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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Marci von Kronland, Johannes Marcus </author> <title>De proportione motus seu, Regula sphygmica ad celeritatem et tarditatem pulsuum</title> <date>1639</date> <place>Prague</place> <translator/> <lang>la</lang> <cvs_file>marci_propo_01_la_1639</cvs_file> <cvs_version/> <locator>062.xml</locator> </info> <text> <front> </front> <body> <chap> <pb/> <pb/> <figure id="fig1"></figure> <p type="caption"> <s>IOANNES MARCVS MARCI PHIL: & MEDIC: DOCTOR <lb/><emph type="italics"/>et Profe&longs;&longs;or natus Landscron<gap/> Harnumdurarum in Boënua <lb/><gap/>.<emph.end type="italics"/></s></p> <figure></figure> <pb/> <p type="main"> <s><emph type="center"/>DIVO <lb/>FERDINANDO <lb/>TERTIO<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>AUGUSTISSIMO ROMANORUM <lb/>IMPERATORI<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Hungariæ & Bohemiæ Regi &c. <lb/><emph type="italics"/>Dominomeo Clementi&longs;&longs;imo.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Augu&longs;ti&longs;sime Cæ&longs;ar<emph.end type="center"/></s></p> <p type="main"> <s>DVm ut annus hic nouus TUÆ Maje­<lb/>&longs;tatiau&longs;picatus ordiatur, vota conci­<lb/>pio, & à tenuitate meà munu&longs;culum <lb/>TUÆ Maie: gratum eflagito: ecce ti­<lb/>bihuncip&longs;um, qui annum au&longs;picatur, <expan abbr="atq;">atque</expan> &longs;ua in ve <lb/>&longs;tigia reuoluit, motum mihi ultrò, ut Mercurius &longs;it <lb/>& munus, &longs;e offerentem: quid enim in quit extra <lb/>me quæris? in me &longs;unt omnia. Ab&longs;it, in quam ego, <lb/>ut ad Cæ&longs;arem ea<gap/>, qui tam in&longs;tabilis es & infidus, <pb/><expan abbr="atq;">atque</expan> eadem, quæ dare videbaris, rur&longs;um aufers. Nul <lb/>lum, inquit ille periculum ab in&longs;tabilitate: hic enim <lb/>Senex, ut vides, me quadratum fecit: quòd &longs;i tibi ita <lb/>videtur, me vel cubum facias. Benè inquam resha­<lb/>bet, ad Cæ&longs;aremibis: verùm his ego te priùs circu­<lb/>lis illigabo, <expan abbr="at&qacute;">atque</expan>; his lineis ceu virgulis &longs;ub leges Geo­<lb/>metriæ cogam, ut non ni&longs;i ad nutum Cæ&longs;aris mo­<lb/>uearis: &longs;is autem men&longs;ura & &longs;imul cu&longs;tos illius mo <lb/>tus, à quo Regalis vita pendet. Hunc ergo motum <lb/>Augu&longs;ti&longs;sime Cæ&longs;ar modulis geometricis ad&longs;tri­<lb/>ctum, & nunc Medicinæ famulantem ad TUAM <lb/>Maie&longs;tatem tanquam Primum Motorem remitto, <lb/>qui & cores & Sol Imperij & Regnorum, Tu<gap/>que <lb/>benignitatis motu hunc in me motum commour­<lb/>&longs;ti. Motum quidem hunc TUÆ Maie&longs;tati vt Soli <lb/>& Motori, at verò eidem Soli vt illuminatori Iri­<lb/>dem votiuam, gratitudinis & debitæ ob&longs;ervantiæ <lb/>ergo à TUÆ Maie&longs;tatis radijs conceptam hic idem <lb/>annus in proximo dabit: quam huc <expan abbr="u&longs;&qacute;">u&longs;que</expan>; quantum­<lb/>uis con&longs;pici volentem, & &longs;uà pulchritudine ambi­<lb/>tio&longs;am eadem fata, quæ pacem morantur, detinue­<lb/>re: ut nimirum hoc demum anno pace é victorijs <pb/>TUÆ Maie&longs;tatis na&longs;cente & pluuiá &longs;anguinis eju&longs;­<lb/>dem radijs &longs;iccatá, Iris con&longs;picua veluti arcus trium <lb/>phalis TUÆ Maie&longs;tatis &longs;equatur pompam trium­<lb/>phalem. </s></p> <p type="main"> <s>Augu&longs;ti&longs;simæ Maie&longs;tatis Tuæ </s></p> <p type="main"> <s><emph type="center"/>humillimus Servus & Cliens<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Joannes Marcus Marci.<emph.end type="italics"/></s></p> <pb/> <p type="main"> <s><emph type="center"/>Definitiones.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>1.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Contraria dicuntur quæ tollunt, uel impediunt &longs;u­<lb/>um contrarium.<emph.end type="italics"/></s></p> <p type="main"> <s>NAm contrario <gap/> e&longs;t natura, ut &longs;imul e&longs;&longs;e <lb/>non po&longs;sint in uno &longs;ubjecto: necesse ergo unum <lb/>ab altero tolli, aut quò minùs recipiatur in illo <lb/>&longs;ubiecto impediri. <expan abbr="Ita&qacute;">Itaque</expan>; calori frigus contrarium di­<lb/>cunt non totà &longs;uà latitudine, &longs;ed &longs;ecundùm illos gra­<lb/>dus, qui &longs;imul e&longs;&longs;e non po&longs;&longs;u<gap/> in codem &longs;ubjecto <gap/><lb/>quatuor autem gradus caloris cum totidem gradibus <lb/>frigoris non e&longs;&longs;e contrarios, verúm inter &longs;e mi&longs;ceri, <expan abbr="at&qacute;">atque</expan>; <lb/>ex illis ita permixtis temperiem na&longs;ci. Simili modo <lb/>motus motui dicet ut contrarius, qui à termino illius <lb/>idem mobile abducit, <expan abbr="nullam&qacute;">nullamque</expan>; partem viæ &longs;eu acce&longs;­<lb/>&longs;us ad illum terminum habet communem. Vt &longs;i in <lb/>fig: 1 ex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> moveatur, erit motus contrarius, qui ex <lb/>eodem <emph type="italics"/>a<emph.end type="italics"/> idem mobilè in cab ducit. Motus verò ex <emph type="italics"/>a<emph.end type="italics"/> in <lb/><emph type="italics"/>d<emph.end type="italics"/> non erit contrarius ab&longs;olutè, propterea quòd hic mo­<lb/>tus non abducit à termino motus <emph type="italics"/>b,<emph.end type="italics"/> verùm ad hunc in <lb/>omni puncto propiùs accedit: quód &longs;i enim ex <emph type="italics"/>b<emph.end type="italics"/> ducan <lb/>tur lineæ <emph type="italics"/>be. bf. bg,<emph.end type="italics"/> erit linea <emph type="italics"/>bf<emph.end type="italics"/> minor quam <emph type="italics"/>be,<emph.end type="italics"/> & <emph type="italics"/>bg<emph.end type="italics"/> mi <lb/>nor quam <emph type="italics"/>bf.<emph.end type="italics"/> Huju&longs;modi ergo motus dum inter&longs;e <pb/>milc entur, non &longs;e mutuó tollunt ab&longs;olutè, verúm in <lb/>eo in quo &longs;unt &longs;imiles, in motum me dium coale&longs;centes <lb/>vià mediá <expan abbr="vtri&qacute;">vtrique</expan>; termino propinquant: in quantum <lb/>verò contrarij, illam rectitudinem viæ tollunt. Con­<lb/>traria ergo dicuntur quæ tollunt, vel impediunt &longs;uum <lb/>contrarium. </s></p> <p type="main"> <s><emph type="center"/>2.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Similia verò qua augent vel perficiunt &longs;uum &longs;imile.<emph.end type="italics"/></s></p> <p type="main"> <s>VT &longs;i ad motum <emph type="italics"/>ac<emph.end type="italics"/> alius ac cedat impul&longs;us, quiper <lb/>eandem lineam <emph type="italics"/>ac<emph.end type="italics"/> moveat idem mobile, erit hic <lb/>motus illi &longs;imilis, ac proinde eundem dicetur augere, <lb/>quemadmodum calor alium calorem &longs;ibi &longs;imilem: ca­<lb/>lor autem à luce, aut è contra, quia di&longs;similes, non di­<lb/>centur augeri. </s></p> <p type="main"> <s><emph type="center"/>3.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Et mixta à quibus actiones procedunt mixtœ.<emph.end type="italics"/></s></p> <p type="main"> <s>ILlarum nimirum qualitatum, quæ vim habent a­<lb/>gendi, latiùs &longs;umpto nomine actionis, pro qualibet <lb/>actione etiam perfectiuà: <expan abbr="ita&qacute;">itaque</expan>; illa <expan abbr="quo&qacute;">quoque</expan>; mutatio, <lb/>quam dulcoacidum inducit, actio dicetur mixta: <lb/>quem admodum frigus calore temperatum actionem <lb/>efficere èx <expan abbr="utro&qacute;">utroque</expan>; mixtam. Sic orgo motus dicetur <pb/>mixtus, dum inpul&longs;us <expan abbr="ne&qacute;">neque</expan>; in totum &longs;imilis, <expan abbr="ne&qacute;">neque</expan>; in to­<lb/>tum e&longs;t contrarius alteri impul&longs;ui. </s></p> <p type="main"> <s><emph type="center"/>4.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus ab&longs;oluté contrarij, qui idem mòbile ducunt <lb/>ex eodem puncto ad partes oppa&longs;itas ejusdem lineæ rectæ.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>5.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus &longs;ecundum quid contrarij, quiex illopuncto, <lb/>&longs;eù principio motus angulum ducunt majorem a ut minorem recto <lb/>minorem verò duobus rectis.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>6.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus qui exeodempuncto tendunt ad ea&longs;dem <lb/>partes lineæ rectæ inter &longs;e &longs;unt &longs;imiles.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>7.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus qui minori angulo ab&longs;i&longs;tunt magis &longs;unt <lb/>&longs;imiles<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>8.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus per fectè mixti quorum principium e&longs;t an­<lb/>gulus rectus.<emph.end type="italics"/></s></p> <p type="main"> <s>VT &longs;i in fig: 2. ex eodem puncto <emph type="italics"/>a<emph.end type="italics"/> moueatur idem <lb/>mobile &longs;imulin <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>e,<emph.end type="italics"/> dicetur hic motus ab&longs;olutè <lb/>contrarius. Motus verò ex eodem puncto <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>d,<emph.end type="italics"/><lb/>aut in <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>f,<emph.end type="italics"/> quorum hic major, ille minor &longs;it angulo re- <pb/>cto, erunt motus &longs;ecundùm quid contrarij: propterea <lb/>quòd non extoto &longs;e impediunt aut tollunt: contrarie­<lb/>tas enim motus ex acce&longs;&longs;u & rece&longs;&longs;u ad eundem termi­<lb/>num prouenit: motus autem &longs;ecundùm quid contrari; <lb/>dum inter &longs;e mi&longs;centur, licet &longs;uos terminos non a&longs;­<lb/>&longs;equantur, ij&longs;dem tamen continuò fiunt propiores. <lb/>Quia verò lineæ motus quò minori angulo ab&longs;i&longs;tunt, <lb/>eò propiùs accedunt ad terminum, erunt hi motus ma <lb/>gis &longs;imiles: perfecta autem &longs;imilioudo in eadem lineà <lb/>rectà, quæ ad eundem terminum perducit. Motus de­<lb/>mum, quorum principium e&longs;t angulus rectus, quia ex <lb/>illà mixt one propiores quidem fiunt termino motus, <lb/>intervallum autem in fine motus &longs;patio inter principi­<lb/>um & terminum motus e&longs;t æquale, nimirum in fig: 7. <lb/>dicentur motus perfectè mixti: tantùm enim con <lb/>trarij, quantùm &longs;imilitudinis ine&longs;t; </s></p> <p type="main"> <s><emph type="center"/>Po&longs;itiones:<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Simile & æquale auget&longs;uum&longs;imile in eademrati­<lb/>one, totum quidem totum, pars verò partem &longs;ibi æqualem.<emph.end type="italics"/></s></p> <p type="main"> <s>SIt linea <emph type="italics"/>ad<emph.end type="italics"/> æqualis lineæ <emph type="italics"/>ef,<emph.end type="italics"/> & diuidatur bifariam in <lb/><emph type="italics"/>b<emph.end type="italics"/>: quód &longs;i ergo tota linea <emph type="italics"/>ad<emph.end type="italics"/> addatut toti <emph type="italics"/>ef,<emph.end type="italics"/> &longs;icuti tota <lb/> <arrow.to.target n="fig2"></arrow.to.target> <pb/>toti, & &longs;emi&longs;sis &longs;emi&longs;si, & <expan abbr="tri&etilde;s">triens</expan> trienti e&longs;t æqualis, ita to­<lb/>ta totam, & &longs;emi&longs;sis &longs;emi&longs;&longs;em, & triens trientem auge­<lb/>bit in eadem ratione, in quà tota totam. Si ergo &longs;emi&longs;­<lb/>&longs;is <emph type="italics"/>ab<emph.end type="italics"/> addatur toti <emph type="italics"/>ef,<emph.end type="italics"/> quia ut <emph type="italics"/>ad<emph.end type="italics"/> ad <emph type="italics"/>ab,<emph.end type="italics"/> ita <emph type="italics"/>ef<emph.end type="italics"/> æqualis <emph type="italics"/>ad<emph.end type="italics"/><lb/>ad candem <emph type="italics"/>ab,<emph.end type="italics"/> erit augmentum æquale eju&longs;dem &longs;emi&longs;­<lb/>&longs;i: &longs;ola ergo &longs;emi&longs;sis lineæ <emph type="italics"/>ef<emph.end type="italics"/> augetur à &longs;emi&longs;&longs;e lineæ <emph type="italics"/>ad<emph.end type="italics"/><lb/>in eà ratione, in quà tota auget totam. Et quia linea <lb/><emph type="italics"/>ad<emph.end type="italics"/> ad &longs;emi&longs;&longs;em <emph type="italics"/>ab<emph.end type="italics"/> rationem habet duplam, habebit <lb/><expan abbr="quo&qacute;">quoque</expan>, <emph type="italics"/>ef<emph.end type="italics"/> ad illam &longs;emi&longs;&longs;em, hoc e&longs;t ad &longs;uum augmen­<lb/>tum rationem duplam. Simili modo &longs;i augmentum <emph type="italics"/>cd<emph.end type="italics"/><lb/>&longs;it triens lineæ <emph type="italics"/>ad,<emph.end type="italics"/> erit linea <emph type="italics"/>ef<emph.end type="italics"/> ad illud augmentum in <lb/>ratione triplá. Simile ergo & æquale auget &longs;uum &longs;i­<lb/>mile in eadem ratione &c. </s></p> <figure id="fig2"></figure> <p type="main"> <s><emph type="center"/>II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Contrarium æquale tollit vel impedit &longs;uum contra­<lb/>rium in eadem ratione, totum quidem totum, pars verò partem <lb/>&longs;ibi æqualem<emph.end type="italics"/></s></p> <figure></figure> <p type="main"> <s>SIt <emph type="italics"/>ab<emph.end type="italics"/> ip&longs;i <emph type="italics"/>df<emph.end type="italics"/> contrarium & æquale, & diuidantur bi­<lb/>fariam in <emph type="italics"/>c<emph.end type="italics"/> & <emph type="italics"/>e<emph.end type="italics"/>: quia ergo <emph type="italics"/>ab<emph.end type="italics"/> totum e&longs;t æquale ip&longs;i <lb/><emph type="italics"/>df<emph.end type="italics"/> toti, erit <expan abbr="quo&qacute;">quoque</expan> &longs;eni&longs;sis <emph type="italics"/>ef<emph.end type="italics"/> æqualis &longs;emi&longs;si <emph type="italics"/>cb<emph.end type="italics"/>: tollit <lb/>autem <emph type="italics"/>ab<emph.end type="italics"/> totum <emph type="italics"/>df,<emph.end type="italics"/> tollet ergo & <emph type="italics"/>eb<emph.end type="italics"/> totum <emph type="italics"/>ef:<emph.end type="italics"/> quod <lb/>idem de reliquis partibus, <expan abbr="quacun&qacute;">quacunque</expan>; ratione diuidan­<lb/>tur, o&longs;tendemus. Dices calorem & frigus e&longs;&longs;e contra <lb/>ria, <expan abbr="ne&qacute;">neque</expan>; tamen à calore totum frigus, <expan abbr="ne&qacute;">neque</expan>; à frigore to- <pb/>tum calorem tolli & expelli, verùm tantum illorum <lb/>exce&longs;&longs;us: partes verò mutilatas inter &longs;e mi&longs;ceri, & ami­<lb/>cabili &longs;ocietate in eodem &longs;ubjecto coniungj. Verùm <lb/>&longs;i in gradibus remi&longs;sis dee&longs;t illa proprietas contrari­<lb/>orum, <expan abbr="ne&qacute;">neque</expan>; &longs;anè contrarietas inerit. Quidquid tamen <lb/>&longs;it de illis qualitatibus, de quibus alio loco di&longs;&longs;eren­<lb/>dum, con&longs;tat ex illà, quæ in motu e&longs;t contrarietate, &longs;i <lb/>æqualis &longs;it, nullum &longs;e qui motum: &longs;i major, hujus ex­<lb/>ce&longs;&longs;uie&longs;&longs;e æqualem. Con&longs;t tuatur enim in bilance <emph type="italics"/>ab <lb/>c<emph.end type="italics"/> pondus <emph type="italics"/>a<emph.end type="italics"/> 8. lib. quod vectem deprimet impul&longs;u 8, li­<lb/> <arrow.to.target n="fig3"></arrow.to.target><lb/>brali, <expan abbr="at&qacute;">atque</expan>; hujus impul&longs;us non mn ao æquali totidem li­<lb/>brarum ponderis <emph type="italics"/>b<emph.end type="italics"/> impul&longs;u inhibetur. Quòd &longs;i pon­<lb/>dus in <emph type="italics"/>e<emph.end type="italics"/> lib. 5. eundem vectem &longs;ur&longs;um trahat, erit im­<lb/>pul&longs;us in <emph type="italics"/>a<emph.end type="italics"/> lib. 3. pondus ergo &longs;eu impul&longs;usin <emph type="italics"/>e<emph.end type="italics"/> contra­<lb/>rius impul&longs;ui in <emph type="italics"/>a<emph.end type="italics"/> to llit partem ex <emph type="italics"/>a<emph.end type="italics"/> &longs;ibi æqualem. Si­<lb/>mili modo &longs;i duo globi æquali ni&longs;u, & in eadem lineá <lb/>motus centri &longs;ibi occurrentes collidantur, nullus ab il- <pb/>lo contactu erit mótus: major verò impul&longs;us minorem <lb/>reflectet, tantò verò minori velocitate mouebitur à <lb/>contactu, quantò major e&longs;t re&longs;i&longs;tentia minoris: quia <lb/>nimirum impul&longs;us minor à majori tollit partem &longs;ibi <lb/>æqualem, &longs;imul verò occumbit erit ergò exce&longs;&longs;us ma­<lb/>joris principium motus à contactu: & cùm &longs;it agens <lb/>nece&longs;&longs;arium, motum producit &longs;ibi a qualem. Dices in­<lb/>terdum fieri ut duo globi &longs;ibi occurrentes <expan abbr="uter&qacute;">uterque</expan>; re&longs;ili­<lb/>at: <expan abbr="quodnõ">quodnon</expan> ni&longs;i ab æquali impul&longs;u e&longs;&longs;e pote&longs;t; propte <lb/>rea quód motus e&longs;t æqualis exce&longs;&longs;ui majoris. <expan abbr="Re&longs;põdeo">Re&longs;pondeo</expan> <lb/>&longs;i motus, quo <expan abbr="centrũ">centrum</expan> <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>; globi mouetur, &longs;it in ea­<lb/>dem lineà rectà, ab æquali impul&longs;u nunquam re&longs;ilire: <lb/>&longs;i autem motus centri unius &longs;it extra lineam motus al­<lb/>terius, quia lateraliter fit contactus, huju&longs;modi quidem <lb/>motum po&longs;&longs;e re&longs;ilire: verùm non ab&longs;oluté, &longs;ed tancùm <lb/>&longs;ecundùm quid e&longs;&longs;e contrarium. Vt in figurà &longs;ubjectà <lb/>&longs;i centrum<gap/> ex <emph type="italics"/>h,<emph.end type="italics"/> & centrum <emph type="italics"/>b<emph.end type="italics"/> ex <emph type="italics"/>l<emph.end type="italics"/> moucantur in ea­<lb/>dem lineà rectà <emph type="italics"/>h fl<emph.end type="italics"/>: &longs;it autem impul&longs;us ex <emph type="italics"/>a<emph.end type="italics"/> æqualis im <lb/>pul&longs;uiex <emph type="italics"/>b,<emph.end type="italics"/> àcontactu in <emph type="italics"/>f<emph.end type="italics"/> nullus erit motus: propterca <lb/>quód impul&longs;us æquales æqualiter reluctantur, <expan abbr="&longs;e&qacute;">&longs;eque</expan>; im­<lb/>pediunt à motu. Quód &longs;i verò centrum grauitatis <emph type="italics"/>a<emph.end type="italics"/><lb/>ex <emph type="italics"/>c<emph.end type="italics"/> in <emph type="italics"/>a,<emph.end type="italics"/> & centrum grauitatis <emph type="italics"/>b<emph.end type="italics"/> ex <emph type="italics"/>d<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> moueatur; quia <lb/>lineæ motus <gap/> non coincidunt eid em lineæ rectæ, <lb/>dico huju&longs;inodi motum non ab&longs;oluté, &longs;ed &longs;ecundùm <lb/>quid e&longs;&longs;e co<gap/>rarium. Ducantur enim ex puncto con- <pb/>tactus <emph type="italics"/>f<emph.end type="italics"/> lineæ <emph type="italics"/>fg. fe<emph.end type="italics"/> motui centri parallclæ, lineæ nimi­<lb/>rum hypomochlij, extra quas cadunt centra <emph type="italics"/>a<emph.end type="italics"/> & <emph type="italics"/>b:<emph.end type="italics"/> quia <lb/>ergo plaga non ni&longs;i per centrum fit grauitatis, erunt li. <lb/>neæ <emph type="italics"/>fab. fbb<emph.end type="italics"/> lineæ motus à percu&longs;sione: &longs;unt autem li <lb/>neæ <emph type="italics"/>ai.bk<emph.end type="italics"/> lineæ motus centri extra hypomochlium: <lb/> <arrow.to.target n="fig4"></arrow.to.target><lb/>quia ergo lineæ motus <emph type="italics"/>ab. ai,<emph.end type="italics"/> & <emph type="italics"/>bl.bk<emph.end type="italics"/> angulos ducunt <lb/><emph type="italics"/>iah.lbk<emph.end type="italics"/> minores duobus rectis, <expan abbr="erũt">erunt</expan> per defini: 5 motus <lb/>&longs;ecundùm quid contrarij, ac proinde inter &longs;emi&longs;centur <lb/>per prop: 31. Verùm de motu reflexo accuratiùs dice­<lb/>mus à prop: 36. <expan abbr="u&longs;&qacute;">u&longs;que</expan>; ad 40. </s></p> <figure id="fig3"></figure> <figure id="fig4"></figure> <p type="main"> <s><emph type="center"/>III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Mixtarum virium mixtæ &longs;unt actiones inea­<lb/>dem ratione, in quà mi&longs;centur mi&longs;cibilia.<emph.end type="italics"/></s></p> <p type="main"> <s>CVm enim mixtum &longs;it &longs;ua mi&longs;cibilia inter &longs;e unita, & <lb/>nece&longs;&longs;ariò agat, <expan abbr="actionem&qacute;">actionemque</expan>; producat &longs;ibi æqua­<lb/>lem aget &longs;ecundùm &longs;e totum, ac proinde &longs;ecundúm il­<lb/>las partes, quæ in illo toto mi&longs;centur: actio ergo mixta <pb/>quia toti æqualis, habet partes virtuales illis partibus, à <lb/>quibus producitur æquales. </s></p> <p type="main"> <s><emph type="center"/>IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Virtus agendi & actio inter &longs;e &longs;unt æquales, e&longs;t &queacute; <lb/>idem modus incrementi.<emph.end type="italics"/></s></p> <p type="main"> <s>VIrtutem enim agendi magnam aut paruam dici­<lb/>mus, quæ multùm aut parum pote&longs;t agere: <expan abbr="ita&qacute;">itaque</expan>; <lb/>hujus molem ex actionum mole æ&longs;timamus; actionem <lb/>verò ab effectu no&longs;cimus: dupla ergo virtus, quæ actio­<lb/>nem dupló, & tripla quæ triplò majorem, aut magis <lb/>perfectam producit. Et quia virtus naturalis non li­<lb/>berè &longs;ed ex nece&longs;sitate agit, <expan abbr="actionem&qacute;">actionemque</expan>; producit &longs;ibi <lb/>æqualem, erit idem modus incrementi <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>;. </s></p> <p type="main"> <s><emph type="center"/>Lemma,<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Sipunctum æqualiter moueatur inplano motu &longs;i­<lb/>mulrecto & laterali in eadem proportione <expan abbr="utrius&qacute;ue">utriusque</expan> interualli, <lb/>de&longs;cribet illo motu triangulum.<emph.end type="italics"/></s></p> <p type="main"> <s>MOueatur in fig: 3. punctum <emph type="italics"/>a<emph.end type="italics"/> ex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>f<emph.end type="italics"/> per lineam re <lb/>ctam <emph type="italics"/>af<emph.end type="italics"/> æqualiter in longum & latum, ita nimi­<lb/>rum ut in quolibet puncto longitudo excur&longs;us lateralis <lb/>&longs;it æ qualis <expan abbr="lõgitudini">longitudini</expan> motus recti inter idem punctum <pb/>& prin cipium motus, id e&longs;t <emph type="italics"/>ab<emph.end type="italics"/> ip&longs;i <emph type="italics"/>bg,<emph.end type="italics"/> & <emph type="italics"/>ac<emph.end type="italics"/> ip&longs;i <emph type="italics"/>cb,<emph.end type="italics"/> & <emph type="italics"/>ad<emph.end type="italics"/><lb/>ip&longs;i <emph type="italics"/>di,<emph.end type="italics"/> & <emph type="italics"/>ac<emph.end type="italics"/> ip&longs;i <emph type="italics"/>ek,<emph.end type="italics"/> & <emph type="italics"/>af<emph.end type="italics"/> ip&longs;i <emph type="italics"/>fl<emph.end type="italics"/> &longs;it æqualis, dico puncta <lb/><emph type="italics"/>aghikl<emph.end type="italics"/> cadere in latus <emph type="italics"/>al<emph.end type="italics"/> trianguli <emph type="italics"/>alf.<emph.end type="italics"/> Quòd &longs;i enim <lb/>punctum <emph type="italics"/><gap/>u:g<emph.end type="italics"/>: dicatur non in latus <emph type="italics"/>al,<emph.end type="italics"/> &longs;ed extra illud ca­<lb/> <arrow.to.target n="fig5"></arrow.to.target><lb/>dere in <emph type="italics"/>r,<emph.end type="italics"/> ducatur linea <emph type="italics"/>ar,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; angulus <emph type="italics"/>rad<emph.end type="italics"/> major <lb/>angulo <emph type="italics"/>iad.<emph.end type="italics"/> quia ergo latus <emph type="italics"/>dr<emph.end type="italics"/> lateri <emph type="italics"/>da<emph.end type="italics"/> e&longs;t æquale, & an <lb/>gulus <emph type="italics"/>adr<emph.end type="italics"/> rectus, erunt anguli <emph type="italics"/>dar. dra<emph.end type="italics"/> inter &longs;e æqua­<lb/>les, ac proinde &longs;emi&longs;&longs;es anguli recti. Similiter quia <lb/>latus <emph type="italics"/>fl<emph.end type="italics"/> e&longs;t æquale lateri <emph type="italics"/>fa,<emph.end type="italics"/> & angulus <emph type="italics"/>afl<emph.end type="italics"/> re­<lb/>ctus, erunt anguli <emph type="italics"/>fal. fla<emph.end type="italics"/> inter &longs;e æquales; igitur & an­<lb/>gulus <emph type="italics"/>laf<emph.end type="italics"/> angulo <emph type="italics"/>rad<emph.end type="italics"/> erit æqualis pars toti, quod e&longs;t ab­<lb/>&longs;urdum: non ergo punctum <emph type="italics"/>i<emph.end type="italics"/> extra latus <emph type="italics"/>al<emph.end type="italics"/> cadit. Simi­<lb/>li modo o&longs;tendemus non cadere intra il'ud latus: ca­<lb/>det ergò nece&longs;&longs;arió in ip&longs;um latus. Si ergo punctum <lb/>æqualiter moueatur in plano motu &longs;imul recto & late­<lb/>rali in eadem proportione &c. </s></p> <pb/> <figure id="fig5"></figure> <p type="main"> <s><emph type="center"/>V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Perfectio inten&longs;ina augetur eo modo, quo triangu­<lb/>lum &longs;ibi &longs;imile manens.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia per&longs;ectio inten&longs;iua non <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; motu fit, ac pro­<lb/>inde in aliquo tempore: &longs;upponatur illud tempus, <lb/>quo calor verbi gratia perficitur in quo <expan abbr="cun&qacute;">cunque</expan>; gradu, e&longs;­<lb/>le æquale lineæ <emph type="italics"/>af<emph.end type="italics"/>: & diuidatur æqualiter in minuta <emph type="italics"/>ab. <lb/>bc. cd. de. ef<emph.end type="italics"/>: quia ergo in &longs;ingulis minutis majora fiunt <lb/>hujus perfectionis in crementa, &longs;i in primo minuto <emph type="italics"/>ab<emph.end type="italics"/><lb/>perfectio inten&longs;iua &longs;it æqualis <emph type="italics"/>bg,<emph.end type="italics"/> erit in minuto &longs;ecun­<lb/>do <emph type="italics"/>bc<emph.end type="italics"/> major quam <emph type="italics"/>bg,<emph.end type="italics"/> & in tertiò minuto <emph type="italics"/>cd<emph.end type="italics"/> major <lb/>quam <emph type="italics"/>ch:<emph.end type="italics"/> dico huju&longs;modi incrementa e&longs;&longs;e &longs;imilia inter <lb/>&longs;e, ac proinde eo modo augeri, quo triangulum &longs;ibi &longs;i­<lb/>mile manens. Quia enim hæc perfectio continuò au­<lb/>getur, & veluti late&longs;eit ex illo puncto quietis; natura <lb/>autem uniformiter agit, <expan abbr="&longs;ibi&qacute;">&longs;ibique</expan>; &longs;emper e&longs;t &longs;imilis, erunt <lb/><expan abbr="quo&qacute;">quoque</expan>; &longs;imilia incrementa: Sicuti ergo perfectionem <lb/>&longs;ummam in tempore <emph type="italics"/>af<emph.end type="italics"/> æqualem lineæ <emph type="italics"/>fl,<emph.end type="italics"/> ita in hujus <lb/>temporis &longs;emi&longs;&longs;e: perfectionis &longs;emi&longs;&longs;em producet: igi­<lb/>tur ut tempus <emph type="italics"/>af<emph.end type="italics"/> ad perfectionem <emph type="italics"/>fl,<emph.end type="italics"/> ita tempus <emph type="italics"/>ac<emph.end type="italics"/> ad <lb/>perfectionem <emph type="italics"/>ek<emph.end type="italics"/> hoc e&longs;t ut latus <emph type="italics"/>af<emph.end type="italics"/> trianguli <emph type="italics"/>afl<emph.end type="italics"/> ad la­<lb/>tus <emph type="italics"/>fl,<emph.end type="italics"/> ita latus <emph type="italics"/>ae<emph.end type="italics"/> trianguli <emph type="italics"/>aek<emph.end type="italics"/> ad latus <emph type="italics"/>ek;<emph.end type="italics"/> ac proinde <lb/>&longs;imilia erunt triangula <emph type="italics"/>afl. aek.<emph.end type="italics"/> perfectio ergo inten&longs;i­<lb/>ua augetur eo modo, quo <expan abbr="triangulũ">triangulum</expan> &longs;ibi &longs;imile manens. </s></p> <pb/> <p type="main"> <s><emph type="center"/>VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us grauitatis ducetur &longs;ecundum rationem di&longs;tantiæ, <lb/>quam habet centrum grauitatis ab hypomochlio.<emph.end type="italics"/></s></p> <p type="main"> <s>HVjus po&longs;itionis veritatem probat Archimedes in <lb/>libro de æquiponderantibus: & nos in libro de <lb/>Arcu cæle&longs;ti ejus rationem à priori dare enitemur; quæ <lb/>non ni&longs;i ex naturà impul&longs;us priùs explicatà reddi po­<lb/>te&longs;t, hujus ergo demon&longs;trationem &longs;upporentes eà ve­<lb/>luti<emph type="italics"/>j<emph.end type="italics"/>am demon&longs;tratì inpo&longs;terum utemur. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us e&longs;t virtus &longs;eu qualitas, locomotiua, quæ <lb/>non ni&longs;i in tempore, & per &longs;patium mouet finitum.<emph.end type="italics"/></s></p> <p type="main"> <s>IMpul&longs;us dicitur ab impellendo: impellitur autem <lb/>mobile, dum loco &longs;uo expul&longs;um in alium transfer­<lb/>tur, aut &longs;impliciter; aut &longs;ecundúm quid, &longs;eu per com­<lb/>mutationem, dum loco rotius immoto partium loca <lb/>permutantur: quod duobus modis fieri pote&longs;t, incho­<lb/>atiuè, & perfectè. Inchoatiuè dico, quæ &longs;ecundùm nul: <lb/>lam partem &longs;en&longs;ibilem, &longs;ed per atomos in &longs;en&longs;iles vibra <lb/>tione quadam mouentur; cuju&longs;modi &longs;unt corpora &longs;o­<lb/>nora, quæ dum &longs;onant, moru quodam tremulo &longs;ub&longs;ul- <pb/>tant: & <expan abbr="quæcun&qacute;">quæcunque</expan>; corpora minorem habent impul­<lb/>&longs;um, quam ut loco moueantur: ut cùm tellus, aut &longs;a­<lb/>xum malleo percu&longs;lum tremit quidem exillo impul&longs;u, <lb/>&longs;ecundùm nullam verò partem &longs;en&longs;ibilem loco moue­<lb/>tur. Quód &longs;i <expan abbr="ne&qacute;">neque</expan>; &longs;onum edant corpora, <expan abbr="ne&qacute;">neque</expan>; tremu­<lb/>lâ vibratione motum te&longs;tentur, non videntur recipere <lb/>impul&longs;um: ut &longs;i granum milij terræ incidat: minorem <lb/>enim habet proportionem hic impul&longs;us, quam ut ali­<lb/>quam partem locomoueat, aut ab alijs auellat. Tre­<lb/>mor autem a percu&longs;sione videtur non <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; di&longs;tractio­<lb/>ne fieri <expan abbr="atomotũ">atomotum</expan>: <expan abbr="dũ">dum</expan> minor e&longs;t impul&longs;us, quam ut to­<lb/>tum m oueat: major verò quam ilia vis partium unit­<lb/>iua, quàinter &longs;e continuantur. Illa ergo corpora, quæ <lb/>uniones habent &longs;olubiles <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; reunione, fragilia &longs;unt: <lb/>cuju&longs;modivitrum, lapides, gemmæ; quæ iteratis per­<lb/>cu&longs;sionibus, ob plures uniones &longs;olutas, demum fran­<lb/>guntur, & di&longs;siliunt: metalla verò tamet&longs;i tremunt <expan abbr="&longs;o-nant&qacute;">&longs;o­<lb/>nantque</expan>; à percu&longs;sione, ob atomos tamen reunibiles non <lb/>ni&longs;i cùm impetus longiùs abduxit, franguntur. Sic a­<lb/>qua in calice vitreo &longs;ub&longs;ultat, & veluti æ&longs;tu agitur ad <lb/>motum digiti per margines circumacti: motu verò ac <lb/>celerato extra calicem &longs;alit, <expan abbr="&longs;uáq;">&longs;uáque</expan> a&longs;pergine etiam lon­<lb/>giùs ad&longs;tantes attingit. <expan abbr="Ita&qacute;">Itaque</expan>; hic impul&longs;us â principio <lb/>quidem non ni&longs;i &longs;ecundùm quid, & inchoatiuè, &longs;olum <lb/>tremorem inducendo: inde commutatione partium, <pb/>quá in gyrum aguntur, perfectà: demum motu &longs;impli<gap/><lb/>citer mouent. Vt igitur impul&longs;us loco moueat mobi­<lb/>le, nece&longs;&longs;e illam re&longs;i&longs;tentiam, quâin loco &longs;uo aut alieno <lb/>detinetur, &longs;uperate. Secundùm quid autem inchoa­<lb/>tiuè mouetur, cùm æquatis viribus inter&longs;e luctantur <lb/>virtus partium vnitiua & impul&longs;us: quà quidem ratio­<lb/>ne cymbala, cordæ, <expan abbr="at&qacute;">atque</expan>; æra tinnula mouentur. Lapi­<lb/>des verò & quæ fragilia &longs;unt, quia ex impul&longs;u uniones <lb/>&longs;en&longs;im depereunt, <expan abbr="ne&qacute;">neque</expan>; reuniri po&longs;&longs;unt, demum â per <lb/>culsione continuatá pluribus unionibus euer&longs;is, &longs;eu <lb/>quia impul&longs;ui necdum ex&longs;oluto alius &longs;uperuenit im­<lb/>pul&longs;us, franguntur. Manife&longs;tum ergo ex his Impul­<lb/>&longs;um e&longs;&longs;e virtutem finitam, quæ non quamlibet mo­<lb/>lem, &longs;ed finitam loco mouere & impeliere pote&longs;t. Et <lb/>quia motus ex unoloco in alium non ni&longs;i per medium <lb/>interuallum defert mobile, eju&longs;mo di motum non po&longs;­<lb/>&longs;e fieri in in&longs;tanti, &longs;ed in aliquo tempore ita o&longs;tende­<lb/>mus. Moueatur ex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>b,<emph.end type="italics"/> inter quæ mediant partes lo­<lb/><gap/>i <emph type="italics"/>cdefg<emph.end type="italics"/> &c. per quas nece&longs;&longs;arió tran&longs;it in <emph type="italics"/>b<emph.end type="italics"/>; propterea <lb/>quòd nequit medium tran&longs;ilire: quòd &longs;i ergo non ni&longs;i <lb/>in uno momento mouetur ex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>b,<emph.end type="italics"/> erit eodem <expan abbr="mom&etilde;-to">momen­<lb/>to</expan> &longs;imul in <emph type="italics"/>cdef<emph.end type="italics"/> pluribus locis adæquatis, quod nullâ <lb/>ratione fieri pote&longs;t. Simili modo o&longs;tendemus alio <lb/>momento in <emph type="italics"/>g,<emph.end type="italics"/> alio in <emph type="italics"/>f,<emph.end type="italics"/> priús nimirum in parte priori <lb/>quam pe&longs;tcriori motum terminari: pluribus ergo mo- <pb/>mentis mouetur ex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>b,<emph.end type="italics"/> ac proînde motus nece&longs;&longs;ariò <lb/>fit in tempore. Sed <expan abbr="ne&qacute;">neque</expan>; tempore infinito per &longs;pati­<lb/>um mouétur finitum, &longs;inimi: um motus eju&longs;dem &longs;it <lb/>rationis & &longs;ibi &longs;im ilis; nam &longs;i velocitas proportionali­<lb/>ter decre&longs;cat, non repugnat per &longs;patium finitum tem­<lb/>pore moueri infini<gap/>o; ut &longs;i per lineam conchoideos ac­<lb/>ce&longs;&longs;us fiat ad alteram parallelam, &longs;patium interjectum <lb/>nullo in tempore tran&longs;ibit. Moueatur ergo mobile ex <lb/><emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>f<emph.end type="italics"/> inotu æquali quantumuis lento: & &longs;umatur tem­<lb/>pus quodcunq: <emph type="italics"/>ik,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; mobile extra terminum <emph type="italics"/>a,<emph.end type="italics"/> in <lb/>quo quie&longs;cebat. aut igitur in <emph type="italics"/>ik<emph.end type="italics"/> aliquam partem ug: <emph type="italics"/>a <lb/>b,<emph.end type="italics"/> aut in&longs;en<gap/>bile punctum tran&longs;mi&longs;it. Si partem, meti­<lb/>etur hæc &longs;patium <emph type="italics"/>af<emph.end type="italics"/> aliquo numero finito: igitur & <lb/>rempus, quo totum &longs;patium decurrit, erit finitum. Si <lb/> <arrow.to.target n="fig6"></arrow.to.target><lb/>non ni&longs;i punctum: quia tempus diuidi pote&longs;t, tran &longs;i­<lb/>bit in hujus &longs;emi&longs;te interuallum puncto minus, quod <lb/>e&longs;t ab&longs;urdum: non igitur motus æqualis per &longs;patium <lb/>finitum tempore infinito e&longs;&longs;e pote&longs;t. Sed <expan abbr="ne&qacute;">neque</expan>, in tem­<lb/>pore finito per &longs;patium infinitum: <expan abbr="nã&qacute;">nanque</expan>; in &longs;emi&longs;&longs;e tem­<lb/>poris, <expan abbr="at&qacute;">atque</expan>; hujus &longs;emi&longs;&longs;e &c. nunquid &longs;patium <gap/>peram­<lb/>bulabit infinitum? quód &longs;i motus illâ &longs;ectione <expan abbr="demũ">demum</expan> <lb/>terminabit in aliquà parte finitâ, erit <expan abbr="quo&qacute;">quoque</expan>; totum fini­<lb/>tum. Deinde cùm motus incipiat à termino, erit ne­<lb/>ce&longs;latiò finitus. moueatur enim ex <emph type="italics"/>a<emph.end type="italics"/> per &longs;patium <emph type="italics"/>bcde<emph.end type="italics"/> <pb/><emph type="italics"/>f<emph.end type="italics"/> &c. in infimtum in tempore <emph type="italics"/>ghikl<emph.end type="italics"/> finito: igitur par­<lb/>t<gap/>m quidem <emph type="italics"/>b<emph.end type="italics"/> in aliquà parte temporis tran&longs;ibit, quæ <lb/>&longs;it <emph type="italics"/>g<emph.end type="italics"/>; men&longs;urabit proinde tempus aliquo numero fini­<lb/>to: & cúm motum ponamus &longs;imilarem, qui in tempo­<lb/>re æquali partes conficit æquales, totidem partes erunt <lb/>in &longs;patio <emph type="italics"/>bcdef,<emph.end type="italics"/> quotin tempore <emph type="italics"/>ghikl,<emph.end type="italics"/> ac proinde to­<lb/>tum interuallum erit finitum. Igitur impul&longs;us e&longs;t vir­<lb/>tus finita, quænon ni&longs;i in tempore & per &longs;patium mo­<lb/>uet finitum. </s></p> <figure id="fig6"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us e&longs;t agens nece&longs;&longs;arium, <expan abbr="motum&qacute;">motumque</expan>; producit <lb/>&longs;ibi æqualem.<emph.end type="italics"/></s></p> <p type="main"> <s>NEce&longs;&longs;arium dico non &longs;olùm quò ad exercitium a­<lb/>ctus, quo modo omnia agentia, quæ non liberè a­<lb/>gunt, nece&longs;&longs;aria di<gap/>untur; &longs;ed etiam quò ad perfect: o­<lb/>nem actus, hoc e&longs;t agere &longs;ecundúm totum po&longs;&longs;e, &longs;eu <lb/>&longs;ummam perfectionem tribuere &longs;uo effectui: quod <lb/>non faciunt reliqua agentia naturalia, quæ non ni&longs;i à le­<lb/>uibus initijs ad &longs;umma cuadunt incrementa: utma­<lb/>nife&longs;tum in calefactione<gap/> At verò impul&longs;us &longs;tatim à <lb/>principio motum veloci&longs;simum producit: qui demum <lb/>&longs;patij tractudangue&longs;cit & emoritur, Cujus ratio e&longs;t, <pb/>quòd impul&longs;us &longs;it qualitas tran&longs;iens, quæ non pote&longs;t in <lb/>&longs;ubjecto con&longs;eruari <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; motu: quód &longs;i enim mobile <lb/>ad motum concitatum vel uno momento detineas, nul <lb/>lus ex illo contactu &longs;equitur motus: ni&longs;i ergo à princi­<lb/>plo, priu&longs;quam virtus ex&longs;oluatur, agat<gap/>, nunquam &longs;uum <lb/>finem a&longs;&longs;equetur: unde à veloci&longs;simo & &longs;ibi æquali <lb/>motu exor&longs;us, quantùm virium deperit, tant<gap/>m de ce­<lb/>leritate remittit. <expan abbr="Ne&qacute;">Neque</expan>; hic nobis aduer&longs;antur, qui ne­<lb/>&longs;cio quas morulas inducunt, velociùs moueri dicentes <lb/>illud mobile, quod paucioribus morulis quie&longs;cit: nam <lb/>ex illorum <expan abbr="quo&qacute;">quoque</expan>; &longs;ententià impul&longs;us id quod pote&longs;t <lb/>&longs;ummum operatur: & à principio quidem pauciori­<lb/>bus morulis quie&longs;cit, inde veluti ex illo motu la&longs;&longs;atus <lb/>longiora ducit interualla. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us non ni&longs;iper lineam rectam mouet &longs;uum mobile.<emph.end type="italics"/></s></p> <p type="main"> <s>DEmotu quidem, qui procedit à grauitate, nullum <lb/>e&longs;t dubium fieri per lineam rectam: &longs;ed etiam ea, <lb/>quæ proijciuntur &longs;eu manu, &longs;eu machinà, rectitudinem <lb/>&longs;eruare con&longs;tat; tantò enim metam feriunt ictu certio­<lb/>re, quantò minùs principium motus à lineà rectà aber <lb/>rauit. At verò quæ circulariter mouentur, dubitatio­<lb/>nem habent: propterea quòd ex impul&longs;u non per line- <pb/>am rectam, &longs;ed circularem mouen videantur. Nihilo­<lb/>minus etiam in his, quæ circulatiter mouentur, impul­<lb/>&longs;um ad motum rectum inelinare, & non ni&longs;i vi ab hy­<lb/>pomochlio illatà circumagj facile o&longs;tendemus. Ete­<lb/>nim eà ratione mouetur totum, quà illius partes, cúm <lb/>motus totius &longs;it &longs;uarum partium motus: at verò partes <lb/>&longs;ingulæ dum circumaguntur, &longs;i non firmiter cohærent <lb/>&longs;<gap/>o hypomochlio, non in circulum, &longs;ed per lineam re­<lb/>ctam mouentur: quod quidem in illà rotà ver&longs;atili, quà <lb/>gemmæ poliuntur, aut in lapide molari licebit experiri: <lb/>quòd &longs;i enim in illà planitie propè centrum arenam, <lb/>aut quid &longs;imile con&longs;<gap/>tuas, videbis ex illà rotatione <lb/>ad circulos &longs;en&longs;im majores à centro propelli, & demum <lb/>excuti. Obijcies globum fi&longs;tulà &longs;triatà emi&longs;&longs;um velo­<lb/>ci&longs;simè gyrando, & veluti aërem terebrando ad metam <lb/>venire, <expan abbr="ne&qacute;">neque</expan>; ullum punctum, præterquam centrum, per <lb/>lineam rectam, &longs;ed per lineam &longs;piralem moueri: quia <lb/>nimirum ab illis &longs;ulcis, quibus fi&longs;tula interné excaua­<lb/>tur, toto illo tractu reuolutus impul&longs;um colligit circu­<lb/>larem: non igitur impul&longs;us nece&longs;&longs;ariò ducit per lineam <lb/>rectam. Deinde &longs;i quis velociter currendo &longs;agittam ja­<lb/>culetur, aut lapidem proijciat, quantumuis principium <lb/>motus per lineam fiat perpendicularem, non tamen il <lb/>iud mobile per lineam rectam, &longs;ed arcuatim &longs;ur&longs;um elu <lb/>ctatur: propterea quòd non ad idem punctum, â quo <pb/>moueri cepit, fit relap&longs;us, verùm ad procur&longs;um jaculan­<lb/>tis in anteriora profertur. <expan abbr="Ita&qacute;">Itaque</expan>; auem in volatu deijce­<lb/>re volentes, illius volatum tanti&longs;per oculis & arcu in­<lb/>tentis &longs;equuntur, & tum in ip&longs;o motu &longs;agittam ejacu­<lb/>lantur: qui motus non videtur fieri per lineam rectam. <lb/>Vt &longs;i auis ex <emph type="italics"/>b<emph.end type="italics"/> in <emph type="italics"/>f<emph.end type="italics"/> reratur, &longs;agitta per lineas <emph type="italics"/>mb.oc<emph.end type="italics"/> illius <lb/>volacum &longs;ecuta, in lineà demum <emph type="italics"/>ad<emph.end type="italics"/> à neruo excu&longs;&longs;a ean <lb/>dem figet in <emph type="italics"/>g.<emph.end type="italics"/> at verò ex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>g<emph.end type="italics"/> non ni&longs;i arcuatim & per <lb/>lineam inflexam, cuju&longs;modi <emph type="italics"/>ahig<emph.end type="italics"/> euadit: propterea <lb/>quòd motus &longs;agittæ videtur compo&longs;itus ex illo motu, <lb/> <arrow.to.target n="fig7"></arrow.to.target><lb/>quo ad motum arcus, & quo à neruo impul&longs;a mouetur: <lb/>at verò motus, quo cumarcu mouetur, e&longs;t circulatis ha­<lb/>bens centrum in oculo &longs;agittantis: motus ergo ab hoc <pb/>in &longs;agittam de<gap/>iuatus, ac pro<gap/>nde motus ex <expan abbr="utro&qacute;">utroque</expan>; mix­<lb/>tus e<gap/>t circularis. De&longs;cribatur arcus <emph type="italics"/>mn,<emph.end type="italics"/> cujus centrum <lb/>mo<gap/>ulo <emph type="italics"/>l,<emph.end type="italics"/> &longs;emidiameter verò &longs;agitta <emph type="italics"/>al<emph.end type="italics"/>: quæ ubi per ar­<lb/>cum <emph type="italics"/>ma<emph.end type="italics"/> moueri cæpit, ab alio impul&longs;u à neruo deriuato <lb/>perlineam agitur <emph type="italics"/>ad<emph.end type="italics"/>: dico motum ex <expan abbr="utroq;">utroque</expan> mixtum, <lb/>nimirum ex motu <emph type="italics"/>man,<emph.end type="italics"/> & ex motu <emph type="italics"/>ad<emph.end type="italics"/> non po&longs;&longs;e fieri <lb/>pe<gap/> lineam rectam. Sit enim motus in <emph type="italics"/>ad<emph.end type="italics"/> ad motum in <lb/><emph type="italics"/>man,<emph.end type="italics"/> ut linea recta <emph type="italics"/>ap<emph.end type="italics"/> ad arcum <emph type="italics"/>aq<emph.end type="italics"/>: & a&longs;&longs;umatur linea <lb/><emph type="italics"/>qh<emph.end type="italics"/> æqualis lineæ <emph type="italics"/>ap,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; motus compo&longs;itus ex <emph type="italics"/>ap. aq<emph.end type="italics"/><lb/>in <emph type="italics"/>h:<emph.end type="italics"/> &longs;imiliter o&longs;tendemus motum in <emph type="italics"/>i<emph.end type="italics"/> & <emph type="italics"/>g<emph.end type="italics"/> componi ex <lb/>motu recto & circulari: dico per puncta <emph type="italics"/>hig<emph.end type="italics"/> non pol­<lb/>&longs;e d<gap/>ci lineam rectam. Sit enim, &longs;i fieri pote&longs;t, linea <emph type="italics"/>ab <lb/>ig<emph.end type="italics"/> recta, & ex puncto <emph type="italics"/>q<emph.end type="italics"/> ducatur linea tangens circulum <lb/>in <emph type="italics"/>q,<emph.end type="italics"/> quæ <expan abbr="utrim&qacute;">utrimque</expan>; producta &longs;ecet lineas <emph type="italics"/>lf. ld<emph.end type="italics"/> in punctis <lb/>s. u: <expan abbr="erunt&qacute;">eruntque</expan>; lineæ <emph type="italics"/>qs. qu<emph.end type="italics"/> inter &longs;e æquales: quibus ex <lb/>puncto <emph type="italics"/>i<emph.end type="italics"/> ducatur linea parallela <emph type="italics"/>ix,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; angulus <emph type="italics"/>ixq<emph.end type="italics"/> re <lb/>ctus, quia ergo in triangulo <emph type="italics"/>hxi<emph.end type="italics"/> duo anguli <emph type="italics"/>hxi. xhi<emph.end type="italics"/> du­<lb/><gap/>bu; agulis <emph type="italics"/>hqu.qhu<emph.end type="italics"/> trianguli <emph type="italics"/>hqu<emph.end type="italics"/> &longs;unt æquales, <expan abbr="uter&qacute;">uterque</expan>; <lb/><expan abbr="utri&qacute;">utrique</expan>, erunt &longs;imilia inter &longs;e; ac proinde ut <emph type="italics"/>hi<emph.end type="italics"/> ad <emph type="italics"/>hq,<emph.end type="italics"/> ita <lb/><emph type="italics"/>xi<emph.end type="italics"/> ad <emph type="italics"/>qu,<emph.end type="italics"/> hoc e&longs;t ad <emph type="italics"/>qs<emph.end type="italics"/> illi æqualem. e&longs;t autem linea <emph type="italics"/>hx<emph.end type="italics"/><lb/>æqualis lin: æ <emph type="italics"/>hq:<emph.end type="italics"/> igitur & linea <emph type="italics"/>xi<emph.end type="italics"/> erit æqualis lineæ <emph type="italics"/>qs,<emph.end type="italics"/><lb/>quod e&longs;t abiuidum: &longs;equeretur enim lineas <emph type="italics"/>is. xq<emph.end type="italics"/> in <lb/>centro <emph type="italics"/>l<emph.end type="italics"/> concurrentes e&longs;&longs;e parallelas. Re&longs;pondeo ad <lb/>primum, motum glebuli, quo gyrando ad metam va­<lb/>dit, e&longs;&longs;e compo&longs;itum ex impui&longs;u recto, quem ip&longs;i con- <pb/>fert puluis pyrius à tergo incen&longs;us, & eximpul&longs;u latera <lb/>li, quem viarum &longs;eu <expan abbr="eanaliculorũ">eanaliculorum</expan> anfractus globulo e­<lb/>rumpenti conciliant: partes enim globuli prominen­<lb/>tes &longs;ulcis impre&longs;&longs;æ, eo&longs;dem ductus &longs;equendo, illà gyra­<lb/>tione globulum reuoluunt; quem motum adjuuat ig­<lb/>nis eadem <emph type="italics"/>v<emph.end type="italics"/>iá pabulum &longs;equendo, & globulum impel­<lb/>lendo: dico ergo hunc motum partim &longs;imilem illi mo­<lb/>tui, quo rota circumagitur, partim di&longs;similem: propter­<lb/>ea, quòd globulus circa centrum mobile, rota autem <lb/>circa immobile reuoluatur. At verò trochus <lb/>aut turbo, dum gyrando in aëre labitur, motu pror&longs;us <lb/>&longs;imi&longs;i fertur: nam ex impul&longs;u funiculi multis &longs;piris re­<lb/>uoluti & retracti in gyrum agitur cirea mobile cen­<lb/>trum: quod &longs;uà grauitate inter gyrandum de&longs;cendit. <lb/>at verò impul&longs;us, quo rota aut turbo circulariter moue <lb/>tur, &longs;i non impediatur, non circulari, &longs;ed motu recto mo <lb/>uebitur: quemadmodum exemplo illarum rerum, quæ <lb/>ad motum rotæ circumaguntur, o&longs;tendimns. <expan abbr="Ita&qacute;">Itaque</expan>; &longs;i ca­<lb/>tenula conuoluta unà extremitate in illiu<gap/> plano firme­<lb/>tur, videbis ex illâ vortigine &longs;en&longs;im reuolui, & demum <lb/>in lineam tangentem eju&longs;dem circuli extendi. Ita tro <lb/>chus aut turboaquà con&longs;per&longs;us in motu re&longs;iccatur, dum <lb/>aquæ guttulæ ex illo impul&longs;u lineam rectam &longs;equendo <lb/>auelluntur. Simili ergo modo impul&longs;us, qui globu­<lb/>lum reuoluit, &longs;i non impediatur, lateraliter, & per line- <pb/>am rectam mouebit. quod quidem co<gap/>tabit, &longs;i globu­<lb/>lum f<gap/>iabilem &longs;ub&longs;tituas: ex motu enim gyrationis in <lb/>atomos infinitas di&longs;sipabitur. At verò continuitas par­<lb/>tium globuli di&longs;&longs;olui nequit ob firmitatem, <expan abbr="ne&qacute;">neque</expan>; late­<lb/>raliter moueri ob <expan abbr="re&longs;i&longs;t&etilde;tiam">re&longs;i&longs;tentiam</expan> illarum partium, quæ im­<lb/>pul&longs;u contra io aguntur: quòt enim lineæ tangentes, <lb/>tot: dem ine&longs;&longs;e videntur impul&longs;us: <expan abbr="ita&qacute;">itaque</expan>; centrum glo­<lb/>buli tantò magis detinetur in lineà rectà, quantò majori <lb/>velocitate in gyrum mouetur. Dices quam ob rem cr<gap/><lb/>go turbo, dum &longs;uper axe mouetur horizonti parallelo, <lb/>non eandem firmitatem habet &longs;ui centri à partibus cir­<lb/>cumactis? <expan abbr="neq;">neque</expan> enim eidem puncto in&longs;i&longs;tit axis, verùm <lb/>huc illuc incerto motu oberrat. Re&longs;pondeo id ab in <lb/>æquali illarum partium &longs;itu, quibus planum tangit, <lb/>prouenite: cùm non in <expan abbr="pũcto">puncto</expan> fiat <expan abbr="cõtactus">contactus</expan>. quia ergo in <lb/>&longs;uperficie illius plant a&longs;perà & in æquali partes aliæ &longs;unt <lb/>depre&longs;&longs;æ, aliæ prominentes & verruco&longs;æ, nece&longs;&longs;e muta­<lb/>tionem fieri in motu: dum vel &longs;ub&longs;idet in lacunas, vel <lb/>ad tubercula offendit. Ad &longs;ecundam objectionem, di­<lb/>co &longs;agittam circulariter moueri ex illo motu, quo cum <lb/>arcu mouetur; impul&longs;us enima centro detinetur, quò <lb/>minùs per lineam rectam moueat: at verò motus &longs;agit­<lb/>tæ à neruo excu&longs;&longs;æ, quia à nullo detinetur, per lineam fit <lb/>mediam inter tangentem & lineam rectam, &longs;iuè per di­<lb/>a netrum pærallel ogrammi, cujus latera &longs;unt in propor- <pb/>tione illorum motuum. Deinde e&longs;to demus impul&longs;um <lb/>lateraliter abducentem e&longs;&longs;e circularem, non tamen &longs;e­<lb/>quitur motum compo&longs;itum e&longs;&longs;e circularem: nam mo­<lb/>tus quidem compo&longs;itus ex motu recto <emph type="italics"/>ap<emph.end type="italics"/> & circulari <emph type="italics"/>a <lb/>q<emph.end type="italics"/> non in <emph type="italics"/>h,<emph.end type="italics"/> ut &longs;upponebatur, verùm in <emph type="italics"/>y<emph.end type="italics"/> abducit mobile, <lb/>propterea quòd interuallum motus circularis in fine <lb/>motus compo&longs;iti &longs;it æquale arcui <emph type="italics"/><expan abbr="aq.">aque</expan><emph.end type="italics"/> &longs;imiliter dum ex <lb/><emph type="italics"/>y<emph.end type="italics"/> per lineam fertur <emph type="italics"/>yz<emph.end type="italics"/> æqualem lineæ <emph type="italics"/>ap,<emph.end type="italics"/> impul&longs;u cir­<lb/>culari &longs;patium tran&longs;mittit <emph type="italics"/>zt<emph.end type="italics"/> æquale &longs;patio <emph type="italics"/>py<emph.end type="italics"/> &longs;eu arcui <lb/><emph type="italics"/>qs:<emph.end type="italics"/> dico puncta <emph type="italics"/>ayt<emph.end type="italics"/> e&longs;&longs;e in lineàrectà, ac proinde mo­<lb/>tum compo&longs;itum <emph type="italics"/>ayt<emph.end type="italics"/> rectum non verò circularem. <lb/>Ducantur enim diametri <emph type="italics"/>ay. y t:<emph.end type="italics"/> quia ergo an­<lb/>gulus <emph type="italics"/>zyt<emph.end type="italics"/> angulo <emph type="italics"/>pay,<emph.end type="italics"/> hic autem angulo alterno <emph type="italics"/>ayq<emph.end type="italics"/><lb/>e&longs;t æqualis, erit eidem angulus <emph type="italics"/>zyt<emph.end type="italics"/> ad verticem æqua­<lb/>lis, ac proinde linea <emph type="italics"/>ayt<emph.end type="italics"/> recta. Ratio autem quam ob­<lb/>rem impul&longs;us non ni&longs;i per lineam rectam moueat, e&longs;t <lb/>hæc: quia cùm motus &longs;it via ad conjunctionem &longs;eu uni <lb/>onem cum &longs;uo termino, ad quem mouetur, erit non &longs;ui <lb/>&longs;ed finis gratia, ac proînde &longs;icuti nihil deficere, ita nihil <lb/>abundare debet: at verò &longs;icuti in vià rectà nihil de e&longs;t ad <lb/>finem con&longs;equendum, ita omnes reliquæ abundant: a­<lb/>bundare enim dicitur, <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; quo finis pote&longs;t obtineri. <lb/>Deinde cùm impul&longs;us &longs;it agens nece&longs;&longs;arium, habebit & <lb/>actionem & modum agendi determinatum; determi­<lb/>natio autem non ni&longs;i in lineà rectâe&longs;&longs;e pot<gap/>&longs;t, cùm hæc <pb/>&longs;it una, lineæ verò obliquæ infinitæ. Conf<gap/> matur ex <lb/>modo agendi reliquorum agentium naturalium, quæ <lb/>non ni&longs;i per lineas rectas operantur. </s></p> <figure id="fig7"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us in quolibet puncto circuli per lineam fit tangentem.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim motus e&longs;t rectus per pro: 3. talis autem <lb/>e&longs;&longs;e non pote&longs;t in circulo, igitur &longs;i incipiat ab ali­<lb/>quo puncto circuli, cadet immediaté po&longs;t illud pun­<lb/>ctum extra peripheriam illius circuli: non pote&longs;t au­<lb/>tem cadere intra circulum, cadet igitur extra circulum. <lb/>Probatur, punctum circuli immediatè ante contactum <lb/>verbi gratia <emph type="italics"/>a<emph.end type="italics"/> impellit <emph type="italics"/>o<emph.end type="italics"/> ad motum rectum: <expan abbr="punctũ">punctum</expan> ergo <lb/>immediatè po&longs;t illum contactum erit cum duobus pun <lb/>ctis <emph type="italics"/>a<emph.end type="italics"/> & <emph type="italics"/>o<emph.end type="italics"/> in lineà rectà, aut certè ad hujus rectitudinem <lb/>quam proximè fieri pote&longs;t, accedet: at verò intia peri­<lb/>pheriam circuli nullum e&longs;&longs;e pote&longs;t punctum, qu<gap/>dcum <lb/>duobus illis punctis <emph type="italics"/>a<emph.end type="italics"/> & <emph type="italics"/>o<emph.end type="italics"/> &longs;it in lineà rectà, aut ad natu­<lb/>ram lineæ rectæ quam proximè accedat, verum ad ma­<lb/>iorem curuitatem: cùm nece&longs;&longs;ariò &longs;it in peripheria ali­<lb/>cujus circuli minoris. Cad<gap/>t enim, &longs;i fieri pote&longs;t, intra <pb/>circulum illud punctum, per quod ducitur linea recta, <lb/>& &longs;it <emph type="italics"/>b<emph.end type="italics"/>: de&longs;cribatur autem circulus minor <emph type="italics"/>afp<emph.end type="italics"/> tangens <lb/>priorem in <emph type="italics"/>a<emph.end type="italics"/>: quód &longs;i ergo punctunm <emph type="italics"/>b<emph.end type="italics"/> cadit extra pe­<lb/>ripheriam hujùs circuli, erit angulus <emph type="italics"/>bae<emph.end type="italics"/> minor <expan abbr="quid&etilde;">quidem</expan> <lb/>recto, major autem angulo &longs;emicirculi <emph type="italics"/>fae<emph.end type="italics"/> contra prop: <lb/>16. tert: Verùm quia po&longs;&longs;et quis dicere illud punctum <lb/> <arrow.to.target n="fig8"></arrow.to.target><lb/>nece&longs;&longs;ariò cadere intra omnes circulos etiam in infini­<lb/>tum minores, propterea quòd angulus &longs;emicirculi &longs;it <lb/>major quouis angulo acuto: alià ratione îdem o&longs;ten­<lb/>demus. producatur ergo linea <emph type="italics"/>ab<emph.end type="italics"/> <expan abbr="utrim&qacute;">utrimque</expan>; in <emph type="italics"/>g. i<emph.end type="italics"/> &longs;ecans <lb/>circulum in <emph type="italics"/>g,<emph.end type="italics"/> arcus autem <emph type="italics"/>ag<emph.end type="italics"/> diuidatur bifariam in <emph type="italics"/>b,<emph.end type="italics"/> & <lb/>ducatur linea <emph type="italics"/>bal<emph.end type="italics"/>; <expan abbr="erit&qacute;">eritque</expan>; angulus <emph type="italics"/>hab,<emph.end type="italics"/> <expan abbr="at&qacute;">atque</expan>; huic ad verti- <pb/>cem æqualis angulus <emph type="italics"/>ial<emph.end type="italics"/> major angulo contactus <emph type="italics"/>cah,<emph.end type="italics"/><lb/><expan abbr="at&qacute;">atque</expan>; huic æquali angulo <emph type="italics"/>kad<emph.end type="italics"/>: multo ergo major angu­<lb/>lus <emph type="italics"/>gab,<emph.end type="italics"/> <expan abbr="at&qacute;">atque</expan>; angulus <emph type="italics"/>iad<emph.end type="italics"/> angulis contactus <emph type="italics"/>cah. kad<emph.end type="italics"/>: <lb/>puncta ergo circa contactum circuli <emph type="italics"/>a<emph.end type="italics"/> majori inter­<lb/><gap/>a<gap/>lo ab&longs;untà lineà quauis &longs;ecante, quarn à lineà conta­<lb/>ctus, ac cum illis punctis, quæ in linea &longs;unt tangente, <lb/>magis accedunt ad naturam lineæ rectæ, quam cum il­<lb/>lis punctis, quæ in lineà &longs;unt &longs;ecante: motus eigò à con­<lb/>tactu per lineam fit tangentem. Quæ igitur circulari­<lb/>ter mouentur, &longs;i in illà gyratione ab hypomochlio libe­<lb/>rentur, motu deinceps recto feruntur, facto initio mo­<lb/>tus abillo puncto circuli, in quo ab hypomochlio avel­<lb/>luntur. Ita ergo lapis fundà circumactus, ubi ex illà ro­<lb/>tatione impul&longs;um collegit, laxatà habenà au olat motu <lb/>recto per lineam tangentem circuli, cujus &longs;emidiame­<lb/>ter e&longs;t longitudo fundæ. </s></p> <figure id="fig8"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us æqualis eodem velæquali tempore per &longs;patium mouet <lb/>æquate.<emph.end type="italics"/></s></p> <p type="main"> <s>MAgnitudo &longs;eu exten &longs;io ine&longs;t motui non per&longs;e, &longs;ed <lb/>ratione loci in quo fit motus; motum enim mag <lb/>num dicimus, qui magno, paruum qui paruo &longs;patio con <lb/>tinetur; &longs;iuè actu habeat illam exten&longs;ionem, &longs;iuè virtu- <pb/>aliter tantum: ut cùm idem &longs;patium currendo aut am­<lb/>bulando &longs;æpiùs remetimur. Quia verò eju&longs;dem<gap/> aut <lb/>æqualis magnitudinis eadem e&longs;t men&longs;ura: e&longs;t autem <lb/>nen&longs;ura motus tempus: erit <expan abbr="quo&qacute;">quoque</expan>; eju&longs;dem aut æqua­<lb/>lis motus idem tempus. Motus ergo æqualis in tempo­<lb/>re æquali per &longs;patium fit æquale: & cùm impul&longs;us &longs;it <lb/>agens nece&longs;&longs;arium, motu<gap/><expan abbr="n&qacute;">nque</expan>; producat &longs;ibi æqualem, <lb/>per prop: 2. æqualis impul&longs;us in eodem vel æquali tem <lb/>pore per &longs;patium mouebit æquale. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Definitio.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us qui mínori tempore per &longs;patium mouet æquale aut <lb/>majus, dicatur velox<gap/> qui verò majoritempore per &longs;patium mouet <lb/>æquale aut minus, dicatur tardus.<emph.end type="italics"/></s></p> <p type="main"> <s>VT &longs;i mobile <emph type="italics"/>H<emph.end type="italics"/> per &longs;patium <emph type="italics"/>de<emph.end type="italics"/> in tempore <emph type="italics"/>ab<emph.end type="italics"/> minori, <lb/>mobile verò <emph type="italics"/>K<emph.end type="italics"/> per idem &longs;patium <emph type="italics"/>de,<emph.end type="italics"/> aut huic æquale <lb/><emph type="italics"/>fg<emph.end type="italics"/> in tempore <emph type="italics"/>abc<emph.end type="italics"/> májori moueatur <gap/> impul&longs;us quo <emph type="italics"/>H<emph.end type="italics"/><lb/>mouetur velox, quo autem <emph type="italics"/>K<emph.end type="italics"/> mouetur dicetur tardus. <lb/>velociùs enim &longs;patium tra&longs;mitti dicimus, in quo mobi­<lb/> <arrow.to.target n="fig9"></arrow.to.target><lb/>le minùs immoratur, &longs;eu ut Atomi&longs;tæ volunt, in quo <lb/>paucioribus morulis interquie&longs;cit. Quod autem mi- <pb/>nori tempore per &longs;patium æquale, idem <expan abbr="quo&qacute;">quoque</expan>; minori <lb/>tempore per &longs;patium majus mouetur. Diuidatur enim <lb/>exce&longs;&longs;us temporis bifariam in <emph type="italics"/>i<emph.end type="italics"/>: <expan abbr="at&qacute;">atque</expan>; hujus iemifsis <emph type="italics"/>bi<emph.end type="italics"/> ad­<lb/>datur minori <emph type="italics"/>ab,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; tempus compo&longs;itum <emph type="italics"/>abi<emph.end type="italics"/> majus <lb/>quidem minori <emph type="italics"/>ab,<emph.end type="italics"/> minus verò tempore majori <emph type="italics"/>abc.<emph.end type="italics"/> in <lb/>tempore ergò <emph type="italics"/>abi<emph.end type="italics"/> &longs;patium majus quam <emph type="italics"/>de,<emph.end type="italics"/> ac proinde <lb/>in minori tempore &longs;patium majus perambulabit. Eo­<lb/>dem modo o&longs;tendemus, &longs;i quid æquali tempore per <lb/>&longs;patium majus moueatur, idem in minori tempore per <lb/>&longs;patium majus moueri: &longs;i nimirum hujùs exce&longs;&longs;um bi­<lb/>fariam &longs;ecemus: nam &longs;patium illud æquale, <expan abbr="at&qacute;">atque</expan>; hujus <lb/>&longs;emi&longs;&longs;em in minori tempore pertran&longs;ibit. </s></p> <figure id="fig9"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us major eodem vel æqualis tempore per &longs;patium majus, <lb/>minori verò tempore per &longs;patium mouet æquale.<emph.end type="italics"/></s></p> <p type="main"> <s>IMpul&longs;um magnum dicimus non exten&longs;iué, &longs;ed inten <lb/>&longs;iué, cujus perfectionem &longs;equitur velocitas motus. <lb/>quia ergo major velocitas in minori tempote per &longs;pati­<lb/>um mouet æquale aut majus, per defin: impul&longs;us verò <lb/>major majorem velocitatem producit, propterea quòd <lb/>agens &longs;it nece&longs;&longs;arium, <expan abbr="motum&qacute;">motumque</expan>; producat &longs;ibi æqua- <pb/><gap/>em: mouebit &longs;ane eodem vel æquali tempore per &longs;pa­<lb/>tium majus, minori verò tempore per &longs;patium æquale. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Velocitas motus eandem rationem habet quam interualla, rati­<lb/>onem verò &longs;uorum temporum reciprocam.<emph.end type="italics"/></s></p> <p type="main"> <s>SIt velocitas <emph type="italics"/>H<emph.end type="italics"/> dupla velocitatis <emph type="italics"/>K:<emph.end type="italics"/> dico hujus interual <lb/><expan abbr="lũ">lum</expan> in ratione <expan abbr="quo&qacute;">quoque</expan>; e&longs;&longs;e duplà ad illud interuallum, </s></p> <p type="caption"> <s>H. K.<lb/> <arrow.to.target n="fig10"></arrow.to.target><lb/>per quod velocitas &longs;ubdupla eodem vel æquali tempo­<lb/>re mouetur: at verò tempus, quo velocitas dùpla per <lb/>&longs;patium æquale mouetur, in ratione &longs;ubduplá ad tem­<lb/>pus velocitatis minoris, Vt &longs;i velo citas <emph type="italics"/>H<emph.end type="italics"/> in tempore <emph type="italics"/>ab,<emph.end type="italics"/><lb/>velo citas autem <emph type="italics"/>K<emph.end type="italics"/> in tempore <emph type="italics"/>abc<emph.end type="italics"/> per idem &longs;patium <emph type="italics"/>de,<emph.end type="italics"/><lb/>aut illi æquale <emph type="italics"/>fg<emph.end type="italics"/> moueatur, erit ut velocitas <emph type="italics"/>H<emph.end type="italics"/> ad veloci­<lb/>tatem K, ita tempus <emph type="italics"/>abc<emph.end type="italics"/> minoris velocitatis ad <expan abbr="t&etilde;pus">tempus</expan> <emph type="italics"/>ab<emph.end type="italics"/><lb/>majoris velocitatis. Quia enim velocitas motus &longs;umi­<lb/>tur à magnitudine interualli, erit in eadem ratione in <lb/>quâ interuallum, ac proinde velo citas dupla per &longs;pati <lb/>um mouebit duplum. E&longs;t autem tempus men&longs;ura <expan abbr="cu-ju&longs;&qacute;">cu­<lb/>ju&longs;que</expan>; velocitatis, minor <expan abbr="quid&etilde;">quidem</expan> majoris, major autem mi <lb/>noris; quot igitur magnitudines minoris interualli in <pb/>majori, totidem men&longs;uræ velocitatis majoris in men&longs;u­<lb/>râ velocitatis minoris continentur. </s></p> <figure id="fig10"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Velocitas à principio motus per lineam perpendicularem e&longs;t <lb/>æqualis grauitati, minor verò per lineam inclinatam.<emph.end type="italics"/></s></p> <p type="main"> <s>IMpul&longs;us, quó magis impeditur ab alio impul&longs;u, eò mi <lb/>nùs mouet: e&longs;t <expan abbr="aut&etilde;">autem</expan> grauitas impul&longs;us deor&longs;um &longs;eu <lb/>ad mundi centrum mouens; in lineà ergo perpendicu­<lb/>lari quia â nullo impeditur impul&longs;u, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; agens nece&longs;&longs;a<gap/><lb/>rium, motum producet &longs;ibi æqualem, <expan abbr="eritq;">eritque</expan> velocitas <lb/>motus æqualis grauitati. In lineâ verò inclinatâ, quia <lb/>grauitas impeditur ab hypomochlio, mouebit tantò <lb/>minús, quantò magis impeditur, per prop: 14. ac proin­<lb/>de velocitas erit minor grauitate. Veloc ras ergo a prin­<lb/>cipio motus per lineam perpendicularem e&longs;t æqualis <lb/>grauitati, minor verò per lineam inclinatam. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Velocitas continuò augetur in motu naturali, minuitur in motu <lb/>violento.<emph.end type="italics"/></s></p> <p type="main"> <s>GRauia enim quò e<gap/> loco altiori cadunt, majori vi­<lb/>olentià incidunt: violentia autem major eximpul- <pb/>&longs;u majori, qui illo de&longs;cen&longs;u continuò majus ac majus <lb/>capit augmentum. <expan abbr="Ita&qacute;">Itaque</expan>; videmus globos ferreos à ma <lb/>chinà bellicà & vi ignis alti&longs;simè extolli, ut relap&longs;u lon­<lb/>giore impul&longs;um colligant majorem <expan abbr="ictu&qacute;">ictuque</expan>; violentiore <lb/>urbium tecta d ruant. Sic etiam fi&longs;tucis altiùs &longs;ublatis <lb/>palos ad gunt & terræ magis infigunt. Similiter pon­<lb/>dus è filo pendulum, quò magis dimouetur â &longs;ua &longs;tatio­<lb/>ne, majori vi recutrit, & ultra &longs;tarionem procurrit: qui <lb/>excur&longs;us non ad grauitatem, &longs;ed ad impul&longs;um illo re­<lb/>cur&longs;u collectum referri pote&longs;t. At verò impul&longs;us ma­<lb/>jor eodem vel æquali tempore per &longs;patium ma<emph type="italics"/>j<emph.end type="italics"/>us, mi­<lb/>nori verò tempore per &longs;patium æquale aut etiam ma<emph type="italics"/>j<emph.end type="italics"/>us <lb/>mouet per prop: 6. ac proinde per definitionem ma<emph type="italics"/>j<emph.end type="italics"/>o­<lb/>ri velocitate. velocitas ergo continuò augetur in motu <lb/>naturali, quod primò erat demon&longs;trandum. Quæ au­<lb/>tem motu violento mouentur, cuiu&longs;modi pro<emph type="italics"/>j<emph.end type="italics"/>ecta &longs;eu <lb/>manu, &longs;eu machinà, à principio quidem veloci&longs;simè, in­<lb/>de minùs velociter mouentur, impul&longs;u veluti &longs;ene&longs;cen­<lb/>te: quia nimirum hu<emph type="italics"/>j<emph.end type="italics"/>us principium e&longs;t externum, à quo <lb/>in motu <expan abbr="&longs;eparãtur">&longs;eparantur</expan>: virtus autem finita, quæ non ni&longs;i in <lb/>tempore & per &longs;patium mouet finitum: non igitur ex­<lb/>tra illud tempus mouere, ac proinde <expan abbr="ne&qacute;">neque</expan>; in &longs;ubiecto <lb/>con&longs;eruari pote&longs;t. Emoritur autem &longs;eu naturâ &longs;uà, &longs;eu <lb/>quia grauitas contraria hunc &longs;en&longs;im atterit <expan abbr="minuit&qacute;">minuitque</expan>: ad <lb/>cuius decrementum grauitas magis ac magis inuale&longs;cit: <pb/>unde priusquam vincat, motu mixto ferri, demum ubi <lb/>præualuit, reuer&longs;ionem fieri videmus. In motu verò <lb/>naturali principium motus e&longs;t internum, nimirum gra­<lb/>uitas, & qui à grauitate na&longs;citur impul&longs;us: qui cùm &longs;it <lb/>agens nece&longs;&longs;arium, motum producet &longs;ibi æqualem, & <lb/>prius quam finiat hunc motum, continuó ex eadem ra­<lb/>dice alius <expan abbr="at&qacute;">atque</expan>; alius impul&longs;us rena&longs;cens velocitatem mo <lb/>tus continuo augebit incremento. Dices quam ob rem <lb/>ergo grauia, dum in hypomochlio quie&longs;cunt, nihilo ma <lb/>gis grauitant, &longs;i continuo veluti fluxu inde na&longs;citur im­<lb/>pul&longs;us? Re&longs;pondeo impul&longs;um quidem continuo fluxu <lb/>à grauitate rena&longs;ci, verùm quantùm grauitas producit, <lb/>tantundem re&longs;i&longs;tentia & quies violenta in hypomo­<lb/>chlio ab&longs;umi<gap/>: <expan abbr="quou&longs;&qacute;">quou&longs;que</expan>; ergo grauia quie&longs;cunt, idem <lb/>manet impul&longs;us, qui nequit ab q motu in &longs;ubiecto <lb/>con eruari. Qui <expan abbr="opinãtur">opinantur</expan> grauia non à &longs;e ip&longs;is, verùm à <lb/>&longs;uo magnete <gap/> tellure moueri quæ opinio non caret <lb/>probabilitate, dicent <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>; moius principium e&longs;&longs;e <lb/>externum: verùm in his, quæ pro<gap/>ciuntur, in motu &longs;e­<lb/>parari, <expan abbr="at&qacute;">atque</expan> ita &longs;en&longs;im deficere impul&longs;um; ob retractio­<lb/>nem verò magneticam, ubi <emph type="italics"/>j<emph.end type="italics"/>am præualuit, non aliter <lb/>quam à g<gap/>auitate fieri conuer&longs;ionem motus. Quæ au­<lb/>tem moueri dicuntur à grauitate, habere impul&longs;um à <lb/>tellure, <expan abbr="at&qacute;">atque</expan>; eo modo, quo ferrum ad &longs;uum magne­<lb/>tem moueri, at verò velocitatem ex illà tractione con- <pb/>tinuatà na&longs;ci, dum impul&longs;us &longs;ibi ip&longs;i in&longs;tat non aliter <lb/>quam &longs;i à tergo impelleretur. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Incrementa velocitatis eadem ratione fiunt in motu recto & <lb/>inclinato.<emph.end type="italics"/></s></p> <p type="main"> <s>TAmet&longs;i grauitas in lineà inclinatâ deficiat ab ill <lb/>perfectione, quam habet in lineà perpendiculari, <lb/>non tamen eo modo, quo in lineà horizontali quie&longs;cit­<lb/>tota: exce&longs;&longs;us enim illius partis, quæ cum centro extra <lb/>hypomochlium cadit, à nullo impeditur: & cúm &longs;it a­<lb/>gens nece&longs;&longs;arium, motum producit &longs;ibi æqualem. quia <lb/>verò velocitas continuò augetur in de&longs;cen&longs;u, &longs;icuti gra­<lb/>uitas perfecta in lineà perpendiculari &longs;e habet ad &longs;uum <lb/>augmentum, ita grauitas diminuta in lineà inclinatà &longs;e <lb/> <arrow.to.target n="fig11"></arrow.to.target><lb/>habebit ad &longs;uum augmentum. Moueatur enim ex <emph type="italics"/>a<emph.end type="italics"/><lb/>idem mobile per lineam perpendicularem <emph type="italics"/>abc<emph.end type="italics"/> & perli- <pb/>neam inclinatam <emph type="italics"/>ade:<emph.end type="italics"/> quia ergo motus <emph type="italics"/>ad<emph.end type="italics"/> motui <emph type="italics"/>ab,<emph.end type="italics"/> & <lb/>motus <emph type="italics"/>ae<emph.end type="italics"/> motui <emph type="italics"/>ac<emph.end type="italics"/> e&longs;t æqualis ut prop: 13. o&longs;tendemus: <lb/>&longs;unt autem duo triangula <emph type="italics"/>dab. eac<emph.end type="italics"/> &longs;imilia inter &longs;e, erit <lb/>ut <emph type="italics"/>bc<emph.end type="italics"/> ad <emph type="italics"/>ba,<emph.end type="italics"/> ita <emph type="italics"/>de<emph.end type="italics"/> ad <emph type="italics"/>da,<emph.end type="italics"/> incrementa nimirum velocita­<lb/>tis motus in linea perpendiculari & lineà inclinata. In <lb/>crementa ergo velocitatis eadem ratione fiunt &c. </s></p> <figure id="fig11"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Impul&longs;us in quolibet motu &longs;eu recto, &longs;eu inclinato e&longs;t major gra­<lb/>uitate.<emph.end type="italics"/></s></p> <p type="main"> <s>MOtum in quolibet puncto lineæ perpendicularis <lb/>e&longs;&longs;e majorem &longs;uà grauitate nullum e&longs;t dubium: <lb/>nam cùm velocitas cum ip&longs;o motu incipiat augeri, &longs;icu <lb/>ti à principio e&longs;t æqualis grauitati, ita in progre&longs;&longs;u erit <lb/>major grauitate. At verò de motu per lineam inclina­<lb/>tam dubitari pote&longs;t: propterea quód à grauitate fiat im <lb/>pedità, ac proinde minori: id tamen hac ratione o&longs;ten­<lb/>demus. Grauitas in lineà inclinatà eò magis impeditur <lb/>à &longs;uà velocitate, quò magis hæc inclinatur, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; &longs;inus an <lb/>guli inclinationis idem qui grauitatis exce&longs;&longs;us: uti <lb/>prop: 14. o&longs;tendemus: grauitas ergo per lineam perpen­<lb/>dicularem ad grauitatem per lineam inclinatam, ut &longs;i­<lb/>nus totus ad &longs;inum complementi anguli inclinationis, <lb/>ac proinde ut linea <emph type="italics"/>ab<emph.end type="italics"/> ad linea <emph type="italics"/>ad.<emph.end type="italics"/> at verò velocitas in <emph type="italics"/>b<emph.end type="italics"/> <pb/>majorem rationem habet ad velocitatem in aliquo pun <lb/>cto <emph type="italics"/>f,<emph.end type="italics"/> cúm omni magnitudine datà minor a&longs;&longs;umi po&longs;sit: <lb/>e&longs;t autem velocitas in <emph type="italics"/>f<emph.end type="italics"/> major &longs;uà grauitate: erit ergo <lb/>velocitas in <emph type="italics"/>d<emph.end type="italics"/> major <expan abbr="quo&qacute;">quoque</expan>; eadem grauitate, cùm majo­<lb/>rem rationem habeat velocitas in <emph type="italics"/>b<emph.end type="italics"/> ad velocitatem in <emph type="italics"/>f,<emph.end type="italics"/><lb/>quam ad velocitatem in <emph type="italics"/>d.<emph.end type="italics"/> Idem de quouis alio pun­<lb/>cto o&longs;tendemus. impul&longs;us ergo in quolibet motu &longs;eu re <lb/>cto, &longs;eu inclinato e&longs;t major grauitate. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Incrementa velocitatis rationem habent quam temporum <lb/>quadrata.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia virtus locomotiua eo modo augetur, quo tri­<lb/>angulum &longs;ibi &longs;imile manens, per po&longs;it: 5. propte­<lb/>rea quòd hujus augmentum &longs;it petfectio inten&longs;iua; <lb/>cùm ex illo puncto quietis veluti late&longs;cit, angulum con <lb/>&longs;tituit &longs;ui augmenti, ma<emph type="italics"/>j<emph.end type="italics"/>orem minoremuè pro <expan abbr="cuiu&longs;&qacute;">cuiu&longs;que</expan>; <lb/>perfectione, quam obtinet in principio motus, &longs;iuè ex <lb/>naturâ &longs;uâ, &longs;iue ex impedimento: majori enim perfecti­<lb/>oni maior angulus debetur. Sit primùm angulus <emph type="italics"/>nag<emph.end type="italics"/><lb/>&longs;emi&longs;sis anguli recti; tempus verò <emph type="italics"/>ag<emph.end type="italics"/> in minuta <emph type="italics"/>ab. bc. <lb/>cd. de. ef.fg<emph.end type="italics"/> æqualiter diui&longs;um: velocitas ergò motus <lb/>augetur impul&longs;u auge&longs;cente in primo quidem minuto <lb/>in <emph type="italics"/>hb,<emph.end type="italics"/> in 2. in <emph type="italics"/>ic,<emph.end type="italics"/> in 3. in <emph type="italics"/>kd,<emph.end type="italics"/> <expan abbr="at&qacute;">atque</expan>; itæ con&longs;equenter æquatà <pb/>areà illius trianguli rectanguli, cujus longitudo nume­<lb/>rus minutorum, ba&longs;is verò terminus augmenti. Quia <lb/>verò eadem e&longs;t ratio motus & virtutis impul&longs;iuæ, vir­<lb/> <arrow.to.target n="fig12"></arrow.to.target><lb/>tus quidem dupla in eodem aut æquali tempore moue­<lb/>bit per &longs;patium duplum: quòd &longs;i ergo in primo minu­<lb/>ro <emph type="italics"/>ab<emph.end type="italics"/> virtus <emph type="italics"/>a<emph.end type="italics"/> late&longs;cens, cum quà pariter cre&longs;cit veloci­<lb/>tas motus, terminum habet &longs;ui incrementi in <emph type="italics"/>hb,<emph.end type="italics"/> in &longs;e­<lb/>cundo minuto in <emph type="italics"/>ic,<emph.end type="italics"/> in 3. in <emph type="italics"/>kd<emph.end type="italics"/> &c. erit ut triangulum re­<lb/>ctangulum <emph type="italics"/>iac<emph.end type="italics"/> ad triangulum rectangulum <emph type="italics"/>hab,<emph.end type="italics"/> ita <lb/>&longs;patium decur&longs;um in duobus minutis ad &longs;patium decur <lb/>&longs;um in uno minuto; at verò duo triangula <emph type="italics"/>iac.hab<emph.end type="italics"/> &longs;unt <lb/>&longs;emi&longs;&longs;es duorum quadratorum <emph type="italics"/>ipac. hoab.<emph.end type="italics"/> ac pro­<lb/>inde in eàdem ratione, nimirum duplicatà ejus, <expan abbr="quã">quam</expan> ha­<lb/>bent latera <emph type="italics"/>ic.hb<emph.end type="italics"/>: igitur ut quadratum lateris <emph type="italics"/>ic<emph.end type="italics"/> ad qua­<lb/>dratum lateris <emph type="italics"/>hb,<emph.end type="italics"/> ita motus duorum minutorum ad <pb/>motum unius minuti; propterea quòd latus <emph type="italics"/>ca<emph.end type="italics"/> ad latus <lb/><emph type="italics"/>ba<emph.end type="italics"/> eandem habeat rationem, quam latus <emph type="italics"/>ic<emph.end type="italics"/> ad latus <emph type="italics"/>hb,<emph.end type="italics"/><lb/>ac proinde illorum quadrata in eadem <expan abbr="quo&qacute;">quoque</expan>, ratione, <lb/>nimirum duplicata. <expan abbr="Ita&qacute;">Itaque</expan>; &longs;i quadratum lateris <emph type="italics"/>ab,<emph.end type="italics"/> hoc <lb/>e&longs;t primi minuti, &longs;ubtrahas â quadrato <emph type="italics"/>ac<emph.end type="italics"/> &longs;ecundi minu­<lb/>ti, numerus reliquus dabit velo citatem motus in eodem <lb/>minuto: ut &longs;i cubitum unum <emph type="italics"/>vg.<emph.end type="italics"/> perambulet in primo <lb/>minuto, hujus quadratum, ide&longs;t unum, ab illius quadra <lb/>to, id, e&longs;t â quatuor &longs;ubtractum relinquit tria totidem <lb/>cubitorum illi &longs;patio, per quod <emph type="italics"/>a<emph.end type="italics"/> mouetur in minuto 2. <lb/>tribuenda. Similiter quia 3. minutis conficit cubitos 9. <lb/>ablato ex his quadrato &longs;ecundi minuti, numerus reli­<lb/>quus dabit velocitatem 5. cubitorum, qui minuto 3. de­<lb/>bentur. Rur&longs;um â numero 4. minuti in &longs;e ducto, ide&longs;t <lb/>16. ablatis 9. quadrato tertij minuti rem anet numerus 7. <lb/>pro 4. minuto: totidem ergo cubitorum &longs;patium trans­<lb/>mittit mobile <emph type="italics"/>a<emph.end type="italics"/> in minuto quarto. Quód &longs;i angulus <lb/>augmenti major &longs;it aut minor &longs;emi&longs;&longs;e anguli recti, ut <lb/>angulus <emph type="italics"/>qag.<emph.end type="italics"/> aut <emph type="italics"/>rag,<emph.end type="italics"/> quod quidem contingit, cùm vir­<lb/>tus impul&longs;iua magis aut minùs e&longs;t inten&longs;a, tum quidem <lb/>illa virtus magis perfecta ex illo puncto continuò majo <lb/>ra &longs;umit incrementa: eadem tamen demon&longs;tratio, <expan abbr="at&qacute;">atque</expan>; <lb/>eadem e&longs;t proportio <expan abbr="utrobi&qacute;">utrobique</expan>;, proptera quòd parallelo­<lb/>gramma in proportione <expan abbr="quo&qacute;">quoque</expan>; &longs;int duplicatá &longs;uorum <lb/>laterum &longs;imul &longs;umptorum.| </s></p> <pb/> <figure id="fig12"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus per lineam perpendicularem & lineam inclinatam, quo­<lb/>rum terminos conjungit linea recta perpendicularis ad lineam in­<lb/>clinatam, inter &longs;o junt æquales.<emph.end type="italics"/></s></p> <p type="main"> <s>ÆQuales dico non velocitate, quæ minor e&longs;t in lineà <lb/>inclinatà, &longs;ed duratione: hoc e&longs;t &longs;i ex eodem puncto <lb/>incipiat motus <emph type="italics"/>Vg.<emph.end type="italics"/> ex <emph type="italics"/>b,<emph.end type="italics"/> & unum quidem mobile per li­<lb/>neam perpendicularem <emph type="italics"/>ba,<emph.end type="italics"/> alterum verò hulc æquale <lb/>per lineam <emph type="italics"/>bf<emph.end type="italics"/> ad horizontem inclinatam moueatur: a&longs;­<lb/>&longs;umpto quolibet puncto in lineà perpendiculari <emph type="italics"/>Vg. a,<emph.end type="italics"/><lb/>linea ex hoc puncto educta perpendicularis ad lineam <lb/><emph type="italics"/>bf<emph.end type="italics"/> locum terminabit in <emph type="italics"/>f,<emph.end type="italics"/> ad quod mobile eodem tem­<lb/> <arrow.to.target n="fig13"></arrow.to.target> <pb/>pore per lineam <emph type="italics"/>bf,<emph.end type="italics"/> quo alterum mobile per lineam <emph type="italics"/>ba<emph.end type="italics"/><lb/>decurrit. Ducatur enim ex puncto contactus <emph type="italics"/>f<emph.end type="italics"/> linea <emph type="italics"/>fe<emph.end type="italics"/><lb/>parallela lineæ perpendiculari <emph type="italics"/>ba,<emph.end type="italics"/> & producatur in <emph type="italics"/>g<emph.end type="italics"/>; ad <lb/>quam ex centto grauitatis <emph type="italics"/>d<emph.end type="italics"/> educta &longs;it linea perpendicu <lb/>laris <emph type="italics"/>dc,<emph.end type="italics"/> di&longs;tantia nimirum centri à lineà hypomochlij <emph type="italics"/>f <lb/>g:<emph.end type="italics"/> e&longs;t autem linea <emph type="italics"/>df<emph.end type="italics"/> &longs;emidiameter circuli, di&longs;tantia eju&longs;­<lb/>dem centri ab hypochlio, quam obtinet in lineâ perpen <lb/>diculari <emph type="italics"/>ba.<emph.end type="italics"/> quia ergo impul&longs;us augetur in ratione di­<lb/>&longs;tantiæ centri ab hypomochlio, per Po&longs;it: 6. <expan abbr="motũ&qacute;">motunque</expan>; pro <lb/>ducit &longs;ibi æqualem, per prop: 2. velocitas autem motus <lb/>eandem rationem habet quam interualla, per prop: 7. e­<lb/>rit ut <emph type="italics"/>fd<emph.end type="italics"/> impul&longs;us major ad <emph type="italics"/>dc<emph.end type="italics"/> impul&longs;um minorem, ita <lb/>motus in <emph type="italics"/>ba<emph.end type="italics"/> ad motum in <emph type="italics"/>bf:<emph.end type="italics"/> proptorea quód triangula­<lb/><emph type="italics"/>abf.fdc<emph.end type="italics"/> &longs;int &longs;imilia, & linea <emph type="italics"/>dc<emph.end type="italics"/> perpendicularis, ac proinde <lb/>linea <expan abbr="quo&qacute;">quoque</expan>; <emph type="italics"/>af,<emph.end type="italics"/> &longs;imilis lineæ <expan abbr="perp&etilde;diculari">perpendiculari</expan> <emph type="italics"/>dc,<emph.end type="italics"/> perpendi­<lb/>cularis. </s></p> <figure id="fig13"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XIV:<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus per lineam minùs inclinatam e&longs;t velocìor motu per li­<lb/>neam magis inclinatam, in ratione, quam habent &longs;inus complemen­<lb/>ti illarum inclinationum.<emph.end type="italics"/></s></p> <p type="main"> <s>DVcantur ex puncto <emph type="italics"/>a<emph.end type="italics"/> lîneæ <emph type="italics"/>ab. ac. ad. ae. af,<emph.end type="italics"/> & &longs;it li­<lb/>nea <emph type="italics"/>ab<emph.end type="italics"/> horizontalis, linea verò <emph type="italics"/>at<emph.end type="italics"/> perpendicularis, <lb/>reliquæ lineæ ad horizontem inclinatæ: dico idem mo­<lb/>bile o verbi grat: inæqualiter moueri, velociùs quidem <pb/>in lineà <emph type="italics"/>ae<emph.end type="italics"/> minus inclinatà, minùs autem velociter in li­<lb/>neà <emph type="italics"/>ad<emph.end type="italics"/> magis inclinatà, <expan abbr="e&longs;&longs;e&qacute;">e&longs;&longs;eque</expan>; rationem velocitatis in <emph type="italics"/>ae<emph.end type="italics"/><lb/>ad velocitatem in <emph type="italics"/>ad,<emph.end type="italics"/> ut &longs;inus anguli <emph type="italics"/>ats<emph.end type="italics"/> ad &longs;inum angu­<lb/>li <emph type="italics"/>atr.<emph.end type="italics"/> Ex punctis contactus <emph type="italics"/>qrs<emph.end type="italics"/> demittantur lineæ <lb/>perpendiculares <emph type="italics"/>qt.rt.st:<emph.end type="italics"/> & aliæ lineæ perpendiculari <emph type="italics"/>at<emph.end type="italics"/><lb/> <arrow.to.target n="fig14"></arrow.to.target><lb/>parallelæ <emph type="italics"/>qg.rh.si<emph.end type="italics"/> <expan abbr="&longs;ecãtes">&longs;ecantes</expan> mobile in <emph type="italics"/>k. n. u,<emph.end type="italics"/> ex centro au <lb/>tem <emph type="italics"/>o<emph.end type="italics"/> ducantur lineæ perpendiculares ad lineam hypo­<lb/>mochlij <foreign lang="greek">oa. ob. og</foreign>, <expan abbr="erunt&qacute;">eruntque</expan>; lineæ <emph type="italics"/>qg. rh. si<emph.end type="italics"/> lineæ hypo­<lb/>mochlij. Quia verò angulus <emph type="italics"/>tsi,<emph.end type="italics"/> hoc e&longs;t angulus <emph type="italics"/>s<gap/>h<emph.end type="italics"/> ex- <pb/>ternus major e&longs;t angulo <emph type="italics"/>trh<emph.end type="italics"/> interno & oppo&longs;ito, erit an <lb/>gulus <foreign lang="greek">gso</foreign> angulo <foreign lang="greek">bgo</foreign>, & latus <foreign lang="greek">go</foreign> latere <foreign lang="greek">bo</foreign> majus: &longs;unt <lb/>autem latera <foreign lang="greek">go. bo</foreign> di&longs;tantia centri grauitatis. Quia er­<lb/>go maior impul&longs;us in <foreign lang="greek">go</foreign> maiori, quam in <foreign lang="greek">bo</foreign> minori di­<lb/>&longs;tantià; erit per prop: 6. velocior motus in linea <emph type="italics"/>as<emph.end type="italics"/> mi­<lb/>nús inclinatá, quam in lineà <emph type="italics"/>ar<emph.end type="italics"/> magis inclinatà. Quòd <lb/>autem velocitas motus &longs;it in ratione, quam habent cor­<lb/>dæ, &longs;eu &longs;inus complementi inclinationum, ita o&longs;tende­<lb/>mus: quia ut <foreign lang="greek">so</foreign> ad <foreign lang="greek">go</foreign>, ita corda <emph type="italics"/>at<emph.end type="italics"/> ad cordam <emph type="italics"/>as,<emph.end type="italics"/> & ut <emph type="italics"/>rò<emph.end type="italics"/><lb/>æqualis <foreign lang="greek">so</foreign> ad <foreign lang="greek">ob</foreign>, ita eadem corda <emph type="italics"/>at<emph.end type="italics"/> ad cordam <emph type="italics"/>ar:<emph.end type="italics"/> erit <lb/><expan abbr="quo&qacute;">quoque</expan>; ut <foreign lang="greek">og</foreign> ad <foreign lang="greek">ob</foreign>, ita <emph type="italics"/>as<emph.end type="italics"/> ad <emph type="italics"/>ar.<emph.end type="italics"/> at verò ut cordæ <emph type="italics"/>as. ar,<emph.end type="italics"/><lb/>ita illarum &longs;emi&longs;&longs;es <emph type="italics"/>al. am<emph.end type="italics"/> &longs;inus angulorum <emph type="italics"/>apl. apm<emph.end type="italics"/><lb/>qui æquales &longs;unt angulis <emph type="italics"/>ats.atr<emph.end type="italics"/> angulis complementi <lb/>inclinationis, ob parallelas <emph type="italics"/>ts. pl,<emph.end type="italics"/> & <emph type="italics"/>tr. pm.<emph.end type="italics"/> <emph type="italics"/>I<emph.end type="italics"/>gitur ut <foreign lang="greek">og</foreign><lb/>ad <foreign lang="greek">ob</foreign>, ita &longs;inus complementi angulorum inclinationis, <lb/>quod erat o&longs;tendendum. </s></p> <figure id="fig14"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus ex eodem puncto per lineas &longs;ubten&longs;as &longs;unt æquales motui <lb/>per diametrum eju&longs;dem circuli.<emph.end type="italics"/></s></p> <p type="main"> <s>MOueatur ex puncto <emph type="italics"/>b<emph.end type="italics"/> mobile per lineas <emph type="italics"/>bi. bh.bg. <lb/>bf.be<emph.end type="italics"/> ad horizontem inclinatas, hoc e&longs;t per cordas <lb/>arcuum <emph type="italics"/>bes.beh.beg.bef.be<emph.end type="italics"/>: dico eodem tempore per <pb/>cordam <emph type="italics"/>bf,<emph.end type="italics"/> aut <emph type="italics"/>bg,<emph.end type="italics"/> quo per diametrum eiu&longs;dem circuli <lb/><emph type="italics"/>ba<emph.end type="italics"/> motum terminari. Quòd &longs;i enim ex puncto <emph type="italics"/>a<emph.end type="italics"/> du <lb/>cantur lineæ rectæ <emph type="italics"/>af. ag,<emph.end type="italics"/> erunt anguli <emph type="italics"/>afb. agb<emph.end type="italics"/> in &longs;emi­<lb/>cirulo recti; ac proinde ex iam demon&longs;tratis motus in <lb/><emph type="italics"/>ba<emph.end type="italics"/> motui in <emph type="italics"/>bf<emph.end type="italics"/> & <emph type="italics"/>bg<emph.end type="italics"/> duratione æqualis. Simili modo &longs;i <lb/>ex punctis <emph type="italics"/>befg<emph.end type="italics"/> in <emph type="italics"/>a<emph.end type="italics"/> terminetur motus, <expan abbr="erũt">erunt</expan> lineæ <emph type="italics"/>be.bf.<emph.end type="italics"/><lb/> <arrow.to.target n="fig15"></arrow.to.target><lb/><emph type="italics"/>bg<emph.end type="italics"/> perpendiculares ad <emph type="italics"/>ae. af. ag,<emph.end type="italics"/> ac proinde motus in <emph type="italics"/>b <lb/>a<emph.end type="italics"/> motui in <emph type="italics"/>ea. fa. ga<emph.end type="italics"/> æqualis. At verò &longs;i ex alio puncto <lb/>Vg <foreign lang="greek">a</foreign> incipiat motus, <expan abbr="ne&qacute;">neque</expan>; ad idem cum diametro pun­<lb/>ctum terminetut, cuju&longs;modi linea <foreign lang="greek">ab</foreign>, er t motus hujus <lb/>motui in diametro <emph type="italics"/>ba<emph.end type="italics"/> inæqualis. Ducatur enim ex <foreign lang="greek">a</foreign> in <lb/><emph type="italics"/>a<emph.end type="italics"/> linea <foreign lang="greek">a</foreign> <emph type="italics"/>a,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; motus hujus motui <emph type="italics"/>ba,<emph.end type="italics"/> ide&longs;t motui <foreign lang="greek">ad</foreign> <pb/>æqualis: linea verò <foreign lang="greek">gb</foreign> perpendicularis ad <foreign lang="greek">ab</foreign> motum ter­<lb/>minabit in <foreign lang="greek">b</foreign> æqualem motui <foreign lang="greek">ag</foreign>: e&longs;t autem linea <foreign lang="greek">ag</foreign> mi­<lb/>nor quam <foreign lang="greek">ad</foreign> motus ergo in <foreign lang="greek">ag</foreign>, ide&longs;t motus huic æqua <lb/>lis in <foreign lang="greek">ab</foreign> minori fit tempore quam in <foreign lang="greek">a<gap/>. </foreign></s></p> <figure id="fig15"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus grauitatis per lineam magis inclinatam in majori à <lb/>centro di&longs;tantià, tempore verò æquali terminatur.<emph.end type="italics"/></s></p> <p type="main"> <s>MOueatur mobile à puncto <emph type="italics"/>b<emph.end type="italics"/> per lineas <emph type="italics"/>ba. bi. bh. bg <lb/>bf be<emph.end type="italics"/>; dico &longs;olam lineam perpendicularem <emph type="italics"/>ba<emph.end type="italics"/> in <lb/>centro <emph type="italics"/>a,<emph.end type="italics"/> reliquas omnes extracentrum, <expan abbr="at&qacute;">atque</expan>; ex inclina­<lb/>tione majori ad majus interuallum terminari: ut quia <lb/>angulus <emph type="italics"/>abh<emph.end type="italics"/> e&longs;t major angulo <emph type="italics"/>abi,<emph.end type="italics"/> erit terminus mo­<lb/> <arrow.to.target n="fig16"></arrow.to.target> <pb/>tus, quem grauitas inducit in lineâ <emph type="italics"/>bh,<emph.end type="italics"/> remotior à cen­<lb/>tro, quàm in lineâ <emph type="italics"/>bi.<emph.end type="italics"/> Ducantur enim à centro <emph type="italics"/>a<emph.end type="italics"/> lineæ <emph type="italics"/>ai. <lb/>ab<emph.end type="italics"/> perpendiculares ad <emph type="italics"/>bi. bh,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; terminus motus gra­<lb/>uitatis in <emph type="italics"/>i<emph.end type="italics"/> & <emph type="italics"/>h<emph.end type="italics"/> ob breui&longs;simam di&longs;tantiam, quæ e&longs;&longs;e po­<lb/><gap/>e&longs;t in illis lin<gap/>is; quód &longs;i enim ex <emph type="italics"/>i<emph.end type="italics"/> moueatur in <emph type="italics"/>st,<emph.end type="italics"/> quia <lb/>illo progre&longs;&longs;u lineæ à centro ductæ fiunt majores, ma­<lb/>jor enim <emph type="italics"/>as<emph.end type="italics"/> angulo recto <emph type="italics"/>ais<emph.end type="italics"/> &longs;ubten&longs;a quam <emph type="italics"/>ai,<emph.end type="italics"/> mobile <lb/>motu naturali à centro magis abduceretur, quod fieri <lb/>nequit. Quia ergo linea <foreign lang="greek">ag</foreign> major e&longs;t quàm linea <emph type="italics"/>ai,<emph.end type="italics"/><lb/>erit linea <emph type="italics"/>ab<emph.end type="italics"/>c dem multò major: igitur punctum <emph type="italics"/>h<emph.end type="italics"/> ter­<lb/>minus motus in lineà magis inclinatà, majori, punctum <lb/>verò <emph type="italics"/>i<emph.end type="italics"/> terminus motus in lineà minús inclinatâ, minori <lb/>à centro abe&longs;t interuallo. Quia vetò <expan abbr="uter&qacute;">uterque</expan>; motus tam <lb/>per <expan abbr="lineã">lineam</expan> <emph type="italics"/>ai<emph.end type="italics"/> <expan abbr="quã">quam</expan> per <expan abbr="lineã">lineam</expan> <emph type="italics"/>ah<emph.end type="italics"/> e&longs;t æqualis motui per <expan abbr="lineã">lineam</expan> <lb/>perpendicularem <emph type="italics"/>ab,<emph.end type="italics"/> propterea quòd lineæ perpendi­<lb/>culares <emph type="italics"/>as. ah<emph.end type="italics"/> <expan abbr="utrum&qacute;">utrumque</expan>; <expan abbr="motũ">motum</expan> conjungunt per prop: 13. <lb/>erit motus <emph type="italics"/>at<emph.end type="italics"/> motui <emph type="italics"/>ab<emph.end type="italics"/> æqualis, ac proinde in tempore <lb/>æquali. </s></p> <figure id="fig16"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus grauitatis ex eodem puncto per lineas ad horizontem in­<lb/>clinatas in circulum terminatur, cuius diameter e&longs;t di&longs;tantia inter <lb/>illud punctum & mundi <gap/>um.<emph.end type="italics"/></s></p> <p type="main"> <s>MOueatur ex puncto <emph type="italics"/>b<emph.end type="italics"/> mobil<gap/> eju&longs;dem rationis per <lb/>lineàs ad horizontem inclinat as <emph type="italics"/>bi. bh. bg. bf.<emph.end type="italics"/> &c. <pb/>&longs;it autem mundi centrum <emph type="italics"/>a,<emph.end type="italics"/> & linea perpendicularis <emph type="italics"/>ba,<emph.end type="italics"/><lb/>dico motum per lineas <emph type="italics"/>bi. bh. bg. bf<emph.end type="italics"/> &c. in circulum ter <lb/>minari, cujus diameter linea perpendicularis <emph type="italics"/>ab<emph.end type="italics"/> di&longs;tan­<lb/>tia inter <emph type="italics"/>b<emph.end type="italics"/> & mundi centrum <emph type="italics"/>a.<emph.end type="italics"/> Ducantur enim à cen­<lb/>tro <emph type="italics"/>a<emph.end type="italics"/> lineæ <emph type="italics"/>ai.ah.ag.af<emph.end type="italics"/> &c. perpendiculares ad <emph type="italics"/>bi.bh.bg. <lb/>bf,<emph.end type="italics"/> <expan abbr="erunt&qacute;">eruntque</expan>; puncta <emph type="italics"/>i.h.g.f<emph.end type="italics"/> termini motus à grauitate: <lb/> <arrow.to.target n="fig17"></arrow.to.target><lb/>propterea quòd minima &longs;it hæc di&longs;tantia à mundi cen­<lb/>tro <emph type="italics"/>a.<emph.end type="italics"/> Quia verò anguli <emph type="italics"/>aib.ahb.afb<emph.end type="italics"/> &longs;unt recti ean­<lb/>dem habentes ba&longs;im <emph type="italics"/>ab.<emph.end type="italics"/> erunt in eodem &longs;emicirculo <emph type="italics"/>bef <lb/>g hia,<emph.end type="italics"/> cujus diameter linea <emph type="italics"/>ba<emph.end type="italics"/> perpendicularis, di&longs;tantia <lb/>inter <emph type="italics"/>b<emph.end type="italics"/> & mundi centrum. </s></p> <pb/> <figure id="fig17"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Velocitas in fine motus æquali tempore per &longs;patium mouet du­<lb/>plum velocitatis eodem motu collectæ.<emph.end type="italics"/></s></p> <p type="main"> <s>VT in fig: 5. &longs;i velocitas motus <emph type="italics"/>a<emph.end type="italics"/> in tempore <emph type="italics"/>ac<emph.end type="italics"/> conti­<lb/>nuò augeatur; quia hujus augmentum e&longs;t perfe­<lb/>ctio inten&longs;iua, ac proinde eo modo augetur, quo trian­<lb/>gulum &longs;ibi &longs;imile manens per po&longs;it: 5. erit velocitas in <lb/>fine motus, ut ba&longs;is eju&longs;dem trianguli <emph type="italics"/>bc.<emph.end type="italics"/> Moueatur er­<lb/>go hæc velocitas in <emph type="italics"/>e,<emph.end type="italics"/> & &longs;it tempus <emph type="italics"/>ec<emph.end type="italics"/> æquale tempori <lb/><emph type="italics"/>ac,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; velocitas illo motu colecta quadratum <emph type="italics"/>bcde<emph.end type="italics"/><lb/>duplum trianguli <emph type="italics"/>abc,<emph.end type="italics"/> propterea quòd eandem ba&longs;im <lb/><emph type="italics"/>bc,<emph.end type="italics"/> altitudinem verò habet æqualem. Quia ergo virtus <lb/>dupla in eodem vel æquali tempore per &longs;patium mouet <lb/><expan abbr="duplũ">duplum</expan>, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; <expan abbr="ead&etilde;">eadem</expan> ratio velocitatis & interualli, velocitas <lb/>in fine motus eodem vel æquali tempore per &longs;patium <lb/>mouebit duplum &c. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Velocitas in motugrauium collecta ultra &longs;tationem defert mo­<lb/>bile.<emph.end type="italics"/></s></p> <p type="main"> <s>STatio quidem grauium e&longs;t centrum terræ, ponderis <lb/>verò è filo penduli linea perpendicularis, in quà de- <pb/>mum mobile ex illâ agitatione conquie&longs;cit. Quòd &longs;i <lb/>ergo &longs;eu corpus graue ad centrum, &longs;eu perpendiculum <lb/>in &longs;uam &longs;tationem moueatur, non &longs;tatim conquie&longs;cit <lb/>ex hoc motu &longs;iuè in centro, &longs;iuè in lineâ perpendiculari, <lb/>verùm ultra hos limites procurrit & recurrit, <expan abbr="at&qacute;">atque</expan>; eò ma­<lb/>gis, quò circuli majores. Quod quidem in perpendicu­<lb/>lo experientià con&longs;tat: de grauium verò à centro excur <lb/>&longs;u licet nulla experientia habeatur, id tamen &longs;imilitudo <lb/>rationis euincit: non enim minùs contra natu­<lb/>ram grauitatis e&longs;&longs;e videtur in circulo à lineâ &longs;tatio­<lb/>nis, quam in lineâ perpendiculari è centro efferi. Hujus <lb/>autem ratio hæc: quia impul&longs;us in quolibet puncto, ac <lb/>proinde in fine motus e&longs;t major grauitate: per prop: 11. <lb/>e&longs;t autem agens nece&longs;&longs;arium per prop: 2. & non ni&longs;i per <lb/>lineam rectam mouet <expan abbr="&longs;uũ">&longs;uum</expan> mobile per prop: 3. &longs;uperabit <lb/>ergo illam, quâ in centro firmatur, grauitatem, non mi­<lb/>nùs, quæm cùm lapidem &longs;imilis impul&longs;us à centro lon­<lb/>giùs abducit. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Velocitas in motu collecta per æquaba &longs;uo augmento decremen­<lb/>ta in quietem terminatur.<emph.end type="italics"/></s></p> <p type="main"> <s>PErpendiculum liberè dimi&longs;&longs;um in &longs;uam &longs;tationem <lb/>recurrit, <expan abbr="at&qacute;">atque</expan>; eodem motu continuato ultra &longs;tatio- <pb/>nem excurrit. Quòd &longs;i ergo impul&longs;us ex illo recur&longs;u <lb/>collectus aut idem maneat, aut continuò augeatur, quia <lb/>per prop: 18. Velocitas in fine eodem vel æquali tempo­<lb/>re per &longs;patium mouet duplum velocitatis ex illo motu <lb/>collectæ, erit ex cur&longs;us major recur&longs;u: & quia ex quoli­<lb/>bet recur&longs;u magis excurrit, erit motus perpendiculi in­<lb/>finitus. At verò hic motus demum conquie&longs;cit: <expan abbr="nõ">non</expan> ergo <lb/>impul&longs;us augeri, aut idem e&longs;&longs;e pote&longs;t. Et quia per ar­<lb/>cus excurrit & recurrit continuò minores, nece&longs;&longs;e im­<lb/>pul&longs;um minui in illo a&longs;cen&longs;u; quia nimirum inter&longs;e <lb/>mi&longs;centur, & in de&longs;cen&longs;u quidem per eandem lineam <lb/>mouent grauitas & impul&longs;us, quem à grauitate conti­<lb/>nuo fluxu na&longs;ci dicebamus: à &longs;tatione verò grauitas im <lb/>pul&longs;ui reluctatur: quia nimirum contrarius impul&longs;us <lb/>ab eâdem grauitate rena&longs;cens tollit partem &longs;ibi æqua­<lb/>lem, per po&longs;it: 2. <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; motus reliquus æqualis exce&longs;&longs;ui <lb/>majoris ut Prop: 30. dicemus: &longs;icut ergo impul&longs;us conti­<lb/>nuò decre&longs;cit ij&longs;dem, quibus augebatur augmentis, ita <lb/>uelocitas à &longs;ummo augmento ad finem <expan abbr="u&longs;&qacute;">u&longs;que</expan>; motus con­<lb/>tinuò fit minor; &longs;imul verò &longs;umpta æqualis velocitatià <lb/>principio motus ad finem <expan abbr="augm&etilde;ti">augmenti</expan> collectæ: ut &longs;i in &longs;ig: 9. <lb/><expan abbr="perpendiculũ">perpendiculum</expan> <emph type="italics"/>ae<emph.end type="italics"/> ex <emph type="italics"/>e<emph.end type="italics"/> recurrat in <emph type="italics"/>b,<emph.end type="italics"/> & ex <emph type="italics"/>b<emph.end type="italics"/> excurrat in <emph type="italics"/>&longs;i<emph.end type="italics"/> a&longs;­<lb/>&longs;umantur autem arcus <emph type="italics"/>bc. bd,<emph.end type="italics"/> & <emph type="italics"/>be.bf<emph.end type="italics"/> inter &longs;e æquales: <lb/>dico augmentum velocitatis in <emph type="italics"/>e<emph.end type="italics"/> eju&longs;dem decremento <lb/>in <emph type="italics"/>f,<emph.end type="italics"/> & augmentum velocitatis in <emph type="italics"/>c<emph.end type="italics"/> eju&longs;dem decremento <pb/>in <emph type="italics"/>d<emph.end type="italics"/> e&longs;&longs;e æquale. Ducantur enim lineæ tangentes <emph type="italics"/>eg fg,<emph.end type="italics"/><lb/>& <emph type="italics"/>ch. d<gap/>:<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; inclinatio <emph type="italics"/>eg<emph.end type="italics"/> inclinationi <emph type="italics"/>fg,<emph.end type="italics"/> & inclina­<lb/>tio <emph type="italics"/>ch<emph.end type="italics"/> æqualis inclinationi <emph type="italics"/>dh<emph.end type="italics"/>: propterea quòd <expan abbr="ãguli">anguli</expan> <emph type="italics"/>eg<gap/><lb/>fga,<emph.end type="italics"/> & anguli <emph type="italics"/>cha. dha<emph.end type="italics"/> &longs;unt æquales, impul&longs;us ergo gra­<lb/>uitatis in <emph type="italics"/>e<emph.end type="italics"/> eju&longs;dem impul&longs;ui in <emph type="italics"/>f,<emph.end type="italics"/> & impul&longs;us grauitatis <lb/>in <emph type="italics"/>c<emph.end type="italics"/> eju&longs;dem impuliui in <emph type="italics"/>d<emph.end type="italics"/> e&longs;t æqualis, ut con&longs;tat ex <lb/>prop. 14. Quia ergo impul&longs;us æquales in <emph type="italics"/>e<emph.end type="italics"/> quidem & <emph type="italics"/>c<emph.end type="italics"/><lb/>augent, in <emph type="italics"/>f<emph.end type="italics"/> verò & <emph type="italics"/>d<emph.end type="italics"/> minuunt velocitatem motus, <expan abbr="erũt">erunt</expan> <lb/>æqualia velocitatis augmenta eju&longs;dem decremento; ac <lb/>proinde velocitas in motu collecta per æqualia &longs;uo aug <lb/>mento decrementa in quietem terminatur. Obijcies &longs;i <lb/>velocitas excur&longs;us &longs;imul &longs;umpta e&longs;t æqualis velocitati in <lb/>recur&longs;u collectæ, quia velocitas æqualis eodem vel æ­<lb/>quali tempore per &longs;patium mouet æquale, eiunt excur­<lb/>&longs;us & recur&longs;us inter &longs;e æquales: ac proinde motus per­<lb/>pendiculi infinitus. Re&longs;pondent quidam excur&longs;um e&longs;­<lb/>&longs;e minorem recur&longs;u: propterea quód illius motus à fu­<lb/>niculo perturbetur, cujus partes inæqualiter mouen­<lb/>tur: velociùs quidem centro propiores, minùs autem <lb/>velociter à centro remotiores. Dum ergo hæ re&longs;titant, <lb/>& minorum circulorum velocitatem morantur; illæ <lb/>præcurrere fe&longs;tinant: nece&longs;&longs;e ex illà luctâ impul&longs;um mi­<lb/><gap/>ui, ut non ni&longs;i ad minus interuallum &longs;e extendat. Hu­<lb/>jus autem &longs;ignum e&longs;&longs;e illos &longs;inus, in quos funis contor­<lb/>quetur, & veluti fluctuat. Verùm licet in fune, aut ca- <pb/>tenà, cujus partes ex <gap/>e lunt pondero&longs;æ, motus hic undo<gap/><lb/>&longs;us &longs;ibi ip&longs;i &longs;it impedimento: non tamen hæc ratio lo­<lb/>cum habet in perquam &longs;ubtili & tenui&longs;simo filo, cujus <lb/>partes non ex &longs;e, ve<gap/>úm ex impul&longs;u ponderis appen&longs;i <lb/><expan abbr="mou&etilde;tur">mouentur</expan>, <expan abbr="eo&qacute;">eoque</expan>; præci&longs;o aut abrupto à motu <expan abbr="conquie&longs;cũt">conquie&longs;cunt</expan>. <lb/>Deinde &longs;i ratio inæqualium circulorum perturbat il­<lb/>lum motum, quo perpendiculum à &longs;ua &longs;tatione procur <lb/>rit, turbabit <expan abbr="quo&qacute;">quoque</expan>; rationem motus, quam ad &longs;e habent <lb/>recur&longs;us: at verò hæcinæ qualitas nihil ob&longs;tat, quò mi­<lb/>nùs recur&longs;us inter &longs;e &longs;int æquales: nihil ergo ob&longs;tabit, <lb/>quò minùs excur&longs;us <expan abbr="quo&qacute;">quoque</expan>; inter &longs;e &longs;int æquales. Præte­<lb/>rea &longs;i funiculo <expan abbr="põ">pom</expan><gap/>us accedat medio inter <expan abbr="hypemochliũ">hypemochlium</expan> <lb/>loco, motum accelerabit; non igitur ex &longs;e motum aut <lb/>pondus habet: propterea quòd negant maius pondus <lb/>velocitatem augere. At verò &longs;i pars illa fili, quæ ob pon <lb/>dus acce&longs;&longs;orium velociùs mouetur, &longs;uo <expan abbr="quo&qacute;">quoque</expan>; pondere <lb/>mouebatur, fiet &longs;anè, ut continuà hac ponderis noui ac­<lb/>ce&longs;sione velocitas in infinitum augeatur. Dicendum <lb/>ergò excur&longs;um perpendiculi continuò quidem mino­<lb/>rem fieri recur&longs;u; cau&longs;am verò hujus inæqualitatis non <lb/>in funiculo, &longs;ed in naturà circuli, in quo perpendiculum <lb/>mouetur, &longs;itam e&longs;&longs;e. Quia enim velocitas motus conti­<lb/>nuo fluxu augetur à grauitate, quæ ex inclinatione ma­<lb/>iori ob maiorem <expan abbr="viol&etilde;tiã">violentiam</expan> hypomochlii minùs grauitat, <lb/>impul&longs;us, quo perpendiculum recurrit, continuó qui- <pb/>dem maiora &longs;umit incrementa: quia tamen in quolibet <lb/>puncto circuli per lineas fit tangentes, quæ in recur&longs;u <lb/>continuó magis ac magis &longs;unt inclinatæ; erunt in quo­<lb/>libet puncto recur&longs;us minora huius velocitatis incre­<lb/>menta: ita nimirum ut &longs;i arcus &longs;umantur æquales, ma­<lb/>jor &longs;it acce&longs;sio velocitatis in arcu<gap/> primo, quam in arcu <lb/>&longs;ecundo: & velocitas in arcu circuli collecta minor ve­<lb/>locitate in lineà rectà illi arcui æquali, quæ tangens &longs;it <lb/>principii eiu&longs;dem motus circularis. Sicuti verò in re­<lb/>cur&longs;u velocitas continuó & inæqualiter cre&longs;cit, ita in <lb/>excur&longs;u, quia motus violentus, proportionaliter decre­<lb/>&longs;cit, <expan abbr="fiunt&qacute;">fiuntque</expan>; huius decrementa æqualia illius incremen­<lb/>tis, prima nimirum ultimis; propterea quód <expan abbr="utra&qacute;">utraque</expan>; <expan abbr="fiũt">fiunt</expan> <lb/>ab eadem grauitate, quæ à principio excur&longs;us per lineas <lb/>grauitat magis inclin atas. Quòd &longs;i ergo &longs;ola grauitas <lb/>minuat impul&longs;um, quia in æqualibus à &longs;tatione interual <lb/>lis, ob &longs;imilem inclinationem, æqualiter grauitat; erunt <lb/>ut arcus inter &longs;e, ita eiu&longs;dem grauitatis impul&longs;us: & <lb/>quia impul&longs;us contrarius tollit partem &longs;ibi æqualem, <lb/>crunt excur&longs;us & recur&longs;us inter &longs;e æquales. At verò <lb/>quia non &longs;ola grauitas impul&longs;um minuit, &longs;ed etiam in­<lb/>clinatio motus; &longs;icuti enim grauitas extra lineam per­<lb/>pendicularem minùs grauitat, ita impul&longs;us extra line­<lb/>am &longs;ui motus, cuius terminus e&longs;t veluti centrum, mi <lb/>nús impellit &longs;uum mobile: quód &longs;i enim funda lapidem <pb/>excutiat, ad majus feretur interuallum, quam ut æquale <lb/>&longs;it illis rotationibus &longs;imul &longs;umptis, in quasidem lapis <lb/>fundæ alligatus reuoluitur. Quia ergo in illa gyratione <lb/>perpendiculi inclinatio motus continuò & æqualiter <lb/>mutatur, velocitas in excur&longs;u collecta eò minùs moue­<lb/>bit, quó major portio ex illâ inclinatione eidem dece­<lb/>dit. Impul&longs;us ergo æqualis quia magis decre&longs;cit in ex­<lb/>cur&longs;u, quam idem augeatur in recur&longs;u, ad minus moue­<lb/>bit interuallum<gap/> ac proinde excur&longs;us perpenidculi eju&longs;­<lb/>dem recur&longs;ibus erunt minores. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Excur&longs;us grauium à termino motus in circulum terminatur, cu­<lb/>jus &longs;emidiameter e&longs;t di&longs;tantià inter principium motus & mundi <lb/>centrum.<emph.end type="italics"/></s></p> <p type="main"> <s>ATermino motus <emph type="italics"/>a.i.h.g.f.e<emph.end type="italics"/> in lineà perpendiculari, & <lb/>lineis ad horizontem inclinatis producantur lineæ <lb/>excur&longs;<gap/> æquales lineis decur&longs;us, nimirum <emph type="italics"/>ap<emph.end type="italics"/> ip&longs;i <emph type="italics"/>ab, io<emph.end type="italics"/><lb/>verò ip&longs;i <emph type="italics"/>ib<emph.end type="italics"/> æqualis, dico puncta <emph type="italics"/>po<emph.end type="italics"/> e&longs;&longs;e in peripheria cir­<lb/>culi, cujus &longs;emidiameter <emph type="italics"/>ab<emph.end type="italics"/> di&longs;tantia inter principium <lb/>motus & mundi centrum. Ducatur enim linea <emph type="italics"/>ao:<emph.end type="italics"/> quia <lb/>ergo lineæ <emph type="italics"/>bi. io<emph.end type="italics"/> inter &longs;e funt æquales, & anguli <emph type="italics"/>bia. oia<emph.end type="italics"/><lb/>recti, erit angulus <emph type="italics"/>abi<emph.end type="italics"/> angulo <emph type="italics"/>aoi,<emph.end type="italics"/> & latus <emph type="italics"/>ab<emph.end type="italics"/> lateri <emph type="italics"/>ao<emph.end type="italics"/><lb/>æquale: e&longs;t autem linea <emph type="italics"/>ap<emph.end type="italics"/> æqualis eidem <emph type="italics"/>ab,<emph.end type="italics"/> puncta er- <pb/>go <emph type="italics"/>p<gap/><emph.end type="italics"/> &longs;unt in peripherià circuli, cujus centrum <emph type="italics"/>a,<emph.end type="italics"/> à quo <lb/>æqualiter ab&longs;i&longs;tunt illæ lineæ. Simili modo o&longs;tende­<lb/>mus puncta <emph type="italics"/>n.m.l<emph.end type="italics"/> e&longs;&longs;e in peripheriá eju&longs;dem circuli, pro­<lb/> <arrow.to.target n="fig18"></arrow.to.target><lb/>pterea quód lineæ <emph type="italics"/>an. am. al,<emph.end type="italics"/> bafes nimirum æqualium <lb/>triangulorum, &longs;unt æquales lineæ <emph type="italics"/>ab.<emph.end type="italics"/> Excur&longs;us ergo <lb/>grauium à termino motus in <expan abbr="circulũ">circulum</expan> terminantur &c. </s></p> <figure id="fig18"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motusper arcus eju&longs;dem circuli rationem habet, quam &longs;inus an <lb/>guli <gap/>pli illerum angulorum, qui complementa &longs;unt inclinationis <lb/><gap/>.<emph.end type="italics"/></s></p> <p type="main"> <s>AS&longs;umantur arcus <emph type="italics"/>bdi. bdc,<emph.end type="italics"/> & <expan abbr="ducãtur">ducantur</expan> cordæ <emph type="italics"/>bi. bc,<emph.end type="italics"/><lb/><expan abbr="erunt&qacute;">eruntque</expan>; anguli <emph type="italics"/>abi. abc<emph.end type="italics"/> anguli inclinationis corda- <pb/>rum <emph type="italics"/>bi. bc,<emph.end type="italics"/> & horum complementa <emph type="italics"/>bai. ba<gap/>,<emph.end type="italics"/> propterea <lb/>quód anguli <emph type="italics"/><gap/>ib<gap/> acb<emph.end type="italics"/> in &longs;emicirculo &longs;unt recti. Tan­<lb/>gant ergo circulum in punctis <emph type="italics"/>ic<emph.end type="italics"/> lineæ <emph type="italics"/>ib. cg<gap/><emph.end type="italics"/> & ex cen­<lb/>tro <emph type="italics"/>k<emph.end type="italics"/> educantur lineæ <emph type="italics"/>ki. kc<emph.end type="italics"/> perpendiculares ad <emph type="italics"/>ih.cg.<emph.end type="italics"/><lb/>quia ergo anguli <emph type="italics"/>khi. kg<gap/><emph.end type="italics"/> &longs;unt anguli inclinationum, e­<lb/> <arrow.to.target n="fig19"></arrow.to.target><lb/>runt anguli <emph type="italics"/>bki. gkc<emph.end type="italics"/> illorum complementa <gap/> angulo­<lb/>rum verò <emph type="italics"/>bai. bac<emph.end type="italics"/> ad peripheriam dupli<gap/> dico velocita­<lb/>tem <gap/>otus in <emph type="italics"/>i<emph.end type="italics"/> ad velocitatem motus in <emph type="italics"/>c<emph.end type="italics"/> e&longs;&longs;e ut &longs;inum <lb/>angu <gap/> &longs;inum anguli <emph type="italics"/>bk<gap/><emph.end type="italics"/> Quia enim motus in <lb/>quolibet puncto circuli per lineam fit tangentem per <pb/>prop: 4. erit ratio velocitatis in <emph type="italics"/>i<emph.end type="italics"/> & <emph type="italics"/>c<emph.end type="italics"/> quæ velociras e&longs;t <lb/>tangentium <emph type="italics"/>ih. cg<emph.end type="italics"/>: e&longs;t autem velocitas in <emph type="italics"/>ih<emph.end type="italics"/> ad veloci­<lb/>tatem in <emph type="italics"/>cg<emph.end type="italics"/> ut &longs;inus <emph type="italics"/>bl<emph.end type="italics"/> anguli <emph type="italics"/>bki<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>bm<emph.end type="italics"/> anguli <lb/><emph type="italics"/>bkc<emph.end type="italics"/> per prop: 14. velocitas ergo in arcu <emph type="italics"/>ib<emph.end type="italics"/> ad velocita­<lb/>tem in arcu <emph type="italics"/>cb<emph.end type="italics"/> ut &longs;inus anguli <emph type="italics"/>bki<emph.end type="italics"/> ad &longs;inum anguli <emph type="italics"/>bkc,<emph.end type="italics"/><lb/>&longs;inus nimirum anguli dupli illorum angulorum, qui <lb/>complementa &longs;unt inclinationis cordarum <emph type="italics"/>bi.bc,<emph.end type="italics"/> quod <lb/>erat o&longs;tendendum. </s></p> <figure id="fig19"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Perpendiculum per arcus æquales eju&longs;dem circuli inæquali <lb/>tempore mouetur: majori quidem propè &longs;tationem, minori verò per <lb/>arcus, qui magis ab&longs;unt à &longs;tatione.<emph.end type="italics"/></s></p> <p type="main"> <s>SInt duo arcus <emph type="italics"/>bd.d&longs;<emph.end type="italics"/> inter &longs;e æquales<gap/> <expan abbr="at&qacute;">atque</expan> <emph type="italics"/>bd<emph.end type="italics"/> propior, <lb/><emph type="italics"/>df<emph.end type="italics"/> verò remotior à &longs;tatione <emph type="italics"/>b,<emph.end type="italics"/> dico motum in <emph type="italics"/>df<emph.end type="italics"/> e&longs;&longs;e <lb/>velociorem motu in <emph type="italics"/>db.<emph.end type="italics"/> Quia enim motus perarcus e­<lb/>ju&longs;dem circuli rationem habent, quam &longs;inus, per prop. <lb/>22. e&longs;t autem &longs;inus <emph type="italics"/>bg<emph.end type="italics"/> major &longs;inu <emph type="italics"/>bt,<emph.end type="italics"/> erit velocior motus <lb/>in <emph type="italics"/>f<emph.end type="italics"/> quam in d: & quia arcus <emph type="italics"/>bd.df<emph.end type="italics"/> &longs;unt æquales, minori <lb/>tempore mouebitur in arcu <emph type="italics"/>df<emph.end type="italics"/> remotiore, quam in ar­<lb/>cu <emph type="italics"/>bd<emph.end type="italics"/> &longs;tationi propiore per prop. 6. Dices velocitas mo­<lb/>tus ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>d<emph.end type="italics"/> augetur inæqualiter, <expan abbr="fiunt&qacute;">fiuntque</expan>; ad &longs;ingula pun­<lb/>cta minora incrementa; mutatà ergo velocitate non ea- <pb/>dem erit ratio motus. Re&longs;pondeo velocitatem ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>d<emph.end type="italics"/><lb/>inæqualiter quidem augeri, & continuó minora fieri in­<lb/>crementa, per prop: 20. at verò velocitatem ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>d<emph.end type="italics"/> col­<lb/> <arrow.to.target n="fig20"></arrow.to.target><lb/>lectam e&longs;&longs;e majorem velocitate ex <emph type="italics"/>d<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> collectà. Quia <lb/>enim velocitatis ex <emph type="italics"/>d<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> continuò <expan abbr="quo&qacute;">quoque</expan>; minora fiunt <lb/>incrementa; velocitas inde collecta erit minor veloci <lb/>tate ab æqualibus ip&longs;i <emph type="italics"/>d<emph.end type="italics"/> incrementis collectá: at verò <lb/>velocitas in <emph type="italics"/>f<emph.end type="italics"/> majora ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>d<emph.end type="italics"/> &longs;umit incrementa, quam <lb/>ut æqualia &longs;int velocitati in <emph type="italics"/>d:<emph.end type="italics"/> velocitas ergo ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>d<emph.end type="italics"/> col­<lb/>lecta e&longs;t multó major velocitate ex <emph type="italics"/>d<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> co<gap/>lecta, ac pro <lb/>inde minori tempore illos arcus perambulat æquales. </s></p> <pb/> <figure id="fig20"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lemma I.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si a&longs;&longs;umantur arcus in ratione continuà, quam habent &longs;inus <lb/>intercipientes illos arcus, major erit proportio inter arcus po&longs;terio­<lb/>res, quam inter arcus priores.<emph.end type="italics"/></s></p> <p type="main"> <s>SIt arcus <emph type="italics"/>bd,<emph.end type="italics"/> &longs;inu <emph type="italics"/>ab<emph.end type="italics"/> & <emph type="italics"/>cd<emph.end type="italics"/> interceptus, in eadem ratio­<lb/>ne ad arcum <emph type="italics"/>df<emph.end type="italics"/> &longs;inu <emph type="italics"/>cd<emph.end type="italics"/> & <emph type="italics"/>ef<emph.end type="italics"/> interceptum, in quà &longs;i­<lb/>&longs;inus <emph type="italics"/>ab<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>cd:<emph.end type="italics"/> & rur&longs;um arcus <emph type="italics"/>df<emph.end type="italics"/> àd arcum <emph type="italics"/>fh,<emph.end type="italics"/> ut <lb/>&longs;inus <emph type="italics"/>cd<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>ef<emph.end type="italics"/>; dico proportionem tam inter &longs;i­<lb/>nus, quam inter arcus illis &longs;inubus interceptos conti­<lb/>nuò fieri majores, nimirum proportionem &longs;inus <emph type="italics"/>cd<emph.end type="italics"/> ad <lb/>&longs;inum <emph type="italics"/>ef,<emph.end type="italics"/> & arcus <emph type="italics"/>df<emph.end type="italics"/> ad arcum <emph type="italics"/>fh<emph.end type="italics"/> e&longs;&longs;e majorem, quam <lb/>&longs;inus <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>cd,<emph.end type="italics"/> aut arcus <emph type="italics"/>bd<emph.end type="italics"/> ad <emph type="italics"/>df.<emph.end type="italics"/> A&longs;&longs;umatur enim ar­<lb/>cus <emph type="italics"/>bd<emph.end type="italics"/> grad: 9. <expan abbr="erit&qacute;">eritque</expan>; <emph type="italics"/>ab<emph.end type="italics"/> 100000. &longs;inus totus, <emph type="italics"/>cd<emph.end type="italics"/> autem <lb/><emph type="italics"/>98769.<emph.end type="italics"/> &longs;inus grad. <emph type="italics"/>81.<emph.end type="italics"/> quòd &longs;i ergo fiat ut <emph type="italics"/>ab<emph.end type="italics"/> &longs;inus totus ad <lb/>9, ita &longs;inus grad. <emph type="italics"/>81.<emph.end type="italics"/> ad aliud, prodibit arcus 8 in datâ ra­<lb/>tione, quam habet &longs;inus <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>cd,<emph.end type="italics"/> &longs;i minutias omittamus. <lb/>Simili modo &longs;i fiat ut &longs;inus <emph type="italics"/>cd<emph.end type="italics"/> grad: <emph type="italics"/>81<emph.end type="italics"/> ad arcum <emph type="italics"/>df<emph.end type="italics"/> grad. <lb/>8, ita &longs;inus <emph type="italics"/>ef<emph.end type="italics"/> grad.73 ad aliud, prodibit arcus <emph type="italics"/>fh<emph.end type="italics"/> grad. 7. <lb/><expan abbr="at&qacute;">atque</expan>; ita con&longs;equenter inuenientur arcus reliqui, quos di <lb/>co majorem rationem habere ad arcus proximè &longs;equen­<lb/>tes, quam ad hos habeant arcus proximè antecedentes. <lb/>E&longs;t enim major proportio grad. 8 ad 7, quam grad. 9 ad <lb/>8: & grad. 4 ad 3, quam grad. 5 ad 4. <expan abbr="at&qacute;">atque</expan>, eadem e&longs;t ratio <pb/>in arcubus reliquis. Si ergo a&longs;&longs;umantur arcus<gap/>in ratio­<lb/>ne continuâ, quam habent &longs;inus intercipientes illos at­<lb/> <arrow.to.target n="fig21"></arrow.to.target><lb/>cus, major e&longs;t proportio inter arcus po&longs;teriores, quam <lb/>inter arcus priores. </s></p> <figure id="fig21"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lemma II.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si quadrans circuli diuidatur in quot libet arcus æquales, mino­<lb/>res verò quam in ratione &longs;ubtriplá æd &longs;inum totum, habebunt &longs;inus <lb/>proximi intercipientes illos arcus minorem rationem quam duplam.<emph.end type="italics"/></s></p> <p type="main"> <s>IN fig: 6. Diuidatur quadrans circuli bifariam in <emph type="italics"/>h<emph.end type="italics"/> in <lb/>arcum <emph type="italics"/>bh<emph.end type="italics"/> gra: 60, & arcum <emph type="italics"/>ho<emph.end type="italics"/> grad: 30, <expan abbr="erit&qacute;">eritque</expan>, arcus <emph type="italics"/>bh<emph.end type="italics"/><lb/>maior &longs;inu toto: propterea quòd quadrans majo­<lb/>rem ad hunc, quam ad arcum grad. 60 habeat <expan abbr="ration&etilde;">rationem</expan>. <lb/>Quòd &longs;i ergo arcus <emph type="italics"/>bh<emph.end type="italics"/> &longs;ubdiuidatur in alios tres arcus <pb/><emph type="italics"/>bd. df.fh<emph.end type="italics"/> inter &longs;e æquales, minor e<gap/>t proportio &longs;inus <emph type="italics"/>ab<emph.end type="italics"/><lb/>ad arcum <emph type="italics"/>bd<emph.end type="italics"/> quam trip la, habebit ergo ad arcum mino­<lb/>rem, quam &longs;it <emph type="italics"/>bd,<emph.end type="italics"/> rationem triplam, qui &longs;it <emph type="italics"/>bq,<emph.end type="italics"/> <expan abbr="at&qacute;">atque</expan>; hunc <lb/><gap/>ercipiens &longs;inus <emph type="italics"/>pq<emph.end type="italics"/> maior &longs;inu <emph type="italics"/>cd:<emph.end type="italics"/> dico &longs;inus proximos <lb/>intercipientes illos arcus, nimirum <emph type="italics"/>ab<emph.end type="italics"/> & <emph type="italics"/>cd,<emph.end type="italics"/> aut <emph type="italics"/>cd<emph.end type="italics"/> & <emph type="italics"/>ef.<emph.end type="italics"/><lb/>aut <emph type="italics"/>ef<emph.end type="italics"/> & <emph type="italics"/>gh<emph.end type="italics"/> minorem rationem habere quam duplam. <lb/>Erit enim &longs;inus <emph type="italics"/>cd.<emph.end type="italics"/> grad: 70, & &longs;inus <emph type="italics"/>ef<emph.end type="italics"/> grad: 50. & &longs;inus <emph type="italics"/>gh<emph.end type="italics"/><lb/>gtad: 30. at verò &longs;inus totus <emph type="italics"/>ab<emph.end type="italics"/> 100000. ad &longs;inum <emph type="italics"/>cd<emph.end type="italics"/> grad. <lb/>70, nimirum ad 93969, & &longs;inus <emph type="italics"/>ef<emph.end type="italics"/> grad: 50 ad &longs;inum <emph type="italics"/>gh<emph.end type="italics"/><lb/>grad. 30 ide&longs;t. 76604. ad 50000 minorem habet <expan abbr="ration&etilde;">rationem</expan> <lb/>quam duplam. Quodidem de aliis &longs;inubus proximè in­<lb/><gap/>ercipientibus illos arcus æquales, ex tabulis &longs;inuum <lb/>con&longs;tabit. Quia verò &longs;inus propiores minorem ha­<lb/>bent rationem, erit minor proportio <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>pq<emph.end type="italics"/> quam ad <lb/><emph type="italics"/>cd.<emph.end type="italics"/> ac proinde minor quam dupla. <gap/>i ergo quadrans cir­<lb/>culi diuidatur in quotlibet arcus æquales, minores verò <lb/>quam in ratione &longs;ubtriplá ad &longs;inum totum, habebunt &longs;i­<lb/>nus proximi intercipientes illos arcus minorem ratio­<lb/>nem quam duplam. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lemma III.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si a&longs;&longs;umantur arcus in ratione continuá, quam habent &longs;inus <lb/>intercipientes illos arcus, <expan abbr="habeat&qacute;">habeatque</expan>; &longs;inus primus ad arcum interce­<lb/>ptum majorem rationem quam triplam, habebunt &longs;inus proximi ra <lb/>tionem ad &longs;e minorem quam duplam.<emph.end type="italics"/></s></p> <pb/> <p type="main"> <s>VT &longs;i arcus <emph type="italics"/>bd<emph.end type="italics"/> ad arcum <emph type="italics"/>df<emph.end type="italics"/> &longs;it ut &longs;inus <emph type="italics"/>ab<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>cd:<emph.end type="italics"/><lb/>& rur&longs;um ut &longs;inus <emph type="italics"/>cd<emph.end type="italics"/> ad <emph type="italics"/>ef,<emph.end type="italics"/> ita arcus <emph type="italics"/>df<emph.end type="italics"/> ad <emph type="italics"/>fh,<emph.end type="italics"/> habeat <lb/>verò &longs;inus <emph type="italics"/>ab<emph.end type="italics"/> ad arcum <emph type="italics"/>bd<emph.end type="italics"/> majorem rationem quam tri <lb/>plam, dico &longs;inus intercipientes illos arcus rationem ad <lb/>&longs;e habere minorem quam duplam. Quia enim &longs;inus <emph type="italics"/>ab<emph.end type="italics"/><lb/> <arrow.to.target n="fig22"></arrow.to.target><lb/>e&longs;t major &longs;inu <emph type="italics"/>cd<emph.end type="italics"/> erit <expan abbr="quo&qacute;">quoque</expan>; arcus <emph type="italics"/>bd<emph.end type="italics"/> major arcu <emph type="italics"/>df<emph.end type="italics"/>: fiat <lb/>ergo arcus <emph type="italics"/>bd<emph.end type="italics"/> æqualis arcui <emph type="italics"/>ds,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; &longs;inus <emph type="italics"/>rs<emph.end type="italics"/> minor &longs;inu <lb/><emph type="italics"/>cd<emph.end type="italics"/>: e&longs;t autem per Lemma 2. minor proportio eju&longs;dem <lb/>&longs;inus <emph type="italics"/>cd<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>rs<emph.end type="italics"/> quam dupla; multò ergo minor ad <lb/>&longs;inum majorem <emph type="italics"/>ef<emph.end type="italics"/> quam dupla. Quod idem de aliis &longs;i­<lb/>nubus o&longs;tendemus. Si ergo a&longs;&longs;umantur arcus in ratio­<lb/>ne continuà &c. </s></p> <figure id="fig22"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Lemma IV.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si a&longs;&longs;umantur arcus in ratione continuà, quam habent &longs;inus in<emph.end type="italics"/> <pb/><emph type="italics"/>tercipientes illos arcus, <expan abbr="habeat&qacute;">habeatque</expan>; &longs;inus primus ad arcum interce­<lb/>ptum majorem rationem quam triplam, erit &longs;inus &longs;ecundus major <lb/>illo arcu intercepto.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim ut &longs;inus ita arcus intercepti; habent <expan abbr="aut&etilde;">autem</expan> <lb/>&longs;inus proximi rationem ad &longs;e minorem quam du­<lb/>plam, per Lemma 3; habebunt <expan abbr="quo&qacute;">quoque</expan>; arcus minorem <lb/>rationem quam duplam. Et quia ut &longs;inus ad &longs;inum, ita <lb/>arcus ad arcum, erit permutando ut &longs;inus primus ad ar­<lb/>cum primum, ita &longs;inus &longs;ecundus ad arcum &longs;ecundum: <lb/>habet autem &longs;inus primus ad arcum primum majorem <lb/>rationem quam triplam, habebit <expan abbr="quo&qacute;">quoque</expan>; &longs;inus &longs;ecundus <lb/>ad arcum &longs;ecundum majorem rationem quam triplam. <lb/>Quia ergo ad eundem arcum &longs;ecundum majorem rati­<lb/>onem habet &longs;inus &longs;ecundus, quam arcus primus, erit &longs;i­<lb/>nus &longs;ecundus major quam arcus primus, hoc e&longs;t quam <lb/>arcus interceptus. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Perpendiculum ex quolibet puncto eju&longs;dem circuli æquali tem­<lb/>pore recurrit in &longs;uam &longs;tationem.<emph.end type="italics"/></s></p> <p type="main"> <s>IN circulo <emph type="italics"/>tuxb<emph.end type="italics"/> &longs;int duo perpendicula <emph type="italics"/>ab. ad<emph.end type="italics"/> extra &longs;u <lb/>am &longs;tationem <emph type="italics"/>at,<emph.end type="italics"/> <expan abbr="habeat&qacute;">habeatque</expan>; &longs;inus totus <emph type="italics"/>ab<emph.end type="italics"/> ad interual­<lb/>lum <emph type="italics"/>bd<emph.end type="italics"/> majorem rationem quam triplá, dico <expan abbr="utrum&qacute;">utrumque</expan>; <pb/><expan abbr="cod&etilde;">codem</expan> tempore recurrere in <emph type="italics"/>t.<emph.end type="italics"/> Erit enim velocitas in <emph type="italics"/>b<emph.end type="italics"/> ad <lb/>velocitatem in <emph type="italics"/>d,<emph.end type="italics"/> ut &longs;inus <emph type="italics"/>ab<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>cd<emph.end type="italics"/> per prop: 22. <lb/>quòd &longs;i ergo in illo recur&longs;u eadem ratio velocitatis con­<lb/>&longs;taret, aut &longs;imilibus augeretur incrementis, quia major <lb/>proportio arcus <emph type="italics"/>bt<emph.end type="italics"/> ad arcum <emph type="italics"/>dt,<emph.end type="italics"/> quam &longs;inus <emph type="italics"/>ab<emph.end type="italics"/> ad &longs;inum <lb/><emph type="italics"/>cd,<emph.end type="italics"/> quo quidem tempore perpendiculum <emph type="italics"/>ab<emph.end type="italics"/> recurrit <lb/>in <emph type="italics"/>t,<emph.end type="italics"/> eodem perpendiculum <emph type="italics"/>cd<emph.end type="italics"/> procurreret extra <emph type="italics"/>t,<emph.end type="italics"/> tanto <lb/>interuallo, quantus e&longs;t exce&longs;&longs;us hujus proportioni<emph type="italics"/>s.<emph.end type="italics"/> At <lb/>verò quia ad &longs;ingula puncta mutatà &longs;inuum ratione, <lb/>mutatur <expan abbr="quo&qacute;">quoque</expan>; ratio velocitatis: major enim proportio <lb/><emph type="italics"/>cd<emph.end type="italics"/> ad <emph type="italics"/>ef,<emph.end type="italics"/> quam <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>cd<emph.end type="italics"/> per lem: 1. erit <expan abbr="quo&qacute;">quoque</expan>; major pro­<lb/>portio arcus <emph type="italics"/>df<emph.end type="italics"/> ad <emph type="italics"/>fh,<emph.end type="italics"/> quam arcus <emph type="italics"/>bd<emph.end type="italics"/> ad <emph type="italics"/>df.<emph.end type="italics"/> quia ergo <lb/>cum hoc &longs;inuum & arcuum decremento continuó au­<lb/>getur illorum proportio, minuitur verò di&longs;tantia ter­<lb/>minorum motus, nece&longs;&longs;e demum ab&longs;umi & deficere, <expan abbr="il-lo&qacute;">il­<lb/>loque</expan>; deficiente <expan abbr="motũ">motum</expan> coæquari: quod non ni&longs;i in pun­<lb/>cto <emph type="italics"/>t<emph.end type="italics"/> dico po&longs;&longs;e fieri. Concurrat enim, &longs;i fieri pote&longs;t, <lb/><expan abbr="utrum&qacute;">utrumque</expan>; perpendiculum in <emph type="italics"/>q<emph.end type="italics"/> minori, quam <emph type="italics"/>t,<emph.end type="italics"/> interuallo: <lb/>& quia non ante <emph type="italics"/>q<emph.end type="italics"/> fit concur&longs;us, &longs;i perpendiculum <emph type="italics"/>ab<emph.end type="italics"/><lb/>&longs;tatuatur in <emph type="italics"/>m<emph.end type="italics"/>; erit perpendiculum <emph type="italics"/>ad<emph.end type="italics"/> inter <emph type="italics"/>m<emph.end type="italics"/> & <emph type="italics"/>q<emph.end type="italics"/>: &longs;it er­<lb/>go in <emph type="italics"/>o.<emph.end type="italics"/> quia verò ut <emph type="italics"/>lm<emph.end type="italics"/> ad <emph type="italics"/>no,<emph.end type="italics"/> ita velocitas motus in <emph type="italics"/>m<emph.end type="italics"/><lb/>ad velocitatem motus in <emph type="italics"/>o<emph.end type="italics"/>: aut arcus <emph type="italics"/>mo<emph.end type="italics"/> ad arcum <emph type="italics"/>oq<emph.end type="italics"/><lb/>eandem habet rationem, quam &longs;inus <emph type="italics"/>lm<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>no,<emph.end type="italics"/><lb/>aut non eandem, &longs;ed vel maiorem vel minorem: habe­<lb/>at primúm eandem rationem. Dum ergo perpendicu- <pb/>lum <emph type="italics"/>ad<emph.end type="italics"/> mouetur ex <emph type="italics"/>o<emph.end type="italics"/> in <emph type="italics"/>q,<emph.end type="italics"/> perpendiculum <emph type="italics"/>ab<emph.end type="italics"/> ex <emph type="italics"/>m<emph.end type="italics"/> in <emph type="italics"/><gap/><emph.end type="italics"/><lb/>promouebitur: non igitur concur&longs;us fit in <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> Simili mo <lb/>do &longs;i <emph type="italics"/>mo<emph.end type="italics"/> ad <emph type="italics"/>oq<emph.end type="italics"/> majorem habeat rationem, perpendicu­<lb/> <arrow.to.target n="fig23"></arrow.to.target><lb/>lum <emph type="italics"/>ad<emph.end type="italics"/> ex <emph type="italics"/>o<emph.end type="italics"/> majori quam <emph type="italics"/>oq<emph.end type="italics"/> interuallo abducetur. Si <lb/>demum minorem habeat rationem, auferatur pars pro­<lb/>portionalis, <expan abbr="at&qacute;">atque</expan>; rur&longs;um alia, <expan abbr="quou&longs;&qacute;">quou&longs;que</expan>; in <emph type="italics"/>q<emph.end type="italics"/> &longs;it æqualis aut <lb/>minor: & tum rur&longs;um o&longs;tendemus perpendiculum <emph type="italics"/>ad<emph.end type="italics"/><lb/>præcurrere: non igitur concur&longs;us in minori quam <emph type="italics"/>t<emph.end type="italics"/> in­<lb/>teruallo e&longs;&longs;e pote&longs;t. Quód &longs;i autem <emph type="italics"/>ad<emph.end type="italics"/> dicatur præcur­<lb/>rere in <emph type="italics"/>t,<emph.end type="italics"/> erit <emph type="italics"/>ab<emph.end type="italics"/> in aliquo puncto minús remoto, verbi <lb/>gratia<emph type="italics"/>s<emph.end type="italics"/>: igitur cùm <emph type="italics"/>ab<emph.end type="italics"/> ferebatur in <emph type="italics"/>q, ad<emph.end type="italics"/> necdum atti­<lb/>git <emph type="italics"/>t<emph.end type="italics"/>: erit ergo in aliquo puncto inter <emph type="italics"/>t<emph.end type="italics"/> & <emph type="italics"/>q,<emph.end type="italics"/> quod &longs;it <emph type="italics"/>s.<emph.end type="italics"/> Et <lb/>quia ut &longs;inus <emph type="italics"/>pq<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>rs,<emph.end type="italics"/> ita motus in <emph type="italics"/>q<emph.end type="italics"/> ad motum in <lb/><emph type="italics"/>s<emph.end type="italics"/>: e&longs;t autem &longs;inus <emph type="italics"/>pq<emph.end type="italics"/> major quam <emph type="italics"/>rs,<emph.end type="italics"/> erit arcus proporti- <pb/>onalis minor qua <emph type="italics"/>qs:<emph.end type="italics"/> quia verò &longs;inus <emph type="italics"/>rs<emph.end type="italics"/> e&longs;t maior arcu <emph type="italics"/>sq<emph.end type="italics"/><lb/>per Lemma 4. minor autem arcu <emph type="italics"/>ts,<emph.end type="italics"/> erit arcus <emph type="italics"/>ts<emph.end type="italics"/> multò <lb/>major arcu proportionali: po&longs;ito ergo perpendiculo <emph type="italics"/>ab<emph.end type="italics"/><lb/>in <emph type="italics"/>s,<emph.end type="italics"/> perpendiculum <emph type="italics"/>ad<emph.end type="italics"/> necdum e&longs;&longs;e pote&longs;t in <emph type="italics"/>t.<emph.end type="italics"/> Quod <lb/>idem de quouis alio puncto o&longs;tendemus. Quia ergo <lb/>perpendiculum <expan abbr="ne&qacute;">neque</expan>; propiùs concurrere, <expan abbr="ne&qacute;">neque</expan>; præcur­<lb/>rere pote&longs;t, concurret nece&longs;&longs;ariò in <emph type="italics"/>t.<emph.end type="italics"/> Poterit eadem ra­<lb/>tio in hunc modum fieri: motus &longs;e habent ut &longs;inus <expan abbr="at&qacute;">atque</expan>; <lb/>horum interualla, &longs;eu arcus &longs;inubus intercepti: hæc au­<lb/>tem interualla continuò fiunt minora, in puncto verò <lb/><emph type="italics"/>t<emph.end type="italics"/> nulla: igitur & motus continuó minori, in puncto ve­<lb/>rò <emph type="italics"/>t<emph.end type="italics"/> nullo <expan abbr="ab&longs;i&longs;tũt">ab&longs;i&longs;tunt</expan> interuallo, Quòd &longs;i a&longs;&longs;umantur plura <lb/>puncta <emph type="italics"/>b.d. f.h.k.m.<emph.end type="italics"/> &c. eadem vià o&longs;tendemus ex omni­<lb/>bus &longs;imul recurrere in <emph type="italics"/>t<emph.end type="italics"/>: &longs;icuti enim ex <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>d,<emph.end type="italics"/> ita ex <emph type="italics"/>d<emph.end type="italics"/> & <emph type="italics"/>f,<emph.end type="italics"/><lb/>& ex <emph type="italics"/>f<emph.end type="italics"/> & <emph type="italics"/>b, et<emph.end type="italics"/> ex <emph type="italics"/>h<emph.end type="italics"/> & <emph type="italics"/>k<emph.end type="italics"/> &c. æqualis fit recur&longs;us. Perpen­<lb/>diculum ergo ex <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>d<emph.end type="italics"/> æqualiter recurrens recurret <lb/><expan abbr="quo&qacute;">quoque</expan>; æqualiter ex <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>f<emph.end type="italics"/> & <emph type="italics"/>h<emph.end type="italics"/> & <emph type="italics"/>k<emph.end type="italics"/> &c. </s></p> <figure id="fig23"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Exeur&longs;us perpendiculi in eodem circulo à lineà &longs;iationis &longs;unt in­<lb/>ter &longs;e æqualis.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia (in fig: 8.) velocitas in <emph type="italics"/>eb<emph.end type="italics"/> velocitati in <emph type="italics"/>fb,<emph.end type="italics"/> & <lb/>velocitas in <emph type="italics"/>cb<emph.end type="italics"/> e&longs;t æqualis velocitati in <emph type="italics"/>db<emph.end type="italics"/> per prop. <lb/>20. e&longs;t <expan abbr="aut&etilde;">autem</expan> velocitas in <emph type="italics"/>eb<emph.end type="italics"/> ad <expan abbr="velocitat&etilde;">velocitatem</expan> in <emph type="italics"/>cb,<emph.end type="italics"/> ut arcus <emph type="italics"/>e<emph.end type="italics"/> <pb/><emph type="italics"/>b<emph.end type="italics"/> ad arcum <emph type="italics"/>cb:<emph.end type="italics"/> propterea quòd perpendiculum ex <emph type="italics"/>c<emph.end type="italics"/> & <emph type="italics"/>e<emph.end type="italics"/><lb/>æquali tempore recurrit in <emph type="italics"/>b<emph.end type="italics"/> per prop: 24. erit ut arcus <lb/><emph type="italics"/>fb<emph.end type="italics"/> ad arcum <emph type="italics"/>db,<emph.end type="italics"/> ita velocitas excur&longs;us in <emph type="italics"/>fb<emph.end type="italics"/> ad velocita­<lb/>tem excur&longs;us in <emph type="italics"/>db.<emph.end type="italics"/> At verò ut idem arcus <emph type="italics"/>fb<emph.end type="italics"/> ad arcum <lb/><emph type="italics"/>db,<emph.end type="italics"/> ita violentia inclinationum in illis arcubus collecta: <lb/>tollit autem violentia partem impul&longs;us &longs;ibi æqualem <lb/>per pofit: 2. igitur ut arcus <emph type="italics"/>fb<emph.end type="italics"/> ad arcum <emph type="italics"/>db,<emph.end type="italics"/> ita ablatum <lb/>ad ablatum, hoc e&longs;t velocitatis decrementum, & velo­<lb/>citas reliqua ad reliquam velocitatem habet autem ve­<lb/>locitas motus eandem rationem, quam interualla. Quia <lb/>ergo excur&longs;us eandem rationem habent tum ad &longs;e, tum <lb/>ad interualla, quam habent recur&longs;us ad &longs;e, & &longs;ua inter­<lb/>ualla; fiunt autem recur&longs;us eodem vel æquali tempo­<lb/>re, erunt <expan abbr="quo&qacute;">quoque</expan>; excur&longs;us eodem vel æquali tempore, ac <lb/>proinde inter &longs;e æquales. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus per arcus &longs;imiles inæqualium circulorum rationem ha­<lb/>bent quam &longs;inus illorum arcuum.<emph.end type="italics"/></s></p> <p type="main"> <s>AS&longs;umantur duo arcus, in circulo quidem maiori <emph type="italics"/>bd. <lb/>bf,<emph.end type="italics"/> in circulo autem minori <emph type="italics"/>ce.cg<emph.end type="italics"/> inter &longs;e &longs;imiles: di<gap/><lb/>co motum perpendiculi ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> ad motum ex <emph type="italics"/>g<emph.end type="italics"/> in <emph type="italics"/>c,<emph.end type="italics"/> & <lb/>motum ex <emph type="italics"/>d<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> ad motum ex <emph type="italics"/>e<emph.end type="italics"/> in <emph type="italics"/>c<emph.end type="italics"/> eandem rationem <lb/>habere quam &longs;inus illorum arcuum. Tangant enim <pb/><expan abbr="utrum&qacute;">utrumque</expan>; circulum in punctis <emph type="italics"/>f.d.g.e<emph.end type="italics"/> lineæ <emph type="italics"/>fk. di,<emph.end type="italics"/> & <emph type="italics"/>gb eh<emph.end type="italics"/>: <lb/><expan abbr="erit&qacute;">eritque</expan>; angulus <emph type="italics"/>akf<emph.end type="italics"/> angulo <emph type="italics"/>abg,<emph.end type="italics"/> & angulus <emph type="italics"/>aid<emph.end type="italics"/> angulo <emph type="italics"/>ab <lb/>e<emph.end type="italics"/> æqualis: propterea quód anguli <emph type="italics"/>afk. agb,<emph.end type="italics"/> & anguli <emph type="italics"/>ad <lb/><gap/> aeh<emph.end type="italics"/> &longs;int recti, anguli verò <emph type="italics"/>kaf.iad<emph.end type="italics"/> communes: velo­<lb/>citas ergo in <emph type="italics"/>f<emph.end type="italics"/> velocitati in <emph type="italics"/>g,<emph.end type="italics"/> & velocitas in <emph type="italics"/>d<emph.end type="italics"/> velocitat: <lb/> <arrow.to.target n="fig24"></arrow.to.target><lb/>in <emph type="italics"/>e<emph.end type="italics"/> e&longs;t æqualis: igitur ut <emph type="italics"/>f<emph.end type="italics"/> ad <emph type="italics"/>d,<emph.end type="italics"/> ita <emph type="italics"/>g<emph.end type="italics"/> ad <emph type="italics"/>e<emph.end type="italics"/>: &longs;ed ut <emph type="italics"/>f<emph.end type="italics"/> ad <emph type="italics"/>d,<emph.end type="italics"/> ita <lb/>&longs;inus arcu<emph type="italics"/>s fb<emph.end type="italics"/> ad &longs;inum arcus <emph type="italics"/>db<emph.end type="italics"/>; & ut <emph type="italics"/>g<emph.end type="italics"/> ad <emph type="italics"/>e<emph.end type="italics"/> ita &longs;inus ar­<lb/>cus <emph type="italics"/>gc<emph.end type="italics"/> ad &longs;inum arcus <emph type="italics"/>ec<emph.end type="italics"/> per prop. 22. erit ergo permu­<lb/>tando motus in <emph type="italics"/>f<emph.end type="italics"/> ad motum in <emph type="italics"/>g,<emph.end type="italics"/> ut &longs;inus arcus <emph type="italics"/>fb<emph.end type="italics"/> ad &longs;i­<lb/>num arcus <emph type="italics"/>ge<emph.end type="italics"/>; & motus in <emph type="italics"/>d<emph.end type="italics"/> ad motum in <emph type="italics"/>e,<emph.end type="italics"/> ut &longs;inus ar- <pb/>cus <emph type="italics"/>db<emph.end type="italics"/> ad &longs;inum arcus <emph type="italics"/>ec.<emph.end type="italics"/> Motus ergo per arcus &longs;imiles <lb/>inæqualium circulorum rationem habent quam &longs;inus <lb/>illorum arcuum, </s></p> <figure id="fig24"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus in circulo minori e&longs;t velocior m<gap/>tuin circulo majori.<emph.end type="italics"/></s></p> <p type="main"> <s>IN circulo maiori <emph type="italics"/>dfb<emph.end type="italics"/> perpendiculum ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>b,<emph.end type="italics"/> in cir­<lb/>culo verò minori <emph type="italics"/>mgc<emph.end type="italics"/> ex <emph type="italics"/>g<emph.end type="italics"/> in <emph type="italics"/>c<emph.end type="italics"/> moueatur: dico velo­<lb/>ciùs ex <emph type="italics"/>g<emph.end type="italics"/> in <emph type="italics"/>c,<emph.end type="italics"/> quam ex <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> recurrere. Quia enim mo­<lb/>tus in <emph type="italics"/>fb<emph.end type="italics"/> ad motum in <emph type="italics"/>gc,<emph.end type="italics"/> ut &longs;inus <emph type="italics"/>bg<emph.end type="italics"/> ad <expan abbr="&longs;inũ">&longs;inum</expan> <emph type="italics"/>cu<emph.end type="italics"/> per prop: <lb/>25. & ut <emph type="italics"/>bg<emph.end type="italics"/> ad <emph type="italics"/>cu,<emph.end type="italics"/> ita <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>ac,<emph.end type="italics"/> propterea quód lineæ <emph type="italics"/>bg cu<emph.end type="italics"/><lb/>&longs;int parallelæ, & triangula <emph type="italics"/>bag. cau<emph.end type="italics"/> fimilia: e&longs;t autem <lb/>maior linea <emph type="italics"/>ab<emph.end type="italics"/> quam <emph type="italics"/>ac,<emph.end type="italics"/> erit <expan abbr="quo&qacute;">quoque</expan>; <emph type="italics"/>bg<emph.end type="italics"/> maior quam <emph type="italics"/>cu<emph.end type="italics"/><lb/>maior ergo motus ab eadem velocitate in <emph type="italics"/>bg,<emph.end type="italics"/> hoc e&longs;t in <lb/><emph type="italics"/>fb<emph.end type="italics"/> maiori, quam in <emph type="italics"/>cu,<emph.end type="italics"/> hoc e&longs;t in <emph type="italics"/>ge,<emph.end type="italics"/> minori interuallo <lb/>per prop: 5. ac proinde in circulo minori e&longs;t velocior <lb/>motus, hoc e&longs;t minori fit tempore, quam in circulo ma­<lb/>jori. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus circulorum &longs;unt in ratione &longs;uorum temporum, quam ha­<lb/>bent diametri ad &longs;e duplicatam.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim ut &longs;inus <emph type="italics"/>bg<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>cu,<emph.end type="italics"/> ita motus in <emph type="italics"/>fb<emph.end type="italics"/><lb/>ad motum in <emph type="italics"/>gc<emph.end type="italics"/> per prop. 25. e&longs;t autem ut <emph type="italics"/>bg<emph.end type="italics"/> ad <emph type="italics"/>c<gap/><emph.end type="italics"/> <pb/>ita motus in <emph type="italics"/>ab<emph.end type="italics"/> ad motum in <emph type="italics"/>ac<emph.end type="italics"/> propterea quòd motus <lb/><emph type="italics"/>ab<emph.end type="italics"/> motui <emph type="italics"/>bg,<emph.end type="italics"/> & motus <emph type="italics"/>ac<emph.end type="italics"/> motui <emph type="italics"/>cu<emph.end type="italics"/> e&longs;t æqualis per prop: <lb/>13. erit motus in <emph type="italics"/>fb<emph.end type="italics"/> ad motum in <emph type="italics"/>gc,<emph.end type="italics"/> ut motus in <emph type="italics"/>ab<emph.end type="italics"/> ad <lb/>motum in <emph type="italics"/>ac.<emph.end type="italics"/> At verò motus in <emph type="italics"/>ab<emph.end type="italics"/> ad motum in <emph type="italics"/>ac,<emph.end type="italics"/> & <lb/> <arrow.to.target n="fig25"></arrow.to.target><lb/>huius duplum <emph type="italics"/>lb<emph.end type="italics"/> ad <emph type="italics"/>mc<emph.end type="italics"/> rationem habent quam tempo­<lb/>rum quadrata per prop: 12. radices ergo quadratæ li<gap/>­<lb/>arum <emph type="italics"/>bl. cm<emph.end type="italics"/> eandem rationem habent quam tempora <lb/>motus circulorum, ac proinde illorum temporum rati­<lb/>onem habent diametri ad &longs;e duplicatam. </s></p> <pb/> <figure id="fig25"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Fieri pote&longs;t ut arcum circuli majoris minori tempore tran&longs;eat, <lb/>quam arcum circuli minoris.<emph.end type="italics"/></s></p> <p type="main"> <s>AS&longs;umatur in fig: 10. &longs;inus <emph type="italics"/>ou<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>qm<emph.end type="italics"/> in eà rati­<lb/>one, in quà diameter major <emph type="italics"/>ab<emph.end type="italics"/> ad minorem <emph type="italics"/>om,<emph.end type="italics"/> <expan abbr="e-tit&qacute;">e­<lb/>titque</expan>; velocitas in <emph type="italics"/>o<emph.end type="italics"/> ad velocitatem in <emph type="italics"/>q,<emph.end type="italics"/> ut <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>om,<emph.end type="italics"/> hoc <lb/>e&longs;t ut motus <emph type="italics"/>qb<emph.end type="italics"/> in circulo maiori ad motum <emph type="italics"/>tm<emph.end type="italics"/> in cir­<lb/>culo minori. Quód &longs;i ergo &longs;umantur duo arcus <emph type="italics"/>op. qr<emph.end type="italics"/><lb/>inter &longs;e æquales, maior erit proportio motus in <emph type="italics"/>qr<emph.end type="italics"/> ad <lb/>motum in <emph type="italics"/>op,<emph.end type="italics"/> quam ad motu in <emph type="italics"/>tm<emph.end type="italics"/>: velocior ergo mo­<lb/>tus in arcu <emph type="italics"/>op<emph.end type="italics"/> circuli maioris, quam in arcu <emph type="italics"/>tm<emph.end type="italics"/> circuli <lb/>minoris. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ab impul&longs;u contrario & æquali nullus e&longs;t motus: ab impul&longs;u <lb/>verò contrario & inæquali ma<gap/> e&longs;t æqualis exce&longs;&longs;ui majoris.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim contrarium æquale tollit vel impedit &longs;u <lb/>um contrarium in eadem ratione, totum quidem <lb/>totum, pars verò partem &longs;ibi æqualem per po&longs;i: 2. Su­<lb/>blato per contrarium æquale toto impul&longs;u nullus erit <lb/>motus, qui e&longs;&longs;e non pote&longs;t <expan abbr="ab&longs;q;">ab&longs;que</expan> impul&longs;u. Quód'&longs;i ve­<lb/>rò impul&longs;us &longs;int inæquales, quia minor à majori tollit <lb/>partem &longs;ibi æqualem, erit reliquus exce&longs;&longs;us principium <pb/>motus. Ab impul&longs;u ergò contrario & æquali nullus e&longs;t <lb/>motus &c. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus &longs;ecundùm quid contrarij per lineam fiunt mediam, cujus <lb/>interuallam determinat &longs;inus complementi inclmationis, in ratione <lb/>quam habent impul&longs;us.<emph.end type="italics"/></s></p> <p type="main"> <s>VI in fig: 2 &longs;i mobile ex eodem puncto <emph type="italics"/>a<emph.end type="italics"/> moueatur <lb/>per lineas <emph type="italics"/>ab. af,<emph.end type="italics"/> aut per lineas <emph type="italics"/>ab. ad,<emph.end type="italics"/> & &longs;it angulus <lb/><emph type="italics"/>baf<emph.end type="italics"/> major, angulus verò <emph type="italics"/>bad<emph.end type="italics"/> minor recto, erunt hi mo­<lb/>tus per definit: 5. &longs;ecundùm quid contrarij, ac proinde <lb/>in eo in quo &longs;unt contrarij, <expan abbr="tollũt">tollunt</expan> aut <expan abbr="impediũt">impediunt</expan> <expan abbr="&longs;uũ">&longs;uum</expan> con <lb/><expan abbr="tratiũ">tratium</expan>, per definit: 1. impul&longs;us ergo in <emph type="italics"/>af<emph.end type="italics"/> ab impul&longs;u in <emph type="italics"/>ab,<emph.end type="italics"/><lb/>& hic ab impul&longs;u in <emph type="italics"/>af<emph.end type="italics"/> retractus, quia <expan abbr="id&etilde;n">idenn</expan> obile e&longs;&longs;e <expan abbr="nõ">non</expan> <lb/>pote&longs;t in plur bus locis, ac proinde <expan abbr="ne&qacute;">neque</expan>; pluribus moti­<lb/>bus agita<gap/>i, mouebitur motu inter <expan abbr="utrum&qacute;">utrumque</expan>; medio, cu­<lb/>ju&longs;modi linea motus <emph type="italics"/>ad<emph.end type="italics"/>: dico huius lineæ interuallum à, <lb/>lineis extremis <emph type="italics"/>ab. af<emph.end type="italics"/> e&longs;&longs;e &longs;inum complementi angulo­<lb/>rum <emph type="italics"/>faddab,<emph.end type="italics"/> in ratione quam habet impul&longs;us <emph type="italics"/>ab<emph.end type="italics"/> ad im­<lb/>pul&longs;um <emph type="italics"/>af.<emph.end type="italics"/> Quia enim velocitas motus per lineas incli­<lb/>natas e&longs;t in ratione &longs;inus complementi illarum inclina­<lb/>tionum, per prop: 14. ratio autem velocitatis e&longs;t eadem <lb/>quæ impul&longs;us, propterea quòd impul&longs;us e&longs;t agens ne­<lb/>ce&longs;&longs;arium, <expan abbr="motum&qacute;">motumque</expan>; producit &longs;ibi æqualem per prop: 2. <pb/>erit &longs;inus complementi anguli <emph type="italics"/>fad<emph.end type="italics"/> ad &longs;inum comple­<lb/>menti anguli <emph type="italics"/>dab,<emph.end type="italics"/> ut impul&longs;us in <emph type="italics"/>af<emph.end type="italics"/> ad impul&longs;um in <emph type="italics"/>ab,<emph.end type="italics"/><lb/>Motus ergò &longs;ecundùm quid contrarij per lineam fiunt <lb/>mediam, cujus interuallum determinat &longs;inu<emph type="italics"/>s<emph.end type="italics"/> &c. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus perfectè mixtus fit per diametrum parallelogrammi, cu­<lb/>jus latera con&longs;tituit motus &longs;implex: & ex impul&longs;u quidem æquali <lb/>e&longs;t æqualis &longs;emis&longs;i, ex inæquali verò major &longs;emi&longs;&longs;e eju&longs;dem motus.<emph.end type="italics"/></s></p> <p type="main"> <s>MOtum perfectè mixtum con&longs;tituunt motus, quiæ­<lb/>qualiter &longs;unt &longs;imiles & contrarij: tantùm enim hic <lb/> <arrow.to.target n="fig26"></arrow.to.target><lb/>illum auget, quantùm & minuit. Moueatur idem mobi <lb/>le ex <emph type="italics"/>a<emph.end type="italics"/> in <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>c,<emph.end type="italics"/> & &longs;it angulus <emph type="italics"/>bac<emph.end type="italics"/> rectus, <expan abbr="erit&qacute;">eritque</expan>; per defini­<lb/>tionem motus medius incipiens ab angulo recto per­<lb/>fectè mixtus: Dico hunc motum fieri per diametrum <lb/><emph type="italics"/>ad<emph.end type="italics"/> parallelogrammi <emph type="italics"/>abdc,<emph.end type="italics"/> cuius latera <emph type="italics"/>ab. ac<emph.end type="italics"/> &longs;unt mo­<lb/>tus, qui inter le <expan abbr="mi&longs;c&etilde;tur">mi&longs;centur</expan>: & <expan abbr="&longs;iquid&etilde;motus">&longs;iquidemmotus</expan> in <emph type="italics"/>ab<emph.end type="italics"/> &longs;it æqua <lb/>lis motui in <emph type="italics"/>ac,<emph.end type="italics"/> <expan abbr="motũ">motum</expan> <expan abbr="mixtũ">mixtum</expan> in <emph type="italics"/>ad<emph.end type="italics"/> e&longs;&longs;e <expan abbr="æqual&etilde;">æqualem</expan> &longs;emi&longs;si utri <pb/><expan abbr="u&longs;&qacute;">u&longs;que</expan>; motus &longs;imul &longs;umpti: &longs;i <expan abbr="aut&etilde;">autem</expan> motus fuerit inæqualis, <lb/><expan abbr="maior&etilde;">maiorem</expan> &longs;emi&longs;&longs;e. Sit primò motus in <emph type="italics"/>ab<emph.end type="italics"/> æqualis motui in <lb/><emph type="italics"/>ac<emph.end type="italics"/>: & ex <emph type="italics"/>bc<emph.end type="italics"/> termino <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>; motus demittantur lineæ <lb/>perpendiculares <emph type="italics"/>be. ce,<emph.end type="italics"/> &longs;inus æqualium angulorum <emph type="italics"/>cde, <lb/>edb.<emph.end type="italics"/> Quia ergo ut <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>ac,<emph.end type="italics"/> ita &longs;inus complementi <emph type="italics"/>eb<emph.end type="italics"/> ad <lb/><emph type="italics"/>ec,<emph.end type="italics"/> erit diameter <emph type="italics"/>ad<emph.end type="italics"/> linea motus mixti. E&longs;t autem mo­<lb/>tus in <emph type="italics"/>ab<emph.end type="italics"/> & <emph type="italics"/>ac<emph.end type="italics"/> duratione quidem æqualis motui in <emph type="italics"/>ae<emph.end type="italics"/><lb/>perprop: 13. magnitudine verò minor, cujus exce&longs;&longs;us <lb/>quadratum <emph type="italics"/>eb.<emph.end type="italics"/> & <emph type="italics"/>ec,<emph.end type="italics"/> &longs;eu <emph type="italics"/>ae<emph.end type="italics"/> & <emph type="italics"/>ed<emph.end type="italics"/>: at verò duo quadrata <emph type="italics"/>ae. <lb/>ed<emph.end type="italics"/> &longs;unt &longs;emi&longs;sis quadrati <emph type="italics"/>ad,<emph.end type="italics"/> hoc e&longs;t motus in <emph type="italics"/>ab.ac,<emph.end type="italics"/> cui, <lb/>æquale e&longs;t quadratum <emph type="italics"/>ad,<emph.end type="italics"/> propterea quòd <emph type="italics"/>ad<emph.end type="italics"/> &longs;it dupla <lb/><emph type="italics"/>ae<emph.end type="italics"/> aut <emph type="italics"/>ed<emph.end type="italics"/>: igitur motus æqualiter mixtus fit per diame­<lb/>trum parallelo grammi, & ab æquali impul&longs;u e&longs;t æqua­<lb/>lis &longs;emi&longs;si <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>; motus &longs;imul &longs;umpti. Quód &longs;i mo­<lb/>tus &longs;it inæqualis, & <emph type="italics"/>u.g.<emph.end type="italics"/> dupló velocior in <emph type="italics"/>ef<emph.end type="italics"/> quam in <emph type="italics"/>eg,<emph.end type="italics"/><lb/>dico motum mixtum fieri quidem per diametrum <emph type="italics"/>eb,<emph.end type="italics"/><lb/>e&longs;&longs;e autem &longs;emi&longs;&longs;e maiorem. De&longs;cripto enim centro <emph type="italics"/>b<emph.end type="italics"/><lb/>arcu <emph type="italics"/>mn,<emph.end type="italics"/> erit &longs;inus complementi <emph type="italics"/>ik<emph.end type="italics"/> ad &longs;inum comple­<lb/>menti <emph type="italics"/>ip,<emph.end type="italics"/> ut motus in <emph type="italics"/>ef<emph.end type="italics"/> ad motum in <emph type="italics"/>eg,<emph.end type="italics"/> ac proinde di­<lb/>ameter <emph type="italics"/>eh<emph.end type="italics"/> linea motus mixti: ad quam ex punctis <emph type="italics"/>fg<emph.end type="italics"/> du­<lb/>ctæ lineæ perpendiculares <emph type="italics"/>fl. go<emph.end type="italics"/> metientur defectum <lb/>motus in <emph type="italics"/>eh.<emph.end type="italics"/> Quia ergo ex angulo recto <emph type="italics"/>efh<emph.end type="italics"/> linea <emph type="italics"/>fl<emph.end type="italics"/> e&longs;t <lb/>perpendicularis ad ba&longs;im <emph type="italics"/>eh,<emph.end type="italics"/> erit ut <emph type="italics"/>ef<emph.end type="italics"/> ad <emph type="italics"/>fh,<emph.end type="italics"/> ita <emph type="italics"/>el<emph.end type="italics"/> ad <emph type="italics"/>lf,<emph.end type="italics"/> & <lb/><emph type="italics"/>lf<emph.end type="italics"/> ad <emph type="italics"/>lh:<emph.end type="italics"/> ponitur autem quadratum <emph type="italics"/>ef<emph.end type="italics"/> duplum quadrat <lb/><emph type="italics"/>fh,<emph.end type="italics"/> &longs;iue <emph type="italics"/>eg,<emph.end type="italics"/> erit ergo quadratum <emph type="italics"/>fl<emph.end type="italics"/> &longs;imiliter duplu quadra <pb/>ti <emph type="italics"/>lh.<emph.end type="italics"/> quadratum ergo <emph type="italics"/>fh<emph.end type="italics"/> <expan abbr="utriq;">utrique</expan> æquale continebit tria <lb/>quadrata, quorum &longs;ingula &longs;int æqualia quadrato <emph type="italics"/>lh.<emph.end type="italics"/> & <lb/>quia quadratum <emph type="italics"/>ef<emph.end type="italics"/> e&longs;t duplum quadrati <emph type="italics"/>fh,<emph.end type="italics"/> erit quadra­<lb/>tum <emph type="italics"/>eh<emph.end type="italics"/> æquale nouem quadiatis <emph type="italics"/>lh<emph.end type="italics"/> &longs;imul &longs;umptis. At <lb/>verò quadratum <emph type="italics"/>el<emph.end type="italics"/> duplum quadrati <emph type="italics"/>lf<emph.end type="italics"/> erit quadruplum <lb/>quadrati <emph type="italics"/>lh,<emph.end type="italics"/> <expan abbr="a&longs;&longs;umpto&qacute;">a&longs;&longs;umptoque</expan>; quadrato <emph type="italics"/>oo,<emph.end type="italics"/> aut huic æquali <emph type="italics"/>lb<emph.end type="italics"/><lb/>erunt duo quadrata <emph type="italics"/>el. lh<emph.end type="italics"/> &longs;imul &longs;umpta æqualia <expan abbr="quin&qacute;">quinque</expan>; <lb/>quadratis <emph type="italics"/>lh<emph.end type="italics"/>: Maiora ergo quam &longs;emi&longs;sis quadrati <emph type="italics"/>eh,<emph.end type="italics"/><lb/>quòd æquale ponitur nouem quadratis <emph type="italics"/>lb.<emph.end type="italics"/> Igitur mo­<lb/>tus perfectè mixtus fit per diametrum parallelogram­<lb/>mi, cujus latera con&longs;tituit motus &longs;implex &c. </s></p> <figure id="fig26"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus mixtus incipiens ab angulo inajori quam recto, e&longs;t minor <lb/>&longs;emi&longs;&longs;e: incipiens verò ab angulo minori quam recto, major &longs;emi&longs;&longs;s <lb/>motus &longs;imul &longs;umpti.<emph.end type="italics"/></s></p> <p type="main"> <s>SIt primùm in fig: 7. angulus <emph type="italics"/>dae<emph.end type="italics"/> maior recto, & an­<lb/>gulus <emph type="italics"/>bac<emph.end type="italics"/> rectus, <expan abbr="erit&qacute;">eritque</expan>; quadratum <emph type="italics"/>bb<emph.end type="italics"/> æquale qua­<lb/>drato <emph type="italics"/>ab<emph.end type="italics"/>: e&longs;t autem quadratum <emph type="italics"/>db,<emph.end type="italics"/> ex ce&longs;&longs;us nimirum <lb/>motus <emph type="italics"/>ad,<emph.end type="italics"/> quadrato <emph type="italics"/>bh,<emph.end type="italics"/> ac proinde quadrato <emph type="italics"/>ah<emph.end type="italics"/> maius: <lb/>igitur quadratum <emph type="italics"/>ad<emph.end type="italics"/> æquale duobus quadratis <emph type="italics"/>dh. ah<emph.end type="italics"/> ad <lb/>quadratum minus <emph type="italics"/>ah<emph.end type="italics"/> maiorem rationem habet quam <lb/>duplam: motus ergo in <emph type="italics"/>ah<emph.end type="italics"/> mixtus e&longs;t minor &longs;emi&longs;&longs;e <pb/>motus in <emph type="italics"/>ad,<emph.end type="italics"/> <expan abbr="at&qacute;">atque</expan>; illius duplum minus quam motus in <emph type="italics"/>a <lb/>d. ae<emph.end type="italics"/> &longs;imul &longs;umpti. Quòd &longs;i angulus <emph type="italics"/>fag<emph.end type="italics"/> &longs;it minor recto, <lb/>erit latus <emph type="italics"/>fh,<emph.end type="italics"/> & huius quadratum minus quam <emph type="italics"/>ah:<emph.end type="italics"/> mo­<lb/>tus ergo in <emph type="italics"/>af<emph.end type="italics"/> ad motum in <emph type="italics"/>ah<emph.end type="italics"/> minorem rationem ha­<lb/>bet quam duplam, ac proinde motus in <emph type="italics"/>ah<emph.end type="italics"/> major &longs;emi&longs;­<lb/>&longs;e motus in <emph type="italics"/>af,<emph.end type="italics"/> & illius duplum majus quá motus in <emph type="italics"/>af. <lb/>ag<emph.end type="italics"/> &longs;imul &longs;umpti. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus mixtus e&longs;t nece&longs;&longs;arió minor diametro quadrati aut <lb/>parallelogrammi, cujus later<gap/> &longs;ant motus &longs;imple<gap/>.<emph.end type="italics"/></s></p> <p type="main"> <s>NAm motus quidem in <emph type="italics"/>be<emph.end type="italics"/> mixtus (in fig: 4.) e&longs;t du­<lb/>plum quadrati eiu&longs;dem <emph type="italics"/>be<emph.end type="italics"/>: quadratum verò <emph type="italics"/>db<emph.end type="italics"/> ad <lb/>quadratum <emph type="italics"/>be<emph.end type="italics"/> e&longs;t quadruplum. Cau&longs;a verò hujus de­<lb/>&longs;ectus e&longs;t contratietas illorum motuum, ex angulis pro­<lb/>ueniens, cum quibus augotur & minuitur, <expan abbr="quou&longs;q;">quou&longs;que</expan> an­<lb/>gulus late&longs;cens æqualis fi<gap/> duobus rectis, in quo &longs;um­<lb/>ma e&longs;t contrarietas, ac proinde nullus e&longs;&longs;e pote&longs;t motus. <lb/>Angulo verò decre cente augetur &longs;imilitudo motus, <lb/><expan abbr="quou&longs;q;">quou&longs;que</expan> angulo deficiente &longs;int una linea motus, in quà <lb/>perfecta &longs;imilitudo, nulla autem e&longs;t contrarietas. <expan abbr="Itaq;">Itaque</expan> <lb/>motus æqualis motum auget in eadem ratione, totus <lb/>quidem totum, pars verò partem &longs;ibi æqualem per <lb/>po&longs;it. 1. </s></p> <pb/> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Problema I.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Lineam motus mixti, & illius magnitudinem determinare.<emph.end type="italics"/></s></p> <p type="main"> <s>SIt primùm motus <emph type="italics"/><expan abbr="pq.">pque</expan> pr<emph.end type="italics"/> perfectè mixtus, incipiens ab <lb/>angulo recto <emph type="italics"/>qpr<emph.end type="italics"/>: & ex <emph type="italics"/>q<emph.end type="italics"/> & <emph type="italics"/>r<emph.end type="italics"/> ducantur lineæ <emph type="italics"/>qs. rs<emph.end type="italics"/> pa <lb/>rallelæ ad <emph type="italics"/><expan abbr="pq.">pque</expan> pr,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; per prop. 31. motus mixtus in di­<lb/>ametro <emph type="italics"/>ps<emph.end type="italics"/>: ad quam ex termino <expan abbr="utriu&longs;q;">utriu&longs;que</expan> motus <emph type="italics"/>q<emph.end type="italics"/> & <emph type="italics"/>r<emph.end type="italics"/><lb/> <arrow.to.target n="fig27"></arrow.to.target><lb/>demittantur lineæ perpendiculares <emph type="italics"/>qt.ru,<emph.end type="italics"/> <expan abbr="eritq;">eritque</expan> motus <lb/>mixtus ex <emph type="italics"/>pqpr<emph.end type="italics"/> æqualis duobus quadratis <emph type="italics"/>pu.pt.<emph.end type="italics"/> ab&longs;ein- <pb/>datur ergo ex linea <emph type="italics"/>tq<emph.end type="italics"/> productà linea <emph type="italics"/>tx<emph.end type="italics"/> æqualis lineæ <emph type="italics"/>p <lb/>u,<emph.end type="italics"/> & ex puncto <emph type="italics"/>p,<emph.end type="italics"/> interuallo autem <emph type="italics"/>px<emph.end type="italics"/> de&longs;cribatur arcus <lb/><emph type="italics"/>xy,<emph.end type="italics"/> <expan abbr="connectantur&qacute;">connectanturque</expan>; linea <emph type="italics"/>px<emph.end type="italics"/>: dico quadratum <emph type="italics"/>py<emph.end type="italics"/> e&longs;&longs;e <lb/>motum mixtum & duratione æqualem motui <emph type="italics"/><expan abbr="pq.">pque</expan> pr<emph.end type="italics"/> &longs;i­<lb/>mul &longs;umptis. Quia enim quadratum <emph type="italics"/>py<emph.end type="italics"/> quadrato <emph type="italics"/>px,<emph.end type="italics"/><lb/>hoc autem duobus quadratis <emph type="italics"/>pt.tx,<emph.end type="italics"/> &longs;eu <emph type="italics"/>pu<emph.end type="italics"/> e&longs;t æquale: e&longs;t <lb/>autem motus <emph type="italics"/>pt<emph.end type="italics"/> motui <emph type="italics"/>pq,<emph.end type="italics"/> & <emph type="italics"/>pu<emph.end type="italics"/> motui <emph type="italics"/>pr<emph.end type="italics"/> æqualis dura­<lb/>tione per prop: 13. erit motus mixtus in <emph type="italics"/>py<emph.end type="italics"/> &longs;imiliter æ­<lb/>qualis motibus <emph type="italics"/>pq<emph.end type="italics"/> & <emph type="italics"/>pr<emph.end type="italics"/> &longs;imul &longs;umptis. Quòd &longs;i verò <lb/>motus imperfectè mixtus & inæqualis <emph type="italics"/>ab. ac<emph.end type="italics"/> ab angulo <lb/>incipiat maiori aut minori quam recto <emph type="italics"/>bac<emph.end type="italics"/>: a&longs;&longs;uman­<lb/>tur duo puncta <emph type="italics"/>fg<emph.end type="italics"/> æqualiter remota ab <emph type="italics"/>a,<emph.end type="italics"/> à quibus pro­<lb/>tractæ lineæ perpendiculares <emph type="italics"/>fh. gh<emph.end type="italics"/> &longs;e inter&longs;ecent in <emph type="italics"/>h,<emph.end type="italics"/> <expan abbr="e-rit&qacute;">e­<lb/>ritque</expan>; angulus <emph type="italics"/>fhg<emph.end type="italics"/> complementum anguli <emph type="italics"/>bac,<emph.end type="italics"/> & &longs;imul <lb/>&longs;umpti æquales duobus rectis. De&longs;cribatur ergo ex <emph type="italics"/>h<emph.end type="italics"/><lb/>arcus <emph type="italics"/>fig,<emph.end type="italics"/> <expan abbr="&longs;ecetur&qacute;">&longs;eceturque</expan>; bifariam in <emph type="italics"/>i<emph.end type="italics"/> eà ratione, ut &longs;inus <emph type="italics"/>ik<emph.end type="italics"/> ad <lb/>&longs;inum <emph type="italics"/>il<emph.end type="italics"/> &longs;it, ut motus <emph type="italics"/>ab<emph.end type="italics"/> ad motum <emph type="italics"/>ac:<emph.end type="italics"/> dico lineam ex <emph type="italics"/>a<emph.end type="italics"/><lb/>productam in <emph type="italics"/>i<emph.end type="italics"/> e&longs;&longs;e lineam motus mixti. Producatur e­<lb/>nim <emph type="italics"/>fh<emph.end type="italics"/> in <emph type="italics"/>p,<emph.end type="italics"/> <expan abbr="erit&qacute;">eritque</expan>; angulus <emph type="italics"/>fpa<emph.end type="italics"/> complementum anguli <emph type="italics"/>f <lb/>ap,<emph.end type="italics"/> & angulus <emph type="italics"/>aog<emph.end type="italics"/> complementum anguli <emph type="italics"/>oag<emph.end type="italics"/>: duo er­<lb/>go anguli <emph type="italics"/>hpo. aog<emph.end type="italics"/> hoc e&longs;t <emph type="italics"/>hop,<emph.end type="italics"/> &longs;imul &longs;umpti &longs;unt æqua <lb/>les duobus angulis <emph type="italics"/>fhi: thg<emph.end type="italics"/> &longs;imul &longs;umptis, propterea <lb/>quód &longs;int complementa eju&longs;dem anguli <emph type="italics"/>fag,<emph.end type="italics"/> e&longs;t autem <lb/>angulus <emph type="italics"/>hop<emph.end type="italics"/> externus major angulo <emph type="italics"/>iho<emph.end type="italics"/> interno quanti­<lb/>tate anguli <emph type="italics"/>bio,<emph.end type="italics"/> angulus verò <emph type="italics"/>iph<emph.end type="italics"/> internus minor angu- <pb/>lo <emph type="italics"/>ihf<emph.end type="italics"/> externo, quanti are ejul dem anguli <emph type="italics"/>hip:<emph.end type="italics"/> angulu, <lb/>ergo <emph type="italics"/>hop<emph.end type="italics"/> angulo <emph type="italics"/>fhi,<emph.end type="italics"/> & angulus <emph type="italics"/>oph<emph.end type="italics"/> angulo <emph type="italics"/>tho<emph.end type="italics"/> &longs;eu <emph type="italics"/>ihg<emph.end type="italics"/><lb/>e&longs;t æqualis, ac proinde <emph type="italics"/>ik. il<emph.end type="italics"/> &longs;unt &longs;inus complementi an­<lb/>gulorum <emph type="italics"/>iag.e ai.<emph.end type="italics"/> Et quia motus &longs;unt in ratione, quam <lb/>habent &longs;inus complementi inclinationum, erit linea <emph type="italics"/>ai<emph.end type="italics"/><lb/>linea motus mixti ex <emph type="italics"/>ab.ac<emph.end type="italics"/>; ad quam ex termino utriu <expan abbr="&longs;&qacute;">&longs;que</expan>; <lb/>motus <emph type="italics"/>b.c<emph.end type="italics"/> demittantur lineæ perpendiculares <emph type="italics"/>bd.ce:<emph.end type="italics"/><lb/><expan abbr="crũr&qacute;">crurrque</expan>, duo quadrata <emph type="italics"/>ad.ae<emph.end type="italics"/> &longs;imul &longs;umpta motus mix­<lb/>tus: ab&longs;cindatur ergo ex <emph type="italics"/>db<emph.end type="italics"/> producta <emph type="italics"/>dm<emph.end type="italics"/> æqualis <emph type="italics"/>ae,<emph.end type="italics"/> & <lb/>centro <emph type="italics"/>a<emph.end type="italics"/> ducatur arcus <emph type="italics"/>mn,<emph.end type="italics"/> dico quadratum <emph type="italics"/>an<emph.end type="italics"/> e&longs;&longs;e ma­<lb/>gnitudinem motus mixti. Erit enim quadratum <emph type="italics"/>am,<emph.end type="italics"/><lb/>hoc e&longs;t <emph type="italics"/>an,<emph.end type="italics"/> æquale duobus quadratis <emph type="italics"/>ad. dm,<emph.end type="italics"/> &longs;eu <emph type="italics"/>ae,<emph.end type="italics"/> cui <lb/>æqualis &longs;umcbatur <emph type="italics"/>dm.<emph.end type="italics"/> Lineam ergo motus mixti & il­<lb/>lius magnitudinem determinauimus, quod erat facien­<lb/>dum. </s></p> <figure id="fig27"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Mobile &longs;eu impul&longs;u, &longs;eu àgrauitate moueatar, &longs;i planum occur­<lb/>rat, reflectit ab eodem plano per lineam rectam.<emph.end type="italics"/></s></p> <p type="main"> <s>IMpul&longs;us fitdum corpus unum alteri in currit & alli <lb/>dit, &longs;iue <expan abbr="utrum&qacute;">utrumque</expan>, &longs;iue unum ex illis moueatur, <expan abbr="at&qacute;">atque</expan>; eo <lb/>magis mouet & impellit, quò magis ferit & allidit: & <lb/>&longs;iquidem re&longs;i&longs;tentia minor e&longs;t impul&longs;u, in illam partem <lb/>mouer illud mobile, in quam &longs;it plaga, eundem motum <pb/>continuando; velocitate tamen eó minori, quó re&longs;i­<lb/>&longs;tentia e&longs;t majòr. Quód &longs;i re&longs;i&longs;tentia &longs;it major impul­<lb/>&longs;u, eádem velocitate, quà impulit, in partem auer&longs;am re <lb/>pellitur: propterea quód illa plaga æqualem in <expan abbr="utro&qacute;">utroque</expan>; <lb/>mobili impul&longs;um producit. E&longs;t autem major plaga ex <lb/>velociori & magis violento incur&longs;u: igitur ab æquali <lb/>plagá æqualis <expan abbr="quo&qacute;">quoque</expan>; recur&longs;us. Et quia per motum fit <lb/>plaga, mouetur autem mobile ad moturn &longs;ui centri, erit <lb/><expan abbr="quoq;">quoque</expan> plaga ab eodem centro. Sed & re&longs;i&longs;tentia fit â cen <lb/>tro &longs;eu grauitatis, &longs;eu contrarij impul&longs;us: cadem ergo ra <lb/> <arrow.to.target n="fig28"></arrow.to.target><lb/>tione minor re&longs;i&longs;tentia impul&longs;um recipir, quà major ei­<lb/>dem re&longs;i&longs;tit. Vt&longs;i mobile ex <emph type="italics"/>a<emph.end type="italics"/> moueatur à grauitate qui <lb/>dem in <emph type="italics"/>b,<emph.end type="italics"/> ex im pul&longs;u verò in <emph type="italics"/>f<emph.end type="italics"/>aut <emph type="italics"/>c<gap/><emph.end type="italics"/> &longs;it autem major re&longs;i- <pb/>&longs;tentià in <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>f,<emph.end type="italics"/> quam ut loco moueantur ex illo impul­<lb/>&longs;u, minor autem in <emph type="italics"/>c<emph.end type="italics"/>: motus quidem ex <emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>f<emph.end type="italics"/> in <emph type="italics"/>a<emph.end type="italics"/> refle­<lb/>ctit, ex <emph type="italics"/>c<emph.end type="italics"/> verò expul&longs;o illo mobili quie&longs;cit, &longs;i&longs;it æquale: <lb/>eundem verò motum continuat in <emph type="italics"/>d,<emph.end type="italics"/> &longs;i minus &longs;it percu&longs;­<lb/>&longs;um: quia tamen re&longs;i&longs;tentia impul&longs;um minuit, quó ma­<lb/>jor re&longs;i&longs;tentia, eòminor velocitas motus. </s></p> <figure id="fig28"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Mstus in &longs;eip&longs;um reflectit, cùm centrum grauitatis & conta­<lb/>ctus &longs;ant in eádem lineá motus.<emph.end type="italics"/></s></p> <p type="main"> <s>GLobus <emph type="italics"/>a<emph.end type="italics"/> occurrat plano in <emph type="italics"/>b,<emph.end type="italics"/> <expan abbr="&longs;itq;">&longs;itque</expan> centrum grauita <lb/>tis aut impul&longs;us <emph type="italics"/>e<emph.end type="italics"/> in lineà motus <emph type="italics"/>ab<emph.end type="italics"/> perpendicular<emph type="sup"/>i<emph.end type="sup"/><lb/>ad contactum <emph type="italics"/>b,<emph.end type="italics"/> dico hunc motum in &longs;eip&longs;um reflecti. <lb/>Quia enim motus & huius plaga ad motum fit &longs;ui cen­<lb/>tri, erit motus globi <emph type="italics"/>a,<emph.end type="italics"/> & hujus plagain lineâ <emph type="italics"/>ab<emph.end type="italics"/> à centro <lb/><emph type="italics"/>e<emph.end type="italics"/> ductà per contactum: & quia eadem ratione impul­<lb/>&longs;um recipit & impellit, <expan abbr="e&longs;tq;">e&longs;tque</expan> major re&longs;i&longs;tentiain <emph type="italics"/>b<emph.end type="italics"/> quam <lb/>impul&longs;us ex <emph type="italics"/>e,<emph.end type="italics"/> erit motus reflexus in eadem lineà <emph type="italics"/>ab.<emph.end type="italics"/><lb/>Motus ergo in &longs;eip&longs;um reflectit, cúm centrum granita <lb/>tis & contactus &longs;untin eadem lineà motus. Ob<gap/>jcies <lb/>cùm pila percutit planum, eàdem vi percutitur ab illo <lb/>plano: e&longs;t autem à percu&longs;sione æ quali impul&longs;us æqua­<lb/>lis, quó enim violentiùs in cidit, eó magis impetuosé re&longs;i­<lb/>lit ab illá plagà: impul&longs;us ergo, quem pila recipit â pla- <pb/>no, e&longs;t æqualis impul&longs;ui, quo tidem plano allidit. Quia <lb/>verò hi impul&longs;us tendunt in partes oppo&longs;itas eju&longs;dem <lb/>lineærectæ, erunt per definit: 4. contrarij ab&longs;olutè: tol­<lb/>lit autem contrarium æquale &longs;uum contrarium in eà­<lb/>dem ratione, totum quidem totum, pars verò partem <lb/>&longs;ibi æqualem; &longs;ublato ergo per contrarium æquale im­<lb/>pul&longs;u nullus erit motus reflexus, cùm linea motus e&longs;t <lb/>perpendicularis ad illud planum. Quód &longs;i à percu&longs;sio­<lb/>ne in plano, aut globo quie&longs;cente factá morus reflectit, <lb/>quid prohibet ab eodem plano, autglobo, &longs;i motu op­<lb/>po&longs;ito ferantur, & violentià æquali &longs;ibi occurant, à per­<lb/>cu&longs;sione æquali eundem motum reflecti? Vt in hac ob­<lb/> <arrow.to.target n="marg1"></arrow.to.target><lb/>&longs;curitate aliquam lucem con&longs;equamur, quæ non ni&longs;i ex <lb/>naturà impul&longs;us priús cognitá eluce&longs;cit, de quâ in lib: de <lb/>Arcu Cæle&longs;ti latiùs di&longs;&longs;eremus, <expan abbr="notãdum">notandum</expan> hic breuiter 1. <lb/> <arrow.to.target n="marg2"></arrow.to.target><lb/>Impul&longs;um fieri à percu&longs;sione juxta determinationem il­<lb/>lius plagæ, <expan abbr="quã">quam</expan> centrum inducit percutientis, & quam <lb/>centrum recipit percu&longs;si; partes enim mobilis impul­<lb/> <arrow.to.target n="marg3"></arrow.to.target><lb/>&longs;um recipiunt per lineas motui centri parallelas. 2. <expan abbr="Hãc">Hanc</expan> <lb/>plagam, quæ fit à corpore percu&longs;&longs;o, aliter dum quie&longs;cit, <lb/>aliter dum e&longs;t in motu impul&longs;um determinare: quia <lb/>enim plaga ex impul&longs;u, percuffum verò quie&longs;eens nul­<lb/>lum ex &longs;e habet impul&longs;um, verùm à percutiente; eádem <lb/>plaga, quà percutitur, impul&longs;um determinat in percuti <lb/>ente: ab æquali ergo plagà æqualis impul&longs;us. Cum au- <pb/>tem percutitur in motu, quia ex &longs;e impul&longs;um habet, <expan abbr="nõ">non</expan> ex <lb/>illà plagà, quam tecipit à percutiente, &longs;ed quam infert <lb/>impulium determinat; licet ergo illorum corporum, <lb/>quæ violentiá inæquali colliduntur, idem &longs;it contactus, <lb/>non tamen cadem ab <expan abbr="utro&qacute;">utroque</expan>, verùm â majori major, à <lb/> <arrow.to.target n="marg4"></arrow.to.target><lb/>ninori impul&longs;u minor infertur plaga. 3. Corpora per­<lb/>cu&longs;&longs;a alia e&longs;&longs;e mollia, quorum partes percu&longs;sioni <expan abbr="cedũt">cedunt</expan>, <lb/>inter fe verò unitæ <expan abbr="man&etilde;t">manent</expan>; cuju&longs;modi argilla, cera, lana, <lb/>plumbum, &c. Alla dura; & &longs;iquidem percu&longs;sioni nul­<lb/>lo modo cedunt, ab&longs;olutè dura; &longs;i autem percu&longs;sioni ce <lb/>dunt, <expan abbr="ne&qacute;">neque</expan>; partes inter &longs;e unitæ manent, fragilia dicun­<lb/>tur; ut vitrum, te&longs;ta, tophus, &c. Corpora demum ab&longs;o­<lb/>lutè dura alia funt &longs;onora, quorum atomi vibratione <lb/>quadam mouentur, ut propo: 1. dictum; alia &longs;urda, quo <lb/> <arrow.to.target n="marg5"></arrow.to.target><lb/>rum <gap/>atomi nullo aut in&longs;en&longs;ibili motu monentur. 4 <lb/>Impul&longs;um naturà &longs;uà inclinare<gap/>ad motum perfectum, <lb/>quo mobile &longs;ecundúm &longs;e totum locum mutat. Quòd <lb/>&longs;i ergo impul&longs;us, quem plaga inducir, proportionem <lb/>habeat ad illud mobile, eodem quo percutiens motu fe­<lb/>retur: &longs;i autem minor &longs;it impul&longs;us quam ut loco moue­<lb/>atur, habeat vorò idem mobile partes &longs;ragiles, aut in &longs;e <lb/>cedentes, percutiens percu&longs;&longs;um perfora bit, aut excaua­<lb/>bit; it a nimirum &longs;i major &longs;it &longs;oliditas per cu&longs;si, quam ut <lb/>impetus per omnes partes cluctetur, qui non prjus iram <lb/>ponit, quam continuatà illarum partium, cuas perrum- <pb/>pit, vel collidit, re&longs;i&longs;tentia vires ab&longs;umat. Ex huju&longs;m<gap/><lb/>di ergo corporibus nullo modo reflectit motus, ni&longs;i in <lb/>progre&longs;&longs;u, priú&longs;quam exoluatur, occurrant partes <gap/> is <lb/>&longs;olidæ: ita enim pila ubi cal cem dera&longs;it àmuro, ex oc­<lb/>cur&longs;u &longs;axi reflectit: quod non fit &longs;iviá, quà irrupit á &longs;il­<lb/>&longs;urà rur&longs;um coëat, quemadmodum in ligno viridi, cu­<lb/>jus vulnus ex partium fi&longs;&longs;arum coalitu mox &longs;olidatur. <lb/>Corpora autem dura ab&longs;oluté quia <expan abbr="ne&qacute;">neque</expan>; perforantur, <lb/><expan abbr="ne&qacute;">neque</expan>; partes habent percu&longs;sioni cedentes, æqualem reci­<lb/>piunt <expan abbr="at&qacute;">atque</expan>; inferunt plagam, morum verò ex illà plagâ re <lb/>flectunt, <expan abbr="at&qacute;">atque</expan>; eó magis, quó duritie magis præ&longs;tant. In­<lb/>de ergò fit quód vala vitrea aut cry&longs;tallina inæqualiter <lb/>colliduntur, prout illa corpora, ad quæ offendunt, per­<lb/>cu&longs;sioni magis aut minús cedunt: quia nimirum non <lb/>ex illà, quam inferunt, &longs;ed ex illâ, quam recipiunt, plag<gap/><lb/>colliduntur. 5. Impul&longs;um fieri per lineam rectam: & &longs;i­<lb/> <arrow.to.target n="marg6"></arrow.to.target><lb/>cuti grauitas minús mouet, quó magis linea motus ad <lb/>horizontem e&longs;t inclinata, quie&longs;cit verò à motu in lineà <lb/>cidem parallelás ita impul&longs;um ex inclinatione motus <lb/>&longs;en&longs;im minui, & demum in hypomochlio deficere. <lb/>Quòd &longs;i ergo mobile occurrat plano, it a ut cont<gap/>ctus <lb/>&longs;it in lineá motus eiu&longs;dem centri, quia centrum hypo­<lb/>mochlio occurrit, totus ex illà plagà emoritur impul­<lb/>&longs;us; propterea quòd motui quies non minùs e&longs;t contra <lb/>ria, quam motus: at verò &longs;i planum &longs;it inclinatum, in il- <pb/>là tantum parte, quæ hypomochlio occurrit, motus <expan abbr="cõ-quie&longs;cit">con­<lb/>quie&longs;cit</expan>, reliquà parte, quæ cum centro extra hypomo­<lb/>chlium cadit, nihil impedità: impul&longs;us ergo pilæ, cúm <lb/>motus centri e&longs;t perpendicularis ad planum, ubi percui­<lb/>&longs;ic in hypomochlio â motu conquie&longs;cit: at vero <expan abbr="planũ">planum</expan> <lb/>exillà plagà in percutiente nouum determinat im p<emph type="italics"/>h<emph.end type="italics"/>l­<lb/>&longs;um, juxta directionem plagæ, quam infert; à quo <expan abbr="ead&etilde;">eadem</expan>, <lb/>quà venit, vià retroagitur: & &longs;iquidem duritie præ&longs;tat, <lb/>erit plaga & qui hanc &longs;equitur impul&longs;us in <expan abbr="utro&qacute;">utroque</expan>; æqua­<lb/>lis, ac proinde motus reflexus æqualis motui recto: de­<lb/>ficiet autem motus reflexus â motu recto, &longs;i defectu du­<lb/>ritiei minorem recipiat, quam dedit plagam. Quód &longs;i <lb/>ergo duo globi violentiá æquali &longs;ibi occurrant, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; mo­<lb/>tus centri <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>; in eádem lineà rectà; quia tum <expan abbr="uterq;">uterque</expan> <lb/>alteri, non minús quam planum, e&longs;t hypomochlij loco, <lb/>ab illâ communi plagà in <expan abbr="utro&qacute;">utroque</expan>, emoritur, nouus verò <lb/>quo retro aguntur, impul&longs;us regeneratur. Licetverò <lb/>po&longs;it: 2. inficiamur eju&longs;modi globos &longs;ibi occurrentes re <lb/>&longs;ilire, id tamen exem pligratia ad naturam contrarij ma­<lb/>gis explicandam, & ex &longs;uppo&longs;itione, &longs;i nimirum impul­<lb/>lus eâlratione mi&longs;ceantur, à nobis dictum fuit: <gap/>at verò hi <lb/>impul&longs;us non mi&longs;centur, verùm uni abolito alius &longs;uc­<lb/>cedit. Quód &longs;i verò <expan abbr="uter&qacute;">uterque</expan>; globus in motu percutlat vi­<lb/>olentià inæ quali, impul&longs;us quidem minoris, ubi percu&longs;­<lb/>&longs;it majus, ob hypomochlium à motu conquie&longs;cit, im- <pb/>pul&longs;um verò &longs;ioi &longs;imilem & æqualem producit, leu de­<lb/>terminat in majori ex ill<gap/>, quam infert, plagà, hoc e&longs;t <lb/>partem tollit à majori &longs;ibi æqualem. At verò majus, ubi <lb/>percu&longs;sit, non videtur conquie&longs;cere â motu, propterea <lb/>quòd minus non habeat rationem hypomochlij ad ma <lb/>jus, impul&longs;um verò in minori producit &longs;ibi æqualem; ut <lb/>&longs;i min or impul&longs;us ut 3. major ut 7. minor quidem à ma­<lb/>jori tollit partem &longs;ibi æqualem ide&longs;t 3. & &longs;imul obcon­<lb/>trariam in hypomochlio quietem ex&longs;pirat; majus verò <lb/>quia tota vi percutit minus, impullum ut 7. producit ex <lb/>illà plagà, motum autem à percu&longs;sione non ni&longs;i partes 4. <lb/>reliquæ perficiunt. <expan abbr="Ita&qacute;">Itaque</expan>; fit ut ex illà in æquali plagà, ve <lb/>locitate ferantur inæquali, minori quidem majus ob vi­<lb/>res à percu&longs;sione accitas & mutilatas, majori verò mi­<lb/>nus obea&longs;dem vires de integro acqui&longs;itas. Dices inter­<lb/>dum fieri ut inæquali violentià &longs;ibi occurrant duo glo­<lb/>bi, & tamen <expan abbr="uter&qacute;">uterque</expan>; re&longs;iliat. Relpondeo &longs;i contactus <gap/>fi <lb/>at in lineà motus centri, videtur non po&longs;&longs;e fieri ut major <lb/>le&longs;iliat, propterea, quód major violentia non detinetur <lb/>à minori: at veró &longs;iex obliquo &longs;e percutiant, fieri po&longs;&longs;e <lb/>ut etiam ille globus, qui magis percu&longs;sit, re&longs;iliat, aut in <lb/>codem, quo percu&longs;sit, loco con&longs;i&longs;tat. In&longs;tabis hanc &longs;o­<lb/>lutionem non <expan abbr="u&longs;q;">u&longs;que</expan> <expan abbr="quaq;">quaque</expan> experientiæ con&longs;onare: nam <lb/><expan abbr="quomodocunq;">quomodocunque</expan> duo globi inter &longs;e commicantur, <expan abbr="atq;">atque</expan> <lb/>adeò in lineà motus centri &longs;e percutiant violentiâ in- <pb/>æquali, <expan abbr="uter&qacute;">uterque</expan>; re&longs;ilit ab illà plagà, magis qu dem qui mi­<lb/>nus, minùs verò qui magis percu&longs;sit: non igitur exce&longs;­<lb/>&longs;us majoris e&longs;t principium morus reliqui à contactu. <lb/>Vt objectioni & experientiæ &longs;atis fiat, dicendum à quo­<lb/>libet contactu impul&longs;um deficere & ex&longs;pirare, nouum <lb/>verò à percu&longs;sione determinari, qui motu eidem plagæ <lb/>æquali retroagit illud mobile. Cùm enim impul&longs;us â <lb/>percu&longs;sione fiat, juxta determinationem plagæ, quam <lb/>recipir à percutiente, nihil mirum &longs;i â determinatione <lb/>nouâ nouum impul&longs;um <expan abbr="cõ&longs;equatur">con&longs;equatur</expan>: quomodo in acu <lb/>nauticà fieri videmus, quæ quories oppo&longs;itum polum <lb/>tangit, directionem, quà eidem polo &longs;e obuertit, &longs;orti­<lb/>tur nouam. Quod minùs difficulter admittes, &longs;i per­<lb/>pendas quá ratione va&longs;tæ campanæ ingens mugitus, & <lb/>qui hunc&longs;uá vibratione fouet in gyrum actus impul&longs;us <lb/>ex leui&longs;simo tactu repente contice&longs;cat: quid ergo mi­<lb/>rum ex tactu pilæ haud paulo majoris im pul&longs;um cohi­<lb/>beri? In&longs;tabis an igitur globus ligneus, &longs;i ex oppo&longs;ito <lb/>quantumuis motu lento moueatur, repercutiet pilam <lb/>ferream <expan abbr="quacun&qacute;">quacunque</expan>; violentiá irruentem? Ad pleniorem <lb/>hujus <expan abbr="at&qacute;">atque</expan>; aliarum obiectionum &longs;olutionem, notandum <lb/>primò: ut mobile moueatur, non &longs;ufficere quemlibet <lb/>impul&longs;um, &longs;ed proportionatum illi mobili: impul&longs;us e­<lb/>nim, quo globus ligneus ad motum concitatur, haud <lb/>quaquam loco mouebit pilam ferream ejuidem molis <pb/>aut maiorem: at verò &longs;i huius impul&longs;u moueatur glo­<lb/>bus ligneus, motu agit abitur multò velociore. Secundò: <lb/> <arrow.to.target n="marg7"></arrow.to.target><lb/>hanc proportionem motus & impul&longs;us non á mole, &longs;ed <lb/>á grauitate illorum corporum determinari: <expan abbr="ita&qacute;">itaque</expan>; glo­<lb/>bus ligneus major, & glans plumbea minor, &longs;i æquipon­<lb/>derant, ab impul&longs;u æquali æquali velocitate mouentur <lb/>Simili modo &longs;i eandem rationem habeant impul&longs;us <lb/>quam habent pondera, erit velocitas motus æqualis' <lb/>Tertió percu&longs;sionem & quæ hanc &longs;equitur plagam non <lb/> <arrow.to.target n="marg8"></arrow.to.target><lb/>uno in&longs;tanti, &longs;ed in aliquo tempore quantumuis imper­<lb/>ceptibili perfici: cùm enim plaga proueniat non ex &longs;olo <lb/>contactu, &longs;ed ex irruptione violentá, quá veluti pene­<lb/>trat percutiens percu&longs;&longs;um, non e&longs;&longs;e pote&longs;t <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; motu; <lb/>cùm ergo percutiens tangit, necdum e&longs;t plaga, &longs;ed fit; <lb/>cujus &longs;ignum fragor â percu&longs;sione non ni&longs;i in tempore <lb/>proueniens. Sicuti ergo plaga &longs;ua habet incrementa, ita <lb/>determinatio impul&longs;us: & &longs;i quod mobile non totam <lb/>plagam recipit, deficiet <expan abbr="quo&qacute;">quoque</expan>; in eadem ratione impul­<lb/>&longs;us. Quartó: impul&longs;um ex&longs;pirare ubi totam perfecit <lb/> <arrow.to.target n="marg9"></arrow.to.target><lb/>plagam, partem verò non ni&longs;i cum parte emori: re&longs;idu­<lb/>um ergo plagæ &longs;eu impul&longs;us, &longs;i nihil e&longs;t quod recipiat il­<lb/>lam plagam, crit principium motus á percu&longs;sione con­<lb/>cinuati. His &longs;uppo&longs;itis, ita rem tran&longs;igemus: &longs;it ergo. </s></p> <p type="margin"> <s><margin.target id="marg1"></margin.target><emph type="italics"/>R<gap/><emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="marg2"></margin.target><emph type="italics"/>No <lb/>1.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="marg3"></margin.target><emph type="italics"/>2.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="marg4"></margin.target><emph type="italics"/>3<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="marg5"></margin.target><emph type="italics"/>4<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="marg6"></margin.target><emph type="italics"/>5<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="marg7"></margin.target>2</s></p> <p type="margin"> <s><margin.target id="marg8"></margin.target>3</s></p> <p type="margin"> <s><margin.target id="marg9"></margin.target>4</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma I.<emph.end type="italics"/><emph.end type="center"/></s></p> <pb/> <p type="main"> <s><emph type="italics"/>Siglobus alium globum percutiat quie&longs;centem & æqualem, illo <lb/>expul&longs;o quie&longs;cit.<emph.end type="italics"/></s></p> <p type="main"> <s>VT &longs;i duo globi lignei inter &longs;e &longs;int æquales, aut cum a­<lb/>lio quouis globo eju&longs;dem ponderis, <expan abbr="at&qacute;">atque</expan>; hic illum <lb/>percutiat quie&longs;centem; quia impul&longs;us percutientis ad <lb/><expan abbr="utrum&qacute;">utrumque</expan>; globum eandem habet rationem ex notabili <lb/>2. æqualis autem impul&longs;us non ni&longs;i á plag á &longs;it perfectâ, e­<lb/>rit velocitas in percu&longs;&longs;o non ante illam plagam: non er­<lb/>go incipiente plagá præcurret <expan abbr="&longs;e&qacute;">&longs;eque</expan>, auellet à <expan abbr="percuti&etilde;te">percutiente</expan>, <lb/>&longs;ed plagà demum perfectà illam velocitatem con&longs;ecu­<lb/>tus. Et quia ex norabili 4. impul&longs;us, ubi plagam peife­<lb/>cit, ex&longs;pirat; nullam verò plagam inducit globus qùie­<lb/>&longs;cens, propterea quód <expan abbr="ne&qacute;">neque</expan>; irruptio violenta &longs;eu pene­<lb/>tratio fiat ab illo globo, qui eàdem velocitate, quà percu <lb/>titur, &longs;e abducit; quie&longs;cet globus percutiens ab illa, <lb/>quam fecit, plagà. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma II.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si globus major percutiat minorem quie&longs;centem, minori expul&longs;o <lb/>eundem motum continuat major.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim minus pondus æquali celeritate mouetur <lb/>a minori impul&longs;u; illam velocitatem motus qua <lb/>præcurrit <expan abbr="&longs;e&qacute;">&longs;eque</expan>; auellit à percutiente, à minori plagâ con- <pb/>&longs;. quetur, quam ut totum impul&longs;um producat. <gap/>t qu<gap/><lb/>impul&longs;us non ni&longs;i à plagà emoritur; impulius reliquus, <lb/>qui nec dum percu&longs;sit, eundem motum continuabit. <lb/>Habeat enim pondus <emph type="italics"/>de<emph.end type="italics"/> ad pondus <emph type="italics"/>fg<emph.end type="italics"/> eandem <expan abbr="ration&etilde;">rationem</expan>, <lb/>quam habet impul&longs;us maioris <emph type="italics"/>ac<emph.end type="italics"/> ad impul&longs;um minoris <lb/> <arrow.to.target n="fig29"></arrow.to.target><lb/><emph type="italics"/>ab,<emph.end type="italics"/> <expan abbr="percutiat&qacute;">percutiatque</expan>; <emph type="italics"/>de<emph.end type="italics"/> ip&longs;um <emph type="italics"/>fg<emph.end type="italics"/>: quia ergo plagà non ni&longs;i in <lb/>aliquo tempore fit, & &longs;icuti plaga, ita <expan abbr="quo&qacute;">quoque</expan>; impul&longs;us <lb/>&longs;ua habet incrementa, erit impul&longs;us <emph type="italics"/>ab<emph.end type="italics"/> prior impul&longs;u <emph type="italics"/>ac.<emph.end type="italics"/><lb/>e&longs;t autem <emph type="italics"/>ac<emph.end type="italics"/> ad <emph type="italics"/>al,<emph.end type="italics"/> ut <emph type="italics"/>de<emph.end type="italics"/> ad <emph type="italics"/>fg<emph.end type="italics"/>: & permutando <emph type="italics"/>ac<emph.end type="italics"/> ad <emph type="italics"/>de,<emph.end type="italics"/><lb/>ut <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>fg<emph.end type="italics"/>; eadem orgò velocitas in <expan abbr="utro&qacute;">utroque</expan>;. Et quia eá­<lb/>dem velocitato mouentur, nulla à contactu erit plaga. <lb/>Ita ergo pila ferrea dum murum percutit, quia minori <lb/>impul&longs;u, ad motum concirantur partes in muro percu&longs;­<lb/>&longs;æ, illam velocitatem motus, quâ pila ferrea mouetur, <lb/>ab incipiente & necdum perfectà plagà con&longs;equuntur: <lb/>impul&longs;æ esgo motum pilæ anteuertunt, <gap/><expan abbr="uo&qacute;">uoque</expan>; impetua­<lb/>liis in&longs;tant: & &longs;icubi major vis ob&longs;tat, pila à teigo hæ­<lb/>rentes nouo impul&longs;u urget, <expan abbr="quou&longs;&qacute;">quou&longs;que</expan>; illà percu&longs;sione <expan abbr="cõ">com</expan> <lb/>tinuatà totum impul um plaga hauriat & ab&longs;umat <lb/>Quód &longs;i major &longs;it impul&longs;us, quam ut æqualis &longs;it illi pla­<lb/>gæ, quà murum perforat, motum à rupturâ continuar li­<lb/>li<gap/>exce&longs;&longs;ui æqualem. </s></p> <pb/> <figure id="fig29"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma III.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si globus minor percutiat majorem quie&longs;centem, habeat verò <lb/>minorem rationem ad &longs;uum impul&longs;um, quam ad globum majorem, <lb/>expul&longs;o majori minor quie&longs;cit aut reflectit.<emph.end type="italics"/></s></p> <p type="main"> <s>HAbeat globus <emph type="italics"/>a<emph.end type="italics"/> maior ad minorem <emph type="italics"/>b<emph.end type="italics"/> rationem du­<lb/>plam, ide&longs;t grauitas &longs;eu pondus majoris &longs;it duplum <lb/>ponderis minoris; impul&longs;us autem minoris ad eju&longs;dem <lb/>grauitatem in ratione majori quam dupl<gap/>. Quia ergo <lb/>grauitas & impul&longs;us inter &longs;e &longs;unt contraria, erit motus <lb/>æqualis exce&longs;&longs;ui maioris; e&longs;t autem impul&longs;us minoris <lb/>maior grauitate maioris, propterea quód ad grauitatem <lb/>minoris maiorem habeat rationem; erit ergo huius ex­<lb/>ce&longs;&longs;us principium motus maiori. Igitur &longs;i globus mi­<lb/>nor percutiat maiorem, quia ab æquali impul&longs;u minor <lb/>e&longs;t velocitas motus, non ante perfectam plagam auelli <lb/>pote&longs;t à percutiente: & quia à plagà perfectâ emoritur <lb/>impul&longs;us, minori autem velocitate maior &longs;e abducit ab <lb/>illà plagà, quàm irruptio fiat minoris; repercutiet ma <lb/>ior minorem, <expan abbr="erit&qacute;">eritque</expan>; huius plaga ad men&longs;uram illius tar­<lb/>ditatis. Globus ergo minor, ubi percu&longs;sit maiorem, illo <lb/>expul&longs;o reflectit. Quòd &longs;i ob motum velociorem nullà <lb/>à percu&longs;&longs;o inducitur plaga, minor expul&longs;o maiori qui­<lb/>e&longs;cit. </s></p> <pb/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma IV.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si globus minor percutiat majorem quie&longs;centem, habeat verò <lb/>majorem rationem ad &longs;uum impul&longs;um, quam ad globum majorem, <lb/>illo immoto reflectit minor.<emph.end type="italics"/></s></p> <p type="main"> <s>VT &longs;i im pul&longs;us, quo minor globus mouetur, ad illius <lb/>grauitatem &longs;it in ratione duplà; globus veró major <lb/>ad minorem rationem habeat maiorem quam duplam, <lb/>erit impul&longs;us minoris minor grauitate maioris; non er­<lb/>gò <expan abbr="illã">illam</expan> mouere valebit, propterea quód motus ab exce&longs;­<lb/>&longs;u fiat maioris. Quód &longs;i ergo minor globus percutiat <lb/>maiorem, quia ex illà plagà minor e&longs;t impul&longs;us, quam ut <lb/>loco moueat; globus quidem maior à percu&longs;sione qui <lb/>e&longs;cit, minor verò quia à percu&longs;&longs;o quie&longs;cente nouam & <lb/>æqualem illi, quam dedit, plagam recipit, motum refle­<lb/>ctit. Ex iam definitis di&longs;&longs;oluemus & hoc </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Problema I.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Globum in plano quie&longs;centem percutere alio globo <expan abbr="quacun&qacute;">quacunque</expan>; vi­<lb/>olentià, <expan abbr="ne&qacute;">neque</expan>; tamen loco mouere.<emph.end type="italics"/></s></p> <p type="main"> <s>AS &longs;umatur globus <emph type="italics"/>a<emph.end type="italics"/> <expan abbr="cuiu&longs;cunq;">cuiu&longs;cunque</expan> molis & ponderis, eius <lb/>tamen firmitatis, quò totum impetum &longs;ufferre vale­<lb/>at, <expan abbr="ne&qacute;">neque</expan>; di&longs;siliat ex illo ictu: <expan abbr="con&longs;tituatur&qacute;">con&longs;tituaturque</expan>; in plano <emph type="italics"/>AB<emph.end type="italics"/> <pb/>liberè, & <expan abbr="ab&longs;q;">ab&longs;que</expan> ullo neku: <expan abbr="qu&etilde;">quem</expan> percuti volumas ab alio <lb/>globo, æquali tamen aut minori, <expan abbr="quacũ&qacute;">quacunque</expan>; violentia, <expan abbr="at&qacute;">atque</expan>; <lb/>adeò à machinà bellicà efful minato, <expan abbr="ne&qacute;">neque</expan>; tamen &longs;uo &longs;o­<lb/>co moueri. quod quidem nullis machinis, aut retinacu­<lb/>lis, &longs;ed duntaxat unius globi appo&longs;itione con&longs;eque­<lb/> <arrow.to.target n="fig30"></arrow.to.target><lb/>mur, quiiram illius fulminis à globo percu&longs;&longs;o hauriat & <lb/>ab&longs;umat. Appone ergo à tergo alium globum illi æqua <lb/>lem <emph type="italics"/>b,<emph.end type="italics"/> & &longs;it linea motus pilæ ad utrum <expan abbr="&qacute;">que</expan> globum perpen <lb/>dicularis; dico globum <emph type="italics"/>a<emph.end type="italics"/> nulla ratione loco moueria <lb/>globo <emph type="italics"/>d.<emph.end type="italics"/> Quia enim globus <emph type="italics"/>a<emph.end type="italics"/> eodem momento, quo <lb/>percutitur à globo <emph type="italics"/>d,<emph.end type="italics"/> percutit globum <emph type="italics"/>b<emph.end type="italics"/> &longs;ibi æqualem, <lb/>inducet illà percu&longs;sione plagam peifectam, ac proinde <pb/>per Poril: 1. â percu&longs;sione quie&longs;cet. Quòd &longs;i plures glo­<lb/>bi æquales &longs;e <expan abbr="contingãt">contingant</expan> in lineà motus centri, ut <emph type="italics"/>f.g.h.i,<emph.end type="italics"/><lb/>percu&longs;&longs;o <emph type="italics"/>f<emph.end type="italics"/>primo ab æquali <emph type="italics"/>e,<emph.end type="italics"/> ultimus <emph type="italics"/>i<emph.end type="italics"/> mouetur, reliquis <lb/><emph type="italics"/>f.g.h<emph.end type="italics"/> immotis; propterea quód per Pori&longs;. <emph type="italics"/>1.<emph.end type="italics"/> po&longs;terior <lb/>prioris exhaurit plagam. At verò &longs;i unus æqualium po&longs;t <lb/>fe habeat minores <expan abbr="quotcun&qacute;">quotcunque</expan>; ut <emph type="italics"/>o.p.q.<emph.end type="italics"/> percu&longs;&longs;o à <emph type="italics"/>k<emph.end type="italics"/> æqua­<lb/>li <emph type="italics"/>l,<emph.end type="italics"/> omnes cum <emph type="italics"/>l<emph.end type="italics"/> moto mouentur, ut con&longs;tat per Pori&longs;.2. <lb/>Quòd &longs;i demum percu&longs;sio incipiat à minori <emph type="italics"/>q<emph.end type="italics"/> ug: omni­<lb/>bus immotis aut reflexis ultimus mouetur, per Pori&longs;. 3. <lb/>aut &longs;i minor o&longs;t impul&longs;us grauitate, quie&longs;cit, per Pori&longs;. <lb/>4. Eadem vià di&longs;&longs;oluemus hoc </s></p> <figure id="fig30"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Problema II.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Globum in plano quis&longs;centem alio globo <expan abbr="quacun&qacute;">quacunque</expan>; violentià per <lb/>cu&longs;&longs;um, ad impe<gap/>am di&longs;t antiam mouere.<emph.end type="italics"/></s></p> <p type="main"> <s>VT &longs;i globum <emph type="italics"/>b<emph.end type="italics"/> ab alio globo æqualiaut minori <expan abbr="qua-cun&qacute;">qua­<lb/>cunque</expan>; violentiâ percu&longs;&longs;um, ad locum determinatum <lb/>vg: <emph type="italics"/>c<emph.end type="italics"/> mouere velis, <expan abbr="ne&qacute;">neque</expan>; limitem hunc præterire, quan­<lb/>tumuis effræni impetu feratur. In eodem loco, quem <lb/>terminum illi motui præfixi&longs;ti, globum con&longs;titue æqua­<lb/>lem, dico in eodem loco à motu quie&longs;cere globum <emph type="italics"/>b.<emph.end type="italics"/><lb/>Quia enim globum <emph type="italics"/>c<emph.end type="italics"/> quie&longs;centem percutit globus æ­<lb/>qualis <emph type="italics"/>b,<emph.end type="italics"/> per Pori&longs;. <emph type="italics"/>i<emph.end type="italics"/>quie&longs;cet ex illa, quam fecit, plagâ. </s></p> <pb/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma V.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>St duo globi eju&longs;dem molis &longs;eu ponderis &longs;e percutiant in motu, <lb/><expan abbr="uter&qacute;">uterque</expan>; reflectit.<emph.end type="italics"/></s></p> <p type="main"> <s>NAm quia idem pondus <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>;, erit <expan abbr="quoq;">quoque</expan> velocitas <lb/>motus, quam plaga inducit, æqualis; eadem ergo ve­<lb/>locitate reflectit percutiens, quà percu&longs;&longs;um mouebatur. <lb/>Ex quo fit manife&longs;tum illorum velo citatem, quæ in mo <lb/>tu &longs;e percutiunt, à percu&longs;sione permutari: quæ enim ma <lb/>gis percutiunt, minùs; & quæ minùs percutiunt, magis <lb/>impetuo&longs;è reflectunt. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma VI.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Siglobus mæjor in mot<gap/> percutiat minorem, babeat verò minor <lb/>minorem rationem ad &longs;u<gap/>, quam <gap/>lobum majorem, <lb/><expan abbr="uter&qacute;">uterque</expan>; reflectit.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim major e&longs;t impul&longs;us minoris grauitate ma­<lb/>joris, ob minorem hujus quam illius ratio nem, &longs;i m<gap/><lb/>nor percutiat majorem, mouebitur ex illà plagà major: <lb/>reflectit autem & minor à majori, propterea quód à qua <lb/><expan abbr="cun&qacute;">cunque</expan>; hujus plagâ mouetur minor. Igitur &longs;i globus ma­<lb/>jor in motu percutiat minorem &c. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma VII.<emph.end type="italics"/><emph.end type="center"/></s></p> <pb/> <p type="main"> <s><emph type="italics"/>Si globus major in motu percutiat minorem, habeat verò minor <lb/>majorem rationem ad &longs;uum impul&longs;um, quam ad globum majorem, <lb/>minori reflexo motum continuat major.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim minor e&longs;t impul&longs;us minoris grauitate ma <lb/>joris, propterea quòd minorem ad hanc quam adim <lb/>pul&longs;um habeat rationem, non poterit grauitas majoris <lb/>moueri ex impul&longs;u minoris: licet ergo plaga fiat à mi­<lb/>nori, quia tamen minorem producit impul&longs;um, quam <lb/>ut grauitatem majorisloco moueat, non pote&longs;t ex illà <lb/>plagà reflecti major. Quia verò à minori impul&longs;u æqua <lb/>li velocitate mouetur minor, erit velocitas in minori æ­<lb/>qualis velocitati majoris à plagà necdum perfectà: im­<lb/>pul&longs;us ergo reliquus, qui necdum percu&longs;sit, motum con­<lb/>tinuabit. Si ergo globus major in motu percutiat mi­<lb/>norem &c. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Pori&longs;ma VIII.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si globus major in motu percutiat minorem, haléat verò minor <lb/>ad majorem eandem rationem, quam habet ad &longs;uum impul&longs;um, mi­<lb/>nori reflexo quie&longs;cit major.<emph.end type="italics"/></s></p> <p type="main"> <s>MInore m<gap/>quidem globem à majori reflecti con&longs;tat, <lb/>propterea quód ex liujus plagà impul&longs;us quidem æ­<lb/>qualis, maior autem velo citas in minori con&longs;e quatur: àt <lb/>verò globum maiorem â percu&longs;sione quie&longs;cere, cùme- <pb/>andem habet rationem minor ad hunc, quam habet ad <lb/>&longs;uum impul&longs;um, ita o&longs;tendemus: motus non ni&longs;i ab ex­<lb/>ce&longs;&longs;u fit maioris; at verò impul&longs;us ex illà plagà, quam in­<lb/>ducit minor in maiori, non maior &longs;ed æqualis e&longs;t eiu&longs;­<lb/>dem grauitati, ex &longs;uppo&longs;itione; non ergo ex illo impul­<lb/>&longs;u moueri pote&longs;t major. Quia verò à percu&longs;sione exol­<lb/>uitur, minor autem, quam ut mouere po&longs;sit, impul&longs;us <lb/>regeneratur, quie&longs;cet ex illà plagà globus maior. </s></p> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Cùm centrum grauitatis cadit extra lineam hypomochlij, motus <lb/>in illam partem, in quà e&longs;t centrum, reflectit.<emph.end type="italics"/></s></p> <p type="main"> <s>OCcurrat globus <emph type="italics"/>dcg<emph.end type="italics"/> plano <emph type="italics"/>ab<emph.end type="italics"/> non perpen di culari­<lb/>ter, &longs;ed ex obliquo, faciens angulum incidentiæ <emph type="italics"/>adc<emph.end type="italics"/><lb/>acutum, <expan abbr="erit&qacute;">eritque</expan>; linea <emph type="italics"/>cd<emph.end type="italics"/> ducta per contactum linea hypo­<lb/>mochlii, & motui centri parallela, centrum verò <emph type="italics"/>e<emph.end type="italics"/> extra <lb/>lineam hypomochlii: dico ex puncto contactus <emph type="italics"/>a<emph.end type="italics"/> mo­<lb/>tum reflexum fieriin illam partem, in quâ e&longs;t centrum <emph type="italics"/>e.<emph.end type="italics"/><lb/>Quia enim motus & plaga ad motum fit centri: <expan abbr="centrũ">centrum</expan> <lb/>verò <emph type="italics"/>e<emph.end type="italics"/> plano occurrrit per lineam <emph type="italics"/>ed,<emph.end type="italics"/> <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; maior re&longs;i&longs;ten <lb/>tia in plano quam impul&longs;us, erit motus reflexus ad partes <lb/>oppo&longs;itas illi plagæ, ac proinde in partem in quà e&longs;t cen­<lb/>trum. </s></p> <pb/> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus reflexus fit per lineam parallelam illi lineæ, quæ cum lineà <lb/>perpendiculari ad contactum angulum con&longs;tituit in centro, cujus &longs;i­<lb/>nus e&longs;t æqualis interuallo inter centrum grauitatis & lineam hy­<lb/>pomochlij.<emph.end type="italics"/></s></p> <p type="main"> <s>IN eàdem figurà ducatur ex <emph type="italics"/>e<emph.end type="italics"/> centro grauitatis &longs;eu im­<lb/>pul&longs;us linea <emph type="italics"/>ef<emph.end type="italics"/> perpendieularis ad lineam hypomo­<lb/>chlii <emph type="italics"/>cd,<emph.end type="italics"/> & linea <emph type="italics"/>eg<emph.end type="italics"/> faciens cum lineà <emph type="italics"/>dh<emph.end type="italics"/> perpendiculari <lb/>ad contactum in eodem centro <emph type="italics"/>e<emph.end type="italics"/> angulum <emph type="italics"/>heg,<emph.end type="italics"/> cuius &longs;i­<lb/>nus <emph type="italics"/>hg<emph.end type="italics"/> &longs;it æqualis lineæ <emph type="italics"/>fe<emph.end type="italics"/> interuallo inter centrum gra­<lb/> <arrow.to.target n="fig31"></arrow.to.target><lb/>uitatis <gap/> & lineam hypomochlii: dico motum reflexum <lb/>fieri per lineam <emph type="italics"/>di<emph.end type="italics"/> parallelam lineæ <emph type="italics"/>eg.<emph.end type="italics"/> Quia enim cen­<lb/>trum grauitatis, dum &longs;uà mole ferit planum in puncto <emph type="italics"/>d<emph.end type="italics"/> <pb/>per lineam <emph type="italics"/>ed<emph.end type="italics"/> &longs;e ip&longs;um veluti partitur: illa quidem pars <lb/>quæ hypomochlio in&longs;i&longs;tit, at&queacute;; <expan abbr="illãm">illamm</expan> plagam inducit, ea­<lb/>dem vià, quá impulit, & impul&longs;u æquali retro agitur: re­<lb/>liqua verò, quæ cum centro extra hypomochlium ca­<lb/>dit, per lineam fertur <emph type="italics"/>ek<emph.end type="italics"/> parallelam lineæ <emph type="italics"/>db,<emph.end type="italics"/> propterea <lb/>quód hæc &longs;it proxima motui grauitatis ab hypomo­<lb/>chlio impeditæ. Quia ergo motus <emph type="italics"/>eh.ek,<emph.end type="italics"/> quibus cen­<lb/>trum grauitatis agitur, &longs;ecundúm quid &longs;unt contrarii, <lb/>propterea quód angulus <emph type="italics"/>hek<emph.end type="italics"/> &longs;it minor duobus rectis, e­<lb/>rit motus mixtus per lineam mediam inter <emph type="italics"/>eh<emph.end type="italics"/> & <emph type="italics"/>ek,<emph.end type="italics"/> cu­<lb/>jus interuallum determinat &longs;inus complementi inclina­<lb/>cionis, in ratione quam habent impul&longs;us per Prop; 31. e&longs;t <lb/>autem interuallum <emph type="italics"/>fe,<emph.end type="italics"/> hoc e&longs;t &longs;inus <emph type="italics"/>dm<emph.end type="italics"/> anguli <emph type="italics"/>dem,<emph.end type="italics"/> men­<lb/>&longs;ura grauitatis extra hypomochlium; linea vero <emph type="italics"/>fd<emph.end type="italics"/> &longs;inus <lb/>anguli reliqui men &longs;uraillius, quæ hypomoch&longs;io in&longs;i&longs;tit <lb/>grauitatis: &longs;i fiat ut <emph type="italics"/>fd<emph.end type="italics"/> ad <emph type="italics"/>ef,<emph.end type="italics"/> ita <emph type="italics"/>kg<emph.end type="italics"/> &longs;inus complementi an <lb/>guli <emph type="italics"/>heg<emph.end type="italics"/> ad<emph type="italics"/>hg<emph.end type="italics"/> &longs;inum complementi anguli <gap/>it li­<lb/>nea <emph type="italics"/>eg<emph.end type="italics"/> linea motus mixti ex <emph type="italics"/>eh<emph.end type="italics"/> & <emph type="italics"/>e<gap/><emph.end type="italics"/> per Prop: 31. Vel &longs;ic <lb/>motus reflexus fit per lineam <emph type="italics"/>de<emph.end type="italics"/> perpendicu<gap/>arem ad <lb/>contactum; inclinatio autem motus reflexi augetur in <lb/>ratione interualli inter centrum grauitatis & hypomo­<lb/>chlium: Si igitur fiat ut &longs;inus totus nimirum motus re­<lb/>flexus, ad men&longs;uiam hujus interuàll<gap/>, hoc e&longs;t grau<gap/>atem <lb/>extra hypomochlium, ita linea motus <emph type="italics"/>eh<emph.end type="italics"/> &longs;inus nimirum <lb/>anguli <emph type="italics"/>hek,<emph.end type="italics"/> hoc e&longs;t &longs;inus totus ad &longs;inum <emph type="italics"/>hg<emph.end type="italics"/> anguli incli- <pb/>nationis, erit eadem linea <emph type="italics"/>eg<emph.end type="italics"/> motus mixti. Quia ergo <lb/>mobile mouetur ad motum &longs;ui centri, erit motus ex <emph type="italics"/>d<emph.end type="italics"/><lb/>reflexus per lineam parallelam illilineæ, quæ cum lineà <lb/>perpendiculari ad contactum angulum con&longs;tituit in <lb/>centro, cujus &longs;inus e&longs;t æqualis interuallo inter centrum <lb/>grauitatis & lineam hypomochlij. </s></p> <figure id="fig31"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Anguli incidentiæ & reflexionis &longs;unt inter &longs;e æquales.<emph.end type="italics"/></s></p> <p type="main"> <s>QVia enim duo latera <emph type="italics"/>eh.bg<emph.end type="italics"/> trianguli <emph type="italics"/>ehg<emph.end type="italics"/> æqualia <lb/>&longs;unt duobus lateribus <emph type="italics"/>ef. fd<emph.end type="italics"/> trianguli <emph type="italics"/>efd,<emph.end type="italics"/> & angu­<lb/>lus, qui adjacet uni æqualium laterum, rectus, erunt tri­<lb/>angula æqualia, & angulus <emph type="italics"/>fde<emph.end type="italics"/> angulo <emph type="italics"/>heg<emph.end type="italics"/> æqualis: e&longs;t <lb/>autem angulo <emph type="italics"/>heg<emph.end type="italics"/> æqualis angulus <emph type="italics"/>edi<emph.end type="italics"/> ob parallelas <emph type="italics"/>eg. <lb/>di<emph.end type="italics"/>; idem ergo angulus <emph type="italics"/>edi<emph.end type="italics"/> e&longs;t æqualis angulo <emph type="italics"/>fde:<emph.end type="italics"/> &longs;unt <lb/>verò duo <expan abbr="quo&qacute;">quoque</expan>; anguli <emph type="italics"/>a.de.bde<emph.end type="italics"/> inter le æquales, nimi­<lb/>rum recti; ablatis ergo duobus angulis <emph type="italics"/>fde.edi<emph.end type="italics"/> æquali­<lb/>bus, erunt anguli reliqui <emph type="italics"/>adf.bdi,<emph.end type="italics"/> anguli nimirum inci­<lb/>dentiæ & reflexionis inter &longs;e æquales. Priu&longs;quam de mo <lb/>tu reflexo finiamus, unum <expan abbr="at&qacute;">atque</expan>; alterum Problema pro <lb/>corollario adducemus, quorum &longs;olutio magis difficilis <lb/>habetur, ex ijs autem, quæ hactenus &longs;unt demon&longs;trata, <lb/>facilè di&longs;&longs;oluuntur. Sit ergo </s></p> <pb/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Problema<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Tribus globis in <expan abbr="quacun&qacute;">quacunque</expan>; di&longs;tantia extra lineam rectam a&longs;&longs;um <lb/>ptis, punctum determinare in globo &longs;ecundo, à quo reflexus primus <lb/>percutiat tertium.<emph.end type="italics"/></s></p> <p type="main"> <s>IN figurà &longs;ubiectà a&longs;&longs;umantur globi <emph type="italics"/>s.p.r.<emph.end type="italics"/> in di&longs;tanciâ <lb/><emph type="italics"/>sp.pr.rs:<emph.end type="italics"/> <expan abbr="oporteat&qacute;">oporteatque</expan>; in globo <emph type="italics"/>p<emph.end type="italics"/> punctum determina­<lb/>re, ad quod globus <emph type="italics"/>s<emph.end type="italics"/> allidens, <expan abbr="inde&qacute;">indeque</expan>; reflexus percutiat <lb/>globum <emph type="italics"/>r.<emph.end type="italics"/> Tangant illos globos lineæ <emph type="italics"/>ac. bd<emph.end type="italics"/> in punctis <lb/><emph type="italics"/>a.c. b.d,<emph.end type="italics"/> & diuidantur bifariam in punctis <emph type="italics"/>e<emph.end type="italics"/> & <emph type="italics"/>f;<emph.end type="italics"/> à quibus in <lb/>circulum <emph type="italics"/>p<emph.end type="italics"/> excurrant lineæ rectæ <emph type="italics"/>eg.fg.<emph.end type="italics"/> &longs;e interlecantes <lb/>in puncto reflexionis <emph type="italics"/>g,<emph.end type="italics"/> eo modo, quo docent Optici in­<lb/>uento, & producantur <expan abbr="utrin&qacute;">utrinque</expan>; in <emph type="italics"/>k.l,<emph.end type="italics"/> & <emph type="italics"/>h. i;<emph.end type="italics"/> dico <expan abbr="punctũ">punctum</expan> <lb/><emph type="italics"/>g<emph.end type="italics"/> e&longs;&longs;e illud pnnctum, â quo globus <emph type="italics"/>s<emph.end type="italics"/> reflexus percutiat <lb/>globum<emph type="italics"/>r.<emph.end type="italics"/> Quia enim angulus <emph type="italics"/>egd<emph.end type="italics"/> angulo <emph type="italics"/>fgc<emph.end type="italics"/> per con­<lb/>&longs;tructionem, & angulus <emph type="italics"/>egh<emph.end type="italics"/> angulo <emph type="italics"/>fgk<emph.end type="italics"/> ad verticem e&longs;t <lb/>æquali<emph type="italics"/>s<emph.end type="italics"/>; ablatis ex his illis erunt anguli reliqui <emph type="italics"/>hgd. kge<emph.end type="italics"/><lb/>æquales: linea ergo &longs;ubten&longs;a <emph type="italics"/>hg<emph.end type="italics"/> e&longs;t æqualis lineæ <emph type="italics"/>kg.<emph.end type="italics"/> & <lb/>quia linea <emph type="italics"/>fd<emph.end type="italics"/> lineæ <emph type="italics"/>fb,<emph.end type="italics"/> & angulus <emph type="italics"/>dfg<emph.end type="italics"/> e&longs;t æqualis angulo <lb/><emph type="italics"/>bfn,<emph.end type="italics"/> erit corda <emph type="italics"/>gh<emph.end type="italics"/> æqualis cordæ<emph type="italics"/>ni.<emph.end type="italics"/> Similiter o&longs;tende­<lb/>mus cordam <emph type="italics"/>gk<emph.end type="italics"/> æqualem cordæ <emph type="italics"/>ml.<emph.end type="italics"/> Ducatur ergo per <lb/>contactum â centro <emph type="italics"/>p<emph.end type="italics"/> linea <emph type="italics"/>pq,<emph.end type="italics"/> <expan abbr="at&qacute;">atque</expan>, ex <emph type="italics"/>q<emph.end type="italics"/> circulus de­<lb/>&longs;cribatur æqualis circulo<emph type="italics"/>s,<emph.end type="italics"/> tangens priorem in <emph type="italics"/>g,<emph.end type="italics"/> <expan abbr="agatur&qacute;">agaturque</expan>; <lb/>linea <emph type="italics"/>qr<emph.end type="italics"/> parallcla lineæ <emph type="italics"/>gi<emph.end type="italics"/>: quòd &longs;i ergo globus <emph type="italics"/>s<emph.end type="italics"/> motu &longs;ui <pb/>centri de&longs;cribat lineam <emph type="italics"/>sq,<emph.end type="italics"/> de&longs;cribet punctum <emph type="italics"/>m<emph.end type="italics"/> motu <lb/>&longs;imili lineam <emph type="italics"/>mg<emph.end type="italics"/> illi parallelam, <expan abbr="tanget&qacute;">tangetque</expan>; globus <emph type="italics"/>s<emph.end type="italics"/> <expan abbr="globũ">globum</expan> <lb/><emph type="italics"/>p<emph.end type="italics"/> in puncto <emph type="italics"/>g<emph.end type="italics"/>: dico punctum <emph type="italics"/>m<emph.end type="italics"/> ex <emph type="italics"/>g<emph.end type="italics"/> per lineam <emph type="italics"/>gi,<emph.end type="italics"/> cen­<lb/>trum veró <emph type="italics"/>q<emph.end type="italics"/> per lineam <emph type="italics"/>qr<emph.end type="italics"/> illi parallelam reflecti. Erit <lb/>enim <emph type="italics"/>gy<emph.end type="italics"/> linea hypomochlir, ad quam ex <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> cadat linea <lb/> <arrow.to.target n="fig32"></arrow.to.target><lb/>perpendicularis <emph type="italics"/>qt,<emph.end type="italics"/> <expan abbr="at&qacute;">atque</expan>; huic æqualis &longs;umatur in lineâ <lb/>motus centri <emph type="italics"/>qz,<emph.end type="italics"/> à cujus termino <emph type="italics"/>z<emph.end type="italics"/> ducta linea perpendi­<lb/>cularis &longs;ecabit circulum in puncto <emph type="italics"/>x,<emph.end type="italics"/> per quod tran&longs;it li­<lb/>nea motus reflexi per Prop 39. tribus ergò globis extra <lb/>lineam rectam a&longs;&longs;umptis punctum determ inauimus in <pb/>globo &longs;ecundo, à quo reflexus primus tangit tertium: <lb/>quod erat faciendum. Secundum Problema. </s></p> <figure id="fig32"></figure> <p type="main"> <s><emph type="center"/>DE MOTV REFLEXO <expan abbr="LAPILLORũ">LAPILLORum</expan> EX AQVA.<emph.end type="center"/></s></p> <p type="main"> <s>QVi obliquè incidentes illam minimè findunt, <lb/><expan abbr="ne&qacute;ue">neque</expan> merguntur; verùm inde reflexi, <expan abbr="at&qacute;">atque</expan>; ite­<lb/>rum relap&longs;i reciprocà alli&longs;ione, & reli&longs;ione &longs;altu quodam <lb/>progredi videntur. E&longs;t autem prima difficultas, quam <lb/>ob rem huju&longs;modi lapilli, <expan abbr="quacun&qacute;">quacunque</expan>; violentià projecti, <lb/>aquam molli&longs;simam non perrumpant, in quâ etiam pul­<lb/>ui&longs;culus & leui&longs;simæ arenulæ &longs;uà grauitate &longs;idunt. Se­<lb/>cunda quà ratibne â primâ reflexione alias inducant pla­<lb/>gas non perpendiculares: conuer&longs;io enim illa motus vi­<lb/>detur non ni&longs;i â grauitate na&longs;ci, quo modo in omnibus <lb/>projectis fieri con&longs;tat: at verò grauitas non ni&longs;i per line­<lb/>am mouet perpendicularem. In figurà &longs;ubjectà lapillus <lb/>&longs;eu globulus <emph type="italics"/>a<emph.end type="italics"/> â percu&longs;sione obliquà <emph type="italics"/>ba<emph.end type="italics"/> reflectit in <emph type="italics"/>k<emph.end type="italics"/>: in <lb/>de verò non perpendiculariter in <emph type="italics"/>q,<emph.end type="italics"/> verùm obliquè rela­<lb/>bitur in <emph type="italics"/>l,<emph.end type="italics"/> <expan abbr="noua&qacute;">nouaque</expan>; illatà & relatâ plagà reflectit in <emph type="italics"/>m<emph.end type="italics"/>: &longs;imi­<lb/>liter ex <emph type="italics"/>m<emph.end type="italics"/> in <emph type="italics"/>u,<emph.end type="italics"/> & ex <emph type="italics"/>o<emph.end type="italics"/> in <emph type="italics"/>x<emph.end type="italics"/> ad nouam &longs;e ex obliquo vibrat <lb/>plagam. Hujus autem &longs;olutio pendet ex his, quæ de mo <lb/>tu reflexo â nobis &longs;unt dicta. Quia enim percu&longs;sio fit á <lb/>centro, magnitudo autem placæ ab hypomochlio deter­<lb/>minatur; quó enim major pars hypomochlio occurrit, <lb/>có majorem plagam inducit, unde ictus graui&longs;simus per <pb/>pendiculatis; propterea quód cum centro partès omnes <lb/>coincidunt, <expan abbr="at&qacute;">atque</expan>; in illam plagam cooperantur: quó ve­<lb/>rò ictus magis e&longs;t obliquus, có minorem plagam infert. <lb/>Quia ergo lapilli obliquè incidentes non ni&longs;i parte exi­<lb/>guà feriunt, major autem vis extra hypomochlium ca­<lb/>dit. <expan abbr="ob&longs;tat&qacute;">ob&longs;tatque</expan>; quò minùs illa &longs;uo fulcro innitatur; inde fit <lb/>ut non mergantur, <expan abbr="ne&qacute;">neque</expan>; findant quantumuis mollem a­<lb/> <arrow.to.target n="fig33"></arrow.to.target><lb/>quam. In globulo enim <emph type="italics"/>a<emph.end type="italics"/> &longs;ola pars <emph type="italics"/>dic<emph.end type="italics"/> hypomochlio oc <lb/>currit, reliqua <emph type="italics"/>dghci<emph.end type="italics"/> cum centro <emph type="italics"/>a<emph.end type="italics"/> extra hypomochli­<lb/>um cadit, <expan abbr="at&qacute;">atque</expan>; ab illâ plagà idem mobile abducit. Quia <lb/>verò minor e&longs;t plaga, quam ut perrum pat; recipiet à per <lb/>cu&longs;&longs;o æqualem, qua re&longs;iliat, plagam, ac proinde mino <lb/>rem, quam ut impul&longs;um producatilli æqualem, quo cen <lb/>trum mouetur. Motus ergò reflexus e&longs;t mixtus ex motu <pb/>centri <emph type="italics"/>ag<emph.end type="italics"/> à primà, & motu <emph type="italics"/>af<emph.end type="italics"/> à plagâ &longs;ecundà, linea <emph type="italics"/>v<emph.end type="italics"/>erò <lb/>motus teflexi <emph type="italics"/>ah<emph.end type="italics"/> per Prop: 39. quia ergo minor impul&longs;us <lb/>à reflexione, impul&longs;u, quo centrum agitur, deficiet pri­<lb/>ùs, <expan abbr="illo&qacute;">illoque</expan>; deficiente motum continuabit major impul­<lb/>lus; & priu&longs;quam &longs;ui juris &longs;it, lineà motus mixti &longs;inuo­<lb/>&longs;è, quomodo grauia à motu violento, &longs;e abducet; inde <lb/>per tangentem arcus jam deficientis, ac proinde ex obli­<lb/>quo &longs;e deuoluet, ut nouà illatà & relatà plagâ &longs;e rur&longs;um <lb/>attollat. Quia verò illo cur&longs;u & recur&longs;u virtus elangue <lb/>&longs;cit, quantumuis æquali parte feriat, minor tamen â per­<lb/>cu&longs;sione &longs;ecundâ fit plaga, quam ut motus inde reflexus <lb/>&longs;it æqualis primo: inde eigo fit ut à &longs;ecundà percu&longs;sione <lb/>in <emph type="italics"/>d<emph.end type="italics"/> minor &longs;it altitudo motus reflexi in <emph type="italics"/>m<emph.end type="italics"/>; & in <emph type="italics"/>o<emph.end type="italics"/> minor <lb/>quàm in <emph type="italics"/>m,<emph.end type="italics"/> <expan abbr="quou&longs;&qacute;">quou&longs;que</expan>; demum motus centri à percu&longs;sioni­<lb/>bus iteratis exoluatur: aut quia minor in fine altitudo <lb/>motus reflexi, quam diameter illius lapilli &longs;eu globuli, <lb/>ob aquam motui reluctantem ict us emoritur; <expan abbr="at&qacute;">atque</expan>; inde <lb/>fit, quòd in fine motus ab huju&longs;modi lapillis aqua di&longs;per <lb/>gatur: à <emph type="italics"/>p<emph.end type="italics"/> enim in <emph type="italics"/>q<emph.end type="italics"/> reflexus motus, ob altitudinem dia­<lb/>metro minorem, viam incedit <emph type="italics"/>pq<emph.end type="italics"/> ob aquæ grauitatem <lb/>magis impeditam. Non &longs;olúm verò in aquà ex huju&longs;mo­<lb/>di ictu obliquo fiunt repercu&longs;siones, verum in <expan abbr="quocun&qacute;">quocunque</expan>; <lb/>alio plano minùs tamen &longs;en&longs;ibiles: cujus ratio e&longs;t mol­<lb/>lities aquæ, quæ pre&longs;&longs;a rea&longs;&longs;urgit, <expan abbr="ictu&qacute;">ictuque</expan>; geminato ferit. <lb/><expan abbr="Ita&qacute;">Itaque</expan>; videmus pilas lu&longs;orias magis re&longs;ilire, quæ â plagà ce <pb/>duntin &longs;e ip&longs;as, & veluti complanantur, <expan abbr="at&qacute;">atque</expan>; ita plagam <lb/>inducuntlatiorem; mox verò â plagâ impul&longs;u gemina­<lb/>to rea&longs;&longs;urgunt: idem enim fit &longs;i<gap/>è planum, &longs;iuè mobile <lb/>eidem plano alli&longs;um eàratione moucatur. Similes ictus <lb/>repetiti fiunt in cauo &longs;phærico, cujulmodi peluis: ab <lb/> <arrow.to.target n="fig34"></arrow.to.target><lb/>uno enim puncto reflexus globus in alia porro offendit <lb/>& allidit: ut &longs;iglobus ex <emph type="italics"/>l<emph.end type="italics"/> demittatur in peluim <emph type="italics"/>msbp,<emph.end type="italics"/> a <lb/>puncto <emph type="italics"/>m<emph.end type="italics"/> ad angulos reflectit æquales in <emph type="italics"/>n,<emph.end type="italics"/> ex<emph type="italics"/>n<emph.end type="italics"/> verò in <lb/><emph type="italics"/>b,<emph.end type="italics"/> ex <emph type="italics"/>b<emph.end type="italics"/> in <emph type="italics"/>o,<emph.end type="italics"/> tum in <emph type="italics"/>p,<emph.end type="italics"/> à quo extra peluim reflectit in <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> <expan abbr="Id&etilde;">Idem</expan> <lb/>ex <emph type="italics"/>r<emph.end type="italics"/> delap&longs;us in <emph type="italics"/>s<emph.end type="italics"/> maiori angulo reflectens, ob cordas ma <lb/>iores, pauciores inducit plagas. Ex <emph type="italics"/>z<emph.end type="italics"/> demum in <emph type="italics"/>b<emph.end type="italics"/> refle­<lb/>xus quia nullibi offendit, quemadmodum <expan abbr="ne&qacute;">neque</expan>; in linea <lb/>perpendiculari <emph type="italics"/>ab,<emph.end type="italics"/> nullam præterea inducit plagam. <lb/>Tertium Problema. </s></p> <pb/> <figure id="fig33"></figure> <figure id="fig34"></figure> <p type="main"> <s><emph type="center"/>DE REFLEXIONE MOTVS CIRCVLARIS.<emph.end type="center"/></s></p> <p type="main"> <s>VT &longs;i duo globi ab eodem hypomochlio filo &longs;u&longs;pen&longs;i, <lb/>& in &longs;uam &longs;tationem recurrentes &longs;e percutiant in il­<lb/>lo motu. Quia enim hic motus di&longs;cedit à lineà rectà, per <lb/>quam ducit impul&longs;us, nece&longs;&longs;e alio modo reflexionem fi­<lb/>eri, quám in motu recto. Mouetur autem vel unus tan­<lb/>tum, vel <expan abbr="uter&qacute;">uterque</expan>;, ac proinde hic illum percutit aut quies <lb/>centem, aut commotum; & &longs;iquidem percu&longs;sio fiat in <lb/>motu, <expan abbr="uter&qacute;">uterque</expan>; reflectit: Si verò quie&longs;cit alter, interdum <lb/>reflectit ille qui percu&longs;sit, interdum in ip&longs;oictu emori­<lb/>cur motus. Quod qua ratione fiat &longs;ubjectà figurá pate­<lb/>fiet. Percutiant ergò &longs;e duo globi <foreign lang="greek"><gap/>g</foreign> ab eodem hypomo <lb/>chlio <foreign lang="greek">a</foreign> &longs;u&longs;pen&longs;i in ip&longs;o motu, & ducantur lineæ tangen­<lb/>tes <foreign lang="greek">bo. qc</foreign> <expan abbr="at&qacute;">atque</expan>; his parallelæ <foreign lang="greek">yi.yk</foreign> lineæ hypomochlij; in <lb/>lineà autem, <foreign lang="greek">g</foreign> per <expan abbr="utrumq;">utrumque</expan> centrum ductà, & <expan abbr="utrinq;">utrinque</expan> <lb/>protractà &longs;umatur <foreign lang="greek">gp</foreign> æqualis <foreign lang="greek">yl</foreign>, & ex <foreign lang="greek">p</foreign> excitetur li­<lb/>nèa perpendicularis <foreign lang="greek">pm</foreign>, er<gap/><expan abbr="tq;">tque</expan> linea <foreign lang="greek">gm</foreign>, &longs;i nihil impediat, <lb/>linea motus reflexi, per Prop: 39. motus nimirum mix­<lb/>tus ex motu centri <foreign lang="greek">gw</foreign> & motu à percu&longs;ione <foreign lang="greek">gn</foreign>. At verò <lb/>huic motui ob&longs;tat funiculus, à quo globus detinetur, <lb/>quò minùs extra <expan abbr="peripheriã">peripheriam</expan> circuli euagetur. Quia ve­<lb/>rò hic motus àre&longs;lexione & motus à retractione funi­<lb/>culi angulum ducunt <foreign lang="greek">agm</foreign> minorem duobus rectis, erunt <lb/>per definit: 5. &longs;ecundùm quid contrarii, ac proinde inter <pb/>&longs;e mi&longs;cenrut. Motus ergò ex <expan abbr="utroq;">utroque</expan> mixtus à percu&longs;sio­<lb/>ne reflectit. Simili modo o&longs;tendemus globum <gap/> refle­<lb/>cti ex illà plagà. Quòd &longs;i globus <emph type="italics"/>a<emph.end type="italics"/> percutiat globum <emph type="italics"/>b<emph.end type="italics"/><lb/>quie&longs;centem, & minori filo &longs;u&longs;pen&longs;um, erit per Prop: 39 <lb/>linea motus reflexi <emph type="italics"/><expan abbr="aq.">aque</expan><emph.end type="italics"/> & quia hic motus in partes oppo­<lb/> <arrow.to.target n="fig35"></arrow.to.target><lb/>&longs;itas tendit eiu&longs;dem lineæ rectæ, per quam retrahitur ab <lb/>hypomochlio, erunt motus ab&longs;olutè contrarii: globus <lb/>ergò <emph type="italics"/>a<emph.end type="italics"/> &longs;i in illo &longs;itu percutiat <emph type="italics"/>b,<emph.end type="italics"/> â percu&longs;sione quie&longs;cet; <lb/>tantò verò minùs reflectet, quantó maior fuerit angu- <pb/>lus <emph type="italics"/><gap/><expan abbr="aq.">aque</expan><emph.end type="italics"/> Si demum globus <emph type="italics"/>b<emph.end type="italics"/> percutiat globum <emph type="italics"/>a<emph.end type="italics"/> quie­<lb/>&longs;centem & longiori filo &longs;u&longs;pen&longs;um, erit linea motus re­<lb/>flexi <emph type="italics"/>br<emph.end type="italics"/> ad ea&longs;dem partes cum retractione hypomo­<lb/>chlii, propterea quòd linea <emph type="italics"/>bp<emph.end type="italics"/> &longs;it motus centri, linea ve­<lb/>rò <emph type="italics"/>bn<emph.end type="italics"/> motus à percu&longs;sione; globus ergo <emph type="italics"/>b<emph.end type="italics"/> percu&longs;&longs;o glo­<lb/>bo <emph type="italics"/>a<emph.end type="italics"/> reflectet in illo &longs;itu à percu&longs;sione: Eadem via di&longs; <lb/>&longs;oluemus & illam quæ&longs;tionem. </s></p> <figure id="fig35"></figure> <p type="main"> <s><emph type="center"/>DE IN ÆQVALIVM PONDERVM LAPSV<emph.end type="center"/></s></p> <p type="main"> <s>MAgnis motibus & animorum contentionibus a <lb/>gitatam: dum hi quidem rationibus &longs;e tuentur, illi <lb/>verò experientià eos urgent, <expan abbr="errori&longs;&qacute;">errori&longs;que</expan>; marife&longs;ti reos p<gap/><lb/>ragunt. Quorum opinio vulgi applau&longs;u excepta pal­<lb/>mam tulit, judice magis &longs;en&longs;u quam ratione. At verò <lb/>qui opinantur inæqualia pondera æquali ap&longs;u ruere, <lb/>videntur magis id, quod motui per &longs;e ine&longs;t, attendi&longs;&longs;e, <lb/>impedimenta verò motus, quæ ab extra fiunt, veluti du­<lb/>biæ &longs;ortis neglexi&longs;&longs;e. Vt verò hanc litem dirimamus, <lb/>memoriá repetendum id, quod Prop: 37. notabili 4. di­<lb/>ximus, impul&longs;um deficere à plagà perfect<gap/>, partem verò <lb/>hujus cum parte æquali plagæ emori. Secundo â re&longs;i­<lb/>&longs;tentiá majori plagam induci majorem: propterea quòd <lb/>percutiens magis tum immoratur. Tertio omnia cor­<lb/>pora re&longs;i&longs;tere diui&longs;ioni, <expan abbr="at&qacute;">atque</expan>; eó magis, quó major e&longs;t vir- <pb/>tus illarum partium unitiua, ut Prop: 1. dictum: quan­<lb/>tumuis ergo aër naturá &longs;uá &longs;it fluidus, <expan abbr="at&qacute;">atque</expan>; omni <lb/>aurá mobilis, non tamen <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>, violentiá, ac proinde <lb/>non <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; plagà findi pote&longs;t. Quar­<lb/>to majorem diui&longs;ionem fieri à majori plagà; multúm e­<lb/>nim aëris non eadem facilitate mouemus, <expan abbr="ne&qacute;">neque</expan>; eadem <lb/>velocitate parte ferri latiore, quam in mucronem tenua­<lb/>ta hunc penetramus. His &longs;uppo&longs;itis: dico 1. motum qua <lb/>tenus àgrauitate procedit eiu&longs;dem &longs;peciei &longs;eu gradus, eà­<lb/>dem celeritate fieri in omnibus, quantumuis mole, figu <lb/>rà, pondere à &longs;e differant: ratio, quia ut mobile mouea­<lb/>tur, non quilibet impul&longs;us, &longs;ed proportionatus e&longs;&longs;e debet <lb/>ad illud mobile; ab eadem ergo proportione eadem ve­<lb/>locitas motus: at veró impul&longs;us, quo totum mobile mo­<lb/>uetur, eandem rationem habet ad illud mobile, quam &longs;e­<lb/>mi&longs;sis illius impul&longs;us ad &longs;emi&longs;&longs;em, & triens ad trientem <lb/>eju&longs;dem mobilis; eadem ergo velocitas motus. Quod <lb/>idem de qualibet particulá, <expan abbr="quacun&qacute;">quacunque</expan>; factá diui&longs;ione, di­<lb/>cendum; non minùs enim extra illud mobile, quam in <lb/>mobili, & alijs conjunctæ &longs;uo inpul&longs;u mouentur. Dices <lb/>virtus collecta e&longs;t fortior &longs;e ip&longs;à di&longs;per&longs;à: major ergo im <lb/>pul&longs;us in partibus unitis, quam extra illam unionem. Re <lb/>&longs;pondeo illud axioma non in omnibus valere, &longs;ed tan­<lb/>tum in ordine ad actionem, quæ extra illud &longs;ubjectum <lb/>terminatur; ita enim lux alteri conjuncta lumen longi- <pb/>ùs protendit, nihilo ex illa conjunctione luce auctà: ita <lb/>ergo impul&longs;us partium unitarum licet magis percutiat, <lb/>non tamen in ordine ad motum, quo illius &longs;ubjectum <lb/>fertur, magis inuale&longs;cit, quemadmodum cùm plures &longs;i­<lb/>mul vocem attollunt, licet magis audiatur, non tamen <lb/>exaliorum vociferatione &longs;ingulorum clamor facilitatur. <lb/>Plura quæ pro hac &longs;ententià, & <expan abbr="cõtra">contra</expan> afferri po&longs;&longs;unt, &longs;uo <lb/>loco dicemus; nunc verò dato e&longs;&longs;e veram, illam inæqua­<lb/>litatem motus con&longs;tare, <expan abbr="at&qacute;">atque</expan>; ex alià radice na&longs;ci paucis o­<lb/>&longs;tendemus. Dico &longs;ecundò, illam inæqualitatem motus, <lb/>quo inæqualia pondera mouentur, e&longs;&longs;e à medio, in quo <lb/>fit motus; <expan abbr="at&qacute;">atque</expan>; illa corpora, quorum grauitas &longs;eu impul­<lb/>&longs;us majorem rationem habet ad &longs;uam plagam, velociùs <lb/>moueri. Quia enim aër re&longs;i&longs;tit diui&longs;ioni ex notabili 3. <lb/>erit plaga ad men&longs;uram hujus re&longs;i&longs;tentiæ; deficiet ergò <lb/>impul&longs;us, ac proinde velocitas motus in eà ratione, in <lb/>quâ magnitudo plagæ: igitur ut plaga ad plagam, ita ve­<lb/>locitatis decrementum. At verò grauitas illorum cor­<lb/>porum majorem rationem habet, quam illorum plaga: <lb/>&longs;it enim globus <emph type="italics"/>ab<emph.end type="italics"/> ad globum <emph type="italics"/>cd<emph.end type="italics"/> in ratione duplà, <expan abbr="erit&qacute;">eritque</expan>; <lb/>illorum plaga æqualis circulo maximo &longs;uæ &longs;phæræ, pro <lb/>pterea quód plaga inducitur non ni&longs;i à parte inferiore, <lb/>quæ aërem findit, & cui &longs;oli aër re&longs;i&longs;tit: habet autem cir­<lb/>culus maximus &longs;phæræ &longs;eu globi in ratione duplà ad ali­<lb/>am &longs;phæram, minorem rationem, quám duplam, ad hu- <pb/>jus circulum maximum; globus ergo major plagam in­<lb/>ducit minorem, quàm ut &longs;it dupla ad plagam minoris <lb/>globi: ut &longs;i globus major &longs;it duarum lib: erit &longs;emi&longs;sis, id­<lb/>e&longs;t lib: una, æqualis globo minori; hujus verò plaga &longs;e­<lb/>mi&longs;sis plagæ totius minor plagâtotá globi minoris. quia <lb/>ergò plaga tollit partem &longs;ibi æqualem, maius erit decre <lb/>mentum velocitatis in librà unà, dum extra illud totum, <lb/>&longs;eu globum maiorem & per &longs;e, ide&longs;t in globo minori mo <lb/>uetur. Et quia in medio &longs;imilari eadem plaga continu­<lb/>atur, eadem ratio erit decrementi quæ interualli; ut &longs;i in <lb/>toto motu deficiat cubitus unus, deficiet in &longs;emi&longs;&longs;e hu­<lb/>jus motus illius &longs;emi&longs;sis: <expan abbr="atq;">atque</expan> inde ratio con&longs;tat, quam ob <lb/>rem à principio motus inæqualia pondera &longs;imul ferri <lb/><expan abbr="videãtur">videantur</expan>, inde verò magnis à &longs;e di&longs;iungi interuallis. Ma­<lb/>lè ergo rationem huius inæqualitatis petunt à proporti­<lb/>one illorum ponderum, quæ á ratione cre&longs;centis plagæ <lb/>de&longs;umi debet; ablatá enim á grauitate &longs;eu impul&longs;u parte <lb/>æquali &longs;uæ plagæ, reliquus impul&longs;us dabit illam inæqua­<lb/>lem velocitatem. Obiicies fieri non po&longs;sè ut eadem ratio <lb/>maneat plagæ in illo motu inæquali continuatæ, propte <lb/>rea quód aër percu&longs;&longs;us alium percutiat, <expan abbr="viamq;">viamque</expan> eá rati­<lb/>one aperiat ruenti globo, plagæ imminenti &longs;e <expan abbr="&longs;ubduc&etilde;s">&longs;ubducens</expan>, <lb/>non aliter, quám cùm ultro cedentem trudimus: <expan abbr="itaq;">itaque</expan> in <lb/>relap&longs;u globi maioris, quem ignis in &longs;ublime tulit, pri­<lb/>u&longs;quam terram feriat, ab aëris percu&longs;sione hiatum in il- <pb/>lá fieri quidam a&longs;&longs;eucrant. Cùm ergò aër ab illo ictu &longs;e <lb/>&longs;ubducat, nullam inducet plagam, nullum proinde velo­<lb/>citatis decrementum; non aliter quam &longs;i globus per fi&longs;­<lb/>&longs;uram muri tran&longs;uolet muro inoffen&longs;o. Deinde cùm im <lb/>pul&longs;us continuò augeatur, erit continuó minor re&longs;i&longs;ten­<lb/>tia. Re&longs;pondeo aerem quidem impelli & præcurrere, <lb/>verùm minori celeritate, quàm ut plagam effugiat á ter­<lb/>go hærentem; major enim globi impetus, quâm ut ab <lb/>aere fluido recipiatur: unde eadem re&longs;i&longs;tentia in aëre per <lb/>forando, non minús, quàm &longs;i &longs;ecundo flumine elucte­<lb/>mur motu velociori, quàm &longs;it defluxus; non minore­<lb/>nim difficultas in perrumpendo, quam &longs;i in aquà fiat im­<lb/>motà. Deinde licet aër percu&longs;&longs;us à plagà &longs;e &longs;ubducat & <lb/>præcurrat, alius tamen in locum plagæ &longs;e in&longs;undit non <lb/>minori vi findendus: <expan abbr="ne&qacute;">neque</expan>; enim aër di&longs;cerpi pote&longs;t eo <lb/>modo, quo corpora magis den&longs;a, in quibus perruptis cor <lb/>pus magis &longs;ubtile interceptum viam præ&longs;tat faciliorem; <lb/>verùm <expan abbr="quacun&qacute;">quacunque</expan>; plaga incidit, eadem aëris &longs;oliditas per­<lb/>rumpenda. Ad &longs;ecundam rationem, dico velocitatem <lb/>motus continuò quidem augeri, ac proinde illam re&longs;i­<lb/>&longs;tentiam medij auctà velocitate faciliùs perrumpi; pro­<lb/>pterea quód ablatà parte æquali major &longs;it exce&longs;&longs;us reli­<lb/>quus: nego autem â velociori plagà minus e&longs;&longs;e decre­<lb/>mentum. An non velociùs vectem deprimunt libræ 10. <lb/>aut 100, quam libra <emph type="italics"/>1?<emph.end type="italics"/> & tamen granum unum aut deci- <pb/>ma pars grani æqualem partem ex hoc, <expan abbr="at&qacute;">atque</expan>, ex illis tollit. <lb/>Verùm deceptio latet ob exiguitatem decrementi, que­<lb/>madmodum &longs;i ad deprimendum libras 100. unum <expan abbr="at&qacute;">atque</expan>; <lb/>alterum granum apponas. Quia ergò retardatio motus <lb/>e&longs;t à medio, quó medium magis re&longs;i&longs;tit diui&longs;ioni, eó mi­<lb/>nor velocitas motus, major autem exce&longs;&longs;us tarditatis in <lb/>minori: propterea quód auctá re&longs;i&longs;tentiá eadem diffe­<lb/>rentia in minori interuallo. E contra minuitur exce&longs; <lb/>&longs;us in medio magis raro; <expan abbr="ita&qacute;">itaque</expan>; &longs;i detur corpus infinitæ ra­<lb/>ritatis, cuiu&longs;modi vacuum, quia nulla re&longs;i&longs;tentia, nulla <lb/><expan abbr="quo&qacute;">quoque</expan>; erit inæqualitas motus. Quòd autem à &longs;olá re&longs;i­<lb/>&longs;tentià medij procedat inæqualitas motus, ratio manife­<lb/>&longs;ta: idem enim pondus &longs;e ip&longs;o velociús, <expan abbr="at&qacute;">atque</expan>; cum alio <lb/>pondere <expan abbr="quocun&qacute;">quocunque</expan>; exce&longs;&longs;u majori, eádem velocitate de­<lb/>&longs;cendit, &longs;i rationem plagæ & re&longs;i&longs;tentiam medii in illâ <lb/> <arrow.to.target n="fig36"></arrow.to.target><lb/>proportione minuàs. Sit enim vas plumbeum, aut de <lb/>alià materià graui, formá dimidiæ &longs;phæræ, cuju&longs;modi <foreign lang="greek">bgd</foreign> <pb/>habens cauitatem in parte &longs;uperiore, & à plagâ auer&longs;a, <lb/>centrum verò grauitatis in <gap/> ne dum labitur &longs;e inuertat: <lb/>quód &longs;i ergo alium globum <expan abbr="quocun&qacute;">quocunque</expan>; exce&longs;&longs;u leuio­<lb/>rem con&longs;tituas in illà cauitate, eádem cum illo va&longs;e ce­<lb/>leritate fexetur. At verò &longs;i inæqualitas motus e&longs;&longs;et <lb/>àgrauitate, oporteret illud vas magis pondero&longs;um <lb/>præcurrere, globum verò leuiorem attolli, & longo po&longs;t <lb/>tergum interuallo relinqui. Obiicies gra uitas e&longs;t impul <lb/>&longs;us, impul&longs;us verò per Prop: 2. motum producit &longs;ibi æ <lb/>qualem; à majori ergò grauitate major, ac proinde velo­<lb/>cior motus: quòd &longs;i ergò libra una in <expan abbr="quin&qacute;">quinque</expan>; &longs;ecundis <lb/>per &longs;patium mouet cubitorum 100, mouebit hujus du­<lb/>plum in eodem, vel æquali tempore per &longs;patium <expan abbr="duplũ">duplum</expan>. <lb/>Deinde plaga inducitur ex motu; non enim manus à la­<lb/>pide in eà quie&longs;cente, &longs;ed ubi iram ex motu concepit, vul <lb/>neratur: at verò majus pondus æquali lap&longs;u magis vulne <lb/>rat, velocior ergo motus. Re&longs;pondeo grauitatem e&longs;&longs;e <lb/>impul&longs;um, & velocitatem motus in eá ratione, in quá e&longs;t <lb/>grauitas &longs;eu impul&longs;us; dupla ergo grauitas in eodem, vel <lb/>æquali tempore mouebit per &longs;patium duplum. At verò <lb/>cùm inferunt libras duas Vg: plumbi in duplà ferri celeri <lb/>tate ad libram unam, falluntur; propterea quòd illa gra­<lb/>uitas in alio &longs;it &longs;ubiecto, cuius partes omnes æquali gra­<lb/>uitate mouentur: &longs;icuti enim pars extra totum Vg. libra <lb/>una â &longs;ua grauitate mouetur cum tantà velocitate, ita <pb/>partes librarum decem, aut centum in toto unitæ eádem <lb/>velocitate <expan abbr="mou&etilde;tur">mouentur</expan> á &longs;uá <expan abbr="cuiq;">cuique</expan> propria grauitate. Quód <lb/>&longs;i grauitas librarum decem con&longs;tituatur in &longs;ubiecto uni­<lb/>us libræ, tum verò decupla velocitate mouebitur illud <lb/>&longs;ubiectum. Ni&longs;i ergò grauitas magis &longs;it inten&longs;a, nihil <lb/>proficiet ad velocitatem augendam illorum moles. <lb/>Quód autem maior grauitas plagam inducat maiorem, <lb/>ut &longs;i libræ decem percutiant libram unam, huius ratio <lb/>e&longs;t, quia totidem fiunt plagæ, quot in maiori continen­<lb/>tur partes æquales: quemadmodum &longs;i decem ictus &longs;i­<lb/>mul inferantur, aut &longs;i priu&longs;quam vis emoriatur prioris <lb/>plagæ, reliquæ &longs;equantur. Impul&longs;us ergò in illo &longs;ubie­<lb/>cto minori á maiori percu&longs;&longs;o magis e&longs;t inten&longs;us. <expan abbr="Atq;">Atque</expan> <lb/>inde fit, quód globus minor accepta àmaiori plaga præ­<lb/>currat; quód &longs;i enim globos <expan abbr="quoteunq;">quoteunque</expan> eà &longs;erie di&longs;po­<lb/>nas, ut continuò maiorem minor &longs;equatur, percu&longs;&longs;o pri­<lb/>mo videbis qua&longs;i uno impetu omnes ad motum conci­<lb/>tari, verùm celeritate, pro ratione magnitudinis, inæ­<lb/>quali. </s></p> <figure id="fig36"></figure> <p type="main"> <s><emph type="center"/>Propo&longs;itio XXXXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Problema II.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Regulam con&longs;truere ad celeritatem & tarditatem pul&longs;uum <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; <lb/>errore metiendam.<emph.end type="italics"/></s></p> <pb/> <p type="main"> <s>REgula bæc nullo apparatu, &longs;ed. hac arte &longs;implici <lb/>confit &longs;iue ex ligno, &longs;iue ex qualibet alià materià. Hu <lb/>ius longitudo <emph type="italics"/>ab<emph.end type="italics"/> unius cubiti, aut ad placitum: quó enim <lb/>maior, eò plures differentias tarditatis indicabit: nam <lb/>ad velocitatem &longs;ummam indicandam quælibet magni­<lb/>tudo &longs;ufficit. Latitudo verò, quæ cordam &longs;eu filum ca­<lb/>piat cum numerorum notis eidem ad&longs;criptis. Filum <lb/>porro eo modo, quo fidibus aptatur; parte &longs;uperiore <lb/>trochleâ ver&longs;atili conuo lutum, parte verò inferiore fora <lb/>mine tran&longs;mi&longs;&longs;um, globulum habens dependentem, qui <lb/>eidem rectitudinem præ&longs;tat & pondus. Tota longitu­<lb/>do regulæ, quæ continetur inter foramen & trochleam, <lb/>æqualiter &longs;ecetur in partes quotlibet Vg. 60, aut 100. <lb/> <arrow.to.target n="fig37"></arrow.to.target><lb/>quas trochleà laxatâ nodulus <emph type="italics"/>q,<emph.end type="italics"/> globulo interea de&longs;cen- <pb/>dente, percurrit, <expan abbr="fuo&qacute;">fuoque</expan>; contactu quot eju&longs;modi &longs;egmen­<lb/>ta contineat longitudo eju&longs;dem fili cum &longs;uo globulo à <lb/>foramine penduli, o&longs;tendit. Cùm ergo per dictum in­<lb/>&longs;trumentum pul&longs;us celeritatem indagare voles, trochle­<lb/>am ver&longs;ando filum eò <expan abbr="u&longs;&qacute;">u&longs;que</expan>; laxa, dum globulus in <emph type="italics"/>e<emph.end type="italics"/> Vg. <lb/>aut <emph type="italics"/>g<emph.end type="italics"/> de&longs;cendat: quom ex <emph type="italics"/>g,<emph.end type="italics"/> in quo naturaliter à motu <lb/>quie&longs;cit, in <gap/> vel <emph type="italics"/>o<emph.end type="italics"/> dimotum inde recurrere &longs;inas; in­<lb/>terea, dum globulus per arcum <emph type="italics"/>cd<emph.end type="italics"/> ultra <expan abbr="citra&qacute;">citraque</expan>; <emph type="italics"/>g<emph.end type="italics"/> excurnt, <lb/><expan abbr="plure&longs;&qacute;">plure&longs;que</expan>; recur&longs;us facit, agitationem quidem arteriæ ma­<lb/>nu, motum verò perpendiculi vi&longs;u explora, <expan abbr="at&qacute;">atque</expan>; unum <lb/>alteri compara. Quód &longs;i tardior arteriæ motus, perpen­<lb/>diculum trochleá laxatá producas, &longs;i celerior contrahas <lb/>Æquato demum <expan abbr="utriu&longs;&qacute;">utriu&longs;que</expan>; motu, quænam &longs;it celeritatis <lb/>ratio, ex numerorum diui&longs;ione, quem nodulus cur<gap/> filo <lb/>depre&longs;&longs;us indicabit, facilè cogno&longs;ces. Quin & quamli­<lb/>bet mutationem ad &longs;ingula momenta ex collatione ad <lb/>huiu&longs;modi numeros factâ conijcies. Vbi ergo men&longs;u­<lb/>ram pul&longs;us quam maximè naturalis hac vià deprehen­<lb/>des: diui&longs;ionis interuallum, quod nodulus indicabit, <lb/>diligenter nom; ad cuius motum reliquos pul&longs;us com <lb/>parando illorum exce&longs;&longs;us &, defectus facilè obrinebis. <lb/>Porro huiu&longs;modi <gap/> & tardit tem pu. <lb/>&longs;uum <expan abbr="ab&longs;&qacute;">ab&longs;que</expan>; errore meti<gap/>i hac vià o&longs;tendemus. Pul&longs;us in <lb/>ter &longs;e aut &longs;unt æquales, quorum eadem e&longs;t velocitas mo­<lb/>tus, atque i&longs;dem fiunt momentis: aut inæquales, cele- <pb/>ritate & tarditate à &longs;e differentes, <expan abbr="quorũ">quorum</expan> inæqualia &longs;unt <lb/>durationis momenta. Quia ergo motus perpendiculi <lb/>e&longs;t illorum men&longs;ura; erit quidem æqualium pul&longs;uum æ­<lb/>qualis, inæqualium verò inæqualis in ea ratione, in quâ <lb/>velocitas pul&longs;uum. At verò recur&longs;us & excur&longs;us perpen <lb/>diculi ex eadem productione inter &longs;e &longs;unt æquales: pro­<lb/>pterea quód perpendiculum ex quolibet puncto <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>circuli æquali tempore recurrit in &longs;uam &longs;tationem per <lb/>Prop: 24. &longs;unt autem excurius <expan abbr="quo&qacute;">quoque</expan>; inter &longs;e æquales per <lb/>Prop: 25. excur&longs;us ergo & recur&longs;us in unà circulatione <lb/>&longs;imul &longs;umpti &longs;unt æquales excur&longs;ibus & recur&longs;ibus o­<lb/>mnium circulationum &longs;imul <expan abbr="quo&qacute;">quoque</expan>; &longs;umptis: & quia uni <lb/>æqualium pul&longs;uum circulatio a&longs;&longs;umpta e&longs;t æqualis, e­<lb/>runt reliquæ circulationes reliquis pul&longs;ibus æquales. <lb/>Motus ergo perpendiculi ex eádem productione fili <lb/>metitur pul&longs;us inter &longs;e æquales. Quia verò motus per­<lb/>pendiculi per arcus &longs;imiles inæqualium circulorum ra­<lb/>tionem habent ad &longs;e quam &longs;inus illorum arcuum, hoc e&longs;t <lb/>lineæ &longs;ubten&longs;æ arcus dupli, per Prop: 25. ac proinde <lb/>quam habent motus per diametrum illorum circulo­<lb/>rum per Prop: 15. motus autem per diametrum &longs;e habent <lb/>ut quadrata temporum per Prop: 12. Si &longs;umatur radix <lb/>quadrata illius proportionis, quam habent diametri ad <lb/>&longs;e, erunt in eadem ratione tempora motus, in quà radices <lb/>quadratæ: ut &longs;i diameter maioris circuli ad diametrum <pb/>minoris circuli &longs;it quadrupla, huius radix quadrata, duo, <lb/>dabit tempus in ratione duplá: &longs;i ergo motus per dia­<lb/>metrum minoris circuli &longs;it unius minuti, erit motus ma­<lb/>ioris diametri duorum minutorum. Sunt autem pro­<lb/>ductiones fili &longs;emidiametri illorum circulorum, in qui­<lb/>bus perpendiculum mouetur, æquales diui&longs;ionum in­<lb/>teruallis, quæ globulus in productione fili percurrit: ea­<lb/>dem ergo proportio interualli, quæ motus illornm cir­<lb/>culorum. Quia ergo motus inæqualium circulorum <lb/>metiuntur pul&longs;us inæquales, eo&longs;dem metientur diui&longs;io­<lb/>num interualla: ac proinde regulam con&longs;truximus ad <lb/>velocitatem & tarditatem pul&longs;uum <expan abbr="ab&longs;q;">ab&longs;que</expan> errore metien <lb/>dam, quod erat faciendum. </s></p> <figure id="fig37"></figure> <p type="main"> <s><emph type="center"/>Parergon.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Problema.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Horologium con&longs;truere, quod &longs;uo motu tempus numeret diui&longs;um <lb/>in partes minores, quàm tertias unius &longs;ecundi.<emph.end type="italics"/></s></p> <p type="main"> <s>QVanti u&longs;us & utilitatis &longs;it tempus in quàm minimas <lb/>partes diui&longs;um po&longs;&longs;e numerare, <expan abbr="norũt">norunt</expan> A&longs;tronomi, <lb/>& ex conatibus Tychonis Brahe &longs;atis con&longs;tat; qui ad hu <lb/><gap/>u&longs;modi horologia fabricanda nihil intentatum reliquit: <lb/>quam uis huius votum non ni&longs;i ad &longs;ecunda numeranda <pb/>le extendit. Aliquid ampliùs damus: & non modó &longs;e­<lb/>cunda, verùm etiam huius triente minorem partem nu­<lb/>merabimus. Hotologium autem hoc nullis rotulis cir­<lb/>cumagitur, nullis ponderibus libratur; verùm &longs;uâ nati­<lb/>uâ grauitate, à quànu&longs;quam aberrat, ad normam præ­<lb/>&longs;criptam agitatur: illud inquam idem, quod ad celerita­<lb/>tem & tarditatem pul&longs;uum metiendam paulo ante con­<lb/>&longs;truximus. Huius enim pondus à filo pendulum &longs;uo <lb/>motu tempus in quotlibet partes diui&longs;um numerabit. <lb/>Quòd autem hic motus minor e&longs;&longs;e po&longs;sit, quâm tertia <lb/>pars unius &longs;ecundi, ita o&longs;ten demus: agitationes arteriæ, <lb/>cuiu&longs;modi in me ip&longs;o numeraui, &longs;patio unius horæ fi­<lb/>unt 4850. motus autem perpendiculi his æquales fiunt â <lb/>productione fili maiori quàm digitorum 5. Quia ergo <lb/>motus circulorum &longs;unt in ratione &longs;uorum temporum, <lb/>quam habent diametri ad &longs;e duplicatam, per Prop: 28. &longs;i <lb/>&longs;umatur pars nona huius productionis pro &longs;emidiame­<lb/>tro circuli, erit hic motus triplo velocior illo, ac proinde <lb/>huius recur&longs;us &longs;patio horæ unius 14550 multò plures, <lb/>quàm 10800 partes tertiæ unius &longs;ecundi. Et quia hic mo­<lb/>tus bifariam &longs;ecari pote&longs;t in excur&longs;um & recur&longs;um, fient <lb/>&longs;anè &longs;patio unius horæ partes 29100. Horologium ergò <lb/>con&longs;truximus, quod &longs;uo motu tempus numerat diui&longs;um <lb/>in partes minores quàm tertias unius &longs;ecundi. Quia ta­<lb/>men hic motus veloci&longs;simus ob paruitatem circelli mi- <pb/>nùs e&longs;t diuturnus, <gap/>ufficiet filum producere, <expan abbr="quou&longs;&qacute;">quou&longs;que</expan>; mo <lb/>tus perpendiculi &longs;ir æqualis uni&longs;ecundo. Quod quidem <lb/>hac ratione con&longs;equemur: &longs;umatur <expan abbr="quæcun&qacute;">quæcunque</expan>; produ­<lb/>ctio fili, aliquantó tamen longior, quò minùs citò à mo­<lb/>tu conquie&longs;cat: <expan abbr="numerentur&qacute;">numerenturque</expan>; huius excur&longs;us per &longs;pati­<lb/>um unius horæ quadrantis, & &longs;int Vg. 300. <expan abbr="erunt&qacute;">eruntque</expan>; &longs;pa­<lb/>tio horæ unius 1200. Quòd &longs;i ergò fiat ut quadratum <lb/>temporis, nimirum trium &longs;ecundorum, ide&longs;t 9 ad 1, ita <lb/>longitudo fili ad minorem, erit hujus motus æqualis <lb/>uni &longs;ecundo. <lb/> <arrow.to.target n="fig38"></arrow.to.target></s></p> <pb/> <figure id="fig38"></figure> <figure></figure> <p type="main"> <s><emph type="center"/>PRAGÆ.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Typis Ioannis Bilinæ.<emph.end type="center"/></s></p> <figure></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Anno<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>M. DC. XXXIX:<emph.end type="italics"/><emph.end type="center"/></s></p> </chap> </body> <back></back> </text> </archimedes>