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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 12:21:30 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd"> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Marci von Kronland, Johannes Marcus </author> <title>De proportione motus figurarum rectilinearum et circuli quadratura ex motu</title> <date>1648</date> <place>Prague</place> <translator/> <lang>la</lang> <cvs_file>marci_figur_063_la_1648.xml</cvs_file> <cvs_version/> <locator>063.xml</locator> </info> <text> <front><section> <pb xlink:href="063/01/001.jpg"/> <figure id="id.063.01.001.1.jpg" xlink:href="063/01/001/1.jpg"/> <p type="caption"> <s>DE <lb/>PROPORTIONE <lb/>MOTVS <lb/>FIGVRARVM RECTI <lb/>LINEARVM <lb/>ET <lb/>CIRCVLI QVADRATVRA EX <lb/>MOTV <lb/>Authore <lb/>Ioanne Marco Marci Medicinæ <lb/>Doctore et Profe&longs;&longs;ore Primario <lb/>S·C·Mtis· Medico Cubiculario <lb/>et in Reg. Boh<gap/> Phy&longs;ico <lb/>Seniore. <lb/>PRAGÆ <lb/><emph type="italics"/>Ano. 1648.<emph.end type="italics"/></s></p> <pb xlink:href="063/01/002.jpg"/><pb xlink:href="063/01/003.jpg"/> <p type="main"> <s><emph type="center"/>SERENISSIMO <lb/>PRINCIPI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>FERDI­<lb/>NANDO <lb/>IV. <lb/>HVNGARIÆ <lb/>ET BOHEMIAE <lb/>REGI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>ARCHIDVCI <lb/>AVSTRIAE.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>DOMINO MEO CLEMENTISSIMO.<emph.end type="center"/></s></p> <pb xlink:href="063/01/004.jpg"/> <p type="main"> <s><emph type="center"/>SER ENISSIME REX &c.<emph.end type="center"/></s></p> <p type="main"> <s>SOpitis eram &longs;en&longs;ibus; uti contingit his, <lb/>qui &longs;omno premuntur: cùm ecce tibi <foreign lang="greek"><gap/>an ao/zi­<lb/>ston</foreign>! cui nulla certa &longs;pecies, omnia tamen ine&longs;&longs;e, <lb/>ip&longs;um verò cubo inniti videbatur. </s><s>Cui ego: Quis­<lb/>es? Tuus, inquit, Motus: ad&longs;um, ut tibi nun cius <lb/>&longs;im, ad novum Regem, annum au&longs;picaturus no­<lb/>vum. </s><s>Ego verò &longs;uccenfens: ô ignaui&longs;&longs;ime, inquam, <lb/>adeone tui generis es oblitus? quem pridem in Hungariam de&longs;tinaram; <lb/>ut inter applau&longs;us tu <expan abbr="quoq;">quoque</expan> plau&longs;um ferres. </s><s>Siue tardus, ait ille, &longs;iue ve­<lb/>lox &longs;im, non degenero à meis natalibus. </s><s><expan abbr="Namq;">Namque</expan> ira&longs;cor his nouis Zeno­<lb/>nibus, qui me ignauis morulis concidunt. </s><s>Quòd verò nunc tardiùs ad­<lb/>&longs;um; quàm forta&longs;&longs;e velles, tibi, non mihi imputa: qui me de circulo qua­<lb/>dratum feci&longs;ti. </s><s>Quanquam, &longs;i mihi au&longs;cultes, in lucro reponas hanc <lb/>meam tarditatem: quæ non <expan abbr="ab&longs;q;">ab&longs;que</expan> nutu accidit illius Genij, qui Symbo­<lb/>lo Regio in te, <expan abbr="libróq;">libróque</expan> tuo prælu&longs;it. </s><s>E&longs;t enim numerus my&longs;ticus huius <lb/>Anni: quòd 1600 cubos efficiant primi paris 200. <expan abbr="atq;">atque</expan> hi alios cubos <lb/>&longs;ecundos 25. qui numerus e&longs;t quadratus &longs;ecundi imparis. </s><s>At veeò nu­<lb/>merus annorum 48 &longs;ex cubos includit primi paris. </s><s>Anni <expan abbr="demũ">demum</expan> 9 elap&longs;i, <lb/>ex quo adMAGNVM CÆSAREM nuncius fui, <expan abbr="quadratũ">quadratum</expan> ab&longs;oluunt pri­<lb/>mi impatis. </s><s>Quò nimirum &longs;tabilitatem præ&longs;agiant futuri regni: in quo <lb/>&longs;ubPRIMO AVGVSTI NOMINIS QVADRATO Sabbatha orbis aget. </s><lb/><s>Quin <expan abbr="ip&longs;ũ">ip&longs;um</expan> nomen au&longs;picatum FERDINANDVS QVARTVS <lb/>hoc my&longs;terio <expan abbr="numerorũ">numerorum</expan> e&longs;t fœcundum. </s><s>In&longs;unt enim 1016 Et 1000 qui­<lb/>dem cubos efficiunt &longs;ecundi imparis octo: cuius duplum dat numerum <lb/>reliquum 16 & ip&longs;um quadratum &longs;ecundi paris: con&longs;tantem verò du­<lb/>obus cubis primi Paris. </s><s>Cui proinde Quadratura debetur Lunulæ ori­<lb/>entis. </s><s>Et quid inquam ego, Symbolo Regio, <expan abbr="mihi&qacute;">mihique</expan>; <expan abbr="atq;">atque</expan> huic meo libro <lb/>e&longs;t commune? <!--neuer Satz-->Tum ille: non vides, inquit, hunc circulum Symbolo ad­<lb/>&longs;criptum, hunc abacum parallelogrammis in&longs;criptum, hanc demum fi­<lb/>guram &longs;tellatam è triangulis & pentagono contextam? <!--neuer Satz-->quid præter <lb/>has figuras habet tuus liber? <!--neuer Satz-->Neq, temerè inter radios geometricæ &longs;tel­<lb/>læ coru&longs;cat <foreign lang="greek">n(g<gap/></foreign> Symbolum medicinæ: quia nimirum <expan abbr="utriu&longs;q;">utriu&longs;que</expan> &longs;cientiæ <pb xlink:href="063/01/005.jpg"/>gnarum e&longs;&longs;e voluit futurum Vatem: qualem <expan abbr="quoq;">quoque</expan> vitæ humanæ cu&longs;to­<lb/>dem requirit ve&longs;ter Hippocrates. </s><s>Rectè quidem tu hæc, inquam ego: at <lb/>verò huius acerræ <expan abbr="atq;">atque</expan> ignis, quis nam in me typus? Tam citò, refert ille, <lb/>es oblitus! nam alioquin malorum &longs;en&longs;us e&longs;&longs;e &longs;olet diuturnus. </s><s>Ego <lb/>verò dic, amabo te, aio quidnam ex igne mali &longs;um pa&longs;&longs;us? namillud qui­<lb/>dem ego pror&longs;us ignoro. </s><s>Quòd enim non ita pridem <expan abbr="utramq;">utramque</expan> Domum, <lb/>quæ ex hæreditate meâ erant reliquæ, ignis ab&longs;ump&longs;it, tu optimè no&longs;ti <lb/>quàm æquo animo tulerim: leuior enim hæc jactura mihi vi&longs;a; quàm ut <lb/>mentem his a&longs;&longs;uetam turbaret. </s><s>Ad hæc ille: non memini&longs;ti, inquit, <lb/>illâ eadem nocte, quâ Phitomorpho&longs;is tua &longs;ymbolo præludebat, ma­<lb/>num tibi adu&longs;tam? Memini &longs;anè, inquam ego. </s><s>Nam ubi &longs;tudijs fe&longs;&longs;um <lb/>caput in codicem &longs;acrum reclina&longs;&longs;em; dormienti mihi, ne&longs;cio quo pa­<lb/>cto, manus dextra &longs;ubducta, & in ignem candelæ paulo remotioris pro­<lb/>ducta digitum anularem adu&longs;&longs;it: cuius &longs;en&longs;us acer me quidem euigila­<lb/>re fecit, manum verò ut in&longs;anam incu&longs;are. </s><s>Ita quidem tu, ait Motus, à <lb/>veritate aberrans: at verò illa te longè &longs;apientior fuit: quæ a Sapien­<lb/>ti&longs;&longs;imo Genio tum dirigebatur: Vt nimirum etiam hac parte &longs;ymbolum <lb/>impleres. </s><s>Deinde veró quòd igne hoc elementari futurum Vatem initi­<lb/>ari oportebat. </s><s>Vide nunc has plantas, quibus Symbolum in&longs;ignitur. </s><lb/><s>Agno&longs;cis hanc perpetuò virentem <expan abbr="atq;">atque</expan> victricem LAVRVM: quam ignis <lb/>Jouius tuetur inclu&longs;us? hanc PALMAM canenti OLIVÆ &longs;ociatam? effare: <lb/>quid &longs;iles? Agno&longs;cis nunc demum tuam Phitomorpho&longs;in? Ohe quid <lb/>audio, inquam ego! etiamne mentis penetralia tibi patent? quem ego <lb/>rebar &longs;olis corporibus mancipatum. </s><s>Et ubi inquit ille maiores per­<lb/>turbationum motus, quàm in mentibus humanis? At velocitas mentis, <lb/>inquam ego, omni motu corporeo e&longs;t velocior. </s><s>Si ergo ine&longs;t velocitas, <lb/>ait, inerit &longs;anè & motus. </s><s>Quanquam falleris, ratus mentem Corpori <lb/>huic terreno alligatam omni motu corporeo e&longs;&longs;e velociorem: quæ neq, <lb/>huius frigidi Saturni velocitatem ullâ ratione a&longs;&longs;equi valet. </s><s>At COPER­<lb/>NICVS, inquam ego, cum GALILÆO & multâ turbâ &longs;ophorum hanc tibi <lb/><expan abbr="Cœlóq;">Cœlóque</expan> prærogatiuam ademit: qui &longs;olem in medio mundi &longs;tare immo­<lb/>tum, terram verò circumire ju&longs;&longs;it. </s><s>Atqui refert ille, in eo &longs;atis o&longs;ten­<lb/>dunt animi &longs;ui tarditatem: Dum a&longs;&longs;equi non valent hanc meam in cor­<lb/>poribus velocitatem. </s><s>Sed hîs relictis ad tuam Phitomorpho&longs;im me <lb/>conuerto: <expan abbr="neq;">neque</expan> enim abe&longs;&longs;e potui ex illâ motione; dum planta una ex <pb xlink:href="063/01/006.jpg"/>aliâ na&longs;ci videbatur: licet motu velociore, quàm pro tuo voto: cùm <expan abbr="necdũ">necdum</expan> <lb/>&longs;atiato tibi illarum Species &longs;ubducebantur. </s><s>Sed quem fui&longs;&longs;e putas illum <lb/><expan abbr="Hortulanũ">Hortulanum</expan>, qui tibi in horto, ut videbatur, <expan abbr="&longs;urculũ">&longs;urculum</expan> LAVRI cupienti qui­<lb/>dem, <expan abbr="neq;">neque</expan> tamen ob reuerentiarn viri petere au&longs;o, ultro in manus dedit <lb/>cum hoc dicto: <emph type="italics"/>Pote&longs;t cre&longs;cere.<emph.end type="italics"/> Tum ego, ô omni&longs;cie Motus, quando­<lb/>quidem nihil Te latet arcanorum: tu &longs;iquidem omnia audis, <expan abbr="vidé&longs;q;">vidé&longs;que</expan> <lb/>etiam quæ Solem oculati&longs;&longs;imum & maximè auritum fugiunt; dic ob&longs;e­<lb/>cro quid tibi videtur de illis ver&longs;ibus, quos SERENISSIMO <lb/>HVNG: ET BOHEMIÆ REGI FERD: IIII. in felici in au­<lb/>guratione accinebam? rectène illam Phitomorpho&longs;im fui a&longs;&longs;ecutus? <lb/>Ne dubita, ait Motus, idem enim Genius, qui ea &longs;imulachra immi&longs;it, <lb/>eorundem &longs;en&longs;um tibi in&longs;tillauit. </s><s>Quid igitur inquam ego, cunctamur? <lb/>Perge mi Motus, <expan abbr="te&qacute;">teque</expan>; ocy&longs;&longs;imé REGI NOVO &longs;i&longs;te: ut &longs;is & munus, <lb/>& futuræ felicitatis augur. </s><s>Tibi liberum permitto, ut vel circulus, vel <lb/>quadratum, imò & cubus fias: prout REGALI TVTELÆ vide­<lb/>bis ex pedire. </s><s>Quem terrâ <expan abbr="mari&qacute;">marique</expan>; &longs;ecutus, ventos fauentes motu circuli <lb/>velociore incitabis: eo&longs;dem furentes quadrato, aut etiam cubo inhibe­<lb/>bis. </s><s>Faxo lubens, inquit, quod imperas; tu verò boni ominis ergò, in hac <lb/>eadem pagellâ tuos ver&longs;us mihi exhibe: quos ego unâ cum libello mox <lb/>ad ultimum terræ feram. </s></p> <p type="main"> <s><emph type="center"/><foreign lang="greek">FITOMOPFWSIS</foreign><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Planta cadem LAVRVS, PALMA, & pallentis OLIVÆ, <lb/>Vi&longs;a mihi: demumgermina VITIS crant.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>LAVRVS.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>A&longs;&longs;ociare virens Regali LAVRE Coronæ, <lb/>Seruet ut æternus Regia &longs;ceptra viror.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PALMA.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Bella procul, LAVRO nam a&longs;&longs;uetus vincere nouit: <lb/>Victorem victrix non ni&longs;i PALMA decet.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>OLIVA.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Na&longs;citur Imperio defe&longs;&longs;o pinguis OLIVA, <lb/>Hac non fucatæ &longs;ymbola pacis habet.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>VITIS.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Terra Bohema oculos &longs;icca: noua VITIS inumbrat, <lb/>Præterita ignorat, qui bibit inde merum.<emph.end type="italics"/><emph.end type="center"/></s></p></section><section> <pb xlink:href="063/01/007.jpg"/> <p type="main"> <s><emph type="center"/>AD LECTOREM.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>NAturam definit Philo&longs;ophus e&longs;&longs;e principium & cau&longs;am mo­<lb/>tûs & quietis eius, in quo e&longs;t, primùm, per&longs;e, & non &longs;ecundùm <lb/>accidens. </s><s>Quia verò ubi hic de&longs;init, ibi Medicus &longs;uæ Specula­<lb/>tionis principium &longs;umit: cuius obiectum e&longs;t natura humana, qua­<lb/>tenus â &longs;anitate in <expan abbr="morbũ">morbum</expan>, & ex hoc in &longs;anitatem mouetur; nece&longs;&longs;e &longs;anè â Me­<lb/>dico motum haud ignorari. </s><s>Præ&longs;ertim verò cùm inter res non naturales, quas <lb/>Medicina &longs;peculatur, numeretur motus & quies. </s><s>Impo&longs;&longs;ibile enim, ait Hip­<lb/>pocrates, hominem comedentem e&longs;&longs;e &longs;anum, &longs;i non laboret. </s><s>Vbiper laborem in­<lb/>telligit motum corporeum: Cuius diuer&longs;as &longs;pecies recen&longs;et libro: 3. de diætâ. <lb/>quæ tamen ob ignorantiam motûs hoc æuo negliguntur. </s><s>Cùm ergo mihi propo­<lb/>&longs;itum &longs;it, <expan abbr="jámq;">jámque</expan> incœptum habeam tractatum de naturâ humanâ, quatenus e&longs;t <lb/>mobilis quoad utrum&que; motum, videlicet internum & externum: tam in &longs;tatu <lb/>naturali, quàm præter naturam: hoc e&longs;t radicem inve&longs;tigare omnium morbo­<lb/>rum, qui ad imaginationem <expan abbr="motúmq;">motúmque</expan> pertinent: id&que; ex intimis, & recondi­<lb/>tis naturæ principijs (ne&queacute; enim &longs;i paraly&longs;is partem unam plure&longs;ue motu pri­<lb/>uat, &longs;cire licet undo hac affectio pullulet, aut quo pacto eidem occurri po&longs;&longs;it; <lb/>ni&longs;i quid motus, & quâ ratione in nobis fiat, priùs norim) cùm rectum & <lb/>&longs;ui, & obliqui &longs;it index; non videbor ab in&longs;tituto aliena &longs;ecutus; &longs;i habitu Phi­<lb/>lo&longs;ophi a&longs;&longs;umpto, ea principia, â quibus dicendorum veritas pendet, priùs &longs;tabi­<lb/>liam. </s><s>Error &longs;iquidem in his tamei&longs;i paruus, te&longs;te Ari&longs;totele, in progre&longs;&longs;u fit <lb/>magnus. </s><s>Licet verò hunc libellum de proportione motûs figurarum rectilinea­<lb/>rum necdum maturum <expan abbr="judicar&etilde;">judicarem</expan>, qui in lucem prodiret, at&queacute; ulteriore limâ eun­<lb/>dem expolire in animo haberem; docti&longs;&longs;imorum tamen virorum hortatu in a­<lb/>liam mentem fui adductus. </s><s>Inter quos eminet Reuerendi&longs;&longs;imus Pra&longs;ul<emph.end type="italics"/> Ioannes <lb/>Caramuel Lobkowitz; <emph type="italics"/>qui eùm in re litterariâ &longs;it laborio&longs;i&longs;&longs;imus, amicos &longs;ues <lb/>non &longs;init e&longs;&longs;e otio&longs;os. </s><s>Et no&longs;tri &longs;æculi Phœnix P. Athana&longs;ius Kircher, qui & &longs;uo <lb/>& aliorum nomine mihi calcar ad debat. </s><s>Scribit, inquiens, P. Mer&longs;ennus opera <lb/>tua Pari&longs;ijs multùm placere: rogat, ut te incitem ad &longs;imilia plura luci danda. </s><lb/><s>Sed qvid inquies ad Medicum circul: quadratura? Et quid inquam ego ad <lb/>&longs;pon&longs;am calami&longs;trata coma, & cincinni? Quæ verò huic tractatui de e&longs;&longs;e vi­<lb/>dentur; &longs;upplebit liber de Motu & huius efficientibus cau&longs;is Grauitate Leuitate & <lb/>Impul&longs;u; qui proximè <expan abbr="librũ">librum</expan> de<emph.end type="italics"/> Arcu cœle&longs;ti, <emph type="italics"/>qui iam &longs;ub prælo judat, &longs;equetur.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PARS PRIMA.<emph.end type="center"/></s></p> <pb xlink:href="063/01/008.jpg"/> <p type="caption"> <s>IOANNES MARCVS MARCI PHIL: & MEDIC: DOCTOR <lb/><emph type="italics"/>et Profe&longs;&longs;or natus Landscronœ Hermundurorum in Boemta <lb/>anno 1595.13 Iunij.<emph.end type="italics"/></s></p> <figure id="id.063.01.008.1.jpg" xlink:href="063/01/008/1.jpg"/></section> </front> <body> <chap> <pb xlink:href="063/01/009.jpg"/> <p type="main"> <s><emph type="center"/>Re&longs;olutio aliquot dubiorum exlibello <lb/>De <lb/><emph type="italics"/>Proportione motús.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>LIbellus de proportione motus <lb/>ante annos novem in lucem datus, ad plures <lb/>quidem peruenit opinione doctrinæ, & Geo­<lb/>metriæ famâ claros: illorum de &longs;e judicia ac <lb/>cen&longs;uram laturus. </s><s>Ex quorum tamen numero <lb/>unus & alter quod &longs;ciam &longs;ubmurmurauit. </s><s><expan abbr="Atq;">Atque</expan> huic quidem <lb/>minùs arri&longs;it illa proportio inter <expan abbr="motumrectũ">motumrectum</expan> & inclinatum <lb/>ad prop. 13. </s><s>Quam ut di&longs;turbaret, machinâ mirâ, & ingeni­<lb/>osâ, ex affirmatiuâ negatiuam expre&longs;&longs;it. </s><s>Ita enim R. P. Bal­<lb/>tha&longs;ar Conradus Soci: IESV. Philo&longs;. & Mathe&longs;eos Profe&longs;&longs;or, ad <lb/>R. P. Theodorum Moretum Soc: IESV, Mathe&longs;eos <expan abbr="quoq;">quoque</expan> tum <lb/>Profe&longs;&longs;orem, <expan abbr="atq;">atque</expan> Geometram percelebrem. </s><s><emph type="italics"/>Mitto, inquit, R. <lb/>Væ di&longs;cur&longs;um &longs;uper prop. 13. Excellentißimi Domini Doctoris Marci: <lb/>cuius propo&longs;itionis contradictoria e&longs;t hæc.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Motus per lineam perpendicularem & lineam inclinatam, quorum <lb/>terminos coniungit linea recta, perpendicularis ad lineam inclinatam, <lb/>non &longs;unt inter &longs;e æquales.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Sit eadem figura, quæ Doctoris; & intelligantur duo &longs;egmenta <lb/>Sphærica GHF. GIF inter &longs;e æqualia.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Dico non e&longs;&longs;e id, quod Author prop: 13 proponit: videlicet non per <lb/>uenturum globum D eodem tempore in plano inclinato BF, à puncto <lb/>B ad punctum F, quo tempore alius globus eidem æqualis ex codem<emph.end type="italics"/> <pb xlink:href="063/01/010.jpg"/> <arrow.to.target n="fig1"/><lb/><emph type="italics"/>puncto B, ad A perneniret lap&longs;u verticali. </s><s>Cùm enim illa duo &longs;eg­<lb/>menta Sphærica GHF, GIF, habeant centrum grauitatis in lineà <lb/>GF: &longs;it&que; F hypomochlium, æquiponderabunt: quare reliqua tantum <lb/>Sphæræ pars GKFI deor&longs;um producet impul&longs;um: Quare & im­<lb/>pul&longs;us motum &longs;ibi æqualem per prop: 2. Doctoris. </s><s>E&longs;t autem ut pars <lb/>Sphæræ GKFI ad totam Sphæram, ita partis eiu&longs;dem impul&longs;us ad to­<lb/>tius Sphæræ impul&longs;um per propo&longs;: 2. in Archimede promoto: quare & mo­<lb/>tus partis eiu&longs;dem ad motum totius erit in eadem ratione. </s><s>Permutan­<lb/>do ergo & velocitas partis ad velocitatem totius per Propo&longs;.<emph.end type="italics"/> 10. <emph type="italics"/>Doctor: <lb/>ergo et interuallum BF ad interuallum BA, uti pars Sphæræ GK <lb/>FI ad totam Sphæram per prop&longs;: 7. eiu&longs;dem.<emph.end type="italics"/></s></p> <figure id="id.063.01.010.1.jpg" xlink:href="063/01/010/1.jpg"/> <p type="main"> <s><emph type="italics"/>Sed pars GKFI non e&longs;t ad totam Sphæram uti CD ad DF, quod <lb/>certum e&longs;t: & patet ex hoc di&longs;cur&longs;u. </s><s>Fingatur enim mente recta H <lb/>D per verticalem GF diui&longs;a bifariam. tunc &longs;i e&longs;&longs;et ut CD ad FD Sim­<lb/>pla ad duplam, ita reliqua magnitudo (ablatis duobus &longs;egmentis Sphæ­<lb/>ricis illis dictis) ad totam: E&longs;&longs;et etiam tota magnitudo dupla illius <lb/>partis GKFI: quod ad oculum fal&longs;um factâ figurâ apparebit. </s><s>Ergo <lb/>ne&que; interuallum BF ad interuallum BA, uti CD ad DF, quod<emph.end type="italics"/> <pb xlink:href="063/01/011.jpg"/><emph type="italics"/>oportuit demon&longs;trare. </s><s>Motus ergo per lineam &c: Examinet R V. <lb/>hunc di&longs;cur&longs;um; & &longs;i putauerit, etiam Excell: Dno Doctori <lb/>o&longs;tendat. </s><s>Reliquas ip&longs;ius propo&longs;itiones per otium in&longs;piciam.<emph.end type="italics"/> Hæc ille <lb/>doctè &longs;anè ac mode&longs;te. </s><s>Quæ priu&longs;quàm ad incudem <lb/>reuocentur, placet non nihil Lucis addere illi propo&longs;i­<lb/>tioni 13. </s><s>Tum enim facilè di&longs;piciemus, an tela huc, an a­<lb/>liò tendant: et an aliquam partem feriant, <expan abbr="demolianturq;">demolianturque</expan>? <lb/>an tota, ut aiunt, uiâ aberrent. </s><s>In illâ <expan abbr="itaq;">itaque</expan> propo&longs;itione <lb/>a&longs;&longs;ero: Si duo circuli æquales ex eodem principio motûs &longs;imul <lb/>ferantur: hic quidem verticali, ille verò motu inclinato, con­<lb/>tinuò in eà ratione labi, ut ex quolibet puncto motûs vertica­<lb/>lis, ducta linea recta &longs;ecet perpendiculariter alterius motum. </s><lb/><s>Huius Apodixis hæc erant fundamenta. 1. &longs;patia decur&longs;a <lb/>eandem rationem ad &longs;e habere, quam impul&longs;us eiu&longs;dem cor­<lb/>poris vel æqualis: ita nimirum, ut &longs;i moueri demus in tempo­<lb/>re AB, per &longs;patium CD; accipiat verò duplum, virtutis im­<lb/>pul&longs;iuæ, moturum &longs;it eodem tempore AB, per duplum &longs;pa­<lb/>tium CD. </s><s>E&longs;t hæc propo&longs;itio Arlis lib. 6. Phy&longs;. cap: 4. & lib: 1. <lb/>de Cælo cap: 6. & alibi. </s><s>Si inquit tanta grauitas per tantum in <lb/>hoc tempore mouetur; tanta & quod &longs;upere&longs;t in minori mo­<lb/>vebitur: Et rationem, quam grauitates habent, tempora è <lb/>conuer&longs;o habebunt: Vt &longs;i dimidia grauitas in hoc, dupla in di­<lb/>midio huius. </s><s>Vbi grauitas maior pro inten&longs;iuà &longs;umi debet; <lb/>quæ idem &longs;ubiectum perficit. </s><s>At verò &longs;i pars accedat æquè <lb/>grauis; tùm huius vi non intenditur motus. </s><s>Vnde &longs;i <expan abbr="vtraq;">vtraque</expan> <lb/>&longs;eor&longs;im æquali celeritate ferebatur; <expan abbr="neq;">neque</expan>, &longs;i connectantur, <lb/>hæc illam trahet, aut impellet: quemadmodum &longs;i duo manibus <lb/>con&longs;ertis cur&longs;u inæqvali ferantur: velocior enim re&longs;tantem <lb/>trahit & ad motum æquè velocem impellit. </s><s>At &longs;i grauitas illa <lb/>æqualis &longs;uo &longs;ubiecto exui, & alteri in&longs;eri detur; tum &longs;anè gra­<lb/>uitas dupla dicetur ine&longs;&longs;e illi &longs;ubiecto: & cum agat &longs;ecundum &longs;e <pb xlink:href="063/01/012.jpg"/>totam, motum producet &longs;ibi æqualem, hoc e&longs;t duplum. </s><s>Jm­<lb/>meritò hic aliqui turbantur, <expan abbr="hæ&longs;itantq;">hæ&longs;itantque</expan> quia inquiunt, licet <lb/><expan abbr="quandoq;">quandoque</expan> velocius feratur in eodem tempore per &longs;patium du­<lb/>plum, non tamen con&longs;tare an illa virtus Locomotiua &longs;it du­<lb/>pla, an in aliâ proportione. </s><s>Verùm hi naturam grauitatis & <lb/>Impul&longs;us videntur ignorare, illam ceu ex atomis conflantes: <lb/>quæ proinde aliquo numero, aut magnitudine &longs;it men&longs;urabi­<lb/>lis. </s><s>At verò quis qualitates &longs;en&longs;um latentes, & vix ab animo <lb/>per&longs;pici valentes men&longs;urabit? quin ip&longs;am coulis &longs;ubiectam al­<lb/>bedinem quis duplam alteri dabit: Sicuti ergo illas qualita­<lb/>tes non ni&longs;i ex effectu no&longs;cim<emph type="sup"/>9<emph.end type="sup"/>; ita ex huius partitione in partes <lb/>analogas &longs;ecamus: ut dupla &longs;it virtus, quæ effectum producit <lb/>duplum; impul&longs;us ergo &longs;eu grauitas dicetur dupla, quæ mo­<lb/>tum valet producere duplum. </s><s>E&longs;t autem de ratione motus <lb/>habere exten&longs;ionem, & in tempore fieri determinato: & ut <lb/>tanto magis &longs;it perfectus, quanto| minùs temporis in&longs;umit. </s><lb/><s>Semi&longs;&longs;is ergo temporis, perfectionem dabit duplam. & quia in <lb/>altera &longs;emi&longs;&longs;e motum producit æqualem, perfectio dupla, eo­<lb/>dem tempore mouebit per &longs;patium duplum. </s><s>Confirmatur <lb/>ex ijs, quæ po&longs;tea dicam ad quæ&longs;t. de cau&longs;a inæqualis reflexio­<lb/>nis: nimirum motum e&longs;&longs;e plagam continuatam in illo medio, <lb/>in quo fit motus: <expan abbr="atq;">atque</expan> impul&longs;um à plagâ incipientem in aliam <lb/>plagam illi æqualem de&longs;tinari: quâ con&longs;ecutâ motus termi­<lb/>natur. </s><s>Cùm ergo Impul&longs;us &longs;it æqualis plagæ, nece&longs;&longs;e illam <lb/>in motu continuatam plagam huic e&longs;&longs;e æqualem. & quia medi­<lb/>um unius e&longs;t rationis, <expan abbr="neq;">neque</expan> magis in una, quàm aliâ parte re&longs;i­<lb/>&longs;tit, erunt partes medij in eâ ratione, in quâ illarum plaga. </s><lb/><s>Medium ergo duplum ab&longs;umet plagam duplam. </s><s>At verò Pla­<lb/>ga dupla non ni&longs;i ab impul&longs;u æquali, id e&longs;t duplo e&longs;&longs;e pote&longs;t: <lb/>Impul&longs;us ergo duplus per medium mouebit duplum. </s><s>De­<lb/>inde cùm velocitas motûs proueniat à minori re&longs;i&longs;tentia me- <pb xlink:href="063/01/013.jpg"/>dij: acrem enim velociùs, quam aquam findit <expan abbr="id&etilde;">idem</expan> mobile: &longs;i mi­<lb/>nuatur re&longs;i&longs;tentia medij, ut fiat &longs;ub dupla prioris; Idem impul­<lb/>&longs;us habebit velocitatem duplam. </s><s>At verò eadem e&longs;t propor­<lb/>tio, &longs;i manente re&longs;i&longs;tentiâ eiu&longs;dem medij, augeatur Impul&longs;us. </s><lb/><s>Igitur &longs;i impul&longs;us rationem habeat duplam ad alium impul­<lb/>&longs;um, mouebitur in eodem medio velocitate duplâ. </s><s>Et quia <lb/>velocitas maior in minori tempore tran&longs;it idem &longs;patium, velo­<lb/>citas dupla in dimidio tempore tran&longs;ibit. </s><s>Quòd &longs;i necdum <lb/>per&longs;ua&longs;i in hac luce caligant, &longs;it ea po&longs;tulatiloco. nam quæ ad <lb/>huius po&longs;itionem &longs;equuntur, &longs;i firmo nexu, & <emph type="italics"/>ut linum lino<emph.end type="italics"/> co­<lb/>hærent, de veritate &longs;uppo&longs;iti non licebit dubitare: quandoqui­<lb/>dem firmitas operis de &longs;ub&longs;tructionibus fidem facit. </s><s>Igitur <lb/>cùm eadem &longs;it ratio motûs, quæ grauitatis &longs;eu impul&longs;us; erit <lb/>motus verticalis duratione æqualis motui inclinato; Si eo mo­<lb/>do habeant &longs;patia, quo illorum grauitates. </s><s>O&longs;ten&longs;um verò <lb/>illa pro. 13. triangula FCD, ABF e&longs;&longs;e &longs;imilia, & in ratione ho­<lb/>mologa &longs;uorum laterum. latus ergo FD ad DC, ut latus A <lb/>B ad AF. </s><s>E&longs;t autem FD men&longs;ura impul&longs;us in lap&longs;u verticali, <lb/>hoc e&longs;t in AB. </s><s>CD verò men&longs;ura impul&longs;us in BF. propterea <lb/>quód impul&longs;us &longs;eu grauitas per po&longs;it. 6<emph type="sup"/>am<emph.end type="sup"/> augetur in ratione <lb/>di&longs;tantiæ centri à linea hypomochlij. </s><s>Concipitur enim cen­<lb/>trum grauitatis in hypomochlio librari: cuius vectis linea per­<lb/>pendicularis à centro productá Quæ &longs;i æqualis &longs;it radio, tota <lb/>grauitas prominet extra lineam hypomochlij: in plano verò in­<lb/>clinato, quò magis inclinatur, eò propiùs accedit ad lineam <lb/>hypomochlij: & quò minor fit vectis, eò minùs gravitat. </s><s>Pro <lb/>cuius maiori declaratione, Notandum Comparationem in&longs;ti­<lb/>tui grauitatis, non inter partes Circuli, quas linea hypomo­<lb/>chlij bifariam &longs;ecat: cùm non illarum, &longs;ed centri ratione fiat <lb/>impul&longs;us, per quartum Theorema huius: in quo omnium vir­<lb/>tus collecta, in &longs;ingulas &longs;e effundit. </s><s><expan abbr="Itaq;">Itaque</expan> fit ut pars nulla &longs;uo <pb xlink:href="063/01/014.jpg"/>motu, &longs;ed motui Centri parallelo feratur: <expan abbr="ictusq;">ictusque</expan> non huius, <lb/>&longs;ed vi centri accidant grauiores: verùm centrum grauitatis <lb/>ad &longs;e ip&longs;um refertur, quatenus ex inæquali remotione à lineâ <lb/>hypomochlij inæqualiter ponderat. </s><s><expan abbr="Neq;">Neque</expan> enim percu&longs;&longs;io fit <lb/>per lineam verticalem &longs;eu hypomochlij; &longs;ed eam, quæ duci­<lb/>tur à centro grauitatis per contactum, per quartum Theor. </s><lb/><s>Vnde fit ut centrum grauitatis &longs;e ip&longs;o utens ad &longs;e mouendum, <lb/>&longs;ibi præponderet in eâ ratione, in quâ e&longs;t vectis. </s><s>Cùm ergo in <lb/>lap&longs;u verticali nihil occurrat centro, totum vectem grauitas <lb/>obtinet: in plano autem inclinato, linea verticalis ducta per <lb/>contactum inæqualiter hunc &longs;ecat, pro ratione inclinationis. <lb/>et tum centrum grauitatis &longs;e ip&longs;um veluti partitur in eam, quæ <lb/>mouet, & in eam quæ in Hypomochlio quie&longs;cit partem. </s><s>Opor­<lb/>tet enim concipere, quemadmodum &longs;i globus ab alio globo æ­<lb/>quali &longs;it levandus. </s><s>Tum enim &longs;i <expan abbr="uterq;">uterque</expan> æqualiter abe&longs;t à tru­<lb/>tinâ, fit æquilibrium: retractione verò unius, eam rationem <lb/>habet grauitas huius ad grauitatem illius, quam interualla. </s></p> <p type="main"> <s>Obijcies. </s><s>Huic po&longs;itioni aduer&longs;ari ea, quæ propo&longs;. 32. & 33 <lb/>&longs;unt dicta: vbi o&longs;tendi Impul&longs;um eo modo augeri, quo triangu­<lb/>lum &longs;ibi &longs;imile manens: <expan abbr="rationémq;">rationémque</expan> habere &longs;uorum tempo­<lb/>rum, in quibus fiunt, duplicatam. </s><s>Quòd &longs;i ergo radius totus <lb/>FD &longs;it quadratum ab hypomochlio in duo quadrata CD. CF <lb/>divi&longs;um, uti propo&longs;itio illa vult; erit grauitas in DF ad gra­<lb/>uitatem in CD, in ratione duplicatâ eius, quam habet &longs;inus to­<lb/>tus ad &longs;inum complementi inclinationis. & quia motus ratio­<lb/>nem habent, quam impul&longs;us, per quartam po&longs;itionem, erit mo­<lb/>tus in AB ad motum in BF in ratione <expan abbr="quoq;">quoque</expan> duplicatâ. </s><s>Maior <lb/>ergo motus BF, quàm utidem tempus <expan abbr="vtrumq;">vtrumque</expan> metiatur. </s><lb/><s>Hanc obiectionem ut diluamus. </s><s>Aduerte ea, quæ in vecte li­<lb/>brantur, duplicem habere impul&longs;um, &longs;eu grauitatem: aliam <lb/>quidem in ordine ad mundi centrum; aliam verò in ordine ad <pb xlink:href="063/01/015.jpg"/>hypomochlium. </s><s>Differre enim à &longs;e con&longs;tat ex eo, quòd hæc <lb/>augeri pote&longs;t infinitè, nihilo auctâ illâ. </s><s><expan abbr="Neq;">Neque</expan> enim velociùs de­<lb/>&longs;cendit vectis ob remotionem ponderis à lineâ hypomochlij: <lb/><expan abbr="neq;">neque</expan> &longs;i alià trutinâ explores in quouis &longs;itu, magis ponderabit. </s><lb/><s>Propterea quòd hic impul&longs;us hypomochlium, non verò mun­<lb/>di centrum re&longs;picit, quantumuis ab eadem grauitate oriatur. <lb/><expan abbr="Atq;">Atque</expan> hunc impul&longs;um augeri in eà ratione, quam vectis obtinet, <lb/>demon&longs;trat Archimedes in lib. de æquiponderantibus. </s><s>Alio <lb/>modo Grauitas, &longs;eu impul&longs;us in ordine ad motum expenditur <lb/>ab&longs;olutè, <expan abbr="ab&longs;q;">ab&longs;que</expan> ullo re&longs;pectu ad hypomochlium: & tum <lb/>rationem quadrati habere dicimus; cuius latera &longs;int duratio <lb/>motûs. </s><s>Nam cùm in aliquo tempore produci &longs;it nece&longs;&longs;e, <expan abbr="atq;">atque</expan> <lb/>eo modo augeatur, quo triangulum &longs;ibi &longs;imile manens, per po­<lb/>&longs;it. quintam; habebit impul&longs;us hic ad illum, rationem eius, <lb/>quam habent tempora, duplicatam. per propo&longs;. 12. </s></p> <p type="main"> <s>Aduerte &longs;ecundo, duobus modis fieri contactum mobi­<lb/>lis & plani: vno modo, cùm incidit plano: alio modo, cùm la­<lb/>bitur per ip&longs;um. </s><s><expan abbr="Neq;">Neque</expan> eadem ratio <expan abbr="utrobiq;">utrobique</expan>. </s><s>Nam cùm labi­<lb/>tur, & labendo tangit planum, eodem modo videtur &longs;e habe­<lb/>re ad hypomochlium, <expan abbr="eandémq;">eandémque</expan> di&longs;tantiam obtinere centrum <lb/>grauitatis: Manet ergo ratio partis motæ ad quie&longs;centem, <expan abbr="quã">quam</expan> <lb/>linea hypomochlii à principio induxit. </s><s>At verò cùm incidit <lb/>eidem plano, plagam infert, & recipit: vnde reflecti contin­<lb/>git. </s><s>O&longs;ten&longs;um verò prop: 37-Plagam in aliquo tempore <lb/>fieri: à Plaga verò impul&longs;um ex&longs;olui. quam ergo rationem <lb/>habet mora plagæ iam perfectæ ad aliam moram plagæ nec­<lb/>dum perfectæ, candem habet impul&longs;us totus ad illum duplica. <lb/>tum. </s><s>Igitur in ca&longs;u verticali, quia hypomochlium occurrit <lb/>centro, <expan abbr="neq;">neque</expan> percu&longs;&longs;ioni cedit, plagam inducit <expan abbr="perfectã">perfectam</expan>, <expan abbr="totũq">totunq</expan>, <lb/>impul&longs;um ex&longs;oluit. & cùm æqualem à percu&longs;&longs;o recipiat plaga, <lb/>eadem, quâ incidit, viâ retro agitur. </s><s>In occur&longs;u autem plani <pb xlink:href="063/01/016.jpg"/>ad ictum inclinati, quia non per centrum grauitatis &longs;eu impul­<lb/>&longs;us &longs;ecatur à lineà hypomochlij; erit ratio Plagæ, quam habet <lb/>in hypomochlio quies. quæ tantò e&longs;t minor, quantò velociùs <lb/>centrum grauitatis à plagâ &longs;e abducit. </s><s>Quòd &longs;i ergo DF &longs;it <lb/>mora plagæ perfectæ, <expan abbr="atq;">atque</expan> huius impul&longs;us quadratum DF; erit <lb/>DC tempus uelocitatis motus, & huius quadratum impul&longs;us: <lb/>reliquum ergo quadratum FC à percu&longs;&longs;ione &longs;eu plagâ, impul­<lb/>&longs;um dabit à reliquo tempore men&longs;uratum. propterea quod <lb/>quadratum FD &longs;it æquale duobus quadratis CD. CF: ac pro­<lb/>inde mora percu&longs;&longs;ionis complementum CD ad &longs;inum <lb/>totum. </s><s>Eodem modo &longs;i plagam metiamur fientem morâ æ­<lb/>quali C Flateri eiu&longs;dem quadrati, erit huius complementum <lb/>mora impul&longs;us reliqui. </s><s><expan abbr="Atq;">Atque</expan> ex his Soluitur illa dubitatio, <lb/>quam ob rem prop: 13. impul&longs;u & grauitate, <expan abbr="horumq;">horumque</expan> diui&longs;i­<lb/>one utamur ceu lineâ rectâ, aut parallelogrammo: propo&longs;i­<lb/>tione autem 32. & 33 motum comparemus ut quadrata. <lb/>quia nimirum hic impul&longs;um ut fientem, ac proinde iuxta mo­<lb/>dum <expan abbr="men&longs;uramq;">men&longs;uramque</expan> plagæ expendimus. </s><s>Non enim à percu&longs;&longs;i­<lb/>one idem e&longs;t impul&longs;us: &longs;ed illa portio, quæ percu&longs;&longs;it, illi de­<lb/>cedit: Alius verò huic æqualis & oppo&longs;itus à percu&longs;&longs;o rege­<lb/>neratur: & cum reliquo impul&longs;u in ordine ad motum medium <lb/>mi&longs;cetur. </s><s>Nece&longs;&longs;e ergo inter &longs;e conferri, ut illorum tempo­<lb/>rum, in quibus producuntur, quadrata. </s><s>At uerò prop: 13. </s><lb/><s>Impul&longs;um &longs;eu grauitatem in facto e&longs;&longs;e, & à centro grauitatis, in <lb/>quo e&longs;t collecta, &longs;ui replicatione in vectem æqualiter fu&longs;am: <lb/>quam fecat bifariam linea hypomochlij in partem motam & <lb/>quie&longs;centem. </s><s>Hæc autem nullam inducit plagam: verùm <lb/>continuò in hypomochlio quie&longs;cit, & in ordine ad motum pro <lb/>nullâ habetur. </s><s>Vnde augmenta velocitatis motus fiunt <expan abbr="absq;">absque</expan> <lb/>ullo ad eam re&longs;pectu. </s><s><expan abbr="Neq;">Neque</expan> enim motu <expan abbr="inuale&longs;c&etilde;te">inuale&longs;cente</expan> augetur illa <lb/>grauitas in hypomochlio quie&longs;cens: quòd linea hypomochlij <pb xlink:href="063/01/017.jpg"/>non hunc, &longs;ed huius principium partiatur. </s><s>Incrementa enim <lb/>motûs <expan abbr="atq;">atque</expan> impul&longs;ûs per lineam fiunt parallelam illi plano, in <lb/>quo mouetur. </s><s>Quia ergo grauitas mouens impul&longs;um produ­<lb/>cit continue maiorem: non quem &longs;ibi grauitas collegit, &longs;ed <lb/>quem natiuum habet ad grauitatem quie&longs;centem conferri de­<lb/>bet: vt eadem &longs;it proportio vectis, quæ partium gravitatis; <lb/>Quod non ni&longs;i in principio motûs contingit. </s><s>Augetur ergo <lb/>gravitas quie&longs;cens eiu&longs;dem mobilis in eá ratione, quam habet <lb/>reliquum &longs;egmentum vectis, ad di&longs;tantiam centri grauitatis à <lb/>lineâ hypomochlij. </s></p> <p type="main"> <s>His iam definitis: videamus quam vim habeat ille di&longs;cur&longs;us: <lb/>& an contrariâ illatione no&longs;tram po&longs;itionem conuellat. </s><s>Cùm <lb/><expan abbr="itaq;">itaque</expan> a&longs;&longs;umit &longs;egmenta æqvalia GHF. GIF. <emph type="italics"/>I&longs;orrhopa:<emph.end type="italics"/> propte­<lb/>rea, quòd centrum grauitatis habeant in lineà hypomochlij F <lb/>G, ac proinde exce&longs;&longs;um &longs;eu præpondium ine&longs;&longs;e reliquo <expan abbr="&longs;egm&etilde;-to">&longs;egmen­<lb/>to</expan> GKFI, <expan abbr="motúmq;">motúmque</expan> deor&longs;um huius ratione fieri; errat primo <lb/>quòd &longs;upponat eadem ratione moueri partes, <expan abbr="eundémq;">eundémque</expan> dare <lb/>& recipere impul&longs;um in toto exi&longs;tentes, & dum per &longs;e mo­<lb/>uentur: quod à veritate e&longs;t alienum. </s><s>Mouentur enim partes <lb/>virtute &longs;ui centri; <expan abbr="neq;">neque</expan> uno modo omnes, <expan abbr="neq;">neque</expan> &longs;imiliter. </s><s>Nam <lb/>cùm per lineas ferantur motui centri parallelas, remotioribus <lb/>à centro plus ine&longs;t violentiæ: <expan abbr="atq;">atque</expan> <expan abbr="unaquæq;">unaquæque</expan> grauiùs percutit <lb/>in toto, quàm &longs;i per &longs;e moveretur. </s><s>Licet ergo illa &longs;egmenta <lb/>&longs;int æqualia & <emph type="italics"/>I&longs;orrhopa,<emph.end type="italics"/> non <expan abbr="tam&etilde;">tamen</expan> &longs;equitur in toto eandem uim <lb/>obtinere: cùm à centro grauitatis mutari po&longs;&longs;it, &longs;icuti habitu­<lb/>do ad vectem, ita <expan abbr="quoq;">quoque</expan> ratio impul&longs;us. </s><s>Secundò decipi­<lb/>tur, quòd comparationem in&longs;titui velit inter partes mobilis <lb/>circa hypomochlium &longs;itas, nullâ habitâ ratione &longs;itûs, & di&longs;tan­<lb/>tiæ ab hypomochlio: quod magnum e&longs;t erratum. </s><s><expan abbr="Neq;">Neque</expan> enim <lb/>&longs;egmentum GIF, &longs;itu permutato C in I, & contrà, æquipon­<lb/>derabit &longs;egmento GHF, aut &longs;ibi ip&longs;i: quomodo ergo reliquum <pb xlink:href="063/01/018.jpg"/>&longs;egmentum GKFI a&longs;&longs;umit in eâ tatione grauitare, in qua e&longs;t <lb/>pars magnitudinis| Sphæræ? cùm & partium magnitudo ob <lb/>curvitatem circuli, & &longs;itus continuò mutentur. </s><s>Propo&longs;. <expan abbr="aut&etilde;">autem</expan> <lb/>illa &longs;ecunda in Archimede promoto, <emph type="italics"/>impul&longs;um partis in eà ratione <lb/>e&longs;&longs;e ad impul&longs;um totius, in quà ip&longs;a e&longs;t pars magnitudinis<emph.end type="italics"/>, vera e&longs;t, &longs;i <lb/>non ratione &longs;itûs mutetur illa habitudo. </s><s>Sed quidquid &longs;it de <lb/>hac proportione partium ad &longs;e, quam in circulo ignoramus, <lb/>nihil huc facit: ubi centrum grauitatis in eadem magnitudi­<lb/>ne expendimus, & ad &longs;e ip&longs;um comparamus: quatenus in di­<lb/>uer&longs;o &longs;itu & remotione ab hypomochlio inæqualiter ponderat. </s><lb/><s>Deinde verò e&longs;to demus Impul&longs;um diuidi in eâ ratione, in quâ <lb/>magnitudo; vt quæ pars &longs;it molis, eadem &longs;it grauitatis &longs;eu im­<lb/>pul&longs;us: an propterea rectè infert in eadem ratione fieri mo­<lb/>tum? Vt &longs;i &longs;egmentum GHFI &longs;it duplum &longs;egmenti GHF, <lb/>ac proinde grauitatem habeat duplam, duplo velociùs moue­<lb/>ri &longs;it nece&longs;&longs;e: id enim quæ&longs;tione de in æquali ponderum la­<lb/>p&longs;u negamus. </s><s>Malè autem propo&longs;: 10. citat in contrarium: quæ <lb/>ponit impui&longs;um producere motum &longs;ibi æqualem. hoc enim de <lb/>inten&longs;ione, non verò exten&longs;ione grauitatis &longs;eu impul&longs;ûs e&longs;t in­<lb/>telligendum: ita nimirum &longs;i idem mobile accipiat impul&longs;um <lb/>duplum. </s><s>At verò cùm acce&longs;&longs;ione partis nouæ augetur impul­<lb/>&longs;us, nihilo plus virium ad &longs;e mouendum <expan abbr="utraq;">utraque</expan> habet. </s><s>Quòd <lb/>&longs;i ex toto mobili grauitas &longs;eu impul&longs;us colligi po&longs;&longs;it in partem <lb/>V.G tertiam; tum verò pars illa celeritate triplà moueretu. <lb/><!--neuer Satz-->Nam &longs;icuti impul&longs;us magnus in magnam molem receptus ex­<lb/>tenuatur: ita in paruam molem contractus intenditur. </s><s>Atq: <lb/>ex his patet manife&longs;tè, in conuellendâ illâ prop. 13. & fal&longs;a a&longs;­<lb/>&longs;umi, & ex malè a&longs;&longs;umptis vitiosè concludi. </s></p> <p type="main"> <s>Alter fuit R. P. Joannes Ciermans: qui in anno po&longs;itio­<lb/>num Mathematicarum, hebdomade tertiâ, men&longs;is Maij, ita in­<lb/>quit. </s><s><emph type="italics"/>Putat Ioannes Marcus Marci&longs;e&longs;e globum &longs;ummà violentià<emph.end type="italics"/> <pb xlink:href="063/01/019.jpg"/><emph type="italics"/>vel è tormento bellico excu&longs;&longs;um, medio in itinere detinere po&longs;&longs;e immo­<lb/>tum: it a ut ne quidem refiliat. id&queacute; &longs;i &longs;olùm illi alterum globum eius­<lb/>dem ponderis tangendum &longs;i&longs;tat. </s><s>Sed rectè ille impetûs naturam a&longs;&longs;e­<lb/>quutus non e&longs;t<emph.end type="italics"/> Sic ille. </s><s>Verùm hac au&longs;terâ cen&longs;urâ &longs;atis o&longs;ten­<lb/>dit, nimiùm &longs;uis &longs;peculationibus fi&longs;um, experientiam hic negle­<lb/>xi&longs;&longs;e, quam etiam pueri globulis ludentes norunt. </s><s>Quòd &longs;i <lb/>aliquando experiri lubebit, facilè mihi per&longs;uadeo, virum æqui­<lb/>ac veri tenacem, non mitiorem erga &longs;uas de impul&longs;u opinio­<lb/>nes cen&longs;orem futurum. </s></p> <p type="main"> <s>Priu&longs;quam verò finem faciam, placet aliquid lucis addere <lb/>his, quæ de o&longs;cillationibus penduli ibidem &longs;unt dicta. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>An Pendulum æquali tempore recurr at, per arcus maiores <lb/>& minores.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In Prop: 24-a&longs;&longs;umitur motus ex B in D æqualis duratione <lb/>motui ex D in F: propterea quòd BD ad DF &longs;it ut AB ad CD: <lb/> <arrow.to.target n="fig2"/><lb/>& ut AB ad CD, hoc e&longs;t vt AW ad WR, ita per prop. 22. vis <lb/>mouens in B ad uim mouentem in D. e&longs;t enim radius AB æ- <pb xlink:href="063/01/020.jpg"/>qualis radio AW, & &longs;inus CD eiu&longs;dem arcus DW æqualis &longs;i­<lb/>nui WR. </s><s>Verùm licet in principio illorum arcuum ita res <lb/>habeat, in lap&longs;u tamen ob nouas inclinationes, continuò mu­<lb/>tatur illa proportio. </s><s>Vnde incrementa velocitatis, cùm ex a­<lb/>liâ <expan abbr="atq;">atque</expan> aliâ radice na&longs;cantur, non eadem ratione fiunt. </s><lb/><s>Nam &longs;inus AB ad &longs;inum proximum minorem rationem ha­<lb/>bet, quàm CD ad &longs;inum æquè proximum: plus igitur hic <lb/>quàm ibi decedit virtuti motrici. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> fiat BD ad <lb/>DF, ut AB ad CD; hoc e&longs;t, vis movens in B ad uim mouentem <lb/>in D, non eodem tempore agitabitur ex D in F, quo ex B in D; <lb/>verùm per &longs;patium minus, quàm &longs;it DF. </s><s>Nihil tamen officit <lb/>hoc no&longs;træ demon&longs;trationi: quin imò uim affert maiorem. </s><lb/><s>Sit enim arcus ille minor, per quem ex D fit motus D b: & <lb/>ducatur &longs;inus ab. </s><s>Quia <expan abbr="itaq;">itaque</expan> &longs;inus ab e&longs;t maior &longs;inu EF, mi­<lb/>nor verò arcu re&longs;iduo b W; habebit maiorem rationem ad ar­<lb/>cum minorem D b, quam recta EF minor ad arcum maiorem <lb/>DF. </s><s>Igitur per 4. lemma, arcus D b e&longs;t multò minor &longs;inu ab, ac <lb/>proinde arcu reliquo b W. </s><s>Ex quo cùm pars proportionalis <lb/>ab&longs;cindi debeat continuò minor, concludam pendulum non <lb/>priùs ex D quàm ex B attingere W. </s><s><expan abbr="Eadémq;">Eadémque</expan> ratione F non an­<lb/>te D, & H non ante F, ac proinde <expan abbr="neq;">neque</expan> H ante D vel B præcur­<lb/>currere in W. </s></p> <figure id="id.063.01.020.1.jpg" xlink:href="063/01/020/1.jpg"/> <p type="main"> <s>Quod &longs;i dicas, pendulum ex maiori interuallo præcurrere: <lb/>&longs;equitur plura pendula eiu&longs;dem longitudinis, <expan abbr="atq;">atque</expan> in eodem <lb/>Circulo, ex inæqualibus &longs;patijs &longs;imul recurrendo &longs;e percutere <lb/>in motu: quod nemo experitur. </s><s>Ne tamen ullus dubitationi <lb/>locus &longs;uper&longs;it, placet aliâ viâ magis planâ idem demon&longs;trare. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lap&longs;us gravium in plano inclinato, e&longs;t æqualis duratione<emph.end type="italics"/> <pb xlink:href="063/01/021.jpg"/><emph type="italics"/>lap&longs;ui interci&longs;o ab alio plano: quorum terminos connect it <lb/>rect a line a, perpendicularis ad motum interci&longs;um.<emph.end type="italics"/><emph.end type="center"/></s></p> <figure id="id.063.01.021.1.jpg" xlink:href="063/01/021/1.jpg"/> <p type="main"> <s>Moueatur primùm ex A in B per planum AG: ex B verò per <lb/>planum BF. </s><s>Dico motum in BF e&longs;&longs;e æqualem duratione mo­<lb/>tui in BG, quorum terminos connectit recta GF perpendicula­<lb/>ris ad BF. </s><s>Nam impul&longs;us in B e&longs;t maior gravitate, per prop. <lb/>11. <expan abbr="motúmq;">motúmque</expan> producit parallelum plano BG, per propo&longs;itio­<lb/>nem tertiam: propterea quòd â gravitate proveniat extra hy­<lb/>pomochlium con&longs;titutâ. </s><s>Igitur cùm aliud planum occurrit <pb xlink:href="063/01/022.jpg"/>quia rationem habet hypomochlij; &longs;ecabitur impul&longs;us eâ rati­<lb/>one, quâ grauitas verticalis &longs;ecatur à plano inclinato, in par­<lb/>tem motam & quie&longs;centem: ac proinde per propo&longs;itionem <lb/>11. motus interci&longs;us à plano, erit| æqualis duratione reliquo <lb/>motui: qvorum terminos connectit linea recta, perpendicu­<lb/>laris ad motum interci&longs;um. </s></p> <p type="main"> <s><emph type="center"/>LEMMA.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Si in &longs;egmento Circuli ducantur duæ chordæ, angulus <lb/>ab his contentus, erit complementum dimidij anguli eiu&longs;­<lb/>dem arcus ad duos rectos.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In &longs;egmento BF ducantur duæ chordæ BC. CF: dico angu­<lb/>lum BCF ab his contentum e&longs;&longs;e complementum dimidij an­<lb/>guli BOF ad duos rectos. </s><s>Nam duo anguli OFC. OCF &longs;unt <lb/>complementum anguli FOC: duo verò anguli OCB, OBC <lb/>complementum anguli COB. </s><s>Cùm igitur FCB &longs;it &longs;emi&longs;&longs;is <lb/>illorum angulorum; erit complementum dimidij anguli FOB. </s></p> <p type="main"> <s><emph type="center"/>Corollarium.<emph.end type="center"/></s></p> <p type="main"> <s>Sequitur angulum externum FCT e&longs;&longs;e æqualem &longs;emi&longs;&longs;i an­<lb/>guli FOB: propterea quòd <expan abbr="utriu&longs;q;">utriu&longs;que</expan> complementum ad duos <lb/>rectos &longs;it angulus FCB. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lap&longs;us grauium in quædrante Circuli, per duas chordas <lb/>æquatur lap&longs;ui per unæm chordam.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Secetur primùm AF quadrans circuli æqualiter in B: & <pb xlink:href="063/01/023.jpg"/> <arrow.to.target n="fig3"/><lb/>ducantur chordæ AF AB. </s><s>BF: dico lap&longs;um per duas chordas <lb/>AB. BF e&longs;&longs;e æqualem lap&longs;ui per chordam AF. </s><s>Producatur e­<lb/>nim AB in G: & &longs;it AG æqualis chordæ parallelæ FL: Ex F <lb/>autem excitetur linea perpendicularis ad BF: dico hanc pro­<lb/>ductam &longs;ecare AG in G. </s><s>Quia ením arcus FB e&longs;t æqualis ar­<lb/>cui AL, &longs;ubtendet chorda LF, hoc e&longs;t illi æqualis AG grad: 135 <lb/>corda vero AB grad. 45. </s><s>Auferatur AB partium 76, 3668 ex <lb/>AG partium 18477590: <expan abbr="atq;">atque</expan> re&longs;idua BG erit partium <lb/>10823922. </s><s>Et quia per Lemma huius angulus FBG e&longs;t <pb xlink:href="063/01/024.jpg"/>grad. 45. &longs;emi&longs;&longs;is nimirum anguli AOF: Si ad huius logarit­<lb/>mum addatur logaritmus lateris BF, erit aggregatum logarit­<lb/>mus lateris FG, &longs;eu BF partium 7653668. </s><s>Quot nimirum <lb/>partium erat quoq, chorda AB, hoc e&longs;t illi æqualis BF. </s><s>Quòd <lb/>&longs;i <expan abbr="itaq;">itaque</expan> ducatur ex G termino motûs linea perpendicularis ad <lb/>BF, &longs;ecabit eandem in puncto F: ac proinde motus ex B in G <lb/>e&longs;t æqualis duratione motui ex B in F per prim. </s><s>Theorema <lb/>huius. <expan abbr="additóq;">additóque</expan> motu communi ex A in B, lap&longs;us per duas chor­<lb/>das AB. BF æquatur lap&longs;ui per chordam AF: qui per prop. 15. <lb/>erat æqualis duratione lap&longs;ui per chordam LF &longs;eu AG. </s></p> <figure id="id.063.01.024.1.jpg" xlink:href="063/01/024/1.jpg"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>ALITER.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ducatur ex F perpendicularis ad BF: dico hanc productam <lb/>&longs;ecare BG. in G. quod &longs;i non; &longs;ecet &longs;i fieri pote&longs;t, in alio pun. <lb/>cto VG: X vel Z. </s><s>Et quia angulus externus NOL e&longs;t grad: <lb/>45. erit angulus OLF internus grad: 22. prim: 30. & angu­<lb/>lus OLA grad. 67. prim: 30: propterea quod LOA ex hy­<lb/>pothe&longs;i &longs;it grad: 45: <expan abbr="Ablatoq;">Ablatoque</expan> OLF ex OLA, re&longs;iduus FLA, <lb/>hoc e&longs;t illi æqualis FGB grad: 45, ob parallelas nimirum & <lb/>æquales FLGA. </s><s>Cùm <expan abbr="itaq;">itaque</expan> in triangulo FBG rectus &longs;it an­<lb/>gulus ZFB, & angulus FBG per lemma huius grad. 45: erit <lb/><expan abbr="quoq;">quoque</expan> angulus FZB grad 45, ac proinde æqualis angulo FG <lb/>B, externus interno: quod e&longs;t ab&longs;urdum. </s><s><expan abbr="Atq;">Atque</expan> ea­<lb/>dem ratione probabitur linea AG non &longs;ecari à perpendiculari <lb/>XF. </s><s>A&longs;&longs;umatur rur&longs;um arcus AC grad 67; & CF grad 23. pro­<lb/>ducatur autem AC in P &longs;umptâ AP æquali chordæ perallelæ F <lb/>M. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> in F excitetur linea perpendicularis ad FC: <lb/>dico protractam &longs;ecare AP in P. </s><s>Quòd &longs;i non; &longs;ecet, &longs;i fieri <lb/>pote&longs;t, in alio puncto V. G: I. </s><s>Et quia angulus FCI per lemma <lb/>huius, e&longs;tgrad 45 erit <expan abbr="quoq;">quoque</expan> angulus FIC grad 35 Exæquatur <lb/>autem angulus FMA angulo FPA ob lineas parallelas, & æqua- <pb xlink:href="063/01/025.jpg"/>les FM, PA. </s><s>Cùm <expan abbr="itaq;">itaque</expan> angulus OMF &longs;it grad. 33. prim. 30. <lb/>&longs;emi&longs;&longs;is nimirum anguli externi NOM grad. 67: & angulus <lb/>OMA grad: 78. prim: 30; quòd æquales &longs;int arcus AM. FC: <lb/>ablato angulo OMF ex OMA, erit angulus reliquus FMA, <lb/>hoc e&longs;t illi æqualis FPA grad: 45. </s><s>Cùm <expan abbr="itaq;">itaque</expan> angulus FIC &longs;it <lb/><expan abbr="quoq;">quoque</expan> o&longs;ten&longs;us grad. 45, erit angulus FIC externus æqualis <lb/>angulo interno FPI: quod e&longs;t ab&longs;urdum. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lap&longs;us grauium in &longs;egmento <lb/>Circuli minore, quàm grad: 90. e&longs;t velocior per duas chordas, quàm per <lb/>unam chordam.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moueatur graue ex B in F per arcum grad: 45. </s><s>Dico veloci­<lb/>ùs moueri per duas chordas BC. CF, quàm per unam chordam <lb/>BF. </s><s>Supponatur BC æqualis CF: & ducatur FQ parallela BC: <lb/>in productâ verò BC &longs;umatur BT æqualis <expan abbr="Fq.">Fque</expan> erit <expan abbr="itaq;">itaque</expan> BT <lb/>partium 11111400, & BC partium 3901806. </s><s>Quâ ablatâ ex <lb/>BT manet CT partium 7209594. </s><s>Adde Logaritmum huius <lb/>logaritmo anguli CTH grad. 67. prim. 30; qui per lemma e&longs;t <lb/>complementum anguli FCT grad: 22. prim. 30. <expan abbr="eritq;">eritque</expan> aggre­<lb/>gatum logaritmus lateris CH partium 6659688. </s><s>E&longs;t autem <lb/>CH maius latere BC, &longs;eu CF partium 3901806. </s><s>Cùm <expan abbr="itaq;">itaque</expan>, <lb/>motus ex C in H &longs;it æqualis duratione motui ex C in T, per pri: <lb/>theorema huius; erit mot<emph type="sup"/>9<emph.end type="sup"/> in CF minor duratione motu in CH: <lb/>additoque communi motu in BC, motus in BC, CF minor du­<lb/>ratione motu in BT &longs;eu <expan abbr="Fq.">Fque</expan> hoc e&longs;t per prop. 15. illi æquali <lb/>motu in BF. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA IV.<emph.end type="center"/></s></p> <pb xlink:href="063/01/026.jpg"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lap&longs;us grauium in eodem &longs;egmento Circuli per plures <lb/>chordas e&longs;t velocior.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moueatur graue ex Q in F: Dico velociùs labi per chordas Q <lb/>B. BC. CF, quàm per chordas QB. BF. </s><s>Quia enim velociùs de­<lb/>&longs;cendit per duas chordas BC. CF, quàm| per chordam BF per <lb/>Theorema tertium: addito motu communi QB, erit velocior <lb/>lap&longs;us per QB. BC. CF, quàm per QB. BF. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pendulum æquali tempore mouetur per arcum Circuli & <lb/>chordam eidem &longs;ubten&longs;am.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moveatur pendulum TC ex C in B: Dico æquali tempore la­<lb/>bi per arcum CEB, & chordam CB. </s><s>Concipiantur enim per &longs;in­<lb/>gula puncta CGHIK eiu&longs;dem arcus CEB duci tangentes, & <lb/>chordæ his parallelæ BL. BM.BN. BO & c. </s><s>Quia <expan abbr="itaq;">itaque</expan> ex C la­<lb/>bendo in &longs;ingula momenta mutat inclinationem, quam indu­<lb/>cunt lineæ tangentes; erit ratio motûs in his homologa motui <lb/>per chordas parallelas. </s><s>Vt &longs;i labi incipiat per tangentem CD, <lb/>interuallum motûs in hac erit æquale motui per chordam pa­<lb/>rallelam AB. </s><s>Nullus autem fit motus in CD, verùm immedi­<lb/>atè transfertur in alias tangentes. </s><s>Simili modo in GHIK ex <lb/>illâ obliquatione contrahetur motus, in&longs;patia æqualia chordis <lb/>parallelis BL. BM. BN, BO: in EPQRS verò æquatur chor­<lb/>dis BC. BG. BH &c. quæ quidem chordæ &longs;ubten dunt duplum <lb/>illius arcûs, cuiús tangens e&longs;t parallela. </s><s>E&longs;t enim CEB duplum <lb/>arcùs ESB. </s><s><expan abbr="Atq;">Atque</expan> hæc ratio arcûs dupli, continuatur <expan abbr="u&longs;q;">u&longs;que</expan> ad <lb/>tangentem BV. quam ubi attigit pendulum ex C, attingit <expan abbr="quoq;">quoque</expan> <pb xlink:href="063/01/027.jpg"/> <arrow.to.target n="fig4"/><lb/>æquale pondus lap&longs;u verticali ex A. </s><s>Propterea quòd tangenti <lb/>BV nulla in circulo re&longs;pondet ex B ducta chorda parallela. </s><lb/><s>Motus igitur ex C, per arcum CEB e&longs;t æqualis duratione mo­<lb/>tui per chordam AB, hoc e&longs;t per theor 15. motui per chordam <lb/>CB. </s></p> <figure id="id.063.01.027.1.jpg" xlink:href="063/01/027/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA. VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pendulum ex quolihet puncto circuli æquali tempore recur­<lb/>rit in &longs;uam &longs;tationem,<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Quia enim lap&longs;us per arcum CEB e&longs;t æqualis lap&longs;ui per <lb/>chordam CB & lap&longs;us per arcum ESB æquatur lap&longs;ui per chor­<lb/>dam EB per 5 theorema huius. </s><s>Sunt autem lap&longs;us per chordam <lb/>CB & EB inter &longs;e æquales duratione per prop: 15 erit <expan abbr="quoq;">quoque</expan> la­<lb/>p&longs;us per arcum CEB æqualis duratione lap&longs;ui per arcum ESB. <pb xlink:href="063/01/028.jpg"/>Igitur pendulum TC ex C & E æquali tempore recurrit in&longs;u­<lb/>am &longs;tationem TB. </s></p> <p type="main"> <s><emph type="italics"/>Obijcies. </s><s>Lap&longs;us grauium per plures chordas e&longs;t velocior per 4. theo­<lb/>rema. </s><s>Cùm ita&que; in circuli curvaturâ &longs;int chordæ pote&longs;tate infinitæ; <lb/>erit velocior lap&longs;us per arcum, quàm per quotcun&que; numero chordas.<emph.end type="italics"/></s></p> <p type="main"> <s>Videtur hæc ratio moui&longs;&longs;e Galilæum, ut in lib. de Sy&longs;temate <lb/>mundi motum per arcus circuli po&longs;uerit velociorem motu per <lb/>illorum chordas. </s></p> <p type="main"> <s><emph type="italics"/>His, inquit, adde mirabile aliud, &longs;cilicet quòd motus cadentium facti <lb/>per arcus quadrantis AB fiant breuioribus temporibus, quàm illi, qui <lb/>per chordas eorundem arcuum fiunt. </s><s>Et paucis interiectis, mobile, in­<lb/>quit, di&longs;cedens à puncto A minori tempore perueniet ad B, currendo per <lb/>duas chordas AD. DB, quàm per &longs;olam chordam AB. </s><s>Sed breui&longs;simum <lb/>omnium tempus fuerit, &longs;i deciderit per arcum ADB.<emph.end type="italics"/></s></p> <p type="main"> <s>Verùm di&longs;&longs;oluitur hæc obiectio, quòd motus per plures chor­<lb/>das interci&longs;us, <expan abbr="atq;">atque</expan> huius exce&longs;&longs;<emph type="sup"/>9<emph.end type="sup"/> determinetur per lineas, quæ <lb/>à termino motús per unam chordam, cadunt perpendiculari­<lb/>ter ad alias, per primum theorem: & quò plures fuerint chor­<lb/>dæ eò exce&longs;&longs;us penultimæ erit maior. </s><s>At verò in defluxu cir­<lb/>culari, quiâ nullum interuallum inter proximas tangentes, &longs;eu <lb/>chordas illarum parallelas; <expan abbr="neq;">neque</expan> ulla cadit perpendicularis. </s><lb/><s>Vndè &longs;i ex quolibet puncto refluxûs labi in cipiat per chordam, <lb/>erit æqualis duratione re&longs;iduo lap&longs;ui, qui cadit per chordam <lb/>verticalem. </s></p> <p type="main"> <s><emph type="center"/>Ad Propo&longs;itionem Vige&longs;imam.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De causâ decrementi o&longs;cillationum, & an æquales &longs;int <lb/>duratione.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Cau&longs;a decrementi o&longs;cillationum non e&longs;t illa, quam attuli ad <lb/>finem prop. 20. &longs;icuti enim gravitas &longs;e habet ad illas inclinatio- <pb xlink:href="063/01/029.jpg"/>ones in excur&longs;u, ita <expan abbr="quoq;">quoque</expan> in recur&longs;u; vnde non magis decre­<lb/>&longs;cit impul&longs;us, quàm priùs augebatur. </s><s>Verùm cau&longs;a huius de­<lb/>crementi e&longs;t plaga, quam infert pendulum in lap&longs;u. </s><s>Cùm <lb/>enim hæc per ea, quæ habentur ad finem prop: 27. minuat <lb/>impul&longs;um; excur&longs;us à &longs;tatione nece&longs;&longs;ariò fit minor recur&longs;u. </s><lb/><s>Et &longs;i quidem pendulum refluat per medium magis den&longs;um; <lb/>quia plaga maior plus adimit de impul&longs;u, excur&longs;us erunt mi­<lb/>nores: uti manife&longs;tum in o&longs;cillationibus in aquâ factis. </s><s>Quæ­<lb/>quidem in vacuo, &longs;i fieri admittamus, quia nullam inducunt <lb/>plagam, e&longs;&longs;ent interminabiles. </s></p> <p type="main"> <s><emph type="italics"/>Dices. </s><s>St plaga minuit impul&longs;um; cùm inæquales &longs;int plagæ, erit quo&que; <lb/>inæquale decrementum: Non igitur excur&longs;us inter&longs;e, ac proinde ne&que; <lb/>o&longs;cillationes erunt pares duratione. An pror&longs;us æquales &longs;int, videtur du­<lb/>bius Galilæus. </s><s>In lib: enim de &longs;y&longs;temate mundi pagina 444. alterum in­<lb/>quit &longs;ingulare profectò miraculo&longs;um e&longs;t, quòd idem pendulum vibrati­<lb/>ones &longs;uas eâdem frequentiâ, aut minimùm, & in&longs;en&longs;ibiliter qua&longs;i diffe­<lb/>rente faciat: &longs;iue illæ fiant per arcus maximos, &longs;iue minimos eiu&longs;dem <lb/>circumferentiæ.<emph.end type="italics"/></s></p> <p type="main"> <s>Dico nihilominus o&longs;cillationes omnes, quæ per arcus fiunt <lb/>eiu&longs;dem circuli, e&longs;&longs;e æquales duratione. </s><s>Cuius ratio e&longs;t, quòd <lb/>men&longs;ura plagæ &longs;it interuallum &longs;eu arcus, per quem pendulum <lb/>recurrit. </s><s>Igitur quemadmodum &longs;e habent arcus ad &longs;e, ita <lb/><expan abbr="quoq;">quoque</expan> decrementum impul&longs;ûs. et quia impul&longs;us in recur&longs;u col­<lb/>lecti candem <expan abbr="quoq;">quoque</expan> rationem habent, quam arcus per prop <lb/>18. & 30: erunt <expan abbr="quoq;">quoque</expan> impul&longs;us reliqui à plagâ in eadem ratio­<lb/>ne: ac proinde excur&longs;us & inter &longs;e, & cum recur&longs;ibus æquales <lb/>duratione. </s><s>Quia verò per arcus minores minor plaga induci­<lb/>tur: hinc e&longs;t quòd differentia inter excur&longs;um & recur&longs;um con­<lb/>tinuò decre&longs;cit. </s><s>Vnde ratio redditur tam numero&longs;arum o­<lb/>&longs;cillationum: quæ etiam pro tatione circuli maioris, quem <lb/>pendulum de&longs;eribit, augentur. </s><s>Supponamus <expan abbr="itaq;">itaque</expan> grauitatem <pb xlink:href="063/01/030.jpg"/>penduli ad grauitatem aëris e&longs;&longs;e in centuplâ ratione VG: ut <lb/>colligi videtur ex Ari&longs;t. </s><s>Et quia impul&longs;us in recur&longs;u collectus <lb/>æquatur duplo eiu&longs;dem arcus, per prop: 18. erit plagæ pars 200 <lb/>totius impul&longs;us. deficiet ergò in excur&longs;u pars <expan abbr="quoq;">quoque</expan> 200 <lb/>illius arcûs, quem pendulum de&longs;cribit in recur&longs;u. </s><lb/><s>Vndè vice ver&longs;â ex notâ differentiâ inter ex­<lb/>cur&longs;um & recur&longs;um unius o&longs;cillationis, <lb/>habetur nota grauitas aëris &longs;eu <lb/>medij. </s></p> <figure id="id.063.01.030.1.jpg" xlink:href="063/01/030/1.jpg"/> <pb xlink:href="063/01/031.jpg"/> <p type="main"> <s><emph type="center"/>SECVNDA PARS.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>DEFINITIO I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Latera motùs figuræ &longs;unt lineæ parallelæ motùi centri gra­<lb/>uit atis: quas de&longs;cribunt in motu figuræ puncta remoti&longs;si­<lb/>ma à lineâ motùs centri.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>VT &longs;i moueatur Figura ABCD ad motum centri grauitatis <lb/>FH: erunt lineæ AE. CG eidem parallelæ, latera motus <lb/>figuræ: quas de&longs;eribunt AC puncta remoti&longs;&longs;ima à lineâ FH <lb/>motûs centri. </s></p> <figure id="id.063.01.031.1.jpg" xlink:href="063/01/031/1.jpg"/> <pb xlink:href="063/01/032.jpg"/> <p type="main"> <s><emph type="center"/>DEFINITIO II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Semidiameter figuræ motùs e&longs;t line a rect a, â centro grauita­<lb/>tis ad alterutrum latus figuræ motús perpendiculariter <lb/>ducta.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In eadem figura &longs;i|ducatur ex F centro gravitatis, ad alteru­<lb/>trum latus AE linea perpendicularis FA, erit hæc &longs;emidiame­<lb/>ter figuræ motûs: quàm & vectem librationis centri nuncu­<lb/>pamus. </s></p> <p type="main"> <s><emph type="center"/>DEFINITIO III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Grauit as mouens e&longs;t pars grauitatis mobilis; quam cen­<lb/>trum grauitatis &longs;eu mobile retinet in libratione ad &longs;e <lb/>mouendum in plano inclinato.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>DEFINITIO IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Grauitas quie&longs;cens e&longs;t pars grauitatis mobilis; quâ cen­<lb/>trum grauitatis &longs;eu mobile in libratione grauitat <lb/>byp omocblium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>AXIOMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Areæ figuræ eandem rationem ad &longs;e babent, quam illarum <lb/>grauitas.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Cùm grauitas magnitudinem &longs;equatur, hæc autem &longs;it area <lb/>figuræ <expan abbr="cuiu&longs;q;">cuiu&longs;que</expan>; erit grauitas hæc ad illam in ratione, quam areæ <lb/>ad &longs;e habent. </s></p> <pb xlink:href="063/01/033.jpg"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIVM I<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sequitur grauitatem figuræ ad grauitatem partis eandem ra­<lb/>tionem habere, quam area figuræ habet ad illam partem: ut &longs;i <lb/>pars &longs;it tertia figuræ, erit grauitas tota tripla eiu&longs;dem graui­<lb/>tatis. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIVM II<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Et cùm impul&longs;us &longs;equatur grauitatem, erit eadem ratio hu­<lb/>ius, quæ grauitatis. </s><s>Impul&longs;us ergo totus ad impul&longs;um partis <lb/>tertiæ erit <expan abbr="quoq;">quoque</expan> triplus. </s></p> <p type="main"> <s><emph type="center"/>AXIOMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Vectis continet grauitatem mobilis: totus totam; pars verò <lb/>partem proportionælem.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Huius veritas con&longs;tat ex prop. 13. & præmi&longs;sâ eiu&longs;dem de­<lb/>claratione. </s></p> <p type="main"> <s><emph type="center"/>AXIOMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus grauium fit per lineas rect as &longs;e inter&longs;ecantes in <lb/>mundi centro.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>AXIOMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lap&longs;us grauium eiu&longs;dem rationis per lineas verticales <lb/>inter &longs;e &longs;unt æquales.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ex &longs;e inquam; nam illa differentia, quæ accidit moli maio­<lb/>ri ob inæqualem plagam; ad medium refertur. </s><s>Vt con&longs;tat <lb/>ex quæ&longs;tione de inæquali ponderum lap&longs;u. </s></p> <pb xlink:href="063/01/034.jpg"/> <p type="main"> <s><emph type="center"/>AXIOMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Si magnitudo aliam percutiat in motu; & &longs;it contactus in <lb/>lineâ rectâ, qnæ tran&longs;it per illarum centra, expulsâ æquali, <lb/>á motu quie&longs;cit. exclusâ verò minori motum continuabit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>AXIOMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Si plures magnitudines contiguæ & æquales habeant cen­<lb/>tra in unâ lineâ rectâ: & magnitudo uni contiguarum æ­<lb/>qualis percutiat primam, omnibus immotis ultima mouetur.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Percutiat circulus B alium circulum &longs;ibi æqualem A in G: <lb/>aut quadratum C &longs;ibi <expan abbr="quoq;">quoque</expan> æquale in F: Dico circulum B ex­<lb/>pul&longs;is A & C quie&longs;cere à motu. <!--neuer Satz-->Et &longs;i plures circuli contigui <lb/>habeant centra in unâ lineâ rectâ; percu&longs;&longs;o primo ultimus mo­<lb/>uebitur. </s><s><expan abbr="Idemq;">Idemque</expan> futurum, &longs;i loco circuli quadratum illi æqua­<lb/> <arrow.to.target n="fig5"/><lb/>le &longs;ub&longs;tituatur. </s><s>At verò &longs;i A & C &longs;it minus quàm B; ijs expul­<lb/>&longs;is motum continuabit. </s><s>Demon&longs;tratum id à me quò ad glo- <pb xlink:href="063/01/035.jpg"/>bos ad prop. 37. pori&longs;. 1. & 2. problem. 1. in lib. de proport: <lb/>motûs. </s><s>Eadem verò e&longs;t ratio reliquarum magnitudinum: <lb/>&longs;iue eiu&longs;dem, &longs;iue alterius &longs;int figuræ. </s><s>Nam quòd percutiens <lb/>à motu quie&longs;cit, huius ratio e&longs;t æqualitas ponderis: quæ to­<lb/>tam in percu&longs;&longs;o exhaurit plagam. </s><s>Vtverò circulus B ad circu­<lb/>lum &longs;ibi æqualem A, ita idem circulus ad quadratum &longs;ibi æqua­<lb/>le C: <expan abbr="e&longs;tq;">e&longs;tque</expan> contactus <expan abbr="utriusq;">utriusque</expan> in puncto G & F. &longs;icuti ergo A e&longs;t <lb/>hypomochlium totius grauitatis &longs;eu impul&longs;us in B; ita C hy­<lb/>pomochlium e&longs;t eiu&longs;dem grauitatis &longs;eu impul&longs;us. </s><s>Impul&longs;us <lb/>autem æqualis ad magnitudinem æqualem eandem habet <lb/>rationem. </s></p> <figure id="id.063.01.035.1.jpg" xlink:href="063/01/035/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Grauitas mouens partium ìn toto e&longs;t minor grauitate mouente <lb/>extra illud totum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit B grauitas mobilis, & A mundi centrum: <expan abbr="eritq;">eritque</expan> linea BA <lb/>motus centri per 3, Axioma: partium verò HD motus eidem <lb/> <arrow.to.target n="fig6"/> <pb xlink:href="063/01/036.jpg"/>paralleli HF. DE. </s><s>Dico grauitatem mouentem in H. D e&longs;&longs;e <lb/>minorem, quàm &longs;i extra lllud totum mouerentur. </s><s>Cùm enim <lb/>motus H &longs;it linea HA, & motus D linea DA per 3. Axioma; <lb/>erunt HF. DE motus inclinati: </s></p> <figure id="id.063.01.036.1.jpg" xlink:href="063/01/036/1.jpg"/> <p type="main"> <s>Et anguli in clinationum AHF. ADE. </s><s>Igitur pars grauitatis <lb/>H & D in hypomochlio quie&longs;cit: <expan abbr="minorq;">minorque</expan> proinde e&longs;t grauitas <lb/>mouens, quàm &longs;i extra illud totum mouerentur. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIVM I.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sequitur grauitatem mouentem partium à centro magis re­<lb/>motarum e&longs;&longs;e minorem: propterea quòd motus &longs;int magis in­<lb/>clinati. </s><s>Nam angulus AIF externus, hoc e&longs;t illi æqualis A <lb/>DE e&longs;t maior angulo interno AHF. & angulus AKG, hoc e&longs;t <lb/>ADE maior angulo ACK. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIVM II.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Vnde nece&longs;se partes propiores centro, remotiorum; cen­<lb/>trum verò omnium e&longs;&longs;e hypomochlium huius grauitatis quie­<lb/>&longs;centis. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Centrum grauitatis habet impul&longs;um omnium partium grauitati <lb/>æqualem.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Cùm enim moveatur ad motum partium mobilis, habebit <lb/>impul&longs;um illarum grauitati moventi æqualem. </s><s>E&longs;t verò <lb/>idem centrum hypomochlium grauitatis quie&longs;centis in motu <lb/>partium eidem parallelo, per Corollarium 2. quæ cùm augeat <lb/>illius grauitatem, habebit <expan abbr="quoq;">quoque</expan> per po&longs;it. 4. impul&longs;um illi æ­<lb/>qualem. </s></p> <pb xlink:href="063/01/037.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Centrum grauitatis producit impul&longs;um in omnibus partibus mobilis <lb/>illarum magnitudini proportionalem.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Quia grauitas movens partium in toto e&longs;t minor, quàm &longs;i <lb/>per &longs;e, & extra illud totum moveatur, per I. THEOREMA; <lb/>erit <expan abbr="quoq;">quoque</expan> illarum motus minùs velox. </s><s>Mouentur autem æqua­<lb/>li cum centro velocitate: habent igitur à centro illum motum. </s><lb/><s>At verò centrum grauitatis à partibus mobilis, ex &longs;e verò nul­<lb/>lam habet grauitatem; <expan abbr="e&longs;tq;">e&longs;tque</expan> totus impul&longs;us æqualis grauitati <lb/>ex omnibus partibus collectæ per THEOREMA II Igitur ut <lb/>tota magnitudo &longs;eu grauitas ad totum impul&longs;um, ita pars mo­<lb/>bilis ad partem impul&longs;us proportionalem. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIVM<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Quodlibet punctum mobilis non &longs;uâ, &longs;ed vi centri gravita­<lb/>tis mouetur. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Percußuo fit à grauitate &longs;eu impul&longs;u centri, non verò à grauitate <lb/>&longs;eu impul&longs;u partium mobilis.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moueantur duo globi A & B inter&longs;e connexi: <expan abbr="percutiatq;">percutiatque</expan> B <lb/>in motu globum C &longs;ibi æqualem. </s><s>Dico impul&longs;um in C e&longs;&longs;e ma­<lb/>iorem, quàm ut æqualis &longs;it impul&longs;ui ex B: ac proinde illam pla­<lb/>gam ad centrum referri. </s><s>Nam globus B, cùm per &longs;e movetur, <lb/>percu&longs;&longs;o æquali C, & expul&longs;o vltimo D, à motu quie&longs;cit per <lb/>AXIOMA 6. </s><s>At verò B connexus A <expan abbr="utrumq;">utrumque</expan> expellit D & C, <lb/><expan abbr="neq;">neque</expan> eo percu&longs;&longs;o quie&longs;cit; Igitur globus C impul&longs;um habet <pb xlink:href="063/01/038.jpg"/>maiorem, quàm ut æqualis &longs;it impul&longs;ui ex B. </s><s>Cuius quidem ra­<lb/>tio e&longs;t partium nexus: unde globus percu&longs;&longs;us fit hypomochli­<lb/>um non &longs;olùm illius, quæ percu&longs;&longs;it; &longs;ed etiam partium conne­<lb/>xarum, &longs;eu centri gravitatis. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIVM<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Sequitur tantam e&longs;&longs;e plagam, quantum ine&longs;t hypomochlio <lb/>de centro grauitatis.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>THEOREMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Percußio & qui hanc &longs;equitur impul&longs;us, fit per lineam rectam, pro­<lb/>ductam à contactu per centrum corporis percußi.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Cùm enim partes mobilis non &longs;uâ &longs;ed vi centri grauitatis <lb/>moveantur, per Corollarium Theorematis 3; nece&longs;&longs;e priùs cen­<lb/>trum grauitatis &longs;eu mobilis impelli. </s><s>At verò principium im­<lb/>pul&longs;ûs e&longs;t contactus: igitur cùm impul&longs;us non ni&longs;i per lineam <lb/>rectam moveat per prop: 3. </s><s>Via impul&longs;us erit linea recta, pro­<lb/>ducta à contactu per centrum grauitatis &longs;eu corporis percu&longs;&longs;i. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us centri grauitatis totus quie&longs;cit; cùm &longs;emidiameter figuræ mo­<lb/>tûs, vel illius centrum hypomochlio occurrit.<emph.end type="italics"/></s></p> <p type="main"> <s>Cùm enim partes mobilis non &longs;uâ, &longs;ed vi centri graùitatis, & <lb/>ad huius motum moveantur, per Corollarium theorematis 3. <lb/>nece&longs;&longs;e ad huius in hypomochlio quietem quie&longs;cere totum im­<lb/>pul&longs;um. </s></p> <pb xlink:href="063/01/039.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us centri grauitatis totus mouet, cùm huius interuallum ab <lb/>hypomochlio e&longs;t œquale &longs;emidiæmetro figuræ motús.<emph.end type="italics"/></s></p> <p type="main"> <s>Impul&longs;us enim centri grauitatis prohibetur à motu; cùm vel <lb/>ip&longs;um centrum, vel pars aliqua à centro mota in hypomochlio <lb/>quie&longs;cit. </s><s>At verò cùm interuallum centri grauitatis e&longs;t æqua­<lb/>le &longs;emidiametro figuræ motûs; <expan abbr="neq;">neque</expan> ip&longs;um centrum, <expan abbr="neq;">neque</expan> ali­<lb/>qua pars à centro mota in hypomochlio quie&longs;cit: totus igitur <lb/>impul&longs;us movet. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us movens ad totum impul&longs;um rationem habet, quam &longs;egmen­<lb/>tum &longs;emidiametri ab hypomochlic & centro grauitatis interceptum, ad <lb/>&longs;emidiametrum figuræ motûs.<emph.end type="italics"/></s></p> <p type="main"> <s>Cùm hypomochlium &longs;it trutina; <expan abbr="totusq;">totusque</expan> impul&longs;us quie&longs;cat, <lb/>cùm centrum hypomochlio occurrit, per theor. 6 totus verò <lb/>impul&longs;us moveat, cùm huius à centro intervallum e&longs;t æquale <lb/>&longs;emidiametro figuræ motùs per theore: 7. erit impul&longs;us mo­<lb/>uens æqualis &longs;egmento &longs;emidiemetri inter centrum grauitatis <lb/>& <expan abbr="hypomochliũ">hypomochlium</expan> intercepto In figurâ <expan abbr="&longs;equ&etilde;ti">&longs;equenti</expan> BEC &longs;it A <expan abbr="centrũ">centrum</expan> <lb/>grauitatis, DE hypomochlium, & AC &longs;imidiameter æqualis <lb/>toti impul&longs;ui: <expan abbr="eritq;">eritque</expan> DA interuallum centri grauitatis A & <lb/>hypomochlij DE, grauitas mouens centri A. </s><s>Vt enim AD ad <lb/>vectem AC; ita per Axioma 2. ratio impul&longs;ús ex eodem pon­<lb/>dere A appen&longs;o. </s></p> <pb xlink:href="063/01/040.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us quie&longs;cens e&longs;t æqualis reliquo &longs;egmento, quod ab&longs;cindit hy­<lb/>pomochlium à &longs;emidiametro figuræ motûs.<emph.end type="italics"/></s></p> <figure id="id.063.01.040.1.jpg" xlink:href="063/01/040/1.jpg"/> <p type="main"> <s>Quia impulfus mouens & quie&longs;cens &longs;imul &longs;umpti, toti impul­<lb/>&longs;ui, hic autem &longs;emidiametro figuræ motus AC ponitur æqua­<lb/>lis per Axioma 2: E&longs;t veró impul&longs;us movens æqualis uni &longs;e­<lb/>gmento AD per theorema 8. erit <expan abbr="quoq;">quoque</expan> impul&longs;us quie&longs;cens <lb/>æqualis alteri &longs;egmento DC. </s></p> <p type="main"> <s><emph type="center"/>LEMMA.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Centrum grauitatis cuius&que; figuræ rectilineæ invenire.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit primùm in triangulo i&longs;opleuro ABC inquirendum cen­<lb/>trum grauitatis. in quo ex duobus angulis B & C demittantur <lb/>lineæ ad ba&longs;im rectæ BD CE. </s><s>Dico in communi illarum &longs;ecti­<lb/>one F e&longs;&longs;e centrum grauitatis. </s><s>Quia enim recta BD &longs;ecat ba­<lb/>&longs;im mediam; eritineâ centrum grauitatis, per prop. 13 lib. 1 <lb/>Archimedis de æquipond. </s><s>E&longs;t verò idem in recta CE: igitur in <lb/>communi &longs;ectione F. </s></p> <pb xlink:href="063/01/041.jpg"/> <p type="main"> <s>Inquirendum iam &longs;it centrum grauitatis in quadrato GHIK. <lb/>in quo ductis diametris GI. HK; erit per prop. 10. eiu&longs;dem li­<lb/>bri centrum grauitatis in communi &longs;ectione L. </s></p> <p type="main"> <s>Similiratione inveniemus centrum grauitatis in pentagono <lb/>isopleuro. &longs;inimirum ex angulis O & P ducantur lineæ OV. <lb/>PS perpendiculares ad latus oppo&longs;itum. </s><s>Erit enim centrum <lb/> <arrow.to.target n="fig7"/><lb/>grauitatis in communi &longs;ectione T. propterea quòd <expan abbr="vtraq;">vtraque</expan> figu­<lb/>ram &longs;ecat bifariam: uti manife&longs;tum, &longs;i in triangula re&longs;oluatur. </s></p> <figure id="id.063.01.041.1.jpg" xlink:href="063/01/041/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus verticalis figuræ rectilineæ ad motum inclinatum e&longs;t in ratione <lb/>&longs;emidiametri figuræ motûs ad huius &longs;egmentum, quod e&longs;t inter <lb/>centrum figuræ & lineam hypomochlij.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moveatur triangulum OMN in plano OB: & ex puncto N <lb/>ducatur linea hypomochlij NS, parallela lateri motus OQ: <lb/>ex centro autem figuræ P, per proximum Lemma inuento, a­<lb/>gatur PQ perpendicularis ad OQ Dico motum verticalem <lb/>in OQ ad motum inclinatum in OB e&longs;&longs;e, ut PQ ad PR. </s><lb/><s>Quia enim gravitas mouens ex præmi&longs;&longs;is, & per po&longs;it. <!--neuer Satz-->4- de <lb/>prop. motûs, e&longs;t æqualis motui; grauitas antem tota, &longs;eu ver­<lb/>ticaliter movens ad grauitatem mouentem in OB e&longs;t ut PQ <pb xlink:href="063/01/042.jpg"/>ad PR per theorem 8. erit <expan abbr="quoq;">quoque</expan> motus verticalis in OQ ad <lb/>motum inclinatum in OB, ut PQ ad PR. <!--neuer Satz-->hoc e&longs;t ut &longs;emidia­<lb/> <arrow.to.target n="fig8"/><lb/>meter figuræ motûs ad huius &longs;egmentum inter centrum figu­<lb/>ræ P & lineam hypomochlij NS. </s></p> <figure id="id.063.01.042.1.jpg" xlink:href="063/01/042/1.jpg"/> <p type="main"> <s>Simili ratione in quadrato K, ut KZ ad KL: & in pentago­<lb/>nout TV ad TX, ita illorum motus verticalis ad motum incli­<lb/>natum in OB. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Figura rectilinea velociùs mouetur in plano minùs inclinate.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sint duo plana, quorum inclinatio CAK maior, & CAI <lb/>minor: dico in plano CAI minoris inclinationis, motum e&longs;&longs;e ve­<lb/>lociorem. </s><s>Ducantur ex D centro figuræ ad lineas verticales <lb/>AI. AK &longs;emidiametri figuræ motûs DF. DE: & ex angulo A <lb/>CB lineæ hypomochlij CG. CH parallelæ lincis verticalibus <lb/>AI. AK. </s><s>Quia <expan abbr="itaq;">itaque</expan> maior e&longs;t DE quàm DF, & DO minor <lb/>quàm DP; erit re&longs;idua OE maior quàm PF. </s><s>Maior proinde <pb xlink:href="063/01/043.jpg"/>ratio EO maioris ad OD minorem, quàm FP minoris ad PD <lb/>maiorem. </s><s>Et componendo ED ad OD, quàm FD ad PD. </s><s>E&longs;t <lb/> <arrow.to.target n="fig9"/><lb/>autem ut ED ad OD, ita motus verticalis ad motum inclina <lb/>tum in plano CAK. </s><s>Et ut FD ad PD, ita idem motus vertica­<lb/>lis ad motum inclinatum in plano CAI, per theorem 10. </s><s>Cùm <lb/><expan abbr="itaq;">itaque</expan> motus inclinatus in plano CAI &longs;it magis &longs;imilis verticali, <lb/>erit velocior motu inclinato in plano CAK. </s></p> <figure id="id.063.01.043.1.jpg" xlink:href="063/01/043/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Grauitas movens inæqualium & &longs;imilium figurarum in eodem pla­<lb/>no inclinato, e&longs;t inæqualis & æqualiter mouet.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moueantur in plano AC duo triangula ABC maius, & A <lb/>DE minus: & ex angulis EC ducantur lineæ EP. CO paralle­<lb/>læ verticali AQ: lineæ verò FG. CF per illorum centra GF. <lb/>quæ per problema theorem: 1 erunt perpendiculares ad ba&longs;im <lb/>AB <expan abbr="demũ">demum</expan> exij&longs;dem centris FG cadant lineæ FM. GN. perpen­<lb/>diculares ad AQ. <!--neuer Satz-->Quoniam <expan abbr="itaq;">itaque</expan> triangula CFH. EGI, & tri­<lb/>angula CFK. EGL &longs;unt &longs;imilia: erit CF ad EG, ut FH ad GI <pb xlink:href="063/01/044.jpg"/>& FK ad GL. &longs;unt verò & triangula AMF, ANG, <expan abbr="atq;">atque</expan> trian­<lb/>gula AMK. ANL &longs;imilia. </s><s>Igitur ut AM ad AN, ita MF ad <lb/>NG, & MK ad NL: ac proinde re&longs;idua KF ad <expan abbr="re&longs;iduã">re&longs;iduam</expan> LG. <lb/><expan abbr="cùmq;">cùmque</expan> &longs;it ut FK ad GL, ita FH ad GI: & ut eadem FK ad GL, <lb/>ita FM ad GN; erit <expan abbr="quoq;">quoque</expan> FH ad GI, ut FM ad GN. </s><s><expan abbr="Quiàitaq;">Quiàitaque</expan> <lb/>grauitas mouens &longs;eu impul&longs;us ad totum impul&longs;um rationem <lb/>habet, <expan abbr="quã">quam</expan> GI ad GN, & FH ad FM, hoc e&longs;t <expan abbr="&longs;egmentũ">&longs;egmentum</expan> &longs;emidiame­<lb/>tri inter centrum figuræ & hypomochlium, ad &longs;emidiametrum <lb/>figuræ motûs per theo. 3. erit in <expan abbr="utroq;">utroque</expan> triangulo eadem pro­<lb/>portio motûs inclinati ad motum verticalem. </s><s><expan abbr="Cùmq;">Cùmque</expan> mo­<lb/>tus verticales inter &longs;e &longs;int æquales; per Axioma 4. erunt <expan abbr="quoq;">quoque</expan> <lb/>motus inclinati inter &longs;e æquales. </s><s>Et quia FM e&longs;t maior quàm <lb/>GN, erit FH grauitas movens in triangulo ABC maior, quàm <lb/>GI grauitas movens in triangulo ADE. </s></p> <figure id="id.063.01.044.1.jpg" xlink:href="063/01/044/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Grauitas quie&longs;cens inæqualium & &longs;imilium figurarum e&longs;t inæqualis, <lb/>& inæqualiter grauitat.<emph.end type="italics"/><emph.end type="center"/></s></p> <pb xlink:href="063/01/045.jpg"/> <p type="main"> <s>In eadem figurâ, quoniam e&longs;t ut FM ad GN, ita FH ad GI <lb/>per theor. 12. erit <expan abbr="quoq;">quoque</expan> HM ad IN, ut FH ad GI. </s><s>Sed FH <lb/>e&longs;t maior quàm GI per idem theorema: igitur & HM maior <lb/>quam IN. </s><s>Et quia HM <expan abbr="atq;">atque</expan> IN e&longs;t impul&longs;us quie&longs;cens per <lb/>theor. 9. maior granitas quie&longs;cet in triangulo maiori, ac proin­<lb/>de &longs;uum planum magis gravitabit. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>LEMMA I<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Inclinationem plani invenire: in quo &longs;emidiameter figuræ motûs <lb/>&longs;ecetur ab hypomochlio in datâ ratione.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Producatur latus AC in I; & &longs;it AI ad CI in datâ ratione: <lb/>ex I verò per centrum figuræ D agatur linearecta IF: <expan abbr="atq;">atque</expan> huic <lb/>ex angulis C & A parallelæ CE. AH: quas &longs;ecet ad angulos re­<lb/>ctos, linea ex centro ducta DH. </s><s>Dico lineam DH, hoc e&longs;t &longs;emi­<lb/>diametrum figuræ motûs, &longs;ectam e&longs;&longs;e in datâ ratione. </s><s>Ex <lb/>F enim protrahatur linea FK parallela DH; <expan abbr="eritq;">eritque</expan> FK ad FL, <lb/>hoc e&longs;t DH ad DG, ut AF ad EF. </s><s>Sed ut AF ad EF ita AI ad <lb/>CI, hoc e&longs;t in datâ ratione. </s></p> <figure id="id.063.01.045.1.jpg" xlink:href="063/01/045/1.jpg"/> <pb xlink:href="063/01/046.jpg"/> <p type="main"> <s>Aliter breuiùs. ex D centro figuræ ducta DA &longs;ecetur in da­<lb/>tâ ratione in O: per quod agatur linea CE, <expan abbr="atq;">atque</expan> eidem peralle­<lb/>la AH: é centro verò D &longs;emidiameter figuræ motûs DH. </s><s>Di­<lb/>co hanc &longs;ecari à lineâ hypomochlij in eadem ratione. </s><s>Cùm <lb/>enim &longs;imilia &longs;int triangula ADH. ODG: erit DH ad DG, ut <lb/>DA ad DO, hoc e&longs;t in datâ ratione. </s></p> <p type="main"> <s><emph type="center"/>LEMMA II<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Si duabus inæqualibus lineis addantur æquales; maiorem rationem ha­<lb/>bet maior ad minorem, quàm eadem maior aucta ad auctam <lb/>minorem.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Duabus inæqualibus AB. CD addantur æquales BF. DL. </s><lb/><s>Dico AB ad CD maiorem rationem habere, quàm AF ad CL. </s><lb/><s>Fiat enim ut AB ad CD minorem: ita BF ad aliam minorem <lb/>DG. erit ergo <expan abbr="utraq;">utraque</expan> antecedens AF ad <expan abbr="utramq;">utramque</expan> con&longs;equen­<lb/>tem CG, ut AB ad CD. </s><s>Sed AF ad CG maiorem habetra­<lb/>tionem, quàm ad CL: igitur & AB ad CD maiorem habet ra­<lb/>tionem, quà AF ad CL. </s></p> <figure id="id.063.01.046.1.jpg" xlink:href="063/01/046/1.jpg"/> <p type="main"> <s><emph type="center"/>LEMMA III<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Si ex eadem ba&longs;i de&longs;cribantur plures figuræ rectilineæ æqualium late­<lb/>rum; & ex illâ ba&longs;i per illarum centra agatur linea recta; ea quæ <lb/>plura habet latera, centrum magis abducit à ba&longs;i.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>De&longs;cribantur ex eadem communi ba&longs;i AC triangulum A <lb/>BC, quadratum ADEC, & pentagonum AFGHC æquali­<lb/>um laterum: & per illarum centra agatur linea recta <expan abbr="Gq.">Gque</expan> &longs;e­<lb/>cans ba&longs;im AC æqualiter per problema theorem. 1. </s><s>Quia <lb/><expan abbr="itaq;">itaque</expan> altitudo trianguli BQ e&longs;t minor latere BA, hoc e&longs;t QR; <pb xlink:href="063/01/047.jpg"/>di&longs;tantia verò eiu&longs;dem centri à ba&longs;i minor &longs;emi&longs;&longs;e <expan abbr="Bq;">Bque</expan> erit <lb/>KQ &longs;emi&longs;&longs;is RQ, hoc e&longs;t di&longs;tantia centri in quadrato, maior <lb/>quàm <expan abbr="Iq.">Ique</expan> E&longs;t verò di&longs;tantia <expan abbr="quoq;">quoque</expan> centri LQ in pentagono <lb/> <arrow.to.target n="fig10"/><lb/>maior quam <expan abbr="Kq.">Kque</expan> Nam cùm centrum &longs;it in mutuâ &longs;ectione <lb/>GQ <expan abbr="atq;">atque</expan> HS perpendicularis ad FA, <expan abbr="&longs;intq;">&longs;intque</expan> duo anguli LSA. <lb/>LQA recti: & angulus SAQ in pentagono maior recto: erit <lb/>angulus SLQ minor recto: acproinde latus LQ maius latere <lb/>SA, &longs;emi&longs;&longs;e lateris FA &longs;eu RQ, di&longs;tantiâ nimirum centri in <lb/>quadrato. </s></p> <figure id="id.063.01.047.1.jpg" xlink:href="063/01/047/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Fieri pote&longs;t ut maior figura æqualiter & minùs grauitet.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>A&longs;lumantur duo triangula, quorum hoc illius &longs;it duplum. </s><lb/><s>Dico id quod e&longs;t maius, po&longs;&longs;e æqualiter & minùs grauitare. </s><lb/><s>Secetur grauitas minoris triangali bifariam & æqualiter à li­<lb/>neâ hypomochlij, per 1. lemma: <expan abbr="eritq;">eritque</expan> grauitas mouens æqua­<lb/>lis quie&longs;centi, per theorema 8. &longs;ub quadrupla verò ad grauita­<lb/>tem trianguli maioris. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> &longs;emidiameter figuræ <lb/>motûs in triangulo maiori &longs;ecetur <expan abbr="quoq;">quoque</expan> à lineâ hypomochlij <lb/>in eâ ratione, ut grauitas movens ad quie&longs;centem &longs;it quadru- <pb xlink:href="063/01/048.jpg"/>pla, per 1. lemma: erit grauitas quie&longs;cens in <expan abbr="utroq;">utroque</expan> triangulo <lb/>æqualis; ac proinde æqualiter grauitabit. </s><s>At verò &longs;i augeatur <lb/>ratio grauitatis moventis ad quie&longs;centem; quia tum minor <lb/>grauitas quie&longs;cit, minùs <expan abbr="quoq;">quoque</expan> hypomochlium grauitabit. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Figura rectilinea, quæ plura habet latera, velociùs mouetur in <lb/>eodem plano inclinato.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moveatur in eodem plano AN triangulum ABC, & quadra­<lb/>tum AEFC: Dico huius motum e&longs;&longs;e velociorem. </s><s>Secetur <lb/>enim in triangulo ABC &longs;emidiameter figuræ motûs DI â li­<lb/>neâ hypomochlij CL bifariam & æqualiter in L, per primum <lb/>lemma: & ducatur in quadrato AEFC &longs;emidiameter figuræ <lb/>motûs GH: quæ maior erit &longs;emidiametro figuræ motûs DI. </s><lb/><s>Propterea quòd per lemma 3 maior &longs;it GO quàm DO. </s><s>Etad­<lb/>ditâ communi OP maior GP, quàm DP. </s><s>Et quia ut GP ad <lb/> <arrow.to.target n="fig11"/> <pb xlink:href="063/01/049.jpg"/>DP, ita GH ad DI; erit <expan abbr="quoq;">quoque</expan> GH maior quam DI, Dico GK <lb/>ad GH maiorem rationem habere, quàm DL ad DI. </s><s>Quia <lb/>enim HK e&longs;t æqualis IL, erit per lemma 2. maior ratio GK ad <lb/>DL, quàm GH ad DI: & permutando GK ad GH, quàm DL <lb/>ad DI. </s><s>E&longs;t autem ut GK ad GH, & DL ad DI, ita motus in­<lb/>clinati ad motum verticalem per theorem: 8. </s><s>Igitur motus <lb/>quadrati AEFC e&longs;t velocior motu trianguli ABC in eodem <lb/>plano inclinato AN. </s></p> <figure id="id.063.01.049.1.jpg" xlink:href="063/01/049/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Figura rectilinea & æqualis, quæ plura habet latera, minùs gra­<lb/>uitat in eodem plano inclinato.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Nam &longs;emidiameter figuræ motús, hoc e&longs;t grauitas tota, &longs;eca­<lb/>tur ab hypomochlio in eam, quæ mouet, & in eam quæ in hy­<lb/>pomochlio quie&longs;cit, per theorema 9. </s><s>E&longs;t autem maior grauitas <lb/>mouens in figurâ plurilaterâ per theor. 15. minor ergo huius <lb/>pars quie&longs;cit; ac proinde minùs grauitat. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Grauitas eiu&longs;dem parallelogrammi mutato &longs;itu inæqualiter mouet, <lb/>& grauitat in eodem plano inclinato.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moueatur in plano BO <expan abbr="parallelogrãmum">parallelogrammum</expan> ABCD: Dico <lb/>ex mutato laterum &longs;itu inæqualiter moveri: velociùs quidem, <lb/>&longs;i minus latus CD, tardiùs verò &longs;i maius latus BD fiat paralle­<lb/>lum eidem plano BO. </s><s><expan abbr="Educãtur">Educantur</expan> ex angulis CD lineæ hypomo­<lb/>chlij CG. DM: & ex centro figuræ E &longs;emidiametri figuræ motûs <lb/>EF. EK. </s><s>Et quia in duobus triangulis &longs;imilibus MBD. GDC <lb/>maior e&longs;t DB quàm CD; erit <expan abbr="quoq;">quoque</expan> BM maior quàm DG. </s><s>Et <pb xlink:href="063/01/050.jpg"/>&longs;i ducantut <expan abbr="Bq.">Bque</expan> DI perpendiculares ad DM. CG: erit maior <lb/>BQ quàm DI, hoc e&longs;t KL quàm FR. </s><s>Rur&longs;um quia angulus <lb/>ECD e&longs;t maior angulo ECA, hoc e&longs;t illi æquali EDB: pro­<lb/>pterea quòd latus AC &longs;eu BD &longs;it maius latere BA. ablatis æ­<lb/>qualibus angulis GCD. MDB, erit angulus reliquus ECG <lb/>maior angulo reliquo EDM. </s><s><expan abbr="A&longs;&longs;umaturitaq;">A&longs;&longs;umaturitaque</expan> angulo EDM <lb/>æqualis angulus ECS: & ex Ead CS cadat perpendicularis ES: <lb/><expan abbr="eruntq;">eruntque</expan> triangula ECS. EDL &longs;imilia & æqualia. </s><s>Propterea <lb/> <arrow.to.target n="fig12"/><lb/>quòd ba&longs;is EC &longs;it æqualis ba&longs;i ED. </s><s>E&longs;t autem ET maior <lb/>quàm ES, hoc e&longs;t quàm EL: et ER maior quàm ET. </s><s>Igitur ea­<lb/>dem ER maior quàm EL. </s><s>Cùm <expan abbr="itaq;">itaque</expan> maior &longs;itratio grauitatis <lb/>mouentis ER ad quie&longs;centem RF, nimirum maioris ad mino­<lb/>rem, quàm EL ad LK minoris ad maiorem; erit per po&longs;it: 4. <lb/>velocior motus in ER quàm in EL. </s><s>Et quia tum minor gra­<lb/>uitas in hypomochlio quie&longs;cit, minùs <expan abbr="quoq;">quoque</expan> &longs;ubiectum pla­<lb/>num grauitabit. </s></p> <pb xlink:href="063/01/051.jpg"/> <figure id="id.063.01.051.1.jpg" xlink:href="063/01/051/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Fieri pote&longs;t ut idem parallelogrammum mutato &longs;itu moueatur, & <lb/>quie&longs;cat in codem plano inclinato.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>A&longs;&longs;umatur inclinatio plani æqualis angulo EDB: cadetq, <lb/>linea hypomochlij DE in centrum figuræ. </s><s>Et quia tum cen­<lb/>trum grauitatis hypomochlio occurrit, quie&longs;cet <expan abbr="parallelogrã-mum">parallelogran­<lb/>mum</expan> in co &longs;itu, per theorema 6. </s><s>Cùm verò angulus ECD &longs;it <lb/>maior angulo inclinationis EDB; &longs;i ex C ducatur linea hypo. <lb/>mochlij, cadet inter EC. DC: ac proinde centrum figuræ ex­<lb/>tra hypomochlium motum continuabit in eodem plano. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus circuli in eodom plano inclinato e&longs;t velocior motufiguræ <lb/>rectilineæ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moueatur in eodem plano AN circulus GCA, atq, penta­<lb/>gonum BILMN: Dico motum circuli e&longs;&longs;e velociorem. </s><s>A&longs;&longs;u­<lb/>matur radius EA æqualis ON & ducantur lineæ hypomochlij <lb/>AC. NR &longs;ecetur autem &longs;emidiameter figuræ motús OQ bifa­<lb/>riam & æqualiter in P: ut &longs;it OP æqualis <expan abbr="Pq.">Pque</expan> per primum <lb/>lemma: dico EF maioren rationem habere ad FG, quàm OP <lb/>ad OQ Nam quia rectus e&longs;t angulus DAE, & angulus BNO <lb/>&longs;emi&longs;&longs;is anguli pentagoni minor recto: &longs;unt verò anguli DAC. <lb/>BNP ein&longs;dem inclination is ex hypothe&longs;i æquales: erit angu­<lb/>lus reliquus FAE maior angulo reltquo PNO. </s><s>Et quia OP <lb/>per con&longs;tructionem e&longs;t æqua is PQ, &longs;i iungatur recta NQ, erit <lb/>angulus PNQ æqualis angulo ONP, maior verò angulo BNP, <lb/>hoc e&longs;t illi æquali angulo DAF: ac proinde maior <expan abbr="quoq;">quoque</expan> an- <pb xlink:href="063/01/052.jpg"/> <arrow.to.target n="fig13"/><lb/>gulo minori GAF. </s><s>Angulus <expan abbr="itaq;">itaque</expan> FAE quia maior angulo <lb/>ONP &longs;eu PNQ, erit multò maior angulo FAG; & FE ma­<lb/>ior quam FG. maiorem proinde <expan abbr="ration&etilde;">rationem</expan> habet FE ad FG, quàm <lb/>OP ad <expan abbr="Pq.">Pque</expan> Et componendo EF ad EG, quàm OP ad <expan abbr="Oq.">Oque</expan> <lb/><expan abbr="Cùmq;">Cùmque</expan> impul&longs;us movens ad totum impul&longs;um &longs;it ut EF ad EG; <lb/>& ut OP ad OQ, per theor: 8. erit per po&longs;it: 4 velocior motus <lb/>circuli E in eodem plano AN, quàm pentagoni BILMN. </s></p> <figure id="id.063.01.052.1.jpg" xlink:href="063/01/052/1.jpg"/> <p type="main"> <s><emph type="center"/>PROBLEMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motum circuli, & trianguli I&longs;igoni ijsdem loci interuallis terminare.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Moveatur triangulum I&longs;ogonum ABC in plano HK: & <pb xlink:href="063/01/053.jpg"/>ex centro E ducatur &longs;emidiameter figuræ motûs EF: <expan abbr="&longs;itq;">&longs;itque</expan> in­<lb/>veniendum planum, in quo circulus P æquali celeritate feratur. </s><lb/><s>In lineâ verticali HI centro O de&longs;cribatur circulus HMN: cu­<lb/>ius diameter HN &longs;it æqualis &longs;emidiametro figuræ motûs EF: <lb/>& ex puncto H ducatur chorda HM æqualis EG &longs;egmento in­<lb/>ter centrum figuræ & hypomochlium. </s><s>Dico inuentum e&longs;&longs;e <lb/> <arrow.to.target n="fig14"/><lb/>planum HML, in quo idem &longs;it circuli, qui trianguli in plano <lb/>HK motus. </s><s>Nam ut EF ad EG, ita totus impul&longs;us, &longs;eu verti­<lb/>caliter mouens ad impul&longs;um in HK per 8. theor: & per po­<lb/>&longs;itionem 4-motus trianguli in HI ad motum eiu&longs;dem in HK. </s><lb/><s>Et ut HN ad HM, ita motus circuli in HI ad motum eiu&longs;dem in <lb/>HL per prop, 13 de pro por: motûs. </s><s>At verò eandem ratio­<lb/>nem habet HN ad HM, quam EF ad EG per con&longs;tructionem. </s><lb/><s>Igitur motus circuli in HL e&longs;t æqualis motui trianguli in HK. <lb/>motum ergo trianguli i&longs;ogoni ij&longs;dem loci interuallis terminaui­<lb/>mus, quod erat faciendam. </s></p> <pb xlink:href="063/01/054.jpg"/> <figure id="id.063.01.054.1.jpg" xlink:href="063/01/054/1.jpg"/> <p type="main"> <s><emph type="center"/>PROBLEMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Exce&longs;&longs;um, quo motus circuli in eodem plano e&longs;t maior motu trianguli <lb/>I&longs;ogoni, indagare.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In eadem figurâ &longs;umptâ diametro circuli HN æquali EF, <lb/>auferatur à plano HR linea HQ æqualis EG; <expan abbr="eritq;">eritque</expan> motus trian­<lb/>guli in HQ æqualis duratione motui circuli in HM per 1. prop. <lb/>motus verò eiu&longs;dem circuli in plano HR æqualis duratione <lb/>terminatur chordâ HR. per prop. 15. </s><s>Exce&longs;&longs;us ergo, quo mo­<lb/>tus circuli in eodem plano e&longs;t maior motu trianguli, erit æqua­<lb/>lis lineæ QR, quam inquirebamus. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motum figurarum rectilinearum periferiâ eiu&longs;dem circuli <lb/>terminare.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Centro H de&longs;cribatur circulus: ad cuius periferiam eodem <lb/>tempore &longs;it terminandus motus ex H. </s><s>Inueniantur <expan abbr="itaq;">itaque</expan> plana; <lb/>in quibns &longs;emidiameter figuræ motûs in unâ <expan abbr="quâq;">quâque</expan> figurâ recti <lb/>lineâ, &longs;ecetur ab hypomochlio in eadem ratione, in quâ &longs;ecatur <lb/>EF à CD per 1 Lemma. </s><s>Et quia illarum grauitas mouens in <lb/>planis iam inventis eandem rationem habet ad &longs;uum mobile: <lb/>eruntmotus per po&longs;it. 4 æquales, ac proinde ij&longs;dem &longs;patijs, hoc <lb/>e&longs;t periferiâ eiu&longs;dem circuli terminabuntur. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Circulo æquale quadratum ex motu invenire.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Percutiat in motu circulus A alium circulum &longs;ibi æqualem B; <lb/><expan abbr="moveaturq;">moveaturque</expan> ex illa plagâ per &longs;patium DE rur&longs;um idem circu- <pb xlink:href="063/01/055.jpg"/>lus A habens eundem impul&longs;um, percutiat eundem circulum <lb/>B contiguum quadrato C. aut igitur moto C circulus B quie­<lb/>&longs;cet, aut illius motum con&longs;equetur. </s><s>Et &longs;i quidem quie&longs;cet, erit <lb/>per 3. Axioma grauitas in C, ac proinde per 1. Axioma huius <lb/> <arrow.to.target n="fig15"/><lb/>area æqualis circulo B. </s><s>Quòd &longs;i verò ad illius motum move­<lb/>tur; erit quadratum C per idem Axioma minus circulo B. </s><lb/><s>Moveatur <expan abbr="itaq;">itaque</expan> B ex illâ plagâ per &longs;patium ML: & quadratum <lb/>C per &longs;patium HI. </s><s>Supponamus verò HI æquale DE, & du­<lb/>plum &longs;patij ML. </s><s>Cùm <expan abbr="itaq;">itaque</expan> pro men&longs;urâ plagæ minuatur im­<lb/>pul&longs;us, ex demon&longs;tratis ad propo&longs;. 31. & motus eandem ratio­<lb/>nem habeant, quam impul&longs;us, per po&longs;it: 4. &longs;it autem motus in <lb/>DE ad motum in ML duplus; erit <expan abbr="quoq;">quoque</expan> impul&longs;us in A ad reli­<lb/>quum impul&longs;um in B, ac proinde ad impul&longs;um in C duplus. </s><lb/><s>Quia verò quadratum C movetur ab æquali impul&longs;u per &longs;pa­<lb/>tium HI duplum &longs;patij ML, erit <expan abbr="quoq;">quoque</expan> circulus B duplus qua­<lb/>drati C. </s><s>Quòd &longs;i enim &longs;emi&longs;&longs;em circuli moveat idem impul­<lb/>&longs;us, quia tum per Corollarium 2. theorematis 1. impul&longs;um ha­<lb/>bet duplum, movebit per po&longs;it. 4. ad intervallum duplum. hoc <pb xlink:href="063/01/056.jpg"/>e&longs;t HI. </s><s>Igitur &longs;i a&longs;&longs;umatur duplum quadrati C, inventum erit <lb/>quadratum æquale dato circulo B. </s></p> <figure id="id.063.01.056.1.jpg" xlink:href="063/01/056/1.jpg"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Alius modus quadrandi circulum ex motu.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>LEMMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si figuræ maior in motu percutiat minorem, habeat verò &longs;egmentum <lb/>&longs;emidiametri figuræ motûs, quod e&longs;t inter lineam hypomochlij, et extre­<lb/>mum motûs, eandem rationem ad alterum &longs;egmentum, quod e&longs;t inter <lb/>eandem lineam hypomochlij & figuræ centrum, quam habet figura <lb/>minor ad maiorem, motus maioris à percußione erit parallelus lineæ <lb/>rectæ per contactum.<emph.end type="italics"/></s></p> <p type="main"> <s>Percutiat quadratum ABCD circulum H in G. & duca­<lb/>tur linea hypomochlij GI &longs;ecans &longs;emidiametrum figuræ mo­<lb/>tûs AE in F: <expan abbr="&longs;itq;">&longs;itque</expan> AF ad FE, ut circulus H ad quadratum <lb/> <arrow.to.target n="fig16"/><lb/>ABCD: Dico motum quadrati à percu&longs;&longs;ione e&longs;&longs;e parallelum <lb/>lateri AB, hoc e&longs;t lineæ rectæ per contactum G. </s><s>Quia enim <pb xlink:href="063/01/057.jpg"/>ut AF men&longs;ura plagæ ad EF re&longs;iduum impul&longs;um, ita circulus <lb/>H, ad quadratum ABCD: erit permutando AF ad H, ut FE <lb/>ad ABCD: ac proinde per po&longs;it. 4. eadem velocitas motûs in <lb/><expan abbr="utrâq;">utrâque</expan> figurâ. </s><s>Quadratum ergo ABCD nullam à circulo per­<lb/>cu&longs;&longs;o recipit plagam. </s><s>Et quia præpondium e&longs;t in E, propterea <lb/>quòd impul&longs;us in AF defecit ex illâ plagâ; nece&longs;&longs;e librationem <lb/>fieri in G. </s><s>Nequit autem revolui centrum E, ni&longs;ilatus AB &longs;e­<lb/>cet circulum H, aut hic à plagâ velociùs &longs;e abducat. </s><s>Quia ve­<lb/>rò eadem velocitas motûs, nece&longs;&longs;e motum in E per lineam fi­<lb/>eri parallelam lateri AB. </s></p> <figure id="id.063.01.057.1.jpg" xlink:href="063/01/057/1.jpg"/> <p type="main"> <s><emph type="center"/>LEMMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si figura maior in motu percutiat minorem; habeat verò &longs;egmentum <lb/>&longs;emidiametri figuræ motûs, quod e&longs;t inter lineam hypomochlij & extre­<lb/>mum motûs, minorem rationem ad alterum &longs;egmentum, quod e&longs;t inter <lb/>eandem lineam hypomochlij & figuræ centrum, quàm habeat figura <lb/>minor ad maiorem, motus figuræ maioris erit parallelus lineæ mediæ <lb/>inter tangentem circuli, & lineam productam à centro maioris ad con­<lb/>tactum.<emph.end type="italics"/></s></p> <p type="main"> <s>Habeat AF ad FE <expan abbr="minor&etilde;">minorem</expan> <expan abbr="ration&etilde;">rationem</expan>, quàm circulus H ad qua­<lb/>dratum ABCD: dico, motum E figuræ maioris e&longs;&longs;e <expan abbr="parallelũ">parallelum</expan> <lb/>lineæ GK mediæ inter GB & GE. </s><s>Quia enim minorem ra­<lb/>tionem habet AF ad FE, quàm circulus H ad quadratum <lb/>ABCD; & permutando AF ad H, quàm FE ad ABCD, mi­<lb/>nori velocitate movebitur ex illâ plagà circulus H, quàm <lb/>quadratum ABCD: eandem ergo recipit à circulo percu&longs;&longs;o, <lb/>quam dedit plagam. </s><s>Et quia præpondium in E; ob tardita­<lb/>tem motûs circuli ad lineam determinatur parallelam lateri <lb/>AB per 1. Lemma: impul&longs;um verò recipit à circulo H per <pb xlink:href="063/01/058.jpg"/>lineam GE per theorem. 5. <expan abbr="&longs;untq;">&longs;untque</expan> impul&longs;us &longs;ubcontrarij; erit <lb/>motus E per prop. 31. de proportione motûs, parallelus lineæ <lb/>GK mediæ inter GB & GE. </s></p> <p type="main"> <s><emph type="center"/>LEMMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si figura maior in motu percutiat minorem; habeat verò &longs;egmentum <lb/>&longs;emidiametri figuræ motûs, quod e&longs;t inter lineam hypomochlij, & ex­<lb/>tremum motûs, maiorem rationem ad reliquum &longs;egmentum, quod e&longs;t <lb/>inter eandem lineam hypomochlij & figuræ centrum, quàm habeat mi­<lb/>nor figura ad maiorem; motus maioris erit parallelus lineæ mediæ inter <lb/>tangentem circuli & eiu&longs;dem perpendicularem ad contactum.<emph.end type="italics"/></s></p> <p type="main"> <s>Habeat AF ad EF maiorem rationem, quàm circulus H ad <lb/>quadratum ABCD: Dico, huius motum ab illâ plagâ e&longs;&longs;e pa­<lb/>rallelum lineæ mediæ inter GB & GH. </s><s>Quia enim men&longs;ura <lb/>plagæ AF ad re&longs;iduum impul&longs;um in FE maiorem rationem <lb/>habet, quàm circulus H ad quadratum ABCD: & permu­<lb/>tando AF ad H, quàm FE ad ABCD, erit velocior motus <lb/>circuli H, quàm quadrati ABCD. nullam ergo à circulo per­<lb/>cu&longs;&longs;o recipit plagam. </s><s>Et quia præpondium in E, nece&longs;&longs;e libra­<lb/>tionem fieri in G: ac proinde motum in E e&longs;&longs;e parallelum lineæ <lb/>mediæ inter GB & GH. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Circulo æquale quadratum ex motu invenire.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Percutiat quadratum ABCD circulum H, & ex illâ plagà <lb/>moveatur centrum E per lineam parallelam lateri GB: duca­<lb/>tur autem linea hypomochlij FG &longs;ecans &longs;emidiametrum figu­<lb/>ræ motûs AE in F. </s><s><expan abbr="Eritq;">Eritque</expan> per 1. Lemma AF ad FE, ut circulus <pb xlink:href="063/01/059.jpg"/>Had ABCD. </s><s>Hoc e&longs;t permutando ut AF ad H, ita FE ad <lb/>ABCD. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> fiat ut FE ad AF, ita ABCD ad aliud <lb/>quadratum: inventum erit circulo H æquale quadratum. </s></p> <p type="main"> <s>Quòd &longs;i ex illâ plagâ moveatur E per lineam parallelam GK: <lb/>erit per Lemma 2. minor proportio AF ad H, quàm FE ad <lb/>ABCD: <expan abbr="Atq;">Atque</expan> huius motus velocior motu circuli. eandem er <lb/>gò plagam recipit quadratum ABCD, quam infert circulo: <lb/>ac proinde illius impul&longs;us à percu&longs;&longs;ione erit æqualis AE: com <lb/>po&longs;itus nimirum ex plagâ reciprocâ AF & impul&longs;u re&longs;iduo FE. </s><lb/><s>Supponamus verò AE ad AF e&longs;&longs;e ut 6 ad 2, hoc e&longs;t in ratione <lb/>triplâ: &longs;patium verò decur&longs;um ab E ad &longs;patium decur&longs;um ab H <lb/>ut 3 ad 2. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> circulus H accipiat impul&longs;um ut 3. hoc <lb/>e&longs;t additâ &longs;emi&longs;&longs;e, movebitur ad idem intervallum cumquadra­<lb/>to ABCD. </s><s>Et &longs;i fiat ut 6 ad 3, ita ABCD ad aliud, inventum <lb/>erit quadratum circulo H æquale. </s></p> <p type="main"> <s>Demum &longs;i motus quadrati E à percu&longs;&longs;ione fiat parallelus li <lb/>neæ mediæ inter tangentem GB, & perpen dicularem GH; erit <lb/>per Lemma 3 maior proportio AF ad H, quàm FE ad ABCD: <lb/>& motus H velocior motu ABCD. </s><s>Ponamus <expan abbr="itaq;">itaque</expan> interval­<lb/>lum motûs Had interuallum motûs ABCD in &longs;e&longs;qui alterâ <lb/>ratione, hoc e&longs;t ut 3 ad 2: FE autem ad AF ut 4 ad 2. </s><s>Quòd&longs;i <lb/><expan abbr="itaq;">itaque</expan> quadratum ABCD accipiat impul&longs;um ut 6; movebitur <lb/>eadem velocitate, & ad idem intervallum cum circulo H per <lb/>po&longs;it. 5. propterea quòd impul&longs;us eandem rationem habeat ad <lb/>&longs;uum mobile, per corollarium 2. 1 Axiomatis. </s><s>Et &longs;i fiat ut 6 <lb/>ad 2, ita quadratum ABCD ad aliud quadratum, inventum <lb/>erit circulo H æquale quadratum. </s></p> <pb xlink:href="063/01/060.jpg"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>ALIA QVADRATVRA CIRCVLI<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>per motum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>DVcatur à contactu G per centrum figuræ E linea GL æ­<lb/>qualis GB: & ex L ad eam perpendicularis LM &longs;ecans B <lb/>C in M: <expan abbr="eritq;">eritque</expan> LM æqualis BM. </s><s>Si enim iungatur recta BL, <lb/>duo anguli GBL. GLB, ac proinde re&longs;idui MBL. MLB &longs;unt <lb/>æquales. </s><s>Centro <expan abbr="itaq;">itaque</expan> M, interuallo ML de&longs;cribatur arcus LB <lb/>&longs;ecans <expan abbr="lineã">lineam</expan> motûs reflexi GK in O: ex O verò demittantur per <lb/>pendiculares ON. OP. </s><s>Quoniam <expan abbr="itaq;">itaque</expan> punctum G à plagâ re­<lb/>ciprocâ ex H per lineam agitur GL per 5 theorema: impul&longs;us <lb/>verò re&longs;iduus in FE per lineam GB per lemma 2. </s><s><expan abbr="E&longs;tq;">E&longs;tque</expan> motus <lb/>medius GK, erit per problem. propo&longs;itionis 35 de propor. mo­<lb/>tûs, vt OP ad ON, ita impul&longs;us in GB ad impul&longs;um in GL, æ­<lb/>qualem impul&longs;ui in H. </s><s>Et &longs;i quidem ON e&longs;t &longs;emi&longs;&longs;is OP, erit <lb/>impul&longs;us in OP ad impul&longs;um in ON ut 4 ad 2. &longs;upponamus ve­<lb/>rò &longs;patium decur&longs;um ab E, ad &longs;patium decur&longs;um ab H e&longs;&longs;e in <lb/>&longs;e&longs;quialterâ ratione, hoc e&longs;t ut 3 ad 2. </s><s>Igitur &longs;i circulus H acci­<lb/>piat impul&longs;um ut 3, movebitur ad idem interuallum cum qua­<lb/>drato ABCD per corollarium 2 Axiomatis 1 & po&longs;itionem 4. </s><lb/><s>Et&longs;i fiat ut 4 ad 3 ita ABCD ad aliud; inventum erit quadra­<lb/>tum dato circulo H æquale. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLL ARIVM<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Eadem ratione inveniemus quadratum æquale &longs;ectionibus <lb/>conicis, <expan abbr="atq;">atque</expan> adeo illarum fru&longs;tis; &longs;i loco circuli hu­<lb/>iu&longs;modi figuras &longs;ub&longs;tituamus.<emph.end type="center"/></s></p> <pb xlink:href="063/01/061.jpg"/> <p type="main"> <s><emph type="center"/>PARS TERTIA.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>DE MOTV REFLEXO FIGVR ARVM<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>RECTILINEARVM.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>E gi de motu reflexo in lib: de proport: motûs, à prop: 36. ad 40. ve­<lb/>rùm hunc non ni&longs;i in circulo expendi. </s><s>Licet verò in Quadraturâ cir­<lb/>culi motus quo&que; re&longs;texus interueniat; dum ab illatâ plagâ aliò, quàm <lb/>ferebatur, viam capeßit: hic tamen unà hypomochlium mouetur: ne&que; <lb/>huius principium e&longs;t grauitas. </s><s>Nece&longs;&longs;e ergo in figuris quo&que; rectilineis <lb/>hunc motum reflexum, quatenus à grauitate & hypomochlio immoto <lb/>procedit, con&longs;idexare.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>THEOREMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus trianguli I&longs;ogoni ad planum & ba&longs;im perpendicularis, in <lb/>&longs;e ip&longs;um reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>TRiangulo <emph type="italics"/>abc<emph.end type="italics"/> labenti occurrat planum <emph type="italics"/>az:<emph.end type="italics"/> <expan abbr="&longs;itq;">&longs;itque</expan> motus <lb/>centri <emph type="italics"/>d<emph.end type="italics"/> ad illud planum, & ba&longs;im <emph type="italics"/>ab<emph.end type="italics"/> perpendicularis <lb/>dico hunc motum in &longs;e ip&longs;um reflecti. </s><s>Nam in primâ quidem <lb/>figurâ motus centri <expan abbr="atq;">atque</expan> huius plaga e&longs;t in eadem lineâ <emph type="italics"/>dc:<emph.end type="italics"/> da­<lb/>bit ergo plagam perfectam. & quia per eandem lineam <emph type="italics"/>dc<emph.end type="italics"/> re­<lb/>cipit à percu&longs;&longs;o æqualem illi, quam dedit plagam per 5 theor: <lb/>2 partis, motus in &longs;e ip&longs;um reflectit. </s><s>In &longs;ecundâ autem figurâ <lb/>percu&longs;&longs;io fit per idem theor. per lineas <emph type="italics"/>da, df, db;<emph.end type="italics"/> e&longs;tq motus <lb/>centri in lineâ <emph type="italics"/>df:<emph.end type="italics"/> erit ergo motus reflexus à plagâ <emph type="italics"/>df<emph.end type="italics"/> in ea­<lb/>dem lineâ <emph type="italics"/>df.<emph.end type="italics"/> at verò plaga in <emph type="italics"/>ad<emph.end type="italics"/> & <emph type="italics"/>bd<emph.end type="italics"/> centrum <emph type="italics"/>d<emph.end type="italics"/> reper­<lb/>cu&longs;&longs;um in partes agit <emph type="italics"/>dg. de.<emph.end type="italics"/> & quia plaga in <emph type="italics"/>da<emph.end type="italics"/> e&longs;t æqualis <pb xlink:href="063/01/062.jpg"/>plagæ in <emph type="italics"/>db;<emph.end type="italics"/> erit motus quoq in <emph type="italics"/>de<emph.end type="italics"/> æqualis motui in <emph type="italics"/>dg:<emph.end type="italics"/> ac <lb/>proinde per prop: 31 motus medius reflectit per lineam <emph type="italics"/>dc.<emph.end type="italics"/><lb/>Cùm igitur hæc &longs;it via centri, motus trianguli in &longs;e ip&longs;um re­<lb/>flectit. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus trianguli I&longs;ogoni ad planum, non verò ad ba&longs;im perpen­<lb/>dicularis, in partem ba&longs;is maiorem reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Triangulum <emph type="italics"/>abc<emph.end type="italics"/> occurrat plano <emph type="italics"/>az<emph.end type="italics"/> ad angulos rectos: <lb/><expan abbr="&longs;ecetq;">&longs;ecetque</expan> motus centri <emph type="italics"/>d<emph.end type="italics"/> ba&longs;im <emph type="italics"/>ac<emph.end type="italics"/> in duo &longs;egmenta <emph type="italics"/>kc<emph.end type="italics"/> maius, <lb/>& <emph type="italics"/>ka<emph.end type="italics"/> minus: dico motum reflexum fieri in partem <emph type="italics"/>kc<emph.end type="italics"/> &longs;e ­<lb/> <arrow.to.target n="fig17"/><lb/>gmenti maioris. </s><s>Excitetur enim linea hypomochlij <emph type="italics"/>af:<emph.end type="italics"/> quam <lb/>&longs;ecet linea <emph type="italics"/>de<emph.end type="italics"/> à centro perpendicularis quia <expan abbr="itaq;">itaque</expan> vectis e&longs;t <lb/><emph type="italics"/>da;<emph.end type="italics"/> <expan abbr="atq;">atque</expan> huius quadratum, ide&longs;t totam grauitatem, &longs;ecat bi­<lb/>fariam linea hypomochlij, iuxta demon&longs;trata in lib: de propor: <lb/>motûs; &longs;i quadratum <emph type="italics"/>ed<emph.end type="italics"/> fit grauitas mouens centri, erit hu­<lb/>ius complementum quadratum <emph type="italics"/>ae,<emph.end type="italics"/> men&longs;ura percu&longs;sionis &longs;cu <pb xlink:href="063/01/063.jpg"/>plagæ. </s><s>Et quia motus centri fit per lineam <emph type="italics"/>di<emph.end type="italics"/> tangentem cir­<lb/>culi centro <emph type="italics"/>a<emph.end type="italics"/> de&longs;cripti per prop: 4: motus autem reflexus à <lb/>plagâ per lineam <emph type="italics"/>dg<emph.end type="italics"/> per 5 theor. 2 part. &longs;i fiat ut <emph type="italics"/>de<emph.end type="italics"/> ad <emph type="italics"/>ea<emph.end type="italics"/> ita <lb/><emph type="italics"/>di<emph.end type="italics"/> ad <emph type="italics"/>dg,<emph.end type="italics"/> erit per prop: 32 motus medius <emph type="italics"/>dh<emph.end type="italics"/> diameter pa­<lb/>rallelogrammi <emph type="italics"/>aihg:<emph.end type="italics"/> ac proinde motus reflexus in partem <lb/><emph type="italics"/>kc<emph.end type="italics"/> &longs;egmenti maioris. </s></p> <figure id="id.063.01.063.1.jpg" xlink:href="063/01/063/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Quadrati perpendicularis ad planum, &longs;i æqualiter &longs;ecet an­<lb/>gulum, aut latus eiu&longs;dem quadrati, in &longs;e ip&longs;um reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Incidat plano <emph type="italics"/>ax<emph.end type="italics"/> perpendiculariter Quadratum <emph type="italics"/>abcd:<emph.end type="italics"/> <expan abbr="&longs;e-cetq;">&longs;e­<lb/>cetque</expan> motus centri <emph type="italics"/>f<emph.end type="italics"/> latus <emph type="italics"/>ad<emph.end type="italics"/> aut angulum <emph type="italics"/>adc<emph.end type="italics"/> in duas par. <lb/>tes æquales: dico, hunc motum in &longs;e ip&longs;um reflecti. </s><s>Nam in <lb/>primâ figurâ, quia coincidit motus centri, & plaga in eandem <lb/>lineam <emph type="italics"/>fd;<emph.end type="italics"/> erit motus à percu&longs;&longs;ione in viâ centri: ac proinde <lb/>in &longs;e ip&longs;um reflexus. </s><s>Infigurâ autem &longs;ecundâ plaga fit per lineas <lb/><emph type="italics"/>fa. fe. fd.<emph.end type="italics"/> per 4. theorema 2 part: & à plagâ quidem in <emph type="italics"/>fe,<emph.end type="italics"/> quòd <lb/>hæc &longs;it via centri, motus in &longs;e ip&longs;um reflectit: à plagâ verò in <lb/><emph type="italics"/>fa<emph.end type="italics"/> & <emph type="italics"/>fd,<emph.end type="italics"/> in partes oppo&longs;itas <emph type="italics"/>fc. fb<emph.end type="italics"/> agitur centrum grauitatis <lb/>per 1 theor: & quia angulus <emph type="italics"/>bfc<emph.end type="italics"/> e&longs;t minor duobus rectis, ac <lb/>proinde motus in <emph type="italics"/>fc. fb<emph.end type="italics"/> per definit. 4 &longs;ubcontrarij; ob æqua­<lb/>les verò plagas <emph type="italics"/>af. df<emph.end type="italics"/> inter &longs;e æquales; erit per prop: 32 mo­<lb/>tus medius in lineâ <emph type="italics"/>fg.<emph.end type="italics"/> Cùm ergo hæc &longs;it via centri, motus <lb/>Quadrati in &longs;e ip&longs;um reflectit. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Quadrati perpendicularis ad planum, inæqualiter autem <lb/>&longs;ecans angulum &longs;eu ba&longs;im, reflectit in partem &longs;egmenti maioris.<emph.end type="italics"/><emph.end type="center"/></s></p> <pb xlink:href="063/01/064.jpg"/> <p type="main"> <s>Idem Quadratum <emph type="italics"/>abcd<emph.end type="italics"/> occurrat plano <emph type="italics"/>ax<emph.end type="italics"/> ad angulos re­<lb/>ctos, motu centri <emph type="italics"/>e<emph.end type="italics"/> inæqualiter &longs;ecante ba&longs;im <emph type="italics"/>ad<emph.end type="italics"/> in <emph type="italics"/>pd<emph.end type="italics"/> maius, <lb/>& <emph type="italics"/>ap<emph.end type="italics"/> minus &longs;egmentum: dico motum reflecti in illam partem, <lb/>in quâ e&longs;t &longs;egmentum maius <emph type="italics"/>pd.<emph.end type="italics"/> Ductâ enim lineâ hypo­<lb/>mochlij <emph type="italics"/>ag,<emph.end type="italics"/> & à centro ad eam perpendiculari <emph type="italics"/>ef;<emph.end type="italics"/> erit gra­<lb/>uitas mouens centri à percu&longs;&longs;ione quadratum <emph type="italics"/>ef,<emph.end type="italics"/> <expan abbr="atq;">atque</expan> huius <lb/>complementum quadratum <emph type="italics"/>af<emph.end type="italics"/> men&longs;ura plagæ: vectis autem <lb/><emph type="italics"/>ea,<emph.end type="italics"/> cuius quadratum grauitas tota, &longs;eu impul&longs;us. </s><s>Et quia <lb/>plaga fit per lineam <emph type="italics"/>ea;<emph.end type="italics"/> erit motus à percu&longs;&longs;ione in eadem lineâ <lb/><emph type="italics"/>ea:<emph.end type="italics"/> per 5 theor. 2 part: motus autem centri à reliquo impul&longs;u <lb/>in lineâ <emph type="italics"/>ek<emph.end type="italics"/> tangente circuli centro <emph type="italics"/>a<emph.end type="italics"/> de&longs;cripti. </s><s>Quòd &longs;i ergo <lb/>fiat ut <emph type="italics"/>ef<emph.end type="italics"/> motus centri ad <emph type="italics"/>af<emph.end type="italics"/> motum repercu&longs;&longs;um, ita <emph type="italics"/>ek<emph.end type="italics"/> ad <lb/><emph type="italics"/>eh;<emph.end type="italics"/> erit diameter parallelogrammi <emph type="italics"/>ehik<emph.end type="italics"/> motus medius per <lb/>prop: 32 ac proinde motus reflexus in partem &longs;egmenti ma­<lb/>ioris </s></p> <p type="main"> <s><emph type="center"/>THEOREMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Pentagoni perpendicularis ad planum & latus eiusdem, <lb/>in &longs;e ip&longs;um reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Nam in primâ quidem figurâ, quia & motus centri & pla­<lb/>ga tota e&longs;t in lineâ <emph type="italics"/>ef;<emph.end type="italics"/> erit motus reflexus in eadem lineâ <emph type="italics"/>ef.<emph.end type="italics"/><lb/>In &longs;ecundâ autem figurâ lineæ percu&longs;&longs;ionis &longs;unt <emph type="italics"/>fa fg fe:<emph.end type="italics"/><lb/>motus ergò reflexus in <emph type="italics"/>fh. fc. fi.<emph.end type="italics"/> Et quia motus in <emph type="italics"/>fh<emph.end type="italics"/> & <emph type="italics"/>fi<emph.end type="italics"/><lb/>&longs;unt &longs;ub contrarij <expan abbr="atq;">atque</expan> inter &longs;e æquales per defini: 4 erit per <lb/>prop: 32 motus medius linea <emph type="italics"/>fc:<emph.end type="italics"/> ac proinde cùm hæc &longs;it via con­<lb/>tri, motus in &longs;e ip&longs;um reflectit. </s></p> <pb xlink:href="063/01/065.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Pentagoni perpendicularis ad planum, non verò ad latus <lb/>eiu&longs;dem, reflectit in partem &longs;egmenti maioris.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Motus Pentagoni <emph type="italics"/>abcde<emph.end type="italics"/> perpendicularis ad planum &longs;e­<lb/>cet latus <emph type="italics"/>ae<emph.end type="italics"/> in duo &longs;egmenta <emph type="italics"/>le<emph.end type="italics"/> maius, & <emph type="italics"/>al<emph.end type="italics"/> minus: Dico <lb/>à percu&longs;&longs;o illo plano reflecti in partem <emph type="italics"/>le<emph.end type="italics"/> &longs;egmenti maioris. </s><lb/><s>Nam &longs;i excitetur linea hypomochlij <emph type="italics"/>ag,<emph.end type="italics"/> & à centro ducatur li­<lb/>nea <emph type="italics"/>fg<emph.end type="italics"/> ad eam perpendicularis; erit quadratum <emph type="italics"/>fg<emph.end type="italics"/> grauitas <lb/>mouens centri; huius autem complementum quadratum <emph type="italics"/>ag<emph.end type="italics"/><lb/>men&longs;ura plagæ: propterea quòd tota grauitas &longs;it æqualis qua­<lb/>drato <emph type="italics"/>af.<emph.end type="italics"/> Et quia plaga fit per lineam <emph type="italics"/>af,<emph.end type="italics"/> erit motus reflexus in <lb/>eadem lineâ <emph type="italics"/>af:<emph.end type="italics"/> motus autem centri in lineâ <emph type="italics"/>fk<emph.end type="italics"/> tangente cir­<lb/>culi centro <emph type="italics"/>a<emph.end type="italics"/> de&longs;cripti. </s><s>Quòd &longs;i ergo fiat ut <emph type="italics"/>fg<emph.end type="italics"/> ad <emph type="italics"/>ga,<emph.end type="italics"/> ita <emph type="italics"/>fk<emph.end type="italics"/> ad <lb/><emph type="italics"/>fh;<emph.end type="italics"/> erit per prop: 32 motus medius diameter parallelogram­<lb/>mi <emph type="italics"/>faik:<emph.end type="italics"/> ac proinde motus pentagoni reflectit in partem <emph type="italics"/>le<emph.end type="italics"/><lb/>&longs;egmenti maioris. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Trianguli i&longs;ogoni ad ba&longs;im, non verò ad planum perpen­<lb/>dicularis, &longs;i in verticem moueatur, in &longs;e ip&longs;um reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In| 1 figurâ trianguli <emph type="italics"/>efg<emph.end type="italics"/> latus <emph type="italics"/>ef<emph.end type="italics"/> &longs;ecetur à motu eiu&longs;dem <lb/><emph type="italics"/>hg<emph.end type="italics"/> æqualiter: occurrat autem plano <emph type="italics"/>ik<emph.end type="italics"/> motu in <emph type="italics"/>g<emph.end type="italics"/> verticem <lb/>conver&longs;o: Dico hunc motum in &longs;e ip&longs;um reflecti. </s><s>Quia enim <lb/>motus centri & plagæ, quam dat, <expan abbr="recipitq;">recipitque</expan> centrum, e&longs;t in <expan abbr="ead&etilde;">eadem</expan> <lb/>lineâ <emph type="italics"/>hg,<emph.end type="italics"/> erit motus à percu&longs;&longs;ione in eadem lineâ <emph type="italics"/>hg<emph.end type="italics"/> per 1 <lb/>theor: ac proinde motus in &longs;e ip&longs;um reflectit. </s></p> <pb xlink:href="063/01/066.jpg"/> <figure id="id.063.01.066.1.jpg" xlink:href="063/01/066/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Trianguli I&longs;ogoni ad ba&longs;im, non verò ad planum perpendi­<lb/>cularis, &longs;i in ba&longs;im moveatur, uno latere eidem plano par alle­<lb/>lo, ad angulos æquales re&longs;tectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In 2 figurâ moveatur triangulum <emph type="italics"/>bcd<emph.end type="italics"/> in ba&longs;im <emph type="italics"/>cd,<emph.end type="italics"/> &longs;ectam <lb/>bifariam & æqualiter à motu centri in <emph type="italics"/>a.<emph.end type="italics"/> <expan abbr="&longs;itq;">&longs;itque</expan> latus <emph type="italics"/>bd<emph.end type="italics"/> paral­<lb/>lelum plano: Dico in hoc ca&longs;u triangulum <emph type="italics"/>bcd<emph.end type="italics"/> motu reflexo <lb/>angulum con&longs;tituere æqualem illi, quem facit cum eodem pla­<lb/>no huius lap&longs;us. </s><s>Excitetur enim linea hypomochlij <emph type="italics"/>cf,<emph.end type="italics"/> du­<lb/>ctâ lineâ à centro perpendiculari <emph type="italics"/>ai.<emph.end type="italics"/> quia <expan abbr="itaq;">itaque</expan> ex demon&longs;tra­<lb/>tis plaga e&longs;t æqualis quadrato <emph type="italics"/>ci,<emph.end type="italics"/> & grauitas mouens centri <lb/>æqualis quadrato <emph type="italics"/>ai:<emph.end type="italics"/> e&longs;t autem plaga, & qui hanc &longs;equitur mo­<lb/>tus reflexus in lineâ <emph type="italics"/>ac<emph.end type="italics"/> per 1 theor: motus verò centri in lineâ <lb/>tangente circuli centro <emph type="italics"/>c<emph.end type="italics"/> <expan abbr="atq;">atque</expan> interuallo <emph type="italics"/>ac<emph.end type="italics"/> de&longs;cripti, paralle­<lb/>la nimirum plano <emph type="italics"/>eg:<emph.end type="italics"/> &longs;i fiat ut <emph type="italics"/>ci<emph.end type="italics"/> ad <emph type="italics"/>ai,<emph.end type="italics"/> ita <emph type="italics"/>cl<emph.end type="italics"/> ad <emph type="italics"/>cm;<emph.end type="italics"/> erit mo­<lb/>tus medius <emph type="italics"/>cn<emph.end type="italics"/> diameter parallclogrammi <emph type="italics"/>clmn:<emph.end type="italics"/> Dico angu­<lb/>lum <emph type="italics"/>ncm<emph.end type="italics"/> e&longs;&longs;e æqualem angulo <emph type="italics"/>fce.<emph.end type="italics"/> Quia enim recta <emph type="italics"/>ac<emph.end type="italics"/> per <pb xlink:href="063/01/067.jpg"/>centrum e&longs;t perpendicularis ad <emph type="italics"/>eg<emph.end type="italics"/> parallelum ip&longs;i <emph type="italics"/>bd,<emph.end type="italics"/> erunt an­<lb/>guli <emph type="italics"/>ace. acg<emph.end type="italics"/> inter &longs;e æquales. </s><s>Sunt autem triangula <emph type="italics"/>ica. lcn<emph.end type="italics"/><lb/>ex con&longs;tructione &longs;imilia; & angulus <emph type="italics"/>ica<emph.end type="italics"/> æqualis angulo <emph type="italics"/>lcn:<emph.end type="italics"/><lb/>quibus ablatis ex <emph type="italics"/>ace. acg<emph.end type="italics"/> anguli reliqui <emph type="italics"/>ecf. mcn,<emph.end type="italics"/> incidentiæ <lb/>& reflexionis inter &longs;e &longs;unt æquales. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Trianguli I&longs;ogoni &longs;i ne&que; ad planum, ne&que; ad ba&longs;im &longs;it per­<lb/>pendicularis, ad angulos inæquales reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In 3 figurâ triangulum <emph type="italics"/>abc<emph.end type="italics"/> occurrat plano habens latus <emph type="italics"/>ac<emph.end type="italics"/><lb/>eidem parallelum: <expan abbr="&longs;itq;">&longs;itque</expan> Iinea hypomochlij <emph type="italics"/>cd,<emph.end type="italics"/> & linea ad eam <lb/>perpendicularis <emph type="italics"/>ef:<emph.end type="italics"/> <expan abbr="eritq;">eritque</expan> grauitas mouens centri Quadratum <lb/><emph type="italics"/>ef:<emph.end type="italics"/> plaga autem huius complementum quadratum <emph type="italics"/>go.<emph.end type="italics"/> quod <lb/>quidem habetur, &longs;i lineâ <emph type="italics"/>gf<emph.end type="italics"/> &longs;ectâ bifarium in <emph type="italics"/>p,<emph.end type="italics"/> eo centro de­<lb/>&longs;cribatur &longs;emicirculus <emph type="italics"/>gof,<emph.end type="italics"/> <expan abbr="&longs;umaturq;">&longs;umaturque</expan> chorda <emph type="italics"/>fo<emph.end type="italics"/> æqualis <emph type="italics"/>fe:<emph.end type="italics"/> nam <lb/>chorda reliqua <emph type="italics"/>og<emph.end type="italics"/> dabit illud quadratum. propterea quòd gra­<lb/>uitas tota &longs;it quadratum <emph type="italics"/>fg.<emph.end type="italics"/> fiat <expan abbr="itaq;">itaque</expan> ut <emph type="italics"/>fo<emph.end type="italics"/> ad <emph type="italics"/>og,<emph.end type="italics"/> ita <emph type="italics"/>fi<emph.end type="italics"/> ad <emph type="italics"/>fb;<emph.end type="italics"/><lb/>erit motus reflexus in lineâ <emph type="italics"/>fh<emph.end type="italics"/> diametro parallelogrammi <emph type="italics"/>fb hi:<emph.end type="italics"/><lb/>angulus autem reflexionis <emph type="italics"/>ifh:<emph.end type="italics"/> quem dico angulo <emph type="italics"/>acd<emph.end type="italics"/> e&longs;&longs;e in­<lb/>æqualem. </s><s>Quia angulus <emph type="italics"/>age<emph.end type="italics"/> externus c&longs;t maior angulo in­<lb/>terno <emph type="italics"/>ecg,<emph.end type="italics"/> æqualis autem angulo <emph type="italics"/>ofg;<emph.end type="italics"/> propterea quòd <expan abbr="uterq;">uterque</expan> <lb/>a&longs;&longs;umpto angulo communi <emph type="italics"/>ogf<emph.end type="italics"/> facit rectum: e&longs;t verò huic <lb/>angulo æqualis angulus reflexionis <emph type="italics"/>hfi;<emph.end type="italics"/> quòd &longs;imilia &longs;int trian­<lb/>gula <emph type="italics"/>gef: hfi:<emph.end type="italics"/> erit ergo æqualis <expan abbr="quoq;">quoque</expan> angulo externo <emph type="italics"/>age:<emph.end type="italics"/> ac <lb/>proinde maior interno <emph type="italics"/>acd<emph.end type="italics"/> angulo incidentiæ. </s><s>In 4 demum <lb/>figurâ centrum <emph type="italics"/>e<emph.end type="italics"/> cadat intra lineam hypomochlij. cùm igitur <lb/>centrum gravitatis contineatur in hypomochlio, erit plaga per­<lb/>fecta: <expan abbr="atq;">atque</expan> huius lineæ <emph type="italics"/>ea. ef. ec:<emph.end type="italics"/> ac proinde per 1 theor: hu­<lb/>ius motus reflexus in lineâ <emph type="italics"/>eb.<emph.end type="italics"/> <!--neuer Satz-->Quia ergo angulus reflexionis <pb xlink:href="063/01/068.jpg"/><emph type="italics"/>efc,<emph.end type="italics"/> nimirum rectus, maior e&longs;t angulo incidentiæ <emph type="italics"/>dcf;<emph.end type="italics"/> motus <lb/>trianguli in eo &longs;itu ad angulos reflectit inæquales. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Si motus Quadrati obliquè, huius autem diameter ad angulos re­<lb/>ctos &longs;ecet planum; ad angulos æquales reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Motus Quadrati <emph type="italics"/>abcd<emph.end type="italics"/> &longs;ecet obliquè planum <emph type="italics"/>el,<emph.end type="italics"/> diameter <lb/>verò <emph type="italics"/>ag<emph.end type="italics"/> ad angulos rectos: dico motum reflexum ab hoc pla­<lb/>no angulum con&longs;tituere æqualem angulo incidentiæ. </s><s>Sit enim <lb/><emph type="italics"/>ap<emph.end type="italics"/> hypomochlij, & <emph type="italics"/>gh<emph.end type="italics"/> linea ad eam perpendicularis: <expan abbr="eritq;">eritque</expan> <lb/>ex iam demon&longs;tratis <expan abbr="quadratũ">quadratum</expan> <emph type="italics"/>hg<emph.end type="italics"/> motus centri, & <emph type="italics"/>ah<emph.end type="italics"/> eiu&longs;dem <lb/>plaga. </s><s>Et quia percu&longs;sic in <emph type="italics"/>ag,<emph.end type="italics"/> erit motus reflexus in eadem <lb/>hneâ <emph type="italics"/>ag:<emph.end type="italics"/> motus autem centri in lineâ plano <emph type="italics"/>el<emph.end type="italics"/> parallelâ. quòd <lb/>&longs;i <expan abbr="itaq;">itaque</expan> fiat ut <emph type="italics"/>ah<emph.end type="italics"/> ad <emph type="italics"/>hg,<emph.end type="italics"/> ita <emph type="italics"/>af<emph.end type="italics"/> ad <emph type="italics"/>ak,<emph.end type="italics"/> erit motus medius <emph type="italics"/>ai,<emph.end type="italics"/> & an­<lb/>gulus reflexionis <emph type="italics"/>iak:<emph.end type="italics"/> quem dico e&longs;&longs;e æqualem angulo <emph type="italics"/>eap.<emph.end type="italics"/><lb/>Quia enim diameter <emph type="italics"/>ag<emph.end type="italics"/> &longs;ecat planum in <emph type="italics"/>a<emph.end type="italics"/> ad angulos rectos; <lb/>erit angulus <emph type="italics"/>eag<emph.end type="italics"/> æqualis angulo <emph type="italics"/>kag.<emph.end type="italics"/> &longs;unt auté per con&longs;tructio­<lb/>nem &longs;imilia triangula <emph type="italics"/>gha. afi;<emph.end type="italics"/> & angulus <emph type="italics"/>gah<emph.end type="italics"/> æqualis angu­<lb/>lo <emph type="italics"/>fai;<emph.end type="italics"/> igitur angulus reliquus <emph type="italics"/>eap<emph.end type="italics"/> e&longs;t æqualis angulo reliquo <lb/><emph type="italics"/>iak<emph.end type="italics"/> angulus nimirum incidentiæ angulo reflexionis: </s></p> <figure id="id.063.01.068.1.jpg" xlink:href="063/01/068/1.jpg"/> <pb xlink:href="063/01/069.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Si ne&queacute; motus Quadrati, ne&que; huius diameter ad angulos rectos &longs;e­<lb/>cet planum, ad angulos inæquales reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Motus Quadrati <emph type="italics"/>abcd<emph.end type="italics"/> obliquè &longs;ecans planum <emph type="italics"/>gr,<emph.end type="italics"/> habeat <lb/>latus <emph type="italics"/>ad<emph.end type="italics"/> eidem plano parallelum: & &longs;it linea hypomochlij <emph type="italics"/>dg.<emph.end type="italics"/><lb/>ad eam verò perpendicularis <emph type="italics"/>eh;<emph.end type="italics"/> |cuius quadratum grauitas <lb/>movens centri, <expan abbr="atq;">atque</expan> huius complementum quadratum <emph type="italics"/>fi,<emph.end type="italics"/> pla­<lb/>ga eiu&longs;dem centri. </s><s>Quod quidem quadratum in &longs;emicirculo <lb/><emph type="italics"/>fie<emph.end type="italics"/> con&longs;tituit chorda reliqua, in quo chorda <emph type="italics"/>ie<emph.end type="italics"/> &longs;it &longs;umpta æ­<lb/>qualis <emph type="italics"/>eh.<emph.end type="italics"/> Et quia plaga fit per lineas <emph type="italics"/>ea. ef. ed:<emph.end type="italics"/> per 4. theo. 2 part. <lb/>erit per 3 theor: huius, motus reflexus in lineâ <emph type="italics"/>ek;<emph.end type="italics"/> motus <lb/>autem centri in lineâ plano <emph type="italics"/>qr<emph.end type="italics"/> parallelâ, &longs;eu tangente cir culi <lb/>centro <emph type="italics"/>f,<emph.end type="italics"/> & interuallo <emph type="italics"/>fe<emph.end type="italics"/> de&longs;cripti. quòd &longs;i ergo fiat ut <emph type="italics"/>ci<emph.end type="italics"/> ad <lb/><emph type="italics"/>if,<emph.end type="italics"/> ita <emph type="italics"/>em<emph.end type="italics"/> ad <emph type="italics"/>ek,<emph.end type="italics"/> erit per prop: 32 motus medius <emph type="italics"/>el<emph.end type="italics"/> diameter <lb/>parallelogrammi <emph type="italics"/>kelm:<emph.end type="italics"/> dico angulum reflexionis <emph type="italics"/>lem<emph.end type="italics"/> e&longs;&longs;e in <lb/>æqualem angulo <emph type="italics"/>adg.<emph.end type="italics"/> Quia enim angulus <emph type="italics"/>afi<emph.end type="italics"/> externus ma­<lb/>ior e&longs;t angulo interno <emph type="italics"/>adh,<emph.end type="italics"/> æqualis autem angulo <emph type="italics"/>ief<emph.end type="italics"/> per 9. <lb/>theor: <expan abbr="atq;">atque</expan> huic æquatur angulus <emph type="italics"/>lem,<emph.end type="italics"/> propterea quòd &longs;imilia <lb/>&longs;int triangula <emph type="italics"/>ief, mel:<emph.end type="italics"/> erit <expan abbr="quoq;">quoque</expan> æqualis angulo externo <lb/><emph type="italics"/>afi,<emph.end type="italics"/> maior verò angulo interno <emph type="italics"/>fdh<emph.end type="italics"/> angulo nimirum inci­<lb/>dentiæ. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Pentagoni &longs;ecans obliquè planum, &longs;i latus oppo&longs;itum habeat <lb/>eidem plano par allelum, ad angulos æquales reflectit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Pentagonum <emph type="italics"/>abcde<emph.end type="italics"/> habeat latus <emph type="italics"/>cd<emph.end type="italics"/> plano <emph type="italics"/>op<emph.end type="italics"/> parallelum <lb/>& oppo&longs;itum: dico ad angulos reflecti æquales. </s><s>Sit enim <pb xlink:href="063/01/070.jpg"/><emph type="italics"/>ab<emph.end type="italics"/> linea hypomochlij, & <emph type="italics"/>fg<emph.end type="italics"/> ad eam perpendicularis: <expan abbr="eritq;">eritque</expan> ex <lb/>iam demon&longs;tratis <emph type="italics"/>fg<emph.end type="italics"/> grauitas mouens, & <emph type="italics"/>ag<emph.end type="italics"/> plaga eiu&longs;dem <lb/>centri. </s><s>Et quia plaga e&longs;t in lineâ <emph type="italics"/>af;<emph.end type="italics"/> erit motus reflexus in <lb/>eadem lineâ <emph type="italics"/>af.<emph.end type="italics"/> quòd &longs;i ergo fiat ut <emph type="italics"/>ag<emph.end type="italics"/> ad <emph type="italics"/>gf,<emph.end type="italics"/> ita <emph type="italics"/>ah<emph.end type="italics"/> ad <emph type="italics"/>ak,<emph.end type="italics"/> erit <lb/>motus medius in <emph type="italics"/>ai,<emph.end type="italics"/> & angulus reflexûs <emph type="italics"/>iak:<emph.end type="italics"/> quem dico æqua­<lb/>lem angulo incidentiæ <emph type="italics"/>oab.<emph.end type="italics"/> Quia enim angulus <emph type="italics"/>oab<emph.end type="italics"/> e&longs;t æ­<lb/>qualis angulo <emph type="italics"/>afg,<emph.end type="italics"/> propterea quòd <expan abbr="uterq;">uterque</expan> &longs;it complementum <lb/>anguli <emph type="italics"/>fag:<emph.end type="italics"/> angulo autem <emph type="italics"/>gfa<emph.end type="italics"/> æquatur angulus <emph type="italics"/>iak,<emph.end type="italics"/> quòd &longs;i­<lb/>milia &longs;int triangula <emph type="italics"/>agf. iak:<emph.end type="italics"/> erit <expan abbr="quoq;">quoque</expan> angulo <emph type="italics"/>oab<emph.end type="italics"/> idem <lb/>angulus <emph type="italics"/>iak<emph.end type="italics"/> æqualis. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motus Pentagoni &longs;ecans obliquè planum, &longs;i latus, quod tangit pla­<lb/>num eidem &longs;it parallelum, ad angulos inæquales re&longs;le­<lb/>ctit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Motus Pentagoni <emph type="italics"/>abcde<emph.end type="italics"/> incidat obliquè plano <emph type="italics"/>&longs;t<emph.end type="italics"/> habens la­<lb/>tus <emph type="italics"/>ae,<emph.end type="italics"/> quod tangit planum, eidem parallelum: dico hunc mo­<lb/>tum reflecti ad angulos inæquales. </s><s>Excitetur linea hypomo­<lb/>chlij <emph type="italics"/>en,<emph.end type="italics"/> & <emph type="italics"/>fg<emph.end type="italics"/> ad eam perpendicularis: <expan abbr="eritq;">eritque</expan> grauitas tota <expan abbr="qua-dratũ">qua­<lb/>dratum</expan> <emph type="italics"/>fh;<emph.end type="italics"/> grauitas autem mo vens quadratum <emph type="italics"/>fg.<emph.end type="italics"/> dividatur bi­<lb/>fariam linea <emph type="italics"/>hf<emph.end type="italics"/> in <emph type="italics"/>p;<emph.end type="italics"/> <expan abbr="eoq;">eoque</expan> centro circulus de&longs;cribatur <emph type="italics"/>hif.<emph.end type="italics"/><lb/>Quòd &longs;i ergo &longs;umatur chorda <emph type="italics"/>fi<emph.end type="italics"/> æqualis <emph type="italics"/>fg;<emph.end type="italics"/> erit chorda re­<lb/>liqua <emph type="italics"/>hi;<emph.end type="italics"/> <expan abbr="atq;">atque</expan> huius quadratum dabit plagam. </s><s>Et quia plaga <lb/>fit per lineas <emph type="italics"/>fa. fh. fe:<emph.end type="italics"/> erit per 5 theor: huius motus reflexus <lb/>in lineâ <emph type="italics"/>fc,<emph.end type="italics"/> & motus centri in lineâ <emph type="italics"/>fm<emph.end type="italics"/> eidem plano parallelâ. </s><lb/><s>Si ergo fiat ut <emph type="italics"/>fi<emph.end type="italics"/> ad <emph type="italics"/>ih,<emph.end type="italics"/> ita <emph type="italics"/>fm<emph.end type="italics"/> ad <emph type="italics"/>fl;<emph.end type="italics"/> erit motus medius <emph type="italics"/>fk,<emph.end type="italics"/> & <lb/>angulus reflexionis <emph type="italics"/>kfm;<emph.end type="italics"/> quem dico inæqualem angulo in­<lb/>cidentiæ <emph type="italics"/>hen.<emph.end type="italics"/> Quia enim angulus <emph type="italics"/>ahi<emph.end type="italics"/> externus e&longs;t maior <pb xlink:href="063/01/071.jpg"/>angulo interno <emph type="italics"/>hei,<emph.end type="italics"/> æqualis autem angulo <emph type="italics"/>ifh;<emph.end type="italics"/> propterea <lb/>quòd <expan abbr="uterq;">uterque</expan> a&longs;&longs;umpto angulo communi <emph type="italics"/>ihf<emph.end type="italics"/> facit rectum: <lb/>& angulo <emph type="italics"/>ifh<emph.end type="italics"/> e&longs;t æqualis angulus <emph type="italics"/>kfm;<emph.end type="italics"/> erit <expan abbr="quoq;">quoque</expan> æqualis an­<lb/>gulo <emph type="italics"/>ahi,<emph.end type="italics"/> ac proinde maior angulo interno <emph type="italics"/>hei,<emph.end type="italics"/> angulo inci­<lb/>dentiæ. </s></p> <p type="main"> <s><emph type="italics"/>Obijcies. </s><s>Si vectis continet gr auitatem mobilis, totus totam, pars ve­<lb/>rò partem proportionalem per 2 Axioma; et impul&longs;us centri grauitatis <lb/>totus mouet, cùm huius interuallum ab hypomochlio eidem e&longs;t æquale per <lb/>7 theorema 2 partis; neceßè in figurâ 3 theor: 2 huius, cùm tota &longs;emidia­<lb/>meter figuræ motûs &longs;it extra hypomochlium, & non ni&longs;i in puncto tan­<lb/>gat planum AZ; aut nullam, aut in&longs;en&longs;ibilem inferre plagam: non igi­<lb/>tur rectè a&longs;&longs;umebatur ratio plagæ ad reliquum impul&longs;um, quam habet <lb/>quadratum ED ad quadratum EA: &longs;iquidem totum impul&longs;um metitur <lb/>quadratum eiu&longs;dem ED.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo no&longs;tram a&longs;&longs;ertionem veram e&longs;&longs;e, cùm &longs;emidia­<lb/>meter figuræ motûs eâ ratione &longs;ecatur ab hypomochlio, ut re­<lb/>liquus impul&longs;us ab illatâ plaga non prohibeatur à &longs;uo mo­<lb/>tu: at verò hic impul&longs;us cogitur ab hypomochlio ad <expan abbr="motũ">motum</expan> incli­<lb/>natum <emph type="italics"/>di,<emph.end type="italics"/> per tangentem circuli centro <emph type="italics"/>a<emph.end type="italics"/> de&longs;cripti. </s><s>Erit <expan abbr="itaq;">itaque</expan> <lb/>impul&longs;us reliquus in eâratione ad totum impul&longs;um, quam ha­<lb/>bet motus in eiu&longs;modi plano inclinato ad motum verticalem. </s><lb/><s>Ducatur enim <emph type="italics"/>el<emph.end type="italics"/> parallela ip&longs;i <emph type="italics"/>di:<emph.end type="italics"/> <expan abbr="eritq;">eritque</expan> motus verticalis in <lb/><emph type="italics"/>ea<emph.end type="italics"/> ad motum inclinatum in <emph type="italics"/>el,<emph.end type="italics"/> ut quadratum <emph type="italics"/>ea<emph.end type="italics"/> ad quadratum <lb/><emph type="italics"/>el,<emph.end type="italics"/> hoc e&longs;t ut quadratum <emph type="italics"/>da<emph.end type="italics"/> ad quadratum <emph type="italics"/>de:<emph.end type="italics"/> quòd &longs;imilia <lb/>&longs;unt triangula <emph type="italics"/>ael. aed.<emph.end type="italics"/> Et quia quadratum <emph type="italics"/>ad<emph.end type="italics"/> hoc e&longs;t totus <lb/>impul&longs;us æquatur duobus quadratis <emph type="italics"/>de. ae;<emph.end type="italics"/> e&longs;t autem quadra­<lb/>tum <emph type="italics"/>de<emph.end type="italics"/> impul&longs;us movens, erit quadratum <emph type="italics"/>ae<emph.end type="italics"/> impul&longs;us qui­<lb/>e&longs;cens, hoc e&longs;t plaga; quam infert eidem plano <emph type="italics"/>az.<emph.end type="italics"/> Magis er­<lb/>go univer&longs;alis e&longs;t hæc ratio, quàm à &longs;emidiametro figuræ mo- <pb xlink:href="063/01/072.jpg"/>tûs de&longs;umpta. vnde etiam hac ad demon&longs;trationem horum the­<lb/>orematum u&longs;i &longs;umus. </s></p> <p type="main"> <s>Forta&longs;&longs;e verò hanc eandem hypothe&longs;im, in motu proiecto­<lb/>rum, non inconvenienter a&longs;&longs;umere licebit. ut &longs;i quadratum E <lb/>percutiat circulum H per 1 & 2 Lemma probl: 5. quia motus <lb/>centri E à percu&longs;&longs;ione fit parallelus rectæ GB, erit inclinatio <lb/>huius æqualis angulo BGQ, hoc e&longs;t illi ad verticem æquali AGI. </s><lb/><s>Igitur ut GI ad GA, ita motus verticalis ad motum inclina­<lb/>tum. e&longs;t verò ut GI ad GA, ita GE ad FE. propterea quòd &longs;i­<lb/>milia &longs;int triangula GEF. AGI. e&longs;t enim AGE &longs;imile <expan abbr="utriq;">utrique</expan> <lb/>triangulo FGE. FAG, <expan abbr="atq;">atque</expan> idem FAG &longs;imile triangulo AGI. </s><lb/><s>Cùm itaq, FE &longs;it impul&longs;us mouens; totum verò impul&longs;um <lb/>metiatur EG; erit huius exce&longs;&longs;us æqualis plagæ. qui nonni&longs;i <lb/>cùm radius EG e&longs;t æqualis &longs;emidiametro figuræ motûs EA, <lb/>æquatur reliquo &longs;egmento AF. </s><s>Quòd &longs;i verò quis opine­<lb/>tur eandem e&longs;&longs;e rationem motûs proiectorum, & qui pro venit <lb/>à grauitate: propterea quòd &longs;icuti lap&longs;us grauium continuò <lb/>augetur: ita <expan abbr="quoq;">quoque</expan> motus proiectorum continuò minuitur: eo <lb/>videlicet modo, quo triangulum &longs;ibi &longs;imile manens; ac pro­<lb/>inde <expan abbr="utrumq;">utrumque</expan> &longs;ecari ab hypomochlio in duo quadrata: is meo <lb/>quidem iudicio haud improbabiliter ita &longs;entiet. </s><s>Tum <expan abbr="itaq;">itaque</expan> <lb/>&longs;umpto impul&longs;u toto æquali quadrato EG: &longs;i EF quadratum <lb/>&longs;it vis movens; erit FG quadratum plaga, &longs;eu impul&longs;us in hy­<lb/>pomochlio quie&longs;cens. </s><s>Siue tamen hac, &longs;iue illâ hypothe&longs;i uta­<lb/>mur, eadem via erit ad circuli quadraturam. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motum verticalem trianguli I&longs;ogoni à plano reflectere ad an­<lb/>gulum datum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit angulus datus grad. 30. ad quem reflectere oportet mo­<lb/>tum trianguli <emph type="italics"/>abc<emph.end type="italics"/> à plano <emph type="italics"/>az.<emph.end type="italics"/> Ducatur linea verticalis <pb xlink:href="063/01/073.jpg"/><emph type="italics"/>af<emph.end type="italics"/> faciens cum rectâ <emph type="italics"/>ad<emph.end type="italics"/> angulum <emph type="italics"/>fad<emph.end type="italics"/> grad. 30. &longs;emi&longs;&longs;em <lb/>complementi anguli reflexionis. </s><s>Secet autem <emph type="italics"/>ad<emph.end type="italics"/> producta <lb/>latus trianguli <emph type="italics"/>bc<emph.end type="italics"/> ad angulos rectos: dico triangulum <emph type="italics"/>abc<emph.end type="italics"/><lb/>in hoc &longs;itu à lap&longs;u verticali reflecti ad grad-30. </s><s>Ducatur enim <lb/>à centro figuræ recta <emph type="italics"/>de<emph.end type="italics"/> perpendicularis ad <emph type="italics"/>af.<emph.end type="italics"/> Et fiat ut <lb/><emph type="italics"/>ae<emph.end type="italics"/> ad <emph type="italics"/>ed,<emph.end type="italics"/> ita <emph type="italics"/>dg<emph.end type="italics"/> ad <emph type="italics"/>di:<emph.end type="italics"/> <expan abbr="eritq;">eritque</expan> <emph type="italics"/>dh<emph.end type="italics"/> motus centri à reflexi­<lb/>one. </s><s>Cuiex <emph type="italics"/>a<emph.end type="italics"/> ducatur parallela <emph type="italics"/>ac.<emph.end type="italics"/> Quia <expan abbr="itaq;">itaque</expan> angulus <emph type="italics"/>e <lb/>ad<emph.end type="italics"/> e&longs;t grad. 30. per con&longs;tructionem; æqualis autem angulo <emph type="italics"/>g <lb/>dh,<emph.end type="italics"/> hoc e&longs;t illi æquali <emph type="italics"/>dai:<emph.end type="italics"/> erit angulus compo&longs;itus <emph type="italics"/>fai<emph.end type="italics"/> grad: <lb/>60, ac proinde angulus reliquus <emph type="italics"/>caz<emph.end type="italics"/> grad. 30. </s></p> <figure id="id.063.01.073.1.jpg" xlink:href="063/01/073/1.jpg"/> <p type="main"> <s><emph type="center"/>PROBLEMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motum verticalem quadrati à plano reflectere ad angulum datum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Inveniendus &longs;it angulus reflexionis grad. 40. </s><s>Ductâ <emph type="italics"/>ag<emph.end type="italics"/> li­<lb/>neâ hypomochlij, fiat angulus <emph type="italics"/>gae<emph.end type="italics"/> grad: 25. &longs;emi&longs;&longs;is comple­<lb/>menti anguli reflexionis. </s><s>Et ex centro figuræ producatur <emph type="italics"/>ef<emph.end type="italics"/> <pb xlink:href="063/01/074.jpg"/>perpendicularis ad <emph type="italics"/>af.<emph.end type="italics"/> Quòd &longs;i <expan abbr="itaq;">itaque</expan> fiat ut <emph type="italics"/>af<emph.end type="italics"/> ad <emph type="italics"/>fe,<emph.end type="italics"/> ita <emph type="italics"/>eh<emph.end type="italics"/><lb/>ad <emph type="italics"/>ek;<emph.end type="italics"/> erit <emph type="italics"/>ei<emph.end type="italics"/> via motûs reflexi. </s><s>Cui ex <emph type="italics"/>a<emph.end type="italics"/> ducatur paral­<lb/>lcla <emph type="italics"/>ad.<emph.end type="italics"/> Et quia angulus <emph type="italics"/>hei,<emph.end type="italics"/> hoc e&longs;t <emph type="italics"/>ead<emph.end type="italics"/> æquatur angu­<lb/>lo <emph type="italics"/>fae:<emph.end type="italics"/> erit angulus compo&longs;itus <emph type="italics"/>fad<emph.end type="italics"/> grad. 50; & angulus <lb/>re&longs;iduus <emph type="italics"/>dax,<emph.end type="italics"/> nimirum angulus reflexionis grad. 40. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Motum verticalem pentagoni à plano reflectere ad angulum datum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Simili modo in pentagono motum verticalem reflectemus <lb/>ad angulum datum. &longs;i ducatur <emph type="italics"/>ac,<emph.end type="italics"/> verticalis; & angulus <emph type="italics"/>gaf<emph.end type="italics"/><lb/>&longs;iat &longs;emi&longs;&longs;is complementi ad angulum quæ&longs;itum. </s><s>Ductâ enim <lb/>ex <emph type="italics"/>a<emph.end type="italics"/>p arallelâ motui reflexo <emph type="italics"/>fi,<emph.end type="italics"/> erit angulus reliquus à parallelâ, <lb/>& plano contentus æqualis angulo quæ&longs;ito. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De lineâ motûs reflexi, & motu proiectorum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Verùm contra hucus&queacute; dicta de motu reflexo poterit quis dubitare: <lb/>quamobrem hic ex occur&longs;u plani, motus at&que; impul&longs;us figuræ rectilineæ <lb/>&longs;ecetur in duo quadrata: in probl: verò 4 & 5 in duo parallelogramma: <lb/>quorum ba&longs;is communis &longs;it radius, &longs;eu &longs;emidiameter figuræ motûs; alti­<lb/>tudo verò eiu&longs;dem &longs;egmenta.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo hie motum con&longs;iderari naturalem à grauitate: <lb/>quem prop: 12. o&longs;tendi eo modo augeri, quo triangulum &longs;ibi <lb/>&longs;imile manens. </s><s>Cùm <expan abbr="itaq;">itaque</expan> plaga inducatur non <expan abbr="ab&longs;q;">ab&longs;que</expan> ali­<lb/>quâ morulâ; nece&longs;&longs;e et illum impul&longs;um, quem plaga ab&longs;umit, <lb/>& quem centrum gravitatis retinet ad &longs;e librandum, habere <lb/>vim quadrati. </s><s>At verò in quadraturâ circuli motu utimur &longs;i­<lb/>milari: Vnde nece&longs;sè eo modo dividi, quo linea recta, &longs;eu pa­<lb/>rallelogrammum. </s></p> <pb xlink:href="063/01/075.jpg"/> <p type="main"> <s><emph type="italics"/>In&longs;tabis &longs;i totus impul&longs;us, VG trianguli ABC, &longs;ecatur in duo qua­<lb/>drata DE at&queacute; EA: quia motus e&longs;t æqualis impul&longs;ui; erit ut quadra­<lb/>tum DE ad quadratum EA, ita motus centri ad motum reflexum à <lb/>plagâ in DG. maior ita&queacute; DG quàm AE: nimirum in ratione duplica­<lb/>tà eius, quam habet AE ad DE: ac proinde angulus reflexionis minor <lb/>angulo GDH.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo cùm motus augeatur pro ratione impul&longs;ús; hu­<lb/>ius verò incrementa pro ratione illius morulæ, in quâ perfici­<lb/>tur plaga, habeant rationem quadrati; nece&longs;se <expan abbr="quoq;">quoque</expan> motum <lb/>inter &longs;e conferri ut quadrata. </s><s>Quod confirmatur à po&longs;terio­<lb/>ri. </s><s>Con&longs;tat experientiâ, <expan abbr="atq;">atque</expan> omnium a&longs;&longs;en&longs;u pilam reflecti <lb/>ad angulos æquales: hoc autem nullâ ratione fieri pote&longs;t, ni&longs;i <lb/>motus ad &longs;e referantur ut quadrata. </s><s>A&longs;&longs;umatur enim figura <lb/>prop: 39: in quâ angulus incidentiæ CDA æquatur angulo <lb/>reflexionis IAB: dico impul&longs;um, & qui hunc &longs;equitur motum <lb/>centri grauitatis re&longs;iduum à plagâ, eandem rationem habere <lb/>ad motum inde reflexum, quam habet quadratum EF ad qua­<lb/>dratum FD, hoc e&longs;t per prop: 12. illorum durationem e&longs;&longs;e <lb/> <arrow.to.target n="fig18"/> <pb xlink:href="063/01/076.jpg"/>ut EF. FD latera eorundem quadratorum. </s><s>Producatur enim <lb/>linea DE motûs reflexi: <expan abbr="atq;">atque</expan> ip&longs;i DI &longs;umatur parallela EG <lb/>ex G verò demittantur perpendiculares GH. GK. </s><s>Quia <expan abbr="itaq;">itaque</expan> <lb/>recta ED e&longs;t perpendicularis ad AB, & angulus CDA a&longs;&longs;umptus <lb/>æqualis angulo IDB; erit angulus reliquus CDE æqualis angu­<lb/>lo reliquo EDI, hoc e&longs;t illi æquali HEG. & cùm rectus &longs;it <expan abbr="uterq;">uterque</expan> <lb/>angulus EFD. EHG, <expan abbr="atq;">atque</expan> HEG æqualis EDF; erunt triangula EFD. <lb/>GHE &longs;imilia. </s><s>Igitur ut EF ad FD, ita HG, &longs;eu EK ad EH. </s><s><expan abbr="Neq;">Neque</expan> <lb/>verò dicendum in hac demon&longs;tratione circulum committi. &longs;i <lb/>quidem hic ab effectu per experientiam cognito, ea principia <lb/>&longs;tabiliuntur; ex quibus propo&longs;itione 39. aliâ viâ notis hic idem <lb/>effectus tanquam illorum conclu&longs;io infertur, </s></p> <figure id="id.063.01.076.1.jpg" xlink:href="063/01/076/1.jpg"/> <p type="main"> <s><emph type="italics"/>Obijcies. </s><s>Motum reflexum non augeri ea modo, quo triangulum &longs;ibi <lb/>&longs;imile manens: non igitur ad &longs;e referri ut quadrata. </s><s>Et de impul&longs;u <lb/>quidem reflexo videtur manife&longs;tum: Quod hic à percußione oriatur, <lb/>at&queacute; continuò, ex quo cæpit, minuatur. </s><s>Idem verò probatur de impul­<lb/>&longs;u, quem centrum grauitatis retinet ad &longs;e librandum. </s><s>Nam cùm prin­<lb/>cipium huius augmenti &longs;it grauitas, motus verò reflexus fiat in partes <lb/>oppo&longs;itas grauitati; nequit grauitas influere in hunc motum: quin poti­<lb/>us eidem reniti, & grauitando ip&longs;um minuere: uti manife&longs;tum in fine <lb/>motûs reflexi & in arcum &longs;inuati.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo nos hic principia motûs reflexi inter &longs;e confer­<lb/>re: quæ con&longs;tat vim quadrati habere: licet fortè in progre&longs;&longs;u <lb/>mutari contingat illam proportionem. </s><s>An verò grauitas in­<lb/>fluat in motum reflexum dubitari pote&longs;t. </s><s>Nam &longs;i ita, idem <lb/>videtur dicendum de motu proiectorum: nullus proinde mo <lb/>tus rectus. </s><s>At verò &longs;i proiecta non ferantur lineâ rectâ, quâ ra­<lb/>tione ictus certi e&longs;&longs;e po&longs;&longs;unt? et tamen con&longs;tat e&longs;&longs;e inter Scyt­<lb/>has adeo &longs;agittandi peritos, ut pomum vertici impo&longs;itum, aut <pb xlink:href="063/01/077.jpg"/>nummum inter duos digitos contentum excutiant. </s><s>Mulieres <lb/><expan abbr="quoq;">quoque</expan> Balearicæ non priùs cibum &longs;uis filijs præ&longs;tabant, quàm <lb/>iactu fundæ eundem attigi&longs;&longs;ent. </s><s>Et ne remotiora &longs;ectemur, <lb/>an non ictus tormentorum adeo certi; ut globi ab his emi&longs;&longs;i per <lb/>ip&longs;um os tormenti oppo&longs;iti &longs;e inferant? </s></p> <p type="main"> <s>Pro quo notandum ex his, quæ in libro de motu po&longs;tea di­<lb/>centur, <expan abbr="utrumq;">utrumque</expan> <expan abbr="motũ">motum</expan>, videlicet naturalem, & qui ex impul&longs;u <lb/>cau&longs;atur, efficienter quidem à principio interno mobilis; de­<lb/>terminatiuè verò ab ideâ provenire. </s><s>Quæ &longs;i ab extra veniat, <lb/>motum non naturalem; idea verò interna & à principijs e&longs;&longs;en­<lb/>tialibus fluens motum naturalem determinat: <expan abbr="atq;">atque</expan> &longs;i ad mun­<lb/>di centrum dirigat, grauitas nun cupatur. </s><s>Fit autem hic mo­<lb/>tus mediante impul&longs;u: qui cùm nece&longs;&longs;ariò producatur, nece&longs;sè <lb/>hunc in de&longs;cen&longs;u continuò augeri per prop: 10. </s><s>Idea verò ex­<lb/>terna impul&longs;um determinat &longs;imilem vel di&longs;&longs;imilem grauitati. </s><lb/><s>Et &longs;iquidem impul&longs;us accedat &longs;imilis illi, qui prouenit à gravi­<lb/>tate; dico ab <expan abbr="utroq;">utroque</expan> &longs;imul fieri motum: &longs;iue impul&longs;us &longs;it ma­<lb/>ior, &longs;iue minor gravitate. </s><s>Et impul&longs;um quidem maiorem <lb/>grauia incitare videtur manife&longs;tum. </s><s>Quòd ab hoc, non verò <lb/>à gravitate fiant incrementa motûs: qui in omni puncto e&longs;t <lb/>maior gravitate, per prop: 11. </s><s>Idem verò dicendum de im­<lb/>pul&longs;u minori. propterea quòd grauitas non ni&longs;i mediante im­<lb/>pul&longs;u moueat: omnis verò acce&longs;&longs;io impul&longs;ûs auget præexi­<lb/>&longs;tentem, & ad motum incitat velociorem, per po&longs;it: 4. </s><s>Quôd <lb/>&longs;i motus &longs;it non naturalis, cuiu&longs;modi &longs;agittæ, vel erit contrari­<lb/>us ab&longs;olutè; qui nimirum fit per eandem lineam rectam: vel <lb/>&longs;ubcontrarius, angulum continens cum lineâ de&longs;cen&longs;us mino­<lb/>rem duobus rectis. </s><s>Ft prioris quidem generis, &longs;i æqualis &longs;it <lb/>gravitati, nullus omninò fit motus; verùm mobile tum quie­<lb/>&longs;cit. </s><s>Propterea quòd de&longs;cen&longs;us grauium fiat mediante im- <pb xlink:href="063/01/078.jpg"/>pul&longs;u: Impul&longs;us verò contrarius tollat vel impediat &longs;uum <lb/>contrarium in eadem ratione, totus totum; pars verò partem <lb/>proportionalem. </s><s>Igitur &longs;i minor &longs;it impul&longs;us gravitate, abla­<lb/>tâ parte æquali à re&longs;iduâ gravitate fit de&longs;cen&longs;us. </s><s>Quòd &longs;i ve­<lb/>rò maior &longs;it impul&longs;us: erit huius exce&longs;&longs;us principium motûs <lb/>&longs;ur&longs;um. </s><s>At verò impul&longs;us &longs;ubcontrarius, &longs;i angulum conti­<lb/>neat rectum, vel maiorem recto, cùm illius motus à centro ab­<lb/>ducat, nullum impul&longs;um videtur gravitas determinare: Vn­<lb/>de motus ab exce&longs;&longs;u fieri dicendus: <expan abbr="quou&longs;q;">quou&longs;que</expan> æquatio fiat <expan abbr="u-triusq;">u­<lb/>triusque</expan>, tum enim motu mi&longs;to ferri, & in &longs;peciem arcüs &longs;inuari <lb/>videtur. </s><s>Quod quidem &longs;upponere debent, qui dicunt mo­<lb/>tum proiectorum fieri per lineam rectam: quod nullo modo <lb/>e&longs;&longs;et, &longs;i motu mi&longs;to ferrentur ex gravitate <expan abbr="atq;">atque</expan> impul&longs;u. </s><s>Nam <lb/>cùm plaga minuat impul&longs;um, gravitas verò eadem maneat; <lb/>nece&longs;se latera motûs continuò aliam <expan abbr="atq;">atque</expan> aliam rationem ad <lb/>&longs;e habere. </s><s>Cuius ratio e&longs;&longs;e videtur; quòd gravitas nonni&longs;i <lb/>idealiter concurrat ad motum & impul&longs;um: unde per aliam <lb/>ideam fortiorem &longs;uperari & excludi pote&longs;t: ut ad <expan abbr="præ&longs;criptũ">præ&longs;criptum</expan> <lb/>huius, non illius moveatur. </s><s>At verò impul&longs;us &longs;ubcontrarij <lb/>nece&longs;&longs;ariò mi&longs;centur, <expan abbr="actionêsq;">actionêsque</expan> producunt mixtas. </s><s>E&longs;t hæc <lb/>&longs;ententia multùm probabilis, &longs;ed oppo&longs;ita magis placet. </s><s>Nam <lb/>cùm motus proiectorum demum &longs;inuetur manife&longs;tè: id non­<lb/>ni&longs;i ex impul&longs;u gravitatis e&longs;&longs;e pote&longs;t: qui mobile ex illâ lineâ <lb/>rectâ ad centrum abducit. </s><s>At verò hoc contingit non &longs;olùm <lb/>æquatâ gravitate, &longs;ed etiam cùm maior e&longs;t impul&longs;us: Igitur in <lb/>reliquum impul&longs;um, quo moveri cæpit, grauitas influit: ac <lb/>proinde nece&longs;se hunc motum e&longs;&longs;e mi&longs;tum. </s><s>A&longs;&longs;umatur enim <lb/>altitudo &longs;agittæ AC, cùm iam manife&longs;tè incipit declinare à li­<lb/>neâ horizonti parallelâ: cuius motus &longs;inuo&longs;us AFG <expan abbr="eritq;">eritque</expan> AG <lb/>maior quàm AC. </s><s>Dico impul&longs;um e&longs;&longs;e maiorem gravitate. <pb xlink:href="063/01/079.jpg"/> <arrow.to.target n="fig19"/><lb/>Quòd &longs;i enim æqualis eidem e&longs;&longs;et, motus medius fieret per di­<lb/>ametrum AG. minor verò effectus grauitate, motum &longs;inuo­<lb/>&longs;um terminabit inter C & G. quod quidem in &longs;yphonibus <expan abbr="atq;">atque</expan> <lb/>effluxibus aquæ &longs;inuo&longs;is magis licebit experiri. </s></p> <figure id="id.063.01.079.1.jpg" xlink:href="063/01/079/1.jpg"/> <p type="main"> <s><emph type="center"/>Quam proportionem habeat impul&longs;us <lb/>ad gravitatem.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Quod verò obijcitur, &longs;i motus eâ ratione &longs;it mi&longs;tus, cúm plaga mi­<lb/>nuat impul&longs;um, grauitas verò eadem maneat; nunquam ad de&longs;tina­<lb/>tam metam mißilia, quæ ad libellam diriguntur, perventura.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo gravitatem ad impul&longs;um <emph type="italics"/>VG<emph.end type="italics"/> &longs;agittæ, valde exi­<lb/>guam proportionem habere: ac proinde ob in&longs;en&longs;ilem cur­<lb/>vitatem pro lineâ rectâ æ&longs;timari. </s><s>Quod quidem hac ratione <lb/>videtur &longs;uaderi. </s><s>Cùm in lap&longs;u grauium impul&longs;us in omni <lb/>puncto motús &longs;it maior gravitate per prop: 11; <expan abbr="atq;">atque</expan> eo modo <lb/>augeatur, quo triangulum &longs;ibi &longs;imile manens, per prop: 12: <pb xlink:href="063/01/080.jpg"/>habebit rationem duplicatam &longs;uæ longitudinis ad datum tri­<lb/>anguli latus, quod gravitati, VG unius libræ, &longs;it æquale. </s><s>Vt <lb/>&longs;i promovi&longs;&longs;e dicatur eo lap&longs;u prius quidem ad digitos 4. </s><s>In­<lb/>de ad pa&longs;&longs;us 3: habebit impul&longs;us hoc intervallo collectus ad <lb/>illum rationem, quam 1804. ad 1. </s><s>At verò &longs;i pila de&longs;cen­<lb/>dat ad totidem pa&longs;&longs;us; minùs offendit, quàm &longs;i eadem ex illâ <lb/>di&longs;tantiâ proijciatur. </s><s>E&longs;t autem impul&longs;us ab arcu, &longs;eu fundâ <lb/>his muitò vehementior: ut nihil dicam de Cylindro bellico. </s><lb/><s>Deinde dico ab huius modi Iobolis nonignorari hanc motûs <lb/>curvitatem: unde etiam rationem habent di&longs;tantiæ. aliter e­<lb/>nim ex magno, aliter ex parvo intervallo ictum dirigunt: <expan abbr="neq;">neque</expan> <lb/>&longs;olùm intervalli, &longs;ed etiam ictûs vehementiæ modum expen­<lb/>dunt. </s></p> <p type="main"> <s><emph type="italics"/>Dices quamobrem alij alijs feticiùs &longs;copum a&longs;&longs;equuntur: tamct&longs;i ijs­<lb/>dem in&longs;trumentis u&longs;i, eadem&que; collineatione factâ.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo id ex diver&longs;o pupillæ &longs;itu provenire. accidit e­<lb/>nim his, quemadmodum &longs;i quis digito pre&longs;&longs;am loco moveat: <lb/>tum &longs;iquidem alius rei, <expan abbr="atq;">atque</expan> imaginis locus. unde cùm ictum <lb/>dirigant ad locum vi&longs;um, quid mirum à loco verò aberrare. </s><lb/><s>Ita quidem in motu proiectorum; quæ lineam &longs;equuntur ex <lb/>angulo recto, aut recto maiore. </s><s>Quòd &longs;i cum motu vertica­<lb/>li angulum <expan abbr="contineãt">contineant</expan> minorem recto; quia tum mobile fit pro­<lb/>pius centro, videbitur hic gravitas capere augmentum eo la­<lb/>p&longs;u: quod &longs;imilis videatur motui inclinato; in quo velocitas <lb/>continnò augetur, Dico nihilominus eandem e&longs;&longs;e <expan abbr="vtro-biq;">vtro­<lb/>bique</expan> rationem. </s><s>Alia autem e&longs;t ratio motûs inclinati. propterea <lb/>quòd pars gravitatis maneat extra hypomochlium: ac proin <lb/>de impul&longs;um producat &longs;ibi æqualem: qui in de&longs;cen&longs;u conti- <pb xlink:href="063/01/081.jpg"/>nuò augetur. </s><s>In proiectis verò tota gravitas &longs;uperatur ab <lb/>impul&longs;u, <expan abbr="atq;">atque</expan> in lineam trahitur nonnaturalem. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Datâ Proportione impul&longs;ûs ad grauitatem, lineam motûs <lb/>inflexi inuenire.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Data &longs;it proportio impul&longs;ûs ad gravitatem, VG &longs;e&longs;cupla. <lb/>a&longs;&longs;umatur autem recta AB via motûs, ad AC motum verti­<lb/>calem in eadem ratione: & &longs;ecetur AB in &longs;egmenta æqualia <lb/>ALMNOPB. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> maneret eadem proportio im­<lb/>pul&longs;us ad gravitatem, motus medius e&longs;&longs;et diameter parallelo­<lb/>grammi ABDC per prop: 32. </s><s>At verò quia plaga impul­<lb/>&longs;um continuò ab&longs;umit: gravitas verò eadem manet; nece&longs;&longs;e <lb/>continuò mutari hanc proportionem: pro ratione nimirum <lb/>&longs;patij tran&longs;mi&longs;&longs;i Igitur ab&longs;umptâ parte impul&longs;us æquali AL: <lb/>principium motûs reliqui determinat AT diameter parallelo­<lb/>grammi APTC in E. propterea quòd TC &longs;it æqualis re&longs;iduo <lb/>impul&longs;ui LB. </s><s>Rur&longs;um peractâ plagâ æquali AM; erit princi­<lb/>pium motûs in F communi &longs;ectione MF. <expan abbr="atq;">atque</expan> AS lineæ dia­<lb/>gonalis parallelogrammi AOSC, <expan abbr="eademq;">eademque</expan> ratione invenie­<lb/>mus puncta reliqua motûs &longs;inuo&longs;i in GH I&c. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Linea motûs proiectorum non e&longs;t circulus, ne&que; ulla &longs;ectionum <lb/>conicarum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Supponamus primùm e&longs;&longs;e lineam circularem.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Quoniam <expan abbr="itaq;">itaque</expan> triangula APT, AEV &longs;unt&longs;imilia, erit FV ad <lb/>AV, ut AP ad TP. e&longs;t autem TP pars 5 AP per probl: 4 Igitur & <pb xlink:href="063/01/082.jpg"/> <arrow.to.target n="fig20"/><lb/>AV pars quinta EV. </s><s>Et quia quadratum E Ve&longs;t æquale re­<lb/>ctangulo contento AV, <expan abbr="atq;">atque</expan> huius complemento ad diame­<lb/>trum circuli; EV verò a&longs;&longs;umpta partium 10, qualium AV e&longs;t <lb/>2; erithuius complementum partium 50: & tota diameter 52. </s><lb/><s>Rur&longs;um quia CG e&longs;t tripla AC: illius verò quadratum æqua­<lb/>le rectangulo contento AC, atq, huius complemento ad dia­<lb/>metrum circuli; e&longs;t verò quadratum CG partium 900, & AC <lb/>partium 10; erit re&longs;iduum &longs;egmentum partium 90: tota verò <lb/>diameter partium 100. e&longs;t verò eadem <expan abbr="quoq;">quoque</expan> partium 52. </s><lb/><s>Non igitur linea motûs AEF GHI e&longs;t peripheria circuli. </s><lb/><s>Dico <expan abbr="neq;">neque</expan> e&longs;&longs;e parabolam. </s><s>Sit enim &longs;i fieri pote&longs;t, linea para­<lb/>bolæ. erit <expan abbr="itaq;">itaque</expan> ut recta AC ad rectam AV, ita quadratum <lb/>&longs;emiordinatæ CG ad quadratum &longs;emiordinatæ VE. et quia <lb/>CG e&longs;t tripla VE; erit eiu&longs;dem quadratum noncuplum ad illud <lb/>quadratum. </s><s>At verò AC ad AV e&longs;t ut 10 ad 2, hoc e&longs;t quin­<lb/>tupla. non igitur ut AC ad AV, ita quadratum CG ad qua­<lb/>dratum VE: ac proinde linea AE FG &c. non e&longs;t parabola. <pb xlink:href="063/01/083.jpg"/>Sit iam &longs;i fieri pote&longs;t, hyperbole. a&longs;&longs;umatur verò huius diame­<lb/>ter partium 8, qualium AC e&longs;t 10, & AV 2. </s><s>Igitur triangu­<lb/>lum rectangulum contentum AV, & latere compo&longs;ito ex AV <lb/><expan abbr="atq;">atque</expan> diametro figuræ erit partium 20: <expan abbr="triangulũ">triangulum</expan> verò <expan abbr="contentũ">contentum</expan> <lb/>AC <expan abbr="atq;">atque</expan> latere compo&longs;ito ex AC & diametro eiu&longs;dem figuræ, <lb/>partium 180: huius verò ratio ad illud noncupla. e&longs;t autem <lb/>quadratum <expan abbr="quoq;">quoque</expan> &longs;emiordinatæ CG ad quadratum alterius <lb/>&longs;emiordinatæ VE in eadem ratione. propterea quòd latus CG <lb/>&longs;it triplium lateris VE. </s><s>Cùm <expan abbr="itaq;">itaque</expan> eandem rationem ad &longs;e <lb/>habeant rectangula &longs;ub&longs;egmentis axis hyperbolæ, quam habent <lb/>quadrata &longs;emiordinatarum; erit permutando eadem <expan abbr="quoq;">quoque</expan> ra­<lb/>tio rectangulorum &longs;ub &longs;egmentis axis ad quadrata &longs;uarum &longs;e­<lb/>miordinatarum: ac proinde puncta EG in eadem hyperbole. </s><lb/><s>Rur&longs;um verò quoniam AOS. AKF &longs;unt triangula &longs;imilia; <lb/>& AO <expan abbr="quadruplũm">quadruplumm</expan> OS; erit <expan abbr="quoq;">quoque</expan> KF quadruplum AK: <lb/>& AK partium 5, qualium KF e&longs;t 20. triangulum ergo <lb/>rectangulum contentum AK <expan abbr="atq;">atque</expan> latere compo&longs;ito ex AK <lb/>& diametro figuræ erit partium 65: rectangulum verò conten­<lb/>tum AV, & latere compo&longs;ito ex AV <expan abbr="atq;">atque</expan> diametro eiu&longs;dem <lb/>figuræ, partium 20. e&longs;t autem ratio 65 ad 20 minor, quàm &longs;it <lb/>quadrati KF ad quadratum VE: Igitur permutando non ea­<lb/>dem e&longs;t ratio rectangulorum &longs;ub &longs;egmentis axis ad quadrata <lb/>&longs;emiordinatarum: ac proinde puncta EF non continentur in <lb/>lineâ hyperbolæ. </s></p> <figure id="id.063.01.083.1.jpg" xlink:href="063/01/083/1.jpg"/> <p type="main"> <s>Demum <expan abbr="neq;">neque</expan> ellip&longs;in e&longs;&longs;e hanc lineam motûs, ita o&longs;tendo. </s><lb/><s>Producatur AC in Z: quam &longs;ecetperpendicularis IZ. </s><s>Cùm <lb/><expan abbr="itaq;">itaque</expan> in I gravitas fiat æqualis impul&longs;ui; erit IZ maior omni­<lb/>bus rectis, quæ ex lineâ motûs cadunt perpendiculariter ad dia­<lb/>metrum AZ: ac proinde erit &longs;emidiameter figuræ. </s><s>At ve­<lb/>rò IZ æquatur &longs;emidiametro AZ: oportebat verò e&longs;&longs;e in­<lb/>æqualem: non igitur puncta AEFGHI in ellip&longs;i continentur. </s></p> <pb xlink:href="063/01/084.jpg"/> <p type="main"> <s><emph type="center"/>De cau&longs;a inæqualis reflexionis<emph.end type="center"/></s></p> <p type="main"> <s>Suppo&longs;ui hactenus in reflexione figuras rectilineas æqua­<lb/>lem dare & recipere impul&longs;um. quod licet ut plurimum fiat; <lb/>non tamen e&longs;t nece&longs;&longs;arium: &longs;ed <expan abbr="quandoq;">quandoque</expan> percutiens mino­<lb/>rem, <expan abbr="quandoq;">quandoque</expan> nullum recipit impul&longs;um. </s></p> <p type="main"> <s>Et &longs;iquidem totam dedit plgam, <expan abbr="nullamq;">nullamque</expan> recepit, non re­<lb/>flectit: verùm à plagâ conquie&longs;eit. </s><s>Ex parte verò plagæ mo­<lb/>tum continuat centrum gravitatis per lineam tangentem cir­<lb/>culi; cuius centrum e&longs;t contactus, & intervallum di&longs;tantia eiu&longs;­<lb/>dem centri gravitatis. </s><s>At cùm minor e&longs;t plaga à percu&longs;&longs;o, <lb/>mutatur ratio motùs reflexi: propterea, quòd centrum præ­<lb/>dominatur. </s><s>Inæqnaliter autem reflecti corpora, &longs;i materiâ <lb/>differant, quantumvis eandem figuram, & magnitudinem, <lb/>quin et gravitatem habeant, con&longs;tat: &longs;i pila plumbea, ferrea, la­<lb/>pidea, o&longs;&longs;ea, lignea, coriacea ex eadem di&longs;tantiâ terræ, aut pari­<lb/>eti allidatur. </s><s>Cau&longs;a huius inæqualitatis videtur non ni&longs;i ex <lb/>naturâ impul&longs;ûs priùs cognitâ obtineri. </s><s><expan abbr="Neq;">Neque</expan> enim cur inæ­<lb/>qualiter recipiatur, con&longs;tare pote&longs;t; ni&longs;i quid, & quomodo in <lb/>corporibus tecipiatur, con&longs;tet. </s><s>De quo alibi: hic verò non ni­<lb/>&longs;i ea, quæ ad in&longs;titutum facere videntur, delibabo. </s></p> <p type="main"> <s>Notandum ergo primò, &longs;i mobile percutiat aliud, produce­<lb/>re impul&longs;um æqualem illi, quo ip&longs;um movetur: globus enim <lb/>percu&longs;&longs;o æquali, eadem celeritate hunc movet: quod non ni&longs;i <lb/>ab impul&longs;u æquali e&longs;&longs;e pote&longs;t. </s><s>At &longs;i maior aut minor gravi­<lb/>tas ine&longs;t percu&longs;&longs;o, inæqualiter movetur: velociùs quidem cui <lb/>minor, tardiùs cui maior ine&longs;t gravitas. </s><s>Vnde apparet cun­<lb/>dem impul&longs;um in paruo &longs;ubiecto colligi & intendi; in magno <lb/>e&longs;&longs;e remi&longs;&longs;iorem: propterea, quòd alia &longs;it proportio moven­<lb/>tis ad mobile. </s><s>Sed dubitabis an in percu&longs;&longs;o æquali idem &longs;it <pb xlink:href="063/01/085.jpg"/>impul&longs;us. </s><s>Nam &longs;i in lineâ rectâ plures globos di&longs;ponas &longs;ibi con­<lb/>tiguos & æquales; percu&longs;&longs;o primo ultimus movetur, omni­<lb/>bus alijs immotis. </s><s>Si ergo primus in &longs;ecundo, hic in tertio <lb/>producit impul&longs;um æqualem illi, quo ip&longs;e moveretur; &longs;equi­<lb/>tur à plagâ, quæ unum movere pote&longs;t, moveri po&longs;&longs;e quolibet <lb/>&longs;patio <expan abbr="abiunctũ">abiunctum</expan>: <expan abbr="perq;">perque</expan> globos infinitos illam vim extendi, <expan abbr="e&longs;&longs;eq;">e&longs;&longs;eque</expan> <lb/>infinitam. E contra vero, &longs;i illâ &longs;erie continuò dere&longs;cit pla­<lb/>ga; ut minor &longs;it in tertio quàm in &longs;ecundo, et in hoc quàm in <lb/>primo: &longs;int globi numero 20. & &longs;ingulorum pondus librale. <lb/>habebit ergo pIaga 20-minorem rationem ad totum impul­<lb/>&longs;um quàm &longs;ubuigecuplam; hoc e&longs;t quàm habeat gravitas illius <lb/>globi ad omnium grauitatem collectam. impul&longs;us ergo minor, <lb/>quàm ut moveat pondus librarum 20; maior autem quàm &longs;it <lb/>re&longs;i&longs;tentia lib: 10 aut 15; percu&longs;&longs;o primo non movebit ulti­<lb/>mum. </s><s>Nam &longs;i totus impul&longs;us minor e&longs;t grauitate totâ, erit <lb/><expan abbr="quoq;">quoque</expan> pars impul&longs;ûs minor illâ gravitate, quæ in eadem e&longs;t ra­<lb/>tione ad totam gravitatem. </s><s>Et cùm pars 20 impul&longs;us neque­<lb/>at movere pondus lib: 1. <expan abbr="neq;">neque</expan> à plagâ minore quàm &longs;it pars 20 <lb/>movebitur. </s><s>Hoc autem e&longs;t contra experientiam. videmus <lb/>enim quovis numero interpo&longs;itis globis æqualibus ultimum <lb/>moveri ex eadem plagâ, æquali cum primo celeritate. </s><s>Dein­<lb/>de &longs;i plaga decre&longs;cens nequit ultimum movere; &longs;unt verò & in­<lb/>termedij <expan abbr="ab&longs;q;">ab&longs;que</expan> motu; erit plaga infinita in mobili, <expan abbr="ab&longs;q;">ab&longs;que</expan> eo <lb/>quòd ullam partem moveat. </s><s>Augeatur enim numerus glo­<lb/>borum in eâ ratione, in quâ plaga: <expan abbr="eritq;">eritque</expan> impul&longs;us ab ultimâ <lb/>plagâ in eadem ratione, hoc e&longs;t minori, quàm ut movere po&longs;­<lb/>&longs;it ultimum globum. </s><s>Quod cùm à ratione & experientia &longs;it <lb/>alienum, dicendum omnes globos, quantumvis numero <lb/>augeantur, ab hoc impul&longs;u peruadi <expan abbr="Neq;">Neque</expan> &longs;equitur virtutis fi­<lb/>nitæ actionem e&longs;&longs;e in&longs;initam. non enim ab extra, &longs;ed à princi­<lb/>pio interno mobilis producitur impul&longs;us; ut &longs;uo loco o&longs;ten- <pb xlink:href="063/01/086.jpg"/>dam: factâ determinatione à &longs;imili per contactum. </s><s>Quid <lb/>ergo mirum mobilia infinita impul&longs;um coacervare infinitum? <lb/><expan abbr="Atq;">Atque</expan> ex his multa arcana panduntur: cùm tanta &longs;it vis &longs;imili­<lb/>tudinis; ut nullis locorum intervallis definiantur ex eâ na&longs;cen­<lb/>tes amores: <expan abbr="neq;">neque</expan> iam miremur cœle&longs;tes influxus his illicibus <lb/>uno ceu momento trahi. </s><s>Dices Quid &longs;i inæquales &longs;int globi <lb/>& continuò minores: an ab infinito numero erit motus? nam <lb/>&longs;i ita, movebitur &longs;anè ultimus celeritate infinitâ. </s><s>Re&longs;pondeo, <lb/>cùm minor globus eadem celeritate feratur à minori impul&longs;u; <lb/>movebitur ab incipiente, & necdum perfectâ plagâ: ac proin­<lb/>de reliquus impul&longs;us motum maioris continuabit per pori&longs;ma <lb/>2. </s><s>Ex quo illud mirabile; in eodem in&longs;tanti ab uno principio <lb/>motûs fluere infinitos inter &longs;e inæquales. </s><s>Licet verò in infi­<lb/>nito daretur ultimus, negamus tamen hunc celeritate move­<lb/>ri infinitâ: propterea quòd impul&longs;us continuò minuatur iuxta <lb/>decrementum illarum Sphærularum. </s><s>At verò infinitum quis <lb/>terminabit? Cùm ergò dicimus numerum infinitum, &longs;ynca­<lb/>tegorematicè intelligi volumus, quouis dato maiorem: <expan abbr="atq;">atque</expan> <lb/>in hoc &longs;icuti cum numero decre&longs;cit moles, ita velocitas mo­<lb/>tûs augeretur. </s><s>Ii&longs;dem connexa, & à vulgi opinione remota <lb/>&longs;unt hæc. </s></p> <p type="main"> <s><emph type="italics"/>Plagam infinitam dare <expan abbr="absq;">absque</expan> eo, quòd percutiens mo­<lb/>ueatur.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Movere corpus in quâcun&queacute; di&longs;tantiâ, abs&queacute; eo, quòd <lb/>ullus in medio &longs;it motus.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Motum eodem in&longs;tanti producere in infinitum.<emph.end type="italics"/></s></p> <p type="main"> <s>Nihil ergo mirum in&longs;tante motu terræ, priu&longs;quam hæc con- <pb xlink:href="063/01/087.jpg"/>cuti & tremere incipiat, <expan abbr="atq;">atque</expan> etiam eâ immotâ ruere ædificia: <lb/>homines pedibus in&longs;i&longs;tere non valentes collabi & vacillare: fa­<lb/>ctâ enim plagâ in vi&longs;ceribus terræ medijs immotis impetus huc <lb/>&longs;e effundit: quemadmodum percu&longs;sâ muri parte oppo&longs;itâ, ea <lb/>quæ muro hærent, delabuntur. </s><s>Notandum &longs;ecundò. impul­<lb/>&longs;um non recipi uniformiter in mobili; &longs;ed rece&longs;&longs;u à &longs;ummo vi­<lb/>gore, quem infert plaga, &longs;en&longs;im attenuari tam in profundum, <lb/>quàm in latum. </s><s><expan abbr="Itaq;">Itaque</expan> videmus illas partes, quæ ictum exci­<lb/>pere coguntur, præ alijs frangi & collidi: nequaquam à plagâ <lb/>remotiores. </s><s>Quia nimirum cùm <expan abbr="unaquæq;">unaquæque</expan> particula &longs;uo impul­<lb/>&longs;u feratur & incitetur ad motum; dum hæ præcurrere fe&longs;ti­<lb/>nant, illæ ob tarditatem &longs;equi non valent, quà impetus magis <lb/>urget, &longs;i uniones habeant &longs;olubiles, avelli contingit. </s><s>Ita <lb/>quidem in principio motûs, <expan abbr="quou&longs;q;">quou&longs;que</expan> producitur impul&longs;us: <lb/>quam tamen inæqualitatem æquat centrum grauitatis, omni­<lb/>um vim colligendo; cùm ab omnibus urgeatur: <expan abbr="atq;">atque</expan> ita fit, ut <lb/>tardiores incitentur, velociores retardentur: quò eodem <lb/>cum centro gravitatis motu ferantur. </s></p> <p type="main"> <s>Motus ergo centri e&longs;t principium motûs reliquorum: & cùm <lb/>à motu fiat plaga; erit huius motus & ratio in ordine ad cen­<lb/>trum <expan abbr="Itáq;">Itáque</expan> fit utictus perpendicularis omnium &longs;it graui&longs;&longs;imus: <lb/>obliquorum verò tantò vim habeat minorem, quantò magis <lb/>obliquè ferit: eo enim modo habet hic motus, quo grauitas <lb/>in lap&longs;u inclinato. </s><s>Quòd &longs;i ergo corpora eiu&longs;dem molis & <lb/>&longs;oliditatis, percutias ictu latiore <expan abbr="eóq;">eóque</expan> plano; videbis in medio <lb/>plagæ &longs;itas partes priùs frangi, ijs quæ in ambitu &longs;unt <expan abbr="quandoq;">quandoque</expan> <lb/>illæ&longs;is. </s><s>Porro impul&longs;us in mobili, quia à plagâ cæpit, in aliam <lb/>plagam de&longs;tinatur. & &longs;i quidem plagam totam peregit, totus; &longs;i <lb/>partem, in eadem ratione ex&longs;olvitur impul&longs;us, ut con&longs;tat ex <lb/>propo&longs;: 37. </s><s>Quin motus in aëre quid aliud, quàm percu&longs;&longs;io <lb/>& plaga continuata: unde in aëre cra&longs;&longs;iore, licet ab eadem <pb xlink:href="063/01/088.jpg"/>viferatur mobile, minor e&longs;t motus. </s><s>Ita in aquâ ob &longs;olidita <lb/>tem & re&longs;i&longs;tentiam maiorem ad minus intervallum plaga cum <lb/>motu terminatur. </s><s>An igitur licebit ex proportione motûs <lb/>in diver&longs;is <expan abbr="elem&etilde;tis">elementis</expan> coniecturam &longs;umere illorum gravitatis? an <lb/>præter <expan abbr="gravitat&etilde;">gravitatem</expan> tenacitas partium huc facit? utlicet æquè gra­<lb/>ves, non <expan abbr="tam&etilde;">tamen</expan> eadem facilitate findantur: cùm & ab eadem <lb/>gravitate percu&longs;&longs;io fiat inæqualis. </s><s>At verò &longs;i motus e&longs;t per­<lb/>cu&longs;&longs;io continuata; an po&longs;ito vacuo nullus erit motus? an &longs;em­<lb/>per movebitur illud mobile? cùm nihil percuti po&longs;&longs;it, <expan abbr="neq;">neque</expan> ab <lb/>ullo minuatur impul&longs;us. </s><s>Deinde quâ ratione &longs;piritus moven­<lb/>tur, &longs;i nullus illorum e&longs;t tactus? an non nece&longs;&longs;e eâ ratione mo­<lb/>veri, quâ corpora, tran&longs;ito priùs medio? cùm dia&longs;tima &longs;it cor­<lb/>porum, non verò &longs;pirituum: qui neq, &longs;ibi &longs;unt vicini, <expan abbr="neq;">neque</expan> cor­<lb/>poreis ab&longs;unt intervallis: cùm <expan abbr="neq;">neque</expan> loco capiantur. </s></p> <p type="main"> <s>Per accidens tamen moveri videntur, & motum corporeum <lb/>adumbrare, per operationem &longs;en&longs;ibilem in medio factam. </s></p> <p type="main"> <s>Quòd &longs;i ergo &longs;piritus ille, qui pacem hic turbat, velit Roma­<lb/>nos inquietare; non nece&longs;&longs;e hunc per Venetos & loca media <lb/>ire, at &longs;i illam columnam, quam| ferunt Româ huc delatam, <lb/>eò referre velit; celeritatem habebit definitam, et non ni&longs;i per <lb/>loca interiecta movebitur. </s></p> <p type="main"> <s>Notandum Tertio, impul&longs;um alium habere proportionem <lb/>ad mobile loco movendum; alium non: ut licet nulli hæreat, <lb/><expan abbr="in&longs;i&longs;tatq;">in&longs;i&longs;tatque</expan> non tamen ex lllâ percu&longs;&longs;ione ad motum incitari. <lb/><expan abbr="atq;">atque</expan> hic impul&longs;us, <expan abbr="quandoq;">quandoque</expan> totum mobile, <expan abbr="quandoq;">quandoque</expan> non ni&longs;i <lb/>aliquam partem pervadit. </s><s>Et quod attinetilla corpora, quæ <lb/>percu&longs;&longs;a loco moventur, in quâ proportione e&longs;&longs;e debeant, di­<lb/>ctum in porismatis ad prop: 37. </s><s>Dubitatio tamen e&longs;&longs;e pote&longs;t, <lb/>quamobrem percu&longs;&longs;o maiori quie&longs;cente <expan abbr="motoq;">motoque</expan> minus <expan abbr="quan-doq;">quan­<lb/>doque</expan> re&longs;iliat, nam totam dedit plagam; & cùm moveatur ma- <pb xlink:href="063/01/089.jpg"/>ius à plagâ &longs;e abducens, nullam recipere videtur. </s><s>Re&longs;pondeo <lb/>id provenire ex inæqualitate motûs. </s><s>Nam cùm tardiùs con­<lb/>citetur ad motum maius, quàm æquale; in illâ morulâ, priu&longs;­<lb/>quam incipiat moveri, re&longs;i&longs;tit: ac proinde repercu&longs;&longs;io fit æ­<lb/>qualis illi morulæ, quâ veluti hæret in principio motûs. </s><s><expan abbr="Itaq;">Itaque</expan> <lb/>fieri pote&longs;t, ut <expan abbr="quandoq;">quandoque</expan> æquali, <expan abbr="quandoq;">quandoque</expan> minori impul&longs;u re­<lb/>&longs;iliat: nunquam verò motum maioris con&longs;equatur: &longs;icuti <expan abbr="neq;">neque</expan> <lb/>maior percu&longs;&longs;o minori quie&longs;cere pote&longs;t, aut reflecti. </s><s>At ve­<lb/>rò illud mobile, quod percu&longs;&longs;um non movetur, nece&longs;&longs;e illam <lb/>plagam à minori recipere: nam &longs;i ab æquali percutiatur &longs;eu tel­<lb/>lus, &longs;eu planctarum unus, locum &longs;anè mutabit. </s><s>Et &longs;i quidem <lb/>corpus fuerit &longs;onorum, diu re&longs;onat; cuius partes omnes vi­<lb/>bratione quadam commoventur. </s><s>Sonus autem &longs;ibi relictus <lb/>cum illo tremore &longs;en&longs;im minuitur & vane&longs;cit; & non ni&longs;i à <expan abbr="cõ-tactu">con­<lb/>tactu</expan> repentè contice&longs;cit. </s><s>In corporibus autem &longs;urdis, quæ <lb/>percu&longs;&longs;a nihil aut parum &longs;onant, vibratio quidem fit, minùs ta­<lb/>men diuturna: quàm ex impul&longs;u reciprocante fieri ex eo con­<lb/>&longs;tat. <!--neuer Satz-->quòd atomi & corpu&longs;cula minuta in &longs;uperficie illorum <lb/>corporum <expan abbr="eod&etilde;">eodem</expan> tremore convellantur, & incitentur ad <expan abbr="motũ">motum</expan>. </s><lb/><s>Minùs tamen regulariter in his, quàm in corporibus &longs;onoris <lb/>fit reciprocatio motûs &longs;eu impul&longs;us, ob atomos inæqualiter &longs;i­<lb/>tas; à quibus via procur&longs;us & recur&longs;us variè detorquetur. </s></p> <p type="main"> <s>Durat verò impul&longs;us à &longs;uperficie ultimâ &longs;e reducens, <expan abbr="rur&longs;úmq;">rur&longs;úmque</expan> <lb/>excurrens veluti &longs;e ip&longs;um per&longs;equendo, <expan abbr="quou&longs;q;">quou&longs;que</expan> plaga conti­<lb/>nuò decre&longs;cens &longs;e ip&longs;am ab&longs;ump&longs;it. </s><s>Quod quidem in corpo­<lb/>ribus non continuis, cuiu&longs;modi lana, promptè fit ob vias mil­<lb/>le modis interci&longs;as. </s><s>Pi&longs;a verò percu&longs;&longs;o &longs;acco licet conti­<lb/>nua non &longs;int, &longs;onant: propterea, quòd partes &longs;en&longs;ibiles & &longs;o­<lb/>num ex le habentes colliduntur: <expan abbr="itaq;">itaque</expan> legumina quò maiora <lb/><expan abbr="magisq;">magisque</expan> rotunda, magis re&longs;onant. </s><s>Ita quidem in corpore <lb/>habet impul&longs;us: quod licet non mouet localiter, omnes ta- <pb xlink:href="063/01/090.jpg"/>men illius partes pervadit. </s><s>In corpore autem va&longs;tæ molis, <lb/>cuiu&longs;modi tellus, eò <expan abbr="u&longs;q;">u&longs;que</expan> procedit, dum illâ extenuatione <lb/>pror&longs;us in&longs;en&longs;ilis euadat: & cùm nulla e&longs;t reciprocatio, <expan abbr="neq;">neque</expan> <lb/>vibratio contingit. </s><s>Tremere tamen interdum &longs;olum ex in­<lb/>genti plagâ con&longs;tat: cùm partes vehementer pre&longs;&longs;æ rea&longs;&longs;ur­<lb/>gunt. </s><s>At verò <expan abbr="quou&longs;q;">quou&longs;que</expan> una <expan abbr="quæq;">quæque</expan> plaga &longs;e extendat, necdum <lb/>liquet: con&longs;tat &longs;anè longi&longs;&longs;imè protendi: in magnâ enim di­<lb/>&longs;tantiâ auribus terræ admotis &longs;onum etiam non magnum per­<lb/>cipiunt excubitores. </s><s>E&longs;t tamen magna differentia pro qua­<lb/>litate terræ: caverno&longs;a enim <expan abbr="multúmq;">multúmque</expan> aëris continens &longs;o­<lb/>num longiùs protendit, quàm uligino&longs;a & palu&longs;tris: & quæ <lb/>continua e&longs;t ac veluti concatenata, quàm &longs;abulo&longs;a & interci&longs;a. </s></p> <p type="main"> <s>Notandum Quartò, impul&longs;um naturâ &longs;uâ lineam rectam & <lb/>viam &longs;equi percutientis. <expan abbr="itaq;">itaque</expan> &longs;i perpendiculariter incidat pla­<lb/>no motum &longs;eu impul&longs;um producit in directum, &longs;i nihil ob&longs;tat. </s><lb/><s>At cùm re&longs;i&longs;tentia maior e&longs;t ex unâ, quàm aliâ parte: ut cùm <lb/>trabem longiorem percutimus non in centro gravitatis, &longs;ed in <lb/>parte uni extremo propiore: tum motus non fit in directum, <lb/>&longs;ed circularis: cuius centrum alterum extremum quie&longs;cens, <lb/>& à plagâ magis remotum. </s><s>Quòd &longs;i percu&longs;&longs;io fiat in centro: <lb/>tamet&longs;i ad partes remotiores à plagâ minor impul&longs;us &longs;e exten­<lb/>dat; quia tamen centrum gravitatis æquationem inducit; <lb/>omnes æqualiter & in directum moventur. </s><s>In Sphærâ autem <lb/>&longs;eu globo impetus à plagâ in centrum dirigitur, &longs;i moveri de­<lb/>beat: quod alioqui non e&longs;t nece&longs;&longs;arium: <expan abbr="quandoq;">quandoque</expan> enim pla­<lb/>ga ex obliquo illius partem decerpit. </s><s>At &longs;i globus alium <lb/>percutiat <expan abbr="quacunq;">quacunque</expan> ratione, nece&longs;lariò hæc plaga centrum &longs;pe­<lb/>ctat. propterea, quòd <expan abbr="utrumq;">utrumque</expan> centrum <expan abbr="atq;">atque</expan> illorum plaga &longs;it in <lb/>eadem lineâ rectâ. </s><s>Nulla tamen plaga ex obliquo facta to- <pb xlink:href="063/01/091.jpg"/>tum impul&longs;um ab&longs;umit: cùm non tota vis centri percutiat. ne­<lb/>ce&longs;&longs;e ergò mobile ab eiu&longs;modi plagâ motum continuare. </s></p> <p type="main"> <s>Notandum Quintò, hanc differentiam e&longs;&longs;e inter corpora <lb/>percu&longs;&longs;a, quæ ex illâ plagâ moventur, & quæ immota manent. <lb/>quòd hæc ictum recipiant <expan abbr="reddant&qacute;">reddantque</expan>;, nequaquam illa: pro­<lb/>pterea, quòd licet ab his contactus fiat, non tamen etiam pla­<lb/>ga: e&longs;t enim plaga irruptio quædam violenta, & veluti pene­<lb/>tratio: at verò quæ à plagâ moventur, nullam faciunt irrup­<lb/>tionem, &longs;ed à plagâ celeriter &longs;e adducunt: non igitur percu­<lb/>tere dicuntur. </s><s>Immota verò quia percu&longs;&longs;ioni non cedunt <lb/>eadem violentiâ irrumpunt <expan abbr="penetrantq;">penetrantque</expan> in ea, à quibus pene­<lb/>trantur: unde percuti & percutere, & impul&longs;um recipere <expan abbr="da-req;">da­<lb/>reque</expan> dicuntur. </s><s>Qui &longs;ummus e&longs;t in contactu: Inde verò &longs;en­<lb/>&longs;im attenuatur. </s><s>Et in percu&longs;&longs;o quidem ex illâ vibratione de­<lb/>mum conquie&longs;cit: in percutiente verò quia priori e&longs;t contra­<lb/>rius, ip&longs;um retroagit. </s><s>Dices quid &longs;i dicamus impul&longs;um non <lb/>ni&longs;i per contrarium impul&longs;um tolli? Nam &longs;i globus alium per­<lb/>cutiat &longs;ibi æqualem & quie&longs;centem, ex illâ communi plagâ in <lb/><expan abbr="utroq;">utroque</expan> producitur impul&longs;us: qui globum quie&longs;centem loco <lb/>movet. propterea quòd huic motui nihil &longs;it contrarium: alte­<lb/>rum verò ob impul&longs;ûs contrarictatem à motu continet. </s></p> <p type="main"> <s>Re&longs;pondeo, licet hæc ratio &longs;it probabilis, non tamen in alijs <lb/>locum habere. </s><s>Nam cùm maiori immoto minor globus allidi­<lb/>tur, &longs;i æqualem dat <expan abbr="recipit&qacute;">recipitque</expan>; impul&longs;um, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan> hic contrarius pri­<lb/>ori, non re&longs;iliet; verùm à motu conquie&longs;cet. </s></p> <p type="main"> <s>Dices à maiori corpore ictum fieri maiorem; ac proinde ab <lb/>huius exce&longs;&longs;u fieri illum motum. </s><s>Sed contra, quia velocitas <lb/>motûs reflexi non augetur in eá ratione, in quâ illorum cor­<lb/>porum magnitudo. </s><s>Deinde cùm duo globi æquales &longs;e <lb/>percutiunt in motu, <expan abbr="uterq;">uterque</expan> reflectit: oportebat verò <expan abbr="utrumq;">utrumque</expan> <pb xlink:href="063/01/092.jpg"/>quie&longs;cere à motu. </s><s>Dicendum ergo in contactu à plagâ per­<lb/>fectâ impul&longs;um ex&longs;pirare: & &longs;i percu&longs;&longs;um non cedat, &longs;ed re­<lb/>nitatur, alium impul&longs;um &longs;ibi comparare ex illâ plagâ: <expan abbr="cúmq;">cúmque</expan> <lb/>æqualem; cùm ex toto e&longs;t immotum. </s><s>At cùm à plagâ &longs;e ab­<lb/>ducens locum mutat &longs;eu totum, &longs;eu &longs;ecundùm partem, minu­<lb/>itur in eadem ratione hic impul&longs;us. </s><s><expan abbr="Itaq;">Itaque</expan> &longs;i corpus per­<lb/>cu&longs;&longs;um in &longs;e ip&longs;um &longs;idit <expan abbr="ceditq;">ceditque</expan>: ut lana, cera, argilla, plum­<lb/>bum; quia ictus &longs;en&longs;im emoritur, nulla vel exigua fit reper­<lb/>cu&longs;&longs;io. </s><s>Et quia huiu&longs;modi plaga non tota &longs;imul, &longs;ed divi­<lb/>&longs;im recipitur: inde fit, ut impul&longs;us ex ea productus minùs la­<lb/>tè evagetur: idem enim fit quemadmodum &longs;i multæ plagæ <lb/>exiguæ continuarentur. </s><s>At cùm &longs;olidum corpus <expan abbr="firmumq;">firmumque</expan> <lb/>percutitur; quia totam plagam &longs;imul admittit, omnia latè con­<lb/>tremi&longs;cunt. </s><s>Corpus ergo cùm incidit alteri, aut totum dat <lb/>impul&longs;um &longs;imul & confertim; aut in plures veluti plagas hunc <lb/>partitur. </s><s>Et &longs;i ita, non reflectit percutiens. </s><s>Quòd &longs;i ceden­<lb/>do demum renitatur; ut cùm partes compre&longs;&longs;æ nequeunt iam <lb/>premi; pars illa duntaxat plagæ reflectit. </s><s>At cùm totum dat <lb/>impul&longs;um; velloco movetur percu&longs;&longs;um, <expan abbr="idq;">idque</expan> eadem celerita­<lb/>te vel minori: & ab hoc quidem reflectit pro men&longs;urâ illius <lb/>tar ditatis; non autem ab eo, quod celeritate movetur æquali. </s><lb/><s>Immotum demum à plagâ aut in &longs;e ip&longs;o terminat impul­<lb/>&longs;um, aut aliò transfert: ut &longs;i plures globi æquales & <lb/>contigui plagam excipiant. & ab illo quidem, <lb/>non autem ab his reflectit motus. </s></p> <figure id="id.063.01.092.1.jpg" xlink:href="063/01/092/1.jpg"/> <pb xlink:href="063/01/093.jpg"/> <p type="main"> <s><emph type="center"/>PARS QVARTA.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>De percu&longs;sionibus.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>QVID COLLISIO ET FRACTVRA.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>COrpora invicem colli&longs;a aut mutant figuram, aut &longs;unt <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> mutatione. </s><s>Mutatur autem figura partis unius plu­<lb/>riumue ami&longs;&longs;ione, aut <foreign lang="greek">metasta(ses</foreign> & &longs;itu illarum permutato: <lb/><expan abbr="atq;">atque</expan> hæc <foreign lang="greek">piesta\</foreign> dicuntur: quorum &longs;uperficies in pro&longs;undum <lb/>permutatur, nec dividitur, nec ulla particula aliò transfertur; <lb/>quemadmodum fit in aquâ pre&longs;sâ. </s><s>Talia verò &longs;unt Ari&longs;toteli, <lb/>quæ meatus habent vacuos cognati corporis, tamet&longs;i forte <lb/>mollioribus &longs;int pleni, in quos partes pre&longs;&longs;æ recipiantur. </s></p> <p type="main"> <s>Ita enim pila ærea aquâ, aut aëre plena, à vi externâ pre&longs;&longs;a &longs;u­<lb/>perficiem gibbam, aliâ &longs;eu planâ, &longs;eu concavâ permutat: quan­<lb/>quam & in totum &longs;olida ob minores meatus <foreign lang="greek">piesta\</foreign> dicantur. <lb/>quæ &longs;i manentem habeant compre&longs;&longs;ionem, <foreign lang="greek">pilhta</foreign> ut cera, æs, <lb/>plumbum, aurum: <foreign lang="greek">a)pilhta</foreign> verò &longs;unt, quæ à compre&longs;&longs;ione re­<lb/>a&longs;&longs;urgunt. </s><s>At verò quæ in percu&longs;&longs;ione partem unam plure&longs;­<lb/>uè amittunt, <foreign lang="greek">xataxta\ xai\ drausta\</foreign> hoc habent di&longs;crimen: quòd <lb/><foreign lang="greek">xa/tazis</foreign> &longs;it divi&longs;io in partes magnas: ut cùm <expan abbr="lignũ">lignum</expan> aut os fran­<lb/>gimus: <foreign lang="greek">qrau_sis</foreign> verò in partes plures quàm duas: ut in lapi. <lb/>de, te&longs;tà, vitro. </s><s>Cuius rationem reddit Ari&longs;toteles <foreign lang="greek">pollou\s e)/xein <lb/>paralla/tontas po)rous</foreign>; <expan abbr="Itaq;">Itaque</expan> fit ut cùm continui |non &longs;int, &longs;ed <lb/>alternâ permutatione po&longs;iti; facto| initio motûs| non in dire­<lb/>ctum, &longs;ed tortuosè procedat fi&longs;&longs;ura: & plaga una, ob|indi&longs;po. <lb/>&longs;itionem &longs;ubiecti, non unum producat e&longs;&longs;ectum. </s><s>Quem mo- <pb xlink:href="063/01/094.jpg"/>tum &longs;eu impul&longs;um duobus modis fieri docet Ari&longs;toteles, <foreign lang="greek">w)'sei</foreign><lb/>&longs;eu pul&longs;ione: ut cùm à tergo motui in&longs;tamus: & percu&longs;&longs;ione, <lb/>in eo à &longs;e differentes; ut <foreign lang="greek">w)sit</foreign> &longs;it <foreign lang="greek">xi/nhsis a)po) th_s a(/ysews, plh­<lb/>gh\d<gap/> a)po\ th=s fora)s</foreign>. </s><s>De quo an verum &longs;it, dubitamus. </s><s>Nam <lb/>&longs;i plures globi inter &longs;e æquales, & contigui ordine &longs;equantur; <lb/>percu&longs;&longs;o primo ultimus movetur omnibus alijs immotis: ne­<lb/>ce&longs;&longs;e autem hunc à penultimo moveri, habebit ergo plagam <lb/>ex hoc, <expan abbr="ab&longs;q;">ab&longs;que</expan> eo quòd moveatur. </s><s>Plagam enim fieri ex eo <lb/>con&longs;tat: quòd &longs;i ultimo loco pila cry&longs;tallina aut vitrea excipi­<lb/>at hunc motum, frangi contingit. </s><s>Dici tamen pote&longs;t pro Ari­<lb/>&longs;totele, ad plagam inducendam motum e&longs;&longs;e nece&longs;&longs;arium: li­<lb/>cet non plagam totam, &longs;ed huius principium &longs;equatur. </s><s>Hæc <lb/>enim à primo globo incipiens ad ultimum terminatur, & ve­<lb/>luti pro unâ plagâ habetur. </s><s>At verò quamobrem à percu&longs;&longs;i­<lb/>one nonnulla frangi contingat, maior e&longs;t dubitatio. </s><s>Nam <lb/>certum e&longs;t dictas pa&longs;&longs;iones ex impul&longs;u provenire: percutere <lb/>enim e&longs;t producere impul&longs;um, & percuti hunc recipere. </s><s>Si <lb/><expan abbr="itaq;">itaque</expan> impul&longs;us corpora, in quibus recipitur, frangit; oporte­<lb/>bit &longs;anè illam velocitatem motûs con&longs;ecuta, quam affert pla­<lb/>ga, frangi in ip&longs;o motu: quod tamen non fit. </s><s>Va&longs;a enim vi­<lb/>trea, priu&longs;quàm &longs;olidum occurrat, in ip&longs;o lap&longs;u non collidun­<lb/>tur. </s><s>Sed <expan abbr="neq;">neque</expan> percu&longs;&longs;io per &longs;e hunc affectum inducit: idem <lb/>enim e&longs;t &longs;iuè percutiat, &longs;iue percutiatur <foreign lang="greek">qrausto\n</foreign> vitrum enim <lb/>& &longs;axo illi&longs;um, & à &longs;axo alli&longs;um pari facilitate <expan abbr="frãgitur">frangitur</expan>. </s><s>At verò <lb/>cùm pila vitrea aut cry&longs;tallina aliam percutit &longs;ibi æqualem &longs;eu <lb/>ferream, &longs;eu lapideam, non frangitur ex illo ictu quantumvis <lb/>inten&longs;o. </s><s>Videtur ergò huius ratio ex impul&longs;u provenire, non <lb/>ab&longs;olutè, quem habere pote&longs;t quouis dato maiorem, <expan abbr="atq;">atque</expan> adeo <lb/>infinitum <expan abbr="ab&longs;q;">ab&longs;que</expan> ullâ partium colli&longs;ione: &longs;ed ex inæquali mo­<lb/>do hunc recipiendi. </s><s>Propterea quòd partes propiores plagæ <lb/>hunc priùs habeant, <expan abbr="magisq;">magisque</expan> inten&longs;um: qui totus non ni&longs;i in a- <pb xlink:href="063/01/095.jpg"/>liquâ morulâ producitur. </s><s>Vnde fit ut partes priùs <expan abbr="magisq;">magisque</expan> <lb/>percu&longs;&longs;æ, priu&longs;quam æquatio fiat à centro gravitatis, præcur­<lb/>rere fe&longs;tinent: aliæ &longs;equi non valentes mutuâ di&longs;tractione à &longs;e <lb/>divellantur. cùm nimirum maior e&longs;t vis ad movendum, quàm <lb/>illa quies & retentio partium unitiva. </s><s>Frangi enim contingit <lb/>illâ parte, quâ impetus magis urget, aut unio minùs re&longs;i&longs;tit: <lb/><expan abbr="itaq;">itaque</expan> videmus <expan abbr="quandoq;">quandoque</expan> partes à plagâ remotiores præ alijs <lb/>frangi. </s><s>Et quidem <foreign lang="greek">qrausto\n</foreign> in multa fragmenta di&longs;&longs;ilit: ut <lb/>vitrum, cry&longs;tallus, te&longs;ta, lapis: <expan abbr="Idq;">Idque</expan> præter opinionem-<foreign lang="greek">to\ xa­<lb/>taxto\n</foreign> verò minùs fallit de&longs;ignationem: at&que; in duas <expan abbr="plerumq;">plerumque</expan> <lb/>partes ab&longs;cedit; factâ divi&longs;ione in centro plagæ. </s><s>Quæ qui­<lb/>dem <foreign lang="greek">xa/tacis</foreign> magis procedit, cùm plaga longiùs abe&longs;t à parti­<lb/>bus extremis: tum enim partem illam, quæ interiacet, pro ve­<lb/>cte habet: cuius hypomochlium &longs;unt extrema. </s><s><expan abbr="Atq;">Atque</expan> ita fit, <lb/>ut vitro fragili, aut &longs;tramine fu&longs;tem <expan abbr="cra&longs;&longs;ior&etilde;">cra&longs;&longs;iorem</expan> <expan abbr="quandoq;">quandoque</expan> &longs;rangi <lb/>contingat: cùm nimirum maior e&longs;t velocitas motûs, quàm re­<lb/>&longs;i&longs;tentia: <expan abbr="nullaq;">nullaque</expan> à percu&longs;&longs;o recipitur plaga. </s><s>Oppo&longs;ito mo­<lb/>dò habet fractura: cùm hypomochlium e&longs;t in centro plagæ, <lb/>&longs;eu divi&longs;ionis: extrema verò <expan abbr="utrinq;">utrinque</expan> adducuntur. </s><s>Nam in pri­<lb/>ori quidem <foreign lang="greek">xatacq</foreign> duo, hic non ni&longs;i unum e&longs;t hypomochli­<lb/>um. </s><s>Dubitabis ergo, quæ harum fractura &longs;it magis expedita. </s><lb/><s>Dicendum verò impul&longs;um extrema adducentem, ut hypo­<lb/>mochlium medio &longs;it loco, prevalere: quod quidem erit ma­<lb/>nife&longs;tum, &longs;i fu&longs;tem parte mediâ præhen&longs;um ijsdem viribus <lb/>frangere coneris. </s><s>Huius autem ratio: quòd extrema vim <lb/>habeant vectis non impeditam: tantâ ergo acce&longs;&longs;ione auge­<lb/>tur impul&longs;us, quanta huius e&longs;t longitudo: re&longs;i&longs;tentiâ in &longs;olâu­<lb/>nione hypomochlij vim habente. </s><s>At verò cùm extrema hy­<lb/>pomochlio innituntur, & plaga fit in huius centro; impul&longs;us <lb/>quidem augetur ex illa remotione <expan abbr="utrinq;">utrinque</expan> ab hypomochlio: <pb xlink:href="063/01/096.jpg"/>verùm partium unio <expan abbr="utriq;">utrique</expan> re&longs;i&longs;tit & divi&longs;ioni, & vectis de­<lb/>pre&longs;&longs;ioni. </s></p> <p type="main"> <s><emph type="center"/>De Contrafi&longs;&longs;urâ.<emph.end type="center"/></s></p> <p type="main"> <s>Contrafi&longs;&longs;ura e&longs;t rima, &longs;eu fractura cranij in parte à percu&longs;­<lb/>&longs;ione &longs;eu plagâ di&longs;lante: quam Hippoc: propterea, quòd <lb/>ægrum & Medicum <expan abbr="quandoq;">quandoque</expan> latens in perniciem adducat, <lb/><foreign lang="greek">cumfezan</foreign> &longs;eu infortunium vocat. </s><s>Alij re&longs;onitum; quòd opi­<lb/>nentur ab ictu re&longs;ultum fieri in illam partem. </s><s>Di&longs;&longs;ident verò <lb/>à &longs;e: quòd alij non ni&longs;i in parte oppo&longs;itâ rimam agi volunt: <lb/>alij hoc negant. </s><s>Et licet in parte oppo&longs;itâ, & à plagâ aliquo <lb/>modo di&longs;tante fi&longs;&longs;uram admittant; non tamen excedere vo­<lb/>lunt os plagâ affectum. </s><s>Ita Paulus Ægineta, Guido de Cauli­<lb/>aco, Vidus Vidius, & Fallopius. </s><s>Probant ex u&longs;u &longs;uturarum: <lb/>quas eo fine à naturâ factas dicunt; quò impetus plagæ in ijs <lb/>terminetur: ne noxa alias <expan abbr="quoq;">quoque</expan> partes attingat: quod quidem <lb/>erat futurum, &longs;i Cranium continuum <expan abbr="atq;">atque</expan> unio&longs;&longs;e factum fu­<lb/>i&longs;&longs;et. A &longs;uturâ verò impal&longs;um &longs;i&longs;ti, manife&longs;tum in vitro, aut <lb/>ære rupto, deficiente in illam fi&longs;&longs;uram &longs;ono: ita ergo in illis <lb/>iuncturis, quibus pectinatim os cranij coit, emori impul&longs;um <lb/>volunt. </s><s>Verùm hi imperiti videntur eorum, quæ circa im­<lb/>pul&longs;um & motum fiunt. </s><s>Nam globi ordine di&longs;po&longs;iti, <expan abbr="&longs;eq;">&longs;eque</expan> <lb/>tangentes minùs &longs;unt continui, quám cranium in illis &longs;uturis: <lb/>in quibus &longs;i quid ine&longs;t humoris aut &longs;piritûs, reliquo in o&longs;&longs;e hu­<lb/>mori & &longs;piritui continuatur: & tamen à primo globo omnes <lb/>reliqui impul&longs;um recipiunt: quid ergo ob&longs;tat, quò minùs <lb/>cranio percu&longs;&longs;o impetus á plagâ totum pervadat: Sonum <lb/>autem deficere cogunt partes á fi&longs;&longs;urâ inæqualiter prominen­<lb/>tes: dum in illâ vibratione partes oppo&longs;itas tangunt: á conta- <pb xlink:href="063/01/097.jpg"/>ctu enim finiri &longs;onum con&longs;tat. </s><s>Simili ergò modo fit, quem­<lb/>admodum &longs;i lamina incurvetur: quæ &longs;onum edit <expan abbr="quou&longs;q;">quou&longs;que</expan> pars <lb/>reflexa aliam partem tangat. </s><s>At &longs;i vitrum perforetur, nihil <lb/>ob&longs;tat ille hiatus, quò minùs partes reliquæ &longs;onent. </s><s>Deinde <lb/>experientia his adver&longs;atur. </s><s>Nicolaus enim Florentinus &longs;er: 7 <lb/>&longs;um: 2. tract: 4. cap: 1. te&longs;tatur in Re&longs;tiario contrafi&longs;&longs;uram in <lb/>parte oppo&longs;itâ plagæ deprehendi&longs;&longs;e: Et Petrus Paw vidi&longs;&longs;e <lb/>ictum os &longs;ini&longs;trum bregmatis, quo loco lamdoidi iun­<lb/>gitur: fi&longs;&longs;o &longs;yncipitis o&longs;&longs;e dextro, loco ita vicino &longs;uturæ coro­<lb/>nariæ, ut pars rimæ eò &longs;e extenderit. </s><s>Cùm <expan abbr="itaq;">itaque</expan> de facto con­<lb/>&longs;tet, cau&longs;am inquirimus. </s><s>Certum e&longs;t adimpul&longs;um referri à pla­<lb/>gâ provenientem: at cur non in loco plagæ &longs;ed huic oppo&longs;ito, <lb/>à minori & iam attenuato impul&longs;u hoc patitur? <expan abbr="Neq;">Neque</expan> enim di­<lb/>ci pote&longs;t ob debilitatem findi illam partem; quam impetus in­<lb/>venit minùs virium habere ad re&longs;i&longs;tendum: tenuiora enim <lb/><expan abbr="minùsq;">minùsque</expan> firma inter&longs;unt o&longs;&longs;a, inter os &longs;ini&longs;trum bregmatis, & <lb/>os &longs;yncipitis dextrum. </s><s>Qui verò aërem illis cavernulis in­<lb/>clu&longs;um huc accer&longs;unt, ineptam pro &longs;e habent rationem: quia <lb/>nimirum ex ictu commoveatur: & per totam cranij &longs;ub&longs;tan­<lb/>tiam pervagatus, in parte demum oppo&longs;itâ allidatur: <expan abbr="reniténsq;">reniténsque</expan> <lb/>os illud findat. </s><s>Quomodo enim aër in illis mæandris tortuo­<lb/>&longs;is, <expan abbr="atq;">atque</expan> in &longs;e reductis moveri pote&longs;t quantumuis impetuo&longs;us? <lb/>an non mille modis interci&longs;us; dum vel allidit, vel re&longs;ilit, priùs <lb/>deficiet? <!--neuer Satz-->Deinde cùm aër &longs;it mollis & fluidus, nequit illum <lb/>impetum &longs;u&longs;tinere, aut con&longs;ervare: & e&longs;to demus <expan abbr="quacunq;">quacunque</expan> vi­<lb/>olentiâ irrure, <expan abbr="&longs;ibiq;">&longs;ibique</expan> obuiam fieri in parte oppo&longs;itâ: an non <lb/>&longs;uis viribus hac ratione occumbet; dum ip&longs;e &longs;ibi in&longs;tat, & in &longs;e <lb/>ip&longs;um luctatur? <!--neuer Satz-->An ergo dicendum in figura cranij &longs;phæroide <lb/>&longs;itam e&longs;&longs;e cau&longs;am? <!--neuer Satz-->quòd partes ab extra pre&longs;&longs;æ magis &longs;tipen­<lb/>tur, <expan abbr="magisq;">magisque</expan> re&longs;i&longs;tant divi&longs;ioni: à centro autem facto motu à &longs;e <pb xlink:href="063/01/098.jpg"/>diducantur? <!--neuer Satz-->Ita enim fornices onera videmus &longs;u&longs;tinere, & <lb/>contra niti: quòd &longs;i à parte internâ &longs;eu cavâ urgeantur; fati&longs;ce­<lb/>re & di&longs;&longs;olvi. </s><s>Cùm ergo cranium in modum fornicis &longs;it re­<lb/>ductum; non facilè à plagâ ab extra incidente di&longs;&longs;olui pote&longs;t: <lb/>parte verò oppo&longs;itâ, quia impetus extra fertur, <expan abbr="e&longs;tq;">e&longs;tque</expan> à centro <lb/>perpendicularis, nihil mirum di&longs;&longs;olvi illam continuitatem. </s><lb/><s>Accedit quòd impul&longs;us, facto principio motûs à plagâ, non <lb/>conquie&longs;cit in parte oppo&longs;itâ; cuius violentia ad maius inter­<lb/>vallum de&longs;tinatur. </s><s>Cùm <expan abbr="itaq;">itaque</expan> reflecti &longs;it nece&longs;&longs;e: & pars &longs;i­<lb/>ni&longs;tra dextror&longs;um, hæc &longs;ini&longs;tror&longs;um abeat; in illâ motuum con­<lb/>trarietate, partibus à &longs;e divul&longs;is accidit fi&longs;&longs;ura. </s><s>Simili modo <lb/>res habetin vitro à ba&longs;i circulari in conum fa&longs;tigiato. quæ pla­<lb/>no æqualiter alli&longs;a abrumpit pedamentum: propterea, quòd <lb/>impetus à lati&longs;&longs;imâ parte incipiens, <expan abbr="&longs;éq;">&longs;éque</expan> reforbens, ab inter&longs;e­<lb/>ctione in cono factâ, rur&longs;um in diuer&longs;a abit. </s><s>Licet verò im­<lb/>pul&longs;us naturâ &longs;uâ lineam rectam &longs;equatur; pro ratione tamen <lb/>&longs;ubiecti illam rectitudinem variè, <expan abbr="atq;">atque</expan> interdum circulo per­<lb/>mutat. </s><s>Et &longs;i quidem illa corpu&longs;cula, quibus corpora inte­<lb/>xuntur, continuâ &longs;erie &longs;e excipiant, impul&longs;us nullibi offendit: <lb/>&longs;ed per atomos uniformes &longs;e circumagens non ni&longs;ilongâ mo­<lb/>râ con&longs;ene&longs;cit. </s><s>At cùm figurâ & &longs;itu à &longs;e differunt: quia mil­<lb/>le modis di&longs;cerpi contingit, citò emoritur. </s><s><expan abbr="Atq;">Atque</expan> ex his vide­<lb/>tur manife&longs;tum, quâ ratione impul&longs;us à parte cranij percu&longs;sâ <lb/>circumgyrando, <expan abbr="&longs;ibiq;">&longs;ibique</expan> obviam factus in parte oppo&longs;itâ rimam <lb/>agat. </s><s>Quia tamen os cranij non inane & vacuum, &longs;ed cere­<lb/>bro, <expan abbr="multisq;">multisque</expan> va&longs;is in eo contentis e&longs;t refertum, illa &longs;imilitudo <lb/>à vitro de&longs;umpta non videtur hic convenire. </s><s>Et cùm impul­<lb/>&longs;us naturâ &longs;uà rectitudinem &longs;equatur; quid cau&longs;æ quòd in ce­<lb/>rebrum non rectâ feratur; &longs;ed per ambages in o&longs;&longs;e cranij ober­<lb/>rat? Et &longs;i ita; an non nece&longs;&longs;e ex illâ vehementiâ ictûs plura e­<lb/>iu&longs;dem va&longs;a di&longs;cerpi & collidi? Pro quo notandum naturam <pb xlink:href="063/01/099.jpg"/>&longs;apienti&longs;&longs;imam cerebrum non pror&longs;us contiguum feci&longs;&longs;e: ve­<lb/>rùm aliquo interuallo inter os cranij & membranas relicto: <lb/>quò nimirum aëri, quem arteriæ in&longs;pirant, &longs;it locus: <expan abbr="cerebrũ">cerebrum</expan> <lb/>verò dilatari, <expan abbr="rur&longs;umq;">rur&longs;umque</expan> contrahi valeat. </s><s>Quod quidem ab aë­<lb/>ris, <expan abbr="Lunæq;">Lunæque</expan> mutatione, fieri ob&longs;ervamus. </s><s>Turget enim in ple­<lb/>nilunio cerebrum & veluti ebullit per vulnera: è contra in no­<lb/>vilunio &longs;ub&longs;idet, & à cranio notabili abe&longs;t intervallo. </s><s>Cùm <lb/>ergo ita habeat, optimè videtur natura caui&longs;&longs;e; quò minùs no­<lb/>xa eò pertingat: impul&longs;us enim per partes contiguas, non ve­<lb/>rò à &longs;e divul&longs;as propagatur. </s></p> <p type="main"> <s><emph type="italics"/>Dices. </s><s>Cerebrum incubare oßis <foreign lang="greek">sfhnoei/dei</foreign>, & cum cranio per falcem fi­<lb/>bras&queacute; in &longs;uturam productas connecti: nihil ergo ob&longs;tat, quò minùs <lb/>hac viá &longs;e inferat.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo quòd &longs;i percu&longs;&longs;io fiat in illâ parte, quâ cerebrum <lb/>&longs;u&longs;tinetur, <expan abbr="e&longs;tq;">e&longs;tque</expan> contiguum o&longs;&longs;i, non <expan abbr="ab&longs;q;">ab&longs;que</expan> periculo fieri pla­<lb/>gam: unde plenilunij tempore, quòd calva cerebrum attingat, <lb/>eiu&longs;modi ictus &longs;unt lethales. </s><s>At verò illæ fibræ, quibus ce­<lb/>rebrum cranio &longs;e in&longs;erit, & à quibus in æquilibri &longs;itu detine­<lb/>tur, via e&longs;&longs;e non pote&longs;t irruenti plagæ: propterea, quòd hu­<lb/>iu&longs;modi &longs;u&longs;pen&longs;oria, nec dura nec rigida &longs;unt, &longs;ed mollia & <lb/>membrano&longs;a filamenta: quæ tendi & laxari facilè po&longs;&longs;unt: pri­<lb/>u&longs;quam ergo tractio aut pul&longs;io fiat, in illâ relaxatione perit im­<lb/>pul&longs;us. </s><s>Deinde cùm impetus &longs;e gyrando, non ni&longs;i obliquè <lb/>&longs;tringat illa filamenta, erit ten&longs;io æqualis motioni illarum par­<lb/>ticularum, quæ &longs;olo tremore convelluntur: ac proinde in&longs;en­<lb/>&longs;ilis, nullam ergo violentiam adducet partibus medio loco &longs;i­<lb/>tis; quantumuis ictus <expan abbr="quandoq;">quandoque</expan> accidant graves. </s><s>Ita quidem <lb/>res habet in fi&longs;&longs;urâ partis oppo&longs;itæ: an verò alijs <expan abbr="quoq;">quoque</expan> locis <lb/>non quidem oppo&longs;itis, verùm á plagâ aliquo modo di&longs;iunctis <pb xlink:href="063/01/100.jpg"/>contingat, videndum. </s><s>Nam ita fieri opinantur, qui negant <lb/>extra &longs;uturam illius o&longs;&longs;is, in quo recipitur plaga, fi&longs;&longs;uram pro­<lb/>tendi. </s><s>Cuius rationem a&longs;&longs;ignant, quòd pars illa à plagâ affe­<lb/>cta nimis &longs;it robu&longs;ta: ac proinde in partem proximam, quæ ob <lb/>nativam con&longs;titutionem minùs re&longs;i&longs;tere valet, illa violentia <lb/>&longs;e recipiat. </s><s>Et quidem experientia his videtur favere. </s><s><expan abbr="Quan-doq;">Quan­<lb/>doque</expan> enim à percu&longs;&longs;ione <expan abbr="neq;">neque</expan> locum plagæ, <expan abbr="neq;">neque</expan> huic oppo&longs;i <lb/>tum, &longs;ed quemvis alium infe&longs;tari & frangi contingit. cùm ni­<lb/>mirum illarum partium unio minorem vim habet ad quie&longs;cen­<lb/>dum, quàm impetus ad movendum. </s></p> <p type="main"> <s><emph type="center"/>Defortificatione aduer&longs;um ictus <lb/>Tormentorum.<emph.end type="center"/></s></p> <p type="main"> <s>QVia mœnia urbis, ca&longs;telli, aut propugnaculi ictus tormen­<lb/>torum admittere nece&longs;&longs;itas <expan abbr="quandoq;">quandoque</expan> cogit; providen­<lb/>dumquô eiu&longs;modi ictus debilitentnr, minorem eâ ratione <expan abbr="plagã">plagam</expan> <lb/>afferentes. </s><s>Id autem duobus modis a&longs;&longs;equi <expan abbr="cõtingit">contingit</expan>: primo ma­<lb/>iori parte ictûs exclu&longs;â, quem totum vitare nequimus: pars <lb/>enim dimidia, aut tertia minorem noxam dabit, quàm totus. </s><lb/><s>Minuitur autem cùm non ni&longs;i obliquè recipitur. </s><s>Con&longs;ide­<lb/>randum ergo quibus poti&longs;&longs;imum locis urbs ad inua&longs;ionem &longs;it <lb/>opportuna & quâ ab ho&longs;tium tormentis minùs tuta: tum enim <lb/>latus munitionis oppo&longs;itum eiu&longs;modi locis, quantùm fieri li­<lb/>cet, <expan abbr="obliq;">oblique</expan> ducendum: quò ictus recipiat magis obliquos. </s><lb/><s>Quâ verò parte ob &longs;itum <expan abbr="locorũ">locorum</expan> machinæ admoveri neque­<lb/>unt, in directum procurrere pote&longs;t. </s><s>Tum igitur ictus obliquè <lb/>incidentes non ni&longs;i partem plagæ dant à lineâ hypomochlij de­<lb/>finitam: reliquâ parte, quæ necdum percu&longs;&longs;it, reflexâ: & &longs;i <lb/>quidem propugnaculum impetitur; quia latera &longs;ibi oppo&longs;itis, <pb xlink:href="063/01/101.jpg"/>à quibus defenditur, habet parallela, in totum aver&longs;â. </s><s>Per­<lb/>cu&longs;&longs;o autem muro licet in aliam partem reflectat: quia tamen <lb/>ex obliquo ictus fiunt, violentiâ in plures di&longs;tractâ, minùs no­<lb/>xæ inferunt. </s></p> <p type="main"> <s>Secundus modus ut &longs;iue totam plagam, &longs;iue illius partem <lb/>recipere cogantur, id cum minori detrimento & concu&longs;&longs;ione <lb/>fiat. </s></p> <p type="main"> <s>Et cùm ruina proveniat ex &longs;olutâ compage: cùm vel partium <lb/>iuncturæ, quibus muri, turres, & propugnacula &longs;unt &longs;tructa, de <lb/>hi&longs;cunt: vel partes &longs;olidæ ex vehementiâ ictûs fati&longs;cunt; ne­<lb/>ce&longs;&longs;e illam violentiam ita di&longs;pen&longs;are; ut nullâ parte in&longs;igniter <lb/>læsâ pertran&longs;eat; & <expan abbr="neq;">neque</expan> partem &longs;olidam frangat; <expan abbr="neq;">neque</expan> unam <lb/>ab aliâ divellat. </s><s>Hoc autem pendet à duobus: materiâ ni­<lb/>mirum, <expan abbr="at&qacute;">atque</expan>; huius partium &longs;itu. </s><s>In quæ&longs;tione enim de fra­<lb/>cturâ o&longs;tendi in diver&longs;is corporibus inæqualiter recipi impul­<lb/>&longs;um. </s><s>Nam quæ cedendo in plures veluti ictus hunc partiun­<lb/>tur. minùs noxæ &longs;entiunt, <expan abbr="minùsq;">minùsque</expan> latè &longs;e extendit plaga. </s><lb/><s>Talia verò &longs;unt <foreign lang="greek">p<gap/>sta\</foreign> cum lentà vi&longs;ciditate, et quæ percu&longs;&longs;a <lb/>minùs &longs;onant: ob atomos enim inæqualiter po&longs;itas per illas <lb/>ambages di&longs;cerpitur impul&longs;us. </s><s>Saxa ergò, quæ &longs;urda dicun­<lb/>tur, cæteris paribus ad impetum &longs;u&longs;tinendum &longs;unt aptiora. </s><lb/><s>Quod attinet &longs;itum, quia &longs;oliditas muri maior e&longs;t longitudine <lb/>aut cra&longs;&longs;itie &longs;axi; nece&longs;&longs;e plura ordine di&longs;poni, <expan abbr="quou&longs;q;">quou&longs;que</expan> &longs;imul <lb/>iuncta adæquent illam molem. </s><s>Alia ergo &longs;itum extra habent, <lb/><expan abbr="ictus&qacute;">ictusque</expan>; & primum impetum &longs;u&longs;tinent; alia medio locò: alia <lb/>demum parieti interno &longs;unt pro firmamento. </s><s>Nihil hic dico <lb/>de illâ concatenatione, quâ duo &longs;axa uno &longs;uperpo&longs;ito nectun­<lb/>tur, <expan abbr="atq;">atque</expan> unum <expan abbr="quod&qacute;">quodque</expan>; duobus retinaculis firmatur: ut licet <lb/>uno exempto nihil detrimenti reliqua &longs;entiant: quod <expan abbr="quid&etilde;">quidem</expan> <lb/>erat futurum, &longs;i totâ mole æqualibus &longs;ab&longs;ternerentur. </s><s>De <lb/>quibus &longs;apienter, <expan abbr="docte&qacute;">docteque</expan>, Architecti: hic enim Geometram, <pb xlink:href="063/01/102.jpg"/>non Architectum agimus. </s><s>Situm ergo con&longs;ideramus, quate­<lb/>nus impul&longs;us à plagâ ad reliqua, quæ ponè &longs;equuntur, tran&longs;it: <lb/><expan abbr="neq;">neque</expan> enim in &longs;uperficie vis hæc finitur, &longs;ed altè penetrat. </s><lb/><s>Aut igitur æqualia, aut inæqualia: <expan abbr="atq;">atque</expan> hæc maiora, vel mino­<lb/>ra &longs;equuntur. </s><s>Videmus autem hunc ferè modum &longs;ervari: <lb/>ut grandi&longs;&longs;ima &longs;axa &longs;int à fronte; quæ cum maximo impetu <lb/>luctentur: interiora verò tanquam ab ictu iam &longs;ecura negle= <lb/>ctim &longs;trui, ruderibus aut minoribus &longs;axis explendo illa inter­<lb/>valla. quod an rectè fiat dubitamus. </s><s>Nam cra&longs;&longs;itudo muri e&longs;t <lb/>ob firmitatem, quò &longs;axa priùs po&longs;ita à po&longs;terioribus contine­<lb/>antur: nece&longs;&longs;e ergo impetum, quo alioquin &longs;axa à fronte po­<lb/>&longs;ita loco moverentur, &longs;u&longs;tinere. </s><s>At verò quâ ratione impe­<lb/>tum maioris id quod multò e&longs;t minus &longs;u&longs;tinebit? Nam per po­<lb/>ri&longs;: 2 &longs;i maius percutiat minus, <expan abbr="utrũq">utrunq</expan>: loco movetur: propte­<lb/>rea, quòd minus eadem velocitate movetur ex impul&longs;u mino­<lb/>ri. </s><s>Tamet&longs;i ergo partes illæ minores, quæ in muro continen­<lb/>tur, <expan abbr="undiq;">undique</expan> &longs;int conclu&longs;æ; quia tamen totum impetum ferre <lb/>non valent, nec in alias minores hunc exonerare: divelli à <lb/>primis, & po&longs;teriores urgere, <expan abbr="atq;">atque</expan> tum metu vacui aërem &longs;or­<lb/>bendo, etiam magnas compages di&longs;&longs;olui e&longs;t nece&longs;&longs;e. </s></p> <p type="main"> <s><emph type="italics"/>Dices. </s><s>Non eandem rationem videri in muro, ubi omnia per calcem <lb/>glutinantur, & veluti unum fiunt; at&que; illoram corporum, quæ &longs;oluta <lb/>motum & impul&longs;um à &longs;e recipiunt. </s><s>Licet ergo impul&longs;us à maiori &longs;axo <lb/>in minora tran&longs;iens omnia loco moveat; non tamen idem futurum in <lb/>muro; cùm illud gluten non minùs coharere faciat, quàm &longs;i partes <lb/>e&longs;&longs;ent continuæ unius &longs;axi maioris. </s><s>Ita&que; duo globuli cerâ coniuncti <lb/>impul&longs;um &longs;u&longs;tinent duplo maiorem, ne&queacute; à percu&longs;&longs;o primo &longs;ecundus rece­<lb/>dit: quantò ergo minùs calce revincta &longs;axa.<emph.end type="italics"/></s></p> <p type="main"> <s>Po&longs;&longs;et quis re&longs;pondere, cùm &longs;axa minora maioribus <expan abbr="cohæreãt">cohæreant</expan> <pb xlink:href="063/01/103.jpg"/><expan abbr="mediãte">mediante</expan> illo glutine ex calce & arenâ multò levioribus; <expan abbr="nõ">non</expan> po&longs;&longs;e <lb/>eo modo habere, quo partes continui: <expan abbr="ne&qacute;">neque</expan>; per modum unius <lb/>cen&longs;eri in ordine ad impul&longs;um; quem etiam in eodem &longs;ubie­<lb/>cto, ob partium di&longs;crimina, o&longs;tendi inæqualem. </s><s>Etenim vi­<lb/>demus, longè differre hunc <expan abbr="nexũ">nexum</expan> à partium eiu&longs;dem &longs;axi unio­<lb/>ne, di&longs;&longs;oluto à murarijs cæmento: parte enim aver&longs;â etiam le­<lb/>viter percu&longs;sâ, illæ&longs;o &longs;axo, decidunt coagmenta. </s><s><expan abbr="Neq;">Neque</expan> ob&longs;tat <lb/>in muro omnia vincta teneri, quò minùs impetus &longs;imili ratione <lb/><expan abbr="atq;">atque</expan> in &longs;olutis pertran&longs;eat. </s><s>Nam &longs;i pila in plano &longs;eu manu, &longs;eu <lb/>aliâ ratione firmetur, quò minùs moveri po&longs;&longs;it à plagâ; nihilo­<lb/>minùs &longs;ibi contiguam movet. </s><s>Nece&longs;&longs;e ergo matori pericu­<lb/>lo &longs;equi &longs;axa minora, quàm æqualia aut maiora: cùm per æqua­<lb/>lia impetus ad extimum <expan abbr="u&longs;&qacute;">u&longs;que</expan>; æqualiter &longs;e effundat: illisq illæ­<lb/>&longs;is & immotis pertran&longs;eat. </s><s>Vnde &longs;iquid periculi non ni&longs;i inter­<lb/>no parieti creatur, qui facilè refici pote&longs;t. </s><s>Non ita cùm imme­<lb/>diatè minora &longs;equuntur: rece&longs;&longs;u enim à primis extima pericli­<lb/>tantur: <expan abbr="neq;">neque</expan> facilè reparari queunt: unde vi&longs;o periculo magis ab <lb/>ho&longs;te infe&longs;tantur. </s><s>Vt verò quid mihi videatur, dicam: eiu&longs;­<lb/>modi &longs;axa, quæ calce glutinantur, aut &longs;unt partes continuæ e­<lb/>iu&longs;dem molis, aut contiguæ. </s><s>Supponamus primùm e&longs;&longs;e conti­<lb/>guas & plagam incipere à maiori. </s><s>Cùm <expan abbr="itaq;">itaque</expan> maius percutiat <lb/>minus, &longs;i non aliunde motus impediatur, movebitur minus ab <lb/>incipiente & necdum perfectâ plagâ: ad cuius motum &longs;equitur <lb/>maius per pori&longs;ma 2. </s><s>Quòd &longs;i verò ab aliâ vi detineatur ne­<lb/>ce&longs;se totum impul&longs;um maioris recipere. </s><s>Et &longs;i quidem illa vis <lb/>retentiva &longs;it minor impul&longs;u; tum &longs;anè movebitur illud mobile: <lb/>reliqua verò, quia illorum plaga perfecta, à motu conquie&longs;cent. </s><lb/><s>Vnde tota illa vis di&longs;tractiua partem ultimam obtinet. </s><s>Si <lb/>autem à minoribus eadem plaga procedat: quia tum hypo­<lb/>mochlium &longs;ecundi e&longs;t tertium; nece&longs;se non ni&longs;i ultimo moto <lb/>moveri primum: ac proinde impul&longs;um maioris recipere mi- <pb xlink:href="063/01/104.jpg"/>nus. </s><s><expan abbr="Cùmq;">Cùmque</expan> ab æquali plagâ incipiat, erit in <expan abbr="utroq;">utroque</expan> extremo <lb/>impul&longs;us æqualis. </s><s>Et quia maiorem rationem habet ad mi­<lb/>nus, maiori <expan abbr="quoq;">quoque</expan> vi eluctabitur. </s><s>Magis ergo periclitatur, <expan abbr="ma-iorq;">ma­<lb/>iorque</expan> ruina imminet extremo; &longs;iplaga incipiat à percu&longs;&longs;o ma­<lb/>iori. </s><s>Ita quidem &longs;i &longs;axa contigua e&longs;&longs;e demus. </s><s>Quòd &longs;i verò <lb/>continua &longs;int; Dico ab eodem impul&longs;u magis infe&longs;tari mino­<lb/>ra, &longs;i ictum primum excipiant. </s><s>Nam cùm impul&longs;us non re­<lb/>cipiatur uniformiter, verùm à contactu &longs;en&longs;im remi&longs;&longs;o vigore <lb/>&longs;e extendat in latum, & profundum: nece&longs;se illas iuncturas, <lb/>quæ circum &longs;axa &longs;unt minora, ab impul&longs;u magis inten&longs;o perva­<lb/>di: & quia minùs firmo nexu cohærent, quàm reliquum &longs;a­<lb/>xum, à &longs;e divelli. </s><s>Ad rationem verò in oppo&longs;itum factam, <lb/>Re&longs;pondeo, licet minùs firmiter glutinentur inter &longs;e &longs;axa; non <lb/>tamen ob illam inæqualitatem de&longs;inere e&longs;&longs;e continua: alio­<lb/>quin <expan abbr="neq;">neque</expan> idem &longs;axum e&longs;&longs;et continuum: quòd diver&longs;is parti­<lb/>bus inæqualiter frangi contingat. </s><s>De quo tamen accuratiùs <lb/>dicetur in libro de motu: qui propediem in lucem prodibit. </s></p> <p type="main"> <s><emph type="center"/>De Percu&longs;&longs;ione & motu orbiculorum.<emph.end type="center"/></s></p> <p type="main"> <s>ORbiculi &longs;unt figuræ circulares, <expan abbr="utrâq;">utrâque</expan> &longs;uperficie planâ <lb/>& parallelâ terminatæ; &longs;eu portiones cylindri habentes <lb/>partem axis re&longs;ecti minorem &longs;emidiametro circuli. </s><s>E&longs;t autem <lb/>hoc illis commune cum globis; ut ordine di&longs;po&longs;iti, <expan abbr="&longs;ibiq;">&longs;ibique</expan> con­<lb/>tigui eadem ratione moveantur: percu&longs;&longs;o enim primo, &longs;i æ­<lb/>quales &longs;int, medijs immotis ultimus movetur. </s><s><expan abbr="Atq;">Atque</expan> inde ra­<lb/>tio con&longs;tat, quamobrem eiu&longs;modi orbiculis ludentes ab <lb/>eadem plagâ, non eundem effectum con&longs;equantur. </s><s><expan abbr="Quan-doq;">Quan­<lb/>doque</expan> enim ad finem tabulæ orbiculum in&longs;equitur ille, qui <lb/>percu&longs;&longs;it: <expan abbr="quandoq;">quandoque</expan> immotus manet. </s><s>Hoc enim fit ob in æ­<lb/>qualem gravitatem: in&longs;equitur enim maior minorem, non <pb xlink:href="063/01/105.jpg"/>verò &longs;ibi æqualem aut maiorem. </s><s>Suppono verò motum fi­<lb/>eri in lineâ centri: nam &longs;i inclinet; quia non totam dat plagam, <lb/>motum continuabit. </s><s>At verò hoc peculiare habent; quòd <lb/>non tantùm in lineâ rectâ &longs;ibi contigui fiant; &longs;ed etiam illâ &longs;u­<lb/>perficie planâ in &longs;imilitudinem cylindri a&longs;&longs;urgant. </s><s>Quòd &longs;i <lb/>ergo his ita cumulatis illum orbiculum, qui ba&longs;is e&longs;t reliquo­<lb/>rum, percutiat æqualis; eadem ratione movebitur huic con­<lb/>tiguus, quantumvis illorum numerus, qui ba&longs;i incumbunt, au­<lb/>geatur: quin etiam quovis onere accepto impul&longs;um tran&longs;mit­<lb/>tit nihilo minorem. </s><s>At verò ba&longs;is excuti non eadem facilita­<lb/>te pote&longs;t: verùm pro numero orbiculorum, aut oneris appen­<lb/>&longs;i ratione, nece&longs;&longs;e plagam fieri maiorem. </s><s>Quòd &longs;i orbicu­<lb/>lus gravitatem habeat æqualem illi, quâ ba&longs;is gravatur, eadem <lb/>facilitate illam loco movebit: verùm ip&longs;e <expan abbr="quoq;">quoque</expan> tran&longs;ito illo <lb/>hiatu, motum ba&longs;is &longs;equetur. </s><s>Cuius ratio e&longs;&longs;e videtur: <lb/>quòd impul&longs;us nece&longs;&longs;ariò fiat iuxta determinationem plagæ; <lb/>licet &longs;ubiectum non moveatur ex eo impul&longs;u: & &longs;i maior &longs;it <lb/>quàm ut in &longs;ubiecto terminetur, aliud percutit &longs;ibi contiguum. <lb/><expan abbr="Neq;">Neque</expan> minor e&longs;t impul&longs;us &longs;i ab a lienâ gravitate detineatur; non <lb/>enim gravitas ab extra veniens, &longs;ed nativa hunc attenuat: quæ <lb/>multam materiam habet coniunctam. </s><s><expan abbr="Itaq;">Itaque</expan> &longs;i magnitudi­<lb/>ne non verò gravitate &longs;int pares; orbiculus maior à minori <lb/>percu&longs;&longs;us, minorem ex eadem plagâ impul&longs;um reliquis dabit, <lb/>quàm &longs;i ab æquali percutiatur: propterea, quòd in multâ ma­<lb/>teriâ magis hebetatur. </s><s>Vt verò &longs;ubiectum moueatur, ne­<lb/>ce&longs;&longs;e & gravitatem nativam, & impul&longs;um contrarium &longs;upera­<lb/>re. </s><s><expan abbr="Itaq;">Itaque</expan> fit ut ba&longs;is illius cylindri orbiculati motui renita­<lb/>tur: quæ & &longs;uâ & illorum, à quibus premitur, gravitate de­<lb/>tinetur. </s><s>Quòd &longs;i gravitas orbiculi augeatur, ut gravitati illius <lb/>cylindri &longs;it æqualis: quæ tota in ba&longs;im colligitur, <expan abbr="atq;">atque</expan> illius vi <lb/>à motu detinetur, percu&longs;&longs;io tum fit æqualis: <expan abbr="quou&longs;q;">quou&longs;que</expan> nimirum <pb xlink:href="063/01/106.jpg"/>à nexu illorum orbiculorum &longs;e expediat: tum enim motu a­<lb/>gitur multò velociore, quàm &longs;i plaga fiat ab æquali: amotâ e­<lb/>nim illâ re&longs;i&longs;tentiâ, impul&longs;us ad gravitatem multò iam mino­<lb/>rem, maiorem habet exce&longs;&longs;um. </s></p> <p type="main"> <s><emph type="italics"/>Dices. </s><s>Quid &longs;i orbiculi percutiant illum cylindrum, à quo &longs;unt re&longs;ecti; <lb/>an non movebunt alios huic contiguos eadem ratione, quà in cylindro <lb/>orbiculato? <!--neuer Satz-->Videtur enim eadem ratio e&longs;&longs;e illius &longs;egmenti, quod percu­<lb/>titur ab æquali, &longs;iue re&longs;ectum &longs;it, &longs;iue continuum: propterea, quòd ma­<lb/>teria una, ac proinde impul&longs;us, qui viam &longs;equitur plagæ, æqualiter per­<lb/>tran&longs;it. </s><s>Illa autem continuitas non videtur mutare naturam impul­<lb/>&longs;us: qui non ni&longs;i vi dimovetur à lineâ rectâ: at&queacute; eâdem gravitate à <lb/>motu detinetur pars re&longs;ecta & continua: eadem ergo plaga, quæ orbicu­<lb/>lum excludit, cylindrum quo&que; &longs;olidum movebit.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo impul&longs;um, cùm à principio fiat interno mobi­<lb/>lis, totum &longs;ubiectum afficere. </s><s>Licet ergo viam &longs;equatur pla­<lb/>gæ; quia tamen in altum <expan abbr="quoq;">quoque</expan> a&longs;&longs;urgit: quantum virium <lb/>huc confert, tantum decedit plagæ oppo&longs;itæ. </s><s>Percu&longs;&longs;o <lb/><expan abbr="itaq;">itaque</expan> cylindro &longs;olido, minori vi moventur orbiculi contigui; <lb/>decre&longs;cente plagâ pro altitudine cylindri. </s><s>Quòd &longs;i orbiculi <lb/>inter &longs;e glutinentur; quia tum extrema fiunt unum, rationem <lb/>habent continui: unde eadem his, quæ cylindro &longs;olido <lb/>conveniunt. </s><s>Verùm de his cùm &longs;citu digna vi­<lb/>deantur continere. enucleatiùs di&longs;&longs;eren­<lb/>dum. </s></p> <pb xlink:href="063/01/107.jpg"/> <p type="main"> <s><emph type="center"/>DEFINITIO I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Cylindrus &longs;olidus e&longs;t, cuius partes omnes &longs;unt continuæ.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>DEFINITIO II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Cylindrus verò orbiculatus, cuius &longs;egmenta &longs;uut orbiculi, <lb/>&longs;imul iuncti at&queacute; inter &longs;e par alleli.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>DEFINITIO III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ba&longs;is Cylindri, orbiculati e&longs;t orbiculus tangens planum, <lb/>à quo reliqui orbiculi eidem par alleli, centrum in eodem axe <lb/>habentes a&longs;&longs;urgunt.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>DEFINITIO IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Grauitas &longs;ecunda e&longs;t vis ab extra proueniens, quâ mo­<lb/>bile detinetur, quò minùs à grauitate primâ, &longs;eu propriâ <lb/>aut impul&longs;u moveatur.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>THEOREMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si duo orbiculi &longs;imul iuncti & æquales eodem impul&longs;u moueantur <lb/>in plano; ad minus intervallum movetur ba&longs;is.<emph.end type="italics"/></s></p> <p type="main"> <s>DIxi Motum e&longs;&longs;e veluti continuatam ex aëris divi&longs;ione <lb/>plagam: & &longs;i medium &longs;it minùs aptum dividi, minorem <lb/>e&longs;&longs;e motum; qui non ni&longs;i à plagâ perfectâ terminatur. </s><s>Plaga <pb xlink:href="063/01/108.jpg"/>autem fit cùm medium re&longs;i&longs;tit: & quia in vacuo nihil re&longs;i&longs;tit; <lb/>interminabilis e&longs;&longs;et in eo motus: in aquâ verò ob re&longs;i&longs;tentiam <lb/>maiorem, priùs quàm in aëre ab&longs;umitur. </s><s>Re&longs;i&longs;tentia autem <lb/>fit cùm vel divi&longs;io, vel gravitas mobilis, vel retentio ob&longs;tat <lb/>motui. </s><s>Ita ergo per plures chartas aliquo intervallo &longs;eiun­<lb/>ctas tran&longs;it glans plumbea, <expan abbr="quou&longs;q;">quou&longs;que</expan> ex illâ divi&longs;ione continu­<lb/>atâ impetus la&longs;&longs;etur. </s><s>Aut cùm plagam recipit gravitas maior <lb/>à minori: aut cùm mobile à maiori vi detinetur, quò minùs <lb/>motum pro&longs;equi valeat. </s><s>Detinetur autem mobile &longs;eu à gra­<lb/>vitate coniunctâ, &longs;eu vi retentivâ: ut &longs;i Miloni digitum infle­<lb/>ctere, aut pomum illius manu conclu&longs;um extorquere cone­<lb/>mur: illa enim retentio ab impul&longs;u fluente, & veluti librato <lb/>procedit: qui non ni&longs;i à maiori impul&longs;u pote&longs;t &longs;uperari. </s><lb/><s>Cui &longs;imilis videtur retentio ex angu&longs;tiâ loci inducta: ut dum <lb/>clavus in pariete fixus detinetur. </s><s>Quò enim maior angu&longs;tia; <lb/>eò tran&longs;itus magis difficilis, & non ni&longs;i maiori vi &longs;uperandus. <lb/><expan abbr="Itaq;">Itaque</expan> fit, ut licet eiu&longs;modi rima toto illo tractu æqualiter ex­<lb/>currat, impetus tamen priu&longs;quam totam tran&longs;eat, ex&longs;olvatur: <lb/>retentio enim illa continuata non aliter, quàm &longs;i plaga produ­<lb/>ceretur, impul&longs;um atterit & ab&longs;umit: <expan abbr="atq;">atque</expan> eò magis, quò &longs;tri­<lb/>ctura magis coarctat. </s><s><expan abbr="Neq;">Neque</expan> aliâ ratione detineri videtur ba­<lb/>&longs;is à gravitate illorum orbiculorum, qui ba&longs;i incumbunt; fit <lb/>enim compre&longs;&longs;io illi &longs;imilis, quam loci angu&longs;tia inducit. </s><s><expan abbr="Itaq;">Itaque</expan> <lb/>&longs;i augeatur numerus, aut pondus orbiculorum; quia magis <lb/>comprimitur ba&longs;is, non ni&longs;i maiori vi excuti pote&longs;t. </s><s>Cùm <lb/>ergo duo orbiculi &longs;imul iuncti moventur: quia compre&longs;&longs;io fit <lb/>ba&longs;is continuata, licet impul&longs;us, quo ba&longs;is movetur &longs;it æqualis; <lb/>ob illam tamen gravitatem acce&longs;&longs;oriam, priùs terminat mo­<lb/>tum. </s><s>Quam inæqualitatem motûs adiuvare videtur &longs;cabri­<lb/>ties loci, &longs;euplani, quod tan&longs;it: <expan abbr="atq;">atque</expan> inde fit quòd <expan abbr="quandoq;">quandoque</expan> in <lb/>medio motu orbiculi circumaguntur. </s></p> <pb xlink:href="063/01/109.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si duo Orbiculi &longs;imul iuncti & æquales percutiant alium maiorem, <lb/>duobus autem illis &longs;imul &longs;umptis æqualem; orbiculo &longs;uperiori reflexo, <lb/>motum continuat ba&longs;is.<emph.end type="italics"/></s></p> <p type="main"> <s>Difficultas hic e&longs;t, quòd cùm orbiculus &longs;olidus percutit ali­<lb/>um &longs;ibi æqualem, illo moto quie&longs;cit: cur igitur non idem fit, <lb/>cùm duo &longs;imul iuncti, <expan abbr="atq;">atque</expan> eidem æquales hunc percutiunt, ve. <lb/>rùm <expan abbr="uterq;">uterque</expan> inæqualiter movetur ex illâ plagâ? Nam eodem <lb/>impul&longs;u agi videntur: nece&longs;&longs;e ergo eandem inferre plagam: <lb/>Et cùm gravitas orbiculi maioris &longs;it dupla, & impul&longs;um reci­<lb/>piat duplum illius, quo &longs;inguli moventur; erit <expan abbr="quoq;">quoque</expan> eadem ve­<lb/>locitas motûs. </s><s>At verò &longs;i ba&longs;is &longs;equitur motum maioris; ne­<lb/>ce&longs;&longs;e huius motum e&longs;&longs;e velociorem quàm ba&longs;is motum: quæ <lb/>ab incipiente & necdum perfectâ plagâ movetur. </s><s>Et &longs;i refle­<lb/>ctit alter orbiculus: quia peractâ huius plagâ necdum incipit <lb/>moveri maior; velocitatem habebit minorem. </s><s>Pro &longs;olutione <lb/>dico, impul&longs;um in <expan abbr="utroq;">utroque</expan> orbiculo e&longs;&longs;e æqualem. </s><lb/><s>Secundô inæqualiter moveri, <expan abbr="magisq;">magisque</expan> impediri motum ba&longs;is <lb/>per 1 Theor: huius; & cùm impul&longs;um determinet motus; ma <lb/>iori tempore plagam perficiet ba&longs;is. </s><s>Cùm ergò maior orbi­<lb/>culus gravitatem habeat duplam; ad illam velocitatem mo­<lb/>tûs, non ni&longs;i ab impul&longs;u duplo perducitur: perfectâ autem pla­<lb/>gâ unius orbiculi, necdum percu&longs;&longs;it alter: <expan abbr="neq;">neque</expan> igitur ex illâ <lb/>plagâ &longs;e abducit orbiculus maior: ac proinde orbiculus, qui <lb/>iam percu&longs;&longs;it, reflectit. </s><s>Et quia minor e&longs;t velocitas motûs <lb/>ba&longs;is, velociùs movebitur ab <expan abbr="utraq;">utraque</expan> plagâ. igitur ad illam ve­<lb/>locitatem, quâ movetur ba&longs;is, ab incipiente & necdum perfe­<lb/>ctâ huius plagâ perducetur: ac proinde reliquus impul&longs;us mo­<lb/>tum continuabit. </s><s>Idem autem fit, &longs;i alter orbiculus &longs;it paulo <pb xlink:href="063/01/110.jpg"/>levior, &longs;eu ba&longs;is, &longs;eu qui &longs;uperiori loco &longs;itum habet. </s><s>At &longs;i <lb/>magnus &longs;it exce&longs;&longs;us; ut cùm ligneo metallicum adiungimus, <lb/>gravior in omni &longs;itu motum &longs;equitur maioris. </s><s>Quòd &longs;i duo <lb/>orbiculi &longs;imul iuncti <expan abbr="atq;">atque</expan> inter &longs;e æquales deficiant à gravitate <lb/>maioris: minùs quidem movetur ba&longs;is, magis autem reflectit <lb/>alter orbiculus. <!--neuer Satz-->E contra &longs;i gravitas excedit: hic quidem mi­<lb/>nùs reflectit, ille verò motum magis producit. </s><s>Cuius ratio <lb/>e&longs;t, quòd horum impul&longs;us maiorem rationem habet ad orbi­<lb/>culum minùs gravem: igitur cùm à minori plagâ eadem ve­<lb/>locitas motûs &longs;equatur; erit maior impul&longs;us reliquus ad mo­<lb/>tum continuandum. </s><s>Et quia velociùs à plagâ &longs;e abducit, erit <lb/>minor reflexio motûs-Cùm verò impul&longs;us minorem habet <lb/>rationem; non ni&longs;i à maiori plagâ ad motum æquè velocem <lb/>cietur maior, & non ni&longs;i tardè à plagâ &longs;e abducit: magis proin­<lb/>de reflectit motus, minùs autem à reliquo impul&longs;u movetur <lb/>ba&longs;is. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si duos orbiculos &longs;imul iunctos percutiat maior; adminus inter­<lb/>vailum movetur ba&longs;is.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i duo orbiculi &longs;int æquales; quia ab eadem plagâ <lb/>idem e&longs;t impul&longs;us, con&longs;tat per Theor: 1. ad minus intervallum <lb/>moveri ba&longs;im. </s><s>Simili modo cùm orbiculi &longs;unt inæqvales, & <lb/>maiorem gravitatem habet ba&longs;is; ab æquali impul&longs;u minùs <lb/>moveri ba&longs;im. </s><s>At cùm pro ba&longs;i e&longs;t orbiculus minùs ponde­<lb/>ro&longs;us; oportebat quidem hunc ab æquali impul&longs;u velociùs, <lb/>& ad maius intervallum moveri. </s><s>Sed quia detinetur ab aliâ <lb/>gravitate; quò magis premitur, eò motum habet magis im­<lb/>peditum. </s><s>Deinde dicolicet &longs;imul fiat, e&longs;&longs;e tamen inæqua. <pb xlink:href="063/01/111.jpg"/>lem plagam, & qui hanc &longs;equitur impul&longs;um. </s><s>Nam cùm à <lb/>principio eodem motu ferantur; nece&longs;&longs;e à maiori impul&longs;u mo­<lb/>veri graviorem: quò minùs ergo velociter irrumpat, <expan abbr="totúmq;">totúmque</expan> <lb/>impul&longs;um recipiat minor, à graviori detinetur. </s><s>Igitur ba&longs;is <lb/>tum quia minori impul&longs;u agitur, tum quia gravitate aliená de­<lb/>tinetur, ad minus intervallum movetur. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si duo orbiculi &longs;imuliuncti & æquales percutiant alium maiorem <lb/>& immotum; uter&queacute; reflectit.<emph.end type="italics"/></s></p> <p type="main"> <s>Quia enim'minor e&longs;t impul&longs;us, à plagâ illorum orbiculo­<lb/>rum, quàm ut loco moveat maiorem: &longs;iue à gravitate primâ <lb/>&longs;eu propriâ, &longs;iue &longs;ecundâ detineatur: ut cùm ligneus metalli­<lb/>cum, aut alium &longs;ibi quidem &longs;imilem verùm in plano firmatum <lb/>percutit: <expan abbr="neq;">neque</expan> hic à plagâ &longs;e abducit, aut alium contiguum <lb/>movet; recipiet <expan abbr="uterq;">uterque</expan> orbiculus à percu&longs;&longs;o æqualem illi quam <lb/>dedit plagam: igitur cùm impul&longs;us &longs;it agens nece&longs;&longs;arium, <expan abbr="u-terq;">u­<lb/>terque</expan> orbiculus reflectet ex illâ plagâ. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si plures orbiculi &longs;imul iuncti percutiant alium maiorem, & à <lb/>plagâ illâ immotum; ad minus intervallum reflectunt ba&longs;i propiores.<emph.end type="italics"/></s></p> <p type="main"> <s>Cùm omnes orbiculi percutiant, <expan abbr="neq;">neque</expan> ad ullius plagam <lb/>moveatur ille orbiculus: recipient à percu&longs;&longs;o æqualem illi, <lb/>quam <expan abbr="quisq;">quisque</expan> dedit plagam. </s><s>At verò ba&longs;is per Theor: 2. mo­<lb/>tum habet magis impeditum: igitur cùm velocitas motûs de­<lb/>terminet plagam; minor erit huius, quàm reliquorum plaga. <pb xlink:href="063/01/112.jpg"/>Et quia propiores illi ba&longs;i, quæ tangit planum, remotioribus <lb/>&longs;unt pro ba&longs;i; erit minor illorum plaga: ac proinde ad minus <lb/>intervallum reflectunt. </s><s>Idem verò contingit &longs;iue eandem <lb/>habeant gravitatem orbiculi reliqui, &longs;iue præponderet ba&longs;is, <lb/>aut minus &longs;it gravis. </s><s>Nam licet ba&longs;is magis pondero&longs;a ma­<lb/>iorem dat plagam; cùm non ni&longs;i à maiori impul&longs;u moveatur <lb/>eodem cum minoris gravitatis motu: quia tamen in ordine <lb/>ad motum hanc expendimus; <expan abbr="at&qacute;">atque</expan>; in eadem ratione &longs;unt mo­<lb/>tus reflexi, minor autem huius motus; minorem <expan abbr="quoq;">quoque</expan> in or­<lb/>dine ad &longs;uum motum dicetur dare & referre plagam. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si plures orbiculi &longs;imul iuncti & æquales percutiant Cylindrum &longs;oli­<lb/>dum; maiorem impul&longs;um recipiunt partes à ba&longs;i remotiores.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam ba&longs;is quidem minorem dat plagam per 5 theor: e&longs;t au­<lb/>tem orbiculus propior remotioribus pro ba&longs;i: erit ergo maior <lb/>illorum plaga, & à maiori plagâ maior <expan abbr="quoq;">quoque</expan> impul&longs;us. </s><s>Sed <lb/>et ratio vectis huc facere videtur. </s><s>Nam orbiculus ip&longs;o cy­<lb/>lindro utitur pro vecte: <expan abbr="atq;">atque</expan> eò magis, quò plaga fit remoti­<lb/>or à ba&longs;i: cuius hypomochlium e&longs;t planum, in quo cylindrus <lb/>firmatur. </s><s><expan abbr="Itaq;">Itaque</expan> à plagâ in medio aut propè ba&longs;im factâ immo­<lb/>tus manet: &longs;i eandem plagam accipiat in &longs;ummo, invertitur. <lb/><expan abbr="Atq;">Atque</expan> inde ratio con&longs;tat, quamobrem partes cylindri &longs;uperiores <lb/>avertuntur ex illâ plagâ, & celeritate motûs alias antevertunt: <lb/>à ba&longs;i enim cum longitudine cylindri continuò accre&longs;cit pla­<lb/>ga. E contra verò &longs;i plaga fiat propè ba&longs;im, & infra medium, <lb/>non percu&longs;sâ reliquâ parte cylindri; re&longs;upinato vertice mo­<lb/>tum accelerat ba&longs;is. </s><s>Cùm autem cylindrus alium percutit &longs;i­<lb/>bi æqualem: quia omnes partes æqualiter moventur; eandem <pb xlink:href="063/01/113.jpg"/><expan abbr="quoq;">quoque</expan> inferunt plagam. </s><s>Non igitur huius ratione videtur <lb/>differre motus; verùm acceleratio ad partes &longs;ummas ad ve­<lb/>ctem referri debet. </s><s><expan abbr="Atq;">Atque</expan> inde &longs;equitur, cylindrum ab æqua­<lb/>li cylindro percu&longs;&longs;um inæqualiter moveri. </s><s>Et cùm orbiculus <lb/>ad alium &longs;ibi æqualem, eo modo habeat, quo cylindrus; ne­<lb/>ce&longs;&longs;e illâ &longs;ucce&longs;&longs;ione orbiculorum in plano motum deficere. </s><lb/><s>Vtergo cylindrus æqualiter moveatur ab alio cylindro; inæ­<lb/>qualis e&longs;&longs;e debet plaga: & tanto maior propè ba&longs;im, quanto <lb/>in &longs;ummo augetur ratio vectis. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si duo orbiculi &longs;imul iuncti & æquales percutiant alios duos, &longs;imul <lb/>quo&que; iunctos & prioribus æquales, habeant verò à tergo orbiculum ma­<lb/>iorem; immotâ ba&longs;i primâ, movetur ba&longs;i, &longs;ecunda.<emph.end type="italics"/></s></p> <p type="main"> <s>Cùm duo orbiculi æquales &longs;imuliuncti percutiunt alios du­<lb/>os &longs;imul <expan abbr="quoq;">quoque</expan> iunctos & æquales: licet inæqualem afferant <lb/>plagam; quia tamen <expan abbr="uterq;">uterque</expan> orbiculus ex illâ inæquali plagâ &longs;e <lb/>abducit; <expan abbr="qui&longs;q;">qui&longs;que</expan> &longs;uo orbiculo expul&longs;o à motu conquie&longs;cit. </s><lb/><s>At cùm alius orbiculus maior accedit: in quem impetus &longs;e ex­<lb/>onerat illorum orbiculorum: quia inæqualem <expan abbr="atq;">atque</expan> minorem <lb/>á ba&longs;i recipit plagam; per theor: 2. huius, reflexo altero orbi­<lb/>culo movebitur ba&longs;is. </s><s>Cùm igitur hæc ba&longs;is &longs;ecunda à pla­<lb/>gâ &longs;e abducat; quie&longs;cet à percu&longs;&longs;ione ba&longs;is prima. </s><s>Con&longs;tat <lb/>verò illo orbiculo reflexo, reflecti <expan abbr="quoq;">quoque</expan> orbiculum priorem <lb/>huic contiguum. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ba&longs;im cylindri orbiculati percutiat alius orbiculus æqualis; habe-<emph.end type="italics"/> <pb xlink:href="063/01/114.jpg"/><emph type="italics"/>at verò impul&longs;um minorem gravitate &longs;ecundâ; ba&longs;im à cylindro non <lb/>excludet.<emph.end type="italics"/></s></p> <p type="main"> <s>Impul&longs;us, quo orbiculus movetur quantumvis exiguus, <lb/>movere pote&longs;t alium &longs;ibi æqualem: At cùm gravitas huius ab <lb/>aliâ vi detinetur; non ni&longs;i á maiori impul&longs;u, quàm &longs;it illa vis <lb/>motui renitens, moveri pote&longs;t. </s><s>Vt &longs;i globum &longs;tylo affixum <lb/>percutiat globus æqualis; illa quidem plaga non ni&longs;i &longs;tylo fra­<lb/>cto, aut avul&longs;o globum movebit. </s><s>Itaq cùm ba&longs;is cylindri <lb/>plurium acce&longs;&longs;ione gravatur; nece&longs;&longs;e plagam ab orbiculo <lb/>illatam e&longs;&longs;e maiorem illâ acce&longs;&longs;oriâ gravitate: quâ velutí affi­<lb/>gitur plano: non &longs;olùm in principio motûs, &longs;ed toto illo tra­<lb/>ctu, quo ba&longs;is eluctatur. </s><s>Nam cùm huius motus non aliter, <lb/>quàm &longs;i corpus &longs;olidum continuatâ plagâ perrumpat, attera­<lb/>tur: &longs;i minor &longs;it quàm re&longs;i&longs;tentia illo tran&longs;itu coacervata; mi­<lb/>nor <expan abbr="quoq;">quoque</expan> erit plaga: deficiet ergo motus priu&longs;quam ba&longs;is <lb/>pertran&longs;eat. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ba&longs;im cylindri orbiculati percutiat alius orbiculus æqualis; habe­<lb/>at verò impul&longs;um æqualem grauitati &longs;ecundæ; exclu&longs;am ba&longs;im non <lb/>ultra cylindrum movebit.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam quia ba&longs;im percutit alius orbiculus æqualis; habebit <lb/>ex illâ plagâ impul&longs;um æqualem. </s><s>Et quia gravitas &longs;ecunda <lb/>huic e&longs;t contraria, & ex &longs;uppo&longs;itione æqualis; tollet pars qui­<lb/>dem gravitatis huius partem, tota verò gravitas totum im­<lb/>pul&longs;um per po&longs;it: 2 de propor: motûs. </s><s>Cùm igitur gravitas <lb/>&longs;ecunda diametro cylindri terminetur; deficiet impul&longs;us, ubi <lb/>cylindrum exce&longs;&longs;it ba&longs;is. </s><s>Et cùm non <expan abbr="ab&longs;q;">ab&longs;que</expan> impul&longs;u moveatur, <lb/>non ultra cylindrum extendet motum. </s></p> <pb xlink:href="063/01/115.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ba&longs;im cylindri orbiculati percutiat alius orbiculus æqualis; ha­<lb/>beat verò impul&longs;um maiorem gravitate &longs;ecundâ; ba&longs;im cylindro ex­<lb/>clu&longs;am movebit.<emph.end type="italics"/></s></p> <p type="main"> <s>Cùm enim gravitas &longs;ecunda tollat partem &longs;ibi æqualem, <lb/><expan abbr="neq;">neque</expan> ultra cylindrum &longs;e extendat: e&longs;t autem ex &longs;uppo&longs;itione <lb/>impul&longs;us orbiculi, ac proinde ba&longs;is maior gravitate: erit hu­<lb/>ius exce&longs;&longs;us principium motûs reliqui à contactu: ba&longs;is ergo <lb/>ubi cylindrum &longs;uperavit, motum à reliquo impul&longs;u continu­<lb/>abit. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si orbiculus æqualis percutiat ba&longs;im cylindri orbiculati minùs gra­<lb/>vem; habeat verò impul&longs;um minorem gravitate &longs;ecundâ; illam ba­<lb/>&longs;im à cylindro non excludet.<emph.end type="italics"/></s></p> <p type="main"> <s>Quia ba&longs;is a&longs;&longs;umitur habere gravitatem minorem, quàm or­<lb/>biculus; movebitur à minori impul&longs;u quàm idem orbiculus: & <lb/>multò etiam minori quàm &longs;it gravitas &longs;ecunda: non igitur <lb/>tran&longs;ire valebit cylindrum, ni&longs;i à tergo in&longs;tet maiorem habens <lb/>gravitatem. </s><s>At verò huius <expan abbr="quoq;">quoque</expan> impul&longs;us a&longs;&longs;umitur minor <lb/>illâ gravitare &longs;ecundâ; non igitur à cylindro eluctari, <expan abbr="neq;">neque</expan> pro­<lb/>inde ba&longs;im excludere valebit. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si orbiculus æqualis percutiat ba&longs;im cylindri orbiculati minùs gra-<emph.end type="italics"/> <pb xlink:href="063/01/116.jpg"/><emph type="italics"/>vem; habeat verò impul&longs;um æqualem gravitati &longs;ecundæ; exclus à ba&longs;i <lb/>illius locum obtinebit.<emph.end type="italics"/></s></p> <p type="main"> <s>Vt &longs;i orbiculus metallicus ba&longs;im ligneam percutiat: <expan abbr="&longs;itq;">&longs;itque</expan> hu­<lb/>ius impul&longs;us æqualis gravitati &longs;ecundæ, quâ ba&longs;is detinetur à <lb/>cylindro: cuius pars e&longs;t gravitas propria eiu&longs;dem ba&longs;is: dico <lb/>hunc orbiculum exclusâ à cylindro ba&longs;i, illius locum obtinere <lb/>Vt enim ba&longs;is à cylindro excludatur, nece&longs;&longs;e &longs;uperare illam re­<lb/>&longs;i&longs;tentiam, dum in cylindro movetur, à gravitate tum propriâ <lb/>tum alienâ provenientem: quam quidem &longs;imul collectam <lb/>metitur diameter eiu&longs;dem cylindri: propterea quòd ultima <lb/>pars ba&longs;is nece&longs;&longs;ariò per hanc moveatur. </s><s>At verò impul&longs;us, <lb/>quo ba&longs;is urgetur ab orbiculo graviore, a&longs;&longs;umitur æqualis re­<lb/>&longs;i&longs;tentiæ &longs;imul collectæ; in omni ergo puncto motûs cylindri­<lb/>ci e&longs;t maior re&longs;i&longs;tentia: <expan abbr="quou&longs;q;">quou&longs;que</expan> in fine motûs eidem gravita­<lb/>ti fiat æqualis. </s><s>Et quia ba&longs;is per 11 huius non ni&longs;i ab impul&longs;u <lb/>fluente movetur; &longs;uccedet continuò in locum huius orbicu­<lb/>lus movens: ac proinde ba&longs;i à cylindro exclusâ eundem lo­<lb/>cum obtinebit. </s></p> <p type="main"> <s><emph type="italics"/>Dices &longs;i in fine motûs impul&longs;us e&longs;t æqualis gravitati &longs;ccundæ, in omni <lb/>verò puncto motûs maior eadem gravitate, quomodo totus impul&longs;us <lb/>e&longs;&longs;e pote&longs;t æqualis toti gravitati? Nam &longs;i æqualibus addantur inæqua­<lb/>lia, erunt tota inæqualia: at&que; maius ab acceßione maiori.<emph.end type="italics"/></s></p> <p type="main"> <s>Refpondeo illam æquationem non ni&longs;i extrin&longs;ecè termina­<lb/>ri: cùm partes habeant nullâ duratione commen&longs;urabiles. </s><lb/><s>Fit ergo quemadmodum in a&longs;cen&longs;ionibus &longs;ignorum; ut licet <lb/>continuò partes maiores aut minores cooriantur; in fine ta­<lb/>men motûs quadrantes inter &longs;e &longs;int æquales. </s></p> <pb xlink:href="063/01/117.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si orbiculus æqualis percutiat ba&longs;im cylindri orbiculati minùs gra­<lb/>vem; habeat verò impul&longs;um duplo maiorem gravitate &longs;ecunda; ex­<lb/>clusâ à cylindro ba&longs;i pertran&longs;ibit.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i impul&longs;um habeat æqualem gravitati &longs;ecundæ; per <lb/>12 huius, &longs;uccedit in locum ba&longs;is à cylindro exclu&longs;æ: Cùm igi­<lb/>tur eadem gravitate detineatur, quâ ba&longs;is exclu&longs;a; non ni&longs;i ab <lb/>impul&longs;u æquali excludi valebit. </s><s>Vt ergo exclusâ ba&longs;i ip&longs;e <lb/><expan abbr="quoq;">quoque</expan> eluctetur; impul&longs;um habebit duplo maiorem. </s><s>Quòd <lb/>&longs;i verò impul&longs;um habeat illâ gravitate maiorem, minorem ve­<lb/>rò quàm duplum; exclusâ ba&longs;i non totus, &longs;ed pro ratione ex­<lb/>ce&longs;&longs;ûs plus, minuù&longs;uè à cylindro prominebit. </s><s>Priu&longs;quam <lb/>hunc motum orbiculorum finiam; admonere volui, ne quis <lb/>ab uno experimento obiter facto, <expan abbr="neq;">neque</expan> ni&longs;i omnibus propo­<lb/>&longs;itionibus priùs expen&longs;is, facile pronuntiet: cùm hæ inter­<lb/>dum illas limitent. </s><s><expan abbr="Icaq;">Icaque</expan> cùm dico orbiculum, &longs;i alium per­<lb/>cutiat &longs;ibi æqualem, illo expul&longs;o quie&longs;cere; id non pror&longs;us ve­<lb/>ritati con&longs;onum videbitur, &longs;i experimentum fiat in orbiculis <lb/>magis pondero&longs;is: cuiu&longs;modi metallici, ex argento, ferro, ære, <lb/>plumbo, &longs;tanno, auro. </s><s>Percu&longs;&longs;o enim æquali non quie&longs;cunt, <lb/>&longs;ed aliquantulum ex illâ plagâ &longs;equuntur: idq magis minù&longs;ue <lb/>pro ratione ponderis. </s><s>Quod quidem ad finem theor: 6 monui <lb/>Quia nimirum rationem cylindri habent eiu&longs;modi orbiculi: <lb/><expan abbr="magi&longs;q;">magi&longs;que</expan> pondero&longs;us æquivalet cylindro longiori. </s><s><expan abbr="Itaq;">Itaque</expan> diffe­<lb/>rentia plagæ in his maior; quæ in orbiculis levioribus evane­<lb/>&longs;cit, & ob exiguitatem &longs;en&longs;um latet. </s><s>Idem fit in globis magni <lb/>ponderis & molis. </s><s>Quia vel centrum gravitatis non e&longs;t idem <lb/>cum centro molis: vel quòd &longs;uperficiem minùs &longs;phæricam <lb/>habentes non in puncto, &longs;ed parte aliquâ dividuâ &longs;e tangunt, <pb xlink:href="063/01/118.jpg"/>vel quòd plaga aliquantulum inclinet. </s><s>Quin & volubilitas <lb/>&longs;peciem motûs continuati <expan abbr="quandoq;">quandoque</expan> præ&longs;tat. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Orbiculorum in cylindro di&longs;po&longs;itorum quemcun&que; imperatum exclu­<lb/>dere, alijs non exclu&longs;is.<emph.end type="italics"/></s></p> <p type="main"> <s>In cylindro orbiculato AI &longs;it excludenda ba&longs;is A. id con&longs;e­<lb/>quemur cum orbiculo æquali M factâ plagâ per 1 Pori&longs;: At <lb/>&longs;i tertius à ba&longs;i C excludi debeat: appone duos à tergo pla­<lb/> <arrow.to.target n="fig21"/><lb/>gæ KL: & cum tribus orbiculis percute cylindrum: namre­<lb/>liquis immotis tertius ex&longs;iliet: propterea, quòd impetus prio­<lb/>rum in illas anterides &longs;e exonerat. </s><s>Quòd &longs;i artem magis la­<lb/>tere velis; &longs;int orbiculi mole, non etiam pondere æquales. </s><lb/><s>Duobus ergo levioribus tertio æquali &longs;ubiectis, &longs;i percu­<lb/>tiatur cylindrus; quia minor plaga leviorum, non ni&longs;i tertium <lb/>excludes. </s><s>Eodem modo &longs;i quartus, aut quintus po&longs;tuletur; <lb/>cum totidem numero orbiculis <expan abbr="plagã">plagam</expan> induces: uno verô minus <lb/>à tergo cylindri <expan abbr="plagã">plagam</expan> excipies: aut certè totidem leviores, <lb/>quot &longs;upere&longs;&longs;e velis, ultimo &longs;uppone. </s></p> <pb xlink:href="063/01/119.jpg"/> <figure id="id.063.01.119.1.jpg" xlink:href="063/01/119/1.jpg"/> <p type="main"> <s><emph type="center"/>PROBLEMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Orbiculos plures &longs;iòi contiguos à cylindro orbiculato excludere, alijs <lb/>non exclu&longs;is.<emph.end type="italics"/></s></p> <p type="main"> <s>Si à ba&longs;i incipiat numerus orbiculorum; cum totidem per­<lb/>cute: <expan abbr="atq;">atque</expan> eundem numerum à cylindro excludes. </s><s>Quòd &longs;i <lb/>orbiculi intere&longs;&longs;e debent; <expan abbr="totid&etilde;">totidem</expan> à tergo cylindri oppone: tum <lb/>enim à &longs;uâ &longs;tatione dimoventur ex illâ plagâ, quibus núlli or­<lb/>biculi &longs;unt oppo&longs;iti. </s><s>Aut certè totidem leviores &longs;uppone, <lb/>quot cum ba&longs;i reliquos e&longs;&longs;e velis: nullâ enim motione ab his <lb/>factâ, numerum quæ&longs;itum dabit plaga reliqua. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Orbiculos plures non contiguos à cylindro orbiculato excludere, alijs <lb/>non exclu&longs;is.<emph.end type="italics"/></s></p> <p type="main"> <s>Sint orbiculi tres excludendi, nimirum 1. 3 & 5 omnibus <lb/>alijs immotis ex illâ plagâ. </s><s>Quod quidem duobus modis con­<lb/>&longs;equimur. uno, &longs;i orbiculi plagam afferentes &longs;int inæquales: <lb/><expan abbr="levioré&longs;q;">levioré&longs;que</expan> percutiant eos, quos manere volumus. </s><s>Secundo <lb/>modo, &longs;i his æqualiter habentibus &longs;equantur inæquales: <expan abbr="atq;">atque</expan> <lb/>illorum plaga, quos excuti volumus, &longs;e recipiat in minores: <lb/>tum enim per Pori&longs;: 2 motum minoris &longs;equitur maior. </s><s>Cùm <lb/>autem dicimus reliquos orbiculos e&longs;&longs;e <expan abbr="ab&longs;q;">ab&longs;que</expan> motu; de illo in­<lb/>tellige, qui provenit à percu&longs;&longs;ione: nece&longs;&longs;e enim in illa inter­<lb/>valla, à quibus orbiculi &longs;unt eiecti, alios &longs;e recipere à gravita­<lb/>te depre&longs;&longs;os. </s><s>Quòd &longs;i tamen dextrè plaga inferatur, <expan abbr="omne&longs;q;">omne&longs;que</expan> <lb/>orbiculi inter &longs;e &longs;int æquales & ad libellam complanati; <expan abbr="ab&longs;q;">ab&longs;que</expan> <pb xlink:href="063/01/120.jpg"/>&longs;uccu&longs;&longs;ione fit cylindri: qui non ni&longs;i ex inæquali orbiculorum <lb/>lap&longs;u, aut cùm plaga in alios impingit, dilabitur. </s></p> <p type="main"> <s><emph type="italics"/>Verùm dubitatio non levis occurrit. </s><s>Nam &longs;i orbiculi inter &longs;e æqua­<lb/>les & contigui longî &longs;erie di&longs;ponantur in lineâ; rectà; percu&longs;&longs;o primo ul­<lb/>timus movetur non eadem ratione: verùm pro numero orbiculorum <lb/>minùs, quou&longs;&que; omnes à plagâ &longs;int immoti. </s><s>Marce&longs;cit ergo illâ exten­<lb/>&longs;ione impul&longs;us; ne&que; totaplaga in &longs;ingulos propagatur. </s><s>Atqui eadem <lb/>ratio videtur &longs;phærularum: quomodo ergo per infinitas hunc extendi <lb/>volumus, quem in orbiculis cito videmus terminari.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo dici po&longs;&longs;e, quòd &longs;i orbiculi per omnia &longs;int æqua­<lb/>les, in lineâ centri gravitatis &longs;itum habentes, eadem ratione, <lb/>quâ in &longs;phærulis interminabilem fore motum. </s><s>Verùm <lb/>quia illorum centrum non nece&longs;&longs;ariò e&longs;t idem cum centro <lb/>gravitatis; cùm partes habeant à &longs;e differentes: inde fieri ut <lb/>centrum gravitatis <expan abbr="plerumq;">plerumque</expan> &longs;it extra illam lineam, quæ tran­<lb/>&longs;it per centra orbiculorum. </s></p> <p type="main"> <s>Quôd &longs;i ita: non iam una omninm e&longs;t plaga; &longs;ed minor quæ <lb/>percutit obliquè: nece&longs;&longs;e ergo dum eâ ratione mutatur cen­<lb/>trum gravitatis, impul&longs;um minui, ac demum deficere. </s></p> <p type="main"> <s>Re&longs;pondeo &longs;ecundò, illam po&longs;itionem de interminabili mo­<lb/>tu &longs;phærularum non ni&longs;i ut probabilem a&longs;&longs;umi. </s><s>Vt verò <lb/>gratiam ineamus etiam cum his, qui eam aver&longs;antur; videa­<lb/>mus &longs;i quâ ratione hunc motum, omnibus immotis, quæ pro <lb/>fundamento &longs;unt adducta, terminare valeamus. </s><lb/><s>Cùm ergo &longs;phærula prima &longs;ecundam hæc tertiam percutit; <lb/>dico inæqualem fieri plagam. </s><s>Nam quia impul&longs;us inæquali­<lb/>ter recipitur in mobili; prout nimirum partes magis, minù&longs;uè <lb/>ab&longs;unt à plagâ; quæ tamen æquationem habent à centro gra- <pb xlink:href="063/01/121.jpg"/>vitatis; quo omnes æqualiter moventur: minor erit vis in <lb/>centro quàm in loco plagæ. </s><s>Quòd &longs;i enim motui veloci&longs;­<lb/>&longs;imo accedat minùs velox; hunc quidem incitari, illum verò <lb/>retardari contingit. </s><s>Igitur cùm per cu&longs;&longs;io fiat à centro, mi­<lb/>nor erit plaga à &longs;ecundo quàm à primo. </s><s>Ratio in oppo­<lb/>&longs;itum facta ita di&longs;&longs;olvitur. </s><s>Impul&longs;um à plagâ 20 ad totum <lb/>impul&longs;um maiorem rationem habere, quàm &longs;ubvigecuplam. </s><lb/><s>Licet enim plaga &longs;ecunda &longs;it minor quàm prima: non tamen <lb/>illud decrementum e&longs;t æquale magnitudini, quam pertran&longs;it, <lb/>&longs;ed exce&longs;&longs;ui, quo plaga maior e&longs;t æquatione centri gravitatis: <lb/>quæ differentia in eiu&longs;modi &longs;phærulis e&longs;t valde exigua. </s><s><expan abbr="Itaq;">Itaque</expan> <lb/>fit ut globus libr: 20. moveri nequeat à plagâ unius libræ: im­<lb/>pul&longs;us tamen tran&longs;iens per globos librales 20. ultimum mo­<lb/>veat. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si orbiculum tangant plures alij eidem æquales; percutiat verò hunc <lb/>alius orbiculus æqualis, ad intervallum maius quadrante à contactu <lb/>illorum orbiculorum; omnes contigui à percu&longs;&longs;o movebuntur.<emph.end type="italics"/></s></p> <p type="main"> <s>Vt &longs;i orbiculum A tangant alij C. D. E: percutiat verò eun­<lb/>dem A æqualis B in puncto H; cuius intervallum HG, vel <lb/>H L maius <expan abbr="quadrãte">quadrante</expan>: dico omnes contiguos C. D. E moveri <lb/>ex illâ plagâ Ducantur à contactu orbiculorum G & L ip&longs;i AH <lb/>parallelæ GP. LO, &longs;ecantes AT. AK in O & P: dico men&longs;u­<lb/>ram plagæ OK <expan abbr="atq;">atque</expan> TP &longs;imul &longs;umptam e&longs;&longs;e minorem radio <lb/>AK: ac proinde impul&longs;um reliquum à plagâ movere orbicu­<lb/>lum D. </s><s>Ducantur rectæ DE. IL. & quia ut AD ad DE, ita <lb/>AI ad IL; &longs;unt verò AD. DE æquales; erit <expan abbr="quoq;">quoque</expan> AI æqua- <pb xlink:href="063/01/122.jpg"/>lis IL chordæ grad: 60. cuius &longs;inus rectus AO; atq, huius <lb/>complementum OK minus &longs;emi&longs;se radij. </s><s>Quòd &longs;i orbiculus <lb/>E tangat A inter L&K; erit minor huius plaga, quàm OK <lb/>ptopterea quòd DE fiat maior quàm AD, & IL maior quàm <lb/>AI: ac proinde AO maior &longs;inu grad: 60. </s></p> <figure id="id.063.01.122.1.jpg" xlink:href="063/01/122/1.jpg"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Idem verò &longs;equitur, &longs;i orbiculi C. D. E a&longs;&longs;umantur mino­<lb/>res, quàm &longs;it A. propterea quòd hi ex impul&longs;u minori move­<lb/>antur, quàm orbiculi æquales, per pori&longs;: 2. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Tres orbiculos percutere eadem plagâ: qui in motu percutiant alios <lb/>tres quolibet intervallo &longs;eiunctos: â quibus rur&longs;um alij tres percutian­<lb/>tur in quouis &longs;itu.<emph.end type="italics"/></s></p> <p type="main"> <s>Sint tres orbiculi in &longs;ltu <emph type="italics"/>a.b.c:<emph.end type="italics"/> quos alij <emph type="italics"/>g.h.i<emph.end type="italics"/> percutere de­<lb/>bent in motu: à quibus rur&longs;um alij tres <emph type="italics"/>d.e.f<emph.end type="italics"/> percutiantur: a <pb xlink:href="063/01/123.jpg"/>&longs;ingulis &longs;inguli. nempe ab <emph type="italics"/>a<emph.end type="italics"/> ip&longs;um <emph type="italics"/>d,<emph.end type="italics"/> & à <emph type="italics"/>b<emph.end type="italics"/> ip&longs;um <emph type="italics"/>e,<emph.end type="italics"/> at&que; de­<lb/>mum <emph type="italics"/>f<emph.end type="italics"/>à<emph type="italics"/>c.<emph.end type="italics"/> Ducantur per illorum centra rectæ <emph type="italics"/>da. eb. fc:<emph.end type="italics"/> & <lb/>producantur extra circulum in <emph type="italics"/>o.p.q,<emph.end type="italics"/> adintervallum &longs;emidia­<lb/>metri eiu&longs;dem circuli: à quibus ducantur aliæ rectæ <emph type="italics"/>ok. pk. qk<emph.end type="italics"/><lb/>per centrum orbiculi <emph type="italics"/>k<emph.end type="italics"/> maioris. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> orbiculi <emph type="italics"/>g. h. i<emph.end type="italics"/><lb/>contigui orbiculo <emph type="italics"/>k<emph.end type="italics"/> habeant centra in ei&longs;dem lineis <emph type="italics"/>ok. pk. qk:<emph.end type="italics"/><lb/>percutiat verò orbiculum <emph type="italics"/>k<emph.end type="italics"/> alius æqualis, vel maior <emph type="italics"/>l<emph.end type="italics"/> in <emph type="italics"/>w<emph.end type="italics"/>: di­<lb/>co orbiculos <emph type="italics"/>g.h.i<emph.end type="italics"/> ex illâ plagâ percutere orbiculos <emph type="italics"/>a.b.c:<emph.end type="italics"/> ab <lb/>his verò rur&longs;um percuti orbiculos <emph type="italics"/>d.e.f.<emph.end type="italics"/><lb/>Cùm enim orbiculi <emph type="italics"/>g.h.i<emph.end type="italics"/> &longs;int minores quàm <emph type="italics"/>k;<emph.end type="italics"/> movebuntur <lb/>exillâ plagâ per coroll: Theor: 14. </s><s>Et quia percu&longs;&longs;io, & qui <lb/>hanc &longs;equitur impul&longs;us, fit per lineam rectam productam à con­<lb/>tactu per centrum corporis percu&longs;&longs;i per 5 Theor: 2 part: erit <lb/>motus orbiculi <emph type="italics"/>g<emph.end type="italics"/> in lineâ <emph type="italics"/>go.<emph.end type="italics"/> <!--neuer Satz-->Ducatur per contactum linea <emph type="italics"/>rs<emph.end type="italics"/><lb/>parallela ip&longs;i <emph type="italics"/>ok:<emph.end type="italics"/> quæ &longs;i &longs;ecet orbiculum <emph type="italics"/>a,<emph.end type="italics"/> erit linea hypomo­<lb/>chlij, & complementum <emph type="italics"/>os<emph.end type="italics"/> eiu&longs;dem plaga: quæ ex demon&longs;tra­<lb/>tis orbiculum <emph type="italics"/>a<emph.end type="italics"/> movebit per rectam <emph type="italics"/>ad.<emph.end type="italics"/> <!--neuer Satz-->Quòd &longs;i verò recta <lb/><emph type="italics"/>rs<emph.end type="italics"/> cadat extra <expan abbr="utrumq;">utrumque</expan> orbiculum; problema locum non ha­<lb/>bebit. </s><s>Non e&longs;t tamen nece&longs;se per <expan abbr="utrumq;">utrumque</expan> centrum duci li­<lb/>neam rectam; ni&longs;i cùm totum impul&longs;um dare volumus orbi­<lb/>culo percu&longs;lo: &longs;ed &longs;ufficit, &longs;i ex centro unius producta linea re­<lb/>cta tangat, vel &longs;ecet <expan abbr="quacunq;">quacunque</expan> ratione alterum orbiculum. </s><lb/><s>Eadem ratione o&longs;tendemus orbiculos <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>c<emph.end type="italics"/> percuti ab<emph type="italics"/>h<emph.end type="italics"/> & <emph type="italics"/>i:<emph.end type="italics"/><lb/>percutere verò eo&longs;dem <emph type="italics"/>e<emph.end type="italics"/> & <emph type="italics"/>f.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>DE <lb/>PROPORTIONE MOTVS ORBICVLO­<lb/>RVM TAM AD SE, QVAM AD MOTVM <lb/>ORBICVLI CONTIGVI, A QVO <lb/>IMPELLVNTVR.<emph.end type="center"/></s></p> <pb xlink:href="063/01/124.jpg"/> <p type="main"> <s>In quâ proportione &longs;inguli orbiculi ferantur, cùm tres con­<lb/>tigui ab æquali impelluntur, dictum Theor: 14. </s><s>Quòd &longs;i ve­<lb/>rò idem orbiculus non ni&longs;i duos habeat &longs;ibi contiguos; aut ip&longs;i <lb/><expan abbr="quoq;">quoque</expan> erunt contigui inter &longs;e, aut non contigui. &longs;int primùm <lb/>contigui. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si in diametro orbiculi productâ fiat contactus orbiculorum; percu­<lb/>tiat verò hunc alius æqualis in parte oppo&longs;itâ diametri; eo immoto u­<lb/>ter&que; contiguorum movetur.<emph.end type="italics"/></s></p> <p type="main"> <s>Percutiat orbiculum A alius æqualis in F: in cuius diametro <lb/>productâ FQ fiat contactus orbiculorum CD: dico immo­<lb/>to A <expan abbr="utrumq;">utrumque</expan> orbiculum C & D moveri ex illâ plagâ. </s><s>Du­<lb/>catur linea hypomochlij HG: & ad eam perpendicularis AN <lb/>quæ &longs;ecabitur in duo &longs;egmenta æqualia AS. NS. propterea <lb/>quòd AS &longs;it &longs;inus rectus grad: 30. &longs;emi&longs;&longs;is nimirum GI grad: <lb/>60. habebit ergo plaga &longs;emi&longs;&longs;em totius impul&longs;ûs: qui per po­<lb/>&longs;it:4 velocitate feretur &longs;ubduplâ illius velocitatis. quâ orbicu­<lb/>lus A moveretur. </s><s>Quod idem dicendum de orbiculo D. </s><lb/><s>Cùm <expan abbr="itaq;">itaque</expan> duo orbiculi C&D &longs;imul contineant totum impul­<lb/>&longs;um ex A; erit plaga perfecta: ac proinde orbiculus A à motu <lb/>conquie&longs;cet. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si diameter orbiculi producta &longs;ecet unum exorbiculis &longs;ibi contiguis; <lb/>percutiat verò hunc alius æqualis in parte oppo&longs;itâ diametri productæ; <lb/>eo immoto, uter&queacute; orbiculorum eidem contiguorum movebitur.<emph.end type="italics"/></s></p> <pb xlink:href="063/01/125.jpg"/> <p type="main"> <s>Cùm enim TP men&longs;ura plagæ, quam recipit orbiculus C <lb/>ex A, &longs;it minor quàm AP &longs;inus rectus grad: 60; qui reliquum <lb/>impul&longs;um, quo centrum A moveretur à plagâ, metitur; erit ut <lb/>AP ad PT, ita motûs in A admotum in C. </s><s>Quia verò or­<lb/>biculus A percutit æqualem D, occurrit verò eidem in lineâ <lb/>centri; dabit plagam perfectam: ac proinde per 1 pori&longs;ma A <lb/>quidem à motu conquie&longs;cet, D verò eadem velocitate feretur. </s><lb/><s>Verùm licet hypomochlium GP eâ ratione impul&longs;um partia­<lb/>tur; quia tamen <expan abbr="utraq;">utraque</expan> plaga fit &longs;imul; habebit plaga ex A ad <lb/>plagam ex P eam rationem, quam AT ad PT. </s><s>Cùm enim <lb/>vectis &longs;it AT, cuius fulcimentum in T; erit per prop: 3 Gui­<lb/>di Vbaldi, ut AT ad PT, ita gravitas appen&longs;ain A adean­<lb/>dem gravitatem appen&longs;am in P. </s><s>At verò eandem rationem <lb/>habet vis &longs;ur&longs;um impellens, quam gravitas deor&longs;um movens: <lb/>quòd gravitas non ni&longs;i mediante impul&longs;u agat. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> <lb/>totus impul&longs;us in AT &longs;it partium 42; & TP pars &longs;exta AT; <lb/>erit plaga TP in primâ quidem partitione, quam hypomo­<lb/>chlium GP inducit, partium 7: impul&longs;us verò reliquus in A <lb/>partium 35. </s><s>At verò cùm percu&longs;&longs;io geminatur; plaga ex A <lb/>quidem e&longs;t partium 36, ex P verò partium 6. </s></p> <p type="main"> <s>Secet nunc orbiculum D, non per centrum, diameter pro­<lb/>ducta ex puncto medio inter F&H: in quo eundem percutiat <lb/>alius orbiculus æqualis: dico immoto A <expan abbr="utrumq;">utrumque</expan> orbiculum <lb/>C & D moveri. </s><s>Ducantur per contactus GI lineæ hypomo­<lb/>chlij eidem diametro parallelæ: quas &longs;ecent lineæ perpendicu­<lb/>lares ex A. erit itaq, huic quidem æqualis linea ex G perpendi­<lb/>cularis ad eandem diametrum &longs;inus grad: 45. propterea quod <lb/>GI &longs;it grad: 60 <expan abbr="atq;">atque</expan> huius &longs;emi&longs;&longs;is VI grad: 30. cuius comple­<lb/>mentum 2928992 men&longs;ura plagæ in C. </s><s>Rur&longs;um quia FI <lb/>e&longs;t grad: 165; erit &longs;emi&longs;&longs;is &longs;inus rectus grad: 82 pr:30. cuius <pb xlink:href="063/01/126.jpg"/>&longs;inus ver&longs;us 8694738 metitur plagam in D. </s><s>Cùm <expan abbr="itaq;">itaque</expan> to­<lb/>tus impul&longs;us &longs;it partium 10000000, <expan abbr="utraq;">utraque</expan> verò plaga &longs;imul <lb/>&longs;umpta partium 11623730 maior &longs;inu toto; erit plaga per­<lb/>fecta: ac proinde orbiculus A à motu conquie&longs;cet. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLLARIV M.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> &longs;inus totus &longs;ecetur in eâ ratione, quam habet <lb/>numerus maior ad minorem; erit motus in D ad motum in <lb/>C in eadem ratione, quæ paulominor e&longs;t quàm tripla. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si duo orbiculi non contigui tangant alium &longs;ibi æqualem ad inter­<lb/>vallum maius quàm.grad: 60. percutiat verò hunc æqualis in parte op­<lb/>po&longs;itâ; uter&que; unà cum orbiculo percu&longs;&longs;o movebitur.<emph.end type="italics"/></s></p> <p type="main"> <s><expan abbr="Tangãt">Tangant</expan> orbiculum A duo æquales in L & V: percutiat ve­<lb/>rò hunc alius æqualis in H: dico <expan abbr="utrumq;">utrumque</expan> orbiculum unà cum <lb/>A moveri ex illâ plagâ. </s><s>Cùm enim AZ &longs;inus grad: 30 &longs;it &longs;e­<lb/>mi&longs;&longs;is AT; erit plaga huic æqualis. </s><s>Et quia AO e&longs;t &longs;inus <lb/>grad: 60; erit complementum OK partium 1339746 in mi­<lb/>nori ratione, quàm &longs;eptuplâ ad &longs;inum totum. </s><s>E&longs;t autem <lb/>ut AK ad OK, ita plaga ex Aad plagam ex O. </s><s>Quòd &longs;i itaq, to­<lb/>tus impul&longs;us &longs;it partium 12; erit in O plaga minor quàm parti­<lb/>um 2: & <expan abbr="utraq;">utraque</expan> plaga &longs;imul &longs;umpta minor quàm partium 8. <lb/>impul&longs;us ergo reliquus in A maior quàm partium 4. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>COROLL ARIV M.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sequitur quô maius intervallum inter contactus orbiculo­<lb/>rum, eò velociorem e&longs;&longs;e motum orbiculi his contigui: propte- <pb xlink:href="063/01/127.jpg"/>rea quòd impul&longs;us reliquus ad plagam continuò maiorem ha­<lb/>beat rationem. </s><s>Et &longs;icuti ab intervallo grad: 60 incipit motus <lb/>orbiculi A; ita motus contiguorum terminatur, ubi contactus <lb/>non ni&longs;i quadrante circuli abfuerit à plagâ. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Duo puncta in peripheriâ orbiculi aßignare: in quibus orbiculi ei­<lb/>dem contigui eadem cum illo velocitate moveantur.<emph.end type="italics"/></s></p> <p type="main"> <s>Secetur diameter orbiculi TK in &longs;ex partes æquales. </s><s>Sup­<lb/>ponamus verò TP <expan abbr="atq;">atque</expan> OK e&longs;&longs;e eiu&longs;modi &longs;egmenta: à qui­<lb/>bus ducantur lineæ perpendiculares LO. GP. </s><s>Dico in pun­<lb/>ctis L. G orbiculos EC eadem cum A celeritate moveri. </s><lb/><s>Cùm enim PT &longs;it pars tertia AT; habebit plaga in A ad <lb/>plagam in P rationem triplam. </s><s>Quòd &longs;i <expan abbr="itaq;">itaque</expan> impul&longs;us æqua­<lb/>lis AT &longs;it partium 12; erit plagain P partium 4. </s><s>E&longs;t verò <lb/>eidem æqualis plagain O; igitur reliquus impul&longs;us in A erit <lb/><expan abbr="quoq;">quoque</expan> partium 4: ac proinde per po&longs;itionem 4 tres orbiculi A <lb/>CE: eadem velocitate moventur. </s></p> <p type="main"> <s><emph type="center"/>PROBLEMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Duo puncta in peripheriâ orbiculi determinare: à quicus orbiculi <lb/>contigui moveantur, tam ad &longs;e, quàm ad motum orbiculi his contigui <lb/>in datâ ratione.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit proportio data motûs orbiculorum contiguorum tri­<lb/>pla: motûs verò orbiculi reliqui ad unum ex his &longs;e&longs;quialtera. </s><lb/><s>Secetur <expan abbr="itaq;">itaque</expan> &longs;emidiameter AT in &longs;ex partes æquales: & du­<lb/>catur linea à &longs;ecundâ divi&longs;ione, quæ à centro, perpendicularis <lb/>producta ad peripheriam: eritq plaga in A ad illam plagamin <pb xlink:href="063/01/128.jpg"/>fe&longs;quialterâ ratione. </s><s>Rur&longs;um verò &longs;ecentur illa quatuor &longs;e­<lb/>gmenta reliqua in tres partes æquales: & ab ultimâ &longs;ectione, <lb/>quæ ad peripheriam, ducatur perpendicularis: <expan abbr="eritq;">eritque</expan> prior <lb/>plaga ad hanc in ratione triplâ. </s><s>Quòd &longs;i itaq, in alterâ &longs;emidi­<lb/>ametro uni &longs;egmento &longs;umatur æquale; & ducatur perpendicu­<lb/>laris ad peripheriam; inventa erunt duo puncta, à quibus or­<lb/>biculi impul&longs;i moveantur in datâ ratione. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si orbiculum Atangat alius æqualis Q ad intervallum grad: 30 à di­<lb/>ametro HI; percutiat verò eundem æqualis H; motus centri'ex illâ pla­<lb/>gâ non dimovetur à lineâ HI.<emph.end type="italics"/></s></p> <p type="main"> <s>Ductâ ex V perpendiculari VZ: erit AZ &longs;inus rectus grad: <lb/>30, &longs;emi&longs;&longs;is radij AT: motus vero in A æqualis plagæ in V. </s><lb/><s>Producatur TQ parallela VZ: <expan abbr="eritq;">eritque</expan> AQ ad AT, ut AV <lb/>ad AZ & VQ ad TZ. &longs;ed ut AQ ad AT, ita TQ ad VZ: & <lb/>permutando TQ ad VQ, ut VZ ad TZ. </s><s>E&longs;t autem VZ <lb/>&longs;inus rectus grad: 60 maior quàm AZ &longs;inus rectus grad: 30. </s><lb/><s>Cùm <expan abbr="itaq;">itaque</expan> motus in A &longs;it æqualis AZ; erit velocior motus in <lb/>VQ, quo centrum Q à contactu&longs;e abducit, quàm ut aliquod <lb/>punctum inter VT ip&longs;um con&longs;equi valeat. non igitur cen­<lb/>trum A dimovetur à lineâ rectâ AI. </s></p> <p type="main"> <s><emph type="center"/>THEOREMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si orbiculum A tangat alius æqualis C adintervallum grad: 60 à di­<lb/>ametro HI, percutiat verò eundem æqualis in H; motus centri A &longs;it per <lb/>tangentem circuli, cuius centrum e&longs;t contactus orbiculi C.<emph.end type="italics"/></s></p> <p type="main"> <s>Quoniam GP &longs;inus grad: 30 e&longs;t minor &longs;inu AP grad: <pb xlink:href="063/01/129.jpg"/> <arrow.to.target n="fig22"/><lb/>60; habebit hic ad PT maiorem rationem, quàm GP. </s><s>E&longs;t au­<lb/>tem ut GP ad PT, ita TR ad RG: velocior ergo motus cen­<lb/>tri A, <expan abbr="atq;">atque</expan> huius parallelorum inter G & T, quàm &longs;it motus or­<lb/>biculi C, quo à contactu orbiculi A &longs;e abducit: nece&longs;se pro­<lb/>inde centrum A prohibitum à contactu viam proximam &longs;equi: <lb/>hoc e&longs;t per tangentem circuli centro G de&longs;cripti. </s></p> <figure id="id.063.01.129.1.jpg" xlink:href="063/01/129/1.jpg"/> <p type="main"> <s><emph type="center"/>THEOREMA XX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si plures orbiculi tangant alium maiorem; percutiat verò hunc æ­<lb/>qualis; omnes contigui unà cum orbiculo maiore movebuntur.<emph.end type="italics"/></s></p> <p type="main"> <s>Tangant orbiculum <emph type="italics"/>k<emph.end type="italics"/> quotlibet alij minores <emph type="italics"/>g. b. m. i:<emph.end type="italics"/> per­<lb/>cutiat verò hunc æqualis <emph type="italics"/>l:<emph.end type="italics"/> dico orbiculos <emph type="italics"/>g. h. m. i<emph.end type="italics"/> unà cum <lb/>orbiculo <emph type="italics"/>k<emph.end type="italics"/> moveri ex illâ plagâ. </s><s>Erit enim ex demon&longs;tratis <lb/>Theor: 16 ut <emph type="italics"/>kz<emph.end type="italics"/> ad <emph type="italics"/>nz<emph.end type="italics"/> ita plaga orbiculi <emph type="italics"/>h<emph.end type="italics"/> ad plagam orbi­<lb/>culi <emph type="italics"/>m:<emph.end type="italics"/> & ut <emph type="italics"/>nz<emph.end type="italics"/> ad <emph type="italics"/>yz,<emph.end type="italics"/> ita plaga in <emph type="italics"/>m<emph.end type="italics"/> ad plagam in <emph type="italics"/>i.<emph.end type="italics"/> <!--neuer Satz-->Quòd <lb/>idem de plagâ orbiculi <emph type="italics"/>g<emph.end type="italics"/> dicendum. maior <expan abbr="itaq;">itaque</expan> omnibus pla­<lb/>ga e&longs;t in <emph type="italics"/>h.<emph.end type="italics"/> <!--neuer Satz-->Quia verò plaga &longs;equitur impul&longs;um, quo percu- <pb xlink:href="063/01/130.jpg"/>tiens erat moturum; percutit verò <emph type="italics"/>k<emph.end type="italics"/> orbiculum minorem <emph type="italics"/>h;<emph.end type="italics"/><lb/>movebitur hic ab incipiente & necdum perfectâ plagâ: orbicu­<lb/>lus ergo <emph type="italics"/>k<emph.end type="italics"/> per pori&longs;ma 2. motum continuabit. </s><s>Simili modo <lb/>orbiculi. reliqui <emph type="italics"/>g. m. i<emph.end type="italics"/> quia minores quàm <emph type="italics"/>k;<emph.end type="italics"/> movebuntur ab <lb/>impul&longs;u minori: ac proinde à plagâ incipiente: unde huius ex­<lb/>ce&longs;&longs;us erit principium motûs orbiculi <emph type="italics"/>k,<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>DE GYRATIONE ORBICVLI.<emph.end type="center"/></s></p> <p type="main"> <s>Si orbiculus percu&longs;&longs;us alium impellat &longs;ibi contiguum & æ­<lb/>qualem; duplici motu videtur ferri ex illâ plagâ: nimirum <lb/>recto & circulari. </s><s>Nam cùm A movetur per lineam AI, pun­<lb/>ctum H in peripheriâ transfertur in F. K &c. </s><s>Quod quidem <lb/>erit manife&longs;tum &longs;i punctum contactûs aliquo &longs;igno notetur. </s><lb/><s>Huius autem motûs ratio videtur referri ad librationem. </s><s>Nam <lb/>cùm ex plagâ in G dece&longs;&longs;erit impul&longs;us æqualis PT; nece&longs;se <lb/>præpondium fieri in K, <expan abbr="atq;">atque</expan> ita revolui orbiculum circa mobi­<lb/>le centrum A. </s></p> <p type="main"> <s><emph type="italics"/>Obijcies. </s><s>Si ob librationem circumagitur orbiculus, neceße buius mo­<lb/>tum e&longs;&longs;e æqualem plagæ. cui æquatur exce&longs;&longs;us in parte oppo&longs;itâ. igitur quò <lb/>contactui propior diameter, quia tum maior plaga; erit quo&que; circulatio <lb/>maior: quod tamen non fit. </s><s>Verùm quò maius intervallum, eò arcum <lb/>de&longs;cribit maiorem. </s><s>Deinde verò &longs;i duo orbiculi contigui inæqualiter ab­<lb/>&longs;int à diametro, cuiu&longs;modi in LV, circulatio procedit ex H in N. e&longs;t au­<lb/>tem maior plaga in V quàm in L: oportebat ergo hunc motum fieri ex <lb/>H in F, &longs;i illa circulatio proveniret ab exce&longs;&longs;u.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo cùm motus hic circularis fluat ab eodem impul­<lb/>&longs;u, quem retinet centrum ad &longs;e movendum; hic autem acce&longs;&longs;u <lb/>ad diametrum continuò minuatur; nece&longs;sc <expan abbr="quoq;">quoque</expan> circulatio- <pb xlink:href="063/01/131.jpg"/>nem æ&longs;timari minorem. </s><s>Deinde verò cùm per Theor: 19 <lb/>ab intervallo grad: 60 motus centri fiat per tangentem circu­<lb/>li; nece&longs;&longs;e hanc librationem magis augeri. </s><s>Vnde etiam ratio <lb/>petenda, quòd circulatio <expan abbr="quandoq;">quandoque</expan> fiat in partem plagæ maio­<lb/>ris: cùm videlicet duo orbiculi inæqualiter ab&longs;unt à lineâ mo <lb/>tûs centri: Hic enim oppo&longs;ita circulatio prævalet: quam deter­<lb/>minat motus centri per tangentem. </s></p> <p type="main"> <s>Po&longs;&longs;e verò mi&longs;ceri motui recto circularem, manife&longs;tum in <lb/>eodem orbiculo; &longs;i convexâ parte tangat planum. á digito e­<lb/>nim compre&longs;&longs;us & eli&longs;us <expan abbr="quandoq;">quandoque</expan> eidem puncto in&longs;i&longs;tens ro­<lb/>tari, <expan abbr="quandoq;">quandoque</expan> à procur&longs;u recurrere, aut etiam retro agi vide­<lb/> <arrow.to.target n="fig23"/><lb/>tur. </s><s>Quòd &longs;i enim motus circularis fiat æqualis motui recto; <lb/>videbitur orbiculus in eodem puncto A circa immobile cen­<lb/>trum gyrari. </s><s>Dividatur peripheria orbiculi in &longs;ex partes æqua­<lb/>les <emph type="italics"/>abcdef:<emph.end type="italics"/> & &longs;umantur his æqualia &longs;egmenta in lineâ re­<lb/>ctâ <emph type="italics"/>aghikl.<emph.end type="italics"/> Cùm <expan abbr="itaq;">itaque</expan> motus in <emph type="italics"/>ab<emph.end type="italics"/> &longs;it æqualis motui centri <lb/>eiu&longs;dem orbiculi in <emph type="italics"/>ag;<emph.end type="italics"/> gyratio autem non ni&longs;i per contactum <pb xlink:href="063/01/132.jpg"/>fiat eiu&longs;dem plani; nece&longs;se ubi ex <emph type="italics"/>a<emph.end type="italics"/> promovit in <emph type="italics"/>g,<emph.end type="italics"/> ip&longs;um <emph type="italics"/>b<emph.end type="italics"/><lb/>revolui in <emph type="italics"/>a.<emph.end type="italics"/> Similiter ubi <emph type="italics"/>b<emph.end type="italics"/> perventurum eratex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>g,<emph.end type="italics"/> ip&longs;um <lb/><emph type="italics"/>c<emph.end type="italics"/> attinget punctum <emph type="italics"/>a.<emph.end type="italics"/> Quòd &longs;i maior &longs;it motus circuli, quàm <lb/>eiu&longs;dem centri; contingit ip&longs;um retroagi. </s><s>Nam cùm ex <emph type="italics"/>a<emph.end type="italics"/><lb/>movetur in <emph type="italics"/>g;<emph.end type="italics"/> motus in peripheriâ fit per maius <expan abbr="&longs;egmentũ">&longs;egmentum</expan> <emph type="italics"/>ab:<emph.end type="italics"/> ac <lb/>proinde orbiculus tangit planum in puncto medio inter <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>c.<emph.end type="italics"/><lb/>Demum &longs;i maior &longs;it motus centri quàm gyrationis; videbitur <lb/>motus rectus, & punctum <emph type="italics"/>b<emph.end type="italics"/> inter <emph type="italics"/>a<emph.end type="italics"/> & <emph type="italics"/>g.<emph.end type="italics"/> Inde ergo ratio reddi­<lb/>tur; quòd motus centri ab illatâ plagâ deflectat à lineâ rectâ <lb/><emph type="italics"/>AI<emph.end type="italics"/> etiam ante grad: 60. </s><s>Cùm enim motus orbiculi circularis <lb/>in plano firmetur, <expan abbr="eaq;">eaque</expan> ratione motui centri reluctetur; ne­<lb/>ce&longs;&longs;e motum mixtum inde procreari. </s></p> <figure id="id.063.01.132.1.jpg" xlink:href="063/01/132/1.jpg"/> <p type="main"> <s><emph type="center"/>De Levigatione & Politura.<emph.end type="center"/></s></p> <p type="main"> <s>COrpora polita dicuntur, quæ &longs;uperficiem habent illius fi­<lb/>guræ, quâ terminantur, æquabilem: ut in cubo perfectè <lb/>planam, in globo &longs;phæricam. </s><s>His opponitur a&longs;perum &longs;eu &longs;ca­<lb/>brum: cuius &longs;uperficies partes habet inæqualiter &longs;itas, magis <lb/>& minùs depre&longs;&longs;as aut elevatas. </s><s><expan abbr="Neq;">Neque</expan> omnia corpora æqua­<lb/>liter: &longs;ed alia magis, alia minùs, alia nullâ indu&longs;triâ poliuntur: <lb/>ut thophus, pumex, &longs;uber, panni lanei &c. </s><s>Et cùm &longs;cabrities <lb/>&longs;eu inæqualitas à duobus proveniat: cùm vel partes in&longs;unt <lb/>verruco&longs;æ, vel pori &longs;eu cavernulæ &longs;uperficiem perforantes, <lb/>quantum vis &longs;en&longs;um lateant: polituram obtinemus contrariâ <lb/>affectione: verrucarum quidem, & quæ prominent, ablatione: <lb/>&longs;patiorum verò inanium repletione. </s><s>Quòd &longs;i eiu&longs;modi um­<lb/>bilici & verruculæ tolli nequeant: aut lacunæ expleri, impo­<lb/>libile dicetur corpus, Talia &longs;unt <foreign lang="greek">a)pie<gap/>a\</foreign>, & quæ dividi neque­<lb/>unt in partes minimas: quia <expan abbr="neq;">neque</expan> compre&longs;&longs;ioni cedunt ad po­<lb/>rum &longs;olidandum, receptâ in eas vacuitates parte magis pre&longs;sâ: <pb xlink:href="063/01/133.jpg"/>ut vitrum, gemmæ, lapides, <expan abbr="omniaq;">omniaque</expan> <foreign lang="greek">qrau<gap/>a</foreign>: <expan abbr="neq;">neque</expan> pars minima <lb/>re&longs;ecari valet, cuiu&longs;modi e&longs;t thophus. </s><s><expan abbr="Atq;">Atque</expan> illa quidem &longs;olá <lb/>partium ablatione poliuntur: & &longs;i quidem poro&longs;a &longs;int, nullâ <lb/>ratione &longs;uam perfectionem a&longs;&longs;equitur politura: quemadmo­<lb/>dum <expan abbr="neq;">neque</expan> individua in partes minimas: ablatâ enim parte ma­<lb/>iori, quàm &longs;it exce&longs;&longs;us; eadem inæqualitas manet. </s><s>Corporaer­<lb/>go <foreign lang="greek">paxume/rea</foreign> & quæ glutinosâ vi&longs;ciditate tenaciùs cohærent, <lb/>ut cera, pix, tela linea, papyrus; &longs;olâ levigatione proficiunt: <lb/>partibus à compre&longs;&longs;ione in eodem &longs;itu manentibus. </s><s>Vnde <lb/>panni lanei, ob pilos à compre&longs;&longs;ione &longs;urrigentes, non levigan­<lb/>tur. metalla <expan abbr="quoq;">quoque</expan> omnia, <expan abbr="atq;">atque</expan> ligna alia magis, alia minùs le­<lb/>vigationi parent. </s><s>Quæ enim mollia &longs;unt, <expan abbr="neq;">neque</expan> compre&longs;&longs;a <lb/>manent in eo &longs;itu, ut medulla &longs;ambuci, aut &longs;pongia, non levi­<lb/>gantur. </s><s>Nece&longs;&longs;e enim reniti aliquas partes: quibus aliæ inni­<lb/>tantur. </s><s>Quod non fit, &longs;i omnes à compre&longs;&longs;ione moveantur <expan abbr="ce-dantq;">ce­<lb/>dantque</expan> <expan abbr="Itaq;">Itaque</expan> ligna duriora, cuiu&longs;modi hebenus, præ alijs levi­<lb/>gantur. </s><s>Cùm igitur illa corpora vel partium ablatione, vel <lb/>illarum &longs;itu permutato &longs;uperficiem politam con&longs;equantur; <lb/>manife&longs;tum levigationem & polituram non <expan abbr="ab&longs;q;">ab&longs;que</expan> motu & im­<lb/>pul&longs;u fieri. </s><s>Cuiu&longs;modi verò &longs;it motus, & quâ ratione fiat, <lb/>nunc dicam, à levigatione incipiendo. </s></p> <p type="main"> <s><emph type="italics"/>E&longs;t autem levigatio motus reciprocus in &longs;uperficie levigandà, factus <lb/>à corpore polito, non &longs;ine compreßione.<emph.end type="italics"/></s></p> <p type="main"> <s>Ni&longs;i enim corpus levigans &longs;it ter&longs;um & politum; <expan abbr="nequaquã">nequaquam</expan> <lb/>aliam &longs;uperficiem levigare valebit: novâ a&longs;peritate ex illa­<lb/>rum partium inæqualitate inductâ: dum magis quidem pro­<lb/>minentes excavant, & veluti &longs;ulcos incidunt: depre&longs;&longs;æ verò <lb/>tubercula attollunt. </s><s><expan abbr="Itaq;">Itaque</expan> videmus ab eiu&longs;modi &longs;uperficie <lb/>a&longs;pcrâ & hamatâ pannos a&longs;perari & villo&longs;os reddi: quò <expan abbr="filam&etilde;-ta">filamen­<lb/>ta</expan> <expan abbr="atq;">atque</expan> illorum textura magis lateant. </s><s>Deinde &longs;i motus fiat <pb xlink:href="063/01/134.jpg"/><expan abbr="ab&longs;q;">ab&longs;que</expan> compre&longs;&longs;ione, aut non ni&longs;i leviter illam &longs;uperficiem tan­<lb/>gendo; <expan abbr="neq;">neque</expan> lacunæ expleri, <expan abbr="neq;">neque</expan> verruculæ deprimi valebunt. <lb/><expan abbr="Neq;">Neque</expan> motu &longs;implici, <expan abbr="atq;">atque</expan> uno tractu perficitur politura: &longs;ed <lb/>motibus iteratis, & in omnes partes reciprocè factis. </s><s>Et licet <lb/><expan abbr="quandoq;">quandoque</expan> &longs;olâ compre&longs;&longs;ione planities inducatur; non tamen <lb/>levigatio e&longs;t perfecta: ob plures &longs;ulcos, <expan abbr="&longs;triá&longs;q;">&longs;triá&longs;que</expan> à compre&longs;&longs;ione <lb/>relictas: quæ magis in profundum, quàm lateraliter movet. </s><lb/><s>Igitur cùm motus &longs;it cau&longs;a levigationis; quo partes &longs;itum variè <lb/>permutant: & velin locum partium compre&longs;&longs;arum; velin me­<lb/>dias cavitates trasferuntur: motus autem à percu&longs;&longs;ione & à ta­<lb/>ctu fiat; quem ex his motum dicemus levigationem? e&longs;t enim <lb/><foreign lang="greek">w(sit xi/n<gap/>sis a)po\ t<gap/>_s a(/yews</foreign>: cùm movens non ni&longs;i tangendo <lb/>movet: at verò partes levigantes non manent, &longs;ed prætere­<lb/>unt: <expan abbr="continuóq;">continuóque</expan> alias tangunt partes: non igitur <foreign lang="greek">w(/sei</foreign> &longs;eu pul­<lb/>&longs;ione moventur partes levigandæ. </s></p> <p type="main"> <s>Re&longs;pondeo, licet partes continuò mutentur: quia tamen <lb/>aliæ <expan abbr="atq;">atque</expan> aliæ &longs;uccedunt eiu&longs;dem rationis, motum continuan­<lb/>tes; per æquivalentiam idem videri movens. </s><s>E&longs;t autem dif­<lb/>ferétia inter ea, quæ <foreign lang="greek">xi/nhsin</foreign> habent <foreign lang="greek">a)po\ tg_s a(/yews</foreign>, & quæ <foreign lang="greek">a)po\ tg_s <lb/>plhgh_s xino<gap/>_nt<gap/></foreign>: quòd hæc in motu &longs;eparantur à movente: ac <lb/>proinde acceptâ plagâ non &longs;it in pote&longs;tate moventis ille mo­<lb/>tus. </s><s>Quæ verò <foreign lang="greek">a)po\ th_s a(/yews</foreign> moventur; impul&longs;um habent <lb/>fluentem: qui non ni&longs;i illis motis e&longs;&longs;e pote&longs;t: <expan abbr="moxq;">moxque</expan> ubi cæpit, <lb/>ex illo contactu finit: & non ni&longs;i impul&longs;u continuato &longs;ervari <lb/>pote&longs;t. </s><s><expan abbr="Atq;">Atque</expan> inde fit, ut nulla particula inter poliendum, &longs;eu <lb/>levigandum divellatur: cùm motus in ip&longs;a plagâ finiat, <expan abbr="neq;">neque</expan> <lb/>ullus re&longs;tet impul&longs;us. </s><s>Et licet non &longs;ine aliquâ tractione par­<lb/>tes levigatæ extendantur; non tamen e&longs;t motus exce&longs;&longs;ivus: <lb/><expan abbr="neq;">neque</expan> per &longs;e, &longs;ed à compre&longs;&longs;ione na&longs;cens: <expan abbr="Itaq;">Itaque</expan> &longs;i excedat, ut <lb/>dum chartam minùs cautè levigamus; partes divelluntur. </s></p> <p type="main"> <s><emph type="italics"/>Dices. <!--neuer Satz-->A quo ergo partes compreßa detinentur in eo &longs;itu? <!--neuer Satz-->ne&que; enim &longs;o-<emph.end type="italics"/> <pb xlink:href="063/01/135.jpg"/><emph type="italics"/>la<emph.end type="italics"/> <foreign lang="greek">p<gap/>esa\</foreign> <emph type="italics"/>levigantur: ne&que; illa filamenta linteorum & minutuli &longs;locci <lb/>in compreßione uniuntur, &longs;uperficiem unam habentes: verùm contigui <lb/>inter &longs;e manent: ita&queacute; linteis excußis rur&longs;um à &longs;e di&longs;iungi, & &longs;uperfici­<lb/>em hi&longs;pidam reddi videmus.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo illum &longs;itum non ni&longs;i à novo motu turbari: mo­<lb/>tum verò non <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; impul&longs;u advenire. </s><s>Quòd &longs;i ergo partes <lb/><expan abbr="neq;">neque</expan> à &longs;e, <expan abbr="neq;">neque</expan> ab extra habeant principium motûs; nece&longs;&longs;e illá <lb/>&longs;uperficiem, in quam terminavit motus, retinere. </s><s><expan abbr="Itaq;">Itaque</expan> lin­<lb/>tea agitata turbantur: dum ex illo motu impetum concipiunt <lb/>particulæ, ab eo &longs;itu di&longs;trahentem. </s><s>Quæ autem rigidiu&longs;cula <lb/>&longs;unt: quia in &longs;e ip&longs;is habent principium motús; à compre&longs;&longs;io­<lb/>ne eo modo, quo arcus à curvaturá, rea&longs;&longs;urgunt. </s><s>Sicuti ve­<lb/>rò duobus modis levigatio fit; <expan abbr="nimirũ">nimirum</expan> depre&longs;&longs;ione & <expan abbr="cõpre&longs;&longs;io-ne">compre&longs;&longs;io­<lb/>ne</expan> <expan abbr="atomorũ">atomorum</expan>; ita <expan abbr="quoq;">quoque</expan> duobus motibus oppo&longs;itis turbatur: cùm <lb/>vel eriguntur: vel partes pre&longs;&longs;æ retume&longs;cunt. </s><s>Superficié le viga­<lb/>tam &longs;equitur tanquam proprietas &longs;plendor: lucis nimirum uni­<lb/>tæ confertim facta evibratio. </s><s>Nam quæ &longs;uperficiem habent <lb/>a&longs;peram, lucem incidentem di&longs;trahunt & inæqualiter <expan abbr="reflectũt">reflectunt</expan>. <lb/><expan abbr="Neq;">Neque</expan> enim ab aliquâ parte radij uniti, &longs;ed à &longs;e divul&longs;i, <expan abbr="&longs;eq;">&longs;eque</expan> in­<lb/>ter&longs;ecantes in retinam feruntur: &longs;inguli non ni&longs;i luce tenui &longs;en­<lb/>&longs;um afficientes. </s></p> <p type="main"> <s><emph type="italics"/>An igitur inferre licet omnia, quæ luce alienâ re&longs;plendent &longs;uperfici­<lb/>em habere levigatam? Nitent enim margaritæ, conchylia, opera item <lb/>figulina vitreata, avium pennæ, atramentum, picturæ &c. in quibus <lb/>tæmen a&longs;peritatem notamus.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo &longs;plendorem non ni&longs;i ex multâ luce unitâ na&longs;ci: <lb/>multa autem fit ratione &longs;ubiecti. nam &longs;ubiectum magis den­<lb/>&longs;um plus lucis continet. </s><s>Corpus ergo den&longs;i&longs;&longs;imum & &longs;ummè <lb/>politum &longs;plendorem habet &longs;ummum. </s><s><expan abbr="Itaq;">Itaque</expan> aurum perfectè <lb/>levigatum præ omnibus alijs &longs;plendet, <expan abbr="aciemq;">aciemque</expan> oculorum per- <pb xlink:href="063/01/136.jpg"/>cellit: plumbum verò licet alijs metallis magis den&longs;um; quia <lb/>tamen ob partes terreas minùs levigari pote&longs;t, & colori atro <lb/>magis mi&longs;cetur; minùs re&longs;plendet. </s><s>Fieri ergo pote&longs;t ut cor­<lb/>pus den&longs;um, & &longs;i minùs politum, magis &longs;plendeat, quàm rarum, <lb/>& è contra: at &longs;ummè levigatum nece&longs;&longs;ariò &longs;uperat den&longs;i&longs;&longs;i­<lb/>mum; &longs;i pror&longs;us &longs;it impolitum. </s><s>Deinde levigatum &longs;eu poli­<lb/>tum duobus modis &longs;umitur: ab&longs;olutè, & &longs;ecundum quid. </s><lb/><s>Ab&longs;olutè quidem, cuius &longs;uperficies <expan abbr="undiq;">undique</expan> e&longs;t ter&longs;a & æqualis: <lb/>&longs;ecundùm quid autem, quod non totam &longs;uperficiem, &longs;ed tan­<lb/>tum aliquas partes habet levigatas, non continuas inter &longs;e, ve­<lb/>rum partibus &longs;cabris interci&longs;as. </s><s>Multa ergo licet &longs;uperficiem <lb/>habeant &longs;cabram & inæqualem; quia tamen eiu&longs;modi umbili­<lb/>cos continent leves & politos; re&longs;plendent. </s><s>Ita enim figulina <lb/>poliuntur: dum metallicus humot illitus, <expan abbr="atq;">atque</expan> igne lique&longs;cens <lb/>&longs;uperficiem inungit: & demum æqualiter concretus &longs;peciem <lb/>vitri a&longs;&longs;umit. </s><s>Similiter panis humido inunctus, cru&longs;tam in igne <lb/>trahit re&longs;plendentem. </s><s>Ita atramentum &longs;criptorium admi&longs;to <lb/>gummi &longs;plendet. quia ob vi&longs;ciditatem minùs &longs;orbetur humor: <lb/>& partes vitriolicæ ceu vi&longs;co cohærentes, inter &longs;iccandum mi­<lb/>nùs hiulcæ fiunt. </s><s>Colores <expan abbr="quoq;">quoque</expan> & picturæ glutine pellucido <lb/>affu&longs;o, aut permixto &longs;imili ratione re&longs;plendent, </s></p> <p type="main"> <s><emph type="italics"/>Sed dices. aquam e&longs;&longs;e &longs;ummè levigatam, minùs tamen alijs &longs;plendere.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo &longs;plendorem e&longs;&longs;e lucem à &longs;uperficie reflexam: <lb/>ut autem reflecti po&longs;&longs;it nece&longs;&longs;e priùs terminari. </s><s>At verò aquam <lb/>pellucidam lux pertran&longs;it: minùs ergo lucis à reflexione. </s><s>De­<lb/>inde cùm aqua &longs;it fluens & minùs den&longs;a corporibus ex eâ con­<lb/>cretis; minor copia lucis in eâ colligi pote&longs;t. </s><s>Nec refert mul­<lb/>ta corpora e&longs;&longs;e rariora: hæc enim &longs;uam concretionem aëri de­<lb/>bent, qui in his prædominatur: cuiu&longs;modi volucrum pennæ, <lb/>& &longs;ambuci medulla. </s></p> <pb xlink:href="063/01/137.jpg"/> <p type="main"> <s>Hæc de levigatione. </s><s>Politura eundem finem habet; nimi­<lb/>rum &longs;uperficiem erugatam, <expan abbr="ter&longs;ámq;">ter&longs;ámque</expan>: differentia e&longs;t in modo <lb/>& medijs ad hunc finem. </s><s>Nam levigatio deprimit, aut in lacu­<lb/>nas transfert: politura adimit &longs;cabritiem efficientes partes. </s><lb/><s>Quæ quidem differentia in materiâ fundatur: cuius partes <expan abbr="neq;">neque</expan> <lb/>comprimi valent, <expan abbr="neq;">neque</expan> aliò transferri: Talia &longs;unt vitra, gemmæ, <lb/>lapides, <expan abbr="atq;">atque</expan> omnia <foreign lang="greek">a)pies a\ x<gap/>\ qzau<gap/>a\</foreign>. </s><s>Igitur quædam <expan abbr="utroq;">utroque</expan> <lb/>modo hunc finem con&longs;equuntur, & politurâ & levigatione, ut <lb/>metalla: quædam &longs;olâ levigatione, ut papyrus, lintea, cera: <lb/>quædam non ni&longs;i politurâ, ut gemmæ, lapides, vitra. </s><s>Ratio <lb/>e&longs;t quia nequeunt dividi in partes minimas: &longs;iue ob vi&longs;cidita­<lb/>tem magis tenacem, &longs;iue ob cra&longs;&longs;itiem. </s><s><expan abbr="Itaq;">Itaque</expan> fit, ut dum plus, <lb/>minú&longs;ue à &longs;cabritie aufert politura; prior inæqualitas con­<lb/>tinuò aliâ permutetur. </s><s>Deinde levigatio in multis incipit à <lb/>politurâ: cùm nimirum maior e&longs;t a&longs;peritas, quàm compre&longs;&longs;io <lb/>e&longs;&longs;e po&longs;&longs;it; nece&longs;&longs;e ergo illum exce&longs;&longs;um adimi, quò reliqua &longs;u­<lb/>perficies levigationem habeat magis expeditam: ita enim ligna <lb/><expan abbr="atq;">atque</expan> metalla non ni&longs;i ferro acci&longs;a levigantur. </s><s><expan abbr="Neq;">Neque</expan> idem e&longs;t <lb/>modus polituræ in omnibus; <expan abbr="neq;">neque</expan> idem principium. </s><s>Nam pro­<lb/>ut materiâ, & &longs;uperficie magis & minùs a&longs;perâ à &longs;e differunt; <lb/><expan abbr="atq;">atque</expan> ab alijs plus, ab alijs minùs e&longs;t auferendum; ita <expan abbr="quoq;">quoque</expan> in&longs;tru­<lb/>menta varia &longs;unt inventa. </s><s>Saxa enim & marmora malleo de­<lb/>cu&longs;&longs;is, aut ferro acci&longs;is promontorijs æquantur: gemmæ verò <lb/>affricatione adlapidem arenarium molâ circumactum, corti­<lb/>cem priùs, quò ve&longs;tiuntur, exuunt; inde &longs;errâ obtusâ in <expan abbr="&longs;egm&etilde;-ta">&longs;egmen­<lb/>ta</expan> dividuntur: demum arenâ <expan abbr="atq;">atque</expan> huius polline levigantur. </s><lb/><s>Ligna verò &longs;ecuri, cuneo, &longs;errâ finduntur & &longs;ecantur: dein­<lb/>de a&longs;ciâ, tornóve poliuntur. </s><s>E&longs;t autem nobis propo&longs;itum non <lb/>ni&longs;i de motu & impul&longs;u agere, quo &longs;uperficiem politam obti­<lb/>nemus; nequaquam verò de arte poliendi, quæ &longs;uis Magi&longs;tris <lb/>e&longs;t relinquenda. </s><s>Incipiam verò à Lignorum politurâ, utpo- <pb xlink:href="063/01/138.jpg"/>te minùs operosâ. </s><s>Cuius principium <foreign lang="greek">sxi/sis x<gap/>\ tm<gap/>_sis<gap/></foreign>: in eo à <lb/>&longs;e differentes: quòd <foreign lang="greek">sxi/sis</foreign> &longs;it <foreign lang="greek">diai/zesi<gap/>)pi\ to\ pl<gap/>on. sxiset<gap/> ga\z</foreign>, <lb/>inquit Ari&longs;toteles, <foreign lang="greek">o(/tan e)pi\ to\ pl<gap/>on di<gap/>zh_t<gap/> h(' to/ di<gap/>zo_un di<gap/>z<gap/>, xa<gap/><lb/>pzohg<gap/>tai h( diai/zesis</foreign>. </s><s>In fi&longs;&longs;ione ergo e&longs;t maior divi&longs;io, |quàm <lb/>ut ad <expan abbr="illã">illam</expan> <expan abbr="plagã">plagam</expan> referri po&longs;&longs;it: <expan abbr="eamq;">eamque</expan> divi&longs;io anteit. </s><s>E&longs;t enim <foreign lang="greek">xi/­<lb/>nhsis a)po\ th_s a(\yews</foreign>: cùm plaga in&longs;equitur <expan abbr="motũ">motum</expan>: <expan abbr="atq;">atque</expan> impul&longs;um <lb/>habet <expan abbr="fluent&etilde;">fluentem</expan>, & à plagâ in&longs;eparabilem. </s><s>Igitur cùm fi&longs;&longs;ura ultra <lb/>plagam &longs;e extendat, non e&longs;&longs;e pote&longs;t à plagâ. </s><s>Huius autem ra­<lb/>tio. quia <foreign lang="greek">to\ di<gap/>z<gap/>_n</foreign> habet vim cunei: cuius ingre&longs;&longs;u in eam pla­<lb/>gam partes di&longs;trahuntur. </s><s>Et cùm fibræ in longitudinem ex­<lb/>currentes flecti nequeant; nece&longs;&longs;e ultra cuneum agi fi&longs;&longs;uram: <lb/><expan abbr="atq;">atque</expan> eò magis, quò fibras habent rigidiores, & minùs lentas. <lb/><expan abbr="Itaq;">Itaque</expan> ligna duriora magis finduntur, quàm mollia ac lenta: quæ <lb/>magis obliquari & flecti valent. </s><s>Vnde angulo obtu&longs;iore, illa <lb/>verò acutiore, fi&longs;&longs;urâ magis productâ finiunt plagam. </s><s>Cùm <lb/>ergo inci&longs;io fit, ferrum in fi&longs;&longs;urâ conquie&longs;cit: partes verò hu­<lb/>ius ingre&longs;&longs;u di&longs;tractæ, quia flecti nequeunt ob rigiditatem, <expan abbr="neq;">neque</expan> <lb/>comprimi vulneris labra: quemadmodum fit in plumbi &longs;ectu­<lb/>râ, illam rectitudinem &longs;ervantes findunt partes ulteriores: <lb/><foreign lang="greek">sxista/</foreign> autem dicit Ari&longs;toteles <foreign lang="greek">o(/sa xata\ mh_xos e)/x<gap/> t<gap/>s po/z<gap/>s xa<gap/><lb/>su(\s pzosfu/et<gap/> a)llh/lois,<gap/> a)lla\ mh<gap/>xata\ pla/tos</foreign>. </s><s>Eiu&longs;modi &longs;unt li­<lb/>gna ferè omnia fibris in longitudinem proten&longs;is: inter quas <lb/>pori &longs;ub&longs;tantiâ molliori & veluti fungosâ pleni inter&longs;unt, per <lb/>quas agitur fi&longs;&longs;ura: non verò in tran&longs;ver&longs;um per illas fibras, <lb/>in quibus non continuantur eiu&longs;modi pori. </s><s>Plaga autem fit à <lb/>&longs;ectione pro ratione compre&longs;&longs;ionis. </s><s>Igitur ligna, quæ fibras <lb/>habent directas, fi&longs;&longs;uram <expan abbr="quoq;">quoque</expan> agunt rectam: quòd &longs;i tortuosè <lb/>procedant, inæqualiter finduntur: cùm plaga viam &longs;equatur <lb/>mediam inter illas fibras. </s><s>At cùm &longs;errâ dividuntur, à &longs;ectio­<lb/>ne etiam inter fibras ductâ nulla &longs;equitur fi&longs;&longs;ura: quia &longs;erratio <lb/>partes fibro&longs;as non di&longs;trahit, &longs;ed di&longs;continuas facit. </s><s>E&longs;t au- <pb xlink:href="063/01/139.jpg"/>tem &longs;erratio motus compo&longs;itus ex inci&longs;ione & fractione. </s><s><expan abbr="Neq;">Neque</expan> <lb/>enim huius dentes a&longs;periu&longs;culi inter &longs;e &longs;unt paralleli; &longs;ed alter­<lb/>natim ad latus <expan abbr="utrinq;">utrinque</expan> reflexi: quò divi&longs;io ex obliquo facta oc­<lb/>currat plagæ oppo&longs;itæ. </s><s><expan abbr="Itaq;">Itaque</expan> partes quidem medias inciden­<lb/>do, partes verò laterales &longs;uâ a&longs;peritate radendo auferunt: <expan abbr="eaq;">eaque</expan> <lb/>ratione vulneris labra, quo motum habeant liberiorem, adau­<lb/>gent. </s><s>Inci&longs;io enim &longs;implex e&longs;t divi&longs;io continui <expan abbr="ab&longs;q;">ab&longs;que</expan> deper­<lb/>ditione alicuius particulæ: ut cùm pomum per medium &longs;eca­<lb/>mus. </s><s>Differt à &longs;ectione &longs;ci&longs;&longs;ura: quòd hæc &longs;it plaga continu­<lb/>ata; &longs;ectio verò &longs;implex & interrupta: quæ tamen ob <expan abbr="vehem&etilde;-tiam">vehemen­<lb/>tiam</expan> excedere pote&longs;t illam. </s><s><expan abbr="Vtraq;">Vtraque</expan> e&longs;t &longs;olutio unionum, &longs;eu <lb/>di&longs;continuatio cum aliquâ compre&longs;&longs;ione: nece&longs;&longs;e enim quod <lb/>incidit recipi in <expan abbr="illãm">illamm</expan> plagam, <expan abbr="partésq;">partésque</expan> medias comprimi in la­<lb/>tus <expan abbr="utrumq;">utrumque</expan>. </s><s>Corpus &longs;erratile e&longs;t <foreign lang="greek">xataxto\n<gap/> xai\ qzauso\n</foreign> <expan abbr="neq;">neque</expan> enim <lb/>lapides, vitrum, gemmæ &longs;errantur. </s><s>Nam &longs;erra, quâ gemmæ <lb/>mediâ arenâ &longs;ecantur, quia dentibus caret, non ni&longs;i impropriè <lb/>dicitur. </s><s>Igitur lignis in hunc modum &longs;errâ <expan abbr="atq;">atque</expan> &longs;ecuri divi&longs;is, <lb/>aut cultro inci&longs;is a&longs;cia &longs;uccedit: quâ &longs;uperficies a&longs;pera & inæ­<lb/>qualis aufertur. </s><s><expan abbr="E&longs;tq;">E&longs;tque</expan> huius motus idem cum inci&longs;ione; ma­<lb/>gis tamen limitatus, ad men&longs;uram ferri inci&longs;orij ab eâ promi­<lb/>nentis. </s><s>Non enim profundiùs agitur plaga, quàm &longs;it illa fer­<lb/>ri longitudo: quæ contrahi & augeri pro libitu pote&longs;t. </s><s>Im­<lb/>pul&longs;um verò habet fluentem: cùm &longs;it <foreign lang="greek">xi/nhsis a)po\ th_s a(/y<gap/>ws</foreign>: <lb/>quam a&longs;cia manu librata dirigit, impul&longs;um cohibens, quò mi­<lb/>nùs latè evagetur. </s><s>Vnde maioribus a&longs;cijs utuntur in politu­<lb/>râ: quò maior compre&longs;&longs;io à pondere, & à parte huius planâ & <lb/>politâ levigatio &longs;imul fiat. </s><s>Huic &longs;imilis videtur motus à tor­<lb/>no factus: idem enim e&longs;t &longs;eu ferrum incidens, &longs;eu corpus inci­<lb/>dendum moveatur. </s><s>E&longs;t ergo manus veluti a&longs;cia, quæ fulcro <lb/>innixa aciem ferri pro voto inci&longs;uræ libratam &longs;u&longs;tinet: Velo­<lb/>cior tamen huius, quàm a&longs;ciæ motus <expan abbr="atq;">atque</expan> in circulum reductus: <lb/>qualis quidem e&longs;&longs;e nequit a&longs;ciæ motus ad globum poliendum. <pb xlink:href="063/01/140.jpg"/>ita quidem &longs;e habet politura in lignis, & quæ his &longs;unt cogna­<lb/>ta. </s><s>At verò torno poliuntur etiam metalla: nequaquam gem­<lb/>mæ, lapides, aut <expan abbr="vitrũ">vitrum</expan>: quòd hæc <foreign lang="greek">a<gap/>tmhta</foreign> &longs;int <foreign lang="greek">xa<gap/> qzausa\</foreign>: & non <lb/>ni&longs;i in plures partes friantur. </s><s><expan abbr="Itaq;">Itaque</expan> <expan abbr="neq;">neque</expan> a&longs;ciâ aut cultro &longs;ecari <lb/>valent: cùm &longs;ectio in duo terminetur, <expan abbr="unámq;">unámque</expan> particulam ab <lb/>alijs avellat. </s></p> <p type="main"> <s><emph type="italics"/>Sed cur metalla ab a&longs;ciâ non poliuntur, eandem vim cum torno ha­<lb/>bente?<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;pondeo in torno e&longs;&longs;e motum velociorem; quo re&longs;i&longs;ten­<lb/>tia & durities metallorum &longs;uperatur. </s><s>Idem enim e&longs;t cùm tor­<lb/>no circumagitur mobile, quemadmodum &longs;i ferrum celerrimè <lb/>moveretur: ut cùm gemmæ orbiculis circumactis poliuntur. </s><lb/><s>Et licet illarum politura torno fieri videatur; ob motum cir­<lb/>cularem illorum orbiculorum, quibus gemmæ &longs;e affricantes <lb/>atteruntur: e&longs;t tamen longè diver&longs;us <expan abbr="atq;">atque</expan> alius motus. </s><s>Non <lb/>enim orbiculi &longs;eu umbilici, quibus præpilantur cylindri ver­<lb/>&longs;atiles, incidunt: &longs;ed arenulæ his intermixtæ: quo modo in a­<lb/>lijs orbiculis contingit horizonti parallelis. lutum enim arenu­<lb/>latum continuò affu&longs;um &longs;uâ a&longs;peritate radit &longs;uperficiem ma­<lb/>gis eminentem. </s><s>Cùm verò hæc omnia &longs;int <foreign lang="greek">qzausa<gap/></foreign>; erit illo­<lb/>rum divi&longs;io <foreign lang="greek">q<gap/>rau_s s</foreign> non <foreign lang="greek">xa/ta<gap/>is</foreign>. quia non una particula, &longs;ed <lb/>plures &longs;imul avelluntur: illæ nimirum, quæ impul&longs;um <expan abbr="motũq;">motunque</expan> <lb/>recipiunt à plagâ, pro numero arenularum, non unâ. </s><s>Vide­<lb/>tur autem motus compo&longs;itus ex inci&longs;ione & fracturâ: com­<lb/>pre&longs;&longs;ione quidem in profundum: tractu verò in latum agen­<lb/>te plagam, ab impul&longs;u fluente inductam. </s><s>Cùm igitur &longs;it <foreign lang="greek">xi/nh­<lb/>sis a)po\ th_s a(/yews</foreign>, nequaquam altè penetrat; &longs;ed mox à com­<lb/>pre&longs;&longs;ione & contactu impul&longs;us cohibetur. </s><s>Cui accedit humi­<lb/>ditas ex polline arenularum continuò affu&longs;a gemmis, impul­<lb/>&longs;um hebetans: alioquin fragilibus, &longs;i arenulâ &longs;iccâ poliantur. <pb xlink:href="063/01/141.jpg"/>quæ & calorem ex illo motu na&longs;centem, quo corpora tene­<lb/>re&longs;cunt, magisq, fragilia fiunt, obtundit. </s><s>Vnde adamantes, qui­<lb/>bus gemmæ &longs;olidiores poliuntur, ex illâ velocitate motûs &longs;pe­<lb/>ciem carbonis igniti a&longs;&longs;umunt. </s><s>Differentia autem plagæ fit <lb/>pro ratione arenularum: cra&longs;&longs;iores enim & magis duræ ma­<lb/>iora auferunt &longs;egmenta. </s><s><expan abbr="Itaq;">Itaque</expan> gemmas rudiores, <expan abbr="multúmq;">multúmque</expan> <lb/>a&longs;peritatis habentes priùs &longs;axis arenulatis, quæ molis circum­<lb/>aguntur, affricantes, illâ attritione complanant: inde lapide <lb/>&longs;miri in farinam trito poliunt: & magis &longs;ubtili eiu&longs;dem polline <lb/>levigantes, demum perfectionem terrâ tripolitanâ inducunt. </s><lb/><s>Vitra tamen quia molliora, calce &longs;tanni levigantur. </s><lb/><s>Nec differt illarum &longs;ectio per &longs;erram æream edentulam facta, <lb/>lenti&longs;&longs;imo tractu arenulis interfu&longs;is radente: pro <expan abbr="quarũ">quarum</expan> diver&longs;i­<lb/>tate mutatur <expan abbr="quoq;">quoque</expan> vulneris amplitudo. </s><s>Nam cra&longs;&longs;ior arena, <lb/>inquit Plinins, laxioribus &longs;egmentis terit, & plus erodit mar­<lb/>moris, <expan abbr="maiú&longs;q;">maiú&longs;que</expan> opus &longs;cabritia polituræ relinquit. </s><s>Ita &longs;ectæ atte­<lb/>nuantur cru&longs;tæ. </s><s>Duplex ergo incommodum ab arenâ cra&longs;&longs;io­<lb/>re: nam & plus decedit gemmis à plagâ latiore: & &longs;uperficies <lb/>a&longs;pera maiorem in poliendo laborem exigit. </s><s><expan abbr="Itaq;">Itaque</expan> olim <expan abbr="u&longs;q;">u&longs;que</expan> <lb/>ad Æthiopas, & Indos arena petebatur: quarum Æthiopica <lb/>mollior, <expan abbr="nullâq;">nullâque</expan> &longs;cabritie &longs;ecans. </s><s>Nunc verò lapis &longs;miri & <lb/>terra tripolitana in u&longs;um polituræ &longs;ucce&longs;&longs;it. </s><s><expan abbr="Neq;">Neque</expan> &longs;olum gem­<lb/>mæ, marmora, & vitrum arenâ, &longs;eu lapide areno&longs;o poliuntur; <lb/>&longs;ed etiam metalla: cotibus enim ferrum atteri & pulvere &longs;mi­<lb/>ri levigari con&longs;tat. </s><s>Verùm hæc in&longs;uper limam &longs;entiunt: <lb/>quòd gemmis non convenit. </s><s>Tamet&longs;i dicat Plinius nobili­<lb/>um gemmarum &longs;oli Topazio id accidere: reliquas verò coti­<lb/>bus Naxijs poliri. </s><s>Quod quidem de politurâ rudiori & incho­<lb/>atâ intelligendum: <expan abbr="neq;">neque</expan> enim aut reliquas gemmas cotibus: <lb/>aut topazium limâ perfici potui&longs;&longs;e credendum. </s><s>No&longs;tratem <lb/><expan abbr="quoq;">quoque</expan> topazium licet molliorem reliquis gemmis, limam re- <pb xlink:href="063/01/142.jpg"/>&longs;puere experientia docet. </s><s>Fuerit ergo alterius generis Plinij <lb/>gemma à no&longs;trâ: quam in&longs;uo genere virentem, <expan abbr="eiu&longs;q;">eiu&longs;que</expan> <lb/>&longs;imilitudinem, ad porri &longs;uccum dirigi te&longs;tatur: cùm <lb/>no&longs;tra &longs;it coloris aurei. </s><s>Motus, quem lima inducit, e&longs;t <lb/>compo&longs;itus ab inci&longs;ione cancellatâ & <foreign lang="greek">xla/te<gap/></foreign>. </s><s>Nam &longs;ulci præ­<lb/>tenues inci&longs;i ab a&longs;peritate tra&longs;versâ eiu&longs;dem limæ raduntur. </s><lb/><s>Maior ergo durities nobilioribus gemmis ine&longs;t, cuiu&longs;modi a­<lb/>damas, calcedonius, &longs;apphyrus, heliotropia, rubinus: quæ <expan abbr="neq;">neque</expan> <lb/>ferro incidi, <expan abbr="neq;">neque</expan> limâ radi &longs;u&longs;tinent: quam tamen &longs;entiunt <lb/>marmora, lapides, vitrum, & gemmæ ignobiliores. A cotibus <lb/>verò hæc univer&longs;a poliuntur: propterea quòd &longs;uperficiem <lb/>habeant cotes &longs;cabram & areno&longs;am: quæ &longs;i polita, <expan abbr="miniméq;">miniméque</expan> <lb/>friabilis e&longs;&longs;ed, haud quaquam attererentur. </s><s>Mutuâ ergo <lb/>affrictone arenulæ coacervantur: quarum ab&longs;ce&longs;&longs;u minui <lb/>cotes, & demum longo u&longs;u ab&longs;umi con&longs;tat. </s><s>Saxa verò duri­<lb/>ora, quia atteri non valent, <expan abbr="neq;">neque</expan> u&longs;um cotis habent. </s><s>Sed <lb/>quæ&longs;tio hic e&longs;t: quamobrem cotes Naxiæ, & quæ nobis &longs;unt in <lb/>u&longs;u, aquâ; Creticæ verò & Laconicæ, ut Plinius te&longs;tatur, oleo <lb/>temperentur: an eiu&longs;modi cotes naturam habent olei, &longs;eu <lb/>bituminis pinguedinem continentes aquæ incommi&longs;cibilem? <lb/>ine&longs;&longs;e enim quibu&longs;dam lapidibus &longs;uccum pinguem & oleo&longs;um <lb/>con&longs;tat exinflammatione. </s><s>Lapis <expan abbr="quoq;">quoque</expan> nephriticus, quem <lb/>I&longs;adam vocant, multùm pingue&longs;cit inter poliendum: quan­<lb/>quam huius pinguedo non oleo&longs;a, &longs;ed quale gummi, aquæ <lb/>commi&longs;cetur. </s><s>At quomodo ergo cotes Ciliciæ, eodem Plinio <lb/>te&longs;te, oleo & aquâ pollent: an <expan abbr="utramq;">utramque</expan> naturam eo modo <lb/>permi&longs;tam habent, quo &longs;migma? quod & pingue in &longs;e tra­<lb/>hit, & aquâ eluitur. </s><s>Ita cotes ton&longs;trinarum humore non quo­<lb/>uis &longs;ed vi&longs;co&longs;o, cuiu&longs;modi &longs;putum, proficiunt. </s><s>Aquas autem <lb/>in Italiâ repertas, aciem trahentes acerrimo &longs;en&longs;u, minerales <lb/>fui&longs;le credo, eiu&longs;dem naturæ cum aquâ forti. </s><s>Magis tamen <pb xlink:href="063/01/143.jpg"/>mirandum, quod tradit Ferdinandus Corte&longs;ius, in Mexico e&longs;&longs;e <lb/>lapidem coloris flavi; ex quo novaculæ fiant acuti&longs;&longs;imæ: quæ <lb/>non à ferro, aut cote, &longs;ed ex aquâ illam aciem trahant. </s><lb/><s>Videtur autem hæc proprietas innuere huius cognationem <lb/>cum aquâ; <expan abbr="e&longs;&longs;éq;">e&longs;&longs;éque</expan> veluti glaciem ex aquâ concretam: à quâ <lb/>rur&longs;um atteratur & liquefiat: mox tamen ab aëre eo modo, <lb/>quo ovorum cortices, indurari. A motibus iam dictis differt <lb/>terebratio & perforatio: <expan abbr="fitq;">fitque</expan> duobus modis. </s><s>Vt cùm cavi­<lb/>tas inducitur <expan abbr="ab&longs;q;">ab&longs;que</expan> deperditione alicuius particulæ: & cùm <lb/>partes ab illâ cavitate excluduntur. </s><s>Et primo quidem modo <lb/>cavitas fit per compre&longs;&longs;ionem: quam &longs;ola <foreign lang="greek">pies a\</foreign> admittunt, cu­<lb/>iu&longs;modi metalla & ligna: nequaquam verò <foreign lang="greek">ta\ qzau<gap/>a\</foreign> at gem­<lb/>mæ, lapides, vitra. </s><s>Cùm deperditione verò &longs;ub&longs;tantiæ & hæc <lb/>& reliqua omnia cavantur: licet non uno modo omnia. </s><lb/><s>Nam gemmæ quidem & vitra non ni&longs;i politurâ, <expan abbr="&longs;en&longs;imq;">&longs;en&longs;imque</expan> ra­<lb/>dendo perforantur: terebratione verò ligna, metalla, o&longs;&longs;a. </s><lb/><s>Et &longs;icuti terebra figurâ à &longs;e differunt; ita etiam modus perfo­<lb/>randi. </s><s>Alia enim circulo; alia formâ &longs;emilunari terminan­<lb/>tur: alia cochleatim &longs;triata, ab acuto &longs;en&longs;im augentur & late­<lb/>&longs;cunt. </s><s><expan abbr="Atq;">Atque</expan> hæc quidem à perforatione incipiendo <foreign lang="greek">sxi/sei x<gap/>/ <lb/>xla/sei</foreign> terminant motum. </s><s>Dum enim cochlea circum acta <lb/>in partem compre&longs;&longs;am vulnus agit, & quæ à tergo &longs;equitur <lb/>helix, ambit latiore plagâ; in tenues & friabiles lamellas eâ <lb/>ratione &longs;cobinatur helicoides, à plagâ inci&longs;us conus: <expan abbr="eoq;">eoque</expan> in <lb/>helicem cavam recepto terebratio procedit, <expan abbr="quou&longs;q;">quou&longs;que</expan> cochlea <lb/>repleta &longs;cobe, educi & expurgari debeat. </s><s>Videtur autem hic <lb/>ratio vectis intervenire: cuius hypomochlium in centro mo­<lb/>tûs: extrema verò &longs;unt circelli &longs;en&longs;im aucti & in conum late­<lb/>&longs;centes. </s><s>Verùm huiu&longs;modi terebella &longs;uperficiem, quæ am­<lb/>bit plagam, minùs æqualem relinquunt: calorem verò ob <lb/>multiplicem motum adaugent. </s><s><expan abbr="Itaq;">Itaque</expan> minùs apta cranio per- <pb xlink:href="063/01/144.jpg"/>forando: ne huius medulla nimiùm exæ&longs;tuet. </s><s>Quæ autem <lb/>circulo finiunt: quia unâ inci&longs;ione auferunt quidquid inclu­<lb/>ditur illo circulo, <expan abbr="unóq;">unóque</expan> motu &longs;implici peragunt inci&longs;ionem; <lb/>in hunc u&longs;um veniunt. </s><s>Nam cùm in gyrum agitur hic cir­<lb/>culus; <expan abbr="unaquæq;">unaquæque</expan> particula incidit: & cùm aliæ eiu&longs;dem rati­<lb/>onis &longs;equantur; vulnus continuò fit maius: <expan abbr="atq;">atque</expan> eò magis, <lb/>quo <foreign lang="greek">qli/yis</foreign> &longs;eu compre&longs;&longs;io maior, motus autem velocior. </s><lb/><s>Differt ab his terebellum, quo metalla perforantur. </s><s>Stylus e­<lb/>nim <foreign lang="greek">amfi/<gap/>zus</foreign> cylindro infixus veluti torno circumagitur, non <lb/>&longs;ine compre&longs;&longs;ione ad corpus terebrandum. </s><s>Qui motus <lb/>inci&longs;ione perficitur: <expan abbr="duritiémq;">duritiémque</expan> metalli &longs;uperat ob <lb/>illam velocitatem. </s></p> <p type="main"> <s><emph type="center"/>FINIS.<emph.end type="center"/></s></p> <figure id="id.063.01.144.1.jpg" xlink:href="063/01/144/1.jpg"/> <p type="main"> <s><emph type="center"/>PRAGÆ.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Ex Typographia Academica.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Anno 1648.<emph.end type="center"/><lb/> <arrow.to.target n="fig24"/></s></p> <figure id="id.063.01.144.2.jpg" xlink:href="063/01/144/2.jpg"/> </chap> </body><pb xlink:href="063/01/145.jpg"/> <back><section><p type="main"> <s><emph type="center"/>[Errata not transcribed.]<emph.end type="center"/></s></p></section></back> </text> </archimedes>