8634" xml:space="preserve">(alia quidem ratione quam <lb/>nos, ſed videre eum quis dubitet qui oculos fabricatus eſt;) <lb/></s> <s xml:id="echoid-s8635" xml:space="preserve">iiſque delectari, neque præterea quidquam requiri. </s> <s xml:id="echoid-s8636" xml:space="preserve">Nonne <lb/>enim ob hoc ipſum & </s> <s xml:id="echoid-s8637" xml:space="preserve">homines condidit, & </s> <s xml:id="echoid-s8638" xml:space="preserve">quicquid conti-<lb/>net mundus univerſus? </s> <s xml:id="echoid-s8639" xml:space="preserve">Itaque quod præcipue me movet, ut <lb/>rationabile animal in Planetis non deeſſe credam, hoc eſt, <lb/>quod nimia Terræ noſtræ præ cæteris illis eſſet præſtantia ac <lb/>nobilitas, ſi ſola animal haberet tam longe cæteris omnibus <lb/>animalibus, nedum ſtirpibus præcellens; </s> <s xml:id="echoid-s8640" xml:space="preserve">in quo ineſt divi-<lb/>num quiddam; </s> <s xml:id="echoid-s8641" xml:space="preserve">quo cognoſcit, intelligit, res innumeras <lb/>memoria complectitur, veri expendendi judicandique capax <lb/>eſt, cujus denique gratia quicquid terra progenerat paratum <lb/>eſſe videtur. </s> <s xml:id="echoid-s8642" xml:space="preserve">Omnia enim in uſus ſuos vertit. </s> <s xml:id="echoid-s8643" xml:space="preserve">Lignis, la-<lb/>pidibus, metallis, domos exſtruit; </s> <s xml:id="echoid-s8644" xml:space="preserve">Avibus, piſcibus, peco-<lb/>re & </s> <s xml:id="echoid-s8645" xml:space="preserve">herbis veſcitur; </s> <s xml:id="echoid-s8646" xml:space="preserve">Aquæ & </s> <s xml:id="echoid-s8647" xml:space="preserve">ventorum commodis ad na-<lb/>vigandum utitur; </s> <s xml:id="echoid-s8648" xml:space="preserve">ex florum odore pulchriſque coloribus <pb o="662" file="0402" n="421" rhead="CHRISTIANI HUGENII"/> voluptatem percipit. </s> <s xml:id="echoid-s8649" xml:space="preserve">Si nullum in Planetis eſt ejuſmodi ani-<lb/>mal, quid eſſe queat, quod tanti æſtimandum ſit, quove is <lb/>defectus penſetur? </s> <s xml:id="echoid-s8650" xml:space="preserve">Pone in Jove majorem multo animantium <lb/>varietatem; </s> <s xml:id="echoid-s8651" xml:space="preserve">plures arbores, herbas, metalla: </s> <s xml:id="echoid-s8652" xml:space="preserve">nihil erit in <lb/>omnibus his, ob quæ tantum dignitatis accedat iſti mundo, <lb/>ac noſtro propter humani ingenii mirabilem naturam. </s> <s xml:id="echoid-s8653" xml:space="preserve">Hic <lb/>ſi me judicium fallit, fateor me pretia rerum æſtimare ne-<lb/>ſcire.</s> <s xml:id="echoid-s8654" xml:space="preserve"/> </p> <div xml:id="echoid-div413" type="float" level="2" n="24"> <note position="right" xlink:label="note-0401-01" xlink:href="note-0401-01a" xml:space="preserve">In Planetis <lb/>eſſe animan-<lb/>tia, quæ ra-<lb/>tione utan-<lb/>tur.</note> </div> <p> <s xml:id="echoid-s8655" xml:space="preserve">Nec dicat aliquis, tantum malorum ac vitiorum eidem <lb/>humano generi ineſſe, ut merito dubitari poſſit, an, tale <lb/>quodpiam animal Planetariis mundis tribuendo, dignitas iis <lb/> <anchor type="note" xlink:label="note-0402-01a" xlink:href="note-0402-01"/> ornamentumque, an his contraria acceſſura ſint. </s> <s xml:id="echoid-s8656" xml:space="preserve">Primum <lb/>namque non impediunt vitia, majori hominum parti<unsure/> inſita, <lb/>quin ii qui virtutem, ac rectum rationis uſum ſectantur, tan-<lb/>quam pulcherrimum quid præſtantiſſimumque cenſendi ſint. <lb/></s> <s xml:id="echoid-s8657" xml:space="preserve">Præterea credibile eſt, ipſa illa animi vitia, magnæ homi-<lb/>num parti, non ſine ſummo conſilio data eſſe. </s> <s xml:id="echoid-s8658" xml:space="preserve">Cum enim <lb/>Dei voluntate ac providentia talis ſit Tellus, ejuſque inco-<lb/>læ, quales cernimus; </s> <s xml:id="echoid-s8659" xml:space="preserve">abſurdum enim foret exiſtimare omnia <lb/>hæc alia facta eſſe, quam ille voluerit, ſciveritque futura; </s> <s xml:id="echoid-s8660" xml:space="preserve"><lb/>putandum eſt utique non fruſtra multiplicem adeo animo-<lb/>rum diverſitatem mortalibus eſſe inſitam; </s> <s xml:id="echoid-s8661" xml:space="preserve">ſed malorum cum <lb/>bonis miſturam, quæque inde eveniunt infortunia, bella, <lb/>calamitates, eo fine accedere, ut neceſſitate urgente ſtimu-<lb/>loſque admovente, ingenia excitentur, exerceanturque, dum <lb/>quærimus ea quibus ab hoſtibus nos tutemur, quibuſve ma-<lb/>chinis teliſque eos perſequamur: </s> <s xml:id="echoid-s8662" xml:space="preserve">Utque paupertatem ac mi-<lb/>ſeriam depellere conantes, varias artes exquiramus, natu-<lb/>ramque ſcrutemur, ex cujus cognitione deinde auctoris po-<lb/>tentiam prudentiamque admirari neceſſe ſit; </s> <s xml:id="echoid-s8663" xml:space="preserve">quas forſan alias <lb/>pari ſtupore ac beſtiæ præteriiſſemus. </s> <s xml:id="echoid-s8664" xml:space="preserve">Nec enim dubitan-<lb/>dum eſt, ſi in continua pace, omniumque rerum affluentia <lb/>homines ætatem agerent, fieri poſſe ut admodum diu, non <lb/>aliter fere quam bruta animalia, victuri ſint; </s> <s xml:id="echoid-s8665" xml:space="preserve">omnis ſcientiæ <lb/>expertes, pluriumque commodorum ignari, quibus melius <lb/>jucundiuſque vita tranſigitur. </s> <s xml:id="echoid-s8666" xml:space="preserve">Careremus mirifica illa ſcri- <pb o="663" file="0403" n="422" rhead="COSMOTHEOROS, LIB. I."/> bendi arte, niſi ſumma in commerciis belliſque neceſſi@as <lb/>eam extudiſſet. </s> <s xml:id="echoid-s8667" xml:space="preserve">Huic artem navigandi, huic ſerendi debe-<lb/>mus, maximamque partem cæterorum quibus fruimur inven-<lb/>torum; </s> <s xml:id="echoid-s8668" xml:space="preserve">itemque naturæ arcana fere omnia, inter experien-<lb/>dum reperta. </s> <s xml:id="echoid-s8669" xml:space="preserve">Ita ea ipſa propter quæ incuſanda rationis fa-<lb/>cultas videbatur, poſſunt dici ad perficiendam exacuendamque <lb/>eam plurimum prodeſſe. </s> <s xml:id="echoid-s8670" xml:space="preserve">Nam & </s> <s xml:id="echoid-s8671" xml:space="preserve">virtutes ipſæ, fortitudo <lb/>animi, & </s> <s xml:id="echoid-s8672" xml:space="preserve">conſtantia, vix aliter quam in periculis rebuſque <lb/>adverſis apparere poſſunt.</s> <s xml:id="echoid-s8673" xml:space="preserve"/> </p> <div xml:id="echoid-div414" type="float" level="2" n="25"> <note position="left" xlink:label="note-0402-01" xlink:href="note-0402-01a" xml:space="preserve">Non obſtare <lb/>hominum <lb/>vitia quo mi-<lb/>nus decorem <lb/>terræ conci-<lb/>lient.</note> </div> <p> <s xml:id="echoid-s8674" xml:space="preserve">Quod ſi igitur genus animalium rationabile in cæteris Pla-<lb/>netis eſſe cogitemus, quod virtutibus vitiiſque fere iiſdem <lb/>atque homines præditum ſit, id tanti eſſe exiſtimandum eſt, <lb/>ut abſque iis, longè quàm Tellus hæc noſtra viliores futuri <lb/>ſint.</s> <s xml:id="echoid-s8675" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8676" xml:space="preserve">Poſitis vero ejuſmodi Planetarum incolis ratione utenti-<lb/> <anchor type="note" xlink:label="note-0403-01a" xlink:href="note-0403-01"/> bus, quæri adhuc poteſt, anne idem illic, atque apud nos, <lb/>ſit hoc quod rationem vocamus. </s> <s xml:id="echoid-s8677" xml:space="preserve">Quod quidem ita eſſe <lb/>omnino dicendum videtur, neque aliter fieri poſſe; </s> <s xml:id="echoid-s8678" xml:space="preserve">ſive <lb/>uſum rationis in his conſideremus quæ ad mores & </s> <s xml:id="echoid-s8679" xml:space="preserve">æquita-<lb/>tem pertinent, ſive in iis quæ ſpectant ad principia & </s> <s xml:id="echoid-s8680" xml:space="preserve">fun-<lb/>damenta ſcientiarum. </s> <s xml:id="echoid-s8681" xml:space="preserve">Etenim ratio apud nos eſt, quæ ſen-<lb/>ſum juſtitiæ, honeſti, laudis, clementiæ, gratitudinis inge-<lb/>nerat, mala ac bona in univerſum diſcernere docet: </s> <s xml:id="echoid-s8682" xml:space="preserve">quæque <lb/>ad hæc animum diſciplinæ, multorumque inventorum capa-<lb/>cem reddit. </s> <s xml:id="echoid-s8683" xml:space="preserve">Exſtaretne alibi diverſa ab hac ratio? </s> <s xml:id="echoid-s8684" xml:space="preserve">cenſere-<lb/>turque injuſtum aut ſceleſtum in Jove aut Marte, quod apud <lb/>nos juſtum ac præclarum habetur? </s> <s xml:id="echoid-s8685" xml:space="preserve">Certè nec veri ſimile eſt, <lb/>nec omnino poſſibile. </s> <s xml:id="echoid-s8686" xml:space="preserve">Cum enim rationis, qualem hic <lb/>agnoſcimus, ductu opus ſit ad tuendam vitam ac ſocietatem <lb/>(nam & </s> <s xml:id="echoid-s8687" xml:space="preserve">hanc apud Planeticolas reperiri oſtendemus) ſi con-<lb/>traria ejus decretis ſtatuantur, ſequetur ruina ac ſubverſio eo-<lb/>rum, quibus ejuſmodi mens perverſa contigiſſet. </s> <s xml:id="echoid-s8688" xml:space="preserve">At con-<lb/>ſervati<unsure/>o, ut videmus, rerum conditori ubique propoſita eſt. <lb/></s> <s xml:id="echoid-s8689" xml:space="preserve">Verum utur affectiones animi à nobis aliquatenus diverſæ <lb/>ſint@ apud iſtos longinquarum terrarum habitatores, puta in <lb/>his quæ ad amicitiam, iram, odium, honeſtatem, verecun- <pb o="664" file="0404" n="423" rhead="CHRISTIANI HUGENII"/> diam, decorem attinent; </s> <s xml:id="echoid-s8690" xml:space="preserve">non tamen dubitari poteſt, quin <lb/>in veri inveſtigandi ſtudio, judicandis rationum conſequen-<lb/>tiis, ac præſertim in ratiociniis, quæ ad quantitatem ac ma-<lb/>gnitudinem ſpectant, circa quæ Geometria verſatur, (ſi quid <lb/>habent ejuſmodi, quod mox inquiremus) non, inquam, <lb/>dubitari poteſt, quin prorſus ſimilis ſit, eademque via in-<lb/>grediatur illorum ac noſtra ratio; </s> <s xml:id="echoid-s8691" xml:space="preserve">quodque apud nos verum <lb/>eſt, idem ſit in cæteris Planetis. </s> <s xml:id="echoid-s8692" xml:space="preserve">Etſi vis ac facultas in his <lb/>rebus major minorve illorum incolis fortaſſe quam nobis <lb/>contigerit.</s> <s xml:id="echoid-s8693" xml:space="preserve"/> </p> <div xml:id="echoid-div415" type="float" level="2" n="26"> <note position="right" xlink:label="note-0403-01" xlink:href="note-0403-01a" xml:space="preserve">Nec ratio-<lb/>nem in Pla-<lb/>netarum in-<lb/>colis à noſtr@ <lb/>diverſam <lb/>eſſe.</note> </div> <p> <s xml:id="echoid-s8694" xml:space="preserve">Sed jam nimis longè provectum me eſſe ſentio. </s> <s xml:id="echoid-s8695" xml:space="preserve">Ante <lb/>enim diſpiciendum erat de ſenſibus corporeis iſtorum in Pla-<lb/>netis agentium, quibus ſi carerent, vix jam aut vitam, ut <lb/>animalia, ſortiti eſſe videri poſſint, aut habere, in quo ra-<lb/>tionis uſum exerceant. </s> <s xml:id="echoid-s8696" xml:space="preserve">Puto autem oſtendi poſſe probabili-<lb/>bus argumentis & </s> <s xml:id="echoid-s8697" xml:space="preserve">bruta animantia, & </s> <s xml:id="echoid-s8698" xml:space="preserve">quibus ratio ineſt, <lb/>convenire, in his quæ ad ſenſus attinent, cum iis quæ ter-<lb/>ram hanc incolunt. </s> <s xml:id="echoid-s8699" xml:space="preserve">Primùm namque ſi cogitemus quid ſit <lb/> <anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/> in animalibus videndi poteſtas, abſque qua neque paſcendi <lb/>ratio eſſet, nec pericula vitandi; </s> <s xml:id="echoid-s8700" xml:space="preserve">nec denique vita alia quam <lb/>talparum aut lumbricorum; </s> <s xml:id="echoid-s8701" xml:space="preserve">prorſus neceſſe eſſe intelligemus <lb/>ut, ubi ſunt animalia his præſtantiora, ibi & </s> <s xml:id="echoid-s8702" xml:space="preserve">viſu prædita <lb/>ſint. </s> <s xml:id="echoid-s8703" xml:space="preserve">Cum nihil ad vitam vel conſervandam, vel exornan-<lb/>dam æque conducat. </s> <s xml:id="echoid-s8704" xml:space="preserve">Quod ſi vero inſpiciamus mirabilem <lb/>lucis naturam, ſtupendumque artificium, quo ad eam fruen-<lb/>dam oculi comparati ſunt, facile cognoſcemus, perceptionem <lb/>rerum procul diſtantium, cum circumſcriptione formarum, diſ-<lb/> <anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/> crimen intervallorum, non alio modo, quam qui ex viſu ſit, <lb/>inſtitui poſſe. </s> <s xml:id="echoid-s8705" xml:space="preserve">Non enim poteſt hic ſenſus, imo nec alius quiſ-<lb/>quam eorum quos novimus, exiſtere, quam ex motu extrinſecus <lb/>adveniente. </s> <s xml:id="echoid-s8706" xml:space="preserve">Qui motus, ut alibi explicuimus, in efficiendo vi-<lb/>ſu, à Sole proficiſcitur, aut ſtellis inerrantibus, aut igne; </s> <s xml:id="echoid-s8707" xml:space="preserve">quo-<lb/>rum particulæ celerrima agitatione concitæ, circumfuſam cæ-<lb/>leſtem materiam continue pulſant, impelluntque; </s> <s xml:id="echoid-s8708" xml:space="preserve">qui impulſus <lb/>a proximis ad longe diſſitas citiſſime propagetur, fere eo modo <lb/>quo ſonus per aërem. </s> <s xml:id="echoid-s8709" xml:space="preserve">Abſque hoc motu, materiaque æthe- <pb o="665" file="0405" n="424" rhead="COSMOTHEOROS, LIB. I."/> ris qui intermedia cæli ſpatia complet, nec Solem nec ſtellas <lb/>cernere poſſemus; </s> <s xml:id="echoid-s8710" xml:space="preserve">neque etiam alia quæ propiora ſunt cor-<lb/>pora; </s> <s xml:id="echoid-s8711" xml:space="preserve">cum ab his ad nos idem ille motus repercuſſus perve-<lb/>nire debeat. </s> <s xml:id="echoid-s8712" xml:space="preserve">Hic, oculorum ſenſu perceptus, lux appella-<lb/>tur. </s> <s xml:id="echoid-s8713" xml:space="preserve">Inque eo ſenſu mirabile eſt ante omnia, quo pacto ad <lb/>tantam ſubtilitatem perduci potuerit, ut minimâ cæleſtis ma-<lb/>teriæ commotiunculâ afficeretur, ſimulque qua ex parte illa <lb/>oriretur perciperet. </s> <s xml:id="echoid-s8714" xml:space="preserve">Tum quomodo nihil ſeſe mutuo impe-<lb/>diant innumeri ejuſmodi pulſuum proceſſus, ſphæricæque <lb/>ſuperficies, aliæ alias trajicientes. </s> <s xml:id="echoid-s8715" xml:space="preserve">Hæc omnia tam mira ac <lb/>ſubtili ratione conſtituta ſunt, ut nec minimam eorum par-<lb/>tem hominum ingenia excogitare potuiſſent, cum vix etiam <lb/>quomodo ſeſe habeant comprehendere queant. </s> <s xml:id="echoid-s8716" xml:space="preserve">Quid enim <lb/>tam mirabile, quam particulam corporis quandam ita fabri-<lb/>catam eſſe, ut ejus opera animal ſentiat procul poſitorum <lb/>corporum figuram, poſitum, motum quemlibet, diſtantiam; <lb/></s> <s xml:id="echoid-s8717" xml:space="preserve">idque etiam cum colorum varietate, quo diſtinctius ea digno-<lb/>ſceret. </s> <s xml:id="echoid-s8718" xml:space="preserve">Oculi vero præter hæc artificioſiſſima conſtructio, <lb/>quæ perfectam rerum extra poſitarum picturam in cava cho-<lb/>roidis ſuperficie imprimere apta eſt, omnem profecto admi-<lb/>rationem ſuperat, neque eſt in quo manifeſtius Geometriæ <lb/>artem Deus exercuerit. </s> <s xml:id="echoid-s8719" xml:space="preserve">Atque hæc non tantum ſolertia ſum-<lb/>ma inventa & </s> <s xml:id="echoid-s8720" xml:space="preserve">fabricata ſunt, ſed & </s> <s xml:id="echoid-s8721" xml:space="preserve">videntur eſſe ejuſmodi, <lb/>ſi quis propius attendat, ut non alia ratione perfici potue-<lb/>rint quàm hac quam cernimus. </s> <s xml:id="echoid-s8722" xml:space="preserve">Nam neque lux aliter, quàm <lb/>communicato motu per materiam cæleſtem, res longo inter-<lb/>vallo remotas ſenſibus noſtris offerre poterat; </s> <s xml:id="echoid-s8723" xml:space="preserve">nec oculo-<lb/>rum artificio ullum aliud par dari ad diſtinctè referendas <lb/>rerum imagines. </s> <s xml:id="echoid-s8724" xml:space="preserve">Ut valde eos falli arbitrer, ſiqui hæc ea-<lb/>dem multis modis ordinari potuiſſe contendere audeant. </s> <s xml:id="echoid-s8725" xml:space="preserve"><lb/>Quare omnino credibile eſt utrumque iſtud eodem modo ſe <lb/>habere in Planetarum regionibus atque hic; </s> <s xml:id="echoid-s8726" xml:space="preserve">neque aliam eſſe <lb/>iis, quæ illic habitant animantibus, videndi rationem. </s> <s xml:id="echoid-s8727" xml:space="preserve">Ha-<lb/>bebunt igitur oculos; </s> <s xml:id="echoid-s8728" xml:space="preserve">atque etiam binos minimum, quò poſ-<lb/>ſint rerum ante pedes poſitarum diſtantias percipere, ſine <lb/>quo vix tutò ingredi licet. </s> <s xml:id="echoid-s8729" xml:space="preserve">Et hæc quidem ad vitæ uſum ne- <pb o="666" file="0406" n="425" rhead="CHISTIANI HUGENII"/> ceſſariò tribuenda ſunt animantibus Planetarum univerſis fe-<lb/>re. </s> <s xml:id="echoid-s8730" xml:space="preserve">Quæ vero ratione & </s> <s xml:id="echoid-s8731" xml:space="preserve">mente prædita ſunt, cum alias <lb/>quoque ex viſu utilitates capere poſſint, tantò magis con-<lb/>ſentaneum eſt ut tam præclaro munere donata ſint. </s> <s xml:id="echoid-s8732" xml:space="preserve">Nos <lb/>enim colorum pulchritudinem, formarum elegantiam, ac <lb/>concinnitatem viſu percipimus; </s> <s xml:id="echoid-s8733" xml:space="preserve">legimus, ſcribimus, cælum <lb/>& </s> <s xml:id="echoid-s8734" xml:space="preserve">aſtra contemplamur, eorumque curſus, magnitudineſque <lb/>metimur; </s> <s xml:id="echoid-s8735" xml:space="preserve">quæ quatenus ad Planetarum incolas quoque per-<lb/>tineant, paulo poſt videbimus. </s> <s xml:id="echoid-s8736" xml:space="preserve">Nunc illud prius quæra-<lb/>mus an cæteros quoque ſenſus noſtros iis contigiſſe veriſi-<lb/> <anchor type="note" xlink:label="note-0406-01a" xlink:href="note-0406-01"/> mile ſit. </s> <s xml:id="echoid-s8737" xml:space="preserve">Ac de auditu quidem multa ſuadent, ut cunctis, <lb/>quæ illic ſunt, animalibus eum ineſſe credamus. </s> <s xml:id="echoid-s8738" xml:space="preserve">Prodeſt <lb/>enim plurimum ad vitam à periculis tutandam; </s> <s xml:id="echoid-s8739" xml:space="preserve">cum ſonitu <lb/>ac fragore ſæpe imminens infortunium cognoſcatur; </s> <s xml:id="echoid-s8740" xml:space="preserve">præſer-<lb/>tim noctu atque in tenebris, cum oculorum auxilium ere-<lb/>ptum eſt. </s> <s xml:id="echoid-s8741" xml:space="preserve">Videmus præterea ut animalia pleraque vocis ſo-<lb/>no ſui ſimilia advocent, multaque inter ſe ſignificent, nobis <lb/>quidem parum intellecta, ſed plura fortaſſe quam putamus. <lb/></s> <s xml:id="echoid-s8742" xml:space="preserve">A pud ea vero quæ ratione utuntur, ſi cogitemus quam mi-<lb/>rabilis ſit vocis & </s> <s xml:id="echoid-s8743" xml:space="preserve">auditus oportunitas, vix credibile videbi-<lb/>tur tam utilem ſenſum, tantumque loquendi artificium, hu-<lb/>jus Terræ noſtræ, ac noſtri tantum cauſa fuiſſe inventum. </s> <s xml:id="echoid-s8744" xml:space="preserve"><lb/>Quomodo enim illis non multum deſit ad vitæ commoda, <lb/>& </s> <s xml:id="echoid-s8745" xml:space="preserve">felicitatem noſtræ ſimilem, qui tanto beneficio carent: </s> <s xml:id="echoid-s8746" xml:space="preserve"><lb/>aut quanam alia re penſari hoc poſſit? </s> <s xml:id="echoid-s8747" xml:space="preserve">Quod ſi porro conſi-<lb/>deremus, quam pulchre, quamque induſtrie natura hoc ef-<lb/>fecerit, ut idem ille aër, cujus reſpiratione vivimus, cujus <lb/>flatu navigamus, qui, ut volare queant, avibus præſtat; </s> <s xml:id="echoid-s8748" xml:space="preserve">ut, <lb/>inquam, idem ille ad exprimendum proferendumque ſonum <lb/>comparatus ſit; </s> <s xml:id="echoid-s8749" xml:space="preserve">ſonus verò ad formandum, auribuſque in-<lb/>gerendum ſermonem; </s> <s xml:id="echoid-s8750" xml:space="preserve">vix credemus inſignem hunc aëris <lb/>uſum, in terris iſtis longinquis eam neglexiſſe? </s> <s xml:id="echoid-s8751" xml:space="preserve">Eſſe enim <lb/> <anchor type="note" xlink:label="note-0406-02a" xlink:href="note-0406-02"/> illic aërem qui terris incumbat, vix dubitari poteſt, cum nu-<lb/>bes in Jove apparere dixerimus. </s> <s xml:id="echoid-s8752" xml:space="preserve">Sicut enim hæ ex aquæ <lb/>guttulis minimis conſtant, ita ex particulis aquæ ſeorſim vo-<lb/>litantibus magna ex parte formatur aër ille qui propius ter- <pb o="667" file="0407" n="426" rhead="COSMOTHEOROS, LIB. I."/> ram circundat. </s> <s xml:id="echoid-s8753" xml:space="preserve">Quem Planetarum globis adeſſe etiam hoc <lb/>ſuadet, quod reſpirandi ratio, qua vita ſuſtentatur omnium <lb/>quæ hic habemus animantium, videtur omnino ex univerſa-<lb/>lioribus illis naturæ inſtitutis eſſe, velut nutriri ex fructibus <lb/>terræ.</s> <s xml:id="echoid-s8754" xml:space="preserve"/> </p> <div xml:id="echoid-div416" type="float" level="2" n="27"> <note position="left" xlink:label="note-0404-01" xlink:href="note-0404-01a" xml:space="preserve">Nec deeſſe <lb/>@ll@s ſenſus.</note> <note position="left" xlink:label="note-0404-02" xlink:href="note-0404-02a" xml:space="preserve">Nec viſum.</note> <note position="left" xlink:label="note-0406-01" xlink:href="note-0406-01a" xml:space="preserve">Non audi-<lb/>tum,</note> <note position="left" xlink:label="note-0406-02" xlink:href="note-0406-02a" xml:space="preserve">Nec per quem <lb/>ſonus perfe-<lb/>ratur aërem,</note> </div> <p> <s xml:id="echoid-s8755" xml:space="preserve">De ſenſibus autem reliquis animalium ut dicere pergam, <lb/> <anchor type="note" xlink:label="note-0407-01a" xlink:href="note-0407-01"/> eum ſanè qui ex tactu oritur, neceſſitate ſumma datum eſſe <lb/>apparet omnibus iis quæ molli flexilique pelle teguntur, quò <lb/>à lædentibus caveant refugiantque; </s> <s xml:id="echoid-s8756" xml:space="preserve">cum abſque eo vulnera, <lb/>plagas, contuſioneſque crebras acceptura fuerint. </s> <s xml:id="echoid-s8757" xml:space="preserve">In quo <lb/>tam provida natura fuit, ut, ne minimam quidem peilis par-<lb/>ticulam, doloris ſenſu vacare voluerit. </s> <s xml:id="echoid-s8758" xml:space="preserve">Itaque hanc faculta-<lb/>tem, tam neceſſariam ad conſervandam animalium incolumi-<lb/>tatem, omnino credibile eſt etiam planetas inhabitantibus <lb/>inditam eſſe.</s> <s xml:id="echoid-s8759" xml:space="preserve"/> </p> <div xml:id="echoid-div417" type="float" level="2" n="28"> <note position="right" xlink:label="note-0407-01" xlink:href="note-0407-01a" xml:space="preserve">Net tactum,</note> </div> <p> <s xml:id="echoid-s8760" xml:space="preserve">Odoratum vero ac guſtum quis non videt noceſſaria eſſe <lb/> <anchor type="note" xlink:label="note-0407-02a" xlink:href="note-0407-02"/> paſcentibus, quo conducibilia a noxiis, nihilve profuturis <lb/>dignoſcant. </s> <s xml:id="echoid-s8761" xml:space="preserve">Itaque ſi herbis, ſeminibus, aut fortaſſe carni-<lb/>bus quoque in regionibus iſtis animalia alantur; </s> <s xml:id="echoid-s8762" xml:space="preserve">etiam his <lb/>ſenſibus, tam ad cavendum, appetendumque neceſlariis, cre-<lb/>dibile eſt ea non deſtitui.</s> <s xml:id="echoid-s8763" xml:space="preserve"/> </p> <div xml:id="echoid-div418" type="float" level="2" n="29"> <note position="right" xlink:label="note-0407-02" xlink:href="note-0407-02a" xml:space="preserve">Nec odora-<lb/>tum, nec <lb/>guſtum.</note> </div> <p> <s xml:id="echoid-s8764" xml:space="preserve">Scio à nonnullis fuiſſe quæſitum, an non alii præter eos <lb/>quinque quos diximus, naturâ dari potuerint. </s> <s xml:id="echoid-s8765" xml:space="preserve">Quod qui-<lb/>dem ſi concedatur, forſan dubitandum ſit animalium plane-<lb/>tariorum ſenſus longè alios eſſe ac noſtratium. </s> <s xml:id="echoid-s8766" xml:space="preserve">Nec ſanè ob-<lb/>ſtare quidquam videtur quo minus alii extare poſſint perci-<lb/>piendi modi: </s> <s xml:id="echoid-s8767" xml:space="preserve">attamen cum perpendimus ad quos vitæ uſus <lb/> <anchor type="note" xlink:label="note-0407-03a" xlink:href="note-0407-03"/> unuſquiſque eorum, quos habemus, comparati ſint; </s> <s xml:id="echoid-s8768" xml:space="preserve">non <lb/>videtur ſaltem alius quiſquam neceſſarius adjungi potuiſſe. <lb/></s> <s xml:id="echoid-s8769" xml:space="preserve">Nempe effecit providentia ut & </s> <s xml:id="echoid-s8770" xml:space="preserve">propinqua, & </s> <s xml:id="echoid-s8771" xml:space="preserve">longius re-<lb/>mota, qualia eſſent oculis ſentiremus. </s> <s xml:id="echoid-s8772" xml:space="preserve">Rurſus ut non viſa, <lb/>ſive a tergo, ſive in tenebris, auditus exciperet. </s> <s xml:id="echoid-s8773" xml:space="preserve">Item ut <lb/>quæ nec oculi nec aures adeſſe nunciarent, alius tamen ſen-<lb/>ſus qui in naribus eſt præſentiret, idque in canibus mirabili <lb/>ut ſcimus ſubtilitate. </s> <s xml:id="echoid-s8774" xml:space="preserve">Poſtremo effecit ut quæ quatuor iſtos <lb/>ſenſus effugerent, quò minus in corpus impacta nocere poſ- <pb o="668" file="0408" n="427" rhead="CRISTIANI HUGENII"/> ſint, tactu perciperentur. </s> <s xml:id="echoid-s8775" xml:space="preserve">Ita omnibus modis ſaluti conſer-<lb/>vationique animalium conſuluit, nec quidquam amplius addi <lb/>aut deſiderari poſſe videtur; </s> <s xml:id="echoid-s8776" xml:space="preserve">ut proinde planetarum incolis <lb/>vix aliud niſi ſuperfluum largitura fuerit.</s> <s xml:id="echoid-s8777" xml:space="preserve"/> </p> <div xml:id="echoid-div419" type="float" level="2" n="30"> <note position="right" xlink:label="note-0407-03" xlink:href="note-0407-03a" xml:space="preserve">Nec horum <lb/>ſenſus longe <lb/>alios eſſe ac <lb/>noſtratium.</note> </div> <p> <s xml:id="echoid-s8778" xml:space="preserve">Cum autem ex ſingulis ſenſibus, præter utilitatem, volu-<lb/>ptas aliqua ad homines perveniat; </s> <s xml:id="echoid-s8779" xml:space="preserve">velut ex guſtatu in cibis; <lb/></s> <s xml:id="echoid-s8780" xml:space="preserve">ex odoratu in floribus & </s> <s xml:id="echoid-s8781" xml:space="preserve">aromatis; </s> <s xml:id="echoid-s8782" xml:space="preserve">ex viſu in contemplanda <lb/>pulchritudine formarum, & </s> <s xml:id="echoid-s8783" xml:space="preserve">colorum; </s> <s xml:id="echoid-s8784" xml:space="preserve">ex auditu harmonico-<lb/>rum ſonorum; </s> <s xml:id="echoid-s8785" xml:space="preserve">ex tactu in rebus venereis, (niſi peculiaris <lb/>quidam ſenſus hic dicendus eſt) animalibus verò cæteris ex <lb/>quibuſdam horum; </s> <s xml:id="echoid-s8786" xml:space="preserve">nonne dicemus hæc naturæ munera fere <lb/>eodem modo reliquorum Planetarum incolis diſtributa eſſe. </s> <s xml:id="echoid-s8787" xml:space="preserve"><lb/>Certe id quidem ratio poſtulare videtur. </s> <s xml:id="echoid-s8788" xml:space="preserve">Sive enim cogite-<lb/> <anchor type="note" xlink:label="note-0408-01a" xlink:href="note-0408-01"/> mus, quanto in univerſum, propter hæc, jucundior feli-<lb/>ciorque vita reddatur, non debemus maximum ejus bonum <lb/>noſtræ Telluris habitatoribus aſcribere, cæteras tenentibus <lb/>denegare, quaſi res noſtræ rebus illorum multò præferendæ <lb/>ſint. </s> <s xml:id="echoid-s8789" xml:space="preserve">Sive ad voluptates, quæ in cibis capiendis, & </s> <s xml:id="echoid-s8790" xml:space="preserve">in con-<lb/>junctione utriuſque ſexus contingunt, attendamus; </s> <s xml:id="echoid-s8791" xml:space="preserve">intellige-<lb/>mus hæc eſſe neceſſaria quædam veluti providæ naturæ juſſa, <lb/>tacitè cogentis ad conſervandum, propagandumque animan-<lb/>tium genus: </s> <s xml:id="echoid-s8792" xml:space="preserve">vel etiam, in beſtiis quidem, fortaſſe genus <lb/>ipſum propagari, ut utraque illa jucunditate fruatur, ut <lb/>proinde, utroque nomine, in cæteris Planetis eadem repe-<lb/>riri conſentaneum ſit. </s> <s xml:id="echoid-s8793" xml:space="preserve">Equidem cum hæc omnia quanti ſint, <lb/>quantamque utilitatem habeant, conſidero; </s> <s xml:id="echoid-s8794" xml:space="preserve">quamque admi-<lb/>rabile ſit, tale quid, quale eſt voluptas, in rerum natura <lb/>exiſtere; </s> <s xml:id="echoid-s8795" xml:space="preserve">omnino adducor ut credam, non ſoli Telluri no-<lb/>ſtræ, quæ de minoribus planetis unus eſt, rem tantam ob-<lb/>tigiſſe. </s> <s xml:id="echoid-s8796" xml:space="preserve">Et hæc quidem de voluptatibus iis quæ ſenſus cor-<lb/>poreos afficiunt, rationis facultatem aut nihil, aut leviter <lb/>tantùm. </s> <s xml:id="echoid-s8797" xml:space="preserve">Sunt autem homini, præter iſtas, aliæ quoque; <lb/></s> <s xml:id="echoid-s8798" xml:space="preserve">quæ mente tantum, & </s> <s xml:id="echoid-s8799" xml:space="preserve">rationis ſenſu percipiuntur; </s> <s xml:id="echoid-s8800" xml:space="preserve">aliæ cum <lb/>lætitia conjunctæ; </s> <s xml:id="echoid-s8801" xml:space="preserve">aliæ ſeriæ, neque ideo minoris faciendæ; </s> <s xml:id="echoid-s8802" xml:space="preserve"><lb/>velut quæ ex oblectatione ſcientiarum, inventorum, veri-<lb/>que cognitione oriuntur; </s> <s xml:id="echoid-s8803" xml:space="preserve">de quibus omnibus, an ad alio- <pb o="669" file="0409" n="428" rhead="COSMOTHEOROS, LIB. I."/> rum quoque planetarum incolas pertineant, in ſequentibus <lb/>dicendi locus erit.</s> <s xml:id="echoid-s8804" xml:space="preserve"/> </p> <div xml:id="echoid-div420" type="float" level="2" n="31"> <note position="left" xlink:label="note-0408-01" xlink:href="note-0408-01a" xml:space="preserve">Ut nec volu-<lb/>ptatem ex iis <lb/>@rtam.</note> </div> <p> <s xml:id="echoid-s8805" xml:space="preserve">Superſunt alia nunc expendenda quæ in terris illis ſimilia <lb/>eſſe rebus noſtris veriſimile ſit. </s> <s xml:id="echoid-s8806" xml:space="preserve">De Elementis terræ, aëris, <lb/>& </s> <s xml:id="echoid-s8807" xml:space="preserve">aquæ, vidimus jam quàm probabile ſit ea in Planetis cæ-<lb/>teris non deeſſe. </s> <s xml:id="echoid-s8808" xml:space="preserve">Videamus & </s> <s xml:id="echoid-s8809" xml:space="preserve">de igne, qui apud nos qui-<lb/>dem non tam Elementum eſſe dicendus eſt, quam motus <lb/>quidam concitatiſſimus particularum à certis corporibus ab-<lb/>reptarum. </s> <s xml:id="echoid-s8810" xml:space="preserve">Hoc verò, quidquid eſt, etiam Planetarum in-<lb/>colis datum eſſe, multa ſunt quæ veriſimiliter probent. </s> <s xml:id="echoid-s8811" xml:space="preserve">Pri-<lb/> <anchor type="note" xlink:label="note-0409-01a" xlink:href="note-0409-01"/> mum quòd non tam in Terra hac, quàm in Sole, ignis ſe-<lb/>des collocata videatur; </s> <s xml:id="echoid-s8812" xml:space="preserve">ac ſicut, calore Solis, herbæ & </s> <s xml:id="echoid-s8813" xml:space="preserve">ani-<lb/>mantia hic creſcunt ac foventur, ita quoque haud dubie in <lb/>cæteris fiat Planetis. </s> <s xml:id="echoid-s8814" xml:space="preserve">Cum autem intenſior calor ignem ge-<lb/>neret, credibile eſt illic quoque, ac præſertim in Soli pro-<lb/>pinquioribus, eoſdem aut majoris caloris gradus exiſtere, <lb/>eorumque vi ignem. </s> <s xml:id="echoid-s8815" xml:space="preserve">Deinde videmus quam multis modis <lb/>excitetur, velut colligendis Solis radiis, repercuſſu pelvium <lb/>aut ſpeculorum; </s> <s xml:id="echoid-s8816" xml:space="preserve">ferri & </s> <s xml:id="echoid-s8817" xml:space="preserve">ſilicis colliſione; </s> <s xml:id="echoid-s8818" xml:space="preserve">lignorum attritu <lb/>mutuo; </s> <s xml:id="echoid-s8819" xml:space="preserve">herbæ non bene ſiccæ congeſtis acervis; </s> <s xml:id="echoid-s8820" xml:space="preserve">ex fulmi-<lb/>ne; </s> <s xml:id="echoid-s8821" xml:space="preserve">ex montium terræque ſulphureæ incendiis. </s> <s xml:id="echoid-s8822" xml:space="preserve">Quare mi-<lb/>rum eſſet, non aliquo ex iſtis omnibus, in Planetarum ter-<lb/>ris, eum accendi. </s> <s xml:id="echoid-s8823" xml:space="preserve">Cogitemus deinde quanta apud nos ſit <lb/>ignis utilitas, quantaque neceſſitas. </s> <s xml:id="echoid-s8824" xml:space="preserve">Hujus enim beneficio <lb/>frigoris incommoda depellimus in iis regionibus, ubi calor <lb/>Solis minus viget propter radiorum obliquitatem, atque ita <lb/>efficimus ne magna Terrarum pars inculta inhabitataque ma-<lb/>neat; </s> <s xml:id="echoid-s8825" xml:space="preserve">quod in omnibus Planetarum globis, ſive æſtatis hye-<lb/>miſque viciſſitudines ſentiant, ſive perpetuo fruantur æqui-<lb/>noctio, æque neceſſarium eſt remedium; </s> <s xml:id="echoid-s8826" xml:space="preserve">quoniam & </s> <s xml:id="echoid-s8827" xml:space="preserve">in his, <lb/>loca polis viciniora, parum juvari Solis calore certum eſt. <lb/></s> <s xml:id="echoid-s8828" xml:space="preserve">Eodem igne nocti lucem inducimus, diemque velut alterum <lb/>creamus, quo non parum temporis vitæ adjicitur. </s> <s xml:id="echoid-s8829" xml:space="preserve">Itaque <lb/>ob hæc omnia prorſus veriſimile eſt tanta re non ſolos Tel-<lb/>luris incolas frui, ſed omnibus Planetis communiter eſſe con-<lb/>ceſſam.</s> <s xml:id="echoid-s8830" xml:space="preserve"/> </p> <div xml:id="echoid-div421" type="float" level="2" n="32"> <note position="right" xlink:label="note-0409-01" xlink:href="note-0409-01a" xml:space="preserve">Ignem quo-<lb/>que planetis <lb/>communem <lb/>eſſe.</note> </div> <pb o="670" file="0410" n="429" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s8831" xml:space="preserve">Porro quæri poteſt de animalibus, tam ratione utentibus <lb/>quàm brutis; </s> <s xml:id="echoid-s8832" xml:space="preserve">atque etiam de ſtirpibus arboribuſque; </s> <s xml:id="echoid-s8833" xml:space="preserve">an, quæ <lb/>iſthic naſcuntur, noſtris magnitudine reſpondeant. </s> <s xml:id="echoid-s8834" xml:space="preserve">Nam <lb/> <anchor type="note" xlink:label="note-0410-01a" xlink:href="note-0410-01"/> ſi hæc ipſorum globorum mole natura metiatur, eſſent in <lb/>Jove ac Saturno animalia quædam decies aut quindecies al-<lb/>tiora Elephantis, aut tantundem longitudine balænas noſtras <lb/>ſuperantia. </s> <s xml:id="echoid-s8835" xml:space="preserve">Tum illa quæ ratione prædita ſunt, gigantum <lb/>corpora haberent noſtris comparata. </s> <s xml:id="echoid-s8836" xml:space="preserve">Qua quidem in re ni-<lb/>hil video quod vel mirum ſit, vel fieri nequeat. </s> <s xml:id="echoid-s8837" xml:space="preserve">Nulla ta-<lb/>men ratione cogimur ut re ipſa id ita eſſe credamus; </s> <s xml:id="echoid-s8838" xml:space="preserve">quan-<lb/>doquidem in multis rebus apparet non iis menſuræ regulis <lb/>naturam ſe obſtrinxiſſe quæ noſtra opinione convenientiores <lb/>videbantur. </s> <s xml:id="echoid-s8839" xml:space="preserve">Veluti quod ipſorum globorum Planetariorum <lb/>moles nequaquam pro diſtantia eorum a Sole conſtituta ſit, <lb/>cum Mars manifeſto minor ſit Venere, etſi remotior: </s> <s xml:id="echoid-s8840" xml:space="preserve">cum-<lb/>que converſio Jovis, ſuper axe ſuo, 10 horis peragatur; <lb/></s> <s xml:id="echoid-s8841" xml:space="preserve">Telluris vero, tantò minoris, impendat horas 24. </s> <s xml:id="echoid-s8842" xml:space="preserve">Poſſet <lb/>vero dubitari, cum proportionem in his ita negligat Natu-<lb/>ra, an non fortaſſe pumiliones quidam ſint incolæ Planeta-<lb/>rum, aut ranis muribuſve non majores. </s> <s xml:id="echoid-s8843" xml:space="preserve">Sed oſtendam poſt-<lb/>ea cur id nequaquam conſentaneum putandum ſit.</s> <s xml:id="echoid-s8844" xml:space="preserve"/> </p> <div xml:id="echoid-div422" type="float" level="2" n="33"> <note position="left" xlink:label="note-0410-01" xlink:href="note-0410-01a" xml:space="preserve">Magnitudi-<lb/>nem corpo-<lb/>rum in Pla-<lb/>netis exiſten-<lb/>tium ex Pla-<lb/>netarum <lb/>magnitudine <lb/>non recte <lb/>nonjici.</note> </div> <p> <s xml:id="echoid-s8845" xml:space="preserve">Aliud quoque dubium exoriri poſſet, utrum genus unum <lb/>tantum animalium quæ rationem ſortita ſint, an plura in <lb/>Planetis ſingulis reperiantur, & </s> <s xml:id="echoid-s8846" xml:space="preserve">num diſpari rationis vi. </s> <s xml:id="echoid-s8847" xml:space="preserve">Ac <lb/>profecto tale quid in Terra hac noſtra contigiſſe cernimus. <lb/></s> <s xml:id="echoid-s8848" xml:space="preserve">Non de iis nunc dico quæ figuram hominum præferunt; </s> <s xml:id="echoid-s8849" xml:space="preserve"><lb/>(etſi de his quoque id non abſurde dici poſſit) ſed ſi quo-<lb/>rundam è beſtiarum genere, ſenſum intellectumque ſpecte-<lb/>mus; </s> <s xml:id="echoid-s8850" xml:space="preserve">veluti canum, ſimiarum, caſtorum, elephantorum; </s> <s xml:id="echoid-s8851" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0410-02a" xlink:href="note-0410-02"/> imo & </s> <s xml:id="echoid-s8852" xml:space="preserve">avium quarundam, & </s> <s xml:id="echoid-s8853" xml:space="preserve">apicularum, ea talia ſunt, ut <lb/>nequaquam folum genus hominum rationis particeps dicen-<lb/>dum videatur. </s> <s xml:id="echoid-s8854" xml:space="preserve">Apparet enim quoddam hujus inſtar in iſtis <lb/>omnibus, quod, abſque ulla inſtitutione aut experientia, iis <lb/>ineſſe deprehenditur.</s> <s xml:id="echoid-s8855" xml:space="preserve"/> </p> <div xml:id="echoid-div423" type="float" level="2" n="34"> <note position="left" xlink:label="note-0410-02" xlink:href="note-0410-02a" xml:space="preserve">In<unsure/> Planetis <lb/>ut in Terra <lb/>varia eſſe <lb/>animalia <lb/>quibus ratio <lb/>competat.</note> </div> <note position="left" xml:space="preserve">Et inter ea <lb/>Hominibus <lb/>@milia.</note> <p> <s xml:id="echoid-s8856" xml:space="preserve">Attamen dubitari nequit quin longè præcellat hominum <lb/>intelligentia & </s> <s xml:id="echoid-s8857" xml:space="preserve">ingenium, quippe innumeris rebus aptum, <pb o="671" file="0411" n="430" rhead="COSMOTHEOROS, LIB. I."/> c<unsure/>onſilii ad futura capax, præteritorum memoria infinita præ-<lb/>ditum. </s> <s xml:id="echoid-s8858" xml:space="preserve">Quod ingens præſtantiæ diſcrimen ſi perpendamus, <lb/>credemus non ſine ratione, in cæteris quoque planetis, u-<lb/>num quoddam genus prætuliſſe naturam; </s> <s xml:id="echoid-s8859" xml:space="preserve">atque eo magis, <lb/>quod ſi plura forent eadem ingenii ſagacitate, poſſent noce-<lb/>re ſibi invicem, ac de poſſeſſionibus & </s> <s xml:id="echoid-s8860" xml:space="preserve">imperio inter ſe con-<lb/>tendere; </s> <s xml:id="echoid-s8861" xml:space="preserve">quod nunc quoque faciunt nimis frequenter, licet <lb/>unius generis ſint, quæ in Terra hac dominantur. </s> <s xml:id="echoid-s8862" xml:space="preserve">Verum <lb/>hæc utcunque ſe habeant, de iis nunc agamus terrarum iſta-<lb/>rum animalibus, quæ maximè cæteris ratione antecellunt; <lb/></s> <s xml:id="echoid-s8863" xml:space="preserve">quæramuſque an ſciri poſſit, quibus in rebus ejus uſum im-<lb/>pendant, & </s> <s xml:id="echoid-s8864" xml:space="preserve">an habeant etiam artes ſcientiaſque ſuas, velut <lb/>nos in hoc noſtro planeta. </s> <s xml:id="echoid-s8865" xml:space="preserve">Quod quidem, inter ea quæ ad <lb/>naturam eorum attinent, præcipuè expendi meretur. </s> <s xml:id="echoid-s8866" xml:space="preserve">Sed, <lb/>quo melius id fiat, paulò altius exordiendum eſt, vitaque <lb/>& </s> <s xml:id="echoid-s8867" xml:space="preserve">ſtudia hominum attentius inſpicienda.</s> <s xml:id="echoid-s8868" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8869" xml:space="preserve">Ac videtur quidem quatenus providendis procurandiſque <lb/>rebus tantum neceſſariis homines intenti ſunt, ut nempe ab <lb/>aëris injuriis tuti habitent; </s> <s xml:id="echoid-s8870" xml:space="preserve">ut mœnibus incluſi ab inimicis <lb/>ſibi caveant, ut leges condant ad ſecure ac tranquille viven-<lb/>dum; </s> <s xml:id="echoid-s8871" xml:space="preserve">ut liberos educent; </s> <s xml:id="echoid-s8872" xml:space="preserve">victum illis, ſibique parent; </s> <s xml:id="echoid-s8873" xml:space="preserve">in <lb/>his omnibus inquam nihil magnum admodum habere vide-<lb/>tur rationis noſtræ uſus, cujus cauſa nos brutis animantibus <lb/>anteferamus. </s> <s xml:id="echoid-s8874" xml:space="preserve">Namque hæc pleraque iſtorum facilius ſimpli-<lb/>ciuſque efficiunt; </s> <s xml:id="echoid-s8875" xml:space="preserve">aliquibus nihil opus habent. </s> <s xml:id="echoid-s8876" xml:space="preserve">Quin imo <lb/>& </s> <s xml:id="echoid-s8877" xml:space="preserve">virtutis, juſtitiæque ſenſus, propter quem paulo ante ex-<lb/>cellere mentem humanam dicebamus; </s> <s xml:id="echoid-s8878" xml:space="preserve">itemque amicitiæ, gra-<lb/>titudinis, honeſti; </s> <s xml:id="echoid-s8879" xml:space="preserve">quid aliud efficiunt, niſi ut vel vitiis ho-<lb/>minum obſiſtatur, vel vita tranquilla & </s> <s xml:id="echoid-s8880" xml:space="preserve">mutuarum injuria-<lb/>rum expers præſtetur; </s> <s xml:id="echoid-s8881" xml:space="preserve">quod beſtiis ſponte ac naturæ ductu <lb/>contigit. </s> <s xml:id="echoid-s8882" xml:space="preserve">Jam ſi curas multiplices, animi ægritudines, con-<lb/>cupiſcentiam, mortis metum, quæ omnia rationem illam <lb/>noſtram comitantur, ante oculos ponamus; </s> <s xml:id="echoid-s8883" xml:space="preserve">eaque cum vita <lb/>parabili, quieta & </s> <s xml:id="echoid-s8884" xml:space="preserve">innocua beſtiarum comparemus; </s> <s xml:id="echoid-s8885" xml:space="preserve">videri <lb/>poſſint harum plurimæ, ac præſertim ex avium genere, ju-<lb/>cundius agere, & </s> <s xml:id="echoid-s8886" xml:space="preserve">meliore quam homines ſorte frui. </s> <s xml:id="echoid-s8887" xml:space="preserve">Nam <pb o="672" file="0412" n="431" rhead="CHRISTIANI HUGENII"/> quod ad voluptates corporis attinet, haud dubie iis æque ac <lb/>nos afficiuntur, quicquid contradicant novi quidam philo-<lb/>ſophi; </s> <s xml:id="echoid-s8888" xml:space="preserve">qui ſenſum omnem ita auferunt reliquis præter homi-<lb/>nem animantibus, ut pro meris automatis aut neuroſpaſtis <lb/>ea haberi velint; </s> <s xml:id="echoid-s8889" xml:space="preserve">quorum abſurdæ, crudelique ſententiæ, <lb/>miror quenquam accedere poſſe; </s> <s xml:id="echoid-s8890" xml:space="preserve">præſertim cum & </s> <s xml:id="echoid-s8891" xml:space="preserve">voce & </s> <s xml:id="echoid-s8892" xml:space="preserve"><lb/>verberibus fugiendis, & </s> <s xml:id="echoid-s8893" xml:space="preserve">re omni contrarium beſtiæ ipſæ ſi-<lb/>gnificent. </s> <s xml:id="echoid-s8894" xml:space="preserve">Imo vix dubito, quin miro pulchroque illo per <lb/>aëra lapſu aves ſeſe delectari ſentiant; </s> <s xml:id="echoid-s8895" xml:space="preserve">magis etiam ſenſuræ <lb/>ſi intelligerent quantopere lentus ac humilis noſter inceſſus <lb/> <anchor type="note" xlink:label="note-0412-01a" xlink:href="note-0412-01"/> ipſarum pernicitate, ſublimique volatu ſuperetur. </s> <s xml:id="echoid-s8896" xml:space="preserve">Quid igi-<lb/>tur eſt in quo potiſſimum eminet humanæ rationis uſus, fa-<lb/>citque ut antecellamus cæteris animantibus? </s> <s xml:id="echoid-s8897" xml:space="preserve">Nihil æquè pu-<lb/>to ac contemplatio naturæ, Deique operum; </s> <s xml:id="echoid-s8898" xml:space="preserve">tum cultura <lb/>ſcientiarum, quibus conſequimur ut eorum præſtantiam, ma-<lb/>gnitudinemque aliqua ex parte cognoſcamus. </s> <s xml:id="echoid-s8899" xml:space="preserve">Abſque enim <lb/>diſciplinis quid eſſet contemplatio? </s> <s xml:id="echoid-s8900" xml:space="preserve">quamque multum inter-<lb/>eſt inter eos qui Solis pulchritudinem, utilitatemque, & </s> <s xml:id="echoid-s8901" xml:space="preserve">cæ-<lb/>lum ſideribus ornatum otioſe intuentur, alioſque doctiores <lb/>qui curſus iſtorum omnium ſcrutantur: </s> <s xml:id="echoid-s8902" xml:space="preserve">quomodo affixæ, <lb/>quæ dicuntur, ſtellæ à vagis differant, quæque cauſa ſit di-<lb/>verſarum anni tempeſtatum intelligunt: </s> <s xml:id="echoid-s8903" xml:space="preserve">qui denique ſubtili <lb/>ratiocinio magnitudinem Solis ac Planetarum, ſimulque di-<lb/>ſtantiam eorum metiuntur; </s> <s xml:id="echoid-s8904" xml:space="preserve">quantumque item inter eos qui <lb/>animalium varios motus agilitatemque mirantur, & </s> <s xml:id="echoid-s8905" xml:space="preserve">hos qui <lb/>fabricam omnium membrorum, artificioſiſſimamque compa-<lb/>gem, architecturamque in iis ſpeculantur? </s> <s xml:id="echoid-s8906" xml:space="preserve">Quod ſi igitur <lb/>Planetæ reliqui dignitate non cedunt Telluri noſtræ, ut in <lb/>ſuperioribus principii fundamentique loco poſuimus; </s> <s xml:id="echoid-s8907" xml:space="preserve">opor-<lb/>tet ibi animalia exiſtere, quæ non ſolum naturæ opera ſpe-<lb/>ctent & </s> <s xml:id="echoid-s8908" xml:space="preserve">admirentur, ſed quorum ratio in examinandis, in-<lb/>telligendiſque iis occupetur, nec minora quam nos conſecu-<lb/>ta ſit. </s> <s xml:id="echoid-s8909" xml:space="preserve">Itaque non tantum ſidera intuentur, ſed & </s> <s xml:id="echoid-s8910" xml:space="preserve">Aſtrono-<lb/> <anchor type="note" xlink:label="note-0412-02a" xlink:href="note-0412-02"/> miæ ſcientiam excolunt; </s> <s xml:id="echoid-s8911" xml:space="preserve">neque aliud obſtat quo minus hoc <lb/>veriſimile credamus, quam ſuperba illa noſtrarum rerum æ-<lb/>ſtimatio, quæ difficulter ſane deponitur. </s> <s xml:id="echoid-s8912" xml:space="preserve">Scio tamen futu- <pb o="673" file="0413" n="432" rhead="COSMOTHEOROS, LIB. I."/> ros, qui dicant nimis audacter nos iſta Planetarum incolis <lb/>tribuere: </s> <s xml:id="echoid-s8913" xml:space="preserve">multorum quippe veriſimilium accumulatione huc <lb/>eſſe perventum; </s> <s xml:id="echoid-s8914" xml:space="preserve">quorum ſi unum quodpiam contra ſe ha-<lb/>beat, quam poſitum ſit, cadat, velut in vitioſa ædificatio-<lb/>ne, omne quod ſuperſtruximus. </s> <s xml:id="echoid-s8915" xml:space="preserve">Sed ſcire eos velim, hoc <lb/>quod de Aſtronomiæ ſtudio diximus, omiſſis fere omnibus <lb/>hactenus adductis confirmari potuiſſe, atque inde initium <lb/>fieri. </s> <s xml:id="echoid-s8916" xml:space="preserve">Poſtquam enim poſitum fuit Terram hanc inter Pla-<lb/>netas eſſe habendam, neque iis dignitate aut ornatu præfe-<lb/>rendam; </s> <s xml:id="echoid-s8917" xml:space="preserve">quis dicere audeat in ea ſola reperiri, qui ſpecta-<lb/>culo Naturæ, quod unum pulcherrimum ac magnificentiſſi-<lb/>mum eſt, fruantur? </s> <s xml:id="echoid-s8918" xml:space="preserve">aut inter eos quibus hoc contigit, nos <lb/>unos eſſe quibus cæli arcana penitius perfectiuſque perſpecta <lb/>ſint? </s> <s xml:id="echoid-s8919" xml:space="preserve">Ecce igitur & </s> <s xml:id="echoid-s8920" xml:space="preserve">hac breviore via comprobata in Planetis <lb/>Aſtronomiæ cognitio, ex qua & </s> <s xml:id="echoid-s8921" xml:space="preserve">animal rationis compos, <lb/>& </s> <s xml:id="echoid-s8922" xml:space="preserve">pleraque alia quæ præceſſere, illis ineſſe conſequebatur. <lb/></s> <s xml:id="echoid-s8923" xml:space="preserve">Adeo ut, ad priora confirmanda, hæc quoque noviſſima ar-<lb/>gumentatio conducat. </s> <s xml:id="echoid-s8924" xml:space="preserve">Quò vero magis probabile fiat, ſal-<lb/>tem in ſuperioribus Planetis, Jove ac Saturno, Aſtronomiæ <lb/>notitiam non deeſſe, conſiderandum eſt, quod ſi homines <lb/>ad ſidera obſervanda impulit, ut credi par eſt, admiratio & </s> <s xml:id="echoid-s8925" xml:space="preserve"><lb/>pavor in defectibus Solis & </s> <s xml:id="echoid-s8926" xml:space="preserve">Lunæ; </s> <s xml:id="echoid-s8927" xml:space="preserve">multo magis, in utro-<lb/>que hoc Planeta, ea ratio valere debuit, propter cotidianas <lb/>fere Lunarum, crebraſque Solis, quæ illic contingunt, ec-<lb/>lipſes. </s> <s xml:id="echoid-s8928" xml:space="preserve">Ut ſi quis æquè ignorare ponatur quid rerum in Pla-<lb/>netis omnibus geratur, multo veriſimilius dicturus ſit Aſtro-<lb/>nomiam in majoribus illis duobus, quàm in hoc noſtro, vi-<lb/>gere.</s> <s xml:id="echoid-s8929" xml:space="preserve"/> </p> <div xml:id="echoid-div424" type="float" level="2" n="35"> <note position="left" xlink:label="note-0412-01" xlink:href="note-0412-01a" xml:space="preserve">Humanam <lb/>rationem præ <lb/>illa bruto-<lb/>rum præcipue <lb/>eminere in <lb/>contempla-<lb/>tione naturæ.</note> <note position="left" xlink:label="note-0412-02" xlink:href="note-0412-02a" xml:space="preserve">Hinc Plane-<lb/>tarum inco-<lb/>las ſcientias <lb/>excolere, & <lb/>inter eas, <lb/>Aſtrono-<lb/>miam.</note> </div> <p> <s xml:id="echoid-s8930" xml:space="preserve">Poſita autem apud Planeticolas hujus ſcientiæ cognitione <lb/>& </s> <s xml:id="echoid-s8931" xml:space="preserve">uſu, quam multa hinc præterea conſequuntur quæ de vi-<lb/>ta ſtatuque eorum reliquo, præter jam dicta, novas conje-<lb/>cturas afferant?</s> <s xml:id="echoid-s8932" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8933" xml:space="preserve">Primùm enim nulla obſervatio ſiderum, ad motus eorum <lb/> <anchor type="note" xlink:label="note-0413-01a" xlink:href="note-0413-01"/> inveſtigandos, abſque organis inſtitui poteſt; </s> <s xml:id="echoid-s8934" xml:space="preserve">ſive ea è me-<lb/>tallo, ſive è ligno aliave ſolida materia fabricata ſint. </s> <s xml:id="echoid-s8935" xml:space="preserve">Quod <lb/>ut fiat, nec fabrorum inſtrumento, ſerra, aſcia, dolabra, <pb o="674" file="0414" n="433" rhead="CHRISTIANI HUGENII"/> malleo, lima, carere poſſunt; </s> <s xml:id="echoid-s8936" xml:space="preserve">neque hæc habere abſque uſu <lb/>ferri aut æque duri cujuſpiam metalli. </s> <s xml:id="echoid-s8937" xml:space="preserve">Sed & </s> <s xml:id="echoid-s8938" xml:space="preserve">circuli arcus <lb/> <anchor type="note" xlink:label="note-0414-01a" xlink:href="note-0414-01"/> in partes æquales diviſi, aut lineæ rectæ in inæquales, in iſtis <lb/>organis requiruntur. </s> <s xml:id="echoid-s8939" xml:space="preserve">Atque hic jam Geometriæ & </s> <s xml:id="echoid-s8940" xml:space="preserve">numero-<lb/>rum ratio arceſſenda eſt. </s> <s xml:id="echoid-s8941" xml:space="preserve">Sed ante omnia quoque neceſſe <lb/>eſt ut obſervationum memoria ad poſteros tranſmittatur; </s> <s xml:id="echoid-s8942" xml:space="preserve">ut <lb/>tempora & </s> <s xml:id="echoid-s8943" xml:space="preserve">Epochæ annotentur; </s> <s xml:id="echoid-s8944" xml:space="preserve">quæ ſine ſcripto non viden-<lb/>tur explicari poſſe. </s> <s xml:id="echoid-s8945" xml:space="preserve">Oportet igitur ut & </s> <s xml:id="echoid-s8946" xml:space="preserve">ſuam ſcribendi ar-<lb/> <anchor type="note" xlink:label="note-0414-02a" xlink:href="note-0414-02"/> tem habeant, multum fortaſſe diſſimilem noſtræ, qua fere <lb/>omnes populi utuntur, ſed quæ vix ingenioſior, aut ad di-<lb/>ſcendum facilior eſſe queat. </s> <s xml:id="echoid-s8947" xml:space="preserve">Quis enim non videt longe eam <lb/>præferendam eſſe Sinarum innumeris characteribus, multo-<lb/>que magis funiculorum nodis, aut pictis imaginibus, quæ a pud <lb/>barbaros Mexicanos Peruvianoſque in uſu erant. </s> <s xml:id="echoid-s8948" xml:space="preserve">Omnium <lb/>quidem Regionum homines aliquam ſcribendi, aut quoquo <lb/>modo annotandi, artem quæſiviſſe videmus: </s> <s xml:id="echoid-s8949" xml:space="preserve">quò minus mi-<lb/>rum ſit, ſi & </s> <s xml:id="echoid-s8950" xml:space="preserve">Planetarum incolæ, neceſſitate coacti, eam re-<lb/>pererint, ac deinde ad Aſtronomiæ aliarumque diſciplinarum <lb/>ſtudia adhibuerint. </s> <s xml:id="echoid-s8951" xml:space="preserve">Neceſſitas vero ſcripturæ in rebus Aſtro-<lb/>nomicis etiam ea re cognoſcitur, quod cum hypotheſibus va-<lb/>riis, ſiderum motus, quaſi divinandi ſint; </s> <s xml:id="echoid-s8952" xml:space="preserve">eæque hypothe-<lb/>ſes priores in ſequentibus corrigendæ, prout obſervatis & </s> <s xml:id="echoid-s8953" xml:space="preserve"><lb/>Geometriæ ratiociniis vitia earum coarguuntur; </s> <s xml:id="echoid-s8954" xml:space="preserve">nihil horum <lb/>poſteris tradi poteſt, niſi literis conſignatum, figuriſque ex-<lb/>poſitum.</s> <s xml:id="echoid-s8955" xml:space="preserve"/> </p> <div xml:id="echoid-div425" type="float" level="2" n="36"> <note position="right" xlink:label="note-0413-01" xlink:href="note-0413-01a" xml:space="preserve">Et quæ ei in-<lb/>ſerviunt ar-<lb/>tes mechani-<lb/>cas.</note> <note position="left" xlink:label="note-0414-01" xlink:href="note-0414-01a" xml:space="preserve">Ut & Geo-<lb/>metriam, <lb/>Arithmeti-<lb/>cam,</note> <note position="left" xlink:label="note-0414-02" xlink:href="note-0414-02a" xml:space="preserve">Et ſoribendi <lb/>artem.</note> </div> <p> <s xml:id="echoid-s8956" xml:space="preserve">Poſtquam vero omnia hæc jam iis conceſſerimus, longe <lb/>etiamnum præſtantior perfectiorque apud nos erit ſiderum <lb/>ſcientia; </s> <s xml:id="echoid-s8957" xml:space="preserve">vel propter agnitam ſyſtematis univerſi veriſſimam <lb/>formam, vel propter uſum teleſcopiorum, quibus Planeta-<lb/>rum corpora, magnitudineſque & </s> <s xml:id="echoid-s8958" xml:space="preserve">varias formas intuemur; <lb/></s> <s xml:id="echoid-s8959" xml:space="preserve"> <anchor type="note" xlink:label="note-0414-03a" xlink:href="note-0414-03"/> ſuperficiei lunaris montes, montiumque umbras; </s> <s xml:id="echoid-s8960" xml:space="preserve">ſtellarum <lb/>ingentem multitudinem, aliaque plura non alias videnda, <lb/>percipimus. </s> <s xml:id="echoid-s8961" xml:space="preserve">Ut fere neceſſe ſit, niſi rurſus nobis tanquam <lb/>hac parte felicioribus blandiri volumus, etiam illam cogni-<lb/>tionis rerum cæleſtium conſummationem Planeticolis tribue-<lb/>re; </s> <s xml:id="echoid-s8962" xml:space="preserve">itemque videndi aciem, quæ vel noſtram longe exupe- <pb o="675" file="0415" n="434" rhead="COSMOTHEOROS, LIB. I."/> ret, vel lentium vitrearum, aut ſpeculorum adminiculo ſicut <lb/>noſtra, adjuvetur. </s> <s xml:id="echoid-s8963" xml:space="preserve">Quod tamen dicere vereor, ne quis, ex <lb/>hoc uno audacius aſſerto, cætera omnia æſtimanda putet, ac <lb/>riſu digna clamitet.</s> <s xml:id="echoid-s8964" xml:space="preserve"/> </p> <div xml:id="echoid-div426" type="float" level="2" n="37"> <note position="left" xlink:label="note-0414-03" xlink:href="note-0414-03a" xml:space="preserve">Opticam.</note> </div> <p> <s xml:id="echoid-s8965" xml:space="preserve">At non ſine ratione, ut videtur, objiciet quiſpiam, Pla-<lb/>netarios noſtros fortaſſe omni ſubtiliore ſcientia deſtitui, quem-<lb/>admodum Americanæ gentes, priuſquam ad illas Europei pe-<lb/>netraſſent. </s> <s xml:id="echoid-s8966" xml:space="preserve">Quas ſi reſpicimus, itemque in Africa, Aſiaque <lb/> <anchor type="note" xlink:label="note-0415-01a" xlink:href="note-0415-01"/> permultas æque barbaras, videbitur hoc tantum ſummo opi-<lb/>ſici propoſitum fuiſie, ut vita fruantur homines, naturæque <lb/>bonis & </s> <s xml:id="echoid-s8967" xml:space="preserve">voluptatibus contenti ſint, grato animo omnium da-<lb/>torem colentes; </s> <s xml:id="echoid-s8968" xml:space="preserve">ſcientiarum vero inquiſitionem præter natu-<lb/>ram paucos aliquos affectaſſe. </s> <s xml:id="echoid-s8969" xml:space="preserve">Talia vero dicentibus non de-<lb/>eſt quod reſponderi poſſit. </s> <s xml:id="echoid-s8970" xml:space="preserve">Prævidit enim certe Deus ho-<lb/>minum ingenia eo eſſe proceſſura, ut res cæleſtes ſcrutaren-<lb/>tur; </s> <s xml:id="echoid-s8971" xml:space="preserve">ut artes vitæ utiles reperirent; </s> <s xml:id="echoid-s8972" xml:space="preserve">maria quoque navigarent, <lb/>metalla effoderent. </s> <s xml:id="echoid-s8973" xml:space="preserve">Poſſetne enim horum quidquam præter <lb/>mentem infinitæ illius intelligentiæ contingere? </s> <s xml:id="echoid-s8974" xml:space="preserve">Quod ſi præ-<lb/>vidit, etiam hominum generi ea deſtinata ſunt, nec poterunt <lb/>artium & </s> <s xml:id="echoid-s8975" xml:space="preserve">doctrinarum ſtudia, quaſi præter naturam eſſent, <lb/>exiſtimari, quæ in ipſa natura indaganda occupantur. </s> <s xml:id="echoid-s8976" xml:space="preserve">Præ-<lb/>ſertim cum tanta illa cupiditas amorque ſciendi non poſſint <lb/>cenſeri fruſtra hominum animis infixa eſſe. </s> <s xml:id="echoid-s8977" xml:space="preserve">Inſtabunt vero <lb/>rurſus dicentque, de ſiderali ſcientia potiſſimum, ſi ad hanc <lb/>quoque homines nati ſunt, cur tam pauciad eam attendunt? <lb/></s> <s xml:id="echoid-s8978" xml:space="preserve">Primum enim ex quatuor Orbis partibus, ſola fere eſt Euro-<lb/>pa, in qua Aſtronomiæ ſtudia excolantur. </s> <s xml:id="echoid-s8979" xml:space="preserve">Nam Aſtrolo-<lb/>giam divinatricem futurorum, quæ non ſcientia, ſed miſe-<lb/>rum quoddam ac ſæpe noxium delirium eſt, ne nominandam <lb/>quidem hic arbitror. </s> <s xml:id="echoid-s8980" xml:space="preserve">At in Europæ Nationibus non unus è <lb/>centum millibus hæc ſtudia amplectitur aut addiſcere curat. </s> <s xml:id="echoid-s8981" xml:space="preserve"><lb/>Tum ad tempus quod attinet, multa ſæcula effluxiſſe dicent, <lb/>antequam aut Aſtronomiæ, aut Geometriæ, ſine qua illa di-<lb/>ſci non poteſt, ulla rudimenta innoteſcerent. </s> <s xml:id="echoid-s8982" xml:space="preserve">Sciri enim quo <lb/>tempore in Ægypto & </s> <s xml:id="echoid-s8983" xml:space="preserve">Græcia primum exortæ fuerint. </s> <s xml:id="echoid-s8984" xml:space="preserve">Ac <lb/>recte quoque adjicient non adhuc octoginta annos præteriiſ- <pb o="676" file="0416" n="435" rhead="CHRISTIANI HUGENII"/> ſe, ex quo verus ac ſimplex Planetarum motus, rejectis Epi-<lb/>cyclorum figmentis, repertus ſit; </s> <s xml:id="echoid-s8985" xml:space="preserve">atque ita demum Aſtrono-<lb/>nomia cum naturæ cognitione conjuncta. </s> <s xml:id="echoid-s8986" xml:space="preserve">Hiſce ut occurra-<lb/>tur, addam ad ſuperius reſponſum, quod à divina provi-<lb/>dentia petebatur, dubitari non poſſe, quin ea conditione <lb/>homines nati ſint, ut multo temporis decurſu paulatim artes <lb/>diſciplinaſque eruant; </s> <s xml:id="echoid-s8987" xml:space="preserve">nullam enim harum iis ingenitam eſſe, <lb/>aut ſubito à Deo infuſam, & </s> <s xml:id="echoid-s8988" xml:space="preserve">has de quibus nunc agimus, <lb/>omnium eſſe difficillimas remotiſſimaſque: </s> <s xml:id="echoid-s8989" xml:space="preserve">ut magis mirum ſit <lb/>unquam incipere eas potuiſſe, quam tam tarde fuiſſe inſpectas. <lb/></s> <s xml:id="echoid-s8990" xml:space="preserve">Pauci fateor ſingulis ætatibus has curant, aut ad ſe pertinere <lb/>exiſtimant: </s> <s xml:id="echoid-s8991" xml:space="preserve">ſed ſi multorum ſæculorum tempora cogitentur, <lb/>non exiguus fiet illorum Numerus; </s> <s xml:id="echoid-s8992" xml:space="preserve">quos, quemadmodum <lb/>ſibi videntur, reliquis beatiores eſſe quis negaverit? </s> <s xml:id="echoid-s8993" xml:space="preserve">Deni-<lb/>que paucorum induſtria in his rebus exerceri ſatis erat, cum <lb/>inventorum utilitas ad nationes totas genteſque longe porri-<lb/>gatur. </s> <s xml:id="echoid-s8994" xml:space="preserve">Cum igitur hujus Terræ incolis, etſi paucis tantum, <lb/>ad ea percipienda ingenium & </s> <s xml:id="echoid-s8995" xml:space="preserve">aptitudo contigerit; </s> <s xml:id="echoid-s8996" xml:space="preserve">nihiloque <lb/>putandi ſint cæterorum planetarum habitatoribus præſtantio-<lb/>res felicioreſve; </s> <s xml:id="echoid-s8997" xml:space="preserve">manet profecto, quam inveneramus, veri-<lb/>ſimilitudo, ut etiam apud illos reperiantur qui cognitione <lb/>Aſtronomiæ non careant. </s> <s xml:id="echoid-s8998" xml:space="preserve">Nunc ad alia pergamus quæ in-<lb/>de conſequi, neceſſe eſt.</s> <s xml:id="echoid-s8999" xml:space="preserve"/> </p> <div xml:id="echoid-div427" type="float" level="2" n="38"> <note position="right" xlink:label="note-0415-01" xlink:href="note-0415-01a" xml:space="preserve">Has ſcien-<lb/>tias homini <lb/>præter natu-<lb/>ram non eſſe.</note> </div> <p> <s xml:id="echoid-s9000" xml:space="preserve">Oſtendimus quomodo unà cum hac ſcientia, non ſolum <lb/>Geometria & </s> <s xml:id="echoid-s9001" xml:space="preserve">Arithmetice, ſed & </s> <s xml:id="echoid-s9002" xml:space="preserve">Mechanicæ artes, inſtru-<lb/>mentaque incolis Planetarum concedenda ſint. </s> <s xml:id="echoid-s9003" xml:space="preserve">Hic verò <lb/>jam ſponte obvenit ut quæramus, quo pacto inſtrumentis <lb/>illis, Machiniſque, & </s> <s xml:id="echoid-s9004" xml:space="preserve">ad ſidera obſervanda organis uti poſ-<lb/>ſint, aut quomodo literas ducere; </s> <s xml:id="echoid-s9005" xml:space="preserve">quæ omnia nos manuum <lb/>opera exequimur. </s> <s xml:id="echoid-s9006" xml:space="preserve">Itaque neceſſario & </s> <s xml:id="echoid-s9007" xml:space="preserve">manus habebunt, <lb/>vel aliud quodpiam, quod vicem earum fungi poſſit, mem-<lb/> <anchor type="note" xlink:label="note-0416-01a" xlink:href="note-0416-01"/> brum. </s> <s xml:id="echoid-s9008" xml:space="preserve">In quibus hominum generi tantum eſſe præſidii exi-<lb/>ſtimabat è veteribus Philoſophis quidam, ut in iis cauſam <lb/>reponeret omnis eorum ſapientiæ. </s> <s xml:id="echoid-s9009" xml:space="preserve">Qui, ut puto, hoc ſen-<lb/>ſit, abſque manuum opera homines ad cultum animi, re-<lb/>rumque cognitionem non fuiſſe perventuros. </s> <s xml:id="echoid-s9010" xml:space="preserve">Et vere qui- <pb o="677" file="0417" n="436" rhead="COSMOTHEOROS, LIB. I."/> dem ille. </s> <s xml:id="echoid-s9011" xml:space="preserve">Finge enim pro manibus datas fuiſſe ungulas, ut <lb/>equis & </s> <s xml:id="echoid-s9012" xml:space="preserve">bobus; </s> <s xml:id="echoid-s9013" xml:space="preserve">nunquam nec oppida nec domos, licet ra-<lb/>tione inſtructi, ædificaſſent. </s> <s xml:id="echoid-s9014" xml:space="preserve">Nihil de quo loquerentur ha-<lb/>buiſſent, niſi de iis quæ ad pabulum, aut ad conjugium, <lb/>aut ſui tutelam attinent. </s> <s xml:id="echoid-s9015" xml:space="preserve">Omni ſcientia, omnique rerum <lb/>memoria caruiſſent: </s> <s xml:id="echoid-s9016" xml:space="preserve">Denique à beſtiis parum abfuiſſent. <lb/></s> <s xml:id="echoid-s9017" xml:space="preserve">Quodnam porro inſtrumentum æque accommodatum ac ma-<lb/>nus eſſe poſſit ad innumera illa ad quæ nobis uſui ſunt, ob-<lb/>eunda? </s> <s xml:id="echoid-s9018" xml:space="preserve">Elephanti proboſcide mirabiliter utuntur, qua & </s> <s xml:id="echoid-s9019" xml:space="preserve"><lb/>amplecti quidvis & </s> <s xml:id="echoid-s9020" xml:space="preserve">projicere, minutioraque quævis è ſolo <lb/>tollere norunt: </s> <s xml:id="echoid-s9021" xml:space="preserve">unde & </s> <s xml:id="echoid-s9022" xml:space="preserve">manus eorum pars illa dicta eſt, cum <lb/>reipſa ſit in longum productus naſus. </s> <s xml:id="echoid-s9023" xml:space="preserve">Roſtro quoque aves <lb/>pleræque nidos exſtruunt, alimentaque congerunt. </s> <s xml:id="echoid-s9024" xml:space="preserve">Sed ha-<lb/>rum nihil eſt quod non manuum oportunitati longè conce-<lb/>dat. </s> <s xml:id="echoid-s9025" xml:space="preserve">Et eſt ſane, tam illarum quam brachiorum, mirabilis <lb/>quædam machinatio; </s> <s xml:id="echoid-s9026" xml:space="preserve">ut protendi, reduci, inque omnem <lb/>partem moveri poſſint. </s> <s xml:id="echoid-s9027" xml:space="preserve">Tum mirâ induſtriâ inſtituti digi-<lb/>torum ac pollicis articuli, ut nervorum attractu quælibet pre-<lb/>hendant, firmiterque contineant. </s> <s xml:id="echoid-s9028" xml:space="preserve">Ut omittam ſenſum il-<lb/>lum, in extremis digitis, exquiſitiſſimæ ſubtilitatis; </s> <s xml:id="echoid-s9029" xml:space="preserve">quo vel <lb/>in tenebris pleraque corpora internoſcimus. </s> <s xml:id="echoid-s9030" xml:space="preserve">Patet itaque aut <lb/>manus brachiaque, aut aliud quid eorum loco, quod vix <lb/>æque aptum excogitari poteſt, Planetarum populis datum <lb/>eſſe, ne non ſolum nobis, ſed & </s> <s xml:id="echoid-s9031" xml:space="preserve">ſimiarum & </s> <s xml:id="echoid-s9032" xml:space="preserve">ſciurorum ge-<lb/>neri, plus indulſiſſe hac in re natura exiſtimetur.</s> <s xml:id="echoid-s9033" xml:space="preserve"/> </p> <div xml:id="echoid-div428" type="float" level="2" n="39"> <note position="left" xlink:label="note-0416-01" xlink:href="note-0416-01a" xml:space="preserve">Planeticolas <lb/>manus habe-<lb/>@@.</note> </div> <p> <s xml:id="echoid-s9034" xml:space="preserve">De pedibus vero minus etiam dubitabitur, ſi repetamus ea <lb/> <anchor type="note" xlink:label="note-0417-01a" xlink:href="note-0417-01"/> quæ ſupra diſſeruimus de vario animalium inceſſu, qui non <lb/>videtur aliis modis, quam quos ibi recenſuimus, cogitari <lb/>poſſe. </s> <s xml:id="echoid-s9035" xml:space="preserve">Inter eos vero non eſt, qui tam bene Planeticolis <lb/>ratione præditis conveniat, quam quo & </s> <s xml:id="echoid-s9036" xml:space="preserve">nos utamur. </s> <s xml:id="echoid-s9037" xml:space="preserve">Ni-<lb/>ſi forte & </s> <s xml:id="echoid-s9038" xml:space="preserve">volandi facultatem in aliquibus Globorum iſtorum <lb/>acceperunt. </s> <s xml:id="echoid-s9039" xml:space="preserve">Quod minus probabile tamen propter vitam in <lb/>ſocietate degendam, de qua poſtea dicemus.</s> <s xml:id="echoid-s9040" xml:space="preserve"/> </p> <div xml:id="echoid-div429" type="float" level="2" n="40"> <note position="right" xlink:label="note-0417-01" xlink:href="note-0417-01a" xml:space="preserve">Et pedes@</note> </div> <p> <s xml:id="echoid-s9041" xml:space="preserve">Non caret autem veriſimilitudine, erectos oculos, vul-<lb/> <anchor type="note" xlink:label="note-0417-02a" xlink:href="note-0417-02"/> tumque ad ſidera contemplanda iis contigiſſe, quandoqui-<lb/>dem hoc in hominum corpore providentiâ divinâ ſic inſtitu- <pb o="678" file="0418" n="437" rhead="CHISTIANI HUGENII"/> tum videtur, & </s> <s xml:id="echoid-s9042" xml:space="preserve">a Philoſophis merito celebrari ſolet. </s> <s xml:id="echoid-s9043" xml:space="preserve">De <lb/>reliquorum vero membrorum poſitu, ſi ſapientiam artificis <lb/>laude dignam cenſemus, quod oculos in ſuprema corporis <lb/>parte collocaverit; </s> <s xml:id="echoid-s9044" xml:space="preserve">ſordidiora vero membra procul inde, at-<lb/> <anchor type="note" xlink:label="note-0418-01a" xlink:href="note-0418-01"/> que a conſpectu quodammodo removerit; </s> <s xml:id="echoid-s9045" xml:space="preserve">nonne putandum <lb/>eſt eadem fere obſervaſſe illum in formandis iſtorum procul <lb/>habitantium corporibus? </s> <s xml:id="echoid-s9046" xml:space="preserve">Nec enim propterea dicimus figu-<lb/>ram noſtræ ſimilem iis tribuiſſe. </s> <s xml:id="echoid-s9047" xml:space="preserve">Eſt enim infinita quædam <lb/>animo concipienda formarum poſſibilium varietas, qua & </s> <s xml:id="echoid-s9048" xml:space="preserve"><lb/>ſingulæ quæque partes iſtorum corporum à noſtris differre <lb/>queant, & </s> <s xml:id="echoid-s9049" xml:space="preserve">totorum exterior interiorque œconomia. </s> <s xml:id="echoid-s9050" xml:space="preserve">Cer-<lb/>nimus quam aptè & </s> <s xml:id="echoid-s9051" xml:space="preserve">commodè animalium noſtrorum quædam <lb/>lana aut pilis veſtiantur; </s> <s xml:id="echoid-s9052" xml:space="preserve">alia elegantius etiam plumis penniſ-<lb/>que. </s> <s xml:id="echoid-s9053" xml:space="preserve">Quidni iſti in Planetis, quos rationis participes dixi-<lb/>mus, aliqua ſimili ratione tecti ſint? </s> <s xml:id="echoid-s9054" xml:space="preserve">propter quod meliori <lb/>quidem conditione beſtiæ, quam homines, apud nos eſſe <lb/>videntur. </s> <s xml:id="echoid-s9055" xml:space="preserve">Niſi hoc eo fine ſic conſtitutum fuit, ut ipſa nu-<lb/>ditas neceſſitatem hominibus imponeret quærendi ac fabri-<lb/>candi varia operimentorum genera, atque hinc etiam inge-<lb/>nii exercendi materia exiſteret. </s> <s xml:id="echoid-s9056" xml:space="preserve">Et apparet ſane, ex hac <lb/>neceſſitate, non minimam commerciorum, artificiorumque <lb/>mechanicorum occaſionem naſci. </s> <s xml:id="echoid-s9057" xml:space="preserve">Sed & </s> <s xml:id="echoid-s9058" xml:space="preserve">propterea forſan <lb/>nudos homines natura produxit, ut pro arbitrio ſuo tenuius <lb/>denſiuſve amicti incedere poſſint; </s> <s xml:id="echoid-s9059" xml:space="preserve">atque ita ad quaſvis terra-<lb/>rum oras inhabitandas ſeſe componere. </s> <s xml:id="echoid-s9060" xml:space="preserve">Alia vero major <lb/>hac, quam diximus, differentia intelligi poſſet inter corpora <lb/>Planetariorum ac noſtra; </s> <s xml:id="echoid-s9061" xml:space="preserve">cum animalia quædam ita à natura <lb/>formata reperiantur, ut veluti oſſa extrinſecus habeant, car-<lb/>nes introrſum, atque oſſibus incluſas, qualia ſunt cancri, <lb/>aſtacique, & </s> <s xml:id="echoid-s9062" xml:space="preserve">fere etiam teſtudines. </s> <s xml:id="echoid-s9063" xml:space="preserve">Attamen hanc membro-<lb/>rum compagem, & </s> <s xml:id="echoid-s9064" xml:space="preserve">in paucis vilioribus tantum illa ſecuta <lb/>eſt, & </s> <s xml:id="echoid-s9065" xml:space="preserve">Planetarum incolis, quo minus eam tribuam, facit, <lb/>quod ſubtili varioque digitorum uſu carituri eſſent, quo tam <lb/>valde eos opus habere oſtenſum fuit: </s> <s xml:id="echoid-s9066" xml:space="preserve">nam abſurda ſpecie <lb/>non multum alioqui moverer.</s> <s xml:id="echoid-s9067" xml:space="preserve"/> </p> <div xml:id="echoid-div430" type="float" level="2" n="41"> <note position="right" xlink:label="note-0417-02" xlink:href="note-0417-02a" xml:space="preserve">Erectos ocu<unsure/>-<lb/>los, vul-<lb/>tumque.</note> <note position="left" xlink:label="note-0418-01" xlink:href="note-0418-01a" xml:space="preserve">Nec tamen <lb/>binc ſequi <lb/>eorum for-<lb/>mam noſtræ <lb/>plane ſimi-<lb/>lem.</note> </div> <p> <s xml:id="echoid-s9068" xml:space="preserve">Etenim omnino cavendum eſt ab errore vulgi, cum ani-<lb/> <anchor type="note" xlink:label="note-0418-02a" xlink:href="note-0418-02"/> <pb o="679" file="0419" n="438" rhead="COSMOTHEOROS, LIB. I."/> mum rationis capacem non alio in corpore, quam noſtris ſi-<lb/>mili habitare poſſe ſibi perſuadet. </s> <s xml:id="echoid-s9069" xml:space="preserve">Ex quo factum eſt, ut <lb/>populi penè omnes, atque etiam Philoſophi quidam, humanam <lb/>formam diis adſcripſerint; </s> <s xml:id="echoid-s9070" xml:space="preserve">Imo ut, à ſimili perſuaſione, cui-<lb/>dam Chriſtianorum ſectæ nomen inditum fuerit. </s> <s xml:id="echoid-s9071" xml:space="preserve">Hoc vero <lb/>non niſi ab hominum imbecillitate & </s> <s xml:id="echoid-s9072" xml:space="preserve">præjudicata opinione <lb/>proficiſci quis non videt? </s> <s xml:id="echoid-s9073" xml:space="preserve">uti illud quoque, quod eximiæ <lb/>quædam pulchritudo humani corporis eſſe putatur: </s> <s xml:id="echoid-s9074" xml:space="preserve">cum ta-<lb/>men ab opinione & </s> <s xml:id="echoid-s9075" xml:space="preserve">aſſuetudine id totum quoque pendeat, <lb/>affectuque eo, quem cunctis animalibus natura provida in-<lb/>generavit; </s> <s xml:id="echoid-s9076" xml:space="preserve">ut ſui ſimilibus maxime caperentur. </s> <s xml:id="echoid-s9077" xml:space="preserve">Illa verò <lb/>tantum poſſunt, ut non ſine horrore aliquo animal homini <lb/>multum diſſimile conſpectum iri credam, in quo rationis & </s> <s xml:id="echoid-s9078" xml:space="preserve"><lb/>ſermonis uſus reperiretur. </s> <s xml:id="echoid-s9079" xml:space="preserve">Nam ſi tale ſolummodo finga-<lb/>mus aut pingamus, quod, cætera homini ſimile, collum <lb/>quadruplo longius habeat, vel oculos rotundos duploque <lb/>amplius diſtantes; </s> <s xml:id="echoid-s9080" xml:space="preserve">continuo eæ figuræ naſcuntur, quas non <lb/>poſſimus intuentes non averſari, quamvis ratio deformitatis <lb/>nulla reddi queat.</s> <s xml:id="echoid-s9081" xml:space="preserve"/> </p> <div xml:id="echoid-div431" type="float" level="2" n="42"> <note position="left" xlink:label="note-0418-02" xlink:href="note-0418-02a" xml:space="preserve">Quo minus <lb/>@nimus ra-<lb/>tionis capa@e <lb/>etiam alii <lb/>formæ inha-<lb/>bitet, nihil <lb/>impedire.</note> </div> <p> <s xml:id="echoid-s9082" xml:space="preserve">Dixi in ſuperioribus cum de magnitudine agerem incola-<lb/> <anchor type="note" xlink:label="note-0419-01a" xlink:href="note-0419-01"/> rum qui in Planetis ſunt, veriſimile videri non eſſe eos val-<lb/>de exiguos nobiſcum comparatos. </s> <s xml:id="echoid-s9083" xml:space="preserve">Suadet enim hoc primò, <lb/>quod probabile ſit, ſicut corpora hominum ſe habent ad Tel-<lb/>luris magnitudinem, ut peragrare univerſam poſſint, atque <lb/>ita formam molemque ejus cognoſcere; </s> <s xml:id="echoid-s9084" xml:space="preserve">eodem modo <lb/>& </s> <s xml:id="echoid-s9085" xml:space="preserve">in cæteris Planetis incoliſque eorum rationalibus or-<lb/>dinatum eſſe; </s> <s xml:id="echoid-s9086" xml:space="preserve">niſi hac in re, quæ ſane magna eſt, nos <lb/>ipſis rurſus præferre velimus. </s> <s xml:id="echoid-s9087" xml:space="preserve">Deinde cum ſiderum ſcien-<lb/>tiam & </s> <s xml:id="echoid-s9088" xml:space="preserve">obſervationes apud eos exerceri oſtenderimus, ſe-<lb/>quitur ut & </s> <s xml:id="echoid-s9089" xml:space="preserve">corpora nacti ſint lignis metalliſve tractan-<lb/>dis, inque inſtrumenta machinaſque adaptandis, idonea. <lb/></s> <s xml:id="echoid-s9090" xml:space="preserve">Quæ & </s> <s xml:id="echoid-s9091" xml:space="preserve">eo præſtabiliora ſunt quo ampliora. </s> <s xml:id="echoid-s9092" xml:space="preserve">Ac ſane ſi <lb/>homunciones quoſdam, muribus non majores, cogite-<lb/>mus, non poſl<unsure/>ent ii ſiderum animadverſiones, quales re-<lb/>quiruntur, inſtituere; </s> <s xml:id="echoid-s9093" xml:space="preserve">nec inſtrumenta ad eas parare, aut <lb/>diſponere. </s> <s xml:id="echoid-s9094" xml:space="preserve">Itaque omnino vel æquales nobis ponendos eſſe <pb o="680" file="0420" n="439" rhead="CHRISTIANI HUGENII"/> exiſtimo, vel majores, ac præſertim in Jove, Saturnoque, <lb/>quorum Globi tanto Tellurem noſtram ſuperant.</s> <s xml:id="echoid-s9095" xml:space="preserve"/> </p> <div xml:id="echoid-div432" type="float" level="2" n="43"> <note position="right" xlink:label="note-0419-01" xlink:href="note-0419-01a" xml:space="preserve">Planeticolas <lb/>nobis vel æ-<lb/>quales vel <lb/>majores eſſe.</note> </div> <p> <s xml:id="echoid-s9096" xml:space="preserve">Porrò quia, ut diximus, aſtronomiæ ſtudium ſine anno-<lb/>tatione obſervatorum non poteſt procedere, ars vero ſcri-<lb/> <anchor type="note" xlink:label="note-0420-01a" xlink:href="note-0420-01"/> bendi non niſi in ſocietate ratione utentium, & </s> <s xml:id="echoid-s9097" xml:space="preserve">cogentibus <lb/>vitæ neceſſitatibus, inveniri potuit; </s> <s xml:id="echoid-s9098" xml:space="preserve">neque magis ars fabro-<lb/>rum aut fuſoria; </s> <s xml:id="echoid-s9099" xml:space="preserve">ſequitur ex eo (quod ſupra dicebam) & </s> <s xml:id="echoid-s9100" xml:space="preserve"><lb/>ſocietates coli apud Planetarum indigenas, ac mutuas ope-<lb/>ras eos inter ſe præſtare; </s> <s xml:id="echoid-s9101" xml:space="preserve">adeoque hac parte ſimilitudinem <lb/>magnam ibi eſſe noſtratium rerum. </s> <s xml:id="echoid-s9102" xml:space="preserve">Quamobrem & </s> <s xml:id="echoid-s9103" xml:space="preserve">certas <lb/>ſtabileſque ſedes potius quam ambulatoriam vitam iis conve-<lb/>nire dicendum eſt. </s> <s xml:id="echoid-s9104" xml:space="preserve">Quid igitur? </s> <s xml:id="echoid-s9105" xml:space="preserve">an & </s> <s xml:id="echoid-s9106" xml:space="preserve">cætera ſociali vitæ <lb/>propria habebunt? </s> <s xml:id="echoid-s9107" xml:space="preserve">leges, Magiſtratus, tecta, urbes, mer-<lb/>caturas aut rerum permutationes? </s> <s xml:id="echoid-s9108" xml:space="preserve">Certe equidem apud bar-<lb/>baros Americæ & </s> <s xml:id="echoid-s9109" xml:space="preserve">inſularum populos, cum primum ad eos <lb/>perventum eſt, eadem hæc fere jam in uſu erant. </s> <s xml:id="echoid-s9110" xml:space="preserve">At non <lb/>propterea negaverim aliter iſta in Planetis cæteris ſe habere <lb/>poſle quam apud nos; </s> <s xml:id="echoid-s9111" xml:space="preserve">cum ex iis quædam ſint quæ abeſſe <lb/>queant à ſocietate animalium ratione præditorum; </s> <s xml:id="echoid-s9112" xml:space="preserve">eoque <lb/>tantum excogitata, ne ratione male utamur & </s> <s xml:id="echoid-s9113" xml:space="preserve">cum aliorum <lb/>injuria, itaque ſocietas ſolvatur. </s> <s xml:id="echoid-s9114" xml:space="preserve">Poſſunt enim in aliis iſtis <lb/>globis in ea rerum abundantia verſari, ut nihil alieni appe-<lb/>tant, rapiantve. </s> <s xml:id="echoid-s9115" xml:space="preserve">Poſſunt ea eſſe æquitate, ut pacem per-<lb/>petuo colant, nec ſibi invicem inſidientur, aut mortem in-<lb/>ferant; </s> <s xml:id="echoid-s9116" xml:space="preserve">imò ut neque oderint nec iraſcantur; </s> <s xml:id="echoid-s9117" xml:space="preserve">quod ſi eſſet, <lb/>multo quàm nos feliciores putandi ſint. </s> <s xml:id="echoid-s9118" xml:space="preserve">Sed veriſimilius eſt, <lb/>ut quemadmodum apud nos, ſic ibi quoque bonis mala, ſa-<lb/>pientiæ ſtultitia, paci bellum miſceatur, nec deſit egeſtas ar-<lb/>tium magiſtra. </s> <s xml:id="echoid-s9119" xml:space="preserve">Quia & </s> <s xml:id="echoid-s9120" xml:space="preserve">ex his utilitatem aliquam proficiſci <lb/>antea oſtendimus; </s> <s xml:id="echoid-s9121" xml:space="preserve">&</s> <s xml:id="echoid-s9122" xml:space="preserve">, ſi nulla eſſet, tamen nec præferendi <lb/>res illorum rebus noſtris cauſam habemus.</s> <s xml:id="echoid-s9123" xml:space="preserve"/> </p> <div xml:id="echoid-div433" type="float" level="2" n="44"> <note position="left" xlink:label="note-0420-01" xlink:href="note-0420-01a" xml:space="preserve">Eos in ſocie-<lb/>tate vivere.</note> </div> <p> <s xml:id="echoid-s9124" xml:space="preserve">Quod autem nunc dicam, audacius, ſcio, videbitur; </s> <s xml:id="echoid-s9125" xml:space="preserve">nec <lb/> <anchor type="note" xlink:label="note-0420-02a" xlink:href="note-0420-02"/> tamen probabilitate caret. </s> <s xml:id="echoid-s9126" xml:space="preserve">Nempe, ſi in ſocietate (quod <lb/>jam penè obtinuimus) vivant gentes Planetarum; </s> <s xml:id="echoid-s9127" xml:space="preserve">etiam, <lb/>præter commoda inde provenientia, voluptate aliqua tali eas <lb/>affici, quali nos, ex congreſſibus colloquiiſque amicorum, <pb o="681" file="0421" n="440" rhead="COSMOTHEOROS, LIB. I."/> amoribus, jocis, ſpectaculis. </s> <s xml:id="echoid-s9128" xml:space="preserve">Hoc, inquam, probabile eſt, <lb/>quia ſi nihil horum Planeticolis concedamus, ſed ſemper eos <lb/>ſeriò, ac ſine omni hilaritate, aut animi remiſſione agere <lb/>putemus; </s> <s xml:id="echoid-s9129" xml:space="preserve">ingens vitæ condimentum, quoque vix illa carere <lb/>poſſit, iis adempturi ſimus, atque ita noſtram hanc beatio-<lb/>rem facturi; </s> <s xml:id="echoid-s9130" xml:space="preserve">contra quam ratio poſtulat.</s> <s xml:id="echoid-s9131" xml:space="preserve"/> </p> <div xml:id="echoid-div434" type="float" level="2" n="45"> <note position="left" xlink:label="note-0420-02" xlink:href="note-0420-02a" xml:space="preserve">Colloquio-<lb/>rum jucun-<lb/>ditate frui.</note> </div> <p> <s xml:id="echoid-s9132" xml:space="preserve">De reliquis vero occupationibus & </s> <s xml:id="echoid-s9133" xml:space="preserve">ſtudiis illorum ut por-<lb/>rò inquiramus; </s> <s xml:id="echoid-s9134" xml:space="preserve">videndum eſt quænam iſtorum, præter ea <lb/>quæ jam diximus, cum noſtris aliquam ſimilitudinem habe-<lb/>re probabile ſit. </s> <s xml:id="echoid-s9135" xml:space="preserve">Domos ſibi eos conſtruere, ideo vel ma-<lb/>xime credere libet, quòd & </s> <s xml:id="echoid-s9136" xml:space="preserve">pluvias in terris illis cadere <lb/>oſtendimus. </s> <s xml:id="echoid-s9137" xml:space="preserve">Sequebatur enim hoc ex eo, quod in Jovis <lb/>Planeta nubium quidam mutabiles tractus cernuntur; </s> <s xml:id="echoid-s9138" xml:space="preserve">vapo-<lb/>res, aquamque haud dubie continentes: </s> <s xml:id="echoid-s9139" xml:space="preserve">quam aliunde quo-<lb/>que illic non deeſſe argumentis adſtruebamus. </s> <s xml:id="echoid-s9140" xml:space="preserve">Erunt ergo <lb/>& </s> <s xml:id="echoid-s9141" xml:space="preserve">imbres & </s> <s xml:id="echoid-s9142" xml:space="preserve">venti, quia attractum à Sole humorem recidere <lb/>in terram neceſſe eſt; </s> <s xml:id="echoid-s9143" xml:space="preserve">& </s> <s xml:id="echoid-s9144" xml:space="preserve">calore ſoluti vapores ventorum cau-<lb/>ſa ſunt; </s> <s xml:id="echoid-s9145" xml:space="preserve">quorum flatus ex illa nubium Jovialium mutabili fa-<lb/>cie cognoſcitur. </s> <s xml:id="echoid-s9146" xml:space="preserve">Adverſus hoc ergo, ut noctes tuto & </s> <s xml:id="echoid-s9147" xml:space="preserve">quie-<lb/> <anchor type="note" xlink:label="note-0421-01a" xlink:href="note-0421-01"/> te tranſigant (habent enim & </s> <s xml:id="echoid-s9148" xml:space="preserve">noctes & </s> <s xml:id="echoid-s9149" xml:space="preserve">ſomnum proinde, <lb/>uti nos) munire ſe eos, caſaſque ac tuguria ædificare, aut <lb/>ſpecus effodere, veriſimile eſt. </s> <s xml:id="echoid-s9150" xml:space="preserve">Atque eo magis quod omne <lb/>genus animalium, apud nos, exceptis piſcibus, ad ſui tute-<lb/>lam hæc molitur. </s> <s xml:id="echoid-s9151" xml:space="preserve">Cur vero caſas & </s> <s xml:id="echoid-s9152" xml:space="preserve">tuguria, & </s> <s xml:id="echoid-s9153" xml:space="preserve">non domos <lb/>amplas & </s> <s xml:id="echoid-s9154" xml:space="preserve">magnificas Planetarum habitatoribus exſtrui creda-<lb/>mus, niſi quod non poſſumus res noſtras non præ omnibus <lb/>pulchras perfectaſque putare. </s> <s xml:id="echoid-s9155" xml:space="preserve">Qui autem nos? </s> <s xml:id="echoid-s9156" xml:space="preserve">Nempe in <lb/>globulo illo vitam agentes, qui non decies milleſimam par-<lb/>tem globorum Saturni aut Jovis æquet, ſi corporum moles <lb/>inter ſe conferantur. </s> <s xml:id="echoid-s9157" xml:space="preserve">Nulla equidem ratio adferri poteſt cur <lb/>non Architecturæ elegantiam, ſymmetriamque, æque cogni-<lb/>tam habeant in iſtis cæteriſque Planetis, ac nos in noſtro: <lb/></s> <s xml:id="echoid-s9158" xml:space="preserve">nec cur non palatia, turres, pyramideſque alicubi noſtris <lb/>multo altiores ſumptuoſioreſque, nec minori concinnitate <lb/>exædificent. </s> <s xml:id="echoid-s9159" xml:space="preserve">Cumque multiplex ſit hominum in his rebus <lb/>induſtria; </s> <s xml:id="echoid-s9160" xml:space="preserve">ut in cædendis lapidibus, coquenda calce & </s> <s xml:id="echoid-s9161" xml:space="preserve">late- <pb o="682" file="0422" n="441" rhead="CHRISTIANI HUGENII"/> ribus; </s> <s xml:id="echoid-s9162" xml:space="preserve">cum ferro, plumbo, vitro utantur, atque ad orna-<lb/>tum auro quoque; </s> <s xml:id="echoid-s9163" xml:space="preserve">his omnibus nihilo inferiora illic haberi <lb/>viriſimile eſt.</s> <s xml:id="echoid-s9164" xml:space="preserve"/> </p> <div xml:id="echoid-div435" type="float" level="2" n="46"> <note position="right" xlink:label="note-0421-01" xlink:href="note-0421-01a" xml:space="preserve">Domos ad-<lb/>verſus plu-<lb/>viam ex-<lb/>ſtruere.</note> </div> <p> <s xml:id="echoid-s9165" xml:space="preserve">Si vero diviſa eſt illis, ſicuti nobis, Globi ſui ſuperficies, <lb/>ut pars terram, pars maria contineat; </s> <s xml:id="echoid-s9166" xml:space="preserve">uti ex ſupra memora-<lb/>tis Jovis obſervationibus colligi poteſt, quia nubes vix ali-<lb/>ter quam ex maris amplis tractibus enaſcerentur; </s> <s xml:id="echoid-s9167" xml:space="preserve">permagna <lb/>ratio eſt ut & </s> <s xml:id="echoid-s9168" xml:space="preserve">navigare eos putemus. </s> <s xml:id="echoid-s9169" xml:space="preserve">Cum alioqui etiam rem <lb/>tantam, tamque utilem, noſtræ Telluris Globo ſoli non abſ-<lb/>que arrogantia aſcripturi ſimus. </s> <s xml:id="echoid-s9170" xml:space="preserve">Præſertim verò in Jovis Sa-<lb/>turnique maribus commoda eſſet navigatio propter Lunarum <lb/>plurium utrobique copiam; </s> <s xml:id="echoid-s9171" xml:space="preserve">quarum ductu longitudinum men-<lb/>ſuram, quam vocant, quæ nobis non contigit, facile con-<lb/>ſequi poſſint. </s> <s xml:id="echoid-s9172" xml:space="preserve">Quod ſi navium uſum habent, quam multa <lb/>præterea habebunt quæ ad eas pertinent; </s> <s xml:id="echoid-s9173" xml:space="preserve">Vela, anchoras, <lb/>funes, trochleas, gubernacula; </s> <s xml:id="echoid-s9174" xml:space="preserve">& </s> <s xml:id="echoid-s9175" xml:space="preserve">horum uſum poculiarem <lb/>quemadmodum nos; </s> <s xml:id="echoid-s9176" xml:space="preserve">ut vento penè contrario navigetur, in <lb/>contrarias vero partes eodem vento facillime. </s> <s xml:id="echoid-s9177" xml:space="preserve">Nec fort ſſe <lb/>nauticæ pyxidis invento carebunt; </s> <s xml:id="echoid-s9178" xml:space="preserve">ſiquidem motus mate@æ <lb/>magneticæ, quæ terræ globum continue pervadit, eſt ejuſmo-<lb/> <anchor type="note" xlink:label="note-0422-01a" xlink:href="note-0422-01"/> di quid, ut cæteris quoque planetis convenire cenſeri poſſit. <lb/></s> <s xml:id="echoid-s9179" xml:space="preserve">Mechanicæ quidem ſcientia, & </s> <s xml:id="echoid-s9180" xml:space="preserve">Aſtronomiæ, in re navali <lb/>neceſſario requiritur, atque adeo utriuſque harum magiſtra <lb/>Geometria, de qua jam ante aliquid attigimus.</s> <s xml:id="echoid-s9181" xml:space="preserve"/> </p> <div xml:id="echoid-div436" type="float" level="2" n="47"> <note position="left" xlink:label="note-0422-01" xlink:href="note-0422-01a" xml:space="preserve">Navigare, <lb/>adeoque & <lb/>artes, quæ eo <lb/>faciunt ex-<lb/>colere.</note> </div> <p> <s xml:id="echoid-s9182" xml:space="preserve">Exiſtimo autem, etiamſi nec ad iſtas attes nec ad alias quaſ-<lb/>dam reſpiciamus, in quibus vel neceſſitas vel occaſio Geo-<lb/>metriæ inveniendæ initium fecerit, non deeſſe rationes, qui-<lb/> <anchor type="note" xlink:label="note-0422-02a" xlink:href="note-0422-02"/> bus veriſimile fiat ejus notitiam Planetarum incolis obtigiſſe. <lb/></s> <s xml:id="echoid-s9183" xml:space="preserve">Sive enim cognitionis ipſius pretium ac dignitas ſpectetur, in <lb/>qua ſingularis quidam intelligentiæ eſt uſus, ac certa indu-<lb/>bitataque veri comprehenſio, quanta in nullis rebus diſcipli-<lb/>niſve aliis reperitu: </s> <s xml:id="echoid-s9184" xml:space="preserve">ſive quod eſt ejuſmodi natura ſua, ac <lb/>talia ejus axiomata & </s> <s xml:id="echoid-s9185" xml:space="preserve">effata, ut quocunque loco & </s> <s xml:id="echoid-s9186" xml:space="preserve">tempo-<lb/>re, aut quibuſcunque in mundis extet, prorſus eadem ubi-<lb/>que eſſe debeat; </s> <s xml:id="echoid-s9187" xml:space="preserve">videtur omnino non ſolis Telluris noſtræ <lb/>incolis res talis parata aut oblata eſſe. </s> <s xml:id="echoid-s9188" xml:space="preserve">Quid quod figuras <pb o="683" file="0423" n="442" rhead="COSMOTHEOROS, LIB. I."/> Geometricas, velut circulos, triangula, polygona, ſphæ-<lb/>ras, multis modis natura ipſa oculis objicit, ad variaſque <lb/>eorum proprietates indagandas quaſi invitat; </s> <s xml:id="echoid-s9189" xml:space="preserve">in quarum <lb/>contemplatione, etiam extra utilitatem omnem, ſumma eſt <lb/>oblectatio. </s> <s xml:id="echoid-s9190" xml:space="preserve">Quis enim non admiratur, cum diſcit ea quæ <lb/>de circulo in Elementis Euclideis, & </s> <s xml:id="echoid-s9191" xml:space="preserve">Apollonii locis Pla-<lb/>nis docentur? </s> <s xml:id="echoid-s9192" xml:space="preserve">aut quæ de ſphæræ ſuperficie & </s> <s xml:id="echoid-s9193" xml:space="preserve">quadratu-<lb/>ra Parabolæ Archimedes prodidit, aut recentiorum ſubti-<lb/>liſſima inventa? </s> <s xml:id="echoid-s9194" xml:space="preserve">Quorum omnium eadem, & </s> <s xml:id="echoid-s9195" xml:space="preserve">ad diſcen-<lb/>dum æque expoſita, eſt veritas in Saturno, ac Jove, atque <lb/>apud nos, & </s> <s xml:id="echoid-s9196" xml:space="preserve">ex iiſdem ſimpliciſſimis principiis pendens, <lb/>quo facilius credi poteſt pulcherrimi jucundiſſimique ſtudii <lb/>in illis ac cæteris planetis aliquos participes eſſe: </s> <s xml:id="echoid-s9197" xml:space="preserve">Etſi præ-<lb/>cipue hoc ſuadet utilitas quæ ex eo in omnem vitam ema-<lb/>nat. </s> <s xml:id="echoid-s9198" xml:space="preserve">Quod ſi jam eo uſque rei Geometricæ peritos qui in <lb/>Planetis ſunt dicerem, ut & </s> <s xml:id="echoid-s9199" xml:space="preserve">Tabulas Sinuum, & </s> <s xml:id="echoid-s9200" xml:space="preserve">Logari-<lb/>thmos, & </s> <s xml:id="echoid-s9201" xml:space="preserve">calculum Analyticum invenerint; </s> <s xml:id="echoid-s9202" xml:space="preserve">abſona ac pe-<lb/>ne ridicula proferre viderer. </s> <s xml:id="echoid-s9203" xml:space="preserve">Nec tamen quidquam obſtat <lb/>quin horum aliquid reperiſſe potuerint, aut aliquando re-<lb/>perturi ſint; </s> <s xml:id="echoid-s9204" xml:space="preserve">atque etiam his noſtris fortaſſe majora. </s> <s xml:id="echoid-s9205" xml:space="preserve">Non <lb/>debemus enim, ut jam ſæpe diximus, præferre nos ipſos <lb/>ac res noſtras rebus Planeticolarum.</s> <s xml:id="echoid-s9206" xml:space="preserve"/> </p> <div xml:id="echoid-div437" type="float" level="2" n="48"> <note position="left" xlink:label="note-0422-02" xlink:href="note-0422-02a" xml:space="preserve">Ut & Geo-<lb/>mesriam.</note> </div> <p> <s xml:id="echoid-s9207" xml:space="preserve">Cæterum illud quod uniuſmodi & </s> <s xml:id="echoid-s9208" xml:space="preserve">æternum in Geome-<lb/>trica ſcientia ineſſe animadvertimus, ſimiliter quoque in <lb/>Harmonicis inveniri certum eſt; </s> <s xml:id="echoid-s9209" xml:space="preserve">cum conſonantiæ omnes <lb/>conſtanti menſura ac proportione conſtituantur; </s> <s xml:id="echoid-s9210" xml:space="preserve">omnis <lb/>vero phtongorum ordo, omniſque cantus delectatio, etiam <lb/>vocis ſingulæ, in conſonantiis fundata ſit. </s> <s xml:id="echoid-s9211" xml:space="preserve">Quo fit ut <lb/> <anchor type="note" xlink:label="note-0423-01a" xlink:href="note-0423-01"/> apud omnes gentes eadem tonorum intervalla canantur, <lb/>ſive per gradus continuos, ſive ſaltu vox progrediatur. <lb/></s> <s xml:id="echoid-s9212" xml:space="preserve">Imo animal quoddam in terris Americæ reperiri fide di-<lb/>gni auctores narrant, quod ſex muſicos tonos deinceps vo-<lb/>ce exprimat: </s> <s xml:id="echoid-s9213" xml:space="preserve">Ut appareat ipſam naturam immutabili ratio-<lb/>ne eos præſcribere. </s> <s xml:id="echoid-s9214" xml:space="preserve">Quandoquidem igitur quæ huc ſpe-<lb/>ctant, certa quoque & </s> <s xml:id="echoid-s9215" xml:space="preserve">unica, & </s> <s xml:id="echoid-s9216" xml:space="preserve">neceſſaria ratione ſe-<lb/>ſe habent, veriſimile eſt, non minus quam Geometriæ, <pb o="684" file="0424" n="443" rhead="CHRISTIANI HUGENII"/> etiam Muſicæ oblectationem ad plures quam ad nos perti-<lb/>nere. </s> <s xml:id="echoid-s9217" xml:space="preserve">Poſitis enim aliis terris atque animalibus ratione & </s> <s xml:id="echoid-s9218" xml:space="preserve"><lb/>auditu pollentibus, cur tantum his noſtris contigiſſet ea vo-<lb/>luptas, quæ ſola ex ſono percipi poteſt? </s> <s xml:id="echoid-s9219" xml:space="preserve">Neſcio equidem <lb/>quantum apud alios valiturum ſit argumentum, quod hic <lb/>ab unitate, & </s> <s xml:id="echoid-s9220" xml:space="preserve">immutabili natura iſtarum artium petiimus; <lb/></s> <s xml:id="echoid-s9221" xml:space="preserve">mihi non leve aut contemnendum videtur, nec multum ei <lb/>cedere, quo in ſuperioribus uſus ſum, cum videndi facul-<lb/>tatem Planetariis animalibus convenire docui.</s> <s xml:id="echoid-s9222" xml:space="preserve"/> </p> <div xml:id="echoid-div438" type="float" level="2" n="49"> <note position="right" xlink:label="note-0423-01" xlink:href="note-0423-01a" xml:space="preserve">Muſicam@</note> </div> <p> <s xml:id="echoid-s9223" xml:space="preserve">Porro ſi tonis harmonicis & </s> <s xml:id="echoid-s9224" xml:space="preserve">cantu delectentur, vix quo-<lb/>que fieri poteſt quin & </s> <s xml:id="echoid-s9225" xml:space="preserve">inſtrumenta quædam muſica repere-<lb/>rint; </s> <s xml:id="echoid-s9226" xml:space="preserve">quoniam & </s> <s xml:id="echoid-s9227" xml:space="preserve">caſu in hujuſmodi inventa incidere con-<lb/>tingit: </s> <s xml:id="echoid-s9228" xml:space="preserve">velut chordis valide contentis, aëris ſono, canna-<lb/>rum aut cicutarum ſibilo. </s> <s xml:id="echoid-s9229" xml:space="preserve">A quibus initiis, ſicuti ad teſtu-<lb/>dines, citharas, tibias, & </s> <s xml:id="echoid-s9230" xml:space="preserve">organa polyplectra nos perve-<lb/>nimus, ita illi quoque non minus elegantia excogitare po-<lb/>tuerint. </s> <s xml:id="echoid-s9231" xml:space="preserve">Sed quemadmodum certi definitique licet ſint to-<lb/>ni, cantuſque intervalla, tamen apud diverſos populos <lb/>alium atque alium eſſe canendi morem ac normam videmus; <lb/></s> <s xml:id="echoid-s9232" xml:space="preserve"> <anchor type="note" xlink:label="note-0424-01a" xlink:href="note-0424-01"/> ut olim apud Dores, Phrygas, Lydos; </s> <s xml:id="echoid-s9233" xml:space="preserve">noſtra ætate apud <lb/>Gallos, Italos, Perſas: </s> <s xml:id="echoid-s9234" xml:space="preserve">ita fieri poteſt ut ab omnibus his <lb/>longius abeat Planetariorum Harmonice, quamvis illorum <lb/>auribus gratiſſima. </s> <s xml:id="echoid-s9235" xml:space="preserve">Cur vero noſtra rudiorem opinemur <lb/>nulla ratio eſt; </s> <s xml:id="echoid-s9236" xml:space="preserve">neque etiam cur non & </s> <s xml:id="echoid-s9237" xml:space="preserve">chromaticis ſonis, <lb/>& </s> <s xml:id="echoid-s9238" xml:space="preserve">quibuſdam Enarmoniis utantur? </s> <s xml:id="echoid-s9239" xml:space="preserve">cum hemitonia quoque <lb/>natura ſuppeditet, certiſque proportionibus definiat. </s> <s xml:id="echoid-s9240" xml:space="preserve">lmo <lb/>ne minus aſſecuti ſint hiſce in rebus quam nos, etiam plu-<lb/>rium vocum aut chordarum concentus, artificioſaque per-<lb/>miſtio, & </s> <s xml:id="echoid-s9241" xml:space="preserve">diſſonantium tonorum, & </s> <s xml:id="echoid-s9242" xml:space="preserve">tritoni, & </s> <s xml:id="echoid-s9243" xml:space="preserve">diapente <lb/>diminutæ uſus iis fortaſſe concedendus ſit. </s> <s xml:id="echoid-s9244" xml:space="preserve">Scio vix ali-<lb/>quam veriſimilitudinem apud multos hæc habitura, ac mi-<lb/>norem etiam, ſi æque doctos dicamus in Jove aut Venere <lb/>incolentes, ac ſunt ii qui in Gallia, Italiave plurimum hac <lb/>arte excellunt. </s> <s xml:id="echoid-s9245" xml:space="preserve">Et tamen fieri poteſt ut vel illis peritiores <lb/>ſint, ac præcipue in parte Theoretica hujus artis ea perſpe-<lb/>xerint, quæ apud noſtrates hoſce parum hactenus intelle- <pb o="685" file="0425" n="444" rhead="COSMOTHEOROS, LIB. I."/> cta ſunt. </s> <s xml:id="echoid-s9246" xml:space="preserve">Si enim ex noſtris Muſicis quæras, cur conſo-<lb/>nantia diapente poſt aliam ſimilem vitioſe ponatur, dicent <lb/>alii nimiam dulcedinem devitari, quæ ex gratiſſimæ con-<lb/>ſonantiæ iteratione naſcatur: </s> <s xml:id="echoid-s9247" xml:space="preserve">alii varietatem in harmonicis <lb/>ſequendam eſſe. </s> <s xml:id="echoid-s9248" xml:space="preserve">Hæc enim præcipui artis auctores, cum-<lb/>que iis Carteſius, adferunt. </s> <s xml:id="echoid-s9249" xml:space="preserve">At Jovis aut Veneris incola <lb/> <anchor type="note" xlink:label="note-0425-01a" xlink:href="note-0425-01"/> forſan veriorem hanc cauſam demonſtrabit, quod à Dia-<lb/>pentè ad aliam deinceps pergendo, tale quid fiat, ac ſi re-<lb/>pente toni ſtatum immutemus; </s> <s xml:id="echoid-s9250" xml:space="preserve">cum Diapente, unà cum <lb/>interjecto ditoni ſono, (qui ſi deſit, mente ſuppletur) <lb/>toni ſpeciem certo conſtituat: </s> <s xml:id="echoid-s9251" xml:space="preserve">hujuſmodi vero ſubita com-<lb/>mutatio auribus merito injucunda inconditaque judicetur; <lb/></s> <s xml:id="echoid-s9252" xml:space="preserve">cum etiam in univerſum ea plerumque durior accidat, <lb/>(præterquam in tranſitu) quæ fit à tribus ſonis conſonis, <lb/>ad trium aliorum harmoniam, nullo priorum manente. </s> <s xml:id="echoid-s9253" xml:space="preserve"><lb/>Sciet etiam ille idem fortaſſe, quod nemo adhuc animad-<lb/>vertit noſtrorum hominum, cur in nullo vocis unius, plu-<lb/>riumve cantu, tonus ſervari poſſit in eadem altitudine ac <lb/>tenore, niſi conſonantia intervalla pleraque ultro, ac ne-<lb/>mine advertente, ita temperentur, ut à perfectione ſumma <lb/>nonnihil deſciſcant. </s> <s xml:id="echoid-s9254" xml:space="preserve">Et cur optimum ſit hoc tempera-<lb/>mentum in chordarum ſyſtemate, cum ex Diapente quarta <lb/>pars commatis ubique deciditur. </s> <s xml:id="echoid-s9255" xml:space="preserve">Quod idem abſque ſen-<lb/>ſibili diſcrimine effici ex diviſione Diapaſon in partes æqua-<lb/>les 31, indeque Cyclum quendam Harmonicum in ſe re-<lb/>deuntem exiſtere, non ita pridem oſtendimus. </s> <s xml:id="echoid-s9256" xml:space="preserve">Quod ta-<lb/>men Planetarum incolæ ſi perſpexerunt, etiam Logari-<lb/>thmorum numeri iis noti eſſe debebunt.</s> <s xml:id="echoid-s9257" xml:space="preserve"/> </p> <div xml:id="echoid-div439" type="float" level="2" n="50"> <note position="left" xlink:label="note-0424-01" xlink:href="note-0424-01a" xml:space="preserve">Quæ tamen <lb/>à noſtra di-<lb/>verſa eſſe <lb/>poſſet.</note> <note position="right" xlink:label="note-0425-01" xlink:href="note-0425-01a" xml:space="preserve">Cur conſo-<lb/>nantia dia-<lb/>pente poſt <lb/>aliam ſimi-<lb/>lem vitioſe <lb/>ponatur.</note> </div> <p> <s xml:id="echoid-s9258" xml:space="preserve">At de Tono vocis temperando quod dixi, probatio-<lb/> <anchor type="note" xlink:label="note-0425-02a" xlink:href="note-0425-02"/> nem habet non difficilem; </s> <s xml:id="echoid-s9259" xml:space="preserve">quam hic adjungimus, quan-<lb/>doquidem jam aliquid præter ſomnia noſtra venditare cœ-<lb/>pimus. </s> <s xml:id="echoid-s9260" xml:space="preserve">Ajo itaque, ſi quis canat deinceps ſonos, quos <lb/>Muſici notant Literis C, F, D, G, C, per intervalla con-<lb/>ſona, omnino perfecta, alternis voce aſcendens deſcen-<lb/>denſque; </s> <s xml:id="echoid-s9261" xml:space="preserve">jam poſteriorem hunc ſonum C, toto Comma-<lb/>te, quod vocant, inferiorem fore C priore, unde cani cœ- <pb o="686" file="0426" n="445" rhead="CHRISTIANI HUGENII"/> pit. </s> <s xml:id="echoid-s9262" xml:space="preserve">Quia nempe ex rationibus intervallorum iſtorum <lb/>perfectis, quæ ſunt 4 ad 3, 5 ad 6, 4 ad 3, 2 ad 3, com-<lb/>ponitur ratio 160 ad 162, hoc eſt 80 ad 81, quæ eſt Com-<lb/>matis. </s> <s xml:id="echoid-s9263" xml:space="preserve">Ut proinde, ſi novies idem hic cantus repetatur, <lb/>jam propemodum tono majore, cujus ratio 8 ad 9. </s> <s xml:id="echoid-s9264" xml:space="preserve">de-<lb/>ſcendiſſe vocem, tonoque excidiſſe oporteat. </s> <s xml:id="echoid-s9265" xml:space="preserve">Hoc ve-<lb/>rò nequaquam patitur aurium ſenſus, ſed toni ab initio <lb/>ſumpti meminit, eodemque revertitur. </s> <s xml:id="echoid-s9266" xml:space="preserve">Itaque cogimur, <lb/>occulto quodam temperamento uti, intervallaque iſta ca-<lb/>nere imperfecta; </s> <s xml:id="echoid-s9267" xml:space="preserve">ex quo multo minor oritur offenſio. <lb/></s> <s xml:id="echoid-s9268" xml:space="preserve">Atque hujuſmodi moderatione fere ubique cantus indi-<lb/>get; </s> <s xml:id="echoid-s9269" xml:space="preserve">uti colligendis rationibus, quemadmodum hic feci-<lb/>mus, facile cognoſcitur. </s> <s xml:id="echoid-s9270" xml:space="preserve">Et hæc quidem in gratiam ar-<lb/>tis illius ſtudioſorum nec Geometriæ rudium exponere <lb/>placuit. </s> <s xml:id="echoid-s9271" xml:space="preserve">Nunc eo unde diſceſſimus revertimur.</s> <s xml:id="echoid-s9272" xml:space="preserve"/> </p> <div xml:id="echoid-div440" type="float" level="2" n="51"> <note position="right" xlink:label="note-0425-02" xlink:href="note-0425-02a" xml:space="preserve">Demon@tra-<lb/>tio tempera-<lb/>menti in to-<lb/>no vocis ad-<lb/>hibendi.</note> </div> <p> <s xml:id="echoid-s9273" xml:space="preserve">Diximus de artibus inventiſque quibuſdam quæ nobiſ-<lb/>cum communia habere Planeticolas veriſimile ſit; </s> <s xml:id="echoid-s9274" xml:space="preserve">præter <lb/>quæ etiam alia exſtare illic neceſſe eſt, ſive ad uſus & </s> <s xml:id="echoid-s9275" xml:space="preserve"><lb/>commoda vitæ facientia ſive ad delectationem. </s> <s xml:id="echoid-s9276" xml:space="preserve">Hæc ve-<lb/>ro quam multa ſint, quantique facienda, ita optime ra-<lb/>tionem inibimus, ſi plurima illa, quæ apud nos reperiun-<lb/>tur, recenſere & </s> <s xml:id="echoid-s9277" xml:space="preserve">ob oculos ponere libuerit.</s> <s xml:id="echoid-s9278" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9279" xml:space="preserve">Expoſui ſupra animantium fruticumque apud nos gene-<lb/>ra quæ plurimum inter ſe figuris differrent: </s> <s xml:id="echoid-s9280" xml:space="preserve">præter quæ, <lb/>minus diſſimilium, ingens copia reperiatur: </s> <s xml:id="echoid-s9281" xml:space="preserve">dixique nihi-<lb/>lo pauciora utriuſque generis, ut longe diverſa, in Pla-<lb/>netarum terris exſtare putandum. </s> <s xml:id="echoid-s9282" xml:space="preserve">Nunc etiam illud vi-<lb/>deamus, quæ utilitas quæve commoda, tum ex animali-<lb/>bus, tum ex herbis arboribuſque ad nos perveniant, ac <lb/>prorſus veriſimile exiſtimemus non minora ex iis, quæ il-<lb/>lic terrarum inveniuntur, ad incolas ipſarum redundare.</s> <s xml:id="echoid-s9283" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9284" xml:space="preserve">Hic vero operæpretium eſt ut quæ ſint divitiæ noſtræ <lb/>inſpiciamus, quæ multæ magnæque ſunt. </s> <s xml:id="echoid-s9285" xml:space="preserve">Nam, præter-<lb/> <anchor type="note" xlink:label="note-0426-01a" xlink:href="note-0426-01"/> quam quod alimenta nobis arborum fructus herbæque ſup-<lb/>peditent; </s> <s xml:id="echoid-s9286" xml:space="preserve">illæ pomis, nucibus; </s> <s xml:id="echoid-s9287" xml:space="preserve">hæ ſeminibus, foliis, ra-<lb/>dicibus; </s> <s xml:id="echoid-s9288" xml:space="preserve">quodque plurimorum ex his in medicina uſus eſb;</s> <s xml:id="echoid-s9289" xml:space="preserve"> <pb o="687" file="0427" n="446" rhead="COSMOTHEOROS, LIB. I."/> petitur ex arboribus materia qua domos naveſque fabrica-<lb/>mus. </s> <s xml:id="echoid-s9290" xml:space="preserve">Et lino veſtes paramus, excogitatis nendi & </s> <s xml:id="echoid-s9291" xml:space="preserve">texen-<lb/>di artificiis. </s> <s xml:id="echoid-s9292" xml:space="preserve">Ex cannabe, ſpartove, fila ac funiculos tor-<lb/>quemus; </s> <s xml:id="echoid-s9293" xml:space="preserve">ex filis vela ac retia conficimus, ex funiculis ru-<lb/>dentes & </s> <s xml:id="echoid-s9294" xml:space="preserve">funes anchorarios. </s> <s xml:id="echoid-s9295" xml:space="preserve">Florum porro odoribus co-<lb/>loribuſque oblectamur; </s> <s xml:id="echoid-s9296" xml:space="preserve">& </s> <s xml:id="echoid-s9297" xml:space="preserve">quamvis ſint etiam qui nares <lb/>offendant, & </s> <s xml:id="echoid-s9298" xml:space="preserve">noxiæ quædam herbæ inveniantur, tamen <lb/>in iis ſæpe boni quid deliteſcit; </s> <s xml:id="echoid-s9299" xml:space="preserve">vel fortaſſe hoc egit na-<lb/>tura ut comparatione mali bona magis eminerent: </s> <s xml:id="echoid-s9300" xml:space="preserve">quod <lb/>multis in rebus ſecuta videtur. </s> <s xml:id="echoid-s9301" xml:space="preserve">Quanta vero ex anima-<lb/>libus eſt utilitas? </s> <s xml:id="echoid-s9302" xml:space="preserve">Oves lanam ad veſtitum præbent, vac-<lb/>cæ lac; </s> <s xml:id="echoid-s9303" xml:space="preserve">utræque carnes ad veſcendum. </s> <s xml:id="echoid-s9304" xml:space="preserve">Aſinis, camelis, <lb/>equis, ad portandas ſarcinas utimur. </s> <s xml:id="echoid-s9305" xml:space="preserve">His etiam ut nos <lb/>vel inſcenſi vehant, vel curribus juncti pertrahant. </s> <s xml:id="echoid-s9306" xml:space="preserve">Ubi <lb/>egregium illud rotarum inventum occurrit, quod libenter <lb/>Planetarum quoque habitatoribus adſcriberem, cum jam in <lb/>ſocietate eos vivere & </s> <s xml:id="echoid-s9307" xml:space="preserve">domos ædificare pene evicerim. <lb/></s> <s xml:id="echoid-s9308" xml:space="preserve">Utrum vero etiam animalibus pro cibo utantur, an Py-<lb/>thagoræ ſimile dogma ſequantur, non habeo quod affir-<lb/>mem. </s> <s xml:id="echoid-s9309" xml:space="preserve">Apparet quidem hoc homini datum eſſe, ut omni-<lb/>bus iis alatur quæ vel in terra vel in aquis naſcuntur, ſi <lb/>quid nutrimenti contineant; </s> <s xml:id="echoid-s9310" xml:space="preserve">ut herbis, pomis, lacte, o-<lb/>vis, melle, piſcibus, volucrum quadrupedumque plurimo-<lb/>rum carnibus. </s> <s xml:id="echoid-s9311" xml:space="preserve">In quo mirum ſane videri poteſt, animal <lb/>illud rationis compos ita eſſe comparatum, ut cum mul-<lb/>torum aliorum pernicie cædeque vivat. </s> <s xml:id="echoid-s9312" xml:space="preserve">Nec tamen natu-<lb/>ræ præſeripto contrarium hoc eſſe putandum eſt, cum pla-<lb/>cuiſſe ei videamus ut leones, lupi, aliaque rapacia, pe-<lb/>cudes & </s> <s xml:id="echoid-s9313" xml:space="preserve">infirmiora quælibet pabuli loco habeant: </s> <s xml:id="echoid-s9314" xml:space="preserve">aquilæ <lb/>columbas leporeſque prædentur: </s> <s xml:id="echoid-s9315" xml:space="preserve">Piſcium permulti piſcicu-<lb/>los ſe minores devorent. </s> <s xml:id="echoid-s9316" xml:space="preserve">Quin & </s> <s xml:id="echoid-s9317" xml:space="preserve">canum varia genera <lb/>ad venandum nobis largita eſt, ut quæ pedibus noſtris <lb/>perſequi nequiremus, illorum celeritate ac ſagacitate con-<lb/>ſequeremur. </s> <s xml:id="echoid-s9318" xml:space="preserve">Præter omnem vero iſtam ex viventibus her-<lb/>biſque utilitatem, hanc quoque delectationem ex iis nos <lb/>capere voluit rerum conditor, ut varias eorum formas na- <pb o="688" file="0428" n="447" rhead="CRISTIANI HUGENII"/> turaſque & </s> <s xml:id="echoid-s9319" xml:space="preserve">generandi vias contemplaremur; </s> <s xml:id="echoid-s9320" xml:space="preserve">in quibus in-<lb/>finita quædam varietas ac mirabilia multa inſunt, quæ a-<lb/>pud naturæ ſcriptores celebrantur. </s> <s xml:id="echoid-s9321" xml:space="preserve">Imo in ipſis inſectis <lb/>quis non miratur apium cellulas hexagonas, aranearum te-<lb/>las; </s> <s xml:id="echoid-s9322" xml:space="preserve">tum bombycum involucra, ex quibus incredibili in-<lb/>duſtria delicatiſſimam veſtem conficimus, eaque copiâ ut <lb/>naves totæ ea onerentur. </s> <s xml:id="echoid-s9323" xml:space="preserve">Atque hæc quidem de herbarum <lb/>animantiumque genere, quatenus homini proſunt, ſumma-<lb/>tim retuliſſe ſufficiat.</s> <s xml:id="echoid-s9324" xml:space="preserve"/> </p> <div xml:id="echoid-div441" type="float" level="2" n="52"> <note position="left" xlink:label="note-0426-01" xlink:href="note-0426-01a" xml:space="preserve">Recenſentur <lb/>commoda <lb/>quæ ad nos-<lb/>perveniunt <lb/>ex animali-<lb/>bus, herbis, <lb/>@rboribus.</note> </div> <p> <s xml:id="echoid-s9325" xml:space="preserve">Cogitetur jam porro quanta ſit ejus ſolertia in reperien-<lb/> <anchor type="note" xlink:label="note-0428-01a" xlink:href="note-0428-01"/> dis, effodiendis, explorandis metallis; </s> <s xml:id="echoid-s9326" xml:space="preserve">itemque in funden-<lb/>dis, repurgandis, miſcendis. </s> <s xml:id="echoid-s9327" xml:space="preserve">Quanta in tenuandis auri <lb/>laminis, aut hydrargyro reſolvendis, ut parvo impendio, <lb/>quæcunque voluerimus, auri ſplendorem coloremque in-<lb/>duant. </s> <s xml:id="echoid-s9328" xml:space="preserve">Quam mira ac multiplex ſit ferri utilitas; </s> <s xml:id="echoid-s9329" xml:space="preserve">quam <lb/>quæ ignorarunt nationes, eæ omnium ferè mechanicarum <lb/>artium rudes vixerunt, proque armis, tantum arcus, cla-<lb/>vas, ſudeſque habuerunt. </s> <s xml:id="echoid-s9330" xml:space="preserve">Nos vero & </s> <s xml:id="echoid-s9331" xml:space="preserve">pulverem ex ſul-<lb/>phure & </s> <s xml:id="echoid-s9332" xml:space="preserve">nitro miſtum habemus, varioſque ejus uſus, qui <lb/>an plus juvet an noceat merito dubitari poteſt. </s> <s xml:id="echoid-s9333" xml:space="preserve">Videba-<lb/>tur enim mira ejus vi, ſimulque artificioſa muniendorum <lb/>oppidorum arte, certius præſidium inventum eſſe, quam <lb/>priſcis temporibus fuerit, adverſus hoſtiles impetus: </s> <s xml:id="echoid-s9334" xml:space="preserve">ſed <lb/>& </s> <s xml:id="echoid-s9335" xml:space="preserve">horum ex eo ſimul violentiam creviſſe videmus, & </s> <s xml:id="echoid-s9336" xml:space="preserve">for-<lb/>titudini viribuſque in præliis multo minus nunc locum eſ-<lb/>ſe quam tunc fuerit. </s> <s xml:id="echoid-s9337" xml:space="preserve">Quod enim olim Imperator Græ-<lb/>cus dixiſſe fertur, Periiſſe Virtutem cum Catapultarum, ac <lb/>Baliſtarum inventa exorirentur, idem nunc majori jure que-<lb/>ri poſſumus; </s> <s xml:id="echoid-s9338" xml:space="preserve">ac maxime Bombis, quos vocant, repertis; <lb/></s> <s xml:id="echoid-s9339" xml:space="preserve">quos non mœnibus, nec ſitu oppida arceſve repellere <lb/>poſſunt, ſed quamvis validæ disjiciuntur, ac ſolo æquan-<lb/>tur. </s> <s xml:id="echoid-s9340" xml:space="preserve">Ut, vel ob hoc unum, melius homines ejus pulve-<lb/>ris invento carituros fuiſſe dicendum ſit. </s> <s xml:id="echoid-s9341" xml:space="preserve">Nec tamen pro-<lb/>pterea prætereundum fuit in commemorandis noſtræ Tel-<lb/>luris repertis, cum veriſimile ſit, etiam in cæteris Planetis, <lb/>noxia artificia quædam cum bonis emerſiſſe.</s> <s xml:id="echoid-s9342" xml:space="preserve"/> </p> <div xml:id="echoid-div442" type="float" level="2" n="53"> <note position="left" xlink:label="note-0428-01" xlink:href="note-0428-01a" xml:space="preserve">Ex Metallis,</note> </div> <pb o="689" file="0429" n="448" rhead="COSMOTHEOROS, LIB. I."/> <p> <s xml:id="echoid-s9343" xml:space="preserve">Auſpicatior eſt aquæ & </s> <s xml:id="echoid-s9344" xml:space="preserve">aëris apud nos uſus: </s> <s xml:id="echoid-s9345" xml:space="preserve">quo & </s> <s xml:id="echoid-s9346" xml:space="preserve">na-<lb/> <anchor type="note" xlink:label="note-0429-01a" xlink:href="note-0429-01"/> vigandi ratio conſtat, & </s> <s xml:id="echoid-s9347" xml:space="preserve">vires comparantur, quibus, nullo <lb/>labore noſtro, molas machinaſque verſemus. </s> <s xml:id="echoid-s9348" xml:space="preserve">At hæ quam <lb/>multiplices, quamque ad varias res adhibentur? </s> <s xml:id="echoid-s9349" xml:space="preserve">Nam & </s> <s xml:id="echoid-s9350" xml:space="preserve">fru-<lb/>menta iis comminuimus, & </s> <s xml:id="echoid-s9351" xml:space="preserve">olea exprimimus, & </s> <s xml:id="echoid-s9352" xml:space="preserve">ligna ſeca-<lb/>mus, & </s> <s xml:id="echoid-s9353" xml:space="preserve">pannos tundendo denſamus; </s> <s xml:id="echoid-s9354" xml:space="preserve">& </s> <s xml:id="echoid-s9355" xml:space="preserve">chartis materiam <lb/>conterimus; </s> <s xml:id="echoid-s9356" xml:space="preserve">quarum aliàs quoque pulcherrimum eſt inven-<lb/>tum, cum ex viliſſimis linteorum ſcrutis, tam pulchra folio-<lb/>tum candidiſſimorum copia paretur. </s> <s xml:id="echoid-s9357" xml:space="preserve">His addatur jam præ-<lb/>clarum illud typographiæ inventum, cujus opera artes omnes <lb/>reliquæ, non ſervantur tantum, ſed & </s> <s xml:id="echoid-s9358" xml:space="preserve">comparantur multo <lb/>quam ante facilius. </s> <s xml:id="echoid-s9359" xml:space="preserve">Item ſculpendi pingendique peritia, a <lb/>parvis rudibuſque initiis eo progreſſa, ut nihil elegantius ab <lb/>hominum ingenio profectum eſſe videatur. </s> <s xml:id="echoid-s9360" xml:space="preserve">Ponatur & </s> <s xml:id="echoid-s9361" xml:space="preserve">vi-<lb/>tri excoquendi ſcientia, atque in tot formas ducendi facili-<lb/>tas. </s> <s xml:id="echoid-s9362" xml:space="preserve">Tum ſpeculorum vitreorum politura, hydrargyrique <lb/>ſuper ea inductio. </s> <s xml:id="echoid-s9363" xml:space="preserve">Ac præcipue quoque vitri uſus mirabi-<lb/>lis, in pervidenda rerum natura, poſt teleſcopii microſco-<lb/>piique inventa. </s> <s xml:id="echoid-s9364" xml:space="preserve">Recenſeantur etiam horologiorum auto-<lb/>matôn fabricæ; </s> <s xml:id="echoid-s9365" xml:space="preserve">aliorum tam exilium, ut geſtanti nihil in-<lb/>commodent; </s> <s xml:id="echoid-s9366" xml:space="preserve">aliorum tam exquiſita æqualitate tempus me-<lb/>tientium, ut nihil ſupra optari poſſit, quibus utriſque inven-<lb/>ta noſtra plurimum profuere.</s> <s xml:id="echoid-s9367" xml:space="preserve"/> </p> <div xml:id="echoid-div443" type="float" level="2" n="54"> <note position="right" xlink:label="note-0429-01" xlink:href="note-0429-01a" xml:space="preserve">Ex aqua, & <lb/>aëre, variiſ-<lb/>que artiſi-<lb/>ciis,</note> </div> <p> <s xml:id="echoid-s9368" xml:space="preserve">Multa addere poſſem de multiplici doctrina & </s> <s xml:id="echoid-s9369" xml:space="preserve">rerum na-<lb/> <anchor type="note" xlink:label="note-0429-02a" xlink:href="note-0429-02"/> turæ cognitione quam præter Geometriæ Aſtronomiæque <lb/>ſcientias conſecuti ſumus, atque ea pleraque noſtra ætate: <lb/></s> <s xml:id="echoid-s9370" xml:space="preserve">velut de gravitate aëris ac vi qua compreſſus reſilit. </s> <s xml:id="echoid-s9371" xml:space="preserve">De ſin-<lb/>gularibus Chymicorum experimentis; </s> <s xml:id="echoid-s9372" xml:space="preserve">è quibus liquores in-<lb/>flammabiles, nuperque ultro lucentes, ac levi tractatione ar-<lb/>dentes, prodierunt. </s> <s xml:id="echoid-s9373" xml:space="preserve">De ſanguinis circuitu per arterias ve-<lb/>naſque, qui antea intelligebatur, nuper vero & </s> <s xml:id="echoid-s9374" xml:space="preserve">oculis uſur-<lb/>pari cœpit, adhibito microſcopio, in piſcium quorundam <lb/>caudis extremis. </s> <s xml:id="echoid-s9375" xml:space="preserve">Item de generatione animalium, quod in-<lb/>ventum eſt nulla niſi ex ſimilium ſemine naſci; </s> <s xml:id="echoid-s9376" xml:space="preserve">idque de her-<lb/>bis quoque verum eſſe. </s> <s xml:id="echoid-s9377" xml:space="preserve">Quodque in ſemine marium repe-<lb/>riuntur animalculorum myriades vivaciſſimorum, quæ ipſam <pb o="690" file="0430" n="449" rhead="CHRISTIANI HUGENII COSMOTH. L. I."/> animantium ſobolem eſſe veriſimillimum ſit: </s> <s xml:id="echoid-s9378" xml:space="preserve">res mirabilis, <lb/>atque ab omni ævo incognita.</s> <s xml:id="echoid-s9379" xml:space="preserve"/> </p> <div xml:id="echoid-div444" type="float" level="2" n="55"> <note position="right" xlink:label="note-0429-02" xlink:href="note-0429-02a" xml:space="preserve">Ex iis, qua@ <lb/>noſtra ætate <lb/>inventa <lb/>ſunt.</note> </div> <p> <s xml:id="echoid-s9380" xml:space="preserve">Jam vero poſtquam hæc omnia accumulavimus Telluris <lb/> <anchor type="note" xlink:label="note-0430-01a" xlink:href="note-0430-01"/> incolarum inventa, putemus fieri quidem poſſe, ut quædam <lb/>eorum etiam apud Planetarios extent; </s> <s xml:id="echoid-s9381" xml:space="preserve">credibile tamen eſſe <lb/>maximam partem eorum illis ignorari. </s> <s xml:id="echoid-s9382" xml:space="preserve">At iis quæ non ha-<lb/>bent rependendis æque multa, pulchraque & </s> <s xml:id="echoid-s9383" xml:space="preserve">utilia, admira-<lb/>tione digna iis tributa eſſe oportet. </s> <s xml:id="echoid-s9384" xml:space="preserve">Quanquam igitur ibi <lb/>terrarum aliquos ratione præditos, & </s> <s xml:id="echoid-s9385" xml:space="preserve">Geometras, & </s> <s xml:id="echoid-s9386" xml:space="preserve">Muſi-<lb/>cos reperiri probabilibus argumentis oſtenderimus, & </s> <s xml:id="echoid-s9387" xml:space="preserve">in ſo-<lb/>cietate communitateque viventes, & </s> <s xml:id="echoid-s9388" xml:space="preserve">manibus pedibuſque in-<lb/>ſtructos, tectiſque & </s> <s xml:id="echoid-s9389" xml:space="preserve">mœnibus munitos: </s> <s xml:id="echoid-s9390" xml:space="preserve">non tamen dubi-<lb/>tandum eſt, quin & </s> <s xml:id="echoid-s9391" xml:space="preserve">formæ, & </s> <s xml:id="echoid-s9392" xml:space="preserve">rerum quas agunt novitate, <lb/>mirabile ſupra quam dici poſſit futurum ſit ſpectaculum, ſi <lb/>quis Mercurius, aut potens Genius eò nos deducat. </s> <s xml:id="echoid-s9393" xml:space="preserve">Sed <lb/>cum ejus itineris conficiendi ſpes omnis adempta ſit, id u-<lb/>num tamen, quod poſſumus inveſtigare non pigebit; </s> <s xml:id="echoid-s9394" xml:space="preserve">qua-<lb/>lis nempe cæleſtium rerum facies ſeſe offerat, in unoquoque <lb/>iſtorum globorum vitam agentibus, cum ad eam hoc quo-<lb/>que pertineat. </s> <s xml:id="echoid-s9395" xml:space="preserve">Simul vero & </s> <s xml:id="echoid-s9396" xml:space="preserve">de præſtantia cujuſque, tum <lb/>ob magnitudinem, tum ob adjunctum comitum lunarum nu-<lb/>merum, quædam ſcitu digna referemus, ac ſtellarum deni-<lb/>que inerrantium incredibilem diſtantiam nova ratione in-<lb/>dagabimus. </s> <s xml:id="echoid-s9397" xml:space="preserve">Sed à longa attentaque meditatione requieſce-<lb/>mus hic paulum, finemque huic Libro imponemus.</s> <s xml:id="echoid-s9398" xml:space="preserve"/> </p> <div xml:id="echoid-div445" type="float" level="2" n="56"> <note position="left" xlink:label="note-0430-01" xlink:href="note-0430-01a" xml:space="preserve">Illa omnia <lb/>veriſimiliter <lb/>non extare <lb/>in Planetis, <lb/>ſed aliis æque <lb/>dignis repen-<lb/>di.</note> </div> <figure> <image file="0430-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0430-01"/> </figure> <pb o="691" file="0431" n="450"/> <figure> <image file="0431-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0431-01"/> </figure> </div> <div xml:id="echoid-div447" type="section" level="1" n="157"> <head xml:id="echoid-head218" xml:space="preserve">CHRISTIANI HUGENII <lb/>COSMOTHEOROS, <lb/>SIVE <lb/>De Terris Cœleſtibus, earumque ornatu, <lb/>Conjecturæ. <lb/>AD <lb/>CONSTANTINUM HUGENIUM, <lb/>Fratrem.</head> <head xml:id="echoid-head219" xml:space="preserve">LIBER II.</head> <p> <s xml:id="echoid-s9399" xml:space="preserve">CUm ante annos complures Librum Athanaſii <lb/>Kircheri, qui Iter Ecſtaticum inſcribitur, <lb/>evolverem; </s> <s xml:id="echoid-s9400" xml:space="preserve">in quo de Natura Siderum, re-<lb/>buſque in Planetarum ſuperficie extantibus, <lb/>diſſeritur; </s> <s xml:id="echoid-s9401" xml:space="preserve">mirabar nihil illic adferri eorum <lb/>quæ mihi jam ab illo tempore circa hæc, <lb/>tanquam valde probabilia, occurrebant: </s> <s xml:id="echoid-s9402" xml:space="preserve">ſed longè alia tra-<lb/>di, inania pleraque, & </s> <s xml:id="echoid-s9403" xml:space="preserve">à ratione aliena. </s> <s xml:id="echoid-s9404" xml:space="preserve">Quod magis etiam <lb/>intellexi, cum conſcriptis ſuperioribus idem opus denuo <lb/>percurrerem. </s> <s xml:id="echoid-s9405" xml:space="preserve">Jamque viſum eſt aliquid eſſe conjecturas no-<lb/>ſtras, ac ponderis nonnihil iis accedere, ſi cum Kircheria-<lb/>nis conferantur. </s> <s xml:id="echoid-s9406" xml:space="preserve">Quod ut judicari poſſit, utque appareat <lb/>quàm de his rebus fruſtra philoſophari conentur, qui funda-<lb/>menta unica veriſimilitudinis, quibus uſi ſumus, rejiciunt; <lb/></s> <s xml:id="echoid-s9407" xml:space="preserve">non abs re erit de opere illo quædam annotaſſe.</s> <s xml:id="echoid-s9408" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9409" xml:space="preserve">Is igitur Vir optimus, Genio quodam duce, per cæli <lb/> <anchor type="note" xlink:label="note-0431-01a" xlink:href="note-0431-01"/> <pb o="692" file="0432" n="451" rhead="CHISTIANI HUGENII"/> ſpatia, ſtellaſque ſe circumferri fingens, partim ea quæ ex <lb/>Aſtronomorum ſcriptis hauſerat, partim quæ ipſe de Plane-<lb/>tarum terris meditatus erat, ac vulgo probari poſſe putabat, <lb/>quaſi viſa enarrat. </s> <s xml:id="echoid-s9410" xml:space="preserve">Antequam verò iter longinquum ingre-<lb/>diatur, hæc duo tanquam certo tenenda ſtatuit ſancitque; <lb/></s> <s xml:id="echoid-s9411" xml:space="preserve">nullum videlicet Telluri motum eſſe tribuendum; </s> <s xml:id="echoid-s9412" xml:space="preserve">tum nihil <lb/>in Planetarum globis Deum extare voluiſſe, quod vita aut <lb/>ſenſu præditum ſit, adeoque nec herbas quidem. </s> <s xml:id="echoid-s9413" xml:space="preserve">Itaque, <lb/>relicto Copernici ſyſtemate, Tychonicum ſibi quod ſequa-<lb/>tur deligit. </s> <s xml:id="echoid-s9414" xml:space="preserve">Sed cum ſtellas inerrantes pro totidem Solibus <lb/>habeat, iiſque ſingulis ſuos Planetas circumponat; </s> <s xml:id="echoid-s9415" xml:space="preserve">hoc ipſo <lb/>(quod an ſenſerit neſcio) infinita numero jam exoriuntur <lb/>ei Copernicea ſyſtemata. </s> <s xml:id="echoid-s9416" xml:space="preserve">Quæ quidem perabſurdè, præter <lb/>ſibi proprios motus, univerſa circum Tellurem noſtram, vi-<lb/>ginti quatuor horis, immani celeritate converti facit. </s> <s xml:id="echoid-s9417" xml:space="preserve">Cum-<lb/>que horum maximam partem fateatur extra hominum con-<lb/>ſpectum eſſe remotam, in hoc quoque incidit incommodum, <lb/>ut fruſtra tot Soles lucere dicendi ſint, fruſtraque calorem <lb/>ſuum impertiri tot globis Telluri ſimilibus, elementaque ea-<lb/>dem, (ita enim vult) & </s> <s xml:id="echoid-s9418" xml:space="preserve">cætera omnia habentibus, præter <lb/>ſtirpes & </s> <s xml:id="echoid-s9419" xml:space="preserve">animalia. </s> <s xml:id="echoid-s9420" xml:space="preserve">Atque hinc porro ad alia magis abſona <lb/>delabitur. </s> <s xml:id="echoid-s9421" xml:space="preserve">Nam quia ne Planetarum quidem, qui noſtro <lb/>ſyſtemate continentur, alium ullum reperit uſum, ad diu <lb/>exploſas Aſtrologorum ineptias ſe convertit; </s> <s xml:id="echoid-s9422" xml:space="preserve">& </s> <s xml:id="echoid-s9423" xml:space="preserve">hoc fine tot <lb/>tantaſque corporum moles conditas eſſe vult, ut influxu eo-<lb/>rum vario, certiſque legibus temperato, mundi univerſitas <lb/>conſervetur, incolumiſque perduret: </s> <s xml:id="echoid-s9424" xml:space="preserve">utque præterea in ho-<lb/>minum animos iidem influxus vires ſuas exerceant. </s> <s xml:id="echoid-s9425" xml:space="preserve">Itaque, <lb/>in Aſtrologicæ artis gratiam, in Veneris Planeta jucundam <lb/>pulchramque rerum faciem ſibi oblatam narrat; </s> <s xml:id="echoid-s9426" xml:space="preserve">cum luce <lb/>blanda, undis dulciter fluctuantibus, odoribus ſuaviſſimis, <lb/>atque undique fulgentibus cryſtallis. </s> <s xml:id="echoid-s9427" xml:space="preserve">In Jove auras ſalubres, <lb/>ac ſuaveolentes, aquas limpidiſſimas, terras argentei ſplen-<lb/>doris. </s> <s xml:id="echoid-s9428" xml:space="preserve">Quò nimirum, ab influxu hujus utriuſque ſideris, <lb/>fauſta ac ſalutaria omnia in Terram homineſque deriventur; </s> <s xml:id="echoid-s9429" xml:space="preserve"><lb/>ut vel pulchros & </s> <s xml:id="echoid-s9430" xml:space="preserve">amabiles, vel ad prudentiam & </s> <s xml:id="echoid-s9431" xml:space="preserve">gravita- <pb o="693" file="0433" n="452" rhead="COSMOTHEOROS, LIB. II."/> tem propenſos reddat. </s> <s xml:id="echoid-s9432" xml:space="preserve">In Mercurio neſcio quid ſerenum <lb/>vividumque, unde ingenium ac ſolertia naſcentibus inſinue-<lb/>tur. </s> <s xml:id="echoid-s9433" xml:space="preserve">At in Marte omnia tetra, exitialia, fætida, piceas <lb/>flammas, fumoſque ſe vidiſſe memorat. </s> <s xml:id="echoid-s9434" xml:space="preserve">In Saturno triſtia, <lb/>horrenda, ſquallida, caliginoſa; </s> <s xml:id="echoid-s9435" xml:space="preserve">ur ex his Planetis, (neſcio <lb/>quare Apoteleſmaticis omnibus inviſis) influxus maligni in-<lb/>feſtique mundo ac mortalibus eveniant; </s> <s xml:id="echoid-s9436" xml:space="preserve">niſi tamen beni-<lb/>gniorum illorum radiis corrigi ac mitigari eos contingat. <lb/></s> <s xml:id="echoid-s9437" xml:space="preserve">Hæc nempe & </s> <s xml:id="echoid-s9438" xml:space="preserve">his ſimilia Genio illi cæleſti comes adhærens <lb/>diſcit. </s> <s xml:id="echoid-s9439" xml:space="preserve">Quem & </s> <s xml:id="echoid-s9440" xml:space="preserve">ſerio reſpondere facit cum interrogatur, <lb/>anne aquis, quæ in Veneris Planeta fluunt, Hebræus aut <lb/>Paganus quiſpiam, eo delatus, ritè baptizari queat. </s> <s xml:id="echoid-s9441" xml:space="preserve">Eo-<lb/>dem quoque docente Magiſtro, intelligit cælum ſtelliferum <lb/>non eſſe ex materia ſolida conflatum, ſed liquidum prorſus, <lb/>in quo ſtellæ Soleſve innumeri longè latéque ſpargantur; </s> <s xml:id="echoid-s9442" xml:space="preserve"><lb/>nuſquam alligati, (& </s> <s xml:id="echoid-s9443" xml:space="preserve">hactenus rectè) quique omnes diei <lb/>ſpatio vaſtiſſimos, ut dixi, circuitus peragant. </s> <s xml:id="echoid-s9444" xml:space="preserve">Quo in mo-<lb/>tu, ſi talis foret, non advertit quanta vi illi undique diffu-<lb/>gituri ſint, ob motum circularem tam immenſæ celeritatis. </s> <s xml:id="echoid-s9445" xml:space="preserve"><lb/>Sed ne ſic avolent, inque ſpatia infinita recedant, Intelli-<lb/>gentiæ motrices, credo, impedient. </s> <s xml:id="echoid-s9446" xml:space="preserve">Etenim unicuique <lb/>ſtellæ fixæ, imo & </s> <s xml:id="echoid-s9447" xml:space="preserve">Planetæ, Intelligentias aut Angelos ſuos <lb/>adjungit, qui impellant eos, curſumque moderentur. </s> <s xml:id="echoid-s9448" xml:space="preserve">In <lb/>quo Doctorum quorundam turbam ſequitur, qui vaniſſimum <lb/>Ariſtotelis commentum inconſideratè, invitaque ratione, <lb/>adoptarunt. </s> <s xml:id="echoid-s9449" xml:space="preserve">Iſtos verò beatos Genios labore tanto Coper-<lb/>nicus liberat, ſolius Terræ inducto motu; </s> <s xml:id="echoid-s9450" xml:space="preserve">cujus ſanè neceſ-<lb/>ſitatem, vel ex hoc uno, omnes vident, niſi qui ultrò, ac <lb/>volentes, cæcutiunt. </s> <s xml:id="echoid-s9451" xml:space="preserve">Equidem cogitavi nonnunquam, me-<lb/>liora à Kirchero exſpectari potuiſſe, ſi, quæ ſentiebat, li-<lb/>berè exponere auſus fuiſſet. </s> <s xml:id="echoid-s9452" xml:space="preserve">Sed cum hoc non auderet, <lb/>neſcio cur non in totum illo argumento abſtinere maluerit. </s> <s xml:id="echoid-s9453" xml:space="preserve"><lb/>Sed hunc celeberrimum ſcriptorem jam omittamus: </s> <s xml:id="echoid-s9454" xml:space="preserve">&</s> <s xml:id="echoid-s9455" xml:space="preserve">, quan-<lb/>doquidem nil veriti ſumus, conjecturis noſtris, ſpectatores <lb/>in Planetis ponere, adeamus nunc, uti propoſitum fuerat, <pb o="694" file="0434" n="453" rhead="CHRISTIANI HUGENII"/> ſingulos; </s> <s xml:id="echoid-s9456" xml:space="preserve">& </s> <s xml:id="echoid-s9457" xml:space="preserve">quinam ſint anni eorum, qui dies, quæ denique <lb/>Aſtronomia, deinceps conſideremus.</s> <s xml:id="echoid-s9458" xml:space="preserve"/> </p> <div xml:id="echoid-div447" type="float" level="2" n="1"> <note position="right" xlink:label="note-0431-01" xlink:href="note-0431-01a" xml:space="preserve">Kircheri iter <lb/>extaticum <lb/>examinatur.</note> </div> <p> <s xml:id="echoid-s9459" xml:space="preserve">Itaque, ut ab intimo, & </s> <s xml:id="echoid-s9460" xml:space="preserve">Soli viciniore incipiam, ſcimus <lb/> <anchor type="note" xlink:label="note-0434-01a" xlink:href="note-0434-01"/> Mercurium triplo propius circiter quam Tellurem noſtram <lb/>ad ingens illud ſidus accedere. </s> <s xml:id="echoid-s9461" xml:space="preserve">Cui conſequens eſt ut triplo <lb/>quoque majus id conſpiciant ejus incolæ, ratione diametri, <lb/>lumen vero & </s> <s xml:id="echoid-s9462" xml:space="preserve">calorem ejus ſentiant noncuplo quam nos ma-<lb/>jorem. </s> <s xml:id="echoid-s9463" xml:space="preserve">Nobis proinde intolerabilem, quique accenſurus ſit <lb/>ſiccatas herbas, fœnum ſtramenque, qualia apud nos cre-<lb/>ſcunt. </s> <s xml:id="echoid-s9464" xml:space="preserve">At nihil impedit ita comparata eſſe, quæ ibi vivunt <lb/>animantia, ut optatam temperiem in ardore illo experian-<lb/>tur. </s> <s xml:id="echoid-s9465" xml:space="preserve">Herbas vero eſſe ea natura, ut multo magis vim calo-<lb/>ris perferant. </s> <s xml:id="echoid-s9466" xml:space="preserve">Nec mirum eſſet iſtos Mercurii indigenas pu-<lb/>tare non ferendo frigore nos urgeri, luceque frui exigua, <lb/>qui tanto longius a Sole abſimus. </s> <s xml:id="echoid-s9467" xml:space="preserve">Sicut nos de Saturni co-<lb/>lonis facile nobis perſuademus. </s> <s xml:id="echoid-s9468" xml:space="preserve">Non deeſt verò dubitandi <lb/>ratio, cum à calore vita pendeat, iſque corpori mentique <lb/>vigorem alacritatemque præſtet; </s> <s xml:id="echoid-s9469" xml:space="preserve">an non, propter Solis vi-<lb/>ciniam, Hermopolitæ illi nobis ingenio præſtare putandi <lb/>ſint? </s> <s xml:id="echoid-s9470" xml:space="preserve">Sed quo minus huic cauſæ tribuam facit, quod calidiſ-<lb/>ſimas terræ noſtræ regiones ſortitos, Africæ, Braſiliæque <lb/>populos, nec ſapientia nec induſtria æquare videmus tem-<lb/>peratiorum tractuum incolas; </s> <s xml:id="echoid-s9471" xml:space="preserve">ut vel ex eo perſpicitur, quod <lb/>in omnium ſcientiarum ac fere artium ignoratione verſentur: <lb/></s> <s xml:id="echoid-s9472" xml:space="preserve">cum nec nauticæ rei, qui circum littora incolunt, niſi pere-<lb/>xiguam notitiam habeant. </s> <s xml:id="echoid-s9473" xml:space="preserve">Nollem quoque Jovicolis, Sa-<lb/>turnicoliſque hebetes, plumbeaſque mentes, intelligentiam-<lb/>ve tribuere noſtra minorem, propterea quod tantò longius <lb/>a Sole remoti vivunt; </s> <s xml:id="echoid-s9474" xml:space="preserve">cum uterque globus iſte tam præſtan-<lb/>ti ſit magnitudine, tantoque comitatu ſtipatus feratur. </s> <s xml:id="echoid-s9475" xml:space="preserve">Qua-<lb/>lis porro ſit Mercurialibus Aſtronomia, utque cæteros Pla-<lb/>netas certis temporibus Soli oppoſitos ſpectent, ex figura <lb/>ſyſtematis, priore libro expoſita, perfacile eſt intelligere. </s> <s xml:id="echoid-s9476" xml:space="preserve"><lb/>Atque his oppoſitionum temporibus Venerem ac Tellurem <lb/>præcipuo ſplendore illic effulgere neceſſe eſt. </s> <s xml:id="echoid-s9477" xml:space="preserve">Nam cum <pb o="695" file="0435" n="454" rhead="COSMOTHEOROS, LIB. II."/> deo lucida nobis Venus appareat, quo tempore tenuem <lb/>naſcentis Lunæ faciem refert; </s> <s xml:id="echoid-s9478" xml:space="preserve">oportet eam ſextuplo aut am-<lb/>plius clariorem cerni, cum Soli opponitur, ex Mercurii <lb/>Globo pleno orbe ſpectatam, & </s> <s xml:id="echoid-s9479" xml:space="preserve">minore quoque intervallo <lb/>diſtantem: </s> <s xml:id="echoid-s9480" xml:space="preserve">atque ita tunc non parum diſpellere nocturnas te-<lb/>nebras gentibus iſtis, Lunæ auxilio carentibus. </s> <s xml:id="echoid-s9481" xml:space="preserve">Quænam <lb/>ſint denique apud eos dierum ſpatia, & </s> <s xml:id="echoid-s9482" xml:space="preserve">an varias anni tem-<lb/>peſtates experiantur, incompertum eſt hactenus, quod igno-<lb/>retur an axem diurnæ converſionis ad orbem, quo circa So-<lb/>lem defertur, obliquum habeat, & </s> <s xml:id="echoid-s9483" xml:space="preserve">quanto tempore conver-<lb/>ſio ea peragatur. </s> <s xml:id="echoid-s9484" xml:space="preserve">Neque enim dubitari debet de diebus no-<lb/>ctibuſque eorum cum in Tellure, Marte, Jove ac Saturno <lb/>hæc viciſſitudo certò cognoſcatur. </s> <s xml:id="echoid-s9485" xml:space="preserve">Anni vero ſpatium vix <lb/>quartam partem noſtri æquare illic conſtat.</s> <s xml:id="echoid-s9486" xml:space="preserve"/> </p> <div xml:id="echoid-div448" type="float" level="2" n="2"> <note position="left" xlink:label="note-0434-01" xlink:href="note-0434-01a" xml:space="preserve">Apparens <lb/>qualis ſit <lb/>conſtitutio <lb/>Solis & Pla-<lb/>netarum in <lb/>Mercurio.</note> </div> <p> <s xml:id="echoid-s9487" xml:space="preserve">In Veneris globo poſitis, eadem fere in cælo apparere <lb/> <anchor type="note" xlink:label="note-0435-01a" xlink:href="note-0435-01"/> neceſſe eſt quæ de Mercurio diximus, niſi quod hunc nun-<lb/>quam videt Soli oppoſitum, cum non niſi 38 circiter gradi-<lb/>bus ab eo recedat. </s> <s xml:id="echoid-s9488" xml:space="preserve">Sol vero illis major apparet quam no-<lb/>bis, diametro ſeſcupla, orbe plus quam duplo; </s> <s xml:id="echoid-s9489" xml:space="preserve">quo & </s> <s xml:id="echoid-s9490" xml:space="preserve">bis <lb/>tantum caloris luciſque præbere cum oportet. </s> <s xml:id="echoid-s9491" xml:space="preserve">Itaque pro-<lb/>pius ad noſtræ temperiem Tellus iſta accedit. </s> <s xml:id="echoid-s9492" xml:space="preserve">At annus <lb/>menſibus noſtris ſeptem cum dimidio fere finitur. </s> <s xml:id="echoid-s9493" xml:space="preserve">Noctu <lb/>verò globus hic noſter, in locis Soli oppoſitis, multo luci-<lb/>dior Veneri apparere debet quàm unquam nobis appareat <lb/>Venus; </s> <s xml:id="echoid-s9494" xml:space="preserve">ac tunc Lunam quoque, perpetuum comitem no-<lb/>ſtrum, facile conſpiciunt, ſi modo oculos habent noſtris non <lb/>imbecilliores. </s> <s xml:id="echoid-s9495" xml:space="preserve">Sæpe autem in Venere miratus ſum, cum tu-<lb/>bis longioribus, pedum 45, aut 60, eam inſpicerem Terræ <lb/>propinquam; </s> <s xml:id="echoid-s9496" xml:space="preserve">Lunæque ſemiplenæ ſimilem, aut jam in cor-<lb/>nua curvari incipienti; </s> <s xml:id="echoid-s9497" xml:space="preserve">prorſus æquabili ſplendore ſuperfi-<lb/>ciem ejus perfundi: </s> <s xml:id="echoid-s9498" xml:space="preserve">ut vix dicere audeam, aliquid maculæ <lb/>ſimile, in ea me animadvertiſſe; </s> <s xml:id="echoid-s9499" xml:space="preserve">cujuſmodi in Jove & </s> <s xml:id="echoid-s9500" xml:space="preserve">Mar-<lb/>te manifeſtè notantur, licet orbe multo minore ſeſe offeren-<lb/>tibus. </s> <s xml:id="echoid-s9501" xml:space="preserve">Si enim maria ac terras habet Veneris globus, obſcu-<lb/>riores nobis maris tractus conſpici deberent; </s> <s xml:id="echoid-s9502" xml:space="preserve">terrarum vero <lb/>clariores; </s> <s xml:id="echoid-s9503" xml:space="preserve">ſicuti ex præaltis rupibus inſpectum deſuper ma- <pb o="696" file="0436" n="455" rhead="CHRISTIANI HUGENII"/> re, non perinde, ac adjacentes terræ, lucidum apparet. <lb/></s> <s xml:id="echoid-s9504" xml:space="preserve">Credebam nimium Veneris fulgorem in cauſa eſſe, quo mi-<lb/>nus diverſitas lucis animadverti poſſet. </s> <s xml:id="echoid-s9505" xml:space="preserve">Sed cum fumo infe-<lb/>ciſſem vitrum oculo proximum, ad auferendam, partem ra-<lb/>diorum, nihilo minus æqualis in tota ſuperficie lux viſa eſt. </s> <s xml:id="echoid-s9506" xml:space="preserve"><lb/>An igitur nulla ibi maria, an Solis lucem magis quam apud <lb/>nos aquæ, aut minus terræ repercutiunt? </s> <s xml:id="echoid-s9507" xml:space="preserve">an potius, (quod <lb/>credibilius mihi videtur) denſior ibi, quam in Jove aut Mar-<lb/>te, Vaporum regio à Sole illuſtrata, Veneriſque globum <lb/>circundans, omnem fere illam quam videmus lucem ad nos <lb/>remittit, vixque ſubjectorum ſibi marium terrarumque diſcri-<lb/>men percipi ſinit? </s> <s xml:id="echoid-s9508" xml:space="preserve">Nam certum eſt noſtram quoque atmo-<lb/>ſphæram, ſi Tellurem procul intueri daretur, plurimum ob-<lb/>ſtituram luce ſua, quo minus terræ mariſque tam diverſa cla-<lb/>ritas apparere poſſet, quam quæ cernitur ex edito ſcopulo <lb/>deſpicienti. </s> <s xml:id="echoid-s9509" xml:space="preserve">Eadem ratione qua Lunæ quoque maculas in-<lb/>terdiu minus aperte quam noctu animadverti ſinunt vapores <lb/>iidem; </s> <s xml:id="echoid-s9510" xml:space="preserve">quoniam tunc quoque inter illam oculoſque noſtros <lb/>interpoſiti, Soliſque luce illuſtres, viſui officiunt: </s> <s xml:id="echoid-s9511" xml:space="preserve">noctu <lb/>non item.</s> <s xml:id="echoid-s9512" xml:space="preserve"/> </p> <div xml:id="echoid-div449" type="float" level="2" n="3"> <note position="right" xlink:label="note-0435-01" xlink:href="note-0435-01a" xml:space="preserve">Qualis in <lb/>Venere.</note> </div> <p> <s xml:id="echoid-s9513" xml:space="preserve">At in Marte reliquis diſci partibus obſcuriores, ut jam <lb/> <anchor type="note" xlink:label="note-0436-01a" xlink:href="note-0436-01"/> dixi, maculæ notantur. </s> <s xml:id="echoid-s9514" xml:space="preserve">Ex quarum recurſibus pridem fuit <lb/>obſervatum dies nocteſque illic iiſdem fere quibus apud nos <lb/>intervallis reverti. </s> <s xml:id="echoid-s9515" xml:space="preserve">Hyemem vero æſtatemque exiguo diſcri-<lb/>mine incolæ ſentiunt, eo quod axis diurnæ converſionis pau-<lb/>lum duntaxat ad orbitam Planetæ inclinatur, ut ex motu <lb/>macularum intellectum eſt. </s> <s xml:id="echoid-s9516" xml:space="preserve">Qui autem ex globo illo Tellu-<lb/>rem noſtram intuentur, eodem modo fere, ac Venus no-<lb/>bis, apparere iis debet, formaſque lunaribus ſimiles oſten-<lb/>dere, ſi teleſcopio ſpectetur; </s> <s xml:id="echoid-s9517" xml:space="preserve">nec ultra gradus 48 à Sole eva-<lb/>gari; </s> <s xml:id="echoid-s9518" xml:space="preserve">in cujus diſco etiam conſpici quandoque poſſit, uti & </s> <s xml:id="echoid-s9519" xml:space="preserve"><lb/>Veneris Mercuriique corpuſcula. </s> <s xml:id="echoid-s9520" xml:space="preserve">Et hoc quidem nunquam <lb/>aliàs; </s> <s xml:id="echoid-s9521" xml:space="preserve">Venus raro iis apparere debet, uti nobis Mercurius. <lb/></s> <s xml:id="echoid-s9522" xml:space="preserve">Terræ vero ſolum in Marte nigriore materia conſtare veriſi-<lb/>mile eſt, quàm in Jove, aut etiam Luna noſtra; </s> <s xml:id="echoid-s9523" xml:space="preserve">eoque fieri <lb/>ut rubicundior Mars ſpectetur, nec, pro ratione intervalli <pb o="697" file="0437" n="456" rhead="COSMOTHEOROS, LIB. II."/> quo à Sole abeſt, lucem remittat. </s> <s xml:id="echoid-s9524" xml:space="preserve">Minor verò eſt globus <lb/>ejus quam ſtellæ Veneris, licet à Sole longius diſtans, ut <lb/>jam ſupra animadvertimus. </s> <s xml:id="echoid-s9525" xml:space="preserve">Nec Lunam habet ullam comi-<lb/>tem; </s> <s xml:id="echoid-s9526" xml:space="preserve">atque in eo Telluri noſtræ, quemadmodum & </s> <s xml:id="echoid-s9527" xml:space="preserve">Venus <lb/>& </s> <s xml:id="echoid-s9528" xml:space="preserve">Mercurius, dignitate impar videtur. </s> <s xml:id="echoid-s9529" xml:space="preserve">Lux vero Solis, ca-<lb/>lorque, Marticolis duplo atque interdum triplo quam nobis <lb/>minor ſentitur; </s> <s xml:id="echoid-s9530" xml:space="preserve">nullo tamen, ut credimus, ipſorum incom-<lb/>modo.</s> <s xml:id="echoid-s9531" xml:space="preserve"/> </p> <div xml:id="echoid-div450" type="float" level="2" n="4"> <note position="left" xlink:label="note-0436-01" xlink:href="note-0436-01a" xml:space="preserve">@n Marte.</note> </div> <p> <s xml:id="echoid-s9532" xml:space="preserve">Quod ſi Tellus hæc, propter adjunctam ei Lunam, præ-<lb/> <anchor type="note" xlink:label="note-0437-01a" xlink:href="note-0437-01"/> ſtare cæteris Planetis, quos huc uſque percurri, dicenda eſt; <lb/></s> <s xml:id="echoid-s9533" xml:space="preserve">nam magnitudine nec cedit iis multum, nec ſuperat; </s> <s xml:id="echoid-s9534" xml:space="preserve">quan-<lb/>topere & </s> <s xml:id="echoid-s9535" xml:space="preserve">his tribus, & </s> <s xml:id="echoid-s9536" xml:space="preserve">Telluri ipſi, anteponenda erunt ſide-<lb/>ra Jovis & </s> <s xml:id="echoid-s9537" xml:space="preserve">Saturni. </s> <s xml:id="echoid-s9538" xml:space="preserve">In quibus ſive globorum molem conſi-<lb/>deremus longiſſimè omnium iſtorum corpuſcula excedentem; </s> <s xml:id="echoid-s9539" xml:space="preserve"><lb/>ſive Lunarum quibus ambiuntur multitudinem, prorſus veri-<lb/>ſimile fit has duas primarias habendas eſſe Tellures, inter eas <lb/>quæ circa Solem ſunt: </s> <s xml:id="echoid-s9540" xml:space="preserve">præ quibus reliquæ quatuor ſint mi-<lb/>nimum quidpiam, ac nequaquam cum iis comparandæ. </s> <s xml:id="echoid-s9541" xml:space="preserve">Quan-<lb/> <anchor type="note" xlink:label="note-0437-02a" xlink:href="note-0437-02"/> ta enim ſit differentia, quò rectius animo concipiatur, ſub-<lb/>jicere hic placuit, ſecundum proportiones veras, aut non mul-<lb/>tum à veris abeuntes, tum Tellurem noſtram, cum circum-<lb/>jecta Lunæ orbita, ipſoque in ea Lunæ globulo; </s> <s xml:id="echoid-s9542" xml:space="preserve">tum Jovis <lb/>ac Saturni ſyſtemata: </s> <s xml:id="echoid-s9543" xml:space="preserve">Illud quaternis, hoc quinis Lunis exor-<lb/>natum; </s> <s xml:id="echoid-s9544" xml:space="preserve">quarum quæque in ſua itidem orbita ponuntur. </s> <s xml:id="echoid-s9545" xml:space="preserve">Jo-<lb/>viales Galilæo deberi notum eſt; </s> <s xml:id="echoid-s9546" xml:space="preserve">quæ quanto animi gaudio <lb/>primum illi animadverſæ ſint, facile quivis ſecum reputet. <lb/></s> <s xml:id="echoid-s9547" xml:space="preserve">Saturniarum una nobis obtigit, quæ cæteris clarior eſt, & </s> <s xml:id="echoid-s9548" xml:space="preserve"><lb/>ab extrema proxima. </s> <s xml:id="echoid-s9549" xml:space="preserve">Quam Anno 1655 teleſcopio noſtro <lb/>non ultra duodecim pedes longo, primi deprehendimus. </s> <s xml:id="echoid-s9550" xml:space="preserve">Re-<lb/>liquæ diligentiſſimis Dominici Caſſini obſervationibus patue-<lb/>runt, vitreis orbibus utenti à Joſ. </s> <s xml:id="echoid-s9551" xml:space="preserve">Campano expolitis, pri-<lb/>mùm 36 pedum; </s> <s xml:id="echoid-s9552" xml:space="preserve">deinde totidem ſupra centenos. </s> <s xml:id="echoid-s9553" xml:space="preserve">Tertiam <lb/>enim quintamque vidimus Anno 1672, ipſo monſtrante Caſ-<lb/>ſino, & </s> <s xml:id="echoid-s9554" xml:space="preserve">poſtea ſæpius. </s> <s xml:id="echoid-s9555" xml:space="preserve">Primam, cum ſecunda, ſibi reper-<lb/>tas ſignificavit, miſſis literis, Anno 1684. </s> <s xml:id="echoid-s9556" xml:space="preserve">Hæ vero difficil-<lb/>limè cernuntur, certoque affirmare nequeo mihi conſpectas <pb o="698" file="0438" n="457" rhead="CHRISTIANI HUGENII"/> hactenus. </s> <s xml:id="echoid-s9557" xml:space="preserve">Nec propterea quidquam vereor Clariſſimo Vi-<lb/>ro fidem habere, atque has quoque Saturno ſocias adſcribe-<lb/>re. </s> <s xml:id="echoid-s9558" xml:space="preserve">Imo præter harum numerum alias quoque, vel unam <lb/>vel plures, latere ſuſpicari licet; </s> <s xml:id="echoid-s9559" xml:space="preserve">nec deeſt ratio. </s> <s xml:id="echoid-s9560" xml:space="preserve">Cum enim, <lb/>inter extremas duas, ſpatium amplius pateat quàm pro di-<lb/>ſtantiis cæterarum; </s> <s xml:id="echoid-s9561" xml:space="preserve">poſſet hoc inſidere ſextus ſatelles: </s> <s xml:id="echoid-s9562" xml:space="preserve">vel <lb/>etiam, ultra quintum, alii circumvagari, qui propter ob-<lb/>ſcuritatem nondum ſint viſi: </s> <s xml:id="echoid-s9563" xml:space="preserve">cum ille ipſe quintus, tan-<lb/>tùm in orbitæ ſuæ parte quæ ad occidentem ſpectat, cerna-<lb/>tur, in reliqua nunquam appareat; </s> <s xml:id="echoid-s9564" xml:space="preserve">cujus rei cauſam ſatis in-<lb/>tellectu facilem poſtea adferemus.</s> <s xml:id="echoid-s9565" xml:space="preserve"/> </p> <div xml:id="echoid-div451" type="float" level="2" n="5"> <note position="right" xlink:label="note-0437-01" xlink:href="note-0437-01a" xml:space="preserve">Jovem & <lb/>Saturnum <lb/>reliquis Pla-<lb/>netis longe <lb/>præſtare, <lb/>tam magni-<lb/>tudine, <lb/>quam Luna-<lb/>rum multi-<lb/>tudine.</note> <note position="right" xlink:label="note-0437-02" xlink:href="note-0437-02a" xml:space="preserve">TAB. LIII.</note> </div> <p> <s xml:id="echoid-s9566" xml:space="preserve">Fortaſſe autem, ubi ad ſigna Borea Saturnus revertetur, <lb/>altéque ſupra horizontem attolletur, (nam, quo tempore <lb/>hæc ſcribimus, maximè deprimitur) aliquid circa hæc novi <lb/>obſervare continget, ſi quis tuas tunc lentes, Frater opti-<lb/>me, ad Teleſcopia pedum 170. </s> <s xml:id="echoid-s9567" xml:space="preserve">& </s> <s xml:id="echoid-s9568" xml:space="preserve">210. </s> <s xml:id="echoid-s9569" xml:space="preserve">paratas, ſideribus <lb/>applicet: </s> <s xml:id="echoid-s9570" xml:space="preserve">quibus majores, formæque perfectioris, nullas ha-<lb/>ctenus extare arbitror. </s> <s xml:id="echoid-s9571" xml:space="preserve">Quanquam enim cælo nondum eas <lb/>admovimus, vel propter moliendi difficultatem, vel quod <lb/>diſceſſus tuus ſtudia hæc noſtra conatuſque interrupit: </s> <s xml:id="echoid-s9572" xml:space="preserve">omni <lb/>tamen vitio eas carere certi ſumus, poſt experimenta illa fa-<lb/>ciliora, quæ in ambulacris ſuburbanis ſub noctem inſtitueba-<lb/>mus; </s> <s xml:id="echoid-s9573" xml:space="preserve">inſpectis procul literis, quibus appoſitum erat lumen. <lb/></s> <s xml:id="echoid-s9574" xml:space="preserve">Quorum equidem lubens reminiſcor, ſimulque jucundi la-<lb/>boris noſtri, quem, in elaborandis expoliendiſque vitreis <lb/>hujuſmodi diſcis, impendere unà ſolebamus; </s> <s xml:id="echoid-s9575" xml:space="preserve">excogitatis no-<lb/>vis artificiis machiniſque, ſemperque ulteriora agitantes. </s> <s xml:id="echoid-s9576" xml:space="preserve">Sed <lb/>redeo ad diagrammata ante deſcripta, de quibus aliqua di-<lb/>cenda ſupererant.</s> <s xml:id="echoid-s9577" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9578" xml:space="preserve">Feci in iis Jovialis globi diametrum duarum circiter ter-<lb/> <anchor type="note" xlink:label="note-0438-01a" xlink:href="note-0438-01"/> tiarum ejus diſtantiæ quæ inter nos noſtramque Lunam in-<lb/>terjacet; </s> <s xml:id="echoid-s9579" xml:space="preserve">quandoquidem plus quàm vicies diametrum Terræ <lb/>diameter Jovis continet; </s> <s xml:id="echoid-s9580" xml:space="preserve">Luna autem diſtat à Terra diame-<lb/>tris hujus triginta. </s> <s xml:id="echoid-s9581" xml:space="preserve">Orbitam vero comitis Jovis extremi ad <lb/>noſtræ Lunæ orbitam poſui ſicut 8 {1/2} ad 1, quoniam ejuſmodi <lb/>inter eas proportio re ipſa reperitur. </s> <s xml:id="echoid-s9582" xml:space="preserve">Et hi quidem comi- <pb o="699" file="0439" n="458" rhead="COSMOTHEOROS, LIB. II."/> tes, ſive Lunæ ſingulæ, non videntur Tellure noſtra mino-<lb/>res eſſe, ut ex umbris earum in Jovis diſco ſæpe obſervatis, <lb/>probari poteſt. </s> <s xml:id="echoid-s9583" xml:space="preserve">Sunt autem (ut hoc quoque addamus) pe-<lb/> <anchor type="note" xlink:label="note-0439-01a" xlink:href="note-0439-01"/> riodorum tempora ſub Ecliptica, apud Caſſinum, intimi <lb/>Jovialium dies 1, horæ 18, 28′, 36″. </s> <s xml:id="echoid-s9584" xml:space="preserve">Secundi dies 3, horæ <lb/>13. </s> <s xml:id="echoid-s9585" xml:space="preserve">13′. </s> <s xml:id="echoid-s9586" xml:space="preserve">52″. </s> <s xml:id="echoid-s9587" xml:space="preserve">Tertii dies 7. </s> <s xml:id="echoid-s9588" xml:space="preserve">horæ 3 59′. </s> <s xml:id="echoid-s9589" xml:space="preserve">40″. </s> <s xml:id="echoid-s9590" xml:space="preserve">Quarti dies <lb/>16, horæ 18. </s> <s xml:id="echoid-s9591" xml:space="preserve">5′. </s> <s xml:id="echoid-s9592" xml:space="preserve">6″. </s> <s xml:id="echoid-s9593" xml:space="preserve">Diſtantiæ à centro jovis, comitis inti-<lb/>mi 2 {5/6} Jovis diametrorum. </s> <s xml:id="echoid-s9594" xml:space="preserve">Secundi 4 {1/2}. </s> <s xml:id="echoid-s9595" xml:space="preserve">Tertii 7 {1/6}. </s> <s xml:id="echoid-s9596" xml:space="preserve">Quar-<lb/>ti 12 {2/3}. </s> <s xml:id="echoid-s9597" xml:space="preserve">In Saturniis periodica tempora, intimi, dies 1, ho-<lb/>ræ 21, 18′, 31″. </s> <s xml:id="echoid-s9598" xml:space="preserve">Secundi dies 2, horæ 17, 41′, 27″. </s> <s xml:id="echoid-s9599" xml:space="preserve">Tertii <lb/>dies 4, horæ 13. </s> <s xml:id="echoid-s9600" xml:space="preserve">47′, 16″. </s> <s xml:id="echoid-s9601" xml:space="preserve">Quarti dies 15, horæ 22, 41′, 11″. <lb/></s> <s xml:id="echoid-s9602" xml:space="preserve">Quinti dies 79, horæ 7, 53′. </s> <s xml:id="echoid-s9603" xml:space="preserve">57″. </s> <s xml:id="echoid-s9604" xml:space="preserve">Diſtantiæ à centro Satur-<lb/>ni, diametris Annuli dimenſæ, Comitis intimi, {39/40}. </s> <s xml:id="echoid-s9605" xml:space="preserve">Secun-<lb/>di 1 {1/4}. </s> <s xml:id="echoid-s9606" xml:space="preserve">Tertii 1 {3/4}. </s> <s xml:id="echoid-s9607" xml:space="preserve">Quarti 4, quæ mihi erat 3 {1/2}. </s> <s xml:id="echoid-s9608" xml:space="preserve">Quinti 12. </s> <s xml:id="echoid-s9609" xml:space="preserve"><lb/>omnia magnis laboribus vigiliiſque reperta.</s> <s xml:id="echoid-s9610" xml:space="preserve"/> </p> <div xml:id="echoid-div452" type="float" level="2" n="6"> <note position="left" xlink:label="note-0438-01" xlink:href="note-0438-01a" xml:space="preserve">Proportio <lb/>diametro-<lb/>rum tum <lb/>Jovis, tum <lb/>orbitarum <lb/>ſatellitum <lb/>ejus, ad or-<lb/>bitam Lunæ <lb/>circa Ter-<lb/>ram.</note> <note position="right" xlink:label="note-0439-01" xlink:href="note-0439-01a" xml:space="preserve">Tempora pe-<lb/>riodorum co-<lb/>mitum Jo-<lb/>vialium.</note> </div> <p> <s xml:id="echoid-s9611" xml:space="preserve">Ecquis jam ſyſtemata hæc inſpiciens, atque inter ſe con-<lb/>ferens, non ſtupet ad magnitudinem, ingentemque paratum <lb/>duorum præ exiguo tenuique Telluris noſtræ? </s> <s xml:id="echoid-s9612" xml:space="preserve">aut cui nunc <lb/>in mentem venire poteſt in hac una Solem ambientium, <lb/>omnem ornatum, omnia animalia, omnes qui cœleſtia mi-<lb/>rentur inveniri; </s> <s xml:id="echoid-s9613" xml:space="preserve">in illis vero nihil impoſuiſſe rerum condito-<lb/>rem; </s> <s xml:id="echoid-s9614" xml:space="preserve">nec alio fine tam vaſtas corporum moles o<unsure/>reaſſe, quam <lb/>ut lucem eorum nos homunculi intueremur, curſumque for-<lb/>ſitan inquireremus?</s> <s xml:id="echoid-s9615" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9616" xml:space="preserve">Credo equidem futuros qui falſa aut incerta eſſe dicant, <lb/> <anchor type="note" xlink:label="note-0439-02a" xlink:href="note-0439-02"/> quæ de magnitudine cœleſtium ſpatiorum nobis hic ſumun-<lb/>tur. </s> <s xml:id="echoid-s9617" xml:space="preserve">Scio enim quam difficulter quiſquam adducatur, qui <lb/>orbis Terrarum ſpatia mirari aſſueverit, inque eo tot popu-<lb/>los, urbes, imperia; </s> <s xml:id="echoid-s9618" xml:space="preserve">ut alibi exſtare credat quorum collatio-<lb/>ne hoc totum tam ſit exiguum quam hæ figuræ demonſtrant. <lb/></s> <s xml:id="echoid-s9619" xml:space="preserve">Atqui ex ſummorum hujus ætatis Aſtronomorum ſcriptis ea <lb/>hauſimus, ex quibus iſtæ ſyſtematum inter ſe rationes con-<lb/>ſequantur. </s> <s xml:id="echoid-s9620" xml:space="preserve">Si enim Terra à Sole decem vel undecim mille <lb/>diametris ſuis diſtat, ut Caſſinus in Gallia, apud Anglos <lb/>Flamſtedius colligunt, parallaxium in Marte ſubtiliſſimis <lb/>obſervationibus uſi; </s> <s xml:id="echoid-s9621" xml:space="preserve">cum nos quoque probabili conjectura, <pb o="700" file="0440" n="459" rhead="CHRISTIANI HUGENII"/> duodecim mille diametros invenerimus; </s> <s xml:id="echoid-s9622" xml:space="preserve">erunt & </s> <s xml:id="echoid-s9623" xml:space="preserve">iſtæ orbium <lb/>magnitudines inter ſe ferè quales hic deſcripſimus.</s> <s xml:id="echoid-s9624" xml:space="preserve"/> </p> <div xml:id="echoid-div453" type="float" level="2" n="7"> <note position="right" xlink:label="note-0439-02" xlink:href="note-0439-02a" xml:space="preserve">De hac pro-<lb/>portione ma-<lb/>gnitudinis <lb/>conſtare ex <lb/>recentiorum <lb/>obſervatio-<lb/>nibus.</note> </div> <p> <s xml:id="echoid-s9625" xml:space="preserve">Sed de Jove dicere pergamus, ex quo Sol ſpectatus dia-<lb/> <anchor type="note" xlink:label="note-0440-01a" xlink:href="note-0440-01"/> metrum quintuplo quam apud nos minorem habet; </s> <s xml:id="echoid-s9626" xml:space="preserve">ut pro-<lb/>inde lucis caloriſque illic pars tantum vigeſima quinta ſentiri <lb/>poſſit. </s> <s xml:id="echoid-s9627" xml:space="preserve">Sed ea lux nequaquam debilis putanda eſt, idque <lb/>oſtendit inſignis Jovis per noctem claritas. </s> <s xml:id="echoid-s9628" xml:space="preserve">Tum quod in <lb/>Solis Eclipſibus quæ nobis contingunt, etiamſi nec vigeſima <lb/>quinta pars diſci ejus ſuperſit, ut me videre memini, non <lb/>admodum ſentiatur obſcuratio. </s> <s xml:id="echoid-s9629" xml:space="preserve">Si vero experimento inqui-<lb/>rere libeat quanta ſit illa in Jove Solis lux, ſumatur tubus <lb/>certæ longitudinis, iſque parte altera obturetur, impoſitâ la-<lb/>mellâ in cujus medio foramen ſit rotundum, ea latitudine <lb/>quæ ad tubi longitudinem ſe habeat ut ſubtenſa 6 ſcrupulo-<lb/>rum primorum ad radium, hoc eſt fere ut 1 ad 570. </s> <s xml:id="echoid-s9630" xml:space="preserve">Dein-<lb/>de ad Solem tubus obvertatur, radiique ejus per foramen in-<lb/>greſſi excipiantur parte oppoſita, in chartæ candidæ folium; <lb/></s> <s xml:id="echoid-s9631" xml:space="preserve">nec aliunde eo lux incidere poſſit. </s> <s xml:id="echoid-s9632" xml:space="preserve">Hi radii imaginem Solis <lb/>circulo referent, cujus claritas erit eadem quæ ſerenis diebus <lb/>percipitur à Jovis incolis. </s> <s xml:id="echoid-s9633" xml:space="preserve">Remotâ autem chartâ, ſi eodem <lb/>loco oculus ponatur, videbit hic Solem ea magnitudine ac <lb/>ſplendore, qui in Jovis globo conſiſtenti appareret.</s> <s xml:id="echoid-s9634" xml:space="preserve"/> </p> <div xml:id="echoid-div454" type="float" level="2" n="8"> <note position="left" xlink:label="note-0440-01" xlink:href="note-0440-01a" xml:space="preserve">Quænam ſit <lb/>Solis appa-<lb/>rens magni-<lb/>tudo & lux <lb/>in Jove, & <lb/>quî cognoſci <lb/>queat.</note> </div> <p> <s xml:id="echoid-s9635" xml:space="preserve">Quod ſi in eodem tubo foramen duplo anguſtiori diame-<lb/> <anchor type="note" xlink:label="note-0440-02a" xlink:href="note-0440-02"/> tro ſtatuatur, incidet in chartam, aut in oculum, lux ejuſ-<lb/>modi qualis ad Saturnicolas pervenit. </s> <s xml:id="echoid-s9636" xml:space="preserve">Quæ cum centeſima <lb/>tantum pars ſit noſtræ quam a Sole accipimus, tamen per <lb/>tenebras noctis Saturnum ſatis lucidum nobis oſtendit. </s> <s xml:id="echoid-s9637" xml:space="preserve">In <lb/>utroque vero Planetarum iſtorum, ſi nubilos quandoque <lb/>dies habent, malignam tunc lucem eſſe oportet, ſi noſtris <lb/>oculis judicanda ſit; </s> <s xml:id="echoid-s9638" xml:space="preserve">at illorum habitatoribus talem haud du-<lb/>biè, ut nihil de tenuitate ejus querantur. </s> <s xml:id="echoid-s9639" xml:space="preserve">Sicut in noctuis, <lb/>veſpertilionibuſque utilior gratiorque eſt crepuſculi lux, aut <lb/>quæ in ipſa nocte relinquitur, quam quæ diei tempore aë-<lb/>rem terramque illuſtrat.</s> <s xml:id="echoid-s9640" xml:space="preserve"/> </p> <div xml:id="echoid-div455" type="float" level="2" n="9"> <note position="left" xlink:label="note-0440-02" xlink:href="note-0440-02a" xml:space="preserve">Itidem in <lb/>Saturno.</note> </div> <p> <s xml:id="echoid-s9641" xml:space="preserve">In Jove porro dierum ſpatia, quinque tantum horas no-<lb/> <anchor type="note" xlink:label="note-0440-03a" xlink:href="note-0440-03"/> ſtrates æquare, ac noctes tantundem, admiratione non ca- <pb file="0441" n="460"/> <pb file="0441a" n="461"/> <anchor type="figure" xlink:label="fig-0441a-01a" xlink:href="fig-0441a-01"/> <anchor type="figure" xlink:label="fig-0441a-02a" xlink:href="fig-0441a-02"/> <anchor type="figure" xlink:label="fig-0441a-03a" xlink:href="fig-0441a-03"/> <pb file="0442" n="462"/> <pb file="0443" n="463"/> <pb file="0443a" n="464"/> <anchor type="figure" xlink:label="fig-0443a-01a" xlink:href="fig-0443a-01"/> <anchor type="figure" xlink:label="fig-0443a-02a" xlink:href="fig-0443a-02"/> <pb file="0444" n="465"/> <pb o="705" file="0445" n="466" rhead="COSMOTHEOROS, LIB. II."/> globus; </s> <s xml:id="echoid-s9642" xml:space="preserve">teleſcopioque utentibus multa particulatim conſpi-<lb/>cienda præbeat; </s> <s xml:id="echoid-s9643" xml:space="preserve">plura quoque ac probabiliora de univerſa <lb/>natura ejus, quàm de Planetarum cæterorum conjici poſſe, <lb/>tanto quippe remotiorum. </s> <s xml:id="echoid-s9644" xml:space="preserve">Sed contra evenit ut vix quid-<lb/>quam de Lunæ rebus dicendum reperiam, nimirum quia ejus <lb/>generis Planetam nullum coram intueri contigit; </s> <s xml:id="echoid-s9645" xml:space="preserve">cum in pri-<lb/>mariis illis aliter hoc ſeſe habeat. </s> <s xml:id="echoid-s9646" xml:space="preserve">Sunt enim, ut jam ſatis <lb/>conſtat, generis ejuſdem ac Tellus noſtra, in qua, quid re-<lb/>rum geratur, quidve exſtet, propè intuemur, eoque de cæ-<lb/>teris ſimilia quædam conjectandi ratio ſuppetit.</s> <s xml:id="echoid-s9647" xml:space="preserve"/> </p> <div xml:id="echoid-div456" type="float" level="2" n="10"> <note position="left" xlink:label="note-0440-03" xlink:href="note-0440-03a" xml:space="preserve">@n Jove dies <lb/>eſſe horarum <lb/>quinque.</note> <figure xlink:label="fig-0441a-01" xlink:href="fig-0441a-01a"> <caption xml:id="echoid-caption181" style="it" xml:space="preserve">Pg. 700<lb/>TAB. LIII.</caption> <variables xml:id="echoid-variables98" xml:space="preserve">4 3 2 1 Annu Sat. lus</variables> </figure> <figure xlink:label="fig-0441a-02" xlink:href="fig-0441a-02a"> <variables xml:id="echoid-variables99" xml:space="preserve">4 3 2 1 Jup.</variables> </figure> <figure xlink:label="fig-0441a-03" xlink:href="fig-0441a-03a"> <variables xml:id="echoid-variables100" xml:space="preserve">Luna Tellus</variables> </figure> <figure xlink:label="fig-0443a-01" xlink:href="fig-0443a-01a"> <caption xml:id="echoid-caption182" style="it" xml:space="preserve">Pag. 704.<lb/>TAB. LIV.<lb/>Fig. 1.</caption> <description xml:id="echoid-description131" xml:space="preserve">Satu@@i.</description> <description xml:id="echoid-description132" xml:space="preserve">Jovis.</description> <description xml:id="echoid-description133" xml:space="preserve">Martis.</description> <description xml:id="echoid-description134" xml:space="preserve">Telluris.</description> <description xml:id="echoid-description135" xml:space="preserve">veneris.</description> <description xml:id="echoid-description136" xml:space="preserve">M@rc. ♎ Sol. ♈</description> <variables xml:id="echoid-variables101" xml:space="preserve">VS</variables> </figure> <figure xlink:label="fig-0443a-02" xlink:href="fig-0443a-02a"> <caption xml:id="echoid-caption183" style="it" xml:space="preserve">Fig. 2</caption> <description xml:id="echoid-description137" xml:space="preserve">Saturnus. Tellus. Luna.</description> <variables xml:id="echoid-variables102" xml:space="preserve">A C D R S K M G H T V N L Q Y P E F B</variables> </figure> </div> <p> <s xml:id="echoid-s9648" xml:space="preserve">Illud vero ſine omni dubitatione ſtatuere poſſumus, ejuſ-<lb/> <anchor type="note" xlink:label="note-0445-01a" xlink:href="note-0445-01"/> dem naturæ, ac Luna noſtra, eſſe illas, quæ Jovem ac Sa-<lb/>turnum comitari dictæ ſunt, ſiquidem eodem prorſus modo <lb/>primarios hoſce Planetas circumeunt, ſimulque cum illis cir-<lb/>cum Solem feruntur, perinde ac cum Tellure Luna. </s> <s xml:id="echoid-s9649" xml:space="preserve">Sed <lb/>& </s> <s xml:id="echoid-s9650" xml:space="preserve">aliam utrobique ſimilitudinem intercedere poſtea videbi-<lb/>mus. </s> <s xml:id="echoid-s9651" xml:space="preserve">Quamobrem ſi quid de Lunæ ſtatu conjicere poſſimus, <lb/>(poſſumus autem pauca admodum) idem in quatuor illis cir-<lb/>ca Jovem, & </s> <s xml:id="echoid-s9652" xml:space="preserve">in quinque Saturniis haud multo aliter ſe ha-<lb/>bere putandum erit. </s> <s xml:id="echoid-s9653" xml:space="preserve">Illud ſemper menti infixum tenendo, <lb/>non eſſe illas viliores aut minore ornatu excultas.</s> <s xml:id="echoid-s9654" xml:space="preserve"/> </p> <div xml:id="echoid-div457" type="float" level="2" n="11"> <note position="right" xlink:label="note-0445-01" xlink:href="note-0445-01a" xml:space="preserve">Satellitum <lb/>Saturni & <lb/>Jovis ean-<lb/>dem ac Lu-<lb/>næ rationem <lb/>eſſe.</note> </div> <p> <s xml:id="echoid-s9655" xml:space="preserve">Illud igitur in Luna noſtra apparet, etiam minoribus per-<lb/> <anchor type="note" xlink:label="note-0445-02a" xlink:href="note-0445-02"/> ſpicillis trium quatuorve pedum longitudine, plurimis mon-<lb/>tium tractibus, rurſuſque planis vallibus latiſſimis, ſuperfi-<lb/>ciem ejus diviſam eſſe. </s> <s xml:id="echoid-s9656" xml:space="preserve">Cernuntur enim montium umbræ ea <lb/>parte quam a Sole averſam habent; </s> <s xml:id="echoid-s9657" xml:space="preserve">ac frequenter jugo in <lb/>circulum fere compoſito incluſæ valles quædam minores ani-<lb/>madvertuntur; </s> <s xml:id="echoid-s9658" xml:space="preserve">quarum medio monticuli, unus plureſve rur-<lb/>ſum eminent. </s> <s xml:id="echoid-s9659" xml:space="preserve">Ex qua vallium rotunditate argumentum ſu-<lb/>mebat Keplerus, Lunicolarum, cum ratione operantium, <lb/>immenſas has eſſe molitiones. </s> <s xml:id="echoid-s9660" xml:space="preserve">Sed hoc incredibile prorſus, <lb/>tum ob nimiam earum magnitudinem; </s> <s xml:id="echoid-s9661" xml:space="preserve">tum quod facile na-<lb/>turalibus cauſis cavitates ejuſmodi orbiculares formari poſſint. <lb/></s> <s xml:id="echoid-s9662" xml:space="preserve">Marium vero ſimilitudinem illic nullam, (etſi & </s> <s xml:id="echoid-s9663" xml:space="preserve">ille, & </s> <s xml:id="echoid-s9664" xml:space="preserve">alii <lb/> <anchor type="note" xlink:label="note-0445-03a" xlink:href="note-0445-03"/> plerique omnes contra ſentiunt) reperio. </s> <s xml:id="echoid-s9665" xml:space="preserve">Nam regiones <lb/>planæ ingentes, quæ montoſis multo obſcuriores ſunt; </s> <s xml:id="echoid-s9666" xml:space="preserve">quaſ- <pb o="706" file="0446" n="467" rhead="CHRISTIANI HUGENII"/> que vulgo pro maribus haberi video, & </s> <s xml:id="echoid-s9667" xml:space="preserve">oceanorum nomini-<lb/>bus inſigniri; </s> <s xml:id="echoid-s9668" xml:space="preserve">in his ipſis, longiori teleſcopio inſpectis, ca-<lb/>vitates exiguas ineſſe comperio rotundas, umbris intus ca-<lb/>dentibus; </s> <s xml:id="echoid-s9669" xml:space="preserve">quod maris ſuperficiei convenire nequit: </s> <s xml:id="echoid-s9670" xml:space="preserve">tum ipſi <lb/>campi illi latiores non prorſus æquabilem ſuperficiem præfe-<lb/>runt, cum diligentiùs eas intuemur. </s> <s xml:id="echoid-s9671" xml:space="preserve">Quocirca maria eſſe <lb/>non poſſunt, ſed materia conſtare debent minus candicante, <lb/>quàm quæ eſt in partibus aſperioribus: </s> <s xml:id="echoid-s9672" xml:space="preserve">in quibus rurſus quæ-<lb/> <anchor type="note" xlink:label="note-0446-01a" xlink:href="note-0446-01"/> dam vividiori lumine cæteris præcellunt. </s> <s xml:id="echoid-s9673" xml:space="preserve">Nulli quoque flu-<lb/>vii in Luna ineſſe videntur. </s> <s xml:id="echoid-s9674" xml:space="preserve">Non enim effugerent aciem <lb/>perſpicillorum noſtrorum; </s> <s xml:id="echoid-s9675" xml:space="preserve">ſaltem ſi inter montes aut ripas <lb/> <anchor type="note" xlink:label="note-0446-02a" xlink:href="note-0446-02"/> præaltas, ut noſtri plerique, laberentur. </s> <s xml:id="echoid-s9676" xml:space="preserve">Sed neque nubes <lb/>ullæ ſunt unde pluviæ generentur ad ſuppeditandum fluviis <lb/>humorem. </s> <s xml:id="echoid-s9677" xml:space="preserve">Si enim eſſent, videremus eas nunc has nunc il-<lb/>las Lunæ regiones obtegere, ac viſui noſtro ſubducere, quod <lb/>nequaquam contigit, ſed perpetua apparet ſerenitas.</s> <s xml:id="echoid-s9678" xml:space="preserve"/> </p> <div xml:id="echoid-div458" type="float" level="2" n="12"> <note position="right" xlink:label="note-0445-02" xlink:href="note-0445-02a" xml:space="preserve">Lunam <lb/>montibus & <lb/>vallibus di-<lb/>ſtinctam eſſe.</note> <note position="right" xlink:label="note-0445-03" xlink:href="note-0445-03a" xml:space="preserve">Carere ver@ <lb/>mari.</note> <note position="left" xlink:label="note-0446-01" xlink:href="note-0446-01a" xml:space="preserve">Fluviis,</note> <note position="left" xlink:label="note-0446-02" xlink:href="note-0446-02a" xml:space="preserve">Nubib{us},</note> </div> <p> <s xml:id="echoid-s9679" xml:space="preserve">Porrò nec aëre aut atmoſphæra Lunam cingi, qualis cir-<lb/> <anchor type="note" xlink:label="note-0446-03a" xlink:href="note-0446-03"/> cum Tellurem hanc ambit, manifeſtum eſt. </s> <s xml:id="echoid-s9680" xml:space="preserve">Quia ſi qua ta-<lb/>lis exiſteret, non poſſet extrema Lunæ ora tam præciſe cir-<lb/>cumſcripta ſpectari, quam ſubeunte ſtella aliqua ſæpe ani-<lb/>madverſa eſt; </s> <s xml:id="echoid-s9681" xml:space="preserve">ſed evanida quadam luce, ac velut lanugine <lb/>finiretur, ut omittam vapores atmoſphæræ noſtræ maximam <lb/>partem ex aquæ particulis conſtare; </s> <s xml:id="echoid-s9682" xml:space="preserve">ac proinde, ubi nulla <lb/>ſunt maria aut fluvii, non eſſe unde eorum copia ſurſum e-<lb/>ducatur. </s> <s xml:id="echoid-s9683" xml:space="preserve">Hæc igitur inſignis differentia quæ Lunam inter <lb/>Terramque noſtram reperitur, omnem fere aditum conje-<lb/>cturis obſtruit. </s> <s xml:id="echoid-s9684" xml:space="preserve">Nam ſi maria amneſque ineſſe cernerentur, <lb/>haud leve argumentum eſſet cæterum quoque Terræ ornatum <lb/>ei convenire, veramque adeo eſſe Xenophanis opinionem, <lb/>qui habitari in Luna dicebat, eamque Terram eſſe multarum <lb/>urbium & </s> <s xml:id="echoid-s9685" xml:space="preserve">montium. </s> <s xml:id="echoid-s9686" xml:space="preserve">Nunc vero in ſolo arido, & </s> <s xml:id="echoid-s9687" xml:space="preserve">omnis <lb/>aquæ experte, non videntur neque herbæ, neque animantia <lb/>exſtare poſſe, cum omnibus iſtis humor materiam & </s> <s xml:id="echoid-s9688" xml:space="preserve">alimen-<lb/>ta præſtare debeat.</s> <s xml:id="echoid-s9689" xml:space="preserve"/> </p> <div xml:id="echoid-div459" type="float" level="2" n="13"> <note position="left" xlink:label="note-0446-03" xlink:href="note-0446-03a" xml:space="preserve">Aëre & <lb/>aqua.</note> </div> <p> <s xml:id="echoid-s9690" xml:space="preserve">Anne igitur credendum, tantæ magnitudinis globum in <lb/> <anchor type="note" xlink:label="note-0446-04a" xlink:href="note-0446-04"/> hoc conditum eſſe ut noctu nobis lucem tenuem largiatur, <pb o="707" file="0447" n="468" rhead="COSMOTHEOROS, LIB. II."/> aut æſtus maris cieat? </s> <s xml:id="echoid-s9691" xml:space="preserve">Nemo erit qui pulcherrimo inde ſpe-<lb/>ctaculo fruatur Telluris noſtræ in ſe revolutæ, & </s> <s xml:id="echoid-s9692" xml:space="preserve">nunc cum <lb/>Europa Africam, nunc Aſiam, nunc Americam oſtentantis; <lb/></s> <s xml:id="echoid-s9693" xml:space="preserve">nunc plene, nunc dimidio orbe lucentis? </s> <s xml:id="echoid-s9694" xml:space="preserve">Omnes item quæ <lb/>Jovem ac Saturnum circunſtant Lunæ, æquè inutiles vacuæ-<lb/>que ferentur? </s> <s xml:id="echoid-s9695" xml:space="preserve">Non habeo equidem quod dicam, cum nulla <lb/>ab re ſimili conjectura ſuppetat. </s> <s xml:id="echoid-s9696" xml:space="preserve">Magis tamen probabile vi-<lb/>detur ob corporum præſtantiam, aliquid in ſuperficie eorum <lb/>geri, aliquid creſcere ac vivere, qualecunque tandem id ſit, <lb/>& </s> <s xml:id="echoid-s9697" xml:space="preserve">quantumlibet à rebus noſtris diverſum. </s> <s xml:id="echoid-s9698" xml:space="preserve">Poſſet forſan ſtir-<lb/>pium animaliumque ibi vitam aliud quid, aquæ noſtræ diſſi-<lb/>mile, ſuſtentare. </s> <s xml:id="echoid-s9699" xml:space="preserve">Poſſet exiguus humor in terra, non æque <lb/>ac noſtra, aquam combibente, ſufficere radiis Solis, unde <lb/>rorem educerent, alendis herbis arboribuſque idoneum. </s> <s xml:id="echoid-s9700" xml:space="preserve"><lb/>Quod idem Plutarcho in mentem veniſſe video in eo qui de <lb/>Facie in orbe Lunæ eſt dialogo. </s> <s xml:id="echoid-s9701" xml:space="preserve">Nam neque apud nos, niſi <lb/>ſumma maris ſuperficie, ac tenui veluti pellicula opus eſſet, <lb/>ad humorem terris, ſatiſque ſuppeditandum, quem Solis vis <lb/>elicuiſſet, quique in rorem tantum, non vero in nubes con-<lb/>denſaretur. </s> <s xml:id="echoid-s9702" xml:space="preserve">Sed hæ admodum leves ſunt conjecturæ aut ſu-<lb/>ſpiciones potius, nec aliud habemus ex quo de Lunæ no-<lb/>ſtræ, atque etiam reliquarum natura aliquid colligamus. </s> <s xml:id="echoid-s9703" xml:space="preserve"><lb/>Omnium enim, uti diximus, eadem putanda eſt; </s> <s xml:id="echoid-s9704" xml:space="preserve">idque præ-<lb/> <anchor type="note" xlink:label="note-0447-01a" xlink:href="note-0447-01"/> ter adductam ſuperius rationem, etiam hac alia confirmatur, <lb/>quod ſicuti Luna noſtra eandem perpetuò faciem ad nos ob-<lb/>verſam habet, ita & </s> <s xml:id="echoid-s9705" xml:space="preserve">illæ Joviales ac Saturniæ ad ſuos Plane-<lb/>tas primarios. </s> <s xml:id="echoid-s9706" xml:space="preserve">Mirum videatur hoc ſciri potuiſſe; </s> <s xml:id="echoid-s9707" xml:space="preserve">at non <lb/>erat difficilis conjectura, poſtquam, ut paulo ante dixi, ani-<lb/>madverſum fuit extremam Saturniarum tunc ſolum conſpici, <lb/>cum Planetæ huic ad occidentem poſita eſt; </s> <s xml:id="echoid-s9708" xml:space="preserve">ab oriente ve-<lb/>ro ſemper eam latere. </s> <s xml:id="echoid-s9709" xml:space="preserve">Facile enim perſpicitur id inde eve-<lb/>nire, quod magna ſui parte obſcuriorem ſuperficiem habeat <lb/>hæc Luna; </s> <s xml:id="echoid-s9710" xml:space="preserve">quæ pars obſcurior cum ad nos converſa eſt, <lb/>tunc cerni nequeat præ luminis tenuitate. </s> <s xml:id="echoid-s9711" xml:space="preserve">Cumque ſemper <lb/>in orbitæ ſuæ latere quod orientem ſpectat obſcurata reperia-<lb/>tur, in altero nunquam, manifeſtum indicium eſt eandem <pb o="708" file="0448" n="469" rhead="CHISTIANI HUGENII"/> globuli regionem ſemper Saturnum reſpicere, quoniam ex <lb/>eo illud contingere neceſſe eſt. </s> <s xml:id="echoid-s9712" xml:space="preserve">Quis vero jam dubitet, cum <lb/>& </s> <s xml:id="echoid-s9713" xml:space="preserve">illius omnium remotiſſimæ & </s> <s xml:id="echoid-s9714" xml:space="preserve">noſtræ Lunæ facies ſemper <lb/>eadem ex primario Planeta ſuo ſpectetur, quin idem in cæ-<lb/>teris, quæ circa Jovem ac Saturnum volvuntur, natura effe-<lb/>cerit? </s> <s xml:id="echoid-s9715" xml:space="preserve">Cauſa vero quare id fiat vix aliunde peti poteſt, quam <lb/>quod denſior ponderoſiorque materia ſit Lunarum om@ium <lb/>parte ea, quâ ſemper à Planetis ſuis averſæ ſunt. </s> <s xml:id="echoid-s9716" xml:space="preserve">Sic enim <lb/>ea ipſa pars majore vi à centro circuitus recedere contendet: <lb/></s> <s xml:id="echoid-s9717" xml:space="preserve">cum alioqui, ex motus legibus, eadem ſemper facies non ad <lb/>Planetam, ſed ad fixas ſtellas eaſdem, continue obverti de-<lb/>buerit.</s> <s xml:id="echoid-s9718" xml:space="preserve"/> </p> <div xml:id="echoid-div460" type="float" level="2" n="14"> <note position="left" xlink:label="note-0446-04" xlink:href="note-0446-04a" xml:space="preserve">Hinc de ani-<lb/>m<unsure/>antibus & <lb/>ſtirpib@@ in-<lb/>certiorem <lb/>conjecturam <lb/>eſſe.</note> <note position="right" xlink:label="note-0447-01" xlink:href="note-0447-01a" xml:space="preserve">Jovis ac Sa-<lb/>turni Lunas, <lb/>non ſecus ac <lb/>noſtram Tel-<lb/>luri, ean-<lb/>dem partem <lb/>ſuo Planetæ <lb/>obvertere.</note> </div> <p> <s xml:id="echoid-s9719" xml:space="preserve">Porro ex hoc poſitu Lunarum ad Planetas ſuos mira quæ-<lb/>dam ſpectacula evenire neceſſe eſt eas habitantibus, qui an <lb/>ſint aliqui, ut jam apparuit, multo incertiſſimum eſt; </s> <s xml:id="echoid-s9720" xml:space="preserve">ſed <lb/>quaſi eſſent ponantur. </s> <s xml:id="echoid-s9721" xml:space="preserve">Satis erit autem de noſtræ Lunæ indi-<lb/> <anchor type="note" xlink:label="note-0448-01a" xlink:href="note-0448-01"/> genis dixiſſe. </s> <s xml:id="echoid-s9722" xml:space="preserve">His igitur ſic in duo hemiſphæria globus ejus <lb/>dividitur, ut qui alterum incolunt, ſemper Telluris noſtræ <lb/>conſpectu fruantur, qui reliquum, ſemper eo careant. </s> <s xml:id="echoid-s9723" xml:space="preserve">Niſi <lb/>quod quidam, circa confinia utriuſque agentes, amittant eum <lb/>per vices ac recuperent. </s> <s xml:id="echoid-s9724" xml:space="preserve">Cernunt autem Gæoſcopi illi Tel-<lb/>lurem in æthere pendentem multo majorem quam quanta <lb/>nobis Luna apparet, quippe ferè quadruplo ampliore dia-<lb/>metro. </s> <s xml:id="echoid-s9725" xml:space="preserve">Sed illud mirabile, quod nocte dieque eodem cœli <lb/>loco velut immobilem perpetuo hærere vident; </s> <s xml:id="echoid-s9726" xml:space="preserve">alii recta ſu-<lb/>pra caput defixam, alii certa altitudine ab horizonte diſtan-<lb/>tem, quidam & </s> <s xml:id="echoid-s9727" xml:space="preserve">in ipſo horizonte ſitam: </s> <s xml:id="echoid-s9728" xml:space="preserve">atque interea con-<lb/>vertentem ſe circum axem ſuum, regioneſque quas continet <lb/>univerſas deinceps oſtendentem horarum viginti quatuor ſpa-<lb/>tio, atque eas quoque proinde (quod utinam videre liceret) <lb/>quæ ad utrumque polum nobis incolis adhuc incognitæ ma-<lb/>nent. </s> <s xml:id="echoid-s9729" xml:space="preserve">Præterea & </s> <s xml:id="echoid-s9730" xml:space="preserve">lumine creſcentem eam vident & </s> <s xml:id="echoid-s9731" xml:space="preserve">immi-<lb/>nutam menſtrua periodo, atque ita per vices plenam, dimi-<lb/>diatam, inque cornua tenuatam, eâdem formarum varietate <lb/>quam Lunæ globus nobis exhibet. </s> <s xml:id="echoid-s9732" xml:space="preserve">Sed lucem à Tellure no-<lb/>ſtra accipiunt quindecuplo majorem quam nos ab illa. </s> <s xml:id="echoid-s9733" xml:space="preserve">Adeo <pb o="709" file="0449" n="470" rhead="COSMOTHEOROS, LIB. II."/> ut, in hæmiſphærio meliore, ad nos obverſo noctes inſi-<lb/>gniter claras habeant; </s> <s xml:id="echoid-s9734" xml:space="preserve">nec tamè<unsure/>n cum claritate illa ullus ad <lb/>eos calor manare poteſt, etſi hoc aliter Keplero viſum eſt. <lb/></s> <s xml:id="echoid-s9735" xml:space="preserve">Sol verò ſemel illis oritur ſingulis menſibus noſtris, ſemelque <lb/>occidit, atque ita dies nocteſque, quindecuplo quam nos <lb/>longiores habent, at inter ſe æquales perpetuo æquinoctio. </s> <s xml:id="echoid-s9736" xml:space="preserve"><lb/>Qua dierum longitudine, quandoquidem non amplius ab il-<lb/>lis quam à nobis Sol abeſt, neceſſe eſſet eos, quibus altè ſu-<lb/>pra horizontem aſcendit, æſtu incommodo torreri, ſi cor-<lb/>pora eorum perinde ac noſtra afficiantur. </s> <s xml:id="echoid-s9737" xml:space="preserve">Aſcendit autem <lb/>maximè iis qui circa confinia hemiſphæriorum, quæ diximus, <lb/>incolunt, qui vero inde procul diſtant, ac circa regiones ha-<lb/>bitant polis Lunæ ſuppoſitas, non magis ob longos iſtos dies <lb/>calebunt, quam qui circa Iſlandiam aut novam Zemblam æ-<lb/>ſtivo tempore cetos piſcantur; </s> <s xml:id="echoid-s9738" xml:space="preserve">qui perſæpe frigora ingentia, <lb/>ipſius ſolſtitii tempore, ac trium licet menſium diebus, ex-<lb/>periuntur. </s> <s xml:id="echoid-s9739" xml:space="preserve">Sunt autem poli Lunæ, quos circum ſtellæ fixæ <lb/>converti cernuntur in ea habitantibus, nequaquam iidem <lb/>qui nobis, neque etiam cum Eclipticæ polis conveniunt, ſed <lb/>his circunferuntur, quinque gradibus ſemper diſtantes, id-<lb/>que periodo annorum novendecim. </s> <s xml:id="echoid-s9740" xml:space="preserve">Anni autem ſpatium <lb/>idem illic quod nobis; </s> <s xml:id="echoid-s9741" xml:space="preserve">quod motu fixarum metiuntur ac re-<lb/>verſione earum ad Solem. </s> <s xml:id="echoid-s9742" xml:space="preserve">Idque iis perfacile eſt, cum diei <lb/>tempore, non minus quam noctu, ſtellas conſpiciunt, nihil <lb/>impediente Solis claritate; </s> <s xml:id="echoid-s9743" xml:space="preserve">quoniam, ut ſupra oſtenſum eſt, <lb/>nullam vaporum ſphæram habent; </s> <s xml:id="echoid-s9744" xml:space="preserve">ſine qua & </s> <s xml:id="echoid-s9745" xml:space="preserve">nos interdiu <lb/>cœlum ſideribus plenum aſpiceremus. </s> <s xml:id="echoid-s9746" xml:space="preserve">Nec vero nubes quo-<lb/>que ullæ unquam obſtant obſervantibus, adeo ut curſus Pla-<lb/>netarum melius quàm nos inveſtigare poſſint; </s> <s xml:id="echoid-s9747" xml:space="preserve">ſed tamen dif-<lb/>ficilius multò verum ſyſtema reperire. </s> <s xml:id="echoid-s9748" xml:space="preserve">Quoniam incipienti-<lb/>bus ſtare Terra ſua videri debuit, in quo eos longius quàm <lb/>nos error abduxit.</s> <s xml:id="echoid-s9749" xml:space="preserve"/> </p> <div xml:id="echoid-div461" type="float" level="2" n="15"> <note position="left" xlink:label="note-0448-01" xlink:href="note-0448-01a" xml:space="preserve">Lunæ incolis <lb/>ſi qui ſint, <lb/>qualis appa-<lb/>ritura ſit cæ-<lb/>lorum conſti-<lb/>tutio, die-<lb/>rum ratio, <lb/>&c.</note> </div> <p> <s xml:id="echoid-s9750" xml:space="preserve">Hæc omnia vero ad Jovis & </s> <s xml:id="echoid-s9751" xml:space="preserve">Saturni Lunas referuntur, <lb/>quibus idem quod nobis Tellus eſt, ſui ſunt primarii Plane-<lb/> <anchor type="note" xlink:label="note-0449-01a" xlink:href="note-0449-01"/> tæ. </s> <s xml:id="echoid-s9752" xml:space="preserve">Singula autem diei noctiſque ſpatia ſimul ſumpta, cu-<lb/>juſque Lunæ periodus metitur, quas ſupra annotavimus.</s> <s xml:id="echoid-s9753" xml:space="preserve"> <pb o="710" file="0450" n="471" rhead="CHRISTIANI HUGENII"/> Quo fit ut Saturni quintam incolentibus, cujus periodus die-<lb/>rum noſtrorum erat 80, eveniant ſui dies nocteſque noſtris <lb/>quadraginta æquales. </s> <s xml:id="echoid-s9754" xml:space="preserve">Iiſdem vero, propter Saturni revo-<lb/>lutionem tricennalem, fiunt æſtates hyemeſque ſingulæ an-<lb/>norum noſtrorum quindecim. </s> <s xml:id="echoid-s9755" xml:space="preserve">Itaque tum propter tam lon-<lb/>ga frigora, tamque longos ſomnos vigiliaſque; </s> <s xml:id="echoid-s9756" xml:space="preserve">etiamſi nil <lb/>aliud eſſet, plane aliam quam apud nos vitam illic fore ma-<lb/>nifeſtum eſt.</s> <s xml:id="echoid-s9757" xml:space="preserve"/> </p> <div xml:id="echoid-div462" type="float" level="2" n="16"> <note position="right" xlink:label="note-0449-01" xlink:href="note-0449-01a" xml:space="preserve">Quod ad Jo-<lb/>vis & Satur-<lb/>ni Lunas fa-<lb/>cile transfer-<lb/>re eſt@</note> </div> <p> <s xml:id="echoid-s9758" xml:space="preserve">Explicuimus igitur hactenus, quæ ad Planetas primarios <lb/>ſecundarioſque Solem circundantes ſpectant. </s> <s xml:id="echoid-s9759" xml:space="preserve">Hinc vero <lb/>priuſquam ad Solem ipſum & </s> <s xml:id="echoid-s9760" xml:space="preserve">ſtellas fixas, tertium nempe <lb/>genus cœleſtium corporum, pergamus, operæ pretium vide-<lb/>tur, ut magnitudinem, ac magnificentiam totius Solaris <lb/>mundi, aliqua ratione, atque evidentius quam hactenus fa-<lb/>ctum ſit, exprimamus. </s> <s xml:id="echoid-s9761" xml:space="preserve">Quod quidem ſchemate in foliis hiſ-<lb/>ce deſcripto haudquaquam poſſumus, propter parvitatem <lb/>corporum Planetariorum ad vaſtiſſimas orbitas ſuas collato-<lb/>rum. </s> <s xml:id="echoid-s9762" xml:space="preserve">Sed verbis ſupplebitur quod deſcriptione perfici ne-<lb/>quit. </s> <s xml:id="echoid-s9763" xml:space="preserve">Itaque repetitâ figurâ quam ſuperioris libri initio po-<lb/> <anchor type="note" xlink:label="note-0450-01a" xlink:href="note-0450-01"/> ſuimus, cogitetur ei ſimilis ac proportione reſpondens, ſed <lb/>quæ deſcripta ſit in ampliſſima politiſſimaque areæ cujuſdam <lb/>planitie, cujus extremus circulus, Saturni orbem referens, <lb/>trecentos ſexaginta pedes ſemidiametro contineat. </s> <s xml:id="echoid-s9764" xml:space="preserve">In cujus <lb/>deinde circunferentia globus Saturni cum ſuo ponatur An-<lb/>nulo, quantus in figura altera cernitur, ubi Solis & </s> <s xml:id="echoid-s9765" xml:space="preserve">Plane-<lb/> <anchor type="note" xlink:label="note-0450-02a" xlink:href="note-0450-02"/> tarum ſunt corpora. </s> <s xml:id="echoid-s9766" xml:space="preserve">Cæterique ſimiliter globi in ſua quiſ-<lb/>que orbita collocentur; </s> <s xml:id="echoid-s9767" xml:space="preserve">inque medio omnium Sol qua ma-<lb/>gnitudine ibi deſignatur, quatuor nempe pollicum diametro. <lb/></s> <s xml:id="echoid-s9768" xml:space="preserve">Ita Telluris circuitus, quem magnum orbem vocant Aſtro-<lb/>nomi, ſemidiametrum ſortietur pedum triginta & </s> <s xml:id="echoid-s9769" xml:space="preserve">ſex. </s> <s xml:id="echoid-s9770" xml:space="preserve">In <lb/>quo Tellus ipſa milii grano non major circunferri cogitanda <lb/>eſt; </s> <s xml:id="echoid-s9771" xml:space="preserve">eique comes Luna, vix punctum viſibile ſuperans, in <lb/>circello paulo plus quam duos pollices lato; </s> <s xml:id="echoid-s9772" xml:space="preserve">velut in adſcri-<lb/>pto hic diagrammate. </s> <s xml:id="echoid-s9773" xml:space="preserve">In quo linea A B circunferentiæ par-<lb/> <anchor type="note" xlink:label="note-0450-03a" xlink:href="note-0450-03"/> tem refert ejus, quam diximus, Telluris orbitæ, cujus tri-<lb/>ginta & </s> <s xml:id="echoid-s9774" xml:space="preserve">ſex pedes continet ſemidiameter. </s> <s xml:id="echoid-s9775" xml:space="preserve">In ea Tellus eſt <pb o="711" file="0451" n="472" rhead="COSMOTHEOROS, LIB. II."/> circellus C: </s> <s xml:id="echoid-s9776" xml:space="preserve">Lunæ vero circum eam via, circulus D E; </s> <s xml:id="echoid-s9777" xml:space="preserve">in <lb/>quo Lunæ corpuſculum quale ad D expreſſum eſt.</s> <s xml:id="echoid-s9778" xml:space="preserve"/> </p> <div xml:id="echoid-div463" type="float" level="2" n="17"> <note position="left" xlink:label="note-0450-01" xlink:href="note-0450-01a" xml:space="preserve">TAB. LIV. <lb/>fig. 1. <lb/>Solaris mun-<lb/>di ſecundum <lb/>veram pro-<lb/>portionem <lb/>d@ſcriptio.</note> <note position="left" xlink:label="note-0450-02" xlink:href="note-0450-02a" xml:space="preserve">TAB. LV. <lb/>fig. 1.</note> <note position="left" xlink:label="note-0450-03" xlink:href="note-0450-03a" xml:space="preserve">TAB. LV. <lb/>fig. 2.</note> </div> <p> <s xml:id="echoid-s9779" xml:space="preserve">Saturniarum vero Lunarum exterior in circulo feretur cu-<lb/>jus ſemidiameter pollicum 29. </s> <s xml:id="echoid-s9780" xml:space="preserve">Jovialium item exterior in <lb/>minore aliquanto, cujus ſemid. </s> <s xml:id="echoid-s9781" xml:space="preserve">poll. </s> <s xml:id="echoid-s9782" xml:space="preserve">19 {1/4}.</s> <s xml:id="echoid-s9783" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9784" xml:space="preserve">Sic demum habebitur germanus & </s> <s xml:id="echoid-s9785" xml:space="preserve">omni proportione perfe-<lb/>ctus ſolaris Regiæ typus, in quo jam Tellus duodecim mille <lb/>diametris ſuis à Sole aberit. </s> <s xml:id="echoid-s9786" xml:space="preserve">Cujus ſpatii amplitudo ſi mil-<lb/>liarium numero deſignanda ſit, plus quam ſeptemdecim mil-<lb/>liones, ut vocant, milliarium Germanicorum comprehen-<lb/>det. </s> <s xml:id="echoid-s9787" xml:space="preserve">Sed melius fortaſſe hanc vaſtitatem animo concipiemus, <lb/>ſi motus cujuſdam celeritate eam metiamur, Heſiodi Poëtæ <lb/>exemplo; </s> <s xml:id="echoid-s9788" xml:space="preserve">qui altitudinem cœli, & </s> <s xml:id="echoid-s9789" xml:space="preserve">Tartari profunditatem <lb/>æquis ſpatiis definiens, novem dierum noctiumque lapſu, fer-<lb/>ream incudem è cœlo dimiſſam, ad terram decimâ perveni-<lb/>re ſcripſit; </s> <s xml:id="echoid-s9790" xml:space="preserve">ac tanto quoque tempore è Terræ ſuperficie ca-<lb/>dentem ad Tartara ferri. </s> <s xml:id="echoid-s9791" xml:space="preserve">Nos vero non incudis lapſum ſed <lb/>continuam potius celeritatem globi ex majore tormento emiſ-<lb/>ſi huc adhibebimus; </s> <s xml:id="echoid-s9792" xml:space="preserve">quem ſingulis horæ ſecundis ſcrupulis, <lb/>ſive arteriæ pulſibus, centum circiter hexapedas conficere <lb/>experimentis compertum eſt, quæ in Baliſticis Merſennus <lb/>commemorat; </s> <s xml:id="echoid-s9793" xml:space="preserve">cum ſonus eo tempore ad centenas octogenas <lb/>extendatur.</s> <s xml:id="echoid-s9794" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9795" xml:space="preserve">Ajo igitur, ſi ex Terra ad Solem tanta illa celeritate glo-<lb/> <anchor type="note" xlink:label="note-0451-01a" xlink:href="note-0451-01"/> bus continuè feratur, fere annos 25 eſſe inſumpturum ante-<lb/>quam iter hoc peragat. </s> <s xml:id="echoid-s9796" xml:space="preserve">Ut proinde à Jove ad Solem 125 <lb/>annis opus habeat, à Saturno 250. </s> <s xml:id="echoid-s9797" xml:space="preserve">Et hic quidem calculus <lb/>ex menſura Terræ diametri pendet, qui ex probatioribus Gal-<lb/>lorum obſervationibus eſt hexapedarum Pariſienſium 6538594. <lb/></s> <s xml:id="echoid-s9798" xml:space="preserve">cum gradus unus circuli maximi efficiat hexapedas 57060. </s> <s xml:id="echoid-s9799" xml:space="preserve"><lb/>Quanta itaque ſint iſtorum orbium ſpatia, quamque exilis, <lb/>eorum reſpectu, Telluris globulus, in quo tam multa homi-<lb/>nes molimur, tantum navigamus, tot bella gerimus, ex his <lb/>intelligitur. </s> <s xml:id="echoid-s9800" xml:space="preserve">Quod utinam diſcant cogitentque Reges & </s> <s xml:id="echoid-s9801" xml:space="preserve">Mo-<lb/>narchæ noſtri; </s> <s xml:id="echoid-s9802" xml:space="preserve">ut ſciant quantilla in re laborent cum de an-<lb/>gulo aliquo terræ occupando totis viribus, magno multorum <pb o="712" file="0452" n="473" rhead="CHRISTIANI HUGENII"/> malo, contendunt. </s> <s xml:id="echoid-s9803" xml:space="preserve">Sed ad noſtra revertamur, ac de Sole <lb/>videamus, cujus jam ſimul ad Planetas & </s> <s xml:id="echoid-s9804" xml:space="preserve">eorum orbitas ma-<lb/>gnitudinem ampla illa deſcriptio, quam expoſuimus, demon-<lb/>ſtrat.</s> <s xml:id="echoid-s9805" xml:space="preserve"/> </p> <div xml:id="echoid-div464" type="float" level="2" n="18"> <note position="right" xlink:label="note-0451-01" xlink:href="note-0451-01a" xml:space="preserve">Immenſitas <lb/>intervallo-<lb/>rum inter So-<lb/>lem & Pla-<lb/>netas, illu-<lb/>ſtratur com-<lb/>paratione <lb/>cum motu <lb/>globi è tor-<lb/>mento emiſſ@.</note> </div> <p> <s xml:id="echoid-s9806" xml:space="preserve">In hoc igitur ipſo Sole non improbabile quibuſdam viſum <lb/> <anchor type="note" xlink:label="note-0452-01a" xlink:href="note-0452-01"/> eſt animalia vivere poſſe. </s> <s xml:id="echoid-s9807" xml:space="preserve">Sed cum multo magis etiam, quam <lb/>in Lunis, conjectura omnis hic def@ciat, neſcio qua ratione <lb/>id ita eſſe opinati ſint. </s> <s xml:id="echoid-s9808" xml:space="preserve">Non enim adhuc planè compertum <lb/>eſt, utrum dura an liquida ſit vaſti illius globi materies; </s> <s xml:id="echoid-s9809" xml:space="preserve">etſi <lb/>propter lucis naturam, quam aliàs explicui, magis veriſimile <lb/>ſit liquidam eſſe; </s> <s xml:id="echoid-s9810" xml:space="preserve">quod etiam perfecta rotunditas ejus, lu-<lb/>menque per totam ſuperficiem æqualiter diffuſum ſuadere vi-<lb/>detur. </s> <s xml:id="echoid-s9811" xml:space="preserve">Nam exigua quædam in diſci circunferentia apparens <lb/>inæqualitas, quæ teleſcopiis, nec tamen ſemper, cernitur, <lb/>& </s> <s xml:id="echoid-s9812" xml:space="preserve">ex qua miros undarum fluctus, flammarumque eructatio-<lb/>nes, nonnulli ſibi fingunt, nihil aliud eſt quàm vaporum <lb/>prope Terram noſtram tremula agitatio, quæ & </s> <s xml:id="echoid-s9813" xml:space="preserve">ſtellas noctu <lb/>ſcintillare ſacit. </s> <s xml:id="echoid-s9814" xml:space="preserve">Neque ego ſaculas illas, quas unà cum ma-<lb/> <anchor type="note" xlink:label="note-0452-02a" xlink:href="note-0452-02"/> culis fere omnes celebrant, unquam videre potui, etſi has <lb/>ſæpius ſpectaverim; </s> <s xml:id="echoid-s9815" xml:space="preserve">ac valde dubito an aliquid in Sole, ipſo <lb/>Sole lucidius appareat. </s> <s xml:id="echoid-s9816" xml:space="preserve">Invenio enim, fideliores obſervatio-<lb/>nes conſulens, non niſi in nubeculis illis ſubfuſcis, quæ ma-<lb/>culas plerumque circundant, aliquando ſolæ feruntur, pun-<lb/>cta quædam clariora interdum notari, quæ non mirum eſ-<lb/>ſet, propter obſcuritatis illius viciniam, ſplendidiora quam <lb/>ſint videri. </s> <s xml:id="echoid-s9817" xml:space="preserve">Summum quidem in Sole calorem, fervorem-<lb/> <anchor type="note" xlink:label="note-0452-03a" xlink:href="note-0452-03"/> que eſſe, certo credendum eſt, in quo nihil omnino noſtro-<lb/>rum corporum ſimile vivere poſſit, aut momento ſupereſſe. <lb/></s> <s xml:id="echoid-s9818" xml:space="preserve">Itaque aliud genus viventium animo concipiendum eſſet, lon-<lb/>geque ab omni natura eorum quæ unquam vidimus, aut co-<lb/>gitavimus, diverſum. </s> <s xml:id="echoid-s9819" xml:space="preserve">Quod ſere idem eſt ac ſi dicamus ni-<lb/>hil hic conjectando nos conſequi poſſe. </s> <s xml:id="echoid-s9820" xml:space="preserve">Eſt quidem tam <lb/>præſtans, tantæque molis corpus haud dubiè maxima ratio-<lb/>ne, ac propter inſignem uſum aliquem creatum. </s> <s xml:id="echoid-s9821" xml:space="preserve">Sed an <lb/>non apparet jam abunde utilitas ejus in mirabili illa lucis ca-<lb/>loriſque in totum Planetarum circumſtantium chorum effu- <pb file="0453" n="474"/> <pb file="0453a" n="475"/> <anchor type="figure" xlink:label="fig-0453a-01a" xlink:href="fig-0453a-01"/> <anchor type="figure" xlink:label="fig-0453a-02a" xlink:href="fig-0453a-02"/> <pb file="0454" n="476"/> <pb o="713" file="0455" n="477" rhead="COSMOTHEOROS, LIB. II."/> ſione; </s> <s xml:id="echoid-s9822" xml:space="preserve">ex qua univerſo animantium generi non vita ſolum <lb/>conſtat, ſed & </s> <s xml:id="echoid-s9823" xml:space="preserve">jucunda ut ſit efficitur? </s> <s xml:id="echoid-s9824" xml:space="preserve">Idque non in exiguis <lb/>ſolum, qualis Tellus noſtra, ſed & </s> <s xml:id="echoid-s9825" xml:space="preserve">in tanto majoribus Jo-<lb/>vis & </s> <s xml:id="echoid-s9826" xml:space="preserve">Saturni globis, quorum non eſt contemptibilis ad So-<lb/>lem collata magnitudo. </s> <s xml:id="echoid-s9827" xml:space="preserve">Hæc quidem tanta ſunt ut nihil mi-<lb/>rum ſit eorum gratia duntaxat Solem eſſe conditum. </s> <s xml:id="echoid-s9828" xml:space="preserve">Nam <lb/>quod Keplerus opinabatur, aliud quoque illi delegatum eſ-<lb/>ſe munus, ut nempe omnium circum ambientium Planetarum <lb/>motus in ſuis orbibus incitaret, propria ſua circa axem con-<lb/>verſione, quod in Epitome ſyſtematis Copernicei multis <lb/>comprobare conatur, non poſſum aſſentiri, propter ea quæ <lb/>in ſequentibus dicentur.</s> <s xml:id="echoid-s9829" xml:space="preserve"/> </p> <div xml:id="echoid-div465" type="float" level="2" n="19"> <note position="left" xlink:label="note-0452-01" xlink:href="note-0452-01a" xml:space="preserve">In Sole <lb/>omnem con-<lb/>jecturam de-<lb/>ficere.</note> <note position="left" xlink:label="note-0452-02" xlink:href="note-0452-02a" xml:space="preserve">Faculas So-<lb/>lares incertas <lb/>videri.</note> <note position="left" xlink:label="note-0452-03" xlink:href="note-0452-03a" xml:space="preserve">Propter calo-<lb/>rem nulla il-<lb/>lic vivere <lb/>copora noſtris <lb/>ſimilia.</note> <figure xlink:label="fig-0453a-01" xlink:href="fig-0453a-01a"> <caption xml:id="echoid-caption184" style="it" xml:space="preserve">Pag. 712.<lb/>TAB. LV.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables103" xml:space="preserve">Sol.<lb/>Sat.<lb/>Jup.<lb/>Mars<lb/>Tellus<lb/>Venus<lb/>Merc.</variables> </figure> <figure xlink:label="fig-0453a-02" xlink:href="fig-0453a-02a"> <caption xml:id="echoid-caption185" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables104" xml:space="preserve">D A C B E</variables> </figure> </div> <p> <s xml:id="echoid-s9830" xml:space="preserve">Solem ex ſtellis inerrantibus unam eſſe, ante Teleſcopii <lb/> <anchor type="note" xlink:label="note-0455-01a" xlink:href="note-0455-01"/> inventionem, adverſari videbatur Copernici ſententiæ; </s> <s xml:id="echoid-s9831" xml:space="preserve">quia <lb/>cum ſtellæ, quæ dicuntur primæ magnitudinis, cenſerentur <lb/>trium ſcrupulorum diametro, eſſentque ſecundum Coperni-<lb/>cum tam procul remotæ, ut totus ille Orbis magnus, quo Terra <lb/>defertur, velut puncti inſtar eſſet ad ſphæram affixarum com-<lb/>paratus; </s> <s xml:id="echoid-s9832" xml:space="preserve">quandoquidem toto anni tempore, etſi locum Ter-<lb/>ra mutaret, nihil mutari cernerentur ſtellarum diſtantiæ; </s> <s xml:id="echoid-s9833" xml:space="preserve">ſe-<lb/>quebatur ſingulas earum, quæ cæteris clariores apparent, ma-<lb/>jores eſſe toto illo magni orbis ambitu: </s> <s xml:id="echoid-s9834" xml:space="preserve">quod abſurdum erat. <lb/></s> <s xml:id="echoid-s9835" xml:space="preserve">Atque hoc, ut palmarium contra Copernici doctrinam ar-<lb/>gumentum, Tycho Braheus objectabat. </s> <s xml:id="echoid-s9836" xml:space="preserve">Sed poſtquam ra-<lb/>dios ſtellarum nudo viſu apparentes, Teleſcopia ſuſtu-<lb/>lerunt; </s> <s xml:id="echoid-s9837" xml:space="preserve">(quod ita optimè faciunt, ſi lens oculo proxi-<lb/>ma flammæ afflatu obſcuretur) atque ita haud aliter eas <lb/>ac puncta lucentia ſpectandas præbuerunt; </s> <s xml:id="echoid-s9838" xml:space="preserve">prorſus ſubla-<lb/>ta quoque eſt ea difficultas, nec quidquam jam impedit quo <lb/>minus ſtellæ iſtæ pro totidem Solibus habeantur. </s> <s xml:id="echoid-s9839" xml:space="preserve">Idque eo <lb/>probabilius redditur, quod conſtet propria luce ſua eas lu-<lb/>cere: </s> <s xml:id="echoid-s9840" xml:space="preserve">tanta enim eſt diſtantia, ut à Sole illam mutuari ne-<lb/>quaquam poſſint. </s> <s xml:id="echoid-s9841" xml:space="preserve">Singulas vero Sole minores non eſſe ni-<lb/>hil credi vetat, cum ex tam immenſo intervallo tam vividum <lb/>lumen fundant. </s> <s xml:id="echoid-s9842" xml:space="preserve">Hanc itaque ſententiam nunc paſſim tenent <lb/>qui Copernici ſyſtema amplectuntur. </s> <s xml:id="echoid-s9843" xml:space="preserve">Qui recte quoque hoc <lb/> <anchor type="note" xlink:label="note-0455-02a" xlink:href="note-0455-02"/> <pb o="714" file="0456" n="478" rhead="CHRISTIANI HUGENII"/> ſtatuunt, non in una eademque ſuperſicie hærere ſtellas iſtas; <lb/></s> <s xml:id="echoid-s9844" xml:space="preserve"> <anchor type="note" xlink:label="note-0456-01a" xlink:href="note-0456-01"/> tum quod nulla ratio hoc ſuadeat, tum quod in eandem <lb/>ſphæram Sol, qui earum una eſt, referri nequeat. </s> <s xml:id="echoid-s9845" xml:space="preserve">Itaque <lb/>veriùs eſſe ſpargi eas per vaſta cœli ſpatia, quantumque à <lb/>Terra aut Sole ad proximas interjacet, tantum circiter ab his <lb/>eſſe ad ſequentes, atque inde rurſus ad alias, continuo pro-<lb/>greſſu.</s> <s xml:id="echoid-s9846" xml:space="preserve"/> </p> <div xml:id="echoid-div466" type="float" level="2" n="20"> <note position="right" xlink:label="note-0455-01" xlink:href="note-0455-01a" xml:space="preserve">Stellas fixas <lb/>totidem eſſe <lb/>Soles.</note> <note position="right" xlink:label="note-0455-02" xlink:href="note-0455-02a" xml:space="preserve">Eas ſpargi <lb/>per vaſta cal@</note> <note position="left" xlink:label="note-0456-01" xlink:href="note-0456-01a" xml:space="preserve">ſpatia, & <lb/>alias ab aliis, <lb/>ut proximas <lb/>à Sole remo-<lb/>veri.</note> </div> <p> <s xml:id="echoid-s9847" xml:space="preserve">Scio etiam hic aliud ſentire Keplerum, in ea, quam di-<lb/>ximus, Epitome. </s> <s xml:id="echoid-s9848" xml:space="preserve">Quamquam enim exiſtimet tota cæli pro-<lb/>funditate ſtellas diſſeminatas eſſe, vult tamen Solem hunc <lb/> <anchor type="note" xlink:label="note-0456-02a" xlink:href="note-0456-02"/> noſtrum multo amplius ſpatium circa ſe habere, quaſi ſphæ-<lb/>ram vacuam, ſupra quam conſertius ſtellis cœlum inci-<lb/>piat. </s> <s xml:id="echoid-s9849" xml:space="preserve">Putabat enim alioqui futurum ut paucæ tantum ſtel-<lb/>læ numerarentur nobis, eæque ſumma magnitudinis diverſi-<lb/>tate: </s> <s xml:id="echoid-s9850" xml:space="preserve">nam cum omnium maximæ tam appareant parvæ, ut <lb/>vix inſtrumentis poſſint notari aut menſurari, conſequens <lb/>eſſe ut quæ duplo aut triplo &</s> <s xml:id="echoid-s9851" xml:space="preserve">c. </s> <s xml:id="echoid-s9852" xml:space="preserve">diſtarent longius, duplo <lb/>& </s> <s xml:id="echoid-s9853" xml:space="preserve">triplo appareant minores, poſitis æqualibus ipſis veris <lb/>magnitudinibus; </s> <s xml:id="echoid-s9854" xml:space="preserve">citoque veniatur ad eas quæ penitus fiant <lb/>inſenſibiles: </s> <s xml:id="echoid-s9855" xml:space="preserve">atque ita pauciſſimas viſum iri ſtellas, eaſque <lb/>in maxima differentia; </s> <s xml:id="echoid-s9856" xml:space="preserve">cum contra amplius quam mille ob-<lb/>ſerventur, nec magnitudine ita multum diverſæ. </s> <s xml:id="echoid-s9857" xml:space="preserve">Sed ex his <lb/>nequaquam id quod ille intendit evincitur; </s> <s xml:id="echoid-s9858" xml:space="preserve">ac præcipue in <lb/>eo deceptus ſuit, quod non advertit ignium, & </s> <s xml:id="echoid-s9859" xml:space="preserve">flammæ <lb/>eam eſſe naturam, ut ex maximis intervallis cerni poſſint, <lb/>iiſque unde alia corpora, æque exiguis angulis comprehen-<lb/>ſa, prorſus evaneſcant. </s> <s xml:id="echoid-s9860" xml:space="preserve">Quod vel lucernæ comprobant, quæ <lb/>per urbium noſtrarum vicos noctu incenduntur. </s> <s xml:id="echoid-s9861" xml:space="preserve">Quæ cum <lb/>ad centenos pedes inter ſe diſtent, tamen earum viginti & </s> <s xml:id="echoid-s9862" xml:space="preserve"><lb/>plures, in continua ſerie magis magiſque remotas, numera-<lb/>re licet, etſi viceſimæ flammula vix 6 ſecundorum ſcrupulo-<lb/>rum angulo conſpiciatur. </s> <s xml:id="echoid-s9863" xml:space="preserve">Idem vero multo magis fieri ne-<lb/>ceſſe eſt in eximia illa ſtellarum luce; </s> <s xml:id="echoid-s9864" xml:space="preserve">adeo ut nihil mirum ſit, <lb/>ad mille aut duo millia earum, oculis notari poſſe; </s> <s xml:id="echoid-s9865" xml:space="preserve">Teleſco-<lb/>piis vero adhibitis, etiam vigecuplo plures deprehendi. </s> <s xml:id="echoid-s9866" xml:space="preserve">Sed <lb/>ſuberat ratio, cur Keplerus Solem præ reliquis ſtellis præci- <pb o="715" file="0457" n="479" rhead="COSMOTHEOROS, LIB. II."/> puum quid habere cuperet; </s> <s xml:id="echoid-s9867" xml:space="preserve">circumque eum eſſe unicum, in <lb/>Natura, Planetarum ſyſtema, idque mundi medio ſitum. <lb/></s> <s xml:id="echoid-s9868" xml:space="preserve">Hiſce nimirum opus habebat ad confirmandum myſterium <lb/>Coſmographicum ſuum, quo certis quibuſdam proportioni-<lb/>bus reſpondere volebat Planetarum à Sole diſtantias diame-<lb/>tris ſphærarum, quæ corporibus polyedris Euclideis inſcri-<lb/>buntur & </s> <s xml:id="echoid-s9869" xml:space="preserve">circunſcribuntur ſingulis. </s> <s xml:id="echoid-s9870" xml:space="preserve">Quod tum demum veri-<lb/>ſimile videri poterat, ſi in mundo univerſo unus tantum eſ-<lb/>ſet circa Solem aberrantium ſiderum chorus, adeoque & </s> <s xml:id="echoid-s9871" xml:space="preserve">Sol <lb/>ipſe ſolus ſui generis.</s> <s xml:id="echoid-s9872" xml:space="preserve"/> </p> <div xml:id="echoid-div467" type="float" level="2" n="21"> <note position="left" xlink:label="note-0456-02" xlink:href="note-0456-02a" xml:space="preserve">Nec Solem <lb/>præ cæteris <lb/>eminere con-<lb/>tra Keple-<lb/>rum notatur.</note> </div> <p> <s xml:id="echoid-s9873" xml:space="preserve">Sed myſterium illud totum, ſi bene perpendatur, ſomnium <lb/>quoddam ex Pythagoræ aut Platonis Philoſophia enatum eſ-<lb/>ſe apparet. </s> <s xml:id="echoid-s9874" xml:space="preserve">Nec proportiones ſatis quadrant, ut ipſe quoque <lb/>auctor agnoſcit; </s> <s xml:id="echoid-s9875" xml:space="preserve">ſed, cur hoc ita ſit, alias cauſas plane frivolas <lb/>comminiſcitur. </s> <s xml:id="echoid-s9876" xml:space="preserve">Idem levioribus etiam argumentis probat ex-<lb/>tremam mundi ſuperficiem, ſtellas omnes continentem, ſphæ-<lb/>ricæ eſſe figuræ; </s> <s xml:id="echoid-s9877" xml:space="preserve">ac numerum præterea earum neceſſariò eſſe fi-<lb/>nitum, exeo quod ſingularum finita ſit magnitudo. </s> <s xml:id="echoid-s9878" xml:space="preserve">Illud vero <lb/>vaniſſimum, quod à Sole, ad ſuperficiem cavam ſphæræ fixa-<lb/>rum, definit ſpatium ſexies centena millia Terræ diametrorum. <lb/></s> <s xml:id="echoid-s9879" xml:space="preserve">Quoniam ſcilicet, ſicut Solis diameter ad diametrum Orbitæ <lb/>Saturni; </s> <s xml:id="echoid-s9880" xml:space="preserve">quos inter ſe eſſe ſtatuit ut 1 ad 2000; </s> <s xml:id="echoid-s9881" xml:space="preserve">ita ſit hic dia-<lb/>meter ad illum ſphæræ fixarum interioris; </s> <s xml:id="echoid-s9882" xml:space="preserve">quod nulla ratio-<lb/>ne nititur. </s> <s xml:id="echoid-s9883" xml:space="preserve">Atque hæc quidem Viro ſummi ingenii, magno-<lb/>que Aſtronomiæ inſtauratori excidiſſe mirum eſt. </s> <s xml:id="echoid-s9884" xml:space="preserve">Nos vero <lb/>unà cum præcipuis noſtræ ætatis Philoſophis, ne dubitemus <lb/>eandem ſtellarum earum & </s> <s xml:id="echoid-s9885" xml:space="preserve">Solis naturam exiſtimare. </s> <s xml:id="echoid-s9886" xml:space="preserve">Ex quo <lb/> <anchor type="note" xlink:label="note-0457-01a" xlink:href="note-0457-01"/> jam mundi idea multo major naſcitur, quam quæ ex hacte-<lb/>nus traditis percipiebatur. </s> <s xml:id="echoid-s9887" xml:space="preserve">Quid enim nunc prohibet, quin <lb/>unamquamque ex ſtellis hiſce, ſive Solibus, haud aliter ac <lb/>Sol noſter, circum ſe Planetas habere putemus, quæ rurſus <lb/>ſuis Lunis ſtipatæ ſint? </s> <s xml:id="echoid-s9888" xml:space="preserve">Imo hoc ita ſe habere, manifeſta <lb/>ecce ratio ſuadet. </s> <s xml:id="echoid-s9889" xml:space="preserve">Etenim ſi cogitatione in cœli regionibus <lb/>nos ponamus, non minus à Sole, quàm fixis ſtellis, remo-<lb/>tos; </s> <s xml:id="echoid-s9890" xml:space="preserve">nihil quicquam diſcriminis haſce inter atque illum tunc <lb/>eſſemus animadverſuri. </s> <s xml:id="echoid-s9891" xml:space="preserve">Longè enim abeſt ut corpora Pla- <pb o="716" file="0458" n="480" rhead="CHRISTIANI HUGENII"/> netarum, Solem ambientium, conſpecturi ſimus, vel ob te-<lb/>nuiſſimam eorum lucem, vel quod univerſæ, quibus ſerun-<lb/>tur, orbitæ in unum idemque lucidum punctum cum Sole <lb/>confunderentur. </s> <s xml:id="echoid-s9892" xml:space="preserve">Hic igitur poſiti, merito eandem omnium <lb/>ſtellarum rationem naturamque eſſe exiſtimaremus; </s> <s xml:id="echoid-s9893" xml:space="preserve">& </s> <s xml:id="echoid-s9894" xml:space="preserve">ex <lb/>una, propius inſpecta, de cæteris quoque judicari poſſe ni-<lb/>hil ambigeremus. </s> <s xml:id="echoid-s9895" xml:space="preserve">At nunc Dei benignitate, ad unam ex <lb/>ipſis, Solem videlicet noſtrum, admoti ſumus, ac tam pro-<lb/>pe acceſſimus, ut circum eam ſex minores globos converti <lb/>cernamus; </s> <s xml:id="echoid-s9896" xml:space="preserve">& </s> <s xml:id="echoid-s9897" xml:space="preserve">circa horum quoſdam, alios obire ſecunda-<lb/>rios. </s> <s xml:id="echoid-s9898" xml:space="preserve">Cur itaque non eo judicio nunc utamur; </s> <s xml:id="echoid-s9899" xml:space="preserve">ac prorſus <lb/>veriſimile putemus non ſolam hanc ſtellam tali comitatu cin-<lb/>gi, aut aliqua in re cæteris præminere? </s> <s xml:id="echoid-s9900" xml:space="preserve">Neque etiam ſolam <lb/>circum axem ſuum converti; </s> <s xml:id="echoid-s9901" xml:space="preserve">ſed potius cæteras omnes ea-<lb/>dem hæc ſimilia habere? </s> <s xml:id="echoid-s9902" xml:space="preserve">Ergo hac ratione etiam cuncta illa <lb/>quæ in Planetis circumſolaribus ineſſe, ad Terræ noſtræ ſi-<lb/>militudinem diſſeruimus, conſentaneum erit, ut ad innu-<lb/>meros Planetas alios, tot mille Solibus additos, æquè per-<lb/>tinere credamus. </s> <s xml:id="echoid-s9903" xml:space="preserve">Eruntque & </s> <s xml:id="echoid-s9904" xml:space="preserve">illic ſtirpes & </s> <s xml:id="echoid-s9905" xml:space="preserve">animalia, at-<lb/>que etiam ratione inſtructa, quæ cœli convexa mirentur, & </s> <s xml:id="echoid-s9906" xml:space="preserve"><lb/>fidera obſervent, motuſque eorum intelligant; </s> <s xml:id="echoid-s9907" xml:space="preserve">atque omnia <lb/>denique habeant, ſine quibus neque hæc haberi poſſe ſupra <lb/>oſtendimus.</s> <s xml:id="echoid-s9908" xml:space="preserve"/> </p> <div xml:id="echoid-div468" type="float" level="2" n="22"> <note position="right" xlink:label="note-0457-01" xlink:href="note-0457-01a" xml:space="preserve">Nihil impe-<lb/>dire, quo mi-<lb/>nus creda-<lb/>mus circa <lb/>unamquam-<lb/>que ex fixis, <lb/>ut circa So-<lb/>lem, eſſe <lb/>Planetas.</note> </div> <p> <s xml:id="echoid-s9909" xml:space="preserve">Quam mirabilis igitur quamque ſtupenda mundi amplitu-<lb/>do & </s> <s xml:id="echoid-s9910" xml:space="preserve">magnificentia jam mente concipienda eſt. </s> <s xml:id="echoid-s9911" xml:space="preserve">Tot So-<lb/>les, tot Terræ, atque harum unaquæque tot herbis, arbo-<lb/>ribus, animalibus, tot maribus, montibuſque exornata. </s> <s xml:id="echoid-s9912" xml:space="preserve">Et <lb/>erit etiam unde augeatur admiratio, ſi quis ea, quæ de fixa-<lb/>rum ſtellarum diſtantia & </s> <s xml:id="echoid-s9913" xml:space="preserve">multitudine hiſce addimus, per-<lb/>penderit.</s> <s xml:id="echoid-s9914" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9915" xml:space="preserve">Tantam igitur eſſe diſtantiam hanc, ut quæ Solem Ter-<lb/>ramque interjacet, Terræque diametrorum duodecim millia <lb/>continet, ei comparata, exilis plane habenda ſit, non una <lb/>ratione conſtat: </s> <s xml:id="echoid-s9916" xml:space="preserve">atque hac inter cæteras, quod ſi proximæ <lb/>quædam inter ſe ſtellæ notentur, quæ claritate plurimum <lb/>differunt; </s> <s xml:id="echoid-s9917" xml:space="preserve">velut in media caudæ, (quæ duplex eſt) Urſæ <pb o="717" file="0459" n="481" rhead="COSMOTHEOROS, LIB. II."/> majoris; </s> <s xml:id="echoid-s9918" xml:space="preserve">nulla apparentis intervalli earum mutatio animad-<lb/>vertitur, quocunque anni tempore ſpectatarum; </s> <s xml:id="echoid-s9919" xml:space="preserve">quod ta-<lb/>men fieri neceſſe eſſet, propter diverſas viſus poſitiones per <lb/>annui Orbis ambitum, orireturque parallaxis aliqua ſi, ut <lb/>conſentaneum eſt, propior ſit ſtella quæ lucidior apparet. <lb/></s> <s xml:id="echoid-s9920" xml:space="preserve">Qui autem ante nos deſiniendi tam vaſti ſpatii rationem inie-<lb/>runt, nihil certi comprehendere potuerunt, propter nimiam <lb/>obſervationum neceſſariarum ſubtilitatem, quæque omnem <lb/>diligentiam ſuperet. </s> <s xml:id="echoid-s9921" xml:space="preserve">Itaque mihi unica hæc via ſupereſſe vi-<lb/>ſa eſt, quam nunc inſiſtam, qua ſaltem veriſimile quid in re <lb/>tam exploratu ardua conſequamur. </s> <s xml:id="echoid-s9922" xml:space="preserve">Cum ergo ſtellæ ut jam <lb/>diximus, totidem ſint Soles; </s> <s xml:id="echoid-s9923" xml:space="preserve">ſi earum aliquam Soli æqua-<lb/>lem eſſe ſumamus, erit illius tanto major quam Solis diſtan-<lb/>tia, quanto apparens diameter diametro Solis minor erit. </s> <s xml:id="echoid-s9924" xml:space="preserve"><lb/>Sed tam exiguæ apparent ſtellæ, etiam quæ primæ ſunt ma-<lb/> <anchor type="note" xlink:label="note-0459-01a" xlink:href="note-0459-01"/> gnitudinis, atque etiam Teleſcopio ſpectatæ, ut veluti pun-<lb/>cta lucentia ſine viſibili latitudine re@ulgeant. </s> <s xml:id="echoid-s9925" xml:space="preserve">Quo fit ut <lb/>ejuſmodi obſervationibus nulla earum menſura deprehendi <lb/>poſſit. </s> <s xml:id="echoid-s9926" xml:space="preserve">Cum itaque hac non ſuccederet, tentavi qua ratio-<lb/>ne Solis diametrum ita imminuere poſſem ut non majorem <lb/>lucem quam Sirius, aut aliud è clarioribus ſideribus, ad ocu-<lb/>lum mitteret. </s> <s xml:id="echoid-s9927" xml:space="preserve">Ergo occluſi rurſus, ut ſupra, tubi duode-<lb/>cimpedalis vacui aperturam alteram lamella tenuiſſima, cu-<lb/>jus medio tam exiguum effeci foramen, ut Lineæ partem <lb/>duodecimam non ſuperaret, ſive pollicis centeſimam qua-<lb/>drageſimam quartam. </s> <s xml:id="echoid-s9928" xml:space="preserve">Hunc tubum ea parte ad Solem ob-<lb/>verti; </s> <s xml:id="echoid-s9929" xml:space="preserve">altera oculo admovi; </s> <s xml:id="echoid-s9930" xml:space="preserve">qui tunc particulam Solis cerne-<lb/>bat, cujus diameter, ad totius diametrum, erat ut 1 ad 182. <lb/></s> <s xml:id="echoid-s9931" xml:space="preserve">Sed eam particulam multo clariorem comperiebam, quàm <lb/>noctu Sirius apparet. </s> <s xml:id="echoid-s9932" xml:space="preserve">Itaque cum longè magis arctandum <lb/>Solis diametrum viderem, id ita effeci, ut, in perforata e-<lb/>juſmodi lamina, vitreum globulum objicerem minutiſſimum, <lb/>pari circiter diametro ac prius illud foramen habebat; </s> <s xml:id="echoid-s9933" xml:space="preserve">quo <lb/>globulo ad microſcopia antehac uſus fueram. </s> <s xml:id="echoid-s9934" xml:space="preserve">Ita per tubum <lb/>in Solem intuenti, contecto undique capite, ne quid diei <lb/>lux turbaret, non minor ejus claritas quam Sirii videbatur.</s> <s xml:id="echoid-s9935" xml:space="preserve"> <pb o="718" file="0460" n="482" rhead="CHRISTIANI HUGENII"/> Atqui, ex Dioptrices legibus inſtituro calculo, fiebat jam <lb/>Solis diameter {1/152} ejus par@iculæ centeſimæ octogeſimæ ſecun-<lb/>dæ, quam, per foramen exiguum, prius conſpexeram. </s> <s xml:id="echoid-s9936" xml:space="preserve">Du-<lb/>ctis autem in ſe {1/152} & </s> <s xml:id="echoid-s9937" xml:space="preserve">{1/182}, fit {1/27664}. </s> <s xml:id="echoid-s9938" xml:space="preserve">Ergo eou@que contracto <lb/>Sole, vel eouſque remoto, (erit enim effectus idem) ut dia-<lb/>meter ejus ſit {1/27664} ejus, quem in cœlo intuemur, ſupereſt <lb/>illi lux quæ Sirii luci non cedat. </s> <s xml:id="echoid-s9939" xml:space="preserve">Solis vero eouſque remo-<lb/>ti diſtantia erit neceſſario, ad eam quam nunc habet, ut <lb/>27664 ad 1: </s> <s xml:id="echoid-s9940" xml:space="preserve">& </s> <s xml:id="echoid-s9941" xml:space="preserve">diameter paulum excedet 4 ſcr. </s> <s xml:id="echoid-s9942" xml:space="preserve">tertia. </s> <s xml:id="echoid-s9943" xml:space="preserve">Ita-<lb/>que cum æqualis ei Sirius ponatur, ſequitur Sirii quoque <lb/>diametrum totidem eſſe ejuſmodi ſcrupulorum; </s> <s xml:id="echoid-s9944" xml:space="preserve">diſtantiam-<lb/>que itidem, ad eam qua à Sole abſumus, ut 27664 ad 1. <lb/></s> <s xml:id="echoid-s9945" xml:space="preserve">Quod quàm incredibile ſit intervallum, apparebit eadem ra-<lb/>tione, quam in æſtimanda Solis diſtantia adhibuimus. </s> <s xml:id="echoid-s9946" xml:space="preserve">Nam <lb/>ſi 25. </s> <s xml:id="echoid-s9947" xml:space="preserve">annis opus habebat tormenti bellici globus, continua <lb/>velocitate, quanta exploditur, incedens, ut à Terra ad So-<lb/>lem perveniret; </s> <s xml:id="echoid-s9948" xml:space="preserve">jam numerus 27664 vicies quinquies du-<lb/>cendus eſt, atque ita fiunt 691600, adeo ut penè ſeptingen-<lb/>ta annorum millia inſumpturus ſit globus, in tanta celeritate <lb/>ſua, priuſquam ad proximas ſtellarum inerrantium perve-<lb/>niat. </s> <s xml:id="echoid-s9949" xml:space="preserve">Atque ad has ſtellas ſerena nocte oculos circumferen-<lb/>tes, quantum horum judicio comprehendere poſſumus, vix <lb/>aliquot milliaribus ſupra verticem eas exſtare putamus. </s> <s xml:id="echoid-s9950" xml:space="preserve">Quæ-<lb/>ſivi vero de proximis tantum. </s> <s xml:id="echoid-s9951" xml:space="preserve">Cæteræ enim cum, ut jam <lb/>diximus, iis ſpatiis in ulteriora cœli recedant, ut non mino-<lb/>ra ſint deinceps à propioribus ad ſequentes, quàm à Sole <lb/>ad iſtas, quanta immenſitas ſupereſt! Si enim plures quàm <lb/>mille, nudo viſu notantur; </s> <s xml:id="echoid-s9952" xml:space="preserve">teleſcopiis verò decuplo aut vi-<lb/>gecuplo amplius; </s> <s xml:id="echoid-s9953" xml:space="preserve">quomodo ſciri poteſt aut definiri, quanta <lb/>ſit multitudo ulteriorum, quas neque hoc auxilio attingere <lb/>licet: </s> <s xml:id="echoid-s9954" xml:space="preserve">aut quis numerus nimis magnus dicendus eſt, ſi ad Dei <lb/>potentiam ſpectemus? </s> <s xml:id="echoid-s9955" xml:space="preserve">Etenim, ſæpe hæc cogitanti mihi, <lb/>in mentem venit, tanrùm in primis numerorum exordiis cal-<lb/>culos omnes noſtros verſari. </s> <s xml:id="echoid-s9956" xml:space="preserve">Eſſe enim in ſerie eorum in-<lb/>finita, qui non tantùm viginti aut triginta, aut centum, <lb/>aut mille notis ſcribantur in progreſſione noſtra denaria; </s> <s xml:id="echoid-s9957" xml:space="preserve">ſed <pb o="719" file="0461" n="483" rhead="COSMOTHEOROS, LIB. II."/> qui tot characteribus conſtent, quot arenæ grana in tota <lb/>Telluris mole continerentur. </s> <s xml:id="echoid-s9958" xml:space="preserve">Quis verò dicere audeat tali <lb/>numero non majorem eſſe multitudinem ſtellarum inerran-<lb/>tium? </s> <s xml:id="echoid-s9959" xml:space="preserve">Nam longè ulterius progreſſi ſunt, qui infinitam eſſe <lb/>dixerunt; </s> <s xml:id="echoid-s9960" xml:space="preserve">ut Veterum aliqui, atque etiam Jordanus Brunus; <lb/></s> <s xml:id="echoid-s9961" xml:space="preserve">qui pluribus argumentis hoc ſe eviciſſe putat, ſed, ut mihi <lb/>videtur, parum firmis. </s> <s xml:id="echoid-s9962" xml:space="preserve">Nec tamen contrarium quoque per-<lb/>ſpicuis rationibus probari poſſe exiſtimo. </s> <s xml:id="echoid-s9963" xml:space="preserve">Illud conſtat, ſpa-<lb/>tium naturæ univerſæ infinitè undique protendi; </s> <s xml:id="echoid-s9964" xml:space="preserve">at nihil ob-<lb/>ſtat, quin, ultra definitam ſtellarum regionem, res alias in-<lb/>numeras Deus effecerit, à cogitationibus noſtris, æque, ac <lb/>ſedibus, remotas.</s> <s xml:id="echoid-s9965" xml:space="preserve"/> </p> <div xml:id="echoid-div469" type="float" level="2" n="23"> <note position="right" xlink:label="note-0459-01" xlink:href="note-0459-01a" xml:space="preserve">Modus pro-<lb/>babiliter in-<lb/>veſtigandi <lb/>diſtantiam <lb/>fixarum à <lb/>Sole.</note> </div> <p> <s xml:id="echoid-s9966" xml:space="preserve">Quid ſi verò nec innumeras quidem condidit, ſed ultra <lb/>eas vacuum reliquit infinitum; </s> <s xml:id="echoid-s9967" xml:space="preserve">ut totum illud, quod exſtare <lb/>voluit, veluti nihil ſit præ iis quæ producere ejus omnipo-<lb/>tentia potuiſſet? </s> <s xml:id="echoid-s9968" xml:space="preserve">Sed ulteriorem horum inquiſitionem, to-<lb/>tamque illam de infinito difficillimam diſputationem perſequi <lb/>omitto, ne ad tot maximarum rerum comprehenſionem, <lb/>qua jam defuncti ſumus, novus labor accedat. </s> <s xml:id="echoid-s9969" xml:space="preserve">Ea tantùm <lb/>hic ſubjungam, ex quibus, quænam ſit noſtra de toto mun-<lb/>di ſpatio opinio, cognoſcatur; </s> <s xml:id="echoid-s9970" xml:space="preserve">quatenus nempe Solibus ſeu <lb/>ſtellis inerrantibus patet, quibus ſua circumponi plenetaria <lb/>ſyſtemata, probabile eſſe antea oſtendimus.</s> <s xml:id="echoid-s9971" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9972" xml:space="preserve">Exiſtimo itaque unumquemque Solem circumdari vortice <lb/> <anchor type="note" xlink:label="note-0461-01a" xlink:href="note-0461-01"/> quodam materiæ celeriter motæ, ſed qui multum diſſimiles <lb/>ſint Carteſianis illis, tum ſpatii ratione, tum motus genere, <lb/>quo in illis materia agitetur. </s> <s xml:id="echoid-s9973" xml:space="preserve">Ea enim apud Carteſium eſt <lb/>vorticum amplitudo, ut quiſque eorum alios ſe circumſiſten@ <lb/>tes contingat, occurrens ſingulis plana ſuperſicie, veluti cum <lb/>in aqua ſapone imbuta bullarum cumulos pueri inflant: </s> <s xml:id="echoid-s9974" xml:space="preserve">mo-<lb/>veri vero univerſam cujuſque materiam ſtatuit, in partem <lb/>eandem rotando. </s> <s xml:id="echoid-s9975" xml:space="preserve">At hunc motum non parum impediri o-<lb/>porteret, propter anguloſam vorticum ſuperficiem. </s> <s xml:id="echoid-s9976" xml:space="preserve">Deinde <lb/>cum ſit ejuſmodi, ut, velut circa axem cylindri, materia <lb/>tota feratur, exoritur ei poſtea non exigua difficultas, cum <lb/>globoſam Solis formam ex hoc motu deducere conatur: </s> <s xml:id="echoid-s9977" xml:space="preserve">fru- <pb o="720" file="0462" n="484" rhead="CHRISTIANI HUGENII"/> ſtra prorſus, atque iis rationibus, quæ incautis aliquid eſ-<lb/>ſe videantur, cum reipſa nihil explicent. </s> <s xml:id="echoid-s9978" xml:space="preserve">Vu@t præterea <lb/>innatare, ac circunferri cum hac materia ætherea, Plane-<lb/>tas; </s> <s xml:id="echoid-s9979" xml:space="preserve">atque ea ratione videlicet in ſuis orbibus eas retine-<lb/>ri, quòd non majore vi, quàm ipſamet, à centro mo-<lb/>tus recedere conentur. </s> <s xml:id="echoid-s9980" xml:space="preserve">Sed hic ex Aſtronomicis com-<lb/>plura objiciuntur, de quibus aliqua attigimus in diatriba <lb/>de cauſſis gravitatis. </s> <s xml:id="echoid-s9981" xml:space="preserve">Ubi & </s> <s xml:id="echoid-s9982" xml:space="preserve">aliam rationem expoſuimus, <lb/>quæ Planetas intra orbium ſuorum limites contineret. </s> <s xml:id="echoid-s9983" xml:space="preserve">Ea <lb/>eſt gravitas eorum Solem verſus; </s> <s xml:id="echoid-s9984" xml:space="preserve">quæ unde exoriatur oſ-<lb/>tendimus, quamque eo magis miror Carteſium præteriiſ-<lb/>ſe, quod de gravitate, qua corpora in Terram ferun-<lb/>tur, primus ſolito meliora adferre cœpiſſet. </s> <s xml:id="echoid-s9985" xml:space="preserve">Refert Plu-<lb/>tarchus in libro ſupra memorato de Facie in orbe Lu-<lb/>næ, fuiſſe jam olim qui putaret ideo manere Lunam in <lb/>Orbe ſuo, quod vis recedendi à Terra, ob motum cir-<lb/>cularem, inhiberetur pari vi gravitatis, qua ad Terram <lb/>accedere conaretur. </s> <s xml:id="echoid-s9986" xml:space="preserve">Idemque ævo noſtro, non de Lu-<lb/>na tantum, ſed & </s> <s xml:id="echoid-s9987" xml:space="preserve">Planetis cæteris ſtatuit Alphonſus Bo-<lb/>rellus; </s> <s xml:id="echoid-s9988" xml:space="preserve">ut nempe Primariis eorum gravitas eſſet Solem ver-<lb/>ſus; </s> <s xml:id="echoid-s9989" xml:space="preserve">Lunis vero ad Terram, Jovem, ac Saturnum, quos <lb/>comitantur. </s> <s xml:id="echoid-s9990" xml:space="preserve">Multoque diligentius ſubtiliuſque idem nu-<lb/>per explicuit Iſacus Neutonus, & </s> <s xml:id="echoid-s9991" xml:space="preserve">quomodo ex his cau-<lb/>ſis naſcantur Planetarum orbes Elliptici, quos Keplerus <lb/>excogitaverat; </s> <s xml:id="echoid-s9992" xml:space="preserve">in quorum foco altero Sol ponitur. </s> <s xml:id="echoid-s9993" xml:space="preserve">Opor-<lb/>tet autem, ſecundum noſtram de natura gravium ſenten-<lb/>tiam, quò Planetæ ad Solem ſuo pondere inclinent, vor-<lb/>ticem turbinemve materiæ cœleſtis circa eum converti non <lb/>totum in eaſdem partes, ſed ita ut variis motibus, iiſque <lb/>celerrimis in omne latus ſecundum diverſas ſui portiones <lb/>rapiatur, nec tamen dilabi poſſit, propter circunſtantem <lb/>ætherem, qui non tali nec tam celeri motu agitetur. </s> <s xml:id="echoid-s9994" xml:space="preserve">Hu-<lb/>juſmodi vortice gravitatem corporum in Terram, ejuſque <lb/>effectus omnes explicare conati ſumus, in ea, cujus me-<lb/>mini, diatriba. </s> <s xml:id="echoid-s9995" xml:space="preserve">@ademque, ut puto, eſt ratio gravitatis <lb/>Planetarum Solem verſus; </s> <s xml:id="echoid-s9996" xml:space="preserve">& </s> <s xml:id="echoid-s9997" xml:space="preserve">ex his quoque tam Terræ <pb o="721" file="0463" n="485" rhead="COSMOTHEOROS, LIB. II."/> noſtræ, quàm cæterarum, atque etiam Solis, rotunditas <lb/>conſequitur; </s> <s xml:id="echoid-s9998" xml:space="preserve">quæ in Carteſiana hypotheſi tantum habet in-<lb/>commodi.</s> <s xml:id="echoid-s9999" xml:space="preserve"/> </p> <div xml:id="echoid-div470" type="float" level="2" n="24"> <note position="right" xlink:label="note-0461-01" xlink:href="note-0461-01a" xml:space="preserve">Unumquem-<lb/>que Solem <lb/>vortice cingi, <lb/>ſed Carteſia-<lb/>nis multum <lb/>diſſimili; <lb/>contra quem <lb/>pluribus diſ-<lb/>putatur.</note> </div> <p> <s xml:id="echoid-s10000" xml:space="preserve">Porro & </s> <s xml:id="echoid-s10001" xml:space="preserve">ſpatia horum vorticum, ut dixi, multo quam <lb/>ille contractiora pono. </s> <s xml:id="echoid-s10002" xml:space="preserve">Sic enim fere eos ſtatuo in va-<lb/>ſta cœli profunditate diſperſos, quemadmodum turbines <lb/>aquæ exiguos, hinc inde in ſpatioſo lacu ſtagnove, bacu-<lb/>li agitatione, excitatos, ac magnis intervallis totiſque ſta-<lb/>diis diſtantes. </s> <s xml:id="echoid-s10003" xml:space="preserve">Et ſicuti horum motus nequaquam ab u-<lb/>nis ad alios perveniunt, nec proinde ſeſe mutuo impe-<lb/>diunt; </s> <s xml:id="echoid-s10004" xml:space="preserve">ita quoque cœleſtium vorticum motus, circum aſtra <lb/>aut Soles, ſe habere exiſtimo.</s> <s xml:id="echoid-s10005" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10006" xml:space="preserve">Itaque neque alii alios deſtruere poſſunt aut abſorbere, <lb/>quemadmodum finxit Carteſius, cum oſtendere vellet quo-<lb/>modo ſtella aut Sol aliquis vertatur in Planetam. </s> <s xml:id="echoid-s10007" xml:space="preserve">Appa-<lb/>ret autem, cum hæc ſcriberet, non attendiſſe eum ad im-<lb/>menſam ſtellarum inter ſe diſtantiam; </s> <s xml:id="echoid-s10008" xml:space="preserve">idque vel ex hoc <lb/>uno, quod, cum primum Cometes aliquis intra vorticem <lb/>noſtrum, cujus centrum Sol occupat, deſcendit; </s> <s xml:id="echoid-s10009" xml:space="preserve">vult eum <lb/>nobis viſibilem fieri, quod eſt abſurdiſſimum. </s> <s xml:id="echoid-s10010" xml:space="preserve">Quomodo <lb/>enim, ſidus ejuſmodi, quod ex Solis lumine repercuſſo <lb/>tantummodo ſplendet; </s> <s xml:id="echoid-s10011" xml:space="preserve">ut cum pleriſque Philoſophis ipſe <lb/>ſtatuit; </s> <s xml:id="echoid-s10012" xml:space="preserve">quomodo, inquam, poſſet conſpici à tanto in-<lb/>tervallo, quod ſaltem decies millies contineret illud quod <lb/>à Terra ad Solem eſt. </s> <s xml:id="echoid-s10013" xml:space="preserve">Non enim ignorare poterat va-<lb/>ſtiſſimum, circa Solem undique extenſum, ſpatium; </s> <s xml:id="echoid-s10014" xml:space="preserve">cum <lb/>ſciret in Copernici ſyſtemate orbem magnum, hoc eſt, <lb/>orbitam Terræ, velut punctum eſſe cum illo compara-<lb/>tum. </s> <s xml:id="echoid-s10015" xml:space="preserve">Sed tota hæc de Cometarum, atque etiam de Pla-<lb/>netarum, & </s> <s xml:id="echoid-s10016" xml:space="preserve">mundi origine, commentatio apud Carte-<lb/>ſium tam levibus rationibus contexta eſt, ut ſæpe mi-<lb/>rer tantum operæ in talibus concinnandis figmentis eum <lb/>impendere potuiſſe. </s> <s xml:id="echoid-s10017" xml:space="preserve">Mihi magnum quid conſecuti vi-<lb/>debimur, ſi quemadmodum ſeſe habeant res, quæ in <lb/>natura exſtant, intellexerimus; </s> <s xml:id="echoid-s10018" xml:space="preserve">à quo longiſſime etiam <pb o="722" file="0464" n="486" rhead="CHRISTIANI HUGENII COSMOTH. L. II."/> nunc abſumus. </s> <s xml:id="echoid-s10019" xml:space="preserve">Quomodo autem quæque effectæ fue-<lb/>rint, quodque ſunt, eſſe cœperint, id nequaquam hu-<lb/>mano ingenio excogitari, aut conjecturis attingi poſſe, <lb/>exiſtimo.</s> <s xml:id="echoid-s10020" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div472" type="section" level="1" n="158"> <head xml:id="echoid-head220" xml:space="preserve">FINIS.</head> <figure> <image file="0464-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0464-01"/> </figure> <pb file="0465" n="487"/> </div> <div xml:id="echoid-div473" type="section" level="1" n="159"> <head xml:id="echoid-head221" xml:space="preserve">CHRISTIANI HUGENII <lb/>OPERA <lb/>MISCELLANEA.</head> <head xml:id="echoid-head222" style="it" xml:space="preserve"><emph style="sc">Tomus</emph> <emph style="sc">Quartus</emph>.</head> <pb file="0466" n="488"/> </div> <div xml:id="echoid-div474" type="section" level="1" n="160"> <head xml:id="echoid-head223" xml:space="preserve">Tomi quarti contenta.</head> <note position="right" xml:space="preserve"> <lb/>De Ratiociniis in <emph style="sc">Ludo</emph> <emph style="sc">Aleæ</emph>. # 723. <lb/><emph style="sc">Novus</emph> <emph style="sc">Cyclus</emph> <emph style="sc">Harmonicus</emph>. # 745. <lb/>Varia de <emph style="sc">Optica</emph>. # 755. <lb/>Experimenta Phyſica. # 767. <lb/></note> <pb file="0467" n="489"/> </div> <div xml:id="echoid-div475" type="section" level="1" n="161"> <head xml:id="echoid-head224" xml:space="preserve">DE <lb/>RATIOCINIIS <lb/>IN <lb/>LUDO ALEÆ <lb/>AUCTORE <lb/>CHRISTIANO HUGENIO.</head> <pb file="0468" n="490"/> <pb o="725" file="0469" n="491"/> <figure> <image file="0469-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0469-01"/> </figure> </div> <div xml:id="echoid-div476" type="section" level="1" n="162"> <head xml:id="echoid-head225" xml:space="preserve">CHRISTIANUS HUGENIUS <lb/>Clariſſimo Viro, <lb/>D. <emph style="sc">Francisco</emph> <emph style="sc">Schotenio</emph> <lb/>S. D.</head> <p style="it"> <s xml:id="echoid-s10021" xml:space="preserve">CUm in editione elegantiſſimorum ingenii <lb/>tui monumentorum, quam præ manibus <lb/>nunc habes, Vir Clariſſime, id inter cæ-<lb/>tera te ſpectare ſciam, ut varietate re-<lb/>rum, quarum tractationem inſtituiſti, <lb/>oſtendas quam latè ſe protendat divina Analytices ſcien-<lb/>tia, facilè intelligo etiam illa plurimùm propoſito tuo <lb/>inſervire poſſe, quæ de aleæ ratiociniis conſcripſimus; <lb/></s> <s xml:id="echoid-s10022" xml:space="preserve">quantò enim minus rationis terminis comprehendi poſſe <lb/>videbantur, quæ fortuita ſunt atque incerta, tantò <lb/>admirabilior ars cenſebitur, cui iſta quoque ſubjacent. </s> <s xml:id="echoid-s10023" xml:space="preserve"><lb/>Quare cum in tui gratiam primùm illa exponenda ſuſ-<lb/>ceperim, túque digna exiſtimes, quæ ſimul cum ſubti-<lb/>liſſimis tuis inventis in lucem exeant, adeo tibi non re-<lb/>fragabor, ut etiam è re mea eſſe exiſtimem bâc po-<lb/>tiſſimùm ratione ipſa in manus hominum pervenire. </s> <s xml:id="echoid-s10024" xml:space="preserve"><lb/>Quippe cum in re levi ac frivola operam collocaſſe vi-<lb/>deri alioqui poſſem, non tamen prorſus utilitatis expers <lb/>ac nullius pretii cenſebitur, quòd tu veluti inter tua <pb o="726" file="0470" n="492" rhead="PRÆFATIO."/> adoptaveris, nec ſine multo labore è vernacula lingua <lb/>noſtra in Latinam converteris. </s> <s xml:id="echoid-s10025" xml:space="preserve">Quanquam, ſi quis <lb/>penitiù<unsure/>s ea quæ tradimus examinare cœperit, non du-<lb/>bito quin continuò reperturus ſit rem non ut videtur lu-<lb/>dicram agi, ſed pulchræ ſubtiliſſimæque contemplationis <lb/>fundamenta explicari. </s> <s xml:id="echoid-s10026" xml:space="preserve">Et Problemata quidem quæ in <lb/>hoc genere proponuntur, nihilo minus profundæ inda-<lb/>ginis viſum iri confido, quam quæ Diophanti libris <lb/>continentur, voluptatis autem aliquanto plus habitura, <lb/>cum non, ſicut illa, in nuda numerorum conſideratio-<lb/>ne terminentur. </s> <s xml:id="echoid-s10027" xml:space="preserve">Sciendum verò, quod jam pridem <lb/>inter præſtantiſſimos totâ Galliâ Geometras calculus hic <lb/>agitatus fuerit, ne quis indebitam mihi primæ inven-<lb/>tionis gloriam hac in re tribuat. </s> <s xml:id="echoid-s10028" xml:space="preserve">Cæterùm illi, diffi-<lb/>cillimis quibuſque quæſtionibus ſe invicem exercere ſoli-<lb/>ti, methodum ſuam quiſque occultam retinuêre, adeo <lb/>ut a primis elementis univerſam hanc materiam evol-<lb/>vere mihi neceſſe fuerit. </s> <s xml:id="echoid-s10029" xml:space="preserve">Quamobrem ignoro etiam-<lb/>num an eodem mecum principio illi utantur; </s> <s xml:id="echoid-s10030" xml:space="preserve">at in re-<lb/>ſolvendis Problematîs pulchrè nobis convenire ſæpenu-<lb/>mero expertus ſum. </s> <s xml:id="echoid-s10031" xml:space="preserve">Horum Problematum nonnulla in <lb/>ſine operis addidiſſe me invenies, omiſſa tamen analy-<lb/>ſi, cum quod prolixam nimis operam poſcebant, ſi per-<lb/>ſpicuè omnia exequi voluiſſem, tum quod relinquen-<lb/>dum aliquid videbatur exercitationi noſtrorum, ſi qui <lb/>erunt, Lectorum. </s> <s xml:id="echoid-s10032" xml:space="preserve">Vale.</s> <s xml:id="echoid-s10033" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10034" xml:space="preserve">Dat. </s> <s xml:id="echoid-s10035" xml:space="preserve">Hagæ Com. <lb/></s> <s xml:id="echoid-s10036" xml:space="preserve">27 Apr. </s> <s xml:id="echoid-s10037" xml:space="preserve">1657.</s> <s xml:id="echoid-s10038" xml:space="preserve"/> </p> <pb o="727" file="0471" n="493"/> <figure> <image file="0471-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0471-01"/> </figure> </div> <div xml:id="echoid-div477" type="section" level="1" n="163"> <head xml:id="echoid-head226" xml:space="preserve">DE <lb/>RATIOCINIIS <lb/>IN <lb/>LUDO ALEÆ.</head> <p> <s xml:id="echoid-s10039" xml:space="preserve">ETſi luſionum, quas ſola ſors moderatur, in-<lb/>certi ſolent eſſe eventus, attamen in his, <lb/>quanto quis ad vincendum quàm perdendum <lb/>propior ſit, certam ſemper habet determina-<lb/>tionem. </s> <s xml:id="echoid-s10040" xml:space="preserve">Ut ſi quis primo jactu unâ teſſerâ <lb/>ſenarium jacere contendat, incertum quidem <lb/>an vincet; </s> <s xml:id="echoid-s10041" xml:space="preserve">at quantò veriſimilius ſit eum perdere quàm vin-<lb/>cere, reipsâ definitum eſt, calculóque ſubducitur. </s> <s xml:id="echoid-s10042" xml:space="preserve">Ita quo-<lb/>que, ſi cum aliquo certem hâc ratione, ut ternis luſibus con-<lb/>ſtet victoria, atque ego jam unum luſum vicerim, incertum <lb/>adhuc uter noſtrum prior tertii victor ſit evaſurus. </s> <s xml:id="echoid-s10043" xml:space="preserve">Verùm <lb/>quanti expectatio mea, & </s> <s xml:id="echoid-s10044" xml:space="preserve">contra quanti illius, æſtimari de-<lb/>beat, certiſſimo ratiocinio conſequi licet, atque hinc defi-<lb/>nire, ſi ludum uti eſt imperfectum linquere inter nos conve-<lb/>nerit, quantò major portio ejus quod depoſitum eſt mihi <lb/>quàm adverſario meo tribuenda eſſet:</s> <s xml:id="echoid-s10045" xml:space="preserve">vel etiam ſi quis in lo-<lb/>cum ſortemque meam ſuccedere cupiat, quo pretio me eam <lb/>ipſi vendere æquum ſit. </s> <s xml:id="echoid-s10046" xml:space="preserve">Atque hinc innumeræ quæſtiones <lb/>exoriri poſſunt inter duos, tres, pluréſve colluſores. </s> <s xml:id="echoid-s10047" xml:space="preserve">Cum-<lb/>que minimè vulgaris ſit hujuſmodi ſupputatio, & </s> <s xml:id="echoid-s10048" xml:space="preserve">ſæpe utili-<lb/>ter adhibeatur, breviter hîc quâ ratione aut methodo expe-<lb/>dienda ſit exponam, ac deinde etiam, quæ ad aleam ſive teſ-<lb/>ſeras propriè pertinent, explicabo.</s> <s xml:id="echoid-s10049" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10050" xml:space="preserve">Hoc autem utrobique utar fundamento: </s> <s xml:id="echoid-s10051" xml:space="preserve">nimirum, in aleæ <pb o="728" file="0472" n="494" rhead="DE RATIOCINIIS"/> ludo tanti æſtimandam eſſe cujuſque ſortem ſeu expectatio-<lb/>nem ad aliquid obtinendum, quantum ſi habeat, poſſit de-<lb/>nuo ad ſimilem ſortem ſive expectationem pervenire, æquâ <lb/>conditione certans. </s> <s xml:id="echoid-s10052" xml:space="preserve">Ut, exempli gratiâ ſi quis me inſcio <lb/>alterâ manu 3 ſolidos occultet, alterâ 7 ſolidos, mihique <lb/>optionem det ex utra manu ſolidos accipere malim; </s> <s xml:id="echoid-s10053" xml:space="preserve">hoc tan-<lb/>tundem mihi valere dico, ac ſi 5 ſolidi mihi dentur. </s> <s xml:id="echoid-s10054" xml:space="preserve">Quo-<lb/>niam quinque ſolidos habens, denuo eò pervenire poſſum, <lb/>ut æquam expectationem nanciſcar ad 3 vel 7 ſolidos obti-<lb/>nendos: </s> <s xml:id="echoid-s10055" xml:space="preserve">idque æquo luſu contendens.</s> <s xml:id="echoid-s10056" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div478" type="section" level="1" n="164"> <head xml:id="echoid-head227" xml:space="preserve"><emph style="sc">Propositio</emph> I.</head> <head xml:id="echoid-head228" style="it" xml:space="preserve">Si a vel b expectem, quorum utrumvis æquè <lb/>facilè mihi obtingere poſſit, expectatio <lb/>mea dicenda eſt valere {a + b/2}.</head> <p> <s xml:id="echoid-s10057" xml:space="preserve">AD hanc regulam non ſolùm demonſtrandam, verùm et-<lb/>iam primitùs eruendam poſito x pro eo quod æquivalet <lb/>expectationi meæ, oportet me, quum x habeo, rurſus ad <lb/>ſimilem ſortem pervenire poſſe, æquâ conditione certantem. <lb/></s> <s xml:id="echoid-s10058" xml:space="preserve">Ponatur itaque luſus eſſe talis, ut cum altero certem hâc con-<lb/>ditione, ut quiſque deponat x, ac ut victor victo traditurus <lb/>ſit a. </s> <s xml:id="echoid-s10059" xml:space="preserve">Hic autem luſus juſtus eſt, & </s> <s xml:id="echoid-s10060" xml:space="preserve">patet me hâc ratione <lb/>æquam habere ſortem ad obtinendum a, ſi luſum perdam ſci-<lb/>licet; </s> <s xml:id="echoid-s10061" xml:space="preserve">aut 2x— a, ſi vincam: </s> <s xml:id="echoid-s10062" xml:space="preserve">tum enim obtineo 2x, id nem-<lb/>pe quod depoſitum eſt, de quo alteri erogandum eſt a. </s> <s xml:id="echoid-s10063" xml:space="preserve"><lb/>Quod ſi autem 2x — a tantundem valeret atque b, æqua mi-<lb/>hi ſors obtingeret ad a quàm ad b. </s> <s xml:id="echoid-s10064" xml:space="preserve">Pono itaque 2x — a = b, <lb/>& </s> <s xml:id="echoid-s10065" xml:space="preserve">fit x = {a + b/2}, pro valore meæ expectationis. </s> <s xml:id="echoid-s10066" xml:space="preserve">Cujus de-<lb/>monſtratio facilis eſt. </s> <s xml:id="echoid-s10067" xml:space="preserve">Etenim habens {a + b/2} poſſum cum a-<lb/>lio certare, qui etiam {a + b/2} deponere volet, hâc conditione <pb o="729" file="0473" n="495" rhead="IN LUDO ALEÆ."/> ut vincens victo ſit traditurus a. </s> <s xml:id="echoid-s10068" xml:space="preserve">Quâ ratione ſimilis expe-<lb/>ctatio mihi obtinget ad obtinendum a, ſi perdam, aut ad <lb/>obtinendum b, ſi vincam; </s> <s xml:id="echoid-s10069" xml:space="preserve">tum enim obtineo a + b, id nem-<lb/>pe quod depoſitum eſt, alterique inde concedo a.</s> <s xml:id="echoid-s10070" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10071" xml:space="preserve">In numeris. </s> <s xml:id="echoid-s10072" xml:space="preserve">Si ad 3 vel 7 æqua ſors mihi obtingat, tum <lb/>expectatio mea per hanc Propoſitionem valet 5; </s> <s xml:id="echoid-s10073" xml:space="preserve">& </s> <s xml:id="echoid-s10074" xml:space="preserve">certum <lb/>eſt me 5 habentem rurſus ad eandem expectationem perve-<lb/>nire poſſe. </s> <s xml:id="echoid-s10075" xml:space="preserve">Si enim cum alio certans 5 deponam, atque il-<lb/>le ſimiliter 5 deponat, hâc conditione, ut, qui vincit, al-<lb/>teri ſit daturus 3: </s> <s xml:id="echoid-s10076" xml:space="preserve">erit hic luſus omnino juſtus, & </s> <s xml:id="echoid-s10077" xml:space="preserve">patet mi-<lb/>hi æquam obtingere ſortem ad obtinendum 3, ſi perdam, <lb/>aut 7, ſi vincam: </s> <s xml:id="echoid-s10078" xml:space="preserve">quoniam tunc obtineo 10, de quo alteri <lb/>concedo 3.</s> <s xml:id="echoid-s10079" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div479" type="section" level="1" n="165"> <head xml:id="echoid-head229" xml:space="preserve"><emph style="sc">Propositio</emph> II.</head> <head xml:id="echoid-head230" style="it" xml:space="preserve">Si a, b, vel c expectem, quorum unumquodque pari <lb/>facilitate mihi obtingere poſſit, expectatio <lb/>mea æſtimanda eſt {a + b + c/3}.</head> <p> <s xml:id="echoid-s10080" xml:space="preserve">AD quod rurſus inveniendum, ponatur, ut ante, x pro <lb/>valore expectationis meæ. </s> <s xml:id="echoid-s10081" xml:space="preserve">Oportet ergo me, cùm x ha-<lb/>beo, ad eandem expectationem pervenire poſſe juſto luſu. <lb/></s> <s xml:id="echoid-s10082" xml:space="preserve">Ponatur luſus eſſe talis, ut cum duobus aliis ludam hâc con-<lb/>ditione, ut quiſque noſtrum trium deponat x, & </s> <s xml:id="echoid-s10083" xml:space="preserve">ut cum u-<lb/>no hoc pactum aggrediar, ſi ipſe victor evadat, mihi ſit da-<lb/>turus b, & </s> <s xml:id="echoid-s10084" xml:space="preserve">ego ipſi traditurus ſim b, ſi idem mihi obtingat. </s> <s xml:id="echoid-s10085" xml:space="preserve"><lb/>Cum altero autem hanc ineam conditionem, ut ille ludum <lb/>vincens mihi traditurus ſit c, aut ego ipſi ſim daturus c, ſi <lb/>ego vincam. </s> <s xml:id="echoid-s10086" xml:space="preserve">Et patet hunc ludum juſtum eſſe. </s> <s xml:id="echoid-s10087" xml:space="preserve">Æquam <lb/>autem hâc ratione ſortem habebo ad obtinendum b, ſi nimi-<lb/>rum primus vincat, aut c, ſi ſecundus vincat, aut etiam <lb/>3x — b — c ſi ego vincam; </s> <s xml:id="echoid-s10088" xml:space="preserve">tunc enim obtineo 3x, quod depo-<lb/>ſitum eſt, de quo uni concedo b, & </s> <s xml:id="echoid-s10089" xml:space="preserve">alteri c. </s> <s xml:id="echoid-s10090" xml:space="preserve">Quòd ſi <lb/>3x — b — c æquale fuerit ipſi a, eadem mihi obtingeret ex- <pb o="730" file="0474" n="496" rhead="DE RATIOCINIIS"/> pectatio ad obtinendum a, quæ ad b, aut ad c. </s> <s xml:id="echoid-s10091" xml:space="preserve">Pono itaque <lb/>3x — b — c = a, & </s> <s xml:id="echoid-s10092" xml:space="preserve">fit x = {a + b + c/3}, pro valore meæ expecta-<lb/>tionis. </s> <s xml:id="echoid-s10093" xml:space="preserve">Eodem modo invenitur, ſi ad a, b, c, aut d æqua ſors <lb/>mihi obtingat, id tanti valoris eſſe, quanti {a + b + c + d/4}. </s> <s xml:id="echoid-s10094" xml:space="preserve">At-<lb/>que ita porrò.</s> <s xml:id="echoid-s10095" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div480" type="section" level="1" n="166"> <head xml:id="echoid-head231" xml:space="preserve"><emph style="sc">Propositio</emph> III.</head> <head xml:id="echoid-head232" style="it" xml:space="preserve">Si numerus caſuum, quibus mihi eveniet a, ſit p, nu-<lb/>merus autem caſuum quibus mihi eveniet b ſit q, <lb/>ſumendo omnes caſus æquè in proclivi eſſe: <lb/>expectatio mea valebit {pa + pq/p + q}.</head> <p> <s xml:id="echoid-s10096" xml:space="preserve">AD hanc regulam eruendam, ponatur rurſus x pro valore <lb/>expectationis meæ: </s> <s xml:id="echoid-s10097" xml:space="preserve">ergo oportet me, cùm x habeo, ad <lb/>eandem expectationem pervenire poſſe, ut ante, juſto luſu. <lb/></s> <s xml:id="echoid-s10098" xml:space="preserve">Ad hoc autem tot colluſores ſumam, ut unà mecum nume-<lb/>rum ipſius p + q efficiant, quorum deponat quiſque x, ita ut <lb/>depoſitum ſit px + qx, & </s> <s xml:id="echoid-s10099" xml:space="preserve">quiſque ſibi ludat æquâ expecta-<lb/>tione ad vincendum. </s> <s xml:id="echoid-s10100" xml:space="preserve">Porrò cum tot ex hiſce colluſoribus, <lb/>quot indicat numerus q, ſigillatim hoc pactum inibo, ut eo-<lb/>rum qui vincat mihi ſit daturus b, aut ego contra ipſi idem <lb/>b, ſi vincam. </s> <s xml:id="echoid-s10101" xml:space="preserve">Similiter cum reliquis colluſoribus, conſti-<lb/>tuentibus p — 1 ſigillatim hanc conditionem aggrediar, ut eo-<lb/>rum quiſque, qui ludum vincit, mihi ſit daturus a, & </s> <s xml:id="echoid-s10102" xml:space="preserve">ego <lb/>tantundem (a ſcilicet) ipſi, ſi ego vincam. </s> <s xml:id="echoid-s10103" xml:space="preserve">Et patet hunc <lb/>luſum hâ conditione juſtum eſſe, nemine videlicet injuriam <lb/>patiente. </s> <s xml:id="echoid-s10104" xml:space="preserve">Deinde patet me nunc q expectationis habere ad <lb/>b, & </s> <s xml:id="echoid-s10105" xml:space="preserve">p — 1 expectationes ad a, & </s> <s xml:id="echoid-s10106" xml:space="preserve">1 expectationem (me nem-<lb/>pe vincente) ad px + qx - bq - ap + a, tunc enim obtineo <lb/>px + qx, id quod depoſitum eſt, de quo tradere debeo b u-<lb/>nicuique q luſorum, & </s> <s xml:id="echoid-s10107" xml:space="preserve">a unicuique p — 1 luſorum, quæ ſi-<lb/>mul conficiunt bq + pa — a. </s> <s xml:id="echoid-s10108" xml:space="preserve">Si itaque qx + bx - bq - ap + a <pb o="731" file="0475" n="497" rhead="IN LUDO ALEÆ."/> æquale eſſet ipſi a, haberem p expectationes ad a, (quando-<lb/>quidem jam p — 1 expectationes ad id habebam) & </s> <s xml:id="echoid-s10109" xml:space="preserve">q expe-<lb/>ctationes ad b, & </s> <s xml:id="echoid-s10110" xml:space="preserve">ſic ad priorem meam exſpectationem rur-<lb/>ſus perveniſſem. </s> <s xml:id="echoid-s10111" xml:space="preserve">Quocirca porrò px + qx - bq - ap + a = a, <lb/>& </s> <s xml:id="echoid-s10112" xml:space="preserve">fit x = {ap + bq/p + q}, pro valore expectationis meæ, omnino ut <lb/>in initio poſitum fuit.</s> <s xml:id="echoid-s10113" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10114" xml:space="preserve">In numeris. </s> <s xml:id="echoid-s10115" xml:space="preserve">Si 3 mihi expectationes forent ad 13, & </s> <s xml:id="echoid-s10116" xml:space="preserve">2 <lb/>expectationes ad 8, haberem per hanc regulam tantundem <lb/>ac 11. </s> <s xml:id="echoid-s10117" xml:space="preserve">Et facile eſt oſtendere, me, ſi 11 habeam, rurſus ad <lb/>eandem expectationem pervenire poſſe. </s> <s xml:id="echoid-s10118" xml:space="preserve">Ludens enim con-<lb/>tra 4 alios, & </s> <s xml:id="echoid-s10119" xml:space="preserve">quiſque noſtrum quinque deponens II, cum <lb/>duobus ex illis ſigillatim pactum inibo, ut horum qui vincat <lb/>mihi ſit daturus 8, aut ego ipſi idem 8, ſi vincam. </s> <s xml:id="echoid-s10120" xml:space="preserve">Simili-<lb/>ter cum duobus reliquis, ut eorum quiſque, qui ludum vin-<lb/>cit, mihi ſit daturus 13, aut ego ipſi tantundem, ſi ego vin-<lb/>cam. </s> <s xml:id="echoid-s10121" xml:space="preserve">Qui quidem luſus juſtus eſt. </s> <s xml:id="echoid-s10122" xml:space="preserve">Et patet me hoc modo <lb/>duas habere expectationes ad 8, nimirum ſi alteruter eorum, <lb/>qui mihi 8 promiſerunt, vincat, & </s> <s xml:id="echoid-s10123" xml:space="preserve">3 expectationes ad 13, <lb/>nimirum ſi alteruter reliquorum duorum, qui mihi 13 trade-<lb/>re debent, vincat, aut ſi ipſe ludum vincam: </s> <s xml:id="echoid-s10124" xml:space="preserve">ego enim lu-<lb/>dum vincens obtineo depoſitum, id eſt, 55, de quo unicui-<lb/>que duorum tradere debeo 13, & </s> <s xml:id="echoid-s10125" xml:space="preserve">unicuique reliquorum duo-<lb/>rum 8, ita ut & </s> <s xml:id="echoid-s10126" xml:space="preserve">mihi relinquatur 13.</s> <s xml:id="echoid-s10127" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div481" type="section" level="1" n="167"> <head xml:id="echoid-head233" xml:space="preserve"><emph style="sc">Propositio</emph> IV.</head> <head xml:id="echoid-head234" style="it" xml:space="preserve">Ut igitur ad primò propoſitam quæſtionem veniamus, <lb/>nimirum, de facienda diſtributione inter diverſos <lb/>colluſores, quando eorum ſortes inæquales ſunt, <lb/>opus eſt ut a facilioribus incipiamus.</head> <p> <s xml:id="echoid-s10128" xml:space="preserve">SUmpto itaque me cum aliquo certare, hoc pacto: </s> <s xml:id="echoid-s10129" xml:space="preserve">ut qui <lb/>priùs ter vicerit, quod depoſitum eſt, lucretur, & </s> <s xml:id="echoid-s10130" xml:space="preserve">me jam <lb/>bis viciſſe, alterum verò ſemel. </s> <s xml:id="echoid-s10131" xml:space="preserve">Scire cupio, ſi luſum pro- <pb o="732" file="0476" n="498" rhead="DE RATIOCINIIS"/> ſequi non velimus, ſed pecuniam, de qua certamus, prout <lb/>æquum eſt, partiri, quantum ejus mihi obtingeret.</s> <s xml:id="echoid-s10132" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10133" xml:space="preserve">Primò conſiderare oportet luſus, qui utrobique deficiunt. <lb/></s> <s xml:id="echoid-s10134" xml:space="preserve">Certum enim eſt, ſi inter nos convenerit, verbi gratiâ, ut <lb/>quod depoſitum eſt lucretur is, qui priùs vigeſies vicerit, & </s> <s xml:id="echoid-s10135" xml:space="preserve"><lb/>ego decies & </s> <s xml:id="echoid-s10136" xml:space="preserve">novies vicero, at alter decies & </s> <s xml:id="echoid-s10137" xml:space="preserve">octies, tantò <lb/>meliorem fore eo caſu ſortem meam quantò hîc melior eſt, <lb/>ubi à tribus luſibus binos conſequutus ſum, ille verò unum <lb/>duntaxat: </s> <s xml:id="echoid-s10138" xml:space="preserve">quia nimirum utrobique mihi unus tantummodo <lb/>luſus ſed ipſi duo deficiunt.</s> <s xml:id="echoid-s10139" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10140" xml:space="preserve">Porrò ad inveniendum quanta pars utrique debeatur, ad-<lb/>vertendum eſt quid fieret, ſi in luſu pergeremus. </s> <s xml:id="echoid-s10141" xml:space="preserve">Certum <lb/>enim eſt, ſi primum ludum vincerem, me præſcriptum nu-<lb/>merum impleturum & </s> <s xml:id="echoid-s10142" xml:space="preserve">omne depoſitum conſecuturum, id <lb/>quod vocetur a. </s> <s xml:id="echoid-s10143" xml:space="preserve">Quod ſi autem alter primum ludum vince-<lb/>ret, tunc æquata utriuſque ſors foret, (quippe utrique uno <lb/>adhuc deficiente ludo,) adeoque cederet cuique {1/2}a. </s> <s xml:id="echoid-s10144" xml:space="preserve">Mani-<lb/>feſtum autem eſt me æquam habere ſortem ad primum lu-<lb/>dum vincendum aut perdendum, ita ut mihi nunc æqua ſit <lb/>expectatio ad obtinendum a aut {1/2}a: </s> <s xml:id="echoid-s10145" xml:space="preserve">quod ipſum per 1<emph style="super">mam</emph> <lb/>Propoſitionem tantum eſt ac ſi utriuſque ſortis dimidium, id <lb/>eſt, {3/4}a, haberem; </s> <s xml:id="echoid-s10146" xml:space="preserve">& </s> <s xml:id="echoid-s10147" xml:space="preserve">relinquitur alteri meo colluſori {1/4}a, quæ <lb/>ipſius portio ſtatim ab initio eodem modo reperiri potuiſſet. <lb/></s> <s xml:id="echoid-s10148" xml:space="preserve">Unde patet, eum, qui ludum meum in ſe recipere vellet, <lb/>mihi {3/4}a pro eo tradere debere; </s> <s xml:id="echoid-s10149" xml:space="preserve">ac proinde ſemper tria con-<lb/>tra unum deponere eum poſſe, qui unum ludum vincere con-<lb/>tendat, priuſquam alter duos vincat.</s> <s xml:id="echoid-s10150" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div482" type="section" level="1" n="168"> <head xml:id="echoid-head235" xml:space="preserve"><emph style="sc">Propositio</emph> V.</head> <head xml:id="echoid-head236" style="it" xml:space="preserve">Panamus unum mihi deficere ludum & colluſori meo <lb/>tres luſus. Oportet hîc facere diſtributionem.</head> <p> <s xml:id="echoid-s10151" xml:space="preserve">ADvertamus itaque rurſus, in quo eſſemus ſtatu, ſi ego vel <lb/>ipſe primum vinceret luſum. </s> <s xml:id="echoid-s10152" xml:space="preserve">Si ego vincerem, obtinerem <lb/>depoſitum, id eſt, a; </s> <s xml:id="echoid-s10153" xml:space="preserve">quòd ſi autem ille primum ludum vin- <pb o="733" file="0477" n="499" rhead="IN LUDO ALEÆ."/> ceret, deficerent ipſi duo luſus & </s> <s xml:id="echoid-s10154" xml:space="preserve">mihi unus; </s> <s xml:id="echoid-s10155" xml:space="preserve">ac proinde in <lb/>eodem ſtatu eſſemus, qui in præcedenti Propoſitione poſi-<lb/>tus fuit, mihique obtingeret {3/4}a, ut ibi oſtenſum eſt. </s> <s xml:id="echoid-s10156" xml:space="preserve">Ita-<lb/>que pari facilitate vel a mihi obtinget vel {3/4}a, id quod tan-<lb/>tum eſt, per 1<emph style="super">mam</emph> Propoſitionem, ac {7/8}a. </s> <s xml:id="echoid-s10157" xml:space="preserve">Et relinquitur {@/8} a <lb/>colluſori meo; </s> <s xml:id="echoid-s10158" xml:space="preserve">ita ut mea ſors ad ſortem illius ſe habeat, ſic <lb/>ut 7 ad 1.</s> <s xml:id="echoid-s10159" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10160" xml:space="preserve">Quemadmodum autem ad hunc calculum requiſitus eſt <lb/>præcedens, ita rurſus hicce inſervit ſequenti: </s> <s xml:id="echoid-s10161" xml:space="preserve">nimirum, ſi <lb/>ponamus mihi unum ludum deficere & </s> <s xml:id="echoid-s10162" xml:space="preserve">colluſori meo 4<emph style="super">or</emph> lu-<lb/>ſus. </s> <s xml:id="echoid-s10163" xml:space="preserve">Et invenitur eodem modo, mihi deberi {15/16} iſtius quod <lb/>depoſitum eſt, & </s> <s xml:id="echoid-s10164" xml:space="preserve">ipſi {1/16}.</s> <s xml:id="echoid-s10165" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div483" type="section" level="1" n="169"> <head xml:id="echoid-head237" xml:space="preserve"><emph style="sc">Propositio</emph> VI.</head> <head xml:id="echoid-head238" style="it" xml:space="preserve">Ponamus mihi deficere duos luſus & colluſori meo <lb/>tres luſus.</head> <p> <s xml:id="echoid-s10166" xml:space="preserve">FIet itaque primo luſu; </s> <s xml:id="echoid-s10167" xml:space="preserve">vel ut mihi unus luſus deficiat & </s> <s xml:id="echoid-s10168" xml:space="preserve"><lb/>ipſi tres (unde mihi per præcedentem Propoſitionem ob-<lb/>tinget {7/8} a); </s> <s xml:id="echoid-s10169" xml:space="preserve">vel ut cuique noſtrum adhuc duo luſus deficiant, <lb/>unde mihi debebitur {1/2} a, quandoquidem ſic utrique æqua ſors <lb/>futura eſt. </s> <s xml:id="echoid-s10170" xml:space="preserve">Eſt mihi autem æqualis facilitas ad primum lu-<lb/>dum vincendum aut perdendum; </s> <s xml:id="echoid-s10171" xml:space="preserve">ita ut mihi æqua ſit expe-<lb/>ctatio ad obtinendum {7/8}a aut {1/2}a, id quod mihi valet {11/16} a, per <lb/>1<emph style="super">mam</emph> Propoſitionem. </s> <s xml:id="echoid-s10172" xml:space="preserve">Et debentur mihi 11 partes ejus quod <lb/>depoſitum eſt, & </s> <s xml:id="echoid-s10173" xml:space="preserve">colluſori meo 5 partes.</s> <s xml:id="echoid-s10174" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div484" type="section" level="1" n="170"> <head xml:id="echoid-head239" xml:space="preserve"><emph style="sc">Propositio</emph> VII.</head> <head xml:id="echoid-head240" style="it" xml:space="preserve">Ponamus mihi deficere duos luſus & colluſori me@ <lb/>quatuor.</head> <p> <s xml:id="echoid-s10175" xml:space="preserve">FIet itaque, ut, ſi primum ludum vincam, unum ludum <lb/>vincere debeam & </s> <s xml:id="echoid-s10176" xml:space="preserve">alter quatuor; </s> <s xml:id="echoid-s10177" xml:space="preserve">vel, ſi eundem perdam, <lb/>duos & </s> <s xml:id="echoid-s10178" xml:space="preserve">alter tres. </s> <s xml:id="echoid-s10179" xml:space="preserve">Ita ut æqua mihi ſors obtingat ad {15/16}a <pb o="734" file="0478" n="500" rhead="DE RATIOCINIIS"/> aut {1@/16}a, id quod tantum valet ac {13/16}a, per 1<emph style="super">mam</emph> Propoſitio-<lb/>nem. </s> <s xml:id="echoid-s10180" xml:space="preserve">Unde patet, eum meliorem habere ſortem, qui duos <lb/>luſus vincere debet dum alter quatuor, quàm eum, qui unum <lb/>dum alter duos. </s> <s xml:id="echoid-s10181" xml:space="preserve">In hoc enim poſteriori caſu, nimirum ipſius <lb/>1 ad 2, portio mea, per 4<emph style="super">tam</emph> Propoſitionem, eſt {3/4}a, quæ <lb/>minor eſt quàm {13/16}a.</s> <s xml:id="echoid-s10182" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div485" type="section" level="1" n="171"> <head xml:id="echoid-head241" xml:space="preserve"><emph style="sc">Propositio</emph> VIII.</head> <head xml:id="echoid-head242" style="it" xml:space="preserve">Nunc verò ponamus tres eſſe colluſores, quorum pri-<lb/>mo ut & ſecundo unus luſus deficiat, ſed <lb/>tertio duo luſus.</head> <p> <s xml:id="echoid-s10183" xml:space="preserve">UTigitur inveniatur primi pars, rurſus advertendum eſt, <lb/>quid ipſi deberetur, ſi vel ipſe vel alter reliquorum duo-<lb/>rum primum luſum vinceret. </s> <s xml:id="echoid-s10184" xml:space="preserve">Si ipſe vinceret, haberet de-<lb/>poſitum, id quod ſit a. </s> <s xml:id="echoid-s10185" xml:space="preserve">Quòd ſi ſecundus vinceret, primus <lb/>nihil haberet, quoniam ſecundus ſic luſui finem impoſuiſſet. <lb/></s> <s xml:id="echoid-s10186" xml:space="preserve">At ſi tertius vinceret, tunc cuique trium adhuc unus defice-<lb/>ret luſus, ideóque tam primo quàm utrique reliquorum de-<lb/>beretur {1/3}a. </s> <s xml:id="echoid-s10187" xml:space="preserve">Et fit primo una expectatio ad a, una ad o, & </s> <s xml:id="echoid-s10188" xml:space="preserve"><lb/>una ad {1/3}a, (quandoquidem æquè facilè contingere poteſt <lb/>cuique trium ut primum ludum vincat,) quod ipſi tantun-<lb/>dem valet ac {4/9}a, per 2<emph style="super">dam</emph> Propoſitionem. </s> <s xml:id="echoid-s10189" xml:space="preserve">Et fit ſimiliter <lb/>ſecundo {4/9}a, & </s> <s xml:id="echoid-s10190" xml:space="preserve">remanet tertio {1/9}a. </s> <s xml:id="echoid-s10191" xml:space="preserve">Cujus pars ſeparatim <lb/>etiam inveniri potuerat, atque inde reliquorum partes deter-<lb/>minari.</s> <s xml:id="echoid-s10192" xml:space="preserve"/> </p> <pb o="357" file="0479" n="501" rhead="IN LUDO ALEÆ."/> </div> <div xml:id="echoid-div486" type="section" level="1" n="172"> <head xml:id="echoid-head243" xml:space="preserve"><emph style="sc">Propositio</emph> IX.</head> <p style="it"> <s xml:id="echoid-s10193" xml:space="preserve">Ut tot colluſorum, quot quis voluerit, ex quibus uni <lb/>plures & </s> <s xml:id="echoid-s10194" xml:space="preserve">alii pauciores luſus deficiunt, cujuſque pars <lb/>inveniatur, conſiderandum eſt, quid illi, cujus par-<lb/>tem invenire volumus, deberetur, ſi vel ipſe, vel <lb/>quiſlibet reliquorum primum ſequentem ludum vin-<lb/>ceret. </s> <s xml:id="echoid-s10195" xml:space="preserve">Horum autem partes ſi in unam ſummam <lb/>colligantur, & </s> <s xml:id="echoid-s10196" xml:space="preserve">aggregatum per numerum colluſo-<lb/>rum dividatur, quotiens oſtendet unius quæſitam <lb/>partem.</s> <s xml:id="echoid-s10197" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10198" xml:space="preserve">POnamus tres eſſe colluſores A, B, & </s> <s xml:id="echoid-s10199" xml:space="preserve">C, & </s> <s xml:id="echoid-s10200" xml:space="preserve">ipſi A unum <lb/>ludum deficere, ipſi B duos luſus, & </s> <s xml:id="echoid-s10201" xml:space="preserve">ipſi C ſimiliter duos <lb/>luſus Invenire oportet, quid ipſi B, ejus quod depoſitum <lb/>eſt, debeatur. </s> <s xml:id="echoid-s10202" xml:space="preserve">Id quod vocetur q.</s> <s xml:id="echoid-s10203" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10204" xml:space="preserve">Primò examinandum eſt, quid ipſi B deberetur, ſi vel <lb/>ipſe, vel A, vel C primum ſequentem ludum vinceret.</s> <s xml:id="echoid-s10205" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10206" xml:space="preserve">Si A vinceret, ludo finem impoſuiſſet, ac per conſequens <lb/>ipſi B deberetur o. </s> <s xml:id="echoid-s10207" xml:space="preserve">Si ipſe B vinceret, deficeret illi adhuc <lb/>unus luſus, & </s> <s xml:id="echoid-s10208" xml:space="preserve">ipſi A unus luſus, at ipſi C duo luſus. </s> <s xml:id="echoid-s10209" xml:space="preserve">Quo-<lb/>circa ipſi B hoc in caſu deberetur {4/9}q, per 8<emph style="super">vam</emph> Propoſitio-<lb/>nem.</s> <s xml:id="echoid-s10210" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10211" xml:space="preserve">Denique ſi C primum ſequentemludum vinceret, tunc ipſis <lb/>A & </s> <s xml:id="echoid-s10212" xml:space="preserve">C ſingulis unus deficeret luſus, ſed ipſi B duo luſus, ac <lb/>per conſequens ipſi B deberetur {1/9}q, per eandem Propoſitio-<lb/>nem 8<emph style="super">vam</emph>. </s> <s xml:id="echoid-s10213" xml:space="preserve">Nunc autem in unam ſummam colligendum eſt, <lb/>id quod in tribus hiſce caſibus ipſi B deberetur: </s> <s xml:id="echoid-s10214" xml:space="preserve">nimirum, o, <lb/>{4/9}q, {1/9}q: </s> <s xml:id="echoid-s10215" xml:space="preserve">quorum ſumma eſt {5/9}q. </s> <s xml:id="echoid-s10216" xml:space="preserve">Quod ipſum diviſum per 3, <lb/>numerum colluſorum, dat {5/27}q. </s> <s xml:id="echoid-s10217" xml:space="preserve">Quæ ipſius B quæſita pars <lb/>eſt. </s> <s xml:id="echoid-s10218" xml:space="preserve">Demonſtratio autem hujus patet ex 2<emph style="super">da</emph> Propoſitione. <lb/></s> <s xml:id="echoid-s10219" xml:space="preserve">Quoniam enim B æquam habet ſortem ad obtinendum o, <lb/>{4/9}q, vel {1/9}q, habet per 2<emph style="super">dam</emph> Propoſitionem tantundem ac<unsure/> <pb o="736" file="0480" n="502" rhead="DE RATIOCINIIS"/> {o + {4/9}q + {1/9}/3}q, id eſt, {5/27}q. </s> <s xml:id="echoid-s10220" xml:space="preserve">Et certum eſt, hunc diviſorem 3 <lb/>eſſe numerum colluſorum.</s> <s xml:id="echoid-s10221" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10222" xml:space="preserve">Ut autem inveniatur, quid cuipiam debeatur in quolibet <lb/>caſu, videlicet ſi vel ipſe vel aliquis reliquorum primum ſe-<lb/>quentem ludum vincat: </s> <s xml:id="echoid-s10223" xml:space="preserve">oportet ſimpliciores caſus primò in-<lb/>veſtigare, & </s> <s xml:id="echoid-s10224" xml:space="preserve">horum medio ſequentes. </s> <s xml:id="echoid-s10225" xml:space="preserve">Nam ſicut hic ulti-<lb/>mus caſus ſolvi non potuit priuſquam ille octavæ Propoſi-<lb/>tionis calculo ſubductus eſſet, in quo deficientes luſus erant <lb/>1, 1, 2, ita etiam cujuſque pars ſupputari nequit in tali caſu, <lb/>ubi deficientes luſus ſunt 1, 2, 3, quin primùm calculo ſub-<lb/>ductus ſit caſus deficientium luſuum 1, 2, 2, quemadmodum <lb/>jam fecimus, & </s> <s xml:id="echoid-s10226" xml:space="preserve">præterea ille, in quo luſus deficientes ſunt <lb/>1, 1, 3; </s> <s xml:id="echoid-s10227" xml:space="preserve">qui ſimiliter per 8<emph style="super">vam</emph> Propoſitionem ſupputari potuiſ-<lb/>ſet. </s> <s xml:id="echoid-s10228" xml:space="preserve">Atque hoc quidem pacto conſequenter ſupputare licet <lb/>caſus omnes, qui in ſequenti tabula comprehenduntur, & </s> <s xml:id="echoid-s10229" xml:space="preserve"><lb/>infinitos alios.</s> <s xml:id="echoid-s10230" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div487" type="section" level="1" n="173"> <head xml:id="echoid-head244" xml:space="preserve">Tabula pro 3 colluſoribus.</head> <note position="right" xml:space="preserve"> <lb/>Luſus, quiipſis \\ deficiunt. # 1 ÷ 1 ÷ 2 # 1 ÷ 2 ÷ 2 # 1 ÷ 1 ÷ 3 # 1 ÷ 2 ÷ 3 <lb/>Eorum partes. # 4 ÷ 4 ÷ 1 # 17 ÷ 5 ÷ 5 # 13 ÷ 13 ÷ 1 # 19 ÷ 6 ÷ 2 <lb/># 9 # 27 # 27 # 27 <lb/>Luſus, quiipſis \\ deficiunt. # 1 ÷ 1 ÷ 4 # 1 ÷ 1 ÷ 5 # 1 ÷ 2 ÷ 4 # 1 ÷ 2 ÷ 5 <lb/>Eorum partes. # 40 ÷ 40 ÷ 1 # 121 ÷ 121 ÷ 1 # 178 ÷ 58 ÷ 7 # 542 ÷ 179 ÷ 8 <lb/># 81 # 243 # 243 # 729 <lb/>Luſus, quiipſis \\ deficiunt. # 1 ÷ 3 ÷ 3 # 1 ÷ 3 ÷ 4 # 1 ÷ 3 ÷ 5 <lb/>Eorum partes. # 65 ÷ 8 ÷ 8 # 616 ÷ 82 ÷ 31 # 629 ÷ 87 ÷ 13 <lb/># 81 # 729 # 729 <lb/>Luſus, quiipſis \\deſiciunt. # 2 ÷ 2 ÷ 3 # 2 ÷ 2 ÷ 4 # 2 ÷ 2 ÷ 5 # 2 ÷ 3 ÷ 3 # 2 ÷ 3 ÷ 4 # 2 ÷ 3 ÷ 5 <lb/>Eorum partes. # 34 ÷ 34 ÷ 13 # 338 ÷ 338 ÷ 53 # @53 ÷ 353 ÷ 23 # 133 ÷ 55 ÷ 55 # 451 ÷ 195 ÷ 83 # 1433 ÷ 635 ÷ 119 <lb/># 81 # 729 # 729 # 243 # 729 # 2187 <lb/></note> <p> <s xml:id="echoid-s10231" xml:space="preserve">Quod ad teſſeras attinet, deiis hæ quæſtiones proponi poſ-<lb/>ſunt: </s> <s xml:id="echoid-s10232" xml:space="preserve">videlicet, quotâ vice unâ teſſerâ ſenarium jacere pe-<lb/>riclitandum ſit, aut aliquod reliquorum punctorum. </s> <s xml:id="echoid-s10233" xml:space="preserve">Item <pb o="737" file="0481" n="503" rhead="IN LUDO ALEÆ."/> quotâ vice duos ſenarios duabus teſſeris, aut tres ſenarios <lb/>tribus teſſeris jacere ſit tentandum. </s> <s xml:id="echoid-s10234" xml:space="preserve">Et plures aliæ hujuſmo-<lb/>di quæſtiones. </s> <s xml:id="echoid-s10235" xml:space="preserve">Ad quas ſolvendas advertendum eſt.</s> <s xml:id="echoid-s10236" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10237" xml:space="preserve">Primò unius teſſeræ ſex eſſe jactus diverſos, quorum qui-<lb/>vis æquè facilè eveniat. </s> <s xml:id="echoid-s10238" xml:space="preserve">Sumo enim teſſeram habere figu-<lb/>ram cubi perfectam.</s> <s xml:id="echoid-s10239" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10240" xml:space="preserve">Porrò duarum teſſerarum 36 eſſe diverſos jactus, quorum <lb/>ſimiliter quivis æquè facilè obtingere poteſt. </s> <s xml:id="echoid-s10241" xml:space="preserve">Nam ratione <lb/>cujuſque jactus unius teſſeræ poteſt unus ſex jactuum alterius <lb/>teſſeræ ſimul contingere. </s> <s xml:id="echoid-s10242" xml:space="preserve">Et ſexies 6 efficiunt 36 jactus.</s> <s xml:id="echoid-s10243" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10244" xml:space="preserve">Item trium teſſerarum eſſe 216 jactus diverſos. </s> <s xml:id="echoid-s10245" xml:space="preserve">Nam ratio-<lb/>ne cujuſque 36 jactuum duarum teſſerarum poteſt unus ſex <lb/>jactuum, qui in 3<emph style="super">tia</emph> ſunt, evenire. </s> <s xml:id="echoid-s10246" xml:space="preserve">Et ſexies 36 efficiunt <lb/>216 jactus.</s> <s xml:id="echoid-s10247" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10248" xml:space="preserve">Eodem modo patet, quatuor teſſerarum jactus eſſe ſexies <lb/>216, id eſt, 1296; </s> <s xml:id="echoid-s10249" xml:space="preserve">atque ſic ulteriùs jactus quotlibet teſſera-<lb/>rum ſupputari poſſe, ſumendo ſemper pro acce<unsure/>ſſione unius <lb/>teſſeræ ſexies jactus præcedentis.</s> <s xml:id="echoid-s10250" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10251" xml:space="preserve">Porrò notandum, duarum teſſerarum unum duntaxat eſſe <lb/>jactum, qui 2 aut 12 puncta efficiat, duos verò jactus, qui <lb/>3 aut 11 puncta efficiant. </s> <s xml:id="echoid-s10252" xml:space="preserve">Si enim teſſeras vocemus A & </s> <s xml:id="echoid-s10253" xml:space="preserve"><lb/>B, patet, ad 3 puncta jacienda in A unum & </s> <s xml:id="echoid-s10254" xml:space="preserve">in B duo, vel <lb/>in B unum & </s> <s xml:id="echoid-s10255" xml:space="preserve">in A duo puncta reperiri poſſe. </s> <s xml:id="echoid-s10256" xml:space="preserve">Similiter ad <lb/>11 puncta jacienda in A quinque & </s> <s xml:id="echoid-s10257" xml:space="preserve">in B ſex, vel in A ſex <lb/>& </s> <s xml:id="echoid-s10258" xml:space="preserve">in B quinque puncta patêre poſſe. </s> <s xml:id="echoid-s10259" xml:space="preserve">Quatuor punctorum <lb/>tres ſunt jactus, videlicet, ipſius A 1 & </s> <s xml:id="echoid-s10260" xml:space="preserve">B 3 puncta; </s> <s xml:id="echoid-s10261" xml:space="preserve">vel <lb/>ipſius A 3 & </s> <s xml:id="echoid-s10262" xml:space="preserve">B 1 punctum; </s> <s xml:id="echoid-s10263" xml:space="preserve">vel ipſius A 2 & </s> <s xml:id="echoid-s10264" xml:space="preserve">B 2 puncta.</s> <s xml:id="echoid-s10265" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10266" xml:space="preserve">Decem punctorum ſimiliter tres ſunt jactus.</s> <s xml:id="echoid-s10267" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10268" xml:space="preserve">Quinque vel novem punctorum 4<emph style="super">or</emph> ſunt jactus.</s> <s xml:id="echoid-s10269" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10270" xml:space="preserve">Sex vel octo punctorum 5<emph style="super">que</emph> ſunt jactus.</s> <s xml:id="echoid-s10271" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10272" xml:space="preserve">Septem punctorum 6 ſunt jactus.</s> <s xml:id="echoid-s10273" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/># 3 vel 18 # # 1 <lb/># 4 vel 17 # # 3 <lb/># 5 vel 16 # # 6 <lb/>In tribus teſſeris reperiuntur # 6 vel 15 # punctorum # 10 # jactus. <lb/># 7 vel 14 # # 15 <lb/># 8 vel 13 # # 21 <lb/># 9 vel 12 # # 25 <lb/># 10 vel 11 # # 27 <lb/></note> <pb o="738" file="0482" n="504" rhead="DE RATIOCINIIS"/> </div> <div xml:id="echoid-div488" type="section" level="1" n="174"> <head xml:id="echoid-head245" xml:space="preserve"><emph style="sc">Propositio</emph> X.</head> <head xml:id="echoid-head246" style="it" xml:space="preserve">Invenire, quot vicibus ſuſcipere quis poſſit, ut <lb/>unâ teſſerâ 6 puncta jaciat.</head> <p> <s xml:id="echoid-s10274" xml:space="preserve">Si quis primâ vice ſenarium jacere contendat, apparet <lb/>unum eſſe caſum, quo vincat, habeatque id, quod pi-<lb/>gnoris loco depoſitum eſt; </s> <s xml:id="echoid-s10275" xml:space="preserve">quinque verò eſſe caſus, quibus <lb/>perdat, & </s> <s xml:id="echoid-s10276" xml:space="preserve">nihil habeat. </s> <s xml:id="echoid-s10277" xml:space="preserve">Sunt enim 5 jactus contra ipſum, <lb/>& </s> <s xml:id="echoid-s10278" xml:space="preserve">tantùm unus pro ipſo. </s> <s xml:id="echoid-s10279" xml:space="preserve">Quod autem depoſitum eſt voce-<lb/>tur a. </s> <s xml:id="echoid-s10280" xml:space="preserve">Eſt itaque ipſi unica expectatio ad obtinendum a, ſed <lb/>quinque ad obtinendum o; </s> <s xml:id="echoid-s10281" xml:space="preserve">id quod per 2<emph style="super">dam</emph> Propoſitionem <lb/>tantundem valet ac {1/6}a. </s> <s xml:id="echoid-s10282" xml:space="preserve">Et manet pro eo qui ipſi hunc ca-<lb/>ſum offert {5/6}a. </s> <s xml:id="echoid-s10283" xml:space="preserve">Ita ut tantummodo 1 contra 5 deponere poſ-<lb/>ſit, qui primâ vice ſuſcipere velit.</s> <s xml:id="echoid-s10284" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10285" xml:space="preserve">Qui duabus vicibus ſemel ſenarium jacere certet, ſors ejus <lb/>hoc pacto computatur. </s> <s xml:id="echoid-s10286" xml:space="preserve">Si primâ vice 6 jaciat, obtinet a. <lb/></s> <s xml:id="echoid-s10287" xml:space="preserve">Si diverſum eveniat, unus ipſi reſtat jactus, qui ex præce-<lb/>denti tantum valet, quantum {1/6}a. </s> <s xml:id="echoid-s10288" xml:space="preserve">Atqui ut primâ vice 6 ja-<lb/>ciat, unus tantùm caſus eſt, & </s> <s xml:id="echoid-s10289" xml:space="preserve">quinque caſus, quibus diver-<lb/>ſum eveniat. </s> <s xml:id="echoid-s10290" xml:space="preserve">Itaque ab initio unus caſus eſt, qui det ipſi a; </s> <s xml:id="echoid-s10291" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s10292" xml:space="preserve">quinque quident {1/6}a, id quod per 2<emph style="super">dam</emph> Propoſitionem va-<lb/>let {11/36}a. </s> <s xml:id="echoid-s10293" xml:space="preserve">Unde contracertanti luſori cedit reliquum {25/36}a; </s> <s xml:id="echoid-s10294" xml:space="preserve">ad-<lb/>eo ut ſors utriuſque ſive æſtimatio expectationis eam ſervet <lb/>rationem, quam 11 ad 25; </s> <s xml:id="echoid-s10295" xml:space="preserve">id eſt minus quàm 1 ad 2.</s> <s xml:id="echoid-s10296" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10297" xml:space="preserve">Hinc eodem modo calculo ſubducitur, quòd ſors ejus, qui <lb/>tribus vicibus ſemel ſenarium jacere ſuſcipit, ſit futura {91/216}a; <lb/></s> <s xml:id="echoid-s10298" xml:space="preserve">ita ut 91 contra 125 deponere poſſit; </s> <s xml:id="echoid-s10299" xml:space="preserve">id eſt, paulò minus <lb/>quàm 3 ad 4.</s> <s xml:id="echoid-s10300" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10301" xml:space="preserve">Qui quatuor vicibus idem ſuſcipit, ſors ejus eſt {671/1296}a; </s> <s xml:id="echoid-s10302" xml:space="preserve">ita <lb/>ut 671 contra 625 deponere poſſit; </s> <s xml:id="echoid-s10303" xml:space="preserve">id eſt, plùs quam 1 <lb/>ad 1.</s> <s xml:id="echoid-s10304" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10305" xml:space="preserve">Qui quinque vicibus idem ſuſcipit, ſors ejus eſt {4651/7776}a, & </s> <s xml:id="echoid-s10306" xml:space="preserve"><lb/>poteſt 4651 contra 3125 deponere; </s> <s xml:id="echoid-s10307" xml:space="preserve">id eſt, paulò minus quàm <lb/>3 ad 2.</s> <s xml:id="echoid-s10308" xml:space="preserve"/> </p> <pb o="739" file="0483" n="505" rhead="IN LUDO ALEÆ."/> <p> <s xml:id="echoid-s10309" xml:space="preserve">Qui ſex vicibus idem ſuſcipit, ſors ejus eſt {31031/46656}a, & </s> <s xml:id="echoid-s10310" xml:space="preserve">po-<lb/>teſt 31031 contra 15625 deponere; </s> <s xml:id="echoid-s10311" xml:space="preserve">id eſt, paulò minus <lb/>quàm 2 ad 1.</s> <s xml:id="echoid-s10312" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10313" xml:space="preserve">Atque ita conſequenter quilibet jactuum numerus inueniri <lb/>poteſt. </s> <s xml:id="echoid-s10314" xml:space="preserve">Sed licet majori compendio progredi, ut in ſequen-<lb/>ti Propoſitione oſtendetur; </s> <s xml:id="echoid-s10315" xml:space="preserve">ſine quo calculus aliàs multo pro-<lb/>lixior foret.</s> <s xml:id="echoid-s10316" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div489" type="section" level="1" n="175"> <head xml:id="echoid-head247" xml:space="preserve"><emph style="sc">Propositio</emph> XI.</head> <head xml:id="echoid-head248" style="it" xml:space="preserve">Invenire, quot vicibus ſuſcipere quis poſſit, ut dua-<lb/>bus teſſeris 12 puncta jaciat.</head> <p> <s xml:id="echoid-s10317" xml:space="preserve">SI quis primâ vice duos ſenarios jacere contendat, ap-<lb/>paret unum eſſe caſum, quo vincat, id eſt, ad ob-<lb/>tinendum a; </s> <s xml:id="echoid-s10318" xml:space="preserve">& </s> <s xml:id="echoid-s10319" xml:space="preserve">35 eſſe caſus, quibus perdat ſive nihil ha-<lb/>beat, quoniam 36 ſunt jactus. </s> <s xml:id="echoid-s10320" xml:space="preserve">Itaque habet, per 2<emph style="super">dam</emph> <lb/>Propoſitionem, {1/36}a.</s> <s xml:id="echoid-s10321" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10322" xml:space="preserve">Qui duabus vicibus idem ſuſcipit, ſi primâ vice duos <lb/>ſenarios jaciat, obtinebit a; </s> <s xml:id="echoid-s10323" xml:space="preserve">ſi verò primâ vice diver-<lb/>ſum eveniat, unus ipſi reſtat jactus, id quod ipſi, per <lb/>illud quod jam dictum eſt, valet {1/36}a.</s> <s xml:id="echoid-s10324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10325" xml:space="preserve">Atqui ut primâ vice duos ſenarios jaciat, unus tan-<lb/>tum eſt caſus, ſed 35 caſus, quibus diverſum eveniat. <lb/></s> <s xml:id="echoid-s10326" xml:space="preserve">Itaque ab initio unus caſus eſt, qui det ipſi a, & </s> <s xml:id="echoid-s10327" xml:space="preserve">35 <lb/>qui dent {1/36}a; </s> <s xml:id="echoid-s10328" xml:space="preserve">id quod per 2<emph style="super">dam</emph> Propoſitionem valet <lb/>{71/2296}a Et remanet contracertanti {1225/1296}a.</s> <s xml:id="echoid-s10329" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10330" xml:space="preserve">Ex his invenire licet, qualis ſit ei ſors aut pars, qui <lb/>idem ſuſcipit quaternis jactibus, prætereundo caſum eum, <lb/>cùm quis illud ternis jactibus ſuſcipit.</s> <s xml:id="echoid-s10331" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10332" xml:space="preserve">Etenim, qui 4<emph style="super">or</emph> vicibus duos ſenarios jacere conten-<lb/>dit, ſi illud 1<emph style="super">ma</emph> aut 2<emph style="super">da</emph> vice faciat, obtinet a; </s> <s xml:id="echoid-s10333" xml:space="preserve">ſin mi-<lb/>nùs, reſtant ipſi duo jactus, qui per illud quod ſupe-<lb/>riùs dictum eſt, valent {71/1296}a. </s> <s xml:id="echoid-s10334" xml:space="preserve">Sed propter eandem ra- <pb o="740" file="0484" n="506" rhead="DE RATIOCINIIS"/> tionem habet etiam 71 caſus, ut ex duobus primis ja-<lb/>ctibus ſemel duos ſenarios jaciat, contra 1225 caſus, <lb/>quibus diverſum eveniat. </s> <s xml:id="echoid-s10335" xml:space="preserve">Habet itaque ab initio 71 ca-<lb/>ſus, qui ipſi dent a, & </s> <s xml:id="echoid-s10336" xml:space="preserve">1225 caſus, qui dent ipſi {71/1296}a. <lb/></s> <s xml:id="echoid-s10337" xml:space="preserve">Quod ipſi per 2<emph style="super">dam</emph> Propoſitionem valet {178991/1679616}a. </s> <s xml:id="echoid-s10338" xml:space="preserve">Et re-<lb/>manet contracertanti {1500625/1679616}a. </s> <s xml:id="echoid-s10339" xml:space="preserve">Id quod oſtendit eorum <lb/>ſortes eſſe ad ſe invicem, ut 178991 ad 1500625.</s> <s xml:id="echoid-s10340" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10341" xml:space="preserve">E quibus porrò eâdem ratione invenitur expectatio e-<lb/>jus, qui 8 vicibus ſemel duos ſenarios jacere certat. <lb/></s> <s xml:id="echoid-s10342" xml:space="preserve">Ac inde rurſus expectatio ejus, qui idem ſuſcipit 16 vi-<lb/>cibus. </s> <s xml:id="echoid-s10343" xml:space="preserve">Atque ex hujus expectatione, ut etiam ex expe-<lb/>ctatione illius, qui iſtud 8 vicibus ſuſcipit, invenitur ex-<lb/>pectatio ejus, qui illud 24 vicibus in ſe recipit. </s> <s xml:id="echoid-s10344" xml:space="preserve">In <lb/>qua operatione, quoniam præcipuè quæritur in quo nu-<lb/>mero jactuum æqualis ſors incipiat, inter eum qui id ſu-<lb/>ſcipit & </s> <s xml:id="echoid-s10345" xml:space="preserve">eum qui offert, licebit à numeris, qui alioquin <lb/>in immenſum excreſcerent, poſteriores aliquot characte-<lb/>res auferre. </s> <s xml:id="echoid-s10346" xml:space="preserve">Atque ita quidem reperio ei, qui illud 24 <lb/>vicibus ſuſcipit, adhuc aliquid deficere; </s> <s xml:id="echoid-s10347" xml:space="preserve">tumque demum <lb/>eum potiorem conditionem inire, cùm 25 jactibus aggre-<lb/>ditur.</s> <s xml:id="echoid-s10348" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div490" type="section" level="1" n="176"> <head xml:id="echoid-head249" xml:space="preserve"><emph style="sc">Propositio</emph> XII.</head> <head xml:id="echoid-head250" style="it" xml:space="preserve">Invenire quot teſſeris ſuſcipere quis poſſit, ut primâ <lb/>vice duos ſenarios jaciat.</head> <p> <s xml:id="echoid-s10349" xml:space="preserve">HOc autem tantundem eſt, ac ſi quis ſcire velit, quo-<lb/>to jactu quiſpiam unâ teſſerâ ſuſcipere poſſit, ut bis <lb/>ſenarium jaciat. </s> <s xml:id="echoid-s10350" xml:space="preserve">Quòd ſi quis duobus jactibus ſuſcipe-<lb/>ret, obtingeret ei, per ea quæ ante oſtenſa ſunt, {1/36}a. <lb/></s> <s xml:id="echoid-s10351" xml:space="preserve">Qui illud tribus jactibus in ſe reciperet, ſi primus ejus <lb/>jactus ſenarius non foret, haberet adhuc duos jactus, <lb/>quorum uterque ſenarius eſſe deberet, id quod tantun- <pb o="741" file="0485" n="507" rhead="IN LUDO ALEÆ."/> dem valere dictum eſt ac {1/36}a. </s> <s xml:id="echoid-s10352" xml:space="preserve">At verò primo ejus ja-<lb/>ctu exiſtente ſenario, opùs eſt ut ex duobus jactibus <lb/>non niſi ſemel ſenarium jaciat. </s> <s xml:id="echoid-s10353" xml:space="preserve">Quod per 10 Propoſi-<lb/>tionem tantundem valet ac ſi {11/36}a haberet. </s> <s xml:id="echoid-s10354" xml:space="preserve">Atqui cer-<lb/>tum eſt ipſum unum habere caſum, quo primâ vice ſe-<lb/>narium jaciat, & </s> <s xml:id="echoid-s10355" xml:space="preserve">quinque caſus quibus diverſum eve-<lb/>niat. </s> <s xml:id="echoid-s10356" xml:space="preserve">Habet itaque ab initio unum caſum ad {11/36}a, & </s> <s xml:id="echoid-s10357" xml:space="preserve"><lb/>5 caſus ad {1/36}a, id quod per 2<emph style="super">dam</emph> Propoſitionem tan-<lb/>tundem valet ac {16/216}a ſeu {2/27}a. </s> <s xml:id="echoid-s10358" xml:space="preserve">Hoc pacto aſſumendo <lb/>continuè unum jactum ampliùs, invenitur 10 jactibus unâ <lb/>teſſerâ, aut 10 teſſeris primo jactu ſuſcipi poſſe, ut duo <lb/>ſenarii jaciantur, idque cum lucro.</s> <s xml:id="echoid-s10359" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div491" type="section" level="1" n="177"> <head xml:id="echoid-head251" xml:space="preserve"><emph style="sc">Propositio</emph> XIII.</head> <p style="it"> <s xml:id="echoid-s10360" xml:space="preserve">Si cum alio ludam duabus teſſeris unum ſolummodo <lb/>jactum, hâc conditione, ut, ſi ſeptenarius eve-<lb/>niat, ego vincam; </s> <s xml:id="echoid-s10361" xml:space="preserve">at ille, ſi denarius obtingat; <lb/></s> <s xml:id="echoid-s10362" xml:space="preserve">ſi vero quidquam aliud accidat, ut tum id quod <lb/>depoſitum eſt æqualiter dividamus: </s> <s xml:id="echoid-s10363" xml:space="preserve">Invenire qua-<lb/>lis iſtius pars cuique noſtrum debeatur.</s> <s xml:id="echoid-s10364" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10365" xml:space="preserve">QUoniam 36 jactuum, qui duabus teſſeris proveniunt, <lb/>6 jactus exiſtunt ſeptem punctorum, & </s> <s xml:id="echoid-s10366" xml:space="preserve">3 jactus <lb/>decem punctorum, reſtant adhuc 27 jactus, qui ludum <lb/>æquare poſſunt; </s> <s xml:id="echoid-s10367" xml:space="preserve">id quod ſi fiat, cuique noſtrum debe-<lb/>bitur {1/2}a. </s> <s xml:id="echoid-s10368" xml:space="preserve">Verùm ſi id non obtingat, habebo 6 caſus, <lb/>quibus vincam, id eſt, ut a habeam; </s> <s xml:id="echoid-s10369" xml:space="preserve">& </s> <s xml:id="echoid-s10370" xml:space="preserve">3 caſus, qui-<lb/>bus diverſum eveniat, nihilque habeam: </s> <s xml:id="echoid-s10371" xml:space="preserve">id quod per <lb/>2<emph style="super">dam</emph> Propoſitionem tantundem eſt ac ſi tali caſu {2/3}a ha-<lb/>berem. </s> <s xml:id="echoid-s10372" xml:space="preserve">Habeo itaque ab initio 27 caſus ad {1/2}a & </s> <s xml:id="echoid-s10373" xml:space="preserve">9 ca-<lb/>ſus ad {2/3}a, id quod, per 2<emph style="super">dam</emph> Propoſitionem, tantun-<lb/>dem eſt ac {13/24}a. </s> <s xml:id="echoid-s10374" xml:space="preserve">Et remanet contracertanti {11/24}a.</s> <s xml:id="echoid-s10375" xml:space="preserve"/> </p> <pb o="742" file="0486" n="508" rhead="DE RATIOCINIIS"/> </div> <div xml:id="echoid-div492" type="section" level="1" n="178"> <head xml:id="echoid-head252" xml:space="preserve"><emph style="sc">Propositio</emph> XIV.</head> <p style="it"> <s xml:id="echoid-s10376" xml:space="preserve">Si ego & </s> <s xml:id="echoid-s10377" xml:space="preserve">alius duabus teſſeris alternatim jaciamus, <lb/>hâc conditione, ut ego vincam ſimul atque ſe-<lb/>ptenarium jaciam, ille vero quam primùm ſena-<lb/>rium jaciat; </s> <s xml:id="echoid-s10378" xml:space="preserve">ita videlicet, ut ipſi primum ja-<lb/>ctum concedam: </s> <s xml:id="echoid-s10379" xml:space="preserve">invenire rationem meæ ad ipſius <lb/>ſortem.</s> <s xml:id="echoid-s10380" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10381" xml:space="preserve">POnatur, ſortem meam valere x, & </s> <s xml:id="echoid-s10382" xml:space="preserve">id quod depoſi-<lb/>tum eſt vocari a; </s> <s xml:id="echoid-s10383" xml:space="preserve">eritque ſors alterius = a—x. </s> <s xml:id="echoid-s10384" xml:space="preserve">Et <lb/>patet, quandocunque ipſius vices jaciendi revertuntur, ſor-<lb/>tem meam tum rurſus debere eſſe = x. </s> <s xml:id="echoid-s10385" xml:space="preserve">At quandocunque <lb/>meæ vices ſunt ut jaciam, ſors mea pluris æſtimanda eſt. <lb/></s> <s xml:id="echoid-s10386" xml:space="preserve">Ponatur itaque pro ejus valore y. </s> <s xml:id="echoid-s10387" xml:space="preserve">ſam quo niam ex 36 ja-<lb/>ctibus reperiuntur 5 in 2 teſſeris, qui colluſori meo ſena-<lb/>rium dare luſuſque victorem reddere poſſunt; </s> <s xml:id="echoid-s10388" xml:space="preserve">& </s> <s xml:id="echoid-s10389" xml:space="preserve">31 jactus, <lb/>quibus diverſum eveniat, id eſt, qui meas jaciendi vices pro-<lb/>movent: </s> <s xml:id="echoid-s10390" xml:space="preserve">habebo, priuſquam jacit, 5 caſus ad obtinen-<lb/>dum 0, & </s> <s xml:id="echoid-s10391" xml:space="preserve">31 caſus ad obtinendum y. </s> <s xml:id="echoid-s10392" xml:space="preserve">id quod per 3<emph style="super">tiam</emph> Pro-<lb/>poſitionem valet {31 y/36}. </s> <s xml:id="echoid-s10393" xml:space="preserve">Poſuimus autem caſum meum à <lb/>principio eſſe = x. </s> <s xml:id="echoid-s10394" xml:space="preserve">Quocirca erit {31 y/36} = x, adeo-<lb/>que y = {36 x/31}. </s> <s xml:id="echoid-s10395" xml:space="preserve">Deinde poſitum fuit, vicibus meis ve-<lb/>nientibus, ſortem meam valere y. </s> <s xml:id="echoid-s10396" xml:space="preserve">Ego verò jacturus, <lb/>habeo 6 caſus ad obtinendum a, quandoquidem 6 ja-<lb/>crus reperiuntur 7 puncrorum, qui me victorem red-<lb/>dunt; </s> <s xml:id="echoid-s10397" xml:space="preserve">habeoque 30 caſus, quibus vices colluſoris mei <lb/>revertuntur, id eſt, ut mihi obtineam x. </s> <s xml:id="echoid-s10398" xml:space="preserve">id quod per 3<emph style="super">tiam</emph> <pb o="743" file="0487" n="509" rhead="IN LUDO ALEÆ."/> Propoſitionem valet {6 a + 30 x/36}. </s> <s xml:id="echoid-s10399" xml:space="preserve">Hoc autem cum ſit <lb/>= y, erit, invento, ut ante, {36 x/31} = y, {30 x + 6 a/36} <lb/>= {36 x/31}. </s> <s xml:id="echoid-s10400" xml:space="preserve">Unde invenitur x = {31 a/61}, valor meæ ſor-<lb/>tis. </s> <s xml:id="echoid-s10401" xml:space="preserve">Et per conſequens colluſoris mei erit {30 a/61}; </s> <s xml:id="echoid-s10402" xml:space="preserve">ita ut <lb/>ratio ſortis meæ ad illius ſortem ſit, ut 31 ad 30.</s> <s xml:id="echoid-s10403" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div493" type="section" level="1" n="179"> <head xml:id="echoid-head253" style="it" xml:space="preserve">Coronidis loco ſubjungantur ſequentia Problemata.</head> <head xml:id="echoid-head254" xml:space="preserve"><emph style="sc">Problema</emph> I.</head> <p> <s xml:id="echoid-s10404" xml:space="preserve">A & </s> <s xml:id="echoid-s10405" xml:space="preserve">B unà ludunt duabus teſſeris, hâc conditione, <lb/>ut A vincat, ſi ſenarium jaciat, at B ſi ſeptenarium ja-<lb/>ciat. </s> <s xml:id="echoid-s10406" xml:space="preserve">A primò unum jactum inſtituet; </s> <s xml:id="echoid-s10407" xml:space="preserve">deinde B duos <lb/>jactus conſequenter; </s> <s xml:id="echoid-s10408" xml:space="preserve">tum rurſus A duos jactus, atque <lb/>ſic deinceps, donec hic vel ille victor evadat. </s> <s xml:id="echoid-s10409" xml:space="preserve">Quæri-<lb/>tur ratio ſortis ipſius A ad ſortem ipſius B? </s> <s xml:id="echoid-s10410" xml:space="preserve">Reſp. </s> <s xml:id="echoid-s10411" xml:space="preserve">ut <lb/>10355 ad 12276.</s> <s xml:id="echoid-s10412" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div494" type="section" level="1" n="180"> <head xml:id="echoid-head255" xml:space="preserve"><emph style="sc">Problema</emph> II.</head> <p> <s xml:id="echoid-s10413" xml:space="preserve">Tres colluſores A, B & </s> <s xml:id="echoid-s10414" xml:space="preserve">C aſſumentes 12 calculos, <lb/>quorum 4 albi & </s> <s xml:id="echoid-s10415" xml:space="preserve">8 nigri exiſtunt, ludunt hâc condi-<lb/>tione: </s> <s xml:id="echoid-s10416" xml:space="preserve">ut, qui primus ipſorum velatis oculis album cal-<lb/>culum elegerit, vincat; </s> <s xml:id="echoid-s10417" xml:space="preserve">& </s> <s xml:id="echoid-s10418" xml:space="preserve">ut prima electio ſit penes A, <lb/>ſecunda penes B, & </s> <s xml:id="echoid-s10419" xml:space="preserve">tertia penes C, & </s> <s xml:id="echoid-s10420" xml:space="preserve">tum ſequens rur-<lb/>ſus penes A, atque ſic deinceps alternatim. </s> <s xml:id="echoid-s10421" xml:space="preserve">Quæritur, <lb/>quænam futura ſit ratio illorum ſortium?</s> <s xml:id="echoid-s10422" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div495" type="section" level="1" n="181"> <head xml:id="echoid-head256" xml:space="preserve"><emph style="sc">Problema</emph> III.</head> <p> <s xml:id="echoid-s10423" xml:space="preserve">A certat cum B quòd ipſe ex 40 chartis luſoriis, id <lb/>eſt, 10 cujuſque ſpeciei, 4 chartas extracturus ſit; </s> <s xml:id="echoid-s10424" xml:space="preserve">ita <pb o="744" file="0488" n="510" rhead="DE RATIOC. IN LUDO ALEÆ."/> ut ex unaquaque ſpecie habeat unam. </s> <s xml:id="echoid-s10425" xml:space="preserve">Et invenitur ratio <lb/>ſortis A ad ſortem B ut 1000 ad 8139.</s> <s xml:id="echoid-s10426" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div496" type="section" level="1" n="182"> <head xml:id="echoid-head257" xml:space="preserve"><emph style="sc">Problema</emph> IV.</head> <p> <s xml:id="echoid-s10427" xml:space="preserve">Aſſumptis, ut ante, 12 calculis, 4 albis & </s> <s xml:id="echoid-s10428" xml:space="preserve">8 nigris, <lb/>certat A cum B, quòd velatis oculis 7 calculos ex iis <lb/>exempturus ſit, inter quos 3 albi erunt. </s> <s xml:id="echoid-s10429" xml:space="preserve">Quæritur ratio <lb/>ſortis ipſius A ad ſortem ipſius B.</s> <s xml:id="echoid-s10430" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div497" type="section" level="1" n="183"> <head xml:id="echoid-head258" xml:space="preserve"><emph style="sc">Problema</emph> V.</head> <p> <s xml:id="echoid-s10431" xml:space="preserve">A & </s> <s xml:id="echoid-s10432" xml:space="preserve">B aſſumentes ſinguli 12 nummos ludunt tribus <lb/>teſſeris hâc conditione: </s> <s xml:id="echoid-s10433" xml:space="preserve">ut, ſi 11 puncta jaciantur, A <lb/>tradat nummum ipſi B; </s> <s xml:id="echoid-s10434" xml:space="preserve">at ſi 14 puncta jaciantur, B tra-<lb/>dat nummum ipſi A; </s> <s xml:id="echoid-s10435" xml:space="preserve">& </s> <s xml:id="echoid-s10436" xml:space="preserve">ut ille ludum victurus ſit, qui <lb/>primùm omnes habuerit nummos. </s> <s xml:id="echoid-s10437" xml:space="preserve">Et invenitur ratio <lb/>ſortis ipſius A ad ſortem ipſius B, ut 244140625 ad <lb/>282429536481.</s> <s xml:id="echoid-s10438" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div498" type="section" level="1" n="184"> <head xml:id="echoid-head259" xml:space="preserve">FINIS.</head> <figure> <image file="0488-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0488-01"/> </figure> <pb file="0489" n="511"/> </div> <div xml:id="echoid-div499" type="section" level="1" n="185"> <head xml:id="echoid-head260" xml:space="preserve">CHRISTIANI HUGENII <lb/>NOVUS CYCLUS <lb/>HARMONICUS.</head> <pb file="0490" n="512"/> <pb o="747" file="0491" n="513"/> <figure> <image file="0491-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0491-01"/> </figure> </div> <div xml:id="echoid-div500" type="section" level="1" n="186"> <head xml:id="echoid-head261" xml:space="preserve">CHRISTIANI HUGENII <lb/>NOVUS CYCLUS <lb/>HARMONICUS.</head> <head xml:id="echoid-head262" style="it" xml:space="preserve">Litteræ D. Hugenii de Cyclo Harmonico.</head> <p> <s xml:id="echoid-s10439" xml:space="preserve">NOvam tibi mitto de Muſica obſervationem. <lb/></s> <s xml:id="echoid-s10440" xml:space="preserve">Spectat hæc prima Scientiæ illius fundamen-<lb/>ta, id eſt, tonorum, qui in cantu & </s> <s xml:id="echoid-s10441" xml:space="preserve">Inſtru-<lb/>mentorum fabrica obſervantur, determina-<lb/>tionem. </s> <s xml:id="echoid-s10442" xml:space="preserve">Quicunque licet parum verſati ſint <lb/>in parte illa Theoriæ, norunt, quid ſit, quod <lb/>vocatur Temperamentum, quod tonos illos <lb/>moderatur & </s> <s xml:id="echoid-s10443" xml:space="preserve">quam neceſſarium illud ſit in harmonia tubo-<lb/>rum Organi vel chordarum Clavecymbali.</s> <s xml:id="echoid-s10444" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10445" xml:space="preserve">Celebriores auctores, uti Zarlinus & </s> <s xml:id="echoid-s10446" xml:space="preserve">Salinas, illud re-<lb/>ferunt inter inventa pulcherrima & </s> <s xml:id="echoid-s10447" xml:space="preserve">utiliſſima, quæ in muſi-<lb/>cis detegi poſſunt; </s> <s xml:id="echoid-s10448" xml:space="preserve">& </s> <s xml:id="echoid-s10449" xml:space="preserve">uterque contendit ſe primum Tempera-<lb/>mentum hoc ad examen revocaſſe, & </s> <s xml:id="echoid-s10450" xml:space="preserve">demonſtrationibus Ma-<lb/>thematicis conſtituiſſe. </s> <s xml:id="echoid-s10451" xml:space="preserve">Ante illos experientia & </s> <s xml:id="echoid-s10452" xml:space="preserve">neceſſitas il-<lb/>lud quidem aliquo modo jam introduxerant, licet vera men-<lb/>ſura & </s> <s xml:id="echoid-s10453" xml:space="preserve">methodus nondum eſſet cognitæ. </s> <s xml:id="echoid-s10454" xml:space="preserve">Inventio hujus Tem-<lb/>peramenti in cauſa fuit, ut merito negligerentur omnes ve-<lb/>terum Tetrachordarum & </s> <s xml:id="echoid-s10455" xml:space="preserve">Diapaſonorum diviſiones quarum <lb/>pleræquæ abſurdæ & </s> <s xml:id="echoid-s10456" xml:space="preserve">uſus nullius erant in compoſitione pluri-<lb/>marum partium: </s> <s xml:id="echoid-s10457" xml:space="preserve">hac etiam inventione noſtrum ſyſtema to-<lb/>norum magis abundat conſonantiis, magiſque eſt juxta na-<lb/>turam cantus, quam illorum erat.</s> <s xml:id="echoid-s10458" xml:space="preserve"/> </p> <pb o="748" file="0492" n="514" rhead="CHRIST. HUGENII"/> <p> <s xml:id="echoid-s10459" xml:space="preserve">Pono notas eſſe proportiones, in quibus conſiſtunt per-<lb/>fectæ conſonantiæ, ſcilicet: </s> <s xml:id="echoid-s10460" xml:space="preserve">Quintam audiri, quando, poſt <lb/>ſonum editum ex motu chordæ integræ, duæ tertiæ partes <lb/>ipſius agitantur: </s> <s xml:id="echoid-s10461" xml:space="preserve">vel proportionem, a qua conſonantia hæc <lb/>pendet, eſſe ut 3 ad 2; </s> <s xml:id="echoid-s10462" xml:space="preserve">Quartæ proportionem eſſe ut 4 ad 3; <lb/></s> <s xml:id="echoid-s10463" xml:space="preserve">Tertiæ majoris ut 5 ad 4; </s> <s xml:id="echoid-s10464" xml:space="preserve">Tertiæ minoris ut 6 ad 5; </s> <s xml:id="echoid-s10465" xml:space="preserve">Sextæ <lb/>majoris ut 5 ad 3; </s> <s xml:id="echoid-s10466" xml:space="preserve">Sextæ minoris ut 8 ad 5. </s> <s xml:id="echoid-s10467" xml:space="preserve">Et quantum ad <lb/>Temperamentum, iidem autores, quos modo memoravi, nos <lb/>docent, ut hoc applicetur inſtrumentis, conſonantiam <lb/>Quintam minuendam eſſe quarta parte Commatis ut vocant, <lb/>quæ adeo exigua eſt diminutio, ut auris hanc difficulter per-<lb/>cipiat, & </s> <s xml:id="echoid-s10468" xml:space="preserve">nullam inde capiat moleſtiam; </s> <s xml:id="echoid-s10469" xml:space="preserve">cum Comma integrum <lb/>ſit ratio inter tonum integræ chordæ, & </s> <s xml:id="echoid-s10470" xml:space="preserve">hujus tonum <lb/>quando parte {1/81<unsure/>} minuitur. </s> <s xml:id="echoid-s10471" xml:space="preserve">Sequitur inde, Quartam auctam <lb/>eſſe hac ipſâ exigua quantitate. </s> <s xml:id="echoid-s10472" xml:space="preserve">Tertia minor pariter, eſt <lb/>diminuta {1/4} Commatis, & </s> <s xml:id="echoid-s10473" xml:space="preserve">conſequenter Sexta major in tantum <lb/>aucta; </s> <s xml:id="echoid-s10474" xml:space="preserve">ſed Tertia major perfectionem ſervat, ut & </s> <s xml:id="echoid-s10475" xml:space="preserve">ideo etiam <lb/>Sexta minor.</s> <s xml:id="echoid-s10476" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10477" xml:space="preserve">Juxta has conſonantiarum menſuras, diſponuntur omnes <lb/>inſtrumentorum toni, tam Diatonici quam Chromatici, quos <lb/>addidere Muſici, & </s> <s xml:id="echoid-s10478" xml:space="preserve">pariter toni Enarmonici, quando & </s> <s xml:id="echoid-s10479" xml:space="preserve">hi <lb/>ut magis completa ſit ſymphonia adjiciuntur.</s> <s xml:id="echoid-s10480" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10481" xml:space="preserve">Obſervatio autem noſtra hæc eſt, ſi dividatur Octava in <lb/>31 partes æquales, quod fiet quærendo 30 medias propor-<lb/>tionales inter totam chordam (quæ ſumitur pro regula Har-<lb/>monicâ) & </s> <s xml:id="echoid-s10482" xml:space="preserve">ejus dimidium, dabitur in tonis, quos produ-<lb/>cunt longitudines hæ, Syſtema parum admodum ab iſto <lb/>quod modo memoravi ex Temperamento deductum, ita ut <lb/>nemo auri quantumvis ſubtili præditus differentiam perci-<lb/>piet; </s> <s xml:id="echoid-s10483" xml:space="preserve">quod tamen novum Syſtema toto cælo ab alio differet, <lb/>novaſque afferet utilitates tum in Theoriam cum in Praxin.</s> <s xml:id="echoid-s10484" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10485" xml:space="preserve">Salinas inventum hoc dividendi octavam in 31 partes æ-<lb/>quales memorat ſed ut hoc culpet, & </s> <s xml:id="echoid-s10486" xml:space="preserve">P. </s> <s xml:id="echoid-s10487" xml:space="preserve">Merſennus pariter <lb/>poſt ipſum hanc rejicit diviſionem, unde mihi facile fides ha-<lb/>benda erit, me nihil ex illis autoribus mutuatum eſſe; </s> <s xml:id="echoid-s10488" xml:space="preserve">ſed ſi <lb/>& </s> <s xml:id="echoid-s10489" xml:space="preserve">hoc feciſſem, ſatis a me præſtitum crederem, ſi præſtan- <pb o="749" file="0493" n="515" rhead="NOVUS CYCLUS HARMONICUS."/> tiam ejus diviſionis ex Geometriæ principiis demonſtrave-<lb/>rim, & </s> <s xml:id="echoid-s10490" xml:space="preserve">illam vindicaverim adverſus injuſtum duorum illo-<lb/>rum celebrium Scriptorum pronunciatum.</s> <s xml:id="echoid-s10491" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10492" xml:space="preserve">In lib. </s> <s xml:id="echoid-s10493" xml:space="preserve">3. </s> <s xml:id="echoid-s10494" xml:space="preserve">Muſices Salinæ integrum exſtat caput de hac di-<lb/>viſione, cujus inſcriptio eſt, De prava conſtitutione cujuſ-<lb/>dam inſtrumenti, quod in Italia citra quadraginta annos fa-<lb/>bricari cæptum eſt, in quo reperitur omnis tonus in partes <lb/>quinque diviſus; </s> <s xml:id="echoid-s10495" xml:space="preserve">Dicit inſtrumentum hoc nominis Archi-<lb/>cymbali donatum fuiſſe, illudque fuiſſe incerti autoris; </s> <s xml:id="echoid-s10496" xml:space="preserve">Mu-<lb/>ſicos quoſdam admodum idoneos id magni feciſſe, præci-<lb/>pue quia continebat omnia intervalla, omnesque conſonan-<lb/>tias (ut credebant, ait) adſcendendo, & </s> <s xml:id="echoid-s10497" xml:space="preserve">deſcendendo & </s> <s xml:id="echoid-s10498" xml:space="preserve">quia <lb/>poſt certam periodum rediebatur ad eundem ſonum vel æqui-<lb/>pollentem, cum eo a quo initium factum erat; </s> <s xml:id="echoid-s10499" xml:space="preserve">Quia Octava di-<lb/>viſa erat in 31 partes æquales, quas vocabant Dieſas (ſeu tonos <lb/>perfectos) quarum tonus debebat continere 5; </s> <s xml:id="echoid-s10500" xml:space="preserve">magnus ſe-<lb/>mitonus 3; </s> <s xml:id="echoid-s10501" xml:space="preserve">parvus 2; </s> <s xml:id="echoid-s10502" xml:space="preserve">Tertia major 10; </s> <s xml:id="echoid-s10503" xml:space="preserve">Tertia minor 8; </s> <s xml:id="echoid-s10504" xml:space="preserve">Quar-<lb/>ta 13; </s> <s xml:id="echoid-s10505" xml:space="preserve">Quinta 18; </s> <s xml:id="echoid-s10506" xml:space="preserve">Sexta minor 21; </s> <s xml:id="echoid-s10507" xml:space="preserve">Sexta major 23. </s> <s xml:id="echoid-s10508" xml:space="preserve">Sed ad-<lb/>dit, cum tentaſſent tale diſponere inſtrumentum hoc ſonum <lb/>dediſſe admodum ingratum, & </s> <s xml:id="echoid-s10509" xml:space="preserve">qui inſigniter aures præſen-<lb/>tium offendebat, unde concludit, talem diviſionem deviare ab <lb/>omni ratione harmonica, ſive examinetur ſecundum menſu-<lb/>ram juſtarum conſonantiarum ſive Temperamenti. </s> <s xml:id="echoid-s10510" xml:space="preserve">Præter <lb/>experientiam & </s> <s xml:id="echoid-s10511" xml:space="preserve">aliud affert argumentum deductum ex Me-<lb/>thodo in hac diviſione adhibita. </s> <s xml:id="echoid-s10512" xml:space="preserve">P. </s> <s xml:id="echoid-s10513" xml:space="preserve">Merſennus etiam con-<lb/>tendit ſe bene illam refutaviſſe. </s> <s xml:id="echoid-s10514" xml:space="preserve">Erravit uterque, quia illos <lb/>latebat Methodus dividendi Octavam in 31 partes æquales, <lb/>quod probabiliter & </s> <s xml:id="echoid-s10515" xml:space="preserve">ipſi inventores non potuere; </s> <s xml:id="echoid-s10516" xml:space="preserve">requiritur <lb/>enim cognitio Logarithmorum, qui nondum illorum nequi-<lb/>dem Salinæ tempore erant inventi. </s> <s xml:id="echoid-s10517" xml:space="preserve">Tandem novum hoc <lb/>Temperamentum, quod adeo repudiant, omnium præſtantiſ-<lb/>ſimum merito dicitur, omnia habens commoda quæ ipſi tri-<lb/>buebantur; </s> <s xml:id="echoid-s10518" xml:space="preserve">præcipuæ ſimplicitatem quam communicat Theo-<lb/>riæ tonorum, & </s> <s xml:id="echoid-s10519" xml:space="preserve">parum admodum differt ab eo, quo utun-<lb/>tur omnes, ut auris differentiam non percipiat uti computa-<lb/>tione probabo.</s> <s xml:id="echoid-s10520" xml:space="preserve"/> </p> <pb o="750" file="0494" n="516" rhead="CHRIST. HUGENII"/> <p> <s xml:id="echoid-s10521" xml:space="preserve">Dico ergo primum, Quintam noſtræ diviſionis non ſupe-<lb/>rare Quintas Temperamenti quam {1/110} Commatis, quæ diffe-<lb/>rentia, auditu nullo modo percipi poteſt; </s> <s xml:id="echoid-s10522" xml:space="preserve">ſed quæ alioquin <lb/>daret iſtam conſonantiam tanto magis perfectam.</s> <s xml:id="echoid-s10523" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10524" xml:space="preserve">Quartæ conſequenter Temperamenti ordinarii non exce-<lb/>dunt noſtras niſi illâ {1/110} commatis, & </s> <s xml:id="echoid-s10525" xml:space="preserve">hæ tendunt etiam <lb/>tanto magis ad perfectionem.</s> <s xml:id="echoid-s10526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10527" xml:space="preserve">Tertiæ minores ſunt minores {3/110} vel circiter {1/37} Commatis <lb/>quàm in temperamento, & </s> <s xml:id="echoid-s10528" xml:space="preserve">Sextæ majores in tantum exce-<lb/>dunt Sextas majores Temperamenti; </s> <s xml:id="echoid-s10529" xml:space="preserve">utræque fateor recedunt <lb/>à perfecta proportione; </s> <s xml:id="echoid-s10530" xml:space="preserve">ſed videmus differentiam {1/37} Comma-<lb/>tis non eſſe ſenſibilem, nec augere ſenſibiliter {1/4} Commatis, <lb/>qua conſonantiæ in Temperamento aberrant à veris.</s> <s xml:id="echoid-s10531" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10532" xml:space="preserve">Tandem Tertiæ majores ſuperant Tertias Temperamenti, <lb/>quæ ſunt perfectæ, {4/110} vel quaſi {1/28} Commatis, quæ adeo exi-<lb/>gua eſt differentia, ut niſi pro perfectis, non obſtante hoc <lb/>augmento haberi queant; </s> <s xml:id="echoid-s10533" xml:space="preserve">quid enim facit {1/28} Commatis, cum <lb/>{1/4} adeo facile patiamur.</s> <s xml:id="echoid-s10534" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10535" xml:space="preserve">Deducimus ex exiguis omnibus hiſce differentiis, Or-<lb/>ganum aut Clavecymbalum conſtitutum juxta Tempera-<lb/>mentum ordinarium, etiam diſpoſitum eſſe juxta novam <lb/>diviſionem, quantum aure poterit diſcerni. </s> <s xml:id="echoid-s10536" xml:space="preserve">Si vero <lb/>quis ſibi hac in parte magis deſiderat ſatisfacere, & </s> <s xml:id="echoid-s10537" xml:space="preserve">ex-<lb/>actiſſime conſtituere inſtrumentum juxta 31 partes æqua-<lb/>les Octavæ, tantum dividenda eſt monochorda juxta nu-<lb/>meros, qui habentur in ſequenti Tabula, &</s> <s xml:id="echoid-s10538" xml:space="preserve">, poſitâ totâ <lb/>chordâ, æquiſonâ cum C Clavecymbali aut Organi, diſponere <lb/>pariter alias chordas vel tubos, cum ſonis, ex memorata. <lb/></s> <s xml:id="echoid-s10539" xml:space="preserve">diviſione oriundis & </s> <s xml:id="echoid-s10540" xml:space="preserve">qui audiuntur poſito pectine in locis no-<lb/>tatis. </s> <s xml:id="echoid-s10541" xml:space="preserve">Quod veroad Archicymbalum de quo Salinas loquitur, <lb/>ſpectat, dubito, an non habuerit 31 palmulas in quavis octa-<lb/>va; </s> <s xml:id="echoid-s10542" xml:space="preserve">ſed quoniam non poterit ſine confuſione talis abacus ad-<lb/>hiberi præ multitudine palmularum, præſtaret meâ ſententiâ <lb/>diſponere 31 chordas ſimplices pro quavis octavâ, quod po-<lb/>teſt fieri ſine diſficultate; </s> <s xml:id="echoid-s10543" xml:space="preserve">& </s> <s xml:id="echoid-s10544" xml:space="preserve">conſtructis palmulis, quæ ele-<lb/>vant ſubſilia, omnibus æqualis longitudinis, altitudinis, & </s> <s xml:id="echoid-s10545" xml:space="preserve">la- <pb o="751" file="0495" n="517" rhead="NOVUS CYCLUS HARMONICUS."/> titudinis, quæ latitudo ſit {1/5} palmulæ vulgaris, deinde ſuper-<lb/>imponendus erit abacus mobilis, cum claviculis in inferiori <lb/>parte ſingularum palmularum fixis; </s> <s xml:id="echoid-s10546" xml:space="preserve">ſemel ita diſpoſiti hi ut <lb/>chordæ, quæ in quavis octava adhibentur, ſonent, omni-<lb/>bus tranſpoſitionibus inſervient. </s> <s xml:id="echoid-s10547" xml:space="preserve">Quæ ergo fient ſine labore <lb/>per tonos, ſemitonos, & </s> <s xml:id="echoid-s10548" xml:space="preserve">uſque ad quintas partes tonorum; <lb/></s> <s xml:id="echoid-s10549" xml:space="preserve">& </s> <s xml:id="echoid-s10550" xml:space="preserve">certum eſt, omnes tonos & </s> <s xml:id="echoid-s10551" xml:space="preserve">conſonantias in ſingulis tranſ-<lb/>poſitionibus æque accuratas eſſe, quod utile & </s> <s xml:id="echoid-s10552" xml:space="preserve">amœnum erit. </s> <s xml:id="echoid-s10553" xml:space="preserve"><lb/>Olim Pariſiis conſtrui curavi, tales abacos mobiles ſuperim-<lb/>ponendos abacis vulgaribus Clavecymbalorum tranſpoſitio-<lb/>nibus plurimis, licet non omnibus completis, inſervientes; </s> <s xml:id="echoid-s10554" xml:space="preserve"><lb/>quod inventum approbarunt & </s> <s xml:id="echoid-s10555" xml:space="preserve">imitati ſunt periti in muſicis <lb/>varii.</s> <s xml:id="echoid-s10556" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10557" xml:space="preserve">Sed ut quis certus ſit de veritate illorum, quæ ſuperius <lb/>dixi, inſpiciat Tabulam ſequentem cujus contenta ut & </s> <s xml:id="echoid-s10558" xml:space="preserve">uſum <lb/>explicabo.</s> <s xml:id="echoid-s10559" xml:space="preserve"/> </p> <figure> <image file="0495-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0495-01"/> </figure> <pb o="752" file="0496" n="518"/> <note position="right" xml:space="preserve"> <lb/>## Diviſio Octavæ in \\ partes 31. æ- \\ quales. # # # ## Diviſio Octavæ ſecun- \\ dum Temperamen- \\ tum vulgare. <lb/>I. # II. # III. # IV. # V. # VI. <lb/>N 97106450 <lb/>4,6989700043 # 50000 # <emph style="sc">Ut</emph><emph style="super">z</emph> # C<emph style="super">z</emph> # 50000 # 4,6989700043 <lb/>4,7086806493 # 51131 <lb/>4,7183912943 # 52278 <lb/>4,7281019393 # 53469 # Sr # B<emph style="super">x</emph> # 53499 # 4,7283474859 <lb/>4,7378125843 # 54678 <lb/>4,7475232293 # 55914 # <emph style="sc">Sa</emph> # B # 55902 # 4,7474250108 <lb/>4,7572338743 # 57179 # * # * # 57243 # 4,7577249674 <lb/>4,7669445193 # 58471 <lb/>4,7766551643 # 59794 # <emph style="sc">La</emph> # A # 59814 # 4,7768024924 <lb/>4,7863658093 # 61146 <lb/>4,7960764543 # 62528 # * # * # 62500 # 4,7958800173 <lb/>4,8057870993 # 63942 # <emph style="sc">Sol</emph><emph style="super">x</emph> # G<emph style="super">x</emph> # 64000 # 4,8061799740 <lb/>4,8154977443 # 65388 <lb/>4,8252083893 # 66866 # <emph style="sc">Sol</emph> # G # 66874 # 4,8252574989 <lb/>4,8349190343 # 68378 <lb/>4,8446296793 # 69924 <lb/>4,8543403243 # 71506 # <emph style="sc">Fa</emph><emph style="super">x</emph> # F<emph style="super">x</emph> # 71554 # 4,8546349804 <lb/>4,8640509693 # 73122 <lb/>4,8737616143 # 74776 # <emph style="sc">Fa</emph> # F # 74767 # 4,8737125054 <lb/>4,8834722593 # 76467 <lb/>4,8931829043 # 78196 <lb/>4,9028935493 # 79964 # <emph style="sc">Mi</emph> # E # 80000 # 4,9030899870 <lb/>4,9126041943 # 81772 <lb/>4,9223148393 # 83621 # <emph style="sc">Ma</emph> # E<emph style="super">b</emph> # 83592 # 4,9221675119 <lb/>4,9320254843 # 85512 # * # * # 85599 # 4,9324674685 <lb/>4,9417361293 # 87445 <lb/>4,9514467743 # 89422 # <emph style="sc">Re</emph> # D # 89443 # 4,9515449935 <lb/>4,9611574193 # 91444 <lb/>4,9708680643 # 93512 # * # * # 93459 # 4,9706225184 <lb/>4,9805787093 # 95627 # <emph style="sc">Ut</emph><emph style="super">x</emph> # C<emph style="super">x</emph> # 95702 # 4,9809224750 <lb/>4,9902893543 # 97789 <lb/>4,9999999993 # 100000 # <emph style="sc">Ut</emph> # C # 100000 # 5,0000000000 <lb/></note> <pb o="753" file="0497" n="519" rhead="NOVUS CYCLUS HARMONICUS."/> </div> <div xml:id="echoid-div501" type="section" level="1" n="187"> <head xml:id="echoid-head263" style="it" xml:space="preserve">Tabulæ Explicatio.</head> <p> <s xml:id="echoid-s10560" xml:space="preserve">Secunda columna continet numeros qui exprimunt longi-<lb/>tudines Chordarum; </s> <s xml:id="echoid-s10561" xml:space="preserve">quibus habentur 31 intervalla æqualia <lb/>ſecundum novam diviſionem; </s> <s xml:id="echoid-s10562" xml:space="preserve">Chorda tota continet partes <lb/>100000 & </s> <s xml:id="echoid-s10563" xml:space="preserve">50000 ejus dimidium dant Octavam.</s> <s xml:id="echoid-s10564" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10565" xml:space="preserve">A latere in 3<emph style="super">2</emph>. </s> <s xml:id="echoid-s10566" xml:space="preserve">columna ſunt ſyllabæ, quæ nobis in cantu <lb/>inſerviunt, & </s> <s xml:id="echoid-s10567" xml:space="preserve">* pro quibuſdam chordis enarmonicis, qua-<lb/>rum prope ſol<emph style="super">x</emph> eſt maxime neceſſaria.</s> <s xml:id="echoid-s10568" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10569" xml:space="preserve">In 4<emph style="super">a</emph>. </s> <s xml:id="echoid-s10570" xml:space="preserve">columna ſunt litteræ quæ ordinario inſerviunt ad <lb/>deſignandos tonos.</s> <s xml:id="echoid-s10571" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10572" xml:space="preserve">Numeri 2<emph style="super">æ</emph>. </s> <s xml:id="echoid-s10573" xml:space="preserve">columnæ ope numerorum col. </s> <s xml:id="echoid-s10574" xml:space="preserve">1<emph style="super">æ</emph>. </s> <s xml:id="echoid-s10575" xml:space="preserve">detecti fuere, <lb/>hi ſunt illorum logarithmi reſpectivi; </s> <s xml:id="echoid-s10576" xml:space="preserve">ad quos habendos di-<lb/>viſi logarithmum numeri 2. </s> <s xml:id="echoid-s10577" xml:space="preserve">qui eſt 0, 30102999566 per 31. <lb/></s> <s xml:id="echoid-s10578" xml:space="preserve">unde venit numerus N. </s> <s xml:id="echoid-s10579" xml:space="preserve">97106450, quem addidi continuo <lb/>Logarithmo num. </s> <s xml:id="echoid-s10580" xml:space="preserve">50000, qui eſt 4. </s> <s xml:id="echoid-s10581" xml:space="preserve">6989700043. </s> <s xml:id="echoid-s10582" xml:space="preserve">& </s> <s xml:id="echoid-s10583" xml:space="preserve">hiſce <lb/>additionibus detegimus omnes Logarithmos hujus columnæ <lb/>uſque ad maximum 4, 9999999993, qui parum deficiens a <lb/>5,0000000000, (in cujus locum ſubſtitui poteſt) demonſtrat <lb/>bene ſubductum calculum. </s> <s xml:id="echoid-s10584" xml:space="preserve">Qui Logarithmos intelligunt no-<lb/>runt ita ineundam computationem, ſi quærantur 30 numeri <lb/>proportionales inter 100000 & </s> <s xml:id="echoid-s10585" xml:space="preserve">50000.</s> <s xml:id="echoid-s10586" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10587" xml:space="preserve">5a. </s> <s xml:id="echoid-s10588" xml:space="preserve">columna continet in numeris longitudines chordarum <lb/>juxta Temperamentum ordinarium, & </s> <s xml:id="echoid-s10589" xml:space="preserve">in 6<emph style="super">a</emph>. </s> <s xml:id="echoid-s10590" xml:space="preserve">columna ſunt <lb/>eorum numerorum Logarithmi.</s> <s xml:id="echoid-s10591" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10592" xml:space="preserve">Poſſem explicare quomodo eos ſupputaverim, & </s> <s xml:id="echoid-s10593" xml:space="preserve">pariter, <lb/>quomodo id Temperamentum poſſet detegi, ſi nondum eſ-<lb/>ſet repertum; </s> <s xml:id="echoid-s10594" xml:space="preserve">ſed id foret nimis longum, & </s> <s xml:id="echoid-s10595" xml:space="preserve">ſufficiet, de-<lb/>monſtrare methodum examinandi, & </s> <s xml:id="echoid-s10596" xml:space="preserve">numerorum ἀκρίβ{ει}αν, & </s> <s xml:id="echoid-s10597" xml:space="preserve"><lb/>omnia quæ dixi de nova diviſione & </s> <s xml:id="echoid-s10598" xml:space="preserve">de ratione quam ha-<lb/>bet cum Temperamento.</s> <s xml:id="echoid-s10599" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10600" xml:space="preserve">Ponamus, determinandum eſſe, an Quinta Vt, Sol, tem-<lb/>peramenti vulgaris, minor ſit {1/4} Commatis, quam vera Quinta, <lb/>cujus ratio eſt trium ad duo. </s> <s xml:id="echoid-s10601" xml:space="preserve">A log. </s> <s xml:id="echoid-s10602" xml:space="preserve">Vt, qui eſt 5,0000000000 <lb/>ſubſtraho log. </s> <s xml:id="echoid-s10603" xml:space="preserve">Sol qui eſt 4,8252574989, id, quod reſtat <pb o="754" file="0498" n="520" rhead="CHRIST. HUG. NOV. CYCL. HARMON."/> 0,1747425011, exhibet quantitatem Quintæ Temperamenti.</s> <s xml:id="echoid-s10604" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10605" xml:space="preserve">Pariter differentia Logarithmorum 3 & </s> <s xml:id="echoid-s10606" xml:space="preserve">2 quæ in tabulis <lb/>Logarithmorum habetur 1760912594, exhibet quantitatem <lb/>Quintæ perfectæ; </s> <s xml:id="echoid-s10607" xml:space="preserve">hinc ſubſtraho Quintam Temperamenti in-<lb/>venti & </s> <s xml:id="echoid-s10608" xml:space="preserve">reſtat 13487583; </s> <s xml:id="echoid-s10609" xml:space="preserve">id quod debet dare Log. </s> <s xml:id="echoid-s10610" xml:space="preserve">{1/4} Com-<lb/>matis; </s> <s xml:id="echoid-s10611" xml:space="preserve">& </s> <s xml:id="echoid-s10612" xml:space="preserve">hoc verum eſt; </s> <s xml:id="echoid-s10613" xml:space="preserve">nam log. </s> <s xml:id="echoid-s10614" xml:space="preserve">Commatis toni id eſt <lb/>differentia logar. </s> <s xml:id="echoid-s10615" xml:space="preserve">81 & </s> <s xml:id="echoid-s10616" xml:space="preserve">80 eſt 53950319, cujus Quarta pars <lb/>eſt 13487580. </s> <s xml:id="echoid-s10617" xml:space="preserve">Si velimus examinare an Quævis Quinta novæ <lb/>diviſionis ut Re, La, differat a vera {1/4} minus {1/110} Commatis; <lb/></s> <s xml:id="echoid-s10618" xml:space="preserve">debemus a Log. </s> <s xml:id="echoid-s10619" xml:space="preserve">Re, qui eſt 4,9514467743, ſubſtrahere Log. </s> <s xml:id="echoid-s10620" xml:space="preserve"><lb/>La, qui eſt 4,7766551643, reſtat 1747916100, quem ſub-<lb/>traho a Logarithmo veræ Quintæ qui erat 1760912594, reſtat <lb/>12996494, qui minor eſt Log. </s> <s xml:id="echoid-s10621" xml:space="preserve">Quartæ partis Commatis, ſcili-<lb/>cet 13487580; </s> <s xml:id="echoid-s10622" xml:space="preserve">a quo igitur eum ſubtraho & </s> <s xml:id="echoid-s10623" xml:space="preserve">reſtat 491086: </s> <s xml:id="echoid-s10624" xml:space="preserve"><lb/>Debemus nunc videre, quam partem Commatis hoc faciat; </s> <s xml:id="echoid-s10625" xml:space="preserve">id-<lb/>circo divido Log. </s> <s xml:id="echoid-s10626" xml:space="preserve">Commatis, ſcilicet 53950319 per 491086, <lb/>quotiens eſt fere {1/110}, Ita ut pateat, noſtram Quintam non ſu-<lb/>perari {1/4} Commatis a Quinta perfecta, ſed {1/110} Commatis de-<lb/>eſſe. </s> <s xml:id="echoid-s10627" xml:space="preserve">Eodem modo poteſt examinari quidquid ſpectat Tempe-<lb/>ramentum; </s> <s xml:id="echoid-s10628" xml:space="preserve">nihil enim in computationibus Muſicis inſtituen-<lb/>dis Logarithmorum uſui anteponendum.</s> <s xml:id="echoid-s10629" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div502" type="section" level="1" n="188"> <head xml:id="echoid-head264" xml:space="preserve">FINIS.</head> <figure> <image file="0498-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0498-01"/> </figure> <pb file="0499" n="521"/> </div> <div xml:id="echoid-div503" type="section" level="1" n="189"> <head xml:id="echoid-head265" xml:space="preserve">CHRISTIANI HUGENII <lb/>VARIA <lb/>DE <lb/>OPTICA.</head> <pb file="0500" n="522"/> <pb o="757" file="0501" n="523"/> <figure> <image file="0501-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0501-01"/> </figure> </div> <div xml:id="echoid-div504" type="section" level="1" n="190"> <head xml:id="echoid-head266" xml:space="preserve">CHRISTIANI HUGENII <lb/>VARIA <lb/>DE <lb/>OPTICA.</head> <head xml:id="echoid-head267" xml:space="preserve">I.</head> <head xml:id="echoid-head268" style="it" xml:space="preserve">Excerpta ex literis D<emph style="super">ni</emph> Hugenii, Academiæ Regiæ <lb/>Scientiarum Socii, ad Autorem Diarii Eruditoruns <lb/>de Catoptrico conſpicillo D<emph style="super">ni</emph> Newtoni.</head> <p> <s xml:id="echoid-s10630" xml:space="preserve">VT deſideras tibi ſententiam meam exponam <lb/>de Teleſcopio D<emph style="super">ni</emph> Newtoni. </s> <s xml:id="echoid-s10631" xml:space="preserve">Licet nondum <lb/>viderim hujusinventi effectum, illud me poſſe <lb/>aſſerere perſuaſum habeo, pulcrum & </s> <s xml:id="echoid-s10632" xml:space="preserve">ingenio-<lb/>ſum eſſe, proſperumque eventum ſortiturum; <lb/></s> <s xml:id="echoid-s10633" xml:space="preserve">ſi modo ad ſpecula cava formanda detegi <lb/>queat, de quo minime deſperandum puto, <lb/>materia quæ poliri poſſit, ut vivide radios reflectat, ut vitrum. </s> <s xml:id="echoid-s10634" xml:space="preserve"><lb/>Commoda hujus conſpicilli præ illis, in quibus vitra tantum <lb/>adhibentur, ſunt, primo quod ſpeculum cavum, licet Sphæri-<lb/>cum, multo melius colligat radios parallelos in unum punctum, <lb/>quam noſtra vitra Sphærica, uti Geometrice poteſt demon-<lb/>ſtrari. </s> <s xml:id="echoid-s10635" xml:space="preserve">Unde ſequitur duorum conſpicillorum ejuſdem lon-<lb/>gitudinis, quorum unum erit novæ illius inventionis, alte-<lb/>rum ordinario vitro objectivo inſtructum, primum, magnam <lb/>habens aperturam, majori copia collecturum radios ab ob-<lb/>jectis procedentes, licet parvum ſpeculum quoſdam inter- <pb o="758" file="0502" n="524" rhead="CHRIST. HUGENII"/> cipiat, ideoque magis poterit amplificare objecta, quam alte-<lb/>rum. </s> <s xml:id="echoid-s10636" xml:space="preserve">Adeo ut Teleſcopium novum dimidiæ aut tertiæ partis <lb/>longitudinis conſpicilli ordinarii, aut forte etiam minus, æ-<lb/>qualem effectum cum hoc præſtabit.</s> <s xml:id="echoid-s10637" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10638" xml:space="preserve">Secunda utilitas eſt, hoc invento vitari incommodum, <lb/>quod a vitris objectivis ſeparari nequit, mutua nempe dua-<lb/>rum ſuperficierum ad ſe inclinatio. </s> <s xml:id="echoid-s10639" xml:space="preserve">Nam licet hæc parva <lb/>ſit, tamen obeſt radiis, qui vitra ad latera tranſeunt & </s> <s xml:id="echoid-s10640" xml:space="preserve">magis <lb/>oberit, ſi vellemus uti vitris hyperbolicis vel ellipticis quibus <lb/>majores concedendæ forent aperturæ.</s> <s xml:id="echoid-s10641" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10642" xml:space="preserve">Tertiam utilitatem cenſeo, reflexione Speculi metallici non <lb/>perire tot radios quot in vitris, quæ in utraque ſuperficie re-<lb/>flectunt notabilem radiorum quantitatem & </s> <s xml:id="echoid-s10643" xml:space="preserve">multos interci-<lb/>piunt obſcuritate ſuæ materiæ.</s> <s xml:id="echoid-s10644" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10645" xml:space="preserve">Cum præterea difficulter admodum vitrum detegatur quod <lb/>longioribus conſpicillis inſerviri queat, quia ſæpe non ubi-<lb/>que homogeneum eſt, quartum novi Teleſcopii commodum <lb/>erit, in metallo nil præter ſuperficiei perfectionem requiri.</s> <s xml:id="echoid-s10646" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10647" xml:space="preserve">Qui viderunt conſpicillum D<emph style="super">ni</emph> Newtoni, obſervarunt, <lb/>illud aliquâ difficultate dirigi ad objecta; </s> <s xml:id="echoid-s10648" xml:space="preserve">ſed huic malo oc-<lb/>curri poteſt ſatis facile, jungendo huic conſpicillo alterum <lb/>exacte parallelum, per quod primùm quæretur objectum. </s> <s xml:id="echoid-s10649" xml:space="preserve">De-<lb/>ſiderabitur fateor alter obſervator, ſi conſpicillum catoptri-<lb/>cum ſit magnum, quoniam ille, qui obſervat, in illa obſervat <lb/>extremitate, quæ maxime elevata eſt. </s> <s xml:id="echoid-s10650" xml:space="preserve">Sed ad incommodum <lb/>hoc non attendendum eſt, ſi ſpectetur utilitas inventi: </s> <s xml:id="echoid-s10651" xml:space="preserve">Si <lb/>loco ſpeculorum Sphæricorum poſſent obtineri parabolica <lb/>exacte formata & </s> <s xml:id="echoid-s10652" xml:space="preserve">polita, conſpicilla hæc effectum præſtarent <lb/>quod de vitris ellipticis vel hyperbolicis ſperavimus; </s> <s xml:id="echoid-s10653" xml:space="preserve">credo <lb/>autem id in ſpeculis facilius obtineri poſſe.</s> <s xml:id="echoid-s10654" xml:space="preserve"/> </p> <pb o="759" file="0503" n="525" rhead="VARIA DE OPTICA."/> </div> <div xml:id="echoid-div505" type="section" level="1" n="191"> <head xml:id="echoid-head269" xml:space="preserve">II.</head> <head xml:id="echoid-head270" xml:space="preserve">CONSTRUCTIO PROBLEMATIS OPTICI.</head> <head xml:id="echoid-head271" xml:space="preserve">Propoſitio 39 Libri v. Alhazeni, & 22 lib. <emph style="sc">VI</emph>. <lb/>Vitellionis.</head> <p style="it"> <s xml:id="echoid-s10655" xml:space="preserve">Puncta B, C, & </s> <s xml:id="echoid-s10656" xml:space="preserve">circulus E K, cujus centrum eſt A, <lb/> <anchor type="note" xlink:label="note-0503-01a" xlink:href="note-0503-01"/> data ſunt in eodem plano; </s> <s xml:id="echoid-s10657" xml:space="preserve">inveniendum eſt pun-<lb/>ctum K in peripheria circuli, ita ut lineæ B K, C K <lb/>faciant cum linea A K angulos inter ſe æquales.</s> <s xml:id="echoid-s10658" xml:space="preserve"/> </p> <div xml:id="echoid-div505" type="float" level="2" n="1"> <note position="right" xlink:label="note-0503-01" xlink:href="note-0503-01a" xml:space="preserve">TAB. LVI. <lb/>fig. 1.</note> </div> <p> <s xml:id="echoid-s10659" xml:space="preserve">DUctis A B, A C; </s> <s xml:id="echoid-s10660" xml:space="preserve">fiat A C, A F:</s> <s xml:id="echoid-s10661" xml:space="preserve">: A F, A Q; </s> <s xml:id="echoid-s10662" xml:space="preserve">& </s> <s xml:id="echoid-s10663" xml:space="preserve">A B, <lb/>A E:</s> <s xml:id="echoid-s10664" xml:space="preserve">: A E, A P: </s> <s xml:id="echoid-s10665" xml:space="preserve">ſint etiam A P, & </s> <s xml:id="echoid-s10666" xml:space="preserve">A Q, bifariam di-<lb/>viſæ in R & </s> <s xml:id="echoid-s10667" xml:space="preserve">S; </s> <s xml:id="echoid-s10668" xml:space="preserve">in angulo B A C ſint perſecta parallelo-<lb/>gramma P A Q H & </s> <s xml:id="echoid-s10669" xml:space="preserve">A R Z S. </s> <s xml:id="echoid-s10670" xml:space="preserve">In R Z producta ſumantur <lb/>Z Y & </s> <s xml:id="echoid-s10671" xml:space="preserve">Z X, utraque æqualis lineæ quæ poteſt differentiam <lb/>inter quadrata Q S & </s> <s xml:id="echoid-s10672" xml:space="preserve">Z S: </s> <s xml:id="echoid-s10673" xml:space="preserve">fiat X V æqualis X Y & </s> <s xml:id="echoid-s10674" xml:space="preserve">paral-<lb/>lela A B; </s> <s xml:id="echoid-s10675" xml:space="preserve">& </s> <s xml:id="echoid-s10676" xml:space="preserve">lateribus X V & </s> <s xml:id="echoid-s10677" xml:space="preserve">X Y deſcribatur hyperbola, <lb/>quæ tranſibit per puncta Q & </s> <s xml:id="echoid-s10678" xml:space="preserve">H, uti patet per conſtructio-<lb/>nem; </s> <s xml:id="echoid-s10679" xml:space="preserve">hyperbola hæc Q X H occurret circulo in puncto K, <lb/>quod quæritur.</s> <s xml:id="echoid-s10680" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10681" xml:space="preserve">Ductis K O & </s> <s xml:id="echoid-s10682" xml:space="preserve">KI parallelis A C & </s> <s xml:id="echoid-s10683" xml:space="preserve">A B, quarum K I <lb/>occurrit Y X in puncto D. </s> <s xml:id="echoid-s10684" xml:space="preserve">Ob hyperbolam rectangulum <lb/>Y D X æquale eſt quadrato K D ordinatæ, vel quadrato O R; <lb/></s> <s xml:id="echoid-s10685" xml:space="preserve">& </s> <s xml:id="echoid-s10686" xml:space="preserve">rectangulum Y T X æquale eſt quadrato H T, vel quadra-<lb/>to P R; </s> <s xml:id="echoid-s10687" xml:space="preserve">& </s> <s xml:id="echoid-s10688" xml:space="preserve">demto a rectangulo Y T X, rectangulum Y D X <lb/>& </s> <s xml:id="echoid-s10689" xml:space="preserve">a quadrato P R, quadratum O R, ſupererit rectangu-<lb/>lum R D T vel A I Q, quod æquale erit rectangulo A O P. </s> <s xml:id="echoid-s10690" xml:space="preserve"><lb/>P O ergo eſt ad A I, vel O K ipſi æqualem, ut Q I ad <lb/>A O, vel I K; </s> <s xml:id="echoid-s10691" xml:space="preserve">& </s> <s xml:id="echoid-s10692" xml:space="preserve">ductis lineis K P, K Q triangula K O P, <lb/>K I Q erunt ſimilia & </s> <s xml:id="echoid-s10693" xml:space="preserve">ideo æquiangula; </s> <s xml:id="echoid-s10694" xml:space="preserve">idcirco anguli <lb/>A P K, A Q K, qui iidem ſunt, vel ſupplementa angulorum <lb/>æqualium O P K, I Q K erunt inter ſe æquales; </s> <s xml:id="echoid-s10695" xml:space="preserve">ſed per con-<lb/>ſtructionem A B, A E vel A K, :</s> <s xml:id="echoid-s10696" xml:space="preserve">: A K vel A E, A P; </s> <s xml:id="echoid-s10697" xml:space="preserve"><lb/>ideo duo triangula B A K, K A P ſunt ſimilia, & </s> <s xml:id="echoid-s10698" xml:space="preserve">ob ean-<lb/>dem cauſam duo triangula C A K, K A Q, ſunt etiam ſi-<lb/>milia; </s> <s xml:id="echoid-s10699" xml:space="preserve">ideo angulus B K A eſt æqualis angulo A P K, &</s> <s xml:id="echoid-s10700" xml:space="preserve"> <pb o="760" file="0504" n="526" rhead="CHRISTIANI HUGENII"/> angulus C K A æqualis ang. </s> <s xml:id="echoid-s10701" xml:space="preserve">A Q K, ſed demonſtravimus <lb/>angulos A P K & </s> <s xml:id="echoid-s10702" xml:space="preserve">A Q K eſſe æquales, ergo anguli B K A, <lb/>C K A erunt inter ſe æquales Q. </s> <s xml:id="echoid-s10703" xml:space="preserve">E. </s> <s xml:id="echoid-s10704" xml:space="preserve">D.</s> <s xml:id="echoid-s10705" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10706" xml:space="preserve">Si punctum H cadat in circumferentiam circuli punctum <lb/>H erit punctum K, quod quæritur & </s> <s xml:id="echoid-s10707" xml:space="preserve">lineæ H P, K P, K O <lb/>in unicam H P coaleſcunt; </s> <s xml:id="echoid-s10708" xml:space="preserve">& </s> <s xml:id="echoid-s10709" xml:space="preserve">ſimiliter lineæ H Q, K Q, <lb/>K I in unam H Q, & </s> <s xml:id="echoid-s10710" xml:space="preserve">quæ ſuperius demonſtrata ſunt & </s> <s xml:id="echoid-s10711" xml:space="preserve"><lb/>hic locum habent, in quo caſu Hyperbolâ non indigemus.</s> <s xml:id="echoid-s10712" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div507" type="section" level="1" n="192"> <head xml:id="echoid-head272" xml:space="preserve">III.</head> <head xml:id="echoid-head273" xml:space="preserve">ALITER.</head> <head xml:id="echoid-head274" style="it" xml:space="preserve">Dato Speculo Cavo aut Convexo, itemque Oculo & <lb/>Puncto Rei viſæ, invenire Punctum Reflexionis.</head> <p> <s xml:id="echoid-s10713" xml:space="preserve">ESto ſpeculum ex ſphæra quæ Centrum habeat A punctum; <lb/></s> <s xml:id="echoid-s10714" xml:space="preserve"> <anchor type="note" xlink:label="note-0504-01a" xlink:href="note-0504-01"/> oculus vero ſit in B, & </s> <s xml:id="echoid-s10715" xml:space="preserve">punctum viſibile in C, Pla-<lb/>numque ductum per A, B, C, faciat in ſphæra Circulum <lb/>D d, in quo invenienda ſint Reflexionis Puncta. </s> <s xml:id="echoid-s10716" xml:space="preserve">Per tria <lb/>Puncta A, B, C, deſcribatur Circuli Circumferentia, cujus <lb/>ſit Centrum Z; </s> <s xml:id="echoid-s10717" xml:space="preserve">occurrat autem ei producta A E, Perpend. <lb/></s> <s xml:id="echoid-s10718" xml:space="preserve">B C, in R; </s> <s xml:id="echoid-s10719" xml:space="preserve">& </s> <s xml:id="echoid-s10720" xml:space="preserve">ſit duabus R A, O A, tertia Proportionalis <lb/>N A; </s> <s xml:id="echoid-s10721" xml:space="preserve">eritque N M, Parallela B C, altera Aſymptoton. </s> <s xml:id="echoid-s10722" xml:space="preserve"><lb/>Rurſus ſint Proportionales E A, {1/2} A O, A I, & </s> <s xml:id="echoid-s10723" xml:space="preserve">ſumma <lb/>I Y æquali I N, ducatur Y M Parallela A Z; </s> <s xml:id="echoid-s10724" xml:space="preserve">eaque erit <lb/>altera Aſymptotos. </s> <s xml:id="echoid-s10725" xml:space="preserve">Denique ſumptis I X, I S, quæ ſin-<lb/>gulæ poſſint dimidium quadratum A O, una cum quadrato <lb/>A I; </s> <s xml:id="echoid-s10726" xml:space="preserve">erunt Puncta X & </s> <s xml:id="echoid-s10727" xml:space="preserve">S in Hyperbola, aut ſectionibus <lb/>oppoſitis D d, ad inventas Aſymptotos deſcribendis, qua-<lb/>rum interſectiones cum Circumferentia D O, oſtendent <lb/>Puncta Reflexionis quæſita. </s> <s xml:id="echoid-s10728" xml:space="preserve">Conſtructio hæc, in omni <lb/>Caſu, quo Problema ſolidum eſt, locum habet, præter-<lb/>quam in uno, ubi non Hyperbola ſed Parabola deſcriben-<lb/>da eſt; </s> <s xml:id="echoid-s10729" xml:space="preserve">cum nimirum Circumferentia per Puncta A, B, C, <lb/>deſcripta, tangit Rectam A E.</s> <s xml:id="echoid-s10730" xml:space="preserve"/> </p> <div xml:id="echoid-div507" type="float" level="2" n="1"> <note position="left" xlink:label="note-0504-01" xlink:href="note-0504-01a" xml:space="preserve">TAB. LVI. <lb/>fig. 2.</note> </div> <pb o="761" file="0505" n="527" rhead="VARIA DE OPTICA."/> </div> <div xml:id="echoid-div509" type="section" level="1" n="193"> <head xml:id="echoid-head275" xml:space="preserve">IV.</head> <head xml:id="echoid-head276" xml:space="preserve">COMPENDIUM.</head> <p> <s xml:id="echoid-s10731" xml:space="preserve">DUctâ lineâ A T, parallelâ C B, eaque biſecta in V, <lb/> <anchor type="note" xlink:label="note-0505-01a" xlink:href="note-0505-01"/> punctum hoc eſt illud, per quod tranſire debet una <lb/>Hyperbolarum Oppoſitarum, quarum Aſymptoti inventæ <lb/>fuerunt Y M, M N.</s> <s xml:id="echoid-s10732" xml:space="preserve"/> </p> <div xml:id="echoid-div509" type="float" level="2" n="1"> <note position="right" xlink:label="note-0505-01" xlink:href="note-0505-01a" xml:space="preserve">TAB. LVI. <lb/>fig. 3.</note> </div> <p> <s xml:id="echoid-s10733" xml:space="preserve">Sed en Tibi bonam illam Conſtructionem, quæ in omni-<lb/> <anchor type="note" xlink:label="note-0505-02a" xlink:href="note-0505-02"/> bus Caſibus obtinet. </s> <s xml:id="echoid-s10734" xml:space="preserve">Sit Circulus datus E D, cujus Centrum <lb/>eſt A; </s> <s xml:id="echoid-s10735" xml:space="preserve">Puncta data, B & </s> <s xml:id="echoid-s10736" xml:space="preserve">C.</s> <s xml:id="echoid-s10737" xml:space="preserve"/> </p> <div xml:id="echoid-div510" type="float" level="2" n="2"> <note position="right" xlink:label="note-0505-02" xlink:href="note-0505-02a" xml:space="preserve">TAB. LVI. <lb/>fig. 4.</note> </div> <p> <s xml:id="echoid-s10738" xml:space="preserve">Ductis Lineis A B, A C, fiant Proportionales B A (Ra-<lb/>dius Circuli) & </s> <s xml:id="echoid-s10739" xml:space="preserve">F A: </s> <s xml:id="echoid-s10740" xml:space="preserve">Eodem modo C A, (Radius Circuli) <lb/>& </s> <s xml:id="echoid-s10741" xml:space="preserve">G A. </s> <s xml:id="echoid-s10742" xml:space="preserve">Tum jungatur F G, eaque biſecetur in H; </s> <s xml:id="echoid-s10743" xml:space="preserve">& </s> <s xml:id="echoid-s10744" xml:space="preserve">per <lb/>hoc punctum ducantur Lineæ L H K, M H N, ſe invicem <lb/>interſecantes ad Angulos Rectos, quarumque L H K ſit Pa-<lb/>rallela ei quæ biſecat Angulum B A C. </s> <s xml:id="echoid-s10745" xml:space="preserve">Hæ ſunt duæ Aſym-<lb/>ptoti Hyperbolarum deſcribendarum per puncta F & </s> <s xml:id="echoid-s10746" xml:space="preserve">G, <lb/>& </s> <s xml:id="echoid-s10747" xml:space="preserve">quarum una tranſibit etiam per Centrum A, quarum In-<lb/>terſectiones cum Circuli Peripheria notabunt puncta Refle-<lb/>xionis quæſita.</s> <s xml:id="echoid-s10748" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div512" type="section" level="1" n="194"> <head xml:id="echoid-head277" xml:space="preserve">V.</head> <head xml:id="echoid-head278" xml:space="preserve">ALIA SOLUTIO.</head> <p style="it"> <s xml:id="echoid-s10749" xml:space="preserve">Problema Alhazeni.</s> <s xml:id="echoid-s10750" xml:space="preserve">] Dato Circulo, cujus Centrum <lb/> <anchor type="note" xlink:label="note-0505-03a" xlink:href="note-0505-03"/> A, Radius A D, & </s> <s xml:id="echoid-s10751" xml:space="preserve">punctis duobus B, C; </s> <s xml:id="echoid-s10752" xml:space="preserve">inve-<lb/>nire punctum H in Circumferentia Circuli dati, un-<lb/>de ductæ H B, H C, faciant ad Circumferentiam <lb/>Angulos æquales.</s> <s xml:id="echoid-s10753" xml:space="preserve"/> </p> <div xml:id="echoid-div512" type="float" level="2" n="1"> <note position="right" xlink:label="note-0505-03" xlink:href="note-0505-03a" xml:space="preserve">TAB. LVI. <lb/>fig. 5.</note> </div> <p> <s xml:id="echoid-s10754" xml:space="preserve">Ponatur Inventum, ductaque A M recta, quæ bifariam <lb/>ſecet Augulum B A C, ducatur ei perpendicularis H F, <lb/>itemque B M, C L. </s> <s xml:id="echoid-s10755" xml:space="preserve">Jungatur porrò A H, cui perpendicu-<lb/>laris ſit H E; </s> <s xml:id="echoid-s10756" xml:space="preserve">rectiſque B H, H C, occurrat A M in pun-<lb/>ctis K, G.</s> <s xml:id="echoid-s10757" xml:space="preserve"/> </p> <pb o="762" file="0506" n="528" rhead="CHRIST. HUGENII"/> <note position="right" xml:space="preserve"> <lb/>Sit jam # A M = a # Quia ergo æquales Anguli K H E & C H Z, \\ ſive E H G; eſtque E H A angulus re- \\ ctus; erit ut K E ad E G, ita K A ad \\ A G. Quia verò B M ad M K, ut H F \\ ad F K, erit, <lb/># M B = b <lb/># A L = c <lb/># L C = n # ut B M + H F ad H F, ita M F ad F K, <lb/>Radius # A D = d # i.e. b + y:y::a - x:{ay - xy/b + y}. add. F A = x <lb/># A F = x # fit K A = {ay + bx/b + y} <lb/># F H = y <lb/></note> <p> <s xml:id="echoid-s10758" xml:space="preserve">Rurſus, quia C L ad L G, ut H F ad F G, erit permutan-<lb/>do & </s> <s xml:id="echoid-s10759" xml:space="preserve">dividendo C L - H F ad H F, ut L F ad F G, <lb/>n - y:</s> <s xml:id="echoid-s10760" xml:space="preserve">y:</s> <s xml:id="echoid-s10761" xml:space="preserve">:c - x:</s> <s xml:id="echoid-s10762" xml:space="preserve">{cy - xy/n - y}, quâ ablatâ ab A F = x, fit <lb/>G A = {nx - cy/n - y}. </s> <s xml:id="echoid-s10763" xml:space="preserve">Eſt autem E A = {dd/x}, quia Proportiona-<lb/>les F A, A H, A E: </s> <s xml:id="echoid-s10764" xml:space="preserve">Ergo E A - G A, hoc eſt, E G, = {dd/x} -<lb/>{nx + cy/n - y}. </s> <s xml:id="echoid-s10765" xml:space="preserve">Et K A - E A, hoc eſt, K E = {ay + bx/b + y} - {dd/x}</s> </p> <p> <s xml:id="echoid-s10766" xml:space="preserve">Sed diximus, quod K E ad E G, ut K A ad A G; </s> <s xml:id="echoid-s10767" xml:space="preserve">i. </s> <s xml:id="echoid-s10768" xml:space="preserve">e. <lb/></s> <s xml:id="echoid-s10769" xml:space="preserve">{ay + bx/b + y} - {dd/x}:</s> <s xml:id="echoid-s10770" xml:space="preserve">{dd/x} - {nx + cy/n - y}:</s> <s xml:id="echoid-s10771" xml:space="preserve">:{ay + bx/b + y}:</s> <s xml:id="echoid-s10772" xml:space="preserve">{nx - cy/n - y}.</s> <s xml:id="echoid-s10773" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10774" xml:space="preserve">Unde invenitur <lb/>2 anxxy + 2bnx<emph style="super">3</emph> - ddbnx - ddnxy = + naddy + bddx <lb/>- 2acxyy - 2bcxxy + ddbcy + ddcyy - addyy - bddxy <lb/>Eſt autem {2bbc/a}x<emph style="super">3</emph> = {2bbcddx/a} - {2bbcyyx/a}, quia xx = dd - yy. <lb/></s> <s xml:id="echoid-s10775" xml:space="preserve">Et quia n = {bc/a}, ergo {- 2bbcxyy/a} - {ddbcyx/a} - 2acxyy <lb/>+ ddcyy = - addyy = bddxy:</s> <s xml:id="echoid-s10776" xml:space="preserve"> <pb o="763" file="0507" n="529" rhead="VARIA DE OPTICA."/> Et diviſis omnibus per y & </s> <s xml:id="echoid-s10777" xml:space="preserve">ductis in a <lb/>- 2bbcxy - ddbcx - 2aacxy + ddcay = - aaddy - bddax <lb/>abddx - cbddx + acddy + aaddy = 2aacxy + 2bbcxy <lb/>{abddx - cbddx + acddy + aaddy/2aac + 2bbc} = xy, quæ æquatio <lb/>eſt ad Hyperbolam.</s> <s xml:id="echoid-s10778" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10779" xml:space="preserve">Vel quia bc = na, {abddx - anddx + acddy + aaddy/2aac + 2bbc} = xy.</s> <s xml:id="echoid-s10780" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10781" xml:space="preserve">Sit {add/aa + bb} = p; </s> <s xml:id="echoid-s10782" xml:space="preserve">Ergo {pbx - pnx + pcy + pay/2c} = xy.</s> <s xml:id="echoid-s10783" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10784" xml:space="preserve">Unde porrò non difficulter invenitur ſequens Conſtructio: <lb/></s> <s xml:id="echoid-s10785" xml:space="preserve"> <anchor type="note" xlink:label="note-0507-01a" xlink:href="note-0507-01"/> jungantur B A, A C, & </s> <s xml:id="echoid-s10786" xml:space="preserve">applicato ſeorſim ad utramque <lb/>Quadrato Radii A D, fiant inde A P, A Q; </s> <s xml:id="echoid-s10787" xml:space="preserve">& </s> <s xml:id="echoid-s10788" xml:space="preserve">juncto P Q, <lb/>dividatur ipſa bifariam in R, & </s> <s xml:id="echoid-s10789" xml:space="preserve">per punctum R ducantur <lb/>R D, R N, ſeſe ad Rectos Angulos ſecantes, quorumque <lb/>R D, ſit parallela A M, quae dividet bifariam Angulum B A C. <lb/></s> <s xml:id="echoid-s10790" xml:space="preserve">Erunt jam R D, R N Aſymptoti oppoſitarum Hyperbola-<lb/>rum, quarum altera per Centrum A tranſire debet, quæque <lb/>ſecabunt Circumferentiam in punctis H quæſitis. </s> <s xml:id="echoid-s10791" xml:space="preserve">Tranſibunt <lb/>autem Hyperbolæ per Puncta P, Q.</s> <s xml:id="echoid-s10792" xml:space="preserve"/> </p> <div xml:id="echoid-div513" type="float" level="2" n="2"> <note position="right" xlink:label="note-0507-01" xlink:href="note-0507-01a" xml:space="preserve">TAB. LVI. <lb/>fig. 6.</note> </div> <p> <s xml:id="echoid-s10793" xml:space="preserve">Ratio Conſtructionis apparet, ductis P γ, & </s> <s xml:id="echoid-s10794" xml:space="preserve">Q ζ, per-<lb/>pendicularibus in A M. </s> <s xml:id="echoid-s10795" xml:space="preserve">Fit enim A γ = {add/aa + bb} ſive p; </s> <s xml:id="echoid-s10796" xml:space="preserve">& </s> <s xml:id="echoid-s10797" xml:space="preserve"><lb/>A ζ = {ap/c}. </s> <s xml:id="echoid-s10798" xml:space="preserve">Item P γ = {pn/c}, & </s> <s xml:id="echoid-s10799" xml:space="preserve">Q ζ = {pb/c}. </s> <s xml:id="echoid-s10800" xml:space="preserve">Quare A O = {pc + pa/2c}, <lb/>& </s> <s xml:id="echoid-s10801" xml:space="preserve">O R = {pb - pn/2c}. </s> <s xml:id="echoid-s10802" xml:space="preserve">Unde Cætera facilia.</s> <s xml:id="echoid-s10803" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10804" xml:space="preserve">pag. </s> <s xml:id="echoid-s10805" xml:space="preserve">præced. </s> <s xml:id="echoid-s10806" xml:space="preserve">lin. </s> <s xml:id="echoid-s10807" xml:space="preserve">pen. </s> <s xml:id="echoid-s10808" xml:space="preserve">lege - 2acxyy <lb/>lin ult. </s> <s xml:id="echoid-s10809" xml:space="preserve">lege - addyy - bddxy:</s> <s xml:id="echoid-s10810" xml:space="preserve"/> </p> <pb o="764" file="0508" n="530" rhead="CHRIST. HUGENII"/> </div> <div xml:id="echoid-div515" type="section" level="1" n="195"> <head xml:id="echoid-head279" xml:space="preserve">VI.</head> <head xml:id="echoid-head280" style="it" xml:space="preserve">Excerpta ex litteris D<emph style="super">ni</emph>. Hugenii Acad. Reg. ſcient. <lb/>Socii, ad auctorem Diarii Paris. de novo Mi-<lb/>croſcopio ex Hollandia allato.</head> <p> <s xml:id="echoid-s10811" xml:space="preserve">MIcroſcopium hoc ex unico formatur exiguo globulo vi-<lb/>treo, ſimili illis, quibus in Hollandia & </s> <s xml:id="echoid-s10812" xml:space="preserve">Anglia ani-<lb/>malcula fuere obſervata in aqua puteali & </s> <s xml:id="echoid-s10813" xml:space="preserve">pluvia, ut & </s> <s xml:id="echoid-s10814" xml:space="preserve">illa in <lb/>qua piper fuit maceratum, de quibus egiſti in 9. </s> <s xml:id="echoid-s10815" xml:space="preserve">& </s> <s xml:id="echoid-s10816" xml:space="preserve">11. </s> <s xml:id="echoid-s10817" xml:space="preserve">diario <lb/>hujus anni (1678); </s> <s xml:id="echoid-s10818" xml:space="preserve">ſed globuli de quibus nunc ago aliis hiſce <lb/>minores ſunt.</s> <s xml:id="echoid-s10819" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10820" xml:space="preserve">Inter ipſos quos ex Hollandia attuli quidam magnitudine <lb/>non ſuperant arenæ granulum; </s> <s xml:id="echoid-s10821" xml:space="preserve">alii adeo exigui ſunt ut vix vi-<lb/>deri queant, quare objecta in immenſum amplificant; </s> <s xml:id="echoid-s10822" xml:space="preserve">eo enim <lb/>hæc majora apparent quo globuli ſunt minores.</s> <s xml:id="echoid-s10823" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10824" xml:space="preserve">Objectum obſervandum inter vitri laminam & </s> <s xml:id="echoid-s10825" xml:space="preserve">lapidis ſpe-<lb/>cularis lamellam includitur; </s> <s xml:id="echoid-s10826" xml:space="preserve">hæc machinæ adaptantur quæ <lb/>mihi aliis huc uſque uſitatis magis commoda videtur. </s> <s xml:id="echoid-s10827" xml:space="preserve">Exigua <lb/>guttula æquæ, in qua per duos aut tres dies piper fuit mace-<lb/>ratum, ita incluſa, magna videtur natantibus piſciculis bene re-<lb/>pleta piſcina.</s> <s xml:id="echoid-s10828" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10829" xml:space="preserve">Quæ pecularia circa aquam hanc obſervavi, ne quæ in <lb/>diario veſtro ſunt relata iterum memorem, hæc ſunt; </s> <s xml:id="echoid-s10830" xml:space="preserve">ex <lb/>omni pipere non eadem generantur animalcula; </s> <s xml:id="echoid-s10831" xml:space="preserve">piper quod-<lb/>dam animalcula ceteris majora format, ſive hoc a piperis ve-<lb/>tuſtate pendeat, ſive ab alia cauſa, quæ in ſequentibus de-<lb/>tegi poterit.</s> <s xml:id="echoid-s10832" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10833" xml:space="preserve">Similia animalcula ex aliis ſeminibus ut ex ſemine corian-<lb/>dri etiam producuntur.</s> <s xml:id="echoid-s10834" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10835" xml:space="preserve">Hoc idem obſervavi in ſucco Betulæ, ubi per quinque <lb/>aut ſex dies hunc ſervaveram.</s> <s xml:id="echoid-s10836" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10837" xml:space="preserve">Quidam animalcula viderunt in aqua in qua nux aroma-<lb/>tica, aut caſia, maceratæ fuere: </s> <s xml:id="echoid-s10838" xml:space="preserve">& </s> <s xml:id="echoid-s10839" xml:space="preserve">probabile admodum eſt <lb/>in multis aliis rebus ſimilia detegi poſſe.</s> <s xml:id="echoid-s10840" xml:space="preserve"/> </p> <pb o="765" file="0509" n="531" rhead="VARIA DE OPTICA."/> <p> <s xml:id="echoid-s10841" xml:space="preserve">Quis forte defendet animalcula haec corruptione, aut fer-<lb/>mentatione, generari; </s> <s xml:id="echoid-s10842" xml:space="preserve">ſed alia dantur quorum origo ab hac <lb/>diverſa eſt, talia ſunt quæ in animalium ſemine deteguntur, <lb/>quæ cum hoc ipſo nata videntur, & </s> <s xml:id="echoid-s10843" xml:space="preserve">tanta copia obſervan-<lb/>tur ut totum ſemen quaſi ex hiſce conſtet.</s> <s xml:id="echoid-s10844" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10845" xml:space="preserve">Translucida omnia ſunt, celerrime moventur, & </s> <s xml:id="echoid-s10846" xml:space="preserve">ranis, <lb/>antequam horum pedes formentur, ſimilia ſunt.</s> <s xml:id="echoid-s10847" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10848" xml:space="preserve">Hæc animalcula in Hollandia primum fuere obſervata, & </s> <s xml:id="echoid-s10849" xml:space="preserve"><lb/>horum inventio admodum mihi utilis videtur, & </s> <s xml:id="echoid-s10850" xml:space="preserve">quæ opus <lb/>ſuppeditabit illis qui in animalium geneſin inquirunt.</s> <s xml:id="echoid-s10851" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div516" type="section" level="1" n="196"> <head xml:id="echoid-head281" xml:space="preserve">FINIS.</head> <figure> <image file="0509-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0509-01"/> </figure> <pb file="0510" n="532"/> <pb file="0511" n="533"/> </div> <div xml:id="echoid-div517" type="section" level="1" n="197"> <head xml:id="echoid-head282" xml:space="preserve">CHRISTIANI HUGENII <lb/>EXPERIMENTA <lb/>PHYSICA.</head> <pb file="0512" n="534"/> <pb o="769" file="0513" n="535"/> <figure> <image file="0513-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0513-01"/> </figure> </div> <div xml:id="echoid-div518" type="section" level="1" n="198"> <head xml:id="echoid-head283" xml:space="preserve">CHRISTIANI HUGENII <lb/>EXPERIMENTA <lb/>PHYSICA.</head> <head xml:id="echoid-head284" style="it" xml:space="preserve">Excerpta ex literis D<emph style="super">ni</emph> Hugenii, Academiæ regiæ ſcien-<lb/>tiarum Socii, ad Auctorem Diarii Eruditorum, de <lb/>Phænomenis aquæ aëre purgatæ.</head> <p> <s xml:id="echoid-s10852" xml:space="preserve">ANtequam tibi communicarem quæ obſervavi <lb/>de ſuſpenſione aquæ in vacuo, altera vice ex-<lb/>perimenta inſtituere volui, ut de obſervatio-<lb/>nibus olim a me factis certus eſſem, & </s> <s xml:id="echoid-s10853" xml:space="preserve">ut <lb/>melius penetrare poſſem in cauſas miri adeo <lb/>effectus. </s> <s xml:id="echoid-s10854" xml:space="preserve">Primum tibi ipſas referam obſerva-<lb/>tiones, & </s> <s xml:id="echoid-s10855" xml:space="preserve">dein pergam ad conjecturas de ipſarum cauſa.</s> <s xml:id="echoid-s10856" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10857" xml:space="preserve">Experimenta quæ illuſtris Boyleus anno 1661 publicavit, <lb/>cum deſcriptione machinæ Pneumaticæ mihi ab eo tempore <lb/>dederunt occaſionem examinandi hanc materiam; </s> <s xml:id="echoid-s10858" xml:space="preserve">Unum ex <lb/>experimentis illius eſt, ſi tubo vitreo, 4 pedes longo, aqua re-<lb/>pleto, cujus extremitas aperta, everſo tubo, immergitur aqua <lb/>vitro contento, ſuperimponas recipiens vel vas, unde aër <lb/>educitur, poſt exhauſtum aërem, quantum ope machinæ <lb/>fleri poteſt, aqua tubi deſcendet in vitrum, donec ad altitu-<lb/>dinem unius tantum pedis ſuſtineatur, ſuperiore tubi parte <lb/>aquâ & </s> <s xml:id="echoid-s10859" xml:space="preserve">aëre manente vacuâ. </s> <s xml:id="echoid-s10860" xml:space="preserve">Cenſet optime, aquam ad illam <lb/>altitudinem unius pedis ſupra ſuperficiem aquæ in vitro <pb o="770" file="0514" n="536" rhead="CHRISTIANI HUGENII"/> hærere propter aërem in recipiente ſuperſtitem, & </s> <s xml:id="echoid-s10861" xml:space="preserve">qui<unsure/> ope <lb/>machinæ, non ſatis exactæ, annihilari non potuit.</s> <s xml:id="echoid-s10862" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10863" xml:space="preserve">Similem conſtrui machinam curaveram, & </s> <s xml:id="echoid-s10864" xml:space="preserve">licet nondum <lb/>mutaveram quæ in ſequentibus correxi, ita tamen omnia <lb/>adaptaveram, ut inſtituto experimento memorato aqua in <lb/>tubo ad libellam aquæ in vitro deſcenderit; </s> <s xml:id="echoid-s10865" xml:space="preserve">Quare in inſti-<lb/>tuendo experimento non tubo longitudinis memoratæ indi-<lb/>gebam. </s> <s xml:id="echoid-s10866" xml:space="preserve">Adhibui tubum novem pollicum cum globo cavo <lb/> <anchor type="note" xlink:label="note-0514-01a" xlink:href="note-0514-01"/> in extremitate ut patet in hac figurâ.</s> <s xml:id="echoid-s10867" xml:space="preserve"/> </p> <div xml:id="echoid-div518" type="float" level="2" n="1"> <note position="left" xlink:label="note-0514-01" xlink:href="note-0514-01a" xml:space="preserve">TAB. LVI. <lb/>fig. 7.</note> </div> <p> <s xml:id="echoid-s10868" xml:space="preserve">Concipiendum eſt vitrum CC in totum aquâ repleri, ex-<lb/>tremitatemque apertam immergi aquâ in vitro D; </s> <s xml:id="echoid-s10869" xml:space="preserve">utrumque <lb/>obtegitur vaſe B, cujus os apertum applicatur molli cuidam <lb/>cimento, quo lamina AA induitur; </s> <s xml:id="echoid-s10870" xml:space="preserve">hæc perforata eſt in <lb/>medio parvo foramine, per quod exit aër, quando agitatur <lb/>antlia; </s> <s xml:id="echoid-s10871" xml:space="preserve">ſi adhibita aqua frigida experimentum inſtituatur in <lb/>totum vas C evacuatur, donec aqua ad libellam aquæ in vitro <lb/>D deſcendat.</s> <s xml:id="echoid-s10872" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div520" type="section" level="1" n="199"> <head xml:id="echoid-head285" xml:space="preserve">EXPERIMENTUMI.</head> <head xml:id="echoid-head286" style="it" xml:space="preserve">Aqua, ſublatâ aëris preſſione, hæret in tubo.</head> <p> <s xml:id="echoid-s10873" xml:space="preserve">SEd ad finem menſis Decembris ejuſdem anni 1661. </s> <s xml:id="echoid-s10874" xml:space="preserve">cum <lb/>reliquiſſem aquam illam in vacuo per 24 horas, (quo <lb/>plane purgatur bullis aëris quas ejicit, quando adhibetur fri-<lb/>gida) cumque impleviſſem vas C, admiratus vidi, licet opti-<lb/>me eduxerim aërem ex vaſe B, aquam nullatenus deſcen-<lb/>dere ex vaſe, quod plenum remanebat. </s> <s xml:id="echoid-s10875" xml:space="preserve">Non poteram ſu-<lb/>ſpicari ullum in machinâ meâ vitium dari, nec vas B <lb/>non bene fuiſſe obturatum; </s> <s xml:id="echoid-s10876" xml:space="preserve">ſed ut quidquid rei eſſet <lb/>expiſcarer, detraxi phialam C ex vaſe, & </s> <s xml:id="echoid-s10877" xml:space="preserve">poſtquam in <lb/>hac admiſi parvam bullulam aëris, repoſui phialam ut an-<lb/>te; </s> <s xml:id="echoid-s10878" xml:space="preserve">& </s> <s xml:id="echoid-s10879" xml:space="preserve">agitatâ machinâ, vidi tandem omnem aquam de-<lb/>ſcendiſſe prope ad libellam illius quæ erat in vitro D. </s> <s xml:id="echoid-s10880" xml:space="preserve">Id <lb/>mihi perſuaſit, non dari vitium in machinâ & </s> <s xml:id="echoid-s10881" xml:space="preserve">aquam aë-<lb/>re purgatam manere ſuſpenſam; </s> <s xml:id="echoid-s10882" xml:space="preserve">licet vas B fuerit plane <lb/>aëre vacuum vel ad minimum ita, uti erat quando aqua <pb o="771" file="0515" n="537" rhead="EXPERIMENTA PHYSICA."/> nondum aëre purgata deſcendebat ex Phialâ; </s> <s xml:id="echoid-s10883" xml:space="preserve">altera vice ut <lb/>aqua deſcenderet, effeci immiſſâ in collum Phialæ bullâ adeo <lb/>parvâ, ut ægre videri poſſet.</s> <s xml:id="echoid-s10884" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div521" type="section" level="1" n="200"> <head xml:id="echoid-head287" xml:space="preserve">EXPERIMENTUM II.</head> <head xml:id="echoid-head288" style="it" xml:space="preserve">Notabile quid in deſcenſu aquæ aëre purgatæ.</head> <p> <s xml:id="echoid-s10885" xml:space="preserve">SEd alio tempore mihi quid valde memorabile contigit. <lb/></s> <s xml:id="echoid-s10886" xml:space="preserve">Cum non immiſiſſem bullam aëream, poſt exhauſtum aë-<lb/>rem bullam formari obſervavi in inferiori parte colli Phialæ; </s> <s xml:id="echoid-s10887" xml:space="preserve"><lb/>Hæc bulla, continuo magis magiſque aucta, ubi ad magnitudi-<lb/>nem parvi pili pervenit, reliquit vitrum & </s> <s xml:id="echoid-s10888" xml:space="preserve">incepit adſcendere <lb/>in collo Phialæ, ubi autem pervenit ad altitudinem unius <lb/>pollicis ſupra ſuperficiem aquæ in vitro D, non ulterius ad-<lb/>ſcendit, ſed ſubito ſeſe expandit ſuperiora verſus & </s> <s xml:id="echoid-s10889" xml:space="preserve">unico <lb/>momento occupavit totam Phialam, ex qua eodem tempore <lb/>aqua deſcendit per parvum id ſpatium, quod reliquum erat in-<lb/>ter interiorem colli ſuperficiem & </s> <s xml:id="echoid-s10890" xml:space="preserve">bullam, quæ extenſa erat, <lb/>omniſque remanſit infra illam altitudinem unius pollicis, ad <lb/>quam bulla cæpit ſeſe expandere. </s> <s xml:id="echoid-s10891" xml:space="preserve">Omnia hæc porro mihi <lb/>contigerunt inſtituenti experimenta cum tubis duorum pe-<lb/>dum & </s> <s xml:id="echoid-s10892" xml:space="preserve">majoribus, ubi aqua manebat ſuſpenſa eodem modo <lb/>ut in tubo 9 pollicum.</s> <s xml:id="echoid-s10893" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10894" xml:space="preserve">Communicavi hæc experimenta cum ſocietate regia An-<lb/>glicana, ſed noluere ſtatim illis fidem dare, mihi autem in-<lb/>dicarunt veriſimiliter aquam Phialæ non deſcendiſſe, prop-<lb/>ter aërem non bene evacuatum: </s> <s xml:id="echoid-s10895" xml:space="preserve">ſed reſpondi, nullam da-<lb/>ri rationem illud ſuſpectandi ſi attendamus ad illud quod <lb/>notavi in fine experimenti fuiſſe obſervatum; </s> <s xml:id="echoid-s10896" xml:space="preserve">& </s> <s xml:id="echoid-s10897" xml:space="preserve">præterea <lb/>me ex experimentis frequenter repetitis de bona machi-<lb/>næ conditione certiſſimum eſſe. </s> <s xml:id="echoid-s10898" xml:space="preserve">Tandem cum anno 1663 <lb/>eſſem in Angliâ me præſente in conſeſſu Societatis Regiæ <lb/>idem experimentum eodemque cum ſucceſſu inſtitutum eſt, <lb/>licet tubi fuerint quatuor & </s> <s xml:id="echoid-s10899" xml:space="preserve">quinque pedum; </s> <s xml:id="echoid-s10900" xml:space="preserve">dein in men-<lb/>tem venit D<emph style="super">no</emph> Boyle idem inſtituere experimentum, non ad-<lb/>hibitâ antliâ, cum argento vivo in tubo vitreo, cujus extre- <pb o="772" file="0516" n="538" rhead="CHRIST. HUGENII"/> mitas aperta immerſa erat in alio argento vivo; </s> <s xml:id="echoid-s10901" xml:space="preserve">quum primo <lb/>detexiſſet methodum perfecte Mercurium aëre purgandi per <lb/>tres vel quatuor dies. </s> <s xml:id="echoid-s10902" xml:space="preserve">Tandem experimentum ſucceſſit, & </s> <s xml:id="echoid-s10903" xml:space="preserve"><lb/>cum in experimento Torricelliano Mercurius deſcendat ad <lb/>altitudinem 27 vel 28 pollicum ſupra ſuperficiem Mercurii, <lb/>in quo impoſitus eſt tubus, ita a D<emph style="super">no</emph> Boyle, & </s> <s xml:id="echoid-s10904" xml:space="preserve">eodem etiam <lb/>tempore a D<emph style="super">no</emph> Vicecomite Brounker, Præſide Societatis Re-<lb/>giæ Anglicanæ, fuit inſtitutum, ut primo hæreret ad altitudi-<lb/>nem 34 pollicum, dein 52, poſtea 55 & </s> <s xml:id="echoid-s10905" xml:space="preserve">tandem ad altitudinem <lb/>75 pollicum, tubo ſemper manente pleno, & </s> <s xml:id="echoid-s10906" xml:space="preserve">huc uſque non-<lb/>dum notum eſt, quænam ſit omnium maxima altitudo poſſi-<lb/>bilis. </s> <s xml:id="echoid-s10907" xml:space="preserve">D<emph style="super">us</emph> Boyle etiam obſervavit, extracto tubo e Mercurio, <lb/>in quo aperta ejus extremitas erat impoſita, & </s> <s xml:id="echoid-s10908" xml:space="preserve">retento tubo <lb/>non clauſo in aëre libero, Mercurium nihilominus non ce-<lb/>cidiſſe. </s> <s xml:id="echoid-s10909" xml:space="preserve">Porro in his experimentis, ut in illis quæ cum aquâ <lb/>inſtituuntur, minimâ aëris bullâ in tubo, ſive ſponte ſive con-<lb/>cuſſione formatâ, ſubito Mercurium deprimit ad ordinariam <lb/>altitudinem 27 vel 28 pollicum.</s> <s xml:id="echoid-s10910" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div522" type="section" level="1" n="201"> <head xml:id="echoid-head289" xml:space="preserve">EXPERIMENTUM III.</head> <head xml:id="echoid-head290" style="it" xml:space="preserve">Adhibito ſpiritu vini loco aquæ.</head> <p> <s xml:id="echoid-s10911" xml:space="preserve">SEd ut ad experimenta noſtra redeam, non ſolum illa ite-<lb/>rum cum aqua inſtitui, ſed & </s> <s xml:id="echoid-s10912" xml:space="preserve">cum Spiritu vini rectifica-<lb/>to, & </s> <s xml:id="echoid-s10913" xml:space="preserve">inveni, ad purgationem aëris tantum requiri, ut per <lb/>unam horam in vacuo relinquatur, licet producat plus aëris <lb/>quam aqua, uti ex hujus operationis circumſtantiis quas re-<lb/>feram quæque notabiles ſatis ſunt, æſtimari facillime poteſt. <lb/></s> <s xml:id="echoid-s10914" xml:space="preserve">Poſtquam vas B proxime ope antliæ aëre evacuatum eſt, <lb/>magnæ bullæ ſpiritus vini exennt & </s> <s xml:id="echoid-s10915" xml:space="preserve">tanta quidem copia, ut <lb/>pars ejiciatur ſupra oram vitri D, quod pariter accidit in a-<lb/>quâ parumper calefactâ; </s> <s xml:id="echoid-s10916" xml:space="preserve">ſed non in illâ quæ frigida immit-<lb/>titur. </s> <s xml:id="echoid-s10917" xml:space="preserve">Illæ ebullitiones continuo minuuntur, ita ut ex vini <lb/>Spiritu, tantum majores aëris bullæ ad certa temporis in-<lb/>tervalla ſingulatim exeant, & </s> <s xml:id="echoid-s10918" xml:space="preserve">tandem nil omnino emergat.</s> <s xml:id="echoid-s10919" xml:space="preserve"> <pb o="773" file="0517" n="539" rhead="EXPERIMENTA PHYSICA."/> Interim bullæ, quæ aſcenderunt in tubum C, ita dilatantùr <lb/>ut plane impleant globum, ut & </s> <s xml:id="echoid-s10920" xml:space="preserve">totam colli longitudinem, <lb/>ita ut omnis vini ſpiritus fugatus ſit, & </s> <s xml:id="echoid-s10921" xml:space="preserve">ut pariter emittat mul-<lb/>tas magnas bullas per aperturam; </s> <s xml:id="echoid-s10922" xml:space="preserve">quod clare indicat, eſſe <lb/>aërem in lagena, vel aliam materiam elaſticam ut aër, quan-<lb/>doquidem illa fugat ſpiritum vini longe infra ſuperficiem il-<lb/>lius qui eſt in vitro D contentus: </s> <s xml:id="echoid-s10923" xml:space="preserve">etiam ſi iterum aërem in-<lb/>trare ſinas in vas B, ſpiritus vini quidem adſcendet in la-<lb/>genam C, non autem plane illam replebit, ſed ſuperius hæ-<lb/>rebit ſatis notabilis aëris bulla.</s> <s xml:id="echoid-s10924" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div523" type="section" level="1" n="202"> <head xml:id="echoid-head291" xml:space="preserve">EXPERIMENTUM IV.</head> <head xml:id="echoid-head292" style="it" xml:space="preserve">Aër, ex Spiritu vini aut aqua exhauſtus, hæc <lb/>corpora iterum intrat.</head> <p> <s xml:id="echoid-s10925" xml:space="preserve">SEd hic notandum, bullam illam ſi ita relinquatur per ſpatium <lb/>unius vel duarum horarum, ſemper evanuiſſe, & </s> <s xml:id="echoid-s10926" xml:space="preserve">in ſpi-<lb/>ritum vini, unde emerſerat, rediiſle. </s> <s xml:id="echoid-s10927" xml:space="preserve">Expertus etiam ſui, ſi <lb/>immittatur porro bulla veri aëris magnitudine unius piſi, <lb/>etiam illam periiſſe, poſtquam per unam noctem relicta fue-<lb/>rit. </s> <s xml:id="echoid-s10928" xml:space="preserve">Idem quoque accidit in aqua. </s> <s xml:id="echoid-s10929" xml:space="preserve">Sed multo majus tempus <lb/>requiritur, ut evaneſcat bulla.</s> <s xml:id="echoid-s10930" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10931" xml:space="preserve">Quod attinet cauſam præcipui noſtri Phænomeni, ſuſpen-<lb/>ſionis nempe aquæ & </s> <s xml:id="echoid-s10932" xml:space="preserve">Mercurii, hæc eſt quam mihi perſua-<lb/>dere potui huc uſque omnium veriſimillimam.</s> <s xml:id="echoid-s10933" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10934" xml:space="preserve">Præter preſſionem aëris, quæ ſuſtinet Mercurium ad alti-<lb/>tudinem 27 pollicum in experimento Torricelliano, & </s> <s xml:id="echoid-s10935" xml:space="preserve">quam <lb/>dari ex infinitis aliis effectibus, quos videmus, conſtat, con-<lb/>cipio & </s> <s xml:id="echoid-s10936" xml:space="preserve">aliam preſſionem illâ fortiorem materiæ aëre ſub-<lb/>tilioris, quæ haud difficulter penetrat vitrum, aquam, <lb/>Mercurium, & </s> <s xml:id="echoid-s10937" xml:space="preserve">omnia alia corpora quæ aëri impene-<lb/>trabilia obſervamus. </s> <s xml:id="echoid-s10938" xml:space="preserve">Hæc preſſio addita ad aëris preſſio-<lb/>nem poteſt ſuſtinere 75 pollices Mercurii & </s> <s xml:id="echoid-s10939" xml:space="preserve">forte adhuc plu-<lb/>res, quamdiu tantum agit in ſuperficiem inferiorem, vel in <lb/>ſuperficiem Mercurii, in quem aperta tubi extremitas immer- <pb o="774" file="0518" n="540" rhead="CHRIST. HUGENII"/> gitur: </s> <s xml:id="echoid-s10940" xml:space="preserve">ſed quam primum materia hæc poteſt agere etiam ad <lb/>alteram partem (quod evenit ſi tubum concutiendo, vel im-<lb/>mittendo parvam aëris bullam, occaſio detur huic materiæ ef-<lb/>fectum ſuum inchoandi) preſſio illius æqualis erit ab utraque <lb/>parte, ita ut ſola ſuperſit æris preſſio quæ ſuſtinet Mercu-<lb/>r@um ad ordinariam altitudinem 27 pollicum; </s> <s xml:id="echoid-s10941" xml:space="preserve">Eadem de cau-<lb/>ſa in experimento aquæ aëre purgatæ, poſt remotam preſ-<lb/>ſio em aëris evacuando recipiens B, altera illa preſſio ejuſ-<lb/>dem materiæ agit etiam ut antea in ſuperficiem aquæ in vi-<lb/>tro D, & </s> <s xml:id="echoid-s10942" xml:space="preserve">cohibet ne aqua in Phiala C deſcendat. </s> <s xml:id="echoid-s10943" xml:space="preserve">Sed <lb/>ubi minima bulla aëris intrat phialam, materia quam dixi <lb/>tranſire per vitrum & </s> <s xml:id="echoid-s10944" xml:space="preserve">aquam, ſubito inflat bullam, editque <lb/>preſſionem æqualem illi, quæ agit in ſuperficiem aquæ in vi-<lb/>tro D, quare omnis aqua phialæ defluit, & </s> <s xml:id="echoid-s10945" xml:space="preserve">ad libellam cum <lb/>illa, quæ eſt in vitro ſeſe conſtituit.</s> <s xml:id="echoid-s10946" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10947" xml:space="preserve">Quæret forte quiſpiam cur preſſio ejus materiæ non agat in <lb/>aquam ſuſpenſam in phiala C, & </s> <s xml:id="echoid-s10948" xml:space="preserve">in Mercurium in tubo D<emph style="super">ni.</emph> <lb/>Boyle, etiam dum vaſa hæc plena ſunt, quoniam ſuppoſui, <lb/>illam haud difficulter penetrare vitrum æque ac aquam & </s> <s xml:id="echoid-s10949" xml:space="preserve"><lb/>Mercurium? </s> <s xml:id="echoid-s10950" xml:space="preserve">Et quare particulæ ejus materiæ non ſe jun-<lb/>gant, & </s> <s xml:id="echoid-s10951" xml:space="preserve">incipiant preſſionem, cum continuo in aqua & </s> <s xml:id="echoid-s10952" xml:space="preserve">Mer-<lb/>curio eant & </s> <s xml:id="echoid-s10953" xml:space="preserve">redeant, & </s> <s xml:id="echoid-s10954" xml:space="preserve">vitrum non impediat illarum com-<lb/>municationem cum particulis exterioribus.</s> <s xml:id="echoid-s10955" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10956" xml:space="preserve">Ut huic difficultati fiat ſatis, quæ revera maxima eſt, di-<lb/>cam, licet partes ejus materiæ, quam ſuppoſui, tranſitum in-<lb/>veniant inter illas, ex quibus vitrum aqua & </s> <s xml:id="echoid-s10957" xml:space="preserve">Mercurius for-<lb/>mantur, non tamen dari tranſitum ſatis latum ut plures ſimul <lb/>tranſeant, neque ut ſe moveant vi, quæ requiritur ut diſper-<lb/>gantur partes Mercurii vel aquæ, quæ inter ſe habent quen-<lb/>dam nexum. </s> <s xml:id="echoid-s10958" xml:space="preserve">Et ex hoc ipſo nexu ſequitur, (licet ad par-<lb/>tem ſuperficiei interioris vitri, quæ tangit aquam vel Mer-<lb/>curium ſuſpenſum, plures partes premantur actione particu-<lb/>larum ejus materiæ, cum tamen plurimæ partes aquæ aut <lb/>Mercurii non premantur, quæ a partibus vitri teguntur,) par-<lb/>tes aquæ aut Mercurii ſeſe mutuo ſuſtinere, omneſque ſuſpen-<lb/>ſas hærere; </s> <s xml:id="echoid-s10959" xml:space="preserve">quia minor eſt preſſio in ſuperficiem aquæ vel ar- <pb o="775" file="0519" n="541" rhead="EXPERIMENTA PHYSICA."/> genti vivi, quæ contingit vitrum, quam in inferiorem, quæ <lb/>tota expoſita eſt actioni materiæ, a qua ſecunda pendet preſ-<lb/>ſio. </s> <s xml:id="echoid-s10960" xml:space="preserve">Solutionem hanc fateor non mihi ita ſatisfacere, <lb/>ut nullus mihi ſuperſit ſcrupulus; </s> <s xml:id="echoid-s10961" xml:space="preserve">nihilominus tamen per-<lb/>ſuaſum habeo novam illam dari preſſionem, præter aëris preſ-<lb/>ſionem, cum ob experimenta ſupra relata tum ob duo alia <lb/>nunc referenda.</s> <s xml:id="echoid-s10962" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div524" type="section" level="1" n="203"> <head xml:id="echoid-head293" xml:space="preserve">EXPERIMENTUM V.</head> <head xml:id="echoid-head294" style="it" xml:space="preserve">Laminæ metallicæ arcte inter ſe cohærent in vacuo <lb/>licet nihil inter has detur.</head> <p> <s xml:id="echoid-s10963" xml:space="preserve">QUando duæ laminæ metallicæ vel marmoreæ, quarum ſu-<lb/>perficies ſint perfecte planæ, ad ſe mutuo applicantur, <lb/>ita cohærent, ut ſuperiore elevatâ inferior eam inſequatur, <lb/>cujus effectus cauſa tribuitur merito preſſioni aëris in ſuper-<lb/>ficies externas; </s> <s xml:id="echoid-s10964" xml:space="preserve">duæ mihi ſunt laminæ, quarum quævis circi-<lb/>ter eſt magnitudinis pollicis quadrati, quæ ex materiâ con-<lb/>flatæ ſunt, ex qua olim ſpecula formabantur, & </s> <s xml:id="echoid-s10965" xml:space="preserve">quæ ita jun-<lb/>guntur, ut, licet nihil interponas, ſuperior non ſolum alte-<lb/>ram ſuſtineat, ſed interdum etiam pondus trium librarum <lb/>cum inferiori conjunctum; </s> <s xml:id="echoid-s10966" xml:space="preserve">in quo ſtatu perſeverant quamdiu <lb/>deſideraveris.</s> <s xml:id="echoid-s10967" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10968" xml:space="preserve">Has ita junctas & </s> <s xml:id="echoid-s10969" xml:space="preserve">oneratas tribus libris ſuſpendi in machi-<lb/>næ meæ recipiente, quod aëre evacuavi in tantum, ut ſuper-<lb/>ſtes aër non preſſione ſua aquam ad altitudinem unius pol-<lb/>licis ſuſtineret, & </s> <s xml:id="echoid-s10970" xml:space="preserve">nihilominus laminæ meæ ſeparatæ non ſunt.</s> <s xml:id="echoid-s10971" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10972" xml:space="preserve">Idem etiam inſtitui experimentum interpoſito ſpiritu vini <lb/>inter laminas & </s> <s xml:id="echoid-s10973" xml:space="preserve">inveni quod in recipiente vacuo aëre ſuſti-<lb/>nerent ſine ſeparatione eadem pondera, quæ ſuſtinebant re-<lb/>cipiente aëre pleno.</s> <s xml:id="echoid-s10974" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10975" xml:space="preserve">Videtur id mihi manifeſte ſatis indicare magnam ſupereſ-<lb/>ſe preſſionem in recipiente, poſtquam aëre vacuus eſt, & </s> <s xml:id="echoid-s10976" xml:space="preserve"><lb/>hanc non magis quam ipſius aëris preſſionem in dubium vo-<lb/>cari poſſe; </s> <s xml:id="echoid-s10977" xml:space="preserve">ſed & </s> <s xml:id="echoid-s10978" xml:space="preserve">alia ad illius confirmationem.</s> <s xml:id="echoid-s10979" xml:space="preserve"/> </p> <pb o="776" file="0520" n="542" rhead="CHRIST. HUG. EXPERIM. PHYSICA."/> </div> <div xml:id="echoid-div525" type="section" level="1" n="204"> <head xml:id="echoid-head295" xml:space="preserve">EXPERIMENTUM VI.</head> <head xml:id="echoid-head296" style="it" xml:space="preserve">Effectus Siphonis in vacuo.</head> <p> <s xml:id="echoid-s10980" xml:space="preserve">NOtum eſt, effectum ſiphonis crurum inæqualium, quo <lb/>evacuatur aqua vaſis ſupra oram, non amplius tribui fugæ <lb/>vacui, ſed ponderi aëris, qui premendo in ſuperficiem aquæ <lb/>in vaſe, illam cogit ut in ſiphonem adſcendat, dum in alio <lb/>crure illa gravitate ſuâ deſcendit. </s> <s xml:id="echoid-s10981" xml:space="preserve">Inveni methodum qua effi-<lb/>ci, ut aqua e ſiphone fluxerit, poſtquam recipiens vacuum <lb/>erat aëre; </s> <s xml:id="echoid-s10982" xml:space="preserve">& </s> <s xml:id="echoid-s10983" xml:space="preserve">obſervavi effectum, ſi aqua purgata fuerit aëre, <lb/>eodem modo ac extra recipiens procedere.</s> <s xml:id="echoid-s10984" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10985" xml:space="preserve">Breviſſimum ſiphonis crus erat octo pollicum & </s> <s xml:id="echoid-s10986" xml:space="preserve">apertura <lb/>duarum linearum. </s> <s xml:id="echoid-s10987" xml:space="preserve">Nec dubitari debet quin recipiens aëre <lb/>bene vacuum fuerit; </s> <s xml:id="echoid-s10988" xml:space="preserve">nam id poſſum affirmare, tum quoniam <lb/>nullum antliâ aërem ulterius extrahere potui, tum ob alia <lb/>ſigna certiora.</s> <s xml:id="echoid-s10989" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10990" xml:space="preserve">Magis ergo confirmatur noſtra hypotheſis materiam da-<lb/>ri prementem aëre ſubtiliorem. </s> <s xml:id="echoid-s10991" xml:space="preserve">Si quis laborem in ſe ſuſcipere <lb/>vellet quærendi, quanta ſit vis hujus preſſionis quod melius <lb/>fieri non poteſt niſi proſequendo experimenta cum tubis <lb/>Mercurio plenis longioribus, quam quibus uſus eſt D<emph style="super">nus</emph> Boy-<lb/>le, inveniet forte, vim illam ſatis validam eſſe ad efficiendam <lb/>unionem partium vitri & </s> <s xml:id="echoid-s10992" xml:space="preserve">aliorum corporum, quæ nimis ar-<lb/>cte cohærent, quam ut juncta tantum forent per contiguitatem <lb/>& </s> <s xml:id="echoid-s10993" xml:space="preserve">quietem uti voluit Carteſius.</s> <s xml:id="echoid-s10994" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div526" type="section" level="1" n="205"> <head xml:id="echoid-head297" xml:space="preserve">FINIS.</head> <pb file="0521" n="543"/> <pb file="0521a" n="544"/> <figure> <caption xml:id="echoid-caption186" style="it" xml:space="preserve">pag. 776.<lb/><emph style="sc">Tab. lvi</emph>.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables105" xml:space="preserve">B H V C K E T D F X P Z Q I Y O R S A</variables> </figure> <figure> <caption xml:id="echoid-caption187" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables106" xml:space="preserve">D S Y A d I M N d X D O Z B M E C R</variables> </figure> <figure> <caption xml:id="echoid-caption188" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables107" xml:space="preserve">Y T V A M N Z B E C R</variables> </figure> <figure> <caption xml:id="echoid-caption189" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables108" xml:space="preserve">L A M F H N G E D K B C</variables> </figure> <figure> <caption xml:id="echoid-caption190" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables109" xml:space="preserve">h A P O R Q G F D Z H E K L C B M</variables> </figure> <figure> <caption xml:id="echoid-caption191" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables110" xml:space="preserve">D A P r N O e Q K I V F H C L B M</variables> </figure> <figure> <caption xml:id="echoid-caption192" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables111" xml:space="preserve">C B C D A A</variables> </figure> <pb file="0522" n="545"/> <pb file="0523" n="546"/> </div> <div xml:id="echoid-div527" type="section" level="1" n="206"> <head xml:id="echoid-head298" xml:space="preserve">INDEX <lb/>RERUM</head> <head xml:id="echoid-head299" style="it" xml:space="preserve">Quatuor Tomis contentarum.</head> <head xml:id="echoid-head300" xml:space="preserve">TOMUS PRIMUS <lb/>OPERA MECHANICA.</head> <note style="it" position="right" xml:space="preserve"> <lb/><emph style="sc">Horologium</emph>. pag. 1. Hagæ-Com. <lb/># ann. 1658. in quarto. <lb/><emph style="sc">Horologium</emph> <emph style="sc">Oscillatorium</emph>, ſi-<lb/># ve de Motu Pendulorum ad Horo-<lb/># logia aptato Demonſtrationes Geo-<lb/># metricæ. p. 15. Pariſiis 1673. infol. <lb/>Deſcriptio Horologii. # 33. <lb/>Tabula Æquationis Dierum. # 44. <lb/>De Deſcenſu Gravium. # 52. <lb/>De Deſcenſu ſuper plano inclinato. # 62. <lb/>Dato in Cycloide puncto, rectam per il-<lb/># lud ducere quæ Cycloidem tangat. # 69. <lb/>De Motu in Cycloide. # 72. <lb/>In Cycloide cujus axis ad perpendiculum <lb/># erectus eſt, vertice deorſum ſpectante, <lb/># tempora deſcenſus quibus mobile, à <lb/># quocunque in ea puncto dimiſſum, ad <lb/># punctum imum verticis pervenit, ſunt <lb/># inter ſe æqualia; habentque ad tem-<lb/># pus caſus perpendicularis per totum <lb/># axem cycloidis eam rationem, quam <lb/># ſemicircumferentia circuli ad diame-<lb/># trum. # 87. <lb/>De Linearum curvarum evolutione & <lb/># dimenſione. # 89. <lb/>Semicycloidis evolutione, à vertice cœ-<lb/># pta, alia ſemicyclois deſcribitur evo-<lb/># lutæ æqualis & ſimilis, cujus baſis eſt <lb/># in ea recta quæ cycloidem evolutam in <lb/># vertice contingit. # 96. <lb/>Cyclois linea ſui axis, ſive diametri cir-<lb/># culi genitoris, quadrupla eſt. # 97. <lb/>Cujus lineæ evolutione parabola deſcriba-<lb/># tur oſtendere. # 99. <lb/>Rectam lineam invenire æqualem datæ <lb/># portioni curvæ paraboloidis, ejus nem-<lb/># pe in qua quadrata ordinatim appli-<lb/># catarum ad axem, ſunt inter ſe ſicut <lb/># cubi abſciſſarum ad verticem. # 100. <lb/>Conoidis parabolici ſuperficiei curvæ cir-<lb/># culum æqualem invenire. # 102. <lb/>Sphæroidis oblongi ſuperficiei circulum <lb/># æqualem invenire. # 103. <lb/>Sphæroidis lati ſive compreſſi ſuperficiei <lb/># circulum æqualem invenire. # Ibid. <lb/>Conoidis hyperbolici ſuperficiei curvæ cir-<lb/># culum æqualem invenire. # 104. <lb/>Curvæ parabolicæ æqualem rectam li-<lb/># neam invenire. # 105. <lb/>Lineas curvas exhibere quarum evolutio-<lb/># ne ellipſes & hyperbolæ deſcribantur, <lb/># rectaſque invenire iiſdem curvis æ-<lb/># quales. # 107. <lb/>Datâ lineâ curvâ, invenire aliam cujus <lb/></note> <pb file="0524" n="547" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># evolutione illa deſcribatur; & oſten-<lb/># dere quod ex unaquaque curva geome-<lb/># trica, alia curva itidem geometrica <lb/># exiſtat, cui recta linea æqualis dari <lb/># poſſit. # 108. <lb/>De Centro Oſcillationis, ſeu Agitatio-<lb/># nis. # 107. <lb/>Ponderibus quotlibet ad eandem partem <lb/># plani exiſtentibus, ſi à ſingulorum cen-<lb/># tris gravitatis agantur in planum il-<lb/># lud perpendiculares; hæ ſingulæ in ſua <lb/># ponder a ductæ, tantundem ſimul effi-<lb/># cient, ac perpendicularis, à centro <lb/># gravitatis ponderum omnium in pla-<lb/># num idem cadens, ducta in pondera <lb/># omnia. # 123. <lb/>Dato pendulo ex ponderibus quotlibet <lb/># compoſito, ſi ſingula ducantur in qua-<lb/># drata diſtantiarum ſuarum ab axe <lb/># oſcillationis, & ſumma productorum <lb/># dividatur per id quod fit ducendo pon-<lb/># derum ſummam, in diſtantiam centri <lb/># gravitatis communis omnium ab eo-<lb/># dem axe oſcillationis; orietur longi-<lb/># tudo penduli ſimplicis compoſito iſo-<lb/># chroni, ſive diſtantia inter axem & <lb/># centrum oſcillationis ipſius penduli com-<lb/># poſiti # 128. <lb/>Dato pendulo ex quotcunque ponderibus <lb/># æqualibus compoſito; ſi ſumma qua-<lb/># dratorum factorum à diſtantiis, qui-<lb/># bus unumquodque pondus abeſt ab axe <lb/># oſcillationis, applicetur ad diſtan-<lb/># tiam centri gravitatis communis ab <lb/># eodem oſcillationis axe, multiplicem <lb/># ſecundum ipſorum ponder um numerum, <lb/># orietur longitudo penduli ſimplicis com-<lb/># poſito iſochroni. # 131. <lb/>Datâ figurâ planâ & in eodem plano li-<lb/># neâ rectâ, quæ vel ſecet figuram vel <lb/># non, ad quam perpendiculares cadant <lb/># à partio<unsure/>ulis ſingulis minimis & æquæ-<lb/># libus, in quas figura diviſa intelligi-<lb/># tur; invenire ſummam quadr atorum <lb/># ab omnibus iſtis perpendicularibus; ſi-<lb/># ve planum, cujus multiplex, ſecun-<lb/># dum particularum numerum, dictæ <lb/># quadratorum ſummæ æquale ſit. # 135. <lb/>Datis in plano punctis quotlibet; ſi ex <lb/># centro gravitatis eorum circulus quili-<lb/># bet deſcribatur; ducantur autem ab <lb/># omnibus datis punctis, ad punctum <lb/># aliquod in circuli illius circumferentia <lb/># lineæ rectæ; erit ſumma quadratorum <lb/># ab omnibus ſemper eidem plano æqua-<lb/># lis. # 138. <lb/>Si figura plana, vel linea in plano exi-<lb/># ſtens, aliter atque aliter ſuſpendatur <lb/># à punctis, quæ, in eodem plano acce-<lb/># pta, æqualiter à centro gravitatis ſuæ <lb/># diſtent; agitata motu in latus, ſibi <lb/># ipſi iſochrona eſt. # 140. <lb/>Datâ figurâ ſolidâ, & lineâ rectâ in-<lb/># terminatâ, quæ vel extra figuram ca-<lb/># dat, vel per eam tranſeat; diviſâque <lb/># figurâ cogitatu in particulas minimas <lb/># æquales, à quibus omnibus ad datam <lb/># rectam perpendiculares ductæ intelli-<lb/># gantur; invenire ſummam omnium <lb/># quæ ab ipſis fiunt quadratorum, ſive <lb/># planum, cujus multiplex ſecundum <lb/># particularum numerum, dictæ qua-<lb/># dratorum ſummæ æquale ſit. # 142. <lb/>Figura quævis, ſive linea fuerit, ſive ſu-<lb/># perficies, ſive ſolidum; ſi aliter atque <lb/># aliter ſuſpendatur, agiteturque ſuper <lb/># axibus inter ſe parallelis, quique à <lb/># centro gravitatis figuræ æqualiter di-<lb/># ſtent, ſibi ipſi iſochrona eſt. # 148. <lb/>Dato plano, cujus multiplex per nume-<lb/># rum particularum, in quas ſuſpenſa <lb/># figura diviſa intelligitur, æquetur <lb/></note> <pb file="0525" n="548" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># quadratis omnium diſtantiarum ab axe <lb/># oſcillationis; ſi illud applicetur ad re-<lb/># ctam, æqualem diſtantiæ inter axem <lb/># oſcillationis & centrum gravitatis ſuſ-<lb/># penſæ magnitudinis, orietur longitu-<lb/># do penduli ſimplicis ipſi iſochroni. # 149. <lb/>Si ſpatium planum, cujus multiplex ſe-<lb/># cundum numerum particularum ſuſpen-<lb/># ſæ magnitudinis, æquetur quadr atis <lb/># diſtantiarum ab axe gravitatis, axi <lb/># oſcillationis parallelo; id, inquam, <lb/># ſpatium ſi applicetur ad rectam, æ-<lb/># qualem diſtantiæ inter utrumque di-<lb/># ctorum axium, orietur recta æqualis <lb/># intervallo, quo centrum oſcillationis <lb/># inferius eſt centro gravitatis ejuſdem <lb/># magnitudinis. # 150. <lb/>Si magnitudo eadem, nunc brevius nunc <lb/># longius ſuſpenſa, agitetur; erunt, ſic-<lb/># ut diſtantiæ axium oſcillationis à cen-<lb/># tro gravitatis inter ſe, ita contraria <lb/># ratione diſtantiæ centrorum oſcillatio-<lb/># nis ab eodem gravitatis centro. # 153. <lb/>Centrum oſcillationis & punctum ſuſpen-<lb/># ſionis inter ſe convertuntur. # 154. <lb/>Quomodo in figuris planis centra oſcilla-<lb/># tionis inveniantur. # 154. <lb/>Centrum oſcillationis Circuli. 157. Re-<lb/># ctanguli. Ibid. Trianguli iſoſcelis. # 158. <lb/># Parabolæ. Ibid. Sectoris circuli. <lb/># Ibid. Circuli, aliter quam ſupra. <lb/># 160. Peripheriæ circuli. Ibid. Po-<lb/># lygonorum ordinatorum. # 161. <lb/>Quomodo, in ſolidis figuris, oſcillationis <lb/># centra inveniantur. # 165. <lb/>Centrum oſcillationis in Pyramide. # 166. <lb/># Coni. 167. Sphæræ. Ibid. Cylin-<lb/># dri. 168. Conoidis Parabolici. Ibid. <lb/># Conoidis Hyperbolici. 169. dimidii <lb/># Coni. # Ibid. <lb/>Horologiorum motum temperare addito <lb/># pondere exiguo ſecundario, quod ſuper <lb/># virga penduli, certa ratione diviſa, <lb/># ſurſum deorſumque moveri poſſit. # 172. <lb/>Centri oſcillationis rationem haberi non <lb/># poſſe, in pendulis inter Cycloides ſuſ-<lb/># penſis; & quomodo hinc orta diffi-<lb/># cultas tollatur. # 177. <lb/>De menſuræ univer ſalis, & perpetuæ, <lb/># conſtituendæ ratione. # 178. <lb/>Spatium definire, quod gravia, perpen-<lb/># diculariter cadentia, dato tempore <lb/># percurrunt. # 182. <lb/>Secu di Horologii Conſtructio è cir-<lb/># culari pendulorum motu deducta # 185. <lb/>De Vi Centrifuga ex motu circulari, <lb/># Theoremata. # 188. <lb/><emph style="sc">Brevis</emph> <emph style="sc">Institutio de</emph> <emph style="sc">Usu</emph> <emph style="sc">Ho-</emph> <lb/># <emph style="sc">ROLOGIORUM AD</emph> <emph style="sc">Inveniendas</emph> <lb/># <emph style="sc">LONGITUDINES</emph>. 193. Hagæ-Com. <lb/># Belgice edita. <lb/>Reducere horologia ad rectam dierum <lb/># menſuram vel cognoſcere quanto citius <lb/># vel tardius ſpatio 24 horarum mo-<lb/># veantur. # 196. <lb/>Tabula Æquationis Dierum. # 198. <lb/>Ope Horologiorum mari invenire longi-<lb/># tudinem loci in quo verſaris. # 202. <lb/>Mari invenire horam diei. # 204. <lb/>Quomodo ex obſervatione ortus & occa-<lb/># ſus Solis & ex hora horologiorum lon-<lb/># gitudo mari inveniri queat. # 205. <lb/>Excerpta ex literis de ſucceſſu horo-<lb/># logiorum datis Londini {13/23} Januarii <lb/># 1665. # 211. <lb/>Experimentum circa pendula. 213. <lb/># Journal des Sçavans 23. Fevr. # 1665. <lb/><emph style="sc">De</emph> <emph style="sc">Hugeniana</emph> <emph style="sc">Centri oscilla-</emph> <lb/># <emph style="sc">TIONIS</emph> <emph style="sc">Determinatione con-</emph> <lb/># <emph style="sc">TROVERSIA</emph>. # 215. <lb/>Obſervationes Abbatis Catalani in <lb/># propoſitionem, quæ fundamentum <lb/></note> <pb file="0526" n="549" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># eſt 4æ. partis tractatus de Pendulis, <lb/># Hugenii # 217. Journal des Sçavans <lb/># 1682. in initio. <lb/>Domini Abbatis Catalani Examen <lb/># Mathematicum Centri Oſcillatio-<lb/># nis. # 219. ibid. <lb/>Excerpta ex literis Domini Hugenii, <lb/># quibus reſpondet obſervationi Ab-<lb/># batis Catelani in 4<emph style="super">am</emph>. propoſitio-<lb/># nem Tractatus de Centris oſcilla-<lb/># tionis. # 222. ibid. 29. Junii 1682. <lb/>Exceptio Abbatis Catalani ad reſpon-<lb/># ſionem Hugenii. # 225. ibid. 20. Ju-<lb/># lii 1682. <lb/>Objectio Abbatis Catalani contra mo-<lb/># tum Pendulorum in Cycloidibus. <lb/># 227. ibid. 7. Sept. 1682. <lb/>Reſponſio ad objectiones Hugenii <lb/># adverſus methodum Abbatis Cata-<lb/># lani de determinando Centro Oſcil-<lb/># lationis # 228. ibid. 14 Sept. 1682. <lb/>Excerpta ex litteris D. Bernoullii da-<lb/># tis Baſileæ ad Autorem Diarii Pa-<lb/># riſienſis, de Controverſia, inter <lb/># Abbatem Catalanum & Hugenium, <lb/># de Centro oſcillationis. # 230. ibid. <lb/># 24. April. 1684. <lb/>Excerpta ex literis D<emph style="super">ni</emph>. Hugenii ad <lb/># Auctores Diarii Pariſienſis, datis <lb/># Hagæ 8. Junii 1684. quæ conti-<lb/># nent ejus reſponſionem ad exceptio-<lb/># nem D<emph style="super">ni</emph>. Abbatis Catalani, de cen-<lb/># tro Oſcillationis. # 232. ibid. 3. Ju-<lb/># lii 1684. <lb/>Reſponſio D<emph style="super">ni</emph>. Abbatis Catalani ad <lb/># Literas D<emph style="super">ni</emph>. Bernoulli de Contro-<lb/># verſia ſua cum D<emph style="super">no</emph>. Hugenio de <lb/># centro Oſcillationis. # 235. ibid. 11. <lb/># Sept. 1684. <lb/>D<emph style="super">ni</emph>. Bernoulli narratio controverſiæ <lb/># inter D<emph style="super">nos</emph>. Hugenium & Abbatem <lb/># Catalanum agitatæ de centro Oſcil-<lb/># lationis, quæ loco Animadverſio-<lb/># nis eſſe poterit in Reſponfionem <lb/># D<emph style="super">ni</emph>. Catalani. # 237. Acta erudito-<lb/># rum Lips. menſis Julii. 1686. <lb/>Litteræ D<emph style="super">ni</emph>. Marchionis de l’ Hôpi-<lb/># tal ad D<emph style="super">um</emph>. Hugenium, in quibus <lb/># contendit, ſe regulam hujus Au-<lb/># ctoris de Centro Oſcillationis pen-<lb/># duli compoſiti demonſtrare per cau-<lb/># ſam Phyſicam, & reſpondere ſimul <lb/># D<emph style="super">no</emph>. Bernoulli. # 242. Hiſtoire des <lb/># ouvrages des Sçavans. menſis Junii. <lb/># 1690. <lb/>Obſervationes D<emph style="super">ni</emph>. Hugenii in literas <lb/># præcedentes & in relationem D<emph style="super">ni</emph>. <lb/># Bernoulli, cujus in iis fit mentio. <lb/># 246. ibid. <lb/><emph style="sc">Machinæ quæd</emph><emph style="sc">Am et</emph> <emph style="sc">Varia cir-</emph> <lb/># <emph style="sc">CA</emph> <emph style="sc">Mechanicam</emph>. # 249. <lb/>Excerpta ex Litteris Domini Huge-<lb/># genii, novam quandam Inventio-<lb/># nem Horologiorum exactiſſimorum <lb/># ac portatilium concernentibus. # 253. <lb/># Journal des Sçavans. # 25. Febr. 1675. <lb/>Nova Libella, Teleſcopio inſtructa, <lb/># propriam ſecum ferens probationem <lb/># & quæ in unica ſtatione verificatur, <lb/># & rectificatur. # 254. ibid. 9. Jan. <lb/># 1680. <lb/>Rectificationis Libellæ Demonſtra-<lb/># tio. # 258. ibid. 26. Feb. 1680. <lb/>Aſtroſcopia Compendiaria, Tubi Op-<lb/># tici molimine liberata. # 261. Hagæ. <lb/># Com. 1684. in quarto. <lb/>Auctarium. # 273. ibid. <lb/>Excerpta ex literis D<emph style="super">ni</emph>. Hugenii de <lb/># novâ methodo conſtruendi´ Baro-<lb/># metrum. # 276. Journal des Sçavans <lb/># 12. Dec. 1672. <lb/>Nova vis movens mediante pulvere <lb/></note> <pb file="0527" n="550" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># nitrato & aëre. # 280. Divers ouvra-<lb/># ges de Mathmatique & de Phiſique de <lb/># M<emph style="super">rs</emph>. de l’ Academie Rojale des ſcien-<lb/># ces. p. 320. <lb/>Demonſtratio Æquilibrii Bilancis. # 282. <lb/># ibid. p. 113. <lb/>Si ſuper planum Horizontale quod im-<lb/># ponitur lineæ rectæ, quæ id dividit <lb/># in duas partes, applicetur pondus, vis, <lb/># quam illud pondus habebit ad defle-<lb/># ctendum planum partem verſus ad <lb/># quam applicatur erit major, quam ſi <lb/># poſitum ſit prope dictam lineam. # 283. <lb/>Si planum Horizontale; oneratum plu-<lb/># rimis ponderibus, maneat in æquili-<lb/># brio, impoſitum lineæ rectæ, quæ id <lb/># ſecat in duas partes, centrum gravi-<lb/># tatis plani ſic onerati erit in ipſa li-<lb/># neâ rectâ. # 284. <lb/>Duo gravia commenſurabilia appenſa ad <lb/># extremitates brachiorum Libræ erunt <lb/># in æquilibrio, ſi brachia ſint in ratio-<lb/># ne reciproca gravium. # 284. <lb/>De potentiis fila funeſve trahentibus. <lb/># 287. ibid. p. 317. <lb/>Datis poſitione quotlibet punctis; ſive in <lb/># eodem plano fuerint, ſive non: ſi à <lb/># puncto quod eorum commune eſt gra-<lb/># vitatis centrum, ad unumquodque <lb/># datorum fila extendantur, eaque ſin-<lb/># gula trahantur à potentiis quæ ſint <lb/># inter ſe ut filorum longitudines, fiet <lb/># æquilibrium manente nodo communi in <lb/># dicto gravitatis centro. # 288. <lb/>Solutio problematis a G G. Leibni-<lb/># tio propoſiti. # 290 Nouvelles de la <lb/># republique des lettres Menſis Oct. 1687. <lb/>Detegere lineam juxta quam ſi corpus <lb/># deſcendat temporibus æqualibus æqua-<lb/># liter tellurem verſus accedat. # 290. <lb/>Chriſtiani Hugenii, Solutio Proble-<lb/># matis de linea in quam flexile ſe <lb/># pondere proprio curvat. # 291. Acta <lb/># Erud. Lips. menſis Junii 1691. <lb/>Hugenii Annotationes in Librum Pa-<lb/># riſiis 1689. editum, de Manuaria <lb/># Nautica. # 292. Bibliotheque Univer-<lb/># ſelle. menſis Sept. 1693. <lb/>Reſponſum D<emph style="super">ni</emph>. Renaldi ad Domi-<lb/># num Hugenium. # 296. Journal des <lb/># Sçavans 16. & 27. Maji. 1695. <lb/>Exceptio D<emph style="super">ni</emph>. Hugenii ad Reſponſum <lb/># D<emph style="super">ni</emph>. Renaldi. # 305. Hiſtoire des ou-<lb/># vrages des Sçavans. menſis April. <lb/># 1694. <lb/></note> </div> <div xml:id="echoid-div528" type="section" level="1" n="207"> <head xml:id="echoid-head301" xml:space="preserve">TOMUS SECUNDUS. <lb/>OPERA GEOMETRICA.</head> <note style="it" position="right" xml:space="preserve"> <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugen@</emph> <emph style="sc">Theore-</emph> <lb/># <emph style="sc">MATA DE</emph> <emph style="sc">Quadratura</emph> <emph style="sc">Hy-</emph> <lb/># <emph style="sc">PERBOLES</emph>, <emph style="sc">Ellipsis et</emph> <emph style="sc">Circu-</emph> <lb/># <emph style="sc">LI, EX DATO</emph> <emph style="sc">Portionum gra-</emph> <lb/># <emph style="sc">VITATIS CENTRO</emph> # 309. Lugd. Bat. <lb/># 1651. in quarto. <lb/>Omnis hyperboles portio ad triangulum <lb/># inſcriptum, eandem cum ipſa Baſin <lb/># habentem eandemque altitudinem, hanc <lb/># habet rationem; quam ſubſeſquialtera <lb/># duarum, lateris tranſverſi & diametri <lb/># portionis, ad eam quæ ex centro ſe-<lb/># ctionis ducitur ad portionis centrum <lb/># gravitatis. # 323. <lb/>Omnis ellipſis vel circuli portio ad trian-<lb/># gulum inſcriptum, eandem cum ipſa <lb/></note> <pb file="0528" n="551" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># Baſin habentem eandemque altitudi-<lb/># nem, hanc habet rationem; quam <lb/># ſubſeſqu@altera diametri portionis re-<lb/># liquæ, ad eam quæ ex figuræ centro <lb/># ducitur ad centrum gravitatis in por-<lb/># tione. # 324. <lb/>In Semicirculo & quolibet circuli ſectore, <lb/># habet arcus ad duas tertias rectæ ſibi <lb/># ſubtenſæ hanc rationem, quam ſemi-<lb/># diameter ad eam, quæ ex centro du-<lb/># citur ad ſectoris centrum gravitatis. <lb/># 326. <lb/>Ε’ΞΕ´ΤΑΣΙSig; <emph style="sc">Cyclometriæ</emph> Clarif-<lb/># ſimi Viri Gregorii à S. Vincentio, <lb/># S. J. editæ Anno 1647. # 328. ibid. <lb/>Chriſtianii Hugenii ad C. V. Franc. <lb/># Xaverium Ainſcom. S. J. <emph style="sc">Episto-</emph> <lb/># <emph style="sc">L@</emph>, qua diluuntur ea quibus <emph style="sc">E</emph> ξέ-<lb/># {τα}{σι}ς Cyclometriæ Gregorii à Sto. <lb/># Vincentio impugnata fuit. # 341. Ha-<lb/># gæ. Com. 1656. in quarto. <lb/>Epiſtola Carteſii. # 347. <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugenii de</emph> <emph style="sc">Circuli</emph> <lb/># <emph style="sc">Magnitudine</emph> <emph style="sc">Inventa</emph>. Acce-<lb/># dunt ejuſdem Problematum quorun-<lb/># dam illuſtrium Conſtructiones. # 351. <lb/># Lugd. Bat. 1654 in quarto. <lb/>Omnis circulus major eſt polygono æqua-<lb/># lium laterum ſibi inſcripto & triente <lb/># exceſſus quo id polygonum ſuperat a-<lb/># liud inſcriptum ſubduplo laterum nu-<lb/># mero. # 361. <lb/>Omnis circulus minor eſt duabus tertiis <lb/># polygoni æqualium laterum ſibi cir-<lb/># cumſcripti & triente polygoni ſimilis <lb/># inſcripti. # 361. <lb/>Omnis circuli circumferentia major eſt <lb/># perimetro polygoni æqualium laterum <lb/># ſibi inſcripti, & triente exceſſus quo <lb/># perimeter eadem ſuper at perimetrum <lb/># alterius polygoni inſcripti ſubduplo la-<lb/># terum numero. # 362. <lb/>Circulo dato, ſi ad diametri terminum <lb/># contingens ducatur, ducatur autem & <lb/># ab oppoſito diametri termino quæ cir-<lb/># cumferentiam ſecet occurratque tan-<lb/># genti ductæ: erunt interceptæ tangen-<lb/># tis duæ tertiæ cum triente ejus quæ ab <lb/># interſectionis puncto diametro ad an-<lb/># gulos rectos incidet, ſimul arcu abſciſ-<lb/># ſo adjacente majores. # 364. <lb/>Omnis circuli circumferentia minor eſt <lb/># duabus tertiis perimetri polygoni æ-<lb/># qualium laterum ſibi inſcripti & trien-<lb/># te perimetri polygoni ſimilis circum-<lb/># ſcripti. # 365. <lb/>Peripheriæ ad diametrum rationem in-<lb/># venire quamlibet veræ propinquam. <lb/># 366. <lb/>Rectam ſumere peripheriæ dati circuli æ-<lb/># qualem. # 368. <lb/>Dato arcui cuicunque rectam æqualem <lb/># ſumere. # 370. <lb/>Omnis circuli circumferentia minor eſt <lb/># minore duarum mediarum proportio-<lb/># nalium inter perimetros polygonorum <lb/># ſimilium, quorum alterum ordinate <lb/># circulo inſcriptum ſit, alterum cir-<lb/># cumſcriptum. Et circulus minor eſt <lb/># polygono iſtis ſimili cujus ambitus ma-<lb/># jori mediarum æquetur. # 373. <lb/>Si inter productam circuli diametrum & <lb/># circumferentiam recta aptetur radio æ-<lb/># qualis, & producta circulum ſecet, <lb/># occurratque tangenti circulum ad alte-<lb/># rum diametri terminum: Intercipiet <lb/># ea partem tangentis arcu adjacente <lb/># abſciſſo majorem. # 376. <lb/>Si diametro circuli ſemidiameter in dire-<lb/># ctum adjiciatur, & ab adjectæ termino <lb/># recta ducatur quæ circulum ſecet, oc-<lb/># curratque tangenti circulum ad termi-<lb/></note> <pb file="0529" n="552" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># num diametri oppoſitum: Intercipiet <lb/># ea partem tangentis arcu adjacente <lb/># abſciſſo minorem. # 377. <lb/>Portionis circuli centrum gravitatis dia-<lb/># metrum portionis ita dividit, ut pars <lb/># quæ ad verticem reliquâ major ſit, <lb/># minor autem quam ejuſdem ſeſquial-<lb/># tera. # 378. <lb/>Circuli portio ſemicirculo minor ad in-<lb/># ſcriptum triangulum maximum majo-<lb/># rem rationem habet quam ſeſquiter-<lb/># tiam, minorem vero quam diameter <lb/># portionis reliquæ tripla ſeſquitertia ad <lb/># circuli diametrum cum tripla ea, quæ <lb/># à centro circuli pertingit ad portionis <lb/># baſin. # 380. <lb/>Arcus quilibet ſemicircumferentiâ mi-<lb/># nor, major eſt ſuâ ſubtenſâ ſimul & <lb/># triente differentiæ quâ ſubtenſa ſinum <lb/># excedit. Idem verò minor quam ſub-<lb/># tenſa ſimul cum ea quæ ad dictum <lb/># trientem ſeſe habeat, ut quadrupla <lb/># ſubtenſa juncta ſinui ad ſubtenſam du-<lb/># plam cum ſinu triplo. # 382. <lb/>Circumferentiæ ad diametrum rationem <lb/># inveſtigare; & ex datis inſcriptis in <lb/># dato circulo invenire longitudinem ar-<lb/># cuum quibus illæ ſubtenduntur. # 384. <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> <emph style="sc">Illvstrium</emph> <lb/># <emph style="sc">QVORUNDAM</emph> <emph style="sc">Problematum</emph> <emph style="sc">Con-</emph> <lb/># <emph style="sc">STRUCTIONES</emph>. # 388 Ibid. <lb/>Datam ſphæram plano ſecare, ut por-<lb/># tiones inter ſerationem habeant datam <lb/># 388. <lb/>Cubum invenire dati cubi duplum. # 391. <lb/>Datis duabus rectis duas medias propor-<lb/># tionales invenire. # 393 <lb/>Quadrato dat@ & uno latere producto, <lb/># aptare ſub angulo exteriori rectum <lb/># magnitudine datam quæ ad angulum <lb/># @ppoſitum pertineat. # 396. <lb/>Dato quadrato, & duobus contiguis la-<lb/># teribus productis, aptare ſub angulo <lb/># interiori rectam magnitudine datam <lb/># quæ per angulum oppoſitum tranſeat. <lb/># Oportet autem non minorem eſſe da-<lb/># tam quam ſit quadrati diameter du-<lb/># pla. # 397. <lb/>Rhombo dato, & uno latere producto, <lb/># aptare ſub angulo exteriori lineam <lb/># magnitudine datam quæ ad oppoſitum <lb/># angulum pertineat. # 397. <lb/>Rhombo dato & duobus contiguis lateri-<lb/># bus productis, aptare ſub angulo in-<lb/># teriori rectam magnitudine datam quæ <lb/># per oppoſitum angulum tranſeat. O-<lb/># portet autem datam non minorem eſſe <lb/># quam duplam diametri quæ reliquos <lb/># duos rhombi angulos conjungit. # 399. <lb/>In Conchoide linea invenire confinia fle-<lb/># xus contrarii<unsure/>. # 403. <lb/><emph style="sc">Controversia de</emph> <emph style="sc">Circuli et</emph> <lb/># <emph style="sc">Hyperbolæ</emph> <emph style="sc">Quadratura</emph>. # 405. <lb/>Vera Circuli & Hyperbolæ Quadra-<lb/># tura Authore Jocobo Gregorio # 407. <lb/># Patavii 1668. in quarto. <lb/>Serierum terminationes invenire. # 423. <lb/># & 426. <lb/>Ex data quantitate, eodem modo com-<lb/># poſita a duobus terminis convergen-<lb/># tibus cujuſcunque ſeriei convergentis, <lb/># quo componitur ex terminis convergen-<lb/># tibus ejuſdem ſeriei immediate ſequen-<lb/># tibus; ſeriei propoſitæ terminationem <lb/># invenire. # 427. <lb/>Sectorem circuli, ellipſeos, vel hyperbo-<lb/># læ, non eſſe compoſitum analyticè à <lb/># triangulo ad centrum inſcripto & tra-<lb/># pezio circumſcripto. # 429. <lb/>Dato circulo æquale invenire quadra-<lb/># tum. # 446. <lb/>Ex dato ſinu invenire arcum. # 448. <lb/></note> <pb file="0530" n="553" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/>Ex dato arcu invenire ſinum. # 448. <lb/>Invenire quadratum æquale ſpatio hyper-<lb/># bolico contento à curva hyperbolica, <lb/># uno aſymptoto & duabus rectis alteri <lb/># aſymptoto parallelis; quod ſpatium <lb/># æquale eſt ſectori hyperbolico cujus ba-<lb/># ſis eſt eadem curva. # 449. <lb/>Propoſiti cujuſcunque numeri logarith-<lb/># mum invenire. # 452. <lb/>Ex dato logarithmo invenire ejus nume-<lb/># rum. # 459. <lb/>Rectâ per datum punctum in diametro <lb/># ductâ, ſemicirculum in ratione data <lb/># dividere. # 460. <lb/>Hugenii Obſervationes in Librum Ja-<lb/># cobi Gregorii, de Vera Circuli & <lb/># Hyperbolæ Quadratura. # 463. four-<lb/># nal des Sçavans 2. Julii. 1668. <lb/>Invenire per logarithmos dimenſionem <lb/># Spatii Hyperbolici contenti inter Cur-<lb/># vam & unam ex Aſymptotis & duas <lb/># lineas parallelas ad alteram Aſympto-<lb/># ton. # 465. <lb/>Domini Gregorii Reſponſum ad Ani-<lb/># madverſiones Domini Hugenii, in <lb/># ejus Librum, de Vera Circuli & <lb/># Hyperbolæ Quadratura. # 466. Phi-<lb/># loſophical Tranſactions. n. 37. <lb/>Excerpta ex Literis D<emph style="super">ni</emph>. Hugenii de <lb/># Reſponſo, quod D<emph style="super">nus</emph>. Gregorius <lb/># dedit ad Examen Libri, cui Titu-<lb/># lus eſt, Vera Circuli & Hyperbo-<lb/># læ Quadratura. # 472. Journal des <lb/># Sçavans 12. Nov. 1668. <lb/>Excerpta ex Epiſtola D. Jacobi Gre-<lb/># gorii, continente quaſdam ejus <lb/># Conſiderationes, ſuper Epiſtola D. <lb/># Hugenii, impreſſa in Vindicatio-<lb/># nem Examinis ſui Libri, de Vera <lb/># Circuli & Hyperbolæ Quadratura. <lb/># 476. Philoſophical Tranſactions. n. <lb/># 44. <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> <emph style="sc">Geometrica</emph> <lb/># <emph style="sc">Varia</emph>. # 483. <lb/>Conſtructio Loci ad Hyperbolam per <lb/># Aſymptotos. # 485. Divers ouvrages <lb/># de Math. & de Ph. de M<emph style="super">rs</emph>. de l’ <lb/># Acad. R. p. 322. <lb/>Demonſtratio Regulæ de Maximis <lb/># & Minimis. # 490. ibid. p. 326. <lb/>Regula ad inveniendas Tangentes li-<lb/># nearum curvarum. # 498. ibid. p. 330. <lb/>Chriſtiani Hugenii Epiſtola de Cur-<lb/># vis quibuſdam peculiaribus. # 507. Hi-<lb/># ſtoire des ouvrages des Sçavans. men-<lb/># ſis Febr. 1693. <lb/>Invenire lineam rectam æqualem datæ <lb/># portioni lineæ logarithmicæ. # Ibid. <lb/>Alia curva cujus longitudo ope ipſius cur-<lb/># væ menſuratur. # 508. <lb/>Quadratura Hyperboles adhibita curva <lb/># illa. # 508. <lb/>Curvæ ejuſdem deſcriptio mechanica. # 509. <lb/>Varia de catinaria. # 511. <lb/>Quadratura curvæ cujus æquatio eſt <lb/>x<emph style="super">3</emph> + y<emph style="super">3</emph> = xyn. # 514. <lb/>Problema ab Eruditis ſolvendum: a <lb/># Johanne Bernoullio in Actis Li-<lb/># pſienſibus menſis Maji Ann. 1693. <lb/># propoſitum. # 515. <lb/>C. H. Z. De Problemate Bernoullia-<lb/># no in Actis Lipſienſibus propoſi-<lb/># to. # 516. Acta Erud. Lips. menſis <lb/># Oct. 1693. <lb/>C. H. Z. Conſtructio univerſalis Pro-<lb/># blematis a Clariſſimo Viro Joh. Ber-<lb/># noullio propoſiti. # 518. ibid. Sept. <lb/># 1694. <lb/></note> <pb file="0531" n="554" rhead="INDEX RERUM."/> </div> <div xml:id="echoid-div529" type="section" level="1" n="208"> <head xml:id="echoid-head302" xml:space="preserve">TOMUS TERTIUS. <lb/>OPERA ASTRONOMICA.</head> <note style="it" position="right" xml:space="preserve"> <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugenii de</emph> <emph style="sc">Satur-</emph> <lb/># <emph style="sc">NI</emph> <emph style="sc">Luna observatio nova</emph>. # 521. <lb/># Hagæ-Com. 1656. <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> <emph style="sc">Zulichemi@</emph>, <lb/># <emph style="sc">Systema</emph> <emph style="sc">Saturnium</emph>, ſive de Cau-<lb/># ſis mirandorum Saturni Phænome-<lb/># non; & Comite ejus Planeta novo. <lb/># 527. Hagæ Com. 1659. <lb/>Teleſcoporium noſtrorum Deſcriptio. # 537. <lb/>Quædam circa planetas & fixas obſer-<lb/># vata. # 539. <lb/>In Jovis diſco zonæ candicantes. # 539. <lb/>In Marte zona obſcura. # 540. <lb/>Fixarum diametri nulla latitudine. # 540. <lb/>Phænomenon in Orione novum. # 540. <lb/>Circa Saturnum obſervationes. # 541. <lb/>Saturnus brachiorum expers inventus. <lb/># 544. <lb/>Brachia Saturno renata. # 545. <lb/>Brachia Saturni in anſas mutari cœpta. <lb/># 546. <lb/>Anſæ Saturni amplius patefactæ. # 547. <lb/>Comes infra Saturnum tranſire viſus <lb/># 548. <lb/>Idem ſupra Saturnum tranſiens. # 548. <lb/>Lunæ Saturniæ periodus. # 548. <lb/>Eadem periodus accuratius ſupputata. <lb/># 549. <lb/>Menſis Saturnicolarum vera longitudo. <lb/># 551. <lb/>Tabula motus æqualis Lunæ Saturniæ <lb/># in orbita ſua reſpectu fixarum. # 552. <lb/>Locus Lunæ Saturniæ quomodo ſuppute-<lb/># tur. # 553. <lb/>Non eſſe alteram anſarum quandoque al-<lb/># tera minorem. # 559. <lb/>Non etiam unam quam alteram citius <lb/># ad medium Saturni Corpus applicari. <lb/># 559. <lb/>Hevelii hypotheſis circa anſas examina-<lb/># tur. # 560. <lb/>Cauſæ phænomenôn Saturni à Roberval-<lb/># lio excogitatæ. # 561. <lb/>Hodiernæ Siculi circa eadem phænomena <lb/># opinio. # 562. <lb/>Noſtra hypotheſis qua ratione excogita-<lb/># ta. # 564. <lb/>Annulo Saturnum cingi. # 565. <lb/>Eum ad Eclipticæ planum obliquè poſi-<lb/># tum eſſe. # 565. <lb/>Occurritur iis quæ de annulo objici poſſent. <lb/># 566. <lb/>Saturnii major diameter Æquatori pa-<lb/># rallela oſtenditur. # 567. <lb/>Hevelii de inclinatione anſarum contra-<lb/># ria opinio. # 569. <lb/>Ricciolus ſibi contrarius Lib. 7. Almag. <lb/># novi. # 570. <lb/>Ulterior hypotheſeos noſtræ explanatio. <lb/># 570. <lb/>Cauſa phaſeos anſarum latiſſimarum. <lb/># 571. <lb/>Clarior Saturnus propter anſas cernitur. <lb/># 572. <lb/>Lunæ Saturniæ motus apparens ellipti-<lb/># cus. # 572. <lb/>Quomodo anſarum latitudo contrahatur. <lb/># 573. <lb/>Quomodo patul@ videri anſæ deſinant. <lb/># 573. <lb/>Majorem Saturni diametrum neceſſario <lb/># Æquatori parallelam ſemper videri, <lb/># 574. <lb/></note> <pb file="0532" n="555" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/>Rotundæ phaſeos cauſæ. # 575. <lb/>Quid ſit in Saturni diſco Zona nigri-<lb/># cans. # 576. <lb/>Saturnus quando neceſſario rotundus ſpe-<lb/># ctari debeat. # 578. <lb/>Quænam Saturnicolis ſint loca æquino-<lb/># ctiorum. # 580. <lb/>Saturnus bra@hiorum expers futurus quo <lb/># modo cognoſcatur. # 585. <lb/>Quibus temporibus rotunda phaſis rever-<lb/># ſura ſit prædicitur. # 587. <lb/>Quando latiſſimæ omnium anſæ appari-<lb/># turæ ſint. # 588. <lb/>De Saturni magnitudine & à terris di-<lb/># ſtantia. # 589. <lb/>Planetarum diametri. # 589. & ſeq. <lb/>Diametri Solis ad Planetarum diame-<lb/># tros ratio. # 591. <lb/>Ratio obſervandi Planetarum diametros <lb/># apparentes accuratiſſima. # 593. <lb/>Diametri Veneris obſervatio. # 594. <lb/><emph style="sc">Eustachi@ de divinis</emph> <emph style="sc">Septempe-</emph> <lb/># <emph style="sc">DANI</emph> <emph style="sc">Brevis</emph> <emph style="sc">Annotatio in</emph> <emph style="sc">Sy-</emph> <lb/># <emph style="sc">STEMA</emph> <emph style="sc">Saturnium</emph> <emph style="sc">Christiani</emph> <lb/># <emph style="sc">Hugenii</emph> ad Sereniſſimum Princi-<lb/># pem Leopoldum Magni Ducis He-<lb/># truriæ fratrem. # 597. Hagæ Com. 1660. <lb/>Honorati Fabri, S. J. de Saturno Sen-<lb/># tentia. # 610. <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> <emph style="sc">Zulichemii</emph> <lb/># <emph style="sc">Brevis</emph> <emph style="sc">Assertio</emph> <emph style="sc">Systematis</emph> <lb/># <emph style="sc">Saturni@ sui</emph>, ad Sereniſſimum <lb/># Principem Leopoldum ab Hetru-<lb/># ria. # 619. Hagæ@Com. 1660. <lb/>Fabriani Syſtematis examen. # 632. <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugeni@ de</emph> <emph style="sc">Saturni</emph> <lb/># <emph style="sc">Annulo observationes</emph>. # 635. <lb/>Obſervationes in Saturnum Pariſiis <lb/># habitæ in Bibliotheca Regia. # 637. <lb/># Journal des Sçavans 11. Febr. 1669. <lb/>Vera annuli inclinatio. # 637. <lb/>Phaſis rotunda. # 638. <lb/>Excerpta ex literis D. Hugenii, Aca-<lb/># demiæ Regiæ Scientiarum Socii, ad <lb/># Auctorem Diarii Eruditorum de fi-<lb/># gura Planetæ Saturni. # 638. ibid. 12. <lb/># Dec. 1672. <lb/>Brachia Saturno renata. # 638. <lb/>Umbra in diſco Saturni. # 638. <lb/>Saturnus rotundus. # 639. <lb/>Linea æquinoctialis annuli. # 640. <lb/><emph style="sc">Christiani</emph> <emph style="sc">Hugenii</emph> ΚΟΣΜΟΘΕΩ-<lb/># ΡΟΣ, <emph style="sc">SIVE DE</emph> <emph style="sc">Terris</emph> <emph style="sc">Coelesti-</emph> <lb/># <emph style="sc">BUS, EARUMQUE ORNATU</emph>, <emph style="sc">Con-</emph> <lb/># <emph style="sc">JECTURÆ</emph>. # 641. Hagæ@Com. 1698. <lb/>Fuiſſe qui Planetis incolas tribuerint, <lb/># ſed nihil præterea de iis inquiſiviſſe. <lb/># 646. <lb/>Conjecturas haſce S. Scripturis non ad-<lb/># verſari. # 647. <lb/>Conjecturas non eſſe vanas, quia non <lb/># plane certæ. # 649. <lb/>Ad ſapientiam & pietatem facere quæ <lb/># hic tractantur. # 649. <lb/>Copernici Syſtema exponitur. # 650. <lb/>Copernici doctrinam quæ rationes confir-<lb/># ment. # 650. <lb/>Planetarum magnitudinis inter ſe & ad <lb/># Solem ratio. # 651. <lb/>Micrometris præſtare lamellas virgulaſ-<lb/># ve tenues. # 652. <lb/>Tellurem Planetis, & hos Telluri recte <lb/># aſſimilari. # 652. <lb/>Ex ſimilitudine in hiſce recte argumen-<lb/># ta peti # 652. <lb/>Planetas ſolidos eſſe & gravitate polle-<lb/># re. # 653. <lb/>Nec deeſſe illis animalia. # 653. <lb/>Ut nec plantas. # 654. <lb/>Non nimiam in hiſce fingendam diſſimi-<lb/># litudinem. # 655. <lb/>Aquas a Planetis non abeſſe. # 656. <lb/>Noſtræ tamen non prorſus ſimiles. # 657. <lb/></note> <pb file="0533" n="556" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/>Nec alia ratione illic neſci & propagari <lb/># ſtirpes quam apud nos. # 657. <lb/>Idem & de animalibus verum eſſe. # 658. <lb/>Summam animalium varietatem eſſe <lb/>in Planetis. # 660. <lb/>Idem in ſtirpibus locum habere. # 660. <lb/>In Planetis eſſe animantia, quæ ratione <lb/># utantur. # 661. <lb/>Non obſtare hominum vitia quo minus <lb/># decorem terræ concilient. # 662. <lb/>Nec rationem in Planetarum incolis à <lb/># noſtra diverſam eſſe. # 663. <lb/>Nec deeſſe illis ſenſus. # 664. & ſeq. <lb/>Nec horam ſenſus longe alios eſſe ac no-<lb/># ſtratium. # 667. <lb/>Ignem quoque Planetis communem eſſe. <lb/># 669. <lb/>Magnitudinem corporum in Planetis exi-<lb/># ſtentium ex Planetarum magnitudine <lb/># non recte conjici. # 670. <lb/>In Planetis ut in Terra varia eſſe ani-<lb/># malia quibus ratio competat. # 670. <lb/>Et inter ea Hominibus ſimilia. # 670. <lb/>Planetarum incolas ſcientias excolere, & <lb/># inter eas, Aſtronomiam. # 672. <lb/>Et quæ ei inſerviunt artes mechanicas. <lb/># 673. <lb/>Ut & Geometriam, Arithmeticam. # 674. <lb/>Et ſcribendi artem. # 674. <lb/>Opticam. # 674. <lb/>Has ſcientias homini præter naturam <lb/># non eſſe. # 675. <lb/>Planeticolas manus habere. # 676. <lb/>Et pedes. # 677. <lb/>Erectos oculos, vultumque. # 677. <lb/>Nec tamen hinc ſequi eorum formam no-<lb/># ſtræ plane ſimilem. # 678. <lb/>Planeticolas nobis vel æquales vel majo-<lb/># res eſſe. # 679. <lb/>Eos in ſocietate vivere. # 680. <lb/>Colloquiorum jucunditate frui. # 680. <lb/>Domos adverſus pluviam exſtruere. # 681. <lb/>Navigare, adeoque & artes, quæ eo fa-<lb/># ciunt excolere. # 682. <lb/>Muſicam. # 683. <lb/>Quæ tamen à noſtra diverſa eſſe poſſet. <lb/># 684. <lb/>Cur conſonantia diapente poſt aliam ſi-<lb/># milem vitioſe ponatur? # 685. <lb/>Demonſtratio temperamenti in tono vocis <lb/># adhibendi. # 685. <lb/>Recenſentur commoda quæ ad nos perve-<lb/># niunt ex animalibus, herbis, arbori-<lb/># bus. # 686. <lb/>Ex Metallis. # 688. <lb/>Ex aqua, & aëre, variiſque artificiis. <lb/># 689. <lb/>Ex iis, quæ noſtra ætate inventa ſunt. <lb/># 689. <lb/>Illa omnia veriſimiliter non extare in <lb/># Planetis, ſed aliis æque dignis repen-<lb/># di. # 690. <lb/>Chriſtiani Hugenii Coſmotheoros, <lb/># Liber II. # 691. <lb/>Kircheri iter extaticum examinatur. <lb/># 691. <lb/>Apparens qualis ſit conſtitutio Solis & <lb/># Planetarum in Mercurio. # 694. <lb/>Qualis in Venere. # 695. <lb/>In Marte. # 696. <lb/>Jovem & Saturnum reliquis Planetis <lb/># longe præſtare, tam magnitudine, <lb/># quam Lunarum multitudine. # 697. <lb/>Proportio diametrorum tum Jovis, tum <lb/># orbitarum ſatellitum ejus, ad orbitam <lb/># Lunæ circa Terram. # 698. <lb/>Tempora periodorum comitum fovialium. <lb/># 699. <lb/>Quænam ſit Solis apparens magnitudo <lb/># & lux in Jove, & quî cognoſci queat. <lb/># 700. <lb/>Itidem in Saturno. # 700. <lb/></note> <pb file="0534" n="557" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/>In Jove dies eſſe horarum quinque. # 700. <lb/>Et perpetuum æquinoctium. # 701. <lb/>Planeticolis ſtellas fixas eodem modo ap-<lb/># parere, ac nobis. # 702. <lb/>Qualis ſit Planetarum aſpectus, dierum <lb/># ratio in Saturno. # 702. <lb/>Qualis ſit Annuli aſpectus in Saturno. <lb/># 703. <lb/>De Luna pauciora conjici poſſe. # 704. <lb/>Satellitum Saturni & Jovis eandem ac <lb/># Lunæ rationem eſſe. # 705. <lb/>Lunam montibus & vallibus diſtinctam <lb/># eſſe. # 705. <lb/>Carere vero mari. # 705. <lb/>Fluviis. # 706. <lb/>Nubibus. # 706. <lb/>Aëre & aqua. # 706. <lb/>Hinc de animantibus & ſtirpibus incer-<lb/># tiorem conjecturam eſſe. # 707. <lb/>Jovis ac Saturni Lunas, non ſecus ac <lb/># noſtram Telluri, eandem partem ſuo <lb/># Planetæ obvertere. # 707. <lb/>Lunæ incolis ſi qui ſint, qualis appari-<lb/># tura ſit cælorum conſtitutio, dierum <lb/># ratio, & c. # 708. <lb/>Quod ad Jovis & Saturni Lunas facile <lb/># transferre eſt. # 709. <lb/>Solaris mundi ſecundum veram propor-<lb/># tionem deſcriptio. # 710. <lb/>Immenſitas intervallorum inter Solem <lb/># & Planetas illuſtratur comparatio-<lb/># ne cum motu globi è tormento emiſſi. <lb/># 711. <lb/>In Sole omnem conjecturam deficere. <lb/># 712. <lb/>Faculas Solares incertas videri. # 712. <lb/>Propter calorem nulla illic vivere corpo-<lb/># ra noſtris ſimilia. # 712. <lb/>Stellas fixas totidem eſſe Soles. # 713. <lb/>Eas ſpargi per vaſta cæli ſpatia, & a-<lb/># lias ab aliis, ut proximas à Sole re-<lb/># moveri. # 714. <lb/>Nec Solem præ cæteris eminere contra <lb/># Keplerum notatur. # 714. <lb/>Nihil impedire, quo minus credamus cir-<lb/># ca unamquamque ex fixis, ut circa <lb/># Solem, eſſe Planetas. # 715. <lb/>Modus probabiliter inveſtigandi diſtan-<lb/># tiam fixarum à Sole. # 717. <lb/>Unumquemque Solem vortice cingi, ſed <lb/># Carteſianis multum diſſimili; contra <lb/># quem pluribus diſputatur. # 719. <lb/></note> </div> <div xml:id="echoid-div530" type="section" level="1" n="209"> <head xml:id="echoid-head303" xml:space="preserve">TOMUS QUARTUS <lb/>OPERA MISCELLANEA.</head> <note style="it" position="right" xml:space="preserve"> <lb/><emph style="sc">De</emph> <emph style="sc">Ratiociniis in</emph> <emph style="sc">Luddo</emph> A-<lb/># <emph style="sc">LEÆ</emph> # 723. Ad calcem Exerci-<lb/># tationum Math. Schotenii. <lb/>Si a vel b expectem, quorum utrumvis <lb/># æquè facilè mibi obtingere poſſit, ex-<lb/># pectatio mea dicenda eſt valere {a+b/2}. <lb/># 728. <lb/>Si a, b, vel c, expectem, quorum unum-<lb/># quodque pari facilitate mibi obtingere <lb/># poſſit, expectatio mea æſtimanda eſt <lb/># {a+b+c/3}. # 729. <lb/>Si numerus caſuum, quibus mibi eve-<lb/># niet a, ſit p; numerus autem caſuum <lb/># quibus mihi eveniet b, ſit q; ſumendo <lb/># omnes caſus æquè in proclivi eſſe: ex-<lb/># pectatio mea valebit {pa+bq/p+q}. # 730. <lb/>De facienda diſtributione inter diverſos <lb/></note> <pb file="0535" n="558" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># colluſores, quando eorum ſortes inæ-<lb/># quales ſunt. # 731. <lb/>De facienda diſtributione, ſi unus mibi <lb/># deficiat luſus & colluſorimeotres luſus. <lb/># 732. <lb/>Item, ſi mihi deficiant duo luſus & col-<lb/># luſori meo tres luſus. # 733. <lb/>Item, ſi mihi deficiant duo luſus, & col-<lb/># luſori meo quatuor. # 733. <lb/>Item, poſitis tribus colluſoribus, quorum <lb/># primo ut & ſecundo unus luſus defi-<lb/># ciat, ſed tertio duo luſus. # 734. <lb/>Item, poſitistot colluſoribus, quot quis vo-<lb/># luerit, ex quibus uni plures & alii <lb/># pauciores luſus deficiunt. # 735. <lb/>Tabula pro 3 colluſoribus. # 736. <lb/>Invenire, quot vicibus ſuſcipere quis poſ-<lb/># ſit, ut unâ teſſerâ 6 puncta jaciat. <lb/># 738. <lb/>Invenire, quot vicibus ſuſcipere quis poſ-<lb/># ſit, ut duabus teſſeris 12 puncta ja-<lb/># ciat. # 739. <lb/>Invenire quot teſſeris ſuſcipere @quis poſ-<lb/># ſit, ut primâ vice duos ſenarios jaciat. <lb/># 740. <lb/>Si cum alio ludam duabus teſſeris unum <lb/># ſolummodo jactum, hâc conditione, ut, <lb/># ſi ſeptenarius eveniat, ego vincam; <lb/># at ille, ſi denarius obtingat; ſi vero <lb/># quidquam aliud accidat, ut tum id <lb/># quod depoſitum eſt æqualiter divida-<lb/># mus: Invenire qualis iſtius pars cui-<lb/># que noſtrum debeatur. # 741. <lb/>Si ego & alius duabus teſſeris alternatim <lb/># jaciamus, hâc conditione, ut ego vin-<lb/># cam ſimul atque ſeptenarium jaciam, <lb/># ille vero quam primum ſenarium ja-<lb/># ciat; ita videlicet, ut ipſi primum <lb/># jactum concedam: invenire rationem <lb/># meæ ad ipſius ſortem. # 742. <lb/>Coronidis loco ſubjungantur V. Pro-<lb/># blemata. # 743. <lb/><emph style="sc">Novus</emph> <emph style="sc">Cyclus</emph> <emph style="sc">Harmonicus</emph>. # 745. <lb/># Hiſtoire des ouvrages des Sçavans. <lb/># Oct. # 1691. <lb/>Tabula Diviſionis octavæ in partes 31. <lb/># æquales, & diviſionis Octavæ ſe-<lb/># cundum Temperamentum vulgare@ <lb/># 752. <lb/><emph style="sc">Varia de</emph> <emph style="sc">Optica</emph>. # 755. <lb/>Excerpta ex literis D<emph style="super">ni</emph>. Hugenii, A-<lb/># cademiæ Regiæ Scientiarum Socii, <lb/># ad Autorem Diarii Eruditorum de <lb/># Catoptrico conſpicillo D<emph style="super">ni</emph>. Newto-<lb/># ni. # 757. Journal des Sçavans 29. <lb/># Febr. 1672. <lb/>Conſtructio Problematis optici. Pro-<lb/># poſitio 39 Libri V. Alhazeni, & <lb/># 22. Lib. VI. Vitellionis. # 759. <lb/>Datis in eodem plano duobus punctis & cir-<lb/># culo; inveniendum eſt punctum in pe-<lb/># ripheria circuli, ita ut lineæ ex pun-<lb/># cto invento ad puncta data faciant cum <lb/># radio circuli, per punctum inventum <lb/># ductum, angulos inter ſe æquales. # 759. <lb/># Ouvrages de M<emph style="super">rs</emph>. de l’Acad. Roj. <lb/># pag. # 376. <lb/>Aliter. Dato Speculo Cavo aut Conve-<lb/># xo, itemque Oculo & Puncto Rei vi-<lb/># ſæ, invenire Punctum Reflexionis. <lb/># 760. Philoſophical Tranſactions. <lb/># n. 97. <lb/>Compendium. # 761. Ibid. n. 98. <lb/>Alia ſolutio. # 761. <lb/>Problema Alhazeni.] Dato Circulo, & <lb/># punctis duobus, invenire punctum in <lb/># Circumferentia Circuli dati, unde du-<lb/># ctæ lineæ ad puncta data faciant ad <lb/># Circumferentiam Angulos æquales. <lb/># 761. Ibid. n. 98. <lb/>Excerpta ex litteris D<emph style="super">ni</emph>. Hugenii, A-<lb/># cad. Reg. Scient. Socii, ad Aucto-<lb/></note> <pb file="0536" n="559" rhead="INDEX RERUM."/> <note style="it" position="right" xml:space="preserve"> <lb/># rem Diarii Paris. de novo Micro-<lb/># ſcopio ex Hollandia allato. # 764. <lb/># Journal des Sçavans. 15. Aug. 1678. <lb/><emph style="sc">Experimenta</emph> <emph style="sc">Physica</emph>. # 767. <lb/>Excerpta ex literis D<emph style="super">ni</emph>. Hugenii, A-<lb/># cademiæ Regiæ Scientiarum So-<lb/># cii, ad Auctorem Diarii Erudito-<lb/># rum, de Phænomenis aquæ aëre <lb/># purgatæ. # 769. ibid. 25. Jull. 1672. <lb/>Experimentum I. Aqua, ſublatâ aë-<lb/># ris preſſione, bæret in tubo. # 770. <lb/>Experimentum II. Notabile quid in de-<lb/># ſcenſu aquæ aëre purgatæ. # 771. <lb/>Experimentum III. Adhibito ſpiritu vi-<lb/># ni loco aquæ. # 772. <lb/>Experimentum IV. Aër, ex Spiritu <lb/># vini aut aqua exhauſtus, hæc corpora <lb/># iterum intrat. # 773. <lb/>Experimentum V. Laminæ metallicæ <lb/># arcte inter ſe cohærent in vacuo licet <lb/># nihil inter has detur. # 775. <lb/>Experimentum VI. Effectus Siphonis <lb/># in vacuo. # 776. <lb/></note> </div> <div xml:id="echoid-div531" type="section" level="1" n="210"> <head xml:id="echoid-head304" xml:space="preserve">FINIS.</head> <figure> <image file="0536-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0536-01"/> </figure> <pb file="0537" n="560"/> </div> <div xml:id="echoid-div532" type="section" level="1" n="211"> <head xml:id="echoid-head305" xml:space="preserve">CATALOGUS <lb/>QUORUNDAM LIBRORUM, <lb/>Qui apud <emph style="sc">Janssonios</emph> <emph style="sc">Van der</emph> <emph style="sc">Aa</emph>, Bibliopolas <lb/>Lugduni Batavorum, venales proſtant.</head> <p style="it"> <s xml:id="echoid-s10995" xml:space="preserve">Abregé de l’ Hiſtoire generale des Turcs, par Vanel, 12°. </s> <s xml:id="echoid-s10996" xml:space="preserve">4 voll. </s> <s xml:id="echoid-s10997" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s10998" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10999" xml:space="preserve">Admiranda rerum admirabilium Encomia, 12°.</s> <s xml:id="echoid-s11000" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11001" xml:space="preserve">Æſchinis Socratici Dialogi tres, per Horreum, 8°. </s> <s xml:id="echoid-s11002" xml:space="preserve">Gr. </s> <s xml:id="echoid-s11003" xml:space="preserve">& </s> <s xml:id="echoid-s11004" xml:space="preserve">Lat.</s> <s xml:id="echoid-s11005" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11006" xml:space="preserve">Alaric ou Rome vaincuë, Poëme Heroïque, par Monſr. </s> <s xml:id="echoid-s11007" xml:space="preserve">de Scudery 12°. </s> <s xml:id="echoid-s11008" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s11009" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11010" xml:space="preserve"><emph style="sc">Amintas</emph> <emph style="sc">Herderspel van</emph> T. </s> <s xml:id="echoid-s11011" xml:space="preserve"><emph style="sc">Tasso, door</emph> <emph style="sc">Wellekens</emph>, 8°.</s> <s xml:id="echoid-s11012" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11013" xml:space="preserve">Andry de la Generation des Vers dans le Corps de l’ Homme 12°.</s> <s xml:id="echoid-s11014" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11015" xml:space="preserve">Annales de la Cour & </s> <s xml:id="echoid-s11016" xml:space="preserve">de Paris, pour les Annees 1697 & </s> <s xml:id="echoid-s11017" xml:space="preserve">1698. </s> <s xml:id="echoid-s11018" xml:space="preserve">12°.</s> <s xml:id="echoid-s11019" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11020" xml:space="preserve">Annus Sacer Poëticus, ſive ſelecta de Divis Cœleſtibus Epigrammata 12°. </s> <s xml:id="echoid-s11021" xml:space="preserve">2 <lb/>Tomis.</s> <s xml:id="echoid-s11022" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11023" xml:space="preserve">Antonii Jus feudale, cum additamentis Strykii, 4°.</s> <s xml:id="echoid-s11024" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11025" xml:space="preserve">Apologie pour leurs Sereniſſ. </s> <s xml:id="echoid-s11026" xml:space="preserve">Majeſtés Britanniques, contre un infame Libelle in-<lb/>tituté le vray Portrait de Guillaume Henry de Nauſſau, nouvel Abſçalom, nou-<lb/>vel Herode, nouveau Cromwel, nouveau Neron, 12°.</s> <s xml:id="echoid-s11027" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11028" xml:space="preserve">— pour les Proteſtans, ou l’ Auteur juſtifié pleinement leur conduite & </s> <s xml:id="echoid-s11029" xml:space="preserve">leur ſe-<lb/>paration de la Communion de Rome. </s> <s xml:id="echoid-s11030" xml:space="preserve">12°.</s> <s xml:id="echoid-s11031" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11032" xml:space="preserve">Athanaſii Opera, Gr. </s> <s xml:id="echoid-s11033" xml:space="preserve">Lat. </s> <s xml:id="echoid-s11034" xml:space="preserve">fol. </s> <s xml:id="echoid-s11035" xml:space="preserve">2 Voll.</s> <s xml:id="echoid-s11036" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11037" xml:space="preserve">Aurea Bulla Caroli V. </s> <s xml:id="echoid-s11038" xml:space="preserve">Imp. </s> <s xml:id="echoid-s11039" xml:space="preserve">de jure & </s> <s xml:id="echoid-s11040" xml:space="preserve">ordine ſuccedendi in ducatum Medio-<lb/>lanenſem, &</s> <s xml:id="echoid-s11041" xml:space="preserve">c. </s> <s xml:id="echoid-s11042" xml:space="preserve">4°.</s> <s xml:id="echoid-s11043" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11044" xml:space="preserve">Bauhuſii & </s> <s xml:id="echoid-s11045" xml:space="preserve">Cabillavi Epigrammata, 24°.</s> <s xml:id="echoid-s11046" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11047" xml:space="preserve">Bellefontaine Medicine Dogmatique Mechanique, avec la Pharmacopée Rationelle, <lb/>12°. </s> <s xml:id="echoid-s11048" xml:space="preserve">2 voll.</s> <s xml:id="echoid-s11049" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11050" xml:space="preserve"><emph style="sc">Belloste</emph> <emph style="sc">Chirurgien d</emph>‘<emph style="sc">Hôpital</emph>, 12°. </s> <s xml:id="echoid-s11051" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s11052" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11053" xml:space="preserve">Biblia Græca LXX. </s> <s xml:id="echoid-s11054" xml:space="preserve">Interpretum, curâ L. </s> <s xml:id="echoid-s11055" xml:space="preserve">Bos, 4°.</s> <s xml:id="echoid-s11056" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11057" xml:space="preserve">Bidermanni Acroamatum Academicorum Libri III. </s> <s xml:id="echoid-s11058" xml:space="preserve">12°.</s> <s xml:id="echoid-s11059" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11060" xml:space="preserve">Biſſchop Chorus Muſarum, ſive Poëmata, 8°.</s> <s xml:id="echoid-s11061" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11062" xml:space="preserve">Blancardi Opera Medica, Theoretica, Practica & </s> <s xml:id="echoid-s11063" xml:space="preserve">Chirurgica, 4°.</s> <s xml:id="echoid-s11064" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11065" xml:space="preserve">Blegny Art de guerir les Maladies Veneriennes, 12°. </s> <s xml:id="echoid-s11066" xml:space="preserve">3 Tomes avec fig.</s> <s xml:id="echoid-s11067" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11068" xml:space="preserve">Boleſlavii Homo bene moriens 12°.</s> <s xml:id="echoid-s11069" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11070" xml:space="preserve">— Gemmeum monile Animæ Chriſtianæ, 12°.</s> <s xml:id="echoid-s11071" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11072" xml:space="preserve">Bontekoe Metaphyſica, OEconomia Animalis & </s> <s xml:id="echoid-s11073" xml:space="preserve">Geulinx Phyſica vera, 8°.</s> <s xml:id="echoid-s11074" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11075" xml:space="preserve">Borgeſii Enodatio Juris Naturæ, 8°.</s> <s xml:id="echoid-s11076" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11077" xml:space="preserve">Breſmal Circulation des Eaux ou l’ Hydrographie des Minerales d’ Aix & </s> <s xml:id="echoid-s11078" xml:space="preserve">de Spa, 8°.</s> <s xml:id="echoid-s11079" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11080" xml:space="preserve">Bruguier Reponſe au Livre intitulé le Renverſement de la Morale de Jeſu Chriſt, <lb/>par les Erreurs des Calveniſtes. </s> <s xml:id="echoid-s11081" xml:space="preserve">12°.</s> <s xml:id="echoid-s11082" xml:space="preserve"/> </p> <pb file="0538" n="561" rhead="CATALOGUS LIBRORUM."/> <p> <s xml:id="echoid-s11083" xml:space="preserve">Burgundius ad Conſuetudines Flandriæ aliarumque Gentium, 12°.</s> <s xml:id="echoid-s11084" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11085" xml:space="preserve">Camerarii Symbola ac Emblemata Ethico Politica, 8°.</s> <s xml:id="echoid-s11086" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11087" xml:space="preserve">Catalogus Librorum Manuſcriptorum Angliæ & </s> <s xml:id="echoid-s11088" xml:space="preserve">Hiberniæ, fol.</s> <s xml:id="echoid-s11089" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11090" xml:space="preserve">Cherubin Effets de la Force de la Contiguité des Corps, 12°.</s> <s xml:id="echoid-s11091" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11092" xml:space="preserve">Choix des bons Mots, 12°.</s> <s xml:id="echoid-s11093" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11094" xml:space="preserve">Claubergii Opera Omnia Philoſophica, curâ Schalbruchii, 4°.</s> <s xml:id="echoid-s11095" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11096" xml:space="preserve">Clos de Aquis Mineralibus Galliæ & </s> <s xml:id="echoid-s11097" xml:space="preserve">de Mixtis naturalibus, 12°.</s> <s xml:id="echoid-s11098" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11099" xml:space="preserve">Crebilon Oeuvres, 12°.</s> <s xml:id="echoid-s11100" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11101" xml:space="preserve">Defen@e de la Monarchie de Sicile contre les Entrepriſes de la Cour de Rome, 12°.</s> <s xml:id="echoid-s11102" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11103" xml:space="preserve">Deſcription du Chateau de Verſailles, 12°. </s> <s xml:id="echoid-s11104" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s11105" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11106" xml:space="preserve"><emph style="sc">Divorce</emph> <emph style="sc">Celeste par</emph> <emph style="sc">Pallavicin</emph>, 12°.</s> <s xml:id="echoid-s11107" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11108" xml:space="preserve">Donkers Idea Febris petechialis, 8°.</s> <s xml:id="echoid-s11109" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11110" xml:space="preserve">Douſæ Poëmata, 8°.</s> <s xml:id="echoid-s11111" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11112" xml:space="preserve"><emph style="sc">Eeuwigduurende</emph> <emph style="sc">Liefdes</emph> <emph style="sc">Almanak</emph>, 8°.</s> <s xml:id="echoid-s11113" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11114" xml:space="preserve">Epiſtolæ Præſtantium ac Eruditorum Virorum, fol. </s> <s xml:id="echoid-s11115" xml:space="preserve">Charta Maj.</s> <s xml:id="echoid-s11116" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11117" xml:space="preserve">Flâviſſæ Poëticæ ſive Electorum Poëticorum Theſaurus Sacro-Profanus, 8°.</s> <s xml:id="echoid-s11118" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11119" xml:space="preserve">Fleetwood Inſcriptionum Antiquarum Sylloge, 8°.</s> <s xml:id="echoid-s11120" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11121" xml:space="preserve">Gaſſendi Abregé de la Philoſophie, 12°. </s> <s xml:id="echoid-s11122" xml:space="preserve">7 Voll.</s> <s xml:id="echoid-s11123" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11124" xml:space="preserve"><emph style="sc">Gobart</emph> <emph style="sc">Tractatus</emph> <emph style="sc">Philosophicus de</emph> <emph style="sc">Barometro</emph>, 12°. </s> <s xml:id="echoid-s11125" xml:space="preserve">cum fig <lb/>Guide, ou nouvelle Deſcription d’ Amſterdam, 8°.</s> <s xml:id="echoid-s11126" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11127" xml:space="preserve">Hartmanni Praxis Chymiatrica, 8°.</s> <s xml:id="echoid-s11128" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11129" xml:space="preserve">Hartſoeker Principes de Phyſique, 4°.</s> <s xml:id="echoid-s11130" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11131" xml:space="preserve"><emph style="sc">Havercamp de nummis</emph> <emph style="sc">Alexandri</emph> <emph style="sc">Magni</emph>, 4°. </s> <s xml:id="echoid-s11132" xml:space="preserve">cum fig.</s> <s xml:id="echoid-s11133" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11134" xml:space="preserve">Herbinius de Cataractis in Genere, 4°. </s> <s xml:id="echoid-s11135" xml:space="preserve">cum fig.</s> <s xml:id="echoid-s11136" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11137" xml:space="preserve">Heucheri novi Proventus Horti Medici Acad. </s> <s xml:id="echoid-s11138" xml:space="preserve">Vitembergenſis, 4°.</s> <s xml:id="echoid-s11139" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11140" xml:space="preserve">Hildani Obſervationes & </s> <s xml:id="echoid-s11141" xml:space="preserve">Epiſtolæ Chirurgico-Medicæ, 4°. </s> <s xml:id="echoid-s11142" xml:space="preserve">2 Voll.</s> <s xml:id="echoid-s11143" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11144" xml:space="preserve">Hippocratis Aphoriſmi per Almeloveen, 24°.</s> <s xml:id="echoid-s11145" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11146" xml:space="preserve">— Metrica paraphraſi tranſlati ab H. </s> <s xml:id="echoid-s11147" xml:space="preserve">v. </s> <s xml:id="echoid-s11148" xml:space="preserve">Poot. </s> <s xml:id="echoid-s11149" xml:space="preserve">24°.</s> <s xml:id="echoid-s11150" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11151" xml:space="preserve"><emph style="sc">Histoire des</emph> <emph style="sc">Indes</emph> <emph style="sc">Orientales par</emph> M<emph style="super">r</emph>. </s> <s xml:id="echoid-s11152" xml:space="preserve"><emph style="sc">Souchu de</emph> <emph style="sc">Rennefort</emph>, 12°.</s> <s xml:id="echoid-s11153" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11154" xml:space="preserve">— des Iles Antilles de l’ Amerique 4°. </s> <s xml:id="echoid-s11155" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s11156" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11157" xml:space="preserve">Homere Iliade & </s> <s xml:id="echoid-s11158" xml:space="preserve">Odyſſee, par Mad. </s> <s xml:id="echoid-s11159" xml:space="preserve">Dacier, 12°. </s> <s xml:id="echoid-s11160" xml:space="preserve">6 Tomes, avec fig.</s> <s xml:id="echoid-s11161" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11162" xml:space="preserve">— Iliade Poëme avec un diſcours ſur Homere, par M<emph style="super">r</emph>. </s> <s xml:id="echoid-s11163" xml:space="preserve">de la Motte. </s> <s xml:id="echoid-s11164" xml:space="preserve">12°.</s> <s xml:id="echoid-s11165" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11166" xml:space="preserve">Huetius de Interpretatione, 8°.</s> <s xml:id="echoid-s11167" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11168" xml:space="preserve">Huygens Traité de la Lumiere. </s> <s xml:id="echoid-s11169" xml:space="preserve">4°.</s> <s xml:id="echoid-s11170" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11171" xml:space="preserve">Jablonski Diſquiſitio de Lingua Lycaonica, 4°.</s> <s xml:id="echoid-s11172" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11173" xml:space="preserve">Jean danſe mieux que Pierre, Pierre danſe mieux que Jean. </s> <s xml:id="echoid-s11174" xml:space="preserve">12°. </s> <s xml:id="echoid-s11175" xml:space="preserve">5 Voll.</s> <s xml:id="echoid-s11176" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11177" xml:space="preserve">Inſtruction pour reunir les Egliſes pretendues Reformees a l’Egliſe Romaine, par <lb/>M<emph style="super">r</emph>. </s> <s xml:id="echoid-s11178" xml:space="preserve">Comiere. </s> <s xml:id="echoid-s11179" xml:space="preserve">12°.</s> <s xml:id="echoid-s11180" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11181" xml:space="preserve">Jurieu Juſtification de la Morale des Reformez, contre les Accuſations de M<emph style="super">r</emph>. </s> <s xml:id="echoid-s11182" xml:space="preserve">Ar-<lb/>naud, 8° 2 Tomes.</s> <s xml:id="echoid-s11183" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11184" xml:space="preserve">Jus Auſtriacum in ſucceſſione Regnorum Hiſpaniæ vindicatum, 4°.</s> <s xml:id="echoid-s11185" xml:space="preserve"/> </p> <pb file="0539" n="562" rhead="CATALOGUS LIBRORUM."/> <p> <s xml:id="echoid-s11186" xml:space="preserve">Knibbe Manuductio ad Oratoriam Sacram, 8°. </s> <s xml:id="echoid-s11187" xml:space="preserve">2 Voll.</s> <s xml:id="echoid-s11188" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11189" xml:space="preserve">Lamy Elemens de Mathematique, ou traité de la Grandeur en General, 12°.</s> <s xml:id="echoid-s11190" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11191" xml:space="preserve">Leti Critique Hiſtorique, Politique, Morale, Economique & </s> <s xml:id="echoid-s11192" xml:space="preserve">Comique, ſur les <lb/>Loteries Anciennes & </s> <s xml:id="echoid-s11193" xml:space="preserve">Modernes, &</s> <s xml:id="echoid-s11194" xml:space="preserve">c. </s> <s xml:id="echoid-s11195" xml:space="preserve">12°.</s> <s xml:id="echoid-s11196" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11197" xml:space="preserve">Letres Provinciales par Louis Montalte 8°. </s> <s xml:id="echoid-s11198" xml:space="preserve">3 Voll.</s> <s xml:id="echoid-s11199" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11200" xml:space="preserve">— de Fr. </s> <s xml:id="echoid-s11201" xml:space="preserve">Rabelais, avec des Obſervations, 8°.</s> <s xml:id="echoid-s11202" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11203" xml:space="preserve">— du Cardinal Bentivoglio, Ital. </s> <s xml:id="echoid-s11204" xml:space="preserve">franc. </s> <s xml:id="echoid-s11205" xml:space="preserve">12°.</s> <s xml:id="echoid-s11206" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11207" xml:space="preserve">Lipenii Biblotheca Juridica, Struvii, fol.</s> <s xml:id="echoid-s11208" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11209" xml:space="preserve">Logique par M<emph style="super">rs</emph>. </s> <s xml:id="echoid-s11210" xml:space="preserve">de Port Rojal, 12°.</s> <s xml:id="echoid-s11211" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11212" xml:space="preserve">— courte & </s> <s xml:id="echoid-s11213" xml:space="preserve">facile par le S<emph style="super">r</emph>. </s> <s xml:id="echoid-s11214" xml:space="preserve">Du Boisverd. </s> <s xml:id="echoid-s11215" xml:space="preserve">8°.</s> <s xml:id="echoid-s11216" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11217" xml:space="preserve"><emph style="sc">Lyk</emph> <emph style="sc">Predikatie over de</emph> <emph style="sc">Dood van d</emph>‘Hr. </s> <s xml:id="echoid-s11218" xml:space="preserve"><emph style="sc">Mitsen, door</emph> J. </s> <s xml:id="echoid-s11219" xml:space="preserve"><emph style="sc">DE</emph> <emph style="sc">Har-</emph> <lb/><emph style="sc">TOGH</emph>, 4°.</s> <s xml:id="echoid-s11220" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11221" xml:space="preserve">Machiavelli Hiſtoria Florentina, 12°.</s> <s xml:id="echoid-s11222" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11223" xml:space="preserve">Malvezzi Opere Hiſtoriſche e Politiſche, 12°.</s> <s xml:id="echoid-s11224" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11225" xml:space="preserve">— Il Tarquino Superbo, 12°.</s> <s xml:id="echoid-s11226" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11227" xml:space="preserve">Manzini Opere, 12°.</s> <s xml:id="echoid-s11228" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11229" xml:space="preserve">— furori della Gioventu Eſercitii Rhetorici, 12°.</s> <s xml:id="echoid-s11230" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11231" xml:space="preserve">Marechal Francois, 12°.</s> <s xml:id="echoid-s11232" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11233" xml:space="preserve">Maſcampi Epitome Hiſtoriæ Civilis, 8°.</s> <s xml:id="echoid-s11234" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11235" xml:space="preserve">Medulla Juſtinianea, 12°.</s> <s xml:id="echoid-s11236" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11237" xml:space="preserve">Melanges Hiſtoriques, 12°.</s> <s xml:id="echoid-s11238" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11239" xml:space="preserve">Menken de la Charlatanerie des Savans, 8°.</s> <s xml:id="echoid-s11240" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11241" xml:space="preserve">Moïens de reunir les Proteſtans avec l’ Egliſe romaine 12°.</s> <s xml:id="echoid-s11242" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11243" xml:space="preserve">Mollenbrocçius de Arthritide, 8°.</s> <s xml:id="echoid-s11244" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11245" xml:space="preserve">Monavii Bronchotomia, quæ eſt gutturis aperiendi artificioſa ratio, 8°.</s> <s xml:id="echoid-s11246" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11247" xml:space="preserve">Mort Idea Actionis Corporum 12°.</s> <s xml:id="echoid-s11248" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11249" xml:space="preserve"><emph style="sc">Mundi</emph> <emph style="sc">Opera</emph> <emph style="sc">Omnia</emph> <emph style="sc">Medico</emph>-<emph style="sc">Physica</emph>, 8°.</s> <s xml:id="echoid-s11250" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11251" xml:space="preserve">Oeuvres de M<emph style="super">r</emph>. </s> <s xml:id="echoid-s11252" xml:space="preserve">Le Noble 12°. </s> <s xml:id="echoid-s11253" xml:space="preserve">19 Voll.</s> <s xml:id="echoid-s11254" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11255" xml:space="preserve">Oudini Commentarius de Scriptoribus Eccleſiæ Antiquis, fol. </s> <s xml:id="echoid-s11256" xml:space="preserve">3 Voll.</s> <s xml:id="echoid-s11257" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11258" xml:space="preserve">Pacii Analyſis & </s> <s xml:id="echoid-s11259" xml:space="preserve">Iſagoge 8°. </s> <s xml:id="echoid-s11260" xml:space="preserve">2 Voll.</s> <s xml:id="echoid-s11261" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11262" xml:space="preserve">Parladorii Opera Juridica, 4°. </s> <s xml:id="echoid-s11263" xml:space="preserve">2 voll.</s> <s xml:id="echoid-s11264" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11265" xml:space="preserve">Paſſe partout Galant, 12°.</s> <s xml:id="echoid-s11266" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11267" xml:space="preserve"><emph style="sc">Pastor fido</emph>, 32°. </s> <s xml:id="echoid-s11268" xml:space="preserve">Appreſſo P. </s> <s xml:id="echoid-s11269" xml:space="preserve">Mortier.</s> <s xml:id="echoid-s11270" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11271" xml:space="preserve">Petau Hiſtoire Univerſelle, 12°. </s> <s xml:id="echoid-s11272" xml:space="preserve">Paris 5 Tomes.</s> <s xml:id="echoid-s11273" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11274" xml:space="preserve">Pilpay Conſeils & </s> <s xml:id="echoid-s11275" xml:space="preserve">Maximes, 8°.</s> <s xml:id="echoid-s11276" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11277" xml:space="preserve">Plazzonus de Partibus Generationis inſervientibus, 12°.</s> <s xml:id="echoid-s11278" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11279" xml:space="preserve"><emph style="sc">Pluimers</emph> <emph style="sc">Gedichten</emph> 4°. </s> <s xml:id="echoid-s11280" xml:space="preserve"><emph style="sc">TWEEDE DEEL</emph>.</s> <s xml:id="echoid-s11281" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11282" xml:space="preserve">Pourtraits des plus Celebres Profeſſeurs de Leide, 4°.</s> <s xml:id="echoid-s11283" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11284" xml:space="preserve">Preparations pour la S<emph style="super">te</emph>. </s> <s xml:id="echoid-s11285" xml:space="preserve">Cene, avec le Voyage de Bethel, 12°.</s> <s xml:id="echoid-s11286" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11287" xml:space="preserve">Rochefoucault Reflexions ou Sentences & </s> <s xml:id="echoid-s11288" xml:space="preserve">Maximes morales, 12°.</s> <s xml:id="echoid-s11289" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11290" xml:space="preserve">Schiere Doctrina Teſtamentorum & </s> <s xml:id="echoid-s11291" xml:space="preserve">Fœderum omnium Divinorum, 4°.</s> <s xml:id="echoid-s11292" xml:space="preserve"/> </p> <pb file="0540" n="563" rhead="CATALOGUS LIBRORUM."/> <p> <s xml:id="echoid-s11293" xml:space="preserve">Senecæ & </s> <s xml:id="echoid-s11294" xml:space="preserve">Syri Mimi Sententiæ, cum Notis Diverſorum & </s> <s xml:id="echoid-s11295" xml:space="preserve">Gruteri, 8°.</s> <s xml:id="echoid-s11296" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11297" xml:space="preserve"><emph style="sc">Serre dolci</emph> <emph style="sc">Pensieri sopra la</emph> <emph style="sc">Morte</emph>, 8°.</s> <s xml:id="echoid-s11298" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11299" xml:space="preserve"><emph style="sc">Theatre et autres</emph> <emph style="sc">Poesies de</emph> <emph style="sc">Madem</emph>: </s> <s xml:id="echoid-s11300" xml:space="preserve"><emph style="sc">Barbier</emph>, 12°. </s> <s xml:id="echoid-s11301" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s11302" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11303" xml:space="preserve"><emph style="sc">Variolarum</emph> <emph style="sc">Methodus per</emph> <emph style="sc">Transplantationem</emph>, 8°.</s> <s xml:id="echoid-s11304" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11305" xml:space="preserve">Veſlingii Syntagma Anatomicum, ſive Anatomia, 4°. </s> <s xml:id="echoid-s11306" xml:space="preserve">cum fig.</s> <s xml:id="echoid-s11307" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11308" xml:space="preserve"><emph style="sc">Vieussens novum</emph> <emph style="sc">Vasorum</emph> <emph style="sc">Corporis</emph> <emph style="sc">Humani</emph> <emph style="sc">Systema</emph>, 8°.</s> <s xml:id="echoid-s11309" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11310" xml:space="preserve"><emph style="sc">Vignole</emph> <emph style="sc">Regle des cinq</emph> <emph style="sc">Ordres d</emph>‘<emph style="sc">Architecture</emph>, 4°. </s> <s xml:id="echoid-s11311" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s11312" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11313" xml:space="preserve">Voyage de Groenland, par le Pere Cordelier Pierre de Meſange, 12°. </s> <s xml:id="echoid-s11314" xml:space="preserve">2 Voll.</s> <s xml:id="echoid-s11315" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11316" xml:space="preserve">— aux Côtes de Guinée & </s> <s xml:id="echoid-s11317" xml:space="preserve">en Amerique, 12°.</s> <s xml:id="echoid-s11318" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11319" xml:space="preserve">— Le Prudent Voyageur, par L. </s> <s xml:id="echoid-s11320" xml:space="preserve">du May, 12°. </s> <s xml:id="echoid-s11321" xml:space="preserve">3 Voll.</s> <s xml:id="echoid-s11322" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11323" xml:space="preserve"><emph style="sc">Wechwyser der</emph> <emph style="sc">Eenvuldigen, door</emph> J. </s> <s xml:id="echoid-s11324" xml:space="preserve"><emph style="sc">DE</emph> <emph style="sc">Hartogh</emph>, 8°.</s> <s xml:id="echoid-s11325" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11326" xml:space="preserve">Weſtenberg ad Pandectas, 8°.</s> <s xml:id="echoid-s11327" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11328" xml:space="preserve">Dictionaire Engliſch and Frenſch & </s> <s xml:id="echoid-s11329" xml:space="preserve">Frenſch and Englich of Miege 8°.</s> <s xml:id="echoid-s11330" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11331" xml:space="preserve">— Italiaanſch en Nederduyts, door Moſes Giron, 4°.</s> <s xml:id="echoid-s11332" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11333" xml:space="preserve">— Antiquitatum Romanarum & </s> <s xml:id="echoid-s11334" xml:space="preserve">Græcar. </s> <s xml:id="echoid-s11335" xml:space="preserve">per Danetum, in Uſum Delphi-<lb/>ni, 4°.</s> <s xml:id="echoid-s11336" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11337" xml:space="preserve">Grammaire Franceiſe d’un tour nouveau, compoſée en faveur de ceux qui preferent <lb/>la Pratique a la Theorie, par M<emph style="super">r</emph>. </s> <s xml:id="echoid-s11338" xml:space="preserve">Derbaud, 12°. </s> <s xml:id="echoid-s11339" xml:space="preserve">avec fig.</s> <s xml:id="echoid-s11340" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11341" xml:space="preserve">Multi etiam alii apud Eoſdem reperiuntur.</s> <s xml:id="echoid-s11342" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div533" type="section" level="1" n="212"> <head xml:id="echoid-head306" xml:space="preserve">AVIS AU RELIEUR.</head> <p> <s xml:id="echoid-s11343" xml:space="preserve">LE Relieur prendra garde que le papier qui eſt à côté des Figures doit être <lb/>conſervé pour faire deborder les Figures hors du Livre, & </s> <s xml:id="echoid-s11344" xml:space="preserve">les placer <lb/>ſuivant la page marquée ſur chaque planche.</s> <s xml:id="echoid-s11345" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div534" type="section" level="1" n="213"> <head xml:id="echoid-head307" style="it" xml:space="preserve">BERIGT AAN DEN BOEKBINDER.</head> <p style="it"> <s xml:id="echoid-s11346" xml:space="preserve">DEn Boek-binder zy gewaarſchout het papier ter zyde de Figuren niet af te <lb/>ſnyden; </s> <s xml:id="echoid-s11347" xml:space="preserve">maar zodanig in te ſetten, dat de Figuren buyten het Boek uitſlaan: <lb/></s> <s xml:id="echoid-s11348" xml:space="preserve">dezelven moeten geplaaſt werden volgens de paginaas op yder Tabel genoteert.</s> <s xml:id="echoid-s11349" xml:space="preserve"/> </p> <pb file="0541" n="564"/> <pb file="0542" n="565"/> <pb file="0543" n="566"/> <pb file="0544" n="567"/> </div></text> </echo>