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Colored diff for /texts/archimedes/xml/fabri_tract_026_la_1646.xml between version 1.20 and 1.21

version 1.20, 2007/02/02 14:39:10

version 1.21, 2007/02/02 17:59:17

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<s id="N22474">quia &longs;it impetus æqualis impre&longs;&longs;us puncto A ma<lb/>ioris rotæ puncto F minoris, ita vt A & F moueantur æquali motu; </s>

<s id="N2247A">mi<lb/>nor rota conficit duos orbes eo tempore, quo maior vnum conficit, vt <lb/>con&longs;tat ex dictis; quia &longs;uppono. </s>

<s id="N22482">v. g. circulum minoris e&longs;&longs;e &longs;ubduplum; <lb/></s>

<s id="N22483">igitur tempus, quo peragitur maior e&longs;t ad tempus, quo peragitur minor <lb/>in ratione dupla; igitur vt radius AE ad radium FE, quod erat demon<lb/>&longs;trandum. </s>

<s id="N22483">igitur tempus, quo peragitur maior e&longs;t ad tempus, quo peragitur minor <lb/>in ratione dupla; </s>

<s id="N22484">igitur vt radius AE ad radium FE, quod erat demon<lb/>&longs;trandum. </s>

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<s id="N22776">&longs;i verò pellatur idem cylindrus in T. v.g. mouebitur circa cen-<pb pagenum="293" xlink:href="026/01/327.jpg"/>trum, quod e&longs;t inter MH, licèt propiùs accedat ad M, quàm ad H, vt <lb/>con&longs;tat ex calculatione; </s>

<s id="N22783">e&longs;t autem aliquod punctum inter TA, ex quo &longs;i <lb/>pellatur, mouebitur circa punctum H; </s>

<s id="N22789">&longs;i verò a&longs;&longs;umantur alia puncta <lb/>ver&longs;us A, ex quibus pellatur, centra motus, erunt extra BH, ac proinde <lb/>extremitas B pul&longs;a ex B mouetur per arcum BN; </s>

<s id="N22791">pul&longs;a ex A per rectam <lb/>AL; pul&longs;a denique ex punctis, quæ &longs;unt inter BA, per arcus maiorum <lb/>circulorum, cò &longs;anè maiorum, quò propiùs punctum, ex quo pellitur, ac<lb/>cedit ad A. </s>

<s id="N22791">pul&longs;a ex A per rectam <lb/>AL; pul&longs;a denique ex punctis, quæ &longs;unt inter BA, per arcus maiorum <lb/>circulorum, eò &longs;anè maiorum, quò propiùs punctum, ex quo pellitur, ac<lb/>cedit ad A. </s>

<s id="N2279E"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s>

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<s id="N22816"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s>

<s id="N22824"><emph type="italics"/>Si &longs;uperponatur corpus plano rotæ, quæ voluitur in circulo horizontæli, pro<lb/>ùcietur per Tangentem extremam.<emph.end type="italics"/> v.g. &longs;it rota ABCD horizontali pa<lb/>rallela quæ vertatur ab A ver&longs;us B celeri motu, &longs;itque planum eius le<lb/>uigati&longs;&longs;imum; </s>

<s id="N22824"><emph type="italics"/>Si &longs;uperponatur corpus plano rotæ, quæ voluitur in circulo horizontali, pro<lb/>iicietur per Tangentem extremam.<emph.end type="italics"/> v.g. &longs;it rota ABCD horizontali pa<lb/>rallela quæ vertatur ab A ver&longs;us B celeri motu, &longs;itque planum eius le<lb/>uigati&longs;&longs;imum; </s>

<s id="N22835">imponatur globus etiam leuigati&longs;&longs;imus puncto A: </s>

<s id="N22839">dico <lb/>quod proiicietur per Tangentem AF, quia impetus, qui in illo impri<lb/>mitur in puncto F e&longs;t determinatus ad Tangentem A <foreign lang="greek">q</foreign>; </s>

<s id="N22845">&longs;ed non impe<lb/>ditur, quominus habeat &longs;uum motum; </s>

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<s id="N22865"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s>

<s id="N22873"><emph type="italics"/>Si imponatur globus puncto F plani horizontalis rotæ ABCD, non proir<lb/>cietur per Tangentem F<emph.end type="italics"/> <foreign lang="greek">u</foreign> quod primò manife&longs;tis experimentis comproba<lb/>tum e&longs;t. </s>

<s id="N22873"><emph type="italics"/>Si imponatur globus puncto F plani horizontalis rotæ ABCD, non proii<lb/>cietur per Tangentem F<emph.end type="italics"/> <foreign lang="greek">u</foreign> quod primò manife&longs;tis experimentis comproba<lb/>tum e&longs;t. </s>

<s id="N22883">Secundò probatur, quia dum globus his punctis, in quibus re-<pb pagenum="294" xlink:href="026/01/328.jpg"/>cta F <foreign lang="greek">u</foreign> &longs;ecat alios maiores circulos concentricos, ab his punctis nouum <lb/>impetum accipit, ratione cuius debet mutare lineam, quod certum e&longs;t; </s>

<s id="N22892"><lb/>cum autem circuli maiores rotæ moucantur velociùs, quàm FGH, po<lb/>tiori iure mutari debet determinatio currentis globi in prædicto plano; </s>

<s id="N22892"><lb/>cum autem circuli maiores rotæ moueantur velociùs, quàm FGH, po<lb/>tiori iure mutari debet determinatio currentis globi in prædicto plano; </s>

<s id="N22899"><lb/>quænam verò &longs;it hæc linea motus, difficilè dictu e&longs;t; dicemus tamen <lb/>Tomo &longs;equenti, cum de lineis motus. </s>

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<s id="N229F3">inde etiam impetum de&longs;trui certum e&longs;t; <lb/>nec enim &longs;ine re&longs;i&longs;tentia id fieri pote&longs;t. </s>

<s id="N229FB">Ob&longs;eruabis decimo&longs;extò, impetum eundem habere po&longs;&longs;e motum cir<lb/>oularem, & rectum in &longs;ublunaribus, & per accidens determinari tantùm <lb/>ad motum circularem, ratione &longs;cilicet impedimenti, vt con&longs;tat ex dictis. </s>

<s id="N229FB">Ob&longs;eruabis decimo&longs;extò, impetum eundem habere po&longs;&longs;e motum cir<lb/>cularem, & rectum in &longs;ublunaribus, & per accidens determinari tantùm <lb/>ad motum circularem, ratione &longs;cilicet impedimenti, vt con&longs;tat ex dictis. </s>

<s id="N22A04">Ob&longs;eruabis decimo&longs;eptimò, motum rectum accelerari, &longs;ed diu non <lb/>durare; </s>

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<s id="N22B1E">res <lb/>e&longs;t clara in ten&longs;o fune, quomodocunque id fiat, quod hîc non di&longs;cutio; <lb/>compre&longs;&longs;io verò e&longs;t vis illata ab extrin&longs;eco corpori, qua contrahitur eius <lb/>exten&longs;io v.g. in intorto fune. </s>

<s id="N22B2C">Ob&longs;eruabis autem ad ten&longs;ionem, & compre&longs;&longs;ionem requiri, vt &longs;ubla<lb/>ta illa vi extrin&longs;eca, vel impedimento admoto corpus ten&longs;um, vel com<lb/>pre&longs;&longs;um ad pri&longs;tinam <gap/>ten&longs;ionem &longs;e&longs;e reducat; neque di&longs;puto de mo<lb/>do, quo id fieri po&longs;&longs;it, qui alterius loci e&longs;t. </s>

<s id="N22B2C">Ob&longs;eruabis autem ad ten&longs;ionem, & compre&longs;&longs;ionem requiri, vt &longs;ubla<lb/>ta illa vi extrin&longs;eca, vel impedimento admoto corpus ten&longs;um, vel com<lb/>pre&longs;&longs;um ad pri&longs;tinam exten&longs;ionem &longs;e&longs;e reducat; neque di&longs;puto de mo<lb/>do, quo id fieri po&longs;&longs;it, qui alterius loci e&longs;t. </s>

<s id="N22B3A"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/></s>

<s id="N22B49"><emph type="italics"/>Corpus graue funependulum à &longs;uæ quiete, vel è &longs;uo centro grauitatis remo<lb/>tum de&longs;cendit &longs;uâ &longs;pente, iterumque a&longs;cendit, id e&longs;t vibratur<emph.end type="italics"/>; cer<lb/>tum e&longs;t. </s>

<s id="N22B49"><emph type="italics"/>Corpus graue funependulum à &longs;uæ quiete, vel è &longs;uo centro grauitatis remo<lb/>tum de&longs;cendit &longs;uâ &longs;ponte, iterumque a&longs;cendit, id e&longs;t vibratur<emph.end type="italics"/>; cer<lb/>tum e&longs;t. </s>

<s id="N22B58"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/></s>

<s id="N22B67"><emph type="italics"/>Funependula longiera maiore tempore &longs;uam vibrationam conficiunt, bre<lb/>uiora minore<emph.end type="italics"/>; quod etiam certum e&longs;t. </s>

<s id="N22B67"><emph type="italics"/>Funependula longiora maiore tempore &longs;uam vibrationam conficiunt, bre<lb/>uiora minore<emph.end type="italics"/>; quod etiam certum e&longs;t. </s>

<s id="N22B74"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 3.<emph.end type="center"/></s>

<s id="N22B83"><emph type="italics"/>Motus naturalis e&longs;t acceleratus in tempore &longs;en&longs;ibili in proportione nume<lb/>rerum<emph.end type="italics"/> 1.3.5.7. <emph type="italics"/>&c.<emph.end type="italics"/> quod multis explicatum e&longs;t lib. 2. &longs;i verò acceleratio <pb pagenum="299" xlink:href="026/01/333.jpg"/>a&longs;&longs;umatur in &longs;ingulis in&longs;tantibus finitis, e&longs;t iuxta &longs;eriem &longs;implicem nu<lb/>merorum 1. 2. 3. 4. &c. </s>

<s id="N22B83"><emph type="italics"/>Motus naturalis e&longs;t acceleratus in tempore &longs;en&longs;ibili in proportione nume<lb/>rorum<emph.end type="italics"/> 1.3.5.7. <emph type="italics"/>&c.<emph.end type="italics"/> quod multis explicatum e&longs;t lib. 2. &longs;i verò acceleratio <pb pagenum="299" xlink:href="026/01/333.jpg"/>a&longs;&longs;umatur in &longs;ingulis in&longs;tantibus finitis, e&longs;t iuxta &longs;eriem &longs;implicem nu<lb/>merorum 1. 2. 3. 4. &c. </s>

<s id="N22B9C"><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 4.<emph.end type="center"/></s>

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<s id="N22D3A"><emph type="italics"/>In motu funependuli decre&longs;cunt &longs;emper incrementa motus.<emph.end type="italics"/></s>

<s id="N22D41"> Probatur faci<lb/>lè; </s>

<s id="N22D46">quia cùm in &longs;ingulis punctis de&longs;cen&longs;us arcus NE mutetur ratio plani <lb/>inclinati diucr&longs;a ab ea, quæ e&longs;t in puncto <expan abbr="q;">que</expan> &longs;unt enim vt Tangentes; </s>

<s id="N22D46">quia cùm in &longs;ingulis punctis de&longs;cen&longs;us arcus NE mutetur ratio plani <lb/>inclinati diuer&longs;a ab ea, quæ e&longs;t in puncto <expan abbr="q;">que</expan> &longs;unt enim vt Tangentes; </s>

<s id="N22D50"><lb/>certè Tangentes punctorum, quæ propiùs accedunt ad N, accedunt <lb/>etiam propiùs ad perpendicularem deor&longs;um, à qua longiùs recedunt <lb/>Tangentes, quæ accedunt propiùs ad E, vt con&longs;tat; </s>

<s id="N22D59">at qui motus in planis, <lb/>quæ accedunt propiùs ad horizontalem, minor e&longs;t; </s>

<s id="N22D5F">igitur incrementa <lb/>motus quæ in de&longs;cen&longs;u NE accedunt, minora &longs;unt ver&longs;us E, maiora ver<lb/>&longs;us N; igitur decre&longs;cunt, quod erat demon&longs;trandum. </s>

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<s id="N22DD0">Denique compertum e&longs;t à me <lb/>in longiore pendulo motum in arcu e&longs;&longs;e tardiorem, quàm in perpendi<lb/>culo: </s>

<s id="N22DD8">nodus ob&longs;eruationis facilis e&longs;t, nam adhibui AE planum durum <lb/>re&longs;pondens accuratè perpendiculari, cui aliud planum E <foreign lang="greek">b</foreign> ad angulos <lb/>rectos affixum erat <expan abbr="re&longs;põdens">re&longs;pondens</expan> Tangenti; </s>

<s id="N22DE8">tùm demi&longs;&longs;o ex A globulo plum<lb/>beo &longs;imulque alio æquali pendulo &longs;cilicet circa A ex N per NE; </s>

<s id="N22DEE">ex quo <lb/>accidit citiùs auditum e&longs;&longs;e ictum globi cadentis perpendiculariter, quàm <lb/>vibrati per arcum NE: quis autem hoc non videat, &longs;iue fen&longs;um ip&longs;um, <lb/>&longs;iue rationem con&longs;ulat? </s>

<s id="N22DEE">ex quo <lb/>accidit citiùs auditum e&longs;&longs;e ictum globi cadentis perpendiculariter, quàm <lb/>vibrati per arcum NE: quis autem hoc non videat, &longs;iue &longs;en&longs;um ip&longs;um, <lb/>&longs;iue rationem con&longs;ulat? </s>

<s id="N22DF8">fuit meum pendulum 12. pedes longum. </s>

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<s id="N22F15"><emph type="italics"/>Si accipiantur tempora æqualia cum motibus inæqualibus, &longs;patia &longs;unt vt <lb/>ba&longs;es triangulorum<emph.end type="italics"/>; </s>

<s id="N22F20">&longs;it enim tempus AI, quo motu maiore acquiratur ve<lb/>locitas IB, & minore IK; </s>

<s id="N22F26">certè &longs;patia &longs;unt vt triangula BAI, KAI; </s>

<s id="N22F2A"><lb/>&longs;ed hæc &longs;unt vt ba&longs;es BI, KI, immò &longs;unt vt rectangula BA KA; nec <lb/>in his e&longs;t quidquam difficultaris. </s>

<s id="N22F2A"><lb/>&longs;ed hæc &longs;unt vt ba&longs;es BI, KI, immò &longs;unt vt rectangula BA KA; nec <lb/>in his e&longs;t quidquam difficultatis. </s>

<s id="N22F33"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 3.<emph.end type="center"/></s>

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<s id="N22F54">velo<lb/>citas primi e&longs;t ad velocitatem &longs;ecundi in tempore DI, vt DP ad DI, <lb/>vel vt AK ad &longs;inum VK, vel vt IP ad NP, vel vt quadratum IA ad <lb/>rectangulum NA; </s>

<s id="N22F5E">&longs;ed &longs;patia &longs;unt vt velocitates &longs;uppo&longs;itis temporibus <lb/>æqualibus; </s>

<s id="N22F64">igitur &longs;patium, quod percurritur in ip&longs;a perpendiculari e&longs;t <lb/>ad &longs;patium, quod percurritur in inclinata DP temporibus æqualibus, vt <lb/>quadratum IA ad rectangulum NA, vel vt DP ad DI, vel vt DT ad <lb/>DP, quæ omnia con&longs;tant; </s>

<s id="N22F6E">&longs;it autem motus in inclinata FP; certè &longs;pa<lb/>tium acqui&longs;itum in perpendiculari e&longs;t ad &longs;patium acqui&longs;itum in FP, vt <lb/>QZP ad ZI, vel FP ad FY, vel AP ad PR, vel AL ad LX, vell I <lb/>ad PM, vel vt quadratum IA, ad rectangulum MA, vel vt F <foreign lang="greek">d</foreign> ad PF, <lb/>&longs;ed F <foreign lang="greek">d</foreign> e&longs;t æqualis DT, quia cum DP & FP percurrantur temporibus <lb/>æqualibus, <expan abbr="&longs;i&qacute;ue">&longs;ique</expan> eo tempore quo percurritur DP, percurritur DT, & <lb/>eo quo percurritur FP, percurritur. </s>

<s id="N22F6E">&longs;it autem motus in inclinata FP; certè &longs;pa<lb/>tium acqui&longs;itum in perpendiculari e&longs;t ad &longs;patium acqui&longs;itum in FP, vt <lb/>QZP ad ZI, vel FP ad FY, vel AP ad PR, vel AL ad LX, vel PI <lb/>ad PM, vel vt quadratum IA, ad rectangulum MA, vel vt F <foreign lang="greek">d</foreign> ad PF, <lb/>&longs;ed F <foreign lang="greek">d</foreign> e&longs;t æqualis DT, quia cum DP & FP percurrantur temporibus <lb/>æqualibus, <expan abbr="&longs;i&qacute;ue">&longs;ique</expan> eo tempore quo percurritur DP, percurritur DT, & <lb/>eo quo percurritur FP, percurritur. </s>

<s id="N22F8A">F <foreign lang="greek">d</foreign>; certè DT & F <foreign lang="greek">d</foreign> &longs;unt <lb/>quales. </s>

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<s id="N22FBB"><emph type="italics"/>Si &longs;int duæ quantitates in data ratione, & aliæ duæ in data, &longs;ed minore; </s>

<s id="N22FC1">&longs;i &longs;it <lb/>media proportionalis inter duas primas & media inter duas posteriores, &longs;itque <lb/>data noua quantitas ad aliam, vt prima priorum quantitatum ad primam <lb/>mediam proportionalem, &longs;it denique eadem quantitas noua ad aliam vt prima <lb/>po&longs;teriorum quantitatum ad &longs;ecundam mediam proportionalem, certè erit mi<lb/>nor ratio neua quantitatis ad &longs;ecundam que&longs;itam, quàm ad primam<emph.end type="italics"/> v. g. &longs;it <lb/>DE prima quantitas, & LK &longs;ecunda; </s>

<s id="N22FC1">&longs;i &longs;it <lb/>media proportionalis inter duas primas & media inter duas posteriores, &longs;itque <lb/>data noua quantitas ad aliam, vt prima priorum quantitatum ad primam <lb/>mediam proportionalem, &longs;it denique eadem quantitas noua ad aliam vt prima <lb/>po&longs;teriorum quantitatum ad &longs;ecundam mediam proportionalem, certè erit mi<lb/>nor ratio noua quantitatis ad &longs;ecundam que&longs;itam, quàm ad primam<emph.end type="italics"/> v. g. &longs;it <lb/>DE prima quantitas, & LK &longs;ecunda; </s>

<s id="N22FD8">&longs;it KR tertia, VZ quarta; </s>

<s id="N22FDC">&longs;itque <lb/>prima ad &longs;ecundam, vt 4. ad 9. & tertia ad quartam, vt 3. ad 12. certè e&longs;t <lb/>minor ratio tertiæ ad quartam, quàm primæ ad &longs;ecundam; </s>

<s id="N22FE4">inter primam <lb/>& &longs;ecundam &longs;it media proportionalis AC æqualis FH, id e&longs;t <foreign lang="greek">s</foreign>, & &longs;it <lb/>quinta quantitas; </s>

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<s id="N23073"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 6.<emph.end type="center"/></s>

<s id="N23082"><emph type="italics"/>Cognite tempore, quo percurritur &longs;egmentum, lineæ cogno&longs;oi pote&longs;t tempus, que <lb/>aliud &longs;egmentum percurretur motu &longs;cilicet propagate<emph.end type="italics"/>; </s>

<s id="N23082"><emph type="italics"/>Cognite tempore, quo percurritur &longs;egmentum, lineæ cogno&longs;ci pote&longs;t tempus, que <lb/>aliud &longs;egmentum percurretur motu &longs;cilicet propagate<emph.end type="italics"/>; </s>

<s id="N2308D">&longs;it v. g. perpendicu<lb/>laris deor&longs;um DI; </s>

<s id="N23097">&longs;it primum &longs;egmentum DG decur&longs;um tempore AB; </s>

<s id="N2309B"><lb/>&longs;it vt DC ad DH, ita DH ad DI; </s>

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<s id="N230B2"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 7.<emph.end type="center"/></s>

<s id="N230C0"><emph type="italics"/>Cognito tempore, quo persurritur chorda cuiu&longs;libet arcus, cogno&longs;ci pote&longs;t <lb/>quantum &longs;paty eodem tempore percurratur in <expan abbr="perp&etilde;diculari">perpendiculari</expan> & in alia chorda<emph.end type="italics"/>; </s>

<s id="N230C0"><emph type="italics"/>Cognito tempore, quo percurritur chorda cuiu&longs;libet arcus, cogno&longs;ci pote&longs;t <lb/>quantum &longs;paty eodem tempore percurratur in <expan abbr="perp&etilde;diculari">perpendiculari</expan> & in alia chorda<emph.end type="italics"/>; </s>

<s id="N230CF"><lb/> &longs;it chorda EL; </s>

<s id="N230D4">fiat angulus rectus ELM, itemque MDE: </s>

<s id="N230D8">dico quod <lb/>eodem tempore percurretur EL EM ED; </s>

<s id="N230DE">&longs;imiliter fiat angulus re<lb/>ctus EIH, itemque HKE, HQE: dico quod eodem tempore percur<lb/>rentur EI, EH, EK, <expan abbr="Eq;">Eque</expan> idem dico de omnibus aliis chordis, quæ <lb/>omnia con&longs;tant ex his quæ diximus lib.2. & 5. </s>

<s id="N230DE">&longs;imiliter fiat angulus re<lb/>ctus EIH, itemque HKE, HQE: dico quod eodem tempore percur<lb/>rentur EI, EH, EK,EQ. idem dico de omnibus aliis chordis, quæ <lb/>omnia con&longs;tant ex his quæ diximus lib.2. & 5. </s>

<s id="N230EF"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 8.<emph.end type="center"/></s>

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<s id="N231F0">immò <lb/>acqui&longs;ita in tribus EILB e&longs;t æqualis acqui&longs;itæ in EB; </s>

<s id="N231F6">quia acqui&longs;ita in <lb/>EIL e&longs;t æqualis acqui&longs;itæ in EL; </s>

<s id="N231FC">igitur acqui&longs;ita in EILB æqualis <lb/>acqui&longs;itæ in ELB: </s>

<s id="N23202">&longs;ed acqui&longs;ita in ELB e&longs;t æqualis acqui&longs;itæ in EB; igi<lb/>tur acqui&longs;ita in EB æqualis acqui&longs;itæ in EILB idem dico de 5. chordis<gap/><lb/>6.7. atque ita deinceps. </s>

<s id="N2320D">Quod certè mirabile e&longs;t, & qua&longs;i paradoxon; </s>

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<s id="N233B1">&longs;ed IP e&longs;t æqua<lb/>lis BT primæ; </s>

<s id="N233B7">igitur BT, cui etiam e&longs;t æqualis TZ; </s>

<s id="N233BB">igitur TZ æqualis <lb/>P <foreign lang="greek">b</foreign> &longs;ecundæ: </s>

<s id="N233C5">idem dico de aliis tran&longs;uerfis: immò demon&longs;trabimus tom. <lb/></s>

<s id="N233C5">idem dico de aliis tran&longs;uer&longs;is: immò demon&longs;trabimus tom. <lb/></s>

<s id="N233CA"><expan abbr="&longs;eq.">&longs;eque</expan> quadratricem quadrantis, cuius radius &longs;it NA terminari ad punctum <lb/>T, ita vt NT &longs;it ba&longs;is quadratricis, & NA latus; non tamen propterea <lb/>hæc linea &longs;inuum e&longs;t quadratrix, vt demon&longs;trabimus. </s>

<s id="N233D7"><emph type="center"/><emph type="italics"/>Lemma<emph.end type="italics"/> 13.<emph.end type="center"/></s>

<s id="N233E5"><emph type="italics"/>Diuer&longs;æ chordæ acquirunt diuer&longs;am velocitatem pro diuer&longs;a ratione &longs;inuum <lb/>ver&longs;orum &longs;uorum arcuum.<emph.end type="italics"/> v. g. velocitas acqui&longs;ita in chorda AM e&longs;t <lb/>æqualis acqnifitz in &longs;inu ver&longs;o AQ, & acqui&longs;ita in chorda AL æqualis <lb/>acqui&longs;itæ in &longs;inu ver&longs;o AR, atque ita deinceps; donec acqui&longs;ita in AC <lb/>&longs;it æqualis acqui&longs;itæ in &longs;inu toto AB. </s>

<s id="N233E5"><emph type="italics"/>Diuer&longs;æ chordæ acquirunt diuer&longs;am velocitatem pro diuer&longs;a ratione &longs;inuum <lb/>ver&longs;orum &longs;uorum arcuum.<emph.end type="italics"/> v. g. velocitas acqui&longs;ita in chorda AM e&longs;t <lb/>æqualis acqui&longs;itæ in &longs;inu ver&longs;o AQ, & acqui&longs;ita in chorda AL æqualis <lb/>acqui&longs;itæ in &longs;inu ver&longs;o AR, atque ita deinceps; donec acqui&longs;ita in AC <lb/>&longs;it æqualis acqui&longs;itæ in &longs;inu toto AB. </s>

<s id="N233FD">Itaque in chorda quæ ducitur ab A, velocitas cre&longs;cit vt in &longs;inu ver&longs;o <lb/>eiu&longs;dem.v.g. </s>

<s id="N23402">in AM, AL, AK; in chorda verò, quæ ducitur ab aliquo <lb/>puncto arcus AC v&longs;que ad C, cre&longs;cit vt in &longs;inu recto. </s>

<s id="N23407">v.g. velocitas ac<lb/>qui&longs;ita in chorda LC e&longs;t æqualis acqui&longs;itæ in perpendiculari LE, quæ <lb/>e&longs;t &longs;inus rectus arcus LC; item acquiritur æqualis velocitas in duabus <lb/>at que in vna, dum &longs;cilicet communes terminos habeant. </s>

<s id="N23413">v.g. in duabus <pb pagenum="307" xlink:href="026/01/341.jpg"/>AKC acquiritur æqualis acqui&longs;itæ in AC; </s>

<s id="N2341E">nam in AK, A <gap/> acquiritur <lb/>æqualis; </s>

<s id="N2341E">nam in AK, A <foreign lang="greek">w</foreign> acquiritur <lb/>æqualis; </s>

<s id="N23426">tùm etiam in KC, <foreign lang="greek">w</foreign> C; Item in tribus acquititur æqualis ac<lb/>qui&longs;itæ in duabus, atque ita deinceps. </s>

<s id="N23426">tùm etiam in KC, <foreign lang="greek">w</foreign> C; Item in tribus acquiritur æqualis ac<lb/>qui&longs;itæ in duabus, atque ita deinceps. </s>

<s id="N23432">Præterea velocitas acqui&longs;ita in chordis mediis.v.g. </s>

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<s id="N234B5">a&longs;&longs;umatur &longs;ubduplum <lb/>quadrati, ex quo extrahatur radix; </s>

<s id="N234BB">habebitur KD, vel BK 37893. &longs;it <lb/>autem LF 200000. ad 141422. æqualem BF, ita BF ad LH 100000. <lb/>certè tempus per LH e&longs;t ad tempus per BH, vt LH ad BH; </s>

<s id="N234C3">&longs;ed tempus <lb/>per LH e&longs;t ad tempus per LF, vt LH ad 141422.igitur tempus per BH <lb/>e&longs;t ad tempus per HF facto initio motus ex L, vt BH 51764. ad 41422. <lb/>igitur ad tempus per BHF, vt 51764.ad 93186. porrò BH & BK æqua<lb/>li tempore percurruntur; </s>

<s id="N234CF">igitur tempus per BK e&longs;t BH, id e&longs;t 51764. <lb/>cùm autem &longs;patia in eadem linea &longs;int in ratione duplicata temporum; <lb/>certè &longs;patium BK acquifitum tempore 51764.e&longs;t ad &longs;patium acqui&longs;itum <lb/>in BF tempore 93186. vt quadratum 51764. ad quadratum 93186.id e&longs;t, <lb/>vt 2679511696.ad 8676630576.vnde factâ regulâ trium habeo &longs;patium <lb/>decur&longs;um in BF 122702. tempore 93186. &longs;ed tota BF e&longs;t 141422. igitur <lb/>citiùs percurruntur duæ BHF, quàm BF. </s>

<s id="N234CF">igitur tempus per BK e&longs;t BH, id e&longs;t 51764. <lb/>cùm autem &longs;patia in eadem linea &longs;int in ratione duplicata temporum; <lb/>certè &longs;patium BK acqui&longs;itum tempore 51764.e&longs;t ad &longs;patium acqui&longs;itum <lb/>in BF tempore 93186. vt quadratum 51764. ad quadratum 93186.id e&longs;t, <lb/>vt 2679511696.ad 8676630576.vnde factâ regulâ trium habeo &longs;patium <lb/>decur&longs;um in BF 122702. tempore 93186. &longs;ed tota BF e&longs;t 141422. igitur <lb/>citiùs percurruntur duæ BHF, quàm BF. </s>

<s id="N234E1">Præterea &longs;int duæ BGF, BG e&longs;t 100000.&longs;it perpendicularis G 4 cùm <lb/>angulus GB 4.&longs;it grad.30. erit vt 5 G ad GB, ita BG ad B 4. igitur B 4. <lb/>erit 115469. &longs;it 4.3.perpendicularis in BF, quadratum B 4. e&longs;t duplum <lb/>quadrati B 3.igitur B 3. erit 81655. iam verò FN e&longs;t &longs;ecans grad.75. &longs;ci<lb/>licet 386370.igitur GN e&longs;t 334606. detracta &longs;cilicet FG æquali BH; </s>

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<s id="N236A9">tùm &longs;tatim alium; </s>

<s id="N236AD">immò &longs;i numeren<lb/>tur vibrationes vtriu&longs;que eodem tempore plures minoris, maioris verò <lb/>pauciores numerabuntur; </s>

<s id="N236B5">&longs;æpiùs numeraui 11.minores eo tantùm tem<lb/>pore, quo alter, qui mecum erat 10. maiores numerabat, & 40. circiter <lb/>minores dum alter 37.maiores recen&longs;eret; </s>

<s id="N236BD">& certè &longs;i vibratio vtraque <lb/>maior &longs;cilicet, & minor per <expan abbr="e&utilde;dem">eundem</expan> arcum tecurreret, centum minores <lb/>eo ferè tempore agerentur, quo 90.maiores; licèt enim vtraque decre&longs;<lb/>cat, maior tamen decre&longs;cit in maiore proportione, quàm minor, cuius <lb/>rei rationem afferemus infrà. </s>

<s id="N236BD">& certè &longs;i vibratio vtraque <lb/>maior &longs;cilicet, & minor per <expan abbr="e&utilde;dem">eundem</expan> arcum recurreret, centum minores <lb/>eo ferè tempore agerentur, quo 90.maiores; licèt enim vtraque decre&longs;<lb/>cat, maior tamen decre&longs;cit in maiore proportione, quàm minor, cuius <lb/>rei rationem afferemus infrà. </s>

<s id="N236CF">Nec e&longs;t quod aliquis cum Galileo, Baliano, & aliis opponat, omnes <lb/>vibrationes, &longs;iue maiores &longs;int, &longs;iue minores e&longs;&longs;e æquè diuturnas, idque <lb/>manife&longs;tis con&longs;tare experimentis, quibus ego alia certi&longs;&longs;ima experimen<lb/>ta oppono, quibus etiam vltrò a&longs;&longs;entitur P. Mer&longs;ennus, Galileo alioqui <lb/>addicti&longs;&longs;imus, in ver&longs;ione eiu&longs;dem Galilci lib. 1. art. </s>

<s id="N236DA">18. & verò docti <lb/>omnes Galilco &longs;unt addicti&longs;simi; </s>

<s id="N236DA">18. & verò docti <lb/>omnes Galileo &longs;unt addicti&longs;simi; </s>

<s id="N236E0">in qua verò proportione minor vibra<lb/>tio breuiore tempore peragatur, quàm major, difficilè dictu e&longs;t, & vix <lb/>determinari pote&longs;t, ni&longs;i fortè dicatur in ea proportione arcum HF citiùs <lb/>percurri, quàm arcum BHF, in qua duæ chordæ HGF citiùs percur<lb/>runtur, quàm duæ BZF; &longs;ed de his fusè aliàs. </s>

<s id="N236EE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s>

<s id="N236FD"><emph type="italics"/>Velocitates acqui&longs;ita in vihrationibus inæqualibus &longs;unt vt altitudines<emph.end type="italics"/>; </s>

<s id="N236FD"><emph type="italics"/>Velocitates acqui&longs;ita in vibrationibus inæqualibus &longs;unt vt altitudines<emph.end type="italics"/>; </s>

<s id="N23706">&longs;int <lb/>enim vibrationes duæ BF, HF; </s>

<s id="N2370C">dico velocitatem acqui&longs;itam in de&longs;cen-<pb pagenum="310" xlink:href="026/01/344.jpg"/>&longs;u BF e&longs;&longs;e ad acqui&longs;itam in de&longs;cen&longs;u HP, vt vecta AF ad rectam OF, <lb/>quod facilè probatur; </s>

<s id="N23717">quia ex B in F æqualis acquiritur velocitas &longs;iue <lb/>per rectam BF <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> mobile, &longs;iue per duas BHF, &longs;iue per tres BHGF, <lb/>&longs;iue per totum quadrantem BHF; </s>

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<s id="N238E9"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s>

<s id="N238F7"><emph type="italics"/>In diner&longs;is punctis arcus diuer&longs;us impetus producitur.<emph.end type="italics"/></s>

<s id="N238F7"><emph type="italics"/>In diuer&longs;is punctis arcus diuer&longs;us impetus producitur.<emph.end type="italics"/></s>

<s id="N23901">&longs;it enim <lb/>pendulum fune ex centro immobili A; </s>

<s id="N23907">&longs;itque AO horizontalis, AD <pb pagenum="312" xlink:href="026/01/346.jpg"/>perpendicularis; </s>

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<s id="N23A65">Tertiò, &longs;i pondus &longs;it in H vel in L e&longs;t determinatio mixta ex duabus <lb/>inæqualibus, ita vt determinatio potentiæ, quæ e&longs;t applicata in A &longs;it mi<lb/>nor determinatione, quæ e&longs;t à grauitatione ponderis; </s>

<s id="N23A6D">&longs;it enim pondus in <lb/>H, &longs;itque determinatio altera per lineam HA, altera per lineam HG; </s>

<s id="N23A73">&longs;i <lb/>vtraque æqualis e&longs;t, linea determinationis mixtæ non e&longs;&longs;et Tangens HF; </s>

<s id="N23A79"><lb/>nec enim angulus AHG diuidit æqualiter bifariam ip&longs;am HF; atqui <lb/>cum vtraque determinatio e&longs;t æqualis, po&longs;ita quod vtraque linea faciat <lb/>angulum, linea nouæ determinationis facit angulum vtrimque æqualem<gap/><lb/>vt demon&longs;trauimus &longs;uprà. </s>

<s id="N23A87"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/></s>

<s id="N23A96">Quartò hinc colligo, determinationem, quæ e&longs;t à potentia applicata <lb/>in A cre&longs;cere continuè ab O ad D, ita vt in O &longs;it nulla, in D &longs;it maxima, <lb/>id e&longs;t æqualis alteri determinationi propriæ grauitationis; </s>

<s id="N23A9E">in reliquis ve<lb/>rò punctis prima e&longs;t ad &longs;ecundam, vt &longs;inus rectus &longs;uperioris arcus ad &longs;i<lb/>num totum, v.g.&longs;i pondus &longs;it in L, determinatio granitationis e&longs;t ad aliam <lb/>vt LA ad LR, &longs;i &longs;it in H vt HA ad HS, &longs;i &longs;it in O vt OA ad nihil; </s>

<s id="N23A9E">in reliquis ve<lb/>rò punctis prima e&longs;t ad &longs;ecundam, vt &longs;inus rectus &longs;uperioris arcus ad &longs;i<lb/>num totum, v.g.&longs;i pondus &longs;it in L, determinatio grauitationis e&longs;t ad aliam <lb/>vt LA ad LR, &longs;i &longs;it in H vt HA ad HS, &longs;i &longs;it in O vt OA ad nihil; </s>

<s id="N23AA8">&longs;i <lb/>&longs;it in D vt DA ad DA; idem dico de omnibus aliis punctis inter<lb/>mediis. </s>

<s id="N23AB2"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/></s>

<s id="N23AC1">Quintò colligo, impetum grauitationis productum in &longs;ingulis pun<lb/>ctis e&longs;&longs;e ad impetum productum in O, id e&longs;t ad maximum, qui po&longs;&longs;it <pb pagenum="314" xlink:href="026/01/348.jpg"/>produc. </s>

<s id="N23ACB">vno in&longs;tanti ab ip&longs;o corpore grani, vt &longs;inum rectum arcus infe<lb/>rioris ad &longs;inum totum; </s>

<s id="N23AD1">&longs;it enim pondus in L, impetus productus in L <lb/>e&longs;t ad productum in O, vt &longs;inus BL ad LA; </s>

<s id="N23AD7">&longs;it in H, vt &longs;inus HC ad <lb/>HA; </s>

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<s id="N23AEF">hæc verò continuo cre&longs;cit; </s>

<s id="N23AF3">ille totus e&longs;t in O nullus in D; </s>

<s id="N23AF7">hæc <lb/>tota in D, nulla in O; </s>

<s id="N23AFD">ille e&longs;t ad totum, vt &longs;inus arcus inferioris ad &longs;i<lb/>num totum; hæc verò e&longs;t ad toram, &longs;eu maximam, vt &longs;inus arcus &longs;uperio<lb/>ris ad &longs;inum totum. </s>

<s id="N23AFD">ille e&longs;t ad totum, vt &longs;inus arcus inferioris ad &longs;i<lb/>num totum; hæc verò e&longs;t ad totam, &longs;eu maximam, vt &longs;inus arcus &longs;uperio<lb/>ris ad &longs;inum totum. </s>

<s id="N23B07"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/></s>

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<s id="N23BCF">hinc etiam rectè explicatur, in quo po&longs;itum &longs;it illud impe<lb/>dimentum ratione cuius corpus rectà deor&longs;um non tendit; quippè in <lb/>eo tantùm po&longs;itum e&longs;t, quod &longs;it noua determinatio, idem dico de re&longs;i<lb/>&longs;tentia. </s>

<s id="N23BDB">Ob&longs;eruabis autem idem præ&longs;tare funcm affixum in A ratione conti<lb/>nuitatis, & vnionis &longs;uarum partium, quod præ&longs;taret potentia in A fune <lb/>ip&longs;o trahens, vt con&longs;tat, &longs;eu pondus contranitens ex rotula appen&longs;um. </s>

<s id="N23BDB">Ob&longs;eruabis autem idem præ&longs;tare funem affixum in A ratione conti<lb/>nuitatis, & vnionis &longs;uarum partium, quod præ&longs;taret potentia in A fune <lb/>ip&longs;o trahens, vt con&longs;tat, &longs;eu pondus contranitens ex rotula appen&longs;um. </s>

<s id="N23BE4"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 8.<emph.end type="center"/></s>

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<s id="N23C34">licèt determinatio noua per <lb/>Tangentem nouam accedat in &longs;ingulis punctis; </s>

<s id="N23C3A">nihil tamen impetus e&longs;t <lb/>fru&longs;trà; </s>

<s id="N23C40">vt in reflexione dictum e&longs;t, adde quod determinatio prior, nihil <lb/>pror&longs;us confert; </s>

<s id="N23C46">quia tota impeditun à potentia retinente in A immo<lb/>biliter; dixi por &longs;e, quia per accidens propter aliquam ten&longs;ionem chor<lb/>dæ pote&longs;t aliquid de&longs;trui, quæ ten&longs;io e&longs;t pror&longs;us per accidens. </s>

<s id="N23C46">quia tota impeditun à potentia retinente in A immo<lb/>biliter; dixi per &longs;e, quia per accidens propter aliquam ten&longs;ionem chor<lb/>dæ pote&longs;t aliquid de&longs;trui, quæ ten&longs;io e&longs;t pror&longs;us per accidens. </s>

<s id="N23C50"><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 10.<emph.end type="center"/></s>

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<s id="N23C5E">Decimò colligo inde reddi rationem à priori, cur ille motus vibra<lb/>tionis funependuli &longs;it acceleratus; </s>

<s id="N23C64">quia impetus additur &longs;ingulis in&longs;tan<lb/>tibus, & nihil de&longs;truitur; </s>

<s id="N23C6A">immò &longs;i de&longs;trueretur iuxta rationem prædicti <lb/>impedimenti, & pondus e&longs;&longs;et in H, cùm ratio impedimenti &longs;it SO, & <lb/>ratio noui impetus CH æqualis SO; </s>

<s id="N23C72">haud dubiè in H <expan abbr="tant&utilde;dem">tantundem</expan> pro<lb/>ducerctur impetus, quantum de&longs;trueretur; igitur nullum &longs;entiretur pon<lb/>dus in H, quod ab&longs;urdum e&longs;t. </s>

<s id="N23C72">haud dubiè in H <expan abbr="tant&utilde;dem">tantundem</expan> pro<lb/>duceretur impetus, quantum de&longs;trueretur; igitur nullum &longs;entiretur pon<lb/>dus in H, quod ab&longs;urdum e&longs;t. </s>

<s id="N23C80"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s>

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<s id="N23D05"><pb pagenum="316" xlink:href="026/01/350.jpg"/>&longs;i enim toto eo tempore, quo corpus graue cadit, numerentur 6. vibratio<lb/>nes tripedalis funependuli; </s>

<s id="N23D0F">haud dubiè motus ille durauit &longs;ex minutis <lb/>&longs;ecundis; igitur &longs;i præcogno&longs;catur quantum &longs;patij percurratur deor&longs;um, <lb/>dum fluit vnum &longs;ecundum minutum, quod &longs;it.v.g. </s>

<s id="N23D17">&longs;patium pedum 18 6/7 <lb/>hoc po&longs;ito, quadrentur tempora &longs;cilicet, & 6. habeo 1. & 36. iam facio <lb/>regulam trium, &longs;i 1.dat. </s>

<s id="N23D1E">18 6/7 quid dabunt 36. & habeo 619. pedes mi<lb/>nus 1/<gap/>. </s>

<s id="N23D1E">18 6/7 quid dabunt 36. & habeo 619. pedes mi<lb/>nus 1/2. </s>

<s id="N23D27">Ob&longs;eruabis autem dictum fui&longs;&longs;e à me &longs;uprà funependulum tripedale <lb/>peragere &longs;uam integram vibrationem tempore vnius &longs;ecundi minuti; </s>

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<s id="N23E08"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s>

<s id="N23E16"><emph type="italics"/>In a&longs;cen&longs;u vibrationis funependuli de&longs;iruitur impetus<emph.end type="italics"/>; patet, quia de&longs;init <lb/>motus; </s>

<s id="N23E16"><emph type="italics"/>In a&longs;cen&longs;u vibrationis funependuli de&longs;truitur impetus<emph.end type="italics"/>; patet, quia de&longs;init <lb/>motus; </s>

<s id="N23E21">igitur & impetus, ne &longs;it fru&longs;trà; </s>

<s id="N23E25">præterea applicatum e&longs;t princi<lb/>pium de&longs;tructionis impetus; </s>

<s id="N23E2B">igitur de&longs;truitur; antecedens ex dicendis <lb/>infra clari&longs;&longs;imum euadet. </s>

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<s id="N23E4F"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s>

<s id="N23E5D"><emph type="italics"/>Singulis in&longs;tantibus in<gap/>qualiter de&longs;truitur impetus in a&longs;cen&longs;u illo vibratio<lb/>nis<emph.end type="italics"/>; prob. </s>

<s id="N23E5D"><emph type="italics"/>Singulis in&longs;tantibus inæqualiter de&longs;truitur impetus in a&longs;cen&longs;u illo vibratio<lb/>nis<emph.end type="italics"/>; prob. </s>

<s id="N23E6A">quia &longs;ingulis in&longs;tantibus mutatur detetminatio, id e&longs;t ratio <lb/>plani inclinati; </s>

<s id="N23E6A">quia &longs;ingulis in&longs;tantibus mutatur determinatio, id e&longs;t ratio <lb/>plani inclinati; </s>

<s id="N23E70">nam quodlibet punctum arcus, vt &longs;æpè dictum e&longs;t, facit <lb/>planum inclinatum diuer&longs;um; </s>

<s id="N23E76">igitur lineæ vtriu&longs;que determinationis <lb/>faciunt diuer&longs;um angulum; </s>

<s id="N23E7C">igitur determinatio noua mixta diuer&longs;a e&longs;t; <lb/>igitur plùs vel minùs impetus de&longs;truitur, quia plùs vel minùs e&longs;t fru&longs;trà, <lb/>quod ex dicendis patebit. </s>

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<s id="N23E86"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s>

<s id="N23E94"><emph type="italics"/>De&longs;truitur impetus in &longs;ingulis punctis iuxta rationem &longs;inunm rectorum ar<lb/>cuum inferiorum<emph.end type="italics"/> v.g. &longs;it arcus a&longs;cen&longs;us DIO, &longs;itque mobile pendulum in <lb/>H; </s>

<s id="N23E94"><emph type="italics"/>De&longs;truitur impetus in &longs;ingulis punctis iuxta rationem &longs;inuum rectorum ar<lb/>cuum inferiorum<emph.end type="italics"/> v.g. &longs;it arcus a&longs;cen&longs;us DIO, &longs;itque mobile pendulum in <lb/>H; </s>

<s id="N23EA3">impetus qui de&longs;truitur in H, e&longs;t ad impetum qui de&longs;truitur in per<lb/>pendiculari &longs;ur&longs;um (&longs;uppo&longs;ito &longs;cilicet <expan abbr="t&etilde;pore">tempore</expan>) vt &longs;inus HC ad &longs;inum HA; </s>

<s id="N23EAD"><lb/>nam de&longs;truitur in ea ratione, iuxta quam de&longs;trueretur in plano inclinato <lb/>EH; </s>

<s id="N23EB4">&longs;ed in planis inclinatis iuxta prædictam rationem impetum de&longs;trui <lb/>demon&longs;tratum e&longs;t &longs;uo loco; </s>

<s id="N23EBA">adde quod impetus innatus determinat mo<lb/>bile ad lineam deor&longs;um HG, alius verò ad lineam HM; </s>

<s id="N23EC0">atqui &longs;re&longs;&longs;ent <lb/><gap/>o gradus impetus, quorum alter e&longs;&longs;et determinatus per HM, alter per <pb pagenum="318" xlink:href="026/01/352.jpg"/>HGV, motus fieret per HX, &longs;ed HX e&longs;t æqualis HM; </s>

<s id="N23EC0">atqui &longs;i e&longs;&longs;ent <lb/>duo gradus impetus, quorum alter e&longs;&longs;et determinatus per HM, alter per <pb pagenum="318" xlink:href="026/01/352.jpg"/>HGV, motus fieret per HX, &longs;ed HX e&longs;t æqualis HM; </s>

<s id="N23ECC">igitur de&longs;truitur <lb/>&longs;ubduplus impetus, quia e&longs;t fru&longs;trà; </s>

<s id="N23ED2">&longs;ed HC e&longs;t &longs;ubdupla HA: </s>

<s id="N23ED6">præterca <lb/>impetus innatus retrahit mobile per HE minùs, quàm AD iuxta eam <lb/>proportionem, in qua motus per HE e&longs;t minor quàm motus per AD; </s>

<s id="N23ED6">præterea <lb/>impetus innatus retrahit mobile per HE minùs, quàm AD iuxta eam <lb/>proportionem, in qua motus per HE e&longs;t minor quàm motus per AD; </s>

<s id="N23EDE">&longs;ed <lb/>motus per HE e&longs;t ad motum per AD vt HE ad AE, vel vt HC ad HA; </s>

<s id="N23EE4"><lb/>igitur illa vis, quæ retrahit mobile per HE e&longs;t ad eam, qua retrahitur <lb/>per AD vt HC ad HA; </s>

<s id="N23EEB">&longs;ed in eadem proportione de&longs;truitur impetus, <lb/>quo mobile fertur &longs;ur&longs;um, in qua retrahitur deor&longs;um; </s>

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<s id="N23EFB">Dictum e&longs;t eodem tempore; </s>

<s id="N23EFF">nam minori tempore minùs impetus de<lb/>&longs;truitur, plùs verò maiori; </s>

<s id="N23F05">vnde quando comparatur impetus de&longs;tructus <lb/>in plano inclinato &longs;ur&longs;um cum de&longs;tructo in verticaili, &longs;emper intelligi<lb/>tur vtrumque de&longs;trui eodem tempore; </s>

<s id="N23F05">vnde quando comparatur impetus de&longs;tructus <lb/>in plano inclinato &longs;ur&longs;um cum de&longs;tructo in verticali, &longs;emper intelligi<lb/>tur vtrumque de&longs;trui eodem tempore; </s>

<s id="N23F0D">alioquin vitio&longs;a e&longs;&longs;et proportio, <lb/>& comparatio; idem dico de impetu producto, quod de de&longs;tructo. </s>

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<s id="N23F2A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s>

<s id="N23F38"><emph type="italics"/>Totus impetus qui concurrit ad de&longs;cen&longs;um funependuli, non coneurrit ad <lb/>a&longs;cen&longs;um,<emph.end type="italics"/> prob. </s>

<s id="N23F38"><emph type="italics"/>Totus impetus qui concurrit ad de&longs;cen&longs;um funependuli, non concurrit ad <lb/>a&longs;cen&longs;um,<emph.end type="italics"/> prob. </s>

<s id="N23F42">quia impetus innatus non concurrit ad a&longs;cen&longs;um, vt <lb/>con&longs;tat ex dictis alibi; &longs;ed hic concurrit ad de&longs;cen&longs;um. </s>

<s id="N23F4A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s>

<s id="N23F58"><emph type="italics"/>Aliquis etiam gradus impetus concurrit ad a&longs;cen&longs;um, qui non coucurrit <lb/>ad de&longs;cen&longs;nm,<emph.end type="italics"/> probatur, quia vltimo in&longs;tanti de&longs;cen&longs;us aliquid impetus <lb/>noui producitur quantumuis minimi, quia &longs;ingulis in&longs;tantibus motus <lb/>deor&longs;um aliquid impetus accedit; </s>

<s id="N23F58"><emph type="italics"/>Aliquis etiam gradus impetus concurrit ad a&longs;cen&longs;um, qui non concurrit <lb/>ad de&longs;cen&longs;um,<emph.end type="italics"/> probatur, quia vltimo in&longs;tanti de&longs;cen&longs;us aliquid impetus <lb/>noui producitur quantumuis minimi, quia &longs;ingulis in&longs;tantibus motus <lb/>deor&longs;um aliquid impetus accedit; </s>

<s id="N23F67">&longs;ed ille impetus non concurrit ad mo<lb/>tum deor&longs;um; </s>

<s id="N23F6D">quia cum primo illo in&longs;tanti, quo e&longs;t, non concurrat ad <lb/>motum, cumque illud in&longs;tans &longs;it vltimum motus deor&longs;um; </s>

<s id="N23F73">certè ad mo<lb/>tum deor&longs;um non concurrit, &longs;ed ad motum &longs;ur&longs;um concurrit, nam pri<lb/>mo in&longs;tanti, quo e&longs;t, exigit motum pro &longs;equenti; e&longs;t autem &longs;equens <lb/>in&longs;tans primum a&longs;cen&longs;us. </s>

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<s id="N23FCC">Hinc manife&longs;ta ratio, cur funependulum po&longs;t vibrationem de&longs;cen&longs;us <lb/>non perueniat in a&longs;cen&longs;u ad tantam altitudinem; </s>

<s id="N23FD2">nec e&longs;t quod aliqui di<lb/>cant aëra interceptum efficere, ne ad æqualem altitudinem a&longs;cendat, <lb/>cùm aër non minùs re&longs;i&longs;tat de&longs;cen&longs;ui, quàm a&longs;cen&longs;ui; quod quomodo <lb/>&longs;iat, iam alibi explicuimus. </s>

<s id="N23FDE"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s>

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<s id="N24136"><emph type="italics"/>Hinc tamdiu durant minimæ illa vibrationes; </s>

<s id="N2413C">quia &longs;ingulæ minima per<lb/>tione imminuuntur, & maiores è contrariòtam citò decurtantur<emph.end type="italics"/>; </s>

<s id="N2413C">quia &longs;ingulæ minima por<lb/>tione imminuuntur, & maiores è contrariò tam citò decurtantur<emph.end type="italics"/>; </s>

<s id="N24145">cuius re&longs; <lb/>non e&longs;t alia ratio præter eam, quam &longs;uprà adduximus, quæ rem ip&longs;am <lb/>cuincit; e&longs;t tamen in&longs;ignis difficultas, quam paulò po&longs;t di&longs;cutiemus in <lb/>&longs;equenti Schol. <emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s>

<s id="N24145">cuius re&longs; <lb/>non e&longs;t alia ratio præter eam, quam &longs;uprà adduximus, quæ rem ip&longs;am <lb/>euincit; e&longs;t tamen in&longs;ignis difficultas, quam paulò po&longs;t di&longs;cutiemus in <lb/>&longs;equenti Schol. <emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s>

<s id="N2415C"><emph type="italics"/>Hinc ratio, cur minimo &longs;erè cur&longs;u funependulum etiam graui&longs;&longs;imum modi<lb/><gap/>a libratione vibretur<emph.end type="italics"/>; </s>

<s id="N2415C"><emph type="italics"/>Hinc ratio, cur minimo &longs;erè cur&longs;u funependulum etiam graui&longs;&longs;imum modi<lb/>ca libratione vibretur<emph.end type="italics"/>; </s>

<s id="N24168">immò, quod fortè alicui mirum videretur, ip&longs;o an<lb/>helitu graui&longs;&longs;ima pondera moueri po&longs;&longs;unt, quod quiuis facilè probare <lb/>poterit; </s>

<s id="N24170">pro quo diligenter ob&longs;eruandum e&longs;t, vt eo dumtaxat ordine an<lb/>helitus repetatur, quo vibrationes fiunt, ita vt iam euntem molem à <lb/>tergo impellat; vnde accidet, vt repetito tandem anhelitu maiore motu <lb/>funependulum vibretur. </s>

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<s id="N241A8">haud dubiè perueniet tantùm <lb/>in C; </s>

<s id="N241AE">ita vt tantum detrahatur &longs;patij in hoc &longs;ecundo a&longs;cen&longs;u, quantum <lb/>detractum e&longs;t in primo: idem dico de tertio, quarto, &c. </s>

<s id="N241B4">ducantur BH, <pb pagenum="321" xlink:href="026/01/355.jpg"/>CG, D Fparallelæ AI; </s>

<s id="N241BD">cum &longs;patium eo modo decidatur ex areu EI, quo <lb/>ex perpendiculo EA maiori vibrationi detrahitur IH, &longs;ecundæ minori <lb/>HG, tertiæ GF, quartæ FE; igitur plùs detrahitur minoribus, quàm <lb/>maioribus. </s>

<s id="N241BD">cum &longs;patium eo modo decidatur ex area EI, quo <lb/>ex perpendiculo EA maiori vibrationi detrahitur IH, &longs;ecundæ minori <lb/>HG, tertiæ GF, quartæ FE; igitur plùs detrahitur minoribus, quàm <lb/>maioribus. </s>

<s id="N241C9">Re&longs;pondeo, maius &longs;patium percurri &longs;ur&longs;um maiore tempore, quàra <lb/>minus; </s>

<s id="N241C9">Re&longs;pondeo, maius &longs;patium percurri &longs;ur&longs;um maiore tempore, quàm <lb/>minus; </s>

<s id="N241CF">&longs;it enim EA con&longs;tans 36. &longs;patia iuxta nouam progre&longs;&longs;ionem <lb/>arithmeticam, &longs;intque 8. gradus impetus acqui&longs;iti in de&longs;cen&longs;u AE con<lb/>iuncti cum innato: </s>

<s id="N241D7">primo in&longs;tanti, &longs;eu tempore percurrentur tantùm 7. <lb/>&longs;patia, <expan abbr="de&longs;truetur&qacute;ue">de&longs;trueturque</expan> vnus gradus impetus, &longs;ecundo 6. <expan abbr="de&longs;truetur&qacute;ue">de&longs;trueturque</expan> al<lb/>ter gradus impetus; denique tertio 5. quatto 4. &c. </s>

<s id="N241E7">igitur 28. &longs;patia 7. <lb/>in&longs;tantibus; igitur non perueniet in A mobile, &longs;ed conficiet &longs;patium, <lb/>quod erit ad EA, vt 28. ad 36. porrò &longs;i cadat ex 28. acquiret 7. gradus <lb/>impetus præter innatum, quorum ope &longs;ecundo a&longs;cendet ad 21. tertio ad <lb/>15. quartò ad 10. quinto ad 6. &longs;extò ad 3. &longs;eptimo ad 1. igitur &longs;patium <lb/>quod amittit in a&longs;cen&longs;u continet 8. in &longs;ecundo 7. in tertio 6. in quarto <lb/>5. in quinto 4. in &longs;exto 3. in &longs;eptimo 2. igitur e&longs;t maxima inæqualitas, <lb/>quæ pari modo explicari pote&longs;t in progre&longs;&longs;ione Galilei. </s>

<s id="N241FC">Secundò, obijci pote&longs;t: </s>

<s id="N241FF">amitti tantùm &longs;patij &longs;ingulis temporibus<gap/><lb/>quantum acquiritur primo tempore, vel in&longs;tanti, cum impetu innato: &longs;ed <lb/>cum primo ille velocitatis gradu vix intra multos annos conficeretur <lb/>modicum &longs;patium. </s>

<s id="N241FF">amitti tantùm &longs;patij &longs;ingulis temporibus, <lb/>quantum acquiritur primo tempore, vel in&longs;tanti, cum impetu innato: &longs;ed <lb/>cum primo ille velocitatis gradu vix intra multos annos conficeretur <lb/>modicum &longs;patium. </s>

<s id="N2420A">Re&longs;pondeo, &longs;i con&longs;ideretur tantùm illud &longs;patium, <lb/>quod acquiritur primo tempore cum impetu non impedito; </s>

<s id="N24210">haud dubiè <lb/>in&longs;en&longs;ibile e&longs;t, & licèt infinitus ferè repetatur illud idem &longs;patium; </s>

<s id="N24216">haud <lb/>dubiè in&longs;en&longs;ibile manet: vnde &longs;i a&longs;cen&longs;us fiat in 10000. in&longs;tantibus, to<lb/>ties accipi debet illud ip&longs;um &longs;patium, ex quo modicum tantùm re&longs;ultat, <lb/>quod minuitur in &longs;ecundo a&longs;cen&longs;u, itemque in tertio, quarto, &c. </s>

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<s id="N24283">Quintò, hinc vera ratio, cur in &longs;uperiore arcu de&longs;truatur citò impe<lb/>tus; </s>

<s id="N24289">tardiùs verò in in inferiore; </s>

<s id="N2428D">quia, cùm Tangens cuiu&longs;libet puncti ar<lb/>cus &longs;it eius planum, & hæc in arcu &longs;uperiore accedat propiùs ad perpen<lb/>diculum; non mirum e&longs;t, &longs;i cum eo impetu per arcum &longs;uperiorem mo<lb/>ucri non po&longs;&longs;it mobile cò non a&longs;cendat, cuius tantùm ope per inferio<lb/>rem arcum a&longs;cendit. </s>

<s id="N2428D">quia, cùm Tangens cuiu&longs;libet puncti ar<lb/>cus &longs;it eius planum, & hæc in arcu &longs;uperiore accedat propiùs ad perpen<lb/>diculum; non mirum e&longs;t, &longs;i cum eo impetu per arcum &longs;uperiorem mo<lb/>ueri non po&longs;&longs;it mobile cò non a&longs;cendat, cuius tantùm ope per inferio<lb/>rem arcum a&longs;cendit. </s>

<s id="N2429B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s>

<s id="N242A9"><emph type="italics"/>Omnes vibrattiones numerari non po&longs;&longs;unt<emph.end type="italics"/>; </s>

<s id="N242A9"><emph type="italics"/>Omnes vibrationes numerari non po&longs;&longs;unt<emph.end type="italics"/>; </s>

<s id="N242B2">certum e&longs;t, cùm <gap/>nt infinitæ <lb/>ferè in&longs;en&longs;ibiles, nec pote&longs;t &longs;en&longs;u di&longs;cerni, quantus &longs;it arcus minimæ vi<lb/>brationis; </s>

<s id="N242B2">certum e&longs;t, cùm &longs;int infinitæ <lb/>ferè in&longs;en&longs;ibiles, nec pote&longs;t &longs;en&longs;u di&longs;cerni, quantus &longs;it arcus minimæ vi<lb/>brationis; </s>

<s id="N242BC">&longs;i tamen de&longs;trueretur tantùm impetus in a&longs;cen&longs;u ab impetu <lb/>innato, nec ten&longs;io funis, triplex determinatio, re&longs;i&longs;tentia aëris, & diuer&longs;æ <lb/>partes funis, quarum minores vibrationes impediuntur quidquam fa<lb/>cerent, cognita differentia prima vibrationis & &longs;ecunda, fortè cogno&longs;ci <lb/>po&longs;&longs;et numerus vibrationum cognito principio progre&longs;&longs;ionis; </s>

<s id="N242C8">quantus <lb/>verò &longs;it numerus vibrationum, quæ incipiunt à maiore, & quantus illa<lb/>rum quæ incipiunt à minore etiam incertum e&longs;t, v.g. &longs;i funependulum <lb/>AD demittatur ex O, & deinde ex L; </s>

<s id="N242D4">certum e&longs;t quidem e&longs;&longs;e plures vi<lb/>brationes cum demittitur ex O toto eo vibrationum numero, quæ re<lb/>cen&longs;entur, donec perueniatur ad illam vibratiouem, cuius a&longs;cen&longs;us con<lb/>&longs;tat arcu DL; </s>

<s id="N242D4">certum e&longs;t quidem e&longs;&longs;e plures vi<lb/>brationes cum demittitur ex O toto eo vibrationum numero, quæ re<lb/>cen&longs;entur, donec perueniatur ad illam vibrationem, cuius a&longs;cen&longs;us con<lb/>&longs;tat arcu DL; </s>

<s id="N242DE">nam deinceps æqualis erit numerus earum, quæ con&longs;e<lb/>quentur, & earum, quarum prima demittitur ex L, vt patet; </s>

<s id="N242E4">quot verò <lb/>præcedant vibrationes antequam perueniatur ad illam, cuius a&longs;cen&longs;us <lb/>e&longs;t arcus DL; equidem aliqua ob&longs;eruatione affixo &longs;cilicet maiore qua<lb/>drante parieti iu &longs;uos gradus di&longs;tributo cogno&longs;ci pote&longs;t, &longs;ed nunquam <lb/>&longs;atis acurata. </s>

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