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DESpecs 2.0 Autumn 2009
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:14:40 +0200
parents 22d6a63640c6
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1 <pb id="p.0001">
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2 <HEAD>ILLVSTRI ET IN
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3 PRIMIS GENEROSO DOMINO,
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4 DOMINO CAROLO BARONI
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5 Azerotin, Name$tij Ro$icij
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6 Brandei$ij Domino,
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7 MICH. VARRO. S. P. D.</HEAD>
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8 <p><I>CVM ab Ineunte &aelig;tate in Ma-
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9 thematicis me exercui$$em, $em-
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10 per animo meo in h&aelig;$it hoc de$ide-
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11 rium, vt qu&aelig; de Archimede Syra-
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12 cu$ano referuntur, ea &amp; ratione
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13 demon$trare &amp; experimento comprobare po$$em.
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14 Cumque multa in eo genere meditatus e$$em, po$t-
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15 quam ad iuris ciuilis $tudium tran$ii, ac tandem
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16 ad Rempub. nunquam mihi tantum otii contigit,
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17 vt ea in ordinem redigere potuerim. At cum pau-
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18 cis retr&ograve; annis $atis longam peregrinationem per
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19 Sarmatiam $u$cepi$$em, e$$etque vacuus curis a-</I>
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20 <foot><I>A 2</I></foot>
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21 <pb>
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22 <I>nimus, atque ad earum rerum meditationem me
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23 impelleret genius meus, c&oelig;pi aliquid de his $cribe-
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24 re, quantum ip$e iter faciendo meo marte a$$equi
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25 potui cum libris de$tituerer. Scriptum illud cum
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26 in maxim&egrave; ardua &amp; diffic&igrave;li contemplatione ver||
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27 $etur, nec ab eo tempore ex quo ad$olitas occupa-
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28 tiones redii ei extremam manum apponere mihi
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29 licuerit, decreueram inter priuatas meas mu$as
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30 a$$eruare Verebar enim vulgi iudici&utilde;, (cuius ta-
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31 m&etilde; rationem viro politico habend&atilde; e$$e cen$eo) ne-
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32 que $olum vulgi, verum etiam eor&utilde; qui doctorum
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33 nomine gaudent, cum inaudita qu&aelig;dam interdum
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34 qu&aelig;que ab eorum opinione recedere vide&atilde;tur at-
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35 tingam. Cupieb&atilde; etiam in eo argumento vlteri&ugrave;s
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36 progredi, ant&egrave; quam quidquam ederem, atque ea
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37 qu&aelig; in meis aduer$ariis ea de re $par$a habeo, iis
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38 qu&aelig; conge$$er&atilde; $ubiicere, vt iu$tum volum&etilde; fieret.</I>
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39 <p><I>Po$tquam ver&ograve; te di$ce$$um ex hac nostra ci-
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40 uitate parare intellexi, in qua dum ver$atus es.
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41 non $ol&ugrave;m ei ornam&etilde;to fui$ti, $ed etiam optimos
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42 quo$que virtutum tuarum admiratione tibi con-
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43 ciliasti: mihi ver&ograve; peculiariter innumera beneuo-</I>
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44 <pb>
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45 <I>lenti&aelig; $igna exhibui$ti, non potui tantum virum,
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46 me&iacute;que am&atilde;tem $ine aliquo munere aut</I> <G>mnhmosu/nw|</G>
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47 <I>dimittere. Itaque libellum hunc tibi dicare, &amp;
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48 $ub tuo nomine in publicum edere $tatui. Squalli-
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49 dus quidem prodit, &amp; rudis, mole etiam perexi-
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50 guus: $plendidius aliquid &amp; magis elaboratum
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51 postularet tua dignitas. Atqui ea e$t argumenti
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52 illius magnitudo, vt in eo aliquid volui$$e $atis
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53 $it. Militare $an&egrave; munus e$t, e&oacute;que nomine tibi
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54 conueniens: $i quis enim hac cognitione in$tructus
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55 fuerit, non min&ugrave;s ea in re bellica vti poterit,
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56 quam Archimedes in Syracu$is defendendis
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57 aduer$us Marcum Marcellum v$us est. E&aacute;-
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58 que mirari de$inet qu&aelig; de eo Plutarchus nar-
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59 rat. Id lucidius dem&otilde;$trare potui$$em, $i mihi h&aelig;c
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60 vlterius per$equendi otium fui$$et: Jeiuna enim
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61 per $e videntur, ni$i quis $uccum qui $ub corti-
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62 ce latet, eliciat. Malui tamen hoc veluti pro-
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63 gymna$mate alios quibus plus est otii &amp; in-
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64 genii ad eiu$modi inqui$itionem hortari &amp; ac-
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65 cendere, qu&agrave;m pr&aelig;claram illam cognitio-
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66 nem diutius $epultam relinquere cum ne-</I>
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67 <pb>
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68 <I>minem hodie animaduertam qui ei in$udet,
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69 neque mihi $pes vlla affulgeat diuturni otii, cuius
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70 ope in hoc $tadio pergere po&szlig;im: Qu&ograve;d $i fort&egrave;
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71 mihi aliquando contingat, riuos plurimos ex his
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72 fontibus me educturum; c&otilde;fido; quibus multum
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73 c&otilde;modit&agrave;tis rebus humanis accedat. Qu&aelig; enim
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74 hic traduntur, ad motus omnes atque ad omnis ge||
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75 neris virium comparationes accommodari po$-
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76 $unt ad fines propo$itos a$$equendos. Quod qu&agrave;m
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77 lat&egrave; pateat, dici non pote$t. E$t enim eorum v$us
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78 non $ol&ugrave;m in mechanicis, in quibus tamen e$t ma||
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79 ximus, $ed etiam in politicis &amp; &oelig;conomicis: $unt
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80 enim in illis motus, $unt vires &amp; re$istenti&aelig;. In
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81 arte medica &amp; reliquis phy$ices partibus pra-
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82 cticis, quantus $it, nemo fando exprimere po&szlig;it.
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83 Quia vero ea qu&aelig; praxim docent gratiora $unt
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84 ip$a contemplatione, &amp; cau$&aelig; propter effecta in-
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85 quirantur,</I> <G>*a)lfh/stas</G> <I>omnes hortor vt quod ego in-
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86 numeris aliis cur is implicitus addere non po$-
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87 $um, ip$i addant. Abundat nunc Europa pr&aelig;-
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88 $tantibus ingeniis, $i M&oelig;cenates ade$$ent.
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89 Ea autem qu&aelig; pr&aelig;ter h&aelig;c meditatus $um, ac fer-</I>
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90 <pb>
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91 <I>m&egrave; parata habeo, h&aelig;c $unt. Tractatus de iactu.
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92 De continuitate eiu$que $olutione. De c&otilde;den$a-
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93 tione &amp; rarefactione earumque cau$is &amp; effecti-
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94 bus: It&etilde; in Mechanicis, tractatus de variis ma-
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95 chinis ad motus ci&etilde;dos, ac de perfecti&szlig;im&aelig; cuiu$-
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96 que ad id quod propo$itum fuerit moliendum in-
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97 ue$tigatione. Nonnulla eti&atilde; de Rerumpublicar&utilde;
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98 motu tum interno t&utilde; externo notaui, qu&aelig; eod&etilde; or-
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99 dine tradere optarem: quorum omnium principia
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100 hic $i quis diligenter animaduertat tradita $unt.
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101 Si quis mihi in his palm&atilde; pr&aelig;ripuerit, meque an-
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102 teuerterit ei maximas gratias agam. Sunt tamen
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103 inter h&aelig;c qu&aelig;dam qu&aelig; vulg&ograve; pandere nephas est.
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104 Itaque theori&aelig; magis in$i$tendum puto, in qua $i
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105 quis exercitatus fuerit, nullo negotio illam in opus
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106 educere poterit, idque $ine periculo fiet, cum vul-
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107 go non pateat. Alioqui pericul&utilde; e$t, ne $i particu-
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108 laria tradantur iis contenti homines, vt fieri $olet
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109 vniuer$alem cognitionem &amp; cau$arum inqui$i-
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110 tionem negligant, pereatque $cientia. Eaque de
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111 cau$a nihil quicquam de his qu&aelig; fecerat Archi-</I>
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112 <pb>
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113 <I>medes $criptum relinquere voluit. Exi$tim&atilde;s eos
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114 qui in iis qu&aelig;</I> <G>qewrhtikw=s</G> <I>tradiderat, diligentem ope-
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115 ram ponere vell&etilde;t, mult&ograve; maiora quoties opus fo-
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116 ret pr&aelig;$tituros. Equidem mihi per$uadeo non de-
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117 futuros qui varia iudicia de hoc no$tro $cripto fe-
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118 rant &amp; qua$i nouum athletam in aren&agrave;m pro-
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119 deuntem mirentur. Quibus vno verbo re$pon-
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120 $um volo, me nullius vnquam in verba ma-
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121 gistri iura$$e, $ed liberrim&egrave; $emper philo$o-
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122 phatum e$$e: ita vt etiam in principia ab aliis
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123 $tatuta animaduertere mihi licere putauerim.
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124 Quod e&ograve; liberi&ugrave;s feci, po$tquam magnam part&etilde;
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125 vulgo receptarum opinionum fal$am e$$e re ip$a
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126 deprehendi. Omnibus placuero, $i tibi placuero. Si
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127 quid in his ob$curius fuerit, habes clari&szlig;im&utilde; vi-
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128 rum Wence$laum lauinium tuum, ab$tru$ioris
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129 philo$ophi&aelig; indagatorem $umm&utilde;, qui tibi omnia
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130 explicabit. Hoc igitur munu$cul&utilde; vt $erena fr&otilde;-
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131 te $u$cipias rogo. Vale. V I. K al. Jun. Anno
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132 Christi Domini</I> M. D. LXXXIV.
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133 <pb n="1">
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134 <fig>
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135 <HEAD>M. VARRONIS DE
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136 MOTV TRACTATVS.</HEAD>
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137 <HEAD><I>PROBLEMA.</I></HEAD>
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138 <HEAD><I>Data vi datum pondus mouere.</I></HEAD>
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139 <p>Hoc problema prima quidem
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140 fronte mirabile videtur, vt $cilicet
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141 pondus vel maximum viribus ta-
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142 men minimis, vt puta $i dicas vnius
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143 hominis, vel etiam imbecillioris a-
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144 licuius animalis viribus totum ter-
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145 r&aelig; globum loco moueri po$$e. Quod Archimedem
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146 Syracu$anum iacta$$e referunt hoc dicto, <G>do\s pou= stw=|
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147 ta\n ga\n kinw=|</G>. Si quis tamen in Geometrico puluere
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148 ver$atus fuerit, pre$ertim ver&ograve; in iis qu&aelig; ab ip$o in
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149 $uo <G>i)soropikw=n</G> libello trad&utilde;tur. Illud c&aelig;ter&aacute;que eiu$-
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150 modi, ludicra Geometri&aelig;, vt &amp; ip$e facere $olitus di||
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151 citur, appellabit. Ex huius autem problematis expli||
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152 catione, omnium machinarum qu<17> ad motus cien-
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153 dos excogitari po$$unt, ratio pendet. Et$i ver&ograve; tot&utilde;
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154 Geometrica demon$tratione expediri po$$it, quo-
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155 <foot>B</foot>
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156 <pb n="2">
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157 niam tamen h<17>c con$ideratio, qu&aelig; Gr&aelig;cis <G>staqmikh\</G> di||
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158 citur Geometri&aelig; tant&ugrave;m e$t <G>u(pa/llhlos</G> mixta $cili-
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159 cet ex phy$ica &amp; Geometrica eo qu&ograve;d illius $ubie-
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160 ctum $it motus. Ide&ograve; paul&ograve; cra$$iori Minerua &amp;
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161 pr&aelig;ter Geometricam $implicitatem <G>fusikw=s</G> etiam
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162 tractanda e$t: alioqui Archimedicis demon$tratio-
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163 nibus $tandum e$$et, qu&aelig; in eo manc&aelig; $unt, qu&ograve;d il-
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164 l&aelig; propo$itiones qu&aelig; ex phy$icis peti debent, inde-
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165 mon$trat&aelig; manent, nec explicantur, $ed pro confe$-
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166 $is principiorum loco po$tulantur. Eas igitur, qua-
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167 tenus ad huius problematis explicationem faciunt,
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168 hic di$cutiemus. Neque enim Euclidis libr&utilde; de gra-
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169 ui &amp; leui, in quo hoc argumentum per$equi volui$-
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170 $e videtur, integrum habemus. Ari$toteles ver&ograve; in
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171 eo quod ab ip$o $criptum extat de Mechanicis fra-
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172 gmento pr<17>ter $uum morem, c&ugrave;m alio qui in omni-
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173 bus exacti$$imus $it, hanc qu&aelig;$tionem poti&ugrave;s nota-
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174 uit qu&agrave;m explicuit in $exto &amp; $eptimo phy$icorum
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175 libro multa pr&aelig;termi$it. Vt igitur ad rem aggredia-
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176 mur, prim&ugrave;m voces, quibus vtendum e$t, definie-
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177 mus, vt intelligatur quo $en$u eas accipiamus.
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178 <HEAD>DEFINITIO I.</HEAD>
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179 <p>Vis dicitur agendi aut agenti re$i$tendi pot&etilde;tia,
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180 pr&aelig;$ertim ver&ograve; mouendi &amp; mouenti re$i$tendi.
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181 <HEAD>II.</HEAD>
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182 <p>Vis $ubiectum dicitur id quod vis mouet, vel
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183 <pb n="3">
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184 quod &agrave; vi mouetur.
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185 <p>H&icirc;c non agimus de vi primaria qu&aelig; virium o-
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186 mnium principium e$t, mou&eacute;tque omnia, nec tam&etilde;
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187 mouetur, $ed de ea, qu<17> dum mouet, eti&atilde; cum $ub-
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188 iecto, cui ine$t, mouetur.
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189 <p>Subiecta ver&ograve; qu&aelig; nullam vim habent, nec mo-
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190 uentur, nec mouenti re$i$t&utilde;t: $i qua ver&ograve; vis illis ac-
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191 cedat, tum ab ea $e moueri patiuntur.
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192 <p>Et$i autem plura $int virium genera, tot $cilicet,
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193 quot $unt in rerum natura contrariorum, actionem
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194 &amp; pa$$ionem recipientium, vt leue graue, rarum d&etilde;-
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195 $um, plenum vacuum, durum molle, &amp; c&aelig;tera eiu$-
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196 modi, quoniam tamen ea omnia h&icirc;c per$equi no$tri
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197 non e$t in$tituti, c&ugrave;m de ea tant&ugrave;m qua motus fit a-
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198 gere $tatuerimus.
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199 <HEAD>III.</HEAD>
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200 <p>C&ugrave;m de motu h&icirc;c agemus motum ad locum,
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201 quem Gr&aelig;ci <G>fora\n</G> vocant, intelligi volumus.
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202 <HEAD>IIII.</HEAD>
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203 <p>Linea autem recta qu&aelig; e$t ab eo loco &agrave; quo mo-
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204 tus fieri incipit ad illum ad quem tendit. Illius vis
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205 qu&aelig; motum efficit, nutus dicetur.
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206 <p>Ead&etilde; ver&ograve; linea c&otilde;$iderata &agrave; loco ad qu&etilde; t&etilde;dit vis
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207 ad ill&utilde; &agrave; quo motus fieri incipit, contra nut&utilde; dicitur.
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208 <p>Itidem &amp; omnes illi parallel&aelig;.
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209 <p>Qu&aelig; ver&ograve; line&aelig; vel rect&aelig;, vel curu&aelig;, nutus line&atilde;
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210 <foot>B 2</foot>
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211 <pb n="4">
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212 ad angulos in&aelig;quales $ecant, ill&aelig; obliqu&egrave; nutum
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213 ver$us, vel contra nutum ferri dic&utilde;tur, habita ratio-
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214 ne partium, quas $pectant.
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215 <p>Plurim&ugrave;m autem &agrave; $itu corporis humani deno-
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216 minationem accipiunt ill&aelig; partes, vt $ur$um, deor-
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217 $um, dextror$um, $ini$tror$um, ant&egrave; vel pon&egrave; di-
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218 cantur. Qu&aelig; ver&ograve; line&aelig;, nutus lineam ad angulos
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219 rectos $ecant, neque ver$us nutum, neque contra
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220 nutum ferri dicuntur, $ed &aelig;qu&egrave; di$tant &agrave; loco natu-
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221 rali.
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222 <HEAD><I>Virium diui$io.</I></HEAD>
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223 <p>Vis autem omnis aut naturalis e$t, aut non natu-
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224 ralis, aut mixta. Naturalis vis e$t, qua res qu&aelig;libet
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225 natura $ua mouetur, aut mouenti re$i$tit habita ra-
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226 tione tum loci $ui naturalis, tum etiam $itus $uarum
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227 partium. Non naturalis e$t qu&aelig; nec ratione loci $ui
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228 naturalis, nec $itus partium mouet. E$t autem h&aelig;c
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229 duplex, fortuita $cilicet &amp; voluntaria: illa vt vis ven-
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230 torum &amp; $pirituum intellectu carentium, h&aelig;c vt a-
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231 nimalium &amp; $pirituum intelligentium &amp; $imilium.
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232 Mixta dicitur, qu&aelig; partim naturalis e$t, partim non
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233 naturalis.
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234 <p>Locus autem naturalis cuiu$que rei e$t, in quo
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235 exi$tens ip$a quie$cit, &amp; $i inde remota $it, vis ei &agrave; na||
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236 tura in$ita eam illuc impellit. Huius ver&ograve; motus aut
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237 quietis cau$$a nulla alia a$$ignari pote$t pr&aelig;ter pri-
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238 <pb n="5">
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239 mam illam rerum omnium conditricem, qu<17>, ne o-
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240 mnia in omnibus e$$ent $ingulis partibus vniuer$i
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241 $ingula loca attribuit circa qu&aelig; conglobantur, &amp; i-
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242 bi h&aelig;rent. Quicquid enim terreum e$t, in terr&aelig; glo-
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243 bum c&otilde;fluxit. Ita quicquid $oli homogenes in Solis
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244 corpus, lunaria omnia in lunam, &amp; $ic de c&aelig;teris cor||
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245 poribus vniuer$i. Singul&aelig; autem eorum partes $uis
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246 totis adh&aelig;rent, nec inde $ponte mou&etilde;tur: $i ver&ograve; in-
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247 de mot&aelig; fuerint, e&ograve; redire nituntur. Quod enim in
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248 terra fieri videmus, idem &amp; in reliquis corporibus v-
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249 niuer$i fieri dicere po$$umus, $cilicet quod vnita cir||
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250 ca locum $uum naturalem maneant: $i enim partes
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251 eorum $ponte $epararentur, &amp; vi ab ip$is $eparat&aelig;
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252 ad locum $uum non redirent, $tatim tota di$$olue-
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253 rentur.
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254 <p>Vis igitur illa in loco $uo quie$cendi grauitas di-
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255 citur, cuius contrarium e$t leuitas. Res autem nulla
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256 per $e leuis dicitur, $ed t&atilde;t&ugrave;m habita ratione alterius
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257 loci, qu&agrave;m $ui naturalis: vt puta $i pars aliqua Solis
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258 vi quapiam in terram inuecta e$$et, $u&oacute;que arbitrio
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259 committeretur, illa $tatim Solem peteret: ita $i ter-
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260 r&aelig; pars in Solem inuecta e$$et, $u&oacute;que arbitrio com||
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261 mitteretur, $tatim &agrave; $ole euolaret, &amp; $e per c&aelig;lum
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262 terram ver$us proriperet. Vt igitur terr&aelig; partes in
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263 terra graues $unt: $ic Solis partes in $ole graues $unt:
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264 in terra ver&ograve; leues, terr&aelig; ver&ograve; partes in Sole leues
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265 <foot>B 3</foot>
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266 <pb n="6">
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267 $unt. Idem de c&aelig;teris corporibus vniuer$i dici po-
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268 te$t. Neque enim facili&ugrave;s ex globo lun&aelig; particulam
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269 ab$trahas, qu&agrave;m ex terra glebam.
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270 <p>Aer autem nullum proprium habet locum natu||
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271 ralem, $ed vbique e$$e pote$t, c&ugrave;m rerum omnium
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272 locus vniuer$alis e$$e videatur: de quo dubitari po-
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273 te$t an $it infinitus, c&ugrave;m omnia contineat, &agrave; nulla re
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274 ver&ograve; totus comprehendatur, $ed h&aelig;c di$putatio non
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275 e$t huius loci.
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276 <p>Et$i autem nullum proprium habet locum natu||
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277 ralem, neque leuis aut grauis $it: leuis tamen e$$e vi-
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278 detur, c&ugrave;m grauis non $it, nec &aelig;gr&egrave; $e moueri patia-
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279 tur. Itaque $i quo modo ita con$titutus $it, vt rem a-
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280 liquam, qu&ograve; min&ugrave;s ad locum $uum naturalem per-
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281 uenire po$$it, impediat, tum per expre$$ionem $eu
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282 <G>e)/kqliyin</G>, ab ea eiicietur, in ei&uacute;$que locum $uccedet,
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283 vt fit in aqua, c&ugrave;m ei introducitur. Qu&aelig; quidem <G>e)/k-
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284 qliyis</G> iis etiam accidit qu&aelig; loco naturali gaudent,
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285 c&ugrave;m $e$e mutu&ograve; impediunt, ne ad illum ferantur.
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286 Pr&aelig;$ertim ver&ograve; in liquidis locum habet, quorum
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287 partes facilius mouentur, qu&agrave;m rerum coh&aelig;renti&utilde;
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288 &amp; compactarum. Inde accidit vt quicquid liquidis
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289 immergitur, tant&ograve; fiat in illis leuius qu&agrave;m grauius,
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290 e$t moles eiu$dem liquoris ip$orum moli &aelig;qualis.
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291 Sed h&aelig;c de loco naturali fu$i&ugrave;s &agrave; nobis alibi medita-
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292 ta, obiter h&icirc;c attigi$$e $ufficiat, vt intelligatur quid
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293 <pb n="7">
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294 $it vis naturalis.
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295 <p>Vis autem voluntaria nullum certum nutum
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296 habet, $ed illum tant&ugrave;m in quo mouentis voluntas
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297 conquie$cit. C&uacute;mque vis naturalis vnum tant&ugrave;m
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298 nutum habeat, $cilicet &agrave; loco &agrave; naturali requie re-
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299 moto, ad ip$am naturalem requiem, illa infinitos nu||
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300 tus habet &amp; indeterminatos per $e ac voluntatis tan
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301 t&ugrave;m decreto determinabiles.
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302 <p>Fortuita ver&ograve; ip$o tant&ugrave;m ca$u determinabilem
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303 nutum habet: qu&ograve; enim ip$a tendit, eo munere dici-
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304 tur $iue $ur$um, $iue deor$um, $iue ad latera. Itaque
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305 &amp; hic &amp; ille nutus <G>a)dio/ristos</G> dicitur. Quemadmodum
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306 &amp; is quem habent vires ill&aelig;, quibus res &agrave; $itu parti&utilde;
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307 naturali remot&aelig; ad illum redeunt: prout enim ab
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308 eo mot&aelig; $unt, ita ad illum redeunt, prout etiam huc
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309 aut illuc obuer$&aelig; $unt, vt vis arcus aut bali$t&aelig;.
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310 <HEAD>DEFIN. V.</HEAD>
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311 <p>Vires autem contrari&aelig; dicuntur, qu&aelig; contrarios
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312 motus ciere po$$unt, vt ea qu<17> $ur$um mouet &amp; qu<17>
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313 deor$um, &amp; $ic de c&aelig;teris.
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314 <p>Con$ideratur autem in vi qu&atilde;titas, tum eo qu&ograve;d
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315 vis partibus $uis con$tet, in quas in infinitum diuidi
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316 pote$t, &amp; rur$us additione aut multiplicatione au-
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317 geri, tum qu&ograve;d &aelig;qualitatis exce$$us &amp; defectus c&otilde;-
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318 parationem recipiat.
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319 <pb n="8">
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320 <p>Ac quoniam vis e$t mouendi potentia, vis par-
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321 tes erunt qu&aelig; motus partes efficient, &amp; qu&aelig; erit mo||
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322 tus partium, men$ura eadem erit &amp; vis partium. Et
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323 motus quidem propria men$ura e$t linea $eu $pati&utilde;.
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324 Quantum enim res qu&aelig;piam mota, $patij percur-
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325 rit, tant&ugrave;m mota e$$e dicitur. Quoniam ver&ograve; in mo||
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326 mento vel in$tanti quod in$tar puncti e$t, &amp; magni-
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327 tudine caret, nullus motus fieri pote$t, $ed motus o-
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328 mnis in tempore fit. Ide&ograve; ad motus men$uram tem-
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329 pus etiam adhibere oportet. Illud enim cum $patio
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330 vel linea, motus dici facit &aelig;quales aut in&aelig;quales.
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331 <HEAD>DEFIN. VI.</HEAD>
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332 <p>&AElig;quales igitur motus dicuntur, qui &aelig;qualibus
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333 temporibus &aelig;qualia $patia percurrunt.
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334 <p>Qui autem &aelig;qualibus temporibus &aelig;qualia $pa-
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335 tia permeant, illi iidem proportionales $unt: hoc
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336 e$t, qu&aelig; e$t ratio temporis, quo alter eorum fit ad
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337 tempus quo fit alter, eadem e$t $patij quod percurrit
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338 alter ad $patium quod reliquus percurrit. Si enim
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339 duorum quorum $inguli vna hora miliaris vnius i-
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340 ter conficiunt, alter eodem motu per tria miliaria
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341 ferri pergat, alter ver&ograve; per duo t&atilde;t&ugrave;m. Ille tribus ho-
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342 ris ea ab$oluet, hic ver&ograve; duabus, &amp; &aelig;qu&egrave; celeriter
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343 ferri dicentur, licet $patia in&aelig;qualia percurr&atilde;t, quo-
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344 niam illa $unt temporibus proportionalia.
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345 <pb n="9">
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346 <HEAD>VII.</HEAD>
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347 <p>In&aelig;quales autem motus dicuntur, quorum tem-
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348 pora non $unt $patiis proportionalia. Eorum autem
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349 maior ille dicitur, cuius maior erit ratio $patij ad $pa||
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350 tium, qu&agrave;m temporis ad tempus, quibus fiunt illi
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351 motus. Ex quo intelligitur etiam quis minor dica-
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352 tur. Maior igitur dicetur qui celeri&ugrave;s feretur, minor,
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353 qui tardi&ugrave;s.
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354 <HEAD>VIII.</HEAD>
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355 <p>&AElig;quales igitur vires dicentur, qu&aelig; &aelig;qualibus
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356 motibus $ubiecta $ua mouebunt. Maior ver&ograve; qu&aelig;
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357 $ubiectum $uum magis vel celeri&ugrave;s mouebit. Mi-
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358 nor, qu&aelig; min&ugrave;s vel tardi&ugrave;s.
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359 <HEAD><I>Motuum diui$io.</I></HEAD>
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360 <p>Comparantur autem motus $ecundum omnes
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361 comparationis gradus, hoc e$t, &aelig;qualitatem, exce$-
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362 $um &amp; defectum: tum $ibi ip$is, c&ugrave;m totus $uis par-
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363 tibus confertur, tum alij aliis.
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364 <p>Quatenus autem $ibi ip$i motus c&otilde;paratur, aut
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365 e$t <G>o(moiomerh\s</G> $eu &aelig;quabilis aut <G>a)nomoiomerh\s</G>, id e$t in&aelig;-
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366 quabilis. Aequabilis, cuius omnes partes tum mini-
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367 m&aelig; tum maiores tempora habent $patiis per qu&aelig;
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368 cientur proportionalia. In&aelig;quabilis ver&ograve;, in quo illa
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369 non $unt proportionalia.
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370 <p>In&aelig;quabilium autem motuum duo $unt genera,
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371 <foot>C</foot>
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372 <pb n="10">
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373 cre$cens $cilicet &amp; decre$cens, &iacute;dque vel continu&egrave;,
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374 vel per interualla, ordinatim vel <G>a)taktw=s</G>.
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375 <p>Cre$cens dicitur, quando partium po$teriorum
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376 ip$ius maior e$t celeritas, qu&agrave;m priorum: decre$c&etilde;s,
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377 quando minor.
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378 <p>Continu&egrave;, quando nulla pars illius vel minima
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379 alteri parti e$t &aelig;qualis $eu $ibi ip$i &aelig;quabilis.
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380 <p>Per interualla ver&ograve;, quando partes quidem illius
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381 $ibi ip$is &aelig;quabiles $unt, ali&aelig; ver&ograve; aliis comparat&aelig;
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382 in&aelig;quabiles.
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383 <p>Ordinatim, quando incrementum aut decrem&etilde;-
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384 tum illud certa aliqua proportionis progre$$ione
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385 fit, vt $i in motu continu&egrave; cre$cente, diui$o toto mo||
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386 tus $patio in aliquot partes &aelig;quales eadem $it ratio
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387 celeritatis, finis primi $patij ad celeritatem finis $e-
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388 cundi, qu&aelig; finis $ecundi ad finem tertij, &amp; $ic dein-
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389 ceps: vel $i finis $ecundi $patij duplo citi&ugrave;s feratur,
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390 qu&agrave;m finis primi &amp; finis tertij, triplo qu&agrave;m primi:
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391 finis ver&ograve; quarti quadruplo, &amp; $ic deinceps. Idem $i
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392 in quauis alia proportionis progre$$ione illarum
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393 partium celeritas, aliarum ad alias $e habeat. In mo-
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394 tu ver&ograve; per interualla cre$cente augm&etilde;tum illud or-
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395 dinatim fieri dicetur, $i interuallorum proportio a-
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396 liquem progre$$ionis ordinem $eruet: puta $i prim&utilde;
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397 interuallum duplo tardi&ugrave;s $ec&utilde;do moueatur, triplo
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398 ver&ograve; tardi&ugrave;s tertio, &amp;c.
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399 <pb n="11">
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400 <p>Inordinat&egrave; autem cre$cere vel decre$cere dicetur
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401 motus, $i incrementum illud aut decrementum nul||
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402 la certa &amp; ordinata proportione fiat.
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403 <p>Comparantur autem motus alij aliis, &aelig;quabiles
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404 $cilicet &amp; in&aelig;quabiles ordinatim progredientes: de
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405 inordinatorum enim tum inter $e, tum cum aliis c&otilde;||
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406 paratione nihil cert&ograve; $tatui pote$t.
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407 <p>Comparantur igitur &aelig;quabiles cum &aelig;quabili-
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408 bus aut cum in&aelig;quabilibus, ac etiam in&aelig;quabiles
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409 cum in&aelig;quabilibus.
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casties
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410 <p>&AElig;quabiles autem cum &aelig;quabilibus compara-
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411 tionibus $uis partibus corre$pondentibus $unt pro-
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412 portionales. Si enim propon&atilde;tur duo motus &aelig;qua-
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413 biles, quorum alter altero maior $it, qu&aelig; erit ratio
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414 dimidi&aelig; partis vnius ad dimidiam partem alterius,
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415 eadem erit terti&aelig; ad tertiam, &amp; $ic deinceps.
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416 <p>&AElig;quabilium ver&ograve; motuum cum in&aelig;quabilibus,
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417 cum iis $cilicet qui per interualla cre$cunt, compa-
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418 ratio fit tanquam cum pluribus diuer$is motibus &aelig;-
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419 quabilibus, $unt enim interuallorum illor&utilde; motus
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420 &aelig;quabiles. At cum continu&egrave; cre$c&etilde;tibus aut decre-
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421 $centibus difficile e$t comparationis modum defi-
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422 nire, c&ugrave;m ne momento quidem $tabilis maneat par||
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423 tium illarum quantitas. E$t tamen motus &aelig;quabilis
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424 illorum men$ura: tanti enim e$$e dicuntur, quanti
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425 forent, $i in ea celeritatis men$ura ad quam perue-
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426 <foot>C 2</foot>
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427 <pb n="12">
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428 nerunt, &aelig;quabiliter pergerent.
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429 <p>Cre$centes ver&ograve; cum cre$centibus continu&egrave;, c&otilde;-
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430 parati $iquidem eadem progre$$ionis proportione
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431 cre$cant, &aelig;qualia $patia ab eorum principiis nume-
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432 rata, &aelig;qualibus temporibus emen$a habebunt, idem
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433 in decre$centibus. Sed h<17>c hactenus inqui$iui$$e $uf-
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434 ficiat: $i quis enim particulatim omnia exp&etilde;dere vel-
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435 let, in infinitum $e$e extenderet h<17>c partium &amp; pro-
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casties
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436 portionum motuum $upputatio.
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437 <p>Iam videamus qu&aelig; vires quos horum motuum
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438 cieant. Et vis quidem ea qua res qu&aelig;libet ad $itum
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439 naturalem $uarum partium redit, motu c&otilde;tinu&egrave; de-
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440 cre$cente mouet: quo enim propi&ugrave;s ad $itum natu-
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441 ralem partium accedit, eo tardior &amp; languidior e$t.
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442 Arcus enim c&ugrave;m prim&ugrave;m laxari incipit, celeri&ugrave;s
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443 mouetur, qu&agrave;m c&ugrave;m motus $ui finis propior e$t.
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444 <p>Vis ver&ograve; naturalis, qua res qu<17>libet ad loc&utilde; $uum
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445 naturalem tendit, $ubiectum $uum, motu continu&egrave;
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446 &amp; ordinatim cre$cente, mouet. Illius autem motus
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447 cau$a e$t qu&ograve;d facili&ugrave;s id moueatur, quod in motu
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448 e$t, qu&agrave;m quod quie$cit. Vis igitur eadem, $ubiect&utilde;
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449 quod iam in motu e$t premens, illud magis moue-
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450 bit, qu&agrave;m $i quie$cat, &amp; magis motum, magis etiam
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451 mouebit: ita vt eadem vis motione maior fiat, qu&agrave;m
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452 per $e $it. Et h&aelig;c e$t cau$a cur ictus, quo magis ab al-
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453 tero venit, eo vehementior $it. Motus autem huius
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454 <pb n="13">
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455 $patia hanc celeritatis proportionem $eruant, vt qu<17>
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456 e$t ratio totius $patij, per quod fit ille motus ad par-
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457 tem ip$ius (vtriu$que initio inde $umpto, vbi e$t mo||
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458 tus initium) eadem $it celeritatis ad celeritatem.
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459 <fig>
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460 Exempli gratia, $i vis aliqua per lineam ABC
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461 mouerit, $itque AB illius line&aelig; pars, qu&aelig; erit
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diff changeset
462 ratio AC ad AB, eadem erit celeritatis motus
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diff changeset
463 in puncto C ad celeritat&etilde; motus in puncto B.
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464 Cuiu$modi proportio ob$eruatur in paralle-
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diff changeset
465 lis triangulum $ecantibus. Vt enim $e habet
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466 <fig>
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467 AC ad AB, $ic CG ad BF, &amp; vt AD ad
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468 AC, $ic DH ad CG. Itaque $i in $patia ali-
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469 quot &aelig;qualia diuidatur totius motus $pa||
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470 tium, finis $ecundi duplo citi&ugrave;s feretur,
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471 qu&agrave;m finis primi: finis ver&ograve; tertijtriplo
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472 citi&ugrave;s qu&agrave;m finis primi, &amp; $ic deinceps.
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473 Hac autem ratione fit, vt $patiorum illo-
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474 rum initio maxima $it celeritatis differ&etilde;-
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475 tia: progre$$u ver&ograve; $emper minuatur, ac tandem fer-
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476 m&egrave; eadem $it, vt fit in trianguli lateribus, qu&aelig; lon-
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477 gi$$im&egrave; producta &aelig;qu&egrave; di$tare videntur. E&aacute;que e$t
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478 ratio cur Solis &amp; Lun&aelig; radij, etiam$i concurrant (in
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479 ip$orum $cilicet corporibus, aut vltra res quas illu-
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480 $tr&atilde;t) paralleli tam&etilde; appare&atilde;t. Ead&etilde; eti&atilde; cau$a e$t cur
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481 line<17> omnes ad perp&etilde;dicul&utilde; in terr&atilde; cadentes, paral-
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482 lel&aelig; videantur, c&ugrave;m tam&etilde; in centro terr&aelig; c&otilde;currant.
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483 <foot>C 3</foot>
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484 <pb n="14">
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485 <p>Hunc igitur motum vis naturalis efficit, mod&ograve;
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486 nulla quies intercedat.
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487 <p>Vis autem voluntaria motum omnem ciere apta
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488 e$t. Fortuita ver&ograve; inordinatum tant&ugrave;m motum effi-
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489 cit. Et h&aelig;c de variis virium motibus, eor&uacute;mq; pro-
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490 portione &amp; partibus.
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491 <p>E$t ver&ograve; etiam virium partitio qu&aelig;dam, qu&aelig; ex
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492 eorum quibus inh&aelig;r&etilde;t $ubiectorum diui$ione, aut
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493 alterius ad alterum proportione re$ultat. In $ubie-
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494 ctis enim, qu&aelig; <G>o(moiomerh=</G> $unt, vis naturalis &aelig;qualiter
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parents:
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495 per omnes partes diffu$a e$t: ita vt qu&aelig; e$t ratio mo-
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496 lis ad molem $ub eadem figura, ead&etilde; $it vis ad vim:
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497 vt $i globus alicuius metalli, ad alium globum eiu$-
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498 dem metalli mole quadruplus $it, illius quoque p&otilde;-
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499 dus ad huius pondus quadruplum erit.
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500 <p>Hactenus vim c&otilde;$iderauimus quatenus mouet,
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diff changeset
501 $upere$t vt eam con$ideremus, quatenus mouenti
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502 re$i$tit. Habet autem locum re$i$tentia, vbi du&aelig; vi-
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503 res contrari&aelig; committuntur: $i enim vtraque eod&etilde;
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parents:
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504 motu moueat, nulla erit re$i$tentia, $ed altera alteri
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505 addita maiorem vim con$tituet.
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506 <p>Committi autem dicuntur vires, quando ita ap-
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diff changeset
507 plicantur &amp; connect&utilde;tur, vt altera nutu $uo moueri
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diff changeset
508 non po$$it, quin altera contra nutum $uum mouea-
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509 tur. Viribus autem ita connexis, accidit vt altera in
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parents:
diff changeset
510 alterius $ubiectum agat, &amp; altera in $ubiecto $uo exi-
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parents:
diff changeset
511 <pb n="15">
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diff changeset
512 $tens, alteri in altero $ubiecto exi$tenti re$i$tat, ita vt
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diff changeset
513 eorum $ubiecta quodammodo reciproca fiant.
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514 <HEAD>CONCLVSIO I.</HEAD>
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515 <p>Illam autem re$i$tentiam vi momenti in eodem
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diff changeset
516 $ubiecto &aelig;qualem, aut eandem cum ea e$$e, ex ip$ius
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parents:
diff changeset
517 definitione con$tat. E$t enim re$i$tere n&otilde; pati $e mo-
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parents:
diff changeset
518 ueri. Quantum autem vis qu&aelig;libet $ubiect&utilde; $uum
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diff changeset
519 nutu $uo mouet, tantum illud contra nutum $uum
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diff changeset
520 moueri non patitur. Mouendi igitur &amp; mou&etilde;ti re-
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diff changeset
521 $i$tendi potentia in eodem $ubiecto &aelig;quales $unt.
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522 <HEAD>II.</HEAD>
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523 <p>Quemadmodum autem in eodem $ubiecto ma-
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diff changeset
524 ior vis ine$$e dicitur, in quo e$t maioris motus po-
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parents:
diff changeset
525 tentia, $ic maior re$i$tentia erit maioris motus con-
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parents:
diff changeset
526 trarij impatientia. Eadem igitur vis magis mouere
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parents:
diff changeset
527 nitenti c&otilde;tra ip$ius nutum magis re$i$tet, minus ve-
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parents:
diff changeset
528 r&ograve; nitenti min&ugrave;s re$i$tet.
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parents:
diff changeset
529 <HEAD>III.</HEAD>
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530 <p>Et quo maior erit motus contrarius, eo magis re-
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parents:
diff changeset
531 $i$tet: id e$t, quo celerius vis qu&aelig;libet &agrave; nutu $uo re-
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parents:
diff changeset
532 uelletur, eo magis re$i$tet. Duplo igitur citi&ugrave;s re-
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parents:
diff changeset
533 uellenti duplo re$i$tet, triplo citius triplo re$i$tet, &amp;
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parents:
diff changeset
534 $ic <G>a)nalo/gws</G> in omni proportione. Qu&aelig; igitur erit
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parents:
diff changeset
535 ratio motus ad motum, contra nutum eadem erit
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parents:
diff changeset
536 in vna &amp; eadem vi ratio re$i$tenti&aelig; ad re$i$tentiam.
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parents:
diff changeset
537 <p>Si ver&ograve; plures vires comparentur.
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parents:
diff changeset
538 <pb n="16">
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539 <HEAD>IIII.</HEAD>
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casties
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540 <p>&AElig;qualium quidem virium &aelig;qualibus motibus
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541 &aelig;quales erunt re$i$tenti&aelig;. Si enim &aelig;quales vires &aelig;-
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parents:
diff changeset
542 qualiter &agrave; $uis nutibus reuellantur, &aelig;qualiter re$i-
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casties
parents:
diff changeset
543 $tent.
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casties
parents:
diff changeset
544 <HEAD>V.</HEAD>
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casties
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545 <p>In&aelig;qualibus ver&ograve; motibus, earum re$i$tenti<17> in-
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parents:
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546 &aelig;quales erunt, &amp; motuum proportionem $equen-
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casties
parents:
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547 tur. Exempli gratia, $int du&aelig; vires A &amp; B, quarum v-
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548 traque vna hora miliare vnum percurrat, ill&aelig; &aelig;qua-
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diff changeset
549 les erunt. Si igitur vtraque quarta parte vnius milia-
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diff changeset
550 ris, vel dimidio miliari &agrave; $uo nutu eodem tempore
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diff changeset
551 reuellatur, eadem erit vtriu$que illi motui re$i$ten-
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diff changeset
552 tia. Sin ver&ograve; altera quidem quarta parte vnius milia-
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parents:
diff changeset
553 ris &agrave; nutu $uo reuellatur, altera ver&ograve; eodem tempo-
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parents:
diff changeset
554 re integro miliari reuellatur: h&aelig;c re$i$tentia ad il-
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parents:
diff changeset
555 lam quadrupla erit, &amp; $ic de c&aelig;teris motuum pro-
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diff changeset
556 portionibus.
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diff changeset
557 <HEAD>VI.</HEAD>
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diff changeset
558 <p>In&aelig;qualium ver&ograve; virium re$i$tenti&aelig; &aelig;qualibus
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diff changeset
559 motibus ip$arum virium proportionem $equentur.
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diff changeset
560 Quo maior enim e$t vis, eo magis eodem tempore
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diff changeset
561 eod&eacute;mque $patio contra nutum mouenti re$i$tit.
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parents:
diff changeset
562 Ex&etilde;pli gratia, $it vis A quadrupla ad vim B, vis A mo-
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parents:
diff changeset
563 uenti $e vno miliari quadruplo re$i$tet, qu&agrave;m B mo-
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diff changeset
564 uenti $e vno miliari eodem tempore.
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diff changeset
565 <pb n="17">
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diff changeset
566 <p>In&aelig;qualium ver&ograve; virium in&aelig;qualibus motibus
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parents:
diff changeset
567 re$i$tenti&aelig;, quando motus eam inter $e proportio-
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parents:
diff changeset
568 nem $eruabunt, qu&aelig; e$t inter eas <G>a)ntipeponqw=s</G>, $eu
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parents:
diff changeset
569 reciproc&egrave;, &aelig;quales erunt. Sit enim vis A qu&aelig; ad vim
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parents:
diff changeset
570 B eandem habeat ration&etilde; quam C ad D, puta quam
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parents:
diff changeset
571 3 ad 1, aut 4 ad 1, aut qu&aelig;cunque alia $it, moueat&uacute;rq;
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parents:
diff changeset
572 vtraque $patio E contra nutum $uum eodem tem-
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parents:
diff changeset
573 pore, re$i$tentia A ad re$i$tentiam B $e habebit, vt C
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parents:
diff changeset
574 ad D per proximam conclu$ionem. Si ver&ograve; B moue-
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parents:
diff changeset
575 atur eodem tempore $patio aliquo quod $e habeat
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diff changeset
576 ad $patium E vt C ad D, re$i$tentia B huic motui $e
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diff changeset
577 habebit ad re$i$tentiam priori motui, vt C ad D per
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diff changeset
578 quartam conclu$ion&etilde;. Du&aelig; ergo re$i$tenti&aelig; ad ean-
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diff changeset
579 dem, eandem habebunt rationem, ergo &aelig;quales e-
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diff changeset
580 runt per <I>9</I> prop. <I>5.</I> Elem. Eucl.
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diff changeset
581 <HEAD>VIII.</HEAD>
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diff changeset
582 <p>Si ver&ograve; motuum virium in&aelig;qualium proportio
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diff changeset
583 non $it eadem qu&aelig; e$t ip$arum virium reciproc&egrave;, vis
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parents:
diff changeset
584 illius qu&aelig; ad alteram maiorem habebit rationem,
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parents:
diff changeset
585 qu&agrave;m motus alterius ad motum ip$ius, re$i$tentia
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parents:
diff changeset
586 maior erit, altera ver&ograve; minor. Exempli gratia, $i vis
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parents:
diff changeset
587 A $it tripla ad vim B, &amp; motus quo B &agrave; nutu $uo reuel||
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parents:
diff changeset
588 letur, $it minor qu&agrave;m triplus ad motum A, re$i$ten-
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parents:
diff changeset
589 tia A erit maior qu&agrave;m re$i$tentia B. Idem erit, $i ma-
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parents:
diff changeset
590 ior $it ratio A ad B, qu&agrave;m 3 ado, motus ver&ograve; B ad mo-
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diff changeset
591 <foot>D</foot>
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parents:
diff changeset
592 <pb n="18">
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parents:
diff changeset
593 tum A $it triplus. Contr&agrave; ver&ograve; $i A ad B $it tripla: mo-
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parents:
diff changeset
594 tus ver&ograve; B ad motum A $it maior qu&agrave;m triplus, ma-
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parents:
diff changeset
595 ior erit re$i$tentia B qu&agrave;m A. Idem accidet, $i A ad B
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diff changeset
596 ratio minor $it qu&agrave;m 3 ad 1, motus ver&ograve; B ad motum
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diff changeset
597 A $it triplus. Hunc autem exce$$um &amp; defectum n&otilde;
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parents:
diff changeset
598 vlteri&ugrave;s $crutabimur, vt ip$ius quantitatis men$ur&atilde;
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diff changeset
599 a$$equamur: alioqui in infinitum fieret progre$$us.
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diff changeset
600 <p>Ex his colligimus re$i$t&etilde;tias tribus modis &aelig;qua-
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parents:
diff changeset
601 les aut in&aelig;quales dici, per $e $cilicet aut motione, aut
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casties
parents:
diff changeset
602 vtroque modo. Per $e quidem c&ugrave;m $ubiectorum
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parents:
diff changeset
603 vires $unt &aelig;quales (id e$t &aelig;qualiter $ua $ubiecta mo-
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parents:
diff changeset
604 uent) aut in&aelig;quales. Motione ver&ograve; quando per $e
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parents:
diff changeset
605 quidem in &aelig;quales $unt, motione ver&ograve; &aelig;quales fi&utilde;t:
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parents:
diff changeset
606 aut quando per $e &aelig;quales $unt, motione ver&ograve; in&aelig;-
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parents:
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607 quales fi&utilde;t eo, quo dictum e$t, modo. Vtroque mo||
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parents:
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608 do &aelig;quales aut in&aelig;quales dic&utilde;tur, quando tum per
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parents:
diff changeset
609 $e, tum motione tales fiunt. His explicatis videamus
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parents:
diff changeset
610 quis eorum $it effectus, vbi vires committentur.
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diff changeset
611 <HEAD>IX.</HEAD>
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612 <p>Prim&ugrave;m ex prima conclu$ione $equitur qu<17> erit
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diff changeset
613 ratio re$i$tenti&aelig; ad re$i$tentiam, eandem fore re$i-
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parents:
diff changeset
614 $tenti&aelig; ad vim cuius e$t altera re$i$tentia: $unt enim
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parents:
diff changeset
615 vis &amp; re$i$t&etilde;tia in eod&etilde; $ubiecto &aelig;quales. Ergo mul-
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parents:
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616 to magis $equitur, $i re$i$t&etilde;tia altera maior, altera mi||
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617 nor fuerit, illam huius vi maior&etilde; fore, vel illius vim
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parents:
diff changeset
618 hac re$i$tentia maiorem, $i &aelig;qualis &aelig;qualem.
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parents:
diff changeset
619 <pb n="19">
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parents:
diff changeset
620 <HEAD>X</HEAD>
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parents:
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621 <p>Re$i$tentia autem vi contrari&aelig; commi$$a t&atilde;tum
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casties
parents:
diff changeset
622 de ea tollit quanta e$t ip$a re$i$tentia. Sublata autem
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parents:
diff changeset
623 vi tollitur motus: $ublata ver&ograve; re$i$tentia, $i vis ad$it,
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casties
parents:
diff changeset
624 $equitur motus. Vbi igitur vis &amp; re$i$tentia in<17>qua-
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parents:
diff changeset
625 les committentur, $i vis maior $it qu&agrave;m re$i$tentia,
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casties
parents:
diff changeset
626 fiet motus $ecundum vis illius nutum, &amp; contra nu-
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casties
parents:
diff changeset
627 tum vis illius qu&aelig; re$i$tit. Si ver&ograve; re$i$tentia maior
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parents:
diff changeset
628 fuerit, tum ip$a fiet vis mouens, &amp; vim contrariam
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parents:
diff changeset
629 contra nutum ip$ius mouebit, dum ip$a nutu $uo
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casties
parents:
diff changeset
630 mouebitur. Hinc $equuntur duo theoremata, circa
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parents:
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631 qu&aelig; totius huius con$iderationis cardo vertitur.
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632 <HEAD>THEOREMA I.</HEAD>
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633 <p>Duarum virium connexarum, quarum ($i mo-
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634 ueantur) motus erunt ip$is <G>a)ntipeponqw=s</G> proportiona-
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casties
parents:
diff changeset
635 les neutra alteram mouebit, $ed &aelig;quilibrium faci&etilde;t.
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casties
parents:
diff changeset
636 <p>Sit vis A commi$$a cum vi B, $itque vis A ad vim
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diff changeset
637 B ratio per $e, vt C ad D qu&aelig;c&utilde;que illa $it, $iue dupla,
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diff changeset
638 $iue tripla, $iue alia. Sit etiam ead&etilde; ratio motus quo
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diff changeset
639 B mouebitur, $i ita, vt connexa $unt, moueantur) ad
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diff changeset
640 motum quo A mouebitur, qu&aelig; e$t C ad D, dico mo-
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parents:
diff changeset
641 tum non $equuturum, $ed factum iri &aelig;quilibrium.
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parents:
diff changeset
642 Aut enim A &amp; B vires per $e erunt &aelig;quales aut in&aelig;-
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diff changeset
643 quales: $i &aelig;quales, ergo &amp; ip$arum motus &aelig;quales e-
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diff changeset
644 runt: $unt enim ex hypothe$i ip$is proportionales:
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diff changeset
645 <foot>D 2</foot>
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diff changeset
646 <pb n="20">
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parents:
diff changeset
647 ergo per quartam conclu$ionem earum re$i$tenti&aelig;
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diff changeset
648 &aelig;quales erunt, ergo per decimam conclu$ionem n&otilde;
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parents:
diff changeset
649 fiet motus. Si ver&ograve; $unt in&aelig;quales, c&ugrave;m earum mo-
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parents:
diff changeset
650 tus ex hypothe$i $int <G>a)ntipeponqw=s</G> proportionales per
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parents:
diff changeset
651 $eptimam conclu$ionem &aelig;quales erunt etiam re$i-
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parents:
diff changeset
652 $tenti&aelig;. Ergo nec motus fiet: nullo igitur modo fiet
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diff changeset
653 earum motus. Quod demon$trandum erat.
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654 <HEAD>THEOREMA II.</HEAD>
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diff changeset
655 <p>Quarum ver&ograve; ita connexarum ($i moue&atilde;tur) mo||
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parents:
diff changeset
656 tus, ip$is proportionales non erunt: illa alteram mo-
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diff changeset
657 uebit, cuius ad alteram ratio maior erit, qu&agrave;m huius
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diff changeset
658 motus ad illius motum.
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diff changeset
659 <p>Sit vis A cum vi B commi$$a, $itque A ad B ratio
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diff changeset
660 per $e, vt C ad D: ita ver&ograve; connex&aelig; $int, vt $i mouean-
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diff changeset
661 tur, minor $it ratio motus quo B mouebitur ad mo-
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diff changeset
662 tum, quo A mouebitur, quum C ad D dico A motu-
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diff changeset
663 ram B: erit enim per octauam conclu$ionem re$i-
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diff changeset
664 $tentia B minor re$i$tentia A: ergo per nonam con-
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diff changeset
665 clu$ionem re$i$tentia B minor vi A: vis igitur A vim
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diff changeset
666 B mouebit per vndecimam conclu$ionem. Quod
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diff changeset
667 demon$trandum erat.
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668 <p>Hinc po$$emus ad problematis no$tri demon-
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diff changeset
669 $trationem rect&agrave; pergere: ante tamen cra$$i&ugrave;s ali-
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diff changeset
670 quanto h&aelig;c explicanda $unt, vt &agrave; quouis facili&ugrave;s in-
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671 telligantur. Du&aelig; $unt motus men$ur&aelig;, locus $cilicet
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diff changeset
672 &amp; tempus: vtroque igitur, tempore videlicet &amp; loco
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diff changeset
673 <pb n="21">
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parents:
diff changeset
674 maior, minor aut &aelig;qualis dicitur: &amp; quo minori t&etilde;-
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parents:
diff changeset
675 pore idem $patium ab$oluitur, eo maior e$t, vel quo
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parents:
diff changeset
676 maius $patium eodem tempore.
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diff changeset
677 <p>Vt igitur motus magnus dicatur, perinde e$t $i
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diff changeset
678 paruo tempore fiat, aut magno $patio.
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679 <p>Quod autem maiorem mot&utilde; ciere pote$t illud,
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diff changeset
680 vis maior dicitur, qu&ograve;d minorem minor.
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681 <p>Vis autem $eu mouendi potentia in eodem $ub-
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682 iecto certa &amp; finita e$t. Qu<17>libet enim res vi natura-
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diff changeset
683 li pr<17>dita, $i &agrave; loco naturali ab$it, $u&oacute;que arbitrio c&otilde;-
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684 mittatur, certo tempore e&ograve; redit. E$t enim certum
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685 in rerum natura quanto tempore libr&aelig; vnius p&otilde;dus
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686 deor$um $ponte $ua delatum, miliare vnum aut duo
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687 conficiat pro ratione materi<17>, vel quantum $patij v-
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688 na vel duabus horis percurrat. Id ver&ograve; quantum $it,
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689 hominum indu$tria nondum quod $ciam explora-
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diff changeset
690 tum e$t: ali&acirc;s autem id demon$trare conabimur.
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parents:
diff changeset
691 <p>Mouendi ver&ograve; potentia in alieno $ubiecto infi-
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692 nita e$t, hoc e$t, in infinit&utilde; augeri vel minui pote$t,
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693 quoniam in finita auctione &amp; diminutione e$t re$i-
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diff changeset
694 $tentia: tanta enim e$t, quanta e$t in eodem $ubiecto
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diff changeset
695 vis: quant&uacute;$q; motus illius e$t vis alterius re$pectu.
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diff changeset
696 Quo igitur vis qu&aelig; alienum $ubiectum mouere ni-
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diff changeset
697 titur, illud magis mouere nitetur, eo min&ugrave;s illud
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diff changeset
698 mouere poterit, maior enim erit illius re$i$tentia.
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699 Quemadmodum enim quod magis nutu $uo mo-
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700 <foot>D 3</foot>
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701 <pb n="22">
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parents:
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702 uetur, maiorem vim mouendi habet. Ita illud idem
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diff changeset
703 quod magis contra nutum $uum mouetur, maior&etilde;
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704 vim re$i$tendi habet. E contrario ver&ograve; quo vis qu&aelig;-
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parents:
diff changeset
705 libet minorem motum in alieno $ubiecto ciere ni-
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parents:
diff changeset
706 tetur, eo facili&ugrave;s illud mouebit.
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parents:
diff changeset
707 <p>Tarditate igitur motus, re$i$tentia in infinitum
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parents:
diff changeset
708 minui pote$t: minuta ver&ograve; re$i$tentia vis contrari&aelig;
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parents:
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709 effectus augetur, ita vt vis qu&aelig; per $e minima e$t, in
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parents:
diff changeset
710 contrariam cui plurimum diminuta $it, re$i$tentia
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parents:
diff changeset
711 maxim&egrave; agat. Perinde igitur e$t, $i vis mouens ma-
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parents:
diff changeset
712 gna $it, mouenda ver&ograve; parua, ac $i illa celeriter fera-
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parents:
diff changeset
713 tur, h&aelig;c ver&ograve; tard&egrave;: qu&atilde;tum enim vis mouens $i ma-
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parents:
diff changeset
714 gna fuerit in mouendam minorem poterit, tantum
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parents:
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715 vis parua celeriter mota in magnam tard&egrave; motam
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parents:
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716 poterit. Si igitur velimus vt vis parua magnam mo-
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parents:
diff changeset
717 ueat, eas ita collocare oportet, vt quantum magna
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parents:
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718 paruam $uperat, tantum illi de motus celeritate de-
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parents:
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719 trahatur, &amp; aliquid ampli&ugrave;s. Si enim tanta $it tardi-
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parents:
diff changeset
720 tas motus vis vnius, re$pectu motus vis alterius, qu&atilde;-
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parents:
diff changeset
721 ta e$t proportio vis illius, ad hanc non fiet motus: vt
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parents:
diff changeset
722 $i pondus A quatuor librarum cum pondere B libr&aelig;
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parents:
diff changeset
723 vnius committatur: $intque ita connexa, vt dum
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diff changeset
724 A vno $patio mouebitur, B quatuor $patiis mouea-
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parents:
diff changeset
725 tur, ita vt motus A quadruplo tardior $it motu B n&otilde;
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parents:
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726 fiet motus, quia quantum A excedit B pondere tan-
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727 t&ugrave;m deficit motus tarditate.
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diff changeset
728 <pb n="23">
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diff changeset
729 <p>Tantum enim e$t libram vnam quatuor $patiis
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diff changeset
730 moueri, quantum libras quatuor vno $patio eodem
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diff changeset
731 tempore: $i igitur alterutri eorum ita con$titutor&utilde;,
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diff changeset
732 vel momentum vis addatur, id cui additum fuerit,
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diff changeset
733 alterum mouebit. Idem fiet $i A ita connectatur vt
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diff changeset
734 vel momento citi&ugrave;s qu&agrave;m moueri po$itum e$t, mo||
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parents:
diff changeset
735 ueatur vel B tardi&ugrave;s.
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diff changeset
736 <p>Id etiam alia ratione o$tendi pote$t, $i vis A qua-
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parents:
diff changeset
737 drupla $it ad vim B, erunt in A quatuor partes, ip$i B
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diff changeset
738 &aelig;quales. Si igitur B cum $ingulis illis committatur,
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diff changeset
739 ita vt &aelig;qualiter moueantur, non fiet motus: $i ver&ograve;
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parents:
diff changeset
740 alterutri aut ip$i B, aut $ingulis illis partibus vis vel
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parents:
diff changeset
741 motus momentum adiiciatur vel detrahatur, illa cui
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diff changeset
742 adiectum fuerit, aut cui non detractum fuerit, nutu
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diff changeset
743 $uo mouebitur, &amp; aliam contra nutum eius moue-
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diff changeset
744 bit. Addito igitur ip$i B momento, dum vno $patio
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diff changeset
745 mouebitur, $ingulas illas partes vno $patio moue-
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diff changeset
746 bit. Vbi igitur $ingulas $emel mouerit, ip$a quater
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diff changeset
747 mota erit. Dum ver&ograve; $ingul&aelig; $emel mot&aelig; erunt, to-
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diff changeset
748 tum ex illis con$tans $emel motum intelligetur: po-
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diff changeset
749 terit igitur B addito ip$i momento, dum quater mo||
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diff changeset
750 uebitur, $emel totum A mouere.
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751 <p>Tertio modo id ip$um concludere po$$umus. Si
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diff changeset
752 du&aelig; vires &aelig;quales connectantur, ita vt mot&aelig;, &aelig;qua-
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diff changeset
753 liter moueantur, altera in alter&atilde; non aget. Si ver&ograve; ita
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diff changeset
754 connectantur vt mot&aelig; in&aelig;qualiter moueantur,
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diff changeset
755 <pb n="24">
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parents:
diff changeset
756 quantum altera alteram celeritate $uperabit, t&atilde;tum
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diff changeset
757 &amp; vi $uperabit. Vt $i vis A vi B &aelig;qualis $it, ac ita con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
758 nect&atilde;tur, vt B quatuor $patiis moueatur, dum A vno
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
759 $patio mouebitur, B quadruplam vim habebit ad A,
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diff changeset
760 quia eam in motibus $uis proportionem $eruant,
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diff changeset
761 quam $i $eruar&etilde;t arbitrio $uo commi$$&aelig;, B ad A qua-
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diff changeset
762 drupla e$$et. Si igitur illis ita connexis, ip$i A adda-
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diff changeset
763 tur vis triplo maior qu&agrave;m ip$a $it, B illis quatuor re-
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diff changeset
764 $i$tet, nec fiet motus.
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
765 <p>Duarum igitur virium comparatarum, quanto
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diff changeset
766 altera $ubiectum $uum celeri&ugrave;s mouebit, qu&agrave;m al-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
767 tera: tanto illa hanc celeri&ugrave;s mouere poterit, qu&agrave;m
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768 ip$a moueatur, $i illi vis mom&etilde;tum additum fuerit.
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diff changeset
769 Ita vt qu&aelig; erit proportio vis ad vim, eadem $it mo-
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diff changeset
770 tus, quem illa in hac ciere pote$t ad motum quo ip$a
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771 mouebitur. Et &egrave; conuer$o qu&aelig; erit ratio motus ad
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772 motum, eadem erit vis cui additum fuerit momen-
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773 tum ad eam quam ip$a mouere poterit ratio reci-
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774 proc&egrave;. Iam ad problematis no$tri demon$tratio-
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diff changeset
775 nem veniamus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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776 <HEAD><I>Propo$iti problematis demon$tratio.</I></HEAD>
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777 <p>Sit data vis A quantacunque illa $it magna vel
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778 parua: datum ver&ograve; pondus B quantumquantum il-
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779 lud $it, dico me vi A pondus B tollere po$$e. Id $ic de-
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parents:
diff changeset
780 mon$tro.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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781 <pb n="25">
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782 <p>Prim&ugrave;m enim ex doctrina $ecundi l&etilde;matis, quod
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783 inferi&ugrave;s demon$trabitur, $ciam proportionem pro-
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784 xim&egrave; maiorem, qu&agrave;m $it A ad B proportio. Deinde
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785 ex doctrina primi lemmatis ita connectam A &amp; B,
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786 vt quando ambo mouebuntur, nunc $it ratio motus
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787 quo B mouebitur ad mot&utilde; quo A mouebitur, qu&agrave;m
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diff changeset
788 $it A ad B. His peractis $equitur vim A pondus B mo-
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789 turam ex $ecundo $uperi&ugrave;s demon$trato theorema-
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790 te. Quod erat propo$itum.
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791 <HEAD>LEMMA I.</HEAD>
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792 <p>Duas vires ita connectere, vt $i moueantur, ear&utilde;
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793 motus, in data ratione alter ad alterum $e habeant
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794 vires contrari&aelig;, aut medio aliquo, aut per $e ab$que
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795 vllo medio committuntur. Si ab$que medio com-
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796 mittant, eodem motu mouebuntur, maior enim mi||
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797 norem eodem motu, quo ip$a moueri poterit, mo-
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798 uebit: aut &aelig;quilibrium faci&etilde;t, $i &aelig;quales $int: vt $i le-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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799 ue graui committatur, $iquidem leuitas grauitate
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parents:
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800 maior $it, attolletur graue: $in ver&ograve; grauitas maior
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801 $it, leue deprimetur: $i &aelig;qualia $int, non mouebun-
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parents:
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802 tur.
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parents:
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803 <p>Si ver&ograve; medio aliquo connectantur mediorum
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804 varia $unt genera. Aut enim medium e$t flexibile &amp;
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805 <G>o(moiomere\s</G>, vt funis, catena, &amp;c. aut e$t inflexibile, il-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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806 l&uacute;dque aut rectum, aut curuum, vt recta linea vel
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807 curua vel angulus. Atque h&aelig;c omnia aut continua
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808 <foot>E</foot>
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809 <pb n="26">
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parents:
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810 $unt aut diui$a, $implicia aut compo$ita.
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811 <p>Horum autem mediorum opera fit vt vires illis
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812 applicat&aelig;, quarum iidem $unt nutus, contrariis mo-
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813 tibus moueantur. Id aut&etilde; fit c&ugrave;m in mediis illis inter
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parents:
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814 eor&utilde; extrema interiacet quies vna vel plures. Ex&etilde;pli
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casties
parents:
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815 gratia, $i duo pondera funis extremitatibus alligata
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casties
parents:
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816 $int, &amp; funis clauo fixo &amp; immobili incumbat pro-
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817 pter illam quietem inter vtrumq; pondus po$it&atilde; n&otilde;
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parents:
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818 poterit alter&utilde; deor$um moueri, quin alter&utilde; $ur$um
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819 moueatur. Id&etilde; fiet in linea recta, $i enim illius extre-
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820 mitatibus pondera duo annexa $int, &amp; inter ea $it in
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diff changeset
821 illa punct&utilde; aliquod quie$cens, dum alterum ex illis
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diff changeset
822 ponderibus deor$um feretur, alterum a$cendet. P&utilde;-
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diff changeset
823 ctum autem illud quie$cens in linea illa recta, Gr<17>cis
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diff changeset
824 hypomochlium dicitur, e&ograve; qu&ograve;d vecti, quem <G>mo/xlon</G>
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diff changeset
825 vocant, $ubiiciatur. Huius autem hypomochlij, in
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diff changeset
826 recta linea $e vecte collocatio faciet, vt line&aelig; extre-
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diff changeset
827 ma $ecundum datam rationem moueantur. Si enim
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diff changeset
828 recta linea in datam rationem diui$a fuerit, hoc e$t,
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diff changeset
829 vt pars altera ad alter&atilde; eam habeat ration&etilde;, qu&atilde; quis
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diff changeset
830 voluerit. (Quod quid&etilde; quo modo fiat docet Euc.) &amp;
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diff changeset
831 in puncto diui$ionis collocetur hypomochli&utilde;, illius
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diff changeset
832 line&aelig; extrema $ecundum ill&atilde; ration&etilde; mouebuntur:
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diff changeset
833 $iue enim con$ideretur circulor&utilde;, quos illa extrema
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parents:
diff changeset
834 de$cribent, proportio $iue $patium quod illa in linea
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diff changeset
835 perpendiculari notabunt vtroque modo illi motus,
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diff changeset
836 <pb n="27">
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parents:
diff changeset
837 parti&utilde; illar&utilde; proportion&etilde; $eruab&utilde;t. Sit ex&etilde;pli gratia
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parents:
diff changeset
838 linea AB, qu&aelig; in puncto C in datam rationem $ecta
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parents:
diff changeset
839 $it, puta vt pars AC quadrupla $it ad partem CB, mo-
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diff changeset
840 ueat&uacute;rque circa centrum C, punctum A de$cribet cir||
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parents:
diff changeset
841 culum quadrupl&utilde; ad illum quem B
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parents:
diff changeset
842 de$cribet. E$t enim ead&etilde; ratio in cir-
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diff changeset
843 culo diametri ad diametr&utilde;, qu&aelig; e$t
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parents:
diff changeset
844 circunferenti&aelig; ad circunferenti&atilde; (vt
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diff changeset
845 alibi demon$trauimus.) Hac igitur
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diff changeset
846 ratione A puncti motus quadruplus
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847 <fig>
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parents:
diff changeset
848 erit ad puncti B mot&utilde;. Si ver&ograve; ponamus AD perpen-
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diff changeset
849 dicular&etilde; e$$e, &amp; linea AB illi prim&ugrave;m coincid&etilde;s circa
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diff changeset
850 punctum C, moueatur donec A ad D perueniat: tum
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diff changeset
851 eod&etilde; momento B perueniet ad E: motum igitur erit
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852 A in linea perp&etilde;diculari tota circuli maioris diame-
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853 tro, qu&aelig; e$t AD:B ver&ograve; in ead&etilde; linea, minoris t&atilde;t&ugrave;m
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854 circuli diametro mot&utilde; erit, qu<17> e$t BE. Atqui diame||
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855 ter AD quadrupla e$t ad BE, quia ex hypothe$i $emi-
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diff changeset
856 diametror&utilde; illor&utilde; circulor&utilde; proportio e$t, vt 4 ad 1.
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diff changeset
857 Motus igitur p&utilde;cti in linea A perp&etilde;diculari ad mot&utilde;
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858 p&utilde;cti B quadruplus erit: Id&etilde; dicetur $i in data aliqua
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859 alia ratione $ecta $it linea AB. Dem&otilde;$trat&utilde; igitur e$t
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diff changeset
860 quomodo fieri po$$it vt rect&aelig; line<17> extrema $ec&utilde;d&utilde;
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861 dat&atilde; ration&etilde; moueantur. Si igitur illis extremis du&aelig;
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862 vires applicentur, moueb&utilde;tur eod&etilde; ip$o motu: ergo
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863 $ecundum datam vel propo$itam rationem. Quod
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864 a$$umptum erat.
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865 <foot>E 2</foot>
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866 <pb n="28">
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867 <p>Quod autem in vecte demon$tratum e$t, illud e-
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868 tiam in reliquis mediis demon$trandum erit, et$i
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869 lemmati $atisfactum e$t, dum in vno exemplo id
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870 probatum e$t. Ante igitur $ecundum lemma de-
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871 mon$trabimus.
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872 <HEAD>LEMMA II.</HEAD>
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873 <p>Proportionem proxim&egrave; maiorem vel minorem,
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874 qu&agrave;m $it dat&aelig; vis ad datum pondus proportio, de-
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875 terminare vis cuiuslibet quantitas ex motu ciere po||
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876 te$t, metienda e$t. Motum autem ciere pote$t vel in
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diff changeset
877 $ubiecto $uo, vel in alieno. Vis autem men$ura non
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878 $umitur ex eo motu quem in $ubiecto $uo ciere po-
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879 te$t, eo qu&ograve;d licet vis qu&aelig;libet certum motum ha-
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880 beat &amp; determinat&utilde; quo $ubiectum $uum mouet,
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parents:
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881 illius tamen quantitas, vt $upr&agrave; diximus, nondum
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882 demon$trata e$t. Supere$t igitur vt vires motu illo
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883 metiamur, quem in alieno $ubiecto ciere po$$unt, vt
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884 id fiat, qu&aelig;renda nobis $unt $ubiecta qu&aelig; in homi-
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885 num pote$tate $int, &amp; cum vi qualibet committi
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886 po$$int. Omnium autem mobilium $ubiectorum,
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parents:
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887 maxim&egrave; in hominum pote$tate $unt grauia: leuia e-
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888 nim coercere vix po$$umus. Ideo vires illis metiri
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889 $olemus, $ed grauibus qu&aelig; vel vel figuris $uis &amp; com||
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890 page vel va$e aliquo coercentur. Itaque v$us homi-
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891 num certas qua$dam ponderum men$uras $ibi $ta
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892 tui, ponderibus $cilicet quibu$dam certa qu&atilde;titate
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893 <pb n="29">
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894 con$tantibus inditis nominibus, vt e$$ent ponder&utilde;
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895 omnium communes men$ur&aelig;, vt $unt libr&aelig;, vncia,
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896 drachm<17>, &amp;c. quas famo$as men$uras voc&atilde;t. Quem-
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897 admodum igitur numeros numeris, $ic pondera p&otilde;-
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898 deribus metimur. T&atilde;tum enim pondus e$$e dicitur,
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899 quot libras vncias drachmas &aelig;qualiter mouere po-
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900 te$t dempto momento. Nec tantum pondera hoc
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901 modo metimur, $ed etiam alias omnes vires motum
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902 ad locum cientes. Quot enim libras vir aut aliud a-
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903 nimal vel v&etilde;tus aut ignis, aut aliqua alia vis dempto
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904 momento mouere poterit, tot libris illam &aelig;qualem
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905 e$$e dicemus. Si igitur data vel propo$ita vis metien-
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906 da $it, $iquidem naturalis $it, quoniam docuimus
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907 vim naturalem per totum $ubiectum diffu$am e$$e
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908 in rebus homogeneis: id e$t, vt qu&aelig; e$t proportio
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909 molis ad molem, eadem $it ponderis ad pondus: $u-
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910 memus partem aliquam illi homogeneam, aut ex i-
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911 p$o $ubiecto, aut ex alio ip$i homogeneo, e&aacute;mque
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912 famo$a aliqua men$ura metiemur, vtramque $cili-
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913 cet committendo &amp; ob$eruando, quem motum al-
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914 tera in altera ciere po$$it, vbi enim &aelig;quilibrium fa-
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915 cient motibus extremorum, quibus affix&aelig; fuerint,
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916 proportionales erunt per 1 theorema: motus autem
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917 illi linearum dimen$ione quam Geometria docet,
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918 noti erunt, &amp; eorum proportio, nota igitur erit &amp;
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919 virium proportio. Atqui men$ur&aelig; famo$&aelig; nota, per
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920 <foot>E 3</foot>
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921 <pb n="30">
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922 $e e$t quantitas: duorum autem proportione cogni-
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diff changeset
923 ta &amp; alterius quantitate, $tatim &amp; reliqui quantitas
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diff changeset
924 innote$cit per $ecundam da. Eu. nota igitur vis qu&atilde;-
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diff changeset
925 titate qu&aelig; parti illi ine$t, no$cetur &amp; vis qu&atilde;titas qu<17>
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parents:
diff changeset
926 toti inerit: qu&aelig; enim erit ratio molis $ubiecti vis da-
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parents:
diff changeset
927 t&aelig; ad molem particul<17> $umpt<17>, eadem erit vis totius
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diff changeset
928 ad vim partis. H&icirc;c igitur erunt quatuor proportio-
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diff changeset
929 nalia, $cilicet vt moles ad molem: $ic vis ad vim ex
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930 quibus tria nota erunt: moles enim metiri Geome-
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diff changeset
931 tria nos docuit, pr&aelig;terea vis partis, vt demon$traui-
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diff changeset
932 mus, nota e$t, ergo &amp; vis totius quantitas per deci-
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diff changeset
933 mam $eptimam $eptimi Elem. Eucl.
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diff changeset
934 <p>Si ver&ograve; $ubiectum non $it homogeneum, vi ta-
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diff changeset
935 men naturali $it pr&aelig;ditum, $i quidem data vis quam
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diff changeset
936 metiri volumus, ea $itque motum ciere volumus,
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937 tum con$iderabimus qu&aelig; pars in illa, vis minimum
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diff changeset
938 habeat: &amp; ex ea totam ip$ius molem &aelig;$timabimus:
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diff changeset
939 $i ver&ograve; $it vis mouenda, $tatuemus qua$i tota $it ho-
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940 mogenea ip$ius parti, qu&aelig; vis plurimum in $e habe-
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diff changeset
941 bit, &amp; ex ea totam ip$ius molem &aelig;$timabimus. Ita-
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diff changeset
942 que tum illius quanta minima, tum huius quanta
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943 maxima e$$e pote$t, vis quantitas nobis nota erit per
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diff changeset
944 proxim&egrave; demon$tratam rationem. Notis autem vi-
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diff changeset
945 rium quantitatibus, nota erit &amp; earum proportio:
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diff changeset
946 ergo &amp; proportio ip$a proxim&egrave; maior vel mi-
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diff changeset
947 nor: addita enim vel detracta ip$ius denomi-
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948 <pb n="31">
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949 nationi, vnitate erit proxim&egrave; maior vel mi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
950 nor.
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parents:
diff changeset
951 <p>Si ver&ograve; vis data non $it naturalis, voluntari&atilde; qui-
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952 dem ita &aelig;$timare po$$umus, qualis vt plurimum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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953 e$t, &amp; $i quidem ea $it, qua mouere volumus, eam
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954 $tatuemus, quanta minima in eiu$dem generis $ub-
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955 iecto e$$e pote$t, vt $i vim hominis quinquaginta li-
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956 bris &aelig;qualem ponamus, vim equi centum: $i ver&ograve;
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957 ea $it quam mouere volumus, $tatuemus eam qu&atilde;-
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958 ta maxima e$$e pote$t, vt vim hominis 300 libra-
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959 rum, vim equi 500 librarum, &amp; $ic d&eacute; c&aelig;teris: ita vt
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960 nullum $it dubium quin illa minor $it, quam $tatue-
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961 rimus, h&aelig;c ver&ograve; maior.
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parents:
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962 <p>Vis ver&ograve; fortuit&aelig; quantitatem nulla certa con-
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963 iectura a$$equi po$$umus: ita vt qua$cunque ma-
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964 chinas ei aptemus, mod&ograve; moueat, mod&ograve; non mo-
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965 ueat, neque ad no$trum in$titut&utilde; magnoper&egrave; per-
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966 tinet illa inqui$itio: cum fortuitorum nulla $it di$ci-
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967 plina.
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968 <p>His igitur modis virium duarum datarum pro-
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969 portio proxim&egrave; maior nota fiet, quod in $ecundo
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970 lemmate demon$trandum $ump$eramus.
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971 <p>Iam redeamus ad mediorum, quibus vi-
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972 res annectuntur, con$ide-
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973 rationem.
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parents:
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974 <pb n="32">
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parents:
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975 <fig>
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parents:
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976 <p>Docuimus quis $it $implicis ve-
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parents:
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977 ctis effectus, $implex autem ve-
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parents:
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978 ctis $emicirculi conuer$ione $uam
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parents:
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979 operationem ab$ouit, ita vt $i
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parents:
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980 vlteri&ugrave;s F moueatur in in alio $e-
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parents:
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981 micirculo motus prioribus contra-
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parents:
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982 rios cieat: vt exempli gratia, $it
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983 vectis AB, cuius hypomochium
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984 $it C, dum A punctum de$cribet $emicirculum AFD,
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985 motus ille deor$um erit: interea ver&ograve; B punctum
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986 de$cribet $emicirculum BGE a$cendendo: $i ver&ograve; A
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parents:
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987 tran$cendat, D incipiet a$cendere: B ver&ograve; tran$cen-
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988 dens, E de$c&etilde;det: ideo excogitata e$t vectis ratio per-
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989 petua ex plurium vectium $ucce$$ione circa idem
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990 hypomochlium: e$t autem illa tum in ergatis aut $u-
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991 culis, tum in duorum tympanorum homocentrico-
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992 rum, $eu eadem axe transfixorum in planis parallelis
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993 aptatione, quorum $emidiametri $int in eadem pro-
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994 portione qu&aelig; in vecte ad propo$itum motum cien-
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995 dum nece$$aria e$t: centrum ver&ograve; eorum $eu axis fi-
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996 xa $it, ac ita vires aptentur, vt maior minorem, mi-
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997 nor ver&ograve; maiorem tympanum impellat. Quemad-
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998 modum autem horum tympanorum homocentri-
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999 corum opera vectis perpetui ratio inu&etilde;ta e$t, ita eo-
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1000 rum multiplicatione motus, &amp; mouentis &amp; mou&etilde;-
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1001 di, proportio in infinitum augeri pote$t. Cuius rei
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1002 <pb n="33">
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1003 maximus e$t v$us: nec enim materia ad vectem, cu-
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1004 ius longitudo $tadij vnius requireretur, idonea in-
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1005 ueniri po$$et: plurium autem tympanorum propor-
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1006 tionalium aptatione fiet machina tractabilis, cuius
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1007 vis maior erit qu&agrave;m vectis, cuius longitudo $tadij v-
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1008 nius e$$et. Si enim duo tympani homocentrici a-
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1009 ptentur, quorum proportio $it alterius ad alterum,
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1010 decupla, vis qu&aelig; libram vnam &aelig;quabit, vim decem
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1011 libris &aelig;qualem dempto momento mouebit, $i con-
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1012 gru&egrave; illis tympanis aptentur: $i ver&ograve; adhuc duo alij
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1013 tympani fiant, quorum alterius ad alterum decupla
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1014 $it proportio, ac minor illor&utilde; ita aptetur, vt maior&etilde;
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1015 ex prioribus moueat, appendatur deinde minori ex
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1016 prioribus tympanis vis centum libris &aelig;qualis, maio-
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1017 ri ver&ograve; ex po$terioribus vis vni libr&aelig; &aelig;qualis, tum
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1018 h&aelig;c illam mouebit, &amp; $ic in infinitum motus extre-
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1019 morum proportio multiplicari pote$t. Flexibili ve-
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1020 r&ograve; medio quies ita aptari pote$t, vt duo eius extre-
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1021 ma diuer$is motibus moueantur, &amp; quidem $ecun-
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1022 dum datam rationem. Exemplum habemus in orga||
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1023 nis poly$pactis, $eu trochleis, in quibus altero funis
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1024 extremo immobili manente, reliquum funis circa
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1025 plures trochleas conuoluitur, quarum ali&aelig; centris
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1026 immobilibus fix&aelig; $unt, ali&aelig; ver&ograve; ip$is contrari&aelig; cen
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1027 tris mobilibus. Atque ita circa illas conuoluitur fu-
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1028 nis, vt inter eas $it $patium tantum, quanta e$t linea,
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1029 <foot>F</foot>
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1030 <pb n="34">
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1031 per quam motum ciere volumus. Quoties igitur fu-
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1032 nis extremum quod moueri pote$t, trahitur, $ingul<17>
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1033 reuolutiones &aelig;qualiter minuuntur, e&aacute;que ratione,
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1034 di$t&atilde;tia qu&aelig; e$t inter trochleas contrarias minuitur,
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1035 ita $cilicet vt quot $unt reuolutiones, in tot partes
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1036 di$tributum $it motus $patium. Quot igitur reuolu||
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1037 tiones erunt, totuplex erit motus extremitatis funis
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1038 ad motum trochlearum mobilium ver$us fixas. Si i-
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1039 gitur motus i$tius extrema con$tituantur, alterum
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1040 quid&etilde; funis illa extremitas qu&aelig; mouetur, alter&utilde; ve-
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diff changeset
1041 r&ograve; terminus $patij illius, quod e$t &agrave; trochleis fixis ad
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1042 mobiles contrarias: qu&aelig; erit proportio numeri con-
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1043 uolutionum funis ad vnitatem, eadem erit motus,
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1044 quo funis extremum mouebitur ad motum quo al-
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1045 terum extremum mouebitur: pote$t autem in infi-
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1046 nitum augeri conuolutionum numerus, ergo &amp; mo||
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1047 tus illius proportio.
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1048 <p>Angulus autem ad motum ciendum ita ad-
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1049 hibetur. Diximus motus men$uram in nutus li-
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1050 nea $umi, quantum igitur vis aliqua per eam ver$us
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1051 locum naturalem mouetur, tanta e$t, quantum ver&ograve;
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parents:
diff changeset
1052 per eam &agrave; loco naturali reuellitur, tanta e$t eius re$i-
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parents:
diff changeset
1053 $tentia. Quod ver&ograve; per lineam &agrave; loco naturali &aelig;qu&egrave;
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parents:
diff changeset
1054 di$tantem (id e$t, per eam qu&aelig; nutus lineas $ecat ad
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parents:
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1055 angulos rectos) mouetur, illud mouenti non re$i-
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diff changeset
1056 $tit, omnium autem linearum inter illas intercepta-
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1057 <pb n="35">
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parents:
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1058 rum, ac cum illis in earum inter$ectionis puncto c&otilde;-
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parents:
diff changeset
1059 currentium, qu&aelig; obliqu&egrave; nutum ver$us, aut contra
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parents:
diff changeset
1060 nutum ferri dicuntur, quo propi&ugrave;s qu&aelig;libet ad nu-
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parents:
diff changeset
1061 tus lineam accedit, per illam rei mot&aelig; vis aut re$i$t&etilde;-
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parents:
diff changeset
1062 tia maior e$t: qu&ograve; ver&ograve; propi&ugrave;s ad lineam &agrave; loco na-
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parents:
diff changeset
1063 turali &aelig;qui di$tantem accedit, e&ograve; minor e$t. Omni&utilde;
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parents:
diff changeset
1064 autem maxima e$t in nutus linea, &aelig;quidi$tans ver&ograve;
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parents:
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1065 &agrave; loco naturali motui per lineam nutus omnino op||
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parents:
diff changeset
1066 po$ita e$t, obliqu&aelig; ver&ograve; non ita quia $ecundum illas
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parents:
diff changeset
1067 eod&etilde; $patio delata vis propi&ugrave;s ad loc&utilde; natural&etilde; acce-
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parents:
diff changeset
1068 dit, aut ab eo recedit, qu&agrave;m e$$et, c&ugrave;m moueri c&oelig;pit.
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parents:
diff changeset
1069 <fig>
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parents:
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1070 <p>Sit exempli gratia AB linea
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parents:
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1071 nutus, vis alicuius, puta ponde-
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parents:
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1072 ris, $itque A $ur$um &amp; contra nu||
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parents:
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1073 tum: B ver&ograve; deor$um &amp; nutum
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parents:
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1074 ver$us, de$cribat&uacute;rque circulus,
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parents:
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1075 cuius AB $it diameter, qu&agrave;m CD,
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parents:
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1076 alia diameter $ecet ad angulos
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1077 rectos in centro E: omnes igitur
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parents:
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1078 line&aelig; &agrave; centro E ad circunferentiam circuli duct&aelig;,
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1079 qu&aelig; cad&etilde;t intra $emicircul&utilde; CAD, contra nut&utilde; a$c&etilde;-
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parents:
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1080 dere dicentur, quatenus circunferentiam $pectant:
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parents:
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1081 quatenus ver&ograve; centrum $pectant, de$cendere dicen-
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parents:
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1082 tur: c&otilde;tra ver&ograve; omnes in $emicirculo CBD &agrave; centro
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parents:
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1083 ad circunferentiam duct&aelig; de$cendere circunferen-
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parents:
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1084 tiam ver$us, &amp; centrum ver$us a$cendere dicentur:
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1085 <foot>F 2</foot>
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1086 <pb n="36">
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parents:
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1087 ill&aelig; igitur erunt, qu&aelig; obliqu&egrave; nutum ver$us aut con||
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parents:
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1088 tra nutum ferri dic&utilde;tur: linea ver&ograve; CED, neque a$c&etilde;-
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parents:
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1089 det, neque de$cendet: line&aelig; ver&ograve; in ip$am ad angu-
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parents:
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1090 los rectos incidentes nutus line&aelig; erunt, quoniam li-
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parents:
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1091 ne&aelig; AB parallel&aelig; erunt: $i igitur &agrave; centro E duc&atilde;tur
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parents:
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1092 line&aelig; ad circunferentiam inter A &amp; D, puta EF, EG,
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parents:
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1093 EH, quarum EF $it proxima line&aelig; AB:EH ver&ograve; pro-
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parents:
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1094 xima line&aelig; CD, ac per illas moueantur contra nut&utilde;
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1095 tres vires &aelig;quales eodem tempore, ita vt prima per
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1096 lineam EF perueniat ad punctum F: $ecunda ver&ograve;
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1097 per EG perueniat ad G, tertia per EH perueniat ad H,
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1098 dico vis mot&aelig; per EF maiorem fore re$i$tentiam,
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1099 qu&agrave;m illius qu&aelig; per EG aut EH, mouebitur &amp; illius
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1100 qu&aelig; per EG mouebitur, maiorem qu&agrave;m eius qu&aelig;
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1101 per EH mouebitur. Ducantur enim &agrave; punctis FGH
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1102 in lineam ED perpendiculares FK, GL, HM, ill&aelig; er&utilde;t
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1103 nutus line&aelig;: quanta igitur erit FK, tantum vis prima
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1104 mota cen$ebitur, quanta ver&ograve; GH, tantum vis $ecun||
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1105 da: quanta ver&ograve; HM, tantum vis tertia mota cen$ebi-
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1106 tur: at qui qu&aelig; e$t ratio motus ad motum in viribus
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1107 &aelig;qualibus, per quartam c&otilde;clu$ionem huius tracta-
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1108 tus, eadem e$t re$i$tenti&aelig; ad re$i$tentiam: qu&aelig; igitur
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1109 erit ratio linearum illarum perpendicularium, inter
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1110 $e eadem erit &amp; re$i$t&etilde;tiarum virium per lineas ob-
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1111 liquas motarum, &agrave; quibus ill&aelig; perpendiculares du-
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1112 ct&aelig; $unt, atqui quo line&aelig; ill&aelig; perpendiculares pro-
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1113 <pb n="37">
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1114 pi&ugrave;s ad AB circuli diametrum accedunt, e&ograve; $unt ma||
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1115 iores per decimam quartam tertij Elem. Eu. ergo &amp;
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1116 vires per eas lineas delat<17> &agrave; quar&utilde; extremitatibus du||
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1117 c&etilde;tur, maiores re$i$t&etilde;tias habeb&utilde;t: at qui qu&ograve; magis
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1118 ad AB, accedunt eo magis ab ED recedunt: qu&ograve; igi-
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1119 tur magis ad ED accedent, e&ograve; minores erunt re$i$t&etilde;-
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1120 ti&aelig;, hinc $equitur tanquam corollarium. Si du&aelig; vi-
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1121 res per$e &aelig;quales triangulo rectangulo aptentur, cu||
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1122 ius latus alterum, angulum rectum con$titu&etilde;tium:
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1123 $it earum nutus linea, ac per illud vis altera mouea-
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1124 tur, altera ver&ograve; per latus angulo recto oppo$itum,
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1125 qu&aelig; erit ratio huius lateris ad illud, eadem erit re$i-
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1126 $tentia vis illius ad vis huius re$i$t&etilde;tiam. Atqui duo-
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1127 rum illorum laterum proportio in infinitum auge-
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1128 ri vel minui pote$t, ergo &amp; re$i$tentia virium illis ap||
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1129 plicatarum. Hoc igitur modo po$$umus vti angulo
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1130 $eu triangulo ad motus ciendos, nempe eo immoto
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1131 vires per eius latera mouendo.
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1132 <p>Sed &amp; alia ratione eo vti po$$umus, ip$um $cilicet
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1133 triangulum mouendo, qui tunc cuneus dicitur. Vt
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1134 autem eo hac ratione vtamur, vires ita di$ponere o-
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1135 portet, vt altera illarum vni ex lateribus angulum
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1136 rectum con$tituentibus incumbat, altera ver&ograve; lateri
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1137 ip$um $ubtendenti. Illa enim tant&ugrave;m mouebitur,
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1138 quantum latus cui altera incumbit. Sit exempli gra-
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1139 <foot>F 3</foot>
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1140 <pb n="38">
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1141 <fig>
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1142 tia triangulus ABC, cuius angulus B
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1143 rectus $it, latus ver&ograve; illum $ubtendens
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1144 $it AC, incumb&aacute;tque vis D lateri AB,
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1145 vis ver&ograve; E lateri AC, $itque vis D nutus
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1146 linea BC, vis ver&ograve; E nutus $it linea AB, erigat&uacute;rque &agrave;
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1147 puncto C linea CF perp&etilde;dicularis ad BC &aelig;qualis AB,
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1148 &agrave; qua vis E n&otilde; di$cedat. Si triangulum illud in plano
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1149 fixo moueatur, donec AB perueniat ad CF, mota e-
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parents:
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1150 rit vis D nutu $uo tant&ugrave;m, quanta e$t linea BC, vis ve-
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1151 r&ograve; E tantum, quanta e$t linea AB. C&ugrave;m autem po$$it
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1152 in infinitum augeri &amp; minui, laterum illorum pro-
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1153 portio, po$$unt etiam duorum illorum extremor&utilde;
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parents:
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1154 motus in data ratione con$titui. Quanta enim erit
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1155 BC ad AB, tantus erit motus vis D ad motum vis E:
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1156 ergo &amp; in hoc medio primum lemma no$trum de-
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parents:
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1157 mon$tratum e$t.
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parents:
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1158 <p>In hoc autem medij genere hoc diuer$um ab
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parents:
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1159 aliis mediis accidit, qu&ograve;d non tam facil&egrave; vtrin-
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parents:
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1160 que motus eo cietur, ac in illis, in quibus $i virium
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parents:
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1161 proportio momento vel $uperet, vel minor $it pro-
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parents:
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1162 portione motus extremor&utilde; medij, tum motus hinc
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1163 vel inde cietur. Atqui in hoc propter $uperficie-
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1164 rum contactum, quarum pori vel a$peritates $e-
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1165 $e mutu&ograve; $ubingrediuntur, &amp; ita inuicem adh&aelig;-
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parents:
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1166 rent, fit vt &aelig;gri&ugrave;s cieatur motus, facili&ugrave;s ve-
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parents:
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1167 <pb n="39">
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1168 r&ograve; mouebuntur, $i ill&aelig; $uperficies leues fue-
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1169 rint, vt $i pinguibus inungantur, vel ex materia
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parents:
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1170 leui con$tent. Ide&ograve; enim reliqua facili&ugrave;s mo-
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parents:
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1171 uentur, qu&ograve;d circa puncta veluti qu&aelig;dam mo-
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parents:
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1172 ueantur.
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casties
parents:
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1173 <p>Simplicis autem trianguli rectilinei aut cu-
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1174 nei in diuturnis motibus ciendis rarus e$t v$us,
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1175 tum ob illud quod notauimus incommodum ex
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1176 vitio materi&aelig;, quo fit vt in eo qua$$atione opus
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1177 $it, tum etiam qu&ograve;d breui eius operatio termine-
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parents:
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1178 tur. Ideo illo vtimur aut c&ugrave;m $olutionem con-
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1179 tinui molimur, qu&aelig; breui tempore fit, vel
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1180 c&ugrave;m aliquid di$tendere aut aliquid figere volu-
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1181 mus.
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1182 <p>Anguli ver&ograve; curui linei magnus e$t v$us, pr&aelig;-
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parents:
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1183 $ertim helicis cylindric&aelig;. Nihil enim aliud e$t he-
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parents:
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1184 lix qu&agrave;m triangulus curuus: $i enim alteram ex
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casties
parents:
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1185 lineis rectis angulum con$tituentibus, cylindri
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casties
parents:
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1186 ba$i ip$i line&aelig; &aelig;quali obuolueris, reliquam ve-
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1187 r&ograve; $eruato eodem quem con$tituunt angulo, $u-
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1188 per cylindri $uperficiem curuaueris, habebis he-
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1189 licem cylindricam, quam $i iterum $eruato eo-
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1190 dem angulo in rectum extendas, habebis trian-
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1191 gulum rectilineum. H&aelig;c autem helix commo-
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1192 di$$ima e$t, tum qu&ograve;d in parua mole triangulum
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1193 <pb n="40">
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1194 longi$$imum obuolutum contineat tum quod par-
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1195 tes eius omnes $ibi inuicem c&otilde;gruant: omnes enim
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1196 partes helicis cylindric&aelig;, aut circa eundem, aut circa
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diff changeset
1197 &aelig;quales cylindros de$cript&aelig;, $eruato eodem angulo
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diff changeset
1198 $ibi inuicem $uppo$it&aelig; congruunt. Quo fit, vt $i ca-
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1199 ui cylindri interiori $uperficiei impre$$a $it helix, a-
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1200 lia ver&ograve; cylindri $uperficiei connex<17> ip$i cau<17> &aelig;qua-
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diff changeset
1201 li, $eruato eodem qui in illo e$t, angulo, $ibi inui-
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1202 cem &amp; omnes vnius omnibus alterius partibus con||
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1203 gruant.
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1204 <p>Huius autem medij c&ugrave;m plures $int partes, con||
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diff changeset
1205 $tat enim duabus $uperficiebus, pluribus etiam mo||
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1206 dis variari pote$t. In $umma autem eius affectus hic
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diff changeset
1207 e$t, vt cylindri ba$i in $uperficie immobili circum a-
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1208 xem conuer$a, vis mouenda helicem premat: dum
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diff changeset
1209 enim cylindrus circum axem conuertetur, vis mo-
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1210 uenda qualibet reuolutione tant&ugrave;m $ecundum cy-
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1211 lindri longitudinem mouebitur, quanta e$t di$tan-
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1212 tia duarum helicis $pirarum. Qu&aelig; igitur erit propor||
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diff changeset
1213 tio circunferenti&aelig; circuli ba$im cylindri con$titu&etilde;-
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diff changeset
1214 tis ad illam di$t&atilde;tiam, eadem erit motus orbicularis
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diff changeset
1215 cuiu$libet puncti in cylindri $uperficie $ignati ad mo||
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1216 tum rectum vis helicem prementis.
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diff changeset
1217 <p>Illud igitur medium ex duobus compo$itum e$t,
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diff changeset
1218 recto $cilicet &amp; curuo: ita igitur vires illi aptand&aelig;
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diff changeset
1219 $unt, vt eius quam mouere volumus, nutus linea $it
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1220 <pb n="41">
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1221 cylindri longitudo: illa ver&ograve; qua mouere volumus,
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1222 in orbem moueatur, aut $i eius nutus linea recta $it,
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diff changeset
1223 circa cylindrum fluens illum moueat contingendo.
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diff changeset
1224 <p>Et h&aelig;c quidem de mediis in $uo genere con$i$t&etilde;-
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parents:
diff changeset
1225 tibus. Po$$unt autem fieri eorum Syzygi&aelig; vectis $ci-
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diff changeset
1226 licet cum trochleis, helicis cum vecte, aut cum tro-
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parents:
diff changeset
1227 chleis, aut omni&utilde; $imul. Vectis cum trochleis $i fu-
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diff changeset
1228 nis illud extremum, quod mouetur in trochleis, er-
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parents:
diff changeset
1229 gatis aut $uculis inuoluatur: helix cum vecte coni&utilde;-
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diff changeset
1230 getur, $i tympanus circum axem moueatur, $itque i-
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diff changeset
1231 ta denticulatus, vt dentes ip$ius helicem in cylindro
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diff changeset
1232 excauatam ingrediantur, quam machinam helicem
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diff changeset
1233 perpetuam vulg&ograve; vocant, e&ograve; qu&ograve;d eius conuer$io
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1234 perpetua e$$e po$$it, cum helicis $implicis operatio
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1235 non excedat ip$ius longitudinem. Quorum omni&utilde;
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1236 tum inter $e, tum ad $ubiecta mouenda accommo-
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parents:
diff changeset
1237 datio ade&ograve; varia e$t, vt $cripto c&otilde;preh&etilde;di vix po$$it.
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1238 Ex his autem qu&aelig; dicta $unt, mediocris ingenij me-
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1239 chanicus poterit ea prout ip$i nece$$e erit aptare. In
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diff changeset
1240 machinis autem omnibus h&aelig;c cautio adhib&etilde;da e$t,
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1241 vt earum $tructura firma $it, pr&aelig;$ertim ver&ograve; vbi cir-
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1242 cum axes aliquas fit motus, deinde vt vincula, qui-
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1243 bus illis vires affing&utilde;tur, valida $int, illarum enim o-
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parents:
diff changeset
1244 mnium vis in $uo $tatu manendi vtrique vi aptand&aelig;
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parents:
diff changeset
1245 &aelig;qualis e$$e debet: aget enim in illas vis vtraq;, qu&ograve;d
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1246 $i medium firmum non $it, motus in ip$a machina
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1247 <foot>G</foot>
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1248 <pb n="42">
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parents:
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1249 ciebitur. Itaque di$$oluetur, ide&ograve; manca videtur pe-
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parents:
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1250 titio illa Archimedis in hoc problemate <G>do\s pou= stw= ta\n
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parents:
diff changeset
1251 ga\n kinw=</G>, qu&ograve;d locum t&atilde;t&ugrave;m in quo con$i$tat, $ibi da-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
1252 ri po$tulet, c&ugrave;m pr<17>terea vincula quibus terra &agrave; loco
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parents:
diff changeset
1253 $uo naturali remota $u$tineri po$$et, petere debue-
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parents:
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1254 rit: id aut&etilde; vt &amp; ea qu&aelig; de motu in infinitum augen-
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1255 do vel minuendo diximus, ita intelligenda $unt, vt
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parents:
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1256 $ciamus infinita hominum pote$tati, quacunque ar-
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diff changeset
1257 te iuuetur, non $ube$$e: quamuis enim Geometri-
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parents:
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1258 ca con$ideratio in infinitum $e$e ext&etilde;dat, $unt ta-
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parents:
diff changeset
1259 men certi fines, vltra quos natura rerum nos pro-
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diff changeset
1260 gredi non patitur: $unt pr&aelig;terea vitia materi&aelig;,
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parents:
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1261 qu&aelig; Geometra non con$iderat, illa tamen non
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parents:
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1262 ob$tant qu&ograve; min&ugrave;s id quod proponitur, ver&utilde; $it in
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parents:
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1263 intellectu. An ver&ograve; id quod proponitur tale $it, vt in
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1264 opus educi po$$it, con$iderabit Geometer habita ra-
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parents:
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1265 tione circun$tantiarum, pr&aelig;$ertim ver&ograve; temporis,
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1266 quod ip$i pr&aelig;$cribetur, &amp; $umptuum quos facere
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1267 poterit is qui aliquid faciendum proponet, qu&aelig; $i
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1268 abund&egrave; $uppetant, nihil non fieri poterit.
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1269 <p>Vt igitur hunc tractatulum concludamus, ac ve-
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parents:
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1270 lut in $ummam contrahamus: In motibus ciendis
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1271 tria $unt con$ideranda. Vis qua motum ciere volu-
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1272 mus, vis quam mouere volumus, &amp; motum quo
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1273 mouere volumus: duo enim qu&aelig;libet ex illis ter-
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1274 tium determinant. Si enim vi parua vim magnam
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1275 <pb n="43">
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1276 mouere volumus, id nonni$i paruo motu facere
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diff changeset
1277 po$$umus: $i ver&ograve; vim aliquam magno motu mo-
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diff changeset
1278 uere velimus, vi magna mouente ad id opus e$t. Si
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diff changeset
1279 vi parua magnum motum ciere volumus, mini-
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diff changeset
1280 mam vim mouendam e$$e oportet: vt puta, $i libra
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diff changeset
1281 vna centum libras mouere velimus, oportet motum
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diff changeset
1282 illius, motu huius c&etilde;tuplo maiorem e$$e. Si ver&ograve; ve-
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parents:
diff changeset
1283 limus libra vna aliam vim ita mouere, vt ea cen-
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diff changeset
1284 tuplo citi&ugrave;s moueatur, qu&agrave;m libr&aelig; illius pondus, il-
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diff changeset
1285 lam cente$imam tant&ugrave;m libr&aelig; vnius partem e$$e ne-
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diff changeset
1286 ce$$e e$t: $i ver&ograve; libram vnam ita mouere velimus, vt
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diff changeset
1287 centuplo citius moueatur, qu&agrave;m vis qu&aelig; illam mo-
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1288 uebit, vi centum libris maiore ad id opus erit. Neq;
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diff changeset
1289 patitur natura $ibi in his vim fieri: $i enim eiu$modi
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diff changeset
1290 proportio aliquo modo infringi po$$et, $tat<*> da-
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diff changeset
1291 retur <G>au)tw/ma e)nde/lexes</G>, vel vt voc&atilde;t, motus perpetuus
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diff changeset
1292 in perpetua materia.
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parents:
diff changeset
1293 <p>Ex his igitur fundamentis inuent&aelig; $unt machin&aelig;
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diff changeset
1294 omnes, quotquot antehac $unt excogitat<17>, &amp; quot-
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diff changeset
1295 quot po$thac excogitari poterunt, ad ea referentur.
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diff changeset
1296 <p>Itaque $i propo$it&aelig; cuiu$cunque machin&aelig; effe-
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diff changeset
1297 ctum $cire velimus, con$ideranda $unt duo eius
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1298 extrema, quibus vires applicantur: qu&aelig; enim erit
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diff changeset
1299 ratio motus vnius ex illis extremis, ad motum
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1300 alterius eadem erit &amp; virium, qu&aelig; illis extre-
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1301 mis ad motum ciendum applicari poterunt,
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1302 <foot>G 2</foot>
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1303 <pb n="44">
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1304 addito aut dempto momento, vt $i dum alter&utilde; ma-
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1305 chin&aelig; extremum palmo vno mouetur, alterum cen||
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1306 tum palmis moueatur vis qu&aelig;libet huic annexa, al-
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diff changeset
1307 teram alteri annexam centuplam momento min&ugrave;s,
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1308 mouebit: $ed motu centuplo: mouebit autem &amp; re-
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1309 liquas omnes vires, qu&aelig; infra centuplam proportio-
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diff changeset
1310 nem ad eam habebunt: $i ver&ograve; non vim centuplam
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diff changeset
1311 mouere, $ed in data vi motum centuplum ciere veli-
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1312 mus, eam quidem in extremo quod centuplo citi&ugrave;s
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diff changeset
1313 mouetur, locare oportebit, alteram ver&ograve; alteri extre||
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1314 mo centuplo maiorem adhibere nece$$e erit.
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1315 <p>Hinc oriuntur tria problemata, videlicet data vi
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1316 datum pondus mouere, quod iam dem&otilde;$trauimus:
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1317 item data vi datum motum ciere, quod ex pr&aelig;ced&etilde;-
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1318 tis demon$tratione ab$oluitur: tertium, datam vim,
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1319 dato motu mouere, quod quidem facili&ugrave;s demon-
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1320 $tratur, qu&agrave;m ab$oluitur. Scimus enim, vt id fiat,
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1321 vim aliam proportionalem (vt docuimus) requiri,
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1322 quo modo autem illam habere po$$imus, explicare
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1323 hoc opus, hic labor e$t. Non enim quemadmodum
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1324 organa ad motus in data proportione ciendos ha-
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1325 bere po$$umus: ita &amp; vires infinit&aelig; magnitudinis po||
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1326 te$tati no$tr&aelig; $ub$unt.
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1327 <p>Quod enim ad vires in grauitatis &amp; leuitatis gene||
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1328 re con$i$tentes attinet, eas vt moueant ant&egrave; moueri
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diff changeset
1329 oportet &agrave; loco vel $itu naturali, vi aut&etilde; aliqua id fie-
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1330 <pb n="45">
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diff changeset
1331 ri oportet, qu&aelig; iam in promptu $it. Qu&aelig; $i tanta e$t,
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1332 vt po$$it motum propo$itum ciere, fru$tr&agrave; fiat, $i alij
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1333 &agrave; loco naturali remouend&aelig;, qua po$tea ad motum
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1334 vtendum $it, adhibeatur.
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1335 <p>Nullum igitur ex eiu$modi viribus commodum
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parents:
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1336 percipi pote$t, ni$i quis in futurum $ibi pro$piciens,
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parents:
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1337 multa &agrave; loco vel $itu naturali ex otio remoueat, vt
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1338 iis, c&ugrave;m opus erit ad motus ciendos vtatur. Hac ra-
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parents:
diff changeset
1339 tione quantas vires in promptuario habebit, tantos
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parents:
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1340 motus ciere poterit. Pr&aelig;cipuum igitur quod ad hu-
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parents:
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1341 ius problematis con$tructionem pertinet, e$t, vt vi-
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parents:
diff changeset
1342 res qu&agrave;m maximas pote$tati no$tr&aelig; $ubiiciamus, vo||
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parents:
diff changeset
1343 luntariis aut fortuitis viribus naturales pr&aelig;parando,
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1344 aut &agrave; natura pr&aelig;paratas, qu&aelig; pote$tati no$tr&aelig; $ub-
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1345 $unt, accipiendo. Qualia multa $i mortales aduerte-
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1346 rent, fieri po$$ent, vt ali&agrave;s, Deo duce, docebimus.
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parents:
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1347 <p>Earum autem virium qu&aelig; in raritatis &amp; den$ita-
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1348 tis proportione con$i$tunt, $ubiecta plurim&ugrave;m in
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1349 no$tra pote$tate $unt: multa enim $unt naturalia $ub
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parents:
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1350 iecta actu, den$a potentia ver&ograve; rara. Si quod igitur
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1351 eorum potentia proxima $it rari$$imum, ita vt nullo
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1352 negotio actus ille raritatis induci po$$it, concluda-
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1353 t&uacute;rq; loco aliquo angu$to, po$tea inducatur ille a-
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1354 ctus, c&ugrave;m rara maior&etilde; locum occupent, quum den-
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1355 $a, fiet vt locus in omnem partem di$tendatur, illius
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1356 autem partes min&ugrave;s coh&aelig;rentes, tantum impellen-
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parents:
diff changeset
1357 <foot>G 3</foot>
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1358 <pb n="46">
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parents:
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1359 tur, quanta e$t proportio molis rei rarefact&aelig; ad mo-
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parents:
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1360 lem illius c&ugrave;m den$a e$$et: illa autem raritatis po-
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parents:
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1361 tentia proxima e$t in compo$itione $ulphuris
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1362 &amp; nitri: ea igitur &amp; $imilibus $ubiectis,
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parents:
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1363 in data vi datus motus
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casties
parents:
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1364 cieri pote$t.
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1365 <HEAD>FINIS.</HEAD>
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1366 <fig>
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1367 <HEAD>Errata qu&aelig; inter imprimendum irrep$erunt,
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1368 $ic corrigito.</HEAD>
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1369 <p><I>Pagina</I> 7. <I>linea</I> 10, <I>munere, lege nuere. pag.</I> 11. <I>lin.</I> 12, <I>comparationibus, lege comparati, ommi-
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1370 bus. pag.</I> 12. <I>lin.</I> 16. <I>finis, lege fini. pag. eadem. lin. penult. Altero, lege Alto. pag.</I> 20.
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1371 <I>lin.</I> 16. <I>quum</I> C <I>ad</I> L. <I>lege quam. pag.</I> 25. <I>lin.</I> 5. <I>nunc $it ratio, lege minor $it. linea</I> 13. <I>Vi-
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1372 res maiori litera</I> (<I>e$t enim $ectionis initium. lin.</I> 15. <I>mittant, lege mittantur. pag.</I> 27. <I>li.</I> 20.
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1373 <I>puncti in linea</I> A, <I>lege puncti</I> A <I>in linea. pag.</I> 28. <I>lin.</I> 9. <I>vis, lege V is, maiori litera</I> (<I>e$t enim
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1374 $ectionis initium. lin. e. motu ciere, leg. motu quem ciere. lin.</I> 23. <I>metiri $olemus, lege non
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1375 $olemus. lin. vlt. $tatui, lege $tatuit. pag.</I> 32. <I>lin.</I> 4. <I>ab$ouit, lege ab$oluit. lin.</I> 5. <I>dele</I> F. <I>lin.</I>
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1376 12. <I>&amp;</I> 13, <I>pone virgulam po$t</I> D <I>&amp;</I> E. <I>lin.</I> 8. <I>hypomochium, lege hypomochlium. pag.</I> 37. <I>li.</I>
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1377 13. <I>re$i$tentia, lege re$i$tenti&aelig;. pag.</I> 39. <I>lin.</I> 3. <I>leui, lege l&aelig;ui. pag.</I> 40. <I>linea</I> 5. <I>$uppo$it&aelig;, lege
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1378 $uperpo$it&aelig; li.</I> 7. <I>connex&aelig;, lege conuex&aelig;. lin.</I> 13. <I>affectus, lege effectus.</I>