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annotate texts/archimedesOldCVSRepository/archimedes/raw/baldi_mecha_01_la_1621.raw @ 23:5690c4247e05
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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Tue, 14 May 2013 12:45:18 +0200 |
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| rev | line source |
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1 <pb> |
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2 <pb id="p.0002"> |
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3 <HEAD>BERNARDINI</HEAD> |
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4 <HEAD>BALDI VRBINATIS |
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5 GVASTALLÆ AB- |
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6 BATIS |
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7 <I>IN</I></HEAD> |
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8 <HEAD>MECHANICA ARISTOTE |
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9 LIS PROBLEMATA |
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10 EXER CITATIONES:</HEAD> |
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11 <HEAD><I>ADIECTA SVCCINCTA NAR- |
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12 ratione de autoris vita & $criptis.</I></HEAD> |
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13 <fig> |
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14 <HEAD><I><*>NTIAE.</I></HEAD> |
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15 <HEAD>Typis & Sumptibus Viduæ Ioannis Albini.</HEAD> |
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16 <HEAD><*></HEAD> |
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17 <pb> |
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18 <fig> |
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19 <HEAD><I>NOBILISSIMO AC GENE- |
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20 ROSO DOMINO</I></HEAD> |
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21 <HEAD>D. ADAMO PHILIP- |
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22 PO BARONI A CRON- |
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23 BERG, EQVITI, SACRÆ CÆSA- |
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24 REÆ MAIESTATIS, ET SERENISSIMI</HEAD> |
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25 <HEAD>Principis Archiducis Alberti Camerario intimo & c. |
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26 Domino meo gratio$i$$imo.</HEAD> |
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27 <p>Opportune $ub hoc ip$um tem- |
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28 pus, quo in Belgium ad Scre- |
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29 ni$$imos Principes iter ador- |
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30 nat. Nobili$$ima & Genero$a |
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31 Dom. V.^{ra}, prodit no$tris for- |
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32 mis in publicum editus Com- |
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33 mentarius Bernardini Baldi Vrbinatis Gua- |
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34 $tallæ Abbatis in Ari$totelis Mechanica. Is |
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35 virin omni $cientiæ genere, at maxime inMa- |
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36 thematicis di$ciplinis fuit ver$ati$$imus, quod |
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37 multa ab eo præclare $crip ta te$tantur opera, |
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38 ex quibus paucula edita, reliqua vero $pera- |
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39 <foot>):( 2</foot> |
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40 <pb> |
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41 mus $uo temporein publicam luce<*> produ- |
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42 cenda. Cum vero nemini$it ob$curum Nobi- |
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43 li$$imæ ac Genero$æ Dom. V.^{ræ} id $emper |
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44 extiti$$e familiari$$imum, vttum dome$ticum |
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45 otium, tum maxime peregtinationes, quibus |
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46 totam pæne Europam $umma cum laude |
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47 circum$crip$it, tum variarum linguarum per- |
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48 fecto v$u, tum Mathematicarum di$ciplina- |
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49 rum notitia & exercitio redderet iucũdiores, |
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50 nulla me tenet dubitatio quin & Baldum Vr- |
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51 binatem no$tris typis loquentem in hoc iti- |
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52 nere, quod à Deo felici$$imum Nobili$$imæ |
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53 acGenero$æ Dom. V.^{ræ} precor, in $uum comi- |
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54 tatum ac tutelam beneuolo animo $it admi$- |
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55 $ura. Id rogo humillime $imulque precor, vt. |
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56 hanc meam typographiam plurimis iam re- |
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57 tro annis de inclytæ familiæ Cronbergicæ tu- |
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58 tela gloriantem, $uo fauore pro$equatur, vi- |
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59 duæque afflictæ fortunis beneuole ad$piret. |
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60 Sic Deus Nobili$$. & Genero$am Dom. V.^{ram} |
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61 illu$tret omnibus bonis, eamque R.^{mo} & Ill.^{mo} |
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62 Principiac Domino meo Clementi$$imo, D. |
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63 Ioanni Suicardo Archiepi$copo Mogunti- |
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64 no Principi Electori ac per <I>G</I>ermaniam Ar- |
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65 <pb> |
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66 chicancellario &c. patruo $uo optati$$imo |
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67 $aluo florentique redhibeat $aluum $imili- |
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68 ter florentem ac incolumem. Moguntiæ è |
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69 typographeio Viduæ Albinianæ, honori No- |
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70 bili$$imæ ac <I>G</I>enero$æ Dom. Ve$træ perpe- |
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71 tuum dicato. Anno 1621.26.Martij. |
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72 <foot>):( 3</foot> |
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73 <pb> |
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74 <fig> |
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75 <HEAD>PRAEFATIO.</HEAD> |
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76 <p><I>Diligenter legenti mihi quæ$tiones il- |
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77 las, in quibus ea quæ ad Mecha- |
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78 nicam facultatem pertinent, expli- |
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79 cantur, multa in mentem venie- |
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80 bant; & primum quidem eorum, quæibi di<G>c</G>pu- |
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81 tantur, vtilitatem, $ubtilitatem, copiam admi- |
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82 rabar: Tum ex animo dolebam, aureum hunc li- |
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83 bellum propènegligi, & ab iis qui pulcherrimis |
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84 hi$ce $tudiis dant operam, a<02>iduè præ manibus |
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85 non haberi: Multas autem Auctoriip$ihaben- |
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86 das referendas<26> e$$e gratias, qui tam egregiam, |
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87 vtilem & probèin$tructam $upellectilem Archi- |
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88 tectis, Mechanicis, & omnibus ferè Artificibus |
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89 $uppeditauerit. Ari$toetlis nomini a$cribitur |
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90 Commentarius, licet nonnulli, $itne Philo$ophi |
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91 illius præclari<02>imi & acuti<02>imilabor, an non, |
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92 adfirmare $ubdubitauerint. Ari$totelis tamen |
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93 e$$e omnes ferè meliores con$entiunt: <*>dquetum |
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94 ex phra$i, & explicatione, quæ Ari$totelem $a- |
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95 piunt, tum iudicio $ubtilitatis & rationum, qui- |
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96 <pb> |
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97 bus quæ$tiones ip$æ ingenio$i<02>imè diluuntur. Vi- |
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98 detur autem mihi, rem accuratius exploranti, $a- |
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99 tis veri$imile (nullum enim habeo opinionis hu- |
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100 ius a$$ertorem,) $ectionem e$$e hanc, & partem |
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101 quandam eius operis nobili<02>imi, quod idem au- |
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102 ctor De Problematibus edidit, & hanc, ne$cio |
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103 quam ob cau$am; ni$i fortè quod tractatio merè |
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104 Phy$icanon$it, àreliquo corpore di$tractam at- |
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105 que reuul$am. Jd certè quod ad rem facit, probè |
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106 nouimus, Diogenem Laërtium inter cætera Ari- |
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107 $totelici ingenij monumenta Mechanica quoque |
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108 adnumera$$e. Quibus con$ideratis magnopere |
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109 $ubit mirari, cur ij qui po$t Ari$totelem floruêre |
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110 at<16> vixere, Mechanici, Archimedes, Athenæus, |
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111 Heron, Pappus, & cæteri, nullam huius libelli fe- |
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112 cerint commemorationem: & $anè debuerunt; |
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113 ne<26> enim à vero est di<02>imile, ip$os per hunc ali- |
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114 quatenus profeci$$e. Verum enimuero cum inge- |
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115 nuiilli fuerint homines, & nullatenus obtrecta- |
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116 tores, credendum potius est, Comment ariolum i- |
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117 $tud, eorum æuo, paucis cognitum, alicubi in Bi- |
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118 bliothecis latui$$e: etenim cætera quo<16> Ari$tote- |
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119 lis $cripta, po$t vetu$tailla tempora, ante Ale- |
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120 xandrum Aphrodi$ien$em, àmultis fui$$e igno- |
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121 <pb> |
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122 rata non dubitamus. Habemus $iquidem, Stra- |
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123 bone te$te, lib. 13. Ari$totelis, & Theophra$ti bi- |
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124 bliothecam, po$t ip$ius Theophra$ti dece$$um, ad |
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125 Neleum quendam Scep$ium, Cori$cifilium, qui |
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126 eius fuerat auditor, perueni$$e; po$t hæc libros, |
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127 blattis olim, & humore corruptos, Apelliconi Te- |
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128 io venditos, & ab eo Athenas translatos, tum |
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129 Athenis captis in Syllæ pote$tatem deacni$$e, eo$- |
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130 que tandem à Sylla acceptos, Tyrannionem |
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131 Grammaticum, vt potuit meliùs emendatos, |
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132 promulga$$e. Exquibus colligimus, mirum non |
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133 e$$e, Archimedi, Heroni, & alijs qui ante Syllam |
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134 vixêre, fui$$e incognitos. quicquid$it, illud cer- |
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135 tumest, Ari$totelem eorum omnium quide Me- |
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136 chanicis commentaria edidere, e$$e longè vetu- |
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137 $ti<02>imum. Pappus enim Heroneiunior, Athe- |
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138 næus Archimediæqualis, vter<16> enim $ub Mar- |
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139 cello, cui Athenæus $uum de bellicis Machinis |
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140 libellū dedicauit. Archimedes verò circa CXL. |
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141 Olympiadem floruit, quamobrem po$t Ari$tote- |
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142 lem Olympiadas XL. hoc est, annos ferè CLX. |
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143 I$thæc autem con$iderantibus, facile e$t cogno$ce- |
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144 refacultatis huius nobilitatem, at<16> dignit atem; |
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145 quippe quod $ummus Philo$ophus non modo eam |
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146 <pb> |
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147 probauerit, $cd etiam $uis acuti<02>imis lucubra- |
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148 tionibus illu$trauerit. Hanc porro tractationem |
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149 $ubiecto quidem Phy$icam e$$e, demon$tratio- |
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150 nibus verò Geometricam, ip$emet nos docuit |
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151 Ari$toteles, cuius etiam naturæ $unt Per$pecti- |
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152 ua, Specularia, Mu$ica, & cæteræ eiu$dem |
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153 modifacultates, quas quidem $ubalternas Peri- |
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154 patetici appellant. Vitruuius Architecturæ |
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155 membrum, vt ita dicam, & portionem quan- |
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156 dam facit, ait enim Architecturæ partes e$$e tres, |
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157 Ædificationem, Gnomonicam, Machinatio- |
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158 nem. Estautem Architectur â quideminferior, |
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159 paret enim Architecto Mechanicus; attamen $i |
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160 cæteras artes $pectes, Architectonica; hæc enim |
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161 omnesferè $edentariæ, $ellulariæue, quas banau- |
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162 $as Græci appellant, ordine $ubijciuntur, & $a- |
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163 nè lati<02>imos i$ihæc habetfines; præcipuè autem |
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164 circa eam ver$atur cognitionem, eamque inter |
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165 cæterasferè principem, quam dixere Centrobari- |
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166 cam, quæ quidem ad Centri grauitatem, eiu$que |
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167 $peculationem pertinet: quà in $pecie inter vete- |
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168 res primum $ibi vindicauit locum Archimedes, |
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169 mox Heron, deinde Pappus; inter neotericos au-</I> |
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170 <foot><I>):():(</I></foot> |
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171 <pb> |
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172 <I>tem Commandinus, qui librum de Centro gra- |
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173 uitatis $olidorum $crip$it, & po$t eum G. Vbal- |
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174 dus è Marchion. Montis, qui non modò ab- |
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175 $oluti<02>imum Mechanicorum librum cum maxi- |
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176 maingenij $ui laude con$crip$it, $ed & Paraphra- |
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177 $in in librum Æqueponderantium Archimedis |
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178 egregiè concinnauit Centrobaricam hanc, igno- |
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179 tam fui$$e Ari$toteli, $ætis patet. nunquam enim |
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180 in Mechanicis demon$trationibus, quod tamen |
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181 est poti<02>imum, grauitatis centrum nominat, e- |
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182 iu$ue naturam atque vim $peculatur. Diuidi- |
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183 tur autem Mechanice tota, te$te Herone apud |
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184 Pappum libro octauo, in Rationalem, hoc est, |
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185 Theoricam & Chirurgicam, id est, manu ope- |
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186 ratricem, quam Praxim aptè dicere valemus. |
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187 Rationalis, $peculationi & demō$trationibus, ex |
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188 Geometricis, Arithmeticis & Phy$icis rationi- |
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189 bus, dat operam; Chirurgica vero materiam |
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190 tractat, & $e$ein varias artes diffundit, Æræ- |
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191 riam, Lignariam, Sculptoriam, Pictoriam, Æ- |
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192 dificatoriam, Machinariam & Thaumaturgi- |
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193 cam, cæterasque eiu$modi. Machinatoriæ au- |
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194 tem $unt partes Manganaria, qua ingentiæ</I> |
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195 <pb> |
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196 <I>transferuntur pondera, tum ip$a Poliorcetica, |
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197 quæ bellicas Machinas ad vrbium expugnatio- |
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198 nes, quod velip$o nomine profitetur, ædificat. At- |
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199 qui hac dere plura $cribere $uper$edemus, ne a- |
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200 ctum agamus: quis quis enim minutè magis hæc |
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201 cogno$cere de$iderat, is Pappum adeat libro cita- |
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202 to, & Guidum Vbaldumin Præfatione quam |
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203 $uo Mechanicorum Operi præpo$uit. Vt autem |
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204 ad Ari$totelis, de quo egimus, libellum reuerta- |
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205 mur, pauci $unt qui ei ante nos $tilum & operam |
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206 commodauerint: Leonicenus Latinum fecit & |
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207 figuris tum breui<02>imis, & parui $ane ponderis, |
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208 marginalibus adnotatiunculis, in$truxit. Po$t |
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209 hunc Alexander Picolomineus luculenti<02>ima |
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210 Paræphra$i illu$trauit. Modo, vt audio, Simon |
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211 Sticinus Hollanden$is quædam edidit, quæ ad |
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212 nos minime peruenêre. Nos demum, omnium, |
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213 tum $cientia, & ingenio, tum ætate, po$tremi huic |
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214 operi manum admouimus; Con$iderantes enim |
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215 Ari$totelem æly<*>s principijs v$um, ac probati<02>i- |
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216 mipo$t eum fecerint Mechanici, demon$tra$$e, |
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217 morem huiu$ce facultatis $tudio$is ge$turos nos |
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218 fore arbitrati$umus, $i ea$dem illas quæ$tiones</I> |
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219 <foot>):():( 2</foot> |
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220 <pb> |
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221 <I>Mechanicis, hocest, Archimedeis probationi- |
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222 bus confirmaremus; dum per lati<02>imos faculta- |
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223 tis huius campos vægantes, alias quoque i$tis af- |
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224 fines dubitationes introducentes $olueremus. |
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225 quicquidautē$ecerimus profecerimu$ue, Lector |
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226 optime, boni con$ule, & quia fax per manus tra- |
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227 ditur, tu interim de me accipe, vt alijs tradas.</I> |
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228 <pb> |
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229 <HEAD>DE VITA ET SCRI- |
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230 PTIS BERNARDINI |
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231 BALDI VRBINATIS</HEAD> |
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232 <HEAD><I>EX LITERIS FABRITII SCHAR- |
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233 loncini ad Illu$tri<02>imum & Reuerendi<02>imum |
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234 Dominum Lælium Ruinum Epi$copum Bal- |
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235 neoregien$em ex-Nuntium Apo$tolicum |
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236 ad Poloniæ Regem & c.</I></HEAD> |
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237 <p>Natus e$t Bern. Baldus Vrbini nobilibus pa- |
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238 rētibus po$tridie Non. Iunij anno MDLIII. |
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239 Genus traxit, quod me $æpè ab eomemini |
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240 audire, à familia Cantagallina, quæ inter |
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241 Peru$inas illu$tris: hocautem cognomen, |
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242 Baldiaccepto, vtin varietate temporum fit, |
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243 Abauus reliquit, à teneris vnguiculis pietatē erga Deum |
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244 præ$etulit; nam vt mater eius narrabat, $anctorum imagi- |
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245 nes & Altariola non cum lætitia $olum, $ed cum venera- |
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246 tione anniculus intuebatur. Præceptoribus in adole$cen- |
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247 tia v$us fuit laudati$$imis Io. And. Palatio, & Io. Antonio |
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248 Turoneo, qui altero doctior, & Paulo Manutio maxime |
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249 carus ob latinæ & græcæ linguæ peritiam propè $ingula- |
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250 rem: adillorum autem $edulitatem tantum animi ardo- |
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251 rem attulit, tantam ingenij aciudicij vim, vt non tantum |
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252 æqualis $ed omnium vicerit expectationem. Puer adhuc |
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253 Aratiapparitiones Italico carmmered didit. Parens hac |
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254 filij laude & gloria motus anno 1573. eum ad maiorem in- |
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255 genij cultum cape$$endum Patauium mi$it. Hîcin Ema- |
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256 nuelis Margunij familiaritatem $tatim venit, cui porro |
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257 <foot>):():( 3</foot> |
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258 <pb> |
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259 fuit in amotibus. Homeri Iliad. illo Doctore & interpre- |
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260 te diligentius quam feci$$et antea, euoluit. priuato autem |
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261 $tudio Anacreonti, Pindaro, Æ$chyli, Euripidi, Sophocli |
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262 operam dedit, $ed præ cæteris Thcocriti Bucolica triuit, |
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263 ad quod $criptionis genus natura magis ferri videbatur: |
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264 centenos græci alicuius poëtæver$us memoriter tenebat, |
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265 $æpeque habebat in ore, in oratoribus græcis ver$andis |
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266 laborem $ealiquem $entire, in poëtis nullum. Scrip$it Pa- |
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267 tauij libellum de Tormentis Bellicis, & eoruminuentori- |
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268 bus, & cum in Tran$alpinorum amicitias incidi$$et, $ibi |
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269 ducebat dedecori ip$os $ua lingua loquentes non intelli- |
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270 gere. quare incredibili celeritate Gallicam & Germani- |
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271 cam didicit. Pe$tilentia ex co Gymna$io exactus in Pa- |
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272 triam redijt, vbi quin quennium integrum Federico Cō- |
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273 mandino affixus omnes Mathe$eos partes perdidicit, cui |
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274 viro in delinean dis figuris ad Euclidis, Pappi, & Heronis |
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275 monumenta manum commodauit: ex eiu$dem obitu do- |
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276 lorem vix con$olabilem $u$tinuit, $u$ceptoque eius vitam |
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277 $cribendi con$ilio, $ubinde ad omnium Mathematicorum |
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278 vitas con$cribendas animum adplicuit, quod & duode- |
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279 cim annorum $patio præ$titit felici$$imè. cum vero Ma- |
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280 thematicarum di$ciplinarum amore torqueretur, ami$$o |
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281 Commandino Præceptore, amicum nactus fuit præ$tan- |
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282 ti$$imum & $ymmy$tam Guidum Vbaldum è Marchioni- |
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283 bus Montis, in cuius $e con$uetudinem daret: quantum |
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284 profeci$$et, o$tendunt ij commentarij quos anno 1582. in |
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285 Ari$t. Mechanica $crip$it. Vt po$tea à grauioribus $tudijs |
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286 ad amœniora animum abduceret, de re nautica poëma I- |
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287 talicè <*>onfecit. quo ab$oluto Paradoxa multa Mathema- |
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288 tica explicauit. Fama de Baldi virtutibus di$$ipata Ferran- |
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289 dus Gonzaga Molfetræ Princeps & Gua$tallæ Dominus |
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290 cœpit deillo in $uam familiam a$ci$cendo cogitare, vt qui |
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291 ij$dem caperetur artibus, quibus excellere Baldus inci- |
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292 <pb> |
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293 piebat: Itaque opera Curtij Arditij honorifice fuit in au- |
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294 lam euocatus, dum vitam non aulicam viueret totus in |
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295 litteras abditus precibus Ve$pa$iani Gonzagæ Sablonetæ |
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296 Ducis ad explanandos Vitruuij libros adactus fuit. quare |
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297 tūc natus de Verborū Vitruuianorum $ignificatione com- |
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298 mentarius; in quo minime miran dum $i minuta quæ dam |
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299 pro$equutus fuit, quæ viro magno minus e$$e digna vi- |
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300 deantur:illi enim Principi morem ge$$it. $cio dixi$$e ali- |
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301 quando Adrianum Romanum è Polonia reuer$um, vbi |
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302 Vitruuium Palatino cuidam explicauerat, $i commen- |
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303 tarium Baldi in Polonia adhibere potui$$em, aurum quod |
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304 mecum attuli emunxi$$em, quia $atis feci$$em muneri la- |
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305 borenullo. Cum Ferrando hero $uo obueni$$et nece$$i- |
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306 tas Hi$panias adeundi, illud iter $ine Baldo facere $e po$- |
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307 $e non putabat, non tam vt haberet, qui erudito cloquio |
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308 viæ tæ dium leuaret, quam cui po$$et arcana committere, |
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309 atque adeo à quo iuuaretur con$ilio. Vix viæ $e dederant |
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310 cum Baldus grauem in morbum delap$us itinere cogitur |
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311 de$i$tere: Mediolanum proinde diuertit, vbi à S. Carolo |
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312 Borromæo & benignè exceptus, & tamdiu detentus do- |
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313 necvaletudinem recuperaret. Gua$tallam po$tea $e re- |
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314 cepit, vbi cum ab$ente Domino liberiori otio fruerctur, |
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315 libros $ex de Aula eruditi$$imos methodo analytica con- |
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316 $crip$it. alios non commemoro, quod cum otium erit, o- |
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317 mnium $yllabum dabo. Anno 1586. ip$o nihil po$tulante |
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318 eligitur Gua$tallæ Abbas, à quo tempore Iuri Can. Con- |
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319 cilijs, & SS.Patribustotum $e dedit. Hebreæ & Chaldææ |
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320 linguarum di$cendarum triennium po$uit. Anno 1593. no- |
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321 uæ Gnomonices libros quinque compo$uit. in$equenti |
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322 Chaldæam Onkeli paraphra$in in Pentateuchum vertit |
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323 & commentarios adiunxit; quo exantlato labore in Iob |
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324 ex Heb. fonte paraphra$in texuit, quam & $cholijs illu- |
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325 $trauit. Tabulam Etru$cam Eugubinam interptetatus |
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326 <pb> |
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327 fuit:in ca autem diuinatione, vt aiebat, $ubci$iuas vnius |
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328 men$is horas con$ump$it. De Firmamento & aquis egre- |
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329 gic $crip$it. Oeconomiam Tropologicam in S.Matthæum |
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330 Card. Baronius, qui non alia Baldi vidit, vehementer pro- |
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331 babat. Romæ dum viueret, fere ne$ciuit quid gereretur |
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332 in Aulis: Arabicæ enim linguæ cum Io. Bapti$ta Raimon- |
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333 do diligenti$$ime $tuduit, & arcana indu$tria Slauonicæ, |
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334 quam perfecte callebat. Ex Arabico vertit Hortum Geo- |
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335 graphicum Anonymi, quem ante $excentos annos flo- |
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336 rui$$e arbitrabatur. Hunc vero extru$i$$et, vtalios Baldi |
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337 libros, Marcus Vel$erus IIvir Aug. $i eo paulo longior |
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338 huius lucis v$ura contigi$$et. Compo$uit & Dictionarium |
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339 Arabicum. atque cum beati$$imam illam vbertatem in- |
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340 genij a$$idue diffundi nece$$e e$$et, anno 1603. orbem vni- |
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341 uer$um de$eribere aggre$$us fuit; atque ita quidem, vt |
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342 tam quæ ad Hi$toriam, quam quæ ad Geographiam per- |
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343 tinerent complecteretur: Neque illu$trare $olum voluit |
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344 quæ nouerunt antiqui, quemadmodum vi$um Ortelio, |
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345 $ed vel oppidula omnia & pagos, de quibus aliquain po- |
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346 $tremis $criptoribus mentio. & profecto totum opus ad |
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347 vmbilicum perduxit: non dige$$it tamen vniuer$um. qua- |
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348 tuor aut ni fallor quinque tantum Tomi fuerunt ordine |
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349 Alphabetico di$po$iti:$upere$$ent $eptem aut octo di$po- |
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350 nendi, quantum ex chartarum & fa$ciculorum mo<*>e con- |
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351 ijcere licet. Anno 1617. quarto Idus Octob. po$teaquam |
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352 dies 40. vehementi de$tillatione vexatus fui$$et, $piritum |
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353 Deo reddidit Sacramentis Eccle$iæ omnibus rite muni- |
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354 tus. Statura procerus fuit, facie oblonga & acribus oculis, |
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355 colore $ubfu$co. Membrorum ei fuit decens habitudo, & |
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356 compactum corpus. Diebus fe$tis omnlbus $acrum facie- |
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357 bat, ieiunabat bis in hebdomada, eleemo$yni$que paupe- |
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358 res $ubleuabat. In$tudijs $ica$$iduus fuit, vt $æpe & legeret |
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359 & comederet. S.Augu$tinilibros de Ciuitate Dei ter in- |
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360 <pb> |
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361 ter prandium euoluit. Statim à noctis meridie dum ei vi- |
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362 res firmiores e$$ent ad lucubrandum $urgebat. à prandio |
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363 Euclidem Arabice editum, vel libellum aliquem germa- |
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364 nicum aut gallicum in manus $umebat. Suauitate morum |
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365 & mode$tia, etiam $i ceteræ dotes abfui$$ent, quemlibet |
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366 ad amorem $ui allicere potui$$et. Sermo modicus ei fuit, |
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367 itemque cultus. Nullos vnquam honores petijt, qui à |
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368 Clem. 8. ampli$$imi promi$$i fuerant; nullum emolumen- |
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369 tum quæ$iuit $uo ceniu contentus. facile parcendum e$$e |
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370 dicebat, ijs maxime qui in re leui impegi$$ent, quoniam $i |
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371 quos cen$emus optimos, nudos con$piceremus, nullum |
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372 eorum non iudicaremus multis dignum verberibus. Bi- |
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373 bliothecam habuit non locupletem, $ed $electis in$tructã |
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374 codicibus. Verum ire per $ingula longum e$$et. Satis mihi |
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375 de incomparabili Baldi doctrina, & $umma innocentia, ô |
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376 rarum connubium, pauca dixi$$e, quæ for$itan ad imitan- |
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377 dum nimis multa. |
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378 <HEAD>SYLLABVS LIBRORVM</HEAD> |
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379 <HEAD>omnium B.Abb.Baldi.</HEAD> |
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380 <p>Arati apparitiones è gr.in Ital. vertit. |
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381 <p>De Tormentis Bellicis & eorum Inuentoribus lib. |
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382 <p>Heronis automata vertit. |
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383 <p>Vitas omnium Mathematicorum $crip$it, & trib. in Tom. |
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384 2.1.P^{s}.à Thalete ad Chri$tum.2.àChri$toad $ua tem- |
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385 pora. |
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386 <p>Earumdem vitarum Epitomen Chronologicum confecit. |
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387 <p>In Ari$tot. Mechan. Commentar. |
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388 <p>De Renautica Poëmation. |
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389 <p>Paradoxorum Mathematicorum liber. |
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390 <p>De$criptio Palatij Ducum Vrbinarum quod e$t Vrbini. |
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391 <p>Poema cui titulus, Lamus. |
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392 <foot>):():():(</foot> |
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393 <pb> |
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394 <p>Carmina pia, quæ in$cribuntur, Anni Corona. |
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395 <p>De Verborum Vitruuianorum $ignificatione. |
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396 <p>Carmina varia & eclogæ mixtæ. |
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397 <p>Apologi centum, quos $crip$it æmulatus Leonem Bapt. |
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398 Albertum. |
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399 <p>De Humanitate Dialogus qui in$cribitur Go$elinus. |
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400 <p>Compatatio Vitæ Mona$ticæ cum $eculari. |
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401 <p>De Aula libri $ex. |
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402 <p>De felicitate Principis Dialogus. |
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403 <p>De Dignitate Dial. |
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404 <p>Carmina Romana. |
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405 <p>Mo$æi fabulam vertit. |
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406 <p>De Italici carminis natura Dial. qui in$cribitur Ta$$us. |
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407 <p>De vniuer$ali Diluuio poemation. |
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408 <p>Nouæ Gnomonices lib. quin que. |
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409 <p>Hieremiæ Threnos vertit, & ex Heb. fonte annotat. ad- |
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410 iecit. |
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411 <p>Poemation in$eriptum, Deiphobe, quod $erip$it æmula- |
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412 tus Lycophonem in Ca$$andra. |
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413 <p>Scala cœle$tis.1.Sermones pij & carmina. |
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414 <p>Onkeli paraphra$in Chaldæam in Pentateuchum ver- |
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415 tit & vberes commentarios adiecit. |
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416 <p>In Iob Paraphra$is latina ex fonte Heb. additis Scholijs. |
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417 <p>De $camillis imparibus Vitruuij. |
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418 <p>De firmamento & aquis. |
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419 <p>Quincti Calabri Paralipomena vertit. |
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420 <p>Tabulæ Etru$cæ Eugubinæ Interpretatio. |
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421 <p>Oeconomía Tropologicain S.Matthæum. |
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422 <p>Vrbini encomium. |
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423 <p>Horti geographici ex Arab. ver$io. |
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424 <p>Aduer$us Aulam Carmina. |
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425 <p>Luciani de mi$erijs.Aulicorum ver$io. |
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426 <p>Oratio ad Romæ con$eruatores pro antiquitatum eius |
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427 Vrbis cu$todia. |
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428 <pb> |
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429 <p>Vniuer$i orbis geographica & Hi$torica de$criptio con- |
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430 texta ex $eptingentis & eo amplius $criptoribus. |
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431 <p>Fede ici Vrbini Ducis Vita. |
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432 <p>Guidi Vbaldi Vibini Ducis Vita. |
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433 <p>Epigrammaton & Odarum libritres. |
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434 <p>Aliorum Carminum liber. |
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435 <p>Sententiarum moralium liber. |
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436 <p>Dictionarium Arabicum. |
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437 <p>Pro Procopio contra Flauium Blondum. |
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438 <p>Horographium vniucr$ale. |
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439 <p>Epigrammata alia. |
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440 <p>Heronis lib. de Balli$tis conuer$io. |
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441 <p>Exercitationes in Ari$totelis Mechan. |
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442 <p>Templi Ezechielis noua de$criptio. |
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443 <p>Antiquitatum Gua$tallen$ium liber. |
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444 <p>Hi$toriæ $cribendæ leges. |
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445 <p>Etalia quædam. |
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446 <p n=>1</p> |
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447 <fig> |
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448 <HEAD>IN MECHANICA ARISTOTE- |
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449 LIS PROBLEMATA</HEAD> |
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450 <HEAD>EXER CITATIONES.</HEAD> |
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451 <HEAD><I>Mechanices de$criptio, natura, finis.</I></HEAD> |
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452 <p>MECHANICE, facultas quædam e$t, quæ |
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453 naturalimateriâ, Geometricis<13>; demon- |
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454 $trationibus v$a, ex centrobaricâ, & eorū |
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455 quæ ad vectem & libram rediguntur, $pe- |
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456 culatione; humanæ con$ulens nece$$itati, |
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457 commoditati<13>ue, $uapte vi, Naturam i- |
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458 p$am vel $ecundans, vel $uperans, varia, ca<13>ue mirabilia |
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459 operatur. Hac diffinitione de$criptionéue brcuiter ca fe- |
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460 rè omnia complexi $umus, quæ fu$i$$imè ab Ari$totele, |
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461 Pappo, Guido Vbaldo, & alijs hac de re tradita fuêre. |
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462 <HEAD><I>Mechanices Obiectum.</I></HEAD> |
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463 <p>Con$ideratautem Mechanicus Graue & Leue. |
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464 <p>Graue duplex, Naturâ, Violentiâ. |
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465 <p>Graue Naturâ dicitur, quod in$ita propen$ione in |
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466 centrum mundifertnr. Graue autem Violentiâ, quod im- |
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467 pre$$o extrin$ecus pondere ab impellente pellitur. |
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468 <p>Leue contrà, quòd Naturâ à centro fertur. |
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469 <p>Gæterùm quicquid graue e$t, $ecundum punctum |
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470 e$t, quod Grauitatis centrum dicitur, & hoc duplex, vt |
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471 duplex e$t grauitas, Naturæ, Violentiæ. |
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472 <foot>A</foot> |
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473 <p n=>2</p> |
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474 <p>Grauitatis centrum in triplici magnitudine con$i- |
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475 deraripote$t, lineari, planà, $olidâ. |
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476 <p>De centro grauitatis linearum nemo $crip$it, $impli- |
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477 ci$$imi enim illud e$t contemplationis. |
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478 <p>De centro grauitatis linearum egregiè tractauit Ar- |
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479 chimedes in libro Æ queponderantium, & de quadratu- |
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480 ra Parabole, tum in co quem de his quæ vehuntur in- |
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481 $crip$it. |
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482 <p>De centro grauitatis $olidorum ipíemet olim $cri- |
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483 p$erat Archimedes, $ed ea quæ protulit, temporis iniuriâ |
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484 deperdita, fuâ diligentiâ re$tituit Iedericus Commandi- |
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485 nus. |
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486 <p>E$$e autem & Leuitatis centrum in rerum natura, |
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487 palam e$t. Punctum enim illud e$t, $ecundum quod lcuia |
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488 rectà à centro $ur$um feruntur. Huius autem non memi- |
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489 nêre Mechanici, propterea quod aut nihil, aut parum ad |
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490 eorum rem faciat. |
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491 <p>Porro Grauitatis centrum ita definit Heron, & qui |
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492 ab Herone Pappus 1.8. Collectionum Mathematicarum. |
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493 <p>Centrum grauitatis vniu$cuiu$q; corporis e$t pun- |
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494 ctum quod dam intra po$itum, à quo $i graue, mente ap- |
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495 pen$um concipiatur, dum fertur, quie$cit, & $eruat eam |
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496 quam in prin cipio habuit po$itionem; neque in ip$a latio- |
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497 ne circumuertitur. Commandinus verò in lib. de centro |
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498 grauitatis $olidorum hoc pacto: Centrum grauitatis v- |
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499 niu$cuiu$que $olidæ figuræ, e$t punctum illud intra po$i- |
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500 tum, circa quod vndique partes æqualium momentorum |
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501 ad$i$tunt. Sienim per tale centrum ducatur planum, fi- |
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502 guram quomodolibet $ecans, in partes æ què ponderantes |
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503 eam diuidit. Nos verò quàm breui$$imè dicimus: Centrū |
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504 grauitatis, vniu$cuiu$q; magnitudinis punctum e$$e intra |
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505 extraue magnitudinem po$itum, per quod $i plano linea |
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506 punctoue diuidatur, in partes $ecatur æqueponderantes. |
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507 <p n=>3</p> |
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508 <fig> |
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509 <p>Diximus, Magnitudinis vtlineæ, plani $olidi<13>; cen- |
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510 trum complecteremur. Eritigitur, vt in præ$enti figura, li- |
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511 neæ quidem centrum A, plani B, $olidi verò C. quod $i ob- |
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512 ijciat qui$piam, lineam & $uperficiem nullam habere gra- |
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513 uitatem; is $ciat, neq; corpora Mathematica grauitatem |
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514 habere, Mechanicum verò funes, ha$tas, vectes pro lineis |
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515 $umere; tabulas verò, & eiu$modi plana ad $uperficierum |
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516 naturam referre. |
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517 <p>Diximus in$uper, intra extraue. Aliquando enim |
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518 grauitatis centrum extra molem corporis cuius corporis |
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519 centrum e$t, cadit, vtin $equenti figura. |
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520 <fig> |
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521 <p>E$to corpus aliquod |
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522 $uperficiesue ABCDE, |
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523 ducatur linea CF, diuidēs |
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524 figuras in partes hinc inde |
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525 æqueponderantes ABC, |
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526 EDC. Ducatur & GH. |
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527 diuídens item in partes æ- |
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528 queponderantes GCH, & GAB, EDH. $ecentautem |
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529 $eip$as in I. eritigitur centrum I extra figuræ terminos & |
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530 molemip$am. Attamen licet hoc verum $it, intra e$$e dici |
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531 pote$t, quippe quod imaginario quodam, & vtita dicam, |
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532 virtuali ambitu ACDA contineatur. |
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533 <p>Dicebamus, duplex e$$e grauitatis centrum, Natu- |
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534 <foot>A 2</foot> |
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535 <p n=>4</p> |
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536 râ, Violentià: a$firmamus modò, hæc re quidem vnum e$- |
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537 $e, & ratione $olum, non autem reip$a ac$i duo e$$ent con- |
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538 $iderari. |
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539 <fig> |
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540 <p>E$to enim grauitatis na- |
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541 turalis centrum B, corporis A, |
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542 $ecundum quod dimi$$um, $ua- |
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543 pte naturâ cadet in C, $i verò |
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544 corpus violenter impellatur in |
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545 D, aiiud acquiret centrum gra- |
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546 uitatis ex violentia $ecundum |
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547 quam fertur, motum, in D, idē |
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548 autem $untre, nempe vnum B, |
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549 duo autem $i violentia & natura $eor$um con$ideren- |
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550 tur. |
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551 <p>Hæc centra, duo motus $equuntur, rectus vterque, |
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552 Naturalis videlicet, & Violentus. Tertius ex his mixtus, & |
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553 is quidem non rectus, $ed curuus. |
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554 <fig> |
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555 <p>Proijciatur enim violen- |
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556 ter corpus graue A $uperante |
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557 igitur violentia, rectà feretur |
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558 in B; ea autem elangue$cente |
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559 paullatim per curuam & mi- |
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560 xtam lineã $ecetur in C, qua- |
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561 tenus enim ad anteriora fer- |
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562 tur, violentia e$t; quatenus ve- |
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563 rò ad inferiores partes, naturæ. Vbi verò peruenit in C, |
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564 violentiâ ce$$ante, naturâ verò manente, rectà deor$um |
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565 fertur DCD. |
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566 <p>Cæteiùm hæc centra, hi<13>ue motus, naturalis nem- |
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567 pe, & violentus diuer$imode $e habent adinuicem. Sie- |
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568 nim graue corpus externâ vi adhibita, centrum mundi |
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569 ver$us impellatur, adiuuabunt $e inuicem Natura, Vio- |
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570 lentia, Si autem contra, altera alteri re$i$ter, in motibus |
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571 <p n=>5</p> |
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572 autem ad latus, eo magis pugnabunt, quo magis ab infe- |
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573 rioribus ad $uperiora fiet motus. |
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574 <HEAD><I>Mechanices præcipua in$trumenta.</I></HEAD> |
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575 <p>Hic ira con$titutis dicimus, in$trumenta, quibusad |
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576 varias operationes Mechanici vtuntur, e$$e inter $e qui- |
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577 dem diuer$a, multiplicia, & $i varietatem $pectes, penè in- |
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578 numerabilia; quod quamuis verum$it, ea omnia Ari$tote- |
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579 les ad vectem re ducit, & libram: quod etiam G. Vbaldus |
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580 in libris Mechanicoiumfecit. Cæterum qui po$t Ari$to- |
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581 telem floruere Mechanici, omnia ad quinque, quas ap- |
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582 pellant, Potentias, redegêre. Sunt autem ex Herone, Pap- |
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583 po, Guido Vbaldo, qui eos $ecutus e$t, Vectis, Trochlea, |
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584 Axis in Peritrochio, Cuneus, Cochlèa. Videtur autemi- |
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585 p$e G. Vbaldus $extam addere, nempe Libram, de qua & |
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586 primus ip$e Mechanicorum tractatum in $tituit. Verum |
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587 enimuero idem ferè $unt Vectis & Libra, ni$i forte quod |
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588 Libra tunc dicitur, cum brachia $unt æ qualia. Vectis vero |
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589 quomodocun que ea $e habeant; quinque harum Poten- |
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590 tiarū imagines ita ob oculos ponimus. Vectis A. Trochlea |
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591 B, Axisin Peritrochio C. Cuneus D. Cochlea vero E. |
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592 <foot>A 3</foot> |
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593 <p n=>6</p> |
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594 <fig> |
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595 <p>Porro, Cuneum ad libram reducere conatur Ari- |
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596 $toteles, quod facit & G. Vbaldus, qui cò refert & Co- |
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597 chleam, quippe quod nihil aliud $it Cochlea, quàm Cu- |
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598 neus Cylindro inuolutus. Nos autem duas tantùm Po- |
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599 tentias ad vectem reduci po$$e arbitramur, Trochleam |
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600 nempe, & Axem in Peritrochio. Nequaquam autem Cu- |
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601 neum & Cochleam. quod latiùs quidem o$tendemus, |
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602 cùm de Cuneo eritnobis $ermo peculiaris. |
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603 <HEAD><I>De Vecte & Libra $ecundum Ari- |
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604 $totelem.</I></HEAD> |
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605 <p>Ari$toteles in ip$o Mechanicorum ingre$$uita $cri- |
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606 bit, Mirum videri ab exigua virtute magnum pondus mo- |
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607 <p n=>7</p> |
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608 ueri, addito nimirum ponderi pondere, $iquidem & vectis |
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609 e$t pondus. Duplex ergo illi admiratio, $cilicet quòd exi- |
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610 gua potentia moucat ingens pondus, id<13>ue etiam addito |
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611 vectis ip$ius pondere, fiat. Hoc $ecundum adieci$$e vide- |
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612 tur, amplificationis alicuius gratiâ. Erenim quatenus |
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613 ad rem pertinet, $i mouendis ponderibus vectis ip$ius |
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614 pondus compares, nullius ferè e$$e momenti proculdu- |
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615 bio affirmaueris. Sed & illud quoque notandum, aliquan- |
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616 do vectis pondus mouenti auxilium ferre, quod fit vbi |
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617 fulcimento inter potentiam mouentem, & pondus ip$um |
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618 collocato, vectis pars quæ à fulcimento ad potentiam e$t, |
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619 premitur. Tunc enim, vt dicebamus, vectis pondere $uo |
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620 potentiam adiuuat. Contra verò accidit, cum pondus i- |
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621 p$um inter fulcimentum e$t & potentiam vel potentia i- |
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622 p$a inter fulcimentum & pondus. tunc cnim vectis vnâ |
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623 cum pondere attollitur. quæ licet vera $int, non tamen in- |
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624 de $e quitur, vectis pondus, quicquam quod curandum $it, |
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625 in operatione efficere, aut impedire. |
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626 <p>Porrò vectem ita finire po$$umus, longitudinem e$- |
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627 $e quandam inflexibilem, quæ fulcimento dato, datâ po- |
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628 tentiâ datum pondus mouetur. |
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629 <p>Ip$a quoque Libra, vt diximus, vectis e$t: eius autem |
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630 naturæ, vt$emper fulcimentum medium obtineat locum |
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631 inter pondus & pondus. Statera autem merus e$t vectis, $i |
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632 $par$um pro fulcimento; appendiculum verò currens pro |
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633 potentia mouente deputaueris. |
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634 <HEAD><I>De Circulo eiusque natura Ari$totelis doctri- |
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635 na examinata.</I></HEAD> |
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636 <p>Ari$toteles, quicquid mirum in Mechanicis opera- |
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637 tur, id totum admirabili circuli naturæ e$$e tribuen dum |
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638 arbitratur. Aitautem, ab$urdum nullatenus e$$e, $i ex re |
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639 mirabili mirandum quippiam oriatur. In circulo autem |
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640 <p n=>8</p> |
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641 quatuorinueniri qualitates admiratione dignas. Primã, |
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642 quod ex contrarijs con$tituatur, mouente videlicet & |
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643 moto. Secundam, quòd contraria in eius circumferentia |
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644 inueniantur, quippe quæ cum vnica linea $it, concaua $i- |
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645 mul e$t & conuexa. Tettiam, quod contrarijs feratur mo- |
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646 tionibus, antror$um nimirum, retror$um, $ur$um, atque |
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647 deor$um. Quartam, quod vnicâ exi$tente $emidiametro, |
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648 nullum in ca punctum $umi po$$it, æqualis alteri, in latio- |
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649 ne, velocitatis. Sit enim circulus AB, cuius centrum C, |
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650 $emidiameter AC, $umatur autem in ea punctum D, i- |
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651 tem<13>ue punctum E. Erit itaque in ip$a circulatione D |
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652 tardius E, ip$um verò E tardius A, & ita citius id feretur |
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653 $emper, quod remotius à mouente termino accipitur. |
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654 <fig> |
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655 <p>Hæc ex illo, quibus ne vltro a$- |
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656 $en$um præbeamus non vnica de cau- |
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657 $a cohibemur. Dicimus igitur, videri |
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658 nobis, circulum non ex contrarijs cō- |
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659 $titui, puta ex manente & moto, $ed ex |
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660 moto $impliciter. Nulla e$t enim $e- |
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661 midiametri pars, quæ non moueatur. |
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662 Punctum autem, quod $tat, $emidia- |
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663 metri pars nulla e$t. Et $anè cur moto |
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664 $emidiamētro fiat circulus, non ideo accidit, quod alterū |
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665 extremum $tet, alterum verò moueatur:led ideo quòd $e- |
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666 midiameter perpetuò candem $eruct longitudinem. Elli- |
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667 p$is $anè centrum habet, $ed ab eo ad circumferentiam |
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668 quatuor tantùm $emidiametri quomodolibet $umpti du- |
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669 cuntur æ quales. Si quis igitur $emidiametrum daret pro- |
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670 portione cre$centem & decre$centem, $tante altero ex- |
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671 tremoru<*> Ellip$is de$criberetur. Præterea & $piralis li- |
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672 nea, quæ mixta e$t, altero $emidiametri extremo manen- |
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673 te, altero vero moto producitur. Legem itaque circulo |
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674 <p n=>9</p> |
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675 prælcribit, non quidem quòd hæc extremitas $ter, illa ve- |
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676 rò moueatur, $ed quod $ua circulatione $em per $emidia- |
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677 meter eandem $eruet longitudinem, quod vel ex ip$a cir- |
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678 culi definitionc colligitur. |
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679 <p>Ad $ecundum miraculum, $cilicet, quòd in circulo |
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680 circum ferentia, quæ vacua linea e$t, concaua $imul$it, & |
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681 conuexa. Diceret qui$piam id, $i modò mirabile e$t non |
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682 circulari tantum, $ed cui ibet curuæ lineæ primo compe- |
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683 tere, etenim & Elhp$is & Hyperbole, & Parabolc, & $pi- |
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684 ra, tum Cy$$ois, Conchois, & infinitæ aliæ irregulares |
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685 concauæ $imul $unt & conuexæ. Sed & hæcin $uperficie- |
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686 bus quoque de$iderantur. |
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687 <p>Ad tertium, quod contrarijs feratur lationibus, an- |
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688 tror$um, retror$um, $ur$um & deor$um. Dicimus, facilè |
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689 $olui, Nullus enim, re bene per$pectâ, affirmauerit circu- |
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690 lum contrarijs lationibus moueri. |
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691 <fig> |
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692 <p>E$to enim circulus ABCD, |
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693 circa centrum E; ponamus ro- |
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694 tari, & A ver$us B, exempli gra- |
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695 tiâ, antror$um, mouebitur autē |
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696 & B ver$us C, & C ver$us D, tum |
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697 D ver$us A. Non puto quenquã |
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698 dicturum, circulum hunc an- |
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699 tror$um codem tempore, & re- |
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700 tror$um ferri nec $ur$um aut de- |
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701 or$um, $i enim qui$piam per eius circuli circumferentiani |
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702 ambularet, is certè centrum ip$um $emper ad dexteram |
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703 haberet, vel ad $ini$tram, $i ad dexteram, antror$um ibit, $i |
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704 ad $ini$tram, tetror$um. Sed nec $ur$um vel deor$um, e$t |
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705 manife$tum. Nihil autem prohibet eundem motum va- |
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706 rio re$pectu contrarium dici po$$e, id tamen profectò fie- |
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707 rinequaquam pote$t, nempe A moueriver$us B, hoc e$t, |
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708 <foot>B</foot> |
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709 <p n=>10</p> |
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710 antro r$um, & eandem codem tempore ver$us B, id e$t, re- |
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711 tror$um; repugnat enim naturæ. |
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712 <p>De quarto circuli miraculo, ibi erit nobis $ermo, vbi |
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713 ca perpenderimus primò, quæ Philo$ophus de Circuli |
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714 productione di$$erens in medium profert. Sunt autem e- |
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715 iu$modi: |
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716 <p>Circulum quidem duplici notione produci, Natu- |
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717 rali videlicet altera, & altera quæ e$t præter naturam, & |
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718 ideo circularem lineam in ter mixtas computari. |
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719 <p>Motus mixtus ait, vel proportione $eruata fit, aut |
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720 non; Si proportione $eruatâ, rectam lineam; ea verò non |
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721 $eruata, circularem lineam produci. |
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722 <fig> |
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723 <p>E$to enim rectangu- |
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724 lum ABCD, cuius late- |
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725 ra in datâ $int proportio- |
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726 ne, AD cum AB. Mo- |
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727 ueatur A, duplici motu, |
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728 Altero quidem tendens |
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729 in B, altero vero ad mo- |
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730 tum lineæ AB, feratur |
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731 ver$us D, $eruata inte- |
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732 rim laterum proportione. Itaque ponatur ex motu ab A |
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733 ver$us B, perueni$$e in E, ex motu autem quo proportio- |
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734 naliter fertur cum linea AB, facta ip$a AB, in FH, perue- |
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735 ni$$e in G, & EG connectatur. Eritigitur Parallelogram- |
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736 mum AEGF, Parallelogrammo ABCD proportiona- |
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737 le $imile, & circa eandem diametrum AGC. Semperigi- |
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738 tur punctum A $i duabus lationibus feratur, laterum pro- |
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739 portione $eruata, lineam producet rectam, diametrum |
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740 nempe AGC. Et hoc $anè nullam habet dubitationem, |
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741 ex ijs quæ docet Euclides 1. 6. prop. 24. |
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742 <p>His ita demon$tratis hac vti videtur Philo$ophus |
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743 <p n=>11</p> |
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744 argumentatione: Si mixtus motus proportione $emotâ, |
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745 rectam producir, $i nun quam $emota, efficiet circulum; $i |
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746 enim modo $eruaretur, modo non, partim recta partim |
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747 non recta produceretur. Ingenio$a quidem argumenta- |
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748 tio, ni vitium contineret. non enim mixtus motus, qui |
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749 nun quam $eruatâ proportione fit, $emper ci, culum pro- |
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750 ducit, $ed & Elli $im pote$t, & quamlibet aliam lineam, |
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751 cuius nulla pars $it recta. Hanc difficultatem vidit Pico- |
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752 lomineus in $ua Paraphra$i, & eam $oluere conatus e$t, |
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753 $ed quàm bene, aliorum e$to <*>dicium. Cæterùm fal$um |
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754 e$t, a$$erere circulum ex mixto motu nun quam $eruatâ |
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755 proportione produci. $eruat enim a$$iduè mixtus motus |
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756 quo producitur ($i cum mixto motu producere velimus) |
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757 aliquam proportionem, $ed non eandem. |
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758 <fig> |
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759 <p>E$to enim recta AB, cui ad rectos |
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760 angulos AC. Moueatur autem A, ver- |
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761 $us C per lineam AC, & eodem tempo- |
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762 re linea AC, ver$us B, ita tamen, vt $em- |
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763 per ip$i AB, $it perpendicularis. feratur |
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764 autem eâ lege, vt quam proportionem |
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765 habet motus lineæ AC ver$us B, ad mo- |
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766 tum puncti A ve, $us C, eandem habeat |
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767 ip$e motus ab A ver$us C, ad re$iduum |
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768 lineæ AB, demptâ nempe ea parte quam |
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769 peragrauit linea AC mota ver$us B. Sit |
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770 autem, cum AC $uo motu peruenerit |
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771 in D, punctum A, $imiliter $uo motu per eam latum perue- |
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772 nitle in E erit eigo ex mixto motu, non quidem in D, nec |
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773 in E, $ed in F, eritque punctum F in circum ferentia circu- |
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774 li, cuius e$t diameter ip$a linea AB, quod quidem demon- |
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775 $tratur ex conuer$a propo$. 13. lib. 6. Elem. E$t enim AE |
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776 hoce$t DF media proportionalis inter EF, hoc e$t, AD, |
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777 & DB. Iterum $i $iat motus AC in GH, ad motum H per |
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778 <foot>B 2</foot> |
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779 <p n=>12</p> |
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780 lineam AC, v$que in C, vt $e habet proportio AG ad |
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781 GH & GH ad GB, erit ex motu mixto A in H, nempe in |
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782 eiu$dem circuli circum ferentia AFHB. ex quibus ha- |
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783 bemus, circulum ex mixto motu fieri po$$e proportioni- |
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784 bus quidem mediarum $eruatis, $ed nun quam ij$dem. |
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785 <p>Vera hæc pro culdubio $unt; nihilominus, veluti ad |
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786 rectam producendam mixtus motus non e$t nece$$arius, |
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787 licet mixto motu produci po$$it, ita ne que ad circularem, |
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788 & ideo verum non e$$e quod a$$erebat Philo$ophus, cir- |
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789 culum ex mixto motu proportione nun quam $eruatâ ne- |
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790 ce$$ariò produci. |
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791 <p>Conatur po$t hæc Ari$toteles rationem afferre, cur |
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792 circuli partes, quò propiores centro fuerint, eo $int tar- |
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793 diores. Ait autem; $i duobus ab eadem potentia latis hoc |
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794 quidem plus repellatur, illud verò minus, æquum e$t tar- |
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795 diùs id moueri quod plus repellitur, eo quod minus. De- |
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796 trahi autem plus lineam, cuius extremum prepius e$t cen- |
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797 tro illa quæ $uum habet terminum à centro remotiorem. |
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798 <fig> |
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799 <p>E$to, inquit, circulus |
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800 BCDE & alter in eo minor |
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801 MNOP circa idem centrum |
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802 A. Ducantur<13>; Diametrima- |
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803 ioris quidem CD, EB, mino- |
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804 ris verò MO, NP. Itaque vbi |
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805 AB circulata eò peruenerit |
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806 vnde e$t gre$$a, ip$a quoque |
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807 AM eo vnde moueri cœpe- |
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808 rat, perueniet. Tardiùs antem |
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809 fertur AM, quam AD, pro- |
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810 pterea quòd AM à centro |
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811 magis retrahatur quàm ip$a AB. Ducatur igitur ALF & |
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812 à puncto L, ip$i AB perpendicularis L q, cadens in mino- |
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813 <p n=>13</p> |
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814 ri circulo, & rur$us ab codem L ip$i AB, parailela duca- |
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815 tur LS, Ab S verò eidem perpendicularis ST, & ab F i- |
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816 tem FX. Sunt ergo q L, ST, quidem æquales, nempeillæ, |
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817 per qua<*>, $ecundum naturam, mouentur puncta BM. Mo- |
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818 tu verò retractionis ad centrum, hoc e$t, præter naturam, |
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819 plus motum e$t M quàm B. Maior enim e$t M q, ip$a BT, |
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820 quod, ceu notum, $uppo$uit Ari$toteles. nos autem inf. à |
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821 demon$trabimus. Si igitur fiat vt motus præter naturam |
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822 ad motum præter naturam, ita motus $ecūdum naturam, |
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823 ad motum $ecundum naturam, punctum B; cum M fuerit |
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824 in L, non eritin S, $ed in F. tunc enim, vt e$t FX motus $e- |
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825 cundùm naturam ad XB, præter naturam, ita e$t q L $e- |
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826 cundum naturam ad q M præter naturam; $ed BF maior |
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827 e$t ML, ergo proportione $eruatâ, velociùs mouetur B |
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828 quàm M circa idem centrum A. Hæc autem $umma e$t |
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829 eorum quæ præfert Ari$toteles. Cæterùm nos parallelo- |
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830 grammum, quod in figura eius habetur prætermi$imus, |
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831 quippe quod nihil ad eam quæ affertur, demon$tratio- |
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832 nem faciat. |
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833 <p>Modò quod pollicebamur, nempe minorem e$$e |
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834 BT, quàm q M, ita demon$tramus. quoniã ST. ex prop. 13. |
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835 1. 6. media proportionalis e$t inter BT & TE, erit qua- |
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836 dratum TS æquale parallelogrãmo $eu rectangulo BT, |
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837 TE, item, quoniam q L media proportionalis e$t inter |
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838 M q, & q O. erit quadratum q L æquale rectangulo M q, |
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839 q O, æqualia ergo $unt rectangula BTE, M q O, itaque |
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840 reciprocalatera habent proportionalia. quare, vt TE, ad |
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841 q O, ita M q ad TB, $ed TE maior e$t ip$a q O, quippe |
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842 quòd pars $it q O ip$ius TE, maior ergo & M q ip$a TB, |
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843 quod o$tendendum fuerat. |
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844 <p>Cæterùm $ubtilia & ingenio$a i$thæc e$$e non nega- |
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845 mus, & longè faciliori & explicatiori modo veritas hæc |
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846 demon$trari pote$t, reiectis nem peillis, $ecundùm, & pr<17>- |
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847 <foot>B 3</foot> |
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848 <p n=>14</p> |
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849 ter naturam motibus, qui quidē in $implici circulo nece$- |
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850 $ario non cadunt: caderent autem forta$$e, $i de circulo |
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851 res e$$et à pōderibus circumlatis ex $tabili centro de$eri- |
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852 pto, qua de re agit G. Vbaldus in Mechanicis ttactatu de |
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853 libra. tunc enim dici pote$t, pondus quod aliâs rectà ad |
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854 mundi centrum tenderet, à circuli centro in circulatio- |
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855 ne retrahi, $ed hæc ad circuli naturam, quatenus circulus |
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856 e$t, ne quaquam $pectant. |
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857 <fig> |
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858 <p>E$to igitur circum ferentia |
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859 AFBH, cuius centrum C, dia- |
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860 meter ACB, $emidiameter AC. |
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861 $umatur in AC punctum quod- |
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862 libet, D, & centro C, $patio CD, |
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863 circumferentia de$cribatur |
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864 DGEI. Dico punctum A velo- |
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865 cius moueri puncto D eâdem |
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866 circulatione rotato. etenim vt |
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867 diameter ad diametrum, & $emidiameter ad $emidiame- |
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868 trum, ita circumferentia ad circumferentiam: igitur vt |
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869 AC ad CD, ita circumferentia AFHB ad circumferen- |
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870 tiam DGEI. At mota linea CA circa centrum C mo- |
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871 uetur $imul & CD, eodem igitur tempore rotationem |
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872 complent puncta AD, maius ergo $patium eodem tem- |
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873 pore metitur A, ip$a D, quare velocior. Ita igitur $e ha- |
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874 bet velocitas ad velocitatem, vt circumferentia ad cir- |
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875 cumferentiam, & diameter ad diametrum, quare id quod |
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876 mouetur in puncto à centro remotiori, velocius illo mo- |
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877 uetur quod ab eo di$tat minus, quod fuerat |
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878 demon$trandum. |
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879 <p n=>15</p> |
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880 <HEAD>QVÆSTIONES |
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881 MECHANICÆ.</HEAD> |
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882 <HEAD>QVÆSTIO I.</HEAD> |
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883 <HEAD><I>Cur maiores libræ exactiores $int mi- |
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884 noribus?</I></HEAD> |
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885 <p>Prioríbus, ceu fundamentis quibu$dam iactis, oppor- |
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886 tunè ad quæ$tiones proponendas, eas <13>ue diluendas $e |
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887 confert Ari$toteles. Porro in propo$ita quæ$tione vide- |
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888 tur prima fi onte cau$$am quæri de re quæ non e$t: etenim |
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889 quis affirmauerit vnquam, lances quibus Apothecarij & |
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890 Macellarij vtuntur, magnas eas quidem, illis exactiores |
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891 e$$e quibus Gemmatij, atque Argentarij $iliquis, & $eru- |
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892 pulis minuti$$ima appendunt, quæ tamen perexiguæ $unt, |
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893 & $i illis comparentur minimæ? Veruntamen, ita pror$us |
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894 res habet, vt a$$erit Ari$toteles. Non enim propterea |
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895 quòd illæ magnæ $int, hæ verò exiguæ, hæ $unt illis exa- |
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896 ctiores; $ed quoniam magnæ, rudes $unt, minores verò ex- |
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897 qui$ita diligentia elaboratæ, & à materiæ pertina cia libe- |
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898 riores. Cæteris ergo paribus, exactiores e$$e maiores, ex |
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899 Philo$ophimente, ita docebimus. |
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900 <fig> |
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901 <p>E$to libra maior AB, |
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902 cuius fulcimentum C. |
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903 Minor verò libra DE, |
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904 circa idem fulcimētum |
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905 C, vnà cum maiori, ima- |
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906 ginatione, conuer$a. Ap- |
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907 ponatur queduis pon- |
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908 dus maiori libræ in A, |
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909 de clinet<13>; exempli gratiâ in F, erit <13>ue minor libra in G, |
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910 in eadem enim linea $unt CGF. Vnaq; igitur ex eodem |
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911 <p n=>16</p> |
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912 centro C portionem circuli de$cribet GD, AF, erit<13>ue |
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913 ACF $ector circuli, cuius diameter AB, $ed DCG $e- |
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914 ctor circuli, cuius diameter DE. Itaque vt diameter ad |
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915 diametrum, ita portio ad portionem: maior autem dia- |
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916 meter AB diametro DE: maior ergo portio AF, portio- |
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917 ne DG. quod autem maius e$t, minus obtutum fallit, ex- |
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918 qui$itius ita que tractum ex maiori AB quàm ex ip$a mi- |
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919 nori DE cogno $cemus, quod fuerat o$tendendum. |
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920 <p>Cæterùm hac eadem de cau$$a, A$tronomica in- |
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921 $trumenta, puta A$trolabia, Armillæ, & alia eiu$modi, |
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922 quo ampliora eò ex qui$itiora, & certiora probantur. |
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923 <fig> |
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924 <p>E$to enim A- |
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925 $trolabium magnum, |
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926 cuius diameter AB, |
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927 paruum autem CD, |
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928 circa idem centrum |
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929 E. Ducatur à centro |
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930 recta EF tangens ma- |
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931 iorem circulum in F, |
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932 minorē verò $ecãs in |
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933 G, vt igitur GD ad to- |
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934 tum circulum GCD, |
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935 ita FB. ad totum cir- |
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936 culum FAB, vt ergò |
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937 GD ad FB, ita gradus |
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938 $ignati in GD, ad eos qui $ignantur in BF, maiores ergo |
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939 $unt qui in FB, & minutarum partium capaciores. Hinc |
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940 itaque apparet, in$trumēta quælibet quò maiora fuerint, |
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941 eò e$$e & exqui$itiora, quod propo$uerat Ari$toteles, in |
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942 hac quæ$tione de Libra. |
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943 <p>Quod autem addit de fraudibus Purpurariorum, |
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944 inquiens; quamobrem machinánturij qui purpuram ven- |
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945 dunt, vt pēdendo defraudent, dum ad medium, $partum, |
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946 <p n=>17</p> |
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947 non ponentes; tum plumbum in alterutram libræ partem |
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948 infundentes; aut ligni quod ad radicem vergebat, in eam |
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949 quam deferri volunt partem con$tituentes, aut $i nodum |
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950 habucrit, ligni enim grauior ea e$t pars, in qua e$t radix, |
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951 nodus verò radix quæ dam e$t. Hinc quæri po$$et: |
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952 <HEAD><I>Vtrum libræ quæ ponderibus vacuæ æquilibrant, |
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953 omni pror$us careant fraude?</I></HEAD> |
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954 <p>Videri cuipiam po$$et, libras, quæ ponderibus va- |
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955 cuæ, æquilibrant, omm pror$us fraude carere, verunta- |
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956 men ita non e$t, quod diligentiùs (res enim magni mo- |
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957 menti e$t) di$quiremus. |
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958 <fig> |
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959 <p>E$to enim libra AB, ita diui$a |
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960 in C, vt AC $it partium IS, CB ve- |
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961 rò carundem $it 10. apponatur parti |
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962 A lanx ponderans 10, parti vero B |
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963 lanx ponderans 15. ex permutata i- |
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964 gitur proportione libra $u$pen$a in |
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965 C, <17>què ponderabit; $i autem appo- |
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966 natur lanci B $acoma vnciarum 6, & in lance A con$titua- |
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967 tur purpura, quæ ita $e habeat ad vncias 6, vt 10 ad 15, ite- |
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968 rum æqueponderabit, $ed vt 10 ad 15, ita 4 ad 6. Purpura- |
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969 rius ergo fraudulentus, ponens in lance A vncias purpuræ |
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970 4, facto æquilibrio petet pretium vnciarum 6, & ita em- |
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971 ptorem decipiet, quod $anè innuerat, non autem demon- |
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972 $trauerat Ari$toteles. Hæc autem faciliora fient ex ijs, |
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973 quæ in $equentibus quæ $tionibus, vbi de vecte agetur, ex- |
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974 plicabuntur. |
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975 <p>Detegitur autem fraus, $i alternatim $acoma in pon- |
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976 derando, modo huic, modò illi lanci apponatur. Si enim |
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977 in lance A con$tituatur $acoma, in B verò purpura non fit |
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978 æquilibrium. |
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979 <foot>C</foot> |
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980 <p n=>18</p> |
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981 <HEAD>QVÆSTIO II.</HEAD> |
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982 <HEAD><I>Cur, $i $ur $um libræ fulcimentum $it, appo$ito ad alteram partem |
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983 pondere, de $cendat libra, & eo amoto, iterum a$cendat, & ad æqui- |
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984 librium reuertatur. Si verò deor $um fulcimentum fuerit, de- |
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985 pre$$a ad æquilibrium non reuertatur?</I></HEAD> |
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986 <p>Bimembrem proponit Philo$ophus quæ$tionem, quam |
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987 trimembrem debuit, triplici $i quidem loco fulcimen- |
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988 tum aptari pote$t, $uperiori, medio, inferiori. Nos de o- |
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989 mnibus ver ba faciemus. |
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990 <HEAD>Prima Quæ$tionis pars.</HEAD> |
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991 <HEAD><I>De Libra $ur $um fulcimentum habente.</I></HEAD> |
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992 <p>Ari$toteles primam quæ$tionis partem ita $oluit: An |
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993 quia $ur$um parte quidem exi$tente, plus libræ extra per- |
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994 pend culum $it? Spartum enim perpendiculum e$t: quare |
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995 nece$$<*> e$t deor$um ferriid quod plus e$t, donec a$cendat |
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996 qua bifariam libram diuidit ad ip$um perpendiculum, |
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997 cum onus in cum bat ad libræ partem $ur$us raptam. |
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998 <fig> |
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999 <p>Sit libra recta (hoc e$t, in æquilibrio con$tituta) BC, |
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1000 $partum autem AD, |
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1001 fulcimentum autem |
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1002 D, de$uper: $parto au- |
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1003 tem deor$um proie- |
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1004 cto ad M perpendicu- |
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1005 laris erit vbi ADM. |
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1006 Si igitur in ip$o B po- |
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1007 natur onus, erit B qui- |
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1008 dem vbi E, C autem |
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1009 vbi H, quamobrem |
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1010 ea quæ bifariam librã |
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1011 $ecat, primo quidem erit DM, ip$ius perpendiculi; in cū- |
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1012 bente autē onere, erit DG. quare libræ ip$ius EH, quod |
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1013 <p n=>19</p> |
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1014 extra perpendiculum, e$t AM, vbi e$t q P maius e$t dimi- |
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1015 dio. Si igitur amoueatur onus ab E, nece$$e e$t deor$um |
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1016 ferri H, minus e$t enim E: $iquidem igitur habuerit $par- |
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1017 tum $ur$um, propter hoc a$cendit libra. |
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1018 <p>Pe$$imè omnes $chema hoc lineârunt, ita vt difficil- |
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1019 limum $it auctoris inde $en$um a$$equi. Nos autem cla- |
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1020 rius rem ob oculos ponimus. Id ergo $ibi vult Ari$toteles, |
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1021 propterea quòd pars iugi HDG maior e$t parte ED q, |
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1022 eam eleuatam nece$$e e$t de$cendere, & iterum à perpen- |
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1023 diculari ADM bifariam diui$am ad æquilibrium reuer- |
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1024 ti, Po$$umus nos idem $impliciori figura demon$trare. |
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1025 <fig> |
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1026 <p>E$to libra AB, bi- |
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1027 fariam, diui$a in G, |
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1028 fulcimentū verò $ur- |
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1029 $um vbi D, prod<*>ca- |
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1030 tur perpendicularis |
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1031 DC in E. Stante igi- |
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1032 tur libra AB, in æqui- |
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1033 librio æqualis e$t pars |
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1034 CH, ip$i parti CB |
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1035 apponatur pondus in |
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1036 B. Declinabit igitur |
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1037 libra mota circa centrum D, fiat autem in FG, $ecet<13>ue |
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1038 perpendicularem in I. Punctum vero C eodem motu cir- |
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1039 ca idem centrum D erit in H. amoueatur pondus appo$i- |
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1040 tum: Dico libram à $itu FG declinaturam & iterum re- |
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1041 uer$uram in $itum pri$tinum ACB. quoniam enim parti |
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1042 GH, quæ æqualis e$t parti HF, additur pars IH, quæ à |
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1043 perpendiculari e$t v$que ad H, ip$i verò HF eadem pars |
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1044 detrahitur, erit IF minor GI. Superabiturita que IF à |
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1045 GI, de$cendetque FI, a$cendet verò IF, doneciterum li- |
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1046 <foot>C <I>2</I></foot> |
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1047 <p n=>20</p> |
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1048 bra ín partes æquales, vt antea, diuidatur in C, $iat que æ- |
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1049 quili brium. |
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1050 <p>Hæc Philo$ophi demon $tratio e$t vera illa quidem, |
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1051 $ed non ex Mechanicis principijs, hoc e$t, ex centri graui- |
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1052 tatis $pe culatione; nos igitur clariùs rem exponemus, his |
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1053 quæ $equuntur con$ideratis. |
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1054 <p>Si pondus circa $tabile centrum conuertatur, dimi$- |
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1055 $um non $tabit, ni$i $ecundum grauitatis centrum fuerit |
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1056 in perpendiculari, quæ per centrum, circa quod conuer- |
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1057 titur, ad mundi centrum cadit. Stabit autem in ea per- |
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1058 pendiculari in duobus punctis, altero à centro mundi |
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1059 remoti$$imo; altero verò cidem quantum licuerit pro- |
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1060 ximo. |
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1061 <fig> |
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1062 <p>E$to corpus A, cuius graui- |
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1063 tatis centrum B, nixum line<17> in- |
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1064 flexibili BC, cum qua liberè |
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1065 conuertatur circa centrum C. |
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1066 Ducatur autem per mundi cen- |
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1067 trum perpendicularis BCD. |
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1068 Sit igitur primò pondus A $ecū- |
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1069 dum gracilis B centrum, in per- |
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1070 pendiculari ip$a $upra centrum |
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1071 C, puta in B. Moueatur & de$cē- |
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1072 dat in E. Po$t hæc verò in F, hoc |
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1073 e$t iterum in ip$a perpendiculari |
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1074 infra centrum C. De$cribet er- |
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1075 go circulum ex centro C, nem- |
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1076 pe BEF $ecantem perpendicu- |
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1077 larem in duobus punctis oppo- |
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1078 $itis BF, dico, pondus libe è di- |
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1079 <p n=>21</p> |
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1080 mi$$um in duobus tantum punctis $uapte naturâ perman- |
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1081 $urum, BF, in B, primò, quoniam cum corpus ip$um A à |
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1082 perpendiculari, quæ $upei ficiei loco intelligitur ABCD |
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1083 per centrum grauitatis diuidatur, in partes diuiditur æ- |
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1084 queponderantes, quare in neutram partem inclinabit. |
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1085 Stabit igitur erectum, lineæ ip$i fultum, inflexibili BC, |
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1086 quæ nititur puncto C. In E verò non $tabit, quippe quod |
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1087 eo $itu centrum ip$um grauitatis $it extra perpendicula- |
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1088 rem, & ideo extra fulcimentum $tabile C. In F verò ite- |
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1089 rum $tabit, pendens à centro C, propterea quòd & ibi ab |
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1090 eadem perpendiculari diuidatur per grauitatis centrum |
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1091 in partes æqueponderantes. E$t igitur re$pectu B, ip$um |
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1092 punctum C, ful cimentum deor$um, re$pectu verò F, ful- |
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1093 cimentum $ur$um. At quia linea DFCB, à centro mundi, |
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1094 quod e$t extra circulum, BEF, circulum ip$um per cen- |
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1095 trum C $ecat, erit pars eius DF quidem breui$$ima, ip$a |
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1096 verò DB longi$$ima, ex propo$. 8. lib. 3. Elem. Pondus igi- |
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1097 tur A conuer$um $eu liberè motum circa centrum C, in |
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1098 duobus tantum locis perpendicularis lineæ $tabit remo- |
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1099 ti$$imo altero, vt e$t B, altero verò cidem quamproximo, |
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1100 vt e$t F. |
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1101 <p>Hoc idem egregiè demon$trauit G. Vbald. in $uis |
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1102 Mechanicis, Tractatu de Libra prop.1.<*> |
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1103 <p>Ad hæc autem dubitare quis po$$et, cur experientiâ |
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1104 docente, pondera quæ infra fulcimentum habent, vt lan- |
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1105 cea $ari$$aue ad planum horizontis perpendiculariter e- |
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1106 recta, licet eo ca$u grauitatis centrum in ip$a perpendicu- |
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1107 lari con$tituatur, non $tet quidem, $ed altrin$ecus ca- |
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1108 dat? |
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1109 <foot>C 3</foot> |
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1110 <p n=>22</p> |
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1111 <fig> |
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1112 <p>Sit enim horizontis |
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1113 planum AB, cui in puncto |
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1114 C perpendiculariter ere- |
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1115 cta $tatuatur $ari$$a DC, |
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1116 cuius grauitatis centrum |
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1117 E, in ip$a perpendiculari. |
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1118 Stabit ergo, ex præmi$$is, |
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1119 & certè $tare debuit, $ta- |
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1120 ret<13>ue, ni vitium ob$taret |
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1121 materiæ; non $tat autem, |
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1122 quia difficillimum e$t gra- |
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1123 uitatis centrum, $uapte naturâ indiui$ibile, ita ad amu$$im |
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1124 $i$tere, vt in neutram partem à perpendiculari declinet. |
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1125 Hæc igitur ex ijs $peculationibus e$t, quæ ad praxim, ma- |
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1126 teriæ vitio impediente, aut vix aut nun quam rediguntur. |
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1127 <p>Hinc autem ea quæ$tio $oluitur, Cur ij qui $ari$$am |
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1128 erectam digito $ummo $u$tinere conantur, non $tent qui- |
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1129 dem, $ed digiti motu, $ari$$æ motum $equantur. |
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1130 <p>Id certè agit, qui nutantis $ari$$æ, digito, motum $e- |
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1131 quitur; vt in ip$o motu digitum a$$iduè centro grauitatis |
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1132 $ari$$æ $upponat, vnde $it vt nun quam extra fulcimentum |
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1133 permanens, nun quam cadat. |
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1134 <p>Similis huic alia quo que dubitatio $oluitur: Nempe, |
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1135 Cur turbines, quibus pueri ludunt, dum quidem rotan- |
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1136 tur, $tent erecti, rotationevero ce$$ante, cadant. |
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1137 <fig> |
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1138 <p>E$to enim Turbo AB, cu- |
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1139 ius grauitatis centrum C, planum |
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1140 horizontis DE, linea Horizonti |
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1141 perpendicularis ABC, tran$iens |
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1142 per centrum grauitatis C, $it au- |
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1143 tem fulcimentum in B. Itaq; cum |
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1144 centrum grauitatis C $it in ip$a |
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1145 perpendiculari, $tabit ex demon- |
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1146 <p n=>23</p> |
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1147 $tratis, at ex vitio materiæ non $tabit. Modò, vt a$$olet, ra- |
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1148 pido motu rotetur. Dico, Turbinem, motu $eu rotatione |
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1149 durante $tare. ea autem paullatim elangue$cente ín ca- |
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1150 $um vergere; ce$$ante verò penitus cadere. fit enim ex in- |
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1151 æqualitate materiæ, vel operis ruditate, vel aliâ quauis |
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1152 ex cau$$a, grauitatis centrum non e$$e in C, $ed exempli |
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1153 gratiâ vbi F, notentur autem hinc inde Turbinis latera |
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1154 notis GH. Vtique cum F extra perpendicularem fuerit, |
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1155 cadet Turbo ad partem G; at id ne $iat, efficitur velocita- |
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1156 te motus, quo centrum F transfertur in contrariam par- |
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1157 tem, vbi I. non autem cadit ver$us H, quoniam eadem ve- |
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1158 locitate iterum transfertur in F, quamobrem cum huius- |
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1159 cemodi centri a$$idua circa perpendicularem fiat trans- |
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1160 latio, ad nullam partem Turbo cadere pote$t; elangue- |
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1161 $cente verò motu rotans, paullatim in cipit inclinari, do- |
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1162 nec eo penitus ce$$ante, ad eam partem cadit, ad quam à |
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1163 per pendiculari grauitatis centrum vergit. De$cribit au- |
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1164 tem in rotatione grauitatis centrum, quod in medio non |
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1165 e$t paruum circulum, per cuius centrum ip$a perpendi- |
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1166 cula<*>is pertingit. |
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1167 <p>Modò redeuntes ad libram, cuius ful cimentum e$t |
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1168 $ur$um, alio principio, nempe Mechanico, cur depre$$a |
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1169 ad æqualitatem reuertatur, demon$trabimus. |
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1170 <p n=>24</p> |
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1171 <fig> |
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1172 <p>Sit igitur, vt $u- |
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1173 periùs, libra AB, cu- |
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1174 ius centrum grauita- |
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1175 tis C, fulcimentum, |
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1176 verò $ur$um, in D li- |
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1177 bræ quidem in C per- |
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1178 pendiculariter con- |
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1179 iunctum. Perpendi- |
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1180 cularis verò quæ per |
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1181 fulcimentum, & gra- |
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1182 uitatis cētrum tran$- |
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1183 iens ad mundi cen- |
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1184 trum tendit DLE. $tante igitur librâ in $ua æqualitate, e- |
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1185 rit centrum grauitatis C in ip$a perpendiculari infra qui- |
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1186 dem fulcimentum D. Loco verò, mundi centro quàm |
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1187 proximo. Pondus po$t hæc apponatur in B, Declinabit au- |
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1188 tem pars CB, in HF, eleuatâ interim parte AC, in GH. |
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1189 Mota igitur libra tota, circa fulcimentum D mouebitur |
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1190 circa idem centrum, & grauitatis centrum C, de$cribens |
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1191 portionem circuli CH, fi et<13>; C in H, & quoniam H, hoc |
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1192 e$t C, extra per pendicularem fit, amoto pondere, ex lan- |
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1193 ce B, cuius pre$$ione libra declinauerat, centrum grauita- |
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1194 tis per eandem circulì portionem HC, ad perpendicula- |
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1195 rem de$cendet, donec iterum in ea quie$cat, quo ca$u li- |
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1196 bra AB ad æquilibrium reuertetur: quod fuerat demon- |
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1197 $trandum. |
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1198 <p>His ita declaratis, o$tendemus, (quod nullus ante |
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1199 nos animaduertit) harum librarum, quæ fulcimentum |
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1200 habent $ur$um, eam e$$e naturam, vt non à quouis ponde- |
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1201 re appo$ito moueantur, vel penitus declinent. |
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1202 <p>Ij$dem enim $tantibus, addatur quoduis pondus lan- |
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1203 ci B; Itaque $i tale fuerit quod $uperet re$i$tentiam, quam |
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1204 <p n=>25</p> |
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1205 illi facit centrum grauitatis contra naturam elatum in H |
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1206 mouebitur quædam libra. Sin autem tam parui momenti |
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1207 $it, vt eam re$i$tentiam non vincat, $tante circa locum in- |
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1208 fimum centro C, non mouebitur aut $altem parum, ip$a |
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1209 libra. |
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1210 <p>Hinc colligimus $ieri po$$e, libras illas, quæ non <*> |
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1211 quouis, quantumuis paruo pondere declinant, cas fulci- <*> |
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1212 mentum habere $ur$um. |
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1213 <p>His ad dimus, cæteris paribus, re$i$tentiam eò e$$e |
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1214 maiorem, quo minus grauitatis centrum di$tat à fulci- |
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1215 mento $ur$um, circa quod ip$a libra aduertitur. |
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1216 <fig> |
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1217 <p>E$to libra AB, cuius gra- |
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1218 uitatis centrum C, & primò |
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1219 quidem eius ful cimentum |
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1220 $ur$um $it vbi D, itaque $i ap- |
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1221 po$ito pondere de clinauerit |
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1222 libra ad partes B, punctum |
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1223 C, dum a$cendet de$cribet |
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1224 portionem circuli CE. fulciatur iterum $ur$um puncto F, |
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1225 & iterum declinet ad partes B, & iterum punctum C, dum |
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1226 a$cendet, circuli portionem de$cribet CG. E$t autem |
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1227 minor angulus contactus ACE, angulo ACG, magis er- |
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1228 go $ur$um, hoc e$t, ad naturam $ui feretur C, per CG, ex |
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1229 centro F, quàm per CE, ex centro D, quod fuerat de- |
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1230 mon$trandum. |
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1231 <p>Hæc autem re$i$tentia ex eodem fulcimento & eo- |
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1232 dem pondere eo faciliùs $uperabitur, quo longius bra- |
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1233 chium libræ fuerit. |
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1234 <p>E$to enim iterum libra AB, cuius fulcimentum D, |
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1235 centrum grauitatis C, $it & alia libra, cuius brachia bre- |
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1236 uiora EF, idem habens centrum C, & eidem puncto $u- |
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1237 $pen$a D. Dico igitur, eodem pondere appo$ito, faciliùs |
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1238 <foot>D</foot> |
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1239 <p n=>26</p> |
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1240 <fig> |
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1241 declinaturam libram ad |
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1242 partes B, quàm $i idem ap- |
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1243 poneretur in F. Demit- |
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1244 tatur enim, à puncto B |
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1245 horizonti perpendicula- |
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1246 ris BG, & ab F item per- |
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1247 pendicularis FH, Tum |
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1248 iuncta DB, centro D, eo- |
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1249 dem vero $patio DB, circuli portio de$cribatur BI, item |
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1250 iuncta DF eodem centro D, $patio DF, portio circuli de- |
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1251 $cribatu: FK. e$t autem maior DB ip$a DF ex propo$. |
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1252 21. lib. 1. Elem. quare maiotis circuli portio e$t BI quàm |
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1253 FK. Obliquior autem, hoc e$t, à perpendiculariremotior |
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1254 e$t motus per FK quàm per BI. maior $i quidem e$t angu- |
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1255 lus KFH angulo IBG. quod nos ita probamus. Ducatur |
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1256 perpendicularis ip$i DF linea LF contingens circulum |
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1257 FK in F, item ip$i DB, perpendicularis MB, contingens |
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1258 circulum BI in B, & quia angulus contingentiæ maioris |
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1259 circuli minor e$t angulo contingentiæ minoris, erit KFL |
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1260 maior IBM, R ectiautem $unt DFL, DBM, minor ergo |
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1261 DFK re$idua ip$o DBI re$iduo. Maior autem DFC ex |
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1262 iam citata propo$. quã DBC, erit igitur re$iduum CFK, |
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1263 multo minus re$iduo FBI, $ed recti $unt CFH, FBG, ex |
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1264 quibus $i detra hantur CFK, FBI, erit re$iduum KFH, |
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1265 maius re$iduo IBG, plus ergo retra hitur à perpendicula- |
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1266 ri po<*>dus de$cendens per FK quàm per BI, minus igitur |
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1267 præ<*>alebit re$i$tentiæ in C pondus appen$um in F, quàm |
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1268 $i appendatur in B. quod fuerat demon$trandum. |
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1269 <p>Po$$<*>mus & idem quoque aliter o$tendere. |
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1270 <p>Sint enim $eor$um duæ libræ, maior AB, mïnor EF, |
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1271 quàm commune grauitatis centrum C, fulcimentum ve- |
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1272 rò $ur$um D. Producatur perpendicularis DC, in G & fiat |
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1273 CG æqualis CB, CH verò æqualis CF. Sunt igitur duo |
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1274 <p n=>27</p> |
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1275 <fig> |
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1276 vectes DG, DH, quo- |
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1277 rum quidem commu- |
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1278 ne fulcimentum D, |
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1279 pondus verò C, poten- |
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1280 tiæ vbi HG. Sunt au- |
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1281 tem hi vectes cius na- |
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1282 turæ, in quibus pōdus |
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1283 e$t inter fulcimentum |
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1284 & potentiam, itaque |
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1285 vt $e habet DC, ad |
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1286 DG, ita potentia in G |
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1287 ad pondus in C, item vt DC ad DH ita potentia in H ad |
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1288 idem pondus C, $ed minor e$t propo$itio DC, ad DG |
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1289 quàm DC ad DH. minor ergo potentia requiritur in G, |
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1290 hoc e$t, in B, quàm in H, hoc e$t in F. Data igitur ponderis |
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1291 æqualitate faciliùs $uperabitur re$i$tentia C in B, quàm |
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1292 in F: quod o$tendendum fuerat. |
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1293 <p>Ad huius libræ naturam illæ quoque rediguntur, |
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1294 quarum iugum non rectum quidem, $ed curuum, vel ex |
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1295 rectis $ur$um in angulum ad fulcimentum detinentibus, |
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1296 nec refert vtrum curuitas $it circuli portio quælibet, aut |
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1297 ellip$is $ecundum alterum diametrorum; quod ita de- |
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1298 mon$tramus. |
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1299 <fig> |
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1300 <p>E$to libra, cuius iugum |
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1301 curuum angulatūue ABC, |
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1302 cuius fulcimentum B, æqua- |
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1303 lia autem brachia AB, BC, |
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1304 & pondera item vtrinq; ap- |
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1305 pen$a æqualia. Demittatur |
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1306 ex puncto B ad mundi cen- |
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1307 trum perpendicularis BD. |
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1308 Stante igitur libra ABC in |
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1309 æquilibrio, erit eius graui- |
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1310 <foot>D 2</foot> |
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1311 <p n=>28</p> |
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1312 tatis centrum in ip$a perpendiculari BD, puta in E. Ap- |
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1313 ponatur pondus in C, declinabit autem libra, $it autem |
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1314 iuxta po$itionem FBG. Centrum igitur grauitatis E per |
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1315 portionem EH, erit in H. A$cendit ergo centrum graui- |
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1316 tatis in H, hoc e$t, $ur$um, id e$t, contra cius naturam; a- |
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1317 moto igitur pondere ex C, grauitatis centrum extra per- |
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1318 pendicularem con$titutum rur$us de$cendet, & iterum |
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1319 libra ABC ad æquilibrium reuertetur. Hoc idem egre- |
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1320 giè o$tendit G. Vbald. in tractatu de libra, propo$. 4. |
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1321 <p>Hinc ratio pendet earum imaguncularum, quas ex |
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1322 contu$a papyro ligneaue leui materia compingunt, per- |
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1323 <13>ue manus earum ambas, ferreum filum trajicientes, v- |
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1324 trinque plumbea appendunt pondera æqualia, ea quidē |
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1325 lege, vt centrum grauitatis infra pedes imaguncula $ta- |
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1326 tuatur. Tunc enim exten $o filo imponentes ceu funam- |
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1327 bulos per illud, vltrò citro<13>; decurrere faciunt, imagun- |
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1328 cula interim erecta & in neutram partem cadente, quod |
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1329 vt figurâ clarius fiat; |
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1330 <fig> |
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1331 <p>E$to imaguncu- |
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1332 la AB, per cuius ma- |
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1333 nus traij ciatur filum |
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1334 ferreum curuum cū |
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1335 æ qualibus ponderi- |
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1336 bus hinc inde appē- |
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1337 $is CD. Nitatur au- |
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1338 tem pedibus filo HI |
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1339 in <I>B</I>, $it<13>; totìus ma- |
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1340 chinæ grauitatis cē- |
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1341 trum E, $itque per- |
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1342 pēdicularis per gra- |
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1343 uitatis centrū tran$i- |
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1344 ens A<I>B</I> E. Itaque in- |
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1345 clinata imaguncula, & conuer$a circa punctum <I>B</I>, $i de- |
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1346 <p n=>29</p> |
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1347 clinet ad partes I, centrum grauitatis eleuabitur in F. Si |
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1348 verò ad partes H eleuabitur in G. quare cum FG loca |
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1349 $intremotiora à mundi centro, quàm $it E, non $tabit gra- |
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1350 uitatis centrum in punctis FG, $ed ad infimum locum re- |
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1351 uertecur, hoc e$t, in ip$a perpendiculari in E, & imagun- |
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1352 cula ad perpendiculum ip$i H<I>B</I>E filo, hoc e$t, ip$i hori- |
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1353 zonti reuertetur. |
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1354 <p>Hinc etiam Arictum, T e$tudinun<*> <13>ue demolito- |
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1355 riatum Machinarum vis pendet, nempe ex ratione libra- |
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1356 rum, quæ fulcimentum habent $ur$um. |
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1357 <fig> |
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1358 <p>E$to enim Aries A<I>B</I> |
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1359 funi appen$us CD, cu- |
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1360 ius grauitatis centrum, |
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1361 D, perpendicularis verò |
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1362 quæ ad mundi centrum |
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1363 ip$a CDE. Stante igitur |
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1364 in æquilibrio machina, |
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1365 centrum grauitatis erit |
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1366 in ip$a perpendiculari. |
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1367 Applicetur alicubi po- |
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1368 tentia retropellens, eleuabitur igitur centrum grauitatis |
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1369 per circuli portionem DF, cuius $emidiameter e$t CD, |
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1370 $i et<13>ue iuxta po$itionem CF. Aries verò in GFH. Di- |
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1371 mi$$a itaque Machina centrum F vtpote graue, non $tabit, |
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1372 $ed $uapte naturâ reuertetur in D. Quadruplici autem |
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1373 de cau$$a motus Arietis violenti$$imus e$t ex vi naturalis |
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1374 ponderis, quo deor$um fertur, tum velo citate naturalis |
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1375 motus in de$cendendo auctæ, tum ex vi pote<*> tiæ impel- |
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1376 lentis, & naturalem motum adiuuantis, tum ex velocita- |
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1377 te ex motu violento deor$um & antror$um impellente |
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1378 acqui$itâ. Id etiam addimus, eo validiores fore ictus, quò |
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1379 grauior fuerit Machina, & maius $patium, quo retrotra- |
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1380 <foot>D 3</foot> |
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1381 <p n=>30</p> |
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1382 hitur, grauitate ip$a & $patio tum virium vnione opera<*> |
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1383 tionem mirum in modum adiuuantibus. |
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1384 <p>Hæc nos de Libra $ur$um fulcimentum habente, dí- |
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1385 cta voluimus, nunc de ea, cuius fulcimentum deor$um, |
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1386 e$t, verba faciemus. |
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1387 <HEAD>Altera quæ$tionis pars:</HEAD> |
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1388 <HEAD><I>De Libra cuius fulcimentum deor$um e$t.</I></HEAD> |
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1389 <p>Si deor$um fuerit, inquit Ari$toteles, id quod $ub- |
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1390 $tat, contrarium facit illi quæ $ur$um habet, nempe ad æ- |
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1391 quilibrium non reuertitur. Plus enim, ait, dimidio fit li- |
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1392 bræ, quæ deor$um e$t pars, quàm quod perpendiculum |
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1393 $ecet, quapropter non a$cendit. eleuata enim pars leuior |
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1394 e$t. |
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1395 <p>Hæc ille, qui $chemate quo que rem aperit, at eo a- |
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1396 pud interpretes, & Picolomineum Paraphra$tem, ita mē- |
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1397 dosè lineato, vt inde ob$curitas lucis loco, legentibus of- |
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1398 fundatur. Nos, quod & $uprà quo que fecimus, no$tra fi- |
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1399 gurâ, $ole ip$o clariorem, ex Ari$to telis ip$ius mente rem |
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|
1400 totam efficiemus. |
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1401 <fig> |
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1402 <p>Sit libra recta, (hoc |
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1403 e$t, in æquilibrio con- |
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1404 $tituta) vbi NG. Per- |
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1405 pendiculum autem (id |
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1406 e$t, perpendicularis |
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1407 quæ ad mundi centrū) |
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1408 KLM. Bifariam igitur |
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1409 $ecatur NG. impo$ito |
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1410 po$thæc onere in ip$o |
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1411 N, erit quidem N, vbi |
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1412 O. ip$um autem G vbi |
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1413 R. KL autem vbi LP. |
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1414 <p n=>31</p> |
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1415 quare maius e$t KO, quam LR, ip$a parte PKL. Amoto |
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1416 igitur onere nece$$e e$t manere. Incumbit enim onus ex- |
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1417 ce$$us medietatis eius, vbi e$t F. Sen$us e$t igitur, idcirco |
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1418 partem iugi KLO inclinatam, ad æquilibrium non re- |
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1419 uerti, propterea quòd maior $it ip$a KLO pars quæ tra- |
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1420 hit, ip$a<*> RKL, quæ trahitur & eleuatur. |
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1421 <fig> |
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1422 <p>Pote$t hoc idem longè |
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1423 $impliciori themate demon- |
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1424 $trari. E$to enim libra AB, |
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1425 cuius centrum C, fulcimen- |
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1426 tum vero deor$um D, Per- |
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1427 pendicularis per centrum & |
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1428 fulcimentum tran$iens EF. |
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1429 Apponatur pondus in B, de- |
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1430 clinabit<13>; puta ad GH, cen- |
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1431 trum verò C, ex $tabili fulci- |
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1432 mento D, circuli portionem de$cribet CI, libra autem |
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1433 $ecabit EF perpendicularem in K. Æquales autem $unt |
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1434 IG, IH, at ex parte HI de$umpta e$t KI, addita <13>ue ip$i |
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1435 IG, maior e$t ergo tota KG, torâ KH. Non igitur KH |
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1436 habet KG, $ed libra, ni$i impedita fuerit, cum centro C |
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1437 de$cendente per <*>in M, ad ip$am perpendicularem dela- |
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1438 ta, ad in feriorem partem, mutatis vicibus quie$cet, facto |
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1439 nempe fulcimento $ur$um, fiet<13>; horizonti æque di$tans |
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1440 iuxta po$itionem LMN. |
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1441 <p>Demon$tratio quidē e$t hæc, $ed non ex proprijs prin- |
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1442 cipijs Mechanicis, nēpe ex ratione cēt<*>i grauitatis petitâ. |
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1443 Ii$dem enim $tantibus, cū centrum grauitatis C fiat extra |
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1444 perpendicularem, de$cendens ad I, nun quam reuert<*> tur |
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1445 in C, a$cen deret enim; $ed $i liberè circa centrum D con- |
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1446 uerteretur, de$cendens vt dictum e$t per circulum CIM |
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1447 pondus B, fieret in L, A vero in N adepta po$itione |
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1448 LMN. |
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1449 <p n=>32</p> |
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1450 <p>Cur autem huius libræ, quæ aliàs inutilis e$t, memi- |
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1451 nerit Philo$ophus, ea videtur cau$$a, quòd inde vectis vir- |
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1452 tutem eliciat, vt $uo loco videbimus. Id autem valde mi- |
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1453 rum, hominem acuti$$imum nihil pror$us de ea libra egi$- |
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1454 $e, quæ fulcimentum nec $ur$um habet, nec deor$um, $ed |
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1455 in ip$o exqui$itè medio, ita vt centrum grauitatis in ip$o- |
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1456 met fulcimento con$i$tat. Nos igitur de hac quod operæ |
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1457 pretium fuerit, & ad rem, qua de agimus, vtile, in medium |
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1458 proferemus. |
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1459 <HEAD><I>De libra cuius fulcimentum est in medio.</I></HEAD> |
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1460 <p>Dicimus itaque, libram, cuius fulcimentum nec $ur- |
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1461 $um e$t, nec deor$um, $ed pror$us in medio, nempe in ip$o |
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1462 grauitatis centro, vbi brachia & pondera vtrinque appo- |
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1463 $ita fuerint æqualia, $i ab æquilibrio mouentur, quomo- |
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1464 docunque po$ita, $tare nec ab eo, quem adepta e$t, $itu di- |
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1465 moueri. |
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1466 <p>Quæ$tionem hanc perperam tractârunt recentio- |
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1467 res quidam, Hieron. Cardanus, Nicolaus Tartalea, & alij |
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1468 nonnulli, qui Iordani Nemoracij a$$ertiones $unt $ecuti, |
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1469 quorum demon$trationes vel paralogi$mos potiùs egre- |
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1470 giè confutauit in libr. Mechanicor. Tractatu de libra pro- |
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|
1471 po$. 4. Guid. Vbald. ad cuius probati$$ima $cripta Lecto- |
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|
1472 rem ablegamus. fu$i$$imè enim ibi hac de re & ab$oluti$$i- |
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1473 mè agit. Nos autem quidem paucis ea, quæ ad hanc co- |
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1474 gnitionem pertinent, explicabimus. |
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|
1475 <fig> |
|
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1476 <p>E$to enim libra A<I>B</I>, |
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1477 cuius brachia æqualia, |
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1478 & centrum grauitatis |
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1479 in C, brachijs verò |
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1480 AC, C<I>B</I> æqualibus, æ- |
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1481 qualia pondera hinc |
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1482 inde apponãtur. Tum |
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|
1483 <p n=>33</p> |
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1484 fulcimento in medio, hoc e$t, vbi grauitatis centrum C |
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1485 applicato per centrum ip$um C ducatur perpendicularis, |
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1486 quæ ad mundi centrum, DCE, $itque primum libra æ- |
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1487 quedi$tans horizonti, con$tituta. Tum ex altera parte |
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1488 pre$$a moueatur & fiat iuxta po$itionem FCG. Dico eam |
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1489 dimi$$am permanere, etenim cum grauitatis centrum $it |
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1490 in ip$a perpendiculari, in neutram partem verget, $ed nec |
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1491 vergere pote$t, quippe quod non circa fulcimentum ceu |
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1492 centrum motus, moueatur grauitatis centrum, $ed in ip$o |
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1493 $it ful cimento; $itum ergo non mutat. Præterea cum per- |
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1494 pendicularis DCE per grauitatis centrum ducatur, cor- |
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1495 pus ip$um ex ponderibus & libra con$tans ab ea in partes |
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1496 çque ponderantes $ecatur, & ideo ex centri grauitatis dif- |
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1497 finitione, quam protulit Pappus, corpus ip$um centro |
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1498 grauitatis appen$um, dum fertur quie$cit, & $eruat eam, |
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1499 quam à principio habuit po$ition<*>. Et $anè $i partes quo- |
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1500 modo libet librâ per grauitatis centrum diuisâ, $untæ- |
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1501 queponderantes nec trahent inuicem, nec trahentur, $ta- |
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1502 bit ergo libra, & quam adepta fuerat po$itionem, eam $er- |
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1503 uabit. Id tamen non negamus, difficile e$$e libras eiu$ce- |
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1504 modi ex materia fabricare, quippe quod non omnia quæ |
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1505 vera $unt, & euidenti$$imis demon$trationibus patent, |
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1506 commodè ad praxim, ex artis & materiæ imperfectione, |
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1507 reducuntur. |
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1508 <p>Cæterùm harum librarum ea e$t virtus, vt vel mini- |
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|
1509 mo pondere altrin$ecus appo$ito, declinet; quod illis quæ |
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1510 centrum $u $um habent, non euenire, demon$trauimus. |
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1511 <p>Circa hæc po$$et cuipiam oriri Dubium, num chor- |
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1512 dulæ, quibus lances appenduntur, variationem aliquam |
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1513 circa ea quæ demon$trata $unt, inducere valeant. |
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1514 <p>Dicimus nullam inde fieri: E$to enim libra AB, cu- |
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1515 ius centrum & fulcimentum C, ab cuius extremitate A |
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1516 dependeat, funiculus AD, ab alia verò <I>B</I>, funiculus <I>B</I>E, |
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1517 <foot>E</foot> |
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1518 <p n=>34</p> |
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1519 <fig> |
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1520 quibus appen$æ $int æ- |
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1521 qualis ponderis lances |
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1522 DE. Moueatur libra, |
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1523 fiatque in ICH, funi- |
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1524 culi verò in lancibus in |
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1525 IK, HL. $ecet autem fu- |
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1526 niculus IK libram A<I>B</I>, |
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1527 in M, LH verò produ- |
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1528 catur & eandem $ecer |
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1529 in N. quoniam igitur |
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1530 IC, æqualis e$t CH, pa- |
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1531 rallelæ autem KI, LN æquales erūt alterni anguli MIC, |
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1532 NHC, $ed & anguli ad verticem ICH, BCH æquales |
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1533 $unt, quare triangulum IMC, æquale triangulo HNC, |
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1534 & latera lateribus, quæ æqualibus angulis $ubtenduntur. |
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1535 Æqualis e$t igitur linea MC lineæ NC. Itaque $i ponde- |
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1536 ra lancesue, KL mente concipiantur appen$æ in punctis |
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1537 MN, ex brachiorum & ponderum æqualitate æquepon- |
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1538 derabunt. quod fuerat demon$trandum. |
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1539 <HEAD>QVÆSTIO III.</HEAD> |
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1540 <HEAD><I>Cur exiguæ vires (quod etiam à principio dixerat) vecte magna |
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1541 mouent pondera, vectes in$uper onus accipientes, cum facilius |
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1542 $it, minorem mouere grauitatem, minor est au- |
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1543 tem $ine vecte?</I></HEAD> |
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1544 <p>Ari$toteles ita quæ$tionem proponit, vt eam R heto- |
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1545 rico quodam fuco admirabiliorem f<*>ciat. Soluit au- |
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1546 tem hoc pacto, inquiēs, fieri po$$e eam e$$e cau$$am, quod |
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1547 vectis $it libra, eius nempe generis quod fulcimentum ha- |
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1548 bet deor$um, atque id circo in ip$a pre$$ione in partes in- |
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1549 æquales vectem diuidi. |
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1550 <p n=>35</p> |
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1551 <fig> |
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1552 <p>Figura quam ex- |
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1553 hibet, vix ferè quid $i- |
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1554 bi velit explicat. Nos |
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1555 ad eius mētem aliam |
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1556 proponemus eamq; |
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1557 longè clariorem. |
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1558 <p>E$to vectis A<I>B</I>, |
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1559 cuius fulcimentum, |
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1560 deor$um in C, pon- |
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1561 dus D, potentia ex vecte, pondus $u$tinens E. Perpendi- |
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1562 cularis per fulcimentum FCG. Itaque quoniam poten- |
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1563 tia in E non $uperat pondus D, nec ab eo $uperatur, $tat |
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1564 vectis cum potentia Horizonti æquidi$tans, hoc e$t, in æ- |
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1565 quilibrio, vectis autem in puncto C diuiditur in partes æ- |
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1566 queponderantes. Modo præualeat potentia ponderi, & |
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1567 vectem deprimat, fiat autem in LCH, erit igitur <I>B</I>, in L, |
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1568 A in H, D in K, & CF, quæ vectem in partes æque ponde- |
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1569 rantes diuidebat, in CI. Iam igitur non æqueponderant |
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1570 partes, $i quidem pars vectis FCI, aufertur parti HCI, & |
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1571 adiungitur parti ICL, quæ ideo $it pondero$ior, vnde & |
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1572 potentia ad ponderis eleuationem adiuuatur. Eadem i- |
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1573 gitur vtitur hic demon$tratione, quam in explicando ef- |
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1574 fectu libræ, cuius fulcimentum deor$um e$t, adhibuerat. |
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1575 Nec alia de cau$$a, vt $uprà notauimus, videtur eius libræ |
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1576 in $uperiori quæ$tione, con$iderationem introduxi$$e. Et |
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1577 $anè verum e$t quod concludit, Veruntamen minimi e$t |
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1578 momenti ad tantam vim parua illa adiectio, quæ parti ve- |
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1579 ctis depre$$æ in ip$a depre$$ione adiungitur. Aliunde igi- |
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1580 tur tantæ rei cau$$a e$t petenda, quod & nos deinceps fa- |
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1581 ciemus. Videtur autem ip$e quoque Ari$toteles non $ibi |
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1582 pror$us in a$$ignata ratione $atis feci$$e, & ideo $ubiungit: |
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1583 quoniam ab æquali pondere celerius mouetur maior ca- |
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1584 rum quæ à centro $unt duo verò pondera; quod mouet & |
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1585 <foot>E 2</foot> |
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1586 <p n=>36</p> |
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1587 quod mouetur, quod igitur motum pondus ad mouens |
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1588 longitudo patitur ad longitudinem, $emper autem quã- |
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1589 tum ab hypomoch$io (id e$t, fulcimento) di$tabit magis, |
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1590 tanto facilius mouebit. Cau$$a autem <*>, quæ retro com- |
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1591 memorata e$t, quoniam quæ plus à centro di$tat maiotē |
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|
1592 de$cribit circulum. quare ab eadem potentia plus $upera- |
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1593 biturid quod mouetur, quæ plus à fulcimento di$$at. H&<*>cedil;c |
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1594 ille, qui a$$erit duo pondera in vecte con$iderari, Pondus |
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1595 nempe motum, & mouentem Potentiam (hanc enim pō- |
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1596 deris habere vim atq; rationem certum e$t) Vires autem |
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1597 potentiam acquirere ex brachij longitudine, & ex inde |
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1598 con$equenti velocitate, quo enim brachia longiora, eo |
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1599 in extremitate velociora, atque idcirco ita $e habere mo- |
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1600 tum pondus ad potentiam mouentem, vt brachij longi- |
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1601 tudo ad brachij longitudinem: brachia autem vocamus, |
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1602 partes illas vectis, quæ à fulcimento ad vtranque vectis |
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1603 extremitatem pertingunt, & ideo quantum à fulcimento |
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1604 potentia di$tabit magis, eo faciliùs pondus mouebit. |
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1605 <p>Vera vtique & explorati$$ima hæc a$$ertio e$t. Ve- |
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1606 runtamen, cau$$am huiu$ce mirabilis effectus, e$$e velo- |
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1607 citatem, quæ brachij longitudinem con$equitur, non af- |
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1608 firmamus. quæ enim velocitas in re $tante? Stant autem |
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1609 vectis, & libra dum manent in æquilibrio, & nihilo $ecius |
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1610 parua potentia ingens $u$tinet pondus. |
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1611 <p>Dicet ad hæc qui$piam, velocitatem in longiori bra- |
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1612 chio $i non actu, $altem potentiâ e$$e maiorem. At quæ$o |
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1613 quid in re quæ e$t actu, momenti habet potentia? actu e- |
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1614 nim $u$tinet, $u$tinens. Con$equìtur, (id vtique fatemur) |
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1615 nece$$ariò velocitas maior motu brachij maioris; non ta- |
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1616 men cau$$a e$t cur vis loco vbi velocitas maior $it, appo$i- |
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1617 ta magis moueat. Sanè ex velocitate, dum mouentur, pō- |
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1618 dus acquirere corpora, tum proiecta, tum cadentia cer- |
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1619 tum e$t, quod etiam in quæ$tione 19. cum Philo$opho cō- |
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1620 <p n=>37</p> |
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1621 $i derabimus. Sed hoc ex velocitate & motu $it, quæ $unt |
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1622 actu. At brachia in ip$o æquilibrio $u$tinent actu quidem, |
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1623 $ed non mouentur. Cæterum videtur A riftoteles id $ub- |
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1624 odora$$e, quod po$tea Archimedes, Mechanicorum prin- |
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1625 ceps, in propo$. 6. primi Æqueponderantium explicitè |
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1626 protulit & probauit: nempe in æquilibrio ita e$$e pondus |
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1627 ad pondus, vt brachium ad brachium, ratione permutata. |
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1628 <fig> |
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1629 <p>E$to enim vectis |
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1630 AB, quomodolibet |
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1631 fulcimento diui$us in |
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1632 C. appēdatur autem |
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1633 in A, pondus D, in B |
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1634 verò pondus E, ita $e |
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1635 habens ad pondus D, vtip$a AC ad CB. Stabit igitur ve- |
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1636 ctis, & neutram in partem verget, erit enim centrum gra- |
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1637 uitatis in C, diui$o nempe ibi vecte in partes æque ponde- |
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1638 rantes. Hoc po$t Archimedem, & in$ignes illos veteres |
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1639 Mechanicos præclari$$imè demon$trauit G. Vbaldus in |
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1640 Mechanicis, Tractatu de Libra propo$. 6. nec non de Ve- |
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1641 cte propo$. 4. |
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1642 <p>Cæterùm vt aliquid interim, quod no$trum $it, affe- |
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1643 ramus, liceat nobis egregios illos viros interrogare, quæ- |
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1644 nam mirabilis eius effectionis $it cau$$a? Dicent permu- |
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1645 tatam proportionem. Teneo, at nondum acquie$co: pe- |
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1646 tam enim, Cur ea rationis permutatio mirabilem illum |
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1647 effectum pariat. Hoc quod illi non do cent, puto nos, i- |
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1648 gnorantiæ $omno $epultos, $omnia$$e. |
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1649 <fig> |
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1650 <p>Æqualitatem $tatus |
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1651 e$$e cau$$am, nemo, vt |
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1652 puto, inficiabitur. res e$t |
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1653 enim per $e clara. E$to $i- |
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1654 quidem linea quæpiam AB, applicetur extremitati A po- |
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1655 <foot>E 3</foot> |
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1656 <p n=>38</p> |
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1657 tentia quæ dam quæ lineam ad $e trahat ad partes nempe |
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1658 A, Tum in B quædam alia potentia ip$i quæ in A potenti<17>, |
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1659 æqualis, quæ lin eam trahat $imili modo ad partes B. Datâ |
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1660 igitur harum potentiarum æqualitate, linea AB, nec ad |
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1661 partes A, nec ad partes B transferetur, $ed pror$us immo- |
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1662 bilis $tabit. |
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1663 <p>His ita con$titutis, Dico vecte quomodolibet diui$o, |
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1664 ponderibu$que vtrinque appo$itis, permutatâ propor- |
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1665 tione $ibi inuicem re$pondentibus, rem e$$e redactam ad |
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1666 æqualitatem, & inde $tatum fieri, hoc e$t, æquilibrium. |
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1667 <fig> |
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1668 <p>E$to enim vectis AB, quo modo libet diui$us in C, & |
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1669 ip$i quidem C fulcimentum $upponatur. Appendantur |
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1670 quo que vtrinque pondera ex ratione brachiorum AC, |
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1671 CB, $ibi inuicem permutatim re$pondentia, $int<13>; DE. |
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1672 Dico vectem ex æqualitate, in neutram partem inclina- |
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1673 turū, $ed perman$urum in æquilibrio. quoniam enim Pō- |
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1674 dus D idem pote$t quod brachium CB, addatur in dire- |
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1675 ctum ip$i AC, recta AF æqualis ip$i CB, item quoniam |
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1676 Pondus E id pote$t quod brachium AC, rectæ CB ad- |
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1677 datur in directum BG, ip$i AC æqualis. Igitur cum par- |
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1678 tes CA, AF totius FC, æquales $int partibus CB, BG, |
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1679 totius CG, erit totum FC, toti CG æquale. Diui$us ita- |
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1680 <p n=>39</p> |
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1681 que erit vectis FG in partes æquales FC, CG in puncto |
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1682 fulcimenti C. Et quoniam æquale in æquale non agit, |
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1683 $tabit vectis & in neutram partem inclinabit. Rur$um |
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1684 quoniam ad partem FC, duæ $unt brachiorum potentiæ |
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1685 FA, HC, appendantur puncto F, duo pondera H, I, ip$is |
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1686 DE æqualia, item puncto G, alia duo pondera ij$dem DE |
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1687 æqualia KL, iterum æqueponderabit, quippe quod æ- |
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1688 quahbus brachijs FCCG æqualia appen$a $int pondera |
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1689 HI KL. Cur igitur $eruata permutatim brachiorum & |
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1690 ponderum proportione fiat æquilibrium, ex his quæ de- |
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1691 mon$trauimus, clarè patet. |
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1692 <p>Sed forte dicet qui$piam, $i brachia, pondera $unt, |
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1693 vel ponderibus æquipollentia, $u$tinenti duplicabitur |
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1694 pondus. |
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1695 <fig> |
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1696 <p>E$to enim vectis AB, |
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1697 ita diui$us in C, vt pars |
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1698 maior CB minori AC $it |
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1699 in proportione quintu- |
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1700 pla. Appendatur autem |
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1701 in A pondus D, quintuplū |
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1702 ponderi E appen$o in B. Si |
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1703 igitur brachio AC, quod |
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1704 e$t vnum, ad datur pondus |
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1705 D, quod e$t quinque, fi ent $ex, item $i brachio CB, quod |
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1706 e$t quinque, addatur pondus E, quod e$t vnum, fient $ex. |
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1707 Fulcimentum igitur $u$tinebit duodecim, quod e$t ab- |
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1708 $urdum ex ijs quæ clarè demon$trauit G. Vbald. in Me- |
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1709 chan. tractatu de Libra propo$. 5. His re$pondemus, bra- |
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1710 chia quidem operari non pondere, $ed potentiâ, quæ vis |
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1711 quædam e$t, non autem pondus. Et$i & illud verum $it, da- |
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1712 to vecte pondero$o, fulcimentum rum ponderum appen- |
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1713 $orum, tum vectis ip$ius pondus $u$tinere. |
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1714 <p>Iacta huiu$cemodi, quam diximus, æqualitate, $e- |
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1715 <p n=>40</p> |
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1716 quitur nece$$ariò, centrum grauitatis ip$ius vectis cum |
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1717 appen$is ponderibus, ac $i vnum idem<13>ue e$$et corpus |
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1718 cadere in perpen diculari quæ per centrum ip$um & ful- |
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1719 cimentum tran$iens ad mundi centrum pertingit. |
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1720 <HEAD>QVÆSTIO IV.</HEAD> |
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1721 <HEAD><I>Quærit hic Ari$toteles, cur ij qui in nauis medio $unt remiges ma- |
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1722 ximè nauem moueant?</I></HEAD> |
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1723 <p>Ait, ideo forta$$e fieri, quò dremus vectis $it, fulcimen- |
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1724 tum verò $calmus, $tat enim. Pondus autem marei- |
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1725 p$um, quod à remo propellitur, mouens verò ip$um remi- |
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1726 gem, $emper autem plus mouere ponderis quimouet, |
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1727 quo magis di$tatà fulcimento. Ita enim maiorem fieri |
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1728 quæ ex centro; Scalmum verò centrum e$$e. Cæterùmin |
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1729 medio nauis plurimum remi intus e$$e. Ibi enim nauem |
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1730 e$$e lati$$imam. Moueri autem nauim, quoniam appellē- |
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1731 te mariremo, extremū illius quod intus e$t anterius pro- |
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1732 mouctur, cuius motum nauis $equitur, cui $calmus alliga- |
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1733 tur. Vbiautem plurimum maris diuidit remus, eo maximè |
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1734 nece$$e e$$e propelli. Plurimum autem diuidi vbi plurima |
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1735 pars remi à $calmo e$t. Rem facilem, eo quod verbis potu- |
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1736 erit, $chemate non declarauit, nos autem apponemus. |
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1737 <fig> |
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1738 <p>E$to enim nauis AB, mare CD, |
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1739 remorum alter, quiad proram EF, cu- |
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1740 ius $calmus G, alterverò in medio na- |
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1741 uis, HI, circa $calmum K. Ait igitur, |
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1742 remos e$$e vectes, $calmos verò fulci- |
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1743 menta, pondus quod remo, ceu vecte, |
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1744 mouetur mare ip$um. Itaque quoniam |
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1745 nauis lata e$t in medio vbi Scalmus K |
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1746 maior pars KH intra nauim e$t, minor |
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1747 verò KI, extra. Contra autem remiad |
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1748 proram, nempe EF pars minor EG |
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1749 <p n=>41</p> |
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1750 intra nauim, pars verò maior GF extra nauim e$t. Pondus |
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1751 autem cò faciliùs mouctur, quo maior e$t vectis pars, quæ |
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1752 à fulcimento e$t ad mouentem potentiam. |
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1753 <p>Acutè $anè Philo$ophus. Ego autem $i per mode$tiam |
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1754 liceret, dicerem, non quidem e$$e fulcimentum $calmū, |
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1755 $ed mare ip$um, pondus vero nauim, ad locum $calmi, nē- |
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1756 pe inter mouentem potentiam, & fulcimentum po$itum, |
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1757 etenim & eo pacto po$$umus vti vecte, quod ob$eruat & |
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1758 demon$trat G. Vbaldus tractatu de vecte propo$. 2. Erunt |
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1759 igitur in de$cripta figura puncta FI, quæ in mari$unt, ful- |
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1760 cimenta, quibus remorum extrema in ip$aim pul$ione ni- |
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1761 tuntur, pondera verò $eu pondus pluribus vectibus & po- |
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1762 tentijs impul$um nauis ip$a, quæ $calmis e$t annexa. Re$i- |
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1763 $tente igitur mari, cedente autem impul$ionibus $calmo, |
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1764 nauis eo transfertur, quo $calmi ab ip$a potentia mouen- |
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1765 te in anteriorem partem pelluntur. quoniam autem vt |
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1766 FG ad FE ita potentia mouens in E ad pondus motum |
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1767 in G. item vt IK ad IH ita potentia mouens in H ad pon- |
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1768 dus motum in K, maior autem e$t proportio FG ad FE |
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1769 quàm proportio IK ad IH. Maiori indiget potentia vt |
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1770 pellatur pondus in G quàm pondus in K. |
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1771 <p>Hæc certè vti diximus ita $e habent. Philo$ophi au- |
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1772 tem ratio tunc procederet, $i $tante naui immobili, vt fit |
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1773 vbi à Remoræ occulta vi aut ab alio impedimento reti- |
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1774 netur, remiges in ip$o remigandi actu mare pul$arent, |
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1775 Tunc enim verè $calmus fieret fulcimentum, mare autem |
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1776 pondus, remex verò ip$e mouens. |
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1777 <p>Addimus, fal$um videri quod a$$erit Ari$toteles, |
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1778 nempeillos qui in media naui $unt, remiges, maximè na- |
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1779 uim mouere; facilius, melius dixi$$et. Si enim maximè, |
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1780 quod ait, denorat, maximo $patio, & velocius pror$us fal- |
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1781 $um, etenim tardius mouent & minori $patio, quod nos i- |
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1782 ta demon$tramus. |
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1783 <foot>F</foot> |
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1784 <p n=>42</p> |
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1785 <fig> |
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1786 <p>E$to enim Remus AB |
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1787 qui marí fulcitur in B, Scal- |
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1788 mus remi qui ad prorã pup- |
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1789 pimue C, qui in media naui |
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1790 D, maior autem remi pars |
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1791 e$t à $calmo Dad A quami- |
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1792 p$ius C 2d A, Pellantur remi & $tante ceu centro BA, in |
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1793 E. eodem igitur tempore C eritin F, & D in G, $ed maiu<*> |
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1794 e$t $patium CF $patio DG, Ergo vnica impul$ione, plus |
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1795 mouit $calmum, hoc e$t, nauim, potentia ad puppim pro- |
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1796 ramue remigans, quàm ea quæ operatur in media naui vt |
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1797 $entire vid<*>batur ($i modo is e$t eius $en$us) Ari$toteles. |
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1798 Nece$$arium igitur e$t, quodait, maximè intelligendum, |
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1799 faciliùs, Veritatem hanc cogno$centes Triremium præ- |
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1800 fecti robu$tiores quidem remiges ad proram & puppim, |
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1801 inualidiores vcrò circa mediam triremem collocant. |
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1802 <HEAD>QVÆSTIO V.</HEAD> |
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1803 <HEAD><I>Dubitat<*>r, Cur paruum exi$tens gubernaculum, & in extremo |
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1804 nauigio tantas habeat vires, vt ab exiguo temone, & ab hominis |
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1805 vnius viribus alioqui modicè vtentis magnæ nauigiorum |
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1806 moueantur moles?</I></HEAD> |
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1807 <p>AN, inquit, quoniam gubernaculum vectis e$t, onus |
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1808 autem mare, Gubernator vero mouens e$t? Non au- |
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1809 tem $ecundùm latitudinem veluti remus, mare accipit |
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1810 gubernaculum; non enim in ante nauigium mouet, $edi- |
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1811 p$um commotum mare accipiens inclinat obliquè. quo- |
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1812 niam enim pondus e$t mare contrario innixum modo na- |
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1813 uem inclinat. fulcimentum enimin contrarium ver$atur, |
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1814 mare vetò interius, & illud exterius. illud autem $equitur |
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1815 nauis quæ illi e$t alligata & remus quidem $ecundum la- |
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1816 titudinem onus propellens & ab eodem repul$us in re- |
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1817 <p n=>43</p> |
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1818 ctum propellit, Gubernaculum verò, vt obliquum iacet |
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1819 hinc inde in obliquum motionem facit. in extremo autē, |
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1820 non in medio iacet, quoniam mouenti fa cillimum e$t mo- |
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1821 tum moucre: prima enim pars celerrimè fertur, & quo- |
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1822 niam, quemadmodum in ijs quæ feruntur in fine deficit |
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1823 latio, $ic ip$ius continui in finem, imbecillima e$t latio. |
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1824 Imbecillima autem ad expellendum e$t facilis. Propter |
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1825 hæc igitur in puppi gubernaculum ponitur, nec minus, |
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1826 quoniam paruaibi motione facta, multo maior fit in vlti- |
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1827 mo, quia æqualis angulus $emper maiorem ad$pectat, tã- |
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1828 to <13>ue magis, quanto maiores fuerint illæ, quæ continent. |
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1829 Exijs ctiam manife$tum e$t, quam ob cau$$am magis in |
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1830 contrarium procedit nauigium, quam remi ip$ius palmu- |
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1831 la, eadem enim magnitudo ij$dem mota viribus in aëre |
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1832 plus quàm in aqua progreditur. Hæc Philo$ophus, qui |
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1833 haudquaquam ex more $uo, quod duobus ferè poterat, |
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1834 $excentis verbis expo$uit. Licebat enimid tantum dicere, |
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1835 Gubernaculum (ita vocatid totum quod gubernaculo & |
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1836 temone con$tat) e$$e ceuremum, quo nauis non antror- |
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1837 $um, $ed obliquè & ad latus mouetur. quamobrem omnia |
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1838 ferè quæ de Temone dicenda fuerant, de remo loquens |
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1839 proponit. Aitautem. |
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1840 <fig> |
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1841 <p>Sit remus AB, |
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1842 $calmus vero C, remi |
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1843 in nauigio principiū |
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1844 A, palmula autem, |
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1845 quæ in mari B. Si igi- |
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1846 tur A, vbi D transla- |
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1847 tum e$t, non erit B v- |
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1848 bi E. æqualis enim, |
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1849 BE ip$i AD, æquale |
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1850 igitur translatum erit, $ed erat minus. eritigitur vbi F, mi- |
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1851 nor enim BF, ip$a AD, quareip$o GF ip$a DG. Hæc |
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1852 <foot>F 2</foot> |
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1853 <p n=>44</p> |
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1854 demon$tratio licet vera videatur, rei ta men, de qua e$t |
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1855 $ermo, minimè aptatur. Si enim aptaretur in ip$ius remi |
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1856 motu, cum palmula e$$et in F, $calmus ficret in G, excur- |
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1857 reretergo vel $calmus per remum, vel remus per $calmū, |
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1858 facta nempe ciu$modi translatione de C in G, & $ic intra |
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1859 nauim modo e$$et pars remi DC, modò verò GD, quod |
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1860 tamen non $ieri ipsâ experientia docemur. Illud quoque |
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1861 fal$um e$t, nauim ip$am tantum moueri in aëre, quantum |
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1862 e$t $patium AD, hoc e$t, remi extremum quod e$t in naui, |
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1863 $iquidem $calmi motu, non autem manubrij remi, nauis |
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1864 agatur. Aliter igitur res $e habet, & forte hoc pacto. |
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1865 <fig> |
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1866 <p>Sit remus AB, cuíus |
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1867 manubrium A, palmula |
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1868 B, $calmus C. Pellatur an- |
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1869 tror$us A, fiat<13>; in D, tunc |
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1870 $i æqualiter mouerentur |
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1871 manubrium & palmula, i- |
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1872 p$a palmula ficret in G, at |
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1873 minus mouetur: fiet ergo |
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1874 in E. ip$e verò $calmus C |
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1875 translatus erit in F, mota<13>; erit nauis à C in F, non autem |
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1876 ab A in D. P o$uitautem Ari$toteles $calmum ad medium |
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1877 remi, $ed non ad medium collocari $olet, maior enim pars |
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1878 in mare propendet puta HB, quo ca$u translationis $pa- |
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1879 tium fit maius, nempe ab H in I. fit autem motus $calmi ex |
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1880 centris qui $unt in $patio ip$o BE, quatenus autem ad te- |
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1881 monem pertinet, quem remum ait, obliquè puppim ip$am |
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1882 propellentem, ita $e res habet. |
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1883 <p>E$to nauis carina AB, prora A, puppis B, Temonis |
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1884 ala BC, gubernaculum BD, cardo verò fulcimentumue |
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1885 B; factaitaque impul$ione obliquâ gubernaculi à D in E, |
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1886 minor fiet motus in mari à C in F, erit<13>ue temo vbi EGF, |
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1887 <p n=>45</p> |
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1888 <fig> |
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1889 cardo verò vbi G, translata igitur e- |
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1890 rit eo motu, puppis ip$a à B in G. facta |
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1891 itaque paruâ motione puppis ex B in |
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1892 G, prora ip$a quæ longè di$tat à pup- |
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1893 pi B maiori $patio $uperato translata |
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1894 erit in H facta proræ in contrariam |
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1895 partem ab ea quæ facta e$t guberna- |
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1896 culi motione. Porrò quod & in præ- |
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1897 cedente quæ$tione a dnotauimus, lō- |
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1898 gè meliùs procedet demon$tratio $i |
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1899 fulcimentū mare intelligatur, quàm |
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1900 $calmus, neque enim mare ceu pon- |
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1901 dus, $ed $calmus ip$e Temonisuecardo, ponderum in$tar |
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1902 transferuntur. |
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1903 <p>Cæterùm in hac $peculatione liceat nobis aliquan- |
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1904 tulum à Philo$opho di$$entire. Certè $i breuitas Temo- |
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1905 nis, è puppi eminentis, re$pectu longitudinis totius nauis |
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1906 con$ideretur, & parua motio, quæ temone guberna culo- |
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1907 ue moto fit, nullius ferè momenti erit ad eam quæ in pro. |
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1908 ra fit translationem. aliter ergo $e rem habere non dubi- |
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1909 tamus, & quæ$tionis $olutionem aliunde petendam. Na- |
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1910 uinon currentenullum ferè, aut qui vix curandus $it ex |
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1911 gubernaculi conuer$ione nauis ad dextram $ini$tramue |
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1912 motum fieri. at eâ currente maximum, experientiâ doce- |
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1913 mur. Obliqui igitur motus qui validèin puppi $it, cau$$a |
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1914 e$t non quidem ex conuer$ione temonis percu$$io maris, |
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1915 $ed mare ip$um, cuius fluctus naui currente obliquam te- |
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1916 monis alam ad eam partem quæ mari obuertitur, impel- |
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1917 lentes temonem cum puppiad contrariam partem vali- |
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1918 di$$imè transferunt. |
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1919 <p>E$to nauis carina AB, prora B, puppis A, Temo AC, |
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1920 gubernaculum AD; Itaque currentenaui, Temone in- |
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1921 terim & guberna culo in eadem carinæ linea exi$tentibus, |
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1922 <foot>F <I>3</I></foot> |
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1923 <p n=>46</p> |
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1924 <fig> |
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1925 Temo quidem mare $ecat, nulla fa- |
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1926 ctâ in puppi, nauis ad $ini$tram dex- |
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1927 tramue translatione. Si verò mouea- |
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1928 tur gubernaculum à D in E, co moto |
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1929 mouebitur aliquantulum & puppis |
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1930 ad partes E, quod voluit Ari$toteles. |
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1931 Sedminimi, vt diximus, ea res ad tan- |
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1932 tum effectum e$t momenti. Temone |
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1933 autem in obliquum cō$tituto vt AF, |
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1934 naui interim, ventorum aut remorum |
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1935 vi pul$a proram ver$us currente te- |
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1936 monis latus à fluctibus obliquam par- |
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1937 tem alamue in ip$o cur$u ferientibus, |
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1938 in contrariam partem transfertur, ad |
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1939 eam nempe, ad quam ip$um gubernaculum vergit. facta i- |
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1940 gitur nauis ceu circa centrum centraue quæ in carina in- |
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1941 ter puppim proramue con$i derantur A, fertur in G, prora |
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1942 verò in H. ex quibus manife$tè apparet, duo ad nauis ex |
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1943 temone in puppi conuer$ione motionem e$$e ne ce$$aria; |
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1944 Temonis nempe obliquationem, & nauis cur$um, quorū |
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1945 $i alterum $ine altero adhibeatur, nullam fieri quæ alicu- |
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1946 ius momenti $it, nauis conuer$ionem. Illud quoque nota- |
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1947 mus, carinam in nauis conuer$ione vectis in$tar $e habere, |
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1948 cuius pars mota ad puppim, & mouens potentia e$t; fulci- |
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1949 mentum verò circa proram, potentia autem mouens ma- |
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1950 reip$um, temonem in nauis cur$u oblique feriens. Vnde |
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1951 colligimus naues, quo longiores $unt in mouente ad Te- |
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1952 monem adhibita maiori facilitate ad dextram $ini$tram- |
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1953 ue propelli: quod $anè ip$emet con$iderauit Ari$toteles, |
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1954 quì idcirco inquir, in extremo, non autem in medio temo- |
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1955 nem poni eo quod mouenti facilimum $it ab extremo |
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1956 motum mouere. |
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1957 <p>Ex hac no$trâ $peculatione ratio habetur eius ma- |
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1958 <p n=>47</p> |
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1959 chinationis, quâ in magnis fluminibus, ceu Pado, Abdua |
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1960 & $imilibus, Portitores, equos, currus, viatore$<13>; ip$os, è |
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1961 ripa in ripam transferunt. Pulcherrima enim res e$t, & |
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1962 nobis per$pecti$$ima, qui Gua$tallâ re$identiæ olim no- |
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1963 $træ oppido ad Padum, Mantuam pergentes $æpi$$imè ad |
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1964 Ca$trum B<*>rgi Iu$is ea qua diximus machinatione lati$- |
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1965 $imum eiu$dem Padi aluum tran$ie cimus. Habet autem |
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1966 $e hoc pacto. |
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1967 <fig> |
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1968 <p>E$to fluminis citerior |
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1969 ripa AB, vlterior CD. Pon- |
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1970 tones duo tabulis $trati, & v- |
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1971 nà firmiter juncti EF, Temo |
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1972 inter eorum puppes extans |
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1973 GH, locus in ripa $tabilis A, |
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1974 funis, quo pontones, & ma- |
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1975 china tota continetur AI. |
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1976 fluuij decur$us ver$us BD, |
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1977 $tantibus itaque pontonibus |
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1978 ad ripam citeriorem AB, Te- |
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1979 mone in neutrã partem pul- |
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1980 $o, cum aqua decurrens eum |
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1981 re$i$tentem non inueniat, |
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1982 $cinditur quidem ab eo, $ed |
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1983 non propellit, eo autem con- |
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1984 uer$o & in GK con$tituto, a- |
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1985 la eius GK ab aqua defluente propul$a machinam $ecum |
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1986 trahit ver$us ripam CD, factâ motione circa centrum $eu |
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1987 $tabilem locum A, otio$is interim portitoribus, donec per |
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1988 circuli portionem ML deuenerit ad vlteriorem ripam in |
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1989 L. Vnde iterum temone in contrariam partem conuer$o, |
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1990 aquâ $imiliter temonem propellente, per eandem circuli |
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1991 portionem ad ripam citeriorem reuertitur, à qua paullo |
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1992 antè di$ce$$erat. Ex quibus apparet, motus cau$$am non |
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1993 <p n=>48</p> |
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1994 e$$e $olam cam, quæ ab ala temonis fit, aquæ percu$$ionē, |
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1995 vt $en$erat Ari$toteles, $ed currentis a quæ temonis alam |
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1996 ferientis impul$ion<*>m: nihil autem referre, vtrum $tante |
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1997 naui a qua currat, vel câ currente a qua $tet, vt in mari fit, |
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1998 idem enim vtroque modo temo patitur. Vt autem machi- |
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1999 næ huius & totius negotij $pecies facilius animo concipia- |
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2000 tur, $chema hoc $tudio $orum oculis $ubijciemus. |
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2001 <fig> |
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2002 <p>Lembi nauiculæueideo appo$itæ $unt, vt oblongum |
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2003 funem $u$tineant; id etenim nî fieret, aquæ immer$us a- |
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2004 quam $cindens machinæ motum impediret, ideo etiam |
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2005 apponuntur, ne funis madens celeriter maceretur & pu- |
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2006 tre$cat. |
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2007 <p>Huic $peculationi affinis e$t ea, velorum eorum, |
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2008 quæ obliquè ventum, excipientia frumentarijs molis |
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2009 dant motum, item verticillorum ex papyro, quibus con- |
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2010 tra ventum currentes per lu$um pueri vtuntur. vnicum |
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2011 <p n=>49</p> |
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2012 enim horum emnium principium, & eadem, ratio. |
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2013 <p>Diximus enim, Temonem currente naui, lateraliter |
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2014 conuer$um obuios fluctus ex cipientem puppim ip$am ob- |
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2015 liquè in alteram partem transferre. Porrò ea vela, de qui- |
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2016 bus loquimur, ventorum flatibus obliquè oppo$ita can- |
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2017 dem ob cau$$am circulariter agitantur, quodvt figurâ eui- |
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2018 dentius fiat, |
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2019 <fig> |
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2020 <p>E$to velum AB, brachio |
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2021 CE obliquè affixum, ita vt |
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2022 angulus ACE maior $it an- |
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2023 gulo BCE, ventus obliquè |
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2024 velum feriens FG. Itaq; quo- |
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2025 niam ventus in velum obli. |
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2026 quum incidit, elabiturvelum, |
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2027 & circa centrum E vnà cum |
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2028 brachio circumuertitur, in |
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2029 cuius locum $uccedit velum |
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2030 HI, ex qua a$$idua velorum |
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2031 $ucce$$ione, brachiorum & a- |
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2032 xis cui adhærent, rotatio fit |
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2033 perpetua. Sed enim de Te- |
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2034 mone agentes non e$t interim cur de caudis auium pi$ci- |
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2035 umque taceamus, in$tar enim remonum $unt à Naturai- |
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2036 p$a opportunis animalium partibus, po$tremis videlicet, |
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2037 appo$iti, quanquam nec$olum Temonis v$um præ$tent, |
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2038 vt videbimus. |
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2039 <p>E$to pi$cis AB, cuius caput A, cauda verò CB. Hac |
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2040 igitur neutram in partem reflexâ, pi$cis pinnarum motu |
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2041 rectâ in anteriorem partem progreditur. Si autem nece$- |
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2042 $e ei fuerit ad dextram $ini$tram<13>ue conuerti non pote- |
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2043 rit, ni$i cauda ip$a iuuetur. Omnis enim motus progre$$i- |
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2044 uus quiete indiget, nec ab$q; $tabili fulcimento progredi |
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2045 <foot>G</foot> |
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2046 <p n=>50</p> |
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2047 <fig> |
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2048 pote$t, quod in libris de ani- |
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2049 malium ince$$u docetip$e- |
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2050 met Philo$ophus. Sit igitur, |
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2051 pi$cem conuerti velle, & fie- |
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2052 ri capite in D, deflectet illi- |
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2053 co caudam in E, ca<13>; aquam |
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2054 ceu $tabile quippiam feriēs |
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2055 ei<13>ue quod<*>mmodo fultus, |
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2056 reliquum corpus CA refle- |
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2057 ctet in D, $i autem conuerti |
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2058 velit in F, caudam defle ctet in G, & eadem ratione <*> cte- |
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2059 tur in F. Sed & Temonis quoque v$um præ$tat natatili- |
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2060 bus & volatilibus cauda. Sit enim rectus pi$cis, hoc e$t, re- |
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2061 ctâ pergens IKL, caudam obliquet in KM itaque ex a- |
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2062 quæ in ip$o motu colli$ione, eius po$teriora pellentur vbi |
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2063 INO. Hæc itaque nos de Temone, quatenus ad hanc |
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2064 quæ$tionem pertinet, con$idera$$e $it $atis. |
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2065 <HEAD>QVÆSTIO VI.</HEAD> |
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2066 <HEAD><I>Dubitatur, Cur quanto Antenna $ublimior fuerit, ÿ$dem velis, & |
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2067 vento eodem celeriùs ferantur nauigia?</I></HEAD> |
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2068 <p>Soluit Philo$ophus, inquiens: An quia malus quidem |
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2069 $it vectis, fulcimentum verò mali $edes, in qua colloca- |
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2070 tur, pondus autem quod moueri debet, ip$um nauigium: |
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2071 mouens verò is, qui vela tendit $piritus? Si igitur quanto |
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2072 remotior fuerit fulcimentum facilius cadem potentia, & |
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2073 citiùs idem mouet pondus, altius certè $ublatâ antennâ, |
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2074 velum à mali $ede, qu<17> fulcimentum e$t remotius faciens, |
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2075 id efficiet. Hæcille. quæ $ic figurâ explicamus. |
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2076 <p n=>51</p> |
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2077 <fig> |
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2078 <p>E$to nauis AB, malus CD, |
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2079 mali $edes D, locus antennæ |
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2080 $ublimior C, depre$$ior E: ita- |
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2081 que quoniam CD vectis e$t, |
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2082 quo mouens remotior fuerit à |
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2083 fulcimento D, co citiùs & vio- |
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2084 lentiùs pellet, velocius ergo |
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2085 nauis mouebitur antenna in |
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2086 C, quàm in E, con$tituta. |
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2087 <p>Plau$ibilia $unt hæc, at certè per veritatem ip$am, |
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2088 non vera. Rogo, Si fulcimentum dum vectis mouetur, cē- |
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2089 trum e$t, centrum vtique motus erit D. $pirante igitur va- |
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2090 lidè vento inclinabitur malus, fiet<13>; vbi FGD, quæ qui- |
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2091 dem in clinatio vio lentius fiet, vento pellentein F q uàm |
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2092 in G, vtpote puncto à fulcimento remotiore. Impul$o ma- |
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2093 lo, duo nece$$ariò cō$equentur, vel enim ad ip$am $edem |
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2094 D. frangetur vel puppis ip$a circa D punctum conuer$a, |
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2095 vt mali $e quatur motum eleuabitur. Prora verò $ubmer- |
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2096 getur facta naui in HDI. Quod $i qui$piam funem ad ma- |
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2097 li $ummitatem annexam ad ip$am puppim alligauerit in |
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2098 B, impe dietur $anè mali in clinatio ad partes F, & ideo nul- |
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2099 la vis pror$us fiet in D ex vectis ratione. Attamen nihilo |
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2100 $ecius, quo $ublimior fuerit antenna, eo faciliùs à $pirante |
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2101 vento puppis eleuabitur. quatenus igitur malus vectis |
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2102 e$t, hoc tantum quod dicimus operatur. Quod $i contrà |
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2103 obiectum fuerit, experientiam docere, quo $ublimior an- |
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2104 tenna fuerit, eo citiùs nauigium, $piritu flante moueri. |
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2105 Re$pon$io facilis, nempe, mirum non e$$e, $i mali pars $ub- |
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2106 limior validius à vento feriatur. Videmus enim, & turres |
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2107 quo $ublimiores fuerint, eo magis à ventorum impetuo$is |
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2108 flatibus infe$tari, quod $anè ad vectis longitudinem refer- |
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2109 re, e$$et ridiculum. Cætcrùm quod ad puppis faciliorem |
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2110 eleuationem ex mali ip$ius altitudine pertinet, ad vectis |
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2111 <foot>G 2</foot> |
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2112 <p n=>52</p> |
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2113 contemplationem reducimus. e$t enim quæ dam vectium |
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2114 $pecies ab alijs non con$iderata, cuius brachia in angu- |
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2115 lum de$inunt, vtip$e angulus in operatione $it fulcimen- |
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2116 tum. |
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2117 <fig> |
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2118 <p>E$to enim vectis, de quo agimus, |
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2119 ABC, cuius brachia AB, BC. iuncta |
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2120 ad angulum B, $it<13>ue B in operatione |
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2121 fulc mentum. Nec quicquam refert |
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2122 quatenus ad v$um pertinet, vtrum an- |
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2123 gulus ip$e rectus $it, acutus vel obtu- |
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2124 $us. $it autem modò rectus. Ponaturi- |
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2125 gitur pondus aliquod in C, tum po- |
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2126 tentia quædam applicetur in A, qu<17> i- |
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2127 p$am vectis extremitatem A propel- |
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2128 lat in D. erit igitur AB in DB & an- |
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2129 gulo $eruato BC in BE. Pondus igi- |
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2130 tur cum parte vectis BC eleuabitur in E. In hoc autem |
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2131 vectis genere attenditur proportio quam habet AB ad |
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2132 BC. Si enim potentia quæ applicatur in A ita $e habet ad |
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2133 pondus in C vt CB, ip$i BA, fiet æ quilibrium. Si maior |
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2134 autem fuerit proportio potentiæ in A, ad pondus in C, ea |
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2135 quam habet AB ad BC, $uperatâ ponderis re$i$tentiâ fiet |
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2136 motus. Res autem haud aliter $e habet, ac $i producta in |
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2137 F, fieret BF æqualis BC. Tunc enim vectis ad rectitudi- |
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2138 nem, $eruatâ proportione, redigeretur, & ita potentia in |
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2139 A, fulcimento B operaretur in F, vt operabatur in C. |
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2140 <p>Ad huius vectis naturam referuntur fabrorum mal- |
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2141 lei, quibus clauos reuellunt, forcipes item quæ tenaci |
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2142 mor$u clauorum capita vmbellasue apprendentes, vio- |
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2143 lenterè tabulis extrahunt. In malleo itaque $ubtili, vt in |
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2144 figura videre e$t, AB vectis e$t pars quæ à fulcimento ad |
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2145 potentiam; ac verò quæ à fulcimento ad pondus, ponderi |
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2146 <p n=>53</p> |
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2147 <fig> |
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2148 $iquidem æquiparatur re$i- |
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2149 $tentia qu<17> fit in C. I dem ob- |
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2150 $eruamus in forcipe, in quo |
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2151 duo quidem brachia AD, |
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2152 CB, quatenus ad appren$io- |
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2153 nem pertinet, fulcimentum, |
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2154 habentin ip$o cētro $eu ver- |
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2155 rebra, & ideo quo longiores |
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2156 fuerint, eo tenaciùs appre- |
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2157 hendunt & retinent. quate- |
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2158 nus autem ad extractionem, |
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2159 facit, pro vnico forceps totus habetur vecte, cuius quidē |
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2160 pars à potentia ad fulcimentum AB. quæ verò à fulcimē- |
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2161 to ad hoc e$t clauum ip$um qui reuellitur AC. Violenti$- |
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2162 $imè autem extrahunt forcipes, propterea quod maxima |
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2163 $it proportio longitudinis brachij BA, ad eam quæ e$t ab |
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2164 A ad C. |
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2165 <p>His igitur hoc pacto examinatis, ad nauim & malum |
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2166 reuertentes, dicimus, tunc facillimam fieri puppis eleua- |
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2167 tionem, proræ verò demer$ionem, cum maxima fuerit |
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2168 proportio, quam habet altitudo mali, ad eam nauis partē |
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2169 quæ à malo ad ip$am puppis extremitatem, pertingit. |
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2170 Quamobrem prudentes nauium fabri, vt huic difficultati |
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2171 occurrant, malum non in medio quidem nauis, $ed in ter- |
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2172 tia ferè parte longitudinis quæ à prora e$t, puppim ver$us |
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2173 con$tituunt. |
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2174 <fig> |
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2175 <p>E$to enim nauis AB; cuius |
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2176 malus CD: prora A: puppis B; vē- |
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2177 to igitur velum impellente, malū |
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2178 ad partem contrariam vergit, pu- |
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2179 ta in FD. At quoniã ca<*>che$ium |
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2180 funi ad puppim vnitur in B, nauim, |
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2181 hoc e$t, ip$am puppim trahatne- |
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2182 <foot>G 3</foot> |
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2183 <p n=>54</p> |
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2184 ce$$e e$t. non pote$t autem; quoniam $uburræ grauitas & |
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2185 onera, quæ naui impo$ita inter D. & <I>B.</I> grauitatis centrum |
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2186 circa punctum E con$tituunt, quod quidem vi ventorum |
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2187 inclinante malo ab E, in G eleuaretur, quo igitur minor |
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2188 fuerit proportio CD ad DE & maius pondus ip$um cu- |
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2189 ius grauitatis centrum in E minus præualebit potentia |
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2190 pellens in C ad eleu<*>tionem partis nauigij, quæ à mali $e- |
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2191 de ad puppim intercedit, An igitur malus $it vectis, pesve- |
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2192 rò fulcimentum, pondus autem quodvecte mouetur, ipsū |
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2193 nauigium, vt placuit Ari$toteli, & qua item ratione malus |
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2194 in nauim vt vectis operetur, exijs qu<17> dicta $unt, facilè pa- |
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2195 tet. |
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2196 <HEAD>QVÆSTIO VII.</HEAD> |
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2197 <HEAD><I>Quaritur, Cur quando ex puppi nauigare voluerint, non flante ex |
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2198 puppi vento, veli quidem partem, quæ ad gubernatorem vergit, |
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2199 con$tringunt; illam verò quæ proram ver$us e$t, pedem |
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2200 facientes, relaxant?</I></HEAD> |
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2201 <p>Mirabilis huius effe ctionis cau$$am explicat Ari$tote- |
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2202 les. inquit enim, An quia retrahere quidem multo |
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2203 exi$tente vento gubernaculum non pote$t, pauco autem |
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2204 pote$t, quem con$tringunt? propellit igitur quidem ip$e |
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2205 ventus, in puppim verò illum con$tituit gubernaculum, |
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2206 retrahens, & mare compellens: $imul & nautæ ip$i cum |
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2207 vento contendunt; in contrariam enim $e reclinant par- |
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2208 tem. Hæcille. |
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2209 <p>Cuius $en$um breuitate $ubob$curum, mirâ facilita- |
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2210 te explicat Picolomineus. Nos autem vt rem lucidiorem |
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2211 faciamus, $chema, quod necip$e fecit, nec Philo$ophus, |
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2212 proponemus. |
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2213 <p>E$to nauis A <I>B</I>, cuius prora A, puppis verò D, guber- |
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2214 naculum C<I>B</I>, temonis ala <I>B</I>D, veli $inus EF, velum vero |
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2215 ita con$titutum, vt directè ex puppi flantem ventum exci- |
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2216 <p n=>55</p> |
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2217 <fig> |
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2218 piat. Hoc vbi euenerit, naui- |
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2219 gium, rectâ è puppi mouetur |
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2220 in proram; Si autem ventus la- |
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2221 teraliter $pirat, puta à parte |
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2222 G ver$us H & nihilo $ecius na- |
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2223 uigium, ac $i ventus ex pup- |
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2224 pi e$$et antror$um propelle- |
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2225 re volunt, velum quidem obli- |
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2226 quant partem cius infimam, |
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2227 pedem nempe, quæ e$t in F |
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2228 contrahentes, Cornu verò |
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2229 antennæ vbi E, proram ver$us |
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2230 laxantes ventum<13>; ip$um obliquè ex cipientesid efficiūt, |
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2231 vt ventus minus violenter feriat, & minori $ui parte velū |
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2232 impleat, & quoniam ventus velum pellit in partem con- |
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2233 trariam, nempe in H, ip$ivt vento re$i$tant conuer$o gu- |
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2234 bernaculo ex C in L, & temone <I>B</I>D, in <I>B</I>M compellunt |
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2235 proram ad partem à qua ventu<*> ip$e $pirat. Sit igitur inter |
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2236 ventum & temonem pugna, illo proram in dextram, hoc |
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2237 verò eandem in $ini$tram pellente, itaq; cum neuter præ- |
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2238 ualeat, nece$$ario nauis mediam viam, quæ inter vtramq; |
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2239 e$t, $uo cur$u tenet. Nautæ autem ideo in partem nauis |
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2240 AE<I>B</I>, quæ ver$us ventum e$t, $e conferunt, vt vento æqui- |
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2241 librium faciant, ne $cilicetnaui in cōtrariam partem pel- |
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2242 lente $piritu, eam demergat. Cæterùm quod nec Ari$to- |
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2243 teles nec Picolomineus animaduerterunt, velum obli- |
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2244 què con$titutum à vento in anteriora impellitur eandem |
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2245 ob cau$$am, quam retulimus, vbi de temone & velis, qui- |
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2246 bus farin ariæ molæ cōuertuntur, verba faceremus. Quod |
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2247 autem addit Picolomineus rem ad vectem reduci po$$e, |
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2248 non e$t cur $ub $ilentio prætereamus. Ventus, in quit, pon- |
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2249 deris gubernaculum mouentis vicem obtinet; centrum |
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2250 verò (fulcimentum intelligit) in medio nauis e$t, quod ta- |
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2251 <p n=>56</p> |
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2252 men ad proram vergit, vt faciliùs ip$i vento re$i$tere po$- |
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2253 $it. Tunc enim in rectum mouebitur nauis, cum $ibi inui- |
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2254 cem æ quatæ vires, qua$i libramentum con$tituerint. Hæc |
|
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2255 ille, cuius $en$um figurâ propo$itâ facilè aperiemus. |
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2256 <fig> |
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2257 <p>E$to carina AB, cuius prora |
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2258 A, puppis, B temo BC, ventus verò |
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2259 obliquè feriens H. Conuer$us ita- |
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2260 que temo vt in BC vndarum vi cur- |
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2261 rente naui repul$us $it in EF ten- |
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2262 dens ver$us I, quo ca$u prora con- |
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2263 uertitur in D, nempe contra ventū |
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2264 qui $pirat ex H. fit autem conuer- |
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2265 $io circa punctum G, quod fulcimenti locum obtinet. Vē- |
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2266 tus verò ad contrariam partē proram impellit, repugnans |
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2267 Temonis violentiæ contra ip$am proram dirigentis. E$t i- |
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2268 gitur AB, $eu DE carina, in$tar vectis, cuius fulcimentum |
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2269 G, vis mouens mare quo temo EF repellitur, pondus ve- |
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2270 ro, ventus premens in D; quo igitur remotior erittemo à |
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2271 fulcimento G, D autem vbi pondus ei vicinius, eo magis |
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2272 temo venti vim$uperabit. Hæc Picolominei ratio, quam |
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2273 explicauimus, $anè ingenio$a e$t, verum enimuero, quo- |
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2274 niam fulcimentum $ui naturâ $tare debet, hic verò nullã |
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2275 habeat $ta bilitatem, difficultatem patitur. |
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2276 <HEAD>QVÆSTIO VIII.</HEAD> |
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2277 <HEAD><I>Quæritur, Cur ex figuris omnibus rotundæ faciliùs |
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2278 moueantur?</I></HEAD> |
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2279 <p>Trifariam, in quit Ari$toteles, circulum rotari contin- |
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2280 git; Aut $ecundum ab$idem cētro $imul moto, quem- |
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2281 admodum plau$tri vertitur rota; aut circa manens cen- |
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2282 trum, velutitrochleæ puteorum, $tante centro: Autin pa- |
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2283 uimento manente centro, $icuti figuli rota conuertitur. |
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2284 <p n=>57</p> |
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2285 Cau$$am verò explicans, ait, celerrima eiu$modi corpora |
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2286 e$$e, eo quod paruâ $ui parte planum contingunt, vti cir- |
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2287 culus $e cundum punctum, item quoniam non offen$ant: |
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2288 Non offen$andi vero e$$e cau$$am, quod $emotum à terra |
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2289 habeant angulum. Item propterea quod corpus, cui fiunt |
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2290 obuiam, $ecundum pu$illum tangunt. Rectilineo autem |
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2291 aliter euenire, quippe quod rectitudine $uâ, multum pla- |
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2292 ni contingat. Ad hæc, quo nutat pondus eo mouentem |
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2293 mouere. |
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2294 <p>Hæc ferè Philo$ophus, cuius rationes ad eum $olum- |
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2295 modo circularem motum faciunt, qui fit $e cundum ab$i- |
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2296 dem, vt in carrorum rotis v$u venit, nec aptantur rotis fi- |
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2297 gulorum trochlei$<13>ue, cuiu$modi $unt illæ, quæ $upra |
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2298 puteos appenduntur. Nos igitur, ad Ari$totelis mentem, |
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2299 primam rotationis $peciem, quæ e$t $ecundum ab$idem, |
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2300 examinabimus. |
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|
2301 <fig> |
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2302 <p>E$to rota $phæ- |
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2303 raue AB, cuius cen- |
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2304 trum C; Horizontis |
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2305 planum DE; conta- |
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2306 ctus circuli in plano |
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2307 B. perpē dicularis ho- |
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2308 rizonti à puncto cō- |
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2309 tactus B ip$a <I>B</I>CA, |
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2310 tran$iens per centrū |
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2311 C, partes rotæ circa |
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2312 perpendicularem AF<I>B</I>, AG<I>B</I>, angulus contactus G<I>B</I>E. |
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2313 Primo itaque id con$tat, circulum in puncto planum, $eu |
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2314 lineam contingere. At quoniam, vt Mechanici, de circulis |
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2315 roti$<13>ue $eu $phæris agimus materialibus, rectè Philo$o- |
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2316 phus non in puncto planum præ cisè tangere dixit, $ed $e- |
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2317 cundum partem $ui minimam. Angulum porro, quem à |
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2318 terra $emotum dicit, ip$e angulus e$t contingenti<17>. cleua- |
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2319 <foot>H</foot> |
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2320 <p n=>58</p> |
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2321 tur enim ex <I>B</I> in G. Si autem corpus quodpiam in plano |
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2322 fuerit, puta HI in puncto illud tanget ci culus ei occur- |
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2323 rens, exempli gratiâ in K. Hæc igitur accidunt circulari |
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2324 figuræ. In lateratis autem $ecus fit, quippe quænec in pū- |
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2325 cto $eu $ecundum paruam $ui partem, planum tangunt, |
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2326 nec $emotum vt circulus à plano habent angulum, nec |
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2327 impingentes offen diculum in puncto tangunt. Cæterùm |
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2328 poti$$imam facilitatis motus in rotatione quæ fit $ecun- |
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2329 dum ab$idem, e$$e cau$$am dixit, nempe quò nutat pon- |
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2330 dus cò à mouente impelli ac moueri. Primò igitu circu- |
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2331 laris $phæricaue figura in æquilibrio $tat; æquales enim |
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2332 $unt partes quæ circa perpendicularem: ceu $unt AF<I>B</I>, |
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2333 AG<I>B.</I> $i enim impul$us fiat ex parte F, pars oppo$ita nuta- |
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2334 bit, & propendet in pa<*>tem G, & $uo nutu motu<13>; $ecum |
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2335 trahet partem AF<I>B</I>, fiet<13>ue progre$$us. Si enim ducatur |
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2336 FCG diameter, ip$i horizonti æ que di$tans, erit veluti li- |
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2337 bra, cuius pondera vtrinque AF<I>B</I>, AG<I>B</I>, brachia verò |
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2338 æqualia CF, CG. Potentia autem quâ trahitur pellitur- |
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2339 ue ad in$tar ponderis $e habet, quo addito partium alteri, |
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2340 facto <13>ue rece$$u ab æquilibrio, $equetur motus. Putauêre |
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2341 quidam, vt refert Philo$ophus, circularē lineam, ita per- |
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2342 peti motu ver$atumiri, vt manentia, propte<*> contrarium |
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2343 nixum, manent, neque enim circulus in plano contrarium |
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2344 nixum habet, cum $it, veluti dicebamus, in æquilibrio & |
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2345 facilis in vtramuis partem moueri. Veruntamen perpe- |
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2346 tuum e$$e non po$$e horum corporum motum, ea e$t cau$- |
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2347 $a, quod violentum accidat naturæ, & ideo non durabile. |
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2348 Ad hæc, addit Philo$ophus, Maiores circulos ad minores |
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2349 nutum habere quēdam; & nutum maioris ad minoris nu- |
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2350 tum, $e habere vt angulos ad angulos, & diametrū ad dia- |
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2351 metrum. Angulos autem hî c$ectores ip$os vocat; oportet |
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2352 enim circulos tum maiores tum minores circa idein cen- |
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2353 trum e$$e con$titutos. Hæc autem non ab$imili ab eo |
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2354 quod $uprà po$uimus $chemate explicantur. |
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2355 <p n=>59</p> |
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2356 <fig> |
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2357 <p>E$to enim circulus |
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2358 AB circa centrum, C, |
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2359 Horizontis planum DE, |
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2360 tangens circulum in B, |
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2361 linea verò perpendicu- |
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2362 laris per centrum BCA. |
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2363 Sit autem circa idem cē- |
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2364 trum C, minor circulus |
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2365 FG, ducatur <13>ue CH $e- |
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2366 cu<*> minorem circulum in I, tangens verò maiorem in H, |
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2367 con$tituen$<13>ue cum AC linea angulum ACH, duos an- |
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2368 gulos, ex Ari$totelis mente comprehendentem, hoc e$t, |
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2369 duos $ectores ACH, FCI. quoniam igitur $ector $eu an- |
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2370 gulus ACH, $uo $patio $uperat angulum $eu $ectorem |
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2371 FGI, facilè ex nutu quem maior $upra minorem habet, |
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2372 maior ip$e mìnorem mouet. Videtur autem tacitè Philo- |
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2373 $ophus hæc ad vectis naturam referre, cuius altera extre- |
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2374 mitatum in centro $it, altera verò in ab $ide, & ita $e habe- |
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2375 renutum maioris $upra minorem, vt vectis ad vectem, hoc |
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2376 e$t, $emid<*>ameter ad $emidiametrum, $eu $ector ad $ecto- |
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2377 rem, quos quidem $ectores, vt vidimus, angulos appellat. |
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2378 Hæc autem quæ de nutu refert, licet $ubtilia $int, vera e$- |
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2379 $e non videntur. Si enim in figura producatur ad oppo$i- |
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2380 tam partem $emidiameter HC in K $ecans minorem cir- |
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2381 culum in L, duos alios $ectores angulosue habebimus, nē- |
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2382 pe KCB, LCG, ip$is ACHFCI æ quales. Itaq; quan- |
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2383 tum adiuu at motum anguli ACH maioris nutus, in de- |
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|
2384 $cendendo ad partes B, tantundem retardat anguli item |
|
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2385 maioris KCB, contra nutus (vtita appellem) in a$cendē- |
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|
2386 do ad partes A. & $anè quatenus ad reinaturam pertinet |
|
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2387 & ad ip$um æquilibrium, non differunt maiores circuli à |
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|
2388 minoribus, nec $unt maiores minoribus mobiliores, imo |
|
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|
2389 ex ali quaratione minores videntur fore ad motum faci- |
|
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|
2390 <foot>H 2</foot> |
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|
2391 <p n=>60</p> |
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2392 liores, tum quia data materiæ æqualitate $unt leuiores, |
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|
2393 tum etiam quod maior e$t angulus contactus ad planuin |
|
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|
2394 circum ferenti<17> minoris quàm maioris circuli, vt in $ubie- |
|
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|
2395 <fig> |
|
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|
2396 cta figura angulus ABC maior |
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2397 e$t angulo DBC, in materiali i- |
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|
2398 gitur circulo rotaue maiore $ui |
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|
2399 parte tanget planum DB circu- |
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|
2400 lus, ip$o AB. quicquid tamen fit, |
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|
2401 mobiliores $unt maiores circuli |
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|
2402 non quidem ex natura circuli, |
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|
2403 quæ tam in maioribus quàm in |
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2404 ip$is minoribus e$t par, $ed alijs de cau$$is, quas $uo loco |
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2405 examin abimus. |
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2406 <p>Cæterùm vt aliquid de motu qui $e cundum ab$idem |
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2407 fit, ex no$tro penu promamus, Dicimus, Circulos, rota$ue, |
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2408 quæ hoc pacto mouentur, vel per horizontis planum mo- |
|
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|
2409 ueri, vel per accliue, aut decliue. Siautem perhorizontis |
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2410 planum, ideo facilem e$$e motum, quòd nunquam, cæte- |
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2411 ris paribus, centrum grauitatis ip$ius corporis à centro |
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2412 mundi, in ip$a rotatione, fiat remotius. |
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|
2413 <fig> |
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2414 <p>E$to enim planum, |
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2415 horizontis AB, cui circu- |
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2416 lus in$i$tat AD, circa cen- |
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|
2417 trum C, diui$us per centrū |
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|
2418 ip$um à perpendiculari |
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|
2419 ACD; Ducatur autem per |
|
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|
2420 centrum C recta linea ho- |
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2421 rizonti æquidi$tans, ECFG: dum diuidatur circulus vt- |
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2422 <*>unque in partes AH, HF, FI, ID, & CI, CH iungan- |
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2423 tur. Po$th æcintelligatur circulum $ecundum ab$idem |
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2424 moueri ad partes G, erit igitur aliquando punctum H, |
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2425 rangens horizontis planum, tangat autem in K, tum F in |
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2426 <p n=>61</p> |
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2427 L, I in N. D verò in O. Ducantur<13>ue KP, LQ, NR, OS |
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2428 ip$i AC parallelæ horizonti autem perpen diculares. |
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2429 Centrum ergo circuli, quod idem & grauitatis e$t centrū, |
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2430 feretur per rectam CPQRS, $unt enim KP, LQ, NR, |
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2431 OS ip$i AC $emidiam etro æquales, nūquam igitur cen- |
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2432 trum ip$um C in circuli rotatione ab horizontis plano e- |
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2433 leuabitur, nec à mundi centro fietremotius. |
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2434 <p>Hoc autem longè aliter cæteris figuris contingit, |
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2435 quarum motus ideo in æ qualis, quòd non $em per in rota- |
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2436 tione centium grauitatis eandem $eruet à mundi centro |
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2437 di$tantiam. |
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2438 <fig> |
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2439 <p>E$to enim Ellip$is |
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2440 ABCD, cuius cētrum |
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2441 E, diameter longior |
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2442 BED, breuior AEC, |
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2443 Horizontis planum, |
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2444 FCG. locus contactus |
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2445 C perpendicularis à |
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2446 contactu per centrum i- |
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2447 p$a CEA diuidens El- |
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2448 lip$im in partes æquales, & æqueponderantes ABC, |
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2449 ADC. Sumantur in quadrante CD, pūcta HI, tum EH, |
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2450 HI iungantur, eritautem EH longior ip$a EC, tum EI, |
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2451 ip$a EH & ED, p$a EI. Rotetur ellip$is $ecun dum ab$i- |
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2452 dem, fiet igitur punctum H in K, & à puncto K horizonti |
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2453 perpendicularis erigatur KL, quæ fiat æ qualis EH. P o$t |
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2454 hæc punctum I eritin M, & ab M perpen dicularis, æqua- |
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2455 lis EI. rui$us D fiat in O, & ip$i ED, æqualis perpendicu- |
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2456 laris OP. Mota igitur ellip$ià C in K, haud ita difficilis e- |
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2457 rit motus, quippe quod haud multum EH $uperet EC, at |
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2458 difficilior erit translatio in M, difficillima verò in O. Val<*> |
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2459 de enim à $itu E, ibi attollitur grauitatis centrum, a$cen- |
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2460 dens nempe vbi P. Videmus igitur ex his eandem poten- |
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|
2461 <foot>H 3</foot> |
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2462 <p n=>62</p> |
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2463 tiam in mouendo ellip$im, haud pariter $e habere, vt in |
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2464 mouendo circulum. ibi enim centrum grauitatis fertur |
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2465 per æquidi$tantem horizonti, hic verò modò attollitur, |
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|
2466 modò deprimitur, quod $anè mole$tiam & difficultatem |
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2467 facit. Sed idem alijs figuris contingere, & maximè latera- |
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2468 tis, ita docebimus. |
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|
2469 <fig> |
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2470 <p>E$to enim triangulum |
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|
2471 æquilaterum ABC, cuius |
|
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2472 grauitatis centrum E hori- |
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|
2473 zontis planum BD. Demit- |
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2474 tatur à vertice A perpendi- |
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|
2475 cularis horizonti AF tran$- |
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2476 ibit autem per centrum E, |
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|
2477 & bifariam diuidet ba$im |
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|
2478 BC in F. Sunt autem trianguli ABF, ACF, æquales & |
|
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|
2479 æqueponderantes. angulus verò AFC rectus. lungatur |
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2480 EC, erit igitur maior EC, ip$a EF. Rotetur iraque trian- |
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|
2481 gulum circa punctum C, fiat<13>; EC horizonti perpendi- |
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|
2482 cularis, $it<13>ue GH, & per E horizonti parallela ducatur |
|
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2483 EK, moto igitur triangulo, centrum grauitatis E transla- |
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2484 tum erit in H, $ed KC æqualis e$t EF, minor autem ip$a |
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2485 CH, eleuatur ergo centrum grauitatis ab Ein H, nempe |
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2486 $upra K, totum $patium KH. ex qua eleuatione fit in mo- |
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|
2487 tu difficultas. Idem pror$us eadem demon$tratione o$ten- |
|
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|
2488 deretur fieri in quadrato & alijs lateratis figuris. Curigi- |
|
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2489 tur in plano horizontis facillimè circularia, difficile autē |
|
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2490 laterata & quæ inæquales habent $emidiametros, mo- |
|
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|
2491 ueantur, ex dictis clarè patet. |
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2492 <p>Ad hanc quæ$tionem illud quoque facit, cur per de- |
|
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|
2493 cliue planum grauiora corpora, & rotunda maximè; ma- |
|
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|
2494 gno impetu dimi$$a, delabantur. |
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|
2495 <p>E$to enim rota $phæraue aut Cylindrus CD, cuius |
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2496 centrum E, tangens decliue planum AB in D, quæritur |
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|
2497 <p n=>63</p> |
|
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|
2498 cur dimi$$a hæc magno impetu deferantur ad partes B, |
|
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|
2499 Ducatur per grauitatis centrum E ad horizontem, BK |
|
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|
2500 perpendicularis FEL $ecans decliue planum in G, cir- |
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|
2501 cum ferentiam verò in H. opponitur autem EG angulo |
|
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|
2502 recto EDG, maior ergo EG ip$a ED, hoc e$t, EH, inter |
|
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|
2503 <fig> |
|
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|
2504 circumferentiam igitur & pla- |
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2505 num decliue, $patium interce- |
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2506 dit HG. Ducatur item DI ip$i |
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2507 FG æquidi$tans. non tran$ibit |
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2508 igitur per centrum E. minor e- |
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2509 rit igitur diametro CD, quare |
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2510 circulum in partes inæquales |
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2511 $ecabit, & non per grauitatis |
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2512 centrum, quod idem cum ma- |
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2513 gnitudinis $eu figuræ centro $upponitur. Dimi$$a igitur |
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2514 rota, contingit quidem planum decliue in puncto D. At |
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2515 centrum grauitatis premit $e cun dam per lineam perpen- |
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2516 dicularem FG, non $u$tentatur autem in H, quippe quod |
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2517 inter planum & circum ferentiã intercedat $patium HG, |
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2518 nec H locum habeat cui innitatur, corpus autem ita per |
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2519 lineam DI e$t diui$um, vt longè maior $it pars IFCHD |
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2520 ip$a DI, & centrum in ea parte eadat quæ non fulcitur. i. |
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2521 taque $uopte nutu, cum extra ful cimentum $it D & per- |
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2522 pendicularem DI ad inferiores partes rapidè rotans de- |
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2523 labitur. Ducatur autem perpen dicularis GL, parallela |
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2524 MN, & quoniam BN breuior e$t BL, erit MN ip$a GL |
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2525 breuior. E$t igitur punctum M mundi centro propius |
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2526 quàm D & G, quare eò non impedita rota ip$a $uo nutu |
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2527 feretur, nec$tabit donec in fimum locū vbi quie$catnan. |
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2528 ci$catur. Po$$umus etiam Rota $phæraue in plano decliui |
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2529 collocata, datam potentiam inuenire, quæ extremitati |
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2530 diametri ad eam partem quavergit applicata ip$am rotam |
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2531 $phæramue impediatne delabatur. |
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2532 <p n=>64</p> |
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2533 <fig> |
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2534 <p>E$to planum in clinatum |
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2535 AB, cui Rota $phæraue in$i- |
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2536 $tat tangat<13>; illud in C. Rota |
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2537 verò ip$a $phæraue DC, cu- |
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2538 ius centrum E, diameter ve- |
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2539 rò DEC ip$i BA ad punctū |
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2540 contactus C, perpendicula- |
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2541 ris. Ducatur per C ip$i hori- |
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2542 zonti perpendiculatis FCG |
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2543 circulum $ecãs in G tum per |
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2544 E ip$i CG perpendicularis, ip$i verò BF horizonti æ qui- |
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2545 di$tans HEI ceu vectis, cuius fulcimentum I re$pondens |
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2546 ip$i C, pondus verò in E, vbi grauitatis e$t centrum. Ap- |
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2547 plicata igitur potentia in H erit pondus inter fulcimen- |
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2548 tum & potentiam, quare vt IE ad IH ita potentla $u$ti- |
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2549 nens in H ad pondus in E, quod demon$trandum fuerat. |
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2550 <p>Quippiam $imile o$ten dit Pappus 1. 8. prop. 9. alijs |
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2551 tamen $uppo$itis & con$ideratis. Dico præterea, ij$dem |
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2552 $tantibus angulum ECI æqualem e$$e angulo inclinatio- |
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2553 nis CBF. Producatur HI concurrens cum ip$a AB in K, |
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2554 concurret autem propterea, quod CIK rectus $it, ICA |
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2555 minorrecto, & quoniam HK parallela e$t horizonti BF |
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2556 alterni anguli IKC, CBF, æquales erunt. Similes autem |
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2557 $unt ECI, ECK, trianguli, e$t<13>ue ECI angulus æqualis |
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2558 angulo EKC, hoc e$t, ip$i CBF. vnde $equitur, quo mi- |
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2559 nor fuerit inclinationis angulus, eo facilius rotam $phæ- |
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2560 ramue in piano inclinato $u$tineri. quo enim minor fuerit |
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2561 angulus ECI, eo minus latus EI & minor proportio EI |
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2562 ad IH, & ideo minor potentia $u$tinens requiratur in H. |
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2563 Cæterùm accliue & decliue planum nihil differunt ni$i |
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2564 re$pectu. |
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2565 <p>His ita con$ideratis, admonetnos locus, vt pulcher- |
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2566 rimam dubitationem diluamus. Quæritur, Cur maiores |
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2567 <p n=>65</p> |
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2568 rot<17> impingentes, facilius offendicula $uperent quàm mi- |
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2569 nores. Neque enim $atisfacere videtur quod ait Ari$tote- |
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2570 les, ex contactu in puncto eo anguli à plano eleuationeid |
|
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2571 fieri, alijs ergo principijs dubitatio $oluitur. |
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|
2572 <fig> |
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2573 <p>E$to rota quidem maior |
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|
2574 AB, circa centrum C minor |
|
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2575 vero DB circa centrum, E, |
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2576 tãgentes horizontis planum |
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|
2577 in B. Diameter maioris AB, |
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2578 minoris DB, offen diculum, |
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|
2579 horizonti perpendiculare |
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|
2580 FG. Ducatur per F horizonti |
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|
2581 parallela FK $ecans minoris |
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2582 rotæ peripheriam in H, dia- |
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2583 metrum verò AB in K, & à |
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|
2584 puncto H ad planū horizon- |
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|
2585 tis perpendicularis demittatur HI: erit autem HI æqua- |
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2586 lis ip$i offendiculo FG, & iungantur BH, BF. Itaq; quo- |
|
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|
2587 niam BH ab extremo B cadit in triangulum KFB, erit |
|
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|
2588 KHB angulus maior angulo KFB. Parallelæ autem $unt |
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2589 <I>K</I>F, BG, pares ergo anguli <I>K</I>HB, HBG, pares item <I>K</I>FB, |
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2590 FBG, Maior ergo HBI, ip$o FBC. At minoris rotæ gra- |
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2591 uitatis centrum mouetur $ecundum lineam BH, maius |
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2592 verò $ecun dum literam BF, difficilius ergo mouebitur, & |
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2593 $uperabit offen diculum minorrota, quàm maior: quod |
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2594 fuerat demon$trandum. |
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2595 <p>Po$$umus idem o$tendere magis mechanicè, hoc |
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2596 e$t, tem ad vectem reducendo. E$to horizontis planum |
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2597 AB, rota maior CD planum tangens in D. rotæ verò ma- |
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2598 ioris centrum E. Rota verò minor FD, tangens itidem |
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2599 planum in D. rotæ autem centrum G, offendiculi verò re- |
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2600 ctitudo DH. Ducatur per Hip$i AB horizonti æquidi- |
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2601 $tans HI $ecans minorem circulum in K, maiorem verò |
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2602 <foot>I</foot> |
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2603 <p n=>66</p> |
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2604 <fig> |
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2605 in I. Ducantur etiam dia- |
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2606 metri maioris quidem |
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2607 LEM, minoris NGO, |
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2608 Tum à puncto K perpen- |
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2609 dicularis ducatur ad |
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2610 GO, ip$a KP, item à pun- |
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2611 cto I ad EM perpendi- |
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2612 cularis IQ. Dico EQ ad |
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2613 QL, minorem habere |
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2614 proportionem quam GP, |
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2615 ad PN. Connectatur |
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2616 GK, & ei per E parallela |
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2617 ducatur ER, $ecans maiorem circulum in R, & ab Rip$i |
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2618 EM perpen dicularis ducatur RS. quoniam igitur ER |
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2619 parallela e$t ip$i GK, erit GER angulus HGK angulo |
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2620 æqualis. Recti autem $unt HGP, GES reliqui ergo KGP, |
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2621 RES ad inuicem $unt æquales. Sed & ESR, GPK recti |
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2622 $unt, quare ERSGKP anguli æquales $unt, & trianguli |
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2623 GPKESR, per pr. diff. 1.6. $imiles. Vtergo GK hoc e$t |
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2624 GN ad GP, ita ER hoc e$t EL ad ES. Componendo igi- |
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2625 tur vt NP ad PG, ita LS ad SE. quamobrem $i fulcimen- |
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2626 tum e$$etin S, pondus in E, potētia in L, idem $ieret ac fiat |
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2627 fulcimento in P, pondere in G, potentia verò in N con$ti- |
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2628 tuta. & id quidem $i eiu$dem ponderis vtraque rota $up- |
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2629 ponatur. Rur$us quoniam vt DK ad totum circulum DF, |
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2630 ita DR ad totum DC. Minor e$t autem proportio DI ad |
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2631 totum circulum DC, ergo minor e$t DI ip$a DR. Maior |
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2632 ergo MI ip$a MR, maior ergo QI ip$a SR, propius ergo |
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2633 centro E e$t Q ip$o puncto S, minor e$t igitur proportio |
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2634 EG ad LQ quàm ES ad SL. Minor ergo potentia requi- |
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2635 ritur in L ad $u$tinendum pondus E ex fulcimento Q hoc |
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2636 e$t I, quàm requiratur in N ad $u$tinendum pondus G ex |
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2637 fulcimento P, hoc e$t K. Minor ergo potentia requiritur |
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|
2638 <p n=>67</p> |
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2639 ad transferendam maiorem retam CD vltra offendicu- |
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2640 lum IV, hoc e$t, DH, quàm requiratur ad trans ferendam |
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2641 minorem vltra offendiculum KT, hoc e$t HD, quod fue- |
|
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|
2642 rat o$ten dendum. |
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2643 <p>Ad hæc, quæri pote$t, quo pacto plau$trorum rotæ |
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2644 in ip$a plau$tri conuer$ione $e habeant, nempe quæ $it li- |
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|
2645 neailla curua, quam in conuer$ione de$cribunt. |
|
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|
2646 <fig> |
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|
2647 <p>E$to rotarum in |
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2648 plano orbita, dū plau- |
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2649 $trum rectâ procedit |
|
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2650 AB, CD, Sunt autemi- |
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2651 p$æ lineæ, quod o$ten- |
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2652 demus po$tea, æquedi- |
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2653 $tantes. Sit itaque pun- |
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2654 ctum. B illud in quod |
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2655 rota quæ per AB $er- |
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|
2656 tur, eò delata planum |
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2657 tangit. D verò alterius rotæ at que plani contactus. Igitur |
|
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2658 dum plau$tri fit conuer$io, punctum D conuer$ionis fit |
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2659 centrum. Stat enim interim rota & circa lineam conuer- |
|
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2660 titur, quæ å puncto contactus D per rotæ centrum ducta |
|
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2661 horizontis plano e$t perpendicularis. ea autem $tante, ro- |
|
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2662 ta quæ in B circa centrum D $emicirculū pertran$it DEF, |
|
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2663 vbi autem rota B, peruenerit in F, plau$tro iam in oppo$i- |
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2664 tam partem conuer$o, rota quæ e$t in D per lineam DC, |
|
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2665 quæ verò in F per rectam FG mouetur, plau$tri<13>ue fit re- |
|
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|
2666 gre$$us. Et quoniam vel D in ip$a conuer$ione $tat omnino |
|
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2667 nec quicquam progreditur, vt in prima figura, vel non $tat |
|
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2668 vt in $ecunda, quo ca$u portionem parui circuli de$cribit, |
|
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2669 ip$i maiori circulo & exteriori concentricam. Vnde col- |
|
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|
2670 ligimus, Plau$trorum conuer$iones flexione<*><13>ue $emper |
|
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|
2671 circa centrum, & con centricorum circulorum portiones |
|
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2672 fieri, <I>H</I>inc etiam di$cimus, cur veteres, vt ex antiquis co- |
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2673 <foot>I 2</foot> |
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2674 <p n=>68</p> |
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2675 gno$cimus ve$tigijs, circos in quibus cur$us quadrigarum |
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2676 fiebant ea forma quæ apparet, efformauerint. Hoc etiam |
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2677 theorema probamus. |
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2678 <p>Cylindros, quorum ba$es axi $unt perpendiculares, |
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2679 dum in æquato plano conuoluuntur, rectâ incedere & |
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2680 per parallelas, quarum di$tantia axis $eu latoris longitudi- |
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2681 ne præfinitur. |
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2682 <fig> |
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2683 <p>E$to enim Cylin- |
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2684 drus ABCD, cuius a- |
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2685 xis GH, horizōtis pla- |
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2686 no in$i$tens $ecundum |
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2687 latus AB, cui latus op- |
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2688 po$itum & <17>quale CD. |
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2689 Moueatur Cylindrus |
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2690 rotans, donec latus |
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2691 CD, in plano $it vbi EF. De$cribat autem circuli CB lineã |
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2692 BF. Circulo verò AD lineam AE. Dico eas rectas e$$e, & |
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2693 parallelas. Si enim $uperficies ba$ium DA, CB, extendan- |
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2694 tur ita vt horizontis planum $ecent, illud $eca bunt iuxta |
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2695 lineas AE BF, recta ergo e$t vtraque. Sed & parallelas e$$e |
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2696 ad inuicem ita o$tendimus. quoniam $emicirculus AD, |
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2697 æqualis e$t $emicirculo BC, erit linea AE, æqualis lineæ |
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2698 BF, $ed & AB, æqualis e$t ip$i DC, quare & ip$i EF. Oppo- |
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2699 $ita igitur quadrilateri figura ABFE latera æqualia $unt, |
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2700 quare EF æquedi$tat ip$i AB, tum AE ip$i BF, quod fue- |
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2701 rat demon$trandum. |
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2702 <p>Probabimus etiam $i cylindri ba$es axi perpendicu- |
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2703 lares non fuerint, & ideo ellip$es in ip$a rotatione perpla- |
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2704 num, parallelas quidem de$cribere, $ed non rectas. |
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2705 <p>E$to enim Cylindrus ABCD, cuius ba$es ellip$es inuicē |
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2706 æquedi$tãtes, quarum axes longiores AB, CD, Commu- |
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2707 nis autem $ectio cylindri & plani ad axem & horizontem |
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2708 planum perpendicularis EHF. Diuidatur autem $emicir- |
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2709 <p n=>69</p> |
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2710 culus EHF in partes æquales quatuor FI, IH, HG, GE. |
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2711 <fig> |
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2712 Tum per diui$ionum puncta lateri parallel<17>, rectæ ducan- |
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2713 tur KGL, M<I>H</I>N, OIP, quæ quidem cū ba$es AMB, DNC |
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2714 parallelæ $int, eruntinuicem æ quales, cum<13>ue circum- |
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2715 ferentia E<I>H</I>F æquales, eos<13>ue rectos angulos cō$tituent. |
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2716 Ducatur po$t hæc $eor$um recta QR, & eidem perpendi- |
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2717 cularis ST eam $ecans in V. applicetur autem rectæ ST |
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2718 æqualis Cylindri lateri BC, ip$a <G>hz</G>. ita tamen vt punctum |
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2719 E congruat puncto V, $it<13>ue V<G>h</G> æqualis EB, V<G>z</G>verò æ- |
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2720 qualis EC. Tum fiant VX, XY, YZ, Z<G>a</G> æ quales ip$is EG, |
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2721 G<I>H</I>, <I>H</I>I, IF, & per puncta X, Y, Z, <G>a</G> & paralleli ip$i ST du- |
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2722 cantur <G>o a p, n *z c, l g m, k x q</G>, tum & his ex altera parte re- |
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2723 $pondentes parallelæ per puncta <G>b, g, d, e</G>. Sit autem <G>o a</G> æ- |
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2724 qualis AF, <G>a</G> <11> æqualis FD, item <G>e</G> <10>, æqualis EC, <G>e s</G> æqualis |
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2725 EB, $ed & <G>n *z</G> <17>qualis OI, <G>*z c</G> ip$i P, <G>l</G>yi<*> $i MH, y <G>m</G> verò ip$i |
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2726 HN, tū <G>k x</G> ip$i KG. & <G>x q</G>, ip$i GL & ip$is æquales & <17>qua- |
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2727 liter po$itæ ad partes R, aliæ paralle læ aptētur per <G>b, g, d, c</G>, |
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2728 <foot>I 3</foot> |
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2729 <p n=>70</p> |
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2730 quibusita di$po$itis per puncta <G>o, n, l, k, h</G>, item per <G>p, c, m, q, z</G>. |
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2731 ducantur lineæ <G>oh, pz</G>, curuæ quidern & codem pacto a- |
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2732 liæ curuæ illis re$pondentes <G>h <10>, zs</G>, Erunt igitur <G>o, h, <10>, |
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2733 p, z, s</G>, parallelæ quidem eo quod linc<17> quæ inter ip$as du- |
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2734 cuntur, parallelæ $int & æquales, non tamen rectæ illæ, |
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2735 $ed curuæ. Moto igitur Cylindro circulus EHF rectam |
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2736 de$cribet<G>ae</G>, ellip$is verò AMB, curuam <G>ohr</G>, ellip$is au- |
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2737 rem DNC, ip$am curuam <G>pzs</G>. In hoc autē Cylindri mo- |
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2738 tuillud mirabile, velociores nempe, in ip$a rotatione e$$e |
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2739 ellip$es ip$o circulo EHF. Ducatur enim recta<G>o<10></G> quæ oc- |
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2740 currat ip$i VS in S, & <G>oh</G> iungatur, fiet<13>ue triangulum |
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2741 <G>oh</G>S. c$t autem, angulus <G>o</G> S <G>h</G> rectus, maior erg. <G>oh</G> i- |
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2742 p$a <G>o</G> S, $ed recta <G>o</G> S æqualis e$t ip$i<G>an</G>, hoce$t, $emicircu- |
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2743 lo FHE. multo maior e$t autem curua, <G>o, n, l, k, h</G>, ip$a recta |
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2744 <G>oh</G>, $ed eodem tempore quo $emicirculus EHF conficit |
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2745 in rotatione $patiū <G>a</G> V, eodem dimidia ellip$is BMA me- |
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2746 titur curuam <G>onlkh</G>. velocior igitur e$t ellip$is ip$o cir- |
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2747 culo. |
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2748 <p>Hæc quo que $peculatio ad motum qui $ecundum |
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2749 ab$idem fit, manife$tè pertinet. Coni, quorum ba$es cir- |
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2750 culi $unt, $i in plano $ecundum latus rotentur, ba$i circu- |
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2751 lum de$cribunt, cuius centrum immobile coni ip$ius e$t |
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2752 vertex, $emidiameter verò ip$um latus. |
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2753 <fig> |
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2754 <p>E$to conus ABC cu- |
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2755 ius vertex C ba$is AB, axis |
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2756 DC, ba$is verò centrum, |
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2757 D, latus quo planum tan- |
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2758 git BC, $ecatur itaque Co- |
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2759 nus per latus BC & axem |
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2760 DE à plano horizonti per- |
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2761 pendiculari, cuius & coni |
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2762 communis $ectio e$t ABC |
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2763 triangulum, & quoniam coni grauitatis centrum e$t in |
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2764 <p n=>71</p> |
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2765 axe ip$o, conus in partes æque pōderantes $ecatur AEBC, |
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2766 AFBC, $tat ergo conus $ibimet æquili bris. Si autem à po- |
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2767 tentia quadam moueatur, puta ab A ver$us F, trahitur $e- |
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2768 micirculus BEA, à $emicirculo AFB, & ita fit rotatio. Ita- |
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2769 que $i imaginemur, in finitos v$que ad verticem parallelos |
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2770 ba$i cir culos, eorum $emicirculi in ip$o motu & trahent & |
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2771 trahentur; at cum ad verticem circuli de$inant, nec ibi $e- |
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2772 micirculi $int qui trahant & trahantur, motus rotationis |
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2773 pror$us ce$lat & vertex ip$e immobilis fit rotationis cen- |
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2774 trum. Quoniam igitur lateris BC, punctum C $tat, B verò |
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2775 circa ip$um mouetur, in ip$o motu circulus de$cribitur |
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2776 BHIK, cuius $emidiameter BC, & eodem pacto alij cir- |
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2777 culi in cono, qui ba$i HEBF $unt æquedi$tantes, circulos |
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2778 in plano circa idem centrum de$cribent, vt facile videre |
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2779 e$t in obiecto $chemate. Huic $imilem demon $trationem |
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2780 affert Heron in libello Automatum, quem nos Tyrones |
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2781 adhuc vernacule è Græco translatum, Ven<*>e tijs prælo |
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2782 $ubiecimus. |
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2783 <p>Porrò $i conus rotundus pro ba$i ellip$im habeat, |
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2784 $ectionem videlicet per planum axi non perpendiculare, |
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2785 in ip$a rotatione, $tante vertice, ellip$is ba$is, ellip$im de- |
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2786 $cribit in plano, cuius maior diameter à puncto quod co- |
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2787 nivertex e$t, ita diuiditur, vt diametri pars maior æqualis |
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2788 $it lateri maximo; minor verò æqualis lateri minimo. Sed |
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2789 hæc ad aliam pertinent $peculationem. |
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2790 <p>His ita que de moturotundorum, qui circa ab$idem |
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2791 fit, con$ideratis, reliquum e$$et de motu trochlearum, qui |
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2792 circa centrum $it, opportunè agere, $ed cùm in $equenti |
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2793 quæ$tione de hoc $ermonem faciat Philo$ophus, ad ea |
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2794 quæ ibi di$puta buntur, lectorem ablegamus. |
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2795 <p>Modò de tertia motus $pecie nobis erit $ermo; in |
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2796 qua quidem $pecienonnulla perpendemus, quæ omi$it A- |
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2797 ri$toteles. Agitur autem hîc de rotundorum corporum |
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2798 <p n=>72</p> |
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2799 motu, qui fit çirca axem horizonti perpendicularem, axis |
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2800 altera extremitate in codem horizontis plano manente, |
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2801 vti videre e$t in ip$is figulorum rotis. |
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2802 <p>Hanc motus $peciem in extrema quæ$tionis parte |
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2803 cum duabus alijs $peciebus comparans ait, cam quæ in |
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2804 obliquo fit motionem (ita enim hanc, de qua agimus, ap- |
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2805 pellat) ip$am impellere mouentem, hoc e$t, nullum ex$e |
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2806 ad motum propen$ionem habere, nutumue, & omnia illi |
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2807 e$$e à motore, $ecundum verò eam motionem, quæ $upra |
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2808 diametrum e$t, $e ip$um mouere circulum. Dixerat enim, |
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2809 ea referens quæ $uperiùs circa principium de circulo ver- |
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2810 ba faciens, examinauerat, circulum ex duabus fieri latio- |
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2811 nibus, altera præter, altera verò $ecundum naturam, & |
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2812 ideo hanc $emper nutum habere, & ceu continuo motam |
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2813 ab eo moueri quimouet. Videtur autem clarè profiteri, |
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2814 ideo difficiliorem e$$e huius terriæ $peciei motum, eo |
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2815 quòd nutu <*>areat proprio & t<*>m ab alieno, vt ita di- |
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2816 cam, motore, moueatur. |
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2817 <p>Veruntamen motum hunc facilitate alijs illis duo- |
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2818 bus nequaquam cedere, facilè ex $equentibus o$tende- |
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2819 mus. |
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2820 <p>Primo, quia pondus totum rotati corporis, ex graui- |
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2821 tatis centro quodin ip$o axe e$t à plano cuinititur, $u$ti- |
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2822 netur: minima quidem $ui parte axe ip$o tangente planū |
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2823 vndefit, nullam ferè dum rotatur corpus, circa centrum |
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2824 vbi nititur, frictionem partium fieri. Præterea grauitatis |
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2825 centrum $emper $tat, nec minimum quidem in ip$a rota- |
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2826 tione attollitur, quod $anè cum naturæ $it repugnans, dif- |
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2827 ficultatem facit. Ad hæc circa axem ita libratur rota, vt |
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2828 quantumuis exigua potentia alteri parti applicetur, alte- |
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2829 raillico $uperata moueatur. Licet enim propliè ea tantū |
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2830 corpora æquilibrare dicantur, quæ ob ponderis hinc in de |
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2831 <p n=>73</p> |
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2832 æqualitatem horizonti fiunt æquidi$tantes, nihilominus |
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2833 & hic aliquam e$$e æquilibrij $imilitudinem patebit. |
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|
2834 <fig> |
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2835 <p>E$to enim rota ABCD, |
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2836 cuius axis horizonti perpendi- |
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2837 cularis FEG tran$iens per cen- |
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2838 trum E, tangens autem planum |
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2839 in puncto G. Ducatur diame- |
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2840 ter BED, Itaque $i per diame- |
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2841 trum BED, & axem FEG cor- |
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2842 pus diuidatur, eo quòd centrū |
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2843 grauitatis in axe inueniatur, |
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2844 corpus ip$um in duas partes tū |
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2845 mole tum pōdere æquales $ecabitur, nempe BAD, BCD. |
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2846 Nullaigitur adhibita vi extranea $tabit corpus in quodã, |
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2847 vt diximus, æquilibrio. At alteri partium potentiâ quauis |
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2848 licet exigua appo$itâ, puta in C, præualebit pars BCD, & |
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2849 partem BAD vel impellet vel rapiet, alterâ interim eius |
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2850 motui ob$equente. Potentia igitur quæ in C, nullam rem |
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2851 quæ impediat inueniens, velo ci$$imè rotam mouet, quod |
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2852 eo faciliùs velocius <13>ue fit, quo magis rota e$t in motu, e- |
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2853 ius verò diameter maior & potentia mouens à centro re- |
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2854 motior, & $anè motus facilitatē inde cogno$cimus, quòd |
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2855 ip$o impul$ore ab impul$u ce$$ante, diuti$<*>è rota im- |
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2856 pre$$um motum $eruet, nec ni$i po$t longam rotationem |
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|
2857 omnino quie$cat. |
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2858 <p>Cæterùm quia $icco, vtaiunt, pede Ari$toteles quæ |
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2859 ad hunc motum pertinēt pertran$ijt, nos quædam quæ ad |
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2860 hancrem faciunt, diligentiùs expendemus. |
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2861 <p>Quærimus igitur primò; Cur ea quæ hoc pacto ro- |
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2862 tãtur, in ip$a rotatione locum non mutent, ni$i extrin$eca |
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2863 aliqua id fiat ex cau$$a. |
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2864 <p>E$to enim rota aut aliud quippiam rotundum ccu |
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2865 Turbines $unt, quibus pueri ludunt, quod circa axem ho- |
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|
2866 <foot>K</foot> |
|
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|
2867 <p n=>74</p> |
|
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|
2868 <fig> |
|
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2869 rizonti perpendicularem mo- |
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|
2870 ueatur, ABCD, cuius centrum |
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|
2871 E, Diameter AEC. Modò circa |
|
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|
2872 centrum E in finiti imagin entur |
|
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|
2873 circuli, alij alijs minores v$que |
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2874 ad centrū ip$um, vti $unt FGH; |
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2875 ibi enim circuli e$$e de$inunt, |
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2876 vbi nullum amplius e$t $patium. |
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|
2877 Applicetur itaque potentia in |
|
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2878 B, quæ rotam v. geat ver$us A. |
|
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|
2879 codem igitur tempore & in$imul A ver$us D, D ver$us C, |
|
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|
2880 & Cver$us B mouebitur. quantum enim $emicirculorum |
|
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|
2881 à parte CBA tran$it vltra diametrum AEC, tantundem |
|
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|
2882 $emicir culorum, qui $unt ad partem ADC, tran$ibit ad |
|
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|
2883 partes CBA. At vbi de$ierit motus, ibi de$init rotatio; vbi |
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2884 autem de$init $patium, de$init motus, $ed vbi de$inunt cir- |
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|
2885 culi, de$init $patium, quare in centro cum non $int circuli, |
|
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2886 nec $patium ibi de$init motus. nulla enim ade$t ratio, cur |
|
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|
2887 ip$um corpus alio à loco in quo e$t, ex rotatione transfe- |
|
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|
2888 ratur. Statergo rotans, quod fuerat demon$trandum. E$t |
|
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|
2889 autem hæc demon$tratio ei $imilis, quam $uprà retuli- |
|
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|
2890 mus de coni in plano circa verticem rotatione, quam ab |
|
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|
2891 Herone in Automatis excogitatam diximus. |
|
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2892 <p>Addimus in hoc rotationis genere corpus in ip$o- |
|
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|
2893 motu fieri leuius, id<13>ue eo magis, quo rotatio velocior. |
|
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2894 Cau$$a e$t, quod lateralis motus eum motum aliqualiter |
|
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|
2895 impedit, qui ex naturali grauitate fit ad centrum, idcirco |
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|
2896 experientiâ docemur, leui$$imos e$$e turbines, quibus pu- |
|
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2897 eri ludunt, $i manus teneantur palmâ, dum citi$$ima rota- |
|
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|
2898 tione mouentur. |
|
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2899 <p>Ad hæc alia proponitur, & $oluitur quæ$tio, Cur ro- |
|
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|
2900 tunda corpora huic motionis generi $int aptiora. |
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|
2901 <p>Explorati$$imum e$t, corporum, quæ ita mouentur, |
|
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|
2902 <p n=>75</p> |
|
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2903 partes eo e$$e velo ciores, quo magis à centro, circa quod |
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2904 mouentur, fuerint remotiores. maius enim eodem tem- |
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2905 pore $patium pertran$eunt. quo igitur figura ijs partibus, |
|
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2906 quæ longius à centro ab$unt, abundauerit magis, eo faci- |
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2907 lius, & velocius in circulum rotata mouebitur. Modò o- |
|
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2908 $tendemus, circularem cæteras omnes ea qua diximus |
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|
2909 partium à centro remoti$$imarum copiâ abundare. |
|
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|
2910 <fig> |
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2911 <p>E$to triangulum puta æqui- |
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2912 iaterum, ABC circa centrum D. |
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|
2913 Ducantur Catheti per centrum ab |
|
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|
2914 oppo$itis angulis ad oppo$ita late- |
|
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|
2915 ra ADG, BDF, CDE, erunt autem |
|
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2916 lateribus perpendiculares. quoniã |
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2917 igitur latera AD, DB, DC, rectis |
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2918 angulis $ubtenduntur, maiora erūt |
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2919 lateribus DE, DF, DG. tres igitur |
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2920 lineæ in hoc triangulo $unt longi$$imæ DA; DB, DC. tres |
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2921 verò breui$$imæ DE, DG, DF, quamobrem rotato $uper |
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2922 centrum D triangulo, tres tantum partes eius ABC velo- |
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2923 ci$$imæ erunt, tres verò tardi$$imæ E, G, F. Minus igitur a- |
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2924 pta e$t motui huic triangularis figura, quam quadrata, in |
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2925 qua partes à centroremoti$$imè, & ideo veloci$$imè $unt |
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2926 quatuor. Itaq; quo <*>agis laterata figura angulis abunda- |
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2927 bit, eo magis erit ad hunc, & cæteros omnes circulares |
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2928 motus aptior. At circulus infinitas, vt ita dicam, partes à |
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2929 centro remoti$$imas habet, itaque nulla figura e$t circu- |
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2930 lari, in ip$a rotatione, commodior atque velocior. Alia |
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2931 quoque de cau$$a id fit, quod dum circularis figura mo- |
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2932 uetur, nullis eminentibus angulis aërem verberet circū- |
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2933 $tãtem, ex qua verberatione motus impeditus $it tardior. |
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2934 Quæri etiam pote$t, Num axe in clinato, rotæ motus ali- |
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2935 qualiter impediatur? Nos negatiuam partem amplecti- |
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2936 mur. |
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2937 <foot>K 2</foot> |
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2938 <p n=>76</p> |
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2939 <fig> |
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2940 <p>E$to enim tota ABCD, cuius cen- |
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2941 trum E axis inclin itus, circa quem |
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2942 conuertitur EGF. Duobus aute pun- |
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2943 ctis fulcitur GF. Sit autem tum gra- |
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2944 uius tum figuræ centrum E, Perpen- |
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2945 dicularis vero per inferius fulcimen- |
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2946 tum tran$iens HFI. Conuer$a igitur |
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2947 rota, grauitatis centrum $tabit nec à |
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2948 $uo $itu $ur$um deor$umue mouebi- |
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2949 tur. E$t autem axis FEG, ceu vectis in |
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2950 quo pondus in E, potentiæ $u$tinentes GF; non enim hic |
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2951 vt in axe perpendiculari pondus totum ab inferiori fulci- |
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2952 mento $u$tinetur. quo igitur minor erit proportio FE ad |
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2953 FG, eo minori in digebit potentiâ is qui pondus $u$tinet in |
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2954 G. Et hæc $anè ita $e habent, grauitatis çentro in axe ip$o |
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2955 con$tituto, $i enim extra fuerit motus impeditur & moto- |
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2956 re ce$$ante citò quie$cit. E$to enim grauitatis centrum in |
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2957 K. Dum igitur circa axem fit motus, centrum circulatum |
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2958 aliquando erit in L; Secetautem rotæ diameter AC per- |
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2959 pendicularem Hl in M. Porrò à punctis LK ad ip$am per- |
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2960 pēdicularem ducantur ad rectos angulos lineæ LN, KO. |
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2961 Maior e$t autem MK ip$a ML, maior ergo MO, ip$a MN. |
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2962 magisigitur à mundi centro di$tat punctum N puncto O. |
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2963 Centrum ergo grauitatis K $i liberè dimittatur, requie$cet |
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2964 in K & contranaturam transferetur in L. Ce$$ante igitur |
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2965 violentiâ & præualentenaturâ citò rota $uâ $ponte quie- |
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2966 $cet, quod fuerat o$tendendum. |
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|
2967 <HEAD>QVÆSTIO IX.</HEAD> |
|
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2968 <HEAD><I>Quæritur, Cur ea quæ per maiores cir culos tolluntur, & trahuntur |
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2969 faciliùs, & celeriùs moueri contingat, veluti maioribus tro- |
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2970 chleis, & $cytalis $imiliter?</I></HEAD> |
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2971 <p>Re$pondet ad hæc Philo$ophus, forteid cuenire, quo- |
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|
2972 <p n=>77</p> |
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|
2973 niam quanto maior fucrit illa quæ à centro e$t, in æquali |
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|
2974 tempore maius mouetur $patium. quamobrem æquali |
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2975 exi$tente onere idem faciet. Ita enim dixerat de librarū |
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2976 natura, & differentijs agens, maiores minoribus exactio- |
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2977 res e$$e. Circulos verò jibras, in quibus centrum $partum, |
|
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2978 $emidiametri hinc in de æqualia brachia. |
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2979 <p>Quod vltimo loco affirmauit, trochleas e$$e in$tar |
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2980 librarum, verum e$t. Quo d autem dixit, faciliùs & cele- |
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2981 rius mouere maiores libras ijs quæ minores $unt, $i $inipli- |
|
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2982 citer intelligatur, fal$um, quippe quod facilitas motus, in |
|
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2983 tractorijs machinis velo citati $it contraria, quod demon- |
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2984 $trauit Guid. Vbald. in tractatu de Trochlea in 2. Corol- |
|
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|
2985 lario propo$itione vltima. |
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2986 <p>Ad id autem quod dixit, quo maiorēs fuerint tro- |
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2987 chleæ, eo faciliùs mouere, non e$t, vt dicebamus, $impli- |
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2988 citer verum, quod facilè o$tendemus. |
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|
2989 <fig> |
|
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2990 <p>Efto enim trochlea AB circa centrum C, appen$a in |
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|
2991 puncto D, perpendicularis quæ ad mundi centrum DCE, |
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2992 pondera æqualia vtrinque appen$a FG. E$to item alia |
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2993 Trochlea, ea<13>; maior HI, circa centrum K appen$a in L, |
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2994 perpen dicularis, quæ ad mundi centrum LKM, æqualia |
|
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|
2995 <foot>K 3</foot> |
|
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|
2996 <p n=>78</p> |
|
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|
2997 pondera vtrinque appen$a N, O. Dico maiorem Hl ip$a |
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2998 minori DE facilius pondera non mouere, eo quòd $it ma- |
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2999 ior, illa verò difficiliùs, propterea quòd $it minor. Etcnim, |
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3000 quoniam vtraque trochlea per centrum graultatis à per- |
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3001 pendiculari diui ditur, erunt partes DAE, DBE, æque pō- |
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3002 derantes. Eadem ratione ip$æ quoque LHM, L<*>M æquè |
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3003 ponderabunt. Itaque $i quantumuis pu$illa pondera ad- |
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3004 das, vtriq; earum ad alteram partem tolletur æquilibriū, |
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3005 nec minus requiritur pondus vt recedat ab æquilibrio |
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3006 Trochlea minor, quàm maior. Vnico autem verbo con- |
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3007 cludi pote$t di$putatio, tã in minori quàm in maiori, bra- |
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3008 chia $iqui dem bifariam diuiduntur, ergo in vtri$q; eadem |
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3009 brachiorum proportio, & eadem ponderum ratio. |
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3010 <p>Explorati$$ima $unt hæc. Veruntamen cùm res ip$a |
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3011 doceat, verum e$$e quod $cribit Ari$toteles, huius effe- |
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3012 ctus cau$$a aliunde à nobis, nempe à mechanicis princi- |
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3013 pijs, e$t mutuanda. Dico igitur, Axium, circa quos tro- |
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3014 chleæ rotæue conuertuntur ad rotas ip$as, varias habere |
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3015 proportiones. O$tendemus autem rotã illam, trochleam- |
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3016 ue faciliùs moueri, & mouere pondera, quo rotæ diame- |
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3017 ter ad axis diametrum maiorem habuerit proportionem, |
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3018 & ideo fieri po$$e rotam maiorem ad $uum axem minorē |
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3019 habere proportionem quam rotam minorem ad $uum. |
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3020 <fig> |
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3021 <p>E$to enim |
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3022 trochlea AB cir- |
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3023 ca centrum C, |
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3024 cuius diameter |
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3025 DCE $it in ip$a |
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3026 quæ ad mundi |
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3027 centrum perpē- |
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3028 diculari: $it au- |
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3029 tem appen$a in D. Alia $imiliter ei æqualis $it trochlea F |
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3030 G circa centrum H, cuius diameter IHK, conueniens |
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3031 <p n=>79</p> |
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3032 cum perpen diculari quæ ad mundi centrum. appendatur |
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3033 autem in I. Habeant autem & axes, circa quos conuertan- |
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3034 tur. Hi$i æquales fuerint, proportione non mutatâ idem |
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3035 operabuntur. Modò ponanturinæquales, $it<13>ue axis ro- |
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3036 tç AB, cra$$ior axe rotæ FG, $it<13>ue cra$$ioris quidem $emi- |
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3037 diameter CL, $ubtilioris autem HM. Dico per tro chleam |
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3038 FG facilius attolli pondera æqualia quàm per AB, licet |
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3039 altera tro chlearum alteri $it æqualis. Quoniam enim me- |
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3040 chanica corpora $ine materia & pondere non $unt, onera |
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3041 appē$a & trochlearum ip$arum grauitas ex $uperiori par- |
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3042 te prement axes, vbi puncta L, M, quæres, $ecutâ in uicem |
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3043 corporum $olidorum fricatione, motum ip$um trochlea- |
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3044 rum difficiliorem & a$periorem facit. Succedit igitur im- |
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3045 pedimentum loco ponderis. Duos igitur habemus vectes |
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3046 DC, IH, quorum fulcimenta contra ip$a C, H. Pondera |
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3047 verò inter fulcimenta & potentiasin L, M. Intelligantur |
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3048 autem potentiæ applicatæ punctis DI. Igitur ex natura e- |
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3049 iu$modi vectis, in quo pondus inter fulcimentum e$t & |
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3050 potentiam erit vt CL, ad CD, ita potentia in D ad pōdus, |
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3051 hoc e$t, re$i$tentiam fricationis, quæ fit in L. Sed maior |
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3052 e$t proportio CL ad CD quàm HM ad HI. Maior igitur |
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3053 ad $uperandum idem $eu æquale impedimentum poten- |
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3054 tia requiritur in D, qu<*>m in I. Itaque cum vis tota in rota- |
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3055 rum & axium, diametrorum proportione con$i$tat, fieri |
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3056 pote$t, quod dicebamus, minorem trochleam dari, quæ |
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3057 maiorem habeat proportionem ad $uum axem, quàm, |
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3058 maior ad $uum, quo ca$u minor rota facilius imp edimen- |
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3059 tum, quod diximus, ip$a maiori rota $eu trochlea $upera- |
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3060 bit. Veruntamen quoniam ex materia fiunt tum axes tum |
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3061 rotæ, nec rei natura patitur axes $ubtiles, & imbecilles |
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3062 magna pōdera $u$tinere po$$e, idcirco cra$$iores fiunt, quç |
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3063 cra$$itudo cum proportione magis à magnarum rotarum |
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3064 diametris $uperetur; fit hinc maiores rotas datâ axium pa- |
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3065 <p n=>80</p> |
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3066 ritate facilius impe dimentum $uperare quàm minores, & |
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3067 hoc videtur $en$i$$e Philo$ophus in ip$a quæ$tionis huius |
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3068 propo$itione, Hinc aurigæ vulgo axungiâ (quæ inde no- |
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3069 men trahit) axium a$peritates mitigant, vt minor in rotan. |
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3070 do, ex fricatione fiat re$i$tentia. Concludimus igitur, fa- |
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3071 cillimè trochleam illam pondus trahere, quæ cum maxi- |
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3072 ma $it, axem habet minimum, cum<13>ue axungiâ aliaue vn- |
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3073 ctuo $a materia perfu$um. De manubrijs, quæ rotarum a- |
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3074 xibus aptantur, nemo ferè verba fecit; nos igitur de his a- |
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3075 liquid; $iquidem res ad $peculationem, qua de agimus, nē- |
|
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|
3076 pe Mechanicam pertinet. |
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3077 <p>Manubria vectes $unt, & ad vectium naturam redu- |
|
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3078 cuntur, corum $cilicet, in quibus fulcimentum e$tinter |
|
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3079 pondus & potentiam. In his autem attenditur proportio, |
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3080 quam habet manubrij longitudo ad ip$um axis $emidia- |
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3081 metrum, eo enim faciliùs mouent, quo eorum longitudo |
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3082 ad axium $emidia metros proportionem, habuerit ma- |
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3083 iorem. Duabus autem partibus con$tant, alterâ, quæ ab |
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3084 axe ad angulum; quæ verè vectis e$t; alterâ, cui manusi- |
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3085 p$a admouetur, ex qua res tota manubrium dicitur. Fiunt |
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3086 autem manubria hæc vt plurimum amouibilia, $unt tamē |
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3087 ceu rotarum ip$arum partes, & rotis ip$is commodè affi- |
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3088 gerentur, ni$i in rotatione à tran$uer$arijs, quibus rotæ $u- |
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3089 $tinentur, impedimentum fieret. |
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|
3090 <fig> |
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3091 <p>E$to enim rota AB, cu- |
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3092 ius axis E, terebretur autem |
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3093 in F, ibi<13>ue paxillus affigatur |
|
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3094 FK. Sit & alia rota CD, cu- |
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3095 ius axis G, manubrium axi |
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3096 appo$itum GHI. Sint autem |
|
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|
3097 rotæ æquales & axes æqua- |
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3098 les. Sint etiam æqualia ip$a |
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3099 $patia EF, GH, hoc e$t, ma- |
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|
3100 <p n=>81</p> |
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3101 nubrij GHI longitudo. Dico, câdem facilitate moueri AB |
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3102 rotam à potentia in FK, quâ mouetur CB, à potentia po- |
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3103 $ita in HI, datis ip$i nempe potentijs æqualibus. Produca- |
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3104 tur enim IH, v$que adrotæ CD latus in L, & LG ducatur, |
|
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3105 & FE in rota AB iungatur. Erunt igitur FE LG inter $e æ- |
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3106 quales. Sunt autem eorum circulorum $emidiametri, qui |
|
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3107 à punctis FL, in ip$a rotatione de$cribuntur. Ita igitur $e |
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3108 habebit potentia applicata in L ad diametrum $emidia- |
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3109 metrumue axis rotæ CD, vt $e habet potentia applicata |
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3110 in F, ad diametrum $emidiame trumue axis E rotæ AB, $ed |
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3111 $patia $unt æqualia & potentiæ æquales, quare nihil re- |
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3112 fert, vtrum manubrium lateria ffigatur, vel axi à latere ro- |
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3113 tæ $eparatum applicetur. |
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|
3114 <fig> |
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3115 <p>Duplex autem e$t ma- |
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3116 nubriorum forma; altera e- |
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3117 nim rectis partibus con$tat, |
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3118 altera verò curua e$t tota, |
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3119 $ed rectis vtimur vt mani- |
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3120 bus apprendamus, curuis |
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3121 verò vt locum illis appona- |
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3122 mus, & pedis pre$$ione ceu |
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3123 in molis lapideis, quibus |
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3124 gladij acuuntur, fieri a$$olet, conuertantnr. Cur autem |
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3125 manubria hæc curua fiant, ea videtur ratio, ne videlicet |
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3126 manubrij capite $upra centrum in linea quæ per centrum |
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3127 tran$it, cō$tituto, factâ interim pre$$ione motus à centro, |
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3128 ad quod directè fieret pre$$io, impediretur. Curuitas autē |
|
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3129 facilitatem quan dam habet, ex qua factâ modicâ flexione |
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3130 axis caput, dum premitur ab ip$a perpendiculari linea le- |
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3131 niter abducitur. quæ cum ce$$ent in manubrijs quæ manu |
|
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3132 aguntur, ideo alia forma, nempe ex rectis partibus pa$$im |
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3133 fiunt. E$to igiturillud quod ex rectis partibus AB, curuum |
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3134 verò CD, linea verò, $ecundum quam pede fit pre$$io |
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|
3135 <foot>L</foot> |
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|
3136 <p n=>82</p> |
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3137 CDE. Hæc itaque de manubrijs $eu vectibus nos con$i- |
|
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3138 dera$$e $it $atis. |
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3139 <p>Quæriinterim po$$et, Cur duabus datis rotis æqua- |
|
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|
3140 lis magnitudinis in æqualis ponderis, circa æquales axes |
|
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|
3141 con$titutis leuior faciliùs moueatur & citiùs quie$cat; |
|
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3142 grauior verò di$ficilius moucatur & tardiùs ce$$et à mo- |
|
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3143 tu, ea videtur ratio, quod grauior re$i$tens magis cum $u- |
|
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3144 peratur impre$$am vim $u$cipit, & diutiùs retinet, quod |
|
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|
3145 ce$$at in leuiore. |
|
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|
3146 <HEAD>QVÆSTIO X.</HEAD> |
|
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3147 <HEAD><I>Dubitat Ari$toteles, Cur faciliùs, quando $ine pondere est, mouca- |
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3148 tur libra, quàm cum pondus habet. Simili modo rota, & eiu$modi- |
|
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3149 quidpiam, quod grauius quidem est, item quod maius & |
|
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|
3150 grauius minori, & leutori?</I></HEAD> |
|
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3151 <p>Breuiter autem $oluit, ait enim, An quia non $olum in |
|
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3152 contrarium quod graue e$t, $ed in obliquam etiam dif- |
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3153 ficulter mouetur? In contrarium enim ei ad quod vergit |
|
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3154 onus mouere difficile e$t, quo autem vergit, e$t facile. In |
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3155 obliquum autem haudqua quam vergit. Nos quod ip$e |
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3156 non fecit figurâ ip$a appo$itâ rem clariorem faciemus. |
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|
3157 <fig> |
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3158 <p>Efto libra AB, cuius ful- |
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3159 cimentum C, pondera vtrin- |
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3160 que appen$a AB, quorum v- |
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|
3161 trumque ponderet 10. Item |
|
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3162 libra DE, cuius fulcimentum |
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3163 F pondere vero appen$a D, E, |
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3164 ip$is A, B, dimidio leuiora, nē- |
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3165 pe S. Addatur ponderi B pon- |
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3166 dus G, & ponderi E pondus |
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3167 H, quorum $imiliter vtrumq; |
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3168 ponderet S, nutabunt igitur |
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3169 libræ ponderibus appo$itis, & |
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3170 <p n=>83</p> |
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3171 BG $ecetur in K, EH verò in N, grauius e$t autem GB, e$t |
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3172 enim IS, ip$o EH, quod e$t 10. Difficiliùs autem de$cen- |
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3173 det BG, quàm EH. hoc autem ex doctrina Ari$totelis, |
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3174 quia non $olum in contrarium quod graue e$t, $ed in obli- |
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3175 quum etiam difficulter mouetur, in contrarium enim ei |
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3176 ad quod vergit onus mouere difficile e$t, quò autem ver- |
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3177 git facilè in obliquum autem puta per lineas BK, EN non |
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3178 vergit onus. Difficiliùs ergo in obliquum mouebitur pon- |
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3179 dus BG ip$o pondere EH. vtrumque autem in de$cen$u |
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3180 retrahitur nempe à perpendicularibus BI, EM & retra- |
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3181 ctionis quidem anguli $unt æquales & æquales ip$æ retra- |
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3182 ctioncs. Sedgrauius e$t pondus GB. quod autem grauius |
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3183 e$t, violentius de$cēdit eo quod e$t leuius. maiori igitur ni- |
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3184 $u atque impetu cum cætera paria $int, de$cendet pondus |
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3185 BG, ip$o EH, quod è diametro Ari$totelis a$$ertioni e$t |
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3186 contrarium. ex alijs igitur principijs veritas ip$a e$t eruen- |
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3187 da. Dicimus autem id ex proportionum fieri inæqualita- |
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3188 te; quia enim is ad 10. proportionem habet $e$quialteram, |
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3189 10. verò ad 5. duplam, maiorem proportionem habet EH |
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3190 ad oppo$itum pondus D, quàm BG ad pondus A, facilius |
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3191 ergo trahet libra DE lcuior pondus D, quàm ip$a AB, gra- |
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3192 uior pondus A, quod vtique fuerat o$tendendum. Alia |
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3193 quoque cau$$a & hæc accidentalis ad hunc effectum pa- |
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3194 riendum concurrit, axium nempe ad fulcimenta, in qui- |
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3195 bus rotantur, fricatio. quo enim maius e$t pondus cæteris |
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3196 paribus, quod nos in præ cedente quæ$tione demon$tra- |
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3197 uimus, eò maiìor fit ip$a collifio. |
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3198 <p>Porrò huius quoq; $peculationis e$t, Cur æqualia & |
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3199 $imilia corpora in æqualibus $imilibu$<13>ue bafibus con$ti- |
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3200 tuta eodem $imili<13>ue plano fulta, ponderibus tamen in- |
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3201 æqualia, non eâdem facilitate euertantur, $ed horum gra- |
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3202 uiora difficilius. |
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|
3203 <foot>L 2</foot> |
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3204 <p n=>84</p> |
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3205 <fig> |
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3206 <p>Sit enim Pri$ma $eu |
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3207 Cylindrus ABCD, cuius |
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3208 grauitatis centrum E in |
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3209 plano Cl, ba$i fultus CD. |
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3210 Sit & alter Cylindrus |
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3211 FGHI, cuius grauitatis |
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3212 centrum K fultus ba$i HI |
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3213 æqualis quidem & $imilis |
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3214 ip$i AD. Sit autem grauior FGHI, ip$o ABCD. Dico, pari |
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3215 potentiâ vtrumque impellente, facilius euer$um iri Cy- |
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3216 lindrum AD, ip$o Fl. Ducantur EC, KH, & æquales po- |
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3217 tentiæ applicentur punctis BG, pellentes Cylindros ad |
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3218 partes AF. Euer$io autem non fiet donec facta corporis |
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3219 conuer$ione circa puncta CH, grauitatis centra E, K trãs- |
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3220 ferunturin L, M, in ip$is $cilicet perpēdicularibus ACFH. |
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3221 Demit tantur EN, KO, perpendiculares ip$is CD, HF. Et |
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3222 quoniam CNE, HOK anguli recti $unt, erunt EC KH i- |
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3223 p$is EN, KO, maiores, quare & LC, MH ip$is EN KO, ma- |
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3224 iores atto lluntur ergo in ip$a cuer$ione, grauitatum cen- |
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3225 tra E in L, K in M. At quod grauius e$t, difficilius contra |
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3226 $ui naturam mouetur, ideo difficilius euertetur corpus |
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3227 FI, ip$o AD, quod fuerat demon$trandum. |
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3228 <HEAD>QVÆSTIO XI.</HEAD> |
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3229 <HEAD><I>Dubitat Philo$ophus, Cur $uper $cytalas facilius portentur onera |
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3230 quàm $uper currus, cum tamen ij magnas habeant rotas, |
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3231 illæ verò pu$illas?</I></HEAD> |
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3232 <p>Optimè re$pondet dubitationi. An, in quiens, quoniam |
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3233 in $cytalis nulla e$t offen$atio; in curribus verò axis |
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3234 e$t, ad quem offen$ant. De$uper enim illum premunt, & |
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3235 à lateribus. quod autem e$t in $cytalis ad i$thæc duo mo- |
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3236 uetur & inferiori $ub$trato $patio, & onere $uperimpo$i- |
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3237 <p n=>85</p> |
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3238 to, in vtri$que enim ijs <*>euoluitur locis circulus, & motus |
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3239 impellitur. Tam appo$itè paucis verbis veritatem expli- |
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3240 cauit, vt ferè quicquid in$uper ad datur, $uperuacaneum |
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3241 videri po$$it. quicquid tamen $it, ad maiorem claritatem |
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3242 aliquantulum in hac ip$a quæ$tione immorabimur. |
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3243 <p>Rotatas $cytalas proponit hîc Ari$toteles. Coniun- |
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3244 ctasautem e$$e rotas ip$is $cytalis e$t intelligen dum, nem- |
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3245 pe, vt $imul rotæ cum $cytalis conuertantur. Secus enim |
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3246 axium & Rotarum fieret offen$atio, cuius offen$ationis |
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3247 vim & effectum cum nouerit Ari$toteles, vel hoc ip$o lo- |
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3248 co te$te, mirum e$t, nihil de ea egi$$e quæ$tione 9. vbi nos |
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3249 hac de re fu$i$$imè tractauimus. |
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3250 <p>Cæterùm quod de rotatis $cytalis $cribit Philo$o- |
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3251 phus, notandum, à Pappo quidem lib. 8. & à no$tris Me- |
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3252 chanicis pa$$im ab$que rotis Cylindrica $implici videli- |
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3253 cet, & tereti formâ ad v$um adhiberi. E$to igitur Ari- |
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3254 <fig> |
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3255 $totelis quidem $cytala |
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3256 AB, Pappi verò $eu vul- |
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3257 garis, & communis CD. |
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3258 His non modò lapicidæ |
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3259 pa$$im, $ed & nautæ na- |
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3260 uium<13>ue fabri $ubdu- |
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3261 cendis & mari inducen- |
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3262 dis nauibus vtuntur, quod varare dicunt vernaculè, Hi- |
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3263 $panico, vt arbitror, vocabulo. ca enim natio teres lignum |
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3264 baculumue appellat Varam. |
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3265 <p>Quæriautem po$$et, vtra harum formatum $it vti- |
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3266 lior atque commodior? Nos rotatas laudamus magis in |
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3267 plano duro<13>ue $olo, minus enim tangunt & minus offen- |
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3268 $ant; in molliori autem & minus duro proponimus non |
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3269 rotatas, $iquidem rotæ $ui naturâ pondere pre$$æ $olum, |
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3270 facillimè $cindunt & ab$orbentur. |
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3271 <p>Quatenus autem ad v$um pertinet. E$to horizontis |
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3272 <foot>L 3</foot> |
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3273 <p n=>86</p> |
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3274 <fig> |
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3275 planum AB, $cytal<17> du<17> |
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3276 CD, EF, Pondus verò |
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3277 eis impo$itum G, tan- |
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3278 gens ip$as in pūctis CE, |
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3279 $cytalæ autem planum |
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3280 in punctis D, F, Pellatur |
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3281 à potentia quapiam pō- |
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3282 dus Gad anteriora, nē- |
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3283 pe ad partes E. rotabuntur igitur $cytalæ & pars quædam |
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3284 $cytalæ D, in qua $it contactus a$cendet in I, C verò de- |
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3285 $cendetin H, nulla re motum impediente, quippe quòd |
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3286 nulla ponderis $cytalarum, & plani ad inuicem fiat offen- |
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3287 $atio. Præterea cum $cytalarum centra ab horizontis pla- |
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3288 no æqualiter di$tent, pondus quidem horizonti æquidi- |
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3289 $tanter mouetur, & ideo cius centrum grauitatis nequa- |
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3290 quam, in motu qui $it, eleuatur. |
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3291 <p>Cæterùm materiæ imperfectione remota nihil re- |
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3292 fert ad facilitatem, vtrum maioris minorisue diametri |
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3293 $int$cytalæ, vt ea po$ita eo quod maiores circuli faciliùs |
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3294 offendicula $uperent, quod demon$tratum e$t in quæ$tio- |
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3295 ne 8. eo vtiliores $unt $cytalæ, quo cra$$iores. Quatenus |
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3296 autem ad plau$tri naturam $pectat, cuius ad $cytalas Phi- |
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3297 lo$ophus fecit comparationem, vt o$ten da mus difficilius |
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3298 ex eo moueri pondera. |
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3299 <fig> |
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3300 <p>E$to plau$tri rota |
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3301 KL, cuius centrum M, a- |
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3302 xis verò NO circa quem |
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3303 rotaip$a conuertiturKL. |
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3304 Funis quo rota ex axis |
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3305 centro M trahitur MP, |
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3306 pondus vero QR. Quo- |
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3307 niam igitur pondus axem |
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3308 premit in N, axis autem rotæ modiolum in O, & codem, |
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3309 <p n=>87</p> |
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3310 tempore potentia quæ trahitin P, axem admouet modio- |
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3311 lo in parte V. duplex itaque fit ex fricatione $eu offen$a- |
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3312 tione impedimentum, in fra nempe, vbi O, & ad latus vbi |
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3313 V. quæ quidem offen$iones currus motum reddunt diffi- |
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3314 ciliorem, quæ quidem difficultas eo maior erit, quo ma- |
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3315 ior fuerit pondus axem premens, & minor proportio $e- |
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3316 midiametri rotæ KM, ad axis $emidiametrum MO. Cur |
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3317 igitur $cytalis facilius pondera transferantur quam plau- |
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3318 $tris, apertè ex dictis ad Ari$to telis mentem demon$tra- |
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3319 uimus. |
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3320 <p>Cætetùm quod ip$e reticuit, n<*>s dicemus, nempe |
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3321 validi$$imè enormia pondera per $cytalas moueri, $i $cy- |
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3322 talis ip$is vectes adiungantur. Et $anè motus erit tar di$$i- |
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3323 mus, veruntamen tar ditas ip$a facilitate, quæ in de fit, v- |
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3324 berrimè compen$atur. |
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3325 <fig> |
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3326 <p>E$to igitur horizontis planum AB, $cytalæ CD, fo- |
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3327 ramina in $cytalis EFGH, vectes foraminibus in$erti IE, |
|
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3328 KF, LG, MH. Pondus vero $cytalis impo$itum N. Appli- |
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3329 catis igìtur quatuor potentijs extremitatibus vectium I, |
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3330 <I>K</I>, L, M, ij$que in anteriora propul$is, fiet $cytalarum rota- |
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3331 <p n=>88</p> |
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3332 tio, & ponderis N translatio ad anteriores partes B. E$to |
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3333 item $eor$um $cytala PR, cuius centrum Q, vectis eidem |
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3334 per centrum in$ertus O, P, Q, R. facto igitur vectis motu |
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3335 O P Q R fiet ex O; centro autē Q circuli quadrans O T. |
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3336 exi$tente igitur O in T erit P in S. facta quartæ partisip$ius |
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3337 $cytalæ rotatione. Et quoniam ex eodem centro $unt qua- |
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3338 drantes PSOT. erit vt OQ ad QP. ita quadrans OT, ad |
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3339 quadrantem PS. Maxima autem e$t proportio OQ, ad |
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3340 QP. Maxima igitur proportio OT ad PS. Ex magno igitur |
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3341 motu O ad T, paruus $it $cytalæ motus à P in S. tardius i- |
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3342 gitur progreditur $cytala, quæ longioribus vectibus rota- |
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3343 tur, vis tamen maxima, quippe quod vt $o habet QP, hoc |
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3344 e$t, QR ad QO, ita potentia in O ad pondus quod premit |
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3345 in P vel in V. Facillimè ita que pondera vectibus & $cyta- |
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3346 lis per horizontis planum transferri, exi$tis patet. |
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3347 <HEAD>QVAESTIO XII.</HEAD> |
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3348 <HEAD><I>Quæritur, Cur Mi$$ilia longius funda mittantur quam manu, |
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3349 præ$ertim cum proijcienti fundæ pondus addatur lapidis $eu mi$$i- |
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3350 lis ponderi: & minus mi$$ili, manu proiecto, com- |
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3351 prehendatur?</I></HEAD> |
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3352 <p>Soluit Philo$ophus, inquiens, fortè ita fieri, quòd fun- |
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3353 ditor mi$$ile proijciatiam ex funda commotum, $iqui- |
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3354 dem fundam circulo $ubinderotans, iaculatur, ex manu |
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3355 autem à quiete e$t initium. Oinnia autem cum in motu |
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3356 $unt, quàm cum quie$cunt, facilius mouentur. Addit præ- |
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3357 terea, An & ob eam cau$$am e$t, $ed nec minus etiam, quia |
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3358 in fundç v$a manus quidem fit centrum, funda verò quod |
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3359 à centro exit? quantò igitur productius fuerit quod à cen- |
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3360 tro e$t, tanto citiùs mouetur; iactus autem, qui manu fit, |
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3361 fundæ re$pectu breuior e$t. |
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3362 <p>Hæc Philo$ophus. Et $anè perquàm appo$itè, itaq; |
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3363 <p n=>89</p> |
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3364 illi pror$us a$$entirer, ni$i pro comperto habercm, in la ctu |
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3365 qui fundâ fit, non e$$e manum ip$am motus centium, $ed |
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3366 potius partem illam brachij, quæ humero iungitur, & id- |
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3367 co motum eo fieri velociotem, quo longior e$t linea quæ |
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3368 ab humero ad $ummitatem fundæ e$t, ea quæ ab humero |
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3369 ad manum ip$am. Illud quo que mirabilc e$t, quod non |
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3370 ob$etuat Ari$toteles, nempe à funditoribus in ip$o eiacu- |
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3371 landi actu, tardam fieri circa caput fundæ rotationem. |
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3372 Quamobrem con$iderandum e$t, quo pacto fiat à tardi- |
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3373 tate velocitas. Re$pondemus, velocitatem acquirinon ex |
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3374 $implici, quæ circa funditoris caput $it, rotatione, $ed ex |
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3375 co impetu qui fit in ip$a lapidis emi$$ione, qui quidem im- |
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3376 petus $i ante vel po$t illud tempus fiat, quod à funditore |
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3377 captatur, ca$$a pror$us & inualida fit ip$a iaculatio. |
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|
3378 <fig> |
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3379 <p>E$to funda AB, manus |
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3380 B, brachium BC. Vt igitur$e |
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3381 habet CH, ad CB, ita veloci- |
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3382 tas AD ad velocitatem, BE; |
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3383 Vidimus nos pueros, arundi- |
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3384 ni ad caput $ci$$æ, paruos la- |
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3385 pides in$erentes, arundinem- |
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3386 <13>ue manu rotantes longi$$i- |
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3387 mè lapides ip$os proijcere; A- |
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3388 rundo FG, lapis F, manus G, |
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3389 brachium GH. |
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3390 <HEAD>QVÆSTIO XIII.</HEAD> |
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3391 <HEAD><I>Quæritur, Cur circa idemiugum, maiores collopes (vectes $unt, |
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3392 quos alij $cytalas appellant, vt Pappus & Heron) faciliùs quàm mi- |
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3393 nores mouentur: & item $uculæ, quæ graciliores $unt eadem |
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3394 vi quam cra$$iores?</I></HEAD> |
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3395 <p>Ideo hoc fieri po$$e docet Philo$ophus, quòd tamiugū |
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3396 quam $ucula cētrum $it, prominenres autem collopum |
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3397 <foot>M</foot> |
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3398 <p n=>90</p> |
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3399 longitudines eæ lineæ quæ $unt à centro. Celeriùs autem |
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3400 moueri & plus ab eadem vi quæ maiorum $unt circulorū |
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3401 quàm quæ minorum. quippe quod ab ea dem vi plus trã$- |
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3402 feratur illud extremum quod longius à centro di$tat. In |
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3403 gracilioribus verò $uculis datâ collopum paritate plus e$- |
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3404 $e id quod à ligno di$tat. |
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3405 <fig> |
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3406 <p>E$to iugum $ucu- |
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3407 laue maior, AB circa |
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3408 centrum C, minor verò |
|
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3409 circa idem centrū DE. |
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3410 Collops autē AF, pon- |
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3411 dus quod per iugum at- |
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3412 tollitur G. A it igitur A- |
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3413 ri$toteles, $uculas, iu- |
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3414 gaue AB, DE ceu cen- |
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3415 tra e$$e, à quibus extat colops AB, ex maiori quidem, totâ |
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3416 $ui parte BF, ex minori autem EF. quo igitur, ait, longior |
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3417 fuerit collops extans, eo maior, & deo velocior ad partē |
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3418 F per maiorem circulum FH, fiet collopis motus & pon- |
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3419 deris eleuatio, at maior e$t collops EF ip$o BF, facil. us er- |
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3420 go mouebitur pondus per $uculam DE, ex collope EF, ab |
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3421 cadem vi, quam per $uculam AB, & collopem BF. |
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3422 <p>Hæc $en$i$$e videtur Ari$to teles, qui cra$$a, vt aiunt, |
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3423 Minerua rem pulchram & $ubtilem e$t pro$equutus. Di- |
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3424 cimus igitur primò, in$trumentum illud quod Latini $u- |
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3425 culam, id e$t, $ero$ulam, à $tridore arbitror qui in conuer- |
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3426 $ione fit, appellauere, Græci verò <G>o)/non</G>, id e$t, A $inum, quip- |
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3427 pe quod ceu A $inus pondera $u$tineat portetque. Hanc |
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3428 eandem Machinam veteres Mechanici vocauere Axem |
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3429 in Peritrochio, cuius nos imaginem, è Pàppo in 8. Col- |
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3430 lect. Mathematicarum de$umptam in ip$o huius no$trio- |
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3431 peris initio, inter quinque Potentias propo$uimus. Huius |
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3432 vim inter antiquos diligenti$$ime examinauêre Heron, & |
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3433 <p n=>91</p> |
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3434 ip$emet Pappus, inter iuniores verò Guilibaldus co Tra- |
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3435 ctatu quem hac de Potentia Mechanicis $uis in$eruit. |
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3436 Summa e$t, hanc Machinam ad vectem reduci. Nec ve- |
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3437 rum e$t quod $cribit Ari$to teles, iugum $uculamue cen- |
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3438 tra e$$e, hæc enim centrum habent, quod in figura $upe- |
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3439 rius po$ita notatur $igno C. igitur vt $e habet FC, ad CA, |
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3440 ita pondus G ad potentiam in F; e$t autem maior propor- |
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3441 tio FC ad CD, quàm FC, ad CA. faciliùs ergo mouebit |
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3442 potentia quæ in F, pondus in D, quàm eadem potentia F, |
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3443 pondus in A, hoc e$t, G. Huius naturæ $unt quo que Erga- |
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3444 tæ, quas machinas no$tr<*>, Græco luxato vocabulo Arga- |
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3445 nos appellant. Suculæ enim reuera $unt, po$itione tantū |
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3446 ab eis differentes, non enim plano horizontis ergatæ æ- |
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3447 quidi$tant, ceu $uculæ & Axis in Peritrochio, $ed eidem |
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3448 fiunt perpendiculares. Cæterùm facilitatem à velocitate |
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3449 non oriri $uperius demon$trauimus. |
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3450 <HEAD>QVAESTIO XIV.</HEAD> |
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3451 <HEAD><I>Proponitur dubitatio, Cur eiu$dem magnitudinis lignum facilius |
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3452 genu frangatur $i qui$piam æque diductis manibus extrema com- |
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3453 prehendens fregerit, quàm $i iuxta genu. Et $i terræ applicans pede |
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3454 $uperpo$ito manu hinc inde diducta confregerit |
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3455 quàm propè.</I></HEAD> |
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3456 <p>Soluitur à Philo$opho paucis verbis, An quia ibi genu |
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3457 centrum e$t, hic verò ip$e pe? quanto autem remotius |
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3458 à centro fuerit, facilius mouetur quodcunque: Moueri |
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3459 autem quod frangitur nece$$e e$t. |
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3460 <p>E$to lignum quod frangi debet AB, genu vel pedis |
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3461 locus C, manuum latè diductarum $itus DE, minus didu- |
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3462 ctarum FG; ltaque quoniam DE magis à centro C di$tant |
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3463 quàm FG, velo cius mouebuntur puncta DE ip$is FG, er- |
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3464 go inde facilius fiet fractio quam ex FG. Hæcille ex $uis |
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|
3465 <foot>M <I>2</I></foot> |
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|
3466 <p n=>92</p> |
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|
3467 <fig> |
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3468 principijs. Nos dili- |
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3469 gentius, $i fieri poterit, |
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3470 effectus huius cau$$am |
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3471 per$crutemur. E$to igi- |
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|
3472 tur in $ecunda figura |
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|
3473 lignum oblongum AB, |
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3474 cuius medium C, linea |
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|
3475 ducatur CD perpen- |
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3476 dicularis ip$i AB. Ad- |
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3477 moucatur genu pūcto |
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3478 C, manus verò diuari- |
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3479 centur in AB, facta i- |
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|
3480 gitur vtrinque impre$- |
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|
3481 $ione, lignum non frã- |
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3482 getur, ni$i partium in CD coniunctarum $eparatio fiat, |
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3483 $it<13>ue altera in E, altera verò in F, fractum ergo erit lignū, |
|
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3484 & centro C immobili permanente, partes facto angulo |
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3485 GCH erunt in GC, HC: Modò lignum $uæ integritati re- |
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3486 $tituetur, & denuò admoto genu puncto C, manus didu- |
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3487 cantur in I, K, quæ lo ca viciniora $intip$i C, quam AB, Di- |
|
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3488 co hinc difficilius fractionem fieri quam ex AB. Con$ide- |
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3489 ramus enim in integro ligno AB, duos vectes ACD, BCD, |
|
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|
3490 quorum anguli concurrunt in commune fulcimentum C, |
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3491 Sunt autem vectes angulati, & eius naturæ, quam exami- |
|
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|
3492 nauimus in quæ$tiones. E$t igitur re$i$tentia, qua ligni |
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3493 partes vniuntur in D, loco ponderis: $uperanda hæc e$t, vt |
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3494 ligni fiat fractio. Dico id facilius ce$$urum, $i fiat ex pun- |
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3495 ctis A, B, remotioribus quam ex IK, ip$i puncto C propio- |
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3496 ribus: etenim vt AC, ad CD, itare$i$tentia quæ fit in Dad |
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3497 potentiam in A, item vt $e habet IC ad CD, ita re$i$tentia |
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3498 in Dad potentiam in I, $ed minor e$t proportio IC ad CD, |
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3499 quam AC ad CD. ergo facilius potentia quæ e$t in A, re- |
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3500 $i$tentiam $uperabit, quæ e$t in D, quam ea quæ e$t in I, |
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3501 <p n=>93</p> |
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3502 quod fuerat demon$trandum. Idem autem intelligendū |
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3503 e$t de parte CB; eadem enim e$t ratio. Curigitur longio- |
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3504 ra & graciliora ligna facilè frangantur, ex i$tis clare patet: |
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3505 nempe quia maxima e$t proportio longitudinis ad cra$$i- |
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3506 tudinem, cuius quidem cra$$itudinis $patium loco partis |
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3507 illius in vecte $uccedit, quæ pertingit à fulcimento ad pō- |
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3508 dus, hoc e$t, ad ip$am re$i$tentiam. Sed nos hac eadem de |
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3509 re nonnulla in declaranda quæ$tione 16. perpendemus. |
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3510 <HEAD>QVAESTIO XV.</HEAD> |
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3511 <HEAD><I>Proponitur inueftigandum, Cur litterales crocæ (glareas dicunt |
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3512 Latini, velcalculos, quos vmbilicos appellat Cicero lib. 2. de Orat.) |
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3513 rotundâ$int figurâ, cum aliquando ex magnis $int la- |
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3514 pidibus te$tisue?</I></HEAD> |
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3515 <p>A It Philo$ophus, ideo forta$$e fieri, quòd ca quæ à me- |
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3516 dio magis recedunt, in motionibus, celerius feran- |
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3517 tur; me dium e$$e centrum, interuallum vero quæ à cen- |
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3518 tro, $emper autem maiorem ab æ quali motione maiorem |
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3519 de$cribere circulum; quod autem maius in æquali tem- |
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3520 pore $patium tran$it, celerius ferri; quæ autem celerius ex |
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3521 æquali feruntur $patio vehementius impetere, quæ autē |
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3522 impetunt, impeti magis, & ideo quæ magis à centro di- |
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3523 $tant, nece$$e e$$e confringi, quod cum glareæ $eu croc æ |
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3524 patiantur, nece$$ariò rotundas fieri. Hactenus ille, & $anè |
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3525 p<*>obabiliter. Verum enimuerò aliter $eres habere vide- |
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3526 tur: $iquidem enim à rotatione ex maiori à centro di$tan- |
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3527 tia id fieret, maiores quidem glareæ crocæue e$$ent ro- |
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3528 tundiores, at nos non maximas modò, $ed & minimas, |
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3529 ea$<13>uemagis angulis carere, & ad rotunditatem accede- |
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3530 revidemus. Præterea non moueri eas circa centrum pa- |
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3531 lam e$t, imò vt varia $unt figura, ita varijs quo que motio- |
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3532 nibus, ex agitatione moueri. Id $anè explorati$$imum e$t, |
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3533 <foot>M 3</foot> |
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3534 <p n=>94</p> |
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3535 angulos omnes, & emin entias quaslibet in corporibus e$- |
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3536 $e infirmiores, offen$ionibus enim expo$itæ $unt, necre$i- |
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3537 $ten di habent facultatem. Itaque in attritione quæ fit in |
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3538 eorum agitatione perpetua, eminentiæ contunduntur, & |
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3539 $uperficies ip$a paullatim leuigatur. |
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3540 <fig> |
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3541 <p>E$to angulatus lapis ABCD. |
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3542 Dum igitur perpeti motione atq; |
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3543 a$$iduâ ver$atione agitatur, fer- |
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3544 tur<13>ue, eminentiæ anguli<13>ue, vt- |
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3545 pote debiles & imbecilli, conte- |
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3546 runtur, & inde figura fit quædam |
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3547 irregularis, ad primam quidem la- |
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3548 pidis formã accedens, leuistamen |
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3549 & quouis angulo carens, qualis e$t E remotis ABCD, an- |
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3550 gularibus eminentijs. |
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3551 <p>Hanc eandem ob cau$$am, $culptores antequam mar- |
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3552 moribusvltimum læuorem inducant, dentato malleo pri- |
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3553 mum quidem vtuntur, tum demum eminentiores parti- |
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3554 culas radula facilè amouentes $uperficiem ip$am læuem |
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3555 & adæquatam reddunt. |
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3556 <p>Hinc etiam no$trates Architecti, in arcium propu- |
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3557 gnaculis efformandis a cutos angulos deuitãt, vtpote de- |
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3558 biliores, & magis offen$ionibus obnoxios. quod nec Vi- |
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3559 truuium latuit, qui ideo lib. 1. cap. 5. ita $cribit: <I>Turresita<26> |
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3560 rotundæ aut polygoniæ $unt faciendæ, quadrat as enim machinæ |
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3561 celerius di$$ipant; & angulos, Arietes tundendo frangunt, inro- |
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3562 tundationibus autem, vti cuneos adcentrum adigendo lædere non |
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3563 po$$unt.</I> Hæcille. Cur autem no$tri rotundas figuras alias |
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3564 vtiles reijciant, ab ijs petendum qui in ea facultate ver- |
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3565 $antur. Porrò quod ad hanc eandem $peculationem facit, |
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3566 videmus, antiquas $tatuas, vt $æpius auribus, na$o, digitis, |
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3567 manibu$ue atque pedibus carere, quippe quodimbecillæ |
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3568 $int partes, & facilè quouis occur$u mutilentur. Quæo- |
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3569 <p n=>95</p> |
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3570 m<*>a cùm vera $int, nemo, vt arbitror, dixerit, ab$olutè, |
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3571 quod voluit Ari$toteles, id ex rotatione velociori & par- |
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3572 tium à centro remotione, fieri. |
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|
3573 <HEAD>QVAESTIO XVI.</HEAD> |
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3574 <HEAD><I>Dubitatur, quare, quò longiora $unt ligna, <*>ãto imbecilliora fiant, |
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3575 & $itolluntur, inflectuntur magis: tamet$i quod breue est ceu bi- |
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3576 cubit umfuerit, tenue, quod verò cubitorum cen- |
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3577 tum cra$$um?</I></HEAD> |
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3578 <p>Ex $uis principijs $oluit Ari$totelcs. Inquit enim: An |
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3579 quia & vectis & ont s & hypomochlium, id e$t, fulci- |
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3580 mentum in leuando, fit ip$a ligni proceritas? Prior namq; |
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3581 illius pars ceu hypomochlium fit, quod verò in extremo |
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3582 e$t, pondus: quamobrem quanto exten$ius fuerit id quod |
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3583 à fulcimento e$t, in flectinece$$e e$t magis; quo enim plus |
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3584 à fulcimento di$tat, eo magis incuruarinece$$e e$t. Ne- |
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3585 ce$$ariò igitur extrema vectis eleuantur. Si igitur flexilis |
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3586 fuerit vectis, ip$um inflectimagis cum extollitur nece$$e |
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3587 e$t, quod longis accidit lignis, in breuibus autem quod vl- |
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3588 timum e$t, quie$centi hypomochlio depropè fit. Hæc |
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3589 $ubiectâ figurâ ob oculos ponimus. |
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3590 <fig> |
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3591 <p>E$to longum ac fle- |
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3592 xile lignum AB, manu ele- |
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3593 uetur in A, fle ctetur itaq; |
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3594 in B, & declinabit in C. et- |
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3595 enim manus quæ $u$tin et |
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3596 in A, fulcimenti loco $uccedit: longitudo vero AB ponde- |
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3597 ris vices refert, at que vectis, quare quo longius abfuerit à |
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3598 fulcimento, id e$t, manu extremum B, eo magis flectetur; |
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3599 $i autem lignum breuius fuerit, nempe terminatum in D, |
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3600 nequaquam fle ctetur, eò quòd eius extremum D minus à |
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3601 fulcimento quod e$t in A $it remotum. Hæcigitur e$t mēs |
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3602 <p n=>96</p> |
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3603 Ari$totelis, cuius quidem $ententiam non damnamus; |
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3604 quippiam tamen addimus. Dicimus autem materiam, |
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3605 quatenus ad hanc contemplationem $pectat, in duplici |
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3606 e$$e differentia. aut enim rarefactionis & con$tipationis |
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3607 e$t incapax, vt in chalybe videmus, nitro, metallo, mar- |
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3608 more, aut capax quidem, & hæc duplex: Vel enim natura |
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3609 nata e$t ad rectitudinem quandam, vt ar borum flagella |
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3610 virgæque, autnon item, ceu $tannum, plumbum, & cæte- |
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3611 ra eiu$modi. |
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3612 <fig> |
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3613 <p>E$to primò vitreum |
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3614 corpus gracile, procerum, |
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3615 teres AB, manu capiaturin |
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3616 A, itaq. pondere ip$ius cor- |
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3617 poris præualente ad partes |
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3618 B, quia in C puncto, quod |
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3619 circa medium e$t, ex parte |
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3620 $uperiori non fit rarefactio, |
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3621 nec in in feriori con$tipatio, |
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3622 nec interim datur penetra- |
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3623 tio corporum, fit fractio à |
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3624 $uperiori parte, & pars CB à |
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3625 reliqua parte AC, auul$a & |
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3626 $eparata cadit in D, $uccedit autem ip$a $eparatio rarefa- |
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3627 ctioni. Porrò quod materias ha$ce non flexibiles diximus, |
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3628 $ed frangibiles, non ideo negamus vel $en$u docente, ali- |
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3629 quam inijs fieri flexionem. Si autem lignea fuerit mate- |
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3630 ria, ca<13>; flexibilis, vt EF, $i manu eleuetur in E, præualen- |
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3631 te pondere in F flectetur vbi G. ibi enim à parte $uperiori |
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3632 fitrarefactio, ab in feriori verò con$tipatio, & pars GF de- |
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3633 clinabitin H, quæ declinatio eò v$que procedet, quo ra- |
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3634 refactio & con$tipatio competens naturæ illius materiæ, |
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3635 quæ flectitur ad $ummam inten$ionem deuenerint; tunc |
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3636 $ivis maioringruerit, frangetur omnino: $i $ecus factaibi |
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3637 <p n=>97</p> |
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3638 re$i$tentia, vbi rarefactio fit & con$tipatio pe$t inclina- |
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3639 tionem $ur$um ferctur pars in clinata & nutans, tum in |
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3640 contrariam partem tendens reflectetur, vt videre e$t in |
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3641 virga IN. Declinans enim in KL, repellente ea quæ infra |
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3642 K fit materiæ conden$atione, impetu ex de$cen$u a cqui- |
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3643 $ito facta reflexione a$cendit in KM, donec paullatim cir- |
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3644 ca pri$tinam rectitudinem reuertatur, & hic quidem mo- |
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3645 tus vibratio dicitur, agitatioue. Si autem virga plumbea |
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3646 fuerit, naturâ non factâ ad rectitudinem, puta OP, pro- |
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3647 prio vincente pondere, ad partes declinabit QS, fiet<13>; in |
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3648 QR rarefacta, nempe $uperiori parte ea con$tipata infe- |
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3649 riori in Q, nec reflectetur, quippe quòd eius natura con- |
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3650 den$ationem & rarefactionem commodè patiatur, nec |
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3651 facta $it ad rectitudinem. |
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3652 <p>Porrò tripliciter fieri pote$t horum oblongorum |
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3653 corporum eleuatio, nempe vel extremorum alteio, aut $i |
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3654 ambobus, $i vtrinque $u$pen datur, vel alicubi inter extre- |
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3655 ma. De priori modo iam egimus. Modò $u$pendatur in |
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3656 medio vt AB, in C. eo igitur ca$u cum fulcimentum $it in |
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3657 C, vtrinq; fit flexio in D, & E, & id quidem $i materia fle- |
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3658 xionem patitur: $in minus, fractio fit in C. Si autem ab ex- |
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3659 <fig> |
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3660 tremis fiat $u$pen$io, vt in |
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3661 AB, tunc ceu duo vectes |
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3662 fient, quorum fulcimenta in |
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3663 extremis AB. Pondera au- |
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3664 tem communia in medio vbi |
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3665 Cremoti$$ima enim ea pars e$t ab extremis AB. Cedente |
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3666 <fig> |
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3667 igitur materia $uomet pon- |
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3668 deri, $iquidein in flexibilis fu- |
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3669 erit, frangetur, & fiet partiū |
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3670 $eparatio in C, duoque in de |
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3671 corpora AD, BE. Si autem fle- |
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3672 xionis capax, vt AB in po$tre- |
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3673 <foot>N</foot> |
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3674 <p n=>98</p> |
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3675 ma figura, facta ex contrario, nempe in in feriori parte cir- |
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3676 ca C rarefactione, in $uperiori verò conden$atione, pon- |
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3677 dere præualente curuabitur, fiet<13>; lignum quidue aliud |
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3678 huiu$modi, vt ADB, nec amplius pondere $uapte naturâ |
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3679 inferiùs vergente ad rectitudinem reuertetur. |
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3680 <p>Cæterùm cur oblonga & graciliora corpora facilius |
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3681 illis, quæ contrario $e habent modo, frangantur, ex me- |
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3682 chanicis principijs in quæ$tione 14. apertè demon$traui- |
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3683 mus. Modò vt ex hac contemplatione, quæ aliàs inutilis |
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3684 videtur, aliquam vtilitatem capiamus, & ex his quæ con- |
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3685 templabimur, Architecti prudentiotes fiant, i$thæcip$a, |
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3686 de quibus agimus, ad rem ædificatoriam commodè apta- |
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3687 bimus. Transferamus igitur cogitationem ad eam trabiū |
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3688 comp gem, quæ ad tecta $u$tinenda ex tran$uer$ario ar- |
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3689 rectario<13>; $it, & duobus cauterijs, quam no$trià Latinis |
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3690 detorto vocabulo Bi$cauterium dicunt. Per$crutabimur |
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3691 enim, vnde illi tanta ad $u$tin endum vis, & quæ compa- |
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3692 gem hanc con$equantur pa$$iones. quamuis enim fabri |
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3693 meræ praxi, quod vtile e$t efficiant, nos meliorum inge- |
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3694 niorum gratiâ, rei ip$ius cau$$as diligenter examinatas in |
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3695 medium proferemus; nec de hac re tantùm agemus, $ed |
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3696 de Cameris quoque, fornicibus eorum<13>ue vitijs & virtu- |
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3697 tibus quatenus ad Mechanicum pertinet, $ermonem ha- |
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3698 bebimus. Quærimus primo, cur perpendiculariter erect<17> |
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3699 trabes $uperimpo$ita pondera validi$$ime $u$tineant? Et |
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3700 $ane hoc omnes norunt, $ed non per cau$$as. |
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3701 <p>E$to horizontis planum, illud<13>ue $olidi$$imum, & |
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3702 impenetrabile AB, trabs eidem adperpendiculum erecta |
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3703 CD fulta ba$i vbi C grauitatis centrum F. pondus $uper- |
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3704 impo$itum FG, cuius grauitatis centrum H: Sint autem |
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3705 H & E in eadem perpendiculari, quæ ad mun di centrum |
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3706 HEC. Itaque eo quod tum penderis tum trabis centra |
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3707 grauitent in perpendiculari, illa verò fulciatur in C, to- |
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|
3708 <p n=>99</p> |
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|
3709 <fig> |
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3710 tius ponderis moles recumbet |
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3711 in C: non de$cendet autem in I, |
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3712 propterea quod $upponatur i- |
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3713 p$um planum AB, impenetrabi- |
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3714 le. Igitur vt pondus H de$cen- |
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3715 dat in C, alterum duorum e$t |
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3716 nece$$arium, nempe vel trabem |
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3717 $ubiectam comminui, aut eius |
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3718 partes $e$e penetrare, & plura |
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3719 corpora e$$e in eodem loco, pu- |
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3720 ta KC, quorum hoc $ecun dum |
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3721 naturæ penitus repugnat, illud |
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3722 vero primum, penè impo$$ibile. Diuidatur enim trabs in |
|
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3723 partes æquales tres, lineis KL, ip$a igitur KC infima $u$ti- |
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3724 net mediam KL, hæc verò $upremam LD, hæc autem pō- |
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|
3725 dus, ip$um $uperpo$itum in H. Seigitur $u$tinent partes. |
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3726 Sed illud totum partibus con$tat. ergo pondus totum à |
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|
3727 trabe tota, hoc e$t, à $e toto $u$tinetur. |
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3728 <p>Præterea in præcedenti quæ$tione mon$trauimus |
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3729 tunc facilem e$$e gracilis & oblongi ligni fractionem, cū |
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3730 maxima e$t longitudinis ad cra$$itudinem proportio. Hîc |
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3731 verò contrà accidit, etenim MD pars vectis quæ à fulci- |
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|
3732 mento e$t ad potentiam minimam habet proportionem |
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3733 ad rectam DC, quæ à fulcimento ad locum fractionis ex- |
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3734 tenditur, vbi C, quod vt euidentius pateat, |
|
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|
3735 <fig> |
|
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3736 <p>E$to $eor$umtrabs AB, |
|
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|
3737 cuius medium C. Sit autem |
|
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|
3738 pondus D impo$itum pun- |
|
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|
3739 cto C. facilè igitur frange- |
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|
3740 tur lignum AB, propterea |
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|
3741 quòd maxima $it proportio |
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|
3742 AC ad CE; re$i$tentia verò |
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3743 fiat in E, addatur vniatu <13>; |
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3744 <foot>N 2</foot> |
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3745 <p n=>100</p> |
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3746 ligno AB lignum FH. Cra$$ius igitur e$t totum AL, ip$o |
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3747 A<*>, & ideo minor proportio AC ad CG quàm AC, ad |
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3748 CE. Addavur adhuc & IM. Longè itaque difficilius fran- |
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3749 getur in K propterea quòd longè minor $it proportio AC |
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3750 ad CK quàm ciuidem ad CE & CG. His igitur con$ide- |
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3751 ratis, & demon$tratis concludimus, impo$libile e$$e ere- |
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3752 ctam trabem ponderi cedere, & frangi. |
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3753 <p>Dicet autem qui$piam, h<17>c$i vera $unt, quo gracilius |
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3754 fuerit fulcrum, eo validiùs $u$tinebit, & frangetur minus, |
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3755 quod oppido fal$um e$t. Re$pondemus, id non ex propor- |
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3756 tionum naturâ, $ed ex materiæ ip$ius infirmitate fieri. Ita |
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3757 quoque invecte non materiam, quatenus ad vim pertinet, |
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3758 $ed proportiones partium con$ideramus. Vtiumqueigi- |
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3759 turrequiritur ad fulcti validitatem proportio longitudi- |
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3760 nis ad cra$$itudinem debita, & materiæ ip$ius robur & |
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3761 fortitudo. Præterea, quoniam pondus, cuifulcrum re$i- |
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3762 $tit, vel ex natura premit, vel ex violentia, illud quidem |
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3763 per lincam perpen dicularem, quæ ad mundi cētrum, hoc |
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3764 autem lateraliter & diuer$imodè, varia fit fulcrorum di$- |
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3765 po$itio. Cuius rei $umma hæce$t, vt $emper contra impe- |
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|
3766 tum $upponantur. |
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|
3767 <fig> |
|
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3768 <p>E$to enim horizontis planum |
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3769 AB, eidē perpendiculares CADB, |
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3770 ítaque $i naturaliter pondus pre- |
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|
3771 matex C, fulcrum $upponetur AE. |
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3772 Siautem ex F ip$um GE, $i verò ex |
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3773 H, $upponaturiuxta BE. Si verò $e- |
|
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|
3774 cundum I ponderi opponatur KE. |
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|
3775 Hæc nos de arrectarijs fulcrisue; |
|
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|
3776 nunc de tran$uer$arijs, & inclinatis agemus, & primum |
|
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|
3777 de tran$uer$arijs, quatenus ad tectorum trabeationes $pe- |
|
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|
3778 ctat. |
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3779 <p>E$to tran$uer$aria trabs AB, muris vtrinq; fulta CD, |
|
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|
3780 <p n=>101</p> |
|
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|
3781 <fig> |
|
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|
3782 cuius grauitatis centrum |
|
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|
3783 E, in perpēdiculari FEG, |
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|
3784 quæ quidem ad mundi |
|
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|
3785 centrum vergit. Itaq; eo- |
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|
3786 dem tendente grauitatis |
|
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|
3787 contro, $i pondus quod |
|
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|
3788 premit in E, non præua- |
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|
3789 leat vnioni partiū ip$ius |
|
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|
3790 materiæ quæ e$t in E, re$i$tet trabs $uomet ponderi, nec |
|
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|
3791 frangetur. Si autem vel in firmitate materiæ, aut vitio, vel |
|
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|
3792 maxima exiftente proportione AF ad FE, fractio fiet in E, |
|
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|
3793 & $ecutâ partium $epaiatione duæ fient vtrin que trabes |
|
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|
3794 AH, Bl, quorum grauitatis centra KL. Erunt igitur duo |
|
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|
3795 vectes AE, BE, quorum fulcimenta MN, quamobrem $i |
|
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|
3796 proportio EM ad MH ita præualeat, vt pondus quod e $t |
|
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|
3797 in E, $uperet pondus muri O $uperimpo$iti, & item muri |
|
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|
3798 P, corruent quidem trabes, & murorum fiet hinc inde di$- |
|
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|
3799 $ipatio. Si autem non præualuerit ea, quam diximus, pro- |
|
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|
3800 portio, $u$pen$æ remanebunt vtrin que trabes vt AHBI. |
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|
3801 <p>Huic difficultati egregiè occurrunt Architecti, ali- |
|
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|
3802 quando autem hoc modo: |
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|
3803 <fig> |
|
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|
3804 <p>E$to tran$uer$aria |
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|
3805 trabs $uâ gracilitate, alia- |
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|
3806 ue de cau$$a imbecilla |
|
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|
3807 AB, muri quibus vtrinq; |
|
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|
3808 $u$tinetur CD, Trabis i- |
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|
3809 p$ius grauitatis centrum |
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|
3810 G. Itaque adpactis trabi |
|
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|
3811 lignis EF, caprcolos ad- |
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3812 dunt muro vtrinque ful- |
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|
3813 tos CE, DF, corum capita adpactis lignis admouentes EF, |
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3814 $ed & tunc validi$$ima fit colligatio, $i inter E & F capreo- |
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3815 lorum capita inte grum lignum trabi $upponatur EF. Ra- |
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|
3816 <foot>N 3</foot> |
|
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|
3817 <p n=>102</p> |
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|
3818 tio autem validitatis patet; premente enim grauitatis cē- |
|
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3819 tro in G, fulcra hincinde $uccurrunt CE, DF, quæ cum $e- |
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3820 ip$is fieri non valeant breuiora, ne corpori detur penetra- |
|
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3821 tio, re$i$tunt & robu$ti$$imè ip$i ponderi $uperimpo$ito |
|
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|
3822 contranituntur. Videntur autem in hoc opere duo con- |
|
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|
3823 $iderari vectes, GH, GB, quorum fulcimenta EF, potentia |
|
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|
3824 premens vtrinque G. Pondera autem parietum partes ca- |
|
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|
3825 pitibus trabis impo$itæ in A & B. Quoniam igitur parua |
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3826 e$t proportio GE ad EH, parua potentia premens in G, |
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3827 maximè autem pondus in A, fieri non pote$t trabem fran- |
|
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|
3828 gi aut muros vtrin que di$$ipare in AB. Po$$unt etiam to- |
|
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|
3829 tius trabis tres partes con$iderari AE, EF, FB, quarum ful- |
|
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3830 cimenta quatuor A, E, F, B, Diui$o igitur pondere & mul- |
|
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3831 tiplicatis fulcimentis impo$$ibile e$t trabem conuelli & |
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|
3832 vitium facere. |
|
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|
3833 <p>Sed & tectorum contignationes imbecilla<13>; tran$- |
|
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|
3834 uer$aria Mechanici corroborare $olent, additis nempe |
|
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|
3835 arrectaria trabe atque cauterijs. |
|
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|
3836 <fig> |
|
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|
3837 <p>E$to enim tran$- |
|
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|
3838 uer$aria trabs AB |
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|
3839 parietibus vtrinque |
|
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|
3840 fulta I, K, arrectariū |
|
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|
3841 CD. Cauterij vtrin- |
|
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|
3842 que AD, BD, ita |
|
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|
3843 tran$uer$ariæ trabi |
|
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|
3844 in AB, & arrectario |
|
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|
3845 in D in$erti, vt ne- |
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|
3846 quaquam inde ela- |
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|
3847 bi valeant. Tum ferrea fa$cia EF mediam tran$uer$ariam |
|
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|
3848 trabem AB, à parte inferiori ip$i arrectario connectens, |
|
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3849 Debet autem arrectarij pes vbi C, aliquantulum à tran$- |
|
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|
3850 uer$aria trabe di$tare, ne deor$um ex pondere vergente |
|
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|
3851 paululum arrectario ip$am tran$uer$ariam premat. His i- |
|
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|
3852 <p n=>103</p> |
|
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|
3853 gitur ita con$titutis pondus quidem tran$uer$ariæ trabis, |
|
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3854 quod $uapte naturâ premit in medio vbi C, ferrea fa$cia, |
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|
3855 arrectariæ trabi affixa di$tinetur, Arrectariam cauterij $u- |
|
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3856 $tinent, hos verò tran$uer$ariæ capita AB, quibus indun- |
|
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|
3857 tur. Tota igitur eiu$cemodi operis vis in eo con$i$tit, vt |
|
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|
3858 probè cauterij tran$uerlariæ & arrectariæ trabi in$eran- |
|
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|
3859 tur. fixis enim cauteriorum pedibus in AB, non de$cendēt |
|
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|
3860 à partibus $eu capitibus D, ijs verò $tantibus $tabit & arre- |
|
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|
3861 ctarium, quo inde $u$pen$o tran$uer$aria trabs ei ex ferrea |
|
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|
3862 fa$cia alligata nequaquam pendebit. Stabit ergo compa- |
|
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|
3863 ges tota & $uapte vi robu$ti$$imè connexa totius tecti |
|
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|
3864 pondus $u$tinebit. |
|
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|
3865 <p>Quoniam autem v$u venire $olet, cauterios nimia |
|
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|
3866 longitudine debiles, aliquando tum proprio tum extra- |
|
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|
3867 neo cedentes ponderi deor$um vergentes pandare, Ar- |
|
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|
3868 chitecti capreolis hinc inde $uppo$itis, ceu fulcris, huic |
|
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|
3869 medentur infirmitati. |
|
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|
3870 <fig> |
|
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|
3871 <p>Sint enim cauterij |
|
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|
3872 debiles hinc inde AB, |
|
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|
3873 AC, media trabs arre- |
|
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|
3874 ctaria, quam Monachū |
|
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|
3875 dicimus AD. Cauterio- |
|
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|
3876 rum mediæ partes E, F, |
|
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|
3877 in punctis igitur EF, vtpote maximè ab extremis di$tanti- |
|
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|
3878 bus debiles cauterij val de laborant. Itaque $uppo$itis v- |
|
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|
3879 trin que arrectariolis EH, Fl, eorum capitibus E, F, duos |
|
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|
3880 cauteriolos $ibi ip$is ad pedem arrectarij in D, re$i$tentes |
|
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|
3881 apponunt. quibus ita con$titutis nec E, nec F ad partes H, |
|
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|
3882 I, de$cendere valent. Capiatur enim inter EH, quoduis |
|
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|
3883 punctum G, & BG, DG, connectantur, erunt autem BG, |
|
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|
3884 DG ip$is BE ED breuiores ex 21. primi elem. Tuncigitur |
|
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|
3885 punctum E fiet in G cum BE, ED fient in BG, DG, quod |
|
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|
3886 non cedentibus B, D, & $ibi ip$is breuioribus factis parti- |
|
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|
3887 <p n=>104</p> |
|
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|
3888 bus BE, ED, pror$us e$t impo$$ibile. $tabuntigitur in co- |
|
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|
3889 rum rectitudine cauterij AB, AC, nec pandabunt, quod |
|
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|
3890 fieri querebatur. |
|
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|
3891 <p>Hîc autem damnandi veniunt ij, quitran$uer$ariæ |
|
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|
3892 quidem trabis capitibus cauteriorum pedes non in$erūt, |
|
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|
3893 $ed ea vice tran$uer$ariolo quodam medios cauterios v- |
|
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|
3894 trin que connectunt ad in$tar elementi A, quam compa- |
|
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|
3895 gem, capram, appellant. Sint enim cauterij hinc inde AB, |
|
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|
3896 AC, quorum medias partes connectit tran$uer$ariolum, |
|
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|
3897 DE. Dico igitur colligationem i$tam magnopere impro- |
|
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|
3898 bandam. Sunt enim AB, AC vectes, quorum commune |
|
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|
3899 fulcimentum A, potentiæ hinc inde diuaricantes B, C, |
|
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|
3900 pondera inter fulcimentum & potentias DE. quoniami- |
|
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|
3901 gitur vt DH ad AB, ita potentia in B, ad pondus in D, par- |
|
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|
3902 ua quidem potentia, pondus in D di$trahet & $uperabit: |
|
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|
3903 facillima<13>; in de fiet tran$uer$ariolì à capreolis ip$is vtrin- |
|
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|
3904 que reuul$io: Et quoniam centrum quidem e$t A, fact, in |
|
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|
3905 D, E, parua diuaricatione, maxima fit in BC, vtpote parti- |
|
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|
3906 bus ab ip$o centro A quam remotis. Calcitrant igitur li- |
|
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|
3907 beri prope cauteriorum pedes, & muros ip$os $ummos, |
|
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|
3908 non $ine magno operis totius vitio, $ua calcitratione pro- |
|
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|
3909 pellunt. |
|
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|
3910 <p>Hæc nos de trabeationibus, modò ad fornicum ca- |
|
22d6a63640c6
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|
3911 merarum<13>; naturam $tilum transferemus; id enim $uadet |
|
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|
3912 vtilitas, imo & nece$$itas ip$a. Pauci enim ante nos hæc |
|
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|
3913 tractarunt, & $anè his probè non cognitis aut neglectis, |
|
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|
3914 Architecti fabri<13>ue ingentes per$æpe incurrunt, & inex- |
|
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|
3915 plicabiles difficultates. Dicimus igitur primò, coctiles la- |
|
22d6a63640c6
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|
3916 teres, & non cuneatos lapides ad rectam lineam di$po$i. |
|
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|
3917 tos, non $tare. |
|
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|
3918 <p>Sint enim muri vtrinque AC, BD. Ducatur hori- |
|
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|
3919 zonti æquidi$tans CD, iuxta quam lateres lapide$ue non |
|
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|
3920 cuneati, $eriatim collocentur EF. Dicimus amoto arma- |
|
22d6a63640c6
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|
3921 <p n=>105</p> |
|
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|
3922 <fig> |
|
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|
3923 mento, hoc e$t, pro- |
|
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|
3924 hibente ip$o lateres |
|
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|
3925 ruere. Producantur |
|
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|
3926 enim AC in G, BD |
|
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|
3927 verò in H, cum ip$is |
|
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|
3928 CG, DH, æquales |
|
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|
3929 fiant CI, DK, & recta |
|
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|
3930 IK iungatur, crit igi- |
|
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|
3931 tur GD $patium ip$i |
|
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|
3932 CK $patio $imile qui- |
|
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|
3933 dem & æquale, quod |
|
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|
3934 cùm ita $it, nihil prohibet quin tota laterum GD moles in |
|
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|
3935 $patium CK transferatur, & corruat. |
|
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|
3936 <p>Si autem cunei ip$i latere$ue, cuneatim di$po$iti, ita |
|
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|
3937 $int vt ad vnum centrum tendant, licet ad rectam lineam |
|
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|
3938 collocentur, non delabentur, $ed $tabunt; quod ita o$ten- |
|
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|
3939 demus. |
|
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|
3940 <fig> |
|
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|
3941 <p>Sint cunci latere$ue |
|
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|
3942 cuneatim di$po$iti ABCD, |
|
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|
3943 tendentes ad centrum, $eu |
|
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|
3944 commune punctum E, Du- |
|
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|
3945 cantur CAE, DBE, $int<13>ue |
|
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|
3946 muri vtrinque ponderire$i- |
|
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|
3947 $tentes CL, DM, Demitta- |
|
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|
3948 tur perpendicularis, quæ ad |
|
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|
3949 mundi centrum FGE $ecans AB, in G. Tum fiat GK <17>qua- |
|
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|
3950 lis GF & per K ip $i AGB parallela ducatur, HKI claudens |
|
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|
3951 $patium AHIB. Quoniam igitur vt EC, ad EA, ita CD ad |
|
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|
3952 AB per 4. propo$. lib. 6. maior erit CD ip$a AB, & eâdem |
|
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|
3953 de cau$$a maior AB, ip$a HI, & idcirco maius ABDC $pa- |
|
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|
3954 tium, $patio AHIB. Non igitur pote$t linea CD, fieri in |
|
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|
3955 AB, neque AB, in HI, neque $patium totum CABD, tran$- |
|
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|
3956 ferri in $patium AHIB non data (quod naturæ ip$i repu- |
|
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|
3957 <foot>O</foot> |
|
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|
3958 <p n=>106</p> |
|
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|
3959 gnat) corporum penetratione. Stabunt ergo cunei, quod |
|
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|
3960 fuerat demon$trandum. |
|
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|
3961 <p>Verumenimuero, debilis hæc $tructura e$t, & eo de- |
|
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|
3962 bilior, quo vani latitudo fucrit maior, cuneorum verò al- |
|
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|
3963 titudo minor. Idem enim patitur quod epi$tylia in $pecie |
|
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|
3964 Aræos$tyla, quæ, vt$cribit Vitruuius lib. 3. c. 2. propter in- |
|
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|
3965 teruallorum magnitudinem franguntur. Id quoque ha- |
|
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|
3966 bet vitij, quod cunei ita di$po$iti $uo pondere in cumbas |
|
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|
3967 vtrinque violenti$$imè pellant. Vtilis tamen e$$c pote$t |
|
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|
3968 ad portarum & fene$trarum, quæ in medijs muris $unt, & |
|
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|
3969 mediocri vano aperiuntur, $uperliminaria. |
|
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|
3970 <p>Si verò ad minorem circuli portionem curuetur Ca- |
|
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|
3971 mera, vtilior quidem erit $tructura ea ip$a, de qua locuti |
|
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|
3972 $umus; non tamen omninò $ine vitio. |
|
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|
3973 <fig> |
|
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|
3974 <p>E$to fornix ex minori |
|
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|
3975 circuli portione AB, cuius in- |
|
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|
3976 cumbæ AF, BH muris fultæ |
|
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|
3977 AC, BD. Con$tet autem vel |
|
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|
3978 ex lapidibus cuneatis, vel ex |
|
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|
3979 coctilibus lateribus ad E cē- |
|
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|
3980 trum tendentibus. Sit<13>; for- |
|
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|
3981 nicis linea exterior FGH, in- |
|
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|
3982 terior AIB. Ducantur EA, |
|
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|
3983 ED, & producantur in M, N. |
|
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|
3984 Quoniam igitur vt EM ad EA, ita MGN ad AIB, maior e- |
|
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|
3985 rit MGN lineaip$a AIB, quamobrem fieri non pote$t vt |
|
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|
3986 aptetur lineæ AIB, & in eius locum de$cendat. Stabit igi- |
|
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|
3987 tur, in cumbis vtrinque non cedentibus. Validè autem |
|
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|
3988 $peciem hanc, loca quibus incumbit, propellere, ita o- |
|
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|
3989 $tendemus. |
|
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|
3990 <p>Producatur in eadem figura CA in K, & DB in L. |
|
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|
3991 Partes igitur quæ muris ad perpendiculum fulciuntur, |
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3992 $unt AKF, BLH, minimæ illæ quidem, maxima verò pars |
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3993 <p n=>107</p> |
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3994 e$t extra fulcimenta, nempetota AKLB quæ idcircó $uo- |
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3995 pte pondere deor$um vergens & in incumbas vtrinq; pel- |
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3996 lens aperitur, & facillimè vitium facit. Eiu$dem ferè na- |
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3997 turæ ea $pecies e$t, quæ vel ex media, vel ex minori ellip$is |
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3998 $ecundum maiorem diametrum fit $egmento. Vtilior ta- |
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3999 men hæc e$t, præcipuè circa incumbas, propterea quod |
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4000 partes habeat erectiores, & circulari illa de qua egimus, |
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4001 magis fultas. circa medium autem pote$t videri debilior, |
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4002 quippe quod ellip$isibi circulo curuetur minus. |
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4003 <p>Ea verò forma, qua mirum in modum delectati $unt |
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4004 Barbari, qui declinante imperio Italiam inua$erunt, & |
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4005 bonam emendati$$imam<13>ue antiquorum ædificandi ra- |
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4006 tionem deturparunt, ex duobus con$tat circuli portioni- |
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4007 bus, quamobrem Albertus lib. 3. ho$ce arcus, compo$itos, |
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4008 appellat. Circinantur autem hoc pacto, diui$a nempe |
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4009 $ubten$a, in partes tres, ea$que æquales, ponitur circini |
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4010 pes in altero diui$ionum puncto & pars circuli de$cribi- |
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4011 tur, mox in altero puncto circini pede collocato alia cir- |
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4012 culi portio lineatur, quibus arcus ip$e integratur. Appel- |
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4013 lant autem tertium acutum, eo quod ex $ubten$a in tres |
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4014 partes diui$a, arcus non fiatrotundus, $ed in acutum an- |
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4015 gulum ex duabus circuli portionibus de$inens. |
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|
4016 <fig> |
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4017 <p>Sint igitur muri |
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4018 AC, BD, in quibus v- |
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4019 trinque incumbæ KA, |
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4020 BI. Ducatur itaque $ub- |
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4021 ten$a horizonti æquidi- |
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4022 $tans AP, quæ in tres æ- |
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4023 quales partes diuidatur |
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4024 punctis E, F, tum centris |
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4025 EF, circulorum portio- |
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4026 nes de$cribantur hinc |
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4027 AG, HK, inde verò BG, |
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4028 <foot>O 2</foot> |
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4029 <p n=>108</p> |
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4030 IH, ex quibus arcus totus integratur. Vtilis hæc quidem |
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4031 $pecies e$t, licet inuenu$ta, propterea quod haud violen- |
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4032 ter incumbas vtrinque repellat, & in $ummo magnis $u$ti- |
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4033 nendis oneribus $it apta. Producantur CH in N, DB verò |
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4034 in O, $it<13>ue centrum grauitatis AG in L, partis vero BG |
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4035 in M. Quoniam igitur centra hæc ob elatam portionum |
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4036 con$titutionem quam proxima lineis AN, BO, fulcimen- |
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4037 torum fiunt, maximè $u$tinētur, & deor$um potius quam |
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4038 lateraliter in cumbas ip$as premunt. Si quid tamen habēt |
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4039 vitij, illud e$t quod grauitatis centra momentum haben- |
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4040 tia ad interiorem partem ver$us PQ vim faciant, & nifi |
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4041 partes magno $uperimpo$ito pondere comprimantur, |
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4042 partes quæ $unt circa HG, $ur$um pellentes aliquali $ibi |
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4043 rectitudine comparata corruunt, facta nempe circa L, M, |
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4044 coniunctarum partium $eparatione. |
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4045 <p>His hoc pacto explicatis de $emicirculari fornice a- |
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4046 gemus, quæ cæteris omnibus vtilior e$t, & longè pulcher- |
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4047 rima, quamobrem Antiquis Architectis omnibus inpri- |
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|
4048 mis admodum familiaris: |
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|
4049 <fig> |
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4050 <p>E$to vanum |
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4051 ABCD, muris v- |
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|
4052 trinque clau$um. |
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4053 Ducatur per sū- |
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4054 mitates murorū |
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4055 horizonti æqui- |
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|
4056 di$tans recta AD, |
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|
4057 hac bifariam $e- |
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|
4058 cta in E, eodem |
|
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4059 centro E, $patio |
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4060 verò EA $emicir- |
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|
4061 culus de$cribatur |
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|
4062 AFD, concaua |
|
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4063 nempeip$ius for- |
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|
4064 <p n=>109</p> |
|
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|
4065 nicis pars; tum eodem centro, $patio verò EG, circinetur |
|
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|
4066 GHI eiu$dem fornicis pars conuexa. Po$t hæc productis |
|
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4067 lineis BH, CD, in OP, $ecetur fornix tota in tres æquales |
|
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|
4068 partes AGKM, MNLK, NDIL, & KME, LNE iungantur, |
|
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|
4069 $int autem partium ip$arum grauitatis centra QRS. E$t |
|
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|
4070 autem R in ip$a perpendiculari HE. Quoniam igitut |
|
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4071 partium AGKM, DILN, quæ vtrinq; $unt grauitatis cen- |
|
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|
4072 tra QS, in ip$is $unt fulcimentorum lineis OH PD, $uâ |
|
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|
4073 $ponte fulcimentis eas $u$tinentibus partes ip$æ $tabunt. |
|
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|
4074 Pars autem media KMNL deor$um vergente perip$am |
|
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|
4075 HE lineam grauitatis centro, $i parumper vel incumbæ |
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4076 vel partes vtrinque AG<I>K</I>M, DILN cedant, vtpote quæ à |
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4077 fulcimentis e$t remoti$$ima, magno impetu $uopte pon- |
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4078 dere deor$um feretur. quæ igitur in his $emicircularibus |
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4079 fornicibus partes $tabiliores $int, quæ verò ca$ibus obno- |
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4080 xiæ, ex his quæ diximus, clarè patet. |
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4081 <p>Cæterùm cur incumbis manentibus fornix $tet, ea |
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4082 cau$$a e$t, quod partes exteriores G<I>K</I>, <I>K</I>L, LI, maiores $int |
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4083 in ferioribus & oppo$itis AM, MN, NG; quod $uprà de- |
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4084 mon$trauimus. |
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4085 <p>Si quid autem vitij in hac $pecie e$t, illud quidem |
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4086 e$t, quod $ummapars <I>K</I>MNL deor$um vergens magnâ vi |
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4087 partes, quæ vtrinque $unt, repellat, ex qua re $olidarum |
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4088 partium fit $olutio, & inde ruina. |
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4089 <p>Huic difficultati vt occurrerent peritiores Archite- |
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4090 cti, plura excogitârunt remedia. Primum enim parietes |
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4091 hinc inde ita $olidos, cra$$os & firmos faciunt, vt $uapte vi |
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4092 re$i$tentes dimoueri loco nequeant, vel para$tatas addūt |
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4093 vtin figura TX, VY. Præterea & ferrea claui ex incumba |
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4094 in incumbam ducta & vtrinque firmata contrarias partes |
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4095 validi$$imè connectunt, quæ calcitrantes (ita enim lo- |
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4096 quuntur no$trates <I>A</I>rchitecti,) fornicis pedes cohibent, & |
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4097 $olidum ne $oluatur impediunt. qua in $pecie dubitan dū |
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4098 <foot>O 3</foot> |
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4099 <p n=>110</p> |
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4100 e$$et, an optimo loco $it a $it clauis, quæ per centrum? Et |
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4101 $anè videtur, quippe quod circa incumbas impetus fiat |
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4102 maior. Ego autem vtiliusibi poni arbitror, vbi punctaq. |
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4103 5. hoc e$t, in medio tertiarum illarum partium, quæ vtrin- |
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4104 que incumbis in$i$tunt, propterea quod primus impul$us |
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4105 ex media parte quæ impendet, ibi fiat. Rarò tamen boni |
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4106 Architecti eo loco aptare $olent, eo quòd eiu$modi cla- |
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4107 ues vel pulcherrimis ædifi cijs minuant gratiam. Vnde fit |
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4108 vt nunquam $atis laudetur Lucianus ille Benuerardus |
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4109 Lauranen$is Dalmata, qui nullibi apparentes eas po$uit |
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4110 in admirabili illa Vrbini Aula, quam Federico Feltrio, fe- |
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4111 lici$$imo æquè & inuicti$$imo Duci, ædificauit. |
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4112 <p>Tertio denique modo huic infirmitati me dentur, |
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4113 vt videre e$t in $equenti figura, in qua vanum ADBC, mu- |
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4114 ri vtrinque AF, BH, fornix verò FGH. Itaque dum muros |
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4115 <fig> |
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4116 ex$truunt, arre- |
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4117 ctarias trabes, ro- |
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4118 bore aliaue mate- |
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4119 ria firmi$$ima, illis |
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4120 in$erunt, quales |
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4121 $unt IF<I>K</I> LHM, |
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4122 ea proceritate vt |
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4123 futuri fornicis $u- |
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4124 perent $ummita- |
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4125 tem. Con$umma- |
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4126 to enim fornice, |
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4127 nondum tamen, |
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4128 exarmato, tran$- |
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4129 uer$ariam trabē à |
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4130 $ummo fornicis |
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4131 dor$o parumper |
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4132 eminentem in punctis I, L, arrectarijs trabibus validi$$i- |
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4133 mis clauibus connectunt, tum punctis NP, Oq, capreolos |
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4134 <p n=>111</p> |
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4135 tran$uer$ario, & arrectarijs ferreis, clauis affigunt. Qui- |
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4136 bus ita concinnatis, facta fornicis validâ pre$$ione in G, |
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4137 in cumbi$<13>ue F, H, ad exteriora repul$is, AB $patium non |
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4138 fit maius. Repul$is enim in cumbis & muros propelli ne- |
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4139 ce$$e e$t, & cum muris ip$as in$ertas trabes, I<I>K</I>, LM. At va- |
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4140 ricari non po$$unt, nî $ecum trahant puncta PQ, quod fie- |
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4141 ri non pote$t, propterea quod in punctis N, O, validè di$- |
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4142 tineantur. Itaque $patio AB non dilatato nulla fit ip$ius |
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4143 fornicis di$$olutio, quod vtique à principio ceu propo$i- |
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4144 tus finis quærebatur. Sed dicet qui$piam, Nonne pende- |
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4145 bit tran$uer$aria trabs in ip$a di$tractione arrectariorum, |
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4146 pre$$a in punctis N, O? aut parum dicimus, aut nihil. Cum |
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4147 enim PQ proxima $int punctis FH, quæ cum arrectarijs à |
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4148 muro di$tinentur, magna in ijs fit vtrobique re$i$tentia. |
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4149 <p>Rebus igitur ita $e habentibus cum ob$erua$$ent Ar- |
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4150 chitecti, ob enormitatem ponderis fornices in tertia illa |
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4151 <fig> |
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4152 parte quæ $umma e$t |
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4153 laborare, quãtum ter- |
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4154 tijs vtrinque partibus |
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4155 $oliditatis addunt, tan- |
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4156 tundem ex illa parte |
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4157 $uprema demere $olēt, |
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4158 vt videre e$t in $ubie- |
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4159 cta figura, in qua par- |
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4160 tes A, B, $olidæ & cra$- |
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4161 $iores, quibus hærent |
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4162 partes, quæ CE, DG |
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4163 cra$$æ quidem & illæ, |
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4164 tum vero $umma EFG, |
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4165 alijs $ubtilior. Minus |
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4166 igitur grauante ponde- |
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4167 re in F, minor fit ad in cumbas pre$$io, aut $i qua fit, à partiū |
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4168 ACE, BDG $oliditate haud inualidè $u$tinetur. |
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4169 <p n=>112</p> |
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4170 <p>Cæterùm admonet nos locus, vt aliquid de forni- |
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4171 cum di$$olutionibus in medium afferamus: cau$$is enim |
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4172 morborum cognitis, facilius periti medici adhibere $o- |
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4173 lent remedia. |
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4174 <fig> |
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4175 <p>E$to enim $emicircula- |
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4176 ris fornix ABC, cuius cen- |
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4177 trum E, perpendicularis ve- |
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4178 rò quæ per centrum DBE, $e- |
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4179 micirculi ABC, diameter |
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4180 AEC, incumbæ vtrinq; A, C. |
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4181 Itaque $i nulla fiat incumba- |
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4182 rum repul$io, $tabit fornix; $i verò fiat, ruinam facict. |
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4183 <p>Pellanturitaque ad exteriores partes, vt in $ecunda |
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4184 <fig> |
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4185 figura, H in F, & C in G, |
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4186 ex qua pul$ione cum ma- |
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4187 ius fiat $patium quod in- |
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4188 tegro fornice impleba- |
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4189 tur, iam di$tractis vtrinq; |
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4190 fornicis partibus nō im- |
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4191 pletur, Diuiditur igitur |
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4192 locus maior factus in tres partes, quarum hincinde duas |
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4193 replent fornicis partes, tertiam verò quæ media e$t, re- |
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4194 plet in$ertus, ne vacuum detur, aër, vt in figura videre e$t, |
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4195 in qua $olutæ vtrinque fornicis partes HIKF, PMNG, aër |
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4196 autem medius $patium replens IKMN. Diuidantur $in- |
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4197 guli qua drantes FK, GN, in partes tres, quarum duæ $int |
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4198 hincinde FQ, GR, & à centris, quæ $eparatis quadranti- |
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4199 bus facta $untin ST, rectæ ducantur SQV. TRX. Quo- |
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4200 niam igitur tertiæ partes vtrinque VIKQ MNRX pro- |
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4201 pria grauitate depre$$æ, nullum quo $u$tineantur fulci- |
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4202 mentum habent, corruent quidem. Ducantur autem re- |
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4203 ctæ QI, RM, con$tituentes cum ip$is QV, RX pares an- |
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4204 gulos VQI MRX. Itaque centris QR partes QIRM ad |
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4205 <p n=>113</p> |
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4206 inferiores partes deuoluentur, fient<13>ue QI, RM, vbi QZ, |
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4207 RZ. Siautem QI, RM perpendicularibus quæ à punctis |
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4208 QR ad perpendicularem DE ducuntur, fuerint maiores |
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4209 conuenient alicubi in ip$a perpendiculari, & altera alte- |
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4210 ram $u$tinebit; $i autem æquales tangent $e & nihilomi- |
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4211 nus fiet ruina, $i minores nec $e inuicem tangent, & nullà |
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4212 re prohibente deor$um corruent. tangant tautem $e in pū- |
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4213 cto Z. quo pacto igitur fornices incumbis cedentibus in |
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4214 medio aperti, di$$oluãtur & ruinam faciant, ex i$tis patet. |
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4215 <p>Ex demon$tratis qua$i ex con$ectario habemus for- |
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4216 nices quo fuerint cra$$iores dato pari in cum barum $ece$- |
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4217 $u, ruinæ minus e$$e obnoxios quàm tenuiores, hoc e$t, |
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4218 maiori aperitione indigere ad ruinam cra$$iores quam te- |
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4219 nuiores, quod licet ex iam dictis re$ultet, nos tamen cla- |
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4220 rius ex $ubiecto $chemate demon$trabimus. |
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4221 <fig> |
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4222 <p>E$to enim cra$$ioris |
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4223 fornicis pars quidē ABCD, |
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4224 tenuioris EFCD circa idē |
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4225 centrum R. Ducatur au- |
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4226 tem RM, $ecans CD in G. |
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4227 EF in H AB, in M. Centro |
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4228 igitur G fiet euer$io portio- |
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4229 num fornicum, MD, HD, |
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4230 Ducantur GA, GE & producta AD in N ip$i AN perpen- |
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4231 dicularis ducatur GN. quoniam igitur GE cadit in trian- |
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4232 gulo AGN erit ex 21. propo$. lib. 1. elem. GA, maior GE. |
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4233 Corruente igitur maioris fornicis portione MD, recta |
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4234 GA centro G punctum A de$cribet portionem AI, mino- |
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4235 ris interim ex GE, de$cribente EL, at cadenti angulo A |
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4236 occurrit in perpendiculari IK in puncto I angulus oppo- |
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4237 $itæ portionis, O, ip$i autem E cadenti per EL non occur- |
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4238 ret punctum P, cadens per Pq eo quod neutrum eorum |
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4239 pertingat ad perpendicularem IK. Tenuioris ergo forni- |
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|
4240 <foot>P</foot> |
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4241 <p n=>114</p> |
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4242 cis partes è $uis locis auul$æ ex eadem aperitione ruinam |
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4243 facient, quod non contingit partibus cra$$ioris. quod $a- |
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4244 nè fuerat de clarandum. |
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4245 <p>Quæritur adhuc, quare grauiores fornices in $um- |
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4246 mis ædificijs non $ine vitio fiant? |
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4247 <p>E$to ædificium ABGH, cuius vtrinq; muri ABCD, |
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4248 EFGH, maiorum $ummitates AD, EH, mediæ murorum |
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4249 partes KL, fornicum $ummus quidem DIE, medius verò |
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4250 <fig> |
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4251 KML. Dico, magis cedere pul- |
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4252 $os muros $ummos circa DE, |
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4253 quam in medio circa KL. Sunt |
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4254 enim muri BA, GH ceu vectes |
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4255 quidam, quorū extremis par- |
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4256 tibus à fulcimentis BG remo- |
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4257 ti$$imis potentia admouetur, |
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4258 hoc e$t, ip$ius fornicis DIE ad |
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4259 DE in cumbans repul$io; lon- |
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4260 gior e$t autem pars à fulcimē- |
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4261 to ad potentiam AB, ip$a BK. |
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4262 Data igitur paritate potentia- |
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4263 rum plus operabitur ea quæ in |
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4264 D, illa quæ K. facilius crgo re- |
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4265 pellentur muri in DE quàm in |
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4266 KL. Alia quo que ratio intercedit, $iquidem pondus muri |
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4267 $uperioris ADK, premens inferiorem murum KBC, cum |
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4268 $ua grauitate firmiorem, & pul$ionibus minus obnoxium |
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4269 reddit. Difficilius enim propellitur id quod graue e$t quã |
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4270 quod leue, vt nos quæ$tione 10. demon$trauimus. |
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4271 <HEAD>QVÆSTIO XVII.</HEAD> |
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4272 <HEAD><I>Quærit Ari$toteles, Cur paruo exi$tente cuneo magna $cindantur |
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4273 pondera & corporum moles, valida<13>, fiat impre$$io?</I></HEAD> |
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4274 <p>In parua re magnum negotium. Etenim quæ$tio hæc |
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4275 <p n=>115</p> |
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4276 clari$$imorum virorum ingenia magnopere fatigauit. Ex |
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4277 quibus Ari$toteles inter veteres, Guid. Vbald. inter re- |
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4278 centiores ad vectis naturam (ne quid in Mechanicis ad |
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4279 vectem non reduci putaretur) cuneum ip$um trahere co- |
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|
4280 <fig> |
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4281 nati $unt. Nos autem pro |
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4282 veritate certantes, $i in |
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4283 horum $ententiam vltrò |
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4284 non tran$ierimus, multa |
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4285 venia digni à non iniquo |
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4286 iudice exi$timabimur. A- |
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4287 ri$totelis mentem clarè |
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4288 & fusè explicat G. V- |
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4289 bald. in Mechan. vbi de |
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4290 Cuneo peculiariter a- |
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|
4291 git. |
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4292 <p>E$to igitur $cindendum quippiam ABCD, Cuneus |
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4293 EFG, cuius pars HFI $ci$$uræ in$erta HI, facta igitur vali- |
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4294 da percu$$ione in EG, fiet vt cum EG fuerit in NO, H $it v- |
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4295 bi N, A vbi P, itemque I vbi O, D verò vbi Q & facta erit |
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4296 $ci$$io NSO, toti nempe cuneo EFG, æqualis. Vultigitur |
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4297 Ari$toteles, duos in cunco vectes con$iderari EF, GF, quo- |
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4298 rum alterius, nempe EF, fulcimentum $it in H, pondus ve- |
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4299 ro in F; alterius autem, hoc e$t, GF fulcimentum quidem |
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4300 $it in I, pondus verò itidem $it in F. His nequaquam con- |
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4301 $entiens G. Vbald. aliam viam ingreditur. Ait enim EHF |
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4302 vectes quidem e$$e, quorum commune fulcimentum F, |
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4303 potentias verò mouentes in EG. Pondera vtrinque inter |
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4304 fulcimenta & potentias, vbi HI, idem<13>; e$$e ac $i EF, GF, |
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4305 <*>eor$um à cuneo con$iderati in puncto F, adinuicem fulti |
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4306 atque di$tracti pondera pellerent H in NP, I verò in O, |
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4307 Q. Verumenimuerò quoniam cunei angulus non muta- |
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4308 tur, nec vertex ip$e centri vllum pror$us præbet v$um, nec |
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4309 eius latera vtrinque di$tracta ad contrarias partes didu- |
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|
4310 <foot>P <I>2</I></foot> |
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4311 <p n=>116</p> |
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4312 cuntur, vectes in cuneo hoc pacto con$iderare videtur à |
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4313 veritate alienum. Ari$totelis autem $olutionem fal$am e$- |
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4314 $e, clarè patet. quo pacto enim F pellet ex fulcimento Hi- |
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4315 p$am ligni partem OS, & idem F ex fulcimento I pellet |
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4316 oppo$itam partem NS, $i inuicem contendentes extremæ |
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4317 vectium partes in F, altera alteri ne quicquam operentur, |
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4318 e$t impedimento? Et $anè opinionis fal$itas inde patet, |
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4319 quòd videamus materiæ partes $ci$$as, in ip$o $ei$$ionis a- |
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4320 ctu facta di$tractione à cunei vertice nequaquam tangi. |
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4321 At eiu$modi operationes per contactum fieri nulli e$t i- |
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4322 gnotum. Solutio igitur i$ta mco iudicio, tanto Philo$o- |
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|
4323 pho pror$us videtur indigna. |
|
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4324 <p>Porrò G. Vbald. ijs quæ de diuaricatis vectibus in |
|
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|
4325 medium adduxerat non acquie$cens alias quærit cau$$as, |
|
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4326 cur cuneus minoris anguli validiùs $cindat. Id<13>; ex quo- |
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4327 dam lemmate demon$trare conatur, figura autem eius ita |
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4328 ferè $e habet. |
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4329 <fig> |
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4330 <p>E$to cuneus ABC, |
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4331 item alius DEF. Demō- |
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4332 $trauit igitur ex a$$um- |
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4333 pto, quo acutior fuerit |
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4334 angulus BIM, eo facilius |
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4335 pondera moueri, & ideo |
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4336 facilius ceu vecte AB |
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4337 moueri pondus I quàm |
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4338 vecte DE pondus Q. In- |
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4339 geniosè quidem. At ma- |
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4340 gnam hæc apud me ha<*> |
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4341 bent difficultatem. Si e- |
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4342 nim ita $e habet AB, ad BI, vt DE, ad EQ (ip$æ enim DE, |
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4343 EQ $upponuntur æquales) ergo eadem æquali$ue poten- |
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4344 tia æqualiter mouebit pondera I & Q. quod ip$i eiu$dem |
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4345 demon$trationi pror$us concludit contrarium. Nec meo |
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4346 <p n=>117</p> |
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4347 quidem iudicio id $equi videtur, propterea quod ex Pap- |
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4348 po ea quæ in planis inclinatis mouentur, redigantur ad li- |
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4349 bram. Ratio enim valde e$t diuer$a, $iquidem pondera |
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4350 quæ in planis inclinatis mouentur, certa habent fulci- |
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4351 menta & determinatas tum brachiorum tum ponderum |
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4352 proportiones, quæ omnia in cuneo, nec quidem mente |
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4353 concipi po$$e, clarè patet. |
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4354 <p>His igitur difficultatibus con$ideratis, Nos cunei |
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4355 vim, ad alia e$$e principia referendam pro comperto ha- |
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4356 bemus. Ordimur igitur hoc pacto. Cuneo quidem res di- |
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4357 uidi certum e$t. Cæ terùm quæ natura diuidere apta $unt, |
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4358 tria $unt, punctum, linea, $uperficies, Puncto enim linea, |
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4359 lineâ $uperficies, $uperficie autem corpus ip$um diuidi- |
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4360 tur. quæ omnia à Mathematico ab$que materia con$ide- |
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4361 rantur. De diui$ione autem quæ fit ex puncto, nihil agit |
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4362 Mechanicus, qui corporibus quidem vtitur, ad cuius na- |
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4363 turam non trahitur punctum, cuius partes $unt nullæ. At |
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4364 non lineis & $uperficiebus modò corpora diuiduntur, $ed |
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4365 etiam corporibus, quod verum e$t, at ea corpora ad linea- |
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4366 rum & $uperficierum naturam quodammodo aptari faci- |
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4367 lè docebimus. Dicimus igitur, duplicem e$$e Cuneorum |
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4368 $peciem, linearem vnam, $uperficialem alteram. linearem |
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4369 appello, quæ ad lineæ naturam magnopere accedit. Tales |
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4370 $unt orbiculares illæ cu$pides, quibus ad perforandum v- |
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4371 timur, & ideo vernaculè Pantirolos vocamus. Acus item |
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4372 $utorij, & cætera quæ nen $ecus ac linea in punctum de$i- |
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4373 nunt, & imagina<*>iam quandam lineam ceu axem in eo |
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4374 puncto de$inen<*>em continent. Ad lineam quo que refe- |
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4375 runtur lateratæ cu$pides oblongæ, & $ubtiles ceu$ubulæ, |
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4376 claui, en$es, pugiones, & his $imilia, quæ cum adacta vali- |
|
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4377 dam faciant partium $eparationem ad cunei naturam nō |
|
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|
4378 referre magnæ videretur dementiæ. Ettunc quantoma- |
|
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4379 gis corpora hæc ad linearem naturam accedunt, eo ma- |
|
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|
4380 <foot>P 3</foot> |
|
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|
4381 <p n=>118</p> |
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|
4382 gis penetrant. Sed & hocidem in rebus non ab arte, $ed |
|
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4383 ab ip$anatura productis facile e$t cogno$cere. Quis enim |
|
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4384 non experitur, quàm validè culex, infirmi$$imum animal, |
|
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4385 & ea paruitate qua e$t, hominum & cæterorum animaliū, |
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4386 cutes aculeata probo$cide penetret? Id vtique non alia de |
|
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4387 cau$$a fit, quod ad imaginariæ lineæ $ubtilitatem quam, |
|
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4388 proximè accedat. Ve$pæ quoque, Apes, Scorpiones a- |
|
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4389 culeis i$tis ceu linearibus cuneis vtuntur. Nec refert, vt |
|
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4390 diximus, vt um laterati $int, ceu $ubulæ, & claui, vel ro- |
|
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4391 tundi & vtrum plura paucioraue latera habeant, dummo- |
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4392 do in punctum & aculeatam aciem de$inant. Altera por- |
|
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4393 ro cuneorum $pecies $uperficiei naturam $apit, acie $iqui- |
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4394 dem in lineam de$init, quæ $uperficiei e$t terminus, quã. |
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4395 obrem huc ea omnia referuntur, quæ acie ipsâ $cindunt, |
|
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4396 ceu $unt cunei propriè dicti, de quibus hoc loco e$t $er- |
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4397 mo, cultra, en$es, a$ciæ, $ecures, $calpra lata, & cætera e- |
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4398 in$modi, quibus corpora acie $cinduntur. Quidam his ad- |
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4399 dunt $erras, quibus haud pror$us a$$entimur. Etenim alia |
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4400 ratione diuidunt, $icut & limæ $olent, deterendo enim, nō |
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4401 $cindendo ferri, ligni, & marmorum duritiem diuidunt & |
|
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4402 domant. His igitur cō$ideratis, $i daretur ex materia qua- |
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4403 piam in frangibili cuneus, qui maximè ad $uperfi ciei natu- |
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4404 ram accederet, vel paruo labore tenaci$$ima ligna validi$- |
|
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4405 $imè $cinderet, & ideo optimè res gladijs illis diuiditur, |
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4406 qui magis ad $uperficiei naturam accedunt. Ex quibus o- |
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4407 mnibus, nî fallimur, clarè patet, curacutiores angulo cu- |
|
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4408 nei obtu$ioribus facilius $cindant, quæ quidem ratio lon- |
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4409 gè ab ea di$tat, ex qua cæteri ferè omnes Cuneum ad ve- |
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|
4410 ctis naturam referre hactenus contenderunt. |
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4411 <p>Cæterùm vtramque eorum quos diximus, cuneorū |
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4412 $peciem $olerti$$ima cognouit Natura, & ideo quoniam |
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4413 res vel contu$ione vel perforatione, vel $ecatione con$i- |
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4414 ciuntur, triplicem dentium qualitatem dentatis animali- |
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4415 <p n=>119</p> |
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|
4416 <fig> |
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|
4417 bus dedit, Molares, |
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4418 qui & Maxillares ap- |
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|
4419 pellantur, quibus |
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4420 cibus contunditur, |
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|
4421 Canini, quibus fit |
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4422 perforatio, Anterio- |
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4423 res, quibus cibus |
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4424 $cinditur, quos ideo |
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4425 <G>temnikou\s</G>, id e$t, $ecan- |
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|
4426 tes appellant Gr<17>ci. |
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|
4427 <p>Molares KK, |
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|
4428 CaniniL, L, Temni- |
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4429 ci $eu $ecantes M. Cuneus orbicularis lineari$<13>ue AB, in |
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4430 quo axis linea e$t, ad cuius naturam accedit AB cuneus |
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4431 $uperficialis CD, accedens ad $uperficiei naturam, quam |
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4432 vitro imaginamur EFGD, in aciem cunei de$inentem, |
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4433 GD, Lateratus lineari$que cuneus, clauus HI. |
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4434 <p>Cunei autem omnes dupliciter $unt efficaces, vel e- |
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4435 nim malleo, vt in ijs fit, quibus lìgna $cin duntur & $calpris |
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4436 fieri $olet, adiguntur, vel impul$u & pre$$ione, vt in gla- |
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4437 dijs fit, pugionibus, cælatorum $calpris, $ubulis, & cæteris |
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4438 eiu$modi. Quidam etiam $unt, qui licet mallei ictu non |
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4439 adigantur, malleum coniunctum habent, ceu $unt $ecu- |
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4440 res, ligones, A$ciæ, & his $imilia, quæ ex percu$$ione $e- |
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4441 metip$a $cindendis rebus in$erunt & validè penetrant. |
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4442 De vi autem & efficacia ictus $eu percu$$ionis hic $uper- |
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4443 $ed emus aliquid, ea de re, in $equenti quæ$tione verba fa- |
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|
4444 cturi. |
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4445 <p>Multa hîc addere potui$$emus ad Cochleam $pe- |
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4446 ctantia, quippe quòd Cochlea cuneus $it Cylindro inuo- |
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4447 lutus, qui quidem ad mallei, $ed vectis virtute $ibi adiun- |
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4448 ctâ, validi$$imè operatur, & $excentis in$eruit v$ibus. Ve- |
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4449 runtamen cùm de hac $<29>ecie egregiè di$$erat G. Vbaldus, |
|
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|
4450 <p n=>120</p> |
|
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4451 con$ultò hanc di$putationem omittimus; idque hac quo- |
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|
4452 que de cau$$a, quod nihil de cochlea, ac $i eam non noui$- |
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|
4453 $et, locutus $it Ari$toteles. |
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4454 <p>Po$$umus autem in actu $ci$$ionis, quæ cuneo fit, a- |
|
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|
4455 liâ tamen ratione vectem con$iderare, nempe non in cu- |
|
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|
4456 neo quidem, $ed in ip$a re quæ $cinditur. |
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|
4457 <fig> |
|
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|
4458 <p>E$to enim quip- |
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|
4459 piam $ci$$ile ABCD, |
|
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|
4460 cui alteri extremita- |
|
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|
4461 tum, puta BD, cuneus |
|
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|
4462 adigatur EFG, fiatq; |
|
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|
4463 $ci$$io per longitudi- |
|
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|
4464 nem $ecundum lineã |
|
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|
4465 EH. facta igitur ex |
|
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|
4466 cunei ingre$$u partiū $eparatione B, expelletur in I, D ve- |
|
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|
4467 rò in K. fient igitur materiæ $ci$$æ partes AIBH, CKDH, |
|
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|
4468 ceu duo vectes, quorum hinc inde in corpore ip$o fulci- |
|
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|
4469 menta L, M potentiæ vtrinque dilatantes BD, pondus ve- |
|
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|
4470 rò materiæ re$i$tentia, in $eparationis loco vbi N. Duca- |
|
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|
4471 tur NL, quanto itaque BN maiorem habebit proportio- |
|
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|
4472 nem ad LN, eo faciliùs re$i$tentia quæ in N, $uperabitur. |
|
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|
4473 Mutatur autē a$$iduè in ip$a $ci$$ione fulcimentum, & cū |
|
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|
4474 fulcimento ip$a proportio. Pertingente enim $ci$$ione in |
|
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|
4475 O, fulcimētum fit in P. quo ca$u $ci$$ura e$t facilior, quip- |
|
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|
4476 pe quod maiorem habeat proportionem BO ad OP, quã |
|
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|
4477 BN ad NL. Hoc autem experiuntur materiarij, qui primis |
|
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|
4478 ictibus, $ecuriculâ nondum probè adactâ, & nondum fa- |
|
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|
4479 ctâ notabili $ci$$ione difficultatem $entiunt, mox factaiã |
|
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|
4480 $eparatione faoillima paullatim fit materiæ totius $epara- |
|
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|
4481 tio. Hocidem & nos ab$que cunei v$u experimur, cum ba- |
|
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|
4482 culum aut quippiam tale manibus diductis $cin dimus. à |
|
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|
4483 principio enim difficultatem $entimus, deinde ex ea quã |
|
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|
4484 diximus proportionc $ci$$io ip$a fit apprime facilis. Vti- |
|
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|
4485 <p n=>121</p> |
|
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|
4486 mur etiam vecte cuneato ad $cindendum & aperiendum: |
|
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|
4487 adacto enim $ci$$uræ cuneo, id<13>ue manu malleoue, tum |
|
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|
4488 ab altera extremitate pre$$o, valida fit ex vectis vi cōtinui |
|
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|
4489 <fig> |
|
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4490 corporis $eparatio. Ma- |
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4491 teria $ci$$ilis AB $calprū |
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4492 ceu vectis cuneatus CD, |
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4493 cuius fulcimentum, E, |
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4494 pondus verò vbi C, po- |
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4495 tentia vbi D, quo ca$u |
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4496 quo maior e$t proportio |
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4497 DE ad EC, eo e$t ip$a $ci$$io leuior & facilior. |
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4498 <HEAD>QV AESTIO XVIII.</HEAD> |
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4499 <HEAD><I>Quærit hic Ari$toteles, Cur per Trochleas ab exiguapotentia in- |
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4500 gentia moueantur pondera?</I></HEAD> |
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4501 <p>De Trochlea Pappus, & veteres: inter recentiores e- |
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4502 gregiè admodum, vt omnia examinauit in Mechani- |
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4503 cis G. Vbaldus. Nos tamen interim po$t clari$$imos illos |
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4504 viros aliquid quod nouitatem & $ubtilitatem $apiat, de |
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4505 no$tro penu promemus. Et $anè inuentis quidem addere |
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4506 res e$t fa cilis, at quod inuentis addas inuenire haud adeo |
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4507 facile. Sed nos primum Philo$ophi ip$ius dicta ad trutinã |
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4508 reuocemus. Ita autem quæ$tionem proponit; Cur $i qui$- |
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4509 piam Trochleas componens duas, in $ignis duobus, ad $e |
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4510 inuicem iunctis contrario ad Trochleas modo circulo fu- |
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4511 nem circumduxerit, cuius alterum quidem caput tigno- |
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4512 rum appendatur alteri, alterum verò Trochleis $it innixū |
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4513 & à funis initio trahere cœperit, magna trahit pondera, li- |
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4514 cetimbecillium fuerit virium? |
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4515 <p>Ob$euri$$ima expo$itio, & nî res e$$et vulgò per $e |
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4516 nota, de<13>ue ea Vitruuius & Mechanici non egi$$ent, diffi- |
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4517 cile vtique e$$et ex eius verbis $en$um a$$equi. |
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4518 <foot>Q</foot> |
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4519 <p n=>122</p> |
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4520 <p>Tigna $anè voca$$e videtur ea ligna, quæ à Vitruuio |
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4521 Rechami dicuntur, in quibus nempe ip$i in$eruntur orbi- |
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4522 culi. Et$i de tignis eiu$mo di aliud quippiam $entire videa- |
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4523 tur Picolomineus. Græea lectio pro tignis habet <G>cu/la</G>, id |
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4524 e$t, ligna; item vbi Leoniceni ver$io legit, ad $e inuicem |
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4525 iunctis, textus habet <G>snm<*>ai/nousin e(autoi_s e)ranti/ws</G>, hoc e$t, in- |
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4526 uicem ex oppo$ito concurrunt. Certè locum totum ita |
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4527 redderem: Cur $i quis duas Trochleas fecerit, in duobus |
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4528 lignis $ibi ex oppo$ito concurrentibus, ei$<13>ue Trochleis |
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4529 circumpo$uerit funem, cuius alterum caput alteri ligno- |
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4530 rum $it annexum, alterum verò Trochleis cohæreat, vel |
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4531 apponatur. Si quis alterum funis principium trahat, ma- |
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4532 gna trahat pondera, et$i trahens potentia $it exigua? Nos |
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4533 verbis figuram, & figurâ verba ip$a elucidabimus. |
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4534 <fig> |
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4535 <p>Sint duo ligna ex oppo$ito concurrentia, |
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4536 in quibus Trochleæ, hoc e$t, orbiculi AB, fu- |
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4537 nis ductarius DABC, cuius alterum caput re- |
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4538 ligatum e$t ligno trochleæ A, vbi e$t C. Tro- |
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4539 chlea A loco $tabili commendata, vbi E. Pon- |
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4540 dus alteri ligno Trochleæ appen$um F. Tra- |
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4541 cto itaque fune DABC, eleuatur & trahitur |
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4542 pondus F. Ex quibus clarè patet, Philo$ophū |
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4543 propo$ui$$e Trochleam duobus tantum orbi- |
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4544 culis munitam, quod vtique $atis erat ad ex- |
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4545 plicationem. Inquit autem, faciliùs vecte quã |
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4546 manu pondus moueri. Trochleam vero (id |
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4547 e$t, orbiculum; ita enim e$t intelligendum) e$- |
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4548 $e vectem, aut vectis virtute operari. Ita autem |
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4549 videtur argumentari. Si vnicâ Trochleâ plus trahitur |
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4550 quàm manu, multo faci ius & velocius id fiet duobus, |
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4551 quibus plus, vt ip$e ait, quàm in duplici velocitate pon- |
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4552 dus leuabitur. Summa dictorum e$t, ex multiplicatione |
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4553 orbiculorum pondus ip$um imminui, & minori difficul- |
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4554 <p n=>123</p> |
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4555 tate leuari, quod $anè verum e$t. Nos tamen nonnulla cō- |
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4556 $iderabimus. quod ait, vecte facilius moueri pondera |
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4557 quam manu, $emper non e$t verum. Si enim vectis pars |
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4558 quæ à fulcimento ad manum breuior fuerit illâ, quæ à |
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4559 fulcimento ad pondus difficilius vecte pondus mouebi- |
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4560 tur quam manu. Idem quoque accidet, $i eo modo vecte |
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4561 vtamur, quem ob$eruat Guidus Vbald. Tract. de Vecte |
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4562 prop. 3. Po$ita nempe inter fulcimentum & pondus $u$ti- |
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4563 nente potentiâ. Præterea quod a$$eruit Ari$toteles, Tro- |
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4564 chleas ad vectem reduci, verum quidem e$t, $ed aptius di- |
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4565 xi$$et ad libram, etenim vectis vtcunque à ful cimento di- |
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4566 uiditur. Libra verò quod & orbiculis ex centro accidit, |
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4567 $emper bifariam. Ad hæc videtur ille ad orbiculorum |
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4568 multiplicitatem Trochlearum vim referre. Si enim, ait, |
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4569 vnicâ Trochleâ pondus facile trahitur, id multo validius |
|
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4570 pluribus fiet. Veruntamen non ab$olutè ex orbiculorum |
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|
4571 multiplicationeid fieriita o$tendemus. |
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4572 <fig> |
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4573 <p>Sint duæ op- |
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4574 po$itæ line<17> rect<17>, |
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4575 vtpote trabes AB, |
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4576 CD, inuicē æqui- |
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4577 di$tantes & ip$æ |
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4578 $tabiles: $uperiori |
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4579 tres appendantur |
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4580 orbiculi ex pūctis |
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4581 E, F, G, nēpe ML, |
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4582 PQ, TV. inferiori |
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4583 autē duobus pun- |
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4584 ctis IH, nempe |
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4585 NO, RS. Erunti- |
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4586 gitur invniuer$um |
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4587 quinque, indatur pereos funis ductarius KLMNOP |
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4588 QRSTVX, ex cuius extremitate pendeat pondus X, |
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4589 <foot>Q <I>2</I></foot> |
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4590 <p n=>124</p> |
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4591 Trahatur funis in K. Dico ex multiplicatione orbiculorū, |
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4592 trahentipondus nequaquam minui. Sint autem orbicu- |
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4593 lorum diametri, LM, NO, PQ, RS, TV, applicetur poten- |
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4594 tîa in S. Erit igitur ad hoc vt $u$tineat æqualis ponderi X, |
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4595 orbiculi enim TV $emidiametri $unt æquales. Transfe- |
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4596 ratur potētia in q, & ita deinceps donec perueniatur in K, |
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4597 vbi funis ip$ius e$t principium, Idem e$t igitur $eruata $em- |
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4598 per $emidiametrorum æqualitate ac $i potentia quæ e$t in |
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4599 K, applicata intelligatur in T vel in V. vbicunque enim |
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4600 collocetur, ponderi erit æqualis. Nihil igitur rebus ita |
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4601 di$po$itis, orbiculorum multiplicatio ad facilitatem ope- |
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4602 ratur. Alia itaque ratio quærenda e$t, quam non $atis ex- |
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4603 plica$$e videtur Ari$toteles. Probabimus autem, nullam |
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4604 ex $uperioribus orbiculis fieri ponderum imminutionem, |
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4605 $ed totam vim in inferioribus con$i$tere. At nos interim |
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4606 quippiam quod ad rem faciat, proponamus. |
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4607 <fig> |
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4608 <p>E$to punctum A, cuirectæ ap- |
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4609 pendantur lineæ BAC, diui$æ qui- |
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4610 dem in A, $it autem lineæ BA caput |
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4611 B, ip$ius verò CA caput C. Modò |
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4612 intelligantur vnitæ in A, $it<13>ue vni- |
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4613 ca linea à puncto A ceu funiculus |
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4614 dependens BAC; Appendatur capi- |
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4615 ti B pondus B. Capiti vero C, pōdus |
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4616 C, inter $e æqualia. Potentia igitur |
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4617 in A, duo $u$tinebit pondera BC. |
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4618 Pondera verò ex æqualitate æque- |
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4619 ponderabunt. Quod $i B potentia |
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4620 dicatur $u$tinens pondus C, aut C |
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4621 potentia $u$tinens pondus D, vel |
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4622 duæ potentiæ inter $e æquales, nihil |
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4623 refert. Vtcunque enim id $it, fiet æquilibrium. Habemus |
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4624 igitur ex i$tis ad $u$tinendum pondus ex $uperiori parte |
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4625 <p n=>125</p> |
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4626 appen$um potentiam requiri ip$i ponderi æqualem. Ani- |
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4627 mo po$thæc concipiatur alia recta linea DEF, cuius inte- |
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4628 gra longitudo $i exten deretur, e$$et DE, EF. Appendatur |
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4629 in E pondus E æ quale alteri ponderum B vel, C, $int autem |
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4630 duæ potentiæ pondus E $u$tinentes D, F. Vtraque igitur |
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4631 dimidium $u$tinebit ponderis E, $ed potentia quæ $u$ti- |
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4632 nebat pondus B, in C erat ip$i B æqualis, vbi appen$io pon- |
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4633 deris erat in $uperiori parte in A, hîc autem, vbi appen$io |
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4634 e$t in parte in feriori, vtraque potentia dimidium $u$tinet |
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4635 appen$i ponderis. Videmus igitur illam appen$ionem |
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4636 quidem pondus nullatenus imminuere, hanc verò pon- |
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4637 dus ip$um, bifariam diui$um, $u$tinentibus potentijs im- |
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4638 partiri. Hæ cin lineis, Mathematicâ v$i ab$tractione, con- |
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4639 $iderauimus, nunc verò eadem mechanicè perpenda- |
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4640 mus. |
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4641 <fig> |
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4642 <p>Sit igitur |
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4643 punctum A, vt |
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4644 in $e quenti figu- |
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4645 ra clauus paxil- |
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4646 lu$ue, cui appen- |
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4647 $us funiculus |
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4648 BAC, & funicu- |
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4649 li capitibus pon- |
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4650 dera BC, $it quo- |
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4651 que anulus D, |
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4652 per quem traìe- |
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4653 ctus funiculus |
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4654 EDF. Anulo au- |
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4655 tem cōiunctum |
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4656 pondus G. His igiturita con$titutis, eadem demon$tra- |
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4657 buntur quæ $uperius, nempe oportere vt fiat æquilibrium |
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4658 B, C, e$$e æqualia, tum potentias, quæ $unt in EF pondus |
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4659 G inter eas diui$um $u$tinere. Porrò volentes Mechanici |
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4660 <foot>Q 3</foot> |
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4661 <p n=>126</p> |
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4662 funiculos circa paxillum, & anulum ad attollenda & de- |
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4663 primenda pondera mouere incommodè illis vtique $uc- |
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4664 cedebat, clauo & anulo motum difficilem facientibus. |
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4665 Quamobrem vt difficultati occurrerent, ad locum claui |
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4666 clauo ip$i orbiculum circumpo$uerunt, & anuli itidem |
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4667 loco orbiculum aptauerunt. Hæc autem agentes reii- |
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4668 p$ius naturam non mutauerunt, $ed $ibi, vt diximus, ex or- |
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4669 biculis maximam commoditatem atq; facilitatem com- |
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4670 parârunt. |
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4671 <p>Ex his principîjs tota Trochlearum ratio pendet, |
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4672 quæ tamen alia quoque con$ideratione in idem tenden- |
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4673 te examinari pote$t, quod quidem fecere veteres, & ip$e, |
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4674 qui veteres optim è imitatus e$t, Guid. Vbaldus. |
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4675 <p>Vidimus vtique nos, à potentia quæ e$t in B, pondus |
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4676 par $u$tineri in C, Potentiam autem quæ e$t in E dimidiū |
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4677 $u$tinere ponderis quod e$t in G. Nos igiturij$dem in$i- |
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4678 $tentes adiecta libra, vecteue, bifariam diui$o rem ip$am |
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4679 ex $ubiecto diagrammate lucidiorem faciemus. |
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4680 <p>E$to linea quædam $tabilis ceu trabs horizontiæ- |
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4681 quedi$tans AB, cui in A funiculus annectatur AC, cuius |
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4682 extremum C vecti cuidam alligetur CD, in medio diui$o |
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4683 vbi E, tum alteri vectis eiu$dem extremitati D, funiculus |
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4684 nectatur DG, & à puncto E pondus appendatur F. puta li- |
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4685 brarum mille, Tum puncto G in medio vectis HI, funis re- |
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4686 ligetur DG, & ex altero vectis extremo alligato fune HK |
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4687 commendetur lo co $tabili in K, & ab alio capite vectis vbi |
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4688 Iad medium vectis MN, vbi L, funis annectatur lL, tum |
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4689 ex vectis capite M, funis commendetur MO, loco $tabili |
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4690 in O, & alteri capiti N, funis, NP, qui alligetur medio ve- |
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4691 cti QR in P, & ex Q, funis QS. Commendetur loco $tabili |
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4692 in S, & alteri vectis extremo R funis alligetur RT, cui |
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4693 quidem potentia $u$tinens applicetur in T. Dico igitur, |
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4694 <p n=>127</p> |
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|
4695 <fig> |
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4696 rebus ita di$po$itis, |
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4697 potentiam in T ita |
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4698 $e habere ad pondus |
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4699 F, vt vnum ad $ex de- |
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4700 cim, hoc e$t, in pro- |
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|
4701 portione e$$e $ub- |
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4702 $exdecupla. Sunt |
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4703 autem, hic vectes |
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|
4704 quatuor in feriorum |
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|
4705 cubiculorum, loco, |
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|
4706 CD, HI, MN, QR, |
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|
4707 qucrum, centra E, |
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4708 G, L, P. quoniam e- |
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4709 nim A hoc e$t, C, v- |
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|
4710 nà cum potentia G, |
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4711 hoc e$t, D, $u$tinet |
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4712 pondus F alterum, |
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4713 ponderis dimidium |
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4714 $u$tinebit C, alterū |
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4715 vero D. erunt igitur |
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4716 vtrinque libr<17> quin- |
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4717 gentæ. Tum potentia in K, hoc e$t, in H, vna cum poten- |
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4718 tia in L, hoc e$t, in I $u$tinebunt quingenta. Quare vtraq; |
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4719 ducenta quin quaginta, $ed hoc totum bifariam diuiditur |
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4720 inter potentias, O, id e$t, M, & P, id e$t H. erunt igitur v- |
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4721 trinque centum viginti quinque. Ea autem $umma iterū |
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4722 bifariam diuìditur, hoc e$t, inter potentias S, id e$t, Q & |
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4723 T, id e$t, R, quare vtraque $u$tinet $exaginta duo cum di- |
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4724 midio. Sed numerus i$te ad Millenarium ita $e habet vt v- |
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4725 num ad $exdecim. Hinc colligimus, pondus totum inter |
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4726 loca $tabilia diuidi, nempe A, K, O, S, & ip$am potentiam |
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4727 quæ $u$tinet in T, & locis ip$is $tabilibus quindecim par- |
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4728 tes integri ponderis, potentia verò T $extam decimam |
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4729 <p n=>128</p> |
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4730 tantùm commendari. Itaque $i ex puncto V appendere- |
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4731 tur AB, in X potentia, quæ in X $u$tineret mille, minus |
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4732 $exaginta duo cum dimidio, quod quidem à potentia in |
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4733 T $u$tinetur; quod $i alius adderetur orbiculus, & fierent |
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4734 quinque, potentia in T $u$tiner et trige$imam $ecundam |
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4735 partem integri ponderis, hoce$t, dimidium librarum $e- |
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4736 vaginta duarum cum dimidio, nempe triginta & vnam |
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4737 cum quarta parte, $i item textus adderetur, potentia in T |
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4738 $exage$imam partem $u$tineret integri ponderis, hoc e$t, |
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4739 libras quindecim & <*> libræ vnius. Vnde patet clarè pon- |
|
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4740 deris diminutionem fieri ex orbiculis inferioribus, non |
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4741 autem ex$uperioribus, $uperiores autem addi non nece$- |
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4742 $itatis quidem, $ed commoditatis gratiâ: neque enim ab$- |
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4743 que $uperioribus vnico ductario fune fieri po$$et attractio |
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4744 & ponderis ip$ius eleuatio. Hactenus igitur nobis i$thæc |
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4745 de Trochleæ natura & vi po$t alios, con$idera$$e $it $atis. |
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4746 <HEAD>QVÆSTIO XIX.</HEAD> |
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4747 <HEAD><I>Dubitat Philo$ophus, Cur $i quis $uper lignum magnam imponat |
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4748 $ecurim, de$uper<13> magnum adijciat pondus, ligni quippiam quod |
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4749 cur andum $it, non diuidit; $i verò $ecurim extollens percutiat, illud |
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4750 $cindit, cum alioquin multo minus habeat ponder is id quod |
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4751 percutit, quam illud quod$uperiacet |
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4752 & premit?</I></HEAD> |
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4753 <p>Poterat Ari$toteles, nî fallimur, rem breuius & vniuer- |
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4754 $alius proponere. Scilicet cur motus ponderi addat |
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4755 pondus & efficacius ex motu quam ex immoto pondere |
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4756 mota res operetur. Soluitautem. An, inquiens, ideo fit, |
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4757 quia omnia cum motu fiunt, & graue ip$um grauitatis ma- |
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4758 gis a$$umit motum, dum mouetur quam dum quie$cit? |
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4759 Incumbens igitur connatam graui motionem non moue- |
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4760 tur, motum verò & $ecundum hanc mouetur & $ecun- |
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4761 <p n=>129</p> |
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4762 dum eam quæ e$t percutiētis? Hæc præclarè quidem, cæ- |
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4763 tera autem, quæ de cuneoiterat, nempe ad vectem eiuslo- |
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4764 perationem referri $uperius confutauimus. Porrò effe- |
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4765 ctus huius, de quo agitur, di$putatio illuc $pectat, videli- |
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4766 cet ad cadentium atque proiectorum naturam. Ad maio- |
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4767 rem autem rei euidentiam hæc addimus. |
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4768 <fig> |
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4769 <p>E$to libra AB, cu- |
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4770 ius centrum C, libra- |
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4771 ta æqualibus ponde- |
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4772 ribus DE, apponatur |
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4773 ponderi E pondus F, |
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4774 item ponderi D pon- |
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4775 dus G ip$i ponderi F |
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4776 æquale, æquilibrabit |
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4777 itidem, Modò non apponatur $impliciter pondus G $ex |
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4778 ex H in lancem A dimittatur, tunc $anè non æquilibrabit, |
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4779 $ed libram deprimet. Duo enim in pondere dimi$$o con- |
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4780 $iderantur pondera; naturale $cilicet, & quod motu ip$i |
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4781 moto, ponderi e$t acqui$itum. Itaque quo motus fuerit |
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4782 maior, puta $i cadat ex I, grauitas ex maiori motu fiet ma- |
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4783 ior. quod vtique efficacius fieret $i pondus G non dimit- |
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4784 tetur modo remoto prohibente, $ed proijceretur. Tunc |
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4785 enim tria concurrerent, grauitas naturalis, grauitas ac- |
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4786 qui$ita ex naturali motu, & ea quæ naturali adij citur ex |
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4787 violentia. Pondus igitur $ecuri impo$itum & $ecuris ip$ius |
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4788 naturalis grauitas naturali tantum grauitate operantur, |
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4789 & ideo minus efficaciter. Hucautem ea ferè pertinent |
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4790 quæ nos à principio de duobus centris retulimus, natura- |
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4791 lis nempe grauitatis, & acqui$itæ. |
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4792 <p>Cæterùm cur mallei & $ecuris ictus $it violenti$$i- |
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4793 mus, ideo fit quod non ex vnico neque duplici, $ed ex tri- |
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4794 plici grauitate operetur. E$to enim $ecuris A, cuius manu- |
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4795 brium AB, brachium vero $ecuri vtentis BC, erit igitur C |
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4796 <foot>R</foot> |
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4797 <p n=>130</p> |
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4798 <fig> |
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4799 locus vbi humero |
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4800 brachium iungi- |
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4801 tur, motus ip$ius |
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4802 centrum, attollit |
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4803 autem $ecurim is |
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4804 qui percutit, & re- |
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4805 tro ad $capulas re- |
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4806 ducens totis viri- |
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4807 bus ex centro C |
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4808 $ecurim vibrat, |
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4809 portionem circuli |
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4810 de$cribens ADE |
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4811 ictum<13>ue faciens |
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4812 in E. Vires igitur acquirit $ecuris, tum ex naturali grauita- |
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4813 te, cadens ex D, in E, tum ex proprio pondere, tum etiam |
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4814 ex violentia eidem à percutiente impre$$a. Fiunt autem |
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4815 motus tam naturalis quàm violentus eo validiores, quo |
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4816 maius e$t $patium, quo res mota mouetur, id<13>ue pr<17>cipuè |
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4817 cum violentia ip$am $ecundat naturam. Itaque maior fit |
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4818 ictus in E quàm in F, & in F maior quàm in D. Item violen- |
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4819 tius feriret percutiens, $imanubrium e$$et longius, puta |
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4820 BG. Tunc enim maior e$$et circulus GH, & motus tum |
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4821 prolixior, tum velocior. quo igitur longiora habet bra- |
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4822 chia is qui $ecuri malleoue vtitur, data virium paritate, ex |
|
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4823 eadem ratione validius percellit. E$t autem $ecuris, vel |
|
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4824 malleus cuneatus, vel cuneus malleatus manubrio in$er- |
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4825 tus. An autem operetur efficacius cuneus malleo percu$- |
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4826 $us, aut cum manubrio motus, vt fit in $ecuri, data aciei & |
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4827 ponderis æqualitate, difficile e$t determinare. Certè va- |
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4828 lidius, & certius fieri $ci$$ionem ex cuneo & malleo, ea ra- |
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4829 tio e$t, quod cuneus adactus, nec inde remotus eam inte |
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4830 rim $eruat, quam antea fecerat partium $eparationem, |
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4831 <p n=>131</p> |
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4832 quod quidem $ecuri non accidit, quæ adacta ad nouam |
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4833 percu$$ionem faciendam extrahitur. |
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4834 <p>Hoc etiam con$ideramus, $ecuris in circulo motum, |
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4835 ex A in D, e$$e videndum, id e$t, non $ecundum naturam, |
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4836 $ur$um enim fertur quod e$t graue, ex D verq in F mixtū: |
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4837 magis autem ad naturalem accedere qui fit ex F in E. Tar- |
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4838 dior ergo ex A in D, velocior ex D, in F, veloci$$imus ex F |
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4839 in E; quæ dam quæ ad hanc rem faciunt, egregiè con$ide- |
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4840 rat Guid, Vbald. in calce Tractatus, De Cuneo; ip$um |
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4841 con$ule. |
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4842 <p>Ad hæc $uccurrit nobis pulcherrima quæ$tio. Du- |
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4843 bitari enim pote$t, vtrumictus ex en$e e$ficacior $it à par- |
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4844 te quæ e$t circa aciem, aut circa medium en$em, vel pro- |
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4845 pe manubrium capulumue; etenim hinc inde $unt ra- |
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4846 tiones. |
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4847 <p>E$to quidem en$is AB, cuius capulus A, $piculum ve |
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4848 rò B, centrum grauitatis C, pars capulo proxima D. Libra- |
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4849 to itaque gladio tres fiunt circulorum portiones BE, CF, |
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4850 DG, quæritur quo loco ictus $it validior, nempe in E, in F, |
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4851 velin G. Videtur validiorem futurum in E, quippe quod |
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4852 ex maiori $emidiametro AB, maioris $it circuli portio BE, |
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4853 & ideo velociormotus ex B in E. Contra efficaciorem |
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4854 futurum apparet in F, propterea quod ibi ex centro C to- |
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4855 tius fiat grauitatis impre$$io, fieri autem validi$$imum in |
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4856 G, licetibi motus $it tardior inde videtur, quod $icon$ide- |
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4857 retur en$is vt vectis, cuius fulcim entum e$t A, potentia |
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4858 premens in B, ponderis vero loco re$i$tentia rei quæ per- |
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4859 cutitur in D. Maior e$t autem proportio BA, ad AD, quam |
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4860 BA ad AC, & ideo violentior fiet pre$$io ex ictu in D, quã |
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4861 in C. Hi$ce hoc pacto con$ideratis, putarem ictum effica- |
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4862 ciorem fieri in F ex medio C, quam ex extremis & oppo- |
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4863 $itis partibus EG. Licet enim in B velocitas $it maior, dee$t |
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4864 ibi pondus. Si enim en$is iterum vt vectis con$ideretur, e- |
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4865 <foot>R 2</foot> |
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4866 <p n=>132</p> |
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4867 runt AB. duo fulcimenta $u$tinentía pondus in C, vbi gra- |
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4868 uitatis e$t centrum. Si igitur paria fuerint $patia BC, CA, |
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4869 <fig> |
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|
4870 in B erit dìmidium |
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4871 ponderis C, quantum |
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4872 ergo velocitate præ- |
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4873 ualetictus in B, tantū |
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4874 ponderis amittit. D |
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|
4875 verò plus quidem de |
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|
4876 pondere participat, |
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4877 $ed velocitatis habet |
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4878 minimum, in C verò |
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4879 velocitas e$t medio- |
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4880 cris, tota tamen ip$ius |
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4881 ex grauitatis centro |
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4882 ponderis fit impre$- |
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4883 $io. |
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4884 <p>Quidam, quod huc pertinet, vt ex acieip$a quæ lon- |
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4885 gius à capulo abe$t, violenti$$imum facerent ictum, Ar- |
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4886 gentum viuum, quod $ui naturâ graui$$imum quidem e$t |
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4887 & mobili$$imum in canali à manubrio ad verticem exca- |
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4888 uato infundunt, quo in gladij de$cen$u ad verticem velo- |
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4889 ci$$imè delato illuc transfert grauitatem totam, quare |
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4890 tum velocitate tum grauitate concurrentibus ictus fit |
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4891 violenti$$imus & longè validi$$imus. |
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4892 <HEAD>QVAESTIO XX.</HEAD> |
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4893 <HEAD><I>Dubitatur, Cur $tatera qua carnes ponderantur, paruo appendicu- |
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4894 lo, magnatrutinet onera, cum alioqui tota, dimidiata exi$tat |
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4895 libra, altera vero parte $ola$it |
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4896 $tatera?</I></HEAD> |
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4897 <p>Soluit Philo$ophus, inquiens, $tateram $imul, & vectem |
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4898 e$$e & libram, ip$ius verò libræ centra $eu fulcimenta |
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4899 <p n=>133</p> |
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4900 e$$e ibi vbi fit $u$pen$io. Pondera verò hinc in de in lance |
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4901 & appendiculo, loco $cilicet æquipondij, appendiculo |
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4902 $uccedente. Reducit autem demon$trationem ad ea quæ |
|
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4903 $tatuit ip$e Mechanica principia; nem pe ad circulum & |
|
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4904 circuli virtutem. Ait igitur, appendiculum licet parui pō- |
|
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4905 deris $it, ideo maiori ponderi virtute æquari, quod lon- |
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4906 gius à centro, hoc e$t, ab ip$o fulcimento $i$tatur. quic- |
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4907 quid tamen $it, $tateram e$$e vectem, res e$t explorati$- |
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4908 $ima. |
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4909 <fig> |
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4910 <p>E$to igitur $tatera AB, |
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4911 cuius appendiculum cur- |
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4912 rens F, fulcimentum cen- |
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4913 trumue C, lanx quæ cate- |
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4914 na $u$penditur E $patium |
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4915 à loco fulcimenti ad ap- |
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4916 pendiculum CF. quod ve- |
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4917 rò à fulcimento ad cate- |
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4918 nam, ex qua lanx appen- |
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4919 ditur AC. Intelligatur autem & aliud fulcimentum D, $it- |
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4920 <13>ue maius $pacium AD, quam AC. Porrò ita $e habeat |
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4921 pondus in E ad appendiculi F pondus, vt CF $patium, ad |
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4922 $patium AC, quo ca$u $eruata, permutatim, ponderum & |
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4923 brachiorum proportione, fiet <17> quilibrium. Si autem pon- |
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4924 deribus ita con$titutis iterum $u$pendatur in D, non fiet |
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4925 æquilibrium, propterea quod minor $it proportio DF ad |
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4926 DA, ea quæ e$t FC ad CA. Minor ergo e$t proportio FD |
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4927 ad DA, quam ponderis E ad pondus F, & idcirco facta |
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4928 $u$pen$ione præualebit pondus E ponderi F. Ita que vt it e- |
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4929 rum fiat æquilibrium, nece$$e e$t iterū proportiones bra- |
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4930 chiorum $eu $patiorum proportionibus ponderum æqua- |
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4931 re. Transferatur igitur (lancis interim immoto pondeie) |
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4932 ip$um appendiculum in B, fiatque vt FC ad CA, ita BD ad |
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4933 DA. Stabitautem iterum $tatera ad eam redacta quam |
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4934 <foot>R 3</foot> |
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4935 <p n=>134</p> |
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4936 diximus brachiorum & ponderum permutatam propor- |
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4937 tionem. |
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4938 <p>Nos $tateris vtimur ex duplici fulcimento, altero |
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4939 propiori, altero à lance $eu loco, vbi lanx appenditur, re- |
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4940 motiori, illa grauiora appendimus pondera, & non per |
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4941 vncias & libras, $ed per libras tantum & $elibra ponde- |
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4942 ramus; & hoc $tateræ latus eo quod minus minutè $it di- |
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4943 ui$um; vulgo no$trates Gro$$um, hoc e$t, rude & cra$$um |
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4944 appellant. Aliud verò, cum fulcimentum e$t loco appen- |
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4945 $ionis lancis vicinius, & per libras, $elibras & vncias diui- |
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4946 ditur, quo quidem minora appendimus pondera, cò quod |
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4947 exqui$itiorē contineat diui$ionem, $ubtile dicunt. Rectè |
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4948 igitur dicebat Philo$ophus, in $tatera plures e$$e libras, |
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4949 quanquam & ea quoque de cau$$a dici po$$it, quod, quot |
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4950 $unt appen diculi, è locoin locum translationes, totidem |
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4951 ex proportionum variatione fiant libræ. Et hoc quidem |
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4952 $en$i$$e videtur Ari$toteles. |
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4953 <fig> |
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4954 <p>Po$$emus & alio |
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4955 modo $tatera vti, nempe |
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4956 $tabili appendiculo, mo- |
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4957 bili autem fulcimento. |
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4958 E$to enim $tatera AB, |
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4959 cuius lanx C appen$a in |
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4960 A, appendiculum verò |
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4961 $tabile D, appen$um in |
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4962 B, Apponatur ip$i l&acedil;nci |
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4963 C, pondus E. Vnicum ergo fiet corpus CEABD con$tans |
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4964 ex lance, libra & ponderibus. Habet ergo hoc totum gra- |
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4965 uitatis $uæ centrum, quod quidem vbi $it e$t ignotum. Ex |
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4966 illo autem inuento $i corpus totum appendatur, partes æ- |
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4967 queponderabunt. Appendatur autem, puta in G, $it autē |
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4968 grauitatis centrum in H. Quoniam igitur He$t extra ful- |
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4969 cimentum G, declinabit $tateræ pars GA, centro G per |
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4970 <p n=>135</p> |
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4971 circuli portionem Hl, à centro grauitatis in ip$a de$cen- |
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4972 $ione de$criptam. Siautem grauitatis centrum fuerit vbi |
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4973 K, eo quodibi quoque $it extra fulcimentum G, de$cen- |
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4974 detpars GB, de$cribente interim grauitatis centro K, cir- |
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4975 culi portionem KL. ltaque $i $tateram totam eum ponde- |
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4976 ribus trahamus pellamu$q; vltro citro<13>;, immoto appen- |
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4977 diculo eritaliquando fulcimentum in ea linea perpendi- |
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4978 culari velloco ip$o, vbi e$t grauitatis centrum, quo ca$u |
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4979 $tatera$tabit, & tuncita erit diui$a, vt fiat brachiorum & |
|
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4980 ponderum eadem ratio, ordine permutato. Hicautem |
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4981 modusideo non e$t in v$u, quod mole$tum $it libram $eu |
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4982 $tateram cum ponderibus vltro citro<13>ue transferre, quæ |
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4983 difficultas commodè appendiculi mobilitate vitatur. |
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4984 <HEAD>QVAESTIO XXI.</HEAD> |
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4985 <HEAD><I>Quæritur, Cur facilius dentes extrahunt Chirurgi, denti forcipis |
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4986 onere adiecto, quam $i$ola manu vtantur?</I></HEAD> |
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4987 <p>Re$pondet Philo$ophus, An quia ex manu, magis quam |
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4988 ex dentiforcipe lubrius elabitur dens? An ferro id po- |
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4989 tius accidit quam digitis, quoniam vndique dentem non |
|
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4990 comprehendunt, quod mollis facit digitorum caro; ad- |
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4991 hæret enim & complectitur magis. Hæc $ecunda ratio |
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4992 videtur primam de$truere, & contrarium pror$us $enten- |
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4993 tiæ, quæ in problemate proponitur, a$$erere. Si Græca ad |
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4994 verbum reddas ita habent: An magisip$a manu labile e$t |
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4995 ferrum, & ip$um vndique (dentemnempe) non comple- |
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4996 ctitur, caro autem digitorum cum mollis $it, adhæret ma- |
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4997 gis, & vndique congruit. Certè vt $ententia non $it con- |
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4998 traria propo$itioni, Græca ver$io ita videtur concinnan- |
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4999 da: Vel magis è m n<*>bitur, mollis enim e$t digitorum |
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5000 caro, ferrum autem circumplectitur, & h<17>ret magis. quic- |
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5001 quid $it, Græcam lectionem contrarium ei quod quæri- |
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5002 <p n=>136</p> |
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5003 tur, affirmare certum e$t. Picolomineus, Ideo, inquit, di- |
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5004 gitorum caro mollis minus aptè extrahit, quod dentem |
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5005 totum comprehendere non pote$t, quod ferrum ob $uam |
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5006 durítiem & con$tantiam commodi$$imè facit. Sen$um ex |
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5007 mente reddidit, quod ex verbis non poterat. Subiungit |
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5008 denique Ari$toteles, An quia dentiforcipes $int duo con- |
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5009 trarij vectes vnicum habentes fulcimentum, ip$am $cili- |
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5010 cet in $trumenti partium connexionem. Hoc igitur ad ex- |
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5011 tractionem vtuntur^{**}, vt facilius moueant. Figuram hoc |
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5012 pactto proponit Philo$ophus. |
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5013 <fig> |
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5014 <p>E$to dentiforcipis alterum |
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5015 quidem extremum vbi A, alte- |
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5016 rum autem quod extrahit B, ve- |
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5017 ctis vbi ADF, alter vectis, vbi |
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5018 BCE, fulcimentum verò CGD |
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5019 connexio vbi G. Densautem pondus: vtroque igitur ve- |
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5020 cte B, & F $imul comprehendentes mouent, Hæcille. At- |
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5021 tamen rem ip$am $ubtilius con$iderantibus aliter videtur |
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5022 habere, acip$e a$$erat. Et$anè dentisforcipis brachia ve- |
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5023 ctes e$$e, quorum commune fulcimentum e$t in ip$o cen- |
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5024 tro vbi vertebra, nemo negauerit. Dentem autem e$$e |
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5025 pondus, ego quidem ab$olute non dixerim. Pondus autē |
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5026 hîcproprie e$t ip$a dentis durities, cuius re$i$tentia eo fa- |
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5027 cilius $uperatur, quo maior e$t proportio brachiorum à |
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5028 manu ad vertebram, ad partem illam quæ à vertebra e$t |
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5029 ad dentem. At dentis ex con$trictione fractio nihil facit |
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5030 pror$us ad extractionem: id tamen operatur brachio- |
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5031 rum longitudine dentiforceps, quod valide ex vectium |
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5032 oppo$itorum vi dentes con$tringit & extra ctioni commo- |
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5033 dum reddit & facilem. Neque enim totus Dentiforceps |
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5034 hic ceu vectis vnicus operatur, quod fit in forcipibus quas |
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5035 Tenaleas vocamus, quibus è tabulis claui reuelluntur, |
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5036 qua de re nos qu<17>$tione 6. verba fecimus. Quo pacto autē |
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5037 <p n=>137</p> |
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5038 dentis ex Dentiforcipe extractio ad vectem reducatur, |
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5039 $ubtilius e$t perpendendum, neque enim res e$t in <29>ropa- |
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5040 tulo. |
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5041 <p>Dicimus igitur, tum dentem ip$um, tum dentifor- |
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5042 cipem vectes e$$e, varia tamen ratione & $atis $ane diuer- |
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5043 $a. Dens enim fit vectis eius nempe naturæ quæ fulcimen- |
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5044 tum habet in angulo, quo ca$u ip$ius Dentiforcipis partiū, |
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5045 quibus Dens apprehenditur, ea quæ longior e$t poten- |
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5046 tiæ mouentis loco $uccedit, breuior vero fulcimentum |
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5047 facit, Dentis vero re$i$tentia ponderis vices refert. |
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5048 <fig> |
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5049 <p>E$to enim dens qui- |
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5050 dem A, cuius diameter |
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5051 BC, longitudo v$que ad |
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5052 extremas radices CD, |
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5053 pars dentiforcipis breui- |
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5054 or CG, longior BG. Fit |
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5055 ergo vectis BCD, habens |
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5056 $ulcimentum in C. Den- |
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5057 teigitur apprehen$o in BC, & manu dentiforcipe ceu ve- |
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5058 cte ad inferiora compre$$o C, fit fulcimentum centrum- |
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5059 ue. Stante enim puncto C, trahente autem potentia quæ |
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5060 e$t in B, fit motus ip$ius B, per circuli portionem BE, radi- |
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5061 cis vero D, fit motus per DF, & inde ip$ius dentis extra- |
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5062 ctio facilis. Quibus con$ideratis vt rem ad proportiones |
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5063 quatenus fieri pote$t reducamus, dicimus, quo maior fu- |
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5064 erit proportio BC, ad CD, hoc e$t, partis vectis, quæ à ful- |
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5065 cimento ad potentiam ad eam quæ à fulcimento e$t ad |
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5066 pondus, eo facilius fieri dentis auul$ionem, quod vtique |
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5067 demon$trandum fuerat. |
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5068 <p>Porro quod in calce quæ$tionis addit Philo$ophus, |
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5069 Dentes commotos facilius manu extrahi quam in$tru- |
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5070 mento, nulla ratione probat. Ego autem arbitror, huc |
|
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5071 pertinere ea verba, quæ $uperius habentur, videlicet fer- |
|
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|
5072 <foot>S</foot> |
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5073 <p n=>138</p> |
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5074 rum quidem non vndique dentem comprehēdere, quod |
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5075 mollisfacit digitorum caro, quæ id circo adhæret & <*>om- |
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5076 plectitur magis. An autem ita $it, alij videant, nobis enim |
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5077 digito rem o$tendi$$e fuerit $atis. |
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5078 <HEAD>QVÆSTIO XXII.</HEAD> |
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5079 <HEAD><I>Hîc quærit Ari$toteles, Cur nuces ab$que ictu facile confringuntur |
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5080 in$trumentis quæ ad eum faciunt v$um, & hoc licet multum aufe- |
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5081 ratur virium, ce$$ante motu & violentia, quod accidit dum mal- |
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5082 leo confringuntur. Addit præterea, citius fieri confractionem |
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5083 graui, & duro in$trumento ferreo vide- |
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5084 licet quàm ligneo.</I></HEAD> |
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5085 <p>Soluit, inquiens, id fieri quod in$trumentum duobus |
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5086 vectibus con$tet, coëuntibus in connexione $eu verte- |
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5087 bra, & idcirco eo violentius fieri confractionem, quo mi- |
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5088 nus e$t $patium à nuce, quæ frangitur, ad vertebram. ma- |
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5089 ius verò quod à vertebra ad extremitates, quæ confrin- |
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5090 gentis manu comprimuntur. Ait igitur, & id quam oppo- |
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5091 fite, vim ex vectibus ictus loco $uccedere & idem operari. |
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5092 <fig> |
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5093 <p>E$to igitur in $trumentum, |
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5094 de quo agimus CDBF, ex duo- |
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5095 bus vectibus con$tans, quorum |
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5096 alter CAF, alter vero DAB ver- |
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5097 tebra $eu connexio A locus v- |
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5098 bi nux frangitur K, manubria |
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5099 vero BF. quo igitur prolixiores |
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5100 erunt AB, AF, breuiores vero ACAD, violentius fiet cō- |
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5101 fractio. Erit autem nucis re$i$tentia loco ponderis A, ful- |
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5102 cimentum BF loco potentiæ. Itaque nî maior $it propor- |
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5103 tio potentiæ ad re$i$tentiam, quam brachij à potentia ad |
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5104 ful cimentum ad eam partem quæ à fulcimento e$t ad nu- |
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5105 cem, non fiet confractio. eo autem magis $up erabit, quo |
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5106 <p n=>139</p> |
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5107 maior fuerit pars vectis quæ à potentia ad fulcimentum. |
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5108 <p>Quod autem addit Ari$toteles, eo maiorem fieri |
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5109 vectium eleuationem, hoc e$t, in$trumenti aperitionem, |
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5110 quo magis nux quæ frangitur, fuerit propior fulcimento, |
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5111 hoc e$t, ip$i vertebræ, facile o$tenditur ex conuer$a 21. |
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5112 propo$. lib. 1. Elem. $i enim ab extremitatibus vnius line æ |
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5113 ad ea$dem partes con$tituantur duæ line æ maiores con- |
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5114 currentes in angulo, & ab ij$dem extremitatibus duæ a- |
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5115 liæ minores, quæ intra triangulum à maioribus con$titu- |
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5116 tum cadant, maiorem angulum continebunt. At talis e$t |
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5117 angulus qui fit in in $trumento, cum partes vectis à verte- |
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5118 bra adnucem fuerint breuiores. magìs ergo dilatantur |
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5119 vectes, & magis dilatati magis comprimuntur, magis au- |
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5120 tem compre$li validius frangunt, quod dixerat Ari$to- |
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5121 teles. |
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5122 <p>Cæterum & illud quod $cribit, ex grauiori & durio- |
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5123 ri materia in$trumentum citius fractionem facere, quam |
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5124 ex leuiori & minus dura, ex parte quidem materiæ verum |
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5125 e$t, nec pertinet ad proportionem, quæ $ane in huiu$modī |
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5126 in$trumentis formæ ferè habent rationem. Nos hi$ce in- |
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5127 $trumentisnon vtimur. Sunt autem $imilia in$trumentis |
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5128 illis, quibus figuli cretaceas pilas ad chirobali$tarum v$um |
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5129 facere & efformare con$ueuerunt. |
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5130 <HEAD>QVÆSTIO XXIII.</HEAD> |
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5131 <p>Pvlcherrimam proponit hoc loco Philo$ophus con- |
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5132 templationem, eamque ad mixtos motus pertinētem. |
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5133 Mixtorum autem motuum $peculationem antiquis Me- |
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5134 chanicis fui$$e tum vtilem tum etiam familiarem, norunt |
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5135 ij qui norunt quæ de lineis $piralibus Helici$ue, cy$$oidi- |
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5136 bus, conchoidibus & alijs eiu$cemo di $cripta & contem- |
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5137 plata reperiuntur, quibus tum ad duarum mediarum pro- |
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5138 <foot>S 2</foot> |
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5139 <p n=>140</p> |
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5140 portionalium inuentionem, tum ad circuli quadratio- |
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5141 nem vti$olent. Quod autem hîc quærit Ari$toteles, ita $e |
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5142 habet. |
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5143 <HEAD><I>Cur $i duo extrema in Rhombo punct a duabus ferantur lationibus, |
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5144 haudquaquam æqualem vtrumque eorum pertran$it rectam, $ed |
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5145 multo plus alteram? Item cur quod $uper latus fertur, minus per- |
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5146 tran$eat quam ip$um latus. Illudenim diametrum pertran$ire |
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5147 certum est, hoc vero maius lat us, licet hoc vnica, illud au- |
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5148 tem duabus feratur lationibus?</I></HEAD> |
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5149 <p>Difficile hoc intellectu prima fronte, & $ane admi- |
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5150 rabile, itaque in tentam con templationem requirit. Nos |
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5151 primo cum Ari$totele, rem totam explicabimus, tum ali- |
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5152 quid forta$$e non pœnitendum no$tro de promptuario |
|
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5153 proferemus. |
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|
5154 <fig> |
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5155 <p>E$to itaque Rhombus ABCD, |
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5156 cuius latera AB, BD, DC, CA, diame- |
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5157 trorum maior AD, minor BC, $ecan- |
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5158 tes $e inuicem in puncto $eu figuræ |
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5159 centro K. Sunt autē ex ip$ius Rhom- |
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5160 binatura latera æqualia & parallela, |
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5161 Angulorum vero qui maiori diame- |
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5162 tro opponuntur, recto maiores, qui |
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5163 vero minori minores. His igitur con- |
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5164 $ideratis, intelligatur punctum A mo- |
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5165 ueri peculiari & $im plici motu, per li- |
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5166 neam AB, ab A ver$us B, & eodem tē- |
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5167 pore moueri totam lineam AB, ver$us lineam DC, hac ta- |
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5168 men lege, vt $emper eidem DC feratur parallela, & eius |
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5169 alterum extremorum feratur per AC, alterum vero per |
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5170 BD, Intelligatur etiam punctum B moueri eodem tem- |
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5171 pore proprio motu, eoque $implici, per eandem rectam |
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5172 BA, ver$us A, & cum eadem, vt dictum e$t, mota; ferri ver- |
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5173 <p n=>141</p> |
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5174 $us CD. Erunt autem $emper AB puncta in eadem linea |
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5175 quæ mouetur, $ibi inuicem ex contrarijs partibus occur- |
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5176 rentia. Itaque cum ex duobus motibus $emper propor- |
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5177 tionalibus, hoc e$t, laterum proportione $eruata, recta |
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5178 producatur, vt demon$tratum e$t à principio, vbi produ- |
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5179 ctio circuli ex Philo$ophi mente e$t declarata, vtraq; pun- |
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5180 cta quæ ean dem laterum proportionem $eruantia mouē- |
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5181 tur, rectas lineas producēt A quidem AD, B autem ip$am |
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5182 BC. Feraturigitur A, tum mixto tum $implici motu per |
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5183 diametrum AD. B vero quoque tum mixto, tum proprio |
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5184 per diametrum BC, $upponitur autem motus omnes $im- |
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5185 plices, tum punctorum, tum etiam line <17>, à qua puncta ip$a |
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5186 feruntur, æquali velocitate fieri. Illud igitur mirabile e$t, |
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5187 cuius etiam ratio quæritur, quo pacto eodem tempore ea- |
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5188 dem que velocitate latum A quidem totam percurrat AD |
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5189 maiorem, B vero totam BC, eamque longe minorem? |
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5190 Porro nece$$e fuit rem in Rhombo $peculari, non autem |
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5191 in quadrato & altera parte longiori rectangulo, in quibus |
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5192 diametri (quod Rhombo non accidit) $unt æquales. Ima- |
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5193 ginemur igitur A, proprio motu percurri$$e $patium AE, |
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5194 nempe ip$ius AB line æ dimidium. Erit igitur in E, item li- |
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5195 neam totam AB eodem tempore pertran$i$$e dimidia op- |
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5196 po$itarum linearum, ACBD, & e$$e translatam, vbi FKG. |
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5197 Quoniam igitur æquali celeritate lineæ AB extremitas |
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5198 A, translata e$t in F & A, punctum per eam motum in E, e- |
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5199 rit $patium AE, æquale $patio AF. Ductis igitur lineis |
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5200 FKG, EKH lateribus AB, AC æquidi$tantibus, erit figura |
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5201 AEKF. Rhombus $imilis quidem Rhombo ABCD, recta |
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5202 igitur FK æqualis erit oppo$itæ AE. quare A punctum |
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5203 translatum erit ex mixto motu in K. Eodem pacto quoniã |
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5204 punctum B. eadem velocitate mouetur ver$us A, & linea |
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5205 AB ver$us CD, cum B fuerit in E extremum line æ motæ |
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5206 BA, nēpe B eritin G. æquales ergo $unt BE, BG & Rhom- |
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5207 <foot>S <I>3</I></foot> |
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5208 <p n=>142</p> |
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5209 bus EBGK, circa diametrum BKC ip$i Rhombo ABCD |
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5210 $imilis, & ideo GK æqualis oppo$itæ BE & BG æqualis |
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5211 EK. Cum ergo B confecerit $patium BE, erit ex mixto |
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5212 motu in K, $uperato nempe $patio BK, idque eodem tem- |
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5213 pore quo A percurrerat totum $patium AK. Ex æquali i- |
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5214 gitur $implicium motuum velocitate, in æqualia $patia |
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5215 AB puncta pertran$ierunt, quæ res miraculo, cuius dilu- |
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5216 tio quæritur, præbet occa$ionem. |
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5217 <p>Porro quod de dimidijs diametris demon$tratum |
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5218 e$t, po$$umus & de totis eadem ratione concludere, quip- |
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5219 pe quod eadem $it proportio partium ad partes, quæ to- |
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5220 tius ad totum. Hæcigitur prima e$t pars propo$itæ quæ- |
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5221 $tionis. Secunda vero dubitatio ita habet; Nempe mirum |
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5222 videri punctum B, cum peruenerit in C, extremum lineæ |
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5223 BA, videlicet ip$um B, translatum e$$e in D, licet æquali- |
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5224 ter moueantur linea BA, per lineam BD, & punctum B per |
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5225 lineam BA. $itque BC ip$a BD maior. Primam dubitatio- |
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5226 nem hoc pacto $oluit Philo$ophus; A fertur tum proprio, |
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5227 tum alieno motu, hoc e$t, line æ AB ver$us oppo$itam par- |
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5228 tem CD, Itaque cum vterque motus deor$um vergat, mo- |
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5229 tus fit velocior. Contra vero B proprio quidem motu fer- |
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5230 tur ver$us A, hoc e$t, $ur$um, alieno vero, hoc e$t, line æ BA |
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5231 ver$us D, hoc e$t, deor$um, qui motus cum inuicem aduer- |
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5232 $entur, motus ip$e fit tardior, non igitur e$t mirum, A eo- |
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5233 dem tempore maius $patium pertran$ire quam B. |
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5234 <p>Hæc $olutio non modo vera videtur, $ed mirabilis |
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5235 & ip$omet Philo$opho digni$$ima, cui quidem temerariū |
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5236 iudicaremus contradicere, nîin genere ver$aremur, in |
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5237 quo non probabilia quæruntur, $ed demon$trata, $ed ve- |
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5238 ra. Futilem igitur e$$e rationem hanc ip$ius Ari$totelis |
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|
5239 pace, hoc pacto o$tendemus. |
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5240 <p>E$to quadratum ABCD, cuius diametri ACBD $e- |
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5241 cantes $e$e in E, moueatur eodem pacto BA, ver$us CD, |
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|
5242 <p n=>143</p> |
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|
5243 <fig> |
|
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5244 item A, ver$us B, & B ver$us A, ita- |
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|
5245 que punctum A tum proprio tum |
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5246 alieno, hoc e$t lineæ illud deferē- |
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5247 tis motu deor$um trudet, hoc e$t, |
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5248 ver$us CD. Motus ergo velocior |
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5249 erit motu puncti B, quod lationi- |
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5250 bus fertur ferè contrarijs, hoc e$t, |
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5251 ex B ver$us A $ur$um, cum linea |
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5252 autem BA ver$us C deor$um. Ve- |
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5253 locius tamen non mouetur, quip- |
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|
5254 pe quod æquali tempore æquale |
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5255 $patium vtrum que punctum conficiat. Stante igitur cau$- |
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5256 $a $equi debui$$et effectus; non $equitur autem, Ari$tote- |
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5257 lis igitur cau$$a non e$t cau$$a. Rhombo quoque inuer$o |
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5258 idem clarius o$tendemus hoc pacto: Sit Rhombus ABCD, |
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|
5259 <fig> |
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5260 cuius diametri AC, BD $ecan- |
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5261 tes $e$e in E. Mota igitur linea |
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5262 AB ver$us CD, nempe deor$um |
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5263 & A quoque deor$um ver$us B, |
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5264 contra vero B quidem $ur- |
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5265 $um ver$us A, deor$um vero |
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5266 ver$us C, erit B tardior A, $ed |
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5267 contrarium fit, quippe quod |
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5268 longior $it BD, per quam mouetur B ip$a AC, per quam |
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|
5269 mouetur A. |
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5270 <p>His igitur non $atisfacientibus veriorem $i perim- |
|
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5271 becillitatem no$tram licuerit, huius effectus cau$$am in- |
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5272 ue$tigabimus. Rationibus igitur & veritate contra aucto- |
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5273 ritatem & probabilitatem e$t nobis pugnandum: quod & |
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|
5274 intrepide faciemus. |
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5275 <p>Dicimus igitur, in quouis parallelogrammo $itillud |
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5276 qua dratum aut altera parte longius, vel idem Rhombus |
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|
5277 Rhomboi$ue $emper mixtos motus proportione $eruata |
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|
5278 <p n=>144</p> |
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|
5279 fieri per diametros. Cæterum díametrorum ad latera |
|
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|
5280 proportiones e$$e varias (quadratis exceptis, in quibus ea- |
|
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|
5281 dem e$t $emper) explorati$$imum. Illud quoque certum |
|
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5282 e$t, in rectangulis nunquam dari po$$e diametros lateri- |
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|
5283 bus vtcunque captis æquales, $emper enim diametri re- |
|
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|
5284 ctis angulis $ubtruduntur. In Rhombis vero & Rhombo- |
|
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|
5285 idibus diametrorum ad latera proportiones variant. Dari |
|
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|
5286 enim po$$unt diametri lateribus longiores item æquales, |
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|
5287 & lateribus quoque ip$is breuiores. |
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|
5288 <p>Itaque diametrorum & laterum varia adinuicem |
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5289 ratione $e habentibus, attentis proportionibus, mixtorū |
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|
5290 & $implicium motuum diuer$a fiet, & varia comparatio. |
|
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|
5291 in quadratis motus mixtus, qui per diametros $emper ve- |
|
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|
5292 locior erit $implici qui per latera, Idem quoque in altera |
|
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|
5293 parte longiori, in quo mixti quidem motus per diametros |
|
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|
5294 erunt velociores, $implices vero qui per latera, tardiores |
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|
5295 quidē, $ed ex illis tardior qui per latus breuius. In Rhom- |
|
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|
5296 bis autem mixtus motus qui fit per diametros inæqualis. |
|
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|
5297 Velocior enim qui per longiorem diametrum, tardior |
|
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|
5298 quiperbreuiorem. Itaque $implices motus punctorum |
|
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|
5299 per latera ad eum qui fit per diametrosinon eodem pacto |
|
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|
5300 $e habent. Porro cum Rhomboides variæ $int diametrorū |
|
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|
5301 adlatera habitudines, varia quoque dari pote$t propor- |
|
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|
5302 tio. aliquando enim diametri dari po$$unt lateribus maio- |
|
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|
5303 res quando que, alter eorum minor. Si autem Rhombus in |
|
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5304 duos $oluatur triangulos, alter diametrorum datur æqua- |
|
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|
5305 lis æqualibus lateribus æquicrurium triangulorum; itaq; |
|
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5306 in i$tis mixti motus per diametros <17>queveloces erunt $im- |
|
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|
5307 plicibus, qui per latera longiora, velociores autem illis |
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5308 qui per latera breuiora. His igitur hoc pacto non perfun- |
|
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|
5309 ctoriè con$ideratis, facile ex proprijs cau$$is, nî fallimu<*>, |
|
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|
5310 hocce Ari$totelicum & mirabile Problema $oluitur. |
|
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|
5311 <p n=>145</p> |
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|
5312 <fig> |
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|
5313 <p>E$to enim Rhombus ABDC, |
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|
5314 cuius diameter longior AD maior $it |
|
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5315 tum lateribus, tum etiam altera dia- |
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|
5316 metro BC. $ecent autem $e inuicem |
|
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|
5317 diametri in E. Ducatur <13>ue ip$is AB, |
|
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|
5318 CD, parallela FG $ecans longiorem |
|
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|
5319 diametrum AD, in H, breuiorem ve- |
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5320 ro BC in I. & per I ip$is BD AC paral- |
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5321 lela ducatur KIL, Cum ergo B mixto |
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5322 motu per diametrum BC erit in I & |
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5323 A per diametrum AD, mixto $imili- |
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5324 ter motu erit in H, & quia motus mi- |
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5325 xti fiunt per diametros, vt dictum e$t, |
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5326 vt $e habet AD ad BC, ita AE ad EB, per 15. propol. 5. elem. |
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5327 item vt AE ad EB, ita per 4. propo$. 6. AH ad BI. e$t enim |
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5328 IH ip$i AB parallela. Longior e$t autem AH ip$a BI, quip- |
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5329 pe quod AE longior $it ip$a EB. motus igitur mixtus pun- |
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5330 cti A per diametrum AD v$que ad H velocior e$t motu B, |
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5331 per diametrum BC v$quead I. Mota igitur linea AB mo- |
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5332 uebuntur communia eius & diametrorum BC, AD pun- |
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5333 cta, quibus $ecantur $emper diametrorum proportione |
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5334 $eruata. Quibus ita $e habentibus, nil mirum e$t punctum |
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5335 A motum per AD velociorem e$$e mixto motu puncti B, |
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5336 quod per minorem diametrum fertur BC. quod fuerat |
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5337 demon$trandum. quatenus vero ad $ecundam problema- |
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5338 tis partem pertinet, dicimus Propo$itionem non e$$e vni- |
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5339 uer$alem. Si enim Rhombus detur, ex duobus æquilateris |
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5340 triangulis con$tans, breuior diameter lateribus erit <17>qua- |
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5341 lis, quare non mouebitur citius motu $implici punctum |
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5342 per latus ac faciat mixto per minorem diametrum, quod |
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5343 vt mirum propo$uerat A ri$toteles. Si autem latus ip$um |
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5344 breuiori diametro $it lōgius, nec mirum quoque erit $im- |
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5345 plici motu moucri velocius quam mixto, quippe quod, vt |
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5346 <foot>T</foot> |
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5347 <p n=>146</p> |
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5348 dictum e$t, motus i$ti à proportionibus linearum, per quas |
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5349 mouentur, legem velocitatis atque tarditatis accipiant. |
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5350 Hæc igitur nos circa hoc mirabile Ari$totelicum proble- |
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5351 ma con$iderare $it $atis. |
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5352 <HEAD>QVÆSTIO XXIV.</HEAD> |
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5353 <p>Mirabilem aliam quæ$tionem proponit Ari$toteles, |
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5354 quæ itidem ad mixtos motus pertinet. |
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5355 <p><I>Dubitatio est, quam ob cau$$am maior circulus æqualem minori |
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5356 circulo circumuoluitur lineam, quando circa idem centrum fue- |
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5357 rint po$iti. Seor$um autem reuoluti quemadmodum alterius ma- |
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5358 gnitudo ad alterius magnitudinem $e habet, ita & illorum adin- |
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5359 uicem $iunt line æ? Præterea vno etiam & eodem vtri$que exi$ten- |
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5360 te centro. Aliquando quidem tanta $it linea, quam conuoluuntur, |
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5361 quantum minor per $e conuoluitur circulus, quando<16> vero quan- |
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|
5362 tum maior.</I> |
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5363 <p>Hæcille, qui vt prober maiorem circulum in $ua ro- |
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5364 tatione maiorem lineam pertran$ire, minorem vero mi- |
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5365 norem; ait $en$u cogno$ci angulum maioris circuli, id e$t, |
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5366 eius qui maiorem habet circumferentiam, e$$e maiorem, |
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5367 eius vero qui minorem, minorem. Ita autem $e habere cir- |
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5368 cumferentias vt $e habent anguli, & eandem proportionē |
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5369 habere per quas tum maior, tum minor circulus circum- |
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5370 uoluuntur. Ad quorum clariorem intelligentiam ea re- |
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5371 uocare oportet in memoriam, quæ dixit de maiorum cir- |
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5372 culorum ad minores circulos nutu. Hic enim, quod ibi |
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5373 quoque fecerat, $ectorem ip$um angulum appellauit, an- |
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5374 gulum vero maiorem maioris circuli $ectorem, & mino- |
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5375 rem angulum minoris ip$ius circuli $ectorem dixit. Clau- |
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|
5376 dit igitur dicens: quoniam circumferentiæ $e habent vt |
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5377 anguli, hoc e$t, vt $ectores, maior erit circumferentia ma- |
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5378 ioris circuli, & ex con$equenti maior linea, per quam cir- |
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|
5379 <p n=>147</p> |
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5380 cumuoluitur, ea per quam minor. Demon$trationem ve- |
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5381 ro ex $en$u petijt. Satautem erat $i dixi$$et, ita $e habere |
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|
5382 circum ferentias vt $e habent diametri $eu $emidiametri, |
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|
5383 & ideo lineas in rotatione de$criptas inuicem $e habere vt |
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5384 diametros. Ob$curiu$culè, hæc $ua figura o$tendit Ari$to- |
|
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5385 teles. Nos igitur claritatem amantibus, no$tram aliquan- |
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|
5386 to, nî fallimur, clariorem, proponemus. |
|
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|
5387 <fig> |
|
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5388 <p>E$to circulus |
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|
5389 maior ABCD, mi- |
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|
5390 nor FGHI, circa i- |
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|
5391 dem, & commune |
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|
5392 cētrum E. Circum- |
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5393 uoluatur maior ad |
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5394 partes D. Sint autē |
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5395 diametri, maioris |
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5396 quidē AEC, BED, |
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|
5397 minoris verò FEH, |
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5398 GEI, fitque CD, |
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|
5399 quadrans maioris, |
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5400 HI vero minoris circuli. Moto igitur maiori circulo $ecū- |
|
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|
5401 dum ab$idem, cum D fuerit in K erit CK ip$i CD æqualis, |
|
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5402 fiet<13>; DE ex puncto K perpendicularis ip$i CK, erit<13> vbi |
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5403 KO, & quia punctum I e$t in linea DE, erit I facta quadrã- |
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5404 tis rotatione in linea KO vbi L, centrum vero E in ip$a |
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5405 KO, vbi O. Reuoluto igitur qua drante maioris, & confe- |
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5406 cto $patio CK minoris circuli quadrans HI conficiet $pa- |
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5407 tium HL, quod ip$i CK $patio e$t æquale. quod autem in |
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5408 quadrantibus fit, in totis etiam fit circulis. Motus igitur |
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5409 minor circulus circa centrum E, vnica rotatione æquauit |
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5410 $patium rotationis maioris circuli. Mirabile itaque e$t mi- |
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5411 norem circulum eodem tempore & circa idem centrum |
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5412 circumuolutum, lineam pertran$i$$e æqualem circum fe- |
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5413 rentiæ maioris circuli. Nec $ecius admirationem facitro- |
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5414 <foot>T 2</foot> |
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5415 <p n=>148</p> |
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5416 tato minori circulo, maiorem vna circumuolutū lineam |
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5417 metiri circumferentiæ minoris circuli æqualem. Rotetur |
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5418 enim minoris circuli quadrans HI per rectam HL. erit i- |
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5419 gitur punctum I vbi M, æquali exi$tente recta HM, ip$i |
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5420 curuæ HI. Tuncautem facto motu centrum E erit vbi P, |
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5421 exi$tente EP, ip$i HM æquali, demittatur autem ex P per |
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5422 M, ip$is HL CK perpendicularis PMN. Et quoniam in |
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5423 eadem linea $unt DIE, vbi E fuerit in Pleritin M, & D in |
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5424 N. quamobrem rotata quarta minori<*> circuli parte, ma- |
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5425 ioris interim circuli quadrans confecit $patium CN æ- |
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5426 quale ip$i HM, hoc minus circuli quadranti HI, quod vti- |
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5427 que e$t admirabile. |
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5428 <p>Porro cau$$am effectus huius mirifici diligenter quæ- |
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5429 rit Philo$ophus, & inuenram accurate explicat. Occur- |
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5430 rit autem primo ab$urdæ cuidam opinioni. Diceret enim |
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5431 qui$piam, ideo tardius moueri maiorem circulum, ad mo- |
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5432 tum minoris, quod interim dū minor moucretur, aliquas |
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5433 inter rotan dum moras interponeret, minor vero ad mo- |
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5434 tum maioris $patia aliqua tran$iliret, & ita $patiorum fieri |
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5435 ad æquationem. Porro demon$trationem aggre$$urus h<17>c |
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5436 a$$umit principia. Eandem<17>qualemue potentiam, aliquã |
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5437 magnitudinem tardius quidem mouere, aliquam vero |
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5438 celerius. quod autem natum e$t aptum moueri, tardius |
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5439 moueri, $i $imul cum non apto nato moueri, moueatur, |
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5440 quam $i $eparatim moueretur, celerius autem $i non $imul |
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5441 <fig> |
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5442 cum eo moueatur. E$to enim corpus A leue |
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5443 quidem & aptum natum moueri$ur$um, cui |
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5444 connectatur B, aptum natum moueri deor- |
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5445 $um, Si quis igitur mouere conetur corpus A |
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5446 $ur$um difficilius mouebit, & tardius iunctū |
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5447 nempeip$i B, quam $i ab ip$o e$$et $eiūctum. |
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5448 Pr<17>terea quod non$uo, $ed alieno motu mo- |
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5449 uetur, impo$$ibile e$$e plus eo moueri qui |
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5450 <p n=>149</p> |
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5451 mouet, $iquidem non $uo, $ed alieno motu mouctur. Mo- |
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5452 to igitur $uo motu maiori circulo, minor non $uo mouc- |
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5453 tur, $ed motu maioris circuli^{1}, & ideo non plus mouetur |
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5454 quam ille moueatur, mouetur autem maiori $patio quam |
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5455 ex $e moueretur, propterea quod maior $it maioris circu- |
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5456 li, à quo $imul defertur, circumferentia. Item $i minor $uo |
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5457 motu circumuoluatur, maiorem feret $ecum, & ideo non |
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5458 plus in $ua rotatione mouebitur maior, quam ip$e minor |
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5459 circulus moueatur. Summa rei h<17>c e$t, alterum ferriab al- |
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5460 tero & latum ad ferentis $patium moueri. Licet enim al- |
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5461 tero moto, alter interim moueatur, nihilrefert. E$t enim |
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5462 ac $i is qui fertur, nullam habeat motionem, aut $i eam ha- |
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5463 beat, ip$a nequaquam vtatur. quod non fit $i vterque $e- |
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5464 paratim circa proprium centrum moueatur, tunc enim |
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5465 magnusmagnum, paruus vero paruum $patium conficit. |
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5466 Hinc decipi ait Ari$toteles illum, qui putat vtrum que cir- |
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5467 culum per $e $u peridem centrum in rotatione moueri, li- |
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5468 cet enim videatur, re vera non e$t. Id enim vtique certum |
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5469 e$t, cum à maiori circulo minor fertur, circa maioris cen- |
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5470 trum motum fieri. Si vero maior à minori feratur circa mi- |
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5471 noris circuli centrum motum fieri. Hæc ferè Philo$ophi |
|
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5472 e$t mens, cuius $olutionem e$$e certi$$imam, & ex veris |
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5473 cau$$is non dubitamus. |
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5474 <p>Hinc ad aliam eam<13>ue certam a$$ertionem tran$i- |
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5475 mus. Dicimus enim, nullam materialem rotã circa axem |
|
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5476 eidem affixum, dum rotatur, po$$e eundem locum $eruare, |
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5477 ni$i cauum fiat, quod axem ip$um recipiat, in tran$uer$a- |
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5478 rijs quibus rota $u$tinetur & progre$$iuum axis motum |
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5479 impediat. |
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5480 <p>E$to enim rota ABCD, cuius centrum E, diametri |
|
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5481 AEC, BED, e$to alia minor rota GH, item minor KL, tum |
|
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5482 minor NO, & adhuc minor QR, circaidem centrum E. |
|
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|
5483 Rotetur itaque $ecundum ab$idem integri quadrantis |
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5484 <foot>T 3</foot> |
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5485 <p n=>150</p> |
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5486 <fig> |
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5487 $patium CD, eritque |
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5488 D, in F, item $i ex rota |
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5489 GH, ex quadrante |
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5490 HT, erit T in I. Ex a- |
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5491 lijs item minoribus in |
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5492 M, P, S. erit itaq; lon- |
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5493 gi$$imū $patium CF, |
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5494 breui$$imū vero RS, |
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5495 Mota igitur rota cir- |
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5496 ca circulū $eu axem, |
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5497 QR, maior rota $pa. |
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5498 tio mouebitur RS, |
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5499 quod $i intra QR, circa centrum E alij infiniti imaginen- |
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5500 tur circuli, quo propio es centro fuerint, eo maioris rotæ |
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5501 progre$$us erit minor, donec ad centrum deueniatur, vbi |
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5502 cum non $it circulus, nullus fiet progre$$iuus motus, $ed |
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5503 circa ip$um centrum nulla facta loci mutatione rotabi- |
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5504 tur. At cum nulla materialis rota circa lineam punctumue |
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5505 imaginarium conuerti po$$it, ideo axi ferreo alteriu$ue |
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5506 materiæ circa quem & cum quo circumuoluatur rota, ca- |
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5507 uum $emirotundum incidere oportet, in quo in$ertus axis |
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5508 dum conuertitur à loco in quo conuertitur, non recedat. |
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5509 <HEAD>QVÆSTIO XXV.</HEAD> |
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5510 <HEAD><I>Quæritur, Cur lectulorum $pondas $ecundum duplam faciant pro- |
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5511 portionem, hanc quidem $ex pedum, vel paulo ampliorem, illam |
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5512 verotrium. Item cur vectes funesue non $ecundum |
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5513 diametrum extendantur?</I></HEAD> |
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5514 <p>Primam quæ$tionis partem ita diluit Philo$ophus, for- |
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5515 ta$$e tantæ fieri $olitos magnitudinis lectulos vt corpo- |
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5516 ribus $int proportionem habentes, & ideo fieri $ecundum |
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5517 $pondas dupli longitu dine nempe cubitorum quatuor, |
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5518 latitudine vero duorum. |
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5519 <p n=>151</p> |
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5520 <p>No$trates alia vtuntur proportione, $e$quialtera, vi- |
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5521 delicet, quam Græci Hemioliam dicunt, communiter e- |
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5522 nim pedes quatuor latos faciunt plus minu$ue, longos ve- |
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5523 ro circiter $ex. quodideo fit vt in eis duo corpora commo- |
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5524 dius cubare po$$int. Lectuliautem, de quibus loquitur |
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5525 Philo$ophus, ad vnum tantummodo $u$tinendum facti |
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5526 videntur, quicquid tamen $it, nullam ferè habet res ex |
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5527 hac parte dubitationem. |
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5528 <p>Secunda quæ$tionis $ectio ea erat, Curnon $ecundū |
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5529 diametros funes extendantur? Re$tium funiumue in le- |
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5530 ctulis muniendis v$us non e$t apud nos. etenim feretra |
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5531 tantum, $eu $andapilas, quibus defunctorum corpora ef- |
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5532 feruntur, funibus ad ea $u$tinen da inteximus. |
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5533 <p>Cæterum lectos tabulis $eu a$$eribus $ternimus, qui- |
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5534 bus $accos paleis plenos imponimus, $accisvero culcitras, |
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5535 & tormenta, ne tabularum durities cubantes offendat. |
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5536 Atqui in re facili multum labora$$e videtur Ari$toteles, |
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5537 tum etiam ob$cure & inuolute nimis quæ$tionem tracta$- |
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5538 $e. Difficilem enim apud eum habethæc explicationem, |
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5539 tum ea quam diximus de cau$$a, tum etiam quod Græca |
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5540 lectio & Latina ver$io corrupta, vt apparet, præ manibus |
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5541 habeantur. Sane vt veritatem hocloco vindicaret in lu- |
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5542 cem, egregie laborauit Picolomineus nec parum profe- |
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5543 cit. Cæterum curre$tes non $ecundum diametrum extru- |
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5544 dantur, triplicem affert Philo$ophus rationem. Prima e$t |
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5545 vt $pondarum ligna, minus di$trahantur. Secunda, vt pō- |
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5546 dus inde commodius $u$tineatur. Tertia, vt in ip$a textura |
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5547 minus re$tium funiumue ab$umatur. |
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5548 <p>Ad primam, cur exten$is diametraliter funibus $pō- |
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5549 dæ ip$æ di$trahantur di$cindanturue, necillenecalij do- |
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5550 cent. Ego autem demon$trarem hoc pacto. |
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5551 <p>E$to $ponda ABCD, cuius longitudo AB, cra$$itudo |
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5552 AC, in ea foramen vtrin que pertinens EF, re$tis per fora- |
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5553 <p n=>152</p> |
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5554 <fig> |
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5555 men inditus GFE, $itque Epars $eu ca- |
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5556 put exterius, quodnodo in E di$tine- |
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5557 tur. Sit autem $pondæ lignum iuxta |
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5558 longitudinem vt natura a$$olet $ci$$ile. |
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5559 Vis quædam, funeita extento applice- |
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5560 tur in G, qu<17> funem ip$um ad $e violen- |
|
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5561 ter trahat. non di$cindetur idcirco |
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5562 $ponda eo quod non diametraliter fu- |
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5563 nis extendatur. Modo facta capitis G |
|
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5564 translatione in H, trahatur valide fu- |
|
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5565 nis, fiet autem pre$$io valida in F. ibi e- |
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5566 nìm impedimentum facit angulus, ne funisip$a dum tra- |
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5567 hitur, rectitudinem a$$equatur. Itaque vi præualente, li- |
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5568 gno vero $ci$$ili, minus re$i$tente, funis, a$$ecuta rectitudi- |
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5569 ne, fiet in HIE $ci$$a $ponda ad quãtitatem trianguli FIE, |
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5570 quod fuerat demon$trandum. |
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5571 <p>Cur autem funes ab angulo in angulum exten$æ mi- |
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5572 nus commode pondus $u$tineant, $atis patet. quo enim fu- |
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5573 nis lōgior, eodebilior, & pre$$io quæ in medio fit, ea vide- |
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5574 licet parte quæ ab extremis e$t remoti$$ima, magis funem |
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5575 fatigat. Longiores autem funes $unt quæ diametraliter |
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5576 extenduntur. |
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5577 <fig> |
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5578 <p>Quatenus ad tertiã |
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5579 rationem pertinet, hoc |
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5580 pacto funes intexit |
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5581 Philo$oph^{9}. E$to lectu- |
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5582 lus cum $uis $pōdis AB |
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5583 CD, cuius $ponda AD, |
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5584 $it pedum $ex, AB vero |
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5585 triū, Diuidatur AD bi- |
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5586 faríam in E & BC in F. item AE in tres AG, GH, HE & in |
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5587 totidem ED, nempe EL, LM, MD. Similiter medietas al- |
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5588 terius $pōdæ BF in tres partes di$tinguatur BN, NO, OF, |
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5589 <p n=>153</p> |
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5590 & FC $imiliterin tres FI, IK, KC, tum alterofunis capite |
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5591 in ducto per foramen A, ibique probe firmato, indatur per |
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5592 F, inde per I, po$tca per GHK CE, & in E probe alligetur: |
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5593 Erunt igitur funis quatuor partes æquales AF, IG, HK, |
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5594 EC, quibus adijciuntur particulæ cadentes extra, quæ |
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5595 $unt FI, GH, KC. Po$t hæc alterius funis principium per |
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5596 foramen traij citur, quod e$t in angulo B. Deinde per E, in- |
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5597 de per L, N, O, M, D, F & in F probe vincitur, & nodo fa- |
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5598 cto ob$irmatur. Erunt igitur aliæ quatuor alterius funis |
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5599 partes, tum inter$e, tum etiam $upradictis æquales, nem- |
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5600 pe BE, NL, OM, FD, quibus il<*>æ pa iter adijciuntur par- |
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5601 ticulæ, quæ caduat extra, videlicet EL, NO, MD. quoniã |
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5602 igitur quadratis ex BA, AE æquale e$t quadratum BE, erit |
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5603 BE quadratum 18. cuius latus radixue 4 1/3 quam proxime. |
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5604 Sunt autem huius longitud n s funes æquales octo. Ea- |
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5605 rum igitur $imul $umptarum longitudo erit pedum 34 2/3 vel |
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5606 circiter, quibus $i ad dantur pedes $ex funium qui cadunt |
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5607 extra, erit re$tis totius longitudo expan$a pedum 40 2/3 plus |
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5608 minu$ue. Picolomineus vero ait 34 2/3, omi$it enim particu- |
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5609 las illas$ex, quæ, vt diximus, cadunt extra. Idem rationem |
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5610 funium diametraliter exten$arum in idem, ait e$$e longi- |
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5611 tudinis pedum 40 1/2. Hic autem eas quoq; particulas præ- |
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5612 termittit, quæ extra cadunt. Itaque his additis clare pa- |
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5613 tet, plus re$tium in$umi diametraliterip$is, quam latera- |
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5614 liter exten$is. Cæterum ratio, qua Philo$ophus hæc pro- |
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5615 bare conatur, adeo e$t mutila, inuoluta, ob$cura, vt Delio |
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5616 pror$us, vt aiunt, indigeat natatore. Huius loci in ex plica- |
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5617 bilem difficultatem, vidit Picolomineus, qui idcirco at- |
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5618 te$tatus e$t, interpretes in hac exponenda fui$$e halluci- |
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5619 natos. Certe Græca lectio ver$ione ip$a Latina non e$t |
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5620 clarior. Nos interim ne inutilem ferè $peculationem ni- |
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5621 mia diligentia, eaque forta$$e fru$tranea pro$equamur, a- |
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5622 lijs difficultatem hanc di$$oluendam aut ceu Gordij no- |
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5623 <foot>V</foot> |
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5624 <p n=>154</p> |
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5625 dum gladio $cindendo relinquemus. Sed interim $ubit |
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5626 mirari, cur veteres vtiliori modo prætermi$$o, inutilioiē |
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5627 fuerint amplexati. Poterant enim reticulatim hoc per li- |
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5628 neas lateribus æ quidi$tantes intexere. |
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5629 <fig> |
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5630 <p>E$to enim lectulus |
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5631 eiu$dem dimen$ionis |
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5632 ABCD, in cuius latere |
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5633 AD $int foramina quin- |
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5634 que E, F, G, H, I, totidem |
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5635 in latere oppo$ito QP, |
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5636 ONM. Duo vero in la- |
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5637 tere breuiori AB, nempe |
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5638 RS, & toti dem in oppo$ito KL incipiatur exten$io à fora- |
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5639 mine E, per QP, F, GON, HIM & in M funis obfirmetur, |
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5640 tum alterius funis caput in datur $i lib et per K, & inde per |
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5641 S, R, L & in L con$tringatur. Sunt autem omnes EQ, FP, |
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5642 GO, NN, IM, pedum quindecim, quibus $i addantur KS, |
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5643 RL, $inguli pedum $ex erunt pedum xxvii. quibus adiectis |
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5644 particulis extra cadentibus QP, FG, ON, HI, & RS, erit |
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5645 integra $umma pedum xxxii. Vide igitur quantum hinc |
|
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5646 minus in$umatur re$tium quam eo modo, quem proba- |
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5647 uit, & ceu vtiliorem propo$uit Ari$toteles. Præterea vali- |
|
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5648 di$$imum e$t hoc texturæ opus nec ex eo fit vera $ponda- |
|
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5649 rum di$tractio $ci$$ioue, quibus haud parum obnoxia e$t |
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5650 ea ratio, quam præfertip$e Philo$ophus. Concludimusi- |
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5651 gitur, autnos eius verba & $en$um non intellexi$$e, aut |
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5652 veteresip$os, quorum v$um ip$e explicat, rei, quam nos |
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5653 proponimus, naturam & commoditatem (quod ta- |
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5654 men vix credibile e$t) igno- |
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5655 rare. |
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5656 <p n=>155</p> |
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5657 <HEAD>QVÆSTIO XXVI.</HEAD> |
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5658 <HEAD><I>Proponitur à Philo $opho examinandum, Cur difficilius$it, langa |
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5659 ligna ab extremo $uper humeros ferre, quam $ecundum me- |
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5660 dium, æquali exi$tente pondere?</I></HEAD> |
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5661 <p>Dvo hîc con$iderat, vibrationem, & pondus. Ait enim |
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5662 primo fieri po$$c, procera ligna vibratione impedien- |
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5663 te, difficilius ferri. Quærerer autem qui$piam, (ip$e enim |
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5664 id reticet) curvibratio hæc ferenti $it nocua. Nos itaque |
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5665 id expliçare conabimur. |
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5666 <fig> |
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5667 <p>E$to igitur lignum |
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5668 oblongum, flexile, & vt |
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5669 ita dicam, vibrabile |
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5670 AB, imponatur hume- |
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5671 ro<*> eique hæreat in C, |
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5672 manu vero $u$tineatur facta compre$$ione in B. Nuteti- |
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5673 gitur & vibretur, in ip$a vibratione, ad partem A. Sit au- |
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5674 tem centrum grauitatis eius D, Lignum igitur in ip$a vi- |
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5675 bratione de$cendet $ua pre$$us grauitate in E, tum facta |
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5676 ligni con$tipatione in ea parte quæ e$t inferius inter C & |
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5677 D, & inde re$i$tentia, codem fere impetu quo de$cende- |
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5678 rat, repul$um per D, nec enim in $ua rectitudine $tabit, a- |
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5679 $cendet in F, facta iterum materiæ con$ti patione inter C |
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5680 & F. Mouebitur igitur lignum $ua grauitate, motu fre- |
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5681 quenti$$imo, $ur$um deor$um, & is interim qui lignum hu- |
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5682 mero fert, procedit antror$um, impedit igitur motus i$te, |
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5683 qui fit $ur$um deor$um lationem, quæ fit ad anteriora; La- |
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5684 torem ip$um quodammodo retrahens. Siautem medio |
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5685 ligno $upponatur humerus, eo quod vibratio $it minor. |
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5686 breuiores enim partes $unt, quæ à medío ad extrema mi- |
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5687 nus à vibratione remorabitur ferens. |
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5688 <p>Quoniam autem non $ola vibratio in hoc lationis |
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5689 modo, nempe ex ligni extremitate difficultatem facit, ait |
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5690 <foot>V 2</foot> |
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5691 <p n=>156</p> |
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5692 Philo$ophus, forte id fieri, quoniam licet nihil inflecta- |
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5693 tur, neque multam habeat longitu dinem, difficilius tamē |
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5694 $it ad ferendum ab extremo, eo quod facilius eleuetur ex |
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5695 medio quam ab extremis, & ideo $ic ferre $it facilius. |
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5696 Cur autem ex medio facilius eleuetur, cau$$am e$$e ait, |
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5697 quod eleuato medio ligno extrema $e$e inuicem $u$pen- |
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5698 dant, & altera pars alteram bene $ubleuet. Medium enim |
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5699 fieri velut centrum, vbi is $upponit humerum qui cleuat |
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5700 aut fert. Extremorum autem interim altero depre$$o al- |
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5701 terum $u$tolli. Nos interim Mechanicis principijs, quod |
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5702 ip$e non fecit, rem clariorem efficiemus. |
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5703 <p>E$to enim oblongum lignum AB, cui humerus $up- |
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5704 ponatur in B, manus vero premendo $u$tinens in B. $it au- |
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5705 tem ligni pars maxima AC, minima CB, inaioris autem ad |
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5706 minorem proportio exempli gratia $it $excupla. Ad hoc i- |
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5707 gitur vt fiat æquilibrium inter potentiam $u$tinentem in |
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5708 B, & pondus comprimens in A, ita $e habere oportet po- |
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5709 tentiam in B, ad pondus in A, vt $e habet pars ligni AC ad |
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5710 <fig> |
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5711 partem CD. E$to igitur pon- |
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5712 dus in A, puta librarum $ex. |
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5713 Erit igitur potentia quæ in B |
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5714 ad hoc vt $u$tineat librarum |
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5715 triginta $ex, quas $i addas pō- |
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5716 deri in A, fiet humerus in C |
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5717 $u$tinens pondus librarum quadraginta duo. Siautem |
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5718 humerus medio ligno, hoc e$t, in D $upponatur, ad hoc vt |
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5719 fiat æquilibrium, nece$$e erit potentiam in B e$$e æqua- |
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5720 lem ponderi in A, quod e$t $ex, quare humerus $u$tinebit |
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5721 duodecim. Vnde patet, longe difficilius portari lignum |
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5722 ex C extremo, quam ex D medio; quod Mechanice fue- |
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5723 rat demon$tran dum. |
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5724 <p>Po$$umus & aliteridem o$tendere. Intelligatur e- |
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5725 nim ij$dem $uppo$itis, vectem quidem e$$e AB, cuius ful- |
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5726 <p n=>157</p> |
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5727 cimentum quidem B, pondus A, potentia $u$tinens in C, |
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5728 nempe inter fulcimentum & pondus. Res igitur ad eum |
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5729 vectis v$um reducitur, de quo G. Vbaldus tractatu de Ve- |
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5730 cte, propo$. 3. Quare vtille o$ten dit, ita $e habere oporter |
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5731 potentiam $u$tinentem ad pondus, vttotus vectis ad par- |
|
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5732 tem eius quæ à potentia ad fulcimentum. Ita igitur $e ha- |
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5733 bebit pre$$io, quæ fit in C ad pondus in A, vt totus vectis |
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5734 AB ad partem eius CB, quæ à potentia ad fulcimentum. |
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5735 Erit igitur potentia $eptupla ponderi, & ideo $u$tinebit |
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5736 pondus librarum quadraginta duarum. quod fuerat o- |
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5737 $tendendum. |
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5738 <p>Hinc alia quæ$tio huic affinis $oluitur, Cur ha$ta $a- |
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5739 ri$$aue $olo iacens manu ad alteram extremitatum ap- |
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5740 pren$a di$ficillime extollatur? |
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5741 <fig> |
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5742 <p>E$to igitur $ari$$a ha- |
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5743 $taue iacens AB, cuius ex- |
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5744 tremitati A manus ad $u- |
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5745 $tollendum applicetur, $it |
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5746 autem pars quæ digitis capitur AC, quæritur cur pars re- |
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5747 liqua CB difficillime $u$tollatur? Facile dubitatio ex præ- |
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5748 demon$tratis $oluitur. E$t enim C fulcimentum, $upponi- |
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5749 tur enim loco, pugno ad $u$tollendum clau$o, digitus in- |
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5750 dex, potentia autem premens in A, vt $uperet grauitatem |
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5751 CB, e$t manus ip$ius carpus, hoc e$t illa manus ip$ius pars, |
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5752 qua pondus facta $uppre$$ione $u$tollitur. E$tigitur AB |
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5753 vectis, cuius fulcimentum C, pondus B, potentia A, Itaq; |
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5754 quoniam maxima e$t proportio BA ad AC, maximam e$- |
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5755 $e oportet potentiam pondus $u$tollentem in C. |
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5756 <p>Huc etiam illud pertinet, Cur ha$ta $olo iacente, $i |
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5757 alterum extremorum manu $u$tollatur, alterum vero ve- |
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5758 lo ci$$ime $ur$um vibretur, & eodem tempore manus ha- |
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5759 $tæ $ic vibratæ $upponatur, haud magna difficultate ha$tæ |
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|
5760 ad perpendiculum fit erectio. |
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|
5761 <foot>V 3</foot> |
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5762 <p n=>158</p> |
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5763 <fig> |
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5764 <p>Sit enim ha$ta AB, quæ |
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5765 manu ex B capta eleuetur in |
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5766 C, & fiat in AC, tum facta ex |
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5767 C partis A veloci vibratione, |
|
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5768 ip$a extremitas A transferatur |
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5769 in D, $itque vbi CD, tum velo- |
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5770 <*>imanus depre$$ione extremi- |
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5771 tas C transferatur in E, fiatq; |
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5772 EF horizonti perpendicuiaris; |
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5773 quod vbi factum fuerit, erunt |
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5774 in eadem linea quæ ad centrum mundi, manus ip$a quæ |
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5775 $u$tinet, & grauitatis ip$ius centrum G, quare manus ip$a |
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5776 facta vibratione tantum portat, quantum præci$e ip$ius |
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5777 e$t ha$tæ pondus. |
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|
5778 <HEAD>QVAESTIO XXVII.</HEAD> |
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5779 <HEAD><I>Dubitatur, Cur $i valde procerum fuerit idem pondus, difficilius |
|
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5780 $uper humeros ge$tatur, etiam$i medium qui$piam illud fe- |
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|
5781 rat quam $i breuius $it?</I></HEAD> |
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5782 <p>Qvæ$tio hæc $uperiori e$t affinis. Ait autem Philo$o- |
|
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5783 phus, cau$$am non e$$eid, quod in præcedenti quæ- |
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5784 $tione dixerat, $ed vibrationem: quo enim longiora $unt |
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5785 ligna, eo magis eorum extrema vibrantur, debiliora enim |
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5786 $unt & à medio remotiora, quare $uopte pondere facilius |
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|
5787 nutant. Siautem breuiora $int ea cau$$a ce$$ante minor |
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5788 fit aut nulla vibratio, quare breuiora feruntur facilius. |
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5789 Dupliciter autem vibratione ip$a, portans offenditur, |
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5790 tum ex cau$$a quam in $uperiori quæ$tione con$ideraui- |
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5791 mus, nempe quod motus $ur$um deor$um a$$iduus, pro- |
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5792 gredientis motum impediat, tum etiam quod duplici |
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5793 pre$$ione grauetur ferentis humerus, quod Philo$ophus |
|
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|
5794 non animaduertit. |
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5795 <p>Sit enim oblongum lignum AB, quod humero me- |
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|
5796 <p n=>159</p> |
|
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|
5797 <fig> |
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|
5798 dio loco $u$tineatur in C. |
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5799 nutabunt ergo extrema AB, |
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5800 à centro C, valde remota, |
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|
5801 cadent autem $imul A m D, |
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|
5802 & B in E trahere $ecum conantes medium C, quare is qui |
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5803 in C $u$tinet, non modo ligni $u$tinet pondus ex grauita- |
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5804 tis centro quod e$t in C, $ed impetum quoque in ip$a ex- |
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5805 tremorum depre$$ione acqui$itum ex ipia violentia. Illud |
|
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5806 autem $ubtiliter con$ideramus, portantem ex vibratione |
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5807 per inter ualla deprimi & $ubleuari. fiat enim vibratum li- |
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5808 gnum ex contrario motu, vbi FCG. alleuiabit igitur eo |
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5809 ca$u portantem, $iquidem impetus ex motu ip$o acqui$i- |
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5810 tus, medium C trahat ad $uperiora. Itaq; cum e$t in DCE |
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5811 portans plus $u$tinet in ACD, æquale, in FCG minus, |
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5812 quod vtique demon$trandum fuerat. E$t autem quæ$tio |
|
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5813 hæc illi familiaris, quam 16. loco explicauimus. |
|
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|
5814 <HEAD>QVAESTIO XXVIII.</HEAD> |
|
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5815 <HEAD><I>Quæritur, Cur iuxta puteos celonia faciunt eo quo vi$untur mo- |
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5816 do? Ligno enim plumbi adiungunt pondus, cum alioquin vas |
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5817 ip$um & plenum & vacuum pon- |
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5818 dus habeat.</I></HEAD> |
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5819 <p>Re$pondet optime Philo$ophus, hauriendi opus duo- |
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5820 bus temporibus diuidi, nempe dumvasip$um vacuum |
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5821 demittitur, dum que extrahitur plenum: Contingere au- |
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5822 tem, vacuum facile demitti, plenum autem difficulter ex- |
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5823 trahi. Expedire nihilominus tardius, hoc e$t difficilius di- |
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5824 mitti vt fa cilius extrahatur, plumbo nempe coadiuuante, |
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5825 & $ane Philo$ophi $olutio e$t lucidi$$ima. Nos autem luci |
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5826 ip$i lucem aliquam adhuc afferre conabimur. |
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5827 <p>E$to Celomum (Latine Tolenonem appellant) ABC, |
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5828 cuius arrectarium BD, tran$uer$um lignum AC, quod |
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5829 <p n=>160</p> |
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5830 <fig> |
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5831 conuertitur, circa pūctum $eu |
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5832 fulcimentum B, pondus, plum- |
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5833 bumue, vbi A, $itula E, funi ap- |
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5834 pen$a CE. Dico rebus ita con- |
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5835 $titutis difficilem quidem e$$e |
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5836 vacuæ $itulæ demi$$ion em, fa- |
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5837 cile vero eiu$dem extractio- |
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5838 nem. Vectis diui$i, $itulæ, ac |
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5839 ponderis, ad hoc vt fiat æ quili- |
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5840 brium, ca debet e$$e propor- |
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5841 tio, vt quema dmodum $e habet AB ad BC, ita $e habeat |
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5842 plenæ $itulæ pondus E ad ip$um pondus A, $uperabit ergo |
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5843 pondus in A $itulam vacuam in E nec fiet æquilibrium, i- |
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5844 taque vt vacua $itula demittatur, tanta vis adhibenda c$t |
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5845 quantum e$t ip$ius aquæ, qua $itulaimpl<*>tur pondus, quæ |
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5846 vis dum apponitur difficilem, vt dicebamus, efficit $itulæ |
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5847 vacuæ demi$$ionem. Plena vero $itula $it æquilibrium, vn- |
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5848 de quantumuis pu$illa vi adhibita, $itula extrahitur, qua$i |
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5849 ex $emetip$a ponderis appen$i virtute a$cendens. Quan- |
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5850 tum igitur pondus dum vacua demittitur impedit, tan- |
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5851 tundem plena dum extrahitur, adiuuat. Qu<17> cum ita $int, |
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5852 $i paria $unt difficultas in demittendo, & facilitas in ex- |
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5853 trahendo, quæ ratio hoc in negotio vtilitatis? Sane $itula |
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5854 vacua, manu per funem facile demittitur, plena vero dif- |
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5855 ficile extrahitur, v$u autem Celonij res permutãtur. Cor- |
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5856 poris enim proprij pondere, dum premit, adiuuatur de- |
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5857 mittens, qui per funem $implicem extrahendo, ab eodem |
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5858 proprij corporis pondere impediebatur. quod quidem ex |
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5859 corporis pondere, auxilium, ingentem parit in extrahen- |
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5860 do commoditatem. |
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5861 <p>Quippiam $imile accidit, aquas è puteis extrahen- |
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5862 tibus v$u trochleæ. Sit enim trochlea puteo imminens |
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5863 ABCD, cuius centrum E $u$pen$a quidem in A, funis, cui |
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5864 <p n=>161</p> |
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5865 $itula $u$penditur FCABG, $itula vero G. E$t igitur dia- |
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5866 meter CED, in$tar libræ, quare vt fiat æquilibrium nece$- |
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5867 $e e$t capiti funis F, potentiam applicare, quæ $it æqualis |
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|
5868 <fig> |
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5869 pondere $itulæ aqua plenæ, itaque extra- |
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5870 hens proprijs viribus corporīs pondus ad- |
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5871 ijciens facile $itulam aqua plenam extra- |
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5872 hit, ex qua re magna extrahentibus fit |
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5873 commoditas. Patet autem diuer$o modo |
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5874 extrahentes iuuare Celonium. & Tro- |
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5875 chleam, ibi enim corporis mole adiuuatur |
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5876 demittens vacuam, hic vero qui extrahit |
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5877 plenam aqua $itulam. |
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5878 <p>Cæterum Celonij partem BC, qui à |
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5879 fulcimento ad funem longe maiorem e$- |
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5880 $e oportet, ip$a AB, vt $itula in profundum po$$it demitti, |
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5881 quamobremita $e debethabere pondus in A, ad pondus |
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5882 $itulæ plenæ, vt$e habet brachium $eu pars BC, ad par- |
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5883 tem BA. Tunc enim ex permutata proportione efficitur |
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5884 æquilibrium. |
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5885 <p>Illud addimus, nouum non <17>$$e Architectis Mecha- |
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5886 nici$que, tum hominum tum animalium vt commodius |
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5887 machinas moueant, adhibere pondera corporum. Nec e- |
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5888 nim alia ratione mouentur Rotæ illæ, quas ob hanc cau$- |
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5889 $am ambulatorias vocant; quarum v$us ad Mangana, ad |
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5890 extrahendas è puteis aquas, & ad farinarias quoque mo- |
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5891 las agitan das a dhibetur. |
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5892 <p>Porro Tollenonem bellicam Machinam à Celonio |
|
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5893 tum forma tum pote$tate nihil differre, videre e$t a pud |
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5894 veteres Mechanicos, Heronem Byzantium, & alios. apud |
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5895 neotericos vero hac de re agunt Daniel Barbarus in Vi- |
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5896 truuium, & Iu$tus Lip$ius in librum quem de bellicis |
|
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5897 machinis edidit, eleganti$$i- |
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5898 mum. |
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5899 <foot>X</foot> |
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5900 <p n=>162</p> |
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5901 <HEAD>QVAESTIO XXIX.</HEAD> |
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5902 <HEAD><I>Dubitatur, Cur quando $uper ligno, aut huiu$modi quopiam, duo |
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5903 portauerint homines, idem pondus non æqualiter premun- |
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5904 tur, $edille magis cui vicinius fuerit |
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5905 pondus?</I></HEAD> |
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5906 <p>Soluit Ari$toteles, inquiens, lignum e$$e vectem, pon- |
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5907 dus vero fulcimentum; res quæ mouetur is qui ponde- |
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5908 ri e$t proximior: mouens vero qui remotior. Itaque quo |
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5909 magis remotus e$t à pondere, hoc e$t, à fulcimento is qui |
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5910 mouet, eo violentius is premitur qui altera vectis parte |
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5911 eaque breuiori, mouetur. |
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5912 <fig> |
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5913 <p>E$to lignum AB, pondus |
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5914 C appen$um in E, vicinius ex- |
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5915 tremo B quamip$i A, $it autē |
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5916 portãtium alter quidem AF, |
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5917 alter vero BG, Imaginemur |
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5918 itaquelocum E à pondere ita |
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5919 figi & deprimi, vt$ur$um qui- |
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5920 dem ferri nequaquam po$$it, |
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5921 circa vero punctum E, ceu |
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5922 circa centrum fulcimentum- |
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5923 ne ip$um vectem conuerti. Lignum ergo AB vectis: mo- |
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5924 uens potentia A, pars vectis à potentia ad fulcimentum |
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5925 AE pars eiu$dem quæ à fulcimento ad rem motam EB, & |
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5926 quoniam quanto longior e$t pars vectis EA ip$a EB, eo fa- |
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5927 cilius potentia quæ e$t in A, operatur in id quod e$t in B, $i |
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5928 res ad proportiones redigatur, erit potentia in A, ad id |
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5929 quod mouetur $eu premitur in B, vt pars vectis EB ad par- |
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5930 tem EA, $ed maior e$t AEip$a EB, ergo maiorem partem |
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5931 $u$tinet ponderis, & plus premitur is qui in E, & qui mo- |
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5932 uet in A. Hæc fere Philo$ophi e$t $ententia: Picolomi- |
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5933 neus vero Paraphra$tes appo$ite duos vectes in vnico li- |
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5934 <p n=>163</p> |
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5935 gno con$iderat, alterum AB, alterum BA, in primo A e$t |
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5936 mouens B, motum in $ecundo B, mouens A vero motum |
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5937 in quibus vectibus $emper idem & commune fulcimen- |
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5938 tum E. Et quoniam in propo$ito diagrammate breuior e$t |
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5939 pars vectis EB, quæ que à mouente ad fulcimentum, parte |
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5940 illa quæ ab eodem fulcìmento ad rem motam, minus o- |
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5941 peratur B in A, quam A in B, & ideo qui in B mouetur plus |
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5942 premitur, contra vero quia maior e$t pars EA ip$a parte |
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5943 EB, magis operatur qui in A in ip$um B, quam econtra. Et |
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5944 $ane con$ideratio hæc $ubtilis e$t & ingenio$a, & quæ $i |
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5945 recte intelligatur, quatenus ad proportiones & effectum |
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5946 ip$um demon$trandum pertinet, à veritate ip$a non ab- |
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5947 horret, Quicquid tamen $it, Mechanice magis hoc pacto |
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5948 quæ$tio diluetur. Dicimus enim, pondus quidem vere e$- |
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5949 $e pondus, non autem fulcimentum, vt $ibi fingebat Ari- |
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5950 $toteles: lignum vero vectem, duo autem qui pondus $u- |
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5951 $tinent pro duplici fulcimento haberi, vtri$que enim ve- |
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5952 ctis cum appen$o pondere innititur. Pote$t etiam alter |
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5953 eorum pro porentia mouente, alter vero pro fulcimen- |
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5954 to, & $ic vici$$im. E$t autem, quomodocunqueres accipia- |
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5955 tur, pondus inter fulcimentum. & potentiam. Quare ex |
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5956 ijs quæ demon$trauit G. Vbald. de hoc vectis genere lo- |
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5957 quens, vt $e habet AE pars ad AB vectem totum, ita po- |
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5958 tentia quæ $u$tinet in B, ad pondus appen$um in E, & vt |
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5959 BE ad BA ita potentia quæ $u$tinet in A ad pondus quod |
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5960 in E. At minor e$t proportio BE, ad BA, quam AE ad AB, |
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5961 quare magis $uperatur pondus in E à potentia quæ in A, |
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5962 quam à potentia quæ in B, & ideo plus ponderis $u$tinet |
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5963 ferens in B, quam ferens in A, quod fuerat demon$tran- |
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5964 dum. |
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5965 <p>Hinc colligimus, pondere in medio vecte appen$o |
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5966 ferentes æqualiter $u$tinere, propterea quod totius vectis |
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5967 ad partes ip$as proportio $it eadem, vel æqualis. |
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|
5968 <foot>X 2</foot> |
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5969 <p n=>164</p> |
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5970 <p>Pulchre autem dubitari pote$t, an idem pror$us con- |
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5971 tingat, $i alterum eorum qui $u$tinent, $it $tatura quidem |
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5972 procerior, alterv<*>ro humilior. |
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|
5973 <fig> |
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5974 <p>Sit enim vectis AB, in cuius |
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5975 medio pondus H libere appen- |
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5976 $um ex C, alter portantium pro- |
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5977 cerior AD, humilior vero BE. $it |
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5978 autem horizontis planum DE, |
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5979 demittatur à puncto Cad horizō- |
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5980 tem perpendicularis, ip$is vero |
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|
5981 AD, BE, æquidi$tans CF. Tran$i- |
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5982 bit autem per ip$ius ponderis, |
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5983 grauitatis centrum H. Dicoigi- |
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5984 tur, nil referre quatenus ad pondus $u$tinendum perti- |
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5985 net, vtrum portantes $int $tatura pares velne. Ducatur e- |
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5986 nim horizonti æquidi$tans GB, $ecans perpendicularem |
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5987 CF in I. Quoniam igitur AG æquidi$tans e$t ip$i CI erit |
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5988 vt AC ad CB per 4. $exti elem, ita GI ad IB. Sunt ergo GI, |
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5989 IB inter $e æquales. Intelligatur itaque pondus H, $olutū |
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5990 à puncto C appen$um e$$e libere ex puncto I, hoce$t, ex |
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5991 medio vectis GB, æqualiter ergo diui$um erit pondus in- |
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5992 ter portantes, licet alter procerior, alter vero $tatura pu- |
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5993 milior, quod fuerat demon$trandum. |
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5994 <p>Si autem pondus ita vecti alligatum $it vt libere non |
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5995 pendeat, vecte ex vna parte eleuato, ex altera vero de- |
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5996 pre$$o, grauitatis centrum ad eam partem verget quæ |
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5997 magis ab horizonte attollitur, & ad eam ip$am partem |
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5998 vectis à pondere ad $u$tinentem fit breuior. |
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5999 <p>E$to enim vectis AB, cuius medium C, pondus vecti |
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6000 in C alligatum CFG, cuius grauitatis centrum H eorum |
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6001 qui portant procerior AB, humilior BE, horizontis planū |
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6002 DE. Demittatur per centrum H horizonti perpendicu- |
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6003 laris IHK, $ecans vectem quidem in I, horizontis vero pla- |
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6004 <p n=>165</p> |
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6005 <fig> |
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6006 num in K. Po$t hæc intelligatur pon- |
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6007 dus $olutum quidem à puncto C, ap- |
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6008 pen$um vero ex puncto I. Stabitigitur |
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6009 ex definitione centri grauitatis nec $i- |
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6010 tu $uo mouebitur. Dico autem par- |
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6011 tem AI ip$a IB e$$e breuiorem, hoc e$t, |
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6012 punctum I cadere inter C & A. Si e- |
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6013 nim non cadat, vel cadet in C, aut in- |
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6014 ter C & B, cadat autem $i fieri pote$t |
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6015 in C. Eritigitur CHK horizonti perpendicularis, $ed ei- |
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6016 dem perpendicularis AD. Eruntigitur BCK BAD anguli |
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6017 inter $e æquales, $ed ip$i BAD angulo æqualis e$t CIH, |
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6018 quare & BCH ip$i CIH æ qualis erit. Producto igitur la- |
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6019 tere IC trianguli ICH erit exterior angulus æqualis inte- |
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6020 riori ex oppo$ito, quod e$t ab$urdum. non ergo I cadet in |
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6021 C. Eadem autem ratione mon$trabitur non cadereinter |
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6022 CB, cadet ergo inter CA, & ideo minor AI ip$a IB. Itaque |
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6023 vt $e habet BI ad BA, ita potentia in A ad pondus in I, $ed |
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6024 maiorem proportionem habet BI ad BA, quam IA ad AB. |
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6025 Ergo minor potentia requiretur in B quam in A, & $ane |
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6026 pars IB re$pondet potentiæ $u$tinenti in A, at IA potentiæ |
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6027 $u$tinenti in B, minor e$t autem AI ip$a IB, ergo maior po- |
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6028 tentia requiritur in B, quam in A, quod fuerat demon- |
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6029 $trandum. |
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6030 <p>Hocitem concludetur, $i portantes $tatura quidem |
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6031 pares fuerint, $ed per planum ambulent horizonti accliue |
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6032 aut decliue. Si enim pon dus libere pendeat, vectis partiū |
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6033 proportio non mutabitur; $r autem libere non pendeat, |
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6034 is magis laborabit qui in a$cen$u præibit, minus vero qui |
|
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|
6035 in de$cen$u. |
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6036 <p>Hinc quoque Carrucarum ratio pendet, quæ dupli- |
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6037 ci manubrio vnica rota vulgo $unt in v$u, pro vecte enim |
|
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6038 habentur, cuius fulcimentum ad contactum plani & ro- |
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|
6039 <foot>X <I>3</I></foot> |
|
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|
6040 <p n=>166</p> |
|
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|
6041 tæ; potentiæ vero ad extremitatem duplicis manubrij. |
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6042 Reducitur enim ad idem genus vectis, in quo pondus in- |
|
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6043 ter fulcimentum e$t & potentiam. quo igitur minor fue- |
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6044 rit proportio partis vectis quæ à centro grauitatis ad i- |
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6045 p$um fulcimentum, ad totum vectem eo facilius pondus |
|
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|
6046 eieuabitur. |
|
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6047 <p>Cur autem difficilime hæ per accliue horizonti pla- |
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6048 num pellantur, duplici fit de cau$$a, tum quia grauitatis |
|
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6049 centrum ad ip$um portantem $eu pellentem vergit, & id- |
|
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|
6050 co pars quæ a fulcimento ad centrum grauitatis ponderis |
|
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6051 fit maior, tum etiam quoniam ip$um graue contra $ui na- |
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|
6052 turam $ur$us pellitur ferturque. |
|
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|
6053 <p>Quærere ad hæc qui$piam po$$et, Cur Baiuli ma- |
|
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6054 gna ferentes pondera, curui in cedant? Dixeritautem ali- |
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6055 quis, ponderis grauitate eos deprimentis id fieri. Nos au- |
|
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6056 tem duplici item de cau$$aid fieri putamus, tum ea quam |
|
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6057 con$iderauimus, tum etiam alia, nempevt grauitatis cen- |
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6058 trum ip$ius ponderis quod $u$tinent, in perpendiculari |
|
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6059 collocent, ne $i extra ponatur is qui fert à centro extra |
|
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6060 fulcimentum po$ito, ad eam partem ad quam vergit tra- |
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|
6061 hatur, & pondere ip$o opprimatur. |
|
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|
6062 <p>Eadem de cau$$a fit quoque vt ij qui magna ponde- |
|
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|
6063 ra $ini$tro ferunt humero, in dextram partem inclinentur, |
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6064 qui vero dextro, contrario modo $e habeant, æquatur e- |
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6065 nim pondus eo pacto, & grauitatis centrum in ip$a per- |
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6066 pendiculari collocatur. |
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6067 <HEAD>QVÆSTIO XXX.</HEAD> |
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6068 <HEAD><I>Cur a$$urgentes omnes fœmori tibiam ad acutum angulum con$ti- |
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6069 tuamus & pectori thoraciue $imiliter fœmur, quod nî fiat |
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6070 haudquaquam $urgere poterunt?</I></HEAD> |
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6071 <p>Ait Philo$ophus, forte id fieri, quod æqualitas $it o- |
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6072 mnino quietis cau$$a, rectum vero angulum quietis |
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6073 <p n=>167</p> |
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6074 angulum e$$e, & $tationem facere, nec alia de cau$$a $tan- |
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6075 tem ip$i terræ e$$e perpendicularem, & ideo caput & pe- |
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6076 des in eadem linea habere, $edentem vero non item. Tūc |
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6077 autem à $e$$ione $urrectionem fieri, cum caput & pedes in |
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6078 vna linea collocantur, quod $ane fit cum pectus & crura |
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6079 acutum cum ip$o fœmore angulum faciunt. |
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6080 <fig> |
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6081 <p>E$to enim $tans AB hori- |
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6082 zonti IBK perpendicularis, cù- |
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6083 ius caput A, pedes vero B, $edcat |
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6084 modo $itque eius cum capite |
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6085 Thorax CD, fœmur DE, crura |
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6086 EF, $intque CDE, DEF anguli |
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6087 recti, quibus ita con$titutis non |
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6088 $unt in eadem linea caput C & |
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6089 pedes F. Surgere ita que non po- |
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6090 terit $edens, propterea quod |
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6091 partes omnes corporìs non $int |
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6092 horizonti perpendiculares. Ad |
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6093 hocautem vt $urrectio fiat, nece$$e e$t vt $edens retrahat |
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6094 quidem pedes in H, & pectore in clinato acutum cum fœ- |
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6095 more angulum con$tituat GDE, quo ca$u fient in eadem |
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6096 recta linea, eaque horizonti perpendiculari caput in G, |
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6097 & pedes in H, ex cuius $itus natura commoda fiet ab ip$o |
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6098 $edente $urrectio. Hæc fere, licet alijs ab eo verbis expli- |
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6099 cata, ip$ius e$t Philo$ophi $ententia; quæ licet vera $it, non |
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6100 tamen ex proprijs, hoc e$t, Mechanicis principijs e$t peti- |
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6101 ta. quod quidem nos facere conabimur. |
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6102 <p>Dicimus autem primo, $edentem non ideo quie$ce- |
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6103 re, vt$entit Ari$toteles, quod rectus angulus quietis $it |
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6104 cau$$a, $ed propterea quod eius thoracis tum etiam fœ- |
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6105 morum pondus ab ip$a $ede $u$tineantur; crura vero & |
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6106 pedes ideo non laborent, quod partim $u$pen$a $int, par- |
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6107 tim $olo ip$i innitantur. Quare cum corpus totum nec $e |
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6108 <p n=>168</p> |
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6109 $u$tineat, nec à pedibus $u$tineatur, fit quies & la$$itudi- |
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6110 nis alleuatio. Natura autem ideo commodam hominibus |
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6111 $e$$ionem facere volui$$e inde apparet, quod clunes, qui- |
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6112 bus tota $uperior pars, & grauior nititur, carno$am fece- |
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6113 rit, & ceruicalis cuiu$dam in$tar mollem & facilem. Sed |
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6114 nos ad quæ$tionem. |
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6115 <fig> |
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6116 <p>E$to enim $tans AB, cuius caput A, |
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6117 Thorax AC, fœmora CD, crura DB, pe- |
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6118 des vero B, centrum vero grauitatis in i- |
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6119 p$o Thorace E. Modo $edeat, $itque ca- |
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6120 put in F, Thorax FG, fœmora GH, crura |
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6121 HI, pedes I, grauitatis vero centrum vbi |
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6122 K. Producatur recta FG in L, $itque FL |
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6123 horizonti perpendicularis. Centrum er- |
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6124 go grauitatis K fulcitur puncto G, hoc e$t, |
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6125 puncto L, in quo po$teriores pedes ip$ius |
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6126 $edis $olo hærent. efficit autem $edens |
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6127 duos rectos angulos FGH, GHI. Rebus |
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6128 igitur ita di$po$itis $eruatis rectis angulis, non fiet $urre- |
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6129 ctio, & id quidem non ideo quod, vt ait Philo$ophus, æ- |
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6130 qualitas & rectitudo angulorum quietis $it cau$$a, $ed |
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6131 propterea quod centro grauitatis extra pedum fulcimē- |
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6132 tum con$tituto, non habet centrum $tabilem locum cui in |
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6133 actu $urrectionis hæreat, & fulciatur, vnde fit vt $i $edenti |
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6134 $ubtrahatur $edes remoto prohibente, $edens pror$us cor- |
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6135 ruat. Modo retrahat qui $edet crura, & pedes ponat in M, |
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6136 à puncto autem M, horizonti perpendicularis erigatur |
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6137 MN. erit ergo fulcimentum in M, $ed adhuc $urgere non |
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6138 poterit, centro grauitatis adhuc extra lineam MN, quæ |
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6139 per fulcimentum e$t, con$tituto. Reclinetur autem pe- |
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6140 ctus ad anteriora, & cum fœmore acutum angulum faciat |
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6141 $itque vbi GO, erit igitur grauitatis centrum vbi P, hoc |
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6142 e$t, in ip$a perpendiculari NM, fretigitur inde commoda |
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6143 <p n=>169</p> |
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6144 $urrectio, propterea quod in eadem linea facta $int, graui- |
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6145 tatis centrum P, & fulcimentum ip$um M. Acutum vero |
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6146 angulum in $urrectione nece$$arium e$$e clare patet, non |
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6147 autem eff<*>us ip$ius e$$e cau$$am, vt videtur $en$i$$e Ari- |
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6148 $toteles; ni<*>i dicamus, cau$$am e$$e cau$$æ, $iquidem acuti |
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6149 qui fiunt anguli centrum & pedes in eadem linea collo- |
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6150 cant, quicquid tamen $it, nos ideo $urrectionem fieri dici- |
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6151 mus, quod immutatis angulis centrum grauitatis $upra |
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6152 fulcimentum, fulcimento vero $ub ip$o grauitatis centro |
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6153 collocetur, & hæc e$t cau$$a proxima. Hæc nos ad Ari$to- |
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6154 telem. Modo qua$dam alias quæ$tiones, necinutiles $ed |
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6155 & eas non iniucundas quoque proponemus. |
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6156 <p>Primum igitur quærimus, Curhominum & cætero- |
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6157 rum animalium, quæ aliquando erecto corpore incedunt, |
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6158 pedes non quidem breues $int & rotundi, $ed longiores |
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6159 potius, & in inferiorem partem porrecti? Item cur magis |
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6160 ad digitos quam ad cal caneum porrigantur? |
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6161 <fig> |
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6162 <p>E$to <*>homo animalue quodpiam $tans |
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6163 AB, cuius pes CD, pedis pars quæ ad digitos |
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6164 BC. qu<17> vero ad cal caneum BD fœmoris ver- |
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6165 tebra E, centrum vero grauitatis ip$ius cor- |
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6166 poris F. Primum igitur $tatuendum e$t, ho- |
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6167 minem & cætera fere animalia à Natura fa- |
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6168 cta e$$e vt ad anteriora moueantur, & ideo o- |
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6169 mnes fere quod in $enioribus manife$te ap- |
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6170 paret, ad anteriora ex ip$a corporis di$po$i- |
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6171 tione vergant. Itaque dum qui $tat horizon- |
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6172 ti pror$us e$t perpendicularis, grauitatis centrum F in ip$a |
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6173 perpendiculari con$tituitur quæ ad mundi centrum AB, |
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6174 & ideo corporis moles pondu$que fulcitur puncto B. Mo- |
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6175 do fiat ex vertebra E thoracis AE, inclinatio in anteriora, |
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6176 in GE & grauitatis centrum D diluetur in H, & per H per- |
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6177 pendicularis demittatur HI, non erit ** extra pedis ful- |
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6178 <foot>Y</foot> |
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6179 <p n=>170</p> |
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6180 cimentum BC. Stabit ergo qui ita in clinatur, nec corruet: |
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6181 $i autem a dhuc propendeat magis, fiatque in KE, centro |
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6182 grauitatis con$tituto in M, ducatur per M perpendicula- |
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6183 ris ML, quare quoniam linea ML extra pedis fulcimen- |
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6184 tum cadit, corruet qui co pacto inclinatur nec $u$tinebi- |
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6185 tur. Cur igitur natura animalibus qu<17> erecto corpore am- |
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6186 balant, pedes in anteriora porrectos fecerit, hinc clare |
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6187 patet. |
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6188 <p>Hinc etiam ceu con$ectarium habemus, cur homi- |
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6189 nes $i impellantur, magis ad ca$um in po$teriora quam in |
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6190 anteriora $int proni. Necnon etiam cur $imiæ, vr$i, & $i |
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6191 quæ cætera eiu$modi animalia diutius erecto corpore |
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6192 ambulare nequeant, nempe ideo quod eorum corporum |
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6193 moles valde in anteriora propendeat, necita commodo, |
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6194 vt humanis cuenit corporibus, pedum ip$orum ba$ibus |
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6195 fulciantur. |
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6196 <p>Quærereitem haud importune po$$umus, Curgral- |
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6197 latores non $tent erecti, ni$i a$$idue moueantur? Solutio |
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6198 facilis. grallæ etenim duobus tantum punctis $olum tan- |
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6199 gunt, nec porrecti beneficio, quod ambulantibus accidit, |
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6200 vti po$$unt. quamobrem grauitatis centrum fit extra ful- |
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6201 cimentum, & ideo coguntur grallatores a$$iduo motu |
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6202 grauitatis centro fulcimentum $upponere, quod dum fit, |
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6203 à ca$u prohibentur. |
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6204 <p>Pote$t autem id quod fulcitur, tripliciter fulciri, nē- |
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6205 peaut puncto, aut linea, aut $uperficie. |
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6206 <p>Quod puncto fulcitur, nulla reimpediente ad quam- |
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6207 uis partem cadere pote$t, centrum $iquidem, motus, pun- |
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6208 ctum e$t. |
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6209 <p>Quod linea fulcitur ad duas tantum partes, ea$que |
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6210 oppo$itas, habet ca$um. $itillud $uperficies, corpu$ue in |
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6211 latus con$titutum. |
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6212 <p n=>171</p> |
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6213 <fig> |
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6214 <p>E$to horizontis pla- |
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6215 num ABCD, cui ad re- |
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6216 ctos angulos in$i$tat $u- |
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6217 perficies EFGH, $ecun- |
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6218 dum latus FG. Sit autem |
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6219 ip$ius $uperficiei grauita- |
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6220 tis centrum I. à quo ad |
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6221 horizontis planum per- |
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6222 pendicularis demittatur IK. Cadet autem in lineam FG. |
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6223 per propo$. 38. vndecimi elem. & anguli IKG IKF recti e- |
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6224 runt. Itaque $uperficie EFGH circa lineam FKG ceu cir- |
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6225 ca axem mota punctum I peripheriam de$cribet LIM, & |
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6226 $iquidem cadat ad partes CD, grauitatis centrum erit vbi |
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6227 M. Si vero ad partes AB, fiet vbi L. Sunt autem LKM pū- |
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6228 cta in recta LKM, quæ quidem communis $ectio e$t plani |
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6229 horizontis, & plani per IKLM, tran$euntis. |
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6230 <fig> |
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6231 <p>Idem quoque de cor- |
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6232 pore dicimus in latus col- |
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6233 locato. E$to enim cubus |
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6234 LO, cuius grauitatis cen- |
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6235 trum R, latus vero quo ful- |
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6236 citur, NO, Si enim ita col- |
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6237 locetur, vt interna $uperfi- |
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6238 cies LNOQ ad rectos an- |
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6239 gulos horizonti $it con$ti- |
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6240 tuta, demi$$a perpendicu- |
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6241 laris à puncto R, ca det in S, in ip$a linea NSO. Cadente i- |
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6242 gitur corpore fiet motus circa lineam NO, centro graui- |
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6243 tatis interim peripheriam TRV. de$cribente. |
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6244 <p>Hincanimaduertere licet, Cur prouidi$$ima Natu- |
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6245 ranulli animantium vnicum dederit pedem, $ed aut qua- |
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6246 ternos, aut $altem binos, & binos quidem ip$os virtute |
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6247 quaternos, $iquidem in quolibet animantium bipedum |
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6248 <foot>Y 2</foot> |
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6249 <p n=>172</p> |
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6250 pede duo $altem puncta con$iderantur, quibus ip$um ani. |
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6251 mal fulcitur. |
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6252 <fig> |
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6253 <p>Sint enim humani pedis ve- |
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6254 $tigia A, B, C, D, in vtroque igitur |
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6255 duo puncta con$iderantur, A, B, |
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6256 C, D, illa quidem ad digitos, hæc |
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6257 autem ad calcaneum. l<*>em quo- |
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6258 que in auium pedibus ob$erua- |
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6259 tur, ex quibus concludimus, bi- |
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6260 pedum omnium fulcimentum e$- |
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6261 $e quadruplex. Porro quadrupe- |
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6262 dia eo quod tota co<*>poris mole |
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6263 ad in feriora vergant, quatuor ful- |
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6264 cimenta, eaque di$tincta, & commode ab inuicem remo- |
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6265 ta eademmet Natura præparauit. |
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6266 <p>Eadem quoque in artificialibus con$ideramus. Sit |
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6267 enim vas quo dpiam ABC, cuius pes vnicus, i$que rotun- |
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6268 dus BC, grauitatis vero centrum D. Quoniam igitur in |
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6269 pedis ip$ius peripheria, infinita puncta intelligantur, dici |
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6270 quo dammodo pote$t vas ip$um infinitis fere punctis, licet |
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6271 <fig> |
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6272 pesvnicus $it, $u$tineri. Non- |
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6273 nulla autem corpora artifi- |
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6274 cialia. quatuor pedibus $u- |
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6275 $tinentur, vt men$æ quædã, |
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6276 nonnulla etiam tribus, vt |
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6277 tripodes, qui nomen ab ip$o |
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6278 pedum numero $ortiuntur. |
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6279 Sit enim triangulum EFG, |
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6280 cuius centrum grauitatis H, |
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6281 nitatur autem tribus pun- |
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6282 ctis I, K, L, $tabit igitur. Si |
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6283 autem duobus tantum; non $tabit. ducta enim IK $i pun- |
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6284 ctis tantum IK innitatur, con$tituto grauitatis centro |
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|
6285 <p n=>173</p> |
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6286 extra fulcimentum IK, verget cedens ver$<*>s partes, L, Si |
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6287 autem innitatur punctis IL, cadet ad partes K. Sivero ip$is |
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6288 KL, cadet ad partes I.Ex quibus apparet, inanimata cor- |
|
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6289 pora aut vnico pede plurium virtutem habente, aut $al- |
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6290 tem tribus actu, vt $u$tineantur, indigere. |
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6291 <p>Hinc etiam patet, cur $enes, imbecilles, curui, & pe- |
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6292 dibus capti, baculi baculorumue fulcimento egeant, ete- |
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6293 nim cum hi debiles $int, & in anteriorem partem magno- |
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6294 pere propen deant, ne grauitatis centrum extra fulcimen- |
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6295 tum fiat, baculo vel baculis indigent, quibus centrum i- |
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6296 p$um ful ciatur. |
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6297 <p>Cæterum cur duplici genu ingeniculati difficile in |
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6298 eo $itu perman eant, ea cau$$a e$t, quod grauitatis centrum |
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6299 in thorace con$titutum, duobus genibus fulciatur, eo$- |
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6300 que premat. quæ quidem genua eo quod natura apta na- |
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6301 ta non $int, veluti pedes, ad $u$tinendam corporis molem |
|
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6302 laborant, idque eo magis, quod cum o$$ea $int, cutem in- |
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6303 ter o$$ium & plani duritiem con$titutam, accidit arctari, |
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6304 & ideo dolorem & mole$tiam ingeniculatis facere. |
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6305 <p>Siautem vnico tantum genu qui$piam nitatur, dif- |
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6306 ficultatem $entiet longe minorem. Triplici enim fulci- |
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6307 <fig> |
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6308 mento eo ca$u ingeniculatus |
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6309 fulcitur. Sit enim ingenicula- |
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|
6310 tus ABCDE, cuius grauitatis |
|
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6311 centrum F. dextrum vero ge- |
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6312 nu, cuinititur D, $ini$trum ve- |
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6313 ro, quod eleuatur B. Tribus ergo fulcimentis ingenicula- |
|
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6314 tus vt diximus, $u$tinetur, CDE. Diuiditur itaque pondus |
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6315 in tres partes, & ideo $ingulæ minus fatigantur. Magis ta- |
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6316 men laborat punctum D, vtpote illud, cui ad perpendicu- |
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6317 lum F grauitatis centrum innititur. |
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6318 <p>Vtique illud quoque mirabile e$t, Aues dormientes |
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6319 vnico tantum pede fulciri, & quod magis mirum e$t, dor- |
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6320 <foot>Y 3</foot> |
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6321 <p n=>174</p> |
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6322 mientes po$$e, quod vel ip$is vigilantibus e$t difficile. Cur |
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6323 id Natura docente faciant, eam puto e$$e cau$$am, quod |
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6324 dum dormiunt, caput $ini$træ alæ, vt naturali calore iu- |
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6325 uentur, $upponunt, quapropter ad eam partem declinan- |
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6326 tes, vt interim æquilibrium faciant, pedem $ubleuant, & |
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6327 eo ca$u ceu inutilem retrahunt atque $u$pendunt: addita |
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6328 item alia cau$$a, nempe vt pedem ip$um dormientes nati- |
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6329 uo calore confoueant. |
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6330 <p>Quæritur et<*>am, Curij qui inclinantur, vt rē quam- |
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6331 piam à $olo $u$tollant, alterum crurium ad anteriora, nē- |
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6332 pever$us manum ip$am, quam porrigunt, extendant? |
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6333 <fig> |
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6334 <p>E$to enim qui$piam ABCD, |
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6335 cuius crura BC, BD, grauitatis |
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6336 centrum E, vclitautem quippiam |
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6337 à $olo tollere quod $it in F. $it per- |
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6338 pendicularis, quæ pergrauitatis |
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6339 centrum GEH. Dumigitur ad |
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6340 anteriora ínclinatur, centrum a- |
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6341 mouet à perpendiculari, quam- |
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6342 obrem docente Natura, crus BC |
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6343 ad centrum ip$um fulciendum. |
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6344 ad anteriora, hoc e$t, ver$us rem |
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6345 $u$tollendam porrigitur. |
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6346 <p>Huius quoque $peculationis e$t inue$tigare, Cur |
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6347 quadrupedia dum gradiuntur, pedes diametraliter mo- |
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6348 ueant. Cuius rei verba fecitip$e quoque Philo$ophus lib. |
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6349 de animalium in ce$$u cap. 12. Nos autem ad maiorem de- |
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6350 clarationem, quodip$e Phy$icis principijs fecit, mecha- |
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6351 nicis demon$trabimus. |
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6352 <p>Sint duæ in plano parallelæ AB, CD, in quibus qua- |
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6353 drupedis pedes E, F, B, D, quorum EF, anteriores, BD vero |
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6354 po$teriores. iungantur BDEF, eritque EBDF parallelo- |
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6355 grammum altera parte longius, cuius diametri ducantur |
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6356 <p n=>175</p> |
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6357 <fig> |
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6358 ED, BF, $ecantes $e$e in G, vbi & grauitatis |
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6359 centrum. Moto igitur po$teriori $ini$tro pe- |
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6360 de B in K, $i anteriorem E, eodem tempore |
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6361 moueret in I, $tantibus interim DF, ceu ful- |
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6362 cimentis, centrum Gextra fulcimenta $ieret |
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6363 ad partes BE. Caderet igitur ad partes BE. Si |
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6364 autem eodem tempore moueret dextros eo- |
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6365 dem pacto centrum extra fulcimenta po$i- |
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6366 tum caderet ad partes ip$as DF. Si autein |
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6367 moto pede B in K, & eodem tempore F in L, |
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6368 & D in H, E, in I, centrum erit in diametris HI, KL, hoc |
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6369 e$t, vbi M, fultum quidem ab ip$is pedibus K, L, H, I. Hoc |
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6370 igitur pacto transfertur vici$$im cum grauitatis centro $i- |
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6371 mul translatis fulcimentis $e$e diametraliter re$ponden- |
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6372 tibus; quod vtique demon$trandum fuerat. |
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6373 <p>Sane & bipedia quoque alternatim gradiendo gra- |
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6374 uitatis centrum transferunt. Dum enim dextrum crus e- |
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6375 leuatur, centrum $ini$tro fulcitur, & econtra. |
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6376 <p>Naturalia i$thæc $unt; in artifi cialibus autem quæri |
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6377 po$$et, Cur Architecti, Arcium muros non ad perpendi- |
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6378 culum erectos, $ed intror$um inclinatos con$tituant? |
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6379 <fig> |
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6380 <p>Vtique hoc faciunt, vt minus |
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6381 $int ad ruinam proni. E$to enim |
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6382 murus ad interiorem partem ver- |
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6383 gens ABCD, Cuius grauitatis cen- |
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6384 trum E ba$is BC erigatur à puncto |
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6385 B horizonti perpendicularis BF, & |
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6386 ad eundem à centro grauitatis E |
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6387 demittatur EM, tum BE iungatur. |
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6388 Po$t hæc à puncto BG angulum. |
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6389 cum linea horizontis BK faciens recto maiorem. Ita que |
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6390 murus hoc pacto con$titutus ad interiorem partem $uo |
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6391 pondere vergit, cadere autem non pote$t, vel quod viuæ |
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6392 <p n=>176</p> |
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6393 rupi, cui forte hæret, fulciatur, vel anti$tatis, quos no- |
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6394 $trates $perones & contra fortes appellant, innitatur. Sed |
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6395 nec in anteriora corruet, quandoquidem ruinam factu- |
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6396 ras, nece$$e e$t vt grauitatis centrum $ecum trahat in per- |
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6397 pendiculari BF, & demum in eam quæ vltra perpendicu- |
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6398 larem e$t BG, facta nempe circa B, ceu circa centrum, cō- |
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6399 uer$ione. Moueatur autem & ex $emidiametro BE cen- |
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6400 tro B portio circuli de$cribatur EH, quæ $ecet BG in H, |
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6401 & BF in I; Et quia EM $emidiametro BK perpendicularis |
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6402 per B, centrum non tran$it, erit EM ip$a BK, hoc e$t, BI |
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6403 bre<*>ior. Ab$cindatur ex BI, ip$i EM æqualis LB. Eritigi- |
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6404 tur punctum L infra punctum I, hoc e$t, ip$o I, mundi cen- |
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6405 tro propius. Nece$$e igitur erit ad hoc vt murus corruat, |
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6406 centrum grauitatis E facta circa B, conuer$ione aliquan- |
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6407 do fieri in I, vt demum transferri po$$it in H, $ed I remo- |
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6408 tius e$t à mundi centro ip$is E, L, a$cendet igitur graue |
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6409 contra $ui naturam ex E in I, at hoc e$t impo$$ibile; quod |
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6410 fuerat demon$tran dum. |
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6411 <p>Ex his ij$dem principijs alia $oluitur quæ$tio, Cur |
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6412 $cilicet Campanaria turris quæ Pi$is vi$itu<*>, nec non alia |
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6413 Bononiæ in foro prope A $ellorum turrim, quam à nobili |
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6414 olim Cari$endorum familia ex$tructam, Cari$endam vo- |
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6415 cant, cuius meminit & Dantes Poeta $ummus in $ua Co- |
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6416 mœdia. Propendet autem hæc in latus, & ita propendet |
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6417 vt perpendicularis, quæ à $ummo inclinatæ partis in $o- |
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6418 lum demittitur, longe cadat ab ip$a, cui nititur, ba$i, quod |
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6419 $ane mirabile videtur, muros nempe, in ruinam pronos, |
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6420 ruinam non facere. |
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6421 <p>E$to enim turris ABCD, ba$i fulta BC, horizontis |
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6422 planum BCF latera AB, DC, centrum vero grauitatis to- |
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6423 tius molis E. Propendeat autem ad partes DC ex angulo |
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6424 DCF. Ita autem con$tituta intelligatur vt perpendicula- |
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6425 ris ab A, in planum horizontis demi$$a per grauitatis cen- |
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6426 <p n=>177</p> |
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6427 <fig> |
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6428 trum E extra ba$im BC, non cadat, |
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6429 cadat autem in C. Quoniam igitur |
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6430 ABCD moles per E grauitatis cen- |
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6431 trum diuiditur, in partes $ecatur æ- |
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6432 queponderantes, $ed & centrum. |
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6433 grauitatis extra fulcimentum non |
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6434 cadit, quare nec pars ACD, trahet |
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6435 partem ABC, nec centrum extra |
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6436 fulcimentum po$itum locum petet |
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6437 centro mundi viciniorem. Cur igitur Cari$enda $tet, & e- |
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6438 gregia illa turris campanaria quæ Pi$is prope $ummum |
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6439 Templum marmoribus præclare ex$tructa videtur, licet |
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6440 ruinam minentur, $tent æternum, nec cadant, ex his quæ |
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6441 con$iderauimus, liquido patet. |
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6442 <HEAD>QVAESTIO XXXI</HEAD> |
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6443 <HEAD><I>Cur facilius moueatur commotum quam manens, veluti currus |
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6444 commotos citius agitant, quam moueri incipientes?</I></HEAD> |
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6445 <HEAD><I>Hoc quæritur.</I></HEAD> |
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6446 <p>Problema hoc e$t mere Phy$icum; verumtamen quo- |
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6447 niam ad localem motum pertinet, de quo ip$e quoque |
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6448 Mechanicus agit, Hi$ce quæ$tionibus contemplatio hæc |
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6449 inter$eritur. Soluit autem Ari$toteles inquiens, id forta$- |
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6450 $e ea de cau$$a fieri, quod difficillimum $it pondus moue- |
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6451 re, quod in contrarium mouetur. Demit enim quippiam |
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6452 de motoris potentia re$i$tens, licet mouens ip$o moto $it |
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6453 longe potentius atque velocius. nece$$e enim e$$e id tar- |
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6454 dius moueri quod repellitur. Hæc verba licet de ea po- |
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6455 tentia dicta videantur, quæ rem motam in contrariam. |
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6456 partem repellit, nihilominus illi quoque aptantur quæ |
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6457 rem immobilem à principio mouere conatur. e$t enim re- |
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6458 $i$tentia rei quæ à $tatu ad motum transfertur ceu quidã |
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6459 <foot>Z</foot> |
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6460 <p n=>178</p> |
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6461 contrarius motus. Contra autem accidit illi quirem mo- |
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6462 tam mouet in ip$o motu: eo enim ca$u mouens ab ip$o rei |
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6463 motu magnopere iuuatur, cooperatur enim motus moto- |
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6464 ri, in ip$am rem motam operanti. Auget autem res mota |
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6465 quodammodo mouentis potentiam. quod enim à mouen- |
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6466 te pateretur, ex $e ip$a agit res quæ mouetur. |
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6467 <fig> |
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6468 <p>E$to horizontis pla- |
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6469 num AB, cui moles quæ- |
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6470 dam in$i$tat, CD. Modo |
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6471 potentia quædam appli- |
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6472 ceturvbi E, quæ molem in |
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6473 anteriora propellat, id |
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6474 e$t, ver$us B. Primumigitur, quoniam à quiete ad motum |
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6475 fit tran$itus, te$i$tit $ua quiere corpus graue, potentiæ im- |
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6476 pellenti, $uperata demum re$i$tentia moles quæ moueri |
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6477 cœpit, fertur in F & mouetur, quare potentia quæ à prin- |
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6478 cipio re$i$tentiam rei non motæ $uperauerat, pellendo |
|
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6479 rem motam pergens facilius pellit: Duo enim $unt quo- |
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6480 dammodo motores, mouens videlicetip$e, & motus quo |
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6481 res mota mouetur. facilius ergo pelletur ex F in G, quam |
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6482 ex D in F, & ex G in B, quam ex F in G, & eo motus fiet in |
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6483 progre$$u facilior atque in ip$a velocitate velocior, quo |
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6484 magis in ip$a motione mouetur. |
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6485 <p>Hinc $oluitur ea quæ$tio apud P hy$icos difficillima, |
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6486 Cur nempe in motu naturali velocitas v$que augeatur; |
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6487 etenim ibi Naturamouens e$t, atque eadem in$eparabilis |
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6488 à remota, vrgetigitur a$$idue, à principio quidem tar dius, |
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6489 po$t hæc autem ea quam diximus, de cau$$a v$que & v$que |
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6490 velocius. Motus ergo fit in motu, qui motus cum $emper à |
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6491 motore, & motu ip$o augeatur, cre$cit ex progre$$u in im- |
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6492 men$um. Certe cau$$am velocitatis auctæ eam e$$e, quod |
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6493 potentia mouens rem motam in motu ip$o moueat, nemo |
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6494 vtarbitror, inficias ibit, acquirit enim corpus motum pō- |
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6495 <p n=>179</p> |
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6496 dero$itatem quan dam accidentalem, quæ cum ex motu |
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6497 perinde augeatur, ip$um motum faciliorem, eoque velo- |
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6498 ciorem facit. Di$putat hæc & Simplicius lib. 7. Phy$ic. c. |
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6499 11. Ari$totelis de Natura libros exponens. |
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6500 <HEAD>QVAESTIO XXXII.</HEAD> |
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6501 <HEAD><I>Quæritur hic, Cur caquæproijciuntur, ce$$ent |
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6502 à latione?</I></HEAD> |
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6503 <p>Hocitidem problema e$t mere Phy$icum. Ad quod ea |
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6504 pertinent quæ à Philo$opho tractantur libro Natu- |
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6505 ralium 8. & lib. 1. de Cœlo. Tres autem affert $ubdubitan- |
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6506 do rationes, An quia impellens de$init potentia, vel pro- |
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6507 pter retractionem, vel propter rei proiectæ in clinationē, |
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6508 quando ea valentior fuerit quam proijcientis vires? |
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6509 <p>Quicquid dicat Philo$ophus, id vtique explorati$- |
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6510 $imum e$t. Proicctaideo à motu ce$$are, propterea quod |
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6511 impre$$io, cuius impetu & virtute feruntur, non $it proie- |
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6512 ctus quidem naturalis, $ed mere accidentalis & violenta, |
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6513 at nullum accidentale & violentum quodque, non natu- |
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6514 rale e$t, perpetuum e$t. Ce$$at ergo accidentalis illa im- |
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6515 pre$$io, eaque paullatim ce$$ante proiecti motus elan- |
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6516 gue$cit, donec quietem pror$us adipi$catur. Illud quoque |
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6517 notamus, quod à multis vidimus non ob$eruatum, nempe |
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6518 violentum mo<*>m violentia præualente non differre à |
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6519 naturali, & ideo tardiorem e$$e à principio po$t hæc, in i- |
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6520 p$o motu fieri velociotem, remittente demum paullatim |
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6521 impre$$a violentia, tardiorem, donecimpetus, & cum im- |
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6522 petu motus euane$cat, & resip$a mota quietem adipi$ca- |
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6523 tur. Vnde etiam experientia docemur, ictum ex proiectis |
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6524 violentius fieri, $i fiat paullo remotior à principio, & tunc |
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6525 demum effe innocenti$$imum, cum ibi fit, vbi proie ctum |
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6526 ex motu plene acqui$ito, $ummam adeptum e$t velocita- |
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6527 <foot>Z 2</foot> |
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6528 <p n=>180</p> |
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6529 tem. Hin evidemus, vel pueros ip$os, docente Natura cū |
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6530 nuces, vel aliud quippiam, parieti alli$um frangere conã- |
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6531 tur, à pariete moderato aliquo $patio recedere. Si autem |
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6532 eos interroges, curid faciant, re$pondebunt, vtinde ictus |
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6533 valentius fiat atque efficacius. Eleganter ex Simplicij & |
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6534 Alexandri Aphrodi$ien$is doctrina, quæ lucidi$$ima e$t, |
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6535 quæ$tionem hanc in $ua Paraphra$i explicat Picolomi- |
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6536 neus. |
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6537 <HEAD>QVAESTIO XXXIII.</HEAD> |
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6538 <HEAD><I>Dubitatur, Cur proiecta moueantur, licet impellens à proiectis $e- |
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6539 paretur; vel vt verbis Philo$ophi vtar, Cur quippiam non pecu- |
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6540 liarem $ibi fertur lationem impul$ore alioquin |
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6541 non con$equente?</I></HEAD> |
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6542 <p>Soluit, inquiens, an videlicet, quoniam primum, id e$t, |
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6543 impellens ip$e, id efficit vt alterum, nempe proiectum |
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6544 ip$um impellat, illud vero (hoc e$t proiectum) alterum |
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6545 impellat, hoc e$t, aërem ip$um mediumue, quod à proie- |
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6546 cto repelletur. Ce$$are autem motum, cum res eo deue- |
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6547 nit, vt motus eidem à proijciente impre$$us, non po$$it |
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6548 amplius rem proiectam mouere, & itidem rem ip$am, aë- |
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6549 rem videlicet non po$$it amplius repellere. Vel etiam |
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6550 quando ip$ius lati grauitas nutu $uo declinat magis quam |
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6551 impellentis in ante $it potentia. Vtique res per $e $atis cla- |
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6552 ra. etenim motus impre$$us a ccidentalis e$t, quod vero la- |
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6553 tioni violentæ re$i$tit principium, naturale, & ab ip$o mo- |
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6554 to in$eparabile, vincente igitur quod natura e$t, paulla- |
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6555 tim remittitur quod ex accidenti e$t, & indeproiecti fit |
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6556 quies. E$t autem & hoc quoque Problema pure phy$icum, |
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6557 & $uperiori, de quo immediate egimus, perquam familia- |
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6558 re, quamobrem ex ij$dem pror$us $oluitur |
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6559 principijs. |
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6560 <p n=>181</p> |
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6561 <HEAD>QVÆSTIO XXXIV.</HEAD> |
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6562 <HEAD><I>Cur neque parua multum, ne<16> magna nimis longe proijci queunt, |
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6563 $ed proportionem quandam habere oportet proiecta ip$a ad |
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6564 eius vires qui proijcit?</I></HEAD> |
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6565 <p>Pvlchre dubitationem diluit, inquiens, An quia nece$- |
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6566 $e e$t quod proijcitur, & impellitur contraniti ei vnde |
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6567 impellitur. Quod autem magnitudine $ua nihil cedit, aut |
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6568 imbecillitate nihil contranititur, non efficit proiectionē |
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6569 neque impul$ionem. quod enim multo impellentis exce- |
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6570 dit vires, haud quaquam cedit. Quod vero e$t multo im- |
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6571 becillius, nihil contranititur, & impre$$ionem non $u$ci- |
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6572 pit. Aliam quoque adiungit rationem, videlicet, Tantum |
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6573 ferriid quod fertur quantum aëris mouerit ad profundū |
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6574 (hoce$t, ad eam partem aëris remotiorem, ad quam fer- |
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6575 tur) etenim proiectum à principio dum fertur aërem pel- |
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6576 lit, non pellit autem $i nihil mouetur. Accidit igitur vt |
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6577 concludit Philo$ophus, proiectai$thæc contrarijs ex cau- |
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6578 $is minus moueri. quod enim valde paruum e$t nihil mo- |
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6579 uet imbecillitate $ua impediente. quod vero valde ma- |
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6580 gnum e$t, ex contraria cau$$a nihil mouet, nempe quod |
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6581 ob magnitudinem $uam nihil moueatur. Vnde fit pro- |
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6582 portionem inter proiectum & proijcientem e$$e inprimis |
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6583 ad motum, necei$$ariam. Hæc eadem præclare in $ua Pa- |
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6584 raphra$i explicat Picolomineus. |
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6585 <p>Huicnos, de proiectis quæ$tioni, hæc addimus. |
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6586 <p>Cur proiecta corpora non $ibimet ip$is $ecun dum, |
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6587 partes æ quegrauia, $i fuerint irregularis figuræ in ip$o mo- |
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6588 tu, $ecundum grauiorem partem antror$us inuiolento, & |
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6589 deor$um in naturali ferantur, & dum in latione conuer- |
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6590 tuntur, $onitum edant. |
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6591 <p>E$to pila ABCD, cuius centrum E concinnata ex |
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6592 di$pari materia leui, nempe BCD, & graui ABD. non ergo |
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6593 <foot>Z 3</foot> |
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6594 <p n=>182</p> |
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6595 <fig> |
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6596 erit centrū grauitatis & cen- |
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6597 trum molis, $it autem grauita- |
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6598 tis centrum F. De$cendat cor- |
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6599 pus prohibente remoto per |
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6600 rectam AG. Et quoniam gra- |
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6601 uiora deor$um tendunt ma- |
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6602 gis, $i à principio motus gra- |
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6603 uior pars fuerit $upra in ip$o |
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6604 de$cen$u conuertet ir pila, & |
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6605 $itum non $eruabit donec $u- |
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6606 perior pars ea quæ grauior, |
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6607 deor$um fiat, vt videre e$t in |
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6608 pila HIK, cuius centrum e$t G. pars grauior HIK. Si au- |
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6609 tem eadem pila, laterali motu violenter feratur ver$us |
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6610 N, ad eam quoque partem conuertetur pars grauior. fa- |
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6611 cto enim molis $eu magnitudinis centro vbi L, grauior |
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6612 pars fiet in MNO; quæ cunque igitur $unt corporaita cō- |
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6613 $tituta, vt in illis non $it idem molis & grauitatis centrum |
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6614 in ip$a latione conuertentur, & corum pars grauior an- |
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6615 tror$us fiet. Sonitus porro in ip$o motu editi ea e$t cau$$a, |
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6616 quod irregulare corpus à principio incipit conuerti, & in |
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6617 ip$a conuer$ione dum fertur aërem verberat, & ab eodem |
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6618 vici$$im reuerberatur, ex qua reuerberatione fit corporis |
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6619 rotatio dum fertur, & ip$e $onitus, quem Græci <G>roicon</G> |
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6620 Rhœzum appellant. |
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6621 <p>Ad hanc quoque $peculationem pertinet, Cur lapi- |
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6622 des ad $uperfi ciem aquæ proiecti non $tatim demergan- |
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6623 tur, $ed aliquot vicibus a quæ $uperficiem radentes, abea, |
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6624 dem re$iliant. |
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6625 <p>E$to aquæ $uper$icies AB, lapis proiectus C, tangens |
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6626 aquæ $uperficiem in D, & inde re$iliens in E, mox iterum |
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6627 eandem tangens in F, & re$iliens in G, donec violēto mo- |
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6628 tu ce$$ante demergatur. Vtique lapis C, proiectus in D, |
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6629 <p n=>183</p> |
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6630 <fig> |
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6631 ni$i medio den$iori, aqua vi- |
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6632 delicet, repelleretur, pene- |
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6633 traret per D, in H. At eo re$i- |
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6634 $tente, & adhuc vigente im- |
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6635 petu, fertur in E ad angulos |
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6636 fere pares. Dico autem fere, |
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6637 $iquidem maior e$t ADC ip$o EDF, propterea quod vis |
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6638 non $it eadem, $ed minor ea quæ ex D pellit in E. Durante |
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6639 igitur impetu quo pellitur antror$um, fiuntip$æ re$ilitio- |
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6640 nes, & eo ce$$ante, re$ilitiones ce$$ant, & lapis $uapte gra- |
|
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6641 uitate demergitur. |
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6642 <p>Huc quoque $pectat, Cur pila lu$oria in horizontis |
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6643 planum proiecta ad pares re$iliat, angulos nempe rectos? |
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6644 <fig> |
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6645 <p>E$to horizontis planum |
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6646 AB, in quod à puncto C per |
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6647 lineam perpendicularem CE |
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6648 cadat proijciaturue pila DE, |
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6649 cuius grauitatis centrum F. |
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6650 Tangit autem planum in pū- |
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6651 cto E. Perpendicularis ergo |
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6652 EC, circulum DE per centrū |
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6653 $ecat, hoc e$t, in partes æ qua- |
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6654 les & æqueponderantes, $ed |
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6655 dum pila cadit proijciturue, |
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6656 agitin planum horizontis, vbi E, & in eodem puncto re. |
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6657 petitur, quare cum cadens & agens diuidatur in partes æ- |
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6658 quales & æqueponderantes & item repatiens & re$iliens |
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6659 diuidatur item in partes æquales & æquepondetantes, ita |
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6660 re$ilit repatiendo, vti egerat in cadendo, hoc e$t; ad angu- |
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6661 los pares; quod fuerat demon$trandum. Modo $it planū |
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6662 aliquod ita ad horizontem inclinatum, vt GH, & in illud |
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6663 cadat proijciaturue eadem pila. Dico eam ab eodem in- |
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6664 clinato plano ad pares angulos re$ilire non tamen rectos. |
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6665 <p n=>184</p> |
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6666 Vtique pila cadens, planum non tanget in E. e$$et enim |
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6667 GH, vbi AB, Tangat autem in I, & à centro F ad contin- |
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6668 gentiæ punctum I, recta ducatur FI. Erit igitur FI (prop. |
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6669 18. lib. 3. elem.) ip$i GH plano perpendicularis. Ducatur |
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6670 item peri, ip$i EC, parallela IK, $ecans pilæ circumferen- |
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6671 tiam in K. Agit ergo & repatitur pila in puncto Inon æ. |
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6672 qualiter inæquales. etenim $unt partes KDLEI, & IK, eo |
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6673 quod IK $ecet circulum non per centrum. repellitur ergo |
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6674 in repatiendo non æ qualiter, $ed iuxta inæqualitatem ea- |
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6675 rundem partium. Ducatur autem recta in circulo LI æ- |
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6676 qualis ip$i IK. Eritigitur LEI, æ qualis IK, & tota KDLI æ- |
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6677 qualis toti IKDL. Vtigitur actio e$t per de$cen$um iuxta |
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6678 rectam KI, ita e$t repa$$io per a$cen$um ex IL. Dico autem |
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6679 angulos KIH, LIG e$$e æquales & $ingulos recto minores. |
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6680 Connectantur FL, FK. Quoniam igitur IK portio æqualis |
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6681 e$t portioni IEL, & recta LI æqualis rectæ IK, & LF æqua- |
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6682 lis ip$i FK, & FI communis, triangulum LFI, æquale e$t |
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6683 triangulo IFK. Quare & angulus FIL <17>qualis angulo FIK, |
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6684 $ed GIF, HIF recti $unt, ergo re$idui LIG, KIH æquales |
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6685 $untinter $e comparati, & recto minores; quod fuerat o- |
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6686 $tendendum. |
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6687 <p>Hinc colligimus, quo magis planum ab æquidi$tan- |
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6688 tia horizontis rece$$erit, eo pilam in eo proiectam in par- |
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6689 tes in æqualiores diuidi & ad minores ip$i plano angulos |
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6690 re$ilire. Nihil autem refert, vtrum planum, in quod pila |
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6691 cadit, ad horizontem $it inclinatum, vel eodem horizonti |
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6692 æquedi$tante pila non ad perpendiculas, $ed iuxta aliquē |
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6693 angulum in illud proijciatur. Hæc $ane ita ex demon$tra- |
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6694 tione fieri o $tenduntur. Veruntamen quoniam proiecta |
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6695 pila materialis e$t, & ideonecæqualis, nec æqueponde- |
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6696 rans & $ua grauitate re$i$tens, non ad pares ex amu$$i re$i- |
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6697 lit angulos, $ed minores aliquantulum in re$ilitione, re. |
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6698 mittente nimirum vi in ip$a reactione. Et $ane fierinon |
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6699 <p n=>185</p> |
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6700 pote$t, pilam à plano re$ilientem eo peruenire vnde à |
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6701 principio di$ce$$erat; Id enim $i daretur, æterna quoque |
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6702 pilæ ip$ius daretur re$ilitio, & paullatim vi & impetu re- |
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6703 mittente per parua interualla motus e$$et, donecres quæ |
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6704 mouebatur, omnino quie$cat. |
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6705 <HEAD>QVÆSTIO XXXV.</HEAD> |
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6706 <HEAD><I>Quærit hoc vltimo Problemate Ari$toteles, Cur eaquæin vorti- |
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6707 co$is feruntur aquis, ad medium tandem agan- |
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6708 tur omnia?</I></HEAD> |
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6709 <p>Tribus rationibus $oluit; quarum prima e$t: Quicquid |
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6710 fertur, magnitudinem habet, cuius extrema in duo- |
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6711 bus $unt circulis, hoc in minori, illud in maiori. Et quo- |
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6712 niam maior velocior e$t, magnitudo media, non æquali- |
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6713 ter fertur, $ed à maiori quidem pellitur, à minori vero re- |
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6714 trahitur, vnde transuer$us fit magnitudinis motus, & ip$a |
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6715 magnitudo ad interiorem propellitur circulum, itaque |
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6716 eodem pacto, è maiori in minorem propul$a in centrum. |
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6717 tantum fertur, & ibi quie$cit. |
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6718 <fig> |
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6719 <p>E$to vortex AB, cuius cen- |
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6720 trum C, magnitudo quæ fer- |
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6721 tur AD, maior circulus AFB, |
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6722 minor DHEG. Velocitas igi- |
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6723 tur in A maior e$t velocitate |
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6724 quæ in D, magnitudinis ergo |
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6725 extremum A, velocius rapitur |
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6726 in A quam eiu$dem extremum |
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6727 inferius D, in D. Velocitas igi- |
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6728 tur maioris circuli pellit Aver- |
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6729 $us F. tarditas vero minoris cir- |
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6730 culi D retrahitad partes G. conuertitur itaque magnitu- |
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6731 do inter pellentem & retrahentem circulum, donec ex- |
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6732 <foot>Aa</foot> |
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6733 <p n=>186</p> |
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6734 tremitas A in circulo minori fuerit vbi H, D vero vbi I, & |
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6735 ita deinceps eadem ratione vbi KL, donec paullatim fe- |
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6736 raturin centrum C, facto nempe à maiori in minorem cir- |
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6737 culum tran$itu. |
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6738 <p>Secunda ratio ita habet, quia quod fertur, $imili $e |
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6739 habet modo ad omnes circulos propter centrum, hoc e$t, |
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6740 in quouis circulo, qui circa idem centrum fertur. Omnes |
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6741 autem circuli mouentur, centrum vero $tat, nece$$e e$t à |
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6742 motu tandem id quod mouetur ad quietis locum, hoc e$t, |
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6743 in centrum ip$um peruenire. |
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6744 <p>Tertia, quoniam circulorum, qui in vorticibus fiunt, |
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6745 velocitas, & ideo impetus non e$t æqualis, $ed $emper ex- |
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6746 terior e$t interiore velocior & violentior, Æqualis autem |
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6747 $emperin mota magnitudine, grauitas, diuer$imode $e |
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6748 habet ad circulos, à quibus mouetur, & ideo modo vin- |
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6749 citur, modovincit: vincitur autem à velocioribus circulis, |
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6750 vincit autem tardiores. Ita que quoniam $ua grauitatere- |
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6751 $i$tens, maioris circuli motum pror$us non $equitur, ad |
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6752 tardiorem reijcitur, hoc e$t, interiorem, & $ic deinceps, |
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6753 donec tandem centrum ip$umnanci$catur, in quo nec $u- |
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6754 perans, nec $uperata quie$cit. |
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6755 <p>Hæ $unt rationes, licet ob$curi$$ime propo$itæ, qui- |
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6756 bus, vt diximus, vtitur Ari$toteles. acutæ $ane illæ quidē, |
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6757 attamen haudqua quam vltro admittendæ. |
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6758 <p>Primo enim fal$um videtur, quod a$$erit, vortices |
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6759 circulos e$$e, & circaidem centrum fieri atque rotari. Spi- |
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6760 ræ enim potius $unt, quæ ab exteriori parte remotioreq; |
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6761 incipientes $piraliter circumuolutæ, ad intimam tandem |
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6762 partem, quæ media e$t & centri vices gerit, deueniunt. |
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6763 qua veritate cognita, omnis pror$us difficultas tollitur, |
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6764 Cum enim ea quæ feruntur, ab aqua ferantur, aqua vero |
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6765 feratur $piraliter, ea quoque $piraliter ferri, e$t nece$$a- |
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6766 <p n=>187</p> |
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6767 rium. Hæc autem clariora erunt $i quo pacto vortices |
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6768 fiant, qui$piam con$iderauerit. |
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6769 <fig> |
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6770 <p>E$to fluminis cuiu$piam curua |
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6771 eademque profunda ripa ABCD. |
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6772 Aquæ vero moles rapida EFDC, |
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6773 quæ quidem co quod magno impe- |
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6774 tu deferatur in C, ripæ ip$ius naturã |
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6775 $equens turbinatim circum uoluitur, |
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6776 egre$$a autem extra locum $euripam |
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6777 B rotationis principium $ecundans, |
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6778 in $eip$am $piraliter contorquetur, |
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6779 & vorticem efficit GHFIK, cuius |
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6780 quidem centrum e$t vbi K. |
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6781 <p>Alia quoque de cau$$a, ex quie$cente nimirum, & |
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6782 mota aqua fiunt $piræ vorticesue. E$to enim fluminis ripa |
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6783 <fig> |
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6784 ABC, $inum efficiens, qui a quam ex |
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6785 ripæ ip$ius obiectu contineat quie- |
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6786 $centem, Cur$us vero fluminis liber & |
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6787 rectus, $it inter lineas AC, DE. Itaque |
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6788 dum aqua AC rapide fertur ad partes |
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6789 A, quie$centem ABC iuxta lineam. |
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6790 CA lateraliter propellit, & cius qui- |
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6791 dem partem quam tangit, $ecum ra- |
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6792 pit, puta ex F in G. Delata igitur aqua |
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6793 & currente ex F ver$us G quie$cens |
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6794 lateraliter eidem $e$e aliqualiter op- |
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6795 ponit, & currentem repellit ex Gin H. Cœpto itaq; $pirali |
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6796 motu aqua circumuoluitur $ecun dum lineam GHK, do- |
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6797 necperueniatad centrum I, vbi circumuolutæ aquæ par- |
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6798 tes $e$e inuicem tangunt. Porro vortices i$ti $piræue, quod |
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6799 nos per Padum, Abduam, & magna flumina nauigantes |
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6800 ob$eruauimus, non eodem permanent loco, $ed rapientis |
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6801 aquæ motum $ecundantes, paullatim in currentem aquã |
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6802 <foot>Aa 2</foot> |
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6803 <p n=>188</p> |
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6804 delati euane$cunt, fiunt etiam eiu$cemodi vortices nau- |
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6805 tis quidem valde formidabiles etiam in mari, de quibus |
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6806 Poëta libro Æneidos primo. |
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6807 <p>— <I>a$t illam ter fluctus ibidem |
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6808 Torquet agens circum, & rapidus vorat æquore vortex.</I> |
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6809 <p>Sed & idem quoque de vorticibus, qui in fluminibus |
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6810 fiunt libro 7. |
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6811 <p>— <I>hunc inter fluuio Tiberinus amœno |
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6812 Vorticibus rapidis, & multa flauus arena |
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6813 In mare prorumpit.</I> |
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6814 <p>Fiunt autem in mari partim occultis de cau$$is, partim |
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6815 etiam ex violentia aquarum $ibi inuicem obuiantium a- |
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6816 gitatione. Sed nos hi$ce explicatis commode ad ea quæ |
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6817 dixerat Ari$toteles, reuertemur. |
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6818 <p>Dicimus igitur, priman eius rationem haud magni |
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6819 videri ponderis, $iquidem non per circulos actu di$tinctos |
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6820 aqua circumfertur, $ed ip$amet $ua mole tota $imul. |
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6821 <fig> |
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6822 <p>E$to enim vortex AB, cu- |
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6823 ius centrum C, $emidiameter |
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6824 CA, fiatautem rotatio totius a- |
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6825 quæ CA ad partes D, in linea |
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6826 autem AC, $it corpus aliquod a- |
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6827 quæ rotatione circumlatū AE, |
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6828 inter circulos maiorem ADB, |
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6829 minorem EFG. velocius autem |
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6830 mouetur ADB, ip$o EFG, citius |
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6831 ergo fertur pars $uperior ip$ius |
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6832 corporis vbi A, quam inferior |
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6833 vbi E. Atidnec A repellit, nec E retrahit, $iquidem eodem |
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6834 tempore quo A permeauit circulū ADB, codem & E per- |
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6835 currit circulum EFG. Itaq; A reuer$o in A & E, punctum |
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6836 reuer$um erit in E, nulla facta corporis E quoad $itum, |
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6837 muratione quod voluit Ari$toteles. |
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6838 <p n=>189</p> |
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6839 <p>Ad $ecundam vero dicimus, non ideo quod omnes |
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6840 circuli æqualiter circa centrum $erantur, ni$i alia quæpiã |
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6841 extranea vis interce$$erit, quæ ea ab exterioribus circulis |
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6842 pellens agat in medium. |
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6843 <fig> |
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6844 <p>Tertia quoque ratio la- |
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6845 borare videtur. |
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6846 <p>E$to enim vortex AB, |
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6847 cuius centrum C, $it autem |
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6848 corpus aliquod E, cuius na- |
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6849 tura apta $it totationi aliqua- |
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6850 tenus re$i$tere. Quoniam i- |
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6851 gitur eius re$i$tentia aliquã- |
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6852 tulum ab aqua rapiente $u- |
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6853 peratur in ip$a rotatione, par- |
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6854 tim aqu<17> impetum $equetur, |
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6855 partim $uapte natura retardabitur. Quamobrem aqua |
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6856 quæ e$t in A, translata in H, corpus ip$um non erit in H, |
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6857 $ed in G. Tardius igitur corpus quam aqua ip$a, rotatio- |
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6858 nem complebit, non tamen propterea, ni$i alia quæ piam |
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6859 ad$it cau$$a, feretur in medium. |
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6860 <p>Cæterum horum vorticum effectum & cau$$am ob- |
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6861 $eruare licet, $i va$e quopiam aqua pleno aquam ip$am |
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6862 baculo manuue circulariter agitauerimus, fiet enim vor- |
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6863 tex, & $i quippiam quod leue $it, in aquam motam proie- |
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6864 cerimus, ea quam diximus de cau$$a in motum ip$um, hoc |
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6865 e$t, vorticis $piræue, centrum feretur. |
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6866 <p>Hæc nos, vt vera proponimus, & forta$$e decipimur. |
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6867 Certe Philo$opho tantæ auctoritatis contradicere, ma- |
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6868 gnæ videtur audaciæ, aut potius in$aniæ. Quicquid ta- |
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6869 men $it, pro pulcherrima veritate labora$$e, à parte |
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6870 aliqua laudis non fuerit pror$us, vt |
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6871 arbitror, alienum. |
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6872 <foot>Aa 3</foot> |
|
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6873 <p n=>190</p> |
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6874 <HEAD>APPENDIX.</HEAD> |
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6875 <p>Modum inueniendarum duarum mediarum propor- |
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6876 tionalium non tantum vtilem e$$e, $ed pror$us nece$- |
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6877 $arium, illi norunt, qui in Mechanicis di$ciplinis vel parū |
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6878 fuerint ver$ati. Nulla enim alia ratio e$t, qua corpore<17> ma- |
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6879 gnitudines $eruata figura & $imilitudine augeri propor- |
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6880 tionaliter imminuiue po$$int. Quamobrem factum e$t vt |
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6881 in his inueniendis tum vetu$ti$$imo tum etiam in feriori æ- |
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6882 uo, clari$$imi Viri magnopere laborauerint. Plato etenim, |
|
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6883 Eudoxus (cuius modum repudiauit Eutocius) Heron A- |
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6884 lexandrinus, Philon Byzantius, Apollonius, clari$$imi |
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6885 Geometræ, Diocles, Pappus, Sporus, Menæchmus, Ar- |
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6886 chytas Tarentinus, Platoni æqualls: Erato$thenes, & Ni- |
|
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6887 comedes ad has inueniendas varias rationes excogitarūt, |
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6888 quorum omnium modos, & in$trumenta, demon$tratio- |
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6889 ne$q; diligenti$$ime collegit, & in illos Cōmentarios con- |
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6890 iecit idemmet Eutocius, quos eleganti$$imos in Archime- |
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6891 dis libros de Sphæra & Cylin dro $crip$it. Nos autem ijs o- |
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6892 mnibus accurate per$pectis, & diligenti$$ime ponderatis, |
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6893 inuenimus eos fere omnes tentando negotium ab$olue- |
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6894 re, quod $ane laborio$um valde e$t & operantibus permo- |
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6895 le$tum. Itaque cum modum praximue inueni$$emus, ex |
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6896 qua is qui operatur tuti$$ime & facillime ad quæ $itas ip$as |
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6897 medias manu ducitur, hunc pulcherrimæ huius facultatis |
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6898 $tudio $is inuidere nefarium iudicaurmus. Quod $i qui$piã |
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6899 dixerit, Balli$tarum, Catapultarum, Scorpionum, & cæ- |
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6900 terarum eiu$cemodi Machinarum v$um, olim apud nos |
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6901 de$ij$$e, & ideo Problema hoc videri $uperuacaneum, Re- |
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6902 $pondemus, nulla alia ratione æneorum tormentorum pi- |
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6903 las augeri imminuiue $eruata ponderis ratione po$$e, in- |
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6904 numeraque e$$e, quæ vt rite perficiantur, hæc penitus in- |
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6905 digent $peculatione. Nos rem Mechanicis vtilem, Me. |
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6906 <p n=>191</p> |
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6907 chanicis no$tris Exercitationibus annectere, haud im- |
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6908 portunum iudicauimus. Sed tempus e$t, vt his breuiter |
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6909 præfatis, ad rem ip$am explicandã commode accedamus. |
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6910 <HEAD><I>Datis duabus proportionalibus prima, & quarta duas inter eas |
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6911 medias in continua proportione inuenire.</I></HEAD> |
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6912 <p>Esto prima datarum AB, quarta BC, inter quas $ecundã |
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6913 & tertiam oportetinuenire. Ducatur recta DE, cui à |
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6914 puncto F, vtcunque $umpto, perpendicularis demittatur |
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6915 FG, Tum ab F ver$us D duplicetur quarta BC, $itque FH, |
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6916 deinde ab H ip$i FG parallela demittatur HI, & ab HF |
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6917 ab$cindatur HK, ip$ius BC quartæ medietati æqualis. |
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6918 Po$thæc puncto K $patio autem medietati, primæ data- |
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6919 rum æquali, in linea HI notetur punctum L, & ip$i HL |
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6920 fiat æqualis FM, & KM iungatur. His ita con$titutis pare- |
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6921 tur $eor$um $cheda regulaue quæpiam NO, in cuius late- |
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6922 re accipiatur OP, æqualis medietati primæ datarum $eu |
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6923 ip$i KL. Tum regulæ latus aptetur puncto L, extremum |
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6924 vero O, feratur a$$idue per rectam EK, ver$us K, nunquam |
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6925 <fig> |
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6926 <p n=>192</p> |
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6927 interim regulæ latere ON amoto à puncto L, idque do- |
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6928 nec punctum P, obuians incidat in lineam KM, puta vbi |
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6929 Qextremum vero O inueniatur in R, notato igitur in li- |
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6930 nea EK puncto R habebitur, quod quærebatur. Erunti- |
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6931 gitur AB prima, RK $ecunda, QL tertia, BC quarta. |
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6932 <p>Hæc praxis ij$dem prin cipijs demon$tratur, quibus |
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6933 $uam ex Conchoide o$tendit Nicomedes. Conficit ille |
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6934 in$trumentum, ex quo de$cribit Conchoidē, ex qua po$t- |
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6935 ea duas medias venatur. Nos autem nec in$trumentum |
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6936 con$truimus nec Conchoidem de$cribimus, & duabus fe- |
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6937 re lineis rem ab$oluimus, vt nemo fere non dixerit, hoci- |
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6938 $tud quod docemus, à Nicomedea praxi e$$e pror$us a- |
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|
6939 lienum. |
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6940 <p>Sed nos, vt eius, quam o$tendimus, operationis de- |
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6941 mon$tratio habeatur; ip$ius Nicomedis ex Pappi libro 3. |
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6942 propo$. 5. de$umptam in medio afferemus, quippe quod |
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6943 i$thæc ea quam in $uis in Archimedem commentarijs re- |
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6944 fert Eutocius, $it lucidior. |
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6945 <p>Datis duabus rectis lineis CD, DA; duæ mediæ in |
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6946 continua proportione hoc modo a$$umuntur. |
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6947 <p>Compleatur ABCD parallelogrammum, & vtraq; |
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6948 ip$arum AB, BC, bifariam $ecetur in punctis L, E, iuncta- |
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6949 que LD producatur; & occurrat productæ CB, in G, ip$i |
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6950 vero BC ad rectos angulos ducatur EF, & CF iungatur, |
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6951 quæ $it æqualis AL. Iungatur præterea FG & ip$i paralle- |
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6952 la $it CH, eritque angulus KCH, æqualis angulo CGF. |
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6953 Tum à dato puncto F ducatur FHK, qu<17> faciat KH æqua- |
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6954 lem ip$i AL vel CF. Hoc enim per lineam Conchoidem |
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6955 fieri po$$e o$tendit Nicomedes, & iuncta KD producatur, |
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6956 occurratque ip$i BA, productæ in puncto M. Dico vt DC |
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6957 ad CK ita CK ad MA & MA ad AD. Quoniam enim BC |
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6958 bifariam $ecta e$t in E, & ip$i adijcitur CK. Rectangulum |
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6959 BKC per 6. $ecundi: vna cum quadrato ex CE, æquale e$t |
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6960 <p n=>193</p> |
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6961 <fig> |
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6962 quadrato ex EK. commune apponatur ex EF quadratum, |
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6963 ergo rectangulum BKC vna cum quadrato CF æquale |
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6964 e$t quadratis ex KE, EF, hoce$t, quadrato ex FK. Et quo- |
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6965 niam vt MA ad AB, ita e$t MD ad DK, vt autem MD ad |
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6966 DK per 2. $exti, ita BC ad C<I>K</I> erit vt MA ad AB, ita BC |
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6967 ad C<I>K</I>. Atque e$t ip$ius AB dimidi<*> AL, & ip$ius BC, du- |
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6968 pla CG, e$t igitur vt MA ad AL, ita GC ad C<I>K</I>. Sed vt GC |
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6969 ad C<I>K</I>, ita FH ad H<I>K</I> propter lineas parallelas GF, CH. |
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6970 quare & componendo vt ML, ad LA, ita F<I>K</I> ad <I>K</I>H, $ed |
|
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6971 AL ponitur æqualis H<I>K</I>, quoniam & ip$i CF, ergo & ML |
|
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6972 per 9. lib. 5. æqualis erit F<I>K</I>, & quadratum ex ML, æquale |
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6973 quadrato ex F<I>K</I>. e$t autem quadrato ex ML, æquale re- |
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6974 ctangulum BMA vna cum quadrato ex AL & quadrato |
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6975 ex Fk æquale o$ten$um e$t rectangulum BkC vna cum. |
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6976 <foot>Bb</foot> |
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6977 <p n=>194</p> |
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6978 quadrato ex CF, quorum quidem quadratum ex AL æ- |
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6979 quale e$t quadrato ex CF, ponitur enim AL, ip$i CF æ- |
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6980 qualis, ergo reliquum BMA rectangulum æquale e$t reli- |
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6981 quo BkC. Vtigitur MB ad Bk, ita Ck ad MA. Sed vt MD |
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6982 ad Bk, ita DC ad Ck. quare vt DC ad Ck, ita e$t Ck ad |
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6983 MA. vt autem MD ad Bk, ita MA, ad AD. Ergo vt DC, |
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6984 prima, ad Ck $ecundam, ita Ck $ecunda ad MA tertiam, |
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6985 & MA tertia ad AD quartam, quod fuerat demon$tran- |
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6986 dum. Hæc Pappus. Quod autem in no$tra Praxi diximus, |
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6987 QL e$$etertiam, earatio e$t, quod LR vt in prima figura |
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6988 e$t, $it æqualis ip$i LM $ecundæ figuræ, in demon$tratio- |
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6989 ne Pappi, ex quibus demptis QR & LA, quæ $unt æqua- |
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6990 les, reliqua QL primæ figuræ æqualis e$t AM $ecundæ fi- |
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6991 guræ, hoc e$t, ip$i tertiæ proportionali: E$t igitur, vt in pri- |
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6992 ma figura dicebamus, AB prima, kR $ecunda, QL tertia, |
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6993 BC quarta. |
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6994 <p>Vides igitur tu quilegis, nos ex Nicomedis demon- |
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6995 $tratione (quatenus ad praxin pertinet) $uperflua re$eca$- |
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6996 $e, & ab$que Conchoidis in$trumento lineaue rem ip$am |
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6997 confeci$$e, idque non rentantes, vtalij, $ed progre- |
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6998 dientes, & qua$i manuductos quæ$i- |
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6999 tum inue$tiga$$e. |
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7000 <HEAD>FINIS.</HEAD> |