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Removing DESpecs directory which deserted to git
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
| children |
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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:BYCAB3V6.xml</dcterms:identifier> <dcterms:creator identifier="GND:118639749">Huygens, Christiaan</dcterms:creator> <dcterms:title xml:lang="la">Christiani Hugenii opera varia; Bd. 1: Opera mechanica</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> <parameters>despecs=2.0</parameters> <log>{ος} in running head makes problems during xml generation</log> </metadata> <text xml:lang="la" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">@@VM MANT, <lb/>OB MERITA</head> <pb file="0002" n="2"/> <pb file="0003" n="3"/> <handwritten/> <pb file="0004" n="4"/> <handwritten/> <pb file="0005" n="5"/> <pb file="0006" n="6"/> <figure> <description xml:id="echoid-description1" xml:space="preserve">CHRISTIANUS HUGENIUS</description> <description xml:id="echoid-description2" xml:space="preserve">natus 14 Aprilis 1629.</description> <description xml:id="echoid-description3" xml:space="preserve">denatus 8 Junii 1695.</description> <description xml:id="echoid-description4" xml:space="preserve"><emph style="sc">Lugd</emph>. <emph style="sc">Bat.</emph> Apud <emph style="sc">Janssonios</emph> <emph style="sc">Van der</emph> Aa. Bibliopolas.</description> </figure> <pb file="0007" n="7"/> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head2" xml:space="preserve"><emph style="red">CHRISTIANI HUGENII</emph></head> <head xml:id="echoid-head3" xml:space="preserve">ZULICHEMII,</head> <head xml:id="echoid-head4" style="it" xml:space="preserve">Dum viveret Zelemii Toparchæ,</head> <head xml:id="echoid-head5" xml:space="preserve"><emph style="red">OPERA VARIA.</emph></head> <head xml:id="echoid-head6" xml:space="preserve"><emph style="sc">Volumen</emph> <emph style="sc">Primum</emph>.</head> <figure> <image file="0007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0007-01"/> </figure> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head7" xml:space="preserve"><emph style="sc">Lugduni</emph> <emph style="sc">Batavorum</emph>, <lb/>Apud <emph style="red">JANSSONIOS VANDER <emph style="sc">Aa</emph>,</emph> <lb/>Bibliopolas, <emph style="sc">MDCCXXIV</emph>.</head> <handwritten/> <pb file="0008" n="8"/> <handwritten/> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head8" xml:space="preserve">MAX-PLANCK-INSTITUT <lb/>FOR WISSENSCHAFTSGESCHICHTE <lb/>Bibliothek</head> <handwritten/> <pb file="0009" n="9"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head9" xml:space="preserve">ADMONITIO BIBLIOPEGIS, <lb/>ubi locandi ſint Tituli.</head> <head xml:id="echoid-head10" xml:space="preserve">AVIS AU RELIEUR, <lb/>Pour placer les Titres ſuivant les Pages marquées ci-deſſous.</head> <head xml:id="echoid-head11" xml:space="preserve">BERIGT AAN DEN BOEKBINDER, <lb/>Om en alwaar de Titels te plaatſen.</head> <p> <s xml:id="echoid-s1" xml:space="preserve">1. </s> <s xml:id="echoid-s2" xml:space="preserve">Chriſtiani Hugenii Opera Mechanica. </s> <s xml:id="echoid-s3" xml:space="preserve">Tomus Primus. </s> <s xml:id="echoid-s4" xml:space="preserve">(ante) (de-<lb/>vant) voor Pag. </s> <s xml:id="echoid-s5" xml:space="preserve">1. </s> <s xml:id="echoid-s6" xml:space="preserve">(& </s> <s xml:id="echoid-s7" xml:space="preserve">poſt Hugenii Vitam) (apres Hugenii Vi-<lb/>ta) agter Hugenii Vita.</s> <s xml:id="echoid-s8" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9" xml:space="preserve">2. </s> <s xml:id="echoid-s10" xml:space="preserve">Chriſtiani Hugenii a Zulichem, dum viveret Zelhemi Toparchæ, <lb/>Opera Varia. </s> <s xml:id="echoid-s11" xml:space="preserve">Volumen Secundum. </s> <s xml:id="echoid-s12" xml:space="preserve">(ante)(devant) voor Pag. <lb/></s> <s xml:id="echoid-s13" xml:space="preserve">309.</s> <s xml:id="echoid-s14" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15" xml:space="preserve">3. </s> <s xml:id="echoid-s16" xml:space="preserve">Chriſtiani Hugenii Opera Geometrica. </s> <s xml:id="echoid-s17" xml:space="preserve">Tomus Secundus. </s> <s xml:id="echoid-s18" xml:space="preserve">(etiam an-<lb/>te) (auſſi devant) mede voor Pag. </s> <s xml:id="echoid-s19" xml:space="preserve">309. </s> <s xml:id="echoid-s20" xml:space="preserve">(& </s> <s xml:id="echoid-s21" xml:space="preserve">poſt Titulum <lb/>Voluminis Secundi) (& </s> <s xml:id="echoid-s22" xml:space="preserve">après le Titre de Volumen Secun-<lb/>dum) en na de Titel van Volumen Secundum.</s> <s xml:id="echoid-s23" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s24" xml:space="preserve">4. </s> <s xml:id="echoid-s25" xml:space="preserve">Chriſtiani Hugenii Opera Aſtronomica. </s> <s xml:id="echoid-s26" xml:space="preserve">Tomus Tertius. </s> <s xml:id="echoid-s27" xml:space="preserve">(ante) (de-<lb/>vant) voor Pag. </s> <s xml:id="echoid-s28" xml:space="preserve">521.</s> <s xml:id="echoid-s29" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s30" xml:space="preserve">5. </s> <s xml:id="echoid-s31" xml:space="preserve">Chriſtiani Hugenii Opera Miſcellanea. </s> <s xml:id="echoid-s32" xml:space="preserve">Tomus Quartus. </s> <s xml:id="echoid-s33" xml:space="preserve">(ante) (de-<lb/>vant) voor Pag. </s> <s xml:id="echoid-s34" xml:space="preserve">723.</s> <s xml:id="echoid-s35" xml:space="preserve"/> </p> <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-div7" type="section" level="1" n="7"> <head xml:id="echoid-head12" xml:space="preserve">G. J.’s GRAVESANDE <lb/>L. S.</head> <p style="it"> <s xml:id="echoid-s36" xml:space="preserve">ILlos de re litteraria bene mereri, ſem-<lb/>per perſuaſum habui, qui doctorum vi-<lb/>rorum ſcripta diſperſa colligunt; </s> <s xml:id="echoid-s37" xml:space="preserve">& </s> <s xml:id="echoid-s38" xml:space="preserve"><lb/>minoribus, ſeorſim perituris, in ma-<lb/>jori volumine collocatis poſteritatem <lb/>quaſi donant.</s> <s xml:id="echoid-s39" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s40" xml:space="preserve">Hac de cauſa cùm Bibliopolæ bujus Urbis, Jan-<lb/>ſonii vander Aa, mihi declararent, ſe in men-<lb/>te habere Hugenii opus, olim Pariſiis editum, de <lb/>Horologio Oſcillatorio typis mandare, & </s> <s xml:id="echoid-s41" xml:space="preserve">hac de <lb/>re quid ſentirem ego quærerent, Ipſis Auctor fui, <lb/>ut non modo in propoſito perſeverarent, ſed ut <lb/>omnia ante edita ejuſdem Auctoris opera minora buic <lb/>adjungerent.</s> <s xml:id="echoid-s42" xml:space="preserve"/> </p> <pb file="0012" n="12" rhead="PRÆFATIO."/> <p style="it"> <s xml:id="echoid-s43" xml:space="preserve">In me libenter ſuſcepi curam colligendi hæc, & </s> <s xml:id="echoid-s44" xml:space="preserve"><lb/>diſponendi; </s> <s xml:id="echoid-s45" xml:space="preserve">ſchedaſque ſemel & </s> <s xml:id="echoid-s46" xml:space="preserve">altera vice a cor-<lb/>rectore examinatas, ipſe attente relegi, & </s> <s xml:id="echoid-s47" xml:space="preserve">cum <lb/>figuris contuli; </s> <s xml:id="echoid-s48" xml:space="preserve">ſcripta etiam quædam, in Diariis <lb/>non ſub Auctoris oculis edita, ſæpe mendis variis <lb/>purgavi.</s> <s xml:id="echoid-s49" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s50" xml:space="preserve">Opera quæ hic exſtant, in quatuor diſperſi to-<lb/>mos; </s> <s xml:id="echoid-s51" xml:space="preserve">quæ Mechanicam Machinaſque ſpectant to-<lb/>mo primo continentur: </s> <s xml:id="echoid-s52" xml:space="preserve">Geometrica ſecundo: </s> <s xml:id="echoid-s53" xml:space="preserve">A-<lb/>ſtronomica tertio: </s> <s xml:id="echoid-s54" xml:space="preserve">Miſcellanea pauca quarto.</s> <s xml:id="echoid-s55" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s56" xml:space="preserve">Tractatus varios Auctor ſeparatim ediderat; </s> <s xml:id="echoid-s57" xml:space="preserve">ſcri-<lb/>pta minora multa in Diariis diſperſa dantur: </s> <s xml:id="echoid-s58" xml:space="preserve">in-<lb/>ter hæc quædam ſpectant controverſias quæ Aucto-<lb/>ri cum viris doctis intercedere, & </s> <s xml:id="echoid-s59" xml:space="preserve">quæ ad ean-<lb/>dem controverſiam pertinent, aliquando in variis <lb/>regionibus publicata exſtant, partim Gallicè conſcri-<lb/>pta, partim Latinè.</s> <s xml:id="echoid-s60" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s61" xml:space="preserve">In edendis libris, ſeparatim ab Auctore publici <lb/>juris factis, exemplaria manu ipſius recognita, & </s> <s xml:id="echoid-s62" xml:space="preserve"><lb/>in variis locis aucta, & </s> <s xml:id="echoid-s63" xml:space="preserve">emendata, ſecutus ſum; <lb/></s> <s xml:id="echoid-s64" xml:space="preserve">quod etiam referri debet ad ſcripta pauca ex iis quæ <lb/>in Diariis fuere tradita.</s> <s xml:id="echoid-s65" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s66" xml:space="preserve">Exemplaria hæc accepi à Nob. </s> <s xml:id="echoid-s67" xml:space="preserve">D<emph style="super">no</emph> Hugenio Ze-<lb/>lemii Toparchâ, qui patrui Chriſtiani opera hæc <pb file="0013" n="13" rhead="PRÆFATIO."/> cum cura ſervaverat, & </s> <s xml:id="echoid-s68" xml:space="preserve">qui libenter mibi hæc ea <lb/>lege conceſſit, ut abſoluta editione in Bibliotbeca <lb/>publica Acad. </s> <s xml:id="echoid-s69" xml:space="preserve">Lugd. </s> <s xml:id="echoid-s70" xml:space="preserve">Bat. </s> <s xml:id="echoid-s71" xml:space="preserve">jungantur cum ejuſdem <lb/>Auctoris manuſcriptis quæ ibi ſervantur.</s> <s xml:id="echoid-s72" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s73" xml:space="preserve">In Indice, ad calcem libri adjecto, monui ubi ſcri-<lb/>pta ſingula antea edita fuere. </s> <s xml:id="echoid-s74" xml:space="preserve">Tractatus ſepara-<lb/>tos Latine Auctor conſcripſit, præter Inſtitutionem <lb/>de Uſu Horologiorum <anchor type="note" xlink:href="" symbol="*"/>, quæ Belgico ſermone in lu- <anchor type="note" xlink:label="note-0013-01a" xlink:href="note-0013-01"/> cem prodiit. </s> <s xml:id="echoid-s75" xml:space="preserve">Ex Gallicis Diariis excerpta ſcripta <lb/>Gallico ſermone conſcripta dantur. </s> <s xml:id="echoid-s76" xml:space="preserve">Illorum autem <lb/>quæ inter opera Acad. </s> <s xml:id="echoid-s77" xml:space="preserve">Reg. </s> <s xml:id="echoid-s78" xml:space="preserve">Paris. </s> <s xml:id="echoid-s79" xml:space="preserve">Artium & </s> <s xml:id="echoid-s80" xml:space="preserve"><lb/>Scient. </s> <s xml:id="echoid-s81" xml:space="preserve">cum publico communicata fuere, quædam Gal-<lb/>lica ſunt reliqua Latina, nempe;</s> <s xml:id="echoid-s82" xml:space="preserve"/> </p> <div xml:id="echoid-div7" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-0013-01" xlink:href="note-0013-01a" xml:space="preserve">pag. 193.</note> </div> <p> <s xml:id="echoid-s83" xml:space="preserve">De potentiis funeſve trahentibus <anchor type="note" xlink:href="" symbol="*"/>:</s> <s xml:id="echoid-s84" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">pag. 287.</note> <p> <s xml:id="echoid-s85" xml:space="preserve">Conſtructio loci ad Hyperbolam <anchor type="note" xlink:href="" symbol="*"/>:</s> <s xml:id="echoid-s86" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">pag. 485.</note> <p> <s xml:id="echoid-s87" xml:space="preserve">De maximis & </s> <s xml:id="echoid-s88" xml:space="preserve">minimis <anchor type="note" xlink:href="" symbol="*"/>:</s> <s xml:id="echoid-s89" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">pag. 490.</note> <p> <s xml:id="echoid-s90" xml:space="preserve">De inveniendis tangentibus <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s91" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">pag. 498.</note> <p style="it"> <s xml:id="echoid-s92" xml:space="preserve">Ut omnia Latino ſermone exhiberentur, quæ <lb/>Belgicè, aut Gallicè, edita fuere, Latinè reddidit <lb/>Clariſſ. </s> <s xml:id="echoid-s93" xml:space="preserve">Hermanni Ooſterdyk Schacht, Med. </s> <s xml:id="echoid-s94" xml:space="preserve">Profeſſo-<lb/>ris in hac Acad. </s> <s xml:id="echoid-s95" xml:space="preserve">Celeberrimi, Filius Digniſſimus Johan-<lb/>nes Ooſterdyk Schacht, qui ad ſtudia ſubtiliora na- <pb file="0014" n="14" rhead="PRÆFATIO."/> tus, à naturâ acceptum, adhuc dum juvenis, di-<lb/>ligentiâ pulcherrime excoluit ingenium.</s> <s xml:id="echoid-s96" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s97" xml:space="preserve">Si collectioni huic volumen operum poſthumorum <lb/>Hugenii, & </s> <s xml:id="echoid-s98" xml:space="preserve">tractatus duos de Lumine & </s> <s xml:id="echoid-s99" xml:space="preserve">Gravitate, <lb/>addas, omina B. </s> <s xml:id="echoid-s100" xml:space="preserve">L. </s> <s xml:id="echoid-s101" xml:space="preserve">habebis Hugenii opera.</s> <s xml:id="echoid-s102" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s103" xml:space="preserve">Hæc autem quæ hìc deficiunt, eâdem formâ cum <lb/>hiſce, brevi edituri ſunt Bibliopolæ Amſtelodamenſes <lb/>Waesbergii.</s> <s xml:id="echoid-s104" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s105" xml:space="preserve">Dabam Lugd. </s> <s xml:id="echoid-s106" xml:space="preserve">Bat. </s> <s xml:id="echoid-s107" xml:space="preserve">3°. </s> <s xml:id="echoid-s108" xml:space="preserve">Id. </s> <s xml:id="echoid-s109" xml:space="preserve">Mart. </s> <s xml:id="echoid-s110" xml:space="preserve">MDCCXXIV.</s> <s xml:id="echoid-s111" xml:space="preserve"/> </p> <pb file="0015" n="15"/> </div> <div xml:id="echoid-div9" type="section" level="1" n="8"> <head xml:id="echoid-head13" xml:space="preserve">HUGENII VITA.</head> <p> <s xml:id="echoid-s112" xml:space="preserve">CHriſtianus Hugenius, natus eſt Ha-<lb/>gæ Comitum in Hollandia, 14. </s> <s xml:id="echoid-s113" xml:space="preserve">A-<lb/>pril. </s> <s xml:id="echoid-s114" xml:space="preserve">1629. </s> <s xml:id="echoid-s115" xml:space="preserve">Patrem habuit Conſtan-<lb/>tinum Hugenium, Equitem, To-<lb/>parcham Zulichemii, Zelhemi, <lb/>& </s> <s xml:id="echoid-s116" xml:space="preserve">in Monikenlandt, qui tribus Auriacis Principi-<lb/>bus a ſecretis & </s> <s xml:id="echoid-s117" xml:space="preserve">conſiliis fuit. </s> <s xml:id="echoid-s118" xml:space="preserve">Mater erat Su-<lb/>ſanna van Baerle.</s> <s xml:id="echoid-s119" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s120" xml:space="preserve">In ſtudiis Mathematicis integram conſumſit vi-<lb/>tam, non tantum ſpeculationibus deditus, ſed <lb/>harum diſciplinarum ſubtiliſſima ad vitæ uſum re-<lb/>ferens. </s> <s xml:id="echoid-s121" xml:space="preserve">Ab ipſa infantia huic ſtudio applicavit <lb/>animum, vix natus annos novem, ipſo Patre du-<lb/>ce, in Muſicis, Arithmeticis, Geographicis, mi-<lb/>ros, & </s> <s xml:id="echoid-s122" xml:space="preserve">vix credibiles, progreſſus fecit, Latinis <lb/>& </s> <s xml:id="echoid-s123" xml:space="preserve">Græcis litteris interim animum applicans.</s> <s xml:id="echoid-s124" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s125" xml:space="preserve">Anno ætatis decimo tertio quàm ingenium ſtu-<lb/>dio Mechanices eſſet aptum, quod tanta deinde <lb/>hominum utilitate excoluit, in examinandis ma-<lb/>chinis, haſque, quantum infanti liceret, imitan-<lb/>do, demonſtravit.</s> <s xml:id="echoid-s126" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s127" xml:space="preserve">Anno 1644, ſtudium Matheſeos aggreſſus eſt, <lb/>Mathematicumque Belgam Stampioen præcepto-<lb/>rem habuit.</s> <s xml:id="echoid-s128" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s129" xml:space="preserve">Sequenti anno Academiam petiit quæ Leidæ <lb/>eſt apud Batavos. </s> <s xml:id="echoid-s130" xml:space="preserve">Ibi Vinnium jus civile expli-<lb/>cantem audivit, & </s> <s xml:id="echoid-s131" xml:space="preserve">magiſtro Schotenio ſtudium <pb file="0016" n="16" rhead="HUGENII VITA."/> Matheſeos continuavit, ingeniique ad hæc ſtu-<lb/>dia nati varia tunc temporis dedit ſpecimina, <lb/>brevique famam inter Mathematicos, annos ſu-<lb/>perantem, acquiſivit.</s> <s xml:id="echoid-s132" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s133" xml:space="preserve">Studium autem Juridicum Bredæ proſecutus <lb/>eſt annis 1646, 1647, 1648, occaſione ſcho-<lb/>læ illuſtris, tunc temporis ibi erectæ, & </s> <s xml:id="echoid-s134" xml:space="preserve">cu-<lb/>ræ Patris ipſius pro parte commiſſæ.</s> <s xml:id="echoid-s135" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s136" xml:space="preserve">Hagam anno ſequenti redux, Henricum Co-<lb/>mitem Naſſavium ſecutus, Holſatiam, & </s> <s xml:id="echoid-s137" xml:space="preserve">Da-<lb/>niam, inviſit. </s> <s xml:id="echoid-s138" xml:space="preserve">Vehementi tenebatur deſide-<lb/>rio in Sueciam uſque iter ſuum producendi, <lb/>Carteſium ut videret, quod ipſi non licuit, bre-<lb/>vi finitâ Comitis legatione.</s> <s xml:id="echoid-s139" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s140" xml:space="preserve">Anno 1651. </s> <s xml:id="echoid-s141" xml:space="preserve">Tractatum edidit de quadratu-<lb/>ra Hyperboles, Ellipſis, & </s> <s xml:id="echoid-s142" xml:space="preserve">Circuli, ex dato portio-<lb/>num gravitatis centro<anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s143" xml:space="preserve"/> </p> <note symbol="*" position="left" xml:space="preserve">Vide pag. <lb/>309.</note> <p> <s xml:id="echoid-s144" xml:space="preserve">Ut librum hunc perluſtrent Lectores rogo Ma-<lb/>thematicos, & </s> <s xml:id="echoid-s145" xml:space="preserve">videant an non merito, in ipſa <lb/>juventute, inter ſummos Mathematicos relatus <lb/>fuerit Hugenius.</s> <s xml:id="echoid-s146" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s147" xml:space="preserve">Eodem anno & </s> <s xml:id="echoid-s148" xml:space="preserve">ſequentibus varia de refractio-<lb/>nibus & </s> <s xml:id="echoid-s149" xml:space="preserve">Dioptrica conſcripſit, quæ in operibus <lb/>poſthumis edita exſtant.</s> <s xml:id="echoid-s150" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s151" xml:space="preserve">Anno 1655. </s> <s xml:id="echoid-s152" xml:space="preserve">Galliam petiit, & </s> <s xml:id="echoid-s153" xml:space="preserve">Andegavi Do-<lb/>ctor Juris renunciatus eſt.</s> <s xml:id="echoid-s154" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s155" xml:space="preserve">Eodem anno cum fratre Conſtantino vitris for-<lb/>mandis, quæ Teleſcopiis majoribus inſervirent <pb file="0017" n="17" rhead="HUGENII VITA."/> operam dedit. </s> <s xml:id="echoid-s156" xml:space="preserve">Teleſcopium decem pedum con-<lb/>ſtruxit quod, ut ipſe perſuaſum habebat, omnia <lb/>illius temporis ſuperabat. </s> <s xml:id="echoid-s157" xml:space="preserve">Hujus auxilio comi-<lb/>tem Saturni detexit. </s> <s xml:id="echoid-s158" xml:space="preserve">Omnes hujus Planetæ ſa-<lb/>tellites tunc temporis Aſtronomos latebant, & </s> <s xml:id="echoid-s159" xml:space="preserve"><lb/>niſi multis annis ſerius, reliquos quatuor, inter <lb/>quos unus Hugeniano à Saturno remotior, de-<lb/>texit Caſinus.</s> <s xml:id="echoid-s160" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s161" xml:space="preserve">Non ſtatim ſibi novum ſidus cognitum dari <lb/>cum Aſtronomis communicavit, ad quoſdam <lb/>tamen inventum, hiſce verbis & </s> <s xml:id="echoid-s162" xml:space="preserve">litteris invo-<lb/>lutum, miſit.</s> <s xml:id="echoid-s163" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s164" xml:space="preserve">Admovere oculis diſtantia ſidera noſtris <emph style="sc">VVVVVVV <lb/>CCCRRHNBQX.</emph></s> </p> <p> <s xml:id="echoid-s165" xml:space="preserve">Quæ verba cum adjectis litteris ipſo vitro in-<lb/>ſcripſit.</s> <s xml:id="echoid-s166" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s167" xml:space="preserve">Explicatio eſt Saturno Luna ſua circumducitur <lb/>diebus ſexdecim horis quatuor. </s> <s xml:id="echoid-s168" xml:space="preserve">Exactius tamen <lb/>in ſequentibus hujus Lunæ periodum determina-<lb/>vit<anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s169" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">Vide pag. <lb/>551.</note> <p> <s xml:id="echoid-s170" xml:space="preserve">Tranſpoſitione litterarum Ænigma explica-<lb/>tur.</s> <s xml:id="echoid-s171" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s172" xml:space="preserve">Per multos annos vitris formandis quibus no-<lb/>va in cœlis detegeret ſedulam cum fratre impen-<lb/>dit operam, præcipuè ab anno 1681 ad an-<lb/>num 1687, artemque hanc perfectiorem red-<lb/>didere; </s> <s xml:id="echoid-s173" xml:space="preserve">multa ex acuratè admodum elaboratis <lb/>vitris majora conſtruxere Teleſcopia. </s> <s xml:id="echoid-s174" xml:space="preserve">Inter vi- <pb file="0018" n="18" rhead="HUGENII VITA."/> tra hæc duo præ ceteris antecellunt, magnitu-<lb/>dine Teleſcopiorum quibus inſervire debent, &</s> <s xml:id="echoid-s175" xml:space="preserve">, ſi <lb/>Auctori noſtri fidem habeamus, excellentiâ <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s176" xml:space="preserve">ma- <anchor type="note" xlink:label="note-0018-01a" xlink:href="note-0018-01"/> jus deſtinatum erat Teleſcopio ducentorum & </s> <s xml:id="echoid-s177" xml:space="preserve">de-<lb/>cem pedum, alterum Teleſcopio centum & </s> <s xml:id="echoid-s178" xml:space="preserve">ſe-<lb/>ptuaginta. </s> <s xml:id="echoid-s179" xml:space="preserve">Hæc duo nunc poſſidet Anglia. </s> <s xml:id="echoid-s180" xml:space="preserve">Mul-<lb/>ta alia Teleſcopiis, centum pedes excedentibus, <lb/>ut & </s> <s xml:id="echoid-s181" xml:space="preserve">minoribus, inſervientia apud heredes adhuc-<lb/>dum ſuperſunt.</s> <s xml:id="echoid-s182" xml:space="preserve"/> </p> <div xml:id="echoid-div9" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-0018-01" xlink:href="note-0018-01a" xml:space="preserve">Vide pag. <lb/>698.</note> </div> <p> <s xml:id="echoid-s183" xml:space="preserve">Anno 1656 tractatum conſcripſit de ratiociniis <lb/>in ludo aleæ <anchor type="note" xlink:href="" symbol="*"/>, editus hic fuit ad calcem Exercita- <anchor type="note" xlink:label="note-0018-02a" xlink:href="note-0018-02"/> tionum Mathematicarum Schotenii. </s> <s xml:id="echoid-s184" xml:space="preserve">Methodum <lb/>in hoc tractatu demonſtravit ipſam ſortem com-<lb/>putationi Mathematicæ ſubjiciendi, primuſque pu-<lb/>blici juris fecit principia artis poſt illum ad vix <lb/>ſperandam perfectionem prolatæ.</s> <s xml:id="echoid-s185" xml:space="preserve"/> </p> <div xml:id="echoid-div10" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0018-02" xlink:href="note-0018-02a" xml:space="preserve">Vide pag. <lb/>723.</note> </div> <p> <s xml:id="echoid-s186" xml:space="preserve">Anno 1657 primus mortalium tempus exactiſ-<lb/>ſime menſuravit, pendula dum Horologiis appli-<lb/>cavit. </s> <s xml:id="echoid-s187" xml:space="preserve">Ante illum Aſtronomi adhibitis pendulis <lb/>tempus quidem menſurabant, ſed ad exigua in-<lb/>tervalla, cum pendula talia homine indigerent <lb/>qui curaret ut in motu perſeverarent. </s> <s xml:id="echoid-s188" xml:space="preserve">Ipſe au-<lb/>tem ope Horologiorum perpetuum quaſi pendu-<lb/>lis motum communicavit, ponderibus enim Horo-<lb/>logia agitabantur, quæ, non mutata actione in <lb/>Horologia, elevari poterant.</s> <s xml:id="echoid-s189" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s190" xml:space="preserve">Perſuaſum habebat talia Horologia & </s> <s xml:id="echoid-s191" xml:space="preserve">mari uſu <lb/>venire poſſe, & </s> <s xml:id="echoid-s192" xml:space="preserve">nil præter hæc in nave requiri <lb/>ad determinandas longitudines.</s> <s xml:id="echoid-s193" xml:space="preserve"/> </p> <pb file="0019" n="19" rhead="HUGENII VITA."/> <p> <s xml:id="echoid-s194" xml:space="preserve">Notum enim eſt, utilis hujus longitudinum pro-<lb/>blematis diu deſideratam, & </s> <s xml:id="echoid-s195" xml:space="preserve">forte deſiderandam, <lb/>ſolutionem, ab exacta temporis menſura pende-<lb/>re.</s> <s xml:id="echoid-s196" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s197" xml:space="preserve">Non tamen ſatis erat Horologiorum motus le-<lb/>gibus fixis adſtrinxiſſe, ſed ut ipſe notabat, mo-<lb/>tum æquabilem ſervare, navim jactantibus auſtris, <lb/>hoc opus, hic labor erat; </s> <s xml:id="echoid-s198" xml:space="preserve">difficultatem tamen ſu-<lb/>perari poſſe ſemper ſperavit, multa tentavit, & </s> <s xml:id="echoid-s199" xml:space="preserve"><lb/>ad mortem fere uſque nova, ut ad ſcopum per-<lb/>veniret, molitus eſt; </s> <s xml:id="echoid-s200" xml:space="preserve">ſed licet ad hunc pertin-<lb/>gere non potuerit, quo ingenio, qua perſpicacitate, <lb/>rem proſecutus eſt, qui horum operum tomum <lb/>primum perleget dijudicabit; </s> <s xml:id="echoid-s201" xml:space="preserve">non omnia tamen <lb/>tentamina publici juris facta fuere.</s> <s xml:id="echoid-s202" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s203" xml:space="preserve">Anno 1659. </s> <s xml:id="echoid-s204" xml:space="preserve">Syſtema Saturnium <anchor type="note" xlink:href="" symbol="*"/> edidit, in <anchor type="note" xlink:label="note-0019-01a" xlink:href="note-0019-01"/> quo veram cauſam anſarum hujus planetæ tradi-<lb/>dit, quam ante illum nemo ne ſuſpicione qui-<lb/>dem attingere potuerat, hancque invictis firmavit <lb/>argumentis.</s> <s xml:id="echoid-s205" xml:space="preserve"/> </p> <div xml:id="echoid-div11" type="float" level="2" n="3"> <note symbol="*" position="right" xlink:label="note-0019-01" xlink:href="note-0019-01a" xml:space="preserve">Vide pag. <lb/>527.</note> </div> <p> <s xml:id="echoid-s206" xml:space="preserve">Sequenti anno altera vice Galliam petiit, unde <lb/>in Angliam anno 1661. </s> <s xml:id="echoid-s207" xml:space="preserve">profectus eſt. </s> <s xml:id="echoid-s208" xml:space="preserve">lbi artem <lb/>ſuam laborandi vitra demonſtravit, cum inter <lb/>omnes conſtaret, Hugenii Teleſcopia, longitu-<lb/>dinem viginti quatuor pedum tunc temporis non <lb/>excedentia, ceteris omnibus perfectiora eſſe.</s> <s xml:id="echoid-s209" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s210" xml:space="preserve">Novum tunc temporis inventum erat Antlia <lb/>pneumatica, hujus, ab ipſo ex Anglia reduce <pb file="0020" n="20" rhead="HUGENII VITA."/> perfectioris redditæ, auxilio varia inſtituit experi-<lb/>menta <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s211" xml:space="preserve"/> </p> <note symbol="*" position="left" xml:space="preserve">Vide pag. <lb/>765.</note> <p> <s xml:id="echoid-s212" xml:space="preserve">Eodem anno regulas de colliſione corporum <lb/>Elaſticorum detexit, quas eaſdem poſtea etiam <lb/>detexere in Anglia viri celeberrimi Walliſius & </s> <s xml:id="echoid-s213" xml:space="preserve"><lb/>Wrennius, cum quo tamen ultimo contentionem, <lb/>ſuper hoc invento, habuit Hugenius noſter.</s> <s xml:id="echoid-s214" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s215" xml:space="preserve">Anno 1663. </s> <s xml:id="echoid-s216" xml:space="preserve">Lutetiam Pariſiorum iterum pe-<lb/>tiit, & </s> <s xml:id="echoid-s217" xml:space="preserve">cum Patre in Angliam iter ſuſcepit, ubi <lb/>Sociorum numero Regiæ ſocietatis Londinienſis <lb/>adſcriptus eſt.</s> <s xml:id="echoid-s218" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s219" xml:space="preserve">Per paucos tantum ibi ſtetit menſes & </s> <s xml:id="echoid-s220" xml:space="preserve">in Gal-<lb/>liam rediit.</s> <s xml:id="echoid-s221" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s222" xml:space="preserve">Anno 1664 Hagam redux de invento applica-<lb/>tionis pendulorum ad Horologia ipſi cum invidio-<lb/>ſo quodam lis fuit.</s> <s xml:id="echoid-s223" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s224" xml:space="preserve">Hoc tempore, in Gallia, ſtudiorum ſe Mæce-<lb/>natem demonſtrabat vir illuſtris Colbertus; </s> <s xml:id="echoid-s225" xml:space="preserve">cu-<lb/>jus conſilia de ſtudiis promovendis libenter au-<lb/>diebat Galliarum Rex. </s> <s xml:id="echoid-s226" xml:space="preserve">Undique viri ſcientiâ illu-<lb/>ſtres in Galliam vocabantur, inter quos Hugenius.</s> <s xml:id="echoid-s227" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s228" xml:space="preserve">Hic anno 1665, nomine Regis, Colberti lit-<lb/>teris, ut Lutetiam peteret, ibique domicilium <lb/>eligeret, promiſſo largo annuo ſtipendio, oblatâ-<lb/>que habitatione in ædificio ſervandis Regiis bi-<lb/>bliothecis deſtinato, invitatus eſt.</s> <s xml:id="echoid-s229" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s230" xml:space="preserve">Ibi vixit ab anno 1666. </s> <s xml:id="echoid-s231" xml:space="preserve">ad annum 1681. </s> <s xml:id="echoid-s232" xml:space="preserve">Du-<lb/>rante hoc tempore pulcherrima, ſubtiliſſimaque <pb file="0021" n="21" rhead="HUGENII VITA."/> multa, in Mathematicis detexit, variaque ex iis <lb/>operibus conſcripſit, quæ nunc in unum corpus <lb/>collecta, quid in variis Matheſeos partibus præſti-<lb/>terit, ſub oculis ponunt.</s> <s xml:id="echoid-s233" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s234" xml:space="preserve">Præter ipſius jam memorata inventa præclara <lb/>inter alia duo inſigni uſu eminent. </s> <s xml:id="echoid-s235" xml:space="preserve">Libellam <lb/>Teleſcopio munitam ita conſtruxit, ut ipſi præ ce-<lb/>teris fides haberi poſſit <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s236" xml:space="preserve">Inventum aliud ſpectat <anchor type="note" xlink:label="note-0021-01a" xlink:href="note-0021-01"/> temporis menſuram, Horologiis portatilibus fi-<lb/>lum, chalybeum, ſpirale, elaſticum, adapta-<lb/>vit <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s237" xml:space="preserve">quo nunc nullum portatile Horologium <anchor type="note" xlink:label="note-0021-02a" xlink:href="note-0021-02"/> deſtituitur, quo etiam ſublato, accuratiſſimè con-<lb/>ſtructa omnem motus æquabilitatem amittunt.</s> <s xml:id="echoid-s238" xml:space="preserve"/> </p> <div xml:id="echoid-div12" type="float" level="2" n="4"> <note symbol="*" position="right" xlink:label="note-0021-01" xlink:href="note-0021-01a" xml:space="preserve">Vide pag. <lb/>254.</note> <note symbol="*" position="right" xlink:label="note-0021-02" xlink:href="note-0021-02a" xml:space="preserve">Vide pag. <lb/>253.</note> </div> <p> <s xml:id="echoid-s239" xml:space="preserve">Nimium verò ſtudiis Mathematicis deditus, <lb/>menti gratum Corpus non potuit ſuſtinere laborem. <lb/></s> <s xml:id="echoid-s240" xml:space="preserve">Bis Hollandiam hac de cauſa petiit, annis 1670, & </s> <s xml:id="echoid-s241" xml:space="preserve"><lb/>1675, iterumque recuperatâ ſanitate in Galliam <lb/>rediit, ſed tandem valetudini ut conſuleret illi <lb/>in perpetuum dixit vale anno 1681, omnibuſque <lb/>Regis beneficiis nuncium remiſit.</s> <s xml:id="echoid-s242" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s243" xml:space="preserve">Reliquum vitæ curſum iiſdem occupatus ſtudiis <lb/>abſolvit.</s> <s xml:id="echoid-s244" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s245" xml:space="preserve">Anno 1682. </s> <s xml:id="echoid-s246" xml:space="preserve">conſtrui curavit Automaton Pla-<lb/>netarium in quo planetarum motus in plano pul-<lb/>cherrime æmulatus eſt.</s> <s xml:id="echoid-s247" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s248" xml:space="preserve">Machina hæc in operibus poſthumis delineata, <lb/>& </s> <s xml:id="echoid-s249" xml:space="preserve">accuratiſſimè deſcripta, datur. </s> <s xml:id="echoid-s250" xml:space="preserve">Exſtat adhucdum <lb/>apud hæredes.</s> <s xml:id="echoid-s251" xml:space="preserve"/> </p> <pb file="0022" n="22" rhead="HUGENII VITA."/> <p> <s xml:id="echoid-s252" xml:space="preserve">Anno 1689. </s> <s xml:id="echoid-s253" xml:space="preserve">Angliam tertia vice inviſit.</s> <s xml:id="echoid-s254" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s255" xml:space="preserve">Sequenti anno tractatus duos, alterum de Lumi-<lb/>ne, de Gravitate alterum edidit.</s> <s xml:id="echoid-s256" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s257" xml:space="preserve">Coſinotheoros tempore mortis ſub prælo ſudabat, <lb/>editio tamen inchoata tantum erat.</s> <s xml:id="echoid-s258" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s259" xml:space="preserve">Vitam finivit Hagæ Comitum octavo Junii 1695.</s> <s xml:id="echoid-s260" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s261" xml:space="preserve">Scripta omnia legato dedit Bibliothecæ Acade-<lb/>miæ Ordinum Hollandiæ quæ eſt Lugd. </s> <s xml:id="echoid-s262" xml:space="preserve">Bat. </s> <s xml:id="echoid-s263" xml:space="preserve">vi-<lb/>roſque duos, inſignes Mathematicos, Burcherum <lb/>de Volder, in eâdem Acad. </s> <s xml:id="echoid-s264" xml:space="preserve">Philoſophiæ & </s> <s xml:id="echoid-s265" xml:space="preserve">Math. <lb/></s> <s xml:id="echoid-s266" xml:space="preserve">Profeſſorem celebrem, & </s> <s xml:id="echoid-s267" xml:space="preserve">Bernhardum Fullenium, <lb/>in Academiâ Friſiâ Franequeræ Profeſſorem, ro-<lb/>gavit ut ex ſcriptis eligerent, quæ prælo committi <lb/>poſſent, cui petitioni volumen debemus operum <lb/>poſthumorum, anno 1700. </s> <s xml:id="echoid-s268" xml:space="preserve">editorum.</s> <s xml:id="echoid-s269" xml:space="preserve"/> </p> <figure> <image file="0022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0022-01"/> </figure> <pb file="0023" n="23"/> </div> <div xml:id="echoid-div14" type="section" level="1" n="9"> <head xml:id="echoid-head14" xml:space="preserve">CHRISTIANI HUGENII <lb/>OPERA <lb/>MECHANICA. <lb/><emph style="sc">Tomus</emph> <emph style="sc">Primus</emph>.</head> <pb file="0024" n="24"/> </div> <div xml:id="echoid-div15" type="section" level="1" n="10"> <head xml:id="echoid-head15" xml:space="preserve">Tomi primi contenta.</head> <note position="right" xml:space="preserve"> <lb/><emph style="sc">Horologium</emph>. # pag. 1. <lb/><emph style="sc">Horologium</emph> <emph style="sc">Oscillatorium</emph>, ſive de motu pendulorum, ad <lb/>horologia aptato, demonſtrationes Geometricæ. # 15. <lb/><emph style="sc">Brevis</emph> <emph style="sc">Institutio de usu</emph> <emph style="sc">Horologiorum</emph>, ad inveniendas <lb/>longitudines. # 193. <lb/>De Hugeniana centri oſcillationis determinatione <emph style="sc">CONTROVER-<lb/>SIA</emph>. # 215. <lb/><emph style="sc">Machinæ quædam</emph>, & varia circa Mechanicam. # 249. <lb/></note> <pb file="0025" n="25"/> </div> <div xml:id="echoid-div16" type="section" level="1" n="11"> <head xml:id="echoid-head16" xml:space="preserve">CHRISTIANI</head> <head xml:id="echoid-head17" xml:space="preserve">HUGENII <emph style="sc">A</emph> ZULICHEM,</head> <head xml:id="echoid-head18" xml:space="preserve"><emph style="sc">Const</emph>, F. <lb/>HOROLOGIUM.</head> <pb file="0026" n="26"/> <pb file="0027" n="27"/> </div> <div xml:id="echoid-div17" type="section" level="1" n="12"> <head xml:id="echoid-head19" style="it" xml:space="preserve">ILLUSTRISSIMIS AC POTENTISSIMIS</head> <head xml:id="echoid-head20" xml:space="preserve">HOLLANDIAE <lb/>Et <lb/>WESTFRISIAE <lb/>ORDINIBUS</head> <head xml:id="echoid-head21" xml:space="preserve">Dominis ſuis, <lb/><emph style="sc">Christianus</emph> <emph style="sc">Hugenius</emph> à <emph style="sc">Zulighem</emph> <lb/>Felicitatem omnem.</head> <p style="it"> <s xml:id="echoid-s270" xml:space="preserve">PRoditum eſt memoriæ primum Ro-<lb/>mæ ſolare horologium fuiſſe, quod <lb/>è capto Siciliæ oppido quodam, an-<lb/>nis poſt urbem conditam <emph style="sc">CCCCLXXVII</emph>, <lb/>cum cætera præda deportatum ſit, <lb/>locoque publico dedicatum. </s> <s xml:id="echoid-s271" xml:space="preserve">Cui non <lb/>planè ad Latii clima deſcripto, eo-<lb/>que nec lineas horis congruentes ex-<lb/>hibenti, quum neceſſitate tamen & </s> <s xml:id="echoid-s272" xml:space="preserve"><lb/>meliorum penuria undecentum annis <lb/>Pop. </s> <s xml:id="echoid-s273" xml:space="preserve">Romanus paruiſſet, Cenſorem tandem Q. </s> <s xml:id="echoid-s274" xml:space="preserve">Marcium Phi-<lb/>lippum diligentius ordinatum juxta poſuiſſe, idque munus in-<lb/>ter cenſoria opera gratiſſime acceptum. </s> <s xml:id="echoid-s275" xml:space="preserve">Mihi, Proceres Am-<lb/>pliſſimi, rem haud abſimilem nec minore publico bono hodie <lb/>agitanti, ut qui non in una modo urbe, ſed omnium ubivis <lb/>horologiorum inſtabilem motum correxerim, ſimilem quoque ab <lb/>univerſis gratiam expectandam cenſuiſſem atque à civibus ſuis <lb/>Q. </s> <s xml:id="echoid-s276" xml:space="preserve">Marcius reportavit, ſi, quemadmodum res eventusque ii-<lb/>dem ex intervallo redire ſolent, ita priſcus candor & </s> <s xml:id="echoid-s277" xml:space="preserve">ingenui-<lb/>tas in terris aliquando reduces cernerentur. </s> <s xml:id="echoid-s278" xml:space="preserve">Verum bæ cum <lb/>jam diu apud majorem hominum partem deſitæ ſint virtutes, <pb file="0028" n="28" rhead="DEDICATIO."/> contraque impoſtura & </s> <s xml:id="echoid-s279" xml:space="preserve">obtrectatio late omnia obtineant; </s> <s xml:id="echoid-s280" xml:space="preserve">quæ-<lb/>nam fortuna maneret inventum noſtrum, ſimul ac vulgò in-<lb/>noteſcere cœpiſſet, facile equidem prævidi, neque me fefellit <lb/>augurium. </s> <s xml:id="echoid-s281" xml:space="preserve">Ecce enim jam primum in patria hac noſtra eo ex-<lb/>ceſſit quorundam tum audacia tum impudentia, ut nihil in-<lb/>terdicto veſtro deterriti, interpolare acceptum à nobis inven-<lb/>tum, ac dein tanquam novum prorſus, noſtroque etiam, ſi <lb/>diis placet, præſtantius jactare auſi ſint. </s> <s xml:id="echoid-s282" xml:space="preserve">Atque hæc qui co-<lb/>ram & </s> <s xml:id="echoid-s283" xml:space="preserve">ante oculos nobis fieri viderunt, nihilo meliora ab ex-<lb/>teris regionibus imminere crebro admonuerunt. </s> <s xml:id="echoid-s284" xml:space="preserve">Nempe alibi <lb/>quoque exorituros, & </s> <s xml:id="echoid-s285" xml:space="preserve">in gloriolam hanc noſtram involaturos <lb/>homines inique invidos, qui, forte an & </s> <s xml:id="echoid-s286" xml:space="preserve">ſibi ipſis, certe orbi <lb/>univerſo perſuadere conentur, non hæc noſtr atium ingeniis de-<lb/>heri, ſed à ſua ſuorumve alicujus induſtria diu ante profecta <lb/>fuiſſe. </s> <s xml:id="echoid-s287" xml:space="preserve">Cujus rei indignitas cum ad gentem omnem noſtram, <lb/>eoque ad vos etiam, Domini Illuſtriſſimi, ſpectare videretur, <lb/>qui nunquam æquo animo tuleritis inventorum longe præcla-<lb/>riſſimorum, typographiæ inquam & </s> <s xml:id="echoid-s288" xml:space="preserve">telescopii, laudem à Ba-<lb/>tavia veſtra, plagiariorum fraude, averti; </s> <s xml:id="echoid-s289" xml:space="preserve">fateor me non le-<lb/>vi ſtimulo impulſum fuiſſe, ut eidem hujus quoque qualiscun-<lb/>que reperti decus adſererem. </s> <s xml:id="echoid-s290" xml:space="preserve">Itaque eam quæ ſola ad hoc pate-<lb/>re viſa eſt, viam ſecutus, rationem omnem & </s> <s xml:id="echoid-s291" xml:space="preserve">conſtructionem <lb/>novi automati, autor ipſe, paucis deſcribendam & </s> <s xml:id="echoid-s292" xml:space="preserve">in publi-<lb/>cum producendam ſuſcepi; </s> <s xml:id="echoid-s293" xml:space="preserve">exiguo ſanè volumine, ſed quod <lb/>brevius etiam fuiſſet niſi obiter ad ea quoque reſpondendum <lb/>duxiſſem quæ à nonnullis objici mihi, ipſumque artificii noſtri <lb/>fundamentum laceſſcre poſſe, proſpiciebam. </s> <s xml:id="echoid-s294" xml:space="preserve">Hoc vero quicquid <lb/>eſt operæ, quum melioribus auſpiciis lucem aſpicere non poſſet, <lb/>veſtro Illuſtriſſimo Nomini actutelæ, ea qua decet veneratione, <lb/>dicatum commiſſumque venio, neque tam pagellas haſce paucu-<lb/>las, quam inventum ipſum, ut videtur, non incelebre futurum, <lb/>dedico conſecroque. </s> <s xml:id="echoid-s295" xml:space="preserve">Vos pro ſolita benignitate veſtrafavete, ad <lb/>publicam utilitatem, quoquo modo ſtudia ſua referenti, neque <lb/>aliud magis in votis habenti, quam ut majoris momenti in rebus <lb/>eadem poſthac approbare vobis conting at. </s> <s xml:id="echoid-s296" xml:space="preserve">Ita Rempublicam ſub <lb/>imperio veſtro incolumem ſervet, beneque fortunet Deus.</s> <s xml:id="echoid-s297" xml:space="preserve"/> </p> <pb o="5" file="0029" n="29"/> <figure> <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0029-01"/> </figure> </div> <div xml:id="echoid-div18" type="section" level="1" n="13"> <head xml:id="echoid-head22" xml:space="preserve">CHRISTIANI <lb/>HUGENII <emph style="sc">A</emph> ZULICHEM, <lb/><emph style="sc">Const</emph>. F. <lb/>HOROLOGIUM.</head> <p> <s xml:id="echoid-s298" xml:space="preserve">TEmporis dimetiendi rationem novam, quam <lb/>exeunte Anno 1656. </s> <s xml:id="echoid-s299" xml:space="preserve">excogitavimus, pauciſque <lb/>deinde menſibus in patria divulgare inſtituimus, <lb/>etſi dubitandum non erat, propter egregiam uti-<lb/>litatem, brevi longe lateque manaturam, quip-<lb/>pe pluribus jam distractis, ac dimiſſis quaqua-<lb/>verſum novi operis exemplaribus; </s> <s xml:id="echoid-s300" xml:space="preserve">nos tamen haud in-<lb/>viti conſiliis eorum obtemperamus, qui ut ſcripto compre-<lb/>henſam in lucem ederemus, autores fuere. </s> <s xml:id="echoid-s301" xml:space="preserve">cum ut illos de-<lb/>mereamur, ad quos, ob locorum intervalla, tardius fortaſſe <lb/>perventura erat: </s> <s xml:id="echoid-s302" xml:space="preserve">tum ut male feriatorum hominum audaciæ <lb/>obviam eamus, ne, quod ſolenne ipſis eſt, alienis inſidien-<lb/>tur inventis, ac per ſummam injuriam pro ſuis venditent. <lb/></s> <s xml:id="echoid-s303" xml:space="preserve">Quanquam hos, ſi fuerit opus, & </s> <s xml:id="echoid-s304" xml:space="preserve">dati Privilegii tempus <lb/>refellere poſſit, quod à Celſiſſimis Fœderatarum Provincia-<lb/>rum Ordinibus die 16. </s> <s xml:id="echoid-s305" xml:space="preserve">Junii Anno 1657. </s> <s xml:id="echoid-s306" xml:space="preserve">impetratum eſt; </s> <s xml:id="echoid-s307" xml:space="preserve">& </s> <s xml:id="echoid-s308" xml:space="preserve"><lb/>teſtes præterea non pauci, quos de oblato nobis recens in-<lb/>vento ſubinde certiores fecimus. </s> <s xml:id="echoid-s309" xml:space="preserve">Occaſionem ei præbuiſſe <lb/>Aſtronomorum pendula, facile quivis conjiciet, qui non <lb/>neſcierit aliquot jam retro annis hæc uſurpari illis cœpta.</s> <s xml:id="echoid-s310" xml:space="preserve"> <pb o="6" file="0030" n="30" rhead="CHRISTIANI HUGENII"/> Nimirum fallentibus clepſydris automatiſque quibuslibet, <lb/>quæ inter obſervandum adhibere conſueverant, tandem, do-<lb/>cente primum Viro ſagaciſſimo Galileo Galilei, hunc mo-<lb/>dum inierunt, ut è catenula tenui pondus appenſum manu <lb/>impellerent, cujus vibrationibus ſingulis dinumeratis, toti-<lb/>dem colligerentur æqualia temporis momenta. </s> <s xml:id="echoid-s311" xml:space="preserve">Hac metho-<lb/>do Obſervationes Eclipſium ſcrupuloſius quam antea pere-<lb/>gere, Soliſque item diametrum, & </s> <s xml:id="echoid-s312" xml:space="preserve">Stellarum diſtantias di-<lb/>menſi ſunt non infeliciter. </s> <s xml:id="echoid-s313" xml:space="preserve">Sed præterquam quod deficiebat <lb/>neceſſario pendulorum motus, niſi adſtantis opera identidem <lb/>juvaretur, tædioſus inſuper labor evadebat, omnes eorum <lb/>itus redituſque numerantibus; </s> <s xml:id="echoid-s314" xml:space="preserve">cui ſane integris noctibus mi-<lb/>rabili patientia nonnullos invigilaſſe, ipſis prodentibus, con-<lb/>ſtat. </s> <s xml:id="echoid-s315" xml:space="preserve">Nos autem æquabiliſſimum hocce genus motus cernen-<lb/>tes, ac veluti unicum in rerum natura datum, quod ad Me-<lb/>chanicam conſtructionem poſſet traduci, quæſivimus quo <lb/>pacto hoc ipſum breviſſime aſſequi liceret, atque ita reme-<lb/>dium inven@re gemino quod retulimus incommodo. </s> <s xml:id="echoid-s316" xml:space="preserve">Ac mul-<lb/>ta fabricæ varietate animo perpenſa, hanc denique, quam <lb/>deinceps tradituri ſumus, ut cæteris planiorem faciliorem-<lb/>que ſelegimus. </s> <s xml:id="echoid-s317" xml:space="preserve">Qua percepta & </s> <s xml:id="echoid-s318" xml:space="preserve">in publicum porro priva-<lb/>tumque uſum, ſicut jam fieri cœpit, converſa, ad univer-<lb/>ſos quidem hic fructus redundabit, quod horologiorum, <lb/>cum inter ſe, tum cum Sole ipſo, quantus nunquam antehac, <lb/>imo quantus pene optari poſſet, conſenſus animadvertetur. <lb/></s> <s xml:id="echoid-s319" xml:space="preserve">Aſtronomi vero id conſequentur, ut nulla poſthac agitando-<lb/>rum perpendiculorum moleſtia, numerandive ſolicitudine, <lb/>& </s> <s xml:id="echoid-s320" xml:space="preserve">illa omnia exequantur, quorum paulo ante meminimus, <lb/>& </s> <s xml:id="echoid-s321" xml:space="preserve">alia illis ſubtiliora, ipſam puta dierum de meridie in me-<lb/>ridiem inæqualitatem, ſcrutentur; </s> <s xml:id="echoid-s322" xml:space="preserve">quam qui negare audent, <lb/>ratione hactenus magis quam certa experientia refutati ſunt. </s> <s xml:id="echoid-s323" xml:space="preserve"><lb/>Ut jam de Longitudinum, quam vocant, ſcientia dicere <lb/>omittam: </s> <s xml:id="echoid-s324" xml:space="preserve">quæ ſi unquam extitura eſt, deſideratumque tan-<lb/>topere uſum curſui navigantium præbitura, non aliter, quam <lb/>vectis per mare exquiſitiſſimis atque omni errore vacuis ho-<lb/>rologiis, id obtineri poſſe, multi nobiſcum exiſtimant. </s> <s xml:id="echoid-s325" xml:space="preserve">Ve- <pb o="7" file="0031" n="31" rhead="HOROLOGIUM."/> rum hæc res vel ipſi mihi, vel aliis quandoque curæ erit. <lb/></s> <s xml:id="echoid-s326" xml:space="preserve">Nunc automaton noſtrum & </s> <s xml:id="echoid-s327" xml:space="preserve">figura oculis ſubjiciam, & </s> <s xml:id="echoid-s328" xml:space="preserve">figu-<lb/>ram verbis quam potero dilucide explicabo.</s> <s xml:id="echoid-s329" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s330" xml:space="preserve">Præcipuam operis partem binæ laminæ continent oblongæ <lb/> <anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/> atque inter ſe æquales, A B, C D; </s> <s xml:id="echoid-s331" xml:space="preserve">quibus rotarum axes u-<lb/>trinque inſerti ſunt. </s> <s xml:id="echoid-s332" xml:space="preserve">Eæ laminæ lateribus tantum hic ſunt <lb/>conſpicuæ: </s> <s xml:id="echoid-s333" xml:space="preserve">columellas autem quatuor, quibus verſus angu-<lb/>los connexæ ſunt, de induſtria exprimere neglexi, ut ne re-<lb/>liquis officerent. </s> <s xml:id="echoid-s334" xml:space="preserve">Prima rota ſeu tympanum dentatum eſt E, <lb/>cujus axi orbiculus quoque F affixus eſt. </s> <s xml:id="echoid-s335" xml:space="preserve">Huic circumjectus <lb/>funis cum appenſo pondere Δ, eo quem poſtea dicemus mo-<lb/>do. </s> <s xml:id="echoid-s336" xml:space="preserve">Ponderis itaque vi tympanum E vertitur. </s> <s xml:id="echoid-s337" xml:space="preserve">Hoc movet <lb/>proximum tympanum H. </s> <s xml:id="echoid-s338" xml:space="preserve">hoc rotam L, cujus dentes ad inſtar <lb/>ſerræ dentium formati ſunt. </s> <s xml:id="echoid-s339" xml:space="preserve">Hujus prope axem erectus ſtat <lb/>axis M N, cum affixis lamellis ſive auriculis binis, quarum <lb/>alteri occurrunt ſuperiores rotæ L dentes, alteri inferiores, <lb/>idque perpetua viciſſitudine, ita ut non in gyrum axis hic <lb/>circumagatur, ſed reciproco motu, nunc in hanc, nunc in il-<lb/>lam partem libretur, dum interim rota L in orbem vertitur. <lb/></s> <s xml:id="echoid-s340" xml:space="preserve">Quem motum pluribus exponere ſuperſedeo, quod in vul-<lb/>garibus paſſim horologiis reperiatur. </s> <s xml:id="echoid-s341" xml:space="preserve">à quibus equidem huc-<lb/>uſque noſtrum hoc non diſcrepat; </s> <s xml:id="echoid-s342" xml:space="preserve">at plurimum in his quæ <lb/>ſequuntur. </s> <s xml:id="echoid-s343" xml:space="preserve">Axi enim N M infigitur O tympanum, cujus den-<lb/>tibus aptantur dentes rotæ P, ejus generis quas coronarias <lb/>vocant artifices noſtri. </s> <s xml:id="echoid-s344" xml:space="preserve">Nec vero toto ambitu dentata ut ſit <lb/>neceſſe eſt, ſed parte ſuperiori duntaxat. </s> <s xml:id="echoid-s345" xml:space="preserve">quippe tympanum <lb/>O, haud aliter atque axis N M cui cohæret, reciprocam li-<lb/>brationem habet, unde & </s> <s xml:id="echoid-s346" xml:space="preserve">rotam P ſimili motu agitat. </s> <s xml:id="echoid-s347" xml:space="preserve">Cum-<lb/>que major ſit hujus diameter quam tympani O, ſequitur ut <lb/>minori etiam circuitus parte rota quam tympanum dictum <lb/>gyretur. </s> <s xml:id="echoid-s348" xml:space="preserve">quod quo pertineat alibi indicabimus. </s> <s xml:id="echoid-s349" xml:space="preserve">Porro ejuſ-<lb/>dem rotæ P axis trans laminam C D aliquantum extenditur, <lb/>habetque conjunctam clavulam Q R, inferius itidem inflexam <lb/>& </s> <s xml:id="echoid-s350" xml:space="preserve">terebratam ad R, ita ut per foramen hoc laxiuſculum vir-<lb/>gula ænea I T libere transmeet. </s> <s xml:id="echoid-s351" xml:space="preserve">Hæc vero virgula ſuperius <lb/>ad S ſuſpenſa eſt filo S I, ex inferiori parte pondus T. </s> <s xml:id="echoid-s352" xml:space="preserve">ſuſti- <pb o="8" file="0032" n="32" rhead="CHRISTIANI HUGENII"/> nens, quod cochleæ ſubjectæ converſione ſurſum propellitur <lb/>cum opus eſt, vel ulterius deſcendit.</s> <s xml:id="echoid-s353" xml:space="preserve"/> </p> <div xml:id="echoid-div18" type="float" level="2" n="1"> <note position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">TAB. 1.</note> </div> <p> <s xml:id="echoid-s354" xml:space="preserve">Quibus expoſitis ut motus ratio, totiuſque adeo inventi <lb/>percipiatur (nam quæ præterea in Schemate notata apparent, <lb/>poſtea exequemur) advertendum eſt in primis, quod ſi per-<lb/>pendiculum S I T per foramen R trajectum non eſſet, neque <lb/>omnino adeſſet, tunc quidem clavula Q R concitato motu <lb/>ultro citroque jactaretur, vi ponderis Δ, omnes rotas auto-<lb/>mati agitantis. </s> <s xml:id="echoid-s355" xml:space="preserve">Transmiſſa autem virgula I T, cum appenſo <lb/>pondere T per foramen R, impeditur eo dictus clavulæ mo-<lb/>tus, totumque horologium quieſcit, donec pondus T ſemel <lb/>impulſum principium agitationis nanciſcatur. </s> <s xml:id="echoid-s356" xml:space="preserve">Quo facto, <lb/>pendulum quidem S I T oſcillatorio motu fertur juxta planum <lb/>laminæ C D. </s> <s xml:id="echoid-s357" xml:space="preserve">clavula vero Q R, momentum ſentiens ponderis <lb/>Δ, ultro obſequitur penduli motui, ita ut pauliſper etiam <lb/>vibrationibus hunc ſingulis adjuvet. </s> <s xml:id="echoid-s358" xml:space="preserve">Atque hoc modo peren-<lb/>nis efficitur penduli agitatio, quæ niſi illud horologio con-<lb/>junctum foret, brevi deficeret vergeretque ad quietem. </s> <s xml:id="echoid-s359" xml:space="preserve">Ad <lb/>ſingulos autem recurſus penduli, percipientur ictus totidem <lb/>ex appulſu dentium rotæ L ad lamellas M, N. </s> <s xml:id="echoid-s360" xml:space="preserve">Et hæc qui-<lb/>dem de motu automati noſtri, quæ præcipue explicationem <lb/>requirebant, quoniam in eo ſumma totius inventi vertitur.</s> <s xml:id="echoid-s361" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s362" xml:space="preserve">In ſchemate porro tertia lamina eſt Y Z, prioribus paral-<lb/>lela, & </s> <s xml:id="echoid-s363" xml:space="preserve">à lamina A B ſpatio diſtans. </s> <s xml:id="echoid-s364" xml:space="preserve">quo in ſpatio conſpici-<lb/>tur tympanum dentatum V, communem cum rota E axem <lb/>habens. </s> <s xml:id="echoid-s365" xml:space="preserve">Huic congruunt dentes rotæ X, quæ media ſui par-<lb/>te conjunctum habet tubulum cavum Γ ultra laminam Y Z <lb/>prominentem, impoſitumque gerentem horologii indicem <lb/>primarium Λ. </s> <s xml:id="echoid-s366" xml:space="preserve">Ipſi vero Γ tubulo alius itidem cavus introrſus <lb/>conſtitutus eſt, laminæque Y Z conſertus, axis nimirum quo <lb/>rota X volvatur, & </s> <s xml:id="echoid-s367" xml:space="preserve">per quem ſimul transmittatur axis rotæ <lb/>H, cui impoſitus eſt index alius Σ longior indice Λ. </s> <s xml:id="echoid-s368" xml:space="preserve">Is ſe-<lb/>cunda ſcrupula demonſtrat. </s> <s xml:id="echoid-s369" xml:space="preserve">Primorum vero ſcrupulorum ſeu <lb/>minutorum index prioribus illis utriſque multo brevior Ψ, <lb/>extremo axi D V, ultra laminam Y Z producto, affixus eſt. <lb/></s> <s xml:id="echoid-s370" xml:space="preserve">Et hic quidem indiculus, laminæ, Y Z proximus fertur, <pb o="9" file="0033" n="33" rhead="HOROLOGIUM."/> parvo in circello ſingula prima ſcrupula diſtinguens. </s> <s xml:id="echoid-s371" xml:space="preserve">Hoc <lb/>verò ſuperior index horarum Λ convertitur: </s> <s xml:id="echoid-s372" xml:space="preserve">& </s> <s xml:id="echoid-s373" xml:space="preserve">ſupra hunc <lb/>denique Σ index, quem dixi, ſecundorum. </s> <s xml:id="echoid-s374" xml:space="preserve">Hæc autem, <lb/>uti & </s> <s xml:id="echoid-s375" xml:space="preserve">tympanorum omnium diſpoſitio ac dentium numerus, <lb/>cum multimodis variari poſſint, nos hunc unum in exem-<lb/>plum proponere ſatis habuimus, eumque experientia com-<lb/>probatum. </s> <s xml:id="echoid-s376" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s377" xml:space="preserve">dentium multitudinem in ſingulis tym-<lb/>panis deſignabimus, eam quæ huic formæ optime convenire <lb/>viſa eſt. </s> <s xml:id="echoid-s378" xml:space="preserve">I<unsure/>n circumferentia rotarum ſingularum E H ſeptua-<lb/>geni bini ſunt, ſeni in tympanis G & </s> <s xml:id="echoid-s379" xml:space="preserve">K. </s> <s xml:id="echoid-s380" xml:space="preserve">rota L viginti <lb/>quinque habet, tympanum O decem. </s> <s xml:id="echoid-s381" xml:space="preserve">rota P viginti, vel <lb/>tantum partem horum aliquam, quia, ut dixi, totam den-<lb/>tibus inſecari nihil opus eſt. </s> <s xml:id="echoid-s382" xml:space="preserve">Penduli longitudo S I T pedis <lb/>Rhenolandici, qui ad Romanum veterem proxime accedit, <lb/>dextantem circiter æquat, & </s> <s xml:id="echoid-s383" xml:space="preserve">cuique vibrationi ſimplici im-<lb/>pendit ſemiſcrupulum ſecundum. </s> <s xml:id="echoid-s384" xml:space="preserve">ad quam menſuram obſer-<lb/>vationibus ad ſolem vel ad aliud hujus generis horologium <lb/>comparatis non difficile perducitur. </s> <s xml:id="echoid-s385" xml:space="preserve">Ea longitudo rotis ita <lb/>ordinatis convenit: </s> <s xml:id="echoid-s386" xml:space="preserve">& </s> <s xml:id="echoid-s387" xml:space="preserve">exquiſitam motus æqualitatem, quæ-<lb/>que etiam Aſtronomicis uſibus ſufficiat, præſtare valet. <lb/></s> <s xml:id="echoid-s388" xml:space="preserve">Quod ſi tamen concinnitate operis inſuper habita, quadru-<lb/>plo majus pendulum adhibeatur, vel ultra etiam producatur, <lb/>rotis interim majoribus quoque adſumptis, haud dubiè len-<lb/>tioribus oſcillationibus tutius etiam fidemus. </s> <s xml:id="echoid-s389" xml:space="preserve">Et jam in ma-<lb/>gnis publicis Horologiis, egregio ſucceſſu, perpendicula <lb/>ejusmodi prælonga uſurpari vidimus, alibi duodecim, alibi <lb/>vicenûm pedum, cum appenſa ſphæra 25 vel 30 librarum. </s> <s xml:id="echoid-s390" xml:space="preserve"><lb/>Cæterum revertendo ad ea quæ hic poſita fuere, apparet <lb/>quidem, rota E ſemel circumacta, duodecies converti ro-<lb/>tam H. </s> <s xml:id="echoid-s391" xml:space="preserve">centies vero quadragies & </s> <s xml:id="echoid-s392" xml:space="preserve">quater eam quæ ſequitur <lb/>L. </s> <s xml:id="echoid-s393" xml:space="preserve">Quæ cum dentes 25 habeat, 3600 vicibus alternatim im-<lb/>pellit lamellas M. </s> <s xml:id="echoid-s394" xml:space="preserve">N. </s> <s xml:id="echoid-s395" xml:space="preserve">ac totidem recurſus duplices facit pen-<lb/>dulum S I T. </s> <s xml:id="echoid-s396" xml:space="preserve">Cumque 3600 ſcrupula ſecunda, horâ unâ <lb/>contineantur; </s> <s xml:id="echoid-s397" xml:space="preserve">hinc horæ ſpatio rota E ſemel convertetur. </s> <s xml:id="echoid-s398" xml:space="preserve"><lb/>Quamobrem & </s> <s xml:id="echoid-s399" xml:space="preserve">circulus indici Ψ ſubjectus in 60 partes divi-<lb/>ditur, quæ prima ſcrupula ſignificent. </s> <s xml:id="echoid-s400" xml:space="preserve">Rota vero H quia <pb o="10" file="0034" n="34" rhead="CHRISTIANI HUGENII"/> duodecies in hora, hoc eſt, ſemel ſpatio 5 ſcrupulorum pri-<lb/>morum verſatur, unaque index Σ, ideo circulum huic in-<lb/>dici ſuppoſitum in 5 partes primum diſpeſcimus, & </s> <s xml:id="echoid-s401" xml:space="preserve">harum <lb/>deinde ſingulas in 60 minores, quæ ſecunda ſcrupula deno-<lb/>tent. </s> <s xml:id="echoid-s402" xml:space="preserve">Denique index Λ in ſuo circulo duodecim horas diſtin-<lb/>guere debet; </s> <s xml:id="echoid-s403" xml:space="preserve">ac proinde, ut harum tempore ſemel circum-<lb/>eat, tympano V ſex dentes tribuuntur, rotæ X ſeptuageni <lb/>bini.</s> <s xml:id="echoid-s404" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s405" xml:space="preserve">Nunc qua ratione pondera Δ, Ξ, horologio appendan-<lb/>tur docebimus. </s> <s xml:id="echoid-s406" xml:space="preserve">Hæc enim novo artificio ita ordinavimus, <lb/>ut cum ſurſum retrahitur pondus primarium Δ, non pro-<lb/>pterea ceſſet aut ullatenus impediatur horologii curſus. <lb/></s> <s xml:id="echoid-s407" xml:space="preserve">Quod in hac inventione apprime neceſſarium erat, ne par-<lb/>ticula temporis aliqua quotidie ſubduceretur, neve penduli <lb/>motus interea dum pondus attollitur langueſceret. </s> <s xml:id="echoid-s408" xml:space="preserve">Paratur <lb/>itaque funis continuus atque in ſe rediens, extremitatibus <lb/>apte inter ſe connexis. </s> <s xml:id="echoid-s409" xml:space="preserve">Is primo orbiculum F amplexus, <lb/>aculeis ferreis aſperum, quo melius funis inhæreat, parte <lb/>altera trochleæ, cui pondus primarium Δ alligatum eſt, cir-<lb/>cumvolvitur. </s> <s xml:id="echoid-s410" xml:space="preserve">Hinc aſcendens ſuper orbiculo Ω tranſit, ac <lb/>rurſus deſcendens ſuſtinet trochleam alteram cum appenſo <lb/>minori pondere Ξ; </s> <s xml:id="echoid-s411" xml:space="preserve">unde denuo ad F redit. </s> <s xml:id="echoid-s412" xml:space="preserve">Orbiculus Ω <lb/>(quem demonſtrandi gratia hic inter laminas A B, Y Z, <lb/>ſuſpendimus, nam alioqui commodius thecæ quæ toti horo-<lb/>logio circumdatur affigi ſolet) circumferentiam verſus den-<lb/>ticulos habet, ut in rota L, ſerratos, ac deſuper premen-<lb/>tem elaterem Θ, quo fit ut in alteram tantummodo partem <lb/>volvipoſſit, attracto nimirum fune Π, ac pondere propter-<lb/>ea Δ aſcendente: </s> <s xml:id="echoid-s413" xml:space="preserve">Nam contrarium motum elater dentibus <lb/>occurrens prohibet. </s> <s xml:id="echoid-s414" xml:space="preserve">Crenam autem ſecundum circumferen-<lb/>tiam dicti orbiculi ita cavari oportet, ut funem immiſſum <lb/>nonnihil coarctet conſtringatque, quo minus poſſit immo-<lb/>to orbiculo delabi; </s> <s xml:id="echoid-s415" xml:space="preserve">quem in finem etiam pondus Ξ ad-<lb/>hibetur. </s> <s xml:id="echoid-s416" xml:space="preserve">His ſic conſtitutis, ſemper Δ dimidia ſui gra-<lb/>vitate incumbet funi Φ, motumque horologio continuabit <lb/>etiam dum attracto fune Π in altum attollitur. </s> <s xml:id="echoid-s417" xml:space="preserve">Et hactenus <pb o="11" file="0035" n="35" rhead="HOROLOGIUM."/> quidem quæ ad conſtructionem automati pertinent declaravi-<lb/>mus.</s> <s xml:id="echoid-s418" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s419" xml:space="preserve">Reliquum eſt, ut, quantum idem iis omnibus, quæ ad <lb/>hanc diem in uſu fuere, antecellat, perſpicuum facia-<lb/>mus. </s> <s xml:id="echoid-s420" xml:space="preserve">Satis conſtat plurimas in his erroris & </s> <s xml:id="echoid-s421" xml:space="preserve">inæqualita-<lb/>tis cauſas eſſe. </s> <s xml:id="echoid-s422" xml:space="preserve">Nam & </s> <s xml:id="echoid-s423" xml:space="preserve">in diſponendis rite elimandiſque <lb/>tympanis vel leviſſimum peccatum, continuo motus in-<lb/>conſtantia notabilis conſequitur. </s> <s xml:id="echoid-s424" xml:space="preserve">Tum vero & </s> <s xml:id="echoid-s425" xml:space="preserve">ſiccato <lb/>atque evaneſcente oleo, quod axibus addi ſolet, tardius <lb/>horæ procedunt. </s> <s xml:id="echoid-s426" xml:space="preserve">atque ut hæc abſint, varias tamen, an-<lb/>ni tempeſtatum & </s> <s xml:id="echoid-s427" xml:space="preserve">aeris, mutationes horologia ſentiunt, <lb/>imo præſentiunt nonnunquam: </s> <s xml:id="echoid-s428" xml:space="preserve">& </s> <s xml:id="echoid-s429" xml:space="preserve">frigore quidem plerun-<lb/>que pigriora comperiuntur, æſtu plus æquo properant. <lb/></s> <s xml:id="echoid-s430" xml:space="preserve">Penduli vero cum ſit ea vis ac proprietas, ut neceſſario eo-<lb/>dem ſemper tenore feratur, neque ab eo niſi mutata lon-<lb/>gitudine unquam declinet; </s> <s xml:id="echoid-s431" xml:space="preserve">apparet ſane omnia illa quæ <lb/>diximus incommoda invento noſtro penitus nos ſuſtuliſſe, <lb/>adeo ut niſi tale quod interveniat impedimentum, quo <lb/>horologii motus omnis ſiſtatur, nulla jam curſus ejus <lb/>retardatio aut inæqualitas timenda ſit. </s> <s xml:id="echoid-s432" xml:space="preserve">At enimvero non <lb/>nemini duplicem hic dubitandi cauſam oboriri poſſe <lb/>ſcio. </s> <s xml:id="echoid-s433" xml:space="preserve">Primum, quod differre à pendulo libero no-<lb/>ſtrum hoc videatur, quippe ad ſingulas vibrationes vim <lb/>quandam ac nixum clavulæ Q R ſentiens. </s> <s xml:id="echoid-s434" xml:space="preserve">Deinde <lb/>quod, etiamſi penduli ſimplicis proprietates retineat, <lb/>perque omnia æmuletur, hujus tamen ipſius geminæ <lb/>inæqualitates à nonnullis, qui ſubtiliter hæc perquiſive-<lb/>runt, animadverſæ ſint. </s> <s xml:id="echoid-s435" xml:space="preserve">Hìc illud quod de clavulæ im-<lb/>preſſione dicitur verum eſſe non diffitemur. </s> <s xml:id="echoid-s436" xml:space="preserve">Sed leviſſi-<lb/>mam utique hanc eſſe novimus ratione gravitatis T, quæ <lb/>ſic temperatur, ut tantum non deficiat penduli agita-<lb/>tio, ſed quam minimâ, & </s> <s xml:id="echoid-s437" xml:space="preserve">eâdem tamen latitudine per-<lb/>ſeveret. </s> <s xml:id="echoid-s438" xml:space="preserve">Proinde nihilo concitatior aut minus æquabilis <lb/>hic ipſius motus evadit, quam ſi clavulæ prorſus obno-<lb/>xius non eſſet, pendulumque ſimplex S I T, ut adhuc <lb/>fieri conſuevit, manu impelleretur. </s> <s xml:id="echoid-s439" xml:space="preserve">Et hoc quidem ex- <pb o="12" file="0036" n="36" rhead="CHRISTIANI HUGENII"/> perientia optime comprobat. </s> <s xml:id="echoid-s440" xml:space="preserve">Penduli vero ipſius, quas ad-<lb/>notant, binas inæqualitates, alii autem contra pernegant, <lb/>earum alteram admittimus, ſed vix quicquam horologio no-<lb/>ſtro officientem, alteram plane nullam eſſe adſeverare non <lb/>dubitamus. </s> <s xml:id="echoid-s441" xml:space="preserve">Illud itaque vere aſſerunt, non prorſus æquali <lb/>tempore latiores ejusdem penduli ac anguſtiores vibrationes <lb/>tranſire, ſed his illas paulo plus inſumere, quod facili ex-<lb/>perimento demonſtrari poteſt. </s> <s xml:id="echoid-s442" xml:space="preserve">Nam ſi pendula duo, pon-<lb/>dere ac longitudine æqualia, alterum procul à perpendicu-<lb/>lo, alterum parumper dimoveantur, ſimul dimiſſa, non <lb/>diu in partes eaſdem una ferri cernentur, ſed prævertet il-<lb/>lud cujus exiliores erunt recurſus. </s> <s xml:id="echoid-s443" xml:space="preserve">Verum huic inæqualitati <lb/>noſtrum, uti dixi, horologium minus obnoxium eſt, eo <lb/>quod omnes oſcillationes æquali ſpatio à perpendiculo ex-<lb/>currant. </s> <s xml:id="echoid-s444" xml:space="preserve">Nec tamen in totum expers remanſit, ſi minutiſ-<lb/>ſima quæque, ſicut hac in re fieri neceſſe eſt, conſectari ve-<lb/>limus. </s> <s xml:id="echoid-s445" xml:space="preserve">Contingit ſiquidem vel aëris intemperie, vel operis <lb/>vitio aliquo, ut non ſemper pari vi agitetur clavula Q R, <lb/>unde & </s> <s xml:id="echoid-s446" xml:space="preserve">oſcillationes penduli (licet exiguo diſcrimine) cre-<lb/>ſcere ac rurſus imminui neceſſe eſt. </s> <s xml:id="echoid-s447" xml:space="preserve">Cumque ampliores re-<lb/>curſus arctioribus, ſicut modo dicebam, plus temporis im-<lb/>pendant; </s> <s xml:id="echoid-s448" xml:space="preserve">idcirco nonnulla hinc in horologii motu inæquali-<lb/>tas exiſtit. </s> <s xml:id="echoid-s449" xml:space="preserve">Et huic quidem, utut contemptibilis videri poſ-<lb/>ſit, remedium etiam adhibere ſolebamus, quamdiu ita con-<lb/>ſtructa erant horologia, ut majuſcula eſſet penduli agitatio. <lb/></s> <s xml:id="echoid-s450" xml:space="preserve">Poſtmodum vero, ne remedio opus eſſet, effecimus adhi-<lb/>bito tympano O rotaque P: </s> <s xml:id="echoid-s451" xml:space="preserve">quibus hoc conſequimur, ut <lb/>quamlibet anguſtæ ſint penduli vibrationes, neque eo ſecius <lb/>axis M N, quantum neceſſe eſt, reciproco motu converta-<lb/>tur. </s> <s xml:id="echoid-s452" xml:space="preserve">Nam cum tympani O diametro dupla vel tripla pona-<lb/>tur diameter rotæ P, ſequitur ut hujus exigua licet oſcilla-<lb/>tione, illud tamen ſatis magnam circuitus partem abſolvat. </s> <s xml:id="echoid-s453" xml:space="preserve"><lb/>Sic igitur oſcillationibus univerſis exilioribus redditis, etiam-<lb/>ſi harum aliæ alias latitudine quandoque excedant, ſingula-<lb/>rum tamen tempora, experientia teſte, nullo memorabili <lb/>diſcrimine differunt. </s> <s xml:id="echoid-s454" xml:space="preserve">Qua ex re & </s> <s xml:id="echoid-s455" xml:space="preserve">hoc contingit, ut aucto <pb o="13" file="0037" n="37" rhead="HOROLOGIUM."/> vel ad duplum pondere Δ, non propterea penduli motus <lb/>acceleretur, aut horologii curſus alteretur, quod in omni-<lb/>bus aliis hactenus uſitatis ſecus accidit. </s> <s xml:id="echoid-s456" xml:space="preserve">Alteram penduli in-<lb/>æqualitatem, vir Aſtronomiæ ſtudiis clarus, Gothofr. </s> <s xml:id="echoid-s457" xml:space="preserve">Wen-<lb/>delinus, primus & </s> <s xml:id="echoid-s458" xml:space="preserve">ſolus, ut opinor, prodidit; </s> <s xml:id="echoid-s459" xml:space="preserve">expertum <lb/>ſeſe ſcribens, ejuſdem penduli velociores eſſe oſcillationes <lb/>hyemali tempore quam æſtivo idque notabili differentia. <lb/></s> <s xml:id="echoid-s460" xml:space="preserve">Sed quoniam in eo examine arenaria tantum horologia, <lb/>vulgariaque automata ſeſe adhibuiſſe fatetur, cum ſcioteri-<lb/>cis, fortaſſe non nimia cura deſcriptis; </s> <s xml:id="echoid-s461" xml:space="preserve">multi, quam recte <lb/>ſe haberet hæc ipſius obſervatio, dubitarunt. </s> <s xml:id="echoid-s462" xml:space="preserve">Mihi certe <lb/>nihil ejusmodi licuit animadvertere. </s> <s xml:id="echoid-s463" xml:space="preserve">Quin contra, & </s> <s xml:id="echoid-s464" xml:space="preserve">mino-<lb/>ribus horologiis, quibus ſemipedale eſt pendulum, & </s> <s xml:id="echoid-s465" xml:space="preserve">ma-<lb/>joribus, in quibus 24 fere pedes æquat, eandem perpetuo <lb/>longitudinem, brumæ tempore ac æſtate media, convenire <lb/>expertus ſum. </s> <s xml:id="echoid-s466" xml:space="preserve">Quæ longitudo ſaltem ſeptima ſui parte per <lb/>hyemem productior eſſe deberet, ſi Wendelini vera foret <lb/>opinio.</s> <s xml:id="echoid-s467" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s468" xml:space="preserve">Aſſerta igitur & </s> <s xml:id="echoid-s469" xml:space="preserve">hac in parte automati noſtri motus æ-<lb/>quabilitate & </s> <s xml:id="echoid-s470" xml:space="preserve">conſtantia, finem jam deſcriptioni impone-<lb/>mus; </s> <s xml:id="echoid-s471" xml:space="preserve">multa quæ his addi præterea poſſent, artificum indu-<lb/>ſtriæ relinquentes, qui rationem inventi hujus edocti, non <lb/>difficile reperient quo pacto illud varii generis horologiis, <lb/>atque iis etiam quæ pridem ad veterem formam fabricata <lb/>ſunt, applicari queat. </s> <s xml:id="echoid-s472" xml:space="preserve">Nos quidem apud eum, cujus opera <lb/>primum in his fabricandis uſi ſumus, talia quoque confecta <lb/>vidimus, quæ non pondere, ſed elateris vi moverentur. </s> <s xml:id="echoid-s473" xml:space="preserve">In <lb/>quibus, cum antehac pyramide illa æquatoria, chordaque <lb/>huic circumvoluta opus eſſet, quorum ope adæquarentur <lb/>primi ac poſtremi elateris impetus; </s> <s xml:id="echoid-s474" xml:space="preserve">nunc iis omiſſis, ipſi <lb/>tympano, cui elater incluſus eſt, dentes adduntur. </s> <s xml:id="echoid-s475" xml:space="preserve">Nam <lb/>licet hoc modo non æque in fine ac principio vigeat pendu-<lb/>li motus, non tamen eo lentiores ſub finem oſcillationes effi-<lb/>ciuntur, uti ſuperius fuit demonſtratum. </s> <s xml:id="echoid-s476" xml:space="preserve">Elater vero ea <lb/>parte, qua ad centrum convolutus eſt, intenditur, atque <lb/>ita cavetur ne quo temporis momento curſus horologii ſuf- <pb o="14" file="0038" n="38" rhead="CHRISTIANI HUGENII HOROLOGIUM."/> flaminetur. </s> <s xml:id="echoid-s477" xml:space="preserve">Mitto quod & </s> <s xml:id="echoid-s478" xml:space="preserve">ſonitu horas edentia automata <lb/>hujusmodi machinatus eſt, ita ut uno eodemque, ſive pon-<lb/>dere, ſive elatere, pars utraque, tam quæ ad hoc compa-<lb/>rata eſt, quam quæ indicem horologii verſat, moveretur. <lb/></s> <s xml:id="echoid-s479" xml:space="preserve">Etenim hæc omnia ad inventum noſtrum haud aliter ſpe-<lb/>ctant, quam quod occaſionem iis atque opportunitatem <lb/>præbuerit.</s> <s xml:id="echoid-s480" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div20" type="section" level="1" n="14"> <head xml:id="echoid-head23" xml:space="preserve">FINIS.</head> <figure> <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0038-01"/> </figure> <pb file="0039" n="39"/> <pb file="0039a" n="40"/> <figure> <caption xml:id="echoid-caption1" style="it" xml:space="preserve">Pag. 14<lb/>TAB. I.</caption> <variables xml:id="echoid-variables1" xml:space="preserve">Y A S C O I Q P M Σ Λ L K R H N G Γ Ψ X E V D F Ω B Θ Φ Z Π Δ T Ξ</variables> </figure> <pb file="0040" n="41"/> <pb file="0041" n="42"/> </div> <div xml:id="echoid-div21" type="section" level="1" n="15"> <head xml:id="echoid-head24" xml:space="preserve">CHRISTIANI <lb/>HUGENII</head> <head xml:id="echoid-head25" xml:space="preserve">ZULICHEMII, CONST. F.</head> <head xml:id="echoid-head26" xml:space="preserve">HOROLOGIUM <lb/>OSCILLATORIUM.</head> <head xml:id="echoid-head27" xml:space="preserve">SIVE</head> <head xml:id="echoid-head28" xml:space="preserve">DE MOTU PENDULORUM <lb/>AD HOROLOGIA APTATO <lb/>DEMONSTRATIONES <lb/>GEOMETRICÆ</head> <pb file="0042" n="43"/> </div> <div xml:id="echoid-div22" type="section" level="1" n="16"> <head xml:id="echoid-head29" xml:space="preserve">Dividitur liber hic in partes quinque, <lb/>quarum</head> <p> <s xml:id="echoid-s481" xml:space="preserve">Prima Deſcriptionem <emph style="sc">Horologii</emph> <emph style="sc">Oscillatorii</emph> continet.</s> <s xml:id="echoid-s482" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s483" xml:space="preserve">Secunda agit de Deſcenſugravium, & </s> <s xml:id="echoid-s484" xml:space="preserve">motu eorum in Cycloide.</s> <s xml:id="echoid-s485" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s486" xml:space="preserve">Tertia de Evolutione & </s> <s xml:id="echoid-s487" xml:space="preserve">Dimenſione linearum curvarum.</s> <s xml:id="echoid-s488" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s489" xml:space="preserve">Quarta de Centro Oſcillationis ſeu Agitationis.</s> <s xml:id="echoid-s490" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s491" xml:space="preserve">Quinta alterius Horologii conſtructionem, in quo circularis <lb/>eſt penduli motus, exhibet, & </s> <s xml:id="echoid-s492" xml:space="preserve">Theoremata <lb/>de Vi Centrifuga.</s> <s xml:id="echoid-s493" xml:space="preserve"/> </p> <pb file="0043" n="44"/> <figure> <image file="0043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0043-01"/> </figure> </div> <div xml:id="echoid-div23" type="section" level="1" n="17"> <head xml:id="echoid-head30" xml:space="preserve">LUDOVICO XIV, <lb/>FRANCIÆ ET NAVARRÆ <lb/>REGI INCLYTO.</head> <p> <s xml:id="echoid-s494" xml:space="preserve">RENATAM, Rex maxime, <lb/>reſtitutamque hoc ſæculo <lb/>Geometriam, Galliæ præ-<lb/>cipue debemus. </s> <s xml:id="echoid-s495" xml:space="preserve">Hinc enim <lb/>orti, qui magna meliorique <lb/>ſui parte deperditam, ac <lb/>veluti ſepultam, inſtaura-<lb/>runt primi, & </s> <s xml:id="echoid-s496" xml:space="preserve">in lucem re-<lb/>duxerunt. </s> <s xml:id="echoid-s497" xml:space="preserve">Quorum veſti-<lb/>giis inſiſtentes, ita eam deinde, per totam Eu-<lb/>ropam, excoluere viri ſubtiliſſimi, ut pauca <lb/>jam poſterorum induſtriæ ab his relicta videan-<lb/>tur; </s> <s xml:id="echoid-s498" xml:space="preserve">veterum vero inventa longiſſime præter-<lb/>vecti ſint. </s> <s xml:id="echoid-s499" xml:space="preserve">In hac ſcientia, quam ſemper admi-<lb/>ratus ſum & </s> <s xml:id="echoid-s500" xml:space="preserve">amavi plurimum, quandocunque <lb/>ad eam animum applicui, illa mihi præ cæteris <lb/>propoſui inveſtiganda, quæ vel ad vitæ com-<lb/>moda, vel ad Naturæ cognitionem, reperta <pb o="18" file="0044" n="45" rhead="DEDICATIO."/> prodeſſe poſſent. </s> <s xml:id="echoid-s501" xml:space="preserve">Tunc verò optimè operam me <lb/>collocaſſe exiſtimavi, cum in ea incidiſſem, in <lb/>quibus utilitas cum inveniendi difficultate, ac <lb/>ſubtilitate aliqua, conjuncta foret. </s> <s xml:id="echoid-s502" xml:space="preserve">Quod ſi com-<lb/>mendationis nonnihil accerſere muneri noſtro <lb/>permittitur, ne prorſus indignum tua magnitudi-<lb/>ne appareat; </s> <s xml:id="echoid-s503" xml:space="preserve">non alias felicius, quam in hoc Ho-<lb/>rologii invento, utrumque illud me conſecu-<lb/>tum eſſe profiteor. </s> <s xml:id="echoid-s504" xml:space="preserve">Etenim, cum ex parte me-<lb/>chanicum ſit inventum; </s> <s xml:id="echoid-s505" xml:space="preserve">ex parte altera, eaque <lb/>multò præcipua, geometricis principiis conſtet; <lb/></s> <s xml:id="echoid-s506" xml:space="preserve">id quod ad hanc attinet, non levi conamine, <lb/>ex intimis artis receſſibus petendum fuit: </s> <s xml:id="echoid-s507" xml:space="preserve">adeo <lb/>quidem, ut inter omnia, quæ impenſiore ſtu-<lb/>dio hactenus pertractaverim, haud dubie pri-<lb/>mum huic ſpeculationi locum tribuam. </s> <s xml:id="echoid-s508" xml:space="preserve">Quæ-<lb/>nam vero in his ſit utilitas, non eſt quod mul-<lb/>tis, Rex potentiſſime, oſtendere tibi laborem. </s> <s xml:id="echoid-s509" xml:space="preserve"><lb/>Non ſolum enim diutinâ experientiâ comper-<lb/>tum habes, ex quo regiæ tuæ penetralibus reci-<lb/>pi meruere Automata noſtra, quantum, æqua-<lb/>bili horarum demonſtratione, cæteris hujuſmo-<lb/>di machinationibus excellant: </s> <s xml:id="echoid-s510" xml:space="preserve">ſed & </s> <s xml:id="echoid-s511" xml:space="preserve">potiores <lb/>uſus eorum, quibuſque jam inde à principio <lb/>mihi deſtinata fuere, non ignoras. </s> <s xml:id="echoid-s512" xml:space="preserve">Illos ſcili-<lb/>cet, quos & </s> <s xml:id="echoid-s513" xml:space="preserve">in Cæleſtium obſervationibus, & </s> <s xml:id="echoid-s514" xml:space="preserve"><lb/>in Longitudinibus locorum inter navigandum <lb/>dimetiendis, præſtare apta ſunt. </s> <s xml:id="echoid-s515" xml:space="preserve">Tuo enim <lb/>juſſu, non ſemel, per mare vecta fuere Horo- <pb o="19" file="0045" n="46" rhead="DEDICATIO."/> logia noſtra. </s> <s xml:id="echoid-s516" xml:space="preserve">Tuis auſpiciis eadem nec pauca, <lb/>Aſtronomiæ uſibus dicata, viſuntur in præclara <lb/>illa Uraniæ arce, quam inſigni nuper magniſicen-<lb/>tia, quantaque antehac regum nemo, exædiſi-<lb/>candam curaſti. </s> <s xml:id="echoid-s517" xml:space="preserve">Quæ quoties mecum reputo, <lb/>toties de fortuna hujus inventi, quod in tua <lb/>tempora inciderit, non parum mihi gratulari <lb/>ſoleo. </s> <s xml:id="echoid-s518" xml:space="preserve">Nec jam requiret quiſquam, opinor, <lb/>qui quantum tibi illud debeat intelliget, cur <lb/>lucubrationes has, quibus rationem ejus omnem <lb/>deſcriptionemque explicui, auguſto Nomini <lb/>tuo inſcribendas duxerim. </s> <s xml:id="echoid-s519" xml:space="preserve">Ac minus etiam id <lb/>mirabitur, qui mihi, ad hæc atque alia medi-<lb/>tanda, tranquillum otium benignitate tua con-<lb/>tigiſſe didicerit. </s> <s xml:id="echoid-s520" xml:space="preserve">Namque & </s> <s xml:id="echoid-s521" xml:space="preserve">hujus, ut mihi ali-<lb/>quatenus apud te ratio conſtaret, adnitendum <lb/>erat; </s> <s xml:id="echoid-s522" xml:space="preserve">& </s> <s xml:id="echoid-s523" xml:space="preserve">quoquo modo conandum, ut, multis <lb/>continuiſque à te beneficiis affectus, nonnulla <lb/>grati animi ſignificatione defungerer. </s> <s xml:id="echoid-s524" xml:space="preserve">Scio equi-<lb/>dem, rebus maximis, negotiiſque iis intento, <lb/>quæ in illo rerum faſtigio poſitum agitare con-<lb/>venit, haudquaquam tibi liberum eſſe, ut ad <lb/>hujuſmodi contemplationes animum, alioqui <lb/>rerum omnium capacem, advertas. </s> <s xml:id="echoid-s525" xml:space="preserve">Sed non <lb/>ideo minus grata hæc fore, minusve tibi pro-<lb/>batum iri arbitror, Rex auguſtiſſime; </s> <s xml:id="echoid-s526" xml:space="preserve">cui illa <lb/>maxime placere videmus, quæ plurimum publi-<lb/>cè proſunt; </s> <s xml:id="echoid-s527" xml:space="preserve">neque aliud magis curæ eſſe, quam <lb/>ut nova incrementa ſumant optimæ diſcipſinæ, <pb o="20" file="0046" n="47" rhead="DEDICATIO."/> noviſque illuſtrentur inventis. </s> <s xml:id="echoid-s528" xml:space="preserve">Hoc enim ſatis <lb/>declarat eximia illa tua, ac ſingularis, tum in <lb/>ipſis promovendis, tum in his qui cognitione <lb/>earum præminent remunerandis, liberalitas. <lb/></s> <s xml:id="echoid-s529" xml:space="preserve">Quam non immenſæ, ac ſolito majores, bello-<lb/>rum impenſæ quidquam imminuunt: </s> <s xml:id="echoid-s530" xml:space="preserve">non Gal-<lb/>liæ tuæ fines circunſcribunt. </s> <s xml:id="echoid-s531" xml:space="preserve">Ut plane te hoc <lb/>agere appareat, quo non ſolum ſub imperio <lb/>tuo viventes, ſed & </s> <s xml:id="echoid-s532" xml:space="preserve">Orbis univerſus, quacun-<lb/>que beneficio tuo dignus eſt, te regnante, eru-<lb/>ditior, ornatior, felicior evadat. </s> <s xml:id="echoid-s533" xml:space="preserve">Cui veriſſi-<lb/>mæ præclariſſimæque gloriæ tuæ, ita aliquid <lb/>fortaſſe etiam hæc literaria monumenta condu-<lb/>cent; </s> <s xml:id="echoid-s534" xml:space="preserve">ut, ſi viguiſſe hoc tempore ſtudia iſta, <lb/>arteſque, poſteris teſtari poſſint, ſimul illos <lb/>edoceant, tuæ hoc virtuti, atque animi magni-<lb/>tudini, ante omnia acceptum ferendum eſ-<lb/>ſe. </s> <s xml:id="echoid-s535" xml:space="preserve">Lutetiæ Pariſiorum; </s> <s xml:id="echoid-s536" xml:space="preserve"><emph style="sc">XXV</emph>. </s> <s xml:id="echoid-s537" xml:space="preserve">Mart. </s> <s xml:id="echoid-s538" xml:space="preserve">A. </s> <s xml:id="echoid-s539" xml:space="preserve"><lb/><emph style="sc">CIƆIƆCLXX III</emph>.</s> <s xml:id="echoid-s540" xml:space="preserve"/> </p> <pb o="21" file="0047" n="48"/> <figure> <image file="0047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0047-01"/> </figure> </div> <div xml:id="echoid-div24" type="section" level="1" n="18"> <head xml:id="echoid-head31" xml:space="preserve">HADRIANI VALLII <lb/>DAPHNIS, <lb/>ECLOGA.</head> <p> <s xml:id="echoid-s541" xml:space="preserve">Ad Chriſtianum Hugenium Zulichemium, <lb/>Conſtantini F.</s> <s xml:id="echoid-s542" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s543" xml:space="preserve"><emph style="sc">FInitimum</emph> tutela, ſimul jucunda voluptas, <lb/>Dilectæ Phæbo, Sceverinides <anchor type="note" xlink:href="" symbol="*"/> Oceaninæ;</s> <s xml:id="echoid-s544" xml:space="preserve"> Hunc quoque Pierium mihi fortunate laborem: <lb/></s> <s xml:id="echoid-s545" xml:space="preserve">Pervigilem noctem quo carmine duxerit Ancon <lb/>Navita, d<emph style="super">1</emph>cemus: </s> <s xml:id="echoid-s546" xml:space="preserve">veſtro ſic gurgite numquam <lb/>Pan lavet, aut turpes inceſtent æquora Fauni.</s> <s xml:id="echoid-s547" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s548" xml:space="preserve">Te, quem Fama vehit ſuper aurea ſidera curru, <lb/>Ne pigeat nobis aurem præbere faventem, <lb/><emph style="sc">Hugenide</emph>, decus Hugenidum, fratrumque patrisque: <lb/></s> <s xml:id="echoid-s549" xml:space="preserve">Haud indigna tuo ferimus donaria ſenſu, <lb/>Siceliſin aptata modis à vate Batavo <lb/>Mixta Palæphatio commenta Solenſia verſu, <lb/>Teque intertextum tuaque præclara reperta.</s> <s xml:id="echoid-s550" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s551" xml:space="preserve">Jam caput Oceano, ſtipata minoribus aſtris, <lb/>Extulerat radiis fraternis æmula Phæbe, <lb/>Cum reditum molirentur paſtoria pubes, <lb/>Sidere quam pleno conchas legiſſe marinas <lb/>Juverat, hærentesque vadis captare paguros. <lb/></s> <s xml:id="echoid-s552" xml:space="preserve">In celſo tamen advertunt Ancona morantem <lb/>Colle, reum toties promiſſi carminis. </s> <s xml:id="echoid-s553" xml:space="preserve">ipſum <lb/>Theſtylis & </s> <s xml:id="echoid-s554" xml:space="preserve">Corydon, quos cætera turba ſecuti, <lb/>A tergo circumveniunt, cinguntque corona.</s> <s xml:id="echoid-s555" xml:space="preserve"/> </p> <note symbol="*" position="foot" xml:space="preserve">Sceverina, Pagus apud Batavos, mari adjacens.</note> <pb o="22" file="0048" n="49" rhead="DAPHNIS ECLOGA."/> <p> <s xml:id="echoid-s556" xml:space="preserve">Ecquid agat, rogitant blande: </s> <s xml:id="echoid-s557" xml:space="preserve">tum fauſta precantur; <lb/></s> <s xml:id="echoid-s558" xml:space="preserve">Et damnant voti, promiſſaque carmina poſcunt. </s> <s xml:id="echoid-s559" xml:space="preserve"><lb/>Contra ille; </s> <s xml:id="echoid-s560" xml:space="preserve">O Pueri, quid portet craſtinus Eos, <lb/>Sedi explorator: </s> <s xml:id="echoid-s561" xml:space="preserve">turmales agmine mergi, <lb/>Solivaga aut cornix, aut alcyones deſertæ <lb/>Si qua darent mihi ſigna. </s> <s xml:id="echoid-s562" xml:space="preserve">maris cras æquor arandum. </s> <s xml:id="echoid-s563" xml:space="preserve"><lb/>Detinuit nunc uſque fovis clementia ſudi, <lb/>Et picturatus tot circum animalibus æther. </s> <s xml:id="echoid-s564" xml:space="preserve"><lb/>Quæ nos in vitreo miramur monſtra profundo, <lb/>Fert radians æther, vultus formasque natantum. </s> <s xml:id="echoid-s565" xml:space="preserve"><lb/>Cancer ibi eſt, delphinque; </s> <s xml:id="echoid-s566" xml:space="preserve">eſt grandi corpore cetus. </s> <s xml:id="echoid-s567" xml:space="preserve"><lb/>Ad Boream piſces, & </s> <s xml:id="echoid-s568" xml:space="preserve">contemplere ſub Auſtro <lb/>Piſces; </s> <s xml:id="echoid-s569" xml:space="preserve">nuper ubi numero creviſſe feruntur. </s> <s xml:id="echoid-s570" xml:space="preserve"><lb/>Sunt urna, fluviusque, & </s> <s xml:id="echoid-s571" xml:space="preserve">apluſtris comta carina <lb/>Illic. </s> <s xml:id="echoid-s572" xml:space="preserve">quin operis ſimulamina plurima veſtri, <lb/>Luminaque in cœlo pecori debentia nomen. </s> <s xml:id="echoid-s573" xml:space="preserve"><lb/>Sunt hœdi parvæque ſues, materque capella. </s> <s xml:id="echoid-s574" xml:space="preserve"><lb/>Et fuſe ſparſo quæ candet ſemita lacte. </s> <s xml:id="echoid-s575" xml:space="preserve"><lb/>Veſtibulum ſervant, elucens vellere fulvo <lb/>Dux aries, ingensque auratus cornua taurus. </s> <s xml:id="echoid-s576" xml:space="preserve"><lb/>Bini cernunturque canes, pernoxque bubulcus; </s> <s xml:id="echoid-s577" xml:space="preserve"><lb/>Plauſtraque; </s> <s xml:id="echoid-s578" xml:space="preserve">quique auriga ſuis excuſſus habenis. </s> <s xml:id="echoid-s579" xml:space="preserve"><lb/>Stellatum volat alatus per inane caballus: </s> <s xml:id="echoid-s580" xml:space="preserve"><lb/>Ac præſepe ſuum juxta ſtabulantur aſelli. </s> <s xml:id="echoid-s581" xml:space="preserve"><lb/>Illic virgo, manum Cereali inluſtris ariſta, <lb/>Et, tranſmutatus faciem, Pan ipſe renidet; </s> <s xml:id="echoid-s582" xml:space="preserve"><lb/>Daphnin amans veſtrum, ſecretæ rupis in umbra, <lb/>Uranie velut edocuit: </s> <s xml:id="echoid-s583" xml:space="preserve">me ſingula Daphnis. </s> <s xml:id="echoid-s584" xml:space="preserve"><lb/>Singula quæ (carmen quia poſcitis) ordine pandam.</s> <s xml:id="echoid-s585" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s586" xml:space="preserve">Extemplo tentat vocem: </s> <s xml:id="echoid-s587" xml:space="preserve">numerosque modosque <lb/>Perpendens mulcet variis concentibus auras. <lb/></s> <s xml:id="echoid-s588" xml:space="preserve">Tum venti poſuere. </s> <s xml:id="echoid-s589" xml:space="preserve">jacet ſine fluctibus æquor; </s> <s xml:id="echoid-s590" xml:space="preserve"><lb/>Factaque ſunt terris, ſunt facta ſilentia ponto. </s> <s xml:id="echoid-s591" xml:space="preserve"><lb/>Mox interfatur: </s> <s xml:id="echoid-s592" xml:space="preserve">Quod proſperet; </s> <s xml:id="echoid-s593" xml:space="preserve">ab fove magno <lb/>Ordiar: </s> <s xml:id="echoid-s594" xml:space="preserve">ordiri conſuerunt ab fove vates. </s> <s xml:id="echoid-s595" xml:space="preserve"><lb/>Vos, nec enim rerum brevis hic mihi naſcitur ordo, <pb o="23" file="0049" n="50" rhead="DAPHNIS ECLOGA."/> Nocturnum chorea defendite corpore frigus.</s> <s xml:id="echoid-s596" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s597" xml:space="preserve">Inde fovis magni cunas, veteriſque celebrat <lb/>Saturni juſſum crudele, dolumque Cybelles; <lb/></s> <s xml:id="echoid-s598" xml:space="preserve">Ortaque Dictæis Corybantia ſacra latebris: </s> <s xml:id="echoid-s599" xml:space="preserve"><lb/>Ut puero nutrix ſit olentis lecta mariti <lb/>Uxor; </s> <s xml:id="echoid-s600" xml:space="preserve">& </s> <s xml:id="echoid-s601" xml:space="preserve">ipſa recens hædos enixa gemellos; </s> <s xml:id="echoid-s602" xml:space="preserve"><lb/>Queis comitata polum modo lucida ſtella frequentet, <lb/>Quæ prius Oleniis balavit beſtia campis; </s> <s xml:id="echoid-s603" xml:space="preserve"><lb/>Sub pedibusque terat formoſi limen Olympi. </s> <s xml:id="echoid-s604" xml:space="preserve"><lb/>Tantus amor fovis, & </s> <s xml:id="echoid-s605" xml:space="preserve">percepti gratia lactis.</s> <s xml:id="echoid-s606" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s607" xml:space="preserve">Nec tamen hoc niveum manaſſe fluore nitorem, <lb/>In duo ſecta vias, oculis manifeſta videntum, <lb/>Semita quo candet ducens ad tecta Tonantis; <lb/></s> <s xml:id="echoid-s608" xml:space="preserve">Tergeminam ſed noctem productumque canebat <lb/>Alciden mundo; </s> <s xml:id="echoid-s609" xml:space="preserve">Deus immortalis haberi <lb/>Haud pote qui fuerat, ſopitæ parvula mammis <lb/>Labra pater gnati niſi conjugis admovißet: </s> <s xml:id="echoid-s610" xml:space="preserve"><lb/>Quæ, ſimul experrecta, ſimul conterrita, ſurgens <lb/>Uvidulas tenero mammas ſubtraxerit ori, <lb/>Indignata. </s> <s xml:id="echoid-s611" xml:space="preserve">pavimentum tabulataque cæli <lb/>Deciduus maculis ut tunc infecerit albis <lb/>Per convexa ruens in ſe revolubilis humor. </s> <s xml:id="echoid-s612" xml:space="preserve"><lb/>Orbita cycneo nunc unde bifurca colore, <lb/>Ducta per æquales medio diſcrimine partes, <lb/>Cæruleum velut argento ferruminet axem: </s> <s xml:id="echoid-s613" xml:space="preserve"><lb/>Axem, cervices qui quum laſſaret Atlantis, <lb/>Haud gravis Herculeo requierit ſarcina collo; </s> <s xml:id="echoid-s614" xml:space="preserve"><lb/>Atque tot ærumnas quem poſt, maneſque ſubactos, <lb/>Ipſe ſuis ornet jam portio magna triumphis; </s> <s xml:id="echoid-s615" xml:space="preserve"><lb/>Heſperidum contra cuſtodem divitis horti <lb/>Inſurgens Anguem pede nixus; </s> <s xml:id="echoid-s616" xml:space="preserve">apertaque retro <lb/>Terribili rictu nil curans ora Leonis; </s> <s xml:id="echoid-s617" xml:space="preserve"><lb/>Lerneæque audacem Hydræ ſuccurrere Cancrum; </s> <s xml:id="echoid-s618" xml:space="preserve"><lb/>Monſtra novercales teſtantia jugiter iras <lb/>Et fruſtra bacchatum odium funonis iniquæ.</s> <s xml:id="echoid-s619" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s620" xml:space="preserve">Hinc aliam memorat graſſatam fraude novercam;</s> <s xml:id="echoid-s621" xml:space="preserve"> <pb o="24" file="0050" n="51" rhead="DAPHNIS ECLOGA."/> Et tranſmittendi pavidam nimis æquoris Hellen: <lb/></s> <s xml:id="echoid-s622" xml:space="preserve">In thalamos ſit ut illa tuos, Neptune, recepta: </s> <s xml:id="echoid-s623" xml:space="preserve"><lb/>Phryxeumque pecus, fætamque heroibus Argo <lb/>Phaſidos ad fluctus deducit & </s> <s xml:id="echoid-s624" xml:space="preserve">æthera cantu.</s> <s xml:id="echoid-s625" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s626" xml:space="preserve">Nec ſilet Europæ vectoris præmia; </s> <s xml:id="echoid-s627" xml:space="preserve">vel te, <lb/>Bigarum Pelopis perjuri, Myrtile, rector. <lb/></s> <s xml:id="echoid-s628" xml:space="preserve">Myrtoum pelagus ſignaras ante caduco <lb/>Funere; </s> <s xml:id="echoid-s629" xml:space="preserve">ſublimem nunc tollunt cornua Tauri.</s> <s xml:id="echoid-s630" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s631" xml:space="preserve">Haud procul his Hyades notat exardeſcere: </s> <s xml:id="echoid-s632" xml:space="preserve">ſed, quæ <lb/>Sunt Hyades Grajis, Suculas dixiſſe Latinos; <lb/></s> <s xml:id="echoid-s633" xml:space="preserve">Atque duas ſeptem mutaſſe Trionibus Arctos; </s> <s xml:id="echoid-s634" xml:space="preserve"><lb/>Arctophylaca pigro, ſua plauſtra ſequente, Bubulco; </s> <s xml:id="echoid-s635" xml:space="preserve"><lb/>Quando bovem priſco vocitabant more trionem, <lb/>Quod tereret duro proſciſſam vomere terram.</s> <s xml:id="echoid-s636" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s637" xml:space="preserve">Hanc adeo ſortem miſerans, ſuſpiria ducit; <lb/></s> <s xml:id="echoid-s638" xml:space="preserve">Buceriumque genus queſtu compellat inani; </s> <s xml:id="echoid-s639" xml:space="preserve"><lb/>Ah pecus infelix, armentum! ſæcla fuerunt, <lb/>Pondere quum duro neque vos gemeretis aratri, <lb/>Navita nec veſtro vocitaret nomine ſtellas. </s> <s xml:id="echoid-s640" xml:space="preserve"><lb/>Tunc neque ſidus erat terris pia Virgo relictis, <lb/>Quæ Cereale manu ſpicum gerit; </s> <s xml:id="echoid-s641" xml:space="preserve">Icariotis <lb/>Sive ſit Erigone@ cui fida Canicula patrem <lb/>Quærenti indigna monſtravit cæde peremtum; </s> <s xml:id="echoid-s642" xml:space="preserve"><lb/>Atque, comes dominæ, domino comitem Oarioni <lb/>Aſtra minor ſocium majorem repperit inter: </s> <s xml:id="echoid-s643" xml:space="preserve"><lb/>Seu magis Aſtræi ſit ſanguine creta, perenne <lb/>De genitore ſuo quæ nomen contulit aſtris: </s> <s xml:id="echoid-s644" xml:space="preserve"><lb/>Sive ſit antiquæ Themidis juſtiſſima proles, <lb/>Averſata jugo vos aſpectare gravari, <lb/>Tempora dum, pulſis melioribus, ærea ſurgunt: </s> <s xml:id="echoid-s645" xml:space="preserve"><lb/>Sive ſit alma Ceres; </s> <s xml:id="echoid-s646" xml:space="preserve">horrens fugitiva videre <lb/>Vos quoque mactari; </s> <s xml:id="echoid-s647" xml:space="preserve">nil pejor linquit inauſum <lb/>Ferrea dum ſoboles, ipſorum inimica Deorum: </s> <s xml:id="echoid-s648" xml:space="preserve"><lb/>Quos, quaſi de terra (nam Dii coluiſtis & </s> <s xml:id="echoid-s649" xml:space="preserve">illam) <lb/>Sit pepuliſſe parum, tentavit pellere cælo.</s> <s xml:id="echoid-s650" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s651" xml:space="preserve">Tum deteſtatur ſuffultos angue Gigantas;</s> <s xml:id="echoid-s652" xml:space="preserve"> <pb o="25" file="0051" n="52" rhead="DAPHNIS ECLOGA."/> Porphyriona, ſtatu terrentem cuncta minaci; <lb/></s> <s xml:id="echoid-s653" xml:space="preserve">Rhæcumque; </s> <s xml:id="echoid-s654" xml:space="preserve">immanemque Gygen, validumque Mimanta; </s> <s xml:id="echoid-s655" xml:space="preserve"><lb/>Enceladumque; </s> <s xml:id="echoid-s656" xml:space="preserve">manusque rotantem Ægeona centum; </s> <s xml:id="echoid-s657" xml:space="preserve"><lb/>Et, cui par nemo feritate, Typhöea dirum, <lb/>Auſos invaſiſſe Deos tellure fugatos, <lb/>Ac totum magno cælum compleſſe tumultu, <lb/>Undique divulſas jaculantes torviter ornos <lb/>De tumulis cumulorum montibus ex aggeſtis. </s> <s xml:id="echoid-s658" xml:space="preserve"><lb/>Terrigenam ut pubem, Divûm penetralia ſancta <lb/>Rimantem, Superi mentito fallere vultu <lb/>Quæſierint, addit; </s> <s xml:id="echoid-s659" xml:space="preserve">dispertitosque pavore: </s> <s xml:id="echoid-s660" xml:space="preserve"><lb/>Donec apud late ſtagnantis flumina Nili <lb/>Horrificam faciem Pan ſumſerit Ægocerotis; </s> <s xml:id="echoid-s661" xml:space="preserve"><lb/>Ambiguoque ſono Superos animarit ad arma, <lb/>Anguipedesque metu dare terga coëgerit omnes: </s> <s xml:id="echoid-s662" xml:space="preserve"><lb/>Cælo donandos Aſinos auxiſſe timorem <lb/>Congerie vocum, perterricrepoque fragore: </s> <s xml:id="echoid-s663" xml:space="preserve"><lb/>Illa cælicolis nam tempeſtate fuiſſe <lb/>Auxilio Satyros, Silenorumque phalangem, <lb/>Evantes in aſellis cum Bacchæo ululatu, <lb/>Thyrſis armatos, tectos colocynthide parma.</s> <s xml:id="echoid-s664" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s665" xml:space="preserve">Parvus ut interea volucer cum matre Cupido <lb/>Venerit Aſſyrii fugiens Euphratis ad undam; <lb/></s> <s xml:id="echoid-s666" xml:space="preserve">Induerintque gregis (Syriæ poſt numina genti) <lb/>Squammigerum formas, gemini nunc aurea Piſces <lb/>Lumina, ſigniferum Capricorno juncta per orbem, <lb/>Ni fuſa medius ſecernat Aquarius Urna; </s> <s xml:id="echoid-s667" xml:space="preserve"><lb/>Deucalioneos neque non ediſſerit imbres, <lb/>Nectaris aut quanti Ganymedes pocula verſet; </s> <s xml:id="echoid-s668" xml:space="preserve"><lb/>Sive ſit is Cecrops, peplo præſignis Athenæ; </s> <s xml:id="echoid-s669" xml:space="preserve"><lb/>Paſtor Ariſtæus ſeu plena alvearia geſtet, <lb/>Quæ ſubter volitetis apes examine denſo.</s> <s xml:id="echoid-s670" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s671" xml:space="preserve">Qualiter & </s> <s xml:id="echoid-s672" xml:space="preserve">pandus vectarit Ariona Delphin, <lb/>Ac aliter vectum Danaeium Perſea narrat; <lb/></s> <s xml:id="echoid-s673" xml:space="preserve">Cepheaque, Andromedenque, & </s> <s xml:id="echoid-s674" xml:space="preserve">mæſtam Caſſiopeiam; </s> <s xml:id="echoid-s675" xml:space="preserve"><lb/>Inſertumque polo vaſtum Piſtricis hiatum:</s> <s xml:id="echoid-s676" xml:space="preserve"> <pb o="26" file="0052" n="53" rhead="DAPHNIS ECLOGA."/> Quem Phaëthonteus longo ſinuamine propter <lb/>Fulgeat Eridanus declivi proximus Auſtro: <lb/></s> <s xml:id="echoid-s677" xml:space="preserve">Nuper ad occulti Batavos ubi verticis axem <lb/>Intuitos nova ſquammigerum ſimulacra micare: </s> <s xml:id="echoid-s678" xml:space="preserve"><lb/>Sollertes Batavos, imo ſeu gurgite piſcem <lb/>Venari ſit opus, vel in alto ſidera cælo.</s> <s xml:id="echoid-s679" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s680" xml:space="preserve">Tum canit, ut Daphnis ſacra ſub rupe docentem <lb/>Viderit Uranien: </s> <s xml:id="echoid-s681" xml:space="preserve">argutas carmina ſilvas, <lb/>Et repetita cavos ediſcere carmina montes: <lb/></s> <s xml:id="echoid-s682" xml:space="preserve">Ut Chaldæa vetus, mira dulcedine capti, <lb/>Stent auditores circum, & </s> <s xml:id="echoid-s683" xml:space="preserve">Babylonia turba; </s> <s xml:id="echoid-s684" xml:space="preserve"><lb/>Dein quos Graja tulit, quos aut Nilotica tellus, <lb/>Itala quos, ac pulchra ſuo cum Cæſare Roma; </s> <s xml:id="echoid-s685" xml:space="preserve"><lb/>Poſt Arabum de ſtirpe viri & </s> <s xml:id="echoid-s686" xml:space="preserve">regnator Iberus; </s> <s xml:id="echoid-s687" xml:space="preserve"><lb/>Ac tandem quos conſultos Germania miſit <lb/>Aſtrorum cælique, ſuæ qui ſidera terræ; </s> <s xml:id="echoid-s688" xml:space="preserve"><lb/>Inferior nullis ut item neque Gallia deſit; </s> <s xml:id="echoid-s689" xml:space="preserve"><lb/>Gallia magnanimi Regis ſplendore ſuperba, <lb/>Borbonios ignes cui parturit arduus æther:</s> <s xml:id="echoid-s690" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s691" xml:space="preserve">Tum Dea quo Daphnin, Divam quo Daphnis amore <lb/>Complexus, quanti non conſcia Latmia ſaxa: <lb/></s> <s xml:id="echoid-s692" xml:space="preserve">Utque Conon juveni radium donarit, utrimque <lb/>Multo inſignem auro, & </s> <s xml:id="echoid-s693" xml:space="preserve">pellucidulis cryſtallis; </s> <s xml:id="echoid-s694" xml:space="preserve"><lb/>Per quas quod ſpectes, prope fiat; </s> <s xml:id="echoid-s695" xml:space="preserve">& </s> <s xml:id="echoid-s696" xml:space="preserve">augmina ſumat: </s> <s xml:id="echoid-s697" xml:space="preserve"><lb/>Dixerit &</s> <s xml:id="echoid-s698" xml:space="preserve">: Sollers, en, primus quale Batavus <lb/>Munus adornarit; </s> <s xml:id="echoid-s699" xml:space="preserve">ſed Etruſci quo decus Arni <lb/>Eſt Antenorea ſenior Tyrrhenus in urbe <lb/>Regna fovis princeps metatus, ab æthere vobis <lb/>Nunquam nota prius miracula nuntia portans; </s> <s xml:id="echoid-s700" xml:space="preserve"><lb/>Lunaï montes; </s> <s xml:id="echoid-s701" xml:space="preserve">vultus tibi, Phoſphore, ternos; </s> <s xml:id="echoid-s702" xml:space="preserve"><lb/>Quove ſatellitio ſubluſtri nocte vagetur <lb/>Stella Deûm regis per cærula templa ſuperne. </s> <s xml:id="echoid-s703" xml:space="preserve"><lb/>Hoc quoque tu non nota prius miracula prodes: </s> <s xml:id="echoid-s704" xml:space="preserve"><lb/>Hujus erat tibi ſervatus ſollertior uſus; </s> <s xml:id="echoid-s705" xml:space="preserve"><lb/>Arcanumque Chroni mortalibus omne recludes. </s> <s xml:id="echoid-s706" xml:space="preserve"><lb/>Accipe fruſtra olim nobis optabile donum.</s> <s xml:id="echoid-s707" xml:space="preserve"/> </p> <pb o="27" file="0053" n="54" rhead="DAPHNIS ECLOGA."/> <p style="it"> <s xml:id="echoid-s708" xml:space="preserve">Daphnidis ad gratum nomen pernice chorea <lb/>Exſultant alacres Pueri: </s> <s xml:id="echoid-s709" xml:space="preserve">neque ſegnius ipſe <lb/>Proſequitur; </s> <s xml:id="echoid-s710" xml:space="preserve">Geminas imitantia lumina falces <lb/>Hactenus ut vane Saturni credita ſidus <lb/>Oblongo tam diverſa ſub imagine diſco <lb/>Fingere, quando globum teretem teres annulus extra <lb/>Splendet, & </s> <s xml:id="echoid-s711" xml:space="preserve">ambo nigror ſpatii diſterminat intus; <lb/></s> <s xml:id="echoid-s712" xml:space="preserve">Exiguo circum quos erret ſtellula gyro: </s> <s xml:id="echoid-s713" xml:space="preserve"><lb/>Omnia divino quæ fretus munere Daphnis <lb/>Extulerit, non ante novam vulgata per artem: </s> <s xml:id="echoid-s714" xml:space="preserve"><lb/>Adjungitque; </s> <s xml:id="echoid-s715" xml:space="preserve">quod his meritis permulſus, eundem <lb/>In ſua magna Chronus ſit adire ſacraria paſſus: </s> <s xml:id="echoid-s716" xml:space="preserve"><lb/>Heic oculis luſtrarit ut omnia; </s> <s xml:id="echoid-s717" xml:space="preserve">promſerit atque <lb/>Inventum ſubtile ſecandi temporis illinc; </s> <s xml:id="echoid-s718" xml:space="preserve"><lb/>Partes quo minimas ac momina dividat horæ, <lb/>Oſcilla ex tenui ſuſpendens mollia filo: </s> <s xml:id="echoid-s719" xml:space="preserve"><lb/>Id labyrintheos curſus qui dirigat alni, <lb/>Ignarumque viæ ratis haud ſinat eſſe magiſtrum: </s> <s xml:id="echoid-s720" xml:space="preserve"><lb/>Cui neque quotidie tam certus ſpondeat auctor, <lb/>Oceano quantum Titan altiſſimus exſtet; </s> <s xml:id="echoid-s721" xml:space="preserve"><lb/>Ac quibus emergat, queis tunc ſimul occidat oris, <lb/>Daphnidos egregio norint conamine docti.</s> <s xml:id="echoid-s722" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s723" xml:space="preserve">Ille canit: </s> <s xml:id="echoid-s724" xml:space="preserve">chorus in numerum ſua brachia quaſſant, <lb/>Alternoque ſolum pede pulſant. </s> <s xml:id="echoid-s725" xml:space="preserve">at freta ſaltu <lb/>Librabant hilares ſeſe ſuper humida thynni. <lb/></s> <s xml:id="echoid-s726" xml:space="preserve">Auritus leporum populus tunc creditur ultro <lb/>Iliceas liquiſſe domos, carasque quietes <lb/>Vicini nemoris: </s> <s xml:id="echoid-s727" xml:space="preserve">nulloque frequentior unquam <lb/>Caricis arroſor prodiiſſe cuniculus antris <lb/>Tempore narratur; </s> <s xml:id="echoid-s728" xml:space="preserve">narrent ſi vera puellæ <lb/>Littoreæ, quæ ſiccandis cuſtodia paſſim <lb/>Retibus ad ventos expanſis forte ſedebant. </s> <s xml:id="echoid-s729" xml:space="preserve"><lb/>Pectore Nereïdes nudo, laſciva caterva, <lb/>Viſa per incertam Lunam, viſæve putantur, <lb/>Et Triton, Glaucusque, procul ſub luce maligna; </s> <s xml:id="echoid-s730" xml:space="preserve"><lb/>Tuque, cubans juxta ſtratas prope littora phocas, <pb o="28" file="0054" n="55" rhead="DAPHNIS ECLOGA."/> Neptuninarum pecudum fidiſſime cuſtos: <lb/></s> <s xml:id="echoid-s731" xml:space="preserve">Neu quisquam ſeræ meminit decedere nocti. </s> <s xml:id="echoid-s732" xml:space="preserve"><lb/>Interea tenebræ denſantur; </s> <s xml:id="echoid-s733" xml:space="preserve">& </s> <s xml:id="echoid-s734" xml:space="preserve">abdita nimbo <lb/>Cynthia dum latitat, cæli de parte ſerena <lb/>Cinctum non ſolitis proceſſit crinibus aſtrum, <lb/>Prolixumque trahens albore notabile ſyrma. </s> <s xml:id="echoid-s735" xml:space="preserve"><lb/>Mirantur chorus attoniti. </s> <s xml:id="echoid-s736" xml:space="preserve">miratur & </s> <s xml:id="echoid-s737" xml:space="preserve">ipſe; </s> <s xml:id="echoid-s738" xml:space="preserve"><lb/>Præſertim tantum capiti cum demſit honorem, <lb/>Ornatumque ſequacem omnem mox reddita Luna. </s> <s xml:id="echoid-s739" xml:space="preserve"><lb/>Infit &</s> <s xml:id="echoid-s740" xml:space="preserve">: Ad ſua quisque mapalia tendite nota, <lb/>Prodigio nil ſolliciti, curamve foventes. </s> <s xml:id="echoid-s741" xml:space="preserve"><lb/>Inſuetos alias tales cantabimus ignes, <lb/>Et trepidantem nequicquam formidine vulgum.</s> <s xml:id="echoid-s742" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s743" xml:space="preserve">Hæc Ancon: </s> <s xml:id="echoid-s744" xml:space="preserve">mihi viſa tibi quæ digna referri, <lb/><emph style="sc">Hugenide</emph>, decus Hugenidum, cui ſidera curæ, <lb/>Nec Phæbum ac Pimplæ fas eſt contemnere Divas, <lb/>Queis tua tota domus, fratres, genitorque dicati. <lb/></s> <s xml:id="echoid-s745" xml:space="preserve">Sic neque te facies peregrini terreat aſtri, <lb/>Idemve anne alius vario fulgore cometes.</s> <s xml:id="echoid-s746" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s747" xml:space="preserve">A. </s> <s xml:id="echoid-s748" xml:space="preserve"><emph style="sc">CIƆ IƆC LXV</emph>.</s> <s xml:id="echoid-s749" xml:space="preserve"/> </p> <pb file="0055" n="56"/> <figure> <image file="0055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0055-01"/> </figure> </div> <div xml:id="echoid-div25" type="section" level="1" n="19"> <head xml:id="echoid-head32" xml:space="preserve">CHRISTIANI HUGENII</head> <head xml:id="echoid-head33" xml:space="preserve">ZULICHEMII, CONST. F.</head> <head xml:id="echoid-head34" xml:space="preserve">HOROLOGIUM <lb/>OSCILLATORIUM,</head> <head xml:id="echoid-head35" xml:space="preserve">SIVE <lb/>DE MOTU PENDULORUM <lb/>AD HOROLOGIA APTATO</head> <head xml:id="echoid-head36" style="it" xml:space="preserve">Demonſtrationes Geometricæ.</head> <p> <s xml:id="echoid-s750" xml:space="preserve">ANNUS agitur ſextus decimus ex quo fabricam <lb/>horologiorum, tunc recens à nobis inventorum, <lb/>edito libello <anchor type="note" xlink:href="" symbol="*"/> publicam fecimus. </s> <s xml:id="echoid-s751" xml:space="preserve">Ab illo verò <anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/> tempore cùm multa invenerimus ad perfectio-<lb/>nem operis ſpectantia, viſum eſt ea ſingula hoc <lb/>libro exponere. </s> <s xml:id="echoid-s752" xml:space="preserve">Quæ quidem adeo ad perfectionem ejus in-<lb/>venti pertinent, ut potiſſima ejus pars cenſeri poſſint, ac <lb/>velut fundamentum totius mechanicæ hujus, quo prius de- <pb o="30" file="0056" n="57" rhead="CHRISTIANI HUGENII"/> ſtituta erat. </s> <s xml:id="echoid-s753" xml:space="preserve">Menſura enim temporis certa atque æqualis, <lb/>pendulo ſimplici naturâ non inerat, cum latiores excurſus <lb/>anguſtioribus tardiores obſerventur; </s> <s xml:id="echoid-s754" xml:space="preserve">ſed geometria duce di-<lb/>verſam ab ea, ignotamque antea penduli ſuſpenſionem re-<lb/>perimus, animadverſâ lineæ cujusdam curvaturâ, quæ ad <lb/>optatam æqualitatem illi conciliandam mirabili planè ratione <lb/>comparata eſt. </s> <s xml:id="echoid-s755" xml:space="preserve">Quam poſtquam horologiis adhibuimus, <lb/>tam conſtans certusque eorum motus evaſit, ut poſt crebra <lb/>experimenta terra marique capta, manifeſtum jam ſit & </s> <s xml:id="echoid-s756" xml:space="preserve"><lb/>Aſtronomiæ ſtudiis & </s> <s xml:id="echoid-s757" xml:space="preserve">arti Nauticæ plurimùm in iis eſſe <lb/>præſidii. </s> <s xml:id="echoid-s758" xml:space="preserve">Hæc ea eſt linea quam defixus in circumferentia <lb/>currentis rotæ clavus, continua circumvolutione, in aëre <lb/>deſignat; </s> <s xml:id="echoid-s759" xml:space="preserve">à Geometris noſtri ævi cycloidis nomine donata, <lb/>& </s> <s xml:id="echoid-s760" xml:space="preserve">ob alias multas ſui proprietates diligenter expenſa; </s> <s xml:id="echoid-s761" xml:space="preserve">à no-<lb/>bis verò propter eam quam diximus menſurandi temporis <lb/>facultatem, quam nihil tale ſuſpicantes, ac tantùm artis ve-<lb/>ſtigiis inſiſtentes, ineſſe ipſi comperimus. </s> <s xml:id="echoid-s762" xml:space="preserve">Hanc cum jam <lb/>pridem amicis horum intelligentibus notam fecerimus (nam <lb/>non multo poſt primam horologii editionem animadverſa <lb/>fuit) nunc eandem, demonſtratione quàm potuimus accu-<lb/>ratiſſima firmatam, omnibus legendam proponimus. </s> <s xml:id="echoid-s763" xml:space="preserve">Itaque <lb/>in hac tradenda demonſtratione potiſſima pars hujus libri <lb/>verſabitur. </s> <s xml:id="echoid-s764" xml:space="preserve">Ubi primùm neceſſe fuit novis nonnullis demon-<lb/>ſtrationibus ſtabilire & </s> <s xml:id="echoid-s765" xml:space="preserve">promovere ulterius viri maximi Ga-<lb/>lilei de deſcenſu gravium doctrinam, cujus fructus deſidera-<lb/>tiſſimus, atque apex veluti ſummus, hæc ipſa quam inve-<lb/>nimus cycloidis eſt proprietas.</s> <s xml:id="echoid-s766" xml:space="preserve"/> </p> <div xml:id="echoid-div25" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">Vide <lb/>ſupra <lb/>pag. 5.</note> </div> <p> <s xml:id="echoid-s767" xml:space="preserve">Quæ porro ut ad pendulorum uſum aptari poſſet, nova cur-<lb/>varum linearum conſideratio adhibenda fuit, earum ſcilicet <pb o="31" file="0057" n="58" rhead="HOROLOG. OSCILLATOR."/> quæ ſui evolutione alias curvas generant. </s> <s xml:id="echoid-s768" xml:space="preserve">Unde comparatio in-<lb/>ter ſe longitudinis curvarum cum rectis naſcitur, quam ulterius <lb/>etiam quam præſens neceſſitas poſtulabat proſecutus ſum, pro-<lb/>pter theoriæ, ut mihi viſum eſt, elegantiam & </s> <s xml:id="echoid-s769" xml:space="preserve">novitatem.</s> <s xml:id="echoid-s770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s771" xml:space="preserve">Cæterùm ad explicandam Penduli Compoſiti naturam, <lb/>cujus utilitatem in conſtructione horum automatôn demon-<lb/>ſtro, adjungenda fuit Centrorum Oſcillationis contemplatio, <lb/>à pluribus quidem, ſed minus feliciter, hactenus tentata; <lb/></s> <s xml:id="echoid-s772" xml:space="preserve">in qua theoremata complura animadverſione, ni fallor, di-<lb/>gna reperientur, ad figuras lineares, planas, ſolidasque per-<lb/>tinentia. </s> <s xml:id="echoid-s773" xml:space="preserve">Ante hæc omnia vero præmittitur ipſa horologii <lb/>mechanica conſtructio, pendulique applicatio, eâ formâ quæ <lb/>ad uſus aſtronomicos aptiſſima reperta eſt, ad cujus inſtar re-<lb/>liquæ omnes, mutatis quæ opus eſt, facile ordinari poſ-<lb/>ſint.</s> <s xml:id="echoid-s774" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s775" xml:space="preserve">Quia vero contigit egregio hujus inventi ſucceſſu, quod <lb/>fieri plerumque ſolet, quodque futurum prædixeram, ut <lb/>plures ſeſe ejus auctores eſſe cuperent, aut ſi non ſibi ipſis, <lb/>ſuæ tamen nationis alicui potius quàm nobis eum honorem <lb/>tribui vellent, iniquis eorum conatibus tandem aliquando <lb/>occurrendum hic arbitror. </s> <s xml:id="echoid-s776" xml:space="preserve">Nec ſanè aliud fere opponere iis <lb/>neceſſe fuerit præterquam id unum, nempe ante annos ſex-<lb/>decim, cum nec dicto nec ſcripto cujusquam de horologiis <lb/>hujusmodi mentio facta eſſet, aut rumor ullus omnino ferre-<lb/>tur (loquor autem de penduli ſimplicis uſu ad horologia <lb/>translato, nam de Cycloidis additione nemo credo contro-<lb/>verſiam movebit) conſtructionem eorum propria meditatio-<lb/>ne me adinveniſſe & </s> <s xml:id="echoid-s777" xml:space="preserve">perficiendam curaſſe. </s> <s xml:id="echoid-s778" xml:space="preserve">In ſequenti anno, <lb/>qui nempe hujus ſæculi quinquageſimus octavus fuit, deli- <pb o="32" file="0058" n="59" rhead="CHR. HUGENII HOROL. OSCILL."/> neationem automati deſcriptionemque typis vulgaſſe; </s> <s xml:id="echoid-s779" xml:space="preserve">exem-<lb/>plaria, tum operis ipſius, tum libelli, quaquaverſum dimi-<lb/>ſiſſe. </s> <s xml:id="echoid-s780" xml:space="preserve">Nam cum hæc ita omnibus nota ſint, ut nec teſtimo-<lb/>niis eruditorum, nec Bataviæ Ordinum actis, quibus poſ-<lb/>ſent, confirmari opus habeant, facile apparet quid de illis <lb/>exiſtimandum ſit, qui ſeptem poſt annis eandem conſtru-<lb/>ctionem, quaſi à ſe ſuisve amicis profectam, libris ſuis ven-<lb/>ditarunt. </s> <s xml:id="echoid-s781" xml:space="preserve">Qui vero Galileo primas hic deferre conantur, ſi <lb/>tentaſſe eum, non vero perfeciſſe inventum dicant, illius <lb/>magis quam meæ laudi detrahere videntur, quippe qui rem <lb/>eandem, meliore quam ille eventu, inveſtigaverim. </s> <s xml:id="echoid-s782" xml:space="preserve">Cum <lb/>autem vel ab ipſo Galileo, vel à filio ejus, quod nuper vo-<lb/>luit vir quidam eruditus, ad exitum perductum fuiſſe con-<lb/>tendunt, horologiaque ejusmodi re ipſâ exhibita, neſcio <lb/>quomodo ſibi creditum iri ſperent, cum vix veriſimile ſit <lb/>adeo utile inventum ignoratum manere potuiſſe annis totis <lb/>octo, donec à me in lucem ederetur. </s> <s xml:id="echoid-s783" xml:space="preserve">Quod ſi deditâ operâ <lb/>celatum fuiſſe dicant, idem hoc intelligunt à quolibet alio <lb/>poſſe obtendi, qui ſibi originem inventi arrogare cupiat. <lb/></s> <s xml:id="echoid-s784" xml:space="preserve">Itaque probandum quidem id foret, neque eo magis ad me <lb/>tamen quicquam pertineret, niſi unà quoque oſtendatur, id <lb/>quod omnes latebat, mihi ſoli innotuiſſe. </s> <s xml:id="echoid-s785" xml:space="preserve">Et hæc quidem <lb/>neceſſariæ defenſionis cauſa dicenda fuere. </s> <s xml:id="echoid-s786" xml:space="preserve">Nunc ad ipſius <lb/>automati conſtructionem pergamus.</s> <s xml:id="echoid-s787" xml:space="preserve"/> </p> <figure> <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0058-01"/> </figure> <pb file="0059" n="60"/> <figure> <image file="0059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0059-01"/> </figure> </div> <div xml:id="echoid-div27" type="section" level="1" n="20"> <head xml:id="echoid-head37" xml:space="preserve">HOROLOGII OSCILLATORII</head> <head xml:id="echoid-head38" style="it" xml:space="preserve">PARS PRIMA, <lb/>Deſcriptionem ejus continens.</head> <p> <s xml:id="echoid-s788" xml:space="preserve">FIGURA adſcripta horologium à latere inſpiciendum præ-<lb/> <anchor type="note" xlink:label="note-0059-01a" xlink:href="note-0059-01"/> bet, ubi primum laminæ binæ ſunt A A, B B, ſemipe-<lb/>dali aut paulo ultra longitudine, latæ pollices duo & </s> <s xml:id="echoid-s789" xml:space="preserve">ſemis, <lb/>quarum anguli quatuor columellis coaptantur, ut ſesquipol-<lb/>lice inter ſe diſtent. </s> <s xml:id="echoid-s790" xml:space="preserve">His laminis rotarum præcipuarum axes <lb/>utrinque inſeruntur. </s> <s xml:id="echoid-s791" xml:space="preserve">Prima atque infima eſt quæ notatur C, <lb/>dentibus 80 inciſa, cujus axi orbiculus quoque D affixus <lb/>eſt, aculeis ferreis aſper, ut funem cum appenſis ponderi-<lb/>bus contineat, quæ qua ratione ordinentur poſtea dicetur. <lb/></s> <s xml:id="echoid-s792" xml:space="preserve">Ponderis itaque vi rota C vertitur; </s> <s xml:id="echoid-s793" xml:space="preserve">hæc movet proximum <lb/>tympanum E dentium octo, unáque rotam F eodem axe hæ-<lb/>rentem, cui dentes 48. </s> <s xml:id="echoid-s794" xml:space="preserve">Hanc excipit tympanum aliud G, <lb/>& </s> <s xml:id="echoid-s795" xml:space="preserve">in eodem axe rota H, quibus dentium numerus idem qui <lb/>tympano rotæque præcedenti. </s> <s xml:id="echoid-s796" xml:space="preserve">Sed hæc rota ejus eſt generis <lb/>quas à forma coronarias vocant artifices noſtri. </s> <s xml:id="echoid-s797" xml:space="preserve">Hujus den-<lb/>tibus agitatur tympanum I ſimulque rota K, quæ eodem <lb/>axe tenetur, ad perpendiculum erecto. </s> <s xml:id="echoid-s798" xml:space="preserve">Tympano dentes 24; </s> <s xml:id="echoid-s799" xml:space="preserve"><lb/>rotæ 15, atque hi ad inſtar ſerræ dentium inciſi. </s> <s xml:id="echoid-s800" xml:space="preserve">Supra me-<lb/>diam rotam K transverſus jacet axis pinnatus L M, cujus <lb/>extrema ſuſtinent hinc inde gnomones N Q & </s> <s xml:id="echoid-s801" xml:space="preserve">P, ſeorſim <lb/>affixi laminæ B B. </s> <s xml:id="echoid-s802" xml:space="preserve">Notanda vero in gnomone N Q pars de-<lb/>orſum prominens Q, quæ oblongo foramine patens trans-<lb/>mittit axem L M, ſimulque retinet eum quem rotæ K tym-<lb/>panoque I communem eſſe diximus, inferiori ſui parte gno-<lb/>moni R innitentem. </s> <s xml:id="echoid-s803" xml:space="preserve">In lamina B B foramen amplum excava-<lb/>tum eſt, quo ultra ipſam extendatur axis pinnatus L M, qui <lb/>ſubtili cuſpide inſertus gnomoni P, liberius ita movetur quam <lb/>ſi ab ipſa lamina B B ſuſtineretur ſimulque ultra eam pro- <pb o="34" file="0060" n="61" rhead="CHRISTIANI HUGENII"/> mineret, debet enim prominere neceſſario ut affigi poſſit cla-<lb/> <anchor type="note" xlink:label="note-0060-01a" xlink:href="note-0060-01"/> vula S, quæ ſimul cum eo verſationes faciat. </s> <s xml:id="echoid-s804" xml:space="preserve">Eſt autem hic <lb/>motus reciprocus, nunc in hanc nunc in illam partem, <lb/>quum dentes rotæ K alternatim occurrant pinnulis L L, no-<lb/>tâ vulgo ratione, quæque proinde diligentiori explicatione <lb/>non indiget.</s> <s xml:id="echoid-s805" xml:space="preserve"/> </p> <div xml:id="echoid-div27" type="float" level="2" n="1"> <note position="right" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">TAB. II. <lb/>Fig. 1.</note> <note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s806" xml:space="preserve">Porro clavula S, ima ſui parte reflexa ac foramine ob-<lb/>longo terebrata, penduli virgam ferream, cui plumbum X <lb/>affixum eſt, amplectitur. </s> <s xml:id="echoid-s807" xml:space="preserve">Hæc vero virga ſupernè duplici <lb/>filo ſuſpenſa eſt inter geminas lamellas, quarum una T hic <lb/>tantum cernitur; </s> <s xml:id="echoid-s808" xml:space="preserve">itaque alteram figuram juxta deſcripſi-<lb/>mus, quæ utriusque formam flexumque & </s> <s xml:id="echoid-s809" xml:space="preserve">totam hanc <lb/> <anchor type="note" xlink:label="note-0060-02a" xlink:href="note-0060-02"/> ſuſpendendi penduli rationem exprimeret. </s> <s xml:id="echoid-s810" xml:space="preserve">Quanquam de <lb/>vera laminarum iſtarum curvatura pluribus poſtea agendum <lb/>erit.</s> <s xml:id="echoid-s811" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="2" n="2"> <note position="left" xlink:label="note-0060-02" xlink:href="note-0060-02a" xml:space="preserve">TAB. II. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s812" xml:space="preserve">Nunc autem ut de motu horologii dicamus, nam reliquas <lb/>figuræ partes poſtea exequemur, facile equidem apparet & </s> <s xml:id="echoid-s813" xml:space="preserve"><lb/>vi rotarum, à pondere tractarum, perpendiculi V X mo-<lb/>tum ſuſtentari, poſtquam ſemel manu incitatum fuerit; </s> <s xml:id="echoid-s814" xml:space="preserve">& </s> <s xml:id="echoid-s815" xml:space="preserve"><lb/>ſimul perpendiculi ſtatos recurſus rotis univerſis, totique <lb/>adeo horologio movendi legem normamque præſcribere. <lb/></s> <s xml:id="echoid-s816" xml:space="preserve">Clavula enim, quantumvis levi rotarum impulſu acta, non <lb/>tantum obſequitur trahenti perpendiculo, ſed & </s> <s xml:id="echoid-s817" xml:space="preserve">ſingulis re-<lb/>curſibus paulisper ejus motum adjuvat, atque ita perennem <lb/>reddit, qui alioqui ſua ſponte, vel verius occurſu aëris, <lb/>deficeret paulatim, vergeretque ad quietem. </s> <s xml:id="echoid-s818" xml:space="preserve">Rurſus vero, <lb/>quum ejusmodi ſit natura penduli ut eodem ſemper tenore <lb/>feratur, neque ab eo ulla ratione præterquam mutata longi-<lb/>tudine dimoveri poſſit; </s> <s xml:id="echoid-s819" xml:space="preserve">utique poſtquam flexu lamellarum, <lb/>inter quas ſuſpenſum eſt, æqualitatem illam conſequuti fui-<lb/>mus; </s> <s xml:id="echoid-s820" xml:space="preserve">nequaquam permittitur rotæ K, ut nunc citius nunc <lb/>tardius incedat, etſi ſæpe, ut in vulgaribus horologiis, id <lb/>facere conetur; </s> <s xml:id="echoid-s821" xml:space="preserve">ſed neceſſario ſinguli dentes ejus coguntur <lb/>æqualibus tranſire temporibus. </s> <s xml:id="echoid-s822" xml:space="preserve">Hinc vero manifeſtum eſt, <lb/>& </s> <s xml:id="echoid-s823" xml:space="preserve">reliquarum quæ præcedunt rotarum, & </s> <s xml:id="echoid-s824" xml:space="preserve">denique etiam <lb/>indicum æquabiles converſiones effici, cum omnia propor- <pb o="35" file="0061" n="62" rhead="HOROLOG. OSCILLATOR."/> tionaliter moveantur. </s> <s xml:id="echoid-s825" xml:space="preserve">Quamobrem ſiquid in fabrica vi-<lb/> <anchor type="note" xlink:label="note-0061-01a" xlink:href="note-0061-01"/> tii fuerit, vel, ob aëris mutatam temperiem, diffici-<lb/>lius rotarum axes volvantur; </s> <s xml:id="echoid-s826" xml:space="preserve">dummodo non eo usque ut <lb/>omnis horologii motus interrumpatur; </s> <s xml:id="echoid-s827" xml:space="preserve">nulla propter <lb/>hæc inæqualitas aut motus retardatio timenda erit, ſem-<lb/>perque aut rectè tempus metietur aut omnino non metie-<lb/>tur.</s> <s xml:id="echoid-s828" xml:space="preserve"/> </p> <div xml:id="echoid-div29" type="float" level="2" n="3"> <note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO HO</emph>-<lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s829" xml:space="preserve">Indices porro hoc pacto circumaguntur atque ordinantur. <lb/></s> <s xml:id="echoid-s830" xml:space="preserve">Tertia lamina prioribus parallela eſt Y Y, pollicis quarta <lb/>parte diſtans ab ea quæ notatur A A. </s> <s xml:id="echoid-s831" xml:space="preserve">In ea circuli horarii <lb/>deſcripti ſunt centro eodem α quo protenditur axis rotæ C. </s> <s xml:id="echoid-s832" xml:space="preserve"><lb/>Quorum circulorum interior duodecim horarum diviſionem <lb/>habet, alter ſcrupulorum 60. </s> <s xml:id="echoid-s833" xml:space="preserve">Axi vero rotæ C aptatur, ul-<lb/>tra laminam A A, rota ß, tubulo cohærens qui usque ad ε <lb/>continuatur trans laminam Y Y; </s> <s xml:id="echoid-s834" xml:space="preserve">atque ita inſidet axi illi, <lb/>ut una cum illo circumferatur; </s> <s xml:id="echoid-s835" xml:space="preserve">ſine illo tamen, ubi opus <lb/>fuerit, converti poſſit. </s> <s xml:id="echoid-s836" xml:space="preserve">Ad ε index imponitur, horæ ſpa-<lb/>tio circuiturus atque ita ſcrupula prima, ſeu ſexageſimas ho-<lb/>rarum, demonſtraturus. </s> <s xml:id="echoid-s837" xml:space="preserve">Rota vero quam diximus ß, aliam <lb/>rotam, totidem quot ipſa habet dentium, impellit, atque <lb/>una affixum ei tympanum cui dentes ſex, axiculo eorum <lb/>communi hinc laminâ A, inde gnomone δ ſuffulto. </s> <s xml:id="echoid-s838" xml:space="preserve">Hoc <lb/>tandem tympano rota ζ movetur, dentes habens 72, tubu-<lb/>lumque affixum qui & </s> <s xml:id="echoid-s839" xml:space="preserve">ipſe ultra laminam Y ad θ porrigitur, <lb/>paulo citra quam deſinit tubulus rotæ ß, quem intra ſe com-<lb/>plectitur. </s> <s xml:id="echoid-s840" xml:space="preserve">Parte extrema θ apponitur horarius index, brevior <lb/>aliquanto illo quem ſcrupula prima ſignare diximus, cum in-<lb/>teriore gyro ferri debeat. </s> <s xml:id="echoid-s841" xml:space="preserve">Secunda vero ſcrupula ut absque <lb/>errore demonſtrentur, imponitur axi rotæ H, usque ad la-<lb/>minam Y producto, orbis λ, cui circulus in ſexaginta par-<lb/>tes diviſus inſcribitur, inciſoque in laminâ Y foramine <lb/>ad Z, eæ diviſiones, cuspide in ſummo foramine defixâ, <lb/>prætereuntes notantur. </s> <s xml:id="echoid-s842" xml:space="preserve">Hæc vero tota indicum circulo-<lb/>rumque horariorum diſpoſitio ex figura minori (fig. </s> <s xml:id="echoid-s843" xml:space="preserve">3.) </s> <s xml:id="echoid-s844" xml:space="preserve">cla-<lb/>rius perſpicitur, exteriorem horologii formam referen-<lb/>te.</s> <s xml:id="echoid-s845" xml:space="preserve"/> </p> <pb o="36" file="0062" n="63" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s846" xml:space="preserve">Cæterum penduli longitudinem, rotis quemadmodum di-<lb/> <anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/> ximus ordinatis, eam eſſe oportet ut ſcrupula ſecunda ſin-<lb/>gulis recurſibus metiatur, quæ longitudo tripedalis eſt, <lb/>& </s> <s xml:id="echoid-s847" xml:space="preserve">commodè in ſchemate exhiberi non potuit. </s> <s xml:id="echoid-s848" xml:space="preserve">Tripedalem <lb/>dico, non alicujus reſpectu pedis qui apud Europæ gentem <lb/>hanc illamve in uſu ſit, ſed certo æternoque pedis modulo <lb/>ab ipſa hujus penduli longitudine deſumpto, quem <emph style="sc">Pedem</emph> <lb/><emph style="sc">Horarium</emph> in poſterum appellare liceat, ad illam enim <lb/>omnium aliorum pedum menſuræ referri debent quas incor-<lb/>ruptas poſteris tradere voluerimus. </s> <s xml:id="echoid-s849" xml:space="preserve">Neque enim, verbi gra-<lb/>tiâ, ignorabitur unquam venturis ſæculis Pariſini pedis mo-<lb/>dus, dum conſtabit eum ad Pedem Horarium eſſe ut 864 <lb/>ad 881. </s> <s xml:id="echoid-s850" xml:space="preserve">Sed de hujus menſuræ exactiſſima conſtitutione plu-<lb/>ribus agemus in iis quæ de Centro Oſcillationis. </s> <s xml:id="echoid-s851" xml:space="preserve">nunc tem-<lb/>pora converſionum in ſingulis rotis indicibusque obiter de-<lb/>ſignabimus, ut rectè omnia ad dentium ſupra deſcriptorum <lb/>numerum quadrare intelligantur.</s> <s xml:id="echoid-s852" xml:space="preserve"/> </p> <div xml:id="echoid-div30" type="float" level="2" n="4"> <note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Hc-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s853" xml:space="preserve">Ergo una quidem converſione rotæ C, decies circumire <lb/>apparet rotam F, ſexagies vero rotam H, & </s> <s xml:id="echoid-s854" xml:space="preserve">centies vicies <lb/>ſupremam K: </s> <s xml:id="echoid-s855" xml:space="preserve">cui quum dentes ſint quindecim, iisque <lb/>alternatim pulſentur pinnulæ L L, una converſione rotæ K <lb/>numerabuntur ictus 30, quibus reſpondent totidem itus re-<lb/>ditusque penduli V X. </s> <s xml:id="echoid-s856" xml:space="preserve">ideoque converſionibus 120, reſpon-<lb/>debunt oſcillationes ſimplices 3600, qui numerus eſt ſcru-<lb/>pulorum ſecundorum unam horam efficientium. </s> <s xml:id="echoid-s857" xml:space="preserve">Itaque ho-<lb/>ræ tempore ſemel circumit rota C, cumque ea ſimul index <lb/>ad E impoſitus, qui ſcrupula prima demonſtrat. </s> <s xml:id="echoid-s858" xml:space="preserve">Et quo-<lb/>niam eodem temporis ſpatio etiam rota ß, & </s> <s xml:id="echoid-s859" xml:space="preserve">per eam γ, <lb/>convertitur, cum tympanidio ſuo dentium ſex, ad quem <lb/>numerum duodecuplus eſt numerus dentium rotæ ζ, appa-<lb/>ret duodecim demum horis hanc circumduci, totidemque <lb/>indicem illi conjunctum in θ. </s> <s xml:id="echoid-s860" xml:space="preserve">Denique cum rotæ H ſexa-<lb/>ginta converſiones reſpondere oſtenderimus ſingulis conver-<lb/>ſionibus rotæ C, hinc illa, una cum affixo orbe λ, ſexagies <lb/>in ſingulas horas circumferetur, hoc eſt, ſemel unius ſcru-<lb/>puli primi tempore, ideoque partes ſexageſimæ orbiculi λ <pb o="37" file="0063" n="64" rhead="HOROLOG. OSCILLATOR."/> <anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/> ſecunda ſcrupula tranſitu ſuo oſtendent: </s> <s xml:id="echoid-s861" xml:space="preserve">atque ita omnia rectè <lb/>ſe habere manifeſtum erit. </s> <s xml:id="echoid-s862" xml:space="preserve">Pondus X in imo perpendiculo <lb/>trilibre eſt, plumbeum totum, vel ænea ſuperficie plumbum <lb/>continente. </s> <s xml:id="echoid-s863" xml:space="preserve">Nec tantum metalli gravitate ſed & </s> <s xml:id="echoid-s864" xml:space="preserve">figurâ in-<lb/>ſuper proſpiciendum (plurimi enim refert) ut quam mini-<lb/>mum occurſu aëris impedimentum ſentiat. </s> <s xml:id="echoid-s865" xml:space="preserve">Eoque in cylin-<lb/> <anchor type="note" xlink:label="note-0063-02a" xlink:href="note-0063-02"/> dri jacentis oblongi & </s> <s xml:id="echoid-s866" xml:space="preserve">utrinque præacuti formam fingitur, <lb/>qualis cernitur ad a ſchemate horologii minore. </s> <s xml:id="echoid-s867" xml:space="preserve">Quanquam <lb/>in his quæ ad navigationem parantur, forma lentis erectæ <lb/>aptior viſa eſt.</s> <s xml:id="echoid-s868" xml:space="preserve"/> </p> <div xml:id="echoid-div31" type="float" level="2" n="5"> <note position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> <note position="right" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">TAB. II. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s869" xml:space="preserve">Porro eodem ſchemate & </s> <s xml:id="echoid-s870" xml:space="preserve">ponderis alterius b, quo motus <lb/>horologii continuatur, ſuſpendendi ratio expreſſa eſt, quam, <lb/>incognitam prius, inveſtigare nobis neceſſe fuit, ne interim <lb/>dum ſurſum retrahitur pondus iſtud, ceſſaret vel impedire-<lb/>tur aliquatenus horologii curſus, quod hic omnino caven-<lb/>dum erat. </s> <s xml:id="echoid-s871" xml:space="preserve">Paratur itaque funis continuus atque in ſe rediens, <lb/>extremitatibus apte inter ſe connexis. </s> <s xml:id="echoid-s872" xml:space="preserve">Is primum orbiculum <lb/>rotæ infimæ conjunctum, qui in ſchemate majori notatus eſt <lb/>D, amplectitur; </s> <s xml:id="echoid-s873" xml:space="preserve">inde deſcendens, altera ſui parte trochleam <lb/>c, cui pondus b appenſum eſt, ſubit. </s> <s xml:id="echoid-s874" xml:space="preserve">Hinc ſuper orbicu-<lb/>lum d aſcendit, extrinſecus horologio affixum, qui ferreos <lb/>per circumferentiam aculeos habet, atque inſuper ſerratis <lb/>dentibus ita eſt aptatus ut volvatur tracto fune e; </s> <s xml:id="echoid-s875" xml:space="preserve">nequa-<lb/>quam vero in partem contrariam revolvi poſſit. </s> <s xml:id="echoid-s876" xml:space="preserve">Ab hoc or-<lb/>biculo deſcendit funis ad alteram trochleam f, cui pondus <lb/>exiguum g appenditur, quantum ſufficit continendo majori <lb/>b, ne aliter quam revoluto orbiculo deſcendat. </s> <s xml:id="echoid-s877" xml:space="preserve">Namque à <lb/>trochlea f rurſus ad ipſum orbiculum D, unde deſcenderat, <lb/>funis revertitur. </s> <s xml:id="echoid-s878" xml:space="preserve">Quibus ita ſe habentibus, manifeſtum eſt <lb/>ſemper pondus b dimidia ſui gravitate conari ut rotas horo-<lb/>logii circumagat, nec tunc quidem ceſſare cum manu fu-<lb/>nem e trahente aſcendere cogitur; </s> <s xml:id="echoid-s879" xml:space="preserve">adeoque horologii mo-<lb/>tum nusquam interrumpi, nec momentum temporis de-<lb/>perdi.</s> <s xml:id="echoid-s880" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s881" xml:space="preserve">Gravitatis modus in pondere b definiri certo non poteſt, <lb/>ſed quo minor conſervando motui ſuffecerit, eo melius ac- <pb o="38" file="0064" n="65" rhead="CHRISTIANI HUGENII"/> curatiusque fabrefactum automaton arguet. </s> <s xml:id="echoid-s882" xml:space="preserve">In noſtris, quæ <lb/> <anchor type="note" xlink:label="note-0064-01a" xlink:href="note-0064-01"/> optima hactenus habemus, ad ſex libras redactum eſt, po-<lb/>ſita nimirum orbiculi D diametro pollicari fere; </s> <s xml:id="echoid-s883" xml:space="preserve">uti exhibi-<lb/>ta fuit; </s> <s xml:id="echoid-s884" xml:space="preserve">item perpendiculi pondere trilibri, ac totidem pe-<lb/>dum longitudine. </s> <s xml:id="echoid-s885" xml:space="preserve">Quæ longitudo, ut hoc etiam admonea-<lb/>mus, trans capſam horologii dependet, oblongo foramine <lb/>perviam, quantum oſcillationibus peragendis neceſſe eſt, <lb/>Ipſum vero horologium, ad hominis altitudinem ſuſpenſum, <lb/>horis 30 moveri perſeverat.</s> <s xml:id="echoid-s886" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="6"> <note position="left" xlink:label="note-0064-01" xlink:href="note-0064-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s887" xml:space="preserve">Supereſt nunc forma lamellarum deſcribenda inter quas <lb/>perpendiculum affigi diximus, quarumque ad æquabilem <lb/>horologio motum præſtandum vel præcipua eſt opera. </s> <s xml:id="echoid-s888" xml:space="preserve">Abs-<lb/>que his enim Penduli ſimplicis oſcillationes (etſi nonnullis <lb/>aliter viſum eſt) non erunt æque diuturnæ, ſed brevioris <lb/>temporis eæ quæ per minores arcus incedent; </s> <s xml:id="echoid-s889" xml:space="preserve">idque primùm <lb/>experimento hujusmodi facile deprehenditur. </s> <s xml:id="echoid-s890" xml:space="preserve">Si enim fila <lb/>accipiantur ejusdem longitudinis duo, paribusque in parte <lb/>ima ponderibus religatis, utrumque ſeorſim ſuſpendatur, <lb/>tumque alterum eorum procul à linea perpendiculari, alte-<lb/>rum parumper duntaxat extrahatur, ſimulque è manu di-<lb/>mittantur; </s> <s xml:id="echoid-s891" xml:space="preserve">non diu utrumque ſimul in partes easdem ferri <lb/>videbitur, ſed prævertet illud cujus exiliores erunt recur-<lb/>ſus. </s> <s xml:id="echoid-s892" xml:space="preserve">Sed & </s> <s xml:id="echoid-s893" xml:space="preserve">temporum per quoslibet arcus rationes numeris <lb/>definiri poſſunt, certâ ſcientiâ nixis, & </s> <s xml:id="echoid-s894" xml:space="preserve">vero quam libuerit <lb/>propinquis, veluti quod tempus deſcenſus per totum circu-<lb/>li quadrantem eſt ad tempus per arcum minimum fere ut <lb/>34 ad 29. </s> <s xml:id="echoid-s895" xml:space="preserve">Adeo ut nequaquam reſiſtentiæ aëris ea diverſitas <lb/>imputanda ſit, ut quidam voluere, ſed ex ipſa motus natu-<lb/>ra circulique proprietate naſcatur. </s> <s xml:id="echoid-s896" xml:space="preserve">Quod alio quoque argu-<lb/>mento concludi poſſit ex ipſa Penduli iſochroni conſtructio-<lb/>ne, ubi à circulari linea haud parum receditur, uti mox <lb/>patebit.</s> <s xml:id="echoid-s897" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s898" xml:space="preserve">Sed videatur forſan in noſtris horologiis hiſce, ubi eadem <lb/>ſemper eſt oſcillationum latitudo, nullius momenti futura <lb/>quam diximus inæqualitas, adeoque nec correctione ulla <lb/>perpendiculi opus fore. </s> <s xml:id="echoid-s899" xml:space="preserve">Quod ſane ita eſſet ſi latitudo omni- <pb o="39" file="0065" n="66" rhead="HOROLOG. OSCILLATOR."/> um planè eadem conſtanter maneret. </s> <s xml:id="echoid-s900" xml:space="preserve">Sed cum pauxillum <lb/> <anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/> quandoque excedat vel deficiat, ex multis minimis differen-<lb/>tiolis tandem magna ſatis conflatur, idque ita eſſe reipſa <lb/>atque experimentis evincitur. </s> <s xml:id="echoid-s901" xml:space="preserve">Etſi enim eadem ſemper ſit <lb/>ponderis vis, rotæ ſibi proximæ reſpectu, tamen per tot <lb/>alias transdita, quantâcunque curâ limatæ fuerint, non <lb/>ſemper eadem ad perpendiculum usque pervenit. </s> <s xml:id="echoid-s902" xml:space="preserve">Præter-<lb/>quam quod frigore quoque difficilior motus rotarum effici-<lb/>tur; </s> <s xml:id="echoid-s903" xml:space="preserve">itemque evaneſcente aut ſordeſcente quod illis additur <lb/>oleo. </s> <s xml:id="echoid-s904" xml:space="preserve">Sed præcipue inæquales fiunt oſcillationes horologiis <lb/>quæ mari vehuntur, ob jactationem navis continuam, adeo <lb/>ut omnibus quidem in univerſum, ſed his maxime omnium <lb/>remedio opus ſit, quo reciprocationum Penduli latiorum <lb/>anguſtiorumque tempora æqualia evadant.</s> <s xml:id="echoid-s905" xml:space="preserve"/> </p> <div xml:id="echoid-div33" type="float" level="2" n="7"> <note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s906" xml:space="preserve">Ad definiendam ergo lamellarum formam in quibus poſi-<lb/>tum eſt remedium iſtud, in primis Penduli longitudinem <lb/>ſtatuiſſe oportet, quæ facile ex eo habetur, quod ſint inter <lb/>ſe longitudines perpendiculorum, ſicut temporum quæ in <lb/>ſingulos recurſus impenduntur quadrata. </s> <s xml:id="echoid-s907" xml:space="preserve">Adeo ut cum tri-<lb/>bus pedibus definiverimus longitudinem perpendiculi quod <lb/>ſcrupula ſecunda metitur, ejus quarta pars, ſive unciæ no-<lb/>vem debeantur ei quod ſemiſecunda notaturum ſit. </s> <s xml:id="echoid-s908" xml:space="preserve">Item ſi <lb/>Penduli longitudo quæratur, cujus recurſus ſimplices 10000 <lb/>horæ ſpatio peragantur, hoc modo ratio inibitur. </s> <s xml:id="echoid-s909" xml:space="preserve">Penduli <lb/>nempe tripedalis ſcimus 3600 recurſus in horas ſingulas nu-<lb/>merari: </s> <s xml:id="echoid-s910" xml:space="preserve">ergo hujus recurſuum tempora ſingula, majora ſunt <lb/>temporibus Penduli quæſiti, proportione 10000 ad 3600, <lb/>ſive 25 ad 9. </s> <s xml:id="echoid-s911" xml:space="preserve">Quare ut quadratum numeri 25 ad quadra-<lb/>tum 9, hoc eſt, ut 625 ad 81, ita erit longitudo pedum 3 <lb/>ad eam quæ quærebatur, nempe unciarum 4 cum {66/100}.</s> <s xml:id="echoid-s912" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s913" xml:space="preserve">Poſita ergo longitudine perpendiculi, puta pedum trium <lb/>in horologio à nobis propoſito, inde Cycloïs linea, quæ <lb/>curvaturam laminarum T datura eſt, hoc modo deſcribe-<lb/>tur.</s> <s xml:id="echoid-s914" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s915" xml:space="preserve">Super tabula plana affigatur regula A B, ſemidigiti craſſitu-<lb/> <anchor type="note" xlink:label="note-0065-02a" xlink:href="note-0065-02"/> dine. </s> <s xml:id="echoid-s916" xml:space="preserve">Deinde fiat cylindrus C D E eadem illa altitudine, dia- <pb o="40" file="0066" n="67" rhead="CHRISTIANI HUGENII"/> metrum vero baſeos, dimidiæ perpendiculi longitudi-<lb/> <anchor type="note" xlink:label="note-0066-01a" xlink:href="note-0066-01"/> ni, æqualem habens; </s> <s xml:id="echoid-s917" xml:space="preserve">ſitque F G H E faſciola, ſeu <lb/>potius bractea tenuis, affixa regulæ in F, cylindro verò in <lb/>circumferentiæ puncto aliquo E, ita ut partim huic circum-<lb/>voluta ſit, partim extendatur juxta latus regulæ A B. </s> <s xml:id="echoid-s918" xml:space="preserve">Cy-<lb/>lindro autem infixa ſit ferrea cuſpis D I, pauxillum ultra <lb/>baſin inferiorem prominens, atque ita ut circumferentiæ ejus <lb/>exacte reſpondeat.</s> <s xml:id="echoid-s919" xml:space="preserve"/> </p> <div xml:id="echoid-div34" type="float" level="2" n="8"> <note position="right" xlink:label="note-0065-02" xlink:href="note-0065-02a" xml:space="preserve">TAB. III. <lb/>Fig. 1.</note> <note position="left" xlink:label="note-0066-01" xlink:href="note-0066-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s920" xml:space="preserve">His ita ſe habentibus, ſi cylindrus ſecundum regulam A B <lb/>volvatur, bracteolæ tantum F G craſſitudine intercedente, <lb/>eâque ſemper quantum poteſt extensâ, deſcribet cuſpis I in <lb/>ſubjecto tabulæ plano lineam curvam K I, quæ Cyclois vo-<lb/>catur. </s> <s xml:id="echoid-s921" xml:space="preserve">Circulus vero genitor erit C D E, cylindri adhibiti <lb/>baſis. </s> <s xml:id="echoid-s922" xml:space="preserve">Quod ſi jam laminam K L ad regulam A B appli-<lb/>cuerimus; </s> <s xml:id="echoid-s923" xml:space="preserve">exaratâ primum in ea cycloidis portione K I, in-<lb/>vertemus deinde ipſam, & </s> <s xml:id="echoid-s924" xml:space="preserve">in ſuperficie adverſa ſimilem li-<lb/>neam K M, ab eodem puncto K egredientem, incidemus. <lb/></s> <s xml:id="echoid-s925" xml:space="preserve">Tum figuram M K I, accurate ſecundum lineas iſtas, ef-<lb/>formabimus, cui figuræ lamellarum interſtitium aptari opor-<lb/>tet, inter quas perpendiculum ſuſpenditur. </s> <s xml:id="echoid-s926" xml:space="preserve">Sufficiunt au-<lb/>tem ad horologiorum uſum portiones exiguæ arcuum K M, <lb/>K I; </s> <s xml:id="echoid-s927" xml:space="preserve">reliquo flexu inutili futuro, ad quem perpendiculi fi-<lb/>lum accedere non poteſt.</s> <s xml:id="echoid-s928" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s929" xml:space="preserve">Verum, ut mirabilis lineæ natura atque effectus plenius <lb/> <anchor type="note" xlink:label="note-0066-02a" xlink:href="note-0066-02"/> intelligantur, integras ſemicycloides K M, K I, alio ſche-<lb/>mate hic exprimere viſum fuit, inter quas ſuſpenſum agita-<lb/>tumque Pendulum K N P, diametri circuli genitoris du-<lb/>plum, cujuscunque amplitudinis oſcillationes, usque ad <lb/>maximam omnium per arcum M P I, iisdem temporibus <lb/>confecturum ſit: </s> <s xml:id="echoid-s930" xml:space="preserve">atque ita, ut appenſæ ſphæræ P centrum, <lb/>in linea M P I, quæ & </s> <s xml:id="echoid-s931" xml:space="preserve">ipſa cyclois integra eſt, ſemper <lb/>verſetur. </s> <s xml:id="echoid-s932" xml:space="preserve">Quæ proprietas inſignis, neſcio an alii præter hanc <lb/>lineæ data ſit, ut nempe ſe ipſam ſui evolutione deſcribat. <lb/></s> <s xml:id="echoid-s933" xml:space="preserve">Hæc autem quæ dicta ſunt, in ſequentibus, ubi de deſcen-<lb/>ſu gravium, deque evolutione curvarum agemus, ſingula <lb/>demonſtrabuntur.</s> <s xml:id="echoid-s934" xml:space="preserve"/> </p> <div xml:id="echoid-div35" type="float" level="2" n="9"> <note position="left" xlink:label="note-0066-02" xlink:href="note-0066-02a" xml:space="preserve">TAB. III. <lb/>Fig. 2.</note> </div> <pb o="41" file="0067" n="68" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s935" xml:space="preserve">Licebit autem aliter quoque, per inventa puncta, cy-<lb/> <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> cloidem deſignare. </s> <s xml:id="echoid-s936" xml:space="preserve">Deſcribatur circulus diametro A B, quæ <lb/>dimidiæ longitudini perpendiculi æqualis ſit. </s> <s xml:id="echoid-s937" xml:space="preserve">In cujus cir-<lb/>cumferentia ſumptis partibus æqualibus quotlibet, A C, <lb/>C D, D E, E F, A G, G H, H I, I K, jungantur <lb/>G C, H D, I E, K F, quæ erunt inter ſe parallelæ. <lb/></s> <s xml:id="echoid-s938" xml:space="preserve">Deinde arcui A F ſumatur æqualis linea recta L M, eaque <lb/>in partes æquales totidem dividatur quot ſunt in arcu A F, <lb/>earumque partium uni æquales ponantur ſingulæ C N, <lb/>G O in recta C G; </s> <s xml:id="echoid-s939" xml:space="preserve">duabus vero partibus rectæ L M, æ-<lb/>quales fiant ſingulæ D P, H Q in recta D H. </s> <s xml:id="echoid-s940" xml:space="preserve">Tribus ve-<lb/>ro, ſingulæ E R, I S in recta E I; </s> <s xml:id="echoid-s941" xml:space="preserve">atque ita porro ſi par-<lb/>tes plures fuerint acceptæ ac tandem toti L M æquales <lb/>fiant ſingulæ F T, K V in linea extrema F K. </s> <s xml:id="echoid-s942" xml:space="preserve">Jam ſi <lb/>curvæ deſcribantur per puncta A O Q S V, A N P R T, <lb/>hæ rurſus quæſitæ cycloidis partes erunt, inter quas per-<lb/>pendiculum affigi oportet.</s> <s xml:id="echoid-s943" xml:space="preserve"/> </p> <div xml:id="echoid-div36" type="float" level="2" n="10"> <note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>. <lb/><emph style="sc">TAB. III.</emph> <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s944" xml:space="preserve">Recta autem L M æqualis arcui A F invenitur, ſi pri-<lb/>mum duabus rectis, quæ ſemiſſibus arcus A F ſubtendun-<lb/>tur, æqualis ponatur X Z, totius vero arcus ſubtenſæ A F <lb/>æqualis ab eodem termino accipiatur X Y, differentiæque <lb/>Y Z triens Z Δ ad totam X Z adponatur. </s> <s xml:id="echoid-s945" xml:space="preserve">Nam tota X Δ <lb/>toti arcui A F tam prope æqualis erit, ut licet ſextans fue-<lb/>rit circumferentiæ, (neque major hic unquam requiritur) <lb/>non una ſexies milleſima parte ſuæ longitudinis deficiat, <lb/>uti in his, quæ de Circuli Magnitudine antehac ſcripſimus, <lb/>demonſtratum eſt.</s> <s xml:id="echoid-s946" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s947" xml:space="preserve">Explicitis quæ ad horologii fabricam attinent, nunc quo-<lb/>que illud declarandum eſt, quo pacto ad veram horarum <lb/>menſuram componi debeat. </s> <s xml:id="echoid-s948" xml:space="preserve">Ergo primum, an recte ſe ha-<lb/>beat motus ejus, hoc modo examinabitur.</s> <s xml:id="echoid-s949" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s950" xml:space="preserve">Oculo obſervatoris certus eligatur locus, unde ſidera de-<lb/>ſpici poſſint, ſimulque tecta parietesve vicinarum ædium, <lb/>ſic poſita, ut, cum eò appulerint ſtellæ quædam è fixarum <lb/>numero, ſimul videri deſinant. </s> <s xml:id="echoid-s951" xml:space="preserve">Eo loco foramen, ad pu-<lb/>pillæ magnitudinem, conſtituatur, ut ſequentibus diebus, <pb o="42" file="0068" n="69" rhead="CHRISTIANI HUGENII"/> absque errore, oculus ad idem punctum reponi poſſit. </s> <s xml:id="echoid-s952" xml:space="preserve">Jam <lb/> <anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/> ad momentum ipſum, cum ſtellarum aliqua è conſpectu abit, <lb/>notetur tempus horologio indicatum. </s> <s xml:id="echoid-s953" xml:space="preserve">Atque idem poſtero <lb/>die, vel potius aliquot diebus intermiſſis, fiat. </s> <s xml:id="echoid-s954" xml:space="preserve">Quod ſi <lb/>tantum unius diei ſpatium duabus obſervationibus interceſſe-<lb/>rit, oportet in poſtrema obſervatione tempus horologii de-<lb/>ficere ab illo, quod prima obſervatione annotatum fuerat, <lb/>ſcrupulis primis 3, ſecundis 56. </s> <s xml:id="echoid-s955" xml:space="preserve">Ita enim rectè ſe habere <lb/>perpendiculi longitudinem conſtabit; </s> <s xml:id="echoid-s956" xml:space="preserve">quum tanto ſuperetur <lb/>quælibet ſiderum fixorum revolutio à die ſolari mediocri. <lb/></s> <s xml:id="echoid-s957" xml:space="preserve">Mediocri dico, quoniam dies ſolares, de meridie ad meri-<lb/>diem, non omnes inter ſe æquales ſunt, ut mox amplius <lb/>exponetur. </s> <s xml:id="echoid-s958" xml:space="preserve">Si vero poſt plures demum dies obſervatio re-<lb/>petatur, in ſingulos tantundem differentiæ cauſa computan-<lb/>dum erit. </s> <s xml:id="echoid-s959" xml:space="preserve">Sit, exempli gratiâ, in prima obſervatione, ad <lb/>momentum evaneſcentis ſtellæ, adnotata horologii hora 9, <lb/>cum ſcrupulis primis 30, ſecundis 18; </s> <s xml:id="echoid-s960" xml:space="preserve">deinde, ſeptimo <lb/>poſt die, eâdem diſparente ſtellâ, indicet horam 8, cum <lb/>ſcrupulis pr. </s> <s xml:id="echoid-s961" xml:space="preserve">50, ſec. </s> <s xml:id="echoid-s962" xml:space="preserve">24. </s> <s xml:id="echoid-s963" xml:space="preserve">Hæc hora deficit à priore ſcrupu-<lb/>lis pr. </s> <s xml:id="echoid-s964" xml:space="preserve">39, ſecundis 54. </s> <s xml:id="echoid-s965" xml:space="preserve">Quæ, in ſeptem diviſa, dant retar-<lb/>dationem diurnam ſcrupulorum 5′. </s> <s xml:id="echoid-s966" xml:space="preserve">42″. </s> <s xml:id="echoid-s967" xml:space="preserve">Debebat autem eſſe <lb/>ſcrupulorum 3′.</s> <s xml:id="echoid-s968" xml:space="preserve">56″. </s> <s xml:id="echoid-s969" xml:space="preserve">quæ illâ minor eſt ſcrupulis 1′.</s> <s xml:id="echoid-s970" xml:space="preserve">46″. </s> <s xml:id="echoid-s971" xml:space="preserve">Ita-<lb/>que tantundem quotidie deficit horologium à vera, ſeu me-<lb/>dia, dierum menſura.</s> <s xml:id="echoid-s972" xml:space="preserve"/> </p> <div xml:id="echoid-div37" type="float" level="2" n="11"> <note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s973" xml:space="preserve">Cæterum alio quoque modo, ad ſolem, horologii motum <lb/>examinare licebit. </s> <s xml:id="echoid-s974" xml:space="preserve">Sed hic jam inæqualitatis dierum natu-<lb/>ralium ratio habenda erit. </s> <s xml:id="echoid-s975" xml:space="preserve">Sunt enim, ut jam dixi, non <lb/>omnes ejusmodi dies inter ſe æquales; </s> <s xml:id="echoid-s976" xml:space="preserve">& </s> <s xml:id="echoid-s977" xml:space="preserve">quanquam exi-<lb/>guum ſit diſcrimen, tamen plurium dierum intervallo ſæpe <lb/>eo usque excreſcit, ut haudquaquam contemni poſſit. </s> <s xml:id="echoid-s978" xml:space="preserve">Ete-<lb/>nim ſi & </s> <s xml:id="echoid-s979" xml:space="preserve">ſolarium quam perfectiſſime deſcriptum habeatur, <lb/>& </s> <s xml:id="echoid-s980" xml:space="preserve">horologii automati motus ad veriſſimam dierum menſu-<lb/>ram exactus ſit, neque ab ea recedat; </s> <s xml:id="echoid-s981" xml:space="preserve">eveniet tamen neceſ-<lb/>ſario ut, certis anni temporibus, ſæpe horæ quadrante, aut <lb/>etiam ſemihora, inter ſe diſcrepent, ac rurſus ſtatis tempo-<lb/>ribus ultro concordent. </s> <s xml:id="echoid-s982" xml:space="preserve">Hoc enim ita eſſe, ex tabula tem- <pb o="43" file="0069" n="70" rhead="HOROLOG. OSCILLATOR."/> poris æquatoria quam ſubjicimus, intelligetur; </s> <s xml:id="echoid-s983" xml:space="preserve">poſtquam <lb/> <anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> uſum ejus oſtenderimus, qui eſt hujusmodi.</s> <s xml:id="echoid-s984" xml:space="preserve"/> </p> <div xml:id="echoid-div38" type="float" level="2" n="12"> <note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s985" xml:space="preserve">Accipiatur æquatio tabulæ, aſſignata diei qua primum <lb/>cum ſole, ſive cum ſciotherico, horologium ut conveniret <lb/>fecimus. </s> <s xml:id="echoid-s986" xml:space="preserve">Itemque æquatio diei, qua quæritur quam bene <lb/>ad dierum menſuram temperatum ſit. </s> <s xml:id="echoid-s987" xml:space="preserve">Quod ſi jam prior æ-<lb/>quatio major fuerit ſequente, ſuperare debebit hora auto-<lb/>mati horam gnomonis eo, quo inter ſe æquationes iſtæ dif-<lb/>ferunt. </s> <s xml:id="echoid-s988" xml:space="preserve">At ſi poſterioris diei æquatio major inveniatur, erit <lb/>exceſſus penes horam gnomonis, ſive eam quæ ex ſole ob-<lb/>ſervatur. </s> <s xml:id="echoid-s989" xml:space="preserve">Ut ſi, exempli gratiâ, die 5 Martii in eandem <lb/>horam conveniant ſciothericum horologium atque automa-<lb/>ton, cujus diei æquatio invenitur, in tabula, ſcrupulorum <lb/>primorum 3, ſecundorum 11. </s> <s xml:id="echoid-s990" xml:space="preserve">lubeatque ſcire ejusdem menſis <lb/>die 20, an automaton horas æquales rectè metiatur necne: <lb/></s> <s xml:id="echoid-s991" xml:space="preserve">invenietur die poſteriori adſcripta æquatio ſcrupulorum pri-<lb/>morum 7, ſecundorum 27. </s> <s xml:id="echoid-s992" xml:space="preserve">quæ quia ſuperat præcedentem <lb/>ſcrupulis primis 4, ſecundis 16, debebit tanto ſerior eſſe <lb/>hora ſciotherici, quam quæ automato indicatur. </s> <s xml:id="echoid-s993" xml:space="preserve">Unde, ſi <lb/>diverſum reperiatur, facile inde colligetur, quantum in dies <lb/>ſingulos exuperet automaton, aut retardet.</s> <s xml:id="echoid-s994" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s995" xml:space="preserve">In computanda tabula hac duplicem cauſam adhibui, u-<lb/>tramque Aſtronomis notam, Eclipticæ nimirum obliquita-<lb/>tem; </s> <s xml:id="echoid-s996" xml:space="preserve">& </s> <s xml:id="echoid-s997" xml:space="preserve">ſolaris motus anomaliam. </s> <s xml:id="echoid-s998" xml:space="preserve">Quod cum ratio poſtulat, <lb/>tum experientia quoque, his ipſis horologiis ſuperſtructa, <lb/>quæque ſine his nequaquam haberi poterat, evincit; </s> <s xml:id="echoid-s999" xml:space="preserve">quan-<lb/>doquidem, cum æquatione hìc propoſita, obſervationes ſo-<lb/>lis, quas ſæpe per complures menſes, quotidie ad momen-<lb/>tum quo meridianum circulum ſol occuparet, inſtituimus, <lb/>planiſſime conſentire inventæ ſunt.</s> <s xml:id="echoid-s1000" xml:space="preserve"/> </p> <pb o="44" file="0070" n="71" rhead="TABULA ÆQUA."/> <note position="right" xml:space="preserve"> <lb/> ## Dies. ## Januar. ## Febr. ## Mart. ## Apr. ## Maj. ## Jun. <lb/>" # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. <lb/>1 # 10 # 40 # 0 # 32 # 2 # 15 # 11 # 18 # 18 # 32 # 18 # 10 <lb/>2 # 10 # 10 # 0 # 24 # 2 # 28 # 11 # 37 # 18 # 39 # 18 # 1 <lb/>3 # 9 # 41 # 0 # 18 # 2 # 42 # 11 # 56 # 18 # 46 # 17 # 51 <lb/>4 # 9 # 13 # 0 # 13 # 2 # 56 # 12 # 15 # 18 # 53 # 17 # 41 <lb/>5 # 8 # 45 # 0 # 9 # 3 # 11 # 12 # 34 # 18 # 59 # 17 # 30 <lb/>6 # 8 # 17 # 0 # 6 # 3 # 26 # 12 # 53 # 19 # 4 # 17 # 19 <lb/>7 # 7 # 50 # 0 # 3 # 3 # 41 # 13 # 12 # 19 # 9 # 17 # 8 <lb/>8 # 7 # 23 # 0 # 1 # 3 # 56 # 13 # 31 # 19 # 14 # 16 # 57 <lb/>9 # 6 # 58 # 0 # 0 # 4 # 12 # 13 # 49 # 19 # 18 # 16 # 46 <lb/>10 # 6 # 34 # 0 # 0 # 4 # 29 # 14 # 6 # 19 # 22 # 16 # 35 <lb/>11 # 6 # 10 # 0 # 0 # 4 # 46 # 14 # 23 # 19 # 25 # 16 # 24 <lb/>12 # 5 # 47 # 0 # 2 # 5 # 4 # 14 # 39 # 19 # 28 # 16 # 13 <lb/>13 # 5 # 24 # 0 # 4 # 5 # 22 # 14 # 55 # 19 # 29 # 16 # 1 <lb/>14 # 5 # 2 # 0 # 8 # 5 # 40 # 15 # 10 # 19 # 29 # 15 # 49 <lb/>15 # 4 # 41 # 0 # 12 # 5 # 58 # 15 # 25 # 19 # 29 # 15 # 37 <lb/>16 # 4 # 21 # 0 # 16 # 6 # 16 # 15 # 39 # 19 # 28 # 15 # 24 <lb/>17 # 4 # 2 # 0 # 21 # 6 # 33 # 15 # 53 # 19 # 26 # 15 # 11 <lb/>18 # 3 # 44 # 0 # 26 # 6 # 51 # 16 # 7 # 19 # 24 # 14 # 58 <lb/>19 # 3 # 27 # 0 # 32 # 7 # 9 # 16 # 21 # 19 # 21 # 14 # 45 <lb/>20 # 3 # 11 # 0 # 40 # 7 # 27 # 16 # 34 # 19 # 18 # 14 # 32 <lb/>21 # 2 # 55 # 0 # 48 # 7 # 45 # 16 # 47 # 19 # 15 # 14 # 19 <lb/>22 # 2 # 39 # 0 # 57 # 8 # 3 # 16 # 59 # 19 # 11 # 14 # 6 <lb/>23 # 2 # 23 # 1 # 6 # 8 # 22 # 17 # 11 # 19 # 7 # 13 # 53 <lb/>24 # 2 # 7 # 1 # 16 # 8 # 41 # 17 # 22 # 19 # 2 # 13 # 40 <lb/>25 # 1 # 52 # 1 # 26 # 9 # 1 # 17 # 33 # 18 # 57 # 13 # 27 <lb/>26 # 1 # 38 # 1 # 37 # 9 # 21 # 17 # 43 # 18 # 51 # 13 # 15 <lb/>27 # 1 # 25 # 1 # 49 # 9 # 41 # 17 # 53 # 18 # 45 # 13 # 3 <lb/>28 # 1 # 13 # 2 # 2 # 10 # 1 # 18 # 3 # 18 # 39 # 12 # 52 <lb/>29 # 1 # 2 ### 10 # 21 # 18 # 13 # 18 # 33 # 12 # 41 <lb/>30 # 0 # 51 ### 10 # 40 # 18 # 23 # 18 # 26 # 12 # 30 <lb/>31 # 0 # 41 ### 10 # 59 ### 18 # 18 <lb/></note> <pb o="45" file="0071" n="72" rhead="TIONIS DIERUM."/> <note position="right" xml:space="preserve"> <lb/> ## Dies. ## Jul. ## Aug. ## Sept. ## Octob. ## Nov. ## Dec. <lb/>" # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. <lb/>1 # 12 # 19 # 10 # 4 # 16 # 23 # 26 # 30 # 31 # 55 # 25 # 34 <lb/>2 # 12 # 8 # 10 # 8 # 16 # 42 # 26 # 49 # 31 # 55 # 25 # 10 <lb/>3 # 11 # 58 # 10 # 13 # 17 # 1 # 27 # 8 # 31 # 54 # 24 # 45 <lb/>4 # 11 # 48 # 10 # 18 # 17 # 21 # 27 # 26 # 31 # 52 # 24 # 20 <lb/>5 # 11 # 38 # 10 # 23 # 17 # 41 # 27 # 43 # 31 # 50 # 23 # 55 <lb/>6 # 11 # 28 # 10 # 28 # 18 # 1 # 28 # 0 # 31 # 47 # 23 # 30 <lb/>7 # 11 # 18 # 10 # 34 # 18 # 21 # 28 # 16 # 31 # 43 # 23 # 4 <lb/>8 # 11 # 9 # 10 # 41 # 18 # 41 # 28 # 32 # 31 # 37 # 22 # 38 <lb/>9 # 11 # 0 # 10 # 49 # 19 # 1 # 28 # 47 # 31 # 30 # 22 # 11 <lb/>10 # 10 # 52 # 10 # 58 # 19 # 21 # 29 # 2 # 31 # 22 # 21 # 43 <lb/>11 # 10 # 47 # 11 # 7 # 19 # 41 # 29 # 16 # 31 # 13 # 21 # 14 <lb/>12 # 10 # 38 # 11 # 16 # 20 # 1 # 29 # 30 # 31 # 3 # 20 # 44 <lb/>13 # 10 # 31 # 11 # 25 # 20 # 22 # 29 # 43 # 30 # 53 # 20 # 14 <lb/>14 # 10 # 25 # 11 # 36 # 20 # 43 # 29 # 56 # 30 # 43 # 19 # 44 <lb/>15 # 10 # 19 # 11 # 48 # 21 # 4 # 30 # 9 # 30 # 32 # 19 # 14 <lb/>16 # 10 # 13 # 12 # 1 # 21 # 25 # 30 # 22 # 30 # 20 # 18 # 44 <lb/>17 # 10 # 7 # 12 # 14 # 21 # 47 # 30 # 34 # 30 # 8 # 18 # 14 <lb/>18 # 10 # 2 # 12 # 28 # 22 # 9 # 30 # 45 # 29 # 55 # 17 # 44 <lb/>19 # 9 # 58 # 12 # 42 # 22 # 31 # 30 # 55 # 29 # 40 # 17 # 14 <lb/>20 # 9 # 54 # 12 # 57 # 22 # 52 # 31 # 4 # 29 # 23 # 16 # 44 <lb/>21 # 9 # 51 # 13 # 12 # 23 # 13 # 31 # 12 # 29 # 6 # 16 # 14 <lb/>22 # 9 # 49 # 13 # 27 # 23 # 33 # 31 # 19 # 28 # 48 # 15 # 44 <lb/>23 # 9 # 47 # 13 # 43 # 23 # 53 # 31 # 26 # 28 # 30 # 15 # 14 <lb/>24 # 9 # 46 # 13 # 59 # 24 # 13 # 31 # 32 # 28 # 11 # 14 # 43 <lb/>25 # 9 # 46 # 14 # 16 # 24 # 33 # 31 # 38 # 27 # 51 # 14 # 12 <lb/>26 # 9 # 46 # 14 # 33 # 24 # 53 # 31 # 43 # 27 # 30 # 13 # 41 <lb/>27 # 9 # 47 # 14 # 50 # 25 # 13 # 31 # 47 # 27 # 8 # 13 # 10 <lb/>28 # 9 # 49 # 15 # 8 # 25 # 33 # 31 # 50 # 26 # 45 # 12 # 40 <lb/>29 # 9 # 52 # 15 # 26 # 25 # 52 # 31 # 53 # 26 # 22 # 12 # 10 <lb/>30 # 9 # 56 # 15 # 45 # 26 # 11 # 31 # 55 # 25 # 58 # 11 # 40 <lb/>31 # 10 # 0 # 16 # 4 ### 31 # 55 ### 11 # 10 <lb/></note> <pb o="46" file="0072" n="73" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1001" xml:space="preserve">Jam poſtquam utrovis modo eorum quos diximus, ſed <lb/> <anchor type="note" xlink:label="note-0072-01a" xlink:href="note-0072-01"/> priore potius, examen inſtitutum fuerit, ſi multum aberra-<lb/>re à media dierum longitudine horologium reperiatur, adeo <lb/>ut differentia ultra tria quatuorve prima ſcrupula aſcendat, <lb/>remedium adhibebitur aucta aut diminuta ipſius penduli longi-<lb/>tudine. </s> <s xml:id="echoid-s1002" xml:space="preserve">Ubi hæc tenenda eſt regula, tot ſcrupulis primis, <lb/>in ſingulos dies, motum horologii acceleratum aut retarda-<lb/>tum iri, quot {7/10} unius lineæ auferentur pendulo aut adden-<lb/>tur. </s> <s xml:id="echoid-s1003" xml:space="preserve">Cumque ad veram menſuram hoc pacto jam prope re-<lb/> <anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/> ductum erit, reliqua correctio transpoſitione exigui ponde-<lb/>ris Δ, virgæ V V adhærentis, commode peragetur. </s> <s xml:id="echoid-s1004" xml:space="preserve">Id <lb/>pondus lentis formam habet, cujus ſectionem ſecundum <lb/>axem in figura 1. </s> <s xml:id="echoid-s1005" xml:space="preserve">expreſſimus. </s> <s xml:id="echoid-s1006" xml:space="preserve">Et quia tantum viceſimam <lb/>triceſimamve, aut etiam minorem, partem æquat ponderis <lb/>X, hinc fit ut ſat magnis ſpatiis è priore loco diſcedens, <lb/>haud multum tamen perpendiculi motum afficiat, acceleran-<lb/>do nempe quoties verſus mediam virgæ longitudinem attra-<lb/>hitur, retardando cum inde ſurſum aut deorſum movetur. <lb/></s> <s xml:id="echoid-s1007" xml:space="preserve">Ne vero diu punctum illud quærendum ſit quo veriſſimam <lb/>daturum ſit dierum menſuram, diviſimus certa ratione, ex <lb/>motus legibus petita, inferiorem virgæ medietatem, poſito <lb/>nimirum pondere Δ parte quinquageſima ponderis X, pari-<lb/>que gravitate ipſius virgæ V V. </s> <s xml:id="echoid-s1008" xml:space="preserve">Quæ quidem diviſiones fi-<lb/> <anchor type="note" xlink:label="note-0072-03a" xlink:href="note-0072-03"/> gura 4 exhibentur, ubi penduli portio inferior in tres partes <lb/>ſecta cernitur, quarum, quæ infimo loco ponenda, eſt A B. <lb/></s> <s xml:id="echoid-s1009" xml:space="preserve">Punctum A eſt centrum gravitatis ponderis X, à puncto <lb/>autem C, centro oſcillationis, partes ſingulæ, quindecim <lb/>ſcrupulorum ſecundorum differentiam diurnam efficiunt, ubi <lb/>tali intervallo mota fuerit lens Δ. </s> <s xml:id="echoid-s1010" xml:space="preserve">Demonſtratio autem divi-<lb/>ſionumque inventio, dabitur in iis quæ de Centro Oſcilla-<lb/>tionis.</s> <s xml:id="echoid-s1011" xml:space="preserve"/> </p> <div xml:id="echoid-div39" type="float" level="2" n="13"> <note position="left" xlink:label="note-0072-01" xlink:href="note-0072-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">@OLOGII</emph>.</note> <note position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">TAB. II. <lb/>Fig. I.</note> <note position="left" xlink:label="note-0072-03" xlink:href="note-0072-03a" xml:space="preserve">TAB. II. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s1012" xml:space="preserve">Cæterum illorum quoque quæ mari vehuntur, longitu-<lb/>dinum inveſtigandarum gratiâ, formam hic deſcriberemus, <lb/>ſi quænam maxime ad hunc uſum accommodata ſit, æque <lb/>ac in præcedentibus, exploratum determinatumque habere-<lb/>mus; </s> <s xml:id="echoid-s1013" xml:space="preserve">etſi quidem jam nunc eo res deducta ſit, ut parum <pb file="0073" n="74"/> <pb file="0073a" n="75"/> <anchor type="figure" xlink:label="fig-0073a-01a" xlink:href="fig-0073a-01"/> <anchor type="figure" xlink:label="fig-0073a-02a" xlink:href="fig-0073a-02"/> <pb file="0074" n="76"/> <pb o="47" file="0075" n="77" rhead="HOROLOG. OSCILLATOR."/> deeſſe videatur ad perficiendum tantæ utilitatis inventum. <lb/></s> <s xml:id="echoid-s1014" xml:space="preserve"> <anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/> Quid autem & </s> <s xml:id="echoid-s1015" xml:space="preserve">qua fortuna hìc tentatum fuerit, quidve de-<lb/>inceps tentandum reſtet, exponere non pigebit.</s> <s xml:id="echoid-s1016" xml:space="preserve"/> </p> <div xml:id="echoid-div40" type="float" level="2" n="14"> <figure xlink:label="fig-0073a-01" xlink:href="fig-0073a-01a"> <caption xml:id="echoid-caption2" style="it" xml:space="preserve">Pag. 46.<lb/>TAB.II.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables2" xml:space="preserve">A Y B P N Q L L M T λ K 15 Z I 24 H S R G 8 48 F 48 48 8 V E λ C 72 D 30 ß 80 θ ε ε θ ß V γ ζ D C Δ 9 γ 30 δ A B Y X</variables> </figure> <figure xlink:label="fig-0073a-02" xlink:href="fig-0073a-02a"> <caption xml:id="echoid-caption3" style="it" xml:space="preserve">Fig. 2.<lb/>Fig. 4.<lb/>Fig. 3.</caption> <variables xml:id="echoid-variables3" xml:space="preserve">B 2′ 30″ 4″ 3′ 30″ 15″ 4″ 1′ 30″ 15″ 45″ d 30″ 15″ e 15″ c C 2′ 3′ b A a f g</variables> </figure> <note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s1017" xml:space="preserve">Prima duo hujusmodi horologia Britannica navi vecta <lb/>fuere anno 1664, quæ vir nobilis è Scotia nobisque amicus <lb/>ad noſtrorum exemplum fabricari curaverat. </s> <s xml:id="echoid-s1018" xml:space="preserve">Hæc ponderis <lb/>loco laminam chalybeam habebant in ſpiram convolutam, <lb/>cujus vi rotæ circumagerentur, quemadmodum in exiguis <lb/>illis quæ circumferri ſolent automatis adhiberi ſolent. </s> <s xml:id="echoid-s1019" xml:space="preserve">Ut <lb/>autem jactationem navis perferre poſſent, è chalybea pila, <lb/>cylindro æneo incluſa, horologia ſuſpenderat, clavulamque <lb/>quæ penduli motum continuat (erat autem ſemipedali longi-<lb/>tudine pendulum) deorſum productam geminaverat, ut li-<lb/>teræ F inverſæ formam referret; </s> <s xml:id="echoid-s1020" xml:space="preserve">ne videlicet in gyrum <lb/>evagari poſſet penduli motus, unde ceſſationis pericu-<lb/>lum. </s> <s xml:id="echoid-s1021" xml:space="preserve">Navis hæc, cum tribus aliis quas itineris ſocias habue-<lb/>rat, poſtquam in Britanniam reverſa eſt, Præfectus claſſis <lb/>hæc retulit. </s> <s xml:id="echoid-s1022" xml:space="preserve">Se nempe, cum à Guineæ littore ſolviſſet, at-<lb/>que ad inſulam, ſancti Thomæ dictam, perveniſſet, quæ <lb/>æquinoctiali circulo ſubjacet, compoſitis hìc ad ſolem horo-<lb/>logiis, occidentem verſus curſum inſtituiſſe, atque ad ſe-<lb/>ptingenta circiter milliaria continuo tramite progreſſum, tum <lb/>rurſus vento favente Libonoto ad Africæ littora declinaviſ-<lb/>ſe. </s> <s xml:id="echoid-s1023" xml:space="preserve">Cum autem ad ducenta trecentave milliaria eò curſum <lb/>tenuiſſet, magiſtros aliarum navium, veritos ne priuſ-<lb/>quam Africam attigiſſent aquâ ad potum deficerentur, ſua-<lb/>ſiſſe ut ad inſulas Americanas, Barbatorum dictas, aquan-<lb/>di gratiâ deflecteret. </s> <s xml:id="echoid-s1024" xml:space="preserve">Tum ſeſe concilio nauclerorum habito, <lb/>juſſiſque ut Ephemeridas ac ſupputationes ſinguli ſuas pro-<lb/>ferrent, reperiſſe cæterorum calculos à ſuis diverſos abire, <lb/>unius quidem 80 milliaribus, alterius centenis, tertii am-<lb/>plius etiam. </s> <s xml:id="echoid-s1025" xml:space="preserve">Ipſum vero, cum ex horologiorum indicio <lb/>collegiſſet non amplius quam triginta circiter milliaribus ab-<lb/>eſſe inſulam del Fuego dictam, quæ una eſt earum, non <lb/>procul ab Africa diſtantium, quæ à Viridi promontorio no-<lb/>men habent, eamque poſtero die teneri poſſe; </s> <s xml:id="echoid-s1026" xml:space="preserve">confiſum <pb o="48" file="0076" n="78" rhead="CHRISTIANI HUGENII"/> pendulis ſuis eò curſum dirigi imperaſſe, ac die inſequenti <lb/> <anchor type="note" xlink:label="note-0076-01a" xlink:href="note-0076-01"/> ſub meridiem eam ipſam in conſpectum veniſſe inſulam, <lb/>paucisque poſt horis navibus ſtationem præbuiſſe. </s> <s xml:id="echoid-s1027" xml:space="preserve">Et hæc <lb/>quidem ex Præfecti illius relatu.</s> <s xml:id="echoid-s1028" xml:space="preserve"/> </p> <div xml:id="echoid-div41" type="float" level="2" n="15"> <note position="left" xlink:label="note-0076-01" xlink:href="note-0076-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">TLO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s1029" xml:space="preserve">Ab eo vero tempore aliquoties tum Batavorum tum Gallo-<lb/>rum opera, idque Regis Sereniſſimi juſſu, repetita ſuere ex-<lb/>perimenta, vario eventu, ſed ita ut ſæpius negligentia eo-<lb/>rum quibus horologia commiſſa erant quam ipſamet automa-<lb/>ta culpari poſſent. </s> <s xml:id="echoid-s1030" xml:space="preserve">Optimus vero ſucceſſus fuit in Mediter-<lb/>raneo mari, expeditione in Cretam inſulam, quò illuſtriſſi-<lb/>mus Dux Belfortius, Candiæ à Turcis obſeſſæ auxilium la-<lb/>turus, cum Gallorum copiis miſſus erat, ubi & </s> <s xml:id="echoid-s1031" xml:space="preserve">in prælio <lb/>occubuit. </s> <s xml:id="echoid-s1032" xml:space="preserve">Is in ea qua vehebatur navi, horologia hujuſce ex-<lb/>perimenti gratiâ habebat, virumque Aſtronomiæ peritum <lb/>iis præfecerat, è cujus obſervationibus, in ſingulos dies habi-<lb/>tis, longitudines locorum ad quæ in ea profectione aut ap-<lb/>pulerunt naves, aut quæ prætervecti dignoſcere oculis po-<lb/>tuerant, horologiorum operâ exacte dimenſas fuiſſe compe-<lb/>rimus, atque ita ut Geographicis deſcriptionibus quæ melio-<lb/>ris notæ habentur eædemmet longitudinum differentiæ diſi-<lb/>gnatæ reperiantur. </s> <s xml:id="echoid-s1033" xml:space="preserve">Namque inter Toloni portum Candiam-<lb/>que oppidum differentia hor. </s> <s xml:id="echoid-s1034" xml:space="preserve">1. </s> <s xml:id="echoid-s1035" xml:space="preserve">ſcrup. </s> <s xml:id="echoid-s1036" xml:space="preserve">22′; </s> <s xml:id="echoid-s1037" xml:space="preserve">reperta fuit, hoc <lb/>eſt graduum longitudinis 20. </s> <s xml:id="echoid-s1038" xml:space="preserve">ſcrup. </s> <s xml:id="echoid-s1039" xml:space="preserve">30′;</s> <s xml:id="echoid-s1040" xml:space="preserve">. ac rurſus à Candia <lb/>Tolonum revertentibus differentia proxime eadem. </s> <s xml:id="echoid-s1041" xml:space="preserve">qui con-<lb/>ſenſus certiſſimum veritatis eſt indicium.</s> <s xml:id="echoid-s1042" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1043" xml:space="preserve">Inter eundem Toloni portum & </s> <s xml:id="echoid-s1044" xml:space="preserve">inſulam quandam cui Ma-<lb/>retimo nomen eſt, prope promontorium Siciliæ quod Occi-<lb/>dentem ſpectat, Lilybæum olim vocatum, differentia horaria <lb/>obſervata eſt ſcrup. </s> <s xml:id="echoid-s1045" xml:space="preserve">prim. </s> <s xml:id="echoid-s1046" xml:space="preserve">25, ſec. </s> <s xml:id="echoid-s1047" xml:space="preserve">20, quibus reſpondent gra-<lb/>dus longitudinis 6, ſcrup. </s> <s xml:id="echoid-s1048" xml:space="preserve">20′;</s> <s xml:id="echoid-s1049" xml:space="preserve">. Item à Tolono ad inſulam Sa-<lb/>pienza dictam, quæ juxta Peloponneſum eſt Occidentem <lb/>verſus, hora 1, ſcrup. </s> <s xml:id="echoid-s1050" xml:space="preserve">prima 5′;</s> <s xml:id="echoid-s1051" xml:space="preserve">, ſec. </s> <s xml:id="echoid-s1052" xml:space="preserve">45″;</s> <s xml:id="echoid-s1053" xml:space="preserve">, quibus reſpon-<lb/>dent longitudinis gradus 16, ſcrup. </s> <s xml:id="echoid-s1054" xml:space="preserve">26.</s> <s xml:id="echoid-s1055" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1056" xml:space="preserve">Horologia ad ſolem examinata fuerant, mane ad Orien-<lb/>tem, veſpere ad Occidentem, ſupputato ex data poli alti-<lb/>tudine utroque temporis momento. </s> <s xml:id="echoid-s1057" xml:space="preserve">Atque hæc ratio cum na- <pb file="0077" n="79"/> <pb file="0077a" n="80"/> <anchor type="figure" xlink:label="fig-0077a-01a" xlink:href="fig-0077a-01"/> <anchor type="figure" xlink:label="fig-0077a-02a" xlink:href="fig-0077a-02"/> <anchor type="figure" xlink:label="fig-0077a-03a" xlink:href="fig-0077a-03"/> <pb file="0078" n="81"/> <pb o="49" file="0079" n="82" rhead="HOROLOG. OSCILLATOR."/> ves in anchoris ſtant omnium optima videtur, quod, abs-<lb/> <anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/> que inſtrumentorum ope, ſolis oculis eæ obſervationes per-<lb/>agantur.</s> <s xml:id="echoid-s1058" xml:space="preserve"/> </p> <div xml:id="echoid-div42" type="float" level="2" n="16"> <figure xlink:label="fig-0077a-01" xlink:href="fig-0077a-01a"> <caption xml:id="echoid-caption4" style="it" xml:space="preserve">Pag. 48.<lb/>TAB. III.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables4" xml:space="preserve">A B G C K H M D I L E</variables> </figure> <figure xlink:label="fig-0077a-02" xlink:href="fig-0077a-02a"> <caption xml:id="echoid-caption5" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables5" xml:space="preserve">K N M I P</variables> </figure> <figure xlink:label="fig-0077a-03" xlink:href="fig-0077a-03a"> <caption xml:id="echoid-caption6" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables6" xml:space="preserve">A G C N O H D P Q R S I E T V K F B L M X Y Z Δ</variables> </figure> <note position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s1059" xml:space="preserve">Pendulum vero unciarum novem longitudine inerat horolo-<lb/>giis hiſce, pondere ſemiſſis. </s> <s xml:id="echoid-s1060" xml:space="preserve">Rotæ ponderum attractu cir-<lb/>cumagebantur, eademque cum illis theca incluſæ erant quater-<lb/>num pedum longitudine. </s> <s xml:id="echoid-s1061" xml:space="preserve">In ima theca plumbum inſuper <lb/>centum atque amplius librarum additum erat, quo melius <lb/>perpendicularem ſitum ſuſpenſa in navi machina ſervaret.</s> <s xml:id="echoid-s1062" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1063" xml:space="preserve">Quanquam autem æquabilis admodum ſibique conſtans <lb/>automati motus per hæc experimenta comperiebatur, tamen <lb/>alia quoque ratione ulterius illud perficere aggreſſi ſumus, <lb/>quæ erat hujuſmodi. </s> <s xml:id="echoid-s1064" xml:space="preserve">Rotæ illi quæ ſerratos dentes habet, <lb/>penduloque proxima eſt, pondus exiguum ex catenula affa-<lb/>bre conſtructa appendimus, quo ſola ipſa moveretur, reli-<lb/>qua omni machina nihil aliud agente quam ut ſingulis ſemi-<lb/>ſcrupulis horariis plumbum illud exiguum ad priorem alti-<lb/>tudinem reſtitueret; </s> <s xml:id="echoid-s1065" xml:space="preserve">eadem fere ratione atque in conſtructio-<lb/>ne horologii ſuperius expoſita videre eſt, ubi pondus al-<lb/>tero fune attollitur, dum altero gravitatem ſuam horolo-<lb/>gii motui impertit. </s> <s xml:id="echoid-s1066" xml:space="preserve">Quibus ita conſtructis, cum veluti ad <lb/>unicam rotam omnia eſſent redacta, major adhuc quam an-<lb/>tea apparuit horologiorum æqualitas, illudque accidit me-<lb/>moratu dignum, quod cum duo ad hanc formam conſtructa <lb/>ex eodem tigno ſuſpendiſſemus, tignum vero fulcris duobus <lb/>impoſitum eſſet; </s> <s xml:id="echoid-s1067" xml:space="preserve">motus penduli utriuſque ita ictibus adver-<lb/>ſis inter ſe conſenſere, ut nunquam inde vel minimum rece-<lb/>derent, ſed utriuſque ſonus una ſemper exaudiretur: </s> <s xml:id="echoid-s1068" xml:space="preserve">imo <lb/>ſi data opera perturbaretur concordia illa, ſemetipſam brevi <lb/>tempore reduceret. </s> <s xml:id="echoid-s1069" xml:space="preserve">Miratus aliquandiu rem adeo inſolitam, <lb/>inveni denique, inſtituto diligenti examine, à motu ti-<lb/>gni ipſius, licet haudquaquam ſenſibili, cauſam petendam <lb/>eſſe. </s> <s xml:id="echoid-s1070" xml:space="preserve">Nempe pendulorum reciprocationes horologiis, quanto-<lb/>libet pondere gravatis, motum aliquem communicare; </s> <s xml:id="echoid-s1071" xml:space="preserve">hunc <lb/>vero motum, tigno ipſi impreſſum, neceſſario efficere ut <lb/>ſi aliter quam contrariis ad unguem ictibus pendulum u- <pb o="50" file="0080" n="83" rhead="CHRISTIANI HUGENII"/> trumque moveatur, eo tamen neceſſario tandem deveniant, <lb/> <anchor type="note" xlink:label="note-0080-01a" xlink:href="note-0080-01"/> ac tum demum tigni motum penitus interquieſcere. </s> <s xml:id="echoid-s1072" xml:space="preserve">Quæ ta-<lb/>men cauſa non ſatis efficaciæ haberet, niſi & </s> <s xml:id="echoid-s1073" xml:space="preserve">horologiorum <lb/>motus aliunde æquabiliſſimus foret atque inter ſe conſen-<lb/>tiens.</s> <s xml:id="echoid-s1074" xml:space="preserve"/> </p> <div xml:id="echoid-div43" type="float" level="2" n="17"> <note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s1075" xml:space="preserve">Cæterum experimentis in Oceani navigatione habitis, ac <lb/>præſertim procella vehementiore aquas agitante, compertum <lb/>fuit primam ac præcipuam curam de motu horologiorum <lb/>abſque interruptione conſervando habendam eſſe, quod jacta-<lb/>tionem navis tantam ægrè tunc perferre illa animadverſum ſit. <lb/></s> <s xml:id="echoid-s1076" xml:space="preserve">Quamobrem nova denique ratione & </s> <s xml:id="echoid-s1077" xml:space="preserve">penduli formam immuta-<lb/>vimus, & </s> <s xml:id="echoid-s1078" xml:space="preserve">aliter horologia ipſa ſuſpendimus. </s> <s xml:id="echoid-s1079" xml:space="preserve">Pendulum trian-<lb/>guli formam habet, in cujus vertice deorſum ſpectante plum-<lb/>bea lens affixa eſt. </s> <s xml:id="echoid-s1080" xml:space="preserve">Anguli utrique reliqui filis inter laminas <lb/>cycloidales ſuſpenſi ſunt. </s> <s xml:id="echoid-s1081" xml:space="preserve">Baſis clavulam bifurcatam puncto <lb/>ſui medio recipit ab eaque movetur, illa vero ab rota ſerra-<lb/>ta horizonti parallela motum accipit. </s> <s xml:id="echoid-s1082" xml:space="preserve">Motus rotarum omni-<lb/>um non à pondere ſed à chalybea lamina, tympano incluſa, <lb/>principium habet. </s> <s xml:id="echoid-s1083" xml:space="preserve">In figura adjecta pendulum triangulare eſt <lb/> <anchor type="note" xlink:label="note-0080-02a" xlink:href="note-0080-02"/> A B C; </s> <s xml:id="echoid-s1084" xml:space="preserve">lens plumbea B; </s> <s xml:id="echoid-s1085" xml:space="preserve">laminæ cycloidales E D, F G. <lb/></s> <s xml:id="echoid-s1086" xml:space="preserve">Clavula bifurcata H K; </s> <s xml:id="echoid-s1087" xml:space="preserve">rota ſerratis dentibus N, quæ cæ-<lb/>teris horologii rotis inferior eſt. </s> <s xml:id="echoid-s1088" xml:space="preserve">Lenticulæ ad temperandum <lb/>penduli motum L L.</s> <s xml:id="echoid-s1089" xml:space="preserve"/> </p> <div xml:id="echoid-div44" type="float" level="2" n="18"> <note position="left" xlink:label="note-0080-02" xlink:href="note-0080-02a" xml:space="preserve">TAB. IV. <lb/>Fig. 1.</note> </div> <note position="left" xml:space="preserve">TAB. IV. <lb/>Fig. 2.</note> <p> <s xml:id="echoid-s1090" xml:space="preserve">Suſpenſionis modum altera hæc figura exhibet; </s> <s xml:id="echoid-s1091" xml:space="preserve">ubi theca <lb/>A B axibus primum duobus, quorum alter C tantum appa-<lb/>ret, rectangulo ferreo D E inſerta eſt; </s> <s xml:id="echoid-s1092" xml:space="preserve">quod deinde rectan-<lb/>gulum rurſus axibus ſuis F G ferreo gnomone F H K G <lb/>ſuſtinetur, qui contignationi navis immobiliter affixus eſt. </s> <s xml:id="echoid-s1093" xml:space="preserve">in <lb/>ima theca pondus 50 librarum appenſum eſt. </s> <s xml:id="echoid-s1094" xml:space="preserve">Quibus ita ſe <lb/>habentibus, quacunque navis inclinatione perpendicularem <lb/>poſitum ſervat horologium. </s> <s xml:id="echoid-s1095" xml:space="preserve">Axis autem C, cum ſibi oppo-<lb/>ſito, ita collocati ſunt, ut ad rectam lineam reſpondeant <lb/>punctis ſuſpenſionum penduli ejus quod diximus: </s> <s xml:id="echoid-s1096" xml:space="preserve">quo fit ut <lb/>motus ipſius oſcillatorius machinam nequaquam commovere <lb/>poſſit, quo nihil eſt alioqui quod magis penduli motum de-<lb/>ſtruat. </s> <s xml:id="echoid-s1097" xml:space="preserve">Porro axium C C, & </s> <s xml:id="echoid-s1098" xml:space="preserve">F G craſſitudo, quæ polli- <pb o="51" file="0081" n="84" rhead="HOROLOG. OSCILLATOR."/> cem æquat, gravitaſque plumbi inferius appenſi, nimiam <lb/> <anchor type="note" xlink:label="note-0081-01a" xlink:href="note-0081-01"/> movendi libertatem horologio adimunt, faciuntque ut ſi for-<lb/>te ſuccuſſu navis graviore commotum fuerit, continuo ad <lb/>quietem perpendiculumque ſuum revertatur.</s> <s xml:id="echoid-s1099" xml:space="preserve"/> </p> <div xml:id="echoid-div45" type="float" level="2" n="19"> <note position="right" xlink:label="note-0081-01" xlink:href="note-0081-01a" xml:space="preserve"><emph style="sc">Descri-</emph> <lb/><emph style="sc">PTIO</emph> <emph style="sc">Ho-</emph> <lb/><emph style="sc">ROLOGII</emph>.</note> </div> <p> <s xml:id="echoid-s1100" xml:space="preserve">Et hæc quidem ita adaptata machina ut in mare deduca-<lb/>tur experientiæque committatur ſupereſt, quæ & </s> <s xml:id="echoid-s1101" xml:space="preserve">certam pe-<lb/>ne ſucceſſus ſpem præbet, quod iis quæ hactenus inſtituere <lb/>licuit experimentis, multo melius quam priores illæ omnem <lb/>motus diverſitatem perferre reperta ſit.</s> <s xml:id="echoid-s1102" xml:space="preserve"/> </p> <figure> <image file="0081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0081-01"/> </figure> </div> <div xml:id="echoid-div47" type="section" level="1" n="21"> <head xml:id="echoid-head39" xml:space="preserve">HOROLOGII OSCILLATORII</head> <head xml:id="echoid-head40" style="it" xml:space="preserve">PARS SECUNDA.</head> <head xml:id="echoid-head41" style="it" xml:space="preserve">De deſcenſu Gravium & motu eorum in Cycloide.</head> <head xml:id="echoid-head42" xml:space="preserve">HYPOTHESES.</head> <head xml:id="echoid-head43" xml:space="preserve">I.</head> <p style="it"> <s xml:id="echoid-s1103" xml:space="preserve">SI gravitas non eſſet, neque aër motui corporum <lb/>officeret, unumquodque eorum, acceptum ſe-<lb/>mel motum continuaturum velocitate æquabili, ſe-<lb/>cundum lineam rectam.</s> <s xml:id="echoid-s1104" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div48" type="section" level="1" n="22"> <head xml:id="echoid-head44" xml:space="preserve">II.</head> <p style="it"> <s xml:id="echoid-s1105" xml:space="preserve">Nunc vero fieri gravitatis actione, undecunque <lb/>illa oriatur, ut moveantur motu compoſito, ex æ-<lb/>quabili quem habent in hanc vel illam partem, & </s> <s xml:id="echoid-s1106" xml:space="preserve"><lb/>ex motu deorſum à gravitate profecto.</s> <s xml:id="echoid-s1107" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div49" type="section" level="1" n="23"> <head xml:id="echoid-head45" xml:space="preserve">III.</head> <p style="it"> <s xml:id="echoid-s1108" xml:space="preserve">Et horum utrumque ſeorſim conſiderari poſſe, <lb/>neque alterum ab altero impediri.</s> <s xml:id="echoid-s1109" xml:space="preserve"/> </p> <pb o="52" file="0082" n="85" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1110" xml:space="preserve">Ponatur grave C è quiete dimiſſum, certo tempore, <lb/> <anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/> quod dicatur F, vi gravitatis tranſire ſpatium C B. </s> <s xml:id="echoid-s1111" xml:space="preserve">Ac <lb/>rurſus intelligatur idem grave accepiſſe alicunde motum quo, <lb/>ſi nulla eſſet gravitas, transiret pari tempore F motu æqua-<lb/>bili lineam rectam C D. </s> <s xml:id="echoid-s1112" xml:space="preserve">Accedente ergo vi gravitatis non <lb/>perveniet grave ex C in D, dicto tempore F, ſed ad pun-<lb/>ctum aliquod E, recta ſub D ſitum, ita ut ſpatium D E <lb/>ſemper æquetur ſpatio C B, ita enim, & </s> <s xml:id="echoid-s1113" xml:space="preserve">motus æquabilis, <lb/>& </s> <s xml:id="echoid-s1114" xml:space="preserve">is qui à gravitate oritur ſuas partes peragent, altero alte-<lb/>rum non impediente. </s> <s xml:id="echoid-s1115" xml:space="preserve">Quamnam vero lineam, compoſito il-<lb/>lo motu, grave percurrat, cum motus æquabilis non recta <lb/>ſurſum aut deorſum ſed in obliquum tendit, è ſequentibus <lb/>definiri poterit. </s> <s xml:id="echoid-s1116" xml:space="preserve">Cum vero deorſum in perpendiculari con-<lb/>tingit motus æquabilis C D, apparet lineam C D, acce-<lb/>dente motu ex gravitate, augeri recta D E. </s> <s xml:id="echoid-s1117" xml:space="preserve">Item, cum ſur-<lb/>ſum tendit motus æquabilis C D, ipſam C D diminui recta <lb/>D E, ut nempe, peracto tempore F, grave inveniatur <lb/>ſemper in puncto E. </s> <s xml:id="echoid-s1118" xml:space="preserve">Quod ſi, utroque hoc caſu, ſeorſim, <lb/>uti diximus, duos motus conſideremus, alterumque ab al-<lb/>tero nullo modo impediri cogitemus, hinc jam acceleratio-<lb/>nis gravium cadentium cauſam legesque reperire licebit. </s> <s xml:id="echoid-s1119" xml:space="preserve">Et <lb/>primum quidem duo iſta ſimul oſtendemus.</s> <s xml:id="echoid-s1120" xml:space="preserve"/> </p> <div xml:id="echoid-div49" type="float" level="2" n="1"> <note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>. <lb/>TAB. IV. <lb/>Fig. 3.</note> </div> </div> <div xml:id="echoid-div51" type="section" level="1" n="24"> <head xml:id="echoid-head46" xml:space="preserve">PROPOSITIO I.</head> <p style="it"> <s xml:id="echoid-s1121" xml:space="preserve">ÆQualibus temporibus æquales celeritatis par-<lb/>tes gravi cadenti accreſcere, & </s> <s xml:id="echoid-s1122" xml:space="preserve">ſpatia æqua-<lb/>libus temporibus ab initio deſcenſus emenſa, augeri <lb/>continue æquali exceſſu.</s> <s xml:id="echoid-s1123" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1124" xml:space="preserve">Ponatur grave aliquod, ex quiete in A, primo tempore <lb/> <anchor type="note" xlink:label="note-0082-02a" xlink:href="note-0082-02"/> lapſum eſſe per ſpatium A B, atque ubi pervenit in B, ac-<lb/>quiſiviſſe celeritatem qua deinceps, tempore ſecundo, mo-<lb/>tu æquabili, percurrere poſſet ſpatium quoddam B D. </s> <s xml:id="echoid-s1125" xml:space="preserve">Sci-<lb/>mus ergo ſpatium ſecundo tempore peragendum majus fore <lb/>ſpatio B D, quia vel ceſſante in B omni gravitatis actione <pb file="0083" n="86"/> <pb file="0083a" n="87"/> <anchor type="figure" xlink:label="fig-0083a-01a" xlink:href="fig-0083a-01"/> <anchor type="figure" xlink:label="fig-0083a-02a" xlink:href="fig-0083a-02"/> <anchor type="figure" xlink:label="fig-0083a-03a" xlink:href="fig-0083a-03"/> <pb file="0084" n="88"/> <pb o="53" file="0085" n="89" rhead="HOROLOG. OSCILLATOR."/> ſpatium B D percurreretur. </s> <s xml:id="echoid-s1126" xml:space="preserve">Feretur vero motu compoſito <lb/> <anchor type="note" xlink:label="note-0085-01a" xlink:href="note-0085-01"/> ex æquabili quo percurſurum eſſet ſpatium B D, & </s> <s xml:id="echoid-s1127" xml:space="preserve">ex mo-<lb/>tu gravium cadentium, quo deprimi neceſſe eſt per ſpatium <lb/>ipſi A B æquale. </s> <s xml:id="echoid-s1128" xml:space="preserve">Quare ad B D addita D E, æquali A B, <lb/>ſcimus tempore ſecundo grave perventurum ad E.</s> <s xml:id="echoid-s1129" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="1"> <note position="left" xlink:label="note-0082-02" xlink:href="note-0082-02a" xml:space="preserve">TAB. V. <lb/>Fig. 1.</note> <figure xlink:label="fig-0083a-01" xlink:href="fig-0083a-01a"> <caption xml:id="echoid-caption7" style="it" xml:space="preserve">Pag. 52.<lb/>TAB. IV.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables7" xml:space="preserve">N H G E F D C A K L L B</variables> </figure> <figure xlink:label="fig-0083a-02" xlink:href="fig-0083a-02a"> <caption xml:id="echoid-caption8" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables8" xml:space="preserve">A B E F D C L</variables> </figure> <figure xlink:label="fig-0083a-03" xlink:href="fig-0083a-03a"> <caption xml:id="echoid-caption9" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables9" xml:space="preserve">D D D E E E D E C D B E D E D D D E E E</variables> </figure> <note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1130" xml:space="preserve">Quod ſi vero inquiramus quam velocitatem habeat in E, <lb/>in fine ſecundi temporis, eam inveniemus duplam eſſe debe-<lb/>re velocitatis quam habebat in B fine temporis primi. </s> <s xml:id="echoid-s1131" xml:space="preserve">Dixi-<lb/>mus enim moveri compoſito motu ex æquabili cum celerita-<lb/>te acquiſita in B, & </s> <s xml:id="echoid-s1132" xml:space="preserve">ex motu à gravitate producto, qui <lb/>cum tempore ſecundo idem plane ſit ac primo, ideo decur-<lb/>ſu temporis ſecundi æqualem celeritatem gravi contuliſſe <lb/>debet atque in fine primi. </s> <s xml:id="echoid-s1133" xml:space="preserve">Quare cum acquiſitam in fine pri-<lb/>mi temporis celeritatem conſervaverit integram, apparet in <lb/>fine ſecundi temporis bis eam celeritatem ineſſe quam acqui-<lb/>ſiverat in fine temporis primi, ſive duplam.</s> <s xml:id="echoid-s1134" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1135" xml:space="preserve">Quod ſi jam, poſtquam pervenit in E, pergeret deinceps <lb/>tantum moveri celeritate æquabili, quantam illic acquiſivit, <lb/>apparet tempore tertio, prioribus æquali, percurſurum ſpa-<lb/>tium E F, quod duplum futurum ſit ſpatii B D; </s> <s xml:id="echoid-s1136" xml:space="preserve">quia hoc <lb/>percurri diximus dimidia hujus celeritatis, motu æquabili, <lb/>& </s> <s xml:id="echoid-s1137" xml:space="preserve">temporis parte æquali. </s> <s xml:id="echoid-s1138" xml:space="preserve">Accedente autem rurſus gravitatis <lb/>actione, percurret tempore tertio, præter ſpatium E F, <lb/>etiam ſpatium F G, ipſi A B vel D E æquale. </s> <s xml:id="echoid-s1139" xml:space="preserve">Itaque in <lb/>fine tertii temporis grave invenietur in G. </s> <s xml:id="echoid-s1140" xml:space="preserve">Velocitatem vero <lb/>hîc habebit triplam jam ejus quam habebat in B, in fine pri-<lb/>mi temporis: </s> <s xml:id="echoid-s1141" xml:space="preserve">quia præter celeritatem acquiſitam in E, quam <lb/>diximus duplam eſſe acquiſitæ in B, vis gravitatis, tempo-<lb/>ris tertii decurſu, æqualem rurſus atque in fine primi cele-<lb/>ritatem contulit. </s> <s xml:id="echoid-s1142" xml:space="preserve">Quamobrem utraque celeritas, in fine tem-<lb/>poris tertii, triplam celeritatem conſtituet ejus quæ fuerat <lb/>in fine temporis primi.</s> <s xml:id="echoid-s1143" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1144" xml:space="preserve">Eodem modo oſtendetur tempore quarto peragi debere & </s> <s xml:id="echoid-s1145" xml:space="preserve"><lb/>ſpatium G H triplum ſpatii B D, & </s> <s xml:id="echoid-s1146" xml:space="preserve">ſpatium H K ipſi A B <lb/>æquale: </s> <s xml:id="echoid-s1147" xml:space="preserve">velocitatemque in K, in fine quarti temporis, fo-<lb/>re quadruplam ejus quæ fuerat in B, in fine temporis primi.</s> <s xml:id="echoid-s1148" xml:space="preserve"> <pb o="54" file="0086" n="90" rhead="CHRISTIANI HUGENII"/> Atque ita ſpatia quotlibet deinceps conſiderata, quæ æqua-<lb/> <anchor type="note" xlink:label="note-0086-01a" xlink:href="note-0086-01"/> libus temporibus peracta fuerint, æquali exceſſu, qui ipſi <lb/>B D æqualis ſit, creſcere manifeſtum eſt; </s> <s xml:id="echoid-s1149" xml:space="preserve">ſimulque etiam <lb/>velocitates per æqualia tempora æqualiter augeri.</s> <s xml:id="echoid-s1150" xml:space="preserve"/> </p> <div xml:id="echoid-div52" type="float" level="2" n="2"> <note position="left" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> </div> <div xml:id="echoid-div54" type="section" level="1" n="25"> <head xml:id="echoid-head47" xml:space="preserve">PROPOSITIO II.</head> <p style="it"> <s xml:id="echoid-s1151" xml:space="preserve">SPatium peractum certo tempore à gravi, è quie-<lb/>te caſum inchoante, dimidium eſt ejus ſpatii <lb/>quod pari tempore transiret motu æquabili, cum <lb/>velocitate quam acquiſivit ultimo caſus momento.</s> <s xml:id="echoid-s1152" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1153" xml:space="preserve">Ponantur quæ in propoſitione præcedenti, ubi quidem <lb/> <anchor type="note" xlink:label="note-0086-02a" xlink:href="note-0086-02"/> A B erat ſpatium certo tempore, à gravi cadente ex A, per-<lb/>actum. </s> <s xml:id="echoid-s1154" xml:space="preserve">B D vero ſpatium quod pari tempore transiri intel-<lb/>ligebatur celeritate æquabili, quanta acquiſita erat in fine <lb/>primi temporis, ſeu in fine ſpatii A B. </s> <s xml:id="echoid-s1155" xml:space="preserve">Dico itaque ſpatium <lb/>B D duplum eſſe ad A B.</s> <s xml:id="echoid-s1156" xml:space="preserve"/> </p> <div xml:id="echoid-div54" type="float" level="2" n="1"> <note position="left" xlink:label="note-0086-02" xlink:href="note-0086-02a" xml:space="preserve">TAB. V. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s1157" xml:space="preserve">Quum enim ſpatia primis quatuor æqualibus temporibus <lb/>à cadente transmiſſa ſint A B, B E, E G, G H, quorum <lb/>inter ſe certa quædam eſt proportio: </s> <s xml:id="echoid-s1158" xml:space="preserve">ſi eorum temporum du-<lb/>pla tempora ſumamus, ut nempe pro primo tempore jam <lb/>accipiantur duo illa quibus ſpatia A B, B E, peracta fue-<lb/>re; </s> <s xml:id="echoid-s1159" xml:space="preserve">pro ſecundo vero tempore duo reliqua quibus peracta <lb/>fuere ſpatia E G, G K, oportet jam ſpatia A E, E K, <lb/>quæ ſunt æqualibus temporibus à quiete peracta, inter ſe <lb/>eſſe ſicut ſpatia A B, B E, quæ æqualibus item tempori-<lb/>bus à quiete percurrebantur.</s> <s xml:id="echoid-s1160" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1161" xml:space="preserve">Quum igitur ſit ut A B ad B E, ita A E ad E K; </s> <s xml:id="echoid-s1162" xml:space="preserve">& </s> <s xml:id="echoid-s1163" xml:space="preserve"><lb/>convertendo, ut E B ſive D A ad A B ita K E ad E A: <lb/></s> <s xml:id="echoid-s1164" xml:space="preserve">erit quoque, dividendo, D B ad B A ut exceſſus K E ſu-<lb/>pra E A ad E A. </s> <s xml:id="echoid-s1165" xml:space="preserve">Quum ſit autem, ex oſtenſis propoſitione <lb/>præcedenti, K E æqualis tum duplæ A B, tum quintuplæ <lb/>B D: </s> <s xml:id="echoid-s1166" xml:space="preserve">E A vero æqualis tum duplæ A B, tum ſimplici B D; </s> <s xml:id="echoid-s1167" xml:space="preserve"><lb/>apparet dictum exceſſum K E ſupra E A æquari quadruplæ <pb o="55" file="0087" n="91" rhead="HOROLOG. OSCILLATOR."/> B D. </s> <s xml:id="echoid-s1168" xml:space="preserve">Sicut igitur D B ad B A ita erit quadrupla D B ad <lb/> <anchor type="note" xlink:label="note-0087-01a" xlink:href="note-0087-01"/> E A: </s> <s xml:id="echoid-s1169" xml:space="preserve">unde E A quadrupla erit ipſius B A: </s> <s xml:id="echoid-s1170" xml:space="preserve">eadem vero E A <lb/>æquatur, uti diximus, & </s> <s xml:id="echoid-s1171" xml:space="preserve">duplæ A B & </s> <s xml:id="echoid-s1172" xml:space="preserve">ſimplici B D. </s> <s xml:id="echoid-s1173" xml:space="preserve">er-<lb/>go B D duplæ A B æqualis erit; </s> <s xml:id="echoid-s1174" xml:space="preserve">quod erat demonſtran-<lb/>dum.</s> <s xml:id="echoid-s1175" xml:space="preserve"/> </p> <div xml:id="echoid-div55" type="float" level="2" n="2"> <note position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> </div> <div xml:id="echoid-div57" type="section" level="1" n="26"> <head xml:id="echoid-head48" xml:space="preserve">PROPOSITIO III.</head> <p style="it"> <s xml:id="echoid-s1176" xml:space="preserve">SPatia duo, à gravi cadente quibuslibet tempo-<lb/>ribus transmiſſa, quorum utrumque ab initio <lb/>deſcenſus accipiatur, ſunt inter ſe in ratione du-<lb/>plicata eorundem temporum, ſive ut temporum qua-<lb/>drata, ſive etiam ut quadrata celeritatum in fine <lb/>cujusque temporis acquiſitarum.</s> <s xml:id="echoid-s1177" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1178" xml:space="preserve">Quum enim oſtenſum ſit propoſitione antecedenti ſpa-<lb/> <anchor type="note" xlink:label="note-0087-02a" xlink:href="note-0087-02"/> tia A B, B E, E G, G K, quotcunque fuerint, æqualibus <lb/>temporibus à cadente, peracta, creſcere æquali exceſſu, qui <lb/>exceſſus ſit ipſi B D æqualis: </s> <s xml:id="echoid-s1179" xml:space="preserve">Patet nunc, quoniam B D eſt <lb/>dupla A B, ſpatium B E fore triplum A B; </s> <s xml:id="echoid-s1180" xml:space="preserve">E G quintu-<lb/>plum ejuſdem A B; </s> <s xml:id="echoid-s1181" xml:space="preserve">G K ſeptuplum; </s> <s xml:id="echoid-s1182" xml:space="preserve">aliaque deinceps au-<lb/>ctum iri ſecundum progreſſionem numerorum imparium ab <lb/>unitate, 1, 3, 5, 7, 9, &</s> <s xml:id="echoid-s1183" xml:space="preserve">c. </s> <s xml:id="echoid-s1184" xml:space="preserve">cumque quotlibet horum nu-<lb/>merorum, ſeſe conſequentium, ſumma faciat quadratum, <lb/>cujus latus eſt ipſa adſumptorum numerorum multitudo (ve-<lb/>lut ſi tres primi addantur, facient novem, ſi quatuor ſexde-<lb/>cim) ſequitur hinc ſpatia, à gravi cadente tranſmiſſa, quo-<lb/>rum utrumque à principio caſus inchoetur, eſſe inter ſe in <lb/>ratione duplicata temporum quibus caſus duravit, ſi nempe <lb/>tempora commenſurabilia ſumantur.</s> <s xml:id="echoid-s1185" xml:space="preserve"/> </p> <div xml:id="echoid-div57" type="float" level="2" n="1"> <note position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">TAB. V. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s1186" xml:space="preserve">Facile autem & </s> <s xml:id="echoid-s1187" xml:space="preserve">ad tempora incommenſurabilia demonſtra-<lb/> <anchor type="note" xlink:label="note-0087-03a" xlink:href="note-0087-03"/> tio extendetur. </s> <s xml:id="echoid-s1188" xml:space="preserve">Sint enim tempora hujuſmodi, quorum inter <lb/>ſe ratio ea quæ linearum A B, C D. </s> <s xml:id="echoid-s1189" xml:space="preserve">ſpatiaque temporibus <lb/>his tranſmiſſa ſint E, & </s> <s xml:id="echoid-s1190" xml:space="preserve">F, utraque nimirum ab initio de-<lb/>ſcenſus adſumpta. </s> <s xml:id="echoid-s1191" xml:space="preserve">Dico eſſe, ut quadratum A B ad quadra-<lb/>tum C D, ita ſpatium E ad F.</s> <s xml:id="echoid-s1192" xml:space="preserve"/> </p> <div xml:id="echoid-div58" type="float" level="2" n="2"> <note position="right" xlink:label="note-0087-03" xlink:href="note-0087-03a" xml:space="preserve">TAB. V. <lb/>Fig. 2.</note> </div> <pb o="56" file="0088" n="92" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1193" xml:space="preserve">Si enim negetur; </s> <s xml:id="echoid-s1194" xml:space="preserve">habeat primo, ſi poteſt, ſpatium E ad F <lb/> <anchor type="note" xlink:label="note-0088-01a" xlink:href="note-0088-01"/> majorem rationem quam quadratum A B ad quadratum <lb/>C D, nempe eam quam quadratum A B ad quadratum C <lb/>G, ſumta C G minore quam C D, & </s> <s xml:id="echoid-s1195" xml:space="preserve">à C D auferatur <lb/>pars D H, minor quam D G exceſſus C D ſupra C G, <lb/>atque ita ut reliqua H C commenſurabilis ſit ipſi A B; <lb/></s> <s xml:id="echoid-s1196" xml:space="preserve">hoc enim fieri poſſe conſtat. </s> <s xml:id="echoid-s1197" xml:space="preserve">Erit ergo C H major quam <lb/>C G. </s> <s xml:id="echoid-s1198" xml:space="preserve">Atqui ut quadratum temporis A B ad quadratum tem-<lb/>poris C H, ita ſpatium E, quod tempore A B peractum <lb/>eſt, ad ſpatium peractum tempore C H, per ſuperiùs oſten-<lb/>ſa. </s> <s xml:id="echoid-s1199" xml:space="preserve">Hoc vero ſpatio majus eſt illud quod tempore C D per-<lb/>curritur, nempe ſpatium F. </s> <s xml:id="echoid-s1200" xml:space="preserve">ergo ſpatii E ad ſpatium F mi-<lb/>nor eſt ratio quam quadrati A B ad quadratum C H. </s> <s xml:id="echoid-s1201" xml:space="preserve">Sicut <lb/>autem ſpatium E ad F, ita ponebatur eſſe quadratum A B <lb/>ad quadratum C G; </s> <s xml:id="echoid-s1202" xml:space="preserve">ergo minor quoque erit ratio quadrati <lb/>A B ad quadratum C G, quam quadrati A B ad quadra-<lb/>tum C H, ac proinde quadratum C G majus quadrato C <lb/>H; </s> <s xml:id="echoid-s1203" xml:space="preserve">quod eſt abſurdum, quum C H major dicta ſit quam <lb/>C G. </s> <s xml:id="echoid-s1204" xml:space="preserve">Non habet igitur ſpatium E ad F majorem rationem <lb/>quam quadratum A B ad quadratum C D.</s> <s xml:id="echoid-s1205" xml:space="preserve"/> </p> <div xml:id="echoid-div59" type="float" level="2" n="3"> <note position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1206" xml:space="preserve">Habeat jam, ſi poteſt, minorem; </s> <s xml:id="echoid-s1207" xml:space="preserve">ſitque ratio ſpatii E ad <lb/>F eadem quæ quadrati A B ad quadratum C L, ſumptâ C L <lb/>majore quam C D, & </s> <s xml:id="echoid-s1208" xml:space="preserve">à C L auferatur L K minor ex-<lb/>ceſſu L D, quo C D ſuperatur à C L, atque ita <lb/>ut reliqua K C ſit commenſurabilis A B. </s> <s xml:id="echoid-s1209" xml:space="preserve">Quia ergo ut qua-<lb/>dratum temporis A B ad quadratum temporis C K, ita eſt <lb/>ſpatium E, peractum tempore A B, ad ſpatium peractum <lb/>tempore C K. </s> <s xml:id="echoid-s1210" xml:space="preserve">Hoc vero ſpatio minus eſt ſpatium peractum <lb/>tempore C D, nempe ſpatium F. </s> <s xml:id="echoid-s1211" xml:space="preserve">erit proinde ſpatii E ad <lb/>F major ratio quam quadrati A B ad quadratum C K. </s> <s xml:id="echoid-s1212" xml:space="preserve">Sic-<lb/>ut autem ſpatium E ad F, ita ponebatur eſſe quadratum <lb/>A B ad quadratum C L. </s> <s xml:id="echoid-s1213" xml:space="preserve">Ergo major erit ratio quadrati A B <lb/>ad quadratum C L quam ejuſdem quadrati A B ad quadra-<lb/>tum C K, ideoque quadratum C L minus erit quam qu. </s> <s xml:id="echoid-s1214" xml:space="preserve">C K. <lb/></s> <s xml:id="echoid-s1215" xml:space="preserve">quod eſt abſurdum, quum C L major ſit quam C K. </s> <s xml:id="echoid-s1216" xml:space="preserve"><lb/>Ergo neque minorem rationem habet ſpatium E ad F quam <pb o="57" file="0089" n="93" rhead="HOROLOG. OSCILLATOR."/> quadratum A B ad quadratum C D. </s> <s xml:id="echoid-s1217" xml:space="preserve">quare neceſſe eſt ut <lb/> <anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/> eandem habeat. </s> <s xml:id="echoid-s1218" xml:space="preserve">Porro cum celeritates in fine temporum A B, <lb/>C D acquiſitæ ſint inter ſe ſicut ipſamet tempora; </s> <s xml:id="echoid-s1219" xml:space="preserve">apparet <lb/>rationem ſpatiorum E ad F eandem quoque eſſe quæ qua-<lb/>dratorum temporum A B, C D, quibus transmiſſa ſunt. <lb/></s> <s xml:id="echoid-s1220" xml:space="preserve">Itaque conſtat propoſitum.</s> <s xml:id="echoid-s1221" xml:space="preserve"/> </p> <div xml:id="echoid-div60" type="float" level="2" n="4"> <note position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">@RAVIUM</emph>.</note> </div> </div> <div xml:id="echoid-div62" type="section" level="1" n="27"> <head xml:id="echoid-head49" xml:space="preserve">PROPOSITIO IV.</head> <p style="it"> <s xml:id="echoid-s1222" xml:space="preserve">SI grave celeritate ea quam in fine deſcenſus ac-<lb/>quiſivit ſurſum tendere cœperit, fiet ut paribus <lb/>temporis partibus, ſpatia quæ prius ſurſum, ea-<lb/>dem deorſum transeat, adeoque ad eandem unde <lb/>deſcenderat altitudinem aſcendat. </s> <s xml:id="echoid-s1223" xml:space="preserve">Item ut æquali-<lb/>bus temporis partibus æqualia amittat celeritatis <lb/>momenta.</s> <s xml:id="echoid-s1224" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">TAB. V. <lb/>Fig. 1.</note> <p> <s xml:id="echoid-s1225" xml:space="preserve">Sunto enim ut in propoſitione 2, ſpatia quotlibet, æqua-<lb/>libus temporis partibus cadendo è quiete peracta, quorum <lb/>primum A B; </s> <s xml:id="echoid-s1226" xml:space="preserve">ſecundum compoſitum ex B D, quod celeri-<lb/>tate æquabili acquiſita per A B tranſeundum erat, & </s> <s xml:id="echoid-s1227" xml:space="preserve">ex D E <lb/>ipſi A B æquali; </s> <s xml:id="echoid-s1228" xml:space="preserve">tertium compoſitum, ex E F, duplo <lb/>ipſius B D, & </s> <s xml:id="echoid-s1229" xml:space="preserve">ex F G, eidem A B æquali; </s> <s xml:id="echoid-s1230" xml:space="preserve">quartum com-<lb/>poſitum ex G H, triplo ipſius B D, & </s> <s xml:id="echoid-s1231" xml:space="preserve">ex H K ipſi itidem <lb/>A B æquali, atque eadem ratione porro creſcentia, ſi plu-<lb/>ra fuerint. </s> <s xml:id="echoid-s1232" xml:space="preserve">Dico totidem æqualibus temporibus eadem ſpatia <lb/>K G, G E, E B, B A, ſingula ſingulis peragenda eſſe à <lb/>gravi ſurſum tendente, atque incipiente cum celeritate in <lb/>fine deſcenſus K acquiſita.</s> <s xml:id="echoid-s1233" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1234" xml:space="preserve">Brevitatis autem gratia celeritas quæque deſignetur de-<lb/>inceps longitudine ſpatii quod grave motu æquabili, cum <lb/>celeritate illa, atque temporis parte una, quales in deſcen-<lb/>ſu conſideravimus, tranſmiſſurum eſſet.</s> <s xml:id="echoid-s1235" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1236" xml:space="preserve">Itaque ex oſtenſis dicta propoſitione, cum in K grave <lb/>pervenerit, habet celeritatem G H auctam celeritate B D, <pb o="58" file="0090" n="94" rhead="CHRISTIANI HUGENII"/> hoc eſt celeritatem K F, quia K F æquatur ipſis H G, B D, <lb/> <anchor type="note" xlink:label="note-0090-01a" xlink:href="note-0090-01"/> ſunt enim partes ſingulæ H K, F G, æquales ipſi A B, <lb/>ac proinde utraque ſimul ipſi B D, quam eſſe duplam <lb/>A B oſtendimus propoſitione 2. </s> <s xml:id="echoid-s1237" xml:space="preserve">Itaque celeritatem in fine <lb/>deſcenſus K acquiſitam ſurſum convertendo, ſi grave æqua-<lb/>bili motu ferretur, conficeret una temporis parte ſpatium <lb/>K F. </s> <s xml:id="echoid-s1238" xml:space="preserve">Atqui, gravitatis actione accedente, diminuetur <lb/>aſcenſus K F ſpatio F G ipſi A B æquali, ut patet ex di-<lb/>ctis ad hypotheſin initio ſumptam. </s> <s xml:id="echoid-s1239" xml:space="preserve">Ergo parte prima tempo-<lb/>ris aſcendet grave tantum per K G, quo eodem ſpatio parte <lb/>temporis noviſſima deſcenderat. </s> <s xml:id="echoid-s1240" xml:space="preserve">Simul vero & </s> <s xml:id="echoid-s1241" xml:space="preserve">celeritati tan-<lb/>tum deceſſiſſe neceſſe eſt, quantum acquiritur temporis parte <lb/>una deorſum cadendo, hoc eſt celeritatem B D. </s> <s xml:id="echoid-s1242" xml:space="preserve">Itaque gra-<lb/>ve, ubi ad G aſcenderit, habet celeritatem reliquam H G, <lb/>cum initio aſcenſus habuerit celeritatem H G una cum cele-<lb/>ritate B D. </s> <s xml:id="echoid-s1243" xml:space="preserve">Eſt autem ipſi H G æqualis G D; </s> <s xml:id="echoid-s1244" xml:space="preserve">quum æque-<lb/>tur ipſi F E una cum D B, hoc eſt una cum dupla A B, <lb/>hoc eſt una cum duabus F G & </s> <s xml:id="echoid-s1245" xml:space="preserve">E D; </s> <s xml:id="echoid-s1246" xml:space="preserve">Ergo ſi ex G, cum <lb/>celeritate æquabili, quantam illic habet, ſurſum pergeret, <lb/>conficeret una parte temporis ſpatium G D. </s> <s xml:id="echoid-s1247" xml:space="preserve">Accedente au-<lb/>tem gravitatis actione, diminuetur aſcenſus iſte ſpatio D E, <lb/>ipſi A B æquali. </s> <s xml:id="echoid-s1248" xml:space="preserve">Ergo, hac ſecunda parte temporis, aſcendet <lb/>per ſpatium G E, quod ſimili temporis parte etiam cadendo <lb/>tranſierat. </s> <s xml:id="echoid-s1249" xml:space="preserve">Simul autem celeritati tantum deceſſiſſe denuo de-<lb/>bet quantum temporis parte una ex caſu acquiritur, nempe <lb/>celeritas B D. </s> <s xml:id="echoid-s1250" xml:space="preserve">Itaque ubi uſque ad E aſcenderit, habet dun-<lb/>taxat celeritatem F E, quæ nimirum relinquitur quum à <lb/>celeritate G D aufertur celeritas B D. </s> <s xml:id="echoid-s1251" xml:space="preserve">Nam B D, ut jam <lb/>diximus, æqualis eſt duabus D E, F G.</s> <s xml:id="echoid-s1252" xml:space="preserve"/> </p> <div xml:id="echoid-div62" type="float" level="2" n="1"> <note position="left" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1253" xml:space="preserve">Eſt autem ipſi F E æqualis E A, quum F E æquetur ipſi <lb/>B D bis ſumptæ, hoc eſt ipſi B D una cum dupla A B, <lb/>hoc eſt una cum duabus A B, D E. </s> <s xml:id="echoid-s1254" xml:space="preserve">Ergo ſi ex E cum ce-<lb/>leritate æquabili, quantam illic habet, ſurſum pergeret, con-<lb/>fecturum eſſet una temporis parte ſpatium E A. </s> <s xml:id="echoid-s1255" xml:space="preserve">Sed acce-<lb/>dente actione gravitatis, diminuetur aſcenſus iſte ipſo ſpatio <lb/>A B. </s> <s xml:id="echoid-s1256" xml:space="preserve">Proinde hac parte temporis per ſpatium E B tantum <pb o="59" file="0091" n="95" rhead="HOROLOG. OSCILLATOR."/> aſcendet, quod ſimili parte temporis deſcendendo quoque <lb/> <anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/> tranſierat. </s> <s xml:id="echoid-s1257" xml:space="preserve">Hic vero rurſus celeritati tantum deceſſiſſe neceſſe <lb/>eſt quantum una temporis parte cadendo deorſum acquiritur, <lb/>hoc eſt celeritatem B D. </s> <s xml:id="echoid-s1258" xml:space="preserve">Itaque grave, ubi uſque ad B a-<lb/>ſcenderit, habet celeritatem ipſam B D reliquam, cum in E <lb/>habuerit celeritatem F E ipſius B D duplam. </s> <s xml:id="echoid-s1259" xml:space="preserve">Si ergo ex B <lb/>cum celeritate æquabili, quantam illic habet, ſurſum per-<lb/>geret, confecturum eſſet parte una temporis ſpatium æquale <lb/>ipſi D B, hoc eſt duplum A B. </s> <s xml:id="echoid-s1260" xml:space="preserve">Sed accedente gravitatis <lb/>actione, diminuitur aſcenſus iſte ſpatio quod ipſi A B æqua-<lb/>le ſit. </s> <s xml:id="echoid-s1261" xml:space="preserve">Igitur hac parte temporis aſcendet tantummodo per <lb/>ſpatium B A, quod etiam primo deſcenſus tempore trans-<lb/>ierat. </s> <s xml:id="echoid-s1262" xml:space="preserve">Atque in fine quidem extremi temporis hujus neceſſa-<lb/>rio grave in A puncto reperietur. </s> <s xml:id="echoid-s1263" xml:space="preserve">Sed dicetur forſan altius <lb/>aſcendiſſe quam ad A, atque inde eo relapſum eſſe. </s> <s xml:id="echoid-s1264" xml:space="preserve">At hoc <lb/>abſurdum eſſet, cum non poſſit, notu à gravitate profecto, al-<lb/>tius quam unde decidit aſcendere. </s> <s xml:id="echoid-s1265" xml:space="preserve">Porro quum celeritati quam <lb/>in B habebat rurſus deceſſerit celeritas B D, patet jam gra-<lb/>vi in A conſtituto nullam celeritatem ſupereſſe, ac proinde <lb/>non altius excurſurum. </s> <s xml:id="echoid-s1266" xml:space="preserve">Itaque oſtenſum eſt ad eandem unde <lb/>decidit altitudinem perveniſſe, & </s> <s xml:id="echoid-s1267" xml:space="preserve">ſingula ſpatia, quæ æqua-<lb/>libus deſcenſus temporibus tranſmiſerat, eadem totidem a-<lb/>ſcenſus temporibus remenſum eſſe: </s> <s xml:id="echoid-s1268" xml:space="preserve">ſed & </s> <s xml:id="echoid-s1269" xml:space="preserve">æqualibus tempo-<lb/>ribus æqualia ipſi deceſſiſſe celeritatis momenta apparuit. </s> <s xml:id="echoid-s1270" xml:space="preserve">Ergo <lb/>conſtat propoſitum.</s> <s xml:id="echoid-s1271" xml:space="preserve"/> </p> <div xml:id="echoid-div63" type="float" level="2" n="2"> <note position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1272" xml:space="preserve">Quia vero in demonſtratione propoſitionis ſecundæ, ex <lb/>qua pendet præcedens, adſumptum fuit certam quandam eſ-<lb/>ſe proportionem ſpatiorum quæ continuis æqualibus tempo-<lb/>ribus à gravi cadente transeuntur, quæque eadem ſit, quæ-<lb/>cunque æqualia tempora accipiantur; </s> <s xml:id="echoid-s1273" xml:space="preserve">quod quidem & </s> <s xml:id="echoid-s1274" xml:space="preserve">ex <lb/>rei natura ita ſe habere neceſſe eſt, & </s> <s xml:id="echoid-s1275" xml:space="preserve">ſi negetur, fatendum <lb/>fruſtra proportionem iſtorum ſpatiorum inveſtigari. </s> <s xml:id="echoid-s1276" xml:space="preserve">Tamen, <lb/>quia propoſitum etiam absque hoc demonſtrari poteſt, Ga-<lb/>lilei methodum ſequendo, operæ pretium erit demonſtra-<lb/>tionem, ab illo minus perfecte traditam, hic accuratius <lb/>conſcribere. </s> <s xml:id="echoid-s1277" xml:space="preserve">itaque rurſum hic demonſtrabimus.</s> <s xml:id="echoid-s1278" xml:space="preserve"/> </p> <pb o="60" file="0092" n="96" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div65" type="section" level="1" n="28"> <note position="left" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SOENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> <head xml:id="echoid-head50" xml:space="preserve">PROPOSITIO V. </head> <p style="it"> <s xml:id="echoid-s1279" xml:space="preserve">SPatium peractum certo tempore, à gravi è quie-<lb/>te caſum inchoante, dimidium eſſe ejus ſpatii <lb/>quod pari tempore transiret motu æquabili, cum <lb/>celeritate quam acquiſivit ultimo caſus momento.</s> <s xml:id="echoid-s1280" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1281" xml:space="preserve">Sit tempus deſcenſus totius A H, quo tempore mobile <lb/> <anchor type="note" xlink:label="note-0092-02a" xlink:href="note-0092-02"/> peregerit ſpatium quoddam cujus quantitas deſignetur plano P. <lb/></s> <s xml:id="echoid-s1282" xml:space="preserve">ductaque H L perpendiculari ad A H, longitudinis cujus-<lb/>libet, referat illa celeritatem in fine caſus acquiſitam. </s> <s xml:id="echoid-s1283" xml:space="preserve">Dein-<lb/>de completo rectangulo A H L M, intelligatur eo notari <lb/>quantitas ſpatii quod percurreretur tempore A H, cum ce-<lb/>leritate H L. </s> <s xml:id="echoid-s1284" xml:space="preserve">Oſtendendum eſt igitur planum P dimidium <lb/>eſſe rectanguli M H, hoc eſt, ducta diagonali A L, æqua-<lb/>le triangulo A H L.</s> <s xml:id="echoid-s1285" xml:space="preserve"/> </p> <div xml:id="echoid-div65" type="float" level="2" n="1"> <note position="left" xlink:label="note-0092-02" xlink:href="note-0092-02a" xml:space="preserve">TAB. V. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s1286" xml:space="preserve">Si planum P non eſt æquale triangulo A H L, ergo aut <lb/>minus eo erit, aut majus. </s> <s xml:id="echoid-s1287" xml:space="preserve">Sit primo, ſi fieri poteſt, pla-<lb/>num P minus triangulo A H L. </s> <s xml:id="echoid-s1288" xml:space="preserve">dividatur autem A H in tot <lb/>partes æquales A C, C E, E G &</s> <s xml:id="echoid-s1289" xml:space="preserve">c. </s> <s xml:id="echoid-s1290" xml:space="preserve">ut, circumſcriptâ tri-<lb/>angulo A H L figurâ è rectangulis quorum altitudo ſingulis <lb/>diviſionum ipſius A H partibus æquetur, ut ſunt rectangula <lb/>B C, D E, F G, alterâque eidem triangulo inſcriptâ, ex <lb/>rectangulis ejusdem altitudinis, ut ſunt K E, O G &</s> <s xml:id="echoid-s1291" xml:space="preserve">c. </s> <s xml:id="echoid-s1292" xml:space="preserve">ut, <lb/>inquam, exceſſus illius figuræ ſupra hanc, minor ſit exceſ-<lb/>ſu trianguli A H L ſupra planum P. </s> <s xml:id="echoid-s1293" xml:space="preserve">hoc enim fieri poſſe <lb/>perſpicuum eſt, cum totus exceſſus figuræ circumſcriptæ ſu-<lb/>per inſcriptam æquetur rectangulo infimo, baſin habenti H L. <lb/></s> <s xml:id="echoid-s1294" xml:space="preserve">Erit itaque omnino exceſſus ipſius trianguli A H L ſupra <lb/>figuram inſcriptam minor quam ſupra planum P, ac proin-<lb/>de figura triangulo inſcripta major plano P. </s> <s xml:id="echoid-s1295" xml:space="preserve">Porro autem, <lb/>quum recta A H tempus totius deſcenſus referat, ejus par-<lb/>tes æquales A C, C E, E G, æquales temporis illius par-<lb/>tes referent. </s> <s xml:id="echoid-s1296" xml:space="preserve">Cumque celeritates mobilis cadentis creſcant <lb/> <anchor type="note" xlink:label="note-0092-03a" xlink:href="note-0092-03"/> eadem proportione qua tempora deſcenſus <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s1297" xml:space="preserve">ſitque celeritas <pb o="61" file="0093" n="97" rhead="HOROLOG. OSCILLATOR."/> in fine totius temporis acquiſita H L; </s> <s xml:id="echoid-s1298" xml:space="preserve">erit ea, quæ in fine <lb/> <anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/> primæ partis temporis A C acquiretur, C K; </s> <s xml:id="echoid-s1299" xml:space="preserve">quia ut A H <lb/>ad A C, ita H L ad C K. </s> <s xml:id="echoid-s1300" xml:space="preserve">Similiter quæ in fine partis tem-<lb/>poris ſecundæ C E acquiritur, erit E O, atque ita dein-<lb/>ceps. </s> <s xml:id="echoid-s1301" xml:space="preserve">Patet autem, tempore primo A C, ſpatium aliquod à <lb/>mobili transmiſſum eſſe, quod majus ſit nihilo; </s> <s xml:id="echoid-s1302" xml:space="preserve">tempore ve-<lb/>ro ſecundo C E transmiſſum eſſe ſpatium quod majus ſit <lb/>quam K E, quia ſpatium K E transmiſſum fuiſſet tempore <lb/>C E, motu æquabili, cum celeritate C K. </s> <s xml:id="echoid-s1303" xml:space="preserve">habent enim ſpa-<lb/>tia, motu æquabili transacta, rationem compoſitam ex ra-<lb/>tione temporum, & </s> <s xml:id="echoid-s1304" xml:space="preserve">ratione velocitatum, ideoque cum tem-<lb/>pore A H, celeritate æquabili H L percurri poſuerimus ſpa-<lb/>tium M H, ſequitur tempore C E, cum celeritate C K, <lb/>percurri ſpatium K E, quum ratio rectanguli M H ad re-<lb/>ctangulum K E componatur ex rationibus A H ad C E, & </s> <s xml:id="echoid-s1305" xml:space="preserve"><lb/>H L ad C K.</s> <s xml:id="echoid-s1306" xml:space="preserve"/> </p> <div xml:id="echoid-div66" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0092-03" xlink:href="note-0092-03a" xml:space="preserve">Prop. I. <lb/>huj.</note> <note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCEN U</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1307" xml:space="preserve">Quum ergo, ut dixi, ſpatium K E ſit illud quod trans-<lb/>mitteretur tempore C E, cum celeritate æquabili C K, mo-<lb/>bile autem feratur tempore C E motu accelerato, qui jam <lb/>principio hujus temporis habet celeritatem C K; </s> <s xml:id="echoid-s1308" xml:space="preserve">manifeſtum <lb/>eſt iſto accelerato motu, tempore C E, majus ſpatium quam <lb/>K E confecturum. </s> <s xml:id="echoid-s1309" xml:space="preserve">Eadem ratione, tempore tertio E G, ma-<lb/>jus ſpatium conficiet quam O G, quia nempe hoc confectu-<lb/>rum eſſet tempore eodem E G, cum celeritate æquabili E O. <lb/></s> <s xml:id="echoid-s1310" xml:space="preserve">Atque ita deinceps, ſingulis temporis A H partibus, à mo-<lb/>bili majora ſpatia quam ſunt rectangula figuræ inſcriptæ, <lb/>ipſis partibus adjacentia, peragentur. </s> <s xml:id="echoid-s1311" xml:space="preserve">Quare totum ſpatium <lb/>motu accelerato peractum majus erit ipſa figura inſcripta. </s> <s xml:id="echoid-s1312" xml:space="preserve"><lb/>Spatium vero illud æquale poſitum fuit plano P. </s> <s xml:id="echoid-s1313" xml:space="preserve">Itaque fi-<lb/>gura inſcripta minor erit ſpatio P. </s> <s xml:id="echoid-s1314" xml:space="preserve">quod eſt abſurdum; </s> <s xml:id="echoid-s1315" xml:space="preserve">eo-<lb/>dem enim ſpatio major oſtenſa fuit. </s> <s xml:id="echoid-s1316" xml:space="preserve">Non eſt igitur planum <lb/>P minus triangulo A H L. </s> <s xml:id="echoid-s1317" xml:space="preserve">At neque majus eſſe oſtendetur.</s> <s xml:id="echoid-s1318" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1319" xml:space="preserve">Sit enim, ſi poteſt; </s> <s xml:id="echoid-s1320" xml:space="preserve">& </s> <s xml:id="echoid-s1321" xml:space="preserve">dividatur A H in partes æquales, <lb/>atque ad earum altitudinem, inſcripta circumſcriptaque rur-<lb/>ſus, ut ante, ſit triangulo A H L figura ex rectangulis, ita <lb/>ut altera alteram excedat minori exceſſu quam quo planum <pb o="62" file="0094" n="98" rhead="CHRISTIANI HUGENII"/> P ſuperat triangulum A H L, erit igitur neceſſario figura <lb/> <anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/> circumſcripta minor plano P. </s> <s xml:id="echoid-s1322" xml:space="preserve">Conſtat jam, prima temporis <lb/>parte A C, minus ſpatium à mobili transmitti quam ſit B C, <lb/>quia hoc percurreretur eodem tempore A C cum celeritate <lb/>æquabili C K, quam demum in fine temporis A C mobile <lb/>adeptum eſt. </s> <s xml:id="echoid-s1323" xml:space="preserve">Similiter ſecunda parte temporis C E, minus <lb/>ſpatium motu accelerato transmittetur quam ſit D E, quia <lb/>hoc percurreretur eodem tempore C E, cum celeritate æ-<lb/>quabili E O, quam demum in fine temporis C E mobile aſ-<lb/>ſequitur. </s> <s xml:id="echoid-s1324" xml:space="preserve">Atque ita deinceps, ſingulis partibus temporis <lb/>A H, minora ſpatia à mobili trajicientur quam ſunt rectan-<lb/>gula figuræ circumſcriptæ, ipſis partibus adjacentia. </s> <s xml:id="echoid-s1325" xml:space="preserve">Quare <lb/>totum ſpatium motu accelerato peractum, minus erit ipſa fi-<lb/>gura circumſcripta. </s> <s xml:id="echoid-s1326" xml:space="preserve">Spatium vero illud æquale poſitum fuit <lb/>plano P; </s> <s xml:id="echoid-s1327" xml:space="preserve">ergo planum P minus quoque erit figura circum-<lb/>ſcripta. </s> <s xml:id="echoid-s1328" xml:space="preserve">quod eſt abſurdum, cum figura hæc plano P minor <lb/>oſtenſa fuerit. </s> <s xml:id="echoid-s1329" xml:space="preserve">Ergo planum P non majus eſt triangulo A H L, <lb/>ſed nec minus eſſe jam oſtenſum fuit. </s> <s xml:id="echoid-s1330" xml:space="preserve">Ergo æquale ſit neceſ-<lb/>ſe eſt; </s> <s xml:id="echoid-s1331" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s1332" xml:space="preserve"/> </p> <div xml:id="echoid-div67" type="float" level="2" n="3"> <note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1333" xml:space="preserve">Et hæc quidem omnia quæ hactenus demonſtrata ſunt, <lb/>gravibus per plana inclinata deſcendentibus atque aſcenden-<lb/>tibus æque ac perpendiculariter motis convenire ſciendum <lb/>eſt: </s> <s xml:id="echoid-s1334" xml:space="preserve">cum, quæ de effectu gravitatis poſita fuerunt, eadem <lb/>ratione utrobique ſint admittenda.</s> <s xml:id="echoid-s1335" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1336" xml:space="preserve">Hinc vero non difficile jam erit demonſtrare propoſitionem <lb/>ſequentem quam concedi ſibi, ut quodammodo per ſe ma-<lb/>nifeſtam, Galileus poſtulavit. </s> <s xml:id="echoid-s1337" xml:space="preserve">nam demonſtratio illa quam <lb/>poſtea adferre conatus eſt, quæque in poſteriori operum <lb/>ejus editione extat, parum firma meo quidem judicio vide-<lb/>tur. </s> <s xml:id="echoid-s1338" xml:space="preserve">Eſt autem propoſitio hujusmodi.</s> <s xml:id="echoid-s1339" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div69" type="section" level="1" n="29"> <head xml:id="echoid-head51" xml:space="preserve">PROPOSITIO VI.</head> <p style="it"> <s xml:id="echoid-s1340" xml:space="preserve">CEleritates gravium, ſuper diverſis planorum <lb/>inclinationibus deſcendendo acquiſitæ, æquales <lb/>ſunt, ſi planorum elevationes fuerint æquales.</s> <s xml:id="echoid-s1341" xml:space="preserve"/> </p> <pb o="63" file="0095" n="99" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s1342" xml:space="preserve">Elevationem plani vocamus altitudinem ejus ſecundum <lb/> <anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/> perpendiculum. <lb/></s> <s xml:id="echoid-s1343" xml:space="preserve"> <anchor type="note" xlink:label="note-0095-02a" xlink:href="note-0095-02"/> Fig. 4. rum. <anchor type="note" xlink:label="note-0095-03a" xlink:href="note-0095-03"/> ſit aſcendere per totam B C. Ideoque cadens ex F in B, ſi continuet porro motum per B C; quod repercuſſu ad ſu- perficiem obliquam fieri poteſt; aſcendet usque in C, hoc eſt, altius quam unde decidit, quod eſt abſurdum.</s></p> <div xml:id="echoid-div69" type="float" level="2" n="1"> <note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <div xml:id="echoid-div70" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0095-03" xlink:href="note-0095-03a" xml:space="preserve">Prop. 4. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1344" xml:space="preserve">Eodem modo oſtendetur neque per planum A B deciden-<lb/>ti minorem velocitatem acquiri quam per C B. </s> <s xml:id="echoid-s1345" xml:space="preserve">Ergo per <lb/>utraque plana eadem velocitas acquiritur, quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s1346" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1347" xml:space="preserve">Quod ſi vero, pro plano alterutro, ſumatur perpendicu-<lb/>lum ipſum planorum elevationi æquale, per quod decidere <lb/>mobile ponatur, ſic quoque eandem quam per plana incli-<lb/>nata velocitatem ei acquiri conſtat; </s> <s xml:id="echoid-s1348" xml:space="preserve">eadem namque eſt de-<lb/>monſtratio.</s> <s xml:id="echoid-s1349" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1350" xml:space="preserve">Porro hinc jam recte quoque procedet demonſtratio alte-<lb/>rius theorematis Galileani, cui reliqua omnia, quæ de de-<lb/>ſcenſu ſuper planis inclinatis tradidit, ſuperſtruuntur. </s> <s xml:id="echoid-s1351" xml:space="preserve">Nempe</s> </p> </div> <div xml:id="echoid-div72" type="section" level="1" n="30"> <head xml:id="echoid-head52" xml:space="preserve">PROPOSITIO VII.</head> <p style="it"> <s xml:id="echoid-s1352" xml:space="preserve">TEmpora deſcenſuum ſuper planis diverſimode <lb/>inclinatis, ſed quorum eadem eſt elevatio, eſſe <lb/>inter ſe ut planorum longitudines.</s> <s xml:id="echoid-s1353" xml:space="preserve"/> </p> <pb o="64" file="0096" n="100" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1354" xml:space="preserve">Sint plana inclinata A C, A D quorum eadem elevatio <lb/> <anchor type="note" xlink:label="note-0096-01a" xlink:href="note-0096-01"/> A B. </s> <s xml:id="echoid-s1355" xml:space="preserve">dico tempus deſcenſus per planum A C ad tempus <lb/> <anchor type="note" xlink:label="note-0096-02a" xlink:href="note-0096-02"/> deſcenſus per A D eſſe ut longitudo A C ad A D. </s> <s xml:id="echoid-s1356" xml:space="preserve">Eſt enim <lb/>tempus per A C æquale tempori motus æquabilis per ean-<lb/>dem A C, cum celeritate dimidia ejus quæ acquiritur caſu <lb/>per A C <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1357" xml:space="preserve">Similiter tempus per A D eſt æquale tempori <anchor type="note" xlink:label="note-0096-03a" xlink:href="note-0096-03"/> motus æquabilis per ipſam A D, cum dimidia celeritate ejus <lb/>quæ acquiritur caſu per A D. </s> <s xml:id="echoid-s1358" xml:space="preserve">Eſt autem hæc dimidia celeri-<lb/>tas illi dimidiæ celerirati æqualis <anchor type="note" xlink:href="" symbol="*"/>, ideoque dictum tempus <anchor type="note" xlink:label="note-0096-04a" xlink:href="note-0096-04"/> motus æquabilis per A C, ad tempus motus æquabilis per A D, <lb/>erit ut A C ad A D. </s> <s xml:id="echoid-s1359" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1360" xml:space="preserve">tempora ſingulis iſtis æqualia, <lb/>nimirum tempus deſcenſus per A C, ad tempus deſcenſus <lb/>per A D, eandem rationem habebunt, nempe quam A C <lb/>ad A D. </s> <s xml:id="echoid-s1361" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s1362" xml:space="preserve"/> </p> <div xml:id="echoid-div72" type="float" level="2" n="1"> <note position="left" xlink:label="note-0096-01" xlink:href="note-0096-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> <note position="left" xlink:label="note-0096-02" xlink:href="note-0096-02a" xml:space="preserve">TAB. V. <lb/>Fig. 5.</note> <note symbol="*" position="left" xlink:label="note-0096-03" xlink:href="note-0096-03a" xml:space="preserve">Prop. 1. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0096-04" xlink:href="note-0096-04a" xml:space="preserve">Prop. <lb/>præced.</note> </div> <p> <s xml:id="echoid-s1363" xml:space="preserve">Eodem modo oſtendetur & </s> <s xml:id="echoid-s1364" xml:space="preserve">tempus deſcenſus per A C, ad <lb/>tempus caſus per A B perpendicularem, eſſe ut A C ad <lb/>A B longitudine.</s> <s xml:id="echoid-s1365" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div74" type="section" level="1" n="31"> <head xml:id="echoid-head53" xml:space="preserve">PROPOSITIO VIII.</head> <p style="it"> <s xml:id="echoid-s1366" xml:space="preserve">SI ex altitudine eadem deſcendat mobile conti-<lb/>nuato motu per quotlibet ac quælibet plana con-<lb/>tigua, utcunque inclinata; </s> <s xml:id="echoid-s1367" xml:space="preserve">ſemper eandem in fine <lb/>velocitatem acquiret, quæ nimirum æqualis erit ei <lb/>quam acquireret cadendo perpendiculariter ex pa-<lb/>ri altitudine.</s> <s xml:id="echoid-s1368" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1369" xml:space="preserve">Sint plana contigua A B, B C, C D, quorum terminus <lb/> <anchor type="note" xlink:label="note-0096-05a" xlink:href="note-0096-05"/> A, ſupra horizontalem lineam D F per infimum terminum <lb/>D ductam, altitudinem habeat quanta eſt perpendicularis E F. <lb/></s> <s xml:id="echoid-s1370" xml:space="preserve">deſcendatque mobile per plana illa ab A uſque in D. </s> <s xml:id="echoid-s1371" xml:space="preserve">Di-<lb/>co in D eam velocitatem habiturum quam, ex E cadens, ha-<lb/>beret in F.</s> <s xml:id="echoid-s1372" xml:space="preserve"/> </p> <div xml:id="echoid-div74" type="float" level="2" n="1"> <note position="left" xlink:label="note-0096-05" xlink:href="note-0096-05a" xml:space="preserve">TAB VI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s1373" xml:space="preserve">Producta enim C B occurrat rectæ A E in G. </s> <s xml:id="echoid-s1374" xml:space="preserve">Itemque <lb/>D C producta occurrat eidem A E in E. </s> <s xml:id="echoid-s1375" xml:space="preserve">Quoniam itaque <pb file="0097" n="101"/> <pb file="0097a" n="102"/> <anchor type="figure" xlink:label="fig-0097a-01a" xlink:href="fig-0097a-01"/> <anchor type="figure" xlink:label="fig-0097a-02a" xlink:href="fig-0097a-02"/> <anchor type="figure" xlink:label="fig-0097a-03a" xlink:href="fig-0097a-03"/> <anchor type="figure" xlink:label="fig-0097a-04a" xlink:href="fig-0097a-04"/> <anchor type="figure" xlink:label="fig-0097a-05a" xlink:href="fig-0097a-05"/> <pb file="0098" n="103"/> <pb o="65" file="0099" n="104" rhead="HOROLOG. OSCILLATOR."/> per A B deſcendens eandem acquirit velocitatem in termi-<lb/> <anchor type="note" xlink:label="note-0099-01a" xlink:href="note-0099-01"/> no B, atque deſcendens per G B <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s1376" xml:space="preserve">manifeſtum eſt, cum <anchor type="note" xlink:label="note-0099-02a" xlink:href="note-0099-02"/> flexus ad B nihil obſtare motui ponatur, tantam velocitatem <lb/>bahiturum ubi in C pervenerit, quantam ſi per G C planum <lb/>deſcendiſſet; </s> <s xml:id="echoid-s1377" xml:space="preserve">hoc eſt, quantam haberet ex deſcenſu per E C. <lb/></s> <s xml:id="echoid-s1378" xml:space="preserve">Quare & </s> <s xml:id="echoid-s1379" xml:space="preserve">reliquum planum C D eodem modo tranſibit ac ſi <lb/>per E C adveniſſet, ac proinde in D denique parem veloci-<lb/>tatem habebit, ac ſi deſcendiſſet per planum E D, hoc eſt, <lb/>eandem quam ex caſu perpendiculari per E F. </s> <s xml:id="echoid-s1380" xml:space="preserve">quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s1381" xml:space="preserve"/> </p> <div xml:id="echoid-div75" type="float" level="2" n="2"> <figure xlink:label="fig-0097a-01" xlink:href="fig-0097a-01a"> <caption xml:id="echoid-caption10" style="it" xml:space="preserve">Pag. 64.<lb/>TAB. V.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables10" xml:space="preserve">A B D E F G H K</variables> </figure> <figure xlink:label="fig-0097a-02" xlink:href="fig-0097a-02a"> <caption xml:id="echoid-caption11" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables11" xml:space="preserve">C A G H B D K L E F</variables> </figure> <figure xlink:label="fig-0097a-03" xlink:href="fig-0097a-03a"> <caption xml:id="echoid-caption12" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables12" xml:space="preserve">A B M C K D E O F G P H L</variables> </figure> <figure xlink:label="fig-0097a-04" xlink:href="fig-0097a-04a"> <caption xml:id="echoid-caption13" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables13" xml:space="preserve">A C F E B D</variables> </figure> <figure xlink:label="fig-0097a-05" xlink:href="fig-0097a-05a"> <caption xml:id="echoid-caption14" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables14" xml:space="preserve">A C D B</variables> </figure> <note position="right" xlink:label="note-0099-01" xlink:href="note-0099-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> <note symbol="*" position="right" xlink:label="note-0099-02" xlink:href="note-0099-02a" xml:space="preserve">Prop. 6. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1382" xml:space="preserve">Hinc liquet etiam per circuli circumferentiam, vel per cur-<lb/>vam quamlibet lineam deſcendente mobili (nam curvas tan-<lb/>quam ex infinitis rectis compoſitæ eſſent hic conſiderare li-<lb/>cet) ſemper eandem illi velocitatem acquiri ſi ab æquali al-<lb/>titudine deſcenderit: </s> <s xml:id="echoid-s1383" xml:space="preserve">tantamque eam eſſe velocitatem, quan-<lb/>tam caſu perpendiculari ex eadem altitudine adipiſceretur.</s> <s xml:id="echoid-s1384" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div77" type="section" level="1" n="32"> <head xml:id="echoid-head54" xml:space="preserve">PROPOSITIO IX.</head> <p style="it"> <s xml:id="echoid-s1385" xml:space="preserve">SI grave, à deſcenſu, ſurſum convertat motum <lb/>ſuum, aſcendet ad eandem unde venit altitudi-<lb/>nem, per quascunque planas ſuperſicies contiguas, <lb/>& </s> <s xml:id="echoid-s1386" xml:space="preserve">quomodocunque inclinatas, inceſſerit.</s> <s xml:id="echoid-s1387" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1388" xml:space="preserve">Cadat grave ex altitudine A B, & </s> <s xml:id="echoid-s1389" xml:space="preserve">ex puncto B inclinata <lb/> <anchor type="note" xlink:label="note-0099-03a" xlink:href="note-0099-03"/> ſint ſurſum plana B C, C D, D E, quorum extremitas E <lb/>ſit eadem altitudine cum puncto A. </s> <s xml:id="echoid-s1390" xml:space="preserve">Dico ſi mobile, poſt ca-<lb/>ſum per A B, convertat motum ut pergat moveri per dicta <lb/>plana inclinata, perventutum uſque in E.</s> <s xml:id="echoid-s1391" xml:space="preserve"/> </p> <div xml:id="echoid-div77" type="float" level="2" n="1"> <note position="right" xlink:label="note-0099-03" xlink:href="note-0099-03a" xml:space="preserve">TAB. VI. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s1392" xml:space="preserve">Dicatur enim, ſi fieri poteſt, tantum ad G perventurum. <lb/></s> <s xml:id="echoid-s1393" xml:space="preserve">Producantur B C & </s> <s xml:id="echoid-s1394" xml:space="preserve">C D, donec occurrant horizontali G F <lb/>in F & </s> <s xml:id="echoid-s1395" xml:space="preserve">H. </s> <s xml:id="echoid-s1396" xml:space="preserve">Cum igitur mobile, ſuperatis planis B C, C D, <lb/>habeat tantum eam velocitatem quâ poſſit aſcendere per <lb/>D G, vel per D H; </s> <s xml:id="echoid-s1397" xml:space="preserve">nam ad hæc utraque eadem velocitate <lb/>opus eſſe conſtat ex propoſitione 6; </s> <s xml:id="echoid-s1398" xml:space="preserve">Ergo, ſuperato plano <lb/>B C, eam duntaxat habebat qua potuiſſet aſcendere per C H, <pb o="66" file="0100" n="105" rhead="CHRISTIANI HUGENII"/> vel per C F. </s> <s xml:id="echoid-s1399" xml:space="preserve">Ergo in B duntaxat eam qua potuiſſet aſcen-<lb/> <anchor type="note" xlink:label="note-0100-01a" xlink:href="note-0100-01"/> dere per B F, hoc eſt, eandem quam acquireret deſcendendo <lb/>per F B. </s> <s xml:id="echoid-s1400" xml:space="preserve">Atqui in B habet velocitatem qua poteſt aſcende-<lb/>re uſque in A. </s> <s xml:id="echoid-s1401" xml:space="preserve">Ergo illa velocitate quam acquirit grave de-<lb/>ſcendendo per F B, poſſet aſcendere per B A, hoc eſt, al-<lb/>tius quam unde diſceſſerat, quod fieri non poteſt.</s> <s xml:id="echoid-s1402" xml:space="preserve"/> </p> <div xml:id="echoid-div78" type="float" level="2" n="2"> <note position="left" xlink:label="note-0100-01" xlink:href="note-0100-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1403" xml:space="preserve">Eſt autem eadem prorſus demonſtratio quotcunque plana <lb/>fuerint per quæ mobile aſcendat. </s> <s xml:id="echoid-s1404" xml:space="preserve">Unde & </s> <s xml:id="echoid-s1405" xml:space="preserve">ſi infinita fuerit <lb/>planorum multitudo, hoc eſt, ſi ſuperficies aliqua curva <lb/>ponatur, per hanc quoque ad eam ex qua venit altitudinem <lb/>mobile aſſurget.</s> <s xml:id="echoid-s1406" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div80" type="section" level="1" n="33"> <head xml:id="echoid-head55" xml:space="preserve">PROPOSITIO X.</head> <p style="it"> <s xml:id="echoid-s1407" xml:space="preserve">SI mobile cadat perpendiculariter, vel per quam-<lb/>libet ſuperficiem deſcendat, ac rurſus impetu <lb/>concepto per quamlibet aliam feratur ſurſum, ha-<lb/>bebit aſcendendo ac deſcendendo in punctis æque al-<lb/>tis eandem ſemper velocitatem.</s> <s xml:id="echoid-s1408" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1409" xml:space="preserve">Ut ſi mobile ex altitudine A B decidens, motum deinde <lb/> <anchor type="note" xlink:label="note-0100-02a" xlink:href="note-0100-02"/> continuet per ſuperficiem B C D, in qua punctum C ſit <lb/>pari altitudine atque in A B eſt punctum E. </s> <s xml:id="echoid-s1410" xml:space="preserve">Dico in C ean-<lb/>dem velocitatem ineſſe mobili atque in E fuerat.</s> <s xml:id="echoid-s1411" xml:space="preserve"/> </p> <div xml:id="echoid-div80" type="float" level="2" n="1"> <note position="left" xlink:label="note-0100-02" xlink:href="note-0100-02a" xml:space="preserve">TAB. IV. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s1412" xml:space="preserve">Quum enim in C ea velocitas ſuperſit mobili qua porro <lb/>aſcendat usque ad D punctum, æque altum ac A <anchor type="note" xlink:href="" symbol="*"/>: </s> <s xml:id="echoid-s1413" xml:space="preserve">cum- <anchor type="note" xlink:label="note-0100-03a" xlink:href="note-0100-03"/> que & </s> <s xml:id="echoid-s1414" xml:space="preserve">ex deſcenſu per A E velocitatem eam acquirat qua, <lb/>converſo motu, aſcenſurum ſit per C D <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s1415" xml:space="preserve">Patet cum per- <anchor type="note" xlink:label="note-0100-04a" xlink:href="note-0100-04"/> venit ad C aſcendendo, eandem ipſum habere velocitatem, <lb/>quam habebat in E deſcendendo; </s> <s xml:id="echoid-s1416" xml:space="preserve">quod erat demonſtran-<lb/>dum.</s> <s xml:id="echoid-s1417" xml:space="preserve"/> </p> <div xml:id="echoid-div81" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0100-03" xlink:href="note-0100-03a" xml:space="preserve">Prop. <lb/>præced.</note> <note symbol="*" position="left" xlink:label="note-0100-04" xlink:href="note-0100-04a" xml:space="preserve">Prop. <lb/>præced.</note> </div> </div> <div xml:id="echoid-div83" type="section" level="1" n="34"> <head xml:id="echoid-head56" xml:space="preserve">PROPOSITIO XI.</head> <p style="it"> <s xml:id="echoid-s1418" xml:space="preserve">SI mobile per ſuperficiem aliquam deorſum ten-<lb/>dat, ac deinde converſo motu ſurſum per ean- <pb o="67" file="0101" n="106" rhead="HOROLOG. OSCILLATOR."/> dem ſuperficiem vel aliam ſimilem ſimiliter que po-<lb/> <anchor type="note" xlink:label="note-0101-01a" xlink:href="note-0101-01"/> ſitam feratur, æqualibus temporibus per idem ſpa-<lb/>tium deſcendet atque aſcendet.</s> <s xml:id="echoid-s1419" xml:space="preserve"/> </p> <div xml:id="echoid-div83" type="float" level="2" n="1"> <note position="right" xlink:label="note-0101-01" xlink:href="note-0101-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1420" xml:space="preserve">Velut ſi per ſuperficiem A B deſcendat mobile, atque, ubi <lb/> <anchor type="note" xlink:label="note-0101-02a" xlink:href="note-0101-02"/> ad B pervenerit, converſo motu ſurſum per eandem A B, vel <lb/>ei ſimilem & </s> <s xml:id="echoid-s1421" xml:space="preserve">reſpectu plani horizontalis ſimiliter poſitam <lb/>B C, aſcendat, conſtat ex ante demonſtratis, perventurum <lb/>ad eandem ex qua venit altitudinem. </s> <s xml:id="echoid-s1422" xml:space="preserve">Cum autem perpetuo, <lb/>in punctis quorum eadem altitudo, eandem velocitatem ha-<lb/>beat aſcendendo ac deſcendendo <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s1423" xml:space="preserve">apparet eandem lineam <anchor type="note" xlink:label="note-0101-03a" xlink:href="note-0101-03"/> bis eadem velocitate ſingulis ſui partibus percurri: </s> <s xml:id="echoid-s1424" xml:space="preserve">unde & </s> <s xml:id="echoid-s1425" xml:space="preserve"><lb/>tempora utriusque motus æqualia eſſe neceſſe eſt; </s> <s xml:id="echoid-s1426" xml:space="preserve">quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s1427" xml:space="preserve"/> </p> <div xml:id="echoid-div84" type="float" level="2" n="2"> <note position="right" xlink:label="note-0101-02" xlink:href="note-0101-02a" xml:space="preserve">TAB. VI. <lb/>Fig. 4.</note> <note symbol="*" position="right" xlink:label="note-0101-03" xlink:href="note-0101-03a" xml:space="preserve">Prop. <lb/>præced.</note> </div> </div> <div xml:id="echoid-div86" type="section" level="1" n="35"> <head xml:id="echoid-head57" xml:space="preserve">PROPOSITIO XII.</head> <p style="it"> <s xml:id="echoid-s1428" xml:space="preserve">ESto circulus A B C, diametro A C, cui ad an-<lb/> <anchor type="note" xlink:label="note-0101-04a" xlink:href="note-0101-04"/> gulos rectos ſit F G; </s> <s xml:id="echoid-s1429" xml:space="preserve">huic vero occurrat à ter-<lb/>mino diametri A educta A F extra circulum, quæ <lb/>quidem neceſſario ſecabit circumferentiam, puta <lb/>in B. </s> <s xml:id="echoid-s1430" xml:space="preserve">Dico arcum B D, lineis G F, A F inter-<lb/>ceptum, minorem eſſe recta D F.</s> <s xml:id="echoid-s1431" xml:space="preserve"/> </p> <div xml:id="echoid-div86" type="float" level="2" n="1"> <note position="right" xlink:label="note-0101-04" xlink:href="note-0101-04a" xml:space="preserve">TAB. VI. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s1432" xml:space="preserve">Jungatur enim B C, & </s> <s xml:id="echoid-s1433" xml:space="preserve">ducatur ex B puncto tangens cir-<lb/>cumferentiam recta B E, quæ neceſſario occurret rectæ F G <lb/>inter F & </s> <s xml:id="echoid-s1434" xml:space="preserve">D. </s> <s xml:id="echoid-s1435" xml:space="preserve">Eſt igitur angulus B A C in circulo æqualis <lb/>angulo E B C <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1436" xml:space="preserve">quare & </s> <s xml:id="echoid-s1437" xml:space="preserve">angulus F B E, qui una cum <anchor type="note" xlink:label="note-0101-05a" xlink:href="note-0101-05"/> E B C conſtituit angulum rectum F B C, erit æqualis B C A. <lb/></s> <s xml:id="echoid-s1438" xml:space="preserve">Quia autem ſimilia ſunt triangula A B C, A G F, erit & </s> <s xml:id="echoid-s1439" xml:space="preserve"><lb/>angulus F æqualis angulo A C B. </s> <s xml:id="echoid-s1440" xml:space="preserve">Ergo idem angulus F æ-<lb/>qualis angulo F B E. </s> <s xml:id="echoid-s1441" xml:space="preserve">Itaque iſoſceles eſt triangulus F E B, <lb/>habens crura æqualia F E, E B. </s> <s xml:id="echoid-s1442" xml:space="preserve">Addita ergo utrique eo-<lb/>rum recta E D, fiet F D, æqualis duabus B E, E D. </s> <s xml:id="echoid-s1443" xml:space="preserve">Has-<lb/>ce vero duas majores eſſe conſtat arcu B D, iisdem termi- <pb o="68" file="0102" n="107" rhead="CHRISTIANI HUGENII"/> nis intercepto, & </s> <s xml:id="echoid-s1444" xml:space="preserve">in eandem partem cavo. </s> <s xml:id="echoid-s1445" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1446" xml:space="preserve">F D <lb/> <anchor type="note" xlink:label="note-0102-01a" xlink:href="note-0102-01"/> eodem arcu B D major erit: </s> <s xml:id="echoid-s1447" xml:space="preserve">quare conſtat propoſitum.</s> <s xml:id="echoid-s1448" xml:space="preserve"/> </p> <div xml:id="echoid-div87" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0101-05" xlink:href="note-0101-05a" xml:space="preserve">Prop. 32. <lb/>Lib. 3. Eucl.</note> <note position="left" xlink:label="note-0102-01" xlink:href="note-0102-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> </div> <div xml:id="echoid-div89" type="section" level="1" n="36"> <head xml:id="echoid-head58" xml:space="preserve">PROPOSITIO XIII.</head> <p style="it"> <s xml:id="echoid-s1449" xml:space="preserve">IIsdem poſitis, ſi recta A B occurrat ipſi D G in-<lb/> <anchor type="note" xlink:label="note-0102-02a" xlink:href="note-0102-02"/> tra circulum; </s> <s xml:id="echoid-s1450" xml:space="preserve">Dico arcum B D, rectis G D, <lb/>A B interceptum, majorem eſſe recta D F.</s> <s xml:id="echoid-s1451" xml:space="preserve"/> </p> <div xml:id="echoid-div89" type="float" level="2" n="1"> <note position="left" xlink:label="note-0102-02" xlink:href="note-0102-02a" xml:space="preserve">TAB. VI. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s1452" xml:space="preserve">Jungatur enim D C & </s> <s xml:id="echoid-s1453" xml:space="preserve">ducatur arcui D B ſubtenſa D B. <lb/></s> <s xml:id="echoid-s1454" xml:space="preserve">Quoniam ergo angulus A B D æqualis A C D, hoc eſt, <lb/>angulo A D G; </s> <s xml:id="echoid-s1455" xml:space="preserve">angulus autem D F B major angulo A D F, <lb/>ſive A D G; </s> <s xml:id="echoid-s1456" xml:space="preserve">erit idem D F B etiam major D B F. </s> <s xml:id="echoid-s1457" xml:space="preserve">Ergo <lb/>in triangulo D F B latus D B majus latere D F; </s> <s xml:id="echoid-s1458" xml:space="preserve">unde mul-<lb/>to magis arcus D B ſuperabit eandem D F. </s> <s xml:id="echoid-s1459" xml:space="preserve">Quare conſtat <lb/>propoſitum.</s> <s xml:id="echoid-s1460" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div91" type="section" level="1" n="37"> <head xml:id="echoid-head59" xml:space="preserve">PROPOSITIO XIV.</head> <p style="it"> <s xml:id="echoid-s1461" xml:space="preserve">SIt cyclois A B C cujus baſis A C axis B D. <lb/></s> <s xml:id="echoid-s1462" xml:space="preserve">Quomodo autem generetur ex definitione & </s> <s xml:id="echoid-s1463" xml:space="preserve"><lb/>deſcriptione mechanica ſuperius traditis ſatis ma-<lb/>nifeſtum arbitror. </s> <s xml:id="echoid-s1464" xml:space="preserve">Et circa axem B D, circulus <lb/>deſcriptus ſit B G D, & </s> <s xml:id="echoid-s1465" xml:space="preserve">à quolibet puncto E in cy-<lb/>cloide ſumpto agatur E F baſi A C parallela, quæ <lb/>occurrat axi B D in F, ſecetque circumferentiam <lb/>B G D in G, Dico rectam G E arcui G B æqua-<lb/>lem eſſe.</s> <s xml:id="echoid-s1466" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1467" xml:space="preserve">Deſcribatur enim per E punctum circulus L E K ipſi <lb/>B G D æqualis, quique tangat baſin cycloidis in K, & </s> <s xml:id="echoid-s1468" xml:space="preserve">du-<lb/>catur diameter K L. </s> <s xml:id="echoid-s1469" xml:space="preserve">Eſt igitur recta A K arcui E K æqua-<lb/>lis; </s> <s xml:id="echoid-s1470" xml:space="preserve">ſed tota A D æqualis ſemicircumferentiæ K E L; </s> <s xml:id="echoid-s1471" xml:space="preserve">ergo <lb/>K D æqualis arcui E L ſive G B. </s> <s xml:id="echoid-s1472" xml:space="preserve">Eſt autem K D ſive N F <lb/>æqualis E G, quoniam E N æqualis G F, & </s> <s xml:id="echoid-s1473" xml:space="preserve">communis <lb/>utrique N G. </s> <s xml:id="echoid-s1474" xml:space="preserve">Ergo conſtat & </s> <s xml:id="echoid-s1475" xml:space="preserve">G E æqualem eſſe arcui G B.</s> <s xml:id="echoid-s1476" xml:space="preserve"/> </p> <pb file="0103" n="108"/> <pb file="0103a" n="109"/> <figure> <caption xml:id="echoid-caption15" style="it" xml:space="preserve">Pag. 68.<lb/>TAB. VI.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables15" xml:space="preserve">A G E B C D F</variables> </figure> <figure> <caption xml:id="echoid-caption16" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables16" xml:space="preserve">A E F H G D C B</variables> </figure> <figure> <caption xml:id="echoid-caption17" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables17" xml:space="preserve">D A E C B</variables> </figure> <figure> <caption xml:id="echoid-caption18" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables18" xml:space="preserve">A C B</variables> </figure> <figure> <caption xml:id="echoid-caption19" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables19" xml:space="preserve">A B F E D G C</variables> </figure> <figure> <caption xml:id="echoid-caption20" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables20" xml:space="preserve">A D G F B C</variables> </figure> <pb file="0104" n="110"/> <pb o="69" file="0105" n="111" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div92" type="section" level="1" n="38"> <head xml:id="echoid-head60" xml:space="preserve">PROPOSITIO XV.</head> <note position="right" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> <p style="it"> <s xml:id="echoid-s1477" xml:space="preserve">DAto in Cycloide puncto, rectam per illud du-<lb/>cere quæ Cycloidem tangat.</s> <s xml:id="echoid-s1478" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1479" xml:space="preserve">Sit cyclois A B C, & </s> <s xml:id="echoid-s1480" xml:space="preserve">punctum in ea datum B, per quod <lb/> <anchor type="note" xlink:label="note-0105-02a" xlink:href="note-0105-02"/> tangentem ducere oporteat.</s> <s xml:id="echoid-s1481" xml:space="preserve"/> </p> <div xml:id="echoid-div92" type="float" level="2" n="1"> <note position="right" xlink:label="note-0105-02" xlink:href="note-0105-02a" xml:space="preserve">TAB. VII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s1482" xml:space="preserve">Circa axem cycloidis A D deſcribatur circulus genitor <lb/>A E D, & </s> <s xml:id="echoid-s1483" xml:space="preserve">ducatur B E parallela baſi cycloidis, quæ dicto <lb/>circulo occurrat in E, & </s> <s xml:id="echoid-s1484" xml:space="preserve">jungatur A E, cui denique paral-<lb/>lela per B agatur H B N. </s> <s xml:id="echoid-s1485" xml:space="preserve">Dico hanc cycloidem in B con-<lb/>tingere.</s> <s xml:id="echoid-s1486" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1487" xml:space="preserve">Sumatur enim in ea punctum quodlibet, à B diverſum, <lb/>ac primo verſus ſuperiora velut H, & </s> <s xml:id="echoid-s1488" xml:space="preserve">per H ducantur re-<lb/>cta baſi cycloidis parallela, quæ occurrat cycloidi in L, cir-<lb/>culo A E D in K, rectæ A E in M. </s> <s xml:id="echoid-s1489" xml:space="preserve">Quia ergo K L eſt <lb/>æqualis arcui K A, recta autem K M minor arcu K E, erit <lb/>recta M L minor arcu A E, hoc eſt, rectâ E B, ſive M H; <lb/></s> <s xml:id="echoid-s1490" xml:space="preserve">unde apparet punctum H eſſe extra cycloidem.</s> <s xml:id="echoid-s1491" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1492" xml:space="preserve">Deinde in recta H N ſumatur punctum N inferius B, & </s> <s xml:id="echoid-s1493" xml:space="preserve"><lb/>per N agatur, ut ante, baſi parallela, quæ occurrat cycloi-<lb/>di in Q, circulo A E D in O, rectæ A E productæ in P. <lb/></s> <s xml:id="echoid-s1494" xml:space="preserve">Quia ergo O Q, æqualis eſt arcui O A; </s> <s xml:id="echoid-s1495" xml:space="preserve">O P autem major <lb/>arcu O E; </s> <s xml:id="echoid-s1496" xml:space="preserve">erit P Q minor arcu E A, hoc eſt, rectâ E B, <lb/>ſive P N. </s> <s xml:id="echoid-s1497" xml:space="preserve">Unde apparet rurſus punctum N eſſe extra cycloi-<lb/>dem. </s> <s xml:id="echoid-s1498" xml:space="preserve">Cum igitur quodlibet punctum præter B, in recta <lb/>H B N ſumptum, ſit extra cycloidem, conſtat illam in <lb/>puncto B cycloidem contingere; </s> <s xml:id="echoid-s1499" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s1500" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1501" xml:space="preserve">Huic demonſtrationi an locum ſuum hic relinquerem dubi-<lb/>tavi, quod non multum ei abſimilem à clariſſimo Wrennio <lb/>editam inveniam in libro Walliſii de Cycloide. </s> <s xml:id="echoid-s1502" xml:space="preserve">Poteſt autem <lb/>& </s> <s xml:id="echoid-s1503" xml:space="preserve">univerſali conſtructione propoſitum abſolvi, quæ non cy-<lb/>cloidi tantum ſed & </s> <s xml:id="echoid-s1504" xml:space="preserve">aliis curvis, ex cujuſlibet figuræ circum-<lb/>volutione genitis, conveniat; </s> <s xml:id="echoid-s1505" xml:space="preserve">dummodo ſit figura in ean-<lb/>dem partem cava, & </s> <s xml:id="echoid-s1506" xml:space="preserve">ex iis quæ geometricæ vocantur.</s> <s xml:id="echoid-s1507" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1508" xml:space="preserve">Sit enim curva N A B, orta ex circumvolutione figuræ <lb/> <anchor type="note" xlink:label="note-0105-03a" xlink:href="note-0105-03"/> <pb o="70" file="0106" n="112" rhead="CHRISTIANI HUGENII"/> O L ſuper regula L D; </s> <s xml:id="echoid-s1509" xml:space="preserve">deſcribente nempe puncto N, in cir-<lb/> <anchor type="note" xlink:label="note-0106-01a" xlink:href="note-0106-01"/> cumferentia figuræ O L ſumpto. </s> <s xml:id="echoid-s1510" xml:space="preserve">Et oporteat ad punctum cur-<lb/>væ A tangentem ducere. </s> <s xml:id="echoid-s1511" xml:space="preserve">Ducatur recta C A à puncto C, <lb/>ubi figura regulam tangebat cum punctum deſcribens eſſet <lb/>in A: </s> <s xml:id="echoid-s1512" xml:space="preserve">quod punctum contactus ſemper inveniri poteſt, ſiqui-<lb/>dem eo reducitur problema ut duæ rectæ inter ſe parallelæ <lb/>ducendæ ſint, quarum altera tranſeat per punctum deſcri-<lb/>bens in figuræ ambitu datum, altera figuram tangat, quæ-<lb/>que inter ſe diſtent quantum diſtat punctum datum A ab re-<lb/>gula L D: </s> <s xml:id="echoid-s1513" xml:space="preserve">dico ipſam C A occurrere curvæ ad angulos <lb/>rectos, ſive circumferentiam M A F deſcriptam centro C <lb/>radio C A, tangere curvam in puncto A, unde perpendicula-<lb/>ris ad A C, per punctum A, ducta curvam ibidem continget.</s> <s xml:id="echoid-s1514" xml:space="preserve"/> </p> <div xml:id="echoid-div93" type="float" level="2" n="2"> <note position="right" xlink:label="note-0105-03" xlink:href="note-0105-03a" xml:space="preserve">TAB. VII. <lb/>Fig. 3.</note> <note position="left" xlink:label="note-0106-01" xlink:href="note-0106-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1515" xml:space="preserve">Ducatur enim C B primum ad punctum curvæ B, quod <lb/>diſtet ultra punctum A ab regula L D, intelligaturque figu-<lb/>ræ poſitus in B E D, cum punctum deſcribens eſſet in B, <lb/>contactus regulæ in D. </s> <s xml:id="echoid-s1516" xml:space="preserve">& </s> <s xml:id="echoid-s1517" xml:space="preserve">punctum curvæ quod erat in C, <lb/>cum punctum deſcribens eſſet in A, hìc jam ſublatum ſit in <lb/>E; </s> <s xml:id="echoid-s1518" xml:space="preserve">& </s> <s xml:id="echoid-s1519" xml:space="preserve">jungantur E C, E B, tangatque figuram in E recta <lb/>K H, occurrens regulæ in H.</s> <s xml:id="echoid-s1520" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1521" xml:space="preserve">Quia ergo recta C D æqualis eſt curvæ E D; </s> <s xml:id="echoid-s1522" xml:space="preserve">eâdem ve-<lb/>ro curva major eſt utraque ſimul E H, H D; </s> <s xml:id="echoid-s1523" xml:space="preserve">erit E H ma-<lb/>jor quam C H. </s> <s xml:id="echoid-s1524" xml:space="preserve">Unde angulus E C H major quam C E H, <lb/>& </s> <s xml:id="echoid-s1525" xml:space="preserve">proinde E C L minor quam C E K. </s> <s xml:id="echoid-s1526" xml:space="preserve">Atqui addendo an-<lb/>gulum K E B, qui æqualis eſt L C A, ad K E C, fit an-<lb/>gulus C E B: </s> <s xml:id="echoid-s1527" xml:space="preserve">& </s> <s xml:id="echoid-s1528" xml:space="preserve">auferendo ab E C L angulum L C B, fit <lb/>E C B. </s> <s xml:id="echoid-s1529" xml:space="preserve">Ergo angulus C E B major omnino angulo E C B. <lb/></s> <s xml:id="echoid-s1530" xml:space="preserve">Itaque in triangulo C E B, latus C B majus erit quam E B. </s> <s xml:id="echoid-s1531" xml:space="preserve"><lb/>ſed E B æquale patet eſſe C A, cum ſit idemmet ipſum unà <lb/>cum figura transpoſitum. </s> <s xml:id="echoid-s1532" xml:space="preserve">Ergo C B etiam major quam C A, <lb/>hoc eſt, quam C F. </s> <s xml:id="echoid-s1533" xml:space="preserve">unde conſtat punctum B eſſe extra cir-<lb/>cumferentiam M A F.</s> <s xml:id="echoid-s1534" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1535" xml:space="preserve">Sit rurſus punctum N in curva ſumptum inter regulam <lb/>L D & </s> <s xml:id="echoid-s1536" xml:space="preserve">punctum A. </s> <s xml:id="echoid-s1537" xml:space="preserve">Cumque punctum deſcribens eſſet in N, <lb/>ponatur ſitus figuræ fuiſſe in V L, punctumque contactus L, <lb/>punctum verò quod tangebat prius regulam in C, ſit jam <pb o="71" file="0107" n="113" rhead="HOROLOG. OSCILLATOR."/> fublatum in V: </s> <s xml:id="echoid-s1538" xml:space="preserve">& </s> <s xml:id="echoid-s1539" xml:space="preserve">jungantur C N, N V, V C, V L. </s> <s xml:id="echoid-s1540" xml:space="preserve">E-<lb/> <anchor type="note" xlink:label="note-0107-01a" xlink:href="note-0107-01"/> rit ergo V N æqualis C A; </s> <s xml:id="echoid-s1541" xml:space="preserve">imo erit ipſa C A translata in <lb/>V N. </s> <s xml:id="echoid-s1542" xml:space="preserve">Jam quia recta L C æquatur curvæ L V, ac proin-<lb/>de major eſt recta L V, erit in triangulo C L V angulus <lb/>L V C major quam L C V. </s> <s xml:id="echoid-s1543" xml:space="preserve">Quare addito inſuper angulo <lb/>L V N ad L V C, fiet totus N V C major utique quam <lb/>L C V, ac proinde omnino major angulo N C V, qui pars <lb/>eſt L C V. </s> <s xml:id="echoid-s1544" xml:space="preserve">Ergo in triangulo C V N latus C N majus erit <lb/>latere V N, cui æquatur C A, ideoque C N major quo-<lb/>que quam C A, hoc eſt quam C M. </s> <s xml:id="echoid-s1545" xml:space="preserve">Unde apparet pun-<lb/>ctum N cadere extra circulum M A F, qui proinde tanget <lb/>curvam in puncto A. </s> <s xml:id="echoid-s1546" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s1547" xml:space="preserve"/> </p> <div xml:id="echoid-div94" type="float" level="2" n="3"> <note position="right" xlink:label="note-0107-01" xlink:href="note-0107-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1548" xml:space="preserve">Eſt autem eadem quoque tum conſtructio tum demonſtra-<lb/>tio, ſi curva genita ſit à puncto deſcribente, vel intra vel <lb/>extra ambitum figuræ circumvolutæ ſumpto. </s> <s xml:id="echoid-s1549" xml:space="preserve">Niſi quod, <lb/>hoc poſteriori caſu, pars quædam curvæ infra regulam de-<lb/>ſcendit, unde nonnulla in demonſtratione oritur diverſitas.</s> <s xml:id="echoid-s1550" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1551" xml:space="preserve">Sit enim punctum A, per quod tangens ducenda eſt, da-<lb/> <anchor type="note" xlink:label="note-0107-02a" xlink:href="note-0107-02"/> tum in parte curvæ N A B, quæ infra regulam C L de-<lb/>ſcendit, deſcripta nimirum à puncto N extra figuram revo-<lb/>lutam ſumpto, ſed certam poſitionem in eodem ipſius pla-<lb/>no habente. </s> <s xml:id="echoid-s1552" xml:space="preserve">Invento igitur puncto C, ubi figura revoluta <lb/>tangit regulam C D quum punctum deſcribens eſſet in A, <lb/>ducatur recta C A. </s> <s xml:id="echoid-s1553" xml:space="preserve">Dico hanc curvæ N A B occurrere ad <lb/>rectos angulos, ſive circumferentiam radio C A centro C <lb/>deſcriptam tangere curvam N A B in puncto A. </s> <s xml:id="echoid-s1554" xml:space="preserve">Oſtendetur <lb/>autem exterius ipſam contingere, cum in curvæ parte ſupra <lb/>regulam C D poſita interius contingat.</s> <s xml:id="echoid-s1555" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="4"> <note position="right" xlink:label="note-0107-02" xlink:href="note-0107-02a" xml:space="preserve">TAB. VIII. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s1556" xml:space="preserve">Poſitis enim & </s> <s xml:id="echoid-s1557" xml:space="preserve">deſcriptis iisdem omnibus quæ prius, os-<lb/>tenditur rurſus angulus E C H major quam C E H. </s> <s xml:id="echoid-s1558" xml:space="preserve">atqui <lb/>ad E C H addito H C B fit angulus E C B; </s> <s xml:id="echoid-s1559" xml:space="preserve">& </s> <s xml:id="echoid-s1560" xml:space="preserve">à C E H <lb/>auferendo H E B, qui æqualis eſt D C A, fit angulus <lb/>C E B. </s> <s xml:id="echoid-s1561" xml:space="preserve">Ergo E C B major omnino quam C E B. </s> <s xml:id="echoid-s1562" xml:space="preserve">unde in <lb/>triangulo E C B latus E B majus quam C B. </s> <s xml:id="echoid-s1563" xml:space="preserve">ſed ipſi E B <lb/>æqualis eſt C A, ſive C F. </s> <s xml:id="echoid-s1564" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1565" xml:space="preserve">C F major quam C B: <lb/></s> <s xml:id="echoid-s1566" xml:space="preserve">ideoque punctum circumferentiæ F eſt ultra curvam N A B <lb/>à centro remotum.</s> <s xml:id="echoid-s1567" xml:space="preserve"/> </p> <pb o="72" file="0108" n="114" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1568" xml:space="preserve">Item rurſus oſtenditur angulus L V C major L C V. </s> <s xml:id="echoid-s1569" xml:space="preserve">Qua-<lb/> <anchor type="note" xlink:label="note-0108-01a" xlink:href="note-0108-01"/> re C V P, qui cum L V C duos rectos æquat, minor erit <lb/>quam V C D. </s> <s xml:id="echoid-s1570" xml:space="preserve">Atqui addendo ad V C D angulum D C N, <lb/>fit V C N; </s> <s xml:id="echoid-s1571" xml:space="preserve">& </s> <s xml:id="echoid-s1572" xml:space="preserve">auferendo ab C V P angulum P V N, fit <lb/>C V N. </s> <s xml:id="echoid-s1573" xml:space="preserve">Ergo angulus V C N omnino major quam C V N. <lb/></s> <s xml:id="echoid-s1574" xml:space="preserve">In triangulo itaque C V N, latus V N majus erit quam <lb/>C N. </s> <s xml:id="echoid-s1575" xml:space="preserve">Eſt autem ipſi V N æqualis C A ſive C M. </s> <s xml:id="echoid-s1576" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1577" xml:space="preserve"><lb/>C M major quam C N, ideoque punctum circumferentiæ <lb/>M erit ultra curvam N A B à centro C remotum. </s> <s xml:id="echoid-s1578" xml:space="preserve">Itaque <lb/>conſtat circumferentiam M A F tangere curvam in puncto A. </s> <s xml:id="echoid-s1579" xml:space="preserve"><lb/>quod erat demonſtrandum.</s> <s xml:id="echoid-s1580" xml:space="preserve"/> </p> <div xml:id="echoid-div96" type="float" level="2" n="5"> <note position="left" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve"><emph style="sc">De de-</emph> <lb/><emph style="sc">SCENSU</emph> <lb/><emph style="sc">GRAVIUM</emph>.</note> </div> <p> <s xml:id="echoid-s1581" xml:space="preserve">Quod ſi punctum curvæ per quod tangens ducenda eſt, <lb/>ſit illud ipſum ubi regula curvam ſecat, erit tangens quæſi-<lb/>ta ſemper regulæ perpendicularis; </s> <s xml:id="echoid-s1582" xml:space="preserve">ut facile eſſet oſtendere.</s> <s xml:id="echoid-s1583" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div98" type="section" level="1" n="39"> <head xml:id="echoid-head61" xml:space="preserve">PROPOSITIO XVI.</head> <p style="it"> <s xml:id="echoid-s1584" xml:space="preserve">SI circuli circumferentiam, cujus centrum E, ſe-<lb/> <anchor type="note" xlink:label="note-0108-02a" xlink:href="note-0108-02"/> cent rectæ duæ parallelæ A F, B G, quarum <lb/> <anchor type="note" xlink:label="note-0108-03a" xlink:href="note-0108-03"/> utraque ad eandem partem centri transeat, vel <lb/>altera A F per centrum ipſum: </s> <s xml:id="echoid-s1585" xml:space="preserve">& </s> <s xml:id="echoid-s1586" xml:space="preserve">à puncto A, <lb/>quo centro propior circumferentiam ſecat, ducatur <lb/>recta ipſam contingens: </s> <s xml:id="echoid-s1587" xml:space="preserve">dico partem hujus A B, à <lb/>parallela utraque interceptam, minorem eſſe arcu <lb/>A C, ab utraque eadem parallela intercepto.</s> <s xml:id="echoid-s1588" xml:space="preserve"/> </p> <div xml:id="echoid-div98" type="float" level="2" n="1"> <note position="left" xlink:label="note-0108-02" xlink:href="note-0108-02a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <note position="left" xlink:label="note-0108-03" xlink:href="note-0108-03a" xml:space="preserve">TAB. VIII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s1589" xml:space="preserve">Ducatur enim arcui A C ſubtenſa recta A C. </s> <s xml:id="echoid-s1590" xml:space="preserve">Quia ergo <lb/>angulus B A F eſt æqualis ei quem capit portio circuli A H F, <lb/>quæ vel major eſt ſemicirculo vel ſemicirculus, erit proinde <lb/>angulus B A F, vel minor recto vel rectus; </s> <s xml:id="echoid-s1591" xml:space="preserve">ideoque angu-<lb/>lus A B C vel major recto vel rectus. </s> <s xml:id="echoid-s1592" xml:space="preserve">Quare in triangulo <lb/>A B C latus A C, angulo B ſubtenſum, majus erit latere <lb/>A B. </s> <s xml:id="echoid-s1593" xml:space="preserve">ſed idem latus A C minus eſt arcu A C. </s> <s xml:id="echoid-s1594" xml:space="preserve">Ergo omni-<lb/>no & </s> <s xml:id="echoid-s1595" xml:space="preserve">A B arcu A C minor erit.</s> <s xml:id="echoid-s1596" xml:space="preserve"/> </p> <pb file="0109" n="115"/> <pb file="0109a" n="116"/> <figure> <caption xml:id="echoid-caption21" style="it" xml:space="preserve">Pag. 72.<lb/>TAB. VII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables21" xml:space="preserve">L B E N G F A K D C</variables> </figure> <figure> <caption xml:id="echoid-caption22" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables22" xml:space="preserve">A H L K M B E N Q P O C D</variables> </figure> <figure> <caption xml:id="echoid-caption23" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables23" xml:space="preserve">B F A K O N M E V L C H D</variables> </figure> <pb file="0110" n="117"/> <pb o="73" file="0111" n="118" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div100" type="section" level="1" n="40"> <head xml:id="echoid-head62" xml:space="preserve">PROPOSITIO XVII.</head> <note position="right" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <p style="it"> <s xml:id="echoid-s1597" xml:space="preserve">IIsdem poſitis, ſi tertia recta prioribus parallela <lb/> <anchor type="note" xlink:label="note-0111-02a" xlink:href="note-0111-02"/> D K, circulum ſecuerit, quæ ab ea quæ centro <lb/>propior eſt A F, tantundem diſtet quantum hæc à <lb/>reliqua B G: </s> <s xml:id="echoid-s1598" xml:space="preserve">dico partem tangentis in A, à pa-<lb/>rallela ultimo adjecta, & </s> <s xml:id="echoid-s1599" xml:space="preserve">media interceptam, nem-<lb/>pe A D, arcu A C à primis duabus parallelis in-<lb/>tercepto minorem eſſe.</s> <s xml:id="echoid-s1600" xml:space="preserve"/> </p> <div xml:id="echoid-div100" type="float" level="2" n="1"> <note position="right" xlink:label="note-0111-02" xlink:href="note-0111-02a" xml:space="preserve">TAB. VIII. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s1601" xml:space="preserve">Hoc enim patet quum A D ipſi A B æqualis ſit, quam <lb/>antea oſtendimus arcu A C minorem eſſe.</s> <s xml:id="echoid-s1602" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div102" type="section" level="1" n="41"> <head xml:id="echoid-head63" xml:space="preserve">PROPOSITIO XVIII.</head> <p style="it"> <s xml:id="echoid-s1603" xml:space="preserve">SI circulum, cujus centrum E, duæ rectæ paral-<lb/> <anchor type="note" xlink:label="note-0111-03a" xlink:href="note-0111-03"/> lelæ ſecuerint A F, B G; </s> <s xml:id="echoid-s1604" xml:space="preserve">& </s> <s xml:id="echoid-s1605" xml:space="preserve">à puncto B, ubi <lb/>quæ à centro remotior eſt, vel tantundem atque <lb/>altera diſtat, circumferentiæ occurrit, ducatur <lb/>recta circumferentiam tangens: </s> <s xml:id="echoid-s1606" xml:space="preserve">erit pars hujus <lb/>B A, à parallelis intercepta, major arcu ab iis-<lb/>dem parallelis intercepto B C.</s> <s xml:id="echoid-s1607" xml:space="preserve"/> </p> <div xml:id="echoid-div102" type="float" level="2" n="1"> <note position="right" xlink:label="note-0111-03" xlink:href="note-0111-03a" xml:space="preserve">TAB. VIII. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s1608" xml:space="preserve">Ducatur enim in puncto C, recta M C L circumferentiam <lb/>tangens, quæ occurrat tangenti B A in L. </s> <s xml:id="echoid-s1609" xml:space="preserve">In triangulo igi-<lb/>tur A C L, angulus C æqualis eſt angulo M C F, hoc eſt, <lb/>ei quem capit portio circuli C B F. </s> <s xml:id="echoid-s1610" xml:space="preserve">angulus autem A æqua-<lb/>tur angulo quem capit portio circuli B C G, quæ portio <lb/>quum ſit major vel æqualis portioni C B F, quippe quum <lb/>B G vel ulterius diſtet à centro quam C F, vel tantun-<lb/>dem: </s> <s xml:id="echoid-s1611" xml:space="preserve">erit proinde trianguli A C L angulus A minor vel <lb/>æqualis angulo C: </s> <s xml:id="echoid-s1612" xml:space="preserve">& </s> <s xml:id="echoid-s1613" xml:space="preserve">conſequenter latus C L vel minus <lb/>vel æquale lateri A L. </s> <s xml:id="echoid-s1614" xml:space="preserve">Atqui C L una cum L B majores <lb/>ſunt arcu C B. </s> <s xml:id="echoid-s1615" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1616" xml:space="preserve">A L una cum L B, hoc eſt, tan- <pb o="74" file="0112" n="119" rhead="CHRISTIANI HUGENII"/> gens A B, eodem arcu C B major erit. </s> <s xml:id="echoid-s1617" xml:space="preserve">quod erat demon-<lb/> <anchor type="note" xlink:label="note-0112-01a" xlink:href="note-0112-01"/> ſtrandum.</s> <s xml:id="echoid-s1618" xml:space="preserve"/> </p> <div xml:id="echoid-div103" type="float" level="2" n="2"> <note position="left" xlink:label="note-0112-01" xlink:href="note-0112-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> </div> <div xml:id="echoid-div105" type="section" level="1" n="42"> <head xml:id="echoid-head64" xml:space="preserve">PROPOSITIO XIX.</head> <p style="it"> <s xml:id="echoid-s1619" xml:space="preserve">IIsdem poſitis, ſi tertia recta prioribus parallela <lb/> <anchor type="note" xlink:label="note-0112-02a" xlink:href="note-0112-02"/> D K circulum ſecet, quæ tantundem diſtet ab ea <lb/>quæ remotior eſt à centro quantum hæc à reliqua <lb/>A F: </s> <s xml:id="echoid-s1620" xml:space="preserve">Erit pars tangentis in B, à parallela me-<lb/>dia, & </s> <s xml:id="echoid-s1621" xml:space="preserve">ultimo addita D K, intercepta, nimirum <lb/>B D, major arcu B C.</s> <s xml:id="echoid-s1622" xml:space="preserve"/> </p> <div xml:id="echoid-div105" type="float" level="2" n="1"> <note position="left" xlink:label="note-0112-02" xlink:href="note-0112-02a" xml:space="preserve">TAB. VIII. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s1623" xml:space="preserve">Hoc enim manifeſtum eſt cum B D fiat ipſi B A æqualis, <lb/>quam oſtendimus arcu B C majorem eſſe.</s> <s xml:id="echoid-s1624" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div107" type="section" level="1" n="43"> <head xml:id="echoid-head65" xml:space="preserve">PROPOSITIO XX.</head> <p style="it"> <s xml:id="echoid-s1625" xml:space="preserve">SI arcus circuli, ſemicircumferentia minor, A B, <lb/> <anchor type="note" xlink:label="note-0112-03a" xlink:href="note-0112-03"/> in partes quotlibet ſecetur lineis rectis paralle-<lb/>lis, quæ & </s> <s xml:id="echoid-s1626" xml:space="preserve">inter ſe, & </s> <s xml:id="echoid-s1627" xml:space="preserve">cum rectis ſibi parallelis <lb/>per terminos arcus ductis, æqualia intervalla con-<lb/>ſtituant, quales ſunt C D, E F, G H, K L & </s> <s xml:id="echoid-s1628" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1629" xml:space="preserve">ducanturque ad terminum arcus alterutrum A, & </s> <s xml:id="echoid-s1630" xml:space="preserve"><lb/>ad reliqua omnia ſectionum puncta rectæ circumfe-<lb/>rentiam tangentes, omnes in eandem partem, & </s> <s xml:id="echoid-s1631" xml:space="preserve"><lb/>ut unaquæque occurrat proximæ dictarum paralle-<lb/>larum; </s> <s xml:id="echoid-s1632" xml:space="preserve">cujusmodi ſunt tangentes A C, D E, F G, <lb/>H K & </s> <s xml:id="echoid-s1633" xml:space="preserve">c. </s> <s xml:id="echoid-s1634" xml:space="preserve">Dico has tangentes, dempta prima A C, <lb/>ſimul ſumptas, minores eſſe arcu propoſito A B. </s> <s xml:id="echoid-s1635" xml:space="preserve"><lb/>Easdem vero omnes, non omiſſa A C, majores eſſe <lb/>arcu A B diminuto parte extrema N B, hoc eſt, <lb/>majores arcu A N.</s> <s xml:id="echoid-s1636" xml:space="preserve"/> </p> <div xml:id="echoid-div107" type="float" level="2" n="1"> <note position="left" xlink:label="note-0112-03" xlink:href="note-0112-03a" xml:space="preserve">TAB. VIII. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s1637" xml:space="preserve">Ponamus enim primo perallelarum aliquas tranſire ab u- <pb o="75" file="0113" n="120" rhead="HOROLOG. OSCILLATOR."/> traque parte centri Z, & </s> <s xml:id="echoid-s1638" xml:space="preserve">ſit G H, earum quæ ſunt à parte <lb/> <anchor type="note" xlink:label="note-0113-01a" xlink:href="note-0113-01"/> B, centro proxima, vel per ipſum centrum tranſeat. </s> <s xml:id="echoid-s1639" xml:space="preserve">Itaque <lb/>tangentes omnes inter G H & </s> <s xml:id="echoid-s1640" xml:space="preserve">B O comprehenſæ, ut H K, <lb/>L M, N O, ſingulæ ſuis arcubus minores ſunt <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1641" xml:space="preserve">Porro <anchor type="note" xlink:label="note-0113-02a" xlink:href="note-0113-02"/> autem & </s> <s xml:id="echoid-s1642" xml:space="preserve">tangens G F, arcu ſequente F D minor eſt <anchor type="note" xlink:href="" symbol="*"/>, &</s> <s xml:id="echoid-s1643" xml:space="preserve"> <anchor type="note" xlink:label="note-0113-03a" xlink:href="note-0113-03"/> ſimiliter tangens E D arcu D A. </s> <s xml:id="echoid-s1644" xml:space="preserve">Itaque tangentes omnes <lb/>inter B O & </s> <s xml:id="echoid-s1645" xml:space="preserve">C D interjectæ, minores ſunt arcubus B H & </s> <s xml:id="echoid-s1646" xml:space="preserve"><lb/>F A, ac proinde omnino minores arcubus B H, H A, ſive <lb/>arcu B A, quod erat primo oſtendendum.</s> <s xml:id="echoid-s1647" xml:space="preserve"/> </p> <div xml:id="echoid-div108" type="float" level="2" n="2"> <note position="right" xlink:label="note-0113-01" xlink:href="note-0113-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <note symbol="*" position="right" xlink:label="note-0113-02" xlink:href="note-0113-02a" xml:space="preserve">Prop. 16. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0113-03" xlink:href="note-0113-03a" xml:space="preserve">Prop. 17. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1648" xml:space="preserve">Porro jam demonſtrabimus tangentes omnes inter B O & </s> <s xml:id="echoid-s1649" xml:space="preserve">A <lb/>majores eſſe arcu A N. </s> <s xml:id="echoid-s1650" xml:space="preserve">Enimvero parallela G H, vel pro-<lb/>pius centrum Z tranſit quam parallela E F, quam pono <lb/>proximam eſſe earum quæ à parte A tranſeunt, vel erit re-<lb/>motior, vel æque diſtabit.</s> <s xml:id="echoid-s1651" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1652" xml:space="preserve">Quod ſi E F longius à centro vel æque remota eſt ac G H, <lb/>erit tangens F G major arcu ſuo F H, & </s> <s xml:id="echoid-s1653" xml:space="preserve">reliquæ tangen-<lb/>tes verſus A, nimirum E D, C A majores ſingulæ arcubus <lb/> <anchor type="note" xlink:label="note-0113-04a" xlink:href="note-0113-04"/> ſuis <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s1654" xml:space="preserve">adeo ut omnes ſimul G F, E D, C A majores ſint arcu H A. </s> <s xml:id="echoid-s1655" xml:space="preserve">ſed & </s> <s xml:id="echoid-s1656" xml:space="preserve">arcu H L major erit tangens L M <anchor type="note" xlink:href="" symbol="*"/>, &</s> <s xml:id="echoid-s1657" xml:space="preserve"> <anchor type="note" xlink:label="note-0113-05a" xlink:href="note-0113-05"/> arcu L N tangens N O; </s> <s xml:id="echoid-s1658" xml:space="preserve">itaque tangentes omnes, præter <lb/>H K, majores ſimul erunt arcu A N; </s> <s xml:id="echoid-s1659" xml:space="preserve">multoque magis, ac-<lb/>cedente ipſa H K, tangentes omnes inter A & </s> <s xml:id="echoid-s1660" xml:space="preserve">B compre-<lb/>henſæ arcu eodem A N majores erunt.</s> <s xml:id="echoid-s1661" xml:space="preserve"/> </p> <div xml:id="echoid-div109" type="float" level="2" n="3"> <note symbol="*" position="right" xlink:label="note-0113-04" xlink:href="note-0113-04a" xml:space="preserve">Prop. 15. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0113-05" xlink:href="note-0113-05a" xml:space="preserve">Prop. 19. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1662" xml:space="preserve">Si vero G H à centro longius diſtat quam E F, erit tan-<lb/>gens K H major arcu H F <anchor type="note" xlink:href="" symbol="*"/>, & </s> <s xml:id="echoid-s1663" xml:space="preserve">tangens M L ut ante ma- <anchor type="note" xlink:label="note-0113-06a" xlink:href="note-0113-06"/> jor arcu L H, & </s> <s xml:id="echoid-s1664" xml:space="preserve">tangens O N major arcu N L, & </s> <s xml:id="echoid-s1665" xml:space="preserve">omnes <lb/>proinde tangentes O N, M L, K H majores arcu N F. <lb/></s> <s xml:id="echoid-s1666" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1667" xml:space="preserve">tangens E D major eſt arcu ſuo F D <anchor type="note" xlink:href="" symbol="*"/>, & </s> <s xml:id="echoid-s1668" xml:space="preserve">tangens <anchor type="note" xlink:label="note-0113-07a" xlink:href="note-0113-07"/> C A major ſimiliter arcu ſuo D A. </s> <s xml:id="echoid-s1669" xml:space="preserve">Itaque tangentes omnes <lb/>inter B O & </s> <s xml:id="echoid-s1670" xml:space="preserve">A, præter G F, majores erunt arcu N A; <lb/></s> <s xml:id="echoid-s1671" xml:space="preserve">multoque magis tangentes eædem, accedente G F, hoc eſt, <lb/>omnes quæ inter B O & </s> <s xml:id="echoid-s1672" xml:space="preserve">A interjiciuntur, eodem arcu N A <lb/>majores erunt.</s> <s xml:id="echoid-s1673" xml:space="preserve"/> </p> <div xml:id="echoid-div110" type="float" level="2" n="4"> <note symbol="*" position="right" xlink:label="note-0113-06" xlink:href="note-0113-06a" xml:space="preserve">Prop. 19. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0113-07" xlink:href="note-0113-07a" xml:space="preserve">Prop. 11. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1674" xml:space="preserve">Ex his vero etiam demonſtratio manifeſta eſt in caſibus <lb/>aliis, qualiscunque ſemicircumferentiæ arcus accipiatur, <lb/>quippe cum vel eadem ſit ubique, vel pars tantum præce-<lb/>dentis demonſtrationis.</s> <s xml:id="echoid-s1675" xml:space="preserve"/> </p> <pb o="76" file="0114" n="121" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div112" type="section" level="1" n="44"> <head xml:id="echoid-head66" xml:space="preserve">PROPOSITIO XXI.</head> <note position="left" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <p style="it"> <s xml:id="echoid-s1676" xml:space="preserve">SI mobile deſcendat continuato motu per quælibet <lb/>plana inclinata contigua, ac rurſus ex pari al-<lb/>titudine deſcendat per plana totidem contigua, ita <lb/>comparata ut ſingula altitudine reſpondeant ſingu-<lb/>lis priorum planorum, ſed majori quam illa ſint <lb/>inclinatione. </s> <s xml:id="echoid-s1677" xml:space="preserve">Dico tempus deſcenſus per minus in-<lb/>clinata, brevius eſſe tempore deſcenſus per magis <lb/>inclinata.</s> <s xml:id="echoid-s1678" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1679" xml:space="preserve">Sint ſeries duæ planorum inter easdem parallelas horizon-<lb/> <anchor type="note" xlink:label="note-0114-02a" xlink:href="note-0114-02"/> tales comprehenſæ A B C D E, F G H K L, atque ita ut <lb/>bina quæque ſibi correſpondentia plana utriusque ſeriei iisdem <lb/>parallelis horizontalibus includantur; </s> <s xml:id="echoid-s1680" xml:space="preserve">unumquodque vero ſeriei <lb/>F G H K L magis inclinatum ſit ad horizontem quam pla-<lb/>num ſibi altitudine reſpondens ſeriei A B C D E. </s> <s xml:id="echoid-s1681" xml:space="preserve">Dico bre-<lb/>viori tempore abſolvi deſcenſum per A B C D E, quam <lb/>per F G H K L.</s> <s xml:id="echoid-s1682" xml:space="preserve"/> </p> <div xml:id="echoid-div112" type="float" level="2" n="1"> <note position="left" xlink:label="note-0114-02" xlink:href="note-0114-02a" xml:space="preserve">TAB. IX. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s1683" xml:space="preserve">Nam primo quidem tempus deſcenſus per A B, brevius <lb/>eſſe conſtat tempore deſcenſus per F G, quum ſit eadem <lb/>ratio horum temporum quæ rectarum A B ad F G <anchor type="note" xlink:href="" symbol="*"/>, ſitque <anchor type="note" xlink:label="note-0114-03a" xlink:href="note-0114-03"/> A B minor quam F G, propter minorem inclinationem. <lb/></s> <s xml:id="echoid-s1684" xml:space="preserve">Producantur jam ſurſum rectæ C B, H G, occurrantque <lb/>horizontali A F in M & </s> <s xml:id="echoid-s1685" xml:space="preserve">N. </s> <s xml:id="echoid-s1686" xml:space="preserve">Itaque tempus per B C poſt <lb/>A B, æquale eſt tempori per eandem B C poſt M B, cum <lb/>in puncto B eadem celeritas contingat, ſive per A B, ſive <lb/>per M B deſcendenti <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1687" xml:space="preserve">ſimiliterque tempus per G H poſt <anchor type="note" xlink:label="note-0114-04a" xlink:href="note-0114-04"/> F G, æquale erit tempori per eandem G H poſt N G. </s> <s xml:id="echoid-s1688" xml:space="preserve">Eſt <lb/>autem tempus per B C poſt M B ad tempus per G H poſt <lb/>N G, ut B C ad G H longitudine, ſive ut C M ad H N, <lb/>cum hanc rationem habeant & </s> <s xml:id="echoid-s1689" xml:space="preserve">tempora per totas M C, N H, <lb/>& </s> <s xml:id="echoid-s1690" xml:space="preserve">per partes M B, N G <anchor type="note" xlink:href="" symbol="*"/>, ideoque etiam tempora reliqua.</s> <s xml:id="echoid-s1691" xml:space="preserve"> <anchor type="note" xlink:label="note-0114-05a" xlink:href="note-0114-05"/> Eſtque B C, minor quam G H propter minorem inclina-<lb/>tionem. </s> <s xml:id="echoid-s1692" xml:space="preserve">Patet igitur tempus per B C poſt M B ſive poſt <pb file="0115" n="122"/> <pb file="0115a" n="123"/> <anchor type="figure" xlink:label="fig-0115a-01a" xlink:href="fig-0115a-01"/> <anchor type="figure" xlink:label="fig-0115a-02a" xlink:href="fig-0115a-02"/> <anchor type="figure" xlink:label="fig-0115a-03a" xlink:href="fig-0115a-03"/> <anchor type="figure" xlink:label="fig-0115a-04a" xlink:href="fig-0115a-04"/> <anchor type="figure" xlink:label="fig-0115a-05a" xlink:href="fig-0115a-05"/> <anchor type="figure" xlink:label="fig-0115a-06a" xlink:href="fig-0115a-06"/> <pb file="0116" n="124"/> <pb o="77" file="0117" n="125" rhead="HOROLOG. OSCILLATOR."/> A B, brevius eſſe tempore per G H poſt N G ſive poſt F G.</s> <s xml:id="echoid-s1693" xml:space="preserve"/> </p> <div xml:id="echoid-div113" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0114-03" xlink:href="note-0114-03a" xml:space="preserve">Prop. 7. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0114-04" xlink:href="note-0114-04a" xml:space="preserve">Prop. 6. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0114-05" xlink:href="note-0114-05a" xml:space="preserve">Prop. 7. <lb/>huj.</note> <figure xlink:label="fig-0115a-01" xlink:href="fig-0115a-01a"> <caption xml:id="echoid-caption24" style="it" xml:space="preserve">Pag. 76.<lb/>TAB. VIII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables24" xml:space="preserve">O P E V D H C L M N A B F</variables> </figure> <figure xlink:label="fig-0115a-02" xlink:href="fig-0115a-02a"> <caption xml:id="echoid-caption25" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables25" xml:space="preserve">A B C E H G F</variables> </figure> <figure xlink:label="fig-0115a-03" xlink:href="fig-0115a-03a"> <caption xml:id="echoid-caption26" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables26" xml:space="preserve">D A B C E H G K F</variables> </figure> <figure xlink:label="fig-0115a-04" xlink:href="fig-0115a-04a"> <caption xml:id="echoid-caption27" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables27" xml:space="preserve">A L C M B E G F</variables> </figure> <figure xlink:label="fig-0115a-05" xlink:href="fig-0115a-05a"> <caption xml:id="echoid-caption28" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables28" xml:space="preserve">A B C D K F G</variables> </figure> <figure xlink:label="fig-0115a-06" xlink:href="fig-0115a-06a"> <caption xml:id="echoid-caption29" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables29" xml:space="preserve">G E C K H F L D M N A O B Z</variables> </figure> </div> <note position="right" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <p> <s xml:id="echoid-s1694" xml:space="preserve">Similiter oſtendetur, productis D C, K H ſurſum, do-<lb/>nec occurrant horizontali A F in O & </s> <s xml:id="echoid-s1695" xml:space="preserve">P, tempus per <lb/>C D poſt A B C, ſive poſt O C, brevius eſſe tempore per <lb/>H K poſt F G H ſive poſt P H. </s> <s xml:id="echoid-s1696" xml:space="preserve">Ac denique tempus per <lb/>D E poſt A B C D, brevius eſſe tempore per K L poſt <lb/>F G H K. </s> <s xml:id="echoid-s1697" xml:space="preserve">Quare totum tempus deſcenſus per A B C D E, <lb/>brevius erit tempore per F G H K L. </s> <s xml:id="echoid-s1698" xml:space="preserve">quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s1699" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1700" xml:space="preserve">Hinc vero manifeſtum eſt, conſiderando curvas lineas <lb/>tanquam ex innumeris rectis compoſitas, ſi fuerint duæ ſu-<lb/>perficies, ſecundum lineas curvas ejusdem altitudinis incli-<lb/>natæ, quarum in punctis quibuslibet æque altis major ſem-<lb/>per ſit inclinatio unius quam reliquæ, etiam tempore bre-<lb/>viori per minus inclinatam grave deſcenſurum quam per ma-<lb/>gis inclinatam.</s> <s xml:id="echoid-s1701" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1702" xml:space="preserve">Velut ſi ſint duæ ſuperficies inclinatæ ſecundum curvas <lb/> <anchor type="note" xlink:label="note-0117-02a" xlink:href="note-0117-02"/> A B, C D, æqualis altitudinis, quarumque in punctis æ-<lb/>que altis quibuslibet E, F, major ſit inclinatio ipſius C D <lb/>quam A B, hoc eſt, ut recta tangens curvam C D in F, <lb/>magis inclinata ſit ad horizontem, quam quæ curvam A B <lb/>tangit in puncto E. </s> <s xml:id="echoid-s1703" xml:space="preserve">erit tempus deſcenſus per A B brevius <lb/>quam per C D.</s> <s xml:id="echoid-s1704" xml:space="preserve"/> </p> <div xml:id="echoid-div114" type="float" level="2" n="3"> <note position="right" xlink:label="note-0117-02" xlink:href="note-0117-02a" xml:space="preserve">TAB. IX. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s1705" xml:space="preserve">Idemque continget ſi altera linearum recta fuerit: </s> <s xml:id="echoid-s1706" xml:space="preserve">dum-<lb/>modo inclinatio rectæ, quæ ubique eſt eadem, major mi-<lb/>norve fuerit inclinatione curvæ in quolibet ſui puncto.</s> <s xml:id="echoid-s1707" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div116" type="section" level="1" n="45"> <head xml:id="echoid-head67" xml:space="preserve">PROPOSITIO XXII.</head> <p style="it"> <s xml:id="echoid-s1708" xml:space="preserve">SI in Cycloide cujus axis ad perpendiculum erectus <lb/>ſtat, vertice deorſum ſpectante, duæ portiones <lb/>curvæ æqualis altitudinis accipiantur, ſed quarum <lb/>altera propior ſit vertici; </s> <s xml:id="echoid-s1709" xml:space="preserve">erit tempus deſcenſus <lb/>per ſuperiorem, brevius tempore per inferiorem.</s> <s xml:id="echoid-s1710" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1711" xml:space="preserve">Sit Cyclois A B, cujus axis A C ad perpendiculum ere-<lb/> <anchor type="note" xlink:label="note-0117-03a" xlink:href="note-0117-03"/> ctus, vertex A deorſum ſpectet; </s> <s xml:id="echoid-s1712" xml:space="preserve">& </s> <s xml:id="echoid-s1713" xml:space="preserve">accipiantur in ea por- <pb o="78" file="0118" n="126" rhead="CHRISTIANI HUGENII"/> tiones B D & </s> <s xml:id="echoid-s1714" xml:space="preserve">E F, æqualis altitudinis, hoc eſt, ejusmodi <lb/> <anchor type="note" xlink:label="note-0118-01a" xlink:href="note-0118-01"/> ut parallelæ horizontales B C, D H, quæ ſuperiorem por-<lb/>tionem B D includunt, æque inter ſe diſtent ac E G, <lb/>F K, inferiorem partionem E F includentes. </s> <s xml:id="echoid-s1715" xml:space="preserve">Dico tempus <lb/>deſcenſus per curvam B D brevius fore tempore per E F.</s> <s xml:id="echoid-s1716" xml:space="preserve"/> </p> <div xml:id="echoid-div116" type="float" level="2" n="1"> <note position="right" xlink:label="note-0117-03" xlink:href="note-0117-03a" xml:space="preserve">TAB. IX. <lb/>Fig. 3.</note> <note position="left" xlink:label="note-0118-01" xlink:href="note-0118-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> <p> <s xml:id="echoid-s1717" xml:space="preserve">Sumatur enim in B D punctum quodlibet L, & </s> <s xml:id="echoid-s1718" xml:space="preserve">in E F <lb/>punctum M, ita ut eadem ſit altitudo E ſupra M quæ B <lb/>ſupra L. </s> <s xml:id="echoid-s1719" xml:space="preserve">Et deſcripto ſuper axe A C ſemicirculo, occurrant <lb/>ei rectæ horizontales L N, M O, in N & </s> <s xml:id="echoid-s1720" xml:space="preserve">O, & </s> <s xml:id="echoid-s1721" xml:space="preserve">jungan-<lb/>tur N A, O A. </s> <s xml:id="echoid-s1722" xml:space="preserve">Itaque quum punctum N ſit altius puncto <lb/>O, manifeſtum eſt rectam N A minus ad horizontem incli-<lb/>nari quam O A. </s> <s xml:id="echoid-s1723" xml:space="preserve">Eſt autem ipſi N A parallela tangens curvæ <lb/>in L puncto <anchor type="note" xlink:href="" symbol="*"/>, & </s> <s xml:id="echoid-s1724" xml:space="preserve">ipſi O A parallela tangens curvæ in M.</s> <s xml:id="echoid-s1725" xml:space="preserve"> <anchor type="note" xlink:label="note-0118-02a" xlink:href="note-0118-02"/> Ergo curva B D in puncto L minus inclinata eſt quam curva <lb/>E F in puncto M. </s> <s xml:id="echoid-s1726" xml:space="preserve">Quod ſi igitur portio E F, invariata in-<lb/>clinatione, altius extolli intelligatur velut in e f, ita ut in-<lb/>ter eaſdem parallelas cum portione B D comprehendatur, <lb/>invenietur punctum M in m, æquali altitudine cum puncto <lb/>L. </s> <s xml:id="echoid-s1727" xml:space="preserve">eritque etiam inclinatio curvæ e f in puncto m, quæ ea-<lb/>dem eſt inclinationi curvæ E F in M, major inclinatione <lb/>curvæ B D in L. </s> <s xml:id="echoid-s1728" xml:space="preserve">Similiter vero, & </s> <s xml:id="echoid-s1729" xml:space="preserve">in quolibet alio puncto <lb/>curvæ e f, major oſtendetur inclinatio quam curv æ B D <lb/>in puncto æque alto. </s> <s xml:id="echoid-s1730" xml:space="preserve">Itaque tempus deſcenſus per B D bre-<lb/>vius erit tempore per e f<anchor type="note" xlink:href="" symbol="*"/>, ſive, quod idem eſt, per E F.</s> <s xml:id="echoid-s1731" xml:space="preserve"> <anchor type="note" xlink:label="note-0118-03a" xlink:href="note-0118-03"/> quod erat demonſtrandum.</s> <s xml:id="echoid-s1732" xml:space="preserve"/> </p> <div xml:id="echoid-div117" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0118-02" xlink:href="note-0118-02a" xml:space="preserve">Prop. 15. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0118-03" xlink:href="note-0118-03a" xml:space="preserve">Prop. <lb/>præced.</note> </div> </div> <div xml:id="echoid-div119" type="section" level="1" n="46"> <head xml:id="echoid-head68" xml:space="preserve">LEMMA.</head> <p style="it"> <s xml:id="echoid-s1733" xml:space="preserve">ESto circulus diametro A C, quem ſecet ad an-<lb/> <anchor type="note" xlink:label="note-0118-04a" xlink:href="note-0118-04"/> gulos rectos D E, & </s> <s xml:id="echoid-s1734" xml:space="preserve">à termino diametri A e-<lb/>ducta recta A B occurrat circumferentiæ in B, ipſi <lb/>vero D E in F. </s> <s xml:id="echoid-s1735" xml:space="preserve">Dico tres haſce, A B, A D, A F, <lb/>proportionales eſſe.</s> <s xml:id="echoid-s1736" xml:space="preserve"/> </p> <div xml:id="echoid-div119" type="float" level="2" n="1"> <note position="left" xlink:label="note-0118-04" xlink:href="note-0118-04a" xml:space="preserve">TAB. IX. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s1737" xml:space="preserve">Sit enim primo interſectio F intra circulum; </s> <s xml:id="echoid-s1738" xml:space="preserve">& </s> <s xml:id="echoid-s1739" xml:space="preserve">arcui B D <lb/>recta ſubtenſa ducatur. </s> <s xml:id="echoid-s1740" xml:space="preserve">Quia igitur arcus æquales ſunt A E, <pb o="79" file="0119" n="127" rhead="HOROLOG. OSCILLATOR."/> A D, erunt anguli ad circumferentiam ipſis inſiſtentes, <lb/> <anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/> E D A, A B D æquales. </s> <s xml:id="echoid-s1741" xml:space="preserve">Itaque in triangulis A B D, <lb/>A D F, æquales anguli A B D, A D F. </s> <s xml:id="echoid-s1742" xml:space="preserve">Communis au-<lb/>tem utrique eſt angulus ad A. </s> <s xml:id="echoid-s1743" xml:space="preserve">Ergo dicti trianguli ſimiles <lb/>erunt, ideoque B A ad A D ut A D ad A F.</s> <s xml:id="echoid-s1744" xml:space="preserve"/> </p> <div xml:id="echoid-div120" type="float" level="2" n="2"> <note position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> <p> <s xml:id="echoid-s1745" xml:space="preserve">Sit jam punctum interſectionis f extra circulum, & </s> <s xml:id="echoid-s1746" xml:space="preserve">du-<lb/>catur b H parallela D E, quæ occurrat rectæ A D in H. <lb/></s> <s xml:id="echoid-s1747" xml:space="preserve">Itaque ſecundum jam demonſtrata erit ut D A ad A b, ita <lb/>A b ad A H, hoc eſt, ita A f ad A D: </s> <s xml:id="echoid-s1748" xml:space="preserve">Ideoque rurſus <lb/>proportionales erunt A f, A D, A b. </s> <s xml:id="echoid-s1749" xml:space="preserve">Quare conſtat propo-<lb/>ſitum.</s> <s xml:id="echoid-s1750" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div122" type="section" level="1" n="47"> <head xml:id="echoid-head69" xml:space="preserve">PROPOSITIO XXIII.</head> <p style="it"> <s xml:id="echoid-s1751" xml:space="preserve">SIt Cyclois A B C, cujus vertex A deorſum con-<lb/> <anchor type="note" xlink:label="note-0119-02a" xlink:href="note-0119-02"/> verſus ſit, axe A D ad perpendiculum erecto; <lb/></s> <s xml:id="echoid-s1752" xml:space="preserve">ſumptoque in ea quolibet puncto B, ducatur inde <lb/>deorſum recta B I quæ Cycloidem tangat, terminetur-<lb/>que recta horizontali A I. </s> <s xml:id="echoid-s1753" xml:space="preserve">recta vero B F ad axem <lb/>perpendicularis agatur, & </s> <s xml:id="echoid-s1754" xml:space="preserve">diviſa bifariam F A in <lb/>X, ſuper ea deſcribatur ſemicirculus F H A. </s> <s xml:id="echoid-s1755" xml:space="preserve">Du-<lb/>ctâ deinde per punctum quodlibet G in curva B A <lb/>ſumptum, rectâ Σ G parallelâ B F, quæ circum-<lb/>ferentiæ F H A occurrat in H, axi A D in Σ, in-<lb/>telligantur per puncta G & </s> <s xml:id="echoid-s1756" xml:space="preserve">H rectæ tangentes u-<lb/>triusque curvæ, earumque tangentium partes iis-<lb/>dem duabus horizontalibus M S, N T interceptæ <lb/>ſint M N, S T. </s> <s xml:id="echoid-s1757" xml:space="preserve">Iisdemque rectis M S, N T in-<lb/>cludantur tangentis B I pars O P, & </s> <s xml:id="echoid-s1758" xml:space="preserve">axis D A <lb/>pars Q R.</s> <s xml:id="echoid-s1759" xml:space="preserve"/> </p> <div xml:id="echoid-div122" type="float" level="2" n="1"> <note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">TAB. IX. <lb/>Fig. 5.</note> </div> <p style="it"> <s xml:id="echoid-s1760" xml:space="preserve">Quibus ita ſe habentibus, dico tempus quo gra-<lb/>ve percurret rectam M N, celeritate æquabili <pb o="80" file="0120" n="128" rhead="CHRISTIANI HUGENII"/> quanta acquiritur deſcendendo per arcum Cycloi-<lb/> <anchor type="note" xlink:label="note-0120-01a" xlink:href="note-0120-01"/> dis B G, fore ad tempus quo percurretur recta <lb/>O P, celeritate æquabili dimidia ejus quæ acqui-<lb/>ritur deſcendendo per totam tangentem B I, ſicut <lb/>eſt tangens S T ad partem axis Q R.</s> <s xml:id="echoid-s1761" xml:space="preserve"/> </p> <div xml:id="echoid-div123" type="float" level="2" n="2"> <note position="left" xlink:label="note-0120-01" xlink:href="note-0120-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN</emph> <emph style="sc">Cy-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> <p> <s xml:id="echoid-s1762" xml:space="preserve">Deſcribatur enim ſuper axe A D ſemicirculus D V A ſe-<lb/>cans rectam B F in V, & </s> <s xml:id="echoid-s1763" xml:space="preserve">Σ G in Φ, & </s> <s xml:id="echoid-s1764" xml:space="preserve">jungatur A V ſe-<lb/>cans rectas O Q, P R, G Σ in E K & </s> <s xml:id="echoid-s1765" xml:space="preserve">Λ. </s> <s xml:id="echoid-s1766" xml:space="preserve">Jungantur item <lb/>H F, H A, H X & </s> <s xml:id="echoid-s1767" xml:space="preserve">A Φ; </s> <s xml:id="echoid-s1768" xml:space="preserve">quæ poſtrema ſecet rectas O Q, <lb/>P R in punctis Δ & </s> <s xml:id="echoid-s1769" xml:space="preserve">Π.</s> <s xml:id="echoid-s1770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1771" xml:space="preserve">Habet ergo dictum tempus per M N ad tempus per O P, <lb/>rationem eam quæ componitur ex ratione ipſarum linearum <lb/>M N ad O P, & </s> <s xml:id="echoid-s1772" xml:space="preserve">ex ratione celeritatum quibus ipſæ per-<lb/>curruntur, contrarie ſumpta <anchor type="note" xlink:href="" symbol="*"/>, hoc eſt, & </s> <s xml:id="echoid-s1773" xml:space="preserve">ex ratione dimi- <anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/> diæ celeritatis ex B I ſive ex F A, ad celeritatem ex B G, ſive <lb/>ex F Σ <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1774" xml:space="preserve">Atqui tota celeritas ex F A ad celeritatem ex F Σ, <anchor type="note" xlink:label="note-0120-03a" xlink:href="note-0120-03"/> eſt in ſubduplicata ratione longitudinum F A ad F Σ <anchor type="note" xlink:href="" symbol="*"/>, ac <anchor type="note" xlink:label="note-0120-04a" xlink:href="note-0120-04"/> proinde eadem quæ F A ad F H Ergo dimidia celeritas ex <lb/>F A ad celeritatem ex F Σ erit ut F X ad F H. </s> <s xml:id="echoid-s1775" xml:space="preserve">Itaque tem-<lb/>pus dictum per M N ad tempus per O P habebit rationem <lb/>compoſitam ex rationibus M N ad O P, & </s> <s xml:id="echoid-s1776" xml:space="preserve">F X ad F H. <lb/></s> <s xml:id="echoid-s1777" xml:space="preserve">Harum vero prior ratio, nempe M N ad O P, eadem oſten-<lb/>detur quæ F H ad H Σ.</s> <s xml:id="echoid-s1778" xml:space="preserve"/> </p> <div xml:id="echoid-div124" type="float" level="2" n="3"> <note symbol="*" position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">Prop. 5. <lb/>Galil. de <lb/>motu æ-<lb/>quab.</note> <note symbol="*" position="left" xlink:label="note-0120-03" xlink:href="note-0120-03a" xml:space="preserve">Prop. 8. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0120-04" xlink:href="note-0120-04a" xml:space="preserve">Prop. 3. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1779" xml:space="preserve">Eſt enim tangens Cycloidis B I parallela rectæ V A, ſi-<lb/>militerque tangens M G N parallela rectæ Φ A; </s> <s xml:id="echoid-s1780" xml:space="preserve">ac proinde <lb/>recta M N æqualis Δ Π, & </s> <s xml:id="echoid-s1781" xml:space="preserve">O P æqualis E K. </s> <s xml:id="echoid-s1782" xml:space="preserve">Ergo dicta <lb/>ratio rectæ M N ad O P eadem eſt quæ Δ Π ad E K; </s> <s xml:id="echoid-s1783" xml:space="preserve">hoc <lb/>eſt, Δ A ad E A; </s> <s xml:id="echoid-s1784" xml:space="preserve">hoc eſt, Φ A ad Λ A; </s> <s xml:id="echoid-s1785" xml:space="preserve">hoc eſt V A ad <lb/>Φ A <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1786" xml:space="preserve">Eſt autem ut V A ad A Φ ita F A ad A H; </s> <s xml:id="echoid-s1787" xml:space="preserve">nam <anchor type="note" xlink:label="note-0120-05a" xlink:href="note-0120-05"/> quia quadratum V A æquale eſt rectangulo D A F, & </s> <s xml:id="echoid-s1788" xml:space="preserve">qua-<lb/>dratum A Φ æquale rectangulo D A Σ, quæ rectangula ſunt <lb/>inter ſe ut F A ad Σ A, hoc eſt ut quadratum F A ad qua-<lb/>dratum A H, erit proinde & </s> <s xml:id="echoid-s1789" xml:space="preserve">quadratum V A ad quadra-<lb/>tum Φ A ut quadratum F A ad quadratum A H; </s> <s xml:id="echoid-s1790" xml:space="preserve">atque <pb o="81" file="0121" n="129" rhead="HOROLOG. OSCILLATOR."/> etiam V A ad A Φ longitudine, ut F A ad A H. </s> <s xml:id="echoid-s1791" xml:space="preserve">Ratio <lb/> <anchor type="note" xlink:label="note-0121-01a" xlink:href="note-0121-01"/> itaque M N ad O P, eadem erit quæ F A ad A H, hoc <lb/>eſt, propter triangula ſimilia F A H, F H Σ, eadem quæ <lb/>F H ad H Σ, ut dictum fuit. </s> <s xml:id="echoid-s1792" xml:space="preserve">Itaque dicta ratio temporis <lb/>per M N ad tempus per O P, componitur ex rationibus <lb/>F X ad F H & </s> <s xml:id="echoid-s1793" xml:space="preserve">F H ad H Σ, ideoque eadem erit quæ <lb/>F X ſive X H ad H Σ. </s> <s xml:id="echoid-s1794" xml:space="preserve">Sicut autem radius X H ad H Σ, <lb/>ita eſt tangens S T ad rectam Q R; </s> <s xml:id="echoid-s1795" xml:space="preserve">hoc enim facile perſpi-<lb/>citur. </s> <s xml:id="echoid-s1796" xml:space="preserve">Igitur tempus motus qualem diximus per M N, ad <lb/>tempus per O P conſtat eſſe ſicut S T ad Q R. </s> <s xml:id="echoid-s1797" xml:space="preserve">quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s1798" xml:space="preserve"/> </p> <div xml:id="echoid-div125" type="float" level="2" n="4"> <note symbol="*" position="left" xlink:label="note-0120-05" xlink:href="note-0120-05a" xml:space="preserve">Lemma <lb/>@ræced,</note> <note position="right" xlink:label="note-0121-01" xlink:href="note-0121-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> </div> <div xml:id="echoid-div127" type="section" level="1" n="48"> <head xml:id="echoid-head70" xml:space="preserve">PROPOSITIO XXIV.</head> <p style="it"> <s xml:id="echoid-s1799" xml:space="preserve">SIt rurſus ut in præcedenti propoſitione Cyclois <lb/> <anchor type="note" xlink:label="note-0121-02a" xlink:href="note-0121-02"/> A B C, cujus vertex A deorſum ſpectet, axis <lb/>A D ad borizontem erectus ſit; </s> <s xml:id="echoid-s1800" xml:space="preserve">& </s> <s xml:id="echoid-s1801" xml:space="preserve">ſumpto in ea <lb/>quovis puncto B, ducatur inde deorſum recta B Θ <lb/>quæ Cycloidem tangat, occurratque rectæ horizon-<lb/>tali A Θ in Θ: </s> <s xml:id="echoid-s1802" xml:space="preserve">recta vero B F ad axem perpendi-<lb/>cularis agatur, & </s> <s xml:id="echoid-s1803" xml:space="preserve">ſuper F A deſcribatur ſemicir-<lb/>culus F H A. </s> <s xml:id="echoid-s1804" xml:space="preserve">Deinde alia recta G E, parallela <lb/>F B, ſecet Cycloidem in E, rectam B Θ in I, cir-<lb/>cumferentiam F H A in H, & </s> <s xml:id="echoid-s1805" xml:space="preserve">denique axem D A <lb/>in G.</s> <s xml:id="echoid-s1806" xml:space="preserve"/> </p> <div xml:id="echoid-div127" type="float" level="2" n="1"> <note position="right" xlink:label="note-0121-02" xlink:href="note-0121-02a" xml:space="preserve">TAB. X. <lb/>Fig. I.</note> </div> <p style="it"> <s xml:id="echoid-s1807" xml:space="preserve">Dico tempus deſcenſus per arcum Cycloidis B E, <lb/>eſſe ad tempus per tangentem B I cum celeritate di-<lb/>midia ex B Θ, ſicut arcus F H ad rectam F G.</s> <s xml:id="echoid-s1808" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1809" xml:space="preserve">Si enim hoc verum non eſt, habebit tempus per arcum <lb/>B E ad dictum tempus per B I, vel majorem rationem quam <lb/>arcus F H ad rectam F G vel minorem. </s> <s xml:id="echoid-s1810" xml:space="preserve">Habeat primo, ſi <lb/>fieri poteſt, majorem.</s> <s xml:id="echoid-s1811" xml:space="preserve"/> </p> <pb o="82" file="0122" n="130" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1812" xml:space="preserve">Itaque tempus aliquod brevius tempore per B E (ſit hoc <lb/> <anchor type="note" xlink:label="note-0122-01a" xlink:href="note-0122-01"/> tempus Z) erit ad dictum tempus per B I ut arcus F H ad <lb/>rectam F G. </s> <s xml:id="echoid-s1813" xml:space="preserve">Quod ſi jam in Cycloide ſupra punctum B ſu-<lb/>matur punctum aliud N, erit tempus per B E poſt N B, <lb/>brevius tempore per B E. </s> <s xml:id="echoid-s1814" xml:space="preserve">Manifeſtum eſt autem punctum N <lb/>tam propinquum ſumi poſſe ipſi B, ut differentia eorum <lb/>temporum ſit quamlibet exigua, ac proinde ut minor ſit <lb/>ea qua tempus Z ſuperatur à tempore per B E. </s> <s xml:id="echoid-s1815" xml:space="preserve">Sit ita-<lb/>que punctum N ita ſumptum. </s> <s xml:id="echoid-s1816" xml:space="preserve">unde quidem tempus per <lb/>B E poſt N B majus erit tempore Z, majoremque pro-<lb/>inde rationem habebit ad tempus dictum per B I cum di-<lb/>midia celeritate ex B Θ, quam arcus F H ad rectam <lb/>F G. </s> <s xml:id="echoid-s1817" xml:space="preserve">Habeat itaque eam quam arcus F H O ad rectam <lb/>F G.</s> <s xml:id="echoid-s1818" xml:space="preserve"/> </p> <div xml:id="echoid-div128" type="float" level="2" n="2"> <note position="left" xlink:label="note-0122-01" xlink:href="note-0122-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> <p> <s xml:id="echoid-s1819" xml:space="preserve">Dividatur F G in partes æquales F P, P Q, &</s> <s xml:id="echoid-s1820" xml:space="preserve">c. </s> <s xml:id="echoid-s1821" xml:space="preserve">qua-<lb/>rum unaquæque minor ſit altitudine lineæ N B, atque item <lb/>altitudine arcus H O; </s> <s xml:id="echoid-s1822" xml:space="preserve">hoc enim fieri poſſe manifeſtum eſt; <lb/></s> <s xml:id="echoid-s1823" xml:space="preserve">& </s> <s xml:id="echoid-s1824" xml:space="preserve">à punctis diviſionum agantur rectæ, baſi D C parallelæ, <lb/>& </s> <s xml:id="echoid-s1825" xml:space="preserve">ad tangentem B Θ terminatæ P Λ, Q Ξ, &</s> <s xml:id="echoid-s1826" xml:space="preserve">c. </s> <s xml:id="echoid-s1827" xml:space="preserve">Quibus-<lb/>que in punctis hæ ſecant circumferentiam F H, ab iis, <lb/>itemque à puncto H, tangentes ſurſum ducantur usque <lb/>ad proximam quæque parallelam, velut Δ Χ, Γ Σ &</s> <s xml:id="echoid-s1828" xml:space="preserve">c. </s> <s xml:id="echoid-s1829" xml:space="preserve">Si-<lb/>militer vero & </s> <s xml:id="echoid-s1830" xml:space="preserve">à punctis, in quibus dictæ parallelæ Cy-<lb/>cloidi occurrunt, tangentes ſurſum ducantur velut S V, <lb/>T M &</s> <s xml:id="echoid-s1831" xml:space="preserve">c. </s> <s xml:id="echoid-s1832" xml:space="preserve">additâ vero ad rectam F G parte una G R æ-<lb/>quali iis quæ ex diviſione, ductaque R Φ parallelâ ſimi-<lb/>liter ipſi D C, patet eam occurrere circumferentiæ F H A <lb/>inter H & </s> <s xml:id="echoid-s1833" xml:space="preserve">O, quia G R minor eſt altitudine puncti H ſupra <lb/>O. </s> <s xml:id="echoid-s1834" xml:space="preserve">Jam vero ſic porro argumentabimur.</s> <s xml:id="echoid-s1835" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1836" xml:space="preserve">Tempus per tangentem V S cum celeritate æquabili quæ <lb/>acquireretur ex B S, majus eſt tempore motus continue ac-<lb/>celerati per arcum B S poſt N B. </s> <s xml:id="echoid-s1837" xml:space="preserve">Nam celeritas ex B S mi-<lb/>nor eſt celeritate ex N B, propterea quod minor altitudo <lb/>B S quam N B. </s> <s xml:id="echoid-s1838" xml:space="preserve">At celeritas ex B S æquabiliter continuari <lb/>ponitur per tangentem V S, cum celeritas acquiſita ex N B <lb/>continue porro acceleretur per arcum B S, qui arcus minor <pb file="0123" n="131"/> <pb file="0123a" n="132"/> <anchor type="figure" xlink:label="fig-0123a-01a" xlink:href="fig-0123a-01"/> <anchor type="figure" xlink:label="fig-0123a-02a" xlink:href="fig-0123a-02"/> <anchor type="figure" xlink:label="fig-0123a-03a" xlink:href="fig-0123a-03"/> <anchor type="figure" xlink:label="fig-0123a-04a" xlink:href="fig-0123a-04"/> <anchor type="figure" xlink:label="fig-0123a-05a" xlink:href="fig-0123a-05"/> <pb file="0124" n="133"/> <pb o="83" file="0125" n="134" rhead="HOROLOG. OSCILLATOR."/> inſuper eſt tangente V S, omnibusque partibus ſuis magis <lb/> <anchor type="note" xlink:label="note-0125-01a" xlink:href="note-0125-01"/> erectus quam ulla pars tangentis V S. </s> <s xml:id="echoid-s1839" xml:space="preserve">Adeo ut omnino ma-<lb/>jus ſit futurum tempus per tangentem V S cum celeritate ex <lb/>B S, tempore per arcum B S poſt N B. </s> <s xml:id="echoid-s1840" xml:space="preserve">Similiter tempus <lb/>per tangentem M T, cum celeritate ex B T, majus erit <lb/>tempore per arcum S T poſt N S, & </s> <s xml:id="echoid-s1841" xml:space="preserve">tempus per tangen-<lb/>tem Π Y cum celeritate ex B Y, majus tempore per arcum <lb/>T Y poſt N T. </s> <s xml:id="echoid-s1842" xml:space="preserve">Atque ita tempora motuum æquabilium <lb/>per tangentes omnes usque ad infimam quæ tangit cycloi-<lb/>dem in E, cum celeritatibus per ſingulas quantæ acquirun-<lb/>tur cadendo ex B adusque punctum ipſarum contactus, ma-<lb/>jora ſimul erunt tempore per arcum B E poſt N B. </s> <s xml:id="echoid-s1843" xml:space="preserve">Eadem <lb/>vero & </s> <s xml:id="echoid-s1844" xml:space="preserve">minora eſſent, ut nunc oſtendemus.</s> <s xml:id="echoid-s1845" xml:space="preserve"/> </p> <div xml:id="echoid-div129" type="float" level="2" n="3"> <figure xlink:label="fig-0123a-01" xlink:href="fig-0123a-01a"> <caption xml:id="echoid-caption30" style="it" xml:space="preserve">Pag. 82.<lb/>TAB. IX.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables30" xml:space="preserve">AMO FNP B G C H D K L</variables> </figure> <figure xlink:label="fig-0123a-02" xlink:href="fig-0123a-02a"> <caption xml:id="echoid-caption31" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables31" xml:space="preserve">A C E F B D</variables> </figure> <figure xlink:label="fig-0123a-03" xlink:href="fig-0123a-03a"> <caption xml:id="echoid-caption32" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables32" xml:space="preserve">C B e N L m E O M D f F A</variables> </figure> <figure xlink:label="fig-0123a-04" xlink:href="fig-0123a-04a"> <caption xml:id="echoid-caption33" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables33" xml:space="preserve">C B E G F D f H b A</variables> </figure> <figure xlink:label="fig-0123a-05" xlink:href="fig-0123a-05a"> <caption xml:id="echoid-caption34" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables34" xml:space="preserve">C V B E S Δ M O Λ H Φ G Π T N P I</variables> </figure> <note position="right" xlink:label="note-0125-01" xlink:href="note-0125-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> <p> <s xml:id="echoid-s1846" xml:space="preserve">Conſiderentur enim denuo tempora eadem motuum æqua-<lb/>bilium per tangentes cycloidis. </s> <s xml:id="echoid-s1847" xml:space="preserve">Et eſt quidem tempus per <lb/>tangentem V S cum celeritate ex B S, ad tempus per re-<lb/>ctam Β Λ cum celeritate dimidia ex F A, ut tangens cir-<lb/>cumferentiæ Δ Χ ad partem axis F P <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1848" xml:space="preserve">Similiterque tem- <anchor type="note" xlink:label="note-0125-02a" xlink:href="note-0125-02"/> pus per tangentem M T, cum celeritate ex B T, ad tem-<lb/>pus per rectam Λ Ξ cum eadem dimidia celeritate ex F A, <lb/>ut tangens Γ Σ ad rectam P Q. </s> <s xml:id="echoid-s1849" xml:space="preserve">Atque ita deinceps ſingula <lb/>tempora per tangentes cycloidis, quæ ſunt eadem ſupradi-<lb/>ctis, erunt ad tempora motus æquabilis per partes ſibi re-<lb/>ſpondentes rectæ B I cum celeritate dimidia ex B Θ, ſicut <lb/>tangentes circumferentiæ F H, iisdem parallelis compre-<lb/>henſæ, ad partes rectæ F G ipſis reſpondentes.</s> <s xml:id="echoid-s1850" xml:space="preserve"/> </p> <div xml:id="echoid-div130" type="float" level="2" n="4"> <note symbol="*" position="right" xlink:label="note-0125-02" xlink:href="note-0125-02a" xml:space="preserve">Prop. <lb/>præced.</note> </div> <p> <s xml:id="echoid-s1851" xml:space="preserve">Sunt igitur quantitates quædam rectæ F P, P Q, &</s> <s xml:id="echoid-s1852" xml:space="preserve">c. </s> <s xml:id="echoid-s1853" xml:space="preserve">& </s> <s xml:id="echoid-s1854" xml:space="preserve"><lb/>totidem aliæ, tempora ſcilicet quibus percurruntur rectæ <lb/>Β Λ, Λ Ξ &</s> <s xml:id="echoid-s1855" xml:space="preserve">c, motu æquabili cum celeritate dimidia ex <lb/>Β Θ; </s> <s xml:id="echoid-s1856" xml:space="preserve">Et unaquæque quantitas in prioribus ad ſequentem ea-<lb/>dem proportione refertur, qua unaquæque poſteriorum ad <lb/>ſuam ſequentem; </s> <s xml:id="echoid-s1857" xml:space="preserve">ſunt enim utrobique inter ſe æquales. </s> <s xml:id="echoid-s1858" xml:space="preserve">Qui-<lb/>bus autem proportionibus priores quantitates ad alias quas-<lb/>dam, nempe ad tangentes circuli Δ Χ, Γ Σ, &</s> <s xml:id="echoid-s1859" xml:space="preserve">c. </s> <s xml:id="echoid-s1860" xml:space="preserve">referun-<lb/>tur, iisdem proportionibus & </s> <s xml:id="echoid-s1861" xml:space="preserve">eodem ordine poſteriores quo-<lb/>que referuntur ad alias quasdam, nempe ad tempora motus <pb o="84" file="0126" n="135" rhead="CHRISTIANI HUGENII"/> qualem diximus per tangentes cycloidis V S, M T &</s> <s xml:id="echoid-s1862" xml:space="preserve">c. </s> <s xml:id="echoid-s1863" xml:space="preserve">Er-<lb/> <anchor type="note" xlink:label="note-0126-01a" xlink:href="note-0126-01"/> go, ſicut ſe habent omnes ſimul priores ad omnes eas ad <lb/>quas ipſæ referuntur, hoc eſt, ſicut tota F G ad tangentes <lb/>omnes Χ Δ, Γ Σ, &</s> <s xml:id="echoid-s1864" xml:space="preserve">c. </s> <s xml:id="echoid-s1865" xml:space="preserve">ita tempus quo percurritur tota B I <lb/>cum celeritate dimidia ex Β Θ, ad tempora omnia motuum <lb/>quales diximus per tangentes cycloidis V S, M T, &</s> <s xml:id="echoid-s1866" xml:space="preserve">c <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s1867" xml:space="preserve"> <anchor type="note" xlink:label="note-0126-02a" xlink:href="note-0126-02"/> Et invertendo itaque, tempora motuum dictorum per tan-<lb/>gentes cycloidis, ad tempus per rectam B I cum celeritate <lb/>dimidia ex B Θ, eandem rationem habebunt quam dictæ tan-<lb/>gentes omnes circumferentiæ F H ad rectam F G; </s> <s xml:id="echoid-s1868" xml:space="preserve">ac mi-<lb/>norem proinde quam arcus F O ad rectam eandem F G; <lb/></s> <s xml:id="echoid-s1869" xml:space="preserve">quia arcus F Φ, ideoque omnino & </s> <s xml:id="echoid-s1870" xml:space="preserve">arcus F O major eſt <lb/>dictis omnibus arcus F H tangentibus <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1871" xml:space="preserve">Atqui tempus per <anchor type="note" xlink:label="note-0126-03a" xlink:href="note-0126-03"/> B E poſt N B, ad tempus per B I cum celeritate dimidia ex <lb/>B Θ, poſuimus eſſe ut arcus F O ad rectam F G. </s> <s xml:id="echoid-s1872" xml:space="preserve">Ergo <lb/>dicta tempora omnia per tangentes cycloidis minora ſimul <lb/>erunt tempore per B E poſt N B, cum antea majora eſſe os-<lb/>tenſum ſit; </s> <s xml:id="echoid-s1873" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s1874" xml:space="preserve">Itaque tempus per arcum <lb/>cycloidis B E, ad tempus per tangentem B I, cum celerita-<lb/>te dimidia ex Β Θ vel ex F A, non habet majorem rationem <lb/>quam arcus circumferentiæ F H ad rectam F G.</s> <s xml:id="echoid-s1875" xml:space="preserve"/> </p> <div xml:id="echoid-div131" type="float" level="2" n="5"> <note position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <note symbol="*" position="left" xlink:label="note-0126-02" xlink:href="note-0126-02a" xml:space="preserve">Prop. 2. <lb/>Archimedis <lb/>de Sphæ-<lb/>roid. & <lb/>Conoid.</note> <note symbol="*" position="left" xlink:label="note-0126-03" xlink:href="note-0126-03a" xml:space="preserve">Prop. 20. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1876" xml:space="preserve">Habeat jam, ſi poteſt, minorem. </s> <s xml:id="echoid-s1877" xml:space="preserve">Ergo tempus aliquod <lb/>majus tempore per arcum B E, (ſit hoc tempus Z) erit ad <lb/>tempus dictum per B I, ut arcus F H ad rectam F G.</s> <s xml:id="echoid-s1878" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1879" xml:space="preserve">Quod ſi jam ſumatur arcus N M æqualis altitudine cum <lb/> <anchor type="note" xlink:label="note-0126-04a" xlink:href="note-0126-04"/> arcu B E, ſed cujus terminus ſuperior N ſit humilior puncto <lb/>B, erit tempus per arcum N M majus tempore per arcum <lb/>BE <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1880" xml:space="preserve">Manifeſtum autem quod punctum N tam propinquum <anchor type="note" xlink:label="note-0126-05a" xlink:href="note-0126-05"/> ſumi poteſt puncto B, ut differentia dictorum temporum ſit <lb/>quamlibet exigua, ac proinde minor ea qua tempus Z ſupe-<lb/>rat tempus per arcum B E. </s> <s xml:id="echoid-s1881" xml:space="preserve">Sit itaque punctum N ita ſum-<lb/>ptum. </s> <s xml:id="echoid-s1882" xml:space="preserve">Unde quidem tempus per N M minus erit tempore Z, <lb/>habebitque proinde ad dictum tempus per B I, cum dimi-<lb/>dia celeritate ex Β Θ, minorem rationem quam arcus F H ad <lb/>rectam F G. </s> <s xml:id="echoid-s1883" xml:space="preserve">Habeat ergo eam quam arcus L Had rectam F G.</s> <s xml:id="echoid-s1884" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="6"> <note position="left" xlink:label="note-0126-04" xlink:href="note-0126-04a" xml:space="preserve">TAB. X. <lb/>Fig. 2.</note> <note symbol="*" position="left" xlink:label="note-0126-05" xlink:href="note-0126-05a" xml:space="preserve">Prop. 22. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s1885" xml:space="preserve">Dividatur jam F G in partes æquales F P, P Q, &</s> <s xml:id="echoid-s1886" xml:space="preserve">c.</s> <s xml:id="echoid-s1887" xml:space="preserve"> <pb o="85" file="0127" n="136" rhead="HOROLOG. OSCILLATOR."/> quarum unaquæque minor ſit arcus cycloidis B N altitudine, <lb/> <anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/> itemque minor altitudine arcus circumferentiæ F L; </s> <s xml:id="echoid-s1888" xml:space="preserve">& </s> <s xml:id="echoid-s1889" xml:space="preserve">ad-<lb/>ditâ ad F G unâ earum partium G ζ, ducantur à punctis di-<lb/>viſionum rectæ baſi D C parallelæ, & </s> <s xml:id="echoid-s1890" xml:space="preserve">ad tangentem B Θ <lb/>terminatæ, P O, Q K, &</s> <s xml:id="echoid-s1891" xml:space="preserve">c; </s> <s xml:id="echoid-s1892" xml:space="preserve">itemque à puncto ζ recta ζ Ω <lb/>quæ ſecet cycloidem in V, circumferentiam in η; </s> <s xml:id="echoid-s1893" xml:space="preserve">quibus-<lb/>que in punctis ductæ parallelæ ſecant circumferentiam F H, <lb/>ab iis tangentes deorſum ducantur usque ad proximam quæ-<lb/>que parallelam, velut θ Δ, Γ Σ: </s> <s xml:id="echoid-s1894" xml:space="preserve">Quarum infima à puncto <lb/>Η ducta occurrat rectæ ζ Ω in X. </s> <s xml:id="echoid-s1895" xml:space="preserve">Similiter vero & </s> <s xml:id="echoid-s1896" xml:space="preserve">à pun-<lb/>ctis, in quibus dictæ parallelæ occurrunt cycloidi, ducan-<lb/>tur totidem tangentes deorſum, velut S Λ, T Ξ, &</s> <s xml:id="echoid-s1897" xml:space="preserve">c. </s> <s xml:id="echoid-s1898" xml:space="preserve">qua-<lb/>rum infima, tangens nempe à puncto E ducta, occurrat re-<lb/>ctæ ζ Ω in R.</s> <s xml:id="echoid-s1899" xml:space="preserve"/> </p> <div xml:id="echoid-div133" type="float" level="2" n="7"> <note position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> <p> <s xml:id="echoid-s1900" xml:space="preserve">Quia igitur P ζ æqualis eſt F G altitudini arcus B E, <lb/>cui æqualis eſt ex conſtructione altitudo arcus N M, erit & </s> <s xml:id="echoid-s1901" xml:space="preserve"><lb/>P ζ æqualis altitudini arcus N M. </s> <s xml:id="echoid-s1902" xml:space="preserve">Eſt autem recta P O ex <lb/>conſtructione ſuperior termino N. </s> <s xml:id="echoid-s1903" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s1904" xml:space="preserve">ζ Ω, & </s> <s xml:id="echoid-s1905" xml:space="preserve">in ea <lb/>punctum V, ſuperius termino M. </s> <s xml:id="echoid-s1906" xml:space="preserve">Quare, cum arcus S V <lb/>æqualis ſit altitudinis cum arcu N M, ſed termino S ſubli-<lb/>miore quam N, erit tempus per S V brevius tempore per N M<anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s1907" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">Prop. 22. <lb/>huj.</note> <p> <s xml:id="echoid-s1908" xml:space="preserve">Atqui tempus per tangentem S Λ, cum celeritate æqua-<lb/>bili ex B S, brevius eſt tempore deſcenſus accelerati per ar-<lb/>cum S T, incipientis in S. </s> <s xml:id="echoid-s1909" xml:space="preserve">Nam celeritas ex B S, qua to-<lb/>ta S Λ transmiſſa ponitur, æqualis eſt celeritati ex S T <anchor type="note" xlink:href="" symbol="*"/>, <anchor type="note" xlink:label="note-0127-03a" xlink:href="note-0127-03"/> quæ motui per arcum S T in fine demum acquiritur; </s> <s xml:id="echoid-s1910" xml:space="preserve">ipſa-<lb/>que S Λ minor eſt quam S T. </s> <s xml:id="echoid-s1911" xml:space="preserve">Similiter tempus per tangen-<lb/>tem T Ξ, cum celeritate æquabili ex B T, brevius eſt tem-<lb/>pore deſcenſus accelerati per arcum T Y poſt S T; </s> <s xml:id="echoid-s1912" xml:space="preserve">quum <lb/>celeritas ex B T, qua tota T Ξ transmiſſa ponitur, ſit æqua-<lb/>lis celeritati ex S Y, quæ in fine demum acquiritur motui <lb/>dicto per arcum T Y poſt S T; </s> <s xml:id="echoid-s1913" xml:space="preserve">ipſaque T Ξ minor ſit arcu <lb/>T Y. </s> <s xml:id="echoid-s1914" xml:space="preserve">Atque ita tempora omnia motuum æquabilium per <lb/>tangentes cycloidis, cum celeritatibus per ſingulas quantæ <lb/>acquiruntur deſcendendo ex B usque ad punctum ipſarum <lb/>contactus, breviora ſimul erunt tempore deſcenſus accelerati <pb o="86" file="0128" n="137" rhead="CHRISTIANI HUGENII"/> per arcum S V. </s> <s xml:id="echoid-s1915" xml:space="preserve">Eadem vero & </s> <s xml:id="echoid-s1916" xml:space="preserve">longiora eſſent, ut nunc <lb/> <anchor type="note" xlink:label="note-0128-01a" xlink:href="note-0128-01"/> oſtendemus.</s> <s xml:id="echoid-s1917" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="8"> <note symbol="*" position="right" xlink:label="note-0127-03" xlink:href="note-0127-03a" xml:space="preserve">Prop. 8. <lb/>huj.</note> <note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> </div> <p> <s xml:id="echoid-s1918" xml:space="preserve">Eſt enim tempus dictum per tangentem S Λ, cum cele-<lb/>ritate æquabili ex B S, ad tempus per rectam O K cum ce-<lb/>leritate æquabili dimidia ex B Θ, ſicut tangens ſemicirculi <lb/>θ Δ ad rectam P Q <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1919" xml:space="preserve">ſimiliterque tempus per tangentem <anchor type="note" xlink:label="note-0128-02a" xlink:href="note-0128-02"/> Τ Ξ, cum celeritate æquabili ex B T, eſt ad tempus per <lb/>rectam Κ Ψ cum celeritate æquabili dimidia ex B Θ, ut tan-<lb/>gens Γ Σ ad rectam Q Π. </s> <s xml:id="echoid-s1920" xml:space="preserve">Atque ita deinceps ſingula tem-<lb/>pora per tangentes cycloidis, quæ ſunt eadem ſupra dictis, <lb/>erunt ad tempora motus æquabilis per partes ſibi reſponden-<lb/>tes rectæ O Ω, cum celeritate dimidia ex B Θ, ut tangen-<lb/>tes circumferentiæ θ η, iisdem parallelis incluſæ, ad partes <lb/>rectæ P ζ ipſis reſpondentes. </s> <s xml:id="echoid-s1921" xml:space="preserve">Unde, ut in priori parte de-<lb/>monſtrationis, concludetur omnes ſimul rectas P Q, Q Π <lb/>&</s> <s xml:id="echoid-s1922" xml:space="preserve">c. </s> <s xml:id="echoid-s1923" xml:space="preserve">hoc eſt, totam P ζ eſſe ad omnes ſimul tangentes θ Δ, <lb/>Γ Σ, &</s> <s xml:id="echoid-s1924" xml:space="preserve">c. </s> <s xml:id="echoid-s1925" xml:space="preserve">ſicut tempus quo percurritur tota O Ω, cum ce-<lb/>leritate dimidia ex B Θ, ad tempora omnia motuum quales <lb/>diximus per tangentes cycloidis S Λ, T Ξ, &</s> <s xml:id="echoid-s1926" xml:space="preserve">c. </s> <s xml:id="echoid-s1927" xml:space="preserve">Quare & </s> <s xml:id="echoid-s1928" xml:space="preserve"><lb/>convertendo, tempora omnia per tangentes cycloidis, eam <lb/>rationem habebunt ad tempus dictum motus æquabilis per <lb/>rectam Ο Ω, ſive per B I, quam dictæ tangentes omnes ar-<lb/>cus θ η ad rectam P ζ vel F G, ac proinde majorem quam <lb/>arcus L H ad rectam F G; </s> <s xml:id="echoid-s1929" xml:space="preserve">eſt enim arcus θ H, adeoque <lb/> <anchor type="note" xlink:label="note-0128-03a" xlink:href="note-0128-03"/> etiam omnino arcus L H, minor dictis tangentibus arcus θ η <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s1930" xml:space="preserve"> Sed tempus per N M poſuimus ab initio ad idem tempus per <lb/>B I ſe habere ut arcus L H ad rectam F G. </s> <s xml:id="echoid-s1931" xml:space="preserve">Ergo tempus per <lb/>N M, multoque magis tempus per S V, minuserit tempore <lb/>per tangentes cycloidis. </s> <s xml:id="echoid-s1932" xml:space="preserve">Quod eſt abſurdum, cum hoc tempus, <lb/>illo per arcum S V, antea minus oſtenſum fuerit. </s> <s xml:id="echoid-s1933" xml:space="preserve">Patet igi-<lb/>tur tempus per arcum cycloidis B E ad tempus per tangen-<lb/>tem B I cum celeritare æquabili dimidia ex B Θ, non mi-<lb/>norem rationem habere quam arcus F H ad rectam F G. <lb/></s> <s xml:id="echoid-s1934" xml:space="preserve">Sed nec majorem habere oſtenſum fuit. </s> <s xml:id="echoid-s1935" xml:space="preserve">Ergo eandem habeat <lb/>neceſſe eſt. </s> <s xml:id="echoid-s1936" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s1937" xml:space="preserve"/> </p> <div xml:id="echoid-div135" type="float" level="2" n="9"> <note symbol="*" position="left" xlink:label="note-0128-02" xlink:href="note-0128-02a" xml:space="preserve">Prop. <lb/>præced.</note> <note symbol="*" position="left" xlink:label="note-0128-03" xlink:href="note-0128-03a" xml:space="preserve">Prop. 20. <lb/>huj.</note> </div> <pb file="0129" n="138"/> <pb file="0129a" n="139"/> <figure> <caption xml:id="echoid-caption35" style="it" xml:space="preserve">Pag. 86.<lb/>TAB. X.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables35" xml:space="preserve">D C N F X B V P Δ Σ S M Λ Q Γ T Π Ξ Y G H E I R Φ O A Θ</variables> </figure> <figure> <caption xml:id="echoid-caption36" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables36" xml:space="preserve">D C F B P Θ S O N Q L Δ K Γ T Λ Π Σ Y Ψ Ξ G H E I ζ η X V R Ω A M Θ</variables> </figure> <pb file="0130" n="140"/> <pb o="87" file="0131" n="141" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div137" type="section" level="1" n="49"> <head xml:id="echoid-head71" xml:space="preserve">PROPOSITIO XXV.</head> <note position="right" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <p style="it"> <s xml:id="echoid-s1938" xml:space="preserve">IN Cycloide cujus axis ad perpendiculum erectus <lb/>eſt, vertice deorſum ſpectante, tempora deſcen-<lb/>ſus quibus mobile, à quocunque in ea puncto dimis-<lb/>ſum, ad punctum imum verticis pervenit, ſunt in-<lb/>ter ſe æqualia; </s> <s xml:id="echoid-s1939" xml:space="preserve">habentque ad tempus caſus perpen-<lb/>dicularis per totum axem cycloidis eam rationem, <lb/>quam ſemicircumferentia circuli ad diametrum.</s> <s xml:id="echoid-s1940" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1941" xml:space="preserve">Eſto cyclois A B C cujus vertex A deorſum ſpectet, axis <lb/> <anchor type="note" xlink:label="note-0131-02a" xlink:href="note-0131-02"/> vero A D ad perpendiculum erectus ſit, & </s> <s xml:id="echoid-s1942" xml:space="preserve">à puncto quovis <lb/>in cycloide ſumpto, velut B; </s> <s xml:id="echoid-s1943" xml:space="preserve">deſcendat mobile impetu na-<lb/>turali per arcum B A, ſive per ſuperficiem ita inflexam. </s> <s xml:id="echoid-s1944" xml:space="preserve">Di-<lb/>co tempus deſcenſus hujus eſſe ad tempus caſus per axem <lb/>D A, ſicut ſemicircumferentia circuli ad diametrum. </s> <s xml:id="echoid-s1945" xml:space="preserve">Quo <lb/>demonſtrato, etiam tempora deſcenſus, per quoslibet cy-<lb/>cloidis arcus ad A terminatos, inter ſe æqualia eſſe conſta-<lb/>bit.</s> <s xml:id="echoid-s1946" xml:space="preserve"/> </p> <div xml:id="echoid-div137" type="float" level="2" n="1"> <note position="right" xlink:label="note-0131-02" xlink:href="note-0131-02a" xml:space="preserve">TAB. XI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s1947" xml:space="preserve">Deſcribatur ſuper axe D A ſemicirculus, cujus circumfe-<lb/>rentiam ſecet recta B F, baſi D C parallela, in E; </s> <s xml:id="echoid-s1948" xml:space="preserve">junctâ-<lb/>que E A, ducatur ei parallela B G, quæ quidem cycloidem <lb/>tanget in B. </s> <s xml:id="echoid-s1949" xml:space="preserve">Eadem vero occurrat rectæ horizontali per A <lb/>ductæ in G: </s> <s xml:id="echoid-s1950" xml:space="preserve">ſitque etiam ſuper F A deſcriptus ſemicirculus <lb/>F H A.</s> <s xml:id="echoid-s1951" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1952" xml:space="preserve">Eſt igitur, per præcedentem, tempus deſcenſus per ar-<lb/>cum cycloidis B A, ad tempus motus æquabilis per rectam <lb/>B G cum celeritate dimidia ex B G, ſicut arcus ſemicirculi <lb/>F H A ad rectam F A. </s> <s xml:id="echoid-s1953" xml:space="preserve">Tempus vero dicti motus æquabilis <lb/>per B G, æquatur tempori deſcenſus naturaliter accelerati <lb/>per eandem B G, ſive per E A, quæ ipſi parallela eſt & </s> <s xml:id="echoid-s1954" xml:space="preserve"><lb/>æqualis, hoc eſt, tempori deſcenſus accelerati per axem <lb/>D A<anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1955" xml:space="preserve">Itaque tempus per arcum B A, erit quoque ad tem- <anchor type="note" xlink:label="note-0131-03a" xlink:href="note-0131-03"/> pus deſcenſus per axem D A, ut ſemicirculi circumferentia <lb/>F H A ad diametrum F A. </s> <s xml:id="echoid-s1956" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s1957" xml:space="preserve"/> </p> <div xml:id="echoid-div138" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0131-03" xlink:href="note-0131-03a" xml:space="preserve">Prop. 6. <lb/>Galil. de <lb/>motu Accel.</note> </div> <pb o="88" file="0132" n="142" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s1958" xml:space="preserve">Quod ſi tota cycloidis cavitas perfecta ponatur, conſtat <lb/> <anchor type="note" xlink:label="note-0132-01a" xlink:href="note-0132-01"/> mobile, poſtquam per arcum B A deſcenderit, inde conti-<lb/>nuato motu per alterum ipſi æqualem arcum aſcenſurum <anchor type="note" xlink:href="" symbol="*"/>, <anchor type="note" xlink:label="note-0132-02a" xlink:href="note-0132-02"/> atque in eo tantundem temporis atque deſcendendo conſum-<lb/>pturum <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1959" xml:space="preserve">Deinde rurſus per A ad B perventurum, ac ſingu- <anchor type="note" xlink:label="note-0132-03a" xlink:href="note-0132-03"/> larum ejusmodi reciprocationum, in magnis parvisve cycloi-<lb/>dis arcubus peractarum, tempora fore ad tempus caſus per-<lb/>pendicularis per axem D A, ſicut circumferentia circuli tota <lb/>ad diametrum ſuam.</s> <s xml:id="echoid-s1960" xml:space="preserve"/> </p> <div xml:id="echoid-div139" type="float" level="2" n="3"> <note position="left" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve"><emph style="sc">De motu</emph> <lb/><emph style="sc">IN CY-</emph> <lb/><emph style="sc">CLOIDE</emph>.</note> <note symbol="*" position="left" xlink:label="note-0132-02" xlink:href="note-0132-02a" xml:space="preserve">Prop. 9. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0132-03" xlink:href="note-0132-03a" xml:space="preserve">Prop. 11. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div141" type="section" level="1" n="50"> <head xml:id="echoid-head72" xml:space="preserve">PROPOSITIO XXVI.</head> <p style="it"> <s xml:id="echoid-s1961" xml:space="preserve">Iisdem poſitis, ſi ducatur inſuper recta horizonta-<lb/> <anchor type="note" xlink:label="note-0132-04a" xlink:href="note-0132-04"/> lis H I quæ arcum B A ſecet in I, circumferen-<lb/>tiam vero F H A in H: </s> <s xml:id="echoid-s1962" xml:space="preserve">dico tempus per arcum <lb/>B I, ad tempus per arcum I A poſt B I, eam ra-<lb/>tionem habere quam arcus circumferentiæ F H ad <lb/>H A.</s> <s xml:id="echoid-s1963" xml:space="preserve"/> </p> <div xml:id="echoid-div141" type="float" level="2" n="1"> <note position="left" xlink:label="note-0132-04" xlink:href="note-0132-04a" xml:space="preserve">TAB. XI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s1964" xml:space="preserve">Occurrat enim recta H I tangenti B G in K, axi D A in <lb/>L. </s> <s xml:id="echoid-s1965" xml:space="preserve">Eſt itaque tempus per arcum B A, ad tempus motus æ-<lb/>quabilis per B G cum celeritate dimidia ex B G, ſicut arcus <lb/>F H A ad rectam F A <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1966" xml:space="preserve">Tempus autem dicti motus æqua- <anchor type="note" xlink:label="note-0132-05a" xlink:href="note-0132-05"/> bilis per B G, eſt ad tempus motus æquabilis per B K, cum <lb/>eadem celeritate dimidia ex B G, ſicut B G ad B K longi-<lb/>tudine, hoc eſt, ſicut F A ad F L. </s> <s xml:id="echoid-s1967" xml:space="preserve">Et rurſus tempus mo-<lb/>tus æquabilis, cum dicta celeritate, per B K, ad tempus <lb/>per arcum B I, ſicut F L ad arcum F H <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s1968" xml:space="preserve">Igitur ex æ- <anchor type="note" xlink:label="note-0132-06a" xlink:href="note-0132-06"/> quo erit tempus per arcum B A ad tempus per B I, ut ar-<lb/>cus F H A ad F H. </s> <s xml:id="echoid-s1969" xml:space="preserve">Et dividendo, & </s> <s xml:id="echoid-s1970" xml:space="preserve">convertendo, tem-<lb/>pus per B I, ad tempus per I A poſt B I, ut arcus F H <lb/>ad H A. </s> <s xml:id="echoid-s1971" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s1972" xml:space="preserve"/> </p> <div xml:id="echoid-div142" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0132-05" xlink:href="note-0132-05a" xml:space="preserve">Prop. 24. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0132-06" xlink:href="note-0132-06a" xml:space="preserve">Prop. 24. <lb/>huj.</note> </div> <pb file="0133" n="143"/> <figure> <image file="0133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0133-01"/> </figure> <note position="right" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <div xml:id="echoid-div144" type="section" level="1" n="51"> <head xml:id="echoid-head73" xml:space="preserve">HOROLOGII OSCILLATORII</head> <head xml:id="echoid-head74" style="it" xml:space="preserve">PARS TERTIA.</head> <p style="it"> <s xml:id="echoid-s1973" xml:space="preserve">De linearum curvarum evolutione & </s> <s xml:id="echoid-s1974" xml:space="preserve">dimenſione.</s> <s xml:id="echoid-s1975" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div145" type="section" level="1" n="52"> <head xml:id="echoid-head75" xml:space="preserve">DEFINITIONES.</head> <head xml:id="echoid-head76" xml:space="preserve">I.</head> <p style="it"> <s xml:id="echoid-s1976" xml:space="preserve">LINEA in unam partem inflexa vocetur quam <lb/>rectæ omnes tangentes ab eadem parte contin-<lb/>gunt. </s> <s xml:id="echoid-s1977" xml:space="preserve">Si autem portiones quasdam rectas lineas ha-<lb/>buerit, hæ ipſæ productæ pro tangentibus habentur.</s> <s xml:id="echoid-s1978" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div146" type="section" level="1" n="53"> <head xml:id="echoid-head77" xml:space="preserve">II.</head> <p style="it"> <s xml:id="echoid-s1979" xml:space="preserve">Cum autem duæ hujusmodi lineæ ab eodem pun-<lb/>cto egrediuntur, quarum convexitas unius obverſa <lb/>ſit ad cavitatem alterius, quales ſunt in figura <lb/>adſcripta curvæ A B C, A D E, ambæ in ean-<lb/> <anchor type="note" xlink:label="note-0133-02a" xlink:href="note-0133-02"/> dem partem cavæ dicantur.</s> <s xml:id="echoid-s1980" xml:space="preserve"/> </p> <div xml:id="echoid-div146" type="float" level="2" n="1"> <note position="right" xlink:label="note-0133-02" xlink:href="note-0133-02a" xml:space="preserve">TAB. XI. <lb/>Fig. 2.</note> </div> </div> <div xml:id="echoid-div148" type="section" level="1" n="54"> <head xml:id="echoid-head78" xml:space="preserve">III.</head> <p style="it"> <s xml:id="echoid-s1981" xml:space="preserve">Si lineæ, in unam partem cavæ, filum ſeu linea <lb/>flexilis circumplicata intelligatur, & </s> <s xml:id="echoid-s1982" xml:space="preserve">manente una <lb/>fili extremitate illi affixa, altera extremitas ab-<lb/>ducatur, ita ut pars ea quæ ſoluta eſt ſemper ex-<lb/>tenſa maneat; </s> <s xml:id="echoid-s1983" xml:space="preserve">manifeſtum eſt curvam quandam <lb/>aliam hac fili extremitate deſcribi. </s> <s xml:id="echoid-s1984" xml:space="preserve">Vocetur autem <lb/>ea, Deſcripta ex evolutione.</s> <s xml:id="echoid-s1985" xml:space="preserve"/> </p> <pb o="90" file="0134" n="144" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div149" type="section" level="1" n="55"> <head xml:id="echoid-head79" xml:space="preserve">IV.</head> <note position="left" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> <p style="it"> <s xml:id="echoid-s1986" xml:space="preserve">Illa vero cui filum circumplicatum erat, dicatur <lb/>Evoluta. </s> <s xml:id="echoid-s1987" xml:space="preserve">In figura ſuperiori, A B C eſt evoluta, <lb/> <anchor type="note" xlink:label="note-0134-02a" xlink:href="note-0134-02"/> A D E deſcripta ex evolutione A B C, ut nempe <lb/>cum extremitas fili ex A venit in D, pars fili ex-<lb/>tenſa ſit D B recta, reliqua parte B C adhuc ap-<lb/>plicata curvæ A B C. </s> <s xml:id="echoid-s1988" xml:space="preserve">Manifeſtum eſt autem D B <lb/>tangere evolutam in B.</s> <s xml:id="echoid-s1989" xml:space="preserve"/> </p> <div xml:id="echoid-div149" type="float" level="2" n="1"> <note position="left" xlink:label="note-0134-02" xlink:href="note-0134-02a" xml:space="preserve">TAB. XI. <lb/>Fig. 2.</note> </div> </div> <div xml:id="echoid-div151" type="section" level="1" n="56"> <head xml:id="echoid-head80" xml:space="preserve">PROPOSITIOI.</head> <p style="it"> <s xml:id="echoid-s1990" xml:space="preserve">R Ecta omnis, quæ evolutam tangit, occurret li-<lb/>@eæ ex evolutione deſcriptæ ad angulos rectos.</s> <s xml:id="echoid-s1991" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1992" xml:space="preserve">Sit A B evoluta, A H vero quæ ex evolutione illius de-<lb/> <anchor type="note" xlink:label="note-0134-03a" xlink:href="note-0134-03"/> ſcripta eſt. </s> <s xml:id="echoid-s1993" xml:space="preserve">Recta autem F D C, tangens curvam A D in D, <lb/>occurrat in C curvæ A C H. </s> <s xml:id="echoid-s1994" xml:space="preserve">Dico ei occurrere ad angulos <lb/>rectos: </s> <s xml:id="echoid-s1995" xml:space="preserve">hoc eſt, ſi ducatur C E recta perpendicularis C D, <lb/>dico eam in C tangere curvam A C H. </s> <s xml:id="echoid-s1996" xml:space="preserve">Quia enim D C <lb/>tangit evolutam in D, apparet ipſam referre poſitionem fili <lb/>tunc cum ejus extremitas pervenit in C. </s> <s xml:id="echoid-s1997" xml:space="preserve">Quod ſi igitur o-<lb/>ſtenderimus filum, in tota reliqua deſcriptione curvæ A C H, <lb/>nusquam pertingere ad rectam C E præterquam in C pun-<lb/>cto, manifeſtum erit rectam C E ibidem curvam A C H <lb/>contingere.</s> <s xml:id="echoid-s1998" xml:space="preserve"/> </p> <div xml:id="echoid-div151" type="float" level="2" n="1"> <note position="left" xlink:label="note-0134-03" xlink:href="note-0134-03a" xml:space="preserve">TAB. XI. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s1999" xml:space="preserve">Sumatur punctum aliquod in A C præter C, quod ſit H, <lb/>ſitque primo remotius à principio evolutionis A quam pun-<lb/>ctum C, & </s> <s xml:id="echoid-s2000" xml:space="preserve">intelligatur pars libera eſſe H G, cum extremi-<lb/>tate ſua ad H pervenit. </s> <s xml:id="echoid-s2001" xml:space="preserve">Tangit ergo H G lineam A B in G. <lb/></s> <s xml:id="echoid-s2002" xml:space="preserve">Cumque interea dum deſcribitur pars curvæ C H, evolu-<lb/>tus ſit arcus D G, occurret C D à parte D producta ipſi <lb/>H G, ut in F. </s> <s xml:id="echoid-s2003" xml:space="preserve">Ponatur autem G H occurrere rectæ C E <lb/>in E. </s> <s xml:id="echoid-s2004" xml:space="preserve">Quia igitur duæ ſimul D F, F G, majores ſunt quam <lb/>D G, ſive curva ea fuerit ſive recta: </s> <s xml:id="echoid-s2005" xml:space="preserve">fiet addendo utrinque <lb/>rectam D C, ut rectæ C F, F G ſimul majores ſint recta <pb o="91" file="0135" n="145" rhead="HOROLOG. OSCILLATOR."/> C D & </s> <s xml:id="echoid-s2006" xml:space="preserve">ipſa D G. </s> <s xml:id="echoid-s2007" xml:space="preserve">Sed propter evolutionem, apparet utris-<lb/> <anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/> que ſimul, rectæ C D, & </s> <s xml:id="echoid-s2008" xml:space="preserve">lineæ D G, æquari rectam H G. <lb/></s> <s xml:id="echoid-s2009" xml:space="preserve">Ergo duæ ſimul C F, F G majores quoque erunt recta H G.</s> <s xml:id="echoid-s2010" xml:space="preserve"><unsure/> <lb/>& </s> <s xml:id="echoid-s2011" xml:space="preserve">ablata communi F G, erit C F major quam H F. </s> <s xml:id="echoid-s2012" xml:space="preserve">Sed <lb/>F E major eſt quam F C, quia angulus C trianguli F C E <lb/>eſt rectus. </s> <s xml:id="echoid-s2013" xml:space="preserve">Ergo F E omnino major quam F H. </s> <s xml:id="echoid-s2014" xml:space="preserve">Unde ap-<lb/>paret, ab hac quidem parte puncti C, fili extremitatem non <lb/>pertingere ad rectam C E.</s> <s xml:id="echoid-s2015" xml:space="preserve"/> </p> <div xml:id="echoid-div152" type="float" level="2" n="2"> <note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2016" xml:space="preserve">Sit jam punctum H propinquius principio evolutionis A <lb/> <anchor type="note" xlink:label="note-0135-02a" xlink:href="note-0135-02"/> quam punctum C, ſitque fili poſitio H G, tunc cum ejus <lb/>extremitas eſſet in H, & </s> <s xml:id="echoid-s2017" xml:space="preserve">ducantur rectæ D G, D H, qua-<lb/>rum hæc occurrat rectæ C E in E: </s> <s xml:id="echoid-s2018" xml:space="preserve">apparet autem D G re-<lb/>ctam non poſſe eſſe in directum ipſi H G, adeoque H G D <lb/>fore triangulum. </s> <s xml:id="echoid-s2019" xml:space="preserve">Jam quia recta D G vel minor eſt quam <lb/>D K G, vel eadem, ſi nempe evolutæ pars D G recta ſit; <lb/></s> <s xml:id="echoid-s2020" xml:space="preserve">additâ utrique G H, erunt rectæ D G, G H ſimul mino-<lb/>res vel æquales duabus iſtis, ſcilicet D K G & </s> <s xml:id="echoid-s2021" xml:space="preserve">G H, ſive <lb/>his æquali rectæ D C. </s> <s xml:id="echoid-s2022" xml:space="preserve">Duabus autem rectis D G, G H mi-<lb/>nor eſt recta D H. </s> <s xml:id="echoid-s2023" xml:space="preserve">Ergo hæc minor utique erit rectâ D C. </s> <s xml:id="echoid-s2024" xml:space="preserve"><lb/>Sed D E major eſt quam D C, quia in triangulo D C E <lb/>angulus C eſt rectus. </s> <s xml:id="echoid-s2025" xml:space="preserve">Ergo D H multo minor quam D E. </s> <s xml:id="echoid-s2026" xml:space="preserve"><lb/>Situm eſt ergo punctum H, hoc eſt extremitas fili G H, in-<lb/>tra angulum D C E. </s> <s xml:id="echoid-s2027" xml:space="preserve">Unde apparet neque inter A & </s> <s xml:id="echoid-s2028" xml:space="preserve">C us-<lb/>quam illam pertingere ad rectam C E. </s> <s xml:id="echoid-s2029" xml:space="preserve">Ergo C E tangit <lb/>curvam A C in C; </s> <s xml:id="echoid-s2030" xml:space="preserve">ac proinde D C, cui C E ducta eſt <lb/>perpendicularis, occurrit curvæ ad angulos rectos. </s> <s xml:id="echoid-s2031" xml:space="preserve">quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s2032" xml:space="preserve"/> </p> <div xml:id="echoid-div153" type="float" level="2" n="3"> <note position="right" xlink:label="note-0135-02" xlink:href="note-0135-02a" xml:space="preserve">TAB. XII. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s2033" xml:space="preserve">Hinc etiam manifeſtum eſt curvam A H C in partem u-<lb/>nam inflexam eſſe, & </s> <s xml:id="echoid-s2034" xml:space="preserve">in eandem partem cavam ac ipſa A G B, <lb/>cujus evolutione deſcripta eſt. </s> <s xml:id="echoid-s2035" xml:space="preserve">Omnes enim tangentes lineæ <lb/>A H C, cadunt extra ſpatium D G A H C: </s> <s xml:id="echoid-s2036" xml:space="preserve">omnes vero <lb/>tangentes lineæ A G D, intra dictum ſpatium. </s> <s xml:id="echoid-s2037" xml:space="preserve">unde liquet <lb/>cavitatem A H C reſpicere convexitatem A G D.</s> <s xml:id="echoid-s2038" xml:space="preserve"/> </p> <pb o="92" file="0136" n="146" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div155" type="section" level="1" n="57"> <head xml:id="echoid-head81" xml:space="preserve">PROPOSITIO II.</head> <note position="left" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph> <lb/>TAB. XII. <lb/>Fig. 2.</note> <p style="it"> <s xml:id="echoid-s2039" xml:space="preserve">OMnis curva linea terminata, in unam partem <lb/>cava, ut A B D, ut poteſt in tot partes dividi, ut <lb/>ſi ſingulis partibus ſubtenſæ rectæ ducantur, velut <lb/>A B, B C, C D; </s> <s xml:id="echoid-s2040" xml:space="preserve">& </s> <s xml:id="echoid-s2041" xml:space="preserve">à ſingulis item diviſionis <lb/>punctis, ipſaque curvæ extremitate rectæ ducan-<lb/>tur curvam tangentes, ut A N, B O, C P, quæ <lb/>occurrant iis, quæ in proxime ſequentibus diviſionis <lb/>punctis curvæ ad angulos rectos inſiſtunt, quales <lb/>ſunt lineæ B N, C O, D P; </s> <s xml:id="echoid-s2042" xml:space="preserve">ut inquam ſubtenſa <lb/>quæque habeat ad ſibi adjacentem curvæ perpendi-<lb/>cularem, velut A B ad B N, B C ad C O, C D <lb/>ad D P, rationem majorem quavis ratione propo-<lb/>ſita.</s> <s xml:id="echoid-s2043" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2044" xml:space="preserve">Sit enim data ratio lineæ E F ad F G, quæ recto angulo <lb/>ad F jungantur, & </s> <s xml:id="echoid-s2045" xml:space="preserve">ducatur recta G E H.</s> <s xml:id="echoid-s2046" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2047" xml:space="preserve">Intelligatur primo curva A B D in partes tam exiguas ſe-<lb/>cta punctis B, C, ut tangentes quæ ad bina quæque inter <lb/>ſe proxima puncta curvam contingunt, occurrant ſibi mutuo <lb/>ſecundum angulos qui ſinguli majores ſint angulo F E H; <lb/></s> <s xml:id="echoid-s2048" xml:space="preserve">quales ſunt anguli A K B, B L C, C M D. </s> <s xml:id="echoid-s2049" xml:space="preserve">quod quidem <lb/>fieri poſſe evidentius eſt quam ut demonſtratione indigeat. </s> <s xml:id="echoid-s2050" xml:space="preserve"><lb/>Ductis jam ſubtenſis A B, B C, C D, & </s> <s xml:id="echoid-s2051" xml:space="preserve">erectis curvæ <lb/>perpendicularibus B N, C O, D P, quæ occurrant pro-<lb/>ductis A K, B L, C M, in N, O, P: </s> <s xml:id="echoid-s2052" xml:space="preserve">dico rationes ſin-<lb/>gulas rectarum, A B ad B N, B C ad C O, C D ad D P, <lb/>majores eſſe ratione E F ad F G.</s> <s xml:id="echoid-s2053" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2054" xml:space="preserve">Quia enim angulus A K B major eſt angulo H E F, erit <lb/>reſiduus illius ad duos rectos, nimirum angulus N K B, <lb/>minor angulo G E F. </s> <s xml:id="echoid-s2055" xml:space="preserve">Angulus autem B trianguli K B N <lb/>eſt rectus, ſicut & </s> <s xml:id="echoid-s2056" xml:space="preserve">angulus F in triangulo E F G. </s> <s xml:id="echoid-s2057" xml:space="preserve">Ergo <pb file="0137" n="147"/> <pb file="0137a" n="148"/> <anchor type="figure" xlink:label="fig-0137a-01a" xlink:href="fig-0137a-01"/> <anchor type="figure" xlink:label="fig-0137a-02a" xlink:href="fig-0137a-02"/> <anchor type="figure" xlink:label="fig-0137a-03a" xlink:href="fig-0137a-03"/> <pb file="0138" n="149"/> <pb o="93" file="0139" n="150" rhead="HOROLOG. OSCILLATOR."/> major erit ratio K B ad B N quam E F ad F G. </s> <s xml:id="echoid-s2058" xml:space="preserve">Sed A B <lb/> <anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/> major eſt quam K B, quoniam angulus K in triangulo A K B <lb/>eſt obtuſus, eſt enim major angulo H E F qui eſt obtuſus <lb/>ex conſtructione. </s> <s xml:id="echoid-s2059" xml:space="preserve">Ergo ratio A B ad B N major erit ratio-<lb/>ne K B ad B N, ac proinde omnino major ratione E F ad <lb/>F G. </s> <s xml:id="echoid-s2060" xml:space="preserve">Eodem modo & </s> <s xml:id="echoid-s2061" xml:space="preserve">ratio B C ad C O, & </s> <s xml:id="echoid-s2062" xml:space="preserve">C D ad D P, <lb/>major oſtendetur ratione E F ad F G. </s> <s xml:id="echoid-s2063" xml:space="preserve">Itaque conſtat pro-<lb/>poſitum.</s> <s xml:id="echoid-s2064" xml:space="preserve"/> </p> <div xml:id="echoid-div155" type="float" level="2" n="1"> <figure xlink:label="fig-0137a-01" xlink:href="fig-0137a-01a"> <caption xml:id="echoid-caption37" style="it" xml:space="preserve">Pag. 92.<lb/>TAB. XI<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables37" xml:space="preserve">D C F E B L H I K A G</variables> </figure> <figure xlink:label="fig-0137a-02" xlink:href="fig-0137a-02a"> <caption xml:id="echoid-caption38" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables38" xml:space="preserve">E D A B C</variables> </figure> <figure xlink:label="fig-0137a-03" xlink:href="fig-0137a-03a"> <caption xml:id="echoid-caption39" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables39" xml:space="preserve">E H C A D F G B</variables> </figure> <note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph></note> </div> </div> <div xml:id="echoid-div157" type="section" level="1" n="58"> <head xml:id="echoid-head82" xml:space="preserve">PROPOSITIO III.</head> <p style="it"> <s xml:id="echoid-s2065" xml:space="preserve">DUæ curvæ in unam partem inflexæ & </s> <s xml:id="echoid-s2066" xml:space="preserve">in eas-<lb/>dem partes cavæ ex eodem puncto egredi ne-<lb/>queunt, ita ad ſe invicem comparatæ, ut recta <lb/>omnis quæ alteri earum ad angulos rectos occurrit, <lb/>ſimiliter occurrat & </s> <s xml:id="echoid-s2067" xml:space="preserve">reliquæ.</s> <s xml:id="echoid-s2068" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2069" xml:space="preserve">Sint enim, ſi fieri poteſt, hujuſmodi lineæ curvæ A C E, <lb/> <anchor type="note" xlink:label="note-0139-02a" xlink:href="note-0139-02"/> A G K, communem terminum habentes A, & </s> <s xml:id="echoid-s2070" xml:space="preserve">ſumpto in ex-<lb/>teriore illarum puncto quolibet K, ſit inde educta K E recta, <lb/>curvæ A G K occurrens ad angulos rectos, ac proinde <lb/>etiam curvæ A C E.</s> <s xml:id="echoid-s2071" xml:space="preserve"/> </p> <div xml:id="echoid-div157" type="float" level="2" n="1"> <note position="right" xlink:label="note-0139-02" xlink:href="note-0139-02a" xml:space="preserve">TAB. XII. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s2072" xml:space="preserve">Poteſt jam recta quædam ſumi major curva K G A, quæ <lb/>ſit Q. </s> <s xml:id="echoid-s2073" xml:space="preserve">Diviſa autem intelligatur ipſa K G A, ut in propo-<lb/>ſitione antecedenti dictum fuit, in tot partes punctis H G F, <lb/>ut ſubtenſæ ſingulæ K H, H G, G F, F A, ad perpen-<lb/>diculares curvæ ſibi contiguas H M, G N, F O, A P <lb/>majorem rationem habeant quam linea Q ad rectam K E. <lb/></s> <s xml:id="echoid-s2074" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s2075" xml:space="preserve">omnes ſimul dictæ ſubtenſæ ad omnes dictas per-<lb/>pendiculares majorem habebunt rationem quam Q ad K E. </s> <s xml:id="echoid-s2076" xml:space="preserve"><lb/>Producantur autem perpendiculares eædem & </s> <s xml:id="echoid-s2077" xml:space="preserve">occurrant cur-<lb/>væ A C E in D, C, B, nimirum ad angulos rectos ex <lb/>hypotheſi. </s> <s xml:id="echoid-s2078" xml:space="preserve">Erit jam K E minor quam M D. </s> <s xml:id="echoid-s2079" xml:space="preserve">Etenim, ducta <lb/>E L ipſi K E perpendiculari, quoniam K E occurrit lineæ <lb/>curvæ E C A ad angulos rectos, tanget E L curvam A C E, <lb/>occurretque neceſſario rectæ M D inter D & </s> <s xml:id="echoid-s2080" xml:space="preserve">M. </s> <s xml:id="echoid-s2081" xml:space="preserve">Unde <pb o="94" file="0140" n="151" rhead="CHRISTIANI HUGENII"/> cum K E ſit breviſſima omnium quæ cadunt inter parallelas <lb/> <anchor type="note" xlink:label="note-0140-01a" xlink:href="note-0140-01"/> E L, K M, erit ea minor quam M L, ac proinde minor <lb/>quoque omnino quam M D. </s> <s xml:id="echoid-s2082" xml:space="preserve">Eodem modo & </s> <s xml:id="echoid-s2083" xml:space="preserve">H D minor <lb/>oſtendetur quam N C, & </s> <s xml:id="echoid-s2084" xml:space="preserve">G C minor quam O B, & </s> <s xml:id="echoid-s2085" xml:space="preserve">F B <lb/>minor quam P A. </s> <s xml:id="echoid-s2086" xml:space="preserve">Cum ſit ergo P A major quam F B, erunt <lb/>duæ ſimul P A, O F majores quam O B. </s> <s xml:id="echoid-s2087" xml:space="preserve">Item quum O B <lb/>ſit major quam G C, erunt duæ ſimul O B, N G, majo-<lb/>res quam N C. </s> <s xml:id="echoid-s2088" xml:space="preserve">Sed duæ P A, O F majores erant quam O B. <lb/></s> <s xml:id="echoid-s2089" xml:space="preserve">Itaque tres ſimul P A, O F, N G omnino majores erunt <lb/>quam N C. </s> <s xml:id="echoid-s2090" xml:space="preserve">Rurſus, quia N C major quam H D, erunt duæ <lb/>ſimul N C, M H majores quam M D. </s> <s xml:id="echoid-s2091" xml:space="preserve">Unde, ſi loco N C <lb/>ſumantur tres hæ ipſa majores P A, O F, N G, erunt omni-<lb/>no hæ quatuor P A, O F, N G, M H majores quam M D: </s> <s xml:id="echoid-s2092" xml:space="preserve"><lb/>ac proinde eædem quoque omnino majores recta K E, quia <lb/>ipſa M D major erat quam K E. </s> <s xml:id="echoid-s2093" xml:space="preserve">Diximus autem ſubtenſas <lb/>omnes A F, F G, G H, H K majorem rationem habere ad <lb/>omnes perpendiculares P A, O F, N G, M H, quam linea <lb/>Q ad K E. </s> <s xml:id="echoid-s2094" xml:space="preserve">Ergo cum dictis perpendicularibus minor etiam <lb/>ſit K E, habebunt dictæ ſubtenſæ ad K E omnino majorem <lb/>rationem quam Q ad K E. </s> <s xml:id="echoid-s2095" xml:space="preserve">Ergo ſubtenſæ ſimul ſumptæ <lb/>majores erunt rectâ Q. </s> <s xml:id="echoid-s2096" xml:space="preserve">Hæc autem ipſa curvâ A G K major <lb/>ſumpta fuit. </s> <s xml:id="echoid-s2097" xml:space="preserve">Ergo ſubtenſæ A F, F G, G H, H K ſimul <lb/>majores erunt curva A G K cujus partibus ſubtenduntur; </s> <s xml:id="echoid-s2098" xml:space="preserve"><lb/>quod eſt abſurdum, cum ſingulæ ſuis arcubus ſint minores. </s> <s xml:id="echoid-s2099" xml:space="preserve"><lb/>Non igitur poterunt eſſe duæ curvæ lineæ quæ quemadmo-<lb/>dum dictum fuit ſeſe habeant. </s> <s xml:id="echoid-s2100" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s2101" xml:space="preserve"/> </p> <div xml:id="echoid-div158" type="float" level="2" n="2"> <note position="left" xlink:label="note-0140-01" xlink:href="note-0140-01a" xml:space="preserve"><emph style="sc">De linea</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO</emph> <lb/><emph style="sc">NE</emph>.</note> </div> </div> <div xml:id="echoid-div160" type="section" level="1" n="59"> <head xml:id="echoid-head83" xml:space="preserve">PROPOSITIO IV.</head> <p style="it"> <s xml:id="echoid-s2102" xml:space="preserve">SI ab eodem puncto duæ lineæ exeant in partem <lb/>unam inflexæ, & </s> <s xml:id="echoid-s2103" xml:space="preserve">in eandem partem cavæ, ita <lb/>vero mutuo comparatæ ut rectæ omnes, quæ alte-<lb/>ram earum contingunt, alteri occurrant ad angu-<lb/>los rectos; </s> <s xml:id="echoid-s2104" xml:space="preserve">poſterior hæc prioris evolutione, à pun-<lb/>cto communi cœpta, deſcribetur.</s> <s xml:id="echoid-s2105" xml:space="preserve"/> </p> <pb o="95" file="0141" n="152" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s2106" xml:space="preserve">Sunto lineæ A B C, A D E, in partem unam inflexæ, <lb/> <anchor type="note" xlink:label="note-0141-01a" xlink:href="note-0141-01"/> & </s> <s xml:id="echoid-s2107" xml:space="preserve">quarum uttraque in eaſdem partes cava exiſtat, habeant-<lb/>que communem terminum A punctum. </s> <s xml:id="echoid-s2108" xml:space="preserve">Omnes autem rectæ <lb/>tangentes lineam A B C, velut B D, C E, occurrant <lb/> <anchor type="note" xlink:label="note-0141-02a" xlink:href="note-0141-02"/> lineæ A D E ad angulos rectos. </s> <s xml:id="echoid-s2109" xml:space="preserve">Dico evolutione ipſius <lb/>A B C, à termino A incepta, deſcribi A D E.</s> <s xml:id="echoid-s2110" xml:space="preserve"/> </p> <div xml:id="echoid-div160" type="float" level="2" n="1"> <note position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph></note> <note position="right" xlink:label="note-0141-02" xlink:href="note-0141-02a" xml:space="preserve">TAB. XII. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s2111" xml:space="preserve">Si enim fieri poteſt, deſcribatur dicta evolutione alia <lb/>quædam curva A F G. </s> <s xml:id="echoid-s2112" xml:space="preserve">Ergo lineæ rectæ quælibet, evolu-<lb/>tam A B C tangentes, ut B D, C E, occurrent ipſi A F G <lb/>ad angulos rectos<anchor type="note" xlink:href="" symbol="*"/>, puta in F & </s> <s xml:id="echoid-s2113" xml:space="preserve">G. </s> <s xml:id="echoid-s2114" xml:space="preserve">Sed eædem tangentes <anchor type="note" xlink:label="note-0141-03a" xlink:href="note-0141-03"/> etiam ad rectos angulos occurrere ponuntur lineæ A D E. <lb/></s> <s xml:id="echoid-s2115" xml:space="preserve">Sunt igitur lineæ curvæ A D E, A F G, eodem puncto <lb/>A terminatæ, inque partem unam flexæ, & </s> <s xml:id="echoid-s2116" xml:space="preserve">ambæ in ean-<lb/>dem partem cavæ, quippe utraque in eandem atque ipſa <lb/>A B C; </s> <s xml:id="echoid-s2117" xml:space="preserve">nam de linea A D E conſtat ex hypotheſi, de <lb/>A F G vero ex propoſitione prima hujus; </s> <s xml:id="echoid-s2118" xml:space="preserve">& </s> <s xml:id="echoid-s2119" xml:space="preserve">omnes quæ <lb/>uni earum occurrunt ad angulos rectos, etiam alteri ſimili-<lb/>ter occurrunt. </s> <s xml:id="echoid-s2120" xml:space="preserve">quod quidem fieri non poſſe antea oſtenſum <lb/>eſt<anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2121" xml:space="preserve">Quare conſtat ipſam A D E deſcriptum iri evolutione <anchor type="note" xlink:label="note-0141-04a" xlink:href="note-0141-04"/> lineæ A B C. </s> <s xml:id="echoid-s2122" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s2123" xml:space="preserve"/> </p> <div xml:id="echoid-div161" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0141-03" xlink:href="note-0141-03a" xml:space="preserve">Prop. 1. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0141-04" xlink:href="note-0141-04a" xml:space="preserve">Prop. 3. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div163" type="section" level="1" n="60"> <head xml:id="echoid-head84" xml:space="preserve">PROPOSITIO V.</head> <p style="it"> <s xml:id="echoid-s2124" xml:space="preserve">SI Cycloidem recta linea in vertice contingat, ſu-<lb/>per qua, tanquam baſi, alia cyclois priori ſimi-<lb/>lis & </s> <s xml:id="echoid-s2125" xml:space="preserve">æqualis conſtituatur, initium ſumens à pun-<lb/>cto dicti verticis; </s> <s xml:id="echoid-s2126" xml:space="preserve">recta quælibet inferiorem cycloi-<lb/>dem tangens, occurret ſuperioris portioni, ſibi ſu-<lb/>perpoſitæ, ad angulos rectos.</s> <s xml:id="echoid-s2127" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2128" xml:space="preserve">Tangat cycloidem A B C in vertice A recta A G, ſuper <lb/> <anchor type="note" xlink:label="note-0141-05a" xlink:href="note-0141-05"/> qua, tanquam baſi, ſimilis alia cyclois conſtituta ſit A E F, <lb/>cujus vertex F. </s> <s xml:id="echoid-s2129" xml:space="preserve">Cycloidem autem A B C tangat recta B K <lb/>in B. </s> <s xml:id="echoid-s2130" xml:space="preserve">Dico eam productam occurrere cycloidi A E F ad an-<lb/>gulos rectos.</s> <s xml:id="echoid-s2131" xml:space="preserve"/> </p> <div xml:id="echoid-div163" type="float" level="2" n="1"> <note position="right" xlink:label="note-0141-05" xlink:href="note-0141-05a" xml:space="preserve">TAB. XVI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s2132" xml:space="preserve">Deſcribatur enim circa A D, axem cycloidis A B C, cir- <pb o="96" file="0142" n="153" rhead="CHRISTIANI HUGENII"/> culus genitor A H D, cui occurrat B H, baſi parallela, in <lb/> <anchor type="note" xlink:label="note-0142-01a" xlink:href="note-0142-01"/> H, & </s> <s xml:id="echoid-s2133" xml:space="preserve">jungatur H A. </s> <s xml:id="echoid-s2134" xml:space="preserve">Quia ergo B K tangit cycloidem in B, <lb/>conſtat eam parallelam eſſe rectæ H A <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2135" xml:space="preserve">Itaque A H B K parallelogrammum eſt, ac proinde A K æqualis H B, hoc <lb/> <anchor type="note" xlink:label="note-0142-02a" xlink:href="note-0142-02"/> eſt, arcui A H<anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2136" xml:space="preserve">Sit porro jam deſcriptus circulus K M, <anchor type="note" xlink:label="note-0142-03a" xlink:href="note-0142-03"/> genitori circulo, hoc eſt ipſi A H D, æqualis, qui tangat <lb/>baſin A G in K, rectam vero B K productam ſecet in pun-<lb/>cto E. </s> <s xml:id="echoid-s2137" xml:space="preserve">Quia ergo ipſi A H parallela eſt B K E, ac proin-<lb/>de angulus E K A æqualis K A H, manifeſtum eſt B K <lb/>productam abſcindere à circulo K M arcum æqualem ei <lb/>quem à circulo A H D abſcindit recta A H. </s> <s xml:id="echoid-s2138" xml:space="preserve">Itaque arcus <lb/>K E æqualis eſt arcui A H, hoc eſt rectæ H B, hoc eſt <lb/>rectæ K A. </s> <s xml:id="echoid-s2139" xml:space="preserve">Hinc vero ſequitur, ex cycloidis proprietate, <lb/>cum circulus genitor M K tangebat regulam in K, punctum <lb/>deſcribens fuiſſe in E. </s> <s xml:id="echoid-s2140" xml:space="preserve">Itaque recta K E occurrit cycloidi in <lb/>E ad angulos rectos <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2141" xml:space="preserve">Eſt autem K E ipſa B K producta.</s> <s xml:id="echoid-s2142" xml:space="preserve"> <anchor type="note" xlink:label="note-0142-04a" xlink:href="note-0142-04"/> Ergo patet productam B K occurrere cycloidiad angulos re-<lb/>ctos. </s> <s xml:id="echoid-s2143" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s2144" xml:space="preserve"/> </p> <div xml:id="echoid-div164" type="float" level="2" n="2"> <note position="left" xlink:label="note-0142-01" xlink:href="note-0142-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph></note> <note symbol="*" position="left" xlink:label="note-0142-02" xlink:href="note-0142-02a" xml:space="preserve">Propoſ. 15. <lb/>partis 2.</note> <note symbol="*" position="left" xlink:label="note-0142-03" xlink:href="note-0142-03a" xml:space="preserve">Propoſ. 14. <lb/>partis 2.</note> <note symbol="*" position="left" xlink:label="note-0142-04" xlink:href="note-0142-04a" xml:space="preserve">Propoſ. 15. <lb/>partis 2.</note> </div> </div> <div xml:id="echoid-div166" type="section" level="1" n="61"> <head xml:id="echoid-head85" xml:space="preserve">PROPOSITIO VI.</head> <p style="it"> <s xml:id="echoid-s2145" xml:space="preserve">SEmicycloidis evolutione, à vertice cœpta, alia <lb/>ſemicyclois deſcribitur evolutæ æqualis & </s> <s xml:id="echoid-s2146" xml:space="preserve">ſimi-<lb/>lis, cujus baſis eſt in ea recta quæ cycloidem evolu-<lb/>tam in vertice contingit.</s> <s xml:id="echoid-s2147" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2148" xml:space="preserve">Sit ſemicyclois A B C, cui ſuperimpoſita ſit alia ſimilis <lb/> <anchor type="note" xlink:label="note-0142-05a" xlink:href="note-0142-05"/> A E F, quemadmodum in propoſitione præcedenti. </s> <s xml:id="echoid-s2149" xml:space="preserve">Dico, <lb/>ſi linea flexilis, circa ſemicycloidem A B C applicata, evol-<lb/>vatur, incipiendo ab A, eam deſcribere extremitate ſua i-<lb/>pſam ſemicycloidem A E F. </s> <s xml:id="echoid-s2150" xml:space="preserve">Quia enim ex puncto A egredi-<lb/>untur ſemicycloides A B C, A E F, in unam partem in-<lb/>flexæ, & </s> <s xml:id="echoid-s2151" xml:space="preserve">ambæ in eandem cavæ, ac præterea ita comparatæ, <lb/>ut omnes tangentes ſemicycloidis A B C occurrant ſemicy-<lb/>cloidi A E F ad angulos rectos, ſequitur hanc evolutione <lb/>illius, à termino A incepta, deſcribi <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2152" xml:space="preserve">quod erat demon- <anchor type="note" xlink:label="note-0142-06a" xlink:href="note-0142-06"/> ſtrandum.</s> <s xml:id="echoid-s2153" xml:space="preserve"/> </p> <div xml:id="echoid-div166" type="float" level="2" n="1"> <note position="left" xlink:label="note-0142-05" xlink:href="note-0142-05a" xml:space="preserve">TAB. XVI. <lb/>Fig. 1.</note> <note symbol="*" position="left" xlink:label="note-0142-06" xlink:href="note-0142-06a" xml:space="preserve">Propoſ. 4. <lb/>huj.</note> </div> <pb file="0143" n="154"/> <pb file="0143a" n="155"/> <figure> <caption xml:id="echoid-caption40" style="it" xml:space="preserve">Pag. 96.<lb/>TAB. XII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables40" xml:space="preserve">C E H A G K D B</variables> </figure> <figure> <caption xml:id="echoid-caption41" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables41" xml:space="preserve">N O L K B C M P G D A E F H</variables> </figure> <figure> <caption xml:id="echoid-caption42" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables42" xml:space="preserve">N M H G K O F L C D B E P A Q</variables> </figure> <figure> <caption xml:id="echoid-caption43" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables43" xml:space="preserve">A D F E G B C</variables> </figure> <pb file="0144" n="156"/> <pb o="97" file="0145" n="157" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s2154" xml:space="preserve">Et apparet, ſi dimidiam cycloidem, ipſi A B C gemel-<lb/> <anchor type="note" xlink:label="note-0145-01a" xlink:href="note-0145-01"/> lam, contrario ſitu ab altera parte lineæ C G diſponamus, <lb/>velut C N, ejus evolutione, vel etiam dum filum, jam <lb/>extenſum in C F, circa eam replicatur, alteram ſemicy-<lb/>cloidem F N fili extremitate deſcriptum iri, quæ ſimul <lb/>cum priore A E F integram conſtituat.</s> <s xml:id="echoid-s2155" xml:space="preserve"/> </p> <div xml:id="echoid-div167" type="float" level="2" n="2"> <note position="right" xlink:label="note-0145-01" xlink:href="note-0145-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph></note> </div> <p> <s xml:id="echoid-s2156" xml:space="preserve">Atque ex his, & </s> <s xml:id="echoid-s2157" xml:space="preserve">propoſitione 25 de deſcenſu gravium, ma-<lb/>nifeſtum jam eſt quod ſupra in Conſtructione Horologii de <lb/>æquabili penduli motu dictum fuit. </s> <s xml:id="echoid-s2158" xml:space="preserve">Patet enim perpendicu-<lb/>lum, inter laminas binas, ſecundum ſemicycloidem inflexas, <lb/>ſuſpenſum agitatumque, motu ſuo cycloidis arcum deſcri-<lb/>bere, ac proinde æqualibus temporibus quaſlibet ejus reci-<lb/>procationes abſolvi. </s> <s xml:id="echoid-s2159" xml:space="preserve">Non refert enim utrum in ſuperficie, <lb/>ſecundum cycloidem curvata, mobile feratur, an filo alliga-<lb/>tum lineam ipſam in aëre percurrat, cum utrobique eandem <lb/>libertatem, eandemque in omnibus curvæ punctis inclina-<lb/>tionem ad motum habeat.</s> <s xml:id="echoid-s2160" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div169" type="section" level="1" n="62"> <head xml:id="echoid-head86" xml:space="preserve">PROPOSITIO VII.</head> <p style="it"> <s xml:id="echoid-s2161" xml:space="preserve">Cyclois linea ſui axis, ſive diametri circuli ge-<lb/>nitoris, quadrupla eſt.</s> <s xml:id="echoid-s2162" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2163" xml:space="preserve">Repetita enim figura præcedenti: </s> <s xml:id="echoid-s2164" xml:space="preserve">cum poſt totam ſemi-<lb/> <anchor type="note" xlink:label="note-0145-02a" xlink:href="note-0145-02"/> cycloidem A B C evolutam, filum occupet rectam C F, <lb/>quæ dupla eſt A D, proptarea quod axes cycloidum A B C, <lb/>A E F ſunt æquales; </s> <s xml:id="echoid-s2165" xml:space="preserve">apparet ſemicycloidem ipſam A B C, <lb/>filo ſibi circum applicito æqualem, duplam eſſe ſui axis A D, <lb/>ac totam proinde cycloidem axis ſui quadruplam.</s> <s xml:id="echoid-s2166" xml:space="preserve"/> </p> <div xml:id="echoid-div169" type="float" level="2" n="1"> <note position="right" xlink:label="note-0145-02" xlink:href="note-0145-02a" xml:space="preserve">TAB. XVI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s2167" xml:space="preserve">Apparet etiam tangentem B E, quæ refert partem fili ex-<lb/>tenſam, antea curvæ parti B A applicatam, huic ipſi longi-<lb/>tudine æquari. </s> <s xml:id="echoid-s2168" xml:space="preserve">Eſt autem B E dupla ipſius B K, ſive A H, <lb/>quoniam in propoſitione quinta oſtenſum eſt K E ipſi A H <lb/>æqualem eſſe. </s> <s xml:id="echoid-s2169" xml:space="preserve">Itaque pars cycloidis A B rectæ A H, ſive <lb/>B K, dupla erit: </s> <s xml:id="echoid-s2170" xml:space="preserve">exiſtente nimirum B H parallela baſi <lb/>cycloidis: </s> <s xml:id="echoid-s2171" xml:space="preserve">idque ubicunque in ea punctum B ſumptum fue-<lb/>rit.</s> <s xml:id="echoid-s2172" xml:space="preserve"/> </p> <pb o="98" file="0146" n="158" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s2173" xml:space="preserve">Hanc cycloidis dimenſionem primus invenit, via tamen <lb/> <anchor type="note" xlink:label="note-0146-01a" xlink:href="note-0146-01"/> longe alia, eximius geometra Chriſtophorus Wren Anglus, <lb/>eamque deinde eleganti demonſtratione confirmavit, quæ <lb/>edita eſt in libro de cycloide viri clariſſimi Ioannis Walliſij. <lb/></s> <s xml:id="echoid-s2174" xml:space="preserve">De eadem vero linea, alia quoque multa extant pulcherrima <lb/>inventa noſtri temporis mathematicorum, quibus præcipuè <lb/>occaſionem præbuere problemata quædam à Blaſio Paſchalio <lb/>Gallo propoſita, qui in his ſtudiis præcellebat. </s> <s xml:id="echoid-s2175" xml:space="preserve">Is cum ſua, <lb/>tum aliorum inventa recenſens, primum omnium Merſennum <lb/>lineam hanc in rerum natura advertiſſe ait. </s> <s xml:id="echoid-s2176" xml:space="preserve">Primum Roberval-<lb/>lium tangentes ejus definiviſſe, ac plana & </s> <s xml:id="echoid-s2177" xml:space="preserve">ſolida dimenſum eſſe. </s> <s xml:id="echoid-s2178" xml:space="preserve"><lb/>Item centra gravitatis tum plani, tum partium ejus inveniſſe. </s> <s xml:id="echoid-s2179" xml:space="preserve"><lb/>Primum Wrennium curvæ cycloidis æqualem rectam dediſ-<lb/>ſe. </s> <s xml:id="echoid-s2180" xml:space="preserve">Me quoque primum reperiſſe dimenſionem abſolutam por-<lb/>tionis cycloidis, quæ rectâ, baſi parallelâ, abſcinditur per <lb/>punctum axis, quod quarta parte ejus à vertice abeſt. </s> <s xml:id="echoid-s2181" xml:space="preserve">quæ <lb/>nimirum portio æquatur dimidio hexagono æquilatero, intra <lb/>circulum genitorem deſcripto. </s> <s xml:id="echoid-s2182" xml:space="preserve">Seipſum denique ſolidorum <lb/>ac ſemiſolidorum, tam circa baſin quàm circa axem, centra <lb/>gravitatis definiviſſe, itemque partium eorum. </s> <s xml:id="echoid-s2183" xml:space="preserve">Lineæ etiam <lb/>ipſius (Sed hæc poſt acceptam à Wrennio dimenſionem) <lb/>centrum gravitatis inveniſſe, & </s> <s xml:id="echoid-s2184" xml:space="preserve">dimenſionem ſuperficierum <lb/>convexarum, quibus ſolida iſta eorumque partes comprehen-<lb/>duntur; </s> <s xml:id="echoid-s2185" xml:space="preserve">earumque ſuperficierum centra gravitatis. </s> <s xml:id="echoid-s2186" xml:space="preserve">Ac denique <lb/>dimenſionem curvarum cujuſvis cycloidis, tam protractæ quam <lb/>contractæ: </s> <s xml:id="echoid-s2187" xml:space="preserve">hoc eſt earum quæ deſcribuntur à puncto intra <lb/>vel extra circumferentiam circuli genitoris ſumpto. </s> <s xml:id="echoid-s2188" xml:space="preserve">Et ho-<lb/>rum quidem demonſtrationes à Paſchalio ſunt editæ. </s> <s xml:id="echoid-s2189" xml:space="preserve">A qui-<lb/>bus ſuas quoque, de eadem linea, ſubtiliſſimas meditationes <lb/>expoſuit Cl. </s> <s xml:id="echoid-s2190" xml:space="preserve">Walliſius, atque eadem illa omnia ſuo Marte <lb/>ſe reperiſſe, ac problemata à Paſchalio propoſita ſolviſſe con-<lb/>tendit. </s> <s xml:id="echoid-s2191" xml:space="preserve">Quod idem & </s> <s xml:id="echoid-s2192" xml:space="preserve">doctiſſimus Lovera ſibi vindicat. </s> <s xml:id="echoid-s2193" xml:space="preserve">Quan-<lb/>tum vero unicuique debeatur, ex ſcriptis eorum eruditi dijudi-<lb/>cent. </s> <s xml:id="echoid-s2194" xml:space="preserve">Nos propterea tantum præcedentia retulimus, quod ſi-<lb/>lentio prætereunda non videbantur egregia adeo inventa, qui-<lb/>bus factum eſt, ut, ex lineis omnibus, nulla nunc melius aut <pb o="99" file="0147" n="159" rhead="HOROLOG. OSCILLATOR."/> penitiùs quam cyclois cognita ſit. </s> <s xml:id="echoid-s2195" xml:space="preserve">Methodum vero noſtram, <lb/> <anchor type="note" xlink:label="note-0147-01a" xlink:href="note-0147-01"/> qua in hac metienda uſi ſumus, in aliis quoque experiri li-<lb/>buit, de quibus porro nunc agemus.</s> <s xml:id="echoid-s2196" xml:space="preserve"/> </p> <div xml:id="echoid-div170" type="float" level="2" n="2"> <note position="left" xlink:label="note-0146-01" xlink:href="note-0146-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> <note position="right" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUT@@-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> </div> <div xml:id="echoid-div172" type="section" level="1" n="63"> <head xml:id="echoid-head87" xml:space="preserve">PROPOSITIO VIII.</head> <p style="it"> <s xml:id="echoid-s2197" xml:space="preserve">CUjus lineæ evolutione parabola deſcribatur os-<lb/>tendere.</s> <s xml:id="echoid-s2198" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2199" xml:space="preserve">Sit paraboloides A B, cujus axis A D; </s> <s xml:id="echoid-s2200" xml:space="preserve">vertex A; </s> <s xml:id="echoid-s2201" xml:space="preserve">pro-<lb/> <anchor type="note" xlink:label="note-0147-02a" xlink:href="note-0147-02"/> prietas autem iſta, ut ordinatim ad axem applicatâ B D, <lb/>cubus abſciſſæ ad verticem D A æquetur ſolido, baſin ha-<lb/>benti quadratum D B, altitudinem vero æqualem lineæ cui-<lb/>dam datæ M; </s> <s xml:id="echoid-s2202" xml:space="preserve">quæ quidem curva pridem geometris nota <lb/>fuit; </s> <s xml:id="echoid-s2203" xml:space="preserve">& </s> <s xml:id="echoid-s2204" xml:space="preserve">ponatur axi D E juncta in directum A E, quæ ha-<lb/>beat {8/27} ipſius M. </s> <s xml:id="echoid-s2205" xml:space="preserve">Jam ſi filum continuum circa E A B ap-<lb/>plicetur, idque ab E evolvi incipiat, dico deſcriptam ex <lb/>evolutione eſſe parabolam E F, cujus axis E A G, vertex <lb/>E, latus rectum æquale duplæ E A.</s> <s xml:id="echoid-s2206" xml:space="preserve"/> </p> <div xml:id="echoid-div172" type="float" level="2" n="1"> <note position="right" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve">TAB. XIII, <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s2207" xml:space="preserve">Sumpto enim in curva A B puncto quolibet B, ducatur <lb/>quæ in ipſo tangat curvam recta B G, occurrens axi E A <lb/>in G. </s> <s xml:id="echoid-s2208" xml:space="preserve">& </s> <s xml:id="echoid-s2209" xml:space="preserve">ex G ducatur porro G F, quæ ad rectos angulos <lb/>occurrat parabolæ E F in F; </s> <s xml:id="echoid-s2210" xml:space="preserve">& </s> <s xml:id="echoid-s2211" xml:space="preserve">ſit ipſi G F perpendicula-<lb/>ris F H, quæ parabolam in F continget; </s> <s xml:id="echoid-s2212" xml:space="preserve">& </s> <s xml:id="echoid-s2213" xml:space="preserve">denique F K <lb/>ordinatim ad axem E G applicetur.</s> <s xml:id="echoid-s2214" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2215" xml:space="preserve">Eſt igitur K G æqualis dimidio lateri recto, hoc eſt, ipſi <lb/>E A; </s> <s xml:id="echoid-s2216" xml:space="preserve">ac proinde, additâ vel ablatâ utrimque A K, erit <lb/>E K æqualis A G. </s> <s xml:id="echoid-s2217" xml:space="preserve">Eſt autem A G triens ipſius A D, quo-<lb/>niam B G tangit paraboloidem in B: </s> <s xml:id="echoid-s2218" xml:space="preserve">illud enim ex natura <lb/>curvæ hujus facile demonſtrari poteſt. </s> <s xml:id="echoid-s2219" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2220" xml:space="preserve">E K æqualis <lb/>eſt trienti A D: </s> <s xml:id="echoid-s2221" xml:space="preserve">& </s> <s xml:id="echoid-s2222" xml:space="preserve">K H, quæ ex natura parabolæ dupla eſt <lb/>K E, æquabitur duabus tertiis A D. </s> <s xml:id="echoid-s2223" xml:space="preserve">Itaque cubus ex K H <lb/>æqualis eſt {8/27} cubi ex A D, hoc eſt, ſolido baſin habenti <lb/>quadratum D B, altitudinem vero æqualem {8/27} M, hoc eſt, <lb/>ipſi A E. </s> <s xml:id="echoid-s2224" xml:space="preserve">Quamobrem ut quadratum D B ad quadratum <lb/>K H, ita erit K H longitudine ad A E, hoc eſt ad K G. <lb/></s> <s xml:id="echoid-s2225" xml:space="preserve">Erat autem K H æqualis {@/3} A D, hoc eſt ipſi G D. </s> <s xml:id="echoid-s2226" xml:space="preserve">Ergo <pb o="100" file="0148" n="160" rhead="CHRISTIANI HUGENII"/> ut quadratum B D ad quadratum D G ita eſt H K ad K G. <lb/></s> <s xml:id="echoid-s2227" xml:space="preserve"> <anchor type="note" xlink:label="note-0148-01a" xlink:href="note-0148-01"/> Ut autem H K ad K G, ita eſt quadratum F K ad quadra-<lb/>tum K G. </s> <s xml:id="echoid-s2228" xml:space="preserve">Ergo ſicut quadratum B D ad quadratum D G, <lb/>ita quadratum F K ad quadratum K G. </s> <s xml:id="echoid-s2229" xml:space="preserve">Et proinde ſicut <lb/>B D ad D G longitudine, ita F K ad K G. </s> <s xml:id="echoid-s2230" xml:space="preserve">Unde ſequitur <lb/>B G F eſſe lineam rectam. </s> <s xml:id="echoid-s2231" xml:space="preserve">Sed G F occurrit parabolæ E F ad <lb/>angulos rectos. </s> <s xml:id="echoid-s2232" xml:space="preserve">Ergo apparet B G, tangentem paraboloidis, <lb/>productam occurrere eidem parabolæ ad angulos rectos. </s> <s xml:id="echoid-s2233" xml:space="preserve">Idque <lb/>ſimiliter de quavis illius tangente demonſtrabitur. </s> <s xml:id="echoid-s2234" xml:space="preserve">Ergo con-<lb/>ſtat ex evolutione lineæ E A B, à termino E incepta, de-<lb/>ſcribi parabolam E F <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2235" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s2236" xml:space="preserve"/> </p> <div xml:id="echoid-div173" type="float" level="2" n="2"> <note position="left" xlink:label="note-0148-01" xlink:href="note-0148-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <note symbol="*" position="left" xml:space="preserve">Propoſ. 4. <lb/>huj.</note> </div> <div xml:id="echoid-div175" type="section" level="1" n="64"> <head xml:id="echoid-head88" xml:space="preserve">PROPOSITIO IX.</head> <p style="it"> <s xml:id="echoid-s2237" xml:space="preserve">REctam lineam invenire æqualem datæ portioni <lb/>curvæ paraboloidis, ejus nempe in qua qua-<lb/>drata ordinatim applicatarum ad axem, ſunt in-<lb/>ter ſe ſicut cubi abſciſſarum ad verticem.</s> <s xml:id="echoid-s2238" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2239" xml:space="preserve">Quomodo hoc fiat ex prop. </s> <s xml:id="echoid-s2240" xml:space="preserve">præcedenti manifeſtum eſt. <lb/></s> <s xml:id="echoid-s2241" xml:space="preserve"> <anchor type="note" xlink:label="note-0148-03a" xlink:href="note-0148-03"/> Parabola vero E F ad conſtructionem non requiritur, quæ <lb/>ſic peragetur. </s> <s xml:id="echoid-s2242" xml:space="preserve">Data quavis parte paraboloidis hujus A B, cui <lb/>rectam æqualem invenire oporteat, ducatur B G tangens in <lb/>puncto B, quæ occurrat axi A G in G. </s> <s xml:id="echoid-s2243" xml:space="preserve">Tanget autem ſi <lb/>A G fuerit tertia pars A D, inter verticem & </s> <s xml:id="echoid-s2244" xml:space="preserve">ordinatim ap-<lb/>plicatam B D interceptæ. </s> <s xml:id="echoid-s2245" xml:space="preserve">Porro ſumpta A E æquali {8/27} lineæ <lb/>M, quæ latus rectum eſt paraboloidis A B, ducatur E F <lb/>parallela B G, occurratque lineæ A F, quæ parallela eſt <lb/>B D, in F. </s> <s xml:id="echoid-s2246" xml:space="preserve">Jam ſi ad rectam B G addatur N F, exceſſus <lb/>rectæ E F ſupra E A, habebitur recta æqualis curvæ A B. <lb/></s> <s xml:id="echoid-s2247" xml:space="preserve">Cujus demonſtratio ex ante dictis facile perſpicitur.</s> <s xml:id="echoid-s2248" xml:space="preserve"/> </p> <div xml:id="echoid-div175" type="float" level="2" n="1"> <note position="left" xlink:label="note-0148-03" xlink:href="note-0148-03a" xml:space="preserve">TAB. XIII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s2249" xml:space="preserve">Semper ergo curva A B tantum ſuperat tangentem B G, <lb/>quantum recta E F rectam E A.</s> <s xml:id="echoid-s2250" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2251" xml:space="preserve">Rurſus autem hic in lineam incidimus, cujus longitudi-<lb/>nem alii jam ante dimenſi ſunt. </s> <s xml:id="echoid-s2252" xml:space="preserve">Illam nempe quam anno 1659 <lb/>Joh. </s> <s xml:id="echoid-s2253" xml:space="preserve">Heuratius Harlemenſis rectæ æqualem oſtendit, cujus <lb/>demonſtratio poſt commentarios Joh. </s> <s xml:id="echoid-s2254" xml:space="preserve">Schotenii in Carteſii <pb o="101" file="0149" n="161" rhead="HOROLOG. OSCILLATOR."/> Geometriam, eodem anno editam, adjecta eſt. </s> <s xml:id="echoid-s2255" xml:space="preserve">Et ille qui-<lb/> <anchor type="note" xlink:label="note-0149-01a" xlink:href="note-0149-01"/> dem omnium primus curvam lineam, ex earum numero qua-<lb/>rum puncta quælibet geometricè definiuntur, ad hanc men-<lb/>ſuram reduxit, cum ſub idem tempus Cycloidis longitudi-<lb/>nem dediſſet Wrennius, non minus ingenioſo epicheremate.</s> <s xml:id="echoid-s2256" xml:space="preserve"/> </p> <div xml:id="echoid-div176" type="float" level="2" n="2"> <note position="right" xlink:label="note-0149-01" xlink:href="note-0149-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2257" xml:space="preserve">Scio equidem, ab edito Heuratii invento, Doctiſſimum <lb/>Walliſium Wilhelmo Nelio, nobili apud ſuos juveni, idem <lb/>attribuere voluiſſe, in libro de Ciſſoide. </s> <s xml:id="echoid-s2258" xml:space="preserve">Sed mihi, quæ il-<lb/>lic adfert perpendenti, videtur non multum quidem ab in-<lb/>vento illo Nelium abfuiſſe, neque tamen plane id adſecutum <lb/>eſſe. </s> <s xml:id="echoid-s2259" xml:space="preserve">Nam neque ex demonſtratione ejus, quam Walliſius <lb/>affert, apparet illum ſatis perſpexiſſe quænam foret curva <lb/>illa, cujus, ſi conſtrueretur, menſuram datam fore videbat. <lb/></s> <s xml:id="echoid-s2260" xml:space="preserve">Et credibile eſt, ſi ſciviſſet ex earum numero eſſe quæ jam-<lb/>pridem Geometris cognitæ fuerant, vel ipſum, vel alios ejus <lb/>nomine, tam nobile inventum Geometris maturius imperti-<lb/>turos fuiſſe, quod, ſi quod aliud, merebatur ut Archime-<lb/>deum illud εὕρη{κα} exclamarent. </s> <s xml:id="echoid-s2261" xml:space="preserve">Sane ejusdem inventi, tan-<lb/>quam à ſe profecti, etiam Fermatius, Tholoſanus ſenator <lb/>ac Geometra peritiſſimus, demonſtrationes conſcripſit, quæ <lb/>anno 1660 excuſæ ſunt; </s> <s xml:id="echoid-s2262" xml:space="preserve">ſed illæ ſero utique.</s> <s xml:id="echoid-s2263" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2264" xml:space="preserve">Cum vero in his ſimus, etiam de nobis dicere liceat, quid <lb/>ad promovendum tam eximium inventum contulerimus: </s> <s xml:id="echoid-s2265" xml:space="preserve">ſi-<lb/>quidem & </s> <s xml:id="echoid-s2266" xml:space="preserve">Heuratio ut eo perveniret occaſionem præbuimus, & </s> <s xml:id="echoid-s2267" xml:space="preserve"><lb/>dimenſionem curvæ parabolicæ ex hyperbolæ data quadratura, <lb/>quæ Heuratiani inventi pars eſt, ante ipſum atque omnium <lb/>primi reperimus. </s> <s xml:id="echoid-s2268" xml:space="preserve">Etenim ſub finem anni 1657 in hæc duo ſi-<lb/>mul incidimus, curvæ parabolicæ quam dixi dimenſionem, <lb/>& </s> <s xml:id="echoid-s2269" xml:space="preserve">ſuperficiei conoidis parabolici in circulum reductionem. <lb/></s> <s xml:id="echoid-s2270" xml:space="preserve">Cumque Schotenio, aliisque item amicorum, per literas indi-<lb/>caſſemus, duo quædam non vulgaria circa parabolam inven-<lb/>ta nobis ſeſe obtuliſſe, eorumque alterum eſſe conoidicæ ſu-<lb/>perficiei extenſionem in circulum, ille literas eas cum Heu-<lb/>ratio, quo tum familiariter utebatur, communicavit. </s> <s xml:id="echoid-s2271" xml:space="preserve">Huic <lb/>vero, acutiſſimi ingenii viro, non difficile fuit intelligere, <lb/>conoidis iſtius ſuperficiei affinem eſſe dimenſionem ipſius cur- <pb o="102" file="0150" n="162" rhead="CHRISTIANI HUGENII"/> væ parabolicæ. </s> <s xml:id="echoid-s2272" xml:space="preserve">Qua utraque inventa, ulterius inde inveſti-<lb/> <anchor type="note" xlink:label="note-0150-01a" xlink:href="note-0150-01"/> gans, in alias iſtas curvas paraboloides incidit, quibus rectæ <lb/>æquales abſolute inveniuntur.</s> <s xml:id="echoid-s2273" xml:space="preserve"/> </p> <div xml:id="echoid-div177" type="float" level="2" n="3"> <note position="left" xlink:label="note-0150-01" xlink:href="note-0150-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2274" xml:space="preserve">Ac de Conoidis quidem ſuperficie in planum redacta, ne <lb/>quis forte teſtimonium deſideret, pauca hæc adſcribere vi-<lb/>ſum eſt ex literis viri clariſſimi, atque inter præcipuos ho-<lb/>die Geometras cenſendi, Franc. </s> <s xml:id="echoid-s2275" xml:space="preserve">Sluſii, quibus eo ipſo anno <lb/>mihi inventum illud, ac prolixius forte quam pro merito, <lb/>gratulatus eſt. </s> <s xml:id="echoid-s2276" xml:space="preserve">In quibus literis 24. </s> <s xml:id="echoid-s2277" xml:space="preserve">Decemb. </s> <s xml:id="echoid-s2278" xml:space="preserve">anni 1657. </s> <s xml:id="echoid-s2279" xml:space="preserve">da-<lb/>tis, iſta habentur. </s> <s xml:id="echoid-s2280" xml:space="preserve">Duo tantum addo, unum &</s> <s xml:id="echoid-s2281" xml:space="preserve">c. </s> <s xml:id="echoid-s2282" xml:space="preserve">Alterum <lb/>eſt, me has omnes curvas, ipſumque adeo locum linearem in-<lb/>tegrum, nihili pene facere præ invento hoc tuo, quo ſuperfi-<lb/>ciei in conoide parabolico rationem ad circulum ſuæ baſeos de-<lb/>monſtraſti. </s> <s xml:id="echoid-s2283" xml:space="preserve">Hanc pro circuli quadratura pulcherrimam ἀ{πα}-<lb/>{γ@}{γὴ}ν præfero libens iis omnibus, quas ex loco lineari nec pau-<lb/>cas olim deduxi, & </s> <s xml:id="echoid-s2284" xml:space="preserve">quas tecum, ſi ita juſſeris, data occa-<lb/>ſione communicabo.</s> <s xml:id="echoid-s2285" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2286" xml:space="preserve">Anno autem inſequenti etiam ſuperficies conoidum hyper-<lb/>bolicorum & </s> <s xml:id="echoid-s2287" xml:space="preserve">ſphæroidum reperi, quomodo ad circulos re-<lb/>duci poſſent, conſtructionesque eorum problematum, non <lb/>addita tamen demonſtratione, Geometris quibuscum tunc <lb/>literarum commercium habebam, in Gallia Paſchalio aliis-<lb/>que, in Anglia Walliſio impertii, qui non multo poſt ſua <lb/>quoque ſuper his, una cum aliis multis ſubtilibus inventis <lb/>in lucem edidit, fecitque ut noſtris demonſtrationibus per-<lb/>ficiendis ſuperſederem. </s> <s xml:id="echoid-s2288" xml:space="preserve">Quoniam vero non inelegantes viſæ <lb/>ſunt conſtructiones noſtræ, neque adhuc publice extant, <lb/>placet hoc loco illas adſcribere.</s> <s xml:id="echoid-s2289" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div179" type="section" level="1" n="65"> <head xml:id="echoid-head89" style="it" xml:space="preserve">Conoidis parabolici ſuperficiei curvæ circulum <lb/>æqualem invenire.</head> <p> <s xml:id="echoid-s2290" xml:space="preserve">SIt datum conoides cujus ſectio per axem parabola A B C; <lb/></s> <s xml:id="echoid-s2291" xml:space="preserve"> <anchor type="note" xlink:label="note-0150-02a" xlink:href="note-0150-02"/> axis ejus B D, vertex B, diameter baſis A C, quæ ſit axi <lb/>B D ad angulos rectos. </s> <s xml:id="echoid-s2292" xml:space="preserve">Et oporteat ſuperficiei portionis cur-<lb/>væ invenire circulum æqualem.</s> <s xml:id="echoid-s2293" xml:space="preserve"/> </p> <div xml:id="echoid-div179" type="float" level="2" n="1"> <note position="left" xlink:label="note-0150-02" xlink:href="note-0150-02a" xml:space="preserve">TAB. XIII. <lb/>Fig. @.</note> </div> <pb o="103" file="0151" n="163" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s2294" xml:space="preserve">Producto axe à parte verticis, ſumatur B E æqualis B D, <lb/> <anchor type="note" xlink:label="note-0151-01a" xlink:href="note-0151-01"/> & </s> <s xml:id="echoid-s2295" xml:space="preserve">jungatur E A, quæ parabolam A B C in A continget. <lb/></s> <s xml:id="echoid-s2296" xml:space="preserve">Porro ſecetur A D in G, ut ſit A G ad G D ſicut E A ad <lb/>A D. </s> <s xml:id="echoid-s2297" xml:space="preserve">Et utrisque ſimul A E, D G æqualis ſtatuatur recta <lb/>H. </s> <s xml:id="echoid-s2298" xml:space="preserve">Item trienti baſis A C æqualis ſit recta L, & </s> <s xml:id="echoid-s2299" xml:space="preserve">inter H <lb/>& </s> <s xml:id="echoid-s2300" xml:space="preserve">L media proportionalis inveniatur K. </s> <s xml:id="echoid-s2301" xml:space="preserve">qua tanquam radio <lb/>circulus deſcribatur. </s> <s xml:id="echoid-s2302" xml:space="preserve">Is æqualis erit ſuperficiei curvæ conoi-<lb/>dis A B C. </s> <s xml:id="echoid-s2303" xml:space="preserve">Hinc ſequitur, ſi fuerit A E dupla A D, ſu-<lb/>perficiem conoidis curvam ad circulum baſeos fore ut 14 ad <lb/>9. </s> <s xml:id="echoid-s2304" xml:space="preserve">Si A E tripla A D, ut 13 ad 6. </s> <s xml:id="echoid-s2305" xml:space="preserve">ſi A E quadrupla A D, <lb/>ut 14 ad 5. </s> <s xml:id="echoid-s2306" xml:space="preserve">Atque ita ſemper fore ut numerus ad numerum, <lb/>ſi A E ad A D ejusmodi rationem habuerit.</s> <s xml:id="echoid-s2307" xml:space="preserve"/> </p> <div xml:id="echoid-div180" type="float" level="2" n="2"> <note position="right" xlink:label="note-0151-01" xlink:href="note-0151-01a" xml:space="preserve"><emph style="sc">De linea</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> </div> <div xml:id="echoid-div182" type="section" level="1" n="66"> <head xml:id="echoid-head90" style="it" xml:space="preserve">Sphæroidis oblongi ſuperſiciei circulum æqualem <lb/>invenire.</head> <p> <s xml:id="echoid-s2308" xml:space="preserve">ESto ſphæroides oblongum cujus axis A B, centrum C, <lb/> <anchor type="note" xlink:label="note-0151-02a" xlink:href="note-0151-02"/> ſectio per axem ellipſis A D B E, cujus minor diame-<lb/>ter D E.</s> <s xml:id="echoid-s2309" xml:space="preserve"/> </p> <div xml:id="echoid-div182" type="float" level="2" n="1"> <note position="right" xlink:label="note-0151-02" xlink:href="note-0151-02a" xml:space="preserve"><emph style="sc">TAB. XIII.</emph> <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s2310" xml:space="preserve">Ponatur D F æqualis C B, ſeu ponatur F alter focorum <lb/>ellipſeos A D B E, rectæque F D parallela ducatur B G, <lb/>occurrens productæ E D in G. </s> <s xml:id="echoid-s2311" xml:space="preserve">centroque G, radio G B, <lb/>deſcribatur ſuper axe A B arcus circumferentiæ B H A. </s> <s xml:id="echoid-s2312" xml:space="preserve">In-<lb/>terque ſemidiametrum C D & </s> <s xml:id="echoid-s2313" xml:space="preserve">rectam utrisque æqualem, ar-<lb/>cui A H B & </s> <s xml:id="echoid-s2314" xml:space="preserve">diametro D E, media proportionalis ſit recta <lb/>K. </s> <s xml:id="echoid-s2315" xml:space="preserve">Erit hæc radius circuli qui ſuperficiei ſphæroidis A D B E <lb/>æqualis ſit.</s> <s xml:id="echoid-s2316" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div184" type="section" level="1" n="67"> <head xml:id="echoid-head91" style="it" xml:space="preserve">Sphæroidis lati ſive compreſſi ſuperficiei circulum <lb/>æqualem invenire.</head> <p> <s xml:id="echoid-s2317" xml:space="preserve">SIt ſphæroides latum cujus axis A B, centrum C, ſectio <lb/> <anchor type="note" xlink:label="note-0151-03a" xlink:href="note-0151-03"/> per axem ellipſis A D B E.</s> <s xml:id="echoid-s2318" xml:space="preserve"/> </p> <div xml:id="echoid-div184" type="float" level="2" n="1"> <note position="right" xlink:label="note-0151-03" xlink:href="note-0151-03a" xml:space="preserve"><emph style="sc">TAB. XIII</emph>. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s2319" xml:space="preserve">Sit rurſus focorum alteruter F, diviſâque bifariam F C <lb/>in G, intelligatur parabola A G B quæ baſin habeat axem <lb/>A B, verticem vero punctum G. </s> <s xml:id="echoid-s2320" xml:space="preserve">Sitque inter dimatrum D E, <lb/>& </s> <s xml:id="echoid-s2321" xml:space="preserve">rectam curvæ parabolicæ A G B æqualem, media pro- <pb o="104" file="0152" n="164" rhead="CHRISTIANI HUGENII"/> portionalis linea H. </s> <s xml:id="echoid-s2322" xml:space="preserve">Erit hæc radius circuli qui ſuperficiei <lb/> <anchor type="note" xlink:label="note-0152-01a" xlink:href="note-0152-01"/> ſphæroidis propoſiti æqualis ſit.</s> <s xml:id="echoid-s2323" xml:space="preserve"/> </p> <div xml:id="echoid-div185" type="float" level="2" n="2"> <note position="left" xlink:label="note-0152-01" xlink:href="note-0152-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLURIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> </div> <div xml:id="echoid-div187" type="section" level="1" n="68"> <head xml:id="echoid-head92" style="it" xml:space="preserve">Conoidis hyperbolici ſuperficiei curvæ circulum <lb/>æqualem invenire.</head> <p> <s xml:id="echoid-s2324" xml:space="preserve">ESto conoides hyperbolicum cujus axis A B, ſectio per <lb/> <anchor type="note" xlink:label="note-0152-02a" xlink:href="note-0152-02"/> axem hyperbola C A D, cujus latus tranſverſum E A, <lb/>centrum F, latus rectum A G.</s> <s xml:id="echoid-s2325" xml:space="preserve"/> </p> <div xml:id="echoid-div187" type="float" level="2" n="1"> <note position="left" xlink:label="note-0152-02" xlink:href="note-0152-02a" xml:space="preserve">TAB. XIV. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s2326" xml:space="preserve">Sumatur in axe recta A H, æqualis dimidio lateri recto <lb/>A G. </s> <s xml:id="echoid-s2327" xml:space="preserve">& </s> <s xml:id="echoid-s2328" xml:space="preserve">ut H F ad A F longitudine ita, ſit A F ad F K <lb/>potentiâ. </s> <s xml:id="echoid-s2329" xml:space="preserve">Et intelligatur vertice K alia hyperbola deſcripta <lb/>K L M, eodem axe & </s> <s xml:id="echoid-s2330" xml:space="preserve">centro F cum priore, quæque late-<lb/>ra rectum & </s> <s xml:id="echoid-s2331" xml:space="preserve">transverſum illi reciproce proportionalia habeat. <lb/></s> <s xml:id="echoid-s2332" xml:space="preserve">Occurrat autem ipſi producta B C in M, ſitque A L paralle-<lb/>la B C. </s> <s xml:id="echoid-s2333" xml:space="preserve">Erit jam ſicut ſpatium A L M B, tribus rectis lineis <lb/>& </s> <s xml:id="echoid-s2334" xml:space="preserve">curva hyperbolica comprehenſum, ad dimidium quadra-<lb/>tum ex B C, ita ſuperficies conoidis curva ad circulum ba-<lb/>ſeos ſuæ, cujus diameter C D. </s> <s xml:id="echoid-s2335" xml:space="preserve">Unde conſtructio reliqua <lb/>facile abſolvetur, poſitâ hyperbolæ quadraturâ.</s> <s xml:id="echoid-s2336" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2337" xml:space="preserve">Quum igitur conoidis parabolici ſuperficies ad circulum <lb/>redigatur, æque ac ſuperficies ſphæræ, ex notis geometriæ <lb/>regulis; </s> <s xml:id="echoid-s2338" xml:space="preserve">in ſuperficie ſphæroidis oblongi, ut idem fiat, po-<lb/>nendum eſt arcus circumferentiæ longitudinem æquari poſſe <lb/>lineæ rectæ. </s> <s xml:id="echoid-s2339" xml:space="preserve">Ad ſphæroidis vero lati, itemque ad conoidis <lb/>hyperbolici ſuperficiem eadem ratione complanandam, hy-<lb/>perbolæ quadratura requiritur. </s> <s xml:id="echoid-s2340" xml:space="preserve">Nam parabolicæ lineæ lon-<lb/>gitudo, quam in ſphæroide hoc adhibuimus, pendet à qua-<lb/>dratura hyperbolæ, ut mox oſtendemus.</s> <s xml:id="echoid-s2341" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2342" xml:space="preserve">Verum, quod non indignum animadverſione videtur, in-<lb/>venimus absque ulla hyperbolicæ quadraturæ ſuppoſitione, <lb/>circulum æqualem conſtrui ſuperficiei utrique ſimul, ſphæ-<lb/>roidis lati & </s> <s xml:id="echoid-s2343" xml:space="preserve">conoidis hyperbolici.</s> <s xml:id="echoid-s2344" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2345" xml:space="preserve">Dato enim ſphæroide quovis lato, poſſe inveniri conoi-<lb/>des hyperbolicum, vel contra, dato conoide hyperbolico, <lb/>poſſe inveniri ſphæroides latum ejusmodi, ut utriusque ſi- <pb file="0153" n="165"/> <pb file="0153a" n="166"/> <anchor type="figure" xlink:label="fig-0153a-01a" xlink:href="fig-0153a-01"/> <anchor type="figure" xlink:label="fig-0153a-02a" xlink:href="fig-0153a-02"/> <anchor type="figure" xlink:label="fig-0153a-03a" xlink:href="fig-0153a-03"/> <anchor type="figure" xlink:label="fig-0153a-04a" xlink:href="fig-0153a-04"/> <anchor type="figure" xlink:label="fig-0153a-05a" xlink:href="fig-0153a-05"/> <pb file="0154" n="167"/> <pb o="105" file="0155" n="168" rhead="HOROLOG. OSCILLATOR."/> mul ſuperficiei exhibeatur circulus æqualis. </s> <s xml:id="echoid-s2346" xml:space="preserve">cujus exemplum <lb/> <anchor type="note" xlink:label="note-0155-01a" xlink:href="note-0155-01"/> in caſu uno cæteris ſimpliciore ſufficiet attuliſſe.</s> <s xml:id="echoid-s2347" xml:space="preserve"/> </p> <div xml:id="echoid-div188" type="float" level="2" n="2"> <figure xlink:label="fig-0153a-01" xlink:href="fig-0153a-01a"> <caption xml:id="echoid-caption44" style="it" xml:space="preserve">Pag. 104.<lb/>TAB. XIII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables44" xml:space="preserve">H E M A F K G B D</variables> </figure> <figure xlink:label="fig-0153a-02" xlink:href="fig-0153a-02a"> <caption xml:id="echoid-caption45" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables45" xml:space="preserve">A F N E G B D</variables> </figure> <figure xlink:label="fig-0153a-03" xlink:href="fig-0153a-03a"> <caption xml:id="echoid-caption46" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables46" xml:space="preserve">A G D C H E K F B</variables> </figure> <figure xlink:label="fig-0153a-04" xlink:href="fig-0153a-04a"> <caption xml:id="echoid-caption47" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables47" xml:space="preserve">E B H X L D C A G D C</variables> </figure> <figure xlink:label="fig-0153a-05" xlink:href="fig-0153a-05a"> <caption xml:id="echoid-caption48" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables48" xml:space="preserve">A D C G F E B H</variables> </figure> <note position="right" xlink:label="note-0155-01" xlink:href="note-0155-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2348" xml:space="preserve">Sit ſphæroides latum cujus axis S I, ſectio per axem el-<lb/>lipſis S T I K; </s> <s xml:id="echoid-s2349" xml:space="preserve">cujus ellipſis centrum O, axis major T K. <lb/></s> <s xml:id="echoid-s2350" xml:space="preserve"> <anchor type="note" xlink:label="note-0155-02a" xlink:href="note-0155-02"/> ponatur autem ellipſis hæc ejusmodi, ut latus transverſum <lb/>T K habeat ad latus rectum eam rationem, quam linea ſe-<lb/>cundum extremam & </s> <s xml:id="echoid-s2351" xml:space="preserve">mediam rationem ſecta, ad partem ſui <lb/>majorem.</s> <s xml:id="echoid-s2352" xml:space="preserve"/> </p> <div xml:id="echoid-div189" type="float" level="2" n="3"> <note position="right" xlink:label="note-0155-02" xlink:href="note-0155-02a" xml:space="preserve">TAB. XIV. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s2353" xml:space="preserve">Sumatur B C potentia dupla ad S O, item B A potentia <lb/>dupla ad O K. </s> <s xml:id="echoid-s2354" xml:space="preserve">& </s> <s xml:id="echoid-s2355" xml:space="preserve">ſint hæ quatuor continue proportionales <lb/>B C, B A, B F, B E, & </s> <s xml:id="echoid-s2356" xml:space="preserve">ponatur E P æqualis E A. </s> <s xml:id="echoid-s2357" xml:space="preserve">In-<lb/>telligatur jam conoides hyperbolicum Q F. </s> <s xml:id="echoid-s2358" xml:space="preserve">N, cujus axis <lb/>F P; </s> <s xml:id="echoid-s2359" xml:space="preserve">axi adjecta, ſive {1/2} latus transverſum F B; </s> <s xml:id="echoid-s2360" xml:space="preserve">dimidium <lb/>latus rectum æquale B C.</s> <s xml:id="echoid-s2361" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2362" xml:space="preserve">Hujus conoidis ſuperficies curva, unà cum ſuperficie ſphæ-<lb/>roidis S I, æquabitur circulo cujus datus erit radius M L, <lb/>qui nempe poſſit quadratum T K cum duplo quadrato S I.</s> <s xml:id="echoid-s2363" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div191" type="section" level="1" n="69"> <head xml:id="echoid-head93" style="it" xml:space="preserve">Curvæ parabolicæ æqualem rectam lineam <lb/>invenire.</head> <p> <s xml:id="echoid-s2364" xml:space="preserve">SIt parabolæ portio A B C, cujus axis B K, baſis A C <lb/> <anchor type="note" xlink:label="note-0155-03a" xlink:href="note-0155-03"/> axi ad angulos rectos; </s> <s xml:id="echoid-s2365" xml:space="preserve">& </s> <s xml:id="echoid-s2366" xml:space="preserve">oporteat curvæ A B C rectam <lb/>æqualem invenire.</s> <s xml:id="echoid-s2367" xml:space="preserve"/> </p> <div xml:id="echoid-div191" type="float" level="2" n="1"> <note position="right" xlink:label="note-0155-03" xlink:href="note-0155-03a" xml:space="preserve">TAB. XIV. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s2368" xml:space="preserve">Accipiatur baſi dimidiæ A K æqualis recta I E, quæ pro-<lb/>ducatur ad H, ut ſit I H æqualis A G, quæ parabolam in <lb/>puncto baſis A contingens, cum axe producto convenit in G. <lb/></s> <s xml:id="echoid-s2369" xml:space="preserve">Sit jam portio hyperbolæ D E F, vertice E, centro I de-<lb/>ſcriptæ, cujusque diameter ſit E H; </s> <s xml:id="echoid-s2370" xml:space="preserve">baſis vero D H F or-<lb/>dinatim ad diametrum applicata. </s> <s xml:id="echoid-s2371" xml:space="preserve">Latus rectum pro lubitu <lb/>ſumi poteſt. </s> <s xml:id="echoid-s2372" xml:space="preserve">Quod ſi jam ſuper baſi D F intelligatur paral-<lb/>lelogrammum conſtitutum D P Q F, quod portioni D E F <lb/>æquale ſit; </s> <s xml:id="echoid-s2373" xml:space="preserve">ejus latus P Q ita ſecabit diametrum hyperbolæ <lb/>in R, ut R I ſit æqualis curvæ parabolicæ A B, cujus du-<lb/>pla eſt A B C.</s> <s xml:id="echoid-s2374" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2375" xml:space="preserve">Apparet igitur hinc quomodo à quadratura hyperbolæ <lb/>pendeat curvæ parabolicæ menſura, & </s> <s xml:id="echoid-s2376" xml:space="preserve">illa ab hac viciſſim.</s> <s xml:id="echoid-s2377" xml:space="preserve"/> </p> <pb o="106" file="0156" n="169" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s2378" xml:space="preserve">Quæcunque vero problemata ad alterum è duobus hiſce <lb/> <anchor type="note" xlink:label="note-0156-01a" xlink:href="note-0156-01"/> reducuntur, quamlibet veræ proximam ſolutionem per nu-<lb/>meros accipiunt, logarithmorum admirabili invento. </s> <s xml:id="echoid-s2379" xml:space="preserve">Cum <lb/>per hos hyperbolæ quadratura, ut olim invenimus, numeris <lb/>quam proxime explicetur. </s> <s xml:id="echoid-s2380" xml:space="preserve">Eſt autem regula hujusmodi.</s> <s xml:id="echoid-s2381" xml:space="preserve"/> </p> <div xml:id="echoid-div192" type="float" level="2" n="2"> <note position="left" xlink:label="note-0156-01" xlink:href="note-0156-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2382" xml:space="preserve">Sit D A B portio hyperbolæ, cujus aſymptoti C S, C V, <lb/> <anchor type="note" xlink:label="note-0156-02a" xlink:href="note-0156-02"/> ductis D E, B V parallelis aſymptoto S C.</s> <s xml:id="echoid-s2383" xml:space="preserve"/> </p> <div xml:id="echoid-div193" type="float" level="2" n="3"> <note position="left" xlink:label="note-0156-02" xlink:href="note-0156-02a" xml:space="preserve">TAB. XV. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s2384" xml:space="preserve">Accipiatur differentia logarithmorum qui conveniunt nu-<lb/>meris, eandem inter ſe rationem habentibus quam rectæ D E, <lb/>B V; </s> <s xml:id="echoid-s2385" xml:space="preserve">ejusque differentiæ quæratur logarithmus. </s> <s xml:id="echoid-s2386" xml:space="preserve">Cui adda-<lb/>tur logarithmus hic (qui ſemper eſt idem) 0,36221,56887. <lb/></s> <s xml:id="echoid-s2387" xml:space="preserve">Summa erit logarithmus numeri qui ſpatium D E V B A D <lb/>deſignabit, tribus rectis & </s> <s xml:id="echoid-s2388" xml:space="preserve">curva D A B comprehenſi, in <lb/>partibus qualium parallelogrammum D E eſt 100000,00000. </s> <s xml:id="echoid-s2389" xml:space="preserve"><lb/>Unde porro facile quoque habebitur area portionis D A B.</s> <s xml:id="echoid-s2390" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2391" xml:space="preserve">Sit ex. </s> <s xml:id="echoid-s2392" xml:space="preserve">gr. </s> <s xml:id="echoid-s2393" xml:space="preserve">proportio D E ad B V ea quæ 36 ad 5.</s> <s xml:id="echoid-s2394" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>Ab # 1,55630,25008, logar<emph style="super">o</emph>. 36. <lb/>auferatur # 0,69897,00043.logar<emph style="super">us</emph>. 5. <lb/>Erit # 0,85733,24965.differ.logar<emph style="super">orum</emph>. <lb/>Et # 9,93314, 92856.logar<emph style="super">us</emph>.differentiæ. <lb/>Cui addatur # 0,36221,56887.logar<emph style="super">us</emph>.ſemper addendus. <lb/>Fit # 10,29536,49743. logar<emph style="super">us</emph>. ſpatii D E V B A D. <lb/></note> <p> <s xml:id="echoid-s2395" xml:space="preserve">Habebit hujus logarithmi numerus 11 characteres, quum <lb/>characteriſtica ſit 10. </s> <s xml:id="echoid-s2396" xml:space="preserve">Quæratur itaque primo numerus pro-<lb/>xime minor, conveniens invento logarithmo, qui numerus <lb/>eſt 19740. </s> <s xml:id="echoid-s2397" xml:space="preserve">Deinde ex differentia logarithmi ejusdem, & </s> <s xml:id="echoid-s2398" xml:space="preserve">pro-<lb/>xime eum in tabula ſequentis, reliqui characteres eliciantur <lb/>81026, ſcribèndi poſt priores, ut fiat 197408, 10260, addi-<lb/>to ad ſinem zero, ut efficiatur numerus characterum 11. </s> <s xml:id="echoid-s2399" xml:space="preserve">Eſt <lb/>ergo area ſpatii D E V B A D proxime partium 197408, <lb/>10260, qualium partium parallelogrammum D C eſt 100000, <lb/>00000.</s> <s xml:id="echoid-s2400" xml:space="preserve"/> </p> <pb file="0157" n="170"/> <pb file="0157a" n="171"/> <figure> <caption xml:id="echoid-caption49" style="it" xml:space="preserve">Pag. 106.<lb/>TAB. XIV.<lb/>Fig. 2.</caption> <variables xml:id="echoid-variables49" xml:space="preserve">T B M S O I C A F K E L Q P N</variables> </figure> <figure> <caption xml:id="echoid-caption50" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables50" xml:space="preserve">E F K L A G H M C B D</variables> </figure> <figure> <caption xml:id="echoid-caption51" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables51" xml:space="preserve">I G E B P R Q A K C D H F</variables> </figure> <pb file="0158" n="172"/> <pb o="107" file="0159" n="173" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div195" type="section" level="1" n="70"> <head xml:id="echoid-head94" xml:space="preserve">PROPOSITIO X.</head> <note position="right" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> <p style="it"> <s xml:id="echoid-s2401" xml:space="preserve">LIneas curvas exhibere quarum evolutione elli-<lb/>pſes & </s> <s xml:id="echoid-s2402" xml:space="preserve">hyperbolæ deſcribantur, rectasque in-<lb/>venire iisdem curvis æquales.</s> <s xml:id="echoid-s2403" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2404" xml:space="preserve">Sit ellipſis vel hyperbole quælibet A B, cujus axis trans-<lb/> <anchor type="note" xlink:label="note-0159-02a" xlink:href="note-0159-02"/> verſus A C; </s> <s xml:id="echoid-s2405" xml:space="preserve">centrum figuræ D; </s> <s xml:id="echoid-s2406" xml:space="preserve">latus rectum duplum ipſius <lb/>A E. </s> <s xml:id="echoid-s2407" xml:space="preserve">Et ſumpto in ſectione quovis puncto, ut B, applice-<lb/>tur ordinatim ad axem recta B K, & </s> <s xml:id="echoid-s2408" xml:space="preserve">ad dictum punctum B <lb/>tangens ducatur quæ conveniat cum axe in F; </s> <s xml:id="echoid-s2409" xml:space="preserve">ſitque B G <lb/>ipſi F B perpendicularis, axique occurrat in G; </s> <s xml:id="echoid-s2410" xml:space="preserve">& </s> <s xml:id="echoid-s2411" xml:space="preserve">produ-<lb/>catur B G usque ad H, ut B H ad H G habeat rationem <lb/>eam quæ componitur ex rationibus G F ad F K, & </s> <s xml:id="echoid-s2412" xml:space="preserve">A D <lb/>ad D E.</s> <s xml:id="echoid-s2413" xml:space="preserve"/> </p> <div xml:id="echoid-div195" type="float" level="2" n="1"> <note position="right" xlink:label="note-0159-02" xlink:href="note-0159-02a" xml:space="preserve">TAB. XV. <lb/>Fig. 2. & 3.</note> </div> <p> <s xml:id="echoid-s2414" xml:space="preserve">Dico curvam E H M, cujus puncta omnia inveniuntur <lb/>eodem modo quo punctum H, eſſe eam cujus evolu-<lb/>tione, unà cum recta E A, deſcribetur ſectio A B. </s> <s xml:id="echoid-s2415" xml:space="preserve">Ipſam <lb/>autem B H tangere curvam in H, & </s> <s xml:id="echoid-s2416" xml:space="preserve">eſſe toti H E A æqua-<lb/>lem. </s> <s xml:id="echoid-s2417" xml:space="preserve">Quamobrem, ſi ab H B auferatur E A, reliqua recta <lb/>portioni curvæ H E æquabitur. </s> <s xml:id="echoid-s2418" xml:space="preserve">Apparet autem, cum cur-<lb/>væ puncta quævis indifferenter, certaque ratione invenian-<lb/>tur, eſſe eam utrobique ex earum genere, quæ merè geo-<lb/>metricæ cenſentur. </s> <s xml:id="echoid-s2419" xml:space="preserve">Unde & </s> <s xml:id="echoid-s2420" xml:space="preserve">relatio horum omnium puncto-<lb/>rum ad puncta axis A C, æquatione aliqua exprimi poterit, <lb/>quam æquationem ad ſextam dimenſionem aſcendere invenio; <lb/></s> <s xml:id="echoid-s2421" xml:space="preserve">minimumque habere terminorum, ſi fuerit A B hyperbola <lb/>cujus latera transverſum rectumque æqualia. </s> <s xml:id="echoid-s2422" xml:space="preserve">Tunc enim du-<lb/>cta ex quovis curvæ puncto, ut H, ad axem C A N per-<lb/>pendiculari H N; </s> <s xml:id="echoid-s2423" xml:space="preserve">vocatâque A C, a; </s> <s xml:id="echoid-s2424" xml:space="preserve">C N, x; </s> <s xml:id="echoid-s2425" xml:space="preserve">& </s> <s xml:id="echoid-s2426" xml:space="preserve">N H, <lb/>y; </s> <s xml:id="echoid-s2427" xml:space="preserve">erit ſemper cubus ab x x-y y-a a æqualis 27 x x y y a a. </s> <s xml:id="echoid-s2428" xml:space="preserve"><lb/>Sed hoc caſu brevius quoque multo, quam prædicta con-<lb/>ſtructione, curvæ E H M puncta reperiri poſſunt, ut in ſe-<lb/>quentibus oſtendetur.</s> <s xml:id="echoid-s2429" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2430" xml:space="preserve">Cæterum notandum eſt, in ellipſi ſingulos quadrantes ſin-<lb/>gularum linearum evolutione deſcribi; </s> <s xml:id="echoid-s2431" xml:space="preserve">ſicut quadrans A B L <pb o="108" file="0160" n="174" rhead="CHRISTIANI HUGENII"/> evolutione lineæ A E H M, quadrans C L evolutione ſimi-<lb/> <anchor type="note" xlink:label="note-0160-01a" xlink:href="note-0160-01"/> lis huic oppoſitæ C O M. </s> <s xml:id="echoid-s2432" xml:space="preserve">Eſt enim hæc in ſectione utraque <lb/>diverſitas, quod cum principium quidem curvæ E H M, <lb/>tam in ellipſi quam in hyperbola, ſit punctum E, ſumpta <lb/>A E æquali {1/2} lateris recti; </s> <s xml:id="echoid-s2433" xml:space="preserve">in hyperbola in infinitum inde <lb/>dicta linea extenditur, at in ellipſi finitur in puncto axis <lb/>minoris M, ſumpta L M æquali {1/2} lateris recti, ſecundum <lb/>quod poſſunt ordinatim applicatæ ad dictum minorem axem. <lb/></s> <s xml:id="echoid-s2434" xml:space="preserve">Namque hos terminos eſſe hujus curvæ, facile apparebit or-<lb/>tum ejus conſideranti, quodque in ellipſi eſt ſicut A D ad <lb/>D E, ita L M ad M D.</s> <s xml:id="echoid-s2435" xml:space="preserve"/> </p> <div xml:id="echoid-div196" type="float" level="2" n="2"> <note position="left" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2436" xml:space="preserve">Horum autem demonſtrationi non immorabimur, ſed ad <lb/>ipſam methodum tradendam pergemus, qua & </s> <s xml:id="echoid-s2437" xml:space="preserve">hæ curvæ ex <lb/>ſectionibus conicis, & </s> <s xml:id="echoid-s2438" xml:space="preserve">aliæ innumeræ ex aliis quibuſcun-<lb/>que datis inveniuntur.</s> <s xml:id="echoid-s2439" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div198" type="section" level="1" n="71"> <head xml:id="echoid-head95" xml:space="preserve">PROPOSITIO XI.</head> <p> <s xml:id="echoid-s2440" xml:space="preserve">DAtâ lineâ curvâ, invenire aliam cujus evolu-<lb/>tione illa deſcribatur; </s> <s xml:id="echoid-s2441" xml:space="preserve">& </s> <s xml:id="echoid-s2442" xml:space="preserve">oſtendere quod ex <lb/>unaquaque curva geometrica, alia curva itidem <lb/>geometrica exiſtat, cui recta linea æqualis dari <lb/>poſſit.</s> <s xml:id="echoid-s2443" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2444" xml:space="preserve">Sit curva quæpiam, vel pars ejus, in partem unam infle-<lb/> <anchor type="note" xlink:label="note-0160-02a" xlink:href="note-0160-02"/> xa A B F, & </s> <s xml:id="echoid-s2445" xml:space="preserve">recta K L, ad quam puncta omnia referan-<lb/>tur; </s> <s xml:id="echoid-s2446" xml:space="preserve">& </s> <s xml:id="echoid-s2447" xml:space="preserve">oporteat invenire curvam aliam, ut D E, cujus <lb/>evolutione ipſa A B F deſcribatur.</s> <s xml:id="echoid-s2448" xml:space="preserve"/> </p> <div xml:id="echoid-div198" type="float" level="2" n="1"> <note position="left" xlink:label="note-0160-02" xlink:href="note-0160-02a" xml:space="preserve">TAB. XV. <lb/>Fig. 4. & 5.</note> </div> <p> <s xml:id="echoid-s2449" xml:space="preserve">Ponatur jam inventa; </s> <s xml:id="echoid-s2450" xml:space="preserve">& </s> <s xml:id="echoid-s2451" xml:space="preserve">quoniam tangentes omnes curvæ <lb/>D E, neceſſe eſt occurrere lineæ A B F, ex evolutione de-<lb/>ſcriptæ, ad angulos rectos; </s> <s xml:id="echoid-s2452" xml:space="preserve">patet quoque viciſſim eas quæ <lb/>ipſi A B F ad rectos angulos inſiſtunt, ut B D, F E, ta-<lb/>cturas evolutam C D E.</s> <s xml:id="echoid-s2453" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2454" xml:space="preserve">Intelligantur autem puncta B, F, inter ſe proxima; </s> <s xml:id="echoid-s2455" xml:space="preserve">& </s> <s xml:id="echoid-s2456" xml:space="preserve">ſi-<lb/>quidem à parte A evolutio incipere ponatur, ulteriuſque in-<lb/>de diſtet F quam B, etiam contactus E ulterius quam D@di- <pb o="109" file="0161" n="175" rhead="HOROLOG. OSCILLATOR."/> ſtabit ab A; </s> <s xml:id="echoid-s2457" xml:space="preserve">interſectio vero rectarum B D, F E, quæ eſt <lb/> <anchor type="note" xlink:label="note-0161-01a" xlink:href="note-0161-01"/> G, cadet ultra punctum D in recta B D. </s> <s xml:id="echoid-s2458" xml:space="preserve">Nam concurrere <lb/>ipſas B D, F E neceſſe eſt, cum curvæ B F ad partem ca-<lb/>vam inſiſtant rectis angulis.</s> <s xml:id="echoid-s2459" xml:space="preserve"/> </p> <div xml:id="echoid-div199" type="float" level="2" n="2"> <note position="right" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2460" xml:space="preserve">Quanto autem punctum F ipſi B propinquius fuerit, tanto <lb/>propius quoque puncta D, G & </s> <s xml:id="echoid-s2461" xml:space="preserve">E convenire apparet; </s> <s xml:id="echoid-s2462" xml:space="preserve">ideo-<lb/>que, ſi interſtitium B F infinite parvum intelligatur, tria <lb/>dicta puncta pro uno eodemque erunt habenda; </s> <s xml:id="echoid-s2463" xml:space="preserve">ac præterea, <lb/>ductâ rectâ B H, quæ curvam in B tangat, eadem quoque <lb/>pro tangente in F cenſebitur. </s> <s xml:id="echoid-s2464" xml:space="preserve">Sit B O parallela K L, & </s> <s xml:id="echoid-s2465" xml:space="preserve"><lb/>in hanc perpendiculares cadant B K, F L: </s> <s xml:id="echoid-s2466" xml:space="preserve">ſecetque F L <lb/>rectam B O in P, & </s> <s xml:id="echoid-s2467" xml:space="preserve">ſint puncta notata M, N, in quibus <lb/>rectæ, B D, F E, occurrant ipſi K L. </s> <s xml:id="echoid-s2468" xml:space="preserve">Quia igitur ratio <lb/>B G ad G M eſt eadem quæ B O ad M N, data hac dabi-<lb/>tur & </s> <s xml:id="echoid-s2469" xml:space="preserve">illa; </s> <s xml:id="echoid-s2470" xml:space="preserve">& </s> <s xml:id="echoid-s2471" xml:space="preserve">quia recta B M datur magnitudine ac po-<lb/>ſitione, dabitur & </s> <s xml:id="echoid-s2472" xml:space="preserve">punctum G in producta B M, ſive D <lb/>in curva C D E, quia G & </s> <s xml:id="echoid-s2473" xml:space="preserve">D in unum convenire diximus. <lb/></s> <s xml:id="echoid-s2474" xml:space="preserve">Datur autem ratio B O ad M N, ſimpliciter quidem in <lb/>Cycloide, ubi primùm omnium illam inveſtigavimus, inve-<lb/>nimuſque duplam; </s> <s xml:id="echoid-s2475" xml:space="preserve">in aliis vero curvis, quas hactenus exa-<lb/>minavimus, per duarum datarum rationum compoſitionem. </s> <s xml:id="echoid-s2476" xml:space="preserve"><lb/>Nam quia ratio B O ad M N componitur ex rationibus B O <lb/>ad B P, ſive N H ad L H, & </s> <s xml:id="echoid-s2477" xml:space="preserve">ex B P ſive K L ad M N; </s> <s xml:id="echoid-s2478" xml:space="preserve"><lb/>patet, ſi rationes hæ utræque dentur etiam ex iis compoſi-<lb/>tam rationem B O ad M N datum iri. </s> <s xml:id="echoid-s2479" xml:space="preserve">Illas vero dari in o-<lb/>mnibus curvis geometricis, in ſequentibus patebit; </s> <s xml:id="echoid-s2480" xml:space="preserve">ac proin-<lb/>de iis ſemper curvas adſignari poſſe, quarum evolutione de-<lb/>ſcribantur, quæque ideo ad rectas lineas ſint reducibiles.</s> <s xml:id="echoid-s2481" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2482" xml:space="preserve">Ponatur primò parabola eſſe A B F, cujus vertex A, <lb/> <anchor type="note" xlink:label="note-0161-02a" xlink:href="note-0161-02"/> axis A Q. </s> <s xml:id="echoid-s2483" xml:space="preserve">Cum igitur lineæ B M, F N, ſint parabolæ ad <lb/>angulos rectos; </s> <s xml:id="echoid-s2484" xml:space="preserve">ductæque ſint ad axem A Q perpendicula-<lb/>res B K, F L; </s> <s xml:id="echoid-s2485" xml:space="preserve">erunt, ex proprietate parabolæ, ſingulæ <lb/>M K, N L dimidio lateri recto æquales; </s> <s xml:id="echoid-s2486" xml:space="preserve">& </s> <s xml:id="echoid-s2487" xml:space="preserve">ablata commu-<lb/>ni L M, æquales inter ſe K L, M N. </s> <s xml:id="echoid-s2488" xml:space="preserve">Hinc, quum ratio <lb/>B G ad G M componatur ex rationibus N H ad H L, & </s> <s xml:id="echoid-s2489" xml:space="preserve"><lb/>K L ad M N, uti dictum fuit, ſitque earum poſterior ratio <pb o="110" file="0162" n="176" rhead="CHRISTIANI HUGENII"/> æqualitatis; </s> <s xml:id="echoid-s2490" xml:space="preserve">liquet rationem B G ad G M fore eandem quæ N H <lb/> <anchor type="note" xlink:label="note-0162-01a" xlink:href="note-0162-01"/> ad H L; </s> <s xml:id="echoid-s2491" xml:space="preserve">& </s> <s xml:id="echoid-s2492" xml:space="preserve">dividendo, B M ad M G, eandem quæ N L <lb/>ad L H, ſive M K ad K H; </s> <s xml:id="echoid-s2493" xml:space="preserve">nam L H, K H pro eadem <lb/>habentur, propter propinquitatem punctorum B, F. </s> <s xml:id="echoid-s2494" xml:space="preserve">Data <lb/>autem eſt ratio M K ad K H, dato puncto B; </s> <s xml:id="echoid-s2495" xml:space="preserve">quoniam <lb/>tam M K, quam K H dantur magnitudine; </s> <s xml:id="echoid-s2496" xml:space="preserve">nam M K <lb/>æquatur dimidio lateri recto, K H vero duplæ K A. </s> <s xml:id="echoid-s2497" xml:space="preserve">Dataque <lb/>etiam eſt poſitione & </s> <s xml:id="echoid-s2498" xml:space="preserve">magnitudine recta B M. </s> <s xml:id="echoid-s2499" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2500" xml:space="preserve">M G <lb/>data erit, adeoque & </s> <s xml:id="echoid-s2501" xml:space="preserve">punctum G, ſive D, in curva R D E; <lb/></s> <s xml:id="echoid-s2502" xml:space="preserve">quod nempe invenitur productâ B M uſque in G, ut ſit <lb/>B M ad M G ſicut {1/2} lateris recti ad duplam K A.</s> <s xml:id="echoid-s2503" xml:space="preserve"/> </p> <div xml:id="echoid-div200" type="float" level="2" n="3"> <note position="right" xlink:label="note-0161-02" xlink:href="note-0161-02a" xml:space="preserve">TAB. XVI. <lb/>Fig. 2.</note> <note position="left" xlink:label="note-0162-01" xlink:href="note-0162-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph></note> </div> <p> <s xml:id="echoid-s2504" xml:space="preserve">Et ſic quidem, adſumptis in parabola A B F aliis quotli-<lb/>bet punctis præter B, totidem quoque puncta lineæ R D E, <lb/>ſimili ratione, invenientur; </s> <s xml:id="echoid-s2505" xml:space="preserve">atque hoc ipſo lineam R D E <lb/>geometricam eſſe conſtat, unáque proprietas ejus innoteſcit, <lb/>ex qua cæteræ deduci poſſunt. </s> <s xml:id="echoid-s2506" xml:space="preserve">Ut ſi inquirere deinde veli-<lb/>mus, quanam æquatione exprimatur relatio punctorum <lb/>omnium curvæ C D E ad rectam A Q: </s> <s xml:id="echoid-s2507" xml:space="preserve">ducta in hanc perpen-<lb/>diculari D Q, vocatoque latere recto parabolæ A B F, a; <lb/></s> <s xml:id="echoid-s2508" xml:space="preserve">A K, b; </s> <s xml:id="echoid-s2509" xml:space="preserve">A Q, x; </s> <s xml:id="echoid-s2510" xml:space="preserve">Q D, y. </s> <s xml:id="echoid-s2511" xml:space="preserve">Quoniam ratio B M ad M D, <lb/>hoc eſt, K M ad M Q, eſt ea quæ {1/2} a ad 2 b, eſtque ipſa <lb/>K M = {1/2} a, erit & </s> <s xml:id="echoid-s2512" xml:space="preserve">M Q æqualis 2 b. </s> <s xml:id="echoid-s2513" xml:space="preserve">Eſt autem M A = {1/2} <lb/>a + b. </s> <s xml:id="echoid-s2514" xml:space="preserve">ergo A Q ſive x æqualis 3 b + {1/2} a. </s> <s xml:id="echoid-s2515" xml:space="preserve">Unde b = {1/3} x <lb/>-{1/6} a. </s> <s xml:id="echoid-s2516" xml:space="preserve">Porro quoniam, ſicut quadratum M K, hoc eſt, {1/4} a a <lb/>ad quadratum K B, hoc eſt, a b, ita qu. </s> <s xml:id="echoid-s2517" xml:space="preserve">M Q, hoc eſt, <lb/>4 b b ad qu. </s> <s xml:id="echoid-s2518" xml:space="preserve">Q D; </s> <s xml:id="echoid-s2519" xml:space="preserve">erit qu. </s> <s xml:id="echoid-s2520" xml:space="preserve">Q D, ſive y y = {16b<emph style="super">3</emph>/4}. </s> <s xml:id="echoid-s2521" xml:space="preserve">Ubi, ſi in <lb/>locum b ſubſtituatur {1/3} x - {1/6}a, quod illi æquale inventum eſt, <lb/>fiet y y = 16. </s> <s xml:id="echoid-s2522" xml:space="preserve">cub. </s> <s xml:id="echoid-s2523" xml:space="preserve">{1/3} x - {1/6} a diviſis per a. </s> <s xml:id="echoid-s2524" xml:space="preserve">Ac proinde {27/16} a y y <lb/>= cubo ab x - {1/2} a. </s> <s xml:id="echoid-s2525" xml:space="preserve">Accipiatur A R in axe parabolæ = {1/2} a; </s> <s xml:id="echoid-s2526" xml:space="preserve"><lb/>eritque R Q = x - {1/2} a. </s> <s xml:id="echoid-s2527" xml:space="preserve">Curvam igitur C D ejus naturæ eſſe <lb/>liquet, ut ſemper cubus lineæ R Q æquetur parallelepipedo, <lb/>cujus baſis qu. </s> <s xml:id="echoid-s2528" xml:space="preserve">Q D, altitudo {27/16} a; </s> <s xml:id="echoid-s2529" xml:space="preserve">ac proinde ipſam para-<lb/>boloidem eſſe, cujus evolutione deſcribi parabolam A B ſu-<lb/>pra oſtendimus; </s> <s xml:id="echoid-s2530" xml:space="preserve">cujus nimirum paraboloidis latus rectum æ-<lb/>quetur {27/16} lateris recti parabolæ A B. </s> <s xml:id="echoid-s2531" xml:space="preserve">tunc enim hujus latus <lb/>rectum æquale fit {15/27} lateris recti paraboloidis, quemadmo-<lb/>dum ibi fuit deſinitum.</s> <s xml:id="echoid-s2532" xml:space="preserve"/> </p> <pb o="111" file="0163" n="177" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s2533" xml:space="preserve">Quomodo porro ratio O B ad B P, ſive N H ad H L, <lb/> <anchor type="note" xlink:label="note-0163-01a" xlink:href="note-0163-01"/> non tantum cum A B F parabola eſt, ſed etiam alia quæli-<lb/>bet curva geometrica, ſemper inveniri poſſit manifeſtum eſt. <lb/></s> <s xml:id="echoid-s2534" xml:space="preserve">Quoniam tantum recta F H ducenda eſt, quæ curvam in <lb/> <anchor type="note" xlink:label="note-0163-02a" xlink:href="note-0163-02"/> adſumpto puncto F tangat, & </s> <s xml:id="echoid-s2535" xml:space="preserve">F N ipſi F H perpendicu-<lb/>laris: </s> <s xml:id="echoid-s2536" xml:space="preserve">unde N H & </s> <s xml:id="echoid-s2537" xml:space="preserve">H L datæ erunt, ac proinde ratio quo-<lb/>que earum data.</s> <s xml:id="echoid-s2538" xml:space="preserve"/> </p> <div xml:id="echoid-div201" type="float" level="2" n="4"> <note position="right" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> <note position="right" xlink:label="note-0163-02" xlink:href="note-0163-02a" xml:space="preserve">TAB. XV. <lb/>Fig. 4. & 5.</note> </div> <p> <s xml:id="echoid-s2539" xml:space="preserve">At non æque liquet quo pacto ratio K L ad M N innoteſcat, <lb/>quam tamen ſemper quoque reperiri poſſe ſic oſten-demus.</s> <s xml:id="echoid-s2540" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2541" xml:space="preserve">Sint rectæ K T, L V, perpendiculares ſuper K L, ſit-<lb/>que K T æqualis K M, & </s> <s xml:id="echoid-s2542" xml:space="preserve">L V æqualis L N, & </s> <s xml:id="echoid-s2543" xml:space="preserve">ducatur <lb/>V X parallela L N, quæ occurrat ipſi K T in X. </s> <s xml:id="echoid-s2544" xml:space="preserve">Quo-<lb/>niam ergo ſemper eadem eſt differentia duarum L K, N M, <lb/>quæ duarum L N, K M, hoc eſt, quæ duarum L V, K T; <lb/></s> <s xml:id="echoid-s2545" xml:space="preserve">eſt autem differentiæ ipſarum L V, K T æqualis X T, & </s> <s xml:id="echoid-s2546" xml:space="preserve"><lb/>X V ipſi L K; </s> <s xml:id="echoid-s2547" xml:space="preserve">erit proinde N M æqualis duabus ſimul <lb/>V X, X T, vel ei quo V X ipſam X T ſuperat. </s> <s xml:id="echoid-s2548" xml:space="preserve">Atque <lb/>adeo, ſi data fuerit ratio V X ad X T, data quoque erit <lb/>ratio V X ad utramque ſimul V X, X T, vel ad exceſſum V X <lb/>ſupra X T, hoc eſt, data erit ratio V X ſive L K ad N M.</s> <s xml:id="echoid-s2549" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2550" xml:space="preserve">Sciendum eſt autem, quoniam K T ipſi K M, & </s> <s xml:id="echoid-s2551" xml:space="preserve">L V <lb/>ipſi L N, æquales ſumptæ ſunt, locum punctorum T, V, <lb/>fore lineam quandam vel rectam vel curvam datam, ut mox <lb/>oſtendetur. </s> <s xml:id="echoid-s2552" xml:space="preserve">Et ſiquidem ſit linea recta; </s> <s xml:id="echoid-s2553" xml:space="preserve">ut contingit ſi A B F <lb/>coni ſectio fuerit, & </s> <s xml:id="echoid-s2554" xml:space="preserve">K L axis ejus; </s> <s xml:id="echoid-s2555" xml:space="preserve">conſtat rationem V X <lb/>ad X T datam fore, data poſitione ipſius lineæ V T, quæ <lb/>locus eſt puuctorum V, T; </s> <s xml:id="echoid-s2556" xml:space="preserve">ſemperque eandem tunc haberi <lb/>dictam rationem, qualecunque fuerit intervallum K L.</s> <s xml:id="echoid-s2557" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2558" xml:space="preserve">At ſi locus alia linea curva fuerit, diverſa erit ratio V X <lb/>ad X T, prout majus minuſve fuerit intervallum K L. </s> <s xml:id="echoid-s2559" xml:space="preserve">In-<lb/>quirendum eſt autem quænam futura ſit iſta ratio, cum K L <lb/>infinite parvum imaginamur, quoniam & </s> <s xml:id="echoid-s2560" xml:space="preserve">puncta B, F, pro-<lb/>xima invicem poſuimus. </s> <s xml:id="echoid-s2561" xml:space="preserve">Similiter itaque & </s> <s xml:id="echoid-s2562" xml:space="preserve">puncta V, T, <lb/>lineæ curvæ minimam particulam intercipere intelligendum <lb/>eſt; </s> <s xml:id="echoid-s2563" xml:space="preserve">unde recta V T, cum ea quæ in T curvam contingit, <lb/>coincidet. </s> <s xml:id="echoid-s2564" xml:space="preserve">Sit ergo tangens illa T Y; </s> <s xml:id="echoid-s2565" xml:space="preserve">poteſt enim duci quo- <pb o="112" file="0164" n="178" rhead="CHRISTIANI HUGENII"/> niam curva, ad quam ſunt puncta T, V, geometrica eſt. <lb/></s> <s xml:id="echoid-s2566" xml:space="preserve"> <anchor type="note" xlink:label="note-0164-01a" xlink:href="note-0164-01"/> Ratio igitur Y K ad K T data erit, adeoque & </s> <s xml:id="echoid-s2567" xml:space="preserve">V X ad <lb/>X T. </s> <s xml:id="echoid-s2568" xml:space="preserve">ex qua etiam rationem L K ad N M dari oſtendimus.</s> <s xml:id="echoid-s2569" xml:space="preserve"/> </p> <div xml:id="echoid-div202" type="float" level="2" n="5"> <note position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2570" xml:space="preserve">Quænam vero ſit linea ad quam ſunt puncta T, V, in-<lb/>venitur ponendo certum punctum S in recta K L, & </s> <s xml:id="echoid-s2571" xml:space="preserve">vocan-<lb/>do S K, x; </s> <s xml:id="echoid-s2572" xml:space="preserve">K T, y. </s> <s xml:id="echoid-s2573" xml:space="preserve">Nam quia data eſt curva A B F, <lb/>eique B M ad angulos rectos ducta, invenietur inde quanti-<lb/>tas lineæ K M, per methodum tangentium à Carteſio traditam, <lb/>quæ ipſi K T, ſive y æquabitur, & </s> <s xml:id="echoid-s2574" xml:space="preserve">ex ea æquatione, natura <lb/>curvæ T V innoteſcet, ad quam deinde tangens ducenda <lb/>eſt. </s> <s xml:id="echoid-s2575" xml:space="preserve">Sed clariora omnia fient ſequenti exemplo.</s> <s xml:id="echoid-s2576" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2577" xml:space="preserve">Sit A B F paraboloides illa, cui ſuperius rectam æqua-<lb/> <anchor type="note" xlink:label="note-0164-02a" xlink:href="note-0164-02"/> lem invenimus; </s> <s xml:id="echoid-s2578" xml:space="preserve">in qua nempe cubi perpendicularium in <lb/>rectam S K, ſint inter ſe ſicut quadrata ex ipſa S K abſciſ-<lb/>ſarum. </s> <s xml:id="echoid-s2579" xml:space="preserve">Et oporteat invenire curvam C D E cujus evolu-<lb/>tione paraboloides S B F deſcribatur.</s> <s xml:id="echoid-s2580" xml:space="preserve"/> </p> <div xml:id="echoid-div203" type="float" level="2" n="6"> <note position="left" xlink:label="note-0164-02" xlink:href="note-0164-02a" xml:space="preserve">TAB. XVI. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s2581" xml:space="preserve">Hic primum ratio B O ad B P facile invenitur, quia <lb/>tangentem paraboloidis in puncto B duci ſcimus, ſumpta S H <lb/>æquali {1/2} S K. </s> <s xml:id="echoid-s2582" xml:space="preserve">Cui tangenti cum B M ad angulos rectos in-<lb/>ſiſtat, dantur jam lineæ M H, H K, ac proinde earum in-<lb/>ter ſe ratio, quæ eſt eadem quæ O B ad B P.</s> <s xml:id="echoid-s2583" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2584" xml:space="preserve">Ut autem ratio B P, ſive K L ad M N innoteſcat, po-<lb/>nantur ad K L perpendiculares rectæ K T, L V, æquales <lb/>ſingulis K M, L N, ſitque V X parallela L K. </s> <s xml:id="echoid-s2585" xml:space="preserve">Jam quia <lb/>ex duabus ſimul K L, L N, auferendo K M, relinquitur <lb/>M N <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s2586" xml:space="preserve">hoc eſt, auferendo ex duabus X V, V L, ſive</s> </p> <p> <s xml:id="echoid-s2587" xml:space="preserve"> <lb/> <anchor type="note" xlink:label="note-0164-03a" xlink:href="note-0164-03"/> <pb file="0165" n="179"/> <pb file="0165a" n="180"/> <anchor type="figure" xlink:label="fig-0165a-01a" xlink:href="fig-0165a-01"/> <anchor type="figure" xlink:label="fig-0165a-02a" xlink:href="fig-0165a-02"/> <anchor type="figure" xlink:label="fig-0165a-03a" xlink:href="fig-0165a-03"/> <anchor type="figure" xlink:label="fig-0165a-04a" xlink:href="fig-0165a-04"/> <anchor type="figure" xlink:label="fig-0165a-05a" xlink:href="fig-0165a-05"/> <pb file="0166" n="181"/> <pb o="113" file="0167" n="182" rhead="HOROLOG. OSCILLATOR."/> X V, X K, ipſam K T; </s> <s xml:id="echoid-s2588" xml:space="preserve">hinc autem relinqui apparet V X <lb/> <anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/> & </s> <s xml:id="echoid-s2589" xml:space="preserve">X T: </s> <s xml:id="echoid-s2590" xml:space="preserve">erunt igitur hæ duæ V X, X T ipſi M N æqua-<lb/>les, ac proinde ratio K L ad M N eadem quæ V X ad <lb/>duas ſimul V X, X T. </s> <s xml:id="echoid-s2591" xml:space="preserve">Ut autem hæc ratio innoteſcat cum <lb/>intervallum K L eſt minimum; </s> <s xml:id="echoid-s2592" xml:space="preserve">oportet ſecundum prædicta <lb/>inquirere quis ſit locus, ſive linea ad quam ſunt puncta <lb/>T, V. </s> <s xml:id="echoid-s2593" xml:space="preserve">Quod ut fiat ſit latus rectum paraboloidis A B F = a; <lb/></s> <s xml:id="echoid-s2594" xml:space="preserve">S K = x; </s> <s xml:id="echoid-s2595" xml:space="preserve">K T = y.</s> <s xml:id="echoid-s2596" xml:space="preserve"/> </p> <div xml:id="echoid-div204" type="float" level="2" n="7"> <note symbol="*" position="foot" xlink:label="note-0164-03" xlink:href="note-0164-03a" xml:space="preserve">In Exemplari ſuo ad marginem ſcripſit Auctor. ſupponitur hic rectam L N <lb/>majorem eſſe quam K M, quod melius fuerat antea probari, etſi verum eſt. <lb/>Demonſtratio autem haud difficilis eſt, ſit abſciſſa S K = x; perpendicularis K B <lb/>= u; Tatus rectum paraboloidis = a. Quia S H = {1/2} SK, eſt H K = {3/2} S K <lb/>({3/2}x). Propter angulum rectum H B M, triangula rectangula H B K, K B M <lb/>ſimilia ſunt, & H K ({3/2}x), K B (u), K M, ſunt in continua proportione; ergo <lb/>K M = {2uu/3x}, cujus quadratum eſt {4u<emph style="super">4.</emph>/9xx} = {4au<emph style="super">4.</emph>/9axx}; ſed ut notavit auctor ex natu-<lb/>ra Paraboloidis A B F, u<emph style="super">3</emph> = axx; ergo quadratum lineæ K M = {4au<emph style="super">4</emph>/9axx} = {4au<emph style="super">4</emph>/9u<emph style="super">3</emph>} = <lb/>{4/9} a u unde ſequitur ipſam K M, augeri ſi creſcat B K (u). Cum autem L F exce-<lb/>dat B K, L N ſuperabit K M, quod demonſtrandum erat.</note> <figure xlink:label="fig-0165a-01" xlink:href="fig-0165a-01a"> <caption xml:id="echoid-caption52" style="it" xml:space="preserve">Pag. 112.<lb/>TAB. XV.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables52" xml:space="preserve">S D A B C E V</variables> </figure> <figure xlink:label="fig-0165a-02" xlink:href="fig-0165a-02a"> <caption xml:id="echoid-caption53" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables53" xml:space="preserve">F A E B K G H N L D M O C</variables> </figure> <figure xlink:label="fig-0165a-03" xlink:href="fig-0165a-03a"> <caption xml:id="echoid-caption54" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables54" xml:space="preserve">C D F A B K E G N H</variables> </figure> <figure xlink:label="fig-0165a-04" xlink:href="fig-0165a-04a"> <caption xml:id="echoid-caption55" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables55" xml:space="preserve">S M A N B K X T P L F V O C Y D E G H</variables> </figure> <figure xlink:label="fig-0165a-05" xlink:href="fig-0165a-05a"> <caption xml:id="echoid-caption56" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables56" xml:space="preserve">Y H A S B K T X F L V P O M N C D G E</variables> </figure> <note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO</emph>-<lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2597" xml:space="preserve">Quia igitur proportionales ſunt K H, K B, K M, eſt-<lb/>que H K = {1/2} x: </s> <s xml:id="echoid-s2598" xml:space="preserve">K B ex natura paraboloidis æqualis R. <lb/></s> <s xml:id="echoid-s2599" xml:space="preserve">cub. </s> <s xml:id="echoid-s2600" xml:space="preserve">a x x: </s> <s xml:id="echoid-s2601" xml:space="preserve">fiet K M, hoc eſt K T = {2/3} R. </s> <s xml:id="echoid-s2602" xml:space="preserve">cub. </s> <s xml:id="echoid-s2603" xml:space="preserve">a a x = y, <lb/>ac proinde {8/27} a a x = y<emph style="super">3</emph>. </s> <s xml:id="echoid-s2604" xml:space="preserve">Unde patet locum punctorum T, <lb/>V, eſſe paraboloidem illam, quam cubicam vocant geome-<lb/>træ. </s> <s xml:id="echoid-s2605" xml:space="preserve">Cui proinde ad T tangens ducetur, ſumptâ S Y duplâ <lb/>ipſius S K, junctâque Y T. </s> <s xml:id="echoid-s2606" xml:space="preserve">Et jam quidem ratio V X ad <lb/>duas ſimul V X, X T, quam diximus eandem eſſe ac K L <lb/>ad M N, erit ea quæ Y K ad utramque ſimul Y K, K T. </s> <s xml:id="echoid-s2607" xml:space="preserve"><lb/>Hæc autem ratio data eſt, ergo & </s> <s xml:id="echoid-s2608" xml:space="preserve">ratio K L ad M N. </s> <s xml:id="echoid-s2609" xml:space="preserve">Sed <lb/>& </s> <s xml:id="echoid-s2610" xml:space="preserve">rationem O B ad P B datam eſſe oſtenſum eſt. </s> <s xml:id="echoid-s2611" xml:space="preserve">Ergo, <lb/>cum ex duabus hiſce componatur ratio B D ad D M, ut ſu-<lb/>pra patuit, dabitur & </s> <s xml:id="echoid-s2612" xml:space="preserve">hæc; </s> <s xml:id="echoid-s2613" xml:space="preserve">& </s> <s xml:id="echoid-s2614" xml:space="preserve">dividendo, ratio B M ad <lb/>M D; </s> <s xml:id="echoid-s2615" xml:space="preserve">adeoque & </s> <s xml:id="echoid-s2616" xml:space="preserve">punctum D in curva D E.</s> <s xml:id="echoid-s2617" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2618" xml:space="preserve">Ad conſtructionem autem breviſſimam hoc pacto hic per-<lb/>veniemus. </s> <s xml:id="echoid-s2619" xml:space="preserve">K T ſive K M dicta fuit y. </s> <s xml:id="echoid-s2620" xml:space="preserve">Itaque M H erit y <lb/>+ {3/2} x. </s> <s xml:id="echoid-s2621" xml:space="preserve">Et M H ad H K, ſive O B ad B P, ut y + {3/2} x <lb/>ad {3/2} x. </s> <s xml:id="echoid-s2622" xml:space="preserve">ſive, ſumptis omnium duplis, ut 2 y + 3 x ad 3 x. <lb/></s> <s xml:id="echoid-s2623" xml:space="preserve">Deinde quia Y K = 3 x, erit Y K ad Y K + K T, ſi-<lb/>ve per prædicta, K L ad M N, ut 3 x ad 3 x + y. </s> <s xml:id="echoid-s2624" xml:space="preserve">Atqui <lb/>ex rationibus O B ad B P, & </s> <s xml:id="echoid-s2625" xml:space="preserve">K L ad M N, componi di-<lb/>ximus rationem B D ad D M. </s> <s xml:id="echoid-s2626" xml:space="preserve">Ergo ratio B D ad D M erit <lb/>compoſita ex rationibus 2 y + 3 x ad 3 x, & </s> <s xml:id="echoid-s2627" xml:space="preserve">3 x ad 3 x <lb/>+ y; </s> <s xml:id="echoid-s2628" xml:space="preserve">ideoque erit ea quæ 2 y + 3 x ad 3 x + y. </s> <s xml:id="echoid-s2629" xml:space="preserve">& </s> <s xml:id="echoid-s2630" xml:space="preserve">divi-<lb/>dendo, ratio B M ad M D, eadem quæ y ad 3 x + y.</s> <s xml:id="echoid-s2631" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2632" xml:space="preserve">Sit S Z perpendicularis ad S K, eique occurrat M B pro-<lb/>ducta in Z. </s> <s xml:id="echoid-s2633" xml:space="preserve">Quia ergo ratio B M ad M D inventa eſt ea quæ <lb/>y ad y + 3 x, hoc eſt quæ M K ad M K + 3 K S. </s> <s xml:id="echoid-s2634" xml:space="preserve">Sicut au- <pb o="114" file="0168" n="183" rhead="CHRISTIANI HUGENII"/> tem M K ad M K + 3 K S, ita M B ad M B + 3 B Z: <lb/></s> <s xml:id="echoid-s2635" xml:space="preserve"> <anchor type="note" xlink:label="note-0168-01a" xlink:href="note-0168-01"/> erit proinde M B ad M D ut M B ad M B + 3 B Z. </s> <s xml:id="echoid-s2636" xml:space="preserve">Un-<lb/>de liquet M D æqualem ſumendam ipſi M B + 3 B Z. </s> <s xml:id="echoid-s2637" xml:space="preserve">At-<lb/>que ita quotlibet puncta curvæ C D E invenire licebit. </s> <s xml:id="echoid-s2638" xml:space="preserve">Cu-<lb/>jus curvæ portio quælibet ut D S, rectæ D B, quæ para-<lb/>boloidi S A B ad angulos rectos occurrit, æqualis erit. <lb/></s> <s xml:id="echoid-s2639" xml:space="preserve">Conſtat autem geometricam eſſe, & </s> <s xml:id="echoid-s2640" xml:space="preserve">ſi velimus, poſſumus <lb/>æquatione aliqua relationem exprimere punctorum omnium <lb/>ipſius ad puncta axis S K.</s> <s xml:id="echoid-s2641" xml:space="preserve"/> </p> <div xml:id="echoid-div205" type="float" level="2" n="8"> <note position="left" xlink:label="note-0168-01" xlink:href="note-0168-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUMCUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph></note> </div> <p> <s xml:id="echoid-s2642" xml:space="preserve">Simili modo autem, ſi inquiramus in paraboloide illa ſive <lb/> <anchor type="note" xlink:label="note-0168-02a" xlink:href="note-0168-02"/> parabola cubica, in qua cubi ordinatim applicatarum ad <lb/>axem, ſunt inter ſe ſicut portiones axis abſciſſæ, inveniemus <lb/>curvam cujus evolutione deſcribitur, quæque proinde rectæ <lb/>lineæ æquari poterit, nihilo difficiliori conſtructione per pun-<lb/>cta determinari. </s> <s xml:id="echoid-s2643" xml:space="preserve">Nam ſi fuerit illa S A B; </s> <s xml:id="echoid-s2644" xml:space="preserve">axis S M; </s> <s xml:id="echoid-s2645" xml:space="preserve">(di-<lb/>citur autem improprie axis in hac curva, cum forma ejus ſit <lb/>ejusmodi, ut ductâ S Z, quæ ſecet S M ad angulos rectos, <lb/>ea portiones ſimiles curvæ habeat ad partes oppoſitas;) </s> <s xml:id="echoid-s2646" xml:space="preserve">aga-<lb/>rur per punctum quodlibet B, in paraboloide ſumptum, re-<lb/>cta B D, quæ curvam ad angulos rectos ſecet, axique ejus <lb/>occurrat in M, rectæ vero S Z in Z. </s> <s xml:id="echoid-s2647" xml:space="preserve">Deinde ſumatur B D <lb/>æqualis dimidiæ B M, unà cum ſesquialtera B Z. </s> <s xml:id="echoid-s2648" xml:space="preserve">Eritque <lb/>D unum è punctis curvæ quæſitæ R D vel R I, cujus evo-<lb/>lutione, juncta tamen recta quadam R A, deſcribetur para-<lb/>boloides S A B. </s> <s xml:id="echoid-s2649" xml:space="preserve">Sunt autem hic, quod notatu dignum eſt, <lb/>quodque in aliis etiam nonnullis harum paraboloidum con-<lb/>tingit, duæ evolutiones in partes contrarias, quarum utra-<lb/>que à puncto certo A initium capit; </s> <s xml:id="echoid-s2650" xml:space="preserve">ita ut evolutione ipſius <lb/>A R D, in infinitum porro continuatæ, deſcribatur para-<lb/>boloidis pars infinita A B F; </s> <s xml:id="echoid-s2651" xml:space="preserve">evolutione autem totius A R I, <lb/>ſimiliter in infinitum extenſæ, tantum particula A S. </s> <s xml:id="echoid-s2652" xml:space="preserve">Pun-<lb/>ctum autem A definitur, ſumptâ S P quæ ſit ad latus re-<lb/>ctum paraboloidis, ſicut unitas ad radicem quadrato-qua-<lb/>draticam numeri 91125, (is cubus eſt ex 45) applicatâque <lb/>ordinatim P A. </s> <s xml:id="echoid-s2653" xml:space="preserve">Unde porro punctum R, confinium dua-<lb/>rum curvarum R D, R I, invenitur ſicut cætera omnia ha- <pb file="0169" n="184"/> <pb file="0169a" n="185"/> <anchor type="figure" xlink:label="fig-0169a-01a" xlink:href="fig-0169a-01"/> <anchor type="figure" xlink:label="fig-0169a-02a" xlink:href="fig-0169a-02"/> <anchor type="figure" xlink:label="fig-0169a-03a" xlink:href="fig-0169a-03"/> <pb file="0170" n="186"/> <pb o="115" file="0171" n="187" rhead="HOROLOG. OSCILLATOR."/> rum curvarum, hoc eſt, ſicut punctum D modo inventum <lb/> <anchor type="note" xlink:label="note-0171-01a" xlink:href="note-0171-01"/> fuit.</s> <s xml:id="echoid-s2654" xml:space="preserve"/> </p> <div xml:id="echoid-div206" type="float" level="2" n="9"> <note position="left" xlink:label="note-0168-02" xlink:href="note-0168-02a" xml:space="preserve">TAB. XVII. <lb/>Fig. 1.</note> <figure xlink:label="fig-0169a-01" xlink:href="fig-0169a-01a"> <caption xml:id="echoid-caption57" style="it" xml:space="preserve">Pag. 114.<lb/>TAB. XVI.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables57" xml:space="preserve">M F E A K G N H B D C</variables> </figure> <figure xlink:label="fig-0169a-02" xlink:href="fig-0169a-02a"> <caption xml:id="echoid-caption58" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables58" xml:space="preserve">H A K B R P F L O M N D Q G E</variables> </figure> <figure xlink:label="fig-0169a-03" xlink:href="fig-0169a-03a"> <caption xml:id="echoid-caption59" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables59" xml:space="preserve">Y H A S Z X T K B V L P F O C M N D G E</variables> </figure> <note position="right" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO</emph>-<lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2655" xml:space="preserve">Denique, quæcunque fuerit ex paraboloidum genere cur-<lb/>va S A B, ſemper æque facile curvam aliam, cujus evolu-<lb/>tione ipſa deſcribatur, quæque propterea rectæ adæquari <lb/>poſſit, per puncta inveniri comperimus. </s> <s xml:id="echoid-s2656" xml:space="preserve">Atque adeo con-<lb/>ſtructionem univerſalem ſequenti tabella exhibemus; </s> <s xml:id="echoid-s2657" xml:space="preserve">quæ <lb/>quousque libuerit extendi poterit.</s> <s xml:id="echoid-s2658" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/># a x = y<emph style="super">2</emph> # # B M + 2 B Z <lb/># a<emph style="super">2</emph> x = y<emph style="super">3</emph> # # {1/2} B M + {3/2} B Z <lb/>Si # a x<emph style="super">2</emph> = y<emph style="super">3</emph> # Erit # 2 B M + 3 B Z # = # B D. <lb/># a x<emph style="super">3</emph> = y<emph style="super">4</emph> # # 3 B M + 4 B Z <lb/># a<emph style="super">3</emph> x = y<emph style="super">4</emph> # # {1/3} B M + {4/3} B Z <lb/></note> <p> <s xml:id="echoid-s2659" xml:space="preserve">Sit S B parabola, vel paraboloidum aliqua, cujus vertex <lb/> <anchor type="note" xlink:label="note-0171-03a" xlink:href="note-0171-03"/> S; </s> <s xml:id="echoid-s2660" xml:space="preserve">recta S K vel axis, vel axi perpendicularis, ad quam re-<lb/>feruntur æquatione puncta paraboloidis; </s> <s xml:id="echoid-s2661" xml:space="preserve">& </s> <s xml:id="echoid-s2662" xml:space="preserve">ipſa quidem S K <lb/>ſemper ad partem cavam ducta intelligitur; </s> <s xml:id="echoid-s2663" xml:space="preserve">cui perpendicu-<lb/>laris S Z. </s> <s xml:id="echoid-s2664" xml:space="preserve">Ponendo jam S K = x; </s> <s xml:id="echoid-s2665" xml:space="preserve">B K = y, quæ à pun-<lb/>cto quovis curvæ perpendicularis eſt ipſi S K; </s> <s xml:id="echoid-s2666" xml:space="preserve">& </s> <s xml:id="echoid-s2667" xml:space="preserve">latere re-<lb/>cto curvæ = a; </s> <s xml:id="echoid-s2668" xml:space="preserve">prior pars tabellæ, quæ ad ſiniſtram eſt, <lb/>naturam ſingularum paraboloidum ſingulis æquationibus ex-<lb/>plicat. </s> <s xml:id="echoid-s2669" xml:space="preserve">Quibus reſpondent in parte dextra quantitates lineæ <lb/>B D, quæ ſi curvæ S B inſiſtat ad angulos rectos, exhibi-<lb/>tura ſit punctum D in curva quæſita C D. </s> <s xml:id="echoid-s2670" xml:space="preserve">Exempli gratia, <lb/>ſi S B eſt parabola quæ ex coni ſectione fit, ei ſcimus con-<lb/>venire æquationem tabellæ primam, a x = y<emph style="super">2</emph>; </s> <s xml:id="echoid-s2671" xml:space="preserve">cui reſpon-<lb/>det ab altera parte B M + 2 B Z = B D. </s> <s xml:id="echoid-s2672" xml:space="preserve">Unde longitudo <lb/>lineæ B D cognoſcitur, adeoque inventio quotlibet puncto-<lb/>rum curvæ C D. </s> <s xml:id="echoid-s2673" xml:space="preserve">Quam quidem, hoc caſu, paraboloidem <lb/>eſſe ſupra demonſtratum fuit, eam nempe, cujus æquatio <lb/>tertia eſt hujus tabellæ.</s> <s xml:id="echoid-s2674" xml:space="preserve"/> </p> <div xml:id="echoid-div207" type="float" level="2" n="10"> <note position="right" xlink:label="note-0171-03" xlink:href="note-0171-03a" xml:space="preserve">TAB XVII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s2675" xml:space="preserve">Conſtruitur autem tabella hoc pacto, ut B M ſumatur <lb/>multiplex ſecundum numerum qui eſt exponens poteſtatis x <lb/>in æquatione; </s> <s xml:id="echoid-s2676" xml:space="preserve">B Z vero, multiplex ſecundum exponentem <lb/>poteſtatis y; </s> <s xml:id="echoid-s2677" xml:space="preserve">ex his autem utrisque compoſitæ accipiatur <lb/>pars denominata ab exponente poteſtatis a.</s> <s xml:id="echoid-s2678" xml:space="preserve"/> </p> <pb o="116" file="0172" n="188" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s2679" xml:space="preserve">Præter haſce autem paraboloides lineas, alias item inve-<lb/> <anchor type="note" xlink:label="note-0172-01a" xlink:href="note-0172-01"/> nimus, à quibus, non abſimili conſtructione, deducuntur <lb/>curvæ rectis comparabiles. </s> <s xml:id="echoid-s2680" xml:space="preserve">Aſſimilantur autem hyperbolis, <lb/>eo quod aſymptotos ſuas habent, ſed tantum angulum re-<lb/>ctum conſtituentes. </s> <s xml:id="echoid-s2681" xml:space="preserve">Et harum primam quidem ſtatuimus hy-<lb/>perbolam ipſam, quæ eſt è coni ſectione.</s> <s xml:id="echoid-s2682" xml:space="preserve"/> </p> <div xml:id="echoid-div208" type="float" level="2" n="11"> <note position="left" xlink:label="note-0172-01" xlink:href="note-0172-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUM CUR-</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO</emph>-<lb/><emph style="sc">NE</emph>.</note> </div> <p> <s xml:id="echoid-s2683" xml:space="preserve">Reliquarum vero naturam ut explicemus; </s> <s xml:id="echoid-s2684" xml:space="preserve">ſunto P S, S K, <lb/> <anchor type="note" xlink:label="note-0172-02a" xlink:href="note-0172-02"/> aſymptoti curvæ A B, rectum angulum comprehendentes, <lb/>& </s> <s xml:id="echoid-s2685" xml:space="preserve">à curvæ puncto quolibet B ducatur B K parallela P S, <lb/>ſitque S K = x; </s> <s xml:id="echoid-s2686" xml:space="preserve">K B = y. </s> <s xml:id="echoid-s2687" xml:space="preserve">Si igitur hyperbola ſit A B, <lb/>ſcimus rectangulum linearum S K, K B, hoc eſt, rectan-<lb/>gulum x y ſemper eidem quadrato æquale eſſe, quod voce-<lb/>tur a a.</s> <s xml:id="echoid-s2688" xml:space="preserve"/> </p> <div xml:id="echoid-div209" type="float" level="2" n="12"> <note position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">TAB. XVII. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s2689" xml:space="preserve">Proxima vero hyperboloidum erit, in quaſolidum ex qua-<lb/>drato lineæ S K, in altitudinem K B ductum, hoc eſt, ſo-<lb/>lidum x x y, cubo certo æquabitur, qui vocetur a<emph style="super">3</emph>. </s> <s xml:id="echoid-s2690" xml:space="preserve">Atque <lb/>ita innumeræ aliæ hujus generis hyperboloides exiſtunt, qua-<lb/>rum proprietatem ſequens tabella fingulis æquationibus ex-<lb/>hibet, ſimulque rationem conſtruendi curvam D C, cujus <lb/>evolutione quæque generetur.</s> <s xml:id="echoid-s2691" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/># x y = a<emph style="super">2</emph> # # {1/2} B M + {1/2} B Z <lb/># x<emph style="super">2</emph> y = a<emph style="super">3</emph> # # {2/3} B M + {1/3} B Z <lb/>Si # x y<emph style="super">2</emph> = a<emph style="super">3</emph> # Erit # {1/3} B M + {2/3} B Z # = B D <lb/># x<emph style="super">3</emph> y = a<emph style="super">4</emph> # # {3/4} B M + {1/4} B Z <lb/># x y<emph style="super">3</emph> = a<emph style="super">4</emph> # # {1/4} B M + {3/4} B Z <lb/></note> <p> <s xml:id="echoid-s2692" xml:space="preserve">Recta D B M Z curvam A B, ut antea quoque, ſecat <lb/>ad angulos rectos, occurritque aſymptotis S K, S P, in M <lb/>& </s> <s xml:id="echoid-s2693" xml:space="preserve">Z. </s> <s xml:id="echoid-s2694" xml:space="preserve">Si igitur exempli gratia hyperbola fuerit A B, cujus <lb/>æquatio eſt x y = a<emph style="super">2</emph>, ſumetur B D = {1/2} B M + {1/2} B Z, <lb/>quemadmodum tabella præcipit. </s> <s xml:id="echoid-s2695" xml:space="preserve">Eritque punctum Din cur-<lb/>va D C quæſita, cujus alia quotlibet puncta ſic inveniri po-<lb/>terunt, & </s> <s xml:id="echoid-s2696" xml:space="preserve">portio ejus quælibet rectæ lineæ adæquari. </s> <s xml:id="echoid-s2697" xml:space="preserve">Et <lb/>hæc quidem eadem illa eſt curva, cujus relationem ad axem <lb/>hyperbolæ ſuperius æquatione expreſſimus. </s> <s xml:id="echoid-s2698" xml:space="preserve">Conſtructio au-<lb/>tem tabellæ hujus plane eadem eſt quæ ſuperioris.</s> <s xml:id="echoid-s2699" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2700" xml:space="preserve">Cæterum, quoniam tum ad harum curvarum, tum ad ea- <pb o="117" file="0173" n="189" rhead="HOROLOG. OSCILLATOR."/> rum quæ ex paraboloidibus naſcuntur conſtructionem, du-<lb/> <anchor type="note" xlink:label="note-0173-01a" xlink:href="note-0173-01"/> cendæ ſunt lineæ D B Z, quæ ad datum punctum B ſecent <lb/>curvas A B, ſive ipſarum tangentes B H, ad angulos re-<lb/>ctos; </s> <s xml:id="echoid-s2701" xml:space="preserve">dicemus in univerſum quomodo hæ tangentes inve-<lb/>niantur. </s> <s xml:id="echoid-s2702" xml:space="preserve">In æquatione itaque, quæ cujusque curvæ naturam <lb/>explicat, quales æquationes duabus tabellis præcedentibus <lb/>exponuntur, conſiderare oportet quæ ſint exponentes pote-<lb/>ſtatum x & </s> <s xml:id="echoid-s2703" xml:space="preserve">y, & </s> <s xml:id="echoid-s2704" xml:space="preserve">facere ut, ſicut exponens poteſtatis x ad <lb/>exponentem poteſtatis y, ita ſit S K ad K H. </s> <s xml:id="echoid-s2705" xml:space="preserve">Juncta enim <lb/>H B curvam in B continget. </s> <s xml:id="echoid-s2706" xml:space="preserve">Velut in tertia hyperboloide, <lb/>cujus æquatio eſt x y<emph style="super">2</emph> = a<emph style="super">3</emph>: </s> <s xml:id="echoid-s2707" xml:space="preserve">quia exponens poteſtatis x eſt <lb/>1, poteſtatis autem y exponens 2; </s> <s xml:id="echoid-s2708" xml:space="preserve">oportet eſſe ut 1 ad 2 ita <lb/>S K ad K H. </s> <s xml:id="echoid-s2709" xml:space="preserve">Horum autem demonſtrationem noverunt <lb/>analyticæ artis periti, qui jam pridem omnes has lineas con-<lb/>templari cœperunt; </s> <s xml:id="echoid-s2710" xml:space="preserve">& </s> <s xml:id="echoid-s2711" xml:space="preserve">non ſolum paraboloidum iſtarum, <lb/>ſed & </s> <s xml:id="echoid-s2712" xml:space="preserve">ſpatiorum quorundam infinitorum, inter hyperboloi-<lb/>des & </s> <s xml:id="echoid-s2713" xml:space="preserve">aſymptotos interjectorum, plana ſolidaque dimenſi <lb/>ſunt. </s> <s xml:id="echoid-s2714" xml:space="preserve">Quod quidem & </s> <s xml:id="echoid-s2715" xml:space="preserve">nos, facili atque univerſali metho-<lb/>do, expedire poſſemus, ex ſola tangentium proprietate ſum-<lb/>pta demonſtratione. </s> <s xml:id="echoid-s2716" xml:space="preserve">Sed illa non ſunt hujus loci.</s> <s xml:id="echoid-s2717" xml:space="preserve"/> </p> <div xml:id="echoid-div210" type="float" level="2" n="13"> <note position="right" xlink:label="note-0173-01" xlink:href="note-0173-01a" xml:space="preserve"><emph style="sc">De linea-</emph> <lb/><emph style="sc">RUMCUR.</emph> <lb/><emph style="sc">VARUM</emph> <lb/><emph style="sc">EVOLUTIO-</emph> <lb/><emph style="sc">NE.</emph></note> </div> <figure> <image file="0173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0173-01"/> </figure> </div> <div xml:id="echoid-div212" type="section" level="1" n="72"> <head xml:id="echoid-head96" xml:space="preserve">HOROLOGII OSCILLATORII <lb/>PARS QUARTA.</head> <head xml:id="echoid-head97" style="it" xml:space="preserve">De centro Oſcillationis.</head> <p> <s xml:id="echoid-s2718" xml:space="preserve">CEntrorum Oſcillationis, ſeu Agitationis, inveſtigatio-<lb/>nem olim mihi, fere adhuc puero, aliiſque multis, do-<lb/>ctiſſimus Merſennus propoſuit, celebre admodum inter illius <lb/>temporis Geometras problema, prout ex litteris ejus ad me <lb/>datis colligo, nec non ex Carteſii haud pridem editis, qui-<lb/>bus ad Merſennianas ſuper his rebus reſponſum continetur. <lb/></s> <s xml:id="echoid-s2719" xml:space="preserve">Poſtulabat autem centra illa ut invenirem in circuli ſectori- <pb o="118" file="0174" n="190" rhead="CHRISTIANI HUGENII"/> bus, tam ab angulo quam à medio arcu ſuſpenſis, atque in <lb/> <anchor type="note" xlink:label="note-0174-01a" xlink:href="note-0174-01"/> latus agitatis, item in circuli ſegmentis, & </s> <s xml:id="echoid-s2720" xml:space="preserve">in triangulis, <lb/>nunc ex vertice, nunc ex media baſi pendentibus. </s> <s xml:id="echoid-s2721" xml:space="preserve">Quod eo <lb/>redit, ut pendulum ſimplex, hoc eſt, pondus filo appenſum <lb/>reperiatur ea longitudine, ut oſcillationes faciat temporum <lb/>eorundem ac figuræ iſtæ, uti dictum eſt, ſuſpenſæ. </s> <s xml:id="echoid-s2722" xml:space="preserve">Simul <lb/>vero pretium operæ, ſi forte quæſitis ſatisfeciſſem, magnum <lb/>ſane & </s> <s xml:id="echoid-s2723" xml:space="preserve">invidioſum pollicebatur. </s> <s xml:id="echoid-s2724" xml:space="preserve">Sed a nemine id quod deſi-<lb/>derabat tunc obtinuit. </s> <s xml:id="echoid-s2725" xml:space="preserve">Nam me quod attinet, cum nihil re-<lb/>perirem quo vel primus aditus ad contemplationem eam pa-<lb/>teſceret; </s> <s xml:id="echoid-s2726" xml:space="preserve">velut à limine repulſus, longiori inveſtigatione tunc <lb/>quidem abſtinui. </s> <s xml:id="echoid-s2727" xml:space="preserve">Qui vero rem ſeſe confeciſſe ſperabant viri <lb/>inſignes, Carteſius, Honoratus Fabrius, aliique, nequaquam <lb/>ſcopum attigerunt, niſi in paucis quibuſdam facilioribus, <lb/>ſed quorum tamen demonſtrationem nullam idoneam, ut mi-<lb/>hi videtur, attulerunt. </s> <s xml:id="echoid-s2728" xml:space="preserve">Idque comparatione eorum quæ hic <lb/>trademus manifeſtum fore ſpero, ſi quis forte quæ ab illis <lb/>tradita ſunt, cum noſtris hiſce contulerit; </s> <s xml:id="echoid-s2729" xml:space="preserve">quæ quidem & </s> <s xml:id="echoid-s2730" xml:space="preserve"><lb/>certioribus principiis demonſtrata arbitror, & </s> <s xml:id="echoid-s2731" xml:space="preserve">experimentis <lb/>prorſus convenientia reperi. </s> <s xml:id="echoid-s2732" xml:space="preserve">Occaſio vero ad hæc denuo ten-<lb/>tanda, ex pendulorum automati noſtri temperandorum ratio-<lb/>ne oblata eſt, dum pondus mobile, præter id quod in imo <lb/>eſt, illis applico, ut in deſcriptione horologii fuit explica-<lb/>tum. </s> <s xml:id="echoid-s2733" xml:space="preserve">Hinc melioribus auſpiciis atque à prima origine rem <lb/>exorſus, tandem difficultates omnes ſuperavi, nec tantum <lb/>problematum Merſennianorum ſolutionem, ſed alia quoque <lb/>illis difficiliora reperi, & </s> <s xml:id="echoid-s2734" xml:space="preserve">viam denique, qua in lineis, ſu-<lb/>perficiebus, ſolidiſque corporibus certa ratione centrum illud <lb/>inveſtigare liceret. </s> <s xml:id="echoid-s2735" xml:space="preserve">Unde quidem, præter voluptatem inve-<lb/>niendi quæ multum ab aliis quæſita fuerant, cognoſcendique <lb/>in his rebus naturæ leges decretaque, utilitatem quoque <lb/>eam cepi, cujus gratia primo animum ad hæc applicueram, <lb/>reperta illa horologii temperandi ratione facili & </s> <s xml:id="echoid-s2736" xml:space="preserve">expedita. <lb/></s> <s xml:id="echoid-s2737" xml:space="preserve">Acceſſit autem hoc quoque, quod pluris faciendum arbitror, <lb/>ut certæ, ſæculiſque omnibus duraturæ, menſuræ defini-<lb/>tionem abſolutiſſimam per hæc tradere poſſem; </s> <s xml:id="echoid-s2738" xml:space="preserve">qualis eſt ea <lb/>quæ ad finem horum adjecta reperietur.</s> <s xml:id="echoid-s2739" xml:space="preserve"/> </p> <div xml:id="echoid-div212" type="float" level="2" n="1"> <note position="left" xlink:label="note-0174-01" xlink:href="note-0174-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <pb o="119" file="0175" n="191" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div214" type="section" level="1" n="73"> <head xml:id="echoid-head98" xml:space="preserve">DEFINITIONES.</head> <note position="right" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <div xml:id="echoid-div215" type="section" level="1" n="74"> <head xml:id="echoid-head99" xml:space="preserve">I.</head> <p style="it"> <s xml:id="echoid-s2740" xml:space="preserve">PEndulum dicatur figura quælibet gravitate præ-<lb/>dita, ſive linea fuerit, ſive ſuperficies, ſive ſo-<lb/>lidum, ita ſuſpenſa ut circa punctum aliquod, vel <lb/>axem potius, qui plano horizontis parallelus intel-<lb/>ligitur, motum reciprocum vi gravitatis ſuæ con-<lb/>tinuare poſſit.</s> <s xml:id="echoid-s2741" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div216" type="section" level="1" n="75"> <head xml:id="echoid-head100" xml:space="preserve">II.</head> <p style="it"> <s xml:id="echoid-s2742" xml:space="preserve">Axis ille horizontis plano parallelus, circa quem <lb/>penduli motus fieri intelligitur, dicatur axis Oſcil-<lb/>cillationis.</s> <s xml:id="echoid-s2743" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div217" type="section" level="1" n="76"> <head xml:id="echoid-head101" xml:space="preserve">III.</head> <p style="it"> <s xml:id="echoid-s2744" xml:space="preserve">Pendulum ſimplex dicatur quod filo vel linea in-<lb/>flexili, gravitatis experte, conſtare intelligitur, <lb/>ima ſui parte pondus affixum gerente; </s> <s xml:id="echoid-s2745" xml:space="preserve">cujus pon-<lb/>deris gravitas, velut in unum punctum collecta, <lb/>cenſenda eſt.</s> <s xml:id="echoid-s2746" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div218" type="section" level="1" n="77"> <head xml:id="echoid-head102" xml:space="preserve">IV.</head> <p style="it"> <s xml:id="echoid-s2747" xml:space="preserve">Pendulum verò compoſitum, quod pluribus pon-<lb/>deribus conſtat, immutabiles diſtantias ſervantibus, <lb/>tum inter ſe, tum ab axe Oſcillationis. </s> <s xml:id="echoid-s2748" xml:space="preserve">Hinc figura <lb/>quælibet ſuſpenſa, ac gravitate prædita, pendu-<lb/>lum compoſitum dici poteſt, quatenus cogitatu in <lb/>partes quotlibet eſt diviſibilis.</s> <s xml:id="echoid-s2749" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div219" type="section" level="1" n="78"> <head xml:id="echoid-head103" xml:space="preserve">V.</head> <p style="it"> <s xml:id="echoid-s2750" xml:space="preserve">Pendula iſochrona vocentur, quorum Oſcillatio- <pb o="120" file="0176" n="192" rhead="CHRISTIANI HUGENII"/> nes, per arcus ſimiles, æqualibus temporibus per-<lb/> <anchor type="note" xlink:label="note-0176-01a" xlink:href="note-0176-01"/> aguntur.</s> <s xml:id="echoid-s2751" xml:space="preserve"/> </p> <div xml:id="echoid-div219" type="float" level="2" n="1"> <note position="left" xlink:label="note-0176-01" xlink:href="note-0176-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> </div> <div xml:id="echoid-div221" type="section" level="1" n="79"> <head xml:id="echoid-head104" xml:space="preserve">VI.</head> <p style="it"> <s xml:id="echoid-s2752" xml:space="preserve">Planum Oſcillationis dicatur illud, quod per cen-<lb/>trum gravitatis figuræ ſuſpenſæ duci intelligitur, <lb/>ad axem oſcillationis rectum.</s> <s xml:id="echoid-s2753" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div222" type="section" level="1" n="80"> <head xml:id="echoid-head105" xml:space="preserve">VII.</head> <p style="it"> <s xml:id="echoid-s2754" xml:space="preserve">Linea centri, recta quæ per centrum gravitatis <lb/>figuræ ducitur, ad axem oſcillationis perpendicula-<lb/>ris.</s> <s xml:id="echoid-s2755" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div223" type="section" level="1" n="81"> <head xml:id="echoid-head106" xml:space="preserve">VIII.</head> <p style="it"> <s xml:id="echoid-s2756" xml:space="preserve">Linea perpendiculi, recta in plano oſcillationis, <lb/>ducta ab axe oſcillationis, ad horizontis planum per-<lb/>pendicularis.</s> <s xml:id="echoid-s2757" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div224" type="section" level="1" n="82"> <head xml:id="echoid-head107" xml:space="preserve">IX.</head> <p style="it"> <s xml:id="echoid-s2758" xml:space="preserve">Centrum oſcillationis vel agitationis figuræ cu-<lb/>juslibet, dicatur punctum in linea centri, tantum <lb/>ab axe oſcillationis diſtans, quanta eſt longitudo <lb/>penduli ſimplicis quod figuræ iſochronum ſit.</s> <s xml:id="echoid-s2759" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div225" type="section" level="1" n="83"> <head xml:id="echoid-head108" xml:space="preserve">X.</head> <p style="it"> <s xml:id="echoid-s2760" xml:space="preserve">Axis gravitatis, linea quævis recta, per cen-<lb/>trum gravitatis figuræ tranſiens.</s> <s xml:id="echoid-s2761" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div226" type="section" level="1" n="84"> <head xml:id="echoid-head109" xml:space="preserve">XI.</head> <p style="it"> <s xml:id="echoid-s2762" xml:space="preserve">Figura plana, vel linea in plano ſita, in pla-<lb/>num agitari dicatur, cum axis oſcillationis in eo-<lb/>dem cum figura lineave eſt plano.</s> <s xml:id="echoid-s2763" xml:space="preserve"/> </p> <pb o="121" file="0177" n="193" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div227" type="section" level="1" n="85"> <head xml:id="echoid-head110" xml:space="preserve">XII.</head> <note position="right" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> <p style="it"> <s xml:id="echoid-s2764" xml:space="preserve">Eædem vero in latus agitari dicantur, cum axis <lb/>oſcillationis ad figuræ lineæve planum rectus eſt.</s> <s xml:id="echoid-s2765" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div228" type="section" level="1" n="86"> <head xml:id="echoid-head111" xml:space="preserve">XIII.</head> <p style="it"> <s xml:id="echoid-s2766" xml:space="preserve">Quando pondera in rectas lineas duci dicentur, <lb/>id ita eſt intelligendum, ac ſi numeri lineæve, <lb/>quantitates ponderum rationemque inter ſemutuam <lb/>exprimentes, ita ducantur.</s> <s xml:id="echoid-s2767" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div229" type="section" level="1" n="87"> <head xml:id="echoid-head112" xml:space="preserve">HYPOTHESES.</head> <head xml:id="echoid-head113" xml:space="preserve">I.</head> <p style="it"> <s xml:id="echoid-s2768" xml:space="preserve">SI pondera quotlibet, vi gravitatis ſuæ, moveri <lb/>incipiant; </s> <s xml:id="echoid-s2769" xml:space="preserve">non poſſe centrum gravitatis ex ipſis <lb/>compoſitæ altius, quam ubi incipiente motu repe-<lb/>riebatur, aſcendere.</s> <s xml:id="echoid-s2770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2771" xml:space="preserve">Altitudo autem in his ſecundum diſtantiam à plano hori-<lb/>zontali conſideratur, graviaque ponuntur ad hoc planum, <lb/>ſecundum rectas ipſi perpendiculares, deſcendere conari. <lb/></s> <s xml:id="echoid-s2772" xml:space="preserve">Quod idem ab omnibus, qui de centro gravitatis egerunt, <lb/>vel ponitur expreſſe, vel à legentibus ſupplendum eſt, cum abs-<lb/>que eo centri gravitatis conſideratio locum non habeat.</s> <s xml:id="echoid-s2773" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2774" xml:space="preserve">Ipſa vero hypotheſis noſtra quominus ſcrupulum moveat, <lb/>nihil aliud ſibi velle eam oſtendemus, quam quod nemo un-<lb/>quam negavit, gravia nempe ſurſum non ferri. </s> <s xml:id="echoid-s2775" xml:space="preserve">Nam primo, <lb/>ſi unum quodpiam corpus grave proponamus, illud vi gravi-<lb/>tatis ſuæ altius aſcendere non poſſe extra dubium eſt. </s> <s xml:id="echoid-s2776" xml:space="preserve">aſcen-<lb/>dere autem tunc intelligitur ſcilicet, cum ejus centrum gra-<lb/>vitatis aſcendit. </s> <s xml:id="echoid-s2777" xml:space="preserve">Sed & </s> <s xml:id="echoid-s2778" xml:space="preserve">idem de quotlibet ponderibus, in-<lb/>ter ſe per lineas inflexiles conjunctis, concedi neceſſe eſt, <lb/>quoniam nihil vetat ipſa tanquam unum aliquod conſiderari.</s> <s xml:id="echoid-s2779" xml:space="preserve"> <pb o="122" file="0178" n="194" rhead="CHRISTIANI HUGENII"/> Itaque neque horum commune gravitatis centrum ultro aſcen-<lb/> <anchor type="note" xlink:label="note-0178-01a" xlink:href="note-0178-01"/> dere poterit.</s> <s xml:id="echoid-s2780" xml:space="preserve"/> </p> <div xml:id="echoid-div229" type="float" level="2" n="1"> <note position="left" xlink:label="note-0178-01" xlink:href="note-0178-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2781" xml:space="preserve">Quod ſi jam pondera quotlibet non inter fe connexa po-<lb/>nantur, illorum quoque aliquod commune centrum gravita-<lb/>tis eſſe ſcimus. </s> <s xml:id="echoid-s2782" xml:space="preserve">Cujus quidem centri quanta erit altitudo, <lb/>tantam ajo & </s> <s xml:id="echoid-s2783" xml:space="preserve">gravitatis ex omnibus compoſitæ altitudinem <lb/>cenſeri debere; </s> <s xml:id="echoid-s2784" xml:space="preserve">ſiquidem omnia ad eandem illam centri gra-<lb/>vitatis altitudinem deduci poſſunt, nullâ accerſitâ po-<lb/>tentiâ quam quæ ipſis ponderibus ineſt, ſed tantum lineis <lb/>inflexilibus ea pro lubitu conjungendo, ac circa gravitatis <lb/>centrum movendo; </s> <s xml:id="echoid-s2785" xml:space="preserve">ad quod nulla vi neque potentia deter-<lb/>minata opus eſt. </s> <s xml:id="echoid-s2786" xml:space="preserve">Quare, ſicut fieri non poteſt ut pondera <lb/>quædam, in plano eodem horizontali poſita, ſupra illud <lb/>planum, vi gravitatis ſuæ, omnia æqualiter attollantur; </s> <s xml:id="echoid-s2787" xml:space="preserve">ita <lb/>nec quorumlibet ponderum, quomodocunque diſpoſitorum, <lb/>centrum gravitatis ad majorem quam habet altitudinem per-<lb/>venire poterit. </s> <s xml:id="echoid-s2788" xml:space="preserve">Quod autem diximus pondera quælibet, <lb/>nulla adhibita vi, ad planum horizontale, per centrum <lb/>commune gravitatis eorum tranſiens, perduci poſſe, ſic <lb/>oſtendetur.</s> <s xml:id="echoid-s2789" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2790" xml:space="preserve">Sint pondera A, B, C, poſitione data, quorum commu-<lb/> <anchor type="note" xlink:label="note-0178-02a" xlink:href="note-0178-02"/> ne gravitatis centrum ſit D. </s> <s xml:id="echoid-s2791" xml:space="preserve">per quod planum horizontale <lb/>ductum ponatur, cujus ſectio recta E F. </s> <s xml:id="echoid-s2792" xml:space="preserve">Sint jam lineæ in-<lb/>flexiles D A, D B, D C, quæ pondera ſibi invariabiliter <lb/>connectant; </s> <s xml:id="echoid-s2793" xml:space="preserve">quæ porro moveantur, donec A ſit in plano <lb/>E F ad E. </s> <s xml:id="echoid-s2794" xml:space="preserve">Virgis vero omnibus per æquales angulos dela-<lb/>tis, erunt jam B in G, & </s> <s xml:id="echoid-s2795" xml:space="preserve">C in H.</s> <s xml:id="echoid-s2796" xml:space="preserve"/> </p> <div xml:id="echoid-div230" type="float" level="2" n="2"> <note position="left" xlink:label="note-0178-02" xlink:href="note-0178-02a" xml:space="preserve">TAB. XVII. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s2797" xml:space="preserve">Rurſus jam B & </s> <s xml:id="echoid-s2798" xml:space="preserve">C connecti intelligantur virgâ H G, quæ <lb/>ſecet planum E F in F; </s> <s xml:id="echoid-s2799" xml:space="preserve">ubi neceſſario quoque erit centrum <lb/>gravitatis binorum iſtorum ponderum connexorum, cum <lb/>trium, in E, G, H, poſitorum, centrum gravitatis ſit D, <lb/>& </s> <s xml:id="echoid-s2800" xml:space="preserve">ejus quod eſt in E, centrum gravitatis ſit quoque in pla-<lb/>no E D F. </s> <s xml:id="echoid-s2801" xml:space="preserve">Moventur igitur rurſus pondera H, G, ſuper <lb/>puncto F, velut axe, absque vi ulla, ac ſimul utraque ad <lb/>planum E F adducuntur, adeo ut jam tria, quæ prius erant <lb/>in A, B, C, ad ipſam ſui centri gravitatis D altitudinem, <pb file="0179" n="195"/> <pb file="0179a" n="196"/> <anchor type="figure" xlink:label="fig-0179a-01a" xlink:href="fig-0179a-01"/> <anchor type="figure" xlink:label="fig-0179a-02a" xlink:href="fig-0179a-02"/> <anchor type="figure" xlink:label="fig-0179a-03a" xlink:href="fig-0179a-03"/> <anchor type="figure" xlink:label="fig-0179a-04a" xlink:href="fig-0179a-04"/> <pb file="0180" n="197"/> <pb o="123" file="0181" n="198" rhead="HOROLOG. OSCILLATOR."/> ſuo ipſorum æquilibrio, translata appareat. </s> <s xml:id="echoid-s2802" xml:space="preserve">quod erat oſten-<lb/> <anchor type="note" xlink:label="note-0181-01a" xlink:href="note-0181-01"/> dendum. </s> <s xml:id="echoid-s2803" xml:space="preserve">Eademque de quotcunque aliis eſt demonſtratio.</s> <s xml:id="echoid-s2804" xml:space="preserve"/> </p> <div xml:id="echoid-div231" type="float" level="2" n="3"> <figure xlink:label="fig-0179a-01" xlink:href="fig-0179a-01a"> <caption xml:id="echoid-caption60" style="it" xml:space="preserve">Pag. 122<lb/>TAB. XVII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables60" xml:space="preserve">S A P B R M D I</variables> </figure> <figure xlink:label="fig-0179a-02" xlink:href="fig-0179a-02a"> <caption xml:id="echoid-caption61" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables61" xml:space="preserve">H S Z K B C M D</variables> </figure> <figure xlink:label="fig-0179a-03" xlink:href="fig-0179a-03a"> <caption xml:id="echoid-caption62" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables62" xml:space="preserve">P S Z M A B K D H</variables> </figure> <figure xlink:label="fig-0179a-04" xlink:href="fig-0179a-04a"> <caption xml:id="echoid-caption63" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables63" xml:space="preserve">H C A E D F B G</variables> </figure> <note position="right" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve"><emph style="sc">De centr@</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2805" xml:space="preserve">Hæc autem hypotheſis noſtra ad liquida etiam corpora <lb/>valet, ac per eam non ſolum omnia illa, quæ de innatanti-<lb/>bus habet Archimedes, demonſtrari poſſunt, ſed & </s> <s xml:id="echoid-s2806" xml:space="preserve">alia ple-<lb/>raque Mechanicæ theoremata. </s> <s xml:id="echoid-s2807" xml:space="preserve">Et ſanè, ſi hac eadem uti <lb/>ſcirent novorum operum machinatores, qui motum perpe-<lb/>tuum irrito conatu moliuntur, facile ſuos ipſi errores depre-<lb/>henderent, intelligerentque rem eam mechanica ratione haud-<lb/>quaquam poſſibilem eſſe.</s> <s xml:id="echoid-s2808" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div233" type="section" level="1" n="88"> <head xml:id="echoid-head114" xml:space="preserve">II.</head> <p style="it"> <s xml:id="echoid-s2809" xml:space="preserve">Remoto aëris, alioque omni impedimento mani-<lb/>feſto, quemadmodum in ſequentibus demonſtratio-<lb/>nibus id intelligivolumus, centrum gravitatis pen-<lb/>duli agitati, æquales arcus deſcendendo ac aſcen-<lb/>dendo percurrere.</s> <s xml:id="echoid-s2810" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2811" xml:space="preserve">De pendulo ſimplici hoc demonſtratum eſt propoſitione 9 <lb/>de Deſcenſu gravium. </s> <s xml:id="echoid-s2812" xml:space="preserve">Idem vero & </s> <s xml:id="echoid-s2813" xml:space="preserve">de compoſito tenendum <lb/>eſſe declarat experientia; </s> <s xml:id="echoid-s2814" xml:space="preserve">ſiquidem, quæcunque fuerit pen-<lb/>duli figura, æque apta continuando motui reperitur, niſi in <lb/>quantum plus minusve aëris objectu impeditur.</s> <s xml:id="echoid-s2815" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div234" type="section" level="1" n="89"> <head xml:id="echoid-head115" xml:space="preserve">PROPOSITIO I.</head> <p style="it"> <s xml:id="echoid-s2816" xml:space="preserve">POnderibus quotlibet ad eandem partem plani <lb/>exiſtentibus, ſi à ſingulorum centris gravitatis <lb/>agantur in planum illud perpendiculares; </s> <s xml:id="echoid-s2817" xml:space="preserve">hæ ſin-<lb/>gulæ in ſua pondera ductæ, tantundem ſimul effi-<lb/>cient, ac perpendicularis, à centro gravitatis pon-<lb/>derum omnium in planum idem cadens, ducta in <lb/>pondera omnia.</s> <s xml:id="echoid-s2818" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2819" xml:space="preserve">Sint pondera A, B, C, ſita ad eandem partem plani, <lb/> <anchor type="note" xlink:label="note-0181-02a" xlink:href="note-0181-02"/> <pb o="124" file="0182" n="199" rhead="CHRISTIANI HUGENII"/> cujus ſectio recta D F, inque ipſum à ſingulis ponderibus <lb/> <anchor type="note" xlink:label="note-0182-01a" xlink:href="note-0182-01"/> ducantur perpendiculares A D, B E, C F. </s> <s xml:id="echoid-s2820" xml:space="preserve">Sit autem G <lb/>punctum centrum gravitatis ponderum omnium A, B, C, <lb/>à quo ducatur perpendicularis in idem planum G H. </s> <s xml:id="echoid-s2821" xml:space="preserve">Dico <lb/>ſummam productorum, quæ fiunt à ſingulis ponderibus in <lb/>ſuas perpendiculares, æquari producto ab recta G H in <lb/>omnia pondera A, B, C.</s> <s xml:id="echoid-s2822" xml:space="preserve"/> </p> <div xml:id="echoid-div234" type="float" level="2" n="1"> <note position="right" xlink:label="note-0181-02" xlink:href="note-0181-02a" xml:space="preserve">TAB. XVII@ <lb/>Fig. 1.</note> <note position="left" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2823" xml:space="preserve">Intelligantur enim perpendiculares, à ſingulis ponderibus <lb/>eductæ, continuari in alteram partem plani D F, ſintque <lb/>ſingulæ D K, E L, F M, ipſi H G æquales; </s> <s xml:id="echoid-s2824" xml:space="preserve">omnesque <lb/>lineæ, inflexiles virgas referant, ad horizontem parallelas; <lb/></s> <s xml:id="echoid-s2825" xml:space="preserve">& </s> <s xml:id="echoid-s2826" xml:space="preserve">ponantur in K, L, M, gravitates ejusmodi, quæ ſingu-<lb/>læ cum ſibi oppoſitis A, B, C, æquilibrium faciant ad in-<lb/>terſectionem plani D E F. </s> <s xml:id="echoid-s2827" xml:space="preserve">Omnes igitur K, L, M, æqui-<lb/>ponderabunt omnibus A, B, C. </s> <s xml:id="echoid-s2828" xml:space="preserve">Erit autem, ſicut longitu-<lb/>do A D ad D K, ita pondus K ad pondus A, ac proinde <lb/>D A ducta in magnitudinem A, æquabitur D K, ſive G H, <lb/>ductæ in K. </s> <s xml:id="echoid-s2829" xml:space="preserve">Similiter E B in B æquabitur E L, ſive G H, <lb/>in L; </s> <s xml:id="echoid-s2830" xml:space="preserve">& </s> <s xml:id="echoid-s2831" xml:space="preserve">F C in C æquabitur F M, ſive G H, in M. </s> <s xml:id="echoid-s2832" xml:space="preserve">Er-<lb/>go ſumma productorum ex A D in A, B E in B, C F in <lb/>F, æquabitur ſummæ productorum ex G H in omnes <lb/>K, L, M. </s> <s xml:id="echoid-s2833" xml:space="preserve">Quum autem K, L, M, æquiponderent ipſis A, <lb/>B, C, etiam iisdem A, B, C, ex centro ipſorum gravita-<lb/>tis G ſuſpenſis, æquiponderabunt. </s> <s xml:id="echoid-s2834" xml:space="preserve">Unde, cum diſtantia <lb/>G H æqualis ſit ſingulis D K, E L, F M, neceſſe eſt ma-<lb/>gnitudines A, B, C, ſimul ſumptas, æquari ipſis <lb/>K, L, M. </s> <s xml:id="echoid-s2835" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s2836" xml:space="preserve">ſumma productorum ex G H in omnes <lb/>A, B, C, æquabitur productis ex D A in A, E B in B, & </s> <s xml:id="echoid-s2837" xml:space="preserve"><lb/>F C in C. </s> <s xml:id="echoid-s2838" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s2839" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2840" xml:space="preserve">Etſi vero in demonſtratione poſitæ fuerint rectæ A K, B L, <lb/>C M, horizonti parallelæ, & </s> <s xml:id="echoid-s2841" xml:space="preserve">planum ad horizontem ere-<lb/>ctum; </s> <s xml:id="echoid-s2842" xml:space="preserve">patet, ſi omnia ſimul in alium quemlibet ſitum trans-<lb/>ponantur, eandem manere productorum æqualitatem, cum <lb/>rectæ omnes ſint eædem quæ prius. </s> <s xml:id="echoid-s2843" xml:space="preserve">Quare conſtat propo-<lb/>ſitum.</s> <s xml:id="echoid-s2844" xml:space="preserve"/> </p> <pb o="125" file="0183" n="200" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div236" type="section" level="1" n="90"> <head xml:id="echoid-head116" xml:space="preserve">PROPOSITIO II.</head> <note position="right" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph> <lb/>TAB. XVIII. <lb/>Fig. 1.</note> <p style="it"> <s xml:id="echoid-s2845" xml:space="preserve">POſitis quæ prius, ſi pondera omnia A, B, C, <lb/>ſint æqualia; </s> <s xml:id="echoid-s2846" xml:space="preserve">dico ſummam omnium perpendi-<lb/>cularium A D, B E, C F, æquari perpendicula-<lb/>ri, à centro gravitatis ductæ, G H, multiplici <lb/>ſecundum ponderum numerum.</s> <s xml:id="echoid-s2847" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2848" xml:space="preserve">Quum enim ſumma productorum, à ponderibus ſingulis <lb/>in ſuas perpendiculares, æquetur producto ex G H in pon-<lb/>dera omnia; </s> <s xml:id="echoid-s2849" xml:space="preserve">ſitque hìc, propter ponderum æqualitatem, <lb/>ſumma illa productorum æqualis producto ex uno pondere <lb/>in ſummam omnium perpendicularium; </s> <s xml:id="echoid-s2850" xml:space="preserve">itemque productum <lb/>ex G H in pondera omnia, idem quod productum ex pon-<lb/>dere uno in G H, multiplicem ſecundum ponderum nume-<lb/>rum: </s> <s xml:id="echoid-s2851" xml:space="preserve">patet ſummam perpendicularium neceſſario jam æquari <lb/>ipſi G H, multiplici ſecundum ponderum numerum. </s> <s xml:id="echoid-s2852" xml:space="preserve">quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s2853" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div237" type="section" level="1" n="91"> <head xml:id="echoid-head117" xml:space="preserve">PROPOSITIO III.</head> <p style="it"> <s xml:id="echoid-s2854" xml:space="preserve">SI magnitudines quædam deſcendant omnes, vel <lb/>aſcendant, licet inæqualibus intervallis; </s> <s xml:id="echoid-s2855" xml:space="preserve">alti-<lb/>tudines deſcenſus vel aſcenſus cujusque, in ipſam <lb/>magnitudinem ductæ, efficient ſummam producto-<lb/>rum æqualem ei, quæ fit ex altitudine deſcenſus <lb/>vel aſcenſus centri gravitatis omnium magnitudi-<lb/>num, ducta in omnes magnitudines.</s> <s xml:id="echoid-s2856" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2857" xml:space="preserve">Sunto magnitudines A, B, C, quæ ex A, B, C, deſcen-<lb/> <anchor type="note" xlink:label="note-0183-02a" xlink:href="note-0183-02"/> dant in D, E, F; </s> <s xml:id="echoid-s2858" xml:space="preserve">vel ex D, E, F, aſcendant in A, B, C. <lb/></s> <s xml:id="echoid-s2859" xml:space="preserve">Sitque earum centrum gravitatis omnium, dum ſunt in <lb/>A, B, C, eadem altitudine cum puncto G; </s> <s xml:id="echoid-s2860" xml:space="preserve">cum vero ſunt in <lb/>D, E, F, eadem altitudine cum puncto H. </s> <s xml:id="echoid-s2861" xml:space="preserve">Dico ſummam <lb/>productorum ex altitudine A D in A, B E in B, C F in C, <lb/>æquari producto ex G H in omnes A, B, C.</s> <s xml:id="echoid-s2862" xml:space="preserve"/> </p> <div xml:id="echoid-div237" type="float" level="2" n="1"> <note position="right" xlink:label="note-0183-02" xlink:href="note-0183-02a" xml:space="preserve">TAB. XVIII. <lb/>Fig. 2.</note> </div> <pb o="126" file="0184" n="201" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s2863" xml:space="preserve">Intelligatur enim planum horizontale cujus ſectio recta <lb/> <anchor type="note" xlink:label="note-0184-01a" xlink:href="note-0184-01"/> M P, atque in ipſum incidant productæ A D, B E, C F <lb/>& </s> <s xml:id="echoid-s2864" xml:space="preserve">G H, in M, N, O, P.</s> <s xml:id="echoid-s2865" xml:space="preserve"/> </p> <div xml:id="echoid-div238" type="float" level="2" n="2"> <note position="left" xlink:label="note-0184-01" xlink:href="note-0184-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2866" xml:space="preserve">Quia igitur ſumma productorum ex A M in A, B N in B, <lb/>C O in C, æqualis eſt facto ex G P in omnes A, B, C <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s2867" xml:space="preserve"> <anchor type="note" xlink:label="note-0184-02a" xlink:href="note-0184-02"/> Similiterque ſumma productorum ex D M in A, E N in B, <lb/>F O in C, æqualis facto ex H P in omnes A, B, C; </s> <s xml:id="echoid-s2868" xml:space="preserve">ſe-<lb/>quitur & </s> <s xml:id="echoid-s2869" xml:space="preserve">exceſſum priorum productorum ſupra poſteriora, <lb/>æquari facto ex G H in omnes magnitudines A, B, C. </s> <s xml:id="echoid-s2870" xml:space="preserve">Di-<lb/>ctum vero exceſſum æquari manifeſtum eſt productis ex A D <lb/>in A, B E in B, C F in C. </s> <s xml:id="echoid-s2871" xml:space="preserve">Ergo hæc ſimul etiam æqua-<lb/>lia erunt producto ex G H in omnes A, B, C. </s> <s xml:id="echoid-s2872" xml:space="preserve">quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s2873" xml:space="preserve"/> </p> <div xml:id="echoid-div239" type="float" level="2" n="3"> <note symbol="*" position="left" xlink:label="note-0184-02" xlink:href="note-0184-02a" xml:space="preserve">Prop. 1. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div241" type="section" level="1" n="92"> <head xml:id="echoid-head118" xml:space="preserve">PROPOSITIO IV.</head> <p style="it"> <s xml:id="echoid-s2874" xml:space="preserve">SI pendulum è pluribus ponderibus compoſitum, <lb/>atque è quiete dimiſſum, partem quamcunque <lb/>oſcillationis integræ confecerit, atque inde porro <lb/>intelligantur pondera ejus ſingula, relicto communi <lb/>vinculo, celeritates acquiſitas ſurſum convertere, <lb/>ac quousque poſſunt aſcendere; </s> <s xml:id="echoid-s2875" xml:space="preserve">hoc facto, centrum <lb/>gravitatis ex omnibus compoſitæ, ad eandem alti-<lb/>tudinem reverſum erit, quam ante inceptam oſcil-<lb/>lationem obtinebat.</s> <s xml:id="echoid-s2876" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2877" xml:space="preserve">Sit pendulum compoſitum ex ponderibus quotlibet <lb/> <anchor type="note" xlink:label="note-0184-03a" xlink:href="note-0184-03"/> A, B, C, virgæ, vel ſuperficiei pondere carenti, inhæren-<lb/>tibus. </s> <s xml:id="echoid-s2878" xml:space="preserve">Sitque ſuſpenſum ab axe per D punctum ducto, qui <lb/>ad planum, quod hic conſpicitur, perpendicularis intelliga-<lb/>tur. </s> <s xml:id="echoid-s2879" xml:space="preserve">In quo eodem plano etiam centrum gravitatis E, pon-<lb/>derum A, B, C, poſitum ſit; </s> <s xml:id="echoid-s2880" xml:space="preserve">lineaque centri D E, incli-<lb/>netur ad lineam perpendiculi D F, angulo E D F: </s> <s xml:id="echoid-s2881" xml:space="preserve">attra-<lb/>cto, nimirum, eo uſque pendulo. </s> <s xml:id="echoid-s2882" xml:space="preserve">Hinc vero dimitti jam <lb/>ponatur, ac partem quamlibet oſcillationis conficere, ita ut <lb/>pondera A, B, C, perveniant in G, H, K. </s> <s xml:id="echoid-s2883" xml:space="preserve">Unde, reli- <pb o="127" file="0185" n="202" rhead="HOROLOG. OSCILLATOR."/> cto deinceps communi vinculo, ſingula intelligantur acqui-<lb/> <anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/> ſitas celeritates ſurſum convertere, (quod impingendo in pla-<lb/>na quædam inclinata, fieri poterit,) & </s> <s xml:id="echoid-s2884" xml:space="preserve">quouſque poſſunt <lb/>aſcendere, nempe in L, M, N. </s> <s xml:id="echoid-s2885" xml:space="preserve">Quo ubi pervenerint, ſit <lb/>centrum gravitatis omnium punctum P. </s> <s xml:id="echoid-s2886" xml:space="preserve">Dico hoc pari alti-<lb/>tudine eſſe cum puncto E.</s> <s xml:id="echoid-s2887" xml:space="preserve"/> </p> <div xml:id="echoid-div241" type="float" level="2" n="1"> <note position="left" xlink:label="note-0184-03" xlink:href="note-0184-03a" xml:space="preserve">TAB. XVIII. <lb/>Fig. 3. 4.</note> <note position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2888" xml:space="preserve">Nam primum quidem, conſtat P non altius eſſe quam E, <lb/>ex prima ſumptarum hypotheſium. </s> <s xml:id="echoid-s2889" xml:space="preserve">Sed nec humilius fore ſic <lb/>oſtendemus. </s> <s xml:id="echoid-s2890" xml:space="preserve">Sit enim, ſi poteſt, P humilius quam E, & </s> <s xml:id="echoid-s2891" xml:space="preserve"><lb/>intelligantur pondera ex iiſdem, ad quas aſcenderunt, alti-<lb/>tudinibus recidere, quæ ſunt L G, M H, N K. </s> <s xml:id="echoid-s2892" xml:space="preserve">Unde <lb/>quidem eaſdem celeritates ipſis acquiri conſtat, quas habe-<lb/>bant ad aſcendendum ad iſtas altitudines <anchor type="note" xlink:href="" symbol="*"/>, hoc eſt, eas i- <anchor type="note" xlink:label="note-0185-02a" xlink:href="note-0185-02"/> pſas quas acquiſierant motu penduli ex C B A D in <lb/>K H G D. </s> <s xml:id="echoid-s2893" xml:space="preserve">Quare, ſi cum dictis celeritatibus ad virgam ſu-<lb/>perficiemve, cui innexa fuere, nunc referantur, eique ſi-<lb/>mul adhæreſcant, motumque ſecundum inceptos arcus con-<lb/>tinuent; </s> <s xml:id="echoid-s2894" xml:space="preserve">quod fiet, ſi priuſquam virgam attingant, à planis <lb/>inclinatis Q Q repercuſſa intelligantur; </s> <s xml:id="echoid-s2895" xml:space="preserve">abſolvet, hoc modo <lb/>reſtitutum pendulum, oſcillationis partem reliquam, æquè <lb/>ac ſi abſque ulla interruptione motum continuaſſet. </s> <s xml:id="echoid-s2896" xml:space="preserve">Ita ut <lb/>centrum gravitatis penduli, E, arcus æquales E F, F R, <lb/>deſcendendo ac aſcendendo percurrat, ac proinde in R ea-<lb/>dem ac in E altitudine reperiatur. </s> <s xml:id="echoid-s2897" xml:space="preserve">Ponebatur autem E eſſe <lb/>altius quam P centrum gravitatis ponderum in L, M, N, <lb/>poſitorum. </s> <s xml:id="echoid-s2898" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s2899" xml:space="preserve">R altius erit quam P: </s> <s xml:id="echoid-s2900" xml:space="preserve">adeoque ponde-<lb/>rum ex L, M, N, delapſorum centrum gravitatis, altius, <lb/>quam unde deſcenderat, aſcendiſſet. </s> <s xml:id="echoid-s2901" xml:space="preserve">quod eſt abſurdum <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s2902" xml:space="preserve"> <anchor type="note" xlink:label="note-0185-03a" xlink:href="note-0185-03"/> Non igitur centrum gravitatis P humilius eſt quam E. </s> <s xml:id="echoid-s2903" xml:space="preserve">Sed <lb/>nec altius erat. </s> <s xml:id="echoid-s2904" xml:space="preserve">Ergo æque altum ſit neceſſe eſt. </s> <s xml:id="echoid-s2905" xml:space="preserve">quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s2906" xml:space="preserve"/> </p> <div xml:id="echoid-div242" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0185-02" xlink:href="note-0185-02a" xml:space="preserve">Propoſ. 4. <lb/>part. 2.</note> <note symbol="*" position="right" xlink:label="note-0185-03" xlink:href="note-0185-03a" xml:space="preserve">Hypoth. @ <lb/>huj.</note> </div> </div> <div xml:id="echoid-div244" type="section" level="1" n="93"> <head xml:id="echoid-head119" xml:space="preserve">PROPOSITIO V.</head> <p style="it"> <s xml:id="echoid-s2907" xml:space="preserve">DAto pendulo ex ponderibus quotlibet compoſito, <lb/>ſi ſingula ducantur in quadrata diſtantiarum <pb o="128" file="0186" n="203" rhead="CHRISTIANI HUGENII"/> ſuarum ab axe oſcillationis, & </s> <s xml:id="echoid-s2908" xml:space="preserve">ſumma producto-<lb/> <anchor type="note" xlink:label="note-0186-01a" xlink:href="note-0186-01"/> rum dividatur per id quod fit ducendo ponderum <lb/>ſummam, in diſtantiam centri gravitatis commu-<lb/>nis omnium ab eodem axe oſcillationis; </s> <s xml:id="echoid-s2909" xml:space="preserve">orietur lon-<lb/>gitudo penduli ſimplicis compoſito iſochroni, ſive di-<lb/>ſtantia inter axem & </s> <s xml:id="echoid-s2910" xml:space="preserve">centrum oſcillationis ipſius <lb/>penduli compoſiti.</s> <s xml:id="echoid-s2911" xml:space="preserve"/> </p> <div xml:id="echoid-div244" type="float" level="2" n="1"> <note position="left" xlink:label="note-0186-01" xlink:href="note-0186-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2912" xml:space="preserve">Sint pondera pendulum componentia, (quorum nec figura <lb/> <anchor type="note" xlink:label="note-0186-02a" xlink:href="note-0186-02"/> nec magnitudo, ſed gravitas tantum conſideretur), A, B, C, <lb/>ſuſpenſa ab axe, qui per punctum D, ad planum quod conſpi-<lb/>citur, rectus intelligitur. </s> <s xml:id="echoid-s2913" xml:space="preserve">In quo plano ſit quoque eorum cen-<lb/>trum commune gravitatis E; </s> <s xml:id="echoid-s2914" xml:space="preserve">nam pondera in diverſis eſſe ni-<lb/>hil refert. </s> <s xml:id="echoid-s2915" xml:space="preserve">Diſtantia puncti E ab axe, nempe recta E D, vo-<lb/>cetur d. </s> <s xml:id="echoid-s2916" xml:space="preserve">Item ponderis A diſtantia A D, ſit e; </s> <s xml:id="echoid-s2917" xml:space="preserve">B D, f; <lb/></s> <s xml:id="echoid-s2918" xml:space="preserve">C D, g. </s> <s xml:id="echoid-s2919" xml:space="preserve">Ducendo itaque ſingula pondera in quadrata ſua-<lb/>rum diſtantiarum, erit productorum ſumma a e e + b f f <lb/>+ c g g. </s> <s xml:id="echoid-s2920" xml:space="preserve">Et rurſus, ducendo ſummam ponderum in diſtan-<lb/>tiam centri gravitatis omnium, productum æquale erit a d <lb/>+ b d + c d <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2921" xml:space="preserve">Unde, productum prius per hoc dividen- <anchor type="note" xlink:label="note-0186-03a" xlink:href="note-0186-03"/> do, habebitur {a e e + b f f + c @ @/a d + b d + c a}. </s> <s xml:id="echoid-s2922" xml:space="preserve">Cui longitudini ſi æqualis ſta-<lb/>tuatur longitudo penduli ſimplicis F G, quæ etiam x vo-<lb/>cabitur; </s> <s xml:id="echoid-s2923" xml:space="preserve">dico hoc illi compoſito iſochronum eſſe.</s> <s xml:id="echoid-s2924" xml:space="preserve"/> </p> <div xml:id="echoid-div245" type="float" level="2" n="2"> <note position="left" xlink:label="note-0186-02" xlink:href="note-0186-02a" xml:space="preserve">TAB. XIX. <lb/>Fig. 1. 2.</note> <note symbol="*" position="left" xlink:label="note-0186-03" xlink:href="note-0186-03a" xml:space="preserve">Prop. 1. <lb/>h@.</note> </div> <p> <s xml:id="echoid-s2925" xml:space="preserve">Ponantur enim tum pendulum F G, tum linea centri <lb/>D E, æqualibus angulis à linea perpendiculi remota, illud <lb/>ab F H, hæc ab D K, atque inde dimiſſa librari, & </s> <s xml:id="echoid-s2926" xml:space="preserve">in <lb/>recta D E ſumatur D L æqualis F G. </s> <s xml:id="echoid-s2927" xml:space="preserve">Itaque pondus G <lb/>penduli F G, integra oſcillatione arcum G M percurret, <lb/>quem linea perpendiculi F H medium ſecabit. </s> <s xml:id="echoid-s2928" xml:space="preserve">punctum ve-<lb/>ro L arcum illi ſimilem & </s> <s xml:id="echoid-s2929" xml:space="preserve">æqualem L N, quem medium <lb/>dividet D K. </s> <s xml:id="echoid-s2930" xml:space="preserve">Itemque centrum gravitatis E, percurret ſi-<lb/>milem arcum E I. </s> <s xml:id="echoid-s2931" xml:space="preserve">Quod ſi in arcubus G M, N L, ſum-<lb/>ptis punctis quibuslibet, ſimiliter ipſos dividentibus, ut O <lb/>& </s> <s xml:id="echoid-s2932" xml:space="preserve">P, eadem celeritas eſſe oſtendatur ponderis G in O, & </s> <s xml:id="echoid-s2933" xml:space="preserve"><lb/>puncti L in P; </s> <s xml:id="echoid-s2934" xml:space="preserve">conſtabit inde æqualibus temporibus utros- <pb file="0187" n="204"/> <pb file="0187a" n="205"/> <anchor type="figure" xlink:label="fig-0187a-01a" xlink:href="fig-0187a-01"/> <anchor type="figure" xlink:label="fig-0187a-02a" xlink:href="fig-0187a-02"/> <anchor type="figure" xlink:label="fig-0187a-03a" xlink:href="fig-0187a-03"/> <anchor type="figure" xlink:label="fig-0187a-04a" xlink:href="fig-0187a-04"/> <pb file="0188" n="206"/> <pb o="129" file="0189" n="207" rhead="HOROLOG. OSCILLATOR."/> que arcus percurri, ac proinde pendulum F G, pendulo <lb/> <anchor type="note" xlink:label="note-0189-01a" xlink:href="note-0189-01"/> compoſito ex A, B, C, iſochronum eſſe. </s> <s xml:id="echoid-s2935" xml:space="preserve">Oſtendetur au-<lb/>tem hoc modo.</s> <s xml:id="echoid-s2936" xml:space="preserve"/> </p> <div xml:id="echoid-div246" type="float" level="2" n="3"> <figure xlink:label="fig-0187a-01" xlink:href="fig-0187a-01a"> <caption xml:id="echoid-caption64" style="it" xml:space="preserve">Pag. 128.<lb/>TAB. XVIII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables64" xml:space="preserve">A G C B D E H F K I M</variables> </figure> <figure xlink:label="fig-0187a-02" xlink:href="fig-0187a-02a"> <caption xml:id="echoid-caption65" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables65" xml:space="preserve">A C G B E F D H M N O P</variables> </figure> <figure xlink:label="fig-0187a-03" xlink:href="fig-0187a-03a"> <caption xml:id="echoid-caption66" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables66" xml:space="preserve">D L Q A G Q M R E P. Q B F N H Q C Q K Q</variables> </figure> <figure xlink:label="fig-0187a-04" xlink:href="fig-0187a-04a"> <caption xml:id="echoid-caption67" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables67" xml:space="preserve">N Q K C Q D L R E P F A Q G M Q Q H B Q</variables> </figure> <note position="right" xlink:label="note-0189-01" xlink:href="note-0189-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2937" xml:space="preserve">Sit primo, ſi poteſt, major celeritas puncti L, ubi in P <lb/>pervenit, quam ponderis G in O. </s> <s xml:id="echoid-s2938" xml:space="preserve">Conſtatautem, dum pun-<lb/>ctum L percurrit arcum L P, ſimul centrum gravitatis E <lb/>percurrere arcum ſimilem E Q. </s> <s xml:id="echoid-s2939" xml:space="preserve">Ducantur à punctis Q, P, O, <lb/>perpendiculares ſurſum, quæ occurrant ſubtenſis arcuum <lb/>E I, L N, G M, in R, S, Y. </s> <s xml:id="echoid-s2940" xml:space="preserve">& </s> <s xml:id="echoid-s2941" xml:space="preserve">S P vocetur y. </s> <s xml:id="echoid-s2942" xml:space="preserve">Unde, <lb/>cum ſit ut L D, x, ad E D, d, ita S P, y, ad R Q; </s> <s xml:id="echoid-s2943" xml:space="preserve">erit <lb/>R Q æqualis {d y/x}. </s> <s xml:id="echoid-s2944" xml:space="preserve">Jam quia pondus G eam celeritatem ha-<lb/>bet in O, qua valet ad eandem unde deſcendit altitudinem <lb/>aſcendere, nempe per arcum O M, vel perpendicularem <lb/>O Y ipſi P S æqualem; </s> <s xml:id="echoid-s2945" xml:space="preserve">punctum igitur L, ubi in P per-<lb/>venerit, majorem ibi celeritatem habebit, quam qua aſcen-<lb/>ditur ad altitudinem P S. </s> <s xml:id="echoid-s2946" xml:space="preserve">Dum vero L tranſit in P, ſimul <lb/>pondera A, B, C, ſimiles arcus percurrunt ipſi L P, nimirum <lb/>A T, B V, C X. </s> <s xml:id="echoid-s2947" xml:space="preserve">Eſtque puncti L celeritas in P, ad celeri-<lb/>tatem ponderis A in T, quum vinculo eodem contineantur, <lb/>ſicut diſtantia D L ad D A. </s> <s xml:id="echoid-s2948" xml:space="preserve">Sed ut quadratum celeritatis <lb/>puncti L, quam habet in P, ad quadratum celeritatis pun-<lb/>cti A in T, ita eſt altitudo ad quam illa celeritate <lb/>aſcendi poteſt, ad altitudinem quò hac celeritate aſcendi <lb/>poteſt <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s2949" xml:space="preserve">Ergo etiam, ut quadratum diſtantiæ D L, quod <anchor type="note" xlink:label="note-0189-02a" xlink:href="note-0189-02"/> eſt x x, ad quadratum diſtantiæ D A, quod eſt e e, ita eſt <lb/>altitudo quo aſcenditur celeritate puncti L, quum eſt in P, <lb/>(quæ altitudo major dicta eſt quam P S ſive y,) ad altitu-<lb/>dinem quo aſcenditur celeritate ponderis A in T; </s> <s xml:id="echoid-s2950" xml:space="preserve">ſi nempe <lb/>poſtquam in T pervenit, relicto pendulo, ſeorſim motum <lb/>ſuum ſurſum converteret. </s> <s xml:id="echoid-s2951" xml:space="preserve">Quæ proinde altitudo major erit <lb/>quam {e e y/x x}.</s> <s xml:id="echoid-s2952" xml:space="preserve"/> </p> <div xml:id="echoid-div247" type="float" level="2" n="4"> <note symbol="*" position="right" xlink:label="note-0189-02" xlink:href="note-0189-02a" xml:space="preserve">Prop. 3. <lb/>& 4. part. @.</note> </div> <p> <s xml:id="echoid-s2953" xml:space="preserve">Eadem ratione, erit altitudo ad quam aſcenderet pondus <lb/>B, celeritate acquiſita per arcum B V, major quam {f f y/x x}. </s> <s xml:id="echoid-s2954" xml:space="preserve">Et <lb/>altitudo ad quam aſcenderet pondus C, celeritate acquiſita <lb/>per arcum C X, major quam {g g y/x x}. </s> <s xml:id="echoid-s2955" xml:space="preserve">Unde, ductis ſingulis al- <pb o="130" file="0190" n="208" rhead="CHRISTIANI HUGENII"/> titudinibus iſtis in ſua pondera, erit ſumma productorum <lb/> <anchor type="note" xlink:label="note-0190-01a" xlink:href="note-0190-01"/> major quam {a e e y + b f f y + c g g y/x x}. </s> <s xml:id="echoid-s2956" xml:space="preserve">quæ proinde major quoque <lb/>probatur quam {a d y + b d y + c d y/x}. </s> <s xml:id="echoid-s2957" xml:space="preserve">Nam quia poſita eſt longitudo <lb/>x æqualis {a e e + b f f + g g/a d + b d + c d}; </s> <s xml:id="echoid-s2958" xml:space="preserve">erit a d x + b d x + c d x æquale <lb/>a e e + b f f + c g g. </s> <s xml:id="echoid-s2959" xml:space="preserve">Et ductis omnibus in y, & </s> <s xml:id="echoid-s2960" xml:space="preserve">dividen-<lb/>do per x x, erit {a d y + b d y + c d y/x} æquale {a e e y + b f f y + c g g y/x x}. </s> <s xml:id="echoid-s2961" xml:space="preserve">Unde <lb/>quod dictum eſt conſequitur. </s> <s xml:id="echoid-s2962" xml:space="preserve">Eſt autem ſumma iſta produ-<lb/>ctorum æqualis ei, quod fit ducendo altitudinem, ad quam <lb/>aſcendit centrum gravitatis commune ponderum A, B, C, in <lb/>ſummam ipſorum ponderum, a + b + c; </s> <s xml:id="echoid-s2963" xml:space="preserve">ſi nempe ſingu-<lb/>la, uti dictum, ſeorſim quousque poſſunt moveantur. </s> <s xml:id="echoid-s2964" xml:space="preserve">Quan-<lb/>titas vero {a d y + b d y + c d y/x} producitur ex deſcenſu centri gravi-<lb/>tatis eorundem ponderum, (qui deſcenſus eſt R Q, ſive {d y/x}, <lb/>ut ſupra inventum fuit,) in eandem quoque ponderum ſum-<lb/>mam a + b + c. </s> <s xml:id="echoid-s2965" xml:space="preserve">Ergo quum prius productum altero hoc <lb/>majus oſtenſum fuerit, ſequitur aſcenſum centri gravitatis <lb/>ponderum A, B, C, ſi, relicto pendulo ubi pervenere in <lb/>T, V, X, ſingula celeritates acquiſitas ſurſum convertant, <lb/>majorem fore ejusdem centri gravitatis deſcenſu, dum ex <lb/>A, B, C, moventur in T, V, X. </s> <s xml:id="echoid-s2966" xml:space="preserve">quod eſt abſurdum, cum <lb/>dictus aſcenſus deſcenſui æqualis eſſe debeat, per anteceden-<lb/>tem.</s> <s xml:id="echoid-s2967" xml:space="preserve"/> </p> <div xml:id="echoid-div248" type="float" level="2" n="5"> <note position="left" xlink:label="note-0190-01" xlink:href="note-0190-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2968" xml:space="preserve">Eodem modo, ſi dicatur celeritatem puncti L, ubi per-<lb/>venerit in P, minorem eſſe celeritate ponderis G quum in O <lb/>pervenerit; </s> <s xml:id="echoid-s2969" xml:space="preserve">oſtendemus aſcenſum poſſibilem centri gravitatis <lb/>ponderum A, B, C, minorem eſſe quam deſcenſum, quod <lb/>eidem propoſitioni antecedenti repugnat. </s> <s xml:id="echoid-s2970" xml:space="preserve">Quare relinquitur <lb/>ut eadem ſit celeritas puncti L, ad P tranſlati, quæ ponde-<lb/>ris G in O. </s> <s xml:id="echoid-s2971" xml:space="preserve">Unde, ut ſuperius dictum, ſequitur pendulum <lb/>ſimplex F G compoſito ex A, B, C, iſochronum eſſe.</s> <s xml:id="echoid-s2972" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div250" type="section" level="1" n="94"> <head xml:id="echoid-head120" xml:space="preserve">PROPOSITIO VI.</head> <p style="it"> <s xml:id="echoid-s2973" xml:space="preserve">DAto pendulo ex quotcunque ponderibus æqua-<lb/>libus compoſito; </s> <s xml:id="echoid-s2974" xml:space="preserve">ſi ſumma quadratorum facto- <pb o="131" file="0191" n="209" rhead="HOROLOG. OSCILLATOR."/> rum à diſtantiis, quibus unumquodque pondus ab-<lb/> <anchor type="note" xlink:label="note-0191-01a" xlink:href="note-0191-01"/> eſt ab axe oſcillationis, applicetur ad diſtantiam <lb/>centri gravitatis communis ab eodem oſcillationis <lb/>axe, multiplicem ſecundum ipſorum ponderum nu-<lb/>merum, orietur longitudo penduli ſimplicis compo-<lb/>ſito iſochroni.</s> <s xml:id="echoid-s2975" xml:space="preserve"/> </p> <div xml:id="echoid-div250" type="float" level="2" n="1"> <note position="right" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s2976" xml:space="preserve">Sint poſita eadem quæ prius, ſed pondera omnia inter ſe <lb/>æqualia intelligantur, & </s> <s xml:id="echoid-s2977" xml:space="preserve">ſingula dicantur a. </s> <s xml:id="echoid-s2978" xml:space="preserve">Rurſus vero <lb/>nulla eorum magnitudo conſideretur, ſed pro minimis ha-<lb/>beantur, quantum ad extenſionem.</s> <s xml:id="echoid-s2979" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2980" xml:space="preserve">Itaque penduli ſimplicis iſochroni longitudo, per propo-<lb/>ſitionem antecedentem, erit {a e e + a f f + a g g/a d + a d + a d}. </s> <s xml:id="echoid-s2981" xml:space="preserve">Vel, quia quanti-<lb/>tas diviſa ac dividens utraque per a dividitur, fiet nunc ea-<lb/>dem longitudo, {e e + f f + g g/3d}. </s> <s xml:id="echoid-s2982" xml:space="preserve">Quo ſignificatur ſumma quadra-<lb/>torum à diſtantiis ponderum ab axe oſcillationis, applicata <lb/>ad diſtantiam centri gravitatis omnium ab eodem oſcillatio-<lb/>nis axe, multiplicem ſecundum numerum ipſorum ponde-<lb/>rum, qui hic eſt 3. </s> <s xml:id="echoid-s2983" xml:space="preserve">facile enim perſpicitur numerum hunc, <lb/>in quem ducitur diſtantia d, reſpondere neceſſario ipſi pon-<lb/>derum numero. </s> <s xml:id="echoid-s2984" xml:space="preserve">Quare conſtat propoſitum.</s> <s xml:id="echoid-s2985" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2986" xml:space="preserve">Quod ſi pondera æqualia in unam lineam rectam conjun-<lb/>cta ſint, atque ex termino ejus ſuperiore ſuſpenſa; </s> <s xml:id="echoid-s2987" xml:space="preserve">conſtat <lb/>diſtantiam centri gravitatis, ex omnibus compoſitæ, ab axe <lb/>oſcillationis, multiplicem ſecundum ponderum numerum, <lb/>æquari ſummæ diſtantiarum omnium ponderum ab eodem <lb/>oſcillationis axe <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s2988" xml:space="preserve">ac proinde, hoc caſu, habebitur quoque <anchor type="note" xlink:label="note-0191-02a" xlink:href="note-0191-02"/> longitudo penduli ſimplicis, compoſito iſochroni, ſi ſumma <lb/>quadratorum à diſtantiis ponderum ſingulorum ab axe oſcil-<lb/>lationis, dividatur per ſummam earundem omnium diſtan-<lb/>tiarum.</s> <s xml:id="echoid-s2989" xml:space="preserve"/> </p> <div xml:id="echoid-div251" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0191-02" xlink:href="note-0191-02a" xml:space="preserve">Prop. 2. <lb/>huj.</note> </div> <pb o="132" file="0192" n="210" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div253" type="section" level="1" n="95"> <head xml:id="echoid-head121" xml:space="preserve">DEFINITIO XIV.</head> <note position="left" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSOILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> <p style="it"> <s xml:id="echoid-s2990" xml:space="preserve">SI fuerint in eodem plano, figura quædam, & </s> <s xml:id="echoid-s2991" xml:space="preserve">li-<lb/>nea recta quæ ipſam extrinſecus tangat; </s> <s xml:id="echoid-s2992" xml:space="preserve">& </s> <s xml:id="echoid-s2993" xml:space="preserve">per <lb/>ambitum figuræ alia recta, plano ejus perpendicu-<lb/>laris, circumferatur, ſuperficiemque quandam de-<lb/>ſcribat, quæ deinde ſecetur plano per dictam tan-<lb/>gentem ducto & </s> <s xml:id="echoid-s2994" xml:space="preserve">ad dictæ figur æplanum inclinato; <lb/></s> <s xml:id="echoid-s2995" xml:space="preserve">ſolidum comprehenſum à duobus planis iſtis, & </s> <s xml:id="echoid-s2996" xml:space="preserve">par-<lb/>te ſuperficiei deſcriptæ, inter utrumque planum in-<lb/>tercepta, vocetur Cuneus ſuper figura illa, tan-<lb/>quam baſi, abſciſſus.</s> <s xml:id="echoid-s2997" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2998" xml:space="preserve">In ſchemate adjecto, eſt A B E C figura data; </s> <s xml:id="echoid-s2999" xml:space="preserve">recta eam <lb/> <anchor type="note" xlink:label="note-0192-02a" xlink:href="note-0192-02"/> tangens M D; </s> <s xml:id="echoid-s3000" xml:space="preserve">quæ vero per ambitum ejus circumfertur, <lb/>E F; </s> <s xml:id="echoid-s3001" xml:space="preserve">cuneus autem figura ſolida planis A B E C, M F G, <lb/>& </s> <s xml:id="echoid-s3002" xml:space="preserve">parte ſuperficiei, à recta E F deſcriptæ, comprehenſa.</s> <s xml:id="echoid-s3003" xml:space="preserve"/> </p> <div xml:id="echoid-div253" type="float" level="2" n="1"> <note position="left" xlink:label="note-0192-02" xlink:href="note-0192-02a" xml:space="preserve">TAB. XXI. <lb/>Fig. 3.</note> </div> </div> <div xml:id="echoid-div255" type="section" level="1" n="96"> <head xml:id="echoid-head122" xml:space="preserve">DEFINITIO XV.</head> <p style="it"> <s xml:id="echoid-s3004" xml:space="preserve">D Iſtantia inter rectam, per quam cuneus abſciſ-<lb/>ſus eſt, & </s> <s xml:id="echoid-s3005" xml:space="preserve">punctum baſeos, in quod perpen-<lb/>dicularis cadit à cunei centro gravitatis, dicatur <lb/>cunei Subcentrica.</s> <s xml:id="echoid-s3006" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3007" xml:space="preserve">Nempe in figura eadem, ſi K ſit centrum gra-<lb/> <anchor type="note" xlink:label="note-0192-03a" xlink:href="note-0192-03"/> vitatis cunei, recta vero K I ad baſin ejus A B E C <lb/>perpendicularis ducta ſit, & </s> <s xml:id="echoid-s3008" xml:space="preserve">rurſus I M perpen-<lb/>dicularis ad M D; </s> <s xml:id="echoid-s3009" xml:space="preserve">erit I M, quam ſubcentri-<lb/>cam dicimus.</s> <s xml:id="echoid-s3010" xml:space="preserve"/> </p> <div xml:id="echoid-div255" type="float" level="2" n="1"> <note position="left" xlink:label="note-0192-03" xlink:href="note-0192-03a" xml:space="preserve">TAB. XIX. <lb/>Fig. 3.</note> </div> </div> <div xml:id="echoid-div257" type="section" level="1" n="97"> <head xml:id="echoid-head123" xml:space="preserve">PROPOSITIO VII.</head> <p style="it"> <s xml:id="echoid-s3011" xml:space="preserve">CUneus ſuper plana figura qualibet abſciſſus, <lb/>plano inclinato ad angulum ſemirectum, æqua- <pb o="133" file="0193" n="211" rhead="HOROLOG. OSCILLATOR."/> lis eſt ſolido, quod fit ducendo figuram eandem, in <lb/> <anchor type="note" xlink:label="note-0193-01a" xlink:href="note-0193-01"/> altitudinem æqualem diſtantiæ centri gravitatis fi-<lb/>guræ, ab recta per quam abſciſſus eſt cuneus.</s> <s xml:id="echoid-s3012" xml:space="preserve"/> </p> <div xml:id="echoid-div257" type="float" level="2" n="1"> <note position="right" xlink:label="note-0193-01" xlink:href="note-0193-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s3013" xml:space="preserve">Sit, ſuper figura plana A C B, cuneus A B D abſciſſus <lb/> <anchor type="note" xlink:label="note-0193-02a" xlink:href="note-0193-02"/> plano ad angulum ſemirectum inclinato, ac transeunte per <lb/>E E, rectam tangentem figuram A C B, inque ejus plano <lb/>ſitam. </s> <s xml:id="echoid-s3014" xml:space="preserve">Centrum vero gravitatis figuræ ſit F, unde in rectam <lb/>E E ducta ſit perpendicularis F @. </s> <s xml:id="echoid-s3015" xml:space="preserve">Dico cuneum A C B æ-<lb/>qualem eſſe ſolido, quod fit ducendo figuram A C B in al-<lb/>titudinem ipſi F A æqualem.</s> <s xml:id="echoid-s3016" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="2" n="2"> <note position="right" xlink:label="note-0193-02" xlink:href="note-0193-02a" xml:space="preserve">TAB. XI. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s3017" xml:space="preserve">Intelligatur enim figura A C B diviſa in particulas mini-<lb/>mas æquales quarum una G. </s> <s xml:id="echoid-s3018" xml:space="preserve">Itaque conſtat, ſi harum ſin-<lb/>gulæ ducantur in diſtantiam ſuam ab recta E E, ſummam <lb/>productorum fore æqualem ei quod fit ducendo rectam A F <lb/>in particulas omnes <anchor type="note" xlink:href="" symbol="*"/>, hoc eſt, ei quod fit ducendo figuram <anchor type="note" xlink:label="note-0193-03a" xlink:href="note-0193-03"/> ipſam A C B, in altitudinem æqualem A F. </s> <s xml:id="echoid-s3019" xml:space="preserve">Atqui particu-<lb/>læ ſingulæ ut G, in diſtantias ſuas G H ductæ, æquales <lb/>ſunt parallelepipedis, vel prismatibus minimis, ſuper ipſas <lb/>erectis, atque ad ſuperficiem obliquam A D terminatis, qua-<lb/>le eſt G K; </s> <s xml:id="echoid-s3020" xml:space="preserve">quia horum altitudines ipſis diſtantiis G H æ-<lb/>quantur, propter angulum ſemirectum inclinationis plano-<lb/>rum A D & </s> <s xml:id="echoid-s3021" xml:space="preserve">A C B. </s> <s xml:id="echoid-s3022" xml:space="preserve">Patetque ex his parallelepipedis totum <lb/>cuneum A B D componi. </s> <s xml:id="echoid-s3023" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3024" xml:space="preserve">cuneus ipſe æquabitur ſo-<lb/>lido ſuper baſi A C B, altitudinem habenti rectæ F A æ-<lb/>qualem. </s> <s xml:id="echoid-s3025" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s3026" xml:space="preserve"/> </p> <div xml:id="echoid-div259" type="float" level="2" n="3"> <note symbol="*" position="right" xlink:label="note-0193-03" xlink:href="note-0193-03a" xml:space="preserve">Prop. 1. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div261" type="section" level="1" n="98"> <head xml:id="echoid-head124" xml:space="preserve">PROPOSITIO VIII.</head> <p style="it"> <s xml:id="echoid-s3027" xml:space="preserve">SI figuram planam linea recta tangat, diviſaque <lb/>intelligatur figura in particulas minimas æqua-<lb/>les, atque à ſingulis ad rectam illam perpendicula-<lb/>res ductæ: </s> <s xml:id="echoid-s3028" xml:space="preserve">erunt omnium harum quadrata, ſimul <lb/>ſumpta, æqualia rectangulo cuidam, multiplici ſe-<lb/>cundum ipſarum particularum numerum; </s> <s xml:id="echoid-s3029" xml:space="preserve">quod <pb o="134" file="0194" n="212" rhead="CHRISTIANI HUGENII"/> nempe rectangulum fit à diſtantia centri gravitatis <lb/> <anchor type="note" xlink:label="note-0194-01a" xlink:href="note-0194-01"/> figuræ ab eadem recta, & </s> <s xml:id="echoid-s3030" xml:space="preserve">à ſubcentrica cunei, qui <lb/>per illam ſuper figura abſcinditur.</s> <s xml:id="echoid-s3031" xml:space="preserve"/> </p> <div xml:id="echoid-div261" type="float" level="2" n="1"> <note position="left" xlink:label="note-0194-01" xlink:href="note-0194-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> </div> <p> <s xml:id="echoid-s3032" xml:space="preserve">Poſitis enim cæteris omnibus quæ in conſtructione præce-<lb/> <anchor type="note" xlink:label="note-0194-02a" xlink:href="note-0194-02"/> denti, ſit L A cunei A B D ſubcentrica in rectam E E. </s> <s xml:id="echoid-s3033" xml:space="preserve">O-<lb/>portet igitur oſtendere, ſummam quadratorum omnium à di-<lb/>ſtantiis particularum figuræ A C B æquari rectangulo ab <lb/>F A, L A, multiplici ſecundum particularum numerum.</s> <s xml:id="echoid-s3034" xml:space="preserve"/> </p> <div xml:id="echoid-div262" type="float" level="2" n="2"> <note position="left" xlink:label="note-0194-02" xlink:href="note-0194-02a" xml:space="preserve">TAB. XIX. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s3035" xml:space="preserve">Et conſtat quidem ex demonſtratione præcedenti, altitu-<lb/>dines parallelepipedorum ſingulorum, ut G K, æquales eſ-<lb/>ſe diſtantiis particularum, quæ ipſorum baſes ſunt, ut G, <lb/>ab recta A E. </s> <s xml:id="echoid-s3036" xml:space="preserve">Quare, ſi jam parallelepipedum G K ducamus <lb/>in diſtantiam G H, perinde eſt ac ſi particula G ducatur in <lb/>quadratum diſtantiæ G H. </s> <s xml:id="echoid-s3037" xml:space="preserve">Eodemque modo ſe res habet in <lb/>reliquis omnibus. </s> <s xml:id="echoid-s3038" xml:space="preserve">Atqui producta omnia parallelepipedorum <lb/>in diſtantias ſuas ab recta A E, æquantur ſimul producto ex <lb/>cuneo A B D in diſtantiam L A <anchor type="note" xlink:href="" symbol="*"/>, quia cuneus gravitat ſu- <anchor type="note" xlink:label="note-0194-03a" xlink:href="note-0194-03"/> per puncto L. </s> <s xml:id="echoid-s3039" xml:space="preserve">Ergo etiam ſumma productorum à particulis <lb/>ſingulis G, in quadrata ſuarum diſtantiarum ab recta A E, <lb/>æquabitur producto ex cuneo A B D in rectam L A, hoc <lb/>eſt, producto ex figura A C B in rectangulum ab F A, L A. <lb/></s> <s xml:id="echoid-s3040" xml:space="preserve">Nam cuneus A B D, æqualis eſt producto ex figura A C B <lb/>in rectam F A <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3041" xml:space="preserve">Rurſus quia figura A C B æqualis eſt pro- <anchor type="note" xlink:label="note-0194-04a" xlink:href="note-0194-04"/> ducto ex particula una G, in numerum ipſarum particula-<lb/>rum; </s> <s xml:id="echoid-s3042" xml:space="preserve">ſequitur, dictum productum ex figura A C B in re-<lb/>ctangulum ab F A, L A, æquari producto ex particula G <lb/>in rectangulum ab F A, L A, multiplici ſecundum nume-<lb/>rum particularum G. </s> <s xml:id="echoid-s3043" xml:space="preserve">Cui proinde etiam æqualis erit dicta <lb/>ſumma productorum, à particulis ſingulis G in quadrata <lb/>ſuarum diſtantiarum ab recta A E, ſive à particula una G in <lb/>ſummam omnium horum quadratorum. </s> <s xml:id="echoid-s3044" xml:space="preserve">Quare, omiſſa utrin-<lb/>que multiplicatione in particulam G, neceſſe eſt ſummam <lb/>@andem quadratorum æquari rectangulo ab F A, L A, mul-<lb/>tiplici ſecundum numerum particularum in quas figura A C B <lb/>diviſa intelligitur. </s> <s xml:id="echoid-s3045" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s3046" xml:space="preserve"/> </p> <div xml:id="echoid-div263" type="float" level="2" n="3"> <note symbol="*" position="left" xlink:label="note-0194-03" xlink:href="note-0194-03a" xml:space="preserve">Prop. 1. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0194-04" xlink:href="note-0194-04a" xml:space="preserve">Prop. <lb/>præced.</note> </div> <pb o="135" file="0195" n="213" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div265" type="section" level="1" n="99"> <head xml:id="echoid-head125" xml:space="preserve">PROPOSITIO IX.</head> <note position="right" xml:space="preserve"><emph style="sc">De centro-</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> <p style="it"> <s xml:id="echoid-s3047" xml:space="preserve">DAtâ figurâ planâ & </s> <s xml:id="echoid-s3048" xml:space="preserve">in eodem plano lineâ re-<lb/>ctâ, quæ vel ſecet figuram vel non, ad quam <lb/>perpendiculares cadant à particulis ſingulis minimis <lb/>& </s> <s xml:id="echoid-s3049" xml:space="preserve">æqualibus, in quas figura diviſa intelligitur; <lb/></s> <s xml:id="echoid-s3050" xml:space="preserve">invenire ſummam quadratorum ab omnibus iſtis per-<lb/>pendicularibus; </s> <s xml:id="echoid-s3051" xml:space="preserve">ſive planum, cujus multiplex, ſe-<lb/>cundum particularum numerum, dictæ quadrato-<lb/>rum ſummæ æquale ſit.</s> <s xml:id="echoid-s3052" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3053" xml:space="preserve">Sit data figura plana A B C, & </s> <s xml:id="echoid-s3054" xml:space="preserve">in eodem plano recta <lb/> <anchor type="note" xlink:label="note-0195-02a" xlink:href="note-0195-02"/> E D; </s> <s xml:id="echoid-s3055" xml:space="preserve">divisâque figurâ cogitatu in particulas minimas æqua-<lb/>les, intelligantur ab unaquaque earum perpendiculares du-<lb/>ctæ in rectam E D, ſicut à particula F ducta eſt F K. </s> <s xml:id="echoid-s3056" xml:space="preserve">O-<lb/>porteatque invenire ſummam quadratorum ab omnibus iſtis <lb/>perpendicularibus.</s> <s xml:id="echoid-s3057" xml:space="preserve"/> </p> <div xml:id="echoid-div265" type="float" level="2" n="1"> <note position="right" xlink:label="note-0195-02" xlink:href="note-0195-02a" xml:space="preserve">TAB. XIX. <lb/>Fig. 5. 6.</note> </div> <p> <s xml:id="echoid-s3058" xml:space="preserve">Sit datæ E D parallela recta A L, quæ figuram tangat, <lb/>ac tota extra eam poſita ſit. </s> <s xml:id="echoid-s3059" xml:space="preserve">Poteſt autem figuram vel ab ea-<lb/>dem parte ex qua eſt E D, vel à parte oppoſita contingere. <lb/></s> <s xml:id="echoid-s3060" xml:space="preserve">Diſtantia vero centri gravitatis figuræ ab recta A L ſit recta <lb/>G A, ſecans E D in E; </s> <s xml:id="echoid-s3061" xml:space="preserve">& </s> <s xml:id="echoid-s3062" xml:space="preserve">ſubcentrica cunei, ſuper figura <lb/>abſciſſi plano per rectam A L, ſit H A. </s> <s xml:id="echoid-s3063" xml:space="preserve">Dico ſummam qua-<lb/>dratorum quæſitam æquari rectangulo A G H una cum qua-<lb/>drato E G, multiplicibus ſecundum particularum numerum, <lb/>in quas figura diviſa intelligitur.</s> <s xml:id="echoid-s3064" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3065" xml:space="preserve">Occurrat enim F K, ſi opus eſt producta, tangenti A L <lb/>in L puncto. </s> <s xml:id="echoid-s3066" xml:space="preserve">Itaque primum, eo caſu quo recta E D à ſi-<lb/>gura diſtat, & </s> <s xml:id="echoid-s3067" xml:space="preserve">tangens A L ad eandem figuræ partem ducta <lb/>eſt, ſic propoſitum oſtendetur. </s> <s xml:id="echoid-s3068" xml:space="preserve">Summa omnium quadrato-<lb/>rum F K æquatur totidem quadratis K L, una cum bis to-<lb/>tidem rectangulis K L F, & </s> <s xml:id="echoid-s3069" xml:space="preserve">totidem inſuper quadratis L F. <lb/></s> <s xml:id="echoid-s3070" xml:space="preserve">Sed quadrata K L æquantur totidem quadratis E A. </s> <s xml:id="echoid-s3071" xml:space="preserve">Et re-<lb/>ctangula K L F æqualia eſſe conſtat totidem rectangulis <pb o="136" file="0196" n="214" rhead="CHRISTIANI HUGENII"/> E A G, quia omnes F L æquales totidem G A <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3072" xml:space="preserve">Et deni- <anchor type="note" xlink:label="note-0196-01a" xlink:href="note-0196-01"/> que quadrata L F æquantur totidem rectangulis H A G <anchor type="note" xlink:href="" symbol="*"/>, hoc eſt, totidem quadratis A G cum totidem rectangulis <lb/> <anchor type="note" xlink:label="note-0196-02a" xlink:href="note-0196-02"/> A G H. </s> <s xml:id="echoid-s3073" xml:space="preserve">Ergo quadrata omnia F K æqualia erunt totidem <lb/> <anchor type="note" xlink:label="note-0196-03a" xlink:href="note-0196-03"/> quadratis E A, cum totidem duplis rectangulis E A G, at-<lb/>que inſuper totidem quadratis A G cum totidem rectangulis <lb/>A G H. </s> <s xml:id="echoid-s3074" xml:space="preserve">Atqui tria iſta; </s> <s xml:id="echoid-s3075" xml:space="preserve">nempe quadratum E A cum duplo <lb/>rectangulo E A G & </s> <s xml:id="echoid-s3076" xml:space="preserve">quadrato A G; </s> <s xml:id="echoid-s3077" xml:space="preserve">faciunt quadratum E G. <lb/></s> <s xml:id="echoid-s3078" xml:space="preserve">Ergo apparet quadrata omnia F K æquari totidem quadratis <lb/>E G, una cum totidem rectangulis A G H. </s> <s xml:id="echoid-s3079" xml:space="preserve">Quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s3080" xml:space="preserve"/> </p> <div xml:id="echoid-div266" type="float" level="2" n="2"> <note position="left" xlink:label="note-0196-01" xlink:href="note-0196-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="left" xlink:label="note-0196-02" xlink:href="note-0196-02a" xml:space="preserve">Prop. 2. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0196-03" xlink:href="note-0196-03a" xml:space="preserve">Prop. <lb/>præced.</note> </div> <p> <s xml:id="echoid-s3081" xml:space="preserve">Porro in reliquis omnibus caſibus, quadrata omnia F K <lb/> <anchor type="note" xlink:label="note-0196-04a" xlink:href="note-0196-04"/> æquantur totidem quadratis K L, minus bis totidem rectan-<lb/>gulis K L F, plus totidem quadratis L F; </s> <s xml:id="echoid-s3082" xml:space="preserve">hoc eſt, toti-<lb/>dem quadratis E A, minus totidem duplis rectangulis E A G, <lb/>plus totidem quadratis A G, cum totidem rectangulis A G H. <lb/></s> <s xml:id="echoid-s3083" xml:space="preserve">Atqui, omnibus hiſce caſibus, fit quadratum E A, plus qua-<lb/>drato A G, minus duplo rectangulo E A G, æquale qua-<lb/>drato E G. </s> <s xml:id="echoid-s3084" xml:space="preserve">Ergo rurſus quadrata omnia F K æqualia erunt <lb/>totidem quadratis E G, una cum totidem rectangulis A G H. </s> <s xml:id="echoid-s3085" xml:space="preserve"><lb/>Quare conſtat propoſitum.</s> <s xml:id="echoid-s3086" xml:space="preserve"/> </p> <div xml:id="echoid-div267" type="float" level="2" n="3"> <note position="left" xlink:label="note-0196-04" xlink:href="note-0196-04a" xml:space="preserve">TAB. XX. <lb/>Fig. 1. 2.</note> </div> <p> <s xml:id="echoid-s3087" xml:space="preserve">Hinc ſequitur, rectangulum A G H eadem magnitudine <lb/>eſſe, utriusvis cunei ſubcentrica fuerit A H; </s> <s xml:id="echoid-s3088" xml:space="preserve">hoc eſt, ſive <lb/>per hanc, ſive per illam tangentium parallelarum A L ab-<lb/>ſciſſi. </s> <s xml:id="echoid-s3089" xml:space="preserve">Itaque A G unius caſus ad A G alterius, ut H G hu-<lb/>jus ad H G illius. </s> <s xml:id="echoid-s3090" xml:space="preserve">Sicut autem rectæ A G inter ſe, ita in <lb/>utroque caſu cunei per A L abſciſſi, ut colligitur ex prop. </s> <s xml:id="echoid-s3091" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s3092" xml:space="preserve">huj. </s> <s xml:id="echoid-s3093" xml:space="preserve">Ergo ita quoque reciproce G H ad G H.</s> <s xml:id="echoid-s3094" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3095" xml:space="preserve">Apparet etiam, dato figuræ planæ centro gravitatis G, & </s> <s xml:id="echoid-s3096" xml:space="preserve"><lb/>ſubcene<unsure/>rica cunei, per alterutram tangentium parallelarum <lb/>A L abſciſſi, dari quoque cunei, pertangentem alteram A L <lb/>abſciſſi, ſubcentricam.</s> <s xml:id="echoid-s3097" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div269" type="section" level="1" n="100"> <head xml:id="echoid-head126" xml:space="preserve">PROPOSITIO X.</head> <p style="it"> <s xml:id="echoid-s3098" xml:space="preserve">Poſitis quæ in propoſitione præcedenti; </s> <s xml:id="echoid-s3099" xml:space="preserve">ſi data <lb/> <anchor type="note" xlink:label="note-0196-05a" xlink:href="note-0196-05"/> recta E D transeat per G, centrum gravita- <pb file="0197" n="215"/> <pb file="0197a" n="216"/> <anchor type="figure" xlink:label="fig-0197a-01a" xlink:href="fig-0197a-01"/> <anchor type="figure" xlink:label="fig-0197a-02a" xlink:href="fig-0197a-02"/> <anchor type="figure" xlink:label="fig-0197a-03a" xlink:href="fig-0197a-03"/> <anchor type="figure" xlink:label="fig-0197a-04a" xlink:href="fig-0197a-04"/> <anchor type="figure" xlink:label="fig-0197a-05a" xlink:href="fig-0197a-05"/> <anchor type="figure" xlink:label="fig-0197a-06a" xlink:href="fig-0197a-06"/> <pb file="0198" n="217"/> <pb o="137" file="0199" n="218" rhead="HOROLOG. OSCILLATOR."/> tis figuræ A B C; </s> <s xml:id="echoid-s3100" xml:space="preserve">erit ſumma quadratorum à di-<lb/> <anchor type="note" xlink:label="note-0199-01a" xlink:href="note-0199-01"/> ſtantiis particularum, in quas figura diviſa intel-<lb/>ligitur, ab recta E D, æqualis rectangulo ſoli <lb/>A G H, multiplici ſecundum ipſarum particula-<lb/>rum numerum.</s> <s xml:id="echoid-s3101" xml:space="preserve"/> </p> <div xml:id="echoid-div269" type="float" level="2" n="1"> <note position="left" xlink:label="note-0196-05" xlink:href="note-0196-05a" xml:space="preserve">TAB. XX. <lb/>Fig. 3.</note> <figure xlink:label="fig-0197a-01" xlink:href="fig-0197a-01a"> <caption xml:id="echoid-caption68" style="it" xml:space="preserve">Pag. 136.<lb/>TAB. XIX.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables68" xml:space="preserve">D C X B Y E R I Q L S N K P A T<lb/>F G Y M H O</variables> </figure> <figure xlink:label="fig-0197a-02" xlink:href="fig-0197a-02a"> <caption xml:id="echoid-caption69" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables69" xml:space="preserve">X C D A T E R I Q L S N K P B Y</variables> </figure> <figure xlink:label="fig-0197a-03" xlink:href="fig-0197a-03a"> <caption xml:id="echoid-caption70" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables70" xml:space="preserve">F G K C D I E M A B D</variables> </figure> <figure xlink:label="fig-0197a-04" xlink:href="fig-0197a-04a"> <caption xml:id="echoid-caption71" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables71" xml:space="preserve">D K E F L B A H G C E</variables> </figure> <figure xlink:label="fig-0197a-05" xlink:href="fig-0197a-05a"> <caption xml:id="echoid-caption72" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables72" xml:space="preserve">D C K L F E A G H D B</variables> </figure> <figure xlink:label="fig-0197a-06" xlink:href="fig-0197a-06a"> <caption xml:id="echoid-caption73" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables73" xml:space="preserve">C D K F L E H G A D B</variables> </figure> <note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3102" xml:space="preserve">Hoc enim manifeſtum eſt, quum nullum tunc ſit quadra-<lb/>tum E G.</s> <s xml:id="echoid-s3103" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div271" type="section" level="1" n="101"> <head xml:id="echoid-head127" xml:space="preserve">PROPOSITIO XI.</head> <p style="it"> <s xml:id="echoid-s3104" xml:space="preserve">POſitis rurſus cæteris ut in præcedentium penul-<lb/> <anchor type="note" xlink:label="note-0199-02a" xlink:href="note-0199-02"/> tima; </s> <s xml:id="echoid-s3105" xml:space="preserve">ſi D E ſit axis figuræ planæ A B C, in <lb/>duas æquales ſimilesque portiones eam dividens, ſit-<lb/>que inſuper V G diſtantia centri gravitatis dimi-<lb/>diæ figuræ D A D ab recta E D, cunei vero, ſu-<lb/>per ipſam abſciſſi per ipſam E D, ſubcentrica G X; <lb/></s> <s xml:id="echoid-s3106" xml:space="preserve">erit rectangulum X G V æquale rectangulo A G H.</s> <s xml:id="echoid-s3107" xml:space="preserve"/> </p> <div xml:id="echoid-div271" type="float" level="2" n="1"> <note position="right" xlink:label="note-0199-02" xlink:href="note-0199-02a" xml:space="preserve">TAB. XX. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s3108" xml:space="preserve">Eſt enim rectangulum X G V, multiplex ſecundum nu-<lb/>merum particularum figuræ D A D, æquale quadratis omni-<lb/>bus perpendicularium à particulis ejusdem figuræ dimidiæ in <lb/>rectam E D cadentium <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3109" xml:space="preserve">Ac proinde idem rectangulum <anchor type="note" xlink:label="note-0199-03a" xlink:href="note-0199-03"/> X G V, multiplex ſecundum numerum particularum totius <lb/>figuræ A B C, æquale erit quadratis perpendicularium, ab <lb/>omnibus particulis figuræ hujus in rectam E D demiſſarum; <lb/></s> <s xml:id="echoid-s3110" xml:space="preserve">hoc eſt, rectangulo A G H multiplici ſecundum eundem <lb/>particularum numerum, ut conſtat ex propoſ. </s> <s xml:id="echoid-s3111" xml:space="preserve">præcedenti. </s> <s xml:id="echoid-s3112" xml:space="preserve"><lb/>Unde ſequitur rectangula X G V, A G H inter ſe æqualia <lb/>eſſe. </s> <s xml:id="echoid-s3113" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s3114" xml:space="preserve"/> </p> <div xml:id="echoid-div272" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0199-03" xlink:href="note-0199-03a" xml:space="preserve">Prop. @. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div274" type="section" level="1" n="102"> <head xml:id="echoid-head128" xml:space="preserve">PROPOSITIO XII.</head> <p style="it"> <s xml:id="echoid-s3115" xml:space="preserve">DAtis in plano punctis quotlibet; </s> <s xml:id="echoid-s3116" xml:space="preserve">ſi ex centro <lb/>gravitatis eorum circulus quilibet deſcribatur;</s> <s xml:id="echoid-s3117" xml:space="preserve"> <pb o="138" file="0200" n="219" rhead="CHRISTIANI HUGENII"/> ducantur autem ab omnibus datis punctis, ad pun-<lb/> <anchor type="note" xlink:label="note-0200-01a" xlink:href="note-0200-01"/> ctum aliquod in circuli illius circumferentia lineæ <lb/>rectæ erit ſumma quadratorum ab omnibus ſem-<lb/>per eidem plano æqualis.</s> <s xml:id="echoid-s3118" xml:space="preserve"/> </p> <div xml:id="echoid-div274" type="float" level="2" n="1"> <note position="left" xlink:label="note-0200-01" xlink:href="note-0200-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3119" xml:space="preserve">Sint data puncta A B C D: </s> <s xml:id="echoid-s3120" xml:space="preserve">centrumque gravitatis eorum, <lb/> <anchor type="note" xlink:label="note-0200-02a" xlink:href="note-0200-02"/> ſive magnitudinum æqualium ab ipſis ſuſpenſarum, ſit E; <lb/></s> <s xml:id="echoid-s3121" xml:space="preserve">& </s> <s xml:id="echoid-s3122" xml:space="preserve">centro E deſcribatur circulus quilibet F f, in cujus cir-<lb/>cumferentia ſumpto puncto aliquo, ut F, ducantur ad id, <lb/>à datis punctis, rectæ A F, B F, C F, D F. </s> <s xml:id="echoid-s3123" xml:space="preserve">Dico earum <lb/>omnium quadrata, ſimul ſumpta, æqualia eſſe plano cuidam <lb/>dato, ſemperque eidem, ubicunque in circumferentia pun-<lb/>ctum F ſumptum fuerit.</s> <s xml:id="echoid-s3124" xml:space="preserve"/> </p> <div xml:id="echoid-div275" type="float" level="2" n="2"> <note position="left" xlink:label="note-0200-02" xlink:href="note-0200-02a" xml:space="preserve">TAB. XX. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s3125" xml:space="preserve">Ducantur enim rectæ G H, G K, angulum rectum con-<lb/>ſtituentes, & </s> <s xml:id="echoid-s3126" xml:space="preserve">quarum unicuique omnia data puncta ſint po-<lb/>ſita ad eandem partem. </s> <s xml:id="echoid-s3127" xml:space="preserve">Et à ſingulis in utramque harum <lb/>perpendiculares agantur A L, A K; </s> <s xml:id="echoid-s3128" xml:space="preserve">B M, B O; </s> <s xml:id="echoid-s3129" xml:space="preserve">C N, <lb/>C P; </s> <s xml:id="echoid-s3130" xml:space="preserve">D H, D Q. </s> <s xml:id="echoid-s3131" xml:space="preserve">A centro autem gravitatis E, & </s> <s xml:id="echoid-s3132" xml:space="preserve">à pun-<lb/>cto F, in alterutram duarum, G H vel G K, perpendi-<lb/>culares E R, F S. </s> <s xml:id="echoid-s3133" xml:space="preserve">Et item, à datis punctis, in ipſam <lb/>F S perpendiculares A V, B X, C Y, D Z. </s> <s xml:id="echoid-s3134" xml:space="preserve">Et F T per-<lb/>pendicularis in ipſam E R. </s> <s xml:id="echoid-s3135" xml:space="preserve">Porro ſit jam</s> </p> <note position="right" xml:space="preserve"> <lb/>A L = a # A K = e # radius # E F = z <lb/>B M = b # B O = f # # G S = x <lb/>C N = c # C P = g <lb/>D H = d # D Q = h <lb/></note> <p> <s xml:id="echoid-s3136" xml:space="preserve">Quia autem E eſt centrum gravitatis punctorum A, B, C, D; <lb/></s> <s xml:id="echoid-s3137" xml:space="preserve">ſi addantur in unum perpendiculares A L, B M, C N, D H, <lb/>compoſitaque ex omnibus dividatur in tot partes, quot ſunt <lb/>data puncta; </s> <s xml:id="echoid-s3138" xml:space="preserve">earum partium uni æqualis erit E R <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3139" xml:space="preserve">Simili- <anchor type="note" xlink:label="note-0200-04a" xlink:href="note-0200-04"/> terque, divisâ in totidem partes ſummâ perpendicularium <lb/>A K, B O, C P, D Q, earum uni æqualis erit perpendi-<lb/>cularis, ducta ex E in rectam G K, ſive ipſa R G <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3140" xml:space="preserve">Ita- <anchor type="note" xlink:label="note-0200-05a" xlink:href="note-0200-05"/> que, ſi ſumma omnium A L, B M, C N, D H, ſive <lb/>a + b + c + d vocetur l: </s> <s xml:id="echoid-s3141" xml:space="preserve">ſumma vero omnium, A K, B O, <lb/>C P, D Q ſive e + f + g + h, vocetur m: </s> <s xml:id="echoid-s3142" xml:space="preserve">& </s> <s xml:id="echoid-s3143" xml:space="preserve">numerus, <pb o="139" file="0201" n="220" rhead="HOROLOG. OSCILLATOR."/> datorum punctorum multitudinem exprimens, dicatur θ; </s> <s xml:id="echoid-s3144" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-0201-01a" xlink:href="note-0201-01"/> E R = {ι/θ}; </s> <s xml:id="echoid-s3145" xml:space="preserve">& </s> <s xml:id="echoid-s3146" xml:space="preserve">R G = {μ/θ}. </s> <s xml:id="echoid-s3147" xml:space="preserve">Cumque G S ſit x, erit R S ſive <lb/>F T = x - {μ/θ}; </s> <s xml:id="echoid-s3148" xml:space="preserve">vel {μ/θ} - x, ſi G R major quam G S; </s> <s xml:id="echoid-s3149" xml:space="preserve">& </s> <s xml:id="echoid-s3150" xml:space="preserve">ſem-<lb/>per quadratum F T = xx - 2 {xμ/θ} + {μμ/θθ}. </s> <s xml:id="echoid-s3151" xml:space="preserve">quo ablato ab qua-<lb/>drato F E = zz, relinquetur quadratum T E = zz - xx <lb/>+ 2 {xμ/θ} - {μμ/θθ}. </s> <s xml:id="echoid-s3152" xml:space="preserve">Et proinde T E = <emph style="red">zz - xx + 2 {xμ/θ} - {μμ/θθ}</emph>. <lb/></s> <s xml:id="echoid-s3153" xml:space="preserve">Erat autem E R = {ι/θ}. </s> <s xml:id="echoid-s3154" xml:space="preserve">Itaque T R = {ι/θ} + vel - <emph style="red">zz - xx <lb/>+ 2 {xμ/θ} - {μμ/θθ}</emph>. </s> <s xml:id="echoid-s3155" xml:space="preserve">quæ T R, brevitatis gratia, dicatury y. </s> <s xml:id="echoid-s3156" xml:space="preserve">Colli-<lb/>gamus jam porro ſummam quadratorum omnium F A, F B, <lb/>F C, F D. </s> <s xml:id="echoid-s3157" xml:space="preserve">Quadratum A F æquatur quadratis A V, V F. </s> <s xml:id="echoid-s3158" xml:space="preserve"><lb/>Eſt autem A V æqualis differentiæ duarum V K, A K, ſi-<lb/>ve duarum S G, A K; </s> <s xml:id="echoid-s3159" xml:space="preserve">ac proinde A V = x - e vel e - x; </s> <s xml:id="echoid-s3160" xml:space="preserve">& </s> <s xml:id="echoid-s3161" xml:space="preserve"><lb/>qu. </s> <s xml:id="echoid-s3162" xml:space="preserve">A V = xx - 2 ex + ee. </s> <s xml:id="echoid-s3163" xml:space="preserve">V F vero æqualis eſt differen-<lb/>tiæ duarum F S, V S ſive duarum F S, A L; </s> <s xml:id="echoid-s3164" xml:space="preserve">ac proinde <lb/>V F = y - a vel a - y; </s> <s xml:id="echoid-s3165" xml:space="preserve">& </s> <s xml:id="echoid-s3166" xml:space="preserve">qu. </s> <s xml:id="echoid-s3167" xml:space="preserve">V F = yy - 2 ay + aa. </s> <s xml:id="echoid-s3168" xml:space="preserve">Ad-<lb/>ditisque quadratis A V, V F, fit quadratum F A = xx - 2 ex <lb/>+ ee + yy - 2 ay + aa. </s> <s xml:id="echoid-s3169" xml:space="preserve">Eodemque modo invenientur qua-<lb/>drata reliquarum F B, F C, F D; </s> <s xml:id="echoid-s3170" xml:space="preserve">atque omnia ordine diſ-<lb/>poſita erunt hæc; </s> <s xml:id="echoid-s3171" xml:space="preserve"><lb/>qu. </s> <s xml:id="echoid-s3172" xml:space="preserve">F A = xx - 2 ex + ee + yy - 2 ay + aa. </s> <s xml:id="echoid-s3173" xml:space="preserve"><lb/>qu. </s> <s xml:id="echoid-s3174" xml:space="preserve">F B = xx - 2 fx + ff + yy - 2 by + bb. </s> <s xml:id="echoid-s3175" xml:space="preserve"><lb/>qu. </s> <s xml:id="echoid-s3176" xml:space="preserve">F C = xx - 2 gx + gg + yy - 2 cy + cc. </s> <s xml:id="echoid-s3177" xml:space="preserve"><lb/>qu. </s> <s xml:id="echoid-s3178" xml:space="preserve">F D = xx - 2 hx + hh + yy - 2 dy + dd.</s> <s xml:id="echoid-s3179" xml:space="preserve"/> </p> <div xml:id="echoid-div276" type="float" level="2" n="3"> <note symbol="*" position="left" xlink:label="note-0200-04" xlink:href="note-0200-04a" xml:space="preserve">Prop. 2. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0200-05" xlink:href="note-0200-05a" xml:space="preserve">Prop. 2. <lb/>huj.</note> <note position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3180" xml:space="preserve">Horum vero ſumma; </s> <s xml:id="echoid-s3181" xml:space="preserve">ſi ponamus quadrata ee + ff + gg <lb/>+ hh = nn; </s> <s xml:id="echoid-s3182" xml:space="preserve">& </s> <s xml:id="echoid-s3183" xml:space="preserve">quadrata aa + bb, + cc + dd = kk; <lb/></s> <s xml:id="echoid-s3184" xml:space="preserve">erit iſta, θ x x - 2 mx + nn + θ yy - 2 ly + kk. </s> <s xml:id="echoid-s3185" xml:space="preserve">Siquidem <lb/>θ erat numerus datorum punctorum ideoque & </s> <s xml:id="echoid-s3186" xml:space="preserve">quadratorum, <lb/>poſitumque fuerat e + f + g + h = m, & </s> <s xml:id="echoid-s3187" xml:space="preserve">a + b + c + d = l.</s> <s xml:id="echoid-s3188" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3189" xml:space="preserve">In iſta vero ſumma, ſi in terminis θ y y & </s> <s xml:id="echoid-s3190" xml:space="preserve">2 l y, pro y, <lb/>ponatur id cujus loco poſitum erat, nempe {ι/θ} + vel -<lb/><emph style="red">zz - xx + 2 {xμ/θ} - {μμ/θθ}</emph> <lb/>fiet + θ y y = {ιι/θ} + 2 l <emph style="red">zz - xx + 2 {xμ/θ} - {μμ/θθ}</emph> + θ zz - θ xx <lb/>+ 2 x m-{μμ/θ}.</s> <s xml:id="echoid-s3191" xml:space="preserve"> <pb o="140" file="0202" n="221" rhead="CHRISTIANI HUGENII"/> & </s> <s xml:id="echoid-s3192" xml:space="preserve">- 2 ly = - 2 {ιι/θ} - 2 l <emph style="red">zz - xx + 2 {xμ/θ} - {μμ/θθ}</emph>. <lb/></s> <s xml:id="echoid-s3193" xml:space="preserve"> <anchor type="note" xlink:label="note-0202-01a" xlink:href="note-0202-01"/> vel + θ yy = {ιι/θ} - 2 l <emph style="red">zz - xx + 2 {xμ/θ} - {μμ/θθ} + θ zz - θ x x <lb/>xm - {μμ/θ}</emph>. <lb/></s> <s xml:id="echoid-s3194" xml:space="preserve">+ 2 & </s> <s xml:id="echoid-s3195" xml:space="preserve">- 2 ly = - 2{ιι/θ} + 2 l <emph style="red">zz - xx + 2 {xμ/θ} - {μμ/θθ}</emph>. </s> <s xml:id="echoid-s3196" xml:space="preserve"><lb/>Ac proinde, utroque caſu, pro θ y y - 2 l y habebitur - {ιι/θ} <lb/>+ θ z z - θ x x + 2 x m - {μμ/θ}. </s> <s xml:id="echoid-s3197" xml:space="preserve">Quò appoſitis reliquis quantita-<lb/>tibus, ſumma prædicta contentis, θ x x - 2 x m + n n + k k, <lb/>fiet tota ſumma, nempe quadratorum F A, F B, F C, F D, <lb/>= θ z z + nn + k k - {μμ - ιι/θ}. </s> <s xml:id="echoid-s3198" xml:space="preserve">Quod apparet eſſe planum da-<lb/>tum, cum hæ quantitates omnes datæ ſint; </s> <s xml:id="echoid-s3199" xml:space="preserve">ſemperque idem <lb/>reperiri, ubicunque in circumferentia ſumptum fuerit pun-<lb/>ctum F. </s> <s xml:id="echoid-s3200" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s3201" xml:space="preserve"/> </p> <div xml:id="echoid-div277" type="float" level="2" n="4"> <note position="left" xlink:label="note-0202-01" xlink:href="note-0202-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">O@CILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3202" xml:space="preserve">Quod ſi puncta data diverſas gravitates habere ponantur, <lb/>invicem commenſurabiles, ut ſi punctum A ponderet ut 2, <lb/>B ut 3, C ut 4, D ut 7, eorumque reperto gravitatis cen-<lb/>tro, circulus rurſus deſcribatur, ad cujus circumferentiæ <lb/>punctum, à datis punctis rectæ ducantur, ac ſingularum <lb/>quadrata multiplicia ſumantur ſecundum numerum ponderis <lb/>puncti ſui; </s> <s xml:id="echoid-s3203" xml:space="preserve">ut quadratum A F duplum, B F triplum, C F <lb/>quadruplum, D F ſeptuplum; </s> <s xml:id="echoid-s3204" xml:space="preserve">dico rurſus ſummam omnium <lb/>æqualem fore ſpatio dato, ſemperque eidem, ubicunque in <lb/>circumferentia punctum ſumptum fuerit. </s> <s xml:id="echoid-s3205" xml:space="preserve">Patet enim hoc ex <lb/>præcedenti demonſtratione, ſi imaginemur puncta ipſa mul-<lb/>tiplicia ſecundum numeros attributæ cuique gravitatis; </s> <s xml:id="echoid-s3206" xml:space="preserve">quaſi <lb/>nempe in A duo puncta conjuncta ſint, in B tria, in C qua-<lb/>tuor, in D ſeptem, atque illa omnia æqualiter gravia.</s> <s xml:id="echoid-s3207" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div279" type="section" level="1" n="103"> <head xml:id="echoid-head129" xml:space="preserve">PROPOSITIO XIII.</head> <p style="it"> <s xml:id="echoid-s3208" xml:space="preserve">SI figura plana, vel linea in plano exiſtens, ali-<lb/>ter atque aliter ſuſpendatur à punctis, quæ, in <lb/>eodem plano accepta, æqualiter à centro gravitatis <lb/>ſuæ diſtent; </s> <s xml:id="echoid-s3209" xml:space="preserve">agitatamotu in latus, ſibi ipſi iſochrona eſt.</s> <s xml:id="echoid-s3210" xml:space="preserve"/> </p> <pb o="141" file="0203" n="222" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s3211" xml:space="preserve">Sit figura plana, vel linea in plano exiſtens A B C, cu-<lb/> <anchor type="note" xlink:label="note-0203-01a" xlink:href="note-0203-01"/> jus centrum gravitatis D. </s> <s xml:id="echoid-s3212" xml:space="preserve">quo eodem centro, circumferentia <lb/>circuli in eodem plano deſcribatur, E C F. </s> <s xml:id="echoid-s3213" xml:space="preserve">Dico, ſi à quo-<lb/>vis in illa puncto, ut E, C, vel G, ſuſpenſa figura agite-<lb/>tur in latus; </s> <s xml:id="echoid-s3214" xml:space="preserve">ſibi ipſi, ſive eidem pendulo ſimplici, iſochro-<lb/>nam eſſe.</s> <s xml:id="echoid-s3215" xml:space="preserve"/> </p> <div xml:id="echoid-div279" type="float" level="2" n="1"> <note position="right" xlink:label="note-0203-01" xlink:href="note-0203-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>. <lb/>TAB. XX. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s3216" xml:space="preserve">Sit prima ſuſpenſio ex E puncto, quando autem eſt extra <lb/>figuram, ut hic, putandum eſt lineam E H, ex qua figura <lb/>pendet, rigidam eſſe, atque immobiliter ipſi affixam.</s> <s xml:id="echoid-s3217" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3218" xml:space="preserve">Intelligatur figura A B C diviſa in particulas minimas æ-<lb/>quales, à quarum omnium centris gravitatis, ad punctum <lb/>E, rectæ ductæ ſint; </s> <s xml:id="echoid-s3219" xml:space="preserve">quas quidem manifeſtum eſt, quum <lb/>moveatur figura motu in latus, eſſe ad axem agitationis per-<lb/>pendiculares. </s> <s xml:id="echoid-s3220" xml:space="preserve">Harum igitur omnium perpendicularium qua-<lb/>drata, diviſa per rectam E D, multiplicem ſecundum nu-<lb/>merum particularum in quas figura diviſa eſt, efficiunt lon-<lb/>gitudinem penduli ſimplicis, figuræ iſochroni <anchor type="note" xlink:href="" symbol="*"/>, quæ ſit K L.</s> <s xml:id="echoid-s3221" xml:space="preserve"> <anchor type="note" xlink:label="note-0203-02a" xlink:href="note-0203-02"/> Suſpensâ autem figurâ ex puncto G, rurſus longitudo pen-<lb/>duli ſimplicis iſochroni invenitur, dividendo quadrata omnia <lb/>linearum, quæ à particulis figuræ ducuntur ad punctum G, <lb/>per rectam G D, multiplicem ſecundum earundem particu-<lb/>larum numerum <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3222" xml:space="preserve">Quum igitur puncta G & </s> <s xml:id="echoid-s3223" xml:space="preserve">E ſint in cir- <anchor type="note" xlink:label="note-0203-03a" xlink:href="note-0203-03"/> cumferentia deſcripta cetnro D, quod eſt centrum gravitatis <lb/>figuræ A B C, ſive centrum gravitatis punctorum omnium, <lb/>quæ centra ſunt particularum figuræ æqualium; </s> <s xml:id="echoid-s3224" xml:space="preserve">erit proinde <lb/>ſumma quadratorum à lineis, qnæ à dictis particulis ad pun-<lb/>ctum G ducuntur, æqualis ſummæ quadratorum à lineis quæ <lb/>ab iiſdem particulis ducuntur ad punctum E <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3225" xml:space="preserve">Hæ vero <anchor type="note" xlink:label="note-0203-04a" xlink:href="note-0203-04"/> quadratorum ſummæ, utraque ſuſpenſione, applicantur ad <lb/>magnitudines æquales: </s> <s xml:id="echoid-s3226" xml:space="preserve">quippe, in ſuſpenſione ex E, ad re-<lb/>ctam E D, multiplicem ſecundum numerum omnium par-<lb/>ticularum; </s> <s xml:id="echoid-s3227" xml:space="preserve">in ſuſpenſione autem ex G, ad rectam D G, <lb/>multiplicem ſecundum earundem particularum numerum. <lb/></s> <s xml:id="echoid-s3228" xml:space="preserve">Ergo patet, ex applicatione hac poſteriori, quum nempe <lb/>ſuſpenſio eſt ex G, fieri longitudinem penduli iſochroni ean-<lb/>dem atque ex applicatione priori, hoc eſt, eandem ipſi K L.</s> <s xml:id="echoid-s3229" xml:space="preserve"> <pb o="142" file="0204" n="223" rhead="CHRISTIANI HUGENII"/> Eodem modo, ſi ex C, vel alio quovis puncto circumfe-<lb/> <anchor type="note" xlink:label="note-0204-01a" xlink:href="note-0204-01"/> rentiæ E C F, figura ſuſpendatur, eidem pendulo K L iſo-<lb/>chrona eſſe probabitur. </s> <s xml:id="echoid-s3230" xml:space="preserve">Itaque conſtat propoſitum.</s> <s xml:id="echoid-s3231" xml:space="preserve"/> </p> <div xml:id="echoid-div280" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0203-02" xlink:href="note-0203-02a" xml:space="preserve">Prop. 6. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0203-03" xlink:href="note-0203-03a" xml:space="preserve">Prop. 6. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0203-04" xlink:href="note-0203-04a" xml:space="preserve">Prop. <lb/>præced.</note> <note position="left" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> </div> <div xml:id="echoid-div282" type="section" level="1" n="104"> <head xml:id="echoid-head130" xml:space="preserve">PROPOSITIO XIV.</head> <p style="it"> <s xml:id="echoid-s3232" xml:space="preserve">DAtâ figurâ ſolidâ, & </s> <s xml:id="echoid-s3233" xml:space="preserve">lineâ rectâ interminatâ, <lb/>quæ vel extra figuram cadat, vel per eam <lb/>transeat; </s> <s xml:id="echoid-s3234" xml:space="preserve">divisâque figurâ cogitatu in particulas <lb/>minimas æquales, à quibus omnibus ad datam re-<lb/>ctam perpendiculares ductæ intelligantur; </s> <s xml:id="echoid-s3235" xml:space="preserve">invenire <lb/>ſummam omnium quæ ab ipſis fiunt quadratorum, <lb/>ſive planum, cujus multiplex ſecundum particula-<lb/>rum numerum, dictæ quadratorum ſummæ æ-<lb/>quale ſit.</s> <s xml:id="echoid-s3236" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3237" xml:space="preserve">Sit data figura ſolida A B C D, & </s> <s xml:id="echoid-s3238" xml:space="preserve">linea recta quæ, per <lb/> <anchor type="note" xlink:label="note-0204-02a" xlink:href="note-0204-02"/> punctum E tranſiens, ad planum hujus paginæ erecta intel-<lb/>ligatur: </s> <s xml:id="echoid-s3239" xml:space="preserve">quæque vel ſecet figuram, vel tota extra cadat. </s> <s xml:id="echoid-s3240" xml:space="preserve">In-<lb/>tellectoque, à ſingulis particulis minimis æqualibus, ſolidum <lb/>A B C D conſtituentibus, velut F, rectas duci perpendi-<lb/>culares in datam rectam per E, quemadmodum hic F E, <lb/>oporteat omnium quadratorum F E ſummam invenire.</s> <s xml:id="echoid-s3241" xml:space="preserve"/> </p> <div xml:id="echoid-div282" type="float" level="2" n="1"> <note position="left" xlink:label="note-0204-02" xlink:href="note-0204-02a" xml:space="preserve">TAB. XXI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s3242" xml:space="preserve">Secetur figura plano E A C, per dictam datam lineam & </s> <s xml:id="echoid-s3243" xml:space="preserve"><lb/>per centrum gravitatis figuræ ducto. </s> <s xml:id="echoid-s3244" xml:space="preserve">Item aliud planum in-<lb/>telligatur per eandem lineam datam, perque E G, quæ ipſi <lb/>eſt ad angulos rectos.</s> <s xml:id="echoid-s3245" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3246" xml:space="preserve">Conſtat jam, quadratum rectæ cujuſque, quæ à particula <lb/>dictarum aliqua, ad lineam datam per E perpendicularis du-<lb/>citur, ſicut F E, æquari quadratis duarum F G, F H, <lb/>quæ, ab eadem particula, in plana per E G & </s> <s xml:id="echoid-s3247" xml:space="preserve">E C ante di-<lb/>cta, perpendiculares aguntur <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3248" xml:space="preserve">Quare, ſi cognoſcere poſſi- <anchor type="note" xlink:label="note-0204-03a" xlink:href="note-0204-03"/> mus ſummam quadratorum, quæ fiunt ab omnibus perpen-<lb/>dicularibus, quæ à particulis univerſis cadunt in plana dicta <lb/>per E G & </s> <s xml:id="echoid-s3249" xml:space="preserve">E C; </s> <s xml:id="echoid-s3250" xml:space="preserve">habebimus etiam huic æqualem ſum- <pb file="0205" n="224"/> <pb file="0205a" n="225"/> <anchor type="figure" xlink:label="fig-0205a-01a" xlink:href="fig-0205a-01"/> <anchor type="figure" xlink:label="fig-0205a-02a" xlink:href="fig-0205a-02"/> <anchor type="figure" xlink:label="fig-0205a-03a" xlink:href="fig-0205a-03"/> <anchor type="figure" xlink:label="fig-0205a-04a" xlink:href="fig-0205a-04"/> <anchor type="figure" xlink:label="fig-0205a-05a" xlink:href="fig-0205a-05"/> <anchor type="figure" xlink:label="fig-0205a-06a" xlink:href="fig-0205a-06"/> <pb file="0206" n="226"/> <pb o="143" file="0207" n="227" rhead="HOROLOG. OSCILLATOR."/> mam quadratorum à perpendicularibus, quæ ab univerſis <lb/> <anchor type="note" xlink:label="note-0207-01a" xlink:href="note-0207-01"/> iiſdem particulis cadunt in rectam datam per E punctum.</s> <s xml:id="echoid-s3251" xml:space="preserve"/> </p> <div xml:id="echoid-div283" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0204-03" xlink:href="note-0204-03a" xml:space="preserve">47. lib. 1. <lb/>Eucl.</note> <figure xlink:label="fig-0205a-01" xlink:href="fig-0205a-01a"> <caption xml:id="echoid-caption74" style="it" xml:space="preserve">Pag. 142.<lb/>TAB. XX.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables74" xml:space="preserve">D L F K A E G H C L K F D B</variables> </figure> <figure xlink:label="fig-0205a-02" xlink:href="fig-0205a-02a"> <caption xml:id="echoid-caption75" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables75" xml:space="preserve">D F K L C H E G A K F L D B</variables> </figure> <figure xlink:label="fig-0205a-03" xlink:href="fig-0205a-03a"> <caption xml:id="echoid-caption76" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables76" xml:space="preserve">L D C A E H G B L D</variables> </figure> <figure xlink:label="fig-0205a-04" xlink:href="fig-0205a-04a"> <caption xml:id="echoid-caption77" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables77" xml:space="preserve">D L C E A X V G H L D B</variables> </figure> <figure xlink:label="fig-0205a-05" xlink:href="fig-0205a-05a"> <caption xml:id="echoid-caption78" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables78" xml:space="preserve">T F K A V Q Z D E O B X P C Y f I G M L R N S H</variables> </figure> <figure xlink:label="fig-0205a-06" xlink:href="fig-0205a-06a"> <caption xml:id="echoid-caption79" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables79" xml:space="preserve">K E A H C L D F G B</variables> </figure> <note position="right" xlink:label="note-0207-01" xlink:href="note-0207-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3252" xml:space="preserve">Illa vero prior quadratorum ſumma colligetur hoc modo. <lb/></s> <s xml:id="echoid-s3253" xml:space="preserve">Ponatur primò figuram planam dari O Q P, ad latus figu-<lb/>ræ ſolidæ A B C D, ejuſdem cum ipſa altitudinis, quæque <lb/>ſit ejuſmodi, ut ſecta lineis rectis Q Q, R R, quæ reſpon-<lb/>deant planis figuram ſolidam A B C D ſecantibus M M, <lb/>N N, & </s> <s xml:id="echoid-s3254" xml:space="preserve">his parallelis; </s> <s xml:id="echoid-s3255" xml:space="preserve">eadem ſit dictarum linearum inter <lb/>ſe, quæ & </s> <s xml:id="echoid-s3256" xml:space="preserve">planorum horum ratio, ſi nempe ſumantur utrin-<lb/>que quæ in ordine ſibi reſpondent. </s> <s xml:id="echoid-s3257" xml:space="preserve">Ut ſi linea R R ſit ad <lb/>Q Q quemadmodum planum N N ad M M. </s> <s xml:id="echoid-s3258" xml:space="preserve">Quod ſi igitur <lb/>figura plana O Q P, in totidem particulas minimas æquales <lb/>diviſa intelligatur, quot intelliguntur in ſolido A B C D, <lb/>erunt etiam in unoquoque ſegmento figuræ planæ, velut <lb/>Q Q R R, tot numero particulæ, quot ſunt in figuræ ſoli-<lb/>dæ ſegmento M M N N, iſti ſegmento reſpondente; </s> <s xml:id="echoid-s3259" xml:space="preserve">ac <lb/>proinde & </s> <s xml:id="echoid-s3260" xml:space="preserve">ſumma quadratorum, à perpendicularibus o-<lb/>mnium particularum figuræ O Q P in planum E G, æqua-<lb/>bitur ſummæ quadratorum, à perpendicularibus omnium par-<lb/>ticularum figuræ ſolidæ, in idem planum E G productis. </s> <s xml:id="echoid-s3261" xml:space="preserve">Il-<lb/>la autem quadratorum ſumma data erit, ſi dentur in figura <lb/>O Q P, cuneoque illius, quæ propoſ. </s> <s xml:id="echoid-s3262" xml:space="preserve">9. </s> <s xml:id="echoid-s3263" xml:space="preserve">huj. </s> <s xml:id="echoid-s3264" xml:space="preserve">requiri dixi-<lb/>mus. </s> <s xml:id="echoid-s3265" xml:space="preserve">Ergo his datis, dabitur quoque ſumma quadratorum, <lb/>à perpendicularibus quæ, à particulis omnibus ſolidi A B C D, <lb/>ducuntur in planum E G.</s> <s xml:id="echoid-s3266" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3267" xml:space="preserve">Ponatur nunc alia item figura plana S Y T Z, cjuſdem cum <lb/>ſolido A B C D latitudinis, hoc eſt, quam includant plana B Y, <lb/>D Z ſolidum contingentia, ac parallela plano E A C, quæque <lb/>ſit ejuſmodi, ut, ſecta lineis rectis V V, X X &</s> <s xml:id="echoid-s3268" xml:space="preserve">c. </s> <s xml:id="echoid-s3269" xml:space="preserve">quæ reſpon-<lb/>deant planis figuram A B C D ſecantibus, K K, L L, & </s> <s xml:id="echoid-s3270" xml:space="preserve">his <lb/>parallelis, faciat eandem inter ſe rationem linearum harum <lb/>atque illorum planorum, ſi ſumantur quæ ſibi mutuo reſpon-<lb/>dent. </s> <s xml:id="echoid-s3271" xml:space="preserve">Itaque rurſus quadrata ſimul omnia perpendicularium, <lb/>à particulis figuræ S Y T Z in rectam S T cadentium, <lb/>æqualia erunt quadratis omnibus perpendicularium quæ, à <lb/>particulis ſolidi A B C D, ducuntur in planum A C. </s> <s xml:id="echoid-s3272" xml:space="preserve">Illo- <pb o="144" file="0208" n="228" rhead="CHRISTIANI HUGENII"/> rum autem ſumma quadratorum data erit, ſi detur diſtantia <lb/> <anchor type="note" xlink:label="note-0208-01a" xlink:href="note-0208-01"/> centri gravitatis figuræ S Y T Z ab recta B Y vel D Z; <lb/></s> <s xml:id="echoid-s3273" xml:space="preserve">nec non diſtantia indidem centri gravitatis cunei ſui abſciſſi <lb/>plano per eandem rectam <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3274" xml:space="preserve">Vel, figura S Y T Z ordinata <anchor type="note" xlink:label="note-0208-02a" xlink:href="note-0208-02"/> exiſtente, ut S T ſit axis ejus, eadem quadratorum ſumma da-<lb/>bitur, ſi detur diſtantia centri gravitatis figuræ dimidiæ S Z T <lb/>ab axe S T, item centri gravitatis cunei, ſuper eadem di-<lb/>midia figura, abſciſſi plano per axem ducto <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3275" xml:space="preserve">Ergo, his <anchor type="note" xlink:label="note-0208-03a" xlink:href="note-0208-03"/> datis, dabitur quoque ſumma quadratorum à perpendicula-<lb/>ribus quæ, à particulis omnibus ſolidi A B C D, ductæ <lb/>intelliguntur in planum E A C. </s> <s xml:id="echoid-s3276" xml:space="preserve">Invenimus autem & </s> <s xml:id="echoid-s3277" xml:space="preserve">ſum-<lb/>mam quadratorum, à perpendicularibus omnibus in planum <lb/>per E G ductis. </s> <s xml:id="echoid-s3278" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3279" xml:space="preserve">aggregatum utriuſque ſummæ ha-<lb/>bebitur, hoc eſt, per ſuperius oſtenſa, ſumma quadratorum <lb/>perpendicularium quæ, à particulis omnibus ſolidi A B C D, <lb/>cadunt in rectam datam per E tranſeuntem, & </s> <s xml:id="echoid-s3280" xml:space="preserve">ad paginæ <lb/>hujus planum erectam. </s> <s xml:id="echoid-s3281" xml:space="preserve">quod erat faciendum.</s> <s xml:id="echoid-s3282" xml:space="preserve"/> </p> <div xml:id="echoid-div284" type="float" level="2" n="3"> <note position="left" xlink:label="note-0208-01" xlink:href="note-0208-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">O@CILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="left" xlink:label="note-0208-02" xlink:href="note-0208-02a" xml:space="preserve">Prop. 9. <lb/>huj.</note> <note symbol="*" position="left" xlink:label="note-0208-03" xlink:href="note-0208-03a" xml:space="preserve">Prop. 11. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div286" type="section" level="1" n="105"> <head xml:id="echoid-head131" xml:space="preserve">PROPOSITIO XV.</head> <p style="it"> <s xml:id="echoid-s3283" xml:space="preserve">IIsdem poſitis, ſi ſolidum A B C D ſit ejusmodi, ut <lb/> <anchor type="note" xlink:label="note-0208-04a" xlink:href="note-0208-04"/> figura plana S Y T Z, ipſi proportionalis, non ha-<lb/>beat notam diſtantiam centri gravitatis à tangenti-<lb/>bus B Y vel D Z, vel, ut ſubcentrica cunei ſuper ipſa <lb/>abſciſſi, plano per easdem B Y vel D Z, ignoretur; <lb/></s> <s xml:id="echoid-s3284" xml:space="preserve">in figura tamen proportionali, quæ à latere eſt, <lb/>O Q P, detur diſtantia Φ P, qua centrum gravita-<lb/>tis figuræ dimidiæ O P V abeſt ab axe O P; </s> <s xml:id="echoid-s3285" xml:space="preserve">li-<lb/>cebit hinc invenire ſummam quadratorum à diſtan-<lb/>tiis particularum ſolidi A B C D à plano E C. </s> <s xml:id="echoid-s3286" xml:space="preserve">O-<lb/>portet autem ut ſectiones omnes, N N, M M, ſint <lb/>plana ſimilia; </s> <s xml:id="echoid-s3287" xml:space="preserve">utque per omnium centra gravitatis <lb/>transeat planum E C; </s> <s xml:id="echoid-s3288" xml:space="preserve">quemadmodum in prismate, <lb/>pyramide, c<unsure/>ono, conoidibus, multisque aliis figu- <pb o="145" file="0209" n="229" rhead="HOROLOG. OSCILLATOR."/> ris contingit. </s> <s xml:id="echoid-s3289" xml:space="preserve">Atque eorum planorum diſtantias cen-<lb/> <anchor type="note" xlink:label="note-0209-01a" xlink:href="note-0209-01"/> tri gravitatis, ſuper tangentibus axi oſcillationis <lb/>parallelis, datas eſſe neceſſe eſt; </s> <s xml:id="echoid-s3290" xml:space="preserve">uti & </s> <s xml:id="echoid-s3291" xml:space="preserve">ſubcentri-<lb/>cas cuneorum, qui ſuper ipſis abſcinduntur, ductis <lb/>planis per easdem tangentes.</s> <s xml:id="echoid-s3292" xml:space="preserve"/> </p> <div xml:id="echoid-div286" type="float" level="2" n="1"> <note position="left" xlink:label="note-0208-04" xlink:href="note-0208-04a" xml:space="preserve">TAB. XXI. <lb/>Fig. 1. & 2.</note> <note position="right" xlink:label="note-0209-01" xlink:href="note-0209-01a" xml:space="preserve"><emph style="sc">De centr@</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3293" xml:space="preserve">Veluti, ſi maxima dictarum ſectionum ſit B D, & </s> <s xml:id="echoid-s3294" xml:space="preserve">in B <lb/> <anchor type="note" xlink:label="note-0209-02a" xlink:href="note-0209-02"/> intelligatur recta parallela axi E, hoc eſt, erecta ad planum <lb/>quod hic conſpicitur, oportet datam eſſe diſtantiam centri <lb/>gr. </s> <s xml:id="echoid-s3295" xml:space="preserve">ſectionis B D à dicta linea in B, quæ ſit B C; </s> <s xml:id="echoid-s3296" xml:space="preserve">itemque <lb/>ſubcentricam cunei, ſuper ſectione B D abſciſſi, plano du-<lb/>cto per eandem lineam in B, quæ ſubcentrica ſit B K.</s> <s xml:id="echoid-s3297" xml:space="preserve"/> </p> <div xml:id="echoid-div287" type="float" level="2" n="2"> <note position="right" xlink:label="note-0209-02" xlink:href="note-0209-02a" xml:space="preserve">Fig. 2</note> </div> <p> <s xml:id="echoid-s3298" xml:space="preserve">Etenim his datis, divisâque P V bifariam in Δ, ſi fiat <lb/>ſicut Δ P ad P Φ, ita rectangulum B C K ad ſpatium quod-<lb/>dam Z; </s> <s xml:id="echoid-s3299" xml:space="preserve">dico hoc ipſum, multiplex per numerum particu-<lb/>larum ſolidi A B C D, æquari ſummæ quæſitæ quadrato-<lb/>rum, à diſtantiis earundem particularum à plano E C.</s> <s xml:id="echoid-s3300" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3301" xml:space="preserve">Quadrata enim à diſtantiis particularum planæ ſectionis <lb/>B D, à plano E C, quod per centrum gravitatis ſuæ tranſit; <lb/></s> <s xml:id="echoid-s3302" xml:space="preserve">ſive quadrata à diſtantiis particularum ſolidarum ſegmenti <lb/>B N N D à plano eodem, æquari conſtat rectangulo B C K, <lb/>multiplici per numerum dictarum particularum <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3303" xml:space="preserve">Similiter, <anchor type="note" xlink:label="note-0209-03a" xlink:href="note-0209-03"/> ſi planæ ſectionis N N diſtantia centri gravitatis, ab recta <lb/>quæ in N intelligitur axi E parallela, ſit N X; </s> <s xml:id="echoid-s3304" xml:space="preserve">ſubcentrica <lb/>vero cunei ſuper ipſa abſciſſi, plano per eandem rectam, ſit <lb/>N F; </s> <s xml:id="echoid-s3305" xml:space="preserve">erunt quadrata à diſtantiis particularum planarum ſe-<lb/>ctionis N N à plano E C, ſive quadrata à diſtantiis parti-<lb/>cularum ſolidarum ſegmenti N M M N, à plano eodem, <lb/>æqualia rectangulo N X F, multiplici per numerum parti-<lb/>cularum ipſarum ſectionis N N, vel ſegmenti N M M N. <lb/></s> <s xml:id="echoid-s3306" xml:space="preserve">Eſt autem B D diviſa ſimiliter in C & </s> <s xml:id="echoid-s3307" xml:space="preserve">K, atque N N in X <lb/>& </s> <s xml:id="echoid-s3308" xml:space="preserve">F. </s> <s xml:id="echoid-s3309" xml:space="preserve">Ergo rectangulum B C K ad rectangulum N X F, <lb/>ſicut quadratum B D ad quadratum N N.</s> <s xml:id="echoid-s3310" xml:space="preserve"/> </p> <div xml:id="echoid-div288" type="float" level="2" n="3"> <note symbol="*" position="right" xlink:label="note-0209-03" xlink:href="note-0209-03a" xml:space="preserve">Prop. 10. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s3311" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s3312" xml:space="preserve">numerus particularum ſectionis B D, ad nu-<lb/>merum particularum ſectionis N N, ſicut ſectiones ipſæ;</s> <s xml:id="echoid-s3313" xml:space="preserve"> <pb o="146" file="0210" n="230" rhead="CHRISTIANI HUGENII"/> hoc eſt, ſicut quadratum B D ad quadratum N N. </s> <s xml:id="echoid-s3314" xml:space="preserve">Itaque <lb/> <anchor type="note" xlink:label="note-0210-01a" xlink:href="note-0210-01"/> rectangulum B C K, multiplex per numerum particularum <lb/>ſectionis B D, ad rectangulum N X F, multiplex per nu-<lb/>merum particularum ſectionis N N, duplicatam habebit ra-<lb/>tionem quadrati B D ad quadratum N N; </s> <s xml:id="echoid-s3315" xml:space="preserve">hoc eſt, eam <lb/>quam quadratum V V ad quadratum R R, in figura pro-<lb/>portionali. </s> <s xml:id="echoid-s3316" xml:space="preserve">Erit igitur & </s> <s xml:id="echoid-s3317" xml:space="preserve">dicta prior ſumma quadratorum, à <lb/>diſtantiis particularum ſegmenti B N N D à plano E C, ad <lb/>ſummam alteram quadratorum, à diſtantiis particularum ſe-<lb/>gmenti N M M N, ut qu. </s> <s xml:id="echoid-s3318" xml:space="preserve">V V ad qu. </s> <s xml:id="echoid-s3319" xml:space="preserve">R R. </s> <s xml:id="echoid-s3320" xml:space="preserve">Eademque <lb/>ratione oſtendetur, ſummas quadratorum à diſtantiis parti-<lb/>larum in reliquis ſegmentis ſolidi A B C D, eſſe inter ſe in <lb/>ratione quadratorum quæ fiunt à rectis in figura O V V, <lb/>quæ baſi cujusque ſegmenti reſpondent. </s> <s xml:id="echoid-s3321" xml:space="preserve">Quare ſumma qua-<lb/>dratorum, à diſtantiis particularum omnium ſegmentorum <lb/>ſolidi A B C D à plano E C, erit ad ſummam quadrato-<lb/>rum, à diſtantiis particularum ſegmentorum totidem, maxi-<lb/>mo ſegmento æqualium, hoc eſt, cylindri vel prismatis <lb/>B D S S, eandem cum ſolido A B C D baſin altitudinem-<lb/>que habentis, ſicut quadrata omnia rectarum V V, R R, Q Q, <lb/>&</s> <s xml:id="echoid-s3322" xml:space="preserve">c. </s> <s xml:id="echoid-s3323" xml:space="preserve">ad quadrata totidem maximo V V æqualia, hoc eſt, <lb/>ſicut ſolidum rotundum O V V circa axem O P, ad cylin-<lb/>drum V V Ω Ω, qui baſin & </s> <s xml:id="echoid-s3324" xml:space="preserve">altitudinem habeat eandem. <lb/></s> <s xml:id="echoid-s3325" xml:space="preserve">Hanc vero rationem ſolidi O V V ad cylindrum V V Ω Ω, <lb/>componi conſtat ex ratione planorum quorum converſione <lb/>generantur, hoc eſt, ex ratione plani O P V, ad rectangu-<lb/>lum P Ω, & </s> <s xml:id="echoid-s3326" xml:space="preserve">ex ratione diſtantiarum quibus horum plano-<lb/>rum centra gravitatis abſunt ab axe O P; </s> <s xml:id="echoid-s3327" xml:space="preserve">hoc eſt, & </s> <s xml:id="echoid-s3328" xml:space="preserve">ex ra-<lb/>tione P Φ ad P Δ. </s> <s xml:id="echoid-s3329" xml:space="preserve">Et prior quidem harum rationum, nem-<lb/>pe plani O P V ad rectangulum P Ω, eadem eſt quæ ſolidi <lb/>A B C D ad cylindrum vel prisma B D S S, hoc eſt, ea-<lb/>dem quæ numeri particularum ſolidi A B C D, ad nume-<lb/>rum particularum cylindri vel prismatis B D S S. </s> <s xml:id="echoid-s3330" xml:space="preserve">Altera <lb/>vero ratio, nempe P Φ ad P Δ, eſt eadem, ex conſtru-<lb/>ctione, quæ ſpatii Z ad rectangulum B C K. </s> <s xml:id="echoid-s3331" xml:space="preserve">Habebit ita-<lb/>que dicta ſumma quadratorum, à diſtantiis omnium particu- <pb o="147" file="0211" n="231" rhead="HOROLOG. OSCILLATOR."/> larum ſolidi A B C D à plano E C, ad ſummam quadrato-<lb/> <anchor type="note" xlink:label="note-0211-01a" xlink:href="note-0211-01"/> rum, à diſtantiis omnium particularum cylindri vel prisma-<lb/>tis B D S S ab eodem plano, rationem eam quæ componi-<lb/>tur ex ratione numeri particularum ſolidi A B C D, ad nu-<lb/>merum particularum cylindri vel prismatis B D S S, & </s> <s xml:id="echoid-s3332" xml:space="preserve">ex <lb/>ratione ſpatii Z ad rectangulum B C K: </s> <s xml:id="echoid-s3333" xml:space="preserve">hoc eſt, rationem <lb/>quam habet rectangulum Z, multiplex per numerum parti-<lb/>cularum ſolidi A B C D, ad rectangulum B C K, multi-<lb/>plex per numerum particularum cylindri vel prismatis B D S S. <lb/></s> <s xml:id="echoid-s3334" xml:space="preserve">Atqui quarta harum magnitudinum æqualis eſt ſecundæ <lb/>nempe rectangulum B C K, multiplex per numerum parti-<lb/>cularum cylindri vel prismatis B D S S, æquale ſummæ <lb/>quadratorum, à diſtantiis particularum ejusdem prismatis <lb/>vel cylindri B D S S à plano E C; </s> <s xml:id="echoid-s3335" xml:space="preserve">ſiquidem rectangulum <lb/>idem B C K, multiplex per numerum particularum ſegmen-<lb/>ti B N N D, æquatur quadratis diſtantiarum particularum <lb/>ejusdem ſegmenti à plano E C <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3336" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s3337" xml:space="preserve">tertia primæ æ- <anchor type="note" xlink:label="note-0211-02a" xlink:href="note-0211-02"/> quabitur; </s> <s xml:id="echoid-s3338" xml:space="preserve">nempe planum Z, multiplex per numerum parti-<lb/>cularum ſolidi A B C D, ſummæ quadratorum, à diſtantiis <lb/>particularum ſolidi ejusdem A B C D à plano E C <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3339" xml:space="preserve">quod <anchor type="note" xlink:label="note-0211-03a" xlink:href="note-0211-03"/> erat demonſtrandum.</s> <s xml:id="echoid-s3340" xml:space="preserve"/> </p> <div xml:id="echoid-div289" type="float" level="2" n="4"> <note position="left" xlink:label="note-0210-01" xlink:href="note-0210-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note position="right" xlink:label="note-0211-01" xlink:href="note-0211-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="right" xlink:label="note-0211-02" xlink:href="note-0211-02a" xml:space="preserve">Prop. 10. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0211-03" xlink:href="note-0211-03a" xml:space="preserve">Prop. 14. <lb/>lib. 5. Eucl.</note> </div> <p> <s xml:id="echoid-s3341" xml:space="preserve">Notandum vero, quando ſolidum A B D rotundum eſt <lb/>circa axem A C, fieri ſemper rectangulum B C K æquale <lb/>quartæ parti quadrati B C; </s> <s xml:id="echoid-s3342" xml:space="preserve">quoniam ſubcentrica cunei, ab-<lb/>ſciſſi ſuper circulo B D, plano per tangentem in B, nempe <lb/>recta B K, æquatur {5/4} radii B C. </s> <s xml:id="echoid-s3343" xml:space="preserve">Unde, ſi P V æqualis <lb/>poſita ſit B C, ſequitur, faciendo ut Ρ Δ ad Ρ Φ ita rectan-<lb/>gulum B C K, hoc eſt, {1/4} quadrati B C, hoc eſt, qu. </s> <s xml:id="echoid-s3344" xml:space="preserve">Ρ Δ <lb/>ad planum aliud Z, fore hoc rectangulo Δ Ρ Φ æquale. </s> <s xml:id="echoid-s3345" xml:space="preserve">Ac <lb/>proinde tunc ipſum rectangulum Δ Ρ Φ, multiplex ſecun-<lb/>dum numerum particularum ſolidi A B D, æquari ſummæ <lb/>quæſitæ quadratorum à perpendicularibus omnibus, quæ à <lb/>particulis iisdem cadunt in planum E C.</s> <s xml:id="echoid-s3346" xml:space="preserve"/> </p> <pb o="148" file="0212" n="232" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div291" type="section" level="1" n="106"> <head xml:id="echoid-head132" xml:space="preserve">PROPOSITIO XVI.</head> <note position="left" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <p style="it"> <s xml:id="echoid-s3347" xml:space="preserve">FIgura quævis, ſive linea fuerit, ſive ſuperſi-<lb/>cies, ſive ſolidum; </s> <s xml:id="echoid-s3348" xml:space="preserve">ſi aliter at que aliter ſuſpen-<lb/>datur, agiteturque ſuper axibus inter ſe paralle-<lb/>lis, quique à centro gravitatis figuræ æqualiter di-<lb/>ſtent, ſibi ipſi iſochrona eſt.</s> <s xml:id="echoid-s3349" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3350" xml:space="preserve">Proponatur magnitudo quævis, cujus centrum gravitatis <lb/>E punctum, ſitque primo ſuſpenſa ab axe, qui per F intel-<lb/> <anchor type="note" xlink:label="note-0212-02a" xlink:href="note-0212-02"/> ligitur hujus paginæ plano ad angulos rectos. </s> <s xml:id="echoid-s3351" xml:space="preserve">Itaque idem <lb/>planum erit & </s> <s xml:id="echoid-s3352" xml:space="preserve">planum oſcillationis. </s> <s xml:id="echoid-s3353" xml:space="preserve">In quo ſi centro E, ra-<lb/>dio E F, deſcribatur circumferentia F H G, ſumptoque in <lb/>illa puncto quovis, ut H, magnitudo ſecundò ſuſpendi intel-<lb/>ligatur ab axe in hoc puncto infixo, atque agitari, manente <lb/>eodem oſcillationis plano. </s> <s xml:id="echoid-s3354" xml:space="preserve">Dico iſochronam fore ſibi ipſi agi-<lb/>tatæ circa axem in F.</s> <s xml:id="echoid-s3355" xml:space="preserve"/> </p> <div xml:id="echoid-div291" type="float" level="2" n="1"> <note position="left" xlink:label="note-0212-02" xlink:href="note-0212-02a" xml:space="preserve">TAB. XXI. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s3356" xml:space="preserve">Intelligatur enim dividi magnitudo propoſita in particu-<lb/>las minimas æquales. </s> <s xml:id="echoid-s3357" xml:space="preserve">Itaque, quia in utraque illa ſuſpenſio-<lb/>ne idem manet oſcillationis planum, reſpectu partium ma-<lb/>gnitudinis; </s> <s xml:id="echoid-s3358" xml:space="preserve">manifeſtum eſt, ſi ab omnibus particulis, in quas <lb/>diviſa eſt magnitudo, perpendiculares cadere concipiantur <lb/>in dictum oſcillationis planum, illas utraque ſuſpenſione oc-<lb/>currere ipſi in punctis iisdem. </s> <s xml:id="echoid-s3359" xml:space="preserve">Sint autem hæc puncta ea <lb/>quæ apparent in ſpatio A B C D.</s> <s xml:id="echoid-s3360" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3361" xml:space="preserve">Quum igitur E ſit centrum gravitatis magnitudinis pro-<lb/>poſitæ, ipſaque proinde circa axem, qui per E punctum <lb/>erectus eſt ad planum A B C D, quovis ſitu æquilibrium <lb/>ſervet; </s> <s xml:id="echoid-s3362" xml:space="preserve">facile perſpicitur, quod ſi punctis omnibus ante di-<lb/>ctis, quæ in ſpatio A B C D ſignantur, æqualis gravitas <lb/>tribuatur, eorum quoque omnium centrum gravitatis futu-<lb/>rum eſt punctum E. </s> <s xml:id="echoid-s3363" xml:space="preserve">Quod ſi vero, ut fieri poteſt, in pun-<lb/>cta aliqua plures perpendiculares coincidant, illa puncta <lb/>quaſi toties geminata intelligenda ſunt, gravitatesque toties <lb/>multiplices accipiendæ. </s> <s xml:id="echoid-s3364" xml:space="preserve">Atque ita conſideratorum, patet <lb/>rurſus centrum gravitatis eſſe E punctum.</s> <s xml:id="echoid-s3365" xml:space="preserve"/> </p> <pb o="149" file="0213" n="233" rhead="HOROLOG. OSCILLATOR."/> <p> <s xml:id="echoid-s3366" xml:space="preserve">Porrò ſummam quadratorum ab rectis, quæ ducuntur à <lb/> <anchor type="note" xlink:label="note-0213-01a" xlink:href="note-0213-01"/> dictis punctis omnibus ad punctum F, eandem eſſe patet cum <lb/>ſumma quadratorum ab iis rectis, quæ à ſingulis particulis <lb/>magnitudinis propoſitæ ducuntur perpendiculares in axem <lb/>oſcillationis per F transeuntem; </s> <s xml:id="echoid-s3367" xml:space="preserve">quippe cum lineæ ipſæ, <lb/>quarum quadrata intelliguntur, utrobique eandem habeant <lb/>longitudinem. </s> <s xml:id="echoid-s3368" xml:space="preserve">Similiter etiam, cum ſuſpenſio eſt ex axe per <lb/>H, patet ſummam quadratorum ab rectis, quæ ab omnibus <lb/>punctis, in ſpatio A B C D ſignatis, ducuntur ad punctum <lb/>H, eandem eſſe cum ſumma quadratorum, ab iis quæ, à <lb/>particulis omnibus magnitudinis propoſitæ, ducuntur per-<lb/>pendiculares in axem oſcillationis per H transeuntem. </s> <s xml:id="echoid-s3369" xml:space="preserve">Ergo <lb/>utroque caſu, ſi ſumma quadratorum ab rectis quæ, à pun-<lb/>ctis omnibus prædictis, ducuntur ad puncta F vel H; </s> <s xml:id="echoid-s3370" xml:space="preserve">di-<lb/>vidatur per rectas E F vel E H, multiplices ſecundum nu-<lb/>merum particularum in quas magnitudo propoſita diviſa in-<lb/>telligitur, orietur ex applicatione hac longitudo penduli ſim-<lb/>plicis, quod magnitudini ſuſpenſæ ex F vel H iſochronum <lb/>fit. </s> <s xml:id="echoid-s3371" xml:space="preserve">Eſt autem ſumma quadratorum utroque caſu æqualis <anchor type="note" xlink:href="" symbol="*"/>;</s> <s xml:id="echoid-s3372" xml:space="preserve"> <anchor type="note" xlink:label="note-0213-02a" xlink:href="note-0213-02"/> & </s> <s xml:id="echoid-s3373" xml:space="preserve">rectæ quoque E F, E H, inter ſe æquales; </s> <s xml:id="echoid-s3374" xml:space="preserve">& </s> <s xml:id="echoid-s3375" xml:space="preserve">particu-<lb/>larum idem numerus. </s> <s xml:id="echoid-s3376" xml:space="preserve">Ergo, quum & </s> <s xml:id="echoid-s3377" xml:space="preserve">applicatæ quantitates, <lb/>& </s> <s xml:id="echoid-s3378" xml:space="preserve">quibus illæ applicantur, utrobique æquales ſint, etiam <lb/>longitudines ex applicatione ortæ æquales erunt, hoc eſt, <lb/>longitudines pendulorum iſochronorum magnitudini propoſi-<lb/>tæ ſuſpenſæ ex F vel ex H. </s> <s xml:id="echoid-s3379" xml:space="preserve">Quare conſtat propoſitum.</s> <s xml:id="echoid-s3380" xml:space="preserve"/> </p> <div xml:id="echoid-div292" type="float" level="2" n="2"> <note position="right" xlink:label="note-0213-01" xlink:href="note-0213-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="right" xlink:label="note-0213-02" xlink:href="note-0213-02a" xml:space="preserve">Prop. 13. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div294" type="section" level="1" n="107"> <head xml:id="echoid-head133" xml:space="preserve">PROPOSITIO XVII.</head> <p style="it"> <s xml:id="echoid-s3381" xml:space="preserve">DAto plano, cujus multiplex per numerum par-<lb/>ticularum, in quas ſuſpenſa figura diviſa in-<lb/>telligitur, æquetur quadratis omnium diſtantia-<lb/>rum ab axe oſcillationis; </s> <s xml:id="echoid-s3382" xml:space="preserve">ſi illud applicetur ad re-<lb/>ctam, æqualem diſtantiæ inter axem oſcillationis <lb/>& </s> <s xml:id="echoid-s3383" xml:space="preserve">centrum gravitatis ſuſpenſæ magnitudinis, orie-<lb/>tur longitudo penduli ſimplicis ipſi iſochroni.</s> <s xml:id="echoid-s3384" xml:space="preserve"/> </p> <pb o="150" file="0214" n="234" rhead="CHRISTIANI HUGENII"/> <p> <s xml:id="echoid-s3385" xml:space="preserve">Sit figura A B C, cujus centrum gravitatis E, ſuſpenſa <lb/> <anchor type="note" xlink:label="note-0214-01a" xlink:href="note-0214-01"/> ab axe qui, per F punctum ad planum quod conſpicitur, <lb/>erectus ſit. </s> <s xml:id="echoid-s3386" xml:space="preserve">Ponendoque diviſam figuram in particulas mini-<lb/> <anchor type="note" xlink:label="note-0214-02a" xlink:href="note-0214-02"/> mas æquales, à quibus omnibus, in dictum axem, perpen-<lb/>diculares cadere intelligantur: </s> <s xml:id="echoid-s3387" xml:space="preserve">eſto, per ſuperius oſtenſa, <lb/>inventum planum H, cujus multiplex per numerum dicta-<lb/>rum particularum, æquetur quadratis omnibus dictarum <lb/>perpendicularium. </s> <s xml:id="echoid-s3388" xml:space="preserve">Applicatoque plano H ad rectam F E, <lb/>fiat longitudo F G. </s> <s xml:id="echoid-s3389" xml:space="preserve">Dico hanc eſſe longitudinem penduli <lb/>ſimplicis, iſochronas oſcillationes habentis magnitudini <lb/>A B C, agitatæ circa axem per F.</s> <s xml:id="echoid-s3390" xml:space="preserve"/> </p> <div xml:id="echoid-div294" type="float" level="2" n="1"> <note position="left" xlink:label="note-0214-01" xlink:href="note-0214-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note position="left" xlink:label="note-0214-02" xlink:href="note-0214-02a" xml:space="preserve">TAB. XXII. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s3391" xml:space="preserve">Quia enim ſumma quadratorum, à diſtantiis ab axe F, <lb/>applicata ad diſtantiam F E, multiplicem ſecundum par-<lb/>tium numerum, facit longitudinem penduli ſimplicis iſo-<lb/>chroni <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3392" xml:space="preserve">Iſti vero quadratorum ſummæ æquale ponitur pla- <anchor type="note" xlink:label="note-0214-03a" xlink:href="note-0214-03"/> num H, multiplex per eundem particularum numerum. </s> <s xml:id="echoid-s3393" xml:space="preserve">Er-<lb/>go & </s> <s xml:id="echoid-s3394" xml:space="preserve">planum H, multiplex per eundem particularum nu-<lb/>merum, ſi applicetur ad diſtantiam F E, multiplicem ſe-<lb/>cundum particularum numerum; </s> <s xml:id="echoid-s3395" xml:space="preserve">ſive, omiſſa communi mul-<lb/>tiplicitate, ſi planum H applicetur ad diſtantiam F E; </s> <s xml:id="echoid-s3396" xml:space="preserve">o-<lb/>rietur quoque longitudo penduli ſimplicis iſochroni. </s> <s xml:id="echoid-s3397" xml:space="preserve">Quam <lb/>proinde ipſam longitudinem F G eſſe conſtat. </s> <s xml:id="echoid-s3398" xml:space="preserve">quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s3399" xml:space="preserve"/> </p> <div xml:id="echoid-div295" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0214-03" xlink:href="note-0214-03a" xml:space="preserve">Prop. 6. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div297" type="section" level="1" n="108"> <head xml:id="echoid-head134" xml:space="preserve">PROPOSITIO XVIII.</head> <p style="it"> <s xml:id="echoid-s3400" xml:space="preserve">SI ſpatium planum, cujus multiplex ſecundum <lb/>numerum particularum ſuſpenſæ magnitudinis, <lb/>æquetur quadratis diſtantiarum ab axe gravitatis, <lb/>axi oſcillationis parallelo; </s> <s xml:id="echoid-s3401" xml:space="preserve">id, inquam, ſpatium <lb/>ſi applicetur ad rectam, æqualem diſtantiæ inter <lb/>utrumque dictorum axium, orietur recta æqualis <lb/>intervallo, quo centrum oſcillationis inferius eſt <lb/>centro gravitatis ejusdem magnitudinis.</s> <s xml:id="echoid-s3402" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3403" xml:space="preserve">Eſto magnitudo A B C D, cujus centrum gravitatis E; <lb/></s> <s xml:id="echoid-s3404" xml:space="preserve"> <anchor type="note" xlink:label="note-0214-04a" xlink:href="note-0214-04"/> <pb o="151" file="0215" n="235" rhead="HOROLOG. OSCILLATOR."/> quæque ſuſpenſa ab axe, qui per punctum F ad planum hu-<lb/> <anchor type="note" xlink:label="note-0215-01a" xlink:href="note-0215-01"/> jus paginæ erectus intelligitur, habeat centrum oſcillationis <lb/>G. </s> <s xml:id="echoid-s3405" xml:space="preserve">Porrò axi per F intelligatur axis alius, per centrum gra-<lb/>vitatis E transiens, parallelus. </s> <s xml:id="echoid-s3406" xml:space="preserve">Diviſaque magnitudine cogita-<lb/>tu in particulas minimas æquales, ſit quadratis diſtantiarum, <lb/>ab axe dicto per E, æquale planum I, multiplex nempe ſe-<lb/>cundum numerum dictarum particularum; </s> <s xml:id="echoid-s3407" xml:space="preserve">applicatoque pla-<lb/>no I ad diſtantiam F E, fiat recta quædam. </s> <s xml:id="echoid-s3408" xml:space="preserve">Dico eam æ-<lb/>qualem eſſe intervallo E G, quo centrum oſcillationis infe-<lb/>rius eſt centro gravitatis magnitudinis A B C D.</s> <s xml:id="echoid-s3409" xml:space="preserve"/> </p> <div xml:id="echoid-div297" type="float" level="2" n="1"> <note position="left" xlink:label="note-0214-04" xlink:href="note-0214-04a" xml:space="preserve">TAB. XXII. <lb/>Fig. 2.</note> <note position="right" xlink:label="note-0215-01" xlink:href="note-0215-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3410" xml:space="preserve">Ut enim univerſali demonſtratione quod propoſitum eſt <lb/>comprehendamus: </s> <s xml:id="echoid-s3411" xml:space="preserve">intelligatur plana figura, magnitudini <lb/>A B C D analoga, ad latus adpoſita, O Q P; </s> <s xml:id="echoid-s3412" xml:space="preserve">quæ nempe, <lb/>ſecta planis horizontalibus iisdem cum magnitudine A B C D, <lb/>habeat ſegmenta intercepta inter bina quæque plana, in ea-<lb/>dem inter ſe ratione cum ſegmentis dictæ magnitudinis, quæ <lb/>ipſis reſpondent; </s> <s xml:id="echoid-s3413" xml:space="preserve">ſintque ſegmenta ſingula figuræ O Q P, <lb/>diviſa in tot particulas æquales, quot continentur ſegmentis <lb/>ipſis reſpondentibus in figura A B C D. </s> <s xml:id="echoid-s3414" xml:space="preserve">Hæc autem intel-<lb/>ligi poſſunt fieri, qualiscunque fuerit magnitudo A B C D, <lb/>ſive linea, ſive ſuperficies, ſive ſolidum. </s> <s xml:id="echoid-s3415" xml:space="preserve">Semper vero cen-<lb/>trum gravitatis figuræ O Q P, quod ſit T, eadem altitu-<lb/>dine eſſe manifeſtum eſt cum centro gravitatis magnitudinis <lb/>A B C D; </s> <s xml:id="echoid-s3416" xml:space="preserve">ideoque, ſi planum horizontale, per F ductum, <lb/>ſecet lineam centri figuræ O Q P, velut hic in S, æquales <lb/>eſſe diſtantias S T, F E.</s> <s xml:id="echoid-s3417" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3418" xml:space="preserve">Porrò autem conſtat quadrata diſtantiarum, ab axe oſcil-<lb/>lationis F, applicata ad diſtantiam F E, multiplicem ſecun-<lb/>dum numerum particularum, efficere longitudinem penduli <lb/>iſochroni <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s3419" xml:space="preserve">quæ longitudo poſita fuit F G. </s> <s xml:id="echoid-s3420" xml:space="preserve">Illorum vero <anchor type="note" xlink:label="note-0215-02a" xlink:href="note-0215-02"/> quadratorum ſummam, æqualem eſſe perſpicuum eſt, qua-<lb/>dratis diſtantiarum à plano horizontali per F, unà cum qua-<lb/>dratis diſtantiarum à plano verticali F E, per axem F & </s> <s xml:id="echoid-s3421" xml:space="preserve">cen-<lb/>trum gravitatis E ducto <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3422" xml:space="preserve">Atqui quadrata diſtantiarum ma- <anchor type="note" xlink:label="note-0215-03a" xlink:href="note-0215-03"/> gnitudinis A B C D à plano horizontali per F, æquantur <lb/>quadratis diſtantiarum figuræ O Q P ab recta S F. </s> <s xml:id="echoid-s3423" xml:space="preserve">Quæ <pb o="152" file="0216" n="236" rhead="CHRISTIANI HUGENII"/> quadrata (ſi O ſit punctum ſupremum figuræ O Q P, & </s> <s xml:id="echoid-s3424" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0216-01a" xlink:href="note-0216-01"/> O H ſubcentrica cunei ſuper ipſa abſciſſi, plano per rectam <lb/>O V, parallelam S F) æqualia ſunt rectangulo O T H & </s> <s xml:id="echoid-s3425" xml:space="preserve"><lb/>quadrato S T, multiplicibus ſecundum numerum particula-<lb/>rum dictæ figuræ<anchor type="note" xlink:href="" symbol="*"/>, ſive magnitudinis A B C D. </s> <s xml:id="echoid-s3426" xml:space="preserve">Quadrata <anchor type="note" xlink:label="note-0216-02a" xlink:href="note-0216-02"/> vero diſtantiarum magnitudinis A B C D à plano F E, <lb/>quantumcunque axis oſcillationis F diſtet à centro gravita-<lb/>tis E, ſemper eadem ſunt: </s> <s xml:id="echoid-s3427" xml:space="preserve">quæ proinde putemus æquari <lb/>ſpatio Z, multiplici ſecundum numerum particularum ma-<lb/>gnitudinis A B C D.</s> <s xml:id="echoid-s3428" xml:space="preserve"/> </p> <div xml:id="echoid-div298" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0215-02" xlink:href="note-0215-02a" xml:space="preserve">Prop. 6. <lb/>huj.</note> <note symbol="*" position="right" xlink:label="note-0215-03" xlink:href="note-0215-03a" xml:space="preserve">Prop. 47. <lb/>lib. 1. Eucl.</note> <note position="left" xlink:label="note-0216-01" xlink:href="note-0216-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="left" xlink:label="note-0216-02" xlink:href="note-0216-02a" xml:space="preserve">Prop. 9. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s3429" xml:space="preserve">Itaque quoniam quadrata diſtantiarum magnitudinis <lb/>A B C D, ab axe oſcillationis F, æquantur iſtis, quadrato <lb/>nimirum S T, rectangulo O T H, & </s> <s xml:id="echoid-s3430" xml:space="preserve">plano Z, multipli-<lb/>cibus per numerum particularum ejusdem magnitudinis; </s> <s xml:id="echoid-s3431" xml:space="preserve">ſi <lb/>applicentur hæc omnia ad diſtantiam F E ſive S T, orietur <lb/>longitudo F G penduli iſochroni magnitudini A B C D <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s3432" xml:space="preserve"> <anchor type="note" xlink:label="note-0216-03a" xlink:href="note-0216-03"/> Sed ex applicatione quadrati S T ad latus ſuum S T, orie-<lb/>tur ipſa S T, ſive F E. </s> <s xml:id="echoid-s3433" xml:space="preserve">Ergo reliqua E G eſt ea quæ ori-<lb/>tur ex applicatione rectanguli O T H, & </s> <s xml:id="echoid-s3434" xml:space="preserve">plani Z, ad ean-<lb/>dem S T vel F E.</s> <s xml:id="echoid-s3435" xml:space="preserve"/> </p> <div xml:id="echoid-div299" type="float" level="2" n="3"> <note symbol="*" position="left" xlink:label="note-0216-03" xlink:href="note-0216-03a" xml:space="preserve">Prop. 6. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s3436" xml:space="preserve">Quare ſupereſt ut demonſtremus rectangulum O T H, <lb/>cum plano Z, æquari plano I. </s> <s xml:id="echoid-s3437" xml:space="preserve">Tunc enim conſtabit, etiam <lb/>planum I, applicatum ad diſtantiam F E, efficere longitu-<lb/>dinem ipſi E G æqualem. </s> <s xml:id="echoid-s3438" xml:space="preserve">Illud autem ſic oſtendetur. </s> <s xml:id="echoid-s3439" xml:space="preserve">Re-<lb/>ctangulum O T H, multiplex ſecundum numerum particu-<lb/>larum figuræ O Q P, ſive magnitudinis A B C D, æ-<lb/> <anchor type="note" xlink:label="note-0216-04a" xlink:href="note-0216-04"/> quatur quadratis diſtantiarum figuræ ab recta X T <anchor type="note" xlink:href="" symbol="*"/>, quæ per centrum gravitatis T ducitur ipſi S F parallela; </s> <s xml:id="echoid-s3440" xml:space="preserve">ac pro-<lb/>inde etiam quadratis diſtantiarum magnitudinis A B C D, <lb/>à plano horizontali K K, ducto per centrum gravitatis E; <lb/></s> <s xml:id="echoid-s3441" xml:space="preserve">cum diſtantiæ utrobique ſint eædem. </s> <s xml:id="echoid-s3442" xml:space="preserve">At vero planum Z, ſi-<lb/>militer multiplex, æquale poſitum fuit quadratis diſtantia-<lb/>rum magnitudinis A B C D à plano verticali F E. </s> <s xml:id="echoid-s3443" xml:space="preserve">Ac pa-<lb/>tet quidem quadrata hæc diſtantiarum à plano F E, una cum <lb/>dictis quadratis diſtantiarum à plano horizontali per E, æ-<lb/>qualia eſſe quadratis diſtantiarum ab axe gravitatis per E, <pb o="153" file="0217" n="237" rhead="HOROLOG. OSCILLATOR."/> qui ſit axi F parallelus <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3444" xml:space="preserve">Itaque rectangulum O T H una <anchor type="note" xlink:label="note-0217-01a" xlink:href="note-0217-01"/> cum plano Z, multiplicia ſecundum numerum particularum <lb/> <anchor type="note" xlink:label="note-0217-02a" xlink:href="note-0217-02"/> magnitudinis A B C D, æqualia erunt quadratis diſtantia-<lb/>rum ejusdem magnitudinis à dicto axe per E. </s> <s xml:id="echoid-s3445" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3446" xml:space="preserve">planum <lb/>I, multiplex ſecundum eundem particularum numerum, æ-<lb/>quale poſitum fuit iisdem diſtantiarum quadratis. </s> <s xml:id="echoid-s3447" xml:space="preserve">Ergo pla-<lb/>num I æquale eſt rectangulo O T H & </s> <s xml:id="echoid-s3448" xml:space="preserve">plano Z ſimul ſum-<lb/>ptis. </s> <s xml:id="echoid-s3449" xml:space="preserve">quod oſtendendum ſupererat.</s> <s xml:id="echoid-s3450" xml:space="preserve"/> </p> <div xml:id="echoid-div300" type="float" level="2" n="4"> <note symbol="*" position="left" xlink:label="note-0216-04" xlink:href="note-0216-04a" xml:space="preserve">Prop. 10. <lb/>huj.</note> <note position="right" xlink:label="note-0217-01" xlink:href="note-0217-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="right" xlink:label="note-0217-02" xlink:href="note-0217-02a" xml:space="preserve">Prop. 47. <lb/>lib. 1. Eucl.</note> </div> <p> <s xml:id="echoid-s3451" xml:space="preserve">Hinc rurſus manifeſtum fit, quod propoſitione 16 demon-<lb/>ſtratum fuit; </s> <s xml:id="echoid-s3452" xml:space="preserve">nempe magnitudinem quamlibet, ſi aliter at-<lb/>que aliter ſuſpendatur atque agitetur, ab axibus parallelis, <lb/>qui à centro gravitatis ſuæ æqualiter diſtent, ſibi ipſi iſo-<lb/>chronam eſſe.</s> <s xml:id="echoid-s3453" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3454" xml:space="preserve">Sive enim magnitudo A B C D ſuſpendatur ab axe F, ſi-<lb/>ve ab axe L illi parallelo; </s> <s xml:id="echoid-s3455" xml:space="preserve">patet eadem utrobique eſſe qua-<lb/>drata diſtantiarum ab axe per E, qui ſit axibus F vel L pa-<lb/>rallelus. </s> <s xml:id="echoid-s3456" xml:space="preserve">Unde & </s> <s xml:id="echoid-s3457" xml:space="preserve">planum I, cujus multiplex, ſecundum <lb/>numerum particularum, æquatur quadratorum ſummæ, u-<lb/>troque caſu idem erit. </s> <s xml:id="echoid-s3458" xml:space="preserve">Hoc vero planum, applicatum ad di-<lb/>ſtantiam centri gravitatis ab axe oſcillationis, quæ utroque <lb/>caſu eadem ponitur, efficit diſtantiam qua centrum oſcilla-<lb/>tionis inferius eſt centro gravitatis; </s> <s xml:id="echoid-s3459" xml:space="preserve">Ergo etiam hæc diſtan-<lb/>tia utroque caſu eadem erit. </s> <s xml:id="echoid-s3460" xml:space="preserve">Velut ſi, facta ſuſpenſione ex <lb/>L, fuerit dicta diſtantia E Y, erit ipſa æqualis E G; </s> <s xml:id="echoid-s3461" xml:space="preserve">& </s> <s xml:id="echoid-s3462" xml:space="preserve">to-<lb/>ta Y L æqualis G F; </s> <s xml:id="echoid-s3463" xml:space="preserve">adeoque, in ſuſpenſione utraque, <lb/>idem pendulum ſimplex iſochronum fit magnitudini A B C D.</s> <s xml:id="echoid-s3464" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div302" type="section" level="1" n="109"> <head xml:id="echoid-head135" xml:space="preserve">PROPOSITIO XIX.</head> <p style="it"> <s xml:id="echoid-s3465" xml:space="preserve">SI magnitudo eadem, nunc brevius nunc longius <lb/>ſuſpenſa, agitetur; </s> <s xml:id="echoid-s3466" xml:space="preserve">erunt, ſicut diſtantiæ axi-<lb/>um oſcillationis à centro gravitatis inter ſe, ita <lb/>contraria ratione diſtantiæ centrorum oſcillationis <lb/>ab eodem gravitatis centro.</s> <s xml:id="echoid-s3467" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3468" xml:space="preserve">Sit magnitudo, cujus centrum gravitatis A, ſuſpenſa pri-<lb/> <anchor type="note" xlink:label="note-0217-03a" xlink:href="note-0217-03"/> <pb o="154" file="0218" n="238" rhead="CHRISTIANI HUGENII"/> mum atque agitata ab axe in B, deinde vero ab axe in C; <lb/></s> <s xml:id="echoid-s3469" xml:space="preserve"> <anchor type="note" xlink:label="note-0218-01a" xlink:href="note-0218-01"/> ſitque in prima ſuſpenſione centrum oſcillationis D, in po-<lb/>ſteriori vero centrum oſcillationis E. </s> <s xml:id="echoid-s3470" xml:space="preserve">Dico eſſe ut B A ad <lb/>C A ita E A ad D A.</s> <s xml:id="echoid-s3471" xml:space="preserve"/> </p> <div xml:id="echoid-div302" type="float" level="2" n="1"> <note position="right" xlink:label="note-0217-03" xlink:href="note-0217-03a" xml:space="preserve">TAB. XXII. <lb/>Fig. 3.</note> <note position="left" xlink:label="note-0218-01" xlink:href="note-0218-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3472" xml:space="preserve">Quum enim, in ſuſpenſione ex B, efficiatur diſtantia A D, <lb/>qua nempe centrum oſcillationis inferius eſt centro gravita-<lb/>tis, applicando ad diſtantiam B A ſpatium quoddam, cujus <lb/>multiplex ſecundum numerum particularum minimarum æ-<lb/>qualium, in quas magnitudo diviſa intelligitur, æquatur <lb/>quadratis diſtantiarum ab axe per A, parallelo axi in B <anchor type="note" xlink:href="" symbol="*"/>;</s> <s xml:id="echoid-s3473" xml:space="preserve"> <anchor type="note" xlink:label="note-0218-02a" xlink:href="note-0218-02"/> erit proinde rectangulum B A D dicto ſpatio æquale. </s> <s xml:id="echoid-s3474" xml:space="preserve">Item, <lb/>in ſuſpenſione ex C, quum fiat diſtantia A E, applicando <lb/>idem dictum ſpatium ad diſtantiam C A; </s> <s xml:id="echoid-s3475" xml:space="preserve">erit & </s> <s xml:id="echoid-s3476" xml:space="preserve">rectangu-<lb/>lum C A E eidem ſpatio æquale. </s> <s xml:id="echoid-s3477" xml:space="preserve">Itaque æqualia inter ſe re-<lb/>ctangula B A D, C A E; </s> <s xml:id="echoid-s3478" xml:space="preserve">ac proinde ratio B A ad C A <lb/>eadem quæ A E ad A D. </s> <s xml:id="echoid-s3479" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s3480" xml:space="preserve"/> </p> <div xml:id="echoid-div303" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0218-02" xlink:href="note-0218-02a" xml:space="preserve">Prop. <lb/>præced.</note> </div> <p> <s xml:id="echoid-s3481" xml:space="preserve">Hinc patet, dato pendulo ſimplici, quod magnitudini <lb/>ſuſpenſæ iſochronum ſit in una ſuſpenſione, datoque ejus <lb/>centro gravitatis; </s> <s xml:id="echoid-s3482" xml:space="preserve">etiam in alia omni ſuſpenſione, longiori <lb/>vel breviori, dummodo idem maneat planum oſcillationis, <lb/>longitudinem penduli iſochroni datam eſſe.</s> <s xml:id="echoid-s3483" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div305" type="section" level="1" n="110"> <head xml:id="echoid-head136" xml:space="preserve">PROPOSITIO XX.</head> <p style="it"> <s xml:id="echoid-s3484" xml:space="preserve">CEntrum Oſcillationis & </s> <s xml:id="echoid-s3485" xml:space="preserve">punctum ſuſpenſionis <lb/>inter ſe convertuntur.</s> <s xml:id="echoid-s3486" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3487" xml:space="preserve">In figura ſuperiori, quia, poſita ſuſpenſione ex B, cen-<lb/> <anchor type="note" xlink:label="note-0218-03a" xlink:href="note-0218-03"/> trum oſcillationis eſt D; </s> <s xml:id="echoid-s3488" xml:space="preserve">etiam invertendo omnia, ponendo-<lb/>que ſuſpenſionem ex D, erit tunc centrum oſcillationis B. <lb/></s> <s xml:id="echoid-s3489" xml:space="preserve">Hoc enim ex ipſa propoſitione præcedenti manifeſtum eſt.</s> <s xml:id="echoid-s3490" xml:space="preserve"/> </p> <div xml:id="echoid-div305" type="float" level="2" n="1"> <note position="left" xlink:label="note-0218-03" xlink:href="note-0218-03a" xml:space="preserve">TAB. XXII. <lb/>Fig. 3.</note> </div> </div> <div xml:id="echoid-div307" type="section" level="1" n="111"> <head xml:id="echoid-head137" xml:space="preserve">PROPOSITIO XXI.</head> <p style="it"> <s xml:id="echoid-s3491" xml:space="preserve">QUomodo in figuris planis centra oſcillationis in-<lb/>veniantur.</s> <s xml:id="echoid-s3492" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3493" xml:space="preserve">Intellectis quæ hactenus demonſtrata ſunt, facile jam erit <pb file="0219" n="239"/> <pb file="0219a" n="240"/> <anchor type="figure" xlink:label="fig-0219a-01a" xlink:href="fig-0219a-01"/> <anchor type="figure" xlink:label="fig-0219a-02a" xlink:href="fig-0219a-02"/> <anchor type="figure" xlink:label="fig-0219a-03a" xlink:href="fig-0219a-03"/> <pb file="0220" n="241"/> <pb o="155" file="0221" n="242" rhead="HOROLOG. OSCILLATOR."/> in plerisque figuris, quæ in Geometria conſiderari conſueve-<lb/> <anchor type="note" xlink:label="note-0221-01a" xlink:href="note-0221-01"/> runt, definire oſcillationis centra. </s> <s xml:id="echoid-s3494" xml:space="preserve">Atque ut de planis figu-<lb/>ris primum dicamus; </s> <s xml:id="echoid-s3495" xml:space="preserve">duplicem in iis oſcillationis motum <lb/>ſupra definivimus; </s> <s xml:id="echoid-s3496" xml:space="preserve">nempe, vel circa axem in eodem cum <lb/>figura plano jacentem, vel circa eum qui ad figuræ planum <lb/>erectus ſit. </s> <s xml:id="echoid-s3497" xml:space="preserve">Quorum priorem vocavimus agitationem in pla-<lb/>num, alterum agitationem in latus.</s> <s xml:id="echoid-s3498" xml:space="preserve"/> </p> <div xml:id="echoid-div307" type="float" level="2" n="1"> <figure xlink:label="fig-0219a-01" xlink:href="fig-0219a-01a"> <caption xml:id="echoid-caption80" style="it" xml:space="preserve">Pag. 154.<lb/>TAB. XXI.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables80" xml:space="preserve">G E G O A K L Q Q M M H F R R N N B D L K C P S V X Z Y X V T</variables> </figure> <figure xlink:label="fig-0219a-02" xlink:href="fig-0219a-02a"> <caption xml:id="echoid-caption81" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables81" xml:space="preserve">F A D E B C G H</variables> </figure> <figure xlink:label="fig-0219a-03" xlink:href="fig-0219a-03a"> <caption xml:id="echoid-caption82" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables82" xml:space="preserve">G E Ω O Ω S A S Q Q M M R R N X F N V P Φ Δ V B C K D Z</variables> </figure> <note position="right" xlink:label="note-0221-01" xlink:href="note-0221-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3499" xml:space="preserve">Quod ſi priore modo agitetur, nempe circa axem in eo-<lb/> <anchor type="note" xlink:label="note-0221-02a" xlink:href="note-0221-02"/> dem plano jacentem, ſicut figura B C D circa axem E F; <lb/></s> <s xml:id="echoid-s3500" xml:space="preserve">hic, ſi cuneus ſuper figura intelligatur abſciſſus, plano quod <lb/>ita ſecet planum figuræ, ut interſectio, quæ hic eſt D D, <lb/>ſit parallela oſcillationis axi; </s> <s xml:id="echoid-s3501" xml:space="preserve">deturque diſtantia centri gra-<lb/>vitatis figuræ ab hac interſectione, ut hic A D; </s> <s xml:id="echoid-s3502" xml:space="preserve">itemque <lb/>ſubcentrica cunei dicti ſuper eadem interſectione, quæ hic <lb/>ſit D H. </s> <s xml:id="echoid-s3503" xml:space="preserve">Habebitur centrum oſcillationis K, figuræ B D C, <lb/>applicando rectangulum D A H ad diſtantiam F A; </s> <s xml:id="echoid-s3504" xml:space="preserve">quo-<lb/>niam ex applicatione hac orietur diſtantia A K, qua cen-<lb/>trum oſcillationis inferius eſt centro gravitatis. </s> <s xml:id="echoid-s3505" xml:space="preserve">Eſt enim re-<lb/>ctangulum D A H, multiplex ſecundum numerum particu-<lb/>larum figuræ B C D, æquale quadratis diſtantiarum ab re-<lb/>cta B A C, quæ per centrum gravitatis A parallela ducitur <lb/>axi oſcillationis E F<anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3506" xml:space="preserve">Quare, applicando idem rectangu- <anchor type="note" xlink:label="note-0221-03a" xlink:href="note-0221-03"/> lum ad diſtantiam F A, orietur diſtantia A K, qua centrum <lb/>oſcillationis inferius eſt centro gravitatis A <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s3507" xml:space="preserve"/> </p> <div xml:id="echoid-div308" type="float" level="2" n="2"> <note position="right" xlink:label="note-0221-02" xlink:href="note-0221-02a" xml:space="preserve">TAB. XXII. <lb/>Fig. 4. & 5.</note> <note symbol="*" position="right" xlink:label="note-0221-03" xlink:href="note-0221-03a" xml:space="preserve">Prop. 10. <lb/>huj.</note> </div> <note symbol="*" position="right" xml:space="preserve">Prop. 18. <lb/>huj.</note> <p> <s xml:id="echoid-s3508" xml:space="preserve">Hinc manifeſtum eſt, ſi axis oſcillationis ſit D D, fieri <lb/>centrum oſcillationis H punctum; </s> <s xml:id="echoid-s3509" xml:space="preserve">adeoque longitudinem <lb/>D H, penduli ſimplicis iſochroni figuræ B C D, eſſe tunc <lb/>ipſam ſubcentricam cunei, abſciſſi plano per D D, ſuper <lb/>ipſam D D. </s> <s xml:id="echoid-s3510" xml:space="preserve">Quod unum ab aliis ante animad verſum fuit, <lb/>non tamen demonſtratum.</s> <s xml:id="echoid-s3511" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3512" xml:space="preserve">Quomodo autem centra gravitatis cuneorum ſuper figuris <lb/>planis inveniantur, perſequi non eſt inſtituti noſtri, & </s> <s xml:id="echoid-s3513" xml:space="preserve">jam <lb/>in multis nota ſunt. </s> <s xml:id="echoid-s3514" xml:space="preserve">Velut, quod ſi figura B C D ſit circu-<lb/>lus, erit D H æqualis {5/8} diametri. </s> <s xml:id="echoid-s3515" xml:space="preserve">Si rectangulum, erit D H <lb/>. </s> <s xml:id="echoid-s3516" xml:space="preserve">= {2/3} diametri. </s> <s xml:id="echoid-s3517" xml:space="preserve">Unde & </s> <s xml:id="echoid-s3518" xml:space="preserve">ratio apparet cur virga, ſeu linea <lb/>gravitate prædita, altero capite ſuſpenſa, iſochrona ſit pen- <pb o="156" file="0222" n="243" rhead="CHRISTIANI HUGENII"/> dulo longitudinis ſubſesquialteræ. </s> <s xml:id="echoid-s3519" xml:space="preserve">Conſiderando nempe li-<lb/> <anchor type="note" xlink:label="note-0222-01a" xlink:href="note-0222-01"/> neam ejusmodi, ac ſi eſſet rectangulum minimæ latitudinis.</s> <s xml:id="echoid-s3520" xml:space="preserve"/> </p> <div xml:id="echoid-div309" type="float" level="2" n="3"> <note position="left" xlink:label="note-0222-01" xlink:href="note-0222-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3521" xml:space="preserve">Quod ſi figura triangulum fuerit, vertice ſurſum conver-<lb/>ſo, fit D H {3/4} diametri. </s> <s xml:id="echoid-s3522" xml:space="preserve">Si deorſum, {1/2} diametri.</s> <s xml:id="echoid-s3523" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3524" xml:space="preserve">Quod autem propoſitione 16 demonſtratum fuit, id ad hu-<lb/>jusmodi figuræ planæ motum ita pertinere ſciendum. </s> <s xml:id="echoid-s3525" xml:space="preserve">Nem-<lb/>pe, ſi aliam atque aliam poſitionem demus figuræ B C D, <lb/>invertendo eam circa axem B A C, ut vel horizonti paral-<lb/>lela jaceat, vel oblique inclinetur, manente eodem agitatio-<lb/>nis axe F E, etiam longitudo penduli iſochroni F K eadem <lb/>manebit. </s> <s xml:id="echoid-s3526" xml:space="preserve">Hoc enim ex propoſitione illa manifeſtum eſt.</s> <s xml:id="echoid-s3527" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3528" xml:space="preserve">Porro quando figura plana, circa axem ad planum figu-<lb/> <anchor type="note" xlink:label="note-0222-02a" xlink:href="note-0222-02"/> ræ erectum, agitatur; </s> <s xml:id="echoid-s3529" xml:space="preserve">quam vocavimus agitationem in latus; <lb/></s> <s xml:id="echoid-s3530" xml:space="preserve">velut ſi figura B C D moveatur circa axem, qui per pun-<lb/>ctum F intelligitur ad planum D B C erectus; </s> <s xml:id="echoid-s3531" xml:space="preserve">hic jam ha-<lb/>benda eſt ſumma quadratorum a diſtantiis particularum <lb/>omnium ab recta quæ per centrum gravitatis A intelligitur <lb/>axi oſcillationis parallela; </s> <s xml:id="echoid-s3532" xml:space="preserve">ſecundum ea quæ prop. </s> <s xml:id="echoid-s3533" xml:space="preserve">18. </s> <s xml:id="echoid-s3534" xml:space="preserve">ex-<lb/>poſita fuere. </s> <s xml:id="echoid-s3535" xml:space="preserve">Hoc eſt ſumma quadratorum a diſtantiis ab ipſo <lb/>A centro gravitatis, quoniam figura plana eſt. </s> <s xml:id="echoid-s3536" xml:space="preserve">Sive etiam <lb/>ſummæ quadratorum a diſtantiis tam ab recta B A C quam <lb/>ab recta D A. </s> <s xml:id="echoid-s3537" xml:space="preserve">Conſtat enim quadratum rectæ O A, quam <lb/>pono eſſe diſtantiam unius cujusdam particulæ a centro A, <lb/>æquari quadratis diſtantiarum O N, O V, quibus eadem <lb/>particula abeſt a rectis B A C, D A <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3538" xml:space="preserve">Atqui ſumma qua- <anchor type="note" xlink:label="note-0222-03a" xlink:href="note-0222-03"/> dratorum a diſtantiis ab recta B A C æquatur rectangulo <lb/>D A H, ſi D H ſit ſubcentrica cunei ſuper figura abſciſſi <lb/>per tangentem D D, parallelam B A <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3539" xml:space="preserve">item ſumma qua- <anchor type="note" xlink:label="note-0222-04a" xlink:href="note-0222-04"/> dratorum a diſtantiis ab recta D A æquatur rectangulo B A L, <lb/>ſi B L ſit ſubcentrica cunei abſciſſi per tangentem B D pa-<lb/>rallelam A D. </s> <s xml:id="echoid-s3540" xml:space="preserve">Oportetque dari, præter figuræ centrum gra-<lb/>vitatis A, ſubcentricamque H D cunei prioris, etiam ſub-<lb/>centricam L B cunei poſterioris. </s> <s xml:id="echoid-s3541" xml:space="preserve">Ita enim nota erunt rectan-<lb/>gula D A H, B A L, quæ ſimul ſumpta faciunt hic ſpa-<lb/>tium applicandum, quod deinceps etiam rectangulum oſcil-<lb/>lationis vocabitur. </s> <s xml:id="echoid-s3542" xml:space="preserve">Quod nempe, applicatum ad diſtantiam <pb file="0223" n="244"/> <pb file="0223a" n="245"/> <anchor type="figure" xlink:label="fig-0223a-01a" xlink:href="fig-0223a-01"/> <anchor type="figure" xlink:label="fig-0223a-02a" xlink:href="fig-0223a-02"/> <anchor type="figure" xlink:label="fig-0223a-03a" xlink:href="fig-0223a-03"/> <anchor type="figure" xlink:label="fig-0223a-04a" xlink:href="fig-0223a-04"/> <anchor type="figure" xlink:label="fig-0223a-05a" xlink:href="fig-0223a-05"/> <pb file="0224" n="246"/> <pb o="157" file="0225" n="247" rhead="HOROLOG. OSCILLATOR."/> F A, dabit diſtantiam A K, qua centrum oſcillationis K in-<lb/> <anchor type="note" xlink:label="note-0225-01a" xlink:href="note-0225-01"/> ferius eſt centro gravitatis A.</s> <s xml:id="echoid-s3543" xml:space="preserve"/> </p> <div xml:id="echoid-div310" type="float" level="2" n="4"> <note position="left" xlink:label="note-0222-02" xlink:href="note-0222-02a" xml:space="preserve">TAB. XXIII. <lb/>Fig. 1. & 2.</note> <note symbol="*" position="left" xlink:label="note-0222-03" xlink:href="note-0222-03a" xml:space="preserve">Per 47. <lb/>lib. 1. <lb/>Elem.</note> <note symbol="*" position="left" xlink:label="note-0222-04" xlink:href="note-0222-04a" xml:space="preserve">Prop. 10. <lb/>huj.</note> <figure xlink:label="fig-0223a-01" xlink:href="fig-0223a-01a"> <caption xml:id="echoid-caption83" style="it" xml:space="preserve">Pag. 156.<lb/>Fig. 2.</caption> <variables xml:id="echoid-variables83" xml:space="preserve">S F Z V O V L A Q Q M M I R R N N X T X K E K Y H G P B C D</variables> </figure> <figure xlink:label="fig-0223a-02" xlink:href="fig-0223a-02a"> <caption xml:id="echoid-caption84" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables84" xml:space="preserve">F H A E G B C</variables> </figure> <figure xlink:label="fig-0223a-03" xlink:href="fig-0223a-03a"> <caption xml:id="echoid-caption85" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables85" xml:space="preserve">C B A E D</variables> </figure> <figure xlink:label="fig-0223a-04" xlink:href="fig-0223a-04a"> <caption xml:id="echoid-caption86" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables86" xml:space="preserve">E F E D D D V O B A N C K H</variables> </figure> <figure xlink:label="fig-0223a-05" xlink:href="fig-0223a-05a"> <caption xml:id="echoid-caption87" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables87" xml:space="preserve">D D D E F E B A C H K</variables> </figure> <note position="right" xlink:label="note-0225-01" xlink:href="note-0225-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3544" xml:space="preserve">Si vero F A ſit axis figuræ B C D, poteſt, pro cuneo <lb/> <anchor type="note" xlink:label="note-0225-02a" xlink:href="note-0225-02"/> abſciſſo per B D ſuper figura tota, adhiberi cuneus ſuper <lb/>figura dimidia D B M abſciſſus plano per D M. </s> <s xml:id="echoid-s3545" xml:space="preserve">Nam, ſi cunei <lb/>hujus ſubcentrica ſuper D M ſit O A, diſtantia vero centri gr. <lb/></s> <s xml:id="echoid-s3546" xml:space="preserve">figuræ planæ D B M ab eadem D M ſit N A, æquale eſſe <lb/>conſtat rectangulum O A N rectangulo B A L <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3547" xml:space="preserve">Itaque <anchor type="note" xlink:label="note-0225-03a" xlink:href="note-0225-03"/> rectangulum O A N, additum rectangulo D A H, conſti-<lb/>tuet quoque planum applicandum ad diſtantiam F A, ut <lb/>fiat diſtantia A K.</s> <s xml:id="echoid-s3548" xml:space="preserve"/> </p> <div xml:id="echoid-div311" type="float" level="2" n="5"> <note position="right" xlink:label="note-0225-02" xlink:href="note-0225-02a" xml:space="preserve">TAB. XXIII. <lb/>Fig. 1.</note> <note symbol="*" position="right" xlink:label="note-0225-03" xlink:href="note-0225-03a" xml:space="preserve">Prop. 11. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s3549" xml:space="preserve">Et horum quidem manifeſta eſt demonſtratio ex præce-<lb/>dentibus, quippe cum rectangula D A H, B A L, vel <lb/>D A H, O A N, multiplicia ſecundum numerum particu-<lb/>larum figuræ, æqualia ſint quadratis diſtantiarum à centro <lb/>gravitatis A; </s> <s xml:id="echoid-s3550" xml:space="preserve">ſive, quod idem hic eſt, ab axe gravitatis axi <lb/>oſcillationis parallelo; </s> <s xml:id="echoid-s3551" xml:space="preserve">ac proinde rectangula dicta, ad diſtan-<lb/>tiam F A applicata, efficiant longitudinem intervalli A K <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s3552" xml:space="preserve"> <anchor type="note" xlink:label="note-0225-04a" xlink:href="note-0225-04"/> </s> </p> <div xml:id="echoid-div312" type="float" level="2" n="6"> <note symbol="*" position="right" xlink:label="note-0225-04" xlink:href="note-0225-04a" xml:space="preserve">Prop. 18. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div314" type="section" level="1" n="112"> <head xml:id="echoid-head138" style="it" xml:space="preserve">Centrum oſcillationis Circuli.</head> <p> <s xml:id="echoid-s3553" xml:space="preserve">Et in circulo quidem rectangula D A H, B A L, inter <lb/>ſe æqualia eſſe liquet, ſimulque efficere ſemiſſem quadrati à <lb/>ſemidiametro. </s> <s xml:id="echoid-s3554" xml:space="preserve">Unde, ſi fiat ut F A ad ſemidiametrum A B, <lb/>ita hæc ad aliam, ejus dimidium erit diſtantia A K, à cen-<lb/>tro gravitatis ad centrum oſcillationis. </s> <s xml:id="echoid-s3555" xml:space="preserve">Si igitur circulus ab <lb/>axe D, in circumferentia ſumpto, agitetur, erit D K æqua-<lb/>lis tribus quartis diametri D M.</s> <s xml:id="echoid-s3556" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3557" xml:space="preserve">Ad hunc modum & </s> <s xml:id="echoid-s3558" xml:space="preserve">in ſequentibus figuris planis centra o-<lb/>ſcillationis quæſivimus, quæ ſimpliciter adſcripſiſſe ſufficiet-<lb/>Nempe,</s> </p> </div> <div xml:id="echoid-div315" type="section" level="1" n="113"> <head xml:id="echoid-head139" style="it" xml:space="preserve">Centrum oſcillationis Rectanguli.</head> <p> <s xml:id="echoid-s3559" xml:space="preserve">In rectangulo omni, ut C B, ſpatium applicandum, ſive <lb/> <anchor type="note" xlink:label="note-0225-05a" xlink:href="note-0225-05"/> rectangulum oſcillationis, invenitur æquale tertiæ parti qua-<lb/>drati à ſemidiagonio A C. </s> <s xml:id="echoid-s3560" xml:space="preserve">Unde ſequitur, ſi rectangulum <pb o="158" file="0226" n="248" rhead="CHRISTIANI HUGENII"/> ab aliquo angulorum ſuſpendatur, motuque hoc laterali agi-<lb/> <anchor type="note" xlink:label="note-0226-01a" xlink:href="note-0226-01"/> tetur, pendulum illi iſochronum eſſe {2/3} diagonii totius.</s> <s xml:id="echoid-s3561" xml:space="preserve"/> </p> <div xml:id="echoid-div315" type="float" level="2" n="1"> <note position="right" xlink:label="note-0225-05" xlink:href="note-0225-05a" xml:space="preserve">TAB. XXIII. <lb/>Fig. 3.</note> <note position="left" xlink:label="note-0226-01" xlink:href="note-0226-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> </div> <div xml:id="echoid-div317" type="section" level="1" n="114"> <head xml:id="echoid-head140" style="it" xml:space="preserve">Centrum oſcillationis Trianguli iſoſcelis.</head> <p> <s xml:id="echoid-s3562" xml:space="preserve">In triangulo iſoſcele, cujuſmodi C B D, ſpatium appli-<lb/> <anchor type="note" xlink:label="note-0226-02a" xlink:href="note-0226-02"/> candum æquatur parti decimæ octavæ quadrati à diametro <lb/>B E, & </s> <s xml:id="echoid-s3563" xml:space="preserve">vigeſimæ quartæ quadrati baſeos C D. </s> <s xml:id="echoid-s3564" xml:space="preserve">Unde, ſi <lb/>ab angulo baſeos ducatur D G, perpendicularis ſuper latus <lb/>D B, quæ occurrat productæ diametro B E in G; </s> <s xml:id="echoid-s3565" xml:space="preserve">ſitque <lb/>A centrum gravitatis trianguli; </s> <s xml:id="echoid-s3566" xml:space="preserve">diviſoque intervallo G A <lb/>in quatuor partes æquales, una earum A K apponatur ipſi <lb/>B A; </s> <s xml:id="echoid-s3567" xml:space="preserve">erit B K longitudo penduli iſochroni, ſi triangulum <lb/>ſuſpendatur ex vetrice B. </s> <s xml:id="echoid-s3568" xml:space="preserve">Cum autem ex puncto mediæ ba-<lb/>ſis E ſuſpenditur, longitudo penduli iſochroni E K æquabi-<lb/>tur dimidiæ B G.</s> <s xml:id="echoid-s3569" xml:space="preserve"/> </p> <div xml:id="echoid-div317" type="float" level="2" n="1"> <note position="left" xlink:label="note-0226-02" xlink:href="note-0226-02a" xml:space="preserve">TAB.XXIII. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s3570" xml:space="preserve">Atque hinc liquet, triangulum iſoſceles rectangulum, ſi <lb/>ex puncto mediæ baſis ſuſpendatur, iſochronum eſſe pendu-<lb/>lo longitudinem diametro ſuæ æqualem habenti. </s> <s xml:id="echoid-s3571" xml:space="preserve">Similiterque, <lb/>ſi ſuſpendatur ab angulo ſuo recto, eidem pendulo iſochro-<lb/>num eſſe.</s> <s xml:id="echoid-s3572" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div319" type="section" level="1" n="115"> <head xml:id="echoid-head141" style="it" xml:space="preserve">Centrum oſcillationis Parabolæ.</head> <p> <s xml:id="echoid-s3573" xml:space="preserve">In parabolæ portione recta, ſpatium applicandum æqua-<lb/>tur {12/175} quadrati axis, una cum quinta parte quadrati dimi-<lb/>diæ baſis. </s> <s xml:id="echoid-s3574" xml:space="preserve">Cumque parabola ex verticis puncto ſuſpenſa eſt, <lb/>invenitur penduli iſochroni longitudo {5/7} axis, atque inſuper <lb/>{@/3} lateris recti. </s> <s xml:id="echoid-s3575" xml:space="preserve">Cum vero ex puncto mediæ baſis ſuſpenditur, <lb/>erit ea longitudo {4/7} axis, & </s> <s xml:id="echoid-s3576" xml:space="preserve">inſuper {1/2} lateris recti.</s> <s xml:id="echoid-s3577" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div320" type="section" level="1" n="116"> <head xml:id="echoid-head142" style="it" xml:space="preserve">Centrum oſcillationis Sectoris circuli.</head> <p> <s xml:id="echoid-s3578" xml:space="preserve">In circuli ſectore B C D, ſi radius B C vocetur r: </s> <s xml:id="echoid-s3579" xml:space="preserve">ſemi <lb/> <anchor type="note" xlink:label="note-0226-03a" xlink:href="note-0226-03"/> arcus C F, p: </s> <s xml:id="echoid-s3580" xml:space="preserve">ſemiſubtenſa C E, b: </s> <s xml:id="echoid-s3581" xml:space="preserve">fit ſpatium applican-<lb/>dum æquale {1/2} rr - {4b b r r/9 p p}, hoc eſt, dimidio quadrati B C, <lb/>minus quadrato B A; </s> <s xml:id="echoid-s3582" xml:space="preserve">ponendo A eſſe centrum gravitatis ſe-<lb/>ctoris. </s> <s xml:id="echoid-s3583" xml:space="preserve">Tunc enim B A = {2 b r/3 p}. </s> <s xml:id="echoid-s3584" xml:space="preserve">Si autem ſuſpendatur ſector <pb o="159" file="0227" n="249" rhead="HOROLOG. OSCILLATOR."/> ex B, centro circuli ſui, fit pendulum ipſi iſochronum {3 pr/4b}, <lb/> <anchor type="note" xlink:label="note-0227-01a" xlink:href="note-0227-01"/> hoc eſt, trium quartarum rectæ, quæ ſit ad radium B F ut <lb/>arcus C F D ad ſubtenſam C D. </s> <s xml:id="echoid-s3585" xml:space="preserve">Hæc autem inveniuntur <lb/>cognitis ſubcentricis cuneorum; </s> <s xml:id="echoid-s3586" xml:space="preserve">tum illius qui ſuper ſectore <lb/>toto abſcinditur, plano ducto per B K parallelam ſubtenſæ <lb/>C D, cujus cunei ſubcentricam ſuper B K invenimus eſſe <lb/>{3/8} y<unsure/> - {3/8} a + {3 p r/8 b}, vocando a ſinum verſum E F; </s> <s xml:id="echoid-s3587" xml:space="preserve">tum illius. <lb/></s> <s xml:id="echoid-s3588" xml:space="preserve">qui ſuper dimidio ſectore B F C abſcinditur plano per <lb/>B F, cujus nempe cunei ſubcentricam ſuper B F invenimus <lb/>{3/8} b - {3 b r/8 a} + {3 p r/8 a}.</s> <s xml:id="echoid-s3589" xml:space="preserve"/> </p> <div xml:id="echoid-div320" type="float" level="2" n="1"> <note position="left" xlink:label="note-0226-03" xlink:href="note-0226-03a" xml:space="preserve">TAB.XXIII. <lb/>Fig. 5.</note> <note position="right" xlink:label="note-0227-01" xlink:href="note-0227-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3590" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3591" xml:space="preserve">alia via, ſectoris centrum oſcillationis, facilius in-<lb/> <anchor type="note" xlink:label="note-0227-02a" xlink:href="note-0227-02"/> venitur, quæ eſt hujusmodi. </s> <s xml:id="echoid-s3592" xml:space="preserve">Intelligatur ſectoris B C D <lb/>pars minima ſector B C P, qui trianguli loco haberi poteſt. <lb/></s> <s xml:id="echoid-s3593" xml:space="preserve">Quadrata autem, à diſtantiis particularum ejus à puncto B, <lb/>æqualia ſunt quadratis diſtantiarum ab recta B R, bifariam <lb/>ſectorem dividente, una cum quadratis diſtantiarum ab recta <lb/>B Q, quæ ipſi B R eſt ad angulos rectos. </s> <s xml:id="echoid-s3594" xml:space="preserve">Sed, horum <lb/>quadratorum ad illa, ratio quavis data eſt major, quoniam <lb/>angulus C B P minimus; </s> <s xml:id="echoid-s3595" xml:space="preserve">ideoque illa pro nullis habenda <lb/>ſunt.</s> <s xml:id="echoid-s3596" xml:space="preserve"/> </p> <div xml:id="echoid-div321" type="float" level="2" n="2"> <note position="right" xlink:label="note-0227-02" xlink:href="note-0227-02a" xml:space="preserve">TAB.XXIII. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s3597" xml:space="preserve">Poſitâ vero B O duarum tertiarum B R, hoc eſt, poſito <lb/>O centro gravitatis trianguli B C P; </s> <s xml:id="echoid-s3598" xml:space="preserve">& </s> <s xml:id="echoid-s3599" xml:space="preserve">B N trium quar-<lb/>tarum B R: </s> <s xml:id="echoid-s3600" xml:space="preserve">ut nempe N ſit centrum gravitatis cunei, ſu-<lb/>per triangulo B C P abſciſſi plano per B Q. </s> <s xml:id="echoid-s3601" xml:space="preserve">His poſitis, <lb/>conſtat quadrata, à diſtantiis particularum trianguli B C P <lb/>ab recta B Q, æquari rectangulo N B O multiplici ſecun-<lb/>dum particularum ejuſdem trianguli numerum. </s> <s xml:id="echoid-s3602" xml:space="preserve">Itaque rectan-<lb/>gulum N B O, ita multiplex, æquale cenſendum quadratis <lb/>diſtantiarum à puncto B particularum trianguli B C P. </s> <s xml:id="echoid-s3603" xml:space="preserve">Sunt <lb/>autem quadrata diſtantiarum harum, ad quadrata diſtantia-<lb/>rum totius ſectoris B C D, ſicut ſector B C P ad ſectorem <lb/>B C D, hoc eſt, ſicut numerus particularum ſectoris B C P, <lb/>ad numerum particularum ſectoris B C D; </s> <s xml:id="echoid-s3604" xml:space="preserve">hoc enim facile <lb/>intelligitur, eo quod ſector B C D dividatur in ſectores qua-<lb/>lis B C P. </s> <s xml:id="echoid-s3605" xml:space="preserve">Ergo rectangulum N B O, multiplex ſecundum <pb o="160" file="0228" n="250" rhead="CHRISTIANI HUGENII"/> numerum particularum ſecctoris B C D, æquale erit quadra-<lb/> <anchor type="note" xlink:label="note-0228-01a" xlink:href="note-0228-01"/> tis diſtantiarum particularum ejus à puncto B. </s> <s xml:id="echoid-s3606" xml:space="preserve">Ideoque re-<lb/>ctangulum N B O, applicatum ad B A, diſtantiam inter <lb/>ſuſpenſionem & </s> <s xml:id="echoid-s3607" xml:space="preserve">centrum gravitatis ſectoris, dabit longitudi-<lb/>nem penduli iſochroni, cum ſector ex B ſuſpenditur <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3608" xml:space="preserve">Eſt <anchor type="note" xlink:label="note-0228-02a" xlink:href="note-0228-02"/> autem rectangulum N B O = {1/2} r r: </s> <s xml:id="echoid-s3609" xml:space="preserve">diſtantia autem B A, ut <lb/>jam ante diximus, = {2 br/3 p}. </s> <s xml:id="echoid-s3610" xml:space="preserve">Unde, facta applicatione, oritur {3 p r/4 b}, <lb/>longitudo penduli iſochroni, ut ante quoque inventa fuit.</s> <s xml:id="echoid-s3611" xml:space="preserve"/> </p> <div xml:id="echoid-div322" type="float" level="2" n="3"> <note position="left" xlink:label="note-0228-01" xlink:href="note-0228-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="left" xlink:label="note-0228-02" xlink:href="note-0228-02a" xml:space="preserve">Prop. 17. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div324" type="section" level="1" n="117"> <head xml:id="echoid-head143" style="it" xml:space="preserve">Centrum oſcillationis Circuli, aliter quam ſupra.</head> <p> <s xml:id="echoid-s3612" xml:space="preserve">Eodem modo etiam ſimpliciſſime, in circulo, centrum <lb/> <anchor type="note" xlink:label="note-0228-03a" xlink:href="note-0228-03"/> oſcillationis invenire licet. </s> <s xml:id="echoid-s3613" xml:space="preserve">Sit enim circulus G C F, cujus <lb/>centrum B; </s> <s xml:id="echoid-s3614" xml:space="preserve">ſectorque in eo minimus intelligatur B C P, <lb/>ſicut ante in ſectore B C D.</s> <s xml:id="echoid-s3615" xml:space="preserve"/> </p> <div xml:id="echoid-div324" type="float" level="2" n="1"> <note position="left" xlink:label="note-0228-03" xlink:href="note-0228-03a" xml:space="preserve">TAB.XXIV. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s3616" xml:space="preserve">Cum igitur, ſecundum modo expoſita, quadrata, à di-<lb/>ſtantiis particularum ſectoris B C P ad centrum B, æquen-<lb/>tur rectangulo N B O, hoc eſt, dimidio quadrato radii, <lb/>multiplici ſecundum ſectoris ipſius particularum numerum; <lb/></s> <s xml:id="echoid-s3617" xml:space="preserve">circulus autem ex ejusmodi ſectoribus componatur; </s> <s xml:id="echoid-s3618" xml:space="preserve">erunt <lb/>proinde quadrata, à diſtantiis particularum circuli totius ad <lb/>centrum B, æqualia dimidio quadrato radii, multiplici ſe-<lb/>cundum numerum earundem circuli particularum.</s> <s xml:id="echoid-s3619" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3620" xml:space="preserve">Eſt autem B centrum gravitatis circuli. </s> <s xml:id="echoid-s3621" xml:space="preserve">Ergo dictum di-<lb/>midium quadratum radii, hic erit ſpatium applicandum di-<lb/>ſtantiæ inter ſuſpenſionem & </s> <s xml:id="echoid-s3622" xml:space="preserve">centrum B, ut habeatur inter-<lb/>vallum, quo centrum oſcillationis inferius eſt ipſo centro B <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s3623" xml:space="preserve"> <anchor type="note" xlink:label="note-0228-04a" xlink:href="note-0228-04"/> quod & </s> <s xml:id="echoid-s3624" xml:space="preserve">ſupra ita ſe habere oſtendimus.</s> <s xml:id="echoid-s3625" xml:space="preserve"/> </p> <div xml:id="echoid-div325" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0228-04" xlink:href="note-0228-04a" xml:space="preserve">Prop. 18. <lb/>@uj.</note> </div> </div> <div xml:id="echoid-div327" type="section" level="1" n="118"> <head xml:id="echoid-head144" style="it" xml:space="preserve">Centrum oſcillationis Peripheriæ circuli.</head> <p> <s xml:id="echoid-s3626" xml:space="preserve">Facilius etiam, centrum oſcillationis circumferentiæ cir-<lb/> <anchor type="note" xlink:label="note-0228-05a" xlink:href="note-0228-05"/> culi, hoc pacto reperitur. </s> <s xml:id="echoid-s3627" xml:space="preserve">Eſto enim circumferentia deſcri-<lb/>pta centro B, radio B R. </s> <s xml:id="echoid-s3628" xml:space="preserve">Quadratum igitur B R, multi-<lb/>plex ſecundum numerum particularum in quas circumferen-<lb/>tia diviſa intelligitur, æquatur quadratis à diſtantiis omnium <pb file="0229" n="251"/> <pb file="0229a" n="252"/> <anchor type="figure" xlink:label="fig-0229a-01a" xlink:href="fig-0229a-01"/> <anchor type="figure" xlink:label="fig-0229a-02a" xlink:href="fig-0229a-02"/> <anchor type="figure" xlink:label="fig-0229a-03a" xlink:href="fig-0229a-03"/> <anchor type="figure" xlink:label="fig-0229a-04a" xlink:href="fig-0229a-04"/> <anchor type="figure" xlink:label="fig-0229a-05a" xlink:href="fig-0229a-05"/> <anchor type="figure" xlink:label="fig-0229a-06a" xlink:href="fig-0229a-06"/> <anchor type="figure" xlink:label="fig-0229a-07a" xlink:href="fig-0229a-07"/> <anchor type="figure" xlink:label="fig-0229a-08a" xlink:href="fig-0229a-08"/> <pb file="0230" n="253"/> <pb o="161" file="0231" n="254" rhead="HOROLOG. OSCILLATOR."/> earum particularum ad centrum B. </s> <s xml:id="echoid-s3629" xml:space="preserve">Quare quadratum B R <lb/> <anchor type="note" xlink:label="note-0231-01a" xlink:href="note-0231-01"/> erit hic ſpatium applicandum <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3630" xml:space="preserve">Patetque hinc, ſi ſuſpenſio <anchor type="note" xlink:label="note-0231-02a" xlink:href="note-0231-02"/> ſit ex G, puncto circumferentiæ, penduli iſochroni longitu-<lb/>dinem æquari diametro G F.</s> <s xml:id="echoid-s3631" xml:space="preserve"/> </p> <div xml:id="echoid-div327" type="float" level="2" n="1"> <note position="left" xlink:label="note-0228-05" xlink:href="note-0228-05a" xml:space="preserve">TAB.XXIV. <lb/>Fig. 2.</note> <figure xlink:label="fig-0229a-01" xlink:href="fig-0229a-01a"> <caption xml:id="echoid-caption88" style="it" xml:space="preserve">Pag. 160.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables88" xml:space="preserve">F D D @ N A L C H K M</variables> </figure> <figure xlink:label="fig-0229a-02" xlink:href="fig-0229a-02a"> <caption xml:id="echoid-caption89" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables89" xml:space="preserve">D D D F B A L C H K</variables> </figure> <figure xlink:label="fig-0229a-03" xlink:href="fig-0229a-03a"> <caption xml:id="echoid-caption90" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables90" xml:space="preserve">C A B</variables> </figure> <figure xlink:label="fig-0229a-04" xlink:href="fig-0229a-04a"> <caption xml:id="echoid-caption91" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables91" xml:space="preserve">B A K C E D G</variables> </figure> <figure xlink:label="fig-0229a-05" xlink:href="fig-0229a-05a"> <variables xml:id="echoid-variables92" xml:space="preserve">G D E C A K B</variables> </figure> <figure xlink:label="fig-0229a-06" xlink:href="fig-0229a-06a"> <variables xml:id="echoid-variables93" xml:space="preserve">G D K C A B</variables> </figure> <figure xlink:label="fig-0229a-07" xlink:href="fig-0229a-07a"> <caption xml:id="echoid-caption92" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables94" xml:space="preserve">K B K A C E D F</variables> </figure> <figure xlink:label="fig-0229a-08" xlink:href="fig-0229a-08a"> <caption xml:id="echoid-caption93" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables95" xml:space="preserve">Q B Q O N A C E D R P F</variables> </figure> <note position="right" xlink:label="note-0231-01" xlink:href="note-0231-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="right" xlink:label="note-0231-02" xlink:href="note-0231-02a" xml:space="preserve">Prop. 18. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div329" type="section" level="1" n="119"> <head xml:id="echoid-head145" style="it" xml:space="preserve">Centrum oſcillationis Polygonorum ordinatorum.</head> <p> <s xml:id="echoid-s3632" xml:space="preserve">Haud abſimiliter & </s> <s xml:id="echoid-s3633" xml:space="preserve">polygono cuivis ordinato, ut A B C, <lb/> <anchor type="note" xlink:label="note-0231-03a" xlink:href="note-0231-03"/> pendulum iſochronum invenitur. </s> <s xml:id="echoid-s3634" xml:space="preserve">Fit enim, ſpatium appli-<lb/>candum, æquale ſemiſſi quadrati perpendicularis ex centro <lb/>in latus polygoni, una cum vigefima quarta parte quadrati <lb/>lateris. </s> <s xml:id="echoid-s3635" xml:space="preserve">At, ſi perimetro polygoni pendulum iſochronum <lb/>quæratur, fit ſpatium applicandum æquale quadrato perpen-<lb/>dicularis à centro in latus, cum duodecima parte quadrati <lb/>lateris.</s> <s xml:id="echoid-s3636" xml:space="preserve"/> </p> <div xml:id="echoid-div329" type="float" level="2" n="1"> <note position="right" xlink:label="note-0231-03" xlink:href="note-0231-03a" xml:space="preserve">TAB XXIV. <lb/>Fig. 3.</note> </div> </div> <div xml:id="echoid-div331" type="section" level="1" n="120"> <head xml:id="echoid-head146" style="it" xml:space="preserve">Loci plani & ſolidi uſus in hac Theoria.</head> <p> <s xml:id="echoid-s3637" xml:space="preserve">Eſt præterea & </s> <s xml:id="echoid-s3638" xml:space="preserve">Locorum contemplatio in his non injucun-<lb/> <anchor type="note" xlink:label="note-0231-04a" xlink:href="note-0231-04"/> da. </s> <s xml:id="echoid-s3639" xml:space="preserve">Ut ſi propoſitum ſit, dato puncto ſuſpenſionis A, & </s> <s xml:id="echoid-s3640" xml:space="preserve"><lb/>longitudine A B, invenire locum duorum ponderum æqua-<lb/>lium C, D, æqualiter ab A & </s> <s xml:id="echoid-s3641" xml:space="preserve">à perpendiculari A B diſtan-<lb/>tium, quæ agitata circa axem in A, perpendicularem plano <lb/>per A C D, iſochrona ſint pendulo ſimplici longitudinis <lb/>A B.</s> <s xml:id="echoid-s3642" xml:space="preserve"/> </p> <div xml:id="echoid-div331" type="float" level="2" n="1"> <note position="right" xlink:label="note-0231-04" xlink:href="note-0231-04a" xml:space="preserve">TAB.XXIV. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s3643" xml:space="preserve">Ponatur A B = a, ductâque C D, quæ ſecet A B ad <lb/>angulos rectos in E, ſit A E indeterminata = x: </s> <s xml:id="echoid-s3644" xml:space="preserve">E C vel <lb/>E D = y. </s> <s xml:id="echoid-s3645" xml:space="preserve">Ergo quadratum A C = x x + y y. </s> <s xml:id="echoid-s3646" xml:space="preserve">Hoc vero <lb/>multiplex ſecundum numerum particularum ponderum C, D, <lb/>quæ hic minima intelliguntur, æquatur quadratis diſtantia-<lb/>rum earundem particularum ab axe ſuſpenſionis A. </s> <s xml:id="echoid-s3647" xml:space="preserve">Ergo <lb/>quadratum A C, ſive x x + y y, applicatum ad diſtantiam <lb/>A E, quæ nempe eſt inter axem ſuſpenſionis & </s> <s xml:id="echoid-s3648" xml:space="preserve">centrum gra-<lb/>vitatis ponderum C, D, efficiet {xx + yy/x}, longitudinem pen-<lb/>duli iſochroni <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s3649" xml:space="preserve">quam propterea oportet æqualem eſſe A B <anchor type="note" xlink:label="note-0231-05a" xlink:href="note-0231-05"/> ſive a. </s> <s xml:id="echoid-s3650" xml:space="preserve">Itaque {x x + y y/x} = a. </s> <s xml:id="echoid-s3651" xml:space="preserve">Et y y = a x - x x. </s> <s xml:id="echoid-s3652" xml:space="preserve">Unde patet, <lb/>locum punctorum C & </s> <s xml:id="echoid-s3653" xml:space="preserve">D, eſſe circumferentiam circuli, cu- <pb o="162" file="0232" n="255" rhead="CHRISTIANI HUGENII"/> jus centrum F, ubi A B bifariam dividitur, radius autem <lb/> <anchor type="note" xlink:label="note-0232-01a" xlink:href="note-0232-01"/> = {1/2} a, ſive F A. </s> <s xml:id="echoid-s3654" xml:space="preserve">Ergo, ubicunque in circumferentia <lb/>A C B D duo pondera æqualia, æqualiter ab A diſtantia, <lb/>ponentur, ea, ex A agitata, iſochrona erunt pendulo lon-<lb/>gitudinem habenti æqualem diametro A B.</s> <s xml:id="echoid-s3655" xml:space="preserve"/> </p> <div xml:id="echoid-div332" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0231-05" xlink:href="note-0231-05a" xml:space="preserve">Prop. 17. <lb/>huj.</note> <note position="left" xlink:label="note-0232-01" xlink:href="note-0232-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3656" xml:space="preserve">Atque hinc manifeſtum quoque, & </s> <s xml:id="echoid-s3657" xml:space="preserve">circumferentiam <lb/>A C B D, ſi gravitas ei tribuatur, & </s> <s xml:id="echoid-s3658" xml:space="preserve">quamlibet ejus por-<lb/>tionem, æqualiter in A vel B diviſam, & </s> <s xml:id="echoid-s3659" xml:space="preserve">ab axe per A ſuſ-<lb/>penſam, eidem pendulo A B iſochronam eſſe.</s> <s xml:id="echoid-s3660" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3661" xml:space="preserve">Loci vero ſolidi exemplum eſto hujusmodi. </s> <s xml:id="echoid-s3662" xml:space="preserve">Sit A N linea <lb/> <anchor type="note" xlink:label="note-0232-02a" xlink:href="note-0232-02"/> inflexilis ſine pondere. </s> <s xml:id="echoid-s3663" xml:space="preserve">Propoſitumque ſit, ad punctum in <lb/>ea acceptum, ut M, affigere ipſi ad angulos rectos lineam, <lb/>ſeu virgam, pondere præditam O M L, ad M bifariam divi-<lb/>ſam, cujus in latus agitatæ oſcillationes, ex ſuſpenſione A, <lb/>iſochronæ ſint pendulo ſimplici longitudinis A N.</s> <s xml:id="echoid-s3664" xml:space="preserve"/> </p> <div xml:id="echoid-div333" type="float" level="2" n="3"> <note position="left" xlink:label="note-0232-02" xlink:href="note-0232-02a" xml:space="preserve">TAB.XXIV. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s3665" xml:space="preserve">Ducatur O H parallela A N, & </s> <s xml:id="echoid-s3666" xml:space="preserve">A H parallela O M, <lb/>& </s> <s xml:id="echoid-s3667" xml:space="preserve">ſit O R æqualis {2/3} O L. </s> <s xml:id="echoid-s3668" xml:space="preserve">Itaque cunei ſuper recta O L, <lb/>abſciſſi plano per O H ducto, ſubcentrica erit O R. </s> <s xml:id="echoid-s3669" xml:space="preserve">Sed <lb/>cunei alterius ſuper eadem O L, abſciſſi plano per rectam <lb/>A H, (eſt autem cuneus hic nihil aliud quam rectangulum) <lb/>ſubcentrica erit ipſa A M. </s> <s xml:id="echoid-s3670" xml:space="preserve">Quare rectangulum illud, quod <lb/>ſupra Oſcillationis vocavimus, erit ſolum rectangulum O M R. <lb/></s> <s xml:id="echoid-s3671" xml:space="preserve">quod nempe, applicatum ad longitudinem A M, dabit di-<lb/>ſtantiam centri oſcillationis lineæ O L, ex A ſuſpenſæ, in-<lb/>fra punctum M.</s> <s xml:id="echoid-s3672" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3673" xml:space="preserve">Sit jam A N = a: </s> <s xml:id="echoid-s3674" xml:space="preserve">A M = x: </s> <s xml:id="echoid-s3675" xml:space="preserve">M O vel M L = y. </s> <s xml:id="echoid-s3676" xml:space="preserve">Eſt <lb/>ergo rectangulum O M R = {1/3} yy. </s> <s xml:id="echoid-s3677" xml:space="preserve">quo applicato ad A M, fit <lb/>{1 y y/3x}. </s> <s xml:id="echoid-s3678" xml:space="preserve">quæ longitudo itaque ipſi M N æqualis eſſe debebit, <lb/>cum velimus centrum oſcillationis virgæ O L eſſe in N. </s> <s xml:id="echoid-s3679" xml:space="preserve">Fit <lb/>ergo æquatio {1 yy/3x} + x = a. </s> <s xml:id="echoid-s3680" xml:space="preserve">Unde y = <emph style="red">3 a x - 3 x x</emph>. </s> <s xml:id="echoid-s3681" xml:space="preserve">Quod <lb/>ſignificat puncta O & </s> <s xml:id="echoid-s3682" xml:space="preserve">L eſſe ad Ellipſin, cujus axis minor <lb/>A N; </s> <s xml:id="echoid-s3683" xml:space="preserve">latus rectum vero, ſecundum quod poſſunt ordinatim <lb/>ad axem hunc applicatæ, ipſius A N triplum.</s> <s xml:id="echoid-s3684" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3685" xml:space="preserve">Hinc vero manifeſtum fit, cum omnis virga ipſi O L pa-<lb/>rallela, & </s> <s xml:id="echoid-s3686" xml:space="preserve">ad Ellipſin hanc terminata, oſcillationes iſochro- <pb o="163" file="0233" n="256" rhead="HOROLOG. OSCILLATOR."/> nas habeat pendulo ſimplici A N, etiam totum Ellipſeos <lb/> <anchor type="note" xlink:label="note-0233-01a" xlink:href="note-0233-01"/> planum, ex A ſuſpenſum & </s> <s xml:id="echoid-s3687" xml:space="preserve">in latus agitatum, ipſi A N <lb/>pendulo iſochronum fore. </s> <s xml:id="echoid-s3688" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3689" xml:space="preserve">partem Ellipſeos quamli-<lb/>bet, quæ lineis una vel duabus, ad A N perpendicularibus, <lb/>abſcindetur.</s> <s xml:id="echoid-s3690" xml:space="preserve"/> </p> <div xml:id="echoid-div334" type="float" level="2" n="4"> <note position="right" xlink:label="note-0233-01" xlink:href="note-0233-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3691" xml:space="preserve">Cæterum adſcribemus & </s> <s xml:id="echoid-s3692" xml:space="preserve">aliud loci plani exemplum, in <lb/>quo nonnulla notatu digna occurrunt.</s> <s xml:id="echoid-s3693" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3694" xml:space="preserve">Sit virga A B ponderis expers, ſuſpenſa ex A; </s> <s xml:id="echoid-s3695" xml:space="preserve">oporteat-<lb/> <anchor type="note" xlink:label="note-0233-02a" xlink:href="note-0233-02"/> que, ad datum in ea punctum B, affigere triangula duo pa-<lb/>ria, & </s> <s xml:id="echoid-s3696" xml:space="preserve">paribus angulis ab axe A B recedentia, quorum an-<lb/>guli ad B minimi, ſive infinite parvi exiſtimandi, quæque, <lb/>ita ſuſpenſa ab A, oſcillationes iſochronas faciant pendulo <lb/>ſimplici datæ longitudinis A L.</s> <s xml:id="echoid-s3697" xml:space="preserve"/> </p> <div xml:id="echoid-div335" type="float" level="2" n="5"> <note position="right" xlink:label="note-0233-02" xlink:href="note-0233-02a" xml:space="preserve">TAB.XXIV. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s3698" xml:space="preserve">Hic, ducta C G perpendiculari in B G, & </s> <s xml:id="echoid-s3699" xml:space="preserve">ponendo <lb/>A B = a; </s> <s xml:id="echoid-s3700" xml:space="preserve">A L = b; </s> <s xml:id="echoid-s3701" xml:space="preserve">B G = x; </s> <s xml:id="echoid-s3702" xml:space="preserve">C G = y: </s> <s xml:id="echoid-s3703" xml:space="preserve">invenitur æqua-<lb/>tio y = <emph style="red">2 a b - 2 a a - {8/3} a x + {4/3} b x - x x</emph> ex qua patet, baſes <lb/>triangulorum C, & </s> <s xml:id="echoid-s3704" xml:space="preserve">D, quæ baſes hic ut puncta conſide-<lb/>rantur, eſſe ad circuli circumferentiam; </s> <s xml:id="echoid-s3705" xml:space="preserve">quia nempe habetur <lb/>terminus ſimplex - x x.</s> <s xml:id="echoid-s3706" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3707" xml:space="preserve">Licet autem hic animadvertere, quod ſi a ſit nihilo æqua-<lb/> <anchor type="note" xlink:label="note-0233-03a" xlink:href="note-0233-03"/> lis, hoc eſt, ſi punctum, ubi affiguntur trianguli B C, <lb/>B D, ſit idem cum puncto A; </s> <s xml:id="echoid-s3708" xml:space="preserve">tum futura ſit æquatio <lb/>y = <emph style="red">{4/3} b x - x x</emph>. </s> <s xml:id="echoid-s3709" xml:space="preserve">Ac proinde, hoc caſu, ſi ſumatur A O <lb/>= {2/3} b, hoc eſt, = {2/3} A L, centroque O per A circulus de-<lb/>ſcribatur A D N; </s> <s xml:id="echoid-s3710" xml:space="preserve">erunt baſes triangulorum A C, A D, ad <lb/>illius circumferentiam. </s> <s xml:id="echoid-s3711" xml:space="preserve">Cum igitur quælibet duo triangula <lb/>acutiſſima, quæ ex A ad circumferentiam A C N D conſti-<lb/>tuuntur, magnitudine & </s> <s xml:id="echoid-s3712" xml:space="preserve">ſitu ſibi reſpondentia, centrum <lb/>oſcillationis habeant punctum L, poſitâ A L = {3/4} diametri <lb/>A N; </s> <s xml:id="echoid-s3713" xml:space="preserve">cumque circulus totus ex ejusmodi triangulorum pa-<lb/>ribus componatur; </s> <s xml:id="echoid-s3714" xml:space="preserve">uti & </s> <s xml:id="echoid-s3715" xml:space="preserve">portio ejus quælibet, ut A C N D, <lb/>latera A C, A D æqualia habens; </s> <s xml:id="echoid-s3716" xml:space="preserve">manifeſtum eſt, tum cir-<lb/>culi totius, tum portionis qualem diximus, centrum oſcilla-<lb/>tionis eſſe in L.</s> <s xml:id="echoid-s3717" xml:space="preserve"/> </p> <div xml:id="echoid-div336" type="float" level="2" n="6"> <note position="right" xlink:label="note-0233-03" xlink:href="note-0233-03a" xml:space="preserve">TAB. XXV. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s3718" xml:space="preserve">Rurſus, ſi in æquatione inventa ponatur {8/3} a = {4/3} b, ſeu <lb/> <anchor type="note" xlink:label="note-0233-04a" xlink:href="note-0233-04"/> <pb o="164" file="0234" n="257" rhead="CHRISTIANI HUGENII"/> 2 a = b; </s> <s xml:id="echoid-s3719" xml:space="preserve">hoc eſt, ſi triangula affigi intelligantur in B, quod <lb/> <anchor type="note" xlink:label="note-0234-01a" xlink:href="note-0234-01"/> longitudinem A L ſecet bifariam, erit y = <emph style="red">2 a a - x x.</emph> <lb/>quæ æquatio docet, quod ſi centro B, radio qui poſſit du-<lb/>plum B A, circumferentia deſcribatur, ea erit locus baſium <lb/>triangulorum acutiſſimorum B C, B D, quorum nempe, <lb/>ex A ſuſpenſorum, centrum oſcillationis erit L punctum. <lb/></s> <s xml:id="echoid-s3720" xml:space="preserve">Cumque & </s> <s xml:id="echoid-s3721" xml:space="preserve">circulus totus, & </s> <s xml:id="echoid-s3722" xml:space="preserve">ſector ejus quilibet, axem <lb/>habens in recta A L, ex hujusmodi triangulorum paribus <lb/>componatur, manifeſtum eſt & </s> <s xml:id="echoid-s3723" xml:space="preserve">horum, ex A ſuſpenſorum, <lb/>centrum oſcillationis eſſe punctum L.</s> <s xml:id="echoid-s3724" xml:space="preserve"/> </p> <div xml:id="echoid-div337" type="float" level="2" n="7"> <note position="right" xlink:label="note-0233-04" xlink:href="note-0233-04a" xml:space="preserve">TAB. XXV. <lb/>Fig. 2.</note> <note position="left" xlink:label="note-0234-01" xlink:href="note-0234-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3725" xml:space="preserve">Adeoque quilibet circuli ſector, ſuſpenſus à puncto quod <lb/>diſtet, à centro circuli ſui, ſemiſſe lateris quadrati circulo <lb/>inſcripti, pendulum iſochronum habebit toti eidem lateri æ-<lb/>quale. </s> <s xml:id="echoid-s3726" xml:space="preserve">Atque ita, hoc uno caſu, absque poſita dimenſione <lb/>arcus, pendulum ſectori iſochronum invenitur.</s> <s xml:id="echoid-s3727" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3728" xml:space="preserve">Porro, ad univerſalem conſtructionem æquationis primæ, <lb/> <anchor type="note" xlink:label="note-0234-02a" xlink:href="note-0234-02"/> y = <emph style="red">2 a b - 2 a a - {8/3} a x + {4/3} b x - x x</emph>, dividatur A L bifariam <lb/>in E, & </s> <s xml:id="echoid-s3729" xml:space="preserve">adponatur ad B E pars ſui tertia E F; </s> <s xml:id="echoid-s3730" xml:space="preserve">eritque F <lb/>centrum deſcribendi circuli; </s> <s xml:id="echoid-s3731" xml:space="preserve">radius autem F O æqualis ſu-<lb/>mendus ei, quæ poteſt duplum differentiæ quadratorum <lb/>A E, E F.</s> <s xml:id="echoid-s3732" xml:space="preserve"/> </p> <div xml:id="echoid-div338" type="float" level="2" n="8"> <note position="left" xlink:label="note-0234-02" xlink:href="note-0234-02a" xml:space="preserve">TAB.XXV. <lb/>Fig. 3. & 4.</note> </div> <p> <s xml:id="echoid-s3733" xml:space="preserve">Si itaque, ex puncto B, ad deſcriptam circumferentiam <lb/>triangula duo paria acutiſſima conſtituantur, ut B C, B D; <lb/></s> <s xml:id="echoid-s3734" xml:space="preserve">illorum, ex A ſuſpenſorum, centrum oſcillationis erit L. </s> <s xml:id="echoid-s3735" xml:space="preserve"><lb/>Quare & </s> <s xml:id="echoid-s3736" xml:space="preserve">portionis cujuslibet deſcripti circuli, cujus portio-<lb/>nis vertex ſit in B, axis vero in recta A L, quales ſunt u-<lb/>traque C B D; </s> <s xml:id="echoid-s3737" xml:space="preserve">poſita ſuſpenſione ex A; </s> <s xml:id="echoid-s3738" xml:space="preserve">centrum oſcilla-<lb/>tionis idem punctum L eſſe conſtat. </s> <s xml:id="echoid-s3739" xml:space="preserve">Atque adeo etiam cir-<lb/>culi ſegmentorum K O N, K M N, quæ facit recta K B N <lb/>perpendicularis ad A B.</s> <s xml:id="echoid-s3740" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3741" xml:space="preserve">Et hæc quidem de motu laterali planorum, ac linearum, <lb/>animadvertiſſe ſufficiat. </s> <s xml:id="echoid-s3742" xml:space="preserve">Quibus hoc tantum addimus; </s> <s xml:id="echoid-s3743" xml:space="preserve">in-<lb/>ventis centris oſcillationis figurarum rectarum, ſeu quæ æ-<lb/>qualiter ad axem utrinque conſtitutæ ſunt; </s> <s xml:id="echoid-s3744" xml:space="preserve">ut trianguli iſo-<lb/>ſcelis, vel parabolicæ ſectionis rectæ etiam obliquarum, <pb file="0235" n="258"/> <pb file="0235a" n="259"/> <anchor type="figure" xlink:label="fig-0235a-01a" xlink:href="fig-0235a-01"/> <anchor type="figure" xlink:label="fig-0235a-02a" xlink:href="fig-0235a-02"/> <anchor type="figure" xlink:label="fig-0235a-03a" xlink:href="fig-0235a-03"/> <anchor type="figure" xlink:label="fig-0235a-04a" xlink:href="fig-0235a-04"/> <anchor type="figure" xlink:label="fig-0235a-05a" xlink:href="fig-0235a-05"/> <anchor type="figure" xlink:label="fig-0235a-06a" xlink:href="fig-0235a-06"/> <pb file="0236" n="260"/> <pb o="165" file="0237" n="261" rhead="HOROLOG. OSCILLATOR."/> quæ velut luxatione illarum efficiuntur, ut trianguli ſcaleni, <lb/> <anchor type="note" xlink:label="note-0237-01a" xlink:href="note-0237-01"/> & </s> <s xml:id="echoid-s3745" xml:space="preserve">parabolæ non rectæ, centra oſcillationis haberi. </s> <s xml:id="echoid-s3746" xml:space="preserve">Ut ſi, <lb/> <anchor type="note" xlink:label="note-0237-02a" xlink:href="note-0237-02"/> exempli gratia, triangulum B A C iſoſceles, cujus axis <lb/>A D, à puncto E ſuſpenſum intelligatur; </s> <s xml:id="echoid-s3747" xml:space="preserve">ſit vero & </s> <s xml:id="echoid-s3748" xml:space="preserve">aliud <lb/>triangulum ſcalenum F A G, axem eundem habens A D, & </s> <s xml:id="echoid-s3749" xml:space="preserve"><lb/>baſin F G æqualem baſi B C; </s> <s xml:id="echoid-s3750" xml:space="preserve">etiam hoc triangulum, ex E <lb/>ſuſpenſum, priori B A C iſochronum eſſe dico.</s> <s xml:id="echoid-s3751" xml:space="preserve"/> </p> <div xml:id="echoid-div339" type="float" level="2" n="9"> <figure xlink:label="fig-0235a-01" xlink:href="fig-0235a-01a"> <caption xml:id="echoid-caption94" style="it" xml:space="preserve">Pag. 164.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables96" xml:space="preserve">G B O N C R P F</variables> </figure> <figure xlink:label="fig-0235a-02" xlink:href="fig-0235a-02a"> <caption xml:id="echoid-caption95" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables97" xml:space="preserve">G B R F</variables> </figure> <figure xlink:label="fig-0235a-03" xlink:href="fig-0235a-03a"> <caption xml:id="echoid-caption96" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables98" xml:space="preserve">A E C F B</variables> </figure> <figure xlink:label="fig-0235a-04" xlink:href="fig-0235a-04a"> <caption xml:id="echoid-caption97" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables99" xml:space="preserve">A C E D F B</variables> </figure> <figure xlink:label="fig-0235a-05" xlink:href="fig-0235a-05a"> <caption xml:id="echoid-caption98" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables100" xml:space="preserve">A B C G D L</variables> </figure> <figure xlink:label="fig-0235a-06" xlink:href="fig-0235a-06a"> <caption xml:id="echoid-caption99" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables101" xml:space="preserve">H A O M R L N</variables> </figure> <note position="right" xlink:label="note-0237-01" xlink:href="note-0237-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note position="right" xlink:label="note-0237-02" xlink:href="note-0237-02a" xml:space="preserve">TAB.XXVII. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s3752" xml:space="preserve">Quia enim virga, ſeu linea gravis, F G, affixa virgæ ſi-<lb/>ne pondere E D in D, ſitu obliquo, ſuſpenſaque ex E, <lb/>iſochrona eſt virgæ B C, ſimiliter in D affixæ <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s3753" xml:space="preserve">idemque <anchor type="note" xlink:label="note-0237-03a" xlink:href="note-0237-03"/> evenit in virgis cæteris trianguli útriusque, quæ axem A D <lb/>ſecant in iisdem punctis, atque inter ſe æquales ſunt: </s> <s xml:id="echoid-s3754" xml:space="preserve">ne-<lb/>ceſſe eſt tota triangula, quæ ex lineis, ſeu virgis iisdem <lb/>compoſita intelligi poſſunt, iſochrona eſſe. </s> <s xml:id="echoid-s3755" xml:space="preserve">In aliis figuris ſi-<lb/>milis eſt demonſtratio.</s> <s xml:id="echoid-s3756" xml:space="preserve"/> </p> <div xml:id="echoid-div340" type="float" level="2" n="10"> <note symbol="*" position="right" xlink:label="note-0237-03" xlink:href="note-0237-03a" xml:space="preserve">Prop. 16. <lb/>huj.</note> </div> </div> <div xml:id="echoid-div342" type="section" level="1" n="121"> <head xml:id="echoid-head147" xml:space="preserve">PROPOSITIO XXII.</head> <p style="it"> <s xml:id="echoid-s3757" xml:space="preserve">QUomodo, in ſolidis figuris, oſcillationis centra <lb/>inveniantur.</s> <s xml:id="echoid-s3758" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3759" xml:space="preserve">In ſolidis porro figuris facile quoque, per ante demon-<lb/> <anchor type="note" xlink:label="note-0237-04a" xlink:href="note-0237-04"/> ſtrata, centrum oſcillationis invenire licebit. </s> <s xml:id="echoid-s3760" xml:space="preserve">Si enim ſit ſc-<lb/>lidum A B C, ſuſpenſum ab axe, qui, per punctum E, <lb/>intelligitur hujus paginæ plano ad rectos angulos; </s> <s xml:id="echoid-s3761" xml:space="preserve">centrum <lb/>autem gravitatis ſit F: </s> <s xml:id="echoid-s3762" xml:space="preserve">ductis jam per F planis E F D, G F H, <lb/>quorum poſterius ſit horizonti parallelum, alterum vero per <lb/>axem E transeat; </s> <s xml:id="echoid-s3763" xml:space="preserve">inventisque, per propoſitionem 14, ſum-<lb/>mis quadratorum à diſtantiis particularum ſolidi A B C à <lb/>plano G F H, itemque à plano E F D; </s> <s xml:id="echoid-s3764" xml:space="preserve">hoc eſt, inven-<lb/>tis rectangulis utrisque, quæ, multiplicia ſecundum numc-<lb/>rum dictarum particularum, æqualia ſint dictis quadratcrum <lb/>ſummis; </s> <s xml:id="echoid-s3765" xml:space="preserve">rectangula hæc applicata ad diſtantiam E F, qua <lb/>nempe axis ſuſpenſionis diſtat à centro gravitatis, dabunt <lb/>intervallum F K, quo centrum agitationis K inferius eſt <lb/>centro gravitatis F. </s> <s xml:id="echoid-s3766" xml:space="preserve">Hoc enim patet ex propoſitione 18. </s> <s xml:id="echoid-s3767" xml:space="preserve">Da-<lb/>bimus autem & </s> <s xml:id="echoid-s3768" xml:space="preserve">horum exempla aliquot.</s> <s xml:id="echoid-s3769" xml:space="preserve"/> </p> <div xml:id="echoid-div342" type="float" level="2" n="1"> <note position="right" xlink:label="note-0237-04" xlink:href="note-0237-04a" xml:space="preserve">TAB. XXV. <lb/>Fig. 5.</note> </div> <pb o="166" file="0238" n="262" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div344" type="section" level="1" n="122"> <head xml:id="echoid-head148" style="it" xml:space="preserve">Centrum oſcillationis in Pyramide.</head> <note position="left" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <p> <s xml:id="echoid-s3770" xml:space="preserve">Sit primum A B C pyramis, verticem habens A, axem <lb/> <anchor type="note" xlink:label="note-0238-02a" xlink:href="note-0238-02"/> A D, baſin vero quadratum, cujus latus B C. </s> <s xml:id="echoid-s3771" xml:space="preserve">ponaturque <lb/>agitari circa axem qui, per verticem A, ſit hujus paginæ <lb/>plano ad angulos rectos.</s> <s xml:id="echoid-s3772" xml:space="preserve"/> </p> <div xml:id="echoid-div344" type="float" level="2" n="1"> <note position="left" xlink:label="note-0238-02" xlink:href="note-0238-02a" xml:space="preserve">TAB.XXVI. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s3773" xml:space="preserve">Hic figura plana proportionalis O V V, à latere adpo-<lb/>nenda, ſecundum propoſitionem 14, conſtabit ex reſiduis <lb/>parabolicis O P V, quæ nempe ſuperſunt, cum, à rectan-<lb/>gulis Ω P, auferuntur ſemiparabolæ O V Ω, verticem ha-<lb/>bentes O.</s> <s xml:id="echoid-s3774" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3775" xml:space="preserve">Sicut enim inter ſe ſectiones pyramidis B C, N N, ita <lb/>quoque rectæ V V, R R, ipſis in figura plana reſponden-<lb/>tes. </s> <s xml:id="echoid-s3776" xml:space="preserve">& </s> <s xml:id="echoid-s3777" xml:space="preserve">ſicut centrum gravitatis E diſtat, à vertice pyrami-<lb/>dis, tribus quartis axis A D, ita quoque centrum gravita-<lb/>tis F, figuræ O V V, diſtabit tribus quartis diametri O P <lb/>à vertice O.</s> <s xml:id="echoid-s3778" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3779" xml:space="preserve">Intellecto porro horizontali plano N E, per centrum gra-<lb/>vitatis pyramidis A B C, quod idem figuram O V V ſecet <lb/>ſecundum R F; </s> <s xml:id="echoid-s3780" xml:space="preserve">inventâque ſubcentricâ cunei, ſuper figura <lb/>O V V abſciſſi plano per O Ω, quæ ſubcentrica ſit O G, <lb/>(eſt autem {4/5} diametri O P) erit rectangulum O F G, mul-<lb/>tiplex per numerum particularum figuræ O V V, æquale <lb/>quadratis diſtantiarum ab recta R F <anchor type="note" xlink:href="" symbol="*"/>, ac proinde quoque <anchor type="note" xlink:label="note-0238-03a" xlink:href="note-0238-03"/> quadratis diſtantiarum à plano N E, particularum ſolidi <lb/>A B C. </s> <s xml:id="echoid-s3781" xml:space="preserve">Fit autem rectangulum O F G æquale {3/80} quadrati <lb/>O P, vel quadrati A D.</s> <s xml:id="echoid-s3782" xml:space="preserve"/> </p> <div xml:id="echoid-div345" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0238-03" xlink:href="note-0238-03a" xml:space="preserve">Prop. 10. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s3783" xml:space="preserve">Deinde, ad inveniendam ſummam quadratorum à diſtan-<lb/>tiis à plano A D, noſcenda primo ſubcentrica cunei, ſuper <lb/>quadratâ baſi pyramidis B C abſciſſi, plano per rectam quæ <lb/>in B intelligitur axi A parallela; </s> <s xml:id="echoid-s3784" xml:space="preserve">quæ ſubcentrica ſit B K; <lb/></s> <s xml:id="echoid-s3785" xml:space="preserve">eſtque {2/3} B C. </s> <s xml:id="echoid-s3786" xml:space="preserve">Noſcenda item diſtantia centr. </s> <s xml:id="echoid-s3787" xml:space="preserve">gr. </s> <s xml:id="echoid-s3788" xml:space="preserve">dimidiæ fi-<lb/>guræ O P V ab O P; </s> <s xml:id="echoid-s3789" xml:space="preserve">quæ ſit Φ P; </s> <s xml:id="echoid-s3790" xml:space="preserve">eſtque {3/10} P V. </s> <s xml:id="echoid-s3791" xml:space="preserve">Inde, <lb/>diviſà bifariam P V in Δ, ſi fiat ut Δ P ad P Φ, hoc eſt, <lb/>ut 5 ad 3, ita rectangulum B D K, quod eſt {1/12} quadrati <lb/>B C, ad aliud ſpatium Z; </s> <s xml:id="echoid-s3792" xml:space="preserve">erit hoc, multiplex ſecundum <pb file="0239" n="263"/> <pb file="0239a" n="264"/> <anchor type="figure" xlink:label="fig-0239a-01a" xlink:href="fig-0239a-01"/> <anchor type="figure" xlink:label="fig-0239a-02a" xlink:href="fig-0239a-02"/> <anchor type="figure" xlink:label="fig-0239a-03a" xlink:href="fig-0239a-03"/> <anchor type="figure" xlink:label="fig-0239a-04a" xlink:href="fig-0239a-04"/> <anchor type="figure" xlink:label="fig-0239a-05a" xlink:href="fig-0239a-05"/> <pb file="0240" n="265"/> <pb o="167" file="0241" n="266" rhead="HOROLOG. OSCILLATOR."/> numerum particularum ſolidi A B C, æquale quadratis di-<lb/> <anchor type="note" xlink:label="note-0241-01a" xlink:href="note-0241-01"/> ſtantiarum à plano A D <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3793" xml:space="preserve">Apparet autem, fieri ſpatium Z <anchor type="note" xlink:label="note-0241-02a" xlink:href="note-0241-02"/> æquale {1/20} quadrati B C.</s> <s xml:id="echoid-s3794" xml:space="preserve"/> </p> <div xml:id="echoid-div346" type="float" level="2" n="3"> <figure xlink:label="fig-0239a-01" xlink:href="fig-0239a-01a"> <caption xml:id="echoid-caption100" style="it" xml:space="preserve">Pag. 166.<lb/>TAB.XXV.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables102" xml:space="preserve">A O C G D L N</variables> </figure> <figure xlink:label="fig-0239a-02" xlink:href="fig-0239a-02a"> <caption xml:id="echoid-caption101" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables103" xml:space="preserve">A B C G D L N</variables> </figure> <figure xlink:label="fig-0239a-03" xlink:href="fig-0239a-03a"> <caption xml:id="echoid-caption102" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables104" xml:space="preserve">O C D A K B N E F C D L M</variables> </figure> <figure xlink:label="fig-0239a-04" xlink:href="fig-0239a-04a"> <caption xml:id="echoid-caption103" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables105" xml:space="preserve">O A C D F E K B N C L D M</variables> </figure> <figure xlink:label="fig-0239a-05" xlink:href="fig-0239a-05a"> <caption xml:id="echoid-caption104" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables106" xml:space="preserve">E A G F H K B D C</variables> </figure> <note position="right" xlink:label="note-0241-01" xlink:href="note-0241-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> <note symbol="*" position="right" xlink:label="note-0241-02" xlink:href="note-0241-02a" xml:space="preserve">Prop. 15. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s3795" xml:space="preserve">Itaque, totum ſpatium applicandum, æquatur hic {3/80} qua-<lb/>drati A D, cum {1/20} quadrati B C. </s> <s xml:id="echoid-s3796" xml:space="preserve">Unde, ſi ſuſpenſio, ut <lb/>hic, poſita fuerit in A, vertice pyramidis, ideoque diſtan-<lb/>tia, ad quam applicatio facienda, A E æqualis {3/4} A D; </s> <s xml:id="echoid-s3797" xml:space="preserve">fiet <lb/>hinc E S, intervallum quo centrum agitationis inferius eſt <lb/>centro gravitatis, æquale {1/20} A D, atque inſuper {1/15} tertiæ <lb/>proportionalis duabus A D, B C. </s> <s xml:id="echoid-s3798" xml:space="preserve">ſive tota A S æqualis {4/5} <lb/>A D, præter dictam {1/15} tertiæ proportionialis.</s> <s xml:id="echoid-s3799" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div348" type="section" level="1" n="123"> <head xml:id="echoid-head149" style="it" xml:space="preserve">Centrum oſcillationis Coni.</head> <p> <s xml:id="echoid-s3800" xml:space="preserve">Quod ſi A B C conus fuerit, omnia eodem modo @e habe-<lb/>bunt, niſi quod ſpatium Z hic fit æquale rectangulo Δ Ρ Φ <anchor type="note" xlink:href="" symbol="*"/>, <anchor type="note" xlink:label="note-0241-03a" xlink:href="note-0241-03"/> hoc eſt {3/2@} quadrati P V vel B D, ſive {3/80} quadrati B C. <lb/></s> <s xml:id="echoid-s3801" xml:space="preserve">Quare, totum ſpatium applicandum, in cono erit {3/80} qua-<lb/>drati A D, una cum {3/80} quadrati B C. </s> <s xml:id="echoid-s3802" xml:space="preserve">Ac proinde, poſita <lb/>ſuſpenſione ex vertice A, fiet E S, qua centrum agitationis <lb/>inferius eſt centro gravitatis, æqualis {1/20} A D, & </s> <s xml:id="echoid-s3803" xml:space="preserve">{1/20} tertiæ <lb/>proportionalis duabus A D, B C. </s> <s xml:id="echoid-s3804" xml:space="preserve">ſive tota A S æqualis {4/5} <lb/>A D, una cum {1/5} tertiæ proportionalis duabus A D, D B. </s> <s xml:id="echoid-s3805" xml:space="preserve"><lb/>Atque hinc manifeſtum eſt, ſi A D, D B æquales ſint, hoc <lb/>eſt, ſi conus A B C ſit rectangulus, fieri A S æqualem axi <lb/>A D.</s> <s xml:id="echoid-s3806" xml:space="preserve"/> </p> <div xml:id="echoid-div348" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-0241-03" xlink:href="note-0241-03a" xml:space="preserve">Prop. 15. <lb/>huj.</note> </div> <p> <s xml:id="echoid-s3807" xml:space="preserve">Sequitur quoque porro, ex propoſitione 20, conum hunc <lb/>rectangulum, ſi ex D centro baſeos ſuſpendatur, iſochro-<lb/>num fore ſibi ipſi ex vertice A ſuſpenſo, quemadmodum & </s> <s xml:id="echoid-s3808" xml:space="preserve"><lb/>de triangulo rectangulo ſupra oſtenſum fuit.</s> <s xml:id="echoid-s3809" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div350" type="section" level="1" n="124"> <head xml:id="echoid-head150" style="it" xml:space="preserve">Centrum oſcillationis Sphæræ.</head> <p> <s xml:id="echoid-s3810" xml:space="preserve">Si A B C ſit ſphæra, erit figura plana proportionalis, à <lb/> <anchor type="note" xlink:label="note-0241-04a" xlink:href="note-0241-04"/> latere adponenda, O V H, ex parabolis compoſita, qua-<lb/>rum baſis communis O H, æqualis ſphæræ diametro A D. <lb/></s> <s xml:id="echoid-s3811" xml:space="preserve">Sectâ vero ſphærâ planis per centrum E, quorum B C ſit <pb o="168" file="0242" n="267" rhead="CHRISTIANI HUGENII"/> horizonti parallelum, A D vero verticale: </s> <s xml:id="echoid-s3812" xml:space="preserve">ut inveniatur <lb/> <anchor type="note" xlink:label="note-0242-01a" xlink:href="note-0242-01"/> ſumma quadratorum à diſtantiis à plano A D, noſcenda eſt <lb/>diſtantia centri gr. </s> <s xml:id="echoid-s3813" xml:space="preserve">parabolæ O V H ab O H, quæ ſit Φ P, <lb/>eſtque {2/5} V P. </s> <s xml:id="echoid-s3814" xml:space="preserve">Deinde, diviſâ P V bifariam in Δ, conſtat <lb/>rectangulum Δ Ρ Φ, multiplex per numerum particularum <lb/>ſphæræ A B C, æquari quadratis diſtantiarum à plano A D <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s3815" xml:space="preserve"> <anchor type="note" xlink:label="note-0242-02a" xlink:href="note-0242-02"/> Eſt autem rectangulum Δ Ρ Φ æquale {1/5} quadrati P V, vel <lb/>quadrati B E.</s> <s xml:id="echoid-s3816" xml:space="preserve"/> </p> <div xml:id="echoid-div350" type="float" level="2" n="1"> <note position="right" xlink:label="note-0241-04" xlink:href="note-0241-04a" xml:space="preserve">TAB. XXVI. <lb/>Fig. 2.</note> <note position="left" xlink:label="note-0242-01" xlink:href="note-0242-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="left" xlink:label="note-0242-02" xlink:href="note-0242-02a" xml:space="preserve">Prop. 15. <lb/>@n fine.</note> </div> <p> <s xml:id="echoid-s3817" xml:space="preserve">Atqui, quadrata diſtantiarum à plano B C, æqualia eſſe <lb/>liquet quadratis diſtantiarum à plano A D, ac proinde ei-<lb/>dem rectangulo Δ Ρ Φ, multiplici per dictum particularum <lb/>numerum. </s> <s xml:id="echoid-s3818" xml:space="preserve">Ergo ſpatium applicandum, in ſphæra A B C, <lb/>erit duplum rectanguli Δ Ρ Φ; </s> <s xml:id="echoid-s3819" xml:space="preserve">ideoque æquale {2/5} quadrati à <lb/>radio E B.</s> <s xml:id="echoid-s3820" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3821" xml:space="preserve">Itaque, ſi ſphæra ſuſpenſa ſit ex puncto in ſuperficie ſua <lb/>A, erit E S, à centro ſphæræ E ad centrum agitationis S, <lb/>æqualis {2/5} ſemidiametri A E. </s> <s xml:id="echoid-s3822" xml:space="preserve">Totaque A S æqualis {7/10} dia-<lb/>metri A D. </s> <s xml:id="echoid-s3823" xml:space="preserve">Si vero ex puncto alio, ut L, ſphæra ſuſpenſa <lb/>ſit; </s> <s xml:id="echoid-s3824" xml:space="preserve">erit E S æqualis {2/5} tertiæ proportionalis duabus L E, E B.</s> <s xml:id="echoid-s3825" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div352" type="section" level="1" n="125"> <head xml:id="echoid-head151" style="it" xml:space="preserve">Centrum oſcillationis Cylindri.</head> <p> <s xml:id="echoid-s3826" xml:space="preserve">In cylindro, invenimus ſpatium applicandum æquari {@/12} <lb/>quadrati altitudinis, una cum {1/4} quadrati à ſemidiametro ba-<lb/>ſis. </s> <s xml:id="echoid-s3827" xml:space="preserve">Unde, ſi cylindrus à centro baſis ſuperioris ſuſpendatur, <lb/>fit longitudo penduli iſochroni æqualis {2/3} altitudinis, una cum <lb/>ſemiſſe ejus, quæ ſit ad ſemidiametrum baſis ut hæc ad alti-<lb/>tudinem.</s> <s xml:id="echoid-s3828" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div353" type="section" level="1" n="126"> <head xml:id="echoid-head152" style="it" xml:space="preserve">Centrum oſcillationis Conoidis Parabolici.</head> <p> <s xml:id="echoid-s3829" xml:space="preserve">In conoide parabolico, rectangulum oſcillationis eſt {@/18} <lb/>quadrati altitudinis, cum {1/6} quadrati à ſemidiametro baſis. <lb/></s> <s xml:id="echoid-s3830" xml:space="preserve">Unde, ſi à puncto verticis fuerit ſuſpenſum, fit longitudo <lb/>penduli iſochroni {3/4} axis, cum {1/4} ejus quæ ſit ad ſemidiame-<lb/>trum baſis, ſicut hæc ad axem, id eſt, una cum {1/4} lateris re-<lb/>cti parabolæ genitricis.</s> <s xml:id="echoid-s3831" xml:space="preserve"/> </p> <pb o="169" file="0243" n="268" rhead="HOROLOG. OSCILLATOR."/> <note position="right" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <div xml:id="echoid-div354" type="section" level="1" n="127"> <head xml:id="echoid-head153" style="it" xml:space="preserve">Centrum oſcillationis Conoidis Hyperbolici.</head> <p> <s xml:id="echoid-s3832" xml:space="preserve">In conoide quoque hyperbolico centrum oſcillationis inve-<lb/> <anchor type="note" xlink:label="note-0243-02a" xlink:href="note-0243-02"/> niri poteſt. </s> <s xml:id="echoid-s3833" xml:space="preserve">Si enim, exempli gratia, ſit conoides cujus ſe-<lb/>ctio per axem, hyperbola B A B; </s> <s xml:id="echoid-s3834" xml:space="preserve">axem habens A D, la-<lb/>tus tranſverſum A F: </s> <s xml:id="echoid-s3835" xml:space="preserve">erit figura plana ipſi proportionalis <lb/>B K A K B, contenta baſi B B, & </s> <s xml:id="echoid-s3836" xml:space="preserve">parabolicæ lineæ por-<lb/>tionibus ſimilibus A K B, quæ parabolæ per verticem A <lb/>tranſeunt, axemque habent G E, dividentem bifariam latus <lb/>tranſverſum A F, ac parallelum baſi B B. </s> <s xml:id="echoid-s3837" xml:space="preserve">Et hujus quidem <lb/>figuræ B K A K B, centrum gravitatis L, tantum diſtat à <lb/>vertice A, quantum centrum gravitatis conoidis A B B; </s> <s xml:id="echoid-s3838" xml:space="preserve">eſt-<lb/>que axis A D ad A L, ſicut tripla F A cum dupla A D, <lb/>ad duplam F A cum ſesquialtera A D. </s> <s xml:id="echoid-s3839" xml:space="preserve">Deinde & </s> <s xml:id="echoid-s3840" xml:space="preserve">diſtantia <lb/>centri gr. </s> <s xml:id="echoid-s3841" xml:space="preserve">figuræ dimidiæ A D B K, ab A D, inveniri po-<lb/>teſt, atque etiam ſubcentrica cunei ſuper figura B K A K B, <lb/>abſciſſi plano per A P, parallelam B B; </s> <s xml:id="echoid-s3842" xml:space="preserve">hujus inquam cu-<lb/>nei ſubcentrica, ſuper ipſa A P, inveniri quoque poteſt; <lb/></s> <s xml:id="echoid-s3843" xml:space="preserve">atque ex his conſequenter centrum agitationis conoidis, in <lb/>quavis ſuſpenſione; </s> <s xml:id="echoid-s3844" xml:space="preserve">dummodo axis, circa quem movetur, <lb/>ſit baſi conoidis parallelus. </s> <s xml:id="echoid-s3845" xml:space="preserve">Atque invenio quidem, ſi axis <lb/>A D lateri tranſverſo A F æqualis ponatur, ſpatium appli-<lb/>candum æquari {1/20} quadrati A D, cum {31/200<unsure/>} quadrati D B. </s> <s xml:id="echoid-s3846" xml:space="preserve"><lb/>Tunc autem A L eſt {7/10} A D.</s> <s xml:id="echoid-s3847" xml:space="preserve"/> </p> <div xml:id="echoid-div354" type="float" level="2" n="1"> <note position="right" xlink:label="note-0243-02" xlink:href="note-0243-02a" xml:space="preserve">TAB. XXVI. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s3848" xml:space="preserve">Unde, ſi conoides hujuſmodi ex vertice A ſuſpendatur, <lb/>invenitur longitudo penduli iſochroni, A S, æqualis {2/3}{7/5} A D, <lb/>cum {31/140} tertiæ proportionalis duabus A D, D B.</s> <s xml:id="echoid-s3849" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div356" type="section" level="1" n="128"> <head xml:id="echoid-head154" style="it" xml:space="preserve">Centrum oſcillationis dimidii Coni.</head> <p> <s xml:id="echoid-s3850" xml:space="preserve">Denique & </s> <s xml:id="echoid-s3851" xml:space="preserve">in ſolidis dimidiatis quibuſdam, quæ fiunt <lb/> <anchor type="note" xlink:label="note-0243-03a" xlink:href="note-0243-03"/> ſectione per axem, centrum agitationis invenire licebit. </s> <s xml:id="echoid-s3852" xml:space="preserve">Ut <lb/>ſi ſit conus dimidiatus A B C, verticem habens A, diame-<lb/>trum ſemicirculi baſeos B C: </s> <s xml:id="echoid-s3853" xml:space="preserve">ejus quidem centrum gravita-<lb/>tis D notum eſt, quoniam A D eſt {3/4} rectæ A E, ita divi-<lb/>dentis B C in E, ut, ſicut quadrans circumferentiæ circuli <pb o="170" file="0244" n="269" rhead="CHRISTIANI HUGENII"/> ad radium, ita ſint {2/3} C B ad B E. </s> <s xml:id="echoid-s3854" xml:space="preserve">Tunc enim E eſt cen-<lb/> <anchor type="note" xlink:label="note-0244-01a" xlink:href="note-0244-01"/> trum gravitatis ſemicirculi baſeos, ideoque in A E centra <lb/>gravitatis omnium ſegmentorum ſemiconi A B D, baſi pa-<lb/>rallelorum.</s> <s xml:id="echoid-s3855" xml:space="preserve"/> </p> <div xml:id="echoid-div356" type="float" level="2" n="1"> <note position="right" xlink:label="note-0243-03" xlink:href="note-0243-03a" xml:space="preserve">TAB. XXVII. <lb/>Fig. 2.</note> <note position="left" xlink:label="note-0244-01" xlink:href="note-0244-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3856" xml:space="preserve">Et figura quidem porro proportionalis à latere ponenda, <lb/>O V V, eadem eſt quæ in cono toto ſupra deſcripta fuit: <lb/></s> <s xml:id="echoid-s3857" xml:space="preserve">per quam nempe invenietur ſumma quadratorum, à diſtan-<lb/>tiis particularum ſemiconi à plano horizontali N D, per <lb/>centrum gravitatis ducto. </s> <s xml:id="echoid-s3858" xml:space="preserve">Verum quadrata diſtantiarum, à <lb/>plano verticali M D O, ut colligantur, altera quoque figu-<lb/>ra proportionalis S Y Z, ſicut ſupra prop. </s> <s xml:id="echoid-s3859" xml:space="preserve">14. </s> <s xml:id="echoid-s3860" xml:space="preserve">adhibenda <lb/>eſt, cujus nempe ſectiones verticales, exhibeant lineas pro-<lb/>portionales ſectionibus ſibi reſpondentibus in ſemicono A B C. </s> <s xml:id="echoid-s3861" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s3862" xml:space="preserve">hujus figuræ cognoſcenda eſt diſtantia centri gr. </s> <s xml:id="echoid-s3863" xml:space="preserve">F ab S Y, <lb/>quam æqualem eſſe conſtat diſtantiæ D N, centri gr. </s> <s xml:id="echoid-s3864" xml:space="preserve">ſemiconi <lb/>à plano trianguli A B. </s> <s xml:id="echoid-s3865" xml:space="preserve">poſitâque H G ſubcentricâ cunei ab-<lb/>ſciſſi ſuper figura S Z Y, ducto plano per S Y, noſcendum <lb/>eſt rectangulum G F H, cujus nempe multiplex, ſecundum <lb/>numerum particularum ſemiconi A B C, æquabitur quadra-<lb/>tis diſtantiarum ſemiconi in planum M D O. </s> <s xml:id="echoid-s3866" xml:space="preserve">Licebit vero <lb/>cognoſcere rectangulum illud G F H, etiamſi ſubcentricæ <lb/>H G longitudo ignoretur, hoc modo.</s> <s xml:id="echoid-s3867" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3868" xml:space="preserve">Diximus ſupra, cum de cono ageremus, quadrata diſtan-<lb/>tiarum à plano per axem ejus, æquari {3/80} quadrati à diametro <lb/>baſis, ſive {3/20} quadrati à ſemidiametro, multiplicis per nu-<lb/>merum particularum coni totius. </s> <s xml:id="echoid-s3869" xml:space="preserve">Unde & </s> <s xml:id="echoid-s3870" xml:space="preserve">hic, in ſemicono <lb/>A B C, quadrata diſtantiarum à plano A B æqualia erunt <lb/>{3/20} quadrati B C, multiplicis per numerum particularum i-<lb/>pſius ſemiconi. </s> <s xml:id="echoid-s3871" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3872" xml:space="preserve">rectangulum H G F, multiplex per nu-<lb/>merum particularum ſemiconi A B C, æquatur quadratis <lb/>diſtantiarum à plano A B, ut patet ex propoſitione 9. </s> <s xml:id="echoid-s3873" xml:space="preserve">Ergo <lb/>rectangulum H G F æquale {3/20} quadrati B C. </s> <s xml:id="echoid-s3874" xml:space="preserve">Ponendo autem <lb/>A B = a; </s> <s xml:id="echoid-s3875" xml:space="preserve">B C = b; </s> <s xml:id="echoid-s3876" xml:space="preserve">& </s> <s xml:id="echoid-s3877" xml:space="preserve">quadrantem circumferentiæ, radio <lb/>B C deſcriptæ, = q; </s> <s xml:id="echoid-s3878" xml:space="preserve">fit E B = {2 b b/3 q}. </s> <s xml:id="echoid-s3879" xml:space="preserve">Cujus cum N D <lb/>tribus quartis æquetur, fiet proinde N D, ſive G F = {1 b b/2 q @}.</s> <s xml:id="echoid-s3880" xml:space="preserve"> <pb file="0245" n="270"/> <pb file="0245a" n="271"/> <anchor type="figure" xlink:label="fig-0245a-01a" xlink:href="fig-0245a-01"/> <anchor type="figure" xlink:label="fig-0245a-02a" xlink:href="fig-0245a-02"/> <anchor type="figure" xlink:label="fig-0245a-03a" xlink:href="fig-0245a-03"/> <pb file="0246" n="272"/> <pb o="171" file="0247" n="273" rhead="HOROLOG. OSCILLATOR."/> Cujus quadratum auferendo à rectangulo H G F, quod erat <lb/> <anchor type="note" xlink:label="note-0247-01a" xlink:href="note-0247-01"/> {3/20} quadrati B C, fiet rectangulum G F H = {3/20} b b - {1b4/4 q q}. </s> <s xml:id="echoid-s3881" xml:space="preserve">Hoc <lb/>autem rectangulum, multiplex per numerum particularum <lb/>ſemiconi A B C, æquatur quadratis diſtantiarum à plano <lb/>M D O. </s> <s xml:id="echoid-s3882" xml:space="preserve">At quadratis diſtantiarum à plano N D æquantur, <lb/>ut in cono, {3/80} quadrati A B, ſive {3/80} a a, multiplices per <lb/>numerum particularum ſemiconi A B C. </s> <s xml:id="echoid-s3883" xml:space="preserve">Itaque, totum ſpa-<lb/>tium applicandum, æquabitur hic {3/80} a a + {3/20} b b - {1 b 4/4 q q}.</s> <s xml:id="echoid-s3884" xml:space="preserve"/> </p> <div xml:id="echoid-div357" type="float" level="2" n="2"> <figure xlink:label="fig-0245a-01" xlink:href="fig-0245a-01a"> <caption xml:id="echoid-caption105" style="it" xml:space="preserve">Pag. 170.<lb/>TAB. XXVI.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables107" xml:space="preserve">Ω O Ω A Z R F R N E N R G S V P Φ Δ V B D K C</variables> </figure> <figure xlink:label="fig-0245a-02" xlink:href="fig-0245a-02a"> <caption xml:id="echoid-caption106" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables108" xml:space="preserve">L O A V P Φ Δ V B E C S H D</variables> </figure> <figure xlink:label="fig-0245a-03" xlink:href="fig-0245a-03a"> <caption xml:id="echoid-caption107" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables109" xml:space="preserve">F G E G P A P K K L B D B S</variables> </figure> <note position="right" xlink:label="note-0247-01" xlink:href="note-0247-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3885" xml:space="preserve">Unde quidem centrum agitationis invenitur in omni ſuſpen-<lb/>ſione ſemiconi, dummodo ab axe qui ſit parallelus baſi trianguli <lb/>à ſectione A B. </s> <s xml:id="echoid-s3886" xml:space="preserve">Notandum vero, cum figura S Z Y ſit ignotæ <lb/>prorſus naturæ, ſubcentricam tamen G H, cunei ſuper ipſa ab-<lb/>ſciſſi plano per S Y, hinc inveniri. </s> <s xml:id="echoid-s3887" xml:space="preserve">Nam, quia rectangulum H G F <lb/>æquale erat {3/20} b b, ſive quadrati B C, & </s> <s xml:id="echoid-s3888" xml:space="preserve">G F æqualis {1b b/2 q}, <lb/>fit inde G H æqualis {3/10} q.</s> <s xml:id="echoid-s3889" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3890" xml:space="preserve">Porro, etiam ſemicylindri, & </s> <s xml:id="echoid-s3891" xml:space="preserve">ſemiconoidis parabolici, <lb/>centra agitationis inveniri poſſunt, atque aliorum inſuper ſe-<lb/>miſolidorum; </s> <s xml:id="echoid-s3892" xml:space="preserve">quæ aliis inveſtiganda relinquimus.</s> <s xml:id="echoid-s3893" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3894" xml:space="preserve">Quemadmodum autem in figuris planis, ita & </s> <s xml:id="echoid-s3895" xml:space="preserve">hic in ſo-<lb/>lidis figuris locum habet, quod de obliquarum centris agi-<lb/>tationis illic diximus, quæ veluti luxatione rectarum conſti-<lb/>tuuntur, quarum centra oſcillationis non differunt à centris <lb/>oſcillationis rectarum. </s> <s xml:id="echoid-s3896" xml:space="preserve">Sic, ſi coni duo fuerunt A B C, A F G, <lb/> <anchor type="note" xlink:label="note-0247-02a" xlink:href="note-0247-02"/> alter rectus, alter ſcalenus; </s> <s xml:id="echoid-s3897" xml:space="preserve">quorum & </s> <s xml:id="echoid-s3898" xml:space="preserve">diametri & </s> <s xml:id="echoid-s3899" xml:space="preserve">baſes <lb/>æquales; </s> <s xml:id="echoid-s3900" xml:space="preserve">hi ex vertice ſuſpenſi, vel à quibuſcunque axibus, <lb/>æqualiter à centris eorum gravitatis diſtantibus, iſochroni <lb/>erunt; </s> <s xml:id="echoid-s3901" xml:space="preserve">dummodo axis, unde conus ſcalenus ſuſpenſus eſt, <lb/>rectus ſit ad planum trianguli per diametrum, quod planum <lb/>baſi eſt ad angulos rectos.</s> <s xml:id="echoid-s3902" xml:space="preserve"/> </p> <div xml:id="echoid-div358" type="float" level="2" n="3"> <note position="right" xlink:label="note-0247-02" xlink:href="note-0247-02a" xml:space="preserve">TAB. XXVII. <lb/>Fig. 1.</note> </div> <pb o="172" file="0248" n="274" rhead="CHRISTIANI HUGENII"/> <note position="left" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <div xml:id="echoid-div360" type="section" level="1" n="129"> <head xml:id="echoid-head155" xml:space="preserve">PROPOSITIO XXIII.</head> <p style="it"> <s xml:id="echoid-s3903" xml:space="preserve">HOrologiorum motum temperare, addito ponde-<lb/>re exiguo ſecundario, quod ſuper virga pen-<lb/>duli, certa ratione diviſa, ſurſum deorſumque <lb/>moveri poſſit.</s> <s xml:id="echoid-s3904" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3905" xml:space="preserve">Ut hoc expediamus, primo penduli ipſius, ex virga gra-<lb/> <anchor type="note" xlink:label="note-0248-02a" xlink:href="note-0248-02"/> vitate prædita, & </s> <s xml:id="echoid-s3906" xml:space="preserve">appenſo parte ima pondere, compoſiti, <lb/>centrum oſcillationis inveniendum eſt.</s> <s xml:id="echoid-s3907" xml:space="preserve"/> </p> <div xml:id="echoid-div360" type="float" level="2" n="1"> <note position="left" xlink:label="note-0248-02" xlink:href="note-0248-02a" xml:space="preserve">TAB. XXVII. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s3908" xml:space="preserve">Sit virga, cum appenſo pondere, A C, cujus longitudo <lb/>dicatur a. </s> <s xml:id="echoid-s3909" xml:space="preserve">Intelligantur autem, tum virga ipſa, tum pondus <lb/>appenſum C, in particulas minimas æquales diviſa, earum-<lb/>que particularum virga habeat numerum b, pondus vero C <lb/>numerum c, ponendo nempe b ad c, ſicut gravitas virgæ ad <lb/>gravitatem appenſi ponderis. </s> <s xml:id="echoid-s3910" xml:space="preserve">Longitudo igitur penduli ſim-<lb/>plicis, dato iſochroni, habebitur, ſi ſumma quadratorum à <lb/>diſtantiis particularum omnium à puncto ſuſpenſionis A, di-<lb/> <anchor type="note" xlink:label="note-0248-03a" xlink:href="note-0248-03"/> vidatur per ſummam earundem diſtantiarum <anchor type="note" xlink:href="" symbol="*"/>. </s> <s xml:id="echoid-s3911" xml:space="preserve">Secetur A C bifariam in M; </s> <s xml:id="echoid-s3912" xml:space="preserve">tum vero in T, ut A T ſit dupla T C. </s> <s xml:id="echoid-s3913" xml:space="preserve">Quia <lb/>ergo M eſt centrum gravitatis lineæ A C, & </s> <s xml:id="echoid-s3914" xml:space="preserve">A T ſubcen-<lb/>trica cunei ſuper ipſa abſciſſi plano per A D, perpendicu-<lb/>larem ad A C; </s> <s xml:id="echoid-s3915" xml:space="preserve">qui cuneus hîc revera triangulum eſt; </s> <s xml:id="echoid-s3916" xml:space="preserve">erit ſum-<lb/>ma quadratorum, à diſtantiis particularum virgæ à puncto <lb/>A, æqualis rectangulo A M T, una cum quadrato A M; <lb/></s> <s xml:id="echoid-s3917" xml:space="preserve">hoc eſt, rectangulo T A M, multiplici ſecundum numerum <lb/>particularum b; </s> <s xml:id="echoid-s3918" xml:space="preserve">hoc eſt, {1/3} a a b; </s> <s xml:id="echoid-s3919" xml:space="preserve">quia M A eſt {1/2} a, & </s> <s xml:id="echoid-s3920" xml:space="preserve">T A <lb/>{2/3} a, ac proinde rectangulum T A M = {1/3} a a. </s> <s xml:id="echoid-s3921" xml:space="preserve">Summa vero <lb/>quadratorum, à diſtantiis particularum ponderis C ab eo-<lb/>dem puncto A, æquabitur quadrato A C, multiplici ſecun-<lb/>dum numerum particularum ipſius ponderis; </s> <s xml:id="echoid-s3922" xml:space="preserve">hoc eſt, a a c. </s> <s xml:id="echoid-s3923" xml:space="preserve"><lb/>Adeoque ſumma quadratorum omnium, tam à diſtantiis par-<lb/>ticularum virgæ, quam ponderis C, erit {1/3} a a b + a a c.</s> <s xml:id="echoid-s3924" xml:space="preserve"/> </p> <div xml:id="echoid-div361" type="float" level="2" n="2"> <note symbol="*" position="left" xlink:label="note-0248-03" xlink:href="note-0248-03a" xml:space="preserve">Prop. 6. <lb/>huj. in fine.</note> </div> <p> <s xml:id="echoid-s3925" xml:space="preserve">Porro, diſtantiæ omnes particularum virgæ A C à pun-<lb/>cto A, æquantur {1/2} b a; </s> <s xml:id="echoid-s3926" xml:space="preserve">longitudini ſcilicet virgæ ipſius, quæ <pb o="173" file="0249" n="275" rhead="HOROLOG. OSCILLATOR."/> eſt a, multiplici ſecundum ſemiſſem numeri particularum <lb/> <anchor type="note" xlink:label="note-0249-01a" xlink:href="note-0249-01"/> quas continet. </s> <s xml:id="echoid-s3927" xml:space="preserve">Et diſtantiæ omnes particularum ponderis C, <lb/>ab eodem puncto A, ſunt a c. </s> <s xml:id="echoid-s3928" xml:space="preserve">Ita ut ſumma utrarumque di-<lb/>ſtantiarum ſit {1/2} a b + a c. </s> <s xml:id="echoid-s3929" xml:space="preserve">Per quam dividendo ſummam <lb/>quadratorum prius inventam, {1/3} a a b + a a c, fit <lb/>{{1/3} a a b + a a c/{1/2} a b + a c} ſive {{1/3} a b + a c/{1/2} b + c}, longitudo penduli iſochroni. </s> <s xml:id="echoid-s3930" xml:space="preserve">Quæ <lb/>itaque habebitur, ſi fiat, ut dimidia gravitas virgæ, una <lb/>cum gravitate appenſi ponderis, ad trientem gravitatis virgæ, <lb/>una cum gravitate ejuſdem appenſi ponderis, ita longitudo <lb/>A C ad aliam. </s> <s xml:id="echoid-s3931" xml:space="preserve">Oportet autem ſumere longitudinem A C, <lb/>à puncto ſuſpenſionis A ad centrum gravitatis ponderis C; <lb/></s> <s xml:id="echoid-s3932" xml:space="preserve">cum magnitudinis ejus ratio hic non habeatur, ac veluti <lb/>minimum conſideretur.</s> <s xml:id="echoid-s3933" xml:space="preserve"/> </p> <div xml:id="echoid-div362" type="float" level="2" n="3"> <note position="right" xlink:label="note-0249-01" xlink:href="note-0249-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3934" xml:space="preserve">Quod ſi jam, præter pondus C, alterum inſuper D virgæ <lb/> <anchor type="note" xlink:label="note-0249-02a" xlink:href="note-0249-02"/> inhærere intelligatur, cujus gravitas, ſeu particularum nume-<lb/>rus ſit d: </s> <s xml:id="echoid-s3935" xml:space="preserve">diſtantia vero A D ſit f. </s> <s xml:id="echoid-s3936" xml:space="preserve">Ut pendulum ſimplex <lb/>huic ita compoſito iſochronum inveniatur, addenda ſunt ad <lb/>ſummam ſuperiorem quadratorum, quadrata diſtantiarum <lb/>particularum ponderis D à puncto A, quæ quadrata apparet <lb/>eſſe d f f. </s> <s xml:id="echoid-s3937" xml:space="preserve">Adeo ut ſumma omnium jam ſit futura {1/3} a a b + <lb/>a a c + f f d. </s> <s xml:id="echoid-s3938" xml:space="preserve">Item, ad ſummam diſtantiarum, addendæ <lb/>diſtantiæ particularum ponderis D, quæ faciunt d f. </s> <s xml:id="echoid-s3939" xml:space="preserve">Ac ſum-<lb/>ma proinde diſtantiarum omnium erit {1/2} b a + c a + d f; <lb/></s> <s xml:id="echoid-s3940" xml:space="preserve">per quam dividenda eſt iſta quadratorum ſumma, & </s> <s xml:id="echoid-s3941" xml:space="preserve">fit <lb/>{{1/3} a a b + a a c + f f d/{1/2} a b + a c + f d}, longitudo penduli iſochroni.</s> <s xml:id="echoid-s3942" xml:space="preserve"/> </p> <div xml:id="echoid-div363" type="float" level="2" n="4"> <note position="right" xlink:label="note-0249-02" xlink:href="note-0249-02a" xml:space="preserve">TAB. XXVII@ <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s3943" xml:space="preserve">Quod ſi vero, hæc longitudo penduli iſochroni, datæ æqualis <lb/>poſtuletur, quæ ſit p, & </s> <s xml:id="echoid-s3944" xml:space="preserve">reliqua omnia quæ prius data ſint, <lb/>præter diſtantiam A D ſeu f, quæ determinat locum pon-<lb/>deris D: </s> <s xml:id="echoid-s3945" xml:space="preserve">ſitque invenienda hæc diſtantia, id fiet hoc modo. <lb/></s> <s xml:id="echoid-s3946" xml:space="preserve">Nempe, cum poſtuletur {{1/3} a a b + a a c + f f d/{1/2} a b + a c + f d} æquale p, orietur ex <lb/>hac æquatione f f = p f + {{1/2} a b p + c a p - {1/3} a a b - a a c/d}. </s> <s xml:id="echoid-s3947" xml:space="preserve">Et f = {1/2} p <pb o="174" file="0250" n="276" rhead="CHRISTIANI HUGENII"/> <anchor type="note" xlink:label="note-0250-01a" xlink:href="note-0250-01"/> + vel - <emph style="red">{1/4} p p + {1/2} a b p + a a p - {1/3} a a b - a a c/d}</emph>. </s> <s xml:id="echoid-s3948" xml:space="preserve">Ubi animadverten-<lb/>dum, duas eſſe veras radices, ſi {1/2} a b p + c a p minus ſit <lb/>quam {1/3} a a b + a a c; </s> <s xml:id="echoid-s3949" xml:space="preserve">hoc eſt, ſi longitudo p minor ſit quam <lb/>{{1/3} a b + a c/{1/2} b + c}, quæ antea inventa fuit longitudo penduli iſochro-<lb/>ni, ſive diſtantia centri oſcillationis à ſuſpenſione, in pen-<lb/>dulo compoſito ex virga A C & </s> <s xml:id="echoid-s3950" xml:space="preserve">pondere C.</s> <s xml:id="echoid-s3951" xml:space="preserve"/> </p> <div xml:id="echoid-div364" type="float" level="2" n="5"> <note position="left" xlink:label="note-0250-01" xlink:href="note-0250-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3952" xml:space="preserve">Unde patet, ſi velimus efficere, ut, applicato pondere D, <lb/>acceleretur penduli motus; </s> <s xml:id="echoid-s3953" xml:space="preserve">poſſe duobus locis, inter A & </s> <s xml:id="echoid-s3954" xml:space="preserve">C, <lb/>illud diſponi, quorum utrolibet eadem celeritas pendulo <lb/>concilietur: </s> <s xml:id="echoid-s3955" xml:space="preserve">velut in D vel E. </s> <s xml:id="echoid-s3956" xml:space="preserve">Quæ loca æqualiter diſtabunt <lb/>à puncto N, quod abeſt ab A, ſemiſſe longitudinis p, hoc <lb/>eſt, ſemiſſe penduli ſimplicis, cui compoſitum hoc iſochro-<lb/>num poſtulabatur. </s> <s xml:id="echoid-s3957" xml:space="preserve">Apparet autem, quando hæc longitudo p <lb/>tantum exiguo minor ponitur quam A C, etiam punctum N <lb/>exiguo ſuperius eſſe puncto medio virgæ A C.</s> <s xml:id="echoid-s3958" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3959" xml:space="preserve">Porro, ex æquatione ſuperiori, f = {1/2} p + vel -<lb/><emph style="red">{1/4} p p + {1/2} a b p + a c p - {1/3} a a b - a a c/d}</emph> habetur determinatio longitudi-<lb/>nis p. </s> <s xml:id="echoid-s3960" xml:space="preserve">Patet enim, {1/4} p p + {1 a b p + a c p/2 d} non minus eſſe debere <lb/>quam {1 a a b - a a c/3 d}. </s> <s xml:id="echoid-s3961" xml:space="preserve">Unde non debebit eſſe minor quam <lb/>{a/d} <emph style="red">{4/3} b d + 4 c d + b b + 4 b c + 4 c c</emph> @ a b - 2 a c/d}. </s> <s xml:id="echoid-s3962" xml:space="preserve">Quod ſi p æquetur <lb/>huic quantitati, hoc eſt, ſi {1/4} p p + {1 a b p + a c p/2 d} fuerit æquale <lb/>{1 a a b + a a c/3 d}, erit jam, in eadem ſuperiori æquatione, f = {1/2} p, <lb/>hoc eſt, {a/2 d} <emph style="red">{4/3} b d + 4 c d + b b + 4 b c + 4 c c -</emph>{a b - 2 a c/2 d}. </s> <s xml:id="echoid-s3963" xml:space="preserve">Quo <lb/>determinatur diſtantia ponderis D à puncto A, ex qua ma-<lb/>xime omnium acceleret motum penduli.</s> <s xml:id="echoid-s3964" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3965" xml:space="preserve">Atque hæc ad horologiorum uſum ſic porro adhibentur. <lb/></s> <s xml:id="echoid-s3966" xml:space="preserve">Sit, exempli gratia, pendulum horologii, quod ſingulis <lb/>oſcillationibus ſcrupula ſecunda notet. </s> <s xml:id="echoid-s3967" xml:space="preserve">Virgæ autem gravitas <lb/>ſit 50<unsure/> gravitatis appenſi ponderis in imo pendulo: </s> <s xml:id="echoid-s3968" xml:space="preserve">&</s> <s xml:id="echoid-s3969" xml:space="preserve">, præ-<lb/>ter hoc, ſit aliud exiguum pondus mobile ſecundum virgæ <lb/>longitudinem, cujus gravitas eadem ponatur quæ ipſius vir- <pb o="175" file="0251" n="277" rhead="HOROLOG. OSCILLATOR."/> gæ. </s> <s xml:id="echoid-s3970" xml:space="preserve">Quæritur jam, quo loco hoc virgæ imponendum, ut <lb/> <anchor type="note" xlink:label="note-0251-01a" xlink:href="note-0251-01"/> uno ſcrupulo primo acceleretur horologii motus, ſpatio 24 <lb/>horarum. </s> <s xml:id="echoid-s3971" xml:space="preserve">Item, ubi collocandum, ut duorum ſcrupulorum <lb/>primorum ſit acceleratio; </s> <s xml:id="echoid-s3972" xml:space="preserve">item, ut trium, quatuor, atque <lb/>ita porro.</s> <s xml:id="echoid-s3973" xml:space="preserve"/> </p> <div xml:id="echoid-div365" type="float" level="2" n="6"> <note position="right" xlink:label="note-0251-01" xlink:href="note-0251-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s3974" xml:space="preserve">Ductis viginti quatuor horis ſexagies, fiunt 1440, quot <lb/>nempe ſcrupula prima una die continentur. </s> <s xml:id="echoid-s3975" xml:space="preserve">Ex his unum <lb/>aufer, quando unius ſcrupuli acceleratio quæritur: </s> <s xml:id="echoid-s3976" xml:space="preserve">ſuper-<lb/>ſunt 1439. </s> <s xml:id="echoid-s3977" xml:space="preserve">Ratio autem 1440 ad 1439 duplicata, proxime <lb/>eſt ea quæ 1440 ad 1438. </s> <s xml:id="echoid-s3978" xml:space="preserve">Ergo, ſi penduli ſimplicis, ſe-<lb/>cunda ſcrupula notantis, longitudo diviſa intelligatur in par-<lb/>tes æquales 1440, earumque 1438 alii pendulo tribuantur, <lb/>hoc præcedet alterum illud, in 24 horis, uno ſcrupulo pri-<lb/>mo. </s> <s xml:id="echoid-s3979" xml:space="preserve">Adeo ut hic p valeat partes 1438.</s> <s xml:id="echoid-s3980" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3981" xml:space="preserve">Quia autem pendulum horologii, ex virga metallica & </s> <s xml:id="echoid-s3982" xml:space="preserve"><lb/>pondere appenſo compoſitum, iſochronum ponitur pendulo <lb/>ſimplici partium 1440; </s> <s xml:id="echoid-s3983" xml:space="preserve">invenienda primum eſt virgæ illius <lb/>longitudo, ex æquatione ſuperius poſita. </s> <s xml:id="echoid-s3984" xml:space="preserve">Erat nempe <lb/>{{1/3} a b + a c/{1/2} b + c} æquale longitudini penduli ſimplicis, quod iſochro-<lb/>num compoſito ex virga habente longitudinem a, gravita-<lb/>tem b, & </s> <s xml:id="echoid-s3985" xml:space="preserve">pondere affixo cujus gravitas c. </s> <s xml:id="echoid-s3986" xml:space="preserve">Ergo ſi longitu-<lb/>do penduli ſimplicis iſochroni dicatur ſ. </s> <s xml:id="echoid-s3987" xml:space="preserve">Erit {{1/2} b ſ + c ſ/{1/3} b + c} = a. <lb/></s> <s xml:id="echoid-s3988" xml:space="preserve">poſitoque, ut hic, c = 50; </s> <s xml:id="echoid-s3989" xml:space="preserve">b = 1; </s> <s xml:id="echoid-s3990" xml:space="preserve">ſ = 1440; </s> <s xml:id="echoid-s3991" xml:space="preserve">fiet, a = 1444 {4/5}, <lb/>longitudo virgæ.</s> <s xml:id="echoid-s3992" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3993" xml:space="preserve">Jam, quia erat f = {1/2} p + vel @ <emph style="red">{1/4} p p + {1/2} a b p + a c p - {1/3} a a b - a a c/d</emph>, <lb/>fiet f = {1/2} p + vel - <emph style="red">{1/4} p p + 72962 p - 105061210</emph>. <lb/></s> <s xml:id="echoid-s3994" xml:space="preserve">Unde porro, ſi p ſit, uti diximus, partium 1438; </s> <s xml:id="echoid-s3995" xml:space="preserve">invenie-<lb/>tur f = 1331 {1/2}, qualium nempe ſ, ſeu pendulum ſimplex, <lb/>ſecunda ſcrupula oſcillationibus deſignans, continet 1440. </s> <s xml:id="echoid-s3996" xml:space="preserve">Cu-<lb/>jus longitudo ſi pedum trium ſtatuatur, quos horarios voca-<lb/>vimus, habebit f uncias 33, & </s> <s xml:id="echoid-s3997" xml:space="preserve">3 unciarum uncias, quas <lb/>lineas vocant. </s> <s xml:id="echoid-s3998" xml:space="preserve">Vel, auferendo hanc longitudinem f à tota <lb/>trium pedum longitudine, ſupererunt unciæ duæ, lineæ 9, <pb o="176" file="0252" n="278" rhead="CHRISTIANI HUGENII"/> à centro oſcillationis penduli compoſiti ſurſum ſumendæ, ut <lb/> <anchor type="note" xlink:label="note-0252-01a" xlink:href="note-0252-01"/> habeatur locus ponderis D, unius ſcrupuli primi accelera-<lb/>tionem præſtans tempore 24 horarum. </s> <s xml:id="echoid-s3999" xml:space="preserve">Eodem modo reli-<lb/>quas diſtantias, quibus virga dividenda eſt, calculo inveſti-<lb/>gavimus, aliam atque aliam ponendo longitudinem p: </s> <s xml:id="echoid-s4000" xml:space="preserve">eas-<lb/>que ſubjecta tabella exhibemus, ſecundum cujus numeros <lb/>etiam virga penduli diviſa eſt, quæ ſuperius in deſcriptione <lb/>horologii fuit exhibita. </s> <s xml:id="echoid-s4001" xml:space="preserve">Procedunt autem accelerationes diur-<lb/>næ, ut jam illic advertimus, per 15 ſcrupula ſecunda, ſeu <lb/>primorum ſcrupulorum quadrantes. </s> <s xml:id="echoid-s4002" xml:space="preserve">Ex. </s> <s xml:id="echoid-s4003" xml:space="preserve">gr. </s> <s xml:id="echoid-s4004" xml:space="preserve">ſi, pondere mo-<lb/>bili D hærente in parte 73, 4, inveniatur horologium tar-<lb/>dius juſto incedere, in 24 horis, differentiâ 15 ſecundorum <lb/>ſcrupulorum; </s> <s xml:id="echoid-s4005" xml:space="preserve">oportebit ſurſum adducere pondus D, usque <lb/>ad numerum 85, 6, ut corrigatur.</s> <s xml:id="echoid-s4006" xml:space="preserve"/> </p> <div xml:id="echoid-div366" type="float" level="2" n="7"> <note position="left" xlink:label="note-0252-01" xlink:href="note-0252-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <note position="right" xml:space="preserve"> <lb/>Acceleratio horolog@i \\ ſpatio 24 horarum. # Partes, à centro oſc. \\ ſurſum accipiendæ. <lb/>Scrup. pr. Sec. # Lineæ & decima linearum pedis horarii. <lb/>0, 15 # 7, 0 <lb/>0, 30 # 15, 2 <lb/>0, 45 # 23, 7 <lb/>1, 0 # 32, 6 <lb/>1, 15 # 41, 9 <lb/>1, 30 # 51, 7 <lb/>1, 45 # 62, 2 <lb/>2, 0 # 73, 4 <lb/>2, 15 # 85, 6 <lb/>2, 30 # 99, 0 <lb/>2, 45 # 114, 1 <lb/>3, 0 # 131, 8 <lb/>3, 15 # 154, 3 <lb/>3, 30 # 192, 6 <lb/></note> <p> <s xml:id="echoid-s4007" xml:space="preserve">Centrum oſcillationis altius eſt centro gravitatis C parti-<lb/>bus 1, 4.</s> <s xml:id="echoid-s4008" xml:space="preserve"/> </p> <pb o="177" file="0253" n="279" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div368" type="section" level="1" n="130"> <head xml:id="echoid-head156" xml:space="preserve">PROPOSITIO XXIV.</head> <note position="right" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS.</emph></note> <p style="it"> <s xml:id="echoid-s4009" xml:space="preserve">CEntri oſcillationis rationem haberi non poſſe, <lb/>in pendulis inter Cycloides ſuſpenſis; </s> <s xml:id="echoid-s4010" xml:space="preserve">& </s> <s xml:id="echoid-s4011" xml:space="preserve">quomo-<lb/>do hinc orta difficultas tollatur.</s> <s xml:id="echoid-s4012" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4013" xml:space="preserve">Si quis, ſubtili examine, contulerit ea quæ in ſuperiori-<lb/>bus, de pendulo inter cycloides ſuſpenſo, demonſtravimus, <lb/>cum his quæ ad centrum oſcillationis pertinent; </s> <s xml:id="echoid-s4014" xml:space="preserve">videbitur ei <lb/>deeſſe aliquid ad perfectam illam, quam præferimus, oſcil-<lb/>lationum æqualitatem. </s> <s xml:id="echoid-s4015" xml:space="preserve">Ac primo dubitabit, an, ad inveni-<lb/>endum circulum cycloidis genitorem, pendulilongitudo ac-<lb/>cipienda ſit à puncto ſuſpenſionis ad centrum gravitatis ap-<lb/>penſi plumbi, an vero ad centrum oſcillationis; </s> <s xml:id="echoid-s4016" xml:space="preserve">quod, abal-<lb/>tero illo, ſæpe ſenſibili intervallo diſtat, atque eo majore, <lb/>quo major fuerit ſphæra aut lens plumbea. </s> <s xml:id="echoid-s4017" xml:space="preserve">Quid enim, ſi <lb/>ſphæræ diameter quartam, aut tertiam partem, pendulilon-<lb/>gitudinis æquet? </s> <s xml:id="echoid-s4018" xml:space="preserve">Quod ſi ad centrum oſcillationis illam lon-<lb/>gitudinem accipiendam dicamus, non tamen expediet quo <lb/>pacto ea, quæ de centro oſcillationis oſtenſa ſunt, conve-<lb/>niant pendulo continue longitudinem ſuam immutanti, qua-<lb/>le illud quod inter cycloides movetur. </s> <s xml:id="echoid-s4019" xml:space="preserve">Poſſet enim videri, et-<lb/>iam centrum oſcillationis mutari, ad ſingulas diverſas longi-<lb/>tudines; </s> <s xml:id="echoid-s4020" xml:space="preserve">quod tamen hoc modo intelligendum non eſt. </s> <s xml:id="echoid-s4021" xml:space="preserve">Res <lb/>ſane explicatu difficillima, ſi omnimodam ἀκ{ρί}βει{αν} ſecte-<lb/>mur. </s> <s xml:id="echoid-s4022" xml:space="preserve">Nam in demonſtratione temporum æqualium in cycloi-<lb/>de, mobile, per eam delatum, veluti punctum gravitate præ-<lb/>ditum conſideravimus. </s> <s xml:id="echoid-s4023" xml:space="preserve">Sed, ſi ad effectum ſpectemus, non <lb/>magni facienda eſt difficultas hæc; </s> <s xml:id="echoid-s4024" xml:space="preserve">cum ponderis, quo pen-<lb/>dulum conſtat, magnitudo in horologiis tanta non requiratur <lb/>(etſi quo majus eo melius) ut differentia centrorum gravita-<lb/>tis, & </s> <s xml:id="echoid-s4025" xml:space="preserve">oſcillationis, aliquid hic turbare poſſit. </s> <s xml:id="echoid-s4026" xml:space="preserve">Quod ſi ta-<lb/>men effugere prorſus has tricas velimus, id ita conſequemur, <lb/>ſi ſphæram lentemve penduli, circa axem ſuum horizontalem, <lb/>mobilem efficiamus: </s> <s xml:id="echoid-s4027" xml:space="preserve">axis extrema utrinque, virgæ penduli <lb/>imæ, inſerendo: </s> <s xml:id="echoid-s4028" xml:space="preserve">quæ idcirco ut bifida hac parte ſit neceſſe <pb o="178" file="0254" n="280" rhead="CHRISTIANI HUGENII"/> eſt. </s> <s xml:id="echoid-s4029" xml:space="preserve">Fit enim hoc modo, ex motus natura, ut eandem per-<lb/> <anchor type="note" xlink:label="note-0254-01a" xlink:href="note-0254-01"/> petuo poſitionem, reſpectu horizontalis plani, ſphæra pen-<lb/>duli ſervet, atque ita puncta ejus quævis, æque ac cen-<lb/>trum ipſum, cycloides eaſdem percurrant. </s> <s xml:id="echoid-s4030" xml:space="preserve">Unde ceſſat hic <lb/>jam centrorum oſcillationis conſideratio; </s> <s xml:id="echoid-s4031" xml:space="preserve">nec minus perfectam <lb/>temporum æqualitatem tale pendulum conſequitur, quam ſi <lb/>puncto unico omnis ejus gravitas contineretur.</s> <s xml:id="echoid-s4032" xml:space="preserve"/> </p> <div xml:id="echoid-div368" type="float" level="2" n="1"> <note position="left" xlink:label="note-0254-01" xlink:href="note-0254-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> </div> <div xml:id="echoid-div370" type="section" level="1" n="131"> <head xml:id="echoid-head157" xml:space="preserve">PROPOSITIO XXV.</head> <p style="it"> <s xml:id="echoid-s4033" xml:space="preserve">DE menſuræ univerſalis, & </s> <s xml:id="echoid-s4034" xml:space="preserve">perpetuæ, conſti-<lb/>tuendæ ratione.</s> <s xml:id="echoid-s4035" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4036" xml:space="preserve">Certa, ac permanens magnitudinum menſura, quæ nullis <lb/>caſibus obnoxia ſit, nec temporum injuriis, aut longinquita-<lb/>te aboleri aut corrumpi poſſit, res eſt & </s> <s xml:id="echoid-s4037" xml:space="preserve">utiliſſima, & </s> <s xml:id="echoid-s4038" xml:space="preserve">à <lb/>multis pridem quæſita. </s> <s xml:id="echoid-s4039" xml:space="preserve">Quæ ſi priſcis temporibus reperta <lb/>fuiſſet, non tam perplexæ nunc forent, de pedis Romani, <lb/>Græci, Hebræique veteris modulo, diſceptationes. </s> <s xml:id="echoid-s4040" xml:space="preserve">Hæc ve-<lb/>ro menſura, Horologii noſtri opera, facile conſtituitur; </s> <s xml:id="echoid-s4041" xml:space="preserve">cum <lb/>ſine illo nequaquam, aut ægre admodum, haberi poſſit. </s> <s xml:id="echoid-s4042" xml:space="preserve">Etſi <lb/>enim, ſimplici pendulorum oſcillatione, hoc à quibuſdam <lb/>tentatum fuerit, numerando recurſus qui tota cæli conver-<lb/>ſione continentur, vel parte ejus cognita, per fixarum ſtel-<lb/>larum diſtantias, ſecundum aſcenſionem rectam; </s> <s xml:id="echoid-s4043" xml:space="preserve">nec certi-<lb/>tudo eadem hoc modo, quæ adhibitis horologiis, contingit, <lb/>& </s> <s xml:id="echoid-s4044" xml:space="preserve">labor longe eſt moleſtiſſimus ac tædioſiſſimus, propter <lb/>numerandi ſolicitudinem. </s> <s xml:id="echoid-s4045" xml:space="preserve">Quia autem, præter horologia, <lb/>aliquid, ad exactiſſimam hujus menſuræ inquiſitionem, <lb/>etiam centrorum oſcillationis notitia confert; </s> <s xml:id="echoid-s4046" xml:space="preserve">ideo hic de-<lb/>mum, poſt eorum tractationem, hanc determinationem ſub-<lb/>jicimus.</s> <s xml:id="echoid-s4047" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4048" xml:space="preserve">Aptiſſima huic rei ſunt horologia, quorum oſcillationes <lb/>ſingulæ ſecunda ſcrupula, vel eorum ſemiſſes, notant, quæ-<lb/>que indicibus etiam, ad ea demonſtranda, inſtructa ſunt. <lb/></s> <s xml:id="echoid-s4049" xml:space="preserve">Poſtquam enim, ad mediocrem dierum longitudinem, ejuſ- <pb o="179" file="0255" n="281" rhead="HOROLOG. OSCILLATOR."/> modi horologium, fixarum ſtellarum obſervationibus, com-<lb/> <anchor type="note" xlink:label="note-0255-01a" xlink:href="note-0255-01"/> poſitum fuerit, methodo illa quam in horologii deſcriptione <lb/>oſtendimus: </s> <s xml:id="echoid-s4050" xml:space="preserve">aliud pendulum ſimplex, hoc eſt, ſphæra plum-<lb/>bea, aut alia materia gravi conſtans, ex tenui filo religata, <lb/>juxta ſuſpendenda eſt, motuque exiguo impellenda; </s> <s xml:id="echoid-s4051" xml:space="preserve">ac tan-<lb/>tiſper producenda, aut contrahenda fili longitudo, donec <lb/>recurſus ejus, per quadrantem horæ, aut ſemiſſem, una feran-<lb/>tur cum reciprocationibus penduli horologio aptati. </s> <s xml:id="echoid-s4052" xml:space="preserve">Dixi <lb/>autem exiguo motu impellendum pendulum, quia oſcilla-<lb/>tiones exiguæ, puta 5 vel 6 partium, ſatis æqualia tempora <lb/>habent, magnæ vero non item. </s> <s xml:id="echoid-s4053" xml:space="preserve">Tunc, acceptâ menſurâ di-<lb/>ſtantiæ, à puncto ſuſpenſionis ad centrum oſcillationis pen-<lb/>duli ſimplicis; </s> <s xml:id="echoid-s4054" xml:space="preserve">eâque, ſi recurſus ſinguli ſcrupula ſecunda <lb/>valeant, in tres partes divisâ; </s> <s xml:id="echoid-s4055" xml:space="preserve">facient hæ ſingulæ longitudi-<lb/>nem pedis, quem <emph style="sc">Horarium</emph> in ſuperioribus vocavimus: <lb/></s> <s xml:id="echoid-s4056" xml:space="preserve">quique, hoc pacto, non ſolum ubique gentium conſtitui <lb/>poſſit, ſed & </s> <s xml:id="echoid-s4057" xml:space="preserve">venturo ævo redintegrari. </s> <s xml:id="echoid-s4058" xml:space="preserve">Adeo ut & </s> <s xml:id="echoid-s4059" xml:space="preserve">moduli <lb/>pedum omnium aliorum, ſemel ad hunc proportionibus ſuis <lb/>expreſſi, certò quoque in poſterum cognoſci poſſint. </s> <s xml:id="echoid-s4060" xml:space="preserve">Sicut <lb/>jam ſupra, pedem Pariſienſem ad hunc horarium eſſe diximus, <lb/>ut 864 ad 881; </s> <s xml:id="echoid-s4061" xml:space="preserve">quod idem eſt ac ſi, poſito prius pede Pa-<lb/>riſienſi, dicamus tribus hujuſmodi pedibus, cum octo lineis <lb/>& </s> <s xml:id="echoid-s4062" xml:space="preserve">dimidia, conſtitui pendulum ſimplex, cujus oſcillationes <lb/>ſcrupulis ſecundis horariis reſponſuræ ſint. </s> <s xml:id="echoid-s4063" xml:space="preserve">Pes autem Pari-<lb/>ſienſis ad R henanum, quo in Patria noſtra utuntur, ſe habet <lb/>ut 144 ad 139; </s> <s xml:id="echoid-s4064" xml:space="preserve">hoc eſt, quinque lineis ſuis diminutus, al-<lb/>terum illum relinquit. </s> <s xml:id="echoid-s4065" xml:space="preserve">Atque ita & </s> <s xml:id="echoid-s4066" xml:space="preserve">hic pes, & </s> <s xml:id="echoid-s4067" xml:space="preserve">alii quilibet, <lb/>perpetuo duraturas menſuras accipiunt.</s> <s xml:id="echoid-s4068" xml:space="preserve"/> </p> <div xml:id="echoid-div370" type="float" level="2" n="1"> <note position="right" xlink:label="note-0255-01" xlink:href="note-0255-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s4069" xml:space="preserve">Quomodo autem centrum oſcillationis in ſphæra, ex <lb/>qualibet longitudine ſuſpenſa, inveniatur, in ſuperioribus <lb/>demonſtratum eſt. </s> <s xml:id="echoid-s4070" xml:space="preserve">Nempe, ſi fiat ut diſtantia inter punctum <lb/>ſuſpenſionis & </s> <s xml:id="echoid-s4071" xml:space="preserve">ſphæræ centrum, ad ſemidiametrum ejus, <lb/>ita hæc ad aliam; </s> <s xml:id="echoid-s4072" xml:space="preserve">ejus duas quintas, à centro deorſum ac-<lb/>ceptas, terminari in quæſito oſcillationis centro. </s> <s xml:id="echoid-s4073" xml:space="preserve">Facile au-<lb/>tem apparet cur neceſſaria ſit hujus centri conſideratio, ad <lb/>accuratam pedis Horarii conſtitutionem. </s> <s xml:id="echoid-s4074" xml:space="preserve">Nam, ſi à pun- <pb o="180" file="0256" n="282" rhead="CHRISTIANI HUGENII"/> cto ſuſpenſionis ad ſphæræ centrum diſtantia accipiatur, <lb/> <anchor type="note" xlink:label="note-0256-01a" xlink:href="note-0256-01"/> ſphæræ autem magnitudo non definiatur proportione ad fili <lb/>longitudinem, non erit certa menſura penduli cujus recurſus <lb/>ſecunda ſcrupula metiantur; </s> <s xml:id="echoid-s4075" xml:space="preserve">ſed quo major erit ejus ſphæra, <lb/>hoc minor invenietur menſura illa, inter centrum ſphæræ & </s> <s xml:id="echoid-s4076" xml:space="preserve"><lb/>punctum ſuſpenſionis intercepta. </s> <s xml:id="echoid-s4077" xml:space="preserve">Quia in iſochronis pendu-<lb/>lis, centra quidem oſcillationis à punctis ſuſpenſionum æ-<lb/>qualiter diſtant; </s> <s xml:id="echoid-s4078" xml:space="preserve">amplius autem deſcendit centrum oſcilla-<lb/>tionis infra centrum ſphæræ majoris, quam minoris.</s> <s xml:id="echoid-s4079" xml:space="preserve"/> </p> <div xml:id="echoid-div371" type="float" level="2" n="2"> <note position="left" xlink:label="note-0256-01" xlink:href="note-0256-01a" xml:space="preserve"><emph style="sc">De centro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s4080" xml:space="preserve">Hinc neceſſe fuit illis, qui, ante hanc centri oſcillatorii <lb/>determinationem, menſuræ univerſalis conſtituendæ ratio-<lb/>nem inierunt; </s> <s xml:id="echoid-s4081" xml:space="preserve">quod, jam inde à prima Horologii noſtri <lb/>inventione, nobilis illa Societas Regia Anglicana ſibi nego-<lb/>tium ſumpſit, & </s> <s xml:id="echoid-s4082" xml:space="preserve">recentius doctiſſimus Aſtronomus Lugdu-<lb/>nenſis, Gabriel Moutonus; </s> <s xml:id="echoid-s4083" xml:space="preserve">his, inquam, neceſſe fuit deſi-<lb/>gnare globuli ſuſpenſi diametrum, vel proportione certa ad <lb/>fili longitudinem, cujus nempe triceſimam vel aliam partem <lb/>æquaret; </s> <s xml:id="echoid-s4084" xml:space="preserve">vel menſura quadam cognita, ut digiti vel polli-<lb/>cis. </s> <s xml:id="echoid-s4085" xml:space="preserve">Sed hoc poſteriore modo, ponitur jam certi aliquid, <lb/>quod id ipſum eſt quod quærendum eſt: </s> <s xml:id="echoid-s4086" xml:space="preserve">etſi ſcio vix ſenſi-<lb/>bilem errorem fore, dummodo ſphæræ iſtam, quam jam di-<lb/>xi, magnitudinem non multum excedant. </s> <s xml:id="echoid-s4087" xml:space="preserve">Priore autem <lb/>poſſet quidem aliquo pacto res explicari; </s> <s xml:id="echoid-s4088" xml:space="preserve">ſed ita, ut nu-<lb/>merandarum oſcillationum labor ſubeundus ſit, calculoque <lb/>etiam utendum. </s> <s xml:id="echoid-s4089" xml:space="preserve">Quamobrem præſtat, centra oſcillationis <lb/>adhibendo, certam rationem ſequi, nulliſque præter neceſ-<lb/>ſitatem legibus obligari. </s> <s xml:id="echoid-s4090" xml:space="preserve">atque hic jam majoribus ſphæris <lb/>quam exiguis potius utendum, quod illæ occurſu aëris mi-<lb/>nus impediantur.</s> <s xml:id="echoid-s4091" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4092" xml:space="preserve">Cæterum, non ſphæræ tantum ex filo ſuſpenſæ, ſed & </s> <s xml:id="echoid-s4093" xml:space="preserve"><lb/>coni, cylindri, aliaque omnia ſolida, planaque, quorum <lb/>centra oſcillationis ſuperius exhibuimus, ad hanc menſuram <lb/>inveſtigandam, apta ſunt; </s> <s xml:id="echoid-s4094" xml:space="preserve">quoniam, à puncto ſuſpenſionis <lb/>ad centrum oſcillationis, certum idemque omnibus iſochronis <lb/>pendulis eſt intervallum. </s> <s xml:id="echoid-s4095" xml:space="preserve">Neque etiam illa duntaxat horolo-<lb/>gia, quæ ſecunda ſcrupula aut eorum ſemiſſes ſingulis penduli <pb o="181" file="0257" n="283" rhead="HOROLOG. OSCILLATOR."/> recurſibus indicant, ad hæc uſurpare poſſumus; </s> <s xml:id="echoid-s4096" xml:space="preserve">ſed & </s> <s xml:id="echoid-s4097" xml:space="preserve">aliâ <lb/> <anchor type="note" xlink:label="note-0257-01a" xlink:href="note-0257-01"/> quàcunque penduli longitudine inſtructis propoſitum obti-<lb/>nebitur, dummodo ex rotarum proportionibus, ſeu dentium <lb/>numero, cognoſcatur numerus oſcillationum certo tem-<lb/>pore peragendarum. </s> <s xml:id="echoid-s4098" xml:space="preserve">Invento enim pendulo ſimplici, cu-<lb/>jus librationes ſingulæ conveniant vel ſingulis, vel binis <lb/>terniſve recurſibus horologii, conſtabit jam hinc, quot <lb/>penduli illius vices horæ ſpatio tranſigantur. </s> <s xml:id="echoid-s4099" xml:space="preserve">Quarum nume-<lb/>rus ſi quadretur, erit ut quadratum è 3600, numero ſcru-<lb/>pulorum ſecundorum horam unam efficientium, ad qua-<lb/>dratum illius numeri, ita longitudo penduli ſimplicis in-<lb/>venti, (quæ longitudo ſemper à puncto ſuſpenſionis ad <lb/>centrum oſcillationis accipienda eſt) ad longitudinem pen-<lb/>duli illius horarii tripedalis, quod diximus. </s> <s xml:id="echoid-s4100" xml:space="preserve">Hoc enim inde <lb/>conſtat, quod duorum quorumvis pendulorum longitudines <lb/>ſunt inter ſe, ſicut quadrata temporum quibus ſingulæ o-<lb/>ſcillationes tranſeunt; </s> <s xml:id="echoid-s4101" xml:space="preserve">ideoque contrariam rationem habent <lb/>quadratorum à numeris, quos efficiunt oſcillationes æquali-<lb/>bus temporum intervallis peractæ. </s> <s xml:id="echoid-s4102" xml:space="preserve">Nam, cum hactenus ex-<lb/>perientiâ tantum comprobatum fuerit Theorema illud, de <lb/>pendulorum longitudinibus; </s> <s xml:id="echoid-s4103" xml:space="preserve">eas nempe duplicatam habere <lb/>rationem temporum, quibus oſcillationes ſingulæ peragun-<lb/>tur; </s> <s xml:id="echoid-s4104" xml:space="preserve">nunc ejus demonſtratio ex ſuperius traditis manifeſta <lb/>eſt. </s> <s xml:id="echoid-s4105" xml:space="preserve">Cum enim oſtenderimus, ſingulos recurſus penduli, in-<lb/>ter cycloides ſuſpenſi, ad caſum perpendicularem, è dimi-<lb/>dia penduli longitudine, certam rationem habere; </s> <s xml:id="echoid-s4106" xml:space="preserve">eam ſci-<lb/>licet quam circumferentia circuli ad diametrum ſuam; </s> <s xml:id="echoid-s4107" xml:space="preserve">faci-<lb/>le hinc colligitur, tempora oſcillationum in duobus pendulis <lb/>eſſe inter ſe, ſicut tempora deſcenſus perpendicularis ex di-<lb/>midiis eorum altitudinibus. </s> <s xml:id="echoid-s4108" xml:space="preserve">Quæ altitudines dimidiæ, ſive <lb/>etiam totæ, cum habeant rationem duplicatam temporum, <lb/>quibus ipſæ deſcenſu perpendiculari percurruntur <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s4109" xml:space="preserve">eædem <anchor type="note" xlink:label="note-0257-02a" xlink:href="note-0257-02"/> quoque duplicatam rationem habebunt temporum, quæ o-<lb/>ſcillationes ſingulas metiuntur. </s> <s xml:id="echoid-s4110" xml:space="preserve">Ab oſcillationibus autem mini-<lb/>mis penduli, inter cycloides ſuſpenſi, non differunt ſenſi-<lb/>biliter oſcillationes minimæ penduli ſimplicis, cujus eadem <pb o="182" file="0258" n="284" rhead="CHRISTIANI HUGENII"/> ſit longitudo. </s> <s xml:id="echoid-s4111" xml:space="preserve">Itaque & </s> <s xml:id="echoid-s4112" xml:space="preserve">pendulorum ſimplicium longitudi-<lb/> <anchor type="note" xlink:label="note-0258-01a" xlink:href="note-0258-01"/> nes, duplicatam rationem habebunt temporum, quibus o-<lb/>ſcillationes minimæ tranſiguntur.</s> <s xml:id="echoid-s4113" xml:space="preserve"/> </p> <div xml:id="echoid-div372" type="float" level="2" n="3"> <note position="right" xlink:label="note-0257-01" xlink:href="note-0257-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> <note symbol="*" position="right" xlink:label="note-0257-02" xlink:href="note-0257-02a" xml:space="preserve">Prop. 3. <lb/>Part. 2.</note> <note position="left" xlink:label="note-0258-01" xlink:href="note-0258-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s4114" xml:space="preserve">Quod ſi quis oſcillationum numerandarum, quæ horæ aut <lb/>ſemihoræ tempore tranſeunt, laborem non defugiat; </s> <s xml:id="echoid-s4115" xml:space="preserve">horo-<lb/>logiumque adſit, cujus index ſecunda ſcrupula demonſtret; <lb/></s> <s xml:id="echoid-s4116" xml:space="preserve">quæcunque accipiatur penduli ſimplicis longitudo, ejus nu-<lb/>merus oſcillationum, quæ hora una continentur, hoc modo <lb/>cognoſcetur; </s> <s xml:id="echoid-s4117" xml:space="preserve">atque inde longitudo penduli tripedalis, ad <lb/>ſecunda ſcrupula, ut antea, calculo prodibit.</s> <s xml:id="echoid-s4118" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div374" type="section" level="1" n="132"> <head xml:id="echoid-head158" xml:space="preserve">PROPOSITIO XXVI.</head> <p style="it"> <s xml:id="echoid-s4119" xml:space="preserve">SPatium deſinire, quod gravia, perpendiculari-<lb/>ter cadentia, dato tempore percurrunt.</s> <s xml:id="echoid-s4120" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4121" xml:space="preserve">Hanc menſuram quicunque hactenus inveſtigarunt, expe-<lb/>rimenta conſulere neceſſe habuerunt; </s> <s xml:id="echoid-s4122" xml:space="preserve">quibus, prout hacte-<lb/>nus inſtituta fuere, non facile ad exactam determinationem <lb/>pervenitur, propter velocitatem cadentium, ſub finem mo-<lb/>tus acquiſitam. </s> <s xml:id="echoid-s4123" xml:space="preserve">Ex noſtra autem prop. </s> <s xml:id="echoid-s4124" xml:space="preserve">25, de Deſcenſu <lb/>gravium, cognitaque longitudine penduli ad ſecunda ſcru-<lb/>pula, abſque experimento, per certam conſequentiam, <lb/>rem expedire poſſumus. </s> <s xml:id="echoid-s4125" xml:space="preserve">Ac primo quidem ſpatium il-<lb/>lud inquiremus, quod unius ſcrupuli ſecundi tempore grave <lb/>præterlabitur; </s> <s xml:id="echoid-s4126" xml:space="preserve">ex quo quælibet alia deinde colligere lice-<lb/>bit. </s> <s xml:id="echoid-s4127" xml:space="preserve">Quia igitur penduli, ad ſecunda ſcrupula, longi-<lb/>tudinem diximus eſſe pedum Horariorum 3: </s> <s xml:id="echoid-s4128" xml:space="preserve">tempus au-<lb/>tem unius oſcillationis minimæ, eſt ad tempus deſcenſus <lb/>perpendicularis ex dimidia penduli altitudine, ut circumfe-<lb/>rentia circuli ad diametrum, hoc eſt, ut 355 ad 113: </s> <s xml:id="echoid-s4129" xml:space="preserve">ſi <lb/>fiat, ut numerus horum prior ad alterum, ita tempus unius <lb/>ſecundi ſcrupuli, ſive ſexaginta tertiorum, ad aliud; </s> <s xml:id="echoid-s4130" xml:space="preserve">fient <lb/>19″′ {1/@0}, tempus deſcenſus per dimidiam penduli altitudinem, <lb/>quæ nempe eſt pedis unciarum 18. </s> <s xml:id="echoid-s4131" xml:space="preserve">Sicut autem quadrata <lb/>temporum, ita ſunt ſpatia illis temporibus peracta, quemad- <pb o="183" file="0259" n="285" rhead="HOROLOG. OSCILLATOR."/> modum ſuperiori propoſitione fuit oſtenſum. </s> <s xml:id="echoid-s4132" xml:space="preserve">Ergo, ſi fiat ut <lb/> <anchor type="note" xlink:label="note-0259-01a" xlink:href="note-0259-01"/> quadratum ex 19″′ {1/10} ad quadratum ex 60″′, hoc eſt, ut 36481 <lb/>ad 360000, ita 18 unciæ ad aliud, fient ped. </s> <s xml:id="echoid-s4133" xml:space="preserve">14. </s> <s xml:id="echoid-s4134" xml:space="preserve">unc. </s> <s xml:id="echoid-s4135" xml:space="preserve">9. <lb/></s> <s xml:id="echoid-s4136" xml:space="preserve">lin. </s> <s xml:id="echoid-s4137" xml:space="preserve">6, altitudo deſcenſus perpendicularis, tempore unius <lb/>ſecundi. </s> <s xml:id="echoid-s4138" xml:space="preserve">Cum autem pes Horarius ſit ad Pariſienſem, ut <lb/>881 ad 864; </s> <s xml:id="echoid-s4139" xml:space="preserve">erit eadem altitudo, ad hanc menſuram redu-<lb/>cta, proxime pedum 15 & </s> <s xml:id="echoid-s4140" xml:space="preserve">unciæ unius. </s> <s xml:id="echoid-s4141" xml:space="preserve">Atque hæc cum <lb/>accuratiſſimis experimentis noſtris prorſus conveniunt. </s> <s xml:id="echoid-s4142" xml:space="preserve">in <lb/>quibus punctum illud temporis, quo cafus finitur, non au-<lb/>rium aut oculi judicio diſcernitur; </s> <s xml:id="echoid-s4143" xml:space="preserve">quorum neutrum hic ſa-<lb/>tis tutum eſt; </s> <s xml:id="echoid-s4144" xml:space="preserve">ſed ſpatium deſcendendo peractum, alio mo-<lb/>do, quem hic exponere tentabimus, abſque ullo errore cog-<lb/>noſcitur.</s> <s xml:id="echoid-s4145" xml:space="preserve"/> </p> <div xml:id="echoid-div374" type="float" level="2" n="1"> <note position="right" xlink:label="note-0259-01" xlink:href="note-0259-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <p> <s xml:id="echoid-s4146" xml:space="preserve">Penduli, ad parietem tabulamve erectam, ſuſpenſi dimi-<lb/>dia oſcillatio moram temporis, cadendo abſumpti, indi-<lb/>cat. </s> <s xml:id="echoid-s4147" xml:space="preserve">Cujus ſphærula, ut eodem momento ac plumbum caſui <lb/>deſtinatum dimittatur, utraque filo tenui connexa tenentur, <lb/>quod admoto igne inciditur. </s> <s xml:id="echoid-s4148" xml:space="preserve">Sed prius, caſuro plumbo, fu-<lb/>niculus alius adnectitur, ejus longitudinis, ut, cum totus <lb/>exierit à plumbo tractus, nondum ad parietem illidatur pen-<lb/>dulum. </s> <s xml:id="echoid-s4149" xml:space="preserve">Funiculi ejus caput alterum, regulæ chartaceæ, <lb/>aut ex tenui membrana paratæ, cohæret; </s> <s xml:id="echoid-s4150" xml:space="preserve">ita ad parietem ta-<lb/>bulamve applicatæ, ut trahentem funem facile ſequi poſſit, <lb/>rectáque ſecundum longitudinem ſuam deſcendere; </s> <s xml:id="echoid-s4151" xml:space="preserve">eo loci <lb/>tranſiens, quo penduli ſphæra ad tabulam accidet. </s> <s xml:id="echoid-s4152" xml:space="preserve">Abſum-<lb/>pto igitur funiculo toto, pars inſuper regulæ deorſum tra-<lb/>hitur à cadente plumbo, priuſquam pendulum ad tabulam <lb/>pertingat. </s> <s xml:id="echoid-s4153" xml:space="preserve">Quæ quanta ſit pars, ſphæra fuligine leviter in-<lb/>fecta, regulamque præterlabentem ſignans, indicat. </s> <s xml:id="echoid-s4154" xml:space="preserve">Huc <lb/>autem addita funiculi longitudine, ſpatium cadendo emen-<lb/>ſum certò definitum habetur.</s> <s xml:id="echoid-s4155" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4156" xml:space="preserve">Aëris autem occurſum, quaſi nullus eſſet in his intelligi-<lb/>gimus, ut menſura cadentibus corporibus præfixa cum ex-<lb/>perimentis exacte conſentiat. </s> <s xml:id="echoid-s4157" xml:space="preserve">Nec ſane tantus eſt ille, ut in <lb/>altitudinibus his, quò aſcendere datur, ſenſibile diſcrimen <lb/>inducere poſſit; </s> <s xml:id="echoid-s4158" xml:space="preserve">dummodo ſolida corpora è metallo, aut, ſi <pb o="184" file="0260" n="286" rhead="CHR. HUGENII HOROL. OSCILL."/> leviore materia conſtent, mole grandiuſcula accipiantur Le-<lb/> <anchor type="note" xlink:label="note-0260-01a" xlink:href="note-0260-01"/> vitas enim materiæ, in iis quæ cadendo aërem ſecant, ita <lb/>magnitudine corporis penſatur, ut ſphæra lignea, vel etiam <lb/>è ſubere formata, paria faciat cum plumbea: </s> <s xml:id="echoid-s4159" xml:space="preserve">quando nimi-<lb/>rum diameter harum ad plumbeæ diametrum eam rationem <lb/>habuerit, quam gravitas plumbi propria ad ligni ſuberisve <lb/>gravitatem. </s> <s xml:id="echoid-s4160" xml:space="preserve">Tunc enim gravitates ſphærarum erunt inter ſe <lb/>ſicut earum ſuperficies. </s> <s xml:id="echoid-s4161" xml:space="preserve">Veruntamen, ut æquali celeritate, <lb/>quantum ſenſu percipi poteſt, decidant corpora, quæ mul-<lb/>tum intrinſeca gravitate differunt nequaquam opus eſt ut <lb/>proportio illa diametrorum ſervetur. </s> <s xml:id="echoid-s4162" xml:space="preserve">Poſſunt enim inter ſe <lb/>æqualia eſſe, dummodo utraque ſatis magna ſint; </s> <s xml:id="echoid-s4163" xml:space="preserve">aut ex <lb/>non nimia altitudine decidant. </s> <s xml:id="echoid-s4164" xml:space="preserve">Etenim illud quoque hic <lb/>animadvertendum eſt, tantam vel altitudinem eſſe poſſe; <lb/></s> <s xml:id="echoid-s4165" xml:space="preserve">vel, in mediocri etiam altitudine, tantam projecti corporis <lb/>levitatem; </s> <s xml:id="echoid-s4166" xml:space="preserve">ut ob aëris renitentiam, acceleratio motus tan-<lb/>dem ab illa, quam in ſuperioribus demonſtravimus, pro-<lb/>portione plurimum receſſura ſit. </s> <s xml:id="echoid-s4167" xml:space="preserve">Namque in univerſum, <lb/>corpori cuilibet, per aërem aliudve liquidum labenti, certus <lb/>celeritatis modus, pro ratione ponderis ac ſuperſiciei ſuæ, <lb/>conſtitutus eſt; </s> <s xml:id="echoid-s4168" xml:space="preserve">quem excedere, aut potius ad quem perve-<lb/>nire nunquam poſſit. </s> <s xml:id="echoid-s4169" xml:space="preserve">Quæ nempe celeritas ea eſt, quam ſi <lb/>aër, aut liquor ille ſurſum tendens, haberet, ſuſpenſum cor-<lb/>pus idem ſibi innatans ſuſtinere poſſet. </s> <s xml:id="echoid-s4170" xml:space="preserve">Verum de his, alias <lb/>fortaſſe, pluribus agendi occaſio erit.</s> <s xml:id="echoid-s4171" xml:space="preserve"/> </p> <div xml:id="echoid-div375" type="float" level="2" n="2"> <note position="left" xlink:label="note-0260-01" xlink:href="note-0260-01a" xml:space="preserve"><emph style="sc">Decentro</emph> <lb/><emph style="sc">OSCILLA-</emph> <lb/><emph style="sc">TIONIS</emph>.</note> </div> <figure> <image file="0260-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0260-01"/> </figure> <pb o="185" file="0261" n="287"/> <figure> <image file="0261-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0261-01"/> </figure> </div> <div xml:id="echoid-div377" type="section" level="1" n="133"> <head xml:id="echoid-head159" xml:space="preserve">HOROLOGII OSCILLATORII</head> <head xml:id="echoid-head160" style="it" xml:space="preserve">PARS QUINTA.</head> <p style="it"> <s xml:id="echoid-s4172" xml:space="preserve">Conſtructionem aliam, è circulari pendulorum <lb/>motu deductam, continens; </s> <s xml:id="echoid-s4173" xml:space="preserve">& </s> <s xml:id="echoid-s4174" xml:space="preserve">Theoremata <lb/>de Vi Centrifuga.</s> <s xml:id="echoid-s4175" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4176" xml:space="preserve">EST & </s> <s xml:id="echoid-s4177" xml:space="preserve">aliud Oſcillatorii motus genus, præter id quod <lb/>hactenus pertractavimus. </s> <s xml:id="echoid-s4178" xml:space="preserve">Ejuſmodi nempe, quo, per <lb/>circuli ambitum, pendulum pondus circumfertur. </s> <s xml:id="echoid-s4179" xml:space="preserve">Unde ali-<lb/>ud quoque horologii commentum deduximus, eodem fere <lb/>tempore quo prius illud; </s> <s xml:id="echoid-s4180" xml:space="preserve">certoque itidem æquabilitatis prin-<lb/>cipio nixum; </s> <s xml:id="echoid-s4181" xml:space="preserve">ſed cujus uſus minus percrebuit, propter al-<lb/>terius illius conſtructionem, quodammodo ſimpliciorem fa-<lb/>cilioremque. </s> <s xml:id="echoid-s4182" xml:space="preserve">Plura tamen hujus quoque generis de quo nunc <lb/>loquimur, nec ſine ſucceſſu, conſtructa fuere: </s> <s xml:id="echoid-s4183" xml:space="preserve">eſtque in <lb/>his ſingulare illud, quod continuo atque æquabili motu cir-<lb/>cumferri cernitur index poſtremus, qui ſecunda ſcrupula <lb/>deſignat; </s> <s xml:id="echoid-s4184" xml:space="preserve">cum in priore noſtro horologio, omnibuſque aliis, <lb/>ſubſultim quaſi feratur. </s> <s xml:id="echoid-s4185" xml:space="preserve">Item hoc quoque, quod abſque ſtre-<lb/>pitu, ſonoque omni, moveantur hac ratione conſtructa au-<lb/>tomata. </s> <s xml:id="echoid-s4186" xml:space="preserve">quanquam, ad obſervationes aſtronomicas, ſonus <lb/>ad ſingula ſecunda ſcrupula repetitus, utilitate non careat. <lb/></s> <s xml:id="echoid-s4187" xml:space="preserve">Et conſtitueram quidem, deſcriptionem horum cum iis de-<lb/>mum edere, quæ ad motum circularem & </s> <s xml:id="echoid-s4188" xml:space="preserve">Vim Centrifu-<lb/>gam, ita enim eam vocarelibet, attinent; </s> <s xml:id="echoid-s4189" xml:space="preserve">de quo argumen- <pb o="186" file="0262" n="288" rhead="CHRISTIANI HUGENII"/> to plura dicenda habeo, quam quæ hoc tempore exequi va-<lb/> <anchor type="note" xlink:label="note-0262-01a" xlink:href="note-0262-01"/> cet. </s> <s xml:id="echoid-s4190" xml:space="preserve">Sed, ut nova nec inutili ſpeculatione maturius fruantur <lb/>harum rerum ſtudioſi, neve caſu aliquo intercidat, hanc <lb/>quoque partem, præter deſtinatum, cæteris adjunxi, qua <lb/>machinæ hujus fabrica breviter exponitur, ſimulque Theo-<lb/>remata traduntur, ad Vim Centrifugam pertinentia; </s> <s xml:id="echoid-s4191" xml:space="preserve">demon-<lb/> <anchor type="note" xlink:label="note-0262-02a" xlink:href="note-0262-02"/> ſtratione ipſorum in aliud tempus dilata <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s4192" xml:space="preserve"/> </p> <div xml:id="echoid-div377" type="float" level="2" n="1"> <note position="left" xlink:label="note-0262-01" xlink:href="note-0262-01a" xml:space="preserve"><emph style="sc">Secundi</emph> <lb/><emph style="sc">@OROLO-</emph> <lb/><emph style="sc">GII DE-</emph> <lb/><emph style="sc">SORIPTIO</emph>.</note> <note symbol="*" position="left" xlink:label="note-0262-02" xlink:href="note-0262-02a" xml:space="preserve">Vide Au-<lb/>ctoris Opera <lb/>poſthuma <lb/>p. 401. <lb/>& ſeq.</note> </div> </div> <div xml:id="echoid-div379" type="section" level="1" n="134"> <head xml:id="echoid-head161" style="it" xml:space="preserve">Horologii ſecundi conſtructio.</head> <p> <s xml:id="echoid-s4193" xml:space="preserve">Non neceſſarium duxi, ut rotarum, quibus interiora ho-<lb/> <anchor type="note" xlink:label="note-0262-03a" xlink:href="note-0262-03"/> rologii conſtant, diſpoſitionem hic exhiberem; </s> <s xml:id="echoid-s4194" xml:space="preserve">cum ea ab <lb/>artificibus facile ordinari, variiſque modis mutari poſſit; <lb/></s> <s xml:id="echoid-s4195" xml:space="preserve">ſed eam partem explicari ſatis eſſe, quæ motum ejus certa <lb/>ratione moderatur. </s> <s xml:id="echoid-s4196" xml:space="preserve">Cujus partis hic figura expreſſa eſt.</s> <s xml:id="echoid-s4197" xml:space="preserve"/> </p> <div xml:id="echoid-div379" type="float" level="2" n="1"> <note position="left" xlink:label="note-0262-03" xlink:href="note-0262-03a" xml:space="preserve">TAB.XXVII. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s4198" xml:space="preserve">Axis D H ad horizontem erectus intelligendus eſt, ac <lb/>ſuper polis duobus mobilis. </s> <s xml:id="echoid-s4199" xml:space="preserve">Huic ad A affixa eſt lamina, <lb/>latitudine aliqua prædita, curvataque ſecundum lineam <lb/>A B; </s> <s xml:id="echoid-s4200" xml:space="preserve">quæ eſt paraboloides illa de qua oſtendimus, Propoſ. </s> <s xml:id="echoid-s4201" xml:space="preserve">8. <lb/></s> <s xml:id="echoid-s4202" xml:space="preserve">partis 3, evolutione ejus, poſtquam ipſi recta quædam juncta <lb/>fuerit, deſcribi parabolam. </s> <s xml:id="echoid-s4203" xml:space="preserve">Earecta hic eſt A E; </s> <s xml:id="echoid-s4204" xml:space="preserve">parabolam <lb/>vero, ex evolutione totius B A E deſcriptam, refert linea <lb/>E F. </s> <s xml:id="echoid-s4205" xml:space="preserve">Filum curvæ B A applicatum, cujus extremo puncto <lb/>parabola deſcribitur, eſt B G F. </s> <s xml:id="echoid-s4206" xml:space="preserve">Pondus illi affixum F. </s> <s xml:id="echoid-s4207" xml:space="preserve"><lb/>Dum autem axis D H in ſeſe vertitur, filum B G F, in re-<lb/>ctam lineam extenſum, ſphærulam F una circumducit, <lb/>ita ut circulos horizonti parallelos percurrat; </s> <s xml:id="echoid-s4208" xml:space="preserve">qui majores <lb/>minoreſve erunt, prout majori aut minori vi axis D H, <lb/>ab rotis horologii in tympanidium K agentibus, incitabitur: </s> <s xml:id="echoid-s4209" xml:space="preserve"><lb/>ſed ita, ut omnes in ſuperficie conoidis parabolici continean-<lb/>tur. </s> <s xml:id="echoid-s4210" xml:space="preserve">Atque hoc ipſo æqualia ſemper circuitus tempora eva-<lb/>dent, ut ex iis, quæ de hoc motu poſtea dicemus, appa-<lb/>rebit.</s> <s xml:id="echoid-s4211" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4212" xml:space="preserve">Quod ſi circuitus ſingulos, ſecundorum ſcrupulorum ſe-<lb/>miſſes notare velimus, oportet latus rectum parabolæ E F <lb/>eſſe 4{1/2} unciarum pedis Horarii noſtri, hoc eſt dimidium <pb o="187" file="0263" n="289" rhead="HOROLOG. OSCILLATOR."/> longitudinis penduli, cujus ſingulæ oſcillationes ſemiſcru-<lb/> <anchor type="note" xlink:label="note-0263-01a" xlink:href="note-0263-01"/> pulum ſecundum impenderent. </s> <s xml:id="echoid-s4213" xml:space="preserve">Ex parabolæ autem latere <lb/>recto, pendet magnitudo lateris recti paraboloidis A B; <lb/></s> <s xml:id="echoid-s4214" xml:space="preserve">quippe quod illius {27/16} continet: </s> <s xml:id="echoid-s4215" xml:space="preserve">atque item longitudo A E, <lb/>quæ lateris recti parabolæ dimidium eſt. </s> <s xml:id="echoid-s4216" xml:space="preserve">Si vero ſecunda <lb/>ſcrupula unoquoque circuitu expleri deſideremus, quadru-<lb/>pla priorum accipienda ſunt, tum latera recta, tum linea <lb/>A E.</s> <s xml:id="echoid-s4217" xml:space="preserve"/> </p> <div xml:id="echoid-div380" type="float" level="2" n="2"> <note position="right" xlink:label="note-0263-01" xlink:href="note-0263-01a" xml:space="preserve"><emph style="sc">Secundi</emph> <lb/><emph style="sc">HOROLO-</emph> <lb/><emph style="sc">GII DE-</emph> <lb/><emph style="sc">SCRIPTIO</emph>.</note> </div> <p> <s xml:id="echoid-s4218" xml:space="preserve">Porro, etſi filum B G F veluti unicum ac ſimplex hacte-<lb/>nus deſignavimus, ſciendum tamen longe præſtare ut parte <lb/>ſuperiori duplex ſit, ac F versùs in angulum coëat, 20 vel <lb/>30 partium. </s> <s xml:id="echoid-s4219" xml:space="preserve">In quem finem & </s> <s xml:id="echoid-s4220" xml:space="preserve">laminæ A B latitudo ad B <lb/>tanta eſſe debet, quanta iſti filorum divaricationi ſufficit, <lb/>vel & </s> <s xml:id="echoid-s4221" xml:space="preserve">ipſa bifida facienda. </s> <s xml:id="echoid-s4222" xml:space="preserve">Hoc pacto enim motus circularis <lb/>ponderis F, abſque alio ullo adminiculo, continuatur, ac filum <lb/>utrumque ſibi annexum in rectum extendit; </s> <s xml:id="echoid-s4223" xml:space="preserve">quod non face-<lb/>ret, ſi unico tantum filo teneretur. </s> <s xml:id="echoid-s4224" xml:space="preserve">Ubi tamen vim illam ab <lb/>horologii rotis, vel pondere vel alia potentia motis, ad con-<lb/>tinuationem hujus motus circularis requiri ſciendum. </s> <s xml:id="echoid-s4225" xml:space="preserve">Quæ <lb/>nempe vis per tympanidium K ad axem K H pervenit, ac <lb/>minimo niſu, motum ſphæræ F ſemel inditum, conſervat.</s> <s xml:id="echoid-s4226" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4227" xml:space="preserve">Hoc autem quo facilius poſſit, liberrimam axis K H re-<lb/>volutionem eſſe oportet. </s> <s xml:id="echoid-s4228" xml:space="preserve">Quod nulla ratione melius perfici <lb/>compertum, quam ſi, parte ſui ima, durato chalybe con-<lb/>ſtet, ſuppoſitamque habeat adamantis ſuperficiem planam; <lb/></s> <s xml:id="echoid-s4229" xml:space="preserve">cujus minima quævis particula hic ſufficit, ſubter laminam <lb/>perforatam collocanda.</s> <s xml:id="echoid-s4230" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4231" xml:space="preserve">Cæterum in locum fili B G F, qua parte curvæ A B appli-<lb/>cari debet, catenulam tenuem ex auro, aliove metallo, adhi-<lb/>bere licebit, quo melius invariata ſervetur longitudo. </s> <s xml:id="echoid-s4232" xml:space="preserve">Atque <lb/>hoc in priore quoque horologio, ubi pendulum inter cycloi-<lb/>des ſuſpenſum eſt, experti ſumus. </s> <s xml:id="echoid-s4233" xml:space="preserve">Sed ibi flexus catenulæ <lb/>continuus, attritu annulorum, perexiguo licet, non parum <lb/>impedit liberam penduli agitationem.</s> <s xml:id="echoid-s4234" xml:space="preserve"/> </p> <pb o="188" file="0264" n="290" rhead="CHRISTIANI HUGENII"/> </div> <div xml:id="echoid-div382" type="section" level="1" n="135"> <head xml:id="echoid-head162" xml:space="preserve">DE VI CENTRIFUGA</head> <head xml:id="echoid-head163" xml:space="preserve">ex motu circulari, Theoremata.</head> <head xml:id="echoid-head164" xml:space="preserve">I.</head> <p style="it"> <s xml:id="echoid-s4235" xml:space="preserve">SI mobilia duo æqualia, æqualibus temporibus <lb/>circumferentias inæquales percurrant; </s> <s xml:id="echoid-s4236" xml:space="preserve">erit vis <lb/>centrifuga in majori circumferentia, ad eam quæ <lb/>in minori, ſicut ipſæ inter ſe circumferentiæ, vel <lb/>earum diametri.</s> <s xml:id="echoid-s4237" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div383" type="section" level="1" n="136"> <head xml:id="echoid-head165" xml:space="preserve">II.</head> <p style="it"> <s xml:id="echoid-s4238" xml:space="preserve">Si duo mobilia æqualia, æquali celeritate fe-<lb/>rantur, in circumferentiis inæqualibus; </s> <s xml:id="echoid-s4239" xml:space="preserve">erunt eo-<lb/>rum vires centrifugæ in ratione contraria diame-<lb/>trorum.</s> <s xml:id="echoid-s4240" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div384" type="section" level="1" n="137"> <head xml:id="echoid-head166" xml:space="preserve">III.</head> <p style="it"> <s xml:id="echoid-s4241" xml:space="preserve">Si duo mobilia æqualia in circumferentiis æqua-<lb/>libus ferantur, celeritate inæquali, ſed utraque <lb/>motu æquabili, qualem in his omnibus intelligi vo-<lb/>lumus; </s> <s xml:id="echoid-s4242" xml:space="preserve">erit vis centrifuga velocioris, ad vim tar-<lb/>dioris, in ratione duplicata celeritatum.</s> <s xml:id="echoid-s4243" xml:space="preserve"/> </p> <pb file="0265" n="291"/> <pb file="0265a" n="292"/> <figure> <caption xml:id="echoid-caption108" style="it" xml:space="preserve">Pag. 188.<lb/>TAB.XXVII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables110" xml:space="preserve">O V V<lb/>A M N D N B O E C<lb/>E A G B D C F</variables> </figure> <figure> <caption xml:id="echoid-caption109" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables111" xml:space="preserve">S Z G F H Y</variables> </figure> <figure> <caption xml:id="echoid-caption110" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables112" xml:space="preserve">D A D M T C</variables> </figure> <figure> <caption xml:id="echoid-caption111" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables113" xml:space="preserve">A E N D C</variables> </figure> <figure> <caption xml:id="echoid-caption112" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables114" xml:space="preserve">K D B G A F E H</variables> </figure> <pb file="0266" n="293"/> <pb o="189" file="0267" n="294" rhead="HOROLOG. OSCILLATOR."/> <note position="right" xml:space="preserve"><emph style="sc">De vi</emph> <lb/><emph style="sc">Centri-</emph> <lb/><emph style="sc">FUGA</emph>.</note> </div> <div xml:id="echoid-div385" type="section" level="1" n="138"> <head xml:id="echoid-head167" xml:space="preserve">IV.</head> <p style="it"> <s xml:id="echoid-s4244" xml:space="preserve">Si mobilia duo æqualia, in circumferentiis in-<lb/>æqualibus circumlata, vim centrifugam æqua-<lb/>lem habuerint; </s> <s xml:id="echoid-s4245" xml:space="preserve">erit tempus circuitus in majori cir-<lb/>cumferentia, ad tempus circuitus in minori, in <lb/>ſubdupla ratione diametrorum.</s> <s xml:id="echoid-s4246" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div386" type="section" level="1" n="139"> <head xml:id="echoid-head168" xml:space="preserve">V.</head> <p style="it"> <s xml:id="echoid-s4247" xml:space="preserve">Si mobile in circumferentia circuli feratur ea <lb/>celeritate, quam acquirit cadendo ex altitudine, <lb/>quæ ſit quartæ parti diametri æqualis; </s> <s xml:id="echoid-s4248" xml:space="preserve">habebit <lb/>vim centrifugam ſuæ gravitati æqualem; </s> <s xml:id="echoid-s4249" xml:space="preserve">hoc eſt, <lb/>eadem vi funem quo in centro detinetur intendet, <lb/>atque cum ex eo ſuſpenſum eſt.</s> <s xml:id="echoid-s4250" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div387" type="section" level="1" n="140"> <head xml:id="echoid-head169" xml:space="preserve">VI.</head> <p style="it"> <s xml:id="echoid-s4251" xml:space="preserve">In cava ſuperficie conoidis parabolici, quod axem <lb/>ad perpendiculum erectum habeat, circuitus omnes <lb/>mobilis, circumferentias horizonti parallelas per-<lb/>currentis, ſive parvæ ſive magnæ fuerint, æqua-<lb/>libus temporibus peraguntur: </s> <s xml:id="echoid-s4252" xml:space="preserve">quæ tempora ſingula <lb/>æquantur binis oſcillationibus penduli, cujus longi-<lb/>tudo ſit dimidium lateris recti parabolæ genitricis.</s> <s xml:id="echoid-s4253" xml:space="preserve"/> </p> <pb o="190" file="0268" n="295" rhead="CHRISTIANI HUGENII"/> <note position="left" xml:space="preserve"><emph style="sc">De vi</emph> <lb/><emph style="sc">Crentri-</emph> <lb/><emph style="sc">FUGA</emph>.</note> </div> <div xml:id="echoid-div388" type="section" level="1" n="141"> <head xml:id="echoid-head170" xml:space="preserve">VII.</head> <p style="it"> <s xml:id="echoid-s4254" xml:space="preserve">Si mobilia duo, ex filis inæqualibus ſuſpenſa, <lb/>gyrentur ita ut circumferentias horizonti paralle-<lb/>las percurrant, capite altero fili immoto manente; <lb/></s> <s xml:id="echoid-s4255" xml:space="preserve">fuerint autem conorum, quorum ſuperficiem ſila <lb/>hoc motu deſcribunt, altitudines æquales; </s> <s xml:id="echoid-s4256" xml:space="preserve">tempo-<lb/>ra quoque circulationum æqualia erunt.</s> <s xml:id="echoid-s4257" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div389" type="section" level="1" n="142"> <head xml:id="echoid-head171" xml:space="preserve">VIII.</head> <p style="it"> <s xml:id="echoid-s4258" xml:space="preserve">Si mobilia duo, uti prius, motu conico gyren-<lb/>tur, filis æqualibus vel inæqualibus ſuſpenſa; </s> <s xml:id="echoid-s4259" xml:space="preserve">fue-<lb/>rintque conorum altitudines inæquales; </s> <s xml:id="echoid-s4260" xml:space="preserve">erunt tem-<lb/>pora circulationum in ſubduplicata ratione ipſarum <lb/>altitudinum.</s> <s xml:id="echoid-s4261" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div390" type="section" level="1" n="143"> <head xml:id="echoid-head172" xml:space="preserve">IX.</head> <p style="it"> <s xml:id="echoid-s4262" xml:space="preserve">Si pendulum, motu conico latum, circuitus mi-<lb/>nimos faciat; </s> <s xml:id="echoid-s4263" xml:space="preserve">eorum ſingulorum tempora, adtem-<lb/>pus caſus prpendicularis ex dupla penduli altitu-<lb/>dine, eam rationem habent, quam circumferentia <lb/>circuli ad diametrum: </s> <s xml:id="echoid-s4264" xml:space="preserve">ac proinde æqualia ſunt <lb/>tempori duarum oſcillationum lateralium, ejusdem <lb/>penduli, minimarum.</s> <s xml:id="echoid-s4265" xml:space="preserve"/> </p> <pb o="191" file="0269" n="296" rhead="HOROLOG. OSCILLATOR."/> </div> <div xml:id="echoid-div391" type="section" level="1" n="144"> <head xml:id="echoid-head173" xml:space="preserve">X.</head> <note position="right" xml:space="preserve"><emph style="sc">De vi</emph> <lb/><emph style="sc">CENTRE-</emph> <lb/><emph style="sc">FUGA</emph>.</note> <p style="it"> <s xml:id="echoid-s4266" xml:space="preserve">Si mobile in circumferentia feratur, circuitus-<lb/>que ſingulos abſolvat eo tempore, quo pendulum, <lb/>longitudinem ſemidiametri circumferentiæ ejus ha-<lb/>bens, motu conico circuitum minimum abſolveret, <lb/>vel duplicem oſcillationem minimam lateralem: </s> <s xml:id="echoid-s4267" xml:space="preserve">ha-<lb/>bebit vim centrifugam ſuæ gravitati æqualem.</s> <s xml:id="echoid-s4268" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div392" type="section" level="1" n="145"> <head xml:id="echoid-head174" xml:space="preserve">XI.</head> <p style="it"> <s xml:id="echoid-s4269" xml:space="preserve">Penduli cujuslibet, motu conico lati, tempora <lb/>circuitus æqualia erunt tempori caſus perpendicula-<lb/>ris, ex altitudine penduli filo æquali; </s> <s xml:id="echoid-s4270" xml:space="preserve">cum angu-<lb/>lus inclinationis fili, ad planum horizontis, fuerit <lb/>partium 2. </s> <s xml:id="echoid-s4271" xml:space="preserve">ſcrup. </s> <s xml:id="echoid-s4272" xml:space="preserve">54, proxime. </s> <s xml:id="echoid-s4273" xml:space="preserve">Exacte vero, ſi <lb/>anguli dicti ſinus fuerit ad radium, ut quadra-<lb/>tum circulo inſcriptum ad quadratum à circumfe-<lb/>rentia ejus.</s> <s xml:id="echoid-s4274" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div393" type="section" level="1" n="146"> <head xml:id="echoid-head175" xml:space="preserve">XII.</head> <p style="it"> <s xml:id="echoid-s4275" xml:space="preserve">Si pendula duo, pondere æqualia, ſed inæquali <lb/>filorum longitudine, motu conico gyrentur, fue-<lb/>rintque conorum altitudines æquales; </s> <s xml:id="echoid-s4276" xml:space="preserve">erunt vires, <lb/>quibus fila ſua intendent, in eadem ratione quæ <lb/>eſt filorum longitudinis.</s> <s xml:id="echoid-s4277" xml:space="preserve"/> </p> <pb o="192" file="0270" n="297" rhead="CHRIST. HUGENII HOROL. OSCILL."/> <note position="left" xml:space="preserve"><emph style="sc">De vii</emph> <lb/><emph style="sc">Centri-</emph> <lb/><emph style="sc">FUGA</emph>.</note> </div> <div xml:id="echoid-div394" type="section" level="1" n="147"> <head xml:id="echoid-head176" xml:space="preserve">XIII.</head> <p style="it"> <s xml:id="echoid-s4278" xml:space="preserve">Si pendulum ſimplex oſcillatione laterali maxi-<lb/>ma agitetur, hoc eſt, ſi per totum circuli quadran-<lb/>tem deſcendat: </s> <s xml:id="echoid-s4279" xml:space="preserve">ubi ad punctum imum circumfe-<lb/>rentiæ pervenerit, triplo majori vi filum ſuum <lb/>trahet, quam ſi ex illo ſimpliciter ſuſpenſum fo-<lb/>ret.</s> <s xml:id="echoid-s4280" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div395" type="section" level="1" n="148"> <head xml:id="echoid-head177" xml:space="preserve">FINIS.</head> <figure> <image file="0270-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0270-01"/> </figure> <pb file="0271" n="298"/> </div> <div xml:id="echoid-div396" type="section" level="1" n="149"> <head xml:id="echoid-head178" xml:space="preserve">BREVIS <lb/><emph style="bf">INSTITUTIO</emph> <lb/>DE USU <lb/><emph style="bf">HOROLOGIORUM</emph> <lb/>AD INVENIENDAS LONGITUDINES.</head> <pb file="0272" n="299"/> </div> <div xml:id="echoid-div397" type="section" level="1" n="150"> <head xml:id="echoid-head179" style="it" xml:space="preserve">Adr. Metius in Geographicis Inſtitutionibus <lb/>Cap. 4.</head> <p> <s xml:id="echoid-s4281" xml:space="preserve">ET hæc ſane facillima & </s> <s xml:id="echoid-s4282" xml:space="preserve">aptiſſima eſt methodus (ſcilicet qua adhibitis Horologiis longitudines de-<lb/>teguntur) quam acquirere poſſis; </s> <s xml:id="echoid-s4283" xml:space="preserve">niſi quod difficultas & </s> <s xml:id="echoid-s4284" xml:space="preserve">error conſiſtat in irregulari Horolo-<lb/>giorum motu: </s> <s xml:id="echoid-s4285" xml:space="preserve">ideoque diligentes inquiſitores & </s> <s xml:id="echoid-s4286" xml:space="preserve">inventores rerum naturalium id curate, neque <lb/>laboris veſtri vos pæniteat, quo hunc errorem tollere tentetis. </s> <s xml:id="echoid-s4287" xml:space="preserve">Inquirite in hunc verum & </s> <s xml:id="echoid-s4288" xml:space="preserve">æqua-<lb/>bilem naturæ curſum; </s> <s xml:id="echoid-s4289" xml:space="preserve">quo potiti verum lapidem Philoſophorum inveniſtis, neque fortes Nau-<lb/>cleri ad ſcopulos toties offendent.</s> <s xml:id="echoid-s4290" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div398" type="section" level="1" n="151"> <head xml:id="echoid-head180" style="it" xml:space="preserve">Fournier in Hydrographia 1. 12. C. 35.</head> <p> <s xml:id="echoid-s4291" xml:space="preserve">UNde tandem colligo, ſi via inveniri queat ad Horologia perficienda, & </s> <s xml:id="echoid-s4292" xml:space="preserve">laborem ſuſcipere ve-<lb/>limus iis bene utendi, nullam praxim (ad inveniendas longitudines) cum hac comparandam eſſe.</s> <s xml:id="echoid-s4293" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div399" type="section" level="1" n="152"> <head xml:id="echoid-head181" style="it" xml:space="preserve">Didericus Rembrantz a Nierop in Animadverſionibus <lb/>de inveniendis longitudinibus.</head> <p> <s xml:id="echoid-s4294" xml:space="preserve">HOc modo laudabiliora cenſerem nova excogitata Horologia Domini Chriſtiani Hugenii a Zuy-<lb/>lichem, quæ, loco liberamenti, a plumbo pendulo oſcillato moderantur, de quibus per certa <lb/>teſtimonia certus ſum, quod hæc tempus exacte ad hebdomadas, imò ferè ad menſes juſte di-<lb/>metiri queant; </s> <s xml:id="echoid-s4295" xml:space="preserve">Unde confiderem, quod ope horum Horologiorum maximum perciperemug<unsure/> <lb/>commodum, imò quod, niſi agitatio navis impediret, ſatis attingere poſſemus ad inventionem <lb/>longitudinum.</s> <s xml:id="echoid-s4296" xml:space="preserve"/> </p> <pb file="0273" n="300"/> </div> <div xml:id="echoid-div400" type="section" level="1" n="153"> <head xml:id="echoid-head182" xml:space="preserve">BREVIS INSTRUCTIO DE USU HOROLO-<lb/>GIORUM AD INVENIENDAS <lb/>LONGITUDINES.</head> <head xml:id="echoid-head183" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s4297" xml:space="preserve">AD minimum bina nova Horologia O-<lb/>ſcillatoria in navem ferantur: </s> <s xml:id="echoid-s4298" xml:space="preserve">ut ſi al-<lb/>terutrum forte fortunâ vel ex negli-<lb/>gentia quieſcat; </s> <s xml:id="echoid-s4299" xml:space="preserve">vel ſi diuturnitate <lb/>temporis contractis ſordibus, purgan-<lb/>dum ſit, alterum nihilominus movea-<lb/>tur; </s> <s xml:id="echoid-s4300" xml:space="preserve">præſtaret autem 3 vel 4 adhibere <lb/>horologia.</s> <s xml:id="echoid-s4301" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div401" type="section" level="1" n="154"> <head xml:id="echoid-head184" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s4302" xml:space="preserve">Cui cura horologiorum committetur, diſcat a fabro quæ <lb/>ſpectant indices horarum minutarum primarum & </s> <s xml:id="echoid-s4303" xml:space="preserve">ſecunda-<lb/>rum, internas etiam horologiorum partes intelligere, & </s> <s xml:id="echoid-s4304" xml:space="preserve">redu-<lb/>cendi ea methodum.</s> <s xml:id="echoid-s4305" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div402" type="section" level="1" n="155"> <head xml:id="echoid-head185" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s4306" xml:space="preserve">Horologia in navi ſuſpendenda ſunt in loco arcte clauſo, <lb/>ubi tuta ſunt ab humore vel ſordibus, & </s> <s xml:id="echoid-s4307" xml:space="preserve">ne diſturbentur con-<lb/>tactibus: </s> <s xml:id="echoid-s4308" xml:space="preserve">Et ſi locum illum in media navi prope malum <lb/>principem ordinare poſſemus, multum præſtaret, quoniam <lb/>ibi motus minimus eſt.</s> <s xml:id="echoid-s4309" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div403" type="section" level="1" n="156"> <head xml:id="echoid-head186" xml:space="preserve">IV.</head> <p> <s xml:id="echoid-s4310" xml:space="preserve">Antequam horologia in navem inferantur, conabimur ea <lb/>aptare ad rectam dierum menſuram, tum enim uſus eſt facil-<lb/>limus, nulluſque fabris labor eſt ad unum bene adaptatum <lb/>horologium alia accommodare. </s> <s xml:id="echoid-s4311" xml:space="preserve">Sed ſi tamen tempus vel op-<lb/>portuna id præſtandi occaſio defuerit, nihilominus poterunt <lb/>æque certe mari uſurpari, dummodo obſervaveris vel ſcias, <lb/>quanto citius vel tardius ſpatio 24 horarum moveantur, ut <lb/>poſtea docebitur.</s> <s xml:id="echoid-s4312" xml:space="preserve"/> </p> <pb o="196" file="0274" n="301" rhead="CHRIST. HUGENII INSTITUTIO"/> </div> <div xml:id="echoid-div404" type="section" level="1" n="157"> <head xml:id="echoid-head187" style="it" xml:space="preserve">V.</head> <head xml:id="echoid-head188" style="it" xml:space="preserve">Reducere horologia ad rectam dierum menſuram vel cogno-<lb/>ſcere quanto citius vel tardius ſpatio 24 horarum movean-<lb/>tur.</head> <p> <s xml:id="echoid-s4313" xml:space="preserve">Obſerva, dies ab una ad alteram meridiem aliquantulum <lb/>differre quæ cauſa eſt cur horologium, licet prorſus exactum <lb/>& </s> <s xml:id="echoid-s4314" xml:space="preserve">ſecundum mediorum dierum menſuram moveatur, non ſem-<lb/>per cum ſole conveniat. </s> <s xml:id="echoid-s4315" xml:space="preserve">Sed ut hæc inæqualitas æquetur, & </s> <s xml:id="echoid-s4316" xml:space="preserve"><lb/>ſemper ope Horologii ſcire poſſimus, quam horam indicat Sol, <lb/>& </s> <s xml:id="echoid-s4317" xml:space="preserve">conſequenter num horologium ad rectam mediorum die-<lb/>rum menſuram diſpoſitum ſit, conducit ſequens tabula cu-<lb/>jus uſus talis eſt. </s> <s xml:id="echoid-s4318" xml:space="preserve">Quando primum horologium conſtituendum <lb/>eſt, ſubtrahe ex hora ſolari obſervatâ æquationem ejus diei quæ <lb/>in tabula reperitur, & </s> <s xml:id="echoid-s4319" xml:space="preserve">diſpone horologium ad reſiduas horas, <lb/>minuta prima & </s> <s xml:id="echoid-s4320" xml:space="preserve">ſecunda: </s> <s xml:id="echoid-s4321" xml:space="preserve">ubi poſt aliquot dies quæritur <lb/>hora ſolaris, adde ad horam horologii æquationem diei ul-<lb/>timi, & </s> <s xml:id="echoid-s4322" xml:space="preserve">aggregatum erit hora ſolaris, ſi horologium exacte <lb/>fuit adaptatum, ſecundum menſuram dierum mediorum; </s> <s xml:id="echoid-s4323" xml:space="preserve">ve-<lb/>rum ut hæ obſervationes quam certiſſime inſtituantur, & </s> <s xml:id="echoid-s4324" xml:space="preserve"><lb/>horologia ad menſuram adaptentur antequam in navem infe-<lb/>rantur, methodus ſequens aptiſſima eſt.</s> <s xml:id="echoid-s4325" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4326" xml:space="preserve">Duc lineam meridianam in pavimento, cujus operationis <lb/>methodi ſatis notæ ſunt; </s> <s xml:id="echoid-s4327" xml:space="preserve">obſervandum præterea ſummam hic <lb/>non requiri exactitudinem; </s> <s xml:id="echoid-s4328" xml:space="preserve">Porro, directe ſuper lineam meri-<lb/>dianam, ſuſpende 2 fila appenſis infra ponderibus, vel aliter <lb/>verticaliter tendantur, certosque a ſe mutuò diſtent pedes, <lb/>quo plures eo melius.</s> <s xml:id="echoid-s4329" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4330" xml:space="preserve">Ubi dein medietas ſolis videtur exacte ex adverſo ambo-<lb/>rum ſilorum (ad quod requiritur vitrum obſcuri coloris, vel <lb/>in fuligine candelæ denigratum) eo momento horologiorum <lb/>indices diſponendi ſunt, non exacte in 12 horas, ſed tanto ma-<lb/>gis retrorſum, quanta eſt æquatio illius diei in Tabula: </s> <s xml:id="echoid-s4331" xml:space="preserve">ex. </s> <s xml:id="echoid-s4332" xml:space="preserve">gr. <lb/></s> <s xml:id="echoid-s4333" xml:space="preserve">ſi fuerit 22 dies Martii, cujus æquatio in Tabula eſt 8 min. </s> <s xml:id="echoid-s4334" xml:space="preserve"><lb/>3 ſec: </s> <s xml:id="echoid-s4335" xml:space="preserve">hæc ſunt ſubducenda ex 12 horis, & </s> <s xml:id="echoid-s4336" xml:space="preserve">reſiduum erit <lb/>11 horæ 51 min. </s> <s xml:id="echoid-s4337" xml:space="preserve">57 ſec: </s> <s xml:id="echoid-s4338" xml:space="preserve">in quot horas minuta & </s> <s xml:id="echoid-s4339" xml:space="preserve">ſecunda <lb/>diſponendi ſunt indices Horologiorum quam primum ſol me- <pb o="197" file="0275" n="302" rhead="DE USU HOROLOG."/> dius ex adverſo 2 filorum conſpicitur. </s> <s xml:id="echoid-s4340" xml:space="preserve">Dein poſt aliquot dies <lb/>obſervatio rurſus eodem modo eſt inſtituenda, ubi ſol ex <lb/>adverſo filorum cernitur, & </s> <s xml:id="echoid-s4341" xml:space="preserve">ſimiliter notanda hora minu-<lb/>ta & </s> <s xml:id="echoid-s4342" xml:space="preserve">ſecunda horologiorum, quibus adde æquationem ejus <lb/>diei excerptam ex Tabulâ, & </s> <s xml:id="echoid-s4343" xml:space="preserve">ſi aggregatum exacte compo-<lb/>nat 12 horas Horologium ad rectam menſuram accommoda-<lb/>tum eſt. </s> <s xml:id="echoid-s4344" xml:space="preserve">Si vero differat dividenda ſunt minuta & </s> <s xml:id="echoid-s4345" xml:space="preserve">ſecunda <lb/>iſtius differentiæ, per numerum dierum inter utramque ob-<lb/>ſervationem, ad obtinendam quotidianam differentiam.</s> <s xml:id="echoid-s4346" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4347" xml:space="preserve">Supponamus hanc ſecundam obſervationem fieri 30. </s> <s xml:id="echoid-s4348" xml:space="preserve">Mar-<lb/>tii ſcil. </s> <s xml:id="echoid-s4349" xml:space="preserve">octo diebus poſt primam & </s> <s xml:id="echoid-s4350" xml:space="preserve">comperiatur medietate <lb/>ſolis viſa in meridiano ex adverſo 2. </s> <s xml:id="echoid-s4351" xml:space="preserve">filorum, <lb/> <anchor type="note" xlink:label="note-0275-01a" xlink:href="note-0275-01"/> horologium indicare <lb/>Æquatio 30. </s> <s xml:id="echoid-s4352" xml:space="preserve">Martii in tabula eſt <lb/>præbet ſummam</s> </p> <div xml:id="echoid-div404" type="float" level="2" n="1"> <note position="right" xlink:label="note-0275-01" xlink:href="note-0275-01a" xml:space="preserve"> <lb/>h. # m. # ſ. <lb/>11. # 51. # 7. <lb/>0. # 10. # 40. <lb/>12. # 1. # 47. <lb/></note> </div> <p> <s xml:id="echoid-s4353" xml:space="preserve">Si hæc ſumma exacte fuiſſet 12 horarum, horologium re-<lb/>cte dispoſitum fuiſſet, ſed cum excedat 12. </s> <s xml:id="echoid-s4354" xml:space="preserve">horas 1. </s> <s xml:id="echoid-s4355" xml:space="preserve">minuto 47. <lb/></s> <s xml:id="echoid-s4356" xml:space="preserve">ſecundis, tanto citius promotum fuit ſpatio octidui; </s> <s xml:id="echoid-s4357" xml:space="preserve">& </s> <s xml:id="echoid-s4358" xml:space="preserve"><lb/>hoc 1 minutum & </s> <s xml:id="echoid-s4359" xml:space="preserve">47 ſecunda, aut 107 ſecunda, diviſa per <lb/>8. </s> <s xml:id="echoid-s4360" xml:space="preserve">efficiunt 13. </s> <s xml:id="echoid-s4361" xml:space="preserve">{3/8} ſecundorum, differentiam in ſpatio 24. </s> <s xml:id="echoid-s4362" xml:space="preserve">hora-<lb/>rum. </s> <s xml:id="echoid-s4363" xml:space="preserve">Quâ differentià cognitâ, ſi non otium nec animus ſit <lb/>ſuſcipiendi moleſtiam ut adaptetur horologium ad veram men-<lb/>ſuram, neceſſe hoc non eſt; </s> <s xml:id="echoid-s4364" xml:space="preserve">ita enim in navim inferre licet, <lb/>modo prædicta quotidiana differentia annotetur, & </s> <s xml:id="echoid-s4365" xml:space="preserve">ad eam <lb/>nosmet componamus ut ſtatim dicetur.</s> <s xml:id="echoid-s4366" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4367" xml:space="preserve">Sed ſi accuratius Horologium diſponere velimus, removen-<lb/>dum eſt minus pondus penduli parumper deorſum, quo tardius <lb/>movebitur: </s> <s xml:id="echoid-s4368" xml:space="preserve">& </s> <s xml:id="echoid-s4369" xml:space="preserve">tum de novo obſervatio per ſolem inſtituenda <lb/>eſt ut antea; </s> <s xml:id="echoid-s4370" xml:space="preserve">ſi tarde nimis motum fuiſſet, ſupra memora-<lb/>tum pondus parumper ſurſum promovendum fuiſſet, <lb/>ita tamen, ne ſupra punctum medium penduli promoveatur. <lb/></s> <s xml:id="echoid-s4371" xml:space="preserve">eâ quippe gaudet proprietate, quod inde ſurſum promotum <lb/>horologium lentius iterum promoveat, cujus rei in deſcri-<lb/>ptione Horologii datur demonſtratio <anchor type="note" xlink:href="" symbol="*"/>; </s> <s xml:id="echoid-s4372" xml:space="preserve">ut & </s> <s xml:id="echoid-s4373" xml:space="preserve">æquationis tem- <anchor type="note" xlink:label="note-0275-02a" xlink:href="note-0275-02"/> poris, cujus ſolummodo hic docemus uſum. </s> <s xml:id="echoid-s4374" xml:space="preserve">Præter certam <pb o="198" file="0276" n="303" rhead="TABULA ÆQUA."/> <anchor type="note" xlink:label="note-0276-01a" xlink:href="note-0276-01"/> <pb o="199" file="0277" n="304" rhead="TIONIS DIERUM."/> <anchor type="note" xlink:label="note-0277-01a" xlink:href="note-0277-01"/> <pb o="200" file="0278" n="305" rhead="CHRIST. HUGENII INSTITUTIO"/> demonſtrationem etiam revera exactis Pendulis compertum <lb/>eſt, quod ad inæqualitatem dierum ad rectam menſuram re-<lb/>ducendam, æquatio, prout eam hic per præcedentem Ta-<lb/>bulam inſtituimus, exacte cum experientia conveniat, ita ut <lb/>tuto ei confidere liceat. </s> <s xml:id="echoid-s4375" xml:space="preserve">Idque tanti momenti eſt in inve-<lb/>niendis longitudinibus, ut, ſi non fuerit obſervatum, non-<lb/>nunquam ſpatio 3 menſium in calculo errorem committas 7 <lb/>graduum & </s> <s xml:id="echoid-s4376" xml:space="preserve">amplius, ſine ulla tamen horologiorum culpa: <lb/></s> <s xml:id="echoid-s4377" xml:space="preserve">qui gradus ſub Tropicis ultra 100 Germanica milliaria conti-<lb/>nent.</s> <s xml:id="echoid-s4378" xml:space="preserve"/> </p> <div xml:id="echoid-div405" type="float" level="2" n="2"> <note symbol="*" position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">Vide ſu-<lb/>pra pag 172.</note> <note position="right" xlink:label="note-0276-01" xlink:href="note-0276-01a" xml:space="preserve"> <lb/>Dies. ## Fanuar. ## Febr. ## Mart. ## Apr. ## Maj. ## Fun. <lb/>" # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. <lb/>1 # 10 # 40 # 0 # 32 # 2 # 15 # 11 # 18 # 18 # 32 # 18 # 10 <lb/>2 # 10 # 10 # 0 # 24 # 2 # 28 # 11 # 37 # 18 # 39 # 18 # 1 <lb/>3 # 9 # 41 # 0 # 18 # 2 # 42 # 11 # 56 # 18 # 46 # 17 # 51 <lb/>4 # 9 # 13 # 0 # 13 # 2 # 56 # 12 # 15 # 18 # 53 # 17 # 41 <lb/>5 # 8 # 45 # 0 # 9 # 3 # 11 # 12 # 34 # 18 # 59 # 17 # 30 <lb/>6 # 8 # 17 # 0 # 6 # 3 # 26 # 12 # 53 # 19 # 4 # 17 # 19 <lb/>7 # 7 # 50 # 0 # 3 # 3 # 41 # 13 # 12 # 19 # 9 # 17 # 8 <lb/>8 # 7 # 23 # 0 # 1 # 3 # 56 # 13 # 31 # 19 # 14 # 16 # 57 <lb/>9 # 6 # 58 # 0 # 0 # 4 # 12 # 13 # 49 # 19 # 18 # 16 # 46 <lb/>10 # 6 # 34 # 0 # 0 # 4 # 29 # 14 # 6 # 19 # 22 # 16 # 35 <lb/>11 # 6 # 10 # 0 # 0 # 4 # 46 # 14 # 23 # 19 # 25 # 16 # 24 <lb/>12 # 5 # 47 # 0 # 2 # 5 # 4 # 14 # 39 # 19 # 28 # 16 # 13 <lb/>13 # 5 # 24 # 0 # 4 # 5 # 22 # 14 # 55 # 19 # 29 # 16 # 1 <lb/>14 # 5 # 2 # 0 # 8 # 5 # 40 # 15 # 10 # 19 # 29 # 15 # 49 <lb/>15 # 4 # 41 # 0 # 12 # 5 # 58 # 15 # 25 # 19 # 29 # 15 # 37 <lb/>16 # 4 # 21 # 0 # 16 # 6 # 16 # 15 # 39 # 19 # 28 # 15 # 24 <lb/>17 # 4 # 2 # 0 # 21 # 6 # 33 # 15 # 53 # 19 # 26 # 15 # 11 <lb/>18 # 3 # 44 # 0 # 26 # 6 # 51 # 16 # 7 # 19 # 24 # 14 # 58 <lb/>19 # 3 # 27 # 0 # 32 # 7 # 9 # 16 # 21 # 19 # 21 # 14 # 45 <lb/>20 # 3 # 11 # 0 # 40 # 7 # 27 # 16 # 34 # 19 # 18 # 14 # 32 <lb/>21 # 2 # 55 # 0 # 48 # 7 # 45 # 16 # 47 # 19 # 15 # 14 # 19 <lb/>22 # 2 # 39 # 0 # 57 # 8 # 3 # 16 # 59 # 19 # 11 # 14 # 6 <lb/>23 # 2 # 23 # 1 # 6 # 8 # 22 # 17 # 11 # 19 # 7 # 13 # 53 <lb/>24 # 2 # 7 # 1 # 16 # 8 # 41 # 17 # 22 # 19 # 2 # 13 # 40 <lb/>25 # 1 # 52 # 1 # 26 # 9 # 1 # 17 # 33 # 18 # 57 # 13 # 27 <lb/>26 # 1 # 38 # 1 # 37 # 9 # 21 # 17 # 43 # 18 # 51 # 13 # 15 <lb/>27 # 1 # 25 # 1 # 49 # 9 # 41 # 17 # 53 # 18 # 45 # 13 # 3 <lb/>28 # 1 # 13 # 2 # 2 # 10 # 1 # 18 # 3 # 18 # 39 # 12 # 52 <lb/>29 # 1 # 2 # # # 10 # 21 # 18 # 13 # 18 # 33 # 12 # 41 <lb/>30 # 0 # 51 # # # 10 # 40 # 18 # 23 # 18 # 26 # 12 # 30 <lb/>31 # 0 # 41 # # # 10 # 59 # # # 18 # 18 <lb/></note> <note position="right" xlink:label="note-0277-01" xlink:href="note-0277-01a" xml:space="preserve"> <lb/>Dies. ## Jul. ## Aug. ## Sept. ## Octob. ## Nov. ## Dec. <lb/>" # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. # Min. # Sec. <lb/>1 # 12 # 19 # 10 # 4 # 16 # 23 # 26 # 30 # 31 # 55 # 25 # 34 <lb/>2 # 12 # 8 # 10 # 8 # 16 # 42 # 26 # 49 # 31 # 55 # 25 # 10 <lb/>3 # 11 # 58 # 10 # 13 # 17 # 1 # 27 # 8 # 31 # 54 # 24 # 45 <lb/>4 # 11 # 48 # 10 # 18 # 17 # 21 # 27 # 26 # 31 # 52 # 24 # 20 <lb/>5 # 11 # 38 # 10 # 23 # 17 # 41 # 27 # 43 # 31 # 50 # 23 # 55 <lb/>6 # 11 # 28 # 10 # 28 # 18 # 1 # 28 # 0 # 31 # 47 # 23 # 30 <lb/>7 # 11 # 18 # 10 # 34 # 18 # 21 # 28 # 16 # 31 # 43 # 23 # 4 <lb/>8 # 11 # 9 # 10 # 41 # 18 # 41 # 28 # 32 # 31 # 37 # 22 # 38 <lb/>9 # 11 # 0 # 10 # 49 # 19 # 1 # 28 # 47 # 31 # 30 # 22 # 11 <lb/>10 # 10 # 52 # 10 # 58 # 19 # 21 # 29 # 2 # 34 # 22 # 21 # 43 <lb/>11 # 10 # 47 # 11 # 7 # 19 # 41 # 29 # 16 # 31 # 13 # 21 # 14 <lb/>12 # 10 # 38 # 11 # 16 # 20 # 1 # 29 # 30 # 31 # 3 # 20 # 44 <lb/>13 # 10 # 31 # 11 # 25 # 20 # 22 # 29 # 43 # 30 # 53 # 20 # 14 <lb/>14 # 10 # 25 # 11 # 36 # 20 # 43 # 29 # 56 # 30 # 43 # 19 # 44 <lb/>15 # 10 # 19 # 11 # 48 # 21 # 4 # 30 # 9 # 30 # 32 # 19 # 14 <lb/>16 # 10 # 13 # 12 # 1 # 21 # 25 # 30 # 22 # 30 # 20 # 18 # 44 <lb/>17 # 10 # 7 # 12 # 14 # 21 # 47 # 30 # 34 # 30 # 8 # 18 # 14 <lb/>18 # 10 # 2 # 12 # 28 # 22 # 9 # 30 # 45 # 29 # 55 # 17 # 44 <lb/>19 # 9 # 58 # 12 # 42 # 22 # 31 # 30 # 55 # 29 # 40 # 17 # 14 <lb/>20 # 9 # 54 # 12 # 57 # 22 # 52 # 31 # 4 # 29 # 23 # 16 # 44 <lb/>21 # 9 # 51 # 13 # 12 # 23 # 13 # 31 # 12 # 29 # 6 # 16 # 14 <lb/>22 # 9 # 49 # 13 # 27 # 23 # 33 # 31 # 19 # 28 # 48 # 15 # 44 <lb/>23 # 9 # 47 # 13 # 43 # 23 # 53 # 31 # 26 # 28 # 30 # 15 # 14 <lb/>24 # 9 # 46 # 13 # 59 # 24 # 13 # 31 # 32 # 28 # 11 # 14 # 43 <lb/>25 # 9 # 46 # 14 # 16 # 24 # 33 # 31 # 38 # 27 # 51 # 14 # 12 <lb/>26 # 9 # 46 # 14 # 33 # 24 # 53 # 31 # 43 # 27 # 30 # 13 # 41 <lb/>27 # 9 # 47 # 14 # 50 # 25 # 13 # 31 # 47 # 27 # 8 # 13 # 10 <lb/>28 # 9 # 49 # 15 # 8 # 25 # 33 # 31 # 50 # 26 # 45 # 12 # 40 <lb/>29 # 9 # 52 # 15 # 26 # 25 # 52 # 31 # 53 # 26 # 22 # 12 # 10 <lb/>30 # 9 # 56 # 15 # 45 # 26 # 11 # 31 # 55 # 25 # 58 # 11 # 40 <lb/>31 # 10 # 0 # 16 # 4 # # # 31 # 55 # # # 11 # 10 <lb/></note> </div> <p> <s xml:id="echoid-s4379" xml:space="preserve">Oſtenſo, quomodo horologia poſſint adaptari terrâ, vel <lb/>quomodo eorum differentia quotidiana ſit invenienda, pro-<lb/>ximum erit dicere, quomodo idem faciendum ſit in navi fixa <lb/>ad anchoram, cum minime poſſibile ſit idem præſtare in ve-<lb/>lificatione.</s> <s xml:id="echoid-s4380" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4381" xml:space="preserve">Obſervabimus quodam die ortum & </s> <s xml:id="echoid-s4382" xml:space="preserve">occaſum Solis, & </s> <s xml:id="echoid-s4383" xml:space="preserve">in <lb/>utraque obſervatione ubi ejus medietas exacte ſupra Horizon-<lb/>tem apparet, notabimus horam quam indicat horologium, <lb/>& </s> <s xml:id="echoid-s4384" xml:space="preserve">ſupputabimus quot horæ interea fuerint præteritæ, cujus <lb/>numeri dimidio addito ad horam obſervationis matutinæ, <lb/>habebimus horam quam indicavit horologium cum Sol eſ-<lb/>ſet in meridiano: </s> <s xml:id="echoid-s4385" xml:space="preserve">cui addens Tabulæ æquationem iſtius diei, <lb/>ſummam notabimus, & </s> <s xml:id="echoid-s4386" xml:space="preserve">horologium ulterius promoveri pa-<lb/>tiemur. </s> <s xml:id="echoid-s4387" xml:space="preserve">dein quibusdam diebus, elapſis (quo autem plures <lb/>præterierint, eo melius eſt) idem omnino faciemus; </s> <s xml:id="echoid-s4388" xml:space="preserve">Et ſi <lb/>hora hujus ultimi diei ſit eadem, cum ea quæ antea fuerat <lb/>notata, horologium recte accommodatum eſt. </s> <s xml:id="echoid-s4389" xml:space="preserve">Sin vero ma-<lb/>jor vel minor ſit, vel lentius vel celerius movetur, & </s> <s xml:id="echoid-s4390" xml:space="preserve">diffe-<lb/>rentia diviſa per numerum dierum interim dilapſarum dabit <lb/>quotidianam differentiam, quam annotabimus, & </s> <s xml:id="echoid-s4391" xml:space="preserve">ſi velimus <lb/>horologium relinquemus in illo ſtatu; </s> <s xml:id="echoid-s4392" xml:space="preserve">vel alioquin remo-<lb/>vendo minus pondus penduli ut ſupra dictum eſt, melius <lb/>horologium accommodabimus.</s> <s xml:id="echoid-s4393" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4394" xml:space="preserve">Ex. </s> <s xml:id="echoid-s4395" xml:space="preserve">Gr. </s> <s xml:id="echoid-s4396" xml:space="preserve">pone 21. </s> <s xml:id="echoid-s4397" xml:space="preserve">Martii mane cum ſolis medie-<lb/> <anchor type="note" xlink:label="note-0278-01a" xlink:href="note-0278-01"/> tas apparet ſupra horizontem horologium indicare. <lb/></s> <s xml:id="echoid-s4398" xml:space="preserve">Et veſperi ubiSolis medietas latet infra horizontem.</s> <s xml:id="echoid-s4399" xml:space="preserve"/> </p> <div xml:id="echoid-div406" type="float" level="2" n="3"> <note position="right" xlink:label="note-0278-01" xlink:href="note-0278-01a" xml:space="preserve"> <lb/>h. # m. # ſ. <lb/>5. # 30. # 10. <lb/>5. # 20. # 6. <lb/></note> </div> <pb o="201" file="0279" n="306" rhead="DE USU HOROLOG."/> <p> <s xml:id="echoid-s4400" xml:space="preserve">Ut ſcias ope horologii horas inter binas <lb/> <anchor type="note" xlink:label="note-0279-01a" xlink:href="note-0279-01"/> obſervationes elapſas, ſubtrahe horam ortus</s> </p> <div xml:id="echoid-div407" type="float" level="2" n="4"> <note position="right" xlink:label="note-0279-01" xlink:href="note-0279-01a" xml:space="preserve"> <lb/>h. # m. # ſ. <lb/>5. # 30. # 10. <lb/>ex 12. # 0. # 0. <lb/>Reſtant 6. # 29. # 50. <lb/>5. # 20. # 6. <lb/></note> </div> <p> <s xml:id="echoid-s4401" xml:space="preserve">Cui ſi addas horam occaſus</s> </p> <p> <s xml:id="echoid-s4402" xml:space="preserve">Prodeunt horæ inter binas obſervationes <lb/>elapſæ</s> </p> <note position="right" xml:space="preserve"> <lb/>11. # 49. # 56. <lb/>5. # 54. # 58. <lb/>5. # 30. # 10. <lb/></note> <p> <s xml:id="echoid-s4403" xml:space="preserve">Quarum dimidium eſt</s> </p> <p> <s xml:id="echoid-s4404" xml:space="preserve">Quo addito ad horam ortus</s> </p> <p> <s xml:id="echoid-s4405" xml:space="preserve">Prodit hora horologii, cum ſol eſſet in Me-<lb/>ridiano</s> </p> <note position="right" xml:space="preserve"> <lb/>11. # 25. # 8. <lb/>0. # 7. # 45. <lb/>11. # 32. # 53. <lb/></note> <p> <s xml:id="echoid-s4406" xml:space="preserve">Cui addita æquatione 21. </s> <s xml:id="echoid-s4407" xml:space="preserve">Martii</s> </p> <p> <s xml:id="echoid-s4408" xml:space="preserve">Summa eſt</s> </p> <p> <s xml:id="echoid-s4409" xml:space="preserve">Septem diebus poſt ſc. </s> <s xml:id="echoid-s4410" xml:space="preserve">28. </s> <s xml:id="echoid-s4411" xml:space="preserve">Martii obſervetur <lb/>ortus ſolis, quum horologium indicat</s> </p> <note position="right" xml:space="preserve"> <lb/>5. # 19. # 4. <lb/>5. # 25. # 4. <lb/></note> <p> <s xml:id="echoid-s4412" xml:space="preserve">Et occaſus quum indicat</s> </p> <p> <s xml:id="echoid-s4413" xml:space="preserve">Ad habendas horas interim elapſas, ſubtra-<lb/>he horam ortus</s> </p> <note position="right" xml:space="preserve"> <lb/>5. # 19. # 4. <lb/>ex 12. # 0. # 0. <lb/>Reſtant 6. # 40. # 56. <lb/>5. # 25. # 4. <lb/>12. # 6. # 0. <lb/>6. # 3. # 0. <lb/>5. # 19. # 4. <lb/></note> <p> <s xml:id="echoid-s4414" xml:space="preserve">Cui ſi addas horam occaſus</s> </p> <p> <s xml:id="echoid-s4415" xml:space="preserve">Prodeunt horæ interea delapſæ</s> </p> <p> <s xml:id="echoid-s4416" xml:space="preserve">Quarum dimidium eſt</s> </p> <p> <s xml:id="echoid-s4417" xml:space="preserve">Quo addito ad horam ortus</s> </p> <p> <s xml:id="echoid-s4418" xml:space="preserve">Prodit hora horologii cum Sol erat in me-<lb/>ridie</s> </p> <note position="right" xml:space="preserve"> <lb/>11. # 22. # 4. <lb/>0. # 10. # 1. <lb/>11. # 32. # 5. <lb/></note> <p> <s xml:id="echoid-s4419" xml:space="preserve">Cui addita æquatione 28. </s> <s xml:id="echoid-s4420" xml:space="preserve">Martii</s> </p> <p> <s xml:id="echoid-s4421" xml:space="preserve">Summa eſt</s> </p> <p> <s xml:id="echoid-s4422" xml:space="preserve">Quæ ſumma ſi fuiſſet eadem cum priori ſcilicet 11. </s> <s xml:id="echoid-s4423" xml:space="preserve">32. </s> <s xml:id="echoid-s4424" xml:space="preserve">53. <lb/></s> <s xml:id="echoid-s4425" xml:space="preserve">ad rectam menſuram diſpoſitum fuiſſet horologium; </s> <s xml:id="echoid-s4426" xml:space="preserve">ſed cum <lb/>poſterior minor ſit priori, differentiâ exiſtente 49. </s> <s xml:id="echoid-s4427" xml:space="preserve">ſecundo-<lb/>rum, horologium ſpatio 7 dierum tanto tardius fuit promo-<lb/>tum, quæ 49 ſecunda diviſa per 7 numerum dierum, dant <lb/>quotientem 7 ſecunda differentiam diurnam, quâ horolo- <pb o="202" file="0280" n="307" rhead="CHRIST. HUGENII INSTITUTIO"/> gium tardius movetur; </s> <s xml:id="echoid-s4428" xml:space="preserve">poſſumus etiam loco ortus & </s> <s xml:id="echoid-s4429" xml:space="preserve">occaſus <lb/>Solis duas æquales Solis altitudines obſervare ante & </s> <s xml:id="echoid-s4430" xml:space="preserve">poſt me-<lb/>ridiem, & </s> <s xml:id="echoid-s4431" xml:space="preserve">hora horologii annotatâ utriuſque obſervationis <lb/>tempore, eodem modo, ac hic dictum eſt procedemus.</s> <s xml:id="echoid-s4432" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div409" type="section" level="1" n="158"> <head xml:id="echoid-head189" xml:space="preserve">VI.</head> <head xml:id="echoid-head190" style="it" xml:space="preserve">Ope Horologiorum mari invenire longitudinem loci in <lb/>quo verſaris.</head> <p> <s xml:id="echoid-s4433" xml:space="preserve">Singulis horologiis nomina vel ſigna impone ut A. </s> <s xml:id="echoid-s4434" xml:space="preserve">B. </s> <s xml:id="echoid-s4435" xml:space="preserve">C. <lb/></s> <s xml:id="echoid-s4436" xml:space="preserve">& </s> <s xml:id="echoid-s4437" xml:space="preserve">antequam velifices, diſpone eadem ſecundum tempus ob-<lb/>fervatum per Solem in loco, ubi moraris, diminutum æquatio-<lb/>ne ejus diei, quo obſervas; </s> <s xml:id="echoid-s4438" xml:space="preserve">quem annotabis.</s> <s xml:id="echoid-s4439" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4440" xml:space="preserve">Dein ubi in mari verſaris ſi vis ſcire longitudinem loci, in <lb/>quo es; </s> <s xml:id="echoid-s4441" xml:space="preserve">quot gradibus Meridianus loci hujus ſit orientalior vel <lb/>occidentalior meridiano loci illius in quo adaptaſti horologia; <lb/></s> <s xml:id="echoid-s4442" xml:space="preserve">obſerva Solem vel Stellas, ut determines horam, & </s> <s xml:id="echoid-s4443" xml:space="preserve">vide quam <lb/>horam eodem momento indicent horologia; </s> <s xml:id="echoid-s4444" xml:space="preserve">quam horam, ſi ho-<lb/>rologia non fuerint diſpoſita ad rectam menſuram æquabis co-<lb/>gnitâ diurnâ differentiâ, ei porrò addens æquationem præſen-<lb/>tis diei, quo ita habeas horam in loco illo, ubi horologia fue-<lb/>re diſpoſita. </s> <s xml:id="echoid-s4445" xml:space="preserve">Si hæc hora eadem ſit cum illa quæ obſervata <lb/>fuit in loco præſente, conſiſtis ſub eodem Meridiano, ac ubi <lb/>horologia fuere diſpoſita ad Solem: </s> <s xml:id="echoid-s4446" xml:space="preserve">Si vero hora obſervata <lb/>major ſit illâ quam horologia oſtendunt, certus es, te per-<lb/>veniſſe ſub Meridianum Orientaliorem; </s> <s xml:id="echoid-s4447" xml:space="preserve">ſin denique fuerit mi-<lb/>nor, perveniſti ſub Meridianum Occidentaliorem: </s> <s xml:id="echoid-s4448" xml:space="preserve">& </s> <s xml:id="echoid-s4449" xml:space="preserve">com-<lb/>putatis in ſingulas quasque horas differentiæ temporis 15 gra-<lb/>dibus longitudinis vel in ſingula minuta 15 minutis, vel to-<lb/>tidem quadrantibus gradus, cognoſces, quot gradibus dicti <lb/>Meridiani ab invicem diſtent. </s> <s xml:id="echoid-s4450" xml:space="preserve">E. </s> <s xml:id="echoid-s4451" xml:space="preserve">Gr. </s> <s xml:id="echoid-s4452" xml:space="preserve">pone Horologia A. </s> <s xml:id="echoid-s4453" xml:space="preserve">B. </s> <s xml:id="echoid-s4454" xml:space="preserve">C. </s> <s xml:id="echoid-s4455" xml:space="preserve"><lb/>fuiſſe aptata ad Solem in loco ex quo abiiſti, 2 Martii, id <lb/>eſt ad horam obſervatam, ſed in tantum diminutam, quanta <lb/>eſt æquatio 2 Martii, 2 min. </s> <s xml:id="echoid-s4456" xml:space="preserve">28 ſec. </s> <s xml:id="echoid-s4457" xml:space="preserve">& </s> <s xml:id="echoid-s4458" xml:space="preserve">pone horologium A <lb/>fuiſſe dispoſitum ad veram menſuram, ſed B moveri ſingu-<lb/>lis diebus 7 ſecundis tardius, & </s> <s xml:id="echoid-s4459" xml:space="preserve">C ſingulis diebus 12 ſecun-<lb/>dis citius.</s> <s xml:id="echoid-s4460" xml:space="preserve"/> </p> <pb o="203" file="0281" n="308" rhead="DE USU HOROLOG."/> <p> <s xml:id="echoid-s4461" xml:space="preserve">Aliquot diebus poſt E. </s> <s xml:id="echoid-s4462" xml:space="preserve">G. </s> <s xml:id="echoid-s4463" xml:space="preserve">15. </s> <s xml:id="echoid-s4464" xml:space="preserve">Maji, ut detegas longi-<lb/>tudinem loci in quo in mari verſaris; </s> <s xml:id="echoid-s4465" xml:space="preserve">obſerva <lb/> <anchor type="note" xlink:label="note-0281-01a" xlink:href="note-0281-01"/> horam diei, quæ ſit</s> </p> <div xml:id="echoid-div409" type="float" level="2" n="1"> <note position="right" xlink:label="note-0281-01" xlink:href="note-0281-01a" xml:space="preserve"> <lb/>h. # m. # ſ. <lb/>5. # 18. # 10. <lb/>2. # 6. # 0. <lb/>1. # 57. # 22. <lb/></note> </div> <p> <s xml:id="echoid-s4466" xml:space="preserve">Et comperis horologium A indicare</s> </p> <p> <s xml:id="echoid-s4467" xml:space="preserve">Sed horologium B. </s> <s xml:id="echoid-s4468" xml:space="preserve">indicare</s> </p> <p> <s xml:id="echoid-s4469" xml:space="preserve">Sed cum ſingulis diebus 7. </s> <s xml:id="echoid-s4470" xml:space="preserve">ſec. </s> <s xml:id="echoid-s4471" xml:space="preserve">tardius mo-<lb/>veatur, prodeunt ſpatio 74. </s> <s xml:id="echoid-s4472" xml:space="preserve">dierum nempe a <lb/>2. </s> <s xml:id="echoid-s4473" xml:space="preserve">Martii ad 15. </s> <s xml:id="echoid-s4474" xml:space="preserve">Maji</s> </p> <note position="right" xml:space="preserve"> <lb/>0. # 8. # 38. <lb/></note> <p> <s xml:id="echoid-s4475" xml:space="preserve">Quæ addita ad horam a B. </s> <s xml:id="echoid-s4476" xml:space="preserve">demonſtratam <lb/>prodit hora eadem ac indicavit horologium A</s> </p> <note position="right" xml:space="preserve"> <lb/>2. # 6. # 0. <lb/>2. # 20. # 48. <lb/></note> <p> <s xml:id="echoid-s4477" xml:space="preserve">Invenis etiam horologium C. </s> <s xml:id="echoid-s4478" xml:space="preserve">indicare</s> </p> <p> <s xml:id="echoid-s4479" xml:space="preserve">Sed cum moveatur 12. </s> <s xml:id="echoid-s4480" xml:space="preserve">ſec. </s> <s xml:id="echoid-s4481" xml:space="preserve">ſingulis diebus <lb/>citius, habemus ſpatio 74. </s> <s xml:id="echoid-s4482" xml:space="preserve">dierum</s> </p> <note position="right" xml:space="preserve"> <lb/>0. # 14. # 48. <lb/></note> <p> <s xml:id="echoid-s4483" xml:space="preserve">Quæ ſubtracta ab hora a C. </s> <s xml:id="echoid-s4484" xml:space="preserve">indicata, iterum <lb/>prodeunt</s> </p> <note position="right" xml:space="preserve"> <lb/>2. # 6. # 0. <lb/>2. # 6. # 0. <lb/>0. # 19. # 29. <lb/>2. # 25. # 29. <lb/>5. # 18. # 10. <lb/>2. # 52. # 41. <lb/></note> <p> <s xml:id="echoid-s4485" xml:space="preserve">Cum itaque hora horologiorum ſit</s> </p> <p> <s xml:id="echoid-s4486" xml:space="preserve">Adde illi æquationem 15. </s> <s xml:id="echoid-s4487" xml:space="preserve">Maji</s> </p> <p> <s xml:id="echoid-s4488" xml:space="preserve">Prodit hora loci ubi horologia diſpoſita ſunt</s> </p> <p> <s xml:id="echoid-s4489" xml:space="preserve">Sed hora obſervata eſt</s> </p> <p> <s xml:id="echoid-s4490" xml:space="preserve">Excedens priorem</s> </p> <p> <s xml:id="echoid-s4491" xml:space="preserve">Ergo Meridianus loci in quo verſaris 15. <lb/></s> <s xml:id="echoid-s4492" xml:space="preserve">Maji Orientalior eſt, Meridianoloci, in quo <lb/>horologia fuere aptata</s> </p> <note position="right" xml:space="preserve"> <lb/>2. # 52. # 41. <lb/>gr. # min. # ſec. <lb/>43. # 0. # 15. <lb/></note> <p> <s xml:id="echoid-s4493" xml:space="preserve">Quæ horæ in gradus reductæ, ponendo 15. <lb/></s> <s xml:id="echoid-s4494" xml:space="preserve">gradus valere unam horam, prodeunt</s> </p> <p> <s xml:id="echoid-s4495" xml:space="preserve">Verum eſt ex eodem calculo poſſe concludi te eſſe ſub Me-<lb/>ridiano hôc ipſo 180. </s> <s xml:id="echoid-s4496" xml:space="preserve">gradus Orientaliori, quia index hora-<lb/>rius revolutionem ſuam abſolvit ſpatio 12. </s> <s xml:id="echoid-s4497" xml:space="preserve">horarum in ho-<lb/>rologiis, ſed differentia tanta eſt, ut nequeas in ea decipi. <lb/></s> <s xml:id="echoid-s4498" xml:space="preserve">Alioquin enim horologium poſſet conſtrui, cujus index cir-<lb/>cuitum ſuum ſemel abſolveret ſpatio 24 horarum.</s> <s xml:id="echoid-s4499" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4500" xml:space="preserve">Obſervandum quoque hic eſt, cum dico, locum tot gradibus <lb/>Orientaliorem eſſe illo ex quo abiiſti, illud dicieo reſpectu quod <lb/>illuc veneris ad Orientem navigans, verſus quam partem gra-<lb/>dus numerari poſſunt usque ad 360; </s> <s xml:id="echoid-s4501" xml:space="preserve">alioquin enim ſatis no- <pb o="204" file="0282" n="309" rhead="CHRIST. HUGENII INSTITUTIO"/> tum eſt: </s> <s xml:id="echoid-s4502" xml:space="preserve">locum, qui 180. </s> <s xml:id="echoid-s4503" xml:space="preserve">gradus Orientem verſus ab alio <lb/>diſtat, tantum etiam Occidentem verſus inde diſtare: </s> <s xml:id="echoid-s4504" xml:space="preserve">& </s> <s xml:id="echoid-s4505" xml:space="preserve">pa-<lb/>riter, qui 300. </s> <s xml:id="echoid-s4506" xml:space="preserve">gradus Orientaliter ab alio diſtat etiam 60. <lb/></s> <s xml:id="echoid-s4507" xml:space="preserve">gradus Occidentaliter inde diſtare.</s> <s xml:id="echoid-s4508" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div411" type="section" level="1" n="159"> <head xml:id="echoid-head191" xml:space="preserve">VII.</head> <head xml:id="echoid-head192" style="it" xml:space="preserve">Mari invenire horam diei.</head> <p> <s xml:id="echoid-s4509" xml:space="preserve">Quandoquidem pro invenienda longitudine requiritur ut <lb/>hora loci, in quo es, cognita ſit, ut ſupra dictum eſt, di-<lb/>cta hora ſumma exactitudine eſt obſervanda; </s> <s xml:id="echoid-s4510" xml:space="preserve">unumquodque <lb/>enim minutum quo in calculo aberras conſtituit errorem {1/4}. <lb/></s> <s xml:id="echoid-s4511" xml:space="preserve">gradus in longitudine, id eſt, prope Æquatorem 3{1/4}. </s> <s xml:id="echoid-s4512" xml:space="preserve">Germa-<lb/>nicorum milliarium ſed minus ubi longe inde abes:</s> <s xml:id="echoid-s4513" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4514" xml:space="preserve">Quare ad certam horæ inventionem, ne fidas obſervationi <lb/>maximæ Solis altitudinis, ut inde concludas præciſe meridiem <lb/>eſſe vel Solem in Meridiano, niſi inter Tropicos Sol fuerit in <lb/>ipſo puncto Zenith vel ei quam proximus. </s> <s xml:id="echoid-s4515" xml:space="preserve">Nam alias Sol <lb/>exiſtens prope Meridianum aliquamdiu perſeverat ſine ulla <lb/>ſenſibili mutatione altitudinis ſuæ; </s> <s xml:id="echoid-s4516" xml:space="preserve">ideoque altitudo meri-<lb/>dialis idonea ſatis eſt ad latitudinem vel elevationem Poli <lb/>loci alicujus dimetiendam, non tamen ad longitudinem ejus <lb/>exacte inveniendam. </s> <s xml:id="echoid-s4517" xml:space="preserve">Multo minus niti potes pyxidibus nau-<lb/>ticis in accurato meridiei tempore inquirendo.</s> <s xml:id="echoid-s4518" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4519" xml:space="preserve">Neque annuli Aſtronomici vel alia horologia ſolaria certa <lb/>ſatis ſunt in oſtendenda hora ad minuta & </s> <s xml:id="echoid-s4520" xml:space="preserve">ſecunda. </s> <s xml:id="echoid-s4521" xml:space="preserve">Sed me-<lb/>lius eſt obſervare ſolis altitudinem, ubi eſt in Oriente vel <lb/>Occidente, quo autem Orienti aut Occidenti propior eſt, <lb/>eo melius; </s> <s xml:id="echoid-s4522" xml:space="preserve">ibi enim cum eſt, mutatur ejus altitudo ſen-<lb/>ſibiliter magis quam ante vel poſt & </s> <s xml:id="echoid-s4523" xml:space="preserve">ita, ex inventa Poli ele-<lb/>vatione, & </s> <s xml:id="echoid-s4524" xml:space="preserve">nota Solis declinatione, hora poteſt computatione <lb/>detegi, cujus modus ab aliis ſatis deſcriptus eſt; </s> <s xml:id="echoid-s4525" xml:space="preserve">quia tamen <lb/>computatio illa moleſta eſt & </s> <s xml:id="echoid-s4526" xml:space="preserve">nonnulli errores in menſuranda <lb/>altitudine Solis committi poſſunt faciliorem hic methodum <lb/>oſtendam & </s> <s xml:id="echoid-s4527" xml:space="preserve">demonſtrabo.</s> <s xml:id="echoid-s4528" xml:space="preserve"/> </p> <pb o="205" file="0283" n="310" rhead="DE USU HOROLOG."/> </div> <div xml:id="echoid-div412" type="section" level="1" n="160"> <head xml:id="echoid-head193" xml:space="preserve">VIII.</head> <head xml:id="echoid-head194" style="it" xml:space="preserve">Quomodo ex obſervatione ortus & occaſus Solis & ex <lb/>hora horologiorum longitudo mari inveniri <lb/>queat.</head> <p> <s xml:id="echoid-s4529" xml:space="preserve">Hæc ſane methodus meo judicio omnium eſt certiſſim@, <lb/>cum ad eam neque notitia Poli elevationis, neque Solis de-<lb/>clinationis, neque ulla inſtrumenta ad obſervandum requi-<lb/>rantur; </s> <s xml:id="echoid-s4530" xml:space="preserve">cum neque refractio quid nocere poſſit, quoniam <lb/>hæc in ortu & </s> <s xml:id="echoid-s4531" xml:space="preserve">occaſu Solis ejusdem diei parum aut nihil <lb/>differre poteſt.</s> <s xml:id="echoid-s4532" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4533" xml:space="preserve">Sicuti ergo antea docuimus horologia in naviadaptare, & </s> <s xml:id="echoid-s4534" xml:space="preserve"><lb/>obſervare quâ horâ eorundem horologiorum Sol fuerit in <lb/>meridiano; </s> <s xml:id="echoid-s4535" xml:space="preserve">hic eodem modo procedendum eſt, id eſt, in or-<lb/>tu & </s> <s xml:id="echoid-s4536" xml:space="preserve">occaſu ſolis, ubi ejus medietas eſt ſupra horizontem, <lb/>annotabis horam demonſtratam tunc temporis per horolo-<lb/>gia; </s> <s xml:id="echoid-s4537" xml:space="preserve">& </s> <s xml:id="echoid-s4538" xml:space="preserve">licet interea velificando fueris progreſſus, nil re-<lb/>fert, uti poſtea demonſtrabitur: </s> <s xml:id="echoid-s4539" xml:space="preserve">dein computans quot in-<lb/>terea horæ horologiorum dilapſæ ſint, earumque dimidium <lb/>addens ad horam ortus, habebis horam quam horologium <lb/>indicabat quum Sol erat in Meridiano; </s> <s xml:id="echoid-s4540" xml:space="preserve">ad quam addenda eſt <lb/>æquatio iſtius diei ex tabula deſumta; </s> <s xml:id="echoid-s4541" xml:space="preserve">& </s> <s xml:id="echoid-s4542" xml:space="preserve">ſi ſumma æqualis <lb/>ſit 12 horis, fuiſti meridie ſub eodem Meridiano, ſub quo <lb/>horologia ad ſolem fuere adaptata; </s> <s xml:id="echoid-s4543" xml:space="preserve">ſed ſi ſumma excedat 12. <lb/></s> <s xml:id="echoid-s4544" xml:space="preserve">horas, fuiſti meridie ſub occidentaliori meridiano, quam <lb/>loci ejus in quo horologia ſunt diſpoſita; </s> <s xml:id="echoid-s4545" xml:space="preserve">ſed ſi ſumma fuerit <lb/>minor 12. </s> <s xml:id="echoid-s4546" xml:space="preserve">horis fuiſti ſub orientaliori Meridiano, idque toties <lb/>quindecim gradibus, quot horis ſumma fuerit minor vel exceſſe-<lb/>rit 12. </s> <s xml:id="echoid-s4547" xml:space="preserve">horas, prout iſtius rei computationem antea jam docuimus.</s> <s xml:id="echoid-s4548" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4549" xml:space="preserve">E. </s> <s xml:id="echoid-s4550" xml:space="preserve">G. </s> <s xml:id="echoid-s4551" xml:space="preserve">Ponatur Horologia A. </s> <s xml:id="echoid-s4552" xml:space="preserve">& </s> <s xml:id="echoid-s4553" xml:space="preserve">B, ut ante, fuiſſe ad-<lb/>aptata ad Solem in loco ex quo deceſſiſti 2. </s> <s xml:id="echoid-s4554" xml:space="preserve">Martii, id <lb/>eſt ad horam Solis diminutam æquatione iſtius diei ſcil. </s> <s xml:id="echoid-s4555" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s4556" xml:space="preserve">min. </s> <s xml:id="echoid-s4557" xml:space="preserve">28. </s> <s xml:id="echoid-s4558" xml:space="preserve">ſec. </s> <s xml:id="echoid-s4559" xml:space="preserve">Horologio A. </s> <s xml:id="echoid-s4560" xml:space="preserve">ad rectam menſuram redacto; </s> <s xml:id="echoid-s4561" xml:space="preserve"><lb/>B. </s> <s xml:id="echoid-s4562" xml:space="preserve">vero ſingulis diebus 7. </s> <s xml:id="echoid-s4563" xml:space="preserve">ſecundis tardius moto; </s> <s xml:id="echoid-s4564" xml:space="preserve">Poſtea <lb/>ſcire deſiderans longitudinem loci in quem perveniſti, <pb o="206" file="0284" n="311" rhead="CHRIST. HUGENII INSTITUTIO"/> (pone 1. </s> <s xml:id="echoid-s4565" xml:space="preserve">Junii,) obſervetur mane ſol medius <lb/> <anchor type="note" xlink:label="note-0284-01a" xlink:href="note-0284-01"/> ſupra horizontem quando horologium indicat <lb/>Et Veſperi iterum medius Sol infra horizon-<lb/>tem, cum idem horologium indicat</s> </p> <div xml:id="echoid-div412" type="float" level="2" n="1"> <note position="right" xlink:label="note-0284-01" xlink:href="note-0284-01a" xml:space="preserve"> <lb/>h. # m. # ſ. <lb/>2. # 30. # 37. <lb/></note> </div> <note position="right" xml:space="preserve"> <lb/>3. # 9. # 7. <lb/></note> <p> <s xml:id="echoid-s4566" xml:space="preserve">Ad inveniendashoras interea elapſas, ſub-<lb/>trahe horam ortus</s> </p> <note position="right" xml:space="preserve"> <lb/>2. # 30. # 37. <lb/>ex 12. # 0. # 0. <lb/>9. # 29. # 23. <lb/>3. # 9. # 7. <lb/>12. # 38. # 30. <lb/>6. # 19. # 15. <lb/>2. # 30. # 37. <lb/></note> <p> <s xml:id="echoid-s4567" xml:space="preserve">Reliquum eſt</s> </p> <p> <s xml:id="echoid-s4568" xml:space="preserve">Huic adde horam occaſus</s> </p> <p> <s xml:id="echoid-s4569" xml:space="preserve">Prodeunt horæ interea elapſæ</s> </p> <p> <s xml:id="echoid-s4570" xml:space="preserve">Quarum dimidio</s> </p> <p> <s xml:id="echoid-s4571" xml:space="preserve">Addito ad horam ortus</s> </p> <p> <s xml:id="echoid-s4572" xml:space="preserve">Habebis horam Horologii A, quum Sol erat <lb/>in Meridiano</s> </p> <note position="right" xml:space="preserve"> <lb/>8. # 49. # 52. <lb/></note> <p> <s xml:id="echoid-s4573" xml:space="preserve">Eodem modo quæratur hora horologii B, <lb/>cum Sol erat in Meridiano, quæ ſit</s> </p> <note position="right" xml:space="preserve"> <lb/>8. # 38. # 5. <lb/></note> <p> <s xml:id="echoid-s4574" xml:space="preserve">Sed hoc horologium ſingulis diebus 7. </s> <s xml:id="echoid-s4575" xml:space="preserve">ſe-<lb/>cundis tardius motum retardatur ſpatio 101. <lb/></s> <s xml:id="echoid-s4576" xml:space="preserve">dierum a 2. </s> <s xml:id="echoid-s4577" xml:space="preserve">Martii ad 1. </s> <s xml:id="echoid-s4578" xml:space="preserve">Junii</s> </p> <note position="right" xml:space="preserve"> <lb/>0. # 11. # 47. <lb/></note> <p> <s xml:id="echoid-s4579" xml:space="preserve">Quæ propterea ad inventam horam addita, <lb/>datur</s> </p> <note position="right" xml:space="preserve"> <lb/>8. # 49. # 52. <lb/></note> <p> <s xml:id="echoid-s4580" xml:space="preserve">Id eſt, eadem hora, quæ per horologium <lb/>A. </s> <s xml:id="echoid-s4581" xml:space="preserve">inventa eſt; </s> <s xml:id="echoid-s4582" xml:space="preserve">ad quam nunc addita æqua-<lb/>tione 1. </s> <s xml:id="echoid-s4583" xml:space="preserve">Junii</s> </p> <note position="right" xml:space="preserve"> <lb/>0. # 18. # 10. <lb/>9. # 8. # 2. <lb/></note> <p> <s xml:id="echoid-s4584" xml:space="preserve">Prodit</s> </p> <p> <s xml:id="echoid-s4585" xml:space="preserve">Hæc eſt diei hora loci in quo horologia ad-<lb/>aptata ſunt, quæ cum coincidit cum meridie <lb/>loci obſervationis;</s> <s xml:id="echoid-s4586" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4587" xml:space="preserve">Differentia eſt</s> </p> <note position="right" xml:space="preserve"> <lb/>2. # 51. # 58. <lb/></note> <p> <s xml:id="echoid-s4588" xml:space="preserve">Quare hic ultimus Meridianus tanto orienta-<lb/>lior eſt; </s> <s xml:id="echoid-s4589" xml:space="preserve">quibus horis reductis ad gradus, <lb/> <anchor type="note" xlink:label="note-0284-10a" xlink:href="note-0284-10"/> uti ſupra docuimus prodeunt</s> </p> <div xml:id="echoid-div413" type="float" level="2" n="2"> <note position="right" xlink:label="note-0284-10" xlink:href="note-0284-10a" xml:space="preserve"> <lb/>gr. # min. # ſec. <lb/>42. # 59. # 30. <lb/></note> </div> <p> <s xml:id="echoid-s4590" xml:space="preserve">Patet te hoc modo invenire longitudinem loci, in <lb/>quo meridie vel Sole exiſtente in Meridiano fuiſti; </s> <s xml:id="echoid-s4591" xml:space="preserve">quæ <lb/>differt a longitudine loci, in quo obſervas Solis occa- <pb o="207" file="0285" n="312" rhead="DE USU HOROLOG."/> ſum, ſed ſine ſenſibili errore æſtimare potes, quantum pau-<lb/>cis horis progreſſus fueris, vel longitudo mutata fuerit: </s> <s xml:id="echoid-s4592" xml:space="preserve">Po-<lb/>tes etiam, loco obſervationis ortus & </s> <s xml:id="echoid-s4593" xml:space="preserve">occaſus Solis, prius So-<lb/>lis occaſum, veſperi obſervare, & </s> <s xml:id="echoid-s4594" xml:space="preserve">dein proximo mane ortum, <lb/>utroque tempore notando horam Horologiorum; </s> <s xml:id="echoid-s4595" xml:space="preserve">& </s> <s xml:id="echoid-s4596" xml:space="preserve">inde <lb/>computa eodem modo horam in loco diſpoſitionis, quum <lb/>media nox erat in loco obſervationis, & </s> <s xml:id="echoid-s4597" xml:space="preserve">detege differen-<lb/>tiam longitudinis ut ante. </s> <s xml:id="echoid-s4598" xml:space="preserve">Tandem potes quoque loco <lb/>ortus & </s> <s xml:id="echoid-s4599" xml:space="preserve">occaſus Solis obſervare duas æquales ſolis altitu-<lb/>dines ante & </s> <s xml:id="echoid-s4600" xml:space="preserve">poſt meridiem, annotando horam horologio-<lb/>rum, & </s> <s xml:id="echoid-s4601" xml:space="preserve">computando eodem modo, quo diximus de or-<lb/>tu & </s> <s xml:id="echoid-s4602" xml:space="preserve">occaſu; </s> <s xml:id="echoid-s4603" xml:space="preserve">conſiderandum tamen eſt, Solis altitudines <lb/>optime obſervari, quando Orienti vel Occidenti proximus <lb/>eſt, ut antea notatum.</s> <s xml:id="echoid-s4604" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4605" xml:space="preserve">Licet forte quis cenſeat, in praxi hujus methodi, inter ante <lb/>& </s> <s xml:id="echoid-s4606" xml:space="preserve">pomeridianam obſervationem, quieſcentem navem deſide-<lb/>rari, aut quæ parum transferatur; </s> <s xml:id="echoid-s4607" xml:space="preserve">certum tamen eſt, progre-<lb/>diendo nullum ſenſibilem errorem cauſari poſſe, in quan-<lb/>tum interea eundem teneas curſum, æquabili velocitate.</s> <s xml:id="echoid-s4608" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4609" xml:space="preserve">Primum enim, ſi curſum Orientem vel Occidentem ver-<lb/>ſus dirigas, nullus omnino error erit, ſed certo conclu-<lb/>dere poteris, qua longitudine meridie vel mediâ nocte fue-<lb/>ris, unde ergo, uti antea dictum eſt, ſatis exacte æſtimare <lb/>potes, ubi ſis ultimæ obſervationis tempore.</s> <s xml:id="echoid-s4610" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4611" xml:space="preserve">V. </s> <s xml:id="echoid-s4612" xml:space="preserve">G. </s> <s xml:id="echoid-s4613" xml:space="preserve">pone, quod ante meridiana Solis altitudo 10 gra-<lb/>duum obſervata ſit, quum horologia indicant 8. </s> <s xml:id="echoid-s4614" xml:space="preserve">horas & </s> <s xml:id="echoid-s4615" xml:space="preserve"><lb/>pomeridiana æqualis altitudo quum horologia indicant 2. <lb/></s> <s xml:id="echoid-s4616" xml:space="preserve">horas; </s> <s xml:id="echoid-s4617" xml:space="preserve">& </s> <s xml:id="echoid-s4618" xml:space="preserve">quod inter utramque obſervationem æquabili ve-<lb/>locitate Orientem verſus navigaverim, licet inſcius me 1. </s> <s xml:id="echoid-s4619" xml:space="preserve">gra-<lb/>dum in longitudine progreſſum eſſe, id eſt, 1. </s> <s xml:id="echoid-s4620" xml:space="preserve">gradum paral-<lb/>leli juxta quem navigo: </s> <s xml:id="echoid-s4621" xml:space="preserve">Agens jam ſecundum præſcriptam <lb/>regulam, comperio longitudinem 15. </s> <s xml:id="echoid-s4622" xml:space="preserve">gradus Orientem ver-<lb/>ſus, calculum ineundo a loco, in quo Horologia fuere diſpo-<lb/>ſita. </s> <s xml:id="echoid-s4623" xml:space="preserve">quam longitudinem 15. </s> <s xml:id="echoid-s4624" xml:space="preserve">graduum dico eſſe loci, ubi <lb/>meridie fui.</s> <s xml:id="echoid-s4625" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4626" xml:space="preserve">Quod ita demonſtratur. </s> <s xml:id="echoid-s4627" xml:space="preserve">Quoniam locus veſpertinæ obſer- <pb o="208" file="0286" n="313" rhead="CHRIST. HUGENII INSTITUTIO"/> vationis uno gradu Orientalior eſt, quam matutinæ cer-<lb/>tum eſt in loco veſpertinæ obſervationis Solem 4. </s> <s xml:id="echoid-s4628" xml:space="preserve">minuta <lb/>prius perventurum eſſe, ad altitudinem 10. </s> <s xml:id="echoid-s4629" xml:space="preserve">graduum, quam in <lb/>loco matutinæ. </s> <s xml:id="echoid-s4630" xml:space="preserve">Duorum enim locorum ſub eodem paralelo <lb/>ſitorum, quot gradus unus altero Orientalior eſt, totidem <lb/>4. </s> <s xml:id="echoid-s4631" xml:space="preserve">minutis prius in illo obſervantur ſingulæ Solis altitudines; <lb/></s> <s xml:id="echoid-s4632" xml:space="preserve">ideo ſi in priori loco cum navi ſubſtitiſſem, Solem veſperti-<lb/>nâ obſervatione reperiſſem ad altitudinem 10. </s> <s xml:id="echoid-s4633" xml:space="preserve">graduum quum <lb/>horologia indicabant non 2. </s> <s xml:id="echoid-s4634" xml:space="preserve">horas, ſed 2. </s> <s xml:id="echoid-s4635" xml:space="preserve">horas 4. </s> <s xml:id="echoid-s4636" xml:space="preserve">min. </s> <s xml:id="echoid-s4637" xml:space="preserve">ubi <lb/>tum loci ejus longitudinem, juxta regulam, inveniſſem 14 {1/2}. </s> <s xml:id="echoid-s4638" xml:space="preserve"><lb/>gradus Orientem verſus. </s> <s xml:id="echoid-s4639" xml:space="preserve">Sed certum eſt, me in priori temporis <lb/>dimidio inter 2. </s> <s xml:id="echoid-s4640" xml:space="preserve">obſervationes progreſſum fuiſſe {1/2} gradum, <lb/>quoniam ponitur, me toto tempore profeciſſe 1. </s> <s xml:id="echoid-s4641" xml:space="preserve">gradum, <lb/>velocitatemque fuiſſe æquabilem. </s> <s xml:id="echoid-s4642" xml:space="preserve">Eram igitur meridie in <lb/>longitudine 15. </s> <s xml:id="echoid-s4643" xml:space="preserve">graduum Orientem verſus, ſicuti prius erat <lb/>inventum.</s> <s xml:id="echoid-s4644" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4645" xml:space="preserve">Pariter poteſt demonſtrari, progreſſum navis Occiden-<lb/>tem verſus nil obſtare, ſed regulam ſequendo, iterum in-<lb/>venire longitudinem loci, quem meridie præternavigaſti.</s> <s xml:id="echoid-s4646" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4647" xml:space="preserve">Si jam curſus inter 2. </s> <s xml:id="echoid-s4648" xml:space="preserve">obſervationes Meridiem verſus <lb/>vel Septentrionem deſciſcat, imo licet fieret directe Septen-<lb/>trionem vel meridiem verſus, modo ponatur æquabilis velo-<lb/>citas, nullus inde orietur error ſi Solis altitudo ſumatur, <lb/>quando prope Orientem vel Occidentem eſt, quibus in lo-<lb/>cis alibi quoque dictum eſt optime fieri obſervationem. <lb/></s> <s xml:id="echoid-s4649" xml:space="preserve">Ratio hæc eſt, quando 2. </s> <s xml:id="echoid-s4650" xml:space="preserve">loca Septentrionaliter vel Au-<lb/>ſtraliter a ſe mutuò diſtant, & </s> <s xml:id="echoid-s4651" xml:space="preserve">ſolummodo 1. </s> <s xml:id="echoid-s4652" xml:space="preserve">vel 2. </s> <s xml:id="echoid-s4653" xml:space="preserve">gra-<lb/>dus in latitudine differunt, ſi Sol reſpectu unius loci in O-<lb/>riente vel Occidente verſetur, ad certam ſupra horizon-<lb/>tem altitudinem, etiam quam proxime eodem tempore ad <lb/>eandem altitudinem ſupra horizontem alterius loci appare-<lb/>bit. </s> <s xml:id="echoid-s4654" xml:space="preserve">Ita ut comperiam, licet navis inter matutinam & </s> <s xml:id="echoid-s4655" xml:space="preserve"><lb/>veſpertinam obſervationem 2. </s> <s xml:id="echoid-s4656" xml:space="preserve">gradus navigaret, quod ra-<lb/>ro vel nunquam accidit, nullum tamen in longitudine er-<lb/>rorem hinc oriri poſſe, vel tantum paucorum minuto-<lb/>rum.</s> <s xml:id="echoid-s4657" xml:space="preserve"/> </p> <pb o="209" file="0287" n="314" rhead="DE USU HOROLOG."/> <p> <s xml:id="echoid-s4658" xml:space="preserve">Præſcriptis ergo modis, vel per ortum & </s> <s xml:id="echoid-s4659" xml:space="preserve">occaſum Solis, <lb/>vel per occaſum & </s> <s xml:id="echoid-s4660" xml:space="preserve">ortum Solis, vel per 2 æquales Solis altitu-<lb/>dines, tuto ſemper uti poſſumus, non obſtante progreſſu <lb/>navium; </s> <s xml:id="echoid-s4661" xml:space="preserve">quemcunque hæ teneant curſum.</s> <s xml:id="echoid-s4662" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4663" xml:space="preserve">Si autem procul ab Æquatore Septentrionem vel Meri-<lb/>diem verſus naviges, præſertim hyeme, altitudo Solis lente <lb/>mutatur, unde incertæ ſunt obſervationes; </s> <s xml:id="echoid-s4664" xml:space="preserve">ſed iis in locis <lb/>gradus longitudinis ſunt breviores, vel pauciora milliaria <lb/>continent quam prope Æquatorem, ideoque errores in in-<lb/>veniendis longitudinibus eo minus ſenſibiles ſunt.</s> <s xml:id="echoid-s4665" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div415" type="section" level="1" n="161"> <head xml:id="echoid-head195" xml:space="preserve">IX.</head> <p> <s xml:id="echoid-s4666" xml:space="preserve">Potes verò, præſertim in iis oris, quæ procul ab Æqua-<lb/>tore Septentrionem vel Auſtrum verſus remotæ ſunt, vel <lb/>etiam ubicunque velis, præſcriptam regulam ad praxin vo-<lb/>care, obſervando 2 æquales altitudines cognitæ alicujus ſtel-<lb/>læ, quæ multum attollitur ſupra Horizontem. </s> <s xml:id="echoid-s4667" xml:space="preserve">Nam inde <lb/>ſecundum memoratam regulam, diſces quâ horâ Horologio-<lb/>rum in Meridiano fuerit ſtella, & </s> <s xml:id="echoid-s4668" xml:space="preserve">porro cognita ejuſdem <lb/>aſcenſione recta, ut & </s> <s xml:id="echoid-s4669" xml:space="preserve">aſcenſione recta Solis, facile inde ſup-<lb/>putabis horam ſolarem, qua comparatâ cum horâ Horolo-<lb/>giorum, ut ante, habebis longitudinem loci, ubi fuiſti ſtella <lb/>exiſtente in Meridiano.</s> <s xml:id="echoid-s4670" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div416" type="section" level="1" n="162"> <head xml:id="echoid-head196" xml:space="preserve">X.</head> <p> <s xml:id="echoid-s4671" xml:space="preserve">Quando Horologia, quorum aliquamdiu motus fuit accu-<lb/>ratus, ab invicem paululum differunt, prout diuturnitate <lb/>temporis facile accidit, ut unum vel alterum minuto cir-<lb/>citer deficiat, eo in caſu computatio ineunda eſt erit ſe-<lb/>cundum iſtud, quod celerius movetur: </s> <s xml:id="echoid-s4672" xml:space="preserve">niſi noris cauſam <lb/>veriſimilem ob quam citius moveatur; </s> <s xml:id="echoid-s4673" xml:space="preserve">facilius enim Pendu-<lb/>li motus retardatur, quàm acceleratur: </s> <s xml:id="echoid-s4674" xml:space="preserve">nam filum, cui Pen-<lb/>dulum appendet, poterit forſan per violentam navis agitatio-<lb/>nem nonnihil extendi, ſed nequit contrahi.</s> <s xml:id="echoid-s4675" xml:space="preserve"/> </p> <pb o="210" file="0288" n="315" rhead="CHRIST. HUGENII INSTITUTIO"/> </div> <div xml:id="echoid-div417" type="section" level="1" n="163"> <head xml:id="echoid-head197" xml:space="preserve">XI.</head> <p> <s xml:id="echoid-s4676" xml:space="preserve">Quando videndam ſe offert regio cognita, ne negli-<lb/>gas annotare longitudinem illius, quantum exacte fieri <lb/>poterit, ex inventa longitudine loci in quo verſaris; </s> <s xml:id="echoid-s4677" xml:space="preserve">Pri-<lb/>mo ad corrigendas inde mappas marinas, poſtquam longitu-<lb/>do loci fuerit diverſis temporibus comperta eadem, ita ut <lb/>nil amplius de ea dubites. </s> <s xml:id="echoid-s4678" xml:space="preserve">In his enim mappis quantum atti-<lb/>net ad ſitum locorum Orientem & </s> <s xml:id="echoid-s4679" xml:space="preserve">Occidentem verſus, multa ſu-<lb/>perſunt emendanda. </s> <s xml:id="echoid-s4680" xml:space="preserve">Secundo ut ſcias, in proſecutione tui <lb/>itineris, quantum reſpectu loci viſi velificando progreſſus <lb/>fueris ad Orientem vel Occidentem: </s> <s xml:id="echoid-s4681" xml:space="preserve">etiamſi infortun@o, <lb/>vel negligentia, omnia quieſcant Horologia, poteris eadem <lb/>ad motum rurſus adaptare, & </s> <s xml:id="echoid-s4682" xml:space="preserve">diſponere ad horam per So-<lb/>lem compertam. </s> <s xml:id="echoid-s4683" xml:space="preserve">computando porro longitudinem ab ejus-<lb/>dem loci viſi Meridiano. </s> <s xml:id="echoid-s4684" xml:space="preserve">Nam nullatenus teneris certum <lb/>Meridianum alicujus loci cogniti, pro initio computationis <lb/>longitudinum habere, cujus uſus tantum eſt in mappis vel <lb/>Tabulis longitudinum, in quo caſu uſu venit Meridianus <lb/>montis Pici in inſula Teneriffa, vel Meridianus inſularum <lb/>Corvo & </s> <s xml:id="echoid-s4685" xml:space="preserve">Flores, occidentaliſſimarum Azorum vel inſula-<lb/>rum Flandricarum, vel cujusvis alterius loci: </s> <s xml:id="echoid-s4686" xml:space="preserve">& </s> <s xml:id="echoid-s4687" xml:space="preserve">foret egre-<lb/>gium (quod non obtinet) ſi omnes auctores unum eundem-<lb/>que Meridianum pro primo eligerent, ut ſingula loca iis-<lb/>dem gradibus longitudinis pariter ac latitudinis determina-<lb/>rentur: </s> <s xml:id="echoid-s4688" xml:space="preserve">ſed in itinere ſatis eſt longitudinum differentiam ob-<lb/>ſervare, initio computationis facto a Meridiano cujuscunque <lb/>loci.</s> <s xml:id="echoid-s4689" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div418" type="section" level="1" n="164"> <head xml:id="echoid-head198" xml:space="preserve">XII.</head> <p> <s xml:id="echoid-s4690" xml:space="preserve">Si accidat, ut mari medio omnia Horologia quieſcant, <lb/>quam primum fieri poteſt, rurſus eadem ad motum adapta, <pb o="211" file="0289" n="316" rhead="DE USU HOROLOG."/> ut illorum ope ſcias, quantum dein ad Orientem vel Occi-<lb/>dentem progreſſus ſis, quod non parvi eſt momenti, nam <lb/>defectu hujus notitiæ, nonnunquam violentis fluctibus ita <lb/>abriperis, ut, licet vento ſecundo naviges, tamen retror-<lb/>ſum abigaris, cujus rei varia dantur exempla.</s> <s xml:id="echoid-s4691" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div419" type="section" level="1" n="165"> <head xml:id="echoid-head199" xml:space="preserve">FINIS.</head> <figure> <image file="0289-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0289-01"/> </figure> </div> <div xml:id="echoid-div420" type="section" level="1" n="166"> <head xml:id="echoid-head200" xml:space="preserve">EXCERPTA EX LITERIS DATIS <lb/>LONDINI {13/23} JANUARII <lb/>MDCLXV.</head> <p style="it"> <s xml:id="echoid-s4692" xml:space="preserve">IAndem Navarchus Holmius huc advenit, & </s> <s xml:id="echoid-s4693" xml:space="preserve">re-<lb/>latio, quam ipſe mihi communicavit de experimen-<lb/>to, quod de noſtris Horologiis fecit, plane nos cer-<lb/>tos facit de bono, quiſperandus eſt, ſucceſſu; </s> <s xml:id="echoid-s4694" xml:space="preserve">Cum <lb/>ad inſulam St. </s> <s xml:id="echoid-s4695" xml:space="preserve">Thomæ ſub æquatore eſſet, ut inde huc veni-<lb/>ret, coactus fuit, longiſſime Occidentem verſus curſum diri-<lb/>gere, proſperum ut obtineret ventum; </s> <s xml:id="echoid-s4696" xml:space="preserve">cum ergo ſub ejus au-<lb/>ſpiciis quatuor darentur naves, omnes ſimul navigarunt per <lb/>600 milliaria non mutato curſu: </s> <s xml:id="echoid-s4697" xml:space="preserve">cum dein ſecundum ventum <lb/>nacti eſſent quo peterent littora Africæ, eo tendebant, curſum <lb/>ſuum Orientem inter & </s> <s xml:id="echoid-s4698" xml:space="preserve">Septentrionem dirigentes; </s> <s xml:id="echoid-s4699" xml:space="preserve">cumque hoc <lb/>curſu 4 vel 500 milliaria confeciſſent, judicabant navitæ 3 <lb/>navium quæ ſub ejus erant auſpiciis, ſe procul a prædictâ or â <lb/>eſſe, ita ut non ſufficientem aquæ quantitatem haberent, dum <lb/>eo pervenient; </s> <s xml:id="echoid-s4700" xml:space="preserve">Navarchi Holmii navis ſatis h@bebat; </s> <s xml:id="echoid-s4701" xml:space="preserve">ſed <lb/>ubi audiret, quid ſentirent cæteri, qui tribus reliquis præ-<lb/>erant navibus, omnes nautas & </s> <s xml:id="echoid-s4702" xml:space="preserve">gubernatores convocari juſ- <pb o="212" file="0290" n="317" rhead="EXCERPTA EX LIT. CHRIST. HUGENII."/> ſit, ut deliberarent de eo quod faciendum eſſet. </s> <s xml:id="echoid-s4703" xml:space="preserve">Ubi verò vidiſſet <lb/>commentarios, & </s> <s xml:id="echoid-s4704" xml:space="preserve">ephemerides, quas illi, qui præerant 3 iis na-<lb/>vibus proferebant, & </s> <s xml:id="echoid-s4705" xml:space="preserve">conjecturam de loco in quo verſaban-<lb/>tur; </s> <s xml:id="echoid-s4706" xml:space="preserve">cumque poſt longas deliberationes, omnes cenſerent, præ-<lb/>ſtare ad Barbadas trajicere, quam oras Africæ quærere, <lb/>quia ventus longe citius eos illuc transferret, dixit ille: </s> <s xml:id="echoid-s4707" xml:space="preserve">Viri <lb/>computatio veſtra cum noſtra minimè convenit; </s> <s xml:id="echoid-s4708" xml:space="preserve">nam ſecun-<lb/>dum Horologia mea proceſſi, unde concludo, vos errare <lb/>omnes, ſtatuendo nos multò longius Occidentem verſus eſſe pro-<lb/>greſſos, quam revera ſumus, unius error eſt 120 milliaria, alîus <lb/>100, alîus 80: </s> <s xml:id="echoid-s4709" xml:space="preserve">verùm ita Horologiis confido ut hac vice experi-<lb/>mentum facere velim; </s> <s xml:id="echoid-s4710" xml:space="preserve">nam ſecundum meam computationem in-<lb/>ſula del Fuogo (quæ una eſt ex inſulis Capoverdæ ſeu promontorii <lb/>Heſperii,) non ultra 30 milliaria a nobis abeſt; </s> <s xml:id="echoid-s4711" xml:space="preserve">Ibi nobis a-<lb/>quam poſſumus curare, & </s> <s xml:id="echoid-s4712" xml:space="preserve">eo curſum dirigere conſtitui; </s> <s xml:id="echoid-s4713" xml:space="preserve">Si <lb/>vera eſt computatio, multum præſtabit nobis hanc viam in-<lb/>gredi, quam a vobis oſtenſam; </s> <s xml:id="echoid-s4714" xml:space="preserve">ſin evenerit aliter, biduum <lb/>tantum producemus navigationem, ſatisque aquæ interim ſu-<lb/>pererit, ut Barbadas perveniamus; </s> <s xml:id="echoid-s4715" xml:space="preserve">quoniam tanta mihi in <lb/>navi copia eſt, ut vobis defectum veſtrum ſupplere queam; <lb/></s> <s xml:id="echoid-s4716" xml:space="preserve">Addens quem curſum tenere vellet, mandavit ut ſequerentur <lb/>ipſum cæteri. </s> <s xml:id="echoid-s4717" xml:space="preserve">Poſtero mane detegebant iuſulam del Fuogo, & </s> <s xml:id="echoid-s4718" xml:space="preserve"><lb/>opportuno illuc advenêre tempore, uti prædixerat: </s> <s xml:id="echoid-s4719" xml:space="preserve">Cogor hic <lb/>deſinere, communicaturus tecum reliquos omnes caſus peculia-<lb/>res, quos ſcriptis mandare promiſit, quamprimum illos nactus <lb/>fuero.</s> <s xml:id="echoid-s4720" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4721" xml:space="preserve">Nota, hæc Horologia fuiſſe ex primo Pendulorum gene-<lb/>re, nec tam exacta quam recentiora.</s> <s xml:id="echoid-s4722" xml:space="preserve"/> </p> <pb o="213" file="0291" n="318"/> </div> <div xml:id="echoid-div421" type="section" level="1" n="167"> <head xml:id="echoid-head201" xml:space="preserve">EXCERPTA EX LITERIS HAGÆ CO-<lb/>MITUM, DIE XXVI. FEBRUAR <lb/>MDCLXV. DATIS.</head> <p style="it"> <s xml:id="echoid-s4723" xml:space="preserve">CUm per aliquot dies neceſſe mihi foret cubiculum <lb/>tenere, & </s> <s xml:id="echoid-s4724" xml:space="preserve">variis obſervationibus circa duo <lb/>mea novæ fabricæ Horologia pendula tempus im-<lb/>penderem, mirum quendam eorum effectum, & </s> <s xml:id="echoid-s4725" xml:space="preserve">a <lb/>nemine unquam vel cogitandum, detexi. </s> <s xml:id="echoid-s4726" xml:space="preserve">Suſpenſa enim juxta <lb/>ſe invicem ad diſtantiam unius aut duorum pedum, tam accu-<lb/>rate congruebant, ut ſine ullâ variatione pendulorum vibra-<lb/>tiones ſimul peragerentur. </s> <s xml:id="echoid-s4727" xml:space="preserve">Quod cum per aliquod tempus im-<lb/>penſe miratus fuiſſem, tandem reperi ex aliquâ quaſi ſympa-<lb/>thiâ id oriri; </s> <s xml:id="echoid-s4728" xml:space="preserve">ita ut ſi pendula moviſſem vibrationibus diverſis <lb/>& </s> <s xml:id="echoid-s4729" xml:space="preserve">ut ita dicam intermixtis, intra dimidiæ horæ ſpatium con-<lb/>ſona rurſus fierent, & </s> <s xml:id="echoid-s4730" xml:space="preserve">ſibi mutuo, quamdiu motum non tur-<lb/>babam, reſponderent. </s> <s xml:id="echoid-s4731" xml:space="preserve">Remotis deinde à ſe invicem Horologiis, <lb/>& </s> <s xml:id="echoid-s4732" xml:space="preserve">ad diſtantiam 15. </s> <s xml:id="echoid-s4733" xml:space="preserve">pedum ſuſpenſis, uno die quinque ſecun-<lb/>dorum diſcrepantiam animadverti, quæ clare demonſtravit, <lb/>priorem pendulorum convenientiam, uti dixi, à ſympathia <lb/>quadam debuiſſe proficiſci, quæ, meo quidem judicio, unicè <lb/>inſenſili aëris agitationi, per pendulorum motus productæ, ad-<lb/>ſcribi poteſt. </s> <s xml:id="echoid-s4734" xml:space="preserve">Continentur tamen Horologia ſuis Capſis, quarum <lb/>utraque ſi omne plumbum contentum numeres, centum fere li- <pb o="214" file="0292" n="319" rhead="EXCERPTA EX LIT. CHRIST. HUGENII."/> brarum pondus æquat. </s> <s xml:id="echoid-s4735" xml:space="preserve">Et hoc obſervandum, pendula, <lb/>cum congruunt, non parallelis motibus ferri, ſed contra-<lb/>riis, nunc accedendo, nunc recedendo. </s> <s xml:id="echoid-s4736" xml:space="preserve">Quando autem horolo-<lb/>gia ad parvam diſtantiam rurſus à me fuere poſita, pendula, <lb/>priorem convenientiam brevi tempore recuperarunt.</s> <s xml:id="echoid-s4737" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4738" xml:space="preserve">Nec hisce contentus, aſſerem latum tres pedes, craſ-<lb/>ſum unum pollicem, ita interpoſui, ut pars inferior fun-<lb/>dum tangeret, ſuperior Horologia excederet & </s> <s xml:id="echoid-s4739" xml:space="preserve">quaſi ſepararet. <lb/></s> <s xml:id="echoid-s4740" xml:space="preserve">Non turbata tamen fuit motuum concordia, ſed dies noctes <lb/>que perduravit, imò ſi ipſe turbaſſem, brevi inſtaurata fuit. </s> <s xml:id="echoid-s4741" xml:space="preserve"><lb/>Nunc in id incumbo, ut quam accuratiſſime Horologia con-<lb/>cordent, experturus deinde ad quam usque diſtantiam ſympa-<lb/>thia hæc ſeſe exerat; </s> <s xml:id="echoid-s4742" xml:space="preserve">ſed, ut ex obſervatis auguror, non in-<lb/>fra 5. </s> <s xml:id="echoid-s4743" xml:space="preserve">aut 6. </s> <s xml:id="echoid-s4744" xml:space="preserve">pedes ſubſiſtet. </s> <s xml:id="echoid-s4745" xml:space="preserve">Major tamen horum omnium eſt <lb/>expectanda veritas, quam præſtabit diligentia mea, & </s> <s xml:id="echoid-s4746" xml:space="preserve">ac-<lb/>curatior in rei cauſas inquiſitio.</s> <s xml:id="echoid-s4747" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4748" xml:space="preserve">Quicquid ſit, habemus duo Horologia, quæ nunquam inter <lb/>ſe diſcrepant: </s> <s xml:id="echoid-s4749" xml:space="preserve">quod etiamſi mirum ſit, tamen eſt veriſſimum. <lb/></s> <s xml:id="echoid-s4750" xml:space="preserve">Addo, præter Horologia quæ novo hoc invento conſtructæ <lb/>ſunt, nulla alia idem præſtitiſſe; </s> <s xml:id="echoid-s4751" xml:space="preserve">unde ſimul patet, quam <lb/>accurata illa ſint, quæ ad perpetuum conſenſum tantillum re-<lb/>quirunt.</s> <s xml:id="echoid-s4752" xml:space="preserve"/> </p> <figure> <image file="0292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0292-01"/> </figure> <pb file="0293" n="320"/> </div> <div xml:id="echoid-div422" type="section" level="1" n="168"> <head xml:id="echoid-head202" xml:space="preserve">DE <lb/>HUGENIANA <lb/>CENTRI <lb/>OSCILLATIONIS <lb/>DETERMINATIONE <lb/>CONTROVERSIA.</head> <pb file="0294" n="321"/> <pb file="0295" n="322"/> <figure> <image file="0295-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0295-01"/> </figure> </div> <div xml:id="echoid-div423" type="section" level="1" n="169"> <head xml:id="echoid-head203" xml:space="preserve">DE <lb/>HUGENIANA <lb/>CENTRI OSCILLATIONIS <lb/>DETERMINATIONE <lb/>CONTROVERSIA.</head> <head xml:id="echoid-head204" xml:space="preserve">I.</head> <head xml:id="echoid-head205" style="it" xml:space="preserve">Obſervationes Abbatis Catelani in propoſitio-<lb/>nem, quæ fundamentum eſt 4<emph style="super">æ</emph>. partis tra-<lb/>ctatus de Pendulis, Hugenii.</head> <p> <s xml:id="echoid-s4753" xml:space="preserve">DOminus Hugenius in tractatu ſuo de Pendulis, ut nil <lb/>quod ad materiam ſpectat intactum relinqueret, hanc <lb/>diviſit in 5 partes, in quarum 4<emph style="super">a</emph>. </s> <s xml:id="echoid-s4754" xml:space="preserve">fuſe examinat quæ-<lb/>ſtionem de Centro Oſcillationis vel vibrationis. </s> <s xml:id="echoid-s4755" xml:space="preserve">Sed <lb/>cum difficile ſit animo ſemper æqualiter attento abſtruſas ve-<lb/>ritates, quales ſunt Mathematicæ, perpendere; </s> <s xml:id="echoid-s4756" xml:space="preserve">non eſt quod <lb/>miremur, ſi quæſtionem illam non æque accurate, quam <lb/>quidem reliquas ejusdem operis, examinaverit. </s> <s xml:id="echoid-s4757" xml:space="preserve">Principium <lb/>autem, quo nititur totum ejus Syſtema Oſcillationis hoc eſt.</s> <s xml:id="echoid-s4758" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4759" xml:space="preserve">Si Pendulum e pluribus ponderibus compoſitum at que e quie-<lb/>te dimiſſum, partem quamcunque Oſcillationis integræ con-<lb/>fecerit; </s> <s xml:id="echoid-s4760" xml:space="preserve">at que inde porro intelligantur pondera ejus ſingula, <lb/>relicto communi vinculo, celeritates acquiſitas ſurſum con-<lb/>vertere, ac quo usque poſſunt aſcendere; </s> <s xml:id="echoid-s4761" xml:space="preserve">hoc facto centrum <lb/>gravitatis ex omnibus compoſitæ, ad eandem altitudinem re-<lb/>verſum erit, quam ante inceptam Oſcillationem obtinebat <anchor type="note" xlink:href="" symbol="*"/>.</s> <s xml:id="echoid-s4762" xml:space="preserve"/> </p> <note symbol="*" position="right" xml:space="preserve">Vide ſ@-<lb/>pra pag. 126.</note> <p> <s xml:id="echoid-s4763" xml:space="preserve">Ut parum firmam propoſitionem hanc demonſtremus, ſuffi-<lb/>ciet obſervaſſe; </s> <s xml:id="echoid-s4764" xml:space="preserve">vim, quam vocamus gravitatem, longe ali-<lb/>ter agere in pondera inter ſe juncta quam in pondera a ſe in-<lb/>vicem ſeparata. </s> <s xml:id="echoid-s4765" xml:space="preserve">Sint A & </s> <s xml:id="echoid-s4766" xml:space="preserve">B æqualia pondera, quorum <lb/> <anchor type="note" xlink:label="note-0295-02a" xlink:href="note-0295-02"/> <pb o="218" file="0296" n="323" rhead="DE CENTRO OSCILL"/> non conſiderantur neque magnitudo neque figura, quaſi ſin-<lb/>gula in unicum punctum reducta forent; </s> <s xml:id="echoid-s4767" xml:space="preserve">ſi ſeparatim ſuſ-<lb/>penſa ex eodem puncto D, & </s> <s xml:id="echoid-s4768" xml:space="preserve">elevata ad idem planum Ho-<lb/>rizontale D A B, dimittantur usque ad F & </s> <s xml:id="echoid-s4769" xml:space="preserve">G, gravitates <lb/>eorum, ex ratione Mechanica, quæ cum experimentis, & </s> <s xml:id="echoid-s4770" xml:space="preserve">prin-<lb/>cipiis Phyſices congruit, augebuntur in tali ratione, vel quod <lb/>idem eſt, acquirent velocitates tales, ut harum quadrata ſint <lb/>inter ſe ut altitudines A H & </s> <s xml:id="echoid-s4771" xml:space="preserve">B I. </s> <s xml:id="echoid-s4772" xml:space="preserve">unde illa pondera per-<lb/>pendiculariter deſcendunt ad Horizontem.</s> <s xml:id="echoid-s4773" xml:space="preserve"/> </p> <div xml:id="echoid-div423" type="float" level="2" n="1"> <note position="right" xlink:label="note-0295-02" xlink:href="note-0295-02a" xml:space="preserve">TAB XXVIII. <lb/>Pig. 1.</note> </div> <p> <s xml:id="echoid-s4774" xml:space="preserve">Quod ſi dein pondera hæc duo, lineâ aut virgâ inflexi-<lb/>li A B, quam pondere expertem ponimus, conjungamus, <lb/>& </s> <s xml:id="echoid-s4775" xml:space="preserve">ex eodem puncto D, ad memoratas diſtantias D A & </s> <s xml:id="echoid-s4776" xml:space="preserve">D B, <lb/>ſuſpenſa, dimittamus ad F & </s> <s xml:id="echoid-s4777" xml:space="preserve">G ab eâdem, quâ ante, alti-<lb/>tudine. </s> <s xml:id="echoid-s4778" xml:space="preserve">Pendulum ex illis compoſitum, acquiret tantum <lb/>velocitatis quantum ſumma duorum Pendulorum ſimplicium, <lb/>quoniam commune gravitatis centrum E idem, quod antea, <lb/>manebit, & </s> <s xml:id="echoid-s4779" xml:space="preserve">ponderum non mutatur ſitus reſpectu centri Telluris; <lb/></s> <s xml:id="echoid-s4780" xml:space="preserve">ſed partes, in quas tota illa velocitas ſe diſtribuet ponderibus <lb/>A & </s> <s xml:id="echoid-s4781" xml:space="preserve">B, erunt inter ſe ut arcus A F, B G, vel ut radii D F, D G; </s> <s xml:id="echoid-s4782" xml:space="preserve"><lb/>quoniam in hoc caſu ratio inter motus ponderum pendebit <lb/>ab eorum ſitu reſpectu puncti ſuſpenſionis D, quod eſt mo-<lb/>tuum centrum. </s> <s xml:id="echoid-s4783" xml:space="preserve">Triangula autem H A F & </s> <s xml:id="echoid-s4784" xml:space="preserve">I B G, ut & </s> <s xml:id="echoid-s4785" xml:space="preserve"><lb/>triangula A F D, B G D cum ſint ſimilia, latera eorum <lb/>A H & </s> <s xml:id="echoid-s4786" xml:space="preserve">B I, A F & </s> <s xml:id="echoid-s4787" xml:space="preserve">B G, D F & </s> <s xml:id="echoid-s4788" xml:space="preserve">D G ſunt proportio-<lb/>nalia, id eſt datur eadem ratio inter altitudines, unde pon-<lb/>dera A & </s> <s xml:id="echoid-s4789" xml:space="preserve">B deſcendunt, & </s> <s xml:id="echoid-s4790" xml:space="preserve">inter velocitates quas acquirunt <lb/>deſcendendo; </s> <s xml:id="echoid-s4791" xml:space="preserve">ſed altitudines ſunt eædem ac in priori ſup-<lb/>poſitione; </s> <s xml:id="echoid-s4792" xml:space="preserve">ergo velocitates ſunt diverſæ, quoniam illæ altitu-<lb/>dines, quæ ſunt proportionales velocitatibus ponderum ſimul <lb/>appenſorum, non ſunt proportionales niſi quadratis veloci-<lb/>tatum, quando ſunt ſeparata.</s> <s xml:id="echoid-s4793" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4794" xml:space="preserve">Porro ponamus pendulum compoſitum in vibratione ſuâ <lb/>occurrere plano duro D F G, quo rumpatur, ita ut a ſe in-<lb/>vicem pondera ſolvantur, erunt hæc reflexa juxta tangentes ar-<lb/>cuum F A & </s> <s xml:id="echoid-s4795" xml:space="preserve">G B ad altitudines, quæ inter ſe erunt ut quadrata <lb/>velocitatum, quas cadendo acquiſivere, id eſt, ut quadrata <pb o="219" file="0297" n="324" rhead="CONTROVERSIA."/> radiorum D F & </s> <s xml:id="echoid-s4796" xml:space="preserve">D G, vel horum proportionalium A H & </s> <s xml:id="echoid-s4797" xml:space="preserve">B I <lb/>nam ſeparatio ponderum non mutat quantitatem motus eo-<lb/>rum: </s> <s xml:id="echoid-s4798" xml:space="preserve">efficit ut moveantur juxta legem corporum cadentium, <lb/>quæ non inter ſe conjuncta ſunt. </s> <s xml:id="echoid-s4799" xml:space="preserve">Demonſtratur in Me-<lb/>chanicis, altitudinem perpendicularem ad horizontem, un-<lb/>de deſcendit, vel ad quam aſcendit, commune gravitatis <lb/>centrum multorum ponderum, æqualem eſſe ſummæ altitu-<lb/>dinum, quarum reſpectu (gallice par raport auquelles). <lb/></s> <s xml:id="echoid-s4800" xml:space="preserve">pondera deſcendunt vel aſcendunt, diviſæ per eorundem nu-<lb/>merum: </s> <s xml:id="echoid-s4801" xml:space="preserve">ſed probavimus pondera, quæ ſeparantur, rupto <lb/>pendulo percuſſione in planum oſcillationi illius oppoſitum, <lb/>iterum aſcenſura eſſe, ad altitudines diverſas ab iis, unde <lb/>deſcenderunt, & </s> <s xml:id="echoid-s4802" xml:space="preserve">quidem tales, ut ſummæ ad utramque partem æ-<lb/>quales eſſe nequeant; </s> <s xml:id="echoid-s4803" xml:space="preserve">nam ultimæ altitudines ſemper habent pro ra-<lb/>dicibus quantitates primis proportionales, & </s> <s xml:id="echoid-s4804" xml:space="preserve">præterea eandem, quam <lb/>eorum radices componentes ſummam, quæ exprimit totam celerita-<lb/>tem penduli A B; </s> <s xml:id="echoid-s4805" xml:space="preserve">ſi ergo diverſas illas ſummas ſeparatim divi-<lb/>damus per numerum ponderum, habebimus altitudinem ad <lb/>quam centrum commune gravitatis iterum aſcendit, diver-<lb/>ſam ab illa, unde deſcendit; </s> <s xml:id="echoid-s4806" xml:space="preserve">quoniam ſunt partes aliquotæ <lb/>ſimiles quantitatum inæqualium.</s> <s xml:id="echoid-s4807" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4808" xml:space="preserve">Propoſitio igitur Domini Hugenii falſa eſt, & </s> <s xml:id="echoid-s4809" xml:space="preserve">conſequenter <lb/>quidquid concluſit circa centrum Oſcillationis corruit; </s> <s xml:id="echoid-s4810" xml:space="preserve">vera <lb/>autem Mathematica hujus quæſtionis ſolutio hæc eſt.</s> <s xml:id="echoid-s4811" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div425" type="section" level="1" n="170"> <head xml:id="echoid-head206" xml:space="preserve">II.</head> <head xml:id="echoid-head207" style="it" xml:space="preserve">Domini Abbatis Catelani Examen Ma-<lb/>thematicum Centri Oſcillationis.</head> <p> <s xml:id="echoid-s4812" xml:space="preserve">QUæſtio de determinando centro Oſcillationis, ſi bene in-<lb/>tellecta fuerit, haud difficilis eſt. </s> <s xml:id="echoid-s4813" xml:space="preserve">Centrum Oſcillationis <lb/>vocatur punctum mobile in Pendulo ad talem ab axe, <lb/>vel centro ſuſpenſionis, diſtantiam, ut ſi omnes aliæ Penduli <lb/>partes deſtruerentur, illa ſola pergeret in vibrationibus ut <lb/>antea; </s> <s xml:id="echoid-s4814" xml:space="preserve">id eſt eodem tempore ac totum Pendulum; </s> <s xml:id="echoid-s4815" xml:space="preserve">Quod ita <lb/>non fiet cum aliis partibus ſingulis ſeparatim ſumtis; </s> <s xml:id="echoid-s4816" xml:space="preserve">nam <pb o="220" file="0298" n="325" rhead="DE CENTRO OSCILL."/> quæ axi viciniores ſunt, breviores & </s> <s xml:id="echoid-s4817" xml:space="preserve">frequentiores vibrationes <lb/>peragent quam remotiores, ſi arcus ſimiles deſcribant, & </s> <s xml:id="echoid-s4818" xml:space="preserve">aër non <lb/>reſiſtat.</s> <s xml:id="echoid-s4819" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4820" xml:space="preserve">Cujus rei ratio eſt, quod viciniores deſcribant arcus mi-<lb/>nores & </s> <s xml:id="echoid-s4821" xml:space="preserve">acquirant celeritates majores reſpectu arcuum quam <lb/>remotiores: </s> <s xml:id="echoid-s4822" xml:space="preserve">nam arcus ſunt proportionales quadratis, & </s> <s xml:id="echoid-s4823" xml:space="preserve"><lb/>velocitates radicibus eorum; </s> <s xml:id="echoid-s4824" xml:space="preserve">quo autem radices minores ſunt <lb/>inter ſe, eo majores ſunt reſpectu quadratorum ſuorum.</s> <s xml:id="echoid-s4825" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4826" xml:space="preserve">Cum in Pendulo omnes partes niſi ſimul, propter earum <lb/>conjunctionem, moveri nequeant, vibratio minus diſtantium <lb/>ab axe ita retardata eſt a vibratione remotiorum, & </s> <s xml:id="echoid-s4827" xml:space="preserve">vibratio <lb/>remotiorum ita accelerata eſt a vibratione aliarum, ut inter <lb/>illas detur compenſatio velocitatum proportionalis arcubus <lb/>quos deſcribunt; </s> <s xml:id="echoid-s4828" xml:space="preserve">ita ut tempus vibrationis totius Penduli <lb/>medium ſit inter tempus, quo Oſcillationem peragunt ejus <lb/>partes a ſe invicem ſolutæ, ut ſit æquale ſummæ illorum <lb/>temporum, diviſæ per numerum partium, quas ut Mathema-<lb/>tice & </s> <s xml:id="echoid-s4829" xml:space="preserve">exactiſſime procedamus conſideramus, ac ſi reductæ <lb/>eſſent in puncta.</s> <s xml:id="echoid-s4830" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4831" xml:space="preserve">Conſtat experimentis, & </s> <s xml:id="echoid-s4832" xml:space="preserve">per Philoſophiam Carteſianam <lb/>demonſtrari poteſt, omnia gravia tellurem verſus cadere in <lb/>temporibus quæ ſunt in ratione ſubduplicatâ, vel ſicuti radi-<lb/>ces, altitudinum, unde deſcendunt, ſi verticaliter deſcendant, <lb/>quod etiam & </s> <s xml:id="echoid-s4833" xml:space="preserve">ex principiis Galilæi demonſtrari poteſt, ſi cadant per <lb/>arcus ſimiles, qui incipiunt omnes in eodem plano.</s> <s xml:id="echoid-s4834" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4835" xml:space="preserve">Hæ altitudines in Pendulis, quæ deſcribunt arcus ſimiles cir-<lb/>ca axem, quocum forrmant idem planum, ſunt inter ſe ut diſtantiæ <lb/>ab axe, circa quem moventur.</s> <s xml:id="echoid-s4836" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4837" xml:space="preserve">Propoſita ergo quæſtio eo redit, ut dividamus, per nume-<lb/>rum partium Penduli, ſummam radicum diſtantiarum partium <lb/>ab axe. </s> <s xml:id="echoid-s4838" xml:space="preserve">vel generaliter ſummam linearum rectarum quæ repræſentant <lb/>tempora vibrationium partium ſeparatim ſumtarum, ut habeamus lineam <lb/>rectam, quæ ſit menſura temporis, quo vibrationes ſuas per-<lb/>agit Pendulum, cujus conſequenter quadratum vel 3a. </s> <s xml:id="echoid-s4839" xml:space="preserve">propor-<lb/>tionalis erit diſtantia inter axem & </s> <s xml:id="echoid-s4840" xml:space="preserve">centrum Oſcillationis.</s> <s xml:id="echoid-s4841" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4842" xml:space="preserve">Applicatio hujus principii tribus magnitudinibus quas ha-<lb/>bet Geometria pro objecto ſatis facilis eſt.</s> <s xml:id="echoid-s4843" xml:space="preserve"/> </p> <pb o="221" file="0299" n="326" rhead="CONTROVERSIA."/> <p> <s xml:id="echoid-s4844" xml:space="preserve">1. </s> <s xml:id="echoid-s4845" xml:space="preserve">Ad determinandum centrum Oſcillationis lineæ rectæ ſuſ-<lb/>penſæ ex axe, debemus illam concipere, diviſam in partes æ-<lb/>quales infinite parvas, vel in omnibus ſuis punctis. </s> <s xml:id="echoid-s4846" xml:space="preserve">Para-<lb/>bola dein ſuper maximâ lineæ ab axe diſtantiâ deſcribenda <lb/>eſt, cujus vertex ſit punctum axis in quod terminatur hæc di-<lb/>ſtantia & </s> <s xml:id="echoid-s4847" xml:space="preserve">parameter linea quæ eſt unitas reſpectu ejusdem di-<lb/>ſtantiæ. </s> <s xml:id="echoid-s4848" xml:space="preserve">E quovis lineæ puncto ducenda eſt axi parallela quæ <lb/>occurrat parabolæ, ejusque applicata fiat, ſumma omnium <lb/>applicatarum ſimilium eſt æqualis rectangulo, cujus altitudo <lb/>eſt linea propoſita, & </s> <s xml:id="echoid-s4849" xml:space="preserve">baſis radix diſtantiæ inter axem & </s> <s xml:id="echoid-s4850" xml:space="preserve">cen-<lb/>trum Oſcillationis quæſitum; </s> <s xml:id="echoid-s4851" xml:space="preserve">nam ſumma illa eſt Parabola, vel Para-<lb/>bolæ portio, cujus Diameter eſt linea data, & </s> <s xml:id="echoid-s4852" xml:space="preserve">Parameter tertia proportio-<lb/>nalis illi lineæ & </s> <s xml:id="echoid-s4853" xml:space="preserve">maximæ ab axe diſtantiæ; </s> <s xml:id="echoid-s4854" xml:space="preserve">vel 4a. </s> <s xml:id="echoid-s4855" xml:space="preserve">proportionalis poſitis <lb/>hiſcc tribus, linea, maxima diſtantia, & </s> <s xml:id="echoid-s4856" xml:space="preserve">differentia hujus cum minimâ.</s> <s xml:id="echoid-s4857" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4858" xml:space="preserve">2. </s> <s xml:id="echoid-s4859" xml:space="preserve">Ut habeamus diſtantiam quâ centrum Oſcillationis <lb/>Plani remotum eſt ab axe, debemus concipere partem ſo-<lb/>lidi Parabolici, cujus Parabolæ habeant pro Diametris ma-<lb/>ximas diſtantias inter axem & </s> <s xml:id="echoid-s4860" xml:space="preserve">unamquamque linea-<lb/>rum parallelarum, quæ implent planum. </s> <s xml:id="echoid-s4861" xml:space="preserve">Si ſolidum hoc <lb/>in duas partes æquales ſecetur juxta Axis longitudinem, & </s> <s xml:id="echoid-s4862" xml:space="preserve">dimidia <lb/>pars ſecta fuerit inter applicatas ad diſtantias ab Axe, & </s> <s xml:id="echoid-s4863" xml:space="preserve">inter Plani <lb/>latera, ſegmentum æquale erit Prismati, quod pro baſi habet <lb/>planum, & </s> <s xml:id="echoid-s4864" xml:space="preserve">pro altitudine radicem diſtantiæ axis a centro <lb/>Oſcillationis ejusdem plani. </s> <s xml:id="echoid-s4865" xml:space="preserve">ſi vibratio fiat circa Punctum vel <lb/>ſi, quum fit circa Planum, Pendulum ſit compoſitum e partibus, <lb/>quæ ſint in planis diverſis reſpectu Axis, determinabitur, eâ me-<lb/>thodo quâ diximus, quodvis centrum Oſcillationis partium, quæ <lb/>ſunt in eâdem lineâ rectâ transeunte per punctum ſuſpenſionis, vel in <lb/>eodem plano transeunte per Axem; </s> <s xml:id="echoid-s4866" xml:space="preserve">omnia illa Oſcillationis centra fa-<lb/>cient Pendulum multo ſimplicius & </s> <s xml:id="echoid-s4867" xml:space="preserve">habens idem Oſcillationis cen-<lb/>trum, ac primum. </s> <s xml:id="echoid-s4868" xml:space="preserve">Invenietur Oſcillationis centrum dividendo per <lb/>numerum aliorum Oſcillationis centrorum ſummam linearum recta-<lb/>rum, quæ repræſentant tempora, quibus conficerent peculiares ſuas <lb/>vibrationes. </s> <s xml:id="echoid-s4869" xml:space="preserve">Tempora illa pendent ab arcubus, vel curvarum portio-<lb/>nibus, deſcriptis ab omnibus Oſcillationis centris in vibratione Pendu-<lb/>li, qui arcus conſiderari debent ſinguli velut infinita plana diverſi-<lb/>mode ad Horizontem inclinata.</s> <s xml:id="echoid-s4870" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4871" xml:space="preserve">3. </s> <s xml:id="echoid-s4872" xml:space="preserve">Quod ad ſolida attinet, concipe illa dividi in parallelas <pb o="222" file="0300" n="327" rhead="DE CENTRO OSCILL."/> inter ſe ſuperficies. </s> <s xml:id="echoid-s4873" xml:space="preserve">& </s> <s xml:id="echoid-s4874" xml:space="preserve">ad axem perpendiculares; </s> <s xml:id="echoid-s4875" xml:space="preserve">formari debet, ſecun-<lb/>dâ ſectione, planum vel ſuperficies curva diſtantiarum inter il-<lb/>lorum centra Oſcillationis & </s> <s xml:id="echoid-s4876" xml:space="preserve">axem †, circa cujus puncta mo-<lb/>ventur. </s> <s xml:id="echoid-s4877" xml:space="preserve">Sic in ſummâ centrorum, quæ ab unâ parte rectas lineas ter-<lb/>minant, ex quibus planum hoc, vel ſuperficies illa curva, compoſita <lb/>eſt, habetur Pendulum magis ſimplex quam ſolidum, & </s> <s xml:id="echoid-s4878" xml:space="preserve">cujus vibra-<lb/>tio æque diuturna eſt. </s> <s xml:id="echoid-s4879" xml:space="preserve">Centrum Oſcillationis novi illius Penduli de-<lb/>terminabitur transferendo omnia illa centra Oſcillationis particularia <lb/>ad Axem qui eſt eorum numerus, & </s> <s xml:id="echoid-s4880" xml:space="preserve">ponendo illum axem ita mo-<lb/>veri, ut puncta ejus percurrant eosdem arcus ac centra. </s> <s xml:id="echoid-s4881" xml:space="preserve">Si ſoli-<lb/>da propoſita ſint Prismata recta, habebunt eadem Oſcil-<lb/>lationis centra, ac eorum baſes, ſi hæ fuerint perpendicula-<lb/>res ad axem.</s> <s xml:id="echoid-s4882" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4883" xml:space="preserve">Sic centrum Oſcillationis ſolidi pendet a centris Oſcillationis certa-<lb/>rum ſuperficierum motarum circa punctum, quarum commune O-<lb/>ſcillationis centrum eſt centrum Oſcillationis lineæ rectæ motæ circa <lb/>aliam rectam vel curvam; </s> <s xml:id="echoid-s4884" xml:space="preserve">ita ut non requirantur aliæ regulæ pro cor-<lb/>poribus quam pro lineis & </s> <s xml:id="echoid-s4885" xml:space="preserve">ſuperficiebus.</s> <s xml:id="echoid-s4886" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div426" type="section" level="1" n="171"> <head xml:id="echoid-head208" xml:space="preserve"><emph style="sp">MONITUM.</emph></head> <p style="it"> <s xml:id="echoid-s4887" xml:space="preserve">Obſervationes Abbatis Catelani primum editæ fuere in <lb/>25 diario Pariſienſi anni 1681, & </s> <s xml:id="echoid-s4888" xml:space="preserve">examen Mathematicum in <lb/>29 diario ejusdem anni. </s> <s xml:id="echoid-s4889" xml:space="preserve">Scripta ambo in primo diario <lb/>ſequentis anni iterum extant, cum monitu varia defici in <lb/>prima editione, quam ſolam Hugenius viderat cum reſpon-<lb/>dit; </s> <s xml:id="echoid-s4890" xml:space="preserve">& </s> <s xml:id="echoid-s4891" xml:space="preserve">in qua non reperiuntur ea quæ minori charactere hic <lb/>edita ſunt. </s> <s xml:id="echoid-s4892" xml:space="preserve">Sola verba ſequentia in prima dantur & </s> <s xml:id="echoid-s4893" xml:space="preserve">in ſe-<lb/>cunda fuere omiſſa poſt †. </s> <s xml:id="echoid-s4894" xml:space="preserve">centrum Oſcillationis hujus plani <lb/>coincidet cum centro Oſcillationis ſolidorum illorum.</s> <s xml:id="echoid-s4895" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div427" type="section" level="1" n="172"> <head xml:id="echoid-head209" xml:space="preserve">III.</head> <head xml:id="echoid-head210" style="it" xml:space="preserve">Excerpta ex literis Domini Hugenii, quibus re-<lb/>ſpondet obſervationi Abbatis Catelani in 4<emph style="super">am</emph>. pro-<lb/>poſitionem Tractatus de centris Oſcillationis.</head> <p> <s xml:id="echoid-s4896" xml:space="preserve">ADmiratus vidi, Theoriam meam de centro Oſcillationis <lb/>oppugnari, contra quam per 9 annos, a quibus typis <pb o="223" file="0301" n="328" rhead="CONTROVERSIA."/> mandata fuit, nemo quid protulit; </s> <s xml:id="echoid-s4897" xml:space="preserve">ſed conſiderata refutatione, <lb/>qua Abbas Catelanus 4<emph style="super">am</emph>. </s> <s xml:id="echoid-s4898" xml:space="preserve">meam propoſitionem aggreditur, <lb/>non vidi, quod ullatenus me feriat. </s> <s xml:id="echoid-s4899" xml:space="preserve">nam ut paucis di-<lb/>cam, in quo fallitur; </s> <s xml:id="echoid-s4900" xml:space="preserve">negat, datis duabus lineis & </s> <s xml:id="echoid-s4901" xml:space="preserve">præter <lb/>has, duabus aliis, quæ diverſam quam primæ inter ſe ratio-<lb/>nem habent, ſummam duarum ultimarum æqualem unquam <lb/>fore ſummæ duarum priorum.</s> <s xml:id="echoid-s4902" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4903" xml:space="preserve">Concipe priores 5 & </s> <s xml:id="echoid-s4904" xml:space="preserve">10 pedum, & </s> <s xml:id="echoid-s4905" xml:space="preserve">alteras 3 & </s> <s xml:id="echoid-s4906" xml:space="preserve">12 & </s> <s xml:id="echoid-s4907" xml:space="preserve">vi-<lb/>de num harum ſumma æque ac illarum non ſit 15: </s> <s xml:id="echoid-s4908" xml:space="preserve">ut autem <lb/>pateat errorem ejus inde oriri, utar eodem, quod ille propo-<lb/>ſuit, exemplo.</s> <s xml:id="echoid-s4909" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4910" xml:space="preserve">A & </s> <s xml:id="echoid-s4911" xml:space="preserve">B ſunt duo pondera applicata virgæ vel lineæ D B, <lb/> <anchor type="note" xlink:label="note-0301-01a" xlink:href="note-0301-01"/> quæ conſiderari debet ut inflexibilis & </s> <s xml:id="echoid-s4912" xml:space="preserve">ſine pondere; </s> <s xml:id="echoid-s4913" xml:space="preserve">quæ <lb/>libere notetur circa punctum D: </s> <s xml:id="echoid-s4914" xml:space="preserve">tale Pendulum compoſitum <lb/>voco e ponderibus A & </s> <s xml:id="echoid-s4915" xml:space="preserve">B; </s> <s xml:id="echoid-s4916" xml:space="preserve">ſi hoc peragat partem vibrationis, <lb/>Ex. </s> <s xml:id="echoid-s4917" xml:space="preserve">Gr. </s> <s xml:id="echoid-s4918" xml:space="preserve">usque ad D F G, & </s> <s xml:id="echoid-s4919" xml:space="preserve">occurrat plano, ad quod <lb/>frangatur, ut pondera a lineâ inflexili ſeparentur, & </s> <s xml:id="echoid-s4920" xml:space="preserve">tendat <lb/>ſurſum eorum unumquodque cum velocitate acquiſita, ad <lb/>maximam quam poteſt altitudinem, velut ad L & </s> <s xml:id="echoid-s4921" xml:space="preserve">M, ſuper <lb/>planis inclinatis ſi velimus, quæ tangant arcus A F, B G; <lb/></s> <s xml:id="echoid-s4922" xml:space="preserve">dico commune centrum gravitatis ponderum A & </s> <s xml:id="echoid-s4923" xml:space="preserve">B quæ aſcen-<lb/>dunt in L & </s> <s xml:id="echoid-s4924" xml:space="preserve">M tunc ad eandem fore altitudinem, ac erat <lb/>in E, ante vibrationem inchoatam.</s> <s xml:id="echoid-s4925" xml:space="preserve"/> </p> <div xml:id="echoid-div427" type="float" level="2" n="1"> <note position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">TAB. XXVIII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s4926" xml:space="preserve">Abbas Catelanus ut falſam hanc probet propoſitionem, <lb/>demonſtrat, altitudines, ad quas duo pondera ſoluta aſcen-<lb/>dunt, ut hic N L, O M, diverſas eſſe ab iis unde de-<lb/>ſcenderunt, ſcilicet A H, B I. </s> <s xml:id="echoid-s4927" xml:space="preserve">id quod veriſſimum eſt ex ra-<lb/>tione ab ipſo datâ, quod alteræ ſint inter ſe ut lineæ D F, <lb/>D G, alteræ vero ut quadrata harum linearum; </s> <s xml:id="echoid-s4928" xml:space="preserve">ſi ergo divi-<lb/>damus, inquit diverſas illas ſummas per numerum illorum <lb/>ponderum, id eſt, ſi ſumamus dimidium linearum L N, M O, <lb/>& </s> <s xml:id="echoid-s4929" xml:space="preserve">dimidium linearum A H, B I. </s> <s xml:id="echoid-s4930" xml:space="preserve">habebimus ab una parte al-<lb/>titudinem ad quam centrum commune gravitatis aſcendit, & </s> <s xml:id="echoid-s4931" xml:space="preserve"><lb/>ab altera altitudinem unde deſcendit: </s> <s xml:id="echoid-s4932" xml:space="preserve">id verum eſt, per divi-<lb/>ſionem has duas altitudines detegi. </s> <s xml:id="echoid-s4933" xml:space="preserve">ſed minime concedo, duas <lb/>ſummas diviſas differre inter ſe; </s> <s xml:id="echoid-s4934" xml:space="preserve">quod Abbas Catelanus pro- <pb o="224" file="0302" n="329" rhead="DE CENTRO OSCILL."/> bare nequit; </s> <s xml:id="echoid-s4935" xml:space="preserve">neque igitur, duas inventas altitudines centri <lb/>gravitatis inæquales eſſe, id quod in concluſione contendit, <lb/>nam licet altitudines L N, M O, diverſam habeant rationem <lb/>inter ſe quam altitudines A H, B I, non ſequitur ſummas <lb/>primarum & </s> <s xml:id="echoid-s4936" xml:space="preserve">ſecundarum differre.</s> <s xml:id="echoid-s4937" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4938" xml:space="preserve">Poſſem præter hunc alium locum obſervare, ubi fallitur <lb/>Abbas Catelanus, ſed non hærebo, quoniam id, quod pro-<lb/>fert non ſpectat ea quibus me aggreditur. </s> <s xml:id="echoid-s4939" xml:space="preserve">Unicum verbum <lb/>addam de examine ejus Mathematico, ut vocat, de centro <lb/>Oſcillationis edito in diario 15. </s> <s xml:id="echoid-s4940" xml:space="preserve">Dec. </s> <s xml:id="echoid-s4941" xml:space="preserve">1681. </s> <s xml:id="echoid-s4942" xml:space="preserve">ubi contendit ſe <lb/>inveniſſe regulam hanc generalem, ſcilicet, per nume-<lb/>rum partium Penduli dividi debere ſummam radicum diſtan-<lb/>tiarum partium ab axe; </s> <s xml:id="echoid-s4943" xml:space="preserve">ut habeamus lineam rectam, quæ ſit <lb/>menſura temporis quo vibrationem peragit illud Pendulum, <lb/>cujus conſequenter quadratum vel 3<emph style="super">a</emph>. </s> <s xml:id="echoid-s4944" xml:space="preserve">proportionalis erit di-<lb/>ſtantia inter axem & </s> <s xml:id="echoid-s4945" xml:space="preserve">centrum Oſcillationis.</s> <s xml:id="echoid-s4946" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">TAB. XXVIII. <lb/>Fig. 3.</note> <p> <s xml:id="echoid-s4947" xml:space="preserve">Relicto omni peculiari examine, ſatis erit ad vitium re-<lb/>gulæ detegendum, notaſſe, juxta hoc principium, duas li-<lb/>neas graves ut A B, B C, junctas & </s> <s xml:id="echoid-s4948" xml:space="preserve">inter ſe angulum quem-<lb/>cunque efficientes, ſuſpenſas in B ſemper idem Oſcillatio-<lb/>nis centrum habere; </s> <s xml:id="echoid-s4949" xml:space="preserve">ideoque vibrationes ſemper fore æque ve-<lb/>loces, uti facile intelligunt hi, qui in hiſce materiis parum ſunt <lb/>verſati; </s> <s xml:id="echoid-s4950" xml:space="preserve">ſed & </s> <s xml:id="echoid-s4951" xml:space="preserve">illi videbunt, quod æqualitas illa vibratio-<lb/>num locum habere nequeat, quoniam augendo angulo tan-<lb/>dem duæ lineæ ſimul junctæ unicam efficiunt rectam a B c, <lb/>cujus vibrationes forent æque diuturnæ cum vibrationibus <lb/>anguli A B C, linea vero recta in puncto medio ſuſpenſa, <lb/>nullas peragit vibrationes, aut ſaltem velocitate infinite exi-<lb/>gua movetur.</s> <s xml:id="echoid-s4952" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4953" xml:space="preserve">Porro credo Abbati Catelano ſatis difficile fore, ſuâ regulâ, <lb/>centrum Oſcillationis in quibusvis particularibus figuris, etiam <lb/>ſimpliciſſimis, determinate, ſed, ſi id forte perficiat, inveniet, <lb/>ſuam Theoriam cum experientiâ nunquam convenire, & </s> <s xml:id="echoid-s4954" xml:space="preserve"><lb/>meam ſemper ſummâ exactitudine eidem reſpondere, ſi mo-<lb/>do experimenta exactiſſimè inſtituta fuerint.</s> <s xml:id="echoid-s4955" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4956" xml:space="preserve">Non poſſum hac occaſione ſilentio præterire, P. </s> <s xml:id="echoid-s4957" xml:space="preserve">Dechales, <pb o="225" file="0303" n="330" rhead="CONTROVERSIA."/> in quodam magni ſui Curſus Mathematici loco, memorando <lb/>experimentum cum Pendulo compoſito ex duobus ponderi-<lb/>bus inſtitutum, in quo computans rationem non habuit <lb/>(ut debebat) ponderis baculi, cui pondera erant applicata, <lb/>immerito regulas noſtras de determinando centro gravitatis <lb/>culpare, quaſi non convenirent cum iis quæ ille revera obſer-<lb/>varat.</s> <s xml:id="echoid-s4958" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div429" type="section" level="1" n="173"> <head xml:id="echoid-head211" xml:space="preserve">IV.</head> <head xml:id="echoid-head212" style="it" xml:space="preserve">Exceptio Abbatis Catelani ad reſponſionem <lb/>Hugenii.</head> <p> <s xml:id="echoid-s4959" xml:space="preserve">NOn debebat Hugenius a ſuo principio ſeparare conſe-<lb/>quentiam, ut hanc aliter intelligat, quam ego in ſcri-<lb/>ptis meis. </s> <s xml:id="echoid-s4960" xml:space="preserve">Prorſus deberem oblitus eſſe Arithmetices, ſi ab-<lb/>ſolute negarem, ut me facere contendit, quod 4 magnitudi-<lb/>nes inæquales poſſint efficere 2 ſummas æquales, quum nil a-<lb/>liud concludo in ſcriptis, præterquam quod propoſitio Hu-<lb/>genii vera eſſe nequeat, niſi pars ſit æqualis toti. </s> <s xml:id="echoid-s4961" xml:space="preserve">Ut hoc <lb/>melius pateat, debemus hic proferre propoſitionem illam ge-<lb/>neralem propriis terminis.</s> <s xml:id="echoid-s4962" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4963" xml:space="preserve">Si Pendulum e pluribus ponderibus compoſitum, at que e quie-<lb/>te demiſſum, partem quamcunque Oſcillationis integræ confece-<lb/>rit, atque inde porro intelligantur pondera ejus ſingula, reli-<lb/>cto communi vinculo, celeritates acquiſitas ſurſum convertere, <lb/>ac quousque poſſunt aſcendere; </s> <s xml:id="echoid-s4964" xml:space="preserve">hoc facto, centrum gravitatis <lb/>ex omnibus compoſitæ, ad eandem altitudinem reverſum erit, <lb/>quam ante inceptam Oſcillationem obtinebat.</s> <s xml:id="echoid-s4965" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4966" xml:space="preserve">Cum hæc propoſitio terminis generaliſſimis concepta ſit, <lb/>ita ut numerus ponderum, ſitus eorum, duratio Vibrationis, <lb/>ipſam non mutent, pono, Exempli cauſâ, Pendulum compoſi-<lb/>tum e ponderibus duo bus æqualibus, & </s> <s xml:id="echoid-s4967" xml:space="preserve">inter ſe ad quam li-<lb/>buerit a ſe invicem diſtantiam junctis. </s> <s xml:id="echoid-s4968" xml:space="preserve">Conſidero porro al-<lb/>titudines, quæ ſunt proportionales quadratis velocitatum in <lb/>duobus Pendulis ſimplicibus, eſſe inter ſe ut velocitates in <pb o="226" file="0304" n="331" rhead="DE CENTRO OSCILL"/> Pendulo compoſito. </s> <s xml:id="echoid-s4969" xml:space="preserve">Nam eandem habent proportionem, <lb/>quam arcus deſcripti a duobus ponderibus æqualibus, ex <lb/>quibus Pendulum formatur; </s> <s xml:id="echoid-s4970" xml:space="preserve">duo illi arcus ſunt ſpatia, quæ <lb/>duo pondera percurrunt, eodem tempore, velocitatibus, <lb/>quæ neceſſario ſunt ipſis ſpatiis proportionales.</s> <s xml:id="echoid-s4971" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4972" xml:space="preserve">Celeritas totalis Penduli compoſiti, quæ inter partes dis-<lb/>tribuitur proportionaliter ad arcus, quos ipſæ deſcribunt, <lb/>ſemper æqualis eſt ſummæ celeritatum, quas eædem partes <lb/>acquirerent, ſi a ſe invicem fuiſſent ſejunctæ, & </s> <s xml:id="echoid-s4973" xml:space="preserve">omnes ſepa-<lb/>ratim ex iisdem altitudinibus & </s> <s xml:id="echoid-s4974" xml:space="preserve">ad easdem ab axe, diſtantias <lb/>deſcendiſſent. </s> <s xml:id="echoid-s4975" xml:space="preserve">Altitudines ſemper ſunt ut quadrata velocitatum, <lb/>ſive pondera ſeparatim adſcendant, ſive deſcendant. </s> <s xml:id="echoid-s4976" xml:space="preserve">Omni-<lb/>bus his bene intellectis facile patet, ad hanc propoſitionem <lb/>redire quæſtionem. </s> <s xml:id="echoid-s4977" xml:space="preserve">Si habeamus du@s magnitudines inæquales <lb/>a a & </s> <s xml:id="echoid-s4978" xml:space="preserve">b b, ſummam radicum ipſarum a † b, & </s> <s xml:id="echoid-s4979" xml:space="preserve">quadrata <lb/>partium illius ſummæ, quæ ſint proportionales dictis magni-<lb/>tudinibus, quæque adeo communem denominatorem habeant <lb/>a a † b b, & </s> <s xml:id="echoid-s4980" xml:space="preserve">numeratores diverſos a<emph style="super">3</emph> † a a b & </s> <s xml:id="echoid-s4981" xml:space="preserve">b<emph style="super">3</emph> † a b b, <lb/>demonſtrare, ſummam harum duarum magnitudinum, quæ <lb/>altitudines, unde duo pondera æqualia Pendulo alligata <lb/>dimittuntur, repræſentant, non eſſe æqualem ſummæ quadra-<lb/>torum illarum partium, quæ altitudines exhibent, ad quas <lb/>duo pondera, poſtquam percuſſione fuerint ſeparata, redeunt, <lb/>niſi minor ex hiſce magnitudinibus a a & </s> <s xml:id="echoid-s4982" xml:space="preserve">b b ſit æqualis majo-<lb/>ri, id eſt, quia iſtæ magnitudines in quæſtione propoſitâ ſem-<lb/>per inæquales ſunt, niſi pars æqualis ſit, toti.</s> <s xml:id="echoid-s4983" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4984" xml:space="preserve">Maxime ſenſibilis hujus veritatis demonſtratio eſt compa-<lb/>ratio terminorum quæſtionis per regulas Algebraicas, id <lb/>quod examinandum relinquo iis, qui uſum illarum regula-<lb/>rum norunt. </s> <s xml:id="echoid-s4985" xml:space="preserve">Quod rem ipſam ſpectat, nullius eſt momenti; <lb/></s> <s xml:id="echoid-s4986" xml:space="preserve">ſive centrum Mathematicum Oſcillationis bene ſive male de-<lb/>terminatum ſit, inventio Penduli nec minus utilis homini-<lb/>bus, nec minus auctore ſuo digna eſt.</s> <s xml:id="echoid-s4987" xml:space="preserve"/> </p> <pb o="227" file="0305" n="332" rhead="CONTROVERSIA."/> </div> <div xml:id="echoid-div430" type="section" level="1" n="174"> <head xml:id="echoid-head213" xml:space="preserve">V.</head> <head xml:id="echoid-head214" style="it" xml:space="preserve">Objectio Abbatis Catelani contra motum <lb/>Pendulorum in Cycloidibus.</head> <p> <s xml:id="echoid-s4988" xml:space="preserve">Si vis gravitatis ageret in corpora tanquam in puncta Ma-<lb/>thematica, vel ſi ſpatium contentum ſub Cycloide eſſet <lb/>diviſibile in infinitas alias Cycloides, ſimiles & </s> <s xml:id="echoid-s4989" xml:space="preserve">parallelas, <lb/>quidam Geometræ revera demonſtraſſent, uti contendunt, <lb/>pendula illam debere deſcribere curvam ut Vibrationes, æ-<lb/>qualibus temporibus, peragant; </s> <s xml:id="echoid-s4990" xml:space="preserve">ſed nulla datur pars in cor-<lb/>pore gravi, quale eſt Pendulum ex cupro vel plumbo, quæ <lb/>non æque ac centrum Tellurem verſus propellatur magis mi-<lb/>nusve pro inclinatione, juxta quam movetur; </s> <s xml:id="echoid-s4991" xml:space="preserve">Præterea <lb/>ſpatium, quod continet Cyclois, non poteſt repleri infini-<lb/>tis aliis Cycloidibus ſimilibus, quoniam tripla circuli ſuperfi-<lb/>cies æqualis eſſet duplo quadrato diametri. </s> <s xml:id="echoid-s4992" xml:space="preserve">Latet igitur@adhuc <lb/>Geometras, quamnam lineam curvam deſcribat Pendulum, <lb/>cujus Vibrationes ſunt iſochronæ. </s> <s xml:id="echoid-s4993" xml:space="preserve">Conſequentia hæc patet, <lb/>ſi conſideremus, in corpore Pendulo, quando centrum vel <lb/>alia quælibet pars Cycloidem percurrit, partes viciniores <lb/>Axi, vel inde remotiores, eodem tempore, deſcribere lineas <lb/>curvas inter ſe ſimiles, ſed quæ non ſunt Cycloides; </s> <s xml:id="echoid-s4994" xml:space="preserve">ut <lb/>patet ex dictis, & </s> <s xml:id="echoid-s4995" xml:space="preserve">quia in quovis arcu perpendiculares <lb/>ductæ e tangentibus ſuis ad tangentes Cycloidis ſunt æqua-<lb/>les. </s> <s xml:id="echoid-s4996" xml:space="preserve">Omnes itaque partes non habent æqualem inclinationem <lb/>in deſcenſu, neque Tellurem verſus cum eâdem velocitatis <lb/>proportione propelluntur; </s> <s xml:id="echoid-s4997" xml:space="preserve">unde ſequitur, Vibrationem to-<lb/>tius Penduli, quæ neceſſario pendet a Vibrationibus, quas <lb/>peragunt partes ejus ſeparatim ſumtæ, differre a Penduli <lb/>Vibratione, ſi reductum foret ad illam partem, quæ move-<lb/>tur in Cycloide.</s> <s xml:id="echoid-s4998" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4999" xml:space="preserve">Videtur huic potius cauſæ, quam craſſitudini fili, cui <lb/>pondus alligatur, adſcribendam eſſe artificum praxin, quos <lb/>experientia cogit curvaturam a Cycloide diverſam laminis <lb/>tribuere, inter quas ſuſpendunt Pendulum.</s> <s xml:id="echoid-s5000" xml:space="preserve"/> </p> <pb o="228" file="0306" n="333" rhead="DE CENTRO OSCILL."/> <p> <s xml:id="echoid-s5001" xml:space="preserve">Non tamen mihi animus eſt, hic abſolute oppugnare ſen-<lb/>tentiam illorum, qui credunt corpora gravia moveri, veluti <lb/>puncta, quæ in Cycloide, ad Horizontem perpendiculari, <lb/>peragunt ſuas Oſcillationes, temporibus æqualibus, a qua-<lb/>cunque dimittantur altitudine. </s> <s xml:id="echoid-s5002" xml:space="preserve">Contendo tantummodo, il-<lb/>lud nondum eſſe demonſtratum, niſi alterutrum horum pro-<lb/>betur, vel quod curvæ parallelæ Cycloidi eandem habent <lb/>proprietatem quantum ad motum corporum, licet non ſint <lb/>Cycloides, vel quod inæqualitas temporis, quod brevius eſt <lb/>in parallelis, interioribus & </s> <s xml:id="echoid-s5003" xml:space="preserve">Cycloide viciniores Axi, ita <lb/>moderetur contraria inæqualitate temporis, quod majus eſt <lb/>in parallelis exterioribus & </s> <s xml:id="echoid-s5004" xml:space="preserve">ipſâ curvâ remotioribus ab Axe, <lb/>ut compenſatio inæqualitatum ambarum in Cycloide detur, <lb/>quæ tanquam medium locum tenet inter omnes curvas ipſi <lb/>parallelas. </s> <s xml:id="echoid-s5005" xml:space="preserve">Geometræ hanc difficultatem examinabunt ſi di-<lb/>gnam ſuâ attentione judicent; </s> <s xml:id="echoid-s5006" xml:space="preserve">nec, niſi poſtquam eorum ſen-<lb/>tentiam novero, obſervationes meas hac de re dare potero.</s> <s xml:id="echoid-s5007" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div431" type="section" level="1" n="175"> <head xml:id="echoid-head215" xml:space="preserve">VI.</head> <head xml:id="echoid-head216" style="it" xml:space="preserve">Reſponſio ad objectiones Hugenii adverſus me-<lb/>thodum Abbatis Catelani de determinan-<lb/>do Centro Oſcillationis.</head> <p> <s xml:id="echoid-s5008" xml:space="preserve">HUgenius propoſuit objectionem adverſus propoſitionem <lb/>deductam ex principio, a me, ad determinandum Mathe-<lb/>matice centrum Oſcillationis Penduli, propoſito; </s> <s xml:id="echoid-s5009" xml:space="preserve">ſed de-<lb/>buit examinare id, quod præcedit locum, quem e ſcri-<lb/>ptis meis profert, nec generalem habere regulam ad <lb/>caſum particularem tantum accommodatam, quem elegi, ut <lb/>uterer exemplo ſimpliciſſimo & </s> <s xml:id="echoid-s5010" xml:space="preserve">facillimo; </s> <s xml:id="echoid-s5011" xml:space="preserve">ſcilicet quando <lb/>Pendula compoſita ſunt ex partibus, quæ deſcribunt arcus <lb/>ſimiles circa Axem, quocum faciunt idem planum; </s> <s xml:id="echoid-s5012" xml:space="preserve">tum <lb/>enim diſtantiæ ab illo Axe ſunt radii arcuum, qui habent <lb/>eandem inter ſe proportionem, ac perpendiculares ad Hori-<lb/>zontem, vel ſinus, qui ſunt altitudines, a quibus ſingulæ <lb/>partes in Oſcillatione deſcendunt. </s> <s xml:id="echoid-s5013" xml:space="preserve">Itaque cum Pendula, de <pb o="229" file="0307" n="334" rhead="CONTROVERSIA."/> quibus Hugenius loquitur, ut probet propoſitionem meam <lb/>falſam, ſint anguli rectilinei agitati circa verticem, non <lb/>habentes requiſitam conditionem, me non feriunt. </s> <s xml:id="echoid-s5014" xml:space="preserve">Si <lb/>concipiamus tales angulos moveri circa Axem, per verti-<lb/>ces illorum transeuntem, patet ſummas diſtantiarum Axis ab <lb/>omnibus punctis linearum, quæ Pendula componunt, eſſe <lb/>inæquales, prout illæ lineæ efficiunt cum Axe angulos ma-<lb/>gis minusve acutos. </s> <s xml:id="echoid-s5015" xml:space="preserve">Et meâ regulâ detego ſummas diſtantia-<lb/>rum eſſe æquales Parabolis habentibus pro Diametro maxi-<lb/>mam ab Axe diſtantiam, & </s> <s xml:id="echoid-s5016" xml:space="preserve">pro Parametro 4<emph style="super">am</emph> proportiona-<lb/>lem poſitis hiſce tribus, linea datâ, quæ eadem eſt in quo-<lb/>vis Pendulo, maximâ diſtantiâ, quæ variat pro variis angu-<lb/>lis, & </s> <s xml:id="echoid-s5017" xml:space="preserve">unitate; </s> <s xml:id="echoid-s5018" xml:space="preserve">unde ſequitur, tempus Oſcillationis valere {2/3} ma-<lb/>ximæ ab Axe diſtantiæ, & </s> <s xml:id="echoid-s5019" xml:space="preserve">non in omni caſu idem eſſe; </s> <s xml:id="echoid-s5020" xml:space="preserve">tanto <lb/>enim brevius eſt, quanto angulus eſt obtuſior, id eſt, quan-<lb/>to Pendulum magis Axi vicinum eſt.</s> <s xml:id="echoid-s5021" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5022" xml:space="preserve">Si Hugenius deſiderat propoſitionem quæ conveniat Pen-<lb/>dulis, circa punctum motis, mutanda tantum erunt verba <lb/>quædam in principio Pendulorum habentibum Axem; </s> <s xml:id="echoid-s5023" xml:space="preserve">loco <lb/>radices diſtantiarum illarum, legendum ſummæ linearum re-<lb/>ctarum, quæ repræſentant tempora Oſcillationum omnium par-<lb/>tium ſeparatim ſumtarum.</s> <s xml:id="echoid-s5024" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5025" xml:space="preserve">Hoc modo propoſitio inſerviet ambobus caſibus. </s> <s xml:id="echoid-s5026" xml:space="preserve">Sed res <lb/>melius intelligitur per generale Principium, quod propoſui <lb/>& </s> <s xml:id="echoid-s5027" xml:space="preserve">ita ſe habet. </s> <s xml:id="echoid-s5028" xml:space="preserve">In eodem Pendulo, cum omnes partes niſi <lb/>ſimul moveri nequeant, propter ſuam conjunctionem, vibra-<lb/>tio minus diſtantium ab Axe, vel puncto ſuſpenſionis, ita <lb/>retardatur a vibratione remotiorum, & </s> <s xml:id="echoid-s5029" xml:space="preserve">reciproce Oſcillatio <lb/>remotiorum ita acceleratur per Oſcillationem aliarum, ut de-<lb/>tur inter illas compenſatio celeritatum proportionalis arcubus, <lb/>vel cur varum portionibus, quas deſcribunt, ita ut tempus Oſcil-<lb/>lationis totius Penduli ſit medium inter tempora Vibratio-<lb/>num, quas peragerent illæ partes, ſi non inter ſe forent con-<lb/>junctæ, id eſt, ut ſit æquale ſummæ temporum illorum diviſæ <lb/>per numerum partium, quas debemus conſiderare ut æquales <lb/>& </s> <s xml:id="echoid-s5030" xml:space="preserve">infinite parvas.</s> <s xml:id="echoid-s5031" xml:space="preserve"/> </p> <pb o="230" file="0308" n="335" rhead="DE CENTRO OSCILL."/> <p> <s xml:id="echoid-s5032" xml:space="preserve">Demonſtrare potero in ſequentibus, non adeo difficile eſſe, <lb/>ut quidem Hugenio videtur, accommodare illud Principium <lb/>ad particulares magnitudinum Geometricarum ſpecies ſuſpen-<lb/>ſarum ex Axe vel puncto.</s> <s xml:id="echoid-s5033" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5034" xml:space="preserve">Quod ad experimenta attinet, paratus ſum demonſtrare hæc <lb/>ita non poſſe inſtitui, ut perfecte conveniant cum regulis ſim-<lb/>plicibus & </s> <s xml:id="echoid-s5035" xml:space="preserve">generalibus, quæ deducuntur e principiis Mathe-<lb/>maticis, eandem ob cauſam ob quam generalis regula, ſta-<lb/>biliri nequit, certa, & </s> <s xml:id="echoid-s5036" xml:space="preserve">conſtans, in caſibus particularibus, qui <lb/>dependent a pluribus cauſis, non exacte notis.</s> <s xml:id="echoid-s5037" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div432" type="section" level="1" n="176"> <head xml:id="echoid-head217" xml:space="preserve">VII.</head> <head xml:id="echoid-head218" style="it" xml:space="preserve">Excerpta ex litteris D. Bernoullii datis Baſileæ ad <lb/>Autorem Diarii Pariſienſis, de Controverſia, <lb/>inter Abbatem Catelanum & Hugenium, <lb/>de Centro Oſcillationis.</head> <p> <s xml:id="echoid-s5038" xml:space="preserve">QUum nondum obſervaverim, Hugenium reſpondiſſe, ad <lb/>exceptionem Abbatis Catelani, quæ ſpectabat primariam <lb/>ejus de centro Oſcillationis propoſitionem, te haud <lb/>ægre laturum credo, ſi verbulum ad ejus defenſionem ad te <lb/>ſcribam. </s> <s xml:id="echoid-s5039" xml:space="preserve">Quicquid D. </s> <s xml:id="echoid-s5040" xml:space="preserve">Catelanus diſputat, eo redit ut probet, <lb/>ſummam radicum duarum magnitudinum quarumvis non poſſe <lb/>in duas partes ita dividi, ut proportionales ſint ad magnitu-<lb/>dines datas, utque ſumma quadratorum ipſorum ſit æqualis <lb/>ſummæ magnitudinum. </s> <s xml:id="echoid-s5041" xml:space="preserve">Id vero neutiquam in dubium ab <lb/>Hugenio vocatur, qui tantum affirmat, ſummam harum ma-<lb/>gnitudinum poſſe eſſe æqualem ſummæ duarum aliarum, quæ <lb/>quadratis priorum proportionales ſunt. </s> <s xml:id="echoid-s5042" xml:space="preserve">Quod & </s> <s xml:id="echoid-s5043" xml:space="preserve">veritati con-<lb/>ſonum eſt.</s> <s xml:id="echoid-s5044" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5045" xml:space="preserve">Atque ut oſtendam, controverſiæ omnis cardinem hic <lb/>verti, utar eodem exemplo de duobus ponderibus æquali-<lb/>bus, & </s> <s xml:id="echoid-s5046" xml:space="preserve">quidem poſitis numeris, ut veritates hæ abſtractæ <lb/>ſenſui magis obviæ fiant.</s> <s xml:id="echoid-s5047" xml:space="preserve"/> </p> <pb o="231" file="0309" n="336" rhead="CONTROVERSIA."/> <p> <s xml:id="echoid-s5048" xml:space="preserve">Sint A & </s> <s xml:id="echoid-s5049" xml:space="preserve">B duo corpora ex Axe D ſuſpenſa ita, ut unius <lb/> <anchor type="note" xlink:label="note-0309-01a" xlink:href="note-0309-01"/> diſtantia ab Axe quadruplo major ſit alterius diſtantiâ.</s> <s xml:id="echoid-s5050" xml:space="preserve"/> </p> <div xml:id="echoid-div432" type="float" level="2" n="1"> <note position="right" xlink:label="note-0309-01" xlink:href="note-0309-01a" xml:space="preserve">TAB. XXVIII. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s5051" xml:space="preserve">Adeoque ſi altitudo perpendicularis B I, ex qua deſcendit <lb/>corpus B deſcribendo arcum B G, ponatur quatuor pedum, <lb/>altera A H, unde corpus A delabitur, unius pedis erit. <lb/></s> <s xml:id="echoid-s5052" xml:space="preserve">Celeritates igitur, quas ſeparatim cadendo acquirunt, quo-<lb/>niam ſunt ut radices altitudinum, ſe habent ut 2 ad 1. </s> <s xml:id="echoid-s5053" xml:space="preserve">Summa <lb/>3, quæ totalem Penduli celeritatem manifeſtat, quando pro-<lb/>portionaliter ad altitudines, ſive ad arcus B G & </s> <s xml:id="echoid-s5054" xml:space="preserve">A F divi-<lb/>ditur, dat gradus celeritatis, quos obtinent pondera, quan-<lb/>do conjunctim in tabulam D G decidunt, videlicet {12/5} & </s> <s xml:id="echoid-s5055" xml:space="preserve">{3/5}, <lb/>quorum quadrata ſunt {144/25} & </s> <s xml:id="echoid-s5056" xml:space="preserve">{9/25}, unde quæ prodit ſumma, <lb/>ſane a ſumma altitudinum, e quibus pondera dimittuntur, <lb/>differt. </s> <s xml:id="echoid-s5057" xml:space="preserve">Veruntamen hæc quadrata proportionem ſolummodo <lb/>altitudinum O M & </s> <s xml:id="echoid-s5058" xml:space="preserve">N L, ad quas pondera, dum a tabula <lb/>reſiliunt, adſcendunt, non ipſas altitudines exprimunt; </s> <s xml:id="echoid-s5059" xml:space="preserve">quas <lb/>inter ratio quidem eſſe poteſt, quæ eſt inter {144/25} & </s> <s xml:id="echoid-s5060" xml:space="preserve">{9/25}, hoc eſt <lb/>inter 16 & </s> <s xml:id="echoid-s5061" xml:space="preserve">1, dum ipſa ſumma eſt quinque, quæ eſt ſum-<lb/>ma altitudinum B I & </s> <s xml:id="echoid-s5062" xml:space="preserve">A H unde pondera delapſa ſunt.</s> <s xml:id="echoid-s5063" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5064" xml:space="preserve">Nam ſi ponamus altitudinem O M 4{17/1@} pedum eſſe, & </s> <s xml:id="echoid-s5065" xml:space="preserve">alte-<lb/>ram N L {5/17}, O M ſe habebit ad N L, ut 16 ad 1; </s> <s xml:id="echoid-s5066" xml:space="preserve">& </s> <s xml:id="echoid-s5067" xml:space="preserve">O M <lb/>† N L erit æqualis B I † A H. </s> <s xml:id="echoid-s5068" xml:space="preserve">Idcirco centrum gravitatis <lb/>commune ponderum A & </s> <s xml:id="echoid-s5069" xml:space="preserve">B, ubi in L, M pervenere, erit <lb/>ad eandem altitudinem, quam obtinebat ante Oſcillationis ini-<lb/>tium. </s> <s xml:id="echoid-s5070" xml:space="preserve">Id clare ex inſpectione figuræ apparet. </s> <s xml:id="echoid-s5071" xml:space="preserve">Pondus enim <lb/>M tantum ſupra lineam Horizontalem B D elevatur, quan-<lb/>tum L infra eam deprimitur, videlicet {12/17} unius pedis; </s> <s xml:id="echoid-s5072" xml:space="preserve">ſe-<lb/>quitur hinc in triangulis ſimilibus M P Q & </s> <s xml:id="echoid-s5073" xml:space="preserve">L Q R latera <lb/>M Q & </s> <s xml:id="echoid-s5074" xml:space="preserve">Q L eſſe æqualia, hoc eſt medium lineæ M L, quæ <lb/>duo pondera conjungit, eſſe in interſectione lineæ Horizon-<lb/>talis.</s> <s xml:id="echoid-s5075" xml:space="preserve"/> </p> <figure> <image file="0309-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0309-01"/> </figure> <pb o="232" file="0310" n="337" rhead="DE CENTRO OSCILL."/> </div> <div xml:id="echoid-div434" type="section" level="1" n="177"> <head xml:id="echoid-head219" xml:space="preserve">VIII.</head> <head xml:id="echoid-head220" style="it" xml:space="preserve">Excerpta ex literis D<emph style="super">ni</emph>. Hugenii ad Auctores Diarii <lb/>Pariſienſis, datis Hagæ 8. Funii 1684. quæ <lb/>continent ejus reſponſionem ad exceptio-<lb/>nem D<emph style="super">ni</emph>. Abbatis Catelani, de cen-<lb/>tro Oſcillationis.</head> <p> <s xml:id="echoid-s5076" xml:space="preserve">HUc usque diſtuli reſpondere ad exceptionem D<emph style="super">ni</emph>. </s> <s xml:id="echoid-s5077" xml:space="preserve">Abbatis <lb/>Catelani, & </s> <s xml:id="echoid-s5078" xml:space="preserve">fere omnis noſtræ controverſiæ oblitus eram, <lb/>quoniam non audiveram, ullum ex iis qui tales ponderant <lb/>quæſtiones in illius partes iviſſe. </s> <s xml:id="echoid-s5079" xml:space="preserve">Sed cum ex amicis quidam <lb/>deſiderarent, ut Geometris facilius redderem litis noſtræ exa-<lb/>men; </s> <s xml:id="echoid-s5080" xml:space="preserve">& </s> <s xml:id="echoid-s5081" xml:space="preserve">eo impedirem quo minus illi, qui hanc norunt, <lb/>ſilentium meum reprehendant, vos rogo ut diariis veſtris, ſe-<lb/>quentia velletis inſerere, cum quibusdam veſtrum jam du-<lb/>dum communicata.</s> <s xml:id="echoid-s5082" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5083" xml:space="preserve">Abbas Catelanus, viſâ meâ reſponſione ad primam obſer-<lb/>vationem, ſuoque errore cognito, credidit ſe illam poſſe <lb/>diſſimulare, ſi diceret, obſervationes ſuas ex vitioſo auto-<lb/>grapho editas eſſe, in quo non ſolum quædam verba dee-<lb/>rant, ſed etiam ſex ſeptemve ordine ſequentes lineæ; </s> <s xml:id="echoid-s5084" xml:space="preserve">cumque <lb/>illæ ſuppletæ ſint in ſecunda Editione, ubi addidit, & </s> <s xml:id="echoid-s5085" xml:space="preserve">qui-<lb/>dem tales, ut ſummæ cum ſex aliis lineis, propoſitam obje-<lb/>ctionem omnino mutavit.</s> <s xml:id="echoid-s5086" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5087" xml:space="preserve">Non opportunum tamen duxit hac de re lectorem mone-<lb/>re, ne quidem in ſuâ exceptione, in qua poſitâ hac muta-<lb/>tione ratiocinatur. </s> <s xml:id="echoid-s5088" xml:space="preserve">Cùm enim antea ſe demonſtraturum pro-<lb/>miſerat, propoſitionem meam 4<emph style="super">am</emph>. </s> <s xml:id="echoid-s5089" xml:space="preserve">de centris Oſcillationum <lb/>falſam eſſe niſi pars ſit æqualis toti, in præſens ut illud <lb/>probet, non tantum ponit axioma hoc irrefragabile totum <lb/>eſſe majus parte, ſed præter hoc pro vero habet quoddam, <lb/>quod ſibi finxit, de motu Pendulorum, principium.</s> <s xml:id="echoid-s5090" xml:space="preserve"/> </p> <pb o="233" file="0311" n="338" rhead="CONTROVERSIA."/> <p> <s xml:id="echoid-s5091" xml:space="preserve">Rem ita ſe habere oſtendam, & </s> <s xml:id="echoid-s5092" xml:space="preserve">ut mutatam ejus objectio-<lb/>nem ſolvam, demonſtrabo principium, quod ponit, ve-<lb/>rum eſſe non poſſe. </s> <s xml:id="echoid-s5093" xml:space="preserve">Etiam falſum eſſe oſtendam alterum ejus <lb/>principium generale, quo utitur in ſuâ verâ ſolutione Mathe-<lb/>matica Problematis de centris Oſcillationis, & </s> <s xml:id="echoid-s5094" xml:space="preserve">tandem ambo <lb/>hæc principia ſibi mutuò contrariare: </s> <s xml:id="echoid-s5095" xml:space="preserve">non deſpero fore ut <lb/>ipſe Abbas Catelanus mecum conveniat, ſi ad ſequentia at-<lb/>tendat.</s> <s xml:id="echoid-s5096" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5097" xml:space="preserve">Quæſtio ſecundum illum ad hanc propoſitionem redit. </s> <s xml:id="echoid-s5098" xml:space="preserve">Si <lb/>habeamus duas magnitudines inæquales a a & </s> <s xml:id="echoid-s5099" xml:space="preserve">b b, & </s> <s xml:id="echoid-s5100" xml:space="preserve">ſummam <lb/>harum radicum dividamus in duas partes, quæ ſint inter ſe <lb/>ut a a ad b b, quæ partes ideo ſunt {a3+a a b/a a+b b} & </s> <s xml:id="echoid-s5101" xml:space="preserve">{b3+a b b/a a+b b}, <lb/>ut facile per Algebram invenitur, demonſtrare, ſummam <lb/>magnitudinum a a & </s> <s xml:id="echoid-s5102" xml:space="preserve">b b, quæ repræſentant altitudines, un-<lb/>de deſcendunt pondera duo æqualia eidem Pendulo alligata, <lb/>non poſſe æquari ſummæ quadratorum partium {a3+a a b/a a + b b} & </s> <s xml:id="echoid-s5103" xml:space="preserve"><lb/>{b3+a b b/a a+b b}, quarum quadrata repræſentant altitudines, ad quas <lb/>pondera, percuſſione ſeparata, redeunt, niſi pars a a æque-<lb/>tur b b; </s> <s xml:id="echoid-s5104" xml:space="preserve">id eſt (quoniam quantitates in quæſtione propoſita <lb/>ſunt inæquales) niſi pars æqualis ſit toti.</s> <s xml:id="echoid-s5105" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5106" xml:space="preserve">Hæc eſt propoſitio Abbatis Catelani, quam tantum cla-<lb/>rius exprimere conatus ſum, quâ demonſtratâ, ut facile fit <lb/>comparando duas illas ſummas per calculum Algebraicum, <lb/>contendit, fundamentalem meam de centris Oſcillationis <lb/>propoſitionem ruere.</s> <s xml:id="echoid-s5107" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5108" xml:space="preserve">Sed etiam relictâ Algebrâ demonſtrari poteſt illius propo-<lb/>ſitio; </s> <s xml:id="echoid-s5109" xml:space="preserve">nam ſi ponatur a a æquale eſſe 1; </s> <s xml:id="echoid-s5110" xml:space="preserve">& </s> <s xml:id="echoid-s5111" xml:space="preserve">b b æquale 4; <lb/></s> <s xml:id="echoid-s5112" xml:space="preserve">ſumma radicum a + b eſt 3, & </s> <s xml:id="echoid-s5113" xml:space="preserve">partes proportionales hujus <lb/>ſummæ ſunt {3/5} & </s> <s xml:id="echoid-s5114" xml:space="preserve">{12/5}, faciunt enim junctim {15/5} hoc eſt 3. </s> <s xml:id="echoid-s5115" xml:space="preserve">Qua-<lb/>drata earundem partium ſunt {9/52} & </s> <s xml:id="echoid-s5116" xml:space="preserve">{144/25}. </s> <s xml:id="echoid-s5117" xml:space="preserve">Hoc igitur ſolum re-<lb/>ſtaret demonſtrandum, ſummam 1. </s> <s xml:id="echoid-s5118" xml:space="preserve">& </s> <s xml:id="echoid-s5119" xml:space="preserve">4. </s> <s xml:id="echoid-s5120" xml:space="preserve">non eſſe æqualem <lb/>ſummæ, quæ prodit ex additione {9/25} ad {144/25}, ſive 5 & </s> <s xml:id="echoid-s5121" xml:space="preserve">6{3/25} non <lb/>eſſe æqualia inter ſe; </s> <s xml:id="echoid-s5122" xml:space="preserve">quod ſane per ſe clarum eſt.</s> <s xml:id="echoid-s5123" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5124" xml:space="preserve">Vera ergo eſſet Abbatis Propoſitio niſi affirmaret quadra-<lb/>ta partium {a3+a a b/a a + b b} & </s> <s xml:id="echoid-s5125" xml:space="preserve">{b3 + a b b/a a + b b}, quæ hic ſunt {9/25} & </s> <s xml:id="echoid-s5126" xml:space="preserve">{144/25}, re- <pb o="234" file="0312" n="339" rhead="DE CENTRO OSCILL"/> præſentare altitudines, ad quas pondera ſejuncta redeunt. <lb/></s> <s xml:id="echoid-s5127" xml:space="preserve">Non diſſentiet & </s> <s xml:id="echoid-s5128" xml:space="preserve">facile probari poteſt illud deductum eſſe <lb/>ex Principio, quod ſibi finxit, & </s> <s xml:id="echoid-s5129" xml:space="preserve">pro fundamento habet pro-<lb/>poſitionis ſuæ; </s> <s xml:id="echoid-s5130" xml:space="preserve">ſcilicet, celeritatem totalem Penduli compoſiti, <lb/>quæ inter partes diſtribuitur proportionaliter ad arcus, ques <lb/>ipſæ deſcribunt, ſemper æqualem eſſe ſummæ celeritatum, quas <lb/>eædem partes acquiſiviſſent, ſi ſejunctæ ſingulæ ſeparatim ex <lb/>iisdem altitudinibus, & </s> <s xml:id="echoid-s5131" xml:space="preserve">in eadem diſtantia ab Axe deſcendiſ-<lb/>ſent. </s> <s xml:id="echoid-s5132" xml:space="preserve">Ponit ergo, ut me refellat, principium hoc, quod fal-<lb/>ſum contendo; </s> <s xml:id="echoid-s5133" xml:space="preserve">in demonſtratione computationem memora-<lb/>tam ſequar.</s> <s xml:id="echoid-s5134" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5135" xml:space="preserve">D<emph style="super">nus</emph> Abbas novit & </s> <s xml:id="echoid-s5136" xml:space="preserve">concedit, detegi altitudinem, unde <lb/>commune ponderum gravitatis centrum deſcendit, ſi divi-<lb/>damus ſummam altitudinum 1. </s> <s xml:id="echoid-s5137" xml:space="preserve">& </s> <s xml:id="echoid-s5138" xml:space="preserve">4. </s> <s xml:id="echoid-s5139" xml:space="preserve">(unde duo pondera ſi-<lb/>mul alligata deſcenderunt) per 2 numerum ponderum, quæ <lb/>ergo eſt {5/2}. </s> <s xml:id="echoid-s5140" xml:space="preserve">Concedit pariter, dari altitudinem, ad quam re-<lb/>vertitur commune eorum gravitatis centrum, ſcilicet {153/50}, vel <lb/>3{3/503}, ſi per numerum ponderum duo, dividamus ſummam al-<lb/>titudinum {9/25} & </s> <s xml:id="echoid-s5141" xml:space="preserve">{144/25}, ad quas pondera percuſſione ſeparata re-<lb/>deunt.</s> <s xml:id="echoid-s5142" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5143" xml:space="preserve">Centrum ergo gravitatis revertetur altius quam unde de-<lb/>ſcenderat, quantum 3{3/50} excedit 2{1/2}, quod primario adverſa-<lb/>tur Mechanices Principio. </s> <s xml:id="echoid-s5144" xml:space="preserve">Hoc ſi D<emph style="super">nu</emph>. </s> <s xml:id="echoid-s5145" xml:space="preserve">Abbas efficere poſ-<lb/>ſit, detectum habebit perpetuum mobile: </s> <s xml:id="echoid-s5146" xml:space="preserve">Quum ergo ejus <lb/>Principium ex quo falſa ſequitur concluſio, falſum ſit, ex-<lb/>inde nil quo mea labefactetur Propoſitio, poteſt inferri <lb/>vel deduci. </s> <s xml:id="echoid-s5147" xml:space="preserve">Quod ad alterum ejus Principium attinet, <lb/>quod pro fundamento habet regulæ generalis de determi-<lb/>nandis centris Oſcillationis, in eundem inducit errorem. <lb/></s> <s xml:id="echoid-s5148" xml:space="preserve">Hoc Principium eſt, tempus Vibrationis Penduli compoſi-<lb/>ti eſſe medium inter tempora Vibrationum partium, id eſt, <lb/>æquale eſſe ſummæ illorum temporum, diviſæ per numerum <lb/>partium. </s> <s xml:id="echoid-s5149" xml:space="preserve">In Pendulo, quale conſideravimus, ubi ponderum <lb/>diſtantiæ, a puncto ſuſpenſionis, ſunt 1 & </s> <s xml:id="echoid-s5150" xml:space="preserve">4, ſi ponamus <lb/>tempus minoris ex partibus ſeparatis eſſe unum, (unde ſe-<lb/>quitur, tempus alterius partis ſeparatim agitatæ eſſe duo;)</s> <s xml:id="echoid-s5151" xml:space="preserve"> <pb o="235" file="0313" n="340" rhead="CONTROVERSIA."/> ſecundum ejus principium, ſumma illorum temporum, quæ <lb/>eſt tria, diviſa per duo, numerum partium, erit tempus <lb/>Penduli compoſiti, ſcilicet {3/2}. </s> <s xml:id="echoid-s5152" xml:space="preserve">Hiſce poſitis, nil præterea po-<lb/>nendo præter id quod concedit D<emph style="super">us</emph>. </s> <s xml:id="echoid-s5153" xml:space="preserve">Abbas, deteguntur alti-<lb/>tudines, ad quas revertuntur pondera, poſtquam a Pendulo <lb/>compoſito ſunt ſejuncta; </s> <s xml:id="echoid-s5154" xml:space="preserve">nempe {4/9} & </s> <s xml:id="echoid-s5155" xml:space="preserve">{64/9}; </s> <s xml:id="echoid-s5156" xml:space="preserve">quarum ſumma {68/9}, <lb/>diviſa per duo, numerum ponderum, dat {34/9}, vel 3{7/9}, alti-<lb/>tudinem, ad quam adſcendit commune gravitatis centrum, <lb/>quæ etiam ſuperat {5/2} vel 2{1/2} unde demonſtravimus centrum <lb/>deſcendiſſe. </s> <s xml:id="echoid-s5157" xml:space="preserve">Non addo methodum computationis quæ facilis <lb/>eſt. </s> <s xml:id="echoid-s5158" xml:space="preserve">D<emph style="super">nus</emph>. </s> <s xml:id="echoid-s5159" xml:space="preserve">Abbas ergo dum quærit principium bis male divi-<lb/>navit; </s> <s xml:id="echoid-s5160" xml:space="preserve">proprie enim divinare eſt, ratiocinia deducere ex <lb/>principiis, quæ levem tantum veri ſpeciem præ ſe ferunt; <lb/></s> <s xml:id="echoid-s5161" xml:space="preserve">& </s> <s xml:id="echoid-s5162" xml:space="preserve">verum eſſet, quæſtionem de centro Oſcillationis non dif-<lb/>ficulter reſolvi, ſi, ut ipſe fecit, tantum ponendum foret id, <lb/>quod ſtatim rem quæſitam determinat.</s> <s xml:id="echoid-s5163" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5164" xml:space="preserve">De cætero, contraria eſſe inter ſe ambo principia clarum <lb/>eſt; </s> <s xml:id="echoid-s5165" xml:space="preserve">quoniam ut patet, ex his diverſæ ſequuntur concluſio-<lb/>nes, altitudines enim, ad quas centrum commune gravitatis <lb/>aſcendit, diverſæ ex utroque deteguntur, nempe 3 {3/50}, 3 {7/9}.</s> <s xml:id="echoid-s5166" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5167" xml:space="preserve">Unum hoc addam, ut reſpondeam difficultati, quam D<emph style="super">nus</emph>. </s> <s xml:id="echoid-s5168" xml:space="preserve">Ab-<lb/>bas propoſuit, contra motum in Cycloide <anchor type="note" xlink:href="" symbol="*"/>, hanc difficulta- <anchor type="note" xlink:label="note-0313-01a" xlink:href="note-0313-01"/> tem ſolutam dari in ipſo meo tractatu de centro Oſcillationis <lb/>in cujus propoſitione 24 explicavi, quomodo efficiqueat, ut <lb/>omnia ponderum Penduli puncta moveantur in Cycloidibus <lb/>æqualibus; </s> <s xml:id="echoid-s5169" xml:space="preserve">licet in praxi minime neceſſaria ſit correctio hæc.</s> <s xml:id="echoid-s5170" xml:space="preserve"/> </p> <div xml:id="echoid-div434" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-0313-01" xlink:href="note-0313-01a" xml:space="preserve">Vide ſu-<lb/>pra pag. 227.</note> </div> </div> <div xml:id="echoid-div436" type="section" level="1" n="178"> <head xml:id="echoid-head221" xml:space="preserve">IX.</head> <head xml:id="echoid-head222" style="it" xml:space="preserve">Reſponſio D<emph style="super">ni</emph>. Abbatis Catelani ad literas D<emph style="super">ni</emph>. <lb/>Bernoulli de Controverſia ſua cum D<emph style="super">no</emph>. Hu-<lb/>genio de centro Oſcillationis <anchor type="note" xlink:href="" symbol="*"/>.</head> <note symbol="*" position="right" xml:space="preserve">Vide ſu-<lb/>pra pag. 23@.</note> <p> <s xml:id="echoid-s5171" xml:space="preserve">UT reſpondeam ad has literas, idem repetam, quo utitur <lb/>D<emph style="super">nus</emph>. </s> <s xml:id="echoid-s5172" xml:space="preserve">Bernoulli, exemplum Penduli compoſiti ex pon-<lb/>deribus æqualibus eidem virgæ applicatis, & </s> <s xml:id="echoid-s5173" xml:space="preserve">ex eodem centro <pb o="236" file="0314" n="341" rhead="DE CENTRO OSCILL."/> ſuſpenſis, a quo pondus unum quater magis quam alterum <lb/>diſtat; </s> <s xml:id="echoid-s5174" xml:space="preserve">ita ut altitudines perpendiculares, unde deſcendunt <lb/>ſint ut 1 ad 4.</s> <s xml:id="echoid-s5175" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5176" xml:space="preserve">Convenit inter nos de proportione inter has altitudines, & </s> <s xml:id="echoid-s5177" xml:space="preserve"><lb/>de ſumma velocitatum, quas illa pondera acquirerent, ſi ſe-<lb/>paratim ab iis altitudinibus caderent; </s> <s xml:id="echoid-s5178" xml:space="preserve">ſed contendimus de <lb/>exprimendis his altitudinibus reſpectu ſpatii, quod ſit com-<lb/>munis earum menſura, & </s> <s xml:id="echoid-s5179" xml:space="preserve">pro unitate habeatur. </s> <s xml:id="echoid-s5180" xml:space="preserve">Cum omni-<lb/>bus, qui ante me de ſimilibus quæſtionibus ſcripſerunt, po-<lb/>no veros numeros, quibus exprimuntur altitudines, eſſe qua-<lb/>drata ipſorum numerorum qui velocitates deſignant, in illis <lb/>caſibus, in quibus inter altitudines & </s> <s xml:id="echoid-s5181" xml:space="preserve">inter velocitates non <lb/>alia datur ratio, præter generalem experientiâ detectam.</s> <s xml:id="echoid-s5182" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5183" xml:space="preserve">Patet autem ex numeris quos in computatione mea detexi, <lb/>(cum 9 & </s> <s xml:id="echoid-s5184" xml:space="preserve">144 ducta in {1/25} pedis, id eſt 6 pedes cum 1 digito <lb/>5 lineis & </s> <s xml:id="echoid-s5185" xml:space="preserve">quod excedit, differant cum 1 pede & </s> <s xml:id="echoid-s5186" xml:space="preserve">4 pedibus, <lb/>aut 5 pedibus,) ſummas altitudinum, ad quas adſcendunt pon-<lb/>dera in exemplo propoſito, non eſſe æqualem ſummæ alti-<lb/>tudinum unde deſcendunt, quam æqualitatem D<emph style="super">us</emph>. </s> <s xml:id="echoid-s5187" xml:space="preserve">Huge-<lb/>nius ponit in generali propoſitione, quâ utitur pro princi-<lb/>pio in ſuo tractatu de centro Oſcillationis.</s> <s xml:id="echoid-s5188" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5189" xml:space="preserve">D<emph style="super">us</emph>. </s> <s xml:id="echoid-s5190" xml:space="preserve">Bernoulli reſpondet, quadrata numerorum, qui <lb/>exprimunt velocitates ponderum, tantum exprimere pro-<lb/>portionem altitudinum, ad quas revertuntur poſt illo-<lb/>rum ſeparationem, & </s> <s xml:id="echoid-s5191" xml:space="preserve">non ipſas altitudines, quæ qui-<lb/>dem inter ſe habere poſſunt rationem, {144/25} ad {9/25} dum ta-<lb/>men harum ſumma eſt 5, quæ eſt ſumma altitudinum, <lb/>unde pondera deſcenderunt. </s> <s xml:id="echoid-s5192" xml:space="preserve">Nam altitudines, ad quas <lb/>revertuntur ſeparata, ſunt juxta illum 4{12/17} & </s> <s xml:id="echoid-s5193" xml:space="preserve">{5/17}, quarum <lb/>ſumma valet 5, ut ſumma primarum altitudinum 1. </s> <s xml:id="echoid-s5194" xml:space="preserve">& </s> <s xml:id="echoid-s5195" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s5196" xml:space="preserve">Iterum reſpondere non difficile erit. </s> <s xml:id="echoid-s5197" xml:space="preserve">Rogatum velim <lb/>D<emph style="super">um</emph>. </s> <s xml:id="echoid-s5198" xml:space="preserve">Bernoulli, dum contendit, proportionem quadra-<lb/>torum numerorum, qui velocitates exprimunt, tantum <lb/>conſiderandam eſſe, quâ motus lege, & </s> <s xml:id="echoid-s5199" xml:space="preserve">per quodnam <lb/>principium Mechanicum pondera, de quibus agimus, po-<lb/>tius redeant ad altitudines, quas ille notavit, &</s> <s xml:id="echoid-s5200" xml:space="preserve"> <pb o="237" file="0315" n="342" rhead="CONTROVERSIA."/> quæ cum ipſius ſententia congruunt, quam ad illarum propor-<lb/>tionales 5{11/17} & </s> <s xml:id="echoid-s5201" xml:space="preserve">{6/17}, quarum ſumma eſt 6; </s> <s xml:id="echoid-s5202" xml:space="preserve">vel ad 3{13/17} & </s> <s xml:id="echoid-s5203" xml:space="preserve">{4/12} qua-<lb/>rum ſumma eſt 4; </s> <s xml:id="echoid-s5204" xml:space="preserve">vel ad innumeras alias ſimiles, quæ ha-<lb/>beant inter ſe eandem proportionem, {144/25} ad {9/25}, ſed ex qui-<lb/>bus altitudo centri gravitatis reverſi major minorve, quam <lb/>unde deſcendit deducitur. </s> <s xml:id="echoid-s5205" xml:space="preserve">Certè pondera non revertentur ad <lb/>omnes altitudines proportionales quadratis velocitatum, quas <lb/>deſcendendo acquiſiverunt, quoniam gravitas gradatim re-<lb/>tardat, & </s> <s xml:id="echoid-s5206" xml:space="preserve">tandem deſtruit velocitates, cum quibus reflectun-<lb/>tur pondera.</s> <s xml:id="echoid-s5207" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5208" xml:space="preserve">Quid ergo eveniet? </s> <s xml:id="echoid-s5209" xml:space="preserve">rogo D<emph style="super">um</emph>. </s> <s xml:id="echoid-s5210" xml:space="preserve">Bernoulli ut illud explica-<lb/>re velit? </s> <s xml:id="echoid-s5211" xml:space="preserve">An Natura per ſe incerta de eo quod ſibi agendum <lb/>eſt, pro voluntate D<emph style="super">ni</emph>. </s> <s xml:id="echoid-s5212" xml:space="preserve">Bernouilli actura eſt? </s> <s xml:id="echoid-s5213" xml:space="preserve">Cum ipſius ve-<lb/>niâ, hac de re dubitabo, donec ſententiam ſuam ſolidis argu-<lb/>mentis, ex Phyſica deductis probet. </s> <s xml:id="echoid-s5214" xml:space="preserve">Interim credo me jure <lb/>concludere, rationibus quas affert pro D. </s> <s xml:id="echoid-s5215" xml:space="preserve">Hugenio hoc con-<lb/>firmari, propoſitionem hujus generalem & </s> <s xml:id="echoid-s5216" xml:space="preserve">fundamentalem <lb/>de centris Oſcillationis non eſſe omni exceptione majorem.</s> <s xml:id="echoid-s5217" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div437" type="section" level="1" n="179"> <head xml:id="echoid-head223" xml:space="preserve">X.</head> <head xml:id="echoid-head224" style="it" xml:space="preserve">Dn. Bernouillii narratio controverſiæ inter Dn. <lb/>Hugenium & Abbatem Catelanum agitatæ <lb/>de Centro Oſcillationis, quæ loco Anim-<lb/>adverſionis eſſe poterit in Reſpon-<lb/>ſionem Dn. Catelani.</head> <head xml:id="echoid-head225" style="it" xml:space="preserve">Excerpta ex Litteris Dn. Bernoullii <lb/>Lipſiam miſſis.</head> <p> <s xml:id="echoid-s5218" xml:space="preserve">Menſe Septembri A. </s> <s xml:id="echoid-s5219" xml:space="preserve">1681. </s> <s xml:id="echoid-s5220" xml:space="preserve">Abbas Catelanus propoſitio-<lb/>nem quandam tractatus Cl. </s> <s xml:id="echoid-s5221" xml:space="preserve">Hugenii, quem de Horo- <pb o="238" file="0316" n="343" rhead="DE CENTRO OSCILL."/> logio Oſcillatorio inſcripſerat, adortus eſt, formata contra il-<lb/>lam objectione; </s> <s xml:id="echoid-s5222" xml:space="preserve">in qua quia mentem ſuam minus feliciter <lb/>expreſſit, anſam dedit iſti controverſiæ, quæ hucusque fere <lb/>inter illos viguit. </s> <s xml:id="echoid-s5223" xml:space="preserve">Verum quidem eſt, eum initio A. </s> <s xml:id="echoid-s5224" xml:space="preserve">1682. <lb/></s> <s xml:id="echoid-s5225" xml:space="preserve">objectioni ſuæ, additis paucis lineis, variationem quandam in-<lb/>duxiſſe; </s> <s xml:id="echoid-s5226" xml:space="preserve">ſed quoniam ejus partes ſatis adhuc male cohæren-<lb/>tes reliquit, eam in mente Lectoris ſui excitavit opinionem, <lb/>quaſi perſuaſum haberet, ſummas altitudinum, e quibus pon-<lb/>dera alicujus penduli junctim deſcendunt, & </s> <s xml:id="echoid-s5227" xml:space="preserve">ad quas poſt-<lb/>modum ſeparatim aſcendunt, inæquales eſſe debere hanc ſo-<lb/>lam ob cauſam, quod priores altitudines ſint proportionales <lb/>ipſis ponderum celeritatibus, poſteriores vero non niſi quadra-<lb/>tis iſtarum celeritatum. </s> <s xml:id="echoid-s5228" xml:space="preserve">Quare etiam Hugenius, id unicum <lb/>Catelano ſcrupulum movere ratus, reſpondere abſtinuit, us-<lb/>que in menſem Junium, quo tandem calamum arripuit, ac <lb/>exemplo duorum numerorum 5 & </s> <s xml:id="echoid-s5229" xml:space="preserve">10, duorumque aliorum 3 <lb/>& </s> <s xml:id="echoid-s5230" xml:space="preserve">12 breviter monſtravit, fieri utique poſſe, ut binæ quan-<lb/>titates eandem cum binis aliis conficiant ſummam, etiamſi di-<lb/>verſam ab illis rationem habeant, neque tum temporis in <lb/>dubium revocavit πρῶ{το}ν Catelani {ψε}ῦδ{ος} quod tamen in pri-<lb/>ma jam objectionis ſuæ impreſſione manifeſte ſatis prodide-<lb/>rat, dum ſuppoſuit: </s> <s xml:id="echoid-s5231" xml:space="preserve">Pendulum ex duobus ponderibus compo-<lb/>ſitum, eandem acquirere celeritatem, quantam acquir at ſum-<lb/>ma pendulorum ſimplicium; </s> <s xml:id="echoid-s5232" xml:space="preserve">id vero ſicco pede præteriit Hu-<lb/>genius, vel quod non penetrarit ſtatim, ob nullam periodo-<lb/>rum connexionem, quorſum falſa iſta Catelani ſuppoſitio <lb/>tenderet, vel potius quod illi ceu veriſimili admodum tum <lb/>ipſemet adſtipularetur. </s> <s xml:id="echoid-s5233" xml:space="preserve">Catelanus interea Hugeniano reſpon-<lb/>ſo non contentus, excepit 20 Julii 1682, ac terminis Alge-<lb/>braicis rem aggreſſus eſt, eodem innixus fundamento: </s> <s xml:id="echoid-s5234" xml:space="preserve">Quod <lb/>totalis celeritas penduli compoſiti æquet ſummam celeritatum <lb/>partium ejus ſeparatarum. </s> <s xml:id="echoid-s5235" xml:space="preserve">Quo facto controverſia iſta ultra <lb/>annum ſopita jacuit. </s> <s xml:id="echoid-s5236" xml:space="preserve">Me quod ſpectabat, cui Hugenii liber <lb/>tum nondum viſus, nedum lectus fuerat, ſcopum alium non <lb/>habebam, quam illuſtrare ejus reſponſionem, remque exa-<lb/>minare, qualiter ab ipſo examinata, atque in Actis recenſita <pb o="239" file="0317" n="344" rhead="CONTROVERSIA."/> fuerat. </s> <s xml:id="echoid-s5237" xml:space="preserve">Animadvertens itaque, Catelani principium ab Hu-<lb/>genio non refutatum eſſe, & </s> <s xml:id="echoid-s5238" xml:space="preserve">ego illud intactum reliqui, <lb/>ſufficere mihi ratus, ſi Hugenianum reſponſum ſimpliciter <lb/>applicarem ad præſentem controverſiam, propoſito eum in <lb/>finem exemplo penduli, e duobus æqualibus ponderibus <lb/>compoſiti; </s> <s xml:id="echoid-s5239" xml:space="preserve">ubi innuere ſaltem volui, quod ſuppoſito pro <lb/>totali ejus celeritate numero ternario (quicquid ſtatuatur de <lb/>celeritatibus utriusque ſeparatim ſpectati ponderis, dummo-<lb/>do eæ ſint in ratione 2 ad 1) quadrata {144/25} & </s> <s xml:id="echoid-s5240" xml:space="preserve">{9/25} ex mente Hu-<lb/>genii ſignificare debeant non niſi rationem altitudinum, ad <lb/>quas aſcendant ſeparata pondera, minime vero ipſas altitu-<lb/>dines (quod ipſe quoque poſtmodum indigitavit Hugenius <lb/>in ſecunda reſponſione, 8 Jun. </s> <s xml:id="echoid-s5241" xml:space="preserve">1684. </s> <s xml:id="echoid-s5242" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/>) partim quoniam ce- <anchor type="note" xlink:label="note-0317-01a" xlink:href="note-0317-01"/> leritates atque altitudines, utpote quantitates heterogeneæ, <lb/>ſe mutuo menſurare non poſſunt; </s> <s xml:id="echoid-s5243" xml:space="preserve">partim etiam, quia ipſe <lb/>Catelanus urgere ſaltem videbatur, altitudines eſſe propor-<lb/>tionales quadratis, vel ſicut quadrata celeritatum; </s> <s xml:id="echoid-s5244" xml:space="preserve">tametſi in <lb/>proxime ſequenti calculo quadrata iſta pro ipſis altitudini-<lb/>bus adhibuerit. </s> <s xml:id="echoid-s5245" xml:space="preserve">Comparato mihi paulo poſt, & </s> <s xml:id="echoid-s5246" xml:space="preserve">perlecto <lb/>Hugenii libro, animadvertebam, Propoſitionem controver-<lb/>ſam ex priore Hypotheſium, quas Auctor initio ſtabiliverat, <lb/>adeo evidenter inferri, ut neutra infringi poſſit, quin ſimul <lb/>evertatur altera; </s> <s xml:id="echoid-s5247" xml:space="preserve">quocirca judicabam, ſi Catelano falſa fuiſ-<lb/>ſet viſa propoſitio, eum potius ipſam adoriri debuiſſe Hy-<lb/>potheſin, magnumque illud inibi contentum Principium Me-<lb/>chanicum. </s> <s xml:id="echoid-s5248" xml:space="preserve">Verum enim vero cum hujus principii veritatem <lb/>nullo jure in dubium revocare poſſem, atque ſimul etiam ſe-<lb/>riem ratiocinii a Catelano ſatis confuſe propoſiti evolvere <lb/>cœpiſſem, errorem ejus detexi illico, falſamque cognovi eſ-<lb/>ſe, qua nitebatur, regulam, nimirum: </s> <s xml:id="echoid-s5249" xml:space="preserve">Celeritatem totalem <lb/>penduli compoſiti æqualem eſſe ſummæ celeritatum partium ejus <lb/>ſeparatarum. </s> <s xml:id="echoid-s5250" xml:space="preserve">Atque ut oſtendam, animadverſum mihi fuiſ-<lb/>ſe errorem, priusquam Hugenii epiſtola de 8. </s> <s xml:id="echoid-s5251" xml:space="preserve">Jun. </s> <s xml:id="echoid-s5252" xml:space="preserve">lucem a-<lb/>ſpexiſſet, afferam hic cauſam phyſicam, omiſſam ab Huge-<lb/>nio, qua fit, ut penduli compoſiti celeritas perpetuo minor <lb/>ſit celeritate partium ejus ſeparatarum: </s> <s xml:id="echoid-s5253" xml:space="preserve">Ponamus majoris evi- <pb o="240" file="0318" n="345" rhead="DE CENTRO OSCILL."/> dentiæ ergo, pondera penduli A & </s> <s xml:id="echoid-s5254" xml:space="preserve">B in linea inflexili D B <lb/> <anchor type="note" xlink:label="note-0318-01a" xlink:href="note-0318-01"/> libere hinc inde moveri poſſe, ſic ut linea hæc, dum rotatur <lb/>circa axem D, quamvis ſecum rapiat pondera, non tamen <lb/>impediat deſcenſum illorum in linea recta centrum Terræ <lb/>verſus. </s> <s xml:id="echoid-s5255" xml:space="preserve">Quo poſito, conſtat, utrumlibet pondus ſigillatim <lb/>dimiſſum, eadem celeritate latum iri, qua ferretur absque <lb/>virga D B, utpote nec a virga, nec ab ejus axe ullo modo <lb/>impeditum; </s> <s xml:id="echoid-s5256" xml:space="preserve">id eſt ſi pondus A absque virga certo tempore <lb/>conficit ſpatium A H, & </s> <s xml:id="echoid-s5257" xml:space="preserve">pondus B ſpatium æquale B N, <lb/>utrumque etiam cum virga, ſed ſigillatim, dimiſſum eodem <lb/>tempore idem ſpatium A H & </s> <s xml:id="echoid-s5258" xml:space="preserve">B N conficiet. </s> <s xml:id="echoid-s5259" xml:space="preserve">Conſtat in-<lb/>ſuper, quod ſi gravitas in utrumque pondus ageret viri-<lb/>bus, quæ proportionatæ forent ipſorum reſpectivis ab axe <lb/>diſtantiis, virga nullum adhuc ipſorum deſcenſui afferret im-<lb/>pedimentum, propterea quoniam exacto certo tempore unum <lb/>eorum reperiretur in H & </s> <s xml:id="echoid-s5260" xml:space="preserve">alterum in I, vel prius in L, po-<lb/>ſterius in N, ſive absque virga, ſive cum virga, ſive ſigil-<lb/>latim ſive conjunctim dimitterentur. </s> <s xml:id="echoid-s5261" xml:space="preserve">Verum enim vero <lb/>quoniam gravitas in utrumque pondus agit viribus æquali-<lb/>bus, ſic ut pondera eodem tempore æqualia ſpatia A H & </s> <s xml:id="echoid-s5262" xml:space="preserve"><lb/>B N transigere annitantur, & </s> <s xml:id="echoid-s5263" xml:space="preserve">tamen interea pondus A jun-<lb/>ctim dimiſſum, ob inflexilem virgam, nequit pertingere niſi <lb/>ad L, dum pondus B jam eſt in N, hinc ſequitur, gravita-<lb/>tis vim in pondere A non eſſe exhauſtam; </s> <s xml:id="echoid-s5264" xml:space="preserve">adeoque reſiduum <lb/>harum virium, ex una parte urgere debere corpus B, ex al-<lb/>tera ipſum axem D, eundemque premendo aliquam ſui par-<lb/>tem ibidem inſumere & </s> <s xml:id="echoid-s5265" xml:space="preserve">deperdere; </s> <s xml:id="echoid-s5266" xml:space="preserve">ſiquidem virga hocce <lb/>caſu inſtar vectis conſiderari poſſit, prout extra dubium eſt; <lb/></s> <s xml:id="echoid-s5267" xml:space="preserve">quod ſi corpus B infinite tarde moveri, id eſt, firmum & </s> <s xml:id="echoid-s5268" xml:space="preserve"><lb/>ſtabile eſſe intelligatur ſicut axis D, corpus A partem ſui <lb/>ponderis æque in axem D atque in corpus B transferret. </s> <s xml:id="echoid-s5269" xml:space="preserve"><lb/>Ex hactenus dictis colligere proclive eſt, ſi quis examinare <lb/>vellet, quantam partem celeritatis ſuæ pondus A in premen-<lb/>do axe D conſumere debeat, eum exinde, imitando Dn. </s> <s xml:id="echoid-s5270" xml:space="preserve">Ca-<lb/>telani ratiocinium, veritatem aut falſitatem Hugenianæ Hy-<lb/>potheſeos, inque hac fundatæ Propoſitionis, detegere poſſe.</s> <s xml:id="echoid-s5271" xml:space="preserve"> <pb o="241" file="0319" n="346" rhead="CONTROVERSIA."/> Rogantur hac occaſione eruditi, ut examinent, qualem le-<lb/>gem communicationis celeritatum obſervent corpora mota, <lb/>quæ ex una parte innituntur firmo fulcimento, ex altera alii <lb/>corpori itidem, ſed tardius moto: </s> <s xml:id="echoid-s5272" xml:space="preserve">ſi namque celeritatis ex-<lb/>ceſſus, qui hinc inde communicandus eſt, in eadem ratione <lb/>diſtribueretur, in qua diſtribuitur onus aliquod, quod vecti <lb/>duobus ſuſtento fulcris impoſitum eſt, nimirum in ratione <lb/>reciproca diſtantiarum mobilis a fulcris, tum imitando ra-<lb/>tiocinium Dn. </s> <s xml:id="echoid-s5273" xml:space="preserve">Catelani, deprehenderemus, ſummam altitu-<lb/>dinum, ad quas aſcendunt ſeparata penduli pondera, viciſ-<lb/>ſim nunc minorem eſſe ſumma altitudinum, e quibus antea <lb/>conjunctim deſcenderant, quod iterum Hugenianam Propo-<lb/>ſitionem everteret.</s> <s xml:id="echoid-s5274" xml:space="preserve"/> </p> <div xml:id="echoid-div437" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-0317-01" xlink:href="note-0317-01a" xml:space="preserve">Vide ſu-<lb/>pra pag. 232.</note> <note position="left" xlink:label="note-0318-01" xlink:href="note-0318-01a" xml:space="preserve">TAB. XXVIII. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s5275" xml:space="preserve">En calculum: </s> <s xml:id="echoid-s5276" xml:space="preserve">Eſto altitudo A L = 1 ped.</s> <s xml:id="echoid-s5277" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5278" xml:space="preserve">Altitudo B N = 4 ped.</s> <s xml:id="echoid-s5279" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>Celeritas ponderis A acquiſita in puncto L, ubi deſcendit <lb/>ſeparatim # = 1 <lb/>Celeritas ponderis B acquiſita in puncto N, quando cadit <lb/>ſeparatim # = 2 <lb/>Celeritas ponderis A acquiſita in puncto L, quando deſcen-<lb/>dit conjunctim # = x <lb/>Igitur Exceſſus celeritatis ponderis A, qui tam in axem, <lb/>quam in pondus B redundat # = 1 - x <lb/>Et pars hujus exceſſus, quæ ſoli ponderi B communicatur <lb/># = {1/4} - {1/2} x <lb/>Tota ergo celeritas ponderis B in puncto N cum conjun-<lb/>ctim cadit # = 2 {1/4} - {1/4} x <lb/></note> <p> <s xml:id="echoid-s5280" xml:space="preserve">Atqui vero 2 {1/4} - {1/4} x, x:</s> <s xml:id="echoid-s5281" xml:space="preserve">: 4, 1. </s> <s xml:id="echoid-s5282" xml:space="preserve">Igitur x = {9/17} & </s> <s xml:id="echoid-s5283" xml:space="preserve">4 x = {36/17} eorum-<lb/>que quadrata {81/289} & </s> <s xml:id="echoid-s5284" xml:space="preserve">{1296/289} quorum ſumma 4 {13/17} minor eſt 1 + 4 = 5. <lb/></s> <s xml:id="echoid-s5285" xml:space="preserve">Antequam finiam, in favorem Dn. </s> <s xml:id="echoid-s5286" xml:space="preserve">Catelani hoc monebo, <lb/>quod etiamſi commune gravitatis centrum juxta illum altius <lb/>aſcendere deberet, quam deſcendit, nondum tamen ſequa-<lb/>tur, repertum fore motum perpetuum, ut ſibi perſuadet Ill. </s> <s xml:id="echoid-s5287" xml:space="preserve"><lb/>Hugenius; </s> <s xml:id="echoid-s5288" xml:space="preserve">quoniam in iſtis abſtrahi ſolet ab aëris reſiſtentia, <lb/>a diminutione celeritatis, quæ neceſſario ſequitur disruptio-<lb/>nem vinculi, quo connectebantur partes penduli, aliorum- <pb o="242" file="0320" n="347" rhead="DE CENTRO OSCILL."/> que obſtaculorum; </s> <s xml:id="echoid-s5289" xml:space="preserve">prout ipſa quoque hæc aëris reſiſtentia <lb/>in cauſa eſt, cur ſimplex pendulum motum ſuum non con-<lb/>tinuet, ut maxime in Hypotheſi Hugeniana ad eandem <lb/>aſcendere debeat altitudinem, a qua deſcendit.</s> <s xml:id="echoid-s5290" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div439" type="section" level="1" n="180"> <head xml:id="echoid-head226" style="it" xml:space="preserve">XI. <lb/>Litteræ D<emph style="super">ni</emph> Marchionis de l’Hôpital ad D<emph style="super">um</emph> Huge-<lb/>nium, in quibus contendit, ſeregulam hujus Au-<lb/>ctoris de Centro oſcillationis penduli compoſiti <lb/>demonſtrare per cauſam Phyſicam, & re-<lb/>ſpondere ſimul D<emph style="super">no</emph> Bernoulli.</head> <p> <s xml:id="echoid-s5291" xml:space="preserve">Ante aliquot annos, cum admiratione legi eruditum tuum <lb/>de centris Oſcillationis tractatum, & </s> <s xml:id="echoid-s5292" xml:space="preserve">pleniſſime mihi <lb/>perſuaſum eſt veras eſſe demonſtrationes tuas. </s> <s xml:id="echoid-s5293" xml:space="preserve">Interea cum <lb/>Acta Lipſienſia nuper mihi ad manus venerint, inveni in actis <lb/>menſis Julii anni 1686. </s> <s xml:id="echoid-s5294" xml:space="preserve">relationem tuæ hac de re cum D<emph style="super">no</emph> <lb/>Abbate Cateleno controverſiæ, à D<emph style="super">no</emph> Bernoulli factam, qui <lb/>tuam ſententiam amplectitur, uti certe debent, quicunque <lb/>aliquem inter Geometras locum ſe tenere contendunt. </s> <s xml:id="echoid-s5295" xml:space="preserve">Sed <lb/>attonitus vidi, finem ratiocinii quo utitur, contrarium eſſe <lb/>tuis demonſtrationibus: </s> <s xml:id="echoid-s5296" xml:space="preserve">unde ad illud ſedulo examinandum <lb/>perductus fui, & </s> <s xml:id="echoid-s5297" xml:space="preserve">cognovi, quod utatur principio veriſſimo, <lb/>licet in eodem applicando fallatur; </s> <s xml:id="echoid-s5298" xml:space="preserve">illud enim principium, <lb/>uti demonſtrabo, ducit ad eandem veritatem, quam probaſti <lb/>in 5<emph style="super">a</emph> tua propoſitione.</s> <s xml:id="echoid-s5299" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5300" xml:space="preserve">Sit D A B virga inflexilis & </s> <s xml:id="echoid-s5301" xml:space="preserve">ſine gravitate, mobilis circa <lb/> <anchor type="note" xlink:label="note-0320-01a" xlink:href="note-0320-01"/> punctum fixum D, ad quam annexa ſint 2 pondera æqualia <lb/>A & </s> <s xml:id="echoid-s5302" xml:space="preserve">B; </s> <s xml:id="echoid-s5303" xml:space="preserve">& </s> <s xml:id="echoid-s5304" xml:space="preserve">ſit diſtantia B D a puncto fixo quadrupla di-<lb/>ſtantiæ A D, quæritur longitudo D G penduli ſimplicis <lb/>iſochroni, id eſt, quod movetur cum eâdem celeritate ac pen-<lb/>dulum compoſitum.</s> <s xml:id="echoid-s5305" xml:space="preserve"/> </p> <div xml:id="echoid-div439" type="float" level="2" n="1"> <note position="left" xlink:label="note-0320-01" xlink:href="note-0320-01a" xml:space="preserve">TAB. XXVIII. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s5306" xml:space="preserve">Ad ſolvendum hoc problema, conſidero velocitates, cum <lb/>quibus corpora A & </s> <s xml:id="echoid-s5307" xml:space="preserve">B incipiunt deſcendere in primo inſtan- <pb o="243" file="0321" n="348" rhead="CONTROVERSIA."/> ti caſus ſui, vel potius, ſpatia, quæ percurrunt eodem tem-<lb/>pore, utut parvo; </s> <s xml:id="echoid-s5308" xml:space="preserve">hoc ſenſu ſumo 1<unsure/> pro velocitate, quacum <lb/>omne corpus grave ſive magnum ſive parvum incipit deſcen-<lb/>dere ſuper planis æqualiter inclinatis; </s> <s xml:id="echoid-s5309" xml:space="preserve">nam, ut ſatis notum <lb/>eſt, illa velocitas eſt æqualis in omnibus corporibus.</s> <s xml:id="echoid-s5310" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5311" xml:space="preserve">Concipio etiam, quantitatem motus corporis initio deſcen-<lb/>ſus ſui oriri ex maſſâ multiplicatâ per illam primam velocitatem; <lb/></s> <s xml:id="echoid-s5312" xml:space="preserve">His ſuppoſitis, conſtat, quod corpus A conetur deſcendere <lb/>cum eâdem celeritate, quâ corpus B; </s> <s xml:id="echoid-s5313" xml:space="preserve">quod cum non poſ-<lb/>ſit, quoniam virgæ junctum eſt in puncto A, cujus veloci-<lb/>tas tantum eſt 4ta pars velocitatis B, debet accelerare mo-<lb/>tum corporis B in pendulo compoſito. </s> <s xml:id="echoid-s5314" xml:space="preserve">Tota ergo difficultas <lb/>conſiſtit in rite determinandâ motûs augmentatione; </s> <s xml:id="echoid-s5315" xml:space="preserve">quod <lb/>hoc pacto facio.</s> <s xml:id="echoid-s5316" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5317" xml:space="preserve">Sit x quantitas motus corporis A in pendulo compoſito, <lb/>reſiduus exceſſus quantitatis ejus motus erit ergo A - x, vis <lb/>hæc applicata in A exerit actionem in punctum fixum D & </s> <s xml:id="echoid-s5318" xml:space="preserve"><lb/>corpus B, quod conſiderari debet ut immobile ipſius reſpe-<lb/>ctu (quoniam clarum eſt, corpus B debere cenſeri immobile <lb/>reſpectu illius exceſſus) & </s> <s xml:id="echoid-s5319" xml:space="preserve">conſequenter virga B D debet <lb/>conſiderari ut vectis cujus extremitates ſuſtinentur in B & </s> <s xml:id="echoid-s5320" xml:space="preserve">D. <lb/></s> <s xml:id="echoid-s5321" xml:space="preserve">habebimus ergo B D, 4. </s> <s xml:id="echoid-s5322" xml:space="preserve">ad A D, 1. </s> <s xml:id="echoid-s5323" xml:space="preserve">ut A - x ad {1/4} A -{1/4} x portio-<lb/>nem exceſſus quantitatis motus corporis A, quæ communi-<lb/>catur corpori B; </s> <s xml:id="echoid-s5324" xml:space="preserve">ita ut quantitas motus corporis B in pen-<lb/>dulo compoſito ſit B + {1/4} A - {1/4} x id eſt {5/4} A - {1/4} x; </s> <s xml:id="echoid-s5325" xml:space="preserve">Jam vero <lb/>propter virgam inflexilem D B velocitas corporis B in pen-<lb/>dulo compoſito debet neceſſario eſſe quadrupla velocita-<lb/>tis corporis A, & </s> <s xml:id="echoid-s5326" xml:space="preserve">conſequenter etiam quantitas ejus motus, <lb/>quoniam corpora ſunt æqualia; </s> <s xml:id="echoid-s5327" xml:space="preserve">unde ſequitur æqualia eſſe <lb/>4 x & </s> <s xml:id="echoid-s5328" xml:space="preserve">{5/4} A - {1/4} x unde deducitur valor ipſius x, {5/17} A, qui ex-<lb/>primit quantitatem motus corporis A in pendulo compoſito; </s> <s xml:id="echoid-s5329" xml:space="preserve"><lb/>Jam ſi fiat, ut {5/17}, velocitas corporis A in pendulo compoſi-<lb/>to, eſt ad 1, velocitatem, omnium corporum gravium in <lb/>extremitate pendulorum ſimplicium, ita D A, 1; </s> <s xml:id="echoid-s5330" xml:space="preserve">eſt ad <lb/>D G, {17/5<unsure/>}, erit hæc longitudo penduli ſimplicis iſochroni; </s> <s xml:id="echoid-s5331" xml:space="preserve">ſi <lb/>enim ſpatia ſunt inter ſe ut velocitates, tempora æqualia <lb/>ſunt.</s> <s xml:id="echoid-s5332" xml:space="preserve"/> </p> <pb o="244" file="0322" n="349" rhead="DE CENTRO OSCILL"/> <p> <s xml:id="echoid-s5333" xml:space="preserve">Si addamus ad pendulum compoſitum D A B novum pon-<lb/> <anchor type="note" xlink:label="note-0322-01a" xlink:href="note-0322-01"/> dus C æquale cuivis ponderi A & </s> <s xml:id="echoid-s5334" xml:space="preserve">B, ita ut D C duplum <lb/>ſit D A, conſiderari debent pondera A & </s> <s xml:id="echoid-s5335" xml:space="preserve">B tanquam ap-<lb/>penſa ad G commune ſuum Oſcillationis centrum, in extre-<lb/>mitate penduli ſimplicis D G; </s> <s xml:id="echoid-s5336" xml:space="preserve">& </s> <s xml:id="echoid-s5337" xml:space="preserve">tunc ponendo x, quanti-<lb/>tatem motus corporis C in pendulo compoſito D C G, ha-<lb/>bebimus C-x pro reſiduo exceſſu quantitatis motus corporis <lb/>C. </s> <s xml:id="echoid-s5338" xml:space="preserve">Quantitas hæc reſidua applicata in C exercet vim in pun-<lb/>ctum fixum D, & </s> <s xml:id="echoid-s5339" xml:space="preserve">punctum G, quod conſidero ut fixum <lb/>ipſius reſpectu; </s> <s xml:id="echoid-s5340" xml:space="preserve">habebimus ergo D G, {17/5}, eſt ad D C, 2, ut C-x <lb/>eſt ad {10 C - 10 x/17}, portionem ejus exceſſus, quæ diſtribuitur in <lb/>G; </s> <s xml:id="echoid-s5341" xml:space="preserve">unde ſequitur, quantitatem motus corporum A & </s> <s xml:id="echoid-s5342" xml:space="preserve">B in <lb/>pendulo compoſito D A C B futuram {5/17} A + {20/17} B + {10 C - 10 x/17} <lb/>id eſt {35 C - 10 x/17}; </s> <s xml:id="echoid-s5343" xml:space="preserve">Ob virgam autem inflexilem D B, velocitas <lb/>corporis A in pendulo compoſito erit neceſſario dimidia ve-<lb/>locitatis corporis C, & </s> <s xml:id="echoid-s5344" xml:space="preserve">velocitas corporis B erit dupla ve-<lb/>locitatis corporis C, & </s> <s xml:id="echoid-s5345" xml:space="preserve">eædem quoque inter motus quanti-<lb/>tates rationes dantur, cum tria corpora ſint æqualia inter ſe; <lb/></s> <s xml:id="echoid-s5346" xml:space="preserve">datur ergo æqualitas inter 2 x + {1/2} x & </s> <s xml:id="echoid-s5347" xml:space="preserve">{35 C - 10 x/17}, unde de-<lb/>ducitur quantitas {2/7} x = C, exprimens quantitatem motus <lb/>corporis C in pendulo compoſito D A C B. </s> <s xml:id="echoid-s5348" xml:space="preserve">Jam ſi fiat, ut <lb/>{2/3}, velocitas corporis C in pendulo compoſito, eſt ad 1, ve-<lb/>locitatem cujusvis corporis gravis in extremitate penduli ſim-<lb/>plicis; </s> <s xml:id="echoid-s5349" xml:space="preserve">ita D C, 2. </s> <s xml:id="echoid-s5350" xml:space="preserve">eſt ad D E, 3, erit hæc longitudo pen-<lb/>duli ſimplicis iſochroni.</s> <s xml:id="echoid-s5351" xml:space="preserve"/> </p> <div xml:id="echoid-div440" type="float" level="2" n="2"> <note position="left" xlink:label="note-0322-01" xlink:href="note-0322-01a" xml:space="preserve">TAB. XXVIII. <lb/>Fig. 7.</note> </div> <p> <s xml:id="echoid-s5352" xml:space="preserve">Si pondera A, B, C eſſent inæqualia, inveniretur ſemper, <lb/>ſequendo hoc ratiocinium, centrum Oſcillationis. </s> <s xml:id="echoid-s5353" xml:space="preserve">ita ut hæc <lb/>methodus ſit generalis, quicunque ſit ponderum numerus, <lb/>& </s> <s xml:id="echoid-s5354" xml:space="preserve">quæcunque eorundem inæqualitas. </s> <s xml:id="echoid-s5355" xml:space="preserve">Oſtendendum nunc, <lb/>methodum hanc etiam uſu venire, ſi pondera ſint ad puncti <lb/>fixi partem utramque diſpoſita.</s> <s xml:id="echoid-s5356" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5357" xml:space="preserve">Sit pendulum compoſitum A D B mobile circa punctum <lb/>fixum D, & </s> <s xml:id="echoid-s5358" xml:space="preserve">oneratum ponderibus æqualibus A & </s> <s xml:id="echoid-s5359" xml:space="preserve">B, ſitque <lb/>D B quadrupla ipſius D A, patet quod corpus A debeat <lb/>retardare motum corporis B in pendulo compoſito; </s> <s xml:id="echoid-s5360" xml:space="preserve">& </s> <s xml:id="echoid-s5361" xml:space="preserve">ut <pb o="245" file="0323" n="350" rhead="CONTROVERSIA."/> præciſe inveniatur quantum retardet, voco x quantitatem <lb/>motus corporis B in pendulo compoſito A D B, & </s> <s xml:id="echoid-s5362" xml:space="preserve">conſe-<lb/>quenter exceſſus reſiduus quantitatis ejus motus erit B x; <lb/></s> <s xml:id="echoid-s5363" xml:space="preserve">ſed ob virgam B A, velocitas corporis A debet neceſſario <lb/>eſſe quarta pars velocitatis corporis B: </s> <s xml:id="echoid-s5364" xml:space="preserve">quantitas ergo ejus <lb/>motûs in pendulo compoſito erit {1/4} x (cum enim corpora A <lb/>& </s> <s xml:id="echoid-s5365" xml:space="preserve">B ſint æqualia quantitates motus ſunt proportionales ve-<lb/>locitatibus. </s> <s xml:id="echoid-s5366" xml:space="preserve">Illa autem motus quantitas produci non potuit, <lb/>niſi per reſiduum exceſſum quantitatis motus corporis B. </s> <s xml:id="echoid-s5367" xml:space="preserve">Pa-<lb/>tet ergo, quod ille exceſſus B - x debeat ſuperare quantitatem <lb/>motus corporis A inferiora verſus, & </s> <s xml:id="echoid-s5368" xml:space="preserve">eidem præterea imprimere <lb/>quantitatem motus {1/4} x ſuperiora verſus; </s> <s xml:id="echoid-s5369" xml:space="preserve">id eſt quod debeat <lb/>agere in corpus A ac ſi vis A + {1/4} x immediate applicita in <lb/>A, illud ſurſum propelleret. </s> <s xml:id="echoid-s5370" xml:space="preserve">Sed vis B x ob punctum fixum D <lb/>agit in A acſi vis 4 B - 4 x immediate applicita in A, illud ſurſum <lb/>propelleret; </s> <s xml:id="echoid-s5371" xml:space="preserve">dabitur ergo æqualitas inter 4 B - 4 x & </s> <s xml:id="echoid-s5372" xml:space="preserve">A + {1/4} x; </s> <s xml:id="echoid-s5373" xml:space="preserve"><lb/>unde deducitur quantitas x = {12/17} B, quæ præciſe exprimit <lb/>quantitatem motus corporis B in pendulo compoſito A D B, <lb/>Porro ſi fiat, ut {12/17}, velocitas corporis B in pendulo compo-<lb/>ſito, eſt ad 1, velocitatem corporis omnis gravis in extremi-<lb/>tate penduli ſimplicis, ita D B, 4. </s> <s xml:id="echoid-s5374" xml:space="preserve">eſt ad D G {17/3}, erit hæc <lb/>longitudo penduli ſimplicis Iſochroni.</s> <s xml:id="echoid-s5375" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5376" xml:space="preserve">Facile concluditur ex his omnibus, principium D<emph style="super">ni</emph> Ber-<lb/>noulli eſſe verum; </s> <s xml:id="echoid-s5377" xml:space="preserve">ſed eundem falli in concluſione quam in-<lb/>de deducit; </s> <s xml:id="echoid-s5378" xml:space="preserve">quoniam conſiderat velocitates acquiſitas corpo-<lb/>rum A & </s> <s xml:id="echoid-s5379" xml:space="preserve">B, cum conſiderare deberet, uti nos fecimus illo-<lb/>rum velocitates incipientes & </s> <s xml:id="echoid-s5380" xml:space="preserve">præterea motus quantitates. <lb/></s> <s xml:id="echoid-s5381" xml:space="preserve">Alias enim nunquam poſſet applicari hoc principium, quod <lb/>non differt à principio, quod obtinet circa vectem, quum <lb/>corpora ſunt inæqualia; </s> <s xml:id="echoid-s5382" xml:space="preserve">Adeo ut credam me plane petitioni <lb/>ejus ſatis feciſſe, Rogantur hac occaſione eruditi & </s> <s xml:id="echoid-s5383" xml:space="preserve">c.</s> <s xml:id="echoid-s5384" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5385" xml:space="preserve">Vides, quo pacto diverſæ viæ ducant ad cognitio-<lb/>nem ejusdem veritatis; </s> <s xml:id="echoid-s5386" xml:space="preserve">nolim tamen meam tuæ æquiparare, <lb/>quæ ultra omnem comparationem eruditior eſt & </s> <s xml:id="echoid-s5387" xml:space="preserve">magis Geo-<lb/>metrica; </s> <s xml:id="echoid-s5388" xml:space="preserve">Interim ſi exiſtimes, non inutile fore demonſtrare <lb/>rationes Phyſicas, quas hic affero, perfecte convenire cum <pb o="246" file="0324" n="351" rhead="DE CENTRO OSCILL."/> tuis demonſtrationibus, idque inſervire poſſe tollendo <lb/>D<emph style="super">ni</emph> Bernoulli dubio, per me licet publici juris fieri has lit-<lb/>teras, ſed ſimul peto, ut iisdem obſervationes tuas ſubjun-<lb/>gere digneris, & </s> <s xml:id="echoid-s5389" xml:space="preserve">perſuaſum habeas me a judicio tuo non <lb/>provocaturum, quod procul dubio doctum ſimul clariſſimum <lb/>& </s> <s xml:id="echoid-s5390" xml:space="preserve">æquiſſimum futurum eſt: </s> <s xml:id="echoid-s5391" xml:space="preserve">ſum ex animo & </s> <s xml:id="echoid-s5392" xml:space="preserve">c.</s> <s xml:id="echoid-s5393" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div442" type="section" level="1" n="181"> <head xml:id="echoid-head227" xml:space="preserve">XII.</head> <head xml:id="echoid-head228" style="it" xml:space="preserve">Obſervationes D<emph style="super">ni</emph> Hugenii in liter as præcedentes & <lb/>in relationem D<emph style="super">ni</emph> Bernoulli, cujus in iis fit mentio.</head> <p> <s xml:id="echoid-s5394" xml:space="preserve">Semper credidi difficile eſſe inventu centrum Oſcillationis <lb/>alia methodo, quam quâ ipſe uſus ſum; </s> <s xml:id="echoid-s5395" xml:space="preserve">neminem quo-<lb/>que vidi, qui id proſpero ſucceſſu tentârit, ſive reſpectu ſo-<lb/>lutionis generalis, ſive in caſu pendulorum compoſitorum, <lb/>quorum pondera ſunt in lineâ recta cum puncto ſuſpenſionis. <lb/></s> <s xml:id="echoid-s5396" xml:space="preserve">Hunc caſum D<emph style="super">nus</emph> Marchio de l’Hôpital ſibi poſt plures alias <lb/>propoſuit, & </s> <s xml:id="echoid-s5397" xml:space="preserve">primus, quod vere poſſum dicere, ſperatum <lb/>ſortitus eſt eventum; </s> <s xml:id="echoid-s5398" xml:space="preserve">nam D<emph style="super">i</emph>. </s> <s xml:id="echoid-s5399" xml:space="preserve">Walliſius, Mariotte, & </s> <s xml:id="echoid-s5400" xml:space="preserve">Pa-<lb/>ter Deſchales quæſiverunt tantum centrum Percuſſionis, nec <lb/>potuerunt demonſtrare idem eſſe cum Centro Oſcillationis, <lb/>licet id revera ita ſe habeat.</s> <s xml:id="echoid-s5401" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5402" xml:space="preserve">Cæterum licet demonſtratio Domini Marchionis valida ſit <lb/>& </s> <s xml:id="echoid-s5403" xml:space="preserve">legitima, & </s> <s xml:id="echoid-s5404" xml:space="preserve">naturæ rei congrua videatur, multa ta-<lb/>men continet quæ Lectorem aliquantum detinere queant; <lb/></s> <s xml:id="echoid-s5405" xml:space="preserve">ut, quando conſiderat quantitatem motus corporis in primo <lb/>initio caſus, & </s> <s xml:id="echoid-s5406" xml:space="preserve">quum diſtinguit & </s> <s xml:id="echoid-s5407" xml:space="preserve">dividit, reſiduum motum <lb/>corporis A, ſcilicet quem haberet cum ſeparatim caderet, <lb/>præ illo quem haberet, ſi deſcenderet veluti pars penduli <lb/>compoſiti, & </s> <s xml:id="echoid-s5408" xml:space="preserve">tandem, quum in pendulo trium ponderum <lb/>vult A & </s> <s xml:id="echoid-s5409" xml:space="preserve">B tanquam fixa conſiderari in G, centro Oſcilla-<lb/>tionis illorum. </s> <s xml:id="echoid-s5410" xml:space="preserve">cum hæc omnia non ſint prorſus evidentia, <lb/>patet viam, quam ingreſſus eſt Marchio, ſatis eſſe difficilem, <lb/>& </s> <s xml:id="echoid-s5411" xml:space="preserve">accuratum valde ratiocinium fuiſſe requiſitum, ne hic de-<lb/>viaretur; </s> <s xml:id="echoid-s5412" xml:space="preserve">D<emph style="super">us</emph> Bernoulli in ſua relatione controverſiarum me <lb/>inter & </s> <s xml:id="echoid-s5413" xml:space="preserve">Abbatem Catelanum, de qua in ſequentibus ali- <pb o="247" file="0325" n="352" rhead="CONTROVERSIA."/> quid obſervabo, eandem viam fuerat ſecutus, ſed cum nec <lb/>illam ad finem usque ſequi potuerit, novo inde ratiocinio <lb/>ejusdem difficultas colligitur.</s> <s xml:id="echoid-s5414" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5415" xml:space="preserve">Obſtrictus ſum D<emph style="super">o</emph>. </s> <s xml:id="echoid-s5416" xml:space="preserve">Bernoulli, quod ſemper in hac con-<lb/>troverſiâ a meis partibus ſteterit adverſus D<emph style="super">um</emph> Abbatem Ca-<lb/>telanum. </s> <s xml:id="echoid-s5417" xml:space="preserve">Interea non potui concipere, quo pacto, poſt-<lb/>quam dixit propoſitionem meam fundamentalem de centro <lb/>Oſcillationis pendere a magno illo principio, ſcilicet quod <lb/>commune centrum gravitatis plurium ponderum non poſſit a-<lb/>ſcendere altius per gravitatis eorum effectum, quam unde de-<lb/>ſcendit, in ſequentibus vertat contra me quoddam ratioci-<lb/>nium, quemadmodum ipſe confitetur, incertum, ac ſi poſ-<lb/>ſet in dubium vocare veritatem hujus ipſius propoſitionis; <lb/></s> <s xml:id="echoid-s5418" xml:space="preserve">cum potius deberet concludere, ſe erraſſe in ſuo ratiocinio.</s> <s xml:id="echoid-s5419" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5420" xml:space="preserve">Ad id vero, quod mihi imputat, me in prima reſponſio-<lb/>ne non refutaſſe falſum D<emph style="super">ni</emph> Abbatis principium, idque in <lb/>ultima non refelliſſe per cauſam ejus Phyſicam, reſpondeo me <lb/>in prima reſponſione credidiſſe, ſufficere, ſi evidens vitium <lb/>oſtenderem in ratiocinio, quod mihi opponitur, licet ulte-<lb/>rius hanc materiam non examinarem; </s> <s xml:id="echoid-s5421" xml:space="preserve">In exceptione autem <lb/>8. </s> <s xml:id="echoid-s5422" xml:space="preserve">Junii 1684. </s> <s xml:id="echoid-s5423" xml:space="preserve">pari jure cum D<emph style="super">no</emph>. </s> <s xml:id="echoid-s5424" xml:space="preserve">Bernoulli poſſem perten-<lb/>dere me id principium refutaſſe per ſuam cauſam Phyſi-<lb/>cam, quoniam oſtendi, repugnare illud magno principio <lb/>naturali quod corpora gravia ſponte non poſſint aſcendere.</s> <s xml:id="echoid-s5425" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5426" xml:space="preserve">Credo enim, in hoc æque conſiſtere cauſam Phyſicam hujus <lb/>phænomeni, ſcilicet quod in pendulo compoſito pondera A & </s> <s xml:id="echoid-s5427" xml:space="preserve"><lb/>B, cum deſcenderint junctim ad partem infimam vibrationum <lb/>ſuarum, non acquirant ſimul tantam velocitatem, quantam <lb/>acquiſiviſſent ſeparatim ex iisdem altitudinibus cadentia, quam <lb/>in eo, quod pondus A conſumat partem ſui motus agendo in <lb/>punctum fixum F, juxta demonſtrationem D<emph style="super">ni</emph>. </s> <s xml:id="echoid-s5428" xml:space="preserve">Bernoulli & </s> <s xml:id="echoid-s5429" xml:space="preserve"><lb/>D<emph style="super">ni</emph> Marchionis de l’Hoſpital; </s> <s xml:id="echoid-s5430" xml:space="preserve">Et ad hoc credendum ex eo ad-<lb/>ducor, quod ſæpe pereat pars motus, licet hunc in aliquo affe-<lb/>ctu edendo conſumi, affirmare non poſſumus, ut in multis caſi-<lb/>bus percuſſionis duorum corporum durorum, juxta id quod ob- <pb o="248" file="0326" n="353" rhead="DE CENTRO OSCILL. CONTROVERSIA."/> ſervavi, quum in lucem ederem leges ejusmodi motuum in <lb/>diariò Pariſienſi 1669 menſis Februarii, ita ut minime pro lege <lb/>naturæ habendum ſit, eandem motus quantitatem ſemper con-<lb/>ſervari, niſi alicui impendatur & </s> <s xml:id="echoid-s5431" xml:space="preserve">conſumatur, ſed hæc con-<lb/>ſtans lex eſt, corpora ſervare vim ſuam adſcendentem, & </s> <s xml:id="echoid-s5432" xml:space="preserve">idcir-<lb/>co ſummam quadratorum velocitatum illorum ſemper manere <lb/>eandem; </s> <s xml:id="echoid-s5433" xml:space="preserve">Hoc autem non ſolum obtinet in ponderibus pen-<lb/>dulorum, & </s> <s xml:id="echoid-s5434" xml:space="preserve">percuſſione corporum durorum, ut ibidem ob-<lb/>ſervavi, ſed in multis quoque aliis mechanicis experimentis.</s> <s xml:id="echoid-s5435" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5436" xml:space="preserve">Demonſtraveram admittendo principium Abbatis Catelani, <lb/>vim aſcendentem ponderum penduli augeri, & </s> <s xml:id="echoid-s5437" xml:space="preserve">ita commune <lb/>eorum gravitatis centrum altius poſſe reverti quam unde de-<lb/>ſcenderat, unde inferebam, hoc poſito inventum fore perpetu-<lb/>um mobile. </s> <s xml:id="echoid-s5438" xml:space="preserve">D<emph style="super">us</emph>. </s> <s xml:id="echoid-s5439" xml:space="preserve">Bernoulli non concedit hanc conſequentiam <lb/>ob aëris obſtaculum, & </s> <s xml:id="echoid-s5440" xml:space="preserve">quædam alia, quæ effectum impedi-<lb/>rent: </s> <s xml:id="echoid-s5441" xml:space="preserve">ſed conſiderare debuiſſet, cum altitudo illa major, <lb/>quam acquirit centrum gravitatis, ſemper determinata ſit quan-<lb/>titas, & </s> <s xml:id="echoid-s5442" xml:space="preserve">effectus obſtaculorum non ſit determinatus, imo mi-<lb/>nui magis magisque poſſit, facile conſtrui poſſe machinam, <lb/>in qua commodum ex elevatione centri gravitatis deriva-<lb/>tum, præponderaret impedimento obſtaculorum; </s> <s xml:id="echoid-s5443" xml:space="preserve">ſed revera<unsure/> <lb/>i<unsure/>d experiri neceſſe erit nunquam.</s> <s xml:id="echoid-s5444" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div443" type="section" level="1" n="182"> <head xml:id="echoid-head229" xml:space="preserve">FINIS.</head> <figure> <image file="0326-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0326-01"/> </figure> <pb file="0327" n="354"/> <pb file="0327a" n="355"/> <figure> <caption xml:id="echoid-caption113" style="it" xml:space="preserve">Pag. 248.<lb/>TAB. XXVIII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables115" xml:space="preserve">B A E D H F I G</variables> </figure> <figure> <caption xml:id="echoid-caption114" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables116" xml:space="preserve">M B A E D L N H F O I G</variables> </figure> <figure> <caption xml:id="echoid-caption115" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables117" xml:space="preserve">O P M I B G Q N L R H A F D</variables> </figure> <figure> <caption xml:id="echoid-caption116" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables118" xml:space="preserve">B A D L N H I</variables> </figure> <figure> <caption xml:id="echoid-caption117" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables119" xml:space="preserve">a B c A C</variables> </figure> <figure> <caption xml:id="echoid-caption118" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables120" xml:space="preserve">D A C B E G</variables> </figure> <figure> <caption xml:id="echoid-caption119" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables121" xml:space="preserve">D A G B</variables> </figure> <pb file="0328" n="356"/> <pb file="0329" n="357"/> </div> <div xml:id="echoid-div444" type="section" level="1" n="183"> <head xml:id="echoid-head230" xml:space="preserve"><emph style="bf">MACHINÆ</emph> <lb/>QUÆDAM, <lb/>ET <lb/><emph style="bf">VARIA</emph> <lb/>CIRCA <lb/><emph style="bf">MECHANICAM.</emph></head> <pb file="0330" n="358"/> <pb file="0331" n="359"/> <figure> <image file="0331-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0331-01"/> </figure> </div> <div xml:id="echoid-div445" type="section" level="1" n="184"> <head xml:id="echoid-head231" xml:space="preserve">MACHINÆ <lb/>QUÆDAM, <lb/>ET <lb/>VARIA <lb/>CIRCA <lb/>MECHANICAM.</head> <head xml:id="echoid-head232" xml:space="preserve">I.</head> <head xml:id="echoid-head233" style="it" xml:space="preserve">Excerpta ex Literis Domini Hugenii, novam quan-<lb/>dam Inventionem Horologiorum exactiſſino-<lb/>rum ac portatilium concernentibus.</head> <p> <s xml:id="echoid-s5445" xml:space="preserve">CUm invenerim artificium diu deſideratum quo fiant <lb/>Horologia & </s> <s xml:id="echoid-s5446" xml:space="preserve">exactiſſima ſimul & </s> <s xml:id="echoid-s5447" xml:space="preserve">portatilia, credo <lb/>rem gratam me facturum publico, ſi id communicem. <lb/></s> <s xml:id="echoid-s5448" xml:space="preserve">Quamobrem mitto Tibi & </s> <s xml:id="echoid-s5449" xml:space="preserve">deſcriptionem & </s> <s xml:id="echoid-s5450" xml:space="preserve">picturam formæ, <lb/>continentem id quod in hoc invento habetur ſingulare, ut <lb/>inter cæteras novitates ſcientiarum, hanc quoque poſſis in-<lb/>ſerere, ſiquidem ita collubuerit, Ephemeridibus Tuis.</s> <s xml:id="echoid-s5451" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5452" xml:space="preserve">Horologia formâ modicâ ad hunc modum conſtructa, <lb/>erunt horarum indices portatiles quam exactiſſimi, ſub ma-<lb/>jori forma autem ubique utiliter adhibebuntur, ac ſpeciatim <lb/>in longitudinibus terrâ marique inveniendis, ſiquidem ipſo-<lb/>rum motus regitur à principio quodam æquabilitatis, non <lb/>aliter ac aliorum cum pendulis, per cycloidem quippe ita cor-<lb/>rectorum ut nullum vecturæ genus illa poſſit ſiſtere aut ſuf-<lb/>flaminare.</s> <s xml:id="echoid-s5453" xml:space="preserve"/> </p> <pb o="254" file="0332" n="360" rhead="MACHINARUM"/> <p> <s xml:id="echoid-s5454" xml:space="preserve">Arcanum inventionis conſiſtit in pinna quadam ſpirali quæ <lb/>altera ſui extremitate interiore affixa eſt haſtæ animulæ ſeu <lb/>raſtri æquilibris, ſed majoris ac ponderoſioris quam pro ſo-<lb/>lito, ac ſupra ſuos cardines ultro citroque mobilis; </s> <s xml:id="echoid-s5455" xml:space="preserve">alterâ <lb/>vero extremitate cohæret particulæ cuidam ſupra Horologii <lb/>ſuperius tegmen eminenti: </s> <s xml:id="echoid-s5456" xml:space="preserve">quæque vibrato ſemel Horologii <lb/>libramento, ſpiras ſuas alternis comprimit ac relaxat, ac ac-<lb/>cedente ſibi parvulo adjumento ab Horologii rotis veniente, <lb/>raſtri, ſeu æquilibrii, motum conſervat, ita quidem, ut, licet <lb/>majores aut minores faciat excurſus, ejus reciprocationes <lb/>tamen una alteri ſint tempore prorſus æquales. </s> <s xml:id="echoid-s5457" xml:space="preserve">In figura, <lb/> <anchor type="note" xlink:label="note-0332-01a" xlink:href="note-0332-01"/> ſuperius Horologii planum eſt A B, circulus æquilibrii vel <lb/>libramentum circulare C D, hujusque axis ſeu haſta E F: <lb/></s> <s xml:id="echoid-s5458" xml:space="preserve">Pinna ſeu elaterium in ſpiram contortum G H M, ferrumi-<lb/>natum ad haſtam æquilibrii in M, & </s> <s xml:id="echoid-s5459" xml:space="preserve">lamellam ſupra planum <lb/>Horologii ſupremum in G, ita quidem ut ſpiræ omnes elaterii <lb/>è duobus iſtis ſuſtentaculis in aëre ſuſpenſæ hæreant, infernè <lb/>nihil attingentes: </s> <s xml:id="echoid-s5460" xml:space="preserve">N O P Q eſt quædam coronis aut fultura <lb/>in qua vertitur alter cardo æquilibrii ſeu libramenti: </s> <s xml:id="echoid-s5461" xml:space="preserve">R S eſt <lb/>una ex rotis dentatis Horologii, habens motum quendam <lb/>libratorium ab occurrente rota proxima ſibi impreſſum. </s> <s xml:id="echoid-s5462" xml:space="preserve">Et <lb/>hæc rota R S implicatur tympano T, ad axem ſeu haſtam <lb/>libramenti firmato, cujus adeo motus hoc medio conſervatur <lb/>quantum opus eſt.</s> <s xml:id="echoid-s5463" xml:space="preserve"/> </p> <div xml:id="echoid-div445" type="float" level="2" n="1"> <note position="left" xlink:label="note-0332-01" xlink:href="note-0332-01a" xml:space="preserve">TAB.XXIX. <lb/>Fig. 1.</note> </div> </div> <div xml:id="echoid-div447" type="section" level="1" n="185"> <head xml:id="echoid-head234" xml:space="preserve">II.</head> <head xml:id="echoid-head235" style="it" xml:space="preserve">Nova Libella, Teleſcopio inſtructa, propriam ſecum <lb/>ferens probationem, & quæ in unica ſtatione <lb/>verificatur, & rectificatur.</head> <p> <s xml:id="echoid-s5464" xml:space="preserve">Hujus inſtrumenti præcipua pars eſt Teleſcopium A B, <lb/> <anchor type="note" xlink:label="note-0332-02a" xlink:href="note-0332-02"/> unius, duorum, aut plurium pedum longitudinis, pro-<lb/>ut ab eo plures exſpectantur effectus: </s> <s xml:id="echoid-s5465" xml:space="preserve">ex duobus, quatuorve <pb o="255" file="0333" n="361" rhead="DESCRIPTIONES."/> convexis vitris modo ordinario, ſat ſuperque cognito, per-<lb/>ficitur; </s> <s xml:id="echoid-s5466" xml:space="preserve">duo ad inverſa objecta videnda, quatuor ad ea <lb/>erigenda inſerviunt. </s> <s xml:id="echoid-s5467" xml:space="preserve">Tubus ex Bractea, aut alio metallo <lb/>ad Cylindri formam componitur, tranſitque in annulum aut <lb/>Cylindrum minorem Cupreum C, qui ipſum in medio in-<lb/>cludit & </s> <s xml:id="echoid-s5468" xml:space="preserve">illi conferuminatur. </s> <s xml:id="echoid-s5469" xml:space="preserve">Cum annulo duo plana & </s> <s xml:id="echoid-s5470" xml:space="preserve">ſimi-<lb/>lia cohærent brachia D & </s> <s xml:id="echoid-s5471" xml:space="preserve">E, unum in ſuperiori parte, alte-<lb/>rum in inferiori, utriusque longitudo eſt fere quartæ partis <lb/>longitudinis Teleſcopii, adeo ut tota Machina Crucem imi-<lb/>tetur. </s> <s xml:id="echoid-s5472" xml:space="preserve">In brachiorum extremitatibus duplicia fila, per parvu-<lb/>los annulos transeuntia, & </s> <s xml:id="echoid-s5473" xml:space="preserve">inter laminas inſerta cum brachiis <lb/>cohærent. </s> <s xml:id="echoid-s5474" xml:space="preserve">Harum laminarum altera brachio affixa eſt, dum <lb/>altera ſeparari poteſt ut fila inter illas inſerantur. </s> <s xml:id="echoid-s5475" xml:space="preserve">Annulo <lb/>crux ſuſpenditur ex hamo F, dum ope alterius annuli cruci, <lb/>uti poſtmodum notabitur, annectitur pondus ejusdem circi-<lb/>ter gravitatis cum illa, & </s> <s xml:id="echoid-s5476" xml:space="preserve">in pyxide G incluſum. </s> <s xml:id="echoid-s5477" xml:space="preserve">Ex hac <lb/>pyxide ſolus egreditur ponderis uncus. </s> <s xml:id="echoid-s5478" xml:space="preserve">Spatium in pyxide <lb/>vacuum oleo nucum, lini, aut alio quocunque quod fri-<lb/>gore non coaleſcit repletur, quo motus ponderis & </s> <s xml:id="echoid-s5479" xml:space="preserve">Teleſco-<lb/>pii illico reprimuntur.</s> <s xml:id="echoid-s5480" xml:space="preserve"/> </p> <div xml:id="echoid-div447" type="float" level="2" n="1"> <note position="left" xlink:label="note-0332-02" xlink:href="note-0332-02a" xml:space="preserve">TAB.XXIX. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s5481" xml:space="preserve">In interiori Teleſcopii parte filum ſericum horizontaliter <lb/>ad vitri objectivi focum expanſum eſt, ſive unum, duo, aut <lb/>tria adſint ocularia; </s> <s xml:id="echoid-s5482" xml:space="preserve">hoc filum ope cochleæ, quæ circum-<lb/>vertitur in foramine H in Teleſcopii tubo, elevatur, & </s> <s xml:id="echoid-s5483" xml:space="preserve"><lb/>deprimitur. </s> <s xml:id="echoid-s5484" xml:space="preserve">Methodus diſponendi filum poſtea explica-<lb/>bitur. </s> <s xml:id="echoid-s5485" xml:space="preserve">I levis admodum ex Cupro annulus eſt, non majoris <lb/>ponderis, quam {1/8@} aut {1/100} partis ponderis ipſius crucis, & </s> <s xml:id="echoid-s5486" xml:space="preserve">qui <lb/>juxta Teleſcopii tubum movetur, & </s> <s xml:id="echoid-s5487" xml:space="preserve">ubi libuerit hæret; </s> <s xml:id="echoid-s5488" xml:space="preserve">inſuper <lb/>ſi Crux non in æquilibrio eſt, alter annulus in interiori Tele-<lb/>ſcopii parte ſufficientis ponderis apponitur, ut æquilibrium <lb/>detur, id eſt, ut Teleſcopii tubus Horizonti parallelus <lb/>fiat, in quo tamen ſumma cura haud requiritur.</s> <s xml:id="echoid-s5489" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5490" xml:space="preserve">Plana Crux ex ligno Machinæ ſuſpenſioni inſervit, ad <lb/>hoc ſuperne uncus F habetur, & </s> <s xml:id="echoid-s5491" xml:space="preserve">in brachiorum altero furca <lb/>K majorem Teleſcopii motum lateralem reprimens, ita ut <lb/>per ſpatium ſemilineæ, id eſt {1/24} pollicis, tantum agitari poſſit.</s> <s xml:id="echoid-s5492" xml:space="preserve"/> </p> <pb o="256" file="0334" n="362" rhead="MACHINARUM"/> <p> <s xml:id="echoid-s5493" xml:space="preserve">Pyxis plumbum, & </s> <s xml:id="echoid-s5494" xml:space="preserve">oleum continens eidem Cruci anne-<lb/>ctitur, ad latera, & </s> <s xml:id="echoid-s5495" xml:space="preserve">fundum incluſa; </s> <s xml:id="echoid-s5496" xml:space="preserve">ut autem ventus a Libel-<lb/>la arceatur, cava crux L planæ ligneæ cruci cum duobus <lb/>aut tribus hamis annectitur, adeo ut integra tum conficiatur <lb/>pyxis.</s> <s xml:id="echoid-s5497" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5498" xml:space="preserve">Ut uſui libella accommodetur, ac rectificetur, ut dictum <lb/>ſuſpenditur, plumbo non infernè annexo, & </s> <s xml:id="echoid-s5499" xml:space="preserve">in quoddam <lb/>diſtans objectum collineatur, attendendo ad punctum quod <lb/>Horizontale filum obtegit, filum enim diſtincte, ut obje-<lb/>ctum ipſum obſervari poteſt. </s> <s xml:id="echoid-s5500" xml:space="preserve">Deinde plumbum addatur, id <lb/>eſt annulo inferiori jungatur, ſi tum Horizontale filum eidem <lb/>objecti puncto reſpondeat, conſtabit crucis centrum gravita-<lb/>tis exacte dari in recta linea quæ duo ſuſpenſionis puncta <lb/>conjungit, id eſt quæ tranſit per puncta quibus fila brachiis <lb/>annectuntur. </s> <s xml:id="echoid-s5501" xml:space="preserve">Hæc eſt primaria requiſita præparatio: </s> <s xml:id="echoid-s5502" xml:space="preserve">verum <lb/>ſi hoc minime reperitur, res facilè perficitur ope annuli I, <lb/>obſervando, annulum vitrum objectivum verſus promoveri <lb/>debere, ſi Teleſcopium declinat, dum pondus annexum <lb/>eſt; </s> <s xml:id="echoid-s5503" xml:space="preserve">contra retrahi debere, ſi Teleſcopium poſt annexum <lb/>pondus elevetur.</s> <s xml:id="echoid-s5504" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5505" xml:space="preserve">Ubi ergo ita conſtituta eſt Machina, ut ad idem punctum <lb/>appenſo & </s> <s xml:id="echoid-s5506" xml:space="preserve">ſublato pondere viſum dirigamus, invertenda eſt, <lb/>ſuſpendendo eam ex annulo inferiori, & </s> <s xml:id="echoid-s5507" xml:space="preserve">plumbum brachio <lb/>alteri annectendo; </s> <s xml:id="echoid-s5508" xml:space="preserve">quia citius motum ſiſtere jubet, & </s> <s xml:id="echoid-s5509" xml:space="preserve">quia <lb/>hoc ad reliqua quæ ſuperſunt perficienda plurimum conducit.</s> <s xml:id="echoid-s5510" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5511" xml:space="preserve">Quod ſi tunc filum in Teleſcopio, uti antea collineando, <lb/>punctum idem ac in præcedenti obſervatione obtegat, pun-<lb/>ctum hoc in eodem plano Horizontali cum centro tubi Te-<lb/>leſcopii exiſtere certiffimum eſt, quemadmodum demonſtra-<lb/>bitur.</s> <s xml:id="echoid-s5512" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5513" xml:space="preserve">Verum ſi filum punctum idem collimando non obtegat, <lb/>elevando & </s> <s xml:id="echoid-s5514" xml:space="preserve">declinando filum, opecochleæ huic uſui adapta-<lb/>tæ, eo poterit reduci, elevando nempe filum ſi ad punctum <lb/>magis elevatum viſus dirigatur, & </s> <s xml:id="echoid-s5515" xml:space="preserve">contra, Teleſcopium <lb/>poſt ſingulas fili mutationes invertendo.</s> <s xml:id="echoid-s5516" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5517" xml:space="preserve">His peractis Inſtrumentum perfecte erit rectificatum, nec <pb o="257" file="0335" n="363" rhead="DESCRIPTIONES."/> obſtat (quod notabile eſt) ſi lentium axes non per harum <lb/>centra transeant, aut ſi ipſæ non exacte in recta linea col-<lb/>locentur; </s> <s xml:id="echoid-s5518" xml:space="preserve">& </s> <s xml:id="echoid-s5519" xml:space="preserve">conſequenter ſecure Machinâ utemur, dummo-<lb/>do nulla ſuperveniat mutatio; </s> <s xml:id="echoid-s5520" xml:space="preserve">filum namque Horizontale <lb/>cujuscunque objecti, ad quem fiet collimatio, indicabit lo-<lb/>cum qui eſt in eodem plano Horizontali cum centro Te-<lb/>leſcopii; </s> <s xml:id="echoid-s5521" xml:space="preserve">ſi autem mutatio quædam detur, in ſingulis obſer-<lb/>vationibus detegi poteſt, primo cum plumbo annexo, dein-<lb/>de ſine plumbo collineando, & </s> <s xml:id="echoid-s5522" xml:space="preserve">tandem Teleſcopium in-<lb/>vertendo. </s> <s xml:id="echoid-s5523" xml:space="preserve">Et hoc utique præcipuum eſt, quo Libella præ <lb/>cæteris gaudet commodum, quod in uſu erroris periculum <lb/>nullum detur.</s> <s xml:id="echoid-s5524" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5525" xml:space="preserve">Pes, quo Machina innitatur, eſt rotunda ferri, aut bra-<lb/>cteæ lamella modice concava, cui annectuntur tres baculi <lb/>longitudinis circiter trium pedum cum ſemiſſe, pyxis huic <lb/>lamellæ impoſita hanc in tribus punctis, tangit, quare facile <lb/>movetur, & </s> <s xml:id="echoid-s5526" xml:space="preserve">ita conſtituitur, juvante cavitate lamellæ, ut <lb/>plumbum in pyxide ſua libero fruatur motu, quod videtur <lb/>trans aperturam M in operculo ligneo.</s> <s xml:id="echoid-s5527" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5528" xml:space="preserve">Gravitas plumbi, ut pyxis firmiter pedi inhæreat, inſervit, <lb/>verum ſi firmius eam ſuſtentari velimus in lamellæ cavæ me-<lb/>dio foramen f@at.</s> <s xml:id="echoid-s5529" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5530" xml:space="preserve">Si totum pondus in pyxide G recludi nolumus, tertia vel <lb/>quarta tantum pars huic includatur & </s> <s xml:id="echoid-s5531" xml:space="preserve">reliquum eidem ferreæ <lb/>caudæ annectatur, ſed extra pyxidem; </s> <s xml:id="echoid-s5532" xml:space="preserve">primo tunc obſerva-<lb/>bitur cum minori pondere ſolo, pyxide contento, tum cum <lb/>altero deſuper addito, in conſtituendo filo Horizontali am-<lb/>bobus utentum.</s> <s xml:id="echoid-s5533" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5534" xml:space="preserve">Hac methodo Teleſcopii agitationes promptiſſime, in <lb/>omnibus obſervationibus quæ ad rectificandam Machinam in-<lb/>ſtituuntur, ceſſant; </s> <s xml:id="echoid-s5535" xml:space="preserve">ſi autem nullum in quibusdam obſerva-<lb/>tionibus imponatur pondus, hæ motiones difficilius ſiſtuntur.</s> <s xml:id="echoid-s5536" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5537" xml:space="preserve">Uncus F, cui Libella appenditur, ſimpliciter ligneæ pla-<lb/>næ cruci annecti poteſt, licet hic annexus apparet annulo, <lb/>qui ope cochleæ adhærentis annulo, quo Machina defertur, <lb/>elevari & </s> <s xml:id="echoid-s5538" xml:space="preserve">deprimi poteſt.</s> <s xml:id="echoid-s5539" xml:space="preserve"/> </p> <pb o="258" file="0336" n="364" rhead="MACHINARUM"/> <p> <s xml:id="echoid-s5540" xml:space="preserve">Quantum hoc proſit, experitur in ipſius tranſlatione, nam <lb/>tum crucis fila laxari queunt curando, ut ſupra furcam K, & </s> <s xml:id="echoid-s5541" xml:space="preserve"><lb/>brachium exiguum curvatum R, deſcendat Teleſcopium, <lb/>ne quidem thecam ligneam aperiendo.</s> <s xml:id="echoid-s5542" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5543" xml:space="preserve">Ne oleum pyxide G contentum ex hac defluat, dum Li-<lb/>bella in itineribus confertur, pyxidis hujus foramen ipſo <lb/>pondere incluſo claudi poteſt; </s> <s xml:id="echoid-s5544" xml:space="preserve">curabitur ad hoc, ut pondus <lb/>illud deſuper planum admodum ſit, & </s> <s xml:id="echoid-s5545" xml:space="preserve">detineatur juxta <lb/>pyxidis operculum ope annuli cochleati S.</s> <s xml:id="echoid-s5546" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5547" xml:space="preserve">Tubus N, illum in magno repræſentat, cui in interiori <lb/> <anchor type="note" xlink:label="note-0336-01a" xlink:href="note-0336-01"/> Teleſcopii parte Horizontale filum cohæret, continet ela-<lb/>ſtrum O P furcæ Q annexum, cui ſericum filum cera affi-<lb/>gitur; </s> <s xml:id="echoid-s5548" xml:space="preserve">hoc elaſtrum furcam ad fruſtum bracteæ T trahit, <lb/>quod ingreditur cochlea, quæ reſpondet foramini H in tubo <lb/>Teleſcopii, quo dato foramine etiam modice tubus N verti <lb/>poteſt, ut filum exacte fiat Horizontale, de quo judicatur <lb/>per Teleſcopium videndo.</s> <s xml:id="echoid-s5549" xml:space="preserve"/> </p> <div xml:id="echoid-div448" type="float" level="2" n="2"> <note position="left" xlink:label="note-0336-01" xlink:href="note-0336-01a" xml:space="preserve">TAB.XXIX. <lb/>Fig. 3.</note> </div> </div> <div xml:id="echoid-div450" type="section" level="1" n="186"> <head xml:id="echoid-head236" style="it" xml:space="preserve">Rectificationis Libellæ Demonſtratio.</head> <p> <s xml:id="echoid-s5550" xml:space="preserve">Rectificationis requiſitum primum fuit hoc; </s> <s xml:id="echoid-s5551" xml:space="preserve">ut centrum <lb/>gravitatis ſuſpenſæ crucis in recta linea per puncta, qui-<lb/>bus fila brachiis annectuntur, transeunte daretur. </s> <s xml:id="echoid-s5552" xml:space="preserve">Ut hujus <lb/>præparationis neceſſitas concipiatur, ſciri oportet, non ſufficere <lb/>ſi Teleſcopium ex utroque annulo ſucceſſive ſuſpenſum ad <lb/>idem objecti punctum dirigatur, hoc enim fieri poteſt, licet <lb/>objecti punctum multum infra, aut ſupra Horizontale pla-<lb/>num inveniatur. </s> <s xml:id="echoid-s5553" xml:space="preserve">Sit A B Cylindri Teleſcopiani axis; </s> <s xml:id="echoid-s5554" xml:space="preserve">ſit <lb/> <anchor type="note" xlink:label="note-0336-02a" xlink:href="note-0336-02"/> C I ſuſpenſionis, aut nexuum filorum linea, quarum linea-<lb/>rum longitudinum hic non habetur ratio, quoniam, cujus-<lb/>cunque magnitudinis Teleſcopium ſit, nihil hoc ad ſuſpen-<lb/>ſi corporis ſitum conferre, notum eſt: </s> <s xml:id="echoid-s5555" xml:space="preserve">ponimus A B, C I <lb/>ſe mutuo ad angulos perfectè rectos ſecare in puncto H. </s> <s xml:id="echoid-s5556" xml:space="preserve">De-<lb/>ſuper centrum gravitatis crucis in E, ponatur in axe A B, <lb/>magis tamen verſus B, quam punctum H; </s> <s xml:id="echoid-s5557" xml:space="preserve">Cruce itaque ſu-<lb/>ſpenſa in C, linea quæ a C ad centrum terræ tendit erit C F, <pb o="259" file="0337" n="365" rhead="DESCRIPTIONES."/> ita ut axis A B declinaturus ſit infra Horizontale planum, <lb/>cui C E perpendicularis eſt, & </s> <s xml:id="echoid-s5558" xml:space="preserve">cum quo efficiet angulum æ-<lb/>qualem angulo H C E; </s> <s xml:id="echoid-s5559" xml:space="preserve">ſi vero viſus radius A B filum Hori-<lb/>zontale, & </s> <s xml:id="echoid-s5560" xml:space="preserve">vitri objectivi B centrum pertranſiens continuetur <lb/>in recta linea ad objecti punctum usque; </s> <s xml:id="echoid-s5561" xml:space="preserve">iſtud punctum infra <lb/>Horizontale planum tunc eſſe futurum, evidens eſt; </s> <s xml:id="echoid-s5562" xml:space="preserve">In-<lb/>vertendo interim Teleſcopium, & </s> <s xml:id="echoid-s5563" xml:space="preserve">hoc per I ſuſpendendo, <lb/>ita tamen, ut extremitas B ad eandem partem remaneat, fa-<lb/>cile patet eundem ſitum, quem ſuſpenſum per C habebat, <lb/>eum acquirere; </s> <s xml:id="echoid-s5564" xml:space="preserve">quia, directionis linea denuo per pun-<lb/>ctum E tranſibit: </s> <s xml:id="echoid-s5565" xml:space="preserve">igitur juxta filum Horizontale, uti antea <lb/>collineabimus ad idem objecti punctum; </s> <s xml:id="echoid-s5566" xml:space="preserve">licet minime æqua <lb/>ſit Libella.</s> <s xml:id="echoid-s5567" xml:space="preserve"/> </p> <div xml:id="echoid-div450" type="float" level="2" n="1"> <note position="left" xlink:label="note-0336-02" xlink:href="note-0336-02a" xml:space="preserve">TAB.XXIX. <lb/>Fig. 4.</note> </div> <p> <s xml:id="echoid-s5568" xml:space="preserve">Per primariam rectificationis partem, defectus hic detegi-<lb/>tur, & </s> <s xml:id="echoid-s5569" xml:space="preserve">corrigitur: </s> <s xml:id="echoid-s5570" xml:space="preserve">nam primo ſi gravitatis Crucis centrum <lb/>exiſtit in H, directionis linea erit C I, & </s> <s xml:id="echoid-s5571" xml:space="preserve">conſtat, pon-<lb/>dere in I annectendo, crucis ſitum non mutari; </s> <s xml:id="echoid-s5572" xml:space="preserve">& </s> <s xml:id="echoid-s5573" xml:space="preserve">idcirco <lb/>per Teleſcopium ad idem punctum, collineabimus; </s> <s xml:id="echoid-s5574" xml:space="preserve">ſed ſi cen-<lb/>trum gravitatis crucis ſit in E, & </s> <s xml:id="echoid-s5575" xml:space="preserve">pondus in I appendatur, <lb/>extremitas B elevabitur, & </s> <s xml:id="echoid-s5576" xml:space="preserve">ideo viſus per Teleſcopium ad <lb/>punctum magis elevatum, quam ante, dirigitur. </s> <s xml:id="echoid-s5577" xml:space="preserve">Quod <lb/>aperte videtur ducendo lineam I E, eamque dividendo in <lb/>K, ita ut pars I K ſit ad K E, uti crucis pondus eſt ad <lb/>pondus in I annexum; </s> <s xml:id="echoid-s5578" xml:space="preserve">nam commune centrum gravitatis erit <lb/>K, & </s> <s xml:id="echoid-s5579" xml:space="preserve">C K directionis linea: </s> <s xml:id="echoid-s5580" xml:space="preserve">& </s> <s xml:id="echoid-s5581" xml:space="preserve">angulus K C E æqualis <lb/>erit illi, quo elevabitur Axis A B. </s> <s xml:id="echoid-s5582" xml:space="preserve">Quia linea C E ſupra <lb/>C K tali angulo elevatur, & </s> <s xml:id="echoid-s5583" xml:space="preserve">quia A B cum C E eundem <lb/>quem ante efficit angulum.</s> <s xml:id="echoid-s5584" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5585" xml:space="preserve">Ut vero clare pateat, appendendo pondus in I ſufficienter <lb/>detegi, an centrum gravitatis crucis detur extra ſuſpenſionis <lb/>lineam; </s> <s xml:id="echoid-s5586" xml:space="preserve">dico, angulum K C E, poſito pondere appenſo æ-<lb/>quale ponderi ipſius crucis, ſenſibiliter æquari {2/3} anguli I C E, <lb/>qui æqualis eſt angulo quo Axis A B, & </s> <s xml:id="echoid-s5587" xml:space="preserve">ideo quoque Vi-<lb/>ſus Radius, magis deprimitur verſus partem B, quam fe-<lb/>ciſſet, ſi centrum gravitatis crucis in H fuiſſet; </s> <s xml:id="echoid-s5588" xml:space="preserve">nam ducendo <lb/>K L parallelam ad E H, in duas partes æquales dividet H I, &</s> <s xml:id="echoid-s5589" xml:space="preserve"> <pb o="260" file="0338" n="366" rhead="MACHINARUM"/> H N valebit {2/3} lineæ L K, aſt L K eſt dimidium H E; <lb/></s> <s xml:id="echoid-s5590" xml:space="preserve">ergo H N eſt {1/3} ex H E, & </s> <s xml:id="echoid-s5591" xml:space="preserve">N E ideo {2/3} ex H E, ſed uti <lb/>E N eſt ad E H, ita ſenſibiliter angulus E C N eſt ad an-<lb/>gulum E C H quia exigui ſunt, id eſt E C K ad E C I.</s> <s xml:id="echoid-s5592" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5593" xml:space="preserve">Cum autem angulus E C K illi æqualis eſt, quo Tele-<lb/>ſcopium elevatur addendo pondus in I, ſequitur exiguum <lb/>pondus P verſus H retro trahendum eſſe ut magis elevetur <lb/>Teleſcopium, ita tamen ut prima elevatio dupla ſit ſecundæ, <lb/>quia angulus K C E duplus eſt K C I; </s> <s xml:id="echoid-s5594" xml:space="preserve">& </s> <s xml:id="echoid-s5595" xml:space="preserve">tunc directionis li-<lb/>nea erit C I, in qua neceſſario gravitatis centrum crucis exi-<lb/>ſtet, quandoquidem centrum gravitatis ponderis in I appen-<lb/>ſi, in hac datur, uti & </s> <s xml:id="echoid-s5596" xml:space="preserve">centrum gravitatis commune ejus-<lb/>dem ponderis & </s> <s xml:id="echoid-s5597" xml:space="preserve">crucis, cujus pars eſt exiguum pondus P. <lb/></s> <s xml:id="echoid-s5598" xml:space="preserve">Si annectendo pondus in I Teleſcopii extremitas B depri-<lb/>matur, dimidia ſua parte augenda erit depreſſio, quod eo-<lb/>dem modo demonſtratur. </s> <s xml:id="echoid-s5599" xml:space="preserve">Hæc angulorum cognitio, ad pri-<lb/>mam Libellæ præparationem faciliorem reddendam uſu venit.</s> <s xml:id="echoid-s5600" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5601" xml:space="preserve">Quod aliam verificationis partem ſpectat, ubi centrum <lb/>gravitatis crucis eſt in C I, ſuſpenſionum linea, ex ante ex-<lb/>plicatis ſequitur, hanc lineam eſſe perpendicularem ad Ho-<lb/>rizontem, ſive crux per C, ſive per I ſuſpendatur, & </s> <s xml:id="echoid-s5602" xml:space="preserve">ſive <lb/>illi ad infra pondus annectatur, ſive crux ſola ſuſpendatur.</s> <s xml:id="echoid-s5603" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5604" xml:space="preserve">Tandem, poſitâ brachiorum ut & </s> <s xml:id="echoid-s5605" xml:space="preserve">filorum æqualitate, <lb/> <anchor type="note" xlink:label="note-0338-01a" xlink:href="note-0338-01"/> certiſſimum eſt, centrum Cylindri Teleſcopii, quod ſit H, <lb/>ad eandem in utraque ſuſpenſione altitudinem dari: </s> <s xml:id="echoid-s5606" xml:space="preserve">ſint ita-<lb/>que D H M, E H P, axis Cylindri in una, & </s> <s xml:id="echoid-s5607" xml:space="preserve">altera ſuſ-<lb/>penſione, ponendo primo, quod differentibus gaudeat po-<lb/>ſitionibus, ſit O objecti punctum, ad quod juxta filum Ho-<lb/>rizontale collineamus, & </s> <s xml:id="echoid-s5608" xml:space="preserve">O M, O P, Radii luminis, qui a <lb/>puncto O tendunt ad centrum aperturæ vitri objectivi, & </s> <s xml:id="echoid-s5609" xml:space="preserve"><lb/>inde æque ac omnes alii radii, qui a puncto O ad vitrum <lb/>objectivum perveniunt, Horizontali filo obviam proficiſcun-<lb/>tur, ſive filum iſtud Teleſcopii axem pertranseat, ſive non; <lb/></s> <s xml:id="echoid-s5610" xml:space="preserve">hoc namque per Dioptricæ leges ſequitur, cum filum pun-<lb/>ctum O tegere videatur, & </s> <s xml:id="echoid-s5611" xml:space="preserve">utrumque diſtincte appareat. </s> <s xml:id="echoid-s5612" xml:space="preserve">Ductis <lb/>lineis rectis H O, M P, ultima parallela erit C I, quia H M, <pb o="261" file="0339" n="367" rhead="DESCRIPTIONES."/> H P ſunt æquales, ac æqualiter inclinatæ ad C I. </s> <s xml:id="echoid-s5613" xml:space="preserve">Ergo an-<lb/>guli M, P, trianguli M H P ſunt æquales; </s> <s xml:id="echoid-s5614" xml:space="preserve">aſt angulos H M O, <lb/>H P O etiam eſſe æquales certum eſt, nullo habito reſpectu <lb/>ad id, quod radiis O M, O P intra Teleſcopium accidit, <lb/>nec ſi vitri objectivi axis per hujus centrum transeat, id eſt, <lb/>an maximam ſuam in centro habeat craſſitudinem. </s> <s xml:id="echoid-s5615" xml:space="preserve">Anguli <lb/>igitur M, P, trianguli M O P ſimiliter æquales ſunt, & </s> <s xml:id="echoid-s5616" xml:space="preserve"><lb/>triangulum hoc eſt Iſosceles, uti M P H; </s> <s xml:id="echoid-s5617" xml:space="preserve">recta idcirco H O <lb/>ſecabit M P ad angulos rectos; </s> <s xml:id="echoid-s5618" xml:space="preserve">ſed M P parallela erat C I; <lb/></s> <s xml:id="echoid-s5619" xml:space="preserve">O H ergo perpendicularis eſt C I, & </s> <s xml:id="echoid-s5620" xml:space="preserve">idcirco punctum O <lb/>in eodem plano Horizontali cum centro H Teleſcopii, quod <lb/>probandum erat.</s> <s xml:id="echoid-s5621" xml:space="preserve"/> </p> <div xml:id="echoid-div451" type="float" level="2" n="2"> <note position="left" xlink:label="note-0338-01" xlink:href="note-0338-01a" xml:space="preserve">TAB.XXIX. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s5622" xml:space="preserve">Si nunc centrum vitri objectivi M, P, in utroque caſu <lb/>detur in eodem puncto ut S, recta H S perpendicularis erit <lb/>C I, quandoquidem anguli C H S, I H S tum æquales <lb/>ſint, propter Teleſcopii inverſionem, verum quia S O tendit <lb/>ad idem punctum O in ambabus ſuſpenſionibus, neceſſario <lb/>erit in eadem recta linea cum H S, ſi enim lineæ hæ angu-<lb/>lum conficerent, angulus iſte ſuperne in una ſuſpenſione eſ-<lb/>ſet dum in altera ſitum contrarium haberet, ſicque juxta <lb/>filum ad duo diſtincta puncta collinearemus, cum ad uni-<lb/>cum punctum id fieri poſuimus; </s> <s xml:id="echoid-s5623" xml:space="preserve">ſed integra igitur linea O S H <lb/>eſt perpendicularis ad C I, ideoque punctum O in eodem <lb/>datur plano Horizontali cum puncto H.</s> <s xml:id="echoid-s5624" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div453" type="section" level="1" n="187"> <head xml:id="echoid-head237" style="it" xml:space="preserve">III. <lb/>Aſtroſcopia Compendiaria, Tubi Optici <lb/>molimine liberata.</head> <p> <s xml:id="echoid-s5625" xml:space="preserve">QUod pleriſque omnibus accidit novis inventis, ut, à par-<lb/>vis orta initiis, cura & </s> <s xml:id="echoid-s5626" xml:space="preserve">tractatione hominum auctiora fiant <lb/>ac perfectiora, id vel præcipue, in admirando illo pro-<lb/>ferendi viſus artificio, uſu veniſſe animadvertimus. </s> <s xml:id="echoid-s5627" xml:space="preserve">Notum <lb/>eſt enim quàm fuerit à prima origine tenue ac pene nihili, <lb/>cum rudimenta ejus quædam, in Portæ Neapolitani libris, <pb o="262" file="0340" n="368" rhead="MACHINARUM"/> obſcure expoſita conſpicerentur; </s> <s xml:id="echoid-s5628" xml:space="preserve">quibus tantum præcelluere <lb/>noſtratium hominum conatus, ut non ſane immerito primi <lb/>ejus inventores haberentur. </s> <s xml:id="echoid-s5629" xml:space="preserve">Hos vero rurſus longiſſime præ-<lb/>vertit Galilæus, tot tantiſque rebus, tubi ſui opera, in cæ-<lb/>lo deprehenſis, quarum nihil quidquam ante ipſum fuerat <lb/>perceptum. </s> <s xml:id="echoid-s5630" xml:space="preserve">Videbatur nihil præſtantius iis, quæ ſibi para-<lb/>verat, organis repertum iri. </s> <s xml:id="echoid-s5631" xml:space="preserve">At, ſi nunc in vitam redeat, <lb/>quis dubitet quin ſuis ipſe multo præpoſiturus ſit ea quæ de-<lb/>inde exſtiterunt; </s> <s xml:id="echoid-s5632" xml:space="preserve">tum noſtra, quibus Saturni planetæ veras <lb/>figuras annulumque primi conſpeximus; </s> <s xml:id="echoid-s5633" xml:space="preserve">tum magis etiam, <lb/>quæ his ſucceſſerunt Italica, ab egregiis artificibus elaborata. <lb/></s> <s xml:id="echoid-s5634" xml:space="preserve">Quibus uſus Vir Clariſſimus Dominicus Caſſinus, alia in-<lb/>ſuper nova phænomena cœlo deduxit; </s> <s xml:id="echoid-s5635" xml:space="preserve">planetariorum glo-<lb/>borum in ſeſe revolutiones, comiteſque Saturni duos, præ-<lb/>ter eum quem nos repereramus, reliquis manifeſtiorem.</s> <s xml:id="echoid-s5636" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5637" xml:space="preserve">Quod ſi attendamus quibus acceſſionibus in tantum hæc ars <lb/>continue creverit, nihil aliud reperiemus niſi auctam tubo-<lb/>rum longitudinem, lenteſque, quas vocant, vitreas in ſphæ-<lb/>ræ majoris convexitatem diligentius conformatas. </s> <s xml:id="echoid-s5638" xml:space="preserve">Etſi enim <lb/>modos quoſdam alios, compendiaque inveſtigaverint viri ſub-<lb/>tiliſſimi; </s> <s xml:id="echoid-s5639" xml:space="preserve">jam conicarum ſectionum præſcriptis figuris, quæ <lb/>vitro inducerentur; </s> <s xml:id="echoid-s5640" xml:space="preserve">jam ſpeculorum reflexionibus radios lu-<lb/>cis colligendo; </s> <s xml:id="echoid-s5641" xml:space="preserve">certum eſt hæc omnia vel fruſtra fuiſſe, vel <lb/>votis & </s> <s xml:id="echoid-s5642" xml:space="preserve">expectatione longe minora, ob cauſas quas expo-<lb/>nere non eſt hujus loci; </s> <s xml:id="echoid-s5643" xml:space="preserve">unamque adeo rationem, qua profi-<lb/>ceretur, hactenus eſſe relictam, tuborum productionem. </s> <s xml:id="echoid-s5644" xml:space="preserve">Et <lb/>ſane, quanto magis rei ipſius naturam intueor, tanto pro-<lb/>pius eſt ut exiſtimem, nihil alia via ne impoſterum quidem <lb/>eſſe ſperandum.</s> <s xml:id="echoid-s5645" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5646" xml:space="preserve">Optime igitur operam ſuam ij collocaſſe videntur, qui pa-<lb/>randis tubi majoris lentibus incubuerunt. </s> <s xml:id="echoid-s5647" xml:space="preserve">Quorum diligentiæ <lb/>ſucceſſus hac in parte non defuit. </s> <s xml:id="echoid-s5648" xml:space="preserve">Sed aliunde non exiguum <lb/>oblatum fuit incommodum, nimia, tuborum longiorum gra-<lb/>vitas ac moles; </s> <s xml:id="echoid-s5649" xml:space="preserve">quibus movendis neceſſario machinæ in auxi-<lb/>lium advocandæ fuerunt. </s> <s xml:id="echoid-s5650" xml:space="preserve">Hæ vero & </s> <s xml:id="echoid-s5651" xml:space="preserve">in iis quæ nunc extant, <lb/>pedum triginta aut quadraginta, longitudinibus difficile con- <pb file="0341" n="369"/> <pb file="0341a" n="370"/> <anchor type="figure" xlink:label="fig-0341a-01a" xlink:href="fig-0341a-01"/> <anchor type="figure" xlink:label="fig-0341a-02a" xlink:href="fig-0341a-02"/> <anchor type="figure" xlink:label="fig-0341a-03a" xlink:href="fig-0341a-03"/> <anchor type="figure" xlink:label="fig-0341a-04a" xlink:href="fig-0341a-04"/> <anchor type="figure" xlink:label="fig-0341a-05a" xlink:href="fig-0341a-05"/> <pb file="0342" n="371"/> <pb o="263" file="0343" n="372" rhead="DESCRIPTIONES."/> ſtruuntur tractanturque; </s> <s xml:id="echoid-s5652" xml:space="preserve">&</s> <s xml:id="echoid-s5653" xml:space="preserve">, ſi ulterius progrediendum ſit, <lb/>multo plus exhibituræ ſint negotii. </s> <s xml:id="echoid-s5654" xml:space="preserve">Adeo ut hic velut obex <lb/>quidam fixus fuiſſe videatur ad majora tendentibus. </s> <s xml:id="echoid-s5655" xml:space="preserve">Quare <lb/>rem inprimis gratam me facturum arbitror hæc ſtudia colen-<lb/>tibus, ſyderumque obſervationi intentis, ſi, quod nuper in-<lb/>veni, oſtendero qua ratione impedimentum illud omne ac <lb/>tædium tollatur; </s> <s xml:id="echoid-s5656" xml:space="preserve">magnoque temporis, operæ & </s> <s xml:id="echoid-s5657" xml:space="preserve">ſumptuum <lb/>compendio, maxima quæque teleſcopia ad hæc ſpectacula <lb/>adhibeantur. </s> <s xml:id="echoid-s5658" xml:space="preserve">Scio inter cætera quæ in hunc finem propoſita <lb/>fuere, hoc quoque, quod hic adferimus, aliis in mentem <lb/>jam à multis annis veniſſe, ut ſine tubo lentes diſponerentur; <lb/></s> <s xml:id="echoid-s5659" xml:space="preserve">ſed quod volebant efficere eos nequiiſſe, niſi machinatione <lb/>quadam difficili nimium, quæque propterea adhuc exitum <lb/>non habuerit. </s> <s xml:id="echoid-s5660" xml:space="preserve">Nos autem quæ docebimus, reipſa utilia eſſe <lb/>invenimus, idque magno commodo noſtro quotidie experi-<lb/>mur. </s> <s xml:id="echoid-s5661" xml:space="preserve">Ea vero ſic ſe habent.</s> <s xml:id="echoid-s5662" xml:space="preserve"/> </p> <div xml:id="echoid-div453" type="float" level="2" n="1"> <figure xlink:label="fig-0341a-01" xlink:href="fig-0341a-01a"> <caption xml:id="echoid-caption120" style="it" xml:space="preserve">Pag. 262.<lb/>TAB.XXIX.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables122" xml:space="preserve">P E O D C Q H M G N B S R T F</variables> </figure> <figure xlink:label="fig-0341a-02" xlink:href="fig-0341a-02a"> <caption xml:id="echoid-caption121" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables123" xml:space="preserve">C A H N E P B L K I</variables> </figure> <figure xlink:label="fig-0341a-03" xlink:href="fig-0341a-03a"> <caption xml:id="echoid-caption122" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables124" xml:space="preserve">N Q O P T</variables> </figure> <figure xlink:label="fig-0341a-04" xlink:href="fig-0341a-04a"> <caption xml:id="echoid-caption123" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables125" xml:space="preserve">F D I C A B H K E R S G</variables> </figure> <figure xlink:label="fig-0341a-05" xlink:href="fig-0341a-05a"> <caption xml:id="echoid-caption124" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables126" xml:space="preserve">L M C M E H O D P I</variables> </figure> </div> <p> <s xml:id="echoid-s5663" xml:space="preserve">Loco patente & </s> <s xml:id="echoid-s5664" xml:space="preserve">undique aperto, malus in terram defigi-<lb/>tur, ad perpendiculum erectus. </s> <s xml:id="echoid-s5665" xml:space="preserve">Noſter, quo primum uſi ſu-<lb/>mus, pedum quinquaginta altitudinem habebat; </s> <s xml:id="echoid-s5666" xml:space="preserve">teleſcopiis <lb/>nempe pedum 70 & </s> <s xml:id="echoid-s5667" xml:space="preserve">amplius ſuffecturus, quanquam non in <lb/>omni ſyderum ſupra horizontem aſcenſu. </s> <s xml:id="echoid-s5668" xml:space="preserve">Deberet enim non <lb/>multo infra totam teleſcopii longitudinem produci. </s> <s xml:id="echoid-s5669" xml:space="preserve">Hujus, <lb/>priuſquam erigatur, latus unum dolabra complanatur at-<lb/>que ibi regulæ binæ affiguntur inter ſe parallelæ, ac ſeſqui-<lb/>pollice diſtantes, itaque canalem efficientes, interius paulo <lb/>latiorem, qui à ſummo malo ad imum fere pertingat, reli-<lb/>quis tantum pedibus tribus vacuis. </s> <s xml:id="echoid-s5670" xml:space="preserve">Præterea in ipſo mali ca-<lb/>cumine, orbiculus imponitur, circum axem mobilis, inque <lb/>eum funis ducitur dupla mali longitudine, craſſitudine mi-<lb/>nimi digiti dimidia. </s> <s xml:id="echoid-s5671" xml:space="preserve">Utque eo, ſi forte opusſit, aſcendi poſ-<lb/>ſit, triangula lignea æqualibus ſpatiis defiguntur, quibus <lb/>ſcandentis pedes inſiſtant. </s> <s xml:id="echoid-s5672" xml:space="preserve">Ita demum paratus malus erigitur, <lb/>parte ea, qua terra tegendus, illita pice, circumdataque <lb/>arena, quo minus putredine corrumpatur. </s> <s xml:id="echoid-s5673" xml:space="preserve">Uſus mali eſt, <lb/>ut lens major ejus opera in altum tollatur quouſque opus eſt; <lb/></s> <s xml:id="echoid-s5674" xml:space="preserve">quod fit hoc modo.</s> <s xml:id="echoid-s5675" xml:space="preserve"/> </p> <pb o="264" file="0344" n="373" rhead="MACHINARUM"/> <p> <s xml:id="echoid-s5676" xml:space="preserve">Aſſerculus bipedalis uno latere ita inciditur, ut intra cana-<lb/>lem, quem diximus, liberrime moveri queat. </s> <s xml:id="echoid-s5677" xml:space="preserve">Hujus medio <lb/>affigitur brachium itidem ligneum, pedem unum à malo ex-<lb/>ſtans, cujus in extremo aliud ſesquipedale, media item ſui <lb/>parte, conjungitur rectis angulis. </s> <s xml:id="echoid-s5678" xml:space="preserve">Utrumque vero Horizon-<lb/>ti parallelum extenditur. </s> <s xml:id="echoid-s5679" xml:space="preserve">Huic tranſverſo brachio lens impo-<lb/>nitur ea qua dicemus ratione, atque omnia ſurſum adducuntur, <lb/>adnexis aſſerculi extremis ad funem ante demonſtratum; </s> <s xml:id="echoid-s5680" xml:space="preserve">qui <lb/>ab imo malo ad ſummum aſcendens, ac ſuper orbiculum trans-<lb/>iens, inde deſcendit rurſus ac, priuſquam terram attingat, <lb/>in ſui ipſius caput alterum innectitur. </s> <s xml:id="echoid-s5681" xml:space="preserve">Habet autem funis is <lb/>adjectum plumbum, pondere æquali quantum eſt brachii mo-<lb/>bilis cum lente impoſita; </s> <s xml:id="echoid-s5682" xml:space="preserve">eoque loco deligatum, ut ad ſum-<lb/>mum malum pertingat, cum lens in imo conſiſtit. </s> <s xml:id="echoid-s5683" xml:space="preserve">Ita hæc <lb/>facillime ad eam quæ requiritur altitudinem erigitur &</s> <s xml:id="echoid-s5684" xml:space="preserve">, omiſ-<lb/>ſo fune, ſponte ibi ſuſpenſa manet. </s> <s xml:id="echoid-s5685" xml:space="preserve">Forma plumbi parte u-<lb/>traque in coni apicem deſinit, ne obhæreat ad triangula quæ <lb/>per malum defixa diximus.</s> <s xml:id="echoid-s5686" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5687" xml:space="preserve">Cæterum lens hæc teleſcopii major collocatur aptaturque <lb/>hoc modo. </s> <s xml:id="echoid-s5688" xml:space="preserve">Primum in annulum ſeu cylindrum cavum, è <lb/>ferri bractea fabricatum, ipſa includitur, longum digitos <lb/>quaternos. </s> <s xml:id="echoid-s5689" xml:space="preserve">Huic cylindro, ſive alteri potius in quem hic in-<lb/>ſeritur, bacillus pedalis, digiti craſſitudine, extrinſecus ſe-<lb/>cundum latus affigitur, totus in partem unam prominens. <lb/></s> <s xml:id="echoid-s5690" xml:space="preserve">Hæc omnia globulo æneo inſiſtunt, avellanæ nucis magni-<lb/>tudine, qui bacillo cohæret, inque ſubjecto ſui moduli ca-<lb/>vo liberrime volvitur; </s> <s xml:id="echoid-s5691" xml:space="preserve">ita tamen ut excidere nequeat. </s> <s xml:id="echoid-s5692" xml:space="preserve">Ca-<lb/>vum partibus duabus conſtat, quæ, ſuper pediculo tereti, <lb/>cochlea junguntur adſtringunturque, ſed ita ut globulum ni-<lb/>hil prorſus premant. </s> <s xml:id="echoid-s5693" xml:space="preserve">Lens igitur, cum bacillo ſibi adfixo, <lb/>hoc modo mobilis efficitur. </s> <s xml:id="echoid-s5694" xml:space="preserve">Quæ porro ut æqualiter librata <lb/>conſiſtat, pondus unius libræ circiter infra bacillum appen-<lb/>ditur, filo æneo craſſiore ſemipedali conjunctum atque infi-<lb/>xum. </s> <s xml:id="echoid-s5695" xml:space="preserve">Cujus flexu facile ita pondus temperatur, ut centrum <lb/>commune, ſuæ lentisque gravitatis, cum centro Sphærulæ <lb/>conveniat, atque hoc pacto quocunque poſitu lens ſuſpenſa <pb o="265" file="0345" n="374" rhead="DESCRIPTIONES."/> maneat, attactuque leviſſimo moveatur. </s> <s xml:id="echoid-s5696" xml:space="preserve">Qua in re potiſſima <lb/>verſatur inventi pars. </s> <s xml:id="echoid-s5697" xml:space="preserve">Pede enim globuli in foramen transverſi <lb/>brachii, quod ſupra deſignavimus, immiſſo, (duo autem vel <lb/>plura ejusmodi foramina fiunt, ut in omnem cæli partem <lb/>commode lens obverti poſſit) filum vel funiculus tenuiſſi-<lb/>mus bacillo, ſive caudæ extremæ, illigatur; </s> <s xml:id="echoid-s5698" xml:space="preserve">juncturus nem-<lb/>pe lentem majorem cum ea quæ oculo proxima ponitur, ac <lb/>proinde futuri teleſcopii longitudinem æquans, vel potius <lb/>paulo excedens. </s> <s xml:id="echoid-s5699" xml:space="preserve">Hinc, ubi ſublata ad malum fuerit lens, <lb/>quocunque id filum, manu leviter tractum, circumferetur, <lb/>lentem una movebit, eamque hoc modo ad aſtrum quodcun-<lb/>que recta opponet. </s> <s xml:id="echoid-s5700" xml:space="preserve">Quod certe absque hoc libramento fieri <lb/>non poſſet. </s> <s xml:id="echoid-s5701" xml:space="preserve">Cæterum ut extento filo cauda ſeu bacillus, <lb/>quem lenti adpoſuimus, parallelus fiat, quod omnino neceſ-<lb/>ſe eſt, inſigitur parti ejus extremæ ſtylus æreus digiti longi-<lb/>tudine, cui deorſum flexo, donec cuſpide ſua tantundem ac <lb/>centrum globuli infra bacillum deſcendat, ita demum filum, <lb/>quod diximus, adnectitur. </s> <s xml:id="echoid-s5702" xml:space="preserve">Cur autem ſtylo flexili hic uta-<lb/>mur poſtea dicetur.</s> <s xml:id="echoid-s5703" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5704" xml:space="preserve">Jam vero & </s> <s xml:id="echoid-s5705" xml:space="preserve">de oculari lente explicandum, quomodo cum <lb/>priore componatur; </s> <s xml:id="echoid-s5706" xml:space="preserve">quod multis verbis non indiget, ſiqui-<lb/>dem eadem fere omnia, quæ in majori lente, obſervanda ſunt. <lb/></s> <s xml:id="echoid-s5707" xml:space="preserve">Similiter enim tubo, ſeu cylindro brevi, hæc quoque in-<lb/>cluditur; </s> <s xml:id="echoid-s5708" xml:space="preserve">item bacillo ſeu caudæ conjungitur; </s> <s xml:id="echoid-s5709" xml:space="preserve">quæ porro <lb/>globulum ſuum cui innitatur habet. </s> <s xml:id="echoid-s5710" xml:space="preserve">Sed hujus loco axiculus <lb/>transverſus adhiberi poteſt. </s> <s xml:id="echoid-s5711" xml:space="preserve">Infra bacillum vero pondus exi-<lb/>guum rurſus appenditur, quanto opus eſt ad faciendum li-<lb/>bramentum. </s> <s xml:id="echoid-s5712" xml:space="preserve">Porro capulus, globulum vel axiculum ferens, <lb/>manu obſervatoris apprehenditur; </s> <s xml:id="echoid-s5713" xml:space="preserve">bacillus lentem majorem <lb/>ſublime poſitam verſus, directus eſt, filo eidem, quod <lb/>inde deſcendit, illigatus. </s> <s xml:id="echoid-s5714" xml:space="preserve">Adducta vero manu, contento-<lb/>que leviter filo, parallelas inter ſe fieri lentes perſpicuum eſt. </s> <s xml:id="echoid-s5715" xml:space="preserve"><lb/>At non eodem modo, bacilli hujus extrema parte, filum ad-<lb/>nectitur, ac ſuperiori illi, qui lentem majorem dirigit, ſed <lb/>per foramen trajectum, inde verticillo involvitur, cujusmo-<lb/>di ſunt quibus teſtudinum chordas intendunt; </s> <s xml:id="echoid-s5716" xml:space="preserve">qui verticillus <pb o="266" file="0346" n="375" rhead="MACHINARUM"/> medio bacillo à latere infixus eſt. </s> <s xml:id="echoid-s5717" xml:space="preserve">Hujus converſione, inter <lb/>obſervandum, fili longitudo producitur contrahiturve, do-<lb/>nec intervallum inter lentem utramque, oculo ſpectatoris <lb/>exacte conveniat, poſtquam antea prope verum fuerit reper-<lb/>tum, quod eſt facillimum.</s> <s xml:id="echoid-s5718" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5719" xml:space="preserve">Cæterum, quo poſſit obſervator immotam detinere lentem <lb/> <anchor type="note" xlink:label="note-0346-01a" xlink:href="note-0346-01"/> ſibi proximam, quod apprime neceſſe eſt, fulcrum quoddam <lb/>præſto eſt è levi materia compactum, duobus pedibus inſi-<lb/>ſtens, ac ſuperiori parte transverſum habens baculum, cui <lb/>brachia utraque, ſive ſtantis ſive ſedentis, innitantur; </s> <s xml:id="echoid-s5720" xml:space="preserve">dum <lb/>altera manus, quomodo diximus, lentem ſuſtinet. </s> <s xml:id="echoid-s5721" xml:space="preserve">Multoque <lb/>expeditior eſt hæc ratio, atque ad uſum accommodatior, <lb/>quam ſi tertius pes fulcro accedat, inque ipſum lens ocula-<lb/>ris imponatur.</s> <s xml:id="echoid-s5722" xml:space="preserve"/> </p> <div xml:id="echoid-div454" type="float" level="2" n="2"> <note position="left" xlink:label="note-0346-01" xlink:href="note-0346-01a" xml:space="preserve">Vide Aucta-<lb/>@um. p@g. <lb/>274.</note> </div> <p> <s xml:id="echoid-s5723" xml:space="preserve">Ut vero noctu, atque in tenebris, ſtellæ quævis teleſco-<lb/>pio noſtro facile reperiantur, lumine utimur laternæ incluſo, <lb/>quales jam vulgo notæ ſunt, vitri convexi vel ſpeculi opera <lb/>longe lucem projicientes. </s> <s xml:id="echoid-s5724" xml:space="preserve">Hujus radiis ad malum lentemque <lb/>in eo hærentem directis, ubi circulus ipſam continens con-<lb/>ſpectus fuerit, facile eo transfertur viſus, ut ſtella ipſi me-<lb/>dia lente tegatur, ſimulque admota lente minori, per u-<lb/>tramque ſe ſpectandam præbeat. </s> <s xml:id="echoid-s5725" xml:space="preserve">Ac ſane multo citius hoc per-<lb/>agitur, quam factum ſit hactenus teleſcopiis tubo inſtructis. <lb/></s> <s xml:id="echoid-s5726" xml:space="preserve">Adeo ut hoc quoque nomine longe præſtet nova hæc obſer-<lb/>vandi ratio. </s> <s xml:id="echoid-s5727" xml:space="preserve">Lunam vero contemplari volentibus, lucerna ni-<lb/>hil opus eſt, quod ipſius aſtri luce lens conſpici poſſit. </s> <s xml:id="echoid-s5728" xml:space="preserve">Sed <lb/>hic ob diſci lunaris amplitudinem; </s> <s xml:id="echoid-s5729" xml:space="preserve">ne partem quampiam in-<lb/>tuenti, ab alia parte lux, aliaque via quam per majorem <lb/>lentem, ad oculum accidat; </s> <s xml:id="echoid-s5730" xml:space="preserve">circulus papyraceus lenti huic <lb/>circumponitur, paulo majore quam dupla diametro ad eum <lb/>quo tota Luna tegeretur. </s> <s xml:id="echoid-s5731" xml:space="preserve">Quod niſi fiat, dilutiores apparent <lb/>umbræ tractuſque ii qui, cæteris obſcuriores, in ejus globo <lb/>conſpici ſolent. </s> <s xml:id="echoid-s5732" xml:space="preserve">Atque ita jam teleſcopii noſtri aërii ratio-<lb/>nem omnem & </s> <s xml:id="echoid-s5733" xml:space="preserve">apparatum explicuimus, non ſane operoſum; </s> <s xml:id="echoid-s5734" xml:space="preserve"><lb/>filoque illo, velut Ariadnæo, unde hactenus inventus non <lb/>erat, exitum reperimus.</s> <s xml:id="echoid-s5735" xml:space="preserve"/> </p> <pb o="267" file="0347" n="376" rhead="DESCRIPTIONES."/> <p> <s xml:id="echoid-s5736" xml:space="preserve">Cæterum quo clarius ea, quæ diximus, intelligantur, <lb/>delineationem hic ſubjicimus, in qua</s> </p> <p style="it"> <s xml:id="echoid-s5737" xml:space="preserve">Malus eſt, a b.</s> <s xml:id="echoid-s5738" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">TAB.XXX.</note> <p style="it"> <s xml:id="echoid-s5739" xml:space="preserve">Aſſerculus in canali mobilis, c d.</s> <s xml:id="echoid-s5740" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5741" xml:space="preserve">Brachium ipſi ad angulos rectos infixum, e.</s> <s xml:id="echoid-s5742" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5743" xml:space="preserve">Baculus tranſverſus in quem lens imponitur, f f.</s> <s xml:id="echoid-s5744" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5745" xml:space="preserve">Funis in ſe rediens, g g.</s> <s xml:id="echoid-s5746" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5747" xml:space="preserve">Plumbum funi innexum, h.</s> <s xml:id="echoid-s5748" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5749" xml:space="preserve">Orbiculus in ſummo malo, a.</s> <s xml:id="echoid-s5750" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5751" xml:space="preserve">Cylindrus cavus lentem primariam continens, i.</s> <s xml:id="echoid-s5752" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5753" xml:space="preserve">Bacillus cylindro affixus, k l.</s> <s xml:id="echoid-s5754" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5755" xml:space="preserve">Globulus æneus bacillo hærens & </s> <s xml:id="echoid-s5756" xml:space="preserve">in ſubjecto cavo volubilis, m.</s> <s xml:id="echoid-s5757" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5758" xml:space="preserve">Plumbum filo æneo junctum, n.</s> <s xml:id="echoid-s5759" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5760" xml:space="preserve">Stylus brevis ac flexilis, extremo bacillo inſertus, l.</s> <s xml:id="echoid-s5761" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5762" xml:space="preserve">Tubulus minorem ſeu ocularem lentem ferens, o.</s> <s xml:id="echoid-s5763" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5764" xml:space="preserve">Bacillus tubulo affixus, p.</s> <s xml:id="echoid-s5765" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5766" xml:space="preserve">Axiculus mobilis, q.</s> <s xml:id="echoid-s5767" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5768" xml:space="preserve">Capulus manu tenendus, r.</s> <s xml:id="echoid-s5769" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5770" xml:space="preserve">Glans plumbea, ſ.</s> <s xml:id="echoid-s5771" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5772" xml:space="preserve">Verticillus cui filum involvitur, t.</s> <s xml:id="echoid-s5773" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5774" xml:space="preserve">Pinnulæ decuſſatim poſitæ, atque ita foramen efficientes quo filum <lb/>trajicitur, u.</s> <s xml:id="echoid-s5775" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5776" xml:space="preserve">Filum tenue bombycinum, l u.</s> <s xml:id="echoid-s5777" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5778" xml:space="preserve">Fulcrum cui ſpectator innititur, x.</s> <s xml:id="echoid-s5779" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5780" xml:space="preserve">Laterna, y.</s> <s xml:id="echoid-s5781" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5782" xml:space="preserve">Triangula per malum diſpoſita, quibus conſcendi poſſit, <lb/>omiſſa ſunt, ne figuram obſcuriorem redderent.</s> <s xml:id="echoid-s5783" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5784" xml:space="preserve">Supereſt ut nonnulla, quæ fortaſſe nondum expertis ſcru-<lb/>pulum injicere poſſent, paulo accuratius examinemus. </s> <s xml:id="echoid-s5785" xml:space="preserve">Vere-<lb/>buntur primum ne, ſubſidente filo quod ad utramque len-<lb/>tem pertingit, flexus ejus, quanquam exiguus, in magnis <lb/>tamen illis, pedum centum aut ducentorum, longitudinibus <lb/>impediat poſitum earum parallelum. </s> <s xml:id="echoid-s5786" xml:space="preserve">Et profecto, ſi fune <lb/>graviore opus foret, non parum noceret curvatura ejus, <lb/>nullaque fere tendendi vehementia ſuperari poſſet hoc incom-<lb/>modum. </s> <s xml:id="echoid-s5787" xml:space="preserve">Nunc verò, ſuſpenſa librataque lente majori ut à no- <pb o="268" file="0348" n="377" rhead="MACHINARUM"/> bis factum eſt, leviſſimi tantum fili bombycini tractueam di-<lb/>rigimus; </s> <s xml:id="echoid-s5788" xml:space="preserve">cujus pondus in pedes quinquaginta ſemidrachmam <lb/>non ſuperat; </s> <s xml:id="echoid-s5789" xml:space="preserve">quodque idem appenſas libras ſeptem ſuſtinet <lb/>priuſquam rumpatur. </s> <s xml:id="echoid-s5790" xml:space="preserve">Quare flexus ejus neque in hac, ne-<lb/>que in multo majori lentium diſtantia quidquam officit, etſi <lb/>non niſi modica vi trahatur, duabus tribuſve æquipollente li-<lb/>bris; </s> <s xml:id="echoid-s5791" xml:space="preserve">utique cum geometrica perfectio nequaquam hic requi-<lb/>ratur, ut cuilibet experto notum.</s> <s xml:id="echoid-s5792" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5793" xml:space="preserve">Etenim certum eſt eadem ratione, qua funis fune levior eſt, <lb/>vim tenſionis diminui, qua utrumque ad rectam lineam æqua-<lb/>liter accedat. </s> <s xml:id="echoid-s5794" xml:space="preserve">Ut proinde funiculus quinquaginta pedes lon-<lb/>gus, atque unciam pendens, vi librarum quadraginta octo o-<lb/>pus habeat, ubi filum noſtrum, longitudine pari, non niſi tri-<lb/>bus libris indigebit. </s> <s xml:id="echoid-s5795" xml:space="preserve">Atque hoc per ſe clarius eſt quam ut <lb/>demonſtratione comprobetur. </s> <s xml:id="echoid-s5796" xml:space="preserve">Idem enim eſt prorſus cum <lb/>ſexdecim funiculi ſemidrachmales trahuntur ſinguli trium li-<lb/>brarum pondere, atque cum uncialem funiculum ſimul com-<lb/>ponentes, is à conjunctis itidem ſexdecies ternis libris conten-<lb/>ditur.</s> <s xml:id="echoid-s5797" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5798" xml:space="preserve">Sed ulterius quoque hæc, quæ ad fili flexum attinent, <lb/>geometriæ rationibus, experimentiſque expendi poſſunt. <lb/></s> <s xml:id="echoid-s5799" xml:space="preserve">Nempe contentum ſilum, flexu illo exiguo, parabolicam li-<lb/>neam tam prope exprimit, ut pro vera abſque errore habea-<lb/>tur. </s> <s xml:id="echoid-s5800" xml:space="preserve">Cujus parabolæ profunditatem, in longitudine pedum <lb/>centum quinquaginta, invenimus pedis unius circiter quar-<lb/>tam partem; </s> <s xml:id="echoid-s5801" xml:space="preserve">cum filum horizonti parallelum tenderetur, nec <lb/> <anchor type="note" xlink:label="note-0348-01a" xlink:href="note-0348-01"/> niſi vi librarum duarum & </s> <s xml:id="echoid-s5802" xml:space="preserve">ſemis. </s> <s xml:id="echoid-s5803" xml:space="preserve">Sit fili parabola a b c, pro-<lb/>funditas ejus d b, ducta nimirum recta a d c. </s> <s xml:id="echoid-s5804" xml:space="preserve">Porro tangant <lb/>parabolam rectæ a e, c f: </s> <s xml:id="echoid-s5805" xml:space="preserve">quibus occurrant c e, a f, paral-<lb/>lelæ d b. </s> <s xml:id="echoid-s5806" xml:space="preserve">Intuenti igitur ex a puncto, ſecundum rectam a e, <lb/>notatum fuit ſpatium c e fieri pedis unius; </s> <s xml:id="echoid-s5807" xml:space="preserve">unde fit d b pe-<lb/>dis quarta pars. </s> <s xml:id="echoid-s5808" xml:space="preserve">Ipſi vero c e æquale eſt a f. </s> <s xml:id="echoid-s5809" xml:space="preserve">Itaque lentem <lb/>in c poſitam ita trahit filum c b a, ut non ad oculum, qui <lb/>eſt in a, ſed ad f punctum directe oppoſita ſit. </s> <s xml:id="echoid-s5810" xml:space="preserve">Ut proinde <lb/>pedis unius intervallo à vero loco oculus abſit: </s> <s xml:id="echoid-s5811" xml:space="preserve">quod in illa <lb/>pedum 150 diſtantia nihil obeſſe poteſt. </s> <s xml:id="echoid-s5812" xml:space="preserve">Fit enim angulus <pb file="0349" n="378"/> <pb file="0349a" n="379"/> <anchor type="figure" xlink:label="fig-0349a-01a" xlink:href="fig-0349a-01"/> <pb file="0350" n="380"/> <pb o="269" file="0351" n="381" rhead="DESCRIPTIONES."/> deflexionis c a e vel a c f tantum duarum quintarum unius <lb/>gradus; </s> <s xml:id="echoid-s5813" xml:space="preserve">adeo ut remedio, quod tamen dabimus, non ſit <lb/>opus. </s> <s xml:id="echoid-s5814" xml:space="preserve">Sumpta autem g h diſtantia prioris dupla, ſeu pe-<lb/>dum trecentorum, ut filum incurvum ſit g b h, erit cavita-<lb/>tis menſura k b, prioris d b quadrupla quidem, ſed angulus <lb/>deflexionis tantummodo duplus, hoc eſt, {4/5} unius gradus; <lb/></s> <s xml:id="echoid-s5815" xml:space="preserve">ut facile perſpicitur, ducta tangente g l, quæ cum perpen-<lb/>diculari h l conveniat. </s> <s xml:id="echoid-s5816" xml:space="preserve">Ipſa enim h l quadrupla erit ad k b <lb/>ſive c e; </s> <s xml:id="echoid-s5817" xml:space="preserve">diſtantia vero g h ad a c erat dupla. </s> <s xml:id="echoid-s5818" xml:space="preserve">Quare angu-<lb/>lus deflexionis h g l, antea inventi c a e, duplus cenſeri po-<lb/>teſt.</s> <s xml:id="echoid-s5819" xml:space="preserve"/> </p> <div xml:id="echoid-div455" type="float" level="2" n="3"> <note position="left" xlink:label="note-0348-01" xlink:href="note-0348-01a" xml:space="preserve">TAB.XXXI. <lb/>Fig. 1.</note> <figure xlink:label="fig-0349a-01" xlink:href="fig-0349a-01a"> <caption xml:id="echoid-caption125" style="it" xml:space="preserve">Pag. 268.<lb/>TAB. XXX.</caption> <variables xml:id="echoid-variables127" xml:space="preserve">a a I L K M g N l O c k P Q T S Q V T S R f f e n l d h g b</variables> </figure> </div> <p> <s xml:id="echoid-s5820" xml:space="preserve">Hæc verò ſcrupulorum 48, aberratio nullius adhuc mo-<lb/>menti eſt, neque neglecta nocebit. </s> <s xml:id="echoid-s5821" xml:space="preserve">Attamen, quo minus <lb/>cauſandi locus hic ſuperſit, oſtendam jam quænam adhiberi <lb/>poſſit correctio, atque ejusmodi quidem ut, una opera, <lb/>omnem aliam lentis declinationem reſtituat.</s> <s xml:id="echoid-s5822" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5823" xml:space="preserve">Igitur ſemel ab initio, ad ſuperiorem lentis magnæ præ-<lb/>parationem, hoc quod dicemus, adjungatur. </s> <s xml:id="echoid-s5824" xml:space="preserve">Nempe lente <lb/>quemadmodum præcepimus librata, atque ad oculi altitudi-<lb/>nem defixa, filum caudæ adnexum manu altera capiatur, <lb/>eaque oculo admoveatur; </s> <s xml:id="echoid-s5825" xml:space="preserve">altera lucernam juxta teneat. <lb/></s> <s xml:id="echoid-s5826" xml:space="preserve">Tum paulatim recedendo, extentumque filum producendo, <lb/>obſervetur an duplex flammæ imago circa mediam lentem <lb/>appareat, ab utraque nimirum ſuperficie ejus reflexa. </s> <s xml:id="echoid-s5827" xml:space="preserve">Id ſi <lb/>contingat ubi jam tota fili longitudo exierit, quanta nimi-<lb/>rum futuro teleſcopio debetur, indicio eſt rectiſſime lentem <lb/>ad oculum converti. </s> <s xml:id="echoid-s5828" xml:space="preserve">Quod ſi altera tantum flammæ reflexio <lb/>conſpiciatur, male collocata erit, ſin neutra, pejus. </s> <s xml:id="echoid-s5829" xml:space="preserve">Hic ve-<lb/>ro jam remedium adhibebitur, ubi cognitum fuerit in quam <lb/>partem lens declinet. </s> <s xml:id="echoid-s5830" xml:space="preserve">Stylus enim æneus extremæ caudæ ad-<lb/>jectus, filumque innexum habens, in partem eandem pa-<lb/>rumper flectendus eſt; </s> <s xml:id="echoid-s5831" xml:space="preserve">ac rurſus, ut ante, lucernæ reflexio <lb/>tentanda; </s> <s xml:id="echoid-s5832" xml:space="preserve">idque ita repetitis vicibus faciendum, quoad u-<lb/>traque flammulæ imago in unum convenire conſpiciatur. </s> <s xml:id="echoid-s5833" xml:space="preserve"><lb/>Tenſione autem fili utendum mediocri, qualem ſupra defi-<lb/>nivimus, duarum aut trium librarum vim referente, eique <pb o="270" file="0352" n="382" rhead="MACHINARUM"/> quatenus licet adſueſcendum. </s> <s xml:id="echoid-s5834" xml:space="preserve">Hoc modo correcta ſemel len-<lb/>tis poſitio ad omnes obſervationes valebit. </s> <s xml:id="echoid-s5835" xml:space="preserve">Neque hic ſubti-<lb/>liter nimium objiciat quiſquam quod obliquo fili aſcenſu, <lb/>cum ad aſtra dirigitur, paulo minor efficitur flexus ejus à <lb/>gravitate ortus, quam cum filum idem horizonti parallelum <lb/>extenditur. </s> <s xml:id="echoid-s5836" xml:space="preserve">Eſt enim differentia hæc perexigua, præſertim <lb/>in tanta fili levitate; </s> <s xml:id="echoid-s5837" xml:space="preserve">& </s> <s xml:id="echoid-s5838" xml:space="preserve">lentium paralleliſmus, ut jam dixi-<lb/>mus, ad geometriæ leges exactus non requiritur.</s> <s xml:id="echoid-s5839" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5840" xml:space="preserve">Multo magis ventus obeſſe dicendus foret, filum ſinuans <lb/>atque in latus impellens, præſertim in magnis, quas dixi-<lb/>mus, longitudinibus; </s> <s xml:id="echoid-s5841" xml:space="preserve">niſi quod tubis quoque idem ventus <lb/>adverſus eſt, qui concuſſu ejus tremunt ac vacillant, magno <lb/>ſpectantis incommodo; </s> <s xml:id="echoid-s5842" xml:space="preserve">ut propterea ſæpe obſervationibus <lb/>ſuperſedendum fuerit. </s> <s xml:id="echoid-s5843" xml:space="preserve">Sed quo æquiore animo hæc diſpen-<lb/>dia feramus, ſciendum eſt, flantibus ventis, ſemper fere <lb/>aëris pelluciditatem adeo turbari, etiamſi ſerenus videatur, <lb/>ut hoc uno omnis teleſcopiorum proſpectus impediatur; <lb/></s> <s xml:id="echoid-s5844" xml:space="preserve">quod exercitatis ignotum eſſe nequit. </s> <s xml:id="echoid-s5845" xml:space="preserve">Imo & </s> <s xml:id="echoid-s5846" xml:space="preserve">tranquillo in-<lb/>terdum ac prorſus ſereno cœlo, ſcintillantibus cum maxime <lb/>ſideribus, fruſtra tamen teleſcopia adhibentur; </s> <s xml:id="echoid-s5847" xml:space="preserve">humido vapo-<lb/>re quodam aërem obſidente, quo fit ut ad Planetarum corpo-<lb/>ra reſpicienti, undatio quædam tremula & </s> <s xml:id="echoid-s5848" xml:space="preserve">fluctuans omnem <lb/>viſus aciem intercipiat. </s> <s xml:id="echoid-s5849" xml:space="preserve">Poſſetque, ubi hoc accidit, ipſa len-<lb/>tium bonitas ſuſpecta eſſe, niſi alio tempore ac puriore cœlo <lb/>fuiſſet cognita. </s> <s xml:id="echoid-s5850" xml:space="preserve">Idem vapor, ut hoc quoque obiter admo-<lb/>neam, non raro, lenti majori adhæreſcens, radiorum lucis <lb/>partem avertit: </s> <s xml:id="echoid-s5851" xml:space="preserve">cui malo, calefacto modice ad ignem vitro, <lb/>occurritur.</s> <s xml:id="echoid-s5852" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5853" xml:space="preserve">Videamus nunc & </s> <s xml:id="echoid-s5854" xml:space="preserve">illud quod de illuſtranda lente eadem <lb/>diximus ad malum ſubrecta. </s> <s xml:id="echoid-s5855" xml:space="preserve">Quæ ſi valde procul diſtet, <lb/>puta ad ducentorum & </s> <s xml:id="echoid-s5856" xml:space="preserve">amplius pedum intervallum, vix vi-<lb/>detur tantum luminis, ut ab obſervatore cerni poſſit, acce-<lb/>ptura, etiamſi lucerna convexo vitreo juvetur, uti præcepi-<lb/>mus. </s> <s xml:id="echoid-s5857" xml:space="preserve">Sed hic intendere amplius lumen licebit, vel aucto lu-<lb/>cernæ ipſius ellychnio, vel latiori lente adhibita leniusque <lb/>convexa, quæ lucem transmiſſam, etiamſi pari quantitate <pb o="271" file="0353" n="383" rhead="DESCRIPTIONES."/> accipiat, minus tamen diffundet, longiusque proinde ejacu-<lb/>labitur.</s> <s xml:id="echoid-s5858" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5859" xml:space="preserve">Quantum igitur ad hæc attinet, nihil admodum referre <lb/>liquet quænam fuerit teleſcopii longitudo, ſed æque facile <lb/>qualiacunque in uſum deduci. </s> <s xml:id="echoid-s5860" xml:space="preserve">Aliquod tantum diſcrimen in <lb/>varia mali altitudine poſitum eſſe. </s> <s xml:id="echoid-s5861" xml:space="preserve">Cujus quidem parandæ <lb/>plures modi ſuppetunt. </s> <s xml:id="echoid-s5862" xml:space="preserve">Poſſumus enim, uno ſtatuto malo, <lb/>alium ejus opera duplo altiorem juxta attollere, ac ſimul fir-<lb/>miorem reddere, transverſis fibulis utrumque conſerendo. <lb/></s> <s xml:id="echoid-s5863" xml:space="preserve">Ac firmiſſima quidem fuerit compages hujusmodi, ſi duo <lb/>mali humiliores, cum tertio duplæ altitudinis, binis ternis-<lb/>ve pedibus inter ſe diſtent, in triangulum diſpoſiti, atque <lb/>uti diximus religati. </s> <s xml:id="echoid-s5864" xml:space="preserve">Qua ratione facile ad centum pedum <lb/>altitudinem perveniemus. </s> <s xml:id="echoid-s5865" xml:space="preserve">Ad multo majores vero, vel vali-<lb/>diori malorum ac trabium ſubſtructione utendo, vel ad tur-<lb/>rim aut ædificii altioris angulum inferiora ligna applicando; </s> <s xml:id="echoid-s5866" xml:space="preserve"><lb/>ita ut nihil tamen obſtet, quo minus, ab imo ad ſummum, <lb/>lens primaria adducatur, per continuum canaliculum, uti <lb/>diximus, aſcendens. </s> <s xml:id="echoid-s5867" xml:space="preserve">Sed & </s> <s xml:id="echoid-s5868" xml:space="preserve">ſuper turri aut domus culmine <lb/>erigi malus poteſt, ut ibi adſtet is cui funis cura demandata <lb/>eſt, ad evehendam demittendamve lentem.</s> <s xml:id="echoid-s5869" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5870" xml:space="preserve">Nec vero præpropera aut ſupervacua cura hæc à nobis a-<lb/>gitari quis putet, quod veriſimile non ſit his altitudinibus <lb/>opus fore. </s> <s xml:id="echoid-s5871" xml:space="preserve">Ecce enim, dum hæc ſcribo, Caſſini literis certior <lb/>fio, lentes quatuor, quarum maxima teleſcopio pedum cen-<lb/>tum quadraginta deſtinata ſit, à Joſepho Campano, eaſque <lb/>præſtantiſſimas Romæ eſſe perfectas, & </s> <s xml:id="echoid-s5872" xml:space="preserve">ad magnum Galliæ <lb/>Regem miſſas. </s> <s xml:id="echoid-s5873" xml:space="preserve">Etſi enim ad cœleſtium obſervationem non-<lb/>dum fuere admotæ, non dubitandum tamen interdiu inſtitu-<lb/>tum fuiſſe earum examen, in atriis porticibuſve prælongis <lb/>unde lux excluſa eſſet. </s> <s xml:id="echoid-s5874" xml:space="preserve">Nunc vero, hoc noſtro invento, uti-<lb/>litas ſua tum his lentibus, tum ſi quæ has longitudine exce-<lb/>dentes prodeant, conſtabit.</s> <s xml:id="echoid-s5875" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5876" xml:space="preserve">Quod ſi cogitemus quibus modis teleſcopiorum efficaciam <lb/>alii augere ſtuduerint; </s> <s xml:id="echoid-s5877" xml:space="preserve">quæ fruſtra illi quæſiverunt, ea nos <lb/>levi hac opera conſecutos eſſe videri poſſit. </s> <s xml:id="echoid-s5878" xml:space="preserve">Sive enim figu- <pb o="272" file="0354" n="384" rhead="MACHINARUM"/> ris lentium hyperbolicis ellipticiſve, ut Carteſius, ſive ſpe-<lb/>culis cavis, ut Neutonus, ſive alia quavis ratione id aggreſſi <lb/>ſint, huc omnia redibant ut brevioribus teleſcopiis, ac mi-<lb/>nori molimine uſurpandis, multum amplificarentur res viſæ. <lb/></s> <s xml:id="echoid-s5879" xml:space="preserve">Nam neque accurata illa ac ſcrupuloſa ſuperficierum forma-<lb/>tio devitari poterat, neque etiam lentium ſpeculorumve <lb/>magnitudo. </s> <s xml:id="echoid-s5880" xml:space="preserve">quoniam obſcuritate nimia, quicquid machinati <lb/>fuerimus, inutile reddi neceſſe eſt, niſi pro ratione perce-<lb/>pti augmenti creſcant aperturæ quibus primum lux ſubintrat. </s> <s xml:id="echoid-s5881" xml:space="preserve"><lb/>Nos vero longitudines quidem non imminuimus, ſed ne obeſ-<lb/>ſent effecimus, quod fere eodem redit.</s> <s xml:id="echoid-s5882" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5883" xml:space="preserve">Si quis vero jam requirat quouſque & </s> <s xml:id="echoid-s5884" xml:space="preserve">quo operæ pretio <lb/>extendi porro teleſcopia poſſe exiſtimem, & </s> <s xml:id="echoid-s5885" xml:space="preserve">num productis <lb/>longe ultra modum eorum quæ paulo ante diximus, ſpe-<lb/>randum ſit adhuc decuplo propius ad lunam cæteraque <lb/>aſtra nos acceſſuros, quam quo triginta pedes habentibus pro-<lb/>ceſſimus; </s> <s xml:id="echoid-s5886" xml:space="preserve">quibus tanti itineris partes centum quadraginta <lb/>novem, una duntaxat reliqua, confectæ ſunt: </s> <s xml:id="echoid-s5887" xml:space="preserve">reſpondebo <lb/>me certos quidem arti terminos præfinire non poſſe; </s> <s xml:id="echoid-s5888" xml:space="preserve">huc ta-<lb/>men, quo dixi, nec maximo hominum conatu perventum <lb/>iri. </s> <s xml:id="echoid-s5889" xml:space="preserve">multoque minus futurum, quod aliqui videntur non de-<lb/>ſperaſſe, ut lunam ac Planetas cæteros velut è propinquo <lb/>inſpiciamus, & </s> <s xml:id="echoid-s5890" xml:space="preserve">utrum animalibus habitentur, an præter <lb/>vaſtas ſolitudines nihil habeant, viſu penetremus. </s> <s xml:id="echoid-s5891" xml:space="preserve">Primum <lb/>enim, in parandis lentibus, ſcio quantopere creſcat cum ma-<lb/>gnitudine formandi difficultas; </s> <s xml:id="echoid-s5892" xml:space="preserve">ipſiusque inveniendi vitri <lb/>quod vitiis iis careat, quæ maxime huic operi infeſta ſunt. <lb/></s> <s xml:id="echoid-s5893" xml:space="preserve">Quanto enim ulterius radii colligentur, tanto magis hæc vi-<lb/>tia ſe prodant neceſſe eſt. </s> <s xml:id="echoid-s5894" xml:space="preserve">Conſtat præterea, ut jam iſta ni-<lb/>hil obſtent, non amplificari res viſas, niſi pro ratione diame-<lb/>trorum aperturæ lentis exterioris. </s> <s xml:id="echoid-s5895" xml:space="preserve">Quæ diametri nequaquam <lb/>creſcunt cum teleſcopiorum longitudine; </s> <s xml:id="echoid-s5896" xml:space="preserve">ſed, quantum vi-<lb/>deo, rationem longitudinum ſubduplam ſequuntur. </s> <s xml:id="echoid-s5897" xml:space="preserve">Adeo <lb/>ut data apertura pollicum trium, in teleſcopio triginta pe-<lb/>des longo; </s> <s xml:id="echoid-s5898" xml:space="preserve">quantam circiter experientia concedi ſinit; </s> <s xml:id="echoid-s5899" xml:space="preserve">alia, <lb/>ad trecentos pedes, non niſi novem unciarum & </s> <s xml:id="echoid-s5900" xml:space="preserve">ſemis ſit <pb o="273" file="0355" n="385" rhead="DESCRIPTIONES."/> ſutura, ac propterea tantum triplo majora fere omnia ſint <lb/>apparitura, prægrandi hoc teleſcopio, quam illo pedum <lb/>tricenum. </s> <s xml:id="echoid-s5901" xml:space="preserve">At ſi decuplo exceſſu idem ſuperandum ſit, jam <lb/>ter mille pedum longitudine opus erit, quo quidem nulla <lb/>humana ope perveniri poſſe, vel ſolius altitudinis cauſa, <lb/>manifeſtum eſt.</s> <s xml:id="echoid-s5902" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5903" xml:space="preserve">Sane majores multo forent, & </s> <s xml:id="echoid-s5904" xml:space="preserve">majori proportione creſce-<lb/>rent, eæ, quas diximus, aperturæ, ſi nihil aliud obſtaret <lb/>quam figuræ ſphæricæ parum idonea, in colligendis radiis, <lb/>curvatura. </s> <s xml:id="echoid-s5905" xml:space="preserve">Nunc vero alia quædam, ex ipſa refractionis na-<lb/>tura, oritur radiorum aberratio, quam ante annos aliquot <lb/>Neutonus egregiis quibusdam experimentis & </s> <s xml:id="echoid-s5906" xml:space="preserve">priſmatum <lb/>vitreorum coloribus comprobavit. </s> <s xml:id="echoid-s5907" xml:space="preserve">Hæc vero & </s> <s xml:id="echoid-s5908" xml:space="preserve">ipſa leges <lb/>ſuas habet, quibus, ſi recte eas perſpicio, ſubdupla illa, <lb/>quam dixi, aperturarum ad longitudines ratio colligitur.</s> <s xml:id="echoid-s5909" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div457" type="section" level="1" n="188"> <head xml:id="echoid-head238" xml:space="preserve">AUCTARIUM.</head> <p style="it"> <s xml:id="echoid-s5910" xml:space="preserve">VIdebatur jam perfecta abſolutaque omnibas numeris nova <lb/>Aſtroſcopia noſtra; </s> <s xml:id="echoid-s5911" xml:space="preserve">typisque excuſa, nondum tamen edita <lb/>erat; </s> <s xml:id="echoid-s5912" xml:space="preserve">cum ſecundis cogitationibus, ut fit, alia quædam nobis <lb/>in mentem venere, quibus ea melior commodiorque fieret. </s> <s xml:id="echoid-s5913" xml:space="preserve">Quæ <lb/>cum auctarii vice hic adponere viſum ſit, ſimul hoc monemus, <lb/>ut, ſicut poſterius reperta fuere, ita ultimo loco, poſtquam <lb/>reliqua deſcriptio ac delineatio percepta fuerit, legantur.</s> <s xml:id="echoid-s5914" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5915" xml:space="preserve">Cum primum ſpectatores invento noſtro, ac Planetis na-<lb/>cti ſumus, teleſcopicis obſervationibus minus aſſuetos, docuit <lb/>experientia, eos quidem per ſe difficilius ſtellæ conſpectum <lb/>conſequi; </s> <s xml:id="echoid-s5916" xml:space="preserve">ſicut antehac quoque, ubi in grandiores tubos <lb/>inciderant, eveniebat. </s> <s xml:id="echoid-s5917" xml:space="preserve">Quod autem hic fieri ſolitum, ut, re-<lb/>perto prius ſidere, ac manente tubo, tantummodo oculum <lb/>ei ſpectator juſſus admoveret, id non perinde nobis nunc <lb/>imitari licebat; </s> <s xml:id="echoid-s5918" xml:space="preserve">cum lens oculo proxima, ubi defigeretur, <lb/>non haberet. </s> <s xml:id="echoid-s5919" xml:space="preserve">Itaque hic quoque ratio fuit excogitanda, qua <lb/>poſitum ſuum ſervaret ocularis lens. </s> <s xml:id="echoid-s5920" xml:space="preserve">Quod quidem præſtiti- <pb o="274" file="0356" n="386" rhead="MACHINARUM"/> mus machinæ exiguæ opera, quæ fulcro bipedi, in deſcri-<lb/>ptione deſignato affigitur; </s> <s xml:id="echoid-s5921" xml:space="preserve">ut in figura adjecta videre eſt.</s> <s xml:id="echoid-s5922" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5923" xml:space="preserve">Tranſverſarii namque in ſummo fulcro pars eſt a a. </s> <s xml:id="echoid-s5924" xml:space="preserve">Rhom-<lb/>bus plicatilis ex ære b b, binis lateribus ad duplam longitu-<lb/> <anchor type="note" xlink:label="note-0356-01a" xlink:href="note-0356-01"/> dinem productis. </s> <s xml:id="echoid-s5925" xml:space="preserve">Longitudo laterum pollices 5 {1/2}, latitudo <lb/>paulo major pollice dimidio; </s> <s xml:id="echoid-s5926" xml:space="preserve">craſſitudo parte ejus decima. <lb/></s> <s xml:id="echoid-s5927" xml:space="preserve">Hunc rhombum tranſverſarii medio applicitum tenet cochlea <lb/>ferrea f, ſuppoſita æris vel ferri particulâ g, ac præterea or-<lb/>biculo ex ære tenui, leniter convexo, cujus preſſu lentus æ-<lb/>quabiliſque efficitur motus rhombi ac diductio. </s> <s xml:id="echoid-s5928" xml:space="preserve">Porro ex an-<lb/>gulo ejus ſuperiore, axis ſeu columella prominet c, per-<lb/>pendiculariter inſiſtens, longitudine ſesqui pollicis. </s> <s xml:id="echoid-s5929" xml:space="preserve">Cujus <lb/>capite altero lamella mobilis adhæret, 4. </s> <s xml:id="echoid-s5930" xml:space="preserve">pollices longa, di-<lb/>midium lata; </s> <s xml:id="echoid-s5931" xml:space="preserve">quæ hic conſpici nequit, quippe tecta capulo <lb/>ligneo d, paris longitudinis, cui conſerta eſt. </s> <s xml:id="echoid-s5932" xml:space="preserve">Huic demum <lb/>capulo, plano ac parte anteriori leviter inciſo, inſeritur la-<lb/>mella altera ænea e, quæ ſuper axiculo mobili bacillum ſu-<lb/>ſtinet, cum affixa oculari lente, tubulo ſuo incluſa. </s> <s xml:id="echoid-s5933" xml:space="preserve">Ut au-<lb/>tem thombus cum impoſito onere æqualiter libretur ſuper <lb/>axe f, adjiciuntur in productis lateribus extremis pondera <lb/>paria h h, quantis ad hoc opus eſt.</s> <s xml:id="echoid-s5934" xml:space="preserve"/> </p> <div xml:id="echoid-div457" type="float" level="2" n="1"> <note position="left" xlink:label="note-0356-01" xlink:href="note-0356-01a" xml:space="preserve">TAB.XXXI. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s5935" xml:space="preserve">Quibus ita ſe habentibus, quocumque perducta fuerit ob-<lb/>fervantis manu lens ocularis, capulo d ſemper deorſum con-<lb/>verſo, ibi ſponte ſua conſiſtit; </s> <s xml:id="echoid-s5936" xml:space="preserve">atque ita, invento ſidere, <lb/>facile imperitior ſpectator in prioris locum ſuccedit, eodem-<lb/>que fruitur ſpectaculo. </s> <s xml:id="echoid-s5937" xml:space="preserve">Facit enim funiculus utramque len-<lb/>tem conjungens, ut poſitum ſuum fulcrum ſervet, ſpecta-<lb/>torem verſus reclinans, etſi duobus tantum pedibus inſiſtat; <lb/></s> <s xml:id="echoid-s5938" xml:space="preserve">ſimulque fulcri pondere, eorumque quæ ipſi impoſita do-<lb/>cuimus, idem funiculus intenditur; </s> <s xml:id="echoid-s5939" xml:space="preserve">adeo ut nihil aptius com-<lb/>modiusve hac in re optari queat.</s> <s xml:id="echoid-s5940" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5941" xml:space="preserve">Altitudo fulcri eſt pedum 4. </s> <s xml:id="echoid-s5942" xml:space="preserve">poll. </s> <s xml:id="echoid-s5943" xml:space="preserve">9. </s> <s xml:id="echoid-s5944" xml:space="preserve">Gravitas ejus libra-<lb/>tum 2 {3/4}. </s> <s xml:id="echoid-s5945" xml:space="preserve">Lentis ocularis, cum tubulo & </s> <s xml:id="echoid-s5946" xml:space="preserve">bacillo, gravitas libra <lb/>dimidia. </s> <s xml:id="echoid-s5947" xml:space="preserve">Rhombi cum ponderibus h h, libræ 2 {1/4}. </s> <s xml:id="echoid-s5948" xml:space="preserve">Quæ pro-<lb/>ptetea adſcribo, ut conſtructionem noſtram, experientia <lb/>comprobatam, eo facilius cuivis imitari liceat.</s> <s xml:id="echoid-s5949" xml:space="preserve"/> </p> <pb o="275" file="0357" n="387" rhead="DESCRIPTIONES."/> <p> <s xml:id="echoid-s5950" xml:space="preserve">Nunc vero aliud præterea addemus, quo perfectior evadat <lb/>hæc noſtra obſervandi ratio. </s> <s xml:id="echoid-s5951" xml:space="preserve">quod licet, omiſſum, nihil <lb/>plerumque noceret, curioſo tamen ſyderum inſpectori ne-<lb/>quaquam eſt negligendum. </s> <s xml:id="echoid-s5952" xml:space="preserve">Nempe cum Saturni comites il-<lb/>los Caſſinianos diligentius requirerem, eoſque difficulter ad-<lb/>ſequerer, præſertim noctibus non admodum obſcuris, intel-<lb/>lexi in cauſa eſſe lucem tenuem quandam, ab aëre ad ocu-<lb/>lum manantem; </s> <s xml:id="echoid-s5953" xml:space="preserve">non eam quæ per lentem majorem advenit, <lb/>ſed quæ extrinſecus circum latera præterlabitur. </s> <s xml:id="echoid-s5954" xml:space="preserve">Huic impor-<lb/>tunæ luculæ excludendæ, nonnihil quidem conducere ſcie-<lb/>bam, ſi circulum illum papyraceum, quo in luna obſervan-<lb/>da utebar, etiam hic lenti majori circumponerem. </s> <s xml:id="echoid-s5955" xml:space="preserve">Sed aliud <lb/>efficacius remedium, circa hæc occupato incidit, priori il-<lb/>li jungendum; </s> <s xml:id="echoid-s5956" xml:space="preserve">ut nempe, perforatæ laminæ oppoſitu, ocu-<lb/>li pupilla arctaretur, quæ alioqui per tenebras late patere ſo-<lb/>let. </s> <s xml:id="echoid-s5957" xml:space="preserve">Cujus ſimul ac experimentum feci, jam clare tres Satur-<lb/>ni Lunulas conſpexi; </s> <s xml:id="echoid-s5958" xml:space="preserve">cum amoto exiguo foramine media il-<lb/>la noſtra tantum cerneretur. </s> <s xml:id="echoid-s5959" xml:space="preserve">Quia vero, ita reductâ pupil-<lb/>lâ, minus facile propoſitum ſydus inveſtigatur, quam cum <lb/>tota patet, idcirco orbiculum illum perforatum, ac ſemipol-<lb/>licem latum, brachiolo quodam mobili, ac Græco Λ hæren-<lb/>tem ſimili, cui in figura hac adſcriptum eſt k, ita conjun-<lb/>ximus tubuli fundo per quem lens ocularis inſpicitur, qui-<lb/>que latiori foramine pervius eſt, ut non ante quam hoc fora-<lb/>mine ſydus inventum fuerit, ſuperinducatur alterum illud an-<lb/>guſtius.</s> <s xml:id="echoid-s5960" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5961" xml:space="preserve">Credidiſſet fortaſſe aliquis hac oculi contractione non pa-<lb/>rum viſum obſcurari. </s> <s xml:id="echoid-s5962" xml:space="preserve">cum tamen certum ſit, ſi diameter exi-<lb/>gui foraminis, ad diametrum aperturæ lentis majoris eam ra-<lb/>tionem habeat, quam habent inter ſe focorum utriuſque diſtan-<lb/>tiæ, nihilo obſcurius teleſcopio ejuſmodi omnia cerni, quam <lb/>ſi apertus ac liber oculus relinquatur. </s> <s xml:id="echoid-s5963" xml:space="preserve">Sed præſtat duplicare <lb/>tantillam hanc latitudinem, vel paulo etiam augere ampli-<lb/>us, quo minus difficilis ſit rei videndæ inquiſitio, nec ni-<lb/>mium cito inventa ſtella elabatur, ob mundi converſionem <lb/>diurnam. </s> <s xml:id="echoid-s5964" xml:space="preserve">Nobis in teleſcopio 34 pedes longo, foraminuli dia- <pb o="276" file="0358" n="388" rhead="MACHINARUM"/> meter decimam ſextam circiter pollicis partem habet. </s> <s xml:id="echoid-s5965" xml:space="preserve">Ipſum <lb/>vero duos pollices cum dimidio ab oculari lente abeſt, quan-<lb/>ta eſt præciſe in hac lente foci diſtantia. </s> <s xml:id="echoid-s5966" xml:space="preserve">Quod diligenter cu-<lb/>randum, quia alias non poterit amplum ſpatium, ut ſolet, <lb/>uno obtutu comprehendi. </s> <s xml:id="echoid-s5967" xml:space="preserve">Facile autem deltoidis brachii fle-<lb/>xu, qui quidem in ſchemate noſtro conſpici nequit, quan-<lb/>tum opus eſt, lamellaperforata removetur, quæ nobis ſemi-<lb/>pollice à tubuli fundo extat.</s> <s xml:id="echoid-s5968" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5969" xml:space="preserve">Porro circulus lenti magnæ circumdatus, teleſcopii partem <lb/>longitudinis quadrageſimam quintam circiter diametro æquet. <lb/></s> <s xml:id="echoid-s5970" xml:space="preserve">Cujus circuli objectu quia paulo impeditiorem reddi neceſſe <lb/>erat aſtri inveſtigationem, viſum fuit imponere bacillo, ſeu <lb/>caudæ lentis ocularis, ſtylum m, perpendiculariter erectum; </s> <s xml:id="echoid-s5971" xml:space="preserve"><lb/>cujus apex tantundem ſupra axem lentium attollitur quantus <lb/>eſt circuli illius ſemidiameter. </s> <s xml:id="echoid-s5972" xml:space="preserve">Hinc enim fit, ut ſi oculum <lb/>prius ibi collocemus, unde cum ſummomargine circuli in ean-<lb/>dem rectam lineam ſtella conveniat; </s> <s xml:id="echoid-s5973" xml:space="preserve">tumque, apprehenſo ca-<lb/>pulo d, moveamus lentem ocularem cum adjuncto bacillo, do-<lb/>nec in eandem quoque rectam quadret extremum ſtyli m; </s> <s xml:id="echoid-s5974" xml:space="preserve"><lb/>fit inquam ut, ad tubulum ocularem viſum referenti, ſtella <lb/>eadem per teleſcopium ſeſe conſpiciendam det, vel certe pa-<lb/>rum abſit. </s> <s xml:id="echoid-s5975" xml:space="preserve">Uſu vero & </s> <s xml:id="echoid-s5976" xml:space="preserve">exercitatione tum hæc, tum cæte-<lb/>ra quæ ad hanc obſervandi rationem pertinent, facilia <lb/>fiunt.</s> <s xml:id="echoid-s5977" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div459" type="section" level="1" n="189"> <head xml:id="echoid-head239" xml:space="preserve">IV.</head> <head xml:id="echoid-head240" style="it" xml:space="preserve">Excerpta ex literis D<emph style="super">ni</emph> Hugenii de novâ <lb/>methodo conſtruendi Barometrum.</head> <p> <s xml:id="echoid-s5978" xml:space="preserve">Quod novam meam methodum Barometri ſpectat, noſti, <lb/>diverſas aëris atmoſphæræ preſſiones multo magis fore <lb/>viſibiles, & </s> <s xml:id="echoid-s5979" xml:space="preserve">faciliores diſtinctu, ſi in tubo 30 pedes alto <lb/>fiat barometrum ope aquæ quam ſunt in barometris vulgari-<lb/>bus, quæ cum Hydrargyro conſtruuntur. </s> <s xml:id="echoid-s5980" xml:space="preserve">Cum enim maxima <lb/>diverſitas ſit circiter duorum pollicum in barometris vul- <pb file="0359" n="389"/> <pb file="0359a" n="390"/> <anchor type="figure" xlink:label="fig-0359a-01a" xlink:href="fig-0359a-01"/> <anchor type="figure" xlink:label="fig-0359a-02a" xlink:href="fig-0359a-02"/> <pb file="0360" n="391"/> <pb o="277" file="0361" n="392" rhead="DESCRIPTIONES."/> garibus, in novo hoc barometro erit viginti octo pollicum <lb/>id eſt decies & </s> <s xml:id="echoid-s5981" xml:space="preserve">quater major erit, aliæque variationes auge-<lb/>buntur eadem ratione, quæ ipſa datur inter Mercurii & </s> <s xml:id="echoid-s5982" xml:space="preserve"><lb/>aquæ gravitates ſpecificas.</s> <s xml:id="echoid-s5983" xml:space="preserve"/> </p> <div xml:id="echoid-div459" type="float" level="2" n="1"> <figure xlink:label="fig-0359a-01" xlink:href="fig-0359a-01a"> <caption xml:id="echoid-caption126" xml:space="preserve">Pag. 276.<lb/>TAB.XXXI.<lb/>Fig. 2.</caption> <variables xml:id="echoid-variables128" xml:space="preserve">a a m f k b e @ b a g a f b b h</variables> </figure> <figure xlink:label="fig-0359a-02" xlink:href="fig-0359a-02a"> <caption xml:id="echoid-caption127" style="it" xml:space="preserve">Fig. 1.</caption> <variables xml:id="echoid-variables129" xml:space="preserve">h g k h d a b c f e l</variables> </figure> </div> <p> <s xml:id="echoid-s5984" xml:space="preserve">Sed uti difficile eſt bene diſponere hæc barometra ob ma-<lb/>gnam tubi altitudinem, quæ impeditetiam, quo minus com-<lb/>modè locari poſſint, aut de loco in locum transferri. </s> <s xml:id="echoid-s5985" xml:space="preserve">Cogita-<lb/>vi, quo pacto conſtrui poſſet barometrum mediocris magni-<lb/>tudinis & </s> <s xml:id="echoid-s5986" xml:space="preserve">portatile, cujus effectus quam proxime idem eſſet <lb/>ac aliorum illorum magnorum barometrorum, & </s> <s xml:id="echoid-s5987" xml:space="preserve">duas diver-<lb/>ſas conſtructiones inveni.</s> <s xml:id="echoid-s5988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5989" xml:space="preserve">Prima eſt, ut fiat tubus vitreus A B quatuor pedum cum <lb/> <anchor type="note" xlink:label="note-0361-01a" xlink:href="note-0361-01"/> ſemiſſe, qui clauſus ſit in extremitate A, & </s> <s xml:id="echoid-s5990" xml:space="preserve">cujus cavitas <lb/>ſit circiter duarum linearum; </s> <s xml:id="echoid-s5991" xml:space="preserve">requiritur ut latior ſit in loco <lb/>medio, ubi detur quaſi pyxis cylindrica C D, cujus altitu-<lb/>do ſit circiter unius pollicis, & </s> <s xml:id="echoid-s5992" xml:space="preserve">Diameter E E 14 vel 15 li-<lb/>nearum, id eſt ſepties vel octies major diametro tubi; </s> <s xml:id="echoid-s5993" xml:space="preserve">in-<lb/>funditur in apertam extremitatem tantum aquæ, quantum <lb/>requiritur ad replendum dimidium receptaculi C D cum <lb/>partis ſuperioris tubi dimidio C F; </s> <s xml:id="echoid-s5994" xml:space="preserve">Porro repletur quid-<lb/>quid ſupereſt Mercurio; </s> <s xml:id="echoid-s5995" xml:space="preserve">qui etiam infunditur in vas G ad al-<lb/>titudinem ſemipollicis, deinde immergitur in hunc extre-<lb/>mitas tubi B; </s> <s xml:id="echoid-s5996" xml:space="preserve">tum pro parte exit Mercurius, & </s> <s xml:id="echoid-s5997" xml:space="preserve">qui ſu-<lb/>per eſt manet ad altitudinem E E; </s> <s xml:id="echoid-s5998" xml:space="preserve">aqua, quæ ſupernatat, <lb/>deſcendit usque in F, relinquens reliquum tubi F A aëre <lb/>vacuum, & </s> <s xml:id="echoid-s5999" xml:space="preserve">ſuperficies aquæ, adſcendendo & </s> <s xml:id="echoid-s6000" xml:space="preserve">deſcendendo <lb/>denotat diverſum atmoſphæræ pondus, gradibus fere æqua-<lb/>libus iis quibus aqua illud denotat in barometro 32 pe-<lb/>dum.</s> <s xml:id="echoid-s6001" xml:space="preserve"/> </p> <div xml:id="echoid-div460" type="float" level="2" n="2"> <note position="right" xlink:label="note-0361-01" xlink:href="note-0361-01a" xml:space="preserve">TAB.XXXII. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s6002" xml:space="preserve">Altera conſtructio partim ſimilis eſt priori, ſed multo me-<lb/> <anchor type="note" xlink:label="note-0361-02a" xlink:href="note-0361-02"/> lior eſt; </s> <s xml:id="echoid-s6003" xml:space="preserve">detur tubus in medio incurvatus H M N, duæ in <lb/>hoc tubo requiruntur pixides cylindricæ æquales K & </s> <s xml:id="echoid-s6004" xml:space="preserve">M, <lb/>quarum una, ſc. </s> <s xml:id="echoid-s6005" xml:space="preserve">K, quæ eſt ad tubi extremitatem, hermetice <lb/>ſit clauſa ſuperius, & </s> <s xml:id="echoid-s6006" xml:space="preserve">M, quæ eſt paululum ſupra curvatu-<lb/>ram, ſit utrinque aperta in locis in quibus cum tubo jungi-<lb/>tur; </s> <s xml:id="echoid-s6007" xml:space="preserve">longitudo crurum determinata eſt per diſtantiam pyxi- <pb o="278" file="0362" n="393" rhead="MACHINARUM"/> dum K, M, quæ ſit circitcr 27{1/2} pollicum <anchor type="note" xlink:href="" symbol="*"/>, menſurando diſtan- tiam inter harum media. </s> <s xml:id="echoid-s6008" xml:space="preserve">Altitudo cujusvis pyxidis eſt præter-<lb/>propter unius pollicis cum ſemiſſe; </s> <s xml:id="echoid-s6009" xml:space="preserve">diameter interior unius polli-<lb/>cis vel 15 linearum; </s> <s xml:id="echoid-s6010" xml:space="preserve">diameter cavitatis reliqui tubi {1/10} vel {1/12} ejusdem <lb/>magnitudinis. </s> <s xml:id="echoid-s6011" xml:space="preserve">Primo infunditur ſolus Mercurius in tubum per <lb/>aperturam N, ut fiat barometrum vulgare ex iis quæ inferius re-<lb/>curvata ſunt. </s> <s xml:id="echoid-s6012" xml:space="preserve">Infundendum vel tollendum eſt Hydrargyrum <lb/>donec ſuperficies dentur circiter in medio pyxidum K & </s> <s xml:id="echoid-s6013" xml:space="preserve">M, <lb/>ſi tempore quo fit hæc operatio aër mediæ ſit gravitatis, id <lb/>eft in barometris vulgaribus Mercurius ſit ad altitudi-<lb/>nem 27{1/2} pollicum; </s> <s xml:id="echoid-s6014" xml:space="preserve">alioquin enim ſi preſſio aëris ſit major vel <lb/>minor ordinariâ, ad hocattendendum, computando pro uno <lb/>pollice variationis, in vulgari barometro, lineam unam cum <lb/>ſemiſſe variationis in quavis pyxide: </s> <s xml:id="echoid-s6015" xml:space="preserve">Poſtquam Mercurius <lb/>bene aëre depurgatus erit, ita ut nullus detur in pyxide K <lb/>infundetur per aperturam N quidam liquor, qui hieme non <lb/>congelatur, & </s> <s xml:id="echoid-s6016" xml:space="preserve">qui nequit diſſolvere Mercurium; </s> <s xml:id="echoid-s6017" xml:space="preserve">E. </s> <s xml:id="echoid-s6018" xml:space="preserve">G. </s> <s xml:id="echoid-s6019" xml:space="preserve">aqua <lb/>communis mixta cum {1/8} aquæ fortis: </s> <s xml:id="echoid-s6020" xml:space="preserve">ſpiritus vini quidem <lb/>poſſidet has duas qualitates, ſed non conveniret barometro, <lb/>quia per calorem dilatatur. </s> <s xml:id="echoid-s6021" xml:space="preserve">Hoc etiam referri debet ad pri-<lb/>mam formam Barometri.</s> <s xml:id="echoid-s6022" xml:space="preserve"/> </p> <div xml:id="echoid-div461" type="float" level="2" n="3"> <note position="right" xlink:label="note-0361-02" xlink:href="note-0361-02a" xml:space="preserve">TAB. XXXII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s6023" xml:space="preserve">Quod attinet ad liquoris quantitatem, debet illa adſcendere ad <lb/>pedem unum circiter in tubum B C poſitâ mediâ aëris preſſione.</s> <s xml:id="echoid-s6024" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6025" xml:space="preserve">Diſpoſito ita hoc Barometro videbimus, maximam diffe-<lb/>rentiam preſſionis aëris, quæ notabitur per ſuperficiem li-<lb/>quoris in tubo M N, fore propemodum 22 pollicum, ſi <lb/>diametri pyxidum cylindricarum, ſint decies majores diametro <lb/>tubi. </s> <s xml:id="echoid-s6026" xml:space="preserve">Et ut inveniamus quantum differentiæ quas indicatæ ba-<lb/>rometrum hoc noſtrum excedantillas, quas poteſt indicare ba-<lb/>rometrum vulgare, generalis eſt regula, differentias noſtrinovi <lb/>Barometri eſſe ad differentias Barometri vulgaris, ut decies & </s> <s xml:id="echoid-s6027" xml:space="preserve"><lb/>quater quadratum diametri pyxidum ad idem quadratum plus <lb/>vicies octies quadratum diametri tubi, qui aquam continet; <lb/></s> <s xml:id="echoid-s6028" xml:space="preserve">& </s> <s xml:id="echoid-s6029" xml:space="preserve">hinc ſequitur, cujuſcunque magnitudinis ſint pyxides, <lb/>maximas differentias non poſſe excedere 28 pollices, quan-<lb/>doquidem differentiæ Barometrorum ordinariorum non ex-<lb/>cedunt duos pollices.</s> <s xml:id="echoid-s6030" xml:space="preserve"/> </p> <note symbol="*" position="foot" xml:space="preserve">Pollices hi ſunt partes duodecimæ pedis Regii Gallici.</note> <pb o="279" file="0363" n="394" rhead="DESCRIPTIONES."/> <p> <s xml:id="echoid-s6031" xml:space="preserve">Ut transferatur commode hoc Barometrum, aſſeri adjun-<lb/>gendum eſt. </s> <s xml:id="echoid-s6032" xml:space="preserve">vel Thecâ includendum; </s> <s xml:id="echoid-s6033" xml:space="preserve">& </s> <s xml:id="echoid-s6034" xml:space="preserve">notari debent in <lb/>ligno diviſiones æquales ad indicandas altitudinum differen-<lb/>tias, quæ augebuntur eadem ratione, in quâ minuetur at-<lb/>moſphæræ pondus.</s> <s xml:id="echoid-s6035" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6036" xml:space="preserve">Sic parvæ mutationes, in pondere atmoſphæræ, & </s> <s xml:id="echoid-s6037" xml:space="preserve">quæ <lb/>non percipiuntur in Barometris ordinariis, in his ſenſibiles <lb/>fiunt.</s> <s xml:id="echoid-s6038" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6039" xml:space="preserve">Exempli gratiâ ſi ferantur in Turrim de Noſtre Dame vel <lb/>Montmartre videbimus fuperficiem aquæ deſcendere in primo <lb/>Barometro quosdam pollices, & </s> <s xml:id="echoid-s6040" xml:space="preserve">tantum in altero adſcende-<lb/>re; </s> <s xml:id="echoid-s6041" xml:space="preserve">ſi ferantur in domum elevatam tantum 50 pedes, & </s> <s xml:id="echoid-s6042" xml:space="preserve">porro <lb/>inde deſcendamus, habebimus notabilem mutationem {1/2} pro-<lb/>pemodum pollicis, ita ut poſſimus ope hujus inſtrumenti ſa-<lb/>tis bene menſurare diverſam altitudinem montium a ſe invi-<lb/>cem diſſitorum, & </s> <s xml:id="echoid-s6043" xml:space="preserve">regionum, quarum ſitus non ſinit, ut has <lb/>metiamur aliter. </s> <s xml:id="echoid-s6044" xml:space="preserve">Si mutationes temporum ope Barometrorum, <lb/>prævideri poſſint uti ſperandum videtur, certum eſt, quod <lb/>illa, quæ hoc modo conſtructa ſunt, magnam utilitatem <lb/>habebunt præ aliis, quæ adhibita ſunt huc usque.</s> <s xml:id="echoid-s6045" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6046" xml:space="preserve">Verum eſt quod quædam novorum horum Barometrorum <lb/>ſenſibiliter quodammodo mutentur ex calore vel frigore aëris <lb/>exterioris, quamcunque etiam ad@ibeamus operam ad illa <lb/>interiori aëre purganda; </s> <s xml:id="echoid-s6047" xml:space="preserve">ſed Barometra ordinaria ſunt etiam <lb/>eidem varietati obnoxia, & </s> <s xml:id="echoid-s6048" xml:space="preserve">ſi in noſtris magis conſpicua ſit, <lb/>hoc inde venit, quod multo majores differentias indicent, <lb/>quam Barometra vulgaria: </s> <s xml:id="echoid-s6049" xml:space="preserve">ſed ut occurramus huic malo, <lb/>quod plane nobis impedimento eſſet, ſi vellemus metiri alti-<lb/>tudines, poſſumus Thermoſcopium includere in parte Baro-<lb/>metri aëre vacuâ, & </s> <s xml:id="echoid-s6050" xml:space="preserve">efficere calefaciendo aërem, qui <lb/>utrumque cingit, ut Thermometrum redeat ad eandem no-<lb/>tam in utraque operatione; </s> <s xml:id="echoid-s6051" xml:space="preserve">& </s> <s xml:id="echoid-s6052" xml:space="preserve">hac viâ certi erimus, quod <lb/>aër externus nullam inducat mutationem Barometro, & </s> <s xml:id="echoid-s6053" xml:space="preserve">quod <lb/>omnis varietas, quæ ibi apparebit, oriatur ex diverſâ atmo-<lb/>ſphæræ gravitate.</s> <s xml:id="echoid-s6054" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6055" xml:space="preserve">Dixi poſteriorem conſtructionem alterâ meliorem eſſe, non <pb o="280" file="0364" n="395" rhead="MACHINARUM"/> ſolum, quia ultimum Barometrum multo minoris voluminis <lb/>eſt, ſed &</s> <s xml:id="echoid-s6056" xml:space="preserve">, quia obſervavi, quod in priori parum aëris, <lb/>quem aqua exhalat in vacuo, pedetentim augeatur temporis <lb/>diuturnitate; </s> <s xml:id="echoid-s6057" xml:space="preserve">cui vitio certum eſt Barometrum 32 pedum, <lb/>de quo ſupra dixi, pariter obnoxium eſſe; </s> <s xml:id="echoid-s6058" xml:space="preserve">Et, ut huic ma-<lb/>lo occurratur, quærendus eſt liquor, qui non producit aë-<lb/>rem, ut faciunt aqua & </s> <s xml:id="echoid-s6059" xml:space="preserve">ſpiritus vini. </s> <s xml:id="echoid-s6060" xml:space="preserve">Sed patet, quod po-<lb/>ſterius noſtrum Barometrum hoc vitio non laboret, quoniam <lb/>aqua in vacuo non eſt incluſa: </s> <s xml:id="echoid-s6061" xml:space="preserve">Quod ſi percipiamus aquam, <lb/>quæ eſt in poſteriore Barometro, vapores emittere, tantum <lb/>ſuperius infundenda eſt gutta olei, quod frigore non inſpiſ-<lb/>ſatur, quodque calore non emittit vapores, ut oleum amy-<lb/>gdali dulcis.</s> <s xml:id="echoid-s6062" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div463" type="section" level="1" n="190"> <head xml:id="echoid-head241" xml:space="preserve">V.</head> <head xml:id="echoid-head242" style="it" xml:space="preserve">Nova vis movens mediante pulvere nitrato <lb/>& aëre.</head> <p> <s xml:id="echoid-s6063" xml:space="preserve">Deſideratum a longo tempore eſt artificium quo vis pul-<lb/>veris pyrii aliis, quam quibus huc uſque inſerviit, uſibus <lb/>applicari poſſet, in quibus omnibus vis admodum ſubita re-<lb/>quiritur, qualem obſervamus in exploſione bombardæ & </s> <s xml:id="echoid-s6064" xml:space="preserve">ſclo-<lb/>peti & </s> <s xml:id="echoid-s6065" xml:space="preserve">disruptione cuniculorum: </s> <s xml:id="echoid-s6066" xml:space="preserve">ſi impetus ille promtus poſſet <lb/>moderari & </s> <s xml:id="echoid-s6067" xml:space="preserve">reduci ad vim magis placidam, maximam in me-<lb/>chanicis utilitatem haberet, & </s> <s xml:id="echoid-s6068" xml:space="preserve">in multis occaſionibus, in qui-<lb/>bus vim hominum, equorum, venti, aliarumque potentia-<lb/>rum adhibemus, inſerviret. </s> <s xml:id="echoid-s6069" xml:space="preserve">Ad hunc effectum excogitavi ma-<lb/>chinam, hic delineatam. </s> <s xml:id="echoid-s6070" xml:space="preserve">Hanc non propono, acſi in omni-<lb/>bus partibus perfecta foret, ſedtanquam inventum, quod cum <lb/>pro parte ſucceſſit, poterit ad majorem proferri perfectionem.</s> <s xml:id="echoid-s6071" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6072" xml:space="preserve">A A cylindrus cavus eſt, intus bene politus, & </s> <s xml:id="echoid-s6073" xml:space="preserve">ubique <lb/> <anchor type="note" xlink:label="note-0364-01a" xlink:href="note-0364-01"/> æqualis magnitudinis; </s> <s xml:id="echoid-s6074" xml:space="preserve">B eſt embolus in ſuperiori parte Cy-<lb/>lyndri, & </s> <s xml:id="echoid-s6075" xml:space="preserve">qui in hoc poteſt moveri; </s> <s xml:id="echoid-s6076" xml:space="preserve">in C C cylindrus eſt per-<lb/>foratus duobus foraminibus, quorum diametri circiter ſunt {1/4} <lb/>diametri cylindri; </s> <s xml:id="echoid-s6077" xml:space="preserve">tubi D, D, corii madidi & </s> <s xml:id="echoid-s6078" xml:space="preserve">mollis, fir- <pb o="281" file="0365" n="396" rhead="DESCRIPTIONES."/> miter alligati ſunt duobus cylindris minoribus, qui cum majo-<lb/>ri cohærent, & </s> <s xml:id="echoid-s6079" xml:space="preserve">circumdant foramina; </s> <s xml:id="echoid-s6080" xml:space="preserve">tubus unus exhibetur <lb/>pendens, alter extenſus. </s> <s xml:id="echoid-s6081" xml:space="preserve">In fundo cylindri cum hoc conjun-<lb/>gitur ope cochleæ interpoſito annulo coriaceo pyxis H; </s> <s xml:id="echoid-s6082" xml:space="preserve">ita <lb/>ut exacte claudat apperturam cylindri; </s> <s xml:id="echoid-s6083" xml:space="preserve">E E ſunt retinacula, <lb/>quæ cylindrum in inferiori parte conjungunt cum theca, <lb/>qua includitur, ſed quæ hic non eſt exhibita, ne turbetur <lb/>figura: </s> <s xml:id="echoid-s6084" xml:space="preserve">E F eſt funis annexus embolo B, circumponitur <lb/>trochlea G, & </s> <s xml:id="echoid-s6085" xml:space="preserve">inſervit ad movendum id, quod ei applica-<lb/>tur.</s> <s xml:id="echoid-s6086" xml:space="preserve"/> </p> <div xml:id="echoid-div463" type="float" level="2" n="1"> <note position="left" xlink:label="note-0364-01" xlink:href="note-0364-01a" xml:space="preserve">TAB. XXXII. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s6087" xml:space="preserve">Obtecto parvâ aquæ quantitate embolo, qui ſuperius fir-<lb/>miter debet conſiſtere, ne poſſit exire ex cylindro, immittitur <lb/>in ciſtam H parum pulveris tormentarii cum parva quantitate <lb/>igniarii Germanici accenſi, & </s> <s xml:id="echoid-s6088" xml:space="preserve">clauditur bene ciſta mediante <lb/>ſua cochleâ.</s> <s xml:id="echoid-s6089" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6090" xml:space="preserve">Pulvis ille paulo poſt accenſus implet cylindrum flammâ <lb/>& </s> <s xml:id="echoid-s6091" xml:space="preserve">fugat aërem per tubos coriaceos C D, C D, qui exten-<lb/>duntur, quique ſtatim clauduntur iterum ab aëre exteriore; <lb/></s> <s xml:id="echoid-s6092" xml:space="preserve">ita ut cylindrus maneat aëre vacuus, ſaltem maximâ parte; </s> <s xml:id="echoid-s6093" xml:space="preserve"><lb/>porro embolus B eſt adactus per preſſionem aëris, qui ſupra <lb/>gravitat ad deſcendendum, & </s> <s xml:id="echoid-s6094" xml:space="preserve">ſic trahit funem F F pariter <lb/>illud quod ei appenditur.</s> <s xml:id="echoid-s6095" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6096" xml:space="preserve">Quantitas ejus preſſionis cognita eſt, & </s> <s xml:id="echoid-s6097" xml:space="preserve">determinata, per <lb/>gravitatem aëris, & </s> <s xml:id="echoid-s6098" xml:space="preserve">per magnitudinem diametri emboli, qui, <lb/>ſi ſit unius pedis, ita premitur, ut ſuſtineat pondus 1800 <lb/>circiter librarum; </s> <s xml:id="echoid-s6099" xml:space="preserve">poſito, quod cylindrus totus fiat aëre va-<lb/>cuus, quod huc uſque efficere non potui; </s> <s xml:id="echoid-s6100" xml:space="preserve">etiam experi-<lb/>menta eo reſpectu in magnis & </s> <s xml:id="echoid-s6101" xml:space="preserve">parvis cylindris non eodem <lb/>modo proceſſere.</s> <s xml:id="echoid-s6102" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6103" xml:space="preserve">Cylindrus diametri 2{1/2} pollicum, & </s> <s xml:id="echoid-s6104" xml:space="preserve">20 poll. </s> <s xml:id="echoid-s6105" xml:space="preserve">longus, pon-<lb/>dere 6 granorum pulveris aëre, evacuatus fuit {5/6} partibus; <lb/></s> <s xml:id="echoid-s6106" xml:space="preserve">in cylindro ejusdem latitudinis, ſed longitudinis 44 poll.</s> <s xml:id="echoid-s6107" xml:space="preserve">, <lb/>requiruntur 36 grana pulveris ut ejiciantur {4/6} aëris; </s> <s xml:id="echoid-s6108" xml:space="preserve">Et in <lb/>cylindro diametri unius pedis, & </s> <s xml:id="echoid-s6109" xml:space="preserve">3{1/2} altitudinis, 1{1/2} drachma <lb/>pulveris ad fugandum {1/2} aëris; </s> <s xml:id="echoid-s6110" xml:space="preserve">& </s> <s xml:id="echoid-s6111" xml:space="preserve">duplicata pulveris quanti-<lb/>tate vix magis evacuatus fuit cylindrus.</s> <s xml:id="echoid-s6112" xml:space="preserve"/> </p> <pb o="282" file="0366" n="397" rhead="VARIA CIRCA"/> <p> <s xml:id="echoid-s6113" xml:space="preserve">Aër vero ille qui ſupereſt in cylindro, impedit magnam <lb/>partem effectus, quem exereret machina, ſi omnis aër pror-<lb/>ſus exhauriretur; </s> <s xml:id="echoid-s6114" xml:space="preserve">ut ſatis videre eſt, & </s> <s xml:id="echoid-s6115" xml:space="preserve">ſimul determinari <lb/>computatione poteſt. </s> <s xml:id="echoid-s6116" xml:space="preserve">Ideo deberet examinari, quæ ratio in-<lb/>ter altitudinem & </s> <s xml:id="echoid-s6117" xml:space="preserve">diametrum cylindri ſit optima in hâc ma-<lb/>china ad h@nc maxime evacuandam adhibitâ minima quan-<lb/>tum poſſet pulveris quantitate, nam licet totus cylindrus <lb/>non evacuetur, vis hujus preſſionis nihilominus magnum <lb/>ederet effectum.</s> <s xml:id="echoid-s6118" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6119" xml:space="preserve">Poterit hæc inſervire non tantum elevationi magnorum <lb/>ponderum quorumcunque & </s> <s xml:id="echoid-s6120" xml:space="preserve">aquarum. </s> <s xml:id="echoid-s6121" xml:space="preserve">Sed etiam ad pro-<lb/>jicienda globos & </s> <s xml:id="echoid-s6122" xml:space="preserve">ſagittas magna vi, juxta methodum bali-<lb/>ſtarum veterum.</s> <s xml:id="echoid-s6123" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6124" xml:space="preserve">Ulterius, cum propter cylindri convexitatem, non ſit neceſſe, <lb/>ut ſit valde ſolidus ad reſiſtendum preſſioni aëris externi, cer-<lb/>tum eſt totam machinam exigui ponderis eſſe poſſe, quæ levi-<lb/>tas conjuncta cum magna vi, quam habet, poterit forte uſui <lb/>eſſe ad effectus edendos quos huc uſque impoſſibiles duximus.</s> <s xml:id="echoid-s6125" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div465" type="section" level="1" n="191"> <head xml:id="echoid-head243" xml:space="preserve">VI.</head> <head xml:id="echoid-head244" style="it" xml:space="preserve">Demonſtratio Æquilibrii bilancis.</head> <p> <s xml:id="echoid-s6126" xml:space="preserve">In demonſtratione, quam Archimedes dedit de propoſitione <lb/> <anchor type="note" xlink:label="note-0366-01a" xlink:href="note-0366-01"/> fundamentali Mechanices, tacite ponit quid, de quo jure <lb/>aliquo poſſumus dubitare; </s> <s xml:id="echoid-s6127" xml:space="preserve">eſt autem hoc, ſi plura pondera <lb/>æqualia annexa ſint libræ ad diſtantias æquales a ſe invicem; <lb/></s> <s xml:id="echoid-s6128" xml:space="preserve">ſive omnia ſint ad eandem partem puncti ſuſpenſionis, ſive quæ-<lb/>dam transferantur ad patrem oppoſitam, ut in hac figura, ubi <lb/>punctum ſuſpenſionis eſt A, pondera habere eandem vim <lb/>ad deflectendam libram quam ſi forent omnia ſuſpenſa in pun-<lb/>cto, ubi eſt commune eorum centrum gravitatis, ut hic eſt <lb/>punctum B. </s> <s xml:id="echoid-s6129" xml:space="preserve">adeo ut ſi ſeparatim ſuſpenſa in æquilibrio fo-<lb/>rent cum contrario pondere C, hoc etiam obtineret ſuſpen-<lb/>@is omnibus ponderibus in puncto B, vel eorum loco pon-<lb/>dus D, quod æquat omnium gravitatem.</s> <s xml:id="echoid-s6130" xml:space="preserve"/> </p> <div xml:id="echoid-div465" type="float" level="2" n="1"> <note position="left" xlink:label="note-0366-01" xlink:href="note-0366-01a" xml:space="preserve">TAB. XXXII. <lb/>Fig. 4.</note> </div> <pb o="283" file="0367" n="398" rhead="MECHANICAM."/> <p> <s xml:id="echoid-s6131" xml:space="preserve">Quidam Geometræ parumper mutando hanc demonſtra-<lb/>tionem tentarunt defectum minus ſenſibilem reddere, ſed <lb/>in totum fuiſſe ſublatum mihi non videtur. </s> <s xml:id="echoid-s6132" xml:space="preserve">Igitur conatus <lb/>ſum alio modo eandem propoſitionem demonſtrare uti ſequi-<lb/>tur.</s> <s xml:id="echoid-s6133" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6134" xml:space="preserve">1°. </s> <s xml:id="echoid-s6135" xml:space="preserve">Poſtulatur cum Archimede, duo pondera æqualia <lb/>appenſa extremitatibus brachiorum æqualium libræ fore in <lb/>æquilibrio.</s> <s xml:id="echoid-s6136" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6137" xml:space="preserve">2°. </s> <s xml:id="echoid-s6138" xml:space="preserve">Poſitis ponderibus æqualibus, & </s> <s xml:id="echoid-s6139" xml:space="preserve">brachiis libræ, cui <lb/>appenſa ſunt, inæqualibus, illam inclinari ad latus brachii <lb/>longioris.</s> <s xml:id="echoid-s6140" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6141" xml:space="preserve">3°. </s> <s xml:id="echoid-s6142" xml:space="preserve">Poſtulatur poſſe concipi, lineas & </s> <s xml:id="echoid-s6143" xml:space="preserve">plana, de quibus <lb/>loquimur in hac demonſtratione, inflexilia & </s> <s xml:id="echoid-s6144" xml:space="preserve">ſine gravitate <lb/>eſſe.</s> <s xml:id="echoid-s6145" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6146" xml:space="preserve">PROP. </s> <s xml:id="echoid-s6147" xml:space="preserve">I. </s> <s xml:id="echoid-s6148" xml:space="preserve">Si ſuper planum Horizontale quod imponitur <lb/>lineæ rectæ, quæ id dividit in duaspartes, applicetur pondus, <lb/>vis, quam illud pondus habebit ad deflectendum planum par-<lb/>tem verſus ad quam applicatur erit major, quam ſi poſitum <lb/>ſit prope dictam lineam.</s> <s xml:id="echoid-s6149" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6150" xml:space="preserve">Sit planum Horizontale A B impoſitum lineæ rectæ <lb/> <anchor type="note" xlink:label="note-0367-01a" xlink:href="note-0367-01"/> C D, & </s> <s xml:id="echoid-s6151" xml:space="preserve">cui applicetur pondus E, cujus diſtantia a C D li-<lb/>neâ perpendiculari E H menſuratur; </s> <s xml:id="echoid-s6152" xml:space="preserve">& </s> <s xml:id="echoid-s6153" xml:space="preserve">cui porro applice-<lb/>tur idem pondus in F, ita ut diſtantia F H minor ſit quam <lb/>E H, dico, quod habebit plus virium ad planum deflecten-<lb/>dum, ſi ſit applicatum in E quam in F.</s> <s xml:id="echoid-s6154" xml:space="preserve"/> </p> <div xml:id="echoid-div466" type="float" level="2" n="2"> <note position="right" xlink:label="note-0367-01" xlink:href="note-0367-01a" xml:space="preserve">TAB. XXXII. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s6155" xml:space="preserve">Nam producta recta E F H in G & </s> <s xml:id="echoid-s6156" xml:space="preserve">poſitis H G & </s> <s xml:id="echoid-s6157" xml:space="preserve">H F <lb/>æqualibus, certum eſt, pondus æquale illi, de quo locuti <lb/>ſumus, applicatum in G in æquilibrio futurum cum altero <lb/>in F, propter æqualia brachia F H, H G.</s> <s xml:id="echoid-s6158" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6159" xml:space="preserve">Sed pondus tranſlatum ex F in E deflectet planum, quo-<lb/>niam plano exiſtente ſine gravitate effectus idem eſt ac in <lb/>bilance brachiorum inæqualium quæ æqualibus ponderibus <lb/>gravantur, idem ergo pondus poſitum in E plus virium ha-<lb/>bet ad planum deflectendum quam ſi eſt in F; </s> <s xml:id="echoid-s6160" xml:space="preserve">Q.</s> <s xml:id="echoid-s6161" xml:space="preserve">E.</s> <s xml:id="echoid-s6162" xml:space="preserve">D.</s> <s xml:id="echoid-s6163" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6164" xml:space="preserve">PROP. </s> <s xml:id="echoid-s6165" xml:space="preserve">II. </s> <s xml:id="echoid-s6166" xml:space="preserve">Si planum Horizont@le; </s> <s xml:id="echoid-s6167" xml:space="preserve">oneratum plurimis pon-<lb/>deribus, maneat in æqualibrio, impoſitum lineæ rectæ, quæ <pb o="284" file="0368" n="399" rhead="VARIA CIRCA"/> id ſecat in duas partes, centrum gravitatis plani ſic onerati <lb/>erit in ipſa lineâ rectâ.</s> <s xml:id="echoid-s6168" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6169" xml:space="preserve">Sit planum Horizontale A B oneratum ponderibus C C, <lb/> <anchor type="note" xlink:label="note-0368-01a" xlink:href="note-0368-01"/> D D & </s> <s xml:id="echoid-s6170" xml:space="preserve">quod manet in æquilibrio, impoſitum rectæ E F; <lb/></s> <s xml:id="echoid-s6171" xml:space="preserve">dico centrum ejus gravitatis eſſe in illa linea E F; </s> <s xml:id="echoid-s6172" xml:space="preserve">nam poſi-<lb/>to, ſi fieri poteſt, centrum gravitatis eſſe alibi in puncto G, <lb/>ducatur per id punctum recta H K parallela ipſi E F.</s> <s xml:id="echoid-s6173" xml:space="preserve"/> </p> <div xml:id="echoid-div467" type="float" level="2" n="3"> <note position="left" xlink:label="note-0368-01" xlink:href="note-0368-01a" xml:space="preserve">TAB. XXXII. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s6174" xml:space="preserve">Tunc ergo, quia planum fultum in puncto G, manet <lb/>in ſuo ſitu Horizontali, debent, ducta linea recta qua-<lb/>cunque in plano per punctum G, pondera ad utramque par-<lb/>tem lineæ eſſe in æquilibrio.</s> <s xml:id="echoid-s6175" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6176" xml:space="preserve">Idcirco pondera C C facient æquilibrium cum ponderibus <lb/>D D, quando planum fulcitur a recta H K: </s> <s xml:id="echoid-s6177" xml:space="preserve">id quod fieri <lb/>nequit, quoniam manet in æquilibrio fultum a recta E F; <lb/></s> <s xml:id="echoid-s6178" xml:space="preserve">nam patet, omnes diſtantias ponderum ad unam partem eſſe <lb/>diminutas, ſcilicet ponderum C C, & </s> <s xml:id="echoid-s6179" xml:space="preserve">conſequenter etiam <lb/>effectus gravitatis eorum; </s> <s xml:id="echoid-s6180" xml:space="preserve">& </s> <s xml:id="echoid-s6181" xml:space="preserve">diſtantias ponderum oppoſito-<lb/>rum D D eſſe auctas, & </s> <s xml:id="echoid-s6182" xml:space="preserve">eodem tempore effectum eorum <lb/>gravitatis, adeo ut ultima pondera deflexura ſint planum ad <lb/>ſuam partem, & </s> <s xml:id="echoid-s6183" xml:space="preserve">multo magis ſi unum vel plura pondera <lb/>C C ſint ad alteram partem lineæ H K; </s> <s xml:id="echoid-s6184" xml:space="preserve">Centrum ergo gra-<lb/>vitatis plani onerati erit in linea E F. </s> <s xml:id="echoid-s6185" xml:space="preserve">Q. </s> <s xml:id="echoid-s6186" xml:space="preserve">E. </s> <s xml:id="echoid-s6187" xml:space="preserve">D.</s> <s xml:id="echoid-s6188" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6189" xml:space="preserve">PROP. </s> <s xml:id="echoid-s6190" xml:space="preserve">III. </s> <s xml:id="echoid-s6191" xml:space="preserve">Duo gravia commenſur abilia appenſa ad extre-<lb/>mitates brachiorum Libræ erunt in æquilibrio, ſi brachia ſint <lb/>in ratione reciproca gravium.</s> <s xml:id="echoid-s6192" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6193" xml:space="preserve">Sint gravia commenſurabilia A & </s> <s xml:id="echoid-s6194" xml:space="preserve">B, quorum A ſit ma-<lb/> <anchor type="note" xlink:label="note-0368-02a" xlink:href="note-0368-02"/> jus; </s> <s xml:id="echoid-s6195" xml:space="preserve">& </s> <s xml:id="echoid-s6196" xml:space="preserve">libra C D E, cujus brachium D E ſit ad D C, ut <lb/>grave A ad grave B; </s> <s xml:id="echoid-s6197" xml:space="preserve">dico, libram eſſe in æquilibrio appenſo <lb/>A ad extremum C, & </s> <s xml:id="echoid-s6198" xml:space="preserve">B ad extremum E, ſi C E ſuſtineatur in D.</s> <s xml:id="echoid-s6199" xml:space="preserve"/> </p> <div xml:id="echoid-div468" type="float" level="2" n="4"> <note position="left" xlink:label="note-0368-02" xlink:href="note-0368-02a" xml:space="preserve">TAB. XXXII. <lb/>Fig. 7.</note> </div> <p> <s xml:id="echoid-s6200" xml:space="preserve">Concipiatur planum parallelum ad horizontem tranſiens <lb/>per lineam C E, in eo plano ſint ductæ per puncta E <lb/>& </s> <s xml:id="echoid-s6201" xml:space="preserve">C rectæ L E G, K C M perpendiculares ad C E; </s> <s xml:id="echoid-s6202" xml:space="preserve">fiat <lb/>ulterius E F æquale C D, & </s> <s xml:id="echoid-s6203" xml:space="preserve">ducantur G F K, M D L <lb/>quæ cum C E angulos ſemirectos efficiunt & </s> <s xml:id="echoid-s6204" xml:space="preserve">ſeſe mutuò ad <lb/>angulos rectos ſecant in N; </s> <s xml:id="echoid-s6205" xml:space="preserve">illæ lineæ neceſſario occurrunt <lb/>duabus prioribus, quas duximus per E & </s> <s xml:id="echoid-s6206" xml:space="preserve">C; </s> <s xml:id="echoid-s6207" xml:space="preserve">ponamus pun- <pb o="285" file="0369" n="400" rhead="MECHANICAM."/> cta occurſus eſſe G, K & </s> <s xml:id="echoid-s6208" xml:space="preserve">M, L; </s> <s xml:id="echoid-s6209" xml:space="preserve">manifeſtum eſt, E G æquale <lb/>eſſe E F, & </s> <s xml:id="echoid-s6210" xml:space="preserve">C K, C F; </s> <s xml:id="echoid-s6211" xml:space="preserve">uti etiam G K, M L, ſe mutuo <lb/>ſecare in medio in puncto N; </s> <s xml:id="echoid-s6212" xml:space="preserve">& </s> <s xml:id="echoid-s6213" xml:space="preserve">triangula G N L, <lb/>K N M eſſe ſimilia & </s> <s xml:id="echoid-s6214" xml:space="preserve">æqualia: </s> <s xml:id="echoid-s6215" xml:space="preserve">ſumatur E H æquale E G, & </s> <s xml:id="echoid-s6216" xml:space="preserve"><lb/>C O æquale C K; </s> <s xml:id="echoid-s6217" xml:space="preserve">tum, quia E D eſt ad D C ut pondus A <lb/>ad B, patet quod lineæ E D & </s> <s xml:id="echoid-s6218" xml:space="preserve">D C, ſint commenſurabiles, ut <lb/>& </s> <s xml:id="echoid-s6219" xml:space="preserve">H G & </s> <s xml:id="echoid-s6220" xml:space="preserve">K O, cum inter ſe ſint ut E F ad F C, id eſt ut <lb/>C D ad D E. </s> <s xml:id="echoid-s6221" xml:space="preserve">Sint ergo K O & </s> <s xml:id="echoid-s6222" xml:space="preserve">H G diviſæ in partes æqua-<lb/>les per maximam earum communem menſuram, & </s> <s xml:id="echoid-s6223" xml:space="preserve">quantita-<lb/>tes A & </s> <s xml:id="echoid-s6224" xml:space="preserve">B pariter diviſæ; </s> <s xml:id="echoid-s6225" xml:space="preserve">idcirco habebuntur tot partes <lb/>ponderis A, quot habentur partes in linea K O, & </s> <s xml:id="echoid-s6226" xml:space="preserve">tot par-<lb/>tes ponderis B, quot habentur partes in linea H G, quæ <lb/>partes ponderum, æquales inter ſe, ſint ſingulæ appenſæ in me-<lb/>dio unius ex partibus linearum K O, H G.</s> <s xml:id="echoid-s6227" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6228" xml:space="preserve">Jam demonſtrabimus, quod gravibus ita diſpoſitis, pla-<lb/>num maneat in æquilibrio, quando fulcitur a puncto D, un-<lb/>de veritas propoſitionis erit manifeſta; </s> <s xml:id="echoid-s6229" xml:space="preserve">quoniam concipere <lb/>poſſumus omnes partes plani eſſe ſublatas & </s> <s xml:id="echoid-s6230" xml:space="preserve">ſolas lineas K O, <lb/>H G, oneratas ponderibus æqualibus ipſis A & </s> <s xml:id="echoid-s6231" xml:space="preserve">B, ſuſtineri <lb/>in extremitatibus libræ C & </s> <s xml:id="echoid-s6232" xml:space="preserve">E, nam cum planum ſit ſine <lb/>gravitate, partes ſublatæ non poſſunt mutare æquilibrium.</s> <s xml:id="echoid-s6233" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6234" xml:space="preserve">Ad demonſtrandum igitur, æquilibrium plani, ut di-<lb/>ctum eſt gravati, dari in puncto D, ſint ductæ ex quovis <lb/>pondere perpendiculares ad lineam L M, quantum neceſſe <lb/>eſt, productam, uti R S, Z I, T V, X Y. </s> <s xml:id="echoid-s6235" xml:space="preserve">Perpendi-<lb/>culares T V & </s> <s xml:id="echoid-s6236" xml:space="preserve">R S, quæ ducuntur a ponderibus ma-<lb/>xime vicinis punctis G & </s> <s xml:id="echoid-s6237" xml:space="preserve">K erunt inter ſe æquales; </s> <s xml:id="echoid-s6238" xml:space="preserve">nam tri-<lb/>angula G N L, K N M ſunt æqualia & </s> <s xml:id="echoid-s6239" xml:space="preserve">ſimilia, uti di-<lb/>ctum eſt, & </s> <s xml:id="echoid-s6240" xml:space="preserve">latera G L & </s> <s xml:id="echoid-s6241" xml:space="preserve">K M ſunt etiam æqualia inter ſe, ut <lb/>& </s> <s xml:id="echoid-s6242" xml:space="preserve">intervalla G T & </s> <s xml:id="echoid-s6243" xml:space="preserve">K R, quæ ſingula æqualia ſunt dimidio u-<lb/>nius ex partibus æqualibus in quas diviſæ ſunt lineæ H G, K O; <lb/></s> <s xml:id="echoid-s6244" xml:space="preserve">unde patet lineas T V, R S, etiam fore æquales, uti di-<lb/>ctum eſt; </s> <s xml:id="echoid-s6245" xml:space="preserve">Tunc ſi fulciatur planum in linea L M Q, <lb/>pondus T in æquilibrio erit cum pondere R.</s> <s xml:id="echoid-s6246" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6247" xml:space="preserve">Pariter, ob æqualitatem perpendicularium X Y & </s> <s xml:id="echoid-s6248" xml:space="preserve">Z I, <lb/>pondus X erit in æquilibrio cum pondere Z, & </s> <s xml:id="echoid-s6249" xml:space="preserve">ſic con- <pb o="286" file="0370" n="401" rhead="VARIA CIRCA"/> ſequenter omnia pondera lineæ G H cum tot ponderi-<lb/>bus ſumtis poſt K in linea K O; </s> <s xml:id="echoid-s6250" xml:space="preserve">id eſt, ſi ſumatur pars <lb/>K P æqualis lineæ G H, pondera appenſa inter K <lb/>& </s> <s xml:id="echoid-s6251" xml:space="preserve">P, æquiponderabunt cum omnibus ponderibus lineæ <lb/>G H.</s> <s xml:id="echoid-s6252" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6253" xml:space="preserve">Si ergo pondera reliqua in linea P O etiam faciant æqui-<lb/>librium unum cum altero in plano fulto a linea L M Q, <lb/>ſequetur planum oneratum omnibus ponderibus manſurum in <lb/>æquilibrio ſuper eandam illam lineam.</s> <s xml:id="echoid-s6254" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6255" xml:space="preserve">Æquilibrium autem ponderum reliquorum ita invenitur: </s> <s xml:id="echoid-s6256" xml:space="preserve">cum <lb/>ſit K O = 2 C F; </s> <s xml:id="echoid-s6257" xml:space="preserve">& </s> <s xml:id="echoid-s6258" xml:space="preserve">K P = H G, id eſt 2 C D, erit <lb/>P O = 2 D F; </s> <s xml:id="echoid-s6259" xml:space="preserve">ſed M O = D F; </s> <s xml:id="echoid-s6260" xml:space="preserve">quoniam C M = C D; <lb/></s> <s xml:id="echoid-s6261" xml:space="preserve">ergo M P eſt dimidium P O; </s> <s xml:id="echoid-s6262" xml:space="preserve">Adeo ut lineâ P O, quæ con-<lb/>tinet numerum partium, quibus K O ſuperat H G, in 2 par-<lb/>tes æquales dividatur per rectam L M Q, manifeſtum ergo <lb/>eſt æqualem numerum ponderum illorum quæ continet illa <lb/>linea P O dari ad partem utramque puncti M, & </s> <s xml:id="echoid-s6263" xml:space="preserve">ſimiliter <lb/>diſponi; </s> <s xml:id="echoid-s6264" xml:space="preserve">ideo ſi numerus, illorum ponderum ſit impar, illud <lb/>quod in medio eſt, erit in puncto M, unde ſequitur, ſingulas <lb/>perpendiculares quas duximus ab iiſdem ponderibus ad lineam <lb/>L M Q æquales eſſe ſibi reſpondentibus, & </s> <s xml:id="echoid-s6265" xml:space="preserve">conſequenter pon-<lb/>dera eſſe in æquilibrio, quando planum fulcitur a linea L M Q; </s> <s xml:id="echoid-s6266" xml:space="preserve"><lb/>quod cum ita ſit demonſtratum de aliis ponderibus linearum <lb/>P K & </s> <s xml:id="echoid-s6267" xml:space="preserve">H G, ſequitur planum cum omnibus ponderibus man-<lb/>ſurum in æquilibrio fultum a linea L M Q; </s> <s xml:id="echoid-s6268" xml:space="preserve">Centrum er-<lb/>go gravitatis plani ſic onerati eſt in illa linea; </s> <s xml:id="echoid-s6269" xml:space="preserve">ſed centrum <lb/>gravitatis etiam eſt in linea C E, quoniam evidens eſt pla-<lb/>num etiam futurum in æquilibrio ſi in hac linea ſuſtinea-<lb/>tur, Erit ergo centrum gravitatis punctum commune illis <lb/>duabus lineis L M Q & </s> <s xml:id="echoid-s6270" xml:space="preserve">C E, ſcilicet punctum D in quo <lb/>ſi planum ſuſtineatur manet in æquilibrio. </s> <s xml:id="echoid-s6271" xml:space="preserve">patet ergo, veritas <lb/>Theorematis.</s> <s xml:id="echoid-s6272" xml:space="preserve"/> </p> <pb file="0371" n="402"/> <pb file="0371a" n="403"/> <figure> <caption xml:id="echoid-caption128" style="it" xml:space="preserve">Pag. 286.<lb/>TAB.XXXII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables130" xml:space="preserve">A E C E E D B G</variables> </figure> <figure> <caption xml:id="echoid-caption129" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables131" xml:space="preserve">H N K M</variables> </figure> <figure> <caption xml:id="echoid-caption130" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables132" xml:space="preserve">B A D C</variables> </figure> <figure> <caption xml:id="echoid-caption131" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables133" xml:space="preserve">A E E C H D G B</variables> </figure> <figure> <caption xml:id="echoid-caption132" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables134" xml:space="preserve">A C C C C H G K E F D D D D</variables> </figure> <figure> <caption xml:id="echoid-caption133" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables135" xml:space="preserve">G F F B D D C D A F A E E H</variables> </figure> <figure> <caption xml:id="echoid-caption134" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables136" xml:space="preserve">K L R Z Y H V N S P A C E B X T M G Q O</variables> </figure> <pb file="0372" n="404"/> <pb o="287" file="0373" n="405" rhead="MECHANICAM."/> </div> <div xml:id="echoid-div470" type="section" level="1" n="192"> <head xml:id="echoid-head245" xml:space="preserve">VII.</head> <head xml:id="echoid-head246" style="it" xml:space="preserve">De potentiis fila funesve trahentibus.</head> <p style="it"> <s xml:id="echoid-s6273" xml:space="preserve">PROP.</s> <s xml:id="echoid-s6274" xml:space="preserve">I. </s> <s xml:id="echoid-s6275" xml:space="preserve">Si puuctum A trahatur a filis duobus A B, A C <lb/> <anchor type="note" xlink:label="note-0373-01a" xlink:href="note-0373-01"/> angulum B A C facientibus, ſintque potentiæ trahentes ut fi-<lb/>@orum ipſorum A B, A C, longitudines multiplices ſecundum <lb/>numeros datos N & </s> <s xml:id="echoid-s6276" xml:space="preserve">O; </s> <s xml:id="echoid-s6277" xml:space="preserve">juncta vero B C dividatur in E, <lb/>ut ſit reciprocè C E ad E B ſicut numerus N ad O, & </s> <s xml:id="echoid-s6278" xml:space="preserve">jun-<lb/>gatur A E: </s> <s xml:id="echoid-s6279" xml:space="preserve">dico filis A B, A C ita trahentibus, æquipol-<lb/>lere filum A E tractu@ à potentia quæ ſit ut longitudo A E <lb/>multiplex ſecundum numerum æqualem utriſque N & </s> <s xml:id="echoid-s6280" xml:space="preserve">O.</s> <s xml:id="echoid-s6281" xml:space="preserve"/> </p> <div xml:id="echoid-div470" type="float" level="2" n="1"> <note position="right" xlink:label="note-0373-01" xlink:href="note-0373-01a" xml:space="preserve">TAB. XXXIII. <lb/>Fig. 1.</note> </div> <p> <s xml:id="echoid-s6282" xml:space="preserve">Producantur enim A B, A C ad F & </s> <s xml:id="echoid-s6283" xml:space="preserve">G, ut ſit <lb/>A F multiplex A B ſecundùm numerum N, & </s> <s xml:id="echoid-s6284" xml:space="preserve">A G multi-<lb/>plex A C ſecundùm numerum O; </s> <s xml:id="echoid-s6285" xml:space="preserve">junctæque F G occur-<lb/>rat A E producta in H, & </s> <s xml:id="echoid-s6286" xml:space="preserve">ſint B K, C L parallelæ A H.</s> <s xml:id="echoid-s6287" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6288" xml:space="preserve">Quia ergo F Had H K ut F A ad A B, hoc eſt, ut nu-<lb/>merus N ad unitatem; </s> <s xml:id="echoid-s6289" xml:space="preserve">H K vero ad H L ut B E ad E C, <lb/>hoc eſt, ut numerus O ad numerum N: </s> <s xml:id="echoid-s6290" xml:space="preserve">erit, inproportio-<lb/>ne turbata F H ad H L, ut numerus O ad unitatem, hoc eſt <lb/>ut G A ad A C, ſive ut G H ad H L. </s> <s xml:id="echoid-s6291" xml:space="preserve">Itaque F H ad <lb/>H L ut G H ad H L, ac proinde F H æqualis H G.</s> <s xml:id="echoid-s6292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6293" xml:space="preserve">Sit jam A H continuata uſque in P, ut ſint æquales A H, <lb/>H P, & </s> <s xml:id="echoid-s6294" xml:space="preserve">jungantur G P, F P: </s> <s xml:id="echoid-s6295" xml:space="preserve">eritque F A G P paralle-<lb/>logrammum, ad cujus diametrum P A ducantur F Q, G R, <lb/>parallelæ B C. </s> <s xml:id="echoid-s6296" xml:space="preserve">Manifeſtum igitur eſt fieri triangula ſimilia <lb/>& </s> <s xml:id="echoid-s6297" xml:space="preserve">æqualia F P Q, G A R, quorum latera inter ſe æqualia <lb/>P Q, R A. </s> <s xml:id="echoid-s6298" xml:space="preserve">Eſt autem A E ad A R ut A C ad A G, hoceſt, <lb/>ut unitas ad numerum O. </s> <s xml:id="echoid-s6299" xml:space="preserve">Eadem verò A E ad A Q ut <lb/>A B ad A F, hoc eſt, ut unitas ad numerum N. </s> <s xml:id="echoid-s6300" xml:space="preserve">Ergo erit <lb/>A E ad utramque ſimul A Q, A R, ſive A Q, Q P, hoc eſt, <lb/>ad A P, ut unitas ad utrumque ſimul numerum N & </s> <s xml:id="echoid-s6301" xml:space="preserve">O.</s> <s xml:id="echoid-s6302" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6303" xml:space="preserve">Cùm ergo potentiæ fila A B, A C trahentes, ſint ut A F, <lb/>A G, quibus æquipollet attractio per filum A E à potentia <lb/>quæ ſit ut A P, ex theoremate Mechanico ſatis noto, ma-<lb/>nifeſta eſt propoſiti veritas.</s> <s xml:id="echoid-s6304" xml:space="preserve"/> </p> <pb o="288" file="0374" n="406" rhead="VARIA CIRCA"/> <p style="it"> <s xml:id="echoid-s6305" xml:space="preserve">PROP. </s> <s xml:id="echoid-s6306" xml:space="preserve">II. </s> <s xml:id="echoid-s6307" xml:space="preserve">Datis poſitione quotlibet punctis; </s> <s xml:id="echoid-s6308" xml:space="preserve">ſive in eodem <lb/>plano fuerint, ſive non: </s> <s xml:id="echoid-s6309" xml:space="preserve">ſi à puncto quod eorum commune eſt <lb/>gravitatis centrum, ad unum quodque datorum fila extendan-<lb/>tur, eaque ſingula trahantur à potentiis quæ ſint inter ſe ut <lb/>filorum longitudines, fiet æquilibrium manente nodo commu-<lb/>ni in dicto gravitatis centro.</s> <s xml:id="echoid-s6310" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6311" xml:space="preserve">Sint data puncta A, B, C, D, E, quæ vel in eodem <lb/> <anchor type="note" xlink:label="note-0374-01a" xlink:href="note-0374-01"/> plano vel aliter utcunque collocata intelligantur: </s> <s xml:id="echoid-s6312" xml:space="preserve">attributâ <lb/>autem ſingulis æquali gravitate, conſtat commune eorum <lb/>gravitatis centrum inveniri hoc modo.</s> <s xml:id="echoid-s6313" xml:space="preserve"/> </p> <div xml:id="echoid-div471" type="float" level="2" n="2"> <note position="left" xlink:label="note-0374-01" xlink:href="note-0374-01a" xml:space="preserve">TAB. XXXIII. <lb/>Fig. 2.</note> </div> <p> <s xml:id="echoid-s6314" xml:space="preserve">Jungantur nempe duo quælibet datorum punctorum rectâ <lb/>A B, quâ bifariam ſectâ in F, erit hoc centrum gravitatis <lb/>punctorum A, B. </s> <s xml:id="echoid-s6315" xml:space="preserve">Ducatur deinde ad punctum aliud C re-<lb/>cta F C quæ ſecetur in G, ut ſit C G dupla G F; </s> <s xml:id="echoid-s6316" xml:space="preserve">& </s> <s xml:id="echoid-s6317" xml:space="preserve">erit <lb/>G centrum gravitatis punctorum trium A, B, C. </s> <s xml:id="echoid-s6318" xml:space="preserve">Rurſus <lb/>ducatur ad aliud punctum recta G D, ſeceturque in H, ut <lb/>ſit D H tripla H G, & </s> <s xml:id="echoid-s6319" xml:space="preserve">fiet H centrum gravitatis punctorum <lb/>quatuor A, B, C, D.</s> <s xml:id="echoid-s6320" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6321" xml:space="preserve">Similiterque ductâ H E ad punctum quintum E, ſectâque <lb/>in K, ut K E ſit quadrupla K H, erit K centrum gravi-<lb/>tatis punctorum quinque A, B, C, D, E. </s> <s xml:id="echoid-s6322" xml:space="preserve">Ac ſimili ratione <lb/>quotcunque punctorum centrum gravitatis invenire licebit.</s> <s xml:id="echoid-s6323" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6324" xml:space="preserve">Porro extentis filis à puncto K ad A, B, C, D, E, <lb/>quæ trahantur ſingula à potentiis quæ ſint inter ſe ut ipſæ <lb/>longitudines K A, K B, K C, K D, K E: </s> <s xml:id="echoid-s6325" xml:space="preserve">dico fieri æ-<lb/>quilibrium manente nodo communi in K. </s> <s xml:id="echoid-s6326" xml:space="preserve">Ducantur enim à <lb/>centris gravitatis inventis F, G, H, ad centrum gravitatis <lb/>omnium punctorum K, rectæ F K, G K, H K. </s> <s xml:id="echoid-s6327" xml:space="preserve">Itaque con-<lb/>ſtat filis A K, B K, punctum K trahentibus cum potentiis quæ <lb/>ſint ut longitudines eorum filorum, æquipollere filum F K, <lb/>tractum à potentia quæ ſit ut dupla longitudo F K. </s> <s xml:id="echoid-s6328" xml:space="preserve">Rurſus <lb/>verò duobus his, filo F K trahenti cum potentia quæ ſit ut <lb/>dupla F K, & </s> <s xml:id="echoid-s6329" xml:space="preserve">filo C K trahenti cum potentia quæ ſit ut <lb/>ſimplex longitudo C K, æquipollet filum G K tractum à <lb/>potentia quæ ſit ut tripla K G per præcedentem: </s> <s xml:id="echoid-s6330" xml:space="preserve">ergo <lb/>filum G K ita tractum æquipollet filis tribus K A, K B, <pb o="289" file="0375" n="407" rhead="MECHANICAM."/> K C. </s> <s xml:id="echoid-s6331" xml:space="preserve">Similiter verò duobus his, filo G K tracto à potentia <lb/>quæ ſit ut tripla G K, & </s> <s xml:id="echoid-s6332" xml:space="preserve">filo D K tracto à potentia quæ ſit <lb/>ut ſimplex longitudo D K, æquipollet filum H K tractum <lb/>à potentia quæ ſit ut quadrupla H K. </s> <s xml:id="echoid-s6333" xml:space="preserve">Ergo hoc æquipollet <lb/>filis onmibus K A, K B, K C, K D, punctum K uti di-<lb/>ctum eſt trahentibus. </s> <s xml:id="echoid-s6334" xml:space="preserve">Atqui filo K H in directum opponitur <lb/>filum K E tractum à potentia quæ eſt ut longitudo K E, id <lb/>eſt ut quadrupla K H. </s> <s xml:id="echoid-s6335" xml:space="preserve">Ergo cum filis K E, K H, in partes <lb/>directè oppoſitas trahentibus cum potentiis æqualibus, pun-<lb/>ctum K neceſſario in locum ſuum ſervatum ſit, ſequitur & </s> <s xml:id="echoid-s6336" xml:space="preserve">filis <lb/>K A, K B, K C, K D, uti dictum eſt trahentibus & </s> <s xml:id="echoid-s6337" xml:space="preserve">ex <lb/>alia parte filo K E nodum reſtare immotum. </s> <s xml:id="echoid-s6338" xml:space="preserve">Quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s6339" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6340" xml:space="preserve">Poſſunt autem & </s> <s xml:id="echoid-s6341" xml:space="preserve">binorum quorumque punctorum centra <lb/>gravitatis primò deſignari, & </s> <s xml:id="echoid-s6342" xml:space="preserve">per hæc deinceps centra gra-<lb/>vitatis quaternorum, & </s> <s xml:id="echoid-s6343" xml:space="preserve">per hæc octonorum & </s> <s xml:id="echoid-s6344" xml:space="preserve">ſic porro; <lb/></s> <s xml:id="echoid-s6345" xml:space="preserve">qua ratione ſimplicior plerumque efficitur demonſtratio, ac <lb/>præſertim ſi datorum punctorum numerus fuerit pariter <lb/>par.</s> <s xml:id="echoid-s6346" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6347" xml:space="preserve">Ut ſi quatuor data fuerint A, B, C, D; </s> <s xml:id="echoid-s6348" xml:space="preserve">ſive in eodem <lb/> <anchor type="note" xlink:label="note-0375-01a" xlink:href="note-0375-01"/> plano, ſive non: </s> <s xml:id="echoid-s6349" xml:space="preserve">junctis A B, C D, diviſisque bifariam in <lb/>E & </s> <s xml:id="echoid-s6350" xml:space="preserve">F; </s> <s xml:id="echoid-s6351" xml:space="preserve">ductâque inde F E, quæ rurſus bifariam ſecetur in <lb/>G; </s> <s xml:id="echoid-s6352" xml:space="preserve">conſtat G eſſe centrum gravitatis punctorum A, B, C, <lb/>D. </s> <s xml:id="echoid-s6353" xml:space="preserve">Quod ſi jam nodus G trahatur filis G A, G B, G C, <lb/>G D, à potentiis quæ ſint inter ſe ut hæ ipſæ filorum longi-<lb/>tudines; </s> <s xml:id="echoid-s6354" xml:space="preserve">dico fieri æquilibrium.</s> <s xml:id="echoid-s6355" xml:space="preserve"/> </p> <div xml:id="echoid-div472" type="float" level="2" n="3"> <note position="right" xlink:label="note-0375-01" xlink:href="note-0375-01a" xml:space="preserve">TAB. XXXIII. <lb/>Fig. 3.</note> </div> <p> <s xml:id="echoid-s6356" xml:space="preserve">Conſtat enim filis G A, G B, æquipollere filum G E <lb/>tractum à potentia quæ ſit ut dupla G E; </s> <s xml:id="echoid-s6357" xml:space="preserve">filis verò G C, <lb/>G D, æquipollere filum G F tractum à potentia quæ ſit ut <lb/>dupla G F. </s> <s xml:id="echoid-s6358" xml:space="preserve">Cum ergo G E, G F æquales ſint, unamque <lb/>lineam rectam efficiant, eodem modo nodus G trahitur, ac <lb/>ſi traheretur à potentiis æqualibus per fila G E, G F. </s> <s xml:id="echoid-s6359" xml:space="preserve">Un-<lb/>de immotum manere neceſſe eſt.</s> <s xml:id="echoid-s6360" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6361" xml:space="preserve">Conſtat verò ſi puncta A, B, C, D non ſint in eodem <lb/>plano, fore G centrum gravitatis pyramidis cujus anguli hæc <lb/>ipſa quatuor puncta; </s> <s xml:id="echoid-s6362" xml:space="preserve">cum in omni pyramide idem ſit cen- <pb o="290" file="0376" n="408" rhead="VARIA CIRCA"/> trum gravitatis ipſius ſolidi & </s> <s xml:id="echoid-s6363" xml:space="preserve">quatuor punctorum angularium, <lb/>uti oſtendere facillimum eſt. </s> <s xml:id="echoid-s6364" xml:space="preserve">Et hinc patet veritas theorema-<lb/>tis Robervalliani, Si à centro gravitatis pyramidis fila ten-<lb/>dantur ad quatuor angulos, quæ trahantur à potentiis quæ <lb/>ſint inter ſe ut filorum ipſorum longitudines, fieri æquilibrium <lb/>manente nodo in dicto gravitatis centro.</s> <s xml:id="echoid-s6365" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div474" type="section" level="1" n="193"> <head xml:id="echoid-head247" xml:space="preserve">VIII.</head> <head xml:id="echoid-head248" style="it" xml:space="preserve">Solitio problematis a G G. Leibnitio propoſiti in <lb/>diario (cui titulus Nouvelles de la Republi-<lb/>que des Lettres) menſis Sept. 1687.</head> <head xml:id="echoid-head249" xml:space="preserve">PROBLEMA.</head> <p> <s xml:id="echoid-s6366" xml:space="preserve">Detegere lineam juxta quam ſi corpus deſcendat tem-<lb/>poribus æqualibus æqualiter tellurem verſus accedat.</s> <s xml:id="echoid-s6367" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div475" type="section" level="1" n="194"> <head xml:id="echoid-head250" xml:space="preserve"><emph style="sc">Solutio</emph>.</head> <p> <s xml:id="echoid-s6368" xml:space="preserve">Impoſſibile eſt problema ſi requiratur ut corpus motum in <lb/>tali linea inchoet a quiete.</s> <s xml:id="echoid-s6369" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">TAB.XXXIII. <lb/>Fig. 4.</note> <p> <s xml:id="echoid-s6370" xml:space="preserve">Sed ſi ponamus corpus quandam, quantumvis exiguam, ve-<lb/>locitatem habere ex. </s> <s xml:id="echoid-s6371" xml:space="preserve">gr. </s> <s xml:id="echoid-s6372" xml:space="preserve">quam acquirit cadendo ab altitu-<lb/>dine A B, quæſito ſatisfacit curva B C, cujus hæc eſt pro-<lb/>prietas ut cubus altitudinis B D æqualis ſit quadrato perpen-<lb/>dicularis C D ad A B continuatam ducto in {9/4} A B.</s> <s xml:id="echoid-s6373" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6374" xml:space="preserve">Præter curvam hanc B C, in numeræ aliæ dantur ejuſdem <lb/>generis, & </s> <s xml:id="echoid-s6375" xml:space="preserve">quæ ſacile deteguntur, in quibus corpus etiam, <lb/>temporibus æqualibus, æqualiter, ſed lentius quam per B C, <lb/>ad Tellurem accedit.</s> <s xml:id="echoid-s6376" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6377" xml:space="preserve">Si B D ſit dupla ipſius B A, tempus deſcenſus per cur-<lb/>væ partem B C aquale erit tempori caſus per A B.</s> <s xml:id="echoid-s6378" xml:space="preserve"/> </p> <pb o="291" file="0377" n="409" rhead="MECHANICAM."/> </div> <div xml:id="echoid-div476" type="section" level="1" n="195"> <head xml:id="echoid-head251" xml:space="preserve">IX.</head> <head xml:id="echoid-head252" style="it" xml:space="preserve">Chriſtiani Hugenii, Solutio Problematis de <lb/>linea in quam flexile ſe pondere pro-<lb/>prio curvat.</head> <p> <s xml:id="echoid-s6379" xml:space="preserve">Si Catena C V A ſuſpenſa ſit ex filis F C, E A utrin-<lb/> <anchor type="note" xlink:label="note-0377-01a" xlink:href="note-0377-01"/> que annexis, ac gravitate carentibus, itaut capita C & </s> <s xml:id="echoid-s6380" xml:space="preserve">A <lb/>ſint pari altitudine, deturque Angulus inclinationis filorum <lb/>productorum C G A, & </s> <s xml:id="echoid-s6381" xml:space="preserve">catenæ totius poſitus, cujus vertex <lb/>ſit V, axis V B.</s> <s xml:id="echoid-s6382" xml:space="preserve"/> </p> <div xml:id="echoid-div476" type="float" level="2" n="1"> <note position="right" xlink:label="note-0377-01" xlink:href="note-0377-01a" xml:space="preserve">TAB.XXXIII. <lb/>Fig. 5.</note> </div> <p> <s xml:id="echoid-s6383" xml:space="preserve">1. </s> <s xml:id="echoid-s6384" xml:space="preserve">Licebit hinc invenire tangentem in dato quovis catenæ <lb/>puncto. </s> <s xml:id="echoid-s6385" xml:space="preserve">Velut ſi punctum datum ſit L, unde ducta appli-<lb/>cata L H dividat æqualiter axem B V. </s> <s xml:id="echoid-s6386" xml:space="preserve">Jam ſi angulus C G A <lb/>ſit 60°, erit inclinanda a puncto A ad axem recta A W, æ-<lb/>qualis {1/2} A B, cui ducta parallela L R, tanget curvam in pun-<lb/>cto L. </s> <s xml:id="echoid-s6387" xml:space="preserve">Item ſi latera G B, B A, A G ſint partium 3, 4, 5, <lb/>erit A W ponenda partium 4 {1/2}.</s> <s xml:id="echoid-s6388" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6389" xml:space="preserve">2. </s> <s xml:id="echoid-s6390" xml:space="preserve">Invenitur porrò & </s> <s xml:id="echoid-s6391" xml:space="preserve">recta linea catenæ æqualis, vel da-<lb/>tæ cuilibet ejus portioni. </s> <s xml:id="echoid-s6392" xml:space="preserve">Semper enim dato angulo C G A, <lb/>data erit ratio axis B V ad curvam V A. </s> <s xml:id="echoid-s6393" xml:space="preserve">Velut ſi latera <lb/>G B, B A, A G ſint ut 3, 4, 5, erit curva V A tripla <lb/>axis V B.</s> <s xml:id="echoid-s6394" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6395" xml:space="preserve">3. </s> <s xml:id="echoid-s6396" xml:space="preserve">Item definitur radius curvitatis in vertice V, hoc eſt, <lb/>ſemidiameter circuli maximi, qui per verticem hunc deſcri-<lb/>ptus totus intra curvam cadat. </s> <s xml:id="echoid-s6397" xml:space="preserve">Nam ſi angulus C G A ſit 60°, <lb/>erit radius curvitatis ipſi axi B V æqualis. </s> <s xml:id="echoid-s6398" xml:space="preserve">Sin vero angulus <lb/>C G A ſit rectus, erit radius curvitatis æqualis curvæ V A.</s> <s xml:id="echoid-s6399" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6400" xml:space="preserve">4. </s> <s xml:id="echoid-s6401" xml:space="preserve">Poterit & </s> <s xml:id="echoid-s6402" xml:space="preserve">circulus æqualis inveniri ſuperficiei conoidis, <lb/>ex revolutione catenæ circa axem ſuum. </s> <s xml:id="echoid-s6403" xml:space="preserve">Ita ſi angulus C G A <lb/>ſit 60°, erit ſuperficies conoidis ex catena C V A genita æ-<lb/>qualis circulo, cujus radius poſſit duplum rectangulum <lb/>B V G.</s> <s xml:id="echoid-s6404" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6405" xml:space="preserve">5. </s> <s xml:id="echoid-s6406" xml:space="preserve">Inveniuntur etiam puncta quotlibet curvæ K N, cujus <lb/>evolutione, una cum recta K V, radio curvitatis in verti- <pb o="292" file="0378" n="410" rhead="VARIA CIRCA"/> ce, curva V A deſcribitur; </s> <s xml:id="echoid-s6407" xml:space="preserve">atque evolutæ ipſius K N lon-<lb/>gitudo. </s> <s xml:id="echoid-s6408" xml:space="preserve">Velut ſi angulus C G A fuerit 60°, erit K N tripla <lb/>axis B V. </s> <s xml:id="echoid-s6409" xml:space="preserve">Si vero latera G B, B A, A G ſint ut 3, 4, 5, <lb/>erit illa {9/4} axis B V.</s> <s xml:id="echoid-s6410" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6411" xml:space="preserve">6. </s> <s xml:id="echoid-s6412" xml:space="preserve">Præterea ſpatii N K V A N quadratura datur. </s> <s xml:id="echoid-s6413" xml:space="preserve">Poſi-<lb/>to enim angulo C G A 60°, erit ſpatium illud æquale rectan-<lb/>gulo ex axe B V, & </s> <s xml:id="echoid-s6414" xml:space="preserve">ea quæ poteſt triplum quadratum <lb/>ejusdem B V. </s> <s xml:id="echoid-s6415" xml:space="preserve">Si vero latera G B, B A, A G ſint ut 3, 4, 5, <lb/>erit idem ſpatium æquale ſeptuplo quadrato B V, cum par-<lb/>te octava.</s> <s xml:id="echoid-s6416" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6417" xml:space="preserve">7. </s> <s xml:id="echoid-s6418" xml:space="preserve">Porro puncta quotlibet catenæ inveniri poſſunt, poſi-<lb/>ta quadratura curvæ alterutrius harum: </s> <s xml:id="echoid-s6419" xml:space="preserve">x x y y = a<emph style="super">4</emph> — a a y y, <lb/>vel x x y y = 4a<emph style="super">4</emph> — x<emph style="super">4</emph>. </s> <s xml:id="echoid-s6420" xml:space="preserve">Vel etiam data diſtantia centri gravi-<lb/>tatis ab axe, in portionibus planis, quas abſcindunt rectæ <lb/>axi parallelæ in curva harum priore. </s> <s xml:id="echoid-s6421" xml:space="preserve">Quadratura autem hu-<lb/>jus curvæ pendet a ſummis ſecantium arcuum per minima <lb/>æqualiter creſcentium: </s> <s xml:id="echoid-s6422" xml:space="preserve">quæ ſummæ ex Tabulis ſinuum egre-<lb/>gio quodam adhibito compendio inveniuntur quamlibet pro-<lb/>xime. </s> <s xml:id="echoid-s6423" xml:space="preserve">Hinc ex. </s> <s xml:id="echoid-s6424" xml:space="preserve">gr. </s> <s xml:id="echoid-s6425" xml:space="preserve">inventum, quod ſi angulus C G A ſit <lb/>rectus, & </s> <s xml:id="echoid-s6426" xml:space="preserve">ponatur axis B V partium 10000; </s> <s xml:id="echoid-s6427" xml:space="preserve">erit B A, <lb/>21279, non una minus. </s> <s xml:id="echoid-s6428" xml:space="preserve">Curva autem V A, per ſuperius <lb/>indicata cognoſcitur hic eſſe partium 24142, non una minus.</s> <s xml:id="echoid-s6429" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6430" xml:space="preserve">In his omnibus non niſi ad caſus ſingulares ſolutiones pro-<lb/>blematum dedi, vitandæ prolixitatis ſtudio & </s> <s xml:id="echoid-s6431" xml:space="preserve">quoniam non <lb/>dubito quin regulas univerſales Viri docti affatim ſint exhi-<lb/>bituri. </s> <s xml:id="echoid-s6432" xml:space="preserve">Quod ſi tamen aliquæ ex noſtris requirentur, eas lu-<lb/>benter mittam. </s> <s xml:id="echoid-s6433" xml:space="preserve">Ac jam pridem omnes apud Clariſſimum Vi-<lb/>rum G. </s> <s xml:id="echoid-s6434" xml:space="preserve">G. </s> <s xml:id="echoid-s6435" xml:space="preserve">Leibnitium involucro quodam obtectas depoſui.</s> <s xml:id="echoid-s6436" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div478" type="section" level="1" n="196"> <head xml:id="echoid-head253" xml:space="preserve">X.</head> <head xml:id="echoid-head254" style="it" xml:space="preserve">Hugenii Annotationes in librum Pariſiis 1689. <lb/>editum, de Manuaria Nautica.</head> <p> <s xml:id="echoid-s6437" xml:space="preserve">Auctor hujus librieſt D. </s> <s xml:id="echoid-s6438" xml:space="preserve">Renaldus (M. </s> <s xml:id="echoid-s6439" xml:space="preserve">Renau, Ingenieur ge-<lb/>neral de Marine), ſumma cura & </s> <s xml:id="echoid-s6440" xml:space="preserve">methodo conſcriptus eſt, <pb o="293" file="0379" n="411" rhead="MECHANICAM."/> & </s> <s xml:id="echoid-s6441" xml:space="preserve">auctoris peritia in Geometricis, & </s> <s xml:id="echoid-s6442" xml:space="preserve">analyſi in hoc patet; </s> <s xml:id="echoid-s6443" xml:space="preserve">nulla <lb/>ponuntur principia, quæ vera non fatear, & </s> <s xml:id="echoid-s6444" xml:space="preserve">integra ſi Theo-<lb/>ria inde legitime deducta foret, nihil in opere culpandum <lb/>eſſet; </s> <s xml:id="echoid-s6445" xml:space="preserve">hoc tamen deficiente, utile credidi de notabili quem <lb/>notavi errorem, monere, cum enim ſpectet ad maximam partem <lb/>regularum, quæ in hoc libro Nautis præſcribuntur poſſet hos <lb/>in maximos, & </s> <s xml:id="echoid-s6446" xml:space="preserve">periculoſos admodum errores inducere.</s> <s xml:id="echoid-s6447" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6448" xml:space="preserve">Initium faciam memorando quæ in Art. </s> <s xml:id="echoid-s6449" xml:space="preserve">1. </s> <s xml:id="echoid-s6450" xml:space="preserve">cap. </s> <s xml:id="echoid-s6451" xml:space="preserve">2. </s> <s xml:id="echoid-s6452" xml:space="preserve">conti-<lb/> <anchor type="note" xlink:label="note-0379-01a" xlink:href="note-0379-01"/> nentur, in quo Auctor navem H B M ponit, in qua linea <lb/>recta D C Veli poſitionem repræſentat, quod tanquam pla-<lb/>na ſuperficies concipitur, perpendiculariter ſuper iſtam li-<lb/>neam elevatam; </s> <s xml:id="echoid-s6453" xml:space="preserve">A B eſt directio venti qui velum propellit; <lb/></s> <s xml:id="echoid-s6454" xml:space="preserve">B G eſt perpendicularis ad D C; </s> <s xml:id="echoid-s6455" xml:space="preserve">G K eſt perpendicularis <lb/>ad B K, carinam navis productam; </s> <s xml:id="echoid-s6456" xml:space="preserve">G E A eſt arcus circuli <lb/>centro B deſcripti; </s> <s xml:id="echoid-s6457" xml:space="preserve">B K G eſt circuli peripheria cujus Diame-<lb/>ter eſt B G.</s> <s xml:id="echoid-s6458" xml:space="preserve"/> </p> <div xml:id="echoid-div478" type="float" level="2" n="1"> <note position="right" xlink:label="note-0379-01" xlink:href="note-0379-01a" xml:space="preserve">TAB. XXXIII. <lb/>Fig. 6.</note> </div> <p> <s xml:id="echoid-s6459" xml:space="preserve">Veriſſimum eſt quod Auctor notat, ſuperficiem C D a vento <lb/>A B propelli juxta lineam B G, ita ut navis per B G ad pun-<lb/>ctum G tenderet, ſi nullibi magis quam ad proram aqua re-<lb/>ſiſteret. </s> <s xml:id="echoid-s6460" xml:space="preserve">Addit, navem iſtam lineam percurrendo directe <lb/>progredi per B K, & </s> <s xml:id="echoid-s6461" xml:space="preserve">ad latus per K G; </s> <s xml:id="echoid-s6462" xml:space="preserve">ſed cum navis ma-<lb/>jori difficultate aquam lateribus, quam prorâ ſecet, non <lb/>poterit juxta directionem K G per integram hanc lineam <lb/>progredi, ſed deerit pars, quæ ſequetur rationem exceſſus quo <lb/>difficultas ſecandi aquam ad latus ſuperat difficultatem qua <lb/>navis hanc prorâ ſecat; </s> <s xml:id="echoid-s6463" xml:space="preserve">ex. </s> <s xml:id="echoid-s6464" xml:space="preserve">gr. </s> <s xml:id="echoid-s6465" xml:space="preserve">ſi difficultas ſecandi aquam <lb/>ad latus ſe habeat ad difficultatem qua prora illam ſecat ut <lb/>decem ad unum, ſi fiat K G ad K L ut 10 ad 1, & </s> <s xml:id="echoid-s6466" xml:space="preserve">ducatur <lb/>B L, dicit Auctor navem moveri per B L, hancque lineam <lb/>percurrere eo tempore quo potuiſſet in G pervenire, ſi reſi-<lb/>ſtentia ab omni parte fuiſſet æqualis.</s> <s xml:id="echoid-s6467" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6468" xml:space="preserve">Auctorem huc uſque nos fuiſſe ſecutos ſufficiat. </s> <s xml:id="echoid-s6469" xml:space="preserve">Con-<lb/>tendo ipſius errorem in eo dari, quod dicat navem per-<lb/>venire ex B in L eodem tempore, quo perveniſſet ex B in G. <lb/></s> <s xml:id="echoid-s6470" xml:space="preserve">Nam ſi deviatio nullam eſſe ponamus, ut ambages remo-<lb/>veantur, certiſſimum eſt navem, juxta Auctorem, progredi <pb o="294" file="0380" n="412" rhead="VARIA CIRCA"/> ex B in K, eodem tempore, quo pergeret ex B in G, ſi <lb/>undequaque aquam eadem facilitate ſecaret, aut directe per <lb/>B G, æque ac per B K progrediretur.</s> <s xml:id="echoid-s6471" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6472" xml:space="preserve">In his ſic ratiocinatum fuiſſe auctorem videtur, ſcilicet, <lb/>ſi in motu ex B in G, navis feratur ad latus per K G & </s> <s xml:id="echoid-s6473" xml:space="preserve">di-<lb/>recte progrediatur per B K, oportet, ut ſublato motu per <lb/>K G, motus per B K ſuperſit, quo motu linea B K percur-<lb/>rebatur eodem tempore quo B G.</s> <s xml:id="echoid-s6474" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6475" xml:space="preserve">Sed notandum erat, licet motus navis per B G poſſit con-<lb/>cipi tanquam compoſitus ex motibus per B K, & </s> <s xml:id="echoid-s6476" xml:space="preserve">K G, <lb/>inde non ſequi ſi in re ipſa tantum ſuperſit motus per B K <lb/>(ſive figura ipſius navis in cauſa ſit, ſive hæc cohæreat <lb/>cum fune infinito B R, perpendiculari ad B M) ventum <lb/>qui navem ex B in G tranſtuliſſet, hanc æquali tempore <lb/>poſſe transferre ex B in K. </s> <s xml:id="echoid-s6477" xml:space="preserve">Ut enim determinemus viam <lb/>juxta B K percurſam, vim propellentem debemus determi-<lb/>nare, & </s> <s xml:id="echoid-s6478" xml:space="preserve">attendere ad reſiſtentiam quam ex actione aquæ pa-<lb/>titur navis.</s> <s xml:id="echoid-s6479" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6480" xml:space="preserve">Conſtat autem in Mechanicis, vim, qua velum D C, na-<lb/>vem pellit per B K, eſſe ad vim qua idem velum, & </s> <s xml:id="echoid-s6481" xml:space="preserve">in ea-<lb/>dem poſitione reſpectu venti, illam pelleret per B G, uti <lb/>B K ad B G: </s> <s xml:id="echoid-s6482" xml:space="preserve">ut Auctor ipſe ponit in his, quæ ſcripſit <lb/>de impreſſionibus aquæ in gubernaculum Art. </s> <s xml:id="echoid-s6483" xml:space="preserve">5. </s> <s xml:id="echoid-s6484" xml:space="preserve">hujus ſecun-<lb/>di Cap. </s> <s xml:id="echoid-s6485" xml:space="preserve">Sed celeritates quoque eſſent ſicuti B K ad B G; <lb/></s> <s xml:id="echoid-s6486" xml:space="preserve">quia point Auctor lineas æqualibus temporibus percurri. </s> <s xml:id="echoid-s6487" xml:space="preserve"><lb/>Vires ergo forent ut celeritates, quod impoſſibile eſt, dictis-<lb/>que Auctoris repugnat in 13° Art. </s> <s xml:id="echoid-s6488" xml:space="preserve">primi Cap. </s> <s xml:id="echoid-s6489" xml:space="preserve">ubi dicit, <lb/>ut corpus diverſis velocitatibus in fluido moveatur, requiri <lb/>vires in ratione quadratorum celeritatum: </s> <s xml:id="echoid-s6490" xml:space="preserve">lineæ ergo B K, <lb/>B G, non percurruntur æquali tempore. </s> <s xml:id="echoid-s6491" xml:space="preserve">Ut autem determi-<lb/>netur, ſpatium juxta B K percurſum continuanda eſt B K <lb/>in S, ita ut B S ſit media proportionalis inter B K, B G. </s> <s xml:id="echoid-s6492" xml:space="preserve"><lb/>Tum B S erit ſpatium, quod eo tempore permeabit navis <lb/>quo pergeret per B G; </s> <s xml:id="echoid-s6493" xml:space="preserve">ſi aquam juxta hanc directionem <lb/>eadem facilitate ſecaret. </s> <s xml:id="echoid-s6494" xml:space="preserve">nam quadrata celeritatum per <pb o="295" file="0381" n="413" rhead="MECHANICAM."/> B G, & </s> <s xml:id="echoid-s6495" xml:space="preserve">B S, & </s> <s xml:id="echoid-s6496" xml:space="preserve">conſequenter etiam aquæ reſiſtentiæ ſunt <lb/>inter ſe ut B G ad B K; </s> <s xml:id="echoid-s6497" xml:space="preserve">aſt, uti modo oſtendi, virium ratio <lb/>eſt etiam ut B G ad B K; </s> <s xml:id="echoid-s6498" xml:space="preserve">vires igitur ſunt ut reſiſtentiæ, & </s> <s xml:id="echoid-s6499" xml:space="preserve"><lb/>etiam ut quadrata velocitatum. </s> <s xml:id="echoid-s6500" xml:space="preserve">Hæ ergo ſunt velocitates, <lb/>quæ ſunt ut B G ad B S, quas Navis in ambobus motibus <lb/>acquirere debet ſecundum ipſam Auctoris ſtatim memoratam, <lb/>nec in dubium revocandam, regulam. </s> <s xml:id="echoid-s6501" xml:space="preserve">Non ergo ut credidit <lb/>Auctor circumferentia circuli B K G determinat ſpatia a navi <lb/>permeanda in diverſis carinæ poſitionibus, manente eadem veli <lb/>C D poſitione reſpectu directionis venti, ſed determinantur <lb/>hæc ſpatia curvâ B I S G T, cujus puncta eodem modo ac S, <lb/>facile inveniuntur. </s> <s xml:id="echoid-s6502" xml:space="preserve">Hic autem notandum eſt, ſpatia quæ hac <lb/>curvâ deteguntur, eo magis abiis differre, quæ Auctor adhibitâ <lb/>circumferentiâ B K G determinat, quo angulus quem carina <lb/>cum venti directione efficit acutior eſt; </s> <s xml:id="echoid-s6503" xml:space="preserve">ita juxta B N navis <lb/>progredietur per B I, quæ dupla eſt ipſius B N circulo <lb/>inſcriptæ, ſi B N ſit {1/4} B G; </s> <s xml:id="echoid-s6504" xml:space="preserve">& </s> <s xml:id="echoid-s6505" xml:space="preserve">tripla ſi B N ſit {1/9} B G.</s> <s xml:id="echoid-s6506" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6507" xml:space="preserve">Error a me notatus in toto fere tractatus reliquo locum ha-<lb/>bet, quo varia labefactantur Theoremata quæ de cætero ele-<lb/>gantia videntur. </s> <s xml:id="echoid-s6508" xml:space="preserve">Quale eſt inter alia hoc. </s> <s xml:id="echoid-s6509" xml:space="preserve">Dato O B A angu-<lb/>lo veli cum vento, carinæ ſitum, quo in adverſum venti ma-<lb/>xime progreditur navis, determinari dividendo æqualiter in <lb/>duas partes complementum O B E anguli dati; </s> <s xml:id="echoid-s6510" xml:space="preserve">unde Au-<lb/>ctor deducit, ponendo quod deviatio nulla ſit, carinæ & </s> <s xml:id="echoid-s6511" xml:space="preserve"><lb/>veli ſitum in hoc caſu omnium maxime utilem dari, quando <lb/>angulus quem carina cum vento efficit eſt 60 gr.</s> <s xml:id="echoid-s6512" xml:space="preserve">, & </s> <s xml:id="echoid-s6513" xml:space="preserve">angulus <lb/>venti cum velo 30 gr.</s> <s xml:id="echoid-s6514" xml:space="preserve">, quod a vero abeſt; </s> <s xml:id="echoid-s6515" xml:space="preserve">nam per Regu-<lb/>lam, quam veram novi detego, quando venti & </s> <s xml:id="echoid-s6516" xml:space="preserve">carinæ an-<lb/>gulus eſt graduum 60, navem celerius moveri, ideoque in <lb/>adverſum venti magis progredi, ſi angulus veli & </s> <s xml:id="echoid-s6517" xml:space="preserve">venti. </s> <s xml:id="echoid-s6518" xml:space="preserve">38 <lb/>gr. </s> <s xml:id="echoid-s6519" xml:space="preserve">23′. </s> <s xml:id="echoid-s6520" xml:space="preserve">fiat, quam ſi angulus hicce foret 30 graduum. </s> <s xml:id="echoid-s6521" xml:space="preserve">Regu-<lb/>la qua detego veli ſitum, ut navis omnium celerrime movea-<lb/>tur, ubi carinæ cum vento angulus datus eſt, talis eſt <lb/>x<emph style="super">4</emph> = a a x x + {1/3} p p x x — {4/9} a a p p. </s> <s xml:id="echoid-s6522" xml:space="preserve">ſcilicet, ſi x deſignat ſi-<lb/>num O Q anguli veli cum vento, a radius B A, p ſinus F P <lb/>anguli carinæ cum vento. </s> <s xml:id="echoid-s6523" xml:space="preserve">Et congruit hgc regula cum illa quam <pb o="296" file="0382" n="414" rhead="VARIA CIRCA"/> D. </s> <s xml:id="echoid-s6524" xml:space="preserve">Fatio antea invenit, cum aliis pulcherrimis circa hanc <lb/>materiam, ut percepi in tabula quadam, in qua quorun-<lb/>dam iſtorum angulorum rationes deſignarat. </s> <s xml:id="echoid-s6525" xml:space="preserve">Duas veras ra-<lb/>dices continet æquatio hæc inſervientes duobus caſibus, in <lb/>quibus carina cum venti linea eundem angulum efficit, ut-<lb/>pote, quando navis vento ſecundo aut adverſo utitur.</s> <s xml:id="echoid-s6526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6527" xml:space="preserve">Cæterum quin vera ſit noſtra regula non poterit, D. <lb/></s> <s xml:id="echoid-s6528" xml:space="preserve">Renaldus dubitare, cum per eam angulus gubernaculi cum <lb/>carina, quo navis omnium celerrime circumvolvitur, idem <lb/>detegatur, quem determinavit in capite 7°. </s> <s xml:id="echoid-s6529" xml:space="preserve">Quod ipſius in-<lb/>ventum certe utiliſſimum eſt. </s> <s xml:id="echoid-s6530" xml:space="preserve">Ponendo enim p = a, id eſt <lb/>ponendo venti lineam ad carinam perpendicularem, noſtrâ <lb/>regulâ habetur x = V {2/3} a a, quem ille detexit ſinum an-<lb/>guli carinæ aut directionis motus aquæ cum gubernaculo, <lb/>quod ſic neceſſario ſeſe habere facile patet.</s> <s xml:id="echoid-s6531" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6532" xml:space="preserve">Licet Theoria hæc poſt a me indicatam correctionem dif-<lb/>ficilior evadat, quam in tractatu D<emph style="super">ni</emph> Renaldi, percipio nihil-<lb/>ominus regulam detegi poſſe, qua & </s> <s xml:id="echoid-s6533" xml:space="preserve">navis & </s> <s xml:id="echoid-s6534" xml:space="preserve">veli ſitus de-<lb/>terminarentur, ut in adverſum venti omnium maxime pro-<lb/>grediatur, ſed computatio nimium longa foret, quare hanc <lb/>nunc non aggredior. </s> <s xml:id="echoid-s6535" xml:space="preserve">Quibus addendum navis deviationem <lb/>in hac computatione non conſiderari, ex qua conſideratione <lb/>maxima daretur difficultas. </s> <s xml:id="echoid-s6536" xml:space="preserve">Quia non modo cum auctore at-<lb/>tendendum eſt ad relationem inter reſiſtentias quas patitur <lb/>navis proram verſus & </s> <s xml:id="echoid-s6537" xml:space="preserve">ad latus, ſed etiam ad actionem venti <lb/>in ipſam navem, præcipue in hujus latus; </s> <s xml:id="echoid-s6538" xml:space="preserve">ita ut ex unica ob-<lb/>ſervatione quæ ad deviationem ſpectant non poſſent deduci.</s> <s xml:id="echoid-s6539" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div480" type="section" level="1" n="197"> <head xml:id="echoid-head255" xml:space="preserve">XI.</head> <head xml:id="echoid-head256" style="it" xml:space="preserve">Reſponſum D<emph style="super">ni</emph> Renaldi ad Dominum <lb/>Hugenium.</head> <p> <s xml:id="echoid-s6540" xml:space="preserve">D<emph style="super">nus</emph> Hugenius hæc ponit, licet motus navis per B G poſſit <lb/>concipi tanquam compoſitus ex motibus per B K & </s> <s xml:id="echoid-s6541" xml:space="preserve">K G, inde <lb/>non ſequi, ſi re ipſa tantum ſuperſit motus per B K, ventum, <lb/>qui navem ex B in G iranſtuliſſet, hanc æquali tempore poſſe <pb o="297" file="0383" n="415" rhead="MECHANICAM."/> transferre ex Bin K. </s> <s xml:id="echoid-s6542" xml:space="preserve">Et primo in eo mihi videtur falli. </s> <s xml:id="echoid-s6543" xml:space="preserve">Nam <lb/>ut navis poſſit ferri ex B in G tempore determinato, neceſſe <lb/>eſt ut reapſe celeritatem habeat, quâ eodem tempore ſecun-<lb/>dum determinationem B K, lineam B K, & </s> <s xml:id="echoid-s6544" xml:space="preserve">ſecundum deter-<lb/>minationem K G, lineam K G poſſit percurrere. </s> <s xml:id="echoid-s6545" xml:space="preserve">Et ne hoc <lb/>in dubium vocari poſſit, concipiamus navim pelli ſecundum <lb/>determinationem B K, vi quâ certo tempore ex B poſſit <lb/>pervenire in K; </s> <s xml:id="echoid-s6546" xml:space="preserve">concipiamus eandem ſimul pelli ſecundum <lb/>K G, vi quâ eodem tempore poſſit percurrere lineam K G: <lb/></s> <s xml:id="echoid-s6547" xml:space="preserve">cum duæ hæ vires nec ſibi contrarientur nec etiam concur-<lb/>rant (eſt enim B K perpendicularis ipſi K G) neceſſe eſt <lb/>ut harum unicuique navis ex toto obſecundet; </s> <s xml:id="echoid-s6548" xml:space="preserve">& </s> <s xml:id="echoid-s6549" xml:space="preserve">per con-<lb/>ſequens celeritas, quam ſingulis momentis habebit ſecundum <lb/>B K, erit ad celeritatem, quam iisdem momentis habebit, <lb/>ſecundum K G, ut B K ad K G; </s> <s xml:id="echoid-s6550" xml:space="preserve">& </s> <s xml:id="echoid-s6551" xml:space="preserve">ita navis utrique vi ſatis-<lb/>faciens, movebitur per B G, & </s> <s xml:id="echoid-s6552" xml:space="preserve">tempore determinato perveniet <lb/>in G; </s> <s xml:id="echoid-s6553" xml:space="preserve">Etideo ſi in effectu ipſi relinquatur ſolus motus per B K, <lb/>vis quæcunque, quâ pelleretur ex B in G, illam tempore <lb/>æquali pellet ex B in K. </s> <s xml:id="echoid-s6554" xml:space="preserve">Inutilem enim reddendo illam im-<lb/>preſſionis partem, quæ requiritur ut eodem tempore navis <lb/>percurrat K G, neque augetur, ut diximus, neque minui-<lb/>tur celeritas per B K. </s> <s xml:id="echoid-s6555" xml:space="preserve">Fateor, ſi angulus B K G eſſet acu-<lb/>tus, vim peculiarem quâ pelleretur navis ſecundum K G, <lb/>aliquid detracturam celeritati, quam haberet ſecundum B K <lb/>utpote ſibi contrariæ; </s> <s xml:id="echoid-s6556" xml:space="preserve">& </s> <s xml:id="echoid-s6557" xml:space="preserve">contra, ſi angulus B K G foret obtu-<lb/>ſus, eandem celeritatem utpote cum altera concurrentem, eſſe <lb/>augendam; </s> <s xml:id="echoid-s6558" xml:space="preserve">ſed cum angulus B K G rectus ſit, vis illa ce-<lb/>leritatem navis ſecundum B K neque auget, neque minuit.</s> <s xml:id="echoid-s6559" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6560" xml:space="preserve">Addit D<emph style="super">nus</emph> Hugenius; </s> <s xml:id="echoid-s6561" xml:space="preserve">Ut enim determinemus viam ju-<lb/>xta B K percurſam, vim propellentem debemus determina-<lb/>re, & </s> <s xml:id="echoid-s6562" xml:space="preserve">attendere ad reſiſtentiam quam ex actione aquæ patitur <lb/>navis. </s> <s xml:id="echoid-s6563" xml:space="preserve">Statim oſtendi relationes celeritatum in variis de-<lb/>terminationibus ſibi invicem perpendicularibus, ſufficere ad <lb/>detegendam viam, quam navis eſt ſecutura; </s> <s xml:id="echoid-s6564" xml:space="preserve">nec per con-<lb/>ſequens ad id requiri relationem virium, nec reſiſtentiarum; <lb/></s> <s xml:id="echoid-s6565" xml:space="preserve">ſed cum dictæ celeritates à viribus dependeant, idem ex <pb o="298" file="0384" n="416" rhead="VARIA CIRCA"/> relatione virium facile probabo.</s> <s xml:id="echoid-s6566" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6567" xml:space="preserve">Demonſtravi Articulo 13° Cap. </s> <s xml:id="echoid-s6568" xml:space="preserve">1. </s> <s xml:id="echoid-s6569" xml:space="preserve">Theoriæ manuariæ <lb/>nauticæ, de quo Articulo D<emph style="super">nus</emph> Hugenius mecum conſentit, <lb/>vires navem propellentes eſſe inter ſe ut quadrata veloci-<lb/>tatum; </s> <s xml:id="echoid-s6570" xml:space="preserve">& </s> <s xml:id="echoid-s6571" xml:space="preserve">propterea vis requiſita, ut navis certo tempo-<lb/>re ſecundum determinationem B K conficiat B K, eſt ad <lb/>vim, quâ ſecundum determinationem K G conficiat K G, <lb/>ut quadratum B K, ad quadratum K G; </s> <s xml:id="echoid-s6572" xml:space="preserve">unde ſequi-<lb/>tur, navem, ſi ſecundum duas illas determinationes ſimul <lb/>pelleretur, habituram vim duabus illis viribus æqualem; <lb/></s> <s xml:id="echoid-s6573" xml:space="preserve">cum ſcilicet neutra neutri quicquam vel addat vel detra-<lb/>hat; </s> <s xml:id="echoid-s6574" xml:space="preserve">per conſequens vis illa exprimetur per quadratum ipſius <lb/>B G, quod æquale eſt quadratis B K & </s> <s xml:id="echoid-s6575" xml:space="preserve">K G; </s> <s xml:id="echoid-s6576" xml:space="preserve">& </s> <s xml:id="echoid-s6577" xml:space="preserve">ita na-<lb/>vis habebit celeritatem ex illa vi ortam, id eſt, dicto <lb/>tempore quantitatem B G peragrabit. </s> <s xml:id="echoid-s6578" xml:space="preserve">Et propterea, ſi na-<lb/>vis pelleretur ſecundum B G, vi quæ exprimitur per qua-<lb/>dratam B G, perveniret in G eodem tempore, quo perve-<lb/>niret in K, ſi pelleretur ſecundum B K, vi quam exprimit <lb/>quadratum B K.</s> <s xml:id="echoid-s6579" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6580" xml:space="preserve">Pergit D<emph style="super">nus</emph> Hugenius hoc modo; </s> <s xml:id="echoid-s6581" xml:space="preserve">Conſtat autem in me-<lb/>chanicis, vim, quâ velum D C pellit navem per B K, eſſe <lb/>ad vim, quâ idem velum, & </s> <s xml:id="echoid-s6582" xml:space="preserve">in eadem poſitione reſpectu <lb/>venti, navem pelleret per B G, ut B K ad B G. </s> <s xml:id="echoid-s6583" xml:space="preserve">Non <lb/>ego fateor id ex regulis Mechanices ſequi; </s> <s xml:id="echoid-s6584" xml:space="preserve">è contra certum <lb/>eſt, virium illarum relationem inter ſe eſſe ut quadratum <lb/>B K ad quadratum B G, non vero ut B K ad B G; </s> <s xml:id="echoid-s6585" xml:space="preserve">& </s> <s xml:id="echoid-s6586" xml:space="preserve">ut <lb/>omne hic dubium eximatur, concipiamus aërem ſecundum <lb/>lineam A B duplo citius moveri uno tempore quam alte-<lb/>ro. </s> <s xml:id="echoid-s6587" xml:space="preserve">Quando duplo citius movebitur quadruplo fortius in <lb/>velum impinget, quoniam unaquæque particula duplo for-<lb/>tius impingit propter velocitatem duplam, propter quam <lb/>etiam duplo plures particulæ eodem tempore impingunt. <lb/></s> <s xml:id="echoid-s6588" xml:space="preserve">Quare ſi velocitas ſit dupla, & </s> <s xml:id="echoid-s6589" xml:space="preserve">maſſa itidem dupla, vis ſeu <lb/>potentia eſt quadrupla. </s> <s xml:id="echoid-s6590" xml:space="preserve">Si tripla foret velocitas, unaquæ-<lb/>que particula triplo fortius impingeret, quia tripla eſſet ve-<lb/>locitas; </s> <s xml:id="echoid-s6591" xml:space="preserve">& </s> <s xml:id="echoid-s6592" xml:space="preserve">ſimul quia tripla eſſet velocitas, triplo plures <pb o="299" file="0385" n="417" rhead="MECHANICAM."/> particulæ ſimul impingerent; </s> <s xml:id="echoid-s6593" xml:space="preserve">unde triplâ exiſtente velocitate, <lb/>maſſâ etidem triplâ, potentia aut vis erit noncupla; </s> <s xml:id="echoid-s6594" xml:space="preserve">ex quo <lb/>patet maſſam augeri in eadem ratione, qua velocitas auge-<lb/>tur; </s> <s xml:id="echoid-s6595" xml:space="preserve">& </s> <s xml:id="echoid-s6596" xml:space="preserve">cum unaquæque pars etiam fortius impingat in ratio-<lb/>ne auctæ velocitatis, potentia aut vis venti in velum, eſt in <lb/>ratione duplicata celeritatum venti, id eſt, in ratione qua-<lb/>dratorum velocitatum venti in velum. </s> <s xml:id="echoid-s6597" xml:space="preserve">Agnoſcit hocce prin-<lb/>cipium Cl. </s> <s xml:id="echoid-s6598" xml:space="preserve">Hugenius; </s> <s xml:id="echoid-s6599" xml:space="preserve">reſtat ergo tantum, ut illud ap-<lb/>plicemus.</s> <s xml:id="echoid-s6600" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6601" xml:space="preserve">Prima applicatio oſtendet, quare vis venti in velum, cum <lb/>ventus velo perpendicularis eſt, ſeſe habeat ad vim ejuſdem <lb/>venti in velum, quando illud inclinatum vento opponitur, <lb/>ut quadratum radii ad quadratum ſinûs anguli incidentiæ; <lb/></s> <s xml:id="echoid-s6602" xml:space="preserve">aut, quod idem eſt, cur vires ejuſdem venti in vela varia in-<lb/>clinatione ipſi obtenſa, ſint inter ſe in ratione quadratorum <lb/>ſinuum angulorum incidentiæ, quod demonſtravi Articulis <lb/>7. </s> <s xml:id="echoid-s6603" xml:space="preserve">8. </s> <s xml:id="echoid-s6604" xml:space="preserve">& </s> <s xml:id="echoid-s6605" xml:space="preserve">9. </s> <s xml:id="echoid-s6606" xml:space="preserve">Cap. </s> <s xml:id="echoid-s6607" xml:space="preserve">1.</s> <s xml:id="echoid-s6608" xml:space="preserve">, & </s> <s xml:id="echoid-s6609" xml:space="preserve">quod etiam hoc modo nunc demonſtro. </s> <s xml:id="echoid-s6610" xml:space="preserve"><lb/>Probatum dedi in Theoria Manuariæ Nauticæ, Artic. </s> <s xml:id="echoid-s6611" xml:space="preserve">6. </s> <s xml:id="echoid-s6612" xml:space="preserve">Cap. </s> <s xml:id="echoid-s6613" xml:space="preserve"><lb/>1. </s> <s xml:id="echoid-s6614" xml:space="preserve">corpus motum ab A in B non occurrere ſuperficiei C D <lb/>niſi ſecundum determinationem ſuam A V, ponendo ſcilicet <lb/>A V perpendicularem ipſi D C productæ, & </s> <s xml:id="echoid-s6615" xml:space="preserve">in illam ſuper-<lb/>ficiem nullam vim exſerere niſi ſecundum hanc determinatio-<lb/>nem; </s> <s xml:id="echoid-s6616" xml:space="preserve">quod agnoſcit D<emph style="super">nus</emph> Hugenius. </s> <s xml:id="echoid-s6617" xml:space="preserve">Hoc poſito, ventus A B <lb/>in velum non agit, niſi ſecundum hanc determinationem, id <lb/>eſt, cum velocitate A V. </s> <s xml:id="echoid-s6618" xml:space="preserve">Si velum C D vento A B per-<lb/>pendiculare eſſet, ventus in velum ageret velocitate A B; </s> <s xml:id="echoid-s6619" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s6620" xml:space="preserve">per conſequens ex principio quod ſtatim adſtruxi, vis <lb/>cum qua ventus in velum ageret, ſi vento eſſet perpendicula-<lb/>re, eſt ad vim venti in velum D, quod inclinatè vento ob-<lb/>tenditur, ut quadratum A B ad quadratum A V, id eſt, <lb/>ut quadratum radii ad quadratum ſinûs anguli incidentiæ.</s> <s xml:id="echoid-s6621" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6622" xml:space="preserve">Secunda applicatio inſervit ſolvendæ quæſtioni, de qua <lb/>lis eſt inter D<emph style="super">num</emph> Hugenium & </s> <s xml:id="echoid-s6623" xml:space="preserve">me, id eſt oſtendit, quod, <lb/>velo conſtituto in ſitu C D, & </s> <s xml:id="echoid-s6624" xml:space="preserve">navi in ſitu B K, vis, qua <lb/>ventus ope veli, navim ſecundum B G propellit, ſit ad vim, <lb/>qua idem ventus, ope ejuſdem veli navim propellit ſecun- <pb o="300" file="0386" n="418" rhead="VARIA CIRCA"/> dum B K, ut quadratum B G ad quadratum B K, & </s> <s xml:id="echoid-s6625" xml:space="preserve">non <lb/>quemadmodum ſuſtinet Hugenius, ut B G ad B K.</s> <s xml:id="echoid-s6626" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6627" xml:space="preserve">Quod ut pateat, concipiamus ventum in velum impin-<lb/>gere cum velocitate B G; </s> <s xml:id="echoid-s6628" xml:space="preserve">quoniam in illud impingit ſolum-<lb/>modo per motum, qui dirigitur ſecundum B G ſpectan-<lb/>dus eſt ventus, tanquam latus ex B in G velocitate B G. <lb/></s> <s xml:id="echoid-s6629" xml:space="preserve">ſed quando illa velocitate fertur ex B in G, fertur veloci-<lb/>tate B K ſecundum determinationem B K, & </s> <s xml:id="echoid-s6630" xml:space="preserve">velocitate <lb/>K G ſecundum determinationem K G; </s> <s xml:id="echoid-s6631" xml:space="preserve">quare ex iis quæ ſu-<lb/>perius a me dicta ſunt, vis qua navis pellitur ſecundum B G <lb/>eſt ad vim qua pellitur ſecundum B K, ut quadratum B G <lb/>ad quadratum B K; </s> <s xml:id="echoid-s6632" xml:space="preserve">& </s> <s xml:id="echoid-s6633" xml:space="preserve">ad vim, qua pellitur ſecundum K G, <lb/>ut quadratum B G ad quadratum K G. </s> <s xml:id="echoid-s6634" xml:space="preserve">Obſervandum au-<lb/>tem eſt, eandem vim venti in velum, id eſt, vim totalem ſe-<lb/>cundum B G, quam vim exprimit quadratum B G, divi-<lb/>viſam eſſe ſecundum B K, & </s> <s xml:id="echoid-s6635" xml:space="preserve">K G, in duas partes, quarum <lb/>ſumma vim totalem adæquat; </s> <s xml:id="echoid-s6636" xml:space="preserve">& </s> <s xml:id="echoid-s6637" xml:space="preserve">illa vis, potentia, ſeu <lb/>motus, quibus tribus una eademque res ſignificatur nomi-<lb/>nibus, nullum vel augmentum vel imminutionem accipit, <lb/>ex noſtris motum conſiderandi modis, qui motus non in ſola <lb/>corporum velocitate conſiſtit, ſed ex eorundem maſſis quo-<lb/>que conſtituitur. </s> <s xml:id="echoid-s6638" xml:space="preserve">Ergo potentia, vis, ſeu motus, eſt pro-<lb/>ductum quadrati celeritatis per maſſam. </s> <s xml:id="echoid-s6639" xml:space="preserve">Quare potentia ex <lb/>duabus aliis potentiis conflata, iis æqualis eſt, dummodo unius <lb/>determinatio determinationi alterius ſit perpendicularis, quia <lb/>eo in caſu duæ illæ potentiæ, nec quicquam addere, nec <lb/>detrahere quicquam altera alteri, poſſunt, duabus deter-<lb/>minationibus, uti diximus, nihil oppoſiti habentibus. </s> <s xml:id="echoid-s6640" xml:space="preserve">Inde <lb/>fit ut potentia ſecundum B K eadem manere poſſit, idem-<lb/>que per conſequens illius effectus, licet in infinitum augea-<lb/>tur ſeu minuatur potentia ſecundum K G. </s> <s xml:id="echoid-s6641" xml:space="preserve">Eo in caſu ſola <lb/>potentia totalis B G mutationem patietur, quia ſemper æ-<lb/>qualis erit ſummæ potentiarum, ex quibus producta fue-<lb/>rit.</s> <s xml:id="echoid-s6642" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6643" xml:space="preserve">Ex omnibus, quæ a me nunc dicta ſunt, ſequitur, na-<lb/>vem H B M, ſi pellatur ſecundum B G ope veli D C, &</s> <s xml:id="echoid-s6644" xml:space="preserve"> <pb o="301" file="0387" n="419" rhead="MECHANICAM."/> ſi velocitas, quâ movetur ad latus, ſit ad velocitatem quâ <lb/>directè progreditur, ut G K ad L K, ſequitur, inquam, <lb/>navem juxta B L proceſſuram, & </s> <s xml:id="echoid-s6645" xml:space="preserve">in L perventuram eo-<lb/>dem tempore, quo peryeniſſet in G, ſi ab omni parte aquas <lb/>ſecaret eadem facilitate, quâ ſecat has à parte proræ; </s> <s xml:id="echoid-s6646" xml:space="preserve">& </s> <s xml:id="echoid-s6647" xml:space="preserve">ſi, <lb/>quemadmodum poſuit D<emph style="super">nus</emph> Hugenius, navis fune B R in-<lb/>finite longo, & </s> <s xml:id="echoid-s6648" xml:space="preserve">ipſi B K perpendiculari adſtringere-<lb/>tur, ad tollendum omnem motum ſecundum determi-<lb/>nationem K G, ſequeretur etiam perventuram navim ad <lb/>punctum K eodem tempore quo perveniſſet ad punctum <lb/>G; </s> <s xml:id="echoid-s6649" xml:space="preserve">quod erat in quæſtione, & </s> <s xml:id="echoid-s6650" xml:space="preserve">quod erat demonſtran-<lb/>dum.</s> <s xml:id="echoid-s6651" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6652" xml:space="preserve">Si vera foret mechanices regula, quam memorat D<emph style="super">nus</emph> Huge-<lb/>nius, ſcilicet vim cum qua velum D C navim propellit ſe-<lb/>cundum B K eſſe ad vim, cum qua idem velum navem ſe-<lb/>cundum B G propellit, ut B K ad B G, non ſolum navis <lb/>non perveniret in K eodem tempore, quo perveniſſet in G, <lb/>poſitis circumſtantiis, de quibus ſupra, ſed & </s> <s xml:id="echoid-s6653" xml:space="preserve">navis aquam <lb/>æqualiter ab omni parte ſecans, & </s> <s xml:id="echoid-s6654" xml:space="preserve">a velo D C, ſecundum <lb/>B G ipſi perpendicularem propulſa, non ferretur juxta li-<lb/>neam B G: </s> <s xml:id="echoid-s6655" xml:space="preserve">nam ex eadem mechanices regula, vis, qua fer-<lb/>retur navis juxta B K ope veli, foret ad vim qua ferretur <lb/>ſecundum K G, ut B K ad K G, & </s> <s xml:id="echoid-s6656" xml:space="preserve">velocitates navis eſſent <lb/>inter ſe ut radices virium. </s> <s xml:id="echoid-s6657" xml:space="preserve">Ergo velocitates quæ ex illis viri-<lb/>bus orirentur, ſcilicet velocitas quam ſingulis momentis ac-<lb/>quiſiviſſet navis in motu juxta B K, eſſet ad velocitatem <lb/>quam iiſdem momentis haberet ex motu juxta K G, ut ra-<lb/>dix B K ad radicem K G. </s> <s xml:id="echoid-s6658" xml:space="preserve">Sed ut navis moveatur ſecun-<lb/>dum B G, quando inæquales ſunt velocitates inter ſe, qua-<lb/>les hic a nobis ponuntur, requiritur, ut ſint inter ſe ſin-<lb/>gulis momentis, ut B K ad K G, & </s> <s xml:id="echoid-s6659" xml:space="preserve">non ut eorundem radi-<lb/>ces. </s> <s xml:id="echoid-s6660" xml:space="preserve">Ergo navis non ferretur juxta B G, quod eſt abſurdum; <lb/></s> <s xml:id="echoid-s6661" xml:space="preserve">nam cum vis totalis, quæ navem impellit, ſit juxta B G, <lb/>ponendo navem ab omni parte æqualiter aquam ſecare, <lb/>non poteſt non illam lineam ſequi.</s> <s xml:id="echoid-s6662" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6663" xml:space="preserve">Ita pergit Cl. </s> <s xml:id="echoid-s6664" xml:space="preserve">Hugenius, Error à me notatus in toto <pb o="302" file="0388" n="420" rhead="VARIA CIRCA"/> fere tractatus reliquo locum habet, quo varia labefactan-<lb/>tur theoremata, quæ decætero elegantia videntur. </s> <s xml:id="echoid-s6665" xml:space="preserve">Quale eſt <lb/>inter alia hoc. </s> <s xml:id="echoid-s6666" xml:space="preserve">Dato O B A angulo veli cum vento, Carinæ ſi-<lb/>tum, quo in adverſum venti maxime progreditur navis, de-<lb/>terminari dividendo æqualiter in duas partes complementum <lb/>O B E anguli dati. </s> <s xml:id="echoid-s6667" xml:space="preserve">Unde Auctor deducit &</s> <s xml:id="echoid-s6668" xml:space="preserve">c.</s> <s xml:id="echoid-s6669" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6670" xml:space="preserve">Siquidem, quem errorem credidit D<emph style="super">nus</emph> Hugenius, errorem <lb/>non eſſe oſtenderim, intacta remanent tractatus mei theo-<lb/>remata.</s> <s xml:id="echoid-s6671" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6672" xml:space="preserve">Addit; </s> <s xml:id="echoid-s6673" xml:space="preserve">Cæterum, quin vera ſit noſtra regula non poterit <lb/>dubitare D<emph style="super">nus</emph> Renaldus; </s> <s xml:id="echoid-s6674" xml:space="preserve">cum per eam angulus gubernaculi, quo <lb/>navis omnium clerrime circumvolvitur, idem detegatur, quem <lb/>Cap. </s> <s xml:id="echoid-s6675" xml:space="preserve">7. </s> <s xml:id="echoid-s6676" xml:space="preserve">determinavit. </s> <s xml:id="echoid-s6677" xml:space="preserve">Quod certe ipſius inventum vtiliſſimum <lb/>eſt. </s> <s xml:id="echoid-s6678" xml:space="preserve">Ponendo enim p = a, id eſt ponendo venti lineam ad carinam <lb/>perpendicularem, noſtrâ regula habetur x = V {2/3} a a, quem il-<lb/>le detexit ſinum anguli carinæ aut directionis motus aquæ cum <lb/>gubernaculo, quod ſic neceſſario ſeſe habere facile patet.</s> <s xml:id="echoid-s6679" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6680" xml:space="preserve">Et hic à D<emph style="super">no</emph> Hugenio diſſentire cogor, & </s> <s xml:id="echoid-s6681" xml:space="preserve">ab ipſo meo <lb/>conſequenter tractatu, in quo error datur maximus, ſtatim <lb/>à me, detectâ prius ingenue ejuſdem cauſâ, oſtendendus. <lb/></s> <s xml:id="echoid-s6682" xml:space="preserve">Compoſueram primo libru meum, pro vero ponendo prin-<lb/>cipium falſum, à P. </s> <s xml:id="echoid-s6683" xml:space="preserve">Pardies, art. </s> <s xml:id="echoid-s6684" xml:space="preserve">118. </s> <s xml:id="echoid-s6685" xml:space="preserve">ſcientiæ virium motri-<lb/>cium prolatum; </s> <s xml:id="echoid-s6686" xml:space="preserve">præter quod nil continet de navis motu tracta-<lb/>tus illius, & </s> <s xml:id="echoid-s6687" xml:space="preserve">quod nec ipſum ulli rei applicatur; </s> <s xml:id="echoid-s6688" xml:space="preserve">imonullam <lb/>auctor viam aperit ſolvendi vel unam, quæ ſpectat theoriam <lb/>Manuariæ Nauticæ, propoſitionem. </s> <s xml:id="echoid-s6689" xml:space="preserve">Cum jam ſub prælo ſu-<lb/>darent ultimæ libri I. </s> <s xml:id="echoid-s6690" xml:space="preserve">paginæ; </s> <s xml:id="echoid-s6691" xml:space="preserve">principii memorati falſita-<lb/>tem animadverti, & </s> <s xml:id="echoid-s6692" xml:space="preserve">quoniam per omnes tractatûs propoſi-<lb/>tiones diſperſum, falſas reddebat reſolutiones omnes, totam <lb/>editionem ſuppreſſi. </s> <s xml:id="echoid-s6693" xml:space="preserve">Solidiori poſtea fundamento nixus eaſ-<lb/>dem denuo ſolvi, præloque ſubjeci, hæ ſunt quæ in lucem <lb/>ſunteditæ. </s> <s xml:id="echoid-s6694" xml:space="preserve">Sed aliis diſtentus negotiis, vocem vis, loco velo-<lb/>citatis, quæ erat ſubſtituenda, in demonſtratione Art. </s> <s xml:id="echoid-s6695" xml:space="preserve">5. </s> <s xml:id="echoid-s6696" xml:space="preserve">Cap. </s> <s xml:id="echoid-s6697" xml:space="preserve"><lb/>2. </s> <s xml:id="echoid-s6698" xml:space="preserve">incogitanter reliqui; </s> <s xml:id="echoid-s6699" xml:space="preserve">quod D<emph style="super">nus</emph> Hugenius non attendit. </s> <s xml:id="echoid-s6700" xml:space="preserve"><lb/>Fateor me, in Art. </s> <s xml:id="echoid-s6701" xml:space="preserve">6. </s> <s xml:id="echoid-s6702" xml:space="preserve">Cap. </s> <s xml:id="echoid-s6703" xml:space="preserve">1. </s> <s xml:id="echoid-s6704" xml:space="preserve">promiſcue vocibus virium & </s> <s xml:id="echoid-s6705" xml:space="preserve"><lb/>velocitatum uti, ſed unius tantum corporis in vacuo motum <pb o="303" file="0389" n="421" rhead="MECHANICAM."/> ibi conſidero, & </s> <s xml:id="echoid-s6706" xml:space="preserve">in eo caſu velocitas & </s> <s xml:id="echoid-s6707" xml:space="preserve">vis eandem ſem-<lb/>per relationem habent. </s> <s xml:id="echoid-s6708" xml:space="preserve">Capitis autem 7. </s> <s xml:id="echoid-s6709" xml:space="preserve">errorem, poſt al-<lb/>teram tantum libri mei editionem animadverti, neque tunc <lb/>illum corrigere per negotia mihi licuit. </s> <s xml:id="echoid-s6710" xml:space="preserve">Falſitatem vero ita <lb/>demonftro.</s> <s xml:id="echoid-s6711" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6712" xml:space="preserve">Centro B deſcribatur circulus A D R E C, & </s> <s xml:id="echoid-s6713" xml:space="preserve">repræſen-<lb/> <anchor type="note" xlink:label="note-0389-01a" xlink:href="note-0389-01"/> tet linea A B carinam navis, B R vero carinam productam. <lb/></s> <s xml:id="echoid-s6714" xml:space="preserve">Supra B R, tanquam diametrum deſcribatur ſemicirculus <lb/>B G R, & </s> <s xml:id="echoid-s6715" xml:space="preserve">ſimiliter ſupra A B ſemicirculus A V B. </s> <s xml:id="echoid-s6716" xml:space="preserve">Sit B D <lb/>certus gubernaculi ſitus, & </s> <s xml:id="echoid-s6717" xml:space="preserve">B C gubernaculum productum; </s> <s xml:id="echoid-s6718" xml:space="preserve"><lb/>B E perpendicularis ſupra A B; </s> <s xml:id="echoid-s6719" xml:space="preserve">B G & </s> <s xml:id="echoid-s6720" xml:space="preserve">A V perpendicu-<lb/>lares ſupra D C; </s> <s xml:id="echoid-s6721" xml:space="preserve">& </s> <s xml:id="echoid-s6722" xml:space="preserve">G H perpendicularis ad B E. </s> <s xml:id="echoid-s6723" xml:space="preserve">Po-<lb/>nendo alium gubernaculi ſitum, qualis eſt B d producta in <lb/>c; </s> <s xml:id="echoid-s6724" xml:space="preserve">ducantur in iisdem circumſtantiis lineæ B g, A u, & </s> <s xml:id="echoid-s6725" xml:space="preserve"><lb/>g h. </s> <s xml:id="echoid-s6726" xml:space="preserve">Si navis antrorſum moveatur ſecundum lineam B A, <lb/>anguli A B C, & </s> <s xml:id="echoid-s6727" xml:space="preserve">A B c æquales angulis G B E, & </s> <s xml:id="echoid-s6728" xml:space="preserve">g B E, <lb/>ſunt anguli incidentiæ aquæ in gubernaculum. </s> <s xml:id="echoid-s6729" xml:space="preserve">Unde ſequi-<lb/>tur, aquam, poſito gubernaculo in ſitu B D, impingi ſe-<lb/>cundum determinationem & </s> <s xml:id="echoid-s6730" xml:space="preserve">cum velocitate A V, & </s> <s xml:id="echoid-s6731" xml:space="preserve">per con-<lb/>ſequens cum vi, quam exprimit quadratum ipſius A V. </s> <s xml:id="echoid-s6732" xml:space="preserve">Et <lb/>quoniam velocitas navis ſeſe habet tantum ut radix ipſius vis, <lb/>propter aquæ reſiſtentiam, propellitur navis ſecundum deter-<lb/>minationem B G, velocitate quæ exprimitur per B G, quia <lb/>B G eſt æqualis & </s> <s xml:id="echoid-s6733" xml:space="preserve">parallela ipſi A V. </s> <s xml:id="echoid-s6734" xml:space="preserve">Sed quando propel-<lb/>litur navis ſecundum B G velocitate B G, propellitur ſecun-<lb/>dum B E velocitate B H. </s> <s xml:id="echoid-s6735" xml:space="preserve">Si gubernaculum foret in alio ſi-<lb/>tu B d, iisdem ratiociniis probaretur, navem pellendam fo-<lb/>re ſecundum B E velocitate B h. </s> <s xml:id="echoid-s6736" xml:space="preserve">Sed quando majori velo-<lb/>citate navis pellitur ſecundum B E, citius etiam convertitur. </s> <s xml:id="echoid-s6737" xml:space="preserve"><lb/>Quare ſi B G, quæ perpendicularis eſt ad gubernaculi ſitum, <lb/>ſecet ſemicirculum B G R bifariam, id eſt, ſi angulus G B E, <lb/>æqualis angulo incidentiæ A B C, ſit 45. </s> <s xml:id="echoid-s6738" xml:space="preserve">graduum, G H <lb/>perpendicularis ipſi B E, erit tangens ſemicirculi. </s> <s xml:id="echoid-s6739" xml:space="preserve">Ergo G H <lb/>quæ exprimit celeritatem, quâ navis pellitur ſecundum B E, <lb/>eſt omnium maxima. </s> <s xml:id="echoid-s6740" xml:space="preserve">Nam ſi gubernaculum conſtituatur in <lb/>alio ſitu, ut in B d, tunc B g ipſi perpendicularis, ſecabit <pb o="304" file="0390" n="422" rhead="VARIA CIRCA"/> ſemicirculum in g, unde ſi ducatur perpendicularis g h, erit <lb/>illa propior puncto B, quam ipſa G h, & </s> <s xml:id="echoid-s6741" xml:space="preserve">navis pelletur <lb/>ſecundum B E velocitate B h, quæ minor erit quam B H. <lb/></s> <s xml:id="echoid-s6742" xml:space="preserve">Quare, ut navis citiſſime convertatur, neceſſe eſt ut vectis <lb/>gubernaculi B C cum carinâ navis faciat angulum 45. </s> <s xml:id="echoid-s6743" xml:space="preserve">gra-<lb/>duum, non autem, ſicut Cap. </s> <s xml:id="echoid-s6744" xml:space="preserve">VII. </s> <s xml:id="echoid-s6745" xml:space="preserve">Theoriæ dicitur, angu-<lb/>lum graduum circiter 55.</s> <s xml:id="echoid-s6746" xml:space="preserve"/> </p> <div xml:id="echoid-div480" type="float" level="2" n="1"> <note position="right" xlink:label="note-0389-01" xlink:href="note-0389-01a" xml:space="preserve">TAB. XXXIII. <lb/>Fig. 7.</note> </div> <p style="it"> <s xml:id="echoid-s6747" xml:space="preserve">Concludit Cl. </s> <s xml:id="echoid-s6748" xml:space="preserve">Hugenius hiſce verbis, licet theoria hæc <lb/>poſt indicatam à me correctionem difficilior evadat, quam in <lb/>tractatu D<emph style="super">ni</emph> Renaldi, percipio mhilominus regulam detegi <lb/>poſſe, qua & </s> <s xml:id="echoid-s6749" xml:space="preserve">navis & </s> <s xml:id="echoid-s6750" xml:space="preserve">veli ſitus determinarentur, ut in ad-<lb/>verſum venti omnium maxime progrediatur, ſed computatio <lb/>nimium longa foret quare hanc nunc non aggredior.</s> <s xml:id="echoid-s6751" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6752" xml:space="preserve">In meo certe tractatu facillima foret determinatio ſitûs <lb/> <anchor type="note" xlink:label="note-0390-01a" xlink:href="note-0390-01"/> navis velique, non tantum ſi in adverſum venti omnium <lb/>maxime ſit progrediendum, ſed & </s> <s xml:id="echoid-s6753" xml:space="preserve">conveniens cuicun-<lb/>que viæ inſtituendæ, ſi deviatio navis à calculo ſeclu-<lb/>deretur; </s> <s xml:id="echoid-s6754" xml:space="preserve">quod ut pateat, ſit linea venti A B, & </s> <s xml:id="echoid-s6755" xml:space="preserve">ſit <lb/>data via B K, cum vento conſtituens angulum quencunque <lb/>A B K. </s> <s xml:id="echoid-s6756" xml:space="preserve">Ut inveniatur, quo veli ſitu, navis illâ viâ citiſſime <lb/>poſſit progredi, ſi ſcilicet nulla ſit deviatio; </s> <s xml:id="echoid-s6757" xml:space="preserve">ſumatur B R <lb/>tanquam diameter, & </s> <s xml:id="echoid-s6758" xml:space="preserve">centro M deſcribatur ſemicirculus B G V; <lb/></s> <s xml:id="echoid-s6759" xml:space="preserve">ab eodem puncto M ducatur M G viæ B K parallela; </s> <s xml:id="echoid-s6760" xml:space="preserve">a pun-<lb/>cto B ad punctum G ducatur B G, & </s> <s xml:id="echoid-s6761" xml:space="preserve">tandem ducatur D B C <lb/>perpendicularis ipſi B G. </s> <s xml:id="echoid-s6762" xml:space="preserve">Dico D C repræſentare veli ſitum, <lb/>quo per viam B K navis omnium citiſſime poſſit progredi. </s> <s xml:id="echoid-s6763" xml:space="preserve"><lb/>Ad id probandum, ducatur G K perpendicularis ipſi B K. </s> <s xml:id="echoid-s6764" xml:space="preserve"><lb/>Ex omnibus quæ ſuperius dicta ſunt, patet ventum A B <lb/>propellere navem ope veli D C, ſecundum B G velocitate <lb/>B G, eodem tempore quo eandem propelleret ſecundum B K <lb/>velocitate B K. </s> <s xml:id="echoid-s6765" xml:space="preserve">Et quoniam G K eſt perpendicularis ipſi B K, <lb/>erit itidem perpendicularis ipſi M G. </s> <s xml:id="echoid-s6766" xml:space="preserve">Ergo G K eſt tangens <lb/>ſemicirculi. </s> <s xml:id="echoid-s6767" xml:space="preserve">Ergo quotcunque ducantur perpendiculares ex <lb/>aliis circumferentiæ ſemicirculi punctis, ſupra B K, cadent <lb/>omnes inter B & </s> <s xml:id="echoid-s6768" xml:space="preserve">K. </s> <s xml:id="echoid-s6769" xml:space="preserve">Ergo navis celeritas in veli ſitu D C <lb/>erit omnium maxima. </s> <s xml:id="echoid-s6770" xml:space="preserve">Addo, velum D C ſecare bifariam <pb o="305" file="0391" n="423" rhead="MECHANICAM."/> angulum A B K, qui eſt angulus venti cum viâ. </s> <s xml:id="echoid-s6771" xml:space="preserve">Ad quod <lb/>demonſtrandum ducatur M N perpendicularis ipſi B G. </s> <s xml:id="echoid-s6772" xml:space="preserve">ſe-<lb/>cat M N angulum B M G angulo A B K æqualem, <lb/>bifariam; </s> <s xml:id="echoid-s6773" xml:space="preserve">& </s> <s xml:id="echoid-s6774" xml:space="preserve">quoniam M N eſt parallela ipſi D C, angulus <lb/>B M N dimidius anguli B M G, qui æqualis eſt angulo <lb/>A B K, æqualis eſt angulo A B C, qui ipſe per conſe-<lb/>quens æqualis eſt dimidio angulo A B K. </s> <s xml:id="echoid-s6775" xml:space="preserve">Unde ſequitur, <lb/>veli ſitum ſemper bifariam debere ſecare angulum venti & </s> <s xml:id="echoid-s6776" xml:space="preserve"><lb/>viæ, & </s> <s xml:id="echoid-s6777" xml:space="preserve">cum agitur de progrediendo in adverſum venti, de-<lb/>bere illum cum velo conſtituere angulum 30. </s> <s xml:id="echoid-s6778" xml:space="preserve">graduum, pro-<lb/>ramque itidem cum velo angulum 30. </s> <s xml:id="echoid-s6779" xml:space="preserve">grad.</s> <s xml:id="echoid-s6780" xml:space="preserve">; quia, ut de-<lb/>monſtratum fuit in Theoria Manuariæ Nauticæ, quocunque <lb/>in ſitu reſpectu venti diſponatur velum, neceſſe eſt ut ejus <lb/>complementum prora bifariam ſecet. </s> <s xml:id="echoid-s6781" xml:space="preserve">Et exinde apparet, quo-<lb/>cunque in ſitu conſtituatur prora, quæ ſemper cum via con-<lb/>gruit; </s> <s xml:id="echoid-s6782" xml:space="preserve">quia ponitur navem non deviare, neceſſe eſſe ut angulum <lb/>quem cum prora, ventus conſtituit, velum bifariam ſecet. <lb/></s> <s xml:id="echoid-s6783" xml:space="preserve">Unde ſequitur angulum rectum debere à prora & </s> <s xml:id="echoid-s6784" xml:space="preserve">velo in <lb/>tres partes æquales ſecari, id eſt, ventum cum velo 30. </s> <s xml:id="echoid-s6785" xml:space="preserve"><lb/>gradus conſtituere, proram itidem cum velo 30. </s> <s xml:id="echoid-s6786" xml:space="preserve">gradus, <lb/>& </s> <s xml:id="echoid-s6787" xml:space="preserve">cum vento per conſequens 60.</s> <s xml:id="echoid-s6788" xml:space="preserve"/> </p> <div xml:id="echoid-div481" type="float" level="2" n="2"> <note position="left" xlink:label="note-0390-01" xlink:href="note-0390-01a" xml:space="preserve">TAB.XXXIII. <lb/>Fig. 8.</note> </div> </div> <div xml:id="echoid-div483" type="section" level="1" n="198"> <head xml:id="echoid-head257" xml:space="preserve">XII.</head> <head xml:id="echoid-head258" style="it" xml:space="preserve">Exceptio D<emph style="super">ni</emph> Hugenii ad Reſponſum <lb/>D<emph style="super">ni</emph> Renaldi.</head> <p> <s xml:id="echoid-s6789" xml:space="preserve">Mihi evidens videbatur illud quod in obſervationibus de <lb/>errore primario, qui reperitur in tractatu de manuaria nauti-<lb/>ca D<emph style="super">ni</emph> Renaldi, propoſueram, & </s> <s xml:id="echoid-s6790" xml:space="preserve">cenſuerunt viri in re Ma-<lb/>thematicâ verſatiſſimi, argumentum meum omni exceptione <lb/>majus eſſe; </s> <s xml:id="echoid-s6791" xml:space="preserve">idcirco neque credideram, velle ipſum quidpiam <lb/>reſpondere, quo ſuam confirmaret Theoriam; </s> <s xml:id="echoid-s6792" xml:space="preserve">interim ex iis, <lb/>quæ in lucem edidit, apparet eum minime de ſuo errore mecum <lb/>ſentire; </s> <s xml:id="echoid-s6793" xml:space="preserve">& </s> <s xml:id="echoid-s6794" xml:space="preserve">quandoquidem rationibus utitur, e quibus haud <lb/>ita facile ſe expedire poſſent illi, qui non ſatis hæcce o-<lb/>mnia excuſſerunt, obſtrictum me credidi ad demonſtrandum <pb o="306" file="0392" n="424" rhead="VARIA CIRCA"/> majori evidentiâ, quam antea, ejus Theoriam ſuſtineri non <lb/>poſſe, niſi principia mechanices, jam dudum ſtabilita, quo-<lb/>rumque veritatem negare nec auderet, nec vellet, evertan-<lb/>tur.</s> <s xml:id="echoid-s6795" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6796" xml:space="preserve">Ne inutiliter protraham controverſiam noſtram reſponden-<lb/>do argumentis quæ D<emph style="super">us</emph> Renaldus mihi objicit, oſtendam tan-<lb/>tum illum, ut jam obſervaram, erraſſe in propoſitione, qua <lb/>nititur tota ejus Theoria, deinde paucis indicabo, quid <lb/>huic errori anſam dederit.</s> <s xml:id="echoid-s6797" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6798" xml:space="preserve">Ut oſtendam, quinam ſit quæſtionis ſtatus, hic maximam <lb/> <anchor type="note" xlink:label="note-0392-01a" xlink:href="note-0392-01"/> partem eorum, quæ noſtras figuras ſpectant, repeto, <lb/>ſcil; </s> <s xml:id="echoid-s6799" xml:space="preserve">H M eſt navis carina; </s> <s xml:id="echoid-s6800" xml:space="preserve">C D velum; </s> <s xml:id="echoid-s6801" xml:space="preserve">A B linea venti <lb/>velum inflantis; </s> <s xml:id="echoid-s6802" xml:space="preserve">B G eſt perpendicularis ad C D; </s> <s xml:id="echoid-s6803" xml:space="preserve">G K per-<lb/>pendicularis ad B K, quæ indicat carinam prolongatam; <lb/></s> <s xml:id="echoid-s6804" xml:space="preserve">continuo G B ad Z, & </s> <s xml:id="echoid-s6805" xml:space="preserve">M H ad X.</s> <s xml:id="echoid-s6806" xml:space="preserve"/> </p> <div xml:id="echoid-div483" type="float" level="2" n="1"> <note position="left" xlink:label="note-0392-01" xlink:href="note-0392-01a" xml:space="preserve">TAB. XXXIII. <lb/>Fig. 9.</note> </div> <p> <s xml:id="echoid-s6807" xml:space="preserve">D<emph style="super">us</emph> Renaldus in Theoria cap. </s> <s xml:id="echoid-s6808" xml:space="preserve">II. </s> <s xml:id="echoid-s6809" xml:space="preserve">Art. </s> <s xml:id="echoid-s6810" xml:space="preserve">1. </s> <s xml:id="echoid-s6811" xml:space="preserve">dicit, ſi pona-<lb/>tur navem undique eâdem, qua puppis, facilitate aquam <lb/>findere, illam ita a vento propelli, ut progreſſura ſit per re-<lb/>ctam B G, quod eſt verum; </s> <s xml:id="echoid-s6812" xml:space="preserve">ſed ſi juxta ſitum carinæ navis <lb/>tantum progredi queat in recta B K, vel ſi funis B R per-<lb/>pendicularis ad B K, & </s> <s xml:id="echoid-s6813" xml:space="preserve">cujus longitudo cenſetur infinita, <lb/>eam cogat inſiſtere in via B K, contendit, velo & </s> <s xml:id="echoid-s6814" xml:space="preserve">vento <lb/>iiſdem ac antea manentibus, navem percurſuram ſpatium B K, <lb/>eodem tempore quo percurriſſet B G; </s> <s xml:id="echoid-s6815" xml:space="preserve">ego autem defendo, <lb/>quod percurreret ſpatium B S, medium proportionale in-<lb/>ter B K & </s> <s xml:id="echoid-s6816" xml:space="preserve">B G; </s> <s xml:id="echoid-s6817" xml:space="preserve">Et hic eſt controverſiæ noſtræ ſtatus.</s> <s xml:id="echoid-s6818" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6819" xml:space="preserve">In probatione, quam affert in reſponſione ſuâ, loco venti <lb/>A B oblique cadentis in velum C D, ſubſtituit ventum Z B, <lb/>qui in id perpendiculariter agit; </s> <s xml:id="echoid-s6820" xml:space="preserve">quod poteſt, & </s> <s xml:id="echoid-s6821" xml:space="preserve">minime <lb/>controverſiam mutat, quum certum ſit, quod quocunque <lb/>ſenſu ventus in velum C D cadat, conetur navem propelle-<lb/>re per viam B G perpendicularem ad C D; </s> <s xml:id="echoid-s6822" xml:space="preserve">neque ullius <lb/>uſus ſunt, quas habet D. </s> <s xml:id="echoid-s6823" xml:space="preserve">Renaldus, diverſas conſiderationes <lb/>de motu venti. </s> <s xml:id="echoid-s6824" xml:space="preserve">Ratiocinando dein invenit, quod vis, quâ <lb/>navis pellitur a vento ſecundum B G ope veli, ſit ad vim, <lb/>qua pellitur ab eodem vento ope ejuſdem veli ſecundum B K, <pb o="307" file="0393" n="425" rhead="MECHANICAM."/> ut quadratum B G ad quadratum B K, non vero ut B G <lb/>ad B K, quod ego contendo, & </s> <s xml:id="echoid-s6825" xml:space="preserve">inde tota res pendet.</s> <s xml:id="echoid-s6826" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6827" xml:space="preserve">Ut determinemus cujus ſententia veraſit, concipiamus pla-<lb/>num, in quo noſtra figura exiſtit, ita ad Horizontem erectum, <lb/>ut linea B G ad eundem ſit perpendicularis, & </s> <s xml:id="echoid-s6828" xml:space="preserve">ſit R B X funis <lb/>affixus in R, ad quem in B alligatum & </s> <s xml:id="echoid-s6829" xml:space="preserve">ſuſpenſum ſit pon-<lb/>dus Q: </s> <s xml:id="echoid-s6830" xml:space="preserve">Ponamus ulterius portionem B X perpendicularem <lb/>ad R B a manu retineri in X; </s> <s xml:id="echoid-s6831" xml:space="preserve">clarum eſt, id exacte re-<lb/>præſentare caſum navis, de quo eſt quæſtio; </s> <s xml:id="echoid-s6832" xml:space="preserve">nam loco ven-<lb/>ti, qui impingendo in velum C D propellat navim ſecun-<lb/>dum B G, hic habemus pondus Q quod trahit punctum B <lb/>juxta B G; </s> <s xml:id="echoid-s6833" xml:space="preserve">& </s> <s xml:id="echoid-s6834" xml:space="preserve">funis B R, qui cenſetur infinitæ longitudinis, <lb/>& </s> <s xml:id="echoid-s6835" xml:space="preserve">impedit, quo minus navis aliam plagam versùs, quam <lb/>B K poſſit progredi, hic eundem edit effectum reſpectu no-<lb/>di B.</s> <s xml:id="echoid-s6836" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6837" xml:space="preserve">Utergo vis, qua ventus propellit navem ſecundum B G, eſt <lb/>ad vim, qua hanc propellit ſecundum B K, ita eſt pondus Q <lb/>ad gravitatem, quæ agit in manum quæ in X cohibet, quò <lb/>minus nodus B moveatur verſus B K; </s> <s xml:id="echoid-s6838" xml:space="preserve">illa enim gravitas æ-<lb/>quipollet vi, qua nodus ſecundum B K trahitur. </s> <s xml:id="echoid-s6839" xml:space="preserve">Cum au-<lb/>tem G K parallela ſit ipſi B R, certum eſt ex notiſſimis Me-<lb/>chanicæ regulis, quod pondus Q ſit ad id quod retinet chor-<lb/>dam B X, vel ad gravitatem quæ in X agit in manum ut <lb/>B G ad B K. </s> <s xml:id="echoid-s6840" xml:space="preserve">Vis ergo, qua navim propellit ventus ſecun-<lb/>dum B G, eſt ad vim, qua propulſa eſt verſus B K, ut <lb/>B G ad B K non vero, ut illarum linearum quadrata, ut <lb/>D<emph style="super">nus</emph> Renaldus contendit.</s> <s xml:id="echoid-s6841" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6842" xml:space="preserve">Ponamus porro, navem H M, quæ undas eadem undi-<lb/>que facilitate ſecat, propelli per ventum Z B vel A B <lb/>(perinde enim eſt) quo naviget certo tempore per B G, <lb/>velo exiſtente in C D, & </s> <s xml:id="echoid-s6843" xml:space="preserve">determinandum eſſe, quan-<lb/>tum progreſſura ſit ſecundum B K æquali tempore eodem <lb/>vento, eodemque veli ſitu. </s> <s xml:id="echoid-s6844" xml:space="preserve">Cum celeritates navis in dua-<lb/>bus illis viis tales debeant eſſe, ut reſiſtentiæ, quas in-<lb/>ferit ipſi aqua, ſint ut vires, quibus propellitur (nam in eo <lb/>caſu ſolo motu æquabili progreditur), & </s> <s xml:id="echoid-s6845" xml:space="preserve">cum reſiſtentiæ <pb o="308" file="0394" n="426" rhead="VARIA CIRCA MECHANICAM."/> ſint ut quadrata celeritatum, ideo requiritur ut quadrata ce-<lb/>leritatum ſint ut vires, id eſt ut G B ad B K; </s> <s xml:id="echoid-s6846" xml:space="preserve">& </s> <s xml:id="echoid-s6847" xml:space="preserve">conſequen-<lb/>ter ut velocitates ſint ut G B ad B S; </s> <s xml:id="echoid-s6848" xml:space="preserve">quandoquidem <lb/>quadrata G B & </s> <s xml:id="echoid-s6849" xml:space="preserve">B S ſunt ut lineæ G B, B K per conſtructio-<lb/>nem. </s> <s xml:id="echoid-s6850" xml:space="preserve">Probavi ergo ex ordinariis Mechanices principiis, id, <lb/>quod in obſervatione mea propoſueram.</s> <s xml:id="echoid-s6851" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6852" xml:space="preserve">Superfluum foret alia argumenta D<emph style="super">ni</emph> Renaldi examinare, <lb/>quibus vult hanc eandem, quam refutavi propoſitionem, con-<lb/>firmare; </s> <s xml:id="echoid-s6853" xml:space="preserve">ſolummodo indicabo originem erroris, qui in iis oc-<lb/>currit, præſertim naſci ex eo, quod in Articulo 7. </s> <s xml:id="echoid-s6854" xml:space="preserve">Cap. </s> <s xml:id="echoid-s6855" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s6856" xml:space="preserve">Theoriæ ſuæ concludat, vires relativas materiæ fluidæ ad <lb/>ſuperficies diverſimode inclinatas eſſe inter ſe, ut quadrata ſi-<lb/>nuum angulorum incidentiæ; </s> <s xml:id="echoid-s6857" xml:space="preserve">debuiſſet addere ad ſuperficies æ-<lb/>quales diverſimode inclinatas; </s> <s xml:id="echoid-s6858" xml:space="preserve">cujus vocis æquales paulo ante in <lb/>eodem Articulo pag. </s> <s xml:id="echoid-s6859" xml:space="preserve">7. </s> <s xml:id="echoid-s6860" xml:space="preserve">quoque non meminerat; </s> <s xml:id="echoid-s6861" xml:space="preserve">quod ſi ſup-<lb/>pleatur, tum demonſtratio optime congruit cum concluſione <lb/>cumque veris principiis P: </s> <s xml:id="echoid-s6862" xml:space="preserve">Pardies in Art. </s> <s xml:id="echoid-s6863" xml:space="preserve">118. </s> <s xml:id="echoid-s6864" xml:space="preserve">Tracta-<lb/>tus de Viribus moventibus. </s> <s xml:id="echoid-s6865" xml:space="preserve">Pater hic tantum in eodem illo <lb/>Articulo deceptus fuit, quod ignorarit vel ſaltem non recor-<lb/>datus fuerit, reſiſtentias aquæ in corpus eſſe ut quadrata ve-<lb/>locitatum ejus corporis; </s> <s xml:id="echoid-s6866" xml:space="preserve">ideo enim p. </s> <s xml:id="echoid-s6867" xml:space="preserve">225. </s> <s xml:id="echoid-s6868" xml:space="preserve">facit a f ad a u in <lb/>duplicatâ ratione b o ad m p cum ſimpliciter facere debuerit <lb/>a f ad a u, ut b o ad m p.</s> <s xml:id="echoid-s6869" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6870" xml:space="preserve">Quod attinet ad utiliſſimum Gubernaculi ſitum, D<emph style="super">nus</emph> Re-<lb/>naldus ſe ipſum nullo jure culpat, & </s> <s xml:id="echoid-s6871" xml:space="preserve">dum quæ primo deter-<lb/>minaverat corrigere conatur, male ratiocinatur; </s> <s xml:id="echoid-s6872" xml:space="preserve">in reſponſio-<lb/>ne p. </s> <s xml:id="echoid-s6873" xml:space="preserve">303, nam tantum determinandum eſt, in quonam Gu-<lb/>bernaculi ſitu aqua id propulſura ſit maximâ vi, juxta per-<lb/>pendicularem ad carinam; </s> <s xml:id="echoid-s6874" xml:space="preserve">unde neceſſario ſequetur maxima <lb/>puppis velocitas juxta illam perpendicularem. </s> <s xml:id="echoid-s6875" xml:space="preserve">Errat etiam, <lb/>quum vult, p. </s> <s xml:id="echoid-s6876" xml:space="preserve">25 Theoriæ ſuæ de Manuaria Nautica, legi <lb/>velocitas loco vis.</s> <s xml:id="echoid-s6877" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6878" xml:space="preserve">Quod ſupereſt, obſervo, totam hanc Theoriam, ut edi-<lb/>derat, veram fore ſi reſiſtentiæ aquæ eſſent ut velocitates <lb/>navis, ſunt autem ut quadrata illarum velocitatum.</s> <s xml:id="echoid-s6879" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div485" type="section" level="1" n="199"> <head xml:id="echoid-head259" xml:space="preserve">FINIS.</head> <pb file="0395" n="427"/> <pb file="0395a" n="428"/> <figure> <caption xml:id="echoid-caption135" style="it" xml:space="preserve">Pag. 308.<lb/>TAB.XXXIII.<lb/>Fig. 1.</caption> <variables xml:id="echoid-variables137" xml:space="preserve">P F Q K H L R G B E C N O 3 A 2</variables> </figure> <figure> <caption xml:id="echoid-caption136" style="it" xml:space="preserve">Fig. 8.</caption> <variables xml:id="echoid-variables138" xml:space="preserve">R G M K N D B V C A</variables> </figure> <figure> <caption xml:id="echoid-caption137" style="it" xml:space="preserve">Fig. 7.</caption> <variables xml:id="echoid-variables139" xml:space="preserve">R d D G g B h H E V C u A c</variables> </figure> <figure> <caption xml:id="echoid-caption138" style="it" xml:space="preserve">Fig. 2.</caption> <variables xml:id="echoid-variables140" xml:space="preserve">B F G C H A K D E</variables> </figure> <figure> <caption xml:id="echoid-caption139" style="it" xml:space="preserve">Fig. 4.</caption> <variables xml:id="echoid-variables141" xml:space="preserve">A B G F E C D</variables> </figure> <figure> <caption xml:id="echoid-caption140" style="it" xml:space="preserve">Fig. 6.</caption> <variables xml:id="echoid-variables142" xml:space="preserve">T G D H B E M L N C K I S P F V R Q O A</variables> </figure> <figure> <caption xml:id="echoid-caption141" style="it" xml:space="preserve">Fig. 3.</caption> <variables xml:id="echoid-variables143" xml:space="preserve">A E G B D F C</variables> </figure> <figure> <caption xml:id="echoid-caption142" style="it" xml:space="preserve">Fig. 5.</caption> <variables xml:id="echoid-variables144" xml:space="preserve">N K F E C B A H L V W R G</variables> </figure> <figure> <caption xml:id="echoid-caption143" style="it" xml:space="preserve">Fig. 9.</caption> <variables xml:id="echoid-variables145" xml:space="preserve">Z R A X H C B D M K S Q G</variables> </figure> <pb file="0396" n="429"/> <pb file="0397" n="430"/> <pb file="0398" n="431"/> <pb file="0399" n="432"/> <pb file="0400" n="433"/> </div></text> </echo>