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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <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></translator> <lang>la</lang> <cvs_file>marci_figur_063_la_1648.xml</cvs_file> <cvs_version></cvs_version> <locator>063.xml</locator> </info> <text> <front><section> <pb xlink:href="063/01/001.jpg"></pb> <figure id="id.063.01.001.1.jpg" xlink:href="063/01/001/1.jpg"></figure> <p type="caption"> <s>DE <lb></lb>PROPORTIONE <lb></lb>MOTVS <lb></lb>FIGVRARVM RECTI <lb></lb>LINEARVM <lb></lb>ET <lb></lb>CIRCVLI QVADRATVRA EX <lb></lb>MOTV <lb></lb>Authore <lb></lb>Ioanne Marco Marci Medicinæ <lb></lb>Doctore et Profeſſore Primario <lb></lb>S·C·Mtis· Medico Cubiculario <lb></lb>et in Reg. Boh<gap></gap> Phyſico <lb></lb>Seniore. <lb></lb>PRAGÆ <lb></lb><emph type="italics"></emph>Ano. 1648.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="063/01/002.jpg"></pb><pb xlink:href="063/01/003.jpg"></pb> <p type="main"> <s><emph type="center"></emph>SERENISSIMO <lb></lb>PRINCIPI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>FERDI<lb></lb>NANDO <lb></lb>IV. <lb></lb>HVNGARIÆ <lb></lb>ET BOHEMIAE <lb></lb>REGI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>ARCHIDVCI <lb></lb>AVSTRIAE.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DOMINO MEO CLEMENTISSIMO.<emph.end type="center"></emph.end></s></p> <pb xlink:href="063/01/004.jpg"></pb> <p type="main"> <s><emph type="center"></emph>SER ENISSIME REX &c.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>SOpitis eram ſenſibus; uti contingit his, <lb></lb>qui ſomno premuntur: cùm ecce tibi <foreign lang="grc"><gap></gap>αν αόζι<lb></lb>στον</foreign>! cui nulla certa ſpecies, omnia tamen ineſſe, <lb></lb>ipſum verò cubo inniti videbatur. </s><s>Cui ego: Quis<lb></lb>es? Tuus, inquit, Motus: adſum, ut tibi nun cius <lb></lb>ſim, ad novum Regem, annum auſpicaturus no<lb></lb>vum. </s><s>Ego verò ſuccenfens: ô ignauiſſime, inquam, <lb></lb>adeone tui generis es oblitus? quem pridem in Hungariam deſtinaram; <lb></lb>ut inter applauſus tu <expan abbr="quoq;">quoque</expan> plauſum ferres. </s><s>Siue tardus, ait ille, ſiue ve<lb></lb>lox ſim, non degenero à meis natalibus. </s><s><expan abbr="Namq;">Namque</expan> iraſcor his nouis Zeno<lb></lb>nibus, qui me ignauis morulis concidunt. </s><s>Quòd verò nunc tardiùs ad<lb></lb>ſum; quàm fortaſſe velles, tibi, non mihi imputa: qui me de circulo qua<lb></lb>dratum feciſti. </s><s>Quanquam, ſi mihi auſcultes, in lucro reponas hanc <lb></lb>meam tarditatem: quæ non <expan abbr="abſq;">abſque</expan> nutu accidit illius Genij, qui Symbo<lb></lb>lo Regio in te, <expan abbr="libróq;">libróque</expan> tuo præluſit. </s><s>Eſt enim numerus myſticus huius <lb></lb>Anni: quòd 1600 cubos efficiant primi paris 200. <expan abbr="atq;">atque</expan> hi alios cubos <lb></lb>ſecundos 25. qui numerus eſt quadratus ſecundi imparis. </s><s>At veeò nu<lb></lb>merus annorum 48 ſex cubos includit primi paris. </s><s>Anni <expan abbr="demũ">demum</expan> 9 elapſi, <lb></lb>ex quo adMAGNVM CÆSAREM nuncius fui, <expan abbr="quadratũ">quadratum</expan> abſoluunt pri<lb></lb>mi impatis. </s><s>Quò nimirum ſtabilitatem præſagiant futuri regni: in quo <lb></lb>ſubPRIMO AVGVSTI NOMINIS QVADRATO Sabbatha orbis aget. </s><lb></lb><s>Quin <expan abbr="ipſũ">ipſum</expan> nomen auſpicatum FERDINANDVS QVARTVS <lb></lb>hoc myſterio <expan abbr="numerorũ">numerorum</expan> eſt fœcundum. </s><s>Inſunt enim 1016 Et 1000 qui<lb></lb>dem cubos efficiunt ſecundi imparis octo: cuius duplum dat numerum <lb></lb>reliquum 16 & ipſum quadratum ſecundi paris: conſtantem verò du<lb></lb>obus cubis primi Paris. </s><s>Cui proinde Quadratura debetur Lunulæ ori<lb></lb>entis. </s><s>Et quid inquam ego, Symbolo Regio, <expan abbr="mihiq́">mihique</expan>; <expan abbr="atq;">atque</expan> huic meo libro <lb></lb>eſt commune? Tum ille: non vides, inquit, hunc circulum Symbolo ad<lb></lb>ſcriptum, hunc abacum parallelogrammis inſcriptum, hanc demum fi<lb></lb>guram ſtellatam è triangulis & pentagono contextam? quid præter <lb></lb>has figuras habet tuus liber? Neq, temerè inter radios geometricæ ſtel<lb></lb>læ coruſcat <foreign lang="grc">ν̔γ<gap></gap></foreign> Symbolum medicinæ: quia nimirum <expan abbr="utriuſq;">utriuſque</expan> ſcientiæ <pb xlink:href="063/01/005.jpg"></pb>gnarum eſſe voluit futurum Vatem: qualem <expan abbr="quoq;">quoque</expan> vitæ humanæ cuſto<lb></lb>dem requirit veſter Hippocrates. </s><s>Rectè quidem tu hæc, inquam ego: at <lb></lb>verò huius acerræ <expan abbr="atq;">atque</expan> ignis, quis nam in me typus? Tam citò, refert ille, <lb></lb>es oblitus! nam alioquin malorum ſenſus eſſe ſolet diuturnus. </s><s>Ego <lb></lb>verò dic, amabo te, aio quidnam ex igne mali ſum paſſus? namillud qui<lb></lb>dem ego prorſus ignoro. </s><s>Quòd enim non ita pridem <expan abbr="utramq;">utramque</expan> Domum, <lb></lb>quæ ex hæreditate meâ erant reliquæ, ignis abſumpſit, tu optimè noſti <lb></lb>quàm æquo animo tulerim: leuior enim hæc jactura mihi viſa; quàm ut <lb></lb>mentem his aſſuetam turbaret. </s><s>Ad hæc ille: non meminiſti, inquit, <lb></lb>illâ eadem nocte, quâ Phitomorphoſis tua ſymbolo præludebat, ma<lb></lb>num tibi aduſtam? Memini ſanè, inquam ego. </s><s>Nam ubi ſtudijs feſſum <lb></lb>caput in codicem ſacrum reclinaſſem; dormienti mihi, neſcio quo pa<lb></lb>cto, manus dextra ſubducta, & in ignem candelæ paulo remotioris pro<lb></lb>ducta digitum anularem aduſſit: cuius ſenſus acer me quidem euigila<lb></lb>re fecit, manum verò ut inſanam incuſare. </s><s>Ita quidem tu, ait Motus, à <lb></lb>veritate aberrans: at verò illa te longè ſapientior fuit: quæ a Sapien<lb></lb>tiſſimo Genio tum dirigebatur: Vt nimirum etiam hac parte ſymbolum <lb></lb>impleres. </s><s>Deinde veró quòd igne hoc elementari futurum Vatem initi<lb></lb>ari oportebat. </s><s>Vide nunc has plantas, quibus Symbolum inſignitur. </s><lb></lb><s>Agnoſcis hanc perpetuò virentem <expan abbr="atq;">atque</expan> victricem LAVRVM: quam ignis <lb></lb>Jouius tuetur incluſus? hanc PALMAM canenti OLIVÆ ſociatam? effare: <lb></lb>quid ſiles? Agnoſcis nunc demum tuam Phitomorphoſin? Ohe quid <lb></lb>audio, inquam ego! etiamne mentis penetralia tibi patent? quem ego <lb></lb>rebar ſolis corporibus mancipatum. </s><s>Et ubi inquit ille maiores per<lb></lb>turbationum motus, quàm in mentibus humanis? At velocitas mentis, <lb></lb>inquam ego, omni motu corporeo eſt velocior. </s><s>Si ergo ineft velocitas, <lb></lb>ait, inerit ſanè & motus. </s><s>Quanquam falleris, ratus mentem Corpori <lb></lb>huic terreno alligatam omni motu corporeo eſſe velociorem: quæ neq, <lb></lb>huius frigidi Saturni velocitatem ullâ ratione aſſequi valet. </s><s>At COPER<lb></lb>NICVS, inquam ego, cum GALILÆO & multâ turbâ ſophorum hanc tibi <lb></lb><expan abbr="Cœlóq;">Cœlóque</expan> prærogatiuam ademit: qui ſolem in medio mundi ſtare immo<lb></lb>tum, terram verò circumire juſſit. </s><s>Atqui refert ille, in eo ſatis oſten<lb></lb>dunt animi ſui tarditatem: Dum aſſequi non valent hanc meam in cor<lb></lb>poribus velocitatem. </s><s>Sed hîs relictis ad tuam Phitomorphoſim me <lb></lb>conuerto: <expan abbr="neq;">neque</expan> enim abeſſe potui ex illâ motione; dum planta una ex <pb xlink:href="063/01/006.jpg"></pb>aliâ naſci videbatur: licet motu velociore, quàm pro tuo voto: cùm <expan abbr="necdũ">necdum</expan> <lb></lb>ſatiato tibi illarum Species ſubducebantur. </s><s>Sed quem fuiſſe putas illum <lb></lb><expan abbr="Hortulanũ">Hortulanum</expan>, qui tibi in horto, ut videbatur, <expan abbr="ſurculũ">ſurculum</expan> LAVRI cupienti qui<lb></lb>dem, <expan abbr="neq;">neque</expan> tamen ob reuerentiarn viri petere auſo, ultro in manus dedit <lb></lb>cum hoc dicto: <emph type="italics"></emph>Poteſt creſcere.<emph.end type="italics"></emph.end> Tum ego, ô omniſcie Motus, quando<lb></lb>quidem nihil Te latet arcanorum: tu ſiquidem omnia audis, <expan abbr="vidéſq;">vidéſque</expan> <lb></lb>etiam quæ Solem oculatiſſimum & maximè auritum fugiunt; dic obſe<lb></lb>cro quid tibi videtur de illis verſibus, quos SERENISSIMO <lb></lb>HVNG: ET BOHEMIÆ REGI FERD: IIII. in felici in au<lb></lb>guratione accinebam? rectène illam Phitomorphoſim fui aſſecutus? <lb></lb>Ne dubita, ait Motus, idem enim Genius, qui ea ſimulachra immiſit, <lb></lb>eorundem ſenſum tibi inſtillauit. </s><s>Quid igitur inquam ego, cunctamur? <lb></lb>Perge mi Motus, <expan abbr="teq́">teque</expan>; ocyſſimé REGI NOVO ſiſte: ut ſis & munus, <lb></lb>& futuræ felicitatis augur. </s><s>Tibi liberum permitto, ut vel circulus, vel <lb></lb>quadratum, imò & cubus fias: prout REGALI TVTELÆ vide<lb></lb>bis ex pedire. </s><s>Quem terrâ <expan abbr="mariq́">marique</expan>; ſecutus, ventos fauentes motu circuli <lb></lb>velociore incitabis: eoſdem furentes quadrato, aut etiam cubo inhibe<lb></lb>bis. </s><s>Faxo lubens, inquit, quod imperas; tu verò boni ominis ergò, in hac <lb></lb>eadem pagellâ tuos verſus mihi exhibe: quos ego unâ cum libello mox <lb></lb>ad ultimum terræ feram. </s></p> <p type="main"> <s><emph type="center"></emph><foreign lang="grc">FITOMOPFWSIS</foreign><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Planta cadem LAVRVS, PALMA, & pallentis OLIVÆ, <lb></lb>Viſa mihi: demumgermina VITIS crant.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LAVRVS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Aſſociare virens Regali LAVRE Coronæ, <lb></lb>Seruet ut æternus Regia ſceptra viror.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PALMA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Bella procul, LAVRO nam aſſuetus vincere nouit: <lb></lb>Victorem victrix non niſi PALMA decet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>OLIVA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Naſcitur Imperio defeſſo pinguis OLIVA, <lb></lb>Hac non fucatæ ſymbola pacis habet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>VITIS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Terra Bohema oculos ſicca: noua VITIS inumbrat, <lb></lb>Præterita ignorat, qui bibit inde merum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p></section><section> <pb xlink:href="063/01/007.jpg"></pb> <p type="main"> <s><emph type="center"></emph>AD LECTOREM.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>NAturam definit Philoſophus eſſe principium & cauſam mo<lb></lb>tûs & quietis eius, in quo eſt, primùm, perſe, & non ſecundùm <lb></lb>accidens. </s><s>Quia verò ubi hic deſinit, ibi Medicus ſuæ Specula<lb></lb>tionis principium ſumit: cuius obiectum eſt natura humana, qua<lb></lb>tenus â ſanitate in <expan abbr="morbũ">morbum</expan>, & ex hoc in ſanitatem mouetur; neceſſe ſanè â Me<lb></lb>dico motum haud ignorari. </s><s>Præſertim verò cùm inter res non naturales, quas <lb></lb>Medicina ſpeculatur, numeretur motus & quies. </s><s>Impoſſibile enim, ait Hip<lb></lb>pocrates, hominem comedentem eſſe ſanum, ſi non laboret. </s><s>Vbiper laborem in<lb></lb>telligit motum corporeum: Cuius diuerſas ſpecies recenſet libro: 3. de diætâ. <lb></lb>quæ tamen ob ignorantiam motûs hoc æuo negliguntur. </s><s>Cùm ergo mihi propo<lb></lb>ſitum ſit, <expan abbr="jámq;">jámque</expan> incœptum habeam tractatum de naturâ humanâ, quatenus eſt <lb></lb>mobilis quoad utrum〈que〉 motum, videlicet internum & externum: tam in ſtatu <lb></lb>naturali, quàm præter naturam: hoc eſt radicem inveſtigare omnium morbo<lb></lb>rum, qui ad imaginationem <expan abbr="motúmq;">motúmque</expan> pertinent: id〈que〉 ex intimis, & recondi<lb></lb>tis naturæ principijs (ne〈qué〉 enim ſi paralyſis partem unam plureſue motu pri<lb></lb>uat, ſcire licet undo hac affectio pullulet, aut quo pacto eidem occurri poſſit; <lb></lb>niſi quid motus, & quâ ratione in nobis fiat, priùs norim) cùm rectum & <lb></lb>ſui, & obliqui ſit index; non videbor ab inſtituto aliena ſecutus; ſi habitu Phi<lb></lb>loſophi aſſumpto, ea principia, â quibus dicendorum veritas pendet, priùs ſtabi<lb></lb>liam. </s><s>Error ſiquidem in his tameiſi paruus, teſte Ariſtotele, in progreſſu fit <lb></lb>magnus. </s><s>Licet verò hunc libellum de proportione motûs figurarum rectilinea<lb></lb>rum necdum maturum <expan abbr="judicarẽ">judicarem</expan>, qui in lucem prodiret, at〈qué〉 ulteriore limâ eun<lb></lb>dem expolire in animo haberem; doctiſſimorum tamen virorum hortatu in a<lb></lb>liam mentem fui adductus. </s><s>Inter quos eminet Reuerendiſſimus Praſul<emph.end type="italics"></emph.end> Ioannes <lb></lb>Caramuel Lobkowitz; <emph type="italics"></emph>qui eùm in re litterariâ ſit laborioſiſſimus, amicos ſues <lb></lb>non ſinit eſſe otioſos. </s><s>Et noſtri ſæculi Phœnix P. Athanaſius Kircher, qui & ſuo <lb></lb>& aliorum nomine mihi calcar ad debat. </s><s>Scribit, inquiens, P. Merſennus opera <lb></lb>tua Pariſijs multùm placere: rogat, ut te incitem ad ſimilia plura luci danda. </s><lb></lb><s>Sed qvid inquies ad Medicum circul: quadratura? Et quid inquam ego ad <lb></lb>ſponſam calamiſtrata coma, & cincinni? Quæ verò huic tractatui de eſſe vi<lb></lb>dentur; ſupplebit liber de Motu & huius efficientibus cauſis Grauitate Leuitate & <lb></lb>Impulſu; qui proximè <expan abbr="librũ">librum</expan> de<emph.end type="italics"></emph.end> Arcu cœleſti, <emph type="italics"></emph>qui iam ſub prælo judat, ſequetur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PARS PRIMA.<emph.end type="center"></emph.end></s></p> <pb xlink:href="063/01/008.jpg"></pb> <p type="caption"> <s>IOANNES MARCVS MARCI PHIL: & MEDIC: DOCTOR <lb></lb><emph type="italics"></emph>et Profeſſor natus Landscronœ Hermundurorum in Boemta <lb></lb>anno 1595.13 Iunij.<emph.end type="italics"></emph.end></s></p> <figure id="id.063.01.008.1.jpg" xlink:href="063/01/008/1.jpg"></figure></section> </front> <body> <chap> <pb xlink:href="063/01/009.jpg"></pb> <p type="main"> <s><emph type="center"></emph>Reſolutio aliquot dubiorum exlibello <lb></lb>De <lb></lb><emph type="italics"></emph>Proportione motús.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>LIbellus de proportione motus <lb></lb>ante annos novem in lucem datus, ad plures <lb></lb>quidem peruenit opinione doctrinæ, & Geo<lb></lb>metriæ famâ claros: illorum de ſe judicia ac <lb></lb>cenſuram laturus. </s><s>Ex quorum tamen numero <lb></lb>unus & alter quod ſciam ſubmurmurauit. </s><s><expan abbr="Atq;">Atque</expan> huic quidem <lb></lb>minùs arriſit illa proportio inter <expan abbr="motumrectũ">motumrectum</expan> & inclinatum <lb></lb>ad prop. 13. </s><s>Quam ut diſturbaret, machinâ mirâ, & ingeni<lb></lb>osâ, ex affirmatiuâ negatiuam expreſſit. </s><s>Ita enim R. P. Bal<lb></lb>thaſar Conradus Soci: IESV. Philoſ. & Matheſeos Profeſſor, ad <lb></lb>R. P. Theodorum Moretum Soc: IESV, Matheſeos <expan abbr="quoq;">quoque</expan> tum <lb></lb>Profeſſorem, <expan abbr="atq;">atque</expan> Geometram percelebrem. </s><s><emph type="italics"></emph>Mitto, inquit, R. <lb></lb>Væ diſcurſum ſuper prop. 13. Excellentißimi Domini Doctoris Marci: <lb></lb>cuius propoſitionis contradictoria eſt hæc.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Motus per lineam perpendicularem & lineam inclinatam, quorum <lb></lb>terminos coniungit linea recta, perpendicularis ad lineam inclinatam, <lb></lb>non ſunt inter ſe æquales.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Sit eadem figura, quæ Doctoris; & intelligantur duo ſegmenta <lb></lb>Sphærica GHF. GIF inter ſe æqualia.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Dico non eſſe id, quod Author prop: 13 proponit: videlicet non per <lb></lb>uenturum globum D eodem tempore in plano inclinato BF, à puncto <lb></lb>B ad punctum F, quo tempore alius globus eidem æqualis ex codem<emph.end type="italics"></emph.end> <pb xlink:href="063/01/010.jpg"></pb> <arrow.to.target n="fig1"></arrow.to.target><lb></lb><emph type="italics"></emph>puncto B, ad A perneniret lapſu verticali. </s><s>Cùm enim illa duo ſeg<lb></lb>menta Sphærica GHF, GIF, habeant centrum grauitatis in lineà <lb></lb>GF: ſit〈que〉 F hypomochlium, æquiponderabunt: quare reliqua tantum <lb></lb>Sphæræ pars GKFI deorſum producet impulſum: Quare & im<lb></lb>pulſus motum ſibi æqualem per prop: 2. Doctoris. </s><s>Eſt autem ut pars <lb></lb>Sphæræ GKFI ad totam Sphæram, ita partis eiuſdem impulſus ad to<lb></lb>tius Sphæræ impulſum per propoſ: 2. in Archimede promoto: quare & mo<lb></lb>tus partis eiuſdem ad motum totius erit in eadem ratione. </s><s>Permutan<lb></lb>do ergo & velocitas partis ad velocitatem totius per Propoſ.<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>Doctor: <lb></lb>ergo et interuallum BF ad interuallum BA, uti pars Sphæræ GK <lb></lb>FI ad totam Sphæram per propſ: 7. eiuſdem.<emph.end type="italics"></emph.end></s></p> <figure id="id.063.01.010.1.jpg" xlink:href="063/01/010/1.jpg"></figure> <p type="main"> <s><emph type="italics"></emph>Sed pars GKFI non eſt ad totam Sphæram uti CD ad DF, quod <lb></lb>certum eſt: & patet ex hoc diſcurſu. </s><s>Fingatur enim mente recta H <lb></lb>D per verticalem GF diuiſa bifariam. tunc ſi eſſet ut CD ad FD Sim<lb></lb>pla ad duplam, ita reliqua magnitudo (ablatis duobus ſegmentis Sphæ<lb></lb>ricis illis dictis) ad totam: Eſſet etiam tota magnitudo dupla illius <lb></lb>partis GKFI: quod ad oculum falſum factâ figurâ apparebit. </s><s>Ergo <lb></lb>ne〈que〉 interuallum BF ad interuallum BA, uti CD ad DF, quod<emph.end type="italics"></emph.end> <pb xlink:href="063/01/011.jpg"></pb><emph type="italics"></emph>oportuit demonſtrare. </s><s>Motus ergo per lineam &c: Examinet R V. <lb></lb>hunc diſcurſum; & ſi putauerit, etiam Excell: Dno Doctori <lb></lb>oſtendat. </s><s>Reliquas ipſius propoſitiones per otium inſpiciam.<emph.end type="italics"></emph.end> Hæc ille <lb></lb>doctè ſanè ac modeſte. </s><s>Quæ priuſquàm ad incudem <lb></lb>reuocentur, placet non nihil Lucis addere illi propoſi<lb></lb>tioni 13. </s><s>Tum enim facilè diſpiciemus, an tela huc, an a<lb></lb>liò tendant: et an aliquam partem feriant, <expan abbr="demolianturq;">demolianturque</expan>? <lb></lb>an tota, ut aiunt, uiâ aberrent. </s><s>In illâ <expan abbr="itaq;">itaque</expan> propoſitione <lb></lb>aſſero: Si duo circuli æquales ex eodem principio motûs ſimul <lb></lb>ferantur: hic quidem verticali, ille verò motu inclinato, con<lb></lb>tinuò in eà ratione labi, ut ex quolibet puncto motûs vertica<lb></lb>lis, ducta linea recta ſecet perpendiculariter alterius motum. </s><lb></lb><s>Huius Apodixis hæc erant fundamenta. 1. ſpatia decurſa <lb></lb>eandem rationem ad ſe habere, quam impulſus eiuſdem cor<lb></lb>poris vel æqualis: ita nimirum, ut ſi moueri demus in tempo<lb></lb>re AB, per ſpatium CD; accipiat verò duplum, virtutis im<lb></lb>pulſiuæ, moturum ſit eodem tempore AB, per duplum ſpa<lb></lb>tium CD. </s><s>Eſt hæc propoſitio Arlis lib. 6. Phyſ. cap: 4. & lib: 1. <lb></lb>de Cælo cap: 6. & alibi. </s><s>Si inquit tanta grauitas per tantum in <lb></lb>hoc tempore mouetur; tanta & quod ſupereſt in minori mo<lb></lb>vebitur: Et rationem, quam grauitates habent, tempora è <lb></lb>conuerſo habebunt: Vt ſi dimidia grauitas in hoc, dupla in di<lb></lb>midio huius. </s><s>Vbi grauitas maior pro intenſiuà ſumi debet; <lb></lb>quæ idem ſubiectum perficit. </s><s>At verò ſi pars accedat æquè <lb></lb>grauis; tùm huius vi non intenditur motus. </s><s>Vnde ſi <expan abbr="vtraq;">vtraque</expan> <lb></lb>ſeorſim æquali celeritate ferebatur; <expan abbr="neq;">neque</expan>, ſi connectantur, <lb></lb>hæc illam trahet, aut impellet: quemadmodum ſi duo manibus <lb></lb>conſertis curſu inæqvali ferantur: velocior enim reſtantem <lb></lb>trahit & ad motum æquè velocem impellit. </s><s>At ſi grauitas illa <lb></lb>æqualis ſuo ſubiecto exui, & alteri inſeri detur; tum ſanè gra<lb></lb>uitas dupla dicetur ineſſe illi ſubiecto: & cum agat ſecundum ſe <pb xlink:href="063/01/012.jpg"></pb>totam, motum producet ſibi æqualem, hoc eſt duplum. </s><s>Jm<lb></lb>meritò hic aliqui turbantur, <expan abbr="hæſitantq;">hæſitantque</expan> quia inquiunt, licet <lb></lb><expan abbr="quandoq;">quandoque</expan> velocius feratur in eodem tempore per ſpatium du<lb></lb>plum, non tamen conſtare an illa virtus Locomotiua ſit du<lb></lb>pla, an in aliâ proportione. </s><s>Verùm hi naturam grauitatis & <lb></lb>Impulſus videntur ignorare, illam ceu ex atomis conflantes: <lb></lb>quæ proinde aliquo numero, aut magnitudine ſit menſurabi<lb></lb>lis. </s><s>At verò quis qualitates ſenſum latentes, & vix ab animo <lb></lb>perſpici valentes menſurabit? quin ipſam coulis ſubiectam al<lb></lb>bedinem quis duplam alteri dabit: Sicuti ergo illas qualita<lb></lb>tes non niſi ex effectu noſcim<emph type="sup"></emph>9<emph.end type="sup"></emph.end>; ita ex huius partitione in partes <lb></lb>analogas ſecamus: ut dupla ſit virtus, quæ effectum producit <lb></lb>duplum; impulſus ergo ſeu grauitas dicetur dupla, quæ mo<lb></lb>tum valet producere duplum. </s><s>Eſt autem de ratione motus <lb></lb>habere extenſionem, & in tempore fieri determinato: & ut <lb></lb>tanto magis ſit perfectus, quanto| minùs temporis inſumit. </s><lb></lb><s>Semiſſis ergo temporis, perfectionem dabit duplam. & quia in <lb></lb>altera ſemiſſe motum producit æqualem, perfectio dupla, eo<lb></lb>dem tempore mouebit per ſpatium duplum. </s><s>Confirmatur <lb></lb>ex ijs, quæ poſtea dicam ad quæſt. de cauſa inæqualis reflexio<lb></lb>nis: nimirum motum eſſe plagam continuatam in illo medio, <lb></lb>in quo fit motus: <expan abbr="atq;">atque</expan> impulſum à plagâ incipientem in aliam <lb></lb>plagam illi æqualem deſtinari: quâ conſecutâ motus termi<lb></lb>natur. </s><s>Cùm ergo Impulſus ſit æqualis plagæ, neceſſe illam <lb></lb>in motu continuatam plagam huic eſſe æqualem. & quia medi<lb></lb>um unius eſt rationis, <expan abbr="neq;">neque</expan> magis in una, quàm aliâ parte reſi<lb></lb>ſtit, erunt partes medij in eâ ratione, in quâ illarum plaga. </s><lb></lb><s>Medium ergo duplum abſumet plagam duplam. </s><s>At verò Pla<lb></lb>ga dupla non niſi ab impulſu æquali, id eſt duplo eſſe poteſt: <lb></lb>Impulſus ergo duplus per medium mouebit duplum. </s><s>De<lb></lb>inde cùm velocitas motûs proueniat à minori reſiſtentia me- <pb xlink:href="063/01/013.jpg"></pb>dij: acrem enim velociùs, quam aquam findit <expan abbr="idẽ">idem</expan> mobile: ſi mi<lb></lb>nuatur reſiſtentia medij, ut fiat ſub dupla prioris; Idem impul<lb></lb>ſus habebit velocitatem duplam. </s><s>At verò eadem eſt propor<lb></lb>tio, ſi manente reſiſtentiâ eiuſdem medij, augeatur Impulſus. </s><lb></lb><s>Igitur ſi impulſus rationem habeat duplam ad alium impul<lb></lb>ſum, mouebitur in eodem medio velocitate duplâ. </s><s>Et quia <lb></lb>velocitas maior in minori tempore tranſit idem ſpatium, velo<lb></lb>citas dupla in dimidio tempore tranſibit. </s><s>Quòd ſi necdum <lb></lb>perſuaſi in hac luce caligant, ſit ea poſtulatiloco. nam quæ ad <lb></lb>huius poſitionem ſequuntur, ſi firmo nexu, & <emph type="italics"></emph>ut linum lino<emph.end type="italics"></emph.end> co<lb></lb>hærent, de veritate ſuppoſiti non licebit dubitare: quandoqui<lb></lb>dem firmitas operis de ſubſtructionibus fidem facit. </s><s>Igitur <lb></lb>cùm eadem ſit ratio motûs, quæ grauitatis ſeu impulſus; erit <lb></lb>motus verticalis duratione æqualis motui inclinato; Si eo mo<lb></lb>do habeant ſpatia, quo illorum grauitates. </s><s>Oſtenſum verò <lb></lb>illa pro. 13. triangula FCD, ABF eſſe ſimilia, & in ratione ho<lb></lb>mologa ſuorum laterum. latus ergo FD ad DC, ut latus A <lb></lb>B ad AF. </s><s>Eſt autem FD menſura impulſus in lapſu verticali, <lb></lb>hoc eſt in AB. </s><s>CD verò menſura impulſus in BF. propterea <lb></lb>quód impulſus ſeu grauitas per poſit. 6<emph type="sup"></emph>am<emph.end type="sup"></emph.end> augetur in ratione <lb></lb>diſtantiæ centri à linea hypomochlij. </s><s>Concipitur enim cen<lb></lb>trum grauitatis in hypomochlio librari: cuius vectis linea per<lb></lb>pendicularis à centro productá Quæ ſi æqualis ſit radio, tota <lb></lb>grauitas prominet extra lineam hypomochlij: in plano verò in<lb></lb>clinato, quò magis inclinatur, eò propiùs accedit ad lineam <lb></lb>hypomochlij: & quò minor fit vectis, eò minùs gravitat. </s><s>Pro <lb></lb>cuius maiori declaratione, Notandum Comparationem inſti<lb></lb>tui grauitatis, non inter partes Circuli, quas linea hypomo<lb></lb>chlij bifariam ſecat: cùm non illarum, ſed centri ratione fiat <lb></lb>impulſus, per quartum Theorema huius: in quo omnium vir<lb></lb>tus collecta, in ſingulas ſe effundit. </s><s><expan abbr="Itaq;">Itaque</expan> fit ut pars nulla ſuo <pb xlink:href="063/01/014.jpg"></pb>motu, ſed motui Centri parallelo feratur: <expan abbr="ictusq;">ictusque</expan> non huius, <lb></lb>ſed vi centri accidant grauiores: verùm centrum grauitatis <lb></lb>ad ſe ipſum refertur, quatenus ex inæquali remotione à lineâ <lb></lb>hypomochlij inæqualiter ponderat. </s><s><expan abbr="Neq;">Neque</expan> enim percuſſio fit <lb></lb>per lineam verticalem ſeu hypomochlij; ſed eam, quæ duci<lb></lb>tur à centro grauitatis per contactum, per quartum Theor. </s><lb></lb><s>Vnde fit ut centrum grauitatis ſe ipſo utens ad ſe mouendum, <lb></lb>ſibi præponderet in eâ ratione, in quâ eſt vectis. </s><s>Cùm ergo in <lb></lb>lapſu verticali nihil occurrat centro, totum vectem grauitas <lb></lb>obtinet: in plano autem inclinato, linea verticalis ducta per <lb></lb>contactum inæqualiter hunc ſecat, pro ratione inclinationis. <lb></lb>et tum centrum grauitatis ſe ipſum veluti partitur in eam, quæ <lb></lb>mouet, & in eam quæ in Hypomochlio quieſcit partem. </s><s>Opor<lb></lb>tet enim concipere, quemadmodum ſi globus ab alio globo æ<lb></lb>quali ſit levandus. </s><s>Tum enim ſi <expan abbr="uterq;">uterque</expan> æqualiter abeſt à tru<lb></lb>tinâ, fit æquilibrium: retractione verò unius, eam rationem <lb></lb>habet grauitas huius ad grauitatem illius, quam interualla. </s></p> <p type="main"> <s>Obijcies. </s><s>Huic poſitioni aduerſari ea, quæ propoſ. 32. & 33 <lb></lb>ſunt dicta: vbi oſtendi Impulſum eo modo augeri, quo triangu<lb></lb>lum ſibi ſimile manens: <expan abbr="rationémq;">rationémque</expan> habere ſuorum tempo<lb></lb>rum, in quibus fiunt, duplicatam. </s><s>Quòd ſi ergo radius totus <lb></lb>FD ſit quadratum ab hypomochlio in duo quadrata CD. CF <lb></lb>diviſum, uti propoſitio illa vult; erit grauitas in DF ad gra<lb></lb>uitatem in CD, in ratione duplicatâ eius, quam habet ſinus to<lb></lb>tus ad ſinum complementi inclinationis. & quia motus ratio<lb></lb>nem habent, quam impulſus, per quartam poſitionem, erit mo<lb></lb>tus in AB ad motum in BF in ratione <expan abbr="quoq;">quoque</expan> duplicatâ. </s><s>Maior <lb></lb>ergo motus BF, quàm utidem tempus <expan abbr="vtrumq;">vtrumque</expan> metiatur. </s><lb></lb><s>Hanc obiectionem ut diluamus. </s><s>Aduerte ea, quæ in vecte li<lb></lb>brantur, duplicem habere impulſum, ſeu grauitatem: aliam <lb></lb>quidem in ordine ad mundi centrum; aliam verò in ordine ad <pb xlink:href="063/01/015.jpg"></pb>hypomochlium. </s><s>Differre enim à ſe conſtat ex eo, quòd hæc <lb></lb>augeri poteſt infinitè, nihilo auctâ illâ. </s><s><expan abbr="Neq;">Neque</expan> enim velociùs de<lb></lb>ſcendit vectis ob remotionem ponderis à lineâ hypomochlij: <lb></lb><expan abbr="neq;">neque</expan> ſi alià trutinâ explores in quouis ſitu, magis ponderabit. </s><lb></lb><s>Propterea quòd hic impulſus hypomochlium, non verò mun<lb></lb>di centrum reſpicit, quantumuis ab eadem grauitate oriatur. <lb></lb><expan abbr="Atq;">Atque</expan> hunc impulſum augeri in eà ratione, quam vectis obtinet, <lb></lb>demonſtrat Archimedes in lib. de æquiponderantibus. </s><s>Alio <lb></lb>modo Grauitas, ſeu impulſus in ordine ad motum expenditur <lb></lb>abſolutè, <expan abbr="abſq;">abſque</expan> ullo reſpectu ad hypomochlium: & tum <lb></lb>rationem quadrati habere dicimus; cuius latera ſint duratio <lb></lb>motûs. </s><s>Nam cùm in aliquo tempore produci ſit neceſſe, <expan abbr="atq;">atque</expan> <lb></lb>eo modo augeatur, quo triangulum ſibi ſimile manens, per po<lb></lb>ſit. quintam; habebit impulſus hic ad illum, rationem eius, <lb></lb>quam habent tempora, duplicatam. per propoſ. 12. </s></p> <p type="main"> <s>Aduerte ſecundo, duobus modis fieri contactum mobi<lb></lb>lis & plani: vno modo, cùm incidit plano: alio modo, cùm la<lb></lb>bitur per ipſum. </s><s><expan abbr="Neq;">Neque</expan> eadem ratio <expan abbr="utrobiq;">utrobique</expan>. </s><s>Nam cùm labi<lb></lb>tur, & labendo tangit planum, eodem modo videtur ſe habe<lb></lb>re ad hypomochlium, <expan abbr="eandémq;">eandémque</expan> diſtantiam obtinere centrum <lb></lb>grauitatis: Manet ergo ratio partis motæ ad quieſcentem, <expan abbr="quã">quam</expan> <lb></lb>linea hypomochlii à principio induxit. </s><s>At verò cùm incidit <lb></lb>eidem plano, plagam infert, & recipit: vnde reflecti contin<lb></lb>git. </s><s>Oſtenſum verò prop: 37-Plagam in aliquo tempore <lb></lb>fieri: à Plaga verò impulſum exſolui. quam ergo rationem <lb></lb>habet mora plagæ iam perfectæ ad aliam moram plagæ nec<lb></lb>dum perfectæ, candem habet impulſus totus ad illum duplica. <lb></lb>tum. </s><s>Igitur in caſu verticali, quia hypomochlium occurrit <lb></lb>centro, <expan abbr="neq;">neque</expan> percuſſioni cedit, plagam inducit <expan abbr="perfectã">perfectam</expan>, <expan abbr="totũq">totunq</expan>, <lb></lb>impulſum exſoluit. & cùm æqualem à percuſſo recipiat plaga, <lb></lb>eadem, quâ incidit, viâ retro agitur. </s><s>In occurſu autem plani <pb xlink:href="063/01/016.jpg"></pb>ad ictum inclinati, quia non per centrum grauitatis ſeu impul<lb></lb>ſus ſecatur à lineà hypomochlij; erit ratio Plagæ, quam habet <lb></lb>in hypomochlio quies. quæ tantò eſt minor, quantò velociùs <lb></lb>centrum grauitatis à plagâ ſe abducit. </s><s>Quòd ſi ergo DF ſit <lb></lb>mora plagæ perfectæ, <expan abbr="atq;">atque</expan> huius impulſus quadratum DF; erit <lb></lb>DC tempus uelocitatis motus, & huius quadratum impulſus: <lb></lb>reliquum ergo quadratum FC à percuſſione ſeu plagâ, impul<lb></lb>ſum dabit à reliquo tempore menſuratum. propterea quod <lb></lb>quadratum FD ſit æquale duobus quadratis CD. CF: ac pro<lb></lb>inde mora percuſſionis complementum CD ad ſinum <lb></lb>totum. </s><s>Eodem modo ſi plagam metiamur fientem morâ æ<lb></lb>quali C Flateri eiuſdem quadrati, erit huius complementum <lb></lb>mora impulſus reliqui. </s><s><expan abbr="Atq;">Atque</expan> ex his Soluitur illa dubitatio, <lb></lb>quam ob rem prop: 13. impulſu & grauitate, <expan abbr="horumq;">horumque</expan> diuiſi<lb></lb>one utamur ceu lineâ rectâ, aut parallelogrammo: propoſi<lb></lb>tione autem 32. & 33 motum comparemus ut quadrata. <lb></lb>quia nimirum hic impulſum ut fientem, ac proinde iuxta mo<lb></lb>dum <expan abbr="menſuramq;">menſuramque</expan> plagæ expendimus. </s><s>Non enim à percuſſi<lb></lb>one idem eſt impulſus: ſed illa portio, quæ percuſſit, illi de<lb></lb>cedit: Alius verò huic æqualis & oppoſitus à percuſſo rege<lb></lb>neratur: & cum reliquo impulſu in ordine ad motum medium <lb></lb>miſcetur. </s><s>Neceſſe ergo inter ſe conferri, ut illorum tempo<lb></lb>rum, in quibus producuntur, quadrata. </s><s>At uerò prop: 13. </s><lb></lb><s>Impulſum ſeu grauitatem in facto eſſe, & à centro grauitatis, in <lb></lb>quo eſt collecta, ſui replicatione in vectem æqualiter fuſam: <lb></lb>quam fecat bifariam linea hypomochlij in partem motam & <lb></lb>quieſcentem. </s><s>Hæc autem nullam inducit plagam: verùm <lb></lb>continuò in hypomochlio quieſcit, & in ordine ad motum pro <lb></lb>nullâ habetur. </s><s>Vnde augmenta velocitatis motus fiunt <expan abbr="absq;">absque</expan> <lb></lb>ullo ad eam reſpectu. </s><s><expan abbr="Neq;">Neque</expan> enim motu <expan abbr="inualeſcẽte">inualeſcente</expan> augetur illa <lb></lb>grauitas in hypomochlio quieſcens: quòd linea hypomochlij <pb xlink:href="063/01/017.jpg"></pb>non hunc, ſed huius principium partiatur. </s><s>Incrementa enim <lb></lb>motûs <expan abbr="atq;">atque</expan> impulſûs per lineam fiunt parallelam illi plano, in <lb></lb>quo mouetur. </s><s>Quia ergo grauitas mouens impulſum produ<lb></lb>cit continue maiorem: non quem ſibi grauitas collegit, ſed <lb></lb>quem natiuum habet ad grauitatem quieſcentem conferri de<lb></lb>bet: vt eadem ſit proportio vectis, quæ partium gravitatis; <lb></lb>Quod non niſi in principio motûs contingit. </s><s>Augetur ergo <lb></lb>gravitas quieſcens eiuſdem mobilis in eá ratione, quam habet <lb></lb>reliquum ſegmentum vectis, ad diſtantiam centri grauitatis à <lb></lb>lineâ hypomochlij. </s></p> <p type="main"> <s>His iam definitis: videamus quam vim habeat ille diſcurſus: <lb></lb>& an contrariâ illatione noſtram poſitionem conuellat. </s><s>Cùm <lb></lb><expan abbr="itaq;">itaque</expan> aſſumit ſegmenta æqvalia GHF. GIF. <emph type="italics"></emph>Iſorrhopa:<emph.end type="italics"></emph.end> propte<lb></lb>rea, quòd centrum grauitatis habeant in lineà hypomochlij F <lb></lb>G, ac proinde exceſſum ſeu præpondium ineſſe reliquo <expan abbr="ſegmẽ-to">ſegmen<lb></lb>to</expan> GKFI, <expan abbr="motúmq;">motúmque</expan> deorſum huius ratione fieri; errat primo <lb></lb>quòd ſupponat eadem ratione moueri partes, <expan abbr="eundémq;">eundémque</expan> dare <lb></lb>& recipere impulſum in toto exiſtentes, & dum per ſe mo<lb></lb>uentur: quod à veritate eſt alienum. </s><s>Mouentur enim partes <lb></lb>virtute ſui centri; <expan abbr="neq;">neque</expan> uno modo omnes, <expan abbr="neq;">neque</expan> ſimiliter. </s><s>Nam <lb></lb>cùm per lineas ferantur motui centri parallelas, remotioribus <lb></lb>à centro plus ineſt violentiæ: <expan abbr="atq;">atque</expan> <expan abbr="unaquæq;">unaquæque</expan> grauiùs percutit <lb></lb>in toto, quàm ſi per ſe moveretur. </s><s>Licet ergo illa ſegmenta <lb></lb>ſint æqualia & <emph type="italics"></emph>Iſorrhopa,<emph.end type="italics"></emph.end> non <expan abbr="tamẽ">tamen</expan> ſequitur in toto eandem uim <lb></lb>obtinere: cùm à centro grauitatis mutari poſſit, ſicuti habitu<lb></lb>do ad vectem, ita <expan abbr="quoq;">quoque</expan> ratio impulſus. </s><s>Secundò decipi<lb></lb>tur, quòd comparationem inſtitui velit inter partes mobilis <lb></lb>circa hypomochlium ſitas, nullâ habitâ ratione ſitûs, & diſtan<lb></lb>tiæ ab hypomochlio: quod magnum eſt erratum. </s><s><expan abbr="Neq;">Neque</expan> enim <lb></lb>ſegmentum GIF, ſitu permutato C in I, & contrà, æquipon<lb></lb>derabit ſegmento GHF, aut ſibi ipſi: quomodo ergo reliquum <pb xlink:href="063/01/018.jpg"></pb>ſegmentum GKFI aſſumit in eâ tatione grauitare, in qua eſt <lb></lb>pars magnitudinis| Sphæræ? cùm & partium magnitudo ob <lb></lb>curvitatem circuli, & ſitus continuò mutentur. </s><s>Propoſ. <expan abbr="autẽ">autem</expan> <lb></lb>illa ſecunda in Archimede promoto, <emph type="italics"></emph>impulſum partis in eà ratione <lb></lb>eſſe ad impulſum totius, in quà ipſa eſt pars magnitudinis<emph.end type="italics"></emph.end>, vera eſt, ſi <lb></lb>non ratione ſitûs mutetur illa habitudo. </s><s>Sed quidquid ſit de <lb></lb>hac proportione partium ad ſe, quam in circulo ignoramus, <lb></lb>nihil huc facit: ubi centrum grauitatis in eadem magnitudi<lb></lb>ne expendimus, & ad ſe ipſum comparamus: quatenus in di<lb></lb>uerſo ſitu & remotione ab hypomochlio inæqualiter ponderat. </s><lb></lb><s>Deinde verò efto demus Impulſum diuidi in eâ ratione, in quâ <lb></lb>magnitudo; vt quæ pars ſit molis, eadem ſit grauitatis ſeu im<lb></lb>pulſus: an propterea rectè infert in eadem ratione fieri mo<lb></lb>tum? Vt ſi ſegmentum GHFI ſit duplum ſegmenti GHF, <lb></lb>ac proinde grauitatem habeat duplam, duplo velociùs moue<lb></lb>ri ſit neceſſe: id enim quæſtione de in æquali ponderum la<lb></lb>pſu negamus. </s><s>Malè autem propoſ: 10. citat in contrarium: quæ <lb></lb>ponit impuiſum producere motum ſibi æqualem. hoc enim de <lb></lb>intenſione, non verò extenſione grauitatis ſeu impulſûs eſt in<lb></lb>telligendum: ita nimirum ſi idem mobile accipiat impulſum <lb></lb>duplum. </s><s>At verò cùm acceſſione partis nouæ augetur impul<lb></lb>ſus, nihilo plus virium ad ſe mouendum <expan abbr="utraq;">utraque</expan> habet. </s><s>Quòd <lb></lb>ſi ex toto mobili grauitas ſeu impulſus colligi poſſit in partem <lb></lb>V.G tertiam; tum verò pars illa celeritate triplà moueretu. <lb></lb>Nam ſicuti impulſus magnus in magnam molem receptus ex<lb></lb>tenuatur: ita in paruam molem contractus intenditur. </s><s>Atq: <lb></lb>ex his patet manifeſtè, in conuellendâ illâ prop. 13. & falſa aſ<lb></lb>ſumi, & ex malè aſſumptis vitiosè concludi. </s></p> <p type="main"> <s>Alter fuit R. P. Joannes Ciermans: qui in anno poſitio<lb></lb>num Mathematicarum, hebdomade tertiâ, menſis Maij, ita in<lb></lb>quit. </s><s><emph type="italics"></emph>Putat Ioannes Marcus Marciſeſe globum ſummà violentià<emph.end type="italics"></emph.end> <pb xlink:href="063/01/019.jpg"></pb><emph type="italics"></emph>vel è tormento bellico excuſſum, medio in itinere detinere poſſe immo<lb></lb>tum: it a ut ne quidem refiliat. id〈qué〉 ſi ſolùm illi alterum globum eius<lb></lb>dem ponderis tangendum ſiſtat. </s><s>Sed rectè ille impetûs naturam aſſe<lb></lb>quutus non eſt<emph.end type="italics"></emph.end> Sic ille. </s><s>Verùm hac auſterâ cenſurâ ſatis oſten<lb></lb>dit, nimiùm ſuis ſpeculationibus fiſum, experientiam hic negle<lb></lb>xiſſe, quam etiam pueri globulis ludentes norunt. </s><s>Quòd ſi <lb></lb>aliquando experiri lubebit, facilè mihi perſuadeo, virum æqui<lb></lb>ac veri tenacem, non mitiorem erga ſuas de impulſu opinio<lb></lb>nes cenſorem futurum. </s></p> <p type="main"> <s>Priuſquam verò finem faciam, placet aliquid lucis addere <lb></lb>his, quæ de oſcillationibus penduli ibidem ſunt dicta. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>An Pendulum æquali tempore recurr at, per arcus maiores <lb></lb>& minores.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In Prop: 24-aſſumitur motus ex B in D æqualis duratione <lb></lb>motui ex D in F: propterea quòd BD ad DF ſit ut AB ad CD: <lb></lb> <arrow.to.target n="fig2"></arrow.to.target><lb></lb>& ut AB ad CD, hoc eſt vt AW ad WR, ita per prop. 22. vis <lb></lb>mouens in B ad uim mouentem in D. eſt enim radius AB æ- <pb xlink:href="063/01/020.jpg"></pb>qualis radio AW, & ſinus CD eiuſdem arcus DW æqualis ſi<lb></lb>nui WR. </s><s>Verùm licet in principio illorum arcuum ita res <lb></lb>habeat, in lapſu tamen ob nouas inclinationes, continuò mu<lb></lb>tatur illa proportio. </s><s>Vnde incrementa velocitatis, cùm ex a<lb></lb>liâ <expan abbr="atq;">atque</expan> aliâ radice naſcantur, non eadem ratione fiunt. </s><lb></lb><s>Nam ſinus AB ad ſinum proximum minorem rationem ha<lb></lb>bet, quàm CD ad ſinum æquè proximum: plus igitur hic <lb></lb>quàm ibi decedit virtuti motrici. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> fiat BD ad <lb></lb>DF, ut AB ad CD; hoc eſt, vis movens in B ad uim mouentem <lb></lb>in D, non eodem tempore agitabitur ex D in F, quo ex B in D; <lb></lb>verùm per ſpatium minus, quàm ſit DF. </s><s>Nihil tamen officit <lb></lb>hoc noſtræ demonſtrationi: quin imò uim affert maiorem. </s><lb></lb><s>Sit enim arcus ille minor, per quem ex D fit motus D b: & <lb></lb>ducatur ſinus ab. </s><s>Quia <expan abbr="itaq;">itaque</expan> ſinus ab eſt maior ſinu EF, mi<lb></lb>nor verò arcu reſiduo b W; habebit maiorem rationem ad ar<lb></lb>cum minorem D b, quam recta EF minor ad arcum maiorem <lb></lb>DF. </s><s>Igitur per 4. lemma, arcus D b eſt multò minor ſinu ab, ac <lb></lb>proinde arcu reliquo b W. </s><s>Ex quo cùm pars proportionalis <lb></lb>abſcindi debeat continuò minor, concludam pendulum non <lb></lb>priùs ex D quàm ex B attingere W. </s><s><expan abbr="Eadémq;">Eadémque</expan> ratione F non an<lb></lb>te D, & H non ante F, ac proinde <expan abbr="neq;">neque</expan> H ante D vel B præcur<lb></lb>currere in W. </s></p> <figure id="id.063.01.020.1.jpg" xlink:href="063/01/020/1.jpg"></figure> <p type="main"> <s>Quod ſi dicas, pendulum ex maiori interuallo præcurrere: <lb></lb>ſequitur plura pendula eiuſdem longitudinis, <expan abbr="atq;">atque</expan> in eodem <lb></lb>Circulo, ex inæqualibus ſpatijs ſimul recurrendo ſe percutere <lb></lb>in motu: quod nemo experitur. </s><s>Ne tamen ullus dubitationi <lb></lb>locus ſuperſit, placet aliâ viâ magis planâ idem demonſtrare. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Lapſus gravium in plano inclinato, eſt æqualis duratione<emph.end type="italics"></emph.end> <pb xlink:href="063/01/021.jpg"></pb><emph type="italics"></emph>lapſui interciſo ab alio plano: quorum terminos connect it <lb></lb>rect a line a, perpendicularis ad motum interciſum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <figure id="id.063.01.021.1.jpg" xlink:href="063/01/021/1.jpg"></figure> <p type="main"> <s>Moueatur primùm ex A in B per planum AG: ex B verò per <lb></lb>planum BF. </s><s>Dico motum in BF eſſe æqualem duratione mo<lb></lb>tui in BG, quorum terminos connectit recta GF perpendicula<lb></lb>ris ad BF. </s><s>Nam impulſus in B eſt maior gravitate, per prop. <lb></lb>11. <expan abbr="motúmq;">motúmque</expan> producit parallelum plano BG, per propoſitio<lb></lb>nem tertiam: propterea quòd â gravitate proveniat extra hy<lb></lb>pomochlium conſtitutâ. </s><s>Igitur cùm aliud planum occurrit <pb xlink:href="063/01/022.jpg"></pb>quia rationem habet hypomochlij; ſecabitur impulſus eâ rati<lb></lb>one, quâ grauitas verticalis ſecatur à plano inclinato, in par<lb></lb>tem motam & quieſcentem: ac proinde per propoſitionem <lb></lb>11. motus interciſus à plano, erit| æqualis duratione reliquo <lb></lb>motui: qvorum terminos connectit linea recta, perpendicu<lb></lb>laris ad motum interciſum. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Si in ſegmento Circuli ducantur duæ chordæ, angulus <lb></lb>ab his contentus, erit complementum dimidij anguli eiuſ<lb></lb>dem arcus ad duos rectos.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In ſegmento BF ducantur duæ chordæ BC. CF: dico angu<lb></lb>lum BCF ab his contentum eſſe complementum dimidij an<lb></lb>guli BOF ad duos rectos. </s><s>Nam duo anguli OFC. OCF ſunt <lb></lb>complementum anguli FOC: duo verò anguli OCB, OBC <lb></lb>complementum anguli COB. </s><s>Cùm igitur FCB ſit ſemiſſis <lb></lb>illorum angulorum; erit complementum dimidij anguli FOB. </s></p> <p type="main"> <s><emph type="center"></emph>Corollarium.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sequitur angulum externum FCT eſſe æqualem ſemiſſi an<lb></lb>guli FOB: propterea quòd <expan abbr="utriuſq;">utriuſque</expan> complementum ad duos <lb></lb>rectos ſit angulus FCB. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Lapſus grauium in quædrante Circuli, per duas chordas <lb></lb>æquatur lapſui per unæm chordam.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Secetur primùm AF quadrans circuli æqualiter in B: & <pb xlink:href="063/01/023.jpg"></pb> <arrow.to.target n="fig3"></arrow.to.target><lb></lb>ducantur chordæ AF AB. </s><s>BF: dico lapſum per duas chordas <lb></lb>AB. BF eſſe æqualem lapſui per chordam AF. </s><s>Producatur e<lb></lb>nim AB in G: & ſit AG æqualis chordæ parallelæ FL: Ex F <lb></lb>autem excitetur linea perpendicularis ad BF: dico hanc pro<lb></lb>ductam ſecare AG in G. </s><s>Quia ením arcus FB eſt æqualis ar<lb></lb>cui AL, ſubtendet chorda LF, hoc eſt illi æqualis AG grad: 135 <lb></lb>corda vero AB grad. 45. </s><s>Auferatur AB partium 76, 3668 ex <lb></lb>AG partium 18477590: <expan abbr="atq;">atque</expan> reſidua BG erit partium <lb></lb>10823922. </s><s>Et quia per Lemma huius angulus FBG eſt <pb xlink:href="063/01/024.jpg"></pb>grad. 45. ſemiſſis nimirum anguli AOF: Si ad huius logarit<lb></lb>mum addatur logaritmus lateris BF, erit aggregatum logarit<lb></lb>mus lateris FG, ſeu BF partium 7653668. </s><s>Quot nimirum <lb></lb>partium erat quoq, chorda AB, hoc eſt illi æqualis BF. </s><s>Quòd <lb></lb>ſi <expan abbr="itaq;">itaque</expan> ducatur ex G termino motûs linea perpendicularis ad <lb></lb>BF, ſecabit eandem in puncto F: ac proinde motus ex B in G <lb></lb>eſt æqualis duratione motui ex B in F per prim. </s><s>Theorema <lb></lb>huius. <expan abbr="additóq;">additóque</expan> motu communi ex A in B, lapſus per duas chor<lb></lb>das AB. BF æquatur lapſui per chordam AF: qui per prop. 15. <lb></lb>erat æqualis duratione lapſui per chordam LF ſeu AG. </s></p> <figure id="id.063.01.024.1.jpg" xlink:href="063/01/024/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>ALITER.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ducatur ex F perpendicularis ad BF: dico hanc productam <lb></lb>ſecare BG. in G. quod ſi non; ſecet ſi fieri poteſt, in alio pun. <lb></lb>cto VG: X vel Z. </s><s>Et quia angulus externus NOL eſt grad: <lb></lb>45. erit angulus OLF internus grad: 22. prim: 30. & angu<lb></lb>lus OLA grad. 67. prim: 30: propterea quod LOA ex hy<lb></lb>potheſi ſit grad: 45: <expan abbr="Ablatoq;">Ablatoque</expan> OLF ex OLA, reſiduus FLA, <lb></lb>hoc eſt illi æqualis FGB grad: 45, ob parallelas nimirum & <lb></lb>æquales FLGA. </s><s>Cùm <expan abbr="itaq;">itaque</expan> in triangulo FBG rectus ſit an<lb></lb>gulus ZFB, & angulus FBG per lemma huius grad. 45: erit <lb></lb><expan abbr="quoq;">quoque</expan> angulus FZB grad 45, ac proinde æqualis angulo FG <lb></lb>B, externus interno: quod eſt abſurdum. </s><s><expan abbr="Atq;">Atque</expan> ea<lb></lb>dem ratione probabitur linea AG non ſecari à perpendiculari <lb></lb>XF. </s><s>Aſſumatur rurſum arcus AC grad 67; & CF grad 23. pro<lb></lb>ducatur autem AC in P ſumptâ AP æquali chordæ perallelæ F <lb></lb>M. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> in F excitetur linea perpendicularis ad FC: <lb></lb>dico protractam ſecare AP in P. </s><s>Quòd ſi non; ſecet, ſi fieri <lb></lb>poteſt, in alio puncto V. G: I. </s><s>Et quia angulus FCI per lemma <lb></lb>huius, eſtgrad 45 erit <expan abbr="quoq;">quoque</expan> angulus FIC grad 35 Exæquatur <lb></lb>autem angulus FMA angulo FPA ob lineas parallelas, & æqua- <pb xlink:href="063/01/025.jpg"></pb>les FM, PA. </s><s>Cùm <expan abbr="itaq;">itaque</expan> angulus OMF ſit grad. 33. prim. 30. <lb></lb>ſemiſſis nimirum anguli externi NOM grad. 67: & angulus <lb></lb>OMA grad: 78. prim: 30; quòd æquales ſint arcus AM. FC: <lb></lb>ablato angulo OMF ex OMA, erit angulus reliquus FMA, <lb></lb>hoc eſt illi æqualis FPA grad: 45. </s><s>Cùm <expan abbr="itaq;">itaque</expan> angulus FIC ſit <lb></lb><expan abbr="quoq;">quoque</expan> oſtenſus grad. 45, erit angulus FIC externus æqualis <lb></lb>angulo interno FPI: quod eſt abſurdum. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Lapſus grauium in ſegmento <lb></lb>Circuli minore, quàm grad: 90. eſt velocior per duas chordas, quàm per <lb></lb>unam chordam.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moueatur graue ex B in F per arcum grad: 45. </s><s>Dico veloci<lb></lb>ùs moueri per duas chordas BC. CF, quàm per unam chordam <lb></lb>BF. </s><s>Supponatur BC æqualis CF: & ducatur FQ parallela BC: <lb></lb>in productâ verò BC ſumatur BT æqualis <expan abbr="Fq.">Fque</expan> erit <expan abbr="itaq;">itaque</expan> BT <lb></lb>partium 11111400, & BC partium 3901806. </s><s>Quâ ablatâ ex <lb></lb>BT manet CT partium 7209594. </s><s>Adde Logaritmum huius <lb></lb>logaritmo anguli CTH grad. 67. prim. 30; qui per lemma eſt <lb></lb>complementum anguli FCT grad: 22. prim. 30. <expan abbr="eritq;">eritque</expan> aggre<lb></lb>gatum logaritmus lateris CH partium 6659688. </s><s>Eſt autem <lb></lb>CH maius latere BC, ſeu CF partium 3901806. </s><s>Cùm <expan abbr="itaq;">itaque</expan>, <lb></lb>motus ex C in H ſit æqualis duratione motui ex C in T, per pri: <lb></lb>theorema huius; erit mot<emph type="sup"></emph>9<emph.end type="sup"></emph.end> in CF minor duratione motu in CH: <lb></lb>additoque communi motu in BC, motus in BC, CF minor du<lb></lb>ratione motu in BT ſeu <expan abbr="Fq.">Fque</expan> hoc eſt per prop. 15. illi æquali <lb></lb>motu in BF. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA IV.<emph.end type="center"></emph.end></s></p> <pb xlink:href="063/01/026.jpg"></pb> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Lapſus grauium in eodem ſegmento Circuli per plures <lb></lb>chordas eſt velocior.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moueatur graue ex Q in F: Dico velociùs labi per chordas Q <lb></lb>B. BC. CF, quàm per chordas QB. BF. </s><s>Quia enim velociùs de<lb></lb>ſcendit per duas chordas BC. CF, quàm| per chordam BF per <lb></lb>Theorema tertium: addito motu communi QB, erit velocior <lb></lb>lapſus per QB. BC. CF, quàm per QB. BF. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Pendulum æquali tempore mouetur per arcum Circuli & <lb></lb>chordam eidem ſubtenſam.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moveatur pendulum TC ex C in B: Dico æquali tempore la<lb></lb>bi per arcum CEB, & chordam CB. </s><s>Concipiantur enim per ſin<lb></lb>gula puncta CGHIK eiuſdem arcus CEB duci tangentes, & <lb></lb>chordæ his parallelæ BL. BM.BN. BO & c. </s><s>Quia <expan abbr="itaq;">itaque</expan> ex C la<lb></lb>bendo in ſingula momenta mutat inclinationem, quam indu<lb></lb>cunt lineæ tangentes; erit ratio motûs in his homologa motui <lb></lb>per chordas parallelas. </s><s>Vt ſi labi incipiat per tangentem CD, <lb></lb>interuallum motûs in hac erit æquale motui per chordam pa<lb></lb>rallelam AB. </s><s>Nullus autem fit motus in CD, verùm immedi<lb></lb>atè transfertur in alias tangentes. </s><s>Simili modo in GHIK ex <lb></lb>illâ obliquatione contrahetur motus, inſpatia æqualia chordis <lb></lb>parallelis BL. BM. BN, BO: in EPQRS verò æquatur chor<lb></lb>dis BC. BG. BH &c. quæ quidem chordæ ſubten dunt duplum <lb></lb>illius arcûs, cuiús tangens eſt parallela. </s><s>Eſt enim CEB duplum <lb></lb>arcùs ESB. </s><s><expan abbr="Atq;">Atque</expan> hæc ratio arcûs dupli, continuatur <expan abbr="uſq;">uſque</expan> ad <lb></lb>tangentem BV. quam ubi attigit pendulum ex C, attingit <expan abbr="quoq;">quoque</expan> <pb xlink:href="063/01/027.jpg"></pb> <arrow.to.target n="fig4"></arrow.to.target><lb></lb>æquale pondus lapſu verticali ex A. </s><s>Propterea quòd tangenti <lb></lb>BV nulla in circulo reſpondet ex B ducta chorda parallela. </s><lb></lb><s>Motus igitur ex C, per arcum CEB eſt æqualis duratione mo<lb></lb>tui per chordam AB, hoc eſt per theor 15. motui per chordam <lb></lb>CB. </s></p> <figure id="id.063.01.027.1.jpg" xlink:href="063/01/027/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA. VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Pendulum ex quolihet puncto circuli æquali tempore recur<lb></lb>rit in ſuam ſtationem,<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Quia enim lapſus per arcum CEB eſt æqualis lapſui per <lb></lb>chordam CB & lapſus per arcum ESB æquatur lapſui per chor<lb></lb>dam EB per 5 theorema huius. </s><s>Sunt autem lapſus per chordam <lb></lb>CB & EB inter ſe æquales duratione per prop: 15 erit <expan abbr="quoq;">quoque</expan> la<lb></lb>pſus per arcum CEB æqualis duratione lapſui per arcum ESB. <pb xlink:href="063/01/028.jpg"></pb>Igitur pendulum TC ex C & E æquali tempore recurrit inſu<lb></lb>am ſtationem TB. </s></p> <p type="main"> <s><emph type="italics"></emph>Obijcies. </s><s>Lapſus grauium per plures chordas eſt velocior per 4. theo<lb></lb>rema. </s><s>Cùm ita〈que〉 in circuli curvaturâ ſint chordæ poteſtate infinitæ; <lb></lb>erit velocior lapſus per arcum, quàm per quotcun〈que〉 numero chordas.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Videtur hæc ratio mouiſſe Galilæum, ut in lib. de Syſtemate <lb></lb>mundi motum per arcus circuli poſuerit velociorem motu per <lb></lb>illorum chordas. </s></p> <p type="main"> <s><emph type="italics"></emph>His, inquit, adde mirabile aliud, ſcilicet quòd motus cadentium facti <lb></lb>per arcus quadrantis AB fiant breuioribus temporibus, quàm illi, qui <lb></lb>per chordas eorundem arcuum fiunt. </s><s>Et paucis interiectis, mobile, in<lb></lb>quit, diſcedens à puncto A minori tempore perueniet ad B, currendo per <lb></lb>duas chordas AD. DB, quàm per ſolam chordam AB. </s><s>Sed breuiſsimum <lb></lb>omnium tempus fuerit, ſi deciderit per arcum ADB.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Verùm diſſoluitur hæc obiectio, quòd motus per plures chor<lb></lb>das interciſus, <expan abbr="atq;">atque</expan> huius exceſſ<emph type="sup"></emph>9<emph.end type="sup"></emph.end> determinetur per lineas, quæ <lb></lb>à termino motús per unam chordam, cadunt perpendiculari<lb></lb>ter ad alias, per primum theorem: & quò plures fuerint chor<lb></lb>dæ eò exceſſus penultimæ erit maior. </s><s>At verò in defluxu cir<lb></lb>culari, quiâ nullum interuallum inter proximas tangentes, ſeu <lb></lb>chordas illarum parallelas; <expan abbr="neq;">neque</expan> ulla cadit perpendicularis. </s><lb></lb><s>Vndè ſi ex quolibet puncto refluxûs labi in cipiat per chordam, <lb></lb>erit æqualis duratione reſiduo lapſui, qui cadit per chordam <lb></lb>verticalem. </s></p> <p type="main"> <s><emph type="center"></emph>Ad Propoſitionem Vigeſimam.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De causâ decrementi oſcillationum, & an æquales ſint <lb></lb>duratione.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cauſa decrementi oſcillationum non eſt illa, quam attuli ad <lb></lb>finem prop. 20. ſicuti enim gravitas ſe habet ad illas inclinatio- <pb xlink:href="063/01/029.jpg"></pb>ones in excurſu, ita <expan abbr="quoq;">quoque</expan> in recurſu; vnde non magis decre<lb></lb>ſcit impulſus, quàm priùs augebatur. </s><s>Verùm cauſa huius de<lb></lb>crementi eſt plaga, quam infert pendulum in lapſu. </s><s>Cùm <lb></lb>enim hæc per ea, quæ habentur ad finem prop: 27. minuat <lb></lb>impulſum; excurſus à ſtatione neceſſariò fit minor recurſu. </s><lb></lb><s>Et ſi quidem pendulum refluat per medium magis denſum; <lb></lb>quia plaga maior plus adimit de impulſu, excurſus erunt mi<lb></lb>nores: uti manifeſtum in oſcillationibus in aquâ factis. </s><s>Quæ<lb></lb>quidem in vacuo, ſi fieri admittamus, quia nullam inducunt <lb></lb>plagam, eſſent interminabiles. </s></p> <p type="main"> <s><emph type="italics"></emph>Dices. </s><s>St plaga minuit impulſum; cùm inæquales ſint plagæ, erit quo〈que〉 <lb></lb>inæquale decrementum: Non igitur excurſus interſe, ac proinde ne〈que〉 <lb></lb>oſcillationes erunt pares duratione. An prorſus æquales ſint, videtur du<lb></lb>bius Galilæus. </s><s>In lib: enim de ſyſtemate mundi pagina 444. alterum in<lb></lb>quit ſingulare profectò miraculoſum eſt, quòd idem pendulum vibrati<lb></lb>ones ſuas eâdem frequentiâ, aut minimùm, & inſenſibiliter quaſi diffe<lb></lb>rente faciat: ſiue illæ fiant per arcus maximos, ſiue minimos eiuſdem <lb></lb>circumferentiæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Dico nihilominus oſcillationes omnes, quæ per arcus fiunt <lb></lb>eiuſdem circuli, eſſe æquales duratione. </s><s>Cuius ratio eſt, quòd <lb></lb>menſura plagæ ſit interuallum ſeu arcus, per quem pendulum <lb></lb>recurrit. </s><s>Igitur quemadmodum ſe habent arcus ad ſe, ita <lb></lb><expan abbr="quoq;">quoque</expan> decrementum impulſûs. et quia impulſus in recurſu col<lb></lb>lecti candem <expan abbr="quoq;">quoque</expan> rationem habent, quam arcus per prop <lb></lb>18. & 30: erunt <expan abbr="quoq;">quoque</expan> impulſus reliqui à plagâ in eadem ratio<lb></lb>ne: ac proinde excurſus & inter ſe, & cum recurſibus æquales <lb></lb>duratione. </s><s>Quia verò per arcus minores minor plaga induci<lb></lb>tur: hinc eſt quòd differentia inter excurſum & recurſum con<lb></lb>tinuò decreſcit. </s><s>Vnde ratio redditur tam numeroſarum o<lb></lb>ſcillationum: quæ etiam pro tatione circuli maioris, quem <lb></lb>pendulum deſeribit, augentur. </s><s>Supponamus <expan abbr="itaq;">itaque</expan> grauitatem <pb xlink:href="063/01/030.jpg"></pb>penduli ad grauitatem aëris eſſe in centuplâ ratione VG: ut <lb></lb>colligi videtur ex Ariſt. </s><s>Et quia impulſus in recurſu collectus <lb></lb>æquatur duplo eiuſdem arcus, per prop: 18. erit plagæ pars 200 <lb></lb>totius impulſus. deficiet ergò in excurſu pars <expan abbr="quoq;">quoque</expan> 200 <lb></lb>illius arcûs, quem pendulum deſcribit in recurſu. </s><lb></lb><s>Vndè vice verſâ ex notâ differentiâ inter ex<lb></lb>curſum & recurſum unius oſcillationis, <lb></lb>habetur nota grauitas aëris ſeu <lb></lb>medij. </s></p> <figure id="id.063.01.030.1.jpg" xlink:href="063/01/030/1.jpg"></figure> <pb xlink:href="063/01/031.jpg"></pb> <p type="main"> <s><emph type="center"></emph>SECVNDA PARS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Latera motùs figuræ ſunt lineæ parallelæ motùi centri gra<lb></lb>uit atis: quas deſcribunt in motu figuræ puncta remotiſsi<lb></lb>ma à lineâ motùs centri.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>VT ſi moueatur Figura ABCD ad motum centri grauitatis <lb></lb>FH: erunt lineæ AE. CG eidem parallelæ, latera motus <lb></lb>figuræ: quas deſeribunt AC puncta remotiſſima à lineâ FH <lb></lb>motûs centri. </s></p> <figure id="id.063.01.031.1.jpg" xlink:href="063/01/031/1.jpg"></figure> <pb xlink:href="063/01/032.jpg"></pb> <p type="main"> <s><emph type="center"></emph>DEFINITIO II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Semidiameter figuræ motùs eſt line a rect a, â centro grauita<lb></lb>tis ad alterutrum latus figuræ motús perpendiculariter <lb></lb>ducta.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In eadem figura ſi|ducatur ex F centro gravitatis, ad alteru<lb></lb>trum latus AE linea perpendicularis FA, erit hæc ſemidiame<lb></lb>ter figuræ motûs: quàm & vectem librationis centri nuncu<lb></lb>pamus. </s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Grauit as mouens eſt pars grauitatis mobilis; quam cen<lb></lb>trum grauitatis ſeu mobile retinet in libratione ad ſe <lb></lb>mouendum in plano inclinato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Grauitas quieſcens eſt pars grauitatis mobilis; quâ cen<lb></lb>trum grauitatis ſeu mobile in libratione grauitat <lb></lb>byp omocblium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>AXIOMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Areæ figuræ eandem rationem ad ſe babent, quam illarum <lb></lb>grauitas.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cùm grauitas magnitudinem ſequatur, hæc autem ſit area <lb></lb>figuræ <expan abbr="cuiuſq;">cuiuſque</expan>; erit grauitas hæc ad illam in ratione, quam areæ <lb></lb>ad ſe habent. </s></p> <pb xlink:href="063/01/033.jpg"></pb> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIVM I<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sequitur grauitatem figuræ ad grauitatem partis eandem ra<lb></lb>tionem habere, quam area figuræ habet ad illam partem: ut ſi <lb></lb>pars ſit tertia figuræ, erit grauitas tota tripla eiuſdem graui<lb></lb>tatis. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIVM II<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Et cùm impulſus ſequatur grauitatem, erit eadem ratio hu<lb></lb>ius, quæ grauitatis. </s><s>Impulſus ergo totus ad impulſum partis <lb></lb>tertiæ erit <expan abbr="quoq;">quoque</expan> triplus. </s></p> <p type="main"> <s><emph type="center"></emph>AXIOMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Vectis continet grauitatem mobilis: totus totam; pars verò <lb></lb>partem proportionælem.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Huius veritas conſtat ex prop. 13. & præmiſsâ eiuſdem de<lb></lb>claratione. </s></p> <p type="main"> <s><emph type="center"></emph>AXIOMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus grauium fit per lineas rect as ſe interſecantes in <lb></lb>mundi centro.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>AXIOMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Lapſus grauium eiuſdem rationis per lineas verticales <lb></lb>inter ſe ſunt æquales.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ex ſe inquam; nam illa differentia, quæ accidit moli maio<lb></lb>ri ob inæqualem plagam; ad medium refertur. </s><s>Vt conſtat <lb></lb>ex quæſtione de inæquali ponderum lapſu. </s></p> <pb xlink:href="063/01/034.jpg"></pb> <p type="main"> <s><emph type="center"></emph>AXIOMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Si magnitudo aliam percutiat in motu; & ſit contactus in <lb></lb>lineâ rectâ, qnæ tranſit per illarum centra, expulsâ æquali, <lb></lb>á motu quieſcit. exclusâ verò minori motum continuabit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>AXIOMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Si plures magnitudines contiguæ & æquales habeant cen<lb></lb>tra in unâ lineâ rectâ: & magnitudo uni contiguarum æ<lb></lb>qualis percutiat primam, omnibus immotis ultima mouetur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Percutiat circulus B alium circulum ſibi æqualem A in G: <lb></lb>aut quadratum C ſibi <expan abbr="quoq;">quoque</expan> æquale in F: Dico circulum B ex<lb></lb>pulſis A & C quieſcere à motu. Et ſi plures circuli contigui <lb></lb>habeant centra in unâ lineâ rectâ; percuſſo primo ultimus mo<lb></lb>uebitur. </s><s><expan abbr="Idemq;">Idemque</expan> futurum, ſi loco circuli quadratum illi æqua<lb></lb> <arrow.to.target n="fig5"></arrow.to.target><lb></lb>le ſubſtituatur. </s><s>At verò ſi A & C ſit minus quàm B; ijs expul<lb></lb>ſis motum continuabit. </s><s>Demonſtratum id à me quò ad glo- <pb xlink:href="063/01/035.jpg"></pb>bos ad prop. 37. poriſ. 1. & 2. problem. 1. in lib. de proport: <lb></lb>motûs. </s><s>Eadem verò eſt ratio reliquarum magnitudinum: <lb></lb>ſiue eiuſdem, ſiue alterius ſint figuræ. </s><s>Nam quòd percutiens <lb></lb>à motu quieſcit, huius ratio eſt æqualitas ponderis: quæ to<lb></lb>tam in percuſſo exhaurit plagam. </s><s>Vtverò circulus B ad circu<lb></lb>lum ſibi æqualem A, ita idem circulus ad quadratum ſibi æqua<lb></lb>le C: <expan abbr="eſtq;">eſtque</expan> contactus <expan abbr="utriusq;">utriusque</expan> in puncto G & F. ſicuti ergo A eſt <lb></lb>hypomochlium totius grauitatis ſeu impulſus in B; ita C hy<lb></lb>pomochlium eſt eiuſdem grauitatis ſeu impulſus. </s><s>Impulſus <lb></lb>autem æqualis ad magnitudinem æqualem eandem habet <lb></lb>rationem. </s></p> <figure id="id.063.01.035.1.jpg" xlink:href="063/01/035/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Grauitas mouens partium ìn toto eſt minor grauitate mouente <lb></lb>extra illud totum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sit B grauitas mobilis, & A mundi centrum: <expan abbr="eritq;">eritque</expan> linea BA <lb></lb>motus centri per 3, Axioma: partium verò HD motus eidem <lb></lb> <arrow.to.target n="fig6"></arrow.to.target> <pb xlink:href="063/01/036.jpg"></pb>paralleli HF. DE. </s><s>Dico grauitatem mouentem in H. D eſſe <lb></lb>minorem, quàm ſi extra lllud totum mouerentur. </s><s>Cùm enim <lb></lb>motus H ſit linea HA, & motus D linea DA per 3. Axioma; <lb></lb>erunt HF. DE motus inclinati: </s></p> <figure id="id.063.01.036.1.jpg" xlink:href="063/01/036/1.jpg"></figure> <p type="main"> <s>Et anguli in clinationum AHF. ADE. </s><s>Igitur pars grauitatis <lb></lb>H & D in hypomochlio quieſcit: <expan abbr="minorq;">minorque</expan> proinde eſt grauitas <lb></lb>mouens, quàm ſi extra illud totum mouerentur. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIVM I.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sequitur grauitatem mouentem partium à centro magis re<lb></lb>motarum eſſe minorem: propterea quòd motus ſint magis in<lb></lb>clinati. </s><s>Nam angulus AIF externus, hoc eſt illi æqualis A <lb></lb>DE eſt maior angulo interno AHF. & angulus AKG, hoc eſt <lb></lb>ADE maior angulo ACK. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIVM II.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Vnde neceſse partes propiores centro, remotiorum; cen<lb></lb>trum verò omnium eſſe hypomochlium huius grauitatis quie<lb></lb>ſcentis. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Centrum grauitatis habet impulſum omnium partium grauitati <lb></lb>æqualem.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cùm enim moveatur ad motum partium mobilis, habebit <lb></lb>impulſum illarum grauitati moventi æqualem. </s><s>Eſt verò <lb></lb>idem centrum hypomochlium grauitatis quieſcentis in motu <lb></lb>partium eidem parallelo, per Corollarium 2. quæ cùm augeat <lb></lb>illius grauitatem, habebit <expan abbr="quoq;">quoque</expan> per poſit. 4. impulſum illi æ<lb></lb>qualem. </s></p> <pb xlink:href="063/01/037.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Centrum grauitatis producit impulſum in omnibus partibus mobilis <lb></lb>illarum magnitudini proportionalem.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Quia grauitas movens partium in toto eſt minor, quàm ſi <lb></lb>per ſe, & extra illud totum moveatur, per I. THEOREMA; <lb></lb>erit <expan abbr="quoq;">quoque</expan> illarum motus minùs velox. </s><s>Mouentur autem æqua<lb></lb>li cum centro velocitate: habent igitur à centro illum motum. </s><lb></lb><s>At verò centrum grauitatis à partibus mobilis, ex ſe verò nul<lb></lb>lam habet grauitatem; <expan abbr="eſtq;">eſtque</expan> totus impulſus æqualis grauitati <lb></lb>ex omnibus partibus collectæ per THEOREMA II Igitur ut <lb></lb>tota magnitudo ſeu grauitas ad totum impulſum, ita pars mo<lb></lb>bilis ad partem impulſus proportionalem. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIVM<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Quodlibet punctum mobilis non ſuâ, ſed vi centri gravita<lb></lb>tis mouetur. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Percußuo fit à grauitate ſeu impulſu centri, non verò à grauitate <lb></lb>ſeu impulſu partium mobilis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moueantur duo globi A & B interſe connexi: <expan abbr="percutiatq;">percutiatque</expan> B <lb></lb>in motu globum C ſibi æqualem. </s><s>Dico impulſum in C eſſe ma<lb></lb>iorem, quàm ut æqualis ſit impulſui ex B: ac proinde illam pla<lb></lb>gam ad centrum referri. </s><s>Nam globus B, cùm per ſe movetur, <lb></lb>percuſſo æquali C, & expulſo vltimo D, à motu quieſcit per <lb></lb>AXIOMA 6. </s><s>At verò B connexus A <expan abbr="utrumq;">utrumque</expan> expellit D & C, <lb></lb><expan abbr="neq;">neque</expan> eo percuſſo quieſcit; Igitur globus C impulſum habet <pb xlink:href="063/01/038.jpg"></pb>maiorem, quàm ut æqualis ſit impulſui ex B. </s><s>Cuius quidem ra<lb></lb>tio eſt partium nexus: unde globus percuſſus fit hypomochli<lb></lb>um non ſolùm illius, quæ percuſſit; ſed etiam partium conne<lb></lb>xarum, ſeu centri gravitatis. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIVM<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>Sequitur tantam eſſe plagam, quantum ineſt hypomochlio <lb></lb>de centro grauitatis.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Percußio & qui hanc ſequitur impulſus, fit per lineam rectam, pro<lb></lb>ductam à contactu per centrum corporis percußi.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cùm enim partes mobilis non ſuâ ſed vi centri grauitatis <lb></lb>moveantur, per Corollarium Theorematis 3; neceſſe priùs cen<lb></lb>trum grauitatis ſeu mobilis impelli. </s><s>At verò principium im<lb></lb>pulſûs eſt contactus: igitur cùm impulſus non niſi per lineam <lb></lb>rectam moveat per prop: 3. </s><s>Via impulſus erit linea recta, pro<lb></lb>ducta à contactu per centrum grauitatis ſeu corporis percuſſi. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Impulſus centri grauitatis totus quieſcit; cùm ſemidiameter figuræ mo<lb></lb>tûs, vel illius centrum hypomochlio occurrit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Cùm enim partes mobilis non ſuâ, ſed vi centri graùitatis, & <lb></lb>ad huius motum moveantur, per Corollarium theorematis 3. <lb></lb>neceſſe ad huius in hypomochlio quietem quieſcere totum im<lb></lb>pulſum. </s></p> <pb xlink:href="063/01/039.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Impulſus centri grauitatis totus mouet, cùm huius interuallum ab <lb></lb>hypomochlio eſt œquale ſemidiæmetro figuræ motús.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Impulſus enim centri grauitatis prohibetur à motu; cùm vel <lb></lb>ipſum centrum, vel pars aliqua à centro mota in hypomochlio <lb></lb>quieſcit. </s><s>At verò cùm interuallum centri grauitatis eſt æqua<lb></lb>le ſemidiametro figuræ motûs; <expan abbr="neq;">neque</expan> ipſum centrum, <expan abbr="neq;">neque</expan> ali<lb></lb>qua pars à centro mota in hypomochlio quieſcit: totus igitur <lb></lb>impulſus movet. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Impulſus movens ad totum impulſum rationem habet, quam ſegmen<lb></lb>tum ſemidiametri ab hypomochlic & centro grauitatis interceptum, ad <lb></lb>ſemidiametrum figuræ motûs.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Cùm hypomochlium ſit trutina; <expan abbr="totusq;">totusque</expan> impulſus quieſcat, <lb></lb>cùm centrum hypomochlio occurrit, per theor. 6 totus verò <lb></lb>impulſus moveat, cùm huius à centro intervallum eſt æquale <lb></lb>ſemidiametro figuræ motùs per theore: 7. erit impulſus mo<lb></lb>uens æqualis ſegmento ſemidiemetri inter centrum grauitatis <lb></lb>& <expan abbr="hypomochliũ">hypomochlium</expan> intercepto In figurâ <expan abbr="ſequẽti">ſequenti</expan> BEC ſit A <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis, DE hypomochlium, & AC ſimidiameter æqualis <lb></lb>toti impulſui: <expan abbr="eritq;">eritque</expan> DA interuallum centri grauitatis A & <lb></lb>hypomochlij DE, grauitas mouens centri A. </s><s>Vt enim AD ad <lb></lb>vectem AC; ita per Axioma 2. ratio impulſús ex eodem pon<lb></lb>dere A appenſo. </s></p> <pb xlink:href="063/01/040.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Impulſus quieſcens eſt æqualis reliquo ſegmento, quod abſcindit hy<lb></lb>pomochlium à ſemidiametro figuræ motûs.<emph.end type="italics"></emph.end></s></p> <figure id="id.063.01.040.1.jpg" xlink:href="063/01/040/1.jpg"></figure> <p type="main"> <s>Quia impulfus mouens & quieſcens ſimul ſumpti, toti impul<lb></lb>ſui, hic autem ſemidiametro figuræ motus AC ponitur æqua<lb></lb>lis per Axioma 2: Eſt veró impulſus movens æqualis uni ſe<lb></lb>gmento AD per theorema 8. erit <expan abbr="quoq;">quoque</expan> impulſus quieſcens <lb></lb>æqualis alteri ſegmento DC. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Centrum grauitatis cuius〈que〉 figuræ rectilineæ invenire.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sit primùm in triangulo iſopleuro ABC inquirendum cen<lb></lb>trum grauitatis. in quo ex duobus angulis B & C demittantur <lb></lb>lineæ ad baſim rectæ BD CE. </s><s>Dico in communi illarum ſecti<lb></lb>one F eſſe centrum grauitatis. </s><s>Quia enim recta BD ſecat ba<lb></lb>ſim mediam; eritineâ centrum grauitatis, per prop. 13 lib. 1 <lb></lb>Archimedis de æquipond. </s><s>Eſt verò idem in recta CE: igitur in <lb></lb>communi ſectione F. </s></p> <pb xlink:href="063/01/041.jpg"></pb> <p type="main"> <s>Inquirendum iam ſit centrum grauitatis in quadrato GHIK. <lb></lb>in quo ductis diametris GI. HK; erit per prop. 10. eiuſdem li<lb></lb>bri centrum grauitatis in communi ſectione L. </s></p> <p type="main"> <s>Similiratione inveniemus centrum grauitatis in pentagono <lb></lb>isopleuro. ſinimirum ex angulis O & P ducantur lineæ OV. <lb></lb>PS perpendiculares ad latus oppoſitum. </s><s>Erit enim centrum <lb></lb> <arrow.to.target n="fig7"></arrow.to.target><lb></lb>grauitatis in communi ſectione T. propterea quòd <expan abbr="vtraq;">vtraque</expan> figu<lb></lb>ram ſecat bifariam: uti manifeſtum, ſi in triangula reſoluatur. </s></p> <figure id="id.063.01.041.1.jpg" xlink:href="063/01/041/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus verticalis figuræ rectilineæ ad motum inclinatum eſt in ratione <lb></lb>ſemidiametri figuræ motûs ad huius ſegmentum, quod eſt inter <lb></lb>centrum figuræ & lineam hypomochlij.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moveatur triangulum OMN in plano OB: & ex puncto N <lb></lb>ducatur linea hypomochlij NS, parallela lateri motus OQ: <lb></lb>ex centro autem figuræ P, per proximum Lemma inuento, a<lb></lb>gatur PQ perpendicularis ad OQ Dico motum verticalem <lb></lb>in OQ ad motum inclinatum in OB eſſe, ut PQ ad PR. </s><lb></lb><s>Quia enim gravitas mouens ex præmiſſis, & per poſit. 4- de <lb></lb>prop. motûs, eſt æqualis motui; grauitas antem tota, ſeu ver<lb></lb>ticaliter movens ad grauitatem mouentem in OB eſt ut PQ <pb xlink:href="063/01/042.jpg"></pb>ad PR per theorem 8. erit <expan abbr="quoq;">quoque</expan> motus verticalis in OQ ad <lb></lb>motum inclinatum in OB, ut PQ ad PR. hoc eſt ut ſemidia<lb></lb> <arrow.to.target n="fig8"></arrow.to.target><lb></lb>meter figuræ motûs ad huius ſegmentum inter centrum figu<lb></lb>ræ P & lineam hypomochlij NS. </s></p> <figure id="id.063.01.042.1.jpg" xlink:href="063/01/042/1.jpg"></figure> <p type="main"> <s>Simili ratione in quadrato K, ut KZ ad KL: & in pentago<lb></lb>nout TV ad TX, ita illorum motus verticalis ad motum incli<lb></lb>natum in OB. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Figura rectilinea velociùs mouetur in plano minùs inclinate.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sint duo plana, quorum inclinatio CAK maior, & CAI <lb></lb>minor: dico in plano CAI minoris inclinationis, motum eſſe ve<lb></lb>lociorem. </s><s>Ducantur ex D centro figuræ ad lineas verticales <lb></lb>AI. AK ſemidiametri figuræ motûs DF. DE: & ex angulo A <lb></lb>CB lineæ hypomochlij CG. CH parallelæ lincis verticalibus <lb></lb>AI. AK. </s><s>Quia <expan abbr="itaq;">itaque</expan> maior eſt DE quàm DF, & DO minor <lb></lb>quàm DP; erit reſidua OE maior quàm PF. </s><s>Maior proinde <pb xlink:href="063/01/043.jpg"></pb>ratio EO maioris ad OD minorem, quàm FP minoris ad PD <lb></lb>maiorem. </s><s>Et componendo ED ad OD, quàm FD ad PD. </s><s>Eſt <lb></lb> <arrow.to.target n="fig9"></arrow.to.target><lb></lb>autem ut ED ad OD, ita motus verticalis ad motum inclina <lb></lb>tum in plano CAK. </s><s>Et ut FD ad PD, ita idem motus vertica<lb></lb>lis ad motum inclinatum in plano CAI, per theorem 10. </s><s>Cùm <lb></lb><expan abbr="itaq;">itaque</expan> motus inclinatus in plano CAI ſit magis ſimilis verticali, <lb></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"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Grauitas movens inæqualium & ſimilium figurarum in eodem pla<lb></lb>no inclinato, eſt inæqualis & æqualiter mouet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moueantur in plano AC duo triangula ABC maius, & A <lb></lb>DE minus: & ex angulis EC ducantur lineæ EP. CO paralle<lb></lb>læ verticali AQ: lineæ verò FG. CF per illorum centra GF. <lb></lb>quæ per problema theorem: 1 erunt perpendiculares ad baſim <lb></lb>AB <expan abbr="demũ">demum</expan> exijſdem centris FG cadant lineæ FM. GN. perpen<lb></lb>diculares ad AQ. Quoniam <expan abbr="itaq;">itaque</expan> triangula CFH. EGI, & tri<lb></lb>angula CFK. EGL ſunt ſimilia: erit CF ad EG, ut FH ad GI <pb xlink:href="063/01/044.jpg"></pb>& FK ad GL. ſunt verò & triangula AMF, ANG, <expan abbr="atq;">atque</expan> trian<lb></lb>gula AMK. ANL ſimilia. </s><s>Igitur ut AM ad AN, ita MF ad <lb></lb>NG, & MK ad NL: ac proinde reſidua KF ad <expan abbr="reſiduã">reſiduam</expan> LG. <lb></lb><expan abbr="cùmq;">cùmque</expan> ſit ut FK ad GL, ita FH ad GI: & ut eadem FK ad GL, <lb></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></lb>grauitas mouens ſeu impulſus ad totum impulſum rationem <lb></lb>habet, <expan abbr="quã">quam</expan> GI ad GN, & FH ad FM, hoc eſt <expan abbr="ſegmentũ">ſegmentum</expan> ſemidiame<lb></lb>tri inter centrum figuræ & hypomochlium, ad ſemidiametrum <lb></lb>figuræ motûs per theo. 3. erit in <expan abbr="utroq;">utroque</expan> triangulo eadem pro<lb></lb>portio motûs inclinati ad motum verticalem. </s><s><expan abbr="Cùmq;">Cùmque</expan> mo<lb></lb>tus verticales inter ſe ſint æquales; per Axioma 4. erunt <expan abbr="quoq;">quoque</expan> <lb></lb>motus inclinati inter ſe æquales. </s><s>Et quia FM eſt maior quàm <lb></lb>GN, erit FH grauitas movens in triangulo ABC maior, quàm <lb></lb>GI grauitas movens in triangulo ADE. </s></p> <figure id="id.063.01.044.1.jpg" xlink:href="063/01/044/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Grauitas quieſcens inæqualium & ſimilium figurarum eſt inæqualis, <lb></lb>& inæqualiter grauitat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <pb xlink:href="063/01/045.jpg"></pb> <p type="main"> <s>In eadem figurâ, quoniam eſt ut FM ad GN, ita FH ad GI <lb></lb>per theor. 12. erit <expan abbr="quoq;">quoque</expan> HM ad IN, ut FH ad GI. </s><s>Sed FH <lb></lb>eſt maior quàm GI per idem theorema: igitur & HM maior <lb></lb>quam IN. </s><s>Et quia HM <expan abbr="atq;">atque</expan> IN eſt impulſus quieſcens per <lb></lb>theor. 9. maior granitas quieſcet in triangulo maiori, ac proin<lb></lb>de ſuum planum magis gravitabit. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>LEMMA I<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Inclinationem plani invenire: in quo ſemidiameter figuræ motûs <lb></lb>ſecetur ab hypomochlio in datâ ratione.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Producatur latus AC in I; & ſit AI ad CI in datâ ratione: <lb></lb>ex I verò per centrum figuræ D agatur linearecta IF: <expan abbr="atq;">atque</expan> huic <lb></lb>ex angulis C & A parallelæ CE. AH: quas ſecet ad angulos re<lb></lb>ctos, linea ex centro ducta DH. </s><s>Dico lineam DH, hoc eſt ſemi<lb></lb>diametrum figuræ motûs, ſectam eſſe in datâ ratione. </s><s>Ex <lb></lb>F enim protrahatur linea FK parallela DH; <expan abbr="eritq;">eritque</expan> FK ad FL, <lb></lb>hoc eſt DH ad DG, ut AF ad EF. </s><s>Sed ut AF ad EF ita AI ad <lb></lb>CI, hoc eſt in datâ ratione. </s></p> <figure id="id.063.01.045.1.jpg" xlink:href="063/01/045/1.jpg"></figure> <pb xlink:href="063/01/046.jpg"></pb> <p type="main"> <s>Aliter breuiùs. ex D centro figuræ ducta DA ſecetur in da<lb></lb>tâ ratione in O: per quod agatur linea CE, <expan abbr="atq;">atque</expan> eidem peralle<lb></lb>la AH: é centro verò D ſemidiameter figuræ motûs DH. </s><s>Di<lb></lb>co hanc ſecari à lineâ hypomochlij in eadem ratione. </s><s>Cùm <lb></lb>enim ſimilia ſint triangula ADH. ODG: erit DH ad DG, ut <lb></lb>DA ad DO, hoc eſt in datâ ratione. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA II<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Si duabus inæqualibus lineis addantur æquales; maiorem rationem ha<lb></lb>bet maior ad minorem, quàm eadem maior aucta ad auctam <lb></lb>minorem.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Duabus inæqualibus AB. CD addantur æquales BF. DL. </s><lb></lb><s>Dico AB ad CD maiorem rationem habere, quàm AF ad CL. </s><lb></lb><s>Fiat enim ut AB ad CD minorem: ita BF ad aliam minorem <lb></lb>DG. erit ergo <expan abbr="utraq;">utraque</expan> antecedens AF ad <expan abbr="utramq;">utramque</expan> conſequen<lb></lb>tem CG, ut AB ad CD. </s><s>Sed AF ad CG maiorem habetra<lb></lb>tionem, quàm ad CL: igitur & AB ad CD maiorem habet ra<lb></lb>tionem, quà AF ad CL. </s></p> <figure id="id.063.01.046.1.jpg" xlink:href="063/01/046/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>LEMMA III<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Si ex eadem baſi deſcribantur plures figuræ rectilineæ æqualium late<lb></lb>rum; & ex illâ baſi per illarum centra agatur linea recta; ea quæ <lb></lb>plura habet latera, centrum magis abducit à baſi.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Deſcribantur ex eadem communi baſi AC triangulum A <lb></lb>BC, quadratum ADEC, & pentagonum AFGHC æquali<lb></lb>um laterum: & per illarum centra agatur linea recta <expan abbr="Gq.">Gque</expan> ſe<lb></lb>cans baſim AC æqualiter per problema theorem. 1. </s><s>Quia <lb></lb><expan abbr="itaq;">itaque</expan> altitudo trianguli BQ eſt minor latere BA, hoc eſt QR; <pb xlink:href="063/01/047.jpg"></pb>diſtantia verò eiuſdem centri à baſi minor ſemiſſe <expan abbr="Bq;">Bque</expan> erit <lb></lb>KQ ſemiſſis RQ, hoc eſt diſtantia centri in quadrato, maior <lb></lb>quàm <expan abbr="Iq.">Ique</expan> Eſt verò diſtantia <expan abbr="quoq;">quoque</expan> centri LQ in pentagono <lb></lb> <arrow.to.target n="fig10"></arrow.to.target><lb></lb>maior quam <expan abbr="Kq.">Kque</expan> Nam cùm centrum ſit in mutuâ ſectione <lb></lb>GQ <expan abbr="atq;">atque</expan> HS perpendicularis ad FA, <expan abbr="ſintq;">ſintque</expan> duo anguli LSA. <lb></lb>LQA recti: & angulus SAQ in pentagono maior recto: erit <lb></lb>angulus SLQ minor recto: acproinde latus LQ maius latere <lb></lb>SA, ſemiſſe lateris FA ſeu RQ, diſtantiâ nimirum centri in <lb></lb>quadrato. </s></p> <figure id="id.063.01.047.1.jpg" xlink:href="063/01/047/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Fieri poteſt ut maior figura æqualiter & minùs grauitet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Aſlumantur duo triangula, quorum hoc illius ſit duplum. </s><lb></lb><s>Dico id quod eſt maius, poſſe æqualiter & minùs grauitare. </s><lb></lb><s>Secetur grauitas minoris triangali bifariam & æqualiter à li<lb></lb>neâ hypomochlij, per 1. lemma: <expan abbr="eritq;">eritque</expan> grauitas mouens æqua<lb></lb>lis quieſcenti, per theorema 8. ſub quadrupla verò ad grauita<lb></lb>tem trianguli maioris. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> ſemidiameter figuræ <lb></lb>motûs in triangulo maiori ſecetur <expan abbr="quoq;">quoque</expan> à lineâ hypomochlij <lb></lb>in eâ ratione, ut grauitas movens ad quieſcentem ſit quadru- <pb xlink:href="063/01/048.jpg"></pb>pla, per 1. lemma: erit grauitas quieſcens in <expan abbr="utroq;">utroque</expan> triangulo <lb></lb>æqualis; ac proinde æqualiter grauitabit. </s><s>At verò ſi augeatur <lb></lb>ratio grauitatis moventis ad quieſcentem; quia tum minor <lb></lb>grauitas quieſcit, minùs <expan abbr="quoq;">quoque</expan> hypomochlium grauitabit. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Figura rectilinea, quæ plura habet latera, velociùs mouetur in <lb></lb>eodem plano inclinato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moveatur in eodem plano AN triangulum ABC, & quadra<lb></lb>tum AEFC: Dico huius motum eſſe velociorem. </s><s>Secetur <lb></lb>enim in triangulo ABC ſemidiameter figuræ motûs DI â li<lb></lb>neâ hypomochlij CL bifariam & æqualiter in L, per primum <lb></lb>lemma: & ducatur in quadrato AEFC ſemidiameter figuræ <lb></lb>motûs GH: quæ maior erit ſemidiametro figuræ motûs DI. </s><lb></lb><s>Propterea quòd per lemma 3 maior ſit GO quàm DO. </s><s>Etad<lb></lb>ditâ communi OP maior GP, quàm DP. </s><s>Et quia ut GP ad <lb></lb> <arrow.to.target n="fig11"></arrow.to.target> <pb xlink:href="063/01/049.jpg"></pb>DP, ita GH ad DI; erit <expan abbr="quoq;">quoque</expan> GH maior quam DI, Dico GK <lb></lb>ad GH maiorem rationem habere, quàm DL ad DI. </s><s>Quia <lb></lb>enim HK eſt æqualis IL, erit per lemma 2. maior ratio GK ad <lb></lb>DL, quàm GH ad DI: & permutando GK ad GH, quàm DL <lb></lb>ad DI. </s><s>Eſt autem ut GK ad GH, & DL ad DI, ita motus in<lb></lb>clinati ad motum verticalem per theorem: 8. </s><s>Igitur motus <lb></lb>quadrati AEFC eſt velocior motu trianguli ABC in eodem <lb></lb>plano inclinato AN. </s></p> <figure id="id.063.01.049.1.jpg" xlink:href="063/01/049/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA XVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Figura rectilinea & æqualis, quæ plura habet latera, minùs gra<lb></lb>uitat in eodem plano inclinato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Nam ſemidiameter figuræ motús, hoc eſt grauitas tota, ſeca<lb></lb>tur ab hypomochlio in eam, quæ mouet, & in eam quæ in hy<lb></lb>pomochlio quieſcit, per theorema 9. </s><s>Eſt autem maior grauitas <lb></lb>mouens in figurâ plurilaterâ per theor. 15. minor ergo huius <lb></lb>pars quieſcit; ac proinde minùs grauitat. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Grauitas eiuſdem parallelogrammi mutato ſitu inæqualiter mouet, <lb></lb>& grauitat in eodem plano inclinato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moueatur in plano BO <expan abbr="parallelogrãmum">parallelogrammum</expan> ABCD: Dico <lb></lb>ex mutato laterum ſitu inæqualiter moveri: velociùs quidem, <lb></lb>ſi minus latus CD, tardiùs verò ſi maius latus BD fiat paralle<lb></lb>lum eidem plano BO. </s><s><expan abbr="Educãtur">Educantur</expan> ex angulis CD lineæ hypomo<lb></lb>chlij CG. DM: & ex centro figuræ E ſemidiametri figuræ motûs <lb></lb>EF. EK. </s><s>Et quia in duobus triangulis ſimilibus MBD. GDC <lb></lb>maior eſt DB quàm CD; erit <expan abbr="quoq;">quoque</expan> BM maior quàm DG. </s><s>Et <pb xlink:href="063/01/050.jpg"></pb>ſi ducantut <expan abbr="Bq.">Bque</expan> DI perpendiculares ad DM. CG: erit maior <lb></lb>BQ quàm DI, hoc eſt KL quàm FR. </s><s>Rurſum quia angulus <lb></lb>ECD eſt maior angulo ECA, hoc eſt illi æquali EDB: pro<lb></lb>pterea quòd latus AC ſeu BD ſit maius latere BA. ablatis æ<lb></lb>qualibus angulis GCD. MDB, erit angulus reliquus ECG <lb></lb>maior angulo reliquo EDM. </s><s><expan abbr="Aſſumaturitaq;">Aſſumaturitaque</expan> angulo EDM <lb></lb>æqualis angulus ECS: & ex Ead CS cadat perpendicularis ES: <lb></lb><expan abbr="eruntq;">eruntque</expan> triangula ECS. EDL ſimilia & æqualia. </s><s>Propterea <lb></lb> <arrow.to.target n="fig12"></arrow.to.target><lb></lb>quòd baſis EC ſit æqualis baſi ED. </s><s>Eſt autem ET maior <lb></lb>quàm ES, hoc eſt quàm EL: et ER maior quàm ET. </s><s>Igitur ea<lb></lb>dem ER maior quàm EL. </s><s>Cùm <expan abbr="itaq;">itaque</expan> maior ſitratio grauitatis <lb></lb>mouentis ER ad quieſcentem RF, nimirum maioris ad mino<lb></lb>rem, quàm EL ad LK minoris ad maiorem; erit per poſit: 4. <lb></lb>velocior motus in ER quàm in EL. </s><s>Et quia tum minor gra<lb></lb>uitas in hypomochlio quieſcit, minùs <expan abbr="quoq;">quoque</expan> ſubiectum pla<lb></lb>num grauitabit. </s></p> <pb xlink:href="063/01/051.jpg"></pb> <figure id="id.063.01.051.1.jpg" xlink:href="063/01/051/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Fieri poteſt ut idem parallelogrammum mutato ſitu moueatur, & <lb></lb>quieſcat in codem plano inclinato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Aſſumatur inclinatio plani æqualis angulo EDB: cadetq, <lb></lb>linea hypomochlij DE in centrum figuræ. </s><s>Et quia tum cen<lb></lb>trum grauitatis hypomochlio occurrit, quieſcet <expan abbr="parallelogrã-mum">parallelogran<lb></lb>mum</expan> in co ſitu, per theorema 6. </s><s>Cùm verò angulus ECD ſit <lb></lb>maior angulo inclinationis EDB; ſi ex C ducatur linea hypo. <lb></lb>mochlij, cadet inter EC. DC: ac proinde centrum figuræ ex<lb></lb>tra hypomochlium motum continuabit in eodem plano. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus circuli in eodom plano inclinato eſt velocior motufiguræ <lb></lb>rectilineæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moueatur in eodem plano AN circulus GCA, atq, penta<lb></lb>gonum BILMN: Dico motum circuli eſſe velociorem. </s><s>Aſſu<lb></lb>matur radius EA æqualis ON & ducantur lineæ hypomochlij <lb></lb>AC. NR ſecetur autem ſemidiameter figuræ motús OQ bifa<lb></lb>riam & æqualiter in P: ut ſit OP æqualis <expan abbr="Pq.">Pque</expan> per primum <lb></lb>lemma: dico EF maioren rationem habere ad FG, quàm OP <lb></lb>ad OQ Nam quia rectus eſt angulus DAE, & angulus BNO <lb></lb>ſemiſſis anguli pentagoni minor recto: ſunt verò anguli DAC. <lb></lb>BNP einſdem inclination is ex hypotheſi æquales: erit angu<lb></lb>lus reliquus FAE maior angulo reltquo PNO. </s><s>Et quia OP <lb></lb>per conſtructionem eſt æqua is PQ, ſi iungatur recta NQ, erit <lb></lb>angulus PNQ æqualis angulo ONP, maior verò angulo BNP, <lb></lb>hoc eſt illi æquali angulo DAF: ac proinde maior <expan abbr="quoq;">quoque</expan> an- <pb xlink:href="063/01/052.jpg"></pb> <arrow.to.target n="fig13"></arrow.to.target><lb></lb>gulo minori GAF. </s><s>Angulus <expan abbr="itaq;">itaque</expan> FAE quia maior angulo <lb></lb>ONP ſeu PNQ, erit multò maior angulo FAG; & FE ma<lb></lb>ior quam FG. maiorem proinde <expan abbr="rationẽ">rationem</expan> habet FE ad FG, quàm <lb></lb>OP ad <expan abbr="Pq.">Pque</expan> Et componendo EF ad EG, quàm OP ad <expan abbr="Oq.">Oque</expan> <lb></lb><expan abbr="Cùmq;">Cùmque</expan> impulſus movens ad totum impulſum ſit ut EF ad EG; <lb></lb>& ut OP ad OQ, per theor: 8. erit per poſit: 4 velocior motus <lb></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"></figure> <p type="main"> <s><emph type="center"></emph>PROBLEMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motum circuli, & trianguli Iſigoni ijsdem loci interuallis terminare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Moveatur triangulum Iſogonum ABC in plano HK: & <pb xlink:href="063/01/053.jpg"></pb>ex centro E ducatur ſemidiameter figuræ motûs EF: <expan abbr="ſitq;">ſitque</expan> in<lb></lb>veniendum planum, in quo circulus P æquali celeritate feratur. </s><lb></lb><s>In lineâ verticali HI centro O deſcribatur circulus HMN: cu<lb></lb>ius diameter HN ſit æqualis ſemidiametro figuræ motûs EF: <lb></lb>& ex puncto H ducatur chorda HM æqualis EG ſegmento in<lb></lb>ter centrum figuræ & hypomochlium. </s><s>Dico inuentum eſſe <lb></lb> <arrow.to.target n="fig14"></arrow.to.target><lb></lb>planum HML, in quo idem ſit circuli, qui trianguli in plano <lb></lb>HK motus. </s><s>Nam ut EF ad EG, ita totus impulſus, ſeu verti<lb></lb>caliter mouens ad impulſum in HK per 8. theor: & per po<lb></lb>ſitionem 4-motus trianguli in HI ad motum eiuſdem in HK. </s><lb></lb><s>Et ut HN ad HM, ita motus circuli in HI ad motum eiuſdem in <lb></lb>HL per prop, 13 de pro por: motûs. </s><s>At verò eandem ratio<lb></lb>nem habet HN ad HM, quam EF ad EG per conſtructionem. </s><lb></lb><s>Igitur motus circuli in HL eſt æqualis motui trianguli in HK. <lb></lb>motum ergo trianguli iſogoni ijſdem loci interuallis terminaui<lb></lb>mus, quod erat faciendam. </s></p> <pb xlink:href="063/01/054.jpg"></pb> <figure id="id.063.01.054.1.jpg" xlink:href="063/01/054/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>PROBLEMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Exceſſum, quo motus circuli in eodem plano eſt maior motu trianguli <lb></lb>Iſogoni, indagare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In eadem figurâ ſumptâ diametro circuli HN æquali EF, <lb></lb>auferatur à plano HR linea HQ æqualis EG; <expan abbr="eritq;">eritque</expan> motus trian<lb></lb>guli in HQ æqualis duratione motui circuli in HM per 1. prop. <lb></lb>motus verò eiuſdem circuli in plano HR æqualis duratione <lb></lb>terminatur chordâ HR. per prop. 15. </s><s>Exceſſus ergo, quo mo<lb></lb>tus circuli in eodem plano eſt maior motu trianguli, erit æqua<lb></lb>lis lineæ QR, quam inquirebamus. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motum figurarum rectilinearum periferiâ eiuſdem circuli <lb></lb>terminare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Centro H deſcribatur circulus: ad cuius periferiam eodem <lb></lb>tempore ſit terminandus motus ex H. </s><s>Inueniantur <expan abbr="itaq;">itaque</expan> plana; <lb></lb>in quibns ſemidiameter figuræ motûs in unâ <expan abbr="quâq;">quâque</expan> figurâ recti <lb></lb>lineâ, ſecetur ab hypomochlio in eadem ratione, in quâ ſecatur <lb></lb>EF à CD per 1 Lemma. </s><s>Et quia illarum grauitas mouens in <lb></lb>planis iam inventis eandem rationem habet ad ſuum mobile: <lb></lb>eruntmotus per poſit. 4 æquales, ac proinde ijſdem ſpatijs, hoc <lb></lb>eſt periferiâ eiuſdem circuli terminabuntur. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Circulo æquale quadratum ex motu invenire.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Percutiat in motu circulus A alium circulum ſibi æqualem B; <lb></lb><expan abbr="moveaturq;">moveaturque</expan> ex illa plagâ per ſpatium DE rurſum idem circu- <pb xlink:href="063/01/055.jpg"></pb>lus A habens eundem impulſum, percutiat eundem circulum <lb></lb>B contiguum quadrato C. aut igitur moto C circulus B quie<lb></lb>ſcet, aut illius motum conſequetur. </s><s>Et ſi quidem quieſcet, erit <lb></lb>per 3. Axioma grauitas in C, ac proinde per 1. Axioma huius <lb></lb> <arrow.to.target n="fig15"></arrow.to.target><lb></lb>area æqualis circulo B. </s><s>Quòd ſi verò ad illius motum move<lb></lb>tur; erit quadratum C per idem Axioma minus circulo B. </s><lb></lb><s>Moveatur <expan abbr="itaq;">itaque</expan> B ex illâ plagâ per ſpatium ML: & quadratum <lb></lb>C per ſpatium HI. </s><s>Supponamus verò HI æquale DE, & du<lb></lb>plum ſpatij ML. </s><s>Cùm <expan abbr="itaq;">itaque</expan> pro menſurâ plagæ minuatur im<lb></lb>pulſus, ex demonſtratis ad propoſ. 31. & motus eandem ratio<lb></lb>nem habeant, quam impulſus, per poſit: 4. ſit autem motus in <lb></lb>DE ad motum in ML duplus; erit <expan abbr="quoq;">quoque</expan> impulſus in A ad reli<lb></lb>quum impulſum in B, ac proinde ad impulſum in C duplus. </s><lb></lb><s>Quia verò quadratum C movetur ab æquali impulſu per ſpa<lb></lb>tium HI duplum ſpatij ML, erit <expan abbr="quoq;">quoque</expan> circulus B duplus qua<lb></lb>drati C. </s><s>Quòd ſi enim ſemiſſem circuli moveat idem impul<lb></lb>ſus, quia tum per Corollarium 2. theorematis 1. impulſum ha<lb></lb>bet duplum, movebit per poſit. 4. ad intervallum duplum. hoc <pb xlink:href="063/01/056.jpg"></pb>eſt HI. </s><s>Igitur ſi aſſumatur duplum quadrati C, inventum erit <lb></lb>quadratum æquale dato circulo B. </s></p> <figure id="id.063.01.056.1.jpg" xlink:href="063/01/056/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Alius modus quadrandi circulum ex motu.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si figuræ maior in motu percutiat minorem, habeat verò ſegmentum <lb></lb>ſemidiametri figuræ motûs, quod eſt inter lineam hypomochlij, et extre<lb></lb>mum motûs, eandem rationem ad alterum ſegmentum, quod eſt inter <lb></lb>eandem lineam hypomochlij & figuræ centrum, quam habet figura <lb></lb>minor ad maiorem, motus maioris à percußione erit parallelus lineæ <lb></lb>rectæ per contactum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Percutiat quadratum ABCD circulum H in G. & duca<lb></lb>tur linea hypomochlij GI ſecans ſemidiametrum figuræ mo<lb></lb>tûs AE in F: <expan abbr="ſitq;">ſitque</expan> AF ad FE, ut circulus H ad quadratum <lb></lb> <arrow.to.target n="fig16"></arrow.to.target><lb></lb>ABCD: Dico motum quadrati à percuſſione eſſe parallelum <lb></lb>lateri AB, hoc eſt lineæ rectæ per contactum G. </s><s>Quia enim <pb xlink:href="063/01/057.jpg"></pb>ut AF menſura plagæ ad EF reſiduum impulſum, ita circulus <lb></lb>H, ad quadratum ABCD: erit permutando AF ad H, ut FE <lb></lb>ad ABCD: ac proinde per poſit. 4. eadem velocitas motûs in <lb></lb><expan abbr="utrâq;">utrâque</expan> figurâ. </s><s>Quadratum ergo ABCD nullam à circulo per<lb></lb>cuſſo recipit plagam. </s><s>Et quia præpondium eſt in E, propterea <lb></lb>quòd impulſus in AF defecit ex illâ plagâ; neceſſe librationem <lb></lb>fieri in G. </s><s>Nequit autem revolui centrum E, niſilatus AB ſe<lb></lb>cet circulum H, aut hic à plagâ velociùs ſe abducat. </s><s>Quia ve<lb></lb>rò eadem velocitas motûs, neceſſe motum in E per lineam fi<lb></lb>eri parallelam lateri AB. </s></p> <figure id="id.063.01.057.1.jpg" xlink:href="063/01/057/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>LEMMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si figura maior in motu percutiat minorem; habeat verò ſegmentum <lb></lb>ſemidiametri figuræ motûs, quod eſt inter lineam hypomochlij & extre<lb></lb>mum motûs, minorem rationem ad alterum ſegmentum, quod eſt inter <lb></lb>eandem lineam hypomochlij & figuræ centrum, quàm habeat figura <lb></lb>minor ad maiorem, motus figuræ maioris erit parallelus lineæ mediæ <lb></lb>inter tangentem circuli, & lineam productam à centro maioris ad con<lb></lb>tactum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Habeat AF ad FE <expan abbr="minorẽ">minorem</expan> <expan abbr="rationẽ">rationem</expan>, quàm circulus H ad qua<lb></lb>dratum ABCD: dico, motum E figuræ maioris eſſe <expan abbr="parallelũ">parallelum</expan> <lb></lb>lineæ GK mediæ inter GB & GE. </s><s>Quia enim minorem ra<lb></lb>tionem habet AF ad FE, quàm circulus H ad quadratum <lb></lb>ABCD; & permutando AF ad H, quàm FE ad ABCD, mi<lb></lb>nori velocitate movebitur ex illâ plagà circulus H, quàm <lb></lb>quadratum ABCD: eandem ergo recipit à circulo percuſſo, <lb></lb>quam dedit plagam. </s><s>Et quia præpondium in E; ob tardita<lb></lb>tem motûs circuli ad lineam determinatur parallelam lateri <lb></lb>AB per 1. Lemma: impulſum verò recipit à circulo H per <pb xlink:href="063/01/058.jpg"></pb>lineam GE per theorem. 5. <expan abbr="ſuntq;">ſuntque</expan> impulſus ſubcontrarij; erit <lb></lb>motus E per prop. 31. de proportione motûs, parallelus lineæ <lb></lb>GK mediæ inter GB & GE. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si figura maior in motu percutiat minorem; habeat verò ſegmentum <lb></lb>ſemidiametri figuræ motûs, quod eſt inter lineam hypomochlij, & ex<lb></lb>tremum motûs, maiorem rationem ad reliquum ſegmentum, quod eſt <lb></lb>inter eandem lineam hypomochlij & figuræ centrum, quàm habeat mi<lb></lb>nor figura ad maiorem; motus maioris erit parallelus lineæ mediæ inter <lb></lb>tangentem circuli & eiuſdem perpendicularem ad contactum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Habeat AF ad EF maiorem rationem, quàm circulus H ad <lb></lb>quadratum ABCD: Dico, huius motum ab illâ plagâ eſſe pa<lb></lb>rallelum lineæ mediæ inter GB & GH. </s><s>Quia enim menſura <lb></lb>plagæ AF ad reſiduum impulſum in FE maiorem rationem <lb></lb>habet, quàm circulus H ad quadratum ABCD: & permu<lb></lb>tando AF ad H, quàm FE ad ABCD, erit velocior motus <lb></lb>circuli H, quàm quadrati ABCD. nullam ergo à circulo per<lb></lb>cuſſo recipit plagam. </s><s>Et quia præpondium in E, neceſſe libra<lb></lb>tionem fieri in G: ac proinde motum in E eſſe parallelum lineæ <lb></lb>mediæ inter GB & GH. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Circulo æquale quadratum ex motu invenire.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Percutiat quadratum ABCD circulum H, & ex illâ plagà <lb></lb>moveatur centrum E per lineam parallelam lateri GB: duca<lb></lb>tur autem linea hypomochlij FG ſecans ſemidiametrum figu<lb></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"></pb>Had ABCD. </s><s>Hoc eſt permutando ut AF ad H, ita FE ad <lb></lb>ABCD. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> fiat ut FE ad AF, ita ABCD ad aliud <lb></lb>quadratum: inventum erit circulo H æquale quadratum. </s></p> <p type="main"> <s>Quòd ſi ex illâ plagâ moveatur E per lineam parallelam GK: <lb></lb>erit per Lemma 2. minor proportio AF ad H, quàm FE ad <lb></lb>ABCD: <expan abbr="Atq;">Atque</expan> huius motus velocior motu circuli. eandem er <lb></lb>gò plagam recipit quadratum ABCD, quam infert circulo: <lb></lb>ac proinde illius impulſus à percuſſione erit æqualis AE: com <lb></lb>poſitus nimirum ex plagâ reciprocâ AF & impulſu reſiduo FE. </s><lb></lb><s>Supponamus verò AE ad AF eſſe ut 6 ad 2, hoc eſt in ratione <lb></lb>triplâ: ſpatium verò decurſum ab E ad ſpatium decurſum ab H <lb></lb>ut 3 ad 2. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> circulus H accipiat impulſum ut 3. hoc <lb></lb>eſt additâ ſemiſſe, movebitur ad idem intervallum cumquadra<lb></lb>to ABCD. </s><s>Et ſi fiat ut 6 ad 3, ita ABCD ad aliud, inventum <lb></lb>erit quadratum circulo H æquale. </s></p> <p type="main"> <s>Demum ſi motus quadrati E à percuſſione fiat parallelus li <lb></lb>neæ mediæ inter tangentem GB, & perpen dicularem GH; erit <lb></lb>per Lemma 3 maior proportio AF ad H, quàm FE ad ABCD: <lb></lb>& motus H velocior motu ABCD. </s><s>Ponamus <expan abbr="itaq;">itaque</expan> interval<lb></lb>lum motûs Had interuallum motûs ABCD in ſeſqui alterâ <lb></lb>ratione, hoc eſt ut 3 ad 2: FE autem ad AF ut 4 ad 2. </s><s>Quòdſi <lb></lb><expan abbr="itaq;">itaque</expan> quadratum ABCD accipiat impulſum ut 6; movebitur <lb></lb>eadem velocitate, & ad idem intervallum cum circulo H per <lb></lb>poſit. 5. propterea quòd impulſus eandem rationem habeat ad <lb></lb>ſuum mobile, per corollarium 2. 1 Axiomatis. </s><s>Et ſi fiat ut 6 <lb></lb>ad 2, ita quadratum ABCD ad aliud quadratum, inventum <lb></lb>erit circulo H æquale quadratum. </s></p> <pb xlink:href="063/01/060.jpg"></pb> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>ALIA QVADRATVRA CIRCVLI<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>per motum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>DVcatur à contactu G per centrum figuræ E linea GL æ<lb></lb>qualis GB: & ex L ad eam perpendicularis LM ſecans B <lb></lb>C in M: <expan abbr="eritq;">eritque</expan> LM æqualis BM. </s><s>Si enim iungatur recta BL, <lb></lb>duo anguli GBL. GLB, ac proinde reſidui MBL. MLB ſunt <lb></lb>æquales. </s><s>Centro <expan abbr="itaq;">itaque</expan> M, interuallo ML deſcribatur arcus LB <lb></lb>ſecans <expan abbr="lineã">lineam</expan> motûs reflexi GK in O: ex O verò demittantur per <lb></lb>pendiculares ON. OP. </s><s>Quoniam <expan abbr="itaq;">itaque</expan> punctum G à plagâ re<lb></lb>ciprocâ ex H per lineam agitur GL per 5 theorema: impulſus <lb></lb>verò reſiduus in FE per lineam GB per lemma 2. </s><s><expan abbr="Eſtq;">Eſtque</expan> motus <lb></lb>medius GK, erit per problem. propoſitionis 35 de propor. mo<lb></lb>tûs, vt OP ad ON, ita impulſus in GB ad impulſum in GL, æ<lb></lb>qualem impulſui in H. </s><s>Et ſi quidem ON eſt ſemiſſis OP, erit <lb></lb>impulſus in OP ad impulſum in ON ut 4 ad 2. ſupponamus ve<lb></lb>rò ſpatium decurſum ab E, ad ſpatium decurſum ab H eſſe in <lb></lb>ſeſquialterâ ratione, hoc eſt ut 3 ad 2. </s><s>Igitur ſi circulus H acci<lb></lb>piat impulſum ut 3, movebitur ad idem interuallum cum qua<lb></lb>drato ABCD per corollarium 2 Axiomatis 1 & poſitionem 4. </s><lb></lb><s>Etſi fiat ut 4 ad 3 ita ABCD ad aliud; inventum erit quadra<lb></lb>tum dato circulo H æquale. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLL ARIVM<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>Eadem ratione inveniemus quadratum æquale ſectionibus <lb></lb>conicis, <expan abbr="atq;">atque</expan> adeo illarum fruſtis; ſi loco circuli hu<lb></lb>iuſmodi figuras ſubſtituamus.<emph.end type="center"></emph.end></s></p> <pb xlink:href="063/01/061.jpg"></pb> <p type="main"> <s><emph type="center"></emph>PARS TERTIA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>DE MOTV REFLEXO FIGVR ARVM<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>RECTILINEARVM.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>E gi de motu reflexo in lib: de proport: motûs, à prop: 36. ad 40. ve<lb></lb>rùm hunc non niſi in circulo expendi. </s><s>Licet verò in Quadraturâ cir<lb></lb>culi motus quo〈que〉 reſtexus interueniat; dum ab illatâ plagâ aliò, quàm <lb></lb>ferebatur, viam capeßit: hic tamen unà hypomochlium mouetur: ne〈que〉 <lb></lb>huius principium eſt grauitas. </s><s>Neceſſe ergo in figuris quo〈que〉 rectilineis <lb></lb>hunc motum reflexum, quatenus à grauitate & hypomochlio immoto <lb></lb>procedit, conſidexare.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus trianguli Iſogoni ad planum & baſim perpendicularis, in <lb></lb>ſe ipſum reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>TRiangulo <emph type="italics"></emph>abc<emph.end type="italics"></emph.end> labenti occurrat planum <emph type="italics"></emph>az:<emph.end type="italics"></emph.end> <expan abbr="ſitq;">ſitque</expan> motus <lb></lb>centri <emph type="italics"></emph>d<emph.end type="italics"></emph.end> ad illud planum, & baſim <emph type="italics"></emph>ab<emph.end type="italics"></emph.end> perpendicularis <lb></lb>dico hunc motum in ſe ipſum reflecti. </s><s>Nam in primâ quidem <lb></lb>figurâ motus centri <expan abbr="atq;">atque</expan> huius plaga eſt in eadem lineâ <emph type="italics"></emph>dc:<emph.end type="italics"></emph.end> da<lb></lb>bit ergo plagam perfectam. & quia per eandem lineam <emph type="italics"></emph>dc<emph.end type="italics"></emph.end> re<lb></lb>cipit à percuſſo æqualem illi, quam dedit plagam per 5 theor: <lb></lb>2 partis, motus in ſe ipſum reflectit. </s><s>In ſecundâ autem figurâ <lb></lb>percuſſio fit per idem theor. per lineas <emph type="italics"></emph>da, df, db;<emph.end type="italics"></emph.end> eſtq motus <lb></lb>centri in lineâ <emph type="italics"></emph>df:<emph.end type="italics"></emph.end> erit ergo motus reflexus à plagâ <emph type="italics"></emph>df<emph.end type="italics"></emph.end> in ea<lb></lb>dem lineâ <emph type="italics"></emph>df.<emph.end type="italics"></emph.end> at verò plaga in <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> & <emph type="italics"></emph>bd<emph.end type="italics"></emph.end> centrum <emph type="italics"></emph>d<emph.end type="italics"></emph.end> reper<lb></lb>cuſſum in partes agit <emph type="italics"></emph>dg. de.<emph.end type="italics"></emph.end> & quia plaga in <emph type="italics"></emph>da<emph.end type="italics"></emph.end> eſt æqualis <pb xlink:href="063/01/062.jpg"></pb>plagæ in <emph type="italics"></emph>db;<emph.end type="italics"></emph.end> erit motus quoq in <emph type="italics"></emph>de<emph.end type="italics"></emph.end> æqualis motui in <emph type="italics"></emph>dg:<emph.end type="italics"></emph.end> ac <lb></lb>proinde per prop: 31 motus medius reflectit per lineam <emph type="italics"></emph>dc.<emph.end type="italics"></emph.end><lb></lb>Cùm igitur hæc ſit via centri, motus trianguli in ſe ipſum re<lb></lb>flectit. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus trianguli Iſogoni ad planum, non verò ad baſim perpen<lb></lb>dicularis, in partem baſis maiorem reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Triangulum <emph type="italics"></emph>abc<emph.end type="italics"></emph.end> occurrat plano <emph type="italics"></emph>az<emph.end type="italics"></emph.end> ad angulos rectos: <lb></lb><expan abbr="ſecetq;">ſecetque</expan> motus centri <emph type="italics"></emph>d<emph.end type="italics"></emph.end> baſim <emph type="italics"></emph>ac<emph.end type="italics"></emph.end> in duo ſegmenta <emph type="italics"></emph>kc<emph.end type="italics"></emph.end> maius, <lb></lb>& <emph type="italics"></emph>ka<emph.end type="italics"></emph.end> minus: dico motum reflexum fieri in partem <emph type="italics"></emph>kc<emph.end type="italics"></emph.end> ſe <lb></lb> <arrow.to.target n="fig17"></arrow.to.target><lb></lb>gmenti maioris. </s><s>Excitetur enim linea hypomochlij <emph type="italics"></emph>af:<emph.end type="italics"></emph.end> quam <lb></lb>ſecet linea <emph type="italics"></emph>de<emph.end type="italics"></emph.end> à centro perpendicularis quia <expan abbr="itaq;">itaque</expan> vectis eſt <lb></lb><emph type="italics"></emph>da;<emph.end type="italics"></emph.end> <expan abbr="atq;">atque</expan> huius quadratum, ideſt totam grauitatem, ſecat bi<lb></lb>fariam linea hypomochlij, iuxta demonſtrata in lib: de propor: <lb></lb>motûs; ſi quadratum <emph type="italics"></emph>ed<emph.end type="italics"></emph.end> fit grauitas mouens centri, erit hu<lb></lb>ius complementum quadratum <emph type="italics"></emph>ae,<emph.end type="italics"></emph.end> menſura percuſsionis ſcu <pb xlink:href="063/01/063.jpg"></pb>plagæ. </s><s>Et quia motus centri fit per lineam <emph type="italics"></emph>di<emph.end type="italics"></emph.end> tangentem cir<lb></lb>culi centro <emph type="italics"></emph>a<emph.end type="italics"></emph.end> deſcripti per prop: 4: motus autem reflexus à <lb></lb>plagâ per lineam <emph type="italics"></emph>dg<emph.end type="italics"></emph.end> per 5 theor. 2 part. ſi fiat ut <emph type="italics"></emph>de<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ea<emph.end type="italics"></emph.end> ita <lb></lb><emph type="italics"></emph>di<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>dg,<emph.end type="italics"></emph.end> erit per prop: 32 motus medius <emph type="italics"></emph>dh<emph.end type="italics"></emph.end> diameter pa<lb></lb>rallelogrammi <emph type="italics"></emph>aihg:<emph.end type="italics"></emph.end> ac proinde motus reflexus in partem <lb></lb><emph type="italics"></emph>kc<emph.end type="italics"></emph.end> ſegmenti maioris. </s></p> <figure id="id.063.01.063.1.jpg" xlink:href="063/01/063/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Quadrati perpendicularis ad planum, ſi æqualiter ſecet an<lb></lb>gulum, aut latus eiuſdem quadrati, in ſe ipſum reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Incidat plano <emph type="italics"></emph>ax<emph.end type="italics"></emph.end> perpendiculariter Quadratum <emph type="italics"></emph>abcd:<emph.end type="italics"></emph.end> <expan abbr="ſe-cetq;">ſe<lb></lb>cetque</expan> motus centri <emph type="italics"></emph>f<emph.end type="italics"></emph.end> latus <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> aut angulum <emph type="italics"></emph>adc<emph.end type="italics"></emph.end> in duas par. <lb></lb>tes æquales: dico, hunc motum in ſe ipſum reflecti. </s><s>Nam in <lb></lb>primâ figurâ, quia coincidit motus centri, & plaga in eandem <lb></lb>lineam <emph type="italics"></emph>fd;<emph.end type="italics"></emph.end> erit motus à percuſſione in viâ centri: ac proinde <lb></lb>in ſe ipſum reflexus. </s><s>Infigurâ autem ſecundâ plaga fit per lineas <lb></lb><emph type="italics"></emph>fa. fe. fd.<emph.end type="italics"></emph.end> per 4. theorema 2 part: & à plagâ quidem in <emph type="italics"></emph>fe,<emph.end type="italics"></emph.end> quòd <lb></lb>hæc ſit via centri, motus in ſe ipſum reflectit: à plagâ verò in <lb></lb><emph type="italics"></emph>fa<emph.end type="italics"></emph.end> & <emph type="italics"></emph>fd,<emph.end type="italics"></emph.end> in partes oppoſitas <emph type="italics"></emph>fc. fb<emph.end type="italics"></emph.end> agitur centrum grauitatis <lb></lb>per 1 theor: & quia angulus <emph type="italics"></emph>bfc<emph.end type="italics"></emph.end> eſt minor duobus rectis, ac <lb></lb>proinde motus in <emph type="italics"></emph>fc. fb<emph.end type="italics"></emph.end> per definit. 4 ſubcontrarij; ob æqua<lb></lb>les verò plagas <emph type="italics"></emph>af. df<emph.end type="italics"></emph.end> inter ſe æquales; erit per prop: 32 mo<lb></lb>tus medius in lineâ <emph type="italics"></emph>fg.<emph.end type="italics"></emph.end> Cùm ergo hæc ſit via centri, motus <lb></lb>Quadrati in ſe ipſum reflectit. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Quadrati perpendicularis ad planum, inæqualiter autem <lb></lb>ſecans angulum ſeu baſim, reflectit in partem ſegmenti maioris.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <pb xlink:href="063/01/064.jpg"></pb> <p type="main"> <s>Idem Quadratum <emph type="italics"></emph>abcd<emph.end type="italics"></emph.end> occurrat plano <emph type="italics"></emph>ax<emph.end type="italics"></emph.end> ad angulos re<lb></lb>ctos, motu centri <emph type="italics"></emph>e<emph.end type="italics"></emph.end> inæqualiter ſecante baſim <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> in <emph type="italics"></emph>pd<emph.end type="italics"></emph.end> maius, <lb></lb>& <emph type="italics"></emph>ap<emph.end type="italics"></emph.end> minus ſegmentum: dico motum reflecti in illam partem, <lb></lb>in quâ eſt ſegmentum maius <emph type="italics"></emph>pd.<emph.end type="italics"></emph.end> Ductâ enim lineâ hypo<lb></lb>mochlij <emph type="italics"></emph>ag,<emph.end type="italics"></emph.end> & à centro ad eam perpendiculari <emph type="italics"></emph>ef;<emph.end type="italics"></emph.end> erit gra<lb></lb>uitas mouens centri à percuſſione quadratum <emph type="italics"></emph>ef,<emph.end type="italics"></emph.end> <expan abbr="atq;">atque</expan> huius <lb></lb>complementum quadratum <emph type="italics"></emph>af<emph.end type="italics"></emph.end> menſura plagæ: vectis autem <lb></lb><emph type="italics"></emph>ea,<emph.end type="italics"></emph.end> cuius quadratum grauitas tota, ſeu impulſus. </s><s>Et quia <lb></lb>plaga fit per lineam <emph type="italics"></emph>ea;<emph.end type="italics"></emph.end> erit motus à percuſſione in eadem lineâ <lb></lb><emph type="italics"></emph>ea:<emph.end type="italics"></emph.end> per 5 theor. 2 part: motus autem centri à reliquo impulſu <lb></lb>in lineâ <emph type="italics"></emph>ek<emph.end type="italics"></emph.end> tangente circuli centro <emph type="italics"></emph>a<emph.end type="italics"></emph.end> deſcripti. </s><s>Quòd ſi ergo <lb></lb>fiat ut <emph type="italics"></emph>ef<emph.end type="italics"></emph.end> motus centri ad <emph type="italics"></emph>af<emph.end type="italics"></emph.end> motum repercuſſum, ita <emph type="italics"></emph>ek<emph.end type="italics"></emph.end> ad <lb></lb><emph type="italics"></emph>eh;<emph.end type="italics"></emph.end> erit diameter parallelogrammi <emph type="italics"></emph>ehik<emph.end type="italics"></emph.end> motus medius per <lb></lb>prop: 32 ac proinde motus reflexus in partem ſegmenti ma<lb></lb>ioris </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Pentagoni perpendicularis ad planum & latus eiusdem, <lb></lb>in ſe ipſum reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Nam in primâ quidem figurâ, quia & motus centri & pla<lb></lb>ga tota eſt in lineâ <emph type="italics"></emph>ef;<emph.end type="italics"></emph.end> erit motus reflexus in eadem lineâ <emph type="italics"></emph>ef.<emph.end type="italics"></emph.end><lb></lb>In ſecundâ autem figurâ lineæ percuſſionis ſunt <emph type="italics"></emph>fa fg fe:<emph.end type="italics"></emph.end><lb></lb>motus ergò reflexus in <emph type="italics"></emph>fh. fc. fi.<emph.end type="italics"></emph.end> Et quia motus in <emph type="italics"></emph>fh<emph.end type="italics"></emph.end> & <emph type="italics"></emph>fi<emph.end type="italics"></emph.end><lb></lb>ſunt ſub contrarij <expan abbr="atq;">atque</expan> inter ſe æquales per defini: 4 erit per <lb></lb>prop: 32 motus medius linea <emph type="italics"></emph>fc:<emph.end type="italics"></emph.end> ac proinde cùm hæc ſit via con<lb></lb>tri, motus in ſe ipſum reflectit. </s></p> <pb xlink:href="063/01/065.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Pentagoni perpendicularis ad planum, non verò ad latus <lb></lb>eiuſdem, reflectit in partem ſegmenti maioris.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Motus Pentagoni <emph type="italics"></emph>abcde<emph.end type="italics"></emph.end> perpendicularis ad planum ſe<lb></lb>cet latus <emph type="italics"></emph>ae<emph.end type="italics"></emph.end> in duo ſegmenta <emph type="italics"></emph>le<emph.end type="italics"></emph.end> maius, & <emph type="italics"></emph>al<emph.end type="italics"></emph.end> minus: Dico <lb></lb>à percuſſo illo plano reflecti in partem <emph type="italics"></emph>le<emph.end type="italics"></emph.end> ſegmenti maioris. </s><lb></lb><s>Nam ſi excitetur linea hypomochlij <emph type="italics"></emph>ag,<emph.end type="italics"></emph.end> & à centro ducatur li<lb></lb>nea <emph type="italics"></emph>fg<emph.end type="italics"></emph.end> ad eam perpendicularis; erit quadratum <emph type="italics"></emph>fg<emph.end type="italics"></emph.end> grauitas <lb></lb>mouens centri; huius autem complementum quadratum <emph type="italics"></emph>ag<emph.end type="italics"></emph.end><lb></lb>menſura plagæ: propterea quòd tota grauitas ſit æqualis qua<lb></lb>drato <emph type="italics"></emph>af.<emph.end type="italics"></emph.end> Et quia plaga fit per lineam <emph type="italics"></emph>af,<emph.end type="italics"></emph.end> erit motus reflexus in <lb></lb>eadem lineâ <emph type="italics"></emph>af:<emph.end type="italics"></emph.end> motus autem centri in lineâ <emph type="italics"></emph>fk<emph.end type="italics"></emph.end> tangente cir<lb></lb>culi centro <emph type="italics"></emph>a<emph.end type="italics"></emph.end> deſcripti. </s><s>Quòd ſi ergo fiat ut <emph type="italics"></emph>fg<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ga,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>fk<emph.end type="italics"></emph.end> ad <lb></lb><emph type="italics"></emph>fh;<emph.end type="italics"></emph.end> erit per prop: 32 motus medius diameter parallelogram<lb></lb>mi <emph type="italics"></emph>faik:<emph.end type="italics"></emph.end> ac proinde motus pentagoni reflectit in partem <emph type="italics"></emph>le<emph.end type="italics"></emph.end><lb></lb>ſegmenti maioris. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Trianguli iſogoni ad baſim, non verò ad planum perpen<lb></lb>dicularis, ſi in verticem moueatur, in ſe ipſum reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In| 1 figurâ trianguli <emph type="italics"></emph>efg<emph.end type="italics"></emph.end> latus <emph type="italics"></emph>ef<emph.end type="italics"></emph.end> ſecetur à motu eiuſdem <lb></lb><emph type="italics"></emph>hg<emph.end type="italics"></emph.end> æqualiter: occurrat autem plano <emph type="italics"></emph>ik<emph.end type="italics"></emph.end> motu in <emph type="italics"></emph>g<emph.end type="italics"></emph.end> verticem <lb></lb>converſo: Dico hunc motum in ſe ipſum reflecti. </s><s>Quia enim <lb></lb>motus centri & plagæ, quam dat, <expan abbr="recipitq;">recipitque</expan> centrum, eſt in <expan abbr="eadẽ">eadem</expan> <lb></lb>lineâ <emph type="italics"></emph>hg,<emph.end type="italics"></emph.end> erit motus à percuſſione in eadem lineâ <emph type="italics"></emph>hg<emph.end type="italics"></emph.end> per 1 <lb></lb>theor: ac proinde motus in ſe ipſum reflectit. </s></p> <pb xlink:href="063/01/066.jpg"></pb> <figure id="id.063.01.066.1.jpg" xlink:href="063/01/066/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Trianguli Iſogoni ad baſim, non verò ad planum perpendi<lb></lb>cularis, ſi in baſim moveatur, uno latere eidem plano par alle<lb></lb>lo, ad angulos æquales reſtectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In 2 figurâ moveatur triangulum <emph type="italics"></emph>bcd<emph.end type="italics"></emph.end> in baſim <emph type="italics"></emph>cd,<emph.end type="italics"></emph.end> ſectam <lb></lb>bifariam & æqualiter à motu centri in <emph type="italics"></emph>a.<emph.end type="italics"></emph.end> <expan abbr="ſitq;">ſitque</expan> latus <emph type="italics"></emph>bd<emph.end type="italics"></emph.end> paral<lb></lb>lelum plano: Dico in hoc caſu triangulum <emph type="italics"></emph>bcd<emph.end type="italics"></emph.end> motu reflexo <lb></lb>angulum conſtituere æqualem illi, quem facit cum eodem pla<lb></lb>no huius lapſus. </s><s>Excitetur enim linea hypomochlij <emph type="italics"></emph>cf,<emph.end type="italics"></emph.end> du<lb></lb>ctâ lineâ à centro perpendiculari <emph type="italics"></emph>ai.<emph.end type="italics"></emph.end> quia <expan abbr="itaq;">itaque</expan> ex demonſtra<lb></lb>tis plaga eſt æqualis quadrato <emph type="italics"></emph>ci,<emph.end type="italics"></emph.end> & grauitas mouens centri <lb></lb>æqualis quadrato <emph type="italics"></emph>ai:<emph.end type="italics"></emph.end> eſt autem plaga, & qui hanc ſequitur mo<lb></lb>tus reflexus in lineâ <emph type="italics"></emph>ac<emph.end type="italics"></emph.end> per 1 theor: motus verò centri in lineâ <lb></lb>tangente circuli centro <emph type="italics"></emph>c<emph.end type="italics"></emph.end> <expan abbr="atq;">atque</expan> interuallo <emph type="italics"></emph>ac<emph.end type="italics"></emph.end> deſcripti, paralle<lb></lb>la nimirum plano <emph type="italics"></emph>eg:<emph.end type="italics"></emph.end> ſi fiat ut <emph type="italics"></emph>ci<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ai,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>cl<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>cm;<emph.end type="italics"></emph.end> erit mo<lb></lb>tus medius <emph type="italics"></emph>cn<emph.end type="italics"></emph.end> diameter parallclogrammi <emph type="italics"></emph>clmn:<emph.end type="italics"></emph.end> Dico angu<lb></lb>lum <emph type="italics"></emph>ncm<emph.end type="italics"></emph.end> eſſe æqualem angulo <emph type="italics"></emph>fce.<emph.end type="italics"></emph.end> Quia enim recta <emph type="italics"></emph>ac<emph.end type="italics"></emph.end> per <pb xlink:href="063/01/067.jpg"></pb>centrum eſt perpendicularis ad <emph type="italics"></emph>eg<emph.end type="italics"></emph.end> parallelum ipſi <emph type="italics"></emph>bd,<emph.end type="italics"></emph.end> erunt an<lb></lb>guli <emph type="italics"></emph>ace. acg<emph.end type="italics"></emph.end> inter ſe æquales. </s><s>Sunt autem triangula <emph type="italics"></emph>ica. lcn<emph.end type="italics"></emph.end><lb></lb>ex conſtructione ſimilia; & angulus <emph type="italics"></emph>ica<emph.end type="italics"></emph.end> æqualis angulo <emph type="italics"></emph>lcn:<emph.end type="italics"></emph.end><lb></lb>quibus ablatis ex <emph type="italics"></emph>ace. acg<emph.end type="italics"></emph.end> anguli reliqui <emph type="italics"></emph>ecf. mcn,<emph.end type="italics"></emph.end> incidentiæ <lb></lb>& reflexionis inter ſe ſunt æquales. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Trianguli Iſogoni ſi ne〈que〉 ad planum, ne〈que〉 ad baſim ſit per<lb></lb>pendicularis, ad angulos inæquales reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In 3 figurâ triangulum <emph type="italics"></emph>abc<emph.end type="italics"></emph.end> occurrat plano habens latus <emph type="italics"></emph>ac<emph.end type="italics"></emph.end><lb></lb>eidem parallelum: <expan abbr="ſitq;">ſitque</expan> Iinea hypomochlij <emph type="italics"></emph>cd,<emph.end type="italics"></emph.end> & linea ad eam <lb></lb>perpendicularis <emph type="italics"></emph>ef:<emph.end type="italics"></emph.end> <expan abbr="eritq;">eritque</expan> grauitas mouens centri Quadratum <lb></lb><emph type="italics"></emph>ef:<emph.end type="italics"></emph.end> plaga autem huius complementum quadratum <emph type="italics"></emph>go.<emph.end type="italics"></emph.end> quod <lb></lb>quidem habetur, ſi lineâ <emph type="italics"></emph>gf<emph.end type="italics"></emph.end> ſectâ bifarium in <emph type="italics"></emph>p,<emph.end type="italics"></emph.end> eo centro de<lb></lb>ſcribatur ſemicirculus <emph type="italics"></emph>gof,<emph.end type="italics"></emph.end> <expan abbr="ſumaturq;">ſumaturque</expan> chorda <emph type="italics"></emph>fo<emph.end type="italics"></emph.end> æqualis <emph type="italics"></emph>fe:<emph.end type="italics"></emph.end> nam <lb></lb>chorda reliqua <emph type="italics"></emph>og<emph.end type="italics"></emph.end> dabit illud quadratum. propterea quòd gra<lb></lb>uitas tota ſit quadratum <emph type="italics"></emph>fg.<emph.end type="italics"></emph.end> fiat <expan abbr="itaq;">itaque</expan> ut <emph type="italics"></emph>fo<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>og,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>fi<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>fb;<emph.end type="italics"></emph.end><lb></lb>erit motus reflexus in lineâ <emph type="italics"></emph>fh<emph.end type="italics"></emph.end> diametro parallelogrammi <emph type="italics"></emph>fb hi:<emph.end type="italics"></emph.end><lb></lb>angulus autem reflexionis <emph type="italics"></emph>ifh:<emph.end type="italics"></emph.end> quem dico angulo <emph type="italics"></emph>acd<emph.end type="italics"></emph.end> eſſe in<lb></lb>æqualem. </s><s>Quia angulus <emph type="italics"></emph>age<emph.end type="italics"></emph.end> externus cſt maior angulo in<lb></lb>terno <emph type="italics"></emph>ecg,<emph.end type="italics"></emph.end> æqualis autem angulo <emph type="italics"></emph>ofg;<emph.end type="italics"></emph.end> propterea quòd <expan abbr="uterq;">uterque</expan> <lb></lb>aſſumpto angulo communi <emph type="italics"></emph>ogf<emph.end type="italics"></emph.end> facit rectum: eſt verò huic <lb></lb>angulo æqualis angulus reflexionis <emph type="italics"></emph>hfi;<emph.end type="italics"></emph.end> quòd ſimilia ſint trian<lb></lb>gula <emph type="italics"></emph>gef: hfi:<emph.end type="italics"></emph.end> erit ergo æqualis <expan abbr="quoq;">quoque</expan> angulo externo <emph type="italics"></emph>age:<emph.end type="italics"></emph.end> ac <lb></lb>proinde maior interno <emph type="italics"></emph>acd<emph.end type="italics"></emph.end> angulo incidentiæ. </s><s>In 4 demum <lb></lb>figurâ centrum <emph type="italics"></emph>e<emph.end type="italics"></emph.end> cadat intra lineam hypomochlij. cùm igitur <lb></lb>centrum gravitatis contineatur in hypomochlio, erit plaga per<lb></lb>fecta: <expan abbr="atq;">atque</expan> huius lineæ <emph type="italics"></emph>ea. ef. ec:<emph.end type="italics"></emph.end> ac proinde per 1 theor: hu<lb></lb>ius motus reflexus in lineâ <emph type="italics"></emph>eb.<emph.end type="italics"></emph.end> Quia ergo angulus reflexionis <pb xlink:href="063/01/068.jpg"></pb><emph type="italics"></emph>efc,<emph.end type="italics"></emph.end> nimirum rectus, maior eſt angulo incidentiæ <emph type="italics"></emph>dcf;<emph.end type="italics"></emph.end> motus <lb></lb>trianguli in eo ſitu ad angulos reflectit inæquales. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Si motus Quadrati obliquè, huius autem diameter ad angulos re<lb></lb>ctos ſecet planum; ad angulos æquales reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Motus Quadrati <emph type="italics"></emph>abcd<emph.end type="italics"></emph.end> ſecet obliquè planum <emph type="italics"></emph>el,<emph.end type="italics"></emph.end> diameter <lb></lb>verò <emph type="italics"></emph>ag<emph.end type="italics"></emph.end> ad angulos rectos: dico motum reflexum ab hoc pla<lb></lb>no angulum conſtituere æqualem angulo incidentiæ. </s><s>Sit enim <lb></lb><emph type="italics"></emph>ap<emph.end type="italics"></emph.end> hypomochlij, & <emph type="italics"></emph>gh<emph.end type="italics"></emph.end> linea ad eam perpendicularis: <expan abbr="eritq;">eritque</expan> <lb></lb>ex iam demonſtratis <expan abbr="quadratũ">quadratum</expan> <emph type="italics"></emph>hg<emph.end type="italics"></emph.end> motus centri, & <emph type="italics"></emph>ah<emph.end type="italics"></emph.end> eiuſdem <lb></lb>plaga. </s><s>Et quia percuſsic in <emph type="italics"></emph>ag,<emph.end type="italics"></emph.end> erit motus reflexus in eadem <lb></lb>hneâ <emph type="italics"></emph>ag:<emph.end type="italics"></emph.end> motus autem centri in lineâ plano <emph type="italics"></emph>el<emph.end type="italics"></emph.end> parallelâ. quòd <lb></lb>ſi <expan abbr="itaq;">itaque</expan> fiat ut <emph type="italics"></emph>ah<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>hg,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>af<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ak,<emph.end type="italics"></emph.end> erit motus medius <emph type="italics"></emph>ai,<emph.end type="italics"></emph.end> & an<lb></lb>gulus reflexionis <emph type="italics"></emph>iak:<emph.end type="italics"></emph.end> quem dico eſſe æqualem angulo <emph type="italics"></emph>eap.<emph.end type="italics"></emph.end><lb></lb>Quia enim diameter <emph type="italics"></emph>ag<emph.end type="italics"></emph.end> ſecat planum in <emph type="italics"></emph>a<emph.end type="italics"></emph.end> ad angulos rectos; <lb></lb>erit angulus <emph type="italics"></emph>eag<emph.end type="italics"></emph.end> æqualis angulo <emph type="italics"></emph>kag.<emph.end type="italics"></emph.end> ſunt auté per conſtructio<lb></lb>nem ſimilia triangula <emph type="italics"></emph>gha. afi;<emph.end type="italics"></emph.end> & angulus <emph type="italics"></emph>gah<emph.end type="italics"></emph.end> æqualis angu<lb></lb>lo <emph type="italics"></emph>fai;<emph.end type="italics"></emph.end> igitur angulus reliquus <emph type="italics"></emph>eap<emph.end type="italics"></emph.end> eſt æqualis angulo reliquo <lb></lb><emph type="italics"></emph>iak<emph.end type="italics"></emph.end> angulus nimirum incidentiæ angulo reflexionis: </s></p> <figure id="id.063.01.068.1.jpg" xlink:href="063/01/068/1.jpg"></figure> <pb xlink:href="063/01/069.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Si ne〈qué〉 motus Quadrati, ne〈que〉 huius diameter ad angulos rectos ſe<lb></lb>cet planum, ad angulos inæquales reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Motus Quadrati <emph type="italics"></emph>abcd<emph.end type="italics"></emph.end> obliquè ſecans planum <emph type="italics"></emph>gr,<emph.end type="italics"></emph.end> habeat <lb></lb>latus <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> eidem plano parallelum: & ſit linea hypomochlij <emph type="italics"></emph>dg.<emph.end type="italics"></emph.end><lb></lb>ad eam verò perpendicularis <emph type="italics"></emph>eh;<emph.end type="italics"></emph.end> |cuius quadratum grauitas <lb></lb>movens centri, <expan abbr="atq;">atque</expan> huius complementum quadratum <emph type="italics"></emph>fi,<emph.end type="italics"></emph.end> pla<lb></lb>ga eiuſdem centri. </s><s>Quod quidem quadratum in ſemicirculo <lb></lb><emph type="italics"></emph>fie<emph.end type="italics"></emph.end> conſtituit chorda reliqua, in quo chorda <emph type="italics"></emph>ie<emph.end type="italics"></emph.end> ſit ſumpta æ<lb></lb>qualis <emph type="italics"></emph>eh.<emph.end type="italics"></emph.end> Et quia plaga fit per lineas <emph type="italics"></emph>ea. ef. ed:<emph.end type="italics"></emph.end> per 4. theo. 2 part. <lb></lb>erit per 3 theor: huius, motus reflexus in lineâ <emph type="italics"></emph>ek;<emph.end type="italics"></emph.end> motus <lb></lb>autem centri in lineâ plano <emph type="italics"></emph>qr<emph.end type="italics"></emph.end> parallelâ, ſeu tangente cir culi <lb></lb>centro <emph type="italics"></emph>f,<emph.end type="italics"></emph.end> & interuallo <emph type="italics"></emph>fe<emph.end type="italics"></emph.end> deſcripti. quòd ſi ergo fiat ut <emph type="italics"></emph>ci<emph.end type="italics"></emph.end> ad <lb></lb><emph type="italics"></emph>if,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>em<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ek,<emph.end type="italics"></emph.end> erit per prop: 32 motus medius <emph type="italics"></emph>el<emph.end type="italics"></emph.end> diameter <lb></lb>parallelogrammi <emph type="italics"></emph>kelm:<emph.end type="italics"></emph.end> dico angulum reflexionis <emph type="italics"></emph>lem<emph.end type="italics"></emph.end> eſſe in <lb></lb>æqualem angulo <emph type="italics"></emph>adg.<emph.end type="italics"></emph.end> Quia enim angulus <emph type="italics"></emph>afi<emph.end type="italics"></emph.end> externus ma<lb></lb>ior eſt angulo interno <emph type="italics"></emph>adh,<emph.end type="italics"></emph.end> æqualis autem angulo <emph type="italics"></emph>ief<emph.end type="italics"></emph.end> per 9. <lb></lb>theor: <expan abbr="atq;">atque</expan> huic æquatur angulus <emph type="italics"></emph>lem,<emph.end type="italics"></emph.end> propterea quòd ſimilia <lb></lb>ſint triangula <emph type="italics"></emph>ief, mel:<emph.end type="italics"></emph.end> erit <expan abbr="quoq;">quoque</expan> æqualis angulo externo <lb></lb><emph type="italics"></emph>afi,<emph.end type="italics"></emph.end> maior verò angulo interno <emph type="italics"></emph>fdh<emph.end type="italics"></emph.end> angulo nimirum inci<lb></lb>dentiæ. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Pentagoni ſecans obliquè planum, ſi latus oppoſitum habeat <lb></lb>eidem plano par allelum, ad angulos æquales reflectit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Pentagonum <emph type="italics"></emph>abcde<emph.end type="italics"></emph.end> habeat latus <emph type="italics"></emph>cd<emph.end type="italics"></emph.end> plano <emph type="italics"></emph>op<emph.end type="italics"></emph.end> parallelum <lb></lb>& oppoſitum: dico ad angulos reflecti æquales. </s><s>Sit enim <pb xlink:href="063/01/070.jpg"></pb><emph type="italics"></emph>ab<emph.end type="italics"></emph.end> linea hypomochlij, & <emph type="italics"></emph>fg<emph.end type="italics"></emph.end> ad eam perpendicularis: <expan abbr="eritq;">eritque</expan> ex <lb></lb>iam demonſtratis <emph type="italics"></emph>fg<emph.end type="italics"></emph.end> grauitas mouens, & <emph type="italics"></emph>ag<emph.end type="italics"></emph.end> plaga eiuſdem <lb></lb>centri. </s><s>Et quia plaga eſt in lineâ <emph type="italics"></emph>af;<emph.end type="italics"></emph.end> erit motus reflexus in <lb></lb>eadem lineâ <emph type="italics"></emph>af.<emph.end type="italics"></emph.end> quòd ſi ergo fiat ut <emph type="italics"></emph>ag<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>gf,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>ah<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ak,<emph.end type="italics"></emph.end> erit <lb></lb>motus medius in <emph type="italics"></emph>ai,<emph.end type="italics"></emph.end> & angulus reflexûs <emph type="italics"></emph>iak:<emph.end type="italics"></emph.end> quem dico æqua<lb></lb>lem angulo incidentiæ <emph type="italics"></emph>oab.<emph.end type="italics"></emph.end> Quia enim angulus <emph type="italics"></emph>oab<emph.end type="italics"></emph.end> eſt æ<lb></lb>qualis angulo <emph type="italics"></emph>afg,<emph.end type="italics"></emph.end> propterea quòd <expan abbr="uterq;">uterque</expan> ſit complementum <lb></lb>anguli <emph type="italics"></emph>fag:<emph.end type="italics"></emph.end> angulo autem <emph type="italics"></emph>gfa<emph.end type="italics"></emph.end> æquatur angulus <emph type="italics"></emph>iak,<emph.end type="italics"></emph.end> quòd ſi<lb></lb>milia ſint triangula <emph type="italics"></emph>agf. iak:<emph.end type="italics"></emph.end> erit <expan abbr="quoq;">quoque</expan> angulo <emph type="italics"></emph>oab<emph.end type="italics"></emph.end> idem <lb></lb>angulus <emph type="italics"></emph>iak<emph.end type="italics"></emph.end> æqualis. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motus Pentagoni ſecans obliquè planum, ſi latus, quod tangit pla<lb></lb>num eidem ſit parallelum, ad angulos inæquales reſle<lb></lb>ctit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Motus Pentagoni <emph type="italics"></emph>abcde<emph.end type="italics"></emph.end> incidat obliquè plano <emph type="italics"></emph>ſt<emph.end type="italics"></emph.end> habens la<lb></lb>tus <emph type="italics"></emph>ae,<emph.end type="italics"></emph.end> quod tangit planum, eidem parallelum: dico hunc mo<lb></lb>tum reflecti ad angulos inæquales. </s><s>Excitetur linea hypomo<lb></lb>chlij <emph type="italics"></emph>en,<emph.end type="italics"></emph.end> & <emph type="italics"></emph>fg<emph.end type="italics"></emph.end> ad eam perpendicularis: <expan abbr="eritq;">eritque</expan> grauitas tota <expan abbr="qua-dratũ">qua<lb></lb>dratum</expan> <emph type="italics"></emph>fh;<emph.end type="italics"></emph.end> grauitas autem mo vens quadratum <emph type="italics"></emph>fg.<emph.end type="italics"></emph.end> dividatur bi<lb></lb>fariam linea <emph type="italics"></emph>hf<emph.end type="italics"></emph.end> in <emph type="italics"></emph>p;<emph.end type="italics"></emph.end> <expan abbr="eoq;">eoque</expan> centro circulus deſcribatur <emph type="italics"></emph>hif.<emph.end type="italics"></emph.end><lb></lb>Quòd ſi ergo ſumatur chorda <emph type="italics"></emph>fi<emph.end type="italics"></emph.end> æqualis <emph type="italics"></emph>fg;<emph.end type="italics"></emph.end> erit chorda re<lb></lb>liqua <emph type="italics"></emph>hi;<emph.end type="italics"></emph.end> <expan abbr="atq;">atque</expan> huius quadratum dabit plagam. </s><s>Et quia plaga <lb></lb>fit per lineas <emph type="italics"></emph>fa. fh. fe:<emph.end type="italics"></emph.end> erit per 5 theor: huius motus reflexus <lb></lb>in lineâ <emph type="italics"></emph>fc,<emph.end type="italics"></emph.end> & motus centri in lineâ <emph type="italics"></emph>fm<emph.end type="italics"></emph.end> eidem plano parallelâ. </s><lb></lb><s>Si ergo fiat ut <emph type="italics"></emph>fi<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ih,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>fm<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>fl;<emph.end type="italics"></emph.end> erit motus medius <emph type="italics"></emph>fk,<emph.end type="italics"></emph.end> & <lb></lb>angulus reflexionis <emph type="italics"></emph>kfm;<emph.end type="italics"></emph.end> quem dico inæqualem angulo in<lb></lb>cidentiæ <emph type="italics"></emph>hen.<emph.end type="italics"></emph.end> Quia enim angulus <emph type="italics"></emph>ahi<emph.end type="italics"></emph.end> externus eſt maior <pb xlink:href="063/01/071.jpg"></pb>angulo interno <emph type="italics"></emph>hei,<emph.end type="italics"></emph.end> æqualis autem angulo <emph type="italics"></emph>ifh;<emph.end type="italics"></emph.end> propterea <lb></lb>quòd <expan abbr="uterq;">uterque</expan> aſſumpto angulo communi <emph type="italics"></emph>ihf<emph.end type="italics"></emph.end> facit rectum: <lb></lb>& angulo <emph type="italics"></emph>ifh<emph.end type="italics"></emph.end> eſt æqualis angulus <emph type="italics"></emph>kfm;<emph.end type="italics"></emph.end> erit <expan abbr="quoq;">quoque</expan> æqualis an<lb></lb>gulo <emph type="italics"></emph>ahi,<emph.end type="italics"></emph.end> ac proinde maior angulo interno <emph type="italics"></emph>hei,<emph.end type="italics"></emph.end> angulo inci<lb></lb>dentiæ. </s></p> <p type="main"> <s><emph type="italics"></emph>Obijcies. </s><s>Si vectis continet gr auitatem mobilis, totus totam, pars ve<lb></lb>rò partem proportionalem per 2 Axioma; et impulſus centri grauitatis <lb></lb>totus mouet, cùm huius interuallum ab hypomochlio eidem eſt æquale per <lb></lb>7 theorema 2 partis; neceßè in figurâ 3 theor: 2 huius, cùm tota ſemidia<lb></lb>meter figuræ motûs ſit extra hypomochlium, & non niſi in puncto tan<lb></lb>gat planum AZ; aut nullam, aut inſenſibilem inferre plagam: non igi<lb></lb>tur rectè aſſumebatur ratio plagæ ad reliquum impulſum, quam habet <lb></lb>quadratum ED ad quadratum EA: ſiquidem totum impulſum metitur <lb></lb>quadratum eiuſdem ED.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo noſtram aſſertionem veram eſſe, cùm ſemidia<lb></lb>meter figuræ motûs eâ ratione ſecatur ab hypomochlio, ut re<lb></lb>liquus impulſus ab illatâ plaga non prohibeatur à ſuo mo<lb></lb>tu: at verò hic impulſus cogitur ab hypomochlio ad <expan abbr="motũ">motum</expan> incli<lb></lb>natum <emph type="italics"></emph>di,<emph.end type="italics"></emph.end> per tangentem circuli centro <emph type="italics"></emph>a<emph.end type="italics"></emph.end> deſcripti. </s><s>Erit <expan abbr="itaq;">itaque</expan> <lb></lb>impulſus reliquus in eâratione ad totum impulſum, quam ha<lb></lb>bet motus in eiuſmodi plano inclinato ad motum verticalem. </s><lb></lb><s>Ducatur enim <emph type="italics"></emph>el<emph.end type="italics"></emph.end> parallela ipſi <emph type="italics"></emph>di:<emph.end type="italics"></emph.end> <expan abbr="eritq;">eritque</expan> motus verticalis in <lb></lb><emph type="italics"></emph>ea<emph.end type="italics"></emph.end> ad motum inclinatum in <emph type="italics"></emph>el,<emph.end type="italics"></emph.end> ut quadratum <emph type="italics"></emph>ea<emph.end type="italics"></emph.end> ad quadratum <lb></lb><emph type="italics"></emph>el,<emph.end type="italics"></emph.end> hoc eſt ut quadratum <emph type="italics"></emph>da<emph.end type="italics"></emph.end> ad quadratum <emph type="italics"></emph>de:<emph.end type="italics"></emph.end> quòd ſimilia <lb></lb>ſunt triangula <emph type="italics"></emph>ael. aed.<emph.end type="italics"></emph.end> Et quia quadratum <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> hoc eſt totus <lb></lb>impulſus æquatur duobus quadratis <emph type="italics"></emph>de. ae;<emph.end type="italics"></emph.end> eſt autem quadra<lb></lb>tum <emph type="italics"></emph>de<emph.end type="italics"></emph.end> impulſus movens, erit quadratum <emph type="italics"></emph>ae<emph.end type="italics"></emph.end> impulſus qui<lb></lb>eſcens, hoc eſt plaga; quam infert eidem plano <emph type="italics"></emph>az.<emph.end type="italics"></emph.end> Magis er<lb></lb>go univerſalis eſt hæc ratio, quàm à ſemidiametro figuræ mo- <pb xlink:href="063/01/072.jpg"></pb>tûs deſumpta. vnde etiam hac ad demonſtrationem horum the<lb></lb>orematum uſi ſumus. </s></p> <p type="main"> <s>Fortaſſe verò hanc eandem hypotheſim, in motu proiecto<lb></lb>rum, non inconvenienter aſſumere licebit. ut ſi quadratum E <lb></lb>percutiat circulum H per 1 & 2 Lemma probl: 5. quia motus <lb></lb>centri E à percuſſione fit parallelus rectæ GB, erit inclinatio <lb></lb>huius æqualis angulo BGQ, hoc eſt illi ad verticem æquali AGI. </s><lb></lb><s>Igitur ut GI ad GA, ita motus verticalis ad motum inclina<lb></lb>tum. eſt verò ut GI ad GA, ita GE ad FE. propterea quòd ſi<lb></lb>milia ſint triangula GEF. AGI. eſt enim AGE ſimile <expan abbr="utriq;">utrique</expan> <lb></lb>triangulo FGE. FAG, <expan abbr="atq;">atque</expan> idem FAG ſimile triangulo AGI. </s><lb></lb><s>Cùm itaq, FE ſit impulſus mouens; totum verò impulſum <lb></lb>metiatur EG; erit huius exceſſus æqualis plagæ. qui nonniſi <lb></lb>cùm radius EG eſt æqualis ſemidiametro figuræ motûs EA, <lb></lb>æquatur reliquo ſegmento AF. </s><s>Quòd ſi verò quis opine<lb></lb>tur eandem eſſe rationem motûs proiectorum, & qui pro venit <lb></lb>à grauitate: propterea quòd ſicuti lapſus grauium continuò <lb></lb>augetur: ita <expan abbr="quoq;">quoque</expan> motus proiectorum continuò minuitur: eo <lb></lb>videlicet modo, quo triangulum ſibi ſimile manens; ac pro<lb></lb>inde <expan abbr="utrumq;">utrumque</expan> ſecari ab hypomochlio in duo quadrata: is meo <lb></lb>quidem iudicio haud improbabiliter ita ſentiet. </s><s>Tum <expan abbr="itaq;">itaque</expan> <lb></lb>ſumpto impulſu toto æquali quadrato EG: ſi EF quadratum <lb></lb>ſit vis movens; erit FG quadratum plaga, ſeu impulſus in hy<lb></lb>pomochlio quieſcens. </s><s>Siue tamen hac, ſiue illâ hypotheſi uta<lb></lb>mur, eadem via erit ad circuli quadraturam. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motum verticalem trianguli Iſogoni à plano reflectere ad an<lb></lb>gulum datum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sit angulus datus grad. 30. ad quem reflectere oportet mo<lb></lb>tum trianguli <emph type="italics"></emph>abc<emph.end type="italics"></emph.end> à plano <emph type="italics"></emph>az.<emph.end type="italics"></emph.end> Ducatur linea verticalis <pb xlink:href="063/01/073.jpg"></pb><emph type="italics"></emph>af<emph.end type="italics"></emph.end> faciens cum rectâ <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> angulum <emph type="italics"></emph>fad<emph.end type="italics"></emph.end> grad. 30. ſemiſſem <lb></lb>complementi anguli reflexionis. </s><s>Secet autem <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> producta <lb></lb>latus trianguli <emph type="italics"></emph>bc<emph.end type="italics"></emph.end> ad angulos rectos: dico triangulum <emph type="italics"></emph>abc<emph.end type="italics"></emph.end><lb></lb>in hoc ſitu à lapſu verticali reflecti ad grad-30. </s><s>Ducatur enim <lb></lb>à centro figuræ recta <emph type="italics"></emph>de<emph.end type="italics"></emph.end> perpendicularis ad <emph type="italics"></emph>af.<emph.end type="italics"></emph.end> Et fiat ut <lb></lb><emph type="italics"></emph>ae<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>ed,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>dg<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>di:<emph.end type="italics"></emph.end> <expan abbr="eritq;">eritque</expan> <emph type="italics"></emph>dh<emph.end type="italics"></emph.end> motus centri à reflexi<lb></lb>one. </s><s>Cuiex <emph type="italics"></emph>a<emph.end type="italics"></emph.end> ducatur parallela <emph type="italics"></emph>ac.<emph.end type="italics"></emph.end> Quia <expan abbr="itaq;">itaque</expan> angulus <emph type="italics"></emph>e <lb></lb>ad<emph.end type="italics"></emph.end> eſt grad. 30. per conſtructionem; æqualis autem angulo <emph type="italics"></emph>g <lb></lb>dh,<emph.end type="italics"></emph.end> hoc eſt illi æquali <emph type="italics"></emph>dai:<emph.end type="italics"></emph.end> erit angulus compoſitus <emph type="italics"></emph>fai<emph.end type="italics"></emph.end> grad: <lb></lb>60, ac proinde angulus reliquus <emph type="italics"></emph>caz<emph.end type="italics"></emph.end> grad. 30. </s></p> <figure id="id.063.01.073.1.jpg" xlink:href="063/01/073/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>PROBLEMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motum verticalem quadrati à plano reflectere ad angulum datum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Inveniendus ſit angulus reflexionis grad. 40. </s><s>Ductâ <emph type="italics"></emph>ag<emph.end type="italics"></emph.end> li<lb></lb>neâ hypomochlij, fiat angulus <emph type="italics"></emph>gae<emph.end type="italics"></emph.end> grad: 25. ſemiſſis comple<lb></lb>menti anguli reflexionis. </s><s>Et ex centro figuræ producatur <emph type="italics"></emph>ef<emph.end type="italics"></emph.end> <pb xlink:href="063/01/074.jpg"></pb>perpendicularis ad <emph type="italics"></emph>af.<emph.end type="italics"></emph.end> Quòd ſi <expan abbr="itaq;">itaque</expan> fiat ut <emph type="italics"></emph>af<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>fe,<emph.end type="italics"></emph.end> ita <emph type="italics"></emph>eh<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>ek;<emph.end type="italics"></emph.end> erit <emph type="italics"></emph>ei<emph.end type="italics"></emph.end> via motûs reflexi. </s><s>Cui ex <emph type="italics"></emph>a<emph.end type="italics"></emph.end> ducatur paral<lb></lb>lcla <emph type="italics"></emph>ad.<emph.end type="italics"></emph.end> Et quia angulus <emph type="italics"></emph>hei,<emph.end type="italics"></emph.end> hoc eſt <emph type="italics"></emph>ead<emph.end type="italics"></emph.end> æquatur angu<lb></lb>lo <emph type="italics"></emph>fae:<emph.end type="italics"></emph.end> erit angulus compoſitus <emph type="italics"></emph>fad<emph.end type="italics"></emph.end> grad. 50; & angulus <lb></lb>reſiduus <emph type="italics"></emph>dax,<emph.end type="italics"></emph.end> nimirum angulus reflexionis grad. 40. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Motum verticalem pentagoni à plano reflectere ad angulum datum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Simili modo in pentagono motum verticalem reflectemus <lb></lb>ad angulum datum. ſi ducatur <emph type="italics"></emph>ac,<emph.end type="italics"></emph.end> verticalis; & angulus <emph type="italics"></emph>gaf<emph.end type="italics"></emph.end><lb></lb>ſiat ſemiſſis complementi ad angulum quæſitum. </s><s>Ductâ enim <lb></lb>ex <emph type="italics"></emph>a<emph.end type="italics"></emph.end>p arallelâ motui reflexo <emph type="italics"></emph>fi,<emph.end type="italics"></emph.end> erit angulus reliquus à parallelâ, <lb></lb>& plano contentus æqualis angulo quæſito. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De lineâ motûs reflexi, & motu proiectorum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Verùm contra hucus〈qué〉 dicta de motu reflexo poterit quis dubitare: <lb></lb>quamobrem hic ex occurſu plani, motus at〈que〉 impulſus figuræ rectilineæ <lb></lb>ſecetur in duo quadrata: in probl: verò 4 & 5 in duo parallelogramma: <lb></lb>quorum baſis communis ſit radius, ſeu ſemidiameter figuræ motûs; alti<lb></lb>tudo verò eiuſdem ſegmenta.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo hie motum conſiderari naturalem à grauitate: <lb></lb>quem prop: 12. oſtendi eo modo augeri, quo triangulum ſibi <lb></lb>ſimile manens. </s><s>Cùm <expan abbr="itaq;">itaque</expan> plaga inducatur non <expan abbr="abſq;">abſque</expan> ali<lb></lb>quâ morulâ; neceſſe et illum impulſum, quem plaga abſumit, <lb></lb>& quem centrum gravitatis retinet ad ſe librandum, habere <lb></lb>vim quadrati. </s><s>At verò in quadraturâ circuli motu utimur ſi<lb></lb>milari: Vnde neceſsè eo modo dividi, quo linea recta, ſeu pa<lb></lb>rallelogrammum. </s></p> <pb xlink:href="063/01/075.jpg"></pb> <p type="main"> <s><emph type="italics"></emph>Inſtabis ſi totus impulſus, VG trianguli ABC, ſecatur in duo qua<lb></lb>drata DE at〈qué〉 EA: quia motus eſt æqualis impulſui; erit ut quadra<lb></lb>tum DE ad quadratum EA, ita motus centri ad motum reflexum à <lb></lb>plagâ in DG. maior ita〈qué〉 DG quàm AE: nimirum in ratione duplica<lb></lb>tà eius, quam habet AE ad DE: ac proinde angulus reflexionis minor <lb></lb>angulo GDH.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo cùm motus augeatur pro ratione impulſús; hu<lb></lb>ius verò incrementa pro ratione illius morulæ, in quâ perfici<lb></lb>tur plaga, habeant rationem quadrati; neceſse <expan abbr="quoq;">quoque</expan> motum <lb></lb>inter ſe conferri ut quadrata. </s><s>Quod confirmatur à poſterio<lb></lb>ri. </s><s>Conſtat experientiâ, <expan abbr="atq;">atque</expan> omnium aſſenſu pilam reflecti <lb></lb>ad angulos æquales: hoc autem nullâ ratione fieri poteſt, niſi <lb></lb>motus ad ſe referantur ut quadrata. </s><s>Aſſumatur enim figura <lb></lb>prop: 39: in quâ angulus incidentiæ CDA æquatur angulo <lb></lb>reflexionis IAB: dico impulſum, & qui hunc ſequitur motum <lb></lb>centri grauitatis reſiduum à plagâ, eandem rationem habere <lb></lb>ad motum inde reflexum, quam habet quadratum EF ad qua<lb></lb>dratum FD, hoc eſt per prop: 12. illorum durationem eſſe <lb></lb> <arrow.to.target n="fig18"></arrow.to.target> <pb xlink:href="063/01/076.jpg"></pb>ut EF. FD latera eorundem quadratorum. </s><s>Producatur enim <lb></lb>linea DE motûs reflexi: <expan abbr="atq;">atque</expan> ipſi DI ſumatur parallela EG <lb></lb>ex G verò demittantur perpendiculares GH. GK. </s><s>Quia <expan abbr="itaq;">itaque</expan> <lb></lb>recta ED eſt perpendicularis ad AB, & angulus CDA aſſumptus <lb></lb>æqualis angulo IDB; erit angulus reliquus CDE æqualis angu<lb></lb>lo reliquo EDI, hoc eſt illi æquali HEG. & cùm rectus ſit <expan abbr="uterq;">uterque</expan> <lb></lb>angulus EFD. EHG, <expan abbr="atq;">atque</expan> HEG æqualis EDF; erunt triangula EFD. <lb></lb>GHE ſimilia. </s><s>Igitur ut EF ad FD, ita HG, ſeu EK ad EH. </s><s><expan abbr="Neq;">Neque</expan> <lb></lb>verò dicendum in hac demonſtratione circulum committi. ſi <lb></lb>quidem hic ab effectu per experientiam cognito, ea principia <lb></lb>ſtabiliuntur; ex quibus propoſitione 39. aliâ viâ notis hic idem <lb></lb>effectus tanquam illorum concluſio infertur, </s></p> <figure id="id.063.01.076.1.jpg" xlink:href="063/01/076/1.jpg"></figure> <p type="main"> <s><emph type="italics"></emph>Obijcies. </s><s>Motum reflexum non augeri ea modo, quo triangulum ſibi <lb></lb>ſimile manens: non igitur ad ſe referri ut quadrata. </s><s>Et de impulſu <lb></lb>quidem reflexo videtur manifeſtum: Quod hic à percußione oriatur, <lb></lb>at〈qué〉 continuò, ex quo cæpit, minuatur. </s><s>Idem verò probatur de impul<lb></lb>ſu, quem centrum grauitatis retinet ad ſe librandum. </s><s>Nam cùm prin<lb></lb>cipium huius augmenti ſit grauitas, motus verò reflexus fiat in partes <lb></lb>oppoſitas grauitati; nequit grauitas influere in hunc motum: quin poti<lb></lb>us eidem reniti, & grauitando ipſum minuere: uti manifeſtum in fine <lb></lb>motûs reflexi & in arcum ſinuati.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo nos hic principia motûs reflexi inter ſe confer<lb></lb>re: quæ conſtat vim quadrati habere: licet fortè in progreſſu <lb></lb>mutari contingat illam proportionem. </s><s>An verò grauitas in<lb></lb>fluat in motum reflexum dubitari poteſt. </s><s>Nam ſi ita, idem <lb></lb>videtur dicendum de motu proiectorum: nullus proinde mo <lb></lb>tus rectus. </s><s>At verò ſi proiecta non ferantur lineâ rectâ, quâ ra<lb></lb>tione ictus certi eſſe poſſunt? et tamen conſtat eſſe inter Scyt<lb></lb>has adeo ſagittandi peritos, ut pomum vertici impoſitum, aut <pb xlink:href="063/01/077.jpg"></pb>nummum inter duos digitos contentum excutiant. </s><s>Mulieres <lb></lb><expan abbr="quoq;">quoque</expan> Balearicæ non priùs cibum ſuis filijs præſtabant, quàm <lb></lb>iactu fundæ eundem attigiſſent. </s><s>Et ne remotiora ſectemur, <lb></lb>an non ictus tormentorum adeo certi; ut globi ab his emiſſi per <lb></lb>ipſum os tormenti oppoſiti ſe inferant? </s></p> <p type="main"> <s>Pro quo notandum ex his, quæ in libro de motu poſtea di<lb></lb>centur, <expan abbr="utrumq;">utrumque</expan> <expan abbr="motũ">motum</expan>, videlicet naturalem, & qui ex impulſu <lb></lb>cauſatur, efficienter quidem à principio interno mobilis; de<lb></lb>terminatiuè verò ab ideâ provenire. </s><s>Quæ ſi ab extra veniat, <lb></lb>motum non naturalem; idea verò interna & à principijs eſſen<lb></lb>tialibus fluens motum naturalem determinat: <expan abbr="atq;">atque</expan> ſi ad mun<lb></lb>di centrum dirigat, grauitas nun cupatur. </s><s>Fit autem hic mo<lb></lb>tus mediante impulſu: qui cùm neceſſariò producatur, neceſsè <lb></lb>hunc in deſcenſu continuò augeri per prop: 10. </s><s>Idea verò ex<lb></lb>terna impulſum determinat ſimilem vel diſſimilem grauitati. </s><lb></lb><s>Et ſiquidem impulſus accedat ſimilis illi, qui prouenit à gravi<lb></lb>tate; dico ab <expan abbr="utroq;">utroque</expan> ſimul fieri motum: ſiue impulſus ſit ma<lb></lb>ior, ſiue minor gravitate. </s><s>Et impulſum quidem maiorem <lb></lb>grauia incitare videtur manifeſtum. </s><s>Quòd ab hoc, non verò <lb></lb>à gravitate fiant incrementa motûs: qui in omni puncto eſt <lb></lb>maior gravitate, per prop: 11. </s><s>Idem verò dicendum de im<lb></lb>pulſu minori. propterea quòd grauitas non niſi mediante im<lb></lb>pulſu moueat: omnis verò acceſſio impulſûs auget præexi<lb></lb>ſtentem, & ad motum incitat velociorem, per poſit: 4. </s><s>Quôd <lb></lb>ſi motus ſit non naturalis, cuiuſmodi ſagittæ, vel erit contrari<lb></lb>us abſolutè; qui nimirum fit per eandem lineam rectam: vel <lb></lb>ſubcontrarius, angulum continens cum lineâ deſcenſus mino<lb></lb>rem duobus rectis. </s><s>Ft prioris quidem generis, ſi æqualis ſit <lb></lb>gravitati, nullus omninò fit motus; verùm mobile tum quie<lb></lb>ſcit. </s><s>Propterea quòd deſcenſus grauium fiat mediante im- <pb xlink:href="063/01/078.jpg"></pb>pulſu: Impulſus verò contrarius tollat vel impediat ſuum <lb></lb>contrarium in eadem ratione, totus totum; pars verò partem <lb></lb>proportionalem. </s><s>Igitur ſi minor ſit impulſus gravitate, abla<lb></lb>tâ parte æquali à reſiduâ gravitate fit deſcenſus. </s><s>Quòd ſi ve<lb></lb>rò maior ſit impulſus: erit huius exceſſus principium motûs <lb></lb>ſurſum. </s><s>At verò impulſus ſubcontrarius, ſi angulum conti<lb></lb>neat rectum, vel maiorem recto, cùm illius motus à centro ab<lb></lb>ducat, nullum impulſum videtur gravitas determinare: Vn<lb></lb>de motus ab exceſſu fieri dicendus: <expan abbr="quouſq;">quouſque</expan> æquatio fiat <expan abbr="u-triusq;">u<lb></lb>triusque</expan>, tum enim motu miſto ferri, & in ſpeciem arcüs ſinuari <lb></lb>videtur. </s><s>Quod quidem ſupponere debent, qui dicunt mo<lb></lb>tum proiectorum fieri per lineam rectam: quod nullo modo <lb></lb>eſſet, ſi motu miſto ferrentur ex gravitate <expan abbr="atq;">atque</expan> impulſu. </s><s>Nam <lb></lb>cùm plaga minuat impulſum, gravitas verò eadem maneat; <lb></lb>neceſse latera motûs continuò aliam <expan abbr="atq;">atque</expan> aliam rationem ad <lb></lb>ſe habere. </s><s>Cuius ratio eſſe videtur; quòd gravitas nonniſi <lb></lb>idealiter concurrat ad motum & impulſum: unde per aliam <lb></lb>ideam fortiorem ſuperari & excludi poteſt: ut ad <expan abbr="præſcriptũ">præſcriptum</expan> <lb></lb>huius, non illius moveatur. </s><s>At verò impulſus ſubcontrarij <lb></lb>neceſſariò miſcentur, <expan abbr="actionêsq;">actionêsque</expan> producunt mixtas. </s><s>Eſt hæc <lb></lb>ſententia multùm probabilis, ſed oppoſita magis placet. </s><s>Nam <lb></lb>cùm motus proiectorum demum ſinuetur manifeſtè: id non<lb></lb>niſi ex impulſu gravitatis eſſe poteſt: qui mobile ex illâ lineâ <lb></lb>rectâ ad centrum abducit. </s><s>At verò hoc contingit non ſolùm <lb></lb>æquatâ gravitate, ſed etiam cùm maior eſt impulſus: Igitur in <lb></lb>reliquum impulſum, quo moveri cæpit, grauitas influit: ac <lb></lb>proinde neceſse hunc motum eſſe miſtum. </s><s>Aſſumatur enim <lb></lb>altitudo ſagittæ AC, cùm iam manifeſtè incipit declinare à li<lb></lb>neâ horizonti parallelâ: cuius motus ſinuoſus AFG <expan abbr="eritq;">eritque</expan> AG <lb></lb>maior quàm AC. </s><s>Dico impulſum eſſe maiorem gravitate. <pb xlink:href="063/01/079.jpg"></pb> <arrow.to.target n="fig19"></arrow.to.target><lb></lb>Quòd ſi enim æqualis eidem eſſet, motus medius fieret per di<lb></lb>ametrum AG. minor verò effectus grauitate, motum ſinuo<lb></lb>ſum terminabit inter C & G. quod quidem in ſyphonibus <expan abbr="atq;">atque</expan> <lb></lb>effluxibus aquæ ſinuoſis magis licebit experiri. </s></p> <figure id="id.063.01.079.1.jpg" xlink:href="063/01/079/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>Quam proportionem habeat impulſus <lb></lb>ad gravitatem.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Quod verò obijcitur, ſi motus eâ ratione ſit miſtus, cúm plaga mi<lb></lb>nuat impulſum, grauitas verò eadem maneat; nunquam ad deſtina<lb></lb>tam metam mißilia, quæ ad libellam diriguntur, perventura.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo gravitatem ad impulſum <emph type="italics"></emph>VG<emph.end type="italics"></emph.end> ſagittæ, valde exi<lb></lb>guam proportionem habere: ac proinde ob inſenſilem cur<lb></lb>vitatem pro lineâ rectâ æſtimari. </s><s>Quod quidem hac ratione <lb></lb>videtur ſuaderi. </s><s>Cùm in lapſu grauium impulſus in omni <lb></lb>puncto motús ſit maior gravitate per prop: 11; <expan abbr="atq;">atque</expan> eo modo <lb></lb>augeatur, quo triangulum ſibi ſimile manens, per prop: 12: <pb xlink:href="063/01/080.jpg"></pb>habebit rationem duplicatam ſuæ longitudinis ad datum tri<lb></lb>anguli latus, quod gravitati, VG unius libræ, ſit æquale. </s><s>Vt <lb></lb>ſi promoviſſe dicatur eo lapſu prius quidem ad digitos 4. </s><s>In<lb></lb>de ad paſſus 3: habebit impulſus hoc intervallo collectus ad <lb></lb>illum rationem, quam 1804. ad 1. </s><s>At verò ſi pila deſcen<lb></lb>dat ad totidem paſſus; minùs offendit, quàm ſi eadem ex illâ <lb></lb>diſtantiâ proijciatur. </s><s>Eſt autem impulſus ab arcu, ſeu fundâ <lb></lb>his muitò vehementior: ut nihil dicam de Cylindro bellico. </s><lb></lb><s>Deinde dico ab huius modi Iobolis nonignorari hanc motûs <lb></lb>curvitatem: unde etiam rationem habent diſtantiæ. aliter e<lb></lb>nim ex magno, aliter ex parvo intervallo ictum dirigunt: <expan abbr="neq;">neque</expan> <lb></lb>ſolùm intervalli, ſed etiam ictûs vehementiæ modum expen<lb></lb>dunt. </s></p> <p type="main"> <s><emph type="italics"></emph>Dices quamobrem alij alijs feticiùs ſcopum aſſequuntur: tamctſi ijs<lb></lb>dem inſtrumentis uſi, eadem〈que〉 collineatione factâ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo id ex diverſo pupillæ ſitu provenire. accidit e<lb></lb>nim his, quemadmodum ſi quis digito preſſam loco moveat: <lb></lb>tum ſiquidem alius rei, <expan abbr="atq;">atque</expan> imaginis locus. unde cùm ictum <lb></lb>dirigant ad locum viſum, quid mirum à loco verò aberrare. </s><lb></lb><s>Ita quidem in motu proiectorum; quæ lineam ſequuntur ex <lb></lb>angulo recto, aut recto maiore. </s><s>Quòd ſi cum motu vertica<lb></lb>li angulum <expan abbr="contineãt">contineant</expan> minorem recto; quia tum mobile fit pro<lb></lb>pius centro, videbitur hic gravitas capere augmentum eo la<lb></lb>pſu: quod ſimilis videatur motui inclinato; in quo velocitas <lb></lb>continnò augetur, Dico nihilominus eandem eſſe <expan abbr="vtro-biq;">vtro<lb></lb>bique</expan> rationem. </s><s>Alia autem eſt ratio motûs inclinati. propterea <lb></lb>quòd pars gravitatis maneat extra hypomochlium: ac proin <lb></lb>de impulſum producat ſibi æqualem: qui in deſcenſu conti- <pb xlink:href="063/01/081.jpg"></pb>nuò augetur. </s><s>In proiectis verò tota gravitas ſuperatur ab <lb></lb>impulſu, <expan abbr="atq;">atque</expan> in lineam trahitur nonnaturalem. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Datâ Proportione impulſûs ad grauitatem, lineam motûs <lb></lb>inflexi inuenire.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Data ſit proportio impulſûs ad gravitatem, VG ſeſcupla. <lb></lb>aſſumatur autem recta AB via motûs, ad AC motum verti<lb></lb>calem in eadem ratione: & ſecetur AB in ſegmenta æqualia <lb></lb>ALMNOPB. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> maneret eadem proportio im<lb></lb>pulſus ad gravitatem, motus medius eſſet diameter parallelo<lb></lb>grammi ABDC per prop: 32. </s><s>At verò quia plaga impul<lb></lb>ſum continuò abſumit: gravitas verò eadem manet; neceſſe <lb></lb>continuò mutari hanc proportionem: pro ratione nimirum <lb></lb>ſpatij tranſmiſſi Igitur abſumptâ parte impulſus æquali AL: <lb></lb>principium motûs reliqui determinat AT diameter parallelo<lb></lb>grammi APTC in E. propterea quòd TC ſit æqualis reſiduo <lb></lb>impulſui LB. </s><s>Rurſum peractâ plagâ æquali AM; erit princi<lb></lb>pium motûs in F communi ſectione MF. <expan abbr="atq;">atque</expan> AS lineæ dia<lb></lb>gonalis parallelogrammi AOSC, <expan abbr="eademq;">eademque</expan> ratione invenie<lb></lb>mus puncta reliqua motûs ſinuoſi in GH I&c. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Linea motûs proiectorum non eſt circulus, ne〈que〉 ulla ſectionum <lb></lb>conicarum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Supponamus primùm eſſe lineam circularem.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Quoniam <expan abbr="itaq;">itaque</expan> triangula APT, AEV ſuntſimilia, erit FV ad <lb></lb>AV, ut AP ad TP. eſt autem TP pars 5 AP per probl: 4 Igitur & <pb xlink:href="063/01/082.jpg"></pb> <arrow.to.target n="fig20"></arrow.to.target><lb></lb>AV pars quinta EV. </s><s>Et quia quadratum E Veſt æquale re<lb></lb>ctangulo contento AV, <expan abbr="atq;">atque</expan> huius complemento ad diame<lb></lb>trum circuli; EV verò aſſumpta partium 10, qualium AV eſt <lb></lb>2; erithuius complementum partium 50: & tota diameter 52. </s><lb></lb><s>Rurſum quia CG eſt tripla AC: illius verò quadratum æqua<lb></lb>le rectangulo contento AC, atq, huius complemento ad dia<lb></lb>metrum circuli; eſt verò quadratum CG partium 900, & AC <lb></lb>partium 10; erit reſiduum ſegmentum partium 90: tota verò <lb></lb>diameter partium 100. eſt verò eadem <expan abbr="quoq;">quoque</expan> partium 52. </s><lb></lb><s>Non igitur linea motûs AEF GHI eſt peripheria circuli. </s><lb></lb><s>Dico <expan abbr="neq;">neque</expan> eſſe parabolam. </s><s>Sit enim ſi fieri poteſt, linea para<lb></lb>bolæ. erit <expan abbr="itaq;">itaque</expan> ut recta AC ad rectam AV, ita quadratum <lb></lb>ſemiordinatæ CG ad quadratum ſemiordinatæ VE. et quia <lb></lb>CG eſt tripla VE; erit eiuſdem quadratum noncuplum ad illud <lb></lb>quadratum. </s><s>At verò AC ad AV eſt ut 10 ad 2, hoc eſt quin<lb></lb>tupla. non igitur ut AC ad AV, ita quadratum CG ad qua<lb></lb>dratum VE: ac proinde linea AE FG &c. non eſt parabola. <pb xlink:href="063/01/083.jpg"></pb>Sit iam ſi fieri poteſt, hyperbole. aſſumatur verò huius diame<lb></lb>ter partium 8, qualium AC eſt 10, & AV 2. </s><s>Igitur triangu<lb></lb>lum rectangulum contentum AV, & latere compoſito ex AV <lb></lb><expan abbr="atq;">atque</expan> diametro figuræ erit partium 20: <expan abbr="triangulũ">triangulum</expan> verò <expan abbr="contentũ">contentum</expan> <lb></lb>AC <expan abbr="atq;">atque</expan> latere compoſito ex AC & diametro eiuſdem figuræ, <lb></lb>partium 180: huius verò ratio ad illud noncupla. eſt autem <lb></lb>quadratum <expan abbr="quoq;">quoque</expan> ſemiordinatæ CG ad quadratum alterius <lb></lb>ſemiordinatæ VE in eadem ratione. propterea quòd latus CG <lb></lb>ſit triplium lateris VE. </s><s>Cùm <expan abbr="itaq;">itaque</expan> eandem rationem ad ſe <lb></lb>habeant rectangula ſubſegmentis axis hyperbolæ, quam habent <lb></lb>quadrata ſemiordinatarum; erit permutando eadem <expan abbr="quoq;">quoque</expan> ra<lb></lb>tio rectangulorum ſub ſegmentis axis ad quadrata ſuarum ſe<lb></lb>miordinatarum: ac proinde puncta EG in eadem hyperbole. </s><lb></lb><s>Rurſum verò quoniam AOS. AKF ſunt triangula ſimilia; <lb></lb>& AO <expan abbr="quadruplũm">quadruplumm</expan> OS; erit <expan abbr="quoq;">quoque</expan> KF quadruplum AK: <lb></lb>& AK partium 5, qualium KF eſt 20. triangulum ergo <lb></lb>rectangulum contentum AK <expan abbr="atq;">atque</expan> latere compoſito ex AK <lb></lb>& diametro figuræ erit partium 65: rectangulum verò conten<lb></lb>tum AV, & latere compoſito ex AV <expan abbr="atq;">atque</expan> diametro eiuſdem <lb></lb>figuræ, partium 20. eſt autem ratio 65 ad 20 minor, quàm ſit <lb></lb>quadrati KF ad quadratum VE: Igitur permutando non ea<lb></lb>dem eſt ratio rectangulorum ſub ſegmentis axis ad quadrata <lb></lb>ſemiordinatarum: ac proinde puncta EF non continentur in <lb></lb>lineâ hyperbolæ. </s></p> <figure id="id.063.01.083.1.jpg" xlink:href="063/01/083/1.jpg"></figure> <p type="main"> <s>Demum <expan abbr="neq;">neque</expan> ellipſin eſſe hanc lineam motûs, ita oſtendo. </s><lb></lb><s>Producatur AC in Z: quam ſecetperpendicularis IZ. </s><s>Cùm <lb></lb><expan abbr="itaq;">itaque</expan> in I gravitas fiat æqualis impulſui; erit IZ maior omni<lb></lb>bus rectis, quæ ex lineâ motûs cadunt perpendiculariter ad dia<lb></lb>metrum AZ: ac proinde erit ſemidiameter figuræ. </s><s>At ve<lb></lb>rò IZ æquatur ſemidiametro AZ: oportebat verò eſſe in<lb></lb>æqualem: non igitur puncta AEFGHI in ellipſi continentur. </s></p> <pb xlink:href="063/01/084.jpg"></pb> <p type="main"> <s><emph type="center"></emph>De cauſa inæqualis reflexionis<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Suppoſui hactenus in reflexione figuras rectilineas æqua<lb></lb>lem dare & recipere impulſum. quod licet ut plurimum fiat; <lb></lb>non tamen eſt neceſſarium: ſed <expan abbr="quandoq;">quandoque</expan> percutiens mino<lb></lb>rem, <expan abbr="quandoq;">quandoque</expan> nullum recipit impulſum. </s></p> <p type="main"> <s>Et ſiquidem totam dedit plgam, <expan abbr="nullamq;">nullamque</expan> recepit, non re<lb></lb>flectit: verùm à plagâ conquieſeit. </s><s>Ex parte verò plagæ mo<lb></lb>tum continuat centrum gravitatis per lineam tangentem cir<lb></lb>culi; cuius centrum eſt contactus, & intervallum diſtantia eiuſ<lb></lb>dem centri gravitatis. </s><s>At cùm minor eſt plaga à percuſſo, <lb></lb>mutatur ratio motùs reflexi: propterea, quòd centrum præ<lb></lb>dominatur. </s><s>Inæqnaliter autem reflecti corpora, ſi materiâ <lb></lb>differant, quantumvis eandem figuram, & magnitudinem, <lb></lb>quin et gravitatem habeant, conſtat: ſi pila plumbea, ferrea, la<lb></lb>pidea, oſſea, lignea, coriacea ex eadem diſtantiâ terræ, aut pari<lb></lb>eti allidatur. </s><s>Cauſa huius inæqualitatis videtur non niſi ex <lb></lb>naturâ impulſûs priùs cognitâ obtineri. </s><s><expan abbr="Neq;">Neque</expan> enim cur inæ<lb></lb>qualiter recipiatur, conſtare poteſt; niſi quid, & quomodo in <lb></lb>corporibus tecipiatur, conſtet. </s><s>De quo alibi: hic verò non ni<lb></lb>ſi ea, quæ ad inſtitutum facere videntur, delibabo. </s></p> <p type="main"> <s>Notandum ergo primò, ſi mobile percutiat aliud, produce<lb></lb>re impulſum æqualem illi, quo ipſum movetur: globus enim <lb></lb>percuſſo æquali, eadem celeritate hunc movet: quod non niſi <lb></lb>ab impulſu æquali eſſe poteſt. </s><s>At ſi maior aut minor gravi<lb></lb>tas ineſt percuſſo, inæqualiter movetur: velociùs quidem cui <lb></lb>minor, tardiùs cui maior ineſt gravitas. </s><s>Vnde apparet cun<lb></lb>dem impulſum in paruo ſubiecto colligi & intendi; in magno <lb></lb>eſſe remiſſiorem: propterea, quòd alia ſit proportio moven<lb></lb>tis ad mobile. </s><s>Sed dubitabis an in percuſſo æquali idem ſit <pb xlink:href="063/01/085.jpg"></pb>impulſus. </s><s>Nam ſi in lineâ rectâ plures globos diſponas ſibi con<lb></lb>tiguos & æquales; percuſſo primo ultimus movetur, omni<lb></lb>bus alijs immotis. </s><s>Si ergo primus in ſecundo, hic in tertio <lb></lb>producit impulſum æqualem illi, quo ipſe moveretur; ſequi<lb></lb>tur à plagâ, quæ unum movere poteſt, moveri poſſe quolibet <lb></lb>ſpatio <expan abbr="abiunctũ">abiunctum</expan>: <expan abbr="perq;">perque</expan> globos infinitos illam vim extendi, <expan abbr="eſſeq;">eſſeque</expan> <lb></lb>infinitam. E contra vero, ſi illâ ſerie continuò dereſcit pla<lb></lb>ga; ut minor ſit in tertio quàm in ſecundo, et in hoc quàm in <lb></lb>primo: ſint globi numero 20. & ſingulorum pondus librale. <lb></lb>habebit ergo pIaga 20-minorem rationem ad totum impul<lb></lb>ſum quàm ſubuigecuplam; hoc eſt quàm habeat gravitas illius <lb></lb>globi ad omnium grauitatem collectam. impulſus ergo minor, <lb></lb>quàm ut moveat pondus librarum 20; maior autem quàm ſit <lb></lb>reſiſtentia lib: 10 aut 15; percuſſo primo non movebit ulti<lb></lb>mum. </s><s>Nam ſi totus impulſus minor eſt grauitate totâ, erit <lb></lb><expan abbr="quoq;">quoque</expan> pars impulſûs minor illâ gravitate, quæ in eadem eſt ra<lb></lb>tione ad totam gravitatem. </s><s>Et cùm pars 20 impulſus neque<lb></lb>at movere pondus lib: 1. <expan abbr="neq;">neque</expan> à plagâ minore quàm ſit pars 20 <lb></lb>movebitur. </s><s>Hoc autem eſt contra experientiam. videmus <lb></lb>enim quovis numero interpoſitis globis æqualibus ultimum <lb></lb>moveri ex eadem plagâ, æquali cum primo celeritate. </s><s>Dein<lb></lb>de ſi plaga decreſcens nequit ultimum movere; ſunt verò & in<lb></lb>termedij <expan abbr="abſq;">abſque</expan> motu; erit plaga infinita in mobili, <expan abbr="abſq;">abſque</expan> eo <lb></lb>quòd ullam partem moveat. </s><s>Augeatur enim numerus glo<lb></lb>borum in eâ ratione, in quâ plaga: <expan abbr="eritq;">eritque</expan> impulſus ab ultimâ <lb></lb>plagâ in eadem ratione, hoc eſt minori, quàm ut movere poſ<lb></lb>ſit ultimum globum. </s><s>Quod cùm à ratione & experientia ſit <lb></lb>alienum, dicendum omnes globos, quantumvis numero <lb></lb>augeantur, ab hoc impulſu peruadi <expan abbr="Neq;">Neque</expan> ſequitur virtutis fi<lb></lb>nitæ actionem eſſe inſinitam. non enim ab extra, ſed à princi<lb></lb>pio interno mobilis producitur impulſus; ut ſuo loco oſten- <pb xlink:href="063/01/086.jpg"></pb>dam: factâ determinatione à ſimili per contactum. </s><s>Quid <lb></lb>ergo mirum mobilia infinita impulſum coacervare infinitum? <lb></lb><expan abbr="Atq;">Atque</expan> ex his multa arcana panduntur: cùm tanta ſit vis ſimili<lb></lb>tudinis; ut nullis locorum intervallis definiantur ex eâ naſcen<lb></lb>tes amores: <expan abbr="neq;">neque</expan> iam miremur cœleſtes influxus his illicibus <lb></lb>uno ceu momento trahi. </s><s>Dices Quid ſi inæquales ſint globi <lb></lb>& continuò minores: an ab infinito numero erit motus? nam <lb></lb>ſi ita, movebitur ſanè ultimus celeritate infinitâ. </s><s>Reſpondeo, <lb></lb>cùm minor globus eadem celeritate feratur à minori impulſu; <lb></lb>movebitur ab incipiente, & necdum perfectâ plagâ: ac proin<lb></lb>de reliquus impulſus motum maioris continuabit per poriſma <lb></lb>2. </s><s>Ex quo illud mirabile; in eodem inſtanti ab uno principio <lb></lb>motûs fluere infinitos inter ſe inæquales. </s><s>Licet verò in infi<lb></lb>nito daretur ultimus, negamus tamen hunc celeritate move<lb></lb>ri infinitâ: propterea quòd impulſus continuò minuatur iuxta <lb></lb>decrementum illarum Sphærularum. </s><s>At verò infinitum quis <lb></lb>terminabit? Cùm ergò dicimus numerum infinitum, ſynca<lb></lb>tegorematicè intelligi volumus, quouis dato maiorem: <expan abbr="atq;">atque</expan> <lb></lb>in hoc ſicuti cum numero decreſcit moles, ita velocitas mo<lb></lb>tûs augeretur. </s><s>Iiſdem connexa, & à vulgi opinione remota <lb></lb>ſunt hæc. </s></p> <p type="main"> <s><emph type="italics"></emph>Plagam infinitam dare <expan abbr="absq;">absque</expan> eo, quòd percutiens mo<lb></lb>ueatur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Movere corpus in quâcun〈qué〉 diſtantiâ, abs〈qué〉 eo, quòd <lb></lb>ullus in medio ſit motus.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Motum eodem inſtanti producere in infinitum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nihil ergo mirum inſtante motu terræ, priuſquam hæc con- <pb xlink:href="063/01/087.jpg"></pb>cuti & tremere incipiat, <expan abbr="atq;">atque</expan> etiam eâ immotâ ruere ædificia: <lb></lb>homines pedibus inſiſtere non valentes collabi & vacillare: fa<lb></lb>ctâ enim plagâ in viſceribus terræ medijs immotis impetus huc <lb></lb>ſe effundit: quemadmodum percuſsâ muri parte oppoſitâ, ea <lb></lb>quæ muro hærent, delabuntur. </s><s>Notandum ſecundò. impul<lb></lb>ſum non recipi uniformiter in mobili; ſed receſſu à ſummo vi<lb></lb>gore, quem infert plaga, ſenſim attenuari tam in profundum, <lb></lb>quàm in latum. </s><s><expan abbr="Itaq;">Itaque</expan> videmus illas partes, quæ ictum exci<lb></lb>pere coguntur, præ alijs frangi & collidi: nequaquam à plagâ <lb></lb>remotiores. </s><s>Quia nimirum cùm <expan abbr="unaquæq;">unaquæque</expan> particula ſuo impul<lb></lb>ſu feratur & incitetur ad motum; dum hæ præcurrere feſti<lb></lb>nant, illæ ob tarditatem ſequi non valent, quà impetus magis <lb></lb>urget, ſi uniones habeant ſolubiles, avelli contingit. </s><s>Ita <lb></lb>quidem in principio motûs, <expan abbr="quouſq;">quouſque</expan> producitur impulſus: <lb></lb>quam tamen inæqualitatem æquat centrum grauitatis, omni<lb></lb>um vim colligendo; cùm ab omnibus urgeatur: <expan abbr="atq;">atque</expan> ita fit, ut <lb></lb>tardiores incitentur, velociores retardentur: quò eodem <lb></lb>cum centro gravitatis motu ferantur. </s></p> <p type="main"> <s>Motus ergo centri eſt principium motûs reliquorum: & cùm <lb></lb>à motu fiat plaga; erit huius motus & ratio in ordine ad cen<lb></lb>trum <expan abbr="Itáq;">Itáque</expan> fit utictus perpendicularis omnium ſit grauiſſimus: <lb></lb>obliquorum verò tantò vim habeat minorem, quantò magis <lb></lb>obliquè ferit: eo enim modo habet hic motus, quo grauitas <lb></lb>in lapſu inclinato. </s><s>Quòd ſi ergo corpora eiuſdem molis & <lb></lb>ſoliditatis, percutias ictu latiore <expan abbr="eóq;">eóque</expan> plano; videbis in medio <lb></lb>plagæ ſitas partes priùs frangi, ijs quæ in ambitu ſunt <expan abbr="quandoq;">quandoque</expan> <lb></lb>illæſis. </s><s>Porro impulſus in mobili, quia à plagâ cæpit, in aliam <lb></lb>plagam deſtinatur. & ſi quidem plagam totam peregit, totus; ſi <lb></lb>partem, in eadem ratione exſolvitur impulſus, ut conſtat ex <lb></lb>propoſ: 37. </s><s>Quin motus in aëre quid aliud, quàm percuſſio <lb></lb>& plaga continuata: unde in aëre craſſiore, licet ab eadem <pb xlink:href="063/01/088.jpg"></pb>viferatur mobile, minor eſt motus. </s><s>Ita in aquâ ob ſolidita <lb></lb>tem & reſiſtentiam maiorem ad minus intervallum plaga cum <lb></lb>motu terminatur. </s><s>An igitur licebit ex proportione motûs <lb></lb>in diverſis <expan abbr="elemẽtis">elementis</expan> coniecturam ſumere illorum gravitatis? an <lb></lb>præter <expan abbr="gravitatẽ">gravitatem</expan> tenacitas partium huc facit? utlicet æquè gra<lb></lb>ves, non <expan abbr="tamẽ">tamen</expan> eadem facilitate findantur: cùm & ab eadem <lb></lb>gravitate percuſſio fiat inæqualis. </s><s>At verò ſi motus eſt per<lb></lb>cuſſio continuata; an poſito vacuo nullus erit motus? an ſem<lb></lb>per movebitur illud mobile? cùm nihil percuti poſſit, <expan abbr="neq;">neque</expan> ab <lb></lb>ullo minuatur impulſus. </s><s>Deinde quâ ratione ſpiritus moven<lb></lb>tur, ſi nullus illorum eſt tactus? an non neceſſe eâ ratione mo<lb></lb>veri, quâ corpora, tranſito priùs medio? cùm diaſtima ſit cor<lb></lb>porum, non verò ſpirituum: qui neq, ſibi ſunt vicini, <expan abbr="neq;">neque</expan> cor<lb></lb>poreis abſunt intervallis: cùm <expan abbr="neq;">neque</expan> loco capiantur. </s></p> <p type="main"> <s>Per accidens tamen moveri videntur, & motum corporeum <lb></lb>adumbrare, per operationem ſenſibilem in medio factam. </s></p> <p type="main"> <s>Quòd ſi ergo ſpiritus ille, qui pacem hic turbat, velit Roma<lb></lb>nos inquietare; non neceſſe hunc per Venetos & loca media <lb></lb>ire, at ſi illam columnam, quam| ferunt Româ huc delatam, <lb></lb>eò referre velit; celeritatem habebit definitam, et non niſi per <lb></lb>loca interiecta movebitur. </s></p> <p type="main"> <s>Notandum Tertio, impulſum alium habere proportionem <lb></lb>ad mobile loco movendum; alium non: ut licet nulli hæreat, <lb></lb><expan abbr="inſiſtatq;">inſiſtatque</expan> non tamen ex lllâ percuſſione ad motum incitari. <lb></lb><expan abbr="atq;">atque</expan> hic impulſus, <expan abbr="quandoq;">quandoque</expan> totum mobile, <expan abbr="quandoq;">quandoque</expan> non niſi <lb></lb>aliquam partem pervadit. </s><s>Et quod attinetilla corpora, quæ <lb></lb>percuſſa loco moventur, in quâ proportione eſſe debeant, di<lb></lb>ctum in porismatis ad prop: 37. </s><s>Dubitatio tamen eſſe poteſt, <lb></lb>quamobrem percuſſo maiori quieſcente <expan abbr="motoq;">motoque</expan> minus <expan abbr="quan-doq;">quan<lb></lb>doque</expan> reſiliat, nam totam dedit plagam; & cùm moveatur ma- <pb xlink:href="063/01/089.jpg"></pb>ius à plagâ ſe abducens, nullam recipere videtur. </s><s>Reſpondeo <lb></lb>id provenire ex inæqualitate motûs. </s><s>Nam cùm tardiùs con<lb></lb>citetur ad motum maius, quàm æquale; in illâ morulâ, priuſ<lb></lb>quam incipiat moveri, reſiſtit: ac proinde repercuſſio fit æ<lb></lb>qualis illi morulæ, quâ veluti hæret in principio motûs. </s><s><expan abbr="Itaq;">Itaque</expan> <lb></lb>fieri poteſt, ut <expan abbr="quandoq;">quandoque</expan> æquali, <expan abbr="quandoq;">quandoque</expan> minori impulſu re<lb></lb>ſiliat: nunquam verò motum maioris conſequatur: ſicuti <expan abbr="neq;">neque</expan> <lb></lb>maior percuſſo minori quieſcere poteſt, aut reflecti. </s><s>At ve<lb></lb>rò illud mobile, quod percuſſum non movetur, neceſſe illam <lb></lb>plagam à minori recipere: nam ſi ab æquali percutiatur ſeu tel<lb></lb>lus, ſeu planctarum unus, locum ſanè mutabit. </s><s>Et ſi quidem <lb></lb>corpus fuerit ſonorum, diu reſonat; cuius partes omnes vi<lb></lb>bratione quadam commoventur. </s><s>Sonus autem ſibi relictus <lb></lb>cum illo tremore ſenſim minuitur & vaneſcit; & non niſi à <expan abbr="cõ-tactu">con<lb></lb>tactu</expan> repentè conticeſcit. </s><s>In corporibus autem ſurdis, quæ <lb></lb>percuſſa nihil aut parum ſonant, vibratio quidem fit, minùs ta<lb></lb>men diuturna: quàm ex impulſu reciprocante fieri ex eo con<lb></lb>ſtat. quòd atomi & corpuſcula minuta in ſuperficie illorum <lb></lb>corporum <expan abbr="eodẽ">eodem</expan> tremore convellantur, & incitentur ad <expan abbr="motũ">motum</expan>. </s><lb></lb><s>Minùs tamen regulariter in his, quàm in corporibus ſonoris <lb></lb>fit reciprocatio motûs ſeu impulſus, ob atomos inæqualiter ſi<lb></lb>tas; à quibus via procurſus & recurſus variè detorquetur. </s></p> <p type="main"> <s>Durat verò impulſus à ſuperficie ultimâ ſe reducens, <expan abbr="rurſúmq;">rurſúmque</expan> <lb></lb>excurrens veluti ſe ipſum perſequendo, <expan abbr="quouſq;">quouſque</expan> plaga conti<lb></lb>nuò decreſcens ſe ipſam abſumpſit. </s><s>Quod quidem in corpo<lb></lb>ribus non continuis, cuiuſmodi lana, promptè fit ob vias mil<lb></lb>le modis interciſas. </s><s>Piſa verò percuſſo ſacco licet conti<lb></lb>nua non ſint, ſonant: propterea, quòd partes ſenſibiles & ſo<lb></lb>num ex le habentes colliduntur: <expan abbr="itaq;">itaque</expan> legumina quò maiora <lb></lb><expan abbr="magisq;">magisque</expan> rotunda, magis reſonant. </s><s>Ita quidem in corpore <lb></lb>habet impulſus: quod licet non mouet localiter, omnes ta- <pb xlink:href="063/01/090.jpg"></pb>men illius partes pervadit. </s><s>In corpore autem vaſtæ molis, <lb></lb>cuiuſmodi tellus, eò <expan abbr="uſq;">uſque</expan> procedit, dum illâ extenuatione <lb></lb>prorſus inſenſilis euadat: & cùm nulla eſt reciprocatio, <expan abbr="neq;">neque</expan> <lb></lb>vibratio contingit. </s><s>Tremere tamen interdum ſolum ex in<lb></lb>genti plagâ conſtat: cùm partes vehementer preſſæ reaſſur<lb></lb>gunt. </s><s>At verò <expan abbr="quouſq;">quouſque</expan> una <expan abbr="quæq;">quæque</expan> plaga ſe extendat, necdum <lb></lb>liquet: conſtat ſanè longiſſimè protendi: in magnâ enim di<lb></lb>ſtantiâ auribus terræ admotis ſonum etiam non magnum per<lb></lb>cipiunt excubitores. </s><s>Eſt tamen magna differentia pro qua<lb></lb>litate terræ: cavernoſa enim <expan abbr="multúmq;">multúmque</expan> aëris continens ſo<lb></lb>num longiùs protendit, quàm uliginoſa & paluſtris: & quæ <lb></lb>continua eſt ac veluti concatenata, quàm ſabuloſa & interciſa. </s></p> <p type="main"> <s>Notandum Quartò, impulſum naturâ ſuâ lineam rectam & <lb></lb>viam ſequi percutientis. <expan abbr="itaq;">itaque</expan> ſi perpendiculariter incidat pla<lb></lb>no motum ſeu impulſum producit in directum, ſi nihil obſtat. </s><lb></lb><s>At cùm reſiſtentia maior eſt ex unâ, quàm aliâ parte: ut cùm <lb></lb>trabem longiorem percutimus non in centro gravitatis, ſed in <lb></lb>parte uni extremo propiore: tum motus non fit in directum, <lb></lb>ſed circularis: cuius centrum alterum extremum quieſcens, <lb></lb>& à plagâ magis remotum. </s><s>Quòd ſi percuſſio fiat in centro: <lb></lb>tametſi ad partes remotiores à plagâ minor impulſus ſe exten<lb></lb>dat; quia tamen centrum gravitatis æquationem inducit; <lb></lb>omnes æqualiter & in directum moventur. </s><s>In Sphærâ autem <lb></lb>ſeu globo impetus à plagâ in centrum dirigitur, ſi moveri de<lb></lb>beat: quod alioqui non eſt neceſſarium: <expan abbr="quandoq;">quandoque</expan> enim pla<lb></lb>ga ex obliquo illius partem decerpit. </s><s>At ſi globus alium <lb></lb>percutiat <expan abbr="quacunq;">quacunque</expan> ratione, neceſlariò hæc plaga centrum ſpe<lb></lb>ctat. propterea, quòd <expan abbr="utrumq;">utrumque</expan> centrum <expan abbr="atq;">atque</expan> illorum plaga ſit in <lb></lb>eadem lineâ rectâ. </s><s>Nulla tamen plaga ex obliquo facta to- <pb xlink:href="063/01/091.jpg"></pb>tum impulſum abſumit: cùm non tota vis centri percutiat. ne<lb></lb>ceſſe ergò mobile ab eiuſmodi plagâ motum continuare. </s></p> <p type="main"> <s>Notandum Quintò, hanc differentiam eſſe inter corpora <lb></lb>percuſſa, quæ ex illâ plagâ moventur, & quæ immota manent. <lb></lb>quòd hæc ictum recipiant <expan abbr="reddantq́">reddantque</expan>;, nequaquam illa: pro<lb></lb>pterea, quòd licet ab his contactus fiat, non tamen etiam pla<lb></lb>ga: eſt enim plaga irruptio quædam violenta, & veluti pene<lb></lb>tratio: at verò quæ à plagâ moventur, nullam faciunt irrup<lb></lb>tionem, ſed à plagâ celeriter ſe adducunt: non igitur percu<lb></lb>tere dicuntur. </s><s>Immota verò quia percuſſioni non cedunt <lb></lb>eadem violentiâ irrumpunt <expan abbr="penetrantq;">penetrantque</expan> in ea, à quibus pene<lb></lb>trantur: unde percuti & percutere, & impulſum recipere <expan abbr="da-req;">da<lb></lb>reque</expan> dicuntur. </s><s>Qui ſummus eſt in contactu: Inde verò ſen<lb></lb>ſim attenuatur. </s><s>Et in percuſſo quidem ex illâ vibratione de<lb></lb>mum conquieſcit: in percutiente verò quia priori eſt contra<lb></lb>rius, ipſum retroagit. </s><s>Dices quid ſi dicamus impulſum non <lb></lb>niſi per contrarium impulſum tolli? Nam ſi globus alium per<lb></lb>cutiat ſibi æqualem & quieſcentem, ex illâ communi plagâ in <lb></lb><expan abbr="utroq;">utroque</expan> producitur impulſus: qui globum quieſcentem loco <lb></lb>movet. propterea quòd huic motui nihil ſit contrarium: alte<lb></lb>rum verò ob impulſûs contrarictatem à motu continet. </s></p> <p type="main"> <s>Reſpondeo, licet hæc ratio ſit probabilis, non tamen in alijs <lb></lb>locum habere. </s><s>Nam cùm maiori immoto minor globus allidi<lb></lb>tur, ſi æqualem dat <expan abbr="recipitq́">recipitque</expan>; impulſum, <expan abbr="eſtq́">eſtque</expan> hic contrarius pri<lb></lb>ori, non reſiliet; verùm à motu conquieſcet. </s></p> <p type="main"> <s>Dices à maiori corpore ictum fieri maiorem; ac proinde ab <lb></lb>huius exceſſu fieri illum motum. </s><s>Sed contra, quia velocitas <lb></lb>motûs reflexi non augetur in eá ratione, in quâ illorum cor<lb></lb>porum magnitudo. </s><s>Deinde cùm duo globi æquales ſe <lb></lb>percutiunt in motu, <expan abbr="uterq;">uterque</expan> reflectit: oportebat verò <expan abbr="utrumq;">utrumque</expan> <pb xlink:href="063/01/092.jpg"></pb>quieſcere à motu. </s><s>Dicendum ergo in contactu à plagâ per<lb></lb>fectâ impulſum exſpirare: & ſi percuſſum non cedat, ſed re<lb></lb>nitatur, alium impulſum ſibi comparare ex illâ plagâ: <expan abbr="cúmq;">cúmque</expan> <lb></lb>æqualem; cùm ex toto eſt immotum. </s><s>At cùm à plagâ ſe ab<lb></lb>ducens locum mutat ſeu totum, ſeu ſecundùm partem, minu<lb></lb>itur in eadem ratione hic impulſus. </s><s><expan abbr="Itaq;">Itaque</expan> ſi corpus per<lb></lb>cuſſum in ſe ipſum ſidit <expan abbr="ceditq;">ceditque</expan>: ut lana, cera, argilla, plum<lb></lb>bum; quia ictus ſenſim emoritur, nulla vel exigua fit reper<lb></lb>cuſſio. </s><s>Et quia huiuſmodi plaga non tota ſimul, ſed divi<lb></lb>ſim recipitur: inde fit, ut impulſus ex ea productus minùs la<lb></lb>tè evagetur: idem enim fit quemadmodum ſi multæ plagæ <lb></lb>exiguæ continuarentur. </s><s>At cùm ſolidum corpus <expan abbr="firmumq;">firmumque</expan> <lb></lb>percutitur; quia totam plagam ſimul admittit, omnia latè con<lb></lb>tremiſcunt. </s><s>Corpus ergo cùm incidit alteri, aut totum dat <lb></lb>impulſum ſimul & confertim; aut in plures veluti plagas hunc <lb></lb>partitur. </s><s>Et ſi ita, non reflectit percutiens. </s><s>Quòd ſi ceden<lb></lb>do demum renitatur; ut cùm partes compreſſæ nequeunt iam <lb></lb>premi; pars illa duntaxat plagæ reflectit. </s><s>At cùm totum dat <lb></lb>impulſum; velloco movetur percuſſum, <expan abbr="idq;">idque</expan> eadem celerita<lb></lb>te vel minori: & ab hoc quidem reflectit pro menſurâ illius <lb></lb>tar ditatis; non autem ab eo, quod celeritate movetur æquali. </s><lb></lb><s>Immotum demum à plagâ aut in ſe ipſo terminat impul<lb></lb>ſum, aut aliò transfert: ut ſi plures globi æquales & <lb></lb>contigui plagam excipiant. & ab illo quidem, <lb></lb>non autem ab his reflectit motus. </s></p> <figure id="id.063.01.092.1.jpg" xlink:href="063/01/092/1.jpg"></figure> <pb xlink:href="063/01/093.jpg"></pb> <p type="main"> <s><emph type="center"></emph>PARS QVARTA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>De percuſsionibus.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>QVID COLLISIO ET FRACTVRA.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>COrpora invicem colliſa aut mutant figuram, aut ſunt <lb></lb><expan abbr="abſq;">abſque</expan> mutatione. </s><s>Mutatur autem figura partis unius plu<lb></lb>riumue amiſſione, aut <foreign lang="grc">μεταστἁσες</foreign> & ſitu illarum permutato: <lb></lb><expan abbr="atq;">atque</expan> hæc <foreign lang="grc">πιεστὰ</foreign> dicuntur: quorum ſuperficies in proſundum <lb></lb>permutatur, nec dividitur, nec ulla particula aliò transfertur; <lb></lb>quemadmodum fit in aquâ preſsâ. </s><s>Talia verò ſunt Ariſtoteli, <lb></lb>quæ meatus habent vacuos cognati corporis, tametſi forte <lb></lb>mollioribus ſint pleni, in quos partes preſſæ recipiantur. </s></p> <p type="main"> <s>Ita enim pila ærea aquâ, aut aëre plena, à vi externâ preſſa ſu<lb></lb>perficiem gibbam, aliâ ſeu planâ, ſeu concavâ permutat: quan<lb></lb>quam & in totum ſolida ob minores meatus <foreign lang="grc">πιεστὰ</foreign> dicantur. <lb></lb>quæ ſi manentem habeant compreſſionem, <foreign lang="grc">πιλητα</foreign> ut cera, æs, <lb></lb>plumbum, aurum: <foreign lang="grc">ἀπιλητα</foreign> verò ſunt, quæ à compreſſione re<lb></lb>aſſurgunt. </s><s>At verò quæ in percuſſione partem unam plureſ<lb></lb>uè amittunt, <foreign lang="grc">χαταχτὰ χαὶ δραυστὰ</foreign> hoc habent diſcrimen: quòd <lb></lb><foreign lang="grc">χάταζις</foreign> ſit diviſio in partes magnas: ut cùm <expan abbr="lignũ">lignum</expan> aut os fran<lb></lb>gimus: <foreign lang="grc">θραῡσις</foreign> verò in partes plures quàm duas: ut in lapi. <lb></lb>de, teſtà, vitro. </s><s>Cuius rationem reddit Ariſtoteles <foreign lang="grc">πολλοὺς ἔχειν <lb></lb>παραλλάτοντας πὀρους</foreign>; <expan abbr="Itaq;">Itaque</expan> fit ut cùm continui |non ſint, ſed <lb></lb>alternâ permutatione poſiti; facto| initio motûs| non in dire<lb></lb>ctum, ſed tortuosè procedat fiſſura: & plaga una, ob|indiſpo. <lb></lb>ſitionem ſubiecti, non unum producat eſſectum. </s><s>Quem mo- <pb xlink:href="063/01/094.jpg"></pb>tum ſeu impulſum duobus modis fieri docet Ariſtoteles, <foreign lang="grc">ὠ'σει</foreign><lb></lb>ſeu pulſione: ut cùm à tergo motui inſtamus: & percuſſione, <lb></lb>in eo à ſe differentes; ut <foreign lang="grc">ὠσιτ</foreign> ſit <foreign lang="grc">χίνησις ἀπὀ τη̄ς ἅψσεως, πλη<lb></lb>γὴδ<gap></gap> ἀπὸ τῆς φορἀς</foreign>. </s><s>De quo an verum ſit, dubitamus. </s><s>Nam <lb></lb>ſi plures globi inter ſe æquales, & contigui ordine ſequantur; <lb></lb>percuſſo primo ultimus movetur omnibus alijs immotis: ne<lb></lb>ceſſe autem hunc à penultimo moveri, habebit ergo plagam <lb></lb>ex hoc, <expan abbr="abſq;">abſque</expan> eo quòd moveatur. </s><s>Plagam enim fieri ex eo <lb></lb>conſtat: quòd ſi ultimo loco pila cryſtallina aut vitrea excipi<lb></lb>at hunc motum, frangi contingit. </s><s>Dici tamen poteſt pro Ari<lb></lb>ſtotele, ad plagam inducendam motum eſſe neceſſarium: li<lb></lb>cet non plagam totam, ſed huius principium ſequatur. </s><s>Hæc <lb></lb>enim à primo globo incipiens ad ultimum terminatur, & ve<lb></lb>luti pro unâ plagâ habetur. </s><s>At verò quamobrem à percuſſi<lb></lb>one nonnulla frangi contingat, maior eſt dubitatio. </s><s>Nam <lb></lb>certum eſt dictas paſſiones ex impulſu provenire: percutere <lb></lb>enim eſt producere impulſum, & percuti hunc recipere. </s><s>Si <lb></lb><expan abbr="itaq;">itaque</expan> impulſus corpora, in quibus recipitur, frangit; oporte<lb></lb>bit ſanè illam velocitatem motûs conſecuta, quam affert pla<lb></lb>ga, frangi in ipſo motu: quod tamen non fit. </s><s>Vaſa enim vi<lb></lb>trea, priuſquàm ſolidum occurrat, in ipſo lapſu non collidun<lb></lb>tur. </s><s>Sed <expan abbr="neq;">neque</expan> percuſſio per ſe hunc affectum inducit: idem <lb></lb>enim eſt ſiuè percutiat, ſiue percutiatur <foreign lang="grc">θραυστὸν</foreign> vitrum enim <lb></lb>& ſaxo illiſum, & à ſaxo alliſum pari facilitate <expan abbr="frãgitur">frangitur</expan>. </s><s>At verò <lb></lb>cùm pila vitrea aut cryſtallina aliam percutit ſibi æqualem ſeu <lb></lb>ferream, ſeu lapideam, non frangitur ex illo ictu quantumvis <lb></lb>intenſo. </s><s>Videtur ergò huius ratio ex impulſu provenire, non <lb></lb>abſolutè, quem habere poteſt quouis dato maiorem, <expan abbr="atq;">atque</expan> adeo <lb></lb>infinitum <expan abbr="abſq;">abſque</expan> ullâ partium colliſione: ſed ex inæquali mo<lb></lb>do hunc recipiendi. </s><s>Propterea quòd partes propiores plagæ <lb></lb>hunc priùs habeant, <expan abbr="magisq;">magisque</expan> intenſum: qui totus non niſi in a- <pb xlink:href="063/01/095.jpg"></pb>liquâ morulâ producitur. </s><s>Vnde fit ut partes priùs <expan abbr="magisq;">magisque</expan> <lb></lb>percuſſæ, priuſquam æquatio fiat à centro gravitatis, præcur<lb></lb>rere feſtinent: aliæ ſequi non valentes mutuâ diſtractione à ſe <lb></lb>divellantur. cùm nimirum maior eſt vis ad movendum, quàm <lb></lb>illa quies & retentio partium unitiva. </s><s>Frangi enim contingit <lb></lb>illâ parte, quâ impetus magis urget, aut unio minùs reſiſtit: <lb></lb><expan abbr="itaq;">itaque</expan> videmus <expan abbr="quandoq;">quandoque</expan> partes à plagâ remotiores præ alijs <lb></lb>frangi. </s><s>Et quidem <foreign lang="grc">θραυστὸν</foreign> in multa fragmenta diſſilit: ut <lb></lb>vitrum, cryſtallus, teſta, lapis: <expan abbr="Idq;">Idque</expan> præter opinionem-<foreign lang="grc">τὸ χα<lb></lb>ταχτὸν</foreign> verò minùs fallit deſignationem: at〈que〉 in duas <expan abbr="plerumq;">plerumque</expan> <lb></lb>partes abſcedit; factâ diviſione in centro plagæ. </s><s>Quæ qui<lb></lb>dem <foreign lang="grc">χάταξις</foreign> magis procedit, cùm plaga longiùs abeſt à parti<lb></lb>bus extremis: tum enim partem illam, quæ interiacet, pro ve<lb></lb>cte habet: cuius hypomochlium ſunt extrema. </s><s><expan abbr="Atq;">Atque</expan> ita fit, <lb></lb>ut vitro fragili, aut ſtramine fuſtem <expan abbr="craſſiorẽ">craſſiorem</expan> <expan abbr="quandoq;">quandoque</expan> ſrangi <lb></lb>contingat: cùm nimirum maior eft velocitas motûs, quàm re<lb></lb>ſiſtentia: <expan abbr="nullaq;">nullaque</expan> à percuſſo recipitur plaga. </s><s>Oppoſito mo<lb></lb>dò habet fractura: cùm hypomochlium eſt in centro plagæ, <lb></lb>ſeu diviſionis: extrema verò <expan abbr="utrinq;">utrinque</expan> adducuntur. </s><s>Nam in pri<lb></lb>ori quidem <foreign lang="grc">χαταξθ</foreign> duo, hic non niſi unum eſt hypomochli<lb></lb>um. </s><s>Dubitabis ergo, quæ harum fractura ſit magis expedita. </s><lb></lb><s>Dicendum verò impulſum extrema adducentem, ut hypo<lb></lb>mochlium medio ſit loco, prevalere: quod quidem erit ma<lb></lb>nifeſtum, ſi fuſtem parte mediâ præhenſum ijsdem viribus <lb></lb>frangere coneris. </s><s>Huius autem ratio: quòd extrema vim <lb></lb>habeant vectis non impeditam: tantâ ergo acceſſione auge<lb></lb>tur impulſus, quanta huius eſt longitudo: reſiſtentiâ in ſolâu<lb></lb>nione hypomochlij vim habente. </s><s>At verò cùm extrema hy<lb></lb>pomochlio innituntur, & plaga fit in huius centro; impulſus <lb></lb>quidem augetur ex illa remotione <expan abbr="utrinq;">utrinque</expan> ab hypomochlio: <pb xlink:href="063/01/096.jpg"></pb>verùm partium unio <expan abbr="utriq;">utrique</expan> reſiſtit & diviſioni, & vectis de<lb></lb>preſſioni. </s></p> <p type="main"> <s><emph type="center"></emph>De Contrafiſſurâ.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Contrafiſſura eſt rima, ſeu fractura cranij in parte à percuſ<lb></lb>ſione ſeu plagâ diſlante: quam Hippoc: propterea, quòd <lb></lb>ægrum & Medicum <expan abbr="quandoq;">quandoque</expan> latens in perniciem adducat, <lb></lb><foreign lang="grc">ξυμφεζαν</foreign> ſeu infortunium vocat. </s><s>Alij reſonitum; quòd opi<lb></lb>nentur ab ictu reſultum fieri in illam partem. </s><s>Diſſident verò <lb></lb>à ſe: quòd alij non niſi in parte oppoſitâ rimam agi volunt: <lb></lb>alij hoc negant. </s><s>Et licet in parte oppoſitâ, & à plagâ aliquo <lb></lb>modo diſtante fiſſuram admittant; non tamen excedere vo<lb></lb>lunt os plagâ affectum. </s><s>Ita Paulus Ægineta, Guido de Cauli<lb></lb>aco, Vidus Vidius, & Fallopius. </s><s>Probant ex uſu ſuturarum: <lb></lb>quas eo fine à naturâ factas dicunt; quò impetus plagæ in ijs <lb></lb>terminetur: ne noxa alias <expan abbr="quoq;">quoque</expan> partes attingat: quod quidem <lb></lb>erat futurum, ſi Cranium continuum <expan abbr="atq;">atque</expan> unioſſe factum fu<lb></lb>iſſet. A ſuturâ verò impalſum ſiſti, manifeſtum in vitro, aut <lb></lb>ære rupto, deficiente in illam fiſſuram ſono: ita ergo in illis <lb></lb>iuncturis, quibus pectinatim os cranij coit, emori impulſum <lb></lb>volunt. </s><s>Verùm hi imperiti videntur eorum, quæ circa im<lb></lb>pulſum & motum fiunt. </s><s>Nam globi ordine diſpoſiti, <expan abbr="ſeq;">ſeque</expan> <lb></lb>tangentes minùs ſunt continui, quám cranium in illis ſuturis: <lb></lb>in quibus ſi quid ineſt humoris aut ſpiritûs, reliquo in oſſe hu<lb></lb>mori & ſpiritui continuatur: & tamen à primo globo omnes <lb></lb>reliqui impulſum recipiunt: quid ergo obſtat, quò minùs <lb></lb>cranio percuſſo impetus á plagâ totum pervadat: Sonum <lb></lb>autem deficere cogunt partes á fiſſurâ inæqualiter prominen<lb></lb>tes: dum in illâ vibratione partes oppoſitas tangunt: á conta- <pb xlink:href="063/01/097.jpg"></pb>ctu enim finiri ſonum conſtat. </s><s>Simili ergò modo fit, quem<lb></lb>admodum ſi lamina incurvetur: quæ ſonum edit <expan abbr="quouſq;">quouſque</expan> pars <lb></lb>reflexa aliam partem tangat. </s><s>At ſi vitrum perforetur, nihil <lb></lb>obſtat ille hiatus, quò minùs partes reliquæ ſonent. </s><s>Deinde <lb></lb>experientia his adverſatur. </s><s>Nicolaus enim Florentinus ſer: 7 <lb></lb>ſum: 2. tract: 4. cap: 1. teſtatur in Reſtiario contrafiſſuram in <lb></lb>parte oppoſitâ plagæ deprehendiſſe: Et Petrus Paw vidiſſe <lb></lb>ictum os ſiniſtrum bregmatis, quo loco lamdoidi iun<lb></lb>gitur: fiſſo ſyncipitis oſſe dextro, loco ita vicino ſuturæ coro<lb></lb>nariæ, ut pars rimæ eò ſe extenderit. </s><s>Cùm <expan abbr="itaq;">itaque</expan> de facto con<lb></lb>ſtet, cauſam inquirimus. </s><s>Certum eſt adimpulſum referri à pla<lb></lb>gâ provenientem: at cur non in loco plagæ ſed huic oppoſito, <lb></lb>à minori & iam attenuato impulſu hoc patitur? <expan abbr="Neq;">Neque</expan> enim di<lb></lb>ci poteſt ob debilitatem findi illam partem; quam impetus in<lb></lb>venit minùs virium habere ad reſiſtendum: tenuiora enim <lb></lb><expan abbr="minùsq;">minùsque</expan> firma interſunt oſſa, inter os ſiniſtrum bregmatis, & <lb></lb>os ſyncipitis dextrum. </s><s>Qui verò aërem illis cavernulis in<lb></lb>cluſum huc accerſunt, ineptam pro ſe habent rationem: quia <lb></lb>nimirum ex ictu commoveatur: & per totam cranij ſubſtan<lb></lb>tiam pervagatus, in parte demum oppoſitâ allidatur: <expan abbr="reniténsq;">reniténsque</expan> <lb></lb>os illud findat. </s><s>Quomodo enim aër in illis mæandris tortuo<lb></lb>ſis, <expan abbr="atq;">atque</expan> in ſe reductis moveri poteſt quantumuis impetuoſus? <lb></lb>an non mille modis interciſus; dum vel allidit, vel reſilit, priùs <lb></lb>deficiet? Deinde cùm aër ſit mollis & fluidus, nequit illum <lb></lb>impetum ſuſtinere, aut conſervare: & eſto demus <expan abbr="quacunq;">quacunque</expan> vi<lb></lb>olentiâ irrure, <expan abbr="ſibiq;">ſibique</expan> obuiam fieri in parte oppoſitâ: an non <lb></lb>ſuis viribus hac ratione occumbet; dum ipſe ſibi inſtat, & in ſe <lb></lb>ipſum luctatur? An ergo dicendum in figura cranij ſphæroide <lb></lb>ſitam eſſe cauſam? quòd partes ab extra preſſæ magis ſtipen<lb></lb>tur, <expan abbr="magisq;">magisque</expan> reſiſtant diviſioni: à centro autem facto motu à ſe <pb xlink:href="063/01/098.jpg"></pb>diducantur? Ita enim fornices onera videmus ſuſtinere, & <lb></lb>contra niti: quòd ſi à parte internâ ſeu cavâ urgeantur; fatiſce<lb></lb>re & diſſolvi. </s><s>Cùm ergo cranium in modum fornicis ſit re<lb></lb>ductum; non facilè à plagâ ab extra incidente diſſolui poteſt: <lb></lb>parte verò oppoſitâ, quia impetus extra fertur, <expan abbr="eſtq;">eſtque</expan> à centro <lb></lb>perpendicularis, nihil mirum diſſolvi illam continuitatem. </s><lb></lb><s>Accedit quòd impulſus, facto principio motûs à plagâ, non <lb></lb>conquieſcit in parte oppoſitâ; cuius violentia ad maius inter<lb></lb>vallum deſtinatur. </s><s>Cùm <expan abbr="itaq;">itaque</expan> reflecti ſit neceſſe: & pars ſi<lb></lb>niſtra dextrorſum, hæc ſiniſtrorſum abeat; in illâ motuum con<lb></lb>trarietate, partibus à ſe divulſis accidit fiſſura. </s><s>Simili modo <lb></lb>res habetin vitro à baſi circulari in conum faſtigiato. quæ pla<lb></lb>no æqualiter alliſa abrumpit pedamentum: propterea, quòd <lb></lb>impetus à latiſſimâ parte incipiens, <expan abbr="ſéq;">ſéque</expan> reforbens, ab interſe<lb></lb>ctione in cono factâ, rurſum in diuerſa abit. </s><s>Licet verò im<lb></lb>pulſus naturâ ſuâ lineam rectam ſequatur; pro ratione tamen <lb></lb>ſubiecti illam rectitudinem variè, <expan abbr="atq;">atque</expan> interdum circulo per<lb></lb>mutat. </s><s>Et ſi quidem illa corpuſcula, quibus corpora inte<lb></lb>xuntur, continuâ ſerie ſe excipiant, impulſus nullibi offendit: <lb></lb>ſed per atomos uniformes ſe circumagens non niſilongâ mo<lb></lb>râ conſeneſcit. </s><s>At cùm figurâ & ſitu à ſe differunt: quia mil<lb></lb>le modis diſcerpi contingit, citò emoritur. </s><s><expan abbr="Atq;">Atque</expan> ex his vide<lb></lb>tur manifeſtum, quâ ratione impulſus à parte cranij percuſsâ <lb></lb>circumgyrando, <expan abbr="ſibiq;">ſibique</expan> obviam factus in parte oppoſitâ rimam <lb></lb>agat. </s><s>Quia tamen os cranij non inane & vacuum, ſed cere<lb></lb>bro, <expan abbr="multisq;">multisque</expan> vaſis in eo contentis eſt refertum, illa ſimilitudo <lb></lb>à vitro deſumpta non videtur hic convenire. </s><s>Et cùm impul<lb></lb>ſus naturâ ſuà rectitudinem ſequatur; quid cauſæ quòd in ce<lb></lb>rebrum non rectâ feratur; ſed per ambages in oſſe cranij ober<lb></lb>rat? Et ſi ita; an non neceſſe ex illâ vehementiâ ictûs plura e<lb></lb>iuſdem vaſa diſcerpi & collidi? Pro quo notandum naturam <pb xlink:href="063/01/099.jpg"></pb>ſapientiſſimam cerebrum non prorſus contiguum feciſſe: ve<lb></lb>rùm aliquo interuallo inter os cranij & membranas relicto: <lb></lb>quò nimirum aëri, quem arteriæ inſpirant, ſit locus: <expan abbr="cerebrũ">cerebrum</expan> <lb></lb>verò dilatari, <expan abbr="rurſumq;">rurſumque</expan> contrahi valeat. </s><s>Quod quidem ab aë<lb></lb>ris, <expan abbr="Lunæq;">Lunæque</expan> mutatione, fieri obſervamus. </s><s>Turget enim in ple<lb></lb>nilunio cerebrum & veluti ebullit per vulnera: è contra in no<lb></lb>vilunio ſubſidet, & à cranio notabili abeſt intervallo. </s><s>Cùm <lb></lb>ergo ita habeat, optimè videtur natura cauiſſe; quò minùs no<lb></lb>xa eò pertingat: impulſus enim per partes contiguas, non ve<lb></lb>rò à ſe divulſas propagatur. </s></p> <p type="main"> <s><emph type="italics"></emph>Dices. </s><s>Cerebrum incubare oßis <foreign lang="grc">σφηνοείδει</foreign>, & cum cranio per falcem fi<lb></lb>bras〈qué〉 in ſuturam productas connecti: nihil ergo obſtat, quò minùs <lb></lb>hac viá ſe inferat.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo quòd ſi percuſſio fiat in illâ parte, quâ cerebrum <lb></lb>ſuſtinetur, <expan abbr="eſtq;">eſtque</expan> contiguum oſſi, non <expan abbr="abſq;">abſque</expan> periculo fieri pla<lb></lb>gam: unde plenilunij tempore, quòd calva cerebrum attingat, <lb></lb>eiuſmodi ictus ſunt lethales. </s><s>At verò illæ fibræ, quibus ce<lb></lb>rebrum cranio ſe inſerit, & à quibus in æquilibri ſitu detine<lb></lb>tur, via eſſe non poteſt irruenti plagæ: propterea, quòd hu<lb></lb>iuſmodi ſuſpenſoria, nec dura nec rigida ſunt, ſed mollia & <lb></lb>membranoſa filamenta: quæ tendi & laxari facilè poſſunt: pri<lb></lb>uſquam ergo tractio aut pulſio fiat, in illâ relaxatione perit im<lb></lb>pulſus. </s><s>Deinde cùm impetus ſe gyrando, non niſi obliquè <lb></lb>ſtringat illa filamenta, erit tenſio æqualis motioni illarum par<lb></lb>ticularum, quæ ſolo tremore convelluntur: ac proinde inſen<lb></lb>ſilis, nullam ergo violentiam adducet partibus medio loco ſi<lb></lb>tis; quantumuis ictus <expan abbr="quandoq;">quandoque</expan> accidant graves. </s><s>Ita quidem <lb></lb>res habet in fiſſurâ partis oppoſitæ: an verò alijs <expan abbr="quoq;">quoque</expan> locis <lb></lb>non quidem oppoſitis, verùm á plagâ aliquo modo diſiunctis <pb xlink:href="063/01/100.jpg"></pb>contingat, videndum. </s><s>Nam ita fieri opinantur, qui negant <lb></lb>extra ſuturam illius oſſis, in quo recipitur plaga, fiſſuram pro<lb></lb>tendi. </s><s>Cuius rationem aſſignant, quòd pars illa à plagâ affe<lb></lb>cta nimis ſit robuſta: ac proinde in partem proximam, quæ ob <lb></lb>nativam conſtitutionem minùs reſiſtere valet, illa violentia <lb></lb>ſe recipiat. </s><s>Et quidem experientia his videtur favere. </s><s><expan abbr="Quan-doq;">Quan<lb></lb>doque</expan> enim à percuſſione <expan abbr="neq;">neque</expan> locum plagæ, <expan abbr="neq;">neque</expan> huic oppoſi <lb></lb>tum, ſed quemvis alium infeſtari & frangi contingit. cùm ni<lb></lb>mirum illarum partium unio minorem vim habet ad quieſcen<lb></lb>dum, quàm impetus ad movendum. </s></p> <p type="main"> <s><emph type="center"></emph>Defortificatione aduerſum ictus <lb></lb>Tormentorum.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>QVia mœnia urbis, caſtelli, aut propugnaculi ictus tormen<lb></lb>torum admittere neceſſitas <expan abbr="quandoq;">quandoque</expan> cogit; providen<lb></lb>dumquô eiuſmodi ictus debilitentnr, minorem eâ ratione <expan abbr="plagã">plagam</expan> <lb></lb>afferentes. </s><s>Id autem duobus modis aſſequi <expan abbr="cõtingit">contingit</expan>: primo ma<lb></lb>iori parte ictûs excluſâ, quem totum vitare nequimus: pars <lb></lb>enim dimidia, aut tertia minorem noxam dabit, quàm totus. </s><lb></lb><s>Minuitur autem cùm non niſi obliquè recipitur. </s><s>Conſide<lb></lb>randum ergo quibus potiſſimum locis urbs ad inuaſionem ſit <lb></lb>opportuna & quâ ab hoſtium tormentis minùs tuta: tum enim <lb></lb>latus munitionis oppoſitum eiuſmodi locis, quantùm fieri li<lb></lb>cet, <expan abbr="obliq;">oblique</expan> ducendum: quò ictus recipiat magis obliquos. </s><lb></lb><s>Quâ verò parte ob ſitum <expan abbr="locorũ">locorum</expan> machinæ admoveri neque<lb></lb>unt, in directum procurrere poteſt. </s><s>Tum igitur ictus obliquè <lb></lb>incidentes non niſi partem plagæ dant à lineâ hypomochlij de<lb></lb>finitam: reliquâ parte, quæ necdum percuſſit, reflexâ: & ſi <lb></lb>quidem propugnaculum impetitur; quia latera ſibi oppoſitis, <pb xlink:href="063/01/101.jpg"></pb>à quibus defenditur, habet parallela, in totum averſâ. </s><s>Per<lb></lb>cuſſo autem muro licet in aliam partem reflectat: quia tamen <lb></lb>ex obliquo ictus fiunt, violentiâ in plures diſtractâ, minùs no<lb></lb>xæ inferunt. </s></p> <p type="main"> <s>Secundus modus ut ſiue totam plagam, ſiue illius partem <lb></lb>recipere cogantur, id cum minori detrimento & concuſſione <lb></lb>fiat. </s></p> <p type="main"> <s>Et cùm ruina proveniat ex ſolutâ compage: cùm vel partium <lb></lb>iuncturæ, quibus muri, turres, & propugnacula ſunt ſtructa, de <lb></lb>hiſcunt: vel partes ſolidæ ex vehementiâ ictûs fatiſcunt; ne<lb></lb>ceſſe illam violentiam ita diſpenſare; ut nullâ parte inſigniter <lb></lb>læsâ pertranſeat; & <expan abbr="neq;">neque</expan> partem ſolidam frangat; <expan abbr="neq;">neque</expan> unam <lb></lb>ab aliâ divellat. </s><s>Hoc autem pendet à duobus: materiâ ni<lb></lb>mirum, <expan abbr="atq́">atque</expan>; huius partium ſitu. </s><s>In quæſtione enim de fra<lb></lb>cturâ oſtendi in diverſis corporibus inæqualiter recipi impul<lb></lb>ſum. </s><s>Nam quæ cedendo in plures veluti ictus hunc partiun<lb></lb>tur. minùs noxæ ſentiunt, <expan abbr="minùsq;">minùsque</expan> latè ſe extendit plaga. </s><lb></lb><s>Talia verò ſunt <foreign lang="grc">π<gap></gap>στὰ</foreign> cum lentà viſciditate, et quæ percuſſa <lb></lb>minùs ſonant: ob atomos enim inæqualiter poſitas per illas <lb></lb>ambages diſcerpitur impulſus. </s><s>Saxa ergò, quæ ſurda dicun<lb></lb>tur, cæteris paribus ad impetum ſuſtinendum ſunt aptiora. </s><lb></lb><s>Quod attinet ſitum, quia ſoliditas muri maior eſt longitudine <lb></lb>aut craſſitie ſaxi; neceſſe plura ordine diſponi, <expan abbr="quouſq;">quouſque</expan> ſimul <lb></lb>iuncta adæquent illam molem. </s><s>Alia ergo ſitum extra habent, <lb></lb><expan abbr="ictusq́">ictusque</expan>; & primum impetum ſuſtinent; alia medio locò: alia <lb></lb>demum parieti interno ſunt pro firmamento. </s><s>Nihil hic dico <lb></lb>de illâ concatenatione, quâ duo ſaxa uno ſuperpoſito nectun<lb></lb>tur, <expan abbr="atq;">atque</expan> unum <expan abbr="quodq́">quodque</expan>; duobus retinaculis firmatur: ut licet <lb></lb>uno exempto nihil detrimenti reliqua ſentiant: quod <expan abbr="quidẽ">quidem</expan> <lb></lb>erat futurum, ſi totâ mole æqualibus ſabſternerentur. </s><s>De <lb></lb>quibus ſapienter, <expan abbr="docteq́">docteque</expan>, Architecti: hic enim Geometram, <pb xlink:href="063/01/102.jpg"></pb>non Architectum agimus. </s><s>Situm ergo conſideramus, quate<lb></lb>nus impulſus à plagâ ad reliqua, quæ ponè ſequuntur, tranſit: <lb></lb><expan abbr="neq;">neque</expan> enim in ſuperficie vis hæc finitur, ſed altè penetrat. </s><lb></lb><s>Aut igitur æqualia, aut inæqualia: <expan abbr="atq;">atque</expan> hæc maiora, vel mino<lb></lb>ra ſequuntur. </s><s>Videmus autem hunc ferè modum ſervari: <lb></lb>ut grandiſſima ſaxa ſint à fronte; quæ cum maximo impetu <lb></lb>luctentur: interiora verò tanquam ab ictu iam ſecura negle= <lb></lb>ctim ſtrui, ruderibus aut minoribus ſaxis explendo illa inter<lb></lb>valla. quod an rectè fiat dubitamus. </s><s>Nam craſſitudo muri eſt <lb></lb>ob firmitatem, quò ſaxa priùs poſita à poſterioribus contine<lb></lb>antur: neceſſe ergo impetum, quo alioquin ſaxa à fronte po<lb></lb>ſita loco moverentur, ſuſtinere. </s><s>At verò quâ ratione impe<lb></lb>tum maioris id quod multò eſt minus ſuſtinebit? Nam per po<lb></lb>riſ: 2 ſi maius percutiat minus, <expan abbr="utrũq">utrunq</expan>: loco movetur: propte<lb></lb>rea, quòd minus eadem velocitate movetur ex impulſu mino<lb></lb>ri. </s><s>Tametſi ergo partes illæ minores, quæ in muro continen<lb></lb>tur, <expan abbr="undiq;">undique</expan> ſint concluſæ; quia tamen totum impetum ferre <lb></lb>non valent, nec in alias minores hunc exonerare: divelli à <lb></lb>primis, & poſteriores urgere, <expan abbr="atq;">atque</expan> tum metu vacui aërem ſor<lb></lb>bendo, etiam magnas compages diſſolui eſt neceſſe. </s></p> <p type="main"> <s><emph type="italics"></emph>Dices. </s><s>Non eandem rationem videri in muro, ubi omnia per calcem <lb></lb>glutinantur, & veluti unum fiunt; at〈que〉 illoram corporum, quæ ſoluta <lb></lb>motum & impulſum à ſe recipiunt. </s><s>Licet ergo impulſus à maiori ſaxo <lb></lb>in minora tranſiens omnia loco moveat; non tamen idem futurum in <lb></lb>muro; cùm illud gluten non minùs coharere faciat, quàm ſi partes <lb></lb>eſſent continuæ unius ſaxi maioris. </s><s>Ita〈que〉 duo globuli cerâ coniuncti <lb></lb>impulſum ſuſtinent duplo maiorem, ne〈qué〉 à percuſſo primo ſecundus rece<lb></lb>dit: quantò ergo minùs calce revincta ſaxa.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Poſſet quis reſpondere, cùm ſaxa minora maioribus <expan abbr="cohæreãt">cohæreant</expan> <pb xlink:href="063/01/103.jpg"></pb><expan abbr="mediãte">mediante</expan> illo glutine ex calce & arenâ multò levioribus; <expan abbr="nõ">non</expan> poſſe <lb></lb>eo modo habere, quo partes continui: <expan abbr="neq́">neque</expan>; per modum unius <lb></lb>cenſeri in ordine ad impulſum; quem etiam in eodem ſubie<lb></lb>cto, ob partium diſcrimina, oſtendi inæqualem. </s><s>Etenim vi<lb></lb>demus, longè differre hunc <expan abbr="nexũ">nexum</expan> à partium eiuſdem ſaxi unio<lb></lb>ne, diſſoluto à murarijs cæmento: parte enim averſâ etiam le<lb></lb>viter percuſsâ, illæſo ſaxo, decidunt coagmenta. </s><s><expan abbr="Neq;">Neque</expan> obſtat <lb></lb>in muro omnia vincta teneri, quò minùs impetus ſimili ratione <lb></lb><expan abbr="atq;">atque</expan> in ſolutis pertranſeat. </s><s>Nam ſi pila in plano ſeu manu, ſeu <lb></lb>aliâ ratione firmetur, quò minùs moveri poſſit à plagâ; nihilo<lb></lb>minùs ſibi contiguam movet. </s><s>Neceſſe ergo matori pericu<lb></lb>lo ſequi ſaxa minora, quàm æqualia aut maiora: cùm per æqua<lb></lb>lia impetus ad extimum <expan abbr="uſq́">uſque</expan>; æqualiter ſe effundat: illisq illæ<lb></lb>ſis & immotis pertranſeat. </s><s>Vnde ſiquid periculi non niſi inter<lb></lb>no parieti creatur, qui facilè refici poteſt. </s><s>Non ita cùm imme<lb></lb>diatè minora ſequuntur: receſſu enim à primis extima pericli<lb></lb>tantur: <expan abbr="neq;">neque</expan> facilè reparari queunt: unde viſo periculo magis ab <lb></lb>hoſte infeſtantur. </s><s>Vt verò quid mihi videatur, dicam: eiuſ<lb></lb>modi ſaxa, quæ calce glutinantur, aut ſunt partes continuæ e<lb></lb>iuſdem molis, aut contiguæ. </s><s>Supponamus primùm eſſe conti<lb></lb>guas & plagam incipere à maiori. </s><s>Cùm <expan abbr="itaq;">itaque</expan> maius percutiat <lb></lb>minus, ſi non aliunde motus impediatur, movebitur minus ab <lb></lb>incipiente & necdum perfectâ plagâ: ad cuius motum ſequitur <lb></lb>maius per poriſma 2. </s><s>Quòd ſi verò ab aliâ vi detineatur ne<lb></lb>ceſse totum impulſum maioris recipere. </s><s>Et ſi quidem illa vis <lb></lb>retentiva ſit minor impulſu; tum ſanè movebitur illud mobile: <lb></lb>reliqua verò, quia illorum plaga perfecta, à motu conquieſcent. </s><lb></lb><s>Vnde tota illa vis diſtractiua partem ultimam obtinet. </s><s>Si <lb></lb>autem à minoribus eadem plaga procedat: quia tum hypo<lb></lb>mochlium ſecundi eſt tertium; neceſse non niſi ultimo moto <lb></lb>moveri primum: ac proinde impulſum maioris recipere mi- <pb xlink:href="063/01/104.jpg"></pb>nus. </s><s><expan abbr="Cùmq;">Cùmque</expan> ab æquali plagâ incipiat, erit in <expan abbr="utroq;">utroque</expan> extremo <lb></lb>impulſus æqualis. </s><s>Et quia maiorem rationem habet ad mi<lb></lb>nus, maiori <expan abbr="quoq;">quoque</expan> vi eluctabitur. </s><s>Magis ergo periclitatur, <expan abbr="ma-iorq;">ma<lb></lb>iorque</expan> ruina imminet extremo; ſiplaga incipiat à percuſſo ma<lb></lb>iori. </s><s>Ita quidem ſi ſaxa contigua eſſe demus. </s><s>Quòd ſi verò <lb></lb>continua ſint; Dico ab eodem impulſu magis infeſtari mino<lb></lb>ra, ſi ictum primum excipiant. </s><s>Nam cùm impulſus non re<lb></lb>cipiatur uniformiter, verùm à contactu ſenſim remiſſo vigore <lb></lb>ſe extendat in latum, & profundum: neceſse illas iuncturas, <lb></lb>quæ circum ſaxa ſunt minora, ab impulſu magis intenſo perva<lb></lb>di: & quia minùs firmo nexu cohærent, quàm reliquum ſa<lb></lb>xum, à ſe divelli. </s><s>Ad rationem verò in oppoſitum factam, <lb></lb>Reſpondeo, licet minùs firmiter glutinentur inter ſe ſaxa; non <lb></lb>tamen ob illam inæqualitatem deſinere eſſe continua: alio<lb></lb>quin <expan abbr="neq;">neque</expan> idem ſaxum eſſet continuum: quòd diverſis parti<lb></lb>bus inæqualiter frangi contingat. </s><s>De quo tamen accuratiùs <lb></lb>dicetur in libro de motu: qui propediem in lucem prodibit. </s></p> <p type="main"> <s><emph type="center"></emph>De Percuſſione & motu orbiculorum.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>ORbiculi ſunt figuræ circulares, <expan abbr="utrâq;">utrâque</expan> ſuperficie planâ <lb></lb>& parallelâ terminatæ; ſeu portiones cylindri habentes <lb></lb>partem axis reſecti minorem ſemidiametro circuli. </s><s>Eſt autem <lb></lb>hoc illis commune cum globis; ut ordine diſpoſiti, <expan abbr="ſibiq;">ſibique</expan> con<lb></lb>tigui eadem ratione moveantur: percuſſo enim primo, ſi æ<lb></lb>quales ſint, medijs immotis ultimus movetur. </s><s><expan abbr="Atq;">Atque</expan> inde ra<lb></lb>tio conſtat, quamobrem eiuſmodi orbiculis ludentes ab <lb></lb>eadem plagâ, non eundem effectum conſequantur. </s><s><expan abbr="Quan-doq;">Quan<lb></lb>doque</expan> enim ad finem tabulæ orbiculum inſequitur ille, qui <lb></lb>percuſſit: <expan abbr="quandoq;">quandoque</expan> immotus manet. </s><s>Hoc enim fit ob in æ<lb></lb>qualem gravitatem: inſequitur enim maior minorem, non <pb xlink:href="063/01/105.jpg"></pb>verò ſibi æqualem aut maiorem. </s><s>Suppono verò motum fi<lb></lb>eri in lineâ centri: nam ſi inclinet; quia non totam dat plagam, <lb></lb>motum continuabit. </s><s>At verò hoc peculiare habent; quòd <lb></lb>non tantùm in lineâ rectâ ſibi contigui fiant; ſed etiam illâ ſu<lb></lb>perficie planâ in ſimilitudinem cylindri aſſurgant. </s><s>Quòd ſi <lb></lb>ergo his ita cumulatis illum orbiculum, qui baſis eſt reliquo<lb></lb>rum, percutiat æqualis; eadem ratione movebitur huic con<lb></lb>tiguus, quantumvis illorum numerus, qui baſi incumbunt, au<lb></lb>geatur: quin etiam quovis onere accepto impulſum tranſmit<lb></lb>tit nihilo minorem. </s><s>At verò baſis excuti non eadem facilita<lb></lb>te poteſt: verùm pro numero orbiculorum, aut oneris appen<lb></lb>ſi ratione, neceſſe plagam fieri maiorem. </s><s>Quòd ſi orbicu<lb></lb>lus gravitatem habeat æqualem illi, quâ baſis gravatur, eadem <lb></lb>facilitate illam loco movebit: verùm ipſe <expan abbr="quoq;">quoque</expan> tranſito illo <lb></lb>hiatu, motum baſis ſequetur. </s><s>Cuius ratio eſſe videtur: <lb></lb>quòd impulſus neceſſariò fiat iuxta determinationem plagæ; <lb></lb>licet ſubiectum non moveatur ex eo impulſu: & ſi maior ſit <lb></lb>quàm ut in ſubiecto terminetur, aliud percutit ſibi contiguum. <lb></lb><expan abbr="Neq;">Neque</expan> minor eſt impulſus ſi ab a lienâ gravitate detineatur; non <lb></lb>enim gravitas ab extra veniens, ſed nativa hunc attenuat: quæ <lb></lb>multam materiam habet coniunctam. </s><s><expan abbr="Itaq;">Itaque</expan> ſi magnitudi<lb></lb>ne non verò gravitate ſint pares; orbiculus maior à minori <lb></lb>percuſſus, minorem ex eadem plagâ impulſum reliquis dabit, <lb></lb>quàm ſi ab æquali percutiatur: propterea, quòd in multâ ma<lb></lb>teriâ magis hebetatur. </s><s>Vt verò ſubiectum moueatur, ne<lb></lb>ceſſe & gravitatem nativam, & impulſum contrarium ſupera<lb></lb>re. </s><s><expan abbr="Itaq;">Itaque</expan> fit ut baſis illius cylindri orbiculati motui renita<lb></lb>tur: quæ & ſuâ & illorum, à quibus premitur, gravitate de<lb></lb>tinetur. </s><s>Quòd ſi gravitas orbiculi augeatur, ut gravitati illius <lb></lb>cylindri ſit æqualis: quæ tota in baſim colligitur, <expan abbr="atq;">atque</expan> illius vi <lb></lb>à motu detinetur, percuſſio tum fit æqualis: <expan abbr="quouſq;">quouſque</expan> nimirum <pb xlink:href="063/01/106.jpg"></pb>à nexu illorum orbiculorum ſe expediat: tum enim motu a<lb></lb>gitur multò velociore, quàm ſi plaga fiat ab æquali: amotâ e<lb></lb>nim illâ reſiſtentiâ, impulſus ad gravitatem multò iam mino<lb></lb>rem, maiorem habet exceſſum. </s></p> <p type="main"> <s><emph type="italics"></emph>Dices. </s><s>Quid ſi orbiculi percutiant illum cylindrum, à quo ſunt reſecti; <lb></lb>an non movebunt alios huic contiguos eadem ratione, quà in cylindro <lb></lb>orbiculato? Videtur enim eadem ratio eſſe illius ſegmenti, quod percu<lb></lb>titur ab æquali, ſiue reſectum ſit, ſiue continuum: propterea, quòd ma<lb></lb>teria una, ac proinde impulſus, qui viam ſequitur plagæ, æqualiter per<lb></lb>tranſit. </s><s>Illa autem continuitas non videtur mutare naturam impul<lb></lb>ſus: qui non niſi vi dimovetur à lineâ rectâ: at〈qué〉 eâdem gravitate à <lb></lb>motu detinetur pars reſecta & continua: eadem ergo plaga, quæ orbicu<lb></lb>lum excludit, cylindrum quo〈que〉 ſolidum movebit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo impulſum, cùm à principio fiat interno mobi<lb></lb>lis, totum ſubiectum afficere. </s><s>Licet ergo viam ſequatur pla<lb></lb>gæ; quia tamen in altum <expan abbr="quoq;">quoque</expan> aſſurgit: quantum virium <lb></lb>huc confert, tantum decedit plagæ oppoſitæ. </s><s>Percuſſo <lb></lb><expan abbr="itaq;">itaque</expan> cylindro ſolido, minori vi moventur orbiculi contigui; <lb></lb>decreſcente plagâ pro altitudine cylindri. </s><s>Quòd ſi orbiculi <lb></lb>inter ſe glutinentur; quia tum extrema fiunt unum, rationem <lb></lb>habent continui: unde eadem his, quæ cylindro ſolido <lb></lb>conveniunt. </s><s>Verùm de his cùm ſcitu digna vi<lb></lb>deantur continere. enucleatiùs diſſeren<lb></lb>dum. </s></p> <pb xlink:href="063/01/107.jpg"></pb> <p type="main"> <s><emph type="center"></emph>DEFINITIO I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cylindrus ſolidus eſt, cuius partes omnes ſunt continuæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cylindrus verò orbiculatus, cuius ſegmenta ſuut orbiculi, <lb></lb>ſimul iuncti at〈qué〉 inter ſe par alleli.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Baſis Cylindri, orbiculati eſt orbiculus tangens planum, <lb></lb>à quo reliqui orbiculi eidem par alleli, centrum in eodem axe <lb></lb>habentes aſſurgunt.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Grauitas ſecunda eſt vis ab extra proueniens, quâ mo<lb></lb>bile detinetur, quò minùs à grauitate primâ, ſeu propriâ <lb></lb>aut impulſu moveatur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si duo orbiculi ſimul iuncti & æquales eodem impulſu moueantur <lb></lb>in plano; ad minus intervallum movetur baſis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>DIxi Motum eſſe veluti continuatam ex aëris diviſione <lb></lb>plagam: & ſi medium ſit minùs aptum dividi, minorem <lb></lb>eſſe motum; qui non niſi à plagâ perfectâ terminatur. </s><s>Plaga <pb xlink:href="063/01/108.jpg"></pb>autem fit cùm medium reſiſtit: & quia in vacuo nihil reſiſtit; <lb></lb>interminabilis eſſet in eo motus: in aquâ verò ob reſiſtentiam <lb></lb>maiorem, priùs quàm in aëre abſumitur. </s><s>Reſiſtentia autem <lb></lb>fit cùm vel diviſio, vel gravitas mobilis, vel retentio obſtat <lb></lb>motui. </s><s>Ita ergo per plures chartas aliquo intervallo ſeiun<lb></lb>ctas tranſit glans plumbea, <expan abbr="quouſq;">quouſque</expan> ex illâ diviſione continu<lb></lb>atâ impetus laſſetur. </s><s>Aut cùm plagam recipit gravitas maior <lb></lb>à minori: aut cùm mobile à maiori vi detinetur, quò minùs <lb></lb>motum proſequi valeat. </s><s>Detinetur autem mobile ſeu à gra<lb></lb>vitate coniunctâ, ſeu vi retentivâ: ut ſi Miloni digitum infle<lb></lb>ctere, aut pomum illius manu concluſum extorquere cone<lb></lb>mur: illa enim retentio ab impulſu fluente, & veluti librato <lb></lb>procedit: qui non niſi à maiori impulſu poteſt ſuperari. </s><lb></lb><s>Cui ſimilis videtur retentio ex anguſtiâ loci inducta: ut dum <lb></lb>clavus in pariete fixus detinetur. </s><s>Quò enim maior anguſtia; <lb></lb>eò tranſitus magis difficilis, & non niſi maiori vi ſuperandus. <lb></lb><expan abbr="Itaq;">Itaque</expan> fit, ut licet eiuſmodi rima toto illo tractu æqualiter ex<lb></lb>currat, impetus tamen priuſquam totam tranſeat, exſolvatur: <lb></lb>retentio enim illa continuata non aliter, quàm ſi plaga produ<lb></lb>ceretur, impulſum atterit & abſumit: <expan abbr="atq;">atque</expan> eò magis, quò ſtri<lb></lb>ctura magis coarctat. </s><s><expan abbr="Neq;">Neque</expan> aliâ ratione detineri videtur ba<lb></lb>ſis à gravitate illorum orbiculorum, qui baſi incumbunt; fit <lb></lb>enim compreſſio illi ſimilis, quam loci anguſtia inducit. </s><s><expan abbr="Itaq;">Itaque</expan> <lb></lb>ſi augeatur numerus, aut pondus orbiculorum; quia magis <lb></lb>comprimitur baſis, non niſi maiori vi excuti poteſt. </s><s>Cùm <lb></lb>ergo duo orbiculi ſimul iuncti moventur: quia compreſſio fit <lb></lb>baſis continuata, licet impulſus, quo baſis movetur ſit æqualis; <lb></lb>ob illam tamen gravitatem acceſſoriam, priùs terminat mo<lb></lb>tum. </s><s>Quam inæqualitatem motûs adiuvare videtur ſcabri<lb></lb>ties loci, ſeuplani, quod tanſit: <expan abbr="atq;">atque</expan> inde fit quòd <expan abbr="quandoq;">quandoque</expan> in <lb></lb>medio motu orbiculi circumaguntur. </s></p> <pb xlink:href="063/01/109.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si duo Orbiculi ſimul iuncti & æquales percutiant alium maiorem, <lb></lb>duobus autem illis ſimul ſumptis æqualem; orbiculo ſuperiori reflexo, <lb></lb>motum continuat baſis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Difficultas hic eſt, quòd cùm orbiculus ſolidus percutit ali<lb></lb>um ſibi æqualem, illo moto quieſcit: cur igitur non idem fit, <lb></lb>cùm duo ſimul iuncti, <expan abbr="atq;">atque</expan> eidem æquales hunc percutiunt, ve. <lb></lb>rùm <expan abbr="uterq;">uterque</expan> inæqualiter movetur ex illâ plagâ? Nam eodem <lb></lb>impulſu agi videntur: neceſſe ergo eandem inferre plagam: <lb></lb>Et cùm gravitas orbiculi maioris ſit dupla, & impulſum reci<lb></lb>piat duplum illius, quo ſinguli moventur; erit <expan abbr="quoq;">quoque</expan> eadem ve<lb></lb>locitas motûs. </s><s>At verò ſi baſis ſequitur motum maioris; ne<lb></lb>ceſſe huius motum eſſe velociorem quàm baſis motum: quæ <lb></lb>ab incipiente & necdum perfectâ plagâ movetur. </s><s>Et ſi refle<lb></lb>ctit alter orbiculus: quia peractâ huius plagâ necdum incipit <lb></lb>moveri maior; velocitatem habebit minorem. </s><s>Pro ſolutione <lb></lb>dico, impulſum in <expan abbr="utroq;">utroque</expan> orbiculo eſſe æqualem. </s><lb></lb><s>Secundô inæqualiter moveri, <expan abbr="magisq;">magisque</expan> impediri motum baſis <lb></lb>per 1 Theor: huius; & cùm impulſum determinet motus; ma <lb></lb>iori tempore plagam perficiet baſis. </s><s>Cùm ergò maior orbi<lb></lb>culus gravitatem habeat duplam; ad illam velocitatem mo<lb></lb>tûs, non niſi ab impulſu duplo perducitur: perfectâ autem pla<lb></lb>gâ unius orbiculi, necdum percuſſit alter: <expan abbr="neq;">neque</expan> igitur ex illâ <lb></lb>plagâ ſe abducit orbiculus maior: ac proinde orbiculus, qui <lb></lb>iam percuſſit, reflectit. </s><s>Et quia minor eſt velocitas motûs <lb></lb>baſis, velociùs movebitur ab <expan abbr="utraq;">utraque</expan> plagâ. igitur ad illam ve<lb></lb>locitatem, quâ movetur baſis, ab incipiente & necdum perfe<lb></lb>ctâ huius plagâ perducetur: ac proinde reliquus impulſus mo<lb></lb>tum continuabit. </s><s>Idem autem fit, ſi alter orbiculus ſit paulo <pb xlink:href="063/01/110.jpg"></pb>levior, ſeu baſis, ſeu qui ſuperiori loco ſitum habet. </s><s>At ſi <lb></lb>magnus ſit exceſſus; ut cùm ligneo metallicum adiungimus, <lb></lb>gravior in omni ſitu motum ſequitur maioris. </s><s>Quòd ſi duo <lb></lb>orbiculi ſimul iuncti <expan abbr="atq;">atque</expan> inter ſe æquales deficiant à gravitate <lb></lb>maioris: minùs quidem movetur baſis, magis autem reflectit <lb></lb>alter orbiculus. E contra ſi gravitas excedit: hic quidem mi<lb></lb>nùs reflectit, ille verò motum magis producit. </s><s>Cuius ratio <lb></lb>eſt, quòd horum impulſus maiorem rationem habet ad orbi<lb></lb>culum minùs gravem: igitur cùm à minori plagâ eadem ve<lb></lb>locitas motûs ſequatur; erit maior impulſus reliquus ad mo<lb></lb>tum continuandum. </s><s>Et quia velociùs à plagâ ſe abducit, erit <lb></lb>minor reflexio motûs-Cùm verò impulſus minorem habet <lb></lb>rationem; non niſi à maiori plagâ ad motum æquè velocem <lb></lb>cietur maior, & non niſi tardè à plagâ ſe abducit: magis proin<lb></lb>de reflectit motus, minùs autem à reliquo impulſu movetur <lb></lb>baſis. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si duos orbiculos ſimul iunctos percutiat maior; adminus inter<lb></lb>vailum movetur baſis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi duo orbiculi ſint æquales; quia ab eadem plagâ <lb></lb>idem eſt impulſus, conſtat per Theor: 1. ad minus intervallum <lb></lb>moveri baſim. </s><s>Simili modo cùm orbiculi ſunt inæqvales, & <lb></lb>maiorem gravitatem habet baſis; ab æquali impulſu minùs <lb></lb>moveri baſim. </s><s>At cùm pro baſi eſt orbiculus minùs ponde<lb></lb>roſus; oportebat quidem hunc ab æquali impulſu velociùs, <lb></lb>& ad maius intervallum moveri. </s><s>Sed quia detinetur ab aliâ <lb></lb>gravitate; quò magis premitur, eò motum habet magis im<lb></lb>peditum. </s><s>Deinde dicolicet ſimul fiat, eſſe tamen inæqua. <pb xlink:href="063/01/111.jpg"></pb>lem plagam, & qui hanc ſequitur impulſum. </s><s>Nam cùm à <lb></lb>principio eodem motu ferantur; neceſſe à maiori impulſu mo<lb></lb>veri graviorem: quò minùs ergo velociter irrumpat, <expan abbr="totúmq;">totúmque</expan> <lb></lb>impulſum recipiat minor, à graviori detinetur. </s><s>Igitur baſis <lb></lb>tum quia minori impulſu agitur, tum quia gravitate aliená de<lb></lb>tinetur, ad minus intervallum movetur. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si duo orbiculi ſimuliuncti & æquales percutiant alium maiorem <lb></lb>& immotum; uter〈qué〉 reflectit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Quia enim'minor eſt impulſus, à plagâ illorum orbiculo<lb></lb>rum, quàm ut loco moveat maiorem: ſiue à gravitate primâ <lb></lb>ſeu propriâ, ſiue ſecundâ detineatur: ut cùm ligneus metalli<lb></lb>cum, aut alium ſibi quidem ſimilem verùm in plano firmatum <lb></lb>percutit: <expan abbr="neq;">neque</expan> hic à plagâ ſe abducit, aut alium contiguum <lb></lb>movet; recipiet <expan abbr="uterq;">uterque</expan> orbiculus à percuſſo æqualem illi quam <lb></lb>dedit plagam: igitur cùm impulſus ſit agens neceſſarium, <expan abbr="u-terq;">u<lb></lb>terque</expan> orbiculus reflectet ex illâ plagâ. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si plures orbiculi ſimul iuncti percutiant alium maiorem, & à <lb></lb>plagâ illâ immotum; ad minus intervallum reflectunt baſi propiores.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Cùm omnes orbiculi percutiant, <expan abbr="neq;">neque</expan> ad ullius plagam <lb></lb>moveatur ille orbiculus: recipient à percuſſo æqualem illi, <lb></lb>quam <expan abbr="quisq;">quisque</expan> dedit plagam. </s><s>At verò baſis per Theor: 2. mo<lb></lb>tum habet magis impeditum: igitur cùm velocitas motûs de<lb></lb>terminet plagam; minor erit huius, quàm reliquorum plaga. <pb xlink:href="063/01/112.jpg"></pb>Et quia propiores illi baſi, quæ tangit planum, remotioribus <lb></lb>ſunt pro baſi; erit minor illorum plaga: ac proinde ad minus <lb></lb>intervallum reflectunt. </s><s>Idem verò contingit ſiue eandem <lb></lb>habeant gravitatem orbiculi reliqui, ſiue præponderet baſis, <lb></lb>aut minus ſit gravis. </s><s>Nam licet baſis magis ponderoſa ma<lb></lb>iorem dat plagam; cùm non niſi à maiori impulſu moveatur <lb></lb>eodem cum minoris gravitatis motu: quia tamen in ordine <lb></lb>ad motum hanc expendimus; <expan abbr="atq́">atque</expan>; in eadem ratione ſunt mo<lb></lb>tus reflexi, minor autem huius motus; minorem <expan abbr="quoq;">quoque</expan> in or<lb></lb>dine ad ſuum motum dicetur dare & referre plagam. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si plures orbiculi ſimul iuncti & æquales percutiant Cylindrum ſoli<lb></lb>dum; maiorem impulſum recipiunt partes à baſi remotiores.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam baſis quidem minorem dat plagam per 5 theor: eſt au<lb></lb>tem orbiculus propior remotioribus pro baſi: erit ergo maior <lb></lb>illorum plaga, & à maiori plagâ maior <expan abbr="quoq;">quoque</expan> impulſus. </s><s>Sed <lb></lb>et ratio vectis huc facere videtur. </s><s>Nam orbiculus ipſo cy<lb></lb>lindro utitur pro vecte: <expan abbr="atq;">atque</expan> eò magis, quò plaga fit remoti<lb></lb>or à baſi: cuius hypomochlium eſt planum, in quo cylindrus <lb></lb>firmatur. </s><s><expan abbr="Itaq;">Itaque</expan> à plagâ in medio aut propè baſim factâ immo<lb></lb>tus manet: ſi eandem plagam accipiat in ſummo, invertitur. <lb></lb><expan abbr="Atq;">Atque</expan> inde ratio conſtat, quamobrem partes cylindri ſuperiores <lb></lb>avertuntur ex illâ plagâ, & celeritate motûs alias antevertunt: <lb></lb>à baſi enim cum longitudine cylindri continuò accreſcit pla<lb></lb>ga. E contra verò ſi plaga fiat propè baſim, & infra medium, <lb></lb>non percuſsâ reliquâ parte cylindri; reſupinato vertice mo<lb></lb>tum accelerat baſis. </s><s>Cùm autem cylindrus alium percutit ſi<lb></lb>bi æqualem: quia omnes partes æqualiter moventur; eandem <pb xlink:href="063/01/113.jpg"></pb><expan abbr="quoq;">quoque</expan> inferunt plagam. </s><s>Non igitur huius ratione videtur <lb></lb>differre motus; verùm acceleratio ad partes ſummas ad ve<lb></lb>ctem referri debet. </s><s><expan abbr="Atq;">Atque</expan> inde ſequitur, cylindrum ab æqua<lb></lb>li cylindro percuſſum inæqualiter moveri. </s><s>Et cùm orbiculus <lb></lb>ad alium ſibi æqualem, eo modo habeat, quo cylindrus; ne<lb></lb>ceſſe illâ ſucceſſione orbiculorum in plano motum deficere. </s><lb></lb><s>Vtergo cylindrus æqualiter moveatur ab alio cylindro; inæ<lb></lb>qualis eſſe debet plaga: & tanto maior propè baſim, quanto <lb></lb>in ſummo augetur ratio vectis. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si duo orbiculi ſimul iuncti & æquales percutiant alios duos, ſimul <lb></lb>quo〈que〉 iunctos & prioribus æquales, habeant verò à tergo orbiculum ma<lb></lb>iorem; immotâ baſi primâ, movetur baſi, ſecunda.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Cùm duo orbiculi æquales ſimuliuncti percutiunt alios du<lb></lb>os ſimul <expan abbr="quoq;">quoque</expan> iunctos & æquales: licet inæqualem afferant <lb></lb>plagam; quia tamen <expan abbr="uterq;">uterque</expan> orbiculus ex illâ inæquali plagâ ſe <lb></lb>abducit; <expan abbr="quiſq;">quiſque</expan> ſuo orbiculo expulſo à motu conquieſcit. </s><lb></lb><s>At cùm alius orbiculus maior accedit: in quem impetus ſe ex<lb></lb>onerat illorum orbiculorum: quia inæqualem <expan abbr="atq;">atque</expan> minorem <lb></lb>á baſi recipit plagam; per theor: 2. huius, reflexo altero orbi<lb></lb>culo movebitur baſis. </s><s>Cùm igitur hæc baſis ſecunda à pla<lb></lb>gâ ſe abducat; quieſcet à percuſſione baſis prima. </s><s>Conſtat <lb></lb>verò illo orbiculo reflexo, reflecti <expan abbr="quoq;">quoque</expan> orbiculum priorem <lb></lb>huic contiguum. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si baſim cylindri orbiculati percutiat alius orbiculus æqualis; habe-<emph.end type="italics"></emph.end> <pb xlink:href="063/01/114.jpg"></pb><emph type="italics"></emph>at verò impulſum minorem gravitate ſecundâ; baſim à cylindro non <lb></lb>excludet.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Impulſus, quo orbiculus movetur quantumvis exiguus, <lb></lb>movere poteſt alium ſibi æqualem: At cùm gravitas huius ab <lb></lb>aliâ vi detinetur; non niſi á maiori impulſu, quàm ſit illa vis <lb></lb>motui renitens, moveri poteſt. </s><s>Vt ſi globum ſtylo affixum <lb></lb>percutiat globus æqualis; illa quidem plaga non niſi ſtylo fra<lb></lb>cto, aut avulſo globum movebit. </s><s>Itaq cùm baſis cylindri <lb></lb>plurium acceſſione gravatur; neceſſe plagam ab orbiculo <lb></lb>illatam eſſe maiorem illâ acceſſoriâ gravitate: quâ velutí affi<lb></lb>gitur plano: non ſolùm in principio motûs, ſed toto illo tra<lb></lb>ctu, quo baſis eluctatur. </s><s>Nam cùm huius motus non aliter, <lb></lb>quàm ſi corpus ſolidum continuatâ plagâ perrumpat, attera<lb></lb>tur: ſi minor ſit quàm reſiſtentia illo tranſitu coacervata; mi<lb></lb>nor <expan abbr="quoq;">quoque</expan> erit plaga: deficiet ergo motus priuſquam baſis <lb></lb>pertranſeat. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si baſim cylindri orbiculati percutiat alius orbiculus æqualis; habe<lb></lb>at verò impulſum æqualem grauitati ſecundæ; excluſam baſim non <lb></lb>ultra cylindrum movebit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam quia baſim percutit alius orbiculus æqualis; habebit <lb></lb>ex illâ plagâ impulſum æqualem. </s><s>Et quia gravitas ſecunda <lb></lb>huic eſt contraria, & ex ſuppoſitione æqualis; tollet pars qui<lb></lb>dem gravitatis huius partem, tota verò gravitas totum im<lb></lb>pulſum per poſit: 2 de propor: motûs. </s><s>Cùm igitur gravitas <lb></lb>ſecunda diametro cylindri terminetur; deficiet impulſus, ubi <lb></lb>cylindrum exceſſit baſis. </s><s>Et cùm non <expan abbr="abſq;">abſque</expan> impulſu moveatur, <lb></lb>non ultra cylindrum extendet motum. </s></p> <pb xlink:href="063/01/115.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si baſim cylindri orbiculati percutiat alius orbiculus æqualis; ha<lb></lb>beat verò impulſum maiorem gravitate ſecundâ; baſim cylindro ex<lb></lb>cluſam movebit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Cùm enim gravitas ſecunda tollat partem ſibi æqualem, <lb></lb><expan abbr="neq;">neque</expan> ultra cylindrum ſe extendat: eſt autem ex ſuppoſitione <lb></lb>impulſus orbiculi, ac proinde baſis maior gravitate: erit hu<lb></lb>ius exceſſus principium motûs reliqui à contactu: baſis ergo <lb></lb>ubi cylindrum ſuperavit, motum à reliquo impulſu continu<lb></lb>abit. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si orbiculus æqualis percutiat baſim cylindri orbiculati minùs gra<lb></lb>vem; habeat verò impulſum minorem gravitate ſecundâ; illam ba<lb></lb>ſim à cylindro non excludet.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Quia baſis aſſumitur habere gravitatem minorem, quàm or<lb></lb>biculus; movebitur à minori impulſu quàm idem orbiculus: & <lb></lb>multò etiam minori quàm ſit gravitas ſecunda: non igitur <lb></lb>tranſire valebit cylindrum, niſi à tergo inſtet maiorem habens <lb></lb>gravitatem. </s><s>At verò huius <expan abbr="quoq;">quoque</expan> impulſus aſſumitur minor <lb></lb>illâ gravitare ſecundâ; non igitur à cylindro eluctari, <expan abbr="neq;">neque</expan> pro<lb></lb>inde baſim excludere valebit. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si orbiculus æqualis percutiat baſim cylindri orbiculati minùs gra-<emph.end type="italics"></emph.end> <pb xlink:href="063/01/116.jpg"></pb><emph type="italics"></emph>vem; habeat verò impulſum æqualem gravitati ſecundæ; exclus à baſi <lb></lb>illius locum obtinebit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Vt ſi orbiculus metallicus baſim ligneam percutiat: <expan abbr="ſitq;">ſitque</expan> hu<lb></lb>ius impulſus æqualis gravitati ſecundæ, quâ baſis detinetur à <lb></lb>cylindro: cuius pars eſt gravitas propria eiuſdem baſis: dico <lb></lb>hunc orbiculum exclusâ à cylindro baſi, illius locum obtinere <lb></lb>Vt enim baſis à cylindro excludatur, neceſſe ſuperare illam re<lb></lb>ſiſtentiam, dum in cylindro movetur, à gravitate tum propriâ <lb></lb>tum alienâ provenientem: quam quidem ſimul collectam <lb></lb>metitur diameter eiuſdem cylindri: propterea quòd ultima <lb></lb>pars baſis neceſſariò per hanc moveatur. </s><s>At verò impulſus, <lb></lb>quo baſis urgetur ab orbiculo graviore, aſſumitur æqualis re<lb></lb>ſiſtentiæ ſimul collectæ; in omni ergo puncto motûs cylindri<lb></lb>ci eſt maior reſiſtentia: <expan abbr="quouſq;">quouſque</expan> in fine motûs eidem gravita<lb></lb>ti fiat æqualis. </s><s>Et quia baſis per 11 huius non niſi ab impulſu <lb></lb>fluente movetur; ſuccedet continuò in locum huius orbicu<lb></lb>lus movens: ac proinde baſi à cylindro exclusâ eundem lo<lb></lb>cum obtinebit. </s></p> <p type="main"> <s><emph type="italics"></emph>Dices ſi in fine motûs impulſus eſt æqualis gravitati ſccundæ, in omni <lb></lb>verò puncto motûs maior eadem gravitate, quomodo totus impulſus <lb></lb>eſſe poteſt æqualis toti gravitati? Nam ſi æqualibus addantur inæqua<lb></lb>lia, erunt tota inæqualia: at〈que〉 maius ab acceßione maiori.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Refpondeo illam æquationem non niſi extrinſecè termina<lb></lb>ri: cùm partes habeant nullâ duratione commenſurabiles. </s><lb></lb><s>Fit ergo quemadmodum in aſcenſionibus ſignorum; ut licet <lb></lb>continuò partes maiores aut minores cooriantur; in fine ta<lb></lb>men motûs quadrantes inter ſe ſint æquales. </s></p> <pb xlink:href="063/01/117.jpg"></pb> <p type="main"> <s><emph type="center"></emph>THEOREMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si orbiculus æqualis percutiat baſim cylindri orbiculati minùs gra<lb></lb>vem; habeat verò impulſum duplo maiorem gravitate ſecunda; ex<lb></lb>clusâ à cylindro baſi pertranſibit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi impulſum habeat æqualem gravitati ſecundæ; per <lb></lb>12 huius, ſuccedit in locum baſis à cylindro excluſæ: Cùm igi<lb></lb>tur eadem gravitate detineatur, quâ baſis excluſa; non niſi ab <lb></lb>impulſu æquali excludi valebit. </s><s>Vt ergo exclusâ baſi ipſe <lb></lb><expan abbr="quoq;">quoque</expan> eluctetur; impulſum habebit duplo maiorem. </s><s>Quòd <lb></lb>ſi verò impulſum habeat illâ gravitate maiorem, minorem ve<lb></lb>rò quàm duplum; exclusâ baſi non totus, ſed pro ratione ex<lb></lb>ceſſûs plus, minuùſuè à cylindro prominebit. </s><s>Priuſquam <lb></lb>hunc motum orbiculorum finiam; admonere volui, ne quis <lb></lb>ab uno experimento obiter facto, <expan abbr="neq;">neque</expan> niſi omnibus propo<lb></lb>ſitionibus priùs expenſis, facile pronuntiet: cùm hæ inter<lb></lb>dum illas limitent. </s><s><expan abbr="Icaq;">Icaque</expan> cùm dico orbiculum, ſi alium per<lb></lb>cutiat ſibi æqualem, illo expulſo quieſcere; id non prorſus ve<lb></lb>ritati conſonum videbitur, ſi experimentum fiat in orbiculis <lb></lb>magis ponderoſis: cuiuſmodi metallici, ex argento, ferro, ære, <lb></lb>plumbo, ſtanno, auro. </s><s>Percuſſo enim æquali non quieſcunt, <lb></lb>ſed aliquantulum ex illâ plagâ ſequuntur: idq magis minùſue <lb></lb>pro ratione ponderis. </s><s>Quod quidem ad finem theor: 6 monui <lb></lb>Quia nimirum rationem cylindri habent eiuſmodi orbiculi: <lb></lb><expan abbr="magiſq;">magiſque</expan> ponderoſus æquivalet cylindro longiori. </s><s><expan abbr="Itaq;">Itaque</expan> diffe<lb></lb>rentia plagæ in his maior; quæ in orbiculis levioribus evane<lb></lb>ſcit, & ob exiguitatem ſenſum latet. </s><s>Idem fit in globis magni <lb></lb>ponderis & molis. </s><s>Quia vel centrum gravitatis non eſt idem <lb></lb>cum centro molis: vel quòd ſuperficiem minùs ſphæricam <lb></lb>habentes non in puncto, ſed parte aliquâ dividuâ ſe tangunt, <pb xlink:href="063/01/118.jpg"></pb>vel quòd plaga aliquantulum inclinet. </s><s>Quin & volubilitas <lb></lb>ſpeciem motûs continuati <expan abbr="quandoq;">quandoque</expan> præſtat. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Orbiculorum in cylindro diſpoſitorum quemcun〈que〉 imperatum exclu<lb></lb>dere, alijs non excluſis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In cylindro orbiculato AI ſit excludenda baſis A. id conſe<lb></lb>quemur cum orbiculo æquali M factâ plagâ per 1 Poriſ: At <lb></lb>ſi tertius à baſi C excludi debeat: appone duos à tergo pla<lb></lb> <arrow.to.target n="fig21"></arrow.to.target><lb></lb>gæ KL: & cum tribus orbiculis percute cylindrum: namre<lb></lb>liquis immotis tertius exſiliet: propterea, quòd impetus prio<lb></lb>rum in illas anterides ſe exonerat. </s><s>Quòd ſi artem magis la<lb></lb>tere velis; ſint orbiculi mole, non etiam pondere æquales. </s><lb></lb><s>Duobus ergo levioribus tertio æquali ſubiectis, ſi percu<lb></lb>tiatur cylindrus; quia minor plaga leviorum, non niſi tertium <lb></lb>excludes. </s><s>Eodem modo ſi quartus, aut quintus poſtuletur; <lb></lb>cum totidem numero orbiculis <expan abbr="plagã">plagam</expan> induces: uno verô minus <lb></lb>à tergo cylindri <expan abbr="plagã">plagam</expan> excipies: aut certè totidem leviores, <lb></lb>quot ſupereſſe velis, ultimo ſuppone. </s></p> <pb xlink:href="063/01/119.jpg"></pb> <figure id="id.063.01.119.1.jpg" xlink:href="063/01/119/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>PROBLEMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Orbiculos plures ſiòi contiguos à cylindro orbiculato excludere, alijs <lb></lb>non excluſis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Si à baſi incipiat numerus orbiculorum; cum totidem per<lb></lb>cute: <expan abbr="atq;">atque</expan> eundem numerum à cylindro excludes. </s><s>Quòd ſi <lb></lb>orbiculi intereſſe debent; <expan abbr="totidẽ">totidem</expan> à tergo cylindri oppone: tum <lb></lb>enim à ſuâ ſtatione dimoventur ex illâ plagâ, quibus núlli or<lb></lb>biculi ſunt oppoſiti. </s><s>Aut certè totidem leviores ſuppone, <lb></lb>quot cum baſi reliquos eſſe velis: nullâ enim motione ab his <lb></lb>factâ, numerum quæſitum dabit plaga reliqua. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Orbiculos plures non contiguos à cylindro orbiculato excludere, alijs <lb></lb>non excluſis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sint orbiculi tres excludendi, nimirum 1. 3 & 5 omnibus <lb></lb>alijs immotis ex illâ plagâ. </s><s>Quod quidem duobus modis con<lb></lb>ſequimur. uno, ſi orbiculi plagam afferentes ſint inæquales: <lb></lb><expan abbr="levioréſq;">levioréſque</expan> percutiant eos, quos manere volumus. </s><s>Secundo <lb></lb>modo, ſi his æqualiter habentibus ſequantur inæquales: <expan abbr="atq;">atque</expan> <lb></lb>illorum plaga, quos excuti volumus, ſe recipiat in minores: <lb></lb>tum enim per Poriſ: 2 motum minoris ſequitur maior. </s><s>Cùm <lb></lb>autem dicimus reliquos orbiculos eſſe <expan abbr="abſq;">abſque</expan> motu; de illo in<lb></lb>tellige, qui provenit à percuſſione: neceſſe enim in illa inter<lb></lb>valla, à quibus orbiculi ſunt eiecti, alios ſe recipere à gravita<lb></lb>te depreſſos. </s><s>Quòd ſi tamen dextrè plaga inferatur, <expan abbr="omneſq;">omneſque</expan> <lb></lb>orbiculi inter ſe ſint æquales & ad libellam complanati; <expan abbr="abſq;">abſque</expan> <pb xlink:href="063/01/120.jpg"></pb>ſuccuſſione fit cylindri: qui non niſi ex inæquali orbiculorum <lb></lb>lapſu, aut cùm plaga in alios impingit, dilabitur. </s></p> <p type="main"> <s><emph type="italics"></emph>Verùm dubitatio non levis occurrit. </s><s>Nam ſi orbiculi inter ſe æqua<lb></lb>les & contigui longî ſerie diſponantur in lineâ; rectà; percuſſo primo ul<lb></lb>timus movetur non eadem ratione: verùm pro numero orbiculorum <lb></lb>minùs, quouſ〈que〉 omnes à plagâ ſint immoti. </s><s>Marceſcit ergo illâ exten<lb></lb>ſione impulſus; ne〈que〉 totaplaga in ſingulos propagatur. </s><s>Atqui eadem <lb></lb>ratio videtur ſphærularum: quomodo ergo per infinitas hunc extendi <lb></lb>volumus, quem in orbiculis cito videmus terminari.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo dici poſſe, quòd ſi orbiculi per omnia ſint æqua<lb></lb>les, in lineâ centri gravitatis ſitum habentes, eadem ratione, <lb></lb>quâ in ſphærulis interminabilem fore motum. </s><s>Verùm <lb></lb>quia illorum centrum non neceſſariò eſt idem cum centro <lb></lb>gravitatis; cùm partes habeant à ſe differentes: inde fieri ut <lb></lb>centrum gravitatis <expan abbr="plerumq;">plerumque</expan> ſit extra illam lineam, quæ tran<lb></lb>ſit per centra orbiculorum. </s></p> <p type="main"> <s>Quôd ſi ita: non iam una omninm eſt plaga; ſed minor quæ <lb></lb>percutit obliquè: neceſſe ergo dum eâ ratione mutatur cen<lb></lb>trum gravitatis, impulſum minui, ac demum deficere. </s></p> <p type="main"> <s>Reſpondeo ſecundò, illam poſitionem de interminabili mo<lb></lb>tu ſphærularum non niſi ut probabilem aſſumi. </s><s>Vt verò <lb></lb>gratiam ineamus etiam cum his, qui eam averſantur; videa<lb></lb>mus ſi quâ ratione hunc motum, omnibus immotis, quæ pro <lb></lb>fundamento ſunt adducta, terminare valeamus. </s><lb></lb><s>Cùm ergo ſphærula prima ſecundam hæc tertiam percutit; <lb></lb>dico inæqualem fieri plagam. </s><s>Nam quia impulſus inæquali<lb></lb>ter recipitur in mobili; prout nimirum partes magis, minùſuè <lb></lb>abſunt à plagâ; quæ tamen æquationem habent à centro gra- <pb xlink:href="063/01/121.jpg"></pb>vitatis; quo omnes æqualiter moventur: minor erit vis in <lb></lb>centro quàm in loco plagæ. </s><s>Quòd ſi enim motui velociſ<lb></lb>ſimo accedat minùs velox; hunc quidem incitari, illum verò <lb></lb>retardari contingit. </s><s>Igitur cùm per cuſſio fiat à centro, mi<lb></lb>nor erit plaga à ſecundo quàm à primo. </s><s>Ratio in oppo<lb></lb>ſitum facta ita diſſolvitur. </s><s>Impulſum à plagâ 20 ad totum <lb></lb>impulſum maiorem rationem habere, quàm ſubvigecuplam. </s><lb></lb><s>Licet enim plaga ſecunda ſit minor quàm prima: non tamen <lb></lb>illud decrementum eſt æquale magnitudini, quam pertranſit, <lb></lb>ſed exceſſui, quo plaga maior eſt æquatione centri gravitatis: <lb></lb>quæ differentia in eiuſmodi ſphærulis eſt valde exigua. </s><s><expan abbr="Itaq;">Itaque</expan> <lb></lb>fit ut globus libr: 20. moveri nequeat à plagâ unius libræ: im<lb></lb>pulſus tamen tranſiens per globos librales 20. ultimum mo<lb></lb>veat. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si orbiculum tangant plures alij eidem æquales; percutiat verò hunc <lb></lb>alius orbiculus æqualis, ad intervallum maius quadrante à contactu <lb></lb>illorum orbiculorum; omnes contigui à percuſſo movebuntur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Vt ſi orbiculum A tangant alij C. D. E: percutiat verò eun<lb></lb>dem A æqualis B in puncto H; cuius intervallum HG, vel <lb></lb>H L maius <expan abbr="quadrãte">quadrante</expan>: dico omnes contiguos C. D. E moveri <lb></lb>ex illâ plagâ Ducantur à contactu orbiculorum G & L ipſi AH <lb></lb>parallelæ GP. LO, ſecantes AT. AK in O & P: dico menſu<lb></lb>ram plagæ OK <expan abbr="atq;">atque</expan> TP ſimul ſumptam eſſe minorem radio <lb></lb>AK: ac proinde impulſum reliquum à plagâ movere orbicu<lb></lb>lum D. </s><s>Ducantur rectæ DE. IL. & quia ut AD ad DE, ita <lb></lb>AI ad IL; ſunt verò AD. DE æquales; erit <expan abbr="quoq;">quoque</expan> AI æqua- <pb xlink:href="063/01/122.jpg"></pb>lis IL chordæ grad: 60. cuius ſinus rectus AO; atq, huius <lb></lb>complementum OK minus ſemiſse radij. </s><s>Quòd ſi orbiculus <lb></lb>E tangat A inter L&K; erit minor huius plaga, quàm OK <lb></lb>ptopterea quòd DE fiat maior quàm AD, & IL maior quàm <lb></lb>AI: ac proinde AO maior ſinu grad: 60. </s></p> <figure id="id.063.01.122.1.jpg" xlink:href="063/01/122/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Idem verò ſequitur, ſi orbiculi C. D. E aſſumantur mino<lb></lb>res, quàm ſit A. propterea quòd hi ex impulſu minori move<lb></lb>antur, quàm orbiculi æquales, per poriſ: 2. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Tres orbiculos percutere eadem plagâ: qui in motu percutiant alios <lb></lb>tres quolibet intervallo ſeiunctos: â quibus rurſum alij tres percutian<lb></lb>tur in quouis ſitu.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sint tres orbiculi in ſltu <emph type="italics"></emph>a.b.c:<emph.end type="italics"></emph.end> quos alij <emph type="italics"></emph>g.h.i<emph.end type="italics"></emph.end> percutere de<lb></lb>bent in motu: à quibus rurſum alij tres <emph type="italics"></emph>d.e.f<emph.end type="italics"></emph.end> percutiantur: a <pb xlink:href="063/01/123.jpg"></pb>ſingulis ſinguli. nempe ab <emph type="italics"></emph>a<emph.end type="italics"></emph.end> ipſum <emph type="italics"></emph>d,<emph.end type="italics"></emph.end> & à <emph type="italics"></emph>b<emph.end type="italics"></emph.end> ipſum <emph type="italics"></emph>e,<emph.end type="italics"></emph.end> at〈que〉 de<lb></lb>mum <emph type="italics"></emph>f<emph.end type="italics"></emph.end>à<emph type="italics"></emph>c.<emph.end type="italics"></emph.end> Ducantur per illorum centra rectæ <emph type="italics"></emph>da. eb. fc:<emph.end type="italics"></emph.end> & <lb></lb>producantur extra circulum in <emph type="italics"></emph>o.p.q,<emph.end type="italics"></emph.end> adintervallum ſemidia<lb></lb>metri eiuſdem circuli: à quibus ducantur aliæ rectæ <emph type="italics"></emph>ok. pk. qk<emph.end type="italics"></emph.end><lb></lb>per centrum orbiculi <emph type="italics"></emph>k<emph.end type="italics"></emph.end> maioris. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> orbiculi <emph type="italics"></emph>g. h. i<emph.end type="italics"></emph.end><lb></lb>contigui orbiculo <emph type="italics"></emph>k<emph.end type="italics"></emph.end> habeant centra in eiſdem lineis <emph type="italics"></emph>ok. pk. qk:<emph.end type="italics"></emph.end><lb></lb>percutiat verò orbiculum <emph type="italics"></emph>k<emph.end type="italics"></emph.end> alius æqualis, vel maior <emph type="italics"></emph>l<emph.end type="italics"></emph.end> in <emph type="italics"></emph>w<emph.end type="italics"></emph.end>: di<lb></lb>co orbiculos <emph type="italics"></emph>g.h.i<emph.end type="italics"></emph.end> ex illâ plagâ percutere orbiculos <emph type="italics"></emph>a.b.c:<emph.end type="italics"></emph.end> ab <lb></lb>his verò rurſum percuti orbiculos <emph type="italics"></emph>d.e.f.<emph.end type="italics"></emph.end><lb></lb>Cùm enim orbiculi <emph type="italics"></emph>g.h.i<emph.end type="italics"></emph.end> ſint minores quàm <emph type="italics"></emph>k;<emph.end type="italics"></emph.end> movebuntur <lb></lb>exillâ plagâ per coroll: Theor: 14. </s><s>Et quia percuſſio, & qui <lb></lb>hanc ſequitur impulſus, fit per lineam rectam productam à con<lb></lb>tactu per centrum corporis percuſſi per 5 Theor: 2 part: erit <lb></lb>motus orbiculi <emph type="italics"></emph>g<emph.end type="italics"></emph.end> in lineâ <emph type="italics"></emph>go.<emph.end type="italics"></emph.end> Ducatur per contactum linea <emph type="italics"></emph>rs<emph.end type="italics"></emph.end><lb></lb>parallela ipſi <emph type="italics"></emph>ok:<emph.end type="italics"></emph.end> quæ ſi ſecet orbiculum <emph type="italics"></emph>a,<emph.end type="italics"></emph.end> erit linea hypomo<lb></lb>chlij, & complementum <emph type="italics"></emph>os<emph.end type="italics"></emph.end> eiuſdem plaga: quæ ex demonſtra<lb></lb>tis orbiculum <emph type="italics"></emph>a<emph.end type="italics"></emph.end> movebit per rectam <emph type="italics"></emph>ad.<emph.end type="italics"></emph.end> Quòd ſi verò recta <lb></lb><emph type="italics"></emph>rs<emph.end type="italics"></emph.end> cadat extra <expan abbr="utrumq;">utrumque</expan> orbiculum; problema locum non ha<lb></lb>bebit. </s><s>Non eſt tamen neceſse per <expan abbr="utrumq;">utrumque</expan> centrum duci li<lb></lb>neam rectam; niſi cùm totum impulſum dare volumus orbi<lb></lb>culo percuſlo: ſed ſufficit, ſi ex centro unius producta linea re<lb></lb>cta tangat, vel ſecet <expan abbr="quacunq;">quacunque</expan> ratione alterum orbiculum. </s><lb></lb><s>Eadem ratione oſtendemus orbiculos <emph type="italics"></emph>b<emph.end type="italics"></emph.end> & <emph type="italics"></emph>c<emph.end type="italics"></emph.end> percuti ab<emph type="italics"></emph>h<emph.end type="italics"></emph.end> & <emph type="italics"></emph>i:<emph.end type="italics"></emph.end><lb></lb>percutere verò eoſdem <emph type="italics"></emph>e<emph.end type="italics"></emph.end> & <emph type="italics"></emph>f.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DE <lb></lb>PROPORTIONE MOTVS ORBICVLO<lb></lb>RVM TAM AD SE, QVAM AD MOTVM <lb></lb>ORBICVLI CONTIGVI, A QVO <lb></lb>IMPELLVNTVR.<emph.end type="center"></emph.end></s></p> <pb xlink:href="063/01/124.jpg"></pb> <p type="main"> <s>In quâ proportione ſinguli orbiculi ferantur, cùm tres con<lb></lb>tigui ab æquali impelluntur, dictum Theor: 14. </s><s>Quòd ſi ve<lb></lb>rò idem orbiculus non niſi duos habeat ſibi contiguos; aut ipſi <lb></lb><expan abbr="quoq;">quoque</expan> erunt contigui inter ſe, aut non contigui. ſint primùm <lb></lb>contigui. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si in diametro orbiculi productâ fiat contactus orbiculorum; percu<lb></lb>tiat verò hunc alius æqualis in parte oppoſitâ diametri; eo immoto u<lb></lb>ter〈que〉 contiguorum movetur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Percutiat orbiculum A alius æqualis in F: in cuius diametro <lb></lb>productâ FQ fiat contactus orbiculorum CD: dico immo<lb></lb>to A <expan abbr="utrumq;">utrumque</expan> orbiculum C & D moveri ex illâ plagâ. </s><s>Du<lb></lb>catur linea hypomochlij HG: & ad eam perpendicularis AN <lb></lb>quæ ſecabitur in duo ſegmenta æqualia AS. NS. propterea <lb></lb>quòd AS ſit ſinus rectus grad: 30. ſemiſſis nimirum GI grad: <lb></lb>60. habebit ergo plaga ſemiſſem totius impulſûs: qui per po<lb></lb>ſit:4 velocitate feretur ſubduplâ illius velocitatis. quâ orbicu<lb></lb>lus A moveretur. </s><s>Quod idem dicendum de orbiculo D. </s><lb></lb><s>Cùm <expan abbr="itaq;">itaque</expan> duo orbiculi C&D ſimul contineant totum impul<lb></lb>ſum ex A; erit plaga perfecta: ac proinde orbiculus A à motu <lb></lb>conquieſcet. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si diameter orbiculi producta ſecet unum exorbiculis ſibi contiguis; <lb></lb>percutiat verò hunc alius æqualis in parte oppoſitâ diametri productæ; <lb></lb>eo immoto, uter〈qué〉 orbiculorum eidem contiguorum movebitur.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="063/01/125.jpg"></pb> <p type="main"> <s>Cùm enim TP menſura plagæ, quam recipit orbiculus C <lb></lb>ex A, ſit minor quàm AP ſinus rectus grad: 60; qui reliquum <lb></lb>impulſum, quo centrum A moveretur à plagâ, metitur; erit ut <lb></lb>AP ad PT, ita motûs in A admotum in C. </s><s>Quia verò or<lb></lb>biculus A percutit æqualem D, occurrit verò eidem in lineâ <lb></lb>centri; dabit plagam perfectam: ac proinde per 1 poriſma A <lb></lb>quidem à motu conquieſcet, D verò eadem velocitate feretur. </s><lb></lb><s>Verùm licet hypomochlium GP eâ ratione impulſum partia<lb></lb>tur; quia tamen <expan abbr="utraq;">utraque</expan> plaga fit ſimul; habebit plaga ex A ad <lb></lb>plagam ex P eam rationem, quam AT ad PT. </s><s>Cùm enim <lb></lb>vectis ſit AT, cuius fulcimentum in T; erit per prop: 3 Gui<lb></lb>di Vbaldi, ut AT ad PT, ita gravitas appenſain A adean<lb></lb>dem gravitatem appenſam in P. </s><s>At verò eandem rationem <lb></lb>habet vis ſurſum impellens, quam gravitas deorſum movens: <lb></lb>quòd gravitas non niſi mediante impulſu agat. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> <lb></lb>totus impulſus in AT ſit partium 42; & TP pars ſexta AT; <lb></lb>erit plaga TP in primâ quidem partitione, quam hypomo<lb></lb>chlium GP inducit, partium 7: impulſus verò reliquus in A <lb></lb>partium 35. </s><s>At verò cùm percuſſio geminatur; plaga ex A <lb></lb>quidem eſt partium 36, ex P verò partium 6. </s></p> <p type="main"> <s>Secet nunc orbiculum D, non per centrum, diameter pro<lb></lb>ducta ex puncto medio inter F&H: in quo eundem percutiat <lb></lb>alius orbiculus æqualis: dico immoto A <expan abbr="utrumq;">utrumque</expan> orbiculum <lb></lb>C & D moveri. </s><s>Ducantur per contactus GI lineæ hypomo<lb></lb>chlij eidem diametro parallelæ: quas ſecent lineæ perpendicu<lb></lb>lares ex A. erit itaq, huic quidem æqualis linea ex G perpendi<lb></lb>cularis ad eandem diametrum ſinus grad: 45. propterea quod <lb></lb>GI ſit grad: 60 <expan abbr="atq;">atque</expan> huius ſemiſſis VI grad: 30. cuius comple<lb></lb>mentum 2928992 menſura plagæ in C. </s><s>Rurſum quia FI <lb></lb>eſt grad: 165; erit ſemiſſis ſinus rectus grad: 82 pr:30. cuius <pb xlink:href="063/01/126.jpg"></pb>ſinus verſus 8694738 metitur plagam in D. </s><s>Cùm <expan abbr="itaq;">itaque</expan> to<lb></lb>tus impulſus ſit partium 10000000, <expan abbr="utraq;">utraque</expan> verò plaga ſimul <lb></lb>ſumpta partium 11623730 maior ſinu toto; erit plaga per<lb></lb>fecta: ac proinde orbiculus A à motu conquieſcet. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLLARIV M.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Quòd ſi <expan abbr="itaq;">itaque</expan> ſinus totus ſecetur in eâ ratione, quam habet <lb></lb>numerus maior ad minorem; erit motus in D ad motum in <lb></lb>C in eadem ratione, quæ paulominor eſt quàm tripla. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si duo orbiculi non contigui tangant alium ſibi æqualem ad inter<lb></lb>vallum maius quàm.grad: 60. percutiat verò hunc æqualis in parte op<lb></lb>poſitâ; uter〈que〉 unà cum orbiculo percuſſo movebitur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><expan abbr="Tangãt">Tangant</expan> orbiculum A duo æquales in L & V: percutiat ve<lb></lb>rò hunc alius æqualis in H: dico <expan abbr="utrumq;">utrumque</expan> orbiculum unà cum <lb></lb>A moveri ex illâ plagâ. </s><s>Cùm enim AZ ſinus grad: 30 ſit ſe<lb></lb>miſſis AT; erit plaga huic æqualis. </s><s>Et quia AO eſt ſinus <lb></lb>grad: 60; erit complementum OK partium 1339746 in mi<lb></lb>nori ratione, quàm ſeptuplâ ad ſinum totum. </s><s>Eſt autem <lb></lb>ut AK ad OK, ita plaga ex Aad plagam ex O. </s><s>Quòd ſi itaq, to<lb></lb>tus impulſus ſit partium 12; erit in O plaga minor quàm parti<lb></lb>um 2: & <expan abbr="utraq;">utraque</expan> plaga ſimul ſumpta minor quàm partium 8. <lb></lb>impulſus ergo reliquus in A maior quàm partium 4. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>COROLL ARIV M.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sequitur quô maius intervallum inter contactus orbiculo<lb></lb>rum, eò velociorem eſſe motum orbiculi his contigui: propte- <pb xlink:href="063/01/127.jpg"></pb>rea quòd impulſus reliquus ad plagam continuò maiorem ha<lb></lb>beat rationem. </s><s>Et ſicuti ab intervallo grad: 60 incipit motus <lb></lb>orbiculi A; ita motus contiguorum terminatur, ubi contactus <lb></lb>non niſi quadrante circuli abfuerit à plagâ. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Duo puncta in peripheriâ orbiculi aßignare: in quibus orbiculi ei<lb></lb>dem contigui eadem cum illo velocitate moveantur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Secetur diameter orbiculi TK in ſex partes æquales. </s><s>Sup<lb></lb>ponamus verò TP <expan abbr="atq;">atque</expan> OK eſſe eiuſmodi ſegmenta: à qui<lb></lb>bus ducantur lineæ perpendiculares LO. GP. </s><s>Dico in pun<lb></lb>ctis L. G orbiculos EC eadem cum A celeritate moveri. </s><lb></lb><s>Cùm enim PT ſit pars tertia AT; habebit plaga in A ad <lb></lb>plagam in P rationem triplam. </s><s>Quòd ſi <expan abbr="itaq;">itaque</expan> impulſus æqua<lb></lb>lis AT ſit partium 12; erit plagain P partium 4. </s><s>Eſt verò <lb></lb>eidem æqualis plagain O; igitur reliquus impulſus in A erit <lb></lb><expan abbr="quoq;">quoque</expan> partium 4: ac proinde per poſitionem 4 tres orbiculi A <lb></lb>CE: eadem velocitate moventur. </s></p> <p type="main"> <s><emph type="center"></emph>PROBLEMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Duo puncta in peripheriâ orbiculi determinare: à quicus orbiculi <lb></lb>contigui moveantur, tam ad ſe, quàm ad motum orbiculi his contigui <lb></lb>in datâ ratione.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit proportio data motûs orbiculorum contiguorum tri<lb></lb>pla: motûs verò orbiculi reliqui ad unum ex his ſeſquialtera. </s><lb></lb><s>Secetur <expan abbr="itaq;">itaque</expan> ſemidiameter AT in ſex partes æquales: & du<lb></lb>catur linea à ſecundâ diviſione, quæ à centro, perpendicularis <lb></lb>producta ad peripheriam: eritq plaga in A ad illam plagamin <pb xlink:href="063/01/128.jpg"></pb>feſquialterâ ratione. </s><s>Rurſum verò ſecentur illa quatuor ſe<lb></lb>gmenta reliqua in tres partes æquales: & ab ultimâ ſectione, <lb></lb>quæ ad peripheriam, ducatur perpendicularis: <expan abbr="eritq;">eritque</expan> prior <lb></lb>plaga ad hanc in ratione triplâ. </s><s>Quòd ſi itaq, in alterâ ſemidi<lb></lb>ametro uni ſegmento ſumatur æquale; & ducatur perpendicu<lb></lb>laris ad peripheriam; inventa erunt duo puncta, à quibus or<lb></lb>biculi impulſi moveantur in datâ ratione. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si orbiculum Atangat alius æqualis Q ad intervallum grad: 30 à di<lb></lb>ametro HI; percutiat verò eundem æqualis H; motus centri'ex illâ pla<lb></lb>gâ non dimovetur à lineâ HI.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ductâ ex V perpendiculari VZ: erit AZ ſinus rectus grad: <lb></lb>30, ſemiſſis radij AT: motus vero in A æqualis plagæ in V. </s><lb></lb><s>Producatur TQ parallela VZ: <expan abbr="eritq;">eritque</expan> AQ ad AT, ut AV <lb></lb>ad AZ & VQ ad TZ. ſed ut AQ ad AT, ita TQ ad VZ: & <lb></lb>permutando TQ ad VQ, ut VZ ad TZ. </s><s>Eſt autem VZ <lb></lb>ſinus rectus grad: 60 maior quàm AZ ſinus rectus grad: 30. </s><lb></lb><s>Cùm <expan abbr="itaq;">itaque</expan> motus in A ſit æqualis AZ; erit velocior motus in <lb></lb>VQ, quo centrum Q à contactuſe abducit, quàm ut aliquod <lb></lb>punctum inter VT ipſum conſequi valeat. non igitur cen<lb></lb>trum A dimovetur à lineâ rectâ AI. </s></p> <p type="main"> <s><emph type="center"></emph>THEOREMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si orbiculum A tangat alius æqualis C adintervallum grad: 60 à di<lb></lb>ametro HI, percutiat verò eundem æqualis in H; motus centri A ſit per <lb></lb>tangentem circuli, cuius centrum eſt contactus orbiculi C.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Quoniam GP ſinus grad: 30 eſt minor ſinu AP grad: <pb xlink:href="063/01/129.jpg"></pb> <arrow.to.target n="fig22"></arrow.to.target><lb></lb>60; habebit hic ad PT maiorem rationem, quàm GP. </s><s>Eſt au<lb></lb>tem ut GP ad PT, ita TR ad RG: velocior ergo motus cen<lb></lb>tri A, <expan abbr="atq;">atque</expan> huius parallelorum inter G & T, quàm ſit motus or<lb></lb>biculi C, quo à contactu orbiculi A ſe abducit: neceſse pro<lb></lb>inde centrum A prohibitum à contactu viam proximam ſequi: <lb></lb>hoc eſt per tangentem circuli centro G deſcripti. </s></p> <figure id="id.063.01.129.1.jpg" xlink:href="063/01/129/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>THEOREMA XX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si plures orbiculi tangant alium maiorem; percutiat verò hunc æ<lb></lb>qualis; omnes contigui unà cum orbiculo maiore movebuntur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Tangant orbiculum <emph type="italics"></emph>k<emph.end type="italics"></emph.end> quotlibet alij minores <emph type="italics"></emph>g. b. m. i:<emph.end type="italics"></emph.end> per<lb></lb>cutiat verò hunc æqualis <emph type="italics"></emph>l:<emph.end type="italics"></emph.end> dico orbiculos <emph type="italics"></emph>g. h. m. i<emph.end type="italics"></emph.end> unà cum <lb></lb>orbiculo <emph type="italics"></emph>k<emph.end type="italics"></emph.end> moveri ex illâ plagâ. </s><s>Erit enim ex demonſtratis <lb></lb>Theor: 16 ut <emph type="italics"></emph>kz<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>nz<emph.end type="italics"></emph.end> ita plaga orbiculi <emph type="italics"></emph>h<emph.end type="italics"></emph.end> ad plagam orbi<lb></lb>culi <emph type="italics"></emph>m:<emph.end type="italics"></emph.end> & ut <emph type="italics"></emph>nz<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>yz,<emph.end type="italics"></emph.end> ita plaga in <emph type="italics"></emph>m<emph.end type="italics"></emph.end> ad plagam in <emph type="italics"></emph>i.<emph.end type="italics"></emph.end> Quòd <lb></lb>idem de plagâ orbiculi <emph type="italics"></emph>g<emph.end type="italics"></emph.end> dicendum. maior <expan abbr="itaq;">itaque</expan> omnibus pla<lb></lb>ga eſt in <emph type="italics"></emph>h.<emph.end type="italics"></emph.end> Quia verò plaga ſequitur impulſum, quo percu- <pb xlink:href="063/01/130.jpg"></pb>tiens erat moturum; percutit verò <emph type="italics"></emph>k<emph.end type="italics"></emph.end> orbiculum minorem <emph type="italics"></emph>h;<emph.end type="italics"></emph.end><lb></lb>movebitur hic ab incipiente & necdum perfectâ plagâ: orbicu<lb></lb>lus ergo <emph type="italics"></emph>k<emph.end type="italics"></emph.end> per poriſma 2. motum continuabit. </s><s>Simili modo <lb></lb>orbiculi. reliqui <emph type="italics"></emph>g. m. i<emph.end type="italics"></emph.end> quia minores quàm <emph type="italics"></emph>k;<emph.end type="italics"></emph.end> movebuntur ab <lb></lb>impulſu minori: ac proinde à plagâ incipiente: unde huius ex<lb></lb>ceſſus erit principium motûs orbiculi <emph type="italics"></emph>k,<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>DE GYRATIONE ORBICVLI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si orbiculus percuſſus alium impellat ſibi contiguum & æ<lb></lb>qualem; duplici motu videtur ferri ex illâ plagâ: nimirum <lb></lb>recto & circulari. </s><s>Nam cùm A movetur per lineam AI, pun<lb></lb>ctum H in peripheriâ transfertur in F. K &c. </s><s>Quod quidem <lb></lb>erit manifeſtum ſi punctum contactûs aliquo ſigno notetur. </s><lb></lb><s>Huius autem motûs ratio videtur referri ad librationem. </s><s>Nam <lb></lb>cùm ex plagâ in G deceſſerit impulſus æqualis PT; neceſse <lb></lb>præpondium fieri in K, <expan abbr="atq;">atque</expan> ita revolui orbiculum circa mobi<lb></lb>le centrum A. </s></p> <p type="main"> <s><emph type="italics"></emph>Obijcies. </s><s>Si ob librationem circumagitur orbiculus, neceße buius mo<lb></lb>tum eſſe æqualem plagæ. cui æquatur exceſſus in parte oppoſitâ. igitur quò <lb></lb>contactui propior diameter, quia tum maior plaga; erit quo〈que〉 circulatio <lb></lb>maior: quod tamen non fit. </s><s>Verùm quò maius intervallum, eò arcum <lb></lb>deſcribit maiorem. </s><s>Deinde verò ſi duo orbiculi contigui inæqualiter ab<lb></lb>ſint à diametro, cuiuſmodi in LV, circulatio procedit ex H in N. eſt au<lb></lb>tem maior plaga in V quàm in L: oportebat ergo hunc motum fieri ex <lb></lb>H in F, ſi illa circulatio proveniret ab exceſſu.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo cùm motus hic circularis fluat ab eodem impul<lb></lb>ſu, quem retinet centrum ad ſe movendum; hic autem acceſſu <lb></lb>ad diametrum continuò minuatur; neceſsc <expan abbr="quoq;">quoque</expan> circulatio- <pb xlink:href="063/01/131.jpg"></pb>nem æſtimari minorem. </s><s>Deinde verò cùm per Theor: 19 <lb></lb>ab intervallo grad: 60 motus centri fiat per tangentem circu<lb></lb>li; neceſſe hanc librationem magis augeri. </s><s>Vnde etiam ratio <lb></lb>petenda, quòd circulatio <expan abbr="quandoq;">quandoque</expan> fiat in partem plagæ maio<lb></lb>ris: cùm videlicet duo orbiculi inæqualiter abſunt à lineâ mo <lb></lb>tûs centri: Hic enim oppoſita circulatio prævalet: quam deter<lb></lb>minat motus centri per tangentem. </s></p> <p type="main"> <s>Poſſe verò miſceri motui recto circularem, manifeſtum in <lb></lb>eodem orbiculo; ſi convexâ parte tangat planum. á digito e<lb></lb>nim compreſſus & eliſus <expan abbr="quandoq;">quandoque</expan> eidem puncto inſiſtens ro<lb></lb>tari, <expan abbr="quandoq;">quandoque</expan> à procurſu recurrere, aut etiam retro agi vide<lb></lb> <arrow.to.target n="fig23"></arrow.to.target><lb></lb>tur. </s><s>Quòd ſi enim motus circularis fiat æqualis motui recto; <lb></lb>videbitur orbiculus in eodem puncto A circa immobile cen<lb></lb>trum gyrari. </s><s>Dividatur peripheria orbiculi in ſex partes æqua<lb></lb>les <emph type="italics"></emph>abcdef:<emph.end type="italics"></emph.end> & ſumantur his æqualia ſegmenta in lineâ re<lb></lb>ctâ <emph type="italics"></emph>aghikl.<emph.end type="italics"></emph.end> Cùm <expan abbr="itaq;">itaque</expan> motus in <emph type="italics"></emph>ab<emph.end type="italics"></emph.end> ſit æqualis motui centri <lb></lb>eiuſdem orbiculi in <emph type="italics"></emph>ag;<emph.end type="italics"></emph.end> gyratio autem non niſi per contactum <pb xlink:href="063/01/132.jpg"></pb>fiat eiuſdem plani; neceſse ubi ex <emph type="italics"></emph>a<emph.end type="italics"></emph.end> promovit in <emph type="italics"></emph>g,<emph.end type="italics"></emph.end> ipſum <emph type="italics"></emph>b<emph.end type="italics"></emph.end><lb></lb>revolui in <emph type="italics"></emph>a.<emph.end type="italics"></emph.end> Similiter ubi <emph type="italics"></emph>b<emph.end type="italics"></emph.end> perventurum eratex <emph type="italics"></emph>a<emph.end type="italics"></emph.end> in <emph type="italics"></emph>g,<emph.end type="italics"></emph.end> ipſum <lb></lb><emph type="italics"></emph>c<emph.end type="italics"></emph.end> attinget punctum <emph type="italics"></emph>a.<emph.end type="italics"></emph.end> Quòd ſi maior ſit motus circuli, quàm <lb></lb>eiuſdem centri; contingit ipſum retroagi. </s><s>Nam cùm ex <emph type="italics"></emph>a<emph.end type="italics"></emph.end><lb></lb>movetur in <emph type="italics"></emph>g;<emph.end type="italics"></emph.end> motus in peripheriâ fit per maius <expan abbr="ſegmentũ">ſegmentum</expan> <emph type="italics"></emph>ab:<emph.end type="italics"></emph.end> ac <lb></lb>proinde orbiculus tangit planum in puncto medio inter <emph type="italics"></emph>b<emph.end type="italics"></emph.end> & <emph type="italics"></emph>c.<emph.end type="italics"></emph.end><lb></lb>Demum ſi maior ſit motus centri quàm gyrationis; videbitur <lb></lb>motus rectus, & punctum <emph type="italics"></emph>b<emph.end type="italics"></emph.end> inter <emph type="italics"></emph>a<emph.end type="italics"></emph.end> & <emph type="italics"></emph>g.<emph.end type="italics"></emph.end> Inde ergo ratio reddi<lb></lb>tur; quòd motus centri ab illatâ plagâ deflectat à lineâ rectâ <lb></lb><emph type="italics"></emph>AI<emph.end type="italics"></emph.end> etiam ante grad: 60. </s><s>Cùm enim motus orbiculi circularis <lb></lb>in plano firmetur, <expan abbr="eaq;">eaque</expan> ratione motui centri reluctetur; ne<lb></lb>ceſſe motum mixtum inde procreari. </s></p> <figure id="id.063.01.132.1.jpg" xlink:href="063/01/132/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>De Levigatione & Politura.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>COrpora polita dicuntur, quæ ſuperficiem habent illius fi<lb></lb>guræ, quâ terminantur, æquabilem: ut in cubo perfectè <lb></lb>planam, in globo ſphæricam. </s><s>His opponitur aſperum ſeu ſca<lb></lb>brum: cuius ſuperficies partes habet inæqualiter ſitas, magis <lb></lb>& minùs depreſſas aut elevatas. </s><s><expan abbr="Neq;">Neque</expan> omnia corpora æqua<lb></lb>liter: ſed alia magis, alia minùs, alia nullâ induſtriâ poliuntur: <lb></lb>ut thophus, pumex, ſuber, panni lanei &c. </s><s>Et cùm ſcabrities <lb></lb>ſeu inæqualitas à duobus proveniat: cùm vel partes inſunt <lb></lb>verrucoſæ, vel pori ſeu cavernulæ ſuperficiem perforantes, <lb></lb>quantum vis ſenſum lateant: polituram obtinemus contrariâ <lb></lb>affectione: verrucarum quidem, & quæ prominent, ablatione: <lb></lb>ſpatiorum verò inanium repletione. </s><s>Quòd ſi eiuſmodi um<lb></lb>bilici & verruculæ tolli nequeant: aut lacunæ expleri, impo<lb></lb>libile dicetur corpus, Talia ſunt <foreign lang="grc">ἀπιε<gap></gap>ὰ</foreign>, & quæ dividi neque<lb></lb>unt in partes minimas: quia <expan abbr="neq;">neque</expan> compreſſioni cedunt ad po<lb></lb>rum ſolidandum, receptâ in eas vacuitates parte magis preſsâ: <pb xlink:href="063/01/133.jpg"></pb>ut vitrum, gemmæ, lapides, <expan abbr="omniaq;">omniaque</expan> <foreign lang="grc">θραυ<gap></gap>α</foreign>: <expan abbr="neq;">neque</expan> pars minima <lb></lb>reſecari valet, cuiuſmodi eſt thophus. </s><s><expan abbr="Atq;">Atque</expan> illa quidem ſolá <lb></lb>partium ablatione poliuntur: & ſi quidem poroſa ſint, nullâ <lb></lb>ratione ſuam perfectionem aſſequitur politura: quemadmo<lb></lb>dum <expan abbr="neq;">neque</expan> individua in partes minimas: ablatâ enim parte ma<lb></lb>iori, quàm ſit exceſſus; eadem inæqualitas manet. </s><s>Corporaer<lb></lb>go <foreign lang="grc">παχυμέρεα</foreign> & quæ glutinosâ viſciditate tenaciùs cohærent, <lb></lb>ut cera, pix, tela linea, papyrus; ſolâ levigatione proficiunt: <lb></lb>partibus à compreſſione in eodem ſitu manentibus. </s><s>Vnde <lb></lb>panni lanei, ob pilos à compreſſione ſurrigentes, non levigan<lb></lb>tur. metalla <expan abbr="quoq;">quoque</expan> omnia, <expan abbr="atq;">atque</expan> ligna alia magis, alia minùs le<lb></lb>vigationi parent. </s><s>Quæ enim mollia ſunt, <expan abbr="neq;">neque</expan> compreſſa <lb></lb>manent in eo ſitu, ut medulla ſambuci, aut ſpongia, non levi<lb></lb>gantur. </s><s>Neceſſe enim reniti aliquas partes: quibus aliæ inni<lb></lb>tantur. </s><s>Quod non fit, ſi omnes à compreſſione moveantur <expan abbr="ce-dantq;">ce<lb></lb>dantque</expan> <expan abbr="Itaq;">Itaque</expan> ligna duriora, cuiuſmodi hebenus, præ alijs levi<lb></lb>gantur. </s><s>Cùm igitur illa corpora vel partium ablatione, vel <lb></lb>illarum ſitu permutato ſuperficiem politam conſequantur; <lb></lb>manifeſtum levigationem & polituram non <expan abbr="abſq;">abſque</expan> motu & im<lb></lb>pulſu fieri. </s><s>Cuiuſmodi verò ſit motus, & quâ ratione fiat, <lb></lb>nunc dicam, à levigatione incipiendo. </s></p> <p type="main"> <s><emph type="italics"></emph>Eſt autem levigatio motus reciprocus in ſuperficie levigandà, factus <lb></lb>à corpore polito, non ſine compreßione.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Niſi enim corpus levigans ſit terſum & politum; <expan abbr="nequaquã">nequaquam</expan> <lb></lb>aliam ſuperficiem levigare valebit: novâ aſperitate ex illa<lb></lb>rum partium inæqualitate inductâ: dum magis quidem pro<lb></lb>minentes excavant, & veluti ſulcos incidunt: depreſſæ verò <lb></lb>tubercula attollunt. </s><s><expan abbr="Itaq;">Itaque</expan> videmus ab eiuſmodi ſuperficie <lb></lb>aſpcrâ & hamatâ pannos aſperari & villoſos reddi: quò <expan abbr="filamẽ-ta">filamen<lb></lb>ta</expan> <expan abbr="atq;">atque</expan> illorum textura magis lateant. </s><s>Deinde ſi motus fiat <pb xlink:href="063/01/134.jpg"></pb><expan abbr="abſq;">abſque</expan> compreſſione, aut non niſi leviter illam ſuperficiem tan<lb></lb>gendo; <expan abbr="neq;">neque</expan> lacunæ expleri, <expan abbr="neq;">neque</expan> verruculæ deprimi valebunt. <lb></lb><expan abbr="Neq;">Neque</expan> motu ſimplici, <expan abbr="atq;">atque</expan> uno tractu perficitur politura: ſed <lb></lb>motibus iteratis, & in omnes partes reciprocè factis. </s><s>Et licet <lb></lb><expan abbr="quandoq;">quandoque</expan> ſolâ compreſſione planities inducatur; non tamen <lb></lb>levigatio eſt perfecta: ob plures ſulcos, <expan abbr="ſtriáſq;">ſtriáſque</expan> à compreſſione <lb></lb>relictas: quæ magis in profundum, quàm lateraliter movet. </s><lb></lb><s>Igitur cùm motus ſit cauſa levigationis; quo partes ſitum variè <lb></lb>permutant: & velin locum partium compreſſarum; velin me<lb></lb>dias cavitates trasferuntur: motus autem à percuſſione & à ta<lb></lb>ctu fiat; quem ex his motum dicemus levigationem? eſt enim <lb></lb><foreign lang="grc">ὡσιτ χίν<gap></gap>σις ἀπὸ τ<gap></gap>̄ς ἅψεως</foreign>: cùm movens non niſi tangendo <lb></lb>movet: at verò partes levigantes non manent, ſed prætere<lb></lb>unt: <expan abbr="continuóq;">continuóque</expan> alias tangunt partes: non igitur <foreign lang="grc">ὥσει</foreign> ſeu pul<lb></lb>ſione moventur partes levigandæ. </s></p> <p type="main"> <s>Reſpondeo, licet partes continuò mutentur: quia tamen <lb></lb>aliæ <expan abbr="atq;">atque</expan> aliæ ſuccedunt eiuſdem rationis, motum continuan<lb></lb>tes; per æquivalentiam idem videri movens. </s><s>Eſt autem dif<lb></lb>ferétia inter ea, quæ <foreign lang="grc">χίνησιν</foreign> habent <foreign lang="grc">ἀπὸ τγ̄ς ἅψεως</foreign>, & quæ <foreign lang="grc">ἀπὸ τγ̄ς <lb></lb>πληγη̄ς χινο<gap></gap>̄ντ<gap></gap></foreign>: quòd hæc in motu ſeparantur à movente: ac <lb></lb>proinde acceptâ plagâ non ſit in poteſtate moventis ille mo<lb></lb>tus. </s><s>Quæ verò <foreign lang="grc">ἀπὸ τη̄ς ἅψεως</foreign> moventur; impulſum habent <lb></lb>fluentem: qui non niſi illis motis eſſe poteſt: <expan abbr="moxq;">moxque</expan> ubi cæpit, <lb></lb>ex illo contactu finit: & non niſi impulſu continuato ſervari <lb></lb>poteſt. </s><s><expan abbr="Atq;">Atque</expan> inde fit, ut nulla particula inter poliendum, ſeu <lb></lb>levigandum divellatur: cùm motus in ipſa plagâ finiat, <expan abbr="neq;">neque</expan> <lb></lb>ullus reſtet impulſus. </s><s>Et licet non ſine aliquâ tractione par<lb></lb>tes levigatæ extendantur; non tamen eſt motus exceſſivus: <lb></lb><expan abbr="neq;">neque</expan> per ſe, ſed à compreſſione naſcens: <expan abbr="Itaq;">Itaque</expan> ſi excedat, ut <lb></lb>dum chartam minùs cautè levigamus; partes divelluntur. </s></p> <p type="main"> <s><emph type="italics"></emph>Dices. A quo ergo partes compreßa detinentur in eo ſitu? ne〈que〉 enim ſo-<emph.end type="italics"></emph.end> <pb xlink:href="063/01/135.jpg"></pb><emph type="italics"></emph>la<emph.end type="italics"></emph.end> <foreign lang="grc">π<gap></gap>εσὰ</foreign> <emph type="italics"></emph>levigantur: ne〈que〉 illa filamenta linteorum & minutuli ſlocci <lb></lb>in compreßione uniuntur, ſuperficiem unam habentes: verùm contigui <lb></lb>inter ſe manent: ita〈qué〉 linteis excußis rurſum à ſe diſiungi, & ſuperfici<lb></lb>em hiſpidam reddi videmus.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo illum ſitum non niſi à novo motu turbari: mo<lb></lb>tum verò non <expan abbr="abſq́">abſque</expan>; impulſu advenire. </s><s>Quòd ſi ergo partes <lb></lb><expan abbr="neq;">neque</expan> à ſe, <expan abbr="neq;">neque</expan> ab extra habeant principium motûs; neceſſe illá <lb></lb>ſuperficiem, in quam terminavit motus, retinere. </s><s><expan abbr="Itaq;">Itaque</expan> lin<lb></lb>tea agitata turbantur: dum ex illo motu impetum concipiunt <lb></lb>particulæ, ab eo ſitu diſtrahentem. </s><s>Quæ autem rigidiuſcula <lb></lb>ſunt: quia in ſe ipſis habent principium motús; à compreſſio<lb></lb>ne eo modo, quo arcus à curvaturá, reaſſurgunt. </s><s>Sicuti ve<lb></lb>rò duobus modis levigatio fit; <expan abbr="nimirũ">nimirum</expan> depreſſione & <expan abbr="cõpreſſio-ne">compreſſio<lb></lb>ne</expan> <expan abbr="atomorũ">atomorum</expan>; ita <expan abbr="quoq;">quoque</expan> duobus motibus oppoſitis turbatur: cùm <lb></lb>vel eriguntur: vel partes preſſæ retumeſcunt. </s><s>Superficié le viga<lb></lb>tam ſequitur tanquam proprietas ſplendor: lucis nimirum uni<lb></lb>tæ confertim facta evibratio. </s><s>Nam quæ ſuperficiem habent <lb></lb>aſperam, lucem incidentem diſtrahunt & inæqualiter <expan abbr="reflectũt">reflectunt</expan>. <lb></lb><expan abbr="Neq;">Neque</expan> enim ab aliquâ parte radij uniti, ſed à ſe divulſi, <expan abbr="ſeq;">ſeque</expan> in<lb></lb>terſecantes in retinam feruntur: ſinguli non niſi luce tenui ſen<lb></lb>ſum afficientes. </s></p> <p type="main"> <s><emph type="italics"></emph>An igitur inferre licet omnia, quæ luce alienâ reſplendent ſuperfici<lb></lb>em habere levigatam? Nitent enim margaritæ, conchylia, opera item <lb></lb>figulina vitreata, avium pennæ, atramentum, picturæ &c. in quibus <lb></lb>tæmen aſperitatem notamus.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo ſplendorem non niſi ex multâ luce unitâ naſci: <lb></lb>multa autem fit ratione ſubiecti. nam ſubiectum magis den<lb></lb>ſum plus lucis continet. </s><s>Corpus ergo denſiſſimum & ſummè <lb></lb>politum ſplendorem habet ſummum. </s><s><expan abbr="Itaq;">Itaque</expan> aurum perfectè <lb></lb>levigatum præ omnibus alijs ſplendet, <expan abbr="aciemq;">aciemque</expan> oculorum per- <pb xlink:href="063/01/136.jpg"></pb>cellit: plumbum verò licet alijs metallis magis denſum; quia <lb></lb>tamen ob partes terreas minùs levigari poteſt, & colori atro <lb></lb>magis miſcetur; minùs reſplendet. </s><s>Fieri ergo poteſt ut cor<lb></lb>pus denſum, & ſi minùs politum, magis ſplendeat, quàm rarum, <lb></lb>& è contra: at ſummè levigatum neceſſariò ſuperat denſiſſi<lb></lb>mum; ſi prorſus ſit impolitum. </s><s>Deinde levigatum ſeu poli<lb></lb>tum duobus modis ſumitur: abſolutè, & ſecundum quid. </s><lb></lb><s>Abſolutè quidem, cuius ſuperficies <expan abbr="undiq;">undique</expan> eſt terſa & æqualis: <lb></lb>ſecundùm quid autem, quod non totam ſuperficiem, ſed tan<lb></lb>tum aliquas partes habet levigatas, non continuas inter ſe, ve<lb></lb>rum partibus ſcabris interciſas. </s><s>Multa ergo licet ſuperficiem <lb></lb>habeant ſcabram & inæqualem; quia tamen eiuſmodi umbili<lb></lb>cos continent leves & politos; reſplendent. </s><s>Ita enim figulina <lb></lb>poliuntur: dum metallicus humot illitus, <expan abbr="atq;">atque</expan> igne liqueſcens <lb></lb>ſuperficiem inungit: & demum æqualiter concretus ſpeciem <lb></lb>vitri aſſumit. </s><s>Similiter panis humido inunctus, cruſtam in igne <lb></lb>trahit reſplendentem. </s><s>Ita atramentum ſcriptorium admiſto <lb></lb>gummi ſplendet. quia ob viſciditatem minùs ſorbetur humor: <lb></lb>& partes vitriolicæ ceu viſco cohærentes, inter ſiccandum mi<lb></lb>nùs hiulcæ fiunt. </s><s>Colores <expan abbr="quoq;">quoque</expan> & picturæ glutine pellucido <lb></lb>affuſo, aut permixto ſimili ratione reſplendent, </s></p> <p type="main"> <s><emph type="italics"></emph>Sed dices. aquam eſſe ſummè levigatam, minùs tamen alijs ſplendere.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo ſplendorem eſſe lucem à ſuperficie reflexam: <lb></lb>ut autem reflecti poſſit neceſſe priùs terminari. </s><s>At verò aquam <lb></lb>pellucidam lux pertranſit: minùs ergo lucis à reflexione. </s><s>De<lb></lb>inde cùm aqua ſit fluens & minùs denſa corporibus ex eâ con<lb></lb>cretis; minor copia lucis in eâ colligi poteſt. </s><s>Nec refert mul<lb></lb>ta corpora eſſe rariora: hæc enim ſuam concretionem aëri de<lb></lb>bent, qui in his prædominatur: cuiuſmodi volucrum pennæ, <lb></lb>& ſambuci medulla. </s></p> <pb xlink:href="063/01/137.jpg"></pb> <p type="main"> <s>Hæc de levigatione. </s><s>Politura eundem finem habet; nimi<lb></lb>rum ſuperficiem erugatam, <expan abbr="terſámq;">terſámque</expan>: differentia eſt in modo <lb></lb>& medijs ad hunc finem. </s><s>Nam levigatio deprimit, aut in lacu<lb></lb>nas transfert: politura adimit ſcabritiem efficientes partes. </s><lb></lb><s>Quæ quidem differentia in materiâ fundatur: cuius partes <expan abbr="neq;">neque</expan> <lb></lb>comprimi valent, <expan abbr="neq;">neque</expan> aliò transferri: Talia ſunt vitra, gemmæ, <lb></lb>lapides, <expan abbr="atq;">atque</expan> omnia <foreign lang="grc">ἀπιες ὰ χ<gap></gap>̀ θζαυ<gap></gap>ὰ</foreign>. </s><s>Igitur quædam <expan abbr="utroq;">utroque</expan> <lb></lb>modo hunc finem conſequuntur, & politurâ & levigatione, ut <lb></lb>metalla: quædam ſolâ levigatione, ut papyrus, lintea, cera: <lb></lb>quædam non niſi politurâ, ut gemmæ, lapides, vitra. </s><s>Ratio <lb></lb>eſt quia nequeunt dividi in partes minimas: ſiue ob viſcidita<lb></lb>tem magis tenacem, ſiue ob craſſitiem. </s><s><expan abbr="Itaq;">Itaque</expan> fit, ut dum plus, <lb></lb>minúſue à ſcabritie aufert politura; prior inæqualitas con<lb></lb>tinuò aliâ permutetur. </s><s>Deinde levigatio in multis incipit à <lb></lb>politurâ: cùm nimirum maior eſt aſperitas, quàm compreſſio <lb></lb>eſſe poſſit; neceſſe ergo illum exceſſum adimi, quò reliqua ſu<lb></lb>perficies levigationem habeat magis expeditam: ita enim ligna <lb></lb><expan abbr="atq;">atque</expan> metalla non niſi ferro acciſa levigantur. </s><s><expan abbr="Neq;">Neque</expan> idem eſt <lb></lb>modus polituræ in omnibus; <expan abbr="neq;">neque</expan> idem principium. </s><s>Nam pro<lb></lb>ut materiâ, & ſuperficie magis & minùs aſperâ à ſe differunt; <lb></lb><expan abbr="atq;">atque</expan> ab alijs plus, ab alijs minùs eſt auferendum; ita <expan abbr="quoq;">quoque</expan> inſtru<lb></lb>menta varia ſunt inventa. </s><s>Saxa enim & marmora malleo de<lb></lb>cuſſis, aut ferro acciſis promontorijs æquantur: gemmæ verò <lb></lb>affricatione adlapidem arenarium molâ circumactum, corti<lb></lb>cem priùs, quò veſtiuntur, exuunt; inde ſerrâ obtusâ in <expan abbr="ſegmẽ-ta">ſegmen<lb></lb>ta</expan> dividuntur: demum arenâ <expan abbr="atq;">atque</expan> huius polline levigantur. </s><lb></lb><s>Ligna verò ſecuri, cuneo, ſerrâ finduntur & ſecantur: dein<lb></lb>de aſciâ, tornóve poliuntur. </s><s>Eſt autem nobis propoſitum non <lb></lb>niſi de motu & impulſu agere, quo ſuperficiem politam obti<lb></lb>nemus; nequaquam verò de arte poliendi, quæ ſuis Magiſtris <lb></lb>eſt relinquenda. </s><s>Incipiam verò à Lignorum politurâ, utpo- <pb xlink:href="063/01/138.jpg"></pb>te minùs operosâ. </s><s>Cuius principium <foreign lang="grc">σχίσις χ<gap></gap>̀ τμ<gap></gap>̄σισ<gap></gap></foreign>: in eo à <lb></lb>ſe differentes: quòd <foreign lang="grc">σχίσις</foreign> ſit <foreign lang="grc">διαίζεσι<gap></gap>̓πὶ τὸ πλ<gap></gap>ον. σχισετ<gap></gap> γὰζ</foreign>, <lb></lb>inquit Ariſtoteles, <foreign lang="grc">ὅταν ἐπὶ τὸ πλ<gap></gap>ον δι<gap></gap>ζη̄τ<gap></gap> ἡ' τό δι<gap></gap>ζο̄υν δι<gap></gap>ζ<gap></gap>, χα<gap></gap><lb></lb>πζοηγ<gap></gap>ται ἡ διαίζεσις</foreign>. </s><s>In fiſſione ergo eſt maior diviſio, |quàm <lb></lb>ut ad <expan abbr="illã">illam</expan> <expan abbr="plagã">plagam</expan> referri poſſit: <expan abbr="eamq;">eamque</expan> diviſio anteit. </s><s>Eſt enim <foreign lang="grc">χί<lb></lb>νησις ἀπὸ τη̄ς ἃψεως</foreign>: cùm plaga inſequitur <expan abbr="motũ">motum</expan>: <expan abbr="atq;">atque</expan> impulſum <lb></lb>habet <expan abbr="fluentẽ">fluentem</expan>, & à plagâ inſeparabilem. </s><s>Igitur cùm fiſſura ultra <lb></lb>plagam ſe extendat, non eſſe poteſt à plagâ. </s><s>Huius autem ra<lb></lb>tio. quia <foreign lang="grc">τὸ δι<gap></gap>ζ<gap></gap>̄ν</foreign> habet vim cunei: cuius ingreſſu in eam pla<lb></lb>gam partes diſtrahuntur. </s><s>Et cùm fibræ in longitudinem ex<lb></lb>currentes flecti nequeant; neceſſe ultra cuneum agi fiſſuram: <lb></lb><expan abbr="atq;">atque</expan> eò magis, quò fibras habent rigidiores, & minùs lentas. <lb></lb><expan abbr="Itaq;">Itaque</expan> ligna duriora magis finduntur, quàm mollia ac lenta: quæ <lb></lb>magis obliquari & flecti valent. </s><s>Vnde angulo obtuſiore, illa <lb></lb>verò acutiore, fiſſurâ magis productâ finiunt plagam. </s><s>Cùm <lb></lb>ergo inciſio fit, ferrum in fiſſurâ conquieſcit: partes verò hu<lb></lb>ius ingreſſu diſtractæ, quia flecti nequeunt ob rigiditatem, <expan abbr="neq;">neque</expan> <lb></lb>comprimi vulneris labra: quemadmodum fit in plumbi ſectu<lb></lb>râ, illam rectitudinem ſervantes findunt partes ulteriores: <lb></lb><foreign lang="grc">σχιστά</foreign> autem dicit Ariſtoteles <foreign lang="grc">ὅσα χατὰ μη̄χος ἔχ<gap></gap> τ<gap></gap>ς πόζ<gap></gap>ς χα<gap></gap><lb></lb>σὓς πζοσφύετ<gap></gap> ἀλλήλοις,<gap></gap> ἀλλὰ μη<gap></gap>χατὰ πλάτος</foreign>. </s><s>Eiuſmodi ſunt li<lb></lb>gna ferè omnia fibris in longitudinem protenſis: inter quas <lb></lb>pori ſubſtantiâ molliori & veluti fungosâ pleni interſunt, per <lb></lb>quas agitur fiſſura: non verò in tranſverſum per illas fibras, <lb></lb>in quibus non continuantur eiuſmodi pori. </s><s>Plaga autem fit à <lb></lb>ſectione pro ratione compreſſionis. </s><s>Igitur ligna, quæ fibras <lb></lb>habent directas, fiſſuram <expan abbr="quoq;">quoque</expan> agunt rectam: quòd ſi tortuosè <lb></lb>procedant, inæqualiter finduntur: cùm plaga viam ſequatur <lb></lb>mediam inter illas fibras. </s><s>At cùm ſerrâ dividuntur, à ſectio<lb></lb>ne etiam inter fibras ductâ nulla ſequitur fiſſura: quia ſerratio <lb></lb>partes fibroſas non diſtrahit, ſed diſcontinuas facit. </s><s>Eſt au- <pb xlink:href="063/01/139.jpg"></pb>tem ſerratio motus compoſitus ex inciſione & fractione. </s><s><expan abbr="Neq;">Neque</expan> <lb></lb>enim huius dentes aſperiuſculi inter ſe ſunt paralleli; ſed alter<lb></lb>natim ad latus <expan abbr="utrinq;">utrinque</expan> reflexi: quò diviſio ex obliquo facta oc<lb></lb>currat plagæ oppoſitæ. </s><s><expan abbr="Itaq;">Itaque</expan> partes quidem medias inciden<lb></lb>do, partes verò laterales ſuâ aſperitate radendo auferunt: <expan abbr="eaq;">eaque</expan> <lb></lb>ratione vulneris labra, quo motum habeant liberiorem, adau<lb></lb>gent. </s><s>Inciſio enim ſimplex eſt diviſio continui <expan abbr="abſq;">abſque</expan> deper<lb></lb>ditione alicuius particulæ: ut cùm pomum per medium ſeca<lb></lb>mus. </s><s>Differt à ſectione ſciſſura: quòd hæc ſit plaga continu<lb></lb>ata; ſectio verò ſimplex & interrupta: quæ tamen ob <expan abbr="vehemẽ-tiam">vehemen<lb></lb>tiam</expan> excedere poteſt illam. </s><s><expan abbr="Vtraq;">Vtraque</expan> eſt ſolutio unionum, ſeu <lb></lb>diſcontinuatio cum aliquâ compreſſione: neceſſe enim quod <lb></lb>incidit recipi in <expan abbr="illãm">illamm</expan> plagam, <expan abbr="partésq;">partésque</expan> medias comprimi in la<lb></lb>tus <expan abbr="utrumq;">utrumque</expan>. </s><s>Corpus ſerratile eſt <foreign lang="grc">χαταχτὸν<gap></gap> χαὶ θζαυσὸν</foreign> <expan abbr="neq;">neque</expan> enim <lb></lb>lapides, vitrum, gemmæ ſerrantur. </s><s>Nam ſerra, quâ gemmæ <lb></lb>mediâ arenâ ſecantur, quia dentibus caret, non niſi impropriè <lb></lb>dicitur. </s><s>Igitur lignis in hunc modum ſerrâ <expan abbr="atq;">atque</expan> ſecuri diviſis, <lb></lb>aut cultro inciſis aſcia ſuccedit: quâ ſuperficies aſpera & inæ<lb></lb>qualis aufertur. </s><s><expan abbr="Eſtq;">Eſtque</expan> huius motus idem cum inciſione; ma<lb></lb>gis tamen limitatus, ad menſuram ferri inciſorij ab eâ promi<lb></lb>nentis. </s><s>Non enim profundiùs agitur plaga, quàm ſit illa fer<lb></lb>ri longitudo: quæ contrahi & augeri pro libitu poteſt. </s><s>Im<lb></lb>pulſum verò habet fluentem: cùm ſit <foreign lang="grc">χίνησις ἀπὸ τη̄ς ἅψ<gap></gap>ως</foreign>: <lb></lb>quam aſcia manu librata dirigit, impulſum cohibens, quò mi<lb></lb>nùs latè evagetur. </s><s>Vnde maioribus aſcijs utuntur in politu<lb></lb>râ: quò maior compreſſio à pondere, & à parte huius planâ & <lb></lb>politâ levigatio ſimul fiat. </s><s>Huic ſimilis videtur motus à tor<lb></lb>no factus: idem enim eſt ſeu ferrum incidens, ſeu corpus inci<lb></lb>dendum moveatur. </s><s>Eſt ergo manus veluti aſcia, quæ fulcro <lb></lb>innixa aciem ferri pro voto inciſuræ libratam ſuſtinet: Velo<lb></lb>cior tamen huius, quàm aſciæ motus <expan abbr="atq;">atque</expan> in circulum reductus: <lb></lb>qualis quidem eſſe nequit aſciæ motus ad globum poliendum. <pb xlink:href="063/01/140.jpg"></pb>ita quidem ſe habet politura in lignis, & quæ his ſunt cogna<lb></lb>ta. </s><s>At verò torno poliuntur etiam metalla: nequaquam gem<lb></lb>mæ, lapides, aut <expan abbr="vitrũ">vitrum</expan>: quòd hæc <foreign lang="grc">α<gap></gap>τμητα</foreign> ſint <foreign lang="grc">χα<gap></gap> θζαυσὰ</foreign>: & non <lb></lb>niſi in plures partes friantur. </s><s><expan abbr="Itaq;">Itaque</expan> <expan abbr="neq;">neque</expan> aſciâ aut cultro ſecari <lb></lb>valent: cùm ſectio in duo terminetur, <expan abbr="unámq;">unámque</expan> particulam ab <lb></lb>alijs avellat. </s></p> <p type="main"> <s><emph type="italics"></emph>Sed cur metalla ab aſciâ non poliuntur, eandem vim cum torno ha<lb></lb>bente?<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſpondeo in torno eſſe motum velociorem; quo reſiſten<lb></lb>tia & durities metallorum ſuperatur. </s><s>Idem enim eſt cùm tor<lb></lb>no circumagitur mobile, quemadmodum ſi ferrum celerrimè <lb></lb>moveretur: ut cùm gemmæ orbiculis circumactis poliuntur. </s><lb></lb><s>Et licet illarum politura torno fieri videatur; ob motum cir<lb></lb>cularem illorum orbiculorum, quibus gemmæ ſe affricantes <lb></lb>atteruntur: eſt tamen longè diverſus <expan abbr="atq;">atque</expan> alius motus. </s><s>Non <lb></lb>enim orbiculi ſeu umbilici, quibus præpilantur cylindri ver<lb></lb>ſatiles, incidunt: ſed arenulæ his intermixtæ: quo modo in a<lb></lb>lijs orbiculis contingit horizonti parallelis. lutum enim arenu<lb></lb>latum continuò affuſum ſuâ aſperitate radit ſuperficiem ma<lb></lb>gis eminentem. </s><s>Cùm verò hæc omnia ſint <foreign lang="grc">θζαυσα<gap></gap></foreign>; erit illo<lb></lb>rum diviſio <foreign lang="grc">θ<gap></gap>ραῡς σ</foreign> non <foreign lang="grc">χάτα<gap></gap>ις</foreign>. quia non una particula, ſed <lb></lb>plures ſimul avelluntur: illæ nimirum, quæ impulſum <expan abbr="motũq;">motunque</expan> <lb></lb>recipiunt à plagâ, pro numero arenularum, non unâ. </s><s>Vide<lb></lb>tur autem motus compoſitus ex inciſione & fracturâ: com<lb></lb>preſſione quidem in profundum: tractu verò in latum agen<lb></lb>te plagam, ab impulſu fluente inductam. </s><s>Cùm igitur ſit <foreign lang="grc">χίνη<lb></lb>σις ἀπὸ τη̄ς ἅψεως</foreign>, nequaquam altè penetrat; ſed mox à com<lb></lb>preſſione & contactu impulſus cohibetur. </s><s>Cui accedit humi<lb></lb>ditas ex polline arenularum continuò affuſa gemmis, impul<lb></lb>ſum hebetans: alioquin fragilibus, ſi arenulâ ſiccâ poliantur. <pb xlink:href="063/01/141.jpg"></pb>quæ & calorem ex illo motu naſcentem, quo corpora tene<lb></lb>reſcunt, magisq, fragilia fiunt, obtundit. </s><s>Vnde adamantes, qui<lb></lb>bus gemmæ ſolidiores poliuntur, ex illâ velocitate motûs ſpe<lb></lb>ciem carbonis igniti aſſumunt. </s><s>Differentia autem plagæ fit <lb></lb>pro ratione arenularum: craſſiores enim & magis duræ ma<lb></lb>iora auferunt ſegmenta. </s><s><expan abbr="Itaq;">Itaque</expan> gemmas rudiores, <expan abbr="multúmq;">multúmque</expan> <lb></lb>aſperitatis habentes priùs ſaxis arenulatis, quæ molis circum<lb></lb>aguntur, affricantes, illâ attritione complanant: inde lapide <lb></lb>ſmiri in farinam trito poliunt: & magis ſubtili eiuſdem polline <lb></lb>levigantes, demum perfectionem terrâ tripolitanâ inducunt. </s><lb></lb><s>Vitra tamen quia molliora, calce ſtanni levigantur. </s><lb></lb><s>Nec differt illarum ſectio per ſerram æream edentulam facta, <lb></lb>lentiſſimo tractu arenulis interfuſis radente: pro <expan abbr="quarũ">quarum</expan> diverſi<lb></lb>tate mutatur <expan abbr="quoq;">quoque</expan> vulneris amplitudo. </s><s>Nam craſſior arena, <lb></lb>inquit Plinins, laxioribus ſegmentis terit, & plus erodit mar<lb></lb>moris, <expan abbr="maiúſq;">maiúſque</expan> opus ſcabritia polituræ relinquit. </s><s>Ita ſectæ atte<lb></lb>nuantur cruſtæ. </s><s>Duplex ergo incommodum ab arenâ craſſio<lb></lb>re: nam & plus decedit gemmis à plagâ latiore: & ſuperficies <lb></lb>aſpera maiorem in poliendo laborem exigit. </s><s><expan abbr="Itaq;">Itaque</expan> olim <expan abbr="uſq;">uſque</expan> <lb></lb>ad Æthiopas, & Indos arena petebatur: quarum Æthiopica <lb></lb>mollior, <expan abbr="nullâq;">nullâque</expan> ſcabritie ſecans. </s><s>Nunc verò lapis ſmiri & <lb></lb>terra tripolitana in uſum polituræ ſucceſſit. </s><s><expan abbr="Neq;">Neque</expan> ſolum gem<lb></lb>mæ, marmora, & vitrum arenâ, ſeu lapide arenoſo poliuntur; <lb></lb>ſed etiam metalla: cotibus enim ferrum atteri & pulvere ſmi<lb></lb>ri levigari conſtat. </s><s>Verùm hæc inſuper limam ſentiunt: <lb></lb>quòd gemmis non convenit. </s><s>Tametſi dicat Plinius nobili<lb></lb>um gemmarum ſoli Topazio id accidere: reliquas verò coti<lb></lb>bus Naxijs poliri. </s><s>Quod quidem de politurâ rudiori & incho<lb></lb>atâ intelligendum: <expan abbr="neq;">neque</expan> enim aut reliquas gemmas cotibus: <lb></lb>aut topazium limâ perfici potuiſſe credendum. </s><s>Noſtratem <lb></lb><expan abbr="quoq;">quoque</expan> topazium licet molliorem reliquis gemmis, limam re- <pb xlink:href="063/01/142.jpg"></pb>ſpuere experientia docet. </s><s>Fuerit ergo alterius generis Plinij <lb></lb>gemma à noſtrâ: quam inſuo genere virentem, <expan abbr="eiuſq;">eiuſque</expan> <lb></lb>ſimilitudinem, ad porri ſuccum dirigi teſtatur: cùm <lb></lb>noſtra ſit coloris aurei. </s><s>Motus, quem lima inducit, eſt <lb></lb>compoſitus ab inciſione cancellatâ & <foreign lang="grc">χλάτε<gap></gap></foreign>. </s><s>Nam ſulci præ<lb></lb>tenues inciſi ab aſperitate traſversâ eiuſdem limæ raduntur. </s><lb></lb><s>Maior ergo durities nobilioribus gemmis ineſt, cuiuſmodi a<lb></lb>damas, calcedonius, ſapphyrus, heliotropia, rubinus: quæ <expan abbr="neq;">neque</expan> <lb></lb>ferro incidi, <expan abbr="neq;">neque</expan> limâ radi ſuſtinent: quam tamen ſentiunt <lb></lb>marmora, lapides, vitrum, & gemmæ ignobiliores. A cotibus <lb></lb>verò hæc univerſa poliuntur: propterea quòd ſuperficiem <lb></lb>habeant cotes ſcabram & arenoſam: quæ ſi polita, <expan abbr="miniméq;">miniméque</expan> <lb></lb>friabilis eſſed, haud quaquam attererentur. </s><s>Mutuâ ergo <lb></lb>affrictone arenulæ coacervantur: quarum abſceſſu minui <lb></lb>cotes, & demum longo uſu abſumi conſtat. </s><s>Saxa verò duri<lb></lb>ora, quia atteri non valent, <expan abbr="neq;">neque</expan> uſum cotis habent. </s><s>Sed <lb></lb>quæſtio hic eſt: quamobrem cotes Naxiæ, & quæ nobis ſunt in <lb></lb>uſu, aquâ; Creticæ verò & Laconicæ, ut Plinius teſtatur, oleo <lb></lb>temperentur: an eiuſmodi cotes naturam habent olei, ſeu <lb></lb>bituminis pinguedinem continentes aquæ incommiſcibilem? <lb></lb>ineſſe enim quibuſdam lapidibus ſuccum pinguem & oleoſum <lb></lb>conſtat exinflammatione. </s><s>Lapis <expan abbr="quoq;">quoque</expan> nephriticus, quem <lb></lb>Iſadam vocant, multùm pingueſcit inter poliendum: quan<lb></lb>quam huius pinguedo non oleoſa, ſed quale gummi, aquæ <lb></lb>commiſcetur. </s><s>At quomodo ergo cotes Ciliciæ, eodem Plinio <lb></lb>teſte, oleo & aquâ pollent: an <expan abbr="utramq;">utramque</expan> naturam eo modo <lb></lb>permiſtam habent, quo ſmigma? quod & pingue in ſe tra<lb></lb>hit, & aquâ eluitur. </s><s>Ita cotes tonſtrinarum humore non quo<lb></lb>uis ſed viſcoſo, cuiuſmodi ſputum, proficiunt. </s><s>Aquas autem <lb></lb>in Italiâ repertas, aciem trahentes acerrimo ſenſu, minerales <lb></lb>fuiſle credo, eiuſdem naturæ cum aquâ forti. </s><s>Magis tamen <pb xlink:href="063/01/143.jpg"></pb>mirandum, quod tradit Ferdinandus Corteſius, in Mexico eſſe <lb></lb>lapidem coloris flavi; ex quo novaculæ fiant acutiſſimæ: quæ <lb></lb>non à ferro, aut cote, ſed ex aquâ illam aciem trahant. </s><lb></lb><s>Videtur autem hæc proprietas innuere huius cognationem <lb></lb>cum aquâ; <expan abbr="eſſéq;">eſſéque</expan> veluti glaciem ex aquâ concretam: à quâ <lb></lb>rurſum atteratur & liquefiat: mox tamen ab aëre eo modo, <lb></lb>quo ovorum cortices, indurari. A motibus iam dictis differt <lb></lb>terebratio & perforatio: <expan abbr="fitq;">fitque</expan> duobus modis. </s><s>Vt cùm cavi<lb></lb>tas inducitur <expan abbr="abſq;">abſque</expan> deperditione alicuius particulæ: & cùm <lb></lb>partes ab illâ cavitate excluduntur. </s><s>Et primo quidem modo <lb></lb>cavitas fit per compreſſionem: quam ſola <foreign lang="grc">πιες ὰ</foreign> admittunt, cu<lb></lb>iuſmodi metalla & ligna: nequaquam verò <foreign lang="grc">τὰ θζαυ<gap></gap>ὰ</foreign> at gem<lb></lb>mæ, lapides, vitra. </s><s>Cùm deperditione verò ſubſtantiæ & hæc <lb></lb>& reliqua omnia cavantur: licet non uno modo omnia. </s><lb></lb><s>Nam gemmæ quidem & vitra non niſi politurâ, <expan abbr="ſenſimq;">ſenſimque</expan> ra<lb></lb>dendo perforantur: terebratione verò ligna, metalla, oſſa. </s><lb></lb><s>Et ſicuti terebra figurâ à ſe differunt; ita etiam modus perfo<lb></lb>randi. </s><s>Alia enim circulo; alia formâ ſemilunari terminan<lb></lb>tur: alia cochleatim ſtriata, ab acuto ſenſim augentur & late<lb></lb>ſcunt. </s><s><expan abbr="Atq;">Atque</expan> hæc quidem à perforatione incipiendo <foreign lang="grc">σχίσει χ<gap></gap>́ <lb></lb>χλάσει</foreign> terminant motum. </s><s>Dum enim cochlea circum acta <lb></lb>in partem compreſſam vulnus agit, & quæ à tergo ſequitur <lb></lb>helix, ambit latiore plagâ; in tenues & friabiles lamellas eâ <lb></lb>ratione ſcobinatur helicoides, à plagâ inciſus conus: <expan abbr="eoq;">eoque</expan> in <lb></lb>helicem cavam recepto terebratio procedit, <expan abbr="quouſq;">quouſque</expan> cochlea <lb></lb>repleta ſcobe, educi & expurgari debeat. </s><s>Videtur autem hic <lb></lb>ratio vectis intervenire: cuius hypomochlium in centro mo<lb></lb>tûs: extrema verò ſunt circelli ſenſim aucti & in conum late<lb></lb>ſcentes. </s><s>Verùm huiuſmodi terebella ſuperficiem, quæ am<lb></lb>bit plagam, minùs æqualem relinquunt: calorem verò ob <lb></lb>multiplicem motum adaugent. </s><s><expan abbr="Itaq;">Itaque</expan> minùs apta cranio per- <pb xlink:href="063/01/144.jpg"></pb>forando: ne huius medulla nimiùm exæſtuet. </s><s>Quæ autem <lb></lb>circulo finiunt: quia unâ inciſione auferunt quidquid inclu<lb></lb>ditur illo circulo, <expan abbr="unóq;">unóque</expan> motu ſimplici peragunt inciſionem; <lb></lb>in hunc uſum veniunt. </s><s>Nam cùm in gyrum agitur hic cir<lb></lb>culus; <expan abbr="unaquæq;">unaquæque</expan> particula incidit: & cùm aliæ eiuſdem rati<lb></lb>onis ſequantur; vulnus continuò fit maius: <expan abbr="atq;">atque</expan> eò magis, <lb></lb>quo <foreign lang="grc">θλίψις</foreign> ſeu compreſſio maior, motus autem velocior. </s><lb></lb><s>Differt ab his terebellum, quo metalla perforantur. </s><s>Stylus e<lb></lb>nim <foreign lang="grc">αμφί<gap></gap>ζυς</foreign> cylindro infixus veluti torno circumagitur, non <lb></lb>ſine compreſſione ad corpus terebrandum. </s><s>Qui motus <lb></lb>inciſione perficitur: <expan abbr="duritiémq;">duritiémque</expan> metalli ſuperat ob <lb></lb>illam velocitatem. </s></p> <p type="main"> <s><emph type="center"></emph>FINIS.<emph.end type="center"></emph.end></s></p> <figure id="id.063.01.144.1.jpg" xlink:href="063/01/144/1.jpg"></figure> <p type="main"> <s><emph type="center"></emph>PRAGÆ.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>Ex Typographia Academica.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>Anno 1648.<emph.end type="center"></emph.end><lb></lb> <arrow.to.target n="fig24"></arrow.to.target></s></p> <figure id="id.063.01.144.2.jpg" xlink:href="063/01/144/2.jpg"></figure> </chap> </body><pb xlink:href="063/01/145.jpg"></pb> <back><section><p type="main"> <s><emph type="center"></emph>[Errata not transcribed.]<emph.end type="center"></emph.end></s></p></section></back> </text> </archimedes>