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annotate texts/archimedesOldCVSRepository/archimedes/xml/balia_demot_076_la_1638.xml @ 27:8dce37600d38

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New Special Instructions
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Wed, 30 Jul 2014 15:58:21 +0200
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1 <?xml version="1.0" encoding="UTF-8"?>
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2 <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd">
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3 <archimedes xmlns:xlink="http://www.w3.org/1999/xlink">
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4 <info>
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5 <author>Baliani, Giovanni Baptista</author>
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6 <title>De Motu Naturali Gravium Solidorum Ioannis Baptistae Baliani</title>
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7 <date>1638</date>
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8 <place>Genua</place>
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9 <translator/>
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10 <lang>la</lang>
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11 <cvs_file>balia_demot_076_la_1638.xml</cvs_file>
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12 <cvs_version/>
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13 <locator>076.xml</locator>
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14 </info>
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15 <text>
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16 <front>
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17 <pb xlink:href="076/01/001.jpg"/>
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18 <section>
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19 <p type="head">
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20 <s id="s.000001">
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21 De MOTV<lb/> NATVRALI,<lb/> GRAVIVM SOLIDORYVM<lb/> IOANNIS BAPTISTAE BALIANI<lb/> PATRITII GENVENSIS.</s>
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22 </p>
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23 <p type="head">
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24 <s id="s.000002">
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25 GENVUAE,<lb/> EX Typographia Io: Mariae Farroni, Nicolai Pe&longs;agnij,<lb/> et Petri Franci&longs;ci Barberij, &longs;oc. <lb/>MDCXXXVIII.<lb/> SVPERIORUM PERMISSV.
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26 </s>
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27 </p>
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28 </section>
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29 <pb xlink:href="076/01/002.jpg"/>
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30 <pb xlink:href="076/01/003.jpg"/>
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31 <section>
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32 <p type="head">
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33 <s id="s.000003">PRAEFATIO</s>
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34 </p>
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35 <p>
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36 <s id="s.000004">Mihi quoque, sicut et caeteris hominibus,<lb/> inest sciendi cupiditas, nec grave fuit, usque<lb/> &agrave; primis annis, et aliorum scripta percur-<lb/>rere, et naturales effectus observare, qui<lb/> facile mihi persuaserim, ex bisce fontibus,<lb/> tum scientiam, tum sapientiam in animum<lb/> derivare, si tandem ex effectibus diligentius perspectis, non<lb/> modo ad inde consequentes, sed etiam ad causas, usque ad<lb/> primam deveniat intellectus. </s>
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37 <s id="s.000005">Statui igitur apud me ipsum<lb/> non acquiescere soli relationi plurimorum, etiam doctiorum;<lb/> potuisse siquidem contingere existimavi, ut aliqua laterent,<lb/> etiam in plurimis oculatissimos, vel non plene ab eis explica-<lb/>rentur; &amp; ratus sum non inutilem laborem futurum, si ex<lb/> accuratiori naturae rerum investigatione, &amp; ex affectionum<lb/> inde resultantium deductione, circa quod omnis demonstra-<lb/>tiva scientia versatur, aut scitis adderem aliqua, aut doctiori-<lb/>bus acuerem desiderium addendi plura: hinc factum est, ut exci-<lb/>tata mens ex praecognitis legendo, ad ea, quae se offerebant,<lb/> secundum privatas, aut publicas occupationes pervestiganda,<lb/> converteretur studiosus. </s>
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38 <s id="s.000006">Inter alia dum anno millesimo sex-<lb/>centesimo undecimo, per paucos menses, ex patriae legis prae-<lb/>scripto, Praefectum Arcis Savonae agerem, ex militaribus<lb/> observationibus quae occurrebant, illud maxime depraehendi,<lb/> ferreos, &amp; lapideos tormentorum bellicorum globos, &amp; sic<lb/> corpora gravia, seu eiusdem, seu diversae speciei, in inaequali<lb/> satis Mole, &amp; gravitate, per idem spatium, aequali tempore,<lb/> &amp; motu, naturaliter descendere, idque ita uniformiter, ut<lb/> repetitis experimentis mihi plane constiterit, duos ex prae-<lb/>dictis globis, vel ferreos ambos, vel alterum lapideum al-<lb/>terum plumbeum, eodem plane momento temporis dimissos<lb/> sibi, per spatium quincaginta pedum, etiam si unus esset<lb/> librae unius tantum, alter quincaginta, in indivisibili tem-<lb/>poris momento, subjectum solum ferire, ut unus tantum am-<lb/>borum ictus sensu perciperetur. </s>
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39 <s id="s.000007">Repetebam animo sapien-<lb/><pb xlink:href="076/01/004.jpg"/>tum esse pronunciatum, gravia moveri naturali motu, se-<lb/>cundum gravitatum proportionem; Processi ulterius, &amp; peri-<lb/>culum feci, num forte iuxta eorum sententiam contingeret,<lb/> si corpora dimissa, ejusdem fere essent molis, sed longe di-<lb/>versi ponderis, puta unum plumbeum, cereum alterum; &amp;<lb/> expertus sum in cereo aliquam longiorem moram in descen-<lb/>su, attamen longe infra proportionem gravitatum, globus<lb/> quippe ille cereus, in data distantia quinquaginta pedum de-<lb/>scensus, uno circiter pede distabat a solo, quando plumbeus<lb/> tangebat subjectum planum, objecto aere intermedio ni fal-<lb/>lor, sensibiliter resistente, &amp; impediente motum. </s>
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40 <s id="s.000008">Institi<lb/> adhuc, &amp; globos in gravitate, &amp; in materia inaequales appendi<lb/> funiculis aequalibus, &amp; agitatos animadverti moveri tempore<lb/> aequali, &amp; hoc servare adeo fideliter, ut globus plumbeus dua-<lb/>rum unciarum, alter librarum duarum, ferreus librarum<lb/> 34. </s>
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41 <s id="s.000009">&amp; lapideus quadraginta circiter, nec non, &amp; lapis in-<lb/>formis, quorum funiculi comprehensis ipsorum semidiame-<lb/>tris aequales essent, uno, &amp; eodem temporis spatio moveren-<lb/>tur, &amp; vibrationes easdem numero darent hinc inde, sive mo-<lb/>tus unius globi fieret per aequale spatium, sive per inaequa-<lb/>le, ita ut qui majori impetu jactabatur, &amp; sic majus spatium<lb/> percurrebat, illud tanto velocius pertransiret. </s>
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42 <s id="s.000010">In quibus<lb/> peragendis illud praeter expectationem sese mihi obtulit,<lb/> quod quotiescunque globi penderent ex funiculis inaequalibus,<lb/> ita inaequali motu ferebantur, ut longitudines funiculorum,<lb/> durationibus motuum, in duplicata ratione responderent.<lb/> </s>
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43 </p>
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44 <p>
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45 <s id="s.000011">Porro cum ex praemissis satis superque liqueret, in naturali<lb/> motu gravium, proportionem gravitatum communiter credi-<lb/>tam, non servari; in eam descendi sententiam, ut arbitrater<lb/> fortasse, gravitatem se habere ut agens, materiam vero, seu<lb/> mavis materiale corpus, ut passum, &amp; proinde gravia mo-<lb/>veri juxta proportionem gravitatis ad materiam, &amp; ubi sine<lb/> impedimento naturaliter perpendiculari motu ferantur, mo-<lb/>veri aequaliter, quia ubi plus est gravitatis, plus pariter sit<lb/> materiae, seu materialis quantitatis; si vero accedat aliquid<lb/> resistantiae, regulari motum secundum excessum virtutis agen<lb/><pb xlink:href="076/01/005.jpg"/>tis supra resistentiam passi, seu impedientia motum; qui<lb/> excessus momentum noncupabitur, &amp; quod communiter gra-<lb/>vitati attributum fuit, momento attribui debere, nimirum<lb/> ut sit momentum ad momentum, ut velocitas ad velocita-<lb/>tem; Et hinc fieri posse, ut cognoscamus qua mensura, seu<lb/> proportione corpora gravia naturali motu ferantur super su-<lb/>bjectis planis, si super eis quomodolibet inclinatis, ipsorum<lb/> gravium momenta ubique innotescant, quae majora, aut mi-<lb/>nora videntur censenda, secundum quod magis, aut minus<lb/> super plano quiescunt, &amp; sic secundum majorem, aut mino-<lb/>rem inclinationem plani resistentis; quod demum tali pro-<lb/>portione facile fieri mihi existimandum videtur, juxta quam<lb/> reciproce momentis proportionantur lineae dictorum plano-<lb/>rum, si ambae ductae sint ab eodem puncto ad idem planum<lb/> orizontale; de quo Simon Stevinus l. p. de Statica prop.<lb/> 19. &amp; acutissime Galileus in Mechanica manuscripta, ubi de<lb/> Cochlea, &amp; ego &aelig;liquali experientia compertum habui. </s>
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46 <s id="s.000012">Cae-<lb/>terum si per experientiam Scienta hominibus efficitur, prae-<lb/>dicta de quibus saepius repetitis actibus expertus fui, ut prin-<lb/>cipia scientiae habenda fore censui; in quibus occultae con-<lb/>clusiones delitescant, demonstrationibus duntaxat aperiendae.<lb/> </s>
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47 <s id="s.000013">Rimari caepi; an deprehenderim aliorum erit judicium.<lb/> </s>
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48 <s id="s.000014">Subjecta paucula, quae presens aliquod otium expedire per-<lb/>misit, de motu naturali solidorum gravium, Amice lector<lb/> tibi exhibeo, mox de liquidorum, &amp; deinceps alia plura tam<lb/> parata daturus, si haec placuerint. </s>
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49 <s id="s.000015">Placuit sane mihi, vel<lb/> paucula tibi dare, qui te ejus ingenii esse confidam, ut non<lb/> verba, sed res, easque non mole, sed pondere censeas, felicior<lb/> si de eorum genere existimaveris, quae non mole magna sunt,<lb/> quod si talia non fuerint, quo minora minus defatigabunt,<lb/> sui exilitate, auctoris partus proprios omnino esse probatura.<lb/> </s>
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50 <s id="s.000016">Idioma latinum elegi ut communius. </s>
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51 <s id="s.000017">Praemisi aliqua na-<lb/>turalia principia, sine quibus naturales conclusiones aliunde<lb/> duci posse non video. </s>
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52 <s id="s.000018">Quae ex praedictis experimentis inno<lb/>tuerunt, suppositiones appellare, &amp; a reliquis petitionibus se-<lb/>cernere libuit. </s>
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53 <s id="s.000019">Petitiones illas, quibus quid fieri petimus, con-<lb/><pb xlink:href="076/01/006.jpg"/>structioni deservientes, tanquam factu, &amp; cognitu faciles,<lb/> &amp; proinde supervacaneas, prudens praetermisi; ratus siqui-<lb/>dem nil inde incredulitatis, aut difficultatis derivaturum.<lb/> </s>
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54 <s id="s.000020">Septimum postulatum ea ratione segregavi, quod illud aliquo<lb/> pacto a 22. prop. pendeat, &amp; quod in illo etiamsi veritas<lb/> non deficiat, evidentiam tamen ut in caeteris non agnoscens,<lb/> certis dubia quo quo pacto permiscere noluerim; ut proinde<lb/> plura eorum, quae ex illa deducta sunt, &amp; diversa Methodo<lb/> &amp; attingendo potius, quam demonstrando subjunxerim.<lb/> </s>
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55 <s id="s.000021">Si quae demum minus probata, seu explicata, aut quo<lb/> quo pacto imperfecta reperies, velim te tribue-<lb/>re cuidam naturali meae propensioni, ad no-<lb/>va potius, qualiacumque ea sint, inve-<lb/>nienda, quam inventa<lb/> perficienda.<lb/> </s>
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56 <s id="s.000022">Vale. </s>
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57 </p>
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58 </section>
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59 <pb xlink:href="076/01/007.jpg"/>
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60 </front>
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61 <body>
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62 <chap>
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63 <p type="head">
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64 <s id="s.000023">DEFINITIONES</s>
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65 </p>
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66 <subchap1>
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67 <p>
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68 <s id="s.000024">Pendulum dicimus pondus filo appensum.</s>
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69 </p>
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70 </subchap1>
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71 <subchap1>
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72 <p>
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73 <s id="s.000025">Pendula dicuntur aequalia, seu aequipendula,<lb/> sive inaequalia, quae, &amp; longiora, aut brevio-<lb/>ra, quatenus fila, e quibus dependent, sunt<lb/> aequalia, longiora, aut breviora.<lb/> </s>
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74 </p>
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75 </subchap1>
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76 <subchap1>
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77 <p>
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78 <s id="s.000026">Vibrationes pendulorum sunt eorum motus hinc inde<lb/>
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79 Vibrationes aequales dicimus, quae fiunt per spatia aequalia, &amp; e contra inaequales.<lb/> </s>
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80 </p>
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81 </subchap1>
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82 <subchap1>
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83 <p>
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84 <s id="s.000027">Vibrationes aeque celeres si fiant per spatia<lb/> aequalia tem-<lb/>pore aequali.<lb/> </s>
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85 </p>
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86 </subchap1>
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87 <subchap1>
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88 <p>
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89 <s id="s.000028">Vibrationis diuturnitatem dicimus ipsius Durationem,<lb/> tempus nimirum, quo ipsa vibratio perficitur.<lb/> </s>
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90 </p>
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91 </subchap1>
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92 <subchap1>
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93 <p>
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94 <s id="s.000029">Vibrationes aequediuturnae, sunt, quae fiunt tempore<lb/> aequali, etiamsi per spatia inaequalia, inde diuturnior<lb/> est, quae longiori perficitur tempore.<lb/> </s>
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95 </p>
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96 </subchap1>
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97 <subchap1>
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98 <p>
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99 <s id="s.000030">Vibrationes integras dicimus eas, quae se extendunt<lb/> per integrum semicirculum, se hinc inde moventes<lb/> per circuli quadrantem.<lb/> </s>
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100 </p>
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101 </subchap1>
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102 <subchap1>
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103 <p>
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104 <s id="s.000031">Vibrationis portio est pars arcus, quem ipsa vibratio<lb/> disignant.<lb/> </s>
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105 </p>
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106 </subchap1>
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107 <subchap1>
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108 <p>
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109 <s id="s.000032">Vibrationum similes portiones sunt arcus ipsarum in-<lb/>tercepti inter binas lineas ductas a centro, a quo<lb/> concipiuntur pendula pendere.<lb/> </s>
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110 </p>
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111 </subchap1>
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112 <subchap1>
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113 <p>
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114 <s id="s.000033">Vibrationis portionem priorem decimus eam mini-<lb/>mam portionem, a qua integra vibratio initium habet.<lb/> </s>
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115 </p>
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116 </subchap1>
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117 <subchap1>
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118 <p>
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119 <s id="s.000034">Momentum est excessus virtutis moventis supra mo-<lb/>tus impedimenta.<lb/> </s>
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120 </p>
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121 </subchap1>
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122 </chap>
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123 <pb xlink:href="076/01/008.jpg"/>
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124 <chap>
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125 <p type="head">
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126 <s id="s.000035">SUPPOSITIONES<lb/></s>
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127 </p>
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128 <subchap1>
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129 <p>
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130 <s id="s.000036">PRIMA. Solidorum aequipendulorum cujus-<lb/>cumque gravitatis vibrationes aequales sunt aeque-<lb/>diuturnae.<lb/> </s>
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131 </p>
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132 </subchap1>
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133 <subchap1>
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134 <p>
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135 <s id="s.000037">2 Equipendulorum eorumdem vibrationes sunt aeque-<lb/>diuturnae, etiamsi inaequales.<lb/> </s>
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136 </p>
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137 </subchap1>
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138 <subchap1>
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139 <p>
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140 <s id="s.000038">3 Pendulorum inaequalium longitudines sunt in du-<lb/>plicata ratione diuturnitatum vibrationum, seu ut<lb/> quadrata vibrationum.<lb/> </s>
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141 </p>
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142 </subchap1>
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143 <subchap1>
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144 <p>
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145 <s id="s.000039">4 Momentum gravis super plano inclinato est ad ip-<lb/>sius gravitatem, ut perpendicularis ad inclinatam,<lb/> si ab eodem puncto ducta sint ad idem planum<lb/> orizontale dicta perpendicularis, &amp; dictum planum<lb/> inclinatum, &amp; proinde tali casu proportio gravita&shy;<lb/>tis ad momentum est reciproca proportioni linea-<lb/>rum super quibus grave movetur.<lb/> </s>
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146 </p>
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147 </subchap1>
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148 </chap>
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149 <pb xlink:href="076/01/009.jpg"/>
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150 <chap>
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151 <p type="head">
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152 <s id="s.000040">PETITIONES, SEU POSTULATA<lb/></s>
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153 </p>
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154 <subchap1>
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155 <p>
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156 <s id="s.000041">Pr. Pendulorum inaequalium portiones similes vibra-<lb/>tionum sunt inter se quoad diuturnitatem, ut vibra-<lb/>tiones integrae.<lb/> </s>
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157 </p>
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158 </subchap1>
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159 <subchap1>
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160 <p>
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161 <s id="s.000042">Sint pendula AB, AC; dependentia a puncto A, &amp; ele-<lb/>ventur ad libellam orizontis puncti A, in E, D, de-<lb/>scribentia arcus BD, CE, integrarum vibrationum, &amp; in<lb/> arcubus BD, CE sumantur portiones similes EF, DG, seu<lb/> HI, KL ductis EA, FA, seu HA, IA. </s>
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162 <s id="s.000043">Peto mihi concedi,<lb/> esse pendulorum diuturnitates in arcubus EC, DB, ut in<lb/> portionibus EF, DG, nec non HI, KL, &amp; ita deinceps.<lb/> </s>
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163 </p>
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164 </subchap1>
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165 <subchap1>
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166 <p>
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167 <s id="s.000044">2. Ut est momentum ad momentum solidi gravis, ita<lb/> velocitas ad velocitatem.<lb/> </s>
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168 </p>
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169 </subchap1>
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170 <subchap1>
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171 <p>
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172 <s id="s.000045">Hujusmodi passio communiter attribui solet gravitati sim-<lb/>pliciter, quod eum nimis clare experientiis supra expo-<lb/>sitis nullo pacto congruere possit, momentis attribuenda<lb/> esse visa est, ut in praefatione explicatum fuit.<lb/> </s>
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173 </p>
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174 </subchap1>
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175 <pb xlink:href="076/01/010.jpg"/>
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176 <subchap1>
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177 <p>
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178 <s id="s.000046">3. Portiones minimae peripheriae Circuli concipiende<lb/> sunt, ac si essent lineae rectae.<lb/> </s>
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179 </p>
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180 </subchap1>
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181 <subchap1>
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182 <p>
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183 <s id="s.000047">Quaecumque arcus portio est circularis, attamen si est<lb/> minima portio, tam parum aberrat a linea recta, ut<lb/> non modo quo ad sensum, sed quoad quascunque physicas<lb/> passiones, perinde esse videatur, ac si esset linea recta, id-<lb/>circo ut petitionem admittendam censeo, quemadmodum in-<lb/>mechanicis admittitur illa, quod perpendiculares sunt paral-<lb/>lelae, etiamsi in centro concurrant universi, quatenus eisdem<lb/> sunt passionibus physicis subjectae, ac si vere essent parallelae.<lb/> </s>
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184 </p>
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185 </subchap1>
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186 <subchap1>
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187 <p>
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188 <s id="s.000048">4. Data recta linea, possimus concipere circulum talis<lb/> magnitudinis, cujus portio peripheriae aequalis quo<lb/> ad sensum datae lineae, concipienda sit, ac si esset<lb/> linea recta.<lb/> </s>
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189 </p>
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190 </subchap1>
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191 <subchap1>
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192 <p>
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193 <s id="s.000049">Haec petitio videtur concedenda, quia si concipiamus cir-<lb/>culum, ejusque portionem minimam, ut in praece-<lb/>denti, si fiat ut hujusmodi portio ad datam lineam, ita<lb/> circulus ad alium, portio hujus, datae lineae aequalis erit, &amp;<lb/> similis omnino praedicta minimae portioni, &amp; proinde pa-<lb/>riter concipienda ut linea recta.<lb/> </s>
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194 </p>
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195 </subchap1>
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196 <subchap1>
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197 <p>
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198 <s id="s.000050">5. Solida perpendicula libero motu aeque velociter<lb/> feruntur, &amp; in tali proportione, ac si essent pendula,<lb/> &amp; moverentur in priori portione vibrationum.<lb/> </s>
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199 </p>
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200 </subchap1>
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201 <subchap1>
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202 <p>
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203 <s id="s.000051">Quoniam prior portio non differt sensibiliter a recta, ut in<lb/> tertia petitione, nec etiamsi sit major ut in quarta, iisdem<lb/> physicis passionibus subjicitur, &amp; exinde motibus aequalibus.<lb/> </s>
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204 </p>
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205 </subchap1>
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206 <subchap1>
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207 <p>
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208 <s id="s.000052">6. Solida naturaliter mota super plano inclinato aeque<lb/> velociter moventur ac si essent pendula, &amp; moveren-<lb/>tur in tali portione vibrationum, quae quoad sensum<lb/> <pb xlink:href="076/01/011.jpg"/>esset aequalis, &amp; paralella lineae dicti plani super qua<lb/> dicta solida moverentur.<lb/> </s>
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209 </p>
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210 </subchap1>
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211 <subchap1>
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212 <p>
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213 <s id="s.000053">Non differt a praecedente, nisi quod in illa motus est per-<lb/>pendicularis, in hac inclinatus, in reliquis est par ratio.<lb/> </s>
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214 </p>
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215 </subchap1>
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216 </chap>
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217 <chap>
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218 <p type="head">
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219 <s id="s.000054">PRONUNCIATA<lb/></s>
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220 </p>
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221 <subchap1>
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222 <p>
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223 <s id="s.000055">P. Quae sunt aequidiuturna tertio, sunt aequidiu-<lb/>turna inter se.<lb/> </s>
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224 </p>
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225 </subchap1>
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226 <subchap1>
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227 <p>
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228 <s id="s.000056">2. Quadrata datorum temporum, sunt etiam quadrata<lb/> aliorum datis aequalium.<lb/> </s>
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229 </p>
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230 </subchap1>
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231 <subchap1>
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232 <p>
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233 <s id="s.000057">3. Gravia eadem super planis aequalibus &amp; pariter incli-<lb/>natis, pariter moventur.<lb/> </s>
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234 </p>
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235 </subchap1>
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236 </chap>
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237 <pb xlink:href="076/01/012.jpg"/>
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238 <chap>
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239 <p type="head">
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240 <s id="s.000058">PROPOSITIO PRIMA.<lb/> </s>
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241 </p>
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242 <subchap1>
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243 <p>
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244 <s id="s.000059">Solidi penduli naturaliter moti vibrationes quan-<lb/>tumvis semper minores, sunt aequidiuturnae.<lb/> </s>
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245 </p>
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246 </subchap1>
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247 <p>
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248 <s id="s.000060">Sit solidum A pendulum debite applicatum filo BA, quod<lb/> ab altera parte elevatum naturaliter, postea faciat hinc<lb/> inde vibrationes semper minores, ita ut prior vibratio sit V. G.<lb/> per spatium CD maius, posterior vero per spatium EF minus.<lb/> </s>
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249 </p>
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250 <p>
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251 <s id="s.000061">Dico quod dicta vibrationes erunt aequidiuturnae, ita ut vibra-<lb/>tio per spatium CD sit eiusdem durationis, ac vibratio per<lb/> spatium EF.<lb/> </s>
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252 </p>
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253 <p>
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254 <s id="s.000062">Sit aliud solidum G aequipendulum solido A, debite applica-<lb/>tum filo HG, quod elevetur ab una parte eodem tempore<lb/> minus quam solidum A ita ut sint minores vibrationes soli-<lb/>di G, quam, solidi A, ut sit motus penduli G in initio per<lb/> spatium IK aequale spatio EF.<lb/> </s>
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255 </p>
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256 <p>
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257 <s id="s.000063">Quoniam spatia EF, &amp; IK, sunt aequalia ex suppositione,<lb/> sunt etiam vibrationes EF, &amp; IK, aequidiuturnae<arrow.to.target n="marg1"/>, sed I<lb/>K, &amp; CD sunt pariter aequidiuturnae<arrow.to.target n="marg2"/>, ergo EF, &amp; CD<lb/> sunt etiam aequidiuturnae<arrow.to.target n="marg3"/>. </s>
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258 <s id="s.000064">Quod fuit probandum.<lb/> </s>
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259 </p>
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260 <p type="margin">
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261 <s id="s.000065"><margin.target id="marg1"/>Per pri-<lb/>mam sup-<lb/>positionem.<lb/></s>
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262 <s id="s.000066"><margin.target id="marg2"/>Per secun-<lb/>dam sup-<lb/>positionem.<lb/></s>
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263 <s id="s.000067"><margin.target id="marg3"/>Per pr.<lb/> pron.</s>
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264 </p>
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265 </chap>
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266 <pb xlink:href="076/01/013.jpg"/>
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267 <chap>
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268 <p type="head">
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269 <s id="s.000068">PROPOSITIO II. PROBLEMA PRIMUM. </s>
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270 </p>
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271 <subchap1>
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272 <p>
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273 <s id="s.000069">Pendula constituere, quorum diuturnitates vibra-<lb/>tionum sint in data ratione.<lb/> </s>
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274 </p>
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275 </subchap1>
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276 <p>
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277 <s id="s.000070">Data sit proportio diuturnitatum vibrationum, quam<lb/> volumus esse inter solida A,B; &amp; sit ea, quae est inter<lb/> C, &amp; D; quae est continuo eadem,<arrow.to.target n="marg4"/>a.<lb/></s>
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278 </p>
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279 <p>
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280 <s id="s.000071"> Venanda est longitudo filorum, quibus applicata dicta solida<lb/> producant vibrationes quaesitas. </s>
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281 </p>
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282 <p type="margin">
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283 <s id="s.000072"><margin.target id="marg4"/>Per pr.<lb/> hujus.</s>
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284 </p>
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285 <p>
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286 <s id="s.000073">Sint E F numeri mensurantes proportionem, quae est inter C<lb/> &amp; D, quorum quadrati numeri G &amp; H, Fila IA, KB<lb/> fiant inter se ut G, ad H, &amp; erunt fila quaesita, quibus si<lb/> applicentur solida A, B, producentur diuturnitates vibra-<lb/>tionum quaesita.<lb/> </s>
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287 </p>
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288 <p>
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289 <s id="s.000074">Quoniam ita est IA, ad KB, ut quadratum G numeri me-<lb/>tientis C, ad quadratum H numeri metientis D, erunt C, &amp;<lb/> D diuturnitates vibrationum pendulorum A, &amp; B<arrow.to.target n="marg5"/>; &amp;<lb/> proinde in ratione data. </s>
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290 <s id="s.000075">Quod faciendum fuit.<lb/> </s>
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291 </p>
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292 <p type="margin">
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293 <s id="s.000076"><margin.target id="marg5"/>Per 3.<lb/> suppo.</s>
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294 </p>
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295 </chap>
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296 <pb xlink:href="076/01/014.jpg"/>
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297 <chap>
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298 <p type="head">
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299 <s id="s.000077">PROPOSITIO TERTIA.<lb/> </s>
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300 </p>
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301 <subchap1>
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302 <p>
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303 <s id="s.000078">Lineae descensus gravium, dum naturali motu perpendicula-<lb/>riter feruntur, sunt in duplicata ratione diuturnitatum.<lb/> </s>
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304 </p>
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305 </subchap1>
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306 <p>
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307 <s id="s.000079">Sint LN, KM linea descensus gravium L, K, &amp; sint P<lb/>O ipsorum diuturnitates.<lb/> </s>
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308 </p>
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309 <p>
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310 <s id="s.000080">Dico LN, KM esse in duplicata ratione ipsarum P, O.<lb/> </s>
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311 </p>
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312 <p>
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313 <s id="s.000081">Sint pendula AH, AI, dependentia a puncto A, &amp; eleven-<lb/>tur ad libellam ipsius A usque ad E, B, quae in elevatione<lb/> producant arcus HB, IE, &amp; sint talis longitudinis, ut du-<lb/>cta ACF, secet arcus BC, &amp; EF, portionis minimae, aequa-<lb/>les quo ad sensum lineis LN, KM, &amp; sit S, quadratum<lb/> diuturnitatis P, &amp; T quadratum O, &amp; Q, R, diuturni-<lb/>tates vibrationum BC, &amp; EF.<lb/> </s>
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314 </p>
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315 <p>
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316 <s id="s.000082">Quoniam diuturnitates Q, R sunt aequales diuturnitatibus<lb/> P, O<arrow.to.target n="marg6"/>; S, T, sunt etiam quadrata ipsarum Q, R<arrow.to.target n="marg7"/>, &amp; quia<lb/> vibrationes integrae pendulorum AH, AI sunt ut qua-<lb/>dratum T ad quadratum S<arrow.to.target n="marg8"/>, portiones BC, EF, sunt pa-<lb/>riter inter se ut quadratum T ad quadratum S<arrow.to.target n="marg9"/>, sed<lb/> BC, &amp; EF sunt aequales lineis KM, LN<arrow.to.target n="marg10"/>, ergo etiam K<lb/>M, LN sunt ut quadrata S, T<arrow.to.target n="marg11"/>, &amp; proinde in duplicata<lb/> ratione P, O, temporum seu diuturnitatum earumdem.<lb/> </s>
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317 <s id="s.000083">Quod, &amp;c.<lb/> </s>
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318 </p>
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319 <p type="margin">
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320 <s id="s.000084"><margin.target id="marg6"/>Per 5.<lb/> pet.<lb/></s>
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321 <s id="s.000085"><margin.target id="marg7"/>Per 2.<lb/> pron.<lb/></s>
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322 <s id="s.000086"><margin.target id="marg8"/>Per 3.<lb/> supposit.<lb/></s>
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323 <s id="s.000087"><margin.target id="marg9"/>Per 5.<lb/> petit.<lb/></s>
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324 <s id="s.000088"><margin.target id="marg10"/>Per 3.<lb/> petit.<lb/></s>
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325 <s id="s.000089"><margin.target id="marg11"/>Per 1.<lb/> pron.<lb/></s>
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326 </p>
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327 </chap>
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328 <pb xlink:href="076/01/015.jpg"/>
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329 <chap>
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330 <p type="head">
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331 <s id="s.000090">PROPOSITIO QUARTA. PROBL. II.<lb/> </s>
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332 </p>
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333 <subchap1>
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334 <p>
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335 <s id="s.000091">Data diuturnitate gravis descendentis a data altitudine,<lb/> constituere altitudinem, a qua idem grave cadat in<lb/> data alia diuturnitate.<lb/> </s>
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336 </p>
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337 </subchap1>
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338 <p>
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339 <s id="s.000092">Sit A diuturnitas gravis B, dum cadit in C, &amp; data sit<lb/> diuturnitas quaecumque D.<lb/> </s>
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340 </p>
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341 <p>
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342 <s id="s.000093">Constituenda est alia altitudo, a qua grave descendat iuxta<lb/> diuturnitatem D.<lb/> </s>
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343 </p>
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344 <p>
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345 <s id="s.000094">Fiant E, &amp; F quadrata temporum A, D, &amp; ut F ad E, fiat<lb/> altitudo GH, ad altitudinem datam BC; Dico GH esse al-<lb/>titudinem quaesitam.<lb/> </s>
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346 </p>
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347 <p>
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348 <s id="s.000095">Quoniam BC, &amp; GH sunt in duplicata ratione datarum diu-<lb/>turnitatum A, D, per constructionem; per ipsas gravia B,<lb/> &amp; G cadent in diuturnitatibus A, &amp; D datis<arrow.to.target n="marg12"/>, unde re-<lb/>perta est altitudo GH quaesita. </s>
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349 <s id="s.000096">Quod fuit faciendum.<lb/> </s>
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350 </p>
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351 <p type="margin">
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352 <s id="s.000097"><margin.target id="marg12"/>Per 3.<lb/> hujus.</s>
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353 </p>
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354 </chap>
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355 <pb xlink:href="076/01/016.jpg"/>
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356 <chap>
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357 <p type="head">
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358 <s id="s.000098">PROPOSITIO V. PROBL. III.<lb/> </s>
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359 </p>
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360 <subchap1>
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361 <p>
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362 <s id="s.000099">Data altitudine, a qua descendat grave in nota diutur-<lb/>nitate; perquirere quanta sit diuturnitas, qua descen-<lb/>dat ab alia altitudine data.<lb/> </s>
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363 </p>
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364 </subchap1>
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365 <p>
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366 <s id="s.000100">Sit A altitudo per quam descendat grave diuturnitate B<lb/> nota, &amp; data sit alia altitudo C.<lb/> </s>
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367 </p>
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368 <p>
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369 <s id="s.000101">Oportet reperire quanta sit diuturnitas, qua idem grave de-<lb/>scendat per C.<lb/> </s>
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370 </p>
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371 <p>
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372 <s id="s.000102">Fiat D quadratum diuturnitatis B, &amp; fiat ut A ad C, ita<lb/> quadratum D ad quadratum E, cuius radix F est diutur-<lb/>nitas quaesita. </s>
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373 </p>
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374 <p>
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375 <s id="s.000103">Quoniam A, &amp; C sunt in duplicata ratione diuturnitatum<lb/> B, &amp; F per constructionem, per ipsas gravia descendent<lb/> in diuturnitatibus B, F,<arrow.to.target n="marg13"/> unde F est diuturnitas ipsius C<lb/> quaesita. </s>
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376 <s id="s.000104">Quod faciendum fuit.<lb/> </s>
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377 </p>
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378 <p type="margin">
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379 <s id="s.000105"><margin.target id="marg13"/>Per 3.<lb/> hujus.<lb/></s>
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380 </p>
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381 </chap>
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382 <pb xlink:href="076/01/017.jpg"/>
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383 <chap>
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384 <p type="head">
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385 <s id="s.000106">PROPOSITIO VI. </s>
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386 </p>
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387 <subchap1>
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388 <p>
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389 <s id="s.000107">Gravia naturali motu descendunt semper velocius ea<lb/> ratione, ut temporibus aequalibus descendant per spa-<lb/>tia semper maiora, iuxta proportionem quam ha-<lb/>bent impares numeri ab unitate inter se.<lb/> </s>
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390 </p>
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391 </subchap1>
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392 <p>
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393 <s id="s.000108">Sit grave A quod descendat per lineam ABC, &amp; tempus<lb/> quo descendit ab A in B sit aequale tempori, quo de-<lb/>scendit a B in C, &amp; a C in D.<lb/> </s>
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394 </p>
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395 <p>
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396 <s id="s.000109">Dico quod lineae AB, BC, CD sunt inter se ut 1. 3. 5. &amp;<lb/> sic deinceps.<lb/> </s>
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397 </p>
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398 <p>
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399 <s id="s.000110">Sit G numerus mensurans tempus, quo A descendit in B, &amp;<lb/> H, quo descendit a B in C, &amp; I, quo descendit a C in D,<lb/> quae tempora sunt ex suppositione aequalia, &amp; sit K qua-<lb/>dratum ipsius G, &amp; L quadratum GH, &amp; M quadratum<lb/> totius GHI. </s>
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400 </p>
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401 <p>
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402 <s id="s.000111">Quoniam quadrata K, L, N sunt ut AB, AC, AD<arrow.to.target n="marg14"/>, quae<lb/> quadrata sunt ut 1, 4, 9, sunt itidem AB, AC, AD, ut<lb/> 1. 4. 9. &amp; dividendo AB, BC, CD, ut 1. 3. 5. &amp; sic dein-<lb/>ceps. </s>
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403 <s id="s.000112">Quod probandum fuit.<lb/> </s>
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404 </p>
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405 <p type="margin">
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406 <s id="s.000113"><margin.target id="marg14"/>Per 3.<lb/> hujus.<lb/></s>
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407 </p>
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408 </chap>
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409 <pb xlink:href="076/01/018.jpg"/>
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410 <chap>
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411 <p type="head">
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412 <s id="s.000114">PROPOSITIO VII.<lb/> </s>
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413 </p>
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414 <subchap1>
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415 <p>
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416 <s id="s.000115">Lineae descensus gravium super plano inclinato mo-<lb/>torum, sunt in duplicata ratione diuturnitatum.<lb/> </s>
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417 </p>
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418 </subchap1>
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419 <p>
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420 <s id="s.000116">Sint AB, CD plana pariter inclinata, super quibus mo-<lb/>veantur gravia A, C, &amp; sint EF ipsorum diuturnitates.<lb/> </s>
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421 </p>
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422 <p>
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423 <s id="s.000117">Dico AB, CD, esse in duplicata ratione ipsarum E, F.<lb/> </s>
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424 </p>
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425 <p>
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426 <s id="s.000118">Secetur AB bifariam in G, &amp; erecta GH, perpendiculari<lb/> longissima, fiant pendula HI, HK, quae sint inter se ut A<lb/>B, CD, &amp; eleventur in L, M, describentia arcus LI, KM,<lb/> secantes GH in N, O, &amp; ab N hinc inde secentur arcus N<lb/>P, NQ aequales quo ad sensum rectis GA, GB, &amp; ductis P<lb/>H, QH, secetur pariter arcus LI, in R, S, &amp; intelligan-<lb/>tur arcus PQ, RS, tam parvae curvitatis, ob maximam<lb/> longitudinem pendulorum HI, HK, ut pro rectis habean-<lb/>tur, puta portionis minimae, &amp; proinde aequales rectis A<lb/>B, CD: sit Z quadratum diuturnitatis E, &amp; V, diuturnitatis<lb/> F, &amp; sint XY diuturnitates vibrationum PQ, RS.<lb/> </s>
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427 </p>
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428 <pb xlink:href="076/01/019.jpg"/>
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429 <p>
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430 <s id="s.000119">Quoniam diuturnitates X, Y, sunt aequales diuturnitatibus<lb/> E, F,<arrow.to.target n="marg15"/> sunt etiam Z, V, quadrata ipsarum X, Y<arrow.to.target n="marg16"/>; &amp; quia<lb/> vibrationes integrae pendulorum HI, HK sunt inter se, ut<lb/> quadratum V, ad quadratum Z<arrow.to.target n="marg17"/>, portiones RS, PQ erunt<lb/> etiam inter se ut quadratum V ad quadratum Z<arrow.to.target n="marg18"/>; sed R<lb/>S, PQ aequantur rectis CD, AB,<arrow.to.target n="marg19"/>, ergo, &amp; CD, AB<lb/> sunt ut quadratum V, ad quadratum Z<arrow.to.target n="marg20"/>, &amp; proinde, in<lb/> duplicata ratione ipsarum EF. </s>
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431 <s id="s.000120">Quod, &amp;c.<lb/> </s>
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432 </p>
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433 <p type="margin">
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434 <s id="s.000121"><margin.target id="marg15"/>Per 6.<lb/> petit.<lb/></s>
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435 <s id="s.000122"><margin.target id="marg16"/>Per 2.<lb/> pron.<lb/></s>
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436 <s id="s.000123"><margin.target id="marg17"/>Per 3.<lb/> hujus.<lb/></s>
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437 <s id="s.000124"><margin.target id="marg18"/>Per pr.<lb/> pet.<lb/></s>
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438 <s id="s.000125"><margin.target id="marg19"/>Per 3.<lb/> petit.<lb/></s>
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439 <s id="s.000126"><margin.target id="marg20"/>Per 2.<lb/> pron.<lb/></s>
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440 </p>
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441 <p>
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442 <s id="s.000127">Corolarium.<lb/> </s>
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443 </p>
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444 <p>
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445 <s id="s.000128">Hinc patet esse longitudines planorum per quae gravia fe-<lb/>runtur ut quadrata temporum, &amp; tempora ut radices<lb/> longitudinum planorum.<lb/> </s>
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446 </p>
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447 </chap>
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448 <pb xlink:href="076/01/020.jpg"/>
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449 <chap>
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450 <p type="head">
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451 <s id="s.000129">PROPOSITIO VIII. PROB. IV.<lb/> </s>
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452 </p>
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453 <subchap1>
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454 <p>
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455 <s id="s.000130">Dato plano inclinato, super quo per spatium datum<lb/> grave moveatur in nota diuturnitate, determinare in<lb/> eodem plano spatium per quod dictum grave mo-<lb/>veatur in quavis alia diuturnitate data.<lb/> </s>
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456 </p>
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457 </subchap1>
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458 <p>
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459 <s id="s.000131">Sit A diuturnitas gravis B, dum descendit in C super pla-<lb/>no inclinato BC, &amp; data diuturnitas D.<lb/> </s>
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460 </p>
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461 <p>
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462 <s id="s.000132">Praescribendum est aliud spatium in eodem plano BC, per<lb/> quod idem grave pertranseat in diuturnitate D.<lb/> </s>
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463 </p>
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464 <p>
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465 <s id="s.000133">Fiant E, F quadrata temporum A, D, &amp; ut F ad E fiat BG ad<lb/> BC, Dico BG esse spatium quaesitum.<lb/> </s>
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466 </p>
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467 <p>
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468 <s id="s.000134">Quoniam BC, &amp; BG sunt in duplicata ratione datorum<lb/> temporum A, D per constructionem, per ipsa cadet grave<lb/> B diuturnitatibus A, D datis<arrow.to.target n="marg21"/>, ergo reperta est BG quae-<lb/>sita. </s>
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469 <s id="s.000135">Quod faciendum erat.<lb/> </s>
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470 </p>
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471 <p type="margin">
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472 <s id="s.000136"><margin.target id="marg21"/>Per 6.<lb/> hujus.<lb/></s>
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473 </p>
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474 </chap>
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475 <pb xlink:href="076/01/021.jpg"/>
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476 <chap>
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477 <p type="head">
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478 <s id="s.000137">PROPOSITIO IX. PROB. V.<lb/> </s>
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479 </p>
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480 <subchap1>
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481 <p>
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482 <s id="s.000138">Dato plano inclinato, super quo per spatium datum gra-<lb/>ve moveatur nota diuturnitate; &amp; dato alio spatio<lb/> quocumque; reperire diuturnitatem, qua grave per<lb/> ipsum descendat.<lb/> </s>
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483 </p>
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484 </subchap1>
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485 <p>
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486 <s id="s.000139">Sit A diuturnitas gravis B, dum descendit in C super pla-<lb/>no inclinato BC, &amp; dato alio spatio BG.<lb/> </s>
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487 </p>
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488 <p>
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489 <s id="s.000140">Querendum quanta sit diuturnitas gravis in BG.<lb/> </s>
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490 </p>
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491 <p>
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492 <s id="s.000141">Fiat E quadratum diuturnitatis A, &amp; ut BC ad BG fiat ut<lb/> quadratum E ad quadratum F, cuius radix D erit diutur-<lb/>nitas ipsius BG quaesita.<lb/> </s>
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493 </p>
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494 <p>
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495 <s id="s.000142">Quoniam BC, &amp; BG sunt in duplicata ratione diuturnita-<lb/>tum A, D per constructionem; per ipsa cadunt gravia diu-<lb/>turnitatibus A, D<arrow.to.target n="marg22"/>, unde D est diuturnitas per spatium<lb/> BG quaesita. </s>
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496 <s id="s.000143">Quod faciendum erat.<lb/> </s>
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497 </p>
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498 <p type="margin">
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499 <s id="s.000144"><margin.target id="marg22"/>Per 7.<lb/> hujus.<lb/></s>
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500 </p>
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501 </chap>
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502 <pb xlink:href="076/01/022.jpg"/>
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503 <chap>
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504 <p type="head">
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505 <s id="s.000145">PROPOSITIO X.<lb/> </s>
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506 </p>
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507 <subchap1>
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508 <p>
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509 <s id="s.000146">Gravia descendunt super planis inclinatis per spatia<lb/> semper maiora, iuxta rationem, quam habent im-<lb/>pares numeri successive inter se.<lb/> </s>
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510 </p>
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511 </subchap1>
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512 <p>
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513 <s id="s.000147">Sit grave A, quod descendat super plano ABC inclinato,<lb/> &amp; tempus quo descendit ab A in B sit aequale tempo-<lb/>ri, quo descendit a B in C, &amp; a C in D.<lb/> </s>
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514 </p>
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515 <p>
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516 <s id="s.000148">Dico quod lineae AB, BC, CD sunt inter se ut 1. 3. 5. &amp;. sic<lb/> deinceps. </s>
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517 </p>
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518 <p>
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519 <s id="s.000149">Sit E numerus mensurans tempus, quo A descendit in B, &amp; F<lb/> quo descendit a B in C, &amp; G quo descendit a C in D, quae<lb/> tempora sunt ex suppositione aequalia, &amp; sit H quadratum<lb/> ipsius E, &amp; I quadratum EF, &amp; K quadratum totius EFG.<lb/> </s>
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520 </p>
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521 <p>
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522 <s id="s.000150">Quoniam quadrata HIK sunt ut AB, AC, AD<arrow.to.target n="marg23"/>, quae<lb/> quadrata sunt ut 1. 4. 9. sunt pariter AB, AC, AD, ut<lb/> 1. 4. 9. &amp; dividendo AB, BC, CD, sunt ut 1. 3. 5. &amp; sic<lb/> deinceps. </s>
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523 <s id="s.000151">Quod probandum erat.<lb/> </s>
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524 </p>
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525 <p type="margin">
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526 <s id="s.000152"><margin.target id="marg23"/>Per 7.<lb/> hujus.<lb/></s>
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527 </p>
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528 </chap>
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529 <pb xlink:href="076/01/023.jpg"/>
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530 <chap>
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531 <p type="head">
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532 <s id="s.000153">PROPOSITIO XI.<lb/> </s>
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533 </p>
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534 <subchap1>
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535 <p>
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536 <s id="s.000154">Si Duo gravia descendant alterum super linea perpen-<lb/>diculari, alterum vero super inclinata; proportio ve-<lb/>locitatum est reciproca proportioni linearum.<lb/> </s>
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537 </p>
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538 </subchap1>
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539 <p>
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540 <s id="s.000155">Sit ABC planum normaliter erectum super lineam ori-<lb/>zontalem BC, cuius latus AB sit perpendiculare, &amp;<lb/> AC, inclinatum.<lb/> </s>
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541 </p>
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542 <p>
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543 <s id="s.000156">Dico quod proportio velocitatum solidorum gravium moto-<lb/>rum secundum lineam AB perpendicularem, &amp; AC in-<lb/>clinatum, est ut proportio longitudinis inclinatae AC ad<lb/> longitudinem perpendicularis AB; videlicet ita est longi-<lb/>tudo AB ad longitudinem AC, ut velocitas super AC ad<lb/> velocitatem in AB.<lb/> </s>
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544 </p>
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545 <p>
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546 <s id="s.000157">Quoniam est ut AC ad AB, ita momentum in AB, ad mo-<lb/>mentum in AC<arrow.to.target n="marg24"/>; &amp; ut momentum in AB ad momentum<lb/> in AC, ita velocitas in AB ad velocitatem in AC<arrow.to.target n="marg25"/>; er-<lb/>go est etiam ut AC ad AB, ita velocitas in AB ad velo-<lb/>citatem in AC. </s>
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547 <s id="s.000158">Quod fuit probandum.<lb/> </s>
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548 </p>
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549 <p type="margin">
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550 <s id="s.000159"><margin.target id="marg24"/>Per 4.<lb/> supp.<lb/></s>
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551 <s id="s.000160"><margin.target id="marg25"/>Per 2.<lb/> pet.<lb/></s>
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552 </p>
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553 </chap>
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554 <pb xlink:href="076/01/024.jpg"/>
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555 <chap>
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556 <p type="head">
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557 <s id="s.000161">PROPOSITIO XII.<lb/> </s>
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558 </p>
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559 <subchap1>
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560 <p>
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561 <s id="s.000162">Gravia descendunt super plana diverse inclinata tali<lb/> proportione, ut si velocitas ad velocitatem recipro-<lb/>ca longitudinibus planorum ductorum ab eodem<lb/> puncto, ad idem planum orizontale.<lb/> </s>
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562 </p>
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563 </subchap1>
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564 <p>
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565 <s id="s.000163">Sint F, D plana inclinata ducta ad idem planum orizon-<lb/>tale.<lb/> </s>
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566 </p>
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567 <p>
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568 <s id="s.000164">Dico esse ut planum D ad planum F, ita velocitatem gravis<lb/> ducti super F, ad velocitatem eiusdem ducti super D.<lb/> </s>
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569 </p>
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570 <p>
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571 <s id="s.000165">Ducatur perpendicularis E, &amp; sint B, A, C velocitates gra-<lb/>vium latorum super perpendiculari, &amp; super planis F, D.<lb/> </s>
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572 </p>
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573 <p>
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574 <s id="s.000166">Quoniam est A ad B, ut E ad F, item, &amp; B ad C, ut D, ad<lb/> E<arrow.to.target n="marg26"/>, erit A ad C ut D ad F<arrow.to.target n="marg27"/>, scilicet velocitas gravis su-<lb/>per F ad velocitatem gravis super D, ut longitudo pla-<lb/>ni D ad longitudinem plani F. </s>
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575 <s id="s.000167">Quod fuit probandum.<lb/> </s>
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576 </p>
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577 <p type="margin">
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578 <s id="s.000168"><margin.target id="marg26"/>Per 11.<lb/> hujus.<lb/></s>
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579 <s id="s.000169"><margin.target id="marg27"/>Per 23.<lb/> Quinti.<lb/></s>
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580 </p>
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581 </chap>
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582 <pb xlink:href="076/01/025.jpg"/>
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583 <chap>
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584 <p type="head">
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585 <s id="s.000170">PROPOSITIO XIII. PROBL. VI.<lb/> </s>
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586 </p>
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587 <subchap1>
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588 <p>
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589 <s id="s.000171">Reperire inclinationem plani, super quo grave movea-<lb/>tur tali velocitate quae cum alia super diversa incli-<lb/>natione sit in ratione data.<lb/> </s>
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590 </p>
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591 </subchap1>
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592 <p>
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593 <s id="s.000172">Moveatur grave A super recta AB, seu perpendicula-<lb/>ri, seu inclinata, &amp; data sit proportio C ad D.<lb/> </s>
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594 </p>
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595 <p>
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596 <s id="s.000173">Oportet reperire aliud planum inclinatum, ita ut velocitas<lb/> gravis moti super AB ad velocitatem alterius moti<lb/> super illo reperiendo, sit ut D ad C.<lb/> </s>
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597 </p>
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598 <p>
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599 <s id="s.000174">Producatur BA; &amp; fiat ut C ad D ita BA, ad AE; &amp;<lb/> centro A, intervallo AE describatur circulus, secans BF<lb/> in F; ni secet, problema insolubile est; si secat, ducatur<lb/> AF, quam dico esse planum quaesitum.<lb/> </s>
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600 </p>
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601 <p>
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602 <s id="s.000175">Quoniam ut C ad D, ita AB ad AE, seu AF per constructio-<lb/>nem, erit C velocitas super AF, &amp; D super AB<arrow.to.target n="marg28"/>, unde velo-<lb/>citates super ipsis sunt in ratione data. </s>
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603 <s id="s.000176">Quod faciendum fuit.<lb/> </s>
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604 </p>
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605 <p type="margin">
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606 <s id="s.000177"><margin.target id="marg28"/>Per 12.<lb/> hujus.<lb/></s>
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607 </p>
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608 </chap>
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609 <pb xlink:href="076/01/026.jpg"/>
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610 <chap>
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611 <p type="head">
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612 <s id="s.000178">PROPOSITIO XIV. PROBL. VII.<lb/> </s>
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613 </p>
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614 <subchap1>
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615 <p>
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616 <s id="s.000179">Data linea perpendiculari, per quam grave descendat,<lb/> cui annectatur linea, seu planum declinans; in decli-<lb/>nante reperire punctum, quo grave perveniat eo<lb/> tempore, quo pertransiverit perpendicularem.<lb/> </s>
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617 </p>
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618 </subchap1>
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619 <p>
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620 <s id="s.000180">Sit triangulum ABC orthogonaliter erectum super pla<lb/>no orizontali BC, cuius latus AB intelligatur sit linea perpendicu-<lb/>laris, per quam grave descendat, &amp; latus AC sit planum<lb/> inclinatum.<lb/> </s>
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621 </p>
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622 <p>
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623 <s id="s.000181">Oportet in plano AC reperire punctum quo grave perveniat<lb/> eodem tempore, quo in B.<lb/> </s>
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624 </p>
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625 <p>
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626 <s id="s.000182">Fiat ut AC ad AB, ita AB ad tertiam AD, &amp; D erit pun-<lb/>ctum quaesitum.<lb/> </s>
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627 </p>
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628 <p>
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629 <s id="s.000183">Quoniam ut AC ad AD, ita quadratum AC ad quadra-<lb/>tum AB<arrow.to.target n="marg29"/>, &amp; ut AC ad AD, ita quadratum temporis A<lb/>C ad quadratum temporis AD<arrow.to.target n="marg30"/>, ergo ut quadratum A<lb/>C ad quadratum AB, ita quadratum temporis AC ad qua-<lb/>dratum temporis AD<arrow.to.target n="marg31"/>, ergo ut AC ad AB, ita tempus<lb/> AC ad tempus AD<arrow.to.target n="marg32"/>, sed ut AC ad AB, ita tempus AC ad<lb/> tempus AB<arrow.to.target n="marg33"/>, ergo tempus AB est aequale tempori AD.<lb/> </s>
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630 <s id="s.000184">Quod, &amp;c.<lb/> </s>
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631 </p>
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632 <p type="margin">
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633 <s id="s.000185"><margin.target id="marg29"/>Per 19.<lb/> Sexti.<lb/></s>
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634 <s id="s.000186"><margin.target id="marg30"/>Per cor.<lb/> 7. hujus.<lb/></s>
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635 <s id="s.000187"><margin.target id="marg31"/>Per 11.<lb/> Quinti.<lb/></s>
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636 <s id="s.000188"><margin.target id="marg32"/>Per 22.<lb/> Sexti.<lb/></s>
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637 <s id="s.000189"><margin.target id="marg33"/>Per 11.<lb/> hujus.<lb/></s>
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638 </p>
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639 </chap>
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640 <pb xlink:href="076/01/027.jpg"/>
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641 <chap>
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642 <p type="head">
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643 <s id="s.000190">PROPOSITIO XV.<lb/> </s>
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644 </p>
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645 <subchap1>
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646 <p>
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647 <s id="s.000191">Linea connectens puncta, ad quae duo gravia ab eo-<lb/>dem puncto digressa, quorum alterum perpenden-<lb/>ter, alterum super plano declinante descendat, simul<lb/> perveniunt, est perpendicularis dicto plano declinanti.<lb/> </s>
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648 </p>
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649 </subchap1>
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650 <p>
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651 <s id="s.000192">Descendant simul duo gravia a puncto A primum per-<lb/>pendiculariter in B, secundum super plano inclinato<lb/> AC, tali lege, ut simul perveniant ad puncta BD,<lb/> &amp; ducta sit linea BD.<lb/> </s>
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652 </p>
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653 <p>
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654 <s id="s.000193">Dico quod dicta linea BD est perpendicularis ad AD.<lb/> </s>
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655 </p>
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656 <p>
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657 <s id="s.000194">Fiat AF aequalis datae AB, &amp; AE aequalis AD, &amp; duca-<lb/>tur EF.<lb/> </s>
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658 </p>
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659 <p>
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660 <s id="s.000195">Quoniam ut AD ad AB, ita AB ad AC<arrow.to.target n="marg34"/>, &amp; AD,<lb/> AE, item AB, AF sunt aequales per constructionem, se-<lb/>quitur quod AE ad AF est ut AB ad AC, ergo EF, BC<lb/> sunt parallelae<arrow.to.target n="marg35"/>, unde triangulum AEF, &amp; proin-<lb/>de ABD est simile triangulo ABC<arrow.to.target n="marg36"/>, unde anguli AB<lb/>C, ADB simul recti, &amp; BD perpendicularis ad AD.<lb/> </s>
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661 <s id="s.000196">Quod, &amp;c.<lb/> </s>
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662 </p>
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663 <p type="margin">
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664 <s id="s.000197"><margin.target id="marg34"/>Per 13.<lb/> hujus.<lb/></s>
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665 <s id="s.000198"><margin.target id="marg35"/>Per 2.<lb/> Sexti.<lb/></s>
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666 <s id="s.000199"><margin.target id="marg36"/>Per 4.<lb/> Sexti.<lb/></s>
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667 </p>
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668 </chap>
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669 <pb xlink:href="076/01/028.jpg"/>
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670 <chap>
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671 <p type="head">
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672 <s id="s.000200">PROPOSITIO XVI. PROBL. VIII.<lb/> </s>
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673 </p>
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674 <subchap1>
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675 <p>
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676 <s id="s.000201">Data linea perpendiculari, &amp; plano declinante; reperire<lb/> in perpendiculari producta punctum, quo perveniat<lb/> grave eo tempore, quo pertransit planum inclinatum.<lb/> </s>
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677 </p>
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678 </subchap1>
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679 <p>
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680 <s id="s.000202">Data sit perpendicularis AB, cui connexum planum<lb/> inclinatum AD.<lb/> </s>
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681 </p>
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682 <p>
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683 <s id="s.000203">Oportet in AB producta reperire punctum, quo perveniat<lb/> grave eo tempore, quo pervenit in puncto D.<lb/> </s>
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684 </p>
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685 <p>
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686 <s id="s.000204">In puncto D perpendicularis erigatur ad AD, &amp; protraha-<lb/>tur usquequo coeat cum AB producta in E, &amp; E est pu-<lb/>nctum quaesitum. </s>
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687 </p>
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688 <p>
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689 <s id="s.000205">Quoniam triangula &amp; ADE, AEC sint aequiangula,<lb/> cum anguli ADE, AEC sint aequales, nempe recti, &amp;<lb/> BAD communis<arrow.to.target n="marg37"/>, sunt etiam similia<arrow.to.target n="marg38"/>, ergo ut AC<lb/> ad AE, ita AE ad AD<arrow.to.target n="marg39"/>, sed ut AC ad AD, ita qua-<lb/>dratum AC ad quadratum AE<arrow.to.target n="marg40"/>, &amp; ut AC ad AD,<lb/> ita quadratum temporis AC ad quadratum temporis A<lb/>D<arrow.to.target n="marg41"/>, ergo ut quadratum AC ad quadratum AE ita qua-<lb/>dratum temporis AC ad quadratum temporis AD<arrow.to.target n="marg42"/>, er-<lb/>go ut AC ad AE, ita tempus AC ad tempus AD<arrow.to.target n="marg43"/>, sed<lb/> ut AC ad AE, ita tempus AC ad tempus AE<arrow.to.target n="marg44"/>, ergo<lb/> tempora AE, &amp; AD sunt aequalia. </s>
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690 <s id="s.000206">Quod &amp;c.<lb/> </s>
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691 </p>
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692 <p type="margin">
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693 <s id="s.000207"><margin.target id="marg37"/>Per 32.<lb/> prim.<lb/></s>
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694 <s id="s.000208"><margin.target id="marg38"/>Per 4.<lb/> sexti.<lb/></s>
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695 <s id="s.000209"><margin.target id="marg39"/>Per 4.<lb/> sexti.<lb/></s>
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696 <s id="s.000210"><margin.target id="marg40"/>Per 19.<lb/> Sexti.<lb/></s>
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697 <s id="s.000211"><margin.target id="marg41"/>Per Cor.<lb/> 7. hujus.<lb/></s>
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698 <s id="s.000212"><margin.target id="marg42"/>Per 11.<lb/> Quinti.<lb/></s>
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699 <s id="s.000213"><margin.target id="marg43"/>Per 22.<lb/> sexti.<lb/></s>
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700 <s id="s.000214"><margin.target id="marg44"/>Per 11.<lb/> hujus.<lb/></s>
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701 </p>
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702 </chap>
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703 <pb xlink:href="076/01/029.jpg"/>
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704 <chap>
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705 <p type="head">
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706 <s id="s.000215">PROPOSITIO XVII. PROBL. IX.<lb/> </s>
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707 </p>
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708 <subchap1>
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709 <p>
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710 <s id="s.000216">Dato plano declinante, super quo grave descendat, &amp;<lb/> dato alio plano minus declinante, in hoc reperire<lb/> punctum, quo perveniat mobile eo tempore, quo<lb/> pertransit dictum planum magis declinans.<lb/> </s>
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711 </p>
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712 </subchap1>
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713 <p>
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714 <s id="s.000217">Sint plana AB, AC quorum AC minus inclinatum.<lb/> </s>
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diff changeset
715 </p>
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diff changeset
716 <p>
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diff changeset
717 <s id="s.000218">Oportet in AC reperire punctum, quo grave perveniat,<lb/> quando pervenit in B.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
718 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
719 <p>
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diff changeset
720 <s id="s.000219">Fiat ut AC ad AB ita AB ad AD, &amp; dico D esse punctum<lb/> quaesitum. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
721 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
722 <p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
723 <s id="s.000220">Quoniam ut AC ad AD ita est quadratum AC ad quadra-<lb/>tum AB<arrow.to.target n="marg45"/>, &amp; ut AC ad AD ita quadratum temporis<lb/> AC ad quadratum temporis AD<arrow.to.target n="marg46"/>, ergo ut quadratum A<lb/>C ad quadratum AB, ita quadratum temporis AC ad<lb/> quadratum temporis AD<arrow.to.target n="marg47"/>, Unde AC ad AB ut tempus<lb/> AC ad tempus AD<arrow.to.target n="marg48"/>, sed ut AC ad AB, ita tempus A<lb/>C ad tempus AB<arrow.to.target n="marg49"/>, ergo tempora AB, AD, sunt aequa-<lb/>lia. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
724 <s id="s.000221">Quod, &amp;c. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
725 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
726 <p type="margin">
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diff changeset
727 <s id="s.000222"><margin.target id="marg45"/>Per 19.<lb/> sexti.<lb/></s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
728 <s id="s.000223"><margin.target id="marg46"/>Per cor.<lb/> 7. hujus.<lb/></s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
729 <s id="s.000224"><margin.target id="marg47"/>Per 11.<lb/> Quinti.<lb/></s>
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diff changeset
730 <s id="s.000225"><margin.target id="marg48"/>Per 22.<lb/> sexti.<lb/></s>
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diff changeset
731 <s id="s.000226"><margin.target id="marg49"/>Per 11.<lb/> hujus.<lb/></s>
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diff changeset
732 </p>
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diff changeset
733 </chap>
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diff changeset
734 <pb xlink:href="076/01/030.jpg"/>
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diff changeset
735 <chap>
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736 <p type="head">
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737 <s id="s.000227">PROPOSITIO XVIII. PROBL. X.<lb/> </s>
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diff changeset
738 </p>
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diff changeset
739 <subchap1>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
740 <p>
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diff changeset
741 <s id="s.000228">Datis planis declinantibus ortis ab eodem puncto, re-<lb/>perire in magis declinante punctum quo grave per-<lb/>veniat eo tempore, quo pertransit planum minus<lb/> declinans. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
742 </p>
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diff changeset
743 </subchap1>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
744 <p>
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diff changeset
745 <s id="s.000229">Datum sit planum minus declinans AC, &amp; magis A<lb/>D, terminantia super plano orizontali BD.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
746 </p>
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diff changeset
747 <p>
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diff changeset
748 <s id="s.000230">Oportet in AD producta reperire punctum, quo perveniat<lb/> grave eo tempore, quo pertransivit planum minus decli-<lb/>nans AC.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
749 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
750 <p>
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diff changeset
751 <s id="s.000231">Fiat ut AD ad AC ita AC ad dictam AD productam in<lb/> E, quod est punctum quaesitum.<lb/> </s>
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752 </p>
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753 <p>
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754 <s id="s.000232">Quoniam ut AE ad AD ita est quadratum AC ad quadra-<lb/>tum AD<arrow.to.target n="marg50"/>, sed AE ad AD est ut quadratum temporis<lb/> AE, ad quadratum temporis AD<arrow.to.target n="marg51"/>, ergo ut quadratum<lb/> AC ad quadratum AD, ita quadratum temporis AE ad<lb/> quadratum temporis AD<arrow.to.target n="marg52"/>, unde AC ad AD ut tempus<lb/> AE ad tempus AD<arrow.to.target n="marg53"/>, sed AC ad AD est ut tempus AC<lb/> ad tempus AD<arrow.to.target n="marg54"/>, ergo tempora AE, AC sunt aequalia.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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755 <s id="s.000233">Quod, &amp;c.<lb/> </s>
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756 </p>
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757 <p type="margin">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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758 <s id="s.000234"><margin.target id="marg50"/>Per 19.<lb/> sexti.<lb/></s>
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759 <s id="s.000235"><margin.target id="marg51"/>Per cor.<lb/> 7. hujus.<lb/></s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
760 <s id="s.000236"><margin.target id="marg52"/>Per 11.<lb/> Quinti.<lb/></s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
761 <s id="s.000237"><margin.target id="marg53"/>Per 22.<lb/> sexti.<lb/></s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
762 <s id="s.000238"><margin.target id="marg54"/>Per 11.<lb/> hujus.<lb/></s>
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763 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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764 </chap>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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765 <pb xlink:href="076/01/031.jpg"/>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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766 <chap>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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767 <p type="head">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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768 <s id="s.000239">PROPOSITIO XIX. PROBL. XI.<lb/> </s>
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769 </p>
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770 <subchap1>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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771 <p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
772 <s id="s.000240">Dato motus naturali gravis quomodocumque ad pun-<lb/>ctum datum, reperire seu in perpendiculari, seu in<lb/> plano quomodolibet inclinato punctum, a quo di-<lb/>gressum, perveniat ad idem punctum quo prius, tem-<lb/>pore aequali. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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773 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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774 </subchap1>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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775 <p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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776 <s id="s.000241">Sit AB linea quomodocumque aut perpendicularis, seu<lb/> planum inclinatum; super qua grave descendat in B, &amp;<lb/> data sit quaecumque linea BC, aut perpendicularis, aut<lb/> quomodolibet inclinata, quae cum AB, coeat in B.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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777 </p>
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778 <p>
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779 <s id="s.000242">Oportet in BC reperire punctum, a quo grave digressum per-<lb/>veniat in B tempore quo pervenit ab A in idem B.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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780 </p>
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781 <p>
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782 <s id="s.000243">Ducatur AC orizontalis, &amp; fiat BD tertia proportiona-<lb/>lis ad CB AB<arrow.to.target n="marg55"/>, &amp; D est punctum quaesitum. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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783 <s id="s.000244">Quod ut probetur.<lb/> </s>
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784 </p>
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785 <p type="margin">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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786 <s id="s.000245"><margin.target id="marg55"/>Per 11.<lb/> Sexti.<lb/></s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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787 </p>
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788 <p>
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789 <s id="s.000246">Fiat iterum rectae AC paralella, &amp; aequalis BE, &amp; ducta<lb/> EA, secetur recta BF parallela ipsi AD.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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790 </p>
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791 <p>
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792 <s id="s.000247">Quoniam AF, BD sunt pariter inclinatae, &amp; aequales<arrow.to.target n="marg56"/>, gra-<lb/>via per ipsas aequali tempore moventur<arrow.to.target n="marg57"/>, ergo aequali tem-<lb/>pore ut per AB<arrow.to.target n="marg58"/>, quod, &amp;c. </s>
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793 </p>
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794 <p type="margin">
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795 <s id="s.000248"><margin.target id="marg56"/>Per 33.<lb/> Primi.<lb/></s>
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796 <s id="s.000249"><margin.target id="marg57"/>Per 3.<lb/> pronun.<lb/></s>
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797 <s id="s.000250"><margin.target id="marg58"/>Per pr.<lb/> pron.<lb/></s>
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798 </p>
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799 </chap>
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800 <pb xlink:href="076/01/032.jpg"/>
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801 <chap>
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802 <p type="head">
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803 <s id="s.000251">PROPOSITIO XX. PROBL. XII.<lb/> </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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804 </p>
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805 <subchap1>
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806 <p>
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807 <s id="s.000252">Datis duobus planis diverse inclinatis longitudinis no-<lb/>tae; &amp; nota diuturnitate gravis moti super uno, re-<lb/>perire diuturnitatem si moveatur super alio.<lb/> </s>
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808 </p>
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809 </subchap1>
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810 <p>
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811 <s id="s.000253">Sint plana AB, CD inclinata, &amp; sit data diuturnitas E<lb/> plani AB.<lb/> </s>
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812 </p>
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813 <p>
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814 <s id="s.000254">Oportet reperire diuturnitatem plani CD.<lb/> </s>
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815 </p>
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816 <p>
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817 <s id="s.000255">Fiat AF, paralella, &amp; aequalis datae CD, in qua reperiatur<lb/> punctum G quo perveniat grave, tempore quo in B<arrow.to.target n="marg59"/>, unde<lb/> E est etiam diuturnitas spatii AG, quo dato, &amp; spatio<lb/> AF perquiratur eias diuturnitas, quae sit H<arrow.to.target n="marg60"/>, &amp; dico<lb/> H esse diuturnitatem quae grave descendit in CD.<lb/> </s>
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818 </p>
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819 <p type="margin">
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820 <s id="s.000256"><margin.target id="marg59"/>Per 17.<lb/> hujus.<lb/></s>
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821 <s id="s.000257"><margin.target id="marg60"/>Per 9.<lb/> hujus.<lb/></s>
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822 </p>
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823 <p>
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824 <s id="s.000258">Quoniam E, H sunt diuturnitates gravium descendentium<lb/> in AG, seu AB, &amp; AF, per constructionem, &amp; AF<lb/> est aequalis, &amp; paralella datae CD per constructionem,<lb/> sunt etiam E, H diuturnitates ipsarum AB, &amp; CD<arrow.to.target n="marg61"/>, unde<lb/> reperta est diuturnitas ipsius CD. </s>
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825 <s id="s.000259">Quod, &amp;c.<lb/> </s>
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826 </p>
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827 <p type="margin">
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828 <s id="s.000260"><margin.target id="marg61"/>Per 3.<lb/> pron.<lb/></s>
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829 </p>
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830 </chap>
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831 <pb xlink:href="076/01/033.jpg"/>
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832 <chap>
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833 <p type="head">
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834 <s id="s.000261">PROPOSITIO XXI. PROBL. XIII.<lb/> </s>
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835 </p>
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836 <subchap1>
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837 <p>
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838 <s id="s.000262">Datis duabus diuturnitatibus, quarum prior sit gravis<lb/> moti super plano dato longitudinis notae, &amp; dato<lb/> alio plano diversimode declinante; reperiendum est<lb/> in eo punctum, quo grave perveniat in secunda<lb/> diuturnitate data.<lb/> </s>
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839 </p>
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840 </subchap1>
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841 <p>
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842 <s id="s.000263">Dato plano declinante AB, super quo grave A movea<lb/>tur diuturnitate C, &amp; dato alio plano D declinationis<lb/> quae sit dissimilis declinationi datae AB; data itidem diu-<lb/>turnitate E.<lb/> </s>
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843 </p>
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844 <p>
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845 <s id="s.000264">Oportet reperire in D punctum quo grave perveniat in<lb/> diuturnitate E.<lb/> </s>
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846 </p>
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847 <p>
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848 <s id="s.000265">Ducatur AF parallela ipsi D, in eaque reperiatur pun-<lb/>ctum F, quo grave perveniat tempore quo in B<arrow.to.target n="marg62"/>, &amp; prae-<lb/>scribatur in eadem spatium AG per quod moveatur in<lb/> diuturnitate E<arrow.to.target n="marg63"/>, &amp; fiat DH aequalis ipsi AG, &amp; dico H<lb/> esse punctum quaesitum.<lb/> </s>
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849 </p>
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850 <p type="margin">
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851 <s id="s.000266"><margin.target id="marg62"/>Per 17.<lb/> hujus.<lb/></s>
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852 <s id="s.000267"><margin.target id="marg63"/>Per 8.<lb/> hujus.<lb/></s>
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853 </p>
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854 <p>
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855 <s id="s.000268">Quoniam diuturnitates in AB, AF sunt aequales per con-<lb/>structionem, &amp; C, E sunt diuturnitates super planis AF,<lb/> AG per constructionem, sunt etiam diuturnitates super<lb/> AB, AG, &amp; proinde super DH ipsi AG aequali, &amp;<lb/> paralellae, quod, &amp;c.<lb/> </s>
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856 </p>
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857 </chap>
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858 <pb xlink:href="076/01/034.jpg"/>
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859 <chap>
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860 <p type="head">
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861 <s id="s.000269">PROPOSITIO XXII.<lb/> </s>
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862 </p>
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863 <subchap1>
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864 <p>
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865 <s id="s.000270">Si duo gravia descendunt alterum quidem perpendicu-<lb/>lariter, alterum vero super plano declinante, perve-<lb/>niunt ad idem planum Orizontale tali ratione, ut sit<lb/> eadem proportio inter diuturnitates eorum, quae in-<lb/>ter perpendicularem, &amp; declinantem.<lb/> </s>
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866 </p>
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867 </subchap1>
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868 <p>
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869 <s id="s.000271">Sit linea AB perpendiculariter erecta super plano Ori-<lb/>zontali BC, &amp; AC planum declinans.<lb/> </s>
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870 </p>
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871 <p>
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872 <s id="s.000272">Dico quod diuturnitates gravium descendentium per AB, &amp;<lb/> per AC, sunt ut AB ad AC.<lb/> </s>
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873 </p>
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874 <p>
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875 <s id="s.000273">Ducatur BD normalis ad AC.<lb/> </s>
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876 </p>
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877 <p>
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878 <s id="s.000274">Quoniam est ut AD ad AC ita quadratum temporis AD<lb/> ad quadratum temporis AC<arrow.to.target n="marg64"/>, &amp; tempora AD, &amp; AB<lb/> sunt aequalia<arrow.to.target n="marg65"/>, &amp; proinde eorum quadrata<arrow.to.target n="marg66"/>, ergo ut A<lb/>D, ad AC ita quadratum temporis AB ad quadratum tem-<lb/>poris AC, sed ut AD ad AC ita quadratum AB ad qua-<lb/>dratum AC<arrow.to.target n="marg67"/>, ergo ut quadratum temporis AB ad qua-<lb/>dratum temporis AC, ita quadratum AB ad quadratum<lb/> AC<arrow.to.target n="marg68"/>, sed quia latera sunt inter se ut eorum quadrata<arrow.to.target n="marg69"/>, est<lb/> ut AB ad AC ita tempus AB ad tempus AC. </s>
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879 <s id="s.000275">Quod, &amp;c.<lb/> </s>
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880 </p>
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881 <p type="margin">
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882 <s id="s.000276"><margin.target id="marg64"/>Per cor.<lb/> 7. hujus.<lb/></s>
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883 <s id="s.000277"><margin.target id="marg65"/>Per 15.<lb/> hujus.<lb/></s>
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884 <s id="s.000278"><margin.target id="marg66"/>Per 2.<lb/> pron.<lb/></s>
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885 <s id="s.000279"><margin.target id="marg67"/>Per 19.<lb/> Sexti.<lb/></s>
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886 <s id="s.000280"><margin.target id="marg68"/>Per 22.<lb/> Quinti.<lb/></s>
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887 <s id="s.000281"><margin.target id="marg69"/>Per 24.<lb/> Sexti.<lb/></s>
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888 </p>
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889 </chap>
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890 <pb xlink:href="076/01/035.jpg"/>
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891 <chap>
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892 <p type="head">
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893 <s id="s.000282">PROPOSITIO XXIII.<lb/> </s>
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894 </p>
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895 <subchap1>
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896 <p>
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897 <s id="s.000283">Duo gravia descendentia super planis diversa ratione<lb/> declinantibus, perveniunt ad idem planum orizon-<lb/>tale ea ratione, ut sit eadem proportio inter diutur-<lb/>nitates, quae inter dicta plana si ab eodem puncto ad<lb/> idem planum orizontale producta sint.<lb/> </s>
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898 </p>
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899 </subchap1>
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900 <p>
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901 <s id="s.000284">Datis planis AB, AC declinantibus, ductis ab eodem<lb/> puncto A ad planum orizontale BC.<lb/> </s>
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902 </p>
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903 <p>
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904 <s id="s.000285">Dico quod diuturnitates gravium descendentium per AB, AC<lb/> sint ut AB ad AC.<lb/> </s>
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905 </p>
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906 <p>
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907 <s id="s.000286">Fiat ut AC ad AB ita AB ad AD, ita ut grave perveniat<lb/> in D eodem tempore quo pervenit in B<arrow.to.target n="marg70"/>.<lb/> </s>
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908 </p>
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909 <p type="margin">
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910 <s id="s.000287"><margin.target id="marg70"/>Per 13.<lb/> hujus.<lb/></s>
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911 </p>
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912 <p>
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913 <s id="s.000288">Quoniam est ut AD ad AC, ita quadratum temporis AD<lb/> ad quadratum temporis AC<arrow.to.target n="marg71"/>, &amp; tempora AD, AB<lb/> sunt aequalia<arrow.to.target n="marg72"/>, &amp; proinde eorum quadrata; ergo ut AD<lb/> ad AC ita quadratum temporis AB, ad quadratum tem-<lb/>poris AC<arrow.to.target n="marg73"/>, sed ut AD ad AC, ita quadratum AB ad qua-<lb/>dratum AC<arrow.to.target n="marg74"/>, ergo ut quadratum temporis AB ad quadra-<lb/>tum temporis AC, ita quadratum AB ad quadratum AC,<lb/> ergo ut tempus AB ad tempus AC, ita AB ad AC<arrow.to.target n="marg75"/>.<lb/> </s>
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diff changeset
914 <s id="s.000289">Quod fuit probandum.<lb/> </s>
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915 </p>
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916 <p type="margin">
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917 <s id="s.000290"><margin.target id="marg71"/>Per Cor.<lb/> 7. hujus.<lb/></s>
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918 <s id="s.000291"><margin.target id="marg72"/>Per 17.<lb/> hujus.<lb/></s>
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919 <s id="s.000292"><margin.target id="marg73"/>Per 2.<lb/> pronun.<lb/></s>
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920 <s id="s.000293"><margin.target id="marg74"/>Per 19.<lb/> sexti.<lb/></s>
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921 <s id="s.000294"><margin.target id="marg75"/>Per 22.<lb/> sexti.<lb/></s>
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922 </p>
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923 </chap>
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924 <pb xlink:href="076/01/036.jpg"/>
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925 <chap>
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926 <p type="head">
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927 <s id="s.000295">PROPOSITIO XXIV<lb/></s>
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928 </p>
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929 <subchap1>
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930 <p>
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931 <s id="s.000296">Datis planis, &amp; perpendiculari ad eadem linea orizon-<lb/>tali egressis, quae coeant infra in eodem puncto, gra-<lb/>via super ipsis mota procedunt ea ratione, ut sit ea-<lb/>dem proportion inter diuturnitates, quae inter longi-<lb/>tudines planorum, &amp; dictam perpendicularem.<lb/> </s>
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932 </p>
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933 </subchap1>
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934 <p>
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935 <s id="s.000297">Data sit linea orizontalis AB, in qua initium sumant<lb/> plana declinantia AC, DC, nec non perpendicula-<lb/>ris BC coeuntia in puncto C.<lb/> </s>
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936 </p>
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937 <p>
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938 <s id="s.000298">Dico quod diuturnitates gravium super ipsis motorum, sunt<lb/> ut AC, DC, BC.<lb/> </s>
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939 </p>
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940 <p>
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941 <s id="s.000299">Ducatur CE paralella ipsi AB, &amp; a puncto A ducantur<lb/> paralellae ipsis CB, CD, &amp; sint AE, AF.<lb/> </s>
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942 </p>
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943 <p>
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944 <s id="s.000300">Quoniam diuturnitates super planis AF, AC, sunt ut A<lb/>F, AC<arrow.to.target n="marg76"/>, &amp; super planis eisdem, &amp; perpendiculari A<lb/>E, sunt ut AF, seu AC ad AE<arrow.to.target n="marg77"/>, &amp; AE, AF sunt paralellae<lb/> ipsis CD, CB, &amp; eisdem aequales<arrow.to.target n="marg78"/>, sequitur quod etiam<lb/> super AC, DC, BC diuturnitates sunt iuxta propor-<lb/>tiones longitudinum<arrow.to.target n="marg79"/>, Quod probandum fuit.<lb/> </s>
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945 </p>
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946 <p type="margin">
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947 <s id="s.000301"><margin.target id="marg76"/>Per 23.<lb/> hujus.<lb/></s>
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948 <s id="s.000302"><margin.target id="marg77"/>Per 15.<lb/> hujus.<lb/></s>
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949 <s id="s.000303"><margin.target id="marg78"/>Per 33.<lb/> prim.<lb/></s>
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950 <s id="s.000304"><margin.target id="marg79"/>Per 3.<lb/> pron.<lb/></s>
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951 </p>
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952 </chap>
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953 <pb xlink:href="076/01/037.jpg"/>
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954 <chap>
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955 <p type="head">
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956 <s id="s.000305">PROPOSITIO XXV. </s>
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957 </p>
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958 <subchap1>
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959 <p>
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960 <s id="s.000306">In circulo Orthogonaliter erecto, si a summitate ad<lb/> puncta peripheriae ducantur plana, quo tempore gra-<lb/>ve perpendiculariter inde pervenit ad planum ori-<lb/>zontale; si descendat per dicta plana, eodem perve-<lb/>niet respective ad quodlibet dictorum punctorum<lb/> peripheriae.<lb/> </s>
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961 </p>
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962 </subchap1>
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963 <p>
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964 <s id="s.000307">Sit circulus cuius centrum B, &amp; diameter AC erectus<lb/> super plano orizontali GC, &amp; in eo ducta sint plana de-<lb/>clinantia a puncto A ad puncta peripheriae DEF, &amp; de-<lb/>scendant gravia super dicta plana, &amp; perpendiculariter.<lb/> </s>
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965 </p>
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966 <p>
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967 <s id="s.000308">Dico quod eodem tempore pervenient ad, D, E, F, C.<lb/> </s>
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968 </p>
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969 <p>
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970 <s id="s.000309">Ducantur DC, EC, FC.<lb/> </s>
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971 </p>
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972 <p>
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973 <s id="s.000310">Quoniam puncta praedicta sunt ea, in quae cadunt perpendi-<lb/>cularia ducta a puncto C in AD, AE, AF<arrow.to.target n="marg80"/>, eo perveniunt<lb/> gravia eodem tempore quo in C<arrow.to.target n="marg81"/>. </s>
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974 <s id="s.000311">Quod probandum fuit.<lb/> </s>
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975 </p>
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976 <p type="margin">
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977 <s id="s.000312"><margin.target id="marg80"/>Per 30.<lb/> Tertii.<lb/></s>
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978 <s id="s.000313"><margin.target id="marg81"/>Per 16.<lb/> hujus.<lb/></s>
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979 </p>
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980 </chap>
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981 <pb xlink:href="076/01/038.jpg"/>
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982 <chap>
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983 <p type="head">
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984 <s id="s.000314">PROPOSITIO XXVI.<lb/> </s>
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985 </p>
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986 <subchap1>
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987 <p>
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988 <s id="s.000315">Si in circulo erecto, a puncto inferiori ducantur plana<lb/> ad puncta peripheriae, &amp; a dictis punctis descendant<lb/> gravia super dicta plana eodem tempore quo a puncto su-<lb/>premo descendit aliud grave perpendiculariter; perve-<lb/>nient omnia eodem instanti ad dictum punctum inferius.<lb/> </s>
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989 </p>
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990 </subchap1>
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991 <p>
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992 <s id="s.000316">Sit circulus cuius diameter ABC erectus super plano<lb/> orizontali, quod tangat in C, &amp; a C ducantur plana C<lb/>D, CE, &amp; a punctis, E, D gravia descendant super dicta<lb/> plana, nec non, &amp; a puncto supremo A perpendiculariter.<lb/> </s>
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993 </p>
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994 <p>
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995 <s id="s.000317">Dico quod eodem tempore perveniunt in C.<lb/> </s>
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996 </p>
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997 <p>
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998 <s id="s.000318">A puncto A ducantur AF, AG paralellae ipsis CE, CD,<lb/> &amp; ducantur AF, FC.<lb/> </s>
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999 </p>
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1000 <p>
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1001 <s id="s.000319">Quoniam in triangulis AEC, AFC anguli alterni FAC,<lb/> ACE sint aequales,<arrow.to.target n="marg82"/>, &amp; anguli AFC, AEC sunt etiam<lb/> aequales puta recti<arrow.to.target n="marg83"/>, &amp; basis AC communis, Triangula<lb/> sunt aequalia<arrow.to.target n="marg84"/>, &amp; proinde AF est aequalis CE, quod idem<lb/> probabitur de reliquis, ergo cum AF, CE, &amp; reliquae<lb/> sint paralellae, &amp; aequales, gravia per CE, CD perve-<lb/>nient in C eodem tempore, quo digressa ab A perveniunt<lb/> ad puncta FG, sed haec eodem tempore quo perpendicularis<lb/> pervenit in C<arrow.to.target n="marg85"/>, ergo etiam ea quae per CE, CD. </s>
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1002 <s id="s.000320">Quod, &amp;c.<lb/> </s>
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1003 </p>
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1004 <p type="margin">
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1005 <s id="s.000321"><margin.target id="marg82"/>Per 29.<lb/> primi.<lb/></s>
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1006 <s id="s.000322"><margin.target id="marg83"/>Per 30.<lb/> Tertii.<lb/></s>
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1007 <s id="s.000323"><margin.target id="marg84"/>Per 26.<lb/> primi.<lb/></s>
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1008 <s id="s.000324"><margin.target id="marg85"/>Per 25.<lb/> hujus.<lb/></s>
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1009 </p>
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1010 </chap>
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1011 <pb xlink:href="076/01/039.jpg"/>
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1012 <chap>
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1013 <p type="head">
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1014 <s id="s.000325">POSTULATUM VII<lb/></s>
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1015 </p>
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1016 <p>
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1017 <s id="s.000326">Ductis planis inclinatis, &amp; linea perpendiculari in-<lb/>ter binas paralellas orizontales, Gravia super illis<lb/> mota ubi perveniunt ad paralellam inferiorem ha-<lb/>bent aequales velocitatis gradus; &amp; proinde si ab inde<lb/> infra sortiantur parem inclinationem, aequeveloci-<lb/>ter moventur.<lb/> </s>
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1018 </p>
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1019 <p>
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1020 <s id="s.000327">Videtur probabile. </s>
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1021 <s id="s.000328">Primo quia si diuturnitates sunt longitu-<lb/>dinibus proportionales, ut propositione 22. huius probatum<lb/> fuit, credibile est motus in fine esse aequales.<lb/> </s>
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1022 </p>
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1023 <p>
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1024 <s id="s.000329">Secundo. Argumento ducto ab experientia pendulorum, quae<lb/> quantumvis longiora, aut breviora, &amp; proinde circa fi-<lb/>nem magis, aut minus inclinata, pariter ascendunt, si pa-<lb/>riter descendant.<lb/> </s>
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1025 </p>
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1026 <p>
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1027 <s id="s.000330">Tertio. Quia videmus aquam per siphones rectos, sive obli-<lb/>quos, seu inclinatos ductam, pariter ascendere, si pariter<lb/> descendat. </s>
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1028 <s id="s.000331">Ceterum fateor minorem evidentiam hoc po-<lb/>stulatum caeteris praemissis prae se ferre, quae fuit causa quod<lb/> illud, ut in praefatione, segregaverim, &amp; sequentia, alia<lb/> methodo, tangendo fere tantummodo exposuerim, &amp; a<lb/> pluribus aliis propositionibus, quae hinc deduci facile pos-<lb/>sent, data opera abstinuerim. </s>
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1029 </p>
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1030 </chap>
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1031 <pb xlink:href="076/01/040.jpg"/>
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1032 <chap>
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1033 <p type="head">
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1034 <s id="s.000332">PROPOSITIO XXVII. PROBL. XIV.<lb/> </s>
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1035 </p>
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1036 <subchap1>
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1037 <p>
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1038 <s id="s.000333">Dato gravi moto perpendiculariter per spatium datum<lb/> diuturnitate data, quod perficiat motum super plano<lb/> inclinato per spatium itidem datum; perquirere in<lb/> ipso diuturnitatem.<lb/> </s>
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1039 </p>
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1040 </subchap1>
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1041 <p>
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1042 <s id="s.000334">Moveatur grave A perpendiculariter per spatium AB<lb/> diuturnitate C, &amp; perseveret in motu super spatio B<lb/>D in plano inclinato BD.<lb/> </s>
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1043 </p>
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1044 <p>
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1045 <s id="s.000335">Venanda est diuturnitas eius in ipso BD.<lb/> </s>
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1046 </p>
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1047 <p>
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1048 <s id="s.000336">Producatur DB donec concurrat cum AE orizontaliter du-<lb/>cta ab A in E, &amp; fiat ut AB ad EB, ita diuturnitas C ad<lb/> diuturnitatem G, quae idcirco erit diuturnitas ipsius EB<arrow.to.target n="marg86"/>,<lb/> &amp; sit H quadratum diuturnitatis G, &amp; fiat ut EB ad ED,<lb/> ita quadratum H ad aliud quod sit I a cuius latere K, quod<lb/> est diuturnitas ipsius ED, ablata KL aequali G, erit LM<lb/> reliquum diuturnitas BD quaesita.<lb/> </s>
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1049 </p>
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1050 <p type="margin">
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1051 <s id="s.000337"><margin.target id="marg86"/>Per 22.<lb/> hujus.<lb/></s>
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1052 </p>
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1053 <pb xlink:href="076/01/041.jpg"/>
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1054 <p>
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1055 <s id="s.000338">Quoniam notum est triangulum AEB, cum notus sit angu-<lb/>lus AEB aequalis alterno EDF inclinationis notae, &amp; E<lb/>AB rectus ex constructione, &amp; notum latus AB ex hypo-<lb/>tesi, notum erit etiam latus EB, &amp; quia diuturnitas in pla-<lb/>no BD est eadem ac si motus antecedens esset per EB<arrow.to.target n="marg87"/>, EB,<lb/> &amp; ED sunt in duplicata ratione diuturnitatum G, K ex con-<lb/>structione; unde a K deducta KL aequali G ex constructio-<lb/>ne, remanet LM diuturnitas BD. </s>
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1056 <s id="s.000339">Quod, &amp;c.<lb/> </s>
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1057 </p>
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1058 <p type="margin">
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1059 <s id="s.000340"><margin.target id="marg87"/>Per 7.<lb/> post.<lb/></s>
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1060 </p>
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1061 <p>
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1062 <s id="s.000341">Inde sequitur quod summa diuturnitatum C, &amp; LM, est diutur-<lb/>nitas totius ABD.<lb/> </s>
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1063 </p>
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1064 <p>
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1065 <s id="s.000342">Eadem operatione pariter reperietur diuturnitas BD si BD<lb/> sit perpendicularis, &amp; AB inclinata.<lb/> </s>
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1066 </p>
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1067 <p>
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1068 <s id="s.000343">Item si ambo sint plana inclinata.<lb/> </s>
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1069 </p>
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1070 <p>
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1071 <s id="s.000344">Ducta AD facile reperietur diuturnitas in ipsa si fiat ut ED<lb/> ad AD, ita K ad aliud per 21. hujus.<lb/> </s>
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1072 </p>
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1073 <pb xlink:href="076/01/042.jpg"/>
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1074 <p>
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1075 <s id="s.000345">Ducto alio plano puta DN, reperietur eius diu-<lb/>turnitas. </s>
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1076 </p>
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1077 <p>
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1078 <s id="s.000346">Si fiat ut ED ad OD ita diuturnitas ipsius ED puta L ad diu-<lb/>turnitatem OD, quae sit P, deinde ut OD ad ON ita<lb/> quadratum diuturnitatis P ad aliud quadratum, cuius Ra-<lb/>dix erit diuturnitas ipsius DN.<lb/> </s>
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1079 </p>
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1080 <p>
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1081 <s id="s.000347">Ex his patet quod si addantur plura plana eadem ratione re-<lb/>perientur eius diuturnitates.<lb/> </s>
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1082 </p>
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1083 <pb xlink:href="076/01/043.jpg"/>
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1084 <p>
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1085 <s id="s.000348">Ex his etiam patet quod si in circulo dentur plura, plana verbi<lb/> gratia AB, BC, CD, DE, &amp; data sit diuturnitas super dia-<lb/>metro, dabitur etiam diuturnitas cuiusvis dicto-<lb/>rum AB, BC, CD, DE, &amp; etiam omnium simul.<lb/> </s>
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1086 </p>
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1087 <p>
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1088 <s id="s.000349">Ex his facile etiam cognoscere poteris esse breviorem, diu-<lb/>turnitatem per A, B, C, C, D, E quam per AE.<lb/> </s>
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1089 </p>
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1090 <p>
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1091 <s id="s.000350">
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1092 FINIS.
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1093 <lb/></s>
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1094 </p>
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1095 </chap>
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1096 </body>
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1097 <back/>
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1098 </text>
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1099 </archimedes>