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Colored diff for /texts/archimedes/xml/fabri_tract_026_la_1646.xml between version 1.19 and 1.20

version 1.19, 2007/02/01 04:42:12

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

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<s id="N21C43"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s id="N21C51">Ob&longs;eruabis ex dictis &longs;atis con&longs;tare, quàm temerè mirentur aliqui tan<lb/>tam motuum c&oelig;le&longs;tium celeritatem, cum mocus circularis velocitas in <lb/>infinitum augeri po&longs;&longs;it: Ob&longs;eruabis præterea, &longs;i fortè motus rectus corpo<lb/>rum c&oelig;le&longs;tium præce&longs;&longs;it per aliquot minuta, motum illum, qui deinde <lb/>&longs;ucce&longs;&longs;it, non e&longs;&longs;e perfectè circularem, &longs;ed mixtum, quem aliquando ex<lb/>plicabimus, & ex eo cau&longs;as Apogæi, Perigæi, declinationis, &c. </s>

<s id="N21C51">Ob&longs;eruabis ex dictis &longs;atis con&longs;tare, quàm temerè mirentur aliqui tan<lb/>tam motuum c&oelig;le&longs;tium celeritatem, cum motus circularis velocitas in <lb/>infinitum augeri po&longs;&longs;it: Ob&longs;eruabis præterea, &longs;i fortè motus rectus corpo<lb/>rum c&oelig;le&longs;tium præce&longs;&longs;it per aliquot minuta, motum illum, qui deinde <lb/>&longs;ucce&longs;&longs;it, non e&longs;&longs;e perfectè circularem, &longs;ed mixtum, quem aliquando ex<lb/>plicabimus, & ex eo cau&longs;as Apogæi, Perigæi, declinationis, &c. </s>

<s id="N21C5F">omné&longs;<lb/>que anomalias deducemus &longs;uo loco. </s>

Line 13170

<s id="N21C74"><emph type="italics"/>Rota circulo verticali parallela circa axem mobilis addito minimo im<lb/>petu per &longs;e moueri pote&longs;t<emph.end type="italics"/>; </s>

<s id="N21C7F">&longs;it enim ABCD plano verticali parallela circa <lb/>centrum E volubilis; </s>

<s id="N21C85">&longs;itque in perfecto æquilibrio, & accedat minima <lb/>vis impetus in A v.g. haud dubiè punctum E de&longs;cendet deor&longs;um, alio<lb/>qum maneret æquilibrium, & non maneret: dixi per &longs;e; </s>

<s id="N21C85">&longs;itque in perfecto æquilibrio, & accedat minima <lb/>vis impetus in A v.g. haud dubiè punctum E de&longs;cendet deor&longs;um, alio<lb/>quin maneret æquilibrium, & non maneret: dixi per &longs;e; </s>

<s id="N21C8F">nam cùm non <lb/>po&longs;&longs;it volui circa centrum E, ni&longs;i vel cum mobili axe duobus hinc inde <lb/>lunatis fulcris &longs;u&longs;tentato, vel facto foramine circa axem immobilem, vel <lb/>circa geminos apices conicos immi&longs;&longs;os iu&longs;tis apothecis in plano rotæ <lb/>excauatis, quales videmus in acu magnetica; atqui non pote&longs;t volui rota <lb/>&longs;iue primo, &longs;iue &longs;ecundo, &longs;iue tertio modo voluatur &longs;ine multa compre&longs;<lb/>&longs;ione partium, id e&longs;t, &longs;ine aliquo affrictu, in quo multæ particulæ vnius <lb/>plani cum particulis alterius qua&longs;i pectinatim commi&longs;&longs;æ, motum & im<lb/>petunt &longs;i&longs;tunt. </s>

Line 13178

<s id="N21CB3"><emph type="italics"/>Rota minor in eodem &longs;itu de quo &longs;uprà æquè facilè moueri pote&longs;t, ac maior<emph.end type="italics"/><pb pagenum="282" xlink:href="026/01/316.jpg"/><emph type="italics"/>per &longs;e.<emph.end type="italics"/></s>

<s id="N21CC3"> Probatur primò, quia vtraque minimo impetu moueti pote&longs;t per <lb/>Th. 21. Secundò, quia addita minima vi impetus in F, & minima in A <lb/>tàm facilè maior rota de&longs;cendit, quàm minor, quia æqualiter tollitur <lb/>æquilibrium vtriu&longs;que: dixi per &longs;e, quia maior rota propter maius pon<lb/>dus maiore affrictu motum impedit. </s>

<s id="N21CC3"> Probatur primò, quia vtraque minimo impetu moueri pote&longs;t per <lb/>Th. 21. Secundò, quia addita minima vi impetus in F, & minima in A <lb/>tàm facilè maior rota de&longs;cendit, quàm minor, quia æqualiter tollitur <lb/>æquilibrium vtriu&longs;que: dixi per &longs;e, quia maior rota propter maius pon<lb/>dus maiore affrictu motum impedit. </s>

<s id="N21CD1"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s>

Line 13220

<s id="N21D93"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s>

<s id="N21DA1"><emph type="italics"/>Hinc impetus vnius partis mobilis continui pote&longs;t impetum &longs;imilem produ<lb/>cere in alia parte eiu&longs;aem mobilis<emph.end type="italics"/>; vt patet ex dictis, quod tantùm locum <lb/>habet in motu circulari. </s>

<s id="N21DA1"><emph type="italics"/>Hinc impetus vnius partis mobilis continui pote&longs;t impetum &longs;imilem produ<lb/>cere in alia parte eiu&longs;dem mobilis<emph.end type="italics"/>; vt patet ex dictis, quod tantùm locum <lb/>habet in motu circulari. </s>

<s id="N21DAE">Diceret aliquis, igitur in motu recto etiam lo<lb/>cum habebit. </s>

<s id="N21DB3">Re&longs;pondeo negando, alioqui minima potentia quodlibet <lb/>pondus motu recto moueret etiam nullo adhibito mechanico organo; </s>

<s id="N21DB9"><lb/>quia modo produceretur tantulus impetus in aliqua parte, hic produce<lb/>ret alium, & hic alium, immò vterque &longs;ecundo in&longs;tanti alium produce-<pb pagenum="283" xlink:href="026/01/317.jpg"/>ret: </s>

<s id="N21DC5">e&longs;&longs;et enim cau&longs;a nece&longs;&longs;aria; </s>

<s id="N21DC9">&longs;ed hoc e&longs;t ab&longs;urdum: ratio verò di&longs;pa<lb/><gap/> e&longs;t, quia mobile, quod motu circulari voluitur circa centrum, <lb/>quod e&longs;t in ip&longs;o mobili duplicis mobilis vicem gerit, quorum vnum im<lb/>pedit motum alterius, nec moueri po&longs;&longs;unt, ni&longs;i motibus oppo&longs;itis. </s>

<s id="N21DC9">&longs;ed hoc e&longs;t ab&longs;urdum: ratio verò di&longs;pa<lb/>ritatis e&longs;t, quia mobile, quod motu circulari voluitur circa centrum, <lb/>quod e&longs;t in ip&longs;o mobili duplicis mobilis vicem gerit, quorum vnum im<lb/>pedit motum alterius, nec moueri po&longs;&longs;unt, ni&longs;i motibus oppo&longs;itis. </s>

<s id="N21DD6"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s>

Line 13264

<s id="N21E97"> Primum e&longs;t, faciliùs &longs;u&longs;tineri maius <lb/>pondus, quàm minus. </s>

<s id="N21E9C">Secundum plùs addi ponderis, quò plùs detrahi<lb/>tur. </s>

<s id="N21EA1">Tertium plùs detrahi, quò plùs additur, v.g. &longs;i detrahatur aliqua por<lb/>tio ex &longs;emicirculo BC DE, &longs;emicirculus rotæ oppo&longs;itus de&longs;cendet, ni&longs;i <lb/>&longs;it potentia in CA, qua &longs;u&longs;tineatur; </s>

<s id="N21EAB">& quò maior portio detrahetur po<lb/>rentiæ, maius pondus incumbet; quò minor, minus. </s>

<s id="N21EAB">& quò maior portio detrahetur po<lb/>tentiæ, maius pondus incumbet; quò minor, minus. </s>

<s id="N21EB1">Sed hæc clara <lb/>&longs;unt. </s>

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<s id="N21ECD"> Duplex impetus con<lb/>&longs;iderari pote&longs;t in rota; </s>

<s id="N21ED3">primus e&longs;t productus ad intra accedente, &longs;cilicet <lb/>minima vi ponderis alteri &longs;emicirculo, putâ puncto A, qua po&longs;ita tolla-<pb pagenum="284" xlink:href="026/01/318.jpg"/>tur æquilibrium, quo &longs;ublato &longs;ua &longs;ponte mouetur rota; </s>

<s id="N21EDE">hic autem impe<lb/>tus primò durat in toto de&longs;cen&longs;u quadrantis AD; </s>

<s id="N21EE4">immò acceleratur tan<lb/>tillùm motus, licèt longè minùs, quàm in funependulo propter re&longs;i&longs;ten<lb/>&longs;tentiam &longs;emicirculi oppo&longs;iti contranitentis; </s>

<s id="N21EE4">immò acceleratur tan<lb/>tillùm motus, licèt longè minùs, quàm in funependulo propter re&longs;i&longs;ten<lb/>tiam &longs;emicirculi oppo&longs;iti contranitentis; </s>

<s id="N21EEC">vbi verò A peruenit in D, <lb/>non acceleratur ampliùs motus, &longs;ed tantillùm a&longs;cendit ver&longs;us C &, dein<lb/>de de&longs;cendit, tandemque quie&longs;cit in D paucis confectis vibrationibus; </s>

<s id="N21EF4"><lb/>&longs;ed de hoc cur&longs;u, & recur&longs;u agemus fusè lib. &longs;equenti; </s>

<s id="N21EF9">alter impetus e&longs;t <lb/>productus ab extrin&longs;eco, applicata &longs;cilicet valida potentiá, qui rotam <lb/>agit velociore motu, vt patet, cùm præter impetum ad intra &longs;it etiam im<lb/>petus productus ab extrin&longs;eca cau&longs;a; </s>

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<s id="N221BF"><emph type="italics"/>Potentia applicata talis e&longs;&longs;e debet, vt po&longs;&longs;it imprimere impetum toti rota<emph.end type="italics"/>; </s>

<s id="N221C8"><pb pagenum="287" xlink:href="026/01/321.jpg"/>cumenim non po&longs;&longs;it moueri vna pars rotæ &longs;ine alia; </s>

<s id="N221D0">certè, vel impetus <lb/>imprimitur omnibus, vel nulli per Th.37. lib.1.præ&longs;ertim cùm totus im<lb/>petus, qui rotæ imprimitur, &longs;it ab extrin&longs;eco; nec enim accidit huic rotæ, <lb/>quod alteri, quæ &longs;itum verticalem habet, cuius &longs;emicirculus, cui admouc<lb/>tur potentia per lineam deor&longs;um motu naturali ex parte deor&longs;um fertur, <lb/>vt &longs;upra explicatum e&longs;t. </s>

<s id="N221D0">certè, vel impetus <lb/>imprimitur omnibus, vel nulli per Th.37. lib.1.præ&longs;ertim cùm totus im<lb/>petus, qui rotæ imprimitur, &longs;it ab extrin&longs;eco; nec enim accidit huic rotæ, <lb/>quod alteri, quæ &longs;itum verticalem habet, cuius &longs;emicirculus, cui admoue<lb/>tur potentia per lineam deor&longs;um motu naturali ex parte deor&longs;um fertur, <lb/>vt &longs;upra explicatum e&longs;t. </s>

<s id="N221DE">Hinc totus impetus in rota horizontali produ<lb/>citur ab extrin&longs;eco; hinc ab ea tantùm potentia volui pote&longs;t, quæ tot <lb/>partes impetus pote&longs;t producere, quot &longs;unt nece&longs;&longs;ariæ, vt omnibus parti<lb/>bus plani illius circularis di&longs;tribuantur, iuxta propagationem, quæ motui <lb/>circulari competit. </s>

Line 13436

<s id="N22250"><emph type="italics"/>Hinc faciliùs mouetur rota motu illo circulari, quàm recto<emph.end type="italics"/>; </s>

<s id="N22259">quia &longs;it dia<lb/>meter AC, vt moueatur motu recto per &longs;e debet produci impetus eiu&longs;<lb/>dem perfectionis in omnibus partibus AC, vt con&longs;tat ex dictis lib. 1. &longs;i <lb/>enim motus omnium partium e&longs;t æqualis; </s>

<s id="N22263">igitur & impe<gap/>, at verò, vt <lb/>moueatur motu circulari in plano horizontali facto &longs;cilicet circulo <lb/>ABCD, & admota potentia in A; </s>

<s id="N22263">igitur & impetus, at verò, vt <lb/>moueatur motu circulari in plano horizontali facto &longs;cilicet circulo <lb/>ABCD, & admota potentia in A; </s>

<s id="N2226D">certè impetus qui producitur in A, <lb/>& in C, e&longs;t minor impetu producto in F, & in H; </s>

<s id="N22273">igitur &longs;i producatur <lb/>in A impetus eiu&longs;dem perfectionis ad motum circularem cum eo, qui <lb/>produceretur admotum rectum; </s>

<s id="N2227B">haud dubiè totus impetus productus in <lb/>AC ad motum rectum e&longs;t perfectior toto impetu producto ad circula<lb/>rem; igitur difficiliùs ille, hic faciliùs producitur. </s>

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<s id="N22369"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s>

<s id="N22377"><emph type="italics"/>Hinc quò propiùs ad centrum applicatur potentia, eò maior e&longs;t difficultas <lb/>motiu<emph.end type="italics"/>; </s>

<s id="N22377"><emph type="italics"/>Hinc quò propiùs ad centrum applicatur potentia, eò maior e&longs;t difficultas <lb/>motus<emph.end type="italics"/>; </s>

<s id="N22382">igitur &longs;i applicetur ip&longs;i centro mathematicè con&longs;iderato e&longs;t infi<lb/>nita difficultas; </s>

<s id="N22388">igitur nulla potentia &longs;uperare po&longs;&longs;et hanc difficultatem; </s>

<s id="N2238C"><pb pagenum="289" xlink:href="026/01/323.jpg"/>hinc vt artifices &longs;uas ver&longs;ent rotas &longs;aciliùs, vel maximè curuum manu<lb/>brium adhibent, vel affixo ver&longs;us circumferentiam in plano rotæ clauo <lb/>rotam agunt in orbes; quæ omnia clarè &longs;equuntur ex dictis. </s>

<s id="N2238C"><pb pagenum="289" xlink:href="026/01/323.jpg"/>hinc vt artifices &longs;uas ver&longs;ent rotas faciliùs, vel maximè curuum manu<lb/>brium adhibent, vel affixo ver&longs;us circumferentiam in plano rotæ clauo <lb/>rotam agunt in orbes; quæ omnia clarè &longs;equuntur ex dictis. </s>

<s id="N2239A"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s>

Line 13525

<s id="N2244B"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s>

<s id="N22459"><emph type="italics"/>Minor rota citiùs &longs;uum gyrum ab&longs;oluit, quàm mator,<emph.end type="italics"/> vt dictum e&longs;t &longs;uprà, <lb/>&longs;iue &longs;it in &longs;itu verticali, &longs;iue in &longs;itu horizontali; </s>

<s id="N22459"><emph type="italics"/>Minor rota citiùs &longs;uum gyrum ab&longs;oluit, quàm maior,<emph.end type="italics"/> vt dictum e&longs;t &longs;uprà, <lb/>&longs;iue &longs;it in &longs;itu verticali, &longs;iue in &longs;itu horizontali; </s>

<s id="N22464">&longs;ed non e&longs;t determinata <lb/>proportio, quàm hîc de&longs;ideramus; dico enim tempora motuum e&longs;&longs;e, vt <lb/>radios. </s>

<s id="N2246C">v.g.tempus, quo rota minor FGHI &longs;uum gyrum ab&longs;oluit, e&longs;&longs;e ad <lb/>tempus, quo maior ABCD &longs;uum perficit, vt e&longs;t radius FE ad radium <lb/>AE, quod demon&longs;tro; </s>

<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;ubduplums <lb/>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="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>

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Line 13563

<s id="N22543"> Probatur, quia punctum B velociùs mouebitur, quàm <lb/>A vel H, vt patet experientiâ: </s>

<s id="N22549">ratio e&longs;t, quia minùs impetus producitur <lb/>in toto cylindro BH, applicata potentia in B, quàm in A, quod e&longs;t cen<lb/>trum grauitatis cylindri BA, vt iam o&longs;tendimus Th. 68. 69. BB; </s>

<s id="N22551">porrò <lb/>ratio à priori e&longs;t, quia cùm impetus producatur tantùm ad extra, vt tol<lb/>latur impedimentum motus, vt fusè o&longs;tendimus lib. 1. certè in tantùm <lb/>amouetur impedimentum, in quantum amouetur corpus impediens mo<lb/>tum alterius; </s>

<s id="N2255D">atqui amoucri tantùm pote&longs;t per motum; </s>

<s id="N2255D">atqui amoueri tantùm pote&longs;t per motum; </s>

<s id="N22561">igitur eo motu <lb/>amouetur, quo faciliùs amoueri pote&longs;t, & minore &longs;umptu, vt ita dicam, <lb/>id e&longs;t minore impetu: </s>

<s id="N22569">porrò cum potentia &longs;it determinata ad producen<lb/>dum tabem impetum, immediatè &longs;cilicet, id e&longs;t, in ea parte, cui immedia<lb/>tè admouetur; </s>

<s id="N22571">alicqui &longs;i po&longs;&longs;et minorem, & minorem in infinitum pro<lb/>ducere po&longs;&longs;et etiam immediatè &longs;ine operâ organi mechanici quodlibet <lb/>pondus mouere, quod e&longs;t ab&longs;urdum, de quo iam &longs;uprà; </s>

<s id="N22571">alioqui &longs;i po&longs;&longs;et minorem, & minorem in infinitum pro<lb/>ducere po&longs;&longs;et etiam immediatè &longs;ine operâ organi mechanici quodlibet <lb/>pondus mouere, quod e&longs;t ab&longs;urdum, de quo iam &longs;uprà; </s>

<s id="N22579">&longs;it igitur potentia <lb/>applicata in A, &longs;cilicet in centro grauitatis cylindri BH; </s>

<s id="N2257F">certè producit <lb/>maximum impetum, quem pote&longs;t producere in cylindro BH (&longs;uppono <lb/>enim e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam, & producere perfecti&longs;&longs;imum impetum, <lb/>quem producere po&longs;&longs;it) producit inquam maximum ratione numeri; </s>

<s id="N22589"><lb/>cùm in toto cylindro BH producat impetum eiu&longs;dem perfectionis; </s>

<s id="N2258E">igi<lb/>tur mouetur motu recto; </s>

<s id="N22594">igitur æquali in omnibus partibus; </s>

<s id="N22598">igitur æqua<lb/>lis e&longs;t impetus in omnibus partibus, id e&longs;t, æquè inten&longs;us; </s>

<s id="N2259E">&longs;it autem po-<pb pagenum="291" xlink:href="026/01/325.jpg"/>tentia applicata in B, ita vt in puncto B producatur impetus eiu&longs;dens <lb/>perfectionis, de quo &longs;uprà: </s>

<s id="N2259E">&longs;it autem po-<pb pagenum="291" xlink:href="026/01/325.jpg"/>tentia applicata in B, ita vt in puncto B producatur impetus eiu&longs;dem <lb/>perfectionis, de quo &longs;uprà: </s>

<s id="N225A9">&longs;i mouetur motu circulari circa aliquod cen<lb/>trum v. g. circa centrum H, & punctum B conficiat arcum BD æqua<lb/>lem rectæ b I, vel BL quam æquali tempore B vel A antè percurrebant <lb/>motu recto; </s>

<s id="N225B7">certè totus cylindrus BH acquiret tantùm &longs;patium BHD <lb/>motu circulari circa centrum H; </s>

<s id="N225BD">&longs;ed motu recto acqui&longs;iuit &longs;patium re<lb/>ctanguli BK, quod maius e&longs;t, vt patet; </s>

<s id="N225C3">igitur motus circularis circa H <lb/>cylindri BH e&longs;t ad rectum, vt &longs;ector BHD ad rectangulum BK; </s>

<s id="N225C9">igitur <lb/>facilitas motus circularis e&longs;t ad facilitatem motus recti præ&longs;entis, vt re<lb/>ctangulum BK ad &longs;ectorem BHD; </s>

<s id="N225D1">quænam verò &longs;it hæc propprtio pa<lb/>tet ex Cyclometria, &longs;uppo&longs;itâ ratione Archimedis periphæriæ ad diame<lb/>trum; </s>

<s id="N225D1">quænam verò &longs;it hæc proportio pa<lb/>tet ex Cyclometria, &longs;uppo&longs;itâ ratione Archimedis periphæriæ ad diame<lb/>trum; </s>

<s id="N225D9">igitur cum cylindrus impul&longs;us in B faciliùs moueri po&longs;&longs;it motu <lb/>circulari, quàm recto, vt con&longs;tat ex dictis; </s>

<s id="N225DF">& cùm eo motu moueatur, <lb/>quo faciliùs moueri pote&longs;t; </s>

<s id="N225E5">modò po&longs;&longs;it ad illum determinari, non mirum <lb/>e&longs;t &longs;i eo moueatur, & minor impetus producatur in eodem cylindro <lb/>BH; debet autem e&longs;&longs;e aliquod centrum huius motus, quod determina<lb/>bimus paulò pò&longs;t, po&longs;tquam breuiter exilem quamdam objectionem de <lb/>impetu refutauerimus. </s>

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<s id="N22671"><lb/>quia primò, nullum e&longs;&longs;et munus huius conditionis; nec enim applica<lb/>ret cau&longs;am &longs;ubjecto, nec remoueret vllum impedimentum. </s>

<s id="N22678">Secundò di<lb/>cam &longs;imiliter calorem e&longs;&longs;e conditionem. </s>

<s id="N2267D">Tertiò, dicerem etiam e&longs;&longs;e con<lb/>ditionem ad motum. </s>

<s id="N22682">Quartò, quis dicat corpus graue producere impe-<pb pagenum="292" xlink:href="026/01/326.jpg"/>tum &longs;ur&longs;um immediatè per &longs;e; &longs;ed hæc omittamus, quæ leuia &longs;unt, præ<lb/>&longs;ertim cùm demon&longs;traucrimus luculenter lib.1.impetum produci ab im<lb/>petu, vt &longs;cilicet tollatur impedimentum. </s>

<s id="N22682">Quartò, quis dicat corpus graue producere impe-<pb pagenum="292" xlink:href="026/01/326.jpg"/>tum &longs;ur&longs;um immediatè per &longs;e; &longs;ed hæc omittamus, quæ leuia &longs;unt, præ<lb/>&longs;ertim cùm demon&longs;trauerimus luculenter lib.1.impetum produci ab im<lb/>petu, vt &longs;cilicet tollatur impedimentum. </s>

<s id="N22691"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s>

Line 13639

Line 13640

<s id="N22715">totum autem &longs;patium <lb/>confectum hoc motu erit conflatum ex &longs;ectoribus BMN, & HMO, vt <lb/>patet: </s>

<s id="N2271D">porrò &longs;ector BMN e&longs;t &longs;ubtriplus quadrantis BMR, qui quadrans <lb/>e&longs;t ad priorem BAE, vt 9. ad 4. id e&longs;t, vt quadratum 3. ad quadratum 2. <lb/>vt con&longs;tat; </s>

<s id="N22725">igitur conflatum ex &longs;ectore BMN, & &longs;ectore HMO e&longs;t ad <lb/>quadrantem BAE, vel conflatum ex geminis &longs;ectoribus BAC, HAG <lb/>vt 3 1/3 ad 4. &longs;i autem accipiatur centrum, vel inter MA, vel MH, maius <lb/>erit &longs;patium, vt con&longs;tat ex Geometria; </s>

<s id="N2272F">igitur circa cemrum M e&longs;t mini<lb/>mum &longs;patium; </s>

<s id="N2272F">igitur circa centrum M e&longs;t mini<lb/>mum &longs;patium; </s>

<s id="N22735">igitur minimus motus; </s>

<s id="N22739">igitur minimus impetus; igitur <lb/>maxima facilitas; igitur &longs;i pellatur in B, vertetur circa M, quod hactenus <lb/>non explicatum modò ab aliquo, quod &longs;ciam, verùm etiam ne propo&longs;itum <lb/>quidem fuit. </s>

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Line 13667

<s id="N227C1">&longs;it autem BT æqua<lb/>lis TA; </s>

<s id="N227C7">certè arcus TS erit æqualis arcui BE; </s>

<s id="N227CB">igitur &longs;ector VMB erit <lb/>&longs;ubduplus quadrantis BMR: </s>

<s id="N227D1">&longs;umiliter &longs;ector HMX erit &longs;ubduplus qua<lb/>drantis HMP; </s>

<s id="N227D1">&longs;imiliter &longs;ector HMX erit &longs;ubduplus qua<lb/>drantis HMP; </s>

<s id="N227D7">igitur motus erit, vt aggregatum ex his duobus &longs;ectori<lb/>bus; </s>

<s id="N227DD">&longs;ed cum applicatur potentia in B, motus e&longs;t vt aggregatum ex duo<lb/>bus &longs;ectoribus BMN, HNO; </s>

<s id="N227E3">&longs;it autem quadrans BMR, vt 9. & qua<lb/>drans HMP vt 1. igitur cum applicatur potentia in B, motus e&longs;t ad mo<lb/>tum cum applicatur in T vt 3 1/3 ad 5. igitur & impetus; igitur facilitas <lb/>primi motus e&longs;t ad facilitatem &longs;ecundi, vt 5. ad 3 1/3 igitur in T diffici<lb/>liùs pellitur, quàm in B. </s>

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