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version 1.5, 2002/06/28 13:03:37

version 1.6, 2002/07/09 23:38:53

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]><archimedes> <info> <author>Ceva, Giovanni</author> <title>Geometria motus opusculum geometricum</title> <date>1692</date> <place>Bologna</place> <editor></editor> <publisher></publisher> <translator></translator> <lang>la</lang> <chunk unit="page*">page</chunk><locator>0000000022</locator> </info> <text> <front> </front> <body> <chap> <pb/><figure></figure><pb/>

]><archimedes> <info> <author>Ceva, Giovanni</author> <title>Geometria motus opusculum geometricum</title> <date>1692</date>

<place>Bologna</place> <editor></editor> <publisher></publisher> <translator></translator> <lang>la</lang> <chunk unit="page*">page</chunk><locator>0000000022</locator> </info> <text> <front> </front> <body> <chap> <pb/><figure></figure><pb/>

<s><emph type="center"/>GEOMETRIA<emph.end type="center"/></s>

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<s><emph type="center"/>FERDINANDO <lb/>CAROLO.<emph.end type="center"/></s>

<s><emph type="italics"/>ITerum, Sereni&longs;&longs;ime Princeps, tuis aduolutus <lb/>genibus opu&longs;culum exhibeo, in quo naturam motuum, pleniori <lb/>methodo, quàm puto antea &longs;it actum, geometricè exequor. <lb/>Neceße habui hæc præmittere, quò viam aperirem, & quo<lb/>dammodo alueum &longs;ternerem aquarum doctrinæ, quarum <lb/>argumentum vtili&longs;&longs;imum, & profundæ indaginis iam diu <lb/>meditor. </s>

<s>Neceße habui hæc præmittere, quò viam aperirem, & quo<lb/>dammodo alueum &longs;ternerem aquarum doctrinæ, quarum <lb/>argumentum vtili&longs;&longs;imum, & profundæ indaginis iam diu <lb/>meditor. </s>

<s>Quam arduum &longs;it, & per quas &longs;alebras eun<lb/>dum, vt nouum aliquid luce dignum è latebris naturæ eruarur <lb/>vtinam Cel&longs;itudini tuæ aliquis veritatum non vulgarium <lb/>indagator fidem faceret; &longs;cio equidem, & laboris improbitas <lb/>tangeret benigni&longs;&longs;imum animum tuum, & &longs;imal naturæ inge<lb/>nium &longs;u&longs;piceres, quæ mentibus aliquorum vim inuentricem <lb/>in&longs;eruit, vt eorum iugi cogitatione humanis v&longs;ibus prouide-<emph.end type="italics"/><pb/><emph type="italics"/>ret. </s>

<s>Et verò (&longs;i in hoc genere de me quidquam confiteri decet) <lb/>ni&longs;i aduer&longs;æ valetudinis experimento prudentior factus indo<lb/>lem meam huiu&longs;cemodi &longs;tudijs intemperanter addictam ali<lb/>quot ab hinc annis compe&longs;cuißem; nec non quotidie munus à <lb/>Cel&longs;itudine Tua &longs;ummo cum honore & beneficentia demanda<lb/>tum (adeo vt hoc etiam nomine Te&longs;eruatorem meum appella<lb/>re po&longs;&longs;im) inde me reuoca&longs;&longs;et; eorum, credo equidem, ponderi, <lb/>a&longs;&longs;iduæque contemplationi &longs;uccumbere nece&longs;&longs;e erat. </s>

<s>Vnde au<lb/>tem, Cel&longs;i&longs;&longs;ime dux, huic &longs;cientiæ tanta vis, vt quos &longs;ibi &longs;emet <lb/>adiunxerit, nonni&longs;i altiori ratione queat a &longs;e ip&longs;a dimittere? <lb/>An quod forta&longs;&longs;e vbi animus publicæ vtilitati de&longs;eruire cæpe<lb/>rit, veluti in nataræ concilium admi&longs;&longs;us, &longs;ui quodammodo <lb/>oblitus, propriam humilioremque &longs;edem reui&longs;ere dedignetur; an <lb/>quia, cùm inter cæteras &longs;cientias Geometria demon&longs;trationem, <lb/>hoc e&longs;t veritatem &longs;inceram, & quandam primi veri particu<lb/>lam profiteatur, hinc ne&longs;cio quid diuinum habent &longs;ibi <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>, <lb/>vnde nonni&longs;i Deo impellente, vbi nimirum officia, potiorque <lb/>ratio id po&longs;tulant, ab eius intuitu retrahatur. </s>

<s>An quod forta&longs;&longs;e vbi animus publicæ vtilitati de&longs;eruire cæpe<lb/>rit, veluti in nataræ concilium admi&longs;&longs;us, &longs;ui quodammodo <lb/>oblitus, propriam humilioremque &longs;edem reui&longs;ere dedignetur; an <lb/>quia, cùm inter cæteras &longs;cientias Geometria demon&longs;trationem, <lb/>hoc e&longs;t veritatem &longs;inceram, & quandam primi veri particu<lb/>lam profiteatur, hinc ne&longs;cio quid diuinum habent &longs;ibi <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>, <lb/>vnde nonni&longs;i Deo impellente, vbi nimirum officia, potiorque <lb/>ratio id po&longs;tulant, ab eius intuitu retrahatur. </s>

<s>Hoc equidem <lb/>puto; atque hinc diuina Geometria iure optimo a docti&longs;&longs;imis, & <lb/>clari&longs;&longs;imis viris pa&longs;&longs;im nuncupatur. </s>

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<s>S de <lb/>motu æquab. <lb/>Def.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>

<s>S de <lb/>motu æquab. <lb/></s>

<s>Def.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>

<s><margin.target id="marg9"></margin.target><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>fig.<emph.end type="italics"/> 6. <lb/><emph type="italics"/>Def.<emph.end type="italics"/> 5. <emph type="italics"/>huius.<emph.end type="italics"/></s>

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<s>Quare etiam vnicum re<lb/>ctangulum MH ad circum&longs;criptam figuram AK, BI, CN, <lb/>DG erit in eadem ratione, in quo vnicum tempus per FM <lb/>iuxta imaginem MH ad omnia &longs;imul illa tempora iuxt&atail; <pb pagenum="6"/>imagines, quæ &longs;unt dicta circum&longs;cripta rectangula. </s>

<s>Et <lb/>quoniam figura imaginis e&longs;t acuminata, habetque vi def. <lb/>2. huius, applicatas, quæ &longs;unt in ratione reciproca veloci<lb/>tatum, quibus nempe mobile afficitur in punctis &longs;patij, à <lb/>quibus deducuntur ip&longs;æ applicatæ; hinc fit, vt earum ve<lb/>locitatum, quas mobile habet in decur&longs;u rectæ AB, ea, qu&ecedil; <lb/>in A maxima &longs;it, & quæ in B minima. </s>

<s>Et <lb/>quoniam figura imaginis e&longs;t acuminata, habetque vi def. <lb/></s>

<s>2. huius, applicatas, quæ &longs;unt in ratione reciproca veloci<lb/>tatum, quibus nempe mobile afficitur in punctis &longs;patij, à <lb/>quibus deducuntur ip&longs;æ applicatæ; hinc fit, vt earum ve<lb/>locitatum, quas mobile habet in decur&longs;u rectæ AB, ea, qu&ecedil; <lb/>in A maxima &longs;it, & quæ in B minima. </s>

<s>Eodem modo iuxta <lb/>reliquas imagines BKIC, CIND, DNGE, quæ itidem acu<lb/>minatæ &longs;unt, velocitates in fine decur&longs;uum C, D, E (&longs;unt <lb/>enim omnes versùs A acuminatæ) minimæ erunt, & ma<lb/>ximæ initio dictorum &longs;patiorum. </s>

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<s>lem.<emph.end type="italics"/> 18. <emph type="italics"/>in <lb/>libro de dim. <lb/>parabolæ.<emph.end type="italics"/></s>

<s>lem.<emph.end type="italics"/> 18. <emph type="italics"/>in <lb/>libro de dim. <lb/></s>

<s>parabolæ.<emph.end type="italics"/></s>

<s><margin.target id="marg14"></margin.target><emph type="italics"/>Ax.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/></s>

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<s>SPatia, quæ curruntur iuxta quaslibet homogeneas ve<lb/>locitatum imagines, nectuntur ex rationibus tempo<lb/>rum, ac æquatricum. </s>

<s>Velocitates æquatrices duorum motuum, quorum ima<lb/>gines velocitatum &longs;int ABCD, EFHI ponantur AG, EL. <lb/>Dico &longs;patia, &longs;eu ip&longs;as imagines componi ex ratione tem<lb/>porum AD ad EI; & ex ea æquatricum AE ad EL. </s>

<s>Velocitates æquatrices duorum motuum, quorum ima<lb/>gines velocitatum &longs;int ABCD, EFHI ponantur AG, EL. <lb/></s>

<s>Dico &longs;patia, &longs;eu ip&longs;as imagines componi ex ratione tem<lb/>porum AD ad EI; & ex ea æquatricum AE ad EL. </s>

<s>Nam <lb/>&longs;i motus, qui e&longs;t iuxta imaginem ABCD per&longs;eueret velo<lb/>citate AG, e&longs;&longs;et quidem æquabilis, idemque &longs;patium illa </s>

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<s>Dico rectangu<lb/>lum DF in DE ad figuram GFDEG, eandem habere ratio<lb/>nem ac figura ACBA ad rectangulum AB in BC. </s>

<s>Sint pri<lb/>mùm ABC, FDE anguli recti, & ducta qualibet HI paral<lb/><arrow.to.target n="marg44"></arrow.to.target><lb/>lela BC, &longs;it BAC ad HIA vt DF ad KF, erit ob naturam <lb/>auuer&longs;arum KL ad DE vt BC ad HI; itaque &longs;i ponatur e&longs;&longs;e <lb/>quidam motus ab F in D iuxta imaginem <expan abbr="velocitat&utilde;">velocitatum</expan> BAC, <lb/><arrow.to.target n="marg45"></arrow.to.target><lb/>erit GFDEG imago temporis eiu&longs;dem motus; nam imago <lb/><arrow.to.target n="marg46"></arrow.to.target><lb/>BAC ad imaginem HIA e&longs;t vt &longs;patium DF ad &longs;patium FK <lb/>& velocitas BC ad <expan abbr="velocitat&etilde;">velocitatem</expan> HI vt reciprocè KL ad DE. <lb/>Sit etiam alius motus, &longs;ed æquabilis, cuius imago velocita<lb/>tum æqualis &longs;it, & homogenea ip&longs;i BAC, rectangulum <expan abbr="n&etilde;-pe">nen<lb/>pe</expan> AB in BM, & ideo &longs;i fiat BM ad BC &longs;icut DE ad DN, <lb/>concipiaturque rectangulum FD in DN, erit hoc imago <lb/><arrow.to.target n="marg47"></arrow.to.target><lb/>temporis dicti motus æquabilis, homogenea, & æqualis <lb/>imagini GFDEG; nam <expan abbr="t&etilde;pora">tempora</expan>, &longs;cilicet imagines GFDEG, <lb/><arrow.to.target n="marg48"></arrow.to.target><lb/>FD in DN rectangulum componuntur ex rationibus &longs;pa<lb/><arrow.to.target n="marg49"></arrow.to.target><lb/>tiorum, hoc e&longs;t imaginum velocitatum inter&longs;e æqualium, <lb/>ABM, ACB, & reciproca æquatricum pariter æqualium <lb/>BM, BM. </s>

<s>Sit etiam alius motus, &longs;ed æquabilis, cuius imago velocita<lb/>tum æqualis &longs;it, & homogenea ip&longs;i BAC, rectangulum <expan abbr="n&etilde;-pe">nen<lb/>pe</expan> AB in BM, & ideo &longs;i fiat BM ad BC &longs;icut DE ad DN, <lb/>concipiaturque rectangulum FD in DN, erit hoc imago <lb/><arrow.to.target n="marg47"></arrow.to.target><lb/>temporis dicti motus æquabilis, homogenea, & æqualis <lb/>imagini GFDEG; nam <expan abbr="t&etilde;pora">tempora</expan>, &longs;cilicet imagines GFDEG, <lb/><arrow.to.target n="marg48"></arrow.to.target><lb/>FD in DN rectangulum componuntur ex rationibus &longs;pa<lb/><arrow.to.target n="marg49"></arrow.to.target><lb/>tiorum, hoc e&longs;t imaginum velocitatum inter&longs;e æqualium, <lb/>ABM, ACB, & reciproca æquatricum pariter æqualium <lb/>BM, BM. </s>

<s>Cum igitur rectangulum FD in DN æquale &longs;it <lb/><arrow.to.target n="marg50"></arrow.to.target><lb/>imagini, &longs;eu figuræ GFDEG, habebit eadem figur&atail; <lb/>GFDEG ad rectangulum FD in DE eandem rationem, <lb/>quam DN ad DE, hoc e&longs;t quam BC ad BM, &longs;eu quam re<lb/>ctangulum AB in BC ad rectangulum AB in BM, aut ad ei <lb/>æqualem figuram ABC; & conuertendo, manife&longs;tum e&longs;t <lb/>quod propo&longs;uimus, nempe rectangulum FD in DE ad fi<lb/>guram GFDEG habere eandem <expan abbr="ration&etilde;">rationem</expan>, ac figura ACBA <pb pagenum="20"/>ad rectangulum AB in BC. quod erat demon&longs;trandum <lb/>primo loco. </s>

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<s><arrow.to.target n="marg55"></arrow.to.target><lb/>quarum a&longs;lymptoti AE, EL; Sint etiam quæcunque apli<lb/>catæ AB, DC alsymptoto EL æquidi&longs;tantes, & habeat <lb/>DE ad EA eandem rationem v. </s>

<s>g. quam cubus ex AB ad <pb pagenum="24"/><arrow.to.target n="marg56"></arrow.to.target><lb/>cubum DC. </s>

<s>quam cubus ex AB ad <pb pagenum="24"/><arrow.to.target n="marg56"></arrow.to.target><lb/>cubum DC. </s>

<s>Patet &longs;i proponeretur illi auuer&longs;a figur&atail; <lb/><arrow.to.target n="marg57"></arrow.to.target><lb/>FGK, e&longs;&longs;etque AE ad DE vt figura GFK ad figuram IHK <lb/>e&longs;&longs;e etiam FG ad IH vt DC ad AB, e&longs;t autem cubus ex <lb/>DC ad cubum ex AB vt AE ad ED; ergo etiam figur&atail; <lb/>FGK ad IHK (&longs;unt enim FG, IH parallel&ecedil;) habebit ean<lb/>dem rationem, ac cubus ex FG ad cubum ex IH: Itaqu&etail; <lb/>GFK erit comunis parabola, hoc e&longs;t quadratica, &longs;eu <expan abbr="&longs;ec&utilde;-">&longs;ecun<lb/></expan><arrow.to.target n="marg58"></arrow.to.target><lb/>da in &longs;erie infinitarum parabolarum, & ob id eadem GFK <lb/><arrow.to.target n="marg59"></arrow.to.target><lb/>parabola ad rectangulum GF in FK erit vt 2 ad 3, in qua <lb/>ratione &longs;e habebit quoque rectangulum BA in AE ad &longs;pa<lb/>tium infinitè longum & BM, et erit vt 2 ad 1; &longs;cilicet vt ex<lb/>ce&longs;&longs;us exponentis maioris pote&longs;tatis, quæ cubica e&longs;t, &longs;uper <lb/>numerum exponentis, qui hoc ca&longs;u e&longs;t tantùm vnitas ra<lb/>dicis, e&longs;t ad hunc ip&longs;um exponentem, &longs;eu vnitatem lineæ <lb/>indicantem, quod concordat cum propo&longs;ita dictoru&mtail; <lb/>authorum. </s>

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<s>Si &longs;imiles pote&longs;tates applicatarum fuerint in eadem ra<lb/>tione, ac &longs;unt inter&longs;e pote&longs;tates quædam aliæ, & eiu&longs;dem <lb/>gradus diametrorum ab ip&longs;is applicatis ab&longs;ci&longs;&longs;arum v&longs;que <pb pagenum="26"/>ad verticem parabolarum, vel trilineorum; erit rectangu<lb/>lum ad parabolam &longs;ibi in&longs;criptam vt aggregatum <expan abbr="expon&etilde;-tium">exponen<lb/>tium</expan> vtriu&longs;que pote&longs;tatis ad exponentem altioris ip&longs;arum <lb/>pote&longs;tatum parabolæ; & ad trilineum vt aggregatum ex<lb/>ponentium pote&longs;tatum trilinei ad exponentem inferioris <lb/>pote&longs;tatis eiu&longs;demmet trilinei. </s>

<s>Sic enim in expo&longs;ita figu<lb/>ra prædicta, &longs;i e&longs;&longs;et quadratum ex FG ad quadratum ex <lb/>IH, &longs;icut cubus ex FK ad cubum ex IH, e&longs;&longs;et rectangulum <lb/>GF in FK ad figuram GFK (quæ tunc foret trilineum, vt <lb/>5 ad 2; nam vbi pote&longs;tas ab&longs;ci&longs;&longs;arum maior e&longs;t illa applica. <lb/>tarum e&longs;t &longs;emper GF trilineum. </s>

<s>tarum e&longs;t &longs;emper GF trilineum. </s>

<s>Simili modo, &longs;i &longs;it vt qua<lb/>dratum ex FK ad quadratum ex KI ita cubocubus ex FG <lb/>ad cubocubum ex IH; hoc e&longs;t &longs;i &longs;it cubus ex FG ad <expan abbr="cub&utilde;">cubum</expan> <lb/>ex IH, vt linea FK ad KI (tolluntur enim vtrinque ex &longs;imi<lb/>libus &longs;imiles rationes) erit &longs;igura GFK parabola, ad quam <lb/>&longs;ibi circum&longs;criptum rectangulum eandem habebit <expan abbr="ration&etilde;">rationem</expan>, <lb/>quam 4 ad 3, & &longs;ic dicendum erit de omnibus alijs para<lb/>bolis atque trilineis. </s>

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<s>QVam rationem habet rectangulum BAE ad &longs;patium <lb/>& BAE &, eandem habet rectangulum CDE ad </s>

<s><arrow.to.target n="marg70"></arrow.to.target><lb/>&longs;patium & CDE, & permutando erit rectangu<lb/>lum BAE ad CDE, &longs;icut &longs;patium & BAE & ad &longs;patiu&mtail; <lb/>& CDE &; &longs;i igitur in eadem propo&longs;ita hyperbola &longs;it po<lb/>te&longs;tas applicatarum DC, AB quintuplicata ip&longs;ius A ad B, <lb/>& AE ad ED &longs;eptuplicata &longs;it eiu&longs;dem; erit &longs;eptuplicat&atail; <lb/>applicatarum in eadem ratione, ac quintuplicata ab&longs;ci&longs;&longs;a<lb/>rum; &longs;cilicet quadratoquadratocubus ex DC ad &longs;imilem <lb/>pote&longs;tatem ex AB erit vt quadratocubus ex AE ad qua<lb/>dratocubum ex DE, eritque &longs;ic maior pote&longs;tas applicata<lb/>rum, atque adeo componetur rectangulum EAB ad EDC <lb/>ex &longs;eptuplicata ip&longs;ius A ad B, qualis e&longs;t AE ad ED, & &longs;ub<lb/>quintuplicata eiu&longs;dem A ad B, quæ e&longs;t AB ad DC; nimi<lb/>rùm erit rectangulum EAB ad EDC in duplicata tantum <lb/>ratione ip&longs;ius A ad B: quare &longs;patium & BAE & ad id <lb/>& CDE &, quæ &longs;unt inter &longs;e, vt ip&longs;a rectangula, erit vt po<lb/>te&longs;tas ex A, cuius exponens e&longs;t differentia exponentium & <lb/>S pote&longs;tatum hyperbolæ ad &longs;imilem pote&longs;tatem ex B. <lb/>Quod &c. </s>

<s><margin.target id="marg70"></margin.target><emph type="italics"/>Pr.<emph.end type="italics"/> 12. <emph type="italics"/>huius.<emph.end type="italics"/></s>

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<s>Sed linquamus hæc alijs di&longs;<lb/>putanda: &longs;atis nobis &longs;it, methodum no&longs;tram, quoad <expan abbr="no&longs;tr&utilde;">no&longs;trum</expan> <lb/>e&longs;t, demon&longs;trare. </s>

<s>Ijs igitur vt &longs;upra propo&longs;itis, concipia<lb/>tur adhuc tempore CD velocitate FC <expan abbr="&longs;pati&utilde;">&longs;patium</expan> exigi quod<lb/>dam, item aliud tempore EG, velocitateque GI, & &longs;ic per <lb/>omnes qua&longs;cunque applicatas: quæritur, quod &longs;patiu&mtail; <lb/>vltimò exactum e&longs;&longs;et, hoc e&longs;t quam rationem id haberet ad <lb/>illud alterum &longs;patium, quod eodem tempore tran&longs;igitur <lb/>iu ta gene&longs;im HACB, cuius imago temporis CD & B. <lb/>I&longs;ti duo motus in exemplo e&longs;&longs;ent, &longs;i in quodam plano mo<lb/>ueretur formica, dum ip&longs;um planum vna eius extremitate <lb/>immobili circumduceretur, Sic formica difficiliùs <expan abbr="a&longs;c&etilde;de-ret">a&longs;cende<lb/>ret</expan> prout ip&longs;um planum magis ad horizontem erigeretur. <lb/>Iam motus extremitatis plani circumactæ habet gene&longs;im <lb/>ACBH, cuius temporis imago & DCB &, et altera gene&longs;is <lb/>FCBK <gap/>ribueretur motui formicæ, nam vt <expan abbr="dict&utilde;">dictum</expan> e&longs;t varius <lb/>motus formicæ pendet ex latione plani, ideò velocitates <lb/>eiu&longs;dem (nam in plano immobili ponimus æquabiliter fer<lb/>ri) durant ij&longs;dem temporibus, quibus velocitates præcipuæ <lb/>gene&longs;is ACBH. </s>

<s>I&longs;ti duo motus in exemplo e&longs;&longs;ent, &longs;i in quodam plano mo<lb/>ueretur formica, dum ip&longs;um planum vna eius extremitate <lb/>immobili circumduceretur, Sic formica difficiliùs <expan abbr="a&longs;c&etilde;de-ret">a&longs;cende<lb/>ret</expan> prout ip&longs;um planum magis ad horizontem erigeretur. <lb/></s>

<s>Iam motus extremitatis plani circumactæ habet gene&longs;im <lb/>ACBH, cuius temporis imago & DCB &, et altera gene&longs;is <lb/>FCBK <gap/>ribueretur motui formicæ, nam vt <expan abbr="dict&utilde;">dictum</expan> e&longs;t varius <lb/>motus formicæ pendet ex latione plani, ideò velocitates <lb/>eiu&longs;dem (nam in plano immobili ponimus æquabiliter fer<lb/>ri) durant ij&longs;dem temporibus, quibus velocitates præcipuæ <lb/>gene&longs;is ACBH. </s>

<s>Sit denique LMSR imago velocitatum <lb/>iuxta gene&longs;im ACBH, cuius temporis imago CD & B; pa<lb/>tet &longs;i &longs;it MP ad PS &longs;icut imago temporis CDEG ad ima<lb/>ginem & BGE &, fore LM ad PQ vt AC ad OG, & con<lb/>cepta etiam figura MNOTS inter parallelas LMN, RST <lb/>ita vt &longs;it &longs;emper MN ad PO &longs;icut FC ad GI, nec non LM <lb/>ad MN vt AC ad FC. (&longs;unt enim initio motuum in C, aut <lb/>in&longs;tanti M, velocitates gene&longs;ium AC, CF, &longs;cilicet LM, MN; <lb/>& in G, hoc e&longs;t in&longs;tanti P &longs;unt velocitates OC, GI; nimi<lb/>rum QP, PO) vocetur proinde gene&longs;is FCBK &longs;puria, ac <lb/>ad&longs;tricta imaginitemporis & DCB &, cuius imago veloci<lb/>tatum MNTS pariter &longs;puria, homogenea tamen ip&longs;i legiti<lb/><gap/>æ LMSR. </s><pb pagenum="33"/>

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<s>Sit etiam gene<lb/>&longs;is altera illi homogenea, &longs;ed &longs;puria, & ad&longs;tricta imagini <lb/>temporis & DCB &, cuius imago velocitatum &longs;puria, prio<lb/>rique legitimæ homogenea NMST. Dico, &longs;patia iuxta has <lb/>imagines tran&longs;acta e&longs;&longs;e vt ip&longs;æ imagines legitima LMSR <lb/>ad &longs;puriam NMST. </s>

<s>Cum temporis momenta M, P in<lb/>telligantur ex minimis temporibus, quæ proponi po&longs;&longs;unt, <lb/>inter&longs;e æqualibus, & quibus æquabiliter perdurant ve<lb/>locitates, quas mobile &longs;ortitur in aduentu &longs;uo in punctis <lb/>C, G, erit vt velocitas FC ad velocitatem GI &longs;ic inter&longs;e <lb/><arrow.to.target n="marg75"></arrow.to.target><lb/>&longs;patia, quæ i&longs;tis velocitatibus, temporibu&longs;que illis æqua<lb/>libus percurrerentur, in qua ratione e&longs;t etiam NM ad OP. <lb/>Deinde momento M peragerentur &longs;patia proportionalia <lb/>velocitatibus FC, AC, &longs;eu rectis NM, ML, momento <lb/>autem P &longs;patia proportionalia velocitatibus GI, GD, <lb/>in qua ratione e&longs;t etiam OP ad PQ, & &longs;ic deinceps <lb/>procedendo per &longs;ingula temporis MR momenta, adeo <lb/>vt, cum &longs;patium velocitate FC exactum ad id veloci<lb/>tate CA, &longs;it vt NM ad ML, &longs;patium velocitate IG ad id <lb/>exactum velocitate GD &longs;it vt OP ad PQ, & &longs;int præterea <lb/>primæ inter&longs;e, hoc e&longs;t &longs;patia velocitatibus FC, GI tran<lb/>&longs;acta, proportionalia tertijs, &longs;patijs videlicet tran&longs;actis <lb/>velocitatibus ML, PQ ergo vt omnes primæ ad omnes <lb/>tertias quantitates, hoc e&longs;t omnia &longs;patia tran&longs;acta iuxta <lb/>gene&longs;im FCBK ad omnia &longs;patia iuxta gene&longs;im ACB, ita <lb/>erit &longs;umma &longs;ecundarum ad |omnes quartas, &longs;cilicet i&longs;ta <lb/>erit imago NMST ad imaginem LMSR. </s>

<s>Deinde momento M peragerentur &longs;patia proportionalia <lb/>velocitatibus FC, AC, &longs;eu rectis NM, ML, momento <lb/>autem P &longs;patia proportionalia velocitatibus GI, GD, <lb/>in qua ratione e&longs;t etiam OP ad PQ, & &longs;ic deinceps <lb/>procedendo per &longs;ingula temporis MR momenta, adeo <lb/>vt, cum &longs;patium velocitate FC exactum ad id veloci<lb/>tate CA, &longs;it vt NM ad ML, &longs;patium velocitate IG ad id <lb/>exactum velocitate GD &longs;it vt OP ad PQ, & &longs;int præterea <lb/>primæ inter&longs;e, hoc e&longs;t &longs;patia velocitatibus FC, GI tran<lb/>&longs;acta, proportionalia tertijs, &longs;patijs videlicet tran&longs;actis <lb/>velocitatibus ML, PQ ergo vt omnes primæ ad omnes <lb/>tertias quantitates, hoc e&longs;t omnia &longs;patia tran&longs;acta iuxta <lb/>gene&longs;im FCBK ad omnia &longs;patia iuxta gene&longs;im ACB, ita <lb/>erit &longs;umma &longs;ecundarum ad |omnes quartas, &longs;cilicet i&longs;ta <lb/>erit imago NMST ad imaginem LMSR. </s>

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<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="italics"/>Hinc patet graue perpendiculariter, violenterque deiectum <lb/>minimè ad terram venturum aggregato virium, quarum vna <lb/>e&longs;t ab impellente impreßa, altera verò à grauitate <expan abbr="depend&etilde;s">dependens</expan>. <lb/>Nam ex impartita vt celerior fit ca&longs;us, quam vt graue in de<lb/>cur&longs;u &longs;uo po&longs;&longs;it ex acceler atione naturali eum gradum acqui<lb/>rere, quem certè &longs;ponte &longs;ua tantùm de&longs;cendens in fine eiu&longs;dem <lb/>altitudinis adeptum e&longs;&longs;et. </s>

<s>Nam ex impartita vt celerior fit ca&longs;us, quam vt graue in de<lb/>cur&longs;u &longs;uo po&longs;&longs;it ex acceler atione naturali eum gradum acqui<lb/>rere, quem certè &longs;ponte &longs;ua tantùm de&longs;cendens in fine eiu&longs;dem <lb/>altitudinis adeptum e&longs;&longs;et. </s>

<s>Hoc ita verum e&longs;t, vt aliquando <lb/>minimum inter&longs;it, inter impetum ab ambabus cau&longs;is proue<lb/>nientem, & eum, qui a &longs;ola oritur grauitate, quamobrem pa<lb/>rum is proficeret, qui conaretur maiorem impetum componere <lb/>in ca&longs;u grauis, illi nempe adiecta vi, mobile idem in decur&longs;u <lb/>impellente, vltra nat<gap/>am grauitatem, quod tamen fieri haud <lb/>dubiè po&longs;&longs;et, &longs;i ca&longs;us obliquus eßet.<emph.end type="italics"/></s><pb pagenum="36"/>

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<s><margin.target id="marg83"></margin.target><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>prima<emph.end type="italics"/></s>

<s>Similiter dum mobile mouetur tempore DG iuxta ima<lb/>gines DCIG, DCFG, feretur verè &longs;ecundùm imagine&mtail; <lb/><arrow.to.target n="marg84"></arrow.to.target><lb/>FCI ver&longs;us L, quamobrem &longs;i &longs;patium, quod exigeretur <lb/>hac imagine &longs;it RQ, habebit i&longs;tud ad LO eandem rationem, <lb/><arrow.to.target n="marg85"></arrow.to.target><lb/>quam imago CFI ad imaginem ABFG, & ideo ex æquali <lb/>QR ad PR &longs;e habebit vt imago CFI ad imaginem HBC; &longs;i <lb/>igitur ponatur ABFG maior imagine AHIG, demptà co<lb/>muniter AHCFG relinquetur HBC maior imagine CEI, & <lb/>ideo etiam PR maior QR: curritur verò PR versùs R tem<lb/>pore AD, & RQ versùs P tempore DG, ergo toto tempo<lb/>re AG curretur PQ differentia &longs;patiorum PR, <expan abbr="Rq.">Rque</expan> Cum <lb/>verò HBC ad CFI, &longs;it vt PR ad RQ, erit diuidendo vt ex<lb/>ce&longs;&longs;us imaginis HBC &longs;upra imaginem FCI ad imagine&mtail; <lb/>i&longs;tam, ita PQ ad QR, & o&longs;ten&longs;um e&longs;t QR ad LO, &longs;icut ima<lb/>go FCI ad imaginem ABFG, ergo ex æquali exce&longs;&longs;us ima<lb/>ginis HBC &longs;upra imaginem AHIG habebit eandem ratio<lb/>nem ad imaginem AHIG, ac PQ ad LO, at e&longs;t in illa <expan abbr="ead&etilde;">eadem</expan> <lb/>ratione etiam LM ad LO (e&longs;t enim LO ad MO vt imago <lb/>ABFG ad imaginem AHIG) ergo PQ erit æqualis LM, <lb/>atque adeo mobile dum currit vtroque motu, hoc e&longs;t iux<lb/>ta &longs;imul duas imagines propo&longs;itas contrariorum motuum, <lb/>peraget &longs;patium LM versùs O &longs;ecundùm imaginem, quæ <lb/>differentia e&longs;t propo&longs;itarum ABFG, AHIG, tempore AG. <lb/>Quod &c. </s><pb pagenum="38"/>

<s><margin.target id="marg84"></margin.target><emph type="italics"/>Ex prim&atail; <lb/>parte.<emph.end type="italics"/></s>

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<s>Itaque &longs;i mobile dum e&longs;t in A in<lb/>telligatur affectum velocitatibus AD, AC habentibus di<lb/>rectiones ip&longs;as rectas AD, AC, perinde e&longs;&longs;et, ac &longs;i &longs;ola fo<lb/>ret mobili velocitas vnâ cum directione AE. </s>

<s>Eadem ra<lb/>tione AF velocitas habens directionem AF, æquipollebit <lb/>duabus velocitatibus AB, AE iuxta directiones rectas ea&longs;-<pb pagenum="39"/>dem ABAE; hoc æquiualebit tribus AB, AC, AD. Mo<lb/>bile igitur ex affectione trium illorum conatuum, vt &longs;up<lb/>po&longs;itum fuit, nitetur &longs;ecundùm AF velocitate ip&longs;a AF <lb/>Quod &c. </s>

<s>Mo<lb/>bile igitur ex affectione trium illorum conatuum, vt &longs;up<lb/>po&longs;itum fuit, nitetur &longs;ecundùm AF velocitate ip&longs;a AF <lb/>Quod &c. </s>

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<s>Cum, vti diximus, ad de&longs;criptionem lineæ duo tantùm <lb/><arrow.to.target n="marg92"></arrow.to.target><lb/>exigantur, nempe motus, & puncti directio; motus verò po<lb/>te&longs;t e&longs;&longs;e quilibet, &longs;equitur ergo directionem, alteram de <lb/>duobus, &longs;eruari debere. </s>

<s><margin.target id="marg92"></margin.target><emph type="italics"/>Ax.<emph.end type="italics"/> 2. <emph type="italics"/>buius. <lb/>pr.<emph.end type="italics"/> 5. <emph type="italics"/>huius.<emph.end type="italics"/></s>

<s><margin.target id="marg92"></margin.target><emph type="italics"/>Ax.<emph.end type="italics"/> 2. <emph type="italics"/>buius. <lb/></s>

<s>pr.<emph.end type="italics"/> 5. <emph type="italics"/>huius.<emph.end type="italics"/></s>

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<s>Si latera compo&longs;iti motus e&longs;&longs;ent duo tantùm AB, AC. <lb/>Facto parallelogrammo vt dictum e&longs;t, inueniretur pun<lb/>ctum E extremum motus: & <expan abbr="quæcunq;">quæcunque</expan> &longs;it &longs;emita, &longs;eu mo<lb/>tus, pote&longs;t idem E &longs;upponi tanquam extremum alterius la<lb/>teris, adeoque, &longs;i motus con&longs;tet ex tribus lateribus AC, <lb/>AB, AD, perinde &longs;it ac &longs;i foret duorum laterum AE, AD; <lb/>nam AC, AD valent &longs;imul ac &longs;olum AE; cum ita &longs;it, facto <lb/>etiam parallelogrammo EADF ex datis punctis E, A, D, <lb/>habebitur F extremum &longs;emitæ, cuius &longs;unt tria latera CA, <lb/>AD, AB — </s>

<s>Si latera compo&longs;iti motus e&longs;&longs;ent duo tantùm AB, AC. <lb/></s>

<s>Facto parallelogrammo vt dictum e&longs;t, inueniretur pun<lb/>ctum E extremum motus: & <expan abbr="quæcunq;">quæcunque</expan> &longs;it &longs;emita, &longs;eu mo<lb/>tus, pote&longs;t idem E &longs;upponi tanquam extremum alterius la<lb/>teris, adeoque, &longs;i motus con&longs;tet ex tribus lateribus AC, <lb/>AB, AD, perinde &longs;it ac &longs;i foret duorum laterum AE, AD; <lb/>nam AC, AD valent &longs;imul ac &longs;olum AE; cum ita &longs;it, facto <lb/>etiam parallelogrammo EADF ex datis punctis E, A, D, <lb/>habebitur F extremum &longs;emitæ, cuius &longs;unt tria latera CA, <lb/>AD, AB — </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/>PROP. XII. THEOR. VIII.<emph.end type="center"/></s>

<s>CVm imagines velocitatum, iuxta quas curruntur du&ecedil; <lb/><arrow.to.target n="marg103"></arrow.to.target><lb/>rectæ, quæ &longs;int latera compo&longs;iti motus, &longs;unt paral. <lb/>logrammum, & triangulum; tunc &longs;emita compo&longs;iti motus <lb/>erit communis parabola. </s>

<s>CVm imagines velocitatum, iuxta quas curruntur du&ecedil; <lb/><arrow.to.target n="marg103"></arrow.to.target><lb/>rectæ, quæ &longs;int latera compo&longs;iti motus, &longs;unt paral. <lb/></s>

<s>logrammum, & triangulum; tunc &longs;emita compo&longs;iti motus <lb/>erit communis parabola. </s>

<s>Tempore HM curratur latus AC iuxta imaginem velo<lb/>citatum HILM rectangulum, & latus AB iuxta imaginem <lb/><arrow.to.target n="marg104"></arrow.to.target><lb/>triangulum HMN; erit CA ad AB, vt imago <expan abbr="parallelogrã-mum">parallelogran<lb/>mum</expan> HILM ad aliam imaginem triangulum NHM. </s>

<s>Fiat <lb/><arrow.to.target n="marg105"></arrow.to.target><lb/><expan abbr="parellogrãmum">parellogrammum</expan> ACDB erit in D extremum &longs;emitæ com<lb/>po&longs;iti motus, quæ &longs;i ponatur AFC; Dico e&longs;&longs;e parabolam. <lb/>Sumatur in ip&longs;a linea quoduis punctum F, ab ip&longs;o dedu-<pb pagenum="47"/>cta FE parallela AB, vti etiam FG parallela AC, erunt <lb/><arrow.to.target n="marg106"></arrow.to.target><lb/>AE, AG latera compo&longs;iti motus, cuius &longs;emita AF: Con<lb/>cipiatur modò P momentum, quo mobile ade&longs;t in F, & <lb/>ducta OPK parallela alteri HI, vel NL, erit imago MHIL ad <lb/><arrow.to.target n="marg107"></arrow.to.target><lb/><expan abbr="imagin&etilde;">imaginem</expan> PHIK, hoc e&longs;t MH ad HP, vt CA ad AE, &longs;eu vt BD <lb/>ad GF. </s>

<s>Sumatur in ip&longs;a linea quoduis punctum F, ab ip&longs;o dedu-<pb pagenum="47"/>cta FE parallela AB, vti etiam FG parallela AC, erunt <lb/><arrow.to.target n="marg106"></arrow.to.target><lb/>AE, AG latera compo&longs;iti motus, cuius &longs;emita AF: Con<lb/>cipiatur modò P momentum, quo mobile ade&longs;t in F, & <lb/>ducta OPK parallela alteri HI, vel NL, erit imago MHIL ad <lb/><arrow.to.target n="marg107"></arrow.to.target><lb/><expan abbr="imagin&etilde;">imaginem</expan> PHIK, hoc e&longs;t MH ad HP, vt CA ad AE, &longs;eu vt BD <lb/>ad GF. </s>

<s>Pariter erit imago NHM ad <expan abbr="imagin&etilde;">imaginem</expan> OHP, hoc e&longs;t <lb/>quadratum ex MH ad <expan abbr="quadrat&utilde;">quadratum</expan> ex PH; immò id ex BO ad <lb/>illud ex GF, vt BA ad AG; quamobrem punctum F cadet <lb/>in curuam parabolicam communem, cuius diameter AB, <lb/>& ba&longs;is, &longs;eu ordinatim applicata BD, &longs;cilicet AFD erit ip&longs;a <lb/>curua parabolica. </s>

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<s>Hyperbolæ IRS &longs;it centrum H, &longs;emiaxis HI, a&longs;&longs;ymptoti <lb/>HT, NH, et SN parallela HI; tùm ducta HM &longs;ecunda dia<lb/>metro hyperbolæ, intelligatur de&longs;criptio parabolæ AFD; <lb/>itaut duplus axis hyperbolæ, hoc e&longs;t quadruplum ip&longs;ius <lb/>HI ad NT eandem habeat rationem, quam DB ba&longs;is pa<lb/>rabolæ ad BA axim eiu&longs;dem. </s>

<s>Dico quadrilineum HISM <lb/>e&longs;&longs;e imaginem velocitatum, iuxta quam motu compo&longs;ito <lb/>de&longs;cribitur parabola AFD; & cum &longs;it homogenea imagi<lb/><arrow.to.target n="marg109"></arrow.to.target><lb/>nibus HILM, HTM, e&longs;&longs;e quoque rectangulum HDLM ad <lb/>imaginem ip&longs;am HISM vt recta CA ad curuam AFD. <lb/>Fiat rectangulum ACDB, et HM &longs;it tempus, quo curritur <lb/><arrow.to.target n="marg110"></arrow.to.target><lb/>vtrunque latus AB, AC, nempe axis AB motu grauium <lb/>iuxta imaginem triangulum HTM, alterum verò latus AC <lb/><arrow.to.target n="marg111"></arrow.to.target><lb/>æquabili motu iuxta imaginem rectangulum HILM, quod <lb/>quidem erit HILM; etenim AB ad &longs;patium AC e&longs;t vt ima<lb/>go triangulum HMT ad imaginem rectangulum HILM, <lb/>&longs;cilicet e&longs;t vt MT ad duplam HI, vel vt NT ad quadru<lb/>plam HI, quemadmodum po&longs;uimus. </s>

<s>Fiat rectangulum ACDB, et HM &longs;it tempus, quo curritur <lb/><arrow.to.target n="marg110"></arrow.to.target><lb/>vtrunque latus AB, AC, nempe axis AB motu grauium <lb/>iuxta imaginem triangulum HTM, alterum verò latus AC <lb/><arrow.to.target n="marg111"></arrow.to.target><lb/>æquabili motu iuxta imaginem rectangulum HILM, quod <lb/>quidem erit HILM; etenim AB ad &longs;patium AC e&longs;t vt ima<lb/>go triangulum HMT ad imaginem rectangulum HILM, <lb/>&longs;cilicet e&longs;t vt MT ad duplam HI, vel vt NT ad quadru<lb/>plam HI, quemadmodum po&longs;uimus. </s>

<s>Iam mon&longs;trauimus <lb/>lineam, quæ curritur iuxta illas imagines motu compo&longs;ito <lb/>parabolam e&longs;&longs;e, cuius diameter AB, & ba&longs;is BD; & pro<lb/>pterea erit ip&longs;a AFD (nam vnica tantum parabola ex <lb/>datis AB, BD po&longs;itione, ac magnitudine, axi &longs;cilicet, ac <lb/>ba&longs;i dari pote&longs;t) Ducatur nunc à quolibet puncto F dictæ <lb/>parabolæ rectæ FE, FG parallelogrammum con&longs;tituentes <lb/>AEFG; & P &longs;it momentum, quo mobile punctum inueni<lb/><arrow.to.target n="marg112"></arrow.to.target><lb/>tur in F. </s>

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<s>Similiter cum &longs;it, vt BA ad AE, ita CA <lb/>ad AF, etiam BC, EF inter&longs;e æquidi&longs;tabunt, &longs;eu ad hori<lb/>zontem æqualiter inclinabuntur, eruntque in ratione ea<lb/>dem, ac BA ad AE. </s>

<s>Idemque o&longs;tenditur de chordis CD, <lb/>FG, quare ex magni Galilei &longs;ententia de motu naturaliter <lb/>accelerato indubitanter &longs;equitur tempus decur&longs;us &longs;phæræ <lb/>grauis ex B in D per binas chordas BC, CD ad tempus <lb/>decur&longs;us per vnicam BD, e&longs;&longs;e vt tempus decur&longs;us grauis <lb/>&longs;phæræ ex E in G per binas EF, FG ad tempus decur&longs;us <lb/>per vnicam EG: cadem itidem ratione demon&longs;tratur (an<lb/>gulis pariter BAC, CAD bifariam &longs;ectis per rectas, quæ <lb/>&longs;imiles arcus BC, EF, ac CD, FG duas in partes diuidant) <lb/>ex quatuor vtrinque arcuum horum cordis, illas inter&longs;e <pb pagenum="57"/>homologas, &longs;imile&longs;que arcus &longs;ubtendentes ad horizonte m <lb/>e&longs;&longs;e æqualiter inclinatas, ac alteram alteri in ratione ea<lb/>dem, in qua &longs;unt rectæ AB, AE &c: ac propterea ex ea<lb/>dem Galilei &longs;cientia con&longs;tabit vtique, tempus decur&longs;us ex <lb/>B in C &longs;phæræ grauis B per quatuor chordas quatuor par<lb/>tes arcus BCD &longs;ubtendentes ad tempus decur&longs;us per vni<lb/>cam BD, e&longs;&longs;e vt tempus decur&longs;us &longs;phæræ grauis E ex E in <lb/>G per quatuor illis homologas chordas quatuor partes <lb/>arcus EFG pariter &longs;ubtendentes ad tempus decur&longs;us per <lb/>vnicam chordam EG: & hoc &longs;emper ita euenire demon<lb/>&longs;trabitur quantacunque, & maxima fuerit in perpetua an<lb/>gulorum bi&longs;ectione æquèmultiplicitas in vtroque arcu <lb/>talium chordarum homologè &longs;umptarum, ac inter&longs;e pro<lb/>portionalium, æqualiterque ad horizontem inclinatarum: <lb/>Propterquam quòd &longs;emper decur&longs;us ex B in D per aggre<lb/>gatum chordarum omnium in arcu BCD ad tempus de<lb/>cur&longs;us per &longs;olam chordam BD e&longs;&longs;e vt tempus decur&longs;us ex <lb/>E in G per aggregatum totidem chordarum in arcu EFG <lb/>ad tempus decur&longs;us per vnicam chordam EG; adeo vt de<lb/>nique iure optimo educi po&longs;&longs;e videatur, tempus decur&longs;us <lb/>grauis ex B in D per aggregatum infinitarum chordarum <lb/>totum arcum BCD con&longs;tituentium, &longs;eu tempus per ip&longs;um <lb/>arcum BCD ad tempus decur&longs;us per &longs;olam cordam BD <lb/>e&longs;&longs;e vt tempus decur&longs;us grauis ex E in G per aggregatum <lb/>totidem infinitarum chordarum dictis homologè propor<lb/>tionalium, æqualiterque &longs;ingulæ &longs;ingulis ad horizonte&mtail; <lb/>inclinatarum, ac totum arcum EFG conformantium, &longs;iue <lb/>vt tempus per ip&longs;um arcum EFG per &longs;olam chordam EG. <lb/>Quocirca permutando, tempus, decur&longs;us &longs;phæræ grauis B <lb/>per arcum BCD ad tempus decur&longs;us &longs;phæræ grauis E per <lb/>arcum &longs;imilem, &longs;imiliterque po&longs;itum EG erit vt tempus <lb/>decur&longs;us per chordam BD ad tempus decur&longs;us per chor<lb/>dam EG; &longs;ed ex eadem Galilaica &longs;cientia de motu, tempus <lb/>decur&longs;us per chordam BD ad tempus decur&longs;us per æqua-<pb pagenum="58"/>iter inclinatam EG e&longs;t in &longs;ubduplicata ratione ip&longs;aru&mtail; <lb/>chordarum BD, EG; ergo tempus quoque decur&longs;us ex B <lb/>per arcum BCD ad tempus decur&longs;us ex E per arcum EFG <lb/>e&longs;t in eadem &longs;ubduplicata ratione chord&ecedil; BD ad chordam <lb/>EG, quod o&longs;tendendum propo&longs;uimus. </s>

<s>Quocirca permutando, tempus, decur&longs;us &longs;phæræ grauis B <lb/>per arcum BCD ad tempus decur&longs;us &longs;phæræ grauis E per <lb/>arcum &longs;imilem, &longs;imiliterque po&longs;itum EG erit vt tempus <lb/>decur&longs;us per chordam BD ad tempus decur&longs;us per chor<lb/>dam EG; &longs;ed ex eadem Galilaica &longs;cientia de motu, tempus <lb/>decur&longs;us per chordam BD ad tempus decur&longs;us per æqua-<pb pagenum="58"/>iter inclinatam EG e&longs;t in &longs;ubduplicata ratione ip&longs;aru&mtail; <lb/>chordarum BD, EG; ergo tempus quoque decur&longs;us ex B <lb/>per arcum BCD ad tempus decur&longs;us ex E per arcum EFG <lb/>e&longs;t in eadem &longs;ubduplicata ratione chord&ecedil; BD ad chordam <lb/>EG, quod o&longs;tendendum propo&longs;uimus. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="italics"/>Hactenus graui&longs;&longs;imus Vir; &longs;upere&longs;t modò, vt quemadmo<lb/>dum annuimus, veritatem eandem no&longs;tra quoque methodo, <lb/>confirmemus, vt ijs, quibus &longs;atis probat demon&longs;tratio allata, <lb/>&longs;it nostra, quam afferemus, in experimentum traditarum hùc <lb/><expan abbr="v&longs;q;">v&longs;que</expan> rérum; & quibus &longs;ecùs acciderit ex aliqua dubitatione, <lb/>hæc per demon&longs;trationes no&longs;tras pror&longs;us, &longs;<gap/>atimq tollatur. <lb/>Illud etiam admoneo, eam rem non tantum me o&longs;ten&longs;urum,<emph.end type="italics"/><pb pagenum="61"/><emph type="italics"/>vt pulcherrima, <expan abbr="vtilimaq;">vtilimaque</expan> veritas pluribus demon&longs;trationi<lb/>bus aperiatur; verùm potius vt ampli&longs;&longs;ima Methodus, qua tum <lb/>vtemur, aliorum motuum demon&longs;trandorum in exemplum <lb/>veniat.<emph.end type="italics"/></s>

<s>Illud etiam admoneo, eam rem non tantum me o&longs;ten&longs;urum,<emph.end type="italics"/><pb pagenum="61"/><emph type="italics"/>vt pulcherrima, <expan abbr="vtilimaq;">vtilimaque</expan> veritas pluribus demon&longs;trationi<lb/>bus aperiatur; verùm potius vt ampli&longs;&longs;ima Methodus, qua tum <lb/>vtemur, aliorum motuum demon&longs;trandorum in exemplum <lb/>veniat.<emph.end type="italics"/></s>

<s><emph type="center"/>PROP. XVI. THEOR. XII.<emph.end type="center"/></s>

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<s>Sunt enim parallelæ &c. </s>

<s>inter&longs;e tam rectæ CB, <lb/>ML, quàm CA, MH; ideo anguli ACB, HML inter&longs;e <lb/>æquabuntur, & &longs;unt circa eos proportionalia latera, nem. <lb/>pe BC ad CA, vt LM, MH; ergo con&longs;tat propo&longs;itum. </s>

<s>inter&longs;e tam rectæ CB, <lb/>ML, quàm CA, MH; ideo anguli ACB, HML inter&longs;e <lb/>æquabuntur, & &longs;unt circa eos proportionalia latera, nem. <lb/></s>

<s>pe BC ad CA, vt LM, MH; ergo con&longs;tat propo&longs;itum. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><margin.target id="marg135"></margin.target><emph type="italics"/>I<gap/>mma<emph.end type="italics"/> 18. <emph type="italics"/>in <lb/>libro de dim. <lb/>parab. </s>

<s><margin.target id="marg135"></margin.target><emph type="italics"/>I<gap/>mma<emph.end type="italics"/> 18. <emph type="italics"/>in <lb/>libro de dim. <lb/></s>

<s>parab. </s>

<s>Euang. <lb/>T<gap/>rric<gap/>l.<emph.end type="italics"/></s>

<s>Euang. <lb/></s>

<s><emph type="center"/>PROP. XIX. THEOR. XV.<emph.end type="center"/></s>

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<s>PRopo&longs;itis ij&longs;dem &longs;olidis, erunt inter &longs;e, vt momenta fi<lb/>gurarum a quibus &longs;unt, quæ tamen figuræ &longs;u&longs;pen&longs;æ <lb/>&longs;int ex longitudinibus deductis ab ip&longs;arum grauitatu&mtail; <lb/>centris v&longs;que ad coeuntium figurarum communes illas &longs;e<lb/>ctiones. </s>

<s>Figuræ, à quibus &longs;unt &longs;olida, ponantur ABC, GLH, <expan abbr="c&etilde;-">cen<lb/></expan><arrow.to.target n="marg139"></arrow.to.target><lb/>tra grauitatum illarum M, N; axes, &longs;iue communes &longs;ectio<lb/>nes coeuntium binarum inter&longs;e &longs;imilium, ac æqualium fi<lb/>gurarum à quibus dicuntur ip&longs;a &longs;olida; & demum MO, NP <lb/>perpendiculares &longs;int ab ip&longs;is centris ad illas communes &longs;e<lb/>ctiones deductæ CE, HL. Dico, &longs;olidum à plana figur&atail; <lb/>ABC ad &longs;olidum a plana GHL eandem habere rationem, <lb/>ac momentum figuræ ABC pendentis ex MO ad momen<lb/><arrow.to.target n="marg140"></arrow.to.target><lb/>tum alterius figuræ &longs;u&longs;pen&longs;æ ex NP, &longs;unt enim hæc &longs;oli<lb/>da inter&longs;e, vt rotunda, quorum genetrices figuræ ABC, <lb/>GLH circa axes CE, HL, huiu&longs;modi verò &longs;olida &longs;unt vt <lb/><arrow.to.target n="marg141"></arrow.to.target><lb/>momenta propo&longs;ita; ergo &longs;olidum à plana figura ABC ad <lb/>&longs;olidum à plana GLH, erit vt momentum figuræ ABC <lb/>&longs;u&longs;pen&longs;æ ex MO ad momentum GLH pendentis ex NP. <lb/>Quod &c. </s>

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<s>EX datis &longs;patijs accelerato motu confectis, cogniti&longs;<lb/>que primis, aut po&longs;tremis &longs;imilium, &longs;impliciumque <lb/>motuum velocitatibus, reperire tempora ip&longs;orum de<lb/>cur&longs;uum. </s>

<s>Spatia motibus acceleratis exacta &longs;unt C, D, & velo<lb/><arrow.to.target n="marg173"></arrow.to.target><lb/>tates, &longs;eu amplitudines gene&longs;um ponantur e&longs;&longs;e A, B, &longs;cili<lb/>cet A principio motus per C, & B initio motus per D, quæ<lb/>ritur ratio temporum, quibus exiguntur propo&longs;ita &longs;patia. <lb/>Vt A ad B, ita fiat C ad E, & inter E, et D &longs;umatur F me<lb/>dia proportionalis. </s>

<s>Vt A ad B, ita fiat C ad E, & inter E, et D &longs;umatur F me<lb/>dia proportionalis. </s>

<s>Dico ip&longs;a tempora e&longs;&longs;e vt E ad F. <lb/></s>

<s>Dico ip&longs;a tempora e&longs;&longs;e vt E ad F. <lb/>Componuntur &longs;patia acceleratis motibus exacta ex ratio<lb/><arrow.to.target n="marg174"></arrow.to.target><lb/>ne quadratorum temporum, & ex ea amplitudinum, &longs;eu <lb/>homologarum velocitatum in &longs;implicibus motibus, &longs;imili<lb/><arrow.to.target n="marg175"></arrow.to.target><lb/>bu&longs;que &longs;umptarum; & ideo temporum quadrata necten<lb/>tur ex ratione &longs;patiorum C ad D, & ex reciproca ampli-<pb pagenum="74"/>tudinum E ad C; temporum igitur quadrata erunt vt E ad <lb/>D, ip&longs;a verò tempora vt E ad F. </s>

<s>Componuntur &longs;patia acceleratis motibus exacta ex ratio<lb/><arrow.to.target n="marg174"></arrow.to.target><lb/>ne quadratorum temporum, & ex ea amplitudinum, &longs;eu <lb/>homologarum velocitatum in &longs;implicibus motibus, &longs;imili<lb/><arrow.to.target n="marg175"></arrow.to.target><lb/>bu&longs;que &longs;umptarum; & ideo temporum quadrata necten<lb/>tur ex ratione &longs;patiorum C ad D, & ex reciproca ampli-<pb pagenum="74"/>tudinum E ad C; temporum igitur quadrata erunt vt E ad <lb/>D, ip&longs;a verò tempora vt E ad F. </s>

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<s>Dico <lb/>tempora horum de&longs;cen&longs;uum e&longs;&longs;e in ratione &longs;ubduplicat&atail; <lb/>arcuum FI, AC, &longs;eu longitudinum filorum, aut ha&longs;tularum <lb/>FA, LA. </s>

<s>Ducamus quamcumque rectam LBG, erit AB <lb/>ad BC, vt FG ad GI, & cum præterea velocitates pendu<lb/>lorum a quiete in A, F &longs;int æquales, pariterque velocita<lb/>tes æquales a quiete in B, G; erit velocitas in A ad veloci<lb/>tatem in B, vt velocitas in F ad velocitatem in G, quare <lb/>con&longs;ideratis arcubus ABC, FGI, vt altitudines rect&ecedil;, (quæ <lb/>item forent in B, G proportionaliter &longs;ect&ecedil;) gene&longs;um &longs;imi<lb/><arrow.to.target n="marg178"></arrow.to.target><lb/>lium &longs;impliciumque motuum, quarum amplitudines æqua <lb/>les &longs;unt, erunt &longs;patia in acceleratis decur&longs;ubus per FI, AC <lb/>in ratione duplicata temporum, &longs;cilicet ip&longs;i arcus, aut lon<lb/>gitudines LF, LA erunt in ratione duplicata temporu&mtail;. <lb/>Quod &c. </s>

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<s>Sint plana AB, AC eandem eleuationem AD habentia. <lb/>Dico tempus lationis per AC ad id per AB e&longs;&longs;e vt AC ad <lb/>AB. (hæc Torricellij propo&longs;itio, <expan abbr="expo&longs;itioq;">expo&longs;itioque</expan> e&longs;t, hancque <lb/>eandem veritatem ex no&longs;tris principijs demon&longs;trare <expan abbr="vis&utilde;">visum</expan> <lb/>e&longs;t, non vt de re illa dubitemus, immò contrà, quòd de e&atail; <lb/>plenè &longs;atisfacti &longs;imus, ex eo rur&longs;us demon&longs;trandam &longs;u&longs;ce<lb/>pimus, vt exinde methodus no&longs;tra, quàm vera &longs;it, eluce&longs;<lb/>cat) Momentum de&longs;cen&longs;us inplano AC ad id de&longs;cen&longs;us &longs;u<lb/><arrow.to.target n="marg180"></arrow.to.target><lb/>per plano AB e&longs;t vt AB ad AC; &longs;unt autem <expan abbr="de&longs;cendenti&utilde;">de&longs;cendentium</expan> <lb/>grauium, etiam &longs;uper planis inclinatis motus, quos &longs;impli<lb/>ces appellamus, inter &longs;e &longs;imiles, nempe quorum gene&longs;es <lb/><arrow.to.target n="marg181"></arrow.to.target><lb/>&longs;unt rectangula; ergo habebimus &longs;implices gene&longs;es, vnam, <lb/>cuius altitudo AC amplitudoque AB; alteram, cuius am<lb/>plitudo AC, altitudo autem AB; itaque propo&longs;itis &longs;patijs <lb/>AC, AB, primi&longs;que velocitatibus AB, AC, &longs;i fiat AB ad AC <lb/>vt CA ad EA, erit EA ad AB duplicata <expan abbr="t&etilde;porum">temporum</expan>, & ideo <lb/><arrow.to.target n="marg182"></arrow.to.target><lb/>ratio temporum per AC, AB erit CA ad AB. </s>

<s>Sint plana AB, AC eandem eleuationem AD habentia. <lb/></s>

<s>Dico tempus lationis per AC ad id per AB e&longs;&longs;e vt AC ad <lb/>AB. (hæc Torricellij propo&longs;itio, <expan abbr="expo&longs;itioq;">expo&longs;itioque</expan> e&longs;t, hancque <lb/>eandem veritatem ex no&longs;tris principijs demon&longs;trare <expan abbr="vis&utilde;">visum</expan> <lb/>e&longs;t, non vt de re illa dubitemus, immò contrà, quòd de e&atail; <lb/>plenè &longs;atisfacti &longs;imus, ex eo rur&longs;us demon&longs;trandam &longs;u&longs;ce<lb/>pimus, vt exinde methodus no&longs;tra, quàm vera &longs;it, eluce&longs;<lb/>cat) Momentum de&longs;cen&longs;us inplano AC ad id de&longs;cen&longs;us &longs;u<lb/><arrow.to.target n="marg180"></arrow.to.target><lb/>per plano AB e&longs;t vt AB ad AC; &longs;unt autem <expan abbr="de&longs;cendenti&utilde;">de&longs;cendentium</expan> <lb/>grauium, etiam &longs;uper planis inclinatis motus, quos &longs;impli<lb/>ces appellamus, inter &longs;e &longs;imiles, nempe quorum gene&longs;es <lb/><arrow.to.target n="marg181"></arrow.to.target><lb/>&longs;unt rectangula; ergo habebimus &longs;implices gene&longs;es, vnam, <lb/>cuius altitudo AC amplitudoque AB; alteram, cuius am<lb/>plitudo AC, altitudo autem AB; itaque propo&longs;itis &longs;patijs <lb/>AC, AB, primi&longs;que velocitatibus AB, AC, &longs;i fiat AB ad AC <lb/>vt CA ad EA, erit EA ad AB duplicata <expan abbr="t&etilde;porum">temporum</expan>, & ideo <lb/><arrow.to.target n="marg182"></arrow.to.target><lb/>ratio temporum per AC, AB erit CA ad AB. </s>

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<s>Dico &longs;patia temporibus &longs;impli<lb/>cium imaginum, ab extremitatibus &longs;olutis exacta, fore i&ntail; <lb/>ratione longitudinum ip&longs;orum funiculorum. </s>

<s>Iam con&longs;tat CE ad DF e&longs;&longs;e, vt AC ad BD, in qua ratione <lb/>&longs;unt etiam velocitates à quiete, dum pondera &longs;ubduceren<lb/>tur ex E, et F, vel ex alijs punctis quibu&longs;cunque &longs;i æqualia <pb pagenum="80"/>pondera &longs;u&longs;pen&longs;a fui&longs;&longs;ent maioris, vel minoris ponderis, <lb/>&longs;ic enim concipiuntur gene&longs;es &longs;imilium, &longs;impliciumque <lb/>motuum, quarum altitudines æquantur elongationibus <lb/>funiculorum; propterea &longs;patia recur&longs;uum temporibus &longs;im<lb/>plicium motuum exacta, nectentur ex rationibus duplicata <lb/>CE ad DF, hoc e&longs;t AC ad BD, & ex reciproca filorum, <lb/>&longs;cilicet BD ad AC, quæ ratio, vti diximus, e&longs;t reciproc&atail; <lb/>primarum velocitatum, &longs;eu amplitudinum gene&longs;um &longs;impli<lb/>cium, ergo ip&longs;a &longs;patia in reditu filorum ab extremitatibus <lb/>&longs;olutis exacta, erunt vt AC ad BF, &longs;eu vt CE ad DF. <lb/>Quod &c. </s>

<s><emph type="center"/>PROP. XXXIX. THEOR. XXXI.<emph.end type="center"/></s>

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<s>In antecedenti figura illud primum &longs;atis patet, quòd &longs;i <lb/>loco ponderis F &longs;u&longs;pen&longs;um fui&longs;&longs;et pondus aliud grauius, <lb/>aut leuius, prior velocitas in a&longs;cen&longs;u fili, &longs;eu funiculi, aut <lb/>chordæ aucta, vel imminuta fui&longs;&longs;et pro magnitudine pon<lb/>deris &longs;ub&longs;tituti; quamobrem priores velocitates ex inæqua <lb/>litate ponderum eidem chordæ &longs;u&longs;pen&longs;orum dependentes <lb/>forent, vt ip&longs;a pondera; verùm cum &longs;uppo&longs;itis funiculis <lb/>æqualia pondera &longs;u&longs;pen&longs;a veniunt, primæ velocitates &longs;unt <lb/><arrow.to.target n="marg190"></arrow.to.target><lb/>vt longitudines funiculorum, ergo velocitates primæ, cum <lb/>inæqualia &longs;unt pondera, quæ &longs;ubtrahuntur, nectentur ex <lb/>ratione longitudinum funiculorum, & ex ea ponderum <lb/>inæqualium: quæcumque igitur &longs;it tractio DF, gene&longs;es ha<lb/>bebimus &longs;imilium &longs;impliciumque motuum, vnam, cuius al<lb/>titudo CE, & alteram habentem altitudinem DF, & &longs;unt <lb/>earundem gene&longs;um amplitudines, &longs;eu primæ velocitates <lb/>in ratione compo&longs;ita funiculorum AC ad BD, & ponderis <lb/><arrow.to.target n="marg191"></arrow.to.target><lb/>pendentis ex E ad pondus &longs;u&longs;pen&longs;um in F; ergo &longs;patia ac<lb/>celeratis motibus tran&longs;acta temporibus gene&longs;um <expan abbr="&longs;implici&utilde;">&longs;implicium</expan> <pb pagenum="83"/>nectentur ex ratione dublicata elongationum, &longs;iue altitu<lb/>dinum gene&longs;um, & ex duabus rationibus reciprocè &longs;um<lb/>ptis funiculorum AC ad BD, & ponderum E ad F. <lb/>Quod &c. </s>

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<s>La &longs;econda difficoltà è che cia&longs;cuna molla nel &longs;uo re<lb/>&longs;tringer&longs;i, par che cagioni qualche effetto contrario all'in<lb/>tento. </s>

<s>Imperoche, per e&longs;empio, nella molla B il mezzo <lb/>anello, che ri&longs;guarda l'e&longs;tremità A, nello &longs;tringer&longs;i fà ben&longs;i <lb/>il &longs;uo douere, perche il &longs;uo moto è ver&longs;o il centro M; ma l' <lb/>altra metà, che ri&longs;guarda il &longs;udetto centro M, nello &longs;trin<lb/>ger&longs;i, hauendo il &longs;uo moto ver&longs;o A, &longs;i oppone al chiudi<lb/>mento della molla &longs;eguente C; e il &longs;imile dica&longs;i dell' altre. <lb/>A ciò &longs;i è po&longs;to rimedio col far più grandi, e più mafficcie <lb/>le molle più vicine al centro M, accre&longs;cendole, e ingro&longs;&longs;an<lb/>dole di mano in mano opportunamente. </s>

<s>A ciò &longs;i è po&longs;to rimedio col far più grandi, e più mafficcie <lb/>le molle più vicine al centro M, accre&longs;cendole, e ingro&longs;&longs;an<lb/>dole di mano in mano opportunamente. </s>

<s>Quindi ne &longs;egue <lb/>che per la maggior grandezza <expan abbr="cõ&longs;entono">con&longs;entono</expan> egualmente all' <lb/>aprir&longs;i con facilità; ma all' incontro nel &longs;errar&longs;i, per e&longs;&longs;ere <lb/>più ma&longs;&longs;iccie, e di maggior corpo, vengono ad hauere <lb/>maggior momento delle men corpulenti, &longs;uperando co&ntail; <lb/>ciò non &longs;olo il detto moto oppo&longs;to, ma etiandio impri<lb/>mendo maggior moto al ferro dell'arco, con cui &longs;i acco<lb/>muna il moto. </s>

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<s>Hinc &longs;equitur vt funis ex medio <lb/>dum attrahitur in O, aperiantur prædictæ commi&longs;&longs;uræ, &longs;eu <lb/>nodi, & curuentur vtraque brachia, vt in eorum altero ap<lb/>paret punctis notato. </s>

<s>Quilibet ex his nodis arcti&longs;limè &longs;trin<lb/>gitur &longs;upernè, a &longs;uo elaterio, vt videre e&longs;t in L, H, D, C, B. <lb/>Elateria autem quò propinquiora centro M tanto maiora, <lb/>& cra&longs;&longs;iora debent e&longs;&longs;e remotioribus: Hinc fit vt, propter <lb/>molem opportunè auctam, æquè facilè aperiantur, ac cæ<lb/>tera; & vice ver&longs;a, propter cra&longs;&longs;itiem maiorem, &longs;ibi relicta <lb/>validiùs re&longs;tringantur. </s>

<s>Quilibet ex his nodis arcti&longs;limè &longs;trin<lb/>gitur &longs;upernè, a &longs;uo elaterio, vt videre e&longs;t in L, H, D, C, B. <lb/></s>

<s>Elateria autem quò propinquiora centro M tanto maiora, <lb/>& cra&longs;&longs;iora debent e&longs;&longs;e remotioribus: Hinc fit vt, propter <lb/>molem opportunè auctam, æquè facilè aperiantur, ac cæ<lb/>tera; & vice ver&longs;a, propter cra&longs;&longs;itiem maiorem, &longs;ibi relicta <lb/>validiùs re&longs;tringantur. </s>

<s>Cuius rei paulo infra rationem <lb/>dabimus. </s>

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<s>Prima <lb/>e&longs;t, quòd licèt vis &longs;ufficiens in A ad vincendum <expan abbr="æquilibri&utilde;">æquilibrium</expan> <lb/>elaterij B, illa eadem quoque &longs;ufficiat ad vincendum æqui<lb/>librium cæterorum, propter æquales proportiones <expan abbr="vecti&utilde;">vectium</expan>; <lb/>his tamen non ob&longs;tantibus, &longs;i con&longs;ideretur brachium iam <lb/>incuruatum, vt apparet in KLA punctis notato, proportio<lb/>nes illæ cernuntur notabiliter variatæ. </s>

<s>Neque enim pro <lb/>longitudinibus vectium &longs;umi po&longs;&longs;unt longitudines priores, <lb/>&longs;ed loco ip&longs;arum accipiendæ &longs;unt applicatæ arcus, videli<lb/>cet af, ag, ai, ak quarum ak, eidemque propinquiores, <expan abbr="quã-">quan-</expan><pb pagenum="94"/>do arcus incuruatur, breuiores fiunt, quàm e&longs;&longs;ent ante&atail;. <lb/>Re&longs;pondeo, quòd corda ao cùm &longs;it obliquior re&longs;pectu <lb/>longitudinis ae, quàm re&longs;pectu cæterarum centro propin<lb/>quiorum, hinc fit vt, quantùm e&longs;t ex hac ratione, faciliùs <lb/>aperiantur partes propinquiores centro; quamobrem, vtra<lb/>que ratione inuicem temperata, dummodo arcus non &longs;it <lb/>&longs;ummè incuruatus omnes partes aperientur, quantum &longs;a<lb/>tis e&longs;t ad intentum. </s>

<s>Re&longs;pondeo, quòd corda ao cùm &longs;it obliquior re&longs;pectu <lb/>longitudinis ae, quàm re&longs;pectu cæterarum centro propin<lb/>quiorum, hinc fit vt, quantùm e&longs;t ex hac ratione, faciliùs <lb/>aperiantur partes propinquiores centro; quamobrem, vtra<lb/>que ratione inuicem temperata, dummodo arcus non &longs;it <lb/>&longs;ummè incuruatus omnes partes aperientur, quantum &longs;a<lb/>tis e&longs;t ad intentum. </s>

<s>Altera difficultas e&longs;t, quod elaterium quodlibet dum <lb/>re&longs;tringitur videtur ob&longs;tare motui elaterij &longs;equentis. </s>

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<s>in Metropol. </s>

<s>Bonon. <lb/>pro Illu&longs;tri&longs;s. </s>

<s>Bonon. <lb/></s>

<s>pro Illu&longs;tri&longs;s. </s>

<s>& Reverendi&longs;s. </s>

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